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
@@ -1,2493 +1,2453 @@
 (* Title:      HOL/Analysis/Convex.thy
    Author:     L C Paulson, University of Cambridge
    Author:     Robert Himmelmann, TU Muenchen
    Author:     Bogdan Grechuk, University of Edinburgh
    Author:     Armin Heller, TU Muenchen
    Author:     Johannes Hoelzl, TU Muenchen
 *)
 
 section \<open>Convex Sets and Functions\<close>
 
 theory Convex
 imports
   Affine
   "HOL-Library.Set_Algebras"
 begin
 
 subsection \<open>Convex Sets\<close>
 
 definition\<^marker>\<open>tag important\<close> convex :: "'a::real_vector set \<Rightarrow> bool"
   where "convex s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u\<ge>0. \<forall>v\<ge>0. u + v = 1 \<longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> s)"
 
 lemma convexI:
   assumes "\<And>x y u v. x \<in> s \<Longrightarrow> y \<in> s \<Longrightarrow> 0 \<le> u \<Longrightarrow> 0 \<le> v \<Longrightarrow> u + v = 1 \<Longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> s"
   shows "convex s"
   using assms unfolding convex_def by fast
 
 lemma convexD:
   assumes "convex s" and "x \<in> s" and "y \<in> s" and "0 \<le> u" and "0 \<le> v" and "u + v = 1"
   shows "u *\<^sub>R x + v *\<^sub>R y \<in> s"
   using assms unfolding convex_def by fast
 
 lemma convex_alt: "convex s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u. 0 \<le> u \<and> u \<le> 1 \<longrightarrow> ((1 - u) *\<^sub>R x + u *\<^sub>R y) \<in> s)"
   (is "_ \<longleftrightarrow> ?alt")
 proof
   show "convex s" if alt: ?alt
   proof -
     {
       fix x y and u v :: real
       assume mem: "x \<in> s" "y \<in> s"
       assume "0 \<le> u" "0 \<le> v"
       moreover
       assume "u + v = 1"
       then have "u = 1 - v" by auto
       ultimately have "u *\<^sub>R x + v *\<^sub>R y \<in> s"
         using alt [rule_format, OF mem] by auto
     }
     then show ?thesis
       unfolding convex_def by auto
   qed
   show ?alt if "convex s"
     using that by (auto simp: convex_def)
 qed
 
 lemma convexD_alt:
   assumes "convex s" "a \<in> s" "b \<in> s" "0 \<le> u" "u \<le> 1"
   shows "((1 - u) *\<^sub>R a + u *\<^sub>R b) \<in> s"
   using assms unfolding convex_alt by auto
 
 lemma mem_convex_alt:
   assumes "convex S" "x \<in> S" "y \<in> S" "u \<ge> 0" "v \<ge> 0" "u + v > 0"
   shows "((u/(u+v)) *\<^sub>R x + (v/(u+v)) *\<^sub>R y) \<in> S"
   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>Convex Functions on a Set\<close>
 
 definition\<^marker>\<open>tag important\<close> convex_on :: "'a::real_vector set \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> bool"
   where "convex_on s f \<longleftrightarrow>
     (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u\<ge>0. \<forall>v\<ge>0. u + v = 1 \<longrightarrow> f (u *\<^sub>R x + v *\<^sub>R y) \<le> u * f x + v * f y)"
 
 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]
+  from f''_ge0_imp_convex[OF convex_real_interval(3), 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'':
   fixes s::"'a::real_vector set"
   assumes "finite s"
   shows "(\<Sum>x\<in>s. (if y = x then f x else 0) *\<^sub>R x) = (if y\<in>s then (f y) *\<^sub>R y else 0)"
 proof -
   have *: "\<And>x y. (if y = x then f x else (0::real)) *\<^sub>R x = (if x=y then (f x) *\<^sub>R x else 0)"
     by auto
   show ?thesis
     unfolding * using sum.delta[OF assms, of y "\<lambda>x. f x *\<^sub>R x"] by auto
 qed
 
 lemma dist_triangle_eq:
   fixes x y z :: "'a::real_inner"
   shows "dist x z = dist x y + dist y z \<longleftrightarrow>
     norm (x - y) *\<^sub>R (y - z) = norm (y - z) *\<^sub>R (x - y)"
 proof -
   have *: "x - y + (y - z) = x - z" by auto
   show ?thesis unfolding dist_norm norm_triangle_eq[of "x - y" "y - z", unfolded *]
     by (auto simp:norm_minus_commute)
 qed
 
 
 
 
 subsection \<open>Cones\<close>
 
 definition\<^marker>\<open>tag important\<close> cone :: "'a::real_vector set \<Rightarrow> bool"
   where "cone s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>c\<ge>0. c *\<^sub>R x \<in> s)"
 
 lemma cone_empty[intro, simp]: "cone {}"
   unfolding cone_def by auto
 
 lemma cone_univ[intro, simp]: "cone UNIV"
   unfolding cone_def by auto
 
 lemma cone_Inter[intro]: "\<forall>s\<in>f. cone s \<Longrightarrow> cone (\<Inter>f)"
   unfolding cone_def by auto
 
 lemma subspace_imp_cone: "subspace S \<Longrightarrow> cone S"
   by (simp add: cone_def subspace_scale)
 
 
 subsubsection \<open>Conic hull\<close>
 
 lemma cone_cone_hull: "cone (cone hull s)"
   unfolding hull_def by auto
 
 lemma cone_hull_eq: "cone hull s = s \<longleftrightarrow> cone s"
   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
 
 lemma convex_cone:
   "convex s \<and> cone s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. (x + y) \<in> s) \<and> (\<forall>x\<in>s. \<forall>c\<ge>0. (c *\<^sub>R x) \<in> s)"
   (is "?lhs = ?rhs")
 proof -
   {
     fix x y
     assume "x\<in>s" "y\<in>s" and ?lhs
     then have "2 *\<^sub>R x \<in>s" "2 *\<^sub>R y \<in> s"
       unfolding cone_def by auto
     then have "x + y \<in> s"
       using \<open>?lhs\<close>[unfolded convex_def, THEN conjunct1]
       apply (erule_tac x="2*\<^sub>R x" in ballE)
       apply (erule_tac x="2*\<^sub>R y" in ballE)
       apply (erule_tac x="1/2" in allE, simp)
       apply (erule_tac x="1/2" in allE, auto)
       done
   }
   then show ?thesis
     unfolding convex_def cone_def by blast
 qed
 
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>Connectedness of convex sets\<close>
 
-lemma 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%unimportant connected_UNIV[intro]: "connected (UNIV :: 'a::real_normed_vector set)"
 by (simp add: convex_connected)
 
 lemma convex_prod:
   assumes "\<And>i. i \<in> Basis \<Longrightarrow> convex {x. P i x}"
   shows "convex {x. \<forall>i\<in>Basis. P i (x\<bullet>i)}"
   using assms unfolding convex_def
   by (auto simp: inner_add_left)
 
 lemma convex_positive_orthant: "convex {x::'a::euclidean_space. (\<forall>i\<in>Basis. 0 \<le> x\<bullet>i)}"
 by (rule convex_prod) (simp flip: atLeast_def)
 
 subsection \<open>Convex hull\<close>
 
 lemma convex_convex_hull [iff]: "convex (convex hull s)"
   unfolding hull_def
   using convex_Inter[of "{t. convex t \<and> s \<subseteq> t}"]
   by auto
 
 lemma convex_hull_subset:
     "s \<subseteq> convex hull t \<Longrightarrow> convex hull s \<subseteq> convex hull t"
   by (simp add: subset_hull)
 
 lemma convex_hull_eq: "convex hull s = s \<longleftrightarrow> convex s"
   by (metis convex_convex_hull hull_same)
 
 subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Convex hull is "preserved" by a linear function\<close>
 
 lemma convex_hull_linear_image:
   assumes f: "linear f"
   shows "f ` (convex hull s) = convex hull (f ` s)"
 proof
   show "convex hull (f ` s) \<subseteq> f ` (convex hull s)"
     by (intro hull_minimal image_mono hull_subset convex_linear_image assms convex_convex_hull)
   show "f ` (convex hull s) \<subseteq> convex hull (f ` s)"
   proof (unfold image_subset_iff_subset_vimage, rule hull_minimal)
     show "s \<subseteq> f -` (convex hull (f ` s))"
       by (fast intro: hull_inc)
     show "convex (f -` (convex hull (f ` s)))"
       by (intro convex_linear_vimage [OF f] convex_convex_hull)
   qed
 qed
 
 lemma in_convex_hull_linear_image:
   assumes "linear f"
     and "x \<in> convex hull s"
   shows "f x \<in> convex hull (f ` s)"
   using convex_hull_linear_image[OF assms(1)] assms(2) by auto
 
 lemma convex_hull_Times:
   "convex hull (s \<times> t) = (convex hull s) \<times> (convex hull t)"
 proof
   show "convex hull (s \<times> t) \<subseteq> (convex hull s) \<times> (convex hull t)"
     by (intro hull_minimal Sigma_mono hull_subset convex_Times convex_convex_hull)
   have "(x, y) \<in> convex hull (s \<times> t)" if x: "x \<in> convex hull s" and y: "y \<in> convex hull t" for x y
   proof (rule hull_induct [OF x], rule hull_induct [OF y])
     fix x y assume "x \<in> s" and "y \<in> t"
     then show "(x, y) \<in> convex hull (s \<times> t)"
       by (simp add: hull_inc)
   next
     fix x let ?S = "((\<lambda>y. (0, y)) -` (\<lambda>p. (- x, 0) + p) ` (convex hull s \<times> t))"
     have "convex ?S"
       by (intro convex_linear_vimage convex_translation convex_convex_hull,
         simp add: linear_iff)
     also have "?S = {y. (x, y) \<in> convex hull (s \<times> t)}"
       by (auto simp: image_def Bex_def)
     finally show "convex {y. (x, y) \<in> convex hull (s \<times> t)}" .
   next
     show "convex {x. (x, y) \<in> convex hull s \<times> t}"
     proof -
       fix y let ?S = "((\<lambda>x. (x, 0)) -` (\<lambda>p. (0, - y) + p) ` (convex hull s \<times> t))"
       have "convex ?S"
       by (intro convex_linear_vimage convex_translation convex_convex_hull,
         simp add: linear_iff)
       also have "?S = {x. (x, y) \<in> convex hull (s \<times> t)}"
         by (auto simp: image_def Bex_def)
       finally show "convex {x. (x, y) \<in> convex hull (s \<times> t)}" .
     qed
   qed
   then show "(convex hull s) \<times> (convex hull t) \<subseteq> convex hull (s \<times> t)"
     unfolding subset_eq split_paired_Ball_Sigma by blast
 qed
 
 
 subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Stepping theorems for convex hulls of finite sets\<close>
 
 lemma convex_hull_empty[simp]: "convex hull {} = {}"
   by (rule hull_unique) auto
 
 lemma convex_hull_singleton[simp]: "convex hull {a} = {a}"
   by (rule hull_unique) auto
 
 lemma convex_hull_insert:
   fixes S :: "'a::real_vector set"
   assumes "S \<noteq> {}"
   shows "convex hull (insert a S) =
          {x. \<exists>u\<ge>0. \<exists>v\<ge>0. \<exists>b. (u + v = 1) \<and> b \<in> (convex hull S) \<and> (x = u *\<^sub>R a + v *\<^sub>R b)}"
   (is "_ = ?hull")
 proof (intro equalityI hull_minimal subsetI)
   fix x
   assume "x \<in> insert a S"
   then have "\<exists>u\<ge>0. \<exists>v\<ge>0. u + v = 1 \<and> (\<exists>b. b \<in> convex hull S \<and> x = u *\<^sub>R a + v *\<^sub>R b)"
   unfolding insert_iff
   proof
     assume "x = a"
     then show ?thesis
       by (rule_tac x=1 in exI) (use assms hull_subset in fastforce)
   next
     assume "x \<in> S"
     with hull_subset[of S convex] show ?thesis
       by force
   qed
   then show "x \<in> ?hull"
     by simp
 next
   fix x
   assume "x \<in> ?hull"
   then obtain u v b where obt: "u\<ge>0" "v\<ge>0" "u + v = 1" "b \<in> convex hull S" "x = u *\<^sub>R a + v *\<^sub>R b"
     by auto
   have "a \<in> convex hull insert a S" "b \<in> convex hull insert a S"
     using hull_mono[of S "insert a S" convex] hull_mono[of "{a}" "insert a S" convex] and obt(4)
     by auto
   then show "x \<in> convex hull insert a S"
     unfolding obt(5) using obt(1-3)
     by (rule convexD [OF convex_convex_hull])
 next
   show "convex ?hull"
   proof (rule convexI)
     fix x y u v
     assume as: "(0::real) \<le> u" "0 \<le> v" "u + v = 1" and x: "x \<in> ?hull" and y: "y \<in> ?hull"
     from x obtain u1 v1 b1 where
       obt1: "u1\<ge>0" "v1\<ge>0" "u1 + v1 = 1" "b1 \<in> convex hull S" and xeq: "x = u1 *\<^sub>R a + v1 *\<^sub>R b1"
       by auto
     from y obtain u2 v2 b2 where
       obt2: "u2\<ge>0" "v2\<ge>0" "u2 + v2 = 1" "b2 \<in> convex hull S" and yeq: "y = u2 *\<^sub>R a + v2 *\<^sub>R b2"
       by auto
     have *: "\<And>(x::'a) s1 s2. x - s1 *\<^sub>R x - s2 *\<^sub>R x = ((1::real) - (s1 + s2)) *\<^sub>R x"
       by (auto simp: algebra_simps)
     have "\<exists>b \<in> convex hull S. u *\<^sub>R x + v *\<^sub>R y =
       (u * u1) *\<^sub>R a + (v * u2) *\<^sub>R a + (b - (u * u1) *\<^sub>R b - (v * u2) *\<^sub>R b)"
     proof (cases "u * v1 + v * v2 = 0")
       case True
       have *: "\<And>(x::'a) s1 s2. x - s1 *\<^sub>R x - s2 *\<^sub>R x = ((1::real) - (s1 + s2)) *\<^sub>R x"
         by (auto simp: algebra_simps)
       have eq0: "u * v1 = 0" "v * v2 = 0"
         using True mult_nonneg_nonneg[OF \<open>u\<ge>0\<close> \<open>v1\<ge>0\<close>] mult_nonneg_nonneg[OF \<open>v\<ge>0\<close> \<open>v2\<ge>0\<close>]
         by arith+
       then have "u * u1 + v * u2 = 1"
         using as(3) obt1(3) obt2(3) by auto
       then show ?thesis
         using "*" eq0 as obt1(4) xeq yeq by auto
     next
       case False
       have "1 - (u * u1 + v * u2) = (u + v) - (u * u1 + v * u2)"
         using as(3) obt1(3) obt2(3) by (auto simp: field_simps)
       also have "\<dots> = u * (v1 + u1 - u1) + v * (v2 + u2 - u2)"
         using as(3) obt1(3) obt2(3) by (auto simp: field_simps)
       also have "\<dots> = u * v1 + v * v2"
         by simp
       finally have **:"1 - (u * u1 + v * u2) = u * v1 + v * v2" by auto
       let ?b = "((u * v1) / (u * v1 + v * v2)) *\<^sub>R b1 + ((v * v2) / (u * v1 + v * v2)) *\<^sub>R b2"
       have zeroes: "0 \<le> u * v1 + v * v2" "0 \<le> u * v1" "0 \<le> u * v1 + v * v2" "0 \<le> v * v2"
         using as(1,2) obt1(1,2) obt2(1,2) by auto
       show ?thesis
       proof
         show "u *\<^sub>R x + v *\<^sub>R y = (u * u1) *\<^sub>R a + (v * u2) *\<^sub>R a + (?b - (u * u1) *\<^sub>R ?b - (v * u2) *\<^sub>R ?b)"
           unfolding xeq yeq * **
           using False by (auto simp: scaleR_left_distrib scaleR_right_distrib)
         show "?b \<in> convex hull S"
           using False zeroes obt1(4) obt2(4)
           by (auto simp: convexD [OF convex_convex_hull] scaleR_left_distrib scaleR_right_distrib  add_divide_distrib[symmetric]  zero_le_divide_iff)
       qed
     qed
     then obtain b where b: "b \<in> convex hull S" 
        "u *\<^sub>R x + v *\<^sub>R y = (u * u1) *\<^sub>R a + (v * u2) *\<^sub>R a + (b - (u * u1) *\<^sub>R b - (v * u2) *\<^sub>R b)" ..
 
     have u1: "u1 \<le> 1"
       unfolding obt1(3)[symmetric] and not_le using obt1(2) by auto
     have u2: "u2 \<le> 1"
       unfolding obt2(3)[symmetric] and not_le using obt2(2) by auto
     have "u1 * u + u2 * v \<le> max u1 u2 * u + max u1 u2 * v"
     proof (rule add_mono)
       show "u1 * u \<le> max u1 u2 * u" "u2 * v \<le> max u1 u2 * v"
         by (simp_all add: as mult_right_mono)
     qed
     also have "\<dots> \<le> 1"
       unfolding distrib_left[symmetric] and as(3) using u1 u2 by auto
     finally have le1: "u1 * u + u2 * v \<le> 1" .    
     show "u *\<^sub>R x + v *\<^sub>R y \<in> ?hull"
     proof (intro CollectI exI conjI)
       show "0 \<le> u * u1 + v * u2"
         by (simp add: as(1) as(2) obt1(1) obt2(1))
       show "0 \<le> 1 - u * u1 - v * u2"
         by (simp add: le1 diff_diff_add mult.commute)
     qed (use b in \<open>auto simp: algebra_simps\<close>)
   qed
 qed
 
 lemma convex_hull_insert_alt:
    "convex hull (insert a S) =
      (if S = {} then {a}
       else {(1 - u) *\<^sub>R a + u *\<^sub>R x |x u. 0 \<le> u \<and> u \<le> 1 \<and> x \<in> convex hull S})"
   apply (auto simp: convex_hull_insert)
   using diff_eq_eq apply fastforce
   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 convex_hull_subset_span: "(convex hull s) \<subseteq> (span s)"
   by (metis hull_minimal span_superset subspace_imp_convex subspace_span)
 
 lemma convex_hull_subset_affine_hull: "(convex hull s) \<subseteq> (affine hull s)"
   by (metis affine_affine_hull affine_imp_convex hull_minimal hull_subset)
 
 lemma aff_dim_convex_hull:
   fixes S :: "'n::euclidean_space set"
   shows "aff_dim (convex hull S) = aff_dim S"
   using aff_dim_affine_hull[of S] convex_hull_subset_affine_hull[of S]
     hull_subset[of S "convex"] aff_dim_subset[of S "convex hull S"]
     aff_dim_subset[of "convex hull S" "affine hull S"]
   by auto
 
 
 subsection \<open>Caratheodory's theorem\<close>
 
 lemma convex_hull_caratheodory_aff_dim:
   fixes p :: "('a::euclidean_space) set"
   shows "convex hull p =
     {y. \<exists>s u. finite s \<and> s \<subseteq> p \<and> card s \<le> aff_dim p + 1 \<and>
       (\<forall>x\<in>s. 0 \<le> u x) \<and> sum u s = 1 \<and> sum (\<lambda>v. u v *\<^sub>R v) s = y}"
   unfolding convex_hull_explicit set_eq_iff mem_Collect_eq
 proof (intro allI iffI)
   fix y
   let ?P = "\<lambda>n. \<exists>s u. finite s \<and> card s = n \<and> s \<subseteq> p \<and> (\<forall>x\<in>s. 0 \<le> u x) \<and>
     sum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = y"
   assume "\<exists>s u. finite s \<and> s \<subseteq> p \<and> (\<forall>x\<in>s. 0 \<le> u x) \<and> sum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = y"
   then obtain N where "?P N" by auto
   then have "\<exists>n\<le>N. (\<forall>k<n. \<not> ?P k) \<and> ?P n"
     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/Convex_Euclidean_Space.thy b/src/HOL/Analysis/Convex_Euclidean_Space.thy
--- a/src/HOL/Analysis/Convex_Euclidean_Space.thy
+++ b/src/HOL/Analysis/Convex_Euclidean_Space.thy
@@ -1,2280 +1,2274 @@
 (* Title:      HOL/Analysis/Convex_Euclidean_Space.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 on (Normed) Euclidean Spaces\<close>
 
 theory Convex_Euclidean_Space
 imports
   Convex
   Topology_Euclidean_Space
 begin
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>Topological Properties of Convex Sets and Functions\<close>
 
 lemma aff_dim_cball:
   fixes a :: "'n::euclidean_space"
   assumes "e > 0"
   shows "aff_dim (cball a e) = int (DIM('n))"
 proof -
   have "(\<lambda>x. a + x) ` (cball 0 e) \<subseteq> cball a e"
     unfolding cball_def dist_norm by auto
   then have "aff_dim (cball (0 :: 'n::euclidean_space) e) \<le> aff_dim (cball a e)"
     using aff_dim_translation_eq[of a "cball 0 e"]
           aff_dim_subset[of "(+) a ` cball 0 e" "cball a e"]
     by auto
   moreover have "aff_dim (cball (0 :: 'n::euclidean_space) e) = int (DIM('n))"
     using hull_inc[of "(0 :: 'n::euclidean_space)" "cball 0 e"]
       centre_in_cball[of "(0 :: 'n::euclidean_space)"] assms
     by (simp add: dim_cball[of e] aff_dim_zero[of "cball 0 e"])
   ultimately show ?thesis
     using aff_dim_le_DIM[of "cball a e"] by auto
 qed
 
 lemma aff_dim_open:
   fixes S :: "'n::euclidean_space set"
   assumes "open S"
     and "S \<noteq> {}"
   shows "aff_dim S = int (DIM('n))"
 proof -
   obtain x where "x \<in> S"
     using assms by auto
   then obtain e where e: "e > 0" "cball x e \<subseteq> S"
     using open_contains_cball[of S] assms by auto
   then have "aff_dim (cball x e) \<le> aff_dim S"
     using aff_dim_subset by auto
   with e show ?thesis
     using aff_dim_cball[of e x] aff_dim_le_DIM[of S] by auto
 qed
 
 lemma low_dim_interior:
   fixes S :: "'n::euclidean_space set"
   assumes "\<not> aff_dim S = int (DIM('n))"
   shows "interior S = {}"
 proof -
   have "aff_dim(interior S) \<le> aff_dim S"
     using interior_subset aff_dim_subset[of "interior S" S] by auto
   then show ?thesis
     using aff_dim_open[of "interior S"] aff_dim_le_DIM[of S] assms by auto
 qed
 
 corollary empty_interior_lowdim:
   fixes S :: "'n::euclidean_space set"
   shows "dim S < DIM ('n) \<Longrightarrow> interior S = {}"
 by (metis low_dim_interior affine_hull_UNIV dim_affine_hull less_not_refl dim_UNIV)
 
 corollary aff_dim_nonempty_interior:
   fixes S :: "'a::euclidean_space set"
   shows "interior S \<noteq> {} \<Longrightarrow> aff_dim S = DIM('a)"
 by (metis low_dim_interior)
 
 
 subsection \<open>Relative interior of a set\<close>
 
 definition\<^marker>\<open>tag important\<close> "rel_interior S =
   {x. \<exists>T. openin (top_of_set (affine hull S)) T \<and> x \<in> T \<and> T \<subseteq> S}"
 
 lemma rel_interior_mono:
    "\<lbrakk>S \<subseteq> T; affine hull S = affine hull T\<rbrakk>
    \<Longrightarrow> (rel_interior S) \<subseteq> (rel_interior T)"
   by (auto simp: rel_interior_def)
 
 lemma rel_interior_maximal:
    "\<lbrakk>T \<subseteq> S; openin(top_of_set (affine hull S)) T\<rbrakk> \<Longrightarrow> T \<subseteq> (rel_interior S)"
   by (auto simp: rel_interior_def)
 
 lemma rel_interior:
   "rel_interior S = {x \<in> S. \<exists>T. open T \<and> x \<in> T \<and> T \<inter> affine hull S \<subseteq> S}"
   unfolding rel_interior_def[of S] openin_open[of "affine hull S"]
   apply auto
 proof -
   fix x T
   assume *: "x \<in> S" "open T" "x \<in> T" "T \<inter> affine hull S \<subseteq> S"
   then have **: "x \<in> T \<inter> affine hull S"
     using hull_inc by auto
   show "\<exists>Tb. (\<exists>Ta. open Ta \<and> Tb = affine hull S \<inter> Ta) \<and> x \<in> Tb \<and> Tb \<subseteq> S"
     apply (rule_tac x = "T \<inter> (affine hull S)" in exI)
     using * **
     apply auto
     done
 qed
 
 lemma mem_rel_interior: "x \<in> rel_interior S \<longleftrightarrow> (\<exists>T. open T \<and> x \<in> T \<inter> S \<and> T \<inter> affine hull S \<subseteq> S)"
   by (auto simp: rel_interior)
 
 lemma mem_rel_interior_ball:
   "x \<in> rel_interior S \<longleftrightarrow> x \<in> S \<and> (\<exists>e. e > 0 \<and> ball x e \<inter> affine hull S \<subseteq> S)"
   apply (simp add: rel_interior, safe)
   apply (force simp: open_contains_ball)
   apply (rule_tac x = "ball x e" in exI, simp)
   done
 
 lemma rel_interior_ball:
   "rel_interior S = {x \<in> S. \<exists>e. e > 0 \<and> ball x e \<inter> affine hull S \<subseteq> S}"
   using mem_rel_interior_ball [of _ S] by auto
 
 lemma mem_rel_interior_cball:
   "x \<in> rel_interior S \<longleftrightarrow> x \<in> S \<and> (\<exists>e. e > 0 \<and> cball x e \<inter> affine hull S \<subseteq> S)"
   apply (simp add: rel_interior, safe)
   apply (force simp: open_contains_cball)
   apply (rule_tac x = "ball x e" in exI)
   apply (simp add: subset_trans [OF ball_subset_cball], auto)
   done
 
 lemma rel_interior_cball:
   "rel_interior S = {x \<in> S. \<exists>e. e > 0 \<and> cball x e \<inter> affine hull S \<subseteq> S}"
   using mem_rel_interior_cball [of _ S] by auto
 
 lemma rel_interior_empty [simp]: "rel_interior {} = {}"
    by (auto simp: rel_interior_def)
 
 lemma affine_hull_sing [simp]: "affine hull {a :: 'n::euclidean_space} = {a}"
   by (metis affine_hull_eq affine_sing)
 
 lemma rel_interior_sing [simp]:
     fixes a :: "'n::euclidean_space"  shows "rel_interior {a} = {a}"
   apply (auto simp: rel_interior_ball)
   apply (rule_tac x=1 in exI, force)
   done
 
 lemma subset_rel_interior:
   fixes S T :: "'n::euclidean_space set"
   assumes "S \<subseteq> T"
     and "affine hull S = affine hull T"
   shows "rel_interior S \<subseteq> rel_interior T"
   using assms by (auto simp: rel_interior_def)
 
 lemma rel_interior_subset: "rel_interior S \<subseteq> S"
   by (auto simp: rel_interior_def)
 
 lemma rel_interior_subset_closure: "rel_interior S \<subseteq> closure S"
   using rel_interior_subset by (auto simp: closure_def)
 
 lemma interior_subset_rel_interior: "interior S \<subseteq> rel_interior S"
   by (auto simp: rel_interior interior_def)
 
 lemma interior_rel_interior:
   fixes S :: "'n::euclidean_space set"
   assumes "aff_dim S = int(DIM('n))"
   shows "rel_interior S = interior S"
 proof -
   have "affine hull S = UNIV"
     using assms affine_hull_UNIV[of S] by auto
   then show ?thesis
     unfolding rel_interior interior_def by auto
 qed
 
 lemma rel_interior_interior:
   fixes S :: "'n::euclidean_space set"
   assumes "affine hull S = UNIV"
   shows "rel_interior S = interior S"
   using assms unfolding rel_interior interior_def by auto
 
 lemma rel_interior_open:
   fixes S :: "'n::euclidean_space set"
   assumes "open S"
   shows "rel_interior S = S"
   by (metis assms interior_eq interior_subset_rel_interior rel_interior_subset set_eq_subset)
 
-lemma interior_ball [simp]: "interior (ball x e) = ball x e"
-  by (simp add: interior_open)
-
 lemma interior_rel_interior_gen:
   fixes S :: "'n::euclidean_space set"
   shows "interior S = (if aff_dim S = int(DIM('n)) then rel_interior S else {})"
   by (metis interior_rel_interior low_dim_interior)
 
 lemma rel_interior_nonempty_interior:
   fixes S :: "'n::euclidean_space set"
   shows "interior S \<noteq> {} \<Longrightarrow> rel_interior S = interior S"
 by (metis interior_rel_interior_gen)
 
 lemma affine_hull_nonempty_interior:
   fixes S :: "'n::euclidean_space set"
   shows "interior S \<noteq> {} \<Longrightarrow> affine hull S = UNIV"
 by (metis affine_hull_UNIV interior_rel_interior_gen)
 
 lemma rel_interior_affine_hull [simp]:
   fixes S :: "'n::euclidean_space set"
   shows "rel_interior (affine hull S) = affine hull S"
 proof -
   have *: "rel_interior (affine hull S) \<subseteq> affine hull S"
     using rel_interior_subset by auto
   {
     fix x
     assume x: "x \<in> affine hull S"
     define e :: real where "e = 1"
     then have "e > 0" "ball x e \<inter> affine hull (affine hull S) \<subseteq> affine hull S"
       using hull_hull[of _ S] by auto
     then have "x \<in> rel_interior (affine hull S)"
       using x rel_interior_ball[of "affine hull S"] by auto
   }
   then show ?thesis using * by auto
 qed
 
 lemma rel_interior_UNIV [simp]: "rel_interior (UNIV :: ('n::euclidean_space) set) = UNIV"
   by (metis open_UNIV rel_interior_open)
 
 lemma rel_interior_convex_shrink:
   fixes S :: "'a::euclidean_space set"
   assumes "convex S"
     and "c \<in> rel_interior S"
     and "x \<in> S"
     and "0 < e"
     and "e \<le> 1"
   shows "x - e *\<^sub>R (x - c) \<in> rel_interior S"
 proof -
   obtain d where "d > 0" and d: "ball c d \<inter> affine hull S \<subseteq> S"
     using assms(2) unfolding  mem_rel_interior_ball by auto
   {
     fix y
     assume as: "dist (x - e *\<^sub>R (x - c)) y < e * d" "y \<in> affine hull S"
     have *: "y = (1 - (1 - e)) *\<^sub>R ((1 / e) *\<^sub>R y - ((1 - e) / e) *\<^sub>R x) + (1 - e) *\<^sub>R x"
       using \<open>e > 0\<close> by (auto simp: scaleR_left_diff_distrib scaleR_right_diff_distrib)
     have "x \<in> affine hull S"
       using assms hull_subset[of S] by auto
     moreover have "1 / e + - ((1 - e) / e) = 1"
       using \<open>e > 0\<close> left_diff_distrib[of "1" "(1-e)" "1/e"] by auto
     ultimately have **: "(1 / e) *\<^sub>R y - ((1 - e) / e) *\<^sub>R x \<in> affine hull S"
       using as affine_affine_hull[of S] mem_affine[of "affine hull S" y x "(1 / e)" "-((1 - e) / e)"]
       by (simp add: algebra_simps)
     have "dist c ((1 / e) *\<^sub>R y - ((1 - e) / e) *\<^sub>R x) = \<bar>1/e\<bar> * norm (e *\<^sub>R c - y + (1 - e) *\<^sub>R x)"
       unfolding dist_norm norm_scaleR[symmetric]
       apply (rule arg_cong[where f=norm])
       using \<open>e > 0\<close>
       apply (auto simp: euclidean_eq_iff[where 'a='a] field_simps inner_simps)
       done
     also have "\<dots> = \<bar>1/e\<bar> * norm (x - e *\<^sub>R (x - c) - y)"
       by (auto intro!:arg_cong[where f=norm] simp add: algebra_simps)
     also have "\<dots> < d"
       using as[unfolded dist_norm] and \<open>e > 0\<close>
       by (auto simp:pos_divide_less_eq[OF \<open>e > 0\<close>] mult.commute)
     finally have "y \<in> S"
       apply (subst *)
       apply (rule assms(1)[unfolded convex_alt,rule_format])
       apply (rule d[THEN subsetD])
       unfolding mem_ball
       using assms(3-5) **
       apply auto
       done
   }
   then have "ball (x - e *\<^sub>R (x - c)) (e*d) \<inter> affine hull S \<subseteq> S"
     by auto
   moreover have "e * d > 0"
     using \<open>e > 0\<close> \<open>d > 0\<close> by simp
   moreover have c: "c \<in> S"
     using assms rel_interior_subset by auto
   moreover from c have "x - e *\<^sub>R (x - c) \<in> S"
     using convexD_alt[of S x c e]
     apply (simp add: algebra_simps)
     using assms
     apply auto
     done
   ultimately show ?thesis
     using mem_rel_interior_ball[of "x - e *\<^sub>R (x - c)" S] \<open>e > 0\<close> by auto
 qed
 
 lemma interior_real_atLeast [simp]:
   fixes a :: real
   shows "interior {a..} = {a<..}"
 proof -
   {
     fix y
     assume "a < y"
     then have "y \<in> interior {a..}"
       apply (simp add: mem_interior)
       apply (rule_tac x="(y-a)" in exI)
       apply (auto simp: dist_norm)
       done
   }
   moreover
   {
     fix y
     assume "y \<in> interior {a..}"
     then obtain e where e: "e > 0" "cball y e \<subseteq> {a..}"
       using mem_interior_cball[of y "{a..}"] by auto
     moreover from e have "y - e \<in> cball y e"
       by (auto simp: cball_def dist_norm)
     ultimately have "a \<le> y - e" by blast
     then have "a < y" using e by auto
   }
   ultimately show ?thesis by auto
 qed
 
 lemma continuous_ge_on_Ioo:
   assumes "continuous_on {c..d} g" "\<And>x. x \<in> {c<..<d} \<Longrightarrow> g x \<ge> a" "c < d" "x \<in> {c..d}"
   shows "g (x::real) \<ge> (a::real)"
 proof-
   from assms(3) have "{c..d} = closure {c<..<d}" by (rule closure_greaterThanLessThan[symmetric])
   also from assms(2) have "{c<..<d} \<subseteq> (g -` {a..} \<inter> {c..d})" by auto
   hence "closure {c<..<d} \<subseteq> closure (g -` {a..} \<inter> {c..d})" by (rule closure_mono)
   also from assms(1) have "closed (g -` {a..} \<inter> {c..d})"
     by (auto simp: continuous_on_closed_vimage)
   hence "closure (g -` {a..} \<inter> {c..d}) = g -` {a..} \<inter> {c..d}" by simp
   finally show ?thesis using \<open>x \<in> {c..d}\<close> by auto
 qed
 
 lemma interior_real_atMost [simp]:
   fixes a :: real
   shows "interior {..a} = {..<a}"
 proof -
   {
     fix y
     assume "a > y"
     then have "y \<in> interior {..a}"
       apply (simp add: mem_interior)
       apply (rule_tac x="(a-y)" in exI)
       apply (auto simp: dist_norm)
       done
   }
   moreover
   {
     fix y
     assume "y \<in> interior {..a}"
     then obtain e where e: "e > 0" "cball y e \<subseteq> {..a}"
       using mem_interior_cball[of y "{..a}"] by auto
     moreover from e have "y + e \<in> cball y e"
       by (auto simp: cball_def dist_norm)
     ultimately have "a \<ge> y + e" by auto
     then have "a > y" using e by auto
   }
   ultimately show ?thesis by auto
 qed
 
 lemma interior_atLeastAtMost_real [simp]: "interior {a..b} = {a<..<b :: real}"
 proof-
   have "{a..b} = {a..} \<inter> {..b}" by auto
   also have "interior \<dots> = {a<..} \<inter> {..<b}"
     by (simp add: interior_real_atLeast interior_real_atMost)
   also have "\<dots> = {a<..<b}" by auto
   finally show ?thesis .
 qed
 
 lemma interior_atLeastLessThan [simp]:
   fixes a::real shows "interior {a..<b} = {a<..<b}"
   by (metis atLeastLessThan_def greaterThanLessThan_def interior_atLeastAtMost_real interior_Int interior_interior interior_real_atLeast)
 
 lemma interior_lessThanAtMost [simp]:
   fixes a::real shows "interior {a<..b} = {a<..<b}"
   by (metis atLeastAtMost_def greaterThanAtMost_def interior_atLeastAtMost_real interior_Int
             interior_interior interior_real_atLeast)
 
 lemma interior_greaterThanLessThan_real [simp]: "interior {a<..<b} = {a<..<b :: real}"
   by (metis interior_atLeastAtMost_real interior_interior)
 
 lemma frontier_real_atMost [simp]:
   fixes a :: real
   shows "frontier {..a} = {a}"
   unfolding frontier_def by (auto simp: interior_real_atMost)
 
 lemma frontier_real_atLeast [simp]: "frontier {a..} = {a::real}"
   by (auto simp: frontier_def)
 
 lemma frontier_real_greaterThan [simp]: "frontier {a<..} = {a::real}"
   by (auto simp: interior_open frontier_def)
 
 lemma frontier_real_lessThan [simp]: "frontier {..<a} = {a::real}"
   by (auto simp: interior_open frontier_def)
 
 lemma rel_interior_real_box [simp]:
   fixes a b :: real
   assumes "a < b"
   shows "rel_interior {a .. b} = {a <..< b}"
 proof -
   have "box a b \<noteq> {}"
     using assms
     unfolding set_eq_iff
     by (auto intro!: exI[of _ "(a + b) / 2"] simp: box_def)
   then show ?thesis
     using interior_rel_interior_gen[of "cbox a b", symmetric]
     by (simp split: if_split_asm del: box_real add: box_real[symmetric] interior_cbox)
 qed
 
 lemma rel_interior_real_semiline [simp]:
   fixes a :: real
   shows "rel_interior {a..} = {a<..}"
 proof -
   have *: "{a<..} \<noteq> {}"
     unfolding set_eq_iff by (auto intro!: exI[of _ "a + 1"])
   then show ?thesis using interior_real_atLeast interior_rel_interior_gen[of "{a..}"]
     by (auto split: if_split_asm)
 qed
 
 subsubsection \<open>Relative open sets\<close>
 
 definition\<^marker>\<open>tag important\<close> "rel_open S \<longleftrightarrow> rel_interior S = S"
 
 lemma rel_open: "rel_open S \<longleftrightarrow> openin (top_of_set (affine hull S)) S"
   unfolding rel_open_def rel_interior_def
   apply auto
   using openin_subopen[of "top_of_set (affine hull S)" S]
   apply auto
   done
 
 lemma openin_rel_interior: "openin (top_of_set (affine hull S)) (rel_interior S)"
   apply (simp add: rel_interior_def)
   apply (subst openin_subopen, blast)
   done
 
 lemma openin_set_rel_interior:
    "openin (top_of_set S) (rel_interior S)"
 by (rule openin_subset_trans [OF openin_rel_interior rel_interior_subset hull_subset])
 
 lemma affine_rel_open:
   fixes S :: "'n::euclidean_space set"
   assumes "affine S"
   shows "rel_open S"
   unfolding rel_open_def
   using assms rel_interior_affine_hull[of S] affine_hull_eq[of S]
   by metis
 
 lemma affine_closed:
   fixes S :: "'n::euclidean_space set"
   assumes "affine S"
   shows "closed S"
 proof -
   {
     assume "S \<noteq> {}"
     then obtain L where L: "subspace L" "affine_parallel S L"
       using assms affine_parallel_subspace[of S] by auto
     then obtain a where a: "S = ((+) a ` L)"
       using affine_parallel_def[of L S] affine_parallel_commut by auto
     from L have "closed L" using closed_subspace by auto
     then have "closed S"
       using closed_translation a by auto
   }
   then show ?thesis by auto
 qed
 
 lemma closure_affine_hull:
   fixes S :: "'n::euclidean_space set"
   shows "closure S \<subseteq> affine hull S"
   by (intro closure_minimal hull_subset affine_closed affine_affine_hull)
 
 lemma closed_affine_hull [iff]:
   fixes S :: "'n::euclidean_space set"
   shows "closed (affine hull S)"
   by (metis affine_affine_hull affine_closed)
 
 lemma closure_same_affine_hull [simp]:
   fixes S :: "'n::euclidean_space set"
   shows "affine hull (closure S) = affine hull S"
 proof -
   have "affine hull (closure S) \<subseteq> affine hull S"
     using hull_mono[of "closure S" "affine hull S" "affine"]
       closure_affine_hull[of S] hull_hull[of "affine" S]
     by auto
   moreover have "affine hull (closure S) \<supseteq> affine hull S"
     using hull_mono[of "S" "closure S" "affine"] closure_subset by auto
   ultimately show ?thesis by auto
 qed
 
 lemma closure_aff_dim [simp]:
   fixes S :: "'n::euclidean_space set"
   shows "aff_dim (closure S) = aff_dim S"
 proof -
   have "aff_dim S \<le> aff_dim (closure S)"
     using aff_dim_subset closure_subset by auto
   moreover have "aff_dim (closure S) \<le> aff_dim (affine hull S)"
     using aff_dim_subset closure_affine_hull by blast
   moreover have "aff_dim (affine hull S) = aff_dim S"
     using aff_dim_affine_hull by auto
   ultimately show ?thesis by auto
 qed
 
 lemma rel_interior_closure_convex_shrink:
   fixes S :: "_::euclidean_space set"
   assumes "convex S"
     and "c \<in> rel_interior S"
     and "x \<in> closure S"
     and "e > 0"
     and "e \<le> 1"
   shows "x - e *\<^sub>R (x - c) \<in> rel_interior S"
 proof -
   obtain d where "d > 0" and d: "ball c d \<inter> affine hull S \<subseteq> S"
     using assms(2) unfolding mem_rel_interior_ball by auto
   have "\<exists>y \<in> S. norm (y - x) * (1 - e) < e * d"
   proof (cases "x \<in> S")
     case True
     then show ?thesis using \<open>e > 0\<close> \<open>d > 0\<close>
       apply (rule_tac bexI[where x=x], auto)
       done
   next
     case False
     then have x: "x islimpt S"
       using assms(3)[unfolded closure_def] by auto
     show ?thesis
     proof (cases "e = 1")
       case True
       obtain y where "y \<in> S" "y \<noteq> x" "dist y x < 1"
         using x[unfolded islimpt_approachable,THEN spec[where x=1]] by auto
       then show ?thesis
         apply (rule_tac x=y in bexI)
         unfolding True
         using \<open>d > 0\<close>
         apply auto
         done
     next
       case False
       then have "0 < e * d / (1 - e)" and *: "1 - e > 0"
         using \<open>e \<le> 1\<close> \<open>e > 0\<close> \<open>d > 0\<close> by auto
       then obtain y where "y \<in> S" "y \<noteq> x" "dist y x < e * d / (1 - e)"
         using x[unfolded islimpt_approachable,THEN spec[where x="e*d / (1 - e)"]] by auto
       then show ?thesis
         apply (rule_tac x=y in bexI)
         unfolding dist_norm
         using pos_less_divide_eq[OF *]
         apply auto
         done
     qed
   qed
   then obtain y where "y \<in> S" and y: "norm (y - x) * (1 - e) < e * d"
     by auto
   define z where "z = c + ((1 - e) / e) *\<^sub>R (x - y)"
   have *: "x - e *\<^sub>R (x - c) = y - e *\<^sub>R (y - z)"
     unfolding z_def using \<open>e > 0\<close>
     by (auto simp: scaleR_right_diff_distrib scaleR_right_distrib scaleR_left_diff_distrib)
   have zball: "z \<in> ball c d"
     using mem_ball z_def dist_norm[of c]
     using y and assms(4,5)
     by (simp add: norm_minus_commute) (simp add: field_simps)
   have "x \<in> affine hull S"
     using closure_affine_hull assms by auto
   moreover have "y \<in> affine hull S"
     using \<open>y \<in> S\<close> hull_subset[of S] by auto
   moreover have "c \<in> affine hull S"
     using assms rel_interior_subset hull_subset[of S] by auto
   ultimately have "z \<in> affine hull S"
     using z_def affine_affine_hull[of S]
       mem_affine_3_minus [of "affine hull S" c x y "(1 - e) / e"]
       assms
     by simp
   then have "z \<in> S" using d zball by auto
   obtain d1 where "d1 > 0" and d1: "ball z d1 \<le> ball c d"
     using zball open_ball[of c d] openE[of "ball c d" z] by auto
   then have "ball z d1 \<inter> affine hull S \<subseteq> ball c d \<inter> affine hull S"
     by auto
   then have "ball z d1 \<inter> affine hull S \<subseteq> S"
     using d by auto
   then have "z \<in> rel_interior S"
     using mem_rel_interior_ball using \<open>d1 > 0\<close> \<open>z \<in> S\<close> by auto
   then have "y - e *\<^sub>R (y - z) \<in> rel_interior S"
     using rel_interior_convex_shrink[of S z y e] assms \<open>y \<in> S\<close> by auto
   then show ?thesis using * by auto
 qed
 
 lemma rel_interior_eq:
    "rel_interior s = s \<longleftrightarrow> openin(top_of_set (affine hull s)) s"
 using rel_open rel_open_def by blast
 
 lemma rel_interior_openin:
    "openin(top_of_set (affine hull s)) s \<Longrightarrow> rel_interior s = s"
 by (simp add: rel_interior_eq)
 
 lemma rel_interior_affine:
   fixes S :: "'n::euclidean_space set"
   shows  "affine S \<Longrightarrow> rel_interior S = S"
 using affine_rel_open rel_open_def by auto
 
 lemma rel_interior_eq_closure:
   fixes S :: "'n::euclidean_space set"
   shows "rel_interior S = closure S \<longleftrightarrow> affine S"
 proof (cases "S = {}")
   case True
  then show ?thesis
     by auto
 next
   case False show ?thesis
   proof
     assume eq: "rel_interior S = closure S"
     have "S = {} \<or> S = affine hull S"
       apply (rule connected_clopen [THEN iffD1, rule_format])
        apply (simp add: affine_imp_convex convex_connected)
       apply (rule conjI)
        apply (metis eq closure_subset openin_rel_interior rel_interior_subset subset_antisym)
       apply (metis closed_subset closure_subset_eq eq hull_subset rel_interior_subset)
       done
     with False have "affine hull S = S"
       by auto
     then show "affine S"
       by (metis affine_hull_eq)
   next
     assume "affine S"
     then show "rel_interior S = closure S"
       by (simp add: rel_interior_affine affine_closed)
   qed
 qed
 
 
 subsubsection\<^marker>\<open>tag unimportant\<close>\<open>Relative interior preserves under linear transformations\<close>
 
 lemma rel_interior_translation_aux:
   fixes a :: "'n::euclidean_space"
   shows "((\<lambda>x. a + x) ` rel_interior S) \<subseteq> rel_interior ((\<lambda>x. a + x) ` S)"
 proof -
   {
     fix x
     assume x: "x \<in> rel_interior S"
     then obtain T where "open T" "x \<in> T \<inter> S" "T \<inter> affine hull S \<subseteq> S"
       using mem_rel_interior[of x S] by auto
     then have "open ((\<lambda>x. a + x) ` T)"
       and "a + x \<in> ((\<lambda>x. a + x) ` T) \<inter> ((\<lambda>x. a + x) ` S)"
       and "((\<lambda>x. a + x) ` T) \<inter> affine hull ((\<lambda>x. a + x) ` S) \<subseteq> (\<lambda>x. a + x) ` S"
       using affine_hull_translation[of a S] open_translation[of T a] x by auto
     then have "a + x \<in> rel_interior ((\<lambda>x. a + x) ` S)"
       using mem_rel_interior[of "a+x" "((\<lambda>x. a + x) ` S)"] by auto
   }
   then show ?thesis by auto
 qed
 
 lemma rel_interior_translation:
   fixes a :: "'n::euclidean_space"
   shows "rel_interior ((\<lambda>x. a + x) ` S) = (\<lambda>x. a + x) ` rel_interior S"
 proof -
   have "(\<lambda>x. (-a) + x) ` rel_interior ((\<lambda>x. a + x) ` S) \<subseteq> rel_interior S"
     using rel_interior_translation_aux[of "-a" "(\<lambda>x. a + x) ` S"]
       translation_assoc[of "-a" "a"]
     by auto
   then have "((\<lambda>x. a + x) ` rel_interior S) \<supseteq> rel_interior ((\<lambda>x. a + x) ` S)"
     using translation_inverse_subset[of a "rel_interior ((+) a ` S)" "rel_interior S"]
     by auto
   then show ?thesis
     using rel_interior_translation_aux[of a S] by auto
 qed
 
 
 lemma affine_hull_linear_image:
   assumes "bounded_linear f"
   shows "f ` (affine hull s) = affine hull f ` s"
 proof -
   interpret f: bounded_linear f by fact
   have "affine {x. f x \<in> affine hull f ` s}"
     unfolding affine_def
     by (auto simp: f.scaleR f.add affine_affine_hull[unfolded affine_def, rule_format])
   moreover have "affine {x. x \<in> f ` (affine hull s)}"
     using affine_affine_hull[unfolded affine_def, of s]
     unfolding affine_def by (auto simp: f.scaleR [symmetric] f.add [symmetric])
   ultimately show ?thesis
     by (auto simp: hull_inc elim!: hull_induct)
 qed 
 
 
 lemma rel_interior_injective_on_span_linear_image:
   fixes f :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space"
     and S :: "'m::euclidean_space set"
   assumes "bounded_linear f"
     and "inj_on f (span S)"
   shows "rel_interior (f ` S) = f ` (rel_interior S)"
 proof -
   {
     fix z
     assume z: "z \<in> rel_interior (f ` S)"
     then have "z \<in> f ` S"
       using rel_interior_subset[of "f ` S"] by auto
     then obtain x where x: "x \<in> S" "f x = z" by auto
     obtain e2 where e2: "e2 > 0" "cball z e2 \<inter> affine hull (f ` S) \<subseteq> (f ` S)"
       using z rel_interior_cball[of "f ` S"] by auto
     obtain K where K: "K > 0" "\<And>x. norm (f x) \<le> norm x * K"
      using assms Real_Vector_Spaces.bounded_linear.pos_bounded[of f] by auto
     define e1 where "e1 = 1 / K"
     then have e1: "e1 > 0" "\<And>x. e1 * norm (f x) \<le> norm x"
       using K pos_le_divide_eq[of e1] by auto
     define e where "e = e1 * e2"
     then have "e > 0" using e1 e2 by auto
     {
       fix y
       assume y: "y \<in> cball x e \<inter> affine hull S"
       then have h1: "f y \<in> affine hull (f ` S)"
         using affine_hull_linear_image[of f S] assms by auto
       from y have "norm (x-y) \<le> e1 * e2"
         using cball_def[of x e] dist_norm[of x y] e_def by auto
       moreover have "f x - f y = f (x - y)"
         using assms linear_diff[of f x y] linear_conv_bounded_linear[of f] by auto
       moreover have "e1 * norm (f (x-y)) \<le> norm (x - y)"
         using e1 by auto
       ultimately have "e1 * norm ((f x)-(f y)) \<le> e1 * e2"
         by auto
       then have "f y \<in> cball z e2"
         using cball_def[of "f x" e2] dist_norm[of "f x" "f y"] e1 x by auto
       then have "f y \<in> f ` S"
         using y e2 h1 by auto
       then have "y \<in> S"
         using assms y hull_subset[of S] affine_hull_subset_span
           inj_on_image_mem_iff [OF \<open>inj_on f (span S)\<close>]
         by (metis Int_iff span_superset subsetCE)
     }
     then have "z \<in> f ` (rel_interior S)"
       using mem_rel_interior_cball[of x S] \<open>e > 0\<close> x by auto
   }
   moreover
   {
     fix x
     assume x: "x \<in> rel_interior S"
     then obtain e2 where e2: "e2 > 0" "cball x e2 \<inter> affine hull S \<subseteq> S"
       using rel_interior_cball[of S] by auto
     have "x \<in> S" using x rel_interior_subset by auto
     then have *: "f x \<in> f ` S" by auto
     have "\<forall>x\<in>span S. f x = 0 \<longrightarrow> x = 0"
       using assms subspace_span linear_conv_bounded_linear[of f]
         linear_injective_on_subspace_0[of f "span S"]
       by auto
     then obtain e1 where e1: "e1 > 0" "\<forall>x \<in> span S. e1 * norm x \<le> norm (f x)"
       using assms injective_imp_isometric[of "span S" f]
         subspace_span[of S] closed_subspace[of "span S"]
       by auto
     define e where "e = e1 * e2"
     hence "e > 0" using e1 e2 by auto
     {
       fix y
       assume y: "y \<in> cball (f x) e \<inter> affine hull (f ` S)"
       then have "y \<in> f ` (affine hull S)"
         using affine_hull_linear_image[of f S] assms by auto
       then obtain xy where xy: "xy \<in> affine hull S" "f xy = y" by auto
       with y have "norm (f x - f xy) \<le> e1 * e2"
         using cball_def[of "f x" e] dist_norm[of "f x" y] e_def by auto
       moreover have "f x - f xy = f (x - xy)"
         using assms linear_diff[of f x xy] linear_conv_bounded_linear[of f] by auto
       moreover have *: "x - xy \<in> span S"
         using subspace_diff[of "span S" x xy] subspace_span \<open>x \<in> S\<close> xy
           affine_hull_subset_span[of S] span_superset
         by auto
       moreover from * have "e1 * norm (x - xy) \<le> norm (f (x - xy))"
         using e1 by auto
       ultimately have "e1 * norm (x - xy) \<le> e1 * e2"
         by auto
       then have "xy \<in> cball x e2"
         using cball_def[of x e2] dist_norm[of x xy] e1 by auto
       then have "y \<in> f ` S"
         using xy e2 by auto
     }
     then have "f x \<in> rel_interior (f ` S)"
       using mem_rel_interior_cball[of "(f x)" "(f ` S)"] * \<open>e > 0\<close> by auto
   }
   ultimately show ?thesis by auto
 qed
 
 lemma rel_interior_injective_linear_image:
   fixes f :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space"
   assumes "bounded_linear f"
     and "inj f"
   shows "rel_interior (f ` S) = f ` (rel_interior S)"
   using assms rel_interior_injective_on_span_linear_image[of f S]
     subset_inj_on[of f "UNIV" "span S"]
   by auto
 
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>Openness and compactness are preserved by convex hull operation\<close>
 
 lemma open_convex_hull[intro]:
   fixes S :: "'a::real_normed_vector set"
   assumes "open S"
   shows "open (convex hull S)"
 proof (clarsimp simp: open_contains_cball convex_hull_explicit)
   fix T and u :: "'a\<Rightarrow>real"
   assume obt: "finite T" "T\<subseteq>S" "\<forall>x\<in>T. 0 \<le> u x" "sum u T = 1" 
 
   from assms[unfolded open_contains_cball] obtain b
     where b: "\<And>x. x\<in>S \<Longrightarrow> 0 < b x \<and> cball x (b x) \<subseteq> S" by metis
   have "b ` T \<noteq> {}"
     using obt by auto
   define i where "i = b ` T"
   let ?\<Phi> = "\<lambda>y. \<exists>F. finite F \<and> F \<subseteq> S \<and> (\<exists>u. (\<forall>x\<in>F. 0 \<le> u x) \<and> sum u F = 1 \<and> (\<Sum>v\<in>F. u v *\<^sub>R v) = y)"
   let ?a = "\<Sum>v\<in>T. u v *\<^sub>R v"
   show "\<exists>e > 0. cball ?a e \<subseteq> {y. ?\<Phi> y}"
   proof (intro exI subsetI conjI)
     show "0 < Min i"
       unfolding i_def and Min_gr_iff[OF finite_imageI[OF obt(1)] \<open>b ` T\<noteq>{}\<close>]
       using b \<open>T\<subseteq>S\<close> by auto
   next
     fix y
     assume "y \<in> cball ?a (Min i)"
     then have y: "norm (?a - y) \<le> Min i"
       unfolding dist_norm[symmetric] by auto
     { fix x
       assume "x \<in> T"
       then have "Min i \<le> b x"
         by (simp add: i_def obt(1))
       then have "x + (y - ?a) \<in> cball x (b x)"
         using y unfolding mem_cball dist_norm by auto
       moreover have "x \<in> S"
         using \<open>x\<in>T\<close> \<open>T\<subseteq>S\<close> by auto
       ultimately have "x + (y - ?a) \<in> S"
         using y b by blast
     }
     moreover
     have *: "inj_on (\<lambda>v. v + (y - ?a)) T"
       unfolding inj_on_def by auto
     have "(\<Sum>v\<in>(\<lambda>v. v + (y - ?a)) ` T. u (v - (y - ?a)) *\<^sub>R v) = y"
       unfolding sum.reindex[OF *] o_def using obt(4)
       by (simp add: sum.distrib sum_subtractf scaleR_left.sum[symmetric] scaleR_right_distrib)
     ultimately show "y \<in> {y. ?\<Phi> y}"
     proof (intro CollectI exI conjI)
       show "finite ((\<lambda>v. v + (y - ?a)) ` T)"
         by (simp add: obt(1))
       show "sum (\<lambda>v. u (v - (y - ?a))) ((\<lambda>v. v + (y - ?a)) ` T) = 1"
         unfolding sum.reindex[OF *] o_def using obt(4) by auto
     qed (use obt(1, 3) in auto)
   qed
 qed
 
 lemma compact_convex_combinations:
   fixes S T :: "'a::real_normed_vector set"
   assumes "compact S" "compact T"
   shows "compact { (1 - u) *\<^sub>R x + u *\<^sub>R y | x y u. 0 \<le> u \<and> u \<le> 1 \<and> x \<in> S \<and> y \<in> T}"
 proof -
   let ?X = "{0..1} \<times> S \<times> T"
   let ?h = "(\<lambda>z. (1 - fst z) *\<^sub>R fst (snd z) + fst z *\<^sub>R snd (snd z))"
   have *: "{ (1 - u) *\<^sub>R x + u *\<^sub>R y | x y u. 0 \<le> u \<and> u \<le> 1 \<and> x \<in> S \<and> y \<in> T} = ?h ` ?X"
     by force
   have "continuous_on ?X (\<lambda>z. (1 - fst z) *\<^sub>R fst (snd z) + fst z *\<^sub>R snd (snd z))"
     unfolding continuous_on by (rule ballI) (intro tendsto_intros)
   with assms show ?thesis
     by (simp add: * compact_Times compact_continuous_image)
 qed
 
 lemma finite_imp_compact_convex_hull:
   fixes S :: "'a::real_normed_vector set"
   assumes "finite S"
   shows "compact (convex hull S)"
 proof (cases "S = {}")
   case True
   then show ?thesis by simp
 next
   case False
   with assms show ?thesis
   proof (induct rule: finite_ne_induct)
     case (singleton x)
     show ?case by simp
   next
     case (insert x A)
     let ?f = "\<lambda>(u, y::'a). u *\<^sub>R x + (1 - u) *\<^sub>R y"
     let ?T = "{0..1::real} \<times> (convex hull A)"
     have "continuous_on ?T ?f"
       unfolding split_def continuous_on by (intro ballI tendsto_intros)
     moreover have "compact ?T"
       by (intro compact_Times compact_Icc insert)
     ultimately have "compact (?f ` ?T)"
       by (rule compact_continuous_image)
     also have "?f ` ?T = convex hull (insert x A)"
       unfolding convex_hull_insert [OF \<open>A \<noteq> {}\<close>]
       apply safe
       apply (rule_tac x=a in exI, simp)
       apply (rule_tac x="1 - a" in exI, simp, fast)
       apply (rule_tac x="(u, b)" in image_eqI, simp_all)
       done
     finally show "compact (convex hull (insert x A))" .
   qed
 qed
 
 lemma compact_convex_hull:
   fixes S :: "'a::euclidean_space set"
   assumes "compact S"
   shows "compact (convex hull S)"
 proof (cases "S = {}")
   case True
   then show ?thesis using compact_empty by simp
 next
   case False
   then obtain w where "w \<in> S" by auto
   show ?thesis
     unfolding caratheodory[of S]
   proof (induct ("DIM('a) + 1"))
     case 0
     have *: "{x.\<exists>sa. finite sa \<and> sa \<subseteq> S \<and> card sa \<le> 0 \<and> x \<in> convex hull sa} = {}"
       using compact_empty by auto
     from 0 show ?case unfolding * by simp
   next
     case (Suc n)
     show ?case
     proof (cases "n = 0")
       case True
       have "{x. \<exists>T. finite T \<and> T \<subseteq> S \<and> card T \<le> Suc n \<and> x \<in> convex hull T} = S"
         unfolding set_eq_iff and mem_Collect_eq
       proof (rule, rule)
         fix x
         assume "\<exists>T. finite T \<and> T \<subseteq> S \<and> card T \<le> Suc n \<and> x \<in> convex hull T"
         then obtain T where T: "finite T" "T \<subseteq> S" "card T \<le> Suc n" "x \<in> convex hull T"
           by auto
         show "x \<in> S"
         proof (cases "card T = 0")
           case True
           then show ?thesis
             using T(4) unfolding card_0_eq[OF T(1)] by simp
         next
           case False
           then have "card T = Suc 0" using T(3) \<open>n=0\<close> by auto
           then obtain a where "T = {a}" unfolding card_Suc_eq by auto
           then show ?thesis using T(2,4) by simp
         qed
       next
         fix x assume "x\<in>S"
         then show "\<exists>T. finite T \<and> T \<subseteq> S \<and> card T \<le> Suc n \<and> x \<in> convex hull T"
           apply (rule_tac x="{x}" in exI)
           unfolding convex_hull_singleton
           apply auto
           done
       qed
       then show ?thesis using assms by simp
     next
       case False
       have "{x. \<exists>T. finite T \<and> T \<subseteq> S \<and> card T \<le> Suc n \<and> x \<in> convex hull T} =
         {(1 - u) *\<^sub>R x + u *\<^sub>R y | x y u.
           0 \<le> u \<and> u \<le> 1 \<and> x \<in> S \<and> y \<in> {x. \<exists>T. finite T \<and> T \<subseteq> S \<and> card T \<le> n \<and> x \<in> convex hull T}}"
         unfolding set_eq_iff and mem_Collect_eq
       proof (rule, rule)
         fix x
         assume "\<exists>u v c. x = (1 - c) *\<^sub>R u + c *\<^sub>R v \<and>
           0 \<le> c \<and> c \<le> 1 \<and> u \<in> S \<and> (\<exists>T. finite T \<and> T \<subseteq> S \<and> card T \<le> n \<and> v \<in> convex hull T)"
         then obtain u v c T where obt: "x = (1 - c) *\<^sub>R u + c *\<^sub>R v"
           "0 \<le> c \<and> c \<le> 1" "u \<in> S" "finite T" "T \<subseteq> S" "card T \<le> n"  "v \<in> convex hull T"
           by auto
         moreover have "(1 - c) *\<^sub>R u + c *\<^sub>R v \<in> convex hull insert u T"
           apply (rule convexD_alt)
           using obt(2) and convex_convex_hull and hull_subset[of "insert u T" convex]
           using obt(7) and hull_mono[of T "insert u T"]
           apply auto
           done
         ultimately show "\<exists>T. finite T \<and> T \<subseteq> S \<and> card T \<le> Suc n \<and> x \<in> convex hull T"
           apply (rule_tac x="insert u T" in exI)
           apply (auto simp: card_insert_if)
           done
       next
         fix x
         assume "\<exists>T. finite T \<and> T \<subseteq> S \<and> card T \<le> Suc n \<and> x \<in> convex hull T"
         then obtain T where T: "finite T" "T \<subseteq> S" "card T \<le> Suc n" "x \<in> convex hull T"
           by auto
         show "\<exists>u v c. x = (1 - c) *\<^sub>R u + c *\<^sub>R v \<and>
           0 \<le> c \<and> c \<le> 1 \<and> u \<in> S \<and> (\<exists>T. finite T \<and> T \<subseteq> S \<and> card T \<le> n \<and> v \<in> convex hull T)"
         proof (cases "card T = Suc n")
           case False
           then have "card T \<le> n" using T(3) by auto
           then show ?thesis
             apply (rule_tac x=w in exI, rule_tac x=x in exI, rule_tac x=1 in exI)
             using \<open>w\<in>S\<close> and T
             apply (auto intro!: exI[where x=T])
             done
         next
           case True
           then obtain a u where au: "T = insert a u" "a\<notin>u"
             apply (drule_tac card_eq_SucD, auto)
             done
           show ?thesis
           proof (cases "u = {}")
             case True
             then have "x = a" using T(4)[unfolded au] by auto
             show ?thesis unfolding \<open>x = a\<close>
               apply (rule_tac x=a in exI)
               apply (rule_tac x=a in exI)
               apply (rule_tac x=1 in exI)
               using T and \<open>n \<noteq> 0\<close>
               unfolding au
               apply (auto intro!: exI[where x="{a}"])
               done
           next
             case False
             obtain ux vx b where obt: "ux\<ge>0" "vx\<ge>0" "ux + vx = 1"
               "b \<in> convex hull u" "x = ux *\<^sub>R a + vx *\<^sub>R b"
               using T(4)[unfolded au convex_hull_insert[OF False]]
               by auto
             have *: "1 - vx = ux" using obt(3) by auto
             show ?thesis
               apply (rule_tac x=a in exI)
               apply (rule_tac x=b in exI)
               apply (rule_tac x=vx in exI)
               using obt and T(1-3)
               unfolding au and * using card_insert_disjoint[OF _ au(2)]
               apply (auto intro!: exI[where x=u])
               done
           qed
         qed
       qed
       then show ?thesis
         using compact_convex_combinations[OF assms Suc] by simp
     qed
   qed
 qed
 
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>Extremal points of a simplex are some vertices\<close>
 
 lemma dist_increases_online:
   fixes a b d :: "'a::real_inner"
   assumes "d \<noteq> 0"
   shows "dist a (b + d) > dist a b \<or> dist a (b - d) > dist a b"
 proof (cases "inner a d - inner b d > 0")
   case True
   then have "0 < inner d d + (inner a d * 2 - inner b d * 2)"
     apply (rule_tac add_pos_pos)
     using assms
     apply auto
     done
   then show ?thesis
     apply (rule_tac disjI2)
     unfolding dist_norm and norm_eq_sqrt_inner and real_sqrt_less_iff
     apply  (simp add: algebra_simps inner_commute)
     done
 next
   case False
   then have "0 < inner d d + (inner b d * 2 - inner a d * 2)"
     apply (rule_tac add_pos_nonneg)
     using assms
     apply auto
     done
   then show ?thesis
     apply (rule_tac disjI1)
     unfolding dist_norm and norm_eq_sqrt_inner and real_sqrt_less_iff
     apply (simp add: algebra_simps inner_commute)
     done
 qed
 
 lemma norm_increases_online:
   fixes d :: "'a::real_inner"
   shows "d \<noteq> 0 \<Longrightarrow> norm (a + d) > norm a \<or> norm(a - d) > norm a"
   using dist_increases_online[of d a 0] unfolding dist_norm by auto
 
 lemma simplex_furthest_lt:
   fixes S :: "'a::real_inner set"
   assumes "finite S"
   shows "\<forall>x \<in> convex hull S.  x \<notin> S \<longrightarrow> (\<exists>y \<in> convex hull S. norm (x - a) < norm(y - a))"
   using assms
 proof induct
   fix x S
   assume as: "finite S" "x\<notin>S" "\<forall>x\<in>convex hull S. x \<notin> S \<longrightarrow> (\<exists>y\<in>convex hull S. norm (x - a) < norm (y - a))"
   show "\<forall>xa\<in>convex hull insert x S. xa \<notin> insert x S \<longrightarrow>
     (\<exists>y\<in>convex hull insert x S. norm (xa - a) < norm (y - a))"
   proof (intro impI ballI, cases "S = {}")
     case False
     fix y
     assume y: "y \<in> convex hull insert x S" "y \<notin> insert x S"
     obtain u v b where obt: "u\<ge>0" "v\<ge>0" "u + v = 1" "b \<in> convex hull S" "y = u *\<^sub>R x + v *\<^sub>R b"
       using y(1)[unfolded convex_hull_insert[OF False]] by auto
     show "\<exists>z\<in>convex hull insert x S. norm (y - a) < norm (z - a)"
     proof (cases "y \<in> convex hull S")
       case True
       then obtain z where "z \<in> convex hull S" "norm (y - a) < norm (z - a)"
         using as(3)[THEN bspec[where x=y]] and y(2) by auto
       then show ?thesis
         apply (rule_tac x=z in bexI)
         unfolding convex_hull_insert[OF False]
         apply auto
         done
     next
       case False
       show ?thesis
         using obt(3)
       proof (cases "u = 0", case_tac[!] "v = 0")
         assume "u = 0" "v \<noteq> 0"
         then have "y = b" using obt by auto
         then show ?thesis using False and obt(4) by auto
       next
         assume "u \<noteq> 0" "v = 0"
         then have "y = x" using obt by auto
         then show ?thesis using y(2) by auto
       next
         assume "u \<noteq> 0" "v \<noteq> 0"
         then obtain w where w: "w>0" "w<u" "w<v"
           using field_lbound_gt_zero[of u v] and obt(1,2) by auto
         have "x \<noteq> b"
         proof
           assume "x = b"
           then have "y = b" unfolding obt(5)
             using obt(3) by (auto simp: scaleR_left_distrib[symmetric])
           then show False using obt(4) and False by simp
         qed
         then have *: "w *\<^sub>R (x - b) \<noteq> 0" using w(1) by auto
         show ?thesis
           using dist_increases_online[OF *, of a y]
         proof (elim disjE)
           assume "dist a y < dist a (y + w *\<^sub>R (x - b))"
           then have "norm (y - a) < norm ((u + w) *\<^sub>R x + (v - w) *\<^sub>R b - a)"
             unfolding dist_commute[of a]
             unfolding dist_norm obt(5)
             by (simp add: algebra_simps)
           moreover have "(u + w) *\<^sub>R x + (v - w) *\<^sub>R b \<in> convex hull insert x S"
             unfolding convex_hull_insert[OF \<open>S\<noteq>{}\<close>]
           proof (intro CollectI conjI exI)
             show "u + w \<ge> 0" "v - w \<ge> 0"
               using obt(1) w by auto
           qed (use obt in auto)
           ultimately show ?thesis by auto
         next
           assume "dist a y < dist a (y - w *\<^sub>R (x - b))"
           then have "norm (y - a) < norm ((u - w) *\<^sub>R x + (v + w) *\<^sub>R b - a)"
             unfolding dist_commute[of a]
             unfolding dist_norm obt(5)
             by (simp add: algebra_simps)
           moreover have "(u - w) *\<^sub>R x + (v + w) *\<^sub>R b \<in> convex hull insert x S"
             unfolding convex_hull_insert[OF \<open>S\<noteq>{}\<close>]
           proof (intro CollectI conjI exI)
             show "u - w \<ge> 0" "v + w \<ge> 0"
               using obt(1) w by auto
           qed (use obt in auto)
           ultimately show ?thesis by auto
         qed
       qed auto
     qed
   qed auto
 qed (auto simp: assms)
 
 lemma simplex_furthest_le:
   fixes S :: "'a::real_inner set"
   assumes "finite S"
     and "S \<noteq> {}"
   shows "\<exists>y\<in>S. \<forall>x\<in> convex hull S. norm (x - a) \<le> norm (y - a)"
 proof -
   have "convex hull S \<noteq> {}"
     using hull_subset[of S convex] and assms(2) by auto
   then obtain x where x: "x \<in> convex hull S" "\<forall>y\<in>convex hull S. norm (y - a) \<le> norm (x - a)"
     using distance_attains_sup[OF finite_imp_compact_convex_hull[OF \<open>finite S\<close>], of a]
     unfolding dist_commute[of a]
     unfolding dist_norm
     by auto
   show ?thesis
   proof (cases "x \<in> S")
     case False
     then obtain y where "y \<in> convex hull S" "norm (x - a) < norm (y - a)"
       using simplex_furthest_lt[OF assms(1), THEN bspec[where x=x]] and x(1)
       by auto
     then show ?thesis
       using x(2)[THEN bspec[where x=y]] by auto
   next
     case True
     with x show ?thesis by auto
   qed
 qed
 
 lemma simplex_furthest_le_exists:
   fixes S :: "('a::real_inner) set"
   shows "finite S \<Longrightarrow> \<forall>x\<in>(convex hull S). \<exists>y\<in>S. norm (x - a) \<le> norm (y - a)"
   using simplex_furthest_le[of S] by (cases "S = {}") auto
 
 lemma simplex_extremal_le:
   fixes S :: "'a::real_inner set"
   assumes "finite S"
     and "S \<noteq> {}"
   shows "\<exists>u\<in>S. \<exists>v\<in>S. \<forall>x\<in>convex hull S. \<forall>y \<in> convex hull S. norm (x - y) \<le> norm (u - v)"
 proof -
   have "convex hull S \<noteq> {}"
     using hull_subset[of S convex] and assms(2) by auto
   then obtain u v where obt: "u \<in> convex hull S" "v \<in> convex hull S"
     "\<forall>x\<in>convex hull S. \<forall>y\<in>convex hull S. norm (x - y) \<le> norm (u - v)"
     using compact_sup_maxdistance[OF finite_imp_compact_convex_hull[OF assms(1)]]
     by (auto simp: dist_norm)
   then show ?thesis
   proof (cases "u\<notin>S \<or> v\<notin>S", elim disjE)
     assume "u \<notin> S"
     then obtain y where "y \<in> convex hull S" "norm (u - v) < norm (y - v)"
       using simplex_furthest_lt[OF assms(1), THEN bspec[where x=u]] and obt(1)
       by auto
     then show ?thesis
       using obt(3)[THEN bspec[where x=y], THEN bspec[where x=v]] and obt(2)
       by auto
   next
     assume "v \<notin> S"
     then obtain y where "y \<in> convex hull S" "norm (v - u) < norm (y - u)"
       using simplex_furthest_lt[OF assms(1), THEN bspec[where x=v]] and obt(2)
       by auto
     then show ?thesis
       using obt(3)[THEN bspec[where x=u], THEN bspec[where x=y]] and obt(1)
       by (auto simp: norm_minus_commute)
   qed auto
 qed
 
 lemma simplex_extremal_le_exists:
   fixes S :: "'a::real_inner set"
   shows "finite S \<Longrightarrow> x \<in> convex hull S \<Longrightarrow> y \<in> convex hull S \<Longrightarrow>
     \<exists>u\<in>S. \<exists>v\<in>S. norm (x - y) \<le> norm (u - v)"
   using convex_hull_empty simplex_extremal_le[of S]
   by(cases "S = {}") auto
 
 
 subsection \<open>Closest point of a convex set is unique, with a continuous projection\<close>
 
 definition\<^marker>\<open>tag important\<close> closest_point :: "'a::{real_inner,heine_borel} set \<Rightarrow> 'a \<Rightarrow> 'a"
   where "closest_point S a = (SOME x. x \<in> S \<and> (\<forall>y\<in>S. dist a x \<le> dist a y))"
 
 lemma closest_point_exists:
   assumes "closed S"
     and "S \<noteq> {}"
-  shows "closest_point S a \<in> S"
+  shows closest_point_in_set: "closest_point S a \<in> S"
     and "\<forall>y\<in>S. dist a (closest_point S a) \<le> dist a y"
   unfolding closest_point_def
   apply(rule_tac[!] someI2_ex)
   apply (auto intro: distance_attains_inf[OF assms(1,2), of a])
   done
 
-lemma closest_point_in_set: "closed S \<Longrightarrow> S \<noteq> {} \<Longrightarrow> closest_point S a \<in> S"
-  by (meson closest_point_exists)
-
 lemma closest_point_le: "closed S \<Longrightarrow> x \<in> S \<Longrightarrow> dist a (closest_point S a) \<le> dist a x"
   using closest_point_exists[of S] by auto
 
 lemma closest_point_self:
   assumes "x \<in> S"
   shows "closest_point S x = x"
   unfolding closest_point_def
   apply (rule some1_equality, rule ex1I[of _ x])
   using assms
   apply auto
   done
 
 lemma closest_point_refl: "closed S \<Longrightarrow> S \<noteq> {} \<Longrightarrow> closest_point S x = x \<longleftrightarrow> x \<in> S"
   using closest_point_in_set[of S x] closest_point_self[of x S]
   by auto
 
 lemma closer_points_lemma:
   assumes "inner y z > 0"
   shows "\<exists>u>0. \<forall>v>0. v \<le> u \<longrightarrow> norm(v *\<^sub>R z - y) < norm y"
 proof -
   have z: "inner z z > 0"
     unfolding inner_gt_zero_iff using assms by auto
   have "norm (v *\<^sub>R z - y) < norm y"
     if "0 < v" and "v \<le> inner y z / inner z z" for v
     unfolding norm_lt using z assms that
     by (simp add: field_simps inner_diff inner_commute mult_strict_left_mono[OF _ \<open>0<v\<close>])
   then show ?thesis
     using assms z
     by (rule_tac x = "inner y z / inner z z" in exI) auto
 qed
 
 lemma closer_point_lemma:
   assumes "inner (y - x) (z - x) > 0"
   shows "\<exists>u>0. u \<le> 1 \<and> dist (x + u *\<^sub>R (z - x)) y < dist x y"
 proof -
   obtain u where "u > 0"
     and u: "\<forall>v>0. v \<le> u \<longrightarrow> norm (v *\<^sub>R (z - x) - (y - x)) < norm (y - x)"
     using closer_points_lemma[OF assms] by auto
   show ?thesis
     apply (rule_tac x="min u 1" in exI)
     using u[THEN spec[where x="min u 1"]] and \<open>u > 0\<close>
     unfolding dist_norm by (auto simp: norm_minus_commute field_simps)
 qed
 
 lemma any_closest_point_dot:
   assumes "convex S" "closed S" "x \<in> S" "y \<in> S" "\<forall>z\<in>S. dist a x \<le> dist a z"
   shows "inner (a - x) (y - x) \<le> 0"
 proof (rule ccontr)
   assume "\<not> ?thesis"
   then obtain u where u: "u>0" "u\<le>1" "dist (x + u *\<^sub>R (y - x)) a < dist x a"
     using closer_point_lemma[of a x y] by auto
   let ?z = "(1 - u) *\<^sub>R x + u *\<^sub>R y"
   have "?z \<in> S"
     using convexD_alt[OF assms(1,3,4), of u] using u by auto
   then show False
     using assms(5)[THEN bspec[where x="?z"]] and u(3)
     by (auto simp: dist_commute algebra_simps)
 qed
 
 lemma any_closest_point_unique:
   fixes x :: "'a::real_inner"
   assumes "convex S" "closed S" "x \<in> S" "y \<in> S"
     "\<forall>z\<in>S. dist a x \<le> dist a z" "\<forall>z\<in>S. dist a y \<le> dist a z"
   shows "x = y"
   using any_closest_point_dot[OF assms(1-4,5)] and any_closest_point_dot[OF assms(1-2,4,3,6)]
   unfolding norm_pths(1) and norm_le_square
   by (auto simp: algebra_simps)
 
 lemma closest_point_unique:
   assumes "convex S" "closed S" "x \<in> S" "\<forall>z\<in>S. dist a x \<le> dist a z"
   shows "x = closest_point S a"
   using any_closest_point_unique[OF assms(1-3) _ assms(4), of "closest_point S a"]
   using closest_point_exists[OF assms(2)] and assms(3) by auto
 
 lemma closest_point_dot:
   assumes "convex S" "closed S" "x \<in> S"
   shows "inner (a - closest_point S a) (x - closest_point S a) \<le> 0"
   apply (rule any_closest_point_dot[OF assms(1,2) _ assms(3)])
   using closest_point_exists[OF assms(2)] and assms(3)
   apply auto
   done
 
 lemma closest_point_lt:
   assumes "convex S" "closed S" "x \<in> S" "x \<noteq> closest_point S a"
   shows "dist a (closest_point S a) < dist a x"
   apply (rule ccontr)
   apply (rule_tac notE[OF assms(4)])
   apply (rule closest_point_unique[OF assms(1-3), of a])
   using closest_point_le[OF assms(2), of _ a]
   apply fastforce
   done
 
 lemma setdist_closest_point:
     "\<lbrakk>closed S; S \<noteq> {}\<rbrakk> \<Longrightarrow> setdist {a} S = dist a (closest_point S a)"
   apply (rule setdist_unique)
   using closest_point_le
   apply (auto simp: closest_point_in_set)
   done
 
 lemma closest_point_lipschitz:
   assumes "convex S"
     and "closed S" "S \<noteq> {}"
   shows "dist (closest_point S x) (closest_point S y) \<le> dist x y"
 proof -
   have "inner (x - closest_point S x) (closest_point S y - closest_point S x) \<le> 0"
     and "inner (y - closest_point S y) (closest_point S x - closest_point S y) \<le> 0"
     apply (rule_tac[!] any_closest_point_dot[OF assms(1-2)])
     using closest_point_exists[OF assms(2-3)]
     apply auto
     done
   then show ?thesis unfolding dist_norm and norm_le
     using inner_ge_zero[of "(x - closest_point S x) - (y - closest_point S y)"]
     by (simp add: inner_add inner_diff inner_commute)
 qed
 
 lemma continuous_at_closest_point:
   assumes "convex S"
     and "closed S"
     and "S \<noteq> {}"
   shows "continuous (at x) (closest_point S)"
   unfolding continuous_at_eps_delta
   using le_less_trans[OF closest_point_lipschitz[OF assms]] by auto
 
 lemma continuous_on_closest_point:
   assumes "convex S"
     and "closed S"
     and "S \<noteq> {}"
   shows "continuous_on t (closest_point S)"
   by (metis continuous_at_imp_continuous_on continuous_at_closest_point[OF assms])
 
 proposition closest_point_in_rel_interior:
   assumes "closed S" "S \<noteq> {}" and x: "x \<in> affine hull S"
     shows "closest_point S x \<in> rel_interior S \<longleftrightarrow> x \<in> rel_interior S"
 proof (cases "x \<in> S")
   case True
   then show ?thesis
     by (simp add: closest_point_self)
 next
   case False
   then have "False" if asm: "closest_point S x \<in> rel_interior S"
   proof -
     obtain e where "e > 0" and clox: "closest_point S x \<in> S"
                and e: "cball (closest_point S x) e \<inter> affine hull S \<subseteq> S"
       using asm mem_rel_interior_cball by blast
     then have clo_notx: "closest_point S x \<noteq> x"
       using \<open>x \<notin> S\<close> by auto
     define y where "y \<equiv> closest_point S x -
                         (min 1 (e / norm(closest_point S x - x))) *\<^sub>R (closest_point S x - x)"
     have "x - y = (1 - min 1 (e / norm (closest_point S x - x))) *\<^sub>R (x - closest_point S x)"
       by (simp add: y_def algebra_simps)
     then have "norm (x - y) = abs ((1 - min 1 (e / norm (closest_point S x - x)))) * norm(x - closest_point S x)"
       by simp
     also have "\<dots> < norm(x - closest_point S x)"
       using clo_notx \<open>e > 0\<close>
       by (auto simp: mult_less_cancel_right2 field_split_simps)
     finally have no_less: "norm (x - y) < norm (x - closest_point S x)" .
     have "y \<in> affine hull S"
       unfolding y_def
       by (meson affine_affine_hull clox hull_subset mem_affine_3_minus2 subsetD x)
     moreover have "dist (closest_point S x) y \<le> e"
       using \<open>e > 0\<close> by (auto simp: y_def min_mult_distrib_right)
     ultimately have "y \<in> S"
       using subsetD [OF e] by simp
     then have "dist x (closest_point S x) \<le> dist x y"
       by (simp add: closest_point_le \<open>closed S\<close>)
     with no_less show False
       by (simp add: dist_norm)
   qed
   moreover have "x \<notin> rel_interior S"
     using rel_interior_subset False by blast
   ultimately show ?thesis by blast
 qed
 
 
 subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Various point-to-set separating/supporting hyperplane theorems\<close>
 
 lemma supporting_hyperplane_closed_point:
   fixes z :: "'a::{real_inner,heine_borel}"
   assumes "convex S"
     and "closed S"
     and "S \<noteq> {}"
     and "z \<notin> S"
   shows "\<exists>a b. \<exists>y\<in>S. inner a z < b \<and> inner a y = b \<and> (\<forall>x\<in>S. inner a x \<ge> b)"
 proof -
   obtain y where "y \<in> S" and y: "\<forall>x\<in>S. dist z y \<le> dist z x"
     by (metis distance_attains_inf[OF assms(2-3)])
   show ?thesis
   proof (intro exI bexI conjI ballI)
     show "(y - z) \<bullet> z < (y - z) \<bullet> y"
       by (metis \<open>y \<in> S\<close> assms(4) diff_gt_0_iff_gt inner_commute inner_diff_left inner_gt_zero_iff right_minus_eq)
     show "(y - z) \<bullet> y \<le> (y - z) \<bullet> x" if "x \<in> S" for x
     proof (rule ccontr)
       have *: "\<And>u. 0 \<le> u \<and> u \<le> 1 \<longrightarrow> dist z y \<le> dist z ((1 - u) *\<^sub>R y + u *\<^sub>R x)"
         using assms(1)[unfolded convex_alt] and y and \<open>x\<in>S\<close> and \<open>y\<in>S\<close> by auto
       assume "\<not> (y - z) \<bullet> y \<le> (y - z) \<bullet> x"
       then obtain v where "v > 0" "v \<le> 1" "dist (y + v *\<^sub>R (x - y)) z < dist y z"
         using closer_point_lemma[of z y x] by (auto simp: inner_diff)
       then show False
         using *[of v] by (auto simp: dist_commute algebra_simps)
     qed
   qed (use \<open>y \<in> S\<close> in auto)
 qed
 
 lemma separating_hyperplane_closed_point:
   fixes z :: "'a::{real_inner,heine_borel}"
   assumes "convex S"
     and "closed S"
     and "z \<notin> S"
   shows "\<exists>a b. inner a z < b \<and> (\<forall>x\<in>S. inner a x > b)"
 proof (cases "S = {}")
   case True
   then show ?thesis
     by (simp add: gt_ex)
 next
   case False
   obtain y where "y \<in> S" and y: "\<And>x. x \<in> S \<Longrightarrow> dist z y \<le> dist z x"
     by (metis distance_attains_inf[OF assms(2) False])
   show ?thesis
   proof (intro exI conjI ballI)
     show "(y - z) \<bullet> z < inner (y - z) z + (norm (y - z))\<^sup>2 / 2"
       using \<open>y\<in>S\<close> \<open>z\<notin>S\<close> by auto
   next
     fix x
     assume "x \<in> S"
     have "False" if *: "0 < inner (z - y) (x - y)"
     proof -
       obtain u where "u > 0" "u \<le> 1" "dist (y + u *\<^sub>R (x - y)) z < dist y z"
         using * closer_point_lemma by blast
       then show False using y[of "y + u *\<^sub>R (x - y)"] convexD_alt [OF \<open>convex S\<close>]
         using \<open>x\<in>S\<close> \<open>y\<in>S\<close> by (auto simp: dist_commute algebra_simps)
     qed
     moreover have "0 < (norm (y - z))\<^sup>2"
       using \<open>y\<in>S\<close> \<open>z\<notin>S\<close> by auto
     then have "0 < inner (y - z) (y - z)"
       unfolding power2_norm_eq_inner by simp
     ultimately show "(y - z) \<bullet> z + (norm (y - z))\<^sup>2 / 2 < (y - z) \<bullet> x"
       by (force simp: field_simps power2_norm_eq_inner inner_commute inner_diff)
   qed 
 qed
 
 lemma separating_hyperplane_closed_0:
   assumes "convex (S::('a::euclidean_space) set)"
     and "closed S"
     and "0 \<notin> S"
   shows "\<exists>a b. a \<noteq> 0 \<and> 0 < b \<and> (\<forall>x\<in>S. inner a x > b)"
 proof (cases "S = {}")
   case True
   have "(SOME i. i\<in>Basis) \<noteq> (0::'a)"
     by (metis Basis_zero SOME_Basis)
   then show ?thesis
     using True zero_less_one by blast
 next
   case False
   then show ?thesis
     using False using separating_hyperplane_closed_point[OF assms]
     by (metis all_not_in_conv inner_zero_left inner_zero_right less_eq_real_def not_le)
 qed
 
 
 subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Now set-to-set for closed/compact sets\<close>
 
 lemma separating_hyperplane_closed_compact:
   fixes S :: "'a::euclidean_space set"
   assumes "convex S"
     and "closed S"
     and "convex T"
     and "compact T"
     and "T \<noteq> {}"
     and "S \<inter> T = {}"
   shows "\<exists>a b. (\<forall>x\<in>S. inner a x < b) \<and> (\<forall>x\<in>T. inner a x > b)"
 proof (cases "S = {}")
   case True
   obtain b where b: "b > 0" "\<forall>x\<in>T. norm x \<le> b"
     using compact_imp_bounded[OF assms(4)] unfolding bounded_pos by auto
   obtain z :: 'a where z: "norm z = b + 1"
     using vector_choose_size[of "b + 1"] and b(1) by auto
   then have "z \<notin> T" using b(2)[THEN bspec[where x=z]] by auto
   then obtain a b where ab: "inner a z < b" "\<forall>x\<in>T. b < inner a x"
     using separating_hyperplane_closed_point[OF assms(3) compact_imp_closed[OF assms(4)], of z]
     by auto
   then show ?thesis
     using True by auto
 next
   case False
   then obtain y where "y \<in> S" by auto
   obtain a b where "0 < b" "\<forall>x \<in> (\<Union>x\<in> S. \<Union>y \<in> T. {x - y}). b < inner a x"
     using separating_hyperplane_closed_point[OF convex_differences[OF assms(1,3)], of 0]
     using closed_compact_differences[OF assms(2,4)]
     using assms(6) by auto 
   then have ab: "\<forall>x\<in>S. \<forall>y\<in>T. b + inner a y < inner a x"
     apply -
     apply rule
     apply rule
     apply (erule_tac x="x - y" in ballE)
     apply (auto simp: inner_diff)
     done
   define k where "k = (SUP x\<in>T. a \<bullet> x)"
   show ?thesis
     apply (rule_tac x="-a" in exI)
     apply (rule_tac x="-(k + b / 2)" in exI)
     apply (intro conjI ballI)
     unfolding inner_minus_left and neg_less_iff_less
   proof -
     fix x assume "x \<in> T"
     then have "inner a x - b / 2 < k"
       unfolding k_def
     proof (subst less_cSUP_iff)
       show "T \<noteq> {}" by fact
       show "bdd_above ((\<bullet>) a ` T)"
         using ab[rule_format, of y] \<open>y \<in> S\<close>
         by (intro bdd_aboveI2[where M="inner a y - b"]) (auto simp: field_simps intro: less_imp_le)
     qed (auto intro!: bexI[of _ x] \<open>0<b\<close>)
     then show "inner a x < k + b / 2"
       by auto
   next
     fix x
     assume "x \<in> S"
     then have "k \<le> inner a x - b"
       unfolding k_def
       apply (rule_tac cSUP_least)
       using assms(5)
       using ab[THEN bspec[where x=x]]
       apply auto
       done
     then show "k + b / 2 < inner a x"
       using \<open>0 < b\<close> by auto
   qed
 qed
 
 lemma separating_hyperplane_compact_closed:
   fixes S :: "'a::euclidean_space set"
   assumes "convex S"
     and "compact S"
     and "S \<noteq> {}"
     and "convex T"
     and "closed T"
     and "S \<inter> T = {}"
   shows "\<exists>a b. (\<forall>x\<in>S. inner a x < b) \<and> (\<forall>x\<in>T. inner a x > b)"
 proof -
   obtain a b where "(\<forall>x\<in>T. inner a x < b) \<and> (\<forall>x\<in>S. b < inner a x)"
     using separating_hyperplane_closed_compact[OF assms(4-5,1-2,3)] and assms(6)
     by auto
   then show ?thesis
     apply (rule_tac x="-a" in exI)
     apply (rule_tac x="-b" in exI, auto)
     done
 qed
 
 
 subsubsection\<^marker>\<open>tag unimportant\<close> \<open>General case without assuming closure and getting non-strict separation\<close>
 
 lemma separating_hyperplane_set_0:
   assumes "convex S" "(0::'a::euclidean_space) \<notin> S"
   shows "\<exists>a. a \<noteq> 0 \<and> (\<forall>x\<in>S. 0 \<le> inner a x)"
 proof -
   let ?k = "\<lambda>c. {x::'a. 0 \<le> inner c x}"
   have *: "frontier (cball 0 1) \<inter> \<Inter>f \<noteq> {}" if as: "f \<subseteq> ?k ` S" "finite f" for f
   proof -
     obtain c where c: "f = ?k ` c" "c \<subseteq> S" "finite c"
       using finite_subset_image[OF as(2,1)] by auto
     then obtain a b where ab: "a \<noteq> 0" "0 < b" "\<forall>x\<in>convex hull c. b < inner a x"
       using separating_hyperplane_closed_0[OF convex_convex_hull, of c]
       using finite_imp_compact_convex_hull[OF c(3), THEN compact_imp_closed] and assms(2)
       using subset_hull[of convex, OF assms(1), symmetric, of c]
       by force
     then have "\<exists>x. norm x = 1 \<and> (\<forall>y\<in>c. 0 \<le> inner y x)"
       apply (rule_tac x = "inverse(norm a) *\<^sub>R a" in exI)
       using hull_subset[of c convex]
       unfolding subset_eq and inner_scaleR
       by (auto simp: inner_commute del: ballE elim!: ballE)
     then show "frontier (cball 0 1) \<inter> \<Inter>f \<noteq> {}"
       unfolding c(1) frontier_cball sphere_def dist_norm by auto
   qed
   have "frontier (cball 0 1) \<inter> (\<Inter>(?k ` S)) \<noteq> {}"
     apply (rule compact_imp_fip)
     apply (rule compact_frontier[OF compact_cball])
     using * closed_halfspace_ge
     by auto
   then obtain x where "norm x = 1" "\<forall>y\<in>S. x\<in>?k y"
     unfolding frontier_cball dist_norm sphere_def by auto
   then show ?thesis
     by (metis inner_commute mem_Collect_eq norm_eq_zero zero_neq_one)
 qed
 
 lemma separating_hyperplane_sets:
   fixes S T :: "'a::euclidean_space set"
   assumes "convex S"
     and "convex T"
     and "S \<noteq> {}"
     and "T \<noteq> {}"
     and "S \<inter> T = {}"
   shows "\<exists>a b. a \<noteq> 0 \<and> (\<forall>x\<in>S. inner a x \<le> b) \<and> (\<forall>x\<in>T. inner a x \<ge> b)"
 proof -
   from separating_hyperplane_set_0[OF convex_differences[OF assms(2,1)]]
   obtain a where "a \<noteq> 0" "\<forall>x\<in>{x - y |x y. x \<in> T \<and> y \<in> S}. 0 \<le> inner a x"
     using assms(3-5) by force
   then have *: "\<And>x y. x \<in> T \<Longrightarrow> y \<in> S \<Longrightarrow> inner a y \<le> inner a x"
     by (force simp: inner_diff)
   then have bdd: "bdd_above (((\<bullet>) a)`S)"
     using \<open>T \<noteq> {}\<close> by (auto intro: bdd_aboveI2[OF *])
   show ?thesis
     using \<open>a\<noteq>0\<close>
     by (intro exI[of _ a] exI[of _ "SUP x\<in>S. a \<bullet> x"])
        (auto intro!: cSUP_upper bdd cSUP_least \<open>a \<noteq> 0\<close> \<open>S \<noteq> {}\<close> *)
 qed
 
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>More convexity generalities\<close>
 
 lemma convex_closure [intro,simp]:
   fixes S :: "'a::real_normed_vector set"
   assumes "convex S"
   shows "convex (closure S)"
   apply (rule convexI)
   apply (unfold closure_sequential, elim exE)
   apply (rule_tac x="\<lambda>n. u *\<^sub>R xa n + v *\<^sub>R xb n" in exI)
   apply (rule,rule)
   apply (rule convexD [OF assms])
   apply (auto del: tendsto_const intro!: tendsto_intros)
   done
 
 lemma convex_interior [intro,simp]:
   fixes S :: "'a::real_normed_vector set"
   assumes "convex S"
   shows "convex (interior S)"
   unfolding convex_alt Ball_def mem_interior
 proof clarify
   fix x y u
   assume u: "0 \<le> u" "u \<le> (1::real)"
   fix e d
   assume ed: "ball x e \<subseteq> S" "ball y d \<subseteq> S" "0<d" "0<e"
   show "\<exists>e>0. ball ((1 - u) *\<^sub>R x + u *\<^sub>R y) e \<subseteq> S"
   proof (intro exI conjI subsetI)
     fix z
     assume "z \<in> ball ((1 - u) *\<^sub>R x + u *\<^sub>R y) (min d e)"
     then have "(1- u) *\<^sub>R (z - u *\<^sub>R (y - x)) + u *\<^sub>R (z + (1 - u) *\<^sub>R (y - x)) \<in> S"
       apply (rule_tac assms[unfolded convex_alt, rule_format])
       using ed(1,2) and u
       unfolding subset_eq mem_ball Ball_def dist_norm
       apply (auto simp: algebra_simps)
       done
     then show "z \<in> S"
       using u by (auto simp: algebra_simps)
   qed(insert u ed(3-4), auto)
 qed
 
 lemma convex_hull_eq_empty[simp]: "convex hull S = {} \<longleftrightarrow> S = {}"
   using hull_subset[of S convex] convex_hull_empty by auto
 
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>Convex set as intersection of halfspaces\<close>
 
 lemma convex_halfspace_intersection:
   fixes s :: "('a::euclidean_space) set"
   assumes "closed s" "convex s"
   shows "s = \<Inter>{h. s \<subseteq> h \<and> (\<exists>a b. h = {x. inner a x \<le> b})}"
   apply (rule set_eqI, rule)
   unfolding Inter_iff Ball_def mem_Collect_eq
   apply (rule,rule,erule conjE)
 proof -
   fix x
   assume "\<forall>xa. s \<subseteq> xa \<and> (\<exists>a b. xa = {x. inner a x \<le> b}) \<longrightarrow> x \<in> xa"
   then have "\<forall>a b. s \<subseteq> {x. inner a x \<le> b} \<longrightarrow> x \<in> {x. inner a x \<le> b}"
     by blast
   then show "x \<in> s"
     apply (rule_tac ccontr)
     apply (drule separating_hyperplane_closed_point[OF assms(2,1)])
     apply (erule exE)+
     apply (erule_tac x="-a" in allE)
     apply (erule_tac x="-b" in allE, auto)
     done
 qed auto
 
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>Convexity of general and special intervals\<close>
 
 lemma is_interval_convex:
   fixes S :: "'a::euclidean_space set"
   assumes "is_interval S"
   shows "convex S"
 proof (rule convexI)
   fix x y and u v :: real
   assume as: "x \<in> S" "y \<in> S" "0 \<le> u" "0 \<le> v" "u + v = 1"
   then have *: "u = 1 - v" "1 - v \<ge> 0" and **: "v = 1 - u" "1 - u \<ge> 0"
     by auto
   {
     fix a b
     assume "\<not> b \<le> u * a + v * b"
     then have "u * a < (1 - v) * b"
       unfolding not_le using as(4) by (auto simp: field_simps)
     then have "a < b"
       unfolding * using as(4) *(2)
       apply (rule_tac mult_left_less_imp_less[of "1 - v"])
       apply (auto simp: field_simps)
       done
     then have "a \<le> u * a + v * b"
       unfolding * using as(4)
       by (auto simp: field_simps intro!:mult_right_mono)
   }
   moreover
   {
     fix a b
     assume "\<not> u * a + v * b \<le> a"
     then have "v * b > (1 - u) * a"
       unfolding not_le using as(4) by (auto simp: field_simps)
     then have "a < b"
       unfolding * using as(4)
       apply (rule_tac mult_left_less_imp_less)
       apply (auto simp: field_simps)
       done
     then have "u * a + v * b \<le> b"
       unfolding **
       using **(2) as(3)
       by (auto simp: field_simps intro!:mult_right_mono)
   }
   ultimately show "u *\<^sub>R x + v *\<^sub>R y \<in> S"
     apply -
     apply (rule assms[unfolded is_interval_def, rule_format, OF as(1,2)])
     using as(3-) DIM_positive[where 'a='a]
     apply (auto simp: inner_simps)
     done
 qed
 
 lemma is_interval_connected:
   fixes S :: "'a::euclidean_space set"
   shows "is_interval S \<Longrightarrow> connected S"
   using is_interval_convex convex_connected by auto
 
 lemma convex_box [simp]: "convex (cbox a b)" "convex (box a (b::'a::euclidean_space))"
   apply (rule_tac[!] is_interval_convex)+
   using is_interval_box is_interval_cbox
   apply auto
   done
 
 text\<open>A non-singleton connected set is perfect (i.e. has no isolated points). \<close>
 lemma connected_imp_perfect:
   fixes a :: "'a::metric_space"
   assumes "connected S" "a \<in> S" and S: "\<And>x. S \<noteq> {x}"
   shows "a islimpt S"
 proof -
   have False if "a \<in> T" "open T" "\<And>y. \<lbrakk>y \<in> S; y \<in> T\<rbrakk> \<Longrightarrow> y = a" for T
   proof -
     obtain e where "e > 0" and e: "cball a e \<subseteq> T"
       using \<open>open T\<close> \<open>a \<in> T\<close> by (auto simp: open_contains_cball)
     have "openin (top_of_set S) {a}"
       unfolding openin_open using that \<open>a \<in> S\<close> by blast
     moreover have "closedin (top_of_set S) {a}"
       by (simp add: assms)
     ultimately show "False"
       using \<open>connected S\<close> connected_clopen S by blast
   qed
   then show ?thesis
     unfolding islimpt_def by blast
 qed
 
 lemma connected_imp_perfect_aff_dim:
      "\<lbrakk>connected S; aff_dim S \<noteq> 0; a \<in> S\<rbrakk> \<Longrightarrow> a islimpt S"
   using aff_dim_sing connected_imp_perfect by blast
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>On \<open>real\<close>, \<open>is_interval\<close>, \<open>convex\<close> and \<open>connected\<close> are all equivalent\<close>
 
 lemma mem_is_interval_1_I:
   fixes a b c::real
   assumes "is_interval S"
   assumes "a \<in> S" "c \<in> S"
   assumes "a \<le> b" "b \<le> c"
   shows "b \<in> S"
   using assms is_interval_1 by blast
 
 lemma is_interval_connected_1:
   fixes s :: "real set"
   shows "is_interval s \<longleftrightarrow> connected s"
   apply rule
   apply (rule is_interval_connected, assumption)
   unfolding is_interval_1
   apply rule
   apply rule
   apply rule
   apply rule
   apply (erule conjE)
   apply (rule ccontr)       
 proof -
   fix a b x
   assume as: "connected s" "a \<in> s" "b \<in> s" "a \<le> x" "x \<le> b" "x \<notin> s"
   then have *: "a < x" "x < b"
     unfolding not_le [symmetric] by auto
   let ?halfl = "{..<x} "
   let ?halfr = "{x<..}"
   {
     fix y
     assume "y \<in> s"
     with \<open>x \<notin> s\<close> have "x \<noteq> y" by auto
     then have "y \<in> ?halfr \<union> ?halfl" by auto
   }
   moreover have "a \<in> ?halfl" "b \<in> ?halfr" using * by auto
   then have "?halfl \<inter> s \<noteq> {}" "?halfr \<inter> s \<noteq> {}"
     using as(2-3) by auto
   ultimately show False
     apply (rule_tac notE[OF as(1)[unfolded connected_def]])
     apply (rule_tac x = ?halfl in exI)
     apply (rule_tac x = ?halfr in exI, rule)
     apply (rule open_lessThan, rule)
     apply (rule open_greaterThan, auto)
     done
 qed
 
 lemma is_interval_convex_1:
   fixes s :: "real set"
   shows "is_interval s \<longleftrightarrow> convex s"
   by (metis is_interval_convex convex_connected is_interval_connected_1)
 
 lemma connected_compact_interval_1:
      "connected S \<and> compact S \<longleftrightarrow> (\<exists>a b. S = {a..b::real})"
   by (auto simp: is_interval_connected_1 [symmetric] is_interval_compact)
 
 lemma connected_convex_1:
   fixes s :: "real set"
   shows "connected s \<longleftrightarrow> convex s"
   by (metis is_interval_convex convex_connected is_interval_connected_1)
 
 lemma connected_convex_1_gen:
   fixes s :: "'a :: euclidean_space set"
   assumes "DIM('a) = 1"
   shows "connected s \<longleftrightarrow> convex s"
 proof -
   obtain f:: "'a \<Rightarrow> real" where linf: "linear f" and "inj f"
     using subspace_isomorphism[OF subspace_UNIV subspace_UNIV,
         where 'a='a and 'b=real]
     unfolding Euclidean_Space.dim_UNIV
     by (auto simp: assms)
   then have "f -` (f ` s) = s"
     by (simp add: inj_vimage_image_eq)
   then show ?thesis
     by (metis connected_convex_1 convex_linear_vimage linf convex_connected connected_linear_image)
 qed
 
 lemma [simp]:
   fixes r s::real
   shows is_interval_io: "is_interval {..<r}"
     and is_interval_oi: "is_interval {r<..}"
     and is_interval_oo: "is_interval {r<..<s}"
     and is_interval_oc: "is_interval {r<..s}"
     and is_interval_co: "is_interval {r..<s}"
   by (simp_all add: is_interval_convex_1)
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>Another intermediate value theorem formulation\<close>
 
 lemma ivt_increasing_component_on_1:
   fixes f :: "real \<Rightarrow> 'a::euclidean_space"
   assumes "a \<le> b"
     and "continuous_on {a..b} f"
     and "(f a)\<bullet>k \<le> y" "y \<le> (f b)\<bullet>k"
   shows "\<exists>x\<in>{a..b}. (f x)\<bullet>k = y"
 proof -
   have "f a \<in> f ` cbox a b" "f b \<in> f ` cbox a b"
     apply (rule_tac[!] imageI)
     using assms(1)
     apply auto
     done
   then show ?thesis
     using connected_ivt_component[of "f ` cbox a b" "f a" "f b" k y]
     by (simp add: connected_continuous_image assms)
 qed
 
 lemma ivt_increasing_component_1:
   fixes f :: "real \<Rightarrow> 'a::euclidean_space"
   shows "a \<le> b \<Longrightarrow> \<forall>x\<in>{a..b}. continuous (at x) f \<Longrightarrow>
     f a\<bullet>k \<le> y \<Longrightarrow> y \<le> f b\<bullet>k \<Longrightarrow> \<exists>x\<in>{a..b}. (f x)\<bullet>k = y"
   by (rule ivt_increasing_component_on_1) (auto simp: continuous_at_imp_continuous_on)
 
 lemma ivt_decreasing_component_on_1:
   fixes f :: "real \<Rightarrow> 'a::euclidean_space"
   assumes "a \<le> b"
     and "continuous_on {a..b} f"
     and "(f b)\<bullet>k \<le> y"
     and "y \<le> (f a)\<bullet>k"
   shows "\<exists>x\<in>{a..b}. (f x)\<bullet>k = y"
   apply (subst neg_equal_iff_equal[symmetric])
   using ivt_increasing_component_on_1[of a b "\<lambda>x. - f x" k "- y"]
   using assms using continuous_on_minus
   apply auto
   done
 
 lemma ivt_decreasing_component_1:
   fixes f :: "real \<Rightarrow> 'a::euclidean_space"
   shows "a \<le> b \<Longrightarrow> \<forall>x\<in>{a..b}. continuous (at x) f \<Longrightarrow>
     f b\<bullet>k \<le> y \<Longrightarrow> y \<le> f a\<bullet>k \<Longrightarrow> \<exists>x\<in>{a..b}. (f x)\<bullet>k = y"
   by (rule ivt_decreasing_component_on_1) (auto simp: continuous_at_imp_continuous_on)
 
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>A bound within an interval\<close>
 
 lemma convex_hull_eq_real_cbox:
   fixes x y :: real assumes "x \<le> y"
   shows "convex hull {x, y} = cbox x y"
 proof (rule hull_unique)
   show "{x, y} \<subseteq> cbox x y" using \<open>x \<le> y\<close> by auto
   show "convex (cbox x y)"
     by (rule convex_box)
 next
   fix S assume "{x, y} \<subseteq> S" and "convex S"
   then show "cbox x y \<subseteq> S"
     unfolding is_interval_convex_1 [symmetric] is_interval_def Basis_real_def
     by - (clarify, simp (no_asm_use), fast)
 qed
 
 lemma unit_interval_convex_hull:
   "cbox (0::'a::euclidean_space) One = convex hull {x. \<forall>i\<in>Basis. (x\<bullet>i = 0) \<or> (x\<bullet>i = 1)}"
   (is "?int = convex hull ?points")
 proof -
   have One[simp]: "\<And>i. i \<in> Basis \<Longrightarrow> One \<bullet> i = 1"
     by (simp add: inner_sum_left sum.If_cases inner_Basis)
   have "?int = {x. \<forall>i\<in>Basis. x \<bullet> i \<in> cbox 0 1}"
     by (auto simp: cbox_def)
   also have "\<dots> = (\<Sum>i\<in>Basis. (\<lambda>x. x *\<^sub>R i) ` cbox 0 1)"
     by (simp only: box_eq_set_sum_Basis)
   also have "\<dots> = (\<Sum>i\<in>Basis. (\<lambda>x. x *\<^sub>R i) ` (convex hull {0, 1}))"
     by (simp only: convex_hull_eq_real_cbox zero_le_one)
   also have "\<dots> = (\<Sum>i\<in>Basis. convex hull ((\<lambda>x. x *\<^sub>R i) ` {0, 1}))"
     by (simp add: convex_hull_linear_image)
   also have "\<dots> = convex hull (\<Sum>i\<in>Basis. (\<lambda>x. x *\<^sub>R i) ` {0, 1})"
     by (simp only: convex_hull_set_sum)
   also have "\<dots> = convex hull {x. \<forall>i\<in>Basis. x\<bullet>i \<in> {0, 1}}"
     by (simp only: box_eq_set_sum_Basis)
   also have "convex hull {x. \<forall>i\<in>Basis. x\<bullet>i \<in> {0, 1}} = convex hull ?points"
     by simp
   finally show ?thesis .
 qed
 
 text \<open>And this is a finite set of vertices.\<close>
 
 lemma unit_cube_convex_hull:
   obtains S :: "'a::euclidean_space set"
   where "finite S" and "cbox 0 (\<Sum>Basis) = convex hull S"
 proof
   show "finite {x::'a. \<forall>i\<in>Basis. x \<bullet> i = 0 \<or> x \<bullet> i = 1}"
   proof (rule finite_subset, clarify)
     show "finite ((\<lambda>S. \<Sum>i\<in>Basis. (if i \<in> S then 1 else 0) *\<^sub>R i) ` Pow Basis)"
       using finite_Basis by blast
     fix x :: 'a
     assume as: "\<forall>i\<in>Basis. x \<bullet> i = 0 \<or> x \<bullet> i = 1"
     show "x \<in> (\<lambda>S. \<Sum>i\<in>Basis. (if i\<in>S then 1 else 0) *\<^sub>R i) ` Pow Basis"
       apply (rule image_eqI[where x="{i. i\<in>Basis \<and> x\<bullet>i = 1}"])
       using as
        apply (subst euclidean_eq_iff, auto)
       done
   qed
   show "cbox 0 One = convex hull {x. \<forall>i\<in>Basis. x \<bullet> i = 0 \<or> x \<bullet> i = 1}"
     using unit_interval_convex_hull by blast
 qed 
 
 text \<open>Hence any cube (could do any nonempty interval).\<close>
 
 lemma cube_convex_hull:
   assumes "d > 0"
   obtains S :: "'a::euclidean_space set" where
     "finite S" and "cbox (x - (\<Sum>i\<in>Basis. d*\<^sub>Ri)) (x + (\<Sum>i\<in>Basis. d*\<^sub>Ri)) = convex hull S"
 proof -
   let ?d = "(\<Sum>i\<in>Basis. d *\<^sub>R i)::'a"
   have *: "cbox (x - ?d) (x + ?d) = (\<lambda>y. x - ?d + (2 * d) *\<^sub>R y) ` cbox 0 (\<Sum>Basis)"
   proof (intro set_eqI iffI)
     fix y
     assume "y \<in> cbox (x - ?d) (x + ?d)"
     then have "inverse (2 * d) *\<^sub>R (y - (x - ?d)) \<in> cbox 0 (\<Sum>Basis)"
       using assms by (simp add: mem_box inner_simps) (simp add: field_simps)
     with \<open>0 < d\<close> show "y \<in> (\<lambda>y. x - sum ((*\<^sub>R) d) Basis + (2 * d) *\<^sub>R y) ` cbox 0 One"
       by (auto intro: image_eqI[where x= "inverse (2 * d) *\<^sub>R (y - (x - ?d))"])
   next
     fix y
     assume "y \<in> (\<lambda>y. x - ?d + (2 * d) *\<^sub>R y) ` cbox 0 One"
     then obtain z where z: "z \<in> cbox 0 One" "y = x - ?d + (2*d) *\<^sub>R z"
       by auto
     then show "y \<in> cbox (x - ?d) (x + ?d)"
       using z assms by (auto simp: mem_box inner_simps)
   qed
   obtain S where "finite S" "cbox 0 (\<Sum>Basis::'a) = convex hull S"
     using unit_cube_convex_hull by auto
   then show ?thesis
     by (rule_tac that[of "(\<lambda>y. x - ?d + (2 * d) *\<^sub>R y)` S"]) (auto simp: convex_hull_affinity *)
 qed
 
 subsection\<^marker>\<open>tag unimportant\<close>\<open>Representation of any interval as a finite convex hull\<close>
 
 lemma image_stretch_interval:
   "(\<lambda>x. \<Sum>k\<in>Basis. (m k * (x\<bullet>k)) *\<^sub>R k) ` cbox a (b::'a::euclidean_space) =
   (if (cbox a b) = {} then {} else
     cbox (\<Sum>k\<in>Basis. (min (m k * (a\<bullet>k)) (m k * (b\<bullet>k))) *\<^sub>R k::'a)
      (\<Sum>k\<in>Basis. (max (m k * (a\<bullet>k)) (m k * (b\<bullet>k))) *\<^sub>R k))"
 proof cases
   assume *: "cbox a b \<noteq> {}"
   show ?thesis
     unfolding box_ne_empty if_not_P[OF *]
     apply (simp add: cbox_def image_Collect set_eq_iff euclidean_eq_iff[where 'a='a] ball_conj_distrib[symmetric])
     apply (subst choice_Basis_iff[symmetric])
   proof (intro allI ball_cong refl)
     fix x i :: 'a assume "i \<in> Basis"
     with * have a_le_b: "a \<bullet> i \<le> b \<bullet> i"
       unfolding box_ne_empty by auto
     show "(\<exists>xa. x \<bullet> i = m i * xa \<and> a \<bullet> i \<le> xa \<and> xa \<le> b \<bullet> i) \<longleftrightarrow>
         min (m i * (a \<bullet> i)) (m i * (b \<bullet> i)) \<le> x \<bullet> i \<and> x \<bullet> i \<le> max (m i * (a \<bullet> i)) (m i * (b \<bullet> i))"
     proof (cases "m i = 0")
       case True
       with a_le_b show ?thesis by auto
     next
       case False
       then have *: "\<And>a b. a = m i * b \<longleftrightarrow> b = a / m i"
         by (auto simp: field_simps)
       from False have
           "min (m i * (a \<bullet> i)) (m i * (b \<bullet> i)) = (if 0 < m i then m i * (a \<bullet> i) else m i * (b \<bullet> i))"
           "max (m i * (a \<bullet> i)) (m i * (b \<bullet> i)) = (if 0 < m i then m i * (b \<bullet> i) else m i * (a \<bullet> i))"
         using a_le_b by (auto simp: min_def max_def mult_le_cancel_left)
       with False show ?thesis using a_le_b
         unfolding * by (auto simp: le_divide_eq divide_le_eq ac_simps)
     qed
   qed
 qed simp
 
 lemma interval_image_stretch_interval:
   "\<exists>u v. (\<lambda>x. \<Sum>k\<in>Basis. (m k * (x\<bullet>k))*\<^sub>R k) ` cbox a (b::'a::euclidean_space) = cbox u (v::'a::euclidean_space)"
   unfolding image_stretch_interval by auto
 
 lemma cbox_translation: "cbox (c + a) (c + b) = image (\<lambda>x. c + x) (cbox a b)"
   using image_affinity_cbox [of 1 c a b]
   using box_ne_empty [of "a+c" "b+c"]  box_ne_empty [of a b]
   by (auto simp: inner_left_distrib add.commute)
 
 lemma cbox_image_unit_interval:
   fixes a :: "'a::euclidean_space"
   assumes "cbox a b \<noteq> {}"
     shows "cbox a b =
            (+) a ` (\<lambda>x. \<Sum>k\<in>Basis. ((b \<bullet> k - a \<bullet> k) * (x \<bullet> k)) *\<^sub>R k) ` cbox 0 One"
 using assms
 apply (simp add: box_ne_empty image_stretch_interval cbox_translation [symmetric])
 apply (simp add: min_def max_def algebra_simps sum_subtractf euclidean_representation)
 done
 
 lemma closed_interval_as_convex_hull:
   fixes a :: "'a::euclidean_space"
   obtains S where "finite S" "cbox a b = convex hull S"
 proof (cases "cbox a b = {}")
   case True with convex_hull_empty that show ?thesis
     by blast
 next
   case False
   obtain S::"'a set" where "finite S" and eq: "cbox 0 One = convex hull S"
     by (blast intro: unit_cube_convex_hull)
   have lin: "linear (\<lambda>x. \<Sum>k\<in>Basis. ((b \<bullet> k - a \<bullet> k) * (x \<bullet> k)) *\<^sub>R k)"
     by (rule linear_compose_sum) (auto simp: algebra_simps linearI)
   have "finite ((+) a ` (\<lambda>x. \<Sum>k\<in>Basis. ((b \<bullet> k - a \<bullet> k) * (x \<bullet> k)) *\<^sub>R k) ` S)"
     by (rule finite_imageI \<open>finite S\<close>)+
   then show ?thesis
     apply (rule that)
     apply (simp add: convex_hull_translation convex_hull_linear_image [OF lin, symmetric])
     apply (simp add: eq [symmetric] cbox_image_unit_interval [OF False])
     done
 qed
 
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>Bounded convex function on open set is continuous\<close>
 
 lemma convex_on_bounded_continuous:
   fixes S :: "('a::real_normed_vector) set"
   assumes "open S"
     and "convex_on S f"
     and "\<forall>x\<in>S. \<bar>f x\<bar> \<le> b"
   shows "continuous_on S f"
   apply (rule continuous_at_imp_continuous_on)
   unfolding continuous_at_real_range
 proof (rule,rule,rule)
   fix x and e :: real
   assume "x \<in> S" "e > 0"
   define B where "B = \<bar>b\<bar> + 1"
   then have B:  "0 < B""\<And>x. x\<in>S \<Longrightarrow> \<bar>f x\<bar> \<le> B"
     using assms(3) by auto 
   obtain k where "k > 0" and k: "cball x k \<subseteq> S"
     using \<open>x \<in> S\<close> assms(1) open_contains_cball_eq by blast
   show "\<exists>d>0. \<forall>x'. norm (x' - x) < d \<longrightarrow> \<bar>f x' - f x\<bar> < e"
   proof (intro exI conjI allI impI)
     fix y
     assume as: "norm (y - x) < min (k / 2) (e / (2 * B) * k)"
     show "\<bar>f y - f x\<bar> < e"
     proof (cases "y = x")
       case False
       define t where "t = k / norm (y - x)"
       have "2 < t" "0<t"
         unfolding t_def using as False and \<open>k>0\<close>
         by (auto simp:field_simps)
       have "y \<in> S"
         apply (rule k[THEN subsetD])
         unfolding mem_cball dist_norm
         apply (rule order_trans[of _ "2 * norm (x - y)"])
         using as
         by (auto simp: field_simps norm_minus_commute)
       {
         define w where "w = x + t *\<^sub>R (y - x)"
         have "w \<in> S"
           using \<open>k>0\<close> by (auto simp: dist_norm t_def w_def k[THEN subsetD])
         have "(1 / t) *\<^sub>R x + - x + ((t - 1) / t) *\<^sub>R x = (1 / t - 1 + (t - 1) / t) *\<^sub>R x"
           by (auto simp: algebra_simps)
         also have "\<dots> = 0"
           using \<open>t > 0\<close> by (auto simp:field_simps)
         finally have w: "(1 / t) *\<^sub>R w + ((t - 1) / t) *\<^sub>R x = y"
           unfolding w_def using False and \<open>t > 0\<close>
           by (auto simp: algebra_simps)
         have 2: "2 * B < e * t"
           unfolding t_def using \<open>0 < e\<close> \<open>0 < k\<close> \<open>B > 0\<close> and as and False
           by (auto simp:field_simps)
         have "f y - f x \<le> (f w - f x) / t"
           using assms(2)[unfolded convex_on_def,rule_format,of w x "1/t" "(t - 1)/t", unfolded w]
           using \<open>0 < t\<close> \<open>2 < t\<close> and \<open>x \<in> S\<close> \<open>w \<in> S\<close>
           by (auto simp:field_simps)
         also have "... < e"
           using B(2)[OF \<open>w\<in>S\<close>] and B(2)[OF \<open>x\<in>S\<close>] 2 \<open>t > 0\<close> by (auto simp: field_simps)
         finally have th1: "f y - f x < e" .
       }
       moreover
       {
         define w where "w = x - t *\<^sub>R (y - x)"
         have "w \<in> S"
           using \<open>k > 0\<close> by (auto simp: dist_norm t_def w_def k[THEN subsetD])
         have "(1 / (1 + t)) *\<^sub>R x + (t / (1 + t)) *\<^sub>R x = (1 / (1 + t) + t / (1 + t)) *\<^sub>R x"
           by (auto simp: algebra_simps)
         also have "\<dots> = x"
           using \<open>t > 0\<close> by (auto simp:field_simps)
         finally have w: "(1 / (1+t)) *\<^sub>R w + (t / (1 + t)) *\<^sub>R y = x"
           unfolding w_def using False and \<open>t > 0\<close>
           by (auto simp: algebra_simps)
         have "2 * B < e * t"
           unfolding t_def
           using \<open>0 < e\<close> \<open>0 < k\<close> \<open>B > 0\<close> and as and False
           by (auto simp:field_simps)
         then have *: "(f w - f y) / t < e"
           using B(2)[OF \<open>w\<in>S\<close>] and B(2)[OF \<open>y\<in>S\<close>]
           using \<open>t > 0\<close>
           by (auto simp:field_simps)
         have "f x \<le> 1 / (1 + t) * f w + (t / (1 + t)) * f y"
           using assms(2)[unfolded convex_on_def,rule_format,of w y "1/(1+t)" "t / (1+t)",unfolded w]
           using \<open>0 < t\<close> \<open>2 < t\<close> and \<open>y \<in> S\<close> \<open>w \<in> S\<close>
           by (auto simp:field_simps)
         also have "\<dots> = (f w + t * f y) / (1 + t)"
           using \<open>t > 0\<close> by (simp add: add_divide_distrib) 
         also have "\<dots> < e + f y"
           using \<open>t > 0\<close> * \<open>e > 0\<close> by (auto simp: field_simps)
         finally have "f x - f y < e" by auto
       }
       ultimately show ?thesis by auto
     qed (insert \<open>0<e\<close>, auto)
   qed (insert \<open>0<e\<close> \<open>0<k\<close> \<open>0<B\<close>, auto simp: field_simps)
 qed
 
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>Upper bound on a ball implies upper and lower bounds\<close>
 
 lemma convex_bounds_lemma:
   fixes x :: "'a::real_normed_vector"
   assumes "convex_on (cball x e) f"
     and "\<forall>y \<in> cball x e. f y \<le> b"
   shows "\<forall>y \<in> cball x e. \<bar>f y\<bar> \<le> b + 2 * \<bar>f x\<bar>"
   apply rule
 proof (cases "0 \<le> e")
   case True
   fix y
   assume y: "y \<in> cball x e"
   define z where "z = 2 *\<^sub>R x - y"
   have *: "x - (2 *\<^sub>R x - y) = y - x"
     by (simp add: scaleR_2)
   have z: "z \<in> cball x e"
     using y unfolding z_def mem_cball dist_norm * by (auto simp: norm_minus_commute)
   have "(1 / 2) *\<^sub>R y + (1 / 2) *\<^sub>R z = x"
     unfolding z_def by (auto simp: algebra_simps)
   then show "\<bar>f y\<bar> \<le> b + 2 * \<bar>f x\<bar>"
     using assms(1)[unfolded convex_on_def,rule_format, OF y z, of "1/2" "1/2"]
     using assms(2)[rule_format,OF y] assms(2)[rule_format,OF z]
     by (auto simp:field_simps)
 next
   case False
   fix y
   assume "y \<in> cball x e"
   then have "dist x y < 0"
     using False unfolding mem_cball not_le by (auto simp del: dist_not_less_zero)
   then show "\<bar>f y\<bar> \<le> b + 2 * \<bar>f x\<bar>"
     using zero_le_dist[of x y] by auto
 qed
 
 
 subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Hence a convex function on an open set is continuous\<close>
 
 lemma real_of_nat_ge_one_iff: "1 \<le> real (n::nat) \<longleftrightarrow> 1 \<le> n"
   by auto
 
 lemma convex_on_continuous:
   assumes "open (s::('a::euclidean_space) set)" "convex_on s f"
   shows "continuous_on s f"
   unfolding continuous_on_eq_continuous_at[OF assms(1)]
 proof
   note dimge1 = DIM_positive[where 'a='a]
   fix x
   assume "x \<in> s"
   then obtain e where e: "cball x e \<subseteq> s" "e > 0"
     using assms(1) unfolding open_contains_cball by auto
   define d where "d = e / real DIM('a)"
   have "0 < d"
     unfolding d_def using \<open>e > 0\<close> dimge1 by auto
   let ?d = "(\<Sum>i\<in>Basis. d *\<^sub>R i)::'a"
   obtain c
     where c: "finite c"
     and c1: "convex hull c \<subseteq> cball x e"
     and c2: "cball x d \<subseteq> convex hull c"
   proof
     define c where "c = (\<Sum>i\<in>Basis. (\<lambda>a. a *\<^sub>R i) ` {x\<bullet>i - d, x\<bullet>i + d})"
     show "finite c"
       unfolding c_def by (simp add: finite_set_sum)
     have 1: "convex hull c = {a. \<forall>i\<in>Basis. a \<bullet> i \<in> cbox (x \<bullet> i - d) (x \<bullet> i + d)}"
       unfolding box_eq_set_sum_Basis
       unfolding c_def convex_hull_set_sum
       apply (subst convex_hull_linear_image [symmetric])
       apply (simp add: linear_iff scaleR_add_left)
       apply (rule sum.cong [OF refl])
       apply (rule image_cong [OF _ refl])
       apply (rule convex_hull_eq_real_cbox)
       apply (cut_tac \<open>0 < d\<close>, simp)
       done
     then have 2: "convex hull c = {a. \<forall>i\<in>Basis. a \<bullet> i \<in> cball (x \<bullet> i) d}"
       by (simp add: dist_norm abs_le_iff algebra_simps)
     show "cball x d \<subseteq> convex hull c"
       unfolding 2
       by (clarsimp simp: dist_norm) (metis inner_commute inner_diff_right norm_bound_Basis_le)
     have e': "e = (\<Sum>(i::'a)\<in>Basis. d)"
       by (simp add: d_def)
     show "convex hull c \<subseteq> cball x e"
       unfolding 2
       apply clarsimp
       apply (subst euclidean_dist_l2)
       apply (rule order_trans [OF L2_set_le_sum])
       apply (rule zero_le_dist)
       unfolding e'
       apply (rule sum_mono, simp)
       done
   qed
   define k where "k = Max (f ` c)"
   have "convex_on (convex hull c) f"
     apply(rule convex_on_subset[OF assms(2)])
     apply(rule subset_trans[OF c1 e(1)])
     done
   then have k: "\<forall>y\<in>convex hull c. f y \<le> k"
     apply (rule_tac convex_on_convex_hull_bound, assumption)
     by (simp add: k_def c)
   have "e \<le> e * real DIM('a)"
     using e(2) real_of_nat_ge_one_iff by auto
   then have "d \<le> e"
     by (simp add: d_def field_split_simps)
   then have dsube: "cball x d \<subseteq> cball x e"
     by (rule subset_cball)
   have conv: "convex_on (cball x d) f"
     using \<open>convex_on (convex hull c) f\<close> c2 convex_on_subset by blast
   then have "\<forall>y\<in>cball x d. \<bar>f y\<bar> \<le> k + 2 * \<bar>f x\<bar>"
     by (rule convex_bounds_lemma) (use c2 k in blast)
   then have "continuous_on (ball x d) f"
     apply (rule_tac convex_on_bounded_continuous)
     apply (rule open_ball, rule convex_on_subset[OF conv])
     apply (rule ball_subset_cball, force)
     done
   then show "continuous (at x) f"
     unfolding continuous_on_eq_continuous_at[OF open_ball]
     using \<open>d > 0\<close> by auto
 qed
 
 end
diff --git a/src/HOL/Analysis/Equivalence_Lebesgue_Henstock_Integration.thy b/src/HOL/Analysis/Equivalence_Lebesgue_Henstock_Integration.thy
--- a/src/HOL/Analysis/Equivalence_Lebesgue_Henstock_Integration.thy
+++ b/src/HOL/Analysis/Equivalence_Lebesgue_Henstock_Integration.thy
@@ -1,5080 +1,5080 @@
 (*  Title:      HOL/Analysis/Equivalence_Lebesgue_Henstock_Integration.thy
     Author:     Johannes Hölzl, TU München
     Author:     Robert Himmelmann, TU München
     Huge cleanup by LCP
 *)
 
 theory Equivalence_Lebesgue_Henstock_Integration
   imports
     Lebesgue_Measure
     Henstock_Kurzweil_Integration
     Complete_Measure
     Set_Integral
     Homeomorphism
     Cartesian_Euclidean_Space
 begin
 
 lemma le_left_mono: "x \<le> y \<Longrightarrow> y \<le> a \<longrightarrow> x \<le> (a::'a::preorder)"
   by (auto intro: order_trans)
 
 lemma ball_trans:
   assumes "y \<in> ball z q" "r + q \<le> s" shows "ball y r \<subseteq> ball z s"
   using assms by metric
 
 lemma has_integral_implies_lebesgue_measurable_cbox:
   fixes f :: "'a :: euclidean_space \<Rightarrow> real"
   assumes f: "(f has_integral I) (cbox x y)"
   shows "f \<in> lebesgue_on (cbox x y) \<rightarrow>\<^sub>M borel"
 proof (rule cld_measure.borel_measurable_cld)
   let ?L = "lebesgue_on (cbox x y)"
   let ?\<mu> = "emeasure ?L"
   let ?\<mu>' = "outer_measure_of ?L"
   interpret L: finite_measure ?L
   proof
     show "?\<mu> (space ?L) \<noteq> \<infinity>"
       by (simp add: emeasure_restrict_space space_restrict_space emeasure_lborel_cbox_eq)
   qed
 
   show "cld_measure ?L"
   proof
     fix B A assume "B \<subseteq> A" "A \<in> null_sets ?L"
     then show "B \<in> sets ?L"
       using null_sets_completion_subset[OF \<open>B \<subseteq> A\<close>, of lborel]
       by (auto simp add: null_sets_restrict_space sets_restrict_space_iff intro: )
   next
     fix A assume "A \<subseteq> space ?L" "\<And>B. B \<in> sets ?L \<Longrightarrow> ?\<mu> B < \<infinity> \<Longrightarrow> A \<inter> B \<in> sets ?L"
     from this(1) this(2)[of "space ?L"] show "A \<in> sets ?L"
       by (auto simp: Int_absorb2 less_top[symmetric])
   qed auto
   then interpret cld_measure ?L
     .
 
   have content_eq_L: "A \<in> sets borel \<Longrightarrow> A \<subseteq> cbox x y \<Longrightarrow> content A = measure ?L A" for A
     by (subst measure_restrict_space) (auto simp: measure_def)
 
   fix E and a b :: real assume "E \<in> sets ?L" "a < b" "0 < ?\<mu> E" "?\<mu> E < \<infinity>"
   then obtain M :: real where "?\<mu> E = M" "0 < M"
     by (cases "?\<mu> E") auto
   define e where "e = M / (4 + 2 / (b - a))"
   from \<open>a < b\<close> \<open>0<M\<close> have "0 < e"
     by (auto intro!: divide_pos_pos simp: field_simps e_def)
 
   have "e < M / (3 + 2 / (b - a))"
     using \<open>a < b\<close> \<open>0 < M\<close>
     unfolding e_def by (intro divide_strict_left_mono add_strict_right_mono mult_pos_pos) (auto simp: field_simps)
   then have "2 * e < (b - a) * (M - e * 3)"
     using \<open>0<M\<close> \<open>0 < e\<close> \<open>a < b\<close> by (simp add: field_simps)
 
   have e_less_M: "e < M / 1"
     unfolding e_def using \<open>a < b\<close> \<open>0<M\<close> by (intro divide_strict_left_mono) (auto simp: field_simps)
 
   obtain d
     where "gauge d"
       and integral_f: "\<forall>p. p tagged_division_of cbox x y \<and> d fine p \<longrightarrow>
         norm ((\<Sum>(x,k) \<in> p. content k *\<^sub>R f x) - I) < e"
     using \<open>0<e\<close> f unfolding has_integral by auto
 
   define C where "C X m = X \<inter> {x. ball x (1/Suc m) \<subseteq> d x}" for X m
   have "incseq (C X)" for X
     unfolding C_def [abs_def]
     by (intro monoI Collect_mono conj_mono imp_refl le_left_mono subset_ball divide_left_mono Int_mono) auto
 
   { fix X assume "X \<subseteq> space ?L" and eq: "?\<mu>' X = ?\<mu> E"
     have "(SUP m. outer_measure_of ?L (C X m)) = outer_measure_of ?L (\<Union>m. C X m)"
       using \<open>X \<subseteq> space ?L\<close> by (intro SUP_outer_measure_of_incseq \<open>incseq (C X)\<close>) (auto simp: C_def)
     also have "(\<Union>m. C X m) = X"
     proof -
       { fix x
         obtain e where "0 < e" "ball x e \<subseteq> d x"
           using gaugeD[OF \<open>gauge d\<close>, of x] unfolding open_contains_ball by auto
         moreover
         obtain n where "1 / (1 + real n) < e"
           using reals_Archimedean[OF \<open>0<e\<close>] by (auto simp: inverse_eq_divide)
         then have "ball x (1 / (1 + real n)) \<subseteq> ball x e"
           by (intro subset_ball) auto
         ultimately have "\<exists>n. ball x (1 / (1 + real n)) \<subseteq> d x"
           by blast }
       then show ?thesis
         by (auto simp: C_def)
     qed
     finally have "(SUP m. outer_measure_of ?L (C X m)) = ?\<mu> E"
       using eq by auto
     also have "\<dots> > M - e"
       using \<open>0 < M\<close> \<open>?\<mu> E = M\<close> \<open>0<e\<close> by (auto intro!: ennreal_lessI)
     finally have "\<exists>m. M - e < outer_measure_of ?L (C X m)"
       unfolding less_SUP_iff by auto }
   note C = this
 
   let ?E = "{x\<in>E. f x \<le> a}" and ?F = "{x\<in>E. b \<le> f x}"
 
   have "\<not> (?\<mu>' ?E = ?\<mu> E \<and> ?\<mu>' ?F = ?\<mu> E)"
   proof
     assume eq: "?\<mu>' ?E = ?\<mu> E \<and> ?\<mu>' ?F = ?\<mu> E"
     with C[of ?E] C[of ?F] \<open>E \<in> sets ?L\<close>[THEN sets.sets_into_space] obtain ma mb
       where "M - e < outer_measure_of ?L (C ?E ma)" "M - e < outer_measure_of ?L (C ?F mb)"
       by auto
     moreover define m where "m = max ma mb"
     ultimately have M_minus_e: "M - e < outer_measure_of ?L (C ?E m)" "M - e < outer_measure_of ?L (C ?F m)"
       using
         incseqD[OF \<open>incseq (C ?E)\<close>, of ma m, THEN outer_measure_of_mono]
         incseqD[OF \<open>incseq (C ?F)\<close>, of mb m, THEN outer_measure_of_mono]
       by (auto intro: less_le_trans)
     define d' where "d' x = d x \<inter> ball x (1 / (3 * Suc m))" for x
     have "gauge d'"
       unfolding d'_def by (intro gauge_Int \<open>gauge d\<close> gauge_ball) auto
     then obtain p where p: "p tagged_division_of cbox x y" "d' fine p"
       by (rule fine_division_exists)
     then have "d fine p"
       unfolding d'_def[abs_def] fine_def by auto
 
     define s where "s = {(x::'a, k). k \<inter> (C ?E m) \<noteq> {} \<and> k \<inter> (C ?F m) \<noteq> {}}"
     define T where "T E k = (SOME x. x \<in> k \<inter> C E m)" for E k
     let ?A = "(\<lambda>(x, k). (T ?E k, k)) ` (p \<inter> s) \<union> (p - s)"
     let ?B = "(\<lambda>(x, k). (T ?F k, k)) ` (p \<inter> s) \<union> (p - s)"
 
     { fix X assume X_eq: "X = ?E \<or> X = ?F"
       let ?T = "(\<lambda>(x, k). (T X k, k))"
       let ?p = "?T ` (p \<inter> s) \<union> (p - s)"
 
       have in_s: "(x, k) \<in> s \<Longrightarrow> T X k \<in> k \<inter> C X m" for x k
         using someI_ex[of "\<lambda>x. x \<in> k \<inter> C X m"] X_eq unfolding ex_in_conv by (auto simp: T_def s_def)
 
       { fix x k assume "(x, k) \<in> p" "(x, k) \<in> s"
         have k: "k \<subseteq> ball x (1 / (3 * Suc m))"
           using \<open>d' fine p\<close>[THEN fineD, OF \<open>(x, k) \<in> p\<close>] by (auto simp: d'_def)
         then have "x \<in> ball (T X k) (1 / (3 * Suc m))"
           using in_s[OF \<open>(x, k) \<in> s\<close>] by (auto simp: C_def subset_eq dist_commute)
         then have "ball x (1 / (3 * Suc m)) \<subseteq> ball (T X k) (1 / Suc m)"
           by (rule ball_trans) (auto simp: field_split_simps)
         with k in_s[OF \<open>(x, k) \<in> s\<close>] have "k \<subseteq> d (T X k)"
           by (auto simp: C_def) }
       then have "d fine ?p"
         using \<open>d fine p\<close> by (auto intro!: fineI)
       moreover
       have "?p tagged_division_of cbox x y"
       proof (rule tagged_division_ofI)
         show "finite ?p"
           using p(1) by auto
       next
         fix z k assume *: "(z, k) \<in> ?p"
         then consider "(z, k) \<in> p" "(z, k) \<notin> s"
           | x' where "(x', k) \<in> p" "(x', k) \<in> s" "z = T X k"
           by (auto simp: T_def)
         then have "z \<in> k \<and> k \<subseteq> cbox x y \<and> (\<exists>a b. k = cbox a b)"
           using p(1) by cases (auto dest: in_s)
         then show "z \<in> k" "k \<subseteq> cbox x y" "\<exists>a b. k = cbox a b"
           by auto
       next
         fix z k z' k' assume "(z, k) \<in> ?p" "(z', k') \<in> ?p" "(z, k) \<noteq> (z', k')"
         with tagged_division_ofD(5)[OF p(1), of _ k _ k']
         show "interior k \<inter> interior k' = {}"
           by (auto simp: T_def dest: in_s)
       next
         have "{k. \<exists>x. (x, k) \<in> ?p} = {k. \<exists>x. (x, k) \<in> p}"
           by (auto simp: T_def image_iff Bex_def)
         then show "\<Union>{k. \<exists>x. (x, k) \<in> ?p} = cbox x y"
           using p(1) by auto
       qed
       ultimately have I: "norm ((\<Sum>(x,k) \<in> ?p. content k *\<^sub>R f x) - I) < e"
         using integral_f by auto
 
       have "(\<Sum>(x,k) \<in> ?p. content k *\<^sub>R f x) =
         (\<Sum>(x,k) \<in> ?T ` (p \<inter> s). content k *\<^sub>R f x) + (\<Sum>(x,k) \<in> p - s. content k *\<^sub>R f x)"
         using p(1)[THEN tagged_division_ofD(1)]
         by (safe intro!: sum.union_inter_neutral) (auto simp: s_def T_def)
       also have "(\<Sum>(x,k) \<in> ?T ` (p \<inter> s). content k *\<^sub>R f x) = (\<Sum>(x,k) \<in> p \<inter> s. content k *\<^sub>R f (T X k))"
       proof (subst sum.reindex_nontrivial, safe)
         fix x1 x2 k assume 1: "(x1, k) \<in> p" "(x1, k) \<in> s" and 2: "(x2, k) \<in> p" "(x2, k) \<in> s"
           and eq: "content k *\<^sub>R f (T X k) \<noteq> 0"
         with tagged_division_ofD(5)[OF p(1), of x1 k x2 k] tagged_division_ofD(4)[OF p(1), of x1 k]
         show "x1 = x2"
           by (auto simp: content_eq_0_interior)
       qed (use p in \<open>auto intro!: sum.cong\<close>)
       finally have eq: "(\<Sum>(x,k) \<in> ?p. content k *\<^sub>R f x) =
         (\<Sum>(x,k) \<in> p \<inter> s. content k *\<^sub>R f (T X k)) + (\<Sum>(x,k) \<in> p - s. content k *\<^sub>R f x)" .
 
       have in_T: "(x, k) \<in> s \<Longrightarrow> T X k \<in> X" for x k
         using in_s[of x k] by (auto simp: C_def)
 
       note I eq in_T }
     note parts = this
 
     have p_in_L: "(x, k) \<in> p \<Longrightarrow> k \<in> sets ?L" for x k
       using tagged_division_ofD(3, 4)[OF p(1), of x k] by (auto simp: sets_restrict_space)
 
     have [simp]: "finite p"
       using tagged_division_ofD(1)[OF p(1)] .
 
     have "(M - 3*e) * (b - a) \<le> (\<Sum>(x,k) \<in> p \<inter> s. content k) * (b - a)"
     proof (intro mult_right_mono)
       have fin: "?\<mu> (E \<inter> \<Union>{k\<in>snd`p. k \<inter> C X m = {}}) < \<infinity>" for X
         using \<open>?\<mu> E < \<infinity>\<close> by (rule le_less_trans[rotated]) (auto intro!: emeasure_mono \<open>E \<in> sets ?L\<close>)
       have sets: "(E \<inter> \<Union>{k\<in>snd`p. k \<inter> C X m = {}}) \<in> sets ?L" for X
         using tagged_division_ofD(1)[OF p(1)] by (intro sets.Diff \<open>E \<in> sets ?L\<close> sets.finite_Union sets.Int) (auto intro: p_in_L)
       { fix X assume "X \<subseteq> E" "M - e < ?\<mu>' (C X m)"
         have "M - e \<le> ?\<mu>' (C X m)"
           by (rule less_imp_le) fact
         also have "\<dots> \<le> ?\<mu>' (E - (E \<inter> \<Union>{k\<in>snd`p. k \<inter> C X m = {}}))"
         proof (intro outer_measure_of_mono subsetI)
           fix v assume "v \<in> C X m"
           then have "v \<in> cbox x y" "v \<in> E"
             using \<open>E \<subseteq> space ?L\<close> \<open>X \<subseteq> E\<close> by (auto simp: space_restrict_space C_def)
           then obtain z k where "(z, k) \<in> p" "v \<in> k"
             using tagged_division_ofD(6)[OF p(1), symmetric] by auto
           then show "v \<in> E - E \<inter> (\<Union>{k\<in>snd`p. k \<inter> C X m = {}})"
             using \<open>v \<in> C X m\<close> \<open>v \<in> E\<close> by auto
         qed
         also have "\<dots> = ?\<mu> E - ?\<mu> (E \<inter> \<Union>{k\<in>snd`p. k \<inter> C X m = {}})"
           using \<open>E \<in> sets ?L\<close> fin[of X] sets[of X] by (auto intro!: emeasure_Diff)
         finally have "?\<mu> (E \<inter> \<Union>{k\<in>snd`p. k \<inter> C X m = {}}) \<le> e"
           using \<open>0 < e\<close> e_less_M apply (cases "?\<mu> (E \<inter> \<Union>{k\<in>snd`p. k \<inter> C X m = {}})")
           by (auto simp add: \<open>?\<mu> E = M\<close> ennreal_minus ennreal_le_iff2)
         note this }
       note upper_bound = this
 
       have "?\<mu> (E \<inter> \<Union>(snd`(p - s))) =
         ?\<mu> ((E \<inter> \<Union>{k\<in>snd`p. k \<inter> C ?E m = {}}) \<union> (E \<inter> \<Union>{k\<in>snd`p. k \<inter> C ?F m = {}}))"
         by (intro arg_cong[where f="?\<mu>"]) (auto simp: s_def image_def Bex_def)
       also have "\<dots> \<le> ?\<mu> (E \<inter> \<Union>{k\<in>snd`p. k \<inter> C ?E m = {}}) + ?\<mu> (E \<inter> \<Union>{k\<in>snd`p. k \<inter> C ?F m = {}})"
         using sets[of ?E] sets[of ?F] M_minus_e by (intro emeasure_subadditive) auto
       also have "\<dots> \<le> e + ennreal e"
         using upper_bound[of ?E] upper_bound[of ?F] M_minus_e by (intro add_mono) auto
       finally have "?\<mu> E - 2*e \<le> ?\<mu> (E - (E \<inter> \<Union>(snd`(p - s))))"
         using \<open>0 < e\<close> \<open>E \<in> sets ?L\<close> tagged_division_ofD(1)[OF p(1)]
         by (subst emeasure_Diff)
            (auto simp: top_unique simp flip: ennreal_plus
                  intro!: sets.Int sets.finite_UN ennreal_mono_minus intro: p_in_L)
       also have "\<dots> \<le> ?\<mu> (\<Union>x\<in>p \<inter> s. snd x)"
       proof (safe intro!: emeasure_mono subsetI)
         fix v assume "v \<in> E" and not: "v \<notin> (\<Union>x\<in>p \<inter> s. snd x)"
         then have "v \<in> cbox x y"
           using \<open>E \<subseteq> space ?L\<close> by (auto simp: space_restrict_space)
         then obtain z k where "(z, k) \<in> p" "v \<in> k"
           using tagged_division_ofD(6)[OF p(1), symmetric] by auto
         with not show "v \<in> \<Union>(snd ` (p - s))"
           by (auto intro!: bexI[of _ "(z, k)"] elim: ballE[of _ _ "(z, k)"])
       qed (auto intro!: sets.Int sets.finite_UN ennreal_mono_minus intro: p_in_L)
       also have "\<dots> = measure ?L (\<Union>x\<in>p \<inter> s. snd x)"
         by (auto intro!: emeasure_eq_ennreal_measure)
       finally have "M - 2 * e \<le> measure ?L (\<Union>x\<in>p \<inter> s. snd x)"
         unfolding \<open>?\<mu> E = M\<close> using \<open>0 < e\<close> by (simp add: ennreal_minus)
       also have "measure ?L (\<Union>x\<in>p \<inter> s. snd x) = content (\<Union>x\<in>p \<inter> s. snd x)"
         using tagged_division_ofD(1,3,4) [OF p(1)]
         by (intro content_eq_L[symmetric])
            (fastforce intro!: sets.finite_UN UN_least del: subsetI)+
       also have "content (\<Union>x\<in>p \<inter> s. snd x) \<le> (\<Sum>k\<in>p \<inter> s. content (snd k))"
         using p(1) by (auto simp: emeasure_lborel_cbox_eq intro!: measure_subadditive_finite
                             dest!: p(1)[THEN tagged_division_ofD(4)])
       finally show "M - 3 * e \<le> (\<Sum>(x, y)\<in>p \<inter> s. content y)"
         using \<open>0 < e\<close> by (simp add: split_beta)
     qed (use \<open>a < b\<close> in auto)
     also have "\<dots> = (\<Sum>(x,k) \<in> p \<inter> s. content k * (b - a))"
       by (simp add: sum_distrib_right split_beta')
     also have "\<dots> \<le> (\<Sum>(x,k) \<in> p \<inter> s. content k * (f (T ?F k) - f (T ?E k)))"
       using parts(3) by (auto intro!: sum_mono mult_left_mono diff_mono)
     also have "\<dots> = (\<Sum>(x,k) \<in> p \<inter> s. content k * f (T ?F k)) - (\<Sum>(x,k) \<in> p \<inter> s. content k * f (T ?E k))"
       by (auto intro!: sum.cong simp: field_simps sum_subtractf[symmetric])
     also have "\<dots> = (\<Sum>(x,k) \<in> ?B. content k *\<^sub>R f x) - (\<Sum>(x,k) \<in> ?A. content k *\<^sub>R f x)"
       by (subst (1 2) parts) auto
     also have "\<dots> \<le> norm ((\<Sum>(x,k) \<in> ?B. content k *\<^sub>R f x) - (\<Sum>(x,k) \<in> ?A. content k *\<^sub>R f x))"
       by auto
     also have "\<dots> \<le> e + e"
       using parts(1)[of ?E] parts(1)[of ?F] by (intro norm_diff_triangle_le[of _ I]) auto
     finally show False
       using \<open>2 * e < (b - a) * (M - e * 3)\<close> by (auto simp: field_simps)
   qed
   moreover have "?\<mu>' ?E \<le> ?\<mu> E" "?\<mu>' ?F \<le> ?\<mu> E"
     unfolding outer_measure_of_eq[OF \<open>E \<in> sets ?L\<close>, symmetric] by (auto intro!: outer_measure_of_mono)
   ultimately show "min (?\<mu>' ?E) (?\<mu>' ?F) < ?\<mu> E"
     unfolding min_less_iff_disj by (auto simp: less_le)
 qed
 
 lemma has_integral_implies_lebesgue_measurable_real:
   fixes f :: "'a :: euclidean_space \<Rightarrow> real"
   assumes f: "(f has_integral I) \<Omega>"
   shows "(\<lambda>x. f x * indicator \<Omega> x) \<in> lebesgue \<rightarrow>\<^sub>M borel"
 proof -
   define B :: "nat \<Rightarrow> 'a set" where "B n = cbox (- real n *\<^sub>R One) (real n *\<^sub>R One)" for n
   show "(\<lambda>x. f x * indicator \<Omega> x) \<in> lebesgue \<rightarrow>\<^sub>M borel"
   proof (rule measurable_piecewise_restrict)
     have "(\<Union>n. box (- real n *\<^sub>R One) (real n *\<^sub>R One)) \<subseteq> \<Union>(B ` UNIV)"
       unfolding B_def by (intro UN_mono box_subset_cbox order_refl)
     then show "countable (range B)" "space lebesgue \<subseteq> \<Union>(B ` UNIV)"
       by (auto simp: B_def UN_box_eq_UNIV)
   next
     fix \<Omega>' assume "\<Omega>' \<in> range B"
     then obtain n where \<Omega>': "\<Omega>' = B n" by auto
     then show "\<Omega>' \<inter> space lebesgue \<in> sets lebesgue"
       by (auto simp: B_def)
 
     have "f integrable_on \<Omega>"
       using f by auto
     then have "(\<lambda>x. f x * indicator \<Omega> x) integrable_on \<Omega>"
       by (auto simp: integrable_on_def cong: has_integral_cong)
     then have "(\<lambda>x. f x * indicator \<Omega> x) integrable_on (\<Omega> \<union> B n)"
       by (rule integrable_on_superset) auto
     then have "(\<lambda>x. f x * indicator \<Omega> x) integrable_on B n"
       unfolding B_def by (rule integrable_on_subcbox) auto
     then show "(\<lambda>x. f x * indicator \<Omega> x) \<in> lebesgue_on \<Omega>' \<rightarrow>\<^sub>M borel"
       unfolding B_def \<Omega>' by (auto intro: has_integral_implies_lebesgue_measurable_cbox simp: integrable_on_def)
   qed
 qed
 
 lemma has_integral_implies_lebesgue_measurable:
   fixes f :: "'a :: euclidean_space \<Rightarrow> 'b :: euclidean_space"
   assumes f: "(f has_integral I) \<Omega>"
   shows "(\<lambda>x. indicator \<Omega> x *\<^sub>R f x) \<in> lebesgue \<rightarrow>\<^sub>M borel"
 proof (intro borel_measurable_euclidean_space[where 'c='b, THEN iffD2] ballI)
   fix i :: "'b" assume "i \<in> Basis"
   have "(\<lambda>x. (f x \<bullet> i) * indicator \<Omega> x) \<in> borel_measurable (completion lborel)"
     using has_integral_linear[OF f bounded_linear_inner_left, of i]
     by (intro has_integral_implies_lebesgue_measurable_real) (auto simp: comp_def)
   then show "(\<lambda>x. indicator \<Omega> x *\<^sub>R f x \<bullet> i) \<in> borel_measurable (completion lborel)"
     by (simp add: ac_simps)
 qed
 
 subsection \<open>Equivalence Lebesgue integral on \<^const>\<open>lborel\<close> and HK-integral\<close>
 
 lemma has_integral_measure_lborel:
   fixes A :: "'a::euclidean_space set"
   assumes A[measurable]: "A \<in> sets borel" and finite: "emeasure lborel A < \<infinity>"
   shows "((\<lambda>x. 1) has_integral measure lborel A) A"
 proof -
   { fix l u :: 'a
     have "((\<lambda>x. 1) has_integral measure lborel (box l u)) (box l u)"
     proof cases
       assume "\<forall>b\<in>Basis. l \<bullet> b \<le> u \<bullet> b"
       then show ?thesis
         apply simp
         apply (subst has_integral_restrict[symmetric, OF box_subset_cbox])
         apply (subst has_integral_spike_interior_eq[where g="\<lambda>_. 1"])
         using has_integral_const[of "1::real" l u]
         apply (simp_all add: inner_diff_left[symmetric] content_cbox_cases)
         done
     next
       assume "\<not> (\<forall>b\<in>Basis. l \<bullet> b \<le> u \<bullet> b)"
       then have "box l u = {}"
         unfolding box_eq_empty by (auto simp: not_le intro: less_imp_le)
       then show ?thesis
         by simp
     qed }
   note has_integral_box = this
 
   { fix a b :: 'a let ?M = "\<lambda>A. measure lborel (A \<inter> box a b)"
     have "Int_stable  (range (\<lambda>(a, b). box a b))"
       by (auto simp: Int_stable_def box_Int_box)
     moreover have "(range (\<lambda>(a, b). box a b)) \<subseteq> Pow UNIV"
       by auto
     moreover have "A \<in> sigma_sets UNIV (range (\<lambda>(a, b). box a b))"
        using A unfolding borel_eq_box by simp
     ultimately have "((\<lambda>x. 1) has_integral ?M A) (A \<inter> box a b)"
     proof (induction rule: sigma_sets_induct_disjoint)
       case (basic A) then show ?case
         by (auto simp: box_Int_box has_integral_box)
     next
       case empty then show ?case
         by simp
     next
       case (compl A)
       then have [measurable]: "A \<in> sets borel"
         by (simp add: borel_eq_box)
 
       have "((\<lambda>x. 1) has_integral ?M (box a b)) (box a b)"
         by (simp add: has_integral_box)
       moreover have "((\<lambda>x. if x \<in> A \<inter> box a b then 1 else 0) has_integral ?M A) (box a b)"
         by (subst has_integral_restrict) (auto intro: compl)
       ultimately have "((\<lambda>x. 1 - (if x \<in> A \<inter> box a b then 1 else 0)) has_integral ?M (box a b) - ?M A) (box a b)"
         by (rule has_integral_diff)
       then have "((\<lambda>x. (if x \<in> (UNIV - A) \<inter> box a b then 1 else 0)) has_integral ?M (box a b) - ?M A) (box a b)"
         by (rule has_integral_cong[THEN iffD1, rotated 1]) auto
       then have "((\<lambda>x. 1) has_integral ?M (box a b) - ?M A) ((UNIV - A) \<inter> box a b)"
         by (subst (asm) has_integral_restrict) auto
       also have "?M (box a b) - ?M A = ?M (UNIV - A)"
         by (subst measure_Diff[symmetric]) (auto simp: emeasure_lborel_box_eq Diff_Int_distrib2)
       finally show ?case .
     next
       case (union F)
       then have [measurable]: "\<And>i. F i \<in> sets borel"
         by (simp add: borel_eq_box subset_eq)
       have "((\<lambda>x. if x \<in> \<Union>(F ` UNIV) \<inter> box a b then 1 else 0) has_integral ?M (\<Union>i. F i)) (box a b)"
       proof (rule has_integral_monotone_convergence_increasing)
         let ?f = "\<lambda>k x. \<Sum>i<k. if x \<in> F i \<inter> box a b then 1 else 0 :: real"
         show "\<And>k. (?f k has_integral (\<Sum>i<k. ?M (F i))) (box a b)"
           using union.IH by (auto intro!: has_integral_sum simp del: Int_iff)
         show "\<And>k x. ?f k x \<le> ?f (Suc k) x"
           by (intro sum_mono2) auto
         from union(1) have *: "\<And>x i j. x \<in> F i \<Longrightarrow> x \<in> F j \<longleftrightarrow> j = i"
           by (auto simp add: disjoint_family_on_def)
         show "\<And>x. (\<lambda>k. ?f k x) \<longlonglongrightarrow> (if x \<in> \<Union>(F ` UNIV) \<inter> box a b then 1 else 0)"
           apply (auto simp: * sum.If_cases Iio_Int_singleton)
           apply (rule_tac k="Suc xa" in LIMSEQ_offset)
           apply simp
           done
         have *: "emeasure lborel ((\<Union>x. F x) \<inter> box a b) \<le> emeasure lborel (box a b)"
           by (intro emeasure_mono) auto
 
         with union(1) show "(\<lambda>k. \<Sum>i<k. ?M (F i)) \<longlonglongrightarrow> ?M (\<Union>i. F i)"
           unfolding sums_def[symmetric] UN_extend_simps
           by (intro measure_UNION) (auto simp: disjoint_family_on_def emeasure_lborel_box_eq top_unique)
       qed
       then show ?case
         by (subst (asm) has_integral_restrict) auto
     qed }
   note * = this
 
   show ?thesis
   proof (rule has_integral_monotone_convergence_increasing)
     let ?B = "\<lambda>n::nat. box (- real n *\<^sub>R One) (real n *\<^sub>R One) :: 'a set"
     let ?f = "\<lambda>n::nat. \<lambda>x. if x \<in> A \<inter> ?B n then 1 else 0 :: real"
     let ?M = "\<lambda>n. measure lborel (A \<inter> ?B n)"
 
     show "\<And>n::nat. (?f n has_integral ?M n) A"
       using * by (subst has_integral_restrict) simp_all
     show "\<And>k x. ?f k x \<le> ?f (Suc k) x"
       by (auto simp: box_def)
     { fix x assume "x \<in> A"
       moreover have "(\<lambda>k. indicator (A \<inter> ?B k) x :: real) \<longlonglongrightarrow> indicator (\<Union>k::nat. A \<inter> ?B k) x"
         by (intro LIMSEQ_indicator_incseq) (auto simp: incseq_def box_def)
       ultimately show "(\<lambda>k. if x \<in> A \<inter> ?B k then 1 else 0::real) \<longlonglongrightarrow> 1"
         by (simp add: indicator_def UN_box_eq_UNIV) }
 
     have "(\<lambda>n. emeasure lborel (A \<inter> ?B n)) \<longlonglongrightarrow> emeasure lborel (\<Union>n::nat. A \<inter> ?B n)"
       by (intro Lim_emeasure_incseq) (auto simp: incseq_def box_def)
     also have "(\<lambda>n. emeasure lborel (A \<inter> ?B n)) = (\<lambda>n. measure lborel (A \<inter> ?B n))"
     proof (intro ext emeasure_eq_ennreal_measure)
       fix n have "emeasure lborel (A \<inter> ?B n) \<le> emeasure lborel (?B n)"
         by (intro emeasure_mono) auto
       then show "emeasure lborel (A \<inter> ?B n) \<noteq> top"
         by (auto simp: top_unique)
     qed
     finally show "(\<lambda>n. measure lborel (A \<inter> ?B n)) \<longlonglongrightarrow> measure lborel A"
       using emeasure_eq_ennreal_measure[of lborel A] finite
       by (simp add: UN_box_eq_UNIV less_top)
   qed
 qed
 
 lemma nn_integral_has_integral:
   fixes f::"'a::euclidean_space \<Rightarrow> real"
   assumes f: "f \<in> borel_measurable borel" "\<And>x. 0 \<le> f x" "(\<integral>\<^sup>+x. f x \<partial>lborel) = ennreal r" "0 \<le> r"
   shows "(f has_integral r) UNIV"
 using f proof (induct f arbitrary: r rule: borel_measurable_induct_real)
   case (set A)
   then have "((\<lambda>x. 1) has_integral measure lborel A) A"
     by (intro has_integral_measure_lborel) (auto simp: ennreal_indicator)
   with set show ?case
     by (simp add: ennreal_indicator measure_def) (simp add: indicator_def)
 next
   case (mult g c)
   then have "ennreal c * (\<integral>\<^sup>+ x. g x \<partial>lborel) = ennreal r"
     by (subst nn_integral_cmult[symmetric]) (auto simp: ennreal_mult)
   with \<open>0 \<le> r\<close> \<open>0 \<le> c\<close>
   obtain r' where "(c = 0 \<and> r = 0) \<or> (0 \<le> r' \<and> (\<integral>\<^sup>+ x. ennreal (g x) \<partial>lborel) = ennreal r' \<and> r = c * r')"
     by (cases "\<integral>\<^sup>+ x. ennreal (g x) \<partial>lborel" rule: ennreal_cases)
        (auto split: if_split_asm simp: ennreal_mult_top ennreal_mult[symmetric])
   with mult show ?case
     by (auto intro!: has_integral_cmult_real)
 next
   case (add g h)
   then have "(\<integral>\<^sup>+ x. h x + g x \<partial>lborel) = (\<integral>\<^sup>+ x. h x \<partial>lborel) + (\<integral>\<^sup>+ x. g x \<partial>lborel)"
     by (simp add: nn_integral_add)
   with add obtain a b where "0 \<le> a" "0 \<le> b" "(\<integral>\<^sup>+ x. h x \<partial>lborel) = ennreal a" "(\<integral>\<^sup>+ x. g x \<partial>lborel) = ennreal b" "r = a + b"
     by (cases "\<integral>\<^sup>+ x. h x \<partial>lborel" "\<integral>\<^sup>+ x. g x \<partial>lborel" rule: ennreal2_cases)
        (auto simp: add_top nn_integral_add top_add simp flip: ennreal_plus)
   with add show ?case
     by (auto intro!: has_integral_add)
 next
   case (seq U)
   note seq(1)[measurable] and f[measurable]
 
   { fix i x
     have "U i x \<le> f x"
       using seq(5)
       apply (rule LIMSEQ_le_const)
       using seq(4)
       apply (auto intro!: exI[of _ i] simp: incseq_def le_fun_def)
       done }
   note U_le_f = this
 
   { fix i
     have "(\<integral>\<^sup>+x. U i x \<partial>lborel) \<le> (\<integral>\<^sup>+x. f x \<partial>lborel)"
       using seq(2) f(2) U_le_f by (intro nn_integral_mono) simp
     then obtain p where "(\<integral>\<^sup>+x. U i x \<partial>lborel) = ennreal p" "p \<le> r" "0 \<le> p"
       using seq(6) \<open>0\<le>r\<close> by (cases "\<integral>\<^sup>+x. U i x \<partial>lborel" rule: ennreal_cases) (auto simp: top_unique)
     moreover note seq
     ultimately have "\<exists>p. (\<integral>\<^sup>+x. U i x \<partial>lborel) = ennreal p \<and> 0 \<le> p \<and> p \<le> r \<and> (U i has_integral p) UNIV"
       by auto }
   then obtain p where p: "\<And>i. (\<integral>\<^sup>+x. ennreal (U i x) \<partial>lborel) = ennreal (p i)"
     and bnd: "\<And>i. p i \<le> r" "\<And>i. 0 \<le> p i"
     and U_int: "\<And>i.(U i has_integral (p i)) UNIV" by metis
 
   have int_eq: "\<And>i. integral UNIV (U i) = p i" using U_int by (rule integral_unique)
 
   have *: "f integrable_on UNIV \<and> (\<lambda>k. integral UNIV (U k)) \<longlonglongrightarrow> integral UNIV f"
   proof (rule monotone_convergence_increasing)
     show "\<And>k. U k integrable_on UNIV" using U_int by auto
     show "\<And>k x. x\<in>UNIV \<Longrightarrow> U k x \<le> U (Suc k) x" using \<open>incseq U\<close> by (auto simp: incseq_def le_fun_def)
     then show "bounded (range (\<lambda>k. integral UNIV (U k)))"
       using bnd int_eq by (auto simp: bounded_real intro!: exI[of _ r])
     show "\<And>x. x\<in>UNIV \<Longrightarrow> (\<lambda>k. U k x) \<longlonglongrightarrow> f x"
       using seq by auto
   qed
   moreover have "(\<lambda>i. (\<integral>\<^sup>+x. U i x \<partial>lborel)) \<longlonglongrightarrow> (\<integral>\<^sup>+x. f x \<partial>lborel)"
     using seq f(2) U_le_f by (intro nn_integral_dominated_convergence[where w=f]) auto
   ultimately have "integral UNIV f = r"
     by (auto simp add: bnd int_eq p seq intro: LIMSEQ_unique)
   with * show ?case
     by (simp add: has_integral_integral)
 qed
 
 lemma nn_integral_lborel_eq_integral:
   fixes f::"'a::euclidean_space \<Rightarrow> real"
   assumes f: "f \<in> borel_measurable borel" "\<And>x. 0 \<le> f x" "(\<integral>\<^sup>+x. f x \<partial>lborel) < \<infinity>"
   shows "(\<integral>\<^sup>+x. f x \<partial>lborel) = integral UNIV f"
 proof -
   from f(3) obtain r where r: "(\<integral>\<^sup>+x. f x \<partial>lborel) = ennreal r" "0 \<le> r"
     by (cases "\<integral>\<^sup>+x. f x \<partial>lborel" rule: ennreal_cases) auto
   then show ?thesis
     using nn_integral_has_integral[OF f(1,2) r] by (simp add: integral_unique)
 qed
 
 lemma nn_integral_integrable_on:
   fixes f::"'a::euclidean_space \<Rightarrow> real"
   assumes f: "f \<in> borel_measurable borel" "\<And>x. 0 \<le> f x" "(\<integral>\<^sup>+x. f x \<partial>lborel) < \<infinity>"
   shows "f integrable_on UNIV"
 proof -
   from f(3) obtain r where r: "(\<integral>\<^sup>+x. f x \<partial>lborel) = ennreal r" "0 \<le> r"
     by (cases "\<integral>\<^sup>+x. f x \<partial>lborel" rule: ennreal_cases) auto
   then show ?thesis
     by (intro has_integral_integrable[where i=r] nn_integral_has_integral[where r=r] f)
 qed
 
 lemma nn_integral_has_integral_lborel:
   fixes f :: "'a::euclidean_space \<Rightarrow> real"
   assumes f_borel: "f \<in> borel_measurable borel" and nonneg: "\<And>x. 0 \<le> f x"
   assumes I: "(f has_integral I) UNIV"
   shows "integral\<^sup>N lborel f = I"
 proof -
   from f_borel have "(\<lambda>x. ennreal (f x)) \<in> borel_measurable lborel" by auto
   from borel_measurable_implies_simple_function_sequence'[OF this]
   obtain F where F: "\<And>i. simple_function lborel (F i)" "incseq F"
                  "\<And>i x. F i x < top" "\<And>x. (SUP i. F i x) = ennreal (f x)"
     by blast
   then have [measurable]: "\<And>i. F i \<in> borel_measurable lborel"
     by (metis borel_measurable_simple_function)
   let ?B = "\<lambda>i::nat. box (- (real i *\<^sub>R One)) (real i *\<^sub>R One) :: 'a set"
 
   have "0 \<le> I"
     using I by (rule has_integral_nonneg) (simp add: nonneg)
 
   have F_le_f: "enn2real (F i x) \<le> f x" for i x
     using F(3,4)[where x=x] nonneg SUP_upper[of i UNIV "\<lambda>i. F i x"]
     by (cases "F i x" rule: ennreal_cases) auto
   let ?F = "\<lambda>i x. F i x * indicator (?B i) x"
   have "(\<integral>\<^sup>+ x. ennreal (f x) \<partial>lborel) = (SUP i. integral\<^sup>N lborel (\<lambda>x. ?F i x))"
   proof (subst nn_integral_monotone_convergence_SUP[symmetric])
     { fix x
       obtain j where j: "x \<in> ?B j"
         using UN_box_eq_UNIV by auto
 
       have "ennreal (f x) = (SUP i. F i x)"
         using F(4)[of x] nonneg[of x] by (simp add: max_def)
       also have "\<dots> = (SUP i. ?F i x)"
       proof (rule SUP_eq)
         fix i show "\<exists>j\<in>UNIV. F i x \<le> ?F j x"
           using j F(2)
           by (intro bexI[of _ "max i j"])
              (auto split: split_max split_indicator simp: incseq_def le_fun_def box_def)
       qed (auto intro!: F split: split_indicator)
       finally have "ennreal (f x) =  (SUP i. ?F i x)" . }
     then show "(\<integral>\<^sup>+ x. ennreal (f x) \<partial>lborel) = (\<integral>\<^sup>+ x. (SUP i. ?F i x) \<partial>lborel)"
       by simp
   qed (insert F, auto simp: incseq_def le_fun_def box_def split: split_indicator)
   also have "\<dots> \<le> ennreal I"
   proof (rule SUP_least)
     fix i :: nat
     have finite_F: "(\<integral>\<^sup>+ x. ennreal (enn2real (F i x) * indicator (?B i) x) \<partial>lborel) < \<infinity>"
     proof (rule nn_integral_bound_simple_function)
       have "emeasure lborel {x \<in> space lborel. ennreal (enn2real (F i x) * indicator (?B i) x) \<noteq> 0} \<le>
         emeasure lborel (?B i)"
         by (intro emeasure_mono)  (auto split: split_indicator)
       then show "emeasure lborel {x \<in> space lborel. ennreal (enn2real (F i x) * indicator (?B i) x) \<noteq> 0} < \<infinity>"
         by (auto simp: less_top[symmetric] top_unique)
     qed (auto split: split_indicator
               intro!: F simple_function_compose1[where g="enn2real"] simple_function_ennreal)
 
     have int_F: "(\<lambda>x. enn2real (F i x) * indicator (?B i) x) integrable_on UNIV"
       using F(4) finite_F
       by (intro nn_integral_integrable_on) (auto split: split_indicator simp: enn2real_nonneg)
 
     have "(\<integral>\<^sup>+ x. F i x * indicator (?B i) x \<partial>lborel) =
       (\<integral>\<^sup>+ x. ennreal (enn2real (F i x) * indicator (?B i) x) \<partial>lborel)"
       using F(3,4)
       by (intro nn_integral_cong) (auto simp: image_iff eq_commute split: split_indicator)
     also have "\<dots> = ennreal (integral UNIV (\<lambda>x. enn2real (F i x) * indicator (?B i) x))"
       using F
       by (intro nn_integral_lborel_eq_integral[OF _ _ finite_F])
          (auto split: split_indicator intro: enn2real_nonneg)
     also have "\<dots> \<le> ennreal I"
       by (auto intro!: has_integral_le[OF integrable_integral[OF int_F] I] nonneg F_le_f
                simp: \<open>0 \<le> I\<close> split: split_indicator )
     finally show "(\<integral>\<^sup>+ x. F i x * indicator (?B i) x \<partial>lborel) \<le> ennreal I" .
   qed
   finally have "(\<integral>\<^sup>+ x. ennreal (f x) \<partial>lborel) < \<infinity>"
     by (auto simp: less_top[symmetric] top_unique)
   from nn_integral_lborel_eq_integral[OF assms(1,2) this] I show ?thesis
     by (simp add: integral_unique)
 qed
 
 lemma has_integral_iff_emeasure_lborel:
   fixes A :: "'a::euclidean_space set"
   assumes A[measurable]: "A \<in> sets borel" and [simp]: "0 \<le> r"
   shows "((\<lambda>x. 1) has_integral r) A \<longleftrightarrow> emeasure lborel A = ennreal r"
 proof (cases "emeasure lborel A = \<infinity>")
   case emeasure_A: True
   have "\<not> (\<lambda>x. 1::real) integrable_on A"
   proof
     assume int: "(\<lambda>x. 1::real) integrable_on A"
     then have "(indicator A::'a \<Rightarrow> real) integrable_on UNIV"
       unfolding indicator_def[abs_def] integrable_restrict_UNIV .
     then obtain r where "((indicator A::'a\<Rightarrow>real) has_integral r) UNIV"
       by auto
     from nn_integral_has_integral_lborel[OF _ _ this] emeasure_A show False
       by (simp add: ennreal_indicator)
   qed
   with emeasure_A show ?thesis
     by auto
 next
   case False
   then have "((\<lambda>x. 1) has_integral measure lborel A) A"
     by (simp add: has_integral_measure_lborel less_top)
   with False show ?thesis
     by (auto simp: emeasure_eq_ennreal_measure has_integral_unique)
 qed
 
 lemma ennreal_max_0: "ennreal (max 0 x) = ennreal x"
   by (auto simp: max_def ennreal_neg)
 
 lemma has_integral_integral_real:
   fixes f::"'a::euclidean_space \<Rightarrow> real"
   assumes f: "integrable lborel f"
   shows "(f has_integral (integral\<^sup>L lborel f)) UNIV"
 proof -
   from integrableE[OF f] obtain r q
     where "0 \<le> r" "0 \<le> q"
       and r: "(\<integral>\<^sup>+ x. ennreal (max 0 (f x)) \<partial>lborel) = ennreal r"
       and q: "(\<integral>\<^sup>+ x. ennreal (max 0 (- f x)) \<partial>lborel) = ennreal q"
       and f: "f \<in> borel_measurable lborel" and eq: "integral\<^sup>L lborel f = r - q"
     unfolding ennreal_max_0 by auto
   then have "((\<lambda>x. max 0 (f x)) has_integral r) UNIV" "((\<lambda>x. max 0 (- f x)) has_integral q) UNIV"
     using nn_integral_has_integral[OF _ _ r] nn_integral_has_integral[OF _ _ q] by auto
   note has_integral_diff[OF this]
   moreover have "(\<lambda>x. max 0 (f x) - max 0 (- f x)) = f"
     by auto
   ultimately show ?thesis
     by (simp add: eq)
 qed
 
 lemma has_integral_AE:
   assumes ae: "AE x in lborel. x \<in> \<Omega> \<longrightarrow> f x = g x"
   shows "(f has_integral x) \<Omega> = (g has_integral x) \<Omega>"
 proof -
   from ae obtain N
     where N: "N \<in> sets borel" "emeasure lborel N = 0" "{x. \<not> (x \<in> \<Omega> \<longrightarrow> f x = g x)} \<subseteq> N"
     by (auto elim!: AE_E)
   then have not_N: "AE x in lborel. x \<notin> N"
     by (simp add: AE_iff_measurable)
   show ?thesis
   proof (rule has_integral_spike_eq[symmetric])
     show "\<And>x. x\<in>\<Omega> - N \<Longrightarrow> f x = g x" using N(3) by auto
     show "negligible N"
       unfolding negligible_def
     proof (intro allI)
       fix a b :: "'a"
       let ?F = "\<lambda>x::'a. if x \<in> cbox a b then indicator N x else 0 :: real"
       have "integrable lborel ?F = integrable lborel (\<lambda>x::'a. 0::real)"
         using not_N N(1) by (intro integrable_cong_AE) auto
       moreover have "(LINT x|lborel. ?F x) = (LINT x::'a|lborel. 0::real)"
         using not_N N(1) by (intro integral_cong_AE) auto
       ultimately have "(?F has_integral 0) UNIV"
         using has_integral_integral_real[of ?F] by simp
       then show "(indicator N has_integral (0::real)) (cbox a b)"
         unfolding has_integral_restrict_UNIV .
     qed
   qed
 qed
 
 lemma nn_integral_has_integral_lebesgue:
   fixes f :: "'a::euclidean_space \<Rightarrow> real"
   assumes nonneg: "\<And>x. 0 \<le> f x" and I: "(f has_integral I) \<Omega>"
   shows "integral\<^sup>N lborel (\<lambda>x. indicator \<Omega> x * f x) = I"
 proof -
   from I have "(\<lambda>x. indicator \<Omega> x *\<^sub>R f x) \<in> lebesgue \<rightarrow>\<^sub>M borel"
     by (rule has_integral_implies_lebesgue_measurable)
   then obtain f' :: "'a \<Rightarrow> real"
     where [measurable]: "f' \<in> borel \<rightarrow>\<^sub>M borel" and eq: "AE x in lborel. indicator \<Omega> x * f x = f' x"
     by (auto dest: completion_ex_borel_measurable_real)
 
   from I have "((\<lambda>x. abs (indicator \<Omega> x * f x)) has_integral I) UNIV"
     using nonneg by (simp add: indicator_def if_distrib[of "\<lambda>x. x * f y" for y] cong: if_cong)
   also have "((\<lambda>x. abs (indicator \<Omega> x * f x)) has_integral I) UNIV \<longleftrightarrow> ((\<lambda>x. abs (f' x)) has_integral I) UNIV"
     using eq by (intro has_integral_AE) auto
   finally have "integral\<^sup>N lborel (\<lambda>x. abs (f' x)) = I"
     by (rule nn_integral_has_integral_lborel[rotated 2]) auto
   also have "integral\<^sup>N lborel (\<lambda>x. abs (f' x)) = integral\<^sup>N lborel (\<lambda>x. abs (indicator \<Omega> x * f x))"
     using eq by (intro nn_integral_cong_AE) auto
   finally show ?thesis
     using nonneg by auto
 qed
 
 lemma has_integral_iff_nn_integral_lebesgue:
   assumes f: "\<And>x. 0 \<le> f x"
   shows "(f has_integral r) UNIV \<longleftrightarrow> (f \<in> lebesgue \<rightarrow>\<^sub>M borel \<and> integral\<^sup>N lebesgue f = r \<and> 0 \<le> r)" (is "?I = ?N")
 proof
   assume ?I
   have "0 \<le> r"
     using has_integral_nonneg[OF \<open>?I\<close>] f by auto
   then show ?N
     using nn_integral_has_integral_lebesgue[OF f \<open>?I\<close>]
       has_integral_implies_lebesgue_measurable[OF \<open>?I\<close>]
     by (auto simp: nn_integral_completion)
 next
   assume ?N
   then obtain f' where f': "f' \<in> borel \<rightarrow>\<^sub>M borel" "AE x in lborel. f x = f' x"
     by (auto dest: completion_ex_borel_measurable_real)
   moreover have "(\<integral>\<^sup>+ x. ennreal \<bar>f' x\<bar> \<partial>lborel) = (\<integral>\<^sup>+ x. ennreal \<bar>f x\<bar> \<partial>lborel)"
     using f' by (intro nn_integral_cong_AE) auto
   moreover have "((\<lambda>x. \<bar>f' x\<bar>) has_integral r) UNIV \<longleftrightarrow> ((\<lambda>x. \<bar>f x\<bar>) has_integral r) UNIV"
     using f' by (intro has_integral_AE) auto
   moreover note nn_integral_has_integral[of "\<lambda>x. \<bar>f' x\<bar>" r] \<open>?N\<close>
   ultimately show ?I
     using f by (auto simp: nn_integral_completion)
 qed
 
 context
   fixes f::"'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
 begin
 
 lemma has_integral_integral_lborel:
   assumes f: "integrable lborel f"
   shows "(f has_integral (integral\<^sup>L lborel f)) UNIV"
 proof -
   have "((\<lambda>x. \<Sum>b\<in>Basis. (f x \<bullet> b) *\<^sub>R b) has_integral (\<Sum>b\<in>Basis. integral\<^sup>L lborel (\<lambda>x. f x \<bullet> b) *\<^sub>R b)) UNIV"
     using f by (intro has_integral_sum finite_Basis ballI has_integral_scaleR_left has_integral_integral_real) auto
   also have eq_f: "(\<lambda>x. \<Sum>b\<in>Basis. (f x \<bullet> b) *\<^sub>R b) = f"
     by (simp add: fun_eq_iff euclidean_representation)
   also have "(\<Sum>b\<in>Basis. integral\<^sup>L lborel (\<lambda>x. f x \<bullet> b) *\<^sub>R b) = integral\<^sup>L lborel f"
     using f by (subst (2) eq_f[symmetric]) simp
   finally show ?thesis .
 qed
 
 lemma integrable_on_lborel: "integrable lborel f \<Longrightarrow> f integrable_on UNIV"
   using has_integral_integral_lborel by auto
 
 lemma integral_lborel: "integrable lborel f \<Longrightarrow> integral UNIV f = (\<integral>x. f x \<partial>lborel)"
   using has_integral_integral_lborel by auto
 
 end
 
 context
 begin
 
 private lemma has_integral_integral_lebesgue_real:
   fixes f :: "'a::euclidean_space \<Rightarrow> real"
   assumes f: "integrable lebesgue f"
   shows "(f has_integral (integral\<^sup>L lebesgue f)) UNIV"
 proof -
   obtain f' where f': "f' \<in> borel \<rightarrow>\<^sub>M borel" "AE x in lborel. f x = f' x"
     using completion_ex_borel_measurable_real[OF borel_measurable_integrable[OF f]] by auto
   moreover have "(\<integral>\<^sup>+ x. ennreal (norm (f x)) \<partial>lborel) = (\<integral>\<^sup>+ x. ennreal (norm (f' x)) \<partial>lborel)"
     using f' by (intro nn_integral_cong_AE) auto
   ultimately have "integrable lborel f'"
     using f by (auto simp: integrable_iff_bounded nn_integral_completion cong: nn_integral_cong_AE)
   note has_integral_integral_real[OF this]
   moreover have "integral\<^sup>L lebesgue f = integral\<^sup>L lebesgue f'"
     using f' f by (intro integral_cong_AE) (auto intro: AE_completion measurable_completion)
   moreover have "integral\<^sup>L lebesgue f' = integral\<^sup>L lborel f'"
     using f' by (simp add: integral_completion)
   moreover have "(f' has_integral integral\<^sup>L lborel f') UNIV \<longleftrightarrow> (f has_integral integral\<^sup>L lborel f') UNIV"
     using f' by (intro has_integral_AE) auto
   ultimately show ?thesis
     by auto
 qed
 
 lemma has_integral_integral_lebesgue:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes f: "integrable lebesgue f"
   shows "(f has_integral (integral\<^sup>L lebesgue f)) UNIV"
 proof -
   have "((\<lambda>x. \<Sum>b\<in>Basis. (f x \<bullet> b) *\<^sub>R b) has_integral (\<Sum>b\<in>Basis. integral\<^sup>L lebesgue (\<lambda>x. f x \<bullet> b) *\<^sub>R b)) UNIV"
     using f by (intro has_integral_sum finite_Basis ballI has_integral_scaleR_left has_integral_integral_lebesgue_real) auto
   also have eq_f: "(\<lambda>x. \<Sum>b\<in>Basis. (f x \<bullet> b) *\<^sub>R b) = f"
     by (simp add: fun_eq_iff euclidean_representation)
   also have "(\<Sum>b\<in>Basis. integral\<^sup>L lebesgue (\<lambda>x. f x \<bullet> b) *\<^sub>R b) = integral\<^sup>L lebesgue f"
     using f by (subst (2) eq_f[symmetric]) simp
   finally show ?thesis .
 qed
 
 lemma has_integral_integral_lebesgue_on:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes "integrable (lebesgue_on S) f" "S \<in> sets lebesgue"
   shows "(f has_integral (integral\<^sup>L (lebesgue_on S) f)) S"
 proof -
   let ?f = "\<lambda>x. if x \<in> S then f x else 0"
   have "integrable lebesgue (\<lambda>x. indicat_real S x *\<^sub>R f x)"
     using indicator_scaleR_eq_if [of S _ f] assms
   by (metis (full_types) integrable_restrict_space sets.Int_space_eq2)
   then have "integrable lebesgue ?f"
     using indicator_scaleR_eq_if [of S _ f] assms by auto
   then have "(?f has_integral (integral\<^sup>L lebesgue ?f)) UNIV"
     by (rule has_integral_integral_lebesgue)
   then have "(f has_integral (integral\<^sup>L lebesgue ?f)) S"
     using has_integral_restrict_UNIV by blast
   moreover
   have "S \<inter> space lebesgue \<in> sets lebesgue"
     by (simp add: assms)
   then have "(integral\<^sup>L lebesgue ?f) = (integral\<^sup>L (lebesgue_on S) f)"
     by (simp add: integral_restrict_space indicator_scaleR_eq_if)
   ultimately show ?thesis
     by auto
 qed
 
 lemma lebesgue_integral_eq_integral:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes "integrable (lebesgue_on S) f" "S \<in> sets lebesgue"
   shows "integral\<^sup>L (lebesgue_on S) f = integral S f"
   by (metis has_integral_integral_lebesgue_on assms integral_unique)
 
 lemma integrable_on_lebesgue:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   shows "integrable lebesgue f \<Longrightarrow> f integrable_on UNIV"
   using has_integral_integral_lebesgue by auto
 
 lemma integral_lebesgue:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   shows "integrable lebesgue f \<Longrightarrow> integral UNIV f = (\<integral>x. f x \<partial>lebesgue)"
   using has_integral_integral_lebesgue by auto
 
 end
 
 subsection \<open>Absolute integrability (this is the same as Lebesgue integrability)\<close>
 
 translations
 "LBINT x. f" == "CONST lebesgue_integral CONST lborel (\<lambda>x. f)"
 
 translations
 "LBINT x:A. f" == "CONST set_lebesgue_integral CONST lborel A (\<lambda>x. f)"
 
 lemma set_integral_reflect:
   fixes S and f :: "real \<Rightarrow> 'a :: {banach, second_countable_topology}"
   shows "(LBINT x : S. f x) = (LBINT x : {x. - x \<in> S}. f (- x))"
   unfolding set_lebesgue_integral_def
   by (subst lborel_integral_real_affine[where c="-1" and t=0])
      (auto intro!: Bochner_Integration.integral_cong split: split_indicator)
 
 lemma borel_integrable_atLeastAtMost':
   fixes f :: "real \<Rightarrow> 'a::{banach, second_countable_topology}"
   assumes f: "continuous_on {a..b} f"
   shows "set_integrable lborel {a..b} f"
   unfolding set_integrable_def
   by (intro borel_integrable_compact compact_Icc f)
 
 lemma integral_FTC_atLeastAtMost:
   fixes f :: "real \<Rightarrow> 'a :: euclidean_space"
   assumes "a \<le> b"
     and F: "\<And>x. a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> (F has_vector_derivative f x) (at x within {a .. b})"
     and f: "continuous_on {a .. b} f"
   shows "integral\<^sup>L lborel (\<lambda>x. indicator {a .. b} x *\<^sub>R f x) = F b - F a"
 proof -
   let ?f = "\<lambda>x. indicator {a .. b} x *\<^sub>R f x"
   have "(?f has_integral (\<integral>x. ?f x \<partial>lborel)) UNIV"
     using borel_integrable_atLeastAtMost'[OF f]
     unfolding set_integrable_def by (rule has_integral_integral_lborel)
   moreover
   have "(f has_integral F b - F a) {a .. b}"
     by (intro fundamental_theorem_of_calculus ballI assms) auto
   then have "(?f has_integral F b - F a) {a .. b}"
     by (subst has_integral_cong[where g=f]) auto
   then have "(?f has_integral F b - F a) UNIV"
     by (intro has_integral_on_superset[where T=UNIV and S="{a..b}"]) auto
   ultimately show "integral\<^sup>L lborel ?f = F b - F a"
     by (rule has_integral_unique)
 qed
 
 lemma set_borel_integral_eq_integral:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes "set_integrable lborel S f"
   shows "f integrable_on S" "LINT x : S | lborel. f x = integral S f"
 proof -
   let ?f = "\<lambda>x. indicator S x *\<^sub>R f x"
   have "(?f has_integral LINT x : S | lborel. f x) UNIV"
     using assms has_integral_integral_lborel
     unfolding set_integrable_def set_lebesgue_integral_def by blast
   hence 1: "(f has_integral (set_lebesgue_integral lborel S f)) S"
     apply (subst has_integral_restrict_UNIV [symmetric])
     apply (rule has_integral_eq)
     by auto
   thus "f integrable_on S"
     by (auto simp add: integrable_on_def)
   with 1 have "(f has_integral (integral S f)) S"
     by (intro integrable_integral, auto simp add: integrable_on_def)
   thus "LINT x : S | lborel. f x = integral S f"
     by (intro has_integral_unique [OF 1])
 qed
 
 lemma has_integral_set_lebesgue:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes f: "set_integrable lebesgue S f"
   shows "(f has_integral (LINT x:S|lebesgue. f x)) S"
   using has_integral_integral_lebesgue f
   by (fastforce simp add: set_integrable_def set_lebesgue_integral_def indicator_def if_distrib[of "\<lambda>x. x *\<^sub>R f _"] cong: if_cong)
 
 lemma set_lebesgue_integral_eq_integral:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes f: "set_integrable lebesgue S f"
   shows "f integrable_on S" "LINT x:S | lebesgue. f x = integral S f"
   using has_integral_set_lebesgue[OF f] by (auto simp: integral_unique integrable_on_def)
 
 lemma lmeasurable_iff_has_integral:
   "S \<in> lmeasurable \<longleftrightarrow> ((indicator S) has_integral measure lebesgue S) UNIV"
   by (subst has_integral_iff_nn_integral_lebesgue)
      (auto simp: ennreal_indicator emeasure_eq_measure2 borel_measurable_indicator_iff intro!: fmeasurableI)
 
 abbreviation
   absolutely_integrable_on :: "('a::euclidean_space \<Rightarrow> 'b::{banach, second_countable_topology}) \<Rightarrow> 'a set \<Rightarrow> bool"
   (infixr "absolutely'_integrable'_on" 46)
   where "f absolutely_integrable_on s \<equiv> set_integrable lebesgue s f"
 
 
 lemma absolutely_integrable_zero [simp]: "(\<lambda>x. 0) absolutely_integrable_on S"
     by (simp add: set_integrable_def)
 
 lemma absolutely_integrable_on_def:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   shows "f absolutely_integrable_on S \<longleftrightarrow> f integrable_on S \<and> (\<lambda>x. norm (f x)) integrable_on S"
 proof safe
   assume f: "f absolutely_integrable_on S"
   then have nf: "integrable lebesgue (\<lambda>x. norm (indicator S x *\<^sub>R f x))"
     using integrable_norm set_integrable_def by blast
   show "f integrable_on S"
     by (rule set_lebesgue_integral_eq_integral[OF f])
   have "(\<lambda>x. norm (indicator S x *\<^sub>R f x)) = (\<lambda>x. if x \<in> S then norm (f x) else 0)"
     by auto
   with integrable_on_lebesgue[OF nf] show "(\<lambda>x. norm (f x)) integrable_on S"
     by (simp add: integrable_restrict_UNIV)
 next
   assume f: "f integrable_on S" and nf: "(\<lambda>x. norm (f x)) integrable_on S"
   show "f absolutely_integrable_on S"
     unfolding set_integrable_def
   proof (rule integrableI_bounded)
     show "(\<lambda>x. indicator S x *\<^sub>R f x) \<in> borel_measurable lebesgue"
       using f has_integral_implies_lebesgue_measurable[of f _ S] by (auto simp: integrable_on_def)
     show "(\<integral>\<^sup>+ x. ennreal (norm (indicator S x *\<^sub>R f x)) \<partial>lebesgue) < \<infinity>"
       using nf nn_integral_has_integral_lebesgue[of "\<lambda>x. norm (f x)" _ S]
       by (auto simp: integrable_on_def nn_integral_completion)
   qed
 qed
 
 lemma integrable_on_lebesgue_on:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes f: "integrable (lebesgue_on S) f" and S: "S \<in> sets lebesgue"
   shows "f integrable_on S"
 proof -
   have "integrable lebesgue (\<lambda>x. indicator S x *\<^sub>R f x)"
     using S f inf_top.comm_neutral integrable_restrict_space by blast
   then show ?thesis
     using absolutely_integrable_on_def set_integrable_def by blast
 qed
 
 lemma absolutely_integrable_imp_integrable:
   assumes "f absolutely_integrable_on S" "S \<in> sets lebesgue"
   shows "integrable (lebesgue_on S) f"
   by (meson assms integrable_restrict_space set_integrable_def sets.Int sets.top)
 
 lemma absolutely_integrable_on_null [intro]:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   shows "content (cbox a b) = 0 \<Longrightarrow> f absolutely_integrable_on (cbox a b)"
   by (auto simp: absolutely_integrable_on_def)
 
 lemma absolutely_integrable_on_open_interval:
   fixes f :: "'a :: euclidean_space \<Rightarrow> 'b :: euclidean_space"
   shows "f absolutely_integrable_on box a b \<longleftrightarrow>
          f absolutely_integrable_on cbox a b"
   by (auto simp: integrable_on_open_interval absolutely_integrable_on_def)
 
 lemma absolutely_integrable_restrict_UNIV:
   "(\<lambda>x. if x \<in> S then f x else 0) absolutely_integrable_on UNIV \<longleftrightarrow> f absolutely_integrable_on S"
     unfolding set_integrable_def
   by (intro arg_cong2[where f=integrable]) auto
 
 lemma absolutely_integrable_onI:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   shows "f integrable_on S \<Longrightarrow> (\<lambda>x. norm (f x)) integrable_on S \<Longrightarrow> f absolutely_integrable_on S"
   unfolding absolutely_integrable_on_def by auto
 
 lemma nonnegative_absolutely_integrable_1:
   fixes f :: "'a :: euclidean_space \<Rightarrow> real"
   assumes f: "f integrable_on A" and "\<And>x. x \<in> A \<Longrightarrow> 0 \<le> f x"
   shows "f absolutely_integrable_on A"
   by (rule absolutely_integrable_onI [OF f]) (use assms in \<open>simp add: integrable_eq\<close>)
 
 lemma absolutely_integrable_on_iff_nonneg:
   fixes f :: "'a :: euclidean_space \<Rightarrow> real"
   assumes "\<And>x. x \<in> S \<Longrightarrow> 0 \<le> f x" shows "f absolutely_integrable_on S \<longleftrightarrow> f integrable_on S"
 proof -
   { assume "f integrable_on S"
     then have "(\<lambda>x. if x \<in> S then f x else 0) integrable_on UNIV"
       by (simp add: integrable_restrict_UNIV)
     then have "(\<lambda>x. if x \<in> S then f x else 0) absolutely_integrable_on UNIV"
       using \<open>f integrable_on S\<close> absolutely_integrable_restrict_UNIV assms nonnegative_absolutely_integrable_1 by blast
     then have "f absolutely_integrable_on S"
       using absolutely_integrable_restrict_UNIV by blast
   }
   then show ?thesis
     unfolding absolutely_integrable_on_def by auto
 qed
 
 lemma absolutely_integrable_on_scaleR_iff:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   shows
    "(\<lambda>x. c *\<^sub>R f x) absolutely_integrable_on S \<longleftrightarrow>
       c = 0 \<or> f absolutely_integrable_on S"
 proof (cases "c=0")
   case False
   then show ?thesis
   unfolding absolutely_integrable_on_def
   by (simp add: norm_mult)
 qed auto
 
 lemma absolutely_integrable_spike:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes "f absolutely_integrable_on T" and S: "negligible S" "\<And>x. x \<in> T - S \<Longrightarrow> g x = f x"
   shows "g absolutely_integrable_on T"
   using assms integrable_spike
   unfolding absolutely_integrable_on_def by metis
 
 lemma absolutely_integrable_negligible:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes "negligible S"
   shows "f absolutely_integrable_on S"
   using assms by (simp add: absolutely_integrable_on_def integrable_negligible)
 
 lemma absolutely_integrable_spike_eq:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes "negligible S" "\<And>x. x \<in> T - S \<Longrightarrow> g x = f x"
   shows "(f absolutely_integrable_on T \<longleftrightarrow> g absolutely_integrable_on T)"
   using assms by (blast intro: absolutely_integrable_spike sym)
 
 lemma absolutely_integrable_spike_set_eq:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes "negligible {x \<in> S - T. f x \<noteq> 0}" "negligible {x \<in> T - S. f x \<noteq> 0}"
   shows "(f absolutely_integrable_on S \<longleftrightarrow> f absolutely_integrable_on T)"
 proof -
   have "(\<lambda>x. if x \<in> S then f x else 0) absolutely_integrable_on UNIV \<longleftrightarrow>
         (\<lambda>x. if x \<in> T then f x else 0) absolutely_integrable_on UNIV"
   proof (rule absolutely_integrable_spike_eq)
     show "negligible ({x \<in> S - T. f x \<noteq> 0} \<union> {x \<in> T - S. f x \<noteq> 0})"
       by (rule negligible_Un [OF assms])
   qed auto
   with absolutely_integrable_restrict_UNIV show ?thesis
     by blast
 qed
 
 lemma absolutely_integrable_spike_set:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes f: "f absolutely_integrable_on S" and neg: "negligible {x \<in> S - T. f x \<noteq> 0}" "negligible {x \<in> T - S. f x \<noteq> 0}"
   shows "f absolutely_integrable_on T"
   using absolutely_integrable_spike_set_eq f neg by blast
 
 lemma absolutely_integrable_reflect[simp]:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   shows "(\<lambda>x. f(-x)) absolutely_integrable_on cbox (-b) (-a) \<longleftrightarrow> f absolutely_integrable_on cbox a b"
   apply (auto simp: absolutely_integrable_on_def dest: integrable_reflect [THEN iffD1])
   unfolding integrable_on_def by auto
 
 lemma absolutely_integrable_reflect_real[simp]:
   fixes f :: "real \<Rightarrow> 'b::euclidean_space"
   shows "(\<lambda>x. f(-x)) absolutely_integrable_on {-b .. -a} \<longleftrightarrow> f absolutely_integrable_on {a..b::real}"
   unfolding box_real[symmetric] by (rule absolutely_integrable_reflect)
 
 lemma absolutely_integrable_on_subcbox:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   shows "\<lbrakk>f absolutely_integrable_on S; cbox a b \<subseteq> S\<rbrakk> \<Longrightarrow> f absolutely_integrable_on cbox a b"
   by (meson absolutely_integrable_on_def integrable_on_subcbox)
 
 lemma absolutely_integrable_on_subinterval:
   fixes f :: "real \<Rightarrow> 'b::euclidean_space"
   shows "\<lbrakk>f absolutely_integrable_on S; {a..b} \<subseteq> S\<rbrakk> \<Longrightarrow> f absolutely_integrable_on {a..b}"
   using absolutely_integrable_on_subcbox by fastforce
 
 lemma integrable_subinterval:
   fixes f :: "real \<Rightarrow> 'a::euclidean_space"
   assumes "integrable (lebesgue_on {a..b}) f"
     and "{c..d} \<subseteq> {a..b}"
   shows "integrable (lebesgue_on {c..d}) f"
 proof (rule absolutely_integrable_imp_integrable)
   show "f absolutely_integrable_on {c..d}"
   proof -
     have "f integrable_on {c..d}"
       using assms integrable_on_lebesgue_on integrable_on_subinterval by fastforce
     moreover have "(\<lambda>x. norm (f x)) integrable_on {c..d}"
     proof (rule integrable_on_subinterval)
       show "(\<lambda>x. norm (f x)) integrable_on {a..b}"
         by (simp add: assms integrable_on_lebesgue_on)
     qed (use assms in auto)
     ultimately show ?thesis
       by (auto simp: absolutely_integrable_on_def)
   qed
 qed auto
 
 lemma indefinite_integral_continuous_real:
   fixes f :: "real \<Rightarrow> 'a::euclidean_space"
   assumes "integrable (lebesgue_on {a..b}) f"
   shows "continuous_on {a..b} (\<lambda>x. integral\<^sup>L (lebesgue_on {a..x}) f)"
 proof -
   have "f integrable_on {a..b}"
     by (simp add: assms integrable_on_lebesgue_on)
   then have "continuous_on {a..b} (\<lambda>x. integral {a..x} f)"
     using indefinite_integral_continuous_1 by blast
   moreover have "integral\<^sup>L (lebesgue_on {a..x}) f = integral {a..x} f" if "a \<le> x" "x \<le> b" for x
   proof -
     have "{a..x} \<subseteq> {a..b}"
       using that by auto
     then have "integrable (lebesgue_on {a..x}) f"
       using integrable_subinterval assms by blast
     then show "integral\<^sup>L (lebesgue_on {a..x}) f = integral {a..x} f"
       by (simp add: lebesgue_integral_eq_integral)
   qed
   ultimately show ?thesis
     by (metis (no_types, lifting) atLeastAtMost_iff continuous_on_cong)
 qed
 
 lemma lmeasurable_iff_integrable_on: "S \<in> lmeasurable \<longleftrightarrow> (\<lambda>x. 1::real) integrable_on S"
   by (subst absolutely_integrable_on_iff_nonneg[symmetric])
      (simp_all add: lmeasurable_iff_integrable set_integrable_def)
 
 lemma lmeasure_integral_UNIV: "S \<in> lmeasurable \<Longrightarrow> measure lebesgue S = integral UNIV (indicator S)"
   by (simp add: lmeasurable_iff_has_integral integral_unique)
 
 lemma lmeasure_integral: "S \<in> lmeasurable \<Longrightarrow> measure lebesgue S = integral S (\<lambda>x. 1::real)"
   by (fastforce simp add: lmeasure_integral_UNIV indicator_def[abs_def] lmeasurable_iff_integrable_on)
 
 lemma integrable_on_const: "S \<in> lmeasurable \<Longrightarrow> (\<lambda>x. c) integrable_on S"
   unfolding lmeasurable_iff_integrable
   by (metis (mono_tags, lifting) integrable_eq integrable_on_scaleR_left lmeasurable_iff_integrable lmeasurable_iff_integrable_on scaleR_one)
 
 lemma integral_indicator:
   assumes "(S \<inter> T) \<in> lmeasurable"
   shows "integral T (indicator S) = measure lebesgue (S \<inter> T)"
 proof -
   have "integral UNIV (indicator (S \<inter> T)) = integral UNIV (\<lambda>a. if a \<in> S \<inter> T then 1::real else 0)"
     by (meson indicator_def)
   moreover
   have "(indicator (S \<inter> T) has_integral measure lebesgue (S \<inter> T)) UNIV"
     using assms by (simp add: lmeasurable_iff_has_integral)
   ultimately have "integral UNIV (\<lambda>x. if x \<in> S \<inter> T then 1 else 0) = measure lebesgue (S \<inter> T)"
     by (metis (no_types) integral_unique)
   then show ?thesis
     using integral_restrict_Int [of UNIV "S \<inter> T" "\<lambda>x. 1::real"]
     apply (simp add: integral_restrict_Int [symmetric])
     by (meson indicator_def)
 qed
 
 lemma measurable_integrable:
   fixes S :: "'a::euclidean_space set"
   shows "S \<in> lmeasurable \<longleftrightarrow> (indicat_real S) integrable_on UNIV"
   by (auto simp: lmeasurable_iff_integrable absolutely_integrable_on_iff_nonneg [symmetric] set_integrable_def)
 
 lemma integrable_on_indicator:
   fixes S :: "'a::euclidean_space set"
   shows "indicat_real S integrable_on T \<longleftrightarrow> (S \<inter> T) \<in> lmeasurable"
   unfolding measurable_integrable
   unfolding integrable_restrict_UNIV [of T, symmetric]
   by (fastforce simp add: indicator_def elim: integrable_eq)
 
 lemma
   assumes \<D>: "\<D> division_of S"
   shows lmeasurable_division: "S \<in> lmeasurable" (is ?l)
     and content_division: "(\<Sum>k\<in>\<D>. measure lebesgue k) = measure lebesgue S" (is ?m)
 proof -
   { fix d1 d2 assume *: "d1 \<in> \<D>" "d2 \<in> \<D>" "d1 \<noteq> d2"
     then obtain a b c d where "d1 = cbox a b" "d2 = cbox c d"
       using division_ofD(4)[OF \<D>] by blast
     with division_ofD(5)[OF \<D> *]
     have "d1 \<in> sets lborel" "d2 \<in> sets lborel" "d1 \<inter> d2 \<subseteq> (cbox a b - box a b) \<union> (cbox c d - box c d)"
       by auto
     moreover have "(cbox a b - box a b) \<union> (cbox c d - box c d) \<in> null_sets lborel"
       by (intro null_sets.Un null_sets_cbox_Diff_box)
     ultimately have "d1 \<inter> d2 \<in> null_sets lborel"
       by (blast intro: null_sets_subset) }
   then show ?l ?m
     unfolding division_ofD(6)[OF \<D>, symmetric]
     using division_ofD(1,4)[OF \<D>]
     by (auto intro!: measure_Union_AE[symmetric] simp: completion.AE_iff_null_sets Int_def[symmetric] pairwise_def null_sets_def)
 qed
 
 lemma has_measure_limit:
   assumes "S \<in> lmeasurable" "e > 0"
   obtains B where "B > 0"
     "\<And>a b. ball 0 B \<subseteq> cbox a b \<Longrightarrow> \<bar>measure lebesgue (S \<inter> cbox a b) - measure lebesgue S\<bar> < e"
   using assms unfolding lmeasurable_iff_has_integral has_integral_alt'
   by (force simp: integral_indicator integrable_on_indicator)
 
 lemma lmeasurable_iff_indicator_has_integral:
   fixes S :: "'a::euclidean_space set"
   shows "S \<in> lmeasurable \<and> m = measure lebesgue S \<longleftrightarrow> (indicat_real S has_integral m) UNIV"
   by (metis has_integral_iff lmeasurable_iff_has_integral measurable_integrable)
 
 lemma has_measure_limit_iff:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
   shows "S \<in> lmeasurable \<and> m = measure lebesgue S \<longleftrightarrow>
           (\<forall>e>0. \<exists>B>0. \<forall>a b. ball 0 B \<subseteq> cbox a b \<longrightarrow>
             (S \<inter> cbox a b) \<in> lmeasurable \<and> \<bar>measure lebesgue (S \<inter> cbox a b) - m\<bar> < e)" (is "?lhs = ?rhs")
 proof
   assume ?lhs then show ?rhs
     by (meson has_measure_limit fmeasurable.Int lmeasurable_cbox)
 next
   assume RHS [rule_format]: ?rhs
   show ?lhs
     apply (simp add: lmeasurable_iff_indicator_has_integral has_integral' [where i=m])
     using RHS
     by (metis (full_types) integral_indicator integrable_integral integrable_on_indicator)
 qed
 
 subsection\<open>Applications to Negligibility\<close>
 
 lemma negligible_iff_null_sets: "negligible S \<longleftrightarrow> S \<in> null_sets lebesgue"
 proof
   assume "negligible S"
   then have "(indicator S has_integral (0::real)) UNIV"
     by (auto simp: negligible)
   then show "S \<in> null_sets lebesgue"
     by (subst (asm) has_integral_iff_nn_integral_lebesgue)
         (auto simp: borel_measurable_indicator_iff nn_integral_0_iff_AE AE_iff_null_sets indicator_eq_0_iff)
 next
   assume S: "S \<in> null_sets lebesgue"
   show "negligible S"
     unfolding negligible_def
   proof (safe intro!: has_integral_iff_nn_integral_lebesgue[THEN iffD2]
                       has_integral_restrict_UNIV[where s="cbox _ _", THEN iffD1])
     fix a b
     show "(\<lambda>x. if x \<in> cbox a b then indicator S x else 0) \<in> lebesgue \<rightarrow>\<^sub>M borel"
       using S by (auto intro!: measurable_If)
     then show "(\<integral>\<^sup>+ x. ennreal (if x \<in> cbox a b then indicator S x else 0) \<partial>lebesgue) = ennreal 0"
       using S[THEN AE_not_in] by (auto intro!: nn_integral_0_iff_AE[THEN iffD2])
   qed auto
 qed
 
 corollary eventually_ae_filter_negligible:
   "eventually P (ae_filter lebesgue) \<longleftrightarrow> (\<exists>N. negligible N \<and> {x. \<not> P x} \<subseteq> N)"
   by (auto simp: completion.AE_iff_null_sets negligible_iff_null_sets null_sets_completion_subset)
 
 lemma starlike_negligible:
   assumes "closed S"
       and eq1: "\<And>c x. (a + c *\<^sub>R x) \<in> S \<Longrightarrow> 0 \<le> c \<Longrightarrow> a + x \<in> S \<Longrightarrow> c = 1"
     shows "negligible S"
 proof -
   have "negligible ((+) (-a) ` S)"
   proof (subst negligible_on_intervals, intro allI)
     fix u v
     show "negligible ((+) (- a) ` S \<inter> cbox u v)"
       using \<open>closed S\<close> eq1 by (auto simp add: negligible_iff_null_sets algebra_simps
         intro!: closed_translation_subtract starlike_negligible_compact cong: image_cong_simp)
         (metis add_diff_eq diff_add_cancel scale_right_diff_distrib)
   qed
   then show ?thesis
     by (rule negligible_translation_rev)
 qed
 
 lemma starlike_negligible_strong:
   assumes "closed S"
       and star: "\<And>c x. \<lbrakk>0 \<le> c; c < 1; a+x \<in> S\<rbrakk> \<Longrightarrow> a + c *\<^sub>R x \<notin> S"
     shows "negligible S"
 proof -
   show ?thesis
   proof (rule starlike_negligible [OF \<open>closed S\<close>, of a])
     fix c x
     assume cx: "a + c *\<^sub>R x \<in> S" "0 \<le> c" "a + x \<in> S"
     with star have "\<not> (c < 1)" by auto
     moreover have "\<not> (c > 1)"
       using star [of "1/c" "c *\<^sub>R x"] cx by force
     ultimately show "c = 1" by arith
   qed
 qed
 
 lemma negligible_hyperplane:
   assumes "a \<noteq> 0 \<or> b \<noteq> 0" shows "negligible {x. a \<bullet> x = b}"
 proof -
   obtain x where x: "a \<bullet> x \<noteq> b"
     using assms
     apply auto
      apply (metis inner_eq_zero_iff inner_zero_right)
     using inner_zero_right by fastforce
   show ?thesis
     apply (rule starlike_negligible [OF closed_hyperplane, of x])
     using x apply (auto simp: algebra_simps)
     done
 qed
 
 lemma negligible_lowdim:
   fixes S :: "'N :: euclidean_space set"
   assumes "dim S < DIM('N)"
     shows "negligible S"
 proof -
   obtain a where "a \<noteq> 0" and a: "span S \<subseteq> {x. a \<bullet> x = 0}"
     using lowdim_subset_hyperplane [OF assms] by blast
   have "negligible (span S)"
     using \<open>a \<noteq> 0\<close> a negligible_hyperplane by (blast intro: negligible_subset)
   then show ?thesis
     using span_base by (blast intro: negligible_subset)
 qed
 
 proposition negligible_convex_frontier:
   fixes S :: "'N :: euclidean_space set"
   assumes "convex S"
     shows "negligible(frontier S)"
 proof -
   have nf: "negligible(frontier S)" if "convex S" "0 \<in> S" for S :: "'N set"
   proof -
     obtain B where "B \<subseteq> S" and indB: "independent B"
                and spanB: "S \<subseteq> span B" and cardB: "card B = dim S"
       by (metis basis_exists)
     consider "dim S < DIM('N)" | "dim S = DIM('N)"
       using dim_subset_UNIV le_eq_less_or_eq by auto
     then show ?thesis
     proof cases
       case 1
       show ?thesis
         by (rule negligible_subset [of "closure S"])
            (simp_all add: frontier_def negligible_lowdim 1)
     next
       case 2
       obtain a where a: "a \<in> interior S"
         apply (rule interior_simplex_nonempty [OF indB])
           apply (simp add: indB independent_finite)
          apply (simp add: cardB 2)
         apply (metis \<open>B \<subseteq> S\<close> \<open>0 \<in> S\<close> \<open>convex S\<close> insert_absorb insert_subset interior_mono subset_hull)
         done
       show ?thesis
       proof (rule starlike_negligible_strong [where a=a])
         fix c::real and x
         have eq: "a + c *\<^sub>R x = (a + x) - (1 - c) *\<^sub>R ((a + x) - a)"
           by (simp add: algebra_simps)
         assume "0 \<le> c" "c < 1" "a + x \<in> frontier S"
         then show "a + c *\<^sub>R x \<notin> frontier S"
           apply (clarsimp simp: frontier_def)
           apply (subst eq)
           apply (rule mem_interior_closure_convex_shrink [OF \<open>convex S\<close> a, of _ "1-c"], auto)
           done
       qed auto
     qed
   qed
   show ?thesis
   proof (cases "S = {}")
     case True then show ?thesis by auto
   next
     case False
     then obtain a where "a \<in> S" by auto
     show ?thesis
       using nf [of "(\<lambda>x. -a + x) ` S"]
       by (metis \<open>a \<in> S\<close> add.left_inverse assms convex_translation_eq frontier_translation
                 image_eqI negligible_translation_rev)
   qed
 qed
 
 corollary negligible_sphere: "negligible (sphere a e)"
   using frontier_cball negligible_convex_frontier convex_cball
   by (blast intro: negligible_subset)
 
 lemma non_negligible_UNIV [simp]: "\<not> negligible UNIV"
   unfolding negligible_iff_null_sets by (auto simp: null_sets_def)
 
 lemma negligible_interval:
   "negligible (cbox a b) \<longleftrightarrow> box a b = {}" "negligible (box a b) \<longleftrightarrow> box a b = {}"
    by (auto simp: negligible_iff_null_sets null_sets_def prod_nonneg inner_diff_left box_eq_empty
                   not_le emeasure_lborel_cbox_eq emeasure_lborel_box_eq
             intro: eq_refl antisym less_imp_le)
 
 proposition open_not_negligible:
   assumes "open S" "S \<noteq> {}"
   shows "\<not> negligible S"
 proof
   assume neg: "negligible S"
   obtain a where "a \<in> S"
     using \<open>S \<noteq> {}\<close> by blast
   then obtain e where "e > 0" "cball a e \<subseteq> S"
     using \<open>open S\<close> open_contains_cball_eq by blast
   let ?p = "a - (e / DIM('a)) *\<^sub>R One" let ?q = "a + (e / DIM('a)) *\<^sub>R One"
   have "cbox ?p ?q \<subseteq> cball a e"
   proof (clarsimp simp: mem_box dist_norm)
     fix x
     assume "\<forall>i\<in>Basis. ?p \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> ?q \<bullet> i"
     then have ax: "\<bar>(a - x) \<bullet> i\<bar> \<le> e / real DIM('a)" if "i \<in> Basis" for i
       using that by (auto simp: algebra_simps)
     have "norm (a - x) \<le> (\<Sum>i\<in>Basis. \<bar>(a - x) \<bullet> i\<bar>)"
       by (rule norm_le_l1)
     also have "\<dots> \<le> DIM('a) * (e / real DIM('a))"
       by (intro sum_bounded_above ax)
     also have "\<dots> = e"
       by simp
     finally show "norm (a - x) \<le> e" .
   qed
   then have "negligible (cbox ?p ?q)"
     by (meson \<open>cball a e \<subseteq> S\<close> neg negligible_subset)
   with \<open>e > 0\<close> show False
     by (simp add: negligible_interval box_eq_empty algebra_simps field_split_simps mult_le_0_iff)
 qed
 
 lemma negligible_convex_interior:
    "convex S \<Longrightarrow> (negligible S \<longleftrightarrow> interior S = {})"
   apply safe
   apply (metis interior_subset negligible_subset open_interior open_not_negligible)
    apply (metis Diff_empty closure_subset frontier_def negligible_convex_frontier negligible_subset)
   done
 
 lemma measure_eq_0_null_sets: "S \<in> null_sets M \<Longrightarrow> measure M S = 0"
   by (auto simp: measure_def null_sets_def)
 
 lemma negligible_imp_measure0: "negligible S \<Longrightarrow> measure lebesgue S = 0"
   by (simp add: measure_eq_0_null_sets negligible_iff_null_sets)
 
 lemma negligible_iff_emeasure0: "S \<in> sets lebesgue \<Longrightarrow> (negligible S \<longleftrightarrow> emeasure lebesgue S = 0)"
   by (auto simp: measure_eq_0_null_sets negligible_iff_null_sets)
 
 lemma negligible_iff_measure0: "S \<in> lmeasurable \<Longrightarrow> (negligible S \<longleftrightarrow> measure lebesgue S = 0)"
   apply (auto simp: measure_eq_0_null_sets negligible_iff_null_sets)
   by (metis completion.null_sets_outer subsetI)
 
 lemma negligible_imp_sets: "negligible S \<Longrightarrow> S \<in> sets lebesgue"
   by (simp add: negligible_iff_null_sets null_setsD2)
 
 lemma negligible_imp_measurable: "negligible S \<Longrightarrow> S \<in> lmeasurable"
   by (simp add: fmeasurableI_null_sets negligible_iff_null_sets)
 
 lemma negligible_iff_measure: "negligible S \<longleftrightarrow> S \<in> lmeasurable \<and> measure lebesgue S = 0"
   by (fastforce simp: negligible_iff_measure0 negligible_imp_measurable dest: negligible_imp_measure0)
 
 lemma negligible_outer:
   "negligible S \<longleftrightarrow> (\<forall>e>0. \<exists>T. S \<subseteq> T \<and> T \<in> lmeasurable \<and> measure lebesgue T < e)" (is "_ = ?rhs")
 proof
   assume "negligible S" then show ?rhs
     by (metis negligible_iff_measure order_refl)
 next
   assume ?rhs then show "negligible S"
   by (meson completion.null_sets_outer negligible_iff_null_sets)
 qed
 
 lemma negligible_outer_le:
      "negligible S \<longleftrightarrow> (\<forall>e>0. \<exists>T. S \<subseteq> T \<and> T \<in> lmeasurable \<and> measure lebesgue T \<le> e)" (is "_ = ?rhs")
 proof
   assume "negligible S" then show ?rhs
     by (metis dual_order.strict_implies_order negligible_imp_measurable negligible_imp_measure0 order_refl)
 next
   assume ?rhs then show "negligible S"
     by (metis le_less_trans negligible_outer field_lbound_gt_zero)
 qed
 
 lemma negligible_UNIV: "negligible S \<longleftrightarrow> (indicat_real S has_integral 0) UNIV" (is "_=?rhs")
 proof
   assume ?rhs
   then show "negligible S"
     apply (auto simp: negligible_def has_integral_iff integrable_on_indicator)
     by (metis negligible integral_unique lmeasure_integral_UNIV negligible_iff_measure0)
 qed (simp add: negligible)
 
 lemma sets_negligible_symdiff:
    "\<lbrakk>S \<in> sets lebesgue; negligible((S - T) \<union> (T - S))\<rbrakk> \<Longrightarrow> T \<in> sets lebesgue"
   by (metis Diff_Diff_Int Int_Diff_Un inf_commute negligible_Un_eq negligible_imp_sets sets.Diff sets.Un)
 
 lemma lmeasurable_negligible_symdiff:
    "\<lbrakk>S \<in> lmeasurable; negligible((S - T) \<union> (T - S))\<rbrakk> \<Longrightarrow> T \<in> lmeasurable"
   using integrable_spike_set_eq lmeasurable_iff_integrable_on by blast
 
 
 lemma measure_Un3_negligible:
   assumes meas: "S \<in> lmeasurable" "T \<in> lmeasurable" "U \<in> lmeasurable"
   and neg: "negligible(S \<inter> T)" "negligible(S \<inter> U)" "negligible(T \<inter> U)" and V: "S \<union> T \<union> U = V"
 shows "measure lebesgue V = measure lebesgue S + measure lebesgue T + measure lebesgue U"
 proof -
   have [simp]: "measure lebesgue (S \<inter> T) = 0"
     using neg(1) negligible_imp_measure0 by blast
   have [simp]: "measure lebesgue (S \<inter> U \<union> T \<inter> U) = 0"
     using neg(2) neg(3) negligible_Un negligible_imp_measure0 by blast
   have "measure lebesgue V = measure lebesgue (S \<union> T \<union> U)"
     using V by simp
   also have "\<dots> = measure lebesgue S + measure lebesgue T + measure lebesgue U"
     by (simp add: measure_Un3 meas fmeasurable.Un Int_Un_distrib2)
   finally show ?thesis .
 qed
 
 lemma measure_translate_add:
   assumes meas: "S \<in> lmeasurable" "T \<in> lmeasurable"
     and U: "S \<union> ((+)a ` T) = U" and neg: "negligible(S \<inter> ((+)a ` T))"
   shows "measure lebesgue S + measure lebesgue T = measure lebesgue U"
 proof -
   have [simp]: "measure lebesgue (S \<inter> (+) a ` T) = 0"
     using neg negligible_imp_measure0 by blast
   have "measure lebesgue (S \<union> ((+)a ` T)) = measure lebesgue S + measure lebesgue T"
     by (simp add: measure_Un3 meas measurable_translation measure_translation fmeasurable.Un)
   then show ?thesis
     using U by auto
 qed
 
 lemma measure_negligible_symdiff:
   assumes S: "S \<in> lmeasurable"
     and neg: "negligible (S - T \<union> (T - S))"
   shows "measure lebesgue T = measure lebesgue S"
 proof -
   have "measure lebesgue (S - T) = 0"
     using neg negligible_Un_eq negligible_imp_measure0 by blast
   then show ?thesis
     by (metis S Un_commute add.right_neutral lmeasurable_negligible_symdiff measure_Un2 neg negligible_Un_eq negligible_imp_measure0)
 qed
 
 lemma measure_closure:
   assumes "bounded S" and neg: "negligible (frontier S)"
   shows "measure lebesgue (closure S) = measure lebesgue S"
 proof -
   have "measure lebesgue (frontier S) = 0"
     by (metis neg negligible_imp_measure0)
   then show ?thesis
     by (metis assms lmeasurable_iff_integrable_on eq_iff_diff_eq_0 has_integral_interior integrable_on_def integral_unique lmeasurable_interior lmeasure_integral measure_frontier)
 qed
 
 lemma measure_interior:
    "\<lbrakk>bounded S; negligible(frontier S)\<rbrakk> \<Longrightarrow> measure lebesgue (interior S) = measure lebesgue S"
   using measure_closure measure_frontier negligible_imp_measure0 by fastforce
 
 lemma measurable_Jordan:
   assumes "bounded S" and neg: "negligible (frontier S)"
   shows "S \<in> lmeasurable"
 proof -
   have "closure S \<in> lmeasurable"
     by (metis lmeasurable_closure \<open>bounded S\<close>)
   moreover have "interior S \<in> lmeasurable"
     by (simp add: lmeasurable_interior \<open>bounded S\<close>)
   moreover have "interior S \<subseteq> S"
     by (simp add: interior_subset)
   ultimately show ?thesis
     using assms by (metis (full_types) closure_subset completion.complete_sets_sandwich_fmeasurable measure_closure measure_interior)
 qed
 
 lemma measurable_convex: "\<lbrakk>convex S; bounded S\<rbrakk> \<Longrightarrow> S \<in> lmeasurable"
   by (simp add: measurable_Jordan negligible_convex_frontier)
 
 lemma content_cball_conv_ball: "content (cball c r) = content (ball c r)"
 proof -
   have "ball c r - cball c r \<union> (cball c r - ball c r) = sphere c r"
     by auto
   hence "measure lebesgue (cball c r) = measure lebesgue (ball c r)"
     using negligible_sphere[of c r] by (intro measure_negligible_symdiff) simp_all
   thus ?thesis by simp
 qed
 
 subsection\<open>Negligibility of image under non-injective linear map\<close>
 
 lemma negligible_Union_nat:
   assumes "\<And>n::nat. negligible(S n)"
   shows "negligible(\<Union>n. S n)"
 proof -
   have "negligible (\<Union>m\<le>k. S m)" for k
     using assms by blast
   then have 0:  "integral UNIV (indicat_real (\<Union>m\<le>k. S m)) = 0"
     and 1: "(indicat_real (\<Union>m\<le>k. S m)) integrable_on UNIV" for k
     by (auto simp: negligible has_integral_iff)
   have 2: "\<And>k x. indicat_real (\<Union>m\<le>k. S m) x \<le> (indicat_real (\<Union>m\<le>Suc k. S m) x)"
     by (simp add: indicator_def)
   have 3: "\<And>x. (\<lambda>k. indicat_real (\<Union>m\<le>k. S m) x) \<longlonglongrightarrow> (indicat_real (\<Union>n. S n) x)"
     by (force simp: indicator_def eventually_sequentially intro: tendsto_eventually)
   have 4: "bounded (range (\<lambda>k. integral UNIV (indicat_real (\<Union>m\<le>k. S m))))"
     by (simp add: 0)
   have *: "indicat_real (\<Union>n. S n) integrable_on UNIV \<and>
         (\<lambda>k. integral UNIV (indicat_real (\<Union>m\<le>k. S m))) \<longlonglongrightarrow> (integral UNIV (indicat_real (\<Union>n. S n)))"
     by (intro monotone_convergence_increasing 1 2 3 4)
   then have "integral UNIV (indicat_real (\<Union>n. S n)) = (0::real)"
     using LIMSEQ_unique by (auto simp: 0)
   then show ?thesis
     using * by (simp add: negligible_UNIV has_integral_iff)
 qed
 
 
 lemma negligible_linear_singular_image:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'n"
   assumes "linear f" "\<not> inj f"
   shows "negligible (f ` S)"
 proof -
   obtain a where "a \<noteq> 0" "\<And>S. f ` S \<subseteq> {x. a \<bullet> x = 0}"
     using assms linear_singular_image_hyperplane by blast
   then show "negligible (f ` S)"
     by (metis negligible_hyperplane negligible_subset)
 qed
 
 lemma measure_negligible_finite_Union:
   assumes "finite \<F>"
     and meas: "\<And>S. S \<in> \<F> \<Longrightarrow> S \<in> lmeasurable"
     and djointish: "pairwise (\<lambda>S T. negligible (S \<inter> T)) \<F>"
   shows "measure lebesgue (\<Union>\<F>) = (\<Sum>S\<in>\<F>. measure lebesgue S)"
   using assms
 proof (induction)
   case empty
   then show ?case
     by auto
 next
   case (insert S \<F>)
   then have "S \<in> lmeasurable" "\<Union>\<F> \<in> lmeasurable" "pairwise (\<lambda>S T. negligible (S \<inter> T)) \<F>"
     by (simp_all add: fmeasurable.finite_Union insert.hyps(1) insert.prems(1) pairwise_insert subsetI)
   then show ?case
   proof (simp add: measure_Un3 insert)
     have *: "\<And>T. T \<in> (\<inter>) S ` \<F> \<Longrightarrow> negligible T"
       using insert by (force simp: pairwise_def)
     have "negligible(S \<inter> \<Union>\<F>)"
       unfolding Int_Union
       by (rule negligible_Union) (simp_all add: * insert.hyps(1))
     then show "measure lebesgue (S \<inter> \<Union>\<F>) = 0"
       using negligible_imp_measure0 by blast
   qed
 qed
 
 lemma measure_negligible_finite_Union_image:
   assumes "finite S"
     and meas: "\<And>x. x \<in> S \<Longrightarrow> f x \<in> lmeasurable"
     and djointish: "pairwise (\<lambda>x y. negligible (f x \<inter> f y)) S"
   shows "measure lebesgue (\<Union>(f ` S)) = (\<Sum>x\<in>S. measure lebesgue (f x))"
 proof -
   have "measure lebesgue (\<Union>(f ` S)) = sum (measure lebesgue) (f ` S)"
     using assms by (auto simp: pairwise_mono pairwise_image intro: measure_negligible_finite_Union)
   also have "\<dots> = sum (measure lebesgue \<circ> f) S"
     using djointish [unfolded pairwise_def] by (metis inf.idem negligible_imp_measure0 sum.reindex_nontrivial [OF \<open>finite S\<close>])
   also have "\<dots> = (\<Sum>x\<in>S. measure lebesgue (f x))"
     by simp
   finally show ?thesis .
 qed
 
 subsection \<open>Negligibility of a Lipschitz image of a negligible set\<close>
 
 text\<open>The bound will be eliminated by a sort of onion argument\<close>
 lemma locally_Lipschitz_negl_bounded:
   fixes f :: "'M::euclidean_space \<Rightarrow> 'N::euclidean_space"
   assumes MleN: "DIM('M) \<le> DIM('N)" "0 < B" "bounded S" "negligible S"
       and lips: "\<And>x. x \<in> S
                       \<Longrightarrow> \<exists>T. open T \<and> x \<in> T \<and>
                               (\<forall>y \<in> S \<inter> T. norm(f y - f x) \<le> B * norm(y - x))"
   shows "negligible (f ` S)"
   unfolding negligible_iff_null_sets
 proof (clarsimp simp: completion.null_sets_outer)
   fix e::real
   assume "0 < e"
   have "S \<in> lmeasurable"
     using \<open>negligible S\<close> by (simp add: negligible_iff_null_sets fmeasurableI_null_sets)
   then have "S \<in> sets lebesgue"
     by blast
   have e22: "0 < e/2 / (2 * B * real DIM('M)) ^ DIM('N)"
     using \<open>0 < e\<close> \<open>0 < B\<close> by (simp add: field_split_simps)
   obtain T where "open T" "S \<subseteq> T" "(T - S) \<in> lmeasurable"
                  "measure lebesgue (T - S) < e/2 / (2 * B * DIM('M)) ^ DIM('N)"
     using sets_lebesgue_outer_open [OF \<open>S \<in> sets lebesgue\<close> e22]
     by (metis emeasure_eq_measure2 ennreal_leI linorder_not_le)
   then have T: "measure lebesgue T \<le> e/2 / (2 * B * DIM('M)) ^ DIM('N)"
     using \<open>negligible S\<close> by (simp add: measure_Diff_null_set negligible_iff_null_sets)
   have "\<exists>r. 0 < r \<and> r \<le> 1/2 \<and>
             (x \<in> S \<longrightarrow> (\<forall>y. norm(y - x) < r
                        \<longrightarrow> y \<in> T \<and> (y \<in> S \<longrightarrow> norm(f y - f x) \<le> B * norm(y - x))))"
         for x
   proof (cases "x \<in> S")
     case True
     obtain U where "open U" "x \<in> U" and U: "\<And>y. y \<in> S \<inter> U \<Longrightarrow> norm(f y - f x) \<le> B * norm(y - x)"
       using lips [OF \<open>x \<in> S\<close>] by auto
     have "x \<in> T \<inter> U"
       using \<open>S \<subseteq> T\<close> \<open>x \<in> U\<close> \<open>x \<in> S\<close> by auto
     then obtain \<epsilon> where "0 < \<epsilon>" "ball x \<epsilon> \<subseteq> T \<inter> U"
       by (metis \<open>open T\<close> \<open>open U\<close> openE open_Int)
     then show ?thesis
       apply (rule_tac x="min (1/2) \<epsilon>" in exI)
       apply (simp del: divide_const_simps)
       apply (intro allI impI conjI)
        apply (metis dist_commute dist_norm mem_ball subsetCE)
       by (metis Int_iff subsetCE U dist_norm mem_ball norm_minus_commute)
   next
     case False
     then show ?thesis
       by (rule_tac x="1/4" in exI) auto
   qed
   then obtain R where R12: "\<And>x. 0 < R x \<and> R x \<le> 1/2"
                 and RT: "\<And>x y. \<lbrakk>x \<in> S; norm(y - x) < R x\<rbrakk> \<Longrightarrow> y \<in> T"
                 and RB: "\<And>x y. \<lbrakk>x \<in> S; y \<in> S; norm(y - x) < R x\<rbrakk> \<Longrightarrow> norm(f y - f x) \<le> B * norm(y - x)"
     by metis+
   then have gaugeR: "gauge (\<lambda>x. ball x (R x))"
     by (simp add: gauge_def)
   obtain c where c: "S \<subseteq> cbox (-c *\<^sub>R One) (c *\<^sub>R One)" "box (-c *\<^sub>R One:: 'M) (c *\<^sub>R One) \<noteq> {}"
   proof -
     obtain B where B: "\<And>x. x \<in> S \<Longrightarrow> norm x \<le> B"
       using \<open>bounded S\<close> bounded_iff by blast
     show ?thesis
       apply (rule_tac c = "abs B + 1" in that)
       using norm_bound_Basis_le Basis_le_norm
        apply (fastforce simp: box_eq_empty mem_box dest!: B intro: order_trans)+
       done
   qed
   obtain \<D> where "countable \<D>"
      and Dsub: "\<Union>\<D> \<subseteq> cbox (-c *\<^sub>R One) (c *\<^sub>R One)"
      and cbox: "\<And>K. K \<in> \<D> \<Longrightarrow> interior K \<noteq> {} \<and> (\<exists>c d. K = cbox c d)"
      and pw:   "pairwise (\<lambda>A B. interior A \<inter> interior B = {}) \<D>"
      and Ksub: "\<And>K. K \<in> \<D> \<Longrightarrow> \<exists>x \<in> S \<inter> K. K \<subseteq> (\<lambda>x. ball x (R x)) x"
      and exN:  "\<And>u v. cbox u v \<in> \<D> \<Longrightarrow> \<exists>n. \<forall>i \<in> Basis. v \<bullet> i - u \<bullet> i = (2*c) / 2^n"
      and "S \<subseteq> \<Union>\<D>"
     using covering_lemma [OF c gaugeR]  by force
   have "\<exists>u v z. K = cbox u v \<and> box u v \<noteq> {} \<and> z \<in> S \<and> z \<in> cbox u v \<and>
                 cbox u v \<subseteq> ball z (R z)" if "K \<in> \<D>" for K
   proof -
     obtain u v where "K = cbox u v"
       using \<open>K \<in> \<D>\<close> cbox by blast
     with that show ?thesis
       apply (rule_tac x=u in exI)
       apply (rule_tac x=v in exI)
       apply (metis Int_iff interior_cbox cbox Ksub)
       done
   qed
   then obtain uf vf zf
     where uvz: "\<And>K. K \<in> \<D> \<Longrightarrow>
                 K = cbox (uf K) (vf K) \<and> box (uf K) (vf K) \<noteq> {} \<and> zf K \<in> S \<and>
                 zf K \<in> cbox (uf K) (vf K) \<and> cbox (uf K) (vf K) \<subseteq> ball (zf K) (R (zf K))"
     by metis
   define prj1 where "prj1 \<equiv> \<lambda>x::'M. x \<bullet> (SOME i. i \<in> Basis)"
   define fbx where "fbx \<equiv> \<lambda>D. cbox (f(zf D) - (B * DIM('M) * (prj1(vf D - uf D))) *\<^sub>R One::'N)
                                     (f(zf D) + (B * DIM('M) * prj1(vf D - uf D)) *\<^sub>R One)"
   have vu_pos: "0 < prj1 (vf X - uf X)" if "X \<in> \<D>" for X
     using uvz [OF that] by (simp add: prj1_def box_ne_empty SOME_Basis inner_diff_left)
   have prj1_idem: "prj1 (vf X - uf X) = (vf X - uf X) \<bullet> i" if  "X \<in> \<D>" "i \<in> Basis" for X i
   proof -
     have "cbox (uf X) (vf X) \<in> \<D>"
       using uvz \<open>X \<in> \<D>\<close> by auto
     with exN obtain n where "\<And>i. i \<in> Basis \<Longrightarrow> vf X \<bullet> i - uf X \<bullet> i = (2*c) / 2^n"
       by blast
     then show ?thesis
       by (simp add: \<open>i \<in> Basis\<close> SOME_Basis inner_diff prj1_def)
   qed
   have countbl: "countable (fbx ` \<D>)"
     using \<open>countable \<D>\<close> by blast
   have "(\<Sum>k\<in>fbx`\<D>'. measure lebesgue k) \<le> e/2" if "\<D>' \<subseteq> \<D>" "finite \<D>'" for \<D>'
   proof -
     have BM_ge0: "0 \<le> B * (DIM('M) * prj1 (vf X - uf X))" if "X \<in> \<D>'" for X
       using \<open>0 < B\<close> \<open>\<D>' \<subseteq> \<D>\<close> that vu_pos by fastforce
     have "{} \<notin> \<D>'"
       using cbox \<open>\<D>' \<subseteq> \<D>\<close> interior_empty by blast
     have "(\<Sum>k\<in>fbx`\<D>'. measure lebesgue k) \<le> sum (measure lebesgue o fbx) \<D>'"
       by (rule sum_image_le [OF \<open>finite \<D>'\<close>]) (force simp: fbx_def)
     also have "\<dots> \<le> (\<Sum>X\<in>\<D>'. (2 * B * DIM('M)) ^ DIM('N) * measure lebesgue X)"
     proof (rule sum_mono)
       fix X assume "X \<in> \<D>'"
       then have "X \<in> \<D>" using \<open>\<D>' \<subseteq> \<D>\<close> by blast
       then have ufvf: "cbox (uf X) (vf X) = X"
         using uvz by blast
       have "prj1 (vf X - uf X) ^ DIM('M) = (\<Prod>i::'M \<in> Basis. prj1 (vf X - uf X))"
         by (rule prod_constant [symmetric])
       also have "\<dots> = (\<Prod>i\<in>Basis. vf X \<bullet> i - uf X \<bullet> i)"
         apply (rule prod.cong [OF refl])
         by (simp add: \<open>X \<in> \<D>\<close> inner_diff_left prj1_idem)
       finally have prj1_eq: "prj1 (vf X - uf X) ^ DIM('M) = (\<Prod>i\<in>Basis. vf X \<bullet> i - uf X \<bullet> i)" .
       have "uf X \<in> cbox (uf X) (vf X)" "vf X \<in> cbox (uf X) (vf X)"
         using uvz [OF \<open>X \<in> \<D>\<close>] by (force simp: mem_box)+
       moreover have "cbox (uf X) (vf X) \<subseteq> ball (zf X) (1/2)"
         by (meson R12 order_trans subset_ball uvz [OF \<open>X \<in> \<D>\<close>])
       ultimately have "uf X \<in> ball (zf X) (1/2)"  "vf X \<in> ball (zf X) (1/2)"
         by auto
       then have "dist (vf X) (uf X) \<le> 1"
         unfolding mem_ball
         by (metis dist_commute dist_triangle_half_l dual_order.order_iff_strict)
       then have 1: "prj1 (vf X - uf X) \<le> 1"
         unfolding prj1_def dist_norm using Basis_le_norm SOME_Basis order_trans by fastforce
       have 0: "0 \<le> prj1 (vf X - uf X)"
         using \<open>X \<in> \<D>\<close> prj1_def vu_pos by fastforce
       have "(measure lebesgue \<circ> fbx) X \<le> (2 * B * DIM('M)) ^ DIM('N) * content (cbox (uf X) (vf X))"
         apply (simp add: fbx_def content_cbox_cases algebra_simps BM_ge0 \<open>X \<in> \<D>'\<close>)
         apply (simp add: power_mult_distrib \<open>0 < B\<close> prj1_eq [symmetric])
         using MleN 0 1 uvz \<open>X \<in> \<D>\<close>
         apply (fastforce simp add: box_ne_empty power_decreasing)
         done
       also have "\<dots> = (2 * B * DIM('M)) ^ DIM('N) * measure lebesgue X"
         by (subst (3) ufvf[symmetric]) simp
       finally show "(measure lebesgue \<circ> fbx) X \<le> (2 * B * DIM('M)) ^ DIM('N) * measure lebesgue X" .
     qed
     also have "\<dots> = (2 * B * DIM('M)) ^ DIM('N) * sum (measure lebesgue) \<D>'"
       by (simp add: sum_distrib_left)
     also have "\<dots> \<le> e/2"
     proof -
       have div: "\<D>' division_of \<Union>\<D>'"
         apply (auto simp: \<open>finite \<D>'\<close> \<open>{} \<notin> \<D>'\<close> division_of_def)
         using cbox that apply blast
         using pairwise_subset [OF pw \<open>\<D>' \<subseteq> \<D>\<close>] unfolding pairwise_def apply force+
         done
       have le_meaT: "measure lebesgue (\<Union>\<D>') \<le> measure lebesgue T"
       proof (rule measure_mono_fmeasurable)
         show "(\<Union>\<D>') \<in> sets lebesgue"
           using div lmeasurable_division by auto
         have "\<Union>\<D>' \<subseteq> \<Union>\<D>"
           using \<open>\<D>' \<subseteq> \<D>\<close> by blast
         also have "... \<subseteq> T"
         proof (clarify)
           fix x D
           assume "x \<in> D" "D \<in> \<D>"
           show "x \<in> T"
             using Ksub [OF \<open>D \<in> \<D>\<close>]
             by (metis \<open>x \<in> D\<close> Int_iff dist_norm mem_ball norm_minus_commute subsetD RT)
         qed
         finally show "\<Union>\<D>' \<subseteq> T" .
         show "T \<in> lmeasurable"
           using \<open>S \<in> lmeasurable\<close> \<open>S \<subseteq> T\<close> \<open>T - S \<in> lmeasurable\<close> fmeasurable_Diff_D by blast
       qed
       have "sum (measure lebesgue) \<D>' = sum content \<D>'"
         using  \<open>\<D>' \<subseteq> \<D>\<close> cbox by (force intro: sum.cong)
       then have "(2 * B * DIM('M)) ^ DIM('N) * sum (measure lebesgue) \<D>' =
                  (2 * B * real DIM('M)) ^ DIM('N) * measure lebesgue (\<Union>\<D>')"
         using content_division [OF div] by auto
       also have "\<dots> \<le> (2 * B * real DIM('M)) ^ DIM('N) * measure lebesgue T"
         apply (rule mult_left_mono [OF le_meaT])
         using \<open>0 < B\<close>
         apply (simp add: algebra_simps)
         done
       also have "\<dots> \<le> e/2"
         using T \<open>0 < B\<close> by (simp add: field_simps)
       finally show ?thesis .
     qed
     finally show ?thesis .
   qed
   then have e2: "sum (measure lebesgue) \<G> \<le> e/2" if "\<G> \<subseteq> fbx ` \<D>" "finite \<G>" for \<G>
     by (metis finite_subset_image that)
   show "\<exists>W\<in>lmeasurable. f ` S \<subseteq> W \<and> measure lebesgue W < e"
   proof (intro bexI conjI)
     have "\<exists>X\<in>\<D>. f y \<in> fbx X" if "y \<in> S" for y
     proof -
       obtain X where "y \<in> X" "X \<in> \<D>"
         using \<open>S \<subseteq> \<Union>\<D>\<close> \<open>y \<in> S\<close> by auto
       then have y: "y \<in> ball(zf X) (R(zf X))"
         using uvz by fastforce
       have conj_le_eq: "z - b \<le> y \<and> y \<le> z + b \<longleftrightarrow> abs(y - z) \<le> b" for z y b::real
         by auto
       have yin: "y \<in> cbox (uf X) (vf X)" and zin: "(zf X) \<in> cbox (uf X) (vf X)"
         using uvz \<open>X \<in> \<D>\<close> \<open>y \<in> X\<close> by auto
       have "norm (y - zf X) \<le> (\<Sum>i\<in>Basis. \<bar>(y - zf X) \<bullet> i\<bar>)"
         by (rule norm_le_l1)
       also have "\<dots> \<le> real DIM('M) * prj1 (vf X - uf X)"
       proof (rule sum_bounded_above)
         fix j::'M assume j: "j \<in> Basis"
         show "\<bar>(y - zf X) \<bullet> j\<bar> \<le> prj1 (vf X - uf X)"
           using yin zin j
           by (fastforce simp add: mem_box prj1_idem [OF \<open>X \<in> \<D>\<close> j] inner_diff_left)
       qed
       finally have nole: "norm (y - zf X) \<le> DIM('M) * prj1 (vf X - uf X)"
         by simp
       have fle: "\<bar>f y \<bullet> i - f(zf X) \<bullet> i\<bar> \<le> B * DIM('M) * prj1 (vf X - uf X)" if "i \<in> Basis" for i
       proof -
         have "\<bar>f y \<bullet> i - f (zf X) \<bullet> i\<bar> = \<bar>(f y - f (zf X)) \<bullet> i\<bar>"
           by (simp add: algebra_simps)
         also have "\<dots> \<le> norm (f y - f (zf X))"
           by (simp add: Basis_le_norm that)
         also have "\<dots> \<le> B * norm(y - zf X)"
           by (metis uvz RB \<open>X \<in> \<D>\<close> dist_commute dist_norm mem_ball \<open>y \<in> S\<close> y)
         also have "\<dots> \<le> B * real DIM('M) * prj1 (vf X - uf X)"
           using \<open>0 < B\<close> by (simp add: nole)
         finally show ?thesis .
       qed
       show ?thesis
         by (rule_tac x=X in bexI)
            (auto simp: fbx_def prj1_idem mem_box conj_le_eq inner_add inner_diff fle \<open>X \<in> \<D>\<close>)
     qed
     then show "f ` S \<subseteq> (\<Union>D\<in>\<D>. fbx D)" by auto
   next
     have 1: "\<And>D. D \<in> \<D> \<Longrightarrow> fbx D \<in> lmeasurable"
       by (auto simp: fbx_def)
     have 2: "I' \<subseteq> \<D> \<Longrightarrow> finite I' \<Longrightarrow> measure lebesgue (\<Union>D\<in>I'. fbx D) \<le> e/2" for I'
       by (rule order_trans[OF measure_Union_le e2]) (auto simp: fbx_def)
     show "(\<Union>D\<in>\<D>. fbx D) \<in> lmeasurable"
       by (intro fmeasurable_UN_bound[OF \<open>countable \<D>\<close> 1 2])
     have "measure lebesgue (\<Union>D\<in>\<D>. fbx D) \<le> e/2"
       by (intro measure_UN_bound[OF \<open>countable \<D>\<close> 1 2])
     then show "measure lebesgue (\<Union>D\<in>\<D>. fbx D) < e"
       using \<open>0 < e\<close> by linarith
   qed
 qed
 
 proposition negligible_locally_Lipschitz_image:
   fixes f :: "'M::euclidean_space \<Rightarrow> 'N::euclidean_space"
   assumes MleN: "DIM('M) \<le> DIM('N)" "negligible S"
       and lips: "\<And>x. x \<in> S
                       \<Longrightarrow> \<exists>T B. open T \<and> x \<in> T \<and>
                               (\<forall>y \<in> S \<inter> T. norm(f y - f x) \<le> B * norm(y - x))"
     shows "negligible (f ` S)"
 proof -
   let ?S = "\<lambda>n. ({x \<in> S. norm x \<le> n \<and>
                           (\<exists>T. open T \<and> x \<in> T \<and>
                                (\<forall>y\<in>S \<inter> T. norm (f y - f x) \<le> (real n + 1) * norm (y - x)))})"
   have negfn: "f ` ?S n \<in> null_sets lebesgue" for n::nat
     unfolding negligible_iff_null_sets[symmetric]
     apply (rule_tac B = "real n + 1" in locally_Lipschitz_negl_bounded)
     by (auto simp: MleN bounded_iff intro: negligible_subset [OF \<open>negligible S\<close>])
   have "S = (\<Union>n. ?S n)"
   proof (intro set_eqI iffI)
     fix x assume "x \<in> S"
     with lips obtain T B where T: "open T" "x \<in> T"
                            and B: "\<And>y. y \<in> S \<inter> T \<Longrightarrow> norm(f y - f x) \<le> B * norm(y - x)"
       by metis+
     have no: "norm (f y - f x) \<le> (nat \<lceil>max B (norm x)\<rceil> + 1) * norm (y - x)" if "y \<in> S \<inter> T" for y
     proof -
       have "B * norm(y - x) \<le> (nat \<lceil>max B (norm x)\<rceil> + 1) * norm (y - x)"
         by (meson max.cobounded1 mult_right_mono nat_ceiling_le_eq nat_le_iff_add norm_ge_zero order_trans)
       then show ?thesis
         using B order_trans that by blast
     qed
     have "x \<in> ?S (nat (ceiling (max B (norm x))))"
       apply (simp add: \<open>x \<in> S \<close>, rule)
       using real_nat_ceiling_ge max.bounded_iff apply blast
       using T no
       apply (force simp: algebra_simps)
       done
     then show "x \<in> (\<Union>n. ?S n)" by force
   qed auto
   then show ?thesis
     by (rule ssubst) (auto simp: image_Union negligible_iff_null_sets intro: negfn)
 qed
 
 corollary negligible_differentiable_image_negligible:
   fixes f :: "'M::euclidean_space \<Rightarrow> 'N::euclidean_space"
   assumes MleN: "DIM('M) \<le> DIM('N)" "negligible S"
       and diff_f: "f differentiable_on S"
     shows "negligible (f ` S)"
 proof -
   have "\<exists>T B. open T \<and> x \<in> T \<and> (\<forall>y \<in> S \<inter> T. norm(f y - f x) \<le> B * norm(y - x))"
         if "x \<in> S" for x
   proof -
     obtain f' where "linear f'"
       and f': "\<And>e. e>0 \<Longrightarrow>
                   \<exists>d>0. \<forall>y\<in>S. norm (y - x) < d \<longrightarrow>
                               norm (f y - f x - f' (y - x)) \<le> e * norm (y - x)"
       using diff_f \<open>x \<in> S\<close>
       by (auto simp: linear_linear differentiable_on_def differentiable_def has_derivative_within_alt)
     obtain B where "B > 0" and B: "\<forall>x. norm (f' x) \<le> B * norm x"
       using linear_bounded_pos \<open>linear f'\<close> by blast
     obtain d where "d>0"
               and d: "\<And>y. \<lbrakk>y \<in> S; norm (y - x) < d\<rbrakk> \<Longrightarrow>
                           norm (f y - f x - f' (y - x)) \<le> norm (y - x)"
       using f' [of 1] by (force simp:)
     have *: "norm (f y - f x) \<le> (B + 1) * norm (y - x)"
               if "y \<in> S" "norm (y - x) < d" for y
     proof -
       have "norm (f y - f x) -B *  norm (y - x) \<le> norm (f y - f x) - norm (f' (y - x))"
         by (simp add: B)
       also have "\<dots> \<le> norm (f y - f x - f' (y - x))"
         by (rule norm_triangle_ineq2)
       also have "... \<le> norm (y - x)"
         by (rule d [OF that])
       finally show ?thesis
         by (simp add: algebra_simps)
     qed
     show ?thesis
       apply (rule_tac x="ball x d" in exI)
       apply (rule_tac x="B+1" in exI)
       using \<open>d>0\<close>
       apply (auto simp: dist_norm norm_minus_commute intro!: *)
       done
   qed
   with negligible_locally_Lipschitz_image assms show ?thesis by metis
 qed
 
 corollary negligible_differentiable_image_lowdim:
   fixes f :: "'M::euclidean_space \<Rightarrow> 'N::euclidean_space"
   assumes MlessN: "DIM('M) < DIM('N)" and diff_f: "f differentiable_on S"
     shows "negligible (f ` S)"
 proof -
   have "x \<le> DIM('M) \<Longrightarrow> x \<le> DIM('N)" for x
     using MlessN by linarith
   obtain lift :: "'M * real \<Rightarrow> 'N" and drop :: "'N \<Rightarrow> 'M * real" and j :: 'N
     where "linear lift" "linear drop" and dropl [simp]: "\<And>z. drop (lift z) = z"
       and "j \<in> Basis" and j: "\<And>x. lift(x,0) \<bullet> j = 0"
     using lowerdim_embeddings [OF MlessN] by metis
   have "negligible {x. x\<bullet>j = 0}"
     by (metis \<open>j \<in> Basis\<close> negligible_standard_hyperplane)
   then have neg0S: "negligible ((\<lambda>x. lift (x, 0)) ` S)"
     apply (rule negligible_subset)
     by (simp add: image_subsetI j)
   have diff_f': "f \<circ> fst \<circ> drop differentiable_on (\<lambda>x. lift (x, 0)) ` S"
     using diff_f
     apply (clarsimp simp add: differentiable_on_def)
     apply (intro differentiable_chain_within linear_imp_differentiable [OF \<open>linear drop\<close>]
-             linear_imp_differentiable [OF fst_linear])
+             linear_imp_differentiable [OF linear_fst])
     apply (force simp: image_comp o_def)
     done
   have "f = (f o fst o drop o (\<lambda>x. lift (x, 0)))"
     by (simp add: o_def)
   then show ?thesis
     apply (rule ssubst)
     apply (subst image_comp [symmetric])
     apply (metis negligible_differentiable_image_negligible order_refl diff_f' neg0S)
     done
 qed
 
 subsection\<open>Measurability of countable unions and intersections of various kinds.\<close>
 
 lemma
   assumes S: "\<And>n. S n \<in> lmeasurable"
     and leB: "\<And>n. measure lebesgue (S n) \<le> B"
     and nest: "\<And>n. S n \<subseteq> S(Suc n)"
   shows measurable_nested_Union: "(\<Union>n. S n) \<in> lmeasurable"
   and measure_nested_Union:  "(\<lambda>n. measure lebesgue (S n)) \<longlonglongrightarrow> measure lebesgue (\<Union>n. S n)" (is ?Lim)
 proof -
   have 1: "\<And>n. (indicat_real (S n)) integrable_on UNIV"
     using S measurable_integrable by blast
   have 2: "\<And>n x::'a. indicat_real (S n) x \<le> (indicat_real (S (Suc n)) x)"
     by (simp add: indicator_leI nest rev_subsetD)
   have 3: "\<And>x. (\<lambda>n. indicat_real (S n) x) \<longlonglongrightarrow> (indicat_real (\<Union>(S ` UNIV)) x)"
     apply (rule tendsto_eventually)
     apply (simp add: indicator_def)
     by (metis eventually_sequentiallyI lift_Suc_mono_le nest subsetCE)
   have 4: "bounded (range (\<lambda>n. integral UNIV (indicat_real (S n))))"
     using leB by (auto simp: lmeasure_integral_UNIV [symmetric] S bounded_iff)
   have "(\<Union>n. S n) \<in> lmeasurable \<and> ?Lim"
     apply (simp add: lmeasure_integral_UNIV S cong: conj_cong)
     apply (simp add: measurable_integrable)
     apply (rule monotone_convergence_increasing [OF 1 2 3 4])
     done
   then show "(\<Union>n. S n) \<in> lmeasurable" "?Lim"
     by auto
 qed
 
 lemma
   assumes S: "\<And>n. S n \<in> lmeasurable"
     and djointish: "pairwise (\<lambda>m n. negligible (S m \<inter> S n)) UNIV"
     and leB: "\<And>n. (\<Sum>k\<le>n. measure lebesgue (S k)) \<le> B"
   shows measurable_countable_negligible_Union: "(\<Union>n. S n) \<in> lmeasurable"
   and   measure_countable_negligible_Union:    "(\<lambda>n. (measure lebesgue (S n))) sums measure lebesgue (\<Union>n. S n)" (is ?Sums)
 proof -
   have 1: "\<Union> (S ` {..n}) \<in> lmeasurable" for n
     using S by blast
   have 2: "measure lebesgue (\<Union> (S ` {..n})) \<le> B" for n
   proof -
     have "measure lebesgue (\<Union> (S ` {..n})) \<le> (\<Sum>k\<le>n. measure lebesgue (S k))"
       by (simp add: S fmeasurableD measure_UNION_le)
     with leB show ?thesis
       using order_trans by blast
   qed
   have 3: "\<And>n. \<Union> (S ` {..n}) \<subseteq> \<Union> (S ` {..Suc n})"
     by (simp add: SUP_subset_mono)
   have eqS: "(\<Union>n. S n) = (\<Union>n. \<Union> (S ` {..n}))"
     using atLeastAtMost_iff by blast
   also have "(\<Union>n. \<Union> (S ` {..n})) \<in> lmeasurable"
     by (intro measurable_nested_Union [OF 1 2] 3)
   finally show "(\<Union>n. S n) \<in> lmeasurable" .
   have eqm: "(\<Sum>i\<le>n. measure lebesgue (S i)) = measure lebesgue (\<Union> (S ` {..n}))" for n
     using assms by (simp add: measure_negligible_finite_Union_image pairwise_mono)
   have "(\<lambda>n. (measure lebesgue (S n))) sums measure lebesgue (\<Union>n. \<Union> (S ` {..n}))"
     by (simp add: sums_def' eqm atLeast0AtMost) (intro measure_nested_Union [OF 1 2] 3)
   then show ?Sums
     by (simp add: eqS)
 qed
 
 lemma negligible_countable_Union [intro]:
   assumes "countable \<F>" and meas: "\<And>S. S \<in> \<F> \<Longrightarrow> negligible S"
   shows "negligible (\<Union>\<F>)"
 proof (cases "\<F> = {}")
   case False
   then show ?thesis
     by (metis from_nat_into range_from_nat_into assms negligible_Union_nat)
 qed simp
 
 lemma
   assumes S: "\<And>n. (S n) \<in> lmeasurable"
     and djointish: "pairwise (\<lambda>m n. negligible (S m \<inter> S n)) UNIV"
     and bo: "bounded (\<Union>n. S n)"
   shows measurable_countable_negligible_Union_bounded: "(\<Union>n. S n) \<in> lmeasurable"
   and   measure_countable_negligible_Union_bounded:    "(\<lambda>n. (measure lebesgue (S n))) sums measure lebesgue (\<Union>n. S n)" (is ?Sums)
 proof -
   obtain a b where ab: "(\<Union>n. S n) \<subseteq> cbox a b"
     using bo bounded_subset_cbox_symmetric by metis
   then have B: "(\<Sum>k\<le>n. measure lebesgue (S k)) \<le> measure lebesgue (cbox a b)" for n
   proof -
     have "(\<Sum>k\<le>n. measure lebesgue (S k)) = measure lebesgue (\<Union> (S ` {..n}))"
       using measure_negligible_finite_Union_image [OF _ _ pairwise_subset] djointish
       by (metis S finite_atMost subset_UNIV)
     also have "\<dots> \<le> measure lebesgue (cbox a b)"
       apply (rule measure_mono_fmeasurable)
       using ab S by force+
     finally show ?thesis .
   qed
   show "(\<Union>n. S n) \<in> lmeasurable"
     by (rule measurable_countable_negligible_Union [OF S djointish B])
   show ?Sums
     by (rule measure_countable_negligible_Union [OF S djointish B])
 qed
 
 lemma measure_countable_Union_approachable:
   assumes "countable \<D>" "e > 0" and measD: "\<And>d. d \<in> \<D> \<Longrightarrow> d \<in> lmeasurable"
       and B: "\<And>D'. \<lbrakk>D' \<subseteq> \<D>; finite D'\<rbrakk> \<Longrightarrow> measure lebesgue (\<Union>D') \<le> B"
     obtains D' where "D' \<subseteq> \<D>" "finite D'" "measure lebesgue (\<Union>\<D>) - e < measure lebesgue (\<Union>D')"
 proof (cases "\<D> = {}")
   case True
   then show ?thesis
     by (simp add: \<open>e > 0\<close> that)
 next
   case False
   let ?S = "\<lambda>n. \<Union>k \<le> n. from_nat_into \<D> k"
   have "(\<lambda>n. measure lebesgue (?S n)) \<longlonglongrightarrow> measure lebesgue (\<Union>n. ?S n)"
   proof (rule measure_nested_Union)
     show "?S n \<in> lmeasurable" for n
       by (simp add: False fmeasurable.finite_UN from_nat_into measD)
     show "measure lebesgue (?S n) \<le> B" for n
       by (metis (mono_tags, lifting) B False finite_atMost finite_imageI from_nat_into image_iff subsetI)
     show "?S n \<subseteq> ?S (Suc n)" for n
       by force
   qed
   then obtain N where N: "\<And>n. n \<ge> N \<Longrightarrow> dist (measure lebesgue (?S n)) (measure lebesgue (\<Union>n. ?S n)) < e"
     using metric_LIMSEQ_D \<open>e > 0\<close> by blast
   show ?thesis
   proof
     show "from_nat_into \<D> ` {..N} \<subseteq> \<D>"
       by (auto simp: False from_nat_into)
     have eq: "(\<Union>n. \<Union>k\<le>n. from_nat_into \<D> k) = (\<Union>\<D>)"
       using \<open>countable \<D>\<close> False
       by (auto intro: from_nat_into dest: from_nat_into_surj [OF \<open>countable \<D>\<close>])
     show "measure lebesgue (\<Union>\<D>) - e < measure lebesgue (\<Union> (from_nat_into \<D> ` {..N}))"
       using N [OF order_refl]
       by (auto simp: eq algebra_simps dist_norm)
   qed auto
 qed
 
 
 subsection\<open>Negligibility is a local property\<close>
 
 lemma locally_negligible_alt:
     "negligible S \<longleftrightarrow> (\<forall>x \<in> S. \<exists>U. openin (top_of_set S) U \<and> x \<in> U \<and> negligible U)"
      (is "_ = ?rhs")
 proof
   assume "negligible S"
   then show ?rhs
     using openin_subtopology_self by blast
 next
   assume ?rhs
   then obtain U where ope: "\<And>x. x \<in> S \<Longrightarrow> openin (top_of_set S) (U x)"
     and cov: "\<And>x. x \<in> S \<Longrightarrow> x \<in> U x"
     and neg: "\<And>x. x \<in> S \<Longrightarrow> negligible (U x)"
     by metis
   obtain \<F> where "\<F> \<subseteq> U ` S" "countable \<F>" and eq: "\<Union>\<F> = \<Union>(U ` S)"
     using ope by (force intro: Lindelof_openin [of "U ` S" S])
   then have "negligible (\<Union>\<F>)"
     by (metis imageE neg negligible_countable_Union subset_eq)
   with eq have "negligible (\<Union>(U ` S))"
     by metis
   moreover have "S \<subseteq> \<Union>(U ` S)"
     using cov by blast
   ultimately show "negligible S"
     using negligible_subset by blast
 qed
 
 lemma locally_negligible:
      "locally negligible S \<longleftrightarrow> negligible S"
   unfolding locally_def
   apply safe
    apply (meson negligible_subset openin_subtopology_self locally_negligible_alt)
   by (meson negligible_subset openin_imp_subset order_refl)
 
 
 subsection\<open>Integral bounds\<close>
 
 lemma set_integral_norm_bound:
   fixes f :: "_ \<Rightarrow> 'a :: {banach, second_countable_topology}"
   shows "set_integrable M k f \<Longrightarrow> norm (LINT x:k|M. f x) \<le> LINT x:k|M. norm (f x)"
   using integral_norm_bound[of M "\<lambda>x. indicator k x *\<^sub>R f x"] by (simp add: set_lebesgue_integral_def)
 
 lemma set_integral_finite_UN_AE:
   fixes f :: "_ \<Rightarrow> _ :: {banach, second_countable_topology}"
   assumes "finite I"
     and ae: "\<And>i j. i \<in> I \<Longrightarrow> j \<in> I \<Longrightarrow> AE x in M. (x \<in> A i \<and> x \<in> A j) \<longrightarrow> i = j"
     and [measurable]: "\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets M"
     and f: "\<And>i. i \<in> I \<Longrightarrow> set_integrable M (A i) f"
   shows "LINT x:(\<Union>i\<in>I. A i)|M. f x = (\<Sum>i\<in>I. LINT x:A i|M. f x)"
   using \<open>finite I\<close> order_refl[of I]
 proof (induction I rule: finite_subset_induct')
   case (insert i I')
   have "AE x in M. (\<forall>j\<in>I'. x \<in> A i \<longrightarrow> x \<notin> A j)"
   proof (intro AE_ball_countable[THEN iffD2] ballI)
     fix j assume "j \<in> I'"
     with \<open>I' \<subseteq> I\<close> \<open>i \<notin> I'\<close> have "i \<noteq> j" "j \<in> I"
       by auto
     then show "AE x in M. x \<in> A i \<longrightarrow> x \<notin> A j"
       using ae[of i j] \<open>i \<in> I\<close> by auto
   qed (use \<open>finite I'\<close> in \<open>rule countable_finite\<close>)
   then have "AE x\<in>A i in M. \<forall>xa\<in>I'. x \<notin> A xa "
     by auto
   with insert.hyps insert.IH[symmetric]
   show ?case
     by (auto intro!: set_integral_Un_AE sets.finite_UN f set_integrable_UN)
 qed (simp add: set_lebesgue_integral_def)
 
 lemma set_integrable_norm:
   fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
   assumes f: "set_integrable M k f" shows "set_integrable M k (\<lambda>x. norm (f x))"
   using integrable_norm f by (force simp add: set_integrable_def)
 
 lemma absolutely_integrable_bounded_variation:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes f: "f absolutely_integrable_on UNIV"
   obtains B where "\<forall>d. d division_of (\<Union>d) \<longrightarrow> sum (\<lambda>k. norm (integral k f)) d \<le> B"
 proof (rule that[of "integral UNIV (\<lambda>x. norm (f x))"]; safe)
   fix d :: "'a set set" assume d: "d division_of \<Union>d"
   have *: "k \<in> d \<Longrightarrow> f absolutely_integrable_on k" for k
     using f[THEN set_integrable_subset, of k] division_ofD(2,4)[OF d, of k] by auto
   note d' = division_ofD[OF d]
   have "(\<Sum>k\<in>d. norm (integral k f)) = (\<Sum>k\<in>d. norm (LINT x:k|lebesgue. f x))"
     by (intro sum.cong refl arg_cong[where f=norm] set_lebesgue_integral_eq_integral(2)[symmetric] *)
   also have "\<dots> \<le> (\<Sum>k\<in>d. LINT x:k|lebesgue. norm (f x))"
     by (intro sum_mono set_integral_norm_bound *)
   also have "\<dots> = (\<Sum>k\<in>d. integral k (\<lambda>x. norm (f x)))"
     by (intro sum.cong refl set_lebesgue_integral_eq_integral(2) set_integrable_norm *)
   also have "\<dots> \<le> integral (\<Union>d) (\<lambda>x. norm (f x))"
     using integrable_on_subdivision[OF d] assms f unfolding absolutely_integrable_on_def
     by (subst integral_combine_division_topdown[OF _ d]) auto
   also have "\<dots> \<le> integral UNIV (\<lambda>x. norm (f x))"
     using integrable_on_subdivision[OF d] assms unfolding absolutely_integrable_on_def
     by (intro integral_subset_le) auto
   finally show "(\<Sum>k\<in>d. norm (integral k f)) \<le> integral UNIV (\<lambda>x. norm (f x))" .
 qed
 
 lemma absdiff_norm_less:
   assumes "sum (\<lambda>x. norm (f x - g x)) s < e"
     and "finite s"
   shows "\<bar>sum (\<lambda>x. norm(f x)) s - sum (\<lambda>x. norm(g x)) s\<bar> < e"
   unfolding sum_subtractf[symmetric]
   apply (rule le_less_trans[OF sum_abs])
   apply (rule le_less_trans[OF _ assms(1)])
   apply (rule sum_mono)
   apply (rule norm_triangle_ineq3)
   done
 
 proposition bounded_variation_absolutely_integrable_interval:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
   assumes f: "f integrable_on cbox a b"
     and *: "\<And>d. d division_of (cbox a b) \<Longrightarrow> sum (\<lambda>K. norm(integral K f)) d \<le> B"
   shows "f absolutely_integrable_on cbox a b"
 proof -
   let ?f = "\<lambda>d. \<Sum>K\<in>d. norm (integral K f)" and ?D = "{d. d division_of (cbox a b)}"
   have D_1: "?D \<noteq> {}"
     by (rule elementary_interval[of a b]) auto
   have D_2: "bdd_above (?f`?D)"
     by (metis * mem_Collect_eq bdd_aboveI2)
   note D = D_1 D_2
   let ?S = "SUP x\<in>?D. ?f x"
   have *: "\<exists>\<gamma>. gauge \<gamma> \<and>
              (\<forall>p. p tagged_division_of cbox a b \<and>
                   \<gamma> fine p \<longrightarrow>
                   norm ((\<Sum>(x,k) \<in> p. content k *\<^sub>R norm (f x)) - ?S) < e)"
     if e: "e > 0" for e
   proof -
     have "?S - e/2 < ?S" using \<open>e > 0\<close> by simp
     then obtain d where d: "d division_of (cbox a b)" "?S - e/2 < (\<Sum>k\<in>d. norm (integral k f))"
       unfolding less_cSUP_iff[OF D] by auto
     note d' = division_ofD[OF this(1)]
 
     have "\<exists>e>0. \<forall>i\<in>d. x \<notin> i \<longrightarrow> ball x e \<inter> i = {}" for x
     proof -
       have "\<exists>d'>0. \<forall>x'\<in>\<Union>{i \<in> d. x \<notin> i}. d' \<le> dist x x'"
       proof (rule separate_point_closed)
         show "closed (\<Union>{i \<in> d. x \<notin> i})"
           using d' by force
         show "x \<notin> \<Union>{i \<in> d. x \<notin> i}"
           by auto
       qed
       then show ?thesis
         by force
     qed
     then obtain k where k: "\<And>x. 0 < k x" "\<And>i x. \<lbrakk>i \<in> d; x \<notin> i\<rbrakk> \<Longrightarrow> ball x (k x) \<inter> i = {}"
       by metis
     have "e/2 > 0"
       using e by auto
     with Henstock_lemma[OF f]
     obtain \<gamma> where g: "gauge \<gamma>"
       "\<And>p. \<lbrakk>p tagged_partial_division_of cbox a b; \<gamma> fine p\<rbrakk>
                 \<Longrightarrow> (\<Sum>(x,k) \<in> p. norm (content k *\<^sub>R f x - integral k f)) < e/2"
       by (metis (no_types, lifting))
     let ?g = "\<lambda>x. \<gamma> x \<inter> ball x (k x)"
     show ?thesis
     proof (intro exI conjI allI impI)
       show "gauge ?g"
         using g(1) k(1) by (auto simp: gauge_def)
     next
       fix p
       assume "p tagged_division_of (cbox a b) \<and> ?g fine p"
       then have p: "p tagged_division_of cbox a b" "\<gamma> fine p" "(\<lambda>x. ball x (k x)) fine p"
         by (auto simp: fine_Int)
       note p' = tagged_division_ofD[OF p(1)]
       define p' where "p' = {(x,k) | x k. \<exists>i l. x \<in> i \<and> i \<in> d \<and> (x,l) \<in> p \<and> k = i \<inter> l}"
       have gp': "\<gamma> fine p'"
         using p(2) by (auto simp: p'_def fine_def)
       have p'': "p' tagged_division_of (cbox a b)"
       proof (rule tagged_division_ofI)
         show "finite p'"
         proof (rule finite_subset)
           show "p' \<subseteq> (\<lambda>(k, x, l). (x, k \<inter> l)) ` (d \<times> p)"
             by (force simp: p'_def image_iff)
           show "finite ((\<lambda>(k, x, l). (x, k \<inter> l)) ` (d \<times> p))"
             by (simp add: d'(1) p'(1))
         qed
       next
         fix x K
         assume "(x, K) \<in> p'"
         then have "\<exists>i l. x \<in> i \<and> i \<in> d \<and> (x, l) \<in> p \<and> K = i \<inter> l"
           unfolding p'_def by auto
         then obtain i l where il: "x \<in> i" "i \<in> d" "(x, l) \<in> p" "K = i \<inter> l" by blast
         show "x \<in> K" and "K \<subseteq> cbox a b"
           using p'(2-3)[OF il(3)] il by auto
         show "\<exists>a b. K = cbox a b"
           unfolding il using p'(4)[OF il(3)] d'(4)[OF il(2)] by (meson Int_interval)
       next
         fix x1 K1
         assume "(x1, K1) \<in> p'"
         then have "\<exists>i l. x1 \<in> i \<and> i \<in> d \<and> (x1, l) \<in> p \<and> K1 = i \<inter> l"
           unfolding p'_def by auto
         then obtain i1 l1 where il1: "x1 \<in> i1" "i1 \<in> d" "(x1, l1) \<in> p" "K1 = i1 \<inter> l1" by blast
         fix x2 K2
         assume "(x2,K2) \<in> p'"
         then have "\<exists>i l. x2 \<in> i \<and> i \<in> d \<and> (x2, l) \<in> p \<and> K2 = i \<inter> l"
           unfolding p'_def by auto
         then obtain i2 l2 where il2: "x2 \<in> i2" "i2 \<in> d" "(x2, l2) \<in> p" "K2 = i2 \<inter> l2" by blast
         assume "(x1, K1) \<noteq> (x2, K2)"
         then have "interior i1 \<inter> interior i2 = {} \<or> interior l1 \<inter> interior l2 = {}"
           using d'(5)[OF il1(2) il2(2)] p'(5)[OF il1(3) il2(3)]  by (auto simp: il1 il2)
         then show "interior K1 \<inter> interior K2 = {}"
           unfolding il1 il2 by auto
       next
         have *: "\<forall>(x, X) \<in> p'. X \<subseteq> cbox a b"
           unfolding p'_def using d' by blast
         have "y \<in> \<Union>{K. \<exists>x. (x, K) \<in> p'}" if y: "y \<in> cbox a b" for y
         proof -
           obtain x l where xl: "(x, l) \<in> p" "y \<in> l"
             using y unfolding p'(6)[symmetric] by auto
           obtain i where i: "i \<in> d" "y \<in> i"
             using y unfolding d'(6)[symmetric] by auto
           have "x \<in> i"
             using fineD[OF p(3) xl(1)] using k(2) i xl by auto
           then show ?thesis
             unfolding p'_def by (rule_tac X="i \<inter> l" in UnionI) (use i xl in auto)
         qed
         show "\<Union>{K. \<exists>x. (x, K) \<in> p'} = cbox a b"
         proof
           show "\<Union>{k. \<exists>x. (x, k) \<in> p'} \<subseteq> cbox a b"
             using * by auto
         next
           show "cbox a b \<subseteq> \<Union>{k. \<exists>x. (x, k) \<in> p'}"
           proof
             fix y
             assume y: "y \<in> cbox a b"
             obtain x L where xl: "(x, L) \<in> p" "y \<in> L"
               using y unfolding p'(6)[symmetric] by auto
             obtain I where i: "I \<in> d" "y \<in> I"
               using y unfolding d'(6)[symmetric] by auto
             have "x \<in> I"
               using fineD[OF p(3) xl(1)] using k(2) i xl by auto
             then show "y \<in> \<Union>{k. \<exists>x. (x, k) \<in> p'}"
               apply (rule_tac X="I \<inter> L" in UnionI)
               using i xl by (auto simp: p'_def)
           qed
         qed
       qed
 
       then have sum_less_e2: "(\<Sum>(x,K) \<in> p'. norm (content K *\<^sub>R f x - integral K f)) < e/2"
         using g(2) gp' tagged_division_of_def by blast
 
       have "(x, I \<inter> L) \<in> p'" if x: "(x, L) \<in> p" "I \<in> d" and y: "y \<in> I" "y \<in> L"
         for x I L y
       proof -
         have "x \<in> I"
           using fineD[OF p(3) that(1)] k(2)[OF \<open>I \<in> d\<close>] y by auto
         with x have "(\<exists>i l. x \<in> i \<and> i \<in> d \<and> (x, l) \<in> p \<and> I \<inter> L = i \<inter> l)"
           by blast
         then have "(x, I \<inter> L) \<in> p'"
           by (simp add: p'_def)
         with y show ?thesis by auto
       qed
       moreover have "\<exists>y i l. (x, K) = (y, i \<inter> l) \<and> (y, l) \<in> p \<and> i \<in> d \<and> i \<inter> l \<noteq> {}"
         if xK: "(x,K) \<in> p'" for x K
       proof -
         obtain i l where il: "x \<in> i" "i \<in> d" "(x, l) \<in> p" "K = i \<inter> l"
           using xK unfolding p'_def by auto
         then show ?thesis
           using p'(2) by fastforce
       qed
       ultimately have p'alt: "p' = {(x, I \<inter> L) | x I L. (x,L) \<in> p \<and> I \<in> d \<and> I \<inter> L \<noteq> {}}"
         by auto
       have sum_p': "(\<Sum>(x,K) \<in> p'. norm (integral K f)) = (\<Sum>k\<in>snd ` p'. norm (integral k f))"
         apply (subst sum.over_tagged_division_lemma[OF p'',of "\<lambda>k. norm (integral k f)"])
          apply (auto intro: integral_null simp: content_eq_0_interior)
         done
       have snd_p_div: "snd ` p division_of cbox a b"
         by (rule division_of_tagged_division[OF p(1)])
       note snd_p = division_ofD[OF snd_p_div]
       have fin_d_sndp: "finite (d \<times> snd ` p)"
         by (simp add: d'(1) snd_p(1))
 
       have *: "\<And>sni sni' sf sf'. \<lbrakk>\<bar>sf' - sni'\<bar> < e/2; ?S - e/2 < sni; sni' \<le> ?S;
                        sni \<le> sni'; sf' = sf\<rbrakk> \<Longrightarrow> \<bar>sf - ?S\<bar> < e"
         by arith
       show "norm ((\<Sum>(x,k) \<in> p. content k *\<^sub>R norm (f x)) - ?S) < e"
         unfolding real_norm_def
       proof (rule *)
         show "\<bar>(\<Sum>(x,K)\<in>p'. norm (content K *\<^sub>R f x)) - (\<Sum>(x,k)\<in>p'. norm (integral k f))\<bar> < e/2"
           using p'' sum_less_e2 unfolding split_def by (force intro!: absdiff_norm_less)
         show "(\<Sum>(x,k) \<in> p'. norm (integral k f)) \<le>?S"
           by (auto simp: sum_p' division_of_tagged_division[OF p''] D intro!: cSUP_upper)
         show "(\<Sum>k\<in>d. norm (integral k f)) \<le> (\<Sum>(x,k) \<in> p'. norm (integral k f))"
         proof -
           have *: "{k \<inter> l | k l. k \<in> d \<and> l \<in> snd ` p} = (\<lambda>(k,l). k \<inter> l) ` (d \<times> snd ` p)"
             by auto
           have "(\<Sum>K\<in>d. norm (integral K f)) \<le> (\<Sum>i\<in>d. \<Sum>l\<in>snd ` p. norm (integral (i \<inter> l) f))"
           proof (rule sum_mono)
             fix K assume k: "K \<in> d"
             from d'(4)[OF this] obtain u v where uv: "K = cbox u v" by metis
             define d' where "d' = {cbox u v \<inter> l |l. l \<in> snd ` p \<and>  cbox u v \<inter> l \<noteq> {}}"
             have uvab: "cbox u v \<subseteq> cbox a b"
               using d(1) k uv by blast
             have "d' division_of cbox u v"
               unfolding d'_def by (rule division_inter_1 [OF snd_p_div uvab])
             moreover then have "norm (\<Sum>i\<in>d'. integral i f) \<le> (\<Sum>k\<in>d'. norm (integral k f))"
               by (simp add: sum_norm_le)
             ultimately have "norm (integral K f) \<le> sum (\<lambda>k. norm (integral k f)) d'"
               apply (subst integral_combine_division_topdown[of _ _ d'])
                 apply (auto simp: uv intro: integrable_on_subcbox[OF assms(1) uvab])
               done
             also have "\<dots> = (\<Sum>I\<in>{K \<inter> L |L. L \<in> snd ` p}. norm (integral I f))"
             proof -
               have *: "norm (integral I f) = 0"
                 if "I \<in> {cbox u v \<inter> l |l. l \<in> snd ` p}"
                   "I \<notin> {cbox u v \<inter> l |l. l \<in> snd ` p \<and> cbox u v \<inter> l \<noteq> {}}" for I
                 using that by auto
               show ?thesis
                 apply (rule sum.mono_neutral_left)
                   apply (simp add: snd_p(1))
                 unfolding d'_def uv using * by auto
             qed
             also have "\<dots> = (\<Sum>l\<in>snd ` p. norm (integral (K \<inter> l) f))"
             proof -
               have *: "norm (integral (K \<inter> l) f) = 0"
                 if "l \<in> snd ` p" "y \<in> snd ` p" "l \<noteq> y" "K \<inter> l = K \<inter> y" for l y
               proof -
                 have "interior (K \<inter> l) \<subseteq> interior (l \<inter> y)"
                   by (metis Int_lower2 interior_mono le_inf_iff that(4))
                 then have "interior (K \<inter> l) = {}"
                   by (simp add: snd_p(5) that)
                 moreover from d'(4)[OF k] snd_p(4)[OF that(1)]
                 obtain u1 v1 u2 v2
                   where uv: "K = cbox u1 u2" "l = cbox v1 v2" by metis
                 ultimately show ?thesis
                   using that integral_null
                   unfolding uv Int_interval content_eq_0_interior
                   by (metis (mono_tags, lifting) norm_eq_zero)
               qed
               show ?thesis
                 unfolding Setcompr_eq_image
                 apply (rule sum.reindex_nontrivial [unfolded o_def])
                  apply (rule finite_imageI)
                  apply (rule p')
                 using * by auto
             qed
             finally show "norm (integral K f) \<le> (\<Sum>l\<in>snd ` p. norm (integral (K \<inter> l) f))" .
           qed
           also have "\<dots> = (\<Sum>(i,l) \<in> d \<times> snd ` p. norm (integral (i\<inter>l) f))"
             by (simp add: sum.cartesian_product)
           also have "\<dots> = (\<Sum>x \<in> d \<times> snd ` p. norm (integral (case_prod (\<inter>) x) f))"
             by (force simp: split_def intro!: sum.cong)
           also have "\<dots> = (\<Sum>k\<in>{i \<inter> l |i l. i \<in> d \<and> l \<in> snd ` p}. norm (integral k f))"
           proof -
             have eq0: " (integral (l1 \<inter> k1) f) = 0"
               if "l1 \<inter> k1 = l2 \<inter> k2" "(l1, k1) \<noteq> (l2, k2)"
                 "l1 \<in> d" "(j1,k1) \<in> p" "l2 \<in> d" "(j2,k2) \<in> p"
               for l1 l2 k1 k2 j1 j2
             proof -
               obtain u1 v1 u2 v2 where uv: "l1 = cbox u1 u2" "k1 = cbox v1 v2"
                 using \<open>(j1, k1) \<in> p\<close> \<open>l1 \<in> d\<close> d'(4) p'(4) by blast
               have "l1 \<noteq> l2 \<or> k1 \<noteq> k2"
                 using that by auto
               then have "interior k1 \<inter> interior k2 = {} \<or> interior l1 \<inter> interior l2 = {}"
                 by (meson d'(5) old.prod.inject p'(5) that(3) that(4) that(5) that(6))
               moreover have "interior (l1 \<inter> k1) = interior (l2 \<inter> k2)"
                 by (simp add: that(1))
               ultimately have "interior(l1 \<inter> k1) = {}"
                 by auto
               then show ?thesis
                 unfolding uv Int_interval content_eq_0_interior[symmetric] by auto
             qed
             show ?thesis
               unfolding *
               apply (rule sum.reindex_nontrivial [OF fin_d_sndp, symmetric, unfolded o_def])
               apply clarsimp
               by (metis eq0 fst_conv snd_conv)
           qed
           also have "\<dots> = (\<Sum>(x,k) \<in> p'. norm (integral k f))"
           proof -
             have 0: "integral (ia \<inter> snd (a, b)) f = 0"
               if "ia \<inter> snd (a, b) \<notin> snd ` p'" "ia \<in> d" "(a, b) \<in> p" for ia a b
             proof -
               have "ia \<inter> b = {}"
                 using that unfolding p'alt image_iff Bex_def not_ex
                 apply (erule_tac x="(a, ia \<inter> b)" in allE)
                 apply auto
                 done
               then show ?thesis by auto
             qed
             have 1: "\<exists>i l. snd (a, b) = i \<inter> l \<and> i \<in> d \<and> l \<in> snd ` p" if "(a, b) \<in> p'" for a b
               using that
               apply (clarsimp simp: p'_def image_iff)
               by (metis (no_types, hide_lams) snd_conv)
             show ?thesis
               unfolding sum_p'
               apply (rule sum.mono_neutral_right)
                 apply (metis * finite_imageI[OF fin_d_sndp])
               using 0 1 by auto
           qed
           finally show ?thesis .
         qed
         show "(\<Sum>(x,k) \<in> p'. norm (content k *\<^sub>R f x)) = (\<Sum>(x,k) \<in> p. content k *\<^sub>R norm (f x))"
         proof -
           let ?S = "{(x, i \<inter> l) |x i l. (x, l) \<in> p \<and> i \<in> d}"
           have *: "?S = (\<lambda>(xl,i). (fst xl, snd xl \<inter> i)) ` (p \<times> d)"
             by force
           have fin_pd: "finite (p \<times> d)"
             using finite_cartesian_product[OF p'(1) d'(1)] by metis
           have "(\<Sum>(x,k) \<in> p'. norm (content k *\<^sub>R f x)) = (\<Sum>(x,k) \<in> ?S. \<bar>content k\<bar> * norm (f x))"
             unfolding norm_scaleR
             apply (rule sum.mono_neutral_left)
               apply (subst *)
               apply (rule finite_imageI [OF fin_pd])
             unfolding p'alt apply auto
             by fastforce
           also have "\<dots> = (\<Sum>((x,l),i)\<in>p \<times> d. \<bar>content (l \<inter> i)\<bar> * norm (f x))"
           proof -
             have "\<bar>content (l1 \<inter> k1)\<bar> * norm (f x1) = 0"
               if "(x1, l1) \<in> p" "(x2, l2) \<in> p" "k1 \<in> d" "k2 \<in> d"
                 "x1 = x2" "l1 \<inter> k1 = l2 \<inter> k2" "x1 \<noteq> x2 \<or> l1 \<noteq> l2 \<or> k1 \<noteq> k2"
               for x1 l1 k1 x2 l2 k2
             proof -
               obtain u1 v1 u2 v2 where uv: "k1 = cbox u1 u2" "l1 = cbox v1 v2"
                 by (meson \<open>(x1, l1) \<in> p\<close> \<open>k1 \<in> d\<close> d(1) division_ofD(4) p'(4))
               have "l1 \<noteq> l2 \<or> k1 \<noteq> k2"
                 using that by auto
               then have "interior k1 \<inter> interior k2 = {} \<or> interior l1 \<inter> interior l2 = {}"
                 apply (rule disjE)
                 using that p'(5) d'(5) by auto
               moreover have "interior (l1 \<inter> k1) = interior (l2 \<inter> k2)"
                 unfolding that ..
               ultimately have "interior (l1 \<inter> k1) = {}"
                 by auto
               then show "\<bar>content (l1 \<inter> k1)\<bar> * norm (f x1) = 0"
                 unfolding uv Int_interval content_eq_0_interior[symmetric] by auto
             qed
             then show ?thesis
               unfolding *
               apply (subst sum.reindex_nontrivial [OF fin_pd])
               unfolding split_paired_all o_def split_def prod.inject
                apply force+
               done
           qed
           also have "\<dots> = (\<Sum>(x,k) \<in> p. content k *\<^sub>R norm (f x))"
           proof -
             have sumeq: "(\<Sum>i\<in>d. content (l \<inter> i) * norm (f x)) = content l * norm (f x)"
               if "(x, l) \<in> p" for x l
             proof -
               note xl = p'(2-4)[OF that]
               then obtain u v where uv: "l = cbox u v" by blast
               have "(\<Sum>i\<in>d. \<bar>content (l \<inter> i)\<bar>) = (\<Sum>k\<in>d. content (k \<inter> cbox u v))"
                 by (simp add: Int_commute uv)
               also have "\<dots> = sum content {k \<inter> cbox u v| k. k \<in> d}"
               proof -
                 have eq0: "content (k \<inter> cbox u v) = 0"
                   if "k \<in> d" "y \<in> d" "k \<noteq> y" and eq: "k \<inter> cbox u v = y \<inter> cbox u v" for k y
                 proof -
                   from d'(4)[OF that(1)] d'(4)[OF that(2)]
                   obtain \<alpha> \<beta> where \<alpha>: "k \<inter> cbox u v = cbox \<alpha> \<beta>"
                     by (meson Int_interval)
                   have "{} = interior ((k \<inter> y) \<inter> cbox u v)"
                     by (simp add: d'(5) that)
                   also have "\<dots> = interior (y \<inter> (k \<inter> cbox u v))"
                     by auto
                   also have "\<dots> = interior (k \<inter> cbox u v)"
                     unfolding eq by auto
                   finally show ?thesis
                     unfolding \<alpha> content_eq_0_interior ..
                 qed
                 then show ?thesis
                   unfolding Setcompr_eq_image
                   apply (rule sum.reindex_nontrivial [OF \<open>finite d\<close>, unfolded o_def, symmetric])
                   by auto
               qed
               also have "\<dots> = sum content {cbox u v \<inter> k |k. k \<in> d \<and> cbox u v \<inter> k \<noteq> {}}"
                 apply (rule sum.mono_neutral_right)
                 unfolding Setcompr_eq_image
                   apply (rule finite_imageI [OF \<open>finite d\<close>])
                  apply (fastforce simp: inf.commute)+
                 done
               finally show "(\<Sum>i\<in>d. content (l \<inter> i) * norm (f x)) = content l * norm (f x)"
                 unfolding sum_distrib_right[symmetric] real_scaleR_def
                 apply (subst(asm) additive_content_division[OF division_inter_1[OF d(1)]])
                 using xl(2)[unfolded uv] unfolding uv apply auto
                 done
             qed
             show ?thesis
               by (subst sum_Sigma_product[symmetric]) (auto intro!: sumeq sum.cong p' d')
           qed
           finally show ?thesis .
         qed
       qed (rule d)
     qed
   qed
   then show ?thesis
     using absolutely_integrable_onI [OF f has_integral_integrable] has_integral[of _ ?S]
     by blast
 qed
 
 
 lemma bounded_variation_absolutely_integrable:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
   assumes "f integrable_on UNIV"
     and "\<forall>d. d division_of (\<Union>d) \<longrightarrow> sum (\<lambda>k. norm (integral k f)) d \<le> B"
   shows "f absolutely_integrable_on UNIV"
 proof (rule absolutely_integrable_onI, fact)
   let ?f = "\<lambda>d. \<Sum>k\<in>d. norm (integral k f)" and ?D = "{d. d division_of  (\<Union>d)}"
   have D_1: "?D \<noteq> {}"
     by (rule elementary_interval) auto
   have D_2: "bdd_above (?f`?D)"
     by (intro bdd_aboveI2[where M=B] assms(2)[rule_format]) simp
   note D = D_1 D_2
   let ?S = "SUP d\<in>?D. ?f d"
   have "\<And>a b. f integrable_on cbox a b"
     using assms(1) integrable_on_subcbox by blast
   then have f_int: "\<And>a b. f absolutely_integrable_on cbox a b"
     apply (rule bounded_variation_absolutely_integrable_interval[where B=B])
     using assms(2) apply blast
     done
   have "((\<lambda>x. norm (f x)) has_integral ?S) UNIV"
     apply (subst has_integral_alt')
     apply safe
   proof goal_cases
     case (1 a b)
     show ?case
       using f_int[of a b] unfolding absolutely_integrable_on_def by auto
   next
     case prems: (2 e)
     have "\<exists>y\<in>sum (\<lambda>k. norm (integral k f)) ` {d. d division_of \<Union>d}. \<not> y \<le> ?S - e"
     proof (rule ccontr)
       assume "\<not> ?thesis"
       then have "?S \<le> ?S - e"
         by (intro cSUP_least[OF D(1)]) auto
       then show False
         using prems by auto
     qed
     then obtain d K where ddiv: "d division_of \<Union>d" and "K = (\<Sum>k\<in>d. norm (integral k f))"
       "Sup (sum (\<lambda>k. norm (integral k f)) ` {d. d division_of \<Union> d}) - e < K"
       by (auto simp add: image_iff not_le)
     then have d: "Sup (sum (\<lambda>k. norm (integral k f)) ` {d. d division_of \<Union> d}) - e
                   < (\<Sum>k\<in>d. norm (integral k f))"
       by auto
     note d'=division_ofD[OF ddiv]
     have "bounded (\<Union>d)"
       by (rule elementary_bounded,fact)
     from this[unfolded bounded_pos] obtain K where
        K: "0 < K" "\<forall>x\<in>\<Union>d. norm x \<le> K" by auto
     show ?case
     proof (intro conjI impI allI exI)
       fix a b :: 'n
       assume ab: "ball 0 (K + 1) \<subseteq> cbox a b"
       have *: "\<And>s s1. \<lbrakk>?S - e < s1; s1 \<le> s; s < ?S + e\<rbrakk> \<Longrightarrow> \<bar>s - ?S\<bar> < e"
         by arith
       show "norm (integral (cbox a b) (\<lambda>x. if x \<in> UNIV then norm (f x) else 0) - ?S) < e"
         unfolding real_norm_def
       proof (rule * [OF d])
         have "(\<Sum>k\<in>d. norm (integral k f)) \<le> sum (\<lambda>k. integral k (\<lambda>x. norm (f x))) d"
         proof (intro sum_mono)
           fix k assume "k \<in> d"
           with d'(4) f_int show "norm (integral k f) \<le> integral k (\<lambda>x. norm (f x))"
             by (force simp: absolutely_integrable_on_def integral_norm_bound_integral)
         qed
         also have "\<dots> = integral (\<Union>d) (\<lambda>x. norm (f x))"
           apply (rule integral_combine_division_bottomup[OF ddiv, symmetric])
           using absolutely_integrable_on_def d'(4) f_int by blast
         also have "\<dots> \<le> integral (cbox a b) (\<lambda>x. if x \<in> UNIV then norm (f x) else 0)"
         proof -
           have "\<Union>d \<subseteq> cbox a b"
             using K(2) ab by fastforce
           then show ?thesis
             using integrable_on_subdivision[OF ddiv] f_int[of a b] unfolding absolutely_integrable_on_def
             by (auto intro!: integral_subset_le)
         qed
         finally show "(\<Sum>k\<in>d. norm (integral k f))
                       \<le> integral (cbox a b) (\<lambda>x. if x \<in> UNIV then norm (f x) else 0)" .
       next
         have "e/2>0"
           using \<open>e > 0\<close> by auto
         moreover
         have f: "f integrable_on cbox a b" "(\<lambda>x. norm (f x)) integrable_on cbox a b"
           using f_int by (auto simp: absolutely_integrable_on_def)
         ultimately obtain d1 where "gauge d1"
            and d1: "\<And>p. \<lbrakk>p tagged_division_of (cbox a b); d1 fine p\<rbrakk> \<Longrightarrow>
             norm ((\<Sum>(x,k) \<in> p. content k *\<^sub>R norm (f x)) - integral (cbox a b) (\<lambda>x. norm (f x))) < e/2"
           unfolding has_integral_integral has_integral by meson
         obtain d2 where "gauge d2"
           and d2: "\<And>p. \<lbrakk>p tagged_partial_division_of (cbox a b); d2 fine p\<rbrakk> \<Longrightarrow>
             (\<Sum>(x,k) \<in> p. norm (content k *\<^sub>R f x - integral k f)) < e/2"
           by (blast intro: Henstock_lemma [OF f(1) \<open>e/2>0\<close>])
         obtain p where
           p: "p tagged_division_of (cbox a b)" "d1 fine p" "d2 fine p"
           by (rule fine_division_exists [OF gauge_Int [OF \<open>gauge d1\<close> \<open>gauge d2\<close>], of a b])
             (auto simp add: fine_Int)
         have *: "\<And>sf sf' si di. \<lbrakk>sf' = sf; si \<le> ?S; \<bar>sf - si\<bar> < e/2;
                       \<bar>sf' - di\<bar> < e/2\<rbrakk> \<Longrightarrow> di < ?S + e"
           by arith
         have "integral (cbox a b) (\<lambda>x. norm (f x)) < ?S + e"
         proof (rule *)
           show "\<bar>(\<Sum>(x,k)\<in>p. norm (content k *\<^sub>R f x)) - (\<Sum>(x,k)\<in>p. norm (integral k f))\<bar> < e/2"
             unfolding split_def
             apply (rule absdiff_norm_less)
             using d2[of p] p(1,3) apply (auto simp: tagged_division_of_def split_def)
             done
           show "\<bar>(\<Sum>(x,k) \<in> p. content k *\<^sub>R norm (f x)) - integral (cbox a b) (\<lambda>x. norm(f x))\<bar> < e/2"
             using d1[OF p(1,2)] by (simp only: real_norm_def)
           show "(\<Sum>(x,k) \<in> p. content k *\<^sub>R norm (f x)) = (\<Sum>(x,k) \<in> p. norm (content k *\<^sub>R f x))"
             by (auto simp: split_paired_all sum.cong [OF refl])
           show "(\<Sum>(x,k) \<in> p. norm (integral k f)) \<le> ?S"
             using partial_division_of_tagged_division[of p "cbox a b"] p(1)
             apply (subst sum.over_tagged_division_lemma[OF p(1)])
             apply (auto simp: content_eq_0_interior tagged_partial_division_of_def intro!: cSUP_upper2 D)
             done
         qed
         then show "integral (cbox a b) (\<lambda>x. if x \<in> UNIV then norm (f x) else 0) < ?S + e"
           by simp
       qed
     qed (insert K, auto)
   qed
   then show "(\<lambda>x. norm (f x)) integrable_on UNIV"
     by blast
 qed
 
 
 subsection\<open>Outer and inner approximation of measurable sets by well-behaved sets.\<close>
 
 proposition measurable_outer_intervals_bounded:
   assumes "S \<in> lmeasurable" "S \<subseteq> cbox a b" "e > 0"
   obtains \<D>
   where "countable \<D>"
         "\<And>K. K \<in> \<D> \<Longrightarrow> K \<subseteq> cbox a b \<and> K \<noteq> {} \<and> (\<exists>c d. K = cbox c d)"
         "pairwise (\<lambda>A B. interior A \<inter> interior B = {}) \<D>"
         "\<And>u v. cbox u v \<in> \<D> \<Longrightarrow> \<exists>n. \<forall>i \<in> Basis. v \<bullet> i - u \<bullet> i = (b \<bullet> i - a \<bullet> i)/2^n"
         "\<And>K. \<lbrakk>K \<in> \<D>; box a b \<noteq> {}\<rbrakk> \<Longrightarrow> interior K \<noteq> {}"
         "S \<subseteq> \<Union>\<D>" "\<Union>\<D> \<in> lmeasurable" "measure lebesgue (\<Union>\<D>) \<le> measure lebesgue S + e"
 proof (cases "box a b = {}")
   case True
   show ?thesis
   proof (cases "cbox a b = {}")
     case True
     with assms have [simp]: "S = {}"
       by auto
     show ?thesis
     proof
       show "countable {}"
         by simp
     qed (use \<open>e > 0\<close> in auto)
   next
     case False
     show ?thesis
     proof
       show "countable {cbox a b}"
         by simp
       show "\<And>u v. cbox u v \<in> {cbox a b} \<Longrightarrow> \<exists>n. \<forall>i\<in>Basis. v \<bullet> i - u \<bullet> i = (b \<bullet> i - a \<bullet> i)/2 ^ n"
         using False by (force simp: eq_cbox intro: exI [where x=0])
       show "measure lebesgue (\<Union>{cbox a b}) \<le> measure lebesgue S + e"
         using assms by (simp add: sum_content.box_empty_imp [OF True])
     qed (use assms \<open>cbox a b \<noteq> {}\<close> in auto)
   qed
 next
   case False
   let ?\<mu> = "measure lebesgue"
   have "S \<inter> cbox a b \<in> lmeasurable"
     using \<open>S \<in> lmeasurable\<close> by blast
   then have indS_int: "(indicator S has_integral (?\<mu> S)) (cbox a b)"
     by (metis integral_indicator \<open>S \<subseteq> cbox a b\<close> has_integral_integrable_integral inf.orderE integrable_on_indicator)
   with \<open>e > 0\<close> obtain \<gamma> where "gauge \<gamma>" and \<gamma>:
     "\<And>\<D>. \<lbrakk>\<D> tagged_division_of (cbox a b); \<gamma> fine \<D>\<rbrakk> \<Longrightarrow> norm ((\<Sum>(x,K)\<in>\<D>. content(K) *\<^sub>R indicator S x) - ?\<mu> S) < e"
     by (force simp: has_integral)
   have inteq: "integral (cbox a b) (indicat_real S) = integral UNIV (indicator S)"
     using assms by (metis has_integral_iff indS_int lmeasure_integral_UNIV)
   obtain \<D> where \<D>: "countable \<D>"  "\<Union>\<D> \<subseteq> cbox a b"
             and cbox: "\<And>K. K \<in> \<D> \<Longrightarrow> interior K \<noteq> {} \<and> (\<exists>c d. K = cbox c d)"
             and djointish: "pairwise (\<lambda>A B. interior A \<inter> interior B = {}) \<D>"
             and covered: "\<And>K. K \<in> \<D> \<Longrightarrow> \<exists>x \<in> S \<inter> K. K \<subseteq> \<gamma> x"
             and close: "\<And>u v. cbox u v \<in> \<D> \<Longrightarrow> \<exists>n. \<forall>i \<in> Basis. v\<bullet>i - u\<bullet>i = (b\<bullet>i - a\<bullet>i)/2^n"
             and covers: "S \<subseteq> \<Union>\<D>"
     using covering_lemma [of S a b \<gamma>] \<open>gauge \<gamma>\<close> \<open>box a b \<noteq> {}\<close> assms by force
   show ?thesis
   proof
     show "\<And>K. K \<in> \<D> \<Longrightarrow> K \<subseteq> cbox a b \<and> K \<noteq> {} \<and> (\<exists>c d. K = cbox c d)"
       by (meson Sup_le_iff \<D>(2) cbox interior_empty)
     have negl_int: "negligible(K \<inter> L)" if "K \<in> \<D>" "L \<in> \<D>" "K \<noteq> L" for K L
     proof -
       have "interior K \<inter> interior L = {}"
         using djointish pairwiseD that by fastforce
       moreover obtain u v x y where "K = cbox u v" "L = cbox x y"
         using cbox \<open>K \<in> \<D>\<close> \<open>L \<in> \<D>\<close> by blast
       ultimately show ?thesis
         by (simp add: Int_interval box_Int_box negligible_interval(1))
     qed
     have fincase: "\<Union>\<F> \<in> lmeasurable \<and> ?\<mu> (\<Union>\<F>) \<le> ?\<mu> S + e" if "finite \<F>" "\<F> \<subseteq> \<D>" for \<F>
     proof -
       obtain t where t: "\<And>K. K \<in> \<F> \<Longrightarrow> t K \<in> S \<inter> K \<and> K \<subseteq> \<gamma>(t K)"
         using covered \<open>\<F> \<subseteq> \<D>\<close> subsetD by metis
       have "\<forall>K \<in> \<F>. \<forall>L \<in> \<F>. K \<noteq> L \<longrightarrow> interior K \<inter> interior L = {}"
         using that djointish by (simp add: pairwise_def) (metis subsetD)
       with cbox that \<D> have \<F>div: "\<F> division_of (\<Union>\<F>)"
         by (fastforce simp: division_of_def dest: cbox)
       then have 1: "\<Union>\<F> \<in> lmeasurable"
         by blast
       have norme: "\<And>p. \<lbrakk>p tagged_division_of cbox a b; \<gamma> fine p\<rbrakk>
           \<Longrightarrow> norm ((\<Sum>(x,K)\<in>p. content K * indicator S x) - integral (cbox a b) (indicator S)) < e"
         by (auto simp: lmeasure_integral_UNIV assms inteq dest: \<gamma>)
       have "\<forall>x K y L. (x,K) \<in> (\<lambda>K. (t K,K)) ` \<F> \<and> (y,L) \<in> (\<lambda>K. (t K,K)) ` \<F> \<and> (x,K) \<noteq> (y,L) \<longrightarrow>             interior K \<inter> interior L = {}"
         using that djointish  by (clarsimp simp: pairwise_def) (metis subsetD)
       with that \<D> have tagged: "(\<lambda>K. (t K, K)) ` \<F> tagged_partial_division_of cbox a b"
         by (auto simp: tagged_partial_division_of_def dest: t cbox)
       have fine: "\<gamma> fine (\<lambda>K. (t K, K)) ` \<F>"
         using t by (auto simp: fine_def)
       have *: "y \<le> ?\<mu> S \<Longrightarrow> \<bar>x - y\<bar> \<le> e \<Longrightarrow> x \<le> ?\<mu> S + e" for x y
         by arith
       have "?\<mu> (\<Union>\<F>) \<le> ?\<mu> S + e"
       proof (rule *)
         have "(\<Sum>K\<in>\<F>. ?\<mu> (K \<inter> S)) = ?\<mu> (\<Union>C\<in>\<F>. C \<inter> S)"
           apply (rule measure_negligible_finite_Union_image [OF \<open>finite \<F>\<close>, symmetric])
           using \<F>div \<open>S \<in> lmeasurable\<close> apply blast
           unfolding pairwise_def
           by (metis inf.commute inf_sup_aci(3) negligible_Int subsetCE negl_int \<open>\<F> \<subseteq> \<D>\<close>)
         also have "\<dots> = ?\<mu> (\<Union>\<F> \<inter> S)"
           by simp
         also have "\<dots> \<le> ?\<mu> S"
           by (simp add: "1" \<open>S \<in> lmeasurable\<close> fmeasurableD measure_mono_fmeasurable sets.Int)
         finally show "(\<Sum>K\<in>\<F>. ?\<mu> (K \<inter> S)) \<le> ?\<mu> S" .
       next
         have "?\<mu> (\<Union>\<F>) = sum ?\<mu> \<F>"
           by (metis \<F>div content_division)
         also have "\<dots> = (\<Sum>K\<in>\<F>. content K)"
           using \<F>div by (force intro: sum.cong)
         also have "\<dots> = (\<Sum>x\<in>\<F>. content x * indicator S (t x))"
           using t by auto
         finally have eq1: "?\<mu> (\<Union>\<F>) = (\<Sum>x\<in>\<F>. content x * indicator S (t x))" .
         have eq2: "(\<Sum>K\<in>\<F>. ?\<mu> (K \<inter> S)) = (\<Sum>K\<in>\<F>. integral K (indicator S))"
           apply (rule sum.cong [OF refl])
           by (metis integral_indicator \<F>div \<open>S \<in> lmeasurable\<close> division_ofD(4) fmeasurable.Int inf.commute lmeasurable_cbox)
         have "\<bar>\<Sum>(x,K)\<in>(\<lambda>K. (t K, K)) ` \<F>. content K * indicator S x - integral K (indicator S)\<bar> \<le> e"
           using Henstock_lemma_part1 [of "indicator S::'a\<Rightarrow>real", OF _ \<open>e > 0\<close> \<open>gauge \<gamma>\<close> _ tagged fine]
             indS_int norme by auto
         then show "\<bar>?\<mu> (\<Union>\<F>) - (\<Sum>K\<in>\<F>. ?\<mu> (K \<inter> S))\<bar> \<le> e"
           by (simp add: eq1 eq2 comm_monoid_add_class.sum.reindex inj_on_def sum_subtractf)
       qed
       with 1 show ?thesis by blast
     qed
     have "\<Union>\<D> \<in> lmeasurable \<and> ?\<mu> (\<Union>\<D>) \<le> ?\<mu> S + e"
     proof (cases "finite \<D>")
       case True
       with fincase show ?thesis
         by blast
     next
       case False
       let ?T = "from_nat_into \<D>"
       have T: "bij_betw ?T UNIV \<D>"
         by (simp add: False \<D>(1) bij_betw_from_nat_into)
       have TM: "\<And>n. ?T n \<in> lmeasurable"
         by (metis False cbox finite.emptyI from_nat_into lmeasurable_cbox)
       have TN: "\<And>m n. m \<noteq> n \<Longrightarrow> negligible (?T m \<inter> ?T n)"
         by (simp add: False \<D>(1) from_nat_into infinite_imp_nonempty negl_int)
       have TB: "(\<Sum>k\<le>n. ?\<mu> (?T k)) \<le> ?\<mu> S + e" for n
       proof -
         have "(\<Sum>k\<le>n. ?\<mu> (?T k)) = ?\<mu> (\<Union> (?T ` {..n}))"
           by (simp add: pairwise_def TM TN measure_negligible_finite_Union_image)
         also have "?\<mu> (\<Union> (?T ` {..n})) \<le> ?\<mu> S + e"
           using fincase [of "?T ` {..n}"] T by (auto simp: bij_betw_def)
         finally show ?thesis .
       qed
       have "\<Union>\<D> \<in> lmeasurable"
         by (metis lmeasurable_compact T \<D>(2) bij_betw_def cbox compact_cbox countable_Un_Int(1) fmeasurableD fmeasurableI2 rangeI)
       moreover
       have "?\<mu> (\<Union>x. from_nat_into \<D> x) \<le> ?\<mu> S + e"
       proof (rule measure_countable_Union_le [OF TM])
         show "?\<mu> (\<Union>x\<le>n. from_nat_into \<D> x) \<le> ?\<mu> S + e" for n
           by (metis (mono_tags, lifting) False fincase finite.emptyI finite_atMost finite_imageI from_nat_into imageE subsetI)
       qed
       ultimately show ?thesis by (metis T bij_betw_def)
     qed
     then show "\<Union>\<D> \<in> lmeasurable" "measure lebesgue (\<Union>\<D>) \<le> ?\<mu> S + e" by blast+
   qed (use \<D> cbox djointish close covers in auto)
 qed
 
 
 subsection\<open>Transformation of measure by linear maps\<close>
 
 lemma emeasure_lebesgue_ball_conv_unit_ball:
   fixes c :: "'a :: euclidean_space"
   assumes "r \<ge> 0"
   shows "emeasure lebesgue (ball c r) =
            ennreal (r ^ DIM('a)) * emeasure lebesgue (ball (0 :: 'a) 1)"
 proof (cases "r = 0")
   case False
   with assms have r: "r > 0" by auto
   have "emeasure lebesgue ((\<lambda>x. c + x) ` (\<lambda>x. r *\<^sub>R x) ` ball (0 :: 'a) 1) =
           r ^ DIM('a) * emeasure lebesgue (ball (0 :: 'a) 1)"
     unfolding image_image using emeasure_lebesgue_affine[of r c "ball 0 1"] assms
     by (simp add: add_ac)
   also have "(\<lambda>x. r *\<^sub>R x) ` ball 0 1 = ball (0 :: 'a) r"
     using r by (subst ball_scale) auto
   also have "(\<lambda>x. c + x) ` \<dots> = ball c r"
     by (subst image_add_ball) (simp_all add: algebra_simps)
   finally show ?thesis by simp
 qed auto
 
 lemma content_ball_conv_unit_ball:
   fixes c :: "'a :: euclidean_space"
   assumes "r \<ge> 0"
   shows "content (ball c r) = r ^ DIM('a) * content (ball (0 :: 'a) 1)"
 proof -
   have "ennreal (content (ball c r)) = emeasure lebesgue (ball c r)"
     using emeasure_lborel_ball_finite[of c r] by (subst emeasure_eq_ennreal_measure) auto
   also have "\<dots> = ennreal (r ^ DIM('a)) * emeasure lebesgue (ball (0 :: 'a) 1)"
     using assms by (intro emeasure_lebesgue_ball_conv_unit_ball) auto
   also have "\<dots> = ennreal (r ^ DIM('a) * content (ball (0::'a) 1))"
     using emeasure_lborel_ball_finite[of "0::'a" 1] assms
     by (subst emeasure_eq_ennreal_measure) (auto simp: ennreal_mult')
   finally show ?thesis 
     using assms by (subst (asm) ennreal_inj) auto
 qed
 
 lemma measurable_linear_image_interval:
    "linear f \<Longrightarrow> f ` (cbox a b) \<in> lmeasurable"
   by (metis bounded_linear_image linear_linear bounded_cbox closure_bounded_linear_image closure_cbox compact_closure lmeasurable_compact)
 
 proposition measure_linear_sufficient:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'n"
   assumes "linear f" and S: "S \<in> lmeasurable"
     and im: "\<And>a b. measure lebesgue (f ` (cbox a b)) = m * measure lebesgue (cbox a b)"
   shows "f ` S \<in> lmeasurable \<and> m * measure lebesgue S = measure lebesgue (f ` S)"
   using le_less_linear [of 0 m]
 proof
   assume "m < 0"
   then show ?thesis
     using im [of 0 One] by auto
 next
   assume "m \<ge> 0"
   let ?\<mu> = "measure lebesgue"
   show ?thesis
   proof (cases "inj f")
     case False
     then have "?\<mu> (f ` S) = 0"
       using \<open>linear f\<close> negligible_imp_measure0 negligible_linear_singular_image by blast
     then have "m * ?\<mu> (cbox 0 (One)) = 0"
       by (metis False \<open>linear f\<close> cbox_borel content_unit im measure_completion negligible_imp_measure0 negligible_linear_singular_image sets_lborel)
     then show ?thesis
       using \<open>linear f\<close> negligible_linear_singular_image negligible_imp_measure0 False
       by (auto simp: lmeasurable_iff_has_integral negligible_UNIV)
   next
     case True
     then obtain h where "linear h" and hf: "\<And>x. h (f x) = x" and fh: "\<And>x. f (h x) = x"
       using \<open>linear f\<close> linear_injective_isomorphism by blast
     have fBS: "(f ` S) \<in> lmeasurable \<and> m * ?\<mu> S = ?\<mu> (f ` S)"
       if "bounded S" "S \<in> lmeasurable" for S
     proof -
       obtain a b where "S \<subseteq> cbox a b"
         using \<open>bounded S\<close> bounded_subset_cbox_symmetric by metis
       have fUD: "(f ` \<Union>\<D>) \<in> lmeasurable \<and> ?\<mu> (f ` \<Union>\<D>) = (m * ?\<mu> (\<Union>\<D>))"
         if "countable \<D>"
           and cbox: "\<And>K. K \<in> \<D> \<Longrightarrow> K \<subseteq> cbox a b \<and> K \<noteq> {} \<and> (\<exists>c d. K = cbox c d)"
           and intint: "pairwise (\<lambda>A B. interior A \<inter> interior B = {}) \<D>"
         for \<D>
       proof -
         have conv: "\<And>K. K \<in> \<D> \<Longrightarrow> convex K"
           using cbox convex_box(1) by blast
         have neg: "negligible (g ` K \<inter> g ` L)" if "linear g" "K \<in> \<D>" "L \<in> \<D>" "K \<noteq> L"
           for K L and g :: "'n\<Rightarrow>'n"
         proof (cases "inj g")
           case True
           have "negligible (frontier(g ` K \<inter> g ` L) \<union> interior(g ` K \<inter> g ` L))"
           proof (rule negligible_Un)
             show "negligible (frontier (g ` K \<inter> g ` L))"
               by (simp add: negligible_convex_frontier convex_Int conv convex_linear_image that)
           next
             have "\<forall>p N. pairwise p N = (\<forall>Na. (Na::'n set) \<in> N \<longrightarrow> (\<forall>Nb. Nb \<in> N \<and> Na \<noteq> Nb \<longrightarrow> p Na Nb))"
               by (metis pairwise_def)
             then have "interior K \<inter> interior L = {}"
               using intint that(2) that(3) that(4) by presburger
             then show "negligible (interior (g ` K \<inter> g ` L))"
               by (metis True empty_imp_negligible image_Int image_empty interior_Int interior_injective_linear_image that(1))
           qed
           moreover have "g ` K \<inter> g ` L \<subseteq> frontier (g ` K \<inter> g ` L) \<union> interior (g ` K \<inter> g ` L)"
             apply (auto simp: frontier_def)
             using closure_subset contra_subsetD by fastforce+
           ultimately show ?thesis
             by (rule negligible_subset)
         next
           case False
           then show ?thesis
             by (simp add: negligible_Int negligible_linear_singular_image \<open>linear g\<close>)
         qed
         have negf: "negligible ((f ` K) \<inter> (f ` L))"
         and negid: "negligible (K \<inter> L)" if "K \<in> \<D>" "L \<in> \<D>" "K \<noteq> L" for K L
           using neg [OF \<open>linear f\<close>] neg [OF linear_id] that by auto
         show ?thesis
         proof (cases "finite \<D>")
           case True
           then have "?\<mu> (\<Union>x\<in>\<D>. f ` x) = (\<Sum>x\<in>\<D>. ?\<mu> (f ` x))"
             using \<open>linear f\<close> cbox measurable_linear_image_interval negf
             by (blast intro: measure_negligible_finite_Union_image [unfolded pairwise_def])
           also have "\<dots> = (\<Sum>k\<in>\<D>. m * ?\<mu> k)"
             by (metis (no_types, lifting) cbox im sum.cong)
           also have "\<dots> = m * ?\<mu> (\<Union>\<D>)"
             unfolding sum_distrib_left [symmetric]
             by (metis True cbox lmeasurable_cbox measure_negligible_finite_Union [unfolded pairwise_def] negid)
           finally show ?thesis
             by (metis True \<open>linear f\<close> cbox image_Union fmeasurable.finite_UN measurable_linear_image_interval)
         next
           case False
           with \<open>countable \<D>\<close> obtain X :: "nat \<Rightarrow> 'n set" where S: "bij_betw X UNIV \<D>"
             using bij_betw_from_nat_into by blast
           then have eq: "(\<Union>\<D>) = (\<Union>n. X n)" "(f ` \<Union>\<D>) = (\<Union>n. f ` X n)"
             by (auto simp: bij_betw_def)
           have meas: "\<And>K. K \<in> \<D> \<Longrightarrow> K \<in> lmeasurable"
             using cbox by blast
           with S have 1: "\<And>n. X n \<in> lmeasurable"
             by (auto simp: bij_betw_def)
           have 2: "pairwise (\<lambda>m n. negligible (X m \<inter> X n)) UNIV"
             using S unfolding bij_betw_def pairwise_def by (metis injD negid range_eqI)
           have "bounded (\<Union>\<D>)"
             by (meson Sup_least bounded_cbox bounded_subset cbox)
           then have 3: "bounded (\<Union>n. X n)"
             using S unfolding bij_betw_def by blast
           have "(\<Union>n. X n) \<in> lmeasurable"
             by (rule measurable_countable_negligible_Union_bounded [OF 1 2 3])
           with S have f1: "\<And>n. f ` (X n) \<in> lmeasurable"
             unfolding bij_betw_def by (metis assms(1) cbox measurable_linear_image_interval rangeI)
           have f2: "pairwise (\<lambda>m n. negligible (f ` (X m) \<inter> f ` (X n))) UNIV"
             using S unfolding bij_betw_def pairwise_def by (metis injD negf rangeI)
           have "bounded (\<Union>\<D>)"
             by (meson Sup_least bounded_cbox bounded_subset cbox)
           then have f3: "bounded (\<Union>n. f ` X n)"
             using S unfolding bij_betw_def
             by (metis bounded_linear_image linear_linear assms(1) image_Union range_composition)
           have "(\<lambda>n. ?\<mu> (X n)) sums ?\<mu> (\<Union>n. X n)"
             by (rule measure_countable_negligible_Union_bounded [OF 1 2 3])
           have meq: "?\<mu> (\<Union>n. f ` X n) = m * ?\<mu> (\<Union>(X ` UNIV))"
           proof (rule sums_unique2 [OF measure_countable_negligible_Union_bounded [OF f1 f2 f3]])
             have m: "\<And>n. ?\<mu> (f ` X n) = (m * ?\<mu> (X n))"
               using S unfolding bij_betw_def by (metis cbox im rangeI)
             show "(\<lambda>n. ?\<mu> (f ` X n)) sums (m * ?\<mu> (\<Union>(X ` UNIV)))"
               unfolding m
               using measure_countable_negligible_Union_bounded [OF 1 2 3] sums_mult by blast
           qed
           show ?thesis
             using measurable_countable_negligible_Union_bounded [OF f1 f2 f3] meq
             by (auto simp: eq [symmetric])
         qed
       qed
       show ?thesis
         unfolding completion.fmeasurable_measure_inner_outer_le
       proof (intro conjI allI impI)
         fix e :: real
         assume "e > 0"
         have 1: "cbox a b - S \<in> lmeasurable"
           by (simp add: fmeasurable.Diff that)
         have 2: "0 < e / (1 + \<bar>m\<bar>)"
           using \<open>e > 0\<close> by (simp add: field_split_simps abs_add_one_gt_zero)
         obtain \<D>
           where "countable \<D>"
             and cbox: "\<And>K. K \<in> \<D> \<Longrightarrow> K \<subseteq> cbox a b \<and> K \<noteq> {} \<and> (\<exists>c d. K = cbox c d)"
             and intdisj: "pairwise (\<lambda>A B. interior A \<inter> interior B = {}) \<D>"
             and DD: "cbox a b - S \<subseteq> \<Union>\<D>" "\<Union>\<D> \<in> lmeasurable"
             and le: "?\<mu> (\<Union>\<D>) \<le> ?\<mu> (cbox a b - S) + e/(1 + \<bar>m\<bar>)"
           by (rule measurable_outer_intervals_bounded [of "cbox a b - S" a b "e/(1 + \<bar>m\<bar>)"]; use 1 2 pairwise_def in force)
         have meq: "?\<mu> (cbox a b - S) = ?\<mu> (cbox a b) - ?\<mu> S"
           by (simp add: measurable_measure_Diff \<open>S \<subseteq> cbox a b\<close> fmeasurableD that(2))
         show "\<exists>T \<in> lmeasurable. T \<subseteq> f ` S \<and> m * ?\<mu> S - e \<le> ?\<mu> T"
         proof (intro bexI conjI)
           show "f ` (cbox a b) - f ` (\<Union>\<D>) \<subseteq> f ` S"
             using \<open>cbox a b - S \<subseteq> \<Union>\<D>\<close> by force
           have "m * ?\<mu> S - e \<le> m * (?\<mu> S - e / (1 + \<bar>m\<bar>))"
             using \<open>m \<ge> 0\<close> \<open>e > 0\<close> by (simp add: field_simps)
           also have "\<dots> \<le> ?\<mu> (f ` cbox a b) - ?\<mu> (f ` (\<Union>\<D>))"
             using le \<open>m \<ge> 0\<close> \<open>e > 0\<close>
             apply (simp add: im fUD [OF \<open>countable \<D>\<close> cbox intdisj] right_diff_distrib [symmetric])
             apply (rule mult_left_mono; simp add: algebra_simps meq)
             done
           also have "\<dots> = ?\<mu> (f ` cbox a b - f ` \<Union>\<D>)"
             apply (rule measurable_measure_Diff [symmetric])
             apply (simp add: assms(1) measurable_linear_image_interval)
             apply (simp add: \<open>countable \<D>\<close> cbox fUD fmeasurableD intdisj)
              apply (simp add: Sup_le_iff cbox image_mono)
             done
           finally show "m * ?\<mu> S - e \<le> ?\<mu> (f ` cbox a b - f ` \<Union>\<D>)" .
           show "f ` cbox a b - f ` \<Union>\<D> \<in> lmeasurable"
             by (simp add: fUD \<open>countable \<D>\<close> \<open>linear f\<close> cbox fmeasurable.Diff intdisj measurable_linear_image_interval)
         qed
       next
         fix e :: real
         assume "e > 0"
         have em: "0 < e / (1 + \<bar>m\<bar>)"
           using \<open>e > 0\<close> by (simp add: field_split_simps abs_add_one_gt_zero)
         obtain \<D>
           where "countable \<D>"
             and cbox: "\<And>K. K \<in> \<D> \<Longrightarrow> K \<subseteq> cbox a b \<and> K \<noteq> {} \<and> (\<exists>c d. K = cbox c d)"
             and intdisj: "pairwise (\<lambda>A B. interior A \<inter> interior B = {}) \<D>"
             and DD: "S \<subseteq> \<Union>\<D>" "\<Union>\<D> \<in> lmeasurable"
             and le: "?\<mu> (\<Union>\<D>) \<le> ?\<mu> S + e/(1 + \<bar>m\<bar>)"
           by (rule measurable_outer_intervals_bounded [of S a b "e/(1 + \<bar>m\<bar>)"]; use \<open>S \<in> lmeasurable\<close> \<open>S \<subseteq> cbox a b\<close> em in force)
         show "\<exists>U \<in> lmeasurable. f ` S \<subseteq> U \<and> ?\<mu> U \<le> m * ?\<mu> S + e"
         proof (intro bexI conjI)
           show "f ` S \<subseteq> f ` (\<Union>\<D>)"
             by (simp add: DD(1) image_mono)
           have "?\<mu> (f ` \<Union>\<D>) \<le> m * (?\<mu> S + e / (1 + \<bar>m\<bar>))"
             using \<open>m \<ge> 0\<close> le mult_left_mono
             by (auto simp: fUD \<open>countable \<D>\<close> \<open>linear f\<close> cbox fmeasurable.Diff intdisj measurable_linear_image_interval)
           also have "\<dots> \<le> m * ?\<mu> S + e"
             using \<open>m \<ge> 0\<close> \<open>e > 0\<close> by (simp add: fUD [OF \<open>countable \<D>\<close> cbox intdisj] field_simps)
           finally show "?\<mu> (f ` \<Union>\<D>) \<le> m * ?\<mu> S + e" .
           show "f ` \<Union>\<D> \<in> lmeasurable"
             by (simp add: \<open>countable \<D>\<close> cbox fUD intdisj)
         qed
       qed
     qed
     show ?thesis
       unfolding has_measure_limit_iff
     proof (intro allI impI)
       fix e :: real
       assume "e > 0"
       obtain B where "B > 0" and B:
         "\<And>a b. ball 0 B \<subseteq> cbox a b \<Longrightarrow> \<bar>?\<mu> (S \<inter> cbox a b) - ?\<mu> S\<bar> < e / (1 + \<bar>m\<bar>)"
         using has_measure_limit [OF S] \<open>e > 0\<close> by (metis abs_add_one_gt_zero zero_less_divide_iff)
       obtain c d::'n where cd: "ball 0 B \<subseteq> cbox c d"
         by (metis bounded_subset_cbox_symmetric bounded_ball)
       with B have less: "\<bar>?\<mu> (S \<inter> cbox c d) - ?\<mu> S\<bar> < e / (1 + \<bar>m\<bar>)" .
       obtain D where "D > 0" and D: "cbox c d \<subseteq> ball 0 D"
         by (metis bounded_cbox bounded_subset_ballD)
       obtain C where "C > 0" and C: "\<And>x. norm (f x) \<le> C * norm x"
         using linear_bounded_pos \<open>linear f\<close> by blast
       have "f ` S \<inter> cbox a b \<in> lmeasurable \<and>
             \<bar>?\<mu> (f ` S \<inter> cbox a b) - m * ?\<mu> S\<bar> < e"
         if "ball 0 (D*C) \<subseteq> cbox a b" for a b
       proof -
         have "bounded (S \<inter> h ` cbox a b)"
           by (simp add: bounded_linear_image linear_linear \<open>linear h\<close> bounded_Int)
         moreover have Shab: "S \<inter> h ` cbox a b \<in> lmeasurable"
           by (simp add: S \<open>linear h\<close> fmeasurable.Int measurable_linear_image_interval)
         moreover have fim: "f ` (S \<inter> h ` (cbox a b)) = (f ` S) \<inter> cbox a b"
           by (auto simp: hf rev_image_eqI fh)
         ultimately have 1: "(f ` S) \<inter> cbox a b \<in> lmeasurable"
               and 2: "m * ?\<mu> (S \<inter> h ` cbox a b) = ?\<mu> ((f ` S) \<inter> cbox a b)"
           using fBS [of "S \<inter> (h ` (cbox a b))"] by auto
         have *: "\<lbrakk>\<bar>z - m\<bar> < e; z \<le> w; w \<le> m\<rbrakk> \<Longrightarrow> \<bar>w - m\<bar> \<le> e"
           for w z m and e::real by auto
         have meas_adiff: "\<bar>?\<mu> (S \<inter> h ` cbox a b) - ?\<mu> S\<bar> \<le> e / (1 + \<bar>m\<bar>)"
         proof (rule * [OF less])
           show "?\<mu> (S \<inter> cbox c d) \<le> ?\<mu> (S \<inter> h ` cbox a b)"
           proof (rule measure_mono_fmeasurable [OF _ _ Shab])
             have "f ` ball 0 D \<subseteq> ball 0 (C * D)"
               using C \<open>C > 0\<close>
               apply (clarsimp simp: algebra_simps)
               by (meson le_less_trans linordered_comm_semiring_strict_class.comm_mult_strict_left_mono)
             then have "f ` ball 0 D \<subseteq> cbox a b"
               by (metis mult.commute order_trans that)
             have "ball 0 D \<subseteq> h ` cbox a b"
               by (metis \<open>f ` ball 0 D \<subseteq> cbox a b\<close> hf image_subset_iff subsetI)
             then show "S \<inter> cbox c d \<subseteq> S \<inter> h ` cbox a b"
               using D by blast
           next
             show "S \<inter> cbox c d \<in> sets lebesgue"
               using S fmeasurable_cbox by blast
           qed
         next
           show "?\<mu> (S \<inter> h ` cbox a b) \<le> ?\<mu> S"
             by (simp add: S Shab fmeasurableD measure_mono_fmeasurable)
         qed
         have "\<bar>?\<mu> (f ` S \<inter> cbox a b) - m * ?\<mu> S\<bar> \<le> m * e / (1 + \<bar>m\<bar>)"
         proof -
           have mm: "\<bar>m\<bar> = m"
             by (simp add: \<open>0 \<le> m\<close>)
           then have "\<bar>?\<mu> S - ?\<mu> (S \<inter> h ` cbox a b)\<bar> * m \<le> e / (1 + m) * m"
             by (metis (no_types) \<open>0 \<le> m\<close> meas_adiff abs_minus_commute mult_right_mono)
           moreover have "\<forall>r. \<bar>r * m\<bar> = m * \<bar>r\<bar>"
             by (metis \<open>0 \<le> m\<close> abs_mult_pos mult.commute)
           ultimately show ?thesis
             apply (simp add: 2 [symmetric])
             by (metis (no_types) abs_minus_commute mult.commute right_diff_distrib' mm)
         qed
         also have "\<dots> < e"
           using \<open>e > 0\<close> by (cases "m \<ge> 0") (simp_all add: field_simps)
         finally have "\<bar>?\<mu> (f ` S \<inter> cbox a b) - m * ?\<mu> S\<bar> < e" .
         with 1 show ?thesis by auto
       qed
       then show "\<exists>B>0. \<forall>a b. ball 0 B \<subseteq> cbox a b \<longrightarrow>
                          f ` S \<inter> cbox a b \<in> lmeasurable \<and>
                          \<bar>?\<mu> (f ` S \<inter> cbox a b) - m * ?\<mu> S\<bar> < e"
         using \<open>C>0\<close> \<open>D>0\<close> by (metis mult_zero_left real_mult_less_iff1)
     qed
   qed
 qed
 
 subsection\<open>Lemmas about absolute integrability\<close>
 
 lemma absolutely_integrable_linear:
   fixes f :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space"
     and h :: "'n::euclidean_space \<Rightarrow> 'p::euclidean_space"
   shows "f absolutely_integrable_on s \<Longrightarrow> bounded_linear h \<Longrightarrow> (h \<circ> f) absolutely_integrable_on s"
   using integrable_bounded_linear[of h lebesgue "\<lambda>x. indicator s x *\<^sub>R f x"]
   by (simp add: linear_simps[of h] set_integrable_def)
 
 lemma absolutely_integrable_sum:
   fixes f :: "'a \<Rightarrow> 'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
   assumes "finite T" and "\<And>a. a \<in> T \<Longrightarrow> (f a) absolutely_integrable_on S"
   shows "(\<lambda>x. sum (\<lambda>a. f a x) T) absolutely_integrable_on S"
   using assms by induction auto
 
 lemma absolutely_integrable_integrable_bound:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
   assumes le: "\<And>x. x\<in>S \<Longrightarrow> norm (f x) \<le> g x" and f: "f integrable_on S" and g: "g integrable_on S"
   shows "f absolutely_integrable_on S"
     unfolding set_integrable_def
 proof (rule Bochner_Integration.integrable_bound)
   have "g absolutely_integrable_on S"
     unfolding absolutely_integrable_on_def
   proof
     show "(\<lambda>x. norm (g x)) integrable_on S"
       using le norm_ge_zero[of "f _"]
       by (intro integrable_spike_finite[OF _ _ g, of "{}"])
          (auto intro!: abs_of_nonneg intro: order_trans simp del: norm_ge_zero)
   qed fact
   then show "integrable lebesgue (\<lambda>x. indicat_real S x *\<^sub>R g x)"
     by (simp add: set_integrable_def)
   show "(\<lambda>x. indicat_real S x *\<^sub>R f x) \<in> borel_measurable lebesgue"
     using f by (auto intro: has_integral_implies_lebesgue_measurable simp: integrable_on_def)
 qed (use le in \<open>force intro!: always_eventually split: split_indicator\<close>)
 
 corollary absolutely_integrable_on_const [simp]:
   fixes c :: "'a::euclidean_space"
   assumes "S \<in> lmeasurable"
   shows "(\<lambda>x. c) absolutely_integrable_on S"
   by (metis (full_types) assms absolutely_integrable_integrable_bound integrable_on_const order_refl)
 
 lemma absolutely_integrable_continuous:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   shows "continuous_on (cbox a b) f \<Longrightarrow> f absolutely_integrable_on cbox a b"
   using absolutely_integrable_integrable_bound
   by (simp add: absolutely_integrable_on_def continuous_on_norm integrable_continuous)
 
 lemma absolutely_integrable_continuous_real:
   fixes f :: "real \<Rightarrow> 'b::euclidean_space"
   shows "continuous_on {a..b} f \<Longrightarrow> f absolutely_integrable_on {a..b}"
   by (metis absolutely_integrable_continuous box_real(2))
 
 lemma continuous_imp_integrable:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes "continuous_on (cbox a b) f"
   shows "integrable (lebesgue_on (cbox a b)) f"
 proof -
   have "f absolutely_integrable_on cbox a b"
     by (simp add: absolutely_integrable_continuous assms)
   then show ?thesis
     by (simp add: integrable_restrict_space set_integrable_def)
 qed
 
 lemma continuous_imp_integrable_real:
   fixes f :: "real \<Rightarrow> 'b::euclidean_space"
   assumes "continuous_on {a..b} f"
   shows "integrable (lebesgue_on {a..b}) f"
   by (metis assms continuous_imp_integrable interval_cbox)
 
 
 
 subsection \<open>Componentwise\<close>
 
 proposition absolutely_integrable_componentwise_iff:
   shows "f absolutely_integrable_on A \<longleftrightarrow> (\<forall>b\<in>Basis. (\<lambda>x. f x \<bullet> b) absolutely_integrable_on A)"
 proof -
   have *: "(\<lambda>x. norm (f x)) integrable_on A \<longleftrightarrow> (\<forall>b\<in>Basis. (\<lambda>x. norm (f x \<bullet> b)) integrable_on A)"
           if "f integrable_on A"
   proof -
     have 1: "\<And>i. \<lbrakk>(\<lambda>x. norm (f x)) integrable_on A; i \<in> Basis\<rbrakk>
                  \<Longrightarrow> (\<lambda>x. f x \<bullet> i) absolutely_integrable_on A"
       apply (rule absolutely_integrable_integrable_bound [where g = "\<lambda>x. norm(f x)"])
       using Basis_le_norm integrable_component that apply fastforce+
       done
     have 2: "\<forall>i\<in>Basis. (\<lambda>x. \<bar>f x \<bullet> i\<bar>) integrable_on A \<Longrightarrow> f absolutely_integrable_on A"
       apply (rule absolutely_integrable_integrable_bound [where g = "\<lambda>x. \<Sum>i\<in>Basis. norm (f x \<bullet> i)"])
       using norm_le_l1 that apply (force intro: integrable_sum)+
       done
     show ?thesis
       apply auto
        apply (metis (full_types) absolutely_integrable_on_def set_integrable_abs 1)
       apply (metis (full_types) absolutely_integrable_on_def 2)
       done
   qed
   show ?thesis
     unfolding absolutely_integrable_on_def
     by (simp add:  integrable_componentwise_iff [symmetric] ball_conj_distrib * cong: conj_cong)
 qed
 
 lemma absolutely_integrable_componentwise:
   shows "(\<And>b. b \<in> Basis \<Longrightarrow> (\<lambda>x. f x \<bullet> b) absolutely_integrable_on A) \<Longrightarrow> f absolutely_integrable_on A"
   using absolutely_integrable_componentwise_iff by blast
 
 lemma absolutely_integrable_component:
   "f absolutely_integrable_on A \<Longrightarrow> (\<lambda>x. f x \<bullet> (b :: 'b :: euclidean_space)) absolutely_integrable_on A"
   by (drule absolutely_integrable_linear[OF _ bounded_linear_inner_left[of b]]) (simp add: o_def)
 
 
 lemma absolutely_integrable_scaleR_left:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
     assumes "f absolutely_integrable_on S"
   shows "(\<lambda>x. c *\<^sub>R f x) absolutely_integrable_on S"
 proof -
   have "(\<lambda>x. c *\<^sub>R x) o f absolutely_integrable_on S"
     apply (rule absolutely_integrable_linear [OF assms])
     by (simp add: bounded_linear_scaleR_right)
   then show ?thesis
     using assms by blast
 qed
 
 lemma absolutely_integrable_scaleR_right:
   assumes "f absolutely_integrable_on S"
   shows "(\<lambda>x. f x *\<^sub>R c) absolutely_integrable_on S"
   using assms by blast
 
 lemma absolutely_integrable_norm:
   fixes f :: "'a :: euclidean_space \<Rightarrow> 'b :: euclidean_space"
   assumes "f absolutely_integrable_on S"
   shows "(norm o f) absolutely_integrable_on S"
   using assms by (simp add: absolutely_integrable_on_def o_def)
 
 lemma absolutely_integrable_abs:
   fixes f :: "'a :: euclidean_space \<Rightarrow> 'b :: euclidean_space"
   assumes "f absolutely_integrable_on S"
   shows "(\<lambda>x. \<Sum>i\<in>Basis. \<bar>f x \<bullet> i\<bar> *\<^sub>R i) absolutely_integrable_on S"
         (is "?g absolutely_integrable_on S")
 proof -
   have eq: "?g =
         (\<lambda>x. \<Sum>i\<in>Basis. ((\<lambda>y. \<Sum>j\<in>Basis. if j = i then y *\<^sub>R j else 0) \<circ>
                (\<lambda>x. norm(\<Sum>j\<in>Basis. if j = i then (x \<bullet> i) *\<^sub>R j else 0)) \<circ> f) x)"
     by (simp add: sum.delta)
   have *: "(\<lambda>y. \<Sum>j\<in>Basis. if j = i then y *\<^sub>R j else 0) \<circ>
            (\<lambda>x. norm (\<Sum>j\<in>Basis. if j = i then (x \<bullet> i) *\<^sub>R j else 0)) \<circ> f
            absolutely_integrable_on S"
         if "i \<in> Basis" for i
   proof -
     have "bounded_linear (\<lambda>y. \<Sum>j\<in>Basis. if j = i then y *\<^sub>R j else 0)"
       by (simp add: linear_linear algebra_simps linearI)
     moreover have "(\<lambda>x. norm (\<Sum>j\<in>Basis. if j = i then (x \<bullet> i) *\<^sub>R j else 0)) \<circ> f
                    absolutely_integrable_on S"
       unfolding o_def
       apply (rule absolutely_integrable_norm [unfolded o_def])
       using assms \<open>i \<in> Basis\<close>
       apply (auto simp: algebra_simps dest: absolutely_integrable_component[where b=i])
       done
     ultimately show ?thesis
       by (subst comp_assoc) (blast intro: absolutely_integrable_linear)
   qed
   show ?thesis
     apply (rule ssubst [OF eq])
     apply (rule absolutely_integrable_sum)
      apply (force simp: intro!: *)+
     done
 qed
 
 lemma abs_absolutely_integrableI_1:
   fixes f :: "'a :: euclidean_space \<Rightarrow> real"
   assumes f: "f integrable_on A" and "(\<lambda>x. \<bar>f x\<bar>) integrable_on A"
   shows "f absolutely_integrable_on A"
   by (rule absolutely_integrable_integrable_bound [OF _ assms]) auto
 
 
 lemma abs_absolutely_integrableI:
   assumes f: "f integrable_on S" and fcomp: "(\<lambda>x. \<Sum>i\<in>Basis. \<bar>f x \<bullet> i\<bar> *\<^sub>R i) integrable_on S"
   shows "f absolutely_integrable_on S"
 proof -
   have "(\<lambda>x. (f x \<bullet> i) *\<^sub>R i) absolutely_integrable_on S" if "i \<in> Basis" for i
   proof -
     have "(\<lambda>x. \<bar>f x \<bullet> i\<bar>) integrable_on S"
       using assms integrable_component [OF fcomp, where y=i] that by simp
     then have "(\<lambda>x. f x \<bullet> i) absolutely_integrable_on S"
       using abs_absolutely_integrableI_1 f integrable_component by blast
     then show ?thesis
       by (rule absolutely_integrable_scaleR_right)
   qed
   then have "(\<lambda>x. \<Sum>i\<in>Basis. (f x \<bullet> i) *\<^sub>R i) absolutely_integrable_on S"
     by (simp add: absolutely_integrable_sum)
   then show ?thesis
     by (simp add: euclidean_representation)
 qed
 
 
 lemma absolutely_integrable_abs_iff:
    "f absolutely_integrable_on S \<longleftrightarrow>
     f integrable_on S \<and> (\<lambda>x. \<Sum>i\<in>Basis. \<bar>f x \<bullet> i\<bar> *\<^sub>R i) integrable_on S"
     (is "?lhs = ?rhs")
 proof
   assume ?lhs then show ?rhs
     using absolutely_integrable_abs absolutely_integrable_on_def by blast
 next
   assume ?rhs
   moreover
   have "(\<lambda>x. if x \<in> S then \<Sum>i\<in>Basis. \<bar>f x \<bullet> i\<bar> *\<^sub>R i else 0) = (\<lambda>x. \<Sum>i\<in>Basis. \<bar>(if x \<in> S then f x else 0) \<bullet> i\<bar> *\<^sub>R i)"
     by force
   ultimately show ?lhs
     by (simp only: absolutely_integrable_restrict_UNIV [of S, symmetric] integrable_restrict_UNIV [of S, symmetric] abs_absolutely_integrableI)
 qed
 
 lemma absolutely_integrable_max:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
   assumes "f absolutely_integrable_on S" "g absolutely_integrable_on S"
    shows "(\<lambda>x. \<Sum>i\<in>Basis. max (f x \<bullet> i) (g x \<bullet> i) *\<^sub>R i)
             absolutely_integrable_on S"
 proof -
   have "(\<lambda>x. \<Sum>i\<in>Basis. max (f x \<bullet> i) (g x \<bullet> i) *\<^sub>R i) =
         (\<lambda>x. (1/2) *\<^sub>R (f x + g x + (\<Sum>i\<in>Basis. \<bar>f x \<bullet> i - g x \<bullet> i\<bar> *\<^sub>R i)))"
   proof (rule ext)
     fix x
     have "(\<Sum>i\<in>Basis. max (f x \<bullet> i) (g x \<bullet> i) *\<^sub>R i) = (\<Sum>i\<in>Basis. ((f x \<bullet> i + g x \<bullet> i + \<bar>f x \<bullet> i - g x \<bullet> i\<bar>) / 2) *\<^sub>R i)"
       by (force intro: sum.cong)
     also have "... = (1 / 2) *\<^sub>R (\<Sum>i\<in>Basis. (f x \<bullet> i + g x \<bullet> i + \<bar>f x \<bullet> i - g x \<bullet> i\<bar>) *\<^sub>R i)"
       by (simp add: scaleR_right.sum)
     also have "... = (1 / 2) *\<^sub>R (f x + g x + (\<Sum>i\<in>Basis. \<bar>f x \<bullet> i - g x \<bullet> i\<bar> *\<^sub>R i))"
       by (simp add: sum.distrib algebra_simps euclidean_representation)
     finally
     show "(\<Sum>i\<in>Basis. max (f x \<bullet> i) (g x \<bullet> i) *\<^sub>R i) =
          (1 / 2) *\<^sub>R (f x + g x + (\<Sum>i\<in>Basis. \<bar>f x \<bullet> i - g x \<bullet> i\<bar> *\<^sub>R i))" .
   qed
   moreover have "(\<lambda>x. (1 / 2) *\<^sub>R (f x + g x + (\<Sum>i\<in>Basis. \<bar>f x \<bullet> i - g x \<bullet> i\<bar> *\<^sub>R i)))
                  absolutely_integrable_on S"
     apply (intro set_integral_add absolutely_integrable_scaleR_left assms)
     using absolutely_integrable_abs [OF set_integral_diff(1) [OF assms]]
     apply (simp add: algebra_simps)
     done
   ultimately show ?thesis by metis
 qed
 
 corollary absolutely_integrable_max_1:
   fixes f :: "'n::euclidean_space \<Rightarrow> real"
   assumes "f absolutely_integrable_on S" "g absolutely_integrable_on S"
    shows "(\<lambda>x. max (f x) (g x)) absolutely_integrable_on S"
   using absolutely_integrable_max [OF assms] by simp
 
 lemma absolutely_integrable_min:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
   assumes "f absolutely_integrable_on S" "g absolutely_integrable_on S"
    shows "(\<lambda>x. \<Sum>i\<in>Basis. min (f x \<bullet> i) (g x \<bullet> i) *\<^sub>R i)
             absolutely_integrable_on S"
 proof -
   have "(\<lambda>x. \<Sum>i\<in>Basis. min (f x \<bullet> i) (g x \<bullet> i) *\<^sub>R i) =
         (\<lambda>x. (1/2) *\<^sub>R (f x + g x - (\<Sum>i\<in>Basis. \<bar>f x \<bullet> i - g x \<bullet> i\<bar> *\<^sub>R i)))"
   proof (rule ext)
     fix x
     have "(\<Sum>i\<in>Basis. min (f x \<bullet> i) (g x \<bullet> i) *\<^sub>R i) = (\<Sum>i\<in>Basis. ((f x \<bullet> i + g x \<bullet> i - \<bar>f x \<bullet> i - g x \<bullet> i\<bar>) / 2) *\<^sub>R i)"
       by (force intro: sum.cong)
     also have "... = (1 / 2) *\<^sub>R (\<Sum>i\<in>Basis. (f x \<bullet> i + g x \<bullet> i - \<bar>f x \<bullet> i - g x \<bullet> i\<bar>) *\<^sub>R i)"
       by (simp add: scaleR_right.sum)
     also have "... = (1 / 2) *\<^sub>R (f x + g x - (\<Sum>i\<in>Basis. \<bar>f x \<bullet> i - g x \<bullet> i\<bar> *\<^sub>R i))"
       by (simp add: sum.distrib sum_subtractf algebra_simps euclidean_representation)
     finally
     show "(\<Sum>i\<in>Basis. min (f x \<bullet> i) (g x \<bullet> i) *\<^sub>R i) =
          (1 / 2) *\<^sub>R (f x + g x - (\<Sum>i\<in>Basis. \<bar>f x \<bullet> i - g x \<bullet> i\<bar> *\<^sub>R i))" .
   qed
   moreover have "(\<lambda>x. (1 / 2) *\<^sub>R (f x + g x - (\<Sum>i\<in>Basis. \<bar>f x \<bullet> i - g x \<bullet> i\<bar> *\<^sub>R i)))
                  absolutely_integrable_on S"
     apply (intro set_integral_add set_integral_diff absolutely_integrable_scaleR_left assms)
     using absolutely_integrable_abs [OF set_integral_diff(1) [OF assms]]
     apply (simp add: algebra_simps)
     done
   ultimately show ?thesis by metis
 qed
 
 corollary absolutely_integrable_min_1:
   fixes f :: "'n::euclidean_space \<Rightarrow> real"
   assumes "f absolutely_integrable_on S" "g absolutely_integrable_on S"
    shows "(\<lambda>x. min (f x) (g x)) absolutely_integrable_on S"
   using absolutely_integrable_min [OF assms] by simp
 
 lemma nonnegative_absolutely_integrable:
   fixes f :: "'a :: euclidean_space \<Rightarrow> 'b :: euclidean_space"
   assumes "f integrable_on A" and comp: "\<And>x b. \<lbrakk>x \<in> A; b \<in> Basis\<rbrakk> \<Longrightarrow> 0 \<le> f x \<bullet> b"
   shows "f absolutely_integrable_on A"
 proof -
   have "(\<lambda>x. (f x \<bullet> i) *\<^sub>R i) absolutely_integrable_on A" if "i \<in> Basis" for i
   proof -
     have "(\<lambda>x. f x \<bullet> i) integrable_on A"
       by (simp add: assms(1) integrable_component)
     then have "(\<lambda>x. f x \<bullet> i) absolutely_integrable_on A"
       by (metis that comp nonnegative_absolutely_integrable_1)
     then show ?thesis
       by (rule absolutely_integrable_scaleR_right)
   qed
   then have "(\<lambda>x. \<Sum>i\<in>Basis. (f x \<bullet> i) *\<^sub>R i) absolutely_integrable_on A"
     by (simp add: absolutely_integrable_sum)
   then show ?thesis
     by (simp add: euclidean_representation)
 qed
 
 lemma absolutely_integrable_component_ubound:
   fixes f :: "'a :: euclidean_space \<Rightarrow> 'b :: euclidean_space"
   assumes f: "f integrable_on A" and g: "g absolutely_integrable_on A"
       and comp: "\<And>x b. \<lbrakk>x \<in> A; b \<in> Basis\<rbrakk> \<Longrightarrow> f x \<bullet> b \<le> g x \<bullet> b"
   shows "f absolutely_integrable_on A"
 proof -
   have "(\<lambda>x. g x - (g x - f x)) absolutely_integrable_on A"
     apply (rule set_integral_diff [OF g nonnegative_absolutely_integrable])
     using Henstock_Kurzweil_Integration.integrable_diff absolutely_integrable_on_def f g apply blast
     by (simp add: comp inner_diff_left)
   then show ?thesis
     by simp
 qed
 
 lemma absolutely_integrable_component_lbound:
   fixes f :: "'a :: euclidean_space \<Rightarrow> 'b :: euclidean_space"
   assumes f: "f absolutely_integrable_on A" and g: "g integrable_on A"
       and comp: "\<And>x b. \<lbrakk>x \<in> A; b \<in> Basis\<rbrakk> \<Longrightarrow> f x \<bullet> b \<le> g x \<bullet> b"
   shows "g absolutely_integrable_on A"
 proof -
   have "(\<lambda>x. f x + (g x - f x)) absolutely_integrable_on A"
     apply (rule set_integral_add [OF f nonnegative_absolutely_integrable])
     using Henstock_Kurzweil_Integration.integrable_diff absolutely_integrable_on_def f g apply blast
     by (simp add: comp inner_diff_left)
   then show ?thesis
     by simp
 qed
 
 lemma integrable_on_1_iff:
   fixes f :: "'a::euclidean_space \<Rightarrow> real^1"
   shows "f integrable_on S \<longleftrightarrow> (\<lambda>x. f x $ 1) integrable_on S"
   by (auto simp: integrable_componentwise_iff [of f] Basis_vec_def cart_eq_inner_axis)
 
 lemma integral_on_1_eq:
   fixes f :: "'a::euclidean_space \<Rightarrow> real^1"
   shows "integral S f = vec (integral S (\<lambda>x. f x $ 1))"
 by (cases "f integrable_on S") (simp_all add: integrable_on_1_iff vec_eq_iff not_integrable_integral)
 
 lemma absolutely_integrable_on_1_iff:
   fixes f :: "'a::euclidean_space \<Rightarrow> real^1"
   shows "f absolutely_integrable_on S \<longleftrightarrow> (\<lambda>x. f x $ 1) absolutely_integrable_on S"
   unfolding absolutely_integrable_on_def
   by (auto simp: integrable_on_1_iff norm_real)
 
 lemma absolutely_integrable_absolutely_integrable_lbound:
   fixes f :: "'m::euclidean_space \<Rightarrow> real"
   assumes f: "f integrable_on S" and g: "g absolutely_integrable_on S"
     and *: "\<And>x. x \<in> S \<Longrightarrow> g x \<le> f x"
   shows "f absolutely_integrable_on S"
   by (rule absolutely_integrable_component_lbound [OF g f]) (simp add: *)
 
 lemma absolutely_integrable_absolutely_integrable_ubound:
   fixes f :: "'m::euclidean_space \<Rightarrow> real"
   assumes fg: "f integrable_on S" "g absolutely_integrable_on S"
     and *: "\<And>x. x \<in> S \<Longrightarrow> f x \<le> g x"
   shows "f absolutely_integrable_on S"
   by (rule absolutely_integrable_component_ubound [OF fg]) (simp add: *)
 
 lemma has_integral_vec1_I_cbox:
   fixes f :: "real^1 \<Rightarrow> 'a::real_normed_vector"
   assumes "(f has_integral y) (cbox a b)"
   shows "((f \<circ> vec) has_integral y) {a$1..b$1}"
 proof -
   have "((\<lambda>x. f(vec x)) has_integral (1 / 1) *\<^sub>R y) ((\<lambda>x. x $ 1) ` cbox a b)"
   proof (rule has_integral_twiddle)
     show "\<exists>w z::real^1. vec ` cbox u v = cbox w z"
          "content (vec ` cbox u v :: (real^1) set) = 1 * content (cbox u v)" for u v
       unfolding vec_cbox_1_eq
       by (auto simp: content_cbox_if_cart interval_eq_empty_cart)
     show "\<exists>w z. (\<lambda>x. x $ 1) ` cbox u v = cbox w z" for u v :: "real^1"
       using vec_nth_cbox_1_eq by blast
   qed (auto simp: continuous_vec assms)
   then show ?thesis
     by (simp add: o_def)
 qed
 
 lemma has_integral_vec1_I:
   fixes f :: "real^1 \<Rightarrow> 'a::real_normed_vector"
   assumes "(f has_integral y) S"
   shows "(f \<circ> vec has_integral y) ((\<lambda>x. x $ 1) ` S)"
 proof -
   have *: "\<exists>z. ((\<lambda>x. if x \<in> (\<lambda>x. x $ 1) ` S then (f \<circ> vec) x else 0) has_integral z) {a..b} \<and> norm (z - y) < e"
     if int: "\<And>a b. ball 0 B \<subseteq> cbox a b \<Longrightarrow>
                     (\<exists>z. ((\<lambda>x. if x \<in> S then f x else 0) has_integral z) (cbox a b) \<and> norm (z - y) < e)"
       and B: "ball 0 B \<subseteq> {a..b}" for e B a b
   proof -
     have [simp]: "(\<exists>y\<in>S. x = y $ 1) \<longleftrightarrow> vec x \<in> S" for x
       by force
     have B': "ball (0::real^1) B \<subseteq> cbox (vec a) (vec b)"
       using B by (simp add: Basis_vec_def cart_eq_inner_axis [symmetric] mem_box norm_real subset_iff)
     show ?thesis
       using int [OF B'] by (auto simp: image_iff o_def cong: if_cong dest!: has_integral_vec1_I_cbox)
   qed
   show ?thesis
     using assms
     apply (subst has_integral_alt)
     apply (subst (asm) has_integral_alt)
     apply (simp add: has_integral_vec1_I_cbox split: if_split_asm)
     apply (metis vector_one_nth)
     apply (erule all_forward imp_forward asm_rl ex_forward conj_forward)+
     apply (blast intro!: *)
     done
 qed
 
 lemma has_integral_vec1_nth_cbox:
   fixes f :: "real \<Rightarrow> 'a::real_normed_vector"
   assumes "(f has_integral y) {a..b}"
   shows "((\<lambda>x::real^1. f(x$1)) has_integral y) (cbox (vec a) (vec b))"
 proof -
   have "((\<lambda>x::real^1. f(x$1)) has_integral (1 / 1) *\<^sub>R y) (vec ` cbox a b)"
   proof (rule has_integral_twiddle)
     show "\<exists>w z::real. (\<lambda>x. x $ 1) ` cbox u v = cbox w z"
          "content ((\<lambda>x. x $ 1) ` cbox u v) = 1 * content (cbox u v)" for u v::"real^1"
       unfolding vec_cbox_1_eq by (auto simp: content_cbox_if_cart interval_eq_empty_cart)
     show "\<exists>w z::real^1. vec ` cbox u v = cbox w z" for u v :: "real"
       using vec_cbox_1_eq by auto
   qed (auto simp: continuous_vec assms)
   then show ?thesis
     using vec_cbox_1_eq by auto
 qed
 
 lemma has_integral_vec1_D_cbox:
   fixes f :: "real^1 \<Rightarrow> 'a::real_normed_vector"
   assumes "((f \<circ> vec) has_integral y) {a$1..b$1}"
   shows "(f has_integral y) (cbox a b)"
   by (metis (mono_tags, lifting) assms comp_apply has_integral_eq has_integral_vec1_nth_cbox vector_one_nth)
 
 lemma has_integral_vec1_D:
   fixes f :: "real^1 \<Rightarrow> 'a::real_normed_vector"
   assumes "((f \<circ> vec) has_integral y) ((\<lambda>x. x $ 1) ` S)"
   shows "(f has_integral y) S"
 proof -
   have *: "\<exists>z. ((\<lambda>x. if x \<in> S then f x else 0) has_integral z) (cbox a b) \<and> norm (z - y) < e"
     if int: "\<And>a b. ball 0 B \<subseteq> {a..b} \<Longrightarrow>
                              (\<exists>z. ((\<lambda>x. if x \<in> (\<lambda>x. x $ 1) ` S then (f \<circ> vec) x else 0) has_integral z) {a..b} \<and> norm (z - y) < e)"
       and B: "ball 0 B \<subseteq> cbox a b" for e B and a b::"real^1"
   proof -
     have B': "ball 0 B \<subseteq> {a$1..b$1}"
       using B
       apply (simp add: subset_iff Basis_vec_def cart_eq_inner_axis [symmetric] mem_box)
       apply (metis (full_types) norm_real vec_component)
       done
     have eq: "(\<lambda>x. if vec x \<in> S then f (vec x) else 0) = (\<lambda>x. if x \<in> S then f x else 0) \<circ> vec"
       by force
     have [simp]: "(\<exists>y\<in>S. x = y $ 1) \<longleftrightarrow> vec x \<in> S" for x
       by force
     show ?thesis
       using int [OF B'] by (auto simp: image_iff eq cong: if_cong dest!: has_integral_vec1_D_cbox)
 qed
   show ?thesis
     using assms
     apply (subst has_integral_alt)
     apply (subst (asm) has_integral_alt)
     apply (simp add: has_integral_vec1_D_cbox eq_cbox split: if_split_asm, blast)
     apply (intro conjI impI)
      apply (metis vector_one_nth)
     apply (erule thin_rl)
     apply (erule all_forward imp_forward asm_rl ex_forward conj_forward)+
     using * apply blast
     done
 qed
 
 
 lemma integral_vec1_eq:
   fixes f :: "real^1 \<Rightarrow> 'a::real_normed_vector"
   shows "integral S f = integral ((\<lambda>x. x $ 1) ` S) (f \<circ> vec)"
   using has_integral_vec1_I [of f] has_integral_vec1_D [of f]
   by (metis has_integral_iff not_integrable_integral)
 
 lemma absolutely_integrable_drop:
   fixes f :: "real^1 \<Rightarrow> 'b::euclidean_space"
   shows "f absolutely_integrable_on S \<longleftrightarrow> (f \<circ> vec) absolutely_integrable_on (\<lambda>x. x $ 1) ` S"
   unfolding absolutely_integrable_on_def integrable_on_def
 proof safe
   fix y r
   assume "(f has_integral y) S" "((\<lambda>x. norm (f x)) has_integral r) S"
   then show "\<exists>y. (f \<circ> vec has_integral y) ((\<lambda>x. x $ 1) ` S)"
             "\<exists>y. ((\<lambda>x. norm ((f \<circ> vec) x)) has_integral y) ((\<lambda>x. x $ 1) ` S)"
     by (force simp: o_def dest!: has_integral_vec1_I)+
 next
   fix y :: "'b" and r :: "real"
   assume "(f \<circ> vec has_integral y) ((\<lambda>x. x $ 1) ` S)"
          "((\<lambda>x. norm ((f \<circ> vec) x)) has_integral r) ((\<lambda>x. x $ 1) ` S)"
   then show "\<exists>y. (f has_integral y) S"  "\<exists>y. ((\<lambda>x. norm (f x)) has_integral y) S"
     by (force simp: o_def intro: has_integral_vec1_D)+
 qed
 
 subsection \<open>Dominated convergence\<close>
 
 lemma dominated_convergence:
   fixes f :: "nat \<Rightarrow> 'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
   assumes f: "\<And>k. (f k) integrable_on S" and h: "h integrable_on S"
     and le: "\<And>k x. x \<in> S \<Longrightarrow> norm (f k x) \<le> h x"
     and conv: "\<And>x. x \<in> S \<Longrightarrow> (\<lambda>k. f k x) \<longlonglongrightarrow> g x"
   shows "g integrable_on S" "(\<lambda>k. integral S (f k)) \<longlonglongrightarrow> integral S g"
 proof -
   have 3: "h absolutely_integrable_on S"
     unfolding absolutely_integrable_on_def
   proof
     show "(\<lambda>x. norm (h x)) integrable_on S"
     proof (intro integrable_spike_finite[OF _ _ h, of "{}"] ballI)
       fix x assume "x \<in> S - {}" then show "norm (h x) = h x"
         by (metis Diff_empty abs_of_nonneg bot_set_def le norm_ge_zero order_trans real_norm_def)
     qed auto
   qed fact
   have 2: "set_borel_measurable lebesgue S (f k)" for k
     unfolding set_borel_measurable_def
     using f by (auto intro: has_integral_implies_lebesgue_measurable simp: integrable_on_def)
   then have 1: "set_borel_measurable lebesgue S g"
     unfolding set_borel_measurable_def
     by (rule borel_measurable_LIMSEQ_metric) (use conv in \<open>auto split: split_indicator\<close>)
   have 4: "AE x in lebesgue. (\<lambda>i. indicator S x *\<^sub>R f i x) \<longlonglongrightarrow> indicator S x *\<^sub>R g x"
     "AE x in lebesgue. norm (indicator S x *\<^sub>R f k x) \<le> indicator S x *\<^sub>R h x" for k
     using conv le by (auto intro!: always_eventually split: split_indicator)
   have g: "g absolutely_integrable_on S"
     using 1 2 3 4 unfolding set_borel_measurable_def set_integrable_def
     by (rule integrable_dominated_convergence)
   then show "g integrable_on S"
     by (auto simp: absolutely_integrable_on_def)
   have "(\<lambda>k. (LINT x:S|lebesgue. f k x)) \<longlonglongrightarrow> (LINT x:S|lebesgue. g x)"
     unfolding set_borel_measurable_def set_lebesgue_integral_def
     using 1 2 3 4 unfolding set_borel_measurable_def set_lebesgue_integral_def set_integrable_def
     by (rule integral_dominated_convergence)
   then show "(\<lambda>k. integral S (f k)) \<longlonglongrightarrow> integral S g"
     using g absolutely_integrable_integrable_bound[OF le f h]
     by (subst (asm) (1 2) set_lebesgue_integral_eq_integral) auto
 qed
 
 lemma has_integral_dominated_convergence:
   fixes f :: "nat \<Rightarrow> 'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
   assumes "\<And>k. (f k has_integral y k) S" "h integrable_on S"
     "\<And>k. \<forall>x\<in>S. norm (f k x) \<le> h x" "\<forall>x\<in>S. (\<lambda>k. f k x) \<longlonglongrightarrow> g x"
     and x: "y \<longlonglongrightarrow> x"
   shows "(g has_integral x) S"
 proof -
   have int_f: "\<And>k. (f k) integrable_on S"
     using assms by (auto simp: integrable_on_def)
   have "(g has_integral (integral S g)) S"
     by (metis assms(2-4) dominated_convergence(1) has_integral_integral int_f)
   moreover have "integral S g = x"
   proof (rule LIMSEQ_unique)
     show "(\<lambda>i. integral S (f i)) \<longlonglongrightarrow> x"
       using integral_unique[OF assms(1)] x by simp
     show "(\<lambda>i. integral S (f i)) \<longlonglongrightarrow> integral S g"
       by (metis assms(2) assms(3) assms(4) dominated_convergence(2) int_f)
   qed
   ultimately show ?thesis
     by simp
 qed
 
 lemma dominated_convergence_integrable_1:
   fixes f :: "nat \<Rightarrow> 'n::euclidean_space \<Rightarrow> real"
   assumes f: "\<And>k. f k absolutely_integrable_on S"
     and h: "h integrable_on S"
     and normg: "\<And>x. x \<in> S \<Longrightarrow> norm(g x) \<le> (h x)"
     and lim: "\<And>x. x \<in> S \<Longrightarrow> (\<lambda>k. f k x) \<longlonglongrightarrow> g x"
  shows "g integrable_on S"
 proof -
   have habs: "h absolutely_integrable_on S"
     using h normg nonnegative_absolutely_integrable_1 norm_ge_zero order_trans by blast
   let ?f = "\<lambda>n x. (min (max (- h x) (f n x)) (h x))"
   have h0: "h x \<ge> 0" if "x \<in> S" for x
     using normg that by force
   have leh: "norm (?f k x) \<le> h x" if "x \<in> S" for k x
     using h0 that by force
   have limf: "(\<lambda>k. ?f k x) \<longlonglongrightarrow> g x" if "x \<in> S" for x
   proof -
     have "\<And>e y. \<bar>f y x - g x\<bar> < e \<Longrightarrow> \<bar>min (max (- h x) (f y x)) (h x) - g x\<bar> < e"
       using h0 [OF that] normg [OF that] by simp
     then show ?thesis
       using lim [OF that] by (auto simp add: tendsto_iff dist_norm elim!: eventually_mono)
   qed
   show ?thesis
   proof (rule dominated_convergence [of ?f S h g])
     have "(\<lambda>x. - h x) absolutely_integrable_on S"
       using habs unfolding set_integrable_def by auto
     then show "?f k integrable_on S" for k
       by (intro set_lebesgue_integral_eq_integral absolutely_integrable_min_1 absolutely_integrable_max_1 f habs)
   qed (use assms leh limf in auto)
 qed
 
 lemma dominated_convergence_integrable:
   fixes f :: "nat \<Rightarrow> 'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
   assumes f: "\<And>k. f k absolutely_integrable_on S"
     and h: "h integrable_on S"
     and normg: "\<And>x. x \<in> S \<Longrightarrow> norm(g x) \<le> (h x)"
     and lim: "\<And>x. x \<in> S \<Longrightarrow> (\<lambda>k. f k x) \<longlonglongrightarrow> g x"
   shows "g integrable_on S"
   using f
   unfolding integrable_componentwise_iff [of g] absolutely_integrable_componentwise_iff [where f = "f k" for k]
 proof clarify
   fix b :: "'m"
   assume fb [rule_format]: "\<And>k. \<forall>b\<in>Basis. (\<lambda>x. f k x \<bullet> b) absolutely_integrable_on S" and b: "b \<in> Basis"
   show "(\<lambda>x. g x \<bullet> b) integrable_on S"
   proof (rule dominated_convergence_integrable_1 [OF fb h])
     fix x
     assume "x \<in> S"
     show "norm (g x \<bullet> b) \<le> h x"
       using norm_nth_le \<open>x \<in> S\<close> b normg order.trans by blast
     show "(\<lambda>k. f k x \<bullet> b) \<longlonglongrightarrow> g x \<bullet> b"
       using \<open>x \<in> S\<close> b lim tendsto_componentwise_iff by fastforce
   qed (use b in auto)
 qed
 
 lemma dominated_convergence_absolutely_integrable:
   fixes f :: "nat \<Rightarrow> 'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
   assumes f: "\<And>k. f k absolutely_integrable_on S"
     and h: "h integrable_on S"
     and normg: "\<And>x. x \<in> S \<Longrightarrow> norm(g x) \<le> (h x)"
     and lim: "\<And>x. x \<in> S \<Longrightarrow> (\<lambda>k. f k x) \<longlonglongrightarrow> g x"
   shows "g absolutely_integrable_on S"
 proof -
   have "g integrable_on S"
     by (rule dominated_convergence_integrable [OF assms])
   with assms show ?thesis
     by (blast intro:  absolutely_integrable_integrable_bound [where g=h])
 qed
 
 
 proposition integral_countable_UN:
   fixes f :: "real^'m \<Rightarrow> real^'n"
   assumes f: "f absolutely_integrable_on (\<Union>(range s))"
     and s: "\<And>m. s m \<in> sets lebesgue"
   shows "\<And>n. f absolutely_integrable_on (\<Union>m\<le>n. s m)"
     and "(\<lambda>n. integral (\<Union>m\<le>n. s m) f) \<longlonglongrightarrow> integral (\<Union>(s ` UNIV)) f" (is "?F \<longlonglongrightarrow> ?I")
 proof -
   show fU: "f absolutely_integrable_on (\<Union>m\<le>n. s m)" for n
     using assms by (blast intro: set_integrable_subset [OF f])
   have fint: "f integrable_on (\<Union> (range s))"
     using absolutely_integrable_on_def f by blast
   let ?h = "\<lambda>x. if x \<in> \<Union>(s ` UNIV) then norm(f x) else 0"
   have "(\<lambda>n. integral UNIV (\<lambda>x. if x \<in> (\<Union>m\<le>n. s m) then f x else 0))
         \<longlonglongrightarrow> integral UNIV (\<lambda>x. if x \<in> \<Union>(s ` UNIV) then f x else 0)"
   proof (rule dominated_convergence)
     show "(\<lambda>x. if x \<in> (\<Union>m\<le>n. s m) then f x else 0) integrable_on UNIV" for n
       unfolding integrable_restrict_UNIV
       using fU absolutely_integrable_on_def by blast
     show "(\<lambda>x. if x \<in> \<Union>(s ` UNIV) then norm(f x) else 0) integrable_on UNIV"
       by (metis (no_types) absolutely_integrable_on_def f integrable_restrict_UNIV)
     show "\<And>x. (\<lambda>n. if x \<in> (\<Union>m\<le>n. s m) then f x else 0)
          \<longlonglongrightarrow> (if x \<in> \<Union>(s ` UNIV) then f x else 0)"
       by (force intro: tendsto_eventually eventually_sequentiallyI)
   qed auto
   then show "?F \<longlonglongrightarrow> ?I"
     by (simp only: integral_restrict_UNIV)
 qed
 
 
 subsection \<open>Fundamental Theorem of Calculus for the Lebesgue integral\<close>
 
 text \<open>
 
 For the positive integral we replace continuity with Borel-measurability.
 
 \<close>
 
 lemma
   fixes f :: "real \<Rightarrow> real"
   assumes [measurable]: "f \<in> borel_measurable borel"
   assumes f: "\<And>x. x \<in> {a..b} \<Longrightarrow> DERIV F x :> f x" "\<And>x. x \<in> {a..b} \<Longrightarrow> 0 \<le> f x" and "a \<le> b"
   shows nn_integral_FTC_Icc: "(\<integral>\<^sup>+x. ennreal (f x) * indicator {a .. b} x \<partial>lborel) = F b - F a" (is ?nn)
     and has_bochner_integral_FTC_Icc_nonneg:
       "has_bochner_integral lborel (\<lambda>x. f x * indicator {a .. b} x) (F b - F a)" (is ?has)
     and integral_FTC_Icc_nonneg: "(\<integral>x. f x * indicator {a .. b} x \<partial>lborel) = F b - F a" (is ?eq)
     and integrable_FTC_Icc_nonneg: "integrable lborel (\<lambda>x. f x * indicator {a .. b} x)" (is ?int)
 proof -
   have *: "(\<lambda>x. f x * indicator {a..b} x) \<in> borel_measurable borel" "\<And>x. 0 \<le> f x * indicator {a..b} x"
     using f(2) by (auto split: split_indicator)
 
   have F_mono: "a \<le> x \<Longrightarrow> x \<le> y \<Longrightarrow> y \<le> b\<Longrightarrow> F x \<le> F y" for x y
     using f by (intro DERIV_nonneg_imp_nondecreasing[of x y F]) (auto intro: order_trans)
 
   have "(f has_integral F b - F a) {a..b}"
     by (intro fundamental_theorem_of_calculus)
        (auto simp: has_field_derivative_iff_has_vector_derivative[symmetric]
              intro: has_field_derivative_subset[OF f(1)] \<open>a \<le> b\<close>)
   then have i: "((\<lambda>x. f x * indicator {a .. b} x) has_integral F b - F a) UNIV"
     unfolding indicator_def if_distrib[where f="\<lambda>x. a * x" for a]
     by (simp cong del: if_weak_cong del: atLeastAtMost_iff)
   then have nn: "(\<integral>\<^sup>+x. f x * indicator {a .. b} x \<partial>lborel) = F b - F a"
     by (rule nn_integral_has_integral_lborel[OF *])
   then show ?has
     by (rule has_bochner_integral_nn_integral[rotated 3]) (simp_all add: * F_mono \<open>a \<le> b\<close>)
   then show ?eq ?int
     unfolding has_bochner_integral_iff by auto
   show ?nn
     by (subst nn[symmetric])
        (auto intro!: nn_integral_cong simp add: ennreal_mult f split: split_indicator)
 qed
 
 lemma
   fixes f :: "real \<Rightarrow> 'a :: euclidean_space"
   assumes "a \<le> b"
   assumes "\<And>x. a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> (F has_vector_derivative f x) (at x within {a .. b})"
   assumes cont: "continuous_on {a .. b} f"
   shows has_bochner_integral_FTC_Icc:
       "has_bochner_integral lborel (\<lambda>x. indicator {a .. b} x *\<^sub>R f x) (F b - F a)" (is ?has)
     and integral_FTC_Icc: "(\<integral>x. indicator {a .. b} x *\<^sub>R f x \<partial>lborel) = F b - F a" (is ?eq)
 proof -
   let ?f = "\<lambda>x. indicator {a .. b} x *\<^sub>R f x"
   have int: "integrable lborel ?f"
     using borel_integrable_compact[OF _ cont] by auto
   have "(f has_integral F b - F a) {a..b}"
     using assms(1,2) by (intro fundamental_theorem_of_calculus) auto
   moreover
   have "(f has_integral integral\<^sup>L lborel ?f) {a..b}"
     using has_integral_integral_lborel[OF int]
     unfolding indicator_def if_distrib[where f="\<lambda>x. x *\<^sub>R a" for a]
     by (simp cong del: if_weak_cong del: atLeastAtMost_iff)
   ultimately show ?eq
     by (auto dest: has_integral_unique)
   then show ?has
     using int by (auto simp: has_bochner_integral_iff)
 qed
 
 lemma
   fixes f :: "real \<Rightarrow> real"
   assumes "a \<le> b"
   assumes deriv: "\<And>x. a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> DERIV F x :> f x"
   assumes cont: "\<And>x. a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> isCont f x"
   shows has_bochner_integral_FTC_Icc_real:
       "has_bochner_integral lborel (\<lambda>x. f x * indicator {a .. b} x) (F b - F a)" (is ?has)
     and integral_FTC_Icc_real: "(\<integral>x. f x * indicator {a .. b} x \<partial>lborel) = F b - F a" (is ?eq)
 proof -
   have 1: "\<And>x. a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> (F has_vector_derivative f x) (at x within {a .. b})"
     unfolding has_field_derivative_iff_has_vector_derivative[symmetric]
     using deriv by (auto intro: DERIV_subset)
   have 2: "continuous_on {a .. b} f"
     using cont by (intro continuous_at_imp_continuous_on) auto
   show ?has ?eq
     using has_bochner_integral_FTC_Icc[OF \<open>a \<le> b\<close> 1 2] integral_FTC_Icc[OF \<open>a \<le> b\<close> 1 2]
     by (auto simp: mult.commute)
 qed
 
 lemma nn_integral_FTC_atLeast:
   fixes f :: "real \<Rightarrow> real"
   assumes f_borel: "f \<in> borel_measurable borel"
   assumes f: "\<And>x. a \<le> x \<Longrightarrow> DERIV F x :> f x"
   assumes nonneg: "\<And>x. a \<le> x \<Longrightarrow> 0 \<le> f x"
   assumes lim: "(F \<longlongrightarrow> T) at_top"
   shows "(\<integral>\<^sup>+x. ennreal (f x) * indicator {a ..} x \<partial>lborel) = T - F a"
 proof -
   let ?f = "\<lambda>(i::nat) (x::real). ennreal (f x) * indicator {a..a + real i} x"
   let ?fR = "\<lambda>x. ennreal (f x) * indicator {a ..} x"
 
   have F_mono: "a \<le> x \<Longrightarrow> x \<le> y \<Longrightarrow> F x \<le> F y" for x y
     using f nonneg by (intro DERIV_nonneg_imp_nondecreasing[of x y F]) (auto intro: order_trans)
   then have F_le_T: "a \<le> x \<Longrightarrow> F x \<le> T" for x
     by (intro tendsto_lowerbound[OF lim])
        (auto simp: eventually_at_top_linorder)
 
   have "(SUP i. ?f i x) = ?fR x" for x
   proof (rule LIMSEQ_unique[OF LIMSEQ_SUP])
     obtain n where "x - a < real n"
       using reals_Archimedean2[of "x - a"] ..
     then have "eventually (\<lambda>n. ?f n x = ?fR x) sequentially"
       by (auto intro!: eventually_sequentiallyI[where c=n] split: split_indicator)
     then show "(\<lambda>n. ?f n x) \<longlonglongrightarrow> ?fR x"
       by (rule tendsto_eventually)
   qed (auto simp: nonneg incseq_def le_fun_def split: split_indicator)
   then have "integral\<^sup>N lborel ?fR = (\<integral>\<^sup>+ x. (SUP i. ?f i x) \<partial>lborel)"
     by simp
   also have "\<dots> = (SUP i. (\<integral>\<^sup>+ x. ?f i x \<partial>lborel))"
   proof (rule nn_integral_monotone_convergence_SUP)
     show "incseq ?f"
       using nonneg by (auto simp: incseq_def le_fun_def split: split_indicator)
     show "\<And>i. (?f i) \<in> borel_measurable lborel"
       using f_borel by auto
   qed
   also have "\<dots> = (SUP i. ennreal (F (a + real i) - F a))"
     by (subst nn_integral_FTC_Icc[OF f_borel f nonneg]) auto
   also have "\<dots> = T - F a"
   proof (rule LIMSEQ_unique[OF LIMSEQ_SUP])
     have "(\<lambda>x. F (a + real x)) \<longlonglongrightarrow> T"
       apply (rule filterlim_compose[OF lim filterlim_tendsto_add_at_top])
       apply (rule LIMSEQ_const_iff[THEN iffD2, OF refl])
       apply (rule filterlim_real_sequentially)
       done
     then show "(\<lambda>n. ennreal (F (a + real n) - F a)) \<longlonglongrightarrow> ennreal (T - F a)"
       by (simp add: F_mono F_le_T tendsto_diff)
   qed (auto simp: incseq_def intro!: ennreal_le_iff[THEN iffD2] F_mono)
   finally show ?thesis .
 qed
 
 lemma integral_power:
   "a \<le> b \<Longrightarrow> (\<integral>x. x^k * indicator {a..b} x \<partial>lborel) = (b^Suc k - a^Suc k) / Suc k"
 proof (subst integral_FTC_Icc_real)
   fix x show "DERIV (\<lambda>x. x^Suc k / Suc k) x :> x^k"
     by (intro derivative_eq_intros) auto
 qed (auto simp: field_simps simp del: of_nat_Suc)
 
 subsection \<open>Integration by parts\<close>
 
 lemma integral_by_parts_integrable:
   fixes f g F G::"real \<Rightarrow> real"
   assumes "a \<le> b"
   assumes cont_f[intro]: "!!x. a \<le>x \<Longrightarrow> x\<le>b \<Longrightarrow> isCont f x"
   assumes cont_g[intro]: "!!x. a \<le>x \<Longrightarrow> x\<le>b \<Longrightarrow> isCont g x"
   assumes [intro]: "!!x. DERIV F x :> f x"
   assumes [intro]: "!!x. DERIV G x :> g x"
   shows  "integrable lborel (\<lambda>x.((F x) * (g x) + (f x) * (G x)) * indicator {a .. b} x)"
   by (auto intro!: borel_integrable_atLeastAtMost continuous_intros) (auto intro!: DERIV_isCont)
 
 lemma integral_by_parts:
   fixes f g F G::"real \<Rightarrow> real"
   assumes [arith]: "a \<le> b"
   assumes cont_f[intro]: "!!x. a \<le>x \<Longrightarrow> x\<le>b \<Longrightarrow> isCont f x"
   assumes cont_g[intro]: "!!x. a \<le>x \<Longrightarrow> x\<le>b \<Longrightarrow> isCont g x"
   assumes [intro]: "!!x. DERIV F x :> f x"
   assumes [intro]: "!!x. DERIV G x :> g x"
   shows "(\<integral>x. (F x * g x) * indicator {a .. b} x \<partial>lborel)
             =  F b * G b - F a * G a - \<integral>x. (f x * G x) * indicator {a .. b} x \<partial>lborel"
 proof-
   have 0: "(\<integral>x. (F x * g x + f x * G x) * indicator {a .. b} x \<partial>lborel) = F b * G b - F a * G a"
     by (rule integral_FTC_Icc_real, auto intro!: derivative_eq_intros continuous_intros)
       (auto intro!: DERIV_isCont)
 
   have "(\<integral>x. (F x * g x + f x * G x) * indicator {a .. b} x \<partial>lborel) =
     (\<integral>x. (F x * g x) * indicator {a .. b} x \<partial>lborel) + \<integral>x. (f x * G x) * indicator {a .. b} x \<partial>lborel"
     apply (subst Bochner_Integration.integral_add[symmetric])
     apply (auto intro!: borel_integrable_atLeastAtMost continuous_intros)
     by (auto intro!: DERIV_isCont Bochner_Integration.integral_cong split: split_indicator)
 
   thus ?thesis using 0 by auto
 qed
 
 lemma integral_by_parts':
   fixes f g F G::"real \<Rightarrow> real"
   assumes "a \<le> b"
   assumes "!!x. a \<le>x \<Longrightarrow> x\<le>b \<Longrightarrow> isCont f x"
   assumes "!!x. a \<le>x \<Longrightarrow> x\<le>b \<Longrightarrow> isCont g x"
   assumes "!!x. DERIV F x :> f x"
   assumes "!!x. DERIV G x :> g x"
   shows "(\<integral>x. indicator {a .. b} x *\<^sub>R (F x * g x) \<partial>lborel)
             =  F b * G b - F a * G a - \<integral>x. indicator {a .. b} x *\<^sub>R (f x * G x) \<partial>lborel"
   using integral_by_parts[OF assms] by (simp add: ac_simps)
 
 lemma has_bochner_integral_even_function:
   fixes f :: "real \<Rightarrow> 'a :: {banach, second_countable_topology}"
   assumes f: "has_bochner_integral lborel (\<lambda>x. indicator {0..} x *\<^sub>R f x) x"
   assumes even: "\<And>x. f (- x) = f x"
   shows "has_bochner_integral lborel f (2 *\<^sub>R x)"
 proof -
   have indicator: "\<And>x::real. indicator {..0} (- x) = indicator {0..} x"
     by (auto split: split_indicator)
   have "has_bochner_integral lborel (\<lambda>x. indicator {.. 0} x *\<^sub>R f x) x"
     by (subst lborel_has_bochner_integral_real_affine_iff[where c="-1" and t=0])
        (auto simp: indicator even f)
   with f have "has_bochner_integral lborel (\<lambda>x. indicator {0..} x *\<^sub>R f x + indicator {.. 0} x *\<^sub>R f x) (x + x)"
     by (rule has_bochner_integral_add)
   then have "has_bochner_integral lborel f (x + x)"
     by (rule has_bochner_integral_discrete_difference[where X="{0}", THEN iffD1, rotated 4])
        (auto split: split_indicator)
   then show ?thesis
     by (simp add: scaleR_2)
 qed
 
 lemma has_bochner_integral_odd_function:
   fixes f :: "real \<Rightarrow> 'a :: {banach, second_countable_topology}"
   assumes f: "has_bochner_integral lborel (\<lambda>x. indicator {0..} x *\<^sub>R f x) x"
   assumes odd: "\<And>x. f (- x) = - f x"
   shows "has_bochner_integral lborel f 0"
 proof -
   have indicator: "\<And>x::real. indicator {..0} (- x) = indicator {0..} x"
     by (auto split: split_indicator)
   have "has_bochner_integral lborel (\<lambda>x. - indicator {.. 0} x *\<^sub>R f x) x"
     by (subst lborel_has_bochner_integral_real_affine_iff[where c="-1" and t=0])
        (auto simp: indicator odd f)
   from has_bochner_integral_minus[OF this]
   have "has_bochner_integral lborel (\<lambda>x. indicator {.. 0} x *\<^sub>R f x) (- x)"
     by simp
   with f have "has_bochner_integral lborel (\<lambda>x. indicator {0..} x *\<^sub>R f x + indicator {.. 0} x *\<^sub>R f x) (x + - x)"
     by (rule has_bochner_integral_add)
   then have "has_bochner_integral lborel f (x + - x)"
     by (rule has_bochner_integral_discrete_difference[where X="{0}", THEN iffD1, rotated 4])
        (auto split: split_indicator)
   then show ?thesis
     by simp
 qed
 
 lemma has_integral_0_closure_imp_0:
   fixes f :: "'a::euclidean_space \<Rightarrow> real"
   assumes f: "continuous_on (closure S) f"
     and nonneg_interior: "\<And>x. x \<in> S \<Longrightarrow> 0 \<le> f x"
     and pos: "0 < emeasure lborel S"
     and finite: "emeasure lborel S < \<infinity>"
     and regular: "emeasure lborel (closure S) = emeasure lborel S"
     and opn: "open S"
   assumes int: "(f has_integral 0) (closure S)"
   assumes x: "x \<in> closure S"
   shows "f x = 0"
 proof -
   have zero: "emeasure lborel (frontier S) = 0"
     using finite closure_subset regular
     unfolding frontier_def
     by (subst emeasure_Diff) (auto simp: frontier_def interior_open \<open>open S\<close> )
   have nonneg: "0 \<le> f x" if "x \<in> closure S" for x
     using continuous_ge_on_closure[OF f that nonneg_interior] by simp
   have "0 = integral (closure S) f"
     by (blast intro: int sym)
   also
   note intl = has_integral_integrable[OF int]
   have af: "f absolutely_integrable_on (closure S)"
     using nonneg
     by (intro absolutely_integrable_onI intl integrable_eq[OF intl]) simp
   then have "integral (closure S) f = set_lebesgue_integral lebesgue (closure S) f"
     by (intro set_lebesgue_integral_eq_integral(2)[symmetric])
   also have "\<dots> = 0 \<longleftrightarrow> (AE x in lebesgue. indicator (closure S) x *\<^sub>R f x = 0)"
     unfolding set_lebesgue_integral_def
   proof (rule integral_nonneg_eq_0_iff_AE)
     show "integrable lebesgue (\<lambda>x. indicat_real (closure S) x *\<^sub>R f x)"
       by (metis af set_integrable_def)
   qed (use nonneg in \<open>auto simp: indicator_def\<close>)
   also have "\<dots> \<longleftrightarrow> (AE x in lebesgue. x \<in> {x. x \<in> closure S \<longrightarrow> f x = 0})"
     by (auto simp: indicator_def)
   finally have "(AE x in lebesgue. x \<in> {x. x \<in> closure S \<longrightarrow> f x = 0})" by simp
   moreover have "(AE x in lebesgue. x \<in> - frontier S)"
     using zero
     by (auto simp: eventually_ae_filter null_sets_def intro!: exI[where x="frontier S"])
   ultimately have ae: "AE x \<in> S in lebesgue. x \<in> {x \<in> closure S. f x = 0}" (is ?th)
     by eventually_elim (use closure_subset in \<open>auto simp: \<close>)
   have "closed {0::real}" by simp
   with continuous_on_closed_vimage[OF closed_closure, of S f] f
   have "closed (f -` {0} \<inter> closure S)" by blast
   then have "closed {x \<in> closure S. f x = 0}" by (auto simp: vimage_def Int_def conj_commute)
   with \<open>open S\<close> have "x \<in> {x \<in> closure S. f x = 0}" if "x \<in> S" for x using ae that
     by (rule mem_closed_if_AE_lebesgue_open)
   then have "f x = 0" if "x \<in> S" for x using that by auto
   from continuous_constant_on_closure[OF f this \<open>x \<in> closure S\<close>]
   show "f x = 0" .
 qed
 
 lemma has_integral_0_cbox_imp_0:
   fixes f :: "'a::euclidean_space \<Rightarrow> real"
   assumes f: "continuous_on (cbox a b) f"
     and nonneg_interior: "\<And>x. x \<in> box a b \<Longrightarrow> 0 \<le> f x"
   assumes int: "(f has_integral 0) (cbox a b)"
   assumes ne: "box a b \<noteq> {}"
   assumes x: "x \<in> cbox a b"
   shows "f x = 0"
 proof -
   have "0 < emeasure lborel (box a b)"
     using ne x unfolding emeasure_lborel_box_eq
     by (force intro!: prod_pos simp: mem_box algebra_simps)
   then show ?thesis using assms
     by (intro has_integral_0_closure_imp_0[of "box a b" f x])
       (auto simp: emeasure_lborel_box_eq emeasure_lborel_cbox_eq algebra_simps mem_box)
 qed
 
 subsection\<open>Various common equivalent forms of function measurability\<close>
 
 lemma indicator_sum_eq:
   fixes m::real and f :: "'a \<Rightarrow> real"
   assumes "\<bar>m\<bar> \<le> 2 ^ (2*n)" "m/2^n \<le> f x" "f x < (m+1)/2^n" "m \<in> \<int>"
   shows   "(\<Sum>k::real | k \<in> \<int> \<and> \<bar>k\<bar> \<le> 2 ^ (2*n).
             k/2^n * indicator {y. k/2^n \<le> f y \<and> f y < (k+1)/2^n} x)  = m/2^n"
           (is "sum ?f ?S = _")
 proof -
   have "sum ?f ?S = sum (\<lambda>k. k/2^n * indicator {y. k/2^n \<le> f y \<and> f y < (k+1)/2^n} x) {m}"
   proof (rule comm_monoid_add_class.sum.mono_neutral_right)
     show "finite ?S"
       by (rule finite_abs_int_segment)
     show "{m} \<subseteq> {k \<in> \<int>. \<bar>k\<bar> \<le> 2 ^ (2*n)}"
       using assms by auto
     show "\<forall>i\<in>{k \<in> \<int>. \<bar>k\<bar> \<le> 2 ^ (2*n)} - {m}. ?f i = 0"
       using assms by (auto simp: indicator_def Ints_def abs_le_iff field_split_simps)
   qed
   also have "\<dots> = m/2^n"
     using assms by (auto simp: indicator_def not_less)
   finally show ?thesis .
 qed
 
 lemma measurable_on_sf_limit_lemma1:
   fixes f :: "'a::euclidean_space \<Rightarrow> real"
   assumes "\<And>a b. {x \<in> S. a \<le> f x \<and> f x < b} \<in> sets (lebesgue_on S)"
   obtains g where "\<And>n. g n \<in> borel_measurable (lebesgue_on S)"
                   "\<And>n. finite(range (g n))"
                   "\<And>x. (\<lambda>n. g n x) \<longlonglongrightarrow> f x"
 proof
   show "(\<lambda>x. sum (\<lambda>k::real. k/2^n * indicator {y. k/2^n \<le> f y \<and> f y < (k+1)/2^n} x)
                  {k \<in> \<int>. \<bar>k\<bar> \<le> 2 ^ (2*n)}) \<in> borel_measurable (lebesgue_on S)"
         (is "?g \<in> _")  for n
   proof -
     have "\<And>k. \<lbrakk>k \<in> \<int>; \<bar>k\<bar> \<le> 2 ^ (2*n)\<rbrakk>
          \<Longrightarrow> Measurable.pred (lebesgue_on S) (\<lambda>x. k / (2^n) \<le> f x \<and> f x < (k+1) / (2^n))"
       using assms by (force simp: pred_def space_restrict_space)
     then show ?thesis
       by (simp add: field_class.field_divide_inverse)
   qed
   show "finite (range (?g n))" for n
   proof -
     have "range (?g n) \<subseteq> (\<lambda>k. k/2^n) ` {k \<in> \<int>. \<bar>k\<bar> \<le> 2 ^ (2*n)}"
     proof clarify
       fix x
       show "?g n x  \<in> (\<lambda>k. k/2^n) ` {k \<in> \<int>. \<bar>k\<bar> \<le> 2 ^ (2*n)}"
       proof (cases "\<exists>k::real. k \<in> \<int> \<and> \<bar>k\<bar> \<le> 2 ^ (2*n) \<and> k/2^n \<le> (f x) \<and> (f x) < (k+1)/2^n")
         case True
         then show ?thesis
           apply clarify
           by (subst indicator_sum_eq) auto
       next
         case False
         then have "?g n x = 0" by auto
         then show ?thesis by force
       qed
     qed
     moreover have "finite ((\<lambda>k::real. (k/2^n)) ` {k \<in> \<int>. \<bar>k\<bar> \<le> 2 ^ (2*n)})"
       by (simp add: finite_abs_int_segment)
     ultimately show ?thesis
       using finite_subset by blast
   qed
   show "(\<lambda>n. ?g n x) \<longlonglongrightarrow> f x" for x
   proof (rule LIMSEQ_I)
     fix e::real
     assume "e > 0"
     obtain N1 where N1: "\<bar>f x\<bar> < 2 ^ N1"
       using real_arch_pow by fastforce
     obtain N2 where N2: "(1/2) ^ N2 < e"
       using real_arch_pow_inv \<open>e > 0\<close> by force
     have "norm (?g n x - f x) < e" if n: "n \<ge> max N1 N2" for n
     proof -
       define m where "m \<equiv> floor(2^n * (f x))"
       have "1 \<le> \<bar>2^n\<bar> * e"
         using n N2 \<open>e > 0\<close> less_eq_real_def less_le_trans by (fastforce simp add: field_split_simps)
       then have *: "\<lbrakk>x \<le> y; y < x + 1\<rbrakk> \<Longrightarrow> abs(x - y) < \<bar>2^n\<bar> * e" for x y::real
         by linarith
       have "\<bar>2^n\<bar> * \<bar>m/2^n - f x\<bar> = \<bar>2^n * (m/2^n - f x)\<bar>"
         by (simp add: abs_mult)
       also have "\<dots> = \<bar>real_of_int \<lfloor>2^n * f x\<rfloor> - f x * 2^n\<bar>"
         by (simp add: algebra_simps m_def)
       also have "\<dots> < \<bar>2^n\<bar> * e"
         by (rule *; simp add: mult.commute)
       finally have "\<bar>2^n\<bar> * \<bar>m/2^n - f x\<bar> < \<bar>2^n\<bar> * e" .
       then have me: "\<bar>m/2^n - f x\<bar> < e"
         by simp
       have "\<bar>real_of_int m\<bar> \<le> 2 ^ (2*n)"
       proof (cases "f x < 0")
         case True
         then have "-\<lfloor>f x\<rfloor> \<le> \<lfloor>(2::real) ^ N1\<rfloor>"
           using N1 le_floor_iff minus_le_iff by fastforce
         with n True have "\<bar>real_of_int\<lfloor>f x\<rfloor>\<bar> \<le> 2 ^ N1"
           by linarith
         also have "\<dots> \<le> 2^n"
           using n by (simp add: m_def)
         finally have "\<bar>real_of_int \<lfloor>f x\<rfloor>\<bar> * 2^n \<le> 2^n * 2^n"
           by simp
         moreover
         have "\<bar>real_of_int \<lfloor>2^n * f x\<rfloor>\<bar> \<le> \<bar>real_of_int \<lfloor>f x\<rfloor>\<bar> * 2^n"
         proof -
           have "\<bar>real_of_int \<lfloor>2^n * f x\<rfloor>\<bar> = - (real_of_int \<lfloor>2^n * f x\<rfloor>)"
             using True by (simp add: abs_if mult_less_0_iff)
           also have "\<dots> \<le> - (real_of_int (\<lfloor>(2::real) ^ n\<rfloor> * \<lfloor>f x\<rfloor>))"
             using le_mult_floor_Ints [of "(2::real)^n"] by simp
           also have "\<dots> \<le> \<bar>real_of_int \<lfloor>f x\<rfloor>\<bar> * 2^n"
             using True
             by simp
           finally show ?thesis .
         qed
         ultimately show ?thesis
           by (metis (no_types, hide_lams) m_def order_trans power2_eq_square power_even_eq)
       next
         case False
         with n N1 have "f x \<le> 2^n"
           by (simp add: not_less) (meson less_eq_real_def one_le_numeral order_trans power_increasing)
         moreover have "0 \<le> m"
           using False m_def by force
         ultimately show ?thesis
           by (metis abs_of_nonneg floor_mono le_floor_iff m_def of_int_0_le_iff power2_eq_square power_mult real_mult_le_cancel_iff1 zero_less_numeral mult.commute zero_less_power)
       qed
       then have "?g n x = m/2^n"
         by (rule indicator_sum_eq) (auto simp add: m_def field_split_simps, linarith)
       then have "norm (?g n x - f x) = norm (m/2^n - f x)"
         by simp
       also have "\<dots> < e"
         by (simp add: me)
       finally show ?thesis .
     qed
     then show "\<exists>no. \<forall>n\<ge>no. norm (?g n x - f x) < e"
       by blast
   qed
 qed
 
 
 lemma borel_measurable_simple_function_limit:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   shows "f \<in> borel_measurable (lebesgue_on S) \<longleftrightarrow>
          (\<exists>g. (\<forall>n. (g n) \<in> borel_measurable (lebesgue_on S)) \<and>
               (\<forall>n. finite (range (g n))) \<and> (\<forall>x. (\<lambda>n. g n x) \<longlonglongrightarrow> f x))"
 proof -
   have "\<exists>g. (\<forall>n. (g n) \<in> borel_measurable (lebesgue_on S)) \<and>
             (\<forall>n. finite (range (g n))) \<and> (\<forall>x. (\<lambda>n. g n x) \<longlonglongrightarrow> f x)"
        if f: "\<And>a i. i \<in> Basis \<Longrightarrow> {x \<in> S. f x \<bullet> i < a} \<in> sets (lebesgue_on S)"
   proof -
     have "\<exists>g. (\<forall>n. (g n) \<in> borel_measurable (lebesgue_on S)) \<and>
                   (\<forall>n. finite(image (g n) UNIV)) \<and>
                   (\<forall>x. ((\<lambda>n. g n x) \<longlonglongrightarrow> f x \<bullet> i))" if "i \<in> Basis" for i
     proof (rule measurable_on_sf_limit_lemma1 [of S "\<lambda>x. f x \<bullet> i"])
       show "{x \<in> S. a \<le> f x \<bullet> i \<and> f x \<bullet> i < b} \<in> sets (lebesgue_on S)" for a b
       proof -
         have "{x \<in> S. a \<le> f x \<bullet> i \<and> f x \<bullet> i < b} = {x \<in> S. f x \<bullet> i < b} - {x \<in> S. a > f x \<bullet> i}"
           by auto
         also have "\<dots> \<in> sets (lebesgue_on S)"
           using f that by blast
         finally show ?thesis .
       qed
     qed blast
     then obtain g where g:
           "\<And>i n. i \<in> Basis \<Longrightarrow> g i n \<in> borel_measurable (lebesgue_on S)"
           "\<And>i n. i \<in> Basis \<Longrightarrow> finite(range (g i n))"
           "\<And>i x. i \<in> Basis \<Longrightarrow> ((\<lambda>n. g i n x) \<longlonglongrightarrow> f x \<bullet> i)"
       by metis
     show ?thesis
     proof (intro conjI allI exI)
       show "(\<lambda>x. \<Sum>i\<in>Basis. g i n x *\<^sub>R i) \<in> borel_measurable (lebesgue_on S)" for n
         by (intro borel_measurable_sum borel_measurable_scaleR) (auto intro: g)
       show "finite (range (\<lambda>x. \<Sum>i\<in>Basis. g i n x *\<^sub>R i))" for n
       proof -
         have "range (\<lambda>x. \<Sum>i\<in>Basis. g i n x *\<^sub>R i) \<subseteq> (\<lambda>h. \<Sum>i\<in>Basis. h i *\<^sub>R i) ` PiE Basis (\<lambda>i. range (g i n))"
         proof clarify
           fix x
           show "(\<Sum>i\<in>Basis. g i n x *\<^sub>R i) \<in> (\<lambda>h. \<Sum>i\<in>Basis. h i *\<^sub>R i) ` (\<Pi>\<^sub>E i\<in>Basis. range (g i n))"
             by (rule_tac x="\<lambda>i\<in>Basis. g i n x" in image_eqI) auto
         qed
         moreover have "finite(PiE Basis (\<lambda>i. range (g i n)))"
           by (simp add: g finite_PiE)
         ultimately show ?thesis
           by (metis (mono_tags, lifting) finite_surj)
       qed
       show "(\<lambda>n. \<Sum>i\<in>Basis. g i n x *\<^sub>R i) \<longlonglongrightarrow> f x" for x
       proof -
         have "(\<lambda>n. \<Sum>i\<in>Basis. g i n x *\<^sub>R i) \<longlonglongrightarrow> (\<Sum>i\<in>Basis. (f x \<bullet> i) *\<^sub>R i)"
           by (auto intro!: tendsto_sum tendsto_scaleR g)
         moreover have "(\<Sum>i\<in>Basis. (f x \<bullet> i) *\<^sub>R i) = f x"
           using euclidean_representation by blast
         ultimately show ?thesis
           by metis
       qed
     qed
   qed
   moreover have "f \<in> borel_measurable (lebesgue_on S)"
               if meas_g: "\<And>n. g n \<in> borel_measurable (lebesgue_on S)"
                  and fin: "\<And>n. finite (range (g n))"
                  and to_f: "\<And>x. (\<lambda>n. g n x) \<longlonglongrightarrow> f x" for  g
     by (rule borel_measurable_LIMSEQ_metric [OF meas_g to_f])
   ultimately show ?thesis
     using borel_measurable_vimage_halfspace_component_lt by blast
 qed
 
 subsection \<open>Lebesgue sets and continuous images\<close>
 
 proposition lebesgue_regular_inner:
  assumes "S \<in> sets lebesgue"
  obtains K C where "negligible K" "\<And>n::nat. compact(C n)" "S = (\<Union>n. C n) \<union> K"
 proof -
   have "\<exists>T. closed T \<and> T \<subseteq> S \<and> (S - T) \<in> lmeasurable \<and> emeasure lebesgue (S - T) < ennreal ((1/2)^n)" for n
     using sets_lebesgue_inner_closed assms
     by (metis sets_lebesgue_inner_closed zero_less_divide_1_iff zero_less_numeral zero_less_power)
   then obtain C where clo: "\<And>n. closed (C n)" and subS: "\<And>n. C n \<subseteq> S"
     and mea: "\<And>n. (S - C n) \<in> lmeasurable"
     and less: "\<And>n. emeasure lebesgue (S - C n) < ennreal ((1/2)^n)"
     by metis
   have "\<exists>F. (\<forall>n::nat. compact(F n)) \<and> (\<Union>n. F n) = C m" for m::nat
     by (metis clo closed_Union_compact_subsets)
   then obtain D :: "[nat,nat] \<Rightarrow> 'a set" where D: "\<And>m n. compact(D m n)" "\<And>m. (\<Union>n. D m n) = C m"
     by metis
   let ?C = "from_nat_into (\<Union>m. range (D m))"
   have "countable (\<Union>m. range (D m))"
     by blast
   have "range (from_nat_into (\<Union>m. range (D m))) = (\<Union>m. range (D m))"
     using range_from_nat_into by simp
   then have CD: "\<exists>m n. ?C k = D m n"  for k
     by (metis (mono_tags, lifting) UN_iff rangeE range_eqI)
   show thesis
   proof
     show "negligible (S - (\<Union>n. C n))"
     proof (clarsimp simp: negligible_outer_le)
       fix e :: "real"
       assume "e > 0"
       then obtain n where n: "(1/2)^n < e"
         using real_arch_pow_inv [of e "1/2"] by auto
       show "\<exists>T. S - (\<Union>n. C n) \<subseteq> T \<and> T \<in> lmeasurable \<and> measure lebesgue T \<le> e"
       proof (intro exI conjI)
         show "S - (\<Union>n. C n) \<subseteq> S - C n"
           by blast
         show "S - C n \<in> lmeasurable"
           by (simp add: mea)
         show "measure lebesgue (S - C n) \<le> e"
           using less [of n] n
           by (simp add: emeasure_eq_measure2 less_le mea)
       qed
     qed
     show "compact (?C n)" for n
       using CD D by metis
     show "S = (\<Union>n. ?C n) \<union> (S - (\<Union>n. C n))" (is "_ = ?rhs")
     proof
       show "S \<subseteq> ?rhs"
         using D by fastforce
       show "?rhs \<subseteq> S"
         using subS D CD by auto (metis Sup_upper range_eqI subsetCE)
     qed
   qed
 qed
 
 lemma sets_lebesgue_continuous_image:
   assumes T: "T \<in> sets lebesgue" and contf: "continuous_on S f"
     and negim: "\<And>T. \<lbrakk>negligible T; T \<subseteq> S\<rbrakk> \<Longrightarrow> negligible(f ` T)" and "T \<subseteq> S"
  shows "f ` T \<in> sets lebesgue"
 proof -
   obtain K C where "negligible K" and com: "\<And>n::nat. compact(C n)" and Teq: "T = (\<Union>n. C n) \<union> K"
     using lebesgue_regular_inner [OF T] by metis
   then have comf: "\<And>n::nat. compact(f ` C n)"
     by (metis Un_subset_iff Union_upper \<open>T \<subseteq> S\<close> compact_continuous_image contf continuous_on_subset rangeI)
   have "((\<Union>n. f ` C n) \<union> f ` K) \<in> sets lebesgue"
   proof (rule sets.Un)
     have "K \<subseteq> S"
       using Teq \<open>T \<subseteq> S\<close> by blast
     show "(\<Union>n. f ` C n) \<in> sets lebesgue"
     proof (rule sets.countable_Union)
       show "range (\<lambda>n. f ` C n) \<subseteq> sets lebesgue"
         using borel_compact comf by (auto simp: borel_compact)
     qed auto
     show "f ` K \<in> sets lebesgue"
       by (simp add: \<open>K \<subseteq> S\<close> \<open>negligible K\<close> negim negligible_imp_sets)
   qed
   then show ?thesis
     by (simp add: Teq image_Un image_Union)
 qed
 
 lemma differentiable_image_in_sets_lebesgue:
   fixes f :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space"
   assumes S: "S \<in> sets lebesgue" and dim: "DIM('m) \<le> DIM('n)" and f: "f differentiable_on S"
   shows "f`S \<in> sets lebesgue"
 proof (rule sets_lebesgue_continuous_image [OF S])
   show "continuous_on S f"
     by (meson differentiable_imp_continuous_on f)
   show "\<And>T. \<lbrakk>negligible T; T \<subseteq> S\<rbrakk> \<Longrightarrow> negligible (f ` T)"
     using differentiable_on_subset f
     by (auto simp: intro!: negligible_differentiable_image_negligible [OF dim])
 qed auto
 
 lemma sets_lebesgue_on_continuous_image:
   assumes S: "S \<in> sets lebesgue" and X: "X \<in> sets (lebesgue_on S)" and contf: "continuous_on S f"
     and negim: "\<And>T. \<lbrakk>negligible T; T \<subseteq> S\<rbrakk> \<Longrightarrow> negligible(f ` T)"
   shows "f ` X \<in> sets (lebesgue_on (f ` S))"
 proof -
   have "X \<subseteq> S"
     by (metis S X sets.Int_space_eq2 sets_restrict_space_iff)
   moreover have "f ` S \<in> sets lebesgue"
     using S contf negim sets_lebesgue_continuous_image by blast
   moreover have "f ` X \<in> sets lebesgue"
     by (metis S X contf negim sets_lebesgue_continuous_image sets_restrict_space_iff space_restrict_space space_restrict_space2)
   ultimately show ?thesis
     by (auto simp: sets_restrict_space_iff)
 qed
 
 lemma differentiable_image_in_sets_lebesgue_on:
   fixes f :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space"
   assumes S: "S \<in> sets lebesgue" and X: "X \<in> sets (lebesgue_on S)" and dim: "DIM('m) \<le> DIM('n)"
        and f: "f differentiable_on S"
      shows "f ` X \<in> sets (lebesgue_on (f`S))"
 proof (rule sets_lebesgue_on_continuous_image [OF S X])
   show "continuous_on S f"
     by (meson differentiable_imp_continuous_on f)
   show "\<And>T. \<lbrakk>negligible T; T \<subseteq> S\<rbrakk> \<Longrightarrow> negligible (f ` T)"
     using differentiable_on_subset f
     by (auto simp: intro!: negligible_differentiable_image_negligible [OF dim])
 qed
 
 subsection \<open>Affine lemmas\<close>
 
 lemma borel_measurable_affine:
   fixes f :: "'a :: euclidean_space \<Rightarrow> 'b :: euclidean_space"
   assumes f: "f \<in> borel_measurable lebesgue" and "c \<noteq> 0"
   shows "(\<lambda>x. f(t + c *\<^sub>R x)) \<in> borel_measurable lebesgue"
 proof -
   { fix a b
     have "{x. f x \<in> cbox a b} \<in> sets lebesgue"
       using f cbox_borel lebesgue_measurable_vimage_borel by blast
     then have "(\<lambda>x. (x - t) /\<^sub>R c) ` {x. f x \<in> cbox a b} \<in> sets lebesgue"
     proof (rule differentiable_image_in_sets_lebesgue)
       show "(\<lambda>x. (x - t) /\<^sub>R c) differentiable_on {x. f x \<in> cbox a b}"
         unfolding differentiable_on_def differentiable_def
         by (rule \<open>c \<noteq> 0\<close> derivative_eq_intros strip exI | simp)+
     qed auto
     moreover
     have "{x. f(t + c *\<^sub>R x) \<in> cbox a b} = (\<lambda>x. (x-t) /\<^sub>R c) ` {x. f x \<in> cbox a b}"
       using \<open>c \<noteq> 0\<close> by (auto simp: image_def)
     ultimately have "{x. f(t + c *\<^sub>R x) \<in> cbox a b} \<in> sets lebesgue"
       by (auto simp: borel_measurable_vimage_closed_interval) }
   then show ?thesis
     by (subst lebesgue_on_UNIV_eq [symmetric]; auto simp: borel_measurable_vimage_closed_interval)
 qed
     
 lemma lebesgue_integrable_real_affine:
   fixes f :: "real \<Rightarrow> 'a :: euclidean_space"
   assumes f: "integrable lebesgue f" and "c \<noteq> 0"
   shows "integrable lebesgue (\<lambda>x. f(t + c * x))"
 proof -
   have "(\<lambda>x. norm (f x)) \<in> borel_measurable lebesgue"
     by (simp add: borel_measurable_integrable f)
   then show ?thesis
     using assms borel_measurable_affine [of f c]
     unfolding integrable_iff_bounded
     by (subst (asm) nn_integral_real_affine_lebesgue[where c=c and t=t])  (auto simp: ennreal_mult_less_top)
 qed
 
 lemma lebesgue_integrable_real_affine_iff:
   fixes f :: "real \<Rightarrow> 'a :: euclidean_space"
   shows "c \<noteq> 0 \<Longrightarrow> integrable lebesgue (\<lambda>x. f(t + c * x)) \<longleftrightarrow> integrable lebesgue f"
   using lebesgue_integrable_real_affine[of f c t]
         lebesgue_integrable_real_affine[of "\<lambda>x. f(t + c * x)" "1/c" "-t/c"]
   by (auto simp: field_simps)
 
 lemma\<^marker>\<open>tag important\<close> lebesgue_integral_real_affine:
   fixes f :: "real \<Rightarrow> 'a :: euclidean_space" and c :: real
   assumes c: "c \<noteq> 0" shows "(\<integral>x. f x \<partial> lebesgue) = \<bar>c\<bar> *\<^sub>R (\<integral>x. f(t + c * x) \<partial>lebesgue)"
 proof cases
   have "(\<lambda>x. t + c * x) \<in> lebesgue \<rightarrow>\<^sub>M lebesgue"
     using lebesgue_affine_measurable[where c= "\<lambda>x::real. c"] \<open>c \<noteq> 0\<close> by simp
   moreover
   assume "integrable lebesgue f"
   ultimately show ?thesis
     by (subst lebesgue_real_affine[OF c, of t]) (auto simp: integral_density integral_distr)
 next
   assume "\<not> integrable lebesgue f" with c show ?thesis
     by (simp add: lebesgue_integrable_real_affine_iff not_integrable_integral_eq)
 qed
 
 lemma has_bochner_integral_lebesgue_real_affine_iff:
   fixes i :: "'a :: euclidean_space"
   shows "c \<noteq> 0 \<Longrightarrow>
     has_bochner_integral lebesgue f i \<longleftrightarrow>
     has_bochner_integral lebesgue (\<lambda>x. f(t + c * x)) (i /\<^sub>R \<bar>c\<bar>)"
   unfolding has_bochner_integral_iff lebesgue_integrable_real_affine_iff
   by (simp_all add: lebesgue_integral_real_affine[symmetric] divideR_right cong: conj_cong)
 
 lemma has_bochner_integral_reflect_real_lemma[intro]:
   fixes f :: "real \<Rightarrow> 'a::euclidean_space"
   assumes "has_bochner_integral (lebesgue_on {a..b}) f i"
   shows "has_bochner_integral (lebesgue_on {-b..-a}) (\<lambda>x. f(-x)) i"
 proof -
   have eq: "indicat_real {a..b} (- x) *\<^sub>R f(- x) = indicat_real {- b..- a} x *\<^sub>R f(- x)" for x
     by (auto simp: indicator_def)
   have i: "has_bochner_integral lebesgue (\<lambda>x. indicator {a..b} x *\<^sub>R f x) i"
     using assms by (auto simp: has_bochner_integral_restrict_space)
   then have "has_bochner_integral lebesgue (\<lambda>x. indicator {-b..-a} x *\<^sub>R f(-x)) i"
     using has_bochner_integral_lebesgue_real_affine_iff [of "-1" "(\<lambda>x. indicator {a..b} x *\<^sub>R f x)" i 0]
     by (auto simp: eq)
   then show ?thesis
     by (auto simp: has_bochner_integral_restrict_space)
 qed
 
 lemma has_bochner_integral_reflect_real[simp]:
   fixes f :: "real \<Rightarrow> 'a::euclidean_space"
   shows "has_bochner_integral (lebesgue_on {-b..-a}) (\<lambda>x. f(-x)) i \<longleftrightarrow> has_bochner_integral (lebesgue_on {a..b}) f i"
   by (auto simp: dest: has_bochner_integral_reflect_real_lemma)
 
 lemma integrable_reflect_real[simp]:
   fixes f :: "real \<Rightarrow> 'a::euclidean_space"
   shows "integrable (lebesgue_on {-b..-a}) (\<lambda>x. f(-x)) \<longleftrightarrow> integrable (lebesgue_on {a..b}) f"
   by (metis has_bochner_integral_iff has_bochner_integral_reflect_real)
 
 lemma integral_reflect_real[simp]:
   fixes f :: "real \<Rightarrow> 'a::euclidean_space"
   shows "integral\<^sup>L (lebesgue_on {-b .. -a}) (\<lambda>x. f(-x)) = integral\<^sup>L (lebesgue_on {a..b::real}) f"
   using has_bochner_integral_reflect_real [of b a f]
   by (metis has_bochner_integral_iff not_integrable_integral_eq)
 
 subsection\<open>More results on integrability\<close>
 
 lemma integrable_on_all_intervals_UNIV:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::banach"
   assumes intf: "\<And>a b. f integrable_on cbox a b"
     and normf: "\<And>x. norm(f x) \<le> g x" and g: "g integrable_on UNIV"
   shows "f integrable_on UNIV"
 proof -
 have intg: "(\<forall>a b. g integrable_on cbox a b)"
     and gle_e: "\<forall>e>0. \<exists>B>0. \<forall>a b c d.
                     ball 0 B \<subseteq> cbox a b \<and> cbox a b \<subseteq> cbox c d \<longrightarrow>
                     \<bar>integral (cbox a b) g - integral (cbox c d) g\<bar>
                     < e"
     using g
     by (auto simp: integrable_alt_subset [of _ UNIV] intf)
   have le: "norm (integral (cbox a b) f - integral (cbox c d) f) \<le> \<bar>integral (cbox a b) g - integral (cbox c d) g\<bar>"
     if "cbox a b \<subseteq> cbox c d" for a b c d
   proof -
     have "norm (integral (cbox a b) f - integral (cbox c d) f) = norm (integral (cbox c d - cbox a b) f)"
       using intf that by (simp add: norm_minus_commute integral_setdiff)
     also have "\<dots> \<le> integral (cbox c d - cbox a b) g"
     proof (rule integral_norm_bound_integral [OF _ _ normf])
       show "f integrable_on cbox c d - cbox a b" "g integrable_on cbox c d - cbox a b"
         by (meson integrable_integral integrable_setdiff intf intg negligible_setdiff that)+
     qed
     also have "\<dots> = integral (cbox c d) g - integral (cbox a b) g"
       using intg that by (simp add: integral_setdiff)
     also have "\<dots> \<le> \<bar>integral (cbox a b) g - integral (cbox c d) g\<bar>"
       by simp
     finally show ?thesis .
   qed
   show ?thesis
     using gle_e
     apply (simp add: integrable_alt_subset [of _ UNIV] intf)
     apply (erule imp_forward all_forward ex_forward asm_rl)+
     by (meson not_less order_trans le)
 qed
 
 lemma integrable_on_all_intervals_integrable_bound:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::banach"
   assumes intf: "\<And>a b. (\<lambda>x. if x \<in> S then f x else 0) integrable_on cbox a b"
     and normf: "\<And>x. x \<in> S \<Longrightarrow> norm(f x) \<le> g x" and g: "g integrable_on S"
   shows "f integrable_on S"
   using integrable_on_all_intervals_UNIV [OF intf, of "(\<lambda>x. if x \<in> S then g x else 0)"]
   by (simp add: g integrable_restrict_UNIV normf)
 
 lemma measurable_bounded_lemma:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes f: "f \<in> borel_measurable lebesgue" and g: "g integrable_on cbox a b"
     and normf: "\<And>x. x \<in> cbox a b \<Longrightarrow> norm(f x) \<le> g x"
   shows "f integrable_on cbox a b"
 proof -
   have "g absolutely_integrable_on cbox a b"
     by (metis (full_types) add_increasing g le_add_same_cancel1 nonnegative_absolutely_integrable_1 norm_ge_zero normf)
   then have "integrable (lebesgue_on (cbox a b)) g"
     by (simp add: integrable_restrict_space set_integrable_def)
   then have "integrable (lebesgue_on (cbox a b)) f"
   proof (rule Bochner_Integration.integrable_bound)
     show "AE x in lebesgue_on (cbox a b). norm (f x) \<le> norm (g x)"
       by (rule AE_I2) (auto intro: normf order_trans)
   qed (simp add: f measurable_restrict_space1)
   then show ?thesis
     by (simp add: integrable_on_lebesgue_on)
 qed
 
 proposition measurable_bounded_by_integrable_imp_integrable:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes f: "f \<in> borel_measurable (lebesgue_on S)" and g: "g integrable_on S"
     and normf: "\<And>x. x \<in> S \<Longrightarrow> norm(f x) \<le> g x" and S: "S \<in> sets lebesgue"
   shows "f integrable_on S"
 proof (rule integrable_on_all_intervals_integrable_bound [OF _ normf g])
   show "(\<lambda>x. if x \<in> S then f x else 0) integrable_on cbox a b" for a b
   proof (rule measurable_bounded_lemma)
     show "(\<lambda>x. if x \<in> S then f x else 0) \<in> borel_measurable lebesgue"
       by (simp add: S borel_measurable_if f)
     show "(\<lambda>x. if x \<in> S then g x else 0) integrable_on cbox a b"
       by (simp add: g integrable_altD(1))
     show "norm (if x \<in> S then f x else 0) \<le> (if x \<in> S then g x else 0)" for x
       using normf by simp
   qed
 qed
 
 lemma measurable_bounded_by_integrable_imp_lebesgue_integrable:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes f: "f \<in> borel_measurable (lebesgue_on S)" and g: "integrable (lebesgue_on S) g"
     and normf: "\<And>x. x \<in> S \<Longrightarrow> norm(f x) \<le> g x" and S: "S \<in> sets lebesgue"
   shows "integrable (lebesgue_on S) f"
 proof -
   have "f absolutely_integrable_on S"
     by (metis (no_types) S absolutely_integrable_integrable_bound f g integrable_on_lebesgue_on measurable_bounded_by_integrable_imp_integrable normf)
   then show ?thesis
     by (simp add: S integrable_restrict_space set_integrable_def)
 qed
 
 lemma measurable_bounded_by_integrable_imp_integrable_real:
   fixes f :: "'a::euclidean_space \<Rightarrow> real"
   assumes "f \<in> borel_measurable (lebesgue_on S)" "g integrable_on S" "\<And>x. x \<in> S \<Longrightarrow> abs(f x) \<le> g x" "S \<in> sets lebesgue"
   shows "f integrable_on S"
   using measurable_bounded_by_integrable_imp_integrable [of f S g] assms by simp
 
 subsection\<open> Relation between Borel measurability and integrability.\<close>
 
 lemma integrable_imp_measurable_weak:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes "S \<in> sets lebesgue" "f integrable_on S"
   shows "f \<in> borel_measurable (lebesgue_on S)"
   by (metis (mono_tags, lifting) assms has_integral_implies_lebesgue_measurable borel_measurable_restrict_space_iff integrable_on_def sets.Int_space_eq2)
 
 lemma integrable_imp_measurable:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes "f integrable_on S"
   shows "f \<in> borel_measurable (lebesgue_on S)"
 proof -
   have "(UNIV::'a set) \<in> sets lborel"
     by simp
   then show ?thesis
     by (metis (mono_tags, lifting) assms borel_measurable_if_D integrable_imp_measurable_weak integrable_restrict_UNIV lebesgue_on_UNIV_eq sets_lebesgue_on_refl)
 qed
 
 lemma integrable_iff_integrable_on:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes "S \<in> sets lebesgue" "(\<integral>\<^sup>+ x. ennreal (norm (f x)) \<partial>lebesgue_on S) < \<infinity>"
   shows "integrable (lebesgue_on S) f \<longleftrightarrow> f integrable_on S"
   using assms integrable_iff_bounded integrable_imp_measurable integrable_on_lebesgue_on by blast
 
 lemma absolutely_integrable_measurable:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes "S \<in> sets lebesgue"
   shows "f absolutely_integrable_on S \<longleftrightarrow> f \<in> borel_measurable (lebesgue_on S) \<and> integrable (lebesgue_on S) (norm \<circ> f)"
     (is "?lhs = ?rhs")
 proof
   assume L: ?lhs
   then have "f \<in> borel_measurable (lebesgue_on S)"
     by (simp add: absolutely_integrable_on_def integrable_imp_measurable)
   then show ?rhs
     using assms set_integrable_norm [of lebesgue S f] L
     by (simp add: integrable_restrict_space set_integrable_def)
 next
   assume ?rhs then show ?lhs
     using assms integrable_on_lebesgue_on
     by (metis absolutely_integrable_integrable_bound comp_def eq_iff measurable_bounded_by_integrable_imp_integrable)
 qed
 
 lemma absolutely_integrable_measurable_real:
   fixes f :: "'a::euclidean_space \<Rightarrow> real"
   assumes "S \<in> sets lebesgue"
   shows "f absolutely_integrable_on S \<longleftrightarrow>
          f \<in> borel_measurable (lebesgue_on S) \<and> integrable (lebesgue_on S) (\<lambda>x. \<bar>f x\<bar>)"
   by (simp add: absolutely_integrable_measurable assms o_def)
 
 lemma absolutely_integrable_measurable_real':
   fixes f :: "'a::euclidean_space \<Rightarrow> real"
   assumes "S \<in> sets lebesgue"
   shows "f absolutely_integrable_on S \<longleftrightarrow> f \<in> borel_measurable (lebesgue_on S) \<and> (\<lambda>x. \<bar>f x\<bar>) integrable_on S"
   using assms
   apply (auto simp: absolutely_integrable_measurable integrable_on_lebesgue_on)
   apply (simp add: integrable_on_lebesgue_on measurable_bounded_by_integrable_imp_lebesgue_integrable)
   using abs_absolutely_integrableI_1 absolutely_integrable_measurable measurable_bounded_by_integrable_imp_integrable_real by blast
 
 lemma absolutely_integrable_imp_borel_measurable:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes "f absolutely_integrable_on S" "S \<in> sets lebesgue"
   shows "f \<in> borel_measurable (lebesgue_on S)"
   using absolutely_integrable_measurable assms by blast
 
 lemma measurable_bounded_by_integrable_imp_absolutely_integrable:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes "f \<in> borel_measurable (lebesgue_on S)" "S \<in> sets lebesgue"
     and "g integrable_on S" and "\<And>x. x \<in> S \<Longrightarrow> norm(f x) \<le> (g x)"
   shows "f absolutely_integrable_on S"
   using assms absolutely_integrable_integrable_bound measurable_bounded_by_integrable_imp_integrable by blast
 
 proposition negligible_differentiable_vimage:
   fixes f :: "'a \<Rightarrow> 'a::euclidean_space"
   assumes "negligible T"
     and f': "\<And>x. x \<in> S \<Longrightarrow> inj(f' x)"
     and derf: "\<And>x. x \<in> S \<Longrightarrow> (f has_derivative f' x) (at x within S)"
   shows "negligible {x \<in> S. f x \<in> T}"
 proof -
   define U where
     "U \<equiv> \<lambda>n::nat. {x \<in> S. \<forall>y. y \<in> S \<and> norm(y - x) < 1/n
                      \<longrightarrow> norm(y - x) \<le> n * norm(f y - f x)}"
   have "negligible {x \<in> U n. f x \<in> T}" if "n > 0" for n
   proof (subst locally_negligible_alt, clarify)
     fix a
     assume a: "a \<in> U n" and fa: "f a \<in> T"
     define V where "V \<equiv> {x. x \<in> U n \<and> f x \<in> T} \<inter> ball a (1 / n / 2)"
     show "\<exists>V. openin (top_of_set {x \<in> U n. f x \<in> T}) V \<and> a \<in> V \<and> negligible V"
     proof (intro exI conjI)
       have noxy: "norm(x - y) \<le> n * norm(f x - f y)" if "x \<in> V" "y \<in> V" for x y
         using that unfolding U_def V_def mem_Collect_eq Int_iff mem_ball dist_norm
         by (meson norm_triangle_half_r)
       then have "inj_on f V"
         by (force simp: inj_on_def)
       then obtain g where g: "\<And>x. x \<in> V \<Longrightarrow> g(f x) = x"
         by (metis inv_into_f_f)
       have "\<exists>T' B. open T' \<and> f x \<in> T' \<and>
                    (\<forall>y\<in>f ` V \<inter> T \<inter> T'. norm (g y - g (f x)) \<le> B * norm (y - f x))"
         if "f x \<in> T" "x \<in> V" for x
         apply (rule_tac x = "ball (f x) 1" in exI)
         using that noxy by (force simp: g)
       then have "negligible (g ` (f ` V \<inter> T))"
         by (force simp: \<open>negligible T\<close> negligible_Int intro!: negligible_locally_Lipschitz_image)
       moreover have "V \<subseteq> g ` (f ` V \<inter> T)"
         by (force simp: g image_iff V_def)
       ultimately show "negligible V"
         by (rule negligible_subset)
     qed (use a fa V_def that in auto)
   qed
   with negligible_countable_Union have "negligible (\<Union>n \<in> {0<..}. {x. x \<in> U n \<and> f x \<in> T})"
     by auto
   moreover have "{x \<in> S. f x \<in> T} \<subseteq> (\<Union>n \<in> {0<..}. {x. x \<in> U n \<and> f x \<in> T})"
   proof clarsimp
     fix x
     assume "x \<in> S" and "f x \<in> T"
     then obtain inj: "inj(f' x)" and der: "(f has_derivative f' x) (at x within S)"
       using assms by metis
     moreover have "linear(f' x)"
       and eps: "\<And>\<epsilon>. \<epsilon> > 0 \<Longrightarrow> \<exists>\<delta>>0. \<forall>y\<in>S. norm (y - x) < \<delta> \<longrightarrow>
                       norm (f y - f x - f' x (y - x)) \<le> \<epsilon> * norm (y - x)"
       using der by (auto simp: has_derivative_within_alt linear_linear)
     ultimately obtain g where "linear g" and g: "g \<circ> f' x = id"
       using linear_injective_left_inverse by metis
     then obtain B where "B > 0" and B: "\<And>z. B * norm z \<le> norm(f' x z)"
       using linear_invertible_bounded_below_pos \<open>linear (f' x)\<close> by blast
     then obtain i where "i \<noteq> 0" and i: "1 / real i < B"
       by (metis (full_types) inverse_eq_divide real_arch_invD)
     then obtain \<delta> where "\<delta> > 0"
          and \<delta>: "\<And>y. \<lbrakk>y\<in>S; norm (y - x) < \<delta>\<rbrakk> \<Longrightarrow>
                   norm (f y - f x - f' x (y - x)) \<le> (B - 1 / real i) * norm (y - x)"
       using eps [of "B - 1/i"] by auto
     then obtain j where "j \<noteq> 0" and j: "inverse (real j) < \<delta>"
       using real_arch_inverse by blast
     have "norm (y - x)/(max i j) \<le> norm (f y - f x)"
       if "y \<in> S" and less: "norm (y - x) < 1 / (max i j)" for y
     proof -
       have "1 / real (max i j) < \<delta>"
         using j \<open>j \<noteq> 0\<close> \<open>0 < \<delta>\<close>
         by (auto simp: field_split_simps max_mult_distrib_left of_nat_max)
     then have "norm (y - x) < \<delta>"
       using less by linarith
     with \<delta> \<open>y \<in> S\<close> have le: "norm (f y - f x - f' x (y - x)) \<le> B * norm (y - x) - norm (y - x)/i"
       by (auto simp: algebra_simps)
     have *: "\<lbrakk>norm(f - f' - y) \<le> b - c; b \<le> norm y; d \<le> c\<rbrakk> \<Longrightarrow> d \<le> norm(f - f')"
       for b c d and y f f'::'a
       using norm_triangle_ineq3 [of "f - f'" y] by simp
     show ?thesis
       apply (rule * [OF le B])
       using \<open>i \<noteq> 0\<close> \<open>j \<noteq> 0\<close> by (simp add: field_split_simps max_mult_distrib_left of_nat_max less_max_iff_disj)
   qed
   with \<open>x \<in> S\<close> \<open>i \<noteq> 0\<close> \<open>j \<noteq> 0\<close> show "\<exists>n\<in>{0<..}. x \<in> U n"
     by (rule_tac x="max i j" in bexI) (auto simp: field_simps U_def less_max_iff_disj)
 qed
   ultimately show ?thesis
     by (rule negligible_subset)
 qed
 
 lemma absolutely_integrable_Un:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes S: "f absolutely_integrable_on S" and T: "f absolutely_integrable_on T"
   shows "f absolutely_integrable_on (S \<union> T)"
 proof -
   have [simp]: "{x. (if x \<in> A then f x else 0) \<noteq> 0} = {x \<in> A. f x \<noteq> 0}" for A
     by auto
   let ?ST = "{x \<in> S. f x \<noteq> 0} \<inter> {x \<in> T. f x \<noteq> 0}"
   have "?ST \<in> sets lebesgue"
   proof (rule Sigma_Algebra.sets.Int)
     have "f integrable_on S"
       using S absolutely_integrable_on_def by blast
     then have "(\<lambda>x. if x \<in> S then f x else 0) integrable_on UNIV"
       by (simp add: integrable_restrict_UNIV)
     then have borel: "(\<lambda>x. if x \<in> S then f x else 0) \<in> borel_measurable (lebesgue_on UNIV)"
       using integrable_imp_measurable lebesgue_on_UNIV_eq by blast
     then show "{x \<in> S. f x \<noteq> 0} \<in> sets lebesgue"
       unfolding borel_measurable_vimage_open
       by (rule allE [where x = "-{0}"]) auto
   next
     have "f integrable_on T"
       using T absolutely_integrable_on_def by blast
     then have "(\<lambda>x. if x \<in> T then f x else 0) integrable_on UNIV"
       by (simp add: integrable_restrict_UNIV)
     then have borel: "(\<lambda>x. if x \<in> T then f x else 0) \<in> borel_measurable (lebesgue_on UNIV)"
       using integrable_imp_measurable lebesgue_on_UNIV_eq by blast
     then show "{x \<in> T. f x \<noteq> 0} \<in> sets lebesgue"
       unfolding borel_measurable_vimage_open
       by (rule allE [where x = "-{0}"]) auto
   qed
   then have "f absolutely_integrable_on ?ST"
     by (rule set_integrable_subset [OF S]) auto
   then have Int: "(\<lambda>x. if x \<in> ?ST then f x else 0) absolutely_integrable_on UNIV"
     using absolutely_integrable_restrict_UNIV by blast
   have "(\<lambda>x. if x \<in> S then f x else 0) absolutely_integrable_on UNIV"
        "(\<lambda>x. if x \<in> T then f x else 0) absolutely_integrable_on UNIV"
     using S T absolutely_integrable_restrict_UNIV by blast+
   then have "(\<lambda>x. (if x \<in> S then f x else 0) + (if x \<in> T then f x else 0)) absolutely_integrable_on UNIV"
     by (rule set_integral_add)
   then have "(\<lambda>x. ((if x \<in> S then f x else 0) + (if x \<in> T then f x else 0)) - (if x \<in> ?ST then f x else 0)) absolutely_integrable_on UNIV"
     using Int by (rule set_integral_diff)
   then have "(\<lambda>x. if x \<in> S \<union> T then f x else 0) absolutely_integrable_on UNIV"
     by (rule absolutely_integrable_spike) (auto intro: empty_imp_negligible)
   then show ?thesis
     unfolding absolutely_integrable_restrict_UNIV .
 qed
 
 lemma absolutely_integrable_on_combine:
   fixes f :: "real \<Rightarrow> 'a::euclidean_space"
   assumes "f absolutely_integrable_on {a..c}"
     and "f absolutely_integrable_on {c..b}"
     and "a \<le> c"
     and "c \<le> b"
   shows "f absolutely_integrable_on {a..b}"
   by (metis absolutely_integrable_Un assms ivl_disj_un_two_touch(4))
 
 lemma uniform_limit_set_lebesgue_integral_at_top:
   fixes f :: "'a \<Rightarrow> real \<Rightarrow> 'b::{banach, second_countable_topology}"
     and g :: "real \<Rightarrow> real"
   assumes bound: "\<And>x y. x \<in> A \<Longrightarrow> y \<ge> a \<Longrightarrow> norm (f x y) \<le> g y"
   assumes integrable: "set_integrable M {a..} g"
   assumes measurable: "\<And>x. x \<in> A \<Longrightarrow> set_borel_measurable M {a..} (f x)"
   assumes "sets borel \<subseteq> sets M"
   shows   "uniform_limit A (\<lambda>b x. LINT y:{a..b}|M. f x y) (\<lambda>x. LINT y:{a..}|M. f x y) at_top"
 proof (cases "A = {}")
   case False
   then obtain x where x: "x \<in> A" by auto
   have g_nonneg: "g y \<ge> 0" if "y \<ge> a" for y
   proof -
     have "0 \<le> norm (f x y)" by simp
     also have "\<dots> \<le> g y" using bound[OF x that] by simp
     finally show ?thesis .
   qed
 
   have integrable': "set_integrable M {a..} (\<lambda>y. f x y)" if "x \<in> A" for x
     unfolding set_integrable_def
   proof (rule Bochner_Integration.integrable_bound)
     show "integrable M (\<lambda>x. indicator {a..} x * g x)"
       using integrable by (simp add: set_integrable_def)
     show "(\<lambda>y. indicat_real {a..} y *\<^sub>R f x y) \<in> borel_measurable M" using measurable[OF that]
       by (simp add: set_borel_measurable_def)
     show "AE y in M. norm (indicat_real {a..} y *\<^sub>R f x y) \<le> norm (indicat_real {a..} y * g y)"
       using bound[OF that] by (intro AE_I2) (auto simp: indicator_def g_nonneg)
   qed
 
   show ?thesis
   proof (rule uniform_limitI)
     fix e :: real assume e: "e > 0"
     have sets [intro]: "A \<in> sets M" if "A \<in> sets borel" for A
       using that assms by blast
 
     have "((\<lambda>b. LINT y:{a..b}|M. g y) \<longlongrightarrow> (LINT y:{a..}|M. g y)) at_top"
       by (intro tendsto_set_lebesgue_integral_at_top assms sets) auto
     with e obtain b0 :: real where b0: "\<forall>b\<ge>b0. \<bar>(LINT y:{a..}|M. g y) - (LINT y:{a..b}|M. g y)\<bar> < e"
       by (auto simp: tendsto_iff eventually_at_top_linorder dist_real_def abs_minus_commute)
     define b where "b = max a b0"
     have "a \<le> b" by (simp add: b_def)
     from b0 have "\<bar>(LINT y:{a..}|M. g y) - (LINT y:{a..b}|M. g y)\<bar> < e"
       by (auto simp: b_def)
     also have "{a..} = {a..b} \<union> {b<..}" by (auto simp: b_def)
     also have "\<bar>(LINT y:\<dots>|M. g y) - (LINT y:{a..b}|M. g y)\<bar> = \<bar>(LINT y:{b<..}|M. g y)\<bar>"
       using \<open>a \<le> b\<close> by (subst set_integral_Un) (auto intro!: set_integrable_subset[OF integrable])
     also have "(LINT y:{b<..}|M. g y) \<ge> 0"
       using g_nonneg \<open>a \<le> b\<close> unfolding set_lebesgue_integral_def
       by (intro Bochner_Integration.integral_nonneg) (auto simp: indicator_def)
     hence "\<bar>(LINT y:{b<..}|M. g y)\<bar> = (LINT y:{b<..}|M. g y)" by simp
     finally have less: "(LINT y:{b<..}|M. g y) < e" .
 
     have "eventually (\<lambda>b. b \<ge> b0) at_top" by (rule eventually_ge_at_top)
     moreover have "eventually (\<lambda>b. b \<ge> a) at_top" by (rule eventually_ge_at_top)
     ultimately show "eventually (\<lambda>b. \<forall>x\<in>A.
                        dist (LINT y:{a..b}|M. f x y) (LINT y:{a..}|M. f x y) < e) at_top"
     proof eventually_elim
       case (elim b)
       show ?case
       proof
         fix x assume x: "x \<in> A"
         have "dist (LINT y:{a..b}|M. f x y) (LINT y:{a..}|M. f x y) =
                 norm ((LINT y:{a..}|M. f x y) - (LINT y:{a..b}|M. f x y))"
           by (simp add: dist_norm norm_minus_commute)
         also have "{a..} = {a..b} \<union> {b<..}" using elim by auto
         also have "(LINT y:\<dots>|M. f x y) - (LINT y:{a..b}|M. f x y) = (LINT y:{b<..}|M. f x y)"
           using elim x
           by (subst set_integral_Un) (auto intro!: set_integrable_subset[OF integrable'])
         also have "norm \<dots> \<le> (LINT y:{b<..}|M. norm (f x y))" using elim x
           by (intro set_integral_norm_bound set_integrable_subset[OF integrable']) auto
         also have "\<dots> \<le> (LINT y:{b<..}|M. g y)" using elim x bound g_nonneg
           by (intro set_integral_mono set_integrable_norm set_integrable_subset[OF integrable']
                     set_integrable_subset[OF integrable]) auto
         also have "(LINT y:{b<..}|M. g y) \<ge> 0"
           using g_nonneg \<open>a \<le> b\<close> unfolding set_lebesgue_integral_def
           by (intro Bochner_Integration.integral_nonneg) (auto simp: indicator_def)
         hence "(LINT y:{b<..}|M. g y) = \<bar>(LINT y:{b<..}|M. g y)\<bar>" by simp
         also have "\<dots> = \<bar>(LINT y:{a..b} \<union> {b<..}|M. g y) - (LINT y:{a..b}|M. g y)\<bar>"
           using elim by (subst set_integral_Un) (auto intro!: set_integrable_subset[OF integrable])
         also have "{a..b} \<union> {b<..} = {a..}" using elim by auto
         also have "\<bar>(LINT y:{a..}|M. g y) - (LINT y:{a..b}|M. g y)\<bar> < e"
           using b0 elim by blast
         finally show "dist (LINT y:{a..b}|M. f x y) (LINT y:{a..}|M. f x y) < e" .
       qed
     qed
   qed
 qed auto
 
 
 
 subsubsection\<open>Differentiability of inverse function (most basic form)\<close>
 
 proposition has_derivative_inverse_within:
   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::euclidean_space"
   assumes der_f: "(f has_derivative f') (at a within S)"
     and cont_g: "continuous (at (f a) within f ` S) g"
     and "a \<in> S" "linear g'" and id: "g' \<circ> f' = id"
     and gf: "\<And>x. x \<in> S \<Longrightarrow> g(f x) = x"
   shows "(g has_derivative g') (at (f a) within f ` S)"
 proof -
   have [simp]: "g' (f' x) = x" for x
     by (simp add: local.id pointfree_idE)
   have "bounded_linear f'"
     and f': "\<And>e. e>0 \<Longrightarrow> \<exists>d>0. \<forall>y\<in>S. norm (y - a) < d \<longrightarrow>
                         norm (f y - f a - f' (y - a)) \<le> e * norm (y - a)"
     using der_f by (auto simp: has_derivative_within_alt)
   obtain C where "C > 0" and C: "\<And>x. norm (g' x) \<le> C * norm x"
     using linear_bounded_pos [OF \<open>linear g'\<close>] by metis
   obtain B k where "B > 0" "k > 0"
     and Bk: "\<And>x. \<lbrakk>x \<in> S; norm(f x - f a) < k\<rbrakk> \<Longrightarrow> norm(x - a) \<le> B * norm(f x - f a)"
   proof -
     obtain B where "B > 0" and B: "\<And>x. B * norm x \<le> norm (f' x)"
       using linear_inj_bounded_below_pos [of f'] \<open>linear g'\<close> id der_f has_derivative_linear
         linear_invertible_bounded_below_pos by blast
     then obtain d where "d>0"
       and d: "\<And>y. \<lbrakk>y \<in> S; norm (y - a) < d\<rbrakk> \<Longrightarrow>
                     norm (f y - f a - f' (y - a)) \<le> B / 2 * norm (y - a)"
       using f' [of "B/2"] by auto
     then obtain e where "e > 0"
       and e: "\<And>x. \<lbrakk>x \<in> S; norm (f x - f a) < e\<rbrakk> \<Longrightarrow> norm (g (f x) - g (f a)) < d"
       using cont_g by (auto simp: continuous_within_eps_delta dist_norm)
     show thesis
     proof
       show "2/B > 0"
         using \<open>B > 0\<close> by simp
       show "norm (x - a) \<le> 2 / B * norm (f x - f a)"
         if "x \<in> S" "norm (f x - f a) < e" for x
       proof -
         have xa: "norm (x - a) < d"
           using e [OF that] gf by (simp add: \<open>a \<in> S\<close> that)
         have *: "\<lbrakk>norm(y - f') \<le> B / 2 * norm x; B * norm x \<le> norm f'\<rbrakk>
                  \<Longrightarrow> norm y \<ge> B / 2 * norm x" for y f'::'b and x::'a
           using norm_triangle_ineq3 [of y f'] by linarith
         show ?thesis
           using * [OF d [OF \<open>x \<in> S\<close> xa] B] \<open>B > 0\<close> by (simp add: field_simps)
       qed
     qed (use \<open>e > 0\<close> in auto)
   qed
   show ?thesis
     unfolding has_derivative_within_alt
   proof (intro conjI impI allI)
     show "bounded_linear g'"
       using \<open>linear g'\<close> by (simp add: linear_linear)
   next
     fix e :: "real"
     assume "e > 0"
     then obtain d where "d>0"
       and d: "\<And>y. \<lbrakk>y \<in> S; norm (y - a) < d\<rbrakk> \<Longrightarrow>
                     norm (f y - f a - f' (y - a)) \<le> e / (B * C) * norm (y - a)"
       using f' [of "e / (B * C)"] \<open>B > 0\<close> \<open>C > 0\<close> by auto
     have "norm (x - a - g' (f x - f a)) \<le> e * norm (f x - f a)"
       if "x \<in> S" and lt_k: "norm (f x - f a) < k" and lt_dB: "norm (f x - f a) < d/B" for x
     proof -
       have "norm (x - a) \<le> B * norm(f x - f a)"
         using Bk lt_k \<open>x \<in> S\<close> by blast
       also have "\<dots> < d"
         by (metis \<open>0 < B\<close> lt_dB mult.commute pos_less_divide_eq)
       finally have lt_d: "norm (x - a) < d" .
       have "norm (x - a - g' (f x - f a)) \<le> norm(g'(f x - f a - (f' (x - a))))"
         by (simp add: linear_diff [OF \<open>linear g'\<close>] norm_minus_commute)
       also have "\<dots> \<le> C * norm (f x - f a - f' (x - a))"
         using C by blast
       also have "\<dots> \<le> e * norm (f x - f a)"
       proof -
         have "norm (f x - f a - f' (x - a)) \<le> e / (B * C) * norm (x - a)"
           using d [OF \<open>x \<in> S\<close> lt_d] .
         also have "\<dots> \<le> (norm (f x - f a) * e) / C"
           using \<open>B > 0\<close> \<open>C > 0\<close> \<open>e > 0\<close> by (simp add: field_simps Bk lt_k \<open>x \<in> S\<close>)
         finally show ?thesis
           using \<open>C > 0\<close> by (simp add: field_simps)
       qed
     finally show ?thesis .
     qed
     then show "\<exists>d>0. \<forall>y\<in>f ` S.
                norm (y - f a) < d \<longrightarrow>
                norm (g y - g (f a) - g' (y - f a)) \<le> e * norm (y - f a)"
       apply (rule_tac x="min k (d / B)" in exI)
       using \<open>k > 0\<close> \<open>B > 0\<close> \<open>d > 0\<close> \<open>a \<in> S\<close> by (auto simp: gf)
   qed
 qed
 
 end
diff --git a/src/HOL/Analysis/Path_Connected.thy b/src/HOL/Analysis/Path_Connected.thy
--- a/src/HOL/Analysis/Path_Connected.thy
+++ b/src/HOL/Analysis/Path_Connected.thy
@@ -1,4175 +1,4175 @@
 (*  Title:      HOL/Analysis/Path_Connected.thy
     Authors:    LC Paulson and Robert Himmelmann (TU Muenchen), based on material from HOL Light
 *)
 
 section \<open>Path-Connectedness\<close>
 
 theory Path_Connected
 imports
   Starlike
   T1_Spaces
 begin
 
 subsection \<open>Paths and Arcs\<close>
 
 definition\<^marker>\<open>tag important\<close> path :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> bool"
   where "path g \<longleftrightarrow> continuous_on {0..1} g"
 
 definition\<^marker>\<open>tag important\<close> pathstart :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> 'a"
   where "pathstart g = g 0"
 
 definition\<^marker>\<open>tag important\<close> pathfinish :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> 'a"
   where "pathfinish g = g 1"
 
 definition\<^marker>\<open>tag important\<close> path_image :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> 'a set"
   where "path_image g = g ` {0 .. 1}"
 
 definition\<^marker>\<open>tag important\<close> reversepath :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> real \<Rightarrow> 'a"
   where "reversepath g = (\<lambda>x. g(1 - x))"
 
 definition\<^marker>\<open>tag important\<close> joinpaths :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> (real \<Rightarrow> 'a) \<Rightarrow> real \<Rightarrow> 'a"
     (infixr "+++" 75)
   where "g1 +++ g2 = (\<lambda>x. if x \<le> 1/2 then g1 (2 * x) else g2 (2 * x - 1))"
 
 definition\<^marker>\<open>tag important\<close> simple_path :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> bool"
   where "simple_path g \<longleftrightarrow>
      path g \<and> (\<forall>x\<in>{0..1}. \<forall>y\<in>{0..1}. g x = g y \<longrightarrow> x = y \<or> x = 0 \<and> y = 1 \<or> x = 1 \<and> y = 0)"
 
 definition\<^marker>\<open>tag important\<close> arc :: "(real \<Rightarrow> 'a :: topological_space) \<Rightarrow> bool"
   where "arc g \<longleftrightarrow> path g \<and> inj_on g {0..1}"
 
 
 subsection\<^marker>\<open>tag unimportant\<close>\<open>Invariance theorems\<close>
 
 lemma path_eq: "path p \<Longrightarrow> (\<And>t. t \<in> {0..1} \<Longrightarrow> p t = q t) \<Longrightarrow> path q"
   using continuous_on_eq path_def by blast
 
 lemma path_continuous_image: "path g \<Longrightarrow> continuous_on (path_image g) f \<Longrightarrow> path(f \<circ> g)"
   unfolding path_def path_image_def
   using continuous_on_compose by blast
 
 lemma path_translation_eq:
   fixes g :: "real \<Rightarrow> 'a :: real_normed_vector"
   shows "path((\<lambda>x. a + x) \<circ> g) = path g"
 proof -
   have g: "g = (\<lambda>x. -a + x) \<circ> ((\<lambda>x. a + x) \<circ> g)"
     by (rule ext) simp
   show ?thesis
     unfolding path_def
     apply safe
     apply (subst g)
     apply (rule continuous_on_compose)
     apply (auto intro: continuous_intros)
     done
 qed
 
 lemma path_linear_image_eq:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
    assumes "linear f" "inj f"
      shows "path(f \<circ> g) = path g"
 proof -
   from linear_injective_left_inverse [OF assms]
   obtain h where h: "linear h" "h \<circ> f = id"
     by blast
   then have g: "g = h \<circ> (f \<circ> g)"
     by (metis comp_assoc id_comp)
   show ?thesis
     unfolding path_def
     using h assms
     by (metis g continuous_on_compose linear_continuous_on linear_conv_bounded_linear)
 qed
 
 lemma pathstart_translation: "pathstart((\<lambda>x. a + x) \<circ> g) = a + pathstart g"
   by (simp add: pathstart_def)
 
 lemma pathstart_linear_image_eq: "linear f \<Longrightarrow> pathstart(f \<circ> g) = f(pathstart g)"
   by (simp add: pathstart_def)
 
 lemma pathfinish_translation: "pathfinish((\<lambda>x. a + x) \<circ> g) = a + pathfinish g"
   by (simp add: pathfinish_def)
 
 lemma pathfinish_linear_image: "linear f \<Longrightarrow> pathfinish(f \<circ> g) = f(pathfinish g)"
   by (simp add: pathfinish_def)
 
 lemma path_image_translation: "path_image((\<lambda>x. a + x) \<circ> g) = (\<lambda>x. a + x) ` (path_image g)"
   by (simp add: image_comp path_image_def)
 
 lemma path_image_linear_image: "linear f \<Longrightarrow> path_image(f \<circ> g) = f ` (path_image g)"
   by (simp add: image_comp path_image_def)
 
 lemma reversepath_translation: "reversepath((\<lambda>x. a + x) \<circ> g) = (\<lambda>x. a + x) \<circ> reversepath g"
   by (rule ext) (simp add: reversepath_def)
 
 lemma reversepath_linear_image: "linear f \<Longrightarrow> reversepath(f \<circ> g) = f \<circ> reversepath g"
   by (rule ext) (simp add: reversepath_def)
 
 lemma joinpaths_translation:
     "((\<lambda>x. a + x) \<circ> g1) +++ ((\<lambda>x. a + x) \<circ> g2) = (\<lambda>x. a + x) \<circ> (g1 +++ g2)"
   by (rule ext) (simp add: joinpaths_def)
 
 lemma joinpaths_linear_image: "linear f \<Longrightarrow> (f \<circ> g1) +++ (f \<circ> g2) = f \<circ> (g1 +++ g2)"
   by (rule ext) (simp add: joinpaths_def)
 
 lemma simple_path_translation_eq:
   fixes g :: "real \<Rightarrow> 'a::euclidean_space"
   shows "simple_path((\<lambda>x. a + x) \<circ> g) = simple_path g"
   by (simp add: simple_path_def path_translation_eq)
 
 lemma simple_path_linear_image_eq:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes "linear f" "inj f"
     shows "simple_path(f \<circ> g) = simple_path g"
   using assms inj_on_eq_iff [of f]
   by (auto simp: path_linear_image_eq simple_path_def path_translation_eq)
 
 lemma arc_translation_eq:
   fixes g :: "real \<Rightarrow> 'a::euclidean_space"
   shows "arc((\<lambda>x. a + x) \<circ> g) = arc g"
   by (auto simp: arc_def inj_on_def path_translation_eq)
 
 lemma arc_linear_image_eq:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
    assumes "linear f" "inj f"
      shows  "arc(f \<circ> g) = arc g"
   using assms inj_on_eq_iff [of f]
   by (auto simp: arc_def inj_on_def path_linear_image_eq)
 
 
 subsection\<^marker>\<open>tag unimportant\<close>\<open>Basic lemmas about paths\<close>
 
 lemma pathin_iff_path_real [simp]: "pathin euclideanreal g \<longleftrightarrow> path g"
   by (simp add: pathin_def path_def)
 
 lemma continuous_on_path: "path f \<Longrightarrow> t \<subseteq> {0..1} \<Longrightarrow> continuous_on t f"
   using continuous_on_subset path_def by blast
 
 lemma arc_imp_simple_path: "arc g \<Longrightarrow> simple_path g"
   by (simp add: arc_def inj_on_def simple_path_def)
 
 lemma arc_imp_path: "arc g \<Longrightarrow> path g"
   using arc_def by blast
 
 lemma arc_imp_inj_on: "arc g \<Longrightarrow> inj_on g {0..1}"
   by (auto simp: arc_def)
 
 lemma simple_path_imp_path: "simple_path g \<Longrightarrow> path g"
   using simple_path_def by blast
 
 lemma simple_path_cases: "simple_path g \<Longrightarrow> arc g \<or> pathfinish g = pathstart g"
   unfolding simple_path_def arc_def inj_on_def pathfinish_def pathstart_def
   by force
 
 lemma simple_path_imp_arc: "simple_path g \<Longrightarrow> pathfinish g \<noteq> pathstart g \<Longrightarrow> arc g"
   using simple_path_cases by auto
 
 lemma arc_distinct_ends: "arc g \<Longrightarrow> pathfinish g \<noteq> pathstart g"
   unfolding arc_def inj_on_def pathfinish_def pathstart_def
   by fastforce
 
 lemma arc_simple_path: "arc g \<longleftrightarrow> simple_path g \<and> pathfinish g \<noteq> pathstart g"
   using arc_distinct_ends arc_imp_simple_path simple_path_cases by blast
 
 lemma simple_path_eq_arc: "pathfinish g \<noteq> pathstart g \<Longrightarrow> (simple_path g = arc g)"
   by (simp add: arc_simple_path)
 
 lemma path_image_const [simp]: "path_image (\<lambda>t. a) = {a}"
   by (force simp: path_image_def)
 
 lemma path_image_nonempty [simp]: "path_image g \<noteq> {}"
   unfolding path_image_def image_is_empty box_eq_empty
   by auto
 
 lemma pathstart_in_path_image[intro]: "pathstart g \<in> path_image g"
   unfolding pathstart_def path_image_def
   by auto
 
 lemma pathfinish_in_path_image[intro]: "pathfinish g \<in> path_image g"
   unfolding pathfinish_def path_image_def
   by auto
 
 lemma connected_path_image[intro]: "path g \<Longrightarrow> connected (path_image g)"
   unfolding path_def path_image_def
   using connected_continuous_image connected_Icc by blast
 
 lemma compact_path_image[intro]: "path g \<Longrightarrow> compact (path_image g)"
   unfolding path_def path_image_def
   using compact_continuous_image connected_Icc by blast
 
 lemma reversepath_reversepath[simp]: "reversepath (reversepath g) = g"
   unfolding reversepath_def
   by auto
 
 lemma pathstart_reversepath[simp]: "pathstart (reversepath g) = pathfinish g"
   unfolding pathstart_def reversepath_def pathfinish_def
   by auto
 
 lemma pathfinish_reversepath[simp]: "pathfinish (reversepath g) = pathstart g"
   unfolding pathstart_def reversepath_def pathfinish_def
   by auto
 
 lemma pathstart_join[simp]: "pathstart (g1 +++ g2) = pathstart g1"
   unfolding pathstart_def joinpaths_def pathfinish_def
   by auto
 
 lemma pathfinish_join[simp]: "pathfinish (g1 +++ g2) = pathfinish g2"
   unfolding pathstart_def joinpaths_def pathfinish_def
   by auto
 
 lemma path_image_reversepath[simp]: "path_image (reversepath g) = path_image g"
 proof -
   have *: "\<And>g. path_image (reversepath g) \<subseteq> path_image g"
     unfolding path_image_def subset_eq reversepath_def Ball_def image_iff
     by force
   show ?thesis
     using *[of g] *[of "reversepath g"]
     unfolding reversepath_reversepath
     by auto
 qed
 
 lemma path_reversepath [simp]: "path (reversepath g) \<longleftrightarrow> path g"
 proof -
   have *: "\<And>g. path g \<Longrightarrow> path (reversepath g)"
     unfolding path_def reversepath_def
     apply (rule continuous_on_compose[unfolded o_def, of _ "\<lambda>x. 1 - x"])
     apply (auto intro: continuous_intros continuous_on_subset[of "{0..1}"])
     done
   show ?thesis
     using *[of "reversepath g"] *[of g]
     unfolding reversepath_reversepath
     by (rule iffI)
 qed
 
 lemma arc_reversepath:
   assumes "arc g" shows "arc(reversepath g)"
 proof -
   have injg: "inj_on g {0..1}"
     using assms
     by (simp add: arc_def)
   have **: "\<And>x y::real. 1-x = 1-y \<Longrightarrow> x = y"
     by simp
   show ?thesis
     using assms  by (clarsimp simp: arc_def intro!: inj_onI) (simp add: inj_onD reversepath_def **)
 qed
 
 lemma simple_path_reversepath: "simple_path g \<Longrightarrow> simple_path (reversepath g)"
   apply (simp add: simple_path_def)
   apply (force simp: reversepath_def)
   done
 
 lemmas reversepath_simps =
   path_reversepath path_image_reversepath pathstart_reversepath pathfinish_reversepath
 
 lemma path_join[simp]:
   assumes "pathfinish g1 = pathstart g2"
   shows "path (g1 +++ g2) \<longleftrightarrow> path g1 \<and> path g2"
   unfolding path_def pathfinish_def pathstart_def
 proof safe
   assume cont: "continuous_on {0..1} (g1 +++ g2)"
   have g1: "continuous_on {0..1} g1 \<longleftrightarrow> continuous_on {0..1} ((g1 +++ g2) \<circ> (\<lambda>x. x / 2))"
     by (intro continuous_on_cong refl) (auto simp: joinpaths_def)
   have g2: "continuous_on {0..1} g2 \<longleftrightarrow> continuous_on {0..1} ((g1 +++ g2) \<circ> (\<lambda>x. x / 2 + 1/2))"
     using assms
     by (intro continuous_on_cong refl) (auto simp: joinpaths_def pathfinish_def pathstart_def)
   show "continuous_on {0..1} g1" and "continuous_on {0..1} g2"
     unfolding g1 g2
     by (auto intro!: continuous_intros continuous_on_subset[OF cont] simp del: o_apply)
 next
   assume g1g2: "continuous_on {0..1} g1" "continuous_on {0..1} g2"
   have 01: "{0 .. 1} = {0..1/2} \<union> {1/2 .. 1::real}"
     by auto
   {
     fix x :: real
     assume "0 \<le> x" and "x \<le> 1"
     then have "x \<in> (\<lambda>x. x * 2) ` {0..1 / 2}"
       by (intro image_eqI[where x="x/2"]) auto
   }
   note 1 = this
   {
     fix x :: real
     assume "0 \<le> x" and "x \<le> 1"
     then have "x \<in> (\<lambda>x. x * 2 - 1) ` {1 / 2..1}"
       by (intro image_eqI[where x="x/2 + 1/2"]) auto
   }
   note 2 = this
   show "continuous_on {0..1} (g1 +++ g2)"
     using assms
     unfolding joinpaths_def 01
     apply (intro continuous_on_cases closed_atLeastAtMost g1g2[THEN continuous_on_compose2] continuous_intros)
     apply (auto simp: field_simps pathfinish_def pathstart_def intro!: 1 2)
     done
 qed
 
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>Path Images\<close>
 
 lemma bounded_path_image: "path g \<Longrightarrow> bounded(path_image g)"
   by (simp add: compact_imp_bounded compact_path_image)
 
 lemma closed_path_image:
   fixes g :: "real \<Rightarrow> 'a::t2_space"
   shows "path g \<Longrightarrow> closed(path_image g)"
   by (metis compact_path_image compact_imp_closed)
 
 lemma connected_simple_path_image: "simple_path g \<Longrightarrow> connected(path_image g)"
   by (metis connected_path_image simple_path_imp_path)
 
 lemma compact_simple_path_image: "simple_path g \<Longrightarrow> compact(path_image g)"
   by (metis compact_path_image simple_path_imp_path)
 
 lemma bounded_simple_path_image: "simple_path g \<Longrightarrow> bounded(path_image g)"
   by (metis bounded_path_image simple_path_imp_path)
 
 lemma closed_simple_path_image:
   fixes g :: "real \<Rightarrow> 'a::t2_space"
   shows "simple_path g \<Longrightarrow> closed(path_image g)"
   by (metis closed_path_image simple_path_imp_path)
 
 lemma connected_arc_image: "arc g \<Longrightarrow> connected(path_image g)"
   by (metis connected_path_image arc_imp_path)
 
 lemma compact_arc_image: "arc g \<Longrightarrow> compact(path_image g)"
   by (metis compact_path_image arc_imp_path)
 
 lemma bounded_arc_image: "arc g \<Longrightarrow> bounded(path_image g)"
   by (metis bounded_path_image arc_imp_path)
 
 lemma closed_arc_image:
   fixes g :: "real \<Rightarrow> 'a::t2_space"
   shows "arc g \<Longrightarrow> closed(path_image g)"
   by (metis closed_path_image arc_imp_path)
 
 lemma path_image_join_subset: "path_image (g1 +++ g2) \<subseteq> path_image g1 \<union> path_image g2"
   unfolding path_image_def joinpaths_def
   by auto
 
 lemma subset_path_image_join:
   assumes "path_image g1 \<subseteq> s"
     and "path_image g2 \<subseteq> s"
   shows "path_image (g1 +++ g2) \<subseteq> s"
   using path_image_join_subset[of g1 g2] and assms
   by auto
 
 lemma path_image_join:
     "pathfinish g1 = pathstart g2 \<Longrightarrow> path_image(g1 +++ g2) = path_image g1 \<union> path_image g2"
   apply (rule subset_antisym [OF path_image_join_subset])
   apply (auto simp: pathfinish_def pathstart_def path_image_def joinpaths_def image_def)
   apply (drule sym)
   apply (rule_tac x="xa/2" in bexI, auto)
   apply (rule ccontr)
   apply (drule_tac x="(xa+1)/2" in bspec)
   apply (auto simp: field_simps)
   apply (drule_tac x="1/2" in bspec, auto)
   done
 
 lemma not_in_path_image_join:
   assumes "x \<notin> path_image g1"
     and "x \<notin> path_image g2"
   shows "x \<notin> path_image (g1 +++ g2)"
   using assms and path_image_join_subset[of g1 g2]
   by auto
 
 lemma pathstart_compose: "pathstart(f \<circ> p) = f(pathstart p)"
   by (simp add: pathstart_def)
 
 lemma pathfinish_compose: "pathfinish(f \<circ> p) = f(pathfinish p)"
   by (simp add: pathfinish_def)
 
 lemma path_image_compose: "path_image (f \<circ> p) = f ` (path_image p)"
   by (simp add: image_comp path_image_def)
 
 lemma path_compose_join: "f \<circ> (p +++ q) = (f \<circ> p) +++ (f \<circ> q)"
   by (rule ext) (simp add: joinpaths_def)
 
 lemma path_compose_reversepath: "f \<circ> reversepath p = reversepath(f \<circ> p)"
   by (rule ext) (simp add: reversepath_def)
 
 lemma joinpaths_eq:
   "(\<And>t. t \<in> {0..1} \<Longrightarrow> p t = p' t) \<Longrightarrow>
    (\<And>t. t \<in> {0..1} \<Longrightarrow> q t = q' t)
    \<Longrightarrow>  t \<in> {0..1} \<Longrightarrow> (p +++ q) t = (p' +++ q') t"
   by (auto simp: joinpaths_def)
 
 lemma simple_path_inj_on: "simple_path g \<Longrightarrow> inj_on g {0<..<1}"
   by (auto simp: simple_path_def path_image_def inj_on_def less_eq_real_def Ball_def)
 
 
 subsection\<^marker>\<open>tag unimportant\<close>\<open>Simple paths with the endpoints removed\<close>
 
 lemma simple_path_endless:
     "simple_path c \<Longrightarrow> path_image c - {pathstart c,pathfinish c} = c ` {0<..<1}"
   apply (auto simp: simple_path_def path_image_def pathstart_def pathfinish_def Ball_def Bex_def image_def)
   apply (metis eq_iff le_less_linear)
   apply (metis leD linear)
   using less_eq_real_def zero_le_one apply blast
   using less_eq_real_def zero_le_one apply blast
   done
 
 lemma connected_simple_path_endless:
     "simple_path c \<Longrightarrow> connected(path_image c - {pathstart c,pathfinish c})"
 apply (simp add: simple_path_endless)
 apply (rule connected_continuous_image)
 apply (meson continuous_on_subset greaterThanLessThan_subseteq_atLeastAtMost_iff le_numeral_extra(3) le_numeral_extra(4) path_def simple_path_imp_path)
 by auto
 
 lemma nonempty_simple_path_endless:
     "simple_path c \<Longrightarrow> path_image c - {pathstart c,pathfinish c} \<noteq> {}"
   by (simp add: simple_path_endless)
 
 
 subsection\<^marker>\<open>tag unimportant\<close>\<open>The operations on paths\<close>
 
 lemma path_image_subset_reversepath: "path_image(reversepath g) \<le> path_image g"
   by (auto simp: path_image_def reversepath_def)
 
 lemma path_imp_reversepath: "path g \<Longrightarrow> path(reversepath g)"
   apply (auto simp: path_def reversepath_def)
   using continuous_on_compose [of "{0..1}" "\<lambda>x. 1 - x" g]
   apply (auto simp: continuous_on_op_minus)
   done
 
 lemma half_bounded_equal: "1 \<le> x * 2 \<Longrightarrow> x * 2 \<le> 1 \<longleftrightarrow> x = (1/2::real)"
   by simp
 
 lemma continuous_on_joinpaths:
   assumes "continuous_on {0..1} g1" "continuous_on {0..1} g2" "pathfinish g1 = pathstart g2"
     shows "continuous_on {0..1} (g1 +++ g2)"
 proof -
   have *: "{0..1::real} = {0..1/2} \<union> {1/2..1}"
     by auto
   have gg: "g2 0 = g1 1"
     by (metis assms(3) pathfinish_def pathstart_def)
   have 1: "continuous_on {0..1/2} (g1 +++ g2)"
     apply (rule continuous_on_eq [of _ "g1 \<circ> (\<lambda>x. 2*x)"])
     apply (rule continuous_intros | simp add: joinpaths_def assms)+
     done
   have "continuous_on {1/2..1} (g2 \<circ> (\<lambda>x. 2*x-1))"
     apply (rule continuous_on_subset [of "{1/2..1}"])
     apply (rule continuous_intros | simp add: image_affinity_atLeastAtMost_diff assms)+
     done
   then have 2: "continuous_on {1/2..1} (g1 +++ g2)"
     apply (rule continuous_on_eq [of "{1/2..1}" "g2 \<circ> (\<lambda>x. 2*x-1)"])
     apply (rule assms continuous_intros | simp add: joinpaths_def mult.commute half_bounded_equal gg)+
     done
   show ?thesis
     apply (subst *)
     apply (rule continuous_on_closed_Un)
     using 1 2
     apply auto
     done
 qed
 
 lemma path_join_imp: "\<lbrakk>path g1; path g2; pathfinish g1 = pathstart g2\<rbrakk> \<Longrightarrow> path(g1 +++ g2)"
   by (simp)
 
 lemma simple_path_join_loop:
   assumes "arc g1" "arc g2"
           "pathfinish g1 = pathstart g2"  "pathfinish g2 = pathstart g1"
           "path_image g1 \<inter> path_image g2 \<subseteq> {pathstart g1, pathstart g2}"
   shows "simple_path(g1 +++ g2)"
 proof -
   have injg1: "inj_on g1 {0..1}"
     using assms
     by (simp add: arc_def)
   have injg2: "inj_on g2 {0..1}"
     using assms
     by (simp add: arc_def)
   have g12: "g1 1 = g2 0"
    and g21: "g2 1 = g1 0"
    and sb:  "g1 ` {0..1} \<inter> g2 ` {0..1} \<subseteq> {g1 0, g2 0}"
     using assms
     by (simp_all add: arc_def pathfinish_def pathstart_def path_image_def)
   { fix x and y::real
     assume xyI: "x = 1 \<longrightarrow> y \<noteq> 0"
        and xy: "x \<le> 1" "0 \<le> y" " y * 2 \<le> 1" "\<not> x * 2 \<le> 1" "g2 (2 * x - 1) = g1 (2 * y)"
     have g1im: "g1 (2 * y) \<in> g1 ` {0..1} \<inter> g2 ` {0..1}"
       using xy
       apply simp
       apply (rule_tac x="2 * x - 1" in image_eqI, auto)
       done
     have False
       using subsetD [OF sb g1im] xy
       apply auto
       apply (drule inj_onD [OF injg1])
       using g21 [symmetric] xyI
       apply (auto dest: inj_onD [OF injg2])
       done
    } note * = this
   { fix x and y::real
     assume xy: "y \<le> 1" "0 \<le> x" "\<not> y * 2 \<le> 1" "x * 2 \<le> 1" "g1 (2 * x) = g2 (2 * y - 1)"
     have g1im: "g1 (2 * x) \<in> g1 ` {0..1} \<inter> g2 ` {0..1}"
       using xy
       apply simp
       apply (rule_tac x="2 * x" in image_eqI, auto)
       done
     have "x = 0 \<and> y = 1"
       using subsetD [OF sb g1im] xy
       apply auto
       apply (force dest: inj_onD [OF injg1])
       using  g21 [symmetric]
       apply (auto dest: inj_onD [OF injg2])
       done
    } note ** = this
   show ?thesis
     using assms
     apply (simp add: arc_def simple_path_def path_join, clarify)
     apply (simp add: joinpaths_def split: if_split_asm)
     apply (force dest: inj_onD [OF injg1])
     apply (metis *)
     apply (metis **)
     apply (force dest: inj_onD [OF injg2])
     done
 qed
 
 lemma arc_join:
   assumes "arc g1" "arc g2"
           "pathfinish g1 = pathstart g2"
           "path_image g1 \<inter> path_image g2 \<subseteq> {pathstart g2}"
     shows "arc(g1 +++ g2)"
 proof -
   have injg1: "inj_on g1 {0..1}"
     using assms
     by (simp add: arc_def)
   have injg2: "inj_on g2 {0..1}"
     using assms
     by (simp add: arc_def)
   have g11: "g1 1 = g2 0"
    and sb:  "g1 ` {0..1} \<inter> g2 ` {0..1} \<subseteq> {g2 0}"
     using assms
     by (simp_all add: arc_def pathfinish_def pathstart_def path_image_def)
   { fix x and y::real
     assume xy: "x \<le> 1" "0 \<le> y" " y * 2 \<le> 1" "\<not> x * 2 \<le> 1" "g2 (2 * x - 1) = g1 (2 * y)"
     have g1im: "g1 (2 * y) \<in> g1 ` {0..1} \<inter> g2 ` {0..1}"
       using xy
       apply simp
       apply (rule_tac x="2 * x - 1" in image_eqI, auto)
       done
     have False
       using subsetD [OF sb g1im] xy
       by (auto dest: inj_onD [OF injg2])
    } note * = this
   show ?thesis
     apply (simp add: arc_def inj_on_def)
     apply (clarsimp simp add: arc_imp_path assms)
     apply (simp add: joinpaths_def split: if_split_asm)
     apply (force dest: inj_onD [OF injg1])
     apply (metis *)
     apply (metis *)
     apply (force dest: inj_onD [OF injg2])
     done
 qed
 
 lemma reversepath_joinpaths:
     "pathfinish g1 = pathstart g2 \<Longrightarrow> reversepath(g1 +++ g2) = reversepath g2 +++ reversepath g1"
   unfolding reversepath_def pathfinish_def pathstart_def joinpaths_def
   by (rule ext) (auto simp: mult.commute)
 
 
 subsection\<^marker>\<open>tag unimportant\<close>\<open>Some reversed and "if and only if" versions of joining theorems\<close>
 
 lemma path_join_path_ends:
   fixes g1 :: "real \<Rightarrow> 'a::metric_space"
   assumes "path(g1 +++ g2)" "path g2"
     shows "pathfinish g1 = pathstart g2"
 proof (rule ccontr)
   define e where "e = dist (g1 1) (g2 0)"
   assume Neg: "pathfinish g1 \<noteq> pathstart g2"
   then have "0 < dist (pathfinish g1) (pathstart g2)"
     by auto
   then have "e > 0"
     by (metis e_def pathfinish_def pathstart_def)
   then obtain d1 where "d1 > 0"
        and d1: "\<And>x'. \<lbrakk>x'\<in>{0..1}; norm x' < d1\<rbrakk> \<Longrightarrow> dist (g2 x') (g2 0) < e/2"
     using assms(2) unfolding path_def continuous_on_iff
     apply (drule_tac x=0 in bspec, simp)
     by (metis half_gt_zero_iff norm_conv_dist)
   obtain d2 where "d2 > 0"
        and d2: "\<And>x'. \<lbrakk>x'\<in>{0..1}; dist x' (1/2) < d2\<rbrakk>
                       \<Longrightarrow> dist ((g1 +++ g2) x') (g1 1) < e/2"
     using assms(1) \<open>e > 0\<close> unfolding path_def continuous_on_iff
     apply (drule_tac x="1/2" in bspec, simp)
     apply (drule_tac x="e/2" in spec)
     apply (force simp: joinpaths_def)
     done
   have int01_1: "min (1/2) (min d1 d2) / 2 \<in> {0..1}"
     using \<open>d1 > 0\<close> \<open>d2 > 0\<close> by (simp add: min_def)
   have dist1: "norm (min (1 / 2) (min d1 d2) / 2) < d1"
     using \<open>d1 > 0\<close> \<open>d2 > 0\<close> by (simp add: min_def dist_norm)
   have int01_2: "1/2 + min (1/2) (min d1 d2) / 4 \<in> {0..1}"
     using \<open>d1 > 0\<close> \<open>d2 > 0\<close> by (simp add: min_def)
   have dist2: "dist (1 / 2 + min (1 / 2) (min d1 d2) / 4) (1 / 2) < d2"
     using \<open>d1 > 0\<close> \<open>d2 > 0\<close> by (simp add: min_def dist_norm)
   have [simp]: "\<not> min (1 / 2) (min d1 d2) \<le> 0"
     using \<open>d1 > 0\<close> \<open>d2 > 0\<close> by (simp add: min_def)
   have "dist (g2 (min (1 / 2) (min d1 d2) / 2)) (g1 1) < e/2"
        "dist (g2 (min (1 / 2) (min d1 d2) / 2)) (g2 0) < e/2"
     using d1 [OF int01_1 dist1] d2 [OF int01_2 dist2] by (simp_all add: joinpaths_def)
   then have "dist (g1 1) (g2 0) < e/2 + e/2"
     using dist_triangle_half_r e_def by blast
   then show False
     by (simp add: e_def [symmetric])
 qed
 
 lemma path_join_eq [simp]:
   fixes g1 :: "real \<Rightarrow> 'a::metric_space"
   assumes "path g1" "path g2"
     shows "path(g1 +++ g2) \<longleftrightarrow> pathfinish g1 = pathstart g2"
   using assms by (metis path_join_path_ends path_join_imp)
 
 lemma simple_path_joinE:
   assumes "simple_path(g1 +++ g2)" and "pathfinish g1 = pathstart g2"
   obtains "arc g1" "arc g2"
           "path_image g1 \<inter> path_image g2 \<subseteq> {pathstart g1, pathstart g2}"
 proof -
   have *: "\<And>x y. \<lbrakk>0 \<le> x; x \<le> 1; 0 \<le> y; y \<le> 1; (g1 +++ g2) x = (g1 +++ g2) y\<rbrakk>
                \<Longrightarrow> x = y \<or> x = 0 \<and> y = 1 \<or> x = 1 \<and> y = 0"
     using assms by (simp add: simple_path_def)
   have "path g1"
     using assms path_join simple_path_imp_path by blast
   moreover have "inj_on g1 {0..1}"
   proof (clarsimp simp: inj_on_def)
     fix x y
     assume "g1 x = g1 y" "0 \<le> x" "x \<le> 1" "0 \<le> y" "y \<le> 1"
     then show "x = y"
       using * [of "x/2" "y/2"] by (simp add: joinpaths_def split_ifs)
   qed
   ultimately have "arc g1"
     using assms  by (simp add: arc_def)
   have [simp]: "g2 0 = g1 1"
     using assms by (metis pathfinish_def pathstart_def)
   have "path g2"
     using assms path_join simple_path_imp_path by blast
   moreover have "inj_on g2 {0..1}"
   proof (clarsimp simp: inj_on_def)
     fix x y
     assume "g2 x = g2 y" "0 \<le> x" "x \<le> 1" "0 \<le> y" "y \<le> 1"
     then show "x = y"
       using * [of "(x + 1) / 2" "(y + 1) / 2"]
       by (force simp: joinpaths_def split_ifs field_split_simps)
   qed
   ultimately have "arc g2"
     using assms  by (simp add: arc_def)
   have "g2 y = g1 0 \<or> g2 y = g1 1"
        if "g1 x = g2 y" "0 \<le> x" "x \<le> 1" "0 \<le> y" "y \<le> 1" for x y
       using * [of "x / 2" "(y + 1) / 2"] that
       by (auto simp: joinpaths_def split_ifs field_split_simps)
   then have "path_image g1 \<inter> path_image g2 \<subseteq> {pathstart g1, pathstart g2}"
     by (fastforce simp: pathstart_def pathfinish_def path_image_def)
   with \<open>arc g1\<close> \<open>arc g2\<close> show ?thesis using that by blast
 qed
 
 lemma simple_path_join_loop_eq:
   assumes "pathfinish g2 = pathstart g1" "pathfinish g1 = pathstart g2"
     shows "simple_path(g1 +++ g2) \<longleftrightarrow>
              arc g1 \<and> arc g2 \<and> path_image g1 \<inter> path_image g2 \<subseteq> {pathstart g1, pathstart g2}"
 by (metis assms simple_path_joinE simple_path_join_loop)
 
 lemma arc_join_eq:
   assumes "pathfinish g1 = pathstart g2"
     shows "arc(g1 +++ g2) \<longleftrightarrow>
            arc g1 \<and> arc g2 \<and> path_image g1 \<inter> path_image g2 \<subseteq> {pathstart g2}"
            (is "?lhs = ?rhs")
 proof
   assume ?lhs
   then have "simple_path(g1 +++ g2)" by (rule arc_imp_simple_path)
   then have *: "\<And>x y. \<lbrakk>0 \<le> x; x \<le> 1; 0 \<le> y; y \<le> 1; (g1 +++ g2) x = (g1 +++ g2) y\<rbrakk>
                \<Longrightarrow> x = y \<or> x = 0 \<and> y = 1 \<or> x = 1 \<and> y = 0"
     using assms by (simp add: simple_path_def)
   have False if "g1 0 = g2 u" "0 \<le> u" "u \<le> 1" for u
     using * [of 0 "(u + 1) / 2"] that assms arc_distinct_ends [OF \<open>?lhs\<close>]
     by (auto simp: joinpaths_def pathstart_def pathfinish_def split_ifs field_split_simps)
   then have n1: "pathstart g1 \<notin> path_image g2"
     unfolding pathstart_def path_image_def
     using atLeastAtMost_iff by blast
   show ?rhs using \<open>?lhs\<close>
     apply (rule simple_path_joinE [OF arc_imp_simple_path assms])
     using n1 by force
 next
   assume ?rhs then show ?lhs
     using assms
     by (fastforce simp: pathfinish_def pathstart_def intro!: arc_join)
 qed
 
 lemma arc_join_eq_alt:
         "pathfinish g1 = pathstart g2
         \<Longrightarrow> (arc(g1 +++ g2) \<longleftrightarrow>
              arc g1 \<and> arc g2 \<and>
              path_image g1 \<inter> path_image g2 = {pathstart g2})"
 using pathfinish_in_path_image by (fastforce simp: arc_join_eq)
 
 
 subsection\<^marker>\<open>tag unimportant\<close>\<open>The joining of paths is associative\<close>
 
 lemma path_assoc:
     "\<lbrakk>pathfinish p = pathstart q; pathfinish q = pathstart r\<rbrakk>
      \<Longrightarrow> path(p +++ (q +++ r)) \<longleftrightarrow> path((p +++ q) +++ r)"
 by simp
 
 lemma simple_path_assoc:
   assumes "pathfinish p = pathstart q" "pathfinish q = pathstart r"
     shows "simple_path (p +++ (q +++ r)) \<longleftrightarrow> simple_path ((p +++ q) +++ r)"
 proof (cases "pathstart p = pathfinish r")
   case True show ?thesis
   proof
     assume "simple_path (p +++ q +++ r)"
     with assms True show "simple_path ((p +++ q) +++ r)"
       by (fastforce simp add: simple_path_join_loop_eq arc_join_eq path_image_join
                     dest: arc_distinct_ends [of r])
   next
     assume 0: "simple_path ((p +++ q) +++ r)"
     with assms True have q: "pathfinish r \<notin> path_image q"
       using arc_distinct_ends
       by (fastforce simp add: simple_path_join_loop_eq arc_join_eq path_image_join)
     have "pathstart r \<notin> path_image p"
       using assms
       by (metis 0 IntI arc_distinct_ends arc_join_eq_alt empty_iff insert_iff
               pathfinish_in_path_image pathfinish_join simple_path_joinE)
     with assms 0 q True show "simple_path (p +++ q +++ r)"
       by (auto simp: simple_path_join_loop_eq arc_join_eq path_image_join
                dest!: subsetD [OF _ IntI])
   qed
 next
   case False
   { fix x :: 'a
     assume a: "path_image p \<inter> path_image q \<subseteq> {pathstart q}"
               "(path_image p \<union> path_image q) \<inter> path_image r \<subseteq> {pathstart r}"
               "x \<in> path_image p" "x \<in> path_image r"
     have "pathstart r \<in> path_image q"
       by (metis assms(2) pathfinish_in_path_image)
     with a have "x = pathstart q"
       by blast
   }
   with False assms show ?thesis
     by (auto simp: simple_path_eq_arc simple_path_join_loop_eq arc_join_eq path_image_join)
 qed
 
 lemma arc_assoc:
      "\<lbrakk>pathfinish p = pathstart q; pathfinish q = pathstart r\<rbrakk>
       \<Longrightarrow> arc(p +++ (q +++ r)) \<longleftrightarrow> arc((p +++ q) +++ r)"
 by (simp add: arc_simple_path simple_path_assoc)
 
 subsubsection\<^marker>\<open>tag unimportant\<close>\<open>Symmetry and loops\<close>
 
 lemma path_sym:
     "\<lbrakk>pathfinish p = pathstart q; pathfinish q = pathstart p\<rbrakk> \<Longrightarrow> path(p +++ q) \<longleftrightarrow> path(q +++ p)"
   by auto
 
 lemma simple_path_sym:
     "\<lbrakk>pathfinish p = pathstart q; pathfinish q = pathstart p\<rbrakk>
      \<Longrightarrow> simple_path(p +++ q) \<longleftrightarrow> simple_path(q +++ p)"
 by (metis (full_types) inf_commute insert_commute simple_path_joinE simple_path_join_loop)
 
 lemma path_image_sym:
     "\<lbrakk>pathfinish p = pathstart q; pathfinish q = pathstart p\<rbrakk>
      \<Longrightarrow> path_image(p +++ q) = path_image(q +++ p)"
 by (simp add: path_image_join sup_commute)
 
 
 subsection\<open>Subpath\<close>
 
 definition\<^marker>\<open>tag important\<close> subpath :: "real \<Rightarrow> real \<Rightarrow> (real \<Rightarrow> 'a) \<Rightarrow> real \<Rightarrow> 'a::real_normed_vector"
   where "subpath a b g \<equiv> \<lambda>x. g((b - a) * x + a)"
 
 lemma path_image_subpath_gen:
   fixes g :: "_ \<Rightarrow> 'a::real_normed_vector"
   shows "path_image(subpath u v g) = g ` (closed_segment u v)"
   by (auto simp add: closed_segment_real_eq path_image_def subpath_def)
 
 lemma path_image_subpath:
   fixes g :: "real \<Rightarrow> 'a::real_normed_vector"
   shows "path_image(subpath u v g) = (if u \<le> v then g ` {u..v} else g ` {v..u})"
   by (simp add: path_image_subpath_gen closed_segment_eq_real_ivl)
 
 lemma path_image_subpath_commute:
   fixes g :: "real \<Rightarrow> 'a::real_normed_vector"
   shows "path_image(subpath u v g) = path_image(subpath v u g)"
   by (simp add: path_image_subpath_gen closed_segment_eq_real_ivl)
 
 lemma path_subpath [simp]:
   fixes g :: "real \<Rightarrow> 'a::real_normed_vector"
   assumes "path g" "u \<in> {0..1}" "v \<in> {0..1}"
     shows "path(subpath u v g)"
 proof -
   have "continuous_on {0..1} (g \<circ> (\<lambda>x. ((v-u) * x+ u)))"
     apply (rule continuous_intros | simp)+
     apply (simp add: image_affinity_atLeastAtMost [where c=u])
     using assms
     apply (auto simp: path_def continuous_on_subset)
     done
   then show ?thesis
     by (simp add: path_def subpath_def)
 qed
 
 lemma pathstart_subpath [simp]: "pathstart(subpath u v g) = g(u)"
   by (simp add: pathstart_def subpath_def)
 
 lemma pathfinish_subpath [simp]: "pathfinish(subpath u v g) = g(v)"
   by (simp add: pathfinish_def subpath_def)
 
 lemma subpath_trivial [simp]: "subpath 0 1 g = g"
   by (simp add: subpath_def)
 
 lemma subpath_reversepath: "subpath 1 0 g = reversepath g"
   by (simp add: reversepath_def subpath_def)
 
 lemma reversepath_subpath: "reversepath(subpath u v g) = subpath v u g"
   by (simp add: reversepath_def subpath_def algebra_simps)
 
 lemma subpath_translation: "subpath u v ((\<lambda>x. a + x) \<circ> g) = (\<lambda>x. a + x) \<circ> subpath u v g"
   by (rule ext) (simp add: subpath_def)
 
 lemma subpath_image: "subpath u v (f \<circ> g) = f \<circ> subpath u v g"
   by (rule ext) (simp add: subpath_def)
 
 lemma affine_ineq:
   fixes x :: "'a::linordered_idom"
   assumes "x \<le> 1" "v \<le> u"
     shows "v + x * u \<le> u + x * v"
 proof -
   have "(1-x)*(u-v) \<ge> 0"
     using assms by auto
   then show ?thesis
     by (simp add: algebra_simps)
 qed
 
 lemma sum_le_prod1:
   fixes a::real shows "\<lbrakk>a \<le> 1; b \<le> 1\<rbrakk> \<Longrightarrow> a + b \<le> 1 + a * b"
 by (metis add.commute affine_ineq mult.right_neutral)
 
 lemma simple_path_subpath_eq:
   "simple_path(subpath u v g) \<longleftrightarrow>
      path(subpath u v g) \<and> u\<noteq>v \<and>
      (\<forall>x y. x \<in> closed_segment u v \<and> y \<in> closed_segment u v \<and> g x = g y
                 \<longrightarrow> x = y \<or> x = u \<and> y = v \<or> x = v \<and> y = u)"
     (is "?lhs = ?rhs")
 proof (rule iffI)
   assume ?lhs
   then have p: "path (\<lambda>x. g ((v - u) * x + u))"
         and sim: "(\<And>x y. \<lbrakk>x\<in>{0..1}; y\<in>{0..1}; g ((v - u) * x + u) = g ((v - u) * y + u)\<rbrakk>
                   \<Longrightarrow> x = y \<or> x = 0 \<and> y = 1 \<or> x = 1 \<and> y = 0)"
     by (auto simp: simple_path_def subpath_def)
   { fix x y
     assume "x \<in> closed_segment u v" "y \<in> closed_segment u v" "g x = g y"
     then have "x = y \<or> x = u \<and> y = v \<or> x = v \<and> y = u"
       using sim [of "(x-u)/(v-u)" "(y-u)/(v-u)"] p
       by (auto split: if_split_asm simp add: closed_segment_real_eq image_affinity_atLeastAtMost)
         (simp_all add: field_split_simps)
   } moreover
   have "path(subpath u v g) \<and> u\<noteq>v"
     using sim [of "1/3" "2/3"] p
     by (auto simp: subpath_def)
   ultimately show ?rhs
     by metis
 next
   assume ?rhs
   then
   have d1: "\<And>x y. \<lbrakk>g x = g y; u \<le> x; x \<le> v; u \<le> y; y \<le> v\<rbrakk> \<Longrightarrow> x = y \<or> x = u \<and> y = v \<or> x = v \<and> y = u"
    and d2: "\<And>x y. \<lbrakk>g x = g y; v \<le> x; x \<le> u; v \<le> y; y \<le> u\<rbrakk> \<Longrightarrow> x = y \<or> x = u \<and> y = v \<or> x = v \<and> y = u"
    and ne: "u < v \<or> v < u"
    and psp: "path (subpath u v g)"
     by (auto simp: closed_segment_real_eq image_affinity_atLeastAtMost)
   have [simp]: "\<And>x. u + x * v = v + x * u \<longleftrightarrow> u=v \<or> x=1"
     by algebra
   show ?lhs using psp ne
     unfolding simple_path_def subpath_def
     by (fastforce simp add: algebra_simps affine_ineq mult_left_mono crossproduct_eq dest: d1 d2)
 qed
 
 lemma arc_subpath_eq:
   "arc(subpath u v g) \<longleftrightarrow> path(subpath u v g) \<and> u\<noteq>v \<and> inj_on g (closed_segment u v)"
     (is "?lhs = ?rhs")
 proof (rule iffI)
   assume ?lhs
   then have p: "path (\<lambda>x. g ((v - u) * x + u))"
         and sim: "(\<And>x y. \<lbrakk>x\<in>{0..1}; y\<in>{0..1}; g ((v - u) * x + u) = g ((v - u) * y + u)\<rbrakk>
                   \<Longrightarrow> x = y)"
     by (auto simp: arc_def inj_on_def subpath_def)
   { fix x y
     assume "x \<in> closed_segment u v" "y \<in> closed_segment u v" "g x = g y"
     then have "x = y"
       using sim [of "(x-u)/(v-u)" "(y-u)/(v-u)"] p
       by (cases "v = u")
         (simp_all split: if_split_asm add: inj_on_def closed_segment_real_eq image_affinity_atLeastAtMost,
            simp add: field_simps)
   } moreover
   have "path(subpath u v g) \<and> u\<noteq>v"
     using sim [of "1/3" "2/3"] p
     by (auto simp: subpath_def)
   ultimately show ?rhs
     unfolding inj_on_def
     by metis
 next
   assume ?rhs
   then
   have d1: "\<And>x y. \<lbrakk>g x = g y; u \<le> x; x \<le> v; u \<le> y; y \<le> v\<rbrakk> \<Longrightarrow> x = y"
    and d2: "\<And>x y. \<lbrakk>g x = g y; v \<le> x; x \<le> u; v \<le> y; y \<le> u\<rbrakk> \<Longrightarrow> x = y"
    and ne: "u < v \<or> v < u"
    and psp: "path (subpath u v g)"
     by (auto simp: inj_on_def closed_segment_real_eq image_affinity_atLeastAtMost)
   show ?lhs using psp ne
     unfolding arc_def subpath_def inj_on_def
     by (auto simp: algebra_simps affine_ineq mult_left_mono crossproduct_eq dest: d1 d2)
 qed
 
 
 lemma simple_path_subpath:
   assumes "simple_path g" "u \<in> {0..1}" "v \<in> {0..1}" "u \<noteq> v"
   shows "simple_path(subpath u v g)"
   using assms
   apply (simp add: simple_path_subpath_eq simple_path_imp_path)
   apply (simp add: simple_path_def closed_segment_real_eq image_affinity_atLeastAtMost, fastforce)
   done
 
 lemma arc_simple_path_subpath:
     "\<lbrakk>simple_path g; u \<in> {0..1}; v \<in> {0..1}; g u \<noteq> g v\<rbrakk> \<Longrightarrow> arc(subpath u v g)"
   by (force intro: simple_path_subpath simple_path_imp_arc)
 
 lemma arc_subpath_arc:
     "\<lbrakk>arc g; u \<in> {0..1}; v \<in> {0..1}; u \<noteq> v\<rbrakk> \<Longrightarrow> arc(subpath u v g)"
   by (meson arc_def arc_imp_simple_path arc_simple_path_subpath inj_onD)
 
 lemma arc_simple_path_subpath_interior:
     "\<lbrakk>simple_path g; u \<in> {0..1}; v \<in> {0..1}; u \<noteq> v; \<bar>u-v\<bar> < 1\<rbrakk> \<Longrightarrow> arc(subpath u v g)"
     apply (rule arc_simple_path_subpath)
     apply (force simp: simple_path_def)+
     done
 
 lemma path_image_subpath_subset:
     "\<lbrakk>u \<in> {0..1}; v \<in> {0..1}\<rbrakk> \<Longrightarrow> path_image(subpath u v g) \<subseteq> path_image g"
   apply (simp add: closed_segment_real_eq image_affinity_atLeastAtMost path_image_subpath)
   apply (auto simp: path_image_def)
   done  
 
 lemma join_subpaths_middle: "subpath (0) ((1 / 2)) p +++ subpath ((1 / 2)) 1 p = p"
   by (rule ext) (simp add: joinpaths_def subpath_def field_split_simps)
 
 
 subsection\<^marker>\<open>tag unimportant\<close>\<open>There is a subpath to the frontier\<close>
 
 lemma subpath_to_frontier_explicit:
     fixes S :: "'a::metric_space set"
     assumes g: "path g" and "pathfinish g \<notin> S"
     obtains u where "0 \<le> u" "u \<le> 1"
                 "\<And>x. 0 \<le> x \<and> x < u \<Longrightarrow> g x \<in> interior S"
                 "(g u \<notin> interior S)" "(u = 0 \<or> g u \<in> closure S)"
 proof -
   have gcon: "continuous_on {0..1} g"     using g by (simp add: path_def)
   then have com: "compact ({0..1} \<inter> {u. g u \<in> closure (- S)})"
     apply (simp add: Int_commute [of "{0..1}"] compact_eq_bounded_closed closed_vimage_Int [unfolded vimage_def])
     using compact_eq_bounded_closed apply fastforce
     done
   have "1 \<in> {u. g u \<in> closure (- S)}"
     using assms by (simp add: pathfinish_def closure_def)
   then have dis: "{0..1} \<inter> {u. g u \<in> closure (- S)} \<noteq> {}"
     using atLeastAtMost_iff zero_le_one by blast
   then obtain u where "0 \<le> u" "u \<le> 1" and gu: "g u \<in> closure (- S)"
                   and umin: "\<And>t. \<lbrakk>0 \<le> t; t \<le> 1; g t \<in> closure (- S)\<rbrakk> \<Longrightarrow> u \<le> t"
     using compact_attains_inf [OF com dis] by fastforce
   then have umin': "\<And>t. \<lbrakk>0 \<le> t; t \<le> 1; t < u\<rbrakk> \<Longrightarrow>  g t \<in> S"
     using closure_def by fastforce
   { assume "u \<noteq> 0"
     then have "u > 0" using \<open>0 \<le> u\<close> by auto
     { fix e::real assume "e > 0"
       obtain d where "d>0" and d: "\<And>x'. \<lbrakk>x' \<in> {0..1}; dist x' u \<le> d\<rbrakk> \<Longrightarrow> dist (g x') (g u) < e"
         using continuous_onE [OF gcon _ \<open>e > 0\<close>] \<open>0 \<le> _\<close> \<open>_ \<le> 1\<close> atLeastAtMost_iff by auto
       have *: "dist (max 0 (u - d / 2)) u \<le> d"
         using \<open>0 \<le> u\<close> \<open>u \<le> 1\<close> \<open>d > 0\<close> by (simp add: dist_real_def)
       have "\<exists>y\<in>S. dist y (g u) < e"
         using \<open>0 < u\<close> \<open>u \<le> 1\<close> \<open>d > 0\<close>
         by (force intro: d [OF _ *] umin')
     }
     then have "g u \<in> closure S"
       by (simp add: frontier_def closure_approachable)
   }
   then show ?thesis
     apply (rule_tac u=u in that)
     apply (auto simp: \<open>0 \<le> u\<close> \<open>u \<le> 1\<close> gu interior_closure umin)
     using \<open>_ \<le> 1\<close> interior_closure umin apply fastforce
     done
 qed
 
 lemma subpath_to_frontier_strong:
     assumes g: "path g" and "pathfinish g \<notin> S"
     obtains u where "0 \<le> u" "u \<le> 1" "g u \<notin> interior S"
                     "u = 0 \<or> (\<forall>x. 0 \<le> x \<and> x < 1 \<longrightarrow> subpath 0 u g x \<in> interior S)  \<and>  g u \<in> closure S"
 proof -
   obtain u where "0 \<le> u" "u \<le> 1"
              and gxin: "\<And>x. 0 \<le> x \<and> x < u \<Longrightarrow> g x \<in> interior S"
              and gunot: "(g u \<notin> interior S)" and u0: "(u = 0 \<or> g u \<in> closure S)"
     using subpath_to_frontier_explicit [OF assms] by blast
   show ?thesis
     apply (rule that [OF \<open>0 \<le> u\<close> \<open>u \<le> 1\<close>])
     apply (simp add: gunot)
     using \<open>0 \<le> u\<close> u0 by (force simp: subpath_def gxin)
 qed
 
 lemma subpath_to_frontier:
     assumes g: "path g" and g0: "pathstart g \<in> closure S" and g1: "pathfinish g \<notin> S"
     obtains u where "0 \<le> u" "u \<le> 1" "g u \<in> frontier S" "(path_image(subpath 0 u g) - {g u}) \<subseteq> interior S"
 proof -
   obtain u where "0 \<le> u" "u \<le> 1"
              and notin: "g u \<notin> interior S"
              and disj: "u = 0 \<or>
                         (\<forall>x. 0 \<le> x \<and> x < 1 \<longrightarrow> subpath 0 u g x \<in> interior S) \<and> g u \<in> closure S"
     using subpath_to_frontier_strong [OF g g1] by blast
   show ?thesis
     apply (rule that [OF \<open>0 \<le> u\<close> \<open>u \<le> 1\<close>])
     apply (metis DiffI disj frontier_def g0 notin pathstart_def)
     using \<open>0 \<le> u\<close> g0 disj
     apply (simp add: path_image_subpath_gen)
     apply (auto simp: closed_segment_eq_real_ivl pathstart_def pathfinish_def subpath_def)
     apply (rename_tac y)
     apply (drule_tac x="y/u" in spec)
     apply (auto split: if_split_asm)
     done
 qed
 
 lemma exists_path_subpath_to_frontier:
     fixes S :: "'a::real_normed_vector set"
     assumes "path g" "pathstart g \<in> closure S" "pathfinish g \<notin> S"
     obtains h where "path h" "pathstart h = pathstart g" "path_image h \<subseteq> path_image g"
                     "path_image h - {pathfinish h} \<subseteq> interior S"
                     "pathfinish h \<in> frontier S"
 proof -
   obtain u where u: "0 \<le> u" "u \<le> 1" "g u \<in> frontier S" "(path_image(subpath 0 u g) - {g u}) \<subseteq> interior S"
     using subpath_to_frontier [OF assms] by blast
   show ?thesis
     apply (rule that [of "subpath 0 u g"])
     using assms u
     apply (simp_all add: path_image_subpath)
     apply (simp add: pathstart_def)
     apply (force simp: closed_segment_eq_real_ivl path_image_def)
     done
 qed
 
 lemma exists_path_subpath_to_frontier_closed:
     fixes S :: "'a::real_normed_vector set"
     assumes S: "closed S" and g: "path g" and g0: "pathstart g \<in> S" and g1: "pathfinish g \<notin> S"
     obtains h where "path h" "pathstart h = pathstart g" "path_image h \<subseteq> path_image g \<inter> S"
                     "pathfinish h \<in> frontier S"
 proof -
   obtain h where h: "path h" "pathstart h = pathstart g" "path_image h \<subseteq> path_image g"
                     "path_image h - {pathfinish h} \<subseteq> interior S"
                     "pathfinish h \<in> frontier S"
     using exists_path_subpath_to_frontier [OF g _ g1] closure_closed [OF S] g0 by auto
   show ?thesis
     apply (rule that [OF \<open>path h\<close>])
     using assms h
     apply auto
     apply (metis Diff_single_insert frontier_subset_eq insert_iff interior_subset subset_iff)
     done
 qed
 
 
 subsection \<open>Shift Path to Start at Some Given Point\<close>
 
 definition\<^marker>\<open>tag important\<close> shiftpath :: "real \<Rightarrow> (real \<Rightarrow> 'a::topological_space) \<Rightarrow> real \<Rightarrow> 'a"
   where "shiftpath a f = (\<lambda>x. if (a + x) \<le> 1 then f (a + x) else f (a + x - 1))"
 
 lemma shiftpath_alt_def: "shiftpath a f = (\<lambda>x. if x \<le> 1-a then f (a + x) else f (a + x - 1))"
   by (auto simp: shiftpath_def)
 
 lemma pathstart_shiftpath: "a \<le> 1 \<Longrightarrow> pathstart (shiftpath a g) = g a"
   unfolding pathstart_def shiftpath_def by auto
 
 lemma pathfinish_shiftpath:
   assumes "0 \<le> a"
     and "pathfinish g = pathstart g"
   shows "pathfinish (shiftpath a g) = g a"
   using assms
   unfolding pathstart_def pathfinish_def shiftpath_def
   by auto
 
 lemma endpoints_shiftpath:
   assumes "pathfinish g = pathstart g"
     and "a \<in> {0 .. 1}"
   shows "pathfinish (shiftpath a g) = g a"
     and "pathstart (shiftpath a g) = g a"
   using assms
   by (auto intro!: pathfinish_shiftpath pathstart_shiftpath)
 
 lemma closed_shiftpath:
   assumes "pathfinish g = pathstart g"
     and "a \<in> {0..1}"
   shows "pathfinish (shiftpath a g) = pathstart (shiftpath a g)"
   using endpoints_shiftpath[OF assms]
   by auto
 
 lemma path_shiftpath:
   assumes "path g"
     and "pathfinish g = pathstart g"
     and "a \<in> {0..1}"
   shows "path (shiftpath a g)"
 proof -
   have *: "{0 .. 1} = {0 .. 1-a} \<union> {1-a .. 1}"
     using assms(3) by auto
   have **: "\<And>x. x + a = 1 \<Longrightarrow> g (x + a - 1) = g (x + a)"
     using assms(2)[unfolded pathfinish_def pathstart_def]
     by auto
   show ?thesis
     unfolding path_def shiftpath_def *
   proof (rule continuous_on_closed_Un)
     have contg: "continuous_on {0..1} g"
       using \<open>path g\<close> path_def by blast
     show "continuous_on {0..1-a} (\<lambda>x. if a + x \<le> 1 then g (a + x) else g (a + x - 1))"
     proof (rule continuous_on_eq)
       show "continuous_on {0..1-a} (g \<circ> (+) a)"
         by (intro continuous_intros continuous_on_subset [OF contg]) (use \<open>a \<in> {0..1}\<close> in auto)
     qed auto
     show "continuous_on {1-a..1} (\<lambda>x. if a + x \<le> 1 then g (a + x) else g (a + x - 1))"
     proof (rule continuous_on_eq)
       show "continuous_on {1-a..1} (g \<circ> (+) (a - 1))"
         by (intro continuous_intros continuous_on_subset [OF contg]) (use \<open>a \<in> {0..1}\<close> in auto)
     qed (auto simp:  "**" add.commute add_diff_eq)
   qed auto
 qed
 
 lemma shiftpath_shiftpath:
   assumes "pathfinish g = pathstart g"
     and "a \<in> {0..1}"
     and "x \<in> {0..1}"
   shows "shiftpath (1 - a) (shiftpath a g) x = g x"
   using assms
   unfolding pathfinish_def pathstart_def shiftpath_def
   by auto
 
 lemma path_image_shiftpath:
   assumes a: "a \<in> {0..1}"
     and "pathfinish g = pathstart g"
   shows "path_image (shiftpath a g) = path_image g"
 proof -
   { fix x
     assume g: "g 1 = g 0" "x \<in> {0..1::real}" and gne: "\<And>y. y\<in>{0..1} \<inter> {x. \<not> a + x \<le> 1} \<Longrightarrow> g x \<noteq> g (a + y - 1)"
     then have "\<exists>y\<in>{0..1} \<inter> {x. a + x \<le> 1}. g x = g (a + y)"
     proof (cases "a \<le> x")
       case False
       then show ?thesis
         apply (rule_tac x="1 + x - a" in bexI)
         using g gne[of "1 + x - a"] a
         apply (force simp: field_simps)+
         done
     next
       case True
       then show ?thesis
         using g a  by (rule_tac x="x - a" in bexI) (auto simp: field_simps)
     qed
   }
   then show ?thesis
     using assms
     unfolding shiftpath_def path_image_def pathfinish_def pathstart_def
     by (auto simp: image_iff)
 qed
 
 lemma simple_path_shiftpath:
   assumes "simple_path g" "pathfinish g = pathstart g" and a: "0 \<le> a" "a \<le> 1"
     shows "simple_path (shiftpath a g)"
   unfolding simple_path_def
 proof (intro conjI impI ballI)
   show "path (shiftpath a g)"
     by (simp add: assms path_shiftpath simple_path_imp_path)
   have *: "\<And>x y. \<lbrakk>g x = g y; x \<in> {0..1}; y \<in> {0..1}\<rbrakk> \<Longrightarrow> x = y \<or> x = 0 \<and> y = 1 \<or> x = 1 \<and> y = 0"
     using assms by (simp add:  simple_path_def)
   show "x = y \<or> x = 0 \<and> y = 1 \<or> x = 1 \<and> y = 0"
     if "x \<in> {0..1}" "y \<in> {0..1}" "shiftpath a g x = shiftpath a g y" for x y
     using that a unfolding shiftpath_def
     by (force split: if_split_asm dest!: *)
 qed
 
 
 subsection \<open>Straight-Line Paths\<close>
 
 definition\<^marker>\<open>tag important\<close> linepath :: "'a::real_normed_vector \<Rightarrow> 'a \<Rightarrow> real \<Rightarrow> 'a"
   where "linepath a b = (\<lambda>x. (1 - x) *\<^sub>R a + x *\<^sub>R b)"
 
 lemma pathstart_linepath[simp]: "pathstart (linepath a b) = a"
   unfolding pathstart_def linepath_def
   by auto
 
 lemma pathfinish_linepath[simp]: "pathfinish (linepath a b) = b"
   unfolding pathfinish_def linepath_def
   by auto
 
 lemma linepath_inner: "linepath a b x \<bullet> v = linepath (a \<bullet> v) (b \<bullet> v) x"
   by (simp add: linepath_def algebra_simps)
 
 lemma Re_linepath': "Re (linepath a b x) = linepath (Re a) (Re b) x"
   by (simp add: linepath_def)
 
 lemma Im_linepath': "Im (linepath a b x) = linepath (Im a) (Im b) x"
   by (simp add: linepath_def)
 
 lemma linepath_0': "linepath a b 0 = a"
   by (simp add: linepath_def)
 
 lemma linepath_1': "linepath a b 1 = b"
   by (simp add: linepath_def)
 
 lemma continuous_linepath_at[intro]: "continuous (at x) (linepath a b)"
   unfolding linepath_def
   by (intro continuous_intros)
 
 lemma continuous_on_linepath [intro,continuous_intros]: "continuous_on s (linepath a b)"
   using continuous_linepath_at
   by (auto intro!: continuous_at_imp_continuous_on)
 
 lemma path_linepath[iff]: "path (linepath a b)"
   unfolding path_def
   by (rule continuous_on_linepath)
 
 lemma path_image_linepath[simp]: "path_image (linepath a b) = closed_segment a b"
   unfolding path_image_def segment linepath_def
   by auto
 
 lemma reversepath_linepath[simp]: "reversepath (linepath a b) = linepath b a"
   unfolding reversepath_def linepath_def
   by auto
 
 lemma linepath_0 [simp]: "linepath 0 b x = x *\<^sub>R b"
   by (simp add: linepath_def)
 
 lemma linepath_cnj: "cnj (linepath a b x) = linepath (cnj a) (cnj b) x"
   by (simp add: linepath_def)
 
 lemma arc_linepath:
   assumes "a \<noteq> b" shows [simp]: "arc (linepath a b)"
 proof -
   {
     fix x y :: "real"
     assume "x *\<^sub>R b + y *\<^sub>R a = x *\<^sub>R a + y *\<^sub>R b"
     then have "(x - y) *\<^sub>R a = (x - y) *\<^sub>R b"
       by (simp add: algebra_simps)
     with assms have "x = y"
       by simp
   }
   then show ?thesis
     unfolding arc_def inj_on_def
     by (fastforce simp: algebra_simps linepath_def)
 qed
 
 lemma simple_path_linepath[intro]: "a \<noteq> b \<Longrightarrow> simple_path (linepath a b)"
   by (simp add: arc_imp_simple_path)
 
 lemma linepath_trivial [simp]: "linepath a a x = a"
   by (simp add: linepath_def real_vector.scale_left_diff_distrib)
 
 lemma linepath_refl: "linepath a a = (\<lambda>x. a)"
   by auto
 
 lemma subpath_refl: "subpath a a g = linepath (g a) (g a)"
   by (simp add: subpath_def linepath_def algebra_simps)
 
 lemma linepath_of_real: "(linepath (of_real a) (of_real b) x) = of_real ((1 - x)*a + x*b)"
   by (simp add: scaleR_conv_of_real linepath_def)
 
 lemma of_real_linepath: "of_real (linepath a b x) = linepath (of_real a) (of_real b) x"
   by (metis linepath_of_real mult.right_neutral of_real_def real_scaleR_def)
 
 lemma inj_on_linepath:
   assumes "a \<noteq> b" shows "inj_on (linepath a b) {0..1}"
 proof (clarsimp simp: inj_on_def linepath_def)
   fix x y
   assume "(1 - x) *\<^sub>R a + x *\<^sub>R b = (1 - y) *\<^sub>R a + y *\<^sub>R b" "0 \<le> x" "x \<le> 1" "0 \<le> y" "y \<le> 1"
   then have "x *\<^sub>R (a - b) = y *\<^sub>R (a - b)"
     by (auto simp: algebra_simps)
   then show "x=y"
     using assms by auto
 qed
 
 lemma linepath_le_1:
   fixes a::"'a::linordered_idom" shows "\<lbrakk>a \<le> 1; b \<le> 1; 0 \<le> u; u \<le> 1\<rbrakk> \<Longrightarrow> (1 - u) * a + u * b \<le> 1"
   using mult_left_le [of a "1-u"] mult_left_le [of b u] by auto
 
 lemma linepath_in_path:
   shows "x \<in> {0..1} \<Longrightarrow> linepath a b x \<in> closed_segment a b"
   by (auto simp: segment linepath_def)
 
 lemma linepath_image_01: "linepath a b ` {0..1} = closed_segment a b"
   by (auto simp: segment linepath_def)
 
 lemma linepath_in_convex_hull:
     fixes x::real
     assumes a: "a \<in> convex hull s"
         and b: "b \<in> convex hull s"
         and x: "0\<le>x" "x\<le>1"
        shows "linepath a b x \<in> convex hull s"
   apply (rule closed_segment_subset_convex_hull [OF a b, THEN subsetD])
   using x
   apply (auto simp: linepath_image_01 [symmetric])
   done
 
 lemma Re_linepath: "Re(linepath (of_real a) (of_real b) x) = (1 - x)*a + x*b"
   by (simp add: linepath_def)
 
 lemma Im_linepath: "Im(linepath (of_real a) (of_real b) x) = 0"
   by (simp add: linepath_def)
 
 lemma bounded_linear_linepath:
   assumes "bounded_linear f"
   shows   "f (linepath a b x) = linepath (f a) (f b) x"
 proof -
   interpret f: bounded_linear f by fact
   show ?thesis by (simp add: linepath_def f.add f.scale)
 qed
 
 lemma bounded_linear_linepath':
   assumes "bounded_linear f"
   shows   "f \<circ> linepath a b = linepath (f a) (f b)"
   using bounded_linear_linepath[OF assms] by (simp add: fun_eq_iff)
 
 lemma linepath_cnj': "cnj \<circ> linepath a b = linepath (cnj a) (cnj b)"
   by (simp add: linepath_def fun_eq_iff)
 
 lemma differentiable_linepath [intro]: "linepath a b differentiable at x within A"
   by (auto simp: linepath_def)
 
 lemma has_vector_derivative_linepath_within:
     "(linepath a b has_vector_derivative (b - a)) (at x within s)"
 apply (simp add: linepath_def has_vector_derivative_def algebra_simps)
 apply (rule derivative_eq_intros | simp)+
 done
 
 
 subsection\<^marker>\<open>tag unimportant\<close>\<open>Segments via convex hulls\<close>
 
 lemma segments_subset_convex_hull:
     "closed_segment a b \<subseteq> (convex hull {a,b,c})"
     "closed_segment a c \<subseteq> (convex hull {a,b,c})"
     "closed_segment b c \<subseteq> (convex hull {a,b,c})"
     "closed_segment b a \<subseteq> (convex hull {a,b,c})"
     "closed_segment c a \<subseteq> (convex hull {a,b,c})"
     "closed_segment c b \<subseteq> (convex hull {a,b,c})"
 by (auto simp: segment_convex_hull linepath_of_real  elim!: rev_subsetD [OF _ hull_mono])
 
 lemma midpoints_in_convex_hull:
   assumes "x \<in> convex hull s" "y \<in> convex hull s"
     shows "midpoint x y \<in> convex hull s"
 proof -
   have "(1 - inverse(2)) *\<^sub>R x + inverse(2) *\<^sub>R y \<in> convex hull s"
     by (rule convexD_alt) (use assms in auto)
   then show ?thesis
     by (simp add: midpoint_def algebra_simps)
 qed
 
 lemma not_in_interior_convex_hull_3:
   fixes a :: "complex"
   shows "a \<notin> interior(convex hull {a,b,c})"
         "b \<notin> interior(convex hull {a,b,c})"
         "c \<notin> interior(convex hull {a,b,c})"
   by (auto simp: card_insert_le_m1 not_in_interior_convex_hull)
 
 lemma midpoint_in_closed_segment [simp]: "midpoint a b \<in> closed_segment a b"
   using midpoints_in_convex_hull segment_convex_hull by blast
 
 lemma midpoint_in_open_segment [simp]: "midpoint a b \<in> open_segment a b \<longleftrightarrow> a \<noteq> b"
   by (simp add: open_segment_def)
 
 lemma continuous_IVT_local_extremum:
   fixes f :: "'a::euclidean_space \<Rightarrow> real"
   assumes contf: "continuous_on (closed_segment a b) f"
       and "a \<noteq> b" "f a = f b"
   obtains z where "z \<in> open_segment a b"
                   "(\<forall>w \<in> closed_segment a b. (f w) \<le> (f z)) \<or>
                    (\<forall>w \<in> closed_segment a b. (f z) \<le> (f w))"
 proof -
   obtain c where "c \<in> closed_segment a b" and c: "\<And>y. y \<in> closed_segment a b \<Longrightarrow> f y \<le> f c"
     using continuous_attains_sup [of "closed_segment a b" f] contf by auto
   obtain d where "d \<in> closed_segment a b" and d: "\<And>y. y \<in> closed_segment a b \<Longrightarrow> f d \<le> f y"
     using continuous_attains_inf [of "closed_segment a b" f] contf by auto
   show ?thesis
   proof (cases "c \<in> open_segment a b \<or> d \<in> open_segment a b")
     case True
     then show ?thesis
       using c d that by blast
   next
     case False
     then have "(c = a \<or> c = b) \<and> (d = a \<or> d = b)"
       by (simp add: \<open>c \<in> closed_segment a b\<close> \<open>d \<in> closed_segment a b\<close> open_segment_def)
     with \<open>a \<noteq> b\<close> \<open>f a = f b\<close> c d show ?thesis
       by (rule_tac z = "midpoint a b" in that) (fastforce+)
   qed
 qed
 
 text\<open>An injective map into R is also an open map w.r.T. the universe, and conversely. \<close>
 proposition injective_eq_1d_open_map_UNIV:
   fixes f :: "real \<Rightarrow> real"
   assumes contf: "continuous_on S f" and S: "is_interval S"
     shows "inj_on f S \<longleftrightarrow> (\<forall>T. open T \<and> T \<subseteq> S \<longrightarrow> open(f ` T))"
           (is "?lhs = ?rhs")
 proof safe
   fix T
   assume injf: ?lhs and "open T" and "T \<subseteq> S"
   have "\<exists>U. open U \<and> f x \<in> U \<and> U \<subseteq> f ` T" if "x \<in> T" for x
   proof -
     obtain \<delta> where "\<delta> > 0" and \<delta>: "cball x \<delta> \<subseteq> T"
       using \<open>open T\<close> \<open>x \<in> T\<close> open_contains_cball_eq by blast
     show ?thesis
     proof (intro exI conjI)
       have "closed_segment (x-\<delta>) (x+\<delta>) = {x-\<delta>..x+\<delta>}"
         using \<open>0 < \<delta>\<close> by (auto simp: closed_segment_eq_real_ivl)
       also have "\<dots> \<subseteq> S"
         using \<delta> \<open>T \<subseteq> S\<close> by (auto simp: dist_norm subset_eq)
       finally have "f ` (open_segment (x-\<delta>) (x+\<delta>)) = open_segment (f (x-\<delta>)) (f (x+\<delta>))"
         using continuous_injective_image_open_segment_1
         by (metis continuous_on_subset [OF contf] inj_on_subset [OF injf])
       then show "open (f ` {x-\<delta><..<x+\<delta>})"
         using \<open>0 < \<delta>\<close> by (simp add: open_segment_eq_real_ivl)
       show "f x \<in> f ` {x - \<delta><..<x + \<delta>}"
         by (auto simp: \<open>\<delta> > 0\<close>)
       show "f ` {x - \<delta><..<x + \<delta>} \<subseteq> f ` T"
         using \<delta> by (auto simp: dist_norm subset_iff)
     qed
   qed
   with open_subopen show "open (f ` T)"
     by blast
 next
   assume R: ?rhs
   have False if xy: "x \<in> S" "y \<in> S" and "f x = f y" "x \<noteq> y" for x y
   proof -
     have "open (f ` open_segment x y)"
       using R
       by (metis S convex_contains_open_segment is_interval_convex open_greaterThanLessThan open_segment_eq_real_ivl xy)
     moreover
     have "continuous_on (closed_segment x y) f"
       by (meson S closed_segment_subset contf continuous_on_subset is_interval_convex that)
     then obtain \<xi> where "\<xi> \<in> open_segment x y"
                     and \<xi>: "(\<forall>w \<in> closed_segment x y. (f w) \<le> (f \<xi>)) \<or>
                             (\<forall>w \<in> closed_segment x y. (f \<xi>) \<le> (f w))"
       using continuous_IVT_local_extremum [of x y f] \<open>f x = f y\<close> \<open>x \<noteq> y\<close> by blast
     ultimately obtain e where "e>0" and e: "\<And>u. dist u (f \<xi>) < e \<Longrightarrow> u \<in> f ` open_segment x y"
       using open_dist by (metis image_eqI)
     have fin: "f \<xi> + (e/2) \<in> f ` open_segment x y" "f \<xi> - (e/2) \<in> f ` open_segment x y"
       using e [of "f \<xi> + (e/2)"] e [of "f \<xi> - (e/2)"] \<open>e > 0\<close> by (auto simp: dist_norm)
     show ?thesis
       using \<xi> \<open>0 < e\<close> fin open_closed_segment by fastforce
   qed
   then show ?lhs
     by (force simp: inj_on_def)
 qed
 
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>Bounding a point away from a path\<close>
 
 lemma not_on_path_ball:
   fixes g :: "real \<Rightarrow> 'a::heine_borel"
   assumes "path g"
     and z: "z \<notin> path_image g"
   shows "\<exists>e > 0. ball z e \<inter> path_image g = {}"
 proof -
   have "closed (path_image g)"
     by (simp add: \<open>path g\<close> closed_path_image)
   then obtain a where "a \<in> path_image g" "\<forall>y \<in> path_image g. dist z a \<le> dist z y"
     by (auto intro: distance_attains_inf[OF _ path_image_nonempty, of g z])
   then show ?thesis
     by (rule_tac x="dist z a" in exI) (use dist_commute z in auto)
 qed
 
 lemma not_on_path_cball:
   fixes g :: "real \<Rightarrow> 'a::heine_borel"
   assumes "path g"
     and "z \<notin> path_image g"
   shows "\<exists>e>0. cball z e \<inter> (path_image g) = {}"
 proof -
   obtain e where "ball z e \<inter> path_image g = {}" "e > 0"
     using not_on_path_ball[OF assms] by auto
   moreover have "cball z (e/2) \<subseteq> ball z e"
     using \<open>e > 0\<close> by auto
   ultimately show ?thesis
     by (rule_tac x="e/2" in exI) auto
 qed
 
 subsection \<open>Path component\<close>
 
 text \<open>Original formalization by Tom Hales\<close>
 
 definition\<^marker>\<open>tag important\<close> "path_component s x y \<longleftrightarrow>
   (\<exists>g. path g \<and> path_image g \<subseteq> s \<and> pathstart g = x \<and> pathfinish g = y)"
 
 abbreviation\<^marker>\<open>tag important\<close>
   "path_component_set s x \<equiv> Collect (path_component s x)"
 
 lemmas path_defs = path_def pathstart_def pathfinish_def path_image_def path_component_def
 
 lemma path_component_mem:
   assumes "path_component s x y"
   shows "x \<in> s" and "y \<in> s"
   using assms
   unfolding path_defs
   by auto
 
 lemma path_component_refl:
   assumes "x \<in> s"
   shows "path_component s x x"
   unfolding path_defs
   apply (rule_tac x="\<lambda>u. x" in exI)
   using assms
   apply (auto intro!: continuous_intros)
   done
 
 lemma path_component_refl_eq: "path_component s x x \<longleftrightarrow> x \<in> s"
   by (auto intro!: path_component_mem path_component_refl)
 
 lemma path_component_sym: "path_component s x y \<Longrightarrow> path_component s y x"
   unfolding path_component_def
   apply (erule exE)
   apply (rule_tac x="reversepath g" in exI, auto)
   done
 
 lemma path_component_trans:
   assumes "path_component s x y" and "path_component s y z"
   shows "path_component s x z"
   using assms
   unfolding path_component_def
   apply (elim exE)
   apply (rule_tac x="g +++ ga" in exI)
   apply (auto simp: path_image_join)
   done
 
 lemma path_component_of_subset: "s \<subseteq> t \<Longrightarrow> path_component s x y \<Longrightarrow> path_component t x y"
   unfolding path_component_def by auto
 
 lemma path_component_linepath:
     fixes s :: "'a::real_normed_vector set"
     shows "closed_segment a b \<subseteq> s \<Longrightarrow> path_component s a b"
   unfolding path_component_def
   by (rule_tac x="linepath a b" in exI, auto)
 
 subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Path components as sets\<close>
 
 lemma path_component_set:
   "path_component_set s x =
     {y. (\<exists>g. path g \<and> path_image g \<subseteq> s \<and> pathstart g = x \<and> pathfinish g = y)}"
   by (auto simp: path_component_def)
 
 lemma path_component_subset: "path_component_set s x \<subseteq> s"
   by (auto simp: path_component_mem(2))
 
 lemma path_component_eq_empty: "path_component_set s x = {} \<longleftrightarrow> x \<notin> s"
   using path_component_mem path_component_refl_eq
     by fastforce
 
 lemma path_component_mono:
      "s \<subseteq> t \<Longrightarrow> (path_component_set s x) \<subseteq> (path_component_set t x)"
   by (simp add: Collect_mono path_component_of_subset)
 
 lemma path_component_eq:
    "y \<in> path_component_set s x \<Longrightarrow> path_component_set s y = path_component_set s x"
 by (metis (no_types, lifting) Collect_cong mem_Collect_eq path_component_sym path_component_trans)
 
 
 subsection \<open>Path connectedness of a space\<close>
 
 definition\<^marker>\<open>tag important\<close> "path_connected s \<longleftrightarrow>
   (\<forall>x\<in>s. \<forall>y\<in>s. \<exists>g. path g \<and> path_image g \<subseteq> s \<and> pathstart g = x \<and> pathfinish g = y)"
 
 lemma path_connectedin_iff_path_connected_real [simp]:
      "path_connectedin euclideanreal S \<longleftrightarrow> path_connected S"
   by (simp add: path_connectedin path_connected_def path_defs)
 
 lemma path_connected_component: "path_connected s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. path_component s x y)"
   unfolding path_connected_def path_component_def by auto
 
 lemma path_connected_component_set: "path_connected s \<longleftrightarrow> (\<forall>x\<in>s. path_component_set s x = s)"
   unfolding path_connected_component path_component_subset
   using path_component_mem by blast
 
 lemma path_component_maximal:
      "\<lbrakk>x \<in> t; path_connected t; t \<subseteq> s\<rbrakk> \<Longrightarrow> t \<subseteq> (path_component_set s x)"
   by (metis path_component_mono path_connected_component_set)
 
 lemma convex_imp_path_connected:
   fixes s :: "'a::real_normed_vector set"
   assumes "convex s"
   shows "path_connected s"
   unfolding path_connected_def
   using assms convex_contains_segment by fastforce
 
 lemma path_connected_UNIV [iff]: "path_connected (UNIV :: 'a::real_normed_vector set)"
   by (simp add: convex_imp_path_connected)
 
 lemma path_component_UNIV: "path_component_set UNIV x = (UNIV :: 'a::real_normed_vector set)"
   using path_connected_component_set by auto
 
 lemma path_connected_imp_connected:
   assumes "path_connected S"
   shows "connected S"
 proof (rule connectedI)
   fix e1 e2
   assume as: "open e1" "open e2" "S \<subseteq> e1 \<union> e2" "e1 \<inter> e2 \<inter> S = {}" "e1 \<inter> S \<noteq> {}" "e2 \<inter> S \<noteq> {}"
   then obtain x1 x2 where obt:"x1 \<in> e1 \<inter> S" "x2 \<in> e2 \<inter> S"
     by auto
   then obtain g where g: "path g" "path_image g \<subseteq> S" "pathstart g = x1" "pathfinish g = x2"
     using assms[unfolded path_connected_def,rule_format,of x1 x2] by auto
   have *: "connected {0..1::real}"
     by (auto intro!: convex_connected)
   have "{0..1} \<subseteq> {x \<in> {0..1}. g x \<in> e1} \<union> {x \<in> {0..1}. g x \<in> e2}"
     using as(3) g(2)[unfolded path_defs] by blast
   moreover have "{x \<in> {0..1}. g x \<in> e1} \<inter> {x \<in> {0..1}. g x \<in> e2} = {}"
     using as(4) g(2)[unfolded path_defs]
     unfolding subset_eq
     by auto
   moreover have "{x \<in> {0..1}. g x \<in> e1} \<noteq> {} \<and> {x \<in> {0..1}. g x \<in> e2} \<noteq> {}"
     using g(3,4)[unfolded path_defs]
     using obt
     by (simp add: ex_in_conv [symmetric], metis zero_le_one order_refl)
   ultimately show False
     using *[unfolded connected_local not_ex, rule_format,
       of "{0..1} \<inter> g -` e1" "{0..1} \<inter> g -` e2"]
     using continuous_openin_preimage_gen[OF g(1)[unfolded path_def] as(1)]
     using continuous_openin_preimage_gen[OF g(1)[unfolded path_def] as(2)]
     by auto
 qed
 
 lemma open_path_component:
   fixes S :: "'a::real_normed_vector set"
   assumes "open S"
   shows "open (path_component_set S x)"
   unfolding open_contains_ball
 proof
   fix y
   assume as: "y \<in> path_component_set S x"
   then have "y \<in> S"
     by (simp add: path_component_mem(2))
   then obtain e where e: "e > 0" "ball y e \<subseteq> S"
     using assms[unfolded open_contains_ball]
     by auto
 have "\<And>u. dist y u < e \<Longrightarrow> path_component S x u"
       by (metis (full_types) as centre_in_ball convex_ball convex_imp_path_connected e mem_Collect_eq mem_ball path_component_eq path_component_of_subset path_connected_component)
   then show "\<exists>e > 0. ball y e \<subseteq> path_component_set S x"
     using \<open>e>0\<close> by auto
 qed
 
 lemma open_non_path_component:
   fixes S :: "'a::real_normed_vector set"
   assumes "open S"
   shows "open (S - path_component_set S x)"
   unfolding open_contains_ball
 proof
   fix y
   assume y: "y \<in> S - path_component_set S x"
   then obtain e where e: "e > 0" "ball y e \<subseteq> S"
     using assms openE by auto
   show "\<exists>e>0. ball y e \<subseteq> S - path_component_set S x"
   proof (intro exI conjI subsetI DiffI notI)
     show "\<And>x. x \<in> ball y e \<Longrightarrow> x \<in> S"
       using e by blast
     show False if "z \<in> ball y e" "z \<in> path_component_set S x" for z
     proof -
       have "y \<in> path_component_set S z"
         by (meson assms convex_ball convex_imp_path_connected e open_contains_ball_eq open_path_component path_component_maximal that(1))
       then have "y \<in> path_component_set S x"
         using path_component_eq that(2) by blast
       then show False
         using y by blast
     qed
   qed (use e in auto)
 qed
 
 lemma connected_open_path_connected:
   fixes S :: "'a::real_normed_vector set"
   assumes "open S"
     and "connected S"
   shows "path_connected S"
   unfolding path_connected_component_set
 proof (rule, rule, rule path_component_subset, rule)
   fix x y
   assume "x \<in> S" and "y \<in> S"
   show "y \<in> path_component_set S x"
   proof (rule ccontr)
     assume "\<not> ?thesis"
     moreover have "path_component_set S x \<inter> S \<noteq> {}"
       using \<open>x \<in> S\<close> path_component_eq_empty path_component_subset[of S x]
       by auto
     ultimately
     show False
       using \<open>y \<in> S\<close> open_non_path_component[OF assms(1)] open_path_component[OF assms(1)]
       using assms(2)[unfolded connected_def not_ex, rule_format,
         of "path_component_set S x" "S - path_component_set S x"]
       by auto
   qed
 qed
 
 lemma path_connected_continuous_image:
   assumes "continuous_on S f"
     and "path_connected S"
   shows "path_connected (f ` S)"
   unfolding path_connected_def
 proof (rule, rule)
   fix x' y'
   assume "x' \<in> f ` S" "y' \<in> f ` S"
   then obtain x y where x: "x \<in> S" and y: "y \<in> S" and x': "x' = f x" and y': "y' = f y"
     by auto
   from x y obtain g where "path g \<and> path_image g \<subseteq> S \<and> pathstart g = x \<and> pathfinish g = y"
     using assms(2)[unfolded path_connected_def] by fast
   then show "\<exists>g. path g \<and> path_image g \<subseteq> f ` S \<and> pathstart g = x' \<and> pathfinish g = y'"
     unfolding x' y'
     apply (rule_tac x="f \<circ> g" in exI)
     unfolding path_defs
     apply (intro conjI continuous_on_compose continuous_on_subset[OF assms(1)])
     apply auto
     done
 qed
 
 lemma path_connected_translationI:
   fixes a :: "'a :: topological_group_add"
   assumes "path_connected S" shows "path_connected ((\<lambda>x. a + x) ` S)"
   by (intro path_connected_continuous_image assms continuous_intros)
 
 lemma path_connected_translation:
   fixes a :: "'a :: topological_group_add"
   shows "path_connected ((\<lambda>x. a + x) ` S) = path_connected S"
 proof -
   have "\<forall>x y. (+) (x::'a) ` (+) (0 - x) ` y = y"
     by (simp add: image_image)
   then show ?thesis
     by (metis (no_types) path_connected_translationI)
 qed
 
 lemma path_connected_segment [simp]:
     fixes a :: "'a::real_normed_vector"
     shows "path_connected (closed_segment a b)"
   by (simp add: convex_imp_path_connected)
 
 lemma path_connected_open_segment [simp]:
     fixes a :: "'a::real_normed_vector"
     shows "path_connected (open_segment a b)"
   by (simp add: convex_imp_path_connected)
 
 lemma homeomorphic_path_connectedness:
   "S homeomorphic T \<Longrightarrow> path_connected S \<longleftrightarrow> path_connected T"
   unfolding homeomorphic_def homeomorphism_def by (metis path_connected_continuous_image)
 
 lemma path_connected_empty [simp]: "path_connected {}"
   unfolding path_connected_def by auto
 
 lemma path_connected_singleton [simp]: "path_connected {a}"
   unfolding path_connected_def pathstart_def pathfinish_def path_image_def
   apply clarify
   apply (rule_tac x="\<lambda>x. a" in exI)
   apply (simp add: image_constant_conv)
   apply (simp add: path_def)
   done
 
 lemma path_connected_Un:
   assumes "path_connected S"
     and "path_connected T"
     and "S \<inter> T \<noteq> {}"
   shows "path_connected (S \<union> T)"
   unfolding path_connected_component
 proof (intro ballI)
   fix x y
   assume x: "x \<in> S \<union> T" and y: "y \<in> S \<union> T"
   from assms obtain z where z: "z \<in> S" "z \<in> T"
     by auto
   show "path_component (S \<union> T) x y"
     using x y
   proof safe
     assume "x \<in> S" "y \<in> S"
     then show "path_component (S \<union> T) x y"
       by (meson Un_upper1 \<open>path_connected S\<close> path_component_of_subset path_connected_component)
   next
     assume "x \<in> S" "y \<in> T"
     then show "path_component (S \<union> T) x y"
       by (metis z assms(1-2) le_sup_iff order_refl path_component_of_subset path_component_trans path_connected_component)
   next
   assume "x \<in> T" "y \<in> S"
     then show "path_component (S \<union> T) x y"
       by (metis z assms(1-2) le_sup_iff order_refl path_component_of_subset path_component_trans path_connected_component)
   next
     assume "x \<in> T" "y \<in> T"
     then show "path_component (S \<union> T) x y"
       by (metis Un_upper1 assms(2) path_component_of_subset path_connected_component sup_commute)
   qed
 qed
 
 lemma path_connected_UNION:
   assumes "\<And>i. i \<in> A \<Longrightarrow> path_connected (S i)"
     and "\<And>i. i \<in> A \<Longrightarrow> z \<in> S i"
   shows "path_connected (\<Union>i\<in>A. S i)"
   unfolding path_connected_component
 proof clarify
   fix x i y j
   assume *: "i \<in> A" "x \<in> S i" "j \<in> A" "y \<in> S j"
   then have "path_component (S i) x z" and "path_component (S j) z y"
     using assms by (simp_all add: path_connected_component)
   then have "path_component (\<Union>i\<in>A. S i) x z" and "path_component (\<Union>i\<in>A. S i) z y"
     using *(1,3) by (auto elim!: path_component_of_subset [rotated])
   then show "path_component (\<Union>i\<in>A. S i) x y"
     by (rule path_component_trans)
 qed
 
 lemma path_component_path_image_pathstart:
   assumes p: "path p" and x: "x \<in> path_image p"
   shows "path_component (path_image p) (pathstart p) x"
 proof -
   obtain y where x: "x = p y" and y: "0 \<le> y" "y \<le> 1"
     using x by (auto simp: path_image_def)
   show ?thesis
     unfolding path_component_def 
   proof (intro exI conjI)
     have "continuous_on {0..1} (p \<circ> ((*) y))"
       apply (rule continuous_intros)+
       using p [unfolded path_def] y
       apply (auto simp: mult_le_one intro: continuous_on_subset [of _ p])
       done
     then show "path (\<lambda>u. p (y * u))"
       by (simp add: path_def)
     show "path_image (\<lambda>u. p (y * u)) \<subseteq> path_image p"
       using y mult_le_one by (fastforce simp: path_image_def image_iff)
   qed (auto simp: pathstart_def pathfinish_def x)
 qed
 
 lemma path_connected_path_image: "path p \<Longrightarrow> path_connected(path_image p)"
   unfolding path_connected_component
   by (meson path_component_path_image_pathstart path_component_sym path_component_trans)
 
 lemma path_connected_path_component [simp]:
    "path_connected (path_component_set s x)"
 proof -
   { fix y z
     assume pa: "path_component s x y" "path_component s x z"
     then have pae: "path_component_set s x = path_component_set s y"
       using path_component_eq by auto
     have yz: "path_component s y z"
       using pa path_component_sym path_component_trans by blast
     then have "\<exists>g. path g \<and> path_image g \<subseteq> path_component_set s x \<and> pathstart g = y \<and> pathfinish g = z"
       apply (simp add: path_component_def, clarify)
       apply (rule_tac x=g in exI)
       by (simp add: pae path_component_maximal path_connected_path_image pathstart_in_path_image)
   }
   then show ?thesis
     by (simp add: path_connected_def)
 qed
 
 lemma path_component: "path_component S x y \<longleftrightarrow> (\<exists>t. path_connected t \<and> t \<subseteq> S \<and> x \<in> t \<and> y \<in> t)"
   apply (intro iffI)
   apply (metis path_connected_path_image path_defs(5) pathfinish_in_path_image pathstart_in_path_image)
   using path_component_of_subset path_connected_component by blast
 
 lemma path_component_path_component [simp]:
    "path_component_set (path_component_set S x) x = path_component_set S x"
 proof (cases "x \<in> S")
   case True show ?thesis
     apply (rule subset_antisym)
     apply (simp add: path_component_subset)
     by (simp add: True path_component_maximal path_component_refl)
 next
   case False then show ?thesis
     by (metis False empty_iff path_component_eq_empty)
 qed
 
 lemma path_component_subset_connected_component:
    "(path_component_set S x) \<subseteq> (connected_component_set S x)"
 proof (cases "x \<in> S")
   case True show ?thesis
     apply (rule connected_component_maximal)
     apply (auto simp: True path_component_subset path_component_refl path_connected_imp_connected)
     done
 next
   case False then show ?thesis
     using path_component_eq_empty by auto
 qed
 
 
 subsection\<^marker>\<open>tag unimportant\<close>\<open>Lemmas about path-connectedness\<close>
 
 lemma path_connected_linear_image:
   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
   assumes "path_connected S" "bounded_linear f"
     shows "path_connected(f ` S)"
 by (auto simp: linear_continuous_on assms path_connected_continuous_image)
 
 lemma is_interval_path_connected: "is_interval S \<Longrightarrow> path_connected S"
   by (simp add: convex_imp_path_connected is_interval_convex)
 
 lemma path_connected_Ioi[simp]: "path_connected {a<..}" for a :: real
   by (simp add: convex_imp_path_connected)
 
 lemma path_connected_Ici[simp]: "path_connected {a..}" for a :: real
   by (simp add: convex_imp_path_connected)
 
 lemma path_connected_Iio[simp]: "path_connected {..<a}" for a :: real
   by (simp add: convex_imp_path_connected)
 
 lemma path_connected_Iic[simp]: "path_connected {..a}" for a :: real
   by (simp add: convex_imp_path_connected)
 
 lemma path_connected_Ioo[simp]: "path_connected {a<..<b}" for a b :: real
   by (simp add: convex_imp_path_connected)
 
 lemma path_connected_Ioc[simp]: "path_connected {a<..b}" for a b :: real
   by (simp add: convex_imp_path_connected)
 
 lemma path_connected_Ico[simp]: "path_connected {a..<b}" for a b :: real
   by (simp add: convex_imp_path_connected)
 
 lemma path_connectedin_path_image:
   assumes "pathin X g" shows "path_connectedin X (g ` ({0..1}))"
   unfolding pathin_def
 proof (rule path_connectedin_continuous_map_image)
   show "continuous_map (subtopology euclideanreal {0..1}) X g"
     using assms pathin_def by blast
 qed (auto simp: is_interval_1 is_interval_path_connected)
 
 lemma path_connected_space_subconnected:
      "path_connected_space X \<longleftrightarrow>
       (\<forall>x \<in> topspace X. \<forall>y \<in> topspace X. \<exists>S. path_connectedin X S \<and> x \<in> S \<and> y \<in> S)"
   unfolding path_connected_space_def Ball_def
   apply (intro all_cong1 imp_cong refl, safe)
   using path_connectedin_path_image apply fastforce
   by (meson path_connectedin)
 
 lemma connectedin_path_image: "pathin X g \<Longrightarrow> connectedin X (g ` ({0..1}))"
   by (simp add: path_connectedin_imp_connectedin path_connectedin_path_image)
 
 lemma compactin_path_image: "pathin X g \<Longrightarrow> compactin X (g ` ({0..1}))"
   unfolding pathin_def
   by (rule image_compactin [of "top_of_set {0..1}"]) auto
 
 lemma linear_homeomorphism_image:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes "linear f" "inj f"
     obtains g where "homeomorphism (f ` S) S g f"
 using linear_injective_left_inverse [OF assms]
 apply clarify
 apply (rule_tac g=g in that)
 using assms
 apply (auto simp: homeomorphism_def eq_id_iff [symmetric] image_comp comp_def linear_conv_bounded_linear linear_continuous_on)
 done
 
 lemma linear_homeomorphic_image:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes "linear f" "inj f"
     shows "S homeomorphic f ` S"
 by (meson homeomorphic_def homeomorphic_sym linear_homeomorphism_image [OF assms])
 
 lemma path_connected_Times:
   assumes "path_connected s" "path_connected t"
     shows "path_connected (s \<times> t)"
 proof (simp add: path_connected_def Sigma_def, clarify)
   fix x1 y1 x2 y2
   assume "x1 \<in> s" "y1 \<in> t" "x2 \<in> s" "y2 \<in> t"
   obtain g where "path g" and g: "path_image g \<subseteq> s" and gs: "pathstart g = x1" and gf: "pathfinish g = x2"
     using \<open>x1 \<in> s\<close> \<open>x2 \<in> s\<close> assms by (force simp: path_connected_def)
   obtain h where "path h" and h: "path_image h \<subseteq> t" and hs: "pathstart h = y1" and hf: "pathfinish h = y2"
     using \<open>y1 \<in> t\<close> \<open>y2 \<in> t\<close> assms by (force simp: path_connected_def)
   have "path (\<lambda>z. (x1, h z))"
     using \<open>path h\<close>
     apply (simp add: path_def)
     apply (rule continuous_on_compose2 [where f = h])
     apply (rule continuous_intros | force)+
     done
   moreover have "path (\<lambda>z. (g z, y2))"
     using \<open>path g\<close>
     apply (simp add: path_def)
     apply (rule continuous_on_compose2 [where f = g])
     apply (rule continuous_intros | force)+
     done
   ultimately have 1: "path ((\<lambda>z. (x1, h z)) +++ (\<lambda>z. (g z, y2)))"
     by (metis hf gs path_join_imp pathstart_def pathfinish_def)
   have "path_image ((\<lambda>z. (x1, h z)) +++ (\<lambda>z. (g z, y2))) \<subseteq> path_image (\<lambda>z. (x1, h z)) \<union> path_image (\<lambda>z. (g z, y2))"
     by (rule Path_Connected.path_image_join_subset)
   also have "\<dots> \<subseteq> (\<Union>x\<in>s. \<Union>x1\<in>t. {(x, x1)})"
     using g h \<open>x1 \<in> s\<close> \<open>y2 \<in> t\<close> by (force simp: path_image_def)
   finally have 2: "path_image ((\<lambda>z. (x1, h z)) +++ (\<lambda>z. (g z, y2))) \<subseteq> (\<Union>x\<in>s. \<Union>x1\<in>t. {(x, x1)})" .
   show "\<exists>g. path g \<and> path_image g \<subseteq> (\<Union>x\<in>s. \<Union>x1\<in>t. {(x, x1)}) \<and>
             pathstart g = (x1, y1) \<and> pathfinish g = (x2, y2)"
     apply (intro exI conjI)
        apply (rule 1)
       apply (rule 2)
      apply (metis hs pathstart_def pathstart_join)
     by (metis gf pathfinish_def pathfinish_join)
 qed
 
 lemma is_interval_path_connected_1:
   fixes s :: "real set"
   shows "is_interval s \<longleftrightarrow> path_connected s"
 using is_interval_connected_1 is_interval_path_connected path_connected_imp_connected by blast
 
 
 subsection\<^marker>\<open>tag unimportant\<close>\<open>Path components\<close>
 
 lemma Union_path_component [simp]:
    "Union {path_component_set S x |x. x \<in> S} = S"
 apply (rule subset_antisym)
 using path_component_subset apply force
 using path_component_refl by auto
 
 lemma path_component_disjoint:
    "disjnt (path_component_set S a) (path_component_set S b) \<longleftrightarrow>
     (a \<notin> path_component_set S b)"
 apply (auto simp: disjnt_def)
 using path_component_eq apply fastforce
 using path_component_sym path_component_trans by blast
 
 lemma path_component_eq_eq:
    "path_component S x = path_component S y \<longleftrightarrow>
         (x \<notin> S) \<and> (y \<notin> S) \<or> x \<in> S \<and> y \<in> S \<and> path_component S x y"
 apply (rule iffI, metis (no_types) path_component_mem(1) path_component_refl)
 apply (erule disjE, metis Collect_empty_eq_bot path_component_eq_empty)
 apply (rule ext)
 apply (metis path_component_trans path_component_sym)
 done
 
 lemma path_component_unique:
   assumes "x \<in> c" "c \<subseteq> S" "path_connected c"
           "\<And>c'. \<lbrakk>x \<in> c'; c' \<subseteq> S; path_connected c'\<rbrakk> \<Longrightarrow> c' \<subseteq> c"
    shows "path_component_set S x = c"
 apply (rule subset_antisym)
 using assms
 apply (metis mem_Collect_eq subsetCE path_component_eq_eq path_component_subset path_connected_path_component)
 by (simp add: assms path_component_maximal)
 
 lemma path_component_intermediate_subset:
    "path_component_set u a \<subseteq> t \<and> t \<subseteq> u
         \<Longrightarrow> path_component_set t a = path_component_set u a"
 by (metis (no_types) path_component_mono path_component_path_component subset_antisym)
 
 lemma complement_path_component_Union:
   fixes x :: "'a :: topological_space"
   shows "S - path_component_set S x =
          \<Union>({path_component_set S y| y. y \<in> S} - {path_component_set S x})"
 proof -
   have *: "(\<And>x. x \<in> S - {a} \<Longrightarrow> disjnt a x) \<Longrightarrow> \<Union>S - a = \<Union>(S - {a})"
     for a::"'a set" and S
     by (auto simp: disjnt_def)
   have "\<And>y. y \<in> {path_component_set S x |x. x \<in> S} - {path_component_set S x}
             \<Longrightarrow> disjnt (path_component_set S x) y"
     using path_component_disjoint path_component_eq by fastforce
   then have "\<Union>{path_component_set S x |x. x \<in> S} - path_component_set S x =
              \<Union>({path_component_set S y |y. y \<in> S} - {path_component_set S x})"
     by (meson *)
   then show ?thesis by simp
 qed
 
 
 subsection\<open>Path components\<close>
 
 definition path_component_of
   where "path_component_of X x y \<equiv> \<exists>g. pathin X g \<and> g 0 = x \<and> g 1 = y"
 
 abbreviation path_component_of_set
   where "path_component_of_set X x \<equiv> Collect (path_component_of X x)"
 
 definition path_components_of :: "'a topology \<Rightarrow> 'a set set"
   where "path_components_of X \<equiv> path_component_of_set X ` topspace X"
 
 lemma pathin_canon_iff: "pathin (top_of_set T) g \<longleftrightarrow> path g \<and> g ` {0..1} \<subseteq> T"
   by (simp add: path_def pathin_def)
 
 lemma path_component_of_canon_iff [simp]:
   "path_component_of (top_of_set T) a b \<longleftrightarrow> path_component T a b"
   by (simp add: path_component_of_def pathin_canon_iff path_defs)
 
 lemma path_component_in_topspace:
    "path_component_of X x y \<Longrightarrow> x \<in> topspace X \<and> y \<in> topspace X"
   by (auto simp: path_component_of_def pathin_def continuous_map_def)
 
 lemma path_component_of_refl:
    "path_component_of X x x \<longleftrightarrow> x \<in> topspace X"
   apply (auto simp: path_component_in_topspace)
   apply (force simp: path_component_of_def pathin_const)
   done
 
 lemma path_component_of_sym:
   assumes "path_component_of X x y"
   shows "path_component_of X y x"
   using assms
   apply (clarsimp simp: path_component_of_def pathin_def)
   apply (rule_tac x="g \<circ> (\<lambda>t. 1 - t)" in exI)
   apply (auto intro!: continuous_map_compose)
   apply (force simp: continuous_map_in_subtopology continuous_on_op_minus)
   done
 
 lemma path_component_of_sym_iff:
    "path_component_of X x y \<longleftrightarrow> path_component_of X y x"
   by (metis path_component_of_sym)
 
 lemma continuous_map_cases_le:
   assumes contp: "continuous_map X euclideanreal p"
     and contq: "continuous_map X euclideanreal q"
     and contf: "continuous_map (subtopology X {x. x \<in> topspace X \<and> p x \<le> q x}) Y f"
     and contg: "continuous_map (subtopology X {x. x \<in> topspace X \<and> q x \<le> p x}) Y g"
     and fg: "\<And>x. \<lbrakk>x \<in> topspace X; p x = q x\<rbrakk> \<Longrightarrow> f x = g x"
   shows "continuous_map X Y (\<lambda>x. if p x \<le> q x then f x else g x)"
 proof -
   have "continuous_map X Y (\<lambda>x. if q x - p x \<in> {0..} then f x else g x)"
   proof (rule continuous_map_cases_function)
     show "continuous_map X euclideanreal (\<lambda>x. q x - p x)"
       by (intro contp contq continuous_intros)
     show "continuous_map (subtopology X {x \<in> topspace X. q x - p x \<in> euclideanreal closure_of {0..}}) Y f"
       by (simp add: contf)
     show "continuous_map (subtopology X {x \<in> topspace X. q x - p x \<in> euclideanreal closure_of (topspace euclideanreal - {0..})}) Y g"
       by (simp add: contg flip: Compl_eq_Diff_UNIV)
   qed (auto simp: fg)
   then show ?thesis
     by simp
 qed
 
 lemma continuous_map_cases_lt:
   assumes contp: "continuous_map X euclideanreal p"
     and contq: "continuous_map X euclideanreal q"
     and contf: "continuous_map (subtopology X {x. x \<in> topspace X \<and> p x \<le> q x}) Y f"
     and contg: "continuous_map (subtopology X {x. x \<in> topspace X \<and> q x \<le> p x}) Y g"
     and fg: "\<And>x. \<lbrakk>x \<in> topspace X; p x = q x\<rbrakk> \<Longrightarrow> f x = g x"
   shows "continuous_map X Y (\<lambda>x. if p x < q x then f x else g x)"
 proof -
   have "continuous_map X Y (\<lambda>x. if q x - p x \<in> {0<..} then f x else g x)"
   proof (rule continuous_map_cases_function)
     show "continuous_map X euclideanreal (\<lambda>x. q x - p x)"
       by (intro contp contq continuous_intros)
     show "continuous_map (subtopology X {x \<in> topspace X. q x - p x \<in> euclideanreal closure_of {0<..}}) Y f"
       by (simp add: contf)
     show "continuous_map (subtopology X {x \<in> topspace X. q x - p x \<in> euclideanreal closure_of (topspace euclideanreal - {0<..})}) Y g"
       by (simp add: contg flip: Compl_eq_Diff_UNIV)
   qed (auto simp: fg)
   then show ?thesis
     by simp
 qed
 
 lemma path_component_of_trans:
   assumes "path_component_of X x y" and "path_component_of X y z"
   shows "path_component_of X x z"
   unfolding path_component_of_def pathin_def
 proof -
   let ?T01 = "top_of_set {0..1::real}"
   obtain g1 g2 where g1: "continuous_map ?T01 X g1" "x = g1 0" "y = g1 1"
     and g2: "continuous_map ?T01 X g2" "g2 0 = g1 1" "z = g2 1"
     using assms unfolding path_component_of_def pathin_def by blast
   let ?g = "\<lambda>x. if x \<le> 1/2 then (g1 \<circ> (\<lambda>t. 2 * t)) x else (g2 \<circ> (\<lambda>t. 2 * t -1)) x"
   show "\<exists>g. continuous_map ?T01 X g \<and> g 0 = x \<and> g 1 = z"
   proof (intro exI conjI)
     show "continuous_map (subtopology euclideanreal {0..1}) X ?g"
     proof (intro continuous_map_cases_le continuous_map_compose, force, force)
       show "continuous_map (subtopology ?T01 {x \<in> topspace ?T01. x \<le> 1/2}) ?T01 ((*) 2)"
         by (auto simp: continuous_map_in_subtopology continuous_map_from_subtopology)
       have "continuous_map
              (subtopology (top_of_set {0..1}) {x. 0 \<le> x \<and> x \<le> 1 \<and> 1 \<le> x * 2})
              euclideanreal (\<lambda>t. 2 * t - 1)"
         by (intro continuous_intros) (force intro: continuous_map_from_subtopology)
       then show "continuous_map (subtopology ?T01 {x \<in> topspace ?T01. 1/2 \<le> x}) ?T01 (\<lambda>t. 2 * t - 1)"
         by (force simp: continuous_map_in_subtopology)
       show "(g1 \<circ> (*) 2) x = (g2 \<circ> (\<lambda>t. 2 * t - 1)) x" if "x \<in> topspace ?T01" "x = 1/2" for x
         using that by (simp add: g2(2) mult.commute continuous_map_from_subtopology)
     qed (auto simp: g1 g2)
   qed (auto simp: g1 g2)
 qed
 
 lemma path_component_of_mono:
    "\<lbrakk>path_component_of (subtopology X S) x y; S \<subseteq> T\<rbrakk> \<Longrightarrow> path_component_of (subtopology X T) x y"
   unfolding path_component_of_def
   by (metis subsetD pathin_subtopology)
 
 lemma path_component_of:
   "path_component_of X x y \<longleftrightarrow> (\<exists>T. path_connectedin X T \<and> x \<in> T \<and> y \<in> T)"
   apply (auto simp: path_component_of_def)
   using path_connectedin_path_image apply fastforce
   apply (metis path_connectedin)
   done
 
 lemma path_component_of_set:
    "path_component_of X x y \<longleftrightarrow> (\<exists>g. pathin X g \<and> g 0 = x \<and> g 1 = y)"
   by (auto simp: path_component_of_def)
 
 lemma path_component_of_subset_topspace:
    "Collect(path_component_of X x) \<subseteq> topspace X"
   using path_component_in_topspace by fastforce
 
 lemma path_component_of_eq_empty:
    "Collect(path_component_of X x) = {} \<longleftrightarrow> (x \<notin> topspace X)"
   using path_component_in_topspace path_component_of_refl by fastforce
 
 lemma path_connected_space_iff_path_component:
    "path_connected_space X \<longleftrightarrow> (\<forall>x \<in> topspace X. \<forall>y \<in> topspace X. path_component_of X x y)"
   by (simp add: path_component_of path_connected_space_subconnected)
 
 lemma path_connected_space_imp_path_component_of:
    "\<lbrakk>path_connected_space X; a \<in> topspace X; b \<in> topspace X\<rbrakk>
         \<Longrightarrow> path_component_of X a b"
   by (simp add: path_connected_space_iff_path_component)
 
 lemma path_connected_space_path_component_set:
    "path_connected_space X \<longleftrightarrow> (\<forall>x \<in> topspace X. Collect(path_component_of X x) = topspace X)"
   using path_component_of_subset_topspace path_connected_space_iff_path_component by fastforce
 
 lemma path_component_of_maximal:
    "\<lbrakk>path_connectedin X s; x \<in> s\<rbrakk> \<Longrightarrow> s \<subseteq> Collect(path_component_of X x)"
   using path_component_of by fastforce
 
 lemma path_component_of_equiv:
    "path_component_of X x y \<longleftrightarrow> x \<in> topspace X \<and> y \<in> topspace X \<and> path_component_of X x = path_component_of X y"
     (is "?lhs = ?rhs")
 proof
   assume ?lhs
   then show ?rhs
     apply (simp add: fun_eq_iff path_component_in_topspace)
     apply (meson path_component_of_sym path_component_of_trans)
     done
 qed (simp add: path_component_of_refl)
 
 lemma path_component_of_disjoint:
      "disjnt (Collect (path_component_of X x)) (Collect (path_component_of X y)) \<longleftrightarrow>
       ~(path_component_of X x y)"
   by (force simp: disjnt_def path_component_of_eq_empty path_component_of_equiv)
 
 lemma path_component_of_eq:
    "path_component_of X x = path_component_of X y \<longleftrightarrow>
         (x \<notin> topspace X) \<and> (y \<notin> topspace X) \<or>
         x \<in> topspace X \<and> y \<in> topspace X \<and> path_component_of X x y"
   by (metis Collect_empty_eq_bot path_component_of_eq_empty path_component_of_equiv)
 
 lemma path_connectedin_path_component_of:
   "path_connectedin X (Collect (path_component_of X x))"
 proof -
   have "\<And>y. path_component_of X x y
         \<Longrightarrow> path_component_of (subtopology X (Collect (path_component_of X x))) x y"
     by (meson path_component_of path_component_of_maximal path_connectedin_subtopology)
   then show ?thesis
     apply (simp add: path_connectedin_def path_component_of_subset_topspace path_connected_space_iff_path_component)
     by (metis Int_absorb1 mem_Collect_eq path_component_of_equiv path_component_of_subset_topspace topspace_subtopology)
 qed
 
 lemma path_connectedin_euclidean [simp]:
    "path_connectedin euclidean S \<longleftrightarrow> path_connected S"
   by (auto simp: path_connectedin_def path_connected_space_iff_path_component path_connected_component)
 
 lemma path_connected_space_euclidean_subtopology [simp]:
    "path_connected_space(subtopology euclidean S) \<longleftrightarrow> path_connected S"
   using path_connectedin_topspace by force
 
 lemma Union_path_components_of:
      "\<Union>(path_components_of X) = topspace X"
   by (auto simp: path_components_of_def path_component_of_equiv)
 
 lemma path_components_of_maximal:
    "\<lbrakk>C \<in> path_components_of X; path_connectedin X S; ~disjnt C S\<rbrakk> \<Longrightarrow> S \<subseteq> C"
   apply (auto simp: path_components_of_def path_component_of_equiv)
   using path_component_of_maximal path_connectedin_def apply fastforce
   by (meson disjnt_subset2 path_component_of_disjoint path_component_of_equiv path_component_of_maximal)
 
 lemma pairwise_disjoint_path_components_of:
      "pairwise disjnt (path_components_of X)"
   by (auto simp: path_components_of_def pairwise_def path_component_of_disjoint path_component_of_equiv)
 
 lemma complement_path_components_of_Union:
    "C \<in> path_components_of X
         \<Longrightarrow> topspace X - C = \<Union>(path_components_of X - {C})"
   by (metis Diff_cancel Diff_subset Union_path_components_of cSup_singleton diff_Union_pairwise_disjoint insert_subset pairwise_disjoint_path_components_of)
 
 lemma nonempty_path_components_of:
   "C \<in> path_components_of X \<Longrightarrow> (C \<noteq> {})"
   apply (clarsimp simp: path_components_of_def path_component_of_eq_empty)
   by (meson path_component_of_refl)
 
 lemma path_components_of_subset: "C \<in> path_components_of X \<Longrightarrow> C \<subseteq> topspace X"
   by (auto simp: path_components_of_def path_component_of_equiv)
 
 lemma path_connectedin_path_components_of:
    "C \<in> path_components_of X \<Longrightarrow> path_connectedin X C"
   by (auto simp: path_components_of_def path_connectedin_path_component_of)
 
 lemma path_component_in_path_components_of:
   "Collect (path_component_of X a) \<in> path_components_of X \<longleftrightarrow> a \<in> topspace X"
   apply (rule iffI)
   using nonempty_path_components_of path_component_of_eq_empty apply fastforce
   by (simp add: path_components_of_def)
 
 lemma path_connectedin_Union:
   assumes \<A>: "\<And>S. S \<in> \<A> \<Longrightarrow> path_connectedin X S" "\<Inter>\<A> \<noteq> {}"
   shows "path_connectedin X (\<Union>\<A>)"
 proof -
   obtain a where "\<And>S. S \<in> \<A> \<Longrightarrow> a \<in> S"
     using assms by blast
   then have "\<And>x. x \<in> topspace (subtopology X (\<Union>\<A>)) \<Longrightarrow> path_component_of (subtopology X (\<Union>\<A>)) a x"
     apply (simp)
     by (meson Union_upper \<A> path_component_of path_connectedin_subtopology)
   then show ?thesis
     using \<A> unfolding path_connectedin_def
     by (metis Sup_le_iff path_component_of_equiv path_connected_space_iff_path_component)
 qed
 
 lemma path_connectedin_Un:
    "\<lbrakk>path_connectedin X S; path_connectedin X T; S \<inter> T \<noteq> {}\<rbrakk>
     \<Longrightarrow> path_connectedin X (S \<union> T)"
   by (blast intro: path_connectedin_Union [of "{S,T}", simplified])
 
 lemma path_connected_space_iff_components_eq:
   "path_connected_space X \<longleftrightarrow>
     (\<forall>C \<in> path_components_of X. \<forall>C' \<in> path_components_of X. C = C')"
   unfolding path_components_of_def
 proof (intro iffI ballI)
   assume "\<forall>C \<in> path_component_of_set X ` topspace X.
              \<forall>C' \<in> path_component_of_set X ` topspace X. C = C'"
   then show "path_connected_space X"
     using path_component_of_refl path_connected_space_iff_path_component by fastforce
 qed (auto simp: path_connected_space_path_component_set)
 
 lemma path_components_of_eq_empty:
    "path_components_of X = {} \<longleftrightarrow> topspace X = {}"
   using Union_path_components_of nonempty_path_components_of by fastforce
 
 lemma path_components_of_empty_space:
    "topspace X = {} \<Longrightarrow> path_components_of X = {}"
   by (simp add: path_components_of_eq_empty)
 
 lemma path_components_of_subset_singleton:
   "path_components_of X \<subseteq> {S} \<longleftrightarrow>
         path_connected_space X \<and> (topspace X = {} \<or> topspace X = S)"
 proof (cases "topspace X = {}")
   case True
   then show ?thesis
     by (auto simp: path_components_of_empty_space path_connected_space_topspace_empty)
 next
   case False
   have "(path_components_of X = {S}) \<longleftrightarrow> (path_connected_space X \<and> topspace X = S)"
   proof (intro iffI conjI)
     assume L: "path_components_of X = {S}"
     then show "path_connected_space X"
       by (simp add: path_connected_space_iff_components_eq)
     show "topspace X = S"
       by (metis L ccpo_Sup_singleton [of S] Union_path_components_of)
   next
     assume R: "path_connected_space X \<and> topspace X = S"
     then show "path_components_of X = {S}"
       using ccpo_Sup_singleton [of S]
       by (metis False all_not_in_conv insert_iff mk_disjoint_insert path_component_in_path_components_of path_connected_space_iff_components_eq path_connected_space_path_component_set)
   qed
   with False show ?thesis
     by (simp add: path_components_of_eq_empty subset_singleton_iff)
 qed
 
 lemma path_connected_space_iff_components_subset_singleton:
    "path_connected_space X \<longleftrightarrow> (\<exists>a. path_components_of X \<subseteq> {a})"
   by (simp add: path_components_of_subset_singleton)
 
 lemma path_components_of_eq_singleton:
    "path_components_of X = {S} \<longleftrightarrow> path_connected_space X \<and> topspace X \<noteq> {} \<and> S = topspace X"
   by (metis cSup_singleton insert_not_empty path_components_of_subset_singleton subset_singleton_iff)
 
 lemma path_components_of_path_connected_space:
    "path_connected_space X \<Longrightarrow> path_components_of X = (if topspace X = {} then {} else {topspace X})"
   by (simp add: path_components_of_eq_empty path_components_of_eq_singleton)
 
 lemma path_component_subset_connected_component_of:
    "path_component_of_set X x \<subseteq> connected_component_of_set X x"
 proof (cases "x \<in> topspace X")
   case True
   then show ?thesis
     by (simp add: connected_component_of_maximal path_component_of_refl path_connectedin_imp_connectedin path_connectedin_path_component_of)
 next
   case False
   then show ?thesis
     using path_component_of_eq_empty by fastforce
 qed
 
 lemma exists_path_component_of_superset:
   assumes S: "path_connectedin X S" and ne: "topspace X \<noteq> {}"
   obtains C where "C \<in> path_components_of X" "S \<subseteq> C"
 proof (cases "S = {}")
   case True
   then show ?thesis
     using ne path_components_of_eq_empty that by fastforce
 next
   case False
   then obtain a where "a \<in> S"
     by blast
   show ?thesis
   proof
     show "Collect (path_component_of X a) \<in> path_components_of X"
       by (meson \<open>a \<in> S\<close> S subsetD path_component_in_path_components_of path_connectedin_subset_topspace)
     show "S \<subseteq> Collect (path_component_of X a)"
       by (simp add: S \<open>a \<in> S\<close> path_component_of_maximal)
   qed
 qed
 
 lemma path_component_of_eq_overlap:
    "path_component_of X x = path_component_of X y \<longleftrightarrow>
       (x \<notin> topspace X) \<and> (y \<notin> topspace X) \<or>
       Collect (path_component_of X x) \<inter> Collect (path_component_of X y) \<noteq> {}"
   by (metis disjnt_def empty_iff inf_bot_right mem_Collect_eq path_component_of_disjoint path_component_of_eq path_component_of_eq_empty)
 
 lemma path_component_of_nonoverlap:
    "Collect (path_component_of X x) \<inter> Collect (path_component_of X y) = {} \<longleftrightarrow>
     (x \<notin> topspace X) \<or> (y \<notin> topspace X) \<or>
     path_component_of X x \<noteq> path_component_of X y"
   by (metis inf.idem path_component_of_eq_empty path_component_of_eq_overlap)
 
 lemma path_component_of_overlap:
    "Collect (path_component_of X x) \<inter> Collect (path_component_of X y) \<noteq> {} \<longleftrightarrow>
     x \<in> topspace X \<and> y \<in> topspace X \<and> path_component_of X x = path_component_of X y"
   by (meson path_component_of_nonoverlap)
 
 lemma path_components_of_disjoint:
      "\<lbrakk>C \<in> path_components_of X; C' \<in> path_components_of X\<rbrakk> \<Longrightarrow> disjnt C C' \<longleftrightarrow> C \<noteq> C'"
   by (auto simp: path_components_of_def path_component_of_disjoint path_component_of_equiv)
 
 lemma path_components_of_overlap:
     "\<lbrakk>C \<in> path_components_of X; C' \<in> path_components_of X\<rbrakk> \<Longrightarrow> C \<inter> C' \<noteq> {} \<longleftrightarrow> C = C'"
   by (auto simp: path_components_of_def path_component_of_equiv)
 
 lemma path_component_of_unique:
    "\<lbrakk>x \<in> C; path_connectedin X C; \<And>C'. \<lbrakk>x \<in> C'; path_connectedin X C'\<rbrakk> \<Longrightarrow> C' \<subseteq> C\<rbrakk>
         \<Longrightarrow> Collect (path_component_of X x) = C"
   by (meson subsetD eq_iff path_component_of_maximal path_connectedin_path_component_of)
 
 lemma path_component_of_discrete_topology [simp]:
   "Collect (path_component_of (discrete_topology U) x) = (if x \<in> U then {x} else {})"
 proof -
   have "\<And>C'. \<lbrakk>x \<in> C'; path_connectedin (discrete_topology U) C'\<rbrakk> \<Longrightarrow> C' \<subseteq> {x}"
     by (metis path_connectedin_discrete_topology subsetD singletonD)
   then have "x \<in> U \<Longrightarrow> Collect (path_component_of (discrete_topology U) x) = {x}"
     by (simp add: path_component_of_unique)
   then show ?thesis
     using path_component_in_topspace by fastforce
 qed
 
 lemma path_component_of_discrete_topology_iff [simp]:
   "path_component_of (discrete_topology U) x y \<longleftrightarrow> x \<in> U \<and> y=x"
   by (metis empty_iff insertI1 mem_Collect_eq path_component_of_discrete_topology singletonD)
 
 lemma path_components_of_discrete_topology [simp]:
    "path_components_of (discrete_topology U) = (\<lambda>x. {x}) ` U"
   by (auto simp: path_components_of_def image_def fun_eq_iff)
 
 lemma homeomorphic_map_path_component_of:
   assumes f: "homeomorphic_map X Y f" and x: "x \<in> topspace X"
   shows "Collect (path_component_of Y (f x)) = f ` Collect(path_component_of X x)"
 proof -
   obtain g where g: "homeomorphic_maps X Y f g"
     using f homeomorphic_map_maps by blast
   show ?thesis
   proof
     have "Collect (path_component_of Y (f x)) \<subseteq> topspace Y"
       by (simp add: path_component_of_subset_topspace)
     moreover have "g ` Collect(path_component_of Y (f x)) \<subseteq> Collect (path_component_of X (g (f x)))"
       using g x unfolding homeomorphic_maps_def
       by (metis f homeomorphic_imp_surjective_map imageI mem_Collect_eq path_component_of_maximal path_component_of_refl path_connectedin_continuous_map_image path_connectedin_path_component_of)
     ultimately show "Collect (path_component_of Y (f x)) \<subseteq> f ` Collect (path_component_of X x)"
       using g x unfolding homeomorphic_maps_def continuous_map_def image_iff subset_iff
       by metis
     show "f ` Collect (path_component_of X x) \<subseteq> Collect (path_component_of Y (f x))"
     proof (rule path_component_of_maximal)
       show "path_connectedin Y (f ` Collect (path_component_of X x))"
         by (meson f homeomorphic_map_path_connectedness_eq path_connectedin_path_component_of)
     qed (simp add: path_component_of_refl x)
   qed
 qed
 
 lemma homeomorphic_map_path_components_of:
   assumes "homeomorphic_map X Y f"
   shows "path_components_of Y = (image f) ` (path_components_of X)"
   unfolding path_components_of_def homeomorphic_imp_surjective_map [OF assms, symmetric]
   apply safe
   using assms homeomorphic_map_path_component_of apply fastforce
   by (metis assms homeomorphic_map_path_component_of imageI)
 
 
 subsection \<open>Sphere is path-connected\<close>
 
 lemma path_connected_punctured_universe:
   assumes "2 \<le> DIM('a::euclidean_space)"
   shows "path_connected (- {a::'a})"
 proof -
   let ?A = "{x::'a. \<exists>i\<in>Basis. x \<bullet> i < a \<bullet> i}"
   let ?B = "{x::'a. \<exists>i\<in>Basis. a \<bullet> i < x \<bullet> i}"
 
   have A: "path_connected ?A"
     unfolding Collect_bex_eq
   proof (rule path_connected_UNION)
     fix i :: 'a
     assume "i \<in> Basis"
     then show "(\<Sum>i\<in>Basis. (a \<bullet> i - 1)*\<^sub>R i) \<in> {x::'a. x \<bullet> i < a \<bullet> i}"
       by simp
     show "path_connected {x. x \<bullet> i < a \<bullet> i}"
       using convex_imp_path_connected [OF convex_halfspace_lt, of i "a \<bullet> i"]
       by (simp add: inner_commute)
   qed
   have B: "path_connected ?B"
     unfolding Collect_bex_eq
   proof (rule path_connected_UNION)
     fix i :: 'a
     assume "i \<in> Basis"
     then show "(\<Sum>i\<in>Basis. (a \<bullet> i + 1) *\<^sub>R i) \<in> {x::'a. a \<bullet> i < x \<bullet> i}"
       by simp
     show "path_connected {x. a \<bullet> i < x \<bullet> i}"
       using convex_imp_path_connected [OF convex_halfspace_gt, of "a \<bullet> i" i]
       by (simp add: inner_commute)
   qed
   obtain S :: "'a set" where "S \<subseteq> Basis" and "card S = Suc (Suc 0)"
     using ex_card[OF assms]
     by auto
   then obtain b0 b1 :: 'a where "b0 \<in> Basis" and "b1 \<in> Basis" and "b0 \<noteq> b1"
     unfolding card_Suc_eq by auto
   then have "a + b0 - b1 \<in> ?A \<inter> ?B"
     by (auto simp: inner_simps inner_Basis)
   then have "?A \<inter> ?B \<noteq> {}"
     by fast
   with A B have "path_connected (?A \<union> ?B)"
     by (rule path_connected_Un)
   also have "?A \<union> ?B = {x. \<exists>i\<in>Basis. x \<bullet> i \<noteq> a \<bullet> i}"
     unfolding neq_iff bex_disj_distrib Collect_disj_eq ..
   also have "\<dots> = {x. x \<noteq> a}"
     unfolding euclidean_eq_iff [where 'a='a]
     by (simp add: Bex_def)
   also have "\<dots> = - {a}"
     by auto
   finally show ?thesis .
 qed
 
 corollary connected_punctured_universe:
   "2 \<le> DIM('N::euclidean_space) \<Longrightarrow> connected(- {a::'N})"
   by (simp add: path_connected_punctured_universe path_connected_imp_connected)
 
 proposition path_connected_sphere:
   fixes a :: "'a :: euclidean_space"
   assumes "2 \<le> DIM('a)"
   shows "path_connected(sphere a r)"
 proof (cases r "0::real" rule: linorder_cases)
   case less
   then show ?thesis
     by (simp)
 next
   case equal
   then show ?thesis
     by (simp)
 next
   case greater
   then have eq: "(sphere (0::'a) r) = (\<lambda>x. (r / norm x) *\<^sub>R x) ` (- {0::'a})"
     by (force simp: image_iff split: if_split_asm)
   have "continuous_on (- {0::'a}) (\<lambda>x. (r / norm x) *\<^sub>R x)"
     by (intro continuous_intros) auto
   then have "path_connected ((\<lambda>x. (r / norm x) *\<^sub>R x) ` (- {0::'a}))"
     by (intro path_connected_continuous_image path_connected_punctured_universe assms)
   with eq have "path_connected (sphere (0::'a) r)"
     by auto
   then have "path_connected((+) a ` (sphere (0::'a) r))"
     by (simp add: path_connected_translation)
   then show ?thesis
     by (metis add.right_neutral sphere_translation)
 qed
 
 lemma connected_sphere:
     fixes a :: "'a :: euclidean_space"
     assumes "2 \<le> DIM('a)"
       shows "connected(sphere a r)"
   using path_connected_sphere [OF assms]
   by (simp add: path_connected_imp_connected)
 
 
 corollary path_connected_complement_bounded_convex:
     fixes s :: "'a :: euclidean_space set"
     assumes "bounded s" "convex s" and 2: "2 \<le> DIM('a)"
     shows "path_connected (- s)"
 proof (cases "s = {}")
   case True then show ?thesis
     using convex_imp_path_connected by auto
 next
   case False
   then obtain a where "a \<in> s" by auto
   { fix x y assume "x \<notin> s" "y \<notin> s"
     then have "x \<noteq> a" "y \<noteq> a" using \<open>a \<in> s\<close> by auto
     then have bxy: "bounded(insert x (insert y s))"
       by (simp add: \<open>bounded s\<close>)
     then obtain B::real where B: "0 < B" and Bx: "norm (a - x) < B" and By: "norm (a - y) < B"
                           and "s \<subseteq> ball a B"
       using bounded_subset_ballD [OF bxy, of a] by (auto simp: dist_norm)
     define C where "C = B / norm(x - a)"
     { fix u
       assume u: "(1 - u) *\<^sub>R x + u *\<^sub>R (a + C *\<^sub>R (x - a)) \<in> s" and "0 \<le> u" "u \<le> 1"
       have CC: "1 \<le> 1 + (C - 1) * u"
         using \<open>x \<noteq> a\<close> \<open>0 \<le> u\<close> Bx
         by (auto simp add: C_def norm_minus_commute)
       have *: "\<And>v. (1 - u) *\<^sub>R x + u *\<^sub>R (a + v *\<^sub>R (x - a)) = a + (1 + (v - 1) * u) *\<^sub>R (x - a)"
         by (simp add: algebra_simps)
       have "a + ((1 / (1 + C * u - u)) *\<^sub>R x + ((u / (1 + C * u - u)) *\<^sub>R a + (C * u / (1 + C * u - u)) *\<^sub>R x)) =
             (1 + (u / (1 + C * u - u))) *\<^sub>R a + ((1 / (1 + C * u - u)) + (C * u / (1 + C * u - u))) *\<^sub>R x"
         by (simp add: algebra_simps)
       also have "\<dots> = (1 + (u / (1 + C * u - u))) *\<^sub>R a + (1 + (u / (1 + C * u - u))) *\<^sub>R x"
         using CC by (simp add: field_simps)
       also have "\<dots> = x + (1 + (u / (1 + C * u - u))) *\<^sub>R a + (u / (1 + C * u - u)) *\<^sub>R x"
         by (simp add: algebra_simps)
       also have "\<dots> = x + ((1 / (1 + C * u - u)) *\<^sub>R a +
               ((u / (1 + C * u - u)) *\<^sub>R x + (C * u / (1 + C * u - u)) *\<^sub>R a))"
         using CC by (simp add: field_simps) (simp add: add_divide_distrib scaleR_add_left)
       finally have xeq: "(1 - 1 / (1 + (C - 1) * u)) *\<^sub>R a + (1 / (1 + (C - 1) * u)) *\<^sub>R (a + (1 + (C - 1) * u) *\<^sub>R (x - a)) = x"
         by (simp add: algebra_simps)
       have False
         using \<open>convex s\<close>
         apply (simp add: convex_alt)
         apply (drule_tac x=a in bspec)
          apply (rule  \<open>a \<in> s\<close>)
         apply (drule_tac x="a + (1 + (C - 1) * u) *\<^sub>R (x - a)" in bspec)
          using u apply (simp add: *)
         apply (drule_tac x="1 / (1 + (C - 1) * u)" in spec)
         using \<open>x \<noteq> a\<close> \<open>x \<notin> s\<close> \<open>0 \<le> u\<close> CC
         apply (auto simp: xeq)
         done
     }
     then have pcx: "path_component (- s) x (a + C *\<^sub>R (x - a))"
       by (force simp: closed_segment_def intro!: path_component_linepath)
     define D where "D = B / norm(y - a)"  \<comment> \<open>massive duplication with the proof above\<close>
     { fix u
       assume u: "(1 - u) *\<^sub>R y + u *\<^sub>R (a + D *\<^sub>R (y - a)) \<in> s" and "0 \<le> u" "u \<le> 1"
       have DD: "1 \<le> 1 + (D - 1) * u"
         using \<open>y \<noteq> a\<close> \<open>0 \<le> u\<close> By
         by (auto simp add: D_def norm_minus_commute)
       have *: "\<And>v. (1 - u) *\<^sub>R y + u *\<^sub>R (a + v *\<^sub>R (y - a)) = a + (1 + (v - 1) * u) *\<^sub>R (y - a)"
         by (simp add: algebra_simps)
       have "a + ((1 / (1 + D * u - u)) *\<^sub>R y + ((u / (1 + D * u - u)) *\<^sub>R a + (D * u / (1 + D * u - u)) *\<^sub>R y)) =
             (1 + (u / (1 + D * u - u))) *\<^sub>R a + ((1 / (1 + D * u - u)) + (D * u / (1 + D * u - u))) *\<^sub>R y"
         by (simp add: algebra_simps)
       also have "\<dots> = (1 + (u / (1 + D * u - u))) *\<^sub>R a + (1 + (u / (1 + D * u - u))) *\<^sub>R y"
         using DD by (simp add: field_simps)
       also have "\<dots> = y + (1 + (u / (1 + D * u - u))) *\<^sub>R a + (u / (1 + D * u - u)) *\<^sub>R y"
         by (simp add: algebra_simps)
       also have "\<dots> = y + ((1 / (1 + D * u - u)) *\<^sub>R a +
               ((u / (1 + D * u - u)) *\<^sub>R y + (D * u / (1 + D * u - u)) *\<^sub>R a))"
         using DD by (simp add: field_simps) (simp add: add_divide_distrib scaleR_add_left)
       finally have xeq: "(1 - 1 / (1 + (D - 1) * u)) *\<^sub>R a + (1 / (1 + (D - 1) * u)) *\<^sub>R (a + (1 + (D - 1) * u) *\<^sub>R (y - a)) = y"
         by (simp add: algebra_simps)
       have False
         using \<open>convex s\<close>
         apply (simp add: convex_alt)
         apply (drule_tac x=a in bspec)
          apply (rule  \<open>a \<in> s\<close>)
         apply (drule_tac x="a + (1 + (D - 1) * u) *\<^sub>R (y - a)" in bspec)
          using u apply (simp add: *)
         apply (drule_tac x="1 / (1 + (D - 1) * u)" in spec)
         using \<open>y \<noteq> a\<close> \<open>y \<notin> s\<close> \<open>0 \<le> u\<close> DD
         apply (auto simp: xeq)
         done
     }
     then have pdy: "path_component (- s) y (a + D *\<^sub>R (y - a))"
       by (force simp: closed_segment_def intro!: path_component_linepath)
     have pyx: "path_component (- s) (a + D *\<^sub>R (y - a)) (a + C *\<^sub>R (x - a))"
       apply (rule path_component_of_subset [of "sphere a B"])
        using \<open>s \<subseteq> ball a B\<close>
        apply (force simp: ball_def dist_norm norm_minus_commute)
       apply (rule path_connected_sphere [OF 2, of a B, simplified path_connected_component, rule_format])
        using \<open>x \<noteq> a\<close>  using \<open>y \<noteq> a\<close>  B apply (auto simp: dist_norm C_def D_def)
       done
     have "path_component (- s) x y"
       by (metis path_component_trans path_component_sym pcx pdy pyx)
   }
   then show ?thesis
     by (auto simp: path_connected_component)
 qed
 
 lemma connected_complement_bounded_convex:
     fixes s :: "'a :: euclidean_space set"
     assumes "bounded s" "convex s" "2 \<le> DIM('a)"
       shows  "connected (- s)"
   using path_connected_complement_bounded_convex [OF assms] path_connected_imp_connected by blast
 
 lemma connected_diff_ball:
     fixes s :: "'a :: euclidean_space set"
     assumes "connected s" "cball a r \<subseteq> s" "2 \<le> DIM('a)"
       shows "connected (s - ball a r)"
   apply (rule connected_diff_open_from_closed [OF ball_subset_cball])
   using assms connected_sphere
   apply (auto simp: cball_diff_eq_sphere dist_norm)
   done
 
 proposition connected_open_delete:
   assumes "open S" "connected S" and 2: "2 \<le> DIM('N::euclidean_space)"
     shows "connected(S - {a::'N})"
 proof (cases "a \<in> S")
   case True
   with \<open>open S\<close> obtain \<epsilon> where "\<epsilon> > 0" and \<epsilon>: "cball a \<epsilon> \<subseteq> S"
     using open_contains_cball_eq by blast
   have "dist a (a + \<epsilon> *\<^sub>R (SOME i. i \<in> Basis)) = \<epsilon>"
     by (simp add: dist_norm SOME_Basis \<open>0 < \<epsilon>\<close> less_imp_le)
   with \<epsilon> have "\<Inter>{S - ball a r |r. 0 < r \<and> r < \<epsilon>} \<subseteq> {} \<Longrightarrow> False"
     apply (drule_tac c="a + scaleR (\<epsilon>) ((SOME i. i \<in> Basis))" in subsetD)
     by auto
   then have nonemp: "(\<Inter>{S - ball a r |r. 0 < r \<and> r < \<epsilon>}) = {} \<Longrightarrow> False"
     by auto
   have con: "\<And>r. r < \<epsilon> \<Longrightarrow> connected (S - ball a r)"
     using \<epsilon> by (force intro: connected_diff_ball [OF \<open>connected S\<close> _ 2])
   have "x \<in> \<Union>{S - ball a r |r. 0 < r \<and> r < \<epsilon>}" if "x \<in> S - {a}" for x
     apply (rule UnionI [of "S - ball a (min \<epsilon> (dist a x) / 2)"])
      using that \<open>0 < \<epsilon>\<close> apply simp_all
     apply (rule_tac x="min \<epsilon> (dist a x) / 2" in exI)
     apply auto
     done
   then have "S - {a} = \<Union>{S - ball a r | r. 0 < r \<and> r < \<epsilon>}"
     by auto
   then show ?thesis
     by (auto intro: connected_Union con dest!: nonemp)
 next
   case False then show ?thesis
     by (simp add: \<open>connected S\<close>)
 qed
 
 corollary path_connected_open_delete:
   assumes "open S" "connected S" and 2: "2 \<le> DIM('N::euclidean_space)"
     shows "path_connected(S - {a::'N})"
 by (simp add: assms connected_open_delete connected_open_path_connected open_delete)
 
 corollary path_connected_punctured_ball:
    "2 \<le> DIM('N::euclidean_space) \<Longrightarrow> path_connected(ball a r - {a::'N})"
 by (simp add: path_connected_open_delete)
 
 corollary connected_punctured_ball:
    "2 \<le> DIM('N::euclidean_space) \<Longrightarrow> connected(ball a r - {a::'N})"
 by (simp add: connected_open_delete)
 
 corollary connected_open_delete_finite:
   fixes S T::"'a::euclidean_space set"
   assumes S: "open S" "connected S" and 2: "2 \<le> DIM('a)" and "finite T"
   shows "connected(S - T)"
   using \<open>finite T\<close> S
 proof (induct T)
   case empty
   show ?case using \<open>connected S\<close> by simp
 next
   case (insert x F)
   then have "connected (S-F)" by auto
   moreover have "open (S - F)" using finite_imp_closed[OF \<open>finite F\<close>] \<open>open S\<close> by auto
   ultimately have "connected (S - F - {x})" using connected_open_delete[OF _ _ 2] by auto
   thus ?case by (metis Diff_insert)
 qed
 
 lemma sphere_1D_doubleton_zero:
   assumes 1: "DIM('a) = 1" and "r > 0"
   obtains x y::"'a::euclidean_space"
     where "sphere 0 r = {x,y} \<and> dist x y = 2*r"
 proof -
   obtain b::'a where b: "Basis = {b}"
     using 1 card_1_singletonE by blast
   show ?thesis
   proof (intro that conjI)
     have "x = norm x *\<^sub>R b \<or> x = - norm x *\<^sub>R b" if "r = norm x" for x
     proof -
       have xb: "(x \<bullet> b) *\<^sub>R b = x"
         using euclidean_representation [of x, unfolded b] by force
       then have "norm ((x \<bullet> b) *\<^sub>R b) = norm x"
         by simp
       with b have "\<bar>x \<bullet> b\<bar> = norm x"
         using norm_Basis by (simp add: b)
       with xb show ?thesis
         apply (simp add: abs_if split: if_split_asm)
         apply (metis add.inverse_inverse real_vector.scale_minus_left xb)
         done
     qed
     with \<open>r > 0\<close> b show "sphere 0 r = {r *\<^sub>R b, - r *\<^sub>R b}"
       by (force simp: sphere_def dist_norm)
     have "dist (r *\<^sub>R b) (- r *\<^sub>R b) = norm (r *\<^sub>R b + r *\<^sub>R b)"
       by (simp add: dist_norm)
     also have "\<dots> = norm ((2*r) *\<^sub>R b)"
       by (metis mult_2 scaleR_add_left)
     also have "\<dots> = 2*r"
       using \<open>r > 0\<close> b norm_Basis by fastforce
     finally show "dist (r *\<^sub>R b) (- r *\<^sub>R b) = 2*r" .
   qed
 qed
 
 lemma sphere_1D_doubleton:
   fixes a :: "'a :: euclidean_space"
   assumes "DIM('a) = 1" and "r > 0"
   obtains x y where "sphere a r = {x,y} \<and> dist x y = 2*r"
 proof -
   have "sphere a r = (+) a ` sphere 0 r"
     by (metis add.right_neutral sphere_translation)
   then show ?thesis
     using sphere_1D_doubleton_zero [OF assms]
     by (metis (mono_tags, lifting) dist_add_cancel image_empty image_insert that)
 qed
 
 lemma psubset_sphere_Compl_connected:
   fixes S :: "'a::euclidean_space set"
   assumes S: "S \<subset> sphere a r" and "0 < r" and 2: "2 \<le> DIM('a)"
   shows "connected(- S)"
 proof -
   have "S \<subseteq> sphere a r"
     using S by blast
   obtain b where "dist a b = r" and "b \<notin> S"
     using S mem_sphere by blast
   have CS: "- S = {x. dist a x \<le> r \<and> (x \<notin> S)} \<union> {x. r \<le> dist a x \<and> (x \<notin> S)}"
     by auto
   have "{x. dist a x \<le> r \<and> x \<notin> S} \<inter> {x. r \<le> dist a x \<and> x \<notin> S} \<noteq> {}"
     using \<open>b \<notin> S\<close> \<open>dist a b = r\<close> by blast
   moreover have "connected {x. dist a x \<le> r \<and> x \<notin> S}"
     apply (rule connected_intermediate_closure [of "ball a r"])
     using assms by auto
   moreover
   have "connected {x. r \<le> dist a x \<and> x \<notin> S}"
     apply (rule connected_intermediate_closure [of "- cball a r"])
     using assms apply (auto intro: connected_complement_bounded_convex)
     apply (metis ComplI interior_cball interior_closure mem_ball not_less)
     done
   ultimately show ?thesis
     by (simp add: CS connected_Un)
 qed
 
 
 subsection\<open>Every annulus is a connected set\<close>
 
 lemma path_connected_2DIM_I:
   fixes a :: "'N::euclidean_space"
   assumes 2: "2 \<le> DIM('N)" and pc: "path_connected {r. 0 \<le> r \<and> P r}"
   shows "path_connected {x. P(norm(x - a))}"
 proof -
   have "{x. P(norm(x - a))} = (+) a ` {x. P(norm x)}"
     by force
   moreover have "path_connected {x::'N. P(norm x)}"
   proof -
     let ?D = "{x. 0 \<le> x \<and> P x} \<times> sphere (0::'N) 1"
     have "x \<in> (\<lambda>z. fst z *\<^sub>R snd z) ` ?D"
       if "P (norm x)" for x::'N
     proof (cases "x=0")
       case True
       with that show ?thesis
         apply (simp add: image_iff)
         apply (rule_tac x=0 in exI, simp)
         using vector_choose_size zero_le_one by blast
     next
       case False
       with that show ?thesis
         by (rule_tac x="(norm x, x /\<^sub>R norm x)" in image_eqI) auto
     qed
     then have *: "{x::'N. P(norm x)} =  (\<lambda>z. fst z *\<^sub>R snd z) ` ?D"
       by auto
     have "continuous_on ?D (\<lambda>z:: real\<times>'N. fst z *\<^sub>R snd z)"
       by (intro continuous_intros)
     moreover have "path_connected ?D"
       by (metis path_connected_Times [OF pc] path_connected_sphere 2)
     ultimately show ?thesis
       apply (subst *)
       apply (rule path_connected_continuous_image, auto)
       done
   qed
   ultimately show ?thesis
     using path_connected_translation by metis
 qed
 
 proposition path_connected_annulus:
   fixes a :: "'N::euclidean_space"
   assumes "2 \<le> DIM('N)"
   shows "path_connected {x. r1 < norm(x - a) \<and> norm(x - a) < r2}"
         "path_connected {x. r1 < norm(x - a) \<and> norm(x - a) \<le> r2}"
         "path_connected {x. r1 \<le> norm(x - a) \<and> norm(x - a) < r2}"
         "path_connected {x. r1 \<le> norm(x - a) \<and> norm(x - a) \<le> r2}"
   by (auto simp: is_interval_def intro!: is_interval_convex convex_imp_path_connected path_connected_2DIM_I [OF assms])
 
 proposition connected_annulus:
   fixes a :: "'N::euclidean_space"
   assumes "2 \<le> DIM('N::euclidean_space)"
   shows "connected {x. r1 < norm(x - a) \<and> norm(x - a) < r2}"
         "connected {x. r1 < norm(x - a) \<and> norm(x - a) \<le> r2}"
         "connected {x. r1 \<le> norm(x - a) \<and> norm(x - a) < r2}"
         "connected {x. r1 \<le> norm(x - a) \<and> norm(x - a) \<le> r2}"
   by (auto simp: path_connected_annulus [OF assms] path_connected_imp_connected)
 
 
 subsection\<^marker>\<open>tag unimportant\<close>\<open>Relations between components and path components\<close>
 
 lemma open_connected_component:
   fixes s :: "'a::real_normed_vector set"
   shows "open s \<Longrightarrow> open (connected_component_set s x)"
     apply (simp add: open_contains_ball, clarify)
     apply (rename_tac y)
     apply (drule_tac x=y in bspec)
      apply (simp add: connected_component_in, clarify)
     apply (rule_tac x=e in exI)
     by (metis mem_Collect_eq connected_component_eq connected_component_maximal centre_in_ball connected_ball)
 
 corollary open_components:
     fixes s :: "'a::real_normed_vector set"
     shows "\<lbrakk>open u; s \<in> components u\<rbrakk> \<Longrightarrow> open s"
   by (simp add: components_iff) (metis open_connected_component)
 
 lemma in_closure_connected_component:
   fixes s :: "'a::real_normed_vector set"
   assumes x: "x \<in> s" and s: "open s"
   shows "x \<in> closure (connected_component_set s y) \<longleftrightarrow>  x \<in> connected_component_set s y"
 proof -
   { assume "x \<in> closure (connected_component_set s y)"
     moreover have "x \<in> connected_component_set s x"
       using x by simp
     ultimately have "x \<in> connected_component_set s y"
       using s by (meson Compl_disjoint closure_iff_nhds_not_empty connected_component_disjoint disjoint_eq_subset_Compl open_connected_component)
   }
   then show ?thesis
     by (auto simp: closure_def)
 qed
 
 lemma connected_disjoint_Union_open_pick:
   assumes "pairwise disjnt B"
           "\<And>S. S \<in> A \<Longrightarrow> connected S \<and> S \<noteq> {}"
           "\<And>S. S \<in> B \<Longrightarrow> open S"
           "\<Union>A \<subseteq> \<Union>B"
           "S \<in> A"
   obtains T where "T \<in> B" "S \<subseteq> T" "S \<inter> \<Union>(B - {T}) = {}"
 proof -
   have "S \<subseteq> \<Union>B" "connected S" "S \<noteq> {}"
     using assms \<open>S \<in> A\<close> by blast+
   then obtain T where "T \<in> B" "S \<inter> T \<noteq> {}"
     by (metis Sup_inf_eq_bot_iff inf.absorb_iff2 inf_commute)
   have 1: "open T" by (simp add: \<open>T \<in> B\<close> assms)
   have 2: "open (\<Union>(B-{T}))" using assms by blast
   have 3: "S \<subseteq> T \<union> \<Union>(B - {T})" using \<open>S \<subseteq> \<Union>B\<close> by blast
   have "T \<inter> \<Union>(B - {T}) = {}" using \<open>T \<in> B\<close> \<open>pairwise disjnt B\<close>
     by (auto simp: pairwise_def disjnt_def)
   then have 4: "T \<inter> \<Union>(B - {T}) \<inter> S = {}" by auto
-  from connectedD [OF \<open>connected S\<close> 1 2 3 4]
+  from connectedD [OF \<open>connected S\<close> 1 2 4 3]
   have "S \<inter> \<Union>(B-{T}) = {}"
     by (auto simp: Int_commute \<open>S \<inter> T \<noteq> {}\<close>)
   with \<open>T \<in> B\<close> have "S \<subseteq> T"
     using "3" by auto
   show ?thesis
     using \<open>S \<inter> \<Union>(B - {T}) = {}\<close> \<open>S \<subseteq> T\<close> \<open>T \<in> B\<close> that by auto
 qed
 
 lemma connected_disjoint_Union_open_subset:
   assumes A: "pairwise disjnt A" and B: "pairwise disjnt B"
       and SA: "\<And>S. S \<in> A \<Longrightarrow> open S \<and> connected S \<and> S \<noteq> {}"
       and SB: "\<And>S. S \<in> B \<Longrightarrow> open S \<and> connected S \<and> S \<noteq> {}"
       and eq [simp]: "\<Union>A = \<Union>B"
     shows "A \<subseteq> B"
 proof
   fix S
   assume "S \<in> A"
   obtain T where "T \<in> B" "S \<subseteq> T" "S \<inter> \<Union>(B - {T}) = {}"
       apply (rule connected_disjoint_Union_open_pick [OF B, of A])
       using SA SB \<open>S \<in> A\<close> by auto
   moreover obtain S' where "S' \<in> A" "T \<subseteq> S'" "T \<inter> \<Union>(A - {S'}) = {}"
       apply (rule connected_disjoint_Union_open_pick [OF A, of B])
       using SA SB \<open>T \<in> B\<close> by auto
   ultimately have "S' = S"
     by (metis A Int_subset_iff SA \<open>S \<in> A\<close> disjnt_def inf.orderE pairwise_def)
   with \<open>T \<subseteq> S'\<close> have "T \<subseteq> S" by simp
   with \<open>S \<subseteq> T\<close> have "S = T" by blast
   with \<open>T \<in> B\<close> show "S \<in> B" by simp
 qed
 
 lemma connected_disjoint_Union_open_unique:
   assumes A: "pairwise disjnt A" and B: "pairwise disjnt B"
       and SA: "\<And>S. S \<in> A \<Longrightarrow> open S \<and> connected S \<and> S \<noteq> {}"
       and SB: "\<And>S. S \<in> B \<Longrightarrow> open S \<and> connected S \<and> S \<noteq> {}"
       and eq [simp]: "\<Union>A = \<Union>B"
     shows "A = B"
 by (rule subset_antisym; metis connected_disjoint_Union_open_subset assms)
 
 proposition components_open_unique:
  fixes S :: "'a::real_normed_vector set"
   assumes "pairwise disjnt A" "\<Union>A = S"
           "\<And>X. X \<in> A \<Longrightarrow> open X \<and> connected X \<and> X \<noteq> {}"
     shows "components S = A"
 proof -
   have "open S" using assms by blast
   show ?thesis
     apply (rule connected_disjoint_Union_open_unique)
     apply (simp add: components_eq disjnt_def pairwise_def)
     using \<open>open S\<close>
     apply (simp_all add: assms open_components in_components_connected in_components_nonempty)
     done
 qed
 
 
 subsection\<^marker>\<open>tag unimportant\<close>\<open>Existence of unbounded components\<close>
 
 lemma cobounded_unbounded_component:
     fixes s :: "'a :: euclidean_space set"
     assumes "bounded (-s)"
       shows "\<exists>x. x \<in> s \<and> \<not> bounded (connected_component_set s x)"
 proof -
   obtain i::'a where i: "i \<in> Basis"
     using nonempty_Basis by blast
   obtain B where B: "B>0" "-s \<subseteq> ball 0 B"
     using bounded_subset_ballD [OF assms, of 0] by auto
   then have *: "\<And>x. B \<le> norm x \<Longrightarrow> x \<in> s"
     by (force simp: ball_def dist_norm)
   have unbounded_inner: "\<not> bounded {x. inner i x \<ge> B}"
     apply (auto simp: bounded_def dist_norm)
     apply (rule_tac x="x + (max B e + 1 + \<bar>i \<bullet> x\<bar>) *\<^sub>R i" in exI)
     apply simp
     using i
     apply (auto simp: algebra_simps)
     done
   have **: "{x. B \<le> i \<bullet> x} \<subseteq> connected_component_set s (B *\<^sub>R i)"
     apply (rule connected_component_maximal)
     apply (auto simp: i intro: convex_connected convex_halfspace_ge [of B])
     apply (rule *)
     apply (rule order_trans [OF _ Basis_le_norm [OF i]])
     by (simp add: inner_commute)
   have "B *\<^sub>R i \<in> s"
     by (rule *) (simp add: norm_Basis [OF i])
   then show ?thesis
     apply (rule_tac x="B *\<^sub>R i" in exI, clarify)
     apply (frule bounded_subset [of _ "{x. B \<le> i \<bullet> x}", OF _ **])
     using unbounded_inner apply blast
     done
 qed
 
 lemma cobounded_unique_unbounded_component:
     fixes s :: "'a :: euclidean_space set"
     assumes bs: "bounded (-s)" and "2 \<le> DIM('a)"
         and bo: "\<not> bounded(connected_component_set s x)"
                 "\<not> bounded(connected_component_set s y)"
       shows "connected_component_set s x = connected_component_set s y"
 proof -
   obtain i::'a where i: "i \<in> Basis"
     using nonempty_Basis by blast
   obtain B where B: "B>0" "-s \<subseteq> ball 0 B"
     using bounded_subset_ballD [OF bs, of 0] by auto
   then have *: "\<And>x. B \<le> norm x \<Longrightarrow> x \<in> s"
     by (force simp: ball_def dist_norm)
   have ccb: "connected (- ball 0 B :: 'a set)"
     using assms by (auto intro: connected_complement_bounded_convex)
   obtain x' where x': "connected_component s x x'" "norm x' > B"
     using bo [unfolded bounded_def dist_norm, simplified, rule_format]
     by (metis diff_zero norm_minus_commute not_less)
   obtain y' where y': "connected_component s y y'" "norm y' > B"
     using bo [unfolded bounded_def dist_norm, simplified, rule_format]
     by (metis diff_zero norm_minus_commute not_less)
   have x'y': "connected_component s x' y'"
     apply (simp add: connected_component_def)
     apply (rule_tac x="- ball 0 B" in exI)
     using x' y'
     apply (auto simp: ccb dist_norm *)
     done
   show ?thesis
     apply (rule connected_component_eq)
     using x' y' x'y'
     by (metis (no_types, lifting) connected_component_eq_empty connected_component_eq_eq connected_component_idemp connected_component_in)
 qed
 
 lemma cobounded_unbounded_components:
     fixes s :: "'a :: euclidean_space set"
     shows "bounded (-s) \<Longrightarrow> \<exists>c. c \<in> components s \<and> \<not>bounded c"
   by (metis cobounded_unbounded_component components_def imageI)
 
 lemma cobounded_unique_unbounded_components:
     fixes s :: "'a :: euclidean_space set"
     shows  "\<lbrakk>bounded (- s); c \<in> components s; \<not> bounded c; c' \<in> components s; \<not> bounded c'; 2 \<le> DIM('a)\<rbrakk> \<Longrightarrow> c' = c"
   unfolding components_iff
   by (metis cobounded_unique_unbounded_component)
 
 lemma cobounded_has_bounded_component:
   fixes S :: "'a :: euclidean_space set"
   assumes "bounded (- S)" "\<not> connected S" "2 \<le> DIM('a)"
   obtains C where "C \<in> components S" "bounded C"
   by (meson cobounded_unique_unbounded_components connected_eq_connected_components_eq assms)
 
 
 subsection\<open>The \<open>inside\<close> and \<open>outside\<close> of a Set\<close>
 
 text\<^marker>\<open>tag important\<close>\<open>The inside comprises the points in a bounded connected component of the set's complement.
   The outside comprises the points in unbounded connected component of the complement.\<close>
 
 definition\<^marker>\<open>tag important\<close> inside where
   "inside S \<equiv> {x. (x \<notin> S) \<and> bounded(connected_component_set ( - S) x)}"
 
 definition\<^marker>\<open>tag important\<close> outside where
   "outside S \<equiv> -S \<inter> {x. \<not> bounded(connected_component_set (- S) x)}"
 
 lemma outside: "outside S = {x. \<not> bounded(connected_component_set (- S) x)}"
   by (auto simp: outside_def) (metis Compl_iff bounded_empty connected_component_eq_empty)
 
 lemma inside_no_overlap [simp]: "inside S \<inter> S = {}"
   by (auto simp: inside_def)
 
 lemma outside_no_overlap [simp]:
    "outside S \<inter> S = {}"
   by (auto simp: outside_def)
 
 lemma inside_Int_outside [simp]: "inside S \<inter> outside S = {}"
   by (auto simp: inside_def outside_def)
 
 lemma inside_Un_outside [simp]: "inside S \<union> outside S = (- S)"
   by (auto simp: inside_def outside_def)
 
 lemma inside_eq_outside:
    "inside S = outside S \<longleftrightarrow> S = UNIV"
   by (auto simp: inside_def outside_def)
 
 lemma inside_outside: "inside S = (- (S \<union> outside S))"
   by (force simp: inside_def outside)
 
 lemma outside_inside: "outside S = (- (S \<union> inside S))"
   by (auto simp: inside_outside) (metis IntI equals0D outside_no_overlap)
 
 lemma union_with_inside: "S \<union> inside S = - outside S"
   by (auto simp: inside_outside) (simp add: outside_inside)
 
 lemma union_with_outside: "S \<union> outside S = - inside S"
   by (simp add: inside_outside)
 
 lemma outside_mono: "S \<subseteq> T \<Longrightarrow> outside T \<subseteq> outside S"
   by (auto simp: outside bounded_subset connected_component_mono)
 
 lemma inside_mono: "S \<subseteq> T \<Longrightarrow> inside S - T \<subseteq> inside T"
   by (auto simp: inside_def bounded_subset connected_component_mono)
 
 lemma segment_bound_lemma:
   fixes u::real
   assumes "x \<ge> B" "y \<ge> B" "0 \<le> u" "u \<le> 1"
   shows "(1 - u) * x + u * y \<ge> B"
 proof -
   obtain dx dy where "dx \<ge> 0" "dy \<ge> 0" "x = B + dx" "y = B + dy"
     using assms by auto (metis add.commute diff_add_cancel)
   with \<open>0 \<le> u\<close> \<open>u \<le> 1\<close> show ?thesis
     by (simp add: add_increasing2 mult_left_le field_simps)
 qed
 
 lemma cobounded_outside:
   fixes S :: "'a :: real_normed_vector set"
   assumes "bounded S" shows "bounded (- outside S)"
 proof -
   obtain B where B: "B>0" "S \<subseteq> ball 0 B"
     using bounded_subset_ballD [OF assms, of 0] by auto
   { fix x::'a and C::real
     assume Bno: "B \<le> norm x" and C: "0 < C"
     have "\<exists>y. connected_component (- S) x y \<and> norm y > C"
     proof (cases "x = 0")
       case True with B Bno show ?thesis by force
     next
       case False 
       have "closed_segment x (((B + C) / norm x) *\<^sub>R x) \<subseteq> - ball 0 B"
       proof
         fix w
         assume "w \<in> closed_segment x (((B + C) / norm x) *\<^sub>R x)"
         then obtain u where
           w: "w = (1 - u + u * (B + C) / norm x) *\<^sub>R x" "0 \<le> u" "u \<le> 1"
           by (auto simp add: closed_segment_def real_vector_class.scaleR_add_left [symmetric])
         with False B C have "B \<le> (1 - u) * norm x + u * (B + C)"
           using segment_bound_lemma [of B "norm x" "B + C" u] Bno
           by simp
         with False B C show "w \<in> - ball 0 B"
           using distrib_right [of _ _ "norm x"]
           by (simp add: ball_def w not_less)
       qed
       also have "... \<subseteq> -S"
         by (simp add: B)
       finally have "\<exists>T. connected T \<and> T \<subseteq> - S \<and> x \<in> T \<and> ((B + C) / norm x) *\<^sub>R x \<in> T"
         by (rule_tac x="closed_segment x (((B+C)/norm x) *\<^sub>R x)" in exI) simp
       with False B
       show ?thesis
         by (rule_tac x="((B+C)/norm x) *\<^sub>R x" in exI) (simp add: connected_component_def)
     qed
   }
   then show ?thesis
     apply (simp add: outside_def assms)
     apply (rule bounded_subset [OF bounded_ball [of 0 B]])
     apply (force simp: dist_norm not_less bounded_pos)
     done
 qed
 
 lemma unbounded_outside:
     fixes S :: "'a::{real_normed_vector, perfect_space} set"
     shows "bounded S \<Longrightarrow> \<not> bounded(outside S)"
   using cobounded_imp_unbounded cobounded_outside by blast
 
 lemma bounded_inside:
     fixes S :: "'a::{real_normed_vector, perfect_space} set"
     shows "bounded S \<Longrightarrow> bounded(inside S)"
   by (simp add: bounded_Int cobounded_outside inside_outside)
 
 lemma connected_outside:
     fixes S :: "'a::euclidean_space set"
     assumes "bounded S" "2 \<le> DIM('a)"
       shows "connected(outside S)"
   apply (clarsimp simp add: connected_iff_connected_component outside)
   apply (rule_tac s="connected_component_set (- S) x" in connected_component_of_subset)
   apply (metis (no_types) assms cobounded_unbounded_component cobounded_unique_unbounded_component connected_component_eq_eq connected_component_idemp double_complement mem_Collect_eq)
   apply clarify
   apply (metis connected_component_eq_eq connected_component_in)
   done
 
 lemma outside_connected_component_lt:
     "outside S = {x. \<forall>B. \<exists>y. B < norm(y) \<and> connected_component (- S) x y}"
 apply (auto simp: outside bounded_def dist_norm)
 apply (metis diff_0 norm_minus_cancel not_less)
 by (metis less_diff_eq norm_minus_commute norm_triangle_ineq2 order.trans pinf(6))
 
 lemma outside_connected_component_le:
    "outside S =
             {x. \<forall>B. \<exists>y. B \<le> norm(y) \<and>
                          connected_component (- S) x y}"
 apply (simp add: outside_connected_component_lt)
 apply (simp add: Set.set_eq_iff)
 by (meson gt_ex leD le_less_linear less_imp_le order.trans)
 
 lemma not_outside_connected_component_lt:
     fixes S :: "'a::euclidean_space set"
     assumes S: "bounded S" and "2 \<le> DIM('a)"
       shows "- (outside S) = {x. \<forall>B. \<exists>y. B < norm(y) \<and> \<not> connected_component (- S) x y}"
 proof -
   obtain B::real where B: "0 < B" and Bno: "\<And>x. x \<in> S \<Longrightarrow> norm x \<le> B"
     using S [simplified bounded_pos] by auto
   { fix y::'a and z::'a
     assume yz: "B < norm z" "B < norm y"
     have "connected_component (- cball 0 B) y z"
       apply (rule connected_componentI [OF _ subset_refl])
       apply (rule connected_complement_bounded_convex)
       using assms yz
       by (auto simp: dist_norm)
     then have "connected_component (- S) y z"
       apply (rule connected_component_of_subset)
       apply (metis Bno Compl_anti_mono mem_cball_0 subset_iff)
       done
   } note cyz = this
   show ?thesis
     apply (auto simp: outside)
     apply (metis Compl_iff bounded_iff cobounded_imp_unbounded mem_Collect_eq not_le)
     apply (simp add: bounded_pos)
     by (metis B connected_component_trans cyz not_le)
 qed
 
 lemma not_outside_connected_component_le:
     fixes S :: "'a::euclidean_space set"
     assumes S: "bounded S"  "2 \<le> DIM('a)"
       shows "- (outside S) = {x. \<forall>B. \<exists>y. B \<le> norm(y) \<and> \<not> connected_component (- S) x y}"
 apply (auto intro: less_imp_le simp: not_outside_connected_component_lt [OF assms])
 by (meson gt_ex less_le_trans)
 
 lemma inside_connected_component_lt:
     fixes S :: "'a::euclidean_space set"
     assumes S: "bounded S"  "2 \<le> DIM('a)"
       shows "inside S = {x. (x \<notin> S) \<and> (\<forall>B. \<exists>y. B < norm(y) \<and> \<not> connected_component (- S) x y)}"
   by (auto simp: inside_outside not_outside_connected_component_lt [OF assms])
 
 lemma inside_connected_component_le:
     fixes S :: "'a::euclidean_space set"
     assumes S: "bounded S"  "2 \<le> DIM('a)"
       shows "inside S = {x. (x \<notin> S) \<and> (\<forall>B. \<exists>y. B \<le> norm(y) \<and> \<not> connected_component (- S) x y)}"
   by (auto simp: inside_outside not_outside_connected_component_le [OF assms])
 
 lemma inside_subset:
   assumes "connected U" and "\<not> bounded U" and "T \<union> U = - S"
   shows "inside S \<subseteq> T"
 apply (auto simp: inside_def)
 by (metis bounded_subset [of "connected_component_set (- S) _"] connected_component_maximal
        Compl_iff Un_iff assms subsetI)
 
 lemma frontier_not_empty:
   fixes S :: "'a :: real_normed_vector set"
   shows "\<lbrakk>S \<noteq> {}; S \<noteq> UNIV\<rbrakk> \<Longrightarrow> frontier S \<noteq> {}"
     using connected_Int_frontier [of UNIV S] by auto
 
 lemma frontier_eq_empty:
   fixes S :: "'a :: real_normed_vector set"
   shows "frontier S = {} \<longleftrightarrow> S = {} \<or> S = UNIV"
 using frontier_UNIV frontier_empty frontier_not_empty by blast
 
 lemma frontier_of_connected_component_subset:
   fixes S :: "'a::real_normed_vector set"
   shows "frontier(connected_component_set S x) \<subseteq> frontier S"
 proof -
   { fix y
     assume y1: "y \<in> closure (connected_component_set S x)"
        and y2: "y \<notin> interior (connected_component_set S x)"
     have "y \<in> closure S"
       using y1 closure_mono connected_component_subset by blast
     moreover have "z \<in> interior (connected_component_set S x)"
           if "0 < e" "ball y e \<subseteq> interior S" "dist y z < e" for e z
     proof -
       have "ball y e \<subseteq> connected_component_set S y"
         apply (rule connected_component_maximal)
         using that interior_subset mem_ball apply auto
         done
       then show ?thesis
         using y1 apply (simp add: closure_approachable open_contains_ball_eq [OF open_interior])
         by (metis connected_component_eq dist_commute mem_Collect_eq mem_ball mem_interior subsetD \<open>0 < e\<close> y2)
     qed
     then have "y \<notin> interior S"
       using y2 by (force simp: open_contains_ball_eq [OF open_interior])
     ultimately have "y \<in> frontier S"
       by (auto simp: frontier_def)
   }
   then show ?thesis by (auto simp: frontier_def)
 qed
 
 lemma frontier_Union_subset_closure:
   fixes F :: "'a::real_normed_vector set set"
   shows "frontier(\<Union>F) \<subseteq> closure(\<Union>t \<in> F. frontier t)"
 proof -
   have "\<exists>y\<in>F. \<exists>y\<in>frontier y. dist y x < e"
        if "T \<in> F" "y \<in> T" "dist y x < e"
           "x \<notin> interior (\<Union>F)" "0 < e" for x y e T
   proof (cases "x \<in> T")
     case True with that show ?thesis
       by (metis Diff_iff Sup_upper closure_subset contra_subsetD dist_self frontier_def interior_mono)
   next
     case False
     have 1: "closed_segment x y \<inter> T \<noteq> {}" using \<open>y \<in> T\<close> by blast
     have 2: "closed_segment x y - T \<noteq> {}"
       using False by blast
     obtain c where "c \<in> closed_segment x y" "c \<in> frontier T"
        using False connected_Int_frontier [OF connected_segment 1 2] by auto
     then show ?thesis
     proof -
       have "norm (y - x) < e"
         by (metis dist_norm \<open>dist y x < e\<close>)
       moreover have "norm (c - x) \<le> norm (y - x)"
         by (simp add: \<open>c \<in> closed_segment x y\<close> segment_bound(1))
       ultimately have "norm (c - x) < e"
         by linarith
       then show ?thesis
         by (metis (no_types) \<open>c \<in> frontier T\<close> dist_norm that(1))
     qed
   qed
   then show ?thesis
     by (fastforce simp add: frontier_def closure_approachable)
 qed
 
 lemma frontier_Union_subset:
   fixes F :: "'a::real_normed_vector set set"
   shows "finite F \<Longrightarrow> frontier(\<Union>F) \<subseteq> (\<Union>t \<in> F. frontier t)"
 by (rule order_trans [OF frontier_Union_subset_closure])
    (auto simp: closure_subset_eq)
 
 lemma frontier_of_components_subset:
   fixes S :: "'a::real_normed_vector set"
   shows "C \<in> components S \<Longrightarrow> frontier C \<subseteq> frontier S"
   by (metis Path_Connected.frontier_of_connected_component_subset components_iff)
 
 lemma frontier_of_components_closed_complement:
   fixes S :: "'a::real_normed_vector set"
   shows "\<lbrakk>closed S; C \<in> components (- S)\<rbrakk> \<Longrightarrow> frontier C \<subseteq> S"
   using frontier_complement frontier_of_components_subset frontier_subset_eq by blast
 
 lemma frontier_minimal_separating_closed:
   fixes S :: "'a::real_normed_vector set"
   assumes "closed S"
       and nconn: "\<not> connected(- S)"
       and C: "C \<in> components (- S)"
       and conn: "\<And>T. \<lbrakk>closed T; T \<subset> S\<rbrakk> \<Longrightarrow> connected(- T)"
     shows "frontier C = S"
 proof (rule ccontr)
   assume "frontier C \<noteq> S"
   then have "frontier C \<subset> S"
     using frontier_of_components_closed_complement [OF \<open>closed S\<close> C] by blast
   then have "connected(- (frontier C))"
     by (simp add: conn)
   have "\<not> connected(- (frontier C))"
     unfolding connected_def not_not
   proof (intro exI conjI)
     show "open C"
       using C \<open>closed S\<close> open_components by blast
     show "open (- closure C)"
       by blast
     show "C \<inter> - closure C \<inter> - frontier C = {}"
       using closure_subset by blast
     show "C \<inter> - frontier C \<noteq> {}"
       using C \<open>open C\<close> components_eq frontier_disjoint_eq by fastforce
     show "- frontier C \<subseteq> C \<union> - closure C"
       by (simp add: \<open>open C\<close> closed_Compl frontier_closures)
     then show "- closure C \<inter> - frontier C \<noteq> {}"
       by (metis (no_types, lifting) C Compl_subset_Compl_iff \<open>frontier C \<subset> S\<close> compl_sup frontier_closures in_components_subset psubsetE sup.absorb_iff2 sup.boundedE sup_bot.right_neutral sup_inf_absorb)
   qed
   then show False
     using \<open>connected (- frontier C)\<close> by blast
 qed
 
 lemma connected_component_UNIV [simp]:
     fixes x :: "'a::real_normed_vector"
     shows "connected_component_set UNIV x = UNIV"
 using connected_iff_eq_connected_component_set [of "UNIV::'a set"] connected_UNIV
 by auto
 
 lemma connected_component_eq_UNIV:
     fixes x :: "'a::real_normed_vector"
     shows "connected_component_set s x = UNIV \<longleftrightarrow> s = UNIV"
   using connected_component_in connected_component_UNIV by blast
 
 lemma components_UNIV [simp]: "components UNIV = {UNIV :: 'a::real_normed_vector set}"
   by (auto simp: components_eq_sing_iff)
 
 lemma interior_inside_frontier:
     fixes s :: "'a::real_normed_vector set"
     assumes "bounded s"
       shows "interior s \<subseteq> inside (frontier s)"
 proof -
   { fix x y
     assume x: "x \<in> interior s" and y: "y \<notin> s"
        and cc: "connected_component (- frontier s) x y"
     have "connected_component_set (- frontier s) x \<inter> frontier s \<noteq> {}"
       apply (rule connected_Int_frontier, simp)
       apply (metis IntI cc connected_component_in connected_component_refl empty_iff interiorE mem_Collect_eq rev_subsetD x)
       using  y cc
       by blast
     then have "bounded (connected_component_set (- frontier s) x)"
       using connected_component_in by auto
   }
   then show ?thesis
     apply (auto simp: inside_def frontier_def)
     apply (rule classical)
     apply (rule bounded_subset [OF assms], blast)
     done
 qed
 
 lemma inside_empty [simp]: "inside {} = ({} :: 'a :: {real_normed_vector, perfect_space} set)"
   by (simp add: inside_def)
 
 lemma outside_empty [simp]: "outside {} = (UNIV :: 'a :: {real_normed_vector, perfect_space} set)"
 using inside_empty inside_Un_outside by blast
 
 lemma inside_same_component:
    "\<lbrakk>connected_component (- s) x y; x \<in> inside s\<rbrakk> \<Longrightarrow> y \<in> inside s"
   using connected_component_eq connected_component_in
   by (fastforce simp add: inside_def)
 
 lemma outside_same_component:
    "\<lbrakk>connected_component (- s) x y; x \<in> outside s\<rbrakk> \<Longrightarrow> y \<in> outside s"
   using connected_component_eq connected_component_in
   by (fastforce simp add: outside_def)
 
 lemma convex_in_outside:
   fixes s :: "'a :: {real_normed_vector, perfect_space} set"
   assumes s: "convex s" and z: "z \<notin> s"
     shows "z \<in> outside s"
 proof (cases "s={}")
   case True then show ?thesis by simp
 next
   case False then obtain a where "a \<in> s" by blast
   with z have zna: "z \<noteq> a" by auto
   { assume "bounded (connected_component_set (- s) z)"
     with bounded_pos_less obtain B where "B>0" and B: "\<And>x. connected_component (- s) z x \<Longrightarrow> norm x < B"
       by (metis mem_Collect_eq)
     define C where "C = (B + 1 + norm z) / norm (z-a)"
     have "C > 0"
       using \<open>0 < B\<close> zna by (simp add: C_def field_split_simps add_strict_increasing)
     have "\<bar>norm (z + C *\<^sub>R (z-a)) - norm (C *\<^sub>R (z-a))\<bar> \<le> norm z"
       by (metis add_diff_cancel norm_triangle_ineq3)
     moreover have "norm (C *\<^sub>R (z-a)) > norm z + B"
       using zna \<open>B>0\<close> by (simp add: C_def le_max_iff_disj)
     ultimately have C: "norm (z + C *\<^sub>R (z-a)) > B" by linarith
     { fix u::real
       assume u: "0\<le>u" "u\<le>1" and ins: "(1 - u) *\<^sub>R z + u *\<^sub>R (z + C *\<^sub>R (z - a)) \<in> s"
       then have Cpos: "1 + u * C > 0"
         by (meson \<open>0 < C\<close> add_pos_nonneg less_eq_real_def zero_le_mult_iff zero_less_one)
       then have *: "(1 / (1 + u * C)) *\<^sub>R z + (u * C / (1 + u * C)) *\<^sub>R z = z"
         by (simp add: scaleR_add_left [symmetric] field_split_simps)
       then have False
         using convexD_alt [OF s \<open>a \<in> s\<close> ins, of "1/(u*C + 1)"] \<open>C>0\<close> \<open>z \<notin> s\<close> Cpos u
         by (simp add: * field_split_simps)
     } note contra = this
     have "connected_component (- s) z (z + C *\<^sub>R (z-a))"
       apply (rule connected_componentI [OF connected_segment [of z "z + C *\<^sub>R (z-a)"]])
       apply (simp add: closed_segment_def)
       using contra
       apply auto
       done
     then have False
       using zna B [of "z + C *\<^sub>R (z-a)"] C
       by (auto simp: field_split_simps max_mult_distrib_right)
   }
   then show ?thesis
     by (auto simp: outside_def z)
 qed
 
 lemma outside_convex:
   fixes s :: "'a :: {real_normed_vector, perfect_space} set"
   assumes "convex s"
     shows "outside s = - s"
   by (metis ComplD assms convex_in_outside equalityI inside_Un_outside subsetI sup.cobounded2)
 
 lemma outside_singleton [simp]:
   fixes x :: "'a :: {real_normed_vector, perfect_space}"
   shows "outside {x} = -{x}"
   by (auto simp: outside_convex)
 
 lemma inside_convex:
   fixes s :: "'a :: {real_normed_vector, perfect_space} set"
   shows "convex s \<Longrightarrow> inside s = {}"
   by (simp add: inside_outside outside_convex)
 
 lemma inside_singleton [simp]:
   fixes x :: "'a :: {real_normed_vector, perfect_space}"
   shows "inside {x} = {}"
   by (auto simp: inside_convex)
 
 lemma outside_subset_convex:
   fixes s :: "'a :: {real_normed_vector, perfect_space} set"
   shows "\<lbrakk>convex t; s \<subseteq> t\<rbrakk> \<Longrightarrow> - t \<subseteq> outside s"
   using outside_convex outside_mono by blast
 
 lemma outside_Un_outside_Un:
   fixes S :: "'a::real_normed_vector set"
   assumes "S \<inter> outside(T \<union> U) = {}"
   shows "outside(T \<union> U) \<subseteq> outside(T \<union> S)"
 proof
   fix x
   assume x: "x \<in> outside (T \<union> U)"
   have "Y \<subseteq> - S" if "connected Y" "Y \<subseteq> - T" "Y \<subseteq> - U" "x \<in> Y" "u \<in> Y" for u Y
   proof -
     have "Y \<subseteq> connected_component_set (- (T \<union> U)) x"
       by (simp add: connected_component_maximal that)
     also have "\<dots> \<subseteq> outside(T \<union> U)"
       by (metis (mono_tags, lifting) Collect_mono mem_Collect_eq outside outside_same_component x)
     finally have "Y \<subseteq> outside(T \<union> U)" .
     with assms show ?thesis by auto
   qed
   with x show "x \<in> outside (T \<union> S)"
     by (simp add: outside_connected_component_lt connected_component_def) meson
 qed
 
 lemma outside_frontier_misses_closure:
     fixes s :: "'a::real_normed_vector set"
     assumes "bounded s"
     shows  "outside(frontier s) \<subseteq> - closure s"
   unfolding outside_inside Lattices.boolean_algebra_class.compl_le_compl_iff
 proof -
   { assume "interior s \<subseteq> inside (frontier s)"
     hence "interior s \<union> inside (frontier s) = inside (frontier s)"
       by (simp add: subset_Un_eq)
     then have "closure s \<subseteq> frontier s \<union> inside (frontier s)"
       using frontier_def by auto
   }
   then show "closure s \<subseteq> frontier s \<union> inside (frontier s)"
     using interior_inside_frontier [OF assms] by blast
 qed
 
 lemma outside_frontier_eq_complement_closure:
   fixes s :: "'a :: {real_normed_vector, perfect_space} set"
     assumes "bounded s" "convex s"
       shows "outside(frontier s) = - closure s"
 by (metis Diff_subset assms convex_closure frontier_def outside_frontier_misses_closure
           outside_subset_convex subset_antisym)
 
 lemma inside_frontier_eq_interior:
      fixes s :: "'a :: {real_normed_vector, perfect_space} set"
      shows "\<lbrakk>bounded s; convex s\<rbrakk> \<Longrightarrow> inside(frontier s) = interior s"
   apply (simp add: inside_outside outside_frontier_eq_complement_closure)
   using closure_subset interior_subset
   apply (auto simp: frontier_def)
   done
 
 lemma open_inside:
     fixes s :: "'a::real_normed_vector set"
     assumes "closed s"
       shows "open (inside s)"
 proof -
   { fix x assume x: "x \<in> inside s"
     have "open (connected_component_set (- s) x)"
       using assms open_connected_component by blast
     then obtain e where e: "e>0" and e: "\<And>y. dist y x < e \<longrightarrow> connected_component (- s) x y"
       using dist_not_less_zero
       apply (simp add: open_dist)
       by (metis (no_types, lifting) Compl_iff connected_component_refl_eq inside_def mem_Collect_eq x)
     then have "\<exists>e>0. ball x e \<subseteq> inside s"
       by (metis e dist_commute inside_same_component mem_ball subsetI x)
   }
   then show ?thesis
     by (simp add: open_contains_ball)
 qed
 
 lemma open_outside:
     fixes s :: "'a::real_normed_vector set"
     assumes "closed s"
       shows "open (outside s)"
 proof -
   { fix x assume x: "x \<in> outside s"
     have "open (connected_component_set (- s) x)"
       using assms open_connected_component by blast
     then obtain e where e: "e>0" and e: "\<And>y. dist y x < e \<longrightarrow> connected_component (- s) x y"
       using dist_not_less_zero
       apply (simp add: open_dist)
       by (metis Int_iff outside_def connected_component_refl_eq  x)
     then have "\<exists>e>0. ball x e \<subseteq> outside s"
       by (metis e dist_commute outside_same_component mem_ball subsetI x)
   }
   then show ?thesis
     by (simp add: open_contains_ball)
 qed
 
 lemma closure_inside_subset:
     fixes s :: "'a::real_normed_vector set"
     assumes "closed s"
       shows "closure(inside s) \<subseteq> s \<union> inside s"
 by (metis assms closure_minimal open_closed open_outside sup.cobounded2 union_with_inside)
 
 lemma frontier_inside_subset:
     fixes s :: "'a::real_normed_vector set"
     assumes "closed s"
       shows "frontier(inside s) \<subseteq> s"
 proof -
   have "closure (inside s) \<inter> - inside s = closure (inside s) - interior (inside s)"
     by (metis (no_types) Diff_Compl assms closure_closed interior_closure open_closed open_inside)
   moreover have "- inside s \<inter> - outside s = s"
     by (metis (no_types) compl_sup double_compl inside_Un_outside)
   moreover have "closure (inside s) \<subseteq> - outside s"
     by (metis (no_types) assms closure_inside_subset union_with_inside)
   ultimately have "closure (inside s) - interior (inside s) \<subseteq> s"
     by blast
   then show ?thesis
     by (simp add: frontier_def open_inside interior_open)
 qed
 
 lemma closure_outside_subset:
     fixes s :: "'a::real_normed_vector set"
     assumes "closed s"
       shows "closure(outside s) \<subseteq> s \<union> outside s"
   apply (rule closure_minimal, simp)
   by (metis assms closed_open inside_outside open_inside)
 
 lemma frontier_outside_subset:
     fixes s :: "'a::real_normed_vector set"
     assumes "closed s"
       shows "frontier(outside s) \<subseteq> s"
   apply (simp add: frontier_def open_outside interior_open)
   by (metis Diff_subset_conv assms closure_outside_subset interior_eq open_outside sup.commute)
 
 lemma inside_complement_unbounded_connected_empty:
      "\<lbrakk>connected (- s); \<not> bounded (- s)\<rbrakk> \<Longrightarrow> inside s = {}"
   apply (simp add: inside_def)
   by (meson Compl_iff bounded_subset connected_component_maximal order_refl)
 
 lemma inside_bounded_complement_connected_empty:
     fixes s :: "'a::{real_normed_vector, perfect_space} set"
     shows "\<lbrakk>connected (- s); bounded s\<rbrakk> \<Longrightarrow> inside s = {}"
   by (metis inside_complement_unbounded_connected_empty cobounded_imp_unbounded)
 
 lemma inside_inside:
     assumes "s \<subseteq> inside t"
     shows "inside s - t \<subseteq> inside t"
 unfolding inside_def
 proof clarify
   fix x
   assume x: "x \<notin> t" "x \<notin> s" and bo: "bounded (connected_component_set (- s) x)"
   show "bounded (connected_component_set (- t) x)"
   proof (cases "s \<inter> connected_component_set (- t) x = {}")
     case True show ?thesis
       apply (rule bounded_subset [OF bo])
       apply (rule connected_component_maximal)
       using x True apply auto
       done
   next
     case False then show ?thesis
       using assms [unfolded inside_def] x
       apply (simp add: disjoint_iff_not_equal, clarify)
       apply (drule subsetD, assumption, auto)
       by (metis (no_types, hide_lams) ComplI connected_component_eq_eq)
   qed
 qed
 
 lemma inside_inside_subset: "inside(inside s) \<subseteq> s"
   using inside_inside union_with_outside by fastforce
 
 lemma inside_outside_intersect_connected:
       "\<lbrakk>connected t; inside s \<inter> t \<noteq> {}; outside s \<inter> t \<noteq> {}\<rbrakk> \<Longrightarrow> s \<inter> t \<noteq> {}"
   apply (simp add: inside_def outside_def ex_in_conv [symmetric] disjoint_eq_subset_Compl, clarify)
   by (metis (no_types, hide_lams) Compl_anti_mono connected_component_eq connected_component_maximal contra_subsetD double_compl)
 
 lemma outside_bounded_nonempty:
   fixes s :: "'a :: {real_normed_vector, perfect_space} set"
     assumes "bounded s" shows "outside s \<noteq> {}"
   by (metis (no_types, lifting) Collect_empty_eq Collect_mem_eq Compl_eq_Diff_UNIV Diff_cancel
                    Diff_disjoint UNIV_I assms ball_eq_empty bounded_diff cobounded_outside convex_ball
                    double_complement order_refl outside_convex outside_def)
 
 lemma outside_compact_in_open:
     fixes s :: "'a :: {real_normed_vector,perfect_space} set"
     assumes s: "compact s" and t: "open t" and "s \<subseteq> t" "t \<noteq> {}"
       shows "outside s \<inter> t \<noteq> {}"
 proof -
   have "outside s \<noteq> {}"
     by (simp add: compact_imp_bounded outside_bounded_nonempty s)
   with assms obtain a b where a: "a \<in> outside s" and b: "b \<in> t" by auto
   show ?thesis
   proof (cases "a \<in> t")
     case True with a show ?thesis by blast
   next
     case False
       have front: "frontier t \<subseteq> - s"
         using \<open>s \<subseteq> t\<close> frontier_disjoint_eq t by auto
       { fix \<gamma>
         assume "path \<gamma>" and pimg_sbs: "path_image \<gamma> - {pathfinish \<gamma>} \<subseteq> interior (- t)"
            and pf: "pathfinish \<gamma> \<in> frontier t" and ps: "pathstart \<gamma> = a"
         define c where "c = pathfinish \<gamma>"
         have "c \<in> -s" unfolding c_def using front pf by blast
         moreover have "open (-s)" using s compact_imp_closed by blast
         ultimately obtain \<epsilon>::real where "\<epsilon> > 0" and \<epsilon>: "cball c \<epsilon> \<subseteq> -s"
           using open_contains_cball[of "-s"] s by blast
         then obtain d where "d \<in> t" and d: "dist d c < \<epsilon>"
           using closure_approachable [of c t] pf unfolding c_def
           by (metis Diff_iff frontier_def)
         then have "d \<in> -s" using \<epsilon>
           using dist_commute by (metis contra_subsetD mem_cball not_le not_less_iff_gr_or_eq)
         have pimg_sbs_cos: "path_image \<gamma> \<subseteq> -s"
           using pimg_sbs apply (auto simp: path_image_def)
           apply (drule subsetD)
           using \<open>c \<in> - s\<close> \<open>s \<subseteq> t\<close> interior_subset apply (auto simp: c_def)
           done
         have "closed_segment c d \<le> cball c \<epsilon>"
           apply (simp add: segment_convex_hull)
           apply (rule hull_minimal)
           using  \<open>\<epsilon> > 0\<close> d apply (auto simp: dist_commute)
           done
         with \<epsilon> have "closed_segment c d \<subseteq> -s" by blast
         moreover have con_gcd: "connected (path_image \<gamma> \<union> closed_segment c d)"
           by (rule connected_Un) (auto simp: c_def \<open>path \<gamma>\<close> connected_path_image)
         ultimately have "connected_component (- s) a d"
           unfolding connected_component_def using pimg_sbs_cos ps by blast
         then have "outside s \<inter> t \<noteq> {}"
           using outside_same_component [OF _ a]  by (metis IntI \<open>d \<in> t\<close> empty_iff)
       } note * = this
       have pal: "pathstart (linepath a b) \<in> closure (- t)"
         by (auto simp: False closure_def)
       show ?thesis
         by (rule exists_path_subpath_to_frontier [OF path_linepath pal _ *]) (auto simp: b)
   qed
 qed
 
 lemma inside_inside_compact_connected:
     fixes s :: "'a :: euclidean_space set"
     assumes s: "closed s" and t: "compact t" and "connected t" "s \<subseteq> inside t"
       shows "inside s \<subseteq> inside t"
 proof (cases "inside t = {}")
   case True with assms show ?thesis by auto
 next
   case False
   consider "DIM('a) = 1" | "DIM('a) \<ge> 2"
     using antisym not_less_eq_eq by fastforce
   then show ?thesis
   proof cases
     case 1 then show ?thesis
              using connected_convex_1_gen assms False inside_convex by blast
   next
     case 2
     have coms: "compact s"
       using assms apply (simp add: s compact_eq_bounded_closed)
        by (meson bounded_inside bounded_subset compact_imp_bounded)
     then have bst: "bounded (s \<union> t)"
       by (simp add: compact_imp_bounded t)
     then obtain r where "0 < r" and r: "s \<union> t \<subseteq> ball 0 r"
       using bounded_subset_ballD by blast
     have outst: "outside s \<inter> outside t \<noteq> {}"
     proof -
       have "- ball 0 r \<subseteq> outside s"
         apply (rule outside_subset_convex)
         using r by auto
       moreover have "- ball 0 r \<subseteq> outside t"
         apply (rule outside_subset_convex)
         using r by auto
       ultimately show ?thesis
         by (metis Compl_subset_Compl_iff Int_subset_iff bounded_ball inf.orderE outside_bounded_nonempty outside_no_overlap)
     qed
     have "s \<inter> t = {}" using assms
       by (metis disjoint_iff_not_equal inside_no_overlap subsetCE)
     moreover have "outside s \<inter> inside t \<noteq> {}"
       by (meson False assms(4) compact_eq_bounded_closed coms open_inside outside_compact_in_open t)
     ultimately have "inside s \<inter> t = {}"
       using inside_outside_intersect_connected [OF \<open>connected t\<close>, of s]
       by (metis "2" compact_eq_bounded_closed coms connected_outside inf.commute inside_outside_intersect_connected outst)
     then show ?thesis
       using inside_inside [OF \<open>s \<subseteq> inside t\<close>] by blast
   qed
 qed
 
 lemma connected_with_inside:
     fixes s :: "'a :: real_normed_vector set"
     assumes s: "closed s" and cons: "connected s"
       shows "connected(s \<union> inside s)"
 proof (cases "s \<union> inside s = UNIV")
   case True with assms show ?thesis by auto
 next
   case False
   then obtain b where b: "b \<notin> s" "b \<notin> inside s" by blast
   have *: "\<exists>y t. y \<in> s \<and> connected t \<and> a \<in> t \<and> y \<in> t \<and> t \<subseteq> (s \<union> inside s)" if "a \<in> (s \<union> inside s)" for a
   using that proof
     assume "a \<in> s" then show ?thesis
       apply (rule_tac x=a in exI)
       apply (rule_tac x="{a}" in exI, simp)
       done
   next
     assume a: "a \<in> inside s"
     show ?thesis
       apply (rule exists_path_subpath_to_frontier [OF path_linepath [of a b], of "inside s"])
       using a apply (simp add: closure_def)
       apply (simp add: b)
       apply (rule_tac x="pathfinish h" in exI)
       apply (rule_tac x="path_image h" in exI)
       apply (simp add: pathfinish_in_path_image connected_path_image, auto)
       using frontier_inside_subset s apply fastforce
       by (metis (no_types, lifting) frontier_inside_subset insertE insert_Diff interior_eq open_inside pathfinish_in_path_image s subsetCE)
   qed
   show ?thesis
     apply (simp add: connected_iff_connected_component)
     apply (simp add: connected_component_def)
     apply (clarify dest!: *)
     apply (rename_tac u u' t t')
     apply (rule_tac x="(s \<union> t \<union> t')" in exI)
     apply (auto simp: intro!: connected_Un cons)
     done
 qed
 
 text\<open>The proof is virtually the same as that above.\<close>
 lemma connected_with_outside:
     fixes s :: "'a :: real_normed_vector set"
     assumes s: "closed s" and cons: "connected s"
       shows "connected(s \<union> outside s)"
 proof (cases "s \<union> outside s = UNIV")
   case True with assms show ?thesis by auto
 next
   case False
   then obtain b where b: "b \<notin> s" "b \<notin> outside s" by blast
   have *: "\<exists>y t. y \<in> s \<and> connected t \<and> a \<in> t \<and> y \<in> t \<and> t \<subseteq> (s \<union> outside s)" if "a \<in> (s \<union> outside s)" for a
   using that proof
     assume "a \<in> s" then show ?thesis
       apply (rule_tac x=a in exI)
       apply (rule_tac x="{a}" in exI, simp)
       done
   next
     assume a: "a \<in> outside s"
     show ?thesis
       apply (rule exists_path_subpath_to_frontier [OF path_linepath [of a b], of "outside s"])
       using a apply (simp add: closure_def)
       apply (simp add: b)
       apply (rule_tac x="pathfinish h" in exI)
       apply (rule_tac x="path_image h" in exI)
       apply (simp add: pathfinish_in_path_image connected_path_image, auto)
       using frontier_outside_subset s apply fastforce
       by (metis (no_types, lifting) frontier_outside_subset insertE insert_Diff interior_eq open_outside pathfinish_in_path_image s subsetCE)
   qed
   show ?thesis
     apply (simp add: connected_iff_connected_component)
     apply (simp add: connected_component_def)
     apply (clarify dest!: *)
     apply (rename_tac u u' t t')
     apply (rule_tac x="(s \<union> t \<union> t')" in exI)
     apply (auto simp: intro!: connected_Un cons)
     done
 qed
 
 lemma inside_inside_eq_empty [simp]:
     fixes s :: "'a :: {real_normed_vector, perfect_space} set"
     assumes s: "closed s" and cons: "connected s"
       shows "inside (inside s) = {}"
   by (metis (no_types) unbounded_outside connected_with_outside [OF assms] bounded_Un
            inside_complement_unbounded_connected_empty unbounded_outside union_with_outside)
 
 lemma inside_in_components:
      "inside s \<in> components (- s) \<longleftrightarrow> connected(inside s) \<and> inside s \<noteq> {}"
   apply (simp add: in_components_maximal)
   apply (auto intro: inside_same_component connected_componentI)
   apply (metis IntI empty_iff inside_no_overlap)
   done
 
 text\<open>The proof is virtually the same as that above.\<close>
 lemma outside_in_components:
      "outside s \<in> components (- s) \<longleftrightarrow> connected(outside s) \<and> outside s \<noteq> {}"
   apply (simp add: in_components_maximal)
   apply (auto intro: outside_same_component connected_componentI)
   apply (metis IntI empty_iff outside_no_overlap)
   done
 
 lemma bounded_unique_outside:
     fixes s :: "'a :: euclidean_space set"
     shows "\<lbrakk>bounded s; DIM('a) \<ge> 2\<rbrakk> \<Longrightarrow> (c \<in> components (- s) \<and> \<not> bounded c \<longleftrightarrow> c = outside s)"
   apply (rule iffI)
   apply (metis cobounded_unique_unbounded_components connected_outside double_compl outside_bounded_nonempty outside_in_components unbounded_outside)
   by (simp add: connected_outside outside_bounded_nonempty outside_in_components unbounded_outside)
 
 
 subsection\<open>Condition for an open map's image to contain a ball\<close>
 
 proposition ball_subset_open_map_image:
   fixes f :: "'a::heine_borel \<Rightarrow> 'b :: {real_normed_vector,heine_borel}"
   assumes contf: "continuous_on (closure S) f"
       and oint: "open (f ` interior S)"
       and le_no: "\<And>z. z \<in> frontier S \<Longrightarrow> r \<le> norm(f z - f a)"
       and "bounded S" "a \<in> S" "0 < r"
     shows "ball (f a) r \<subseteq> f ` S"
 proof (cases "f ` S = UNIV")
   case True then show ?thesis by simp
 next
   case False
     obtain w where w: "w \<in> frontier (f ` S)"
                and dw_le: "\<And>y. y \<in> frontier (f ` S) \<Longrightarrow> norm (f a - w) \<le> norm (f a - y)"
       apply (rule distance_attains_inf [of "frontier(f ` S)" "f a"])
       using \<open>a \<in> S\<close> by (auto simp: frontier_eq_empty dist_norm False)
     then obtain \<xi> where \<xi>: "\<And>n. \<xi> n \<in> f ` S" and tendsw: "\<xi> \<longlonglongrightarrow> w"
       by (metis Diff_iff frontier_def closure_sequential)
     then have "\<And>n. \<exists>x \<in> S. \<xi> n = f x" by force
     then obtain z where zs: "\<And>n. z n \<in> S" and fz: "\<And>n. \<xi> n = f (z n)"
       by metis
     then obtain y K where y: "y \<in> closure S" and "strict_mono (K :: nat \<Rightarrow> nat)" 
                       and Klim: "(z \<circ> K) \<longlonglongrightarrow> y"
       using \<open>bounded S\<close>
       apply (simp add: compact_closure [symmetric] compact_def)
       apply (drule_tac x=z in spec)
       using closure_subset apply force
       done
     then have ftendsw: "((\<lambda>n. f (z n)) \<circ> K) \<longlonglongrightarrow> w"
       by (metis LIMSEQ_subseq_LIMSEQ fun.map_cong0 fz tendsw)
     have zKs: "\<And>n. (z \<circ> K) n \<in> S" by (simp add: zs)
     have fz: "f \<circ> z = \<xi>"  "(\<lambda>n. f (z n)) = \<xi>"
       using fz by auto
     then have "(\<xi> \<circ> K) \<longlonglongrightarrow> f y"
       by (metis (no_types) Klim zKs y contf comp_assoc continuous_on_closure_sequentially)
     with fz have wy: "w = f y" using fz LIMSEQ_unique ftendsw by auto
     have rle: "r \<le> norm (f y - f a)"
       apply (rule le_no)
       using w wy oint
       by (force simp: imageI image_mono interiorI interior_subset frontier_def y)
     have **: "(b \<inter> (- S) \<noteq> {} \<and> b - (- S) \<noteq> {} \<Longrightarrow> b \<inter> f \<noteq> {})
               \<Longrightarrow> (b \<inter> S \<noteq> {}) \<Longrightarrow> b \<inter> f = {} \<Longrightarrow>
               b \<subseteq> S" for b f and S :: "'b set"
       by blast
     show ?thesis
       apply (rule **)   (*such a horrible mess*)
       apply (rule connected_Int_frontier [where t = "f`S", OF connected_ball])
       using \<open>a \<in> S\<close> \<open>0 < r\<close>
       apply (auto simp: disjoint_iff_not_equal dist_norm)
       by (metis dw_le norm_minus_commute not_less order_trans rle wy)
 qed
 
 
 subsubsection\<open>Special characterizations of classes of functions into and out of R.\<close>
 
 lemma Hausdorff_space_euclidean [simp]: "Hausdorff_space (euclidean :: 'a::metric_space topology)"
 proof -
   have "\<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> disjnt U V"
     if "x \<noteq> y"
     for x y :: 'a
   proof (intro exI conjI)
     let ?r = "dist x y / 2"
     have [simp]: "?r > 0"
       by (simp add: that)
     show "open (ball x ?r)" "open (ball y ?r)" "x \<in> (ball x ?r)" "y \<in> (ball y ?r)"
       by (auto simp add: that)
     show "disjnt (ball x ?r) (ball y ?r)"
       unfolding disjnt_def by (simp add: disjoint_ballI)
   qed
   then show ?thesis
     by (simp add: Hausdorff_space_def)
 qed
 
 proposition embedding_map_into_euclideanreal:
   assumes "path_connected_space X"
   shows "embedding_map X euclideanreal f \<longleftrightarrow>
          continuous_map X euclideanreal f \<and> inj_on f (topspace X)"
   proof safe
   show "continuous_map X euclideanreal f"
     if "embedding_map X euclideanreal f"
     using continuous_map_in_subtopology homeomorphic_imp_continuous_map that
     unfolding embedding_map_def by blast
   show "inj_on f (topspace X)"
     if "embedding_map X euclideanreal f"
     using that homeomorphic_imp_injective_map
     unfolding embedding_map_def by blast
   show "embedding_map X euclideanreal f"
     if cont: "continuous_map X euclideanreal f" and inj: "inj_on f (topspace X)"
   proof -
     obtain g where gf: "\<And>x. x \<in> topspace X \<Longrightarrow> g (f x) = x"
       using inv_into_f_f [OF inj] by auto
     show ?thesis
       unfolding embedding_map_def homeomorphic_map_maps homeomorphic_maps_def
     proof (intro exI conjI)
       show "continuous_map X (top_of_set (f ` topspace X)) f"
         by (simp add: cont continuous_map_in_subtopology)
       let ?S = "f ` topspace X"
       have eq: "{x \<in> ?S. g x \<in> U} = f ` U" if "openin X U" for U
         using openin_subset [OF that] by (auto simp: gf)
       have 1: "g ` ?S \<subseteq> topspace X"
         using eq by blast
       have "openin (top_of_set ?S) {x \<in> ?S. g x \<in> T}"
         if "openin X T" for T
       proof -
         have "T \<subseteq> topspace X"
           by (simp add: openin_subset that)
         have RR: "\<forall>x \<in> ?S \<inter> g -` T. \<exists>d>0. \<forall>x' \<in> ?S \<inter> ball x d. g x' \<in> T"
         proof (clarsimp simp add: gf)
           have pcS: "path_connectedin euclidean ?S"
             using assms cont path_connectedin_continuous_map_image path_connectedin_topspace by blast
           show "\<exists>d>0. \<forall>x'\<in>f ` topspace X \<inter> ball (f x) d. g x' \<in> T"
             if "x \<in> T" for x
           proof -
             have x: "x \<in> topspace X"
               using \<open>T \<subseteq> topspace X\<close> \<open>x \<in> T\<close> by blast
             obtain u v d where "0 < d" "u \<in> topspace X" "v \<in> topspace X"
                          and sub_fuv: "?S \<inter> {f x - d .. f x + d} \<subseteq> {f u..f v}"
             proof (cases "\<exists>u \<in> topspace X. f u < f x")
               case True
               then obtain u where u: "u \<in> topspace X" "f u < f x" ..
               show ?thesis
               proof (cases "\<exists>v \<in> topspace X. f x < f v")
                 case True
                 then obtain v where v: "v \<in> topspace X" "f x < f v" ..
                 show ?thesis
                 proof
                   let ?d = "min (f x - f u) (f v - f x)"
                   show "0 < ?d"
                     by (simp add: \<open>f u < f x\<close> \<open>f x < f v\<close>)
                   show "f ` topspace X \<inter> {f x - ?d..f x + ?d} \<subseteq> {f u..f v}"
                     by fastforce
                 qed (auto simp: u v)
               next
                 case False
                 show ?thesis
                 proof
                   let ?d = "f x - f u"
                   show "0 < ?d"
                     by (simp add: u)
                   show "f ` topspace X \<inter> {f x - ?d..f x + ?d} \<subseteq> {f u..f x}"
                     using x u False by auto
                 qed (auto simp: x u)
               qed
             next
               case False
               note no_u = False
               show ?thesis
               proof (cases "\<exists>v \<in> topspace X. f x < f v")
                 case True
                 then obtain v where v: "v \<in> topspace X" "f x < f v" ..
                 show ?thesis
                 proof
                   let ?d = "f v - f x"
                   show "0 < ?d"
                     by (simp add: v)
                   show "f ` topspace X \<inter> {f x - ?d..f x + ?d} \<subseteq> {f x..f v}"
                     using False by auto
                 qed (auto simp: x v)
               next
                 case False
                 show ?thesis
                 proof
                   show "f ` topspace X \<inter> {f x - 1..f x + 1} \<subseteq> {f x..f x}"
                     using False no_u by fastforce
                 qed (auto simp: x)
               qed
             qed
             then obtain h where "pathin X h" "h 0 = u" "h 1 = v"
               using assms unfolding path_connected_space_def by blast
             obtain C where "compactin X C" "connectedin X C" "u \<in> C" "v \<in> C"
             proof
               show "compactin X (h ` {0..1})"
                 using that by (simp add: \<open>pathin X h\<close> compactin_path_image)
               show "connectedin X (h ` {0..1})"
                 using \<open>pathin X h\<close> connectedin_path_image by blast
             qed (use \<open>h 0 = u\<close> \<open>h 1 = v\<close> in auto)
             have "continuous_map (subtopology euclideanreal (?S \<inter> {f x - d .. f x + d})) (subtopology X C) g"
             proof (rule continuous_inverse_map)
               show "compact_space (subtopology X C)"
                 using \<open>compactin X C\<close> compactin_subspace by blast
               show "continuous_map (subtopology X C) euclideanreal f"
                 by (simp add: cont continuous_map_from_subtopology)
               have "{f u .. f v} \<subseteq> f ` topspace (subtopology X C)"
               proof (rule connected_contains_Icc)
                 show "connected (f ` topspace (subtopology X C))"
                   using connectedin_continuous_map_image [OF cont]
                   by (simp add: \<open>compactin X C\<close> \<open>connectedin X C\<close> compactin_subset_topspace inf_absorb2)
                 show "f u \<in> f ` topspace (subtopology X C)"
                   by (simp add: \<open>u \<in> C\<close> \<open>u \<in> topspace X\<close>)
                 show "f v \<in> f ` topspace (subtopology X C)"
                   by (simp add: \<open>v \<in> C\<close> \<open>v \<in> topspace X\<close>)
               qed
               then show "f ` topspace X \<inter> {f x - d..f x + d} \<subseteq> f ` topspace (subtopology X C)"
                 using sub_fuv by blast
             qed (auto simp: gf)
             then have contg: "continuous_map (subtopology euclideanreal (?S \<inter> {f x - d .. f x + d})) X g"
               using continuous_map_in_subtopology by blast
             have "\<exists>e>0. \<forall>x \<in> ?S \<inter> {f x - d .. f x + d} \<inter> ball (f x) e. g x \<in> T"
               using openin_continuous_map_preimage [OF contg \<open>openin X T\<close>] x \<open>x \<in> T\<close> \<open>0 < d\<close>
               unfolding openin_euclidean_subtopology_iff
               by (force simp: gf dist_commute)
             then obtain e where "e > 0 \<and> (\<forall>x\<in>f ` topspace X \<inter> {f x - d..f x + d} \<inter> ball (f x) e. g x \<in> T)"
               by metis
             with \<open>0 < d\<close> have "min d e > 0" "\<forall>u. u \<in> topspace X \<longrightarrow> \<bar>f x - f u\<bar> < min d e \<longrightarrow> u \<in> T"
               using dist_real_def gf by force+
             then show ?thesis
               by (metis (full_types) Int_iff dist_real_def image_iff mem_ball gf)
           qed
         qed
         then obtain d where d: "\<And>r. r \<in> ?S \<inter> g -` T \<Longrightarrow>
                 d r > 0 \<and> (\<forall>x \<in> ?S \<inter> ball r (d r). g x \<in> T)"
           by metis
         show ?thesis
           unfolding openin_subtopology
         proof (intro exI conjI)
           show "{x \<in> ?S. g x \<in> T} = (\<Union>r \<in> ?S \<inter> g -` T. ball r (d r)) \<inter> f ` topspace X"
             using d by (auto simp: gf)
         qed auto
       qed
       then show "continuous_map (top_of_set ?S) X g"
         by (simp add: continuous_map_def gf)
     qed (auto simp: gf)
   qed
 qed
 
 subsubsection \<open>An injective function into R is a homeomorphism and so an open map.\<close>
 
 lemma injective_into_1d_eq_homeomorphism:
   fixes f :: "'a::topological_space \<Rightarrow> real"
   assumes f: "continuous_on S f" and S: "path_connected S"
   shows "inj_on f S \<longleftrightarrow> (\<exists>g. homeomorphism S (f ` S) f g)"
 proof
   show "\<exists>g. homeomorphism S (f ` S) f g"
     if "inj_on f S"
   proof -
     have "embedding_map (top_of_set S) euclideanreal f"
       using that embedding_map_into_euclideanreal [of "top_of_set S" f] assms by auto
     then show ?thesis
       by (simp add: embedding_map_def) (metis all_closedin_homeomorphic_image f homeomorphism_injective_closed_map that)
   qed
 qed (metis homeomorphism_def inj_onI)
 
 lemma injective_into_1d_imp_open_map:
   fixes f :: "'a::topological_space \<Rightarrow> real"
   assumes "continuous_on S f" "path_connected S" "inj_on f S" "openin (subtopology euclidean S) T"
   shows "openin (subtopology euclidean (f ` S)) (f ` T)"
   using assms homeomorphism_imp_open_map injective_into_1d_eq_homeomorphism by blast
 
 lemma homeomorphism_into_1d:
   fixes f :: "'a::topological_space \<Rightarrow> real"
   assumes "path_connected S" "continuous_on S f" "f ` S = T" "inj_on f S"
   shows "\<exists>g. homeomorphism S T f g"
   using assms injective_into_1d_eq_homeomorphism by blast
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>Rectangular paths\<close>
 
 definition\<^marker>\<open>tag unimportant\<close> rectpath where
   "rectpath a1 a3 = (let a2 = Complex (Re a3) (Im a1); a4 = Complex (Re a1) (Im a3)
                       in linepath a1 a2 +++ linepath a2 a3 +++ linepath a3 a4 +++ linepath a4 a1)"
 
 lemma path_rectpath [simp, intro]: "path (rectpath a b)"
   by (simp add: Let_def rectpath_def)
 
 lemma pathstart_rectpath [simp]: "pathstart (rectpath a1 a3) = a1"
   by (simp add: rectpath_def Let_def)
 
 lemma pathfinish_rectpath [simp]: "pathfinish (rectpath a1 a3) = a1"
   by (simp add: rectpath_def Let_def)
 
 lemma simple_path_rectpath [simp, intro]:
   assumes "Re a1 \<noteq> Re a3" "Im a1 \<noteq> Im a3"
   shows   "simple_path (rectpath a1 a3)"
   unfolding rectpath_def Let_def using assms
   by (intro simple_path_join_loop arc_join arc_linepath)
      (auto simp: complex_eq_iff path_image_join closed_segment_same_Re closed_segment_same_Im)
 
 lemma path_image_rectpath:
   assumes "Re a1 \<le> Re a3" "Im a1 \<le> Im a3"
   shows "path_image (rectpath a1 a3) =
            {z. Re z \<in> {Re a1, Re a3} \<and> Im z \<in> {Im a1..Im a3}} \<union>
            {z. Im z \<in> {Im a1, Im a3} \<and> Re z \<in> {Re a1..Re a3}}" (is "?lhs = ?rhs")
 proof -
   define a2 a4 where "a2 = Complex (Re a3) (Im a1)" and "a4 = Complex (Re a1) (Im a3)"
   have "?lhs = closed_segment a1 a2 \<union> closed_segment a2 a3 \<union>
                   closed_segment a4 a3 \<union> closed_segment a1 a4"
     by (simp_all add: rectpath_def Let_def path_image_join closed_segment_commute
                       a2_def a4_def Un_assoc)
   also have "\<dots> = ?rhs" using assms
     by (auto simp: rectpath_def Let_def path_image_join a2_def a4_def
           closed_segment_same_Re closed_segment_same_Im closed_segment_eq_real_ivl)
   finally show ?thesis .
 qed
 
 lemma path_image_rectpath_subset_cbox:
   assumes "Re a \<le> Re b" "Im a \<le> Im b"
   shows   "path_image (rectpath a b) \<subseteq> cbox a b"
   using assms by (auto simp: path_image_rectpath in_cbox_complex_iff)
 
 lemma path_image_rectpath_inter_box:
   assumes "Re a \<le> Re b" "Im a \<le> Im b"
   shows   "path_image (rectpath a b) \<inter> box a b = {}"
   using assms by (auto simp: path_image_rectpath in_box_complex_iff)
 
 lemma path_image_rectpath_cbox_minus_box:
   assumes "Re a \<le> Re b" "Im a \<le> Im b"
   shows   "path_image (rectpath a b) = cbox a b - box a b"
   using assms by (auto simp: path_image_rectpath in_cbox_complex_iff
                              in_box_complex_iff)
 
 end
diff --git a/src/HOL/Analysis/Polytope.thy b/src/HOL/Analysis/Polytope.thy
--- a/src/HOL/Analysis/Polytope.thy
+++ b/src/HOL/Analysis/Polytope.thy
@@ -1,4005 +1,4005 @@
 section \<open>Faces, Extreme Points, Polytopes, Polyhedra etc\<close>
 
 text\<open>Ported from HOL Light by L C Paulson\<close>
 
 theory Polytope
 imports Cartesian_Euclidean_Space Path_Connected
 begin
 
 subsection \<open>Faces of a (usually convex) set\<close>
 
 definition\<^marker>\<open>tag important\<close> face_of :: "['a::real_vector set, 'a set] \<Rightarrow> bool" (infixr "(face'_of)" 50)
   where
   "T face_of S \<longleftrightarrow>
         T \<subseteq> S \<and> convex T \<and>
         (\<forall>a \<in> S. \<forall>b \<in> S. \<forall>x \<in> T. x \<in> open_segment a b \<longrightarrow> a \<in> T \<and> b \<in> T)"
 
 lemma face_ofD: "\<lbrakk>T face_of S; x \<in> open_segment a b; a \<in> S; b \<in> S; x \<in> T\<rbrakk> \<Longrightarrow> a \<in> T \<and> b \<in> T"
   unfolding face_of_def by blast
 
 lemma face_of_translation_eq [simp]:
     "((+) a ` T face_of (+) a ` S) \<longleftrightarrow> T face_of S"
 proof -
   have *: "\<And>a T S. T face_of S \<Longrightarrow> ((+) a ` T face_of (+) a ` S)"
     apply (simp add: face_of_def Ball_def, clarify)
     by (meson imageI open_segment_translation_eq)
   show ?thesis
     apply (rule iffI)
     apply (force simp: image_comp o_def dest: * [where a = "-a"])
     apply (blast intro: *)
     done
 qed
 
 lemma face_of_linear_image:
   assumes "linear f" "inj f"
     shows "(f ` c face_of f ` S) \<longleftrightarrow> c face_of S"
 by (simp add: face_of_def inj_image_subset_iff inj_image_mem_iff open_segment_linear_image assms)
 
 lemma face_of_refl: "convex S \<Longrightarrow> S face_of S"
   by (auto simp: face_of_def)
 
 lemma face_of_refl_eq: "S face_of S \<longleftrightarrow> convex S"
   by (auto simp: face_of_def)
 
 lemma empty_face_of [iff]: "{} face_of S"
   by (simp add: face_of_def)
 
 lemma face_of_empty [simp]: "S face_of {} \<longleftrightarrow> S = {}"
   by (meson empty_face_of face_of_def subset_empty)
 
 lemma face_of_trans [trans]: "\<lbrakk>S face_of T; T face_of u\<rbrakk> \<Longrightarrow> S face_of u"
   unfolding face_of_def by (safe; blast)
 
 lemma face_of_face: "T face_of S \<Longrightarrow> (f face_of T \<longleftrightarrow> f face_of S \<and> f \<subseteq> T)"
   unfolding face_of_def by (safe; blast)
 
 lemma face_of_subset: "\<lbrakk>F face_of S; F \<subseteq> T; T \<subseteq> S\<rbrakk> \<Longrightarrow> F face_of T"
   unfolding face_of_def by (safe; blast)
 
 lemma face_of_slice: "\<lbrakk>F face_of S; convex T\<rbrakk> \<Longrightarrow> (F \<inter> T) face_of (S \<inter> T)"
   unfolding face_of_def by (blast intro: convex_Int)
 
 lemma face_of_Int: "\<lbrakk>t1 face_of S; t2 face_of S\<rbrakk> \<Longrightarrow> (t1 \<inter> t2) face_of S"
   unfolding face_of_def by (blast intro: convex_Int)
 
 lemma face_of_Inter: "\<lbrakk>A \<noteq> {}; \<And>T. T \<in> A \<Longrightarrow> T face_of S\<rbrakk> \<Longrightarrow> (\<Inter> A) face_of S"
   unfolding face_of_def by (blast intro: convex_Inter)
 
 lemma face_of_Int_Int: "\<lbrakk>F face_of T; F' face_of t'\<rbrakk> \<Longrightarrow> (F \<inter> F') face_of (T \<inter> t')"
   unfolding face_of_def by (blast intro: convex_Int)
 
 lemma face_of_imp_subset: "T face_of S \<Longrightarrow> T \<subseteq> S"
   unfolding face_of_def by blast
 
 proposition face_of_imp_eq_affine_Int:
   fixes S :: "'a::euclidean_space set"
   assumes S: "convex S"  and T: "T face_of S"
   shows "T = (affine hull T) \<inter> S"
 proof -
   have "convex T" using T by (simp add: face_of_def)
   have *: False if x: "x \<in> affine hull T" and "x \<in> S" "x \<notin> T" and y: "y \<in> rel_interior T" for x y
   proof -
     obtain e where "e>0" and e: "cball y e \<inter> affine hull T \<subseteq> T"
       using y by (auto simp: rel_interior_cball)
     have "y \<noteq> x" "y \<in> S" "y \<in> T"
       using face_of_imp_subset rel_interior_subset T that by blast+
     then have zne: "\<And>u. \<lbrakk>u \<in> {0<..<1}; (1 - u) *\<^sub>R y + u *\<^sub>R x \<in> T\<rbrakk> \<Longrightarrow>  False"
       using \<open>x \<in> S\<close> \<open>x \<notin> T\<close> \<open>T face_of S\<close> unfolding face_of_def
       apply clarify
       apply (drule_tac x=x in bspec, assumption)
       apply (drule_tac x=y in bspec, assumption)
       apply (subst (asm) open_segment_commute)
       apply (force simp: open_segment_image_interval image_def)
       done
     have in01: "min (1/2) (e / norm (x - y)) \<in> {0<..<1}"
       using \<open>y \<noteq> x\<close> \<open>e > 0\<close> by simp
     show ?thesis
       apply (rule zne [OF in01])
       apply (rule e [THEN subsetD])
       apply (rule IntI)
         using \<open>y \<noteq> x\<close> \<open>e > 0\<close>
         apply (simp add: cball_def dist_norm algebra_simps)
         apply (simp add: Real_Vector_Spaces.scaleR_diff_right [symmetric] norm_minus_commute min_mult_distrib_right)
       apply (rule mem_affine [OF affine_affine_hull _ x])
       using \<open>y \<in> T\<close>  apply (auto simp: hull_inc)
       done
   qed
   show ?thesis
     apply (rule subset_antisym)
     using assms apply (simp add: hull_subset face_of_imp_subset)
     apply (cases "T={}", simp)
     apply (force simp: rel_interior_eq_empty [symmetric] \<open>convex T\<close> intro: *)
     done
 qed
 
 lemma face_of_imp_closed:
      fixes S :: "'a::euclidean_space set"
      assumes "convex S" "closed S" "T face_of S" shows "closed T"
   by (metis affine_affine_hull affine_closed closed_Int face_of_imp_eq_affine_Int assms)
 
 lemma face_of_Int_supporting_hyperplane_le_strong:
     assumes "convex(S \<inter> {x. a \<bullet> x = b})" and aleb: "\<And>x. x \<in> S \<Longrightarrow> a \<bullet> x \<le> b"
       shows "(S \<inter> {x. a \<bullet> x = b}) face_of S"
 proof -
   have *: "a \<bullet> u = a \<bullet> x" if "x \<in> open_segment u v" "u \<in> S" "v \<in> S" and b: "b = a \<bullet> x"
           for u v x
   proof (rule antisym)
     show "a \<bullet> u \<le> a \<bullet> x"
       using aleb \<open>u \<in> S\<close> \<open>b = a \<bullet> x\<close> by blast
   next
     obtain \<xi> where "b = a \<bullet> ((1 - \<xi>) *\<^sub>R u + \<xi> *\<^sub>R v)" "0 < \<xi>" "\<xi> < 1"
       using \<open>b = a \<bullet> x\<close> \<open>x \<in> open_segment u v\<close> in_segment
       by (auto simp: open_segment_image_interval split: if_split_asm)
     then have "b + \<xi> * (a \<bullet> u) \<le> a \<bullet> u + \<xi> * b"
       using aleb [OF \<open>v \<in> S\<close>] by (simp add: algebra_simps)
     then have "(1 - \<xi>) * b \<le> (1 - \<xi>) * (a \<bullet> u)"
       by (simp add: algebra_simps)
     then have "b \<le> a \<bullet> u"
       using \<open>\<xi> < 1\<close> by auto
     with b show "a \<bullet> x \<le> a \<bullet> u" by simp
   qed
   show ?thesis
     apply (simp add: face_of_def assms)
     using "*" open_segment_commute by blast
 qed
 
 lemma face_of_Int_supporting_hyperplane_ge_strong:
    "\<lbrakk>convex(S \<inter> {x. a \<bullet> x = b}); \<And>x. x \<in> S \<Longrightarrow> a \<bullet> x \<ge> b\<rbrakk>
     \<Longrightarrow> (S \<inter> {x. a \<bullet> x = b}) face_of S"
   using face_of_Int_supporting_hyperplane_le_strong [of S "-a" "-b"] by simp
 
 lemma face_of_Int_supporting_hyperplane_le:
     "\<lbrakk>convex S; \<And>x. x \<in> S \<Longrightarrow> a \<bullet> x \<le> b\<rbrakk> \<Longrightarrow> (S \<inter> {x. a \<bullet> x = b}) face_of S"
   by (simp add: convex_Int convex_hyperplane face_of_Int_supporting_hyperplane_le_strong)
 
 lemma face_of_Int_supporting_hyperplane_ge:
     "\<lbrakk>convex S; \<And>x. x \<in> S \<Longrightarrow> a \<bullet> x \<ge> b\<rbrakk> \<Longrightarrow> (S \<inter> {x. a \<bullet> x = b}) face_of S"
   by (simp add: convex_Int convex_hyperplane face_of_Int_supporting_hyperplane_ge_strong)
 
 lemma face_of_imp_convex: "T face_of S \<Longrightarrow> convex T"
   using face_of_def by blast
 
 lemma face_of_imp_compact:
     fixes S :: "'a::euclidean_space set"
     shows "\<lbrakk>convex S; compact S; T face_of S\<rbrakk> \<Longrightarrow> compact T"
   by (meson bounded_subset compact_eq_bounded_closed face_of_imp_closed face_of_imp_subset)
 
 lemma face_of_Int_subface:
      "\<lbrakk>A \<inter> B face_of A; A \<inter> B face_of B; C face_of A; D face_of B\<rbrakk>
       \<Longrightarrow> (C \<inter> D) face_of C \<and> (C \<inter> D) face_of D"
   by (meson face_of_Int_Int face_of_face inf_le1 inf_le2)
 
 lemma subset_of_face_of:
     fixes S :: "'a::real_normed_vector set"
     assumes "T face_of S" "u \<subseteq> S" "T \<inter> (rel_interior u) \<noteq> {}"
       shows "u \<subseteq> T"
 proof
   fix c
   assume "c \<in> u"
   obtain b where "b \<in> T" "b \<in> rel_interior u" using assms by auto
   then obtain e where "e>0" "b \<in> u" and e: "cball b e \<inter> affine hull u \<subseteq> u"
     by (auto simp: rel_interior_cball)
   show "c \<in> T"
   proof (cases "b=c")
     case True with \<open>b \<in> T\<close> show ?thesis by blast
   next
     case False
     define d where "d = b + (e / norm(b - c)) *\<^sub>R (b - c)"
     have "d \<in> cball b e \<inter> affine hull u"
       using \<open>e > 0\<close> \<open>b \<in> u\<close> \<open>c \<in> u\<close>
       by (simp add: d_def dist_norm hull_inc mem_affine_3_minus False)
     with e have "d \<in> u" by blast
     have nbc: "norm (b - c) + e > 0" using \<open>e > 0\<close>
       by (metis add.commute le_less_trans less_add_same_cancel2 norm_ge_zero)
     then have [simp]: "d \<noteq> c" using False scaleR_cancel_left [of "1 + (e / norm (b - c))" b c]
       by (simp add: algebra_simps d_def) (simp add: field_split_simps)
     have [simp]: "((e - e * e / (e + norm (b - c))) / norm (b - c)) = (e / (e + norm (b - c)))"
       using False nbc
       by (simp add: divide_simps) (simp add: algebra_simps)
     have "b \<in> open_segment d c"
       apply (simp add: open_segment_image_interval)
       apply (simp add: d_def algebra_simps image_def)
       apply (rule_tac x="e / (e + norm (b - c))" in bexI)
       using False nbc \<open>0 < e\<close>
       apply (auto simp: algebra_simps)
       done
     then have "d \<in> T \<and> c \<in> T"
       apply (rule face_ofD [OF \<open>T face_of S\<close>])
       using \<open>d \<in> u\<close>  \<open>c \<in> u\<close> \<open>u \<subseteq> S\<close>  \<open>b \<in> T\<close>  apply auto
       done
     then show ?thesis ..
   qed
 qed
 
 lemma face_of_eq:
     fixes S :: "'a::real_normed_vector set"
     assumes "T face_of S" "u face_of S" "(rel_interior T) \<inter> (rel_interior u) \<noteq> {}"
       shows "T = u"
   apply (rule subset_antisym)
   apply (metis assms disjoint_iff_not_equal face_of_imp_subset rel_interior_subset subsetCE subset_of_face_of)
   by (metis assms disjoint_iff_not_equal face_of_imp_subset rel_interior_subset subset_iff subset_of_face_of)
 
 lemma face_of_disjoint_rel_interior:
       fixes S :: "'a::real_normed_vector set"
       assumes "T face_of S" "T \<noteq> S"
         shows "T \<inter> rel_interior S = {}"
   by (meson assms subset_of_face_of face_of_imp_subset order_refl subset_antisym)
 
 lemma face_of_disjoint_interior:
       fixes S :: "'a::real_normed_vector set"
       assumes "T face_of S" "T \<noteq> S"
         shows "T \<inter> interior S = {}"
 proof -
   have "T \<inter> interior S \<subseteq> rel_interior S"
     by (meson inf_sup_ord(2) interior_subset_rel_interior order.trans)
   thus ?thesis
     by (metis (no_types) Int_greatest assms face_of_disjoint_rel_interior inf_sup_ord(1) subset_empty)
 qed
 
 lemma face_of_subset_rel_boundary:
   fixes S :: "'a::real_normed_vector set"
   assumes "T face_of S" "T \<noteq> S"
     shows "T \<subseteq> (S - rel_interior S)"
 by (meson DiffI assms disjoint_iff_not_equal face_of_disjoint_rel_interior face_of_imp_subset rev_subsetD subsetI)
 
 lemma face_of_subset_rel_frontier:
     fixes S :: "'a::real_normed_vector set"
     assumes "T face_of S" "T \<noteq> S"
       shows "T \<subseteq> rel_frontier S"
   using assms closure_subset face_of_disjoint_rel_interior face_of_imp_subset rel_frontier_def by fastforce
 
 lemma face_of_aff_dim_lt:
   fixes S :: "'a::euclidean_space set"
   assumes "convex S" "T face_of S" "T \<noteq> S"
     shows "aff_dim T < aff_dim S"
 proof -
   have "aff_dim T \<le> aff_dim S"
     by (simp add: face_of_imp_subset aff_dim_subset assms)
   moreover have "aff_dim T \<noteq> aff_dim S"
   proof (cases "T = {}")
     case True then show ?thesis
       by (metis aff_dim_empty \<open>T \<noteq> S\<close>)
   next case False then show ?thesis
     by (metis Set.set_insert assms convex_rel_frontier_aff_dim dual_order.irrefl face_of_imp_convex face_of_subset_rel_frontier insert_not_empty subsetI)
   qed
   ultimately show ?thesis
     by simp
 qed
 
 lemma subset_of_face_of_affine_hull:
     fixes S :: "'a::euclidean_space set"
   assumes T: "T face_of S" and "convex S" "U \<subseteq> S" and dis: "\<not> disjnt (affine hull T) (rel_interior U)"
   shows "U \<subseteq> T"
   apply (rule subset_of_face_of [OF T \<open>U \<subseteq> S\<close>])
   using face_of_imp_eq_affine_Int [OF \<open>convex S\<close> T]
   using rel_interior_subset [of U] dis
   using \<open>U \<subseteq> S\<close> disjnt_def by fastforce
 
 lemma affine_hull_face_of_disjoint_rel_interior:
     fixes S :: "'a::euclidean_space set"
   assumes "convex S" "F face_of S" "F \<noteq> S"
   shows "affine hull F \<inter> rel_interior S = {}"
   by (metis assms disjnt_def face_of_imp_subset order_refl subset_antisym subset_of_face_of_affine_hull)
 
 lemma affine_diff_divide:
     assumes "affine S" "k \<noteq> 0" "k \<noteq> 1" and xy: "x \<in> S" "y /\<^sub>R (1 - k) \<in> S"
       shows "(x - y) /\<^sub>R k \<in> S"
 proof -
   have "inverse(k) *\<^sub>R (x - y) = (1 - inverse k) *\<^sub>R inverse(1 - k) *\<^sub>R y + inverse(k) *\<^sub>R x"
     using assms
     by (simp add: algebra_simps) (simp add: scaleR_left_distrib [symmetric] field_split_simps)
   then show ?thesis
     using \<open>affine S\<close> xy by (auto simp: affine_alt)
 qed
 
 proposition face_of_convex_hulls:
       assumes S: "finite S" "T \<subseteq> S" and disj: "affine hull T \<inter> convex hull (S - T) = {}"
       shows  "(convex hull T) face_of (convex hull S)"
 proof -
   have fin: "finite T" "finite (S - T)" using assms
     by (auto simp: finite_subset)
   have *: "x \<in> convex hull T"
           if x: "x \<in> convex hull S" and y: "y \<in> convex hull S" and w: "w \<in> convex hull T" "w \<in> open_segment x y"
           for x y w
   proof -
     have waff: "w \<in> affine hull T"
       using convex_hull_subset_affine_hull w by blast
     obtain a b where a: "\<And>i. i \<in> S \<Longrightarrow> 0 \<le> a i" and asum: "sum a S = 1" and aeqx: "(\<Sum>i\<in>S. a i *\<^sub>R i) = x"
                  and b: "\<And>i. i \<in> S \<Longrightarrow> 0 \<le> b i" and bsum: "sum b S = 1" and beqy: "(\<Sum>i\<in>S. b i *\<^sub>R i) = y"
       using x y by (auto simp: assms convex_hull_finite)
     obtain u where "(1 - u) *\<^sub>R x + u *\<^sub>R y \<in> convex hull T" "x \<noteq> y" and weq: "w = (1 - u) *\<^sub>R x + u *\<^sub>R y"
                and u01: "0 < u" "u < 1"
       using w by (auto simp: open_segment_image_interval split: if_split_asm)
     define c where "c i = (1 - u) * a i + u * b i" for i
     have cge0: "\<And>i. i \<in> S \<Longrightarrow> 0 \<le> c i"
       using a b u01 by (simp add: c_def)
     have sumc1: "sum c S = 1"
       by (simp add: c_def sum.distrib sum_distrib_left [symmetric] asum bsum)
     have sumci_xy: "(\<Sum>i\<in>S. c i *\<^sub>R i) = (1 - u) *\<^sub>R x + u *\<^sub>R y"
       apply (simp add: c_def sum.distrib scaleR_left_distrib)
       by (simp only: scaleR_scaleR [symmetric] Real_Vector_Spaces.scaleR_right.sum [symmetric] aeqx beqy)
     show ?thesis
     proof (cases "sum c (S - T) = 0")
       case True
       have ci0: "\<And>i. i \<in> (S - T) \<Longrightarrow> c i = 0"
         using True cge0 fin(2) sum_nonneg_eq_0_iff by auto
       have a0: "a i = 0" if "i \<in> (S - T)" for i
         using ci0 [OF that] u01 a [of i] b [of i] that
         by (simp add: c_def Groups.ordered_comm_monoid_add_class.add_nonneg_eq_0_iff)
       have [simp]: "sum a T = 1"
         using assms by (metis sum.mono_neutral_cong_right a0 asum)
       show ?thesis
         apply (simp add: convex_hull_finite \<open>finite T\<close>)
         apply (rule_tac x=a in exI)
         using a0 assms
         apply (auto simp: cge0 a aeqx [symmetric] sum.mono_neutral_right)
         done
     next
       case False
       define k where "k = sum c (S - T)"
       have "k > 0" using False
         unfolding k_def by (metis DiffD1 antisym_conv cge0 sum_nonneg not_less)
       have weq_sumsum: "w = sum (\<lambda>x. c x *\<^sub>R x) T + sum (\<lambda>x. c x *\<^sub>R x) (S - T)"
         by (metis (no_types) add.commute S(1) S(2) sum.subset_diff sumci_xy weq)
       show ?thesis
       proof (cases "k = 1")
         case True
         then have "sum c T = 0"
           by (simp add: S k_def sum_diff sumc1)
         then have [simp]: "sum c (S - T) = 1"
           by (simp add: S sum_diff sumc1)
         have ci0: "\<And>i. i \<in> T \<Longrightarrow> c i = 0"
           by (meson \<open>finite T\<close> \<open>sum c T = 0\<close> \<open>T \<subseteq> S\<close> cge0 sum_nonneg_eq_0_iff subsetCE)
         then have [simp]: "(\<Sum>i\<in>S-T. c i *\<^sub>R i) = w"
           by (simp add: weq_sumsum)
         have "w \<in> convex hull (S - T)"
           apply (simp add: convex_hull_finite fin)
           apply (rule_tac x=c in exI)
           apply (auto simp: cge0 weq True k_def)
           done
         then show ?thesis
           using disj waff by blast
       next
         case False
         then have sumcf: "sum c T = 1 - k"
           by (simp add: S k_def sum_diff sumc1)
         have "(\<Sum>i\<in>T. c i *\<^sub>R i) /\<^sub>R (1 - k) \<in> convex hull T"
           apply (simp add: convex_hull_finite fin)
           apply (rule_tac x="\<lambda>i. inverse (1-k) * c i" in exI)
           apply auto
           apply (metis sumcf cge0 inverse_nonnegative_iff_nonnegative mult_nonneg_nonneg S(2) sum_nonneg subsetCE)
           apply (metis False mult.commute right_inverse right_minus_eq sum_distrib_left sumcf)
           by (metis (mono_tags, lifting) scaleR_right.sum scaleR_scaleR sum.cong)
         with \<open>0 < k\<close>  have "inverse(k) *\<^sub>R (w - sum (\<lambda>i. c i *\<^sub>R i) T) \<in> affine hull T"
           by (simp add: affine_diff_divide [OF affine_affine_hull] False waff convex_hull_subset_affine_hull [THEN subsetD])
         moreover have "inverse(k) *\<^sub>R (w - sum (\<lambda>x. c x *\<^sub>R x) T) \<in> convex hull (S - T)"
           apply (simp add: weq_sumsum convex_hull_finite fin)
           apply (rule_tac x="\<lambda>i. inverse k * c i" in exI)
           using \<open>k > 0\<close> cge0
           apply (auto simp: scaleR_right.sum sum_distrib_left [symmetric] k_def [symmetric])
           done
         ultimately show ?thesis
           using disj by blast
       qed
     qed
   qed
   have [simp]: "convex hull T \<subseteq> convex hull S"
     by (simp add: \<open>T \<subseteq> S\<close> hull_mono)
   show ?thesis
     using open_segment_commute by (auto simp: face_of_def intro: *)
 qed
 
 proposition face_of_convex_hull_insert:
    "\<lbrakk>finite S; a \<notin> affine hull S; T face_of convex hull S\<rbrakk> \<Longrightarrow> T face_of convex hull insert a S"
   apply (rule face_of_trans, blast)
   apply (rule face_of_convex_hulls; force simp: insert_Diff_if)
   done
 
 proposition face_of_affine_trivial:
     assumes "affine S" "T face_of S"
     shows "T = {} \<or> T = S"
 proof (rule ccontr, clarsimp)
   assume "T \<noteq> {}" "T \<noteq> S"
   then obtain a where "a \<in> T" by auto
   then have "a \<in> S"
     using \<open>T face_of S\<close> face_of_imp_subset by blast
   have "S \<subseteq> T"
   proof
     fix b  assume "b \<in> S"
     show "b \<in> T"
     proof (cases "a = b")
       case True with \<open>a \<in> T\<close> show ?thesis by auto
     next
       case False
       then have "a \<in> open_segment (2 *\<^sub>R a - b) b"
         apply (auto simp: open_segment_def closed_segment_def)
         apply (rule_tac x="1/2" in exI)
         apply (simp add: algebra_simps)
         by (simp add: scaleR_2)
       moreover have "2 *\<^sub>R a - b \<in> S"
         by (rule mem_affine [OF \<open>affine S\<close> \<open>a \<in> S\<close> \<open>b \<in> S\<close>, of 2 "-1", simplified])
       moreover note \<open>b \<in> S\<close> \<open>a \<in> T\<close>
       ultimately show ?thesis
         by (rule face_ofD [OF \<open>T face_of S\<close>, THEN conjunct2])
     qed
   qed
   then show False
     using \<open>T \<noteq> S\<close> \<open>T face_of S\<close> face_of_imp_subset by blast
 qed
 
 
 lemma face_of_affine_eq:
    "affine S \<Longrightarrow> (T face_of S \<longleftrightarrow> T = {} \<or> T = S)"
 using affine_imp_convex face_of_affine_trivial face_of_refl by auto
 
 
 proposition Inter_faces_finite_altbound:
     fixes T :: "'a::euclidean_space set set"
     assumes cfaI: "\<And>c. c \<in> T \<Longrightarrow> c face_of S"
     shows "\<exists>F'. finite F' \<and> F' \<subseteq> T \<and> card F' \<le> DIM('a) + 2 \<and> \<Inter>F' = \<Inter>T"
 proof (cases "\<forall>F'. finite F' \<and> F' \<subseteq> T \<and> card F' \<le> DIM('a) + 2 \<longrightarrow> (\<exists>c. c \<in> T \<and> c \<inter> (\<Inter>F') \<subset> (\<Inter>F'))")
   case True
   then obtain c where c:
        "\<And>F'. \<lbrakk>finite F'; F' \<subseteq> T; card F' \<le> DIM('a) + 2\<rbrakk> \<Longrightarrow> c F' \<in> T \<and> c F' \<inter> (\<Inter>F') \<subset> (\<Inter>F')"
     by metis
   define d where "d = rec_nat {c{}} (\<lambda>n r. insert (c r) r)"
   have [simp]: "d 0 = {c {}}"
     by (simp add: d_def)
   have dSuc [simp]: "\<And>n. d (Suc n) = insert (c (d n)) (d n)"
     by (simp add: d_def)
   have dn_notempty: "d n \<noteq> {}" for n
     by (induction n) auto
   have dn_le_Suc: "d n \<subseteq> T \<and> finite(d n) \<and> card(d n) \<le> Suc n" if "n \<le> DIM('a) + 2" for n
   using that
   proof (induction n)
     case 0
     then show ?case by (simp add: c)
   next
     case (Suc n)
     then show ?case by (auto simp: c card_insert_if)
   qed
   have aff_dim_le: "aff_dim(\<Inter>(d n)) \<le> DIM('a) - int n" if "n \<le> DIM('a) + 2" for n
   using that
   proof (induction n)
     case 0
     then show ?case
       by (simp add: aff_dim_le_DIM)
   next
     case (Suc n)
     have fs: "\<Inter>(d (Suc n)) face_of S"
       by (meson Suc.prems cfaI dn_le_Suc dn_notempty face_of_Inter subsetCE)
     have condn: "convex (\<Inter>(d n))"
       using Suc.prems nat_le_linear not_less_eq_eq
       by (blast intro: face_of_imp_convex cfaI convex_Inter dest: dn_le_Suc)
     have fdn: "\<Inter>(d (Suc n)) face_of \<Inter>(d n)"
       by (metis (no_types, lifting) Inter_anti_mono Suc.prems dSuc cfaI dn_le_Suc dn_notempty face_of_Inter face_of_imp_subset face_of_subset subset_iff subset_insertI)
     have ne: "\<Inter>(d (Suc n)) \<noteq> \<Inter>(d n)"
       by (metis (no_types, lifting) Suc.prems Suc_leD c complete_lattice_class.Inf_insert dSuc dn_le_Suc less_irrefl order.trans)
     have *: "\<And>m::int. \<And>d. \<And>d'::int. d < d' \<and> d' \<le> m - n \<Longrightarrow> d \<le> m - of_nat(n+1)"
       by arith
     have "aff_dim (\<Inter>(d (Suc n))) < aff_dim (\<Inter>(d n))"
       by (rule face_of_aff_dim_lt [OF condn fdn ne])
     moreover have "aff_dim (\<Inter>(d n)) \<le> int (DIM('a)) - int n"
       using Suc by auto
     ultimately
     have "aff_dim (\<Inter>(d (Suc n))) \<le> int (DIM('a)) - (n+1)" by arith
     then show ?case by linarith
   qed
   have "aff_dim (\<Inter>(d (DIM('a) + 2))) \<le> -2"
       using aff_dim_le [OF order_refl] by simp
   with aff_dim_geq [of "\<Inter>(d (DIM('a) + 2))"] show ?thesis
     using order.trans by fastforce
 next
   case False
   then show ?thesis
     apply simp
     apply (erule ex_forward)
     by blast
 qed
 
 lemma faces_of_translation:
    "{F. F face_of image (\<lambda>x. a + x) S} = image (image (\<lambda>x. a + x)) {F. F face_of S}"
 apply (rule subset_antisym, clarify)
 apply (auto simp: image_iff)
 apply (metis face_of_imp_subset face_of_translation_eq subset_imageE)
 done
 
 proposition face_of_Times:
   assumes "F face_of S" and "F' face_of S'"
     shows "(F \<times> F') face_of (S \<times> S')"
 proof -
   have "F \<times> F' \<subseteq> S \<times> S'"
     using assms [unfolded face_of_def] by blast
   moreover
   have "convex (F \<times> F')"
     using assms [unfolded face_of_def] by (blast intro: convex_Times)
   moreover
     have "a \<in> F \<and> a' \<in> F' \<and> b \<in> F \<and> b' \<in> F'"
        if "a \<in> S" "b \<in> S" "a' \<in> S'" "b' \<in> S'" "x \<in> F \<times> F'" "x \<in> open_segment (a,a') (b,b')"
        for a b a' b' x
   proof (cases "b=a \<or> b'=a'")
     case True with that show ?thesis
       using assms
       by (force simp: in_segment dest: face_ofD)
   next
     case False with assms [unfolded face_of_def] that show ?thesis
       by (blast dest!: open_segment_PairD)
   qed
   ultimately show ?thesis
     unfolding face_of_def by blast
 qed
 
 corollary face_of_Times_decomp:
     fixes S :: "'a::euclidean_space set" and S' :: "'b::euclidean_space set"
     shows "c face_of (S \<times> S') \<longleftrightarrow> (\<exists>F F'. F face_of S \<and> F' face_of S' \<and> c = F \<times> F')"
      (is "?lhs = ?rhs")
 proof
   assume c: ?lhs
   show ?rhs
   proof (cases "c = {}")
     case True then show ?thesis by auto
   next
     case False
     have 1: "fst ` c \<subseteq> S" "snd ` c \<subseteq> S'"
       using c face_of_imp_subset by fastforce+
     have "convex c"
       using c by (metis face_of_imp_convex)
     have conv: "convex (fst ` c)" "convex (snd ` c)"
-      by (simp_all add: \<open>convex c\<close> convex_linear_image fst_linear snd_linear)
+      by (simp_all add: \<open>convex c\<close> convex_linear_image linear_fst linear_snd)
     have fstab: "a \<in> fst ` c \<and> b \<in> fst ` c"
             if "a \<in> S" "b \<in> S" "x \<in> open_segment a b" "(x,x') \<in> c" for a b x x'
     proof -
       have *: "(x,x') \<in> open_segment (a,x') (b,x')"
         using that by (auto simp: in_segment)
       show ?thesis
         using face_ofD [OF c *] that face_of_imp_subset [OF c] by force
     qed
     have fst: "fst ` c face_of S"
       by (force simp: face_of_def 1 conv fstab)
     have sndab: "a' \<in> snd ` c \<and> b' \<in> snd ` c"
             if "a' \<in> S'" "b' \<in> S'" "x' \<in> open_segment a' b'" "(x,x') \<in> c" for a' b' x x'
     proof -
       have *: "(x,x') \<in> open_segment (x,a') (x,b')"
         using that by (auto simp: in_segment)
       show ?thesis
         using face_ofD [OF c *] that face_of_imp_subset [OF c] by force
     qed
     have snd: "snd ` c face_of S'"
       by (force simp: face_of_def 1 conv sndab)
     have cc: "rel_interior c \<subseteq> rel_interior (fst ` c) \<times> rel_interior (snd ` c)"
-      by (force simp: face_of_Times rel_interior_Times conv fst snd \<open>convex c\<close> fst_linear snd_linear rel_interior_convex_linear_image [symmetric])
+      by (force simp: face_of_Times rel_interior_Times conv fst snd \<open>convex c\<close> linear_fst linear_snd rel_interior_convex_linear_image [symmetric])
     have "c = fst ` c \<times> snd ` c"
       apply (rule face_of_eq [OF c])
       apply (simp_all add: face_of_Times rel_interior_Times conv fst snd)
       using False rel_interior_eq_empty \<open>convex c\<close> cc
       apply blast
       done
     with fst snd show ?thesis by metis
   qed
 next
   assume ?rhs with face_of_Times show ?lhs by auto
 qed
 
 lemma face_of_Times_eq:
     fixes S :: "'a::euclidean_space set" and S' :: "'b::euclidean_space set"
     shows "(F \<times> F') face_of (S \<times> S') \<longleftrightarrow>
            F = {} \<or> F' = {} \<or> F face_of S \<and> F' face_of S'"
 by (auto simp: face_of_Times_decomp times_eq_iff)
 
 lemma hyperplane_face_of_halfspace_le: "{x. a \<bullet> x = b} face_of {x. a \<bullet> x \<le> b}"
 proof -
   have "{x. a \<bullet> x \<le> b} \<inter> {x. a \<bullet> x = b} = {x. a \<bullet> x = b}"
     by auto
   with face_of_Int_supporting_hyperplane_le [OF convex_halfspace_le [of a b], of a b]
   show ?thesis by auto
 qed
 
 lemma hyperplane_face_of_halfspace_ge: "{x. a \<bullet> x = b} face_of {x. a \<bullet> x \<ge> b}"
 proof -
   have "{x. a \<bullet> x \<ge> b} \<inter> {x. a \<bullet> x = b} = {x. a \<bullet> x = b}"
     by auto
   with face_of_Int_supporting_hyperplane_ge [OF convex_halfspace_ge [of b a], of b a]
   show ?thesis by auto
 qed
 
 lemma face_of_halfspace_le:
   fixes a :: "'n::euclidean_space"
   shows "F face_of {x. a \<bullet> x \<le> b} \<longleftrightarrow>
          F = {} \<or> F = {x. a \<bullet> x = b} \<or> F = {x. a \<bullet> x \<le> b}"
      (is "?lhs = ?rhs")
 proof (cases "a = 0")
   case True then show ?thesis
     using face_of_affine_eq affine_UNIV by auto
 next
   case False
   then have ine: "interior {x. a \<bullet> x \<le> b} \<noteq> {}"
     using halfspace_eq_empty_lt interior_halfspace_le by blast
   show ?thesis
   proof
     assume L: ?lhs
     have "F \<noteq> {x. a \<bullet> x \<le> b} \<Longrightarrow> F face_of {x. a \<bullet> x = b}"
       using False
       apply (simp add: frontier_halfspace_le [symmetric] rel_frontier_nonempty_interior [OF ine, symmetric])
       apply (rule face_of_subset [OF L])
       apply (simp add: face_of_subset_rel_frontier [OF L])
       apply (force simp: rel_frontier_def closed_halfspace_le)
       done
     with L show ?rhs
       using affine_hyperplane face_of_affine_eq by blast
   next
     assume ?rhs
     then show ?lhs
       by (metis convex_halfspace_le empty_face_of face_of_refl hyperplane_face_of_halfspace_le)
   qed
 qed
 
 lemma face_of_halfspace_ge:
   fixes a :: "'n::euclidean_space"
   shows "F face_of {x. a \<bullet> x \<ge> b} \<longleftrightarrow>
          F = {} \<or> F = {x. a \<bullet> x = b} \<or> F = {x. a \<bullet> x \<ge> b}"
 using face_of_halfspace_le [of F "-a" "-b"] by simp
 
 subsection\<open>Exposed faces\<close>
 
 text\<open>That is, faces that are intersection with supporting hyperplane\<close>
 
 definition\<^marker>\<open>tag important\<close> exposed_face_of :: "['a::euclidean_space set, 'a set] \<Rightarrow> bool"
                                (infixr "(exposed'_face'_of)" 50)
   where "T exposed_face_of S \<longleftrightarrow>
          T face_of S \<and> (\<exists>a b. S \<subseteq> {x. a \<bullet> x \<le> b} \<and> T = S \<inter> {x. a \<bullet> x = b})"
 
 lemma empty_exposed_face_of [iff]: "{} exposed_face_of S"
   apply (simp add: exposed_face_of_def)
   apply (rule_tac x=0 in exI)
   apply (rule_tac x=1 in exI, force)
   done
 
 lemma exposed_face_of_refl_eq [simp]: "S exposed_face_of S \<longleftrightarrow> convex S"
   apply (simp add: exposed_face_of_def face_of_refl_eq, auto)
   apply (rule_tac x=0 in exI)+
   apply force
   done
 
 lemma exposed_face_of_refl: "convex S \<Longrightarrow> S exposed_face_of S"
   by simp
 
 lemma exposed_face_of:
     "T exposed_face_of S \<longleftrightarrow>
      T face_of S \<and>
      (T = {} \<or> T = S \<or>
       (\<exists>a b. a \<noteq> 0 \<and> S \<subseteq> {x. a \<bullet> x \<le> b} \<and> T = S \<inter> {x. a \<bullet> x = b}))"
 proof (cases "T = {}")
   case True then show ?thesis
     by simp
 next
   case False
   show ?thesis
   proof (cases "T = S")
     case True then show ?thesis
       by (simp add: face_of_refl_eq)
   next
     case False
     with \<open>T \<noteq> {}\<close> show ?thesis
       apply (auto simp: exposed_face_of_def)
       apply (metis inner_zero_left)
       done
   qed
 qed
 
 lemma exposed_face_of_Int_supporting_hyperplane_le:
    "\<lbrakk>convex S; \<And>x. x \<in> S \<Longrightarrow> a \<bullet> x \<le> b\<rbrakk> \<Longrightarrow> (S \<inter> {x. a \<bullet> x = b}) exposed_face_of S"
 by (force simp: exposed_face_of_def face_of_Int_supporting_hyperplane_le)
 
 lemma exposed_face_of_Int_supporting_hyperplane_ge:
    "\<lbrakk>convex S; \<And>x. x \<in> S \<Longrightarrow> a \<bullet> x \<ge> b\<rbrakk> \<Longrightarrow> (S \<inter> {x. a \<bullet> x = b}) exposed_face_of S"
 using exposed_face_of_Int_supporting_hyperplane_le [of S "-a" "-b"] by simp
 
 proposition exposed_face_of_Int:
   assumes "T exposed_face_of S"
       and "u exposed_face_of S"
     shows "(T \<inter> u) exposed_face_of S"
 proof -
   obtain a b where T: "S \<inter> {x. a \<bullet> x = b} face_of S"
                and S: "S \<subseteq> {x. a \<bullet> x \<le> b}"
                and teq: "T = S \<inter> {x. a \<bullet> x = b}"
     using assms by (auto simp: exposed_face_of_def)
   obtain a' b' where u: "S \<inter> {x. a' \<bullet> x = b'} face_of S"
                  and s': "S \<subseteq> {x. a' \<bullet> x \<le> b'}"
                  and ueq: "u = S \<inter> {x. a' \<bullet> x = b'}"
     using assms by (auto simp: exposed_face_of_def)
   have tu: "T \<inter> u face_of S"
     using T teq u ueq by (simp add: face_of_Int)
   have ss: "S \<subseteq> {x. (a + a') \<bullet> x \<le> b + b'}"
     using S s' by (force simp: inner_left_distrib)
   show ?thesis
     apply (simp add: exposed_face_of_def tu)
     apply (rule_tac x="a+a'" in exI)
     apply (rule_tac x="b+b'" in exI)
     using S s'
     apply (fastforce simp: ss inner_left_distrib teq ueq)
     done
 qed
 
 proposition exposed_face_of_Inter:
     fixes P :: "'a::euclidean_space set set"
   assumes "P \<noteq> {}"
       and "\<And>T. T \<in> P \<Longrightarrow> T exposed_face_of S"
     shows "\<Inter>P exposed_face_of S"
 proof -
   obtain Q where "finite Q" and QsubP: "Q \<subseteq> P" "card Q \<le> DIM('a) + 2" and IntQ: "\<Inter>Q = \<Inter>P"
     using Inter_faces_finite_altbound [of P S] assms [unfolded exposed_face_of]
     by force
   show ?thesis
   proof (cases "Q = {}")
     case True then show ?thesis
       by (metis IntQ Inter_UNIV_conv(2) assms(1) assms(2) ex_in_conv)
   next
     case False
     have "Q \<subseteq> {T. T exposed_face_of S}"
       using QsubP assms by blast
     moreover have "Q \<subseteq> {T. T exposed_face_of S} \<Longrightarrow> \<Inter>Q exposed_face_of S"
       using \<open>finite Q\<close> False
       apply (induction Q rule: finite_induct)
       using exposed_face_of_Int apply fastforce+
       done
     ultimately show ?thesis
       by (simp add: IntQ)
   qed
 qed
 
 proposition exposed_face_of_sums:
   assumes "convex S" and "convex T"
       and "F exposed_face_of {x + y | x y. x \<in> S \<and> y \<in> T}"
           (is "F exposed_face_of ?ST")
   obtains k l
     where "k exposed_face_of S" "l exposed_face_of T"
           "F = {x + y | x y. x \<in> k \<and> y \<in> l}"
 proof (cases "F = {}")
   case True then show ?thesis
     using that by blast
 next
   case False
   show ?thesis
   proof (cases "F = ?ST")
     case True then show ?thesis
       using assms exposed_face_of_refl_eq that by blast
   next
     case False
     obtain p where "p \<in> F" using \<open>F \<noteq> {}\<close> by blast
     moreover
     obtain u z where T: "?ST \<inter> {x. u \<bullet> x = z} face_of ?ST"
                  and S: "?ST \<subseteq> {x. u \<bullet> x \<le> z}"
                  and feq: "F = ?ST \<inter> {x. u \<bullet> x = z}"
       using assms by (auto simp: exposed_face_of_def)
     ultimately obtain a0 b0
             where p: "p = a0 + b0" and "a0 \<in> S" "b0 \<in> T" and z: "u \<bullet> p = z"
       by auto
     have lez: "u \<bullet> (x + y) \<le> z" if "x \<in> S" "y \<in> T" for x y
       using S that by auto
     have sef: "S \<inter> {x. u \<bullet> x = u \<bullet> a0} exposed_face_of S"
       apply (rule exposed_face_of_Int_supporting_hyperplane_le [OF \<open>convex S\<close>])
       apply (metis p z add_le_cancel_right inner_right_distrib lez [OF _ \<open>b0 \<in> T\<close>])
       done
     have tef: "T \<inter> {x. u \<bullet> x = u \<bullet> b0} exposed_face_of T"
       apply (rule exposed_face_of_Int_supporting_hyperplane_le [OF \<open>convex T\<close>])
       apply (metis p z add.commute add_le_cancel_right inner_right_distrib lez [OF \<open>a0 \<in> S\<close>])
       done
     have "{x + y |x y. x \<in> S \<and> u \<bullet> x = u \<bullet> a0 \<and> y \<in> T \<and> u \<bullet> y = u \<bullet> b0} \<subseteq> F"
       by (auto simp: feq) (metis inner_right_distrib p z)
     moreover have "F \<subseteq> {x + y |x y. x \<in> S \<and> u \<bullet> x = u \<bullet> a0 \<and> y \<in> T \<and> u \<bullet> y = u \<bullet> b0}"
       apply (auto simp: feq)
       apply (rename_tac x y)
       apply (rule_tac x=x in exI)
       apply (rule_tac x=y in exI, simp)
       using z p \<open>a0 \<in> S\<close> \<open>b0 \<in> T\<close>
       apply clarify
       apply (simp add: inner_right_distrib)
       apply (metis add_le_cancel_right antisym lez [unfolded inner_right_distrib] add.commute)
       done
     ultimately have "F = {x + y |x y. x \<in> S \<inter> {x. u \<bullet> x = u \<bullet> a0} \<and> y \<in> T \<inter> {x. u \<bullet> x = u \<bullet> b0}}"
       by blast
     then show ?thesis
       by (rule that [OF sef tef])
   qed
 qed
 
 proposition exposed_face_of_parallel:
    "T exposed_face_of S \<longleftrightarrow>
          T face_of S \<and>
          (\<exists>a b. S \<subseteq> {x. a \<bullet> x \<le> b} \<and> T = S \<inter> {x. a \<bullet> x = b} \<and>
                 (T \<noteq> {} \<longrightarrow> T \<noteq> S \<longrightarrow> a \<noteq> 0) \<and>
                 (T \<noteq> S \<longrightarrow> (\<forall>w \<in> affine hull S. (w + a) \<in> affine hull S)))"
   (is "?lhs = ?rhs")
 proof
   assume ?lhs then show ?rhs
   proof (clarsimp simp: exposed_face_of_def)
     fix a b
     assume faceS: "S \<inter> {x. a \<bullet> x = b} face_of S" and Ssub: "S \<subseteq> {x. a \<bullet> x \<le> b}" 
     show "\<exists>c d. S \<subseteq> {x. c \<bullet> x \<le> d} \<and>
                 S \<inter> {x. a \<bullet> x = b} = S \<inter> {x. c \<bullet> x = d} \<and>
                 (S \<inter> {x. a \<bullet> x = b} \<noteq> {} \<longrightarrow> S \<inter> {x. a \<bullet> x = b} \<noteq> S \<longrightarrow> c \<noteq> 0) \<and>
                 (S \<inter> {x. a \<bullet> x = b} \<noteq> S \<longrightarrow> (\<forall>w \<in> affine hull S. w + c \<in> affine hull S))"
     proof (cases "affine hull S \<inter> {x. -a \<bullet> x \<le> -b} = {} \<or> affine hull S \<subseteq> {x. - a \<bullet> x \<le> - b}")
       case True
       then show ?thesis
       proof
         assume "affine hull S \<inter> {x. - a \<bullet> x \<le> - b} = {}"
        then show ?thesis
          apply (rule_tac x="0" in exI)
          apply (rule_tac x="1" in exI)
          using hull_subset by fastforce
     next
       assume "affine hull S \<subseteq> {x. - a \<bullet> x \<le> - b}"
       then show ?thesis
          apply (rule_tac x="0" in exI)
          apply (rule_tac x="0" in exI)
         using Ssub hull_subset by fastforce
     qed
   next
     case False
     then obtain a' b' where "a' \<noteq> 0" 
       and le: "affine hull S \<inter> {x. a' \<bullet> x \<le> b'} = affine hull S \<inter> {x. - a \<bullet> x \<le> - b}" 
       and eq: "affine hull S \<inter> {x. a' \<bullet> x = b'} = affine hull S \<inter> {x. - a \<bullet> x = - b}" 
       and mem: "\<And>w. w \<in> affine hull S \<Longrightarrow> w + a' \<in> affine hull S"
       using affine_parallel_slice affine_affine_hull by metis 
     show ?thesis
     proof (intro conjI impI allI ballI exI)
       have *: "S \<subseteq> - (affine hull S \<inter> {x. P x}) \<union> affine hull S \<inter> {x. Q x} \<Longrightarrow> S \<subseteq> {x. \<not> P x \<or> Q x}" 
         for P Q 
         using hull_subset by fastforce  
       have "S \<subseteq> {x. \<not> (a' \<bullet> x \<le> b') \<or> a' \<bullet> x = b'}"
         apply (rule *)
         apply (simp only: le eq)
         using Ssub by auto
       then show "S \<subseteq> {x. - a' \<bullet> x \<le> - b'}"
         by auto 
       show "S \<inter> {x. a \<bullet> x = b} = S \<inter> {x. - a' \<bullet> x = - b'}"
         using eq hull_subset [of S affine] by force
       show "\<lbrakk>S \<inter> {x. a \<bullet> x = b} \<noteq> {}; S \<inter> {x. a \<bullet> x = b} \<noteq> S\<rbrakk> \<Longrightarrow> - a' \<noteq> 0"
         using \<open>a' \<noteq> 0\<close> by auto
       show "w + - a' \<in> affine hull S"
         if "S \<inter> {x. a \<bullet> x = b} \<noteq> S" "w \<in> affine hull S" for w
       proof -
         have "w + 1 *\<^sub>R (w - (w + a')) \<in> affine hull S"
           using affine_affine_hull mem mem_affine_3_minus that(2) by blast
         then show ?thesis  by simp
       qed
     qed
   qed
 qed
 next
   assume ?rhs then show ?lhs
     unfolding exposed_face_of_def by blast
 qed
 
 subsection\<open>Extreme points of a set: its singleton faces\<close>
 
 definition\<^marker>\<open>tag important\<close> extreme_point_of :: "['a::real_vector, 'a set] \<Rightarrow> bool"
                                (infixr "(extreme'_point'_of)" 50)
   where "x extreme_point_of S \<longleftrightarrow>
          x \<in> S \<and> (\<forall>a \<in> S. \<forall>b \<in> S. x \<notin> open_segment a b)"
 
 lemma extreme_point_of_stillconvex:
    "convex S \<Longrightarrow> (x extreme_point_of S \<longleftrightarrow> x \<in> S \<and> convex(S - {x}))"
   by (fastforce simp add: convex_contains_segment extreme_point_of_def open_segment_def)
 
 lemma face_of_singleton:
    "{x} face_of S \<longleftrightarrow> x extreme_point_of S"
 by (fastforce simp add: extreme_point_of_def face_of_def)
 
 lemma extreme_point_not_in_REL_INTERIOR:
     fixes S :: "'a::real_normed_vector set"
     shows "\<lbrakk>x extreme_point_of S; S \<noteq> {x}\<rbrakk> \<Longrightarrow> x \<notin> rel_interior S"
 apply (simp add: face_of_singleton [symmetric])
 apply (blast dest: face_of_disjoint_rel_interior)
 done
 
 lemma extreme_point_not_in_interior:
     fixes S :: "'a::{real_normed_vector, perfect_space} set"
     shows "x extreme_point_of S \<Longrightarrow> x \<notin> interior S"
 apply (case_tac "S = {x}")
 apply (simp add: empty_interior_finite)
 by (meson contra_subsetD extreme_point_not_in_REL_INTERIOR interior_subset_rel_interior)
 
 lemma extreme_point_of_face:
      "F face_of S \<Longrightarrow> v extreme_point_of F \<longleftrightarrow> v extreme_point_of S \<and> v \<in> F"
   by (meson empty_subsetI face_of_face face_of_singleton insert_subset)
 
 lemma extreme_point_of_convex_hull:
    "x extreme_point_of (convex hull S) \<Longrightarrow> x \<in> S"
 apply (simp add: extreme_point_of_stillconvex)
 using hull_minimal [of S "(convex hull S) - {x}" convex]
 using hull_subset [of S convex]
 apply blast
 done
 
 proposition extreme_points_of_convex_hull:
    "{x. x extreme_point_of (convex hull S)} \<subseteq> S"
   using extreme_point_of_convex_hull by auto
 
 lemma extreme_point_of_empty [simp]: "\<not> (x extreme_point_of {})"
   by (simp add: extreme_point_of_def)
 
 lemma extreme_point_of_singleton [iff]: "x extreme_point_of {a} \<longleftrightarrow> x = a"
   using extreme_point_of_stillconvex by auto
 
 lemma extreme_point_of_translation_eq:
    "(a + x) extreme_point_of (image (\<lambda>x. a + x) S) \<longleftrightarrow> x extreme_point_of S"
 by (auto simp: extreme_point_of_def)
 
 lemma extreme_points_of_translation:
    "{x. x extreme_point_of (image (\<lambda>x. a + x) S)} =
     (\<lambda>x. a + x) ` {x. x extreme_point_of S}"
   using extreme_point_of_translation_eq
   by auto (metis (no_types, lifting) image_iff mem_Collect_eq minus_add_cancel)
 
 lemma extreme_points_of_translation_subtract:
    "{x. x extreme_point_of (image (\<lambda>x. x - a) S)} =
     (\<lambda>x. x - a) ` {x. x extreme_point_of S}"
   using extreme_points_of_translation [of "- a" S]
   by simp
 
 lemma extreme_point_of_Int:
    "\<lbrakk>x extreme_point_of S; x extreme_point_of T\<rbrakk> \<Longrightarrow> x extreme_point_of (S \<inter> T)"
 by (simp add: extreme_point_of_def)
 
 lemma extreme_point_of_Int_supporting_hyperplane_le:
    "\<lbrakk>S \<inter> {x. a \<bullet> x = b} = {c}; \<And>x. x \<in> S \<Longrightarrow> a \<bullet> x \<le> b\<rbrakk> \<Longrightarrow> c extreme_point_of S"
 apply (simp add: face_of_singleton [symmetric])
 by (metis face_of_Int_supporting_hyperplane_le_strong convex_singleton)
 
 lemma extreme_point_of_Int_supporting_hyperplane_ge:
    "\<lbrakk>S \<inter> {x. a \<bullet> x = b} = {c}; \<And>x. x \<in> S \<Longrightarrow> a \<bullet> x \<ge> b\<rbrakk> \<Longrightarrow> c extreme_point_of S"
 apply (simp add: face_of_singleton [symmetric])
 by (metis face_of_Int_supporting_hyperplane_ge_strong convex_singleton)
 
 lemma exposed_point_of_Int_supporting_hyperplane_le:
    "\<lbrakk>S \<inter> {x. a \<bullet> x = b} = {c}; \<And>x. x \<in> S \<Longrightarrow> a \<bullet> x \<le> b\<rbrakk> \<Longrightarrow> {c} exposed_face_of S"
 apply (simp add: exposed_face_of_def face_of_singleton)
 apply (force simp: extreme_point_of_Int_supporting_hyperplane_le)
 done
 
 lemma exposed_point_of_Int_supporting_hyperplane_ge:
     "\<lbrakk>S \<inter> {x. a \<bullet> x = b} = {c}; \<And>x. x \<in> S \<Longrightarrow> a \<bullet> x \<ge> b\<rbrakk> \<Longrightarrow> {c} exposed_face_of S"
 using exposed_point_of_Int_supporting_hyperplane_le [of S "-a" "-b" c]
 by simp
 
 lemma extreme_point_of_convex_hull_insert:
    "\<lbrakk>finite S; a \<notin> convex hull S\<rbrakk> \<Longrightarrow> a extreme_point_of (convex hull (insert a S))"
 apply (case_tac "a \<in> S")
 apply (simp add: hull_inc)
 using face_of_convex_hulls [of "insert a S" "{a}"]
 apply (auto simp: face_of_singleton hull_same)
 done
 
 subsection\<open>Facets\<close>
 
 definition\<^marker>\<open>tag important\<close> facet_of :: "['a::euclidean_space set, 'a set] \<Rightarrow> bool"
                     (infixr "(facet'_of)" 50)
   where "F facet_of S \<longleftrightarrow> F face_of S \<and> F \<noteq> {} \<and> aff_dim F = aff_dim S - 1"
 
 lemma facet_of_empty [simp]: "\<not> S facet_of {}"
   by (simp add: facet_of_def)
 
 lemma facet_of_irrefl [simp]: "\<not> S facet_of S "
   by (simp add: facet_of_def)
 
 lemma facet_of_imp_face_of: "F facet_of S \<Longrightarrow> F face_of S"
   by (simp add: facet_of_def)
 
 lemma facet_of_imp_subset: "F facet_of S \<Longrightarrow> F \<subseteq> S"
   by (simp add: face_of_imp_subset facet_of_def)
 
 lemma hyperplane_facet_of_halfspace_le:
    "a \<noteq> 0 \<Longrightarrow> {x. a \<bullet> x = b} facet_of {x. a \<bullet> x \<le> b}"
 unfolding facet_of_def hyperplane_eq_empty
 by (auto simp: hyperplane_face_of_halfspace_ge hyperplane_face_of_halfspace_le
            DIM_positive Suc_leI of_nat_diff aff_dim_halfspace_le)
 
 lemma hyperplane_facet_of_halfspace_ge:
     "a \<noteq> 0 \<Longrightarrow> {x. a \<bullet> x = b} facet_of {x. a \<bullet> x \<ge> b}"
 unfolding facet_of_def hyperplane_eq_empty
 by (auto simp: hyperplane_face_of_halfspace_le hyperplane_face_of_halfspace_ge
            DIM_positive Suc_leI of_nat_diff aff_dim_halfspace_ge)
 
 lemma facet_of_halfspace_le:
     "F facet_of {x. a \<bullet> x \<le> b} \<longleftrightarrow> a \<noteq> 0 \<and> F = {x. a \<bullet> x = b}"
     (is "?lhs = ?rhs")
 proof
   assume c: ?lhs
   with c facet_of_irrefl show ?rhs
     by (force simp: aff_dim_halfspace_le facet_of_def face_of_halfspace_le cong: conj_cong split: if_split_asm)
 next
   assume ?rhs then show ?lhs
     by (simp add: hyperplane_facet_of_halfspace_le)
 qed
 
 lemma facet_of_halfspace_ge:
     "F facet_of {x. a \<bullet> x \<ge> b} \<longleftrightarrow> a \<noteq> 0 \<and> F = {x. a \<bullet> x = b}"
 using facet_of_halfspace_le [of F "-a" "-b"] by simp
 
 subsection \<open>Edges: faces of affine dimension 1\<close> (*FIXME too small subsection, rearrange? *)
 
 definition\<^marker>\<open>tag important\<close> edge_of :: "['a::euclidean_space set, 'a set] \<Rightarrow> bool"  (infixr "(edge'_of)" 50)
   where "e edge_of S \<longleftrightarrow> e face_of S \<and> aff_dim e = 1"
 
 lemma edge_of_imp_subset:
    "S edge_of T \<Longrightarrow> S \<subseteq> T"
 by (simp add: edge_of_def face_of_imp_subset)
 
 subsection\<open>Existence of extreme points\<close>
 
 proposition different_norm_3_collinear_points:
   fixes a :: "'a::euclidean_space"
   assumes "x \<in> open_segment a b" "norm(a) = norm(b)" "norm(x) = norm(b)"
   shows False
 proof -
   obtain u where "norm ((1 - u) *\<^sub>R a + u *\<^sub>R b) = norm b"
              and "a \<noteq> b"
              and u01: "0 < u" "u < 1"
     using assms by (auto simp: open_segment_image_interval if_splits)
   then have "(1 - u) *\<^sub>R a \<bullet> (1 - u) *\<^sub>R a + ((1 - u) * 2) *\<^sub>R a \<bullet> u *\<^sub>R b =
              (1 - u * u) *\<^sub>R (a \<bullet> a)"
     using assms by (simp add: norm_eq algebra_simps inner_commute)
   then have "(1 - u) *\<^sub>R ((1 - u) *\<^sub>R a \<bullet> a + (2 * u) *\<^sub>R  a \<bullet> b) =
              (1 - u) *\<^sub>R ((1 + u) *\<^sub>R (a \<bullet> a))"
     by (simp add: algebra_simps)
   then have "(1 - u) *\<^sub>R (a \<bullet> a) + (2 * u) *\<^sub>R (a \<bullet> b) = (1 + u) *\<^sub>R (a \<bullet> a)"
     using u01 by auto
   then have "a \<bullet> b = a \<bullet> a"
     using u01 by (simp add: algebra_simps)
   then have "a = b"
     using \<open>norm(a) = norm(b)\<close> norm_eq vector_eq by fastforce
   then show ?thesis
     using \<open>a \<noteq> b\<close> by force
 qed
 
 proposition extreme_point_exists_convex:
   fixes S :: "'a::euclidean_space set"
   assumes "compact S" "convex S" "S \<noteq> {}"
   obtains x where "x extreme_point_of S"
 proof -
   obtain x where "x \<in> S" and xsup: "\<And>y. y \<in> S \<Longrightarrow> norm y \<le> norm x"
     using distance_attains_sup [of S 0] assms by auto
   have False if "a \<in> S" "b \<in> S" and x: "x \<in> open_segment a b" for a b
   proof -
     have noax: "norm a \<le> norm x" and nobx: "norm b \<le> norm x" using xsup that by auto
     have "a \<noteq> b"
       using empty_iff open_segment_idem x by auto
     have *: "(1 - u) * na + u * nb < norm x" if "na < norm x"  "nb \<le> norm x" "0 < u" "u < 1" for na nb u
     proof -
       have "(1 - u) * na + u * nb < (1 - u) * norm x + u * nb"
         by (simp add: that)
       also have "... \<le> (1 - u) * norm x + u * norm x"
         by (simp add: that)
       finally have "(1 - u) * na + u * nb < (1 - u) * norm x + u * norm x" .
       then show ?thesis
       using scaleR_collapse [symmetric, of "norm x" u] by auto
     qed
     have "norm x < norm x" if "norm a < norm x"
       using x
       apply (clarsimp simp only: open_segment_image_interval \<open>a \<noteq> b\<close> if_False)
       apply (rule norm_triangle_lt)
       apply (simp add: norm_mult)
       using * [of "norm a" "norm b"] nobx that
         apply blast
       done
     moreover have "norm x < norm x" if "norm b < norm x"
       using x
       apply (clarsimp simp only: open_segment_image_interval \<open>a \<noteq> b\<close> if_False)
       apply (rule norm_triangle_lt)
       apply (simp add: norm_mult)
       using * [of "norm b" "norm a" "1-u" for u] noax that
         apply (simp add: add.commute)
       done
     ultimately have "\<not> (norm a < norm x) \<and> \<not> (norm b < norm x)"
       by auto
     then show ?thesis
       using different_norm_3_collinear_points noax nobx that(3) by fastforce
   qed
   then show ?thesis
     apply (rule_tac x=x in that)
     apply (force simp: extreme_point_of_def \<open>x \<in> S\<close>)
     done
 qed
 
 subsection\<open>Krein-Milman, the weaker form\<close>
 
 proposition Krein_Milman:
   fixes S :: "'a::euclidean_space set"
   assumes "compact S" "convex S"
     shows "S = closure(convex hull {x. x extreme_point_of S})"
 proof (cases "S = {}")
   case True then show ?thesis   by simp
 next
   case False
   have "closed S"
     by (simp add: \<open>compact S\<close> compact_imp_closed)
   have "closure (convex hull {x. x extreme_point_of S}) \<subseteq> S"
     apply (rule closure_minimal [OF hull_minimal \<open>closed S\<close>])
     using assms
     apply (auto simp: extreme_point_of_def)
     done
   moreover have "u \<in> closure (convex hull {x. x extreme_point_of S})"
                 if "u \<in> S" for u
   proof (rule ccontr)
     assume unot: "u \<notin> closure(convex hull {x. x extreme_point_of S})"
     then obtain a b where "a \<bullet> u < b"
           and ab: "\<And>x. x \<in> closure(convex hull {x. x extreme_point_of S}) \<Longrightarrow> b < a \<bullet> x"
       using separating_hyperplane_closed_point [of "closure(convex hull {x. x extreme_point_of S})"]
       by blast
     have "continuous_on S ((\<bullet>) a)"
       by (rule continuous_intros)+
     then obtain m where "m \<in> S" and m: "\<And>y. y \<in> S \<Longrightarrow> a \<bullet> m \<le> a \<bullet> y"
       using continuous_attains_inf [of S "\<lambda>x. a \<bullet> x"] \<open>compact S\<close> \<open>u \<in> S\<close>
       by auto
     define T where "T = S \<inter> {x. a \<bullet> x = a \<bullet> m}"
     have "m \<in> T"
       by (simp add: T_def \<open>m \<in> S\<close>)
     moreover have "compact T"
       by (simp add: T_def compact_Int_closed [OF \<open>compact S\<close> closed_hyperplane])
     moreover have "convex T"
       by (simp add: T_def convex_Int [OF \<open>convex S\<close> convex_hyperplane])
     ultimately obtain v where v: "v extreme_point_of T"
       using extreme_point_exists_convex [of T] by auto
     then have "{v} face_of T"
       by (simp add: face_of_singleton)
     also have "T face_of S"
       by (simp add: T_def m face_of_Int_supporting_hyperplane_ge [OF \<open>convex S\<close>])
     finally have "v extreme_point_of S"
       by (simp add: face_of_singleton)
     then have "b < a \<bullet> v"
       using closure_subset by (simp add: closure_hull hull_inc ab)
     then show False
       using \<open>a \<bullet> u < b\<close> \<open>{v} face_of T\<close> face_of_imp_subset m T_def that by fastforce
   qed
   ultimately show ?thesis
     by blast
 qed
 
 text\<open>Now the sharper form.\<close>
 
 lemma Krein_Milman_Minkowski_aux:
   fixes S :: "'a::euclidean_space set"
   assumes n: "dim S = n" and S: "compact S" "convex S" "0 \<in> S"
     shows "0 \<in> convex hull {x. x extreme_point_of S}"
 using n S
 proof (induction n arbitrary: S rule: less_induct)
   case (less n S) show ?case
   proof (cases "0 \<in> rel_interior S")
     case True with Krein_Milman show ?thesis
       by (metis subsetD convex_convex_hull convex_rel_interior_closure less.prems(2) less.prems(3) rel_interior_subset)
   next
     case False
     have "rel_interior S \<noteq> {}"
       by (simp add: rel_interior_convex_nonempty_aux less)
     then obtain c where c: "c \<in> rel_interior S" by blast
     obtain a where "a \<noteq> 0"
               and le_ay: "\<And>y. y \<in> S \<Longrightarrow> a \<bullet> 0 \<le> a \<bullet> y"
               and less_ay: "\<And>y. y \<in> rel_interior S \<Longrightarrow> a \<bullet> 0 < a \<bullet> y"
       by (blast intro: supporting_hyperplane_rel_boundary intro!: less False)
     have face: "S \<inter> {x. a \<bullet> x = 0} face_of S"
       apply (rule face_of_Int_supporting_hyperplane_ge [OF \<open>convex S\<close>])
       using le_ay by auto
     then have co: "compact (S \<inter> {x. a \<bullet> x = 0})" "convex (S \<inter> {x. a \<bullet> x = 0})"
       using less.prems by (blast intro: face_of_imp_compact face_of_imp_convex)+
     have "a \<bullet> y = 0" if "y \<in> span (S \<inter> {x. a \<bullet> x = 0})" for y
     proof -
       have "y \<in> span {x. a \<bullet> x = 0}"
         by (metis inf.cobounded2 span_mono subsetCE that)
       then show ?thesis
         by (blast intro: span_induct [OF _ subspace_hyperplane])
     qed
     then have "dim (S \<inter> {x. a \<bullet> x = 0}) < n"
       by (metis (no_types) less_ay c subsetD dim_eq_span inf.strict_order_iff
            inf_le1 \<open>dim S = n\<close> not_le rel_interior_subset span_0 span_base)
     then have "0 \<in> convex hull {x. x extreme_point_of (S \<inter> {x. a \<bullet> x = 0})}"
       by (rule less.IH) (auto simp: co less.prems)
     then show ?thesis
       by (metis (mono_tags, lifting) Collect_mono_iff \<open>S \<inter> {x. a \<bullet> x = 0} face_of S\<close> extreme_point_of_face hull_mono subset_iff)
   qed
 qed
 
 
 theorem Krein_Milman_Minkowski:
   fixes S :: "'a::euclidean_space set"
   assumes "compact S" "convex S"
     shows "S = convex hull {x. x extreme_point_of S}"
 proof
   show "S \<subseteq> convex hull {x. x extreme_point_of S}"
   proof
     fix a assume [simp]: "a \<in> S"
     have 1: "compact ((+) (- a) ` S)"
       by (simp add: \<open>compact S\<close> compact_translation_subtract cong: image_cong_simp)
     have 2: "convex ((+) (- a) ` S)"
       by (simp add: \<open>convex S\<close> compact_translation_subtract)
     show a_invex: "a \<in> convex hull {x. x extreme_point_of S}"
       using Krein_Milman_Minkowski_aux [OF refl 1 2]
             convex_hull_translation [of "-a"]
       by (auto simp: extreme_points_of_translation_subtract translation_assoc cong: image_cong_simp)
     qed
 next
   show "convex hull {x. x extreme_point_of S} \<subseteq> S"
   proof -
     have "{a. a extreme_point_of S} \<subseteq> S"
       using extreme_point_of_def by blast
     then show ?thesis
       by (simp add: \<open>convex S\<close> hull_minimal)
   qed
 qed
 
 
 subsection\<open>Applying it to convex hulls of explicitly indicated finite sets\<close>
 
 corollary Krein_Milman_polytope:
   fixes S :: "'a::euclidean_space set"
   shows
    "finite S
        \<Longrightarrow> convex hull S =
            convex hull {x. x extreme_point_of (convex hull S)}"
   by (simp add: Krein_Milman_Minkowski finite_imp_compact_convex_hull)
 
 lemma extreme_points_of_convex_hull_eq:
   fixes S :: "'a::euclidean_space set"
   shows
    "\<lbrakk>compact S; \<And>T. T \<subset> S \<Longrightarrow> convex hull T \<noteq> convex hull S\<rbrakk>
         \<Longrightarrow> {x. x extreme_point_of (convex hull S)} = S"
 by (metis (full_types) Krein_Milman_Minkowski compact_convex_hull convex_convex_hull extreme_points_of_convex_hull psubsetI)
 
 
 lemma extreme_point_of_convex_hull_eq:
   fixes S :: "'a::euclidean_space set"
   shows
    "\<lbrakk>compact S; \<And>T. T \<subset> S \<Longrightarrow> convex hull T \<noteq> convex hull S\<rbrakk>
     \<Longrightarrow> (x extreme_point_of (convex hull S) \<longleftrightarrow> x \<in> S)"
 using extreme_points_of_convex_hull_eq by auto
 
 lemma extreme_point_of_convex_hull_convex_independent:
   fixes S :: "'a::euclidean_space set"
   assumes "compact S" and S: "\<And>a. a \<in> S \<Longrightarrow> a \<notin> convex hull (S - {a})"
   shows "(x extreme_point_of (convex hull S) \<longleftrightarrow> x \<in> S)"
 proof -
   have "convex hull T \<noteq> convex hull S" if "T \<subset> S" for T
   proof -
     obtain a where  "T \<subseteq> S" "a \<in> S" "a \<notin> T" using \<open>T \<subset> S\<close> by blast
     then show ?thesis
       by (metis (full_types) Diff_eq_empty_iff Diff_insert0 S hull_mono hull_subset insert_Diff_single subsetCE)
   qed
   then show ?thesis
     by (rule extreme_point_of_convex_hull_eq [OF \<open>compact S\<close>])
 qed
 
 lemma extreme_point_of_convex_hull_affine_independent:
   fixes S :: "'a::euclidean_space set"
   shows
    "\<not> affine_dependent S
          \<Longrightarrow> (x extreme_point_of (convex hull S) \<longleftrightarrow> x \<in> S)"
 by (metis aff_independent_finite affine_dependent_def affine_hull_convex_hull extreme_point_of_convex_hull_convex_independent finite_imp_compact hull_inc)
 
 text\<open>Elementary proofs exist, not requiring Euclidean spaces and all this development\<close>
 lemma extreme_point_of_convex_hull_2:
   fixes x :: "'a::euclidean_space"
   shows "x extreme_point_of (convex hull {a,b}) \<longleftrightarrow> x = a \<or> x = b"
 proof -
   have "x extreme_point_of (convex hull {a,b}) \<longleftrightarrow> x \<in> {a,b}"
     by (intro extreme_point_of_convex_hull_affine_independent affine_independent_2)
   then show ?thesis
     by simp
 qed
 
 lemma extreme_point_of_segment:
   fixes x :: "'a::euclidean_space"
   shows
    "x extreme_point_of closed_segment a b \<longleftrightarrow> x = a \<or> x = b"
 by (simp add: extreme_point_of_convex_hull_2 segment_convex_hull)
 
 lemma face_of_convex_hull_subset:
   fixes S :: "'a::euclidean_space set"
   assumes "compact S" and T: "T face_of (convex hull S)"
   obtains s' where "s' \<subseteq> S" "T = convex hull s'"
 apply (rule_tac s' = "{x. x extreme_point_of T}" in that)
 using T extreme_point_of_convex_hull extreme_point_of_face apply blast
 by (metis (no_types) Krein_Milman_Minkowski assms compact_convex_hull convex_convex_hull face_of_imp_compact face_of_imp_convex)
 
 
 lemma face_of_convex_hull_aux:
   assumes eq: "x *\<^sub>R p = u *\<^sub>R a + v *\<^sub>R b + w *\<^sub>R c"
     and x: "u + v + w = x" "x \<noteq> 0" and S: "affine S" "a \<in> S" "b \<in> S" "c \<in> S"
   shows "p \<in> S"
 proof -
   have "p = (u *\<^sub>R a + v *\<^sub>R b + w *\<^sub>R c) /\<^sub>R x"
     by (metis \<open>x \<noteq> 0\<close> eq mult.commute right_inverse scaleR_one scaleR_scaleR)
   moreover have "affine hull {a,b,c} \<subseteq> S"
     by (simp add: S hull_minimal)
   moreover have "(u *\<^sub>R a + v *\<^sub>R b + w *\<^sub>R c) /\<^sub>R x \<in> affine hull {a,b,c}"
     apply (simp add: affine_hull_3)
     apply (rule_tac x="u/x" in exI)
     apply (rule_tac x="v/x" in exI)
     apply (rule_tac x="w/x" in exI)
     using x apply (auto simp: algebra_simps field_split_simps)
     done
   ultimately show ?thesis by force
 qed
 
 proposition face_of_convex_hull_insert_eq:
   fixes a :: "'a :: euclidean_space"
   assumes "finite S" and a: "a \<notin> affine hull S"
   shows "(F face_of (convex hull (insert a S)) \<longleftrightarrow>
           F face_of (convex hull S) \<or>
           (\<exists>F'. F' face_of (convex hull S) \<and> F = convex hull (insert a F')))"
          (is "F face_of ?CAS \<longleftrightarrow> _")
 proof safe
   assume F: "F face_of ?CAS"
     and *: "\<nexists>F'. F' face_of convex hull S \<and> F = convex hull insert a F'"
   obtain T where T: "T \<subseteq> insert a S" and FeqT: "F = convex hull T"
     by (metis F \<open>finite S\<close> compact_insert finite_imp_compact face_of_convex_hull_subset)
   show "F face_of convex hull S"
   proof (cases "a \<in> T")
     case True
     have "F = convex hull insert a (convex hull T \<inter> convex hull S)"
     proof
       have "T \<subseteq> insert a (convex hull T \<inter> convex hull S)"
         using T hull_subset by fastforce
       then show "F \<subseteq> convex hull insert a (convex hull T \<inter> convex hull S)"
         by (simp add: FeqT hull_mono)
       show "convex hull insert a (convex hull T \<inter> convex hull S) \<subseteq> F"
         apply (rule hull_minimal)
         using True by (auto simp: \<open>F = convex hull T\<close> hull_inc)
     qed
     moreover have "convex hull T \<inter> convex hull S face_of convex hull S"
       by (metis F FeqT convex_convex_hull face_of_slice hull_mono inf.absorb_iff2 subset_insertI)
     ultimately show ?thesis
       using * by force
   next
     case False
     then show ?thesis
       by (metis FeqT F T face_of_subset hull_mono subset_insert subset_insertI)
   qed
 next
   assume "F face_of convex hull S"
   show "F face_of ?CAS"
     by (simp add: \<open>F face_of convex hull S\<close> a face_of_convex_hull_insert \<open>finite S\<close>)
 next
   fix F
   assume F: "F face_of convex hull S"
   show "convex hull insert a F face_of ?CAS"
   proof (cases "S = {}")
     case True
     then show ?thesis
       using F face_of_affine_eq by auto
   next
     case False
     have anotc: "a \<notin> convex hull S"
       by (metis (no_types) a affine_hull_convex_hull hull_inc)
     show ?thesis
     proof (cases "F = {}")
       case True show ?thesis
         using anotc by (simp add: \<open>F = {}\<close> \<open>finite S\<close> extreme_point_of_convex_hull_insert face_of_singleton)
     next
       case False
       have "convex hull insert a F \<subseteq> ?CAS"
         by (simp add: F a \<open>finite S\<close> convex_hull_subset face_of_convex_hull_insert face_of_imp_subset hull_inc)
       moreover
       have "(\<exists>y v. (1 - ub) *\<^sub>R a + ub *\<^sub>R b = (1 - v) *\<^sub>R a + v *\<^sub>R y \<and>
                    0 \<le> v \<and> v \<le> 1 \<and> y \<in> F) \<and>
             (\<exists>x u. (1 - uc) *\<^sub>R a + uc *\<^sub>R c = (1 - u) *\<^sub>R a + u *\<^sub>R x \<and>
                    0 \<le> u \<and> u \<le> 1 \<and> x \<in> F)"
         if *: "(1 - ux) *\<^sub>R a + ux *\<^sub>R x
                \<in> open_segment ((1 - ub) *\<^sub>R a + ub *\<^sub>R b) ((1 - uc) *\<^sub>R a + uc *\<^sub>R c)"
           and "0 \<le> ub" "ub \<le> 1" "0 \<le> uc" "uc \<le> 1" "0 \<le> ux" "ux \<le> 1"
           and b: "b \<in> convex hull S" and c: "c \<in> convex hull S" and "x \<in> F"
         for b c ub uc ux x
       proof -
         obtain v where ne: "(1 - ub) *\<^sub>R a + ub *\<^sub>R b \<noteq> (1 - uc) *\<^sub>R a + uc *\<^sub>R c"
           and eq: "(1 - ux) *\<^sub>R a + ux *\<^sub>R x =
                     (1 - v) *\<^sub>R ((1 - ub) *\<^sub>R a + ub *\<^sub>R b) + v *\<^sub>R ((1 - uc) *\<^sub>R a + uc *\<^sub>R c)"
           and "0 < v" "v < 1"
           using * by (auto simp: in_segment)
         then have 0: "((1 - ux) - ((1 - v) * (1 - ub) + v * (1 - uc))) *\<^sub>R a +
                       (ux *\<^sub>R x - (((1 - v) * ub) *\<^sub>R b + (v * uc) *\<^sub>R c)) = 0"
           by (auto simp: algebra_simps)
         then have "((1 - ux) - ((1 - v) * (1 - ub) + v * (1 - uc))) *\<^sub>R a =
                    ((1 - v) * ub) *\<^sub>R b + (v * uc) *\<^sub>R c + (-ux) *\<^sub>R x"
           by (auto simp: algebra_simps)
         then have "a \<in> affine hull S" if "1 - ux - ((1 - v) * (1 - ub) + v * (1 - uc)) \<noteq> 0"
           apply (rule face_of_convex_hull_aux)
           using b c that apply (auto simp: algebra_simps)
           using F convex_hull_subset_affine_hull face_of_imp_subset \<open>x \<in> F\<close> apply blast+
           done
         then have "1 - ux - ((1 - v) * (1 - ub) + v * (1 - uc)) = 0"
           using a by blast
         with 0 have equx: "(1 - v) * ub + v * uc = ux"
           and uxx: "ux *\<^sub>R x = (((1 - v) * ub) *\<^sub>R b + (v * uc) *\<^sub>R c)"
           by auto (auto simp: algebra_simps)
         show ?thesis
         proof (cases "uc = 0")
           case True
           then show ?thesis
             using equx 0 \<open>0 \<le> ub\<close> \<open>ub \<le> 1\<close> \<open>v < 1\<close> \<open>x \<in> F\<close>
             apply (auto simp: algebra_simps)
              apply (rule_tac x=x in exI, simp)
              apply (rule_tac x=ub in exI, auto)
              apply (metis add.left_neutral diff_eq_eq less_irrefl mult.commute mult_cancel_right1 real_vector.scale_cancel_left real_vector.scale_left_diff_distrib)
             using \<open>x \<in> F\<close> \<open>uc \<le> 1\<close> apply blast
             done
         next
           case False
           show ?thesis
           proof (cases "ub = 0")
             case True
             then show ?thesis
               using equx 0 \<open>0 \<le> uc\<close> \<open>uc \<le> 1\<close> \<open>0 < v\<close> \<open>x \<in> F\<close> \<open>uc \<noteq> 0\<close> by (force simp: algebra_simps)
           next
             case False
             then have "0 < ub" "0 < uc"
               using \<open>uc \<noteq> 0\<close> \<open>0 \<le> ub\<close> \<open>0 \<le> uc\<close> by auto
             then have "ux \<noteq> 0"
               by (metis \<open>0 < v\<close> \<open>v < 1\<close> diff_ge_0_iff_ge dual_order.strict_implies_order equx leD le_add_same_cancel2 zero_le_mult_iff zero_less_mult_iff)
             have "b \<in> F \<and> c \<in> F"
             proof (cases "b = c")
               case True
               then show ?thesis
                 by (metis \<open>ux \<noteq> 0\<close> equx real_vector.scale_cancel_left scaleR_add_left uxx \<open>x \<in> F\<close>)
             next
               case False
               have "x = (((1 - v) * ub) *\<^sub>R b + (v * uc) *\<^sub>R c) /\<^sub>R ux"
                 by (metis \<open>ux \<noteq> 0\<close> uxx mult.commute right_inverse scaleR_one scaleR_scaleR)
               also have "... = (1 - v * uc / ux) *\<^sub>R b + (v * uc / ux) *\<^sub>R c"
                 using \<open>ux \<noteq> 0\<close> equx apply (auto simp: algebra_simps field_split_simps)
                 by (metis add.commute add_diff_eq add_divide_distrib diff_add_cancel scaleR_add_left)
               finally have "x = (1 - v * uc / ux) *\<^sub>R b + (v * uc / ux) *\<^sub>R c" .
               then have "x \<in> open_segment b c"
                 apply (simp add: in_segment \<open>b \<noteq> c\<close>)
                 apply (rule_tac x="(v * uc) / ux" in exI)
                 using \<open>0 \<le> ux\<close> \<open>ux \<noteq> 0\<close> \<open>0 < uc\<close> \<open>0 < v\<close> \<open>0 < ub\<close> \<open>v < 1\<close> equx
                 apply (force simp: algebra_simps field_split_simps)
                 done
               then show ?thesis
                 by (rule face_ofD [OF F _ b c \<open>x \<in> F\<close>])
             qed
             with \<open>0 \<le> ub\<close> \<open>ub \<le> 1\<close> \<open>0 \<le> uc\<close> \<open>uc \<le> 1\<close> show ?thesis by blast
           qed
         qed
       qed
       moreover have "convex hull F = F"
         by (meson F convex_hull_eq face_of_imp_convex)
       ultimately show ?thesis
         unfolding face_of_def by (fastforce simp: convex_hull_insert_alt \<open>S \<noteq> {}\<close> \<open>F \<noteq> {}\<close>)
     qed
   qed
 qed
 
 lemma face_of_convex_hull_insert2:
   fixes a :: "'a :: euclidean_space"
   assumes S: "finite S" and a: "a \<notin> affine hull S" and F: "F face_of convex hull S"
   shows "convex hull (insert a F) face_of convex hull (insert a S)"
   by (metis F face_of_convex_hull_insert_eq [OF S a])
 
 proposition face_of_convex_hull_affine_independent:
   fixes S :: "'a::euclidean_space set"
   assumes "\<not> affine_dependent S"
     shows "(T face_of (convex hull S) \<longleftrightarrow> (\<exists>c. c \<subseteq> S \<and> T = convex hull c))"
           (is "?lhs = ?rhs")
 proof
   assume ?lhs
   then show ?rhs
     by (meson \<open>T face_of convex hull S\<close> aff_independent_finite assms face_of_convex_hull_subset finite_imp_compact)
 next
   assume ?rhs
   then obtain c where "c \<subseteq> S" and T: "T = convex hull c"
     by blast
   have "affine hull c \<inter> affine hull (S - c) = {}"
     apply (rule disjoint_affine_hull [OF assms \<open>c \<subseteq> S\<close>], auto)
     done
   then have "affine hull c \<inter> convex hull (S - c) = {}"
     using convex_hull_subset_affine_hull by fastforce
   then show ?lhs
     by (metis face_of_convex_hulls \<open>c \<subseteq> S\<close> aff_independent_finite assms T)
 qed
 
 lemma facet_of_convex_hull_affine_independent:
   fixes S :: "'a::euclidean_space set"
   assumes "\<not> affine_dependent S"
     shows "T facet_of (convex hull S) \<longleftrightarrow>
            T \<noteq> {} \<and> (\<exists>u. u \<in> S \<and> T = convex hull (S - {u}))"
           (is "?lhs = ?rhs")
 proof
   assume ?lhs
   then have "T face_of (convex hull S)" "T \<noteq> {}"
         and afft: "aff_dim T = aff_dim (convex hull S) - 1"
     by (auto simp: facet_of_def)
   then obtain c where "c \<subseteq> S" and c: "T = convex hull c"
     by (auto simp: face_of_convex_hull_affine_independent [OF assms])
   then have affs: "aff_dim S = aff_dim c + 1"
     by (metis aff_dim_convex_hull afft eq_diff_eq)
   have "\<not> affine_dependent c"
     using \<open>c \<subseteq> S\<close> affine_dependent_subset assms by blast
   with affs have "card (S - c) = 1"
     apply (simp add: aff_dim_affine_independent [symmetric] aff_dim_convex_hull)
     by (metis aff_dim_affine_independent aff_independent_finite One_nat_def \<open>c \<subseteq> S\<close> add.commute
                 add_diff_cancel_right' assms card_Diff_subset card_mono of_nat_1 of_nat_diff of_nat_eq_iff)
   then obtain u where u: "u \<in> S - c"
     by (metis DiffI \<open>c \<subseteq> S\<close> aff_independent_finite assms cancel_comm_monoid_add_class.diff_cancel
                 card_Diff_subset subsetI subset_antisym zero_neq_one)
   then have u: "S = insert u c"
     by (metis Diff_subset \<open>c \<subseteq> S\<close> \<open>card (S - c) = 1\<close> card_1_singletonE double_diff insert_Diff insert_subset singletonD)
   have "T = convex hull (c - {u})"
     by (metis Diff_empty Diff_insert0 \<open>T facet_of convex hull S\<close> c facet_of_irrefl insert_absorb u)
   with \<open>T \<noteq> {}\<close> show ?rhs
     using c u by auto
 next
   assume ?rhs
   then obtain u where "T \<noteq> {}" "u \<in> S" and u: "T = convex hull (S - {u})"
     by (force simp: facet_of_def)
   then have "\<not> S \<subseteq> {u}"
     using \<open>T \<noteq> {}\<close> u by auto
   have [simp]: "aff_dim (convex hull (S - {u})) = aff_dim (convex hull S) - 1"
     using assms \<open>u \<in> S\<close>
     apply (simp add: aff_dim_convex_hull affine_dependent_def)
     apply (drule bspec, assumption)
     by (metis add_diff_cancel_right' aff_dim_insert insert_Diff [of u S])
   show ?lhs
     apply (subst u)
     apply (simp add: \<open>\<not> S \<subseteq> {u}\<close> facet_of_def face_of_convex_hull_affine_independent [OF assms], blast)
     done
 qed
 
 lemma facet_of_convex_hull_affine_independent_alt:
   fixes S :: "'a::euclidean_space set"
   shows
    "\<not>affine_dependent S
         \<Longrightarrow> (T facet_of (convex hull S) \<longleftrightarrow>
              2 \<le> card S \<and> (\<exists>u. u \<in> S \<and> T = convex hull (S - {u})))"
 apply (simp add: facet_of_convex_hull_affine_independent)
 apply (auto simp: Set.subset_singleton_iff)
 apply (metis Diff_cancel Int_empty_right Int_insert_right_if1  aff_independent_finite card_eq_0_iff card_insert_if card_mono card_subset_eq convex_hull_eq_empty eq_iff equals0D finite_insert finite_subset inf.absorb_iff2 insert_absorb insert_not_empty  not_less_eq_eq numeral_2_eq_2)
 done
 
 lemma segment_face_of:
   assumes "(closed_segment a b) face_of S"
   shows "a extreme_point_of S" "b extreme_point_of S"
 proof -
   have as: "{a} face_of S"
     by (metis (no_types) assms convex_hull_singleton empty_iff extreme_point_of_convex_hull_insert face_of_face face_of_singleton finite.emptyI finite.insertI insert_absorb insert_iff segment_convex_hull)
   moreover have "{b} face_of S"
   proof -
     have "b \<in> convex hull {a} \<or> b extreme_point_of convex hull {b, a}"
       by (meson extreme_point_of_convex_hull_insert finite.emptyI finite.insertI)
     moreover have "closed_segment a b = convex hull {b, a}"
       using closed_segment_commute segment_convex_hull by blast
     ultimately show ?thesis
       by (metis as assms face_of_face convex_hull_singleton empty_iff face_of_singleton insertE)
     qed
   ultimately show "a extreme_point_of S" "b extreme_point_of S"
     using face_of_singleton by blast+
 qed
 
 
 proposition Krein_Milman_frontier:
   fixes S :: "'a::euclidean_space set"
   assumes "convex S" "compact S"
     shows "S = convex hull (frontier S)"
           (is "?lhs = ?rhs")
 proof
   have "?lhs \<subseteq> convex hull {x. x extreme_point_of S}"
     using Krein_Milman_Minkowski assms by blast
   also have "... \<subseteq> ?rhs"
     apply (rule hull_mono)
     apply (auto simp: frontier_def extreme_point_not_in_interior)
     using closure_subset apply (force simp: extreme_point_of_def)
     done
   finally show "?lhs \<subseteq> ?rhs" .
 next
   have "?rhs \<subseteq> convex hull S"
     by (metis Diff_subset \<open>compact S\<close> closure_closed compact_eq_bounded_closed frontier_def hull_mono)
   also have "... \<subseteq> ?lhs"
     by (simp add: \<open>convex S\<close> hull_same)
   finally show "?rhs \<subseteq> ?lhs" .
 qed
 
 subsection\<open>Polytopes\<close>
 
 definition\<^marker>\<open>tag important\<close> polytope where
  "polytope S \<equiv> \<exists>v. finite v \<and> S = convex hull v"
 
 lemma polytope_translation_eq: "polytope (image (\<lambda>x. a + x) S) \<longleftrightarrow> polytope S"
 apply (simp add: polytope_def, safe)
 apply (metis convex_hull_translation finite_imageI translation_galois)
 by (metis convex_hull_translation finite_imageI)
 
 lemma polytope_linear_image: "\<lbrakk>linear f; polytope p\<rbrakk> \<Longrightarrow> polytope(image f p)"
   unfolding polytope_def using convex_hull_linear_image by blast
 
 lemma polytope_empty: "polytope {}"
   using convex_hull_empty polytope_def by blast
 
 lemma polytope_convex_hull: "finite S \<Longrightarrow> polytope(convex hull S)"
   using polytope_def by auto
 
 lemma polytope_Times: "\<lbrakk>polytope S; polytope T\<rbrakk> \<Longrightarrow> polytope(S \<times> T)"
   unfolding polytope_def
   by (metis finite_cartesian_product convex_hull_Times)
 
 lemma face_of_polytope_polytope:
   fixes S :: "'a::euclidean_space set"
   shows "\<lbrakk>polytope S; F face_of S\<rbrakk> \<Longrightarrow> polytope F"
 unfolding polytope_def
 by (meson face_of_convex_hull_subset finite_imp_compact finite_subset)
 
 lemma finite_polytope_faces:
   fixes S :: "'a::euclidean_space set"
   assumes "polytope S"
   shows "finite {F. F face_of S}"
 proof -
   obtain v where "finite v" "S = convex hull v"
     using assms polytope_def by auto
   have "finite ((hull) convex ` {T. T \<subseteq> v})"
     by (simp add: \<open>finite v\<close>)
   moreover have "{F. F face_of S} \<subseteq> ((hull) convex ` {T. T \<subseteq> v})"
     by (metis (no_types, lifting) \<open>finite v\<close> \<open>S = convex hull v\<close> face_of_convex_hull_subset finite_imp_compact image_eqI mem_Collect_eq subsetI)
   ultimately show ?thesis
     by (blast intro: finite_subset)
 qed
 
 lemma finite_polytope_facets:
   assumes "polytope S"
   shows "finite {T. T facet_of S}"
 by (simp add: assms facet_of_def finite_polytope_faces)
 
 lemma polytope_scaling:
   assumes "polytope S"  shows "polytope (image (\<lambda>x. c *\<^sub>R x) S)"
 by (simp add: assms polytope_linear_image)
 
 lemma polytope_imp_compact:
   fixes S :: "'a::real_normed_vector set"
   shows "polytope S \<Longrightarrow> compact S"
 by (metis finite_imp_compact_convex_hull polytope_def)
 
 lemma polytope_imp_convex: "polytope S \<Longrightarrow> convex S"
   by (metis convex_convex_hull polytope_def)
 
 lemma polytope_imp_closed:
   fixes S :: "'a::real_normed_vector set"
   shows "polytope S \<Longrightarrow> closed S"
 by (simp add: compact_imp_closed polytope_imp_compact)
 
 lemma polytope_imp_bounded:
   fixes S :: "'a::real_normed_vector set"
   shows "polytope S \<Longrightarrow> bounded S"
 by (simp add: compact_imp_bounded polytope_imp_compact)
 
 lemma polytope_interval: "polytope(cbox a b)"
   unfolding polytope_def by (meson closed_interval_as_convex_hull)
 
 lemma polytope_sing: "polytope {a}"
   using polytope_def by force
 
 lemma face_of_polytope_insert:
      "\<lbrakk>polytope S; a \<notin> affine hull S; F face_of S\<rbrakk> \<Longrightarrow> F face_of convex hull (insert a S)"
   by (metis (no_types, lifting) affine_hull_convex_hull face_of_convex_hull_insert hull_insert polytope_def)
 
 proposition face_of_polytope_insert2:
   fixes a :: "'a :: euclidean_space"
   assumes "polytope S" "a \<notin> affine hull S" "F face_of S"
   shows "convex hull (insert a F) face_of convex hull (insert a S)"
 proof -
   obtain V where "finite V" "S = convex hull V"
     using assms by (auto simp: polytope_def)
   then have "convex hull (insert a F) face_of convex hull (insert a V)"
     using affine_hull_convex_hull assms face_of_convex_hull_insert2 by blast
   then show ?thesis
     by (metis \<open>S = convex hull V\<close> hull_insert)
 qed
 
 
 subsection\<open>Polyhedra\<close>
 
 definition\<^marker>\<open>tag important\<close> polyhedron where
  "polyhedron S \<equiv>
         \<exists>F. finite F \<and>
             S = \<Inter> F \<and>
             (\<forall>h \<in> F. \<exists>a b. a \<noteq> 0 \<and> h = {x. a \<bullet> x \<le> b})"
 
 lemma polyhedron_Int [intro,simp]:
    "\<lbrakk>polyhedron S; polyhedron T\<rbrakk> \<Longrightarrow> polyhedron (S \<inter> T)"
   apply (simp add: polyhedron_def, clarify)
   apply (rename_tac F G)
   apply (rule_tac x="F \<union> G" in exI, auto)
   done
 
 lemma polyhedron_UNIV [iff]: "polyhedron UNIV"
   unfolding polyhedron_def
   by (rule_tac x="{}" in exI) auto
 
 lemma polyhedron_Inter [intro,simp]:
    "\<lbrakk>finite F; \<And>S. S \<in> F \<Longrightarrow> polyhedron S\<rbrakk> \<Longrightarrow> polyhedron(\<Inter>F)"
 by (induction F rule: finite_induct) auto
 
 
 lemma polyhedron_empty [iff]: "polyhedron ({} :: 'a :: euclidean_space set)"
 proof -
   have "\<exists>a. a \<noteq> 0 \<and>
              (\<exists>b. {x. (SOME i. i \<in> Basis) \<bullet> x \<le> - 1} = {x. a \<bullet> x \<le> b})"
     by (rule_tac x="(SOME i. i \<in> Basis)" in exI) (force simp: SOME_Basis nonzero_Basis)
   moreover have "\<exists>a b. a \<noteq> 0 \<and>
                        {x. - (SOME i. i \<in> Basis) \<bullet> x \<le> - 1} = {x. a \<bullet> x \<le> b}"
       apply (rule_tac x="-(SOME i. i \<in> Basis)" in exI)
       apply (rule_tac x="-1" in exI)
       apply (simp add: SOME_Basis nonzero_Basis)
       done
   ultimately show ?thesis
     unfolding polyhedron_def
     apply (rule_tac x="{{x. (SOME i. i \<in> Basis) \<bullet> x \<le> -1},
                         {x. -(SOME i. i \<in> Basis) \<bullet> x \<le> -1}}" in exI)
     apply force
     done
 qed
 
 lemma polyhedron_halfspace_le:
   fixes a :: "'a :: euclidean_space"
   shows "polyhedron {x. a \<bullet> x \<le> b}"
 proof (cases "a = 0")
   case True then show ?thesis by auto
 next
   case False
   then show ?thesis
     unfolding polyhedron_def
     by (rule_tac x="{{x. a \<bullet> x \<le> b}}" in exI) auto
 qed
 
 lemma polyhedron_halfspace_ge:
   fixes a :: "'a :: euclidean_space"
   shows "polyhedron {x. a \<bullet> x \<ge> b}"
 using polyhedron_halfspace_le [of "-a" "-b"] by simp
 
 lemma polyhedron_hyperplane:
   fixes a :: "'a :: euclidean_space"
   shows "polyhedron {x. a \<bullet> x = b}"
 proof -
   have "{x. a \<bullet> x = b} = {x. a \<bullet> x \<le> b} \<inter> {x. a \<bullet> x \<ge> b}"
     by force
   then show ?thesis
     by (simp add: polyhedron_halfspace_ge polyhedron_halfspace_le)
 qed
 
 lemma affine_imp_polyhedron:
   fixes S :: "'a :: euclidean_space set"
   shows "affine S \<Longrightarrow> polyhedron S"
 by (metis affine_hull_eq polyhedron_Inter polyhedron_hyperplane affine_hull_finite_intersection_hyperplanes [of S])
 
 lemma polyhedron_imp_closed:
   fixes S :: "'a :: euclidean_space set"
   shows "polyhedron S \<Longrightarrow> closed S"
 apply (simp add: polyhedron_def)
 using closed_halfspace_le by fastforce
 
 lemma polyhedron_imp_convex:
   fixes S :: "'a :: euclidean_space set"
   shows "polyhedron S \<Longrightarrow> convex S"
 apply (simp add: polyhedron_def)
 using convex_Inter convex_halfspace_le by fastforce
 
 lemma polyhedron_affine_hull:
   fixes S :: "'a :: euclidean_space set"
   shows "polyhedron(affine hull S)"
 by (simp add: affine_imp_polyhedron)
 
 
 subsection\<open>Canonical polyhedron representation making facial structure explicit\<close>
 
 proposition polyhedron_Int_affine:
   fixes S :: "'a :: euclidean_space set"
   shows "polyhedron S \<longleftrightarrow>
            (\<exists>F. finite F \<and> S = (affine hull S) \<inter> \<Inter>F \<and>
                 (\<forall>h \<in> F. \<exists>a b. a \<noteq> 0 \<and> h = {x. a \<bullet> x \<le> b}))"
         (is "?lhs = ?rhs")
 proof
   assume ?lhs then show ?rhs
     apply (simp add: polyhedron_def)
     apply (erule ex_forward)
     using hull_subset apply force
     done
 next
   assume ?rhs then show ?lhs
     apply clarify
     apply (erule ssubst)
     apply (force intro: polyhedron_affine_hull polyhedron_halfspace_le)
     done
 qed
 
 proposition rel_interior_polyhedron_explicit:
   assumes "finite F"
       and seq: "S = affine hull S \<inter> \<Inter>F"
       and faceq: "\<And>h. h \<in> F \<Longrightarrow> a h \<noteq> 0 \<and> h = {x. a h \<bullet> x \<le> b h}"
       and psub: "\<And>F'. F' \<subset> F \<Longrightarrow> S \<subset> affine hull S \<inter> \<Inter>F'"
     shows "rel_interior S = {x \<in> S. \<forall>h \<in> F. a h \<bullet> x < b h}"
 proof -
   have rels: "\<And>x. x \<in> rel_interior S \<Longrightarrow> x \<in> S"
     by (meson IntE mem_rel_interior)
   moreover have "a i \<bullet> x < b i" if x: "x \<in> rel_interior S" and "i \<in> F" for x i
   proof -
     have fif: "F - {i} \<subset> F"
       using \<open>i \<in> F\<close> Diff_insert_absorb Diff_subset set_insert psubsetI by blast
     then have "S \<subset> affine hull S \<inter> \<Inter>(F - {i})"
       by (rule psub)
     then obtain z where ssub: "S \<subseteq> \<Inter>(F - {i})" and zint: "z \<in> \<Inter>(F - {i})"
                     and "z \<notin> S" and zaff: "z \<in> affine hull S"
       by auto
     have "z \<noteq> x"
       using \<open>z \<notin> S\<close> rels x by blast
     have "z \<notin> affine hull S \<inter> \<Inter>F"
       using \<open>z \<notin> S\<close> seq by auto
     then have aiz: "a i \<bullet> z > b i"
       using faceq zint zaff by fastforce
     obtain e where "e > 0" "x \<in> S" and e: "ball x e \<inter> affine hull S \<subseteq> S"
       using x by (auto simp: mem_rel_interior_ball)
     then have ins: "\<And>y. \<lbrakk>norm (x - y) < e; y \<in> affine hull S\<rbrakk> \<Longrightarrow> y \<in> S"
       by (metis IntI subsetD dist_norm mem_ball)
     define \<xi> where "\<xi> = min (1/2) (e / 2 / norm(z - x))"
     have "norm (\<xi> *\<^sub>R x - \<xi> *\<^sub>R z) = norm (\<xi> *\<^sub>R (x - z))"
       by (simp add: \<xi>_def algebra_simps norm_mult)
     also have "... = \<xi> * norm (x - z)"
       using \<open>e > 0\<close> by (simp add: \<xi>_def)
     also have "... < e"
       using \<open>z \<noteq> x\<close> \<open>e > 0\<close> by (simp add: \<xi>_def min_def field_split_simps norm_minus_commute)
     finally have lte: "norm (\<xi> *\<^sub>R x - \<xi> *\<^sub>R z) < e" .
     have \<xi>_aff: "\<xi> *\<^sub>R z + (1 - \<xi>) *\<^sub>R x \<in> affine hull S"
       by (metis \<open>x \<in> S\<close> add.commute affine_affine_hull diff_add_cancel hull_inc mem_affine zaff)
     have "\<xi> *\<^sub>R z + (1 - \<xi>) *\<^sub>R x \<in> S"
       apply (rule ins [OF _ \<xi>_aff])
       apply (simp add: algebra_simps lte)
       done
     then obtain l where l: "0 < l" "l < 1" and ls: "(l *\<^sub>R z + (1 - l) *\<^sub>R x) \<in> S"
       apply (rule_tac l = \<xi> in that)
       using \<open>e > 0\<close> \<open>z \<noteq> x\<close>  apply (auto simp: \<xi>_def)
       done
     then have i: "l *\<^sub>R z + (1 - l) *\<^sub>R x \<in> i"
       using seq \<open>i \<in> F\<close> by auto
     have "b i * l + (a i \<bullet> x) * (1 - l) < a i \<bullet> (l *\<^sub>R z + (1 - l) *\<^sub>R x)"
       using l by (simp add: algebra_simps aiz)
     also have "\<dots> \<le> b i" using i l
       using faceq mem_Collect_eq \<open>i \<in> F\<close> by blast
     finally have "(a i \<bullet> x) * (1 - l) < b i * (1 - l)"
       by (simp add: algebra_simps)
     with l show ?thesis
       by simp
   qed
   moreover have "x \<in> rel_interior S"
            if "x \<in> S" and less: "\<And>h. h \<in> F \<Longrightarrow> a h \<bullet> x < b h" for x
   proof -
     have 1: "\<And>h. h \<in> F \<Longrightarrow> x \<in> interior h"
       by (metis interior_halfspace_le mem_Collect_eq less faceq)
     have 2: "\<And>y. \<lbrakk>\<forall>h\<in>F. y \<in> interior h; y \<in> affine hull S\<rbrakk> \<Longrightarrow> y \<in> S"
       by (metis IntI Inter_iff contra_subsetD interior_subset seq)
     show ?thesis
       apply (simp add: rel_interior \<open>x \<in> S\<close>)
       apply (rule_tac x="\<Inter>h\<in>F. interior h" in exI)
       apply (auto simp: \<open>finite F\<close> open_INT 1 2)
       done
   qed
   ultimately show ?thesis by blast
 qed
 
 
 lemma polyhedron_Int_affine_parallel:
   fixes S :: "'a :: euclidean_space set"
   shows "polyhedron S \<longleftrightarrow>
          (\<exists>F. finite F \<and>
               S = (affine hull S) \<inter> (\<Inter>F) \<and>
               (\<forall>h \<in> F. \<exists>a b. a \<noteq> 0 \<and> h = {x. a \<bullet> x \<le> b} \<and>
                              (\<forall>x \<in> affine hull S. (x + a) \<in> affine hull S)))"
     (is "?lhs = ?rhs")
 proof
   assume ?lhs
   then obtain F where "finite F" and seq: "S = (affine hull S) \<inter> \<Inter>F"
                   and faces: "\<And>h. h \<in> F \<Longrightarrow> \<exists>a b. a \<noteq> 0 \<and> h = {x. a \<bullet> x \<le> b}"
     by (fastforce simp add: polyhedron_Int_affine)
   then obtain a b where ab: "\<And>h. h \<in> F \<Longrightarrow> a h \<noteq> 0 \<and> h = {x. a h \<bullet> x \<le> b h}"
     by metis
   show ?rhs
   proof -
     have "\<exists>a' b'. a' \<noteq> 0 \<and>
                   affine hull S \<inter> {x. a' \<bullet> x \<le> b'} = affine hull S \<inter> h \<and>
                   (\<forall>w \<in> affine hull S. (w + a') \<in> affine hull S)"
         if "h \<in> F" "\<not>(affine hull S \<subseteq> h)" for h
     proof -
       have "a h \<noteq> 0" and "h = {x. a h \<bullet> x \<le> b h}" "h \<inter> \<Inter>F = \<Inter>F"
         using \<open>h \<in> F\<close> ab by auto
       then have "(affine hull S) \<inter> {x. a h \<bullet> x \<le> b h} \<noteq> {}"
         by (metis (no_types) affine_hull_eq_empty inf.absorb_iff2 inf_assoc inf_bot_right inf_commute seq that(2))
       moreover have "\<not> (affine hull S \<subseteq> {x. a h \<bullet> x \<le> b h})"
         using \<open>h = {x. a h \<bullet> x \<le> b h}\<close> that(2) by blast
       ultimately show ?thesis
         using affine_parallel_slice [of "affine hull S"]
         by (metis \<open>h = {x. a h \<bullet> x \<le> b h}\<close> affine_affine_hull)
     qed
     then obtain a b
          where ab: "\<And>h. \<lbrakk>h \<in> F; \<not> (affine hull S \<subseteq> h)\<rbrakk>
              \<Longrightarrow> a h \<noteq> 0 \<and>
                   affine hull S \<inter> {x. a h \<bullet> x \<le> b h} = affine hull S \<inter> h \<and>
                   (\<forall>w \<in> affine hull S. (w + a h) \<in> affine hull S)"
       by metis
     have seq2: "S = affine hull S \<inter> (\<Inter>h\<in>{h \<in> F. \<not> affine hull S \<subseteq> h}. {x. a h \<bullet> x \<le> b h})"
       by (subst seq) (auto simp: ab INT_extend_simps)
     show ?thesis
       apply (rule_tac x="(\<lambda>h. {x. a h \<bullet> x \<le> b h}) ` {h. h \<in> F \<and> \<not>(affine hull S \<subseteq> h)}" in exI)
       apply (intro conjI seq2)
         using \<open>finite F\<close> apply force
        using ab apply blast
        done
   qed
 next
   assume ?rhs then show ?lhs
     apply (simp add: polyhedron_Int_affine)
     by metis
 qed
 
 
 proposition polyhedron_Int_affine_parallel_minimal:
   fixes S :: "'a :: euclidean_space set"
   shows "polyhedron S \<longleftrightarrow>
          (\<exists>F. finite F \<and>
               S = (affine hull S) \<inter> (\<Inter>F) \<and>
               (\<forall>h \<in> F. \<exists>a b. a \<noteq> 0 \<and> h = {x. a \<bullet> x \<le> b} \<and>
                              (\<forall>x \<in> affine hull S. (x + a) \<in> affine hull S)) \<and>
               (\<forall>F'. F' \<subset> F \<longrightarrow> S \<subset> (affine hull S) \<inter> (\<Inter>F')))"
     (is "?lhs = ?rhs")
 proof
   assume ?lhs
   then obtain f0
            where f0: "finite f0"
                  "S = (affine hull S) \<inter> (\<Inter>f0)"
                    (is "?P f0")
                  "\<forall>h \<in> f0. \<exists>a b. a \<noteq> 0 \<and> h = {x. a \<bullet> x \<le> b} \<and>
                              (\<forall>x \<in> affine hull S. (x + a) \<in> affine hull S)"
                    (is "?Q f0")
     by (force simp: polyhedron_Int_affine_parallel)
   define n where "n = (LEAST n. \<exists>F. card F = n \<and> finite F \<and> ?P F \<and> ?Q F)"
   have nf: "\<exists>F. card F = n \<and> finite F \<and> ?P F \<and> ?Q F"
     apply (simp add: n_def)
     apply (rule LeastI [where k = "card f0"])
     using f0 apply auto
     done
   then obtain F where F: "card F = n" "finite F" and seq: "?P F" and aff: "?Q F"
     by blast
   then have "\<not> (finite g \<and> ?P g \<and> ?Q g)" if "card g < n" for g
     using that by (auto simp: n_def dest!: not_less_Least)
   then have *: "\<not> (?P g \<and> ?Q g)" if "g \<subset> F" for g
     using that \<open>finite F\<close> psubset_card_mono \<open>card F = n\<close>
     by (metis finite_Int inf.strict_order_iff)
   have 1: "\<And>F'. F' \<subset> F \<Longrightarrow> S \<subseteq> affine hull S \<inter> \<Inter>F'"
     by (subst seq) blast
   have 2: "\<And>F'. F' \<subset> F \<Longrightarrow> S \<noteq> affine hull S \<inter> \<Inter>F'"
     apply (frule *)
     by (metis aff subsetCE subset_iff_psubset_eq)
   show ?rhs
     by (metis \<open>finite F\<close> seq aff psubsetI 1 2)
 next
   assume ?rhs then show ?lhs
     by (auto simp: polyhedron_Int_affine_parallel)
 qed
 
 
 lemma polyhedron_Int_affine_minimal:
   fixes S :: "'a :: euclidean_space set"
   shows "polyhedron S \<longleftrightarrow>
          (\<exists>F. finite F \<and> S = (affine hull S) \<inter> \<Inter>F \<and>
               (\<forall>h \<in> F. \<exists>a b. a \<noteq> 0 \<and> h = {x. a \<bullet> x \<le> b}) \<and>
               (\<forall>F'. F' \<subset> F \<longrightarrow> S \<subset> (affine hull S) \<inter> \<Inter>F'))"
 apply (rule iffI)
  apply (force simp: polyhedron_Int_affine_parallel_minimal elim!: ex_forward)
 apply (auto simp: polyhedron_Int_affine elim!: ex_forward)
 done
 
 proposition facet_of_polyhedron_explicit:
   assumes "finite F"
       and seq: "S = affine hull S \<inter> \<Inter>F"
       and faceq: "\<And>h. h \<in> F \<Longrightarrow> a h \<noteq> 0 \<and> h = {x. a h \<bullet> x \<le> b h}"
       and psub: "\<And>F'. F' \<subset> F \<Longrightarrow> S \<subset> affine hull S \<inter> \<Inter>F'"
     shows "c facet_of S \<longleftrightarrow> (\<exists>h. h \<in> F \<and> c = S \<inter> {x. a h \<bullet> x = b h})"
 proof (cases "S = {}")
   case True with psub show ?thesis by force
 next
   case False
   have "polyhedron S"
     apply (simp add: polyhedron_Int_affine)
     apply (rule_tac x=F in exI)
     using assms  apply force
     done
   then have "convex S"
     by (rule polyhedron_imp_convex)
   with False rel_interior_eq_empty have "rel_interior S \<noteq> {}" by blast
   then obtain x where "x \<in> rel_interior S" by auto
   then obtain T where "open T" "x \<in> T" "x \<in> S" "T \<inter> affine hull S \<subseteq> S"
     by (force simp: mem_rel_interior)
   then have xaff: "x \<in> affine hull S" and xint: "x \<in> \<Inter>F"
     using seq hull_inc by auto
   have "rel_interior S = {x \<in> S. \<forall>h\<in>F. a h \<bullet> x < b h}"
     by (rule rel_interior_polyhedron_explicit [OF \<open>finite F\<close> seq faceq psub])
   with \<open>x \<in> rel_interior S\<close>
   have [simp]: "\<And>h. h\<in>F \<Longrightarrow> a h \<bullet> x < b h" by blast
   have *: "(S \<inter> {x. a h \<bullet> x = b h}) facet_of S" if "h \<in> F" for h
   proof -
     have "S \<subset> affine hull S \<inter> \<Inter>(F - {h})"
       using psub that by (metis Diff_disjoint Diff_subset insert_disjoint(2) psubsetI)
     then obtain z where zaff: "z \<in> affine hull S" and zint: "z \<in> \<Inter>(F - {h})" and "z \<notin> S"
       by force
     then have "z \<noteq> x" "z \<notin> h" using seq \<open>x \<in> S\<close> by auto
     have "x \<in> h" using that xint by auto
     then have able: "a h \<bullet> x \<le> b h"
       using faceq that by blast
     also have "... < a h \<bullet> z" using \<open>z \<notin> h\<close> faceq [OF that] xint by auto
     finally have xltz: "a h \<bullet> x < a h \<bullet> z" .
     define l where "l = (b h - a h \<bullet> x) / (a h \<bullet> z - a h \<bullet> x)"
     define w where "w = (1 - l) *\<^sub>R x + l *\<^sub>R z"
     have "0 < l" "l < 1"
       using able xltz \<open>b h < a h \<bullet> z\<close> \<open>h \<in> F\<close>
       by (auto simp: l_def field_split_simps)
     have awlt: "a i \<bullet> w < b i" if "i \<in> F" "i \<noteq> h" for i
     proof -
       have "(1 - l) * (a i \<bullet> x) < (1 - l) * b i"
         by (simp add: \<open>l < 1\<close> \<open>i \<in> F\<close>)
       moreover have "l * (a i \<bullet> z) \<le> l * b i"
         apply (rule mult_left_mono)
         apply (metis Diff_insert_absorb Inter_iff Set.set_insert \<open>h \<in> F\<close> faceq insertE mem_Collect_eq that zint)
         using \<open>0 < l\<close>
         apply simp
         done
       ultimately show ?thesis by (simp add: w_def algebra_simps)
     qed
     have weq: "a h \<bullet> w = b h"
       using xltz unfolding w_def l_def
       by (simp add: algebra_simps) (simp add: field_simps)
     have "w \<in> affine hull S"
       by (simp add: w_def mem_affine xaff zaff)
     moreover have "w \<in> \<Inter>F"
       using \<open>a h \<bullet> w = b h\<close> awlt faceq less_eq_real_def by blast
     ultimately have "w \<in> S"
       using seq by blast
     with weq have "S \<inter> {x. a h \<bullet> x = b h} \<noteq> {}" by blast
     moreover have "S \<inter> {x. a h \<bullet> x = b h} face_of S"
       apply (rule face_of_Int_supporting_hyperplane_le)
       apply (rule \<open>convex S\<close>)
       apply (subst (asm) seq)
       using faceq that apply fastforce
       done
     moreover have "affine hull (S \<inter> {x. a h \<bullet> x = b h}) =
                    (affine hull S) \<inter> {x. a h \<bullet> x = b h}"
     proof
       show "affine hull (S \<inter> {x. a h \<bullet> x = b h}) \<subseteq> affine hull S \<inter> {x. a h \<bullet> x = b h}"
         apply (intro Int_greatest hull_mono Int_lower1)
         apply (metis affine_hull_eq affine_hyperplane hull_mono inf_le2)
         done
     next
       show "affine hull S \<inter> {x. a h \<bullet> x = b h} \<subseteq> affine hull (S \<inter> {x. a h \<bullet> x = b h})"
       proof
         fix y
         assume yaff: "y \<in> affine hull S \<inter> {y. a h \<bullet> y = b h}"
         obtain T where "0 < T"
                  and T: "\<And>j. \<lbrakk>j \<in> F; j \<noteq> h\<rbrakk> \<Longrightarrow> T * (a j \<bullet> y - a j \<bullet> w) \<le> b j - a j \<bullet> w"
         proof (cases "F - {h} = {}")
           case True then show ?thesis
             by (rule_tac T=1 in that) auto
         next
           case False
           then obtain h' where h': "h' \<in> F - {h}" by auto
           let ?body = "(\<lambda>j. if 0 < a j \<bullet> y - a j \<bullet> w
               then (b j - a j \<bullet> w) / (a j \<bullet> y - a j \<bullet> w)
               else 1) ` (F - {h})"
           define inff where "inff = Inf ?body"
           from \<open>finite F\<close> have "finite ?body"
             by blast
           moreover from h' have "?body \<noteq> {}"
             by blast
           moreover have "j > 0" if "j \<in> ?body" for j
           proof -
             from that obtain x where "x \<in> F" and "x \<noteq> h" and *: "j =
               (if 0 < a x \<bullet> y - a x \<bullet> w
                 then (b x - a x \<bullet> w) / (a x \<bullet> y - a x \<bullet> w) else 1)"
               by blast
             with awlt [of x] have "a x \<bullet> w < b x"
               by simp
             with * show ?thesis
               by simp
           qed
           ultimately have "0 < inff"
             by (simp_all add: finite_less_Inf_iff inff_def)
           moreover have "inff * (a j \<bullet> y - a j \<bullet> w) \<le> b j - a j \<bullet> w"
                         if "j \<in> F" "j \<noteq> h" for j
           proof (cases "a j \<bullet> w < a j \<bullet> y")
             case True
             then have "inff \<le> (b j - a j \<bullet> w) / (a j \<bullet> y - a j \<bullet> w)"
               apply (simp add: inff_def)
               apply (rule cInf_le_finite)
               using \<open>finite F\<close> apply blast
               apply (simp add: that split: if_split_asm)
               done
             then show ?thesis
               using \<open>0 < inff\<close> awlt [OF that] mult_strict_left_mono
               by (fastforce simp add: algebra_simps field_split_simps split: if_split_asm)
           next
             case False
             with \<open>0 < inff\<close> have "inff * (a j \<bullet> y - a j \<bullet> w) \<le> 0"
               by (simp add: mult_le_0_iff)
             also have "... < b j - a j \<bullet> w"
               by (simp add: awlt that)
             finally show ?thesis by simp
           qed
           ultimately show ?thesis
             by (blast intro: that)
         qed
         define c where "c = (1 - T) *\<^sub>R w + T *\<^sub>R y"
         have "(1 - T) *\<^sub>R w + T *\<^sub>R y \<in> j" if "j \<in> F" for j
         proof (cases "j = h")
           case True
           have "(1 - T) *\<^sub>R w + T *\<^sub>R y \<in> {x. a h \<bullet> x \<le> b h}"
             using weq yaff by (auto simp: algebra_simps)
           with True faceq [OF that] show ?thesis by metis
         next
           case False
           with T that have "(1 - T) *\<^sub>R w + T *\<^sub>R y \<in> {x. a j \<bullet> x \<le> b j}"
             by (simp add: algebra_simps)
           with faceq [OF that] show ?thesis by simp
         qed
         moreover have "(1 - T) *\<^sub>R w + T *\<^sub>R y \<in> affine hull S"
           apply (rule affine_affine_hull [simplified affine_alt, rule_format])
           apply (simp add: \<open>w \<in> affine hull S\<close>)
           using yaff apply blast
           done
         ultimately have "c \<in> S"
           using seq by (force simp: c_def)
         moreover have "a h \<bullet> c = b h"
           using yaff by (force simp: c_def algebra_simps weq)
         ultimately have caff: "c \<in> affine hull (S \<inter> {y. a h \<bullet> y = b h})"
           by (simp add: hull_inc)
         have waff: "w \<in> affine hull (S \<inter> {y. a h \<bullet> y = b h})"
           using \<open>w \<in> S\<close> weq by (blast intro: hull_inc)
         have yeq: "y = (1 - inverse T) *\<^sub>R w + c /\<^sub>R T"
           using \<open>0 < T\<close> by (simp add: c_def algebra_simps)
         show "y \<in> affine hull (S \<inter> {y. a h \<bullet> y = b h})"
           by (metis yeq affine_affine_hull [simplified affine_alt, rule_format, OF waff caff])
       qed
     qed
     ultimately show ?thesis
       apply (simp add: facet_of_def)
       apply (subst aff_dim_affine_hull [symmetric])
       using  \<open>b h < a h \<bullet> z\<close> zaff
       apply (force simp: aff_dim_affine_Int_hyperplane)
       done
   qed
   show ?thesis
   proof
     show "\<exists>h. h \<in> F \<and> c = S \<inter> {x. a h \<bullet> x = b h} \<Longrightarrow> c facet_of S"
       using * by blast
   next
     assume "c facet_of S"
     then have "c face_of S" "convex c" "c \<noteq> {}" and affc: "aff_dim c = aff_dim S - 1"
       by (auto simp: facet_of_def face_of_imp_convex)
     then obtain x where x: "x \<in> rel_interior c"
       by (force simp: rel_interior_eq_empty)
     then have "x \<in> c"
       by (meson subsetD rel_interior_subset)
     then have "x \<in> S"
       using \<open>c facet_of S\<close> facet_of_imp_subset by blast
     have rels: "rel_interior S = {x \<in> S. \<forall>h\<in>F. a h \<bullet> x < b h}"
       by (rule rel_interior_polyhedron_explicit [OF assms])
     have "c \<noteq> S"
       using \<open>c facet_of S\<close> facet_of_irrefl by blast
     then have "x \<notin> rel_interior S"
       by (metis IntI empty_iff \<open>x \<in> c\<close> \<open>c \<noteq> S\<close> \<open>c face_of S\<close> face_of_disjoint_rel_interior)
     with rels \<open>x \<in> S\<close> obtain i where "i \<in> F" and i: "a i \<bullet> x \<ge> b i"
       by force
     have "x \<in> {u. a i \<bullet> u \<le> b i}"
       by (metis IntD2 InterE \<open>i \<in> F\<close> \<open>x \<in> S\<close> faceq seq)
     then have "a i \<bullet> x \<le> b i" by simp
     then have "a i \<bullet> x = b i" using i by auto
     have "c \<subseteq> S \<inter> {x. a i \<bullet> x = b i}"
       apply (rule subset_of_face_of [of _ S])
         apply (simp add: "*" \<open>i \<in> F\<close> facet_of_imp_face_of)
        apply (simp add: \<open>c face_of S\<close> face_of_imp_subset)
       using \<open>a i \<bullet> x = b i\<close> \<open>x \<in> S\<close> x by blast
     then have cface: "c face_of (S \<inter> {x. a i \<bullet> x = b i})"
       by (meson \<open>c face_of S\<close> face_of_subset inf_le1)
     have con: "convex (S \<inter> {x. a i \<bullet> x = b i})"
       by (simp add: \<open>convex S\<close> convex_Int convex_hyperplane)
     show "\<exists>h. h \<in> F \<and> c = S \<inter> {x. a h \<bullet> x = b h}"
       apply (rule_tac x=i in exI)
       apply (simp add: \<open>i \<in> F\<close>)
       by (metis (no_types) * \<open>i \<in> F\<close> affc facet_of_def less_irrefl face_of_aff_dim_lt [OF con cface])
   qed
 qed
 
 
 lemma face_of_polyhedron_subset_explicit:
   fixes S :: "'a :: euclidean_space set"
   assumes "finite F"
       and seq: "S = affine hull S \<inter> \<Inter>F"
       and faceq: "\<And>h. h \<in> F \<Longrightarrow> a h \<noteq> 0 \<and> h = {x. a h \<bullet> x \<le> b h}"
       and psub: "\<And>F'. F' \<subset> F \<Longrightarrow> S \<subset> affine hull S \<inter> \<Inter>F'"
       and c: "c face_of S" and "c \<noteq> {}" "c \<noteq> S"
    obtains h where "h \<in> F" "c \<subseteq> S \<inter> {x. a h \<bullet> x = b h}"
 proof -
   have "c \<subseteq> S" using \<open>c face_of S\<close>
     by (simp add: face_of_imp_subset)
   have "polyhedron S"
     apply (simp add: polyhedron_Int_affine)
     by (metis \<open>finite F\<close> faceq seq)
   then have "convex S"
     by (simp add: polyhedron_imp_convex)
   then have *: "(S \<inter> {x. a h \<bullet> x = b h}) face_of S" if "h \<in> F" for h
     apply (rule face_of_Int_supporting_hyperplane_le)
     using faceq seq that by fastforce
   have "rel_interior c \<noteq> {}"
     using c \<open>c \<noteq> {}\<close> face_of_imp_convex rel_interior_eq_empty by blast
   then obtain x where "x \<in> rel_interior c" by auto
   have rels: "rel_interior S = {x \<in> S. \<forall>h\<in>F. a h \<bullet> x < b h}"
     by (rule rel_interior_polyhedron_explicit [OF \<open>finite F\<close> seq faceq psub])
   then have xnot: "x \<notin> rel_interior S"
     by (metis IntI \<open>x \<in> rel_interior c\<close> c \<open>c \<noteq> S\<close> contra_subsetD empty_iff face_of_disjoint_rel_interior rel_interior_subset)
   then have "x \<in> S"
     using \<open>c \<subseteq> S\<close> \<open>x \<in> rel_interior c\<close> rel_interior_subset by auto
   then have xint: "x \<in> \<Inter>F"
     using seq by blast
   have "F \<noteq> {}" using assms
     by (metis affine_Int affine_Inter affine_affine_hull ex_in_conv face_of_affine_trivial)
   then obtain i where "i \<in> F" "\<not> (a i \<bullet> x < b i)"
     using \<open>x \<in> S\<close> rels xnot by auto
   with xint have "a i \<bullet> x = b i"
     by (metis eq_iff mem_Collect_eq not_le Inter_iff faceq)
   have face: "S \<inter> {x. a i \<bullet> x = b i} face_of S"
     by (simp add: "*" \<open>i \<in> F\<close>)
   show ?thesis
     apply (rule_tac h = i in that)
      apply (rule \<open>i \<in> F\<close>)
     apply (rule subset_of_face_of [OF face \<open>c \<subseteq> S\<close>])
     using \<open>a i \<bullet> x = b i\<close> \<open>x \<in> rel_interior c\<close> \<open>x \<in> S\<close> apply blast
     done
 qed
 
 text\<open>Initial part of proof duplicates that above\<close>
 proposition face_of_polyhedron_explicit:
   fixes S :: "'a :: euclidean_space set"
   assumes "finite F"
       and seq: "S = affine hull S \<inter> \<Inter>F"
       and faceq: "\<And>h. h \<in> F \<Longrightarrow> a h \<noteq> 0 \<and> h = {x. a h \<bullet> x \<le> b h}"
       and psub: "\<And>F'. F' \<subset> F \<Longrightarrow> S \<subset> affine hull S \<inter> \<Inter>F'"
       and c: "c face_of S" and "c \<noteq> {}" "c \<noteq> S"
     shows "c = \<Inter>{S \<inter> {x. a h \<bullet> x = b h} | h. h \<in> F \<and> c \<subseteq> S \<inter> {x. a h \<bullet> x = b h}}"
 proof -
   let ?ab = "\<lambda>h. {x. a h \<bullet> x = b h}"
   have "c \<subseteq> S" using \<open>c face_of S\<close>
     by (simp add: face_of_imp_subset)
   have "polyhedron S"
     apply (simp add: polyhedron_Int_affine)
     by (metis \<open>finite F\<close> faceq seq)
   then have "convex S"
     by (simp add: polyhedron_imp_convex)
   then have *: "(S \<inter> ?ab h) face_of S" if "h \<in> F" for h
     apply (rule face_of_Int_supporting_hyperplane_le)
     using faceq seq that by fastforce
   have "rel_interior c \<noteq> {}"
     using c \<open>c \<noteq> {}\<close> face_of_imp_convex rel_interior_eq_empty by blast
   then obtain z where z: "z \<in> rel_interior c" by auto
   have rels: "rel_interior S = {z \<in> S. \<forall>h\<in>F. a h \<bullet> z < b h}"
     by (rule rel_interior_polyhedron_explicit [OF \<open>finite F\<close> seq faceq psub])
   then have xnot: "z \<notin> rel_interior S"
     by (metis IntI \<open>z \<in> rel_interior c\<close> c \<open>c \<noteq> S\<close> contra_subsetD empty_iff face_of_disjoint_rel_interior rel_interior_subset)
   then have "z \<in> S"
     using \<open>c \<subseteq> S\<close> \<open>z \<in> rel_interior c\<close> rel_interior_subset by auto
   with seq have xint: "z \<in> \<Inter>F" by blast
   have "open (\<Inter>h\<in>{h \<in> F. a h \<bullet> z < b h}. {w. a h \<bullet> w < b h})"
     by (auto simp: \<open>finite F\<close> open_halfspace_lt open_INT)
   then obtain e where "0 < e"
                  "ball z e \<subseteq> (\<Inter>h\<in>{h \<in> F. a h \<bullet> z < b h}. {w. a h \<bullet> w < b h})"
     by (auto intro: openE [of _ z])
   then have e: "\<And>h. \<lbrakk>h \<in> F; a h \<bullet> z < b h\<rbrakk> \<Longrightarrow> ball z e \<subseteq> {w. a h \<bullet> w < b h}"
     by blast
   have "c \<subseteq> (S \<inter> ?ab h) \<longleftrightarrow> z \<in> S \<inter> ?ab h" if "h \<in> F" for h
   proof
     show "z \<in> S \<inter> ?ab h \<Longrightarrow> c \<subseteq> S \<inter> ?ab h"
       apply (rule subset_of_face_of [of _ S])
       using that \<open>c \<subseteq> S\<close> \<open>z \<in> rel_interior c\<close>
       using facet_of_polyhedron_explicit [OF \<open>finite F\<close> seq faceq psub]
             unfolding facet_of_def
       apply auto
       done
   next
     show "c \<subseteq> S \<inter> ?ab h \<Longrightarrow> z \<in> S \<inter> ?ab h"
       using \<open>z \<in> rel_interior c\<close> rel_interior_subset by force
   qed
   then have **: "{S \<inter> ?ab h | h. h \<in> F \<and> c \<subseteq> S \<and> c \<subseteq> ?ab h} =
                  {S \<inter> ?ab h |h. h \<in> F \<and> z \<in> S \<inter> ?ab h}"
     by blast
   have bsub: "ball z e \<inter> affine hull \<Inter>{S \<inter> ?ab h |h. h \<in> F \<and> a h \<bullet> z = b h}
              \<subseteq> affine hull S \<inter> \<Inter>F \<inter> \<Inter>{?ab h |h. h \<in> F \<and> a h \<bullet> z = b h}"
             if "i \<in> F" and i: "a i \<bullet> z = b i" for i
   proof -
     have sub: "ball z e \<inter> \<Inter>{?ab h |h. h \<in> F \<and> a h \<bullet> z = b h} \<subseteq> j"
              if "j \<in> F" for j
     proof -
       have "a j \<bullet> z \<le> b j" using faceq that xint by auto
       then consider "a j \<bullet> z < b j" | "a j \<bullet> z = b j" by linarith
       then have "\<exists>G. G \<in> {?ab h |h. h \<in> F \<and> a h \<bullet> z = b h} \<and> ball z e \<inter> G \<subseteq> j"
       proof cases
         assume "a j \<bullet> z < b j"
         then have "ball z e \<inter> {x. a i \<bullet> x = b i} \<subseteq> j"
           using e [OF \<open>j \<in> F\<close>] faceq that
           by (fastforce simp: ball_def)
         then show ?thesis
           by (rule_tac x="{x. a i \<bullet> x = b i}" in exI) (force simp: \<open>i \<in> F\<close> i)
       next
         assume eq: "a j \<bullet> z = b j"
         with faceq that show ?thesis
           by (rule_tac x="{x. a j \<bullet> x = b j}" in exI) (fastforce simp add: \<open>j \<in> F\<close>)
       qed
       then show ?thesis  by blast
     qed
     have 1: "affine hull \<Inter>{S \<inter> ?ab h |h. h \<in> F \<and> a h \<bullet> z = b h} \<subseteq> affine hull S"
       apply (rule hull_mono)
       using that \<open>z \<in> S\<close> by auto
     have 2: "affine hull \<Inter>{S \<inter> ?ab h |h. h \<in> F \<and> a h \<bullet> z = b h}
           \<subseteq> \<Inter>{?ab h |h. h \<in> F \<and> a h \<bullet> z = b h}"
       by (rule hull_minimal) (auto intro: affine_hyperplane)
     have 3: "ball z e \<inter> \<Inter>{?ab h |h. h \<in> F \<and> a h \<bullet> z = b h} \<subseteq> \<Inter>F"
       by (iprover intro: sub Inter_greatest)
     have *: "\<lbrakk>A \<subseteq> (B :: 'a set); A \<subseteq> C; E \<inter> C \<subseteq> D\<rbrakk> \<Longrightarrow> E \<inter> A \<subseteq> (B \<inter> D) \<inter> C"
              for A B C D E  by blast
     show ?thesis by (intro * 1 2 3)
   qed
   have "\<exists>h. h \<in> F \<and> c \<subseteq> ?ab h"
     apply (rule face_of_polyhedron_subset_explicit [OF \<open>finite F\<close> seq faceq psub])
     using assms by auto
   then have fac: "\<Inter>{S \<inter> ?ab h |h. h \<in> F \<and> c \<subseteq> S \<inter> ?ab h} face_of S"
     using * by (force simp: \<open>c \<subseteq> S\<close> intro: face_of_Inter)
   have red:
      "(\<And>a. P a \<Longrightarrow> T \<subseteq> S \<inter> \<Inter>{F x |x. P x}) \<Longrightarrow> T \<subseteq> \<Inter>{S \<inter> F x |x. P x}"
      for P T F   by blast
   have "ball z e \<inter> affine hull \<Inter>{S \<inter> ?ab h |h. h \<in> F \<and> a h \<bullet> z = b h}
         \<subseteq> \<Inter>{S \<inter> ?ab h |h. h \<in> F \<and> a h \<bullet> z = b h}"
     apply (rule red)
     apply (metis seq bsub)
     done
   with \<open>0 < e\<close> have zinrel: "z \<in> rel_interior
                     (\<Inter>{S \<inter> ?ab h |h. h \<in> F \<and> z \<in> S \<and> a h \<bullet> z = b h})"
     by (auto simp: mem_rel_interior_ball \<open>z \<in> S\<close>)
   show ?thesis
     apply (rule face_of_eq [OF c fac])
     using z zinrel apply (force simp: **)
     done
 qed
 
 
 subsection\<open>More general corollaries from the explicit representation\<close>
 
 corollary facet_of_polyhedron:
   assumes "polyhedron S" and "c facet_of S"
   obtains a b where "a \<noteq> 0" "S \<subseteq> {x. a \<bullet> x \<le> b}" "c = S \<inter> {x. a \<bullet> x = b}"
 proof -
   obtain F where "finite F" and seq: "S = affine hull S \<inter> \<Inter>F"
              and faces: "\<And>h. h \<in> F \<Longrightarrow> \<exists>a b. a \<noteq> 0 \<and> h = {x. a \<bullet> x \<le> b}"
              and min: "\<And>F'. F' \<subset> F \<Longrightarrow> S \<subset> (affine hull S) \<inter> \<Inter>F'"
     using assms by (simp add: polyhedron_Int_affine_minimal) meson
   then obtain a b where ab: "\<And>h. h \<in> F \<Longrightarrow> a h \<noteq> 0 \<and> h = {x. a h \<bullet> x \<le> b h}"
     by metis
   obtain i where "i \<in> F" and c: "c = S \<inter> {x. a i \<bullet> x = b i}"
     using facet_of_polyhedron_explicit [OF \<open>finite F\<close> seq ab min] assms
     by force
   moreover have ssub: "S \<subseteq> {x. a i \<bullet> x \<le> b i}"
      apply (subst seq)
      using \<open>i \<in> F\<close> ab by auto
   ultimately show ?thesis
     by (rule_tac a = "a i" and b = "b i" in that) (simp_all add: ab)
 qed
 
 corollary face_of_polyhedron:
   assumes "polyhedron S" and "c face_of S" and "c \<noteq> {}" and "c \<noteq> S"
     shows "c = \<Inter>{F. F facet_of S \<and> c \<subseteq> F}"
 proof -
   obtain F where "finite F" and seq: "S = affine hull S \<inter> \<Inter>F"
              and faces: "\<And>h. h \<in> F \<Longrightarrow> \<exists>a b. a \<noteq> 0 \<and> h = {x. a \<bullet> x \<le> b}"
              and min: "\<And>F'. F' \<subset> F \<Longrightarrow> S \<subset> (affine hull S) \<inter> \<Inter>F'"
     using assms by (simp add: polyhedron_Int_affine_minimal) meson
   then obtain a b where ab: "\<And>h. h \<in> F \<Longrightarrow> a h \<noteq> 0 \<and> h = {x. a h \<bullet> x \<le> b h}"
     by metis
   show ?thesis
     apply (subst face_of_polyhedron_explicit [OF \<open>finite F\<close> seq ab min])
     apply (auto simp: assms facet_of_polyhedron_explicit [OF \<open>finite F\<close> seq ab min] cong: Collect_cong)
     done
 qed
 
 lemma face_of_polyhedron_subset_facet:
   assumes "polyhedron S" and "c face_of S" and "c \<noteq> {}" and "c \<noteq> S"
   obtains F where "F facet_of S" "c \<subseteq> F"
 using face_of_polyhedron assms
 by (metis (no_types, lifting) Inf_greatest antisym_conv face_of_imp_subset mem_Collect_eq)
 
 
 lemma exposed_face_of_polyhedron:
   assumes "polyhedron S"
     shows "F exposed_face_of S \<longleftrightarrow> F face_of S"
 proof
   show "F exposed_face_of S \<Longrightarrow> F face_of S"
     by (simp add: exposed_face_of_def)
 next
   assume "F face_of S"
   show "F exposed_face_of S"
   proof (cases "F = {} \<or> F = S")
     case True then show ?thesis
       using \<open>F face_of S\<close> exposed_face_of by blast
   next
     case False
     then have "{g. g facet_of S \<and> F \<subseteq> g} \<noteq> {}"
       by (metis Collect_empty_eq_bot \<open>F face_of S\<close> assms empty_def face_of_polyhedron_subset_facet)
     moreover have "\<And>T. \<lbrakk>T facet_of S; F \<subseteq> T\<rbrakk> \<Longrightarrow> T exposed_face_of S"
       by (metis assms exposed_face_of facet_of_imp_face_of facet_of_polyhedron)
     ultimately have "\<Inter>{fa.
        fa facet_of S \<and> F \<subseteq> fa} exposed_face_of S"
       by (metis (no_types, lifting) mem_Collect_eq exposed_face_of_Inter)
     then show ?thesis
       using False
       apply (subst face_of_polyhedron [OF assms \<open>F face_of S\<close>], auto)
       done
   qed
 qed
 
 lemma face_of_polyhedron_polyhedron:
   fixes S :: "'a :: euclidean_space set"
   assumes "polyhedron S" "c face_of S" shows "polyhedron c"
 by (metis assms face_of_imp_eq_affine_Int polyhedron_Int polyhedron_affine_hull polyhedron_imp_convex)
 
 lemma finite_polyhedron_faces:
   fixes S :: "'a :: euclidean_space set"
   assumes "polyhedron S"
     shows "finite {F. F face_of S}"
 proof -
   obtain F where "finite F" and seq: "S = affine hull S \<inter> \<Inter>F"
              and faces: "\<And>h. h \<in> F \<Longrightarrow> \<exists>a b. a \<noteq> 0 \<and> h = {x. a \<bullet> x \<le> b}"
              and min:   "\<And>F'. F' \<subset> F \<Longrightarrow> S \<subset> (affine hull S) \<inter> \<Inter>F'"
     using assms by (simp add: polyhedron_Int_affine_minimal) meson
   then obtain a b where ab: "\<And>h. h \<in> F \<Longrightarrow> a h \<noteq> 0 \<and> h = {x. a h \<bullet> x \<le> b h}"
     by metis
   have "finite {\<Inter>{S \<inter> {x. a h \<bullet> x = b h} |h. h \<in> F'}| F'. F' \<in> Pow F}"
     by (simp add: \<open>finite F\<close>)
   moreover have "{F. F face_of S} - {{}, S} \<subseteq> {\<Inter>{S \<inter> {x. a h \<bullet> x = b h} |h. h \<in> F'}| F'. F' \<in> Pow F}"
     apply clarify
     apply (rename_tac c)
     apply (drule face_of_polyhedron_explicit [OF \<open>finite F\<close> seq ab min, simplified], simp_all)
     apply (erule ssubst)
     apply (rule_tac x="{h \<in> F. c \<subseteq> S \<inter> {x. a h \<bullet> x = b h}}" in exI, auto)
     done
   ultimately show ?thesis
     by (meson finite.emptyI finite.insertI finite_Diff2 finite_subset)
 qed
 
 lemma finite_polyhedron_exposed_faces:
    "polyhedron S \<Longrightarrow> finite {F. F exposed_face_of S}"
 using exposed_face_of_polyhedron finite_polyhedron_faces by fastforce
 
 lemma finite_polyhedron_extreme_points:
   fixes S :: "'a :: euclidean_space set"
   shows "polyhedron S \<Longrightarrow> finite {v. v extreme_point_of S}"
 apply (simp add: face_of_singleton [symmetric])
 apply (rule finite_subset [OF _ finite_vimageI [OF finite_polyhedron_faces]], auto)
 done
 
 lemma finite_polyhedron_facets:
   fixes S :: "'a :: euclidean_space set"
   shows "polyhedron S \<Longrightarrow> finite {F. F facet_of S}"
 unfolding facet_of_def
 by (blast intro: finite_subset [OF _ finite_polyhedron_faces])
 
 
 proposition rel_interior_of_polyhedron:
   fixes S :: "'a :: euclidean_space set"
   assumes "polyhedron S"
     shows "rel_interior S = S - \<Union>{F. F facet_of S}"
 proof -
   obtain F where "finite F" and seq: "S = affine hull S \<inter> \<Inter>F"
              and faces: "\<And>h. h \<in> F \<Longrightarrow> \<exists>a b. a \<noteq> 0 \<and> h = {x. a \<bullet> x \<le> b}"
              and min: "\<And>F'. F' \<subset> F \<Longrightarrow> S \<subset> (affine hull S) \<inter> \<Inter>F'"
     using assms by (simp add: polyhedron_Int_affine_minimal) meson
   then obtain a b where ab: "\<And>h. h \<in> F \<Longrightarrow> a h \<noteq> 0 \<and> h = {x. a h \<bullet> x \<le> b h}"
     by metis
   have facet: "(c facet_of S) \<longleftrightarrow> (\<exists>h. h \<in> F \<and> c = S \<inter> {x. a h \<bullet> x = b h})" for c
     by (rule facet_of_polyhedron_explicit [OF \<open>finite F\<close> seq ab min])
   have rel: "rel_interior S = {x \<in> S. \<forall>h\<in>F. a h \<bullet> x < b h}"
     by (rule rel_interior_polyhedron_explicit [OF \<open>finite F\<close> seq ab min])
   have "a h \<bullet> x < b h" if "x \<in> S" "h \<in> F" and xnot: "x \<notin> \<Union>{F. F facet_of S}" for x h
   proof -
     have "x \<in> \<Inter>F" using seq that by force
     with \<open>h \<in> F\<close> ab have "a h \<bullet> x \<le> b h" by auto
     then consider "a h \<bullet> x < b h" | "a h \<bullet> x = b h" by linarith
     then show ?thesis
     proof cases
       case 1 then show ?thesis .
     next
       case 2
       have "Collect ((\<in>) x) \<notin> Collect ((\<in>) (\<Union>{A. A facet_of S}))"
         using xnot by fastforce
       then have "F \<notin> Collect ((\<in>) h)"
         using 2 \<open>x \<in> S\<close> facet by blast
       with \<open>h \<in> F\<close> have "\<Inter>F \<subseteq> S \<inter> {x. a h \<bullet> x = b h}" by blast
       with 2 that \<open>x \<in> \<Inter>F\<close> show ?thesis
         apply simp
         apply (drule_tac x="\<Inter>F" in spec)
         apply (simp add: facet)
         apply (drule_tac x=h in spec)
         using seq by auto
       qed
   qed
   moreover have "\<exists>h\<in>F. a h \<bullet> x \<ge> b h" if "x \<in> \<Union>{F. F facet_of S}" for x
     using that by (force simp: facet)
   ultimately show ?thesis
     by (force simp: rel)
 qed
 
 lemma rel_boundary_of_polyhedron:
   fixes S :: "'a :: euclidean_space set"
   assumes "polyhedron S"
     shows "S - rel_interior S = \<Union> {F. F facet_of S}"
 using facet_of_imp_subset by (fastforce simp add: rel_interior_of_polyhedron assms)
 
 lemma rel_frontier_of_polyhedron:
   fixes S :: "'a :: euclidean_space set"
   assumes "polyhedron S"
     shows "rel_frontier S = \<Union> {F. F facet_of S}"
 by (simp add: assms rel_frontier_def polyhedron_imp_closed rel_boundary_of_polyhedron)
 
 lemma rel_frontier_of_polyhedron_alt:
   fixes S :: "'a :: euclidean_space set"
   assumes "polyhedron S"
     shows "rel_frontier S = \<Union> {F. F face_of S \<and> (F \<noteq> S)}"
 apply (rule subset_antisym)
   apply (force simp: rel_frontier_of_polyhedron facet_of_def assms)
 using face_of_subset_rel_frontier by fastforce
 
 
 text\<open>A characterization of polyhedra as having finitely many faces\<close>
 
 proposition polyhedron_eq_finite_exposed_faces:
   fixes S :: "'a :: euclidean_space set"
   shows "polyhedron S \<longleftrightarrow> closed S \<and> convex S \<and> finite {F. F exposed_face_of S}"
          (is "?lhs = ?rhs")
 proof
   assume ?lhs
   then show ?rhs
     by (auto simp: polyhedron_imp_closed polyhedron_imp_convex finite_polyhedron_exposed_faces)
 next
   assume ?rhs
   then have "closed S" "convex S" and fin: "finite {F. F exposed_face_of S}" by auto
   show ?lhs
   proof (cases "S = {}")
     case True then show ?thesis by auto
   next
     case False
     define F where "F = {h. h exposed_face_of S \<and> h \<noteq> {} \<and> h \<noteq> S}"
     have "finite F" by (simp add: fin F_def)
     have hface: "h face_of S"
       and "\<exists>a b. a \<noteq> 0 \<and> S \<subseteq> {x. a \<bullet> x \<le> b} \<and> h = S \<inter> {x. a \<bullet> x = b}"
       if "h \<in> F" for h
       using exposed_face_of F_def that by simp_all auto
     then obtain a b where ab:
       "\<And>h. h \<in> F \<Longrightarrow> a h \<noteq> 0 \<and> S \<subseteq> {x. a h \<bullet> x \<le> b h} \<and> h = S \<inter> {x. a h \<bullet> x = b h}"
       by metis
     have *: "False"
       if paff: "p \<in> affine hull S" and "p \<notin> S"
       and pint: "p \<in> \<Inter>{{x. a h \<bullet> x \<le> b h} |h. h \<in> F}" for p
     proof -
       have "rel_interior S \<noteq> {}"
         by (simp add: \<open>S \<noteq> {}\<close> \<open>convex S\<close> rel_interior_eq_empty)
       then obtain c where c: "c \<in> rel_interior S" by auto
       with rel_interior_subset have "c \<in> S"  by blast
       have ccp: "closed_segment c p \<subseteq> affine hull S"
         by (meson affine_affine_hull affine_imp_convex c closed_segment_subset hull_subset paff rel_interior_subset subsetCE)
       obtain x where xcl: "x \<in> closed_segment c p" and "x \<in> S" and xnot: "x \<notin> rel_interior S"
         using connected_openin [of "closed_segment c p"]
         apply simp
         apply (drule_tac x="closed_segment c p \<inter> rel_interior S" in spec)
         apply (erule impE)
          apply (force simp: openin_rel_interior openin_Int intro: openin_subtopology_Int_subset [OF _ ccp])
         apply (drule_tac x="closed_segment c p \<inter> (- S)" in spec)
         using rel_interior_subset \<open>closed S\<close> c \<open>p \<notin> S\<close> apply blast
         done
       then obtain \<mu> where "0 \<le> \<mu>" "\<mu> \<le> 1" and xeq: "x = (1 - \<mu>) *\<^sub>R c + \<mu> *\<^sub>R p"
         by (auto simp: in_segment)
       show False
       proof (cases "\<mu>=0 \<or> \<mu>=1")
         case True with xeq c xnot \<open>x \<in> S\<close> \<open>p \<notin> S\<close>
         show False by auto
       next
         case False
         then have xos: "x \<in> open_segment c p"
           using \<open>x \<in> S\<close> c open_segment_def that(2) xcl xnot by auto
         have xclo: "x \<in> closure S"
           using \<open>x \<in> S\<close> closure_subset by blast
         obtain d where "d \<noteq> 0"
               and dle: "\<And>y. y \<in> closure S \<Longrightarrow> d \<bullet> x \<le> d \<bullet> y"
               and dless: "\<And>y. y \<in> rel_interior S \<Longrightarrow> d \<bullet> x < d \<bullet> y"
           by (metis supporting_hyperplane_relative_frontier [OF \<open>convex S\<close> xclo xnot])
         have sex: "S \<inter> {y. d \<bullet> y = d \<bullet> x} exposed_face_of S"
           by (simp add: \<open>closed S\<close> dle exposed_face_of_Int_supporting_hyperplane_ge [OF \<open>convex S\<close>])
         have sne: "S \<inter> {y. d \<bullet> y = d \<bullet> x} \<noteq> {}"
           using \<open>x \<in> S\<close> by blast
         have sns: "S \<inter> {y. d \<bullet> y = d \<bullet> x} \<noteq> S"
           by (metis (mono_tags) Int_Collect c subsetD dless not_le order_refl rel_interior_subset)
         obtain h where "h \<in> F" "x \<in> h"
           apply (rule_tac h="S \<inter> {y. d \<bullet> y = d \<bullet> x}" in that)
           apply (simp_all add: F_def sex sne sns \<open>x \<in> S\<close>)
           done
         have abface: "{y. a h \<bullet> y = b h} face_of {y. a h \<bullet> y \<le> b h}"
           using hyperplane_face_of_halfspace_le by blast
         then have "c \<in> h"
           using face_ofD [OF abface xos] \<open>c \<in> S\<close> \<open>h \<in> F\<close> ab pint \<open>x \<in> h\<close> by blast
         with c have "h \<inter> rel_interior S \<noteq> {}" by blast
         then show False
           using \<open>h \<in> F\<close> F_def face_of_disjoint_rel_interior hface by auto
       qed
     qed
     have "S \<subseteq> affine hull S \<inter> \<Inter>{{x. a h \<bullet> x \<le> b h} |h. h \<in> F}"
       using ab by (auto simp: hull_subset)
     moreover have "affine hull S \<inter> \<Inter>{{x. a h \<bullet> x \<le> b h} |h. h \<in> F} \<subseteq> S"
       using * by blast
     ultimately have "S = affine hull S \<inter> \<Inter> {{x. a h \<bullet> x \<le> b h} |h. h \<in> F}" ..
     then show ?thesis
       apply (rule ssubst)
       apply (force intro: polyhedron_affine_hull polyhedron_halfspace_le simp: \<open>finite F\<close>)
       done
   qed
 qed
 
 corollary polyhedron_eq_finite_faces:
   fixes S :: "'a :: euclidean_space set"
   shows "polyhedron S \<longleftrightarrow> closed S \<and> convex S \<and> finite {F. F face_of S}"
          (is "?lhs = ?rhs")
 proof
   assume ?lhs
   then show ?rhs
     by (simp add: finite_polyhedron_faces polyhedron_imp_closed polyhedron_imp_convex)
 next
   assume ?rhs
   then show ?lhs
     by (force simp: polyhedron_eq_finite_exposed_faces exposed_face_of intro: finite_subset)
 qed
 
 lemma polyhedron_linear_image_eq:
   fixes h :: "'a :: euclidean_space \<Rightarrow> 'b :: euclidean_space"
   assumes "linear h" "bij h"
     shows "polyhedron (h ` S) \<longleftrightarrow> polyhedron S"
 proof -
   have *: "{f. P f} = (image h) ` {f. P (h ` f)}" for P
     apply safe
     apply (rule_tac x="inv h ` x" in image_eqI)
     apply (auto simp: \<open>bij h\<close> bij_is_surj image_f_inv_f)
     done
   have "inj h" using bij_is_inj assms by blast
   then have injim: "inj_on ((`) h) A" for A
     by (simp add: inj_on_def inj_image_eq_iff)
   show ?thesis
     using \<open>linear h\<close> \<open>inj h\<close>
     apply (simp add: polyhedron_eq_finite_faces closed_injective_linear_image_eq)
     apply (simp add: * face_of_linear_image [of h _ S, symmetric] finite_image_iff injim)
     done
 qed
 
 lemma polyhedron_negations:
   fixes S :: "'a :: euclidean_space set"
   shows   "polyhedron S \<Longrightarrow> polyhedron(image uminus S)"
   by (subst polyhedron_linear_image_eq)
     (auto simp: bij_uminus intro!: linear_uminus)
 
 subsection\<open>Relation between polytopes and polyhedra\<close>
 
 proposition polytope_eq_bounded_polyhedron:
   fixes S :: "'a :: euclidean_space set"
   shows "polytope S \<longleftrightarrow> polyhedron S \<and> bounded S"
          (is "?lhs = ?rhs")
 proof
   assume ?lhs
   then show ?rhs
     by (simp add: finite_polytope_faces polyhedron_eq_finite_faces
                   polytope_imp_closed polytope_imp_convex polytope_imp_bounded)
 next
   assume ?rhs then show ?lhs
     unfolding polytope_def
     apply (rule_tac x="{v. v extreme_point_of S}" in exI)
     apply (simp add: finite_polyhedron_extreme_points Krein_Milman_Minkowski compact_eq_bounded_closed polyhedron_imp_closed polyhedron_imp_convex)
     done
 qed
 
 lemma polytope_Int:
   fixes S :: "'a :: euclidean_space set"
   shows "\<lbrakk>polytope S; polytope T\<rbrakk> \<Longrightarrow> polytope(S \<inter> T)"
 by (simp add: polytope_eq_bounded_polyhedron bounded_Int)
 
 
 lemma polytope_Int_polyhedron:
   fixes S :: "'a :: euclidean_space set"
   shows "\<lbrakk>polytope S; polyhedron T\<rbrakk> \<Longrightarrow> polytope(S \<inter> T)"
   by (simp add: bounded_Int polytope_eq_bounded_polyhedron)
 
 lemma polyhedron_Int_polytope:
   fixes S :: "'a :: euclidean_space set"
   shows "\<lbrakk>polyhedron S; polytope T\<rbrakk> \<Longrightarrow> polytope(S \<inter> T)"
   by (simp add: bounded_Int polytope_eq_bounded_polyhedron)
 
 lemma polytope_imp_polyhedron:
   fixes S :: "'a :: euclidean_space set"
   shows "polytope S \<Longrightarrow> polyhedron S"
   by (simp add: polytope_eq_bounded_polyhedron)
 
 lemma polytope_facet_exists:
   fixes p :: "'a :: euclidean_space set"
   assumes "polytope p" "0 < aff_dim p"
   obtains F where "F facet_of p"
 proof (cases "p = {}")
   case True with assms show ?thesis by auto
 next
   case False
   then obtain v where "v extreme_point_of p"
     using extreme_point_exists_convex
     by (blast intro: \<open>polytope p\<close> polytope_imp_compact polytope_imp_convex)
   then
   show ?thesis
     by (metis face_of_polyhedron_subset_facet polytope_imp_polyhedron aff_dim_sing
        all_not_in_conv assms face_of_singleton less_irrefl singletonI that)
 qed
 
 lemma polyhedron_interval [iff]: "polyhedron(cbox a b)"
 by (metis polytope_imp_polyhedron polytope_interval)
 
 lemma polyhedron_convex_hull:
   fixes S :: "'a :: euclidean_space set"
   shows "finite S \<Longrightarrow> polyhedron(convex hull S)"
 by (simp add: polytope_convex_hull polytope_imp_polyhedron)
 
 
 subsection\<open>Relative and absolute frontier of a polytope\<close>
 
 lemma rel_boundary_of_convex_hull:
     fixes S :: "'a::euclidean_space set"
     assumes "\<not> affine_dependent S"
       shows "(convex hull S) - rel_interior(convex hull S) = (\<Union>a\<in>S. convex hull (S - {a}))"
 proof -
   have "finite S" by (metis assms aff_independent_finite)
   then consider "card S = 0" | "card S = 1" | "2 \<le> card S" by arith
   then show ?thesis
   proof cases
     case 1 then have "S = {}" by (simp add: \<open>finite S\<close>)
     then show ?thesis by simp
   next
     case 2 show ?thesis
       by (auto intro: card_1_singletonE [OF \<open>card S = 1\<close>])
   next
     case 3
     with assms show ?thesis
       by (auto simp: polyhedron_convex_hull rel_boundary_of_polyhedron facet_of_convex_hull_affine_independent_alt \<open>finite S\<close>)
   qed
 qed
 
 proposition frontier_of_convex_hull:
     fixes S :: "'a::euclidean_space set"
     assumes "card S = Suc (DIM('a))"
       shows "frontier(convex hull S) = \<Union> {convex hull (S - {a}) | a. a \<in> S}"
 proof (cases "affine_dependent S")
   case True
     have [iff]: "finite S"
       using assms using card_infinite by force
     then have ccs: "closed (convex hull S)"
       by (simp add: compact_imp_closed finite_imp_compact_convex_hull)
     { fix x T
       assume "finite T" "T \<subseteq> S" "int (card T) \<le> aff_dim S + 1" "x \<in> convex hull T"
       then have "S \<noteq> T"
         using True \<open>finite S\<close> aff_dim_le_card affine_independent_iff_card by fastforce
       then obtain a where "a \<in> S" "a \<notin> T"
         using \<open>T \<subseteq> S\<close> by blast
       then have "x \<in> (\<Union>a\<in>S. convex hull (S - {a}))"
         using True affine_independent_iff_card [of S]
         apply simp
         apply (metis (no_types, hide_lams) Diff_eq_empty_iff Diff_insert0 \<open>a \<notin> T\<close> \<open>T \<subseteq> S\<close> \<open>x \<in> convex hull T\<close>  hull_mono insert_Diff_single   subsetCE)
         done
     } note * = this
     have 1: "convex hull S \<subseteq> (\<Union> a\<in>S. convex hull (S - {a}))"
       apply (subst caratheodory_aff_dim)
       apply (blast intro: *)
       done
     have 2: "\<Union>((\<lambda>a. convex hull (S - {a})) ` S) \<subseteq> convex hull S"
       by (rule Union_least) (metis (no_types, lifting)  Diff_subset hull_mono imageE)
     show ?thesis using True
       apply (simp add: segment_convex_hull frontier_def)
       using interior_convex_hull_eq_empty [OF assms]
       apply (simp add: closure_closed [OF ccs])
       apply (rule subset_antisym)
       using 1 apply blast
       using 2 apply blast
       done
 next
   case False
   then have "frontier (convex hull S) = (convex hull S) - rel_interior(convex hull S)"
     apply (simp add: rel_boundary_of_convex_hull [symmetric] frontier_def)
     by (metis aff_independent_finite assms closure_convex_hull finite_imp_compact_convex_hull hull_hull interior_convex_hull_eq_empty rel_interior_nonempty_interior)
   also have "... = \<Union>{convex hull (S - {a}) |a. a \<in> S}"
   proof -
     have "convex hull S - rel_interior (convex hull S) = rel_frontier (convex hull S)"
       by (simp add: False aff_independent_finite polyhedron_convex_hull rel_boundary_of_polyhedron rel_frontier_of_polyhedron)
     then show ?thesis
       by (simp add: False rel_frontier_convex_hull_cases)
   qed
   finally show ?thesis .
 qed
 
 subsection\<open>Special case of a triangle\<close>
 
 proposition frontier_of_triangle:
     fixes a :: "'a::euclidean_space"
     assumes "DIM('a) = 2"
     shows "frontier(convex hull {a,b,c}) = closed_segment a b \<union> closed_segment b c \<union> closed_segment c a"
           (is "?lhs = ?rhs")
 proof (cases "b = a \<or> c = a \<or> c = b")
   case True then show ?thesis
     by (auto simp: assms segment_convex_hull frontier_def empty_interior_convex_hull insert_commute card_insert_le_m1 hull_inc insert_absorb)
 next
   case False then have [simp]: "card {a, b, c} = Suc (DIM('a))"
     by (simp add: card_insert Set.insert_Diff_if assms)
   show ?thesis
   proof
     show "?lhs \<subseteq> ?rhs"
       using False
       by (force simp: segment_convex_hull frontier_of_convex_hull insert_Diff_if insert_commute split: if_split_asm)
     show "?rhs \<subseteq> ?lhs"
       using False
       apply (simp add: frontier_of_convex_hull segment_convex_hull)
       apply (intro conjI subsetI)
         apply (rule_tac X="convex hull {a,b}" in UnionI; force simp: Set.insert_Diff_if)
        apply (rule_tac X="convex hull {b,c}" in UnionI; force)
       apply (rule_tac X="convex hull {a,c}" in UnionI; force simp: insert_commute Set.insert_Diff_if)
       done
   qed
 qed
 
 corollary inside_of_triangle:
     fixes a :: "'a::euclidean_space"
     assumes "DIM('a) = 2"
     shows "inside (closed_segment a b \<union> closed_segment b c \<union> closed_segment c a) = interior(convex hull {a,b,c})"
 by (metis assms frontier_of_triangle bounded_empty bounded_insert convex_convex_hull inside_frontier_eq_interior bounded_convex_hull)
 
 corollary interior_of_triangle:
     fixes a :: "'a::euclidean_space"
     assumes "DIM('a) = 2"
     shows "interior(convex hull {a,b,c}) =
            convex hull {a,b,c} - (closed_segment a b \<union> closed_segment b c \<union> closed_segment c a)"
   using interior_subset
   by (force simp: frontier_of_triangle [OF assms, symmetric] frontier_def Diff_Diff_Int)
 
 subsection\<open>Subdividing a cell complex\<close>
 
 lemma subdivide_interval:
   fixes x::real
   assumes "a < \<bar>x - y\<bar>" "0 < a"
   obtains n where "n \<in> \<int>" "x < n * a \<and> n * a < y \<or> y <  n * a \<and> n * a < x"
 proof -
   consider "a + x < y" | "a + y < x"
     using assms by linarith
   then show ?thesis
   proof cases
     case 1
     let ?n = "of_int (floor (x/a)) + 1"
     have x: "x < ?n * a"
       by (meson \<open>0 < a\<close> divide_less_eq floor_eq_iff)
     have "?n * a \<le> a + x"
       apply (simp add: algebra_simps)
       by (metis \<open>0 < a\<close> floor_correct less_irrefl nonzero_mult_div_cancel_left real_mult_le_cancel_iff2 times_divide_eq_right)
     also have "... < y"
       by (rule 1)
     finally have "?n * a < y" .
     with x show ?thesis
       using Ints_1 Ints_add Ints_of_int that by blast
   next
     case 2
     let ?n = "of_int (floor (y/a)) + 1"
     have y: "y < ?n * a"
       by (meson \<open>0 < a\<close> divide_less_eq floor_eq_iff)
     have "?n * a \<le> a + y"
       apply (simp add: algebra_simps)
       by (metis \<open>0 < a\<close> floor_correct less_irrefl nonzero_mult_div_cancel_left real_mult_le_cancel_iff2 times_divide_eq_right)
     also have "... < x"
       by (rule 2)
     finally have "?n * a < x" .
     then show ?thesis
       using Ints_1 Ints_add Ints_of_int that y by blast
   qed
 qed
 
 lemma cell_subdivision_lemma:
   assumes "finite \<F>"
       and "\<And>X. X \<in> \<F> \<Longrightarrow> polytope X"
       and "\<And>X. X \<in> \<F> \<Longrightarrow> aff_dim X \<le> d"
       and "\<And>X Y. \<lbrakk>X \<in> \<F>; Y \<in> \<F>\<rbrakk> \<Longrightarrow> (X \<inter> Y) face_of X \<and> (X \<inter> Y) face_of Y"
       and "finite I"
     shows "\<exists>\<G>. \<Union>\<G> = \<Union>\<F> \<and>
                  finite \<G> \<and>
                  (\<forall>C \<in> \<G>. \<exists>D. D \<in> \<F> \<and> C \<subseteq> D) \<and>
                  (\<forall>C \<in> \<F>. \<forall>x \<in> C. \<exists>D. D \<in> \<G> \<and> x \<in> D \<and> D \<subseteq> C) \<and>
                  (\<forall>X \<in> \<G>. polytope X) \<and>
                  (\<forall>X \<in> \<G>. aff_dim X \<le> d) \<and>
                  (\<forall>X \<in> \<G>. \<forall>Y \<in> \<G>. X \<inter> Y face_of X \<and> X \<inter> Y face_of Y) \<and>
                  (\<forall>X \<in> \<G>. \<forall>x \<in> X. \<forall>y \<in> X. \<forall>a b.
                           (a,b) \<in> I \<longrightarrow> a \<bullet> x \<le> b \<and> a \<bullet> y \<le> b \<or>
                                         a \<bullet> x \<ge> b \<and> a \<bullet> y \<ge> b)"
   using \<open>finite I\<close>
 proof induction
   case empty
   then show ?case
     by (rule_tac x="\<F>" in exI) (auto simp: assms)
 next
   case (insert ab I)
   then obtain \<G> where eq: "\<Union>\<G> = \<Union>\<F>" and "finite \<G>"
                    and sub1: "\<And>C. C \<in> \<G> \<Longrightarrow> \<exists>D. D \<in> \<F> \<and> C \<subseteq> D"
                    and sub2: "\<And>C x. C \<in> \<F> \<and> x \<in> C \<Longrightarrow> \<exists>D. D \<in> \<G> \<and> x \<in> D \<and> D \<subseteq> C"
                    and poly: "\<And>X. X \<in> \<G> \<Longrightarrow> polytope X"
                    and aff: "\<And>X. X \<in> \<G> \<Longrightarrow> aff_dim X \<le> d"
                    and face: "\<And>X Y. \<lbrakk>X \<in> \<G>; Y \<in> \<G>\<rbrakk> \<Longrightarrow> X \<inter> Y face_of X \<and> X \<inter> Y face_of Y"
                    and I: "\<And>X x y a b.  \<lbrakk>X \<in> \<G>; x \<in> X; y \<in> X; (a,b) \<in> I\<rbrakk> \<Longrightarrow>
                                     a \<bullet> x \<le> b \<and> a \<bullet> y \<le> b \<or> a \<bullet> x \<ge> b \<and> a \<bullet> y \<ge> b"
     by (auto simp: that)
   obtain a b where "ab = (a,b)"
     by fastforce
   let ?\<G> = "(\<lambda>X. X \<inter> {x. a \<bullet> x \<le> b}) ` \<G> \<union> (\<lambda>X. X \<inter> {x. a \<bullet> x \<ge> b}) ` \<G>"
   have eqInt: "(S \<inter> Collect P) \<inter> (T \<inter> Collect Q) = (S \<inter> T) \<inter> (Collect P \<inter> Collect Q)" for S T::"'a set" and P Q
     by blast
   show ?case
   proof (intro conjI exI)
     show "\<Union>?\<G> = \<Union>\<F>"
       by (force simp: eq [symmetric])
     show "finite ?\<G>"
       using \<open>finite \<G>\<close> by force
     show "\<forall>X \<in> ?\<G>. polytope X"
       by (force simp: poly polytope_Int_polyhedron polyhedron_halfspace_le polyhedron_halfspace_ge)
     show "\<forall>X \<in> ?\<G>. aff_dim X \<le> d"
       by (auto; metis order_trans aff aff_dim_subset inf_le1)
     show "\<forall>X \<in> ?\<G>. \<forall>x \<in> X. \<forall>y \<in> X. \<forall>a b.
                           (a,b) \<in> insert ab I \<longrightarrow> a \<bullet> x \<le> b \<and> a \<bullet> y \<le> b \<or>
                                                   a \<bullet> x \<ge> b \<and> a \<bullet> y \<ge> b"
       using \<open>ab = (a, b)\<close> I by fastforce
     show "\<forall>X \<in> ?\<G>. \<forall>Y \<in> ?\<G>. X \<inter> Y face_of X \<and> X \<inter> Y face_of Y"
       by (auto simp: eqInt halfspace_Int_eq face_of_Int_Int face face_of_halfspace_le face_of_halfspace_ge)
     show "\<forall>C \<in> ?\<G>. \<exists>D. D \<in> \<F> \<and> C \<subseteq> D"
       using sub1 by force
     show "\<forall>C\<in>\<F>. \<forall>x\<in>C. \<exists>D. D \<in> ?\<G> \<and> x \<in> D \<and> D \<subseteq> C"
     proof (intro ballI)
       fix C z
       assume "C \<in> \<F>" "z \<in> C"
       with sub2 obtain D where D: "D \<in> \<G>" "z \<in> D" "D \<subseteq> C" by blast
       have "D \<in> \<G> \<and> z \<in> D \<inter> {x. a \<bullet> x \<le> b} \<and> D \<inter> {x. a \<bullet> x \<le> b} \<subseteq> C \<or>
             D \<in> \<G> \<and> z \<in> D \<inter> {x. a \<bullet> x \<ge> b} \<and> D \<inter> {x. a \<bullet> x \<ge> b} \<subseteq> C"
         using linorder_class.linear [of "a \<bullet> z" b] D by blast
       then show "\<exists>D. D \<in> ?\<G> \<and> z \<in> D \<and> D \<subseteq> C"
         by blast
     qed
   qed
 qed
 
 
 proposition cell_complex_subdivision_exists:
   fixes \<F> :: "'a::euclidean_space set set"
   assumes "0 < e" "finite \<F>"
       and poly: "\<And>X. X \<in> \<F> \<Longrightarrow> polytope X"
       and aff: "\<And>X. X \<in> \<F> \<Longrightarrow> aff_dim X \<le> d"
       and face: "\<And>X Y. \<lbrakk>X \<in> \<F>; Y \<in> \<F>\<rbrakk> \<Longrightarrow> X \<inter> Y face_of X \<and> X \<inter> Y face_of Y"
   obtains "\<F>'" where "finite \<F>'" "\<Union>\<F>' = \<Union>\<F>" "\<And>X. X \<in> \<F>' \<Longrightarrow> diameter X < e"
                 "\<And>X. X \<in> \<F>' \<Longrightarrow> polytope X" "\<And>X. X \<in> \<F>' \<Longrightarrow> aff_dim X \<le> d"
                 "\<And>X Y. \<lbrakk>X \<in> \<F>'; Y \<in> \<F>'\<rbrakk> \<Longrightarrow> X \<inter> Y face_of X \<and> X \<inter> Y face_of Y"
                 "\<And>C. C \<in> \<F>' \<Longrightarrow> \<exists>D. D \<in> \<F> \<and> C \<subseteq> D"
                 "\<And>C x. C \<in> \<F> \<and> x \<in> C \<Longrightarrow> \<exists>D. D \<in> \<F>' \<and> x \<in> D \<and> D \<subseteq> C"
 proof -
   have "bounded(\<Union>\<F>)"
     by (simp add: \<open>finite \<F>\<close> poly bounded_Union polytope_imp_bounded)
   then obtain B where "B > 0" and B: "\<And>x. x \<in> \<Union>\<F> \<Longrightarrow> norm x < B"
     by (meson bounded_pos_less)
   define C where "C \<equiv> {z \<in> \<int>. \<bar>z * e / 2 / real DIM('a)\<bar> \<le> B}"
   define I where "I \<equiv> \<Union>i \<in> Basis. \<Union>j \<in> C. { (i::'a, j * e / 2 / DIM('a)) }"
   have "finite C"
     using finite_int_segment [of "-B / (e / 2 / DIM('a))" "B / (e / 2 / DIM('a))"]
     apply (simp add: C_def)
     apply (erule rev_finite_subset)
     using \<open>0 < e\<close>
     apply (auto simp: field_split_simps)
     done
   then have "finite I"
     by (simp add: I_def)
   obtain \<F>' where eq: "\<Union>\<F>' = \<Union>\<F>" and "finite \<F>'"
               and poly: "\<And>X. X \<in> \<F>' \<Longrightarrow> polytope X"
               and aff: "\<And>X. X \<in> \<F>' \<Longrightarrow> aff_dim X \<le> d"
               and face: "\<And>X Y. \<lbrakk>X \<in> \<F>'; Y \<in> \<F>'\<rbrakk> \<Longrightarrow> X \<inter> Y face_of X \<and> X \<inter> Y face_of Y"
               and I: "\<And>X x y a b.  \<lbrakk>X \<in> \<F>'; x \<in> X; y \<in> X; (a,b) \<in> I\<rbrakk> \<Longrightarrow>
                                      a \<bullet> x \<le> b \<and> a \<bullet> y \<le> b \<or> a \<bullet> x \<ge> b \<and> a \<bullet> y \<ge> b"
               and sub1: "\<And>C. C \<in> \<F>' \<Longrightarrow> \<exists>D. D \<in> \<F> \<and> C \<subseteq> D"
               and sub2: "\<And>C x. C \<in> \<F> \<and> x \<in> C \<Longrightarrow> \<exists>D. D \<in> \<F>' \<and> x \<in> D \<and> D \<subseteq> C"
     apply (rule exE [OF cell_subdivision_lemma])
     using assms \<open>finite I\<close> apply auto
     done
   show ?thesis
   proof (rule_tac \<F>'="\<F>'" in that)
     show "diameter X < e" if "X \<in> \<F>'" for X
     proof -
       have "diameter X \<le> e/2"
       proof (rule diameter_le)
         show "norm (x - y) \<le> e / 2" if "x \<in> X" "y \<in> X" for x y
         proof -
           have "norm x < B" "norm y < B"
             using B \<open>X \<in> \<F>'\<close> eq that by fastforce+
           have "norm (x - y) \<le> (\<Sum>b\<in>Basis. \<bar>(x-y) \<bullet> b\<bar>)"
             by (rule norm_le_l1)
           also have "... \<le> of_nat (DIM('a)) * (e / 2 / DIM('a))"
           proof (rule sum_bounded_above)
             fix i::'a
             assume "i \<in> Basis"
             then have I': "\<And>z b. \<lbrakk>z \<in> C; b = z * e / (2 * real DIM('a))\<rbrakk> \<Longrightarrow> i \<bullet> x \<le> b \<and> i \<bullet> y \<le> b \<or> i \<bullet> x \<ge> b \<and> i \<bullet> y \<ge> b"
               using I \<open>X \<in> \<F>'\<close> that
               by (fastforce simp: I_def)
             show "\<bar>(x - y) \<bullet> i\<bar> \<le> e / 2 / real DIM('a)"
             proof (rule ccontr)
               assume "\<not> \<bar>(x - y) \<bullet> i\<bar> \<le> e / 2 / real DIM('a)"
               then have xyi: "\<bar>i \<bullet> x - i \<bullet> y\<bar> > e / 2 / real DIM('a)"
                 by (simp add: inner_commute inner_diff_right)
               obtain n where "n \<in> \<int>" and n: "i \<bullet> x < n * (e / 2 / real DIM('a)) \<and> n * (e / 2 / real DIM('a)) < i \<bullet> y \<or> i \<bullet> y < n * (e / 2 / real DIM('a)) \<and> n * (e / 2 / real DIM('a)) < i \<bullet> x"
                 using subdivide_interval [OF xyi] DIM_positive \<open>0 < e\<close>
                 by (auto simp: zero_less_divide_iff)
               have "\<bar>i \<bullet> x\<bar> < B"
                 by (metis \<open>i \<in> Basis\<close> \<open>norm x < B\<close> inner_commute norm_bound_Basis_lt)
               have "\<bar>i \<bullet> y\<bar> < B"
                 by (metis \<open>i \<in> Basis\<close> \<open>norm y < B\<close> inner_commute norm_bound_Basis_lt)
               have *: "\<bar>n * e\<bar> \<le> B * (2 * real DIM('a))"
                       if "\<bar>ix\<bar> < B" "\<bar>iy\<bar> < B"
                          and ix: "ix * (2 * real DIM('a)) < n * e"
                          and iy: "n * e < iy * (2 * real DIM('a))" for ix iy
               proof (rule abs_leI)
                 have "iy * (2 * real DIM('a)) \<le> B * (2 * real DIM('a))"
                   by (rule mult_right_mono) (use \<open>\<bar>iy\<bar> < B\<close> in linarith)+
                 then show "n * e \<le> B * (2 * real DIM('a))"
                   using iy by linarith
               next
                 have "- ix * (2 * real DIM('a)) \<le> B * (2 * real DIM('a))"
                   by (rule mult_right_mono) (use \<open>\<bar>ix\<bar> < B\<close> in linarith)+
                 then show "- (n * e) \<le> B * (2 * real DIM('a))"
                   using ix by linarith
               qed
               have "n \<in> C"
                 using \<open>n \<in> \<int>\<close> n  by (auto simp: C_def divide_simps intro: * \<open>\<bar>i \<bullet> x\<bar> < B\<close> \<open>\<bar>i \<bullet> y\<bar> < B\<close>)
               show False
                 using  I' [OF \<open>n \<in> C\<close> refl] n  by auto
             qed
           qed
           also have "... = e / 2"
             by simp
           finally show ?thesis .
         qed
       qed (use \<open>0 < e\<close> in force)
       also have "... < e"
         by (simp add: \<open>0 < e\<close>)
       finally show ?thesis .
     qed
   qed (auto simp: eq poly aff face sub1 sub2 \<open>finite \<F>'\<close>)
 qed
 
 
 subsection\<open>Simplexes\<close>
 
 text\<open>The notion of n-simplex for integer \<^term>\<open>n \<ge> -1\<close>\<close>
 
 definition\<^marker>\<open>tag important\<close> simplex :: "int \<Rightarrow> 'a::euclidean_space set \<Rightarrow> bool" (infix "simplex" 50)
   where "n simplex S \<equiv> \<exists>C. \<not> affine_dependent C \<and> int(card C) = n + 1 \<and> S = convex hull C"
 
 lemma simplex:
     "n simplex S \<longleftrightarrow> (\<exists>C. finite C \<and>
                        \<not> affine_dependent C \<and>
                        int(card C) = n + 1 \<and>
                        S = convex hull C)"
   by (auto simp add: simplex_def intro: aff_independent_finite)
 
 lemma simplex_convex_hull:
    "\<not> affine_dependent C \<and> int(card C) = n + 1 \<Longrightarrow> n simplex (convex hull C)"
   by (auto simp add: simplex_def)
 
 lemma convex_simplex: "n simplex S \<Longrightarrow> convex S"
   by (metis convex_convex_hull simplex_def)
 
 lemma compact_simplex: "n simplex S \<Longrightarrow> compact S"
   unfolding simplex
   using finite_imp_compact_convex_hull by blast
 
 lemma closed_simplex: "n simplex S \<Longrightarrow> closed S"
   by (simp add: compact_imp_closed compact_simplex)
 
 lemma simplex_imp_polytope:
    "n simplex S \<Longrightarrow> polytope S"
   unfolding simplex_def polytope_def
   using aff_independent_finite by blast
 
 lemma simplex_imp_polyhedron:
    "n simplex S \<Longrightarrow> polyhedron S"
   by (simp add: polytope_imp_polyhedron simplex_imp_polytope)
 
 lemma simplex_dim_ge: "n simplex S \<Longrightarrow> -1 \<le> n"
   by (metis (no_types, hide_lams) aff_dim_geq affine_independent_iff_card diff_add_cancel diff_diff_eq2 simplex_def)
 
 lemma simplex_empty [simp]: "n simplex {} \<longleftrightarrow> n = -1"
 proof
   assume "n simplex {}"
   then show "n = -1"
     unfolding simplex by (metis card_empty convex_hull_eq_empty diff_0 diff_eq_eq of_nat_0)
 next
   assume "n = -1" then show "n simplex {}"
     by (fastforce simp: simplex)
 qed
 
 lemma simplex_minus_1 [simp]: "-1 simplex S \<longleftrightarrow> S = {}"
   by (metis simplex cancel_comm_monoid_add_class.diff_cancel card_0_eq diff_minus_eq_add of_nat_eq_0_iff simplex_empty)
 
 
 lemma aff_dim_simplex:
    "n simplex S \<Longrightarrow> aff_dim S = n"
   by (metis simplex add.commute add_diff_cancel_left' aff_dim_convex_hull affine_independent_iff_card)
 
 lemma zero_simplex_sing: "0 simplex {a}"
   apply (simp add: simplex_def)
   by (metis affine_independent_1 card_empty card_insert_disjoint convex_hull_singleton empty_iff finite.emptyI)
 
 lemma simplex_sing [simp]: "n simplex {a} \<longleftrightarrow> n = 0"
   using aff_dim_simplex aff_dim_sing zero_simplex_sing by blast
 
 lemma simplex_zero: "0 simplex S \<longleftrightarrow> (\<exists>a. S = {a})"
 apply (auto simp: )
   using aff_dim_eq_0 aff_dim_simplex by blast
 
 lemma one_simplex_segment: "a \<noteq> b \<Longrightarrow> 1 simplex closed_segment a b"
   apply (simp add: simplex_def)
   apply (rule_tac x="{a,b}" in exI)
   apply (auto simp: segment_convex_hull)
   done
 
 lemma simplex_segment_cases:
    "(if a = b then 0 else 1) simplex closed_segment a b"
   by (auto simp: one_simplex_segment)
 
 lemma simplex_segment:
    "\<exists>n. n simplex closed_segment a b"
   using simplex_segment_cases by metis
 
 lemma polytope_lowdim_imp_simplex:
   assumes "polytope P" "aff_dim P \<le> 1"
   obtains n where "n simplex P"
 proof (cases "P = {}")
   case True
   then show ?thesis
     by (simp add: that)
 next
   case False
   then show ?thesis
     by (metis assms compact_convex_collinear_segment collinear_aff_dim polytope_imp_compact polytope_imp_convex simplex_segment_cases that)
 qed
 
 lemma simplex_insert_dimplus1:
   fixes n::int
   assumes "n simplex S" and a: "a \<notin> affine hull S"
   shows "(n+1) simplex (convex hull (insert a S))"
 proof -
   obtain C where C: "finite C" "\<not> affine_dependent C" "int(card C) = n+1" and S: "S = convex hull C"
     using assms unfolding simplex by force
   show ?thesis
     unfolding simplex
   proof (intro exI conjI)
       have "aff_dim S = n"
         using aff_dim_simplex assms(1) by blast
       moreover have "a \<notin> affine hull C"
         using S a affine_hull_convex_hull by blast
       moreover have "a \<notin> C"
           using S a hull_inc by fastforce
       ultimately show "\<not> affine_dependent (insert a C)"
         by (simp add: C S aff_dim_convex_hull aff_dim_insert affine_independent_iff_card)
   next
     have "a \<notin> C"
       using S a hull_inc by fastforce
     then show "int (card (insert a C)) = n + 1 + 1"
       by (simp add: C)
   next
     show "convex hull insert a S = convex hull (insert a C)"
       by (simp add: S convex_hull_insert_segments)
   qed (use C in auto)
 qed
 
 subsection \<open>Simplicial complexes and triangulations\<close>
 
 definition\<^marker>\<open>tag important\<close> simplicial_complex where
  "simplicial_complex \<C> \<equiv>
         finite \<C> \<and>
         (\<forall>S \<in> \<C>. \<exists>n. n simplex S) \<and>
         (\<forall>F S. S \<in> \<C> \<and> F face_of S \<longrightarrow> F \<in> \<C>) \<and>
         (\<forall>S S'. S \<in> \<C> \<and> S' \<in> \<C>
                 \<longrightarrow> (S \<inter> S') face_of S \<and> (S \<inter> S') face_of S')"
 
 definition\<^marker>\<open>tag important\<close> triangulation where
  "triangulation \<T> \<equiv>
         finite \<T> \<and>
         (\<forall>T \<in> \<T>. \<exists>n. n simplex T) \<and>
         (\<forall>T T'. T \<in> \<T> \<and> T' \<in> \<T>
                 \<longrightarrow> (T \<inter> T') face_of T \<and> (T \<inter> T') face_of T')"
 
 
 subsection\<open>Refining a cell complex to a simplicial complex\<close>
 
 proposition convex_hull_insert_Int_eq:
   fixes z :: "'a :: euclidean_space"
   assumes z: "z \<in> rel_interior S"
       and T: "T \<subseteq> rel_frontier S"
       and U: "U \<subseteq> rel_frontier S"
       and "convex S" "convex T" "convex U"
   shows "convex hull (insert z T) \<inter> convex hull (insert z U) = convex hull (insert z (T \<inter> U))"
     (is "?lhs = ?rhs")
 proof
   show "?lhs \<subseteq> ?rhs"
   proof (cases "T={} \<or> U={}")
     case True then show ?thesis by auto
   next
     case False
     then have "T \<noteq> {}" "U \<noteq> {}" by auto
     have TU: "convex (T \<inter> U)"
       by (simp add: \<open>convex T\<close> \<open>convex U\<close> convex_Int)
     have "(\<Union>x\<in>T. closed_segment z x) \<inter> (\<Union>x\<in>U. closed_segment z x)
           \<subseteq> (if T \<inter> U = {} then {z} else \<Union>((closed_segment z) ` (T \<inter> U)))" (is "_ \<subseteq> ?IF")
     proof clarify
       fix x t u
       assume xt: "x \<in> closed_segment z t"
         and xu: "x \<in> closed_segment z u"
         and "t \<in> T" "u \<in> U"
       then have ne: "t \<noteq> z" "u \<noteq> z"
         using T U z unfolding rel_frontier_def by blast+
       show "x \<in> ?IF"
       proof (cases "x = z")
         case True then show ?thesis by auto
       next
         case False
         have t: "t \<in> closure S"
           using T \<open>t \<in> T\<close> rel_frontier_def by auto
         have u: "u \<in> closure S"
           using U \<open>u \<in> U\<close> rel_frontier_def by auto
         show ?thesis
         proof (cases "t = u")
           case True
           then show ?thesis
             using \<open>t \<in> T\<close> \<open>u \<in> U\<close> xt by auto
         next
           case False
           have tnot: "t \<notin> closed_segment u z"
           proof -
             have "t \<in> closure S - rel_interior S"
               using T \<open>t \<in> T\<close> rel_frontier_def by blast
             then have "t \<notin> open_segment z u"
               by (meson DiffD2 rel_interior_closure_convex_segment [OF \<open>convex S\<close> z u] subsetD)
             then show ?thesis
               by (simp add: \<open>t \<noteq> u\<close> \<open>t \<noteq> z\<close> open_segment_commute open_segment_def)
           qed
           moreover have "u \<notin> closed_segment z t"
             using rel_interior_closure_convex_segment [OF \<open>convex S\<close> z t] \<open>u \<in> U\<close> \<open>u \<noteq> z\<close>
               U [unfolded rel_frontier_def] tnot
             by (auto simp: closed_segment_eq_open)
           ultimately
           have "\<not>(between (t,u) z | between (u,z) t | between (z,t) u)" if "x \<noteq> z"
             using that xt xu
             apply (simp add: between_mem_segment [symmetric])
             by (metis between_commute between_trans_2 between_antisym)
           then have "\<not> collinear {t, z, u}" if "x \<noteq> z"
             by (auto simp: that collinear_between_cases between_commute)
           moreover have "collinear {t, z, x}"
             by (metis closed_segment_commute collinear_2 collinear_closed_segment collinear_triples ends_in_segment(1) insert_absorb insert_absorb2 xt)
           moreover have "collinear {z, x, u}"
             by (metis closed_segment_commute collinear_2 collinear_closed_segment collinear_triples ends_in_segment(1) insert_absorb insert_absorb2 xu)
           ultimately have False
             using collinear_3_trans [of t z x u] \<open>x \<noteq> z\<close> by blast
           then show ?thesis by metis
         qed
       qed
     qed
     then show ?thesis
       using False \<open>convex T\<close> \<open>convex U\<close> TU
       by (simp add: convex_hull_insert_segments hull_same split: if_split_asm)
   qed
   show "?rhs \<subseteq> ?lhs"
     by (metis inf_greatest hull_mono inf.cobounded1 inf.cobounded2 insert_mono)
 qed
 
 lemma simplicial_subdivision_aux:
   assumes "finite \<M>"
       and "\<And>C. C \<in> \<M> \<Longrightarrow> polytope C"
       and "\<And>C. C \<in> \<M> \<Longrightarrow> aff_dim C \<le> of_nat n"
       and "\<And>C F. \<lbrakk>C \<in> \<M>; F face_of C\<rbrakk> \<Longrightarrow> F \<in> \<M>"
       and "\<And>C1 C2. \<lbrakk>C1 \<in> \<M>; C2 \<in> \<M>\<rbrakk> \<Longrightarrow> C1 \<inter> C2 face_of C1 \<and> C1 \<inter> C2 face_of C2"
     shows "\<exists>\<T>. simplicial_complex \<T> \<and>
                 (\<forall>K \<in> \<T>. aff_dim K \<le> of_nat n) \<and>
                 \<Union>\<T> = \<Union>\<M> \<and>
                 (\<forall>C \<in> \<M>. \<exists>F. finite F \<and> F \<subseteq> \<T> \<and> C = \<Union>F) \<and>
                 (\<forall>K \<in> \<T>. \<exists>C. C \<in> \<M> \<and> K \<subseteq> C)"
   using assms
 proof (induction n arbitrary: \<M> rule: less_induct)
   case (less n)
   then have poly\<M>: "\<And>C. C \<in> \<M> \<Longrightarrow> polytope C"
       and aff\<M>:    "\<And>C. C \<in> \<M> \<Longrightarrow> aff_dim C \<le> of_nat n"
       and face\<M>:   "\<And>C F. \<lbrakk>C \<in> \<M>; F face_of C\<rbrakk> \<Longrightarrow> F \<in> \<M>"
       and intface\<M>: "\<And>C1 C2. \<lbrakk>C1 \<in> \<M>; C2 \<in> \<M>\<rbrakk> \<Longrightarrow> C1 \<inter> C2 face_of C1 \<and> C1 \<inter> C2 face_of C2"
     by metis+
   show ?case
   proof (cases "n \<le> 1")
     case True
     have "\<And>s. \<lbrakk>n \<le> 1; s \<in> \<M>\<rbrakk> \<Longrightarrow> \<exists>m. m simplex s"
       using poly\<M> aff\<M> by (force intro: polytope_lowdim_imp_simplex)
     then show ?thesis
       unfolding simplicial_complex_def
       apply (rule_tac x="\<M>" in exI)
       using True by (auto simp: less.prems)
   next
     case False
     define \<S> where "\<S> \<equiv> {C \<in> \<M>. aff_dim C < n}"
     have "finite \<S>" "\<And>C. C \<in> \<S> \<Longrightarrow> polytope C" "\<And>C. C \<in> \<S> \<Longrightarrow> aff_dim C \<le> int (n - 1)"
          "\<And>C F. \<lbrakk>C \<in> \<S>; F face_of C\<rbrakk> \<Longrightarrow> F \<in> \<S>"
          "\<And>C1 C2. \<lbrakk>C1 \<in> \<S>; C2 \<in> \<S>\<rbrakk>  \<Longrightarrow> C1 \<inter> C2 face_of C1 \<and> C1 \<inter> C2 face_of C2"
       using less.prems
       apply (auto simp: \<S>_def)
       by (metis aff_dim_subset face_of_imp_subset less_le not_le)
     with less.IH [of "n-1" \<S>] False
     obtain \<U> where "simplicial_complex \<U>"
            and aff_dim\<U>: "\<And>K. K \<in> \<U> \<Longrightarrow> aff_dim K \<le> int (n - 1)"
            and        "\<Union>\<U> = \<Union>\<S>"
            and fin\<U>:  "\<And>C. C \<in> \<S> \<Longrightarrow> \<exists>F. finite F \<and> F \<subseteq> \<U> \<and> C = \<Union>F"
            and C\<U>:    "\<And>K. K \<in> \<U> \<Longrightarrow> \<exists>C. C \<in> \<S> \<and> K \<subseteq> C"
       by auto
     then have "finite \<U>"
          and simpl\<U>: "\<And>S. S \<in> \<U> \<Longrightarrow> \<exists>n. n simplex S"
          and face\<U>:  "\<And>F S. \<lbrakk>S \<in> \<U>; F face_of S\<rbrakk> \<Longrightarrow> F \<in> \<U>"
          and faceI\<U>: "\<And>S S'. \<lbrakk>S \<in> \<U>; S' \<in> \<U>\<rbrakk> \<Longrightarrow> (S \<inter> S') face_of S \<and> (S \<inter> S') face_of S'"
       by (auto simp: simplicial_complex_def)
     define \<N> where "\<N> \<equiv> {C \<in> \<M>. aff_dim C = n}"
     have "finite \<N>"
       by (simp add: \<N>_def less.prems(1))
     have poly\<N>: "\<And>C. C \<in> \<N> \<Longrightarrow> polytope C"
       and convex\<N>: "\<And>C. C \<in> \<N> \<Longrightarrow> convex C"
       and closed\<N>: "\<And>C. C \<in> \<N> \<Longrightarrow> closed C"
       by (auto simp: \<N>_def poly\<M> polytope_imp_convex polytope_imp_closed)
     have in_rel_interior: "(SOME z. z \<in> rel_interior C) \<in> rel_interior C" if "C \<in> \<N>" for C
         using that poly\<M> polytope_imp_convex rel_interior_aff_dim some_in_eq by (fastforce simp: \<N>_def)
     have *: "\<exists>T. \<not> affine_dependent T \<and> card T \<le> n \<and> aff_dim K < n \<and> K = convex hull T"
       if "K \<in> \<U>" for K
     proof -
       obtain r where r: "r simplex K"
         using \<open>K \<in> \<U>\<close> simpl\<U> by blast
       have "r = aff_dim K"
         using \<open>r simplex K\<close> aff_dim_simplex by blast
       with r
       show ?thesis
         unfolding simplex_def
         using False \<open>\<And>K. K \<in> \<U> \<Longrightarrow> aff_dim K \<le> int (n - 1)\<close> that by fastforce
     qed
     have ahK_C_disjoint: "affine hull K \<inter> rel_interior C = {}"
       if "C \<in> \<N>" "K \<in> \<U>" "K \<subseteq> rel_frontier C" for C K
     proof -
       have "convex C" "closed C"
         by (auto simp: convex\<N> closed\<N> \<open>C \<in> \<N>\<close>)
       obtain F where F: "F face_of C" and "F \<noteq> C" "K \<subseteq> F"
       proof -
         obtain L where "L \<in> \<S>" "K \<subseteq> L"
           using \<open>K \<in> \<U>\<close> C\<U> by blast
         have "K \<le> rel_frontier C"
           by (simp add: \<open>K \<subseteq> rel_frontier C\<close>)
         also have "... \<le> C"
           by (simp add: \<open>closed C\<close> rel_frontier_def subset_iff)
         finally have "K \<subseteq> C" .
         have "L \<inter> C face_of C"
           using \<N>_def \<S>_def \<open>C \<in> \<N>\<close> \<open>L \<in> \<S>\<close> intface\<M> by auto
         moreover have "L \<inter> C \<noteq> C"
           using \<open>C \<in> \<N>\<close> \<open>L \<in> \<S>\<close>
           apply (clarsimp simp: \<N>_def \<S>_def)
           by (metis aff_dim_subset inf_le1 not_le)
         moreover have "K \<subseteq> L \<inter> C"
           using \<open>C \<in> \<N>\<close> \<open>L \<in> \<S>\<close> \<open>K \<subseteq> C\<close> \<open>K \<subseteq> L\<close>
           by (auto simp: \<N>_def \<S>_def)
         ultimately show ?thesis using that by metis
       qed
       have "affine hull F \<inter> rel_interior C = {}"
         by (rule affine_hull_face_of_disjoint_rel_interior [OF \<open>convex C\<close> F \<open>F \<noteq> C\<close>])
       with hull_mono [OF \<open>K \<subseteq> F\<close>]
       show "affine hull K \<inter> rel_interior C = {}"
         by fastforce
     qed
     let ?\<T> = "(\<Union>C \<in> \<N>. \<Union>K \<in> \<U> \<inter> Pow (rel_frontier C).
                      {convex hull (insert (SOME z. z \<in> rel_interior C) K)})"
     have "\<exists>\<T>. simplicial_complex \<T> \<and>
               (\<forall>K \<in> \<T>. aff_dim K \<le> of_nat n) \<and>
               (\<forall>C \<in> \<M>. \<exists>F. F \<subseteq> \<T> \<and> C = \<Union>F) \<and>
               (\<forall>K \<in> \<T>. \<exists>C. C \<in> \<M> \<and> K \<subseteq> C)"
     proof (rule exI, intro conjI ballI)
       show "simplicial_complex (\<U> \<union> ?\<T>)"
         unfolding simplicial_complex_def
       proof (intro conjI impI ballI allI)
         show "finite (\<U> \<union> ?\<T>)"
           using \<open>finite \<U>\<close> \<open>finite \<N>\<close> by simp
         show "\<exists>n. n simplex S" if "S \<in> \<U> \<union> ?\<T>" for S
           using that ahK_C_disjoint in_rel_interior simpl\<U> simplex_insert_dimplus1 by fastforce
         show "F \<in> \<U> \<union> ?\<T>" if S: "S \<in> \<U> \<union> ?\<T> \<and> F face_of S" for F S
         proof -
           have "F \<in> \<U>" if "S \<in> \<U>"
             using S face\<U> that by blast
           moreover have "F \<in> \<U> \<union> ?\<T>"
             if "F face_of S" "C \<in> \<N>" "K \<in> \<U>" and "K \<subseteq> rel_frontier C"
               and S: "S = convex hull insert (SOME z. z \<in> rel_interior C) K" for C K
           proof -
             let ?z = "SOME z. z \<in> rel_interior C"
             have "?z \<in> rel_interior C"
               by (simp add: in_rel_interior \<open>C \<in> \<N>\<close>)
             moreover
             obtain I where "\<not> affine_dependent I" "card I \<le> n" "aff_dim K < int n" "K = convex hull I"
               using * [OF \<open>K \<in> \<U>\<close>] by auto
             ultimately have "?z \<notin> affine hull I"
               using ahK_C_disjoint affine_hull_convex_hull that by blast
             have "compact I" "finite I"
               by (auto simp: \<open>\<not> affine_dependent I\<close> aff_independent_finite finite_imp_compact)
             moreover have "F face_of convex hull insert ?z I"
               by (metis S \<open>F face_of S\<close> \<open>K = convex hull I\<close> convex_hull_eq_empty convex_hull_insert_segments hull_hull)
             ultimately obtain J where "J \<subseteq> insert ?z I" "F = convex hull J"
               using face_of_convex_hull_subset [of "insert ?z I" F] by auto
             show ?thesis
             proof (cases "?z \<in> J")
               case True
               have "F \<in> (\<Union>K\<in>\<U> \<inter> Pow (rel_frontier C). {convex hull insert ?z K})"
               proof
                 have "convex hull (J - {?z}) face_of K"
                   by (metis True \<open>J \<subseteq> insert ?z I\<close> \<open>K = convex hull I\<close> \<open>\<not> affine_dependent I\<close> face_of_convex_hull_affine_independent subset_insert_iff)
                 then have "convex hull (J - {?z}) \<in> \<U>"
                   by (rule face\<U> [OF \<open>K \<in> \<U>\<close>])
                 moreover
                 have "\<And>x. x \<in> convex hull (J - {?z}) \<Longrightarrow> x \<in> rel_frontier C"
                   by (metis True \<open>J \<subseteq> insert ?z I\<close> \<open>K = convex hull I\<close> subsetD hull_mono subset_insert_iff that(4))
                 ultimately show "convex hull (J - {?z}) \<in> \<U> \<inter> Pow (rel_frontier C)" by auto
                 let ?F = "convex hull insert ?z (convex hull (J - {?z}))"
                 have "F \<subseteq> ?F"
                   apply (clarsimp simp: \<open>F = convex hull J\<close>)
                   by (metis True subsetD hull_mono hull_subset subset_insert_iff)
                 moreover have "?F \<subseteq> F"
                   apply (clarsimp simp: \<open>F = convex hull J\<close>)
                   by (metis (no_types, lifting) True convex_hull_eq_empty convex_hull_insert_segments hull_hull insert_Diff)
                 ultimately
                 show "F \<in> {?F}" by auto
               qed
               with \<open>C\<in>\<N>\<close> show ?thesis by auto
             next
               case False
               then have "F \<in> \<U>"
                 using face_of_convex_hull_affine_independent [OF \<open>\<not> affine_dependent I\<close>]
                 by (metis Int_absorb2 Int_insert_right_if0 \<open>F = convex hull J\<close> \<open>J \<subseteq> insert ?z I\<close> \<open>K = convex hull I\<close> face\<U> inf_le2 \<open>K \<in> \<U>\<close>)
               then show "F \<in> \<U> \<union> ?\<T>"
                 by blast
             qed
           qed
           ultimately show ?thesis
             using that by auto
         qed
         have "(S \<inter> S' face_of S) \<and> (S \<inter> S' face_of S')"
           if "S \<in> \<U> \<union> ?\<T>" "S' \<in> \<U> \<union> ?\<T>" for S S'
         proof -
           have symmy: "\<lbrakk>\<And>X Y. R X Y \<Longrightarrow> R Y X;
                         \<And>X Y. \<lbrakk>X \<in> \<U>; Y \<in> \<U>\<rbrakk> \<Longrightarrow> R X Y;
                         \<And>X Y. \<lbrakk>X \<in> \<U>; Y \<in> ?\<T>\<rbrakk> \<Longrightarrow> R X Y;
                         \<And>X Y. \<lbrakk>X \<in> ?\<T>; Y \<in> ?\<T>\<rbrakk> \<Longrightarrow> R X Y\<rbrakk> \<Longrightarrow> R S S'" for R
             using that by (metis (no_types, lifting) Un_iff)
           show ?thesis
           proof (rule symmy)
             show "Y \<inter> X face_of Y \<and> Y \<inter> X face_of X"
               if "X \<inter> Y face_of X \<and> X \<inter> Y face_of Y" for X Y :: "'a set"
               by (simp add: inf_commute that)
           next
             show "X \<inter> Y face_of X \<and> X \<inter> Y face_of Y"
               if "X \<in> \<U>" and "Y \<in> \<U>" for X Y
               by (simp add: faceI\<U> that)
           next
             show "X \<inter> Y face_of X \<and> X \<inter> Y face_of Y"
               if XY: "X \<in> \<U>" "Y \<in> ?\<T>" for X Y
             proof -
               obtain C K
                 where "C \<in> \<N>" "K \<in> \<U>" "K \<subseteq> rel_frontier C"
                 and Y: "Y = convex hull insert (SOME z. z \<in> rel_interior C) K"
                 using XY by blast
               have "convex C"
                 by (simp add: \<open>C \<in> \<N>\<close> convex\<N>)
               have "K \<subseteq> C"
                 by (metis DiffE \<open>C \<in> \<N>\<close> \<open>K \<subseteq> rel_frontier C\<close> closed\<N> closure_closed rel_frontier_def subset_iff)
               let ?z = "(SOME z. z \<in> rel_interior C)"
               have z: "?z \<in> rel_interior C"
                 using \<open>C \<in> \<N>\<close> in_rel_interior by blast
               obtain D where "D \<in> \<S>" "X \<subseteq> D"
                 using C\<U> \<open>X \<in> \<U>\<close> by blast
               have "D \<inter> rel_interior C = (C \<inter> D) \<inter> rel_interior C"
                 using rel_interior_subset by blast
               also have "(C \<inter> D) \<inter> rel_interior C = {}"
               proof (rule face_of_disjoint_rel_interior)
                 show "C \<inter> D face_of C"
                   using \<N>_def \<S>_def \<open>C \<in> \<N>\<close> \<open>D \<in> \<S>\<close> intface\<M> by blast
                 show "C \<inter> D \<noteq> C"
                   by (metis (mono_tags, lifting) Int_lower2 \<N>_def \<S>_def \<open>C \<in> \<N>\<close> \<open>D \<in> \<S>\<close> aff_dim_subset mem_Collect_eq not_le)
               qed
               finally have DC: "D \<inter> rel_interior C = {}" .
               have eq: "X \<inter> convex hull (insert ?z K) = X \<inter> convex hull K"
                 apply (rule Int_convex_hull_insert_rel_exterior [OF \<open>convex C\<close> \<open>K \<subseteq> C\<close> z])
                 using DC by (meson \<open>X \<subseteq> D\<close> disjnt_def disjnt_subset1)
               obtain I where I: "\<not> affine_dependent I"
                          and Keq: "K = convex hull I" and [simp]: "convex hull K = K"
                 using "*" \<open>K \<in> \<U>\<close> by force
               then have "?z \<notin> affine hull I"
                 using ahK_C_disjoint \<open>C \<in> \<N>\<close> \<open>K \<in> \<U>\<close> \<open>K \<subseteq> rel_frontier C\<close> affine_hull_convex_hull z by blast
               have "X \<inter> K face_of K"
                 by (simp add: \<open>K \<in> \<U>\<close> faceI\<U> \<open>X \<in> \<U>\<close>)
               also have "... face_of convex hull insert ?z K"
                 by (metis I Keq \<open>?z \<notin> affine hull I\<close> aff_independent_finite convex_convex_hull face_of_convex_hull_insert face_of_refl hull_insert)
               finally have "X \<inter> K face_of convex hull insert ?z K" .
               then show ?thesis
                 using "*" \<open>K \<in> \<U>\<close> faceI\<U> that(1) by (fastforce simp add: Y eq)
             qed
           next
             show "X \<inter> Y face_of X \<and> X \<inter> Y face_of Y"
               if XY: "X \<in> ?\<T>" "Y \<in> ?\<T>" for X Y
             proof -
               obtain C K D L
                 where "C \<in> \<N>" "K \<in> \<U>" "K \<subseteq> rel_frontier C"
                 and X: "X = convex hull insert (SOME z. z \<in> rel_interior C) K"
                 and "D \<in> \<N>" "L \<in> \<U>" "L \<subseteq> rel_frontier D"
                 and Y: "Y = convex hull insert (SOME z. z \<in> rel_interior D) L"
                 using XY by blast
               let ?z = "(SOME z. z \<in> rel_interior C)"
               have z: "?z \<in> rel_interior C"
                 using \<open>C \<in> \<N>\<close> in_rel_interior by blast
               have "convex C"
                 by (simp add: \<open>C \<in> \<N>\<close> convex\<N>)
               have "convex K"
                 using "*" \<open>K \<in> \<U>\<close> by blast
               have "convex L"
                 by (meson \<open>L \<in> \<U>\<close> convex_simplex simpl\<U>)
               show ?thesis
               proof (cases "D=C")
                 case True
                 then have "L \<subseteq> rel_frontier C"
                   using \<open>L \<subseteq> rel_frontier D\<close> by auto
                 show ?thesis
                   apply (simp add: X Y True)
                   apply (simp add: convex_hull_insert_Int_eq [OF z] \<open>K \<subseteq> rel_frontier C\<close> \<open>L \<subseteq> rel_frontier C\<close> \<open>convex C\<close> \<open>convex K\<close> \<open>convex L\<close>)
                   using face_of_polytope_insert2
                   by (metis "*" IntI \<open>C \<in> \<N>\<close> \<open>K \<in> \<U>\<close> \<open>L \<in> \<U>\<close>\<open>K \<subseteq> rel_frontier C\<close> \<open>L \<subseteq> rel_frontier C\<close> aff_independent_finite ahK_C_disjoint empty_iff faceI\<U> polytope_convex_hull z)
               next
                 case False
                 have "convex D"
                   by (simp add: \<open>D \<in> \<N>\<close> convex\<N>)
                 have "K \<subseteq> C"
                   by (metis DiffE \<open>C \<in> \<N>\<close> \<open>K \<subseteq> rel_frontier C\<close> closed\<N> closure_closed rel_frontier_def subset_eq)
                 have "L \<subseteq> D"
                   by (metis DiffE \<open>D \<in> \<N>\<close> \<open>L \<subseteq> rel_frontier D\<close> closed\<N> closure_closed rel_frontier_def subset_eq)
                 let ?w = "(SOME w. w \<in> rel_interior D)"
                 have w: "?w \<in> rel_interior D"
                   using \<open>D \<in> \<N>\<close> in_rel_interior by blast
                 have "C \<inter> rel_interior D = (D \<inter> C) \<inter> rel_interior D"
                   using rel_interior_subset by blast
                 also have "(D \<inter> C) \<inter> rel_interior D = {}"
                 proof (rule face_of_disjoint_rel_interior)
                   show "D \<inter> C face_of D"
                     using \<N>_def \<open>C \<in> \<N>\<close> \<open>D \<in> \<N>\<close> intface\<M> by blast
                   have "D \<in> \<M> \<and> aff_dim D = int n"
                     using \<N>_def \<open>D \<in> \<N>\<close> by blast
                   moreover have "C \<in> \<M> \<and> aff_dim C = int n"
                     using \<N>_def \<open>C \<in> \<N>\<close> by blast
                   ultimately show "D \<inter> C \<noteq> D"
                     by (metis False face_of_aff_dim_lt inf.idem inf_le1 intface\<M> not_le poly\<M> polytope_imp_convex)
                 qed
                 finally have CD: "C \<inter> (rel_interior D) = {}" .
                 have zKC: "(convex hull insert ?z K) \<subseteq> C"
                   by (metis DiffE \<open>C \<in> \<N>\<close> \<open>K \<subseteq> rel_frontier C\<close> closed\<N> closure_closed convex\<N> hull_minimal insert_subset rel_frontier_def rel_interior_subset subset_iff z)
                 have eq: "convex hull (insert ?z K) \<inter> convex hull (insert ?w L) =
                           convex hull (insert ?z K) \<inter> convex hull L"
                   apply (rule Int_convex_hull_insert_rel_exterior [OF \<open>convex D\<close> \<open>L \<subseteq> D\<close> w])
                   using zKC CD apply (force simp: disjnt_def)
                   done
                 have ch_id: "convex hull K = K" "convex hull L = L"
                   using "*" \<open>K \<in> \<U>\<close> \<open>L \<in> \<U>\<close> hull_same by auto
                 have "convex C"
                   by (simp add: \<open>C \<in> \<N>\<close> convex\<N>)
                 have "convex hull (insert ?z K) \<inter> L = L \<inter> convex hull (insert ?z K)"
                   by blast
                 also have "... = convex hull K \<inter> L"
                 proof (subst Int_convex_hull_insert_rel_exterior [OF \<open>convex C\<close> \<open>K \<subseteq> C\<close> z])
                   have "(C \<inter> D) \<inter> rel_interior C = {}"
                   proof (rule face_of_disjoint_rel_interior)
                     show "C \<inter> D face_of C"
                       using \<N>_def \<open>C \<in> \<N>\<close> \<open>D \<in> \<N>\<close> intface\<M> by blast
                     have "D \<in> \<M>" "aff_dim D = int n"
                       using \<N>_def \<open>D \<in> \<N>\<close> by fastforce+
                     moreover have "C \<in> \<M>" "aff_dim C = int n"
                       using \<N>_def \<open>C \<in> \<N>\<close> by fastforce+
                     ultimately have "aff_dim D + - 1 * aff_dim C \<le> 0"
                       by fastforce
                     then have "\<not> C face_of D"
                       using False \<open>convex D\<close> face_of_aff_dim_lt by fastforce
                     show "C \<inter> D \<noteq> C"
                       using \<open>C \<in> \<M>\<close> \<open>D \<in> \<M>\<close> \<open>\<not> C face_of D\<close> intface\<M> by fastforce
                   qed
                   then have "D \<inter> rel_interior C = {}"
                     by (metis inf.absorb_iff2 inf_assoc inf_sup_aci(1) rel_interior_subset)
                   then show "disjnt L (rel_interior C)"
                     by (meson \<open>L \<subseteq> D\<close> disjnt_def disjnt_subset1)
                 next
                   show "L \<inter> convex hull K = convex hull K \<inter> L"
                     by force
                 qed
                 finally have chKL: "convex hull (insert ?z K) \<inter> L = convex hull K \<inter> L" .
                 have "convex hull insert ?z K \<inter> convex hull L face_of K"
                   by (simp add: \<open>K \<in> \<U>\<close> \<open>L \<in> \<U>\<close> ch_id chKL faceI\<U>)
                 also have "... face_of convex hull insert ?z K"
                 proof -
                   obtain I where I: "\<not> affine_dependent I" "K = convex hull I"
                     using * [OF \<open>K \<in> \<U>\<close>] by auto
                   then have "\<And>a. a \<notin> rel_interior C \<or> a \<notin> affine hull I"
                     using ahK_C_disjoint \<open>C \<in> \<N>\<close> \<open>K \<in> \<U>\<close> \<open>K \<subseteq> rel_frontier C\<close> affine_hull_convex_hull by blast
                   then show ?thesis
                     by (metis I affine_independent_insert face_of_convex_hull_affine_independent hull_insert subset_insertI z)
                 qed
                 finally have 1: "convex hull insert ?z K \<inter> convex hull L face_of convex hull insert ?z K" .
                 have "convex hull insert ?z K \<inter> convex hull L face_of L"
                   by (simp add: \<open>K \<in> \<U>\<close> \<open>L \<in> \<U>\<close> ch_id chKL faceI\<U>)
                 also have "... face_of convex hull insert ?w L"
                 proof -
                   obtain I where I: "\<not> affine_dependent I" "L = convex hull I"
                     using * [OF \<open>L \<in> \<U>\<close>] by auto
                   then have "\<And>a. a \<notin> rel_interior D \<or> a \<notin> affine hull I"
                     using \<open>D \<in> \<N>\<close> \<open>L \<in> \<U>\<close> \<open>L \<subseteq> rel_frontier D\<close> affine_hull_convex_hull ahK_C_disjoint by blast
                   then show ?thesis
                     by (metis I aff_independent_finite convex_convex_hull face_of_convex_hull_insert face_of_refl hull_insert w)
                 qed
                 finally have 2: "convex hull insert ?z K \<inter> convex hull L face_of convex hull insert ?w L" .
                 show ?thesis
                   by (simp add: X Y eq 1 2)
               qed
             qed
           qed
         qed
         then
         show "S \<inter> S' face_of S" "S \<inter> S' face_of S'" if "S \<in> \<U> \<union> ?\<T> \<and> S' \<in> \<U> \<union> ?\<T>" for S S'
           using that by auto
       qed
       show "\<exists>F \<subseteq> \<U> \<union> ?\<T>. C = \<Union>F" if "C \<in> \<M>" for C
       proof (cases "C \<in> \<S>")
         case True
         then show ?thesis
           by (meson UnCI fin\<U> subsetD subsetI)
       next
         case False
         then have "C \<in> \<N>"
           by (simp add: \<N>_def \<S>_def aff\<M> less_le that)
         let ?z = "SOME z. z \<in> rel_interior C"
         have z: "?z \<in> rel_interior C"
           using \<open>C \<in> \<N>\<close> in_rel_interior by blast
         let ?F = "\<Union>K \<in> \<U> \<inter> Pow (rel_frontier C). {convex hull (insert ?z K)}"
         have "?F \<subseteq> ?\<T>"
           using \<open>C \<in> \<N>\<close> by blast
         moreover have "C \<subseteq> \<Union>?F"
         proof
           fix x
           assume "x \<in> C"
           have "convex C"
             using \<open>C \<in> \<N>\<close> convex\<N> by blast
           have "bounded C"
             using \<open>C \<in> \<N>\<close> by (simp add: poly\<M> polytope_imp_bounded that)
           have "polytope C"
             using \<open>C \<in> \<N>\<close> poly\<N> by auto
           have "\<not> (?z = x \<and> C = {?z})"
             using \<open>C \<in> \<N>\<close> aff_dim_sing [of ?z] \<open>\<not> n \<le> 1\<close> by (force simp: \<N>_def)
           then obtain y where y: "y \<in> rel_frontier C" and xzy: "x \<in> closed_segment ?z y"
             and sub: "open_segment ?z y \<subseteq> rel_interior C"
             by (blast intro: segment_to_rel_frontier [OF \<open>convex C\<close> \<open>bounded C\<close> z \<open>x \<in> C\<close>])
           then obtain F where "y \<in> F" "F face_of C" "F \<noteq> C"
             by (auto simp: rel_frontier_of_polyhedron_alt [OF polytope_imp_polyhedron [OF \<open>polytope C\<close>]])
           then obtain \<G> where "finite \<G>" "\<G> \<subseteq> \<U>" "F = \<Union>\<G>"
             by (metis (mono_tags, lifting) \<S>_def \<open>C \<in> \<M>\<close> \<open>convex C\<close> aff\<M> face\<M> face_of_aff_dim_lt fin\<U> le_less_trans mem_Collect_eq not_less)
           then obtain K where "y \<in> K" "K \<in> \<G>"
             using \<open>y \<in> F\<close> by blast
           moreover have x: "x \<in> convex hull {?z,y}"
             using segment_convex_hull xzy by auto
           moreover have "convex hull {?z,y} \<subseteq> convex hull insert ?z K"
             by (metis (full_types) \<open>y \<in> K\<close> hull_mono empty_subsetI insertCI insert_subset)
           moreover have "K \<in> \<U>"
             using \<open>K \<in> \<G>\<close> \<open>\<G> \<subseteq> \<U>\<close> by blast
           moreover have "K \<subseteq> rel_frontier C"
             using \<open>F = \<Union>\<G>\<close> \<open>F \<noteq> C\<close> \<open>F face_of C\<close> \<open>K \<in> \<G>\<close> face_of_subset_rel_frontier by fastforce
           ultimately show "x \<in> \<Union>?F"
             by force
         qed
         moreover
         have "convex hull insert (SOME z. z \<in> rel_interior C) K \<subseteq> C"
           if "K \<in> \<U>" "K \<subseteq> rel_frontier C" for K
         proof (rule hull_minimal)
           show "insert (SOME z. z \<in> rel_interior C) K \<subseteq> C"
             using that \<open>C \<in> \<N>\<close> in_rel_interior rel_interior_subset
             by (force simp: closure_eq rel_frontier_def closed\<N>)
           show "convex C"
             by (simp add: \<open>C \<in> \<N>\<close> convex\<N>)
         qed
         then have "\<Union>?F \<subseteq> C"
           by auto
         ultimately show ?thesis
           by blast
       qed
 
       have "(\<exists>C. C \<in> \<M> \<and> L \<subseteq> C) \<and> aff_dim L \<le> int n"  if "L \<in> \<U> \<union> ?\<T>" for L
         using that
       proof
         assume "L \<in> \<U>"
         then show ?thesis
           using C\<U> \<S>_def "*" by fastforce
       next
         assume "L \<in> ?\<T>"
         then obtain C K where "C \<in> \<N>"
           and L: "L = convex hull insert (SOME z. z \<in> rel_interior C) K"
           and K: "K \<in> \<U>" "K \<subseteq> rel_frontier C"
           by auto
         then have "convex hull C = C"
           by (meson convex\<N> convex_hull_eq)
         then have "convex C"
           by (metis (no_types) convex_convex_hull)
         have "rel_frontier C \<subseteq> C"
           by (metis DiffE closed\<N> \<open>C \<in> \<N>\<close> closure_closed rel_frontier_def subsetI)
         have "K \<subseteq> C"
           using K \<open>rel_frontier C \<subseteq> C\<close> by blast
         have "C \<in> \<M>"
           using \<N>_def \<open>C \<in> \<N>\<close> by auto
         moreover have "L \<subseteq> C"
           using K L \<open>C \<in> \<N>\<close>
           by (metis \<open>K \<subseteq> C\<close> \<open>convex hull C = C\<close> contra_subsetD hull_mono in_rel_interior insert_subset rel_interior_subset)
         ultimately show ?thesis
           using \<open>rel_frontier C \<subseteq> C\<close> \<open>L \<subseteq> C\<close> aff\<M> aff_dim_subset \<open>C \<in> \<M>\<close> dual_order.trans by blast
       qed
       then show "\<exists>C. C \<in> \<M> \<and> L \<subseteq> C" "aff_dim L \<le> int n" if "L \<in> \<U> \<union> ?\<T>" for L
         using that by auto
     qed
     then show ?thesis
       apply (rule ex_forward, safe)
         apply (meson Union_iff subsetCE, fastforce)
       by (meson infinite_super simplicial_complex_def)
   qed
 qed
 
 
 lemma simplicial_subdivision_of_cell_complex_lowdim:
   assumes "finite \<M>"
       and poly: "\<And>C. C \<in> \<M> \<Longrightarrow> polytope C"
       and face: "\<And>C1 C2. \<lbrakk>C1 \<in> \<M>; C2 \<in> \<M>\<rbrakk> \<Longrightarrow> C1 \<inter> C2 face_of C1 \<and> C1 \<inter> C2 face_of C2"
       and aff: "\<And>C. C \<in> \<M> \<Longrightarrow> aff_dim C \<le> d"
   obtains \<T> where "simplicial_complex \<T>" "\<And>K. K \<in> \<T> \<Longrightarrow> aff_dim K \<le> d"
                   "\<Union>\<T> = \<Union>\<M>"
                   "\<And>C. C \<in> \<M> \<Longrightarrow> \<exists>F. finite F \<and> F \<subseteq> \<T> \<and> C = \<Union>F"
                   "\<And>K. K \<in> \<T> \<Longrightarrow> \<exists>C. C \<in> \<M> \<and> K \<subseteq> C"
 proof (cases "d \<ge> 0")
   case True
   then obtain n where n: "d = of_nat n"
     using zero_le_imp_eq_int by blast
   have "\<exists>\<T>. simplicial_complex \<T> \<and>
             (\<forall>K\<in>\<T>. aff_dim K \<le> int n) \<and>
             \<Union>\<T> = \<Union>(\<Union>C\<in>\<M>. {F. F face_of C}) \<and>
             (\<forall>C\<in>\<Union>C\<in>\<M>. {F. F face_of C}.
                 \<exists>F. finite F \<and> F \<subseteq> \<T> \<and> C = \<Union>F) \<and>
             (\<forall>K\<in>\<T>. \<exists>C. C \<in> (\<Union>C\<in>\<M>. {F. F face_of C}) \<and> K \<subseteq> C)"
   proof (rule simplicial_subdivision_aux)
     show "finite (\<Union>C\<in>\<M>. {F. F face_of C})"
       using \<open>finite \<M>\<close> poly polyhedron_eq_finite_faces polytope_imp_polyhedron by fastforce
     show "polytope F" if "F \<in> (\<Union>C\<in>\<M>. {F. F face_of C})" for F
       using poly that face_of_polytope_polytope by blast
     show "aff_dim F \<le> int n" if "F \<in> (\<Union>C\<in>\<M>. {F. F face_of C})" for F
       using that
       by clarify (metis n aff_dim_subset aff face_of_imp_subset order_trans)
     show "F \<in> (\<Union>C\<in>\<M>. {F. F face_of C})"
       if "G \<in> (\<Union>C\<in>\<M>. {F. F face_of C})" and "F face_of G" for F G
       using that face_of_trans by blast
   next
     show "F1 \<inter> F2 face_of F1 \<and> F1 \<inter> F2 face_of F2"
       if "F1 \<in> (\<Union>C\<in>\<M>. {F. F face_of C})" and "F2 \<in> (\<Union>C\<in>\<M>. {F. F face_of C})" for F1 F2
       using that
       by safe (meson face face_of_Int_subface)+
   qed
   moreover
   have "\<Union>(\<Union>C\<in>\<M>. {F. F face_of C}) = \<Union>\<M>"
     using face_of_imp_subset face by blast
   ultimately show ?thesis
     apply clarify
     apply (rule that, assumption+)
        using n apply blast
       apply (simp_all add: poly face_of_refl polytope_imp_convex)
     using face_of_imp_subset by fastforce
 next
   case False
   then have m1: "\<And>C. C \<in> \<M> \<Longrightarrow> aff_dim C = -1"
     by (metis aff aff_dim_empty_eq aff_dim_negative_iff dual_order.trans not_less)
   then have face\<M>: "\<And>F S. \<lbrakk>S \<in> \<M>; F face_of S\<rbrakk> \<Longrightarrow> F \<in> \<M>"
     by (metis aff_dim_empty face_of_empty)
   show ?thesis
   proof
     have "\<And>S. S \<in> \<M> \<Longrightarrow> \<exists>n. n simplex S"
       by (metis (no_types) m1 aff_dim_empty simplex_minus_1)
     then show "simplicial_complex \<M>"
       by (auto simp: simplicial_complex_def \<open>finite \<M>\<close> face intro: face\<M>)
     show "aff_dim K \<le> d" if "K \<in> \<M>" for K
       by (simp add: that aff)
     show "\<exists>F. finite F \<and> F \<subseteq> \<M> \<and> C = \<Union>F" if "C \<in> \<M>" for C
       using \<open>C \<in> \<M>\<close> equals0I by auto
     show "\<exists>C. C \<in> \<M> \<and> K \<subseteq> C" if "K \<in> \<M>" for K
       using \<open>K \<in> \<M>\<close> by blast
   qed auto
 qed
 
 proposition simplicial_subdivision_of_cell_complex:
   assumes "finite \<M>"
       and poly: "\<And>C. C \<in> \<M> \<Longrightarrow> polytope C"
       and face: "\<And>C1 C2. \<lbrakk>C1 \<in> \<M>; C2 \<in> \<M>\<rbrakk> \<Longrightarrow> C1 \<inter> C2 face_of C1 \<and> C1 \<inter> C2 face_of C2"
   obtains \<T> where "simplicial_complex \<T>"
                   "\<Union>\<T> = \<Union>\<M>"
                   "\<And>C. C \<in> \<M> \<Longrightarrow> \<exists>F. finite F \<and> F \<subseteq> \<T> \<and> C = \<Union>F"
                   "\<And>K. K \<in> \<T> \<Longrightarrow> \<exists>C. C \<in> \<M> \<and> K \<subseteq> C"
   by (blast intro: simplicial_subdivision_of_cell_complex_lowdim [OF assms aff_dim_le_DIM])
 
 corollary fine_simplicial_subdivision_of_cell_complex:
   assumes "0 < e" "finite \<M>"
       and poly: "\<And>C. C \<in> \<M> \<Longrightarrow> polytope C"
       and face: "\<And>C1 C2. \<lbrakk>C1 \<in> \<M>; C2 \<in> \<M>\<rbrakk> \<Longrightarrow> C1 \<inter> C2 face_of C1 \<and> C1 \<inter> C2 face_of C2"
   obtains \<T> where "simplicial_complex \<T>"
                   "\<And>K. K \<in> \<T> \<Longrightarrow> diameter K < e"
                   "\<Union>\<T> = \<Union>\<M>"
                   "\<And>C. C \<in> \<M> \<Longrightarrow> \<exists>F. finite F \<and> F \<subseteq> \<T> \<and> C = \<Union>F"
                   "\<And>K. K \<in> \<T> \<Longrightarrow> \<exists>C. C \<in> \<M> \<and> K \<subseteq> C"
 proof -
   obtain \<N> where \<N>: "finite \<N>" "\<Union>\<N> = \<Union>\<M>" 
               and diapoly: "\<And>X. X \<in> \<N> \<Longrightarrow> diameter X < e" "\<And>X. X \<in> \<N> \<Longrightarrow> polytope X"
                and      "\<And>X Y. \<lbrakk>X \<in> \<N>; Y \<in> \<N>\<rbrakk> \<Longrightarrow> X \<inter> Y face_of X \<and> X \<inter> Y face_of Y"
                and \<N>covers: "\<And>C x. C \<in> \<M> \<and> x \<in> C \<Longrightarrow> \<exists>D. D \<in> \<N> \<and> x \<in> D \<and> D \<subseteq> C"
                and \<N>covered: "\<And>C. C \<in> \<N> \<Longrightarrow> \<exists>D. D \<in> \<M> \<and> C \<subseteq> D"
     by (blast intro: cell_complex_subdivision_exists [OF \<open>0 < e\<close> \<open>finite \<M>\<close> poly aff_dim_le_DIM face])
   then obtain \<T> where \<T>: "simplicial_complex \<T>" "\<Union>\<T> = \<Union>\<N>"
                    and \<T>covers: "\<And>C. C \<in> \<N> \<Longrightarrow> \<exists>F. finite F \<and> F \<subseteq> \<T> \<and> C = \<Union>F"
                    and \<T>covered: "\<And>K. K \<in> \<T> \<Longrightarrow> \<exists>C. C \<in> \<N> \<and> K \<subseteq> C"
     using simplicial_subdivision_of_cell_complex [OF \<open>finite \<N>\<close>] by metis
   show ?thesis
   proof
     show "simplicial_complex \<T>"
       by (rule \<T>)
     show "diameter K < e" if "K \<in> \<T>" for K
       by (metis le_less_trans diapoly \<T>covered diameter_subset polytope_imp_bounded that)
     show "\<Union>\<T> = \<Union>\<M>"
       by (simp add: \<N>(2) \<open>\<Union>\<T> = \<Union>\<N>\<close>)
     show "\<exists>F. finite F \<and> F \<subseteq> \<T> \<and> C = \<Union>F" if "C \<in> \<M>" for C
     proof -
       { fix x
         assume "x \<in> C"
         then obtain D where "D \<in> \<T>" "x \<in> D" "D \<subseteq> C"
           using \<N>covers \<open>C \<in> \<M>\<close> \<T>covers by force
         then have "\<exists>X\<in>\<T> \<inter> Pow C. x \<in> X"
           using \<open>D \<in> \<T>\<close> \<open>D \<subseteq> C\<close> \<open>x \<in> D\<close> by blast
       }
       moreover
       have "finite (\<T> \<inter> Pow C)"
         using \<open>simplicial_complex \<T>\<close> simplicial_complex_def by auto
       ultimately show ?thesis
         by (rule_tac x="(\<T> \<inter> Pow C)" in exI) auto
     qed
     show "\<exists>C. C \<in> \<M> \<and> K \<subseteq> C" if "K \<in> \<T>" for K
       by (meson \<N>covered \<T>covered order_trans that)
   qed
 qed
 
 subsection\<open>Some results on cell division with full-dimensional cells only\<close>
 
 lemma convex_Union_fulldim_cells:
   assumes "finite \<S>" and clo: "\<And>C. C \<in> \<S> \<Longrightarrow> closed C" and con: "\<And>C. C \<in> \<S> \<Longrightarrow> convex C"
       and eq: "\<Union>\<S> = U"and  "convex U"
  shows "\<Union>{C \<in> \<S>. aff_dim C = aff_dim U} = U"  (is "?lhs = U")
 proof -
   have "closed U"
     using \<open>finite \<S>\<close> clo eq by blast
   have "?lhs \<subseteq> U"
     using eq by blast
   moreover have "U \<subseteq> ?lhs"
   proof (cases "\<forall>C \<in> \<S>. aff_dim C = aff_dim U")
     case True
     then show ?thesis
       using eq by blast
   next
     case False
     have "closed ?lhs"
       by (simp add: \<open>finite \<S>\<close> clo closed_Union)
     moreover have "U \<subseteq> closure ?lhs"
     proof -
       have "U \<subseteq> closure(\<Inter>{U - C |C. C \<in> \<S> \<and> aff_dim C < aff_dim U})"
       proof (rule Baire [OF \<open>closed U\<close>])
         show "countable {U - C |C. C \<in> \<S> \<and> aff_dim C < aff_dim U}"
           using \<open>finite \<S>\<close> uncountable_infinite by fastforce
         have "\<And>C. C \<in> \<S> \<Longrightarrow> openin (top_of_set U) (U-C)"
           by (metis Sup_upper clo closed_limpt closedin_limpt eq openin_diff openin_subtopology_self)
         then show "openin (top_of_set U) T \<and> U \<subseteq> closure T"
           if "T \<in> {U - C |C. C \<in> \<S> \<and> aff_dim C < aff_dim U}" for T
           using that dense_complement_convex_closed \<open>closed U\<close> \<open>convex U\<close> by auto
       qed
       also have "... \<subseteq> closure ?lhs"
       proof -
         obtain C where "C \<in> \<S>" "aff_dim C < aff_dim U"
           by (metis False Sup_upper aff_dim_subset eq eq_iff not_le)
         have "\<exists>X. X \<in> \<S> \<and> aff_dim X = aff_dim U \<and> x \<in> X"
           if "\<And>V. (\<exists>C. V = U - C \<and> C \<in> \<S> \<and> aff_dim C < aff_dim U) \<Longrightarrow> x \<in> V" for x
         proof -
           have "x \<in> U \<and> x \<in> \<Union>\<S>"
             using \<open>C \<in> \<S>\<close> \<open>aff_dim C < aff_dim U\<close> eq that by blast
           then show ?thesis
             by (metis Diff_iff Sup_upper Union_iff aff_dim_subset dual_order.order_iff_strict eq that)
         qed
         then show ?thesis
           by (auto intro!: closure_mono)
       qed
       finally show ?thesis .
     qed
     ultimately show ?thesis
       using closure_subset_eq by blast
   qed
   ultimately show ?thesis by blast
 qed
 
 proposition fine_triangular_subdivision_of_cell_complex:
   assumes "0 < e" "finite \<M>"
       and poly: "\<And>C. C \<in> \<M> \<Longrightarrow> polytope C"
       and aff: "\<And>C. C \<in> \<M> \<Longrightarrow> aff_dim C = d"
       and face: "\<And>C1 C2. \<lbrakk>C1 \<in> \<M>; C2 \<in> \<M>\<rbrakk> \<Longrightarrow> C1 \<inter> C2 face_of C1 \<and> C1 \<inter> C2 face_of C2"
   obtains \<T> where "triangulation \<T>" "\<And>k. k \<in> \<T> \<Longrightarrow> diameter k < e"
                  "\<And>k. k \<in> \<T> \<Longrightarrow> aff_dim k = d" "\<Union>\<T> = \<Union>\<M>"
                  "\<And>C. C \<in> \<M> \<Longrightarrow> \<exists>f. finite f \<and> f \<subseteq> \<T> \<and> C = \<Union>f"
                  "\<And>k. k \<in> \<T> \<Longrightarrow> \<exists>C. C \<in> \<M> \<and> k \<subseteq> C"
 proof -
   obtain \<T> where "simplicial_complex \<T>"
              and dia\<T>: "\<And>K. K \<in> \<T> \<Longrightarrow> diameter K < e"
              and "\<Union>\<T> = \<Union>\<M>"
              and in\<M>: "\<And>C. C \<in> \<M> \<Longrightarrow> \<exists>F. finite F \<and> F \<subseteq> \<T> \<and> C = \<Union>F"
              and in\<T>: "\<And>K. K \<in> \<T> \<Longrightarrow> \<exists>C. C \<in> \<M> \<and> K \<subseteq> C"
     by (blast intro: fine_simplicial_subdivision_of_cell_complex [OF \<open>e > 0\<close> \<open>finite \<M>\<close> poly face])
   let ?\<T> = "{K \<in> \<T>. aff_dim K = d}"
   show thesis
   proof
     show "triangulation ?\<T>"
       using \<open>simplicial_complex \<T>\<close> by (auto simp: triangulation_def simplicial_complex_def)
     show "diameter L < e" if "L \<in> {K \<in> \<T>. aff_dim K = d}" for L
       using that by (auto simp: dia\<T>)
     show "aff_dim L = d" if "L \<in> {K \<in> \<T>. aff_dim K = d}" for L
       using that by auto
     show "\<exists>F. finite F \<and> F \<subseteq> {K \<in> \<T>. aff_dim K = d} \<and> C = \<Union>F" if "C \<in> \<M>" for C
     proof -
       obtain F where "finite F" "F \<subseteq> \<T>" "C = \<Union>F"
         using in\<M> [OF \<open>C \<in> \<M>\<close>] by auto
       show ?thesis
       proof (intro exI conjI)
         show "finite {K \<in> F. aff_dim K = d}"
           by (simp add: \<open>finite F\<close>)
         show "{K \<in> F. aff_dim K = d} \<subseteq> {K \<in> \<T>. aff_dim K = d}"
           using \<open>F \<subseteq> \<T>\<close> by blast
         have "d = aff_dim C"
           by (simp add: aff that)
         moreover have "\<And>K. K \<in> F \<Longrightarrow> closed K \<and> convex K"
           using \<open>simplicial_complex \<T>\<close> \<open>F \<subseteq> \<T>\<close>
           unfolding simplicial_complex_def by (metis subsetCE \<open>F \<subseteq> \<T>\<close> closed_simplex convex_simplex)
         moreover have "convex (\<Union>F)"
           using \<open>C = \<Union>F\<close> poly polytope_imp_convex that by blast
         ultimately show "C = \<Union>{K \<in> F. aff_dim K = d}"
           by (simp add: convex_Union_fulldim_cells \<open>C = \<Union>F\<close> \<open>finite F\<close>)
       qed
     qed
     then show "\<Union>{K \<in> \<T>. aff_dim K = d} = \<Union>\<M>"
       by auto (meson in\<T> subsetCE)
     show "\<exists>C. C \<in> \<M> \<and> L \<subseteq> C"
       if "L \<in> {K \<in> \<T>. aff_dim K = d}" for L
       using that by (auto simp: in\<T>)
   qed
 qed
 
 end
diff --git a/src/HOL/Analysis/Starlike.thy b/src/HOL/Analysis/Starlike.thy
--- a/src/HOL/Analysis/Starlike.thy
+++ b/src/HOL/Analysis/Starlike.thy
@@ -1,6585 +1,6585 @@
 (* Title:      HOL/Analysis/Starlike.thy
    Author:     L C Paulson, University of Cambridge
    Author:     Robert Himmelmann, TU Muenchen
    Author:     Bogdan Grechuk, University of Edinburgh
    Author:     Armin Heller, TU Muenchen
    Author:     Johannes Hoelzl, TU Muenchen
 *)
 chapter \<open>Unsorted\<close>
 
 theory Starlike
   imports
     Convex_Euclidean_Space
     Line_Segment
 begin
 
 lemma affine_hull_closed_segment [simp]:
      "affine hull (closed_segment a b) = affine hull {a,b}"
   by (simp add: segment_convex_hull)
 
 lemma affine_hull_open_segment [simp]:
     fixes a :: "'a::euclidean_space"
     shows "affine hull (open_segment a b) = (if a = b then {} else affine hull {a,b})"
 by (metis affine_hull_convex_hull affine_hull_empty closure_open_segment closure_same_affine_hull segment_convex_hull)
 
 lemma rel_interior_closure_convex_segment:
   fixes S :: "_::euclidean_space set"
   assumes "convex S" "a \<in> rel_interior S" "b \<in> closure S"
     shows "open_segment a b \<subseteq> rel_interior S"
 proof
   fix x
   have [simp]: "(1 - u) *\<^sub>R a + u *\<^sub>R b = b - (1 - u) *\<^sub>R (b - a)" for u
     by (simp add: algebra_simps)
   assume "x \<in> open_segment a b"
   then show "x \<in> rel_interior S"
     unfolding closed_segment_def open_segment_def  using assms
     by (auto intro: rel_interior_closure_convex_shrink)
 qed
 
 lemma convex_hull_insert_segments:
    "convex hull (insert a S) =
     (if S = {} then {a} else  \<Union>x \<in> convex hull S. closed_segment a x)"
   by (force simp add: convex_hull_insert_alt in_segment)
 
 lemma Int_convex_hull_insert_rel_exterior:
   fixes z :: "'a::euclidean_space"
   assumes "convex C" "T \<subseteq> C" and z: "z \<in> rel_interior C" and dis: "disjnt S (rel_interior C)"
   shows "S \<inter> (convex hull (insert z T)) = S \<inter> (convex hull T)" (is "?lhs = ?rhs")
 proof
   have "T = {} \<Longrightarrow> z \<notin> S"
     using dis z by (auto simp add: disjnt_def)
   then show "?lhs \<subseteq> ?rhs"
   proof (clarsimp simp add: convex_hull_insert_segments)
     fix x y
     assume "x \<in> S" and y: "y \<in> convex hull T" and "x \<in> closed_segment z y"
     have "y \<in> closure C"
       by (metis y \<open>convex C\<close> \<open>T \<subseteq> C\<close> closure_subset contra_subsetD convex_hull_eq hull_mono)
     moreover have "x \<notin> rel_interior C"
       by (meson \<open>x \<in> S\<close> dis disjnt_iff)
     moreover have "x \<in> open_segment z y \<union> {z, y}"
       using \<open>x \<in> closed_segment z y\<close> closed_segment_eq_open by blast
     ultimately show "x \<in> convex hull T"
       using rel_interior_closure_convex_segment [OF \<open>convex C\<close> z]
       using y z by blast
   qed
   show "?rhs \<subseteq> ?lhs"
     by (meson hull_mono inf_mono subset_insertI subset_refl)
 qed
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>Shrinking towards the interior of a convex set\<close>
 
 lemma mem_interior_convex_shrink:
   fixes S :: "'a::euclidean_space set"
   assumes "convex S"
     and "c \<in> interior S"
     and "x \<in> S"
     and "0 < e"
     and "e \<le> 1"
   shows "x - e *\<^sub>R (x - c) \<in> interior S"
 proof -
   obtain d where "d > 0" and d: "ball c d \<subseteq> S"
     using assms(2) unfolding mem_interior by auto
   show ?thesis
     unfolding mem_interior
   proof (intro exI subsetI conjI)
     fix y
     assume "y \<in> ball (x - e *\<^sub>R (x - c)) (e*d)"
     then have as: "dist (x - e *\<^sub>R (x - c)) y < e * d"
       by simp
     have *: "y = (1 - (1 - e)) *\<^sub>R ((1 / e) *\<^sub>R y - ((1 - e) / e) *\<^sub>R x) + (1 - e) *\<^sub>R x"
       using \<open>e > 0\<close> by (auto simp add: scaleR_left_diff_distrib scaleR_right_diff_distrib)
     have "dist c ((1 / e) *\<^sub>R y - ((1 - e) / e) *\<^sub>R x) = \<bar>1/e\<bar> * norm (e *\<^sub>R c - y + (1 - e) *\<^sub>R x)"
       unfolding dist_norm
       unfolding norm_scaleR[symmetric]
       apply (rule arg_cong[where f=norm])
       using \<open>e > 0\<close>
       by (auto simp add: euclidean_eq_iff[where 'a='a] field_simps inner_simps)
     also have "\<dots> = \<bar>1/e\<bar> * norm (x - e *\<^sub>R (x - c) - y)"
       by (auto intro!:arg_cong[where f=norm] simp add: algebra_simps)
     also have "\<dots> < d"
       using as[unfolded dist_norm] and \<open>e > 0\<close>
       by (auto simp add:pos_divide_less_eq[OF \<open>e > 0\<close>] mult.commute)
     finally show "y \<in> S"
       apply (subst *)
       apply (rule assms(1)[unfolded convex_alt,rule_format])
       apply (rule d[unfolded subset_eq,rule_format])
       unfolding mem_ball
       using assms(3-5)
       apply auto
       done
   qed (insert \<open>e>0\<close> \<open>d>0\<close>, auto)
 qed
 
 lemma mem_interior_closure_convex_shrink:
   fixes S :: "'a::euclidean_space set"
   assumes "convex S"
     and "c \<in> interior S"
     and "x \<in> closure S"
     and "0 < e"
     and "e \<le> 1"
   shows "x - e *\<^sub>R (x - c) \<in> interior S"
 proof -
   obtain d where "d > 0" and d: "ball c d \<subseteq> S"
     using assms(2) unfolding mem_interior by auto
   have "\<exists>y\<in>S. norm (y - x) * (1 - e) < e * d"
   proof (cases "x \<in> S")
     case True
     then show ?thesis
       using \<open>e > 0\<close> \<open>d > 0\<close>
       apply (rule_tac bexI[where x=x])
       apply (auto)
       done
   next
     case False
     then have x: "x islimpt S"
       using assms(3)[unfolded closure_def] by auto
     show ?thesis
     proof (cases "e = 1")
       case True
       obtain y where "y \<in> S" "y \<noteq> x" "dist y x < 1"
         using x[unfolded islimpt_approachable,THEN spec[where x=1]] by auto
       then show ?thesis
         apply (rule_tac x=y in bexI)
         unfolding True
         using \<open>d > 0\<close>
         apply auto
         done
     next
       case False
       then have "0 < e * d / (1 - e)" and *: "1 - e > 0"
         using \<open>e \<le> 1\<close> \<open>e > 0\<close> \<open>d > 0\<close> by auto
       then obtain y where "y \<in> S" "y \<noteq> x" "dist y x < e * d / (1 - e)"
         using x[unfolded islimpt_approachable,THEN spec[where x="e*d / (1 - e)"]] by auto
       then show ?thesis
         apply (rule_tac x=y in bexI)
         unfolding dist_norm
         using pos_less_divide_eq[OF *]
         apply auto
         done
     qed
   qed
   then obtain y where "y \<in> S" and y: "norm (y - x) * (1 - e) < e * d"
     by auto
   define z where "z = c + ((1 - e) / e) *\<^sub>R (x - y)"
   have *: "x - e *\<^sub>R (x - c) = y - e *\<^sub>R (y - z)"
     unfolding z_def using \<open>e > 0\<close>
     by (auto simp add: scaleR_right_diff_distrib scaleR_right_distrib scaleR_left_diff_distrib)
   have "z \<in> interior S"
     apply (rule interior_mono[OF d,unfolded subset_eq,rule_format])
     unfolding interior_open[OF open_ball] mem_ball z_def dist_norm using y and assms(4,5)
     by simp (simp add: field_simps norm_minus_commute)
   then show ?thesis
     unfolding *
     using mem_interior_convex_shrink \<open>y \<in> S\<close> assms by blast
 qed
 
 lemma in_interior_closure_convex_segment:
   fixes S :: "'a::euclidean_space set"
   assumes "convex S" and a: "a \<in> interior S" and b: "b \<in> closure S"
     shows "open_segment a b \<subseteq> interior S"
 proof (clarsimp simp: in_segment)
   fix u::real
   assume u: "0 < u" "u < 1"
   have "(1 - u) *\<^sub>R a + u *\<^sub>R b = b - (1 - u) *\<^sub>R (b - a)"
     by (simp add: algebra_simps)
   also have "... \<in> interior S" using mem_interior_closure_convex_shrink [OF assms] u
     by simp
   finally show "(1 - u) *\<^sub>R a + u *\<^sub>R b \<in> interior S" .
 qed
 
 lemma convex_closure_interior:
   fixes S :: "'a::euclidean_space set"
   assumes "convex S" and int: "interior S \<noteq> {}"
   shows "closure(interior S) = closure S"
 proof -
   obtain a where a: "a \<in> interior S"
     using int by auto
   have "closure S \<subseteq> closure(interior S)"
   proof
     fix x
     assume x: "x \<in> closure S"
     show "x \<in> closure (interior S)"
     proof (cases "x=a")
       case True
       then show ?thesis
         using \<open>a \<in> interior S\<close> closure_subset by blast
     next
       case False
       show ?thesis
       proof (clarsimp simp add: closure_def islimpt_approachable)
         fix e::real
         assume xnotS: "x \<notin> interior S" and "0 < e"
         show "\<exists>x'\<in>interior S. x' \<noteq> x \<and> dist x' x < e"
         proof (intro bexI conjI)
           show "x - min (e/2 / norm (x - a)) 1 *\<^sub>R (x - a) \<noteq> x"
             using False \<open>0 < e\<close> by (auto simp: algebra_simps min_def)
           show "dist (x - min (e/2 / norm (x - a)) 1 *\<^sub>R (x - a)) x < e"
             using \<open>0 < e\<close> by (auto simp: dist_norm min_def)
           show "x - min (e/2 / norm (x - a)) 1 *\<^sub>R (x - a) \<in> interior S"
             apply (clarsimp simp add: min_def a)
             apply (rule mem_interior_closure_convex_shrink [OF \<open>convex S\<close> a x])
             using \<open>0 < e\<close> False apply (auto simp: field_split_simps)
             done
         qed
       qed
     qed
   qed
   then show ?thesis
     by (simp add: closure_mono interior_subset subset_antisym)
 qed
 
 lemma closure_convex_Int_superset:
   fixes S :: "'a::euclidean_space set"
   assumes "convex S" "interior S \<noteq> {}" "interior S \<subseteq> closure T"
   shows "closure(S \<inter> T) = closure S"
 proof -
   have "closure S \<subseteq> closure(interior S)"
     by (simp add: convex_closure_interior assms)
   also have "... \<subseteq> closure (S \<inter> T)"
     using interior_subset [of S] assms
     by (metis (no_types, lifting) Int_assoc Int_lower2 closure_mono closure_open_Int_superset inf.orderE open_interior)
   finally show ?thesis
     by (simp add: closure_mono dual_order.antisym)
 qed
 
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>Some obvious but surprisingly hard simplex lemmas\<close>
 
 lemma simplex:
   assumes "finite S"
     and "0 \<notin> S"
   shows "convex hull (insert 0 S) = {y. \<exists>u. (\<forall>x\<in>S. 0 \<le> u x) \<and> sum u S \<le> 1 \<and> sum (\<lambda>x. u x *\<^sub>R x) S = y}"
 proof (simp add: convex_hull_finite set_eq_iff assms, safe)
   fix x and u :: "'a \<Rightarrow> real"
   assume "0 \<le> u 0" "\<forall>x\<in>S. 0 \<le> u x" "u 0 + sum u S = 1"
   then show "\<exists>v. (\<forall>x\<in>S. 0 \<le> v x) \<and> sum v S \<le> 1 \<and> (\<Sum>x\<in>S. v x *\<^sub>R x) = (\<Sum>x\<in>S. u x *\<^sub>R x)"
     by force
 next
   fix x and u :: "'a \<Rightarrow> real"
   assume "\<forall>x\<in>S. 0 \<le> u x" "sum u S \<le> 1"
   then show "\<exists>v. 0 \<le> v 0 \<and> (\<forall>x\<in>S. 0 \<le> v x) \<and> v 0 + sum v S = 1 \<and> (\<Sum>x\<in>S. v x *\<^sub>R x) = (\<Sum>x\<in>S. u x *\<^sub>R x)"
     by (rule_tac x="\<lambda>x. if x = 0 then 1 - sum u S else u x" in exI) (auto simp: sum_delta_notmem assms if_smult)
 qed
 
 lemma substd_simplex:
   assumes d: "d \<subseteq> Basis"
   shows "convex hull (insert 0 d) =
     {x. (\<forall>i\<in>Basis. 0 \<le> x\<bullet>i) \<and> (\<Sum>i\<in>d. x\<bullet>i) \<le> 1 \<and> (\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x\<bullet>i = 0)}"
   (is "convex hull (insert 0 ?p) = ?s")
 proof -
   let ?D = d
   have "0 \<notin> ?p"
     using assms by (auto simp: image_def)
   from d have "finite d"
     by (blast intro: finite_subset finite_Basis)
   show ?thesis
     unfolding simplex[OF \<open>finite d\<close> \<open>0 \<notin> ?p\<close>]
   proof (intro set_eqI; safe)
     fix u :: "'a \<Rightarrow> real"
     assume as: "\<forall>x\<in>?D. 0 \<le> u x" "sum u ?D \<le> 1" 
     let ?x = "(\<Sum>x\<in>?D. u x *\<^sub>R x)"
     have ind: "\<forall>i\<in>Basis. i \<in> d \<longrightarrow> u i = ?x \<bullet> i"
       and notind: "(\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> ?x \<bullet> i = 0)"
       using substdbasis_expansion_unique[OF assms] by blast+
     then have **: "sum u ?D = sum ((\<bullet>) ?x) ?D"
       using assms by (auto intro!: sum.cong)
     show "0 \<le> ?x \<bullet> i" if "i \<in> Basis" for i
       using as(1) ind notind that by fastforce
     show "sum ((\<bullet>) ?x) ?D \<le> 1"
       using "**" as(2) by linarith
     show "?x \<bullet> i = 0" if "i \<in> Basis" "i \<notin> d" for i
       using notind that by blast
   next
     fix x 
     assume "\<forall>i\<in>Basis. 0 \<le> x \<bullet> i" "sum ((\<bullet>) x) ?D \<le> 1" "(\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0)"
     with d show "\<exists>u. (\<forall>x\<in>?D. 0 \<le> u x) \<and> sum u ?D \<le> 1 \<and> (\<Sum>x\<in>?D. u x *\<^sub>R x) = x"
       unfolding substdbasis_expansion_unique[OF assms] 
       by (rule_tac x="inner x" in exI) auto
   qed
 qed
 
 lemma std_simplex:
   "convex hull (insert 0 Basis) =
     {x::'a::euclidean_space. (\<forall>i\<in>Basis. 0 \<le> x\<bullet>i) \<and> sum (\<lambda>i. x\<bullet>i) Basis \<le> 1}"
   using substd_simplex[of Basis] by auto
 
 lemma interior_std_simplex:
   "interior (convex hull (insert 0 Basis)) =
     {x::'a::euclidean_space. (\<forall>i\<in>Basis. 0 < x\<bullet>i) \<and> sum (\<lambda>i. x\<bullet>i) Basis < 1}"
   unfolding set_eq_iff mem_interior std_simplex
 proof (intro allI iffI CollectI; clarify)
   fix x :: 'a
   fix e
   assume "e > 0" and as: "ball x e \<subseteq> {x. (\<forall>i\<in>Basis. 0 \<le> x \<bullet> i) \<and> sum ((\<bullet>) x) Basis \<le> 1}"
   show "(\<forall>i\<in>Basis. 0 < x \<bullet> i) \<and> sum ((\<bullet>) x) Basis < 1"
   proof safe
     fix i :: 'a
     assume i: "i \<in> Basis"
     then show "0 < x \<bullet> i"
       using as[THEN subsetD[where c="x - (e / 2) *\<^sub>R i"]] and \<open>e > 0\<close> 
       by (force simp add: inner_simps)
   next
     have **: "dist x (x + (e / 2) *\<^sub>R (SOME i. i\<in>Basis)) < e" using \<open>e > 0\<close>
       unfolding dist_norm
       by (auto intro!: mult_strict_left_mono simp: SOME_Basis)
     have "\<And>i. i \<in> Basis \<Longrightarrow> (x + (e / 2) *\<^sub>R (SOME i. i\<in>Basis)) \<bullet> i =
       x\<bullet>i + (if i = (SOME i. i\<in>Basis) then e/2 else 0)"
       by (auto simp: SOME_Basis inner_Basis inner_simps)
     then have *: "sum ((\<bullet>) (x + (e / 2) *\<^sub>R (SOME i. i\<in>Basis))) Basis =
       sum (\<lambda>i. x\<bullet>i + (if (SOME i. i\<in>Basis) = i then e/2 else 0)) Basis"
       by (auto simp: intro!: sum.cong)
     have "sum ((\<bullet>) x) Basis < sum ((\<bullet>) (x + (e / 2) *\<^sub>R (SOME i. i\<in>Basis))) Basis"
       using \<open>e > 0\<close> DIM_positive by (auto simp: SOME_Basis sum.distrib *)
     also have "\<dots> \<le> 1"
       using ** as by force
     finally show "sum ((\<bullet>) x) Basis < 1" by auto
   qed 
 next
   fix x :: 'a
   assume as: "\<forall>i\<in>Basis. 0 < x \<bullet> i" "sum ((\<bullet>) x) Basis < 1"
   obtain a :: 'b where "a \<in> UNIV" using UNIV_witness ..
   let ?d = "(1 - sum ((\<bullet>) x) Basis) / real (DIM('a))"
   show "\<exists>e>0. ball x e \<subseteq> {x. (\<forall>i\<in>Basis. 0 \<le> x \<bullet> i) \<and> sum ((\<bullet>) x) Basis \<le> 1}"
   proof (rule_tac x="min (Min (((\<bullet>) x) ` Basis)) D" for D in exI, intro conjI subsetI CollectI)
     fix y
     assume y: "y \<in> ball x (min (Min ((\<bullet>) x ` Basis)) ?d)"
     have "sum ((\<bullet>) y) Basis \<le> sum (\<lambda>i. x\<bullet>i + ?d) Basis"
     proof (rule sum_mono)
       fix i :: 'a
       assume i: "i \<in> Basis"
       have "\<bar>y\<bullet>i - x\<bullet>i\<bar> \<le> norm (y - x)"
         by (metis Basis_le_norm i inner_commute inner_diff_right)
       also have "... < ?d"
         using y by (simp add: dist_norm norm_minus_commute)
       finally have "\<bar>y\<bullet>i - x\<bullet>i\<bar> < ?d" .
       then show "y \<bullet> i \<le> x \<bullet> i + ?d" by auto
     qed
     also have "\<dots> \<le> 1"
       unfolding sum.distrib sum_constant
       by (auto simp add: Suc_le_eq)
     finally show "sum ((\<bullet>) y) Basis \<le> 1" .
     show "(\<forall>i\<in>Basis. 0 \<le> y \<bullet> i)"
     proof safe
       fix i :: 'a
       assume i: "i \<in> Basis"
       have "norm (x - y) < Min (((\<bullet>) x) ` Basis)"
         using y by (auto simp: dist_norm less_eq_real_def)
       also have "... \<le> x\<bullet>i"
         using i by auto
       finally have "norm (x - y) < x\<bullet>i" .
       then show "0 \<le> y\<bullet>i"
         using Basis_le_norm[OF i, of "x - y"] and as(1)[rule_format, OF i]
         by (auto simp: inner_simps)
     qed
   next
     have "Min (((\<bullet>) x) ` Basis) > 0"
       using as by simp
     moreover have "?d > 0"
       using as by (auto simp: Suc_le_eq)
     ultimately show "0 < min (Min ((\<bullet>) x ` Basis)) ((1 - sum ((\<bullet>) x) Basis) / real DIM('a))"
       by linarith
   qed 
 qed
 
 lemma interior_std_simplex_nonempty:
   obtains a :: "'a::euclidean_space" where
     "a \<in> interior(convex hull (insert 0 Basis))"
 proof -
   let ?D = "Basis :: 'a set"
   let ?a = "sum (\<lambda>b::'a. inverse (2 * real DIM('a)) *\<^sub>R b) Basis"
   {
     fix i :: 'a
     assume i: "i \<in> Basis"
     have "?a \<bullet> i = inverse (2 * real DIM('a))"
       by (rule trans[of _ "sum (\<lambda>j. if i = j then inverse (2 * real DIM('a)) else 0) ?D"])
          (simp_all add: sum.If_cases i) }
   note ** = this
   show ?thesis
     apply (rule that[of ?a])
     unfolding interior_std_simplex mem_Collect_eq
   proof safe
     fix i :: 'a
     assume i: "i \<in> Basis"
     show "0 < ?a \<bullet> i"
       unfolding **[OF i] by (auto simp add: Suc_le_eq)
   next
     have "sum ((\<bullet>) ?a) ?D = sum (\<lambda>i. inverse (2 * real DIM('a))) ?D"
       apply (rule sum.cong)
       apply rule
       apply auto
       done
     also have "\<dots> < 1"
       unfolding sum_constant divide_inverse[symmetric]
       by (auto simp add: field_simps)
     finally show "sum ((\<bullet>) ?a) ?D < 1" by auto
   qed
 qed
 
 lemma rel_interior_substd_simplex:
   assumes D: "D \<subseteq> Basis"
   shows "rel_interior (convex hull (insert 0 D)) =
     {x::'a::euclidean_space. (\<forall>i\<in>D. 0 < x\<bullet>i) \<and> (\<Sum>i\<in>D. x\<bullet>i) < 1 \<and> (\<forall>i\<in>Basis. i \<notin> D \<longrightarrow> x\<bullet>i = 0)}"
   (is "rel_interior (convex hull (insert 0 ?p)) = ?s")
 proof -
   have "finite D"
     using D finite_Basis finite_subset by blast
   show ?thesis
   proof (cases "D = {}")
     case True
     then show ?thesis
       using rel_interior_sing using euclidean_eq_iff[of _ 0] by auto
   next
     case False
     have h0: "affine hull (convex hull (insert 0 ?p)) =
       {x::'a::euclidean_space. (\<forall>i\<in>Basis. i \<notin> D \<longrightarrow> x\<bullet>i = 0)}"
       using affine_hull_convex_hull affine_hull_substd_basis assms by auto
     have aux: "\<And>x::'a. \<forall>i\<in>Basis. (\<forall>i\<in>D. 0 \<le> x\<bullet>i) \<and> (\<forall>i\<in>Basis. i \<notin> D \<longrightarrow> x\<bullet>i = 0) \<longrightarrow> 0 \<le> x\<bullet>i"
       by auto
     {
       fix x :: "'a::euclidean_space"
       assume x: "x \<in> rel_interior (convex hull (insert 0 ?p))"
       then obtain e where "e > 0" and
         "ball x e \<inter> {xa. (\<forall>i\<in>Basis. i \<notin> D \<longrightarrow> xa\<bullet>i = 0)} \<subseteq> convex hull (insert 0 ?p)"
         using mem_rel_interior_ball[of x "convex hull (insert 0 ?p)"] h0 by auto
       then have as [rule_format]: "\<And>y. dist x y < e \<and> (\<forall>i\<in>Basis. i \<notin> D \<longrightarrow> y\<bullet>i = 0) \<longrightarrow>
         (\<forall>i\<in>D. 0 \<le> y \<bullet> i) \<and> sum ((\<bullet>) y) D \<le> 1"
         unfolding ball_def unfolding substd_simplex[OF assms] using assms by auto
       have x0: "(\<forall>i\<in>Basis. i \<notin> D \<longrightarrow> x\<bullet>i = 0)"
         using x rel_interior_subset  substd_simplex[OF assms] by auto
       have "(\<forall>i\<in>D. 0 < x \<bullet> i) \<and> sum ((\<bullet>) x) D < 1 \<and> (\<forall>i\<in>Basis. i \<notin> D \<longrightarrow> x\<bullet>i = 0)"
       proof (intro conjI ballI)
         fix i :: 'a
         assume "i \<in> D"
         then have "\<forall>j\<in>D. 0 \<le> (x - (e / 2) *\<^sub>R i) \<bullet> j"
           apply -
           apply (rule as[THEN conjunct1])
           using D \<open>e > 0\<close> x0
           apply (auto simp: dist_norm inner_simps inner_Basis)
           done
         then show "0 < x \<bullet> i"
           using \<open>e > 0\<close> \<open>i \<in> D\<close> D  by (force simp: inner_simps inner_Basis)
       next
         obtain a where a: "a \<in> D"
           using \<open>D \<noteq> {}\<close> by auto
         then have **: "dist x (x + (e / 2) *\<^sub>R a) < e"
           using \<open>e > 0\<close> norm_Basis[of a] D
           unfolding dist_norm
           by auto
         have "\<And>i. i \<in> Basis \<Longrightarrow> (x + (e / 2) *\<^sub>R a) \<bullet> i = x\<bullet>i + (if i = a then e/2 else 0)"
           using a D by (auto simp: inner_simps inner_Basis)
         then have *: "sum ((\<bullet>) (x + (e / 2) *\<^sub>R a)) D =
           sum (\<lambda>i. x\<bullet>i + (if a = i then e/2 else 0)) D"
           using D by (intro sum.cong) auto
         have "a \<in> Basis"
           using \<open>a \<in> D\<close> D by auto
         then have h1: "(\<forall>i\<in>Basis. i \<notin> D \<longrightarrow> (x + (e / 2) *\<^sub>R a) \<bullet> i = 0)"
           using x0 D \<open>a\<in>D\<close> by (auto simp add: inner_add_left inner_Basis)
         have "sum ((\<bullet>) x) D < sum ((\<bullet>) (x + (e / 2) *\<^sub>R a)) D"
           using \<open>e > 0\<close> \<open>a \<in> D\<close> \<open>finite D\<close> by (auto simp add: * sum.distrib)
         also have "\<dots> \<le> 1"
           using ** h1 as[rule_format, of "x + (e / 2) *\<^sub>R a"]
           by auto
         finally show "sum ((\<bullet>) x) D < 1" "\<And>i. i\<in>Basis \<Longrightarrow> i \<notin> D \<longrightarrow> x\<bullet>i = 0"
           using x0 by auto
       qed
     }
     moreover
     {
       fix x :: "'a::euclidean_space"
       assume as: "x \<in> ?s"
       have "\<forall>i. 0 < x\<bullet>i \<or> 0 = x\<bullet>i \<longrightarrow> 0 \<le> x\<bullet>i"
         by auto
       moreover have "\<forall>i. i \<in> D \<or> i \<notin> D" by auto
       ultimately
       have "\<forall>i. (\<forall>i\<in>D. 0 < x\<bullet>i) \<and> (\<forall>i. i \<notin> D \<longrightarrow> x\<bullet>i = 0) \<longrightarrow> 0 \<le> x\<bullet>i"
         by metis
       then have h2: "x \<in> convex hull (insert 0 ?p)"
         using as assms
         unfolding substd_simplex[OF assms] by fastforce
       obtain a where a: "a \<in> D"
         using \<open>D \<noteq> {}\<close> by auto
       let ?d = "(1 - sum ((\<bullet>) x) D) / real (card D)"
       have "0 < card D" using \<open>D \<noteq> {}\<close> \<open>finite D\<close>
         by (simp add: card_gt_0_iff)
       have "Min (((\<bullet>) x) ` D) > 0"
         using as \<open>D \<noteq> {}\<close> \<open>finite D\<close> by (simp)
       moreover have "?d > 0" using as using \<open>0 < card D\<close> by auto
       ultimately have h3: "min (Min (((\<bullet>) x) ` D)) ?d > 0"
         by auto
 
       have "x \<in> rel_interior (convex hull (insert 0 ?p))"
         unfolding rel_interior_ball mem_Collect_eq h0
         apply (rule,rule h2)
         unfolding substd_simplex[OF assms]
         apply (rule_tac x="min (Min (((\<bullet>) x) ` D)) ?d" in exI)
         apply (rule, rule h3)
         apply safe
         unfolding mem_ball
       proof -
         fix y :: 'a
         assume y: "dist x y < min (Min ((\<bullet>) x ` D)) ?d"
         assume y2: "\<forall>i\<in>Basis. i \<notin> D \<longrightarrow> y\<bullet>i = 0"
         have "sum ((\<bullet>) y) D \<le> sum (\<lambda>i. x\<bullet>i + ?d) D"
         proof (rule sum_mono)
           fix i
           assume "i \<in> D"
           with D have i: "i \<in> Basis"
             by auto
           have "\<bar>y\<bullet>i - x\<bullet>i\<bar> \<le> norm (y - x)"
             by (metis i inner_commute inner_diff_right norm_bound_Basis_le order_refl)
           also have "... < ?d"
             by (metis dist_norm min_less_iff_conj norm_minus_commute y)
           finally have "\<bar>y\<bullet>i - x\<bullet>i\<bar> < ?d" .
           then show "y \<bullet> i \<le> x \<bullet> i + ?d" by auto
         qed
         also have "\<dots> \<le> 1"
           unfolding sum.distrib sum_constant  using \<open>0 < card D\<close>
           by auto
         finally show "sum ((\<bullet>) y) D \<le> 1" .
 
         fix i :: 'a
         assume i: "i \<in> Basis"
         then show "0 \<le> y\<bullet>i"
         proof (cases "i\<in>D")
           case True
           have "norm (x - y) < x\<bullet>i"
             using y[unfolded min_less_iff_conj dist_norm, THEN conjunct1]
             using Min_gr_iff[of "(\<bullet>) x ` D" "norm (x - y)"] \<open>0 < card D\<close> \<open>i \<in> D\<close>
             by (simp add: card_gt_0_iff)
           then show "0 \<le> y\<bullet>i"
             using Basis_le_norm[OF i, of "x - y"] and as(1)[rule_format]
             by (auto simp: inner_simps)
         qed (insert y2, auto)
       qed
     }
     ultimately have
       "\<And>x. x \<in> rel_interior (convex hull insert 0 D) \<longleftrightarrow>
         x \<in> {x. (\<forall>i\<in>D. 0 < x \<bullet> i) \<and> sum ((\<bullet>) x) D < 1 \<and> (\<forall>i\<in>Basis. i \<notin> D \<longrightarrow> x \<bullet> i = 0)}"
       by blast
     then show ?thesis by (rule set_eqI)
   qed
 qed
 
 lemma rel_interior_substd_simplex_nonempty:
   assumes "D \<noteq> {}"
     and "D \<subseteq> Basis"
   obtains a :: "'a::euclidean_space"
     where "a \<in> rel_interior (convex hull (insert 0 D))"
 proof -
   let ?D = D
   let ?a = "sum (\<lambda>b::'a::euclidean_space. inverse (2 * real (card D)) *\<^sub>R b) ?D"
   have "finite D"
     apply (rule finite_subset)
     using assms(2)
     apply auto
     done
   then have d1: "0 < real (card D)"
     using \<open>D \<noteq> {}\<close> by auto
   {
     fix i
     assume "i \<in> D"
     have "?a \<bullet> i = inverse (2 * real (card D))"
       apply (rule trans[of _ "sum (\<lambda>j. if i = j then inverse (2 * real (card D)) else 0) ?D"])
       unfolding inner_sum_left
       apply (rule sum.cong)
       using \<open>i \<in> D\<close> \<open>finite D\<close> sum.delta'[of D i "(\<lambda>k. inverse (2 * real (card D)))"]
         d1 assms(2)
       by (auto simp: inner_Basis rev_subsetD[OF _ assms(2)])
   }
   note ** = this
   show ?thesis
     apply (rule that[of ?a])
     unfolding rel_interior_substd_simplex[OF assms(2)] mem_Collect_eq
   proof safe
     fix i
     assume "i \<in> D"
     have "0 < inverse (2 * real (card D))"
       using d1 by auto
     also have "\<dots> = ?a \<bullet> i" using **[of i] \<open>i \<in> D\<close>
       by auto
     finally show "0 < ?a \<bullet> i" by auto
   next
     have "sum ((\<bullet>) ?a) ?D = sum (\<lambda>i. inverse (2 * real (card D))) ?D"
       by (rule sum.cong) (rule refl, rule **)
     also have "\<dots> < 1"
       unfolding sum_constant divide_real_def[symmetric]
       by (auto simp add: field_simps)
     finally show "sum ((\<bullet>) ?a) ?D < 1" by auto
   next
     fix i
     assume "i \<in> Basis" and "i \<notin> D"
     have "?a \<in> span D"
     proof (rule span_sum[of D "(\<lambda>b. b /\<^sub>R (2 * real (card D)))" D])
       {
         fix x :: "'a::euclidean_space"
         assume "x \<in> D"
         then have "x \<in> span D"
           using span_base[of _ "D"] by auto
         then have "x /\<^sub>R (2 * real (card D)) \<in> span D"
           using span_mul[of x "D" "(inverse (real (card D)) / 2)"] by auto
       }
       then show "\<And>x. x\<in>D \<Longrightarrow> x /\<^sub>R (2 * real (card D)) \<in> span D"
         by auto
     qed
     then show "?a \<bullet> i = 0 "
       using \<open>i \<notin> D\<close> unfolding span_substd_basis[OF assms(2)] using \<open>i \<in> Basis\<close> by auto
   qed
 qed
 
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>Relative interior of convex set\<close>
 
 lemma rel_interior_convex_nonempty_aux:
   fixes S :: "'n::euclidean_space set"
   assumes "convex S"
     and "0 \<in> S"
   shows "rel_interior S \<noteq> {}"
 proof (cases "S = {0}")
   case True
   then show ?thesis using rel_interior_sing by auto
 next
   case False
   obtain B where B: "independent B \<and> B \<le> S \<and> S \<le> span B \<and> card B = dim S"
     using basis_exists[of S] by metis
   then have "B \<noteq> {}"
     using B assms \<open>S \<noteq> {0}\<close> span_empty by auto
   have "insert 0 B \<le> span B"
     using subspace_span[of B] subspace_0[of "span B"]
       span_superset by auto
   then have "span (insert 0 B) \<le> span B"
     using span_span[of B] span_mono[of "insert 0 B" "span B"] by blast
   then have "convex hull insert 0 B \<le> span B"
     using convex_hull_subset_span[of "insert 0 B"] by auto
   then have "span (convex hull insert 0 B) \<le> span B"
     using span_span[of B]
       span_mono[of "convex hull insert 0 B" "span B"] by blast
   then have *: "span (convex hull insert 0 B) = span B"
     using span_mono[of B "convex hull insert 0 B"] hull_subset[of "insert 0 B"] by auto
   then have "span (convex hull insert 0 B) = span S"
     using B span_mono[of B S] span_mono[of S "span B"]
       span_span[of B] by auto
   moreover have "0 \<in> affine hull (convex hull insert 0 B)"
     using hull_subset[of "convex hull insert 0 B"] hull_subset[of "insert 0 B"] by auto
   ultimately have **: "affine hull (convex hull insert 0 B) = affine hull S"
     using affine_hull_span_0[of "convex hull insert 0 B"] affine_hull_span_0[of "S"]
       assms hull_subset[of S]
     by auto
   obtain d and f :: "'n \<Rightarrow> 'n" where
     fd: "card d = card B" "linear f" "f ` B = d"
       "f ` span B = {x. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = (0::real)} \<and> inj_on f (span B)"
     and d: "d \<subseteq> Basis"
     using basis_to_substdbasis_subspace_isomorphism[of B,OF _ ] B by auto
   then have "bounded_linear f"
     using linear_conv_bounded_linear by auto
   have "d \<noteq> {}"
     using fd B \<open>B \<noteq> {}\<close> by auto
   have "insert 0 d = f ` (insert 0 B)"
     using fd linear_0 by auto
   then have "(convex hull (insert 0 d)) = f ` (convex hull (insert 0 B))"
     using convex_hull_linear_image[of f "(insert 0 d)"]
       convex_hull_linear_image[of f "(insert 0 B)"] \<open>linear f\<close>
     by auto
   moreover have "rel_interior (f ` (convex hull insert 0 B)) =
     f ` rel_interior (convex hull insert 0 B)"
     apply (rule  rel_interior_injective_on_span_linear_image[of f "(convex hull insert 0 B)"])
     using \<open>bounded_linear f\<close> fd *
     apply auto
     done
   ultimately have "rel_interior (convex hull insert 0 B) \<noteq> {}"
     using rel_interior_substd_simplex_nonempty[OF \<open>d \<noteq> {}\<close> d]
     apply auto
     apply blast
     done
   moreover have "convex hull (insert 0 B) \<subseteq> S"
     using B assms hull_mono[of "insert 0 B" "S" "convex"] convex_hull_eq
     by auto
   ultimately show ?thesis
     using subset_rel_interior[of "convex hull insert 0 B" S] ** by auto
 qed
 
 lemma rel_interior_eq_empty:
   fixes S :: "'n::euclidean_space set"
   assumes "convex S"
   shows "rel_interior S = {} \<longleftrightarrow> S = {}"
 proof -
   {
     assume "S \<noteq> {}"
     then obtain a where "a \<in> S" by auto
     then have "0 \<in> (+) (-a) ` S"
       using assms exI[of "(\<lambda>x. x \<in> S \<and> - a + x = 0)" a] by auto
     then have "rel_interior ((+) (-a) ` S) \<noteq> {}"
       using rel_interior_convex_nonempty_aux[of "(+) (-a) ` S"]
         convex_translation[of S "-a"] assms
       by auto
     then have "rel_interior S \<noteq> {}"
       using rel_interior_translation [of "- a"] by simp
   }
   then show ?thesis by auto
 qed
 
 lemma interior_simplex_nonempty:
   fixes S :: "'N :: euclidean_space set"
   assumes "independent S" "finite S" "card S = DIM('N)"
   obtains a where "a \<in> interior (convex hull (insert 0 S))"
 proof -
   have "affine hull (insert 0 S) = UNIV"
     by (simp add: hull_inc affine_hull_span_0 dim_eq_full[symmetric]
          assms(1) assms(3) dim_eq_card_independent)
   moreover have "rel_interior (convex hull insert 0 S) \<noteq> {}"
     using rel_interior_eq_empty [of "convex hull (insert 0 S)"] by auto
   ultimately have "interior (convex hull insert 0 S) \<noteq> {}"
     by (simp add: rel_interior_interior)
   with that show ?thesis
     by auto
 qed
 
 lemma convex_rel_interior:
   fixes S :: "'n::euclidean_space set"
   assumes "convex S"
   shows "convex (rel_interior S)"
 proof -
   {
     fix x y and u :: real
     assume assm: "x \<in> rel_interior S" "y \<in> rel_interior S" "0 \<le> u" "u \<le> 1"
     then have "x \<in> S"
       using rel_interior_subset by auto
     have "x - u *\<^sub>R (x-y) \<in> rel_interior S"
     proof (cases "0 = u")
       case False
       then have "0 < u" using assm by auto
       then show ?thesis
         using assm rel_interior_convex_shrink[of S y x u] assms \<open>x \<in> S\<close> by auto
     next
       case True
       then show ?thesis using assm by auto
     qed
     then have "(1 - u) *\<^sub>R x + u *\<^sub>R y \<in> rel_interior S"
       by (simp add: algebra_simps)
   }
   then show ?thesis
     unfolding convex_alt by auto
 qed
 
 lemma convex_closure_rel_interior:
   fixes S :: "'n::euclidean_space set"
   assumes "convex S"
   shows "closure (rel_interior S) = closure S"
 proof -
   have h1: "closure (rel_interior S) \<le> closure S"
     using closure_mono[of "rel_interior S" S] rel_interior_subset[of S] by auto
   show ?thesis
   proof (cases "S = {}")
     case False
     then obtain a where a: "a \<in> rel_interior S"
       using rel_interior_eq_empty assms by auto
     { fix x
       assume x: "x \<in> closure S"
       {
         assume "x = a"
         then have "x \<in> closure (rel_interior S)"
           using a unfolding closure_def by auto
       }
       moreover
       {
         assume "x \<noteq> a"
          {
            fix e :: real
            assume "e > 0"
            define e1 where "e1 = min 1 (e/norm (x - a))"
            then have e1: "e1 > 0" "e1 \<le> 1" "e1 * norm (x - a) \<le> e"
              using \<open>x \<noteq> a\<close> \<open>e > 0\<close> le_divide_eq[of e1 e "norm (x - a)"]
              by simp_all
            then have *: "x - e1 *\<^sub>R (x - a) \<in> rel_interior S"
              using rel_interior_closure_convex_shrink[of S a x e1] assms x a e1_def
              by auto
            have "\<exists>y. y \<in> rel_interior S \<and> y \<noteq> x \<and> dist y x \<le> e"
               apply (rule_tac x="x - e1 *\<^sub>R (x - a)" in exI)
               using * e1 dist_norm[of "x - e1 *\<^sub>R (x - a)" x] \<open>x \<noteq> a\<close>
               apply simp
               done
         }
         then have "x islimpt rel_interior S"
           unfolding islimpt_approachable_le by auto
         then have "x \<in> closure(rel_interior S)"
           unfolding closure_def by auto
       }
       ultimately have "x \<in> closure(rel_interior S)" by auto
     }
     then show ?thesis using h1 by auto
   next
     case True
     then have "rel_interior S = {}" by auto
     then have "closure (rel_interior S) = {}"
       using closure_empty by auto
     with True show ?thesis by auto
   qed
 qed
 
 lemma rel_interior_same_affine_hull:
   fixes S :: "'n::euclidean_space set"
   assumes "convex S"
   shows "affine hull (rel_interior S) = affine hull S"
   by (metis assms closure_same_affine_hull convex_closure_rel_interior)
 
 lemma rel_interior_aff_dim:
   fixes S :: "'n::euclidean_space set"
   assumes "convex S"
   shows "aff_dim (rel_interior S) = aff_dim S"
   by (metis aff_dim_affine_hull2 assms rel_interior_same_affine_hull)
 
 lemma rel_interior_rel_interior:
   fixes S :: "'n::euclidean_space set"
   assumes "convex S"
   shows "rel_interior (rel_interior S) = rel_interior S"
 proof -
   have "openin (top_of_set (affine hull (rel_interior S))) (rel_interior S)"
     using openin_rel_interior[of S] rel_interior_same_affine_hull[of S] assms by auto
   then show ?thesis
     using rel_interior_def by auto
 qed
 
 lemma rel_interior_rel_open:
   fixes S :: "'n::euclidean_space set"
   assumes "convex S"
   shows "rel_open (rel_interior S)"
   unfolding rel_open_def using rel_interior_rel_interior assms by auto
 
 lemma convex_rel_interior_closure_aux:
   fixes x y z :: "'n::euclidean_space"
   assumes "0 < a" "0 < b" "(a + b) *\<^sub>R z = a *\<^sub>R x + b *\<^sub>R y"
   obtains e where "0 < e" "e \<le> 1" "z = y - e *\<^sub>R (y - x)"
 proof -
   define e where "e = a / (a + b)"
   have "z = (1 / (a + b)) *\<^sub>R ((a + b) *\<^sub>R z)"
     using assms  by (simp add: eq_vector_fraction_iff)
   also have "\<dots> = (1 / (a + b)) *\<^sub>R (a *\<^sub>R x + b *\<^sub>R y)"
     using assms scaleR_cancel_left[of "1/(a+b)" "(a + b) *\<^sub>R z" "a *\<^sub>R x + b *\<^sub>R y"]
     by auto
   also have "\<dots> = y - e *\<^sub>R (y-x)"
     using e_def
     apply (simp add: algebra_simps)
     using scaleR_left_distrib[of "a/(a+b)" "b/(a+b)" y] assms add_divide_distrib[of a b "a+b"]
     apply auto
     done
   finally have "z = y - e *\<^sub>R (y-x)"
     by auto
   moreover have "e > 0" using e_def assms by auto
   moreover have "e \<le> 1" using e_def assms by auto
   ultimately show ?thesis using that[of e] by auto
 qed
 
 lemma convex_rel_interior_closure:
   fixes S :: "'n::euclidean_space set"
   assumes "convex S"
   shows "rel_interior (closure S) = rel_interior S"
 proof (cases "S = {}")
   case True
   then show ?thesis
     using assms rel_interior_eq_empty by auto
 next
   case False
   have "rel_interior (closure S) \<supseteq> rel_interior S"
     using subset_rel_interior[of S "closure S"] closure_same_affine_hull closure_subset
     by auto
   moreover
   {
     fix z
     assume z: "z \<in> rel_interior (closure S)"
     obtain x where x: "x \<in> rel_interior S"
       using \<open>S \<noteq> {}\<close> assms rel_interior_eq_empty by auto
     have "z \<in> rel_interior S"
     proof (cases "x = z")
       case True
       then show ?thesis using x by auto
     next
       case False
       obtain e where e: "e > 0" "cball z e \<inter> affine hull closure S \<le> closure S"
         using z rel_interior_cball[of "closure S"] by auto
       hence *: "0 < e/norm(z-x)" using e False by auto
       define y where "y = z + (e/norm(z-x)) *\<^sub>R (z-x)"
       have yball: "y \<in> cball z e"
         using y_def dist_norm[of z y] e by auto
       have "x \<in> affine hull closure S"
         using x rel_interior_subset_closure hull_inc[of x "closure S"] by blast
       moreover have "z \<in> affine hull closure S"
         using z rel_interior_subset hull_subset[of "closure S"] by blast
       ultimately have "y \<in> affine hull closure S"
         using y_def affine_affine_hull[of "closure S"]
           mem_affine_3_minus [of "affine hull closure S" z z x "e/norm(z-x)"] by auto
       then have "y \<in> closure S" using e yball by auto
       have "(1 + (e/norm(z-x))) *\<^sub>R z = (e/norm(z-x)) *\<^sub>R x + y"
         using y_def by (simp add: algebra_simps)
       then obtain e1 where "0 < e1" "e1 \<le> 1" "z = y - e1 *\<^sub>R (y - x)"
         using * convex_rel_interior_closure_aux[of "e / norm (z - x)" 1 z x y]
         by (auto simp add: algebra_simps)
       then show ?thesis
         using rel_interior_closure_convex_shrink assms x \<open>y \<in> closure S\<close>
         by auto
     qed
   }
   ultimately show ?thesis by auto
 qed
 
 lemma convex_interior_closure:
   fixes S :: "'n::euclidean_space set"
   assumes "convex S"
   shows "interior (closure S) = interior S"
   using closure_aff_dim[of S] interior_rel_interior_gen[of S]
     interior_rel_interior_gen[of "closure S"]
     convex_rel_interior_closure[of S] assms
   by auto
 
 lemma closure_eq_rel_interior_eq:
   fixes S1 S2 :: "'n::euclidean_space set"
   assumes "convex S1"
     and "convex S2"
   shows "closure S1 = closure S2 \<longleftrightarrow> rel_interior S1 = rel_interior S2"
   by (metis convex_rel_interior_closure convex_closure_rel_interior assms)
 
 lemma closure_eq_between:
   fixes S1 S2 :: "'n::euclidean_space set"
   assumes "convex S1"
     and "convex S2"
   shows "closure S1 = closure S2 \<longleftrightarrow> rel_interior S1 \<le> S2 \<and> S2 \<subseteq> closure S1"
   (is "?A \<longleftrightarrow> ?B")
 proof
   assume ?A
   then show ?B
     by (metis assms closure_subset convex_rel_interior_closure rel_interior_subset)
 next
   assume ?B
   then have "closure S1 \<subseteq> closure S2"
     by (metis assms(1) convex_closure_rel_interior closure_mono)
   moreover from \<open>?B\<close> have "closure S1 \<supseteq> closure S2"
     by (metis closed_closure closure_minimal)
   ultimately show ?A ..
 qed
 
 lemma open_inter_closure_rel_interior:
   fixes S A :: "'n::euclidean_space set"
   assumes "convex S"
     and "open A"
   shows "A \<inter> closure S = {} \<longleftrightarrow> A \<inter> rel_interior S = {}"
   by (metis assms convex_closure_rel_interior open_Int_closure_eq_empty)
 
 lemma rel_interior_open_segment:
   fixes a :: "'a :: euclidean_space"
   shows "rel_interior(open_segment a b) = open_segment a b"
 proof (cases "a = b")
   case True then show ?thesis by auto
 next
   case False then show ?thesis
     apply (simp add: rel_interior_eq openin_open)
     apply (rule_tac x="ball (inverse 2 *\<^sub>R (a + b)) (norm(b - a) / 2)" in exI)
     apply (simp add: open_segment_as_ball)
     done
 qed
 
 lemma rel_interior_closed_segment:
   fixes a :: "'a :: euclidean_space"
   shows "rel_interior(closed_segment a b) =
          (if a = b then {a} else open_segment a b)"
 proof (cases "a = b")
   case True then show ?thesis by auto
 next
   case False then show ?thesis
     by simp
        (metis closure_open_segment convex_open_segment convex_rel_interior_closure
               rel_interior_open_segment)
 qed
 
 lemmas rel_interior_segment = rel_interior_closed_segment rel_interior_open_segment
 
 subsection\<open>The relative frontier of a set\<close>
 
 definition\<^marker>\<open>tag important\<close> "rel_frontier S = closure S - rel_interior S"
 
 lemma rel_frontier_empty [simp]: "rel_frontier {} = {}"
   by (simp add: rel_frontier_def)
 
 lemma rel_frontier_eq_empty:
     fixes S :: "'n::euclidean_space set"
     shows "rel_frontier S = {} \<longleftrightarrow> affine S"
   unfolding rel_frontier_def
   using rel_interior_subset_closure  by (auto simp add: rel_interior_eq_closure [symmetric])
 
 lemma rel_frontier_sing [simp]:
     fixes a :: "'n::euclidean_space"
     shows "rel_frontier {a} = {}"
   by (simp add: rel_frontier_def)
 
 lemma rel_frontier_affine_hull:
   fixes S :: "'a::euclidean_space set"
   shows "rel_frontier S \<subseteq> affine hull S"
 using closure_affine_hull rel_frontier_def by fastforce
 
 lemma rel_frontier_cball [simp]:
     fixes a :: "'n::euclidean_space"
     shows "rel_frontier(cball a r) = (if r = 0 then {} else sphere a r)"
 proof (cases rule: linorder_cases [of r 0])
   case less then show ?thesis
     by (force simp: sphere_def)
 next
   case equal then show ?thesis by simp
 next
   case greater then show ?thesis
     apply simp
     by (metis centre_in_ball empty_iff frontier_cball frontier_def interior_cball interior_rel_interior_gen rel_frontier_def)
 qed
 
 lemma rel_frontier_translation:
   fixes a :: "'a::euclidean_space"
   shows "rel_frontier((\<lambda>x. a + x) ` S) = (\<lambda>x. a + x) ` (rel_frontier S)"
 by (simp add: rel_frontier_def translation_diff rel_interior_translation closure_translation)
 
 lemma rel_frontier_nonempty_interior:
   fixes S :: "'n::euclidean_space set"
   shows "interior S \<noteq> {} \<Longrightarrow> rel_frontier S = frontier S"
 by (metis frontier_def interior_rel_interior_gen rel_frontier_def)
 
 lemma rel_frontier_frontier:
   fixes S :: "'n::euclidean_space set"
   shows "affine hull S = UNIV \<Longrightarrow> rel_frontier S = frontier S"
 by (simp add: frontier_def rel_frontier_def rel_interior_interior)
 
 lemma closest_point_in_rel_frontier:
    "\<lbrakk>closed S; S \<noteq> {}; x \<in> affine hull S - rel_interior S\<rbrakk>
    \<Longrightarrow> closest_point S x \<in> rel_frontier S"
   by (simp add: closest_point_in_rel_interior closest_point_in_set rel_frontier_def)
 
 lemma closed_rel_frontier [iff]:
   fixes S :: "'n::euclidean_space set"
   shows "closed (rel_frontier S)"
 proof -
   have *: "closedin (top_of_set (affine hull S)) (closure S - rel_interior S)"
     by (simp add: closed_subset closedin_diff closure_affine_hull openin_rel_interior)
   show ?thesis
     apply (rule closedin_closed_trans[of "affine hull S" "rel_frontier S"])
     unfolding rel_frontier_def
     using * closed_affine_hull
     apply auto
     done
 qed
 
 lemma closed_rel_boundary:
   fixes S :: "'n::euclidean_space set"
   shows "closed S \<Longrightarrow> closed(S - rel_interior S)"
 by (metis closed_rel_frontier closure_closed rel_frontier_def)
 
 lemma compact_rel_boundary:
   fixes S :: "'n::euclidean_space set"
   shows "compact S \<Longrightarrow> compact(S - rel_interior S)"
 by (metis bounded_diff closed_rel_boundary closure_eq compact_closure compact_imp_closed)
 
 lemma bounded_rel_frontier:
   fixes S :: "'n::euclidean_space set"
   shows "bounded S \<Longrightarrow> bounded(rel_frontier S)"
 by (simp add: bounded_closure bounded_diff rel_frontier_def)
 
 lemma compact_rel_frontier_bounded:
   fixes S :: "'n::euclidean_space set"
   shows "bounded S \<Longrightarrow> compact(rel_frontier S)"
 using bounded_rel_frontier closed_rel_frontier compact_eq_bounded_closed by blast
 
 lemma compact_rel_frontier:
   fixes S :: "'n::euclidean_space set"
   shows "compact S \<Longrightarrow> compact(rel_frontier S)"
 by (meson compact_eq_bounded_closed compact_rel_frontier_bounded)
 
 lemma convex_same_rel_interior_closure:
   fixes S :: "'n::euclidean_space set"
   shows "\<lbrakk>convex S; convex T\<rbrakk>
          \<Longrightarrow> rel_interior S = rel_interior T \<longleftrightarrow> closure S = closure T"
 by (simp add: closure_eq_rel_interior_eq)
 
 lemma convex_same_rel_interior_closure_straddle:
   fixes S :: "'n::euclidean_space set"
   shows "\<lbrakk>convex S; convex T\<rbrakk>
          \<Longrightarrow> rel_interior S = rel_interior T \<longleftrightarrow>
              rel_interior S \<subseteq> T \<and> T \<subseteq> closure S"
 by (simp add: closure_eq_between convex_same_rel_interior_closure)
 
 lemma convex_rel_frontier_aff_dim:
   fixes S1 S2 :: "'n::euclidean_space set"
   assumes "convex S1"
     and "convex S2"
     and "S2 \<noteq> {}"
     and "S1 \<le> rel_frontier S2"
   shows "aff_dim S1 < aff_dim S2"
 proof -
   have "S1 \<subseteq> closure S2"
     using assms unfolding rel_frontier_def by auto
   then have *: "affine hull S1 \<subseteq> affine hull S2"
     using hull_mono[of "S1" "closure S2"] closure_same_affine_hull[of S2] by blast
   then have "aff_dim S1 \<le> aff_dim S2"
     using * aff_dim_affine_hull[of S1] aff_dim_affine_hull[of S2]
       aff_dim_subset[of "affine hull S1" "affine hull S2"]
     by auto
   moreover
   {
     assume eq: "aff_dim S1 = aff_dim S2"
     then have "S1 \<noteq> {}"
       using aff_dim_empty[of S1] aff_dim_empty[of S2] \<open>S2 \<noteq> {}\<close> by auto
     have **: "affine hull S1 = affine hull S2"
        apply (rule affine_dim_equal)
        using * affine_affine_hull
        apply auto
        using \<open>S1 \<noteq> {}\<close> hull_subset[of S1]
        apply auto
        using eq aff_dim_affine_hull[of S1] aff_dim_affine_hull[of S2]
        apply auto
        done
     obtain a where a: "a \<in> rel_interior S1"
       using \<open>S1 \<noteq> {}\<close> rel_interior_eq_empty assms by auto
     obtain T where T: "open T" "a \<in> T \<inter> S1" "T \<inter> affine hull S1 \<subseteq> S1"
        using mem_rel_interior[of a S1] a by auto
     then have "a \<in> T \<inter> closure S2"
       using a assms unfolding rel_frontier_def by auto
     then obtain b where b: "b \<in> T \<inter> rel_interior S2"
       using open_inter_closure_rel_interior[of S2 T] assms T by auto
     then have "b \<in> affine hull S1"
       using rel_interior_subset hull_subset[of S2] ** by auto
     then have "b \<in> S1"
       using T b by auto
     then have False
       using b assms unfolding rel_frontier_def by auto
   }
   ultimately show ?thesis
     using less_le by auto
 qed
 
 lemma convex_rel_interior_if:
   fixes S ::  "'n::euclidean_space set"
   assumes "convex S"
     and "z \<in> rel_interior S"
   shows "\<forall>x\<in>affine hull S. \<exists>m. m > 1 \<and> (\<forall>e. e > 1 \<and> e \<le> m \<longrightarrow> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S)"
 proof -
   obtain e1 where e1: "e1 > 0 \<and> cball z e1 \<inter> affine hull S \<subseteq> S"
     using mem_rel_interior_cball[of z S] assms by auto
   {
     fix x
     assume x: "x \<in> affine hull S"
     {
       assume "x \<noteq> z"
       define m where "m = 1 + e1/norm(x-z)"
       hence "m > 1" using e1 \<open>x \<noteq> z\<close> by auto
       {
         fix e
         assume e: "e > 1 \<and> e \<le> m"
         have "z \<in> affine hull S"
           using assms rel_interior_subset hull_subset[of S] by auto
         then have *: "(1 - e)*\<^sub>R x + e *\<^sub>R z \<in> affine hull S"
           using mem_affine[of "affine hull S" x z "(1-e)" e] affine_affine_hull[of S] x
           by auto
         have "norm (z + e *\<^sub>R x - (x + e *\<^sub>R z)) = norm ((e - 1) *\<^sub>R (x - z))"
           by (simp add: algebra_simps)
         also have "\<dots> = (e - 1) * norm (x-z)"
           using norm_scaleR e by auto
         also have "\<dots> \<le> (m - 1) * norm (x - z)"
           using e mult_right_mono[of _ _ "norm(x-z)"] by auto
         also have "\<dots> = (e1 / norm (x - z)) * norm (x - z)"
           using m_def by auto
         also have "\<dots> = e1"
           using \<open>x \<noteq> z\<close> e1 by simp
         finally have **: "norm (z + e *\<^sub>R x - (x + e *\<^sub>R z)) \<le> e1"
           by auto
         have "(1 - e)*\<^sub>R x+ e *\<^sub>R z \<in> cball z e1"
           using m_def **
           unfolding cball_def dist_norm
           by (auto simp add: algebra_simps)
         then have "(1 - e) *\<^sub>R x+ e *\<^sub>R z \<in> S"
           using e * e1 by auto
       }
       then have "\<exists>m. m > 1 \<and> (\<forall>e. e > 1 \<and> e \<le> m \<longrightarrow> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S )"
         using \<open>m> 1 \<close> by auto
     }
     moreover
     {
       assume "x = z"
       define m where "m = 1 + e1"
       then have "m > 1"
         using e1 by auto
       {
         fix e
         assume e: "e > 1 \<and> e \<le> m"
         then have "(1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S"
           using e1 x \<open>x = z\<close> by (auto simp add: algebra_simps)
         then have "(1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S"
           using e by auto
       }
       then have "\<exists>m. m > 1 \<and> (\<forall>e. e > 1 \<and> e \<le> m \<longrightarrow> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S)"
         using \<open>m > 1\<close> by auto
     }
     ultimately have "\<exists>m. m > 1 \<and> (\<forall>e. e > 1 \<and> e \<le> m \<longrightarrow> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S )"
       by blast
   }
   then show ?thesis by auto
 qed
 
 lemma convex_rel_interior_if2:
   fixes S :: "'n::euclidean_space set"
   assumes "convex S"
   assumes "z \<in> rel_interior S"
   shows "\<forall>x\<in>affine hull S. \<exists>e. e > 1 \<and> (1 - e)*\<^sub>R x + e *\<^sub>R z \<in> S"
   using convex_rel_interior_if[of S z] assms by auto
 
 lemma convex_rel_interior_only_if:
   fixes S :: "'n::euclidean_space set"
   assumes "convex S"
     and "S \<noteq> {}"
   assumes "\<forall>x\<in>S. \<exists>e. e > 1 \<and> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S"
   shows "z \<in> rel_interior S"
 proof -
   obtain x where x: "x \<in> rel_interior S"
     using rel_interior_eq_empty assms by auto
   then have "x \<in> S"
     using rel_interior_subset by auto
   then obtain e where e: "e > 1 \<and> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S"
     using assms by auto
   define y where [abs_def]: "y = (1 - e) *\<^sub>R x + e *\<^sub>R z"
   then have "y \<in> S" using e by auto
   define e1 where "e1 = 1/e"
   then have "0 < e1 \<and> e1 < 1" using e by auto
   then have "z  =y - (1 - e1) *\<^sub>R (y - x)"
     using e1_def y_def by (auto simp add: algebra_simps)
   then show ?thesis
     using rel_interior_convex_shrink[of S x y "1-e1"] \<open>0 < e1 \<and> e1 < 1\<close> \<open>y \<in> S\<close> x assms
     by auto
 qed
 
 lemma convex_rel_interior_iff:
   fixes S :: "'n::euclidean_space set"
   assumes "convex S"
     and "S \<noteq> {}"
   shows "z \<in> rel_interior S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e. e > 1 \<and> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S)"
   using assms hull_subset[of S "affine"]
     convex_rel_interior_if[of S z] convex_rel_interior_only_if[of S z]
   by auto
 
 lemma convex_rel_interior_iff2:
   fixes S :: "'n::euclidean_space set"
   assumes "convex S"
     and "S \<noteq> {}"
   shows "z \<in> rel_interior S \<longleftrightarrow> (\<forall>x\<in>affine hull S. \<exists>e. e > 1 \<and> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S)"
   using assms hull_subset[of S] convex_rel_interior_if2[of S z] convex_rel_interior_only_if[of S z]
   by auto
 
 lemma convex_interior_iff:
   fixes S :: "'n::euclidean_space set"
   assumes "convex S"
   shows "z \<in> interior S \<longleftrightarrow> (\<forall>x. \<exists>e. e > 0 \<and> z + e *\<^sub>R x \<in> S)"
 proof (cases "aff_dim S = int DIM('n)")
   case False
   { assume "z \<in> interior S"
     then have False
       using False interior_rel_interior_gen[of S] by auto }
   moreover
   { assume r: "\<forall>x. \<exists>e. e > 0 \<and> z + e *\<^sub>R x \<in> S"
     { fix x
       obtain e1 where e1: "e1 > 0 \<and> z + e1 *\<^sub>R (x - z) \<in> S"
         using r by auto
       obtain e2 where e2: "e2 > 0 \<and> z + e2 *\<^sub>R (z - x) \<in> S"
         using r by auto
       define x1 where [abs_def]: "x1 = z + e1 *\<^sub>R (x - z)"
       then have x1: "x1 \<in> affine hull S"
         using e1 hull_subset[of S] by auto
       define x2 where [abs_def]: "x2 = z + e2 *\<^sub>R (z - x)"
       then have x2: "x2 \<in> affine hull S"
         using e2 hull_subset[of S] by auto
       have *: "e1/(e1+e2) + e2/(e1+e2) = 1"
         using add_divide_distrib[of e1 e2 "e1+e2"] e1 e2 by simp
       then have "z = (e2/(e1+e2)) *\<^sub>R x1 + (e1/(e1+e2)) *\<^sub>R x2"
         using x1_def x2_def
         apply (auto simp add: algebra_simps)
         using scaleR_left_distrib[of "e1/(e1+e2)" "e2/(e1+e2)" z]
         apply auto
         done
       then have z: "z \<in> affine hull S"
         using mem_affine[of "affine hull S" x1 x2 "e2/(e1+e2)" "e1/(e1+e2)"]
           x1 x2 affine_affine_hull[of S] *
         by auto
       have "x1 - x2 = (e1 + e2) *\<^sub>R (x - z)"
         using x1_def x2_def by (auto simp add: algebra_simps)
       then have "x = z+(1/(e1+e2)) *\<^sub>R (x1-x2)"
         using e1 e2 by simp
       then have "x \<in> affine hull S"
         using mem_affine_3_minus[of "affine hull S" z x1 x2 "1/(e1+e2)"]
           x1 x2 z affine_affine_hull[of S]
         by auto
     }
     then have "affine hull S = UNIV"
       by auto
     then have "aff_dim S = int DIM('n)"
       using aff_dim_affine_hull[of S] by (simp)
     then have False
       using False by auto
   }
   ultimately show ?thesis by auto
 next
   case True
   then have "S \<noteq> {}"
     using aff_dim_empty[of S] by auto
   have *: "affine hull S = UNIV"
     using True affine_hull_UNIV by auto
   {
     assume "z \<in> interior S"
     then have "z \<in> rel_interior S"
       using True interior_rel_interior_gen[of S] by auto
     then have **: "\<forall>x. \<exists>e. e > 1 \<and> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S"
       using convex_rel_interior_iff2[of S z] assms \<open>S \<noteq> {}\<close> * by auto
     fix x
     obtain e1 where e1: "e1 > 1" "(1 - e1) *\<^sub>R (z - x) + e1 *\<^sub>R z \<in> S"
       using **[rule_format, of "z-x"] by auto
     define e where [abs_def]: "e = e1 - 1"
     then have "(1 - e1) *\<^sub>R (z - x) + e1 *\<^sub>R z = z + e *\<^sub>R x"
       by (simp add: algebra_simps)
     then have "e > 0" "z + e *\<^sub>R x \<in> S"
       using e1 e_def by auto
     then have "\<exists>e. e > 0 \<and> z + e *\<^sub>R x \<in> S"
       by auto
   }
   moreover
   {
     assume r: "\<forall>x. \<exists>e. e > 0 \<and> z + e *\<^sub>R x \<in> S"
     {
       fix x
       obtain e1 where e1: "e1 > 0" "z + e1 *\<^sub>R (z - x) \<in> S"
         using r[rule_format, of "z-x"] by auto
       define e where "e = e1 + 1"
       then have "z + e1 *\<^sub>R (z - x) = (1 - e) *\<^sub>R x + e *\<^sub>R z"
         by (simp add: algebra_simps)
       then have "e > 1" "(1 - e)*\<^sub>R x + e *\<^sub>R z \<in> S"
         using e1 e_def by auto
       then have "\<exists>e. e > 1 \<and> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S" by auto
     }
     then have "z \<in> rel_interior S"
       using convex_rel_interior_iff2[of S z] assms \<open>S \<noteq> {}\<close> by auto
     then have "z \<in> interior S"
       using True interior_rel_interior_gen[of S] by auto
   }
   ultimately show ?thesis by auto
 qed
 
 
 subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Relative interior and closure under common operations\<close>
 
 lemma rel_interior_inter_aux: "\<Inter>{rel_interior S |S. S \<in> I} \<subseteq> \<Inter>I"
 proof -
   {
     fix y
     assume "y \<in> \<Inter>{rel_interior S |S. S \<in> I}"
     then have y: "\<forall>S \<in> I. y \<in> rel_interior S"
       by auto
     {
       fix S
       assume "S \<in> I"
       then have "y \<in> S"
         using rel_interior_subset y by auto
     }
     then have "y \<in> \<Inter>I" by auto
   }
   then show ?thesis by auto
 qed
 
 lemma convex_closure_rel_interior_inter:
   assumes "\<forall>S\<in>I. convex (S :: 'n::euclidean_space set)"
     and "\<Inter>{rel_interior S |S. S \<in> I} \<noteq> {}"
   shows "\<Inter>{closure S |S. S \<in> I} \<le> closure (\<Inter>{rel_interior S |S. S \<in> I})"
 proof -
   obtain x where x: "\<forall>S\<in>I. x \<in> rel_interior S"
     using assms by auto
   {
     fix y
     assume "y \<in> \<Inter>{closure S |S. S \<in> I}"
     then have y: "\<forall>S \<in> I. y \<in> closure S"
       by auto
     {
       assume "y = x"
       then have "y \<in> closure (\<Inter>{rel_interior S |S. S \<in> I})"
         using x closure_subset[of "\<Inter>{rel_interior S |S. S \<in> I}"] by auto
     }
     moreover
     {
       assume "y \<noteq> x"
       { fix e :: real
         assume e: "e > 0"
         define e1 where "e1 = min 1 (e/norm (y - x))"
         then have e1: "e1 > 0" "e1 \<le> 1" "e1 * norm (y - x) \<le> e"
           using \<open>y \<noteq> x\<close> \<open>e > 0\<close> le_divide_eq[of e1 e "norm (y - x)"]
           by simp_all
         define z where "z = y - e1 *\<^sub>R (y - x)"
         {
           fix S
           assume "S \<in> I"
           then have "z \<in> rel_interior S"
             using rel_interior_closure_convex_shrink[of S x y e1] assms x y e1 z_def
             by auto
         }
         then have *: "z \<in> \<Inter>{rel_interior S |S. S \<in> I}"
           by auto
         have "\<exists>z. z \<in> \<Inter>{rel_interior S |S. S \<in> I} \<and> z \<noteq> y \<and> dist z y \<le> e"
           apply (rule_tac x="z" in exI)
           using \<open>y \<noteq> x\<close> z_def * e1 e dist_norm[of z y]
           apply simp
           done
       }
       then have "y islimpt \<Inter>{rel_interior S |S. S \<in> I}"
         unfolding islimpt_approachable_le by blast
       then have "y \<in> closure (\<Inter>{rel_interior S |S. S \<in> I})"
         unfolding closure_def by auto
     }
     ultimately have "y \<in> closure (\<Inter>{rel_interior S |S. S \<in> I})"
       by auto
   }
   then show ?thesis by auto
 qed
 
 lemma convex_closure_inter:
   assumes "\<forall>S\<in>I. convex (S :: 'n::euclidean_space set)"
     and "\<Inter>{rel_interior S |S. S \<in> I} \<noteq> {}"
   shows "closure (\<Inter>I) = \<Inter>{closure S |S. S \<in> I}"
 proof -
   have "\<Inter>{closure S |S. S \<in> I} \<le> closure (\<Inter>{rel_interior S |S. S \<in> I})"
     using convex_closure_rel_interior_inter assms by auto
   moreover
   have "closure (\<Inter>{rel_interior S |S. S \<in> I}) \<le> closure (\<Inter>I)"
     using rel_interior_inter_aux closure_mono[of "\<Inter>{rel_interior S |S. S \<in> I}" "\<Inter>I"]
     by auto
   ultimately show ?thesis
     using closure_Int[of I] by auto
 qed
 
 lemma convex_inter_rel_interior_same_closure:
   assumes "\<forall>S\<in>I. convex (S :: 'n::euclidean_space set)"
     and "\<Inter>{rel_interior S |S. S \<in> I} \<noteq> {}"
   shows "closure (\<Inter>{rel_interior S |S. S \<in> I}) = closure (\<Inter>I)"
 proof -
   have "\<Inter>{closure S |S. S \<in> I} \<le> closure (\<Inter>{rel_interior S |S. S \<in> I})"
     using convex_closure_rel_interior_inter assms by auto
   moreover
   have "closure (\<Inter>{rel_interior S |S. S \<in> I}) \<le> closure (\<Inter>I)"
     using rel_interior_inter_aux closure_mono[of "\<Inter>{rel_interior S |S. S \<in> I}" "\<Inter>I"]
     by auto
   ultimately show ?thesis
     using closure_Int[of I] by auto
 qed
 
 lemma convex_rel_interior_inter:
   assumes "\<forall>S\<in>I. convex (S :: 'n::euclidean_space set)"
     and "\<Inter>{rel_interior S |S. S \<in> I} \<noteq> {}"
   shows "rel_interior (\<Inter>I) \<subseteq> \<Inter>{rel_interior S |S. S \<in> I}"
 proof -
   have "convex (\<Inter>I)"
     using assms convex_Inter by auto
   moreover
   have "convex (\<Inter>{rel_interior S |S. S \<in> I})"
     apply (rule convex_Inter)
     using assms convex_rel_interior
     apply auto
     done
   ultimately
   have "rel_interior (\<Inter>{rel_interior S |S. S \<in> I}) = rel_interior (\<Inter>I)"
     using convex_inter_rel_interior_same_closure assms
       closure_eq_rel_interior_eq[of "\<Inter>{rel_interior S |S. S \<in> I}" "\<Inter>I"]
     by blast
   then show ?thesis
     using rel_interior_subset[of "\<Inter>{rel_interior S |S. S \<in> I}"] by auto
 qed
 
 lemma convex_rel_interior_finite_inter:
   assumes "\<forall>S\<in>I. convex (S :: 'n::euclidean_space set)"
     and "\<Inter>{rel_interior S |S. S \<in> I} \<noteq> {}"
     and "finite I"
   shows "rel_interior (\<Inter>I) = \<Inter>{rel_interior S |S. S \<in> I}"
 proof -
   have "\<Inter>I \<noteq> {}"
     using assms rel_interior_inter_aux[of I] by auto
   have "convex (\<Inter>I)"
     using convex_Inter assms by auto
   show ?thesis
   proof (cases "I = {}")
     case True
     then show ?thesis
       using Inter_empty rel_interior_UNIV by auto
   next
     case False
     {
       fix z
       assume z: "z \<in> \<Inter>{rel_interior S |S. S \<in> I}"
       {
         fix x
         assume x: "x \<in> \<Inter>I"
         {
           fix S
           assume S: "S \<in> I"
           then have "z \<in> rel_interior S" "x \<in> S"
             using z x by auto
           then have "\<exists>m. m > 1 \<and> (\<forall>e. e > 1 \<and> e \<le> m \<longrightarrow> (1 - e)*\<^sub>R x + e *\<^sub>R z \<in> S)"
             using convex_rel_interior_if[of S z] S assms hull_subset[of S] by auto
         }
         then obtain mS where
           mS: "\<forall>S\<in>I. mS S > 1 \<and> (\<forall>e. e > 1 \<and> e \<le> mS S \<longrightarrow> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S)" by metis
         define e where "e = Min (mS ` I)"
         then have "e \<in> mS ` I" using assms \<open>I \<noteq> {}\<close> by simp
         then have "e > 1" using mS by auto
         moreover have "\<forall>S\<in>I. e \<le> mS S"
           using e_def assms by auto
         ultimately have "\<exists>e > 1. (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> \<Inter>I"
           using mS by auto
       }
       then have "z \<in> rel_interior (\<Inter>I)"
         using convex_rel_interior_iff[of "\<Inter>I" z] \<open>\<Inter>I \<noteq> {}\<close> \<open>convex (\<Inter>I)\<close> by auto
     }
     then show ?thesis
       using convex_rel_interior_inter[of I] assms by auto
   qed
 qed
 
 lemma convex_closure_inter_two:
   fixes S T :: "'n::euclidean_space set"
   assumes "convex S"
     and "convex T"
   assumes "rel_interior S \<inter> rel_interior T \<noteq> {}"
   shows "closure (S \<inter> T) = closure S \<inter> closure T"
   using convex_closure_inter[of "{S,T}"] assms by auto
 
 lemma convex_rel_interior_inter_two:
   fixes S T :: "'n::euclidean_space set"
   assumes "convex S"
     and "convex T"
     and "rel_interior S \<inter> rel_interior T \<noteq> {}"
   shows "rel_interior (S \<inter> T) = rel_interior S \<inter> rel_interior T"
   using convex_rel_interior_finite_inter[of "{S,T}"] assms by auto
 
 lemma convex_affine_closure_Int:
   fixes S T :: "'n::euclidean_space set"
   assumes "convex S"
     and "affine T"
     and "rel_interior S \<inter> T \<noteq> {}"
   shows "closure (S \<inter> T) = closure S \<inter> T"
 proof -
   have "affine hull T = T"
     using assms by auto
   then have "rel_interior T = T"
     using rel_interior_affine_hull[of T] by metis
   moreover have "closure T = T"
     using assms affine_closed[of T] by auto
   ultimately show ?thesis
     using convex_closure_inter_two[of S T] assms affine_imp_convex by auto
 qed
 
 lemma connected_component_1_gen:
   fixes S :: "'a :: euclidean_space set"
   assumes "DIM('a) = 1"
   shows "connected_component S a b \<longleftrightarrow> closed_segment a b \<subseteq> S"
 unfolding connected_component_def
 by (metis (no_types, lifting) assms subsetD subsetI convex_contains_segment convex_segment(1)
             ends_in_segment connected_convex_1_gen)
 
 lemma connected_component_1:
   fixes S :: "real set"
   shows "connected_component S a b \<longleftrightarrow> closed_segment a b \<subseteq> S"
 by (simp add: connected_component_1_gen)
 
 lemma convex_affine_rel_interior_Int:
   fixes S T :: "'n::euclidean_space set"
   assumes "convex S"
     and "affine T"
     and "rel_interior S \<inter> T \<noteq> {}"
   shows "rel_interior (S \<inter> T) = rel_interior S \<inter> T"
 proof -
   have "affine hull T = T"
     using assms by auto
   then have "rel_interior T = T"
     using rel_interior_affine_hull[of T] by metis
   moreover have "closure T = T"
     using assms affine_closed[of T] by auto
   ultimately show ?thesis
     using convex_rel_interior_inter_two[of S T] assms affine_imp_convex by auto
 qed
 
 lemma convex_affine_rel_frontier_Int:
    fixes S T :: "'n::euclidean_space set"
   assumes "convex S"
     and "affine T"
     and "interior S \<inter> T \<noteq> {}"
   shows "rel_frontier(S \<inter> T) = frontier S \<inter> T"
 using assms
 apply (simp add: rel_frontier_def convex_affine_closure_Int frontier_def)
 by (metis Diff_Int_distrib2 Int_emptyI convex_affine_closure_Int convex_affine_rel_interior_Int empty_iff interior_rel_interior_gen)
 
 lemma rel_interior_convex_Int_affine:
   fixes S :: "'a::euclidean_space set"
   assumes "convex S" "affine T" "interior S \<inter> T \<noteq> {}"
     shows "rel_interior(S \<inter> T) = interior S \<inter> T"
 proof -
   obtain a where aS: "a \<in> interior S" and aT:"a \<in> T"
     using assms by force
   have "rel_interior S = interior S"
     by (metis (no_types) aS affine_hull_nonempty_interior equals0D rel_interior_interior)
   then show ?thesis
     by (metis (no_types) affine_imp_convex assms convex_rel_interior_inter_two hull_same rel_interior_affine_hull)
 qed
 
 lemma closure_convex_Int_affine:
   fixes S :: "'a::euclidean_space set"
   assumes "convex S" "affine T" "rel_interior S \<inter> T \<noteq> {}"
   shows "closure(S \<inter> T) = closure S \<inter> T"
 proof
   have "closure (S \<inter> T) \<subseteq> closure T"
     by (simp add: closure_mono)
   also have "... \<subseteq> T"
     by (simp add: affine_closed assms)
   finally show "closure(S \<inter> T) \<subseteq> closure S \<inter> T"
     by (simp add: closure_mono)
 next
   obtain a where "a \<in> rel_interior S" "a \<in> T"
     using assms by auto
   then have ssT: "subspace ((\<lambda>x. (-a)+x) ` T)" and "a \<in> S"
     using affine_diffs_subspace rel_interior_subset assms by blast+
   show "closure S \<inter> T \<subseteq> closure (S \<inter> T)"
   proof
     fix x  assume "x \<in> closure S \<inter> T"
     show "x \<in> closure (S \<inter> T)"
     proof (cases "x = a")
       case True
       then show ?thesis
         using \<open>a \<in> S\<close> \<open>a \<in> T\<close> closure_subset by fastforce
     next
       case False
       then have "x \<in> closure(open_segment a x)"
         by auto
       then show ?thesis
         using \<open>x \<in> closure S \<inter> T\<close> assms convex_affine_closure_Int by blast
     qed
   qed
 qed
 
 lemma subset_rel_interior_convex:
   fixes S T :: "'n::euclidean_space set"
   assumes "convex S"
     and "convex T"
     and "S \<le> closure T"
     and "\<not> S \<subseteq> rel_frontier T"
   shows "rel_interior S \<subseteq> rel_interior T"
 proof -
   have *: "S \<inter> closure T = S"
     using assms by auto
   have "\<not> rel_interior S \<subseteq> rel_frontier T"
     using closure_mono[of "rel_interior S" "rel_frontier T"] closed_rel_frontier[of T]
       closure_closed[of S] convex_closure_rel_interior[of S] closure_subset[of S] assms
     by auto
   then have "rel_interior S \<inter> rel_interior (closure T) \<noteq> {}"
     using assms rel_frontier_def[of T] rel_interior_subset convex_rel_interior_closure[of T]
     by auto
   then have "rel_interior S \<inter> rel_interior T = rel_interior (S \<inter> closure T)"
     using assms convex_closure convex_rel_interior_inter_two[of S "closure T"]
       convex_rel_interior_closure[of T]
     by auto
   also have "\<dots> = rel_interior S"
     using * by auto
   finally show ?thesis
     by auto
 qed
 
 lemma rel_interior_convex_linear_image:
   fixes f :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space"
   assumes "linear f"
     and "convex S"
   shows "f ` (rel_interior S) = rel_interior (f ` S)"
 proof (cases "S = {}")
   case True
   then show ?thesis
     using assms by auto
 next
   case False
   interpret linear f by fact
   have *: "f ` (rel_interior S) \<subseteq> f ` S"
     unfolding image_mono using rel_interior_subset by auto
   have "f ` S \<subseteq> f ` (closure S)"
     unfolding image_mono using closure_subset by auto
   also have "\<dots> = f ` (closure (rel_interior S))"
     using convex_closure_rel_interior assms by auto
   also have "\<dots> \<subseteq> closure (f ` (rel_interior S))"
     using closure_linear_image_subset assms by auto
   finally have "closure (f ` S) = closure (f ` rel_interior S)"
     using closure_mono[of "f ` S" "closure (f ` rel_interior S)"] closure_closure
       closure_mono[of "f ` rel_interior S" "f ` S"] *
     by auto
   then have "rel_interior (f ` S) = rel_interior (f ` rel_interior S)"
     using assms convex_rel_interior
       linear_conv_bounded_linear[of f] convex_linear_image[of _ S]
       convex_linear_image[of _ "rel_interior S"]
       closure_eq_rel_interior_eq[of "f ` S" "f ` rel_interior S"]
     by auto
   then have "rel_interior (f ` S) \<subseteq> f ` rel_interior S"
     using rel_interior_subset by auto
   moreover
   {
     fix z
     assume "z \<in> f ` rel_interior S"
     then obtain z1 where z1: "z1 \<in> rel_interior S" "f z1 = z" by auto
     {
       fix x
       assume "x \<in> f ` S"
       then obtain x1 where x1: "x1 \<in> S" "f x1 = x" by auto
       then obtain e where e: "e > 1" "(1 - e) *\<^sub>R x1 + e *\<^sub>R z1 \<in> S"
         using convex_rel_interior_iff[of S z1] \<open>convex S\<close> x1 z1 by auto
       moreover have "f ((1 - e) *\<^sub>R x1 + e *\<^sub>R z1) = (1 - e) *\<^sub>R x + e *\<^sub>R z"
         using x1 z1 by (simp add: linear_add linear_scale \<open>linear f\<close>)
       ultimately have "(1 - e) *\<^sub>R x + e *\<^sub>R z \<in> f ` S"
         using imageI[of "(1 - e) *\<^sub>R x1 + e *\<^sub>R z1" S f] by auto
       then have "\<exists>e. e > 1 \<and> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> f ` S"
         using e by auto
     }
     then have "z \<in> rel_interior (f ` S)"
       using convex_rel_interior_iff[of "f ` S" z] \<open>convex S\<close> \<open>linear f\<close>
         \<open>S \<noteq> {}\<close> convex_linear_image[of f S]  linear_conv_bounded_linear[of f]
       by auto
   }
   ultimately show ?thesis by auto
 qed
 
 lemma rel_interior_convex_linear_preimage:
   fixes f :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space"
   assumes "linear f"
     and "convex S"
     and "f -` (rel_interior S) \<noteq> {}"
   shows "rel_interior (f -` S) = f -` (rel_interior S)"
 proof -
   interpret linear f by fact
   have "S \<noteq> {}"
     using assms by auto
   have nonemp: "f -` S \<noteq> {}"
     by (metis assms(3) rel_interior_subset subset_empty vimage_mono)
   then have "S \<inter> (range f) \<noteq> {}"
     by auto
   have conv: "convex (f -` S)"
     using convex_linear_vimage assms by auto
   then have "convex (S \<inter> range f)"
     by (simp add: assms(2) convex_Int convex_linear_image linear_axioms)
   {
     fix z
     assume "z \<in> f -` (rel_interior S)"
     then have z: "f z \<in> rel_interior S"
       by auto
     {
       fix x
       assume "x \<in> f -` S"
       then have "f x \<in> S" by auto
       then obtain e where e: "e > 1" "(1 - e) *\<^sub>R f x + e *\<^sub>R f z \<in> S"
         using convex_rel_interior_iff[of S "f z"] z assms \<open>S \<noteq> {}\<close> by auto
       moreover have "(1 - e) *\<^sub>R f x + e *\<^sub>R f z = f ((1 - e) *\<^sub>R x + e *\<^sub>R z)"
         using \<open>linear f\<close> by (simp add: linear_iff)
       ultimately have "\<exists>e. e > 1 \<and> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> f -` S"
         using e by auto
     }
     then have "z \<in> rel_interior (f -` S)"
       using convex_rel_interior_iff[of "f -` S" z] conv nonemp by auto
   }
   moreover
   {
     fix z
     assume z: "z \<in> rel_interior (f -` S)"
     {
       fix x
       assume "x \<in> S \<inter> range f"
       then obtain y where y: "f y = x" "y \<in> f -` S" by auto
       then obtain e where e: "e > 1" "(1 - e) *\<^sub>R y + e *\<^sub>R z \<in> f -` S"
         using convex_rel_interior_iff[of "f -` S" z] z conv by auto
       moreover have "(1 - e) *\<^sub>R x + e *\<^sub>R f z = f ((1 - e) *\<^sub>R y + e *\<^sub>R z)"
         using \<open>linear f\<close> y by (simp add: linear_iff)
       ultimately have "\<exists>e. e > 1 \<and> (1 - e) *\<^sub>R x + e *\<^sub>R f z \<in> S \<inter> range f"
         using e by auto
     }
     then have "f z \<in> rel_interior (S \<inter> range f)"
       using \<open>convex (S \<inter> (range f))\<close> \<open>S \<inter> range f \<noteq> {}\<close>
         convex_rel_interior_iff[of "S \<inter> (range f)" "f z"]
       by auto
     moreover have "affine (range f)"
       by (simp add: linear_axioms linear_subspace_image subspace_imp_affine)
     ultimately have "f z \<in> rel_interior S"
       using convex_affine_rel_interior_Int[of S "range f"] assms by auto
     then have "z \<in> f -` (rel_interior S)"
       by auto
   }
   ultimately show ?thesis by auto
 qed
 
 lemma rel_interior_Times:
   fixes S :: "'n::euclidean_space set"
     and T :: "'m::euclidean_space set"
   assumes "convex S"
     and "convex T"
   shows "rel_interior (S \<times> T) = rel_interior S \<times> rel_interior T"
 proof -
   { assume "S = {}"
     then have ?thesis
       by auto
   }
   moreover
   { assume "T = {}"
     then have ?thesis
        by auto
   }
   moreover
   {
     assume "S \<noteq> {}" "T \<noteq> {}"
     then have ri: "rel_interior S \<noteq> {}" "rel_interior T \<noteq> {}"
       using rel_interior_eq_empty assms by auto
     then have "fst -` rel_interior S \<noteq> {}"
       using fst_vimage_eq_Times[of "rel_interior S"] by auto
     then have "rel_interior ((fst :: 'n * 'm \<Rightarrow> 'n) -` S) = fst -` rel_interior S"
-      using fst_linear \<open>convex S\<close> rel_interior_convex_linear_preimage[of fst S] by auto
+      using linear_fst \<open>convex S\<close> rel_interior_convex_linear_preimage[of fst S] by auto
     then have s: "rel_interior (S \<times> (UNIV :: 'm set)) = rel_interior S \<times> UNIV"
       by (simp add: fst_vimage_eq_Times)
     from ri have "snd -` rel_interior T \<noteq> {}"
       using snd_vimage_eq_Times[of "rel_interior T"] by auto
     then have "rel_interior ((snd :: 'n * 'm \<Rightarrow> 'm) -` T) = snd -` rel_interior T"
-      using snd_linear \<open>convex T\<close> rel_interior_convex_linear_preimage[of snd T] by auto
+      using linear_snd \<open>convex T\<close> rel_interior_convex_linear_preimage[of snd T] by auto
     then have t: "rel_interior ((UNIV :: 'n set) \<times> T) = UNIV \<times> rel_interior T"
       by (simp add: snd_vimage_eq_Times)
     from s t have *: "rel_interior (S \<times> (UNIV :: 'm set)) \<inter> rel_interior ((UNIV :: 'n set) \<times> T) =
       rel_interior S \<times> rel_interior T" by auto
     have "S \<times> T = S \<times> (UNIV :: 'm set) \<inter> (UNIV :: 'n set) \<times> T"
       by auto
     then have "rel_interior (S \<times> T) = rel_interior ((S \<times> (UNIV :: 'm set)) \<inter> ((UNIV :: 'n set) \<times> T))"
       by auto
     also have "\<dots> = rel_interior (S \<times> (UNIV :: 'm set)) \<inter> rel_interior ((UNIV :: 'n set) \<times> T)"
        apply (subst convex_rel_interior_inter_two[of "S \<times> (UNIV :: 'm set)" "(UNIV :: 'n set) \<times> T"])
        using * ri assms convex_Times
        apply auto
        done
     finally have ?thesis using * by auto
   }
   ultimately show ?thesis by blast
 qed
 
 lemma rel_interior_scaleR:
   fixes S :: "'n::euclidean_space set"
   assumes "c \<noteq> 0"
   shows "((*\<^sub>R) c) ` (rel_interior S) = rel_interior (((*\<^sub>R) c) ` S)"
   using rel_interior_injective_linear_image[of "((*\<^sub>R) c)" S]
     linear_conv_bounded_linear[of "(*\<^sub>R) c"] linear_scaleR injective_scaleR[of c] assms
   by auto
 
 lemma rel_interior_convex_scaleR:
   fixes S :: "'n::euclidean_space set"
   assumes "convex S"
   shows "((*\<^sub>R) c) ` (rel_interior S) = rel_interior (((*\<^sub>R) c) ` S)"
   by (metis assms linear_scaleR rel_interior_convex_linear_image)
 
 lemma convex_rel_open_scaleR:
   fixes S :: "'n::euclidean_space set"
   assumes "convex S"
     and "rel_open S"
   shows "convex (((*\<^sub>R) c) ` S) \<and> rel_open (((*\<^sub>R) c) ` S)"
   by (metis assms convex_scaling rel_interior_convex_scaleR rel_open_def)
 
 lemma convex_rel_open_finite_inter:
   assumes "\<forall>S\<in>I. convex (S :: 'n::euclidean_space set) \<and> rel_open S"
     and "finite I"
   shows "convex (\<Inter>I) \<and> rel_open (\<Inter>I)"
 proof (cases "\<Inter>{rel_interior S |S. S \<in> I} = {}")
   case True
   then have "\<Inter>I = {}"
     using assms unfolding rel_open_def by auto
   then show ?thesis
     unfolding rel_open_def by auto
 next
   case False
   then have "rel_open (\<Inter>I)"
     using assms unfolding rel_open_def
     using convex_rel_interior_finite_inter[of I]
     by auto
   then show ?thesis
     using convex_Inter assms by auto
 qed
 
 lemma convex_rel_open_linear_image:
   fixes f :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space"
   assumes "linear f"
     and "convex S"
     and "rel_open S"
   shows "convex (f ` S) \<and> rel_open (f ` S)"
   by (metis assms convex_linear_image rel_interior_convex_linear_image rel_open_def)
 
 lemma convex_rel_open_linear_preimage:
   fixes f :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space"
   assumes "linear f"
     and "convex S"
     and "rel_open S"
   shows "convex (f -` S) \<and> rel_open (f -` S)"
 proof (cases "f -` (rel_interior S) = {}")
   case True
   then have "f -` S = {}"
     using assms unfolding rel_open_def by auto
   then show ?thesis
     unfolding rel_open_def by auto
 next
   case False
   then have "rel_open (f -` S)"
     using assms unfolding rel_open_def
     using rel_interior_convex_linear_preimage[of f S]
     by auto
   then show ?thesis
     using convex_linear_vimage assms
     by auto
 qed
 
 lemma rel_interior_projection:
   fixes S :: "('m::euclidean_space \<times> 'n::euclidean_space) set"
     and f :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space set"
   assumes "convex S"
     and "f = (\<lambda>y. {z. (y, z) \<in> S})"
   shows "(y, z) \<in> rel_interior S \<longleftrightarrow> (y \<in> rel_interior {y. (f y \<noteq> {})} \<and> z \<in> rel_interior (f y))"
 proof -
   {
     fix y
     assume "y \<in> {y. f y \<noteq> {}}"
     then obtain z where "(y, z) \<in> S"
       using assms by auto
     then have "\<exists>x. x \<in> S \<and> y = fst x"
       apply (rule_tac x="(y, z)" in exI)
       apply auto
       done
     then obtain x where "x \<in> S" "y = fst x"
       by blast
     then have "y \<in> fst ` S"
       unfolding image_def by auto
   }
   then have "fst ` S = {y. f y \<noteq> {}}"
     unfolding fst_def using assms by auto
   then have h1: "fst ` rel_interior S = rel_interior {y. f y \<noteq> {}}"
-    using rel_interior_convex_linear_image[of fst S] assms fst_linear by auto
+    using rel_interior_convex_linear_image[of fst S] assms linear_fst by auto
   {
     fix y
     assume "y \<in> rel_interior {y. f y \<noteq> {}}"
     then have "y \<in> fst ` rel_interior S"
       using h1 by auto
     then have *: "rel_interior S \<inter> fst -` {y} \<noteq> {}"
       by auto
     moreover have aff: "affine (fst -` {y})"
       unfolding affine_alt by (simp add: algebra_simps)
     ultimately have **: "rel_interior (S \<inter> fst -` {y}) = rel_interior S \<inter> fst -` {y}"
       using convex_affine_rel_interior_Int[of S "fst -` {y}"] assms by auto
     have conv: "convex (S \<inter> fst -` {y})"
       using convex_Int assms aff affine_imp_convex by auto
     {
       fix x
       assume "x \<in> f y"
       then have "(y, x) \<in> S \<inter> (fst -` {y})"
         using assms by auto
       moreover have "x = snd (y, x)" by auto
       ultimately have "x \<in> snd ` (S \<inter> fst -` {y})"
         by blast
     }
     then have "snd ` (S \<inter> fst -` {y}) = f y"
       using assms by auto
     then have ***: "rel_interior (f y) = snd ` rel_interior (S \<inter> fst -` {y})"
-      using rel_interior_convex_linear_image[of snd "S \<inter> fst -` {y}"] snd_linear conv
+      using rel_interior_convex_linear_image[of snd "S \<inter> fst -` {y}"] linear_snd conv
       by auto
     {
       fix z
       assume "z \<in> rel_interior (f y)"
       then have "z \<in> snd ` rel_interior (S \<inter> fst -` {y})"
         using *** by auto
       moreover have "{y} = fst ` rel_interior (S \<inter> fst -` {y})"
         using * ** rel_interior_subset by auto
       ultimately have "(y, z) \<in> rel_interior (S \<inter> fst -` {y})"
         by force
       then have "(y,z) \<in> rel_interior S"
         using ** by auto
     }
     moreover
     {
       fix z
       assume "(y, z) \<in> rel_interior S"
       then have "(y, z) \<in> rel_interior (S \<inter> fst -` {y})"
         using ** by auto
       then have "z \<in> snd ` rel_interior (S \<inter> fst -` {y})"
         by (metis Range_iff snd_eq_Range)
       then have "z \<in> rel_interior (f y)"
         using *** by auto
     }
     ultimately have "\<And>z. (y, z) \<in> rel_interior S \<longleftrightarrow> z \<in> rel_interior (f y)"
       by auto
   }
   then have h2: "\<And>y z. y \<in> rel_interior {t. f t \<noteq> {}} \<Longrightarrow>
     (y, z) \<in> rel_interior S \<longleftrightarrow> z \<in> rel_interior (f y)"
     by auto
   {
     fix y z
     assume asm: "(y, z) \<in> rel_interior S"
     then have "y \<in> fst ` rel_interior S"
       by (metis Domain_iff fst_eq_Domain)
     then have "y \<in> rel_interior {t. f t \<noteq> {}}"
       using h1 by auto
     then have "y \<in> rel_interior {t. f t \<noteq> {}}" and "(z \<in> rel_interior (f y))"
       using h2 asm by auto
   }
   then show ?thesis using h2 by blast
 qed
 
 lemma rel_frontier_Times:
   fixes S :: "'n::euclidean_space set"
     and T :: "'m::euclidean_space set"
   assumes "convex S"
     and "convex T"
   shows "rel_frontier S \<times> rel_frontier T \<subseteq> rel_frontier (S \<times> T)"
     by (force simp: rel_frontier_def rel_interior_Times assms closure_Times)
 
 
 subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Relative interior of convex cone\<close>
 
 lemma cone_rel_interior:
   fixes S :: "'m::euclidean_space set"
   assumes "cone S"
   shows "cone ({0} \<union> rel_interior S)"
 proof (cases "S = {}")
   case True
   then show ?thesis
     by (simp add: cone_0)
 next
   case False
   then have *: "0 \<in> S \<and> (\<forall>c. c > 0 \<longrightarrow> (*\<^sub>R) c ` S = S)"
     using cone_iff[of S] assms by auto
   then have *: "0 \<in> ({0} \<union> rel_interior S)"
     and "\<forall>c. c > 0 \<longrightarrow> (*\<^sub>R) c ` ({0} \<union> rel_interior S) = ({0} \<union> rel_interior S)"
     by (auto simp add: rel_interior_scaleR)
   then show ?thesis
     using cone_iff[of "{0} \<union> rel_interior S"] by auto
 qed
 
 lemma rel_interior_convex_cone_aux:
   fixes S :: "'m::euclidean_space set"
   assumes "convex S"
   shows "(c, x) \<in> rel_interior (cone hull ({(1 :: real)} \<times> S)) \<longleftrightarrow>
     c > 0 \<and> x \<in> (((*\<^sub>R) c) ` (rel_interior S))"
 proof (cases "S = {}")
   case True
   then show ?thesis
     by (simp add: cone_hull_empty)
 next
   case False
   then obtain s where "s \<in> S" by auto
   have conv: "convex ({(1 :: real)} \<times> S)"
     using convex_Times[of "{(1 :: real)}" S] assms convex_singleton[of "1 :: real"]
     by auto
   define f where "f y = {z. (y, z) \<in> cone hull ({1 :: real} \<times> S)}" for y
   then have *: "(c, x) \<in> rel_interior (cone hull ({(1 :: real)} \<times> S)) =
     (c \<in> rel_interior {y. f y \<noteq> {}} \<and> x \<in> rel_interior (f c))"
     apply (subst rel_interior_projection[of "cone hull ({(1 :: real)} \<times> S)" f c x])
     using convex_cone_hull[of "{(1 :: real)} \<times> S"] conv
     apply auto
     done
   {
     fix y :: real
     assume "y \<ge> 0"
     then have "y *\<^sub>R (1,s) \<in> cone hull ({1 :: real} \<times> S)"
       using cone_hull_expl[of "{(1 :: real)} \<times> S"] \<open>s \<in> S\<close> by auto
     then have "f y \<noteq> {}"
       using f_def by auto
   }
   then have "{y. f y \<noteq> {}} = {0..}"
     using f_def cone_hull_expl[of "{1 :: real} \<times> S"] by auto
   then have **: "rel_interior {y. f y \<noteq> {}} = {0<..}"
     using rel_interior_real_semiline by auto
   {
     fix c :: real
     assume "c > 0"
     then have "f c = ((*\<^sub>R) c ` S)"
       using f_def cone_hull_expl[of "{1 :: real} \<times> S"] by auto
     then have "rel_interior (f c) = (*\<^sub>R) c ` rel_interior S"
       using rel_interior_convex_scaleR[of S c] assms by auto
   }
   then show ?thesis using * ** by auto
 qed
 
 lemma rel_interior_convex_cone:
   fixes S :: "'m::euclidean_space set"
   assumes "convex S"
   shows "rel_interior (cone hull ({1 :: real} \<times> S)) =
     {(c, c *\<^sub>R x) | c x. c > 0 \<and> x \<in> rel_interior S}"
   (is "?lhs = ?rhs")
 proof -
   {
     fix z
     assume "z \<in> ?lhs"
     have *: "z = (fst z, snd z)"
       by auto
     then have "z \<in> ?rhs"
       using rel_interior_convex_cone_aux[of S "fst z" "snd z"] assms \<open>z \<in> ?lhs\<close> by fastforce
   }
   moreover
   {
     fix z
     assume "z \<in> ?rhs"
     then have "z \<in> ?lhs"
       using rel_interior_convex_cone_aux[of S "fst z" "snd z"] assms
       by auto
   }
   ultimately show ?thesis by blast
 qed
 
 lemma convex_hull_finite_union:
   assumes "finite I"
   assumes "\<forall>i\<in>I. convex (S i) \<and> (S i) \<noteq> {}"
   shows "convex hull (\<Union>(S ` I)) =
     {sum (\<lambda>i. c i *\<^sub>R s i) I | c s. (\<forall>i\<in>I. c i \<ge> 0) \<and> sum c I = 1 \<and> (\<forall>i\<in>I. s i \<in> S i)}"
   (is "?lhs = ?rhs")
 proof -
   have "?lhs \<supseteq> ?rhs"
   proof
     fix x
     assume "x \<in> ?rhs"
     then obtain c s where *: "sum (\<lambda>i. c i *\<^sub>R s i) I = x" "sum c I = 1"
       "(\<forall>i\<in>I. c i \<ge> 0) \<and> (\<forall>i\<in>I. s i \<in> S i)" by auto
     then have "\<forall>i\<in>I. s i \<in> convex hull (\<Union>(S ` I))"
       using hull_subset[of "\<Union>(S ` I)" convex] by auto
     then show "x \<in> ?lhs"
       unfolding *(1)[symmetric]
       apply (subst convex_sum[of I "convex hull \<Union>(S ` I)" c s])
       using * assms convex_convex_hull
       apply auto
       done
   qed
 
   {
     fix i
     assume "i \<in> I"
     with assms have "\<exists>p. p \<in> S i" by auto
   }
   then obtain p where p: "\<forall>i\<in>I. p i \<in> S i" by metis
 
   {
     fix i
     assume "i \<in> I"
     {
       fix x
       assume "x \<in> S i"
       define c where "c j = (if j = i then 1::real else 0)" for j
       then have *: "sum c I = 1"
         using \<open>finite I\<close> \<open>i \<in> I\<close> sum.delta[of I i "\<lambda>j::'a. 1::real"]
         by auto
       define s where "s j = (if j = i then x else p j)" for j
       then have "\<forall>j. c j *\<^sub>R s j = (if j = i then x else 0)"
         using c_def by (auto simp add: algebra_simps)
       then have "x = sum (\<lambda>i. c i *\<^sub>R s i) I"
         using s_def c_def \<open>finite I\<close> \<open>i \<in> I\<close> sum.delta[of I i "\<lambda>j::'a. x"]
         by auto
       then have "x \<in> ?rhs"
         apply auto
         apply (rule_tac x = c in exI)
         apply (rule_tac x = s in exI)
         using * c_def s_def p \<open>x \<in> S i\<close>
         apply auto
         done
     }
     then have "?rhs \<supseteq> S i" by auto
   }
   then have *: "?rhs \<supseteq> \<Union>(S ` I)" by auto
 
   {
     fix u v :: real
     assume uv: "u \<ge> 0 \<and> v \<ge> 0 \<and> u + v = 1"
     fix x y
     assume xy: "x \<in> ?rhs \<and> y \<in> ?rhs"
     from xy obtain c s where
       xc: "x = sum (\<lambda>i. c i *\<^sub>R s i) I \<and> (\<forall>i\<in>I. c i \<ge> 0) \<and> sum c I = 1 \<and> (\<forall>i\<in>I. s i \<in> S i)"
       by auto
     from xy obtain d t where
       yc: "y = sum (\<lambda>i. d i *\<^sub>R t i) I \<and> (\<forall>i\<in>I. d i \<ge> 0) \<and> sum d I = 1 \<and> (\<forall>i\<in>I. t i \<in> S i)"
       by auto
     define e where "e i = u * c i + v * d i" for i
     have ge0: "\<forall>i\<in>I. e i \<ge> 0"
       using e_def xc yc uv by simp
     have "sum (\<lambda>i. u * c i) I = u * sum c I"
       by (simp add: sum_distrib_left)
     moreover have "sum (\<lambda>i. v * d i) I = v * sum d I"
       by (simp add: sum_distrib_left)
     ultimately have sum1: "sum e I = 1"
       using e_def xc yc uv by (simp add: sum.distrib)
     define q where "q i = (if e i = 0 then p i else (u * c i / e i) *\<^sub>R s i + (v * d i / e i) *\<^sub>R t i)"
       for i
     {
       fix i
       assume i: "i \<in> I"
       have "q i \<in> S i"
       proof (cases "e i = 0")
         case True
         then show ?thesis using i p q_def by auto
       next
         case False
         then show ?thesis
           using mem_convex_alt[of "S i" "s i" "t i" "u * (c i)" "v * (d i)"]
             mult_nonneg_nonneg[of u "c i"] mult_nonneg_nonneg[of v "d i"]
             assms q_def e_def i False xc yc uv
           by (auto simp del: mult_nonneg_nonneg)
       qed
     }
     then have qs: "\<forall>i\<in>I. q i \<in> S i" by auto
     {
       fix i
       assume i: "i \<in> I"
       have "(u * c i) *\<^sub>R s i + (v * d i) *\<^sub>R t i = e i *\<^sub>R q i"
       proof (cases "e i = 0")
         case True
         have ge: "u * (c i) \<ge> 0 \<and> v * d i \<ge> 0"
           using xc yc uv i by simp
         moreover from ge have "u * c i \<le> 0 \<and> v * d i \<le> 0"
           using True e_def i by simp
         ultimately have "u * c i = 0 \<and> v * d i = 0" by auto
         with True show ?thesis by auto
       next
         case False
         then have "(u * (c i)/(e i))*\<^sub>R (s i)+(v * (d i)/(e i))*\<^sub>R (t i) = q i"
           using q_def by auto
         then have "e i *\<^sub>R ((u * (c i)/(e i))*\<^sub>R (s i)+(v * (d i)/(e i))*\<^sub>R (t i))
                = (e i) *\<^sub>R (q i)" by auto
         with False show ?thesis by (simp add: algebra_simps)
       qed
     }
     then have *: "\<forall>i\<in>I. (u * c i) *\<^sub>R s i + (v * d i) *\<^sub>R t i = e i *\<^sub>R q i"
       by auto
     have "u *\<^sub>R x + v *\<^sub>R y = sum (\<lambda>i. (u * c i) *\<^sub>R s i + (v * d i) *\<^sub>R t i) I"
       using xc yc by (simp add: algebra_simps scaleR_right.sum sum.distrib)
     also have "\<dots> = sum (\<lambda>i. e i *\<^sub>R q i) I"
       using * by auto
     finally have "u *\<^sub>R x + v *\<^sub>R y = sum (\<lambda>i. (e i) *\<^sub>R (q i)) I"
       by auto
     then have "u *\<^sub>R x + v *\<^sub>R y \<in> ?rhs"
       using ge0 sum1 qs by auto
   }
   then have "convex ?rhs" unfolding convex_def by auto
   then show ?thesis
     using \<open>?lhs \<supseteq> ?rhs\<close> * hull_minimal[of "\<Union>(S ` I)" ?rhs convex]
     by blast
 qed
 
 lemma convex_hull_union_two:
   fixes S T :: "'m::euclidean_space set"
   assumes "convex S"
     and "S \<noteq> {}"
     and "convex T"
     and "T \<noteq> {}"
   shows "convex hull (S \<union> T) =
     {u *\<^sub>R s + v *\<^sub>R t | u v s t. u \<ge> 0 \<and> v \<ge> 0 \<and> u + v = 1 \<and> s \<in> S \<and> t \<in> T}"
   (is "?lhs = ?rhs")
 proof
   define I :: "nat set" where "I = {1, 2}"
   define s where "s i = (if i = (1::nat) then S else T)" for i
   have "\<Union>(s ` I) = S \<union> T"
     using s_def I_def by auto
   then have "convex hull (\<Union>(s ` I)) = convex hull (S \<union> T)"
     by auto
   moreover have "convex hull \<Union>(s ` I) =
     {\<Sum> i\<in>I. c i *\<^sub>R sa i | c sa. (\<forall>i\<in>I. 0 \<le> c i) \<and> sum c I = 1 \<and> (\<forall>i\<in>I. sa i \<in> s i)}"
       apply (subst convex_hull_finite_union[of I s])
       using assms s_def I_def
       apply auto
       done
   moreover have
     "{\<Sum>i\<in>I. c i *\<^sub>R sa i | c sa. (\<forall>i\<in>I. 0 \<le> c i) \<and> sum c I = 1 \<and> (\<forall>i\<in>I. sa i \<in> s i)} \<le> ?rhs"
     using s_def I_def by auto
   ultimately show "?lhs \<subseteq> ?rhs" by auto
   {
     fix x
     assume "x \<in> ?rhs"
     then obtain u v s t where *: "x = u *\<^sub>R s + v *\<^sub>R t \<and> u \<ge> 0 \<and> v \<ge> 0 \<and> u + v = 1 \<and> s \<in> S \<and> t \<in> T"
       by auto
     then have "x \<in> convex hull {s, t}"
       using convex_hull_2[of s t] by auto
     then have "x \<in> convex hull (S \<union> T)"
       using * hull_mono[of "{s, t}" "S \<union> T"] by auto
   }
   then show "?lhs \<supseteq> ?rhs" by blast
 qed
 
 proposition ray_to_rel_frontier:
   fixes a :: "'a::real_inner"
   assumes "bounded S"
       and a: "a \<in> rel_interior S"
       and aff: "(a + l) \<in> affine hull S"
       and "l \<noteq> 0"
   obtains d where "0 < d" "(a + d *\<^sub>R l) \<in> rel_frontier S"
            "\<And>e. \<lbrakk>0 \<le> e; e < d\<rbrakk> \<Longrightarrow> (a + e *\<^sub>R l) \<in> rel_interior S"
 proof -
   have aaff: "a \<in> affine hull S"
     by (meson a hull_subset rel_interior_subset rev_subsetD)
   let ?D = "{d. 0 < d \<and> a + d *\<^sub>R l \<notin> rel_interior S}"
   obtain B where "B > 0" and B: "S \<subseteq> ball a B"
     using bounded_subset_ballD [OF \<open>bounded S\<close>] by blast
   have "a + (B / norm l) *\<^sub>R l \<notin> ball a B"
     by (simp add: dist_norm \<open>l \<noteq> 0\<close>)
   with B have "a + (B / norm l) *\<^sub>R l \<notin> rel_interior S"
     using rel_interior_subset subsetCE by blast
   with \<open>B > 0\<close> \<open>l \<noteq> 0\<close> have nonMT: "?D \<noteq> {}"
     using divide_pos_pos zero_less_norm_iff by fastforce
   have bdd: "bdd_below ?D"
     by (metis (no_types, lifting) bdd_belowI le_less mem_Collect_eq)
   have relin_Ex: "\<And>x. x \<in> rel_interior S \<Longrightarrow>
                     \<exists>e>0. \<forall>x'\<in>affine hull S. dist x' x < e \<longrightarrow> x' \<in> rel_interior S"
     using openin_rel_interior [of S] by (simp add: openin_euclidean_subtopology_iff)
   define d where "d = Inf ?D"
   obtain \<epsilon> where "0 < \<epsilon>" and \<epsilon>: "\<And>\<eta>. \<lbrakk>0 \<le> \<eta>; \<eta> < \<epsilon>\<rbrakk> \<Longrightarrow> (a + \<eta> *\<^sub>R l) \<in> rel_interior S"
   proof -
     obtain e where "e>0"
             and e: "\<And>x'. x' \<in> affine hull S \<Longrightarrow> dist x' a < e \<Longrightarrow> x' \<in> rel_interior S"
       using relin_Ex a by blast
     show thesis
     proof (rule_tac \<epsilon> = "e / norm l" in that)
       show "0 < e / norm l" by (simp add: \<open>0 < e\<close> \<open>l \<noteq> 0\<close>)
     next
       show "a + \<eta> *\<^sub>R l \<in> rel_interior S" if "0 \<le> \<eta>" "\<eta> < e / norm l" for \<eta>
       proof (rule e)
         show "a + \<eta> *\<^sub>R l \<in> affine hull S"
           by (metis (no_types) add_diff_cancel_left' aff affine_affine_hull mem_affine_3_minus aaff)
         show "dist (a + \<eta> *\<^sub>R l) a < e"
           using that by (simp add: \<open>l \<noteq> 0\<close> dist_norm pos_less_divide_eq)
       qed
     qed
   qed
   have inint: "\<And>e. \<lbrakk>0 \<le> e; e < d\<rbrakk> \<Longrightarrow> a + e *\<^sub>R l \<in> rel_interior S"
     unfolding d_def using cInf_lower [OF _ bdd]
     by (metis (no_types, lifting) a add.right_neutral le_less mem_Collect_eq not_less real_vector.scale_zero_left)
   have "\<epsilon> \<le> d"
     unfolding d_def
     apply (rule cInf_greatest [OF nonMT])
     using \<epsilon> dual_order.strict_implies_order le_less_linear by blast
   with \<open>0 < \<epsilon>\<close> have "0 < d" by simp
   have "a + d *\<^sub>R l \<notin> rel_interior S"
   proof
     assume adl: "a + d *\<^sub>R l \<in> rel_interior S"
     obtain e where "e > 0"
              and e: "\<And>x'. x' \<in> affine hull S \<Longrightarrow> dist x' (a + d *\<^sub>R l) < e \<Longrightarrow> x' \<in> rel_interior S"
       using relin_Ex adl by blast
     have "d + e / norm l \<le> Inf {d. 0 < d \<and> a + d *\<^sub>R l \<notin> rel_interior S}"
     proof (rule cInf_greatest [OF nonMT], clarsimp)
       fix x::real
       assume "0 < x" and nonrel: "a + x *\<^sub>R l \<notin> rel_interior S"
       show "d + e / norm l \<le> x"
       proof (cases "x < d")
         case True with inint nonrel \<open>0 < x\<close>
           show ?thesis by auto
       next
         case False
           then have dle: "x < d + e / norm l \<Longrightarrow> dist (a + x *\<^sub>R l) (a + d *\<^sub>R l) < e"
             by (simp add: field_simps \<open>l \<noteq> 0\<close>)
           have ain: "a + x *\<^sub>R l \<in> affine hull S"
             by (metis add_diff_cancel_left' aff affine_affine_hull mem_affine_3_minus aaff)
           show ?thesis
             using e [OF ain] nonrel dle by force
       qed
     qed
     then show False
       using \<open>0 < e\<close> \<open>l \<noteq> 0\<close> by (simp add: d_def [symmetric] field_simps)
   qed
   moreover have "a + d *\<^sub>R l \<in> closure S"
   proof (clarsimp simp: closure_approachable)
     fix \<eta>::real assume "0 < \<eta>"
     have 1: "a + (d - min d (\<eta> / 2 / norm l)) *\<^sub>R l \<in> S"
       apply (rule subsetD [OF rel_interior_subset inint])
       using \<open>l \<noteq> 0\<close> \<open>0 < d\<close> \<open>0 < \<eta>\<close> by auto
     have "norm l * min d (\<eta> / (norm l * 2)) \<le> norm l * (\<eta> / (norm l * 2))"
       by (metis min_def mult_left_mono norm_ge_zero order_refl)
     also have "... < \<eta>"
       using \<open>l \<noteq> 0\<close> \<open>0 < \<eta>\<close> by (simp add: field_simps)
     finally have 2: "norm l * min d (\<eta> / (norm l * 2)) < \<eta>" .
     show "\<exists>y\<in>S. dist y (a + d *\<^sub>R l) < \<eta>"
       apply (rule_tac x="a + (d - min d (\<eta> / 2 / norm l)) *\<^sub>R l" in bexI)
       using 1 2 \<open>0 < d\<close> \<open>0 < \<eta>\<close> apply (auto simp: algebra_simps)
       done
   qed
   ultimately have infront: "a + d *\<^sub>R l \<in> rel_frontier S"
     by (simp add: rel_frontier_def)
   show ?thesis
     by (rule that [OF \<open>0 < d\<close> infront inint])
 qed
 
 corollary ray_to_frontier:
   fixes a :: "'a::euclidean_space"
   assumes "bounded S"
       and a: "a \<in> interior S"
       and "l \<noteq> 0"
   obtains d where "0 < d" "(a + d *\<^sub>R l) \<in> frontier S"
            "\<And>e. \<lbrakk>0 \<le> e; e < d\<rbrakk> \<Longrightarrow> (a + e *\<^sub>R l) \<in> interior S"
 proof -
   have "interior S = rel_interior S"
     using a rel_interior_nonempty_interior by auto
   then have "a \<in> rel_interior S"
     using a by simp
   then show ?thesis
     apply (rule ray_to_rel_frontier [OF \<open>bounded S\<close> _ _ \<open>l \<noteq> 0\<close>])
      using a affine_hull_nonempty_interior apply blast
     by (simp add: \<open>interior S = rel_interior S\<close> frontier_def rel_frontier_def that)
 qed
 
 
 lemma segment_to_rel_frontier_aux:
   fixes x :: "'a::euclidean_space"
   assumes "convex S" "bounded S" and x: "x \<in> rel_interior S" and y: "y \<in> S" and xy: "x \<noteq> y"
   obtains z where "z \<in> rel_frontier S" "y \<in> closed_segment x z"
                    "open_segment x z \<subseteq> rel_interior S"
 proof -
   have "x + (y - x) \<in> affine hull S"
     using hull_inc [OF y] by auto
   then obtain d where "0 < d" and df: "(x + d *\<^sub>R (y-x)) \<in> rel_frontier S"
                   and di: "\<And>e. \<lbrakk>0 \<le> e; e < d\<rbrakk> \<Longrightarrow> (x + e *\<^sub>R (y-x)) \<in> rel_interior S"
     by (rule ray_to_rel_frontier [OF \<open>bounded S\<close> x]) (use xy in auto)
   show ?thesis
   proof
     show "x + d *\<^sub>R (y - x) \<in> rel_frontier S"
       by (simp add: df)
   next
     have "open_segment x y \<subseteq> rel_interior S"
       using rel_interior_closure_convex_segment [OF \<open>convex S\<close> x] closure_subset y by blast
     moreover have "x + d *\<^sub>R (y - x) \<in> open_segment x y" if "d < 1"
       using xy
       apply (auto simp: in_segment)
       apply (rule_tac x="d" in exI)
       using \<open>0 < d\<close> that apply (auto simp: algebra_simps)
       done
     ultimately have "1 \<le> d"
       using df rel_frontier_def by fastforce
     moreover have "x = (1 / d) *\<^sub>R x + ((d - 1) / d) *\<^sub>R x"
       by (metis \<open>0 < d\<close> add.commute add_divide_distrib diff_add_cancel divide_self_if less_irrefl scaleR_add_left scaleR_one)
     ultimately show "y \<in> closed_segment x (x + d *\<^sub>R (y - x))"
       apply (auto simp: in_segment)
       apply (rule_tac x="1/d" in exI)
       apply (auto simp: algebra_simps)
       done
   next
     show "open_segment x (x + d *\<^sub>R (y - x)) \<subseteq> rel_interior S"
       apply (rule rel_interior_closure_convex_segment [OF \<open>convex S\<close> x])
       using df rel_frontier_def by auto
   qed
 qed
 
 lemma segment_to_rel_frontier:
   fixes x :: "'a::euclidean_space"
   assumes S: "convex S" "bounded S" and x: "x \<in> rel_interior S"
       and y: "y \<in> S" and xy: "\<not>(x = y \<and> S = {x})"
   obtains z where "z \<in> rel_frontier S" "y \<in> closed_segment x z"
                   "open_segment x z \<subseteq> rel_interior S"
 proof (cases "x=y")
   case True
   with xy have "S \<noteq> {x}"
     by blast
   with True show ?thesis
     by (metis Set.set_insert all_not_in_conv ends_in_segment(1) insert_iff segment_to_rel_frontier_aux[OF S x] that y)
 next
   case False
   then show ?thesis
     using segment_to_rel_frontier_aux [OF S x y] that by blast
 qed
 
 proposition rel_frontier_not_sing:
   fixes a :: "'a::euclidean_space"
   assumes "bounded S"
     shows "rel_frontier S \<noteq> {a}"
 proof (cases "S = {}")
   case True  then show ?thesis  by simp
 next
   case False
   then obtain z where "z \<in> S"
     by blast
   then show ?thesis
   proof (cases "S = {z}")
     case True then show ?thesis  by simp
   next
     case False
     then obtain w where "w \<in> S" "w \<noteq> z"
       using \<open>z \<in> S\<close> by blast
     show ?thesis
     proof
       assume "rel_frontier S = {a}"
       then consider "w \<notin> rel_frontier S" | "z \<notin> rel_frontier S"
         using \<open>w \<noteq> z\<close> by auto
       then show False
       proof cases
         case 1
         then have w: "w \<in> rel_interior S"
           using \<open>w \<in> S\<close> closure_subset rel_frontier_def by fastforce
         have "w + (w - z) \<in> affine hull S"
           by (metis \<open>w \<in> S\<close> \<open>z \<in> S\<close> affine_affine_hull hull_inc mem_affine_3_minus scaleR_one)
         then obtain e where "0 < e" "(w + e *\<^sub>R (w - z)) \<in> rel_frontier S"
           using \<open>w \<noteq> z\<close>  \<open>z \<in> S\<close> by (metis assms ray_to_rel_frontier right_minus_eq w)
         moreover obtain d where "0 < d" "(w + d *\<^sub>R (z - w)) \<in> rel_frontier S"
           using ray_to_rel_frontier [OF \<open>bounded S\<close> w, of "1 *\<^sub>R (z - w)"]  \<open>w \<noteq> z\<close>  \<open>z \<in> S\<close>
           by (metis add.commute add.right_neutral diff_add_cancel hull_inc scaleR_one)
         ultimately have "d *\<^sub>R (z - w) = e *\<^sub>R (w - z)"
           using \<open>rel_frontier S = {a}\<close> by force
         moreover have "e \<noteq> -d "
           using \<open>0 < e\<close> \<open>0 < d\<close> by force
         ultimately show False
           by (metis (no_types, lifting) \<open>w \<noteq> z\<close> eq_iff_diff_eq_0 minus_diff_eq real_vector.scale_cancel_right real_vector.scale_minus_right scaleR_left.minus)
       next
         case 2
         then have z: "z \<in> rel_interior S"
           using \<open>z \<in> S\<close> closure_subset rel_frontier_def by fastforce
         have "z + (z - w) \<in> affine hull S"
           by (metis \<open>z \<in> S\<close> \<open>w \<in> S\<close> affine_affine_hull hull_inc mem_affine_3_minus scaleR_one)
         then obtain e where "0 < e" "(z + e *\<^sub>R (z - w)) \<in> rel_frontier S"
           using \<open>w \<noteq> z\<close>  \<open>w \<in> S\<close> by (metis assms ray_to_rel_frontier right_minus_eq z)
         moreover obtain d where "0 < d" "(z + d *\<^sub>R (w - z)) \<in> rel_frontier S"
           using ray_to_rel_frontier [OF \<open>bounded S\<close> z, of "1 *\<^sub>R (w - z)"]  \<open>w \<noteq> z\<close>  \<open>w \<in> S\<close>
           by (metis add.commute add.right_neutral diff_add_cancel hull_inc scaleR_one)
         ultimately have "d *\<^sub>R (w - z) = e *\<^sub>R (z - w)"
           using \<open>rel_frontier S = {a}\<close> by force
         moreover have "e \<noteq> -d "
           using \<open>0 < e\<close> \<open>0 < d\<close> by force
         ultimately show False
           by (metis (no_types, lifting) \<open>w \<noteq> z\<close> eq_iff_diff_eq_0 minus_diff_eq real_vector.scale_cancel_right real_vector.scale_minus_right scaleR_left.minus)
       qed
     qed
   qed
 qed
 
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>Convexity on direct sums\<close>
 
 lemma closure_sum:
   fixes S T :: "'a::real_normed_vector set"
   shows "closure S + closure T \<subseteq> closure (S + T)"
   unfolding set_plus_image closure_Times [symmetric] split_def
   by (intro closure_bounded_linear_image_subset bounded_linear_add
     bounded_linear_fst bounded_linear_snd)
 
 lemma rel_interior_sum:
   fixes S T :: "'n::euclidean_space set"
   assumes "convex S"
     and "convex T"
   shows "rel_interior (S + T) = rel_interior S + rel_interior T"
 proof -
   have "rel_interior S + rel_interior T = (\<lambda>(x,y). x + y) ` (rel_interior S \<times> rel_interior T)"
     by (simp add: set_plus_image)
   also have "\<dots> = (\<lambda>(x,y). x + y) ` rel_interior (S \<times> T)"
     using rel_interior_Times assms by auto
   also have "\<dots> = rel_interior (S + T)"
     using fst_snd_linear convex_Times assms
       rel_interior_convex_linear_image[of "(\<lambda>(x,y). x + y)" "S \<times> T"]
     by (auto simp add: set_plus_image)
   finally show ?thesis ..
 qed
 
 lemma rel_interior_sum_gen:
   fixes S :: "'a \<Rightarrow> 'n::euclidean_space set"
   assumes "\<forall>i\<in>I. convex (S i)"
   shows "rel_interior (sum S I) = sum (\<lambda>i. rel_interior (S i)) I"
   apply (subst sum_set_cond_linear[of convex])
   using rel_interior_sum rel_interior_sing[of "0"] assms
   apply (auto simp add: convex_set_plus)
   done
 
 lemma convex_rel_open_direct_sum:
   fixes S T :: "'n::euclidean_space set"
   assumes "convex S"
     and "rel_open S"
     and "convex T"
     and "rel_open T"
   shows "convex (S \<times> T) \<and> rel_open (S \<times> T)"
   by (metis assms convex_Times rel_interior_Times rel_open_def)
 
 lemma convex_rel_open_sum:
   fixes S T :: "'n::euclidean_space set"
   assumes "convex S"
     and "rel_open S"
     and "convex T"
     and "rel_open T"
   shows "convex (S + T) \<and> rel_open (S + T)"
   by (metis assms convex_set_plus rel_interior_sum rel_open_def)
 
 lemma convex_hull_finite_union_cones:
   assumes "finite I"
     and "I \<noteq> {}"
   assumes "\<forall>i\<in>I. convex (S i) \<and> cone (S i) \<and> S i \<noteq> {}"
   shows "convex hull (\<Union>(S ` I)) = sum S I"
   (is "?lhs = ?rhs")
 proof -
   {
     fix x
     assume "x \<in> ?lhs"
     then obtain c xs where
       x: "x = sum (\<lambda>i. c i *\<^sub>R xs i) I \<and> (\<forall>i\<in>I. c i \<ge> 0) \<and> sum c I = 1 \<and> (\<forall>i\<in>I. xs i \<in> S i)"
       using convex_hull_finite_union[of I S] assms by auto
     define s where "s i = c i *\<^sub>R xs i" for i
     {
       fix i
       assume "i \<in> I"
       then have "s i \<in> S i"
         using s_def x assms mem_cone[of "S i" "xs i" "c i"] by auto
     }
     then have "\<forall>i\<in>I. s i \<in> S i" by auto
     moreover have "x = sum s I" using x s_def by auto
     ultimately have "x \<in> ?rhs"
       using set_sum_alt[of I S] assms by auto
   }
   moreover
   {
     fix x
     assume "x \<in> ?rhs"
     then obtain s where x: "x = sum s I \<and> (\<forall>i\<in>I. s i \<in> S i)"
       using set_sum_alt[of I S] assms by auto
     define xs where "xs i = of_nat(card I) *\<^sub>R s i" for i
     then have "x = sum (\<lambda>i. ((1 :: real) / of_nat(card I)) *\<^sub>R xs i) I"
       using x assms by auto
     moreover have "\<forall>i\<in>I. xs i \<in> S i"
       using x xs_def assms by (simp add: cone_def)
     moreover have "\<forall>i\<in>I. (1 :: real) / of_nat (card I) \<ge> 0"
       by auto
     moreover have "sum (\<lambda>i. (1 :: real) / of_nat (card I)) I = 1"
       using assms by auto
     ultimately have "x \<in> ?lhs"
       apply (subst convex_hull_finite_union[of I S])
       using assms
       apply blast
       using assms
       apply blast
       apply rule
       apply (rule_tac x = "(\<lambda>i. (1 :: real) / of_nat (card I))" in exI)
       apply auto
       done
   }
   ultimately show ?thesis by auto
 qed
 
 lemma convex_hull_union_cones_two:
   fixes S T :: "'m::euclidean_space set"
   assumes "convex S"
     and "cone S"
     and "S \<noteq> {}"
   assumes "convex T"
     and "cone T"
     and "T \<noteq> {}"
   shows "convex hull (S \<union> T) = S + T"
 proof -
   define I :: "nat set" where "I = {1, 2}"
   define A where "A i = (if i = (1::nat) then S else T)" for i
   have "\<Union>(A ` I) = S \<union> T"
     using A_def I_def by auto
   then have "convex hull (\<Union>(A ` I)) = convex hull (S \<union> T)"
     by auto
   moreover have "convex hull \<Union>(A ` I) = sum A I"
     apply (subst convex_hull_finite_union_cones[of I A])
     using assms A_def I_def
     apply auto
     done
   moreover have "sum A I = S + T"
     using A_def I_def
     unfolding set_plus_def
     apply auto
     unfolding set_plus_def
     apply auto
     done
   ultimately show ?thesis by auto
 qed
 
 lemma rel_interior_convex_hull_union:
   fixes S :: "'a \<Rightarrow> 'n::euclidean_space set"
   assumes "finite I"
     and "\<forall>i\<in>I. convex (S i) \<and> S i \<noteq> {}"
   shows "rel_interior (convex hull (\<Union>(S ` I))) =
     {sum (\<lambda>i. c i *\<^sub>R s i) I | c s. (\<forall>i\<in>I. c i > 0) \<and> sum c I = 1 \<and>
       (\<forall>i\<in>I. s i \<in> rel_interior(S i))}"
   (is "?lhs = ?rhs")
 proof (cases "I = {}")
   case True
   then show ?thesis
     using convex_hull_empty by auto
 next
   case False
   define C0 where "C0 = convex hull (\<Union>(S ` I))"
   have "\<forall>i\<in>I. C0 \<ge> S i"
     unfolding C0_def using hull_subset[of "\<Union>(S ` I)"] by auto
   define K0 where "K0 = cone hull ({1 :: real} \<times> C0)"
   define K where "K i = cone hull ({1 :: real} \<times> S i)" for i
   have "\<forall>i\<in>I. K i \<noteq> {}"
     unfolding K_def using assms
     by (simp add: cone_hull_empty_iff[symmetric])
   {
     fix i
     assume "i \<in> I"
     then have "convex (K i)"
       unfolding K_def
       apply (subst convex_cone_hull)
       apply (subst convex_Times)
       using assms
       apply auto
       done
   }
   then have convK: "\<forall>i\<in>I. convex (K i)"
     by auto
   {
     fix i
     assume "i \<in> I"
     then have "K0 \<supseteq> K i"
       unfolding K0_def K_def
       apply (subst hull_mono)
       using \<open>\<forall>i\<in>I. C0 \<ge> S i\<close>
       apply auto
       done
   }
   then have "K0 \<supseteq> \<Union>(K ` I)" by auto
   moreover have "convex K0"
     unfolding K0_def
     apply (subst convex_cone_hull)
     apply (subst convex_Times)
     unfolding C0_def
     using convex_convex_hull
     apply auto
     done
   ultimately have geq: "K0 \<supseteq> convex hull (\<Union>(K ` I))"
     using hull_minimal[of _ "K0" "convex"] by blast
   have "\<forall>i\<in>I. K i \<supseteq> {1 :: real} \<times> S i"
     using K_def by (simp add: hull_subset)
   then have "\<Union>(K ` I) \<supseteq> {1 :: real} \<times> \<Union>(S ` I)"
     by auto
   then have "convex hull \<Union>(K ` I) \<supseteq> convex hull ({1 :: real} \<times> \<Union>(S ` I))"
     by (simp add: hull_mono)
   then have "convex hull \<Union>(K ` I) \<supseteq> {1 :: real} \<times> C0"
     unfolding C0_def
     using convex_hull_Times[of "{(1 :: real)}" "\<Union>(S ` I)"] convex_hull_singleton
     by auto
   moreover have "cone (convex hull (\<Union>(K ` I)))"
     apply (subst cone_convex_hull)
     using cone_Union[of "K ` I"]
     apply auto
     unfolding K_def
     using cone_cone_hull
     apply auto
     done
   ultimately have "convex hull (\<Union>(K ` I)) \<supseteq> K0"
     unfolding K0_def
     using hull_minimal[of _ "convex hull (\<Union>(K ` I))" "cone"]
     by blast
   then have "K0 = convex hull (\<Union>(K ` I))"
     using geq by auto
   also have "\<dots> = sum K I"
     apply (subst convex_hull_finite_union_cones[of I K])
     using assms
     apply blast
     using False
     apply blast
     unfolding K_def
     apply rule
     apply (subst convex_cone_hull)
     apply (subst convex_Times)
     using assms cone_cone_hull \<open>\<forall>i\<in>I. K i \<noteq> {}\<close> K_def
     apply auto
     done
   finally have "K0 = sum K I" by auto
   then have *: "rel_interior K0 = sum (\<lambda>i. (rel_interior (K i))) I"
     using rel_interior_sum_gen[of I K] convK by auto
   {
     fix x
     assume "x \<in> ?lhs"
     then have "(1::real, x) \<in> rel_interior K0"
       using K0_def C0_def rel_interior_convex_cone_aux[of C0 "1::real" x] convex_convex_hull
       by auto
     then obtain k where k: "(1::real, x) = sum k I \<and> (\<forall>i\<in>I. k i \<in> rel_interior (K i))"
       using \<open>finite I\<close> * set_sum_alt[of I "\<lambda>i. rel_interior (K i)"] by auto
     {
       fix i
       assume "i \<in> I"
       then have "convex (S i) \<and> k i \<in> rel_interior (cone hull {1} \<times> S i)"
         using k K_def assms by auto
       then have "\<exists>ci si. k i = (ci, ci *\<^sub>R si) \<and> 0 < ci \<and> si \<in> rel_interior (S i)"
         using rel_interior_convex_cone[of "S i"] by auto
     }
     then obtain c s where
       cs: "\<forall>i\<in>I. k i = (c i, c i *\<^sub>R s i) \<and> 0 < c i \<and> s i \<in> rel_interior (S i)"
       by metis
     then have "x = (\<Sum>i\<in>I. c i *\<^sub>R s i) \<and> sum c I = 1"
       using k by (simp add: sum_prod)
     then have "x \<in> ?rhs"
       using k cs by auto
   }
   moreover
   {
     fix x
     assume "x \<in> ?rhs"
     then obtain c s where cs: "x = sum (\<lambda>i. c i *\<^sub>R s i) I \<and>
         (\<forall>i\<in>I. c i > 0) \<and> sum c I = 1 \<and> (\<forall>i\<in>I. s i \<in> rel_interior (S i))"
       by auto
     define k where "k i = (c i, c i *\<^sub>R s i)" for i
     {
       fix i assume "i \<in> I"
       then have "k i \<in> rel_interior (K i)"
         using k_def K_def assms cs rel_interior_convex_cone[of "S i"]
         by auto
     }
     then have "(1::real, x) \<in> rel_interior K0"
       using K0_def * set_sum_alt[of I "(\<lambda>i. rel_interior (K i))"] assms k_def cs
       apply auto
       apply (rule_tac x = k in exI)
       apply (simp add: sum_prod)
       done
     then have "x \<in> ?lhs"
       using K0_def C0_def rel_interior_convex_cone_aux[of C0 1 x]
       by auto
   }
   ultimately show ?thesis by blast
 qed
 
 
 lemma convex_le_Inf_differential:
   fixes f :: "real \<Rightarrow> real"
   assumes "convex_on I f"
     and "x \<in> interior I"
     and "y \<in> I"
   shows "f y \<ge> f x + Inf ((\<lambda>t. (f x - f t) / (x - t)) ` ({x<..} \<inter> I)) * (y - x)"
   (is "_ \<ge> _ + Inf (?F x) * (y - x)")
 proof (cases rule: linorder_cases)
   assume "x < y"
   moreover
   have "open (interior I)" by auto
   from openE[OF this \<open>x \<in> interior I\<close>]
   obtain e where e: "0 < e" "ball x e \<subseteq> interior I" .
   moreover define t where "t = min (x + e / 2) ((x + y) / 2)"
   ultimately have "x < t" "t < y" "t \<in> ball x e"
     by (auto simp: dist_real_def field_simps split: split_min)
   with \<open>x \<in> interior I\<close> e interior_subset[of I] have "t \<in> I" "x \<in> I" by auto
 
   have "open (interior I)" by auto
   from openE[OF this \<open>x \<in> interior I\<close>]
   obtain e where "0 < e" "ball x e \<subseteq> interior I" .
   moreover define K where "K = x - e / 2"
   with \<open>0 < e\<close> have "K \<in> ball x e" "K < x"
     by (auto simp: dist_real_def)
   ultimately have "K \<in> I" "K < x" "x \<in> I"
     using interior_subset[of I] \<open>x \<in> interior I\<close> by auto
 
   have "Inf (?F x) \<le> (f x - f y) / (x - y)"
   proof (intro bdd_belowI cInf_lower2)
     show "(f x - f t) / (x - t) \<in> ?F x"
       using \<open>t \<in> I\<close> \<open>x < t\<close> by auto
     show "(f x - f t) / (x - t) \<le> (f x - f y) / (x - y)"
       using \<open>convex_on I f\<close> \<open>x \<in> I\<close> \<open>y \<in> I\<close> \<open>x < t\<close> \<open>t < y\<close>
       by (rule convex_on_diff)
   next
     fix y
     assume "y \<in> ?F x"
     with order_trans[OF convex_on_diff[OF \<open>convex_on I f\<close> \<open>K \<in> I\<close> _ \<open>K < x\<close> _]]
     show "(f K - f x) / (K - x) \<le> y" by auto
   qed
   then show ?thesis
     using \<open>x < y\<close> by (simp add: field_simps)
 next
   assume "y < x"
   moreover
   have "open (interior I)" by auto
   from openE[OF this \<open>x \<in> interior I\<close>]
   obtain e where e: "0 < e" "ball x e \<subseteq> interior I" .
   moreover define t where "t = x + e / 2"
   ultimately have "x < t" "t \<in> ball x e"
     by (auto simp: dist_real_def field_simps)
   with \<open>x \<in> interior I\<close> e interior_subset[of I] have "t \<in> I" "x \<in> I" by auto
 
   have "(f x - f y) / (x - y) \<le> Inf (?F x)"
   proof (rule cInf_greatest)
     have "(f x - f y) / (x - y) = (f y - f x) / (y - x)"
       using \<open>y < x\<close> by (auto simp: field_simps)
     also
     fix z
     assume "z \<in> ?F x"
     with order_trans[OF convex_on_diff[OF \<open>convex_on I f\<close> \<open>y \<in> I\<close> _ \<open>y < x\<close>]]
     have "(f y - f x) / (y - x) \<le> z"
       by auto
     finally show "(f x - f y) / (x - y) \<le> z" .
   next
     have "open (interior I)" by auto
     from openE[OF this \<open>x \<in> interior I\<close>]
     obtain e where e: "0 < e" "ball x e \<subseteq> interior I" .
     then have "x + e / 2 \<in> ball x e"
       by (auto simp: dist_real_def)
     with e interior_subset[of I] have "x + e / 2 \<in> {x<..} \<inter> I"
       by auto
     then show "?F x \<noteq> {}"
       by blast
   qed
   then show ?thesis
     using \<open>y < x\<close> by (simp add: field_simps)
 qed simp
 
 subsection\<^marker>\<open>tag unimportant\<close>\<open>Explicit formulas for interior and relative interior of convex hull\<close>
 
 lemma at_within_cbox_finite:
   assumes "x \<in> box a b" "x \<notin> S" "finite S"
   shows "(at x within cbox a b - S) = at x"
 proof -
   have "interior (cbox a b - S) = box a b - S"
     using \<open>finite S\<close> by (simp add: interior_diff finite_imp_closed)
   then show ?thesis
     using at_within_interior assms by fastforce
 qed
 
 lemma affine_independent_convex_affine_hull:
   fixes s :: "'a::euclidean_space set"
   assumes "\<not> affine_dependent s" "t \<subseteq> s"
     shows "convex hull t = affine hull t \<inter> convex hull s"
 proof -
   have fin: "finite s" "finite t" using assms aff_independent_finite finite_subset by auto
     { fix u v x
       assume uv: "sum u t = 1" "\<forall>x\<in>s. 0 \<le> v x" "sum v s = 1"
                  "(\<Sum>x\<in>s. v x *\<^sub>R x) = (\<Sum>v\<in>t. u v *\<^sub>R v)" "x \<in> t"
       then have s: "s = (s - t) \<union> t" \<comment> \<open>split into separate cases\<close>
         using assms by auto
       have [simp]: "(\<Sum>x\<in>t. v x *\<^sub>R x) + (\<Sum>x\<in>s - t. v x *\<^sub>R x) = (\<Sum>x\<in>t. u x *\<^sub>R x)"
                    "sum v t + sum v (s - t) = 1"
         using uv fin s
         by (auto simp: sum.union_disjoint [symmetric] Un_commute)
       have "(\<Sum>x\<in>s. if x \<in> t then v x - u x else v x) = 0"
            "(\<Sum>x\<in>s. (if x \<in> t then v x - u x else v x) *\<^sub>R x) = 0"
         using uv fin
         by (subst s, subst sum.union_disjoint, auto simp: algebra_simps sum_subtractf)+
     } note [simp] = this
   have "convex hull t \<subseteq> affine hull t"
     using convex_hull_subset_affine_hull by blast
   moreover have "convex hull t \<subseteq> convex hull s"
     using assms hull_mono by blast
   moreover have "affine hull t \<inter> convex hull s \<subseteq> convex hull t"
     using assms
     apply (simp add: convex_hull_finite affine_hull_finite fin affine_dependent_explicit)
     apply (drule_tac x=s in spec)
     apply (auto simp: fin)
     apply (rule_tac x=u in exI)
     apply (rename_tac v)
     apply (drule_tac x="\<lambda>x. if x \<in> t then v x - u x else v x" in spec)
     apply (force)+
     done
   ultimately show ?thesis
     by blast
 qed
 
 lemma affine_independent_span_eq:
   fixes s :: "'a::euclidean_space set"
   assumes "\<not> affine_dependent s" "card s = Suc (DIM ('a))"
     shows "affine hull s = UNIV"
 proof (cases "s = {}")
   case True then show ?thesis
     using assms by simp
 next
   case False
     then obtain a t where t: "a \<notin> t" "s = insert a t"
       by blast
     then have fin: "finite t" using assms
       by (metis finite_insert aff_independent_finite)
     show ?thesis
     using assms t fin
       apply (simp add: affine_dependent_iff_dependent affine_hull_insert_span_gen)
       apply (rule subset_antisym)
       apply force
       apply (rule Fun.vimage_subsetD)
       apply (metis add.commute diff_add_cancel surj_def)
       apply (rule card_ge_dim_independent)
       apply (auto simp: card_image inj_on_def dim_subset_UNIV)
       done
 qed
 
 lemma affine_independent_span_gt:
   fixes s :: "'a::euclidean_space set"
   assumes ind: "\<not> affine_dependent s" and dim: "DIM ('a) < card s"
     shows "affine hull s = UNIV"
   apply (rule affine_independent_span_eq [OF ind])
   apply (rule antisym)
   using assms
   apply auto
   apply (metis add_2_eq_Suc' not_less_eq_eq affine_dependent_biggerset aff_independent_finite)
   done
 
 lemma empty_interior_affine_hull:
   fixes s :: "'a::euclidean_space set"
   assumes "finite s" and dim: "card s \<le> DIM ('a)"
     shows "interior(affine hull s) = {}"
   using assms
   apply (induct s rule: finite_induct)
   apply (simp_all add:  affine_dependent_iff_dependent affine_hull_insert_span_gen interior_translation)
   apply (rule empty_interior_lowdim)
   by (auto simp: Suc_le_lessD card_image_le dual_order.trans intro!: dim_le_card'[THEN le_less_trans])
 
 lemma empty_interior_convex_hull:
   fixes s :: "'a::euclidean_space set"
   assumes "finite s" and dim: "card s \<le> DIM ('a)"
     shows "interior(convex hull s) = {}"
   by (metis Diff_empty Diff_eq_empty_iff convex_hull_subset_affine_hull
             interior_mono empty_interior_affine_hull [OF assms])
 
 lemma explicit_subset_rel_interior_convex_hull:
   fixes s :: "'a::euclidean_space set"
   shows "finite s
          \<Longrightarrow> {y. \<exists>u. (\<forall>x \<in> s. 0 < u x \<and> u x < 1) \<and> sum u s = 1 \<and> sum (\<lambda>x. u x *\<^sub>R x) s = y}
              \<subseteq> rel_interior (convex hull s)"
   by (force simp add:  rel_interior_convex_hull_union [where S="\<lambda>x. {x}" and I=s, simplified])
 
 lemma explicit_subset_rel_interior_convex_hull_minimal:
   fixes s :: "'a::euclidean_space set"
   shows "finite s
          \<Longrightarrow> {y. \<exists>u. (\<forall>x \<in> s. 0 < u x) \<and> sum u s = 1 \<and> sum (\<lambda>x. u x *\<^sub>R x) s = y}
              \<subseteq> rel_interior (convex hull s)"
   by (force simp add:  rel_interior_convex_hull_union [where S="\<lambda>x. {x}" and I=s, simplified])
 
 lemma rel_interior_convex_hull_explicit:
   fixes s :: "'a::euclidean_space set"
   assumes "\<not> affine_dependent s"
   shows "rel_interior(convex hull s) =
          {y. \<exists>u. (\<forall>x \<in> s. 0 < u x) \<and> sum u s = 1 \<and> sum (\<lambda>x. u x *\<^sub>R x) s = y}"
          (is "?lhs = ?rhs")
 proof
   show "?rhs \<le> ?lhs"
     by (simp add: aff_independent_finite explicit_subset_rel_interior_convex_hull_minimal assms)
 next
   show "?lhs \<le> ?rhs"
   proof (cases "\<exists>a. s = {a}")
     case True then show "?lhs \<le> ?rhs"
       by force
   next
     case False
     have fs: "finite s"
       using assms by (simp add: aff_independent_finite)
     { fix a b and d::real
       assume ab: "a \<in> s" "b \<in> s" "a \<noteq> b"
       then have s: "s = (s - {a,b}) \<union> {a,b}" \<comment> \<open>split into separate cases\<close>
         by auto
       have "(\<Sum>x\<in>s. if x = a then - d else if x = b then d else 0) = 0"
            "(\<Sum>x\<in>s. (if x = a then - d else if x = b then d else 0) *\<^sub>R x) = d *\<^sub>R b - d *\<^sub>R a"
         using ab fs
         by (subst s, subst sum.union_disjoint, auto)+
     } note [simp] = this
     { fix y
       assume y: "y \<in> convex hull s" "y \<notin> ?rhs"
       { fix u T a
         assume ua: "\<forall>x\<in>s. 0 \<le> u x" "sum u s = 1" "\<not> 0 < u a" "a \<in> s"
            and yT: "y = (\<Sum>x\<in>s. u x *\<^sub>R x)" "y \<in> T" "open T"
            and sb: "T \<inter> affine hull s \<subseteq> {w. \<exists>u. (\<forall>x\<in>s. 0 \<le> u x) \<and> sum u s = 1 \<and> (\<Sum>x\<in>s. u x *\<^sub>R x) = w}"
         have ua0: "u a = 0"
           using ua by auto
         obtain b where b: "b\<in>s" "a \<noteq> b"
           using ua False by auto
         obtain e where e: "0 < e" "ball (\<Sum>x\<in>s. u x *\<^sub>R x) e \<subseteq> T"
           using yT by (auto elim: openE)
         with b obtain d where d: "0 < d" "norm(d *\<^sub>R (a-b)) < e"
           by (auto intro: that [of "e / 2 / norm(a-b)"])
         have "(\<Sum>x\<in>s. u x *\<^sub>R x) \<in> affine hull s"
           using yT y by (metis affine_hull_convex_hull hull_redundant_eq)
         then have "(\<Sum>x\<in>s. u x *\<^sub>R x) - d *\<^sub>R (a - b) \<in> affine hull s"
           using ua b by (auto simp: hull_inc intro: mem_affine_3_minus2)
         then have "y - d *\<^sub>R (a - b) \<in> T \<inter> affine hull s"
           using d e yT by auto
         then obtain v where "\<forall>x\<in>s. 0 \<le> v x"
                             "sum v s = 1"
                             "(\<Sum>x\<in>s. v x *\<^sub>R x) = (\<Sum>x\<in>s. u x *\<^sub>R x) - d *\<^sub>R (a - b)"
           using subsetD [OF sb] yT
           by auto
         then have False
           using assms
           apply (simp add: affine_dependent_explicit_finite fs)
           apply (drule_tac x="\<lambda>x. (v x - u x) - (if x = a then -d else if x = b then d else 0)" in spec)
           using ua b d
           apply (auto simp: algebra_simps sum_subtractf sum.distrib)
           done
       } note * = this
       have "y \<notin> rel_interior (convex hull s)"
         using y
         apply (simp add: mem_rel_interior)
         apply (auto simp: convex_hull_finite [OF fs])
         apply (drule_tac x=u in spec)
         apply (auto intro: *)
         done
     } with rel_interior_subset show "?lhs \<le> ?rhs"
       by blast
   qed
 qed
 
 lemma interior_convex_hull_explicit_minimal:
   fixes s :: "'a::euclidean_space set"
   shows
    "\<not> affine_dependent s
         ==> interior(convex hull s) =
              (if card(s) \<le> DIM('a) then {}
               else {y. \<exists>u. (\<forall>x \<in> s. 0 < u x) \<and> sum u s = 1 \<and> (\<Sum>x\<in>s. u x *\<^sub>R x) = y})"
   apply (simp add: aff_independent_finite empty_interior_convex_hull, clarify)
   apply (rule trans [of _ "rel_interior(convex hull s)"])
   apply (simp add: affine_independent_span_gt rel_interior_interior)
   by (simp add: rel_interior_convex_hull_explicit)
 
 lemma interior_convex_hull_explicit:
   fixes s :: "'a::euclidean_space set"
   assumes "\<not> affine_dependent s"
   shows
    "interior(convex hull s) =
              (if card(s) \<le> DIM('a) then {}
               else {y. \<exists>u. (\<forall>x \<in> s. 0 < u x \<and> u x < 1) \<and> sum u s = 1 \<and> (\<Sum>x\<in>s. u x *\<^sub>R x) = y})"
 proof -
   { fix u :: "'a \<Rightarrow> real" and a
     assume "card Basis < card s" and u: "\<And>x. x\<in>s \<Longrightarrow> 0 < u x" "sum u s = 1" and a: "a \<in> s"
     then have cs: "Suc 0 < card s"
       by (metis DIM_positive less_trans_Suc)
     obtain b where b: "b \<in> s" "a \<noteq> b"
     proof (cases "s \<le> {a}")
       case True
       then show thesis
         using cs subset_singletonD by fastforce
     next
       case False
       then show thesis
       by (blast intro: that)
     qed
     have "u a + u b \<le> sum u {a,b}"
       using a b by simp
     also have "... \<le> sum u s"
       apply (rule Groups_Big.sum_mono2)
       using a b u
       apply (auto simp: less_imp_le aff_independent_finite assms)
       done
     finally have "u a < 1"
       using \<open>b \<in> s\<close> u by fastforce
   } note [simp] = this
   show ?thesis
     using assms
     apply (auto simp: interior_convex_hull_explicit_minimal)
     apply (rule_tac x=u in exI)
     apply (auto simp: not_le)
     done
 qed
 
 lemma interior_closed_segment_ge2:
   fixes a :: "'a::euclidean_space"
   assumes "2 \<le> DIM('a)"
     shows  "interior(closed_segment a b) = {}"
 using assms unfolding segment_convex_hull
 proof -
   have "card {a, b} \<le> DIM('a)"
     using assms
     by (simp add: card_insert_if linear not_less_eq_eq numeral_2_eq_2)
   then show "interior (convex hull {a, b}) = {}"
     by (metis empty_interior_convex_hull finite.insertI finite.emptyI)
 qed
 
 lemma interior_open_segment:
   fixes a :: "'a::euclidean_space"
   shows  "interior(open_segment a b) =
                  (if 2 \<le> DIM('a) then {} else open_segment a b)"
 proof (simp add: not_le, intro conjI impI)
   assume "2 \<le> DIM('a)"
   then show "interior (open_segment a b) = {}"
     apply (simp add: segment_convex_hull open_segment_def)
     apply (metis Diff_subset interior_mono segment_convex_hull subset_empty interior_closed_segment_ge2)
     done
 next
   assume le2: "DIM('a) < 2"
   show "interior (open_segment a b) = open_segment a b"
   proof (cases "a = b")
     case True then show ?thesis by auto
   next
     case False
     with le2 have "affine hull (open_segment a b) = UNIV"
       apply simp
       apply (rule affine_independent_span_gt)
       apply (simp_all add: affine_dependent_def insert_Diff_if)
       done
     then show "interior (open_segment a b) = open_segment a b"
       using rel_interior_interior rel_interior_open_segment by blast
   qed
 qed
 
 lemma interior_closed_segment:
   fixes a :: "'a::euclidean_space"
   shows "interior(closed_segment a b) =
                  (if 2 \<le> DIM('a) then {} else open_segment a b)"
 proof (cases "a = b")
   case True then show ?thesis by simp
 next
   case False
   then have "closure (open_segment a b) = closed_segment a b"
     by simp
   then show ?thesis
     by (metis (no_types) convex_interior_closure convex_open_segment interior_open_segment)
 qed
 
 lemmas interior_segment = interior_closed_segment interior_open_segment
 
 lemma closed_segment_eq [simp]:
   fixes a :: "'a::euclidean_space"
   shows "closed_segment a b = closed_segment c d \<longleftrightarrow> {a,b} = {c,d}"
 proof
   assume abcd: "closed_segment a b = closed_segment c d"
   show "{a,b} = {c,d}"
   proof (cases "a=b \<or> c=d")
     case True with abcd show ?thesis by force
   next
     case False
     then have neq: "a \<noteq> b \<and> c \<noteq> d" by force
     have *: "closed_segment c d - {a, b} = rel_interior (closed_segment c d)"
       using neq abcd by (metis (no_types) open_segment_def rel_interior_closed_segment)
     have "b \<in> {c, d}"
     proof -
       have "insert b (closed_segment c d) = closed_segment c d"
         using abcd by blast
       then show ?thesis
         by (metis DiffD2 Diff_insert2 False * insertI1 insert_Diff_if open_segment_def rel_interior_closed_segment)
     qed
     moreover have "a \<in> {c, d}"
       by (metis Diff_iff False * abcd ends_in_segment(1) insertI1 open_segment_def rel_interior_closed_segment)
     ultimately show "{a, b} = {c, d}"
       using neq by fastforce
   qed
 next
   assume "{a,b} = {c,d}"
   then show "closed_segment a b = closed_segment c d"
     by (simp add: segment_convex_hull)
 qed
 
 lemma closed_open_segment_eq [simp]:
   fixes a :: "'a::euclidean_space"
   shows "closed_segment a b \<noteq> open_segment c d"
 by (metis DiffE closed_segment_neq_empty closure_closed_segment closure_open_segment ends_in_segment(1) insertI1 open_segment_def)
 
 lemma open_closed_segment_eq [simp]:
   fixes a :: "'a::euclidean_space"
   shows "open_segment a b \<noteq> closed_segment c d"
 using closed_open_segment_eq by blast
 
 lemma open_segment_eq [simp]:
   fixes a :: "'a::euclidean_space"
   shows "open_segment a b = open_segment c d \<longleftrightarrow> a = b \<and> c = d \<or> {a,b} = {c,d}"
         (is "?lhs = ?rhs")
 proof
   assume abcd: ?lhs
   show ?rhs
   proof (cases "a=b \<or> c=d")
     case True with abcd show ?thesis
       using finite_open_segment by fastforce
   next
     case False
     then have a2: "a \<noteq> b \<and> c \<noteq> d" by force
     with abcd show ?rhs
       unfolding open_segment_def
       by (metis (no_types) abcd closed_segment_eq closure_open_segment)
   qed
 next
   assume ?rhs
   then show ?lhs
     by (metis Diff_cancel convex_hull_singleton insert_absorb2 open_segment_def segment_convex_hull)
 qed
 
 subsection\<^marker>\<open>tag unimportant\<close>\<open>Similar results for closure and (relative or absolute) frontier\<close>
 
 lemma closure_convex_hull [simp]:
   fixes s :: "'a::euclidean_space set"
   shows "compact s ==> closure(convex hull s) = convex hull s"
   by (simp add: compact_imp_closed compact_convex_hull)
 
 lemma rel_frontier_convex_hull_explicit:
   fixes s :: "'a::euclidean_space set"
   assumes "\<not> affine_dependent s"
   shows "rel_frontier(convex hull s) =
          {y. \<exists>u. (\<forall>x \<in> s. 0 \<le> u x) \<and> (\<exists>x \<in> s. u x = 0) \<and> sum u s = 1 \<and> sum (\<lambda>x. u x *\<^sub>R x) s = y}"
 proof -
   have fs: "finite s"
     using assms by (simp add: aff_independent_finite)
   show ?thesis
     apply (simp add: rel_frontier_def finite_imp_compact rel_interior_convex_hull_explicit assms fs)
     apply (auto simp: convex_hull_finite fs)
     apply (drule_tac x=u in spec)
     apply (rule_tac x=u in exI)
     apply force
     apply (rename_tac v)
     apply (rule notE [OF assms])
     apply (simp add: affine_dependent_explicit)
     apply (rule_tac x=s in exI)
     apply (auto simp: fs)
     apply (rule_tac x = "\<lambda>x. u x - v x" in exI)
     apply (force simp: sum_subtractf scaleR_diff_left)
     done
 qed
 
 lemma frontier_convex_hull_explicit:
   fixes s :: "'a::euclidean_space set"
   assumes "\<not> affine_dependent s"
   shows "frontier(convex hull s) =
          {y. \<exists>u. (\<forall>x \<in> s. 0 \<le> u x) \<and> (DIM ('a) < card s \<longrightarrow> (\<exists>x \<in> s. u x = 0)) \<and>
              sum u s = 1 \<and> sum (\<lambda>x. u x *\<^sub>R x) s = y}"
 proof -
   have fs: "finite s"
     using assms by (simp add: aff_independent_finite)
   show ?thesis
   proof (cases "DIM ('a) < card s")
     case True
     with assms fs show ?thesis
       by (simp add: rel_frontier_def frontier_def rel_frontier_convex_hull_explicit [symmetric]
                     interior_convex_hull_explicit_minimal rel_interior_convex_hull_explicit)
   next
     case False
     then have "card s \<le> DIM ('a)"
       by linarith
     then show ?thesis
       using assms fs
       apply (simp add: frontier_def interior_convex_hull_explicit finite_imp_compact)
       apply (simp add: convex_hull_finite)
       done
   qed
 qed
 
 lemma rel_frontier_convex_hull_cases:
   fixes s :: "'a::euclidean_space set"
   assumes "\<not> affine_dependent s"
   shows "rel_frontier(convex hull s) = \<Union>{convex hull (s - {x}) |x. x \<in> s}"
 proof -
   have fs: "finite s"
     using assms by (simp add: aff_independent_finite)
   { fix u a
   have "\<forall>x\<in>s. 0 \<le> u x \<Longrightarrow> a \<in> s \<Longrightarrow> u a = 0 \<Longrightarrow> sum u s = 1 \<Longrightarrow>
             \<exists>x v. x \<in> s \<and>
                   (\<forall>x\<in>s - {x}. 0 \<le> v x) \<and>
                       sum v (s - {x}) = 1 \<and> (\<Sum>x\<in>s - {x}. v x *\<^sub>R x) = (\<Sum>x\<in>s. u x *\<^sub>R x)"
     apply (rule_tac x=a in exI)
     apply (rule_tac x=u in exI)
     apply (simp add: Groups_Big.sum_diff1 fs)
     done }
   moreover
   { fix a u
     have "a \<in> s \<Longrightarrow> \<forall>x\<in>s - {a}. 0 \<le> u x \<Longrightarrow> sum u (s - {a}) = 1 \<Longrightarrow>
             \<exists>v. (\<forall>x\<in>s. 0 \<le> v x) \<and>
                  (\<exists>x\<in>s. v x = 0) \<and> sum v s = 1 \<and> (\<Sum>x\<in>s. v x *\<^sub>R x) = (\<Sum>x\<in>s - {a}. u x *\<^sub>R x)"
     apply (rule_tac x="\<lambda>x. if x = a then 0 else u x" in exI)
     apply (auto simp: sum.If_cases Diff_eq if_smult fs)
     done }
   ultimately show ?thesis
     using assms
     apply (simp add: rel_frontier_convex_hull_explicit)
     apply (simp add: convex_hull_finite fs Union_SetCompr_eq, auto)
     done
 qed
 
 lemma frontier_convex_hull_eq_rel_frontier:
   fixes s :: "'a::euclidean_space set"
   assumes "\<not> affine_dependent s"
   shows "frontier(convex hull s) =
            (if card s \<le> DIM ('a) then convex hull s else rel_frontier(convex hull s))"
   using assms
   unfolding rel_frontier_def frontier_def
   by (simp add: affine_independent_span_gt rel_interior_interior
                 finite_imp_compact empty_interior_convex_hull aff_independent_finite)
 
 lemma frontier_convex_hull_cases:
   fixes s :: "'a::euclidean_space set"
   assumes "\<not> affine_dependent s"
   shows "frontier(convex hull s) =
            (if card s \<le> DIM ('a) then convex hull s else \<Union>{convex hull (s - {x}) |x. x \<in> s})"
 by (simp add: assms frontier_convex_hull_eq_rel_frontier rel_frontier_convex_hull_cases)
 
 lemma in_frontier_convex_hull:
   fixes s :: "'a::euclidean_space set"
   assumes "finite s" "card s \<le> Suc (DIM ('a))" "x \<in> s"
   shows   "x \<in> frontier(convex hull s)"
 proof (cases "affine_dependent s")
   case True
   with assms show ?thesis
     apply (auto simp: affine_dependent_def frontier_def finite_imp_compact hull_inc)
     by (metis card.insert_remove convex_hull_subset_affine_hull empty_interior_affine_hull finite_Diff hull_redundant insert_Diff insert_Diff_single insert_not_empty interior_mono not_less_eq_eq subset_empty)
 next
   case False
   { assume "card s = Suc (card Basis)"
     then have cs: "Suc 0 < card s"
       by (simp)
     with subset_singletonD have "\<exists>y \<in> s. y \<noteq> x"
       by (cases "s \<le> {x}") fastforce+
   } note [dest!] = this
   show ?thesis using assms
     unfolding frontier_convex_hull_cases [OF False] Union_SetCompr_eq
     by (auto simp: le_Suc_eq hull_inc)
 qed
 
 lemma not_in_interior_convex_hull:
   fixes s :: "'a::euclidean_space set"
   assumes "finite s" "card s \<le> Suc (DIM ('a))" "x \<in> s"
   shows   "x \<notin> interior(convex hull s)"
 using in_frontier_convex_hull [OF assms]
 by (metis Diff_iff frontier_def)
 
 lemma interior_convex_hull_eq_empty:
   fixes s :: "'a::euclidean_space set"
   assumes "card s = Suc (DIM ('a))"
   shows   "interior(convex hull s) = {} \<longleftrightarrow> affine_dependent s"
 proof -
   { fix a b
     assume ab: "a \<in> interior (convex hull s)" "b \<in> s" "b \<in> affine hull (s - {b})"
     then have "interior(affine hull s) = {}" using assms
       by (metis DIM_positive One_nat_def Suc_mono card.remove card_infinite empty_interior_affine_hull eq_iff hull_redundant insert_Diff not_less zero_le_one)
     then have False using ab
       by (metis convex_hull_subset_affine_hull equals0D interior_mono subset_eq)
   } then
   show ?thesis
     using assms
     apply auto
     apply (metis UNIV_I affine_hull_convex_hull affine_hull_empty affine_independent_span_eq convex_convex_hull empty_iff rel_interior_interior rel_interior_same_affine_hull)
     apply (auto simp: affine_dependent_def)
     done
 qed
 
 
 subsection \<open>Coplanarity, and collinearity in terms of affine hull\<close>
 
 definition\<^marker>\<open>tag important\<close> coplanar  where
    "coplanar s \<equiv> \<exists>u v w. s \<subseteq> affine hull {u,v,w}"
 
 lemma collinear_affine_hull:
   "collinear s \<longleftrightarrow> (\<exists>u v. s \<subseteq> affine hull {u,v})"
 proof (cases "s={}")
   case True then show ?thesis
     by simp
 next
   case False
   then obtain x where x: "x \<in> s" by auto
   { fix u
     assume *: "\<And>x y. \<lbrakk>x\<in>s; y\<in>s\<rbrakk> \<Longrightarrow> \<exists>c. x - y = c *\<^sub>R u"
     have "\<exists>u v. s \<subseteq> {a *\<^sub>R u + b *\<^sub>R v |a b. a + b = 1}"
       apply (rule_tac x=x in exI)
       apply (rule_tac x="x+u" in exI, clarify)
       apply (erule exE [OF * [OF x]])
       apply (rename_tac c)
       apply (rule_tac x="1+c" in exI)
       apply (rule_tac x="-c" in exI)
       apply (simp add: algebra_simps)
       done
   } moreover
   { fix u v x y
     assume *: "s \<subseteq> {a *\<^sub>R u + b *\<^sub>R v |a b. a + b = 1}"
     have "x\<in>s \<Longrightarrow> y\<in>s \<Longrightarrow> \<exists>c. x - y = c *\<^sub>R (v-u)"
       apply (drule subsetD [OF *])+
       apply simp
       apply clarify
       apply (rename_tac r1 r2)
       apply (rule_tac x="r1-r2" in exI)
       apply (simp add: algebra_simps)
       apply (metis scaleR_left.add)
       done
   } ultimately
   show ?thesis
   unfolding collinear_def affine_hull_2
     by blast
 qed
 
 lemma collinear_closed_segment [simp]: "collinear (closed_segment a b)"
 by (metis affine_hull_convex_hull collinear_affine_hull hull_subset segment_convex_hull)
 
 lemma collinear_open_segment [simp]: "collinear (open_segment a b)"
   unfolding open_segment_def
   by (metis convex_hull_subset_affine_hull segment_convex_hull dual_order.trans
     convex_hull_subset_affine_hull Diff_subset collinear_affine_hull)
 
 lemma collinear_between_cases:
   fixes c :: "'a::euclidean_space"
   shows "collinear {a,b,c} \<longleftrightarrow> between (b,c) a \<or> between (c,a) b \<or> between (a,b) c"
          (is "?lhs = ?rhs")
 proof
   assume ?lhs
   then obtain u v where uv: "\<And>x. x \<in> {a, b, c} \<Longrightarrow> \<exists>c. x = u + c *\<^sub>R v"
     by (auto simp: collinear_alt)
   show ?rhs
     using uv [of a] uv [of b] uv [of c] by (auto simp: between_1)
 next
   assume ?rhs
   then show ?lhs
     unfolding between_mem_convex_hull
     by (metis (no_types, hide_lams) collinear_closed_segment collinear_subset hull_redundant hull_subset insert_commute segment_convex_hull)
 qed
 
 
 lemma subset_continuous_image_segment_1:
   fixes f :: "'a::euclidean_space \<Rightarrow> real"
   assumes "continuous_on (closed_segment a b) f"
   shows "closed_segment (f a) (f b) \<subseteq> image f (closed_segment a b)"
 by (metis connected_segment convex_contains_segment ends_in_segment imageI
            is_interval_connected_1 is_interval_convex connected_continuous_image [OF assms])
 
 lemma continuous_injective_image_segment_1:
   fixes f :: "'a::euclidean_space \<Rightarrow> real"
   assumes contf: "continuous_on (closed_segment a b) f"
       and injf: "inj_on f (closed_segment a b)"
   shows "f ` (closed_segment a b) = closed_segment (f a) (f b)"
 proof
   show "closed_segment (f a) (f b) \<subseteq> f ` closed_segment a b"
     by (metis subset_continuous_image_segment_1 contf)
   show "f ` closed_segment a b \<subseteq> closed_segment (f a) (f b)"
   proof (cases "a = b")
     case True
     then show ?thesis by auto
   next
     case False
     then have fnot: "f a \<noteq> f b"
       using inj_onD injf by fastforce
     moreover
     have "f a \<notin> open_segment (f c) (f b)" if c: "c \<in> closed_segment a b" for c
     proof (clarsimp simp add: open_segment_def)
       assume fa: "f a \<in> closed_segment (f c) (f b)"
       moreover have "closed_segment (f c) (f b) \<subseteq> f ` closed_segment c b"
         by (meson closed_segment_subset contf continuous_on_subset convex_closed_segment ends_in_segment(2) subset_continuous_image_segment_1 that)
       ultimately have "f a \<in> f ` closed_segment c b"
         by blast
       then have a: "a \<in> closed_segment c b"
         by (meson ends_in_segment inj_on_image_mem_iff_alt injf subset_closed_segment that)
       have cb: "closed_segment c b \<subseteq> closed_segment a b"
         by (simp add: closed_segment_subset that)
       show "f a = f c"
       proof (rule between_antisym)
         show "between (f c, f b) (f a)"
           by (simp add: between_mem_segment fa)
         show "between (f a, f b) (f c)"
           by (metis a cb between_antisym between_mem_segment between_triv1 subset_iff)
       qed
     qed
     moreover
     have "f b \<notin> open_segment (f a) (f c)" if c: "c \<in> closed_segment a b" for c
     proof (clarsimp simp add: open_segment_def fnot eq_commute)
       assume fb: "f b \<in> closed_segment (f a) (f c)"
       moreover have "closed_segment (f a) (f c) \<subseteq> f ` closed_segment a c"
         by (meson contf continuous_on_subset ends_in_segment(1) subset_closed_segment subset_continuous_image_segment_1 that)
       ultimately have "f b \<in> f ` closed_segment a c"
         by blast
       then have b: "b \<in> closed_segment a c"
         by (meson ends_in_segment inj_on_image_mem_iff_alt injf subset_closed_segment that)
       have ca: "closed_segment a c \<subseteq> closed_segment a b"
         by (simp add: closed_segment_subset that)
       show "f b = f c"
       proof (rule between_antisym)
         show "between (f c, f a) (f b)"
           by (simp add: between_commute between_mem_segment fb)
         show "between (f b, f a) (f c)"
           by (metis b between_antisym between_commute between_mem_segment between_triv2 that)
       qed
     qed
     ultimately show ?thesis
       by (force simp: closed_segment_eq_real_ivl open_segment_eq_real_ivl split: if_split_asm)
   qed
 qed
 
 lemma continuous_injective_image_open_segment_1:
   fixes f :: "'a::euclidean_space \<Rightarrow> real"
   assumes contf: "continuous_on (closed_segment a b) f"
       and injf: "inj_on f (closed_segment a b)"
     shows "f ` (open_segment a b) = open_segment (f a) (f b)"
 proof -
   have "f ` (open_segment a b) = f ` (closed_segment a b) - {f a, f b}"
     by (metis (no_types, hide_lams) empty_subsetI ends_in_segment image_insert image_is_empty inj_on_image_set_diff injf insert_subset open_segment_def segment_open_subset_closed)
   also have "... = open_segment (f a) (f b)"
     using continuous_injective_image_segment_1 [OF assms]
     by (simp add: open_segment_def inj_on_image_set_diff [OF injf])
   finally show ?thesis .
 qed
 
 lemma collinear_imp_coplanar:
   "collinear s ==> coplanar s"
 by (metis collinear_affine_hull coplanar_def insert_absorb2)
 
 lemma collinear_small:
   assumes "finite s" "card s \<le> 2"
     shows "collinear s"
 proof -
   have "card s = 0 \<or> card s = 1 \<or> card s = 2"
     using assms by linarith
   then show ?thesis using assms
     using card_eq_SucD
     by auto (metis collinear_2 numeral_2_eq_2)
 qed
 
 lemma coplanar_small:
   assumes "finite s" "card s \<le> 3"
     shows "coplanar s"
 proof -
   have "card s \<le> 2 \<or> card s = Suc (Suc (Suc 0))"
     using assms by linarith
   then show ?thesis using assms
     apply safe
     apply (simp add: collinear_small collinear_imp_coplanar)
     apply (safe dest!: card_eq_SucD)
     apply (auto simp: coplanar_def)
     apply (metis hull_subset insert_subset)
     done
 qed
 
 lemma coplanar_empty: "coplanar {}"
   by (simp add: coplanar_small)
 
 lemma coplanar_sing: "coplanar {a}"
   by (simp add: coplanar_small)
 
 lemma coplanar_2: "coplanar {a,b}"
   by (auto simp: card_insert_if coplanar_small)
 
 lemma coplanar_3: "coplanar {a,b,c}"
   by (auto simp: card_insert_if coplanar_small)
 
 lemma collinear_affine_hull_collinear: "collinear(affine hull s) \<longleftrightarrow> collinear s"
   unfolding collinear_affine_hull
   by (metis affine_affine_hull subset_hull hull_hull hull_mono)
 
 lemma coplanar_affine_hull_coplanar: "coplanar(affine hull s) \<longleftrightarrow> coplanar s"
   unfolding coplanar_def
   by (metis affine_affine_hull subset_hull hull_hull hull_mono)
 
 lemma coplanar_linear_image:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
   assumes "coplanar s" "linear f" shows "coplanar(f ` s)"
 proof -
   { fix u v w
     assume "s \<subseteq> affine hull {u, v, w}"
     then have "f ` s \<subseteq> f ` (affine hull {u, v, w})"
       by (simp add: image_mono)
     then have "f ` s \<subseteq> affine hull (f ` {u, v, w})"
       by (metis assms(2) linear_conv_bounded_linear affine_hull_linear_image)
   } then
   show ?thesis
     by auto (meson assms(1) coplanar_def)
 qed
 
 lemma coplanar_translation_imp: "coplanar s \<Longrightarrow> coplanar ((\<lambda>x. a + x) ` s)"
   unfolding coplanar_def
   apply clarify
   apply (rule_tac x="u+a" in exI)
   apply (rule_tac x="v+a" in exI)
   apply (rule_tac x="w+a" in exI)
   using affine_hull_translation [of a "{u,v,w}" for u v w]
   apply (force simp: add.commute)
   done
 
 lemma coplanar_translation_eq: "coplanar((\<lambda>x. a + x) ` s) \<longleftrightarrow> coplanar s"
     by (metis (no_types) coplanar_translation_imp translation_galois)
 
 lemma coplanar_linear_image_eq:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes "linear f" "inj f" shows "coplanar(f ` s) = coplanar s"
 proof
   assume "coplanar s"
   then show "coplanar (f ` s)"
     unfolding coplanar_def
     using affine_hull_linear_image [of f "{u,v,w}" for u v w]  assms
     by (meson coplanar_def coplanar_linear_image)
 next
   obtain g where g: "linear g" "g \<circ> f = id"
     using linear_injective_left_inverse [OF assms]
     by blast
   assume "coplanar (f ` s)"
   then obtain u v w where "f ` s \<subseteq> affine hull {u, v, w}"
     by (auto simp: coplanar_def)
   then have "g ` f ` s \<subseteq> g ` (affine hull {u, v, w})"
     by blast
   then have "s \<subseteq> g ` (affine hull {u, v, w})"
     using g by (simp add: Fun.image_comp)
   then show "coplanar s"
     unfolding coplanar_def
     using affine_hull_linear_image [of g "{u,v,w}" for u v w]  \<open>linear g\<close> linear_conv_bounded_linear
     by fastforce
 qed
 (*The HOL Light proof is simply
     MATCH_ACCEPT_TAC(LINEAR_INVARIANT_RULE COPLANAR_LINEAR_IMAGE));;
 *)
 
 lemma coplanar_subset: "\<lbrakk>coplanar t; s \<subseteq> t\<rbrakk> \<Longrightarrow> coplanar s"
   by (meson coplanar_def order_trans)
 
 lemma affine_hull_3_imp_collinear: "c \<in> affine hull {a,b} \<Longrightarrow> collinear {a,b,c}"
   by (metis collinear_2 collinear_affine_hull_collinear hull_redundant insert_commute)
 
 lemma collinear_3_imp_in_affine_hull: "\<lbrakk>collinear {a,b,c}; a \<noteq> b\<rbrakk> \<Longrightarrow> c \<in> affine hull {a,b}"
   unfolding collinear_def
   apply clarify
   apply (frule_tac x=b in bspec, blast, drule_tac x=a in bspec, blast, erule exE)
   apply (drule_tac x=c in bspec, blast, drule_tac x=a in bspec, blast, erule exE)
   apply (rename_tac y x)
   apply (simp add: affine_hull_2)
   apply (rule_tac x="1 - x/y" in exI)
   apply (simp add: algebra_simps)
   done
 
 lemma collinear_3_affine_hull:
   assumes "a \<noteq> b"
     shows "collinear {a,b,c} \<longleftrightarrow> c \<in> affine hull {a,b}"
 using affine_hull_3_imp_collinear assms collinear_3_imp_in_affine_hull by blast
 
 lemma collinear_3_eq_affine_dependent:
   "collinear{a,b,c} \<longleftrightarrow> a = b \<or> a = c \<or> b = c \<or> affine_dependent {a,b,c}"
 apply (case_tac "a=b", simp)
 apply (case_tac "a=c")
 apply (simp add: insert_commute)
 apply (case_tac "b=c")
 apply (simp add: insert_commute)
 apply (auto simp: affine_dependent_def collinear_3_affine_hull insert_Diff_if)
 apply (metis collinear_3_affine_hull insert_commute)+
 done
 
 lemma affine_dependent_imp_collinear_3:
   "affine_dependent {a,b,c} \<Longrightarrow> collinear{a,b,c}"
 by (simp add: collinear_3_eq_affine_dependent)
 
 lemma collinear_3: "NO_MATCH 0 x \<Longrightarrow> collinear {x,y,z} \<longleftrightarrow> collinear {0, x-y, z-y}"
   by (auto simp add: collinear_def)
 
 lemma collinear_3_expand:
    "collinear{a,b,c} \<longleftrightarrow> a = c \<or> (\<exists>u. b = u *\<^sub>R a + (1 - u) *\<^sub>R c)"
 proof -
   have "collinear{a,b,c} = collinear{a,c,b}"
     by (simp add: insert_commute)
   also have "... = collinear {0, a - c, b - c}"
     by (simp add: collinear_3)
   also have "... \<longleftrightarrow> (a = c \<or> b = c \<or> (\<exists>ca. b - c = ca *\<^sub>R (a - c)))"
     by (simp add: collinear_lemma)
   also have "... \<longleftrightarrow> a = c \<or> (\<exists>u. b = u *\<^sub>R a + (1 - u) *\<^sub>R c)"
     by (cases "a = c \<or> b = c") (auto simp: algebra_simps)
   finally show ?thesis .
 qed
 
 lemma collinear_aff_dim: "collinear S \<longleftrightarrow> aff_dim S \<le> 1"
 proof
   assume "collinear S"
   then obtain u and v :: "'a" where "aff_dim S \<le> aff_dim {u,v}"
     by (metis \<open>collinear S\<close> aff_dim_affine_hull aff_dim_subset collinear_affine_hull)
   then show "aff_dim S \<le> 1"
     using order_trans by fastforce
 next
   assume "aff_dim S \<le> 1"
   then have le1: "aff_dim (affine hull S) \<le> 1"
     by simp
   obtain B where "B \<subseteq> S" and B: "\<not> affine_dependent B" "affine hull S = affine hull B"
     using affine_basis_exists [of S] by auto
   then have "finite B" "card B \<le> 2"
     using B le1 by (auto simp: affine_independent_iff_card)
   then have "collinear B"
     by (rule collinear_small)
   then show "collinear S"
     by (metis \<open>affine hull S = affine hull B\<close> collinear_affine_hull_collinear)
 qed
 
 lemma collinear_midpoint: "collinear{a,midpoint a b,b}"
   apply (auto simp: collinear_3 collinear_lemma)
   apply (drule_tac x="-1" in spec)
   apply (simp add: algebra_simps)
   done
 
 lemma midpoint_collinear:
   fixes a b c :: "'a::real_normed_vector"
   assumes "a \<noteq> c"
     shows "b = midpoint a c \<longleftrightarrow> collinear{a,b,c} \<and> dist a b = dist b c"
 proof -
   have *: "a - (u *\<^sub>R a + (1 - u) *\<^sub>R c) = (1 - u) *\<^sub>R (a - c)"
           "u *\<^sub>R a + (1 - u) *\<^sub>R c - c = u *\<^sub>R (a - c)"
           "\<bar>1 - u\<bar> = \<bar>u\<bar> \<longleftrightarrow> u = 1/2" for u::real
     by (auto simp: algebra_simps)
   have "b = midpoint a c \<Longrightarrow> collinear{a,b,c} "
     using collinear_midpoint by blast
   moreover have "collinear{a,b,c} \<Longrightarrow> b = midpoint a c \<longleftrightarrow> dist a b = dist b c"
     apply (auto simp: collinear_3_expand assms dist_midpoint)
     apply (simp add: dist_norm * assms midpoint_def del: divide_const_simps)
     apply (simp add: algebra_simps)
     done
   ultimately show ?thesis by blast
 qed
 
 lemma between_imp_collinear:
   fixes x :: "'a :: euclidean_space"
   assumes "between (a,b) x"
     shows "collinear {a,x,b}"
 proof (cases "x = a \<or> x = b \<or> a = b")
   case True with assms show ?thesis
     by (auto simp: dist_commute)
 next
   case False with assms show ?thesis
     apply (auto simp: collinear_3 collinear_lemma between_norm)
     apply (drule_tac x="-(norm(b - x) / norm(x - a))" in spec)
     apply (simp add: vector_add_divide_simps real_vector.scale_minus_right [symmetric])
     done
 qed
 
 lemma midpoint_between:
   fixes a b :: "'a::euclidean_space"
   shows "b = midpoint a c \<longleftrightarrow> between (a,c) b \<and> dist a b = dist b c"
 proof (cases "a = c")
   case True then show ?thesis
     by (auto simp: dist_commute)
 next
   case False
   show ?thesis
     apply (rule iffI)
     apply (simp add: between_midpoint(1) dist_midpoint)
     using False between_imp_collinear midpoint_collinear by blast
 qed
 
 lemma collinear_triples:
   assumes "a \<noteq> b"
     shows "collinear(insert a (insert b S)) \<longleftrightarrow> (\<forall>x \<in> S. collinear{a,b,x})"
           (is "?lhs = ?rhs")
 proof safe
   fix x
   assume ?lhs and "x \<in> S"
   then show "collinear {a, b, x}"
     using collinear_subset by force
 next
   assume ?rhs
   then have "\<forall>x \<in> S. collinear{a,x,b}"
     by (simp add: insert_commute)
   then have *: "\<exists>u. x = u *\<^sub>R a + (1 - u) *\<^sub>R b" if "x \<in> (insert a (insert b S))" for x
     using that assms collinear_3_expand by fastforce+
   show ?lhs
     unfolding collinear_def
     apply (rule_tac x="b-a" in exI)
     apply (clarify dest!: *)
     by (metis (no_types, hide_lams) add.commute diff_add_cancel diff_diff_eq2 real_vector.scale_right_diff_distrib scaleR_left.diff)
 qed
 
 lemma collinear_4_3:
   assumes "a \<noteq> b"
     shows "collinear {a,b,c,d} \<longleftrightarrow> collinear{a,b,c} \<and> collinear{a,b,d}"
   using collinear_triples [OF assms, of "{c,d}"] by (force simp:)
 
 lemma collinear_3_trans:
   assumes "collinear{a,b,c}" "collinear{b,c,d}" "b \<noteq> c"
     shows "collinear{a,b,d}"
 proof -
   have "collinear{b,c,a,d}"
     by (metis (full_types) assms collinear_4_3 insert_commute)
   then show ?thesis
     by (simp add: collinear_subset)
 qed
 
 lemma affine_hull_2_alt:
   fixes a b :: "'a::real_vector"
   shows "affine hull {a,b} = range (\<lambda>u. a + u *\<^sub>R (b - a))"
 apply (simp add: affine_hull_2, safe)
 apply (rule_tac x=v in image_eqI)
 apply (simp add: algebra_simps)
 apply (metis scaleR_add_left scaleR_one, simp)
 apply (rule_tac x="1-u" in exI)
 apply (simp add: algebra_simps)
 done
 
 lemma interior_convex_hull_3_minimal:
   fixes a :: "'a::euclidean_space"
   shows "\<lbrakk>\<not> collinear{a,b,c}; DIM('a) = 2\<rbrakk>
          \<Longrightarrow> interior(convex hull {a,b,c}) =
                 {v. \<exists>x y z. 0 < x \<and> 0 < y \<and> 0 < z \<and> x + y + z = 1 \<and>
                             x *\<^sub>R a + y *\<^sub>R b + z *\<^sub>R c = v}"
 apply (simp add: collinear_3_eq_affine_dependent interior_convex_hull_explicit_minimal, safe)
 apply (rule_tac x="u a" in exI, simp)
 apply (rule_tac x="u b" in exI, simp)
 apply (rule_tac x="u c" in exI, simp)
 apply (rename_tac uu x y z)
 apply (rule_tac x="\<lambda>r. (if r=a then x else if r=b then y else if r=c then z else 0)" in exI)
 apply simp
 done
 
 
 subsection\<^marker>\<open>tag unimportant\<close>\<open>Basic lemmas about hyperplanes and halfspaces\<close>
 
 lemma halfspace_Int_eq:
      "{x. a \<bullet> x \<le> b} \<inter> {x. b \<le> a \<bullet> x} = {x. a \<bullet> x = b}"
      "{x. b \<le> a \<bullet> x} \<inter> {x. a \<bullet> x \<le> b} = {x. a \<bullet> x = b}"
   by auto
 
 lemma hyperplane_eq_Ex:
   assumes "a \<noteq> 0" obtains x where "a \<bullet> x = b"
   by (rule_tac x = "(b / (a \<bullet> a)) *\<^sub>R a" in that) (simp add: assms)
 
 lemma hyperplane_eq_empty:
      "{x. a \<bullet> x = b} = {} \<longleftrightarrow> a = 0 \<and> b \<noteq> 0"
   using hyperplane_eq_Ex apply auto[1]
   using inner_zero_right by blast
 
 lemma hyperplane_eq_UNIV:
    "{x. a \<bullet> x = b} = UNIV \<longleftrightarrow> a = 0 \<and> b = 0"
 proof -
   have "UNIV \<subseteq> {x. a \<bullet> x = b} \<Longrightarrow> a = 0 \<and> b = 0"
     apply (drule_tac c = "((b+1) / (a \<bullet> a)) *\<^sub>R a" in subsetD)
     apply simp_all
     by (metis add_cancel_right_right zero_neq_one)
   then show ?thesis by force
 qed
 
 lemma halfspace_eq_empty_lt:
    "{x. a \<bullet> x < b} = {} \<longleftrightarrow> a = 0 \<and> b \<le> 0"
 proof -
   have "{x. a \<bullet> x < b} \<subseteq> {} \<Longrightarrow> a = 0 \<and> b \<le> 0"
     apply (rule ccontr)
     apply (drule_tac c = "((b-1) / (a \<bullet> a)) *\<^sub>R a" in subsetD)
     apply force+
     done
   then show ?thesis by force
 qed
 
 lemma halfspace_eq_empty_gt:
    "{x. a \<bullet> x > b} = {} \<longleftrightarrow> a = 0 \<and> b \<ge> 0"
 using halfspace_eq_empty_lt [of "-a" "-b"]
 by simp
 
 lemma halfspace_eq_empty_le:
    "{x. a \<bullet> x \<le> b} = {} \<longleftrightarrow> a = 0 \<and> b < 0"
 proof -
   have "{x. a \<bullet> x \<le> b} \<subseteq> {} \<Longrightarrow> a = 0 \<and> b < 0"
     apply (rule ccontr)
     apply (drule_tac c = "((b-1) / (a \<bullet> a)) *\<^sub>R a" in subsetD)
     apply force+
     done
   then show ?thesis by force
 qed
 
 lemma halfspace_eq_empty_ge:
    "{x. a \<bullet> x \<ge> b} = {} \<longleftrightarrow> a = 0 \<and> b > 0"
 using halfspace_eq_empty_le [of "-a" "-b"]
 by simp
 
 subsection\<^marker>\<open>tag unimportant\<close>\<open>Use set distance for an easy proof of separation properties\<close>
 
 proposition\<^marker>\<open>tag unimportant\<close> separation_closures:
   fixes S :: "'a::euclidean_space set"
   assumes "S \<inter> closure T = {}" "T \<inter> closure S = {}"
   obtains U V where "U \<inter> V = {}" "open U" "open V" "S \<subseteq> U" "T \<subseteq> V"
 proof (cases "S = {} \<or> T = {}")
   case True with that show ?thesis by auto
 next
   case False
   define f where "f \<equiv> \<lambda>x. setdist {x} T - setdist {x} S"
   have contf: "continuous_on UNIV f"
     unfolding f_def by (intro continuous_intros continuous_on_setdist)
   show ?thesis
   proof (rule_tac U = "{x. f x > 0}" and V = "{x. f x < 0}" in that)
     show "{x. 0 < f x} \<inter> {x. f x < 0} = {}"
       by auto
     show "open {x. 0 < f x}"
       by (simp add: open_Collect_less contf)
     show "open {x. f x < 0}"
       by (simp add: open_Collect_less contf)
     show "S \<subseteq> {x. 0 < f x}"
       apply (clarsimp simp add: f_def setdist_sing_in_set)
       using assms
       by (metis False IntI empty_iff le_less setdist_eq_0_sing_2 setdist_pos_le setdist_sym)
     show "T \<subseteq> {x. f x < 0}"
       apply (clarsimp simp add: f_def setdist_sing_in_set)
       using assms
       by (metis False IntI empty_iff le_less setdist_eq_0_sing_2 setdist_pos_le setdist_sym)
   qed
 qed
 
 lemma separation_normal:
   fixes S :: "'a::euclidean_space set"
   assumes "closed S" "closed T" "S \<inter> T = {}"
   obtains U V where "open U" "open V" "S \<subseteq> U" "T \<subseteq> V" "U \<inter> V = {}"
 using separation_closures [of S T]
 by (metis assms closure_closed disjnt_def inf_commute)
 
 lemma separation_normal_local:
   fixes S :: "'a::euclidean_space set"
   assumes US: "closedin (top_of_set U) S"
       and UT: "closedin (top_of_set U) T"
       and "S \<inter> T = {}"
   obtains S' T' where "openin (top_of_set U) S'"
                       "openin (top_of_set U) T'"
                       "S \<subseteq> S'"  "T \<subseteq> T'"  "S' \<inter> T' = {}"
 proof (cases "S = {} \<or> T = {}")
   case True with that show ?thesis
     using UT US by (blast dest: closedin_subset)
 next
   case False
   define f where "f \<equiv> \<lambda>x. setdist {x} T - setdist {x} S"
   have contf: "continuous_on U f"
     unfolding f_def by (intro continuous_intros)
   show ?thesis
   proof (rule_tac S' = "(U \<inter> f -` {0<..})" and T' = "(U \<inter> f -` {..<0})" in that)
     show "(U \<inter> f -` {0<..}) \<inter> (U \<inter> f -` {..<0}) = {}"
       by auto
     show "openin (top_of_set U) (U \<inter> f -` {0<..})"
       by (rule continuous_openin_preimage [where T=UNIV]) (simp_all add: contf)
   next
     show "openin (top_of_set U) (U \<inter> f -` {..<0})"
       by (rule continuous_openin_preimage [where T=UNIV]) (simp_all add: contf)
   next
     have "S \<subseteq> U" "T \<subseteq> U"
       using closedin_imp_subset assms by blast+
     then show "S \<subseteq> U \<inter> f -` {0<..}" "T \<subseteq> U \<inter> f -` {..<0}"
       using assms False by (force simp add: f_def setdist_sing_in_set intro!: setdist_gt_0_closedin)+
   qed
 qed
 
 lemma separation_normal_compact:
   fixes S :: "'a::euclidean_space set"
   assumes "compact S" "closed T" "S \<inter> T = {}"
   obtains U V where "open U" "compact(closure U)" "open V" "S \<subseteq> U" "T \<subseteq> V" "U \<inter> V = {}"
 proof -
   have "closed S" "bounded S"
     using assms by (auto simp: compact_eq_bounded_closed)
   then obtain r where "r>0" and r: "S \<subseteq> ball 0 r"
     by (auto dest!: bounded_subset_ballD)
   have **: "closed (T \<union> - ball 0 r)" "S \<inter> (T \<union> - ball 0 r) = {}"
     using assms r by blast+
   then show ?thesis
     apply (rule separation_normal [OF \<open>closed S\<close>])
     apply (rule_tac U=U and V=V in that)
     by auto (meson bounded_ball bounded_subset compl_le_swap2 disjoint_eq_subset_Compl)
 qed
 
 subsection\<open>Connectedness of the intersection of a chain\<close>
 
 proposition connected_chain:
   fixes \<F> :: "'a :: euclidean_space set set"
   assumes cc: "\<And>S. S \<in> \<F> \<Longrightarrow> compact S \<and> connected S"
       and linear: "\<And>S T. S \<in> \<F> \<and> T \<in> \<F> \<Longrightarrow> S \<subseteq> T \<or> T \<subseteq> S"
   shows "connected(\<Inter>\<F>)"
 proof (cases "\<F> = {}")
   case True then show ?thesis
     by auto
 next
   case False
   then have cf: "compact(\<Inter>\<F>)"
     by (simp add: cc compact_Inter)
   have False if AB: "closed A" "closed B" "A \<inter> B = {}"
                 and ABeq: "A \<union> B = \<Inter>\<F>" and "A \<noteq> {}" "B \<noteq> {}" for A B
   proof -
     obtain U V where "open U" "open V" "A \<subseteq> U" "B \<subseteq> V" "U \<inter> V = {}"
       using separation_normal [OF AB] by metis
     obtain K where "K \<in> \<F>" "compact K"
       using cc False by blast
     then obtain N where "open N" and "K \<subseteq> N"
       by blast
     let ?\<C> = "insert (U \<union> V) ((\<lambda>S. N - S) ` \<F>)"
     obtain \<D> where "\<D> \<subseteq> ?\<C>" "finite \<D>" "K \<subseteq> \<Union>\<D>"
     proof (rule compactE [OF \<open>compact K\<close>])
       show "K \<subseteq> \<Union>(insert (U \<union> V) ((-) N ` \<F>))"
         using \<open>K \<subseteq> N\<close> ABeq \<open>A \<subseteq> U\<close> \<open>B \<subseteq> V\<close> by auto
       show "\<And>B. B \<in> insert (U \<union> V) ((-) N ` \<F>) \<Longrightarrow> open B"
         by (auto simp:  \<open>open U\<close> \<open>open V\<close> open_Un \<open>open N\<close> cc compact_imp_closed open_Diff)
     qed
     then have "finite(\<D> - {U \<union> V})"
       by blast
     moreover have "\<D> - {U \<union> V} \<subseteq> (\<lambda>S. N - S) ` \<F>"
       using \<open>\<D> \<subseteq> ?\<C>\<close> by blast
     ultimately obtain \<G> where "\<G> \<subseteq> \<F>" "finite \<G>" and Deq: "\<D> - {U \<union> V} = (\<lambda>S. N-S) ` \<G>"
       using finite_subset_image by metis
     obtain J where "J \<in> \<F>" and J: "(\<Union>S\<in>\<G>. N - S) \<subseteq> N - J"
     proof (cases "\<G> = {}")
       case True
       with \<open>\<F> \<noteq> {}\<close> that show ?thesis
         by auto
     next
       case False
       have "\<And>S T. \<lbrakk>S \<in> \<G>; T \<in> \<G>\<rbrakk> \<Longrightarrow> S \<subseteq> T \<or> T \<subseteq> S"
         by (meson \<open>\<G> \<subseteq> \<F>\<close> in_mono local.linear)
       with \<open>finite \<G>\<close> \<open>\<G> \<noteq> {}\<close>
       have "\<exists>J \<in> \<G>. (\<Union>S\<in>\<G>. N - S) \<subseteq> N - J"
       proof induction
         case (insert X \<H>)
         show ?case
         proof (cases "\<H> = {}")
           case True then show ?thesis by auto
         next
           case False
           then have "\<And>S T. \<lbrakk>S \<in> \<H>; T \<in> \<H>\<rbrakk> \<Longrightarrow> S \<subseteq> T \<or> T \<subseteq> S"
             by (simp add: insert.prems)
           with insert.IH False obtain J where "J \<in> \<H>" and J: "(\<Union>Y\<in>\<H>. N - Y) \<subseteq> N - J"
             by metis
           have "N - J \<subseteq> N - X \<or> N - X \<subseteq> N - J"
             by (meson Diff_mono \<open>J \<in> \<H>\<close> insert.prems(2) insert_iff order_refl)
           then show ?thesis
           proof
             assume "N - J \<subseteq> N - X" with J show ?thesis
               by auto
           next
             assume "N - X \<subseteq> N - J"
             with J have "N - X \<union> \<Union> ((-) N ` \<H>) \<subseteq> N - J"
               by auto
             with \<open>J \<in> \<H>\<close> show ?thesis
               by blast
           qed
         qed
       qed simp
       with \<open>\<G> \<subseteq> \<F>\<close> show ?thesis by (blast intro: that)
     qed
     have "K \<subseteq> \<Union>(insert (U \<union> V) (\<D> - {U \<union> V}))"
       using \<open>K \<subseteq> \<Union>\<D>\<close> by auto
     also have "... \<subseteq> (U \<union> V) \<union> (N - J)"
       by (metis (no_types, hide_lams) Deq Un_subset_iff Un_upper2 J Union_insert order_trans sup_ge1)
     finally have "J \<inter> K \<subseteq> U \<union> V"
       by blast
     moreover have "connected(J \<inter> K)"
       by (metis Int_absorb1 \<open>J \<in> \<F>\<close> \<open>K \<in> \<F>\<close> cc inf.orderE local.linear)
     moreover have "U \<inter> (J \<inter> K) \<noteq> {}"
       using ABeq \<open>J \<in> \<F>\<close> \<open>K \<in> \<F>\<close> \<open>A \<noteq> {}\<close> \<open>A \<subseteq> U\<close> by blast
     moreover have "V \<inter> (J \<inter> K) \<noteq> {}"
       using ABeq \<open>J \<in> \<F>\<close> \<open>K \<in> \<F>\<close> \<open>B \<noteq> {}\<close> \<open>B \<subseteq> V\<close> by blast
     ultimately show False
         using connectedD [of "J \<inter> K" U V] \<open>open U\<close> \<open>open V\<close> \<open>U \<inter> V = {}\<close>  by auto
   qed
   with cf show ?thesis
     by (auto simp: connected_closed_set compact_imp_closed)
 qed
 
 lemma connected_chain_gen:
   fixes \<F> :: "'a :: euclidean_space set set"
   assumes X: "X \<in> \<F>" "compact X"
       and cc: "\<And>T. T \<in> \<F> \<Longrightarrow> closed T \<and> connected T"
       and linear: "\<And>S T. S \<in> \<F> \<and> T \<in> \<F> \<Longrightarrow> S \<subseteq> T \<or> T \<subseteq> S"
   shows "connected(\<Inter>\<F>)"
 proof -
   have "\<Inter>\<F> = (\<Inter>T\<in>\<F>. X \<inter> T)"
     using X by blast
   moreover have "connected (\<Inter>T\<in>\<F>. X \<inter> T)"
   proof (rule connected_chain)
     show "\<And>T. T \<in> (\<inter>) X ` \<F> \<Longrightarrow> compact T \<and> connected T"
       using cc X by auto (metis inf.absorb2 inf.orderE local.linear)
     show "\<And>S T. S \<in> (\<inter>) X ` \<F> \<and> T \<in> (\<inter>) X ` \<F> \<Longrightarrow> S \<subseteq> T \<or> T \<subseteq> S"
       using local.linear by blast
   qed
   ultimately show ?thesis
     by metis
 qed
 
 lemma connected_nest:
   fixes S :: "'a::linorder \<Rightarrow> 'b::euclidean_space set"
   assumes S: "\<And>n. compact(S n)" "\<And>n. connected(S n)"
     and nest: "\<And>m n. m \<le> n \<Longrightarrow> S n \<subseteq> S m"
   shows "connected(\<Inter> (range S))"
   apply (rule connected_chain)
   using S apply blast
   by (metis image_iff le_cases nest)
 
 lemma connected_nest_gen:
   fixes S :: "'a::linorder \<Rightarrow> 'b::euclidean_space set"
   assumes S: "\<And>n. closed(S n)" "\<And>n. connected(S n)" "compact(S k)"
     and nest: "\<And>m n. m \<le> n \<Longrightarrow> S n \<subseteq> S m"
   shows "connected(\<Inter> (range S))"
   apply (rule connected_chain_gen [of "S k"])
   using S apply auto
   by (meson le_cases nest subsetCE)
 
 subsection\<open>Proper maps, including projections out of compact sets\<close>
 
 lemma finite_indexed_bound:
   assumes A: "finite A" "\<And>x. x \<in> A \<Longrightarrow> \<exists>n::'a::linorder. P x n"
     shows "\<exists>m. \<forall>x \<in> A. \<exists>k\<le>m. P x k"
 using A
 proof (induction A)
   case empty then show ?case by force
 next
   case (insert a A)
     then obtain m n where "\<forall>x \<in> A. \<exists>k\<le>m. P x k" "P a n"
       by force
     then show ?case
       apply (rule_tac x="max m n" in exI, safe)
       using max.cobounded2 apply blast
       by (meson le_max_iff_disj)
 qed
 
 proposition proper_map:
   fixes f :: "'a::heine_borel \<Rightarrow> 'b::heine_borel"
   assumes "closedin (top_of_set S) K"
       and com: "\<And>U. \<lbrakk>U \<subseteq> T; compact U\<rbrakk> \<Longrightarrow> compact (S \<inter> f -` U)"
       and "f ` S \<subseteq> T"
     shows "closedin (top_of_set T) (f ` K)"
 proof -
   have "K \<subseteq> S"
     using assms closedin_imp_subset by metis
   obtain C where "closed C" and Keq: "K = S \<inter> C"
     using assms by (auto simp: closedin_closed)
   have *: "y \<in> f ` K" if "y \<in> T" and y: "y islimpt f ` K" for y
   proof -
     obtain h where "\<forall>n. (\<exists>x\<in>K. h n = f x) \<and> h n \<noteq> y" "inj h" and hlim: "(h \<longlongrightarrow> y) sequentially"
       using \<open>y \<in> T\<close> y by (force simp: limpt_sequential_inj)
     then obtain X where X: "\<And>n. X n \<in> K \<and> h n = f (X n) \<and> h n \<noteq> y"
       by metis
     then have fX: "\<And>n. f (X n) = h n"
       by metis
     have "compact (C \<inter> (S \<inter> f -` insert y (range (\<lambda>i. f(X(n + i))))))" for n
       apply (rule closed_Int_compact [OF \<open>closed C\<close>])
       apply (rule com)
        using X \<open>K \<subseteq> S\<close> \<open>f ` S \<subseteq> T\<close> \<open>y \<in> T\<close> apply blast
       apply (rule compact_sequence_with_limit)
       apply (simp add: fX add.commute [of n] LIMSEQ_ignore_initial_segment [OF hlim])
       done
     then have comf: "compact {a \<in> K. f a \<in> insert y (range (\<lambda>i. f(X(n + i))))}" for n
       by (simp add: Keq Int_def conj_commute)
     have ne: "\<Inter>\<F> \<noteq> {}"
              if "finite \<F>"
                 and \<F>: "\<And>t. t \<in> \<F> \<Longrightarrow>
                            (\<exists>n. t = {a \<in> K. f a \<in> insert y (range (\<lambda>i. f (X (n + i))))})"
              for \<F>
     proof -
       obtain m where m: "\<And>t. t \<in> \<F> \<Longrightarrow> \<exists>k\<le>m. t = {a \<in> K. f a \<in> insert y (range (\<lambda>i. f (X (k + i))))}"
         apply (rule exE)
         apply (rule finite_indexed_bound [OF \<open>finite \<F>\<close> \<F>], assumption, force)
         done
       have "X m \<in> \<Inter>\<F>"
         using X le_Suc_ex by (fastforce dest: m)
       then show ?thesis by blast
     qed
     have "\<Inter>{{a. a \<in> K \<and> f a \<in> insert y (range (\<lambda>i. f(X(n + i))))} |n. n \<in> UNIV}
                \<noteq> {}"
       apply (rule compact_fip_Heine_Borel)
        using comf apply force
       using ne  apply (simp add: subset_iff del: insert_iff)
       done
     then have "\<exists>x. x \<in> (\<Inter>n. {a \<in> K. f a \<in> insert y (range (\<lambda>i. f (X (n + i))))})"
       by blast
     then show ?thesis
       apply (simp add: image_iff fX)
       by (metis \<open>inj h\<close> le_add1 not_less_eq_eq rangeI range_ex1_eq)
   qed
   with assms closedin_subset show ?thesis
     by (force simp: closedin_limpt)
 qed
 
 
 lemma compact_continuous_image_eq:
   fixes f :: "'a::heine_borel \<Rightarrow> 'b::heine_borel"
   assumes f: "inj_on f S"
   shows "continuous_on S f \<longleftrightarrow> (\<forall>T. compact T \<and> T \<subseteq> S \<longrightarrow> compact(f ` T))"
            (is "?lhs = ?rhs")
 proof
   assume ?lhs then show ?rhs
     by (metis continuous_on_subset compact_continuous_image)
 next
   assume RHS: ?rhs
   obtain g where gf: "\<And>x. x \<in> S \<Longrightarrow> g (f x) = x"
     by (metis inv_into_f_f f)
   then have *: "(S \<inter> f -` U) = g ` U" if "U \<subseteq> f ` S" for U
     using that by fastforce
   have gfim: "g ` f ` S \<subseteq> S" using gf by auto
   have **: "compact (f ` S \<inter> g -` C)" if C: "C \<subseteq> S" "compact C" for C
   proof -
     obtain h where "h C \<in> C \<and> h C \<notin> S \<or> compact (f ` C)"
       by (force simp: C RHS)
     moreover have "f ` C = (f ` S \<inter> g -` C)"
       using C gf by auto
     ultimately show ?thesis
       using C by auto
   qed
   show ?lhs
     using proper_map [OF _ _ gfim] **
     by (simp add: continuous_on_closed * closedin_imp_subset)
 qed
 
 subsection\<^marker>\<open>tag unimportant\<close>\<open>Trivial fact: convexity equals connectedness for collinear sets\<close>
 
 lemma convex_connected_collinear:
   fixes S :: "'a::euclidean_space set"
   assumes "collinear S"
     shows "convex S \<longleftrightarrow> connected S"
 proof
   assume "convex S"
   then show "connected S"
     using convex_connected by blast
 next
   assume S: "connected S"
   show "convex S"
   proof (cases "S = {}")
     case True
     then show ?thesis by simp
   next
     case False
     then obtain a where "a \<in> S" by auto
     have "collinear (affine hull S)"
       by (simp add: assms collinear_affine_hull_collinear)
     then obtain z where "z \<noteq> 0" "\<And>x. x \<in> affine hull S \<Longrightarrow> \<exists>c. x - a = c *\<^sub>R z"
       by (meson \<open>a \<in> S\<close> collinear hull_inc)
     then obtain f where f: "\<And>x. x \<in> affine hull S \<Longrightarrow> x - a = f x *\<^sub>R z"
       by metis
     then have inj_f: "inj_on f (affine hull S)"
       by (metis diff_add_cancel inj_onI)
     have diff: "x - y = (f x - f y) *\<^sub>R z" if x: "x \<in> affine hull S" and y: "y \<in> affine hull S" for x y
     proof -
       have "f x *\<^sub>R z = x - a"
         by (simp add: f hull_inc x)
       moreover have "f y *\<^sub>R z = y - a"
         by (simp add: f hull_inc y)
       ultimately show ?thesis
         by (simp add: scaleR_left.diff)
     qed
     have cont_f: "continuous_on (affine hull S) f"
       apply (clarsimp simp: dist_norm continuous_on_iff diff)
       by (metis \<open>z \<noteq> 0\<close> mult.commute mult_less_cancel_left_pos norm_minus_commute real_norm_def zero_less_mult_iff zero_less_norm_iff)
     then have conn_fS: "connected (f ` S)"
       by (meson S connected_continuous_image continuous_on_subset hull_subset)
     show ?thesis
     proof (clarsimp simp: convex_contains_segment)
       fix x y z
       assume "x \<in> S" "y \<in> S" "z \<in> closed_segment x y"
       have False if "z \<notin> S"
       proof -
         have "f ` (closed_segment x y) = closed_segment (f x) (f y)"
           apply (rule continuous_injective_image_segment_1)
           apply (meson \<open>x \<in> S\<close> \<open>y \<in> S\<close> convex_affine_hull convex_contains_segment hull_inc continuous_on_subset [OF cont_f])
           by (meson \<open>x \<in> S\<close> \<open>y \<in> S\<close> convex_affine_hull convex_contains_segment hull_inc inj_on_subset [OF inj_f])
         then have fz: "f z \<in> closed_segment (f x) (f y)"
           using \<open>z \<in> closed_segment x y\<close> by blast
         have "z \<in> affine hull S"
           by (meson \<open>x \<in> S\<close> \<open>y \<in> S\<close> \<open>z \<in> closed_segment x y\<close> convex_affine_hull convex_contains_segment hull_inc subset_eq)
         then have fz_notin: "f z \<notin> f ` S"
           using hull_subset inj_f inj_onD that by fastforce
         moreover have "{..<f z} \<inter> f ` S \<noteq> {}" "{f z<..} \<inter> f ` S \<noteq> {}"
         proof -
           have "{..<f z} \<inter> f ` {x,y} \<noteq> {}"  "{f z<..} \<inter> f ` {x,y} \<noteq> {}"
             using fz fz_notin \<open>x \<in> S\<close> \<open>y \<in> S\<close>
              apply (auto simp: closed_segment_eq_real_ivl split: if_split_asm)
              apply (metis image_eqI less_eq_real_def)+
             done
           then show "{..<f z} \<inter> f ` S \<noteq> {}" "{f z<..} \<inter> f ` S \<noteq> {}"
             using \<open>x \<in> S\<close> \<open>y \<in> S\<close> by blast+
         qed
         ultimately show False
           using connectedD [OF conn_fS, of "{..<f z}" "{f z<..}"] by force
       qed
       then show "z \<in> S" by meson
     qed
   qed
 qed
 
 lemma compact_convex_collinear_segment_alt:
   fixes S :: "'a::euclidean_space set"
   assumes "S \<noteq> {}" "compact S" "connected S" "collinear S"
   obtains a b where "S = closed_segment a b"
 proof -
   obtain \<xi> where "\<xi> \<in> S" using \<open>S \<noteq> {}\<close> by auto
   have "collinear (affine hull S)"
     by (simp add: assms collinear_affine_hull_collinear)
   then obtain z where "z \<noteq> 0" "\<And>x. x \<in> affine hull S \<Longrightarrow> \<exists>c. x - \<xi> = c *\<^sub>R z"
     by (meson \<open>\<xi> \<in> S\<close> collinear hull_inc)
   then obtain f where f: "\<And>x. x \<in> affine hull S \<Longrightarrow> x - \<xi> = f x *\<^sub>R z"
     by metis
   let ?g = "\<lambda>r. r *\<^sub>R z + \<xi>"
   have gf: "?g (f x) = x" if "x \<in> affine hull S" for x
     by (metis diff_add_cancel f that)
   then have inj_f: "inj_on f (affine hull S)"
     by (metis inj_onI)
   have diff: "x - y = (f x - f y) *\<^sub>R z" if x: "x \<in> affine hull S" and y: "y \<in> affine hull S" for x y
   proof -
     have "f x *\<^sub>R z = x - \<xi>"
       by (simp add: f hull_inc x)
     moreover have "f y *\<^sub>R z = y - \<xi>"
       by (simp add: f hull_inc y)
     ultimately show ?thesis
       by (simp add: scaleR_left.diff)
   qed
   have cont_f: "continuous_on (affine hull S) f"
     apply (clarsimp simp: dist_norm continuous_on_iff diff)
     by (metis \<open>z \<noteq> 0\<close> mult.commute mult_less_cancel_left_pos norm_minus_commute real_norm_def zero_less_mult_iff zero_less_norm_iff)
   then have "connected (f ` S)"
     by (meson \<open>connected S\<close> connected_continuous_image continuous_on_subset hull_subset)
   moreover have "compact (f ` S)"
     by (meson \<open>compact S\<close> compact_continuous_image_eq cont_f hull_subset inj_f)
   ultimately obtain x y where "f ` S = {x..y}"
     by (meson connected_compact_interval_1)
   then have fS_eq: "f ` S = closed_segment x y"
     using \<open>S \<noteq> {}\<close> closed_segment_eq_real_ivl by auto
   obtain a b where "a \<in> S" "f a = x" "b \<in> S" "f b = y"
     by (metis (full_types) ends_in_segment fS_eq imageE)
   have "f ` (closed_segment a b) = closed_segment (f a) (f b)"
     apply (rule continuous_injective_image_segment_1)
      apply (meson \<open>a \<in> S\<close> \<open>b \<in> S\<close> convex_affine_hull convex_contains_segment hull_inc continuous_on_subset [OF cont_f])
     by (meson \<open>a \<in> S\<close> \<open>b \<in> S\<close> convex_affine_hull convex_contains_segment hull_inc inj_on_subset [OF inj_f])
   then have "f ` (closed_segment a b) = f ` S"
     by (simp add: \<open>f a = x\<close> \<open>f b = y\<close> fS_eq)
   then have "?g ` f ` (closed_segment a b) = ?g ` f ` S"
     by simp
   moreover have "(\<lambda>x. f x *\<^sub>R z + \<xi>) ` closed_segment a b = closed_segment a b"
     apply safe
      apply (metis (mono_tags, hide_lams) \<open>a \<in> S\<close> \<open>b \<in> S\<close> convex_affine_hull convex_contains_segment gf hull_inc subsetCE)
     by (metis (mono_tags, lifting) \<open>a \<in> S\<close> \<open>b \<in> S\<close> convex_affine_hull convex_contains_segment gf hull_subset image_iff subsetCE)
   ultimately have "closed_segment a b = S"
     using gf by (simp add: image_comp o_def hull_inc cong: image_cong)
   then show ?thesis
     using that by blast
 qed
 
 lemma compact_convex_collinear_segment:
   fixes S :: "'a::euclidean_space set"
   assumes "S \<noteq> {}" "compact S" "convex S" "collinear S"
   obtains a b where "S = closed_segment a b"
   using assms convex_connected_collinear compact_convex_collinear_segment_alt by blast
 
 
 lemma proper_map_from_compact:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes contf: "continuous_on S f" and imf: "f ` S \<subseteq> T" and "compact S"
           "closedin (top_of_set T) K"
   shows "compact (S \<inter> f -` K)"
 by (rule closedin_compact [OF \<open>compact S\<close>] continuous_closedin_preimage_gen assms)+
 
 lemma proper_map_fst:
   assumes "compact T" "K \<subseteq> S" "compact K"
     shows "compact (S \<times> T \<inter> fst -` K)"
 proof -
   have "(S \<times> T \<inter> fst -` K) = K \<times> T"
     using assms by auto
   then show ?thesis by (simp add: assms compact_Times)
 qed
 
 lemma closed_map_fst:
   fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
   assumes "compact T" "closedin (top_of_set (S \<times> T)) c"
    shows "closedin (top_of_set S) (fst ` c)"
 proof -
   have *: "fst ` (S \<times> T) \<subseteq> S"
     by auto
   show ?thesis
     using proper_map [OF _ _ *] by (simp add: proper_map_fst assms)
 qed
 
 lemma proper_map_snd:
   assumes "compact S" "K \<subseteq> T" "compact K"
     shows "compact (S \<times> T \<inter> snd -` K)"
 proof -
   have "(S \<times> T \<inter> snd -` K) = S \<times> K"
     using assms by auto
   then show ?thesis by (simp add: assms compact_Times)
 qed
 
 lemma closed_map_snd:
   fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
   assumes "compact S" "closedin (top_of_set (S \<times> T)) c"
    shows "closedin (top_of_set T) (snd ` c)"
 proof -
   have *: "snd ` (S \<times> T) \<subseteq> T"
     by auto
   show ?thesis
     using proper_map [OF _ _ *] by (simp add: proper_map_snd assms)
 qed
 
 lemma closedin_compact_projection:
   fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
   assumes "compact S" and clo: "closedin (top_of_set (S \<times> T)) U"
     shows "closedin (top_of_set T) {y. \<exists>x. x \<in> S \<and> (x, y) \<in> U}"
 proof -
   have "U \<subseteq> S \<times> T"
     by (metis clo closedin_imp_subset)
   then have "{y. \<exists>x. x \<in> S \<and> (x, y) \<in> U} = snd ` U"
     by force
   moreover have "closedin (top_of_set T) (snd ` U)"
     by (rule closed_map_snd [OF assms])
   ultimately show ?thesis
     by simp
 qed
 
 
 lemma closed_compact_projection:
   fixes S :: "'a::euclidean_space set"
     and T :: "('a * 'b::euclidean_space) set"
   assumes "compact S" and clo: "closed T"
     shows "closed {y. \<exists>x. x \<in> S \<and> (x, y) \<in> T}"
 proof -
   have *: "{y. \<exists>x. x \<in> S \<and> Pair x y \<in> T} =
         {y. \<exists>x. x \<in> S \<and> Pair x y \<in> ((S \<times> UNIV) \<inter> T)}"
     by auto
   show ?thesis
     apply (subst *)
     apply (rule closedin_closed_trans [OF _ closed_UNIV])
     apply (rule closedin_compact_projection [OF \<open>compact S\<close>])
     by (simp add: clo closedin_closed_Int)
 qed
 
 subsubsection\<^marker>\<open>tag unimportant\<close>\<open>Representing affine hull as a finite intersection of hyperplanes\<close>
 
 proposition\<^marker>\<open>tag unimportant\<close> affine_hull_convex_Int_nonempty_interior:
   fixes S :: "'a::real_normed_vector set"
   assumes "convex S" "S \<inter> interior T \<noteq> {}"
     shows "affine hull (S \<inter> T) = affine hull S"
 proof
   show "affine hull (S \<inter> T) \<subseteq> affine hull S"
     by (simp add: hull_mono)
 next
   obtain a where "a \<in> S" "a \<in> T" and at: "a \<in> interior T"
     using assms interior_subset by blast
   then obtain e where "e > 0" and e: "cball a e \<subseteq> T"
     using mem_interior_cball by blast
   have *: "x \<in> (+) a ` span ((\<lambda>x. x - a) ` (S \<inter> T))" if "x \<in> S" for x
   proof (cases "x = a")
     case True with that span_0 eq_add_iff image_def mem_Collect_eq show ?thesis
       by blast
   next
     case False
     define k where "k = min (1/2) (e / norm (x-a))"
     have k: "0 < k" "k < 1"
       using \<open>e > 0\<close> False by (auto simp: k_def)
     then have xa: "(x-a) = inverse k *\<^sub>R k *\<^sub>R (x-a)"
       by simp
     have "e / norm (x - a) \<ge> k"
       using k_def by linarith
     then have "a + k *\<^sub>R (x - a) \<in> cball a e"
       using \<open>0 < k\<close> False
       by (simp add: dist_norm) (simp add: field_simps)
     then have T: "a + k *\<^sub>R (x - a) \<in> T"
       using e by blast
     have S: "a + k *\<^sub>R (x - a) \<in> S"
       using k \<open>a \<in> S\<close> convexD [OF \<open>convex S\<close> \<open>a \<in> S\<close> \<open>x \<in> S\<close>, of "1-k" k]
       by (simp add: algebra_simps)
     have "inverse k *\<^sub>R k *\<^sub>R (x-a) \<in> span ((\<lambda>x. x - a) ` (S \<inter> T))"
       apply (rule span_mul)
       apply (rule span_base)
       apply (rule image_eqI [where x = "a + k *\<^sub>R (x - a)"])
       apply (auto simp: S T)
       done
     with xa image_iff show ?thesis  by fastforce
   qed
   show "affine hull S \<subseteq> affine hull (S \<inter> T)"
     apply (simp add: subset_hull)
     apply (simp add: \<open>a \<in> S\<close> \<open>a \<in> T\<close> hull_inc affine_hull_span_gen [of a])
     apply (force simp: *)
     done
 qed
 
 corollary affine_hull_convex_Int_open:
   fixes S :: "'a::real_normed_vector set"
   assumes "convex S" "open T" "S \<inter> T \<noteq> {}"
     shows "affine hull (S \<inter> T) = affine hull S"
 using affine_hull_convex_Int_nonempty_interior assms interior_eq by blast
 
 corollary affine_hull_affine_Int_nonempty_interior:
   fixes S :: "'a::real_normed_vector set"
   assumes "affine S" "S \<inter> interior T \<noteq> {}"
     shows "affine hull (S \<inter> T) = affine hull S"
 by (simp add: affine_hull_convex_Int_nonempty_interior affine_imp_convex assms)
 
 corollary affine_hull_affine_Int_open:
   fixes S :: "'a::real_normed_vector set"
   assumes "affine S" "open T" "S \<inter> T \<noteq> {}"
     shows "affine hull (S \<inter> T) = affine hull S"
 by (simp add: affine_hull_convex_Int_open affine_imp_convex assms)
 
 corollary affine_hull_convex_Int_openin:
   fixes S :: "'a::real_normed_vector set"
   assumes "convex S" "openin (top_of_set (affine hull S)) T" "S \<inter> T \<noteq> {}"
     shows "affine hull (S \<inter> T) = affine hull S"
 using assms unfolding openin_open
 by (metis affine_hull_convex_Int_open hull_subset inf.orderE inf_assoc)
 
 corollary affine_hull_openin:
   fixes S :: "'a::real_normed_vector set"
   assumes "openin (top_of_set (affine hull T)) S" "S \<noteq> {}"
     shows "affine hull S = affine hull T"
 using assms unfolding openin_open
 by (metis affine_affine_hull affine_hull_affine_Int_open hull_hull)
 
 corollary affine_hull_open:
   fixes S :: "'a::real_normed_vector set"
   assumes "open S" "S \<noteq> {}"
     shows "affine hull S = UNIV"
 by (metis affine_hull_convex_Int_nonempty_interior assms convex_UNIV hull_UNIV inf_top.left_neutral interior_open)
 
 lemma aff_dim_convex_Int_nonempty_interior:
   fixes S :: "'a::euclidean_space set"
   shows "\<lbrakk>convex S; S \<inter> interior T \<noteq> {}\<rbrakk> \<Longrightarrow> aff_dim(S \<inter> T) = aff_dim S"
 using aff_dim_affine_hull2 affine_hull_convex_Int_nonempty_interior by blast
 
 lemma aff_dim_convex_Int_open:
   fixes S :: "'a::euclidean_space set"
   shows "\<lbrakk>convex S; open T; S \<inter> T \<noteq> {}\<rbrakk> \<Longrightarrow>  aff_dim(S \<inter> T) = aff_dim S"
 using aff_dim_convex_Int_nonempty_interior interior_eq by blast
 
 lemma affine_hull_Diff:
   fixes S:: "'a::real_normed_vector set"
   assumes ope: "openin (top_of_set (affine hull S)) S" and "finite F" "F \<subset> S"
     shows "affine hull (S - F) = affine hull S"
 proof -
   have clo: "closedin (top_of_set S) F"
     using assms finite_imp_closedin by auto
   moreover have "S - F \<noteq> {}"
     using assms by auto
   ultimately show ?thesis
     by (metis ope closedin_def topspace_euclidean_subtopology affine_hull_openin openin_trans)
 qed
 
 lemma affine_hull_halfspace_lt:
   fixes a :: "'a::euclidean_space"
   shows "affine hull {x. a \<bullet> x < r} = (if a = 0 \<and> r \<le> 0 then {} else UNIV)"
 using halfspace_eq_empty_lt [of a r]
 by (simp add: open_halfspace_lt affine_hull_open)
 
 lemma affine_hull_halfspace_le:
   fixes a :: "'a::euclidean_space"
   shows "affine hull {x. a \<bullet> x \<le> r} = (if a = 0 \<and> r < 0 then {} else UNIV)"
 proof (cases "a = 0")
   case True then show ?thesis by simp
 next
   case False
   then have "affine hull closure {x. a \<bullet> x < r} = UNIV"
     using affine_hull_halfspace_lt closure_same_affine_hull by fastforce
   moreover have "{x. a \<bullet> x < r} \<subseteq> {x. a \<bullet> x \<le> r}"
     by (simp add: Collect_mono)
   ultimately show ?thesis using False antisym_conv hull_mono top_greatest
     by (metis affine_hull_halfspace_lt)
 qed
 
 lemma affine_hull_halfspace_gt:
   fixes a :: "'a::euclidean_space"
   shows "affine hull {x. a \<bullet> x > r} = (if a = 0 \<and> r \<ge> 0 then {} else UNIV)"
 using halfspace_eq_empty_gt [of r a]
 by (simp add: open_halfspace_gt affine_hull_open)
 
 lemma affine_hull_halfspace_ge:
   fixes a :: "'a::euclidean_space"
   shows "affine hull {x. a \<bullet> x \<ge> r} = (if a = 0 \<and> r > 0 then {} else UNIV)"
 using affine_hull_halfspace_le [of "-a" "-r"] by simp
 
 lemma aff_dim_halfspace_lt:
   fixes a :: "'a::euclidean_space"
   shows "aff_dim {x. a \<bullet> x < r} =
         (if a = 0 \<and> r \<le> 0 then -1 else DIM('a))"
 by simp (metis aff_dim_open halfspace_eq_empty_lt open_halfspace_lt)
 
 lemma aff_dim_halfspace_le:
   fixes a :: "'a::euclidean_space"
   shows "aff_dim {x. a \<bullet> x \<le> r} =
         (if a = 0 \<and> r < 0 then -1 else DIM('a))"
 proof -
   have "int (DIM('a)) = aff_dim (UNIV::'a set)"
     by (simp)
   then have "aff_dim (affine hull {x. a \<bullet> x \<le> r}) = DIM('a)" if "(a = 0 \<longrightarrow> r \<ge> 0)"
     using that by (simp add: affine_hull_halfspace_le not_less)
   then show ?thesis
     by (force)
 qed
 
 lemma aff_dim_halfspace_gt:
   fixes a :: "'a::euclidean_space"
   shows "aff_dim {x. a \<bullet> x > r} =
         (if a = 0 \<and> r \<ge> 0 then -1 else DIM('a))"
 by simp (metis aff_dim_open halfspace_eq_empty_gt open_halfspace_gt)
 
 lemma aff_dim_halfspace_ge:
   fixes a :: "'a::euclidean_space"
   shows "aff_dim {x. a \<bullet> x \<ge> r} =
         (if a = 0 \<and> r > 0 then -1 else DIM('a))"
 using aff_dim_halfspace_le [of "-a" "-r"] by simp
 
 proposition aff_dim_eq_hyperplane:
   fixes S :: "'a::euclidean_space set"
   shows "aff_dim S = DIM('a) - 1 \<longleftrightarrow> (\<exists>a b. a \<noteq> 0 \<and> affine hull S = {x. a \<bullet> x = b})"
 proof (cases "S = {}")
   case True then show ?thesis
     by (auto simp: dest: hyperplane_eq_Ex)
 next
   case False
   then obtain c where "c \<in> S" by blast
   show ?thesis
   proof (cases "c = 0")
     case True show ?thesis
       using span_zero [of S]
         apply (simp add: aff_dim_eq_dim [of c] affine_hull_span_gen [of c] \<open>c \<in> S\<close> hull_inc dim_eq_hyperplane
           del: One_nat_def)
         apply (auto simp add: \<open>c = 0\<close>)
         done
   next
     case False
     have xc_im: "x \<in> (+) c ` {y. a \<bullet> y = 0}" if "a \<bullet> x = a \<bullet> c" for a x
     proof -
       have "\<exists>y. a \<bullet> y = 0 \<and> c + y = x"
         by (metis that add.commute diff_add_cancel inner_commute inner_diff_left right_minus_eq)
       then show "x \<in> (+) c ` {y. a \<bullet> y = 0}"
         by blast
     qed
     have 2: "span ((\<lambda>x. x - c) ` S) = {x. a \<bullet> x = 0}"
          if "(+) c ` span ((\<lambda>x. x - c) ` S) = {x. a \<bullet> x = b}" for a b
     proof -
       have "b = a \<bullet> c"
         using span_0 that by fastforce
       with that have "(+) c ` span ((\<lambda>x. x - c) ` S) = {x. a \<bullet> x = a \<bullet> c}"
         by simp
       then have "span ((\<lambda>x. x - c) ` S) = (\<lambda>x. x - c) ` {x. a \<bullet> x = a \<bullet> c}"
         by (metis (no_types) image_cong translation_galois uminus_add_conv_diff)
       also have "... = {x. a \<bullet> x = 0}"
         by (force simp: inner_distrib inner_diff_right
              intro: image_eqI [where x="x+c" for x])
       finally show ?thesis .
     qed
     show ?thesis
       apply (simp add: aff_dim_eq_dim [of c] affine_hull_span_gen [of c] \<open>c \<in> S\<close> hull_inc dim_eq_hyperplane
                   del: One_nat_def cong: image_cong_simp, safe)
       apply (fastforce simp add: inner_distrib intro: xc_im)
       apply (force simp: intro!: 2)
       done
   qed
 qed
 
 corollary aff_dim_hyperplane [simp]:
   fixes a :: "'a::euclidean_space"
   shows "a \<noteq> 0 \<Longrightarrow> aff_dim {x. a \<bullet> x = r} = DIM('a) - 1"
 by (metis aff_dim_eq_hyperplane affine_hull_eq affine_hyperplane)
 
 subsection\<^marker>\<open>tag unimportant\<close>\<open>Some stepping theorems\<close>
 
 lemma aff_dim_insert:
   fixes a :: "'a::euclidean_space"
   shows "aff_dim (insert a S) = (if a \<in> affine hull S then aff_dim S else aff_dim S + 1)"
 proof (cases "S = {}")
   case True then show ?thesis
     by simp
 next
   case False
   then obtain x s' where S: "S = insert x s'" "x \<notin> s'"
     by (meson Set.set_insert all_not_in_conv)
   show ?thesis using S
     apply (simp add: hull_redundant cong: aff_dim_affine_hull2)
     apply (simp add: affine_hull_insert_span_gen hull_inc)
     by (force simp add: span_zero insert_commute [of a] hull_inc aff_dim_eq_dim [of x] dim_insert
       cong: image_cong_simp)
 qed
 
 lemma affine_dependent_choose:
   fixes a :: "'a :: euclidean_space"
   assumes "\<not>(affine_dependent S)"
   shows "affine_dependent(insert a S) \<longleftrightarrow> a \<notin> S \<and> a \<in> affine hull S"
         (is "?lhs = ?rhs")
 proof safe
   assume "affine_dependent (insert a S)" and "a \<in> S"
   then show "False"
     using \<open>a \<in> S\<close> assms insert_absorb by fastforce
 next
   assume lhs: "affine_dependent (insert a S)"
   then have "a \<notin> S"
     by (metis (no_types) assms insert_absorb)
   moreover have "finite S"
     using affine_independent_iff_card assms by blast
   moreover have "aff_dim (insert a S) \<noteq> int (card S)"
     using \<open>finite S\<close> affine_independent_iff_card \<open>a \<notin> S\<close> lhs by fastforce
   ultimately show "a \<in> affine hull S"
     by (metis aff_dim_affine_independent aff_dim_insert assms)
 next
   assume "a \<notin> S" and "a \<in> affine hull S"
   show "affine_dependent (insert a S)"
     by (simp add: \<open>a \<in> affine hull S\<close> \<open>a \<notin> S\<close> affine_dependent_def)
 qed
 
 lemma affine_independent_insert:
   fixes a :: "'a :: euclidean_space"
   shows "\<lbrakk>\<not> affine_dependent S; a \<notin> affine hull S\<rbrakk> \<Longrightarrow> \<not> affine_dependent(insert a S)"
   by (simp add: affine_dependent_choose)
 
 lemma subspace_bounded_eq_trivial:
   fixes S :: "'a::real_normed_vector set"
   assumes "subspace S"
     shows "bounded S \<longleftrightarrow> S = {0}"
 proof -
   have "False" if "bounded S" "x \<in> S" "x \<noteq> 0" for x
   proof -
     obtain B where B: "\<And>y. y \<in> S \<Longrightarrow> norm y < B" "B > 0"
       using \<open>bounded S\<close> by (force simp: bounded_pos_less)
     have "(B / norm x) *\<^sub>R x \<in> S"
       using assms subspace_mul \<open>x \<in> S\<close> by auto
     moreover have "norm ((B / norm x) *\<^sub>R x) = B"
       using that B by (simp add: algebra_simps)
     ultimately show False using B by force
   qed
   then have "bounded S \<Longrightarrow> S = {0}"
     using assms subspace_0 by fastforce
   then show ?thesis
     by blast
 qed
 
 lemma affine_bounded_eq_trivial:
   fixes S :: "'a::real_normed_vector set"
   assumes "affine S"
     shows "bounded S \<longleftrightarrow> S = {} \<or> (\<exists>a. S = {a})"
 proof (cases "S = {}")
   case True then show ?thesis
     by simp
 next
   case False
   then obtain b where "b \<in> S" by blast
   with False assms show ?thesis
     apply safe
     using affine_diffs_subspace [OF assms \<open>b \<in> S\<close>]
     apply (metis (no_types, lifting) subspace_bounded_eq_trivial ab_left_minus bounded_translation
                 image_empty image_insert translation_invert)
     apply force
     done
 qed
 
 lemma affine_bounded_eq_lowdim:
   fixes S :: "'a::euclidean_space set"
   assumes "affine S"
     shows "bounded S \<longleftrightarrow> aff_dim S \<le> 0"
 apply safe
 using affine_bounded_eq_trivial assms apply fastforce
 by (metis aff_dim_sing aff_dim_subset affine_dim_equal affine_sing all_not_in_conv assms bounded_empty bounded_insert dual_order.antisym empty_subsetI insert_subset)
 
 
 lemma bounded_hyperplane_eq_trivial_0:
   fixes a :: "'a::euclidean_space"
   assumes "a \<noteq> 0"
   shows "bounded {x. a \<bullet> x = 0} \<longleftrightarrow> DIM('a) = 1"
 proof
   assume "bounded {x. a \<bullet> x = 0}"
   then have "aff_dim {x. a \<bullet> x = 0} \<le> 0"
     by (simp add: affine_bounded_eq_lowdim affine_hyperplane)
   with assms show "DIM('a) = 1"
     by (simp add: le_Suc_eq)
 next
   assume "DIM('a) = 1"
   then show "bounded {x. a \<bullet> x = 0}"
     by (simp add: affine_bounded_eq_lowdim affine_hyperplane assms)
 qed
 
 lemma bounded_hyperplane_eq_trivial:
   fixes a :: "'a::euclidean_space"
   shows "bounded {x. a \<bullet> x = r} \<longleftrightarrow> (if a = 0 then r \<noteq> 0 else DIM('a) = 1)"
 proof (simp add: bounded_hyperplane_eq_trivial_0, clarify)
   assume "r \<noteq> 0" "a \<noteq> 0"
   have "aff_dim {x. y \<bullet> x = 0} = aff_dim {x. a \<bullet> x = r}" if "y \<noteq> 0" for y::'a
     by (metis that \<open>a \<noteq> 0\<close> aff_dim_hyperplane)
   then show "bounded {x. a \<bullet> x = r} = (DIM('a) = Suc 0)"
     by (metis One_nat_def \<open>a \<noteq> 0\<close> affine_bounded_eq_lowdim affine_hyperplane bounded_hyperplane_eq_trivial_0)
 qed
 
 subsection\<^marker>\<open>tag unimportant\<close>\<open>General case without assuming closure and getting non-strict separation\<close>
 
 proposition\<^marker>\<open>tag unimportant\<close> separating_hyperplane_closed_point_inset:
   fixes S :: "'a::euclidean_space set"
   assumes "convex S" "closed S" "S \<noteq> {}" "z \<notin> S"
   obtains a b where "a \<in> S" "(a - z) \<bullet> z < b" "\<And>x. x \<in> S \<Longrightarrow> b < (a - z) \<bullet> x"
 proof -
   obtain y where "y \<in> S" and y: "\<And>u. u \<in> S \<Longrightarrow> dist z y \<le> dist z u"
     using distance_attains_inf [of S z] assms by auto
   then have *: "(y - z) \<bullet> z < (y - z) \<bullet> z + (norm (y - z))\<^sup>2 / 2"
     using \<open>y \<in> S\<close> \<open>z \<notin> S\<close> by auto
   show ?thesis
   proof (rule that [OF \<open>y \<in> S\<close> *])
     fix x
     assume "x \<in> S"
     have yz: "0 < (y - z) \<bullet> (y - z)"
       using \<open>y \<in> S\<close> \<open>z \<notin> S\<close> by auto
     { assume 0: "0 < ((z - y) \<bullet> (x - y))"
       with any_closest_point_dot [OF \<open>convex S\<close> \<open>closed S\<close>]
       have False
         using y \<open>x \<in> S\<close> \<open>y \<in> S\<close> not_less by blast
     }
     then have "0 \<le> ((y - z) \<bullet> (x - y))"
       by (force simp: not_less inner_diff_left)
     with yz have "0 < 2 * ((y - z) \<bullet> (x - y)) + (y - z) \<bullet> (y - z)"
       by (simp add: algebra_simps)
     then show "(y - z) \<bullet> z + (norm (y - z))\<^sup>2 / 2 < (y - z) \<bullet> x"
       by (simp add: field_simps inner_diff_left inner_diff_right dot_square_norm [symmetric])
   qed
 qed
 
 lemma separating_hyperplane_closed_0_inset:
   fixes S :: "'a::euclidean_space set"
   assumes "convex S" "closed S" "S \<noteq> {}" "0 \<notin> S"
   obtains a b where "a \<in> S" "a \<noteq> 0" "0 < b" "\<And>x. x \<in> S \<Longrightarrow> a \<bullet> x > b"
 using separating_hyperplane_closed_point_inset [OF assms]
 by simp (metis \<open>0 \<notin> S\<close>)
 
 
 proposition\<^marker>\<open>tag unimportant\<close> separating_hyperplane_set_0_inspan:
   fixes S :: "'a::euclidean_space set"
   assumes "convex S" "S \<noteq> {}" "0 \<notin> S"
   obtains a where "a \<in> span S" "a \<noteq> 0" "\<And>x. x \<in> S \<Longrightarrow> 0 \<le> a \<bullet> x"
 proof -
   define k where [abs_def]: "k c = {x. 0 \<le> c \<bullet> x}" for c :: 'a
   have *: "span S \<inter> frontier (cball 0 1) \<inter> \<Inter>f' \<noteq> {}"
           if f': "finite f'" "f' \<subseteq> k ` S" for f'
   proof -
     obtain C where "C \<subseteq> S" "finite C" and C: "f' = k ` C"
       using finite_subset_image [OF f'] by blast
     obtain a where "a \<in> S" "a \<noteq> 0"
       using \<open>S \<noteq> {}\<close> \<open>0 \<notin> S\<close> ex_in_conv by blast
     then have "norm (a /\<^sub>R (norm a)) = 1"
       by simp
     moreover have "a /\<^sub>R (norm a) \<in> span S"
       by (simp add: \<open>a \<in> S\<close> span_scale span_base)
     ultimately have ass: "a /\<^sub>R (norm a) \<in> span S \<inter> sphere 0 1"
       by simp
     show ?thesis
     proof (cases "C = {}")
       case True with C ass show ?thesis
         by auto
     next
       case False
       have "closed (convex hull C)"
         using \<open>finite C\<close> compact_eq_bounded_closed finite_imp_compact_convex_hull by auto
       moreover have "convex hull C \<noteq> {}"
         by (simp add: False)
       moreover have "0 \<notin> convex hull C"
         by (metis \<open>C \<subseteq> S\<close> \<open>convex S\<close> \<open>0 \<notin> S\<close> convex_hull_subset hull_same insert_absorb insert_subset)
       ultimately obtain a b
             where "a \<in> convex hull C" "a \<noteq> 0" "0 < b"
                   and ab: "\<And>x. x \<in> convex hull C \<Longrightarrow> a \<bullet> x > b"
         using separating_hyperplane_closed_0_inset by blast
       then have "a \<in> S"
         by (metis \<open>C \<subseteq> S\<close> assms(1) subsetCE subset_hull)
       moreover have "norm (a /\<^sub>R (norm a)) = 1"
         using \<open>a \<noteq> 0\<close> by simp
       moreover have "a /\<^sub>R (norm a) \<in> span S"
         by (simp add: \<open>a \<in> S\<close> span_scale span_base)
       ultimately have ass: "a /\<^sub>R (norm a) \<in> span S \<inter> sphere 0 1"
         by simp
       have aa: "a /\<^sub>R (norm a) \<in> (\<Inter>c\<in>C. {x. 0 \<le> c \<bullet> x})"
         apply (clarsimp simp add: field_split_simps)
         using ab \<open>0 < b\<close>
         by (metis hull_inc inner_commute less_eq_real_def less_trans)
       show ?thesis
         apply (simp add: C k_def)
         using ass aa Int_iff empty_iff by blast
     qed
   qed
   have "(span S \<inter> frontier(cball 0 1)) \<inter> (\<Inter> (k ` S)) \<noteq> {}"
     apply (rule compact_imp_fip)
     apply (blast intro: compact_cball)
     using closed_halfspace_ge k_def apply blast
     apply (metis *)
     done
   then show ?thesis
     unfolding set_eq_iff k_def
     by simp (metis inner_commute norm_eq_zero that zero_neq_one)
 qed
 
 
 lemma separating_hyperplane_set_point_inaff:
   fixes S :: "'a::euclidean_space set"
   assumes "convex S" "S \<noteq> {}" and zno: "z \<notin> S"
   obtains a b where "(z + a) \<in> affine hull (insert z S)"
                 and "a \<noteq> 0" and "a \<bullet> z \<le> b"
                 and "\<And>x. x \<in> S \<Longrightarrow> a \<bullet> x \<ge> b"
 proof -
   from separating_hyperplane_set_0_inspan [of "image (\<lambda>x. -z + x) S"]
   have "convex ((+) (- z) ` S)"
     using \<open>convex S\<close> by simp
   moreover have "(+) (- z) ` S \<noteq> {}"
     by (simp add: \<open>S \<noteq> {}\<close>)
   moreover have "0 \<notin> (+) (- z) ` S"
     using zno by auto
   ultimately obtain a where "a \<in> span ((+) (- z) ` S)" "a \<noteq> 0"
                   and a:  "\<And>x. x \<in> ((+) (- z) ` S) \<Longrightarrow> 0 \<le> a \<bullet> x"
     using separating_hyperplane_set_0_inspan [of "image (\<lambda>x. -z + x) S"]
     by blast
   then have szx: "\<And>x. x \<in> S \<Longrightarrow> a \<bullet> z \<le> a \<bullet> x"
     by (metis (no_types, lifting) imageI inner_minus_right inner_right_distrib minus_add neg_le_0_iff_le neg_le_iff_le real_add_le_0_iff)
   show ?thesis
     apply (rule_tac a=a and b = "a  \<bullet> z" in that, simp_all)
     using \<open>a \<in> span ((+) (- z) ` S)\<close> affine_hull_insert_span_gen apply blast
     apply (simp_all add: \<open>a \<noteq> 0\<close> szx)
     done
 qed
 
 proposition\<^marker>\<open>tag unimportant\<close> supporting_hyperplane_rel_boundary:
   fixes S :: "'a::euclidean_space set"
   assumes "convex S" "x \<in> S" and xno: "x \<notin> rel_interior S"
   obtains a where "a \<noteq> 0"
               and "\<And>y. y \<in> S \<Longrightarrow> a \<bullet> x \<le> a \<bullet> y"
               and "\<And>y. y \<in> rel_interior S \<Longrightarrow> a \<bullet> x < a \<bullet> y"
 proof -
   obtain a b where aff: "(x + a) \<in> affine hull (insert x (rel_interior S))"
                   and "a \<noteq> 0" and "a \<bullet> x \<le> b"
                   and ageb: "\<And>u. u \<in> (rel_interior S) \<Longrightarrow> a \<bullet> u \<ge> b"
     using separating_hyperplane_set_point_inaff [of "rel_interior S" x] assms
     by (auto simp: rel_interior_eq_empty convex_rel_interior)
   have le_ay: "a \<bullet> x \<le> a \<bullet> y" if "y \<in> S" for y
   proof -
     have con: "continuous_on (closure (rel_interior S)) ((\<bullet>) a)"
       by (rule continuous_intros continuous_on_subset | blast)+
     have y: "y \<in> closure (rel_interior S)"
       using \<open>convex S\<close> closure_def convex_closure_rel_interior \<open>y \<in> S\<close>
       by fastforce
     show ?thesis
       using continuous_ge_on_closure [OF con y] ageb \<open>a \<bullet> x \<le> b\<close>
       by fastforce
   qed
   have 3: "a \<bullet> x < a \<bullet> y" if "y \<in> rel_interior S" for y
   proof -
     obtain e where "0 < e" "y \<in> S" and e: "cball y e \<inter> affine hull S \<subseteq> S"
       using \<open>y \<in> rel_interior S\<close> by (force simp: rel_interior_cball)
     define y' where "y' = y - (e / norm a) *\<^sub>R ((x + a) - x)"
     have "y' \<in> cball y e"
       unfolding y'_def using \<open>0 < e\<close> by force
     moreover have "y' \<in> affine hull S"
       unfolding y'_def
       by (metis \<open>x \<in> S\<close> \<open>y \<in> S\<close> \<open>convex S\<close> aff affine_affine_hull hull_redundant
                 rel_interior_same_affine_hull hull_inc mem_affine_3_minus2)
     ultimately have "y' \<in> S"
       using e by auto
     have "a \<bullet> x \<le> a \<bullet> y"
       using le_ay \<open>a \<noteq> 0\<close> \<open>y \<in> S\<close> by blast
     moreover have "a \<bullet> x \<noteq> a \<bullet> y"
       using le_ay [OF \<open>y' \<in> S\<close>] \<open>a \<noteq> 0\<close>
       apply (simp add: y'_def inner_diff dot_square_norm power2_eq_square)
       by (metis \<open>0 < e\<close> add_le_same_cancel1 inner_commute inner_real_def inner_zero_left le_diff_eq norm_le_zero_iff real_mult_le_cancel_iff2)
     ultimately show ?thesis by force
   qed
   show ?thesis
     by (rule that [OF \<open>a \<noteq> 0\<close> le_ay 3])
 qed
 
 lemma supporting_hyperplane_relative_frontier:
   fixes S :: "'a::euclidean_space set"
   assumes "convex S" "x \<in> closure S" "x \<notin> rel_interior S"
   obtains a where "a \<noteq> 0"
               and "\<And>y. y \<in> closure S \<Longrightarrow> a \<bullet> x \<le> a \<bullet> y"
               and "\<And>y. y \<in> rel_interior S \<Longrightarrow> a \<bullet> x < a \<bullet> y"
 using supporting_hyperplane_rel_boundary [of "closure S" x]
 by (metis assms convex_closure convex_rel_interior_closure)
 
 
 subsection\<^marker>\<open>tag unimportant\<close>\<open> Some results on decomposing convex hulls: intersections, simplicial subdivision\<close>
 
 lemma
   fixes s :: "'a::euclidean_space set"
   assumes "\<not> affine_dependent(s \<union> t)"
     shows convex_hull_Int_subset: "convex hull s \<inter> convex hull t \<subseteq> convex hull (s \<inter> t)" (is ?C)
       and affine_hull_Int_subset: "affine hull s \<inter> affine hull t \<subseteq> affine hull (s \<inter> t)" (is ?A)
 proof -
   have [simp]: "finite s" "finite t"
     using aff_independent_finite assms by blast+
     have "sum u (s \<inter> t) = 1 \<and>
           (\<Sum>v\<in>s \<inter> t. u v *\<^sub>R v) = (\<Sum>v\<in>s. u v *\<^sub>R v)"
       if [simp]:  "sum u s = 1"
                  "sum v t = 1"
          and eq: "(\<Sum>x\<in>t. v x *\<^sub>R x) = (\<Sum>x\<in>s. u x *\<^sub>R x)" for u v
     proof -
     define f where "f x = (if x \<in> s then u x else 0) - (if x \<in> t then v x else 0)" for x
     have "sum f (s \<union> t) = 0"
       apply (simp add: f_def sum_Un sum_subtractf)
       apply (simp add: sum.inter_restrict [symmetric] Int_commute)
       done
     moreover have "(\<Sum>x\<in>(s \<union> t). f x *\<^sub>R x) = 0"
       apply (simp add: f_def sum_Un scaleR_left_diff_distrib sum_subtractf)
       apply (simp add: if_smult sum.inter_restrict [symmetric] Int_commute eq
           cong del: if_weak_cong)
       done
     ultimately have "\<And>v. v \<in> s \<union> t \<Longrightarrow> f v = 0"
       using aff_independent_finite assms unfolding affine_dependent_explicit
       by blast
     then have u [simp]: "\<And>x. x \<in> s \<Longrightarrow> u x = (if x \<in> t then v x else 0)"
       by (simp add: f_def) presburger
     have "sum u (s \<inter> t) = sum u s"
       by (simp add: sum.inter_restrict)
     then have "sum u (s \<inter> t) = 1"
       using that by linarith
     moreover have "(\<Sum>v\<in>s \<inter> t. u v *\<^sub>R v) = (\<Sum>v\<in>s. u v *\<^sub>R v)"
       by (auto simp: if_smult sum.inter_restrict intro: sum.cong)
     ultimately show ?thesis
       by force
     qed
     then show ?A ?C
       by (auto simp: convex_hull_finite affine_hull_finite)
 qed
 
 
 proposition\<^marker>\<open>tag unimportant\<close> affine_hull_Int:
   fixes s :: "'a::euclidean_space set"
   assumes "\<not> affine_dependent(s \<union> t)"
     shows "affine hull (s \<inter> t) = affine hull s \<inter> affine hull t"
 apply (rule subset_antisym)
 apply (simp add: hull_mono)
 by (simp add: affine_hull_Int_subset assms)
 
 proposition\<^marker>\<open>tag unimportant\<close> convex_hull_Int:
   fixes s :: "'a::euclidean_space set"
   assumes "\<not> affine_dependent(s \<union> t)"
     shows "convex hull (s \<inter> t) = convex hull s \<inter> convex hull t"
 apply (rule subset_antisym)
 apply (simp add: hull_mono)
 by (simp add: convex_hull_Int_subset assms)
 
 proposition\<^marker>\<open>tag unimportant\<close>
   fixes s :: "'a::euclidean_space set set"
   assumes "\<not> affine_dependent (\<Union>s)"
     shows affine_hull_Inter: "affine hull (\<Inter>s) = (\<Inter>t\<in>s. affine hull t)" (is "?A")
       and convex_hull_Inter: "convex hull (\<Inter>s) = (\<Inter>t\<in>s. convex hull t)" (is "?C")
 proof -
   have "finite s"
     using aff_independent_finite assms finite_UnionD by blast
   then have "?A \<and> ?C" using assms
   proof (induction s rule: finite_induct)
     case empty then show ?case by auto
   next
     case (insert t F)
     then show ?case
     proof (cases "F={}")
       case True then show ?thesis by simp
     next
       case False
       with "insert.prems" have [simp]: "\<not> affine_dependent (t \<union> \<Inter>F)"
         by (auto intro: affine_dependent_subset)
       have [simp]: "\<not> affine_dependent (\<Union>F)"
         using affine_independent_subset insert.prems by fastforce
       show ?thesis
         by (simp add: affine_hull_Int convex_hull_Int insert.IH)
     qed
   qed
   then show "?A" "?C"
     by auto
 qed
 
 proposition\<^marker>\<open>tag unimportant\<close> in_convex_hull_exchange_unique:
   fixes S :: "'a::euclidean_space set"
   assumes naff: "\<not> affine_dependent S" and a: "a \<in> convex hull S"
       and S: "T \<subseteq> S" "T' \<subseteq> S"
       and x:  "x \<in> convex hull (insert a T)"
       and x': "x \<in> convex hull (insert a T')"
     shows "x \<in> convex hull (insert a (T \<inter> T'))"
 proof (cases "a \<in> S")
   case True
   then have "\<not> affine_dependent (insert a T \<union> insert a T')"
     using affine_dependent_subset assms by auto
   then have "x \<in> convex hull (insert a T \<inter> insert a T')"
     by (metis IntI convex_hull_Int x x')
   then show ?thesis
     by simp
 next
   case False
   then have anot: "a \<notin> T" "a \<notin> T'"
     using assms by auto
   have [simp]: "finite S"
     by (simp add: aff_independent_finite assms)
   then obtain b where b0: "\<And>s. s \<in> S \<Longrightarrow> 0 \<le> b s"
                   and b1: "sum b S = 1" and aeq: "a = (\<Sum>s\<in>S. b s *\<^sub>R s)"
     using a by (auto simp: convex_hull_finite)
   have fin [simp]: "finite T" "finite T'"
     using assms infinite_super \<open>finite S\<close> by blast+
   then obtain c c' where c0: "\<And>t. t \<in> insert a T \<Longrightarrow> 0 \<le> c t"
                      and c1: "sum c (insert a T) = 1"
                      and xeq: "x = (\<Sum>t \<in> insert a T. c t *\<^sub>R t)"
                      and c'0: "\<And>t. t \<in> insert a T' \<Longrightarrow> 0 \<le> c' t"
                      and c'1: "sum c' (insert a T') = 1"
                      and x'eq: "x = (\<Sum>t \<in> insert a T'. c' t *\<^sub>R t)"
     using x x' by (auto simp: convex_hull_finite)
   with fin anot
   have sumTT': "sum c T = 1 - c a" "sum c' T' = 1 - c' a"
    and wsumT: "(\<Sum>t \<in> T. c t *\<^sub>R t) = x - c a *\<^sub>R a"
     by simp_all
   have wsumT': "(\<Sum>t \<in> T'. c' t *\<^sub>R t) = x - c' a *\<^sub>R a"
     using x'eq fin anot by simp
   define cc  where "cc \<equiv> \<lambda>x. if x \<in> T then c x else 0"
   define cc' where "cc' \<equiv> \<lambda>x. if x \<in> T' then c' x else 0"
   define dd  where "dd \<equiv> \<lambda>x. cc x - cc' x + (c a - c' a) * b x"
   have sumSS': "sum cc S = 1 - c a" "sum cc' S = 1 - c' a"
     unfolding cc_def cc'_def  using S
     by (simp_all add: Int_absorb1 Int_absorb2 sum_subtractf sum.inter_restrict [symmetric] sumTT')
   have wsumSS: "(\<Sum>t \<in> S. cc t *\<^sub>R t) = x - c a *\<^sub>R a" "(\<Sum>t \<in> S. cc' t *\<^sub>R t) = x - c' a *\<^sub>R a"
     unfolding cc_def cc'_def  using S
     by (simp_all add: Int_absorb1 Int_absorb2 if_smult sum.inter_restrict [symmetric] wsumT wsumT' cong: if_cong)
   have sum_dd0: "sum dd S = 0"
     unfolding dd_def  using S
     by (simp add: sumSS' comm_monoid_add_class.sum.distrib sum_subtractf
                   algebra_simps sum_distrib_right [symmetric] b1)
   have "(\<Sum>v\<in>S. (b v * x) *\<^sub>R v) = x *\<^sub>R (\<Sum>v\<in>S. b v *\<^sub>R v)" for x
     by (simp add: pth_5 real_vector.scale_sum_right mult.commute)
   then have *: "(\<Sum>v\<in>S. (b v * x) *\<^sub>R v) = x *\<^sub>R a" for x
     using aeq by blast
   have "(\<Sum>v \<in> S. dd v *\<^sub>R v) = 0"
     unfolding dd_def using S
     by (simp add: * wsumSS sum.distrib sum_subtractf algebra_simps)
   then have dd0: "dd v = 0" if "v \<in> S" for v
     using naff that \<open>finite S\<close> sum_dd0 unfolding affine_dependent_explicit
     apply (simp only: not_ex)
     apply (drule_tac x=S in spec)
     apply (drule_tac x=dd in spec, simp)
     done
   consider "c' a \<le> c a" | "c a \<le> c' a" by linarith
   then show ?thesis
   proof cases
     case 1
     then have "sum cc S \<le> sum cc' S"
       by (simp add: sumSS')
     then have le: "cc x \<le> cc' x" if "x \<in> S" for x
       using dd0 [OF that] 1 b0 mult_left_mono that
       by (fastforce simp add: dd_def algebra_simps)
     have cc0: "cc x = 0" if "x \<in> S" "x \<notin> T \<inter> T'" for x
       using le [OF \<open>x \<in> S\<close>] that c0
       by (force simp: cc_def cc'_def split: if_split_asm)
     show ?thesis
     proof (simp add: convex_hull_finite, intro exI conjI)
       show "\<forall>x\<in>T \<inter> T'. 0 \<le> (cc(a := c a)) x"
         by (simp add: c0 cc_def)
       show "0 \<le> (cc(a := c a)) a"
         by (simp add: c0)
       have "sum (cc(a := c a)) (insert a (T \<inter> T')) = c a + sum (cc(a := c a)) (T \<inter> T')"
         by (simp add: anot)
       also have "... = c a + sum (cc(a := c a)) S"
         apply simp
         apply (rule sum.mono_neutral_left)
         using \<open>T \<subseteq> S\<close> apply (auto simp: \<open>a \<notin> S\<close> cc0)
         done
       also have "... = c a + (1 - c a)"
         by (metis \<open>a \<notin> S\<close> fun_upd_other sum.cong sumSS')
       finally show "sum (cc(a := c a)) (insert a (T \<inter> T')) = 1"
         by simp
       have "(\<Sum>x\<in>insert a (T \<inter> T'). (cc(a := c a)) x *\<^sub>R x) = c a *\<^sub>R a + (\<Sum>x \<in> T \<inter> T'. (cc(a := c a)) x *\<^sub>R x)"
         by (simp add: anot)
       also have "... = c a *\<^sub>R a + (\<Sum>x \<in> S. (cc(a := c a)) x *\<^sub>R x)"
         apply simp
         apply (rule sum.mono_neutral_left)
         using \<open>T \<subseteq> S\<close> apply (auto simp: \<open>a \<notin> S\<close> cc0)
         done
       also have "... = c a *\<^sub>R a + x - c a *\<^sub>R a"
         by (simp add: wsumSS \<open>a \<notin> S\<close> if_smult sum_delta_notmem)
       finally show "(\<Sum>x\<in>insert a (T \<inter> T'). (cc(a := c a)) x *\<^sub>R x) = x"
         by simp
     qed
   next
     case 2
     then have "sum cc' S \<le> sum cc S"
       by (simp add: sumSS')
     then have le: "cc' x \<le> cc x" if "x \<in> S" for x
       using dd0 [OF that] 2 b0 mult_left_mono that
       by (fastforce simp add: dd_def algebra_simps)
     have cc0: "cc' x = 0" if "x \<in> S" "x \<notin> T \<inter> T'" for x
       using le [OF \<open>x \<in> S\<close>] that c'0
       by (force simp: cc_def cc'_def split: if_split_asm)
     show ?thesis
     proof (simp add: convex_hull_finite, intro exI conjI)
       show "\<forall>x\<in>T \<inter> T'. 0 \<le> (cc'(a := c' a)) x"
         by (simp add: c'0 cc'_def)
       show "0 \<le> (cc'(a := c' a)) a"
         by (simp add: c'0)
       have "sum (cc'(a := c' a)) (insert a (T \<inter> T')) = c' a + sum (cc'(a := c' a)) (T \<inter> T')"
         by (simp add: anot)
       also have "... = c' a + sum (cc'(a := c' a)) S"
         apply simp
         apply (rule sum.mono_neutral_left)
         using \<open>T \<subseteq> S\<close> apply (auto simp: \<open>a \<notin> S\<close> cc0)
         done
       also have "... = c' a + (1 - c' a)"
         by (metis \<open>a \<notin> S\<close> fun_upd_other sum.cong sumSS')
       finally show "sum (cc'(a := c' a)) (insert a (T \<inter> T')) = 1"
         by simp
       have "(\<Sum>x\<in>insert a (T \<inter> T'). (cc'(a := c' a)) x *\<^sub>R x) = c' a *\<^sub>R a + (\<Sum>x \<in> T \<inter> T'. (cc'(a := c' a)) x *\<^sub>R x)"
         by (simp add: anot)
       also have "... = c' a *\<^sub>R a + (\<Sum>x \<in> S. (cc'(a := c' a)) x *\<^sub>R x)"
         apply simp
         apply (rule sum.mono_neutral_left)
         using \<open>T \<subseteq> S\<close> apply (auto simp: \<open>a \<notin> S\<close> cc0)
         done
       also have "... = c a *\<^sub>R a + x - c a *\<^sub>R a"
         by (simp add: wsumSS \<open>a \<notin> S\<close> if_smult sum_delta_notmem)
       finally show "(\<Sum>x\<in>insert a (T \<inter> T'). (cc'(a := c' a)) x *\<^sub>R x) = x"
         by simp
     qed
   qed
 qed
 
 corollary\<^marker>\<open>tag unimportant\<close> convex_hull_exchange_Int:
   fixes a  :: "'a::euclidean_space"
   assumes "\<not> affine_dependent S" "a \<in> convex hull S" "T \<subseteq> S" "T' \<subseteq> S"
   shows "(convex hull (insert a T)) \<inter> (convex hull (insert a T')) =
          convex hull (insert a (T \<inter> T'))"
 apply (rule subset_antisym)
   using in_convex_hull_exchange_unique assms apply blast
   by (metis hull_mono inf_le1 inf_le2 insert_inter_insert le_inf_iff)
 
 lemma Int_closed_segment:
   fixes b :: "'a::euclidean_space"
   assumes "b \<in> closed_segment a c \<or> \<not> collinear{a,b,c}"
     shows "closed_segment a b \<inter> closed_segment b c = {b}"
 proof (cases "c = a")
   case True
   then show ?thesis
     using assms collinear_3_eq_affine_dependent by fastforce
 next
   case False
   from assms show ?thesis
   proof
     assume "b \<in> closed_segment a c"
     moreover have "\<not> affine_dependent {a, c}"
       by (simp)
     ultimately show ?thesis
       using False convex_hull_exchange_Int [of "{a,c}" b "{a}" "{c}"]
       by (simp add: segment_convex_hull insert_commute)
   next
     assume ncoll: "\<not> collinear {a, b, c}"
     have False if "closed_segment a b \<inter> closed_segment b c \<noteq> {b}"
     proof -
       have "b \<in> closed_segment a b" and "b \<in> closed_segment b c"
         by auto
       with that obtain d where "b \<noteq> d" "d \<in> closed_segment a b" "d \<in> closed_segment b c"
         by force
       then have d: "collinear {a, d, b}"  "collinear {b, d, c}"
         by (auto simp:  between_mem_segment between_imp_collinear)
       have "collinear {a, b, c}"
         apply (rule collinear_3_trans [OF _ _ \<open>b \<noteq> d\<close>])
         using d  by (auto simp: insert_commute)
       with ncoll show False ..
     qed
     then show ?thesis
       by blast
   qed
 qed
 
 lemma affine_hull_finite_intersection_hyperplanes:
   fixes s :: "'a::euclidean_space set"
   obtains f where
      "finite f"
      "of_nat (card f) + aff_dim s = DIM('a)"
      "affine hull s = \<Inter>f"
      "\<And>h. h \<in> f \<Longrightarrow> \<exists>a b. a \<noteq> 0 \<and> h = {x. a \<bullet> x = b}"
 proof -
   obtain b where "b \<subseteq> s"
              and indb: "\<not> affine_dependent b"
              and eq: "affine hull s = affine hull b"
     using affine_basis_exists by blast
   obtain c where indc: "\<not> affine_dependent c" and "b \<subseteq> c"
              and affc: "affine hull c = UNIV"
     by (metis extend_to_affine_basis affine_UNIV hull_same indb subset_UNIV)
   then have "finite c"
     by (simp add: aff_independent_finite)
   then have fbc: "finite b" "card b \<le> card c"
     using \<open>b \<subseteq> c\<close> infinite_super by (auto simp: card_mono)
   have imeq: "(\<lambda>x. affine hull x) ` ((\<lambda>a. c - {a}) ` (c - b)) = ((\<lambda>a. affine hull (c - {a})) ` (c - b))"
     by blast
   have card1: "card ((\<lambda>a. affine hull (c - {a})) ` (c - b)) = card (c - b)"
     apply (rule card_image [OF inj_onI])
     by (metis Diff_eq_empty_iff Diff_iff indc affine_dependent_def hull_subset insert_iff)
   have card2: "(card (c - b)) + aff_dim s = DIM('a)"
   proof -
     have aff: "aff_dim (UNIV::'a set) = aff_dim c"
       by (metis aff_dim_affine_hull affc)
     have "aff_dim b = aff_dim s"
       by (metis (no_types) aff_dim_affine_hull eq)
     then have "int (card b) = 1 + aff_dim s"
       by (simp add: aff_dim_affine_independent indb)
     then show ?thesis
       using fbc aff
       by (simp add: \<open>\<not> affine_dependent c\<close> \<open>b \<subseteq> c\<close> aff_dim_affine_independent card_Diff_subset of_nat_diff)
   qed
   show ?thesis
   proof (cases "c = b")
     case True show ?thesis
       apply (rule_tac f="{}" in that)
       using True affc
       apply (simp_all add: eq [symmetric])
       by (metis aff_dim_UNIV aff_dim_affine_hull)
   next
     case False
     have ind: "\<not> affine_dependent (\<Union>a\<in>c - b. c - {a})"
       by (rule affine_independent_subset [OF indc]) auto
     have affeq: "affine hull s = (\<Inter>x\<in>(\<lambda>a. c - {a}) ` (c - b). affine hull x)"
       using \<open>b \<subseteq> c\<close> False
       apply (subst affine_hull_Inter [OF ind, symmetric])
       apply (simp add: eq double_diff)
       done
     have *: "1 + aff_dim (c - {t}) = int (DIM('a))"
             if t: "t \<in> c" for t
     proof -
       have "insert t c = c"
         using t by blast
       then show ?thesis
         by (metis (full_types) add.commute aff_dim_affine_hull aff_dim_insert aff_dim_UNIV affc affine_dependent_def indc insert_Diff_single t)
     qed
     show ?thesis
       apply (rule_tac f = "(\<lambda>x. affine hull x) ` ((\<lambda>a. c - {a}) ` (c - b))" in that)
          using \<open>finite c\<close> apply blast
         apply (simp add: imeq card1 card2)
       apply (simp add: affeq, clarify)
       apply (metis DIM_positive One_nat_def Suc_leI add_diff_cancel_left' of_nat_1 aff_dim_eq_hyperplane of_nat_diff *)
       done
   qed
 qed
 
 lemma affine_hyperplane_sums_eq_UNIV_0:
   fixes S :: "'a :: euclidean_space set"
   assumes "affine S"
      and "0 \<in> S" and "w \<in> S"
      and "a \<bullet> w \<noteq> 0"
    shows "{x + y| x y. x \<in> S \<and> a \<bullet> y = 0} = UNIV"
 proof -
   have "subspace S"
     by (simp add: assms subspace_affine)
   have span1: "span {y. a \<bullet> y = 0} \<subseteq> span {x + y |x y. x \<in> S \<and> a \<bullet> y = 0}"
     apply (rule span_mono)
     using \<open>0 \<in> S\<close> add.left_neutral by force
   have "w \<notin> span {y. a \<bullet> y = 0}"
     using \<open>a \<bullet> w \<noteq> 0\<close> span_induct subspace_hyperplane by auto
   moreover have "w \<in> span {x + y |x y. x \<in> S \<and> a \<bullet> y = 0}"
     using \<open>w \<in> S\<close>
     by (metis (mono_tags, lifting) inner_zero_right mem_Collect_eq pth_d span_base)
   ultimately have span2: "span {y. a \<bullet> y = 0} \<noteq> span {x + y |x y. x \<in> S \<and> a \<bullet> y = 0}"
     by blast
   have "a \<noteq> 0" using assms inner_zero_left by blast
   then have "DIM('a) - 1 = dim {y. a \<bullet> y = 0}"
     by (simp add: dim_hyperplane)
   also have "... < dim {x + y |x y. x \<in> S \<and> a \<bullet> y = 0}"
     using span1 span2 by (blast intro: dim_psubset)
   finally have DIM_lt: "DIM('a) - 1 < dim {x + y |x y. x \<in> S \<and> a \<bullet> y = 0}" .
   have subs: "subspace {x + y| x y. x \<in> S \<and> a \<bullet> y = 0}"
     using subspace_sums [OF \<open>subspace S\<close> subspace_hyperplane] by simp
   moreover have "span {x + y| x y. x \<in> S \<and> a \<bullet> y = 0} = UNIV"
     apply (rule dim_eq_full [THEN iffD1])
     apply (rule antisym [OF dim_subset_UNIV])
     using DIM_lt apply simp
     done
   ultimately show ?thesis
     by (simp add: subs) (metis (lifting) span_eq_iff subs)
 qed
 
 proposition\<^marker>\<open>tag unimportant\<close> affine_hyperplane_sums_eq_UNIV:
   fixes S :: "'a :: euclidean_space set"
   assumes "affine S"
       and "S \<inter> {v. a \<bullet> v = b} \<noteq> {}"
       and "S - {v. a \<bullet> v = b} \<noteq> {}"
     shows "{x + y| x y. x \<in> S \<and> a \<bullet> y = b} = UNIV"
 proof (cases "a = 0")
   case True with assms show ?thesis
     by (auto simp: if_splits)
 next
   case False
   obtain c where "c \<in> S" and c: "a \<bullet> c = b"
     using assms by force
   with affine_diffs_subspace [OF \<open>affine S\<close>]
   have "subspace ((+) (- c) ` S)" by blast
   then have aff: "affine ((+) (- c) ` S)"
     by (simp add: subspace_imp_affine)
   have 0: "0 \<in> (+) (- c) ` S"
     by (simp add: \<open>c \<in> S\<close>)
   obtain d where "d \<in> S" and "a \<bullet> d \<noteq> b" and dc: "d-c \<in> (+) (- c) ` S"
     using assms by auto
   then have adc: "a \<bullet> (d - c) \<noteq> 0"
     by (simp add: c inner_diff_right)
   let ?U = "(+) (c+c) ` {x + y |x y. x \<in> (+) (- c) ` S \<and> a \<bullet> y = 0}"
   have "u + v \<in> (+) (c + c) ` {x + v |x v. x \<in> (+) (- c) ` S \<and> a \<bullet> v = 0}"
               if "u \<in> S" "b = a \<bullet> v" for u v
     apply (rule_tac x="u+v-c-c" in image_eqI)
     apply (simp_all add: algebra_simps)
     apply (rule_tac x="u-c" in exI)
     apply (rule_tac x="v-c" in exI)
     apply (simp add: algebra_simps that c)
     done
   moreover have "\<lbrakk>a \<bullet> v = 0; u \<in> S\<rbrakk>
        \<Longrightarrow> \<exists>x ya. v + (u + c) = x + ya \<and> x \<in> S \<and> a \<bullet> ya = b" for v u
     by (metis add.left_commute c inner_right_distrib pth_d)
   ultimately have "{x + y |x y. x \<in> S \<and> a \<bullet> y = b} = ?U"
     by (fastforce simp: algebra_simps)
   also have "... = range ((+) (c + c))"
     by (simp only: affine_hyperplane_sums_eq_UNIV_0 [OF aff 0 dc adc])
   also have "... = UNIV"
     by simp
   finally show ?thesis .
 qed
 
 lemma aff_dim_sums_Int_0:
   assumes "affine S"
       and "affine T"
       and "0 \<in> S" "0 \<in> T"
     shows "aff_dim {x + y| x y. x \<in> S \<and> y \<in> T} = (aff_dim S + aff_dim T) - aff_dim(S \<inter> T)"
 proof -
   have "0 \<in> {x + y |x y. x \<in> S \<and> y \<in> T}"
     using assms by force
   then have 0: "0 \<in> affine hull {x + y |x y. x \<in> S \<and> y \<in> T}"
     by (metis (lifting) hull_inc)
   have sub: "subspace S"  "subspace T"
     using assms by (auto simp: subspace_affine)
   show ?thesis
     using dim_sums_Int [OF sub] by (simp add: aff_dim_zero assms 0 hull_inc)
 qed
 
 proposition aff_dim_sums_Int:
   assumes "affine S"
       and "affine T"
       and "S \<inter> T \<noteq> {}"
     shows "aff_dim {x + y| x y. x \<in> S \<and> y \<in> T} = (aff_dim S + aff_dim T) - aff_dim(S \<inter> T)"
 proof -
   obtain a where a: "a \<in> S" "a \<in> T" using assms by force
   have aff: "affine ((+) (-a) ` S)"  "affine ((+) (-a) ` T)"
     using affine_translation [symmetric, of "- a"] assms by (simp_all cong: image_cong_simp)
   have zero: "0 \<in> ((+) (-a) ` S)"  "0 \<in> ((+) (-a) ` T)"
     using a assms by auto
   have "{x + y |x y. x \<in> (+) (- a) ` S \<and> y \<in> (+) (- a) ` T} =
       (+) (- 2 *\<^sub>R a) ` {x + y| x y. x \<in> S \<and> y \<in> T}"
     by (force simp: algebra_simps scaleR_2)
   moreover have "(+) (- a) ` S \<inter> (+) (- a) ` T = (+) (- a) ` (S \<inter> T)"
     by auto
   ultimately show ?thesis
     using aff_dim_sums_Int_0 [OF aff zero] aff_dim_translation_eq
     by (metis (lifting))
 qed
 
 lemma aff_dim_affine_Int_hyperplane:
   fixes a :: "'a::euclidean_space"
   assumes "affine S"
     shows "aff_dim(S \<inter> {x. a \<bullet> x = b}) =
              (if S \<inter> {v. a \<bullet> v = b} = {} then - 1
               else if S \<subseteq> {v. a \<bullet> v = b} then aff_dim S
               else aff_dim S - 1)"
 proof (cases "a = 0")
   case True with assms show ?thesis
     by auto
 next
   case False
   then have "aff_dim (S \<inter> {x. a \<bullet> x = b}) = aff_dim S - 1"
             if "x \<in> S" "a \<bullet> x \<noteq> b" and non: "S \<inter> {v. a \<bullet> v = b} \<noteq> {}" for x
   proof -
     have [simp]: "{x + y| x y. x \<in> S \<and> a \<bullet> y = b} = UNIV"
       using affine_hyperplane_sums_eq_UNIV [OF assms non] that  by blast
     show ?thesis
       using aff_dim_sums_Int [OF assms affine_hyperplane non]
       by (simp add: of_nat_diff False)
   qed
   then show ?thesis
     by (metis (mono_tags, lifting) inf.orderE aff_dim_empty_eq mem_Collect_eq subsetI)
 qed
 
 lemma aff_dim_lt_full:
   fixes S :: "'a::euclidean_space set"
   shows "aff_dim S < DIM('a) \<longleftrightarrow> (affine hull S \<noteq> UNIV)"
 by (metis (no_types) aff_dim_affine_hull aff_dim_le_DIM aff_dim_UNIV affine_hull_UNIV less_le)
 
 lemma aff_dim_openin:
   fixes S :: "'a::euclidean_space set"
   assumes ope: "openin (top_of_set T) S" and "affine T" "S \<noteq> {}"
   shows "aff_dim S = aff_dim T"
 proof -
   show ?thesis
   proof (rule order_antisym)
     show "aff_dim S \<le> aff_dim T"
       by (blast intro: aff_dim_subset [OF openin_imp_subset] ope)
   next
     obtain a where "a \<in> S"
       using \<open>S \<noteq> {}\<close> by blast
     have "S \<subseteq> T"
       using ope openin_imp_subset by auto
     then have "a \<in> T"
       using \<open>a \<in> S\<close> by auto
     then have subT': "subspace ((\<lambda>x. - a + x) ` T)"
       using affine_diffs_subspace \<open>affine T\<close> by auto
     then obtain B where Bsub: "B \<subseteq> ((\<lambda>x. - a + x) ` T)" and po: "pairwise orthogonal B"
                     and eq1: "\<And>x. x \<in> B \<Longrightarrow> norm x = 1" and "independent B"
                     and cardB: "card B = dim ((\<lambda>x. - a + x) ` T)"
                     and spanB: "span B = ((\<lambda>x. - a + x) ` T)"
       by (rule orthonormal_basis_subspace) auto
     obtain e where "0 < e" and e: "cball a e \<inter> T \<subseteq> S"
       by (meson \<open>a \<in> S\<close> openin_contains_cball ope)
     have "aff_dim T = aff_dim ((\<lambda>x. - a + x) ` T)"
       by (metis aff_dim_translation_eq)
     also have "... = dim ((\<lambda>x. - a + x) ` T)"
       using aff_dim_subspace subT' by blast
     also have "... = card B"
       by (simp add: cardB)
     also have "... = card ((\<lambda>x. e *\<^sub>R x) ` B)"
       using \<open>0 < e\<close>  by (force simp: inj_on_def card_image)
     also have "... \<le> dim ((\<lambda>x. - a + x) ` S)"
     proof (simp, rule independent_card_le_dim)
       have e': "cball 0 e \<inter> (\<lambda>x. x - a) ` T \<subseteq> (\<lambda>x. x - a) ` S"
         using e by (auto simp: dist_norm norm_minus_commute subset_eq)
       have "(\<lambda>x. e *\<^sub>R x) ` B \<subseteq> cball 0 e \<inter> (\<lambda>x. x - a) ` T"
         using Bsub \<open>0 < e\<close> eq1 subT' \<open>a \<in> T\<close> by (auto simp: subspace_def)
       then show "(\<lambda>x. e *\<^sub>R x) ` B \<subseteq> (\<lambda>x. x - a) ` S"
         using e' by blast
       show "independent ((\<lambda>x. e *\<^sub>R x) ` B)"
         using linear_scale_self \<open>independent B\<close>
         apply (rule linear_independent_injective_image)
         using \<open>0 < e\<close> inj_on_def by fastforce
     qed
     also have "... = aff_dim S"
       using \<open>a \<in> S\<close> aff_dim_eq_dim hull_inc by (force cong: image_cong_simp)
     finally show "aff_dim T \<le> aff_dim S" .
   qed
 qed
 
 lemma dim_openin:
   fixes S :: "'a::euclidean_space set"
   assumes ope: "openin (top_of_set T) S" and "subspace T" "S \<noteq> {}"
   shows "dim S = dim T"
 proof (rule order_antisym)
   show "dim S \<le> dim T"
     by (metis ope dim_subset openin_subset topspace_euclidean_subtopology)
 next
   have "dim T = aff_dim S"
     using aff_dim_openin
     by (metis aff_dim_subspace \<open>subspace T\<close> \<open>S \<noteq> {}\<close> ope subspace_affine)
   also have "... \<le> dim S"
     by (metis aff_dim_subset aff_dim_subspace dim_span span_superset
         subspace_span)
   finally show "dim T \<le> dim S" by simp
 qed
 
 subsection\<open>Lower-dimensional affine subsets are nowhere dense\<close>
 
 proposition dense_complement_subspace:
   fixes S :: "'a :: euclidean_space set"
   assumes dim_less: "dim T < dim S" and "subspace S" shows "closure(S - T) = S"
 proof -
   have "closure(S - U) = S" if "dim U < dim S" "U \<subseteq> S" for U
   proof -
     have "span U \<subset> span S"
       by (metis neq_iff psubsetI span_eq_dim span_mono that)
     then obtain a where "a \<noteq> 0" "a \<in> span S" and a: "\<And>y. y \<in> span U \<Longrightarrow> orthogonal a y"
       using orthogonal_to_subspace_exists_gen by metis
     show ?thesis
     proof
       have "closed S"
         by (simp add: \<open>subspace S\<close> closed_subspace)
       then show "closure (S - U) \<subseteq> S"
         by (simp add: closure_minimal)
       show "S \<subseteq> closure (S - U)"
       proof (clarsimp simp: closure_approachable)
         fix x and e::real
         assume "x \<in> S" "0 < e"
         show "\<exists>y\<in>S - U. dist y x < e"
         proof (cases "x \<in> U")
           case True
           let ?y = "x + (e/2 / norm a) *\<^sub>R a"
           show ?thesis
           proof
             show "dist ?y x < e"
               using \<open>0 < e\<close> by (simp add: dist_norm)
           next
             have "?y \<in> S"
               by (metis \<open>a \<in> span S\<close> \<open>x \<in> S\<close> assms(2) span_eq_iff subspace_add subspace_scale)
             moreover have "?y \<notin> U"
             proof -
               have "e/2 / norm a \<noteq> 0"
                 using \<open>0 < e\<close> \<open>a \<noteq> 0\<close> by auto
               then show ?thesis
                 by (metis True \<open>a \<noteq> 0\<close> a orthogonal_scaleR orthogonal_self real_vector.scale_eq_0_iff span_add_eq span_base)
             qed
             ultimately show "?y \<in> S - U" by blast
           qed
         next
           case False
           with \<open>0 < e\<close> \<open>x \<in> S\<close> show ?thesis by force
         qed
       qed
     qed
   qed
   moreover have "S - S \<inter> T = S-T"
     by blast
   moreover have "dim (S \<inter> T) < dim S"
     by (metis dim_less dim_subset inf.cobounded2 inf.orderE inf.strict_boundedE not_le)
   ultimately show ?thesis
     by force
 qed
 
 corollary\<^marker>\<open>tag unimportant\<close> dense_complement_affine:
   fixes S :: "'a :: euclidean_space set"
   assumes less: "aff_dim T < aff_dim S" and "affine S" shows "closure(S - T) = S"
 proof (cases "S \<inter> T = {}")
   case True
   then show ?thesis
     by (metis Diff_triv affine_hull_eq \<open>affine S\<close> closure_same_affine_hull closure_subset hull_subset subset_antisym)
 next
   case False
   then obtain z where z: "z \<in> S \<inter> T" by blast
   then have "subspace ((+) (- z) ` S)"
     by (meson IntD1 affine_diffs_subspace \<open>affine S\<close>)
   moreover have "int (dim ((+) (- z) ` T)) < int (dim ((+) (- z) ` S))"
 thm aff_dim_eq_dim
     using z less by (simp add: aff_dim_eq_dim_subtract [of z] hull_inc cong: image_cong_simp)
   ultimately have "closure(((+) (- z) ` S) - ((+) (- z) ` T)) = ((+) (- z) ` S)"
     by (simp add: dense_complement_subspace)
   then show ?thesis
     by (metis closure_translation translation_diff translation_invert)
 qed
 
 corollary\<^marker>\<open>tag unimportant\<close> dense_complement_openin_affine_hull:
   fixes S :: "'a :: euclidean_space set"
   assumes less: "aff_dim T < aff_dim S"
       and ope: "openin (top_of_set (affine hull S)) S"
     shows "closure(S - T) = closure S"
 proof -
   have "affine hull S - T \<subseteq> affine hull S"
     by blast
   then have "closure (S \<inter> closure (affine hull S - T)) = closure (S \<inter> (affine hull S - T))"
     by (rule closure_openin_Int_closure [OF ope])
   then show ?thesis
     by (metis Int_Diff aff_dim_affine_hull affine_affine_hull dense_complement_affine hull_subset inf.orderE less)
 qed
 
 corollary\<^marker>\<open>tag unimportant\<close> dense_complement_convex:
   fixes S :: "'a :: euclidean_space set"
   assumes "aff_dim T < aff_dim S" "convex S"
     shows "closure(S - T) = closure S"
 proof
   show "closure (S - T) \<subseteq> closure S"
     by (simp add: closure_mono)
   have "closure (rel_interior S - T) = closure (rel_interior S)"
     apply (rule dense_complement_openin_affine_hull)
     apply (simp add: assms rel_interior_aff_dim)
     using \<open>convex S\<close> rel_interior_rel_open rel_open by blast
   then show "closure S \<subseteq> closure (S - T)"
     by (metis Diff_mono \<open>convex S\<close> closure_mono convex_closure_rel_interior order_refl rel_interior_subset)
 qed
 
 corollary\<^marker>\<open>tag unimportant\<close> dense_complement_convex_closed:
   fixes S :: "'a :: euclidean_space set"
   assumes "aff_dim T < aff_dim S" "convex S" "closed S"
     shows "closure(S - T) = S"
   by (simp add: assms dense_complement_convex)
 
 
 subsection\<^marker>\<open>tag unimportant\<close>\<open>Parallel slices, etc\<close>
 
 text\<open> If we take a slice out of a set, we can do it perpendicularly,
   with the normal vector to the slice parallel to the affine hull.\<close>
 
 proposition\<^marker>\<open>tag unimportant\<close> affine_parallel_slice:
   fixes S :: "'a :: euclidean_space set"
   assumes "affine S"
       and "S \<inter> {x. a \<bullet> x \<le> b} \<noteq> {}"
       and "\<not> (S \<subseteq> {x. a \<bullet> x \<le> b})"
   obtains a' b' where "a' \<noteq> 0"
                    "S \<inter> {x. a' \<bullet> x \<le> b'} = S \<inter> {x. a \<bullet> x \<le> b}"
                    "S \<inter> {x. a' \<bullet> x = b'} = S \<inter> {x. a \<bullet> x = b}"
                    "\<And>w. w \<in> S \<Longrightarrow> (w + a') \<in> S"
 proof (cases "S \<inter> {x. a \<bullet> x = b} = {}")
   case True
   then obtain u v where "u \<in> S" "v \<in> S" "a \<bullet> u \<le> b" "a \<bullet> v > b"
     using assms by (auto simp: not_le)
   define \<eta> where "\<eta> = u + ((b - a \<bullet> u) / (a \<bullet> v - a \<bullet> u)) *\<^sub>R (v - u)"
   have "\<eta> \<in> S"
     by (simp add: \<eta>_def \<open>u \<in> S\<close> \<open>v \<in> S\<close> \<open>affine S\<close> mem_affine_3_minus)
   moreover have "a \<bullet> \<eta> = b"
     using \<open>a \<bullet> u \<le> b\<close> \<open>b < a \<bullet> v\<close>
     by (simp add: \<eta>_def algebra_simps) (simp add: field_simps)
   ultimately have False
     using True by force
   then show ?thesis ..
 next
   case False
   then obtain z where "z \<in> S" and z: "a \<bullet> z = b"
     using assms by auto
   with affine_diffs_subspace [OF \<open>affine S\<close>]
   have sub: "subspace ((+) (- z) ` S)" by blast
   then have aff: "affine ((+) (- z) ` S)" and span: "span ((+) (- z) ` S) = ((+) (- z) ` S)"
     by (auto simp: subspace_imp_affine)
   obtain a' a'' where a': "a' \<in> span ((+) (- z) ` S)" and a: "a = a' + a''"
                   and "\<And>w. w \<in> span ((+) (- z) ` S) \<Longrightarrow> orthogonal a'' w"
     using orthogonal_subspace_decomp_exists [of "(+) (- z) ` S" "a"] by metis
   then have "\<And>w. w \<in> S \<Longrightarrow> a'' \<bullet> (w-z) = 0"
     by (simp add: span_base orthogonal_def)
   then have a'': "\<And>w. w \<in> S \<Longrightarrow> a'' \<bullet> w = (a - a') \<bullet> z"
     by (simp add: a inner_diff_right)
   then have ba'': "\<And>w. w \<in> S \<Longrightarrow> a'' \<bullet> w = b - a' \<bullet> z"
     by (simp add: inner_diff_left z)
   have "\<And>w. w \<in> (+) (- z) ` S \<Longrightarrow> (w + a') \<in> (+) (- z) ` S"
     by (metis subspace_add a' span_eq_iff sub)
   then have Sclo: "\<And>w. w \<in> S \<Longrightarrow> (w + a') \<in> S"
     by fastforce
   show ?thesis
   proof (cases "a' = 0")
     case True
     with a assms True a'' diff_zero less_irrefl show ?thesis
       by auto
   next
     case False
     show ?thesis
       apply (rule_tac a' = "a'" and b' = "a' \<bullet> z" in that)
       apply (auto simp: a ba'' inner_left_distrib False Sclo)
       done
   qed
 qed
 
 lemma diffs_affine_hull_span:
   assumes "a \<in> S"
     shows "{x - a |x. x \<in> affine hull S} = span {x - a |x. x \<in> S}"
 proof -
   have *: "((\<lambda>x. x - a) ` (S - {a})) = {x. x + a \<in> S} - {0}"
     by (auto simp: algebra_simps)
   show ?thesis
     apply (simp add: affine_hull_span2 [OF assms] *)
     apply (auto simp: algebra_simps)
     done
 qed
 
 lemma aff_dim_dim_affine_diffs:
   fixes S :: "'a :: euclidean_space set"
   assumes "affine S" "a \<in> S"
     shows "aff_dim S = dim {x - a |x. x \<in> S}"
 proof -
   obtain B where aff: "affine hull B = affine hull S"
              and ind: "\<not> affine_dependent B"
              and card: "of_nat (card B) = aff_dim S + 1"
     using aff_dim_basis_exists by blast
   then have "B \<noteq> {}" using assms
     by (metis affine_hull_eq_empty ex_in_conv)
   then obtain c where "c \<in> B" by auto
   then have "c \<in> S"
     by (metis aff affine_hull_eq \<open>affine S\<close> hull_inc)
   have xy: "x - c = y - a \<longleftrightarrow> y = x + 1 *\<^sub>R (a - c)" for x y c and a::'a
     by (auto simp: algebra_simps)
   have *: "{x - c |x. x \<in> S} = {x - a |x. x \<in> S}"
     apply safe
     apply (simp_all only: xy)
     using mem_affine_3_minus [OF \<open>affine S\<close>] \<open>a \<in> S\<close> \<open>c \<in> S\<close> apply blast+
     done
   have affS: "affine hull S = S"
     by (simp add: \<open>affine S\<close>)
   have "aff_dim S = of_nat (card B) - 1"
     using card by simp
   also have "... = dim {x - c |x. x \<in> B}"
     by (simp add: affine_independent_card_dim_diffs [OF ind \<open>c \<in> B\<close>])
   also have "... = dim {x - c | x. x \<in> affine hull B}"
      by (simp add: diffs_affine_hull_span \<open>c \<in> B\<close>)
   also have "... = dim {x - a |x. x \<in> S}"
      by (simp add: affS aff *)
    finally show ?thesis .
 qed
 
 lemma aff_dim_linear_image_le:
   assumes "linear f"
     shows "aff_dim(f ` S) \<le> aff_dim S"
 proof -
   have "aff_dim (f ` T) \<le> aff_dim T" if "affine T" for T
   proof (cases "T = {}")
     case True then show ?thesis by (simp add: aff_dim_geq)
   next
     case False
     then obtain a where "a \<in> T" by auto
     have 1: "((\<lambda>x. x - f a) ` f ` T) = {x - f a |x. x \<in> f ` T}"
       by auto
     have 2: "{x - f a| x. x \<in> f ` T} = f ` {x - a| x. x \<in> T}"
       by (force simp: linear_diff [OF assms])
     have "aff_dim (f ` T) = int (dim {x - f a |x. x \<in> f ` T})"
       by (simp add: \<open>a \<in> T\<close> hull_inc aff_dim_eq_dim [of "f a"] 1 cong: image_cong_simp)
     also have "... = int (dim (f ` {x - a| x. x \<in> T}))"
       by (force simp: linear_diff [OF assms] 2)
     also have "... \<le> int (dim {x - a| x. x \<in> T})"
       by (simp add: dim_image_le [OF assms])
     also have "... \<le> aff_dim T"
       by (simp add: aff_dim_dim_affine_diffs [symmetric] \<open>a \<in> T\<close> \<open>affine T\<close>)
     finally show ?thesis .
   qed
   then
   have "aff_dim (f ` (affine hull S)) \<le> aff_dim (affine hull S)"
     using affine_affine_hull [of S] by blast
   then show ?thesis
     using affine_hull_linear_image assms linear_conv_bounded_linear by fastforce
 qed
 
 lemma aff_dim_injective_linear_image [simp]:
   assumes "linear f" "inj f"
     shows "aff_dim (f ` S) = aff_dim S"
 proof (rule antisym)
   show "aff_dim (f ` S) \<le> aff_dim S"
     by (simp add: aff_dim_linear_image_le assms(1))
 next
   obtain g where "linear g" "g \<circ> f = id"
     using assms(1) assms(2) linear_injective_left_inverse by blast
   then have "aff_dim S \<le> aff_dim(g ` f ` S)"
     by (simp add: image_comp)
   also have "... \<le> aff_dim (f ` S)"
     by (simp add: \<open>linear g\<close> aff_dim_linear_image_le)
   finally show "aff_dim S \<le> aff_dim (f ` S)" .
 qed
 
 
 lemma choose_affine_subset:
   assumes "affine S" "-1 \<le> d" and dle: "d \<le> aff_dim S"
   obtains T where "affine T" "T \<subseteq> S" "aff_dim T = d"
 proof (cases "d = -1 \<or> S={}")
   case True with assms show ?thesis
     by (metis aff_dim_empty affine_empty bot.extremum that eq_iff)
 next
   case False
   with assms obtain a where "a \<in> S" "0 \<le> d" by auto
   with assms have ss: "subspace ((+) (- a) ` S)"
     by (simp add: affine_diffs_subspace_subtract cong: image_cong_simp)
   have "nat d \<le> dim ((+) (- a) ` S)"
     by (metis aff_dim_subspace aff_dim_translation_eq dle nat_int nat_mono ss)
   then obtain T where "subspace T" and Tsb: "T \<subseteq> span ((+) (- a) ` S)"
                   and Tdim: "dim T = nat d"
     using choose_subspace_of_subspace [of "nat d" "(+) (- a) ` S"] by blast
   then have "affine T"
     using subspace_affine by blast
   then have "affine ((+) a ` T)"
     by (metis affine_hull_eq affine_hull_translation)
   moreover have "(+) a ` T \<subseteq> S"
   proof -
     have "T \<subseteq> (+) (- a) ` S"
       by (metis (no_types) span_eq_iff Tsb ss)
     then show "(+) a ` T \<subseteq> S"
       using add_ac by auto
   qed
   moreover have "aff_dim ((+) a ` T) = d"
     by (simp add: aff_dim_subspace Tdim \<open>0 \<le> d\<close> \<open>subspace T\<close> aff_dim_translation_eq)
   ultimately show ?thesis
     by (rule that)
 qed
 
 subsection\<open>Paracompactness\<close>
 
 proposition paracompact:
   fixes S :: "'a :: {metric_space,second_countable_topology} set"
   assumes "S \<subseteq> \<Union>\<C>" and opC: "\<And>T. T \<in> \<C> \<Longrightarrow> open T"
   obtains \<C>' where "S \<subseteq> \<Union> \<C>'"
                and "\<And>U. U \<in> \<C>' \<Longrightarrow> open U \<and> (\<exists>T. T \<in> \<C> \<and> U \<subseteq> T)"
                and "\<And>x. x \<in> S
                        \<Longrightarrow> \<exists>V. open V \<and> x \<in> V \<and> finite {U. U \<in> \<C>' \<and> (U \<inter> V \<noteq> {})}"
 proof (cases "S = {}")
   case True with that show ?thesis by blast
 next
   case False
   have "\<exists>T U. x \<in> U \<and> open U \<and> closure U \<subseteq> T \<and> T \<in> \<C>" if "x \<in> S" for x
   proof -
     obtain T where "x \<in> T" "T \<in> \<C>" "open T"
       using assms \<open>x \<in> S\<close> by blast
     then obtain e where "e > 0" "cball x e \<subseteq> T"
       by (force simp: open_contains_cball)
     then show ?thesis
       apply (rule_tac x = T in exI)
       apply (rule_tac x = "ball x e" in exI)
       using  \<open>T \<in> \<C>\<close>
       apply (simp add: closure_minimal)
       using closed_cball closure_minimal by blast
   qed
   then obtain F G where Gin: "x \<in> G x" and oG: "open (G x)"
                     and clos: "closure (G x) \<subseteq> F x" and Fin: "F x \<in> \<C>"
          if "x \<in> S" for x
     by metis
   then obtain \<F> where "\<F> \<subseteq> G ` S" "countable \<F>" "\<Union>\<F> = \<Union>(G ` S)"
     using Lindelof [of "G ` S"] by (metis image_iff)
   then obtain K where K: "K \<subseteq> S" "countable K" and eq: "\<Union>(G ` K) = \<Union>(G ` S)"
     by (metis countable_subset_image)
   with False Gin have "K \<noteq> {}" by force
   then obtain a :: "nat \<Rightarrow> 'a" where "range a = K"
     by (metis range_from_nat_into \<open>countable K\<close>)
   then have odif: "\<And>n. open (F (a n) - \<Union>{closure (G (a m)) |m. m < n})"
     using \<open>K \<subseteq> S\<close> Fin opC by (fastforce simp add:)
   let ?C = "range (\<lambda>n. F(a n) - \<Union>{closure(G(a m)) |m. m < n})"
   have enum_S: "\<exists>n. x \<in> F(a n) \<and> x \<in> G(a n)" if "x \<in> S" for x
   proof -
     have "\<exists>y \<in> K. x \<in> G y" using eq that Gin by fastforce
     then show ?thesis
       using clos K \<open>range a = K\<close> closure_subset by blast
   qed
   have 1: "S \<subseteq> Union ?C"
   proof
     fix x assume "x \<in> S"
     define n where "n \<equiv> LEAST n. x \<in> F(a n)"
     have n: "x \<in> F(a n)"
       using enum_S [OF \<open>x \<in> S\<close>] by (force simp: n_def intro: LeastI)
     have notn: "x \<notin> F(a m)" if "m < n" for m
       using that not_less_Least by (force simp: n_def)
     then have "x \<notin> \<Union>{closure (G (a m)) |m. m < n}"
       using n \<open>K \<subseteq> S\<close> \<open>range a = K\<close> clos notn by fastforce
     with n show "x \<in> Union ?C"
       by blast
   qed
   have 3: "\<exists>V. open V \<and> x \<in> V \<and> finite {U. U \<in> ?C \<and> (U \<inter> V \<noteq> {})}" if "x \<in> S" for x
   proof -
     obtain n where n: "x \<in> F(a n)" "x \<in> G(a n)"
       using \<open>x \<in> S\<close> enum_S by auto
     have "{U \<in> ?C. U \<inter> G (a n) \<noteq> {}} \<subseteq> (\<lambda>n. F(a n) - \<Union>{closure(G(a m)) |m. m < n}) ` atMost n"
     proof clarsimp
       fix k  assume "(F (a k) - \<Union>{closure (G (a m)) |m. m < k}) \<inter> G (a n) \<noteq> {}"
       then have "k \<le> n"
         by auto (metis closure_subset not_le subsetCE)
       then show "F (a k) - \<Union>{closure (G (a m)) |m. m < k}
                  \<in> (\<lambda>n. F (a n) - \<Union>{closure (G (a m)) |m. m < n}) ` {..n}"
         by force
     qed
     moreover have "finite ((\<lambda>n. F(a n) - \<Union>{closure(G(a m)) |m. m < n}) ` atMost n)"
       by force
     ultimately have *: "finite {U \<in> ?C. U \<inter> G (a n) \<noteq> {}}"
       using finite_subset by blast
     show ?thesis
       apply (rule_tac x="G (a n)" in exI)
       apply (intro conjI oG n *)
       using \<open>K \<subseteq> S\<close> \<open>range a = K\<close> apply blast
       done
   qed
   show ?thesis
     apply (rule that [OF 1 _ 3])
     using Fin \<open>K \<subseteq> S\<close> \<open>range a = K\<close>  apply (auto simp: odif)
     done
 qed
 
 corollary paracompact_closedin:
   fixes S :: "'a :: {metric_space,second_countable_topology} set"
   assumes cin: "closedin (top_of_set U) S"
       and oin: "\<And>T. T \<in> \<C> \<Longrightarrow> openin (top_of_set U) T"
       and "S \<subseteq> \<Union>\<C>"
   obtains \<C>' where "S \<subseteq> \<Union> \<C>'"
                and "\<And>V. V \<in> \<C>' \<Longrightarrow> openin (top_of_set U) V \<and> (\<exists>T. T \<in> \<C> \<and> V \<subseteq> T)"
                and "\<And>x. x \<in> U
                        \<Longrightarrow> \<exists>V. openin (top_of_set U) V \<and> x \<in> V \<and>
                                finite {X. X \<in> \<C>' \<and> (X \<inter> V \<noteq> {})}"
 proof -
   have "\<exists>Z. open Z \<and> (T = U \<inter> Z)" if "T \<in> \<C>" for T
     using oin [OF that] by (auto simp: openin_open)
   then obtain F where opF: "open (F T)" and intF: "U \<inter> F T = T" if "T \<in> \<C>" for T
     by metis
   obtain K where K: "closed K" "U \<inter> K = S"
     using cin by (auto simp: closedin_closed)
   have 1: "U \<subseteq> \<Union>(insert (- K) (F ` \<C>))"
     by clarsimp (metis Int_iff Union_iff \<open>U \<inter> K = S\<close> \<open>S \<subseteq> \<Union>\<C>\<close> subsetD intF)
   have 2: "\<And>T. T \<in> insert (- K) (F ` \<C>) \<Longrightarrow> open T"
     using \<open>closed K\<close> by (auto simp: opF)
   obtain \<D> where "U \<subseteq> \<Union>\<D>"
              and D1: "\<And>U. U \<in> \<D> \<Longrightarrow> open U \<and> (\<exists>T. T \<in> insert (- K) (F ` \<C>) \<and> U \<subseteq> T)"
              and D2: "\<And>x. x \<in> U \<Longrightarrow> \<exists>V. open V \<and> x \<in> V \<and> finite {U \<in> \<D>. U \<inter> V \<noteq> {}}"
     by (blast intro: paracompact [OF 1 2])
   let ?C = "{U \<inter> V |V. V \<in> \<D> \<and> (V \<inter> K \<noteq> {})}"
   show ?thesis
   proof (rule_tac \<C>' = "{U \<inter> V |V. V \<in> \<D> \<and> (V \<inter> K \<noteq> {})}" in that)
     show "S \<subseteq> \<Union>?C"
       using \<open>U \<inter> K = S\<close> \<open>U \<subseteq> \<Union>\<D>\<close> K by (blast dest!: subsetD)
     show "\<And>V. V \<in> ?C \<Longrightarrow> openin (top_of_set U) V \<and> (\<exists>T. T \<in> \<C> \<and> V \<subseteq> T)"
       using D1 intF by fastforce
     have *: "{X. (\<exists>V. X = U \<inter> V \<and> V \<in> \<D> \<and> V \<inter> K \<noteq> {}) \<and> X \<inter> (U \<inter> V) \<noteq> {}} \<subseteq>
              (\<lambda>x. U \<inter> x) ` {U \<in> \<D>. U \<inter> V \<noteq> {}}" for V
       by blast
     show "\<exists>V. openin (top_of_set U) V \<and> x \<in> V \<and> finite {X \<in> ?C. X \<inter> V \<noteq> {}}"
          if "x \<in> U" for x
       using D2 [OF that]
       apply clarify
       apply (rule_tac x="U \<inter> V" in exI)
       apply (auto intro: that finite_subset [OF *])
       done
     qed
 qed
 
 corollary\<^marker>\<open>tag unimportant\<close> paracompact_closed:
   fixes S :: "'a :: {metric_space,second_countable_topology} set"
   assumes "closed S"
       and opC: "\<And>T. T \<in> \<C> \<Longrightarrow> open T"
       and "S \<subseteq> \<Union>\<C>"
   obtains \<C>' where "S \<subseteq> \<Union>\<C>'"
                and "\<And>U. U \<in> \<C>' \<Longrightarrow> open U \<and> (\<exists>T. T \<in> \<C> \<and> U \<subseteq> T)"
                and "\<And>x. \<exists>V. open V \<and> x \<in> V \<and>
                                finite {U. U \<in> \<C>' \<and> (U \<inter> V \<noteq> {})}"
   by (rule paracompact_closedin [of UNIV S \<C>]) (auto simp: assms)
 
   
 subsection\<^marker>\<open>tag unimportant\<close>\<open>Closed-graph characterization of continuity\<close>
 
 lemma continuous_closed_graph_gen:
   fixes T :: "'b::real_normed_vector set"
   assumes contf: "continuous_on S f" and fim: "f ` S \<subseteq> T"
     shows "closedin (top_of_set (S \<times> T)) ((\<lambda>x. Pair x (f x)) ` S)"
 proof -
   have eq: "((\<lambda>x. Pair x (f x)) ` S) =(S \<times> T \<inter> (\<lambda>z. (f \<circ> fst)z - snd z) -` {0})"
     using fim by auto
   show ?thesis
     apply (subst eq)
     apply (intro continuous_intros continuous_closedin_preimage continuous_on_subset [OF contf])
     by auto
 qed
 
 lemma continuous_closed_graph_eq:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes "compact T" and fim: "f ` S \<subseteq> T"
   shows "continuous_on S f \<longleftrightarrow>
          closedin (top_of_set (S \<times> T)) ((\<lambda>x. Pair x (f x)) ` S)"
          (is "?lhs = ?rhs")
 proof -
   have "?lhs" if ?rhs
   proof (clarsimp simp add: continuous_on_closed_gen [OF fim])
     fix U
     assume U: "closedin (top_of_set T) U"
     have eq: "(S \<inter> f -` U) = fst ` (((\<lambda>x. Pair x (f x)) ` S) \<inter> (S \<times> U))"
       by (force simp: image_iff)
     show "closedin (top_of_set S) (S \<inter> f -` U)"
       by (simp add: U closedin_Int closedin_Times closed_map_fst [OF \<open>compact T\<close>] that eq)
   qed
   with continuous_closed_graph_gen assms show ?thesis by blast
 qed
 
 lemma continuous_closed_graph:
   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
   assumes "closed S" and contf: "continuous_on S f"
   shows "closed ((\<lambda>x. Pair x (f x)) ` S)"
   apply (rule closedin_closed_trans)
    apply (rule continuous_closed_graph_gen [OF contf subset_UNIV])
   by (simp add: \<open>closed S\<close> closed_Times)
 
 lemma continuous_from_closed_graph:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes "compact T" and fim: "f ` S \<subseteq> T" and clo: "closed ((\<lambda>x. Pair x (f x)) ` S)"
   shows "continuous_on S f"
     using fim clo
     by (auto intro: closed_subset simp: continuous_closed_graph_eq [OF \<open>compact T\<close> fim])
 
 lemma continuous_on_Un_local_open:
   assumes opS: "openin (top_of_set (S \<union> T)) S"
       and opT: "openin (top_of_set (S \<union> T)) T"
       and contf: "continuous_on S f" and contg: "continuous_on T f"
     shows "continuous_on (S \<union> T) f"
   using pasting_lemma [of "{S,T}" "top_of_set (S \<union> T)" id euclidean "\<lambda>i. f" f] contf contg opS opT
   by (simp add: subtopology_subtopology) (metis inf.absorb2 openin_imp_subset)  
 
 lemma continuous_on_cases_local_open:
   assumes opS: "openin (top_of_set (S \<union> T)) S"
       and opT: "openin (top_of_set (S \<union> T)) T"
       and contf: "continuous_on S f" and contg: "continuous_on T g"
       and fg: "\<And>x. x \<in> S \<and> \<not>P x \<or> x \<in> T \<and> P x \<Longrightarrow> f x = g x"
     shows "continuous_on (S \<union> T) (\<lambda>x. if P x then f x else g x)"
 proof -
   have "\<And>x. x \<in> S \<Longrightarrow> (if P x then f x else g x) = f x"  "\<And>x. x \<in> T \<Longrightarrow> (if P x then f x else g x) = g x"
     by (simp_all add: fg)
   then have "continuous_on S (\<lambda>x. if P x then f x else g x)" "continuous_on T (\<lambda>x. if P x then f x else g x)"
     by (simp_all add: contf contg cong: continuous_on_cong)
   then show ?thesis
     by (rule continuous_on_Un_local_open [OF opS opT])
 qed
 
 subsection\<^marker>\<open>tag unimportant\<close>\<open>The union of two collinear segments is another segment\<close>
 
 proposition\<^marker>\<open>tag unimportant\<close> in_convex_hull_exchange:
   fixes a :: "'a::euclidean_space"
   assumes a: "a \<in> convex hull S" and xS: "x \<in> convex hull S"
   obtains b where "b \<in> S" "x \<in> convex hull (insert a (S - {b}))"
 proof (cases "a \<in> S")
   case True
   with xS insert_Diff that  show ?thesis by fastforce
 next
   case False
   show ?thesis
   proof (cases "finite S \<and> card S \<le> Suc (DIM('a))")
     case True
     then obtain u where u0: "\<And>i. i \<in> S \<Longrightarrow> 0 \<le> u i" and u1: "sum u S = 1"
                     and ua: "(\<Sum>i\<in>S. u i *\<^sub>R i) = a"
         using a by (auto simp: convex_hull_finite)
     obtain v where v0: "\<And>i. i \<in> S \<Longrightarrow> 0 \<le> v i" and v1: "sum v S = 1"
                and vx: "(\<Sum>i\<in>S. v i *\<^sub>R i) = x"
       using True xS by (auto simp: convex_hull_finite)
     show ?thesis
     proof (cases "\<exists>b. b \<in> S \<and> v b = 0")
       case True
       then obtain b where b: "b \<in> S" "v b = 0"
         by blast
       show ?thesis
       proof
         have fin: "finite (insert a (S - {b}))"
           using sum.infinite v1 by fastforce
         show "x \<in> convex hull insert a (S - {b})"
           unfolding convex_hull_finite [OF fin] mem_Collect_eq
         proof (intro conjI exI ballI)
           have "(\<Sum>x \<in> insert a (S - {b}). if x = a then 0 else v x) =
                 (\<Sum>x \<in> S - {b}. if x = a then 0 else v x)"
             apply (rule sum.mono_neutral_right)
             using fin by auto
           also have "... = (\<Sum>x \<in> S - {b}. v x)"
             using b False by (auto intro!: sum.cong split: if_split_asm)
           also have "... = (\<Sum>x\<in>S. v x)"
             by (metis \<open>v b = 0\<close> diff_zero sum.infinite sum_diff1 u1 zero_neq_one)
           finally show "(\<Sum>x\<in>insert a (S - {b}). if x = a then 0 else v x) = 1"
             by (simp add: v1)
           show "\<And>x. x \<in> insert a (S - {b}) \<Longrightarrow> 0 \<le> (if x = a then 0 else v x)"
             by (auto simp: v0)
           have "(\<Sum>x \<in> insert a (S - {b}). (if x = a then 0 else v x) *\<^sub>R x) =
                 (\<Sum>x \<in> S - {b}. (if x = a then 0 else v x) *\<^sub>R x)"
             apply (rule sum.mono_neutral_right)
             using fin by auto
           also have "... = (\<Sum>x \<in> S - {b}. v x *\<^sub>R x)"
             using b False by (auto intro!: sum.cong split: if_split_asm)
           also have "... = (\<Sum>x\<in>S. v x *\<^sub>R x)"
             by (metis (no_types, lifting) b(2) diff_zero fin finite.emptyI finite_Diff2 finite_insert scale_eq_0_iff sum_diff1)
           finally show "(\<Sum>x\<in>insert a (S - {b}). (if x = a then 0 else v x) *\<^sub>R x) = x"
             by (simp add: vx)
         qed
       qed (rule \<open>b \<in> S\<close>)
     next
       case False
       have le_Max: "u i / v i \<le> Max ((\<lambda>i. u i / v i) ` S)" if "i \<in> S" for i
         by (simp add: True that)
       have "Max ((\<lambda>i. u i / v i) ` S) \<in> (\<lambda>i. u i / v i) ` S"
         using True v1 by (auto intro: Max_in)
       then obtain b where "b \<in> S" and beq: "Max ((\<lambda>b. u b / v b) ` S) = u b / v b"
         by blast
       then have "0 \<noteq> u b / v b"
         using le_Max beq divide_le_0_iff le_numeral_extra(2) sum_nonpos u1
         by (metis False eq_iff v0)
       then have  "0 < u b" "0 < v b"
         using False \<open>b \<in> S\<close> u0 v0 by force+
       have fin: "finite (insert a (S - {b}))"
         using sum.infinite v1 by fastforce
       show ?thesis
       proof
         show "x \<in> convex hull insert a (S - {b})"
           unfolding convex_hull_finite [OF fin] mem_Collect_eq
         proof (intro conjI exI ballI)
           have "(\<Sum>x \<in> insert a (S - {b}). if x=a then v b / u b else v x - (v b / u b) * u x) =
                 v b / u b + (\<Sum>x \<in> S - {b}. v x - (v b / u b) * u x)"
             using \<open>a \<notin> S\<close> \<open>b \<in> S\<close> True  apply simp
             apply (rule sum.cong, auto)
             done
           also have "... = v b / u b + (\<Sum>x \<in> S - {b}. v x) - (v b / u b) * (\<Sum>x \<in> S - {b}. u x)"
             by (simp add: Groups_Big.sum_subtractf sum_distrib_left)
           also have "... = (\<Sum>x\<in>S. v x)"
             using \<open>0 < u b\<close> True  by (simp add: Groups_Big.sum_diff1 u1 field_simps)
           finally show "sum (\<lambda>x. if x=a then v b / u b else v x - (v b / u b) * u x) (insert a (S - {b})) = 1"
             by (simp add: v1)
           show "0 \<le> (if i = a then v b / u b else v i - v b / u b * u i)"
             if "i \<in> insert a (S - {b})" for i
             using \<open>0 < u b\<close> \<open>0 < v b\<close> v0 [of i] le_Max [of i] beq that False
             by (auto simp: field_simps split: if_split_asm)
           have "(\<Sum>x\<in>insert a (S - {b}). (if x=a then v b / u b else v x - v b / u b * u x) *\<^sub>R x) =
                 (v b / u b) *\<^sub>R a + (\<Sum>x\<in>S - {b}. (v x - v b / u b * u x) *\<^sub>R x)"
             using \<open>a \<notin> S\<close> \<open>b \<in> S\<close> True  apply simp
             apply (rule sum.cong, auto)
             done
           also have "... = (v b / u b) *\<^sub>R a + (\<Sum>x \<in> S - {b}. v x *\<^sub>R x) - (v b / u b) *\<^sub>R (\<Sum>x \<in> S - {b}. u x *\<^sub>R x)"
             by (simp add: Groups_Big.sum_subtractf scaleR_left_diff_distrib sum_distrib_left scale_sum_right)
           also have "... = (\<Sum>x\<in>S. v x *\<^sub>R x)"
             using \<open>0 < u b\<close> True  by (simp add: ua vx Groups_Big.sum_diff1 algebra_simps)
           finally
           show "(\<Sum>x\<in>insert a (S - {b}). (if x=a then v b / u b else v x - v b / u b * u x) *\<^sub>R x) = x"
             by (simp add: vx)
         qed
       qed (rule \<open>b \<in> S\<close>)
     qed
   next
     case False
     obtain T where "finite T" "T \<subseteq> S" and caT: "card T \<le> Suc (DIM('a))" and xT: "x \<in> convex hull T"
       using xS by (auto simp: caratheodory [of S])
     with False obtain b where b: "b \<in> S" "b \<notin> T"
       by (metis antisym subsetI)
     show ?thesis
     proof
       show "x \<in> convex hull insert a (S - {b})"
         using  \<open>T \<subseteq> S\<close> b by (blast intro: subsetD [OF hull_mono xT])
     qed (rule \<open>b \<in> S\<close>)
   qed
 qed
 
 lemma convex_hull_exchange_Union:
   fixes a :: "'a::euclidean_space"
   assumes "a \<in> convex hull S"
   shows "convex hull S = (\<Union>b \<in> S. convex hull (insert a (S - {b})))" (is "?lhs = ?rhs")
 proof
   show "?lhs \<subseteq> ?rhs"
     by (blast intro: in_convex_hull_exchange [OF assms])
   show "?rhs \<subseteq> ?lhs"
   proof clarify
     fix x b
     assume"b \<in> S" "x \<in> convex hull insert a (S - {b})"
     then show "x \<in> convex hull S" if "b \<in> S"
       by (metis (no_types) that assms order_refl hull_mono hull_redundant insert_Diff_single insert_subset subsetCE)
   qed
 qed
 
 lemma Un_closed_segment:
   fixes a :: "'a::euclidean_space"
   assumes "b \<in> closed_segment a c"
     shows "closed_segment a b \<union> closed_segment b c = closed_segment a c"
 proof (cases "c = a")
   case True
   with assms show ?thesis by simp
 next
   case False
   with assms have "convex hull {a, b} \<union> convex hull {b, c} = (\<Union>ba\<in>{a, c}. convex hull insert b ({a, c} - {ba}))"
     by (auto simp: insert_Diff_if insert_commute)
   then show ?thesis
     using convex_hull_exchange_Union
     by (metis assms segment_convex_hull)
 qed
 
 lemma Un_open_segment:
   fixes a :: "'a::euclidean_space"
   assumes "b \<in> open_segment a c"
   shows "open_segment a b \<union> {b} \<union> open_segment b c = open_segment a c"
 proof -
   have b: "b \<in> closed_segment a c"
     by (simp add: assms open_closed_segment)
   have *: "open_segment a c \<subseteq> insert b (open_segment a b \<union> open_segment b c)"
           if "{b,c,a} \<union> open_segment a b \<union> open_segment b c = {c,a} \<union> open_segment a c"
   proof -
     have "insert a (insert c (insert b (open_segment a b \<union> open_segment b c))) = insert a (insert c (open_segment a c))"
       using that by (simp add: insert_commute)
     then show ?thesis
       by (metis (no_types) Diff_cancel Diff_eq_empty_iff Diff_insert2 open_segment_def)
   qed
   show ?thesis
     using Un_closed_segment [OF b]
     apply (simp add: closed_segment_eq_open)
       apply (rule equalityI)
     using assms
      apply (simp add: b subset_open_segment)
       using * by (simp add: insert_commute)
 qed
 
 subsection\<open>Covering an open set by a countable chain of compact sets\<close>
   
 proposition open_Union_compact_subsets:
   fixes S :: "'a::euclidean_space set"
   assumes "open S"
   obtains C where "\<And>n. compact(C n)" "\<And>n. C n \<subseteq> S"
                   "\<And>n. C n \<subseteq> interior(C(Suc n))"
                   "\<Union>(range C) = S"
                   "\<And>K. \<lbrakk>compact K; K \<subseteq> S\<rbrakk> \<Longrightarrow> \<exists>N. \<forall>n\<ge>N. K \<subseteq> (C n)"
 proof (cases "S = {}")
   case True
   then show ?thesis
     by (rule_tac C = "\<lambda>n. {}" in that) auto
 next
   case False
   then obtain a where "a \<in> S"
     by auto
   let ?C = "\<lambda>n. cball a (real n) - (\<Union>x \<in> -S. \<Union>e \<in> ball 0 (1 / real(Suc n)). {x + e})"
   have "\<exists>N. \<forall>n\<ge>N. K \<subseteq> (f n)"
         if "\<And>n. compact(f n)" and sub_int: "\<And>n. f n \<subseteq> interior (f(Suc n))"
             and eq: "\<Union>(range f) = S" and "compact K" "K \<subseteq> S" for f K
   proof -
     have *: "\<forall>n. f n \<subseteq> (\<Union>n. interior (f n))"
       by (meson Sup_upper2 UNIV_I \<open>\<And>n. f n \<subseteq> interior (f (Suc n))\<close> image_iff)
     have mono: "\<And>m n. m \<le> n \<Longrightarrow>f m \<subseteq> f n"
       by (meson dual_order.trans interior_subset lift_Suc_mono_le sub_int)
     obtain I where "finite I" and I: "K \<subseteq> (\<Union>i\<in>I. interior (f i))"
     proof (rule compactE_image [OF \<open>compact K\<close>])
       show "K \<subseteq> (\<Union>n. interior (f n))"
         using \<open>K \<subseteq> S\<close> \<open>\<Union>(f ` UNIV) = S\<close> * by blast
     qed auto
     { fix n
       assume n: "Max I \<le> n"
       have "(\<Union>i\<in>I. interior (f i)) \<subseteq> f n"
         by (rule UN_least) (meson dual_order.trans interior_subset mono I Max_ge [OF \<open>finite I\<close>] n)
       then have "K \<subseteq> f n"
         using I by auto
     }
     then show ?thesis
       by blast
   qed
   moreover have "\<exists>f. (\<forall>n. compact(f n)) \<and> (\<forall>n. (f n) \<subseteq> S) \<and> (\<forall>n. (f n) \<subseteq> interior(f(Suc n))) \<and>
                      ((\<Union>(range f) = S))"
   proof (intro exI conjI allI)
     show "\<And>n. compact (?C n)"
       by (auto simp: compact_diff open_sums)
     show "\<And>n. ?C n \<subseteq> S"
       by auto
     show "?C n \<subseteq> interior (?C (Suc n))" for n
     proof (simp add: interior_diff, rule Diff_mono)
       show "cball a (real n) \<subseteq> ball a (1 + real n)"
         by (simp add: cball_subset_ball_iff)
       have cl: "closed (\<Union>x\<in>- S. \<Union>e\<in>cball 0 (1 / (2 + real n)). {x + e})"
         using assms by (auto intro: closed_compact_sums)
       have "closure (\<Union>x\<in>- S. \<Union>y\<in>ball 0 (1 / (2 + real n)). {x + y})
             \<subseteq> (\<Union>x \<in> -S. \<Union>e \<in> cball 0 (1 / (2 + real n)). {x + e})"
         by (intro closure_minimal UN_mono ball_subset_cball order_refl cl)
       also have "... \<subseteq> (\<Union>x \<in> -S. \<Union>y\<in>ball 0 (1 / (1 + real n)). {x + y})"
         apply (intro UN_mono order_refl)
         apply (simp add: cball_subset_ball_iff field_split_simps)
         done
       finally show "closure (\<Union>x\<in>- S. \<Union>y\<in>ball 0 (1 / (2 + real n)). {x + y})
                     \<subseteq> (\<Union>x \<in> -S. \<Union>y\<in>ball 0 (1 / (1 + real n)). {x + y})" .
     qed
     have "S \<subseteq> \<Union> (range ?C)"
     proof
       fix x
       assume x: "x \<in> S"
       then obtain e where "e > 0" and e: "ball x e \<subseteq> S"
         using assms open_contains_ball by blast
       then obtain N1 where "N1 > 0" and N1: "real N1 > 1/e"
         using reals_Archimedean2
         by (metis divide_less_0_iff less_eq_real_def neq0_conv not_le of_nat_0 of_nat_1 of_nat_less_0_iff)
       obtain N2 where N2: "norm(x - a) \<le> real N2"
         by (meson real_arch_simple)
       have N12: "inverse((N1 + N2) + 1) \<le> inverse(N1)"
         using \<open>N1 > 0\<close> by (auto simp: field_split_simps)
       have "x \<noteq> y + z" if "y \<notin> S" "norm z < 1 / (1 + (real N1 + real N2))" for y z
       proof -
         have "e * real N1 < e * (1 + (real N1 + real N2))"
           by (simp add: \<open>0 < e\<close>)
         then have "1 / (1 + (real N1 + real N2)) < e"
           using N1 \<open>e > 0\<close>
           by (metis divide_less_eq less_trans mult.commute of_nat_add of_nat_less_0_iff of_nat_Suc)
         then have "x - z \<in> ball x e"
           using that by simp
         then have "x - z \<in> S"
           using e by blast
         with that show ?thesis
           by auto
       qed
       with N2 show "x \<in> \<Union> (range ?C)"
         by (rule_tac a = "N1+N2" in UN_I) (auto simp: dist_norm norm_minus_commute)
     qed
     then show "\<Union> (range ?C) = S" by auto
   qed
   ultimately show ?thesis
     using that by metis
 qed
 
 
 subsection\<open>Orthogonal complement\<close>
 
 definition\<^marker>\<open>tag important\<close> orthogonal_comp ("_\<^sup>\<bottom>" [80] 80)
   where "orthogonal_comp W \<equiv> {x. \<forall>y \<in> W. orthogonal y x}"
 
 proposition subspace_orthogonal_comp: "subspace (W\<^sup>\<bottom>)"
   unfolding subspace_def orthogonal_comp_def orthogonal_def
   by (auto simp: inner_right_distrib)
 
 lemma orthogonal_comp_anti_mono:
   assumes "A \<subseteq> B"
   shows "B\<^sup>\<bottom> \<subseteq> A\<^sup>\<bottom>"
 proof
   fix x assume x: "x \<in> B\<^sup>\<bottom>"
   show "x \<in> orthogonal_comp A" using x unfolding orthogonal_comp_def
     by (simp add: orthogonal_def, metis assms in_mono)
 qed
 
 lemma orthogonal_comp_null [simp]: "{0}\<^sup>\<bottom> = UNIV"
   by (auto simp: orthogonal_comp_def orthogonal_def)
 
 lemma orthogonal_comp_UNIV [simp]: "UNIV\<^sup>\<bottom> = {0}"
   unfolding orthogonal_comp_def orthogonal_def
   by auto (use inner_eq_zero_iff in blast)
 
 lemma orthogonal_comp_subset: "U \<subseteq> U\<^sup>\<bottom>\<^sup>\<bottom>"
   by (auto simp: orthogonal_comp_def orthogonal_def inner_commute)
 
 lemma subspace_sum_minimal:
   assumes "S \<subseteq> U" "T \<subseteq> U" "subspace U"
   shows "S + T \<subseteq> U"
 proof
   fix x
   assume "x \<in> S + T"
   then obtain xs xt where "xs \<in> S" "xt \<in> T" "x = xs+xt"
     by (meson set_plus_elim)
   then show "x \<in> U"
     by (meson assms subsetCE subspace_add)
 qed
 
 proposition subspace_sum_orthogonal_comp:
   fixes U :: "'a :: euclidean_space set"
   assumes "subspace U"
   shows "U + U\<^sup>\<bottom> = UNIV"
 proof -
   obtain B where "B \<subseteq> U"
     and ortho: "pairwise orthogonal B" "\<And>x. x \<in> B \<Longrightarrow> norm x = 1"
     and "independent B" "card B = dim U" "span B = U"
     using orthonormal_basis_subspace [OF assms] by metis
   then have "finite B"
     by (simp add: indep_card_eq_dim_span)
   have *: "\<forall>x\<in>B. \<forall>y\<in>B. x \<bullet> y = (if x=y then 1 else 0)"
     using ortho norm_eq_1 by (auto simp: orthogonal_def pairwise_def)
   { fix v
     let ?u = "\<Sum>b\<in>B. (v \<bullet> b) *\<^sub>R b"
     have "v = ?u + (v - ?u)"
       by simp
     moreover have "?u \<in> U"
       by (metis (no_types, lifting) \<open>span B = U\<close> assms subspace_sum span_base span_mul)
     moreover have "(v - ?u) \<in> U\<^sup>\<bottom>"
     proof (clarsimp simp: orthogonal_comp_def orthogonal_def)
       fix y
       assume "y \<in> U"
       with \<open>span B = U\<close> span_finite [OF \<open>finite B\<close>]
       obtain u where u: "y = (\<Sum>b\<in>B. u b *\<^sub>R b)"
         by auto
       have "b \<bullet> (v - ?u) = 0" if "b \<in> B" for b
         using that \<open>finite B\<close>
         by (simp add: * algebra_simps inner_sum_right if_distrib [of "(*)v" for v] inner_commute cong: if_cong)
       then show "y \<bullet> (v - ?u) = 0"
         by (simp add: u inner_sum_left)
     qed
     ultimately have "v \<in> U + U\<^sup>\<bottom>"
       using set_plus_intro by fastforce
   } then show ?thesis
     by auto
 qed
 
 lemma orthogonal_Int_0:
   assumes "subspace U"
   shows "U \<inter> U\<^sup>\<bottom> = {0}"
   using orthogonal_comp_def orthogonal_self
   by (force simp: assms subspace_0 subspace_orthogonal_comp)
 
 lemma orthogonal_comp_self:
   fixes U :: "'a :: euclidean_space set"
   assumes "subspace U"
   shows "U\<^sup>\<bottom>\<^sup>\<bottom> = U"
 proof
   have ssU': "subspace (U\<^sup>\<bottom>)"
     by (simp add: subspace_orthogonal_comp)
   have "u \<in> U" if "u \<in> U\<^sup>\<bottom>\<^sup>\<bottom>" for u
   proof -
     obtain v w where "u = v+w" "v \<in> U" "w \<in> U\<^sup>\<bottom>"
       using subspace_sum_orthogonal_comp [OF assms] set_plus_elim by blast
     then have "u-v \<in> U\<^sup>\<bottom>"
       by simp
     moreover have "v \<in> U\<^sup>\<bottom>\<^sup>\<bottom>"
       using \<open>v \<in> U\<close> orthogonal_comp_subset by blast
     then have "u-v \<in> U\<^sup>\<bottom>\<^sup>\<bottom>"
       by (simp add: subspace_diff subspace_orthogonal_comp that)
     ultimately have "u-v = 0"
       using orthogonal_Int_0 ssU' by blast
     with \<open>v \<in> U\<close> show ?thesis
       by auto
   qed
   then show "U\<^sup>\<bottom>\<^sup>\<bottom> \<subseteq> U"
     by auto
 qed (use orthogonal_comp_subset in auto)
 
 lemma ker_orthogonal_comp_adjoint:
   fixes f :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space"
   assumes "linear f"
   shows "f -` {0} =  (range (adjoint f))\<^sup>\<bottom>"
   apply (auto simp: orthogonal_comp_def orthogonal_def)
   apply (simp add: adjoint_works assms(1) inner_commute)
   by (metis adjoint_works all_zero_iff assms(1) inner_commute)
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>A non-injective linear function maps into a hyperplane.\<close>
 
 lemma linear_surj_adj_imp_inj:
   fixes f :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space"
   assumes "linear f" "surj (adjoint f)"
   shows "inj f"
 proof -
   have "\<exists>x. y = adjoint f x" for y
     using assms by (simp add: surjD)
   then show "inj f"
     using assms unfolding inj_on_def image_def
     by (metis (no_types) adjoint_works euclidean_eqI)
 qed
 
 \<comment> \<open>\<^url>\<open>https://mathonline.wikidot.com/injectivity-and-surjectivity-of-the-adjoint-of-a-linear-map\<close>\<close>
 lemma surj_adjoint_iff_inj [simp]:
   fixes f :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space"
   assumes "linear f"
   shows  "surj (adjoint f) \<longleftrightarrow> inj f"
 proof
   assume "surj (adjoint f)"
   then show "inj f"
     by (simp add: assms linear_surj_adj_imp_inj)
 next
   assume "inj f"
   have "f -` {0} = {0}"
     using assms \<open>inj f\<close> linear_0 linear_injective_0 by fastforce
   moreover have "f -` {0} = range (adjoint f)\<^sup>\<bottom>"
     by (intro ker_orthogonal_comp_adjoint assms)
   ultimately have "range (adjoint f)\<^sup>\<bottom>\<^sup>\<bottom> = UNIV"
     by (metis orthogonal_comp_null)
   then show "surj (adjoint f)"
     using adjoint_linear \<open>linear f\<close>
     by (subst (asm) orthogonal_comp_self)
       (simp add: adjoint_linear linear_subspace_image)
 qed
 
 lemma inj_adjoint_iff_surj [simp]:
   fixes f :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space"
   assumes "linear f"
   shows  "inj (adjoint f) \<longleftrightarrow> surj f"
 proof
   assume "inj (adjoint f)"
   have "(adjoint f) -` {0} = {0}"
     by (metis \<open>inj (adjoint f)\<close> adjoint_linear assms surj_adjoint_iff_inj ker_orthogonal_comp_adjoint orthogonal_comp_UNIV)
   then have "(range(f))\<^sup>\<bottom> = {0}"
     by (metis (no_types, hide_lams) adjoint_adjoint adjoint_linear assms ker_orthogonal_comp_adjoint set_zero)
   then show "surj f"
     by (metis \<open>inj (adjoint f)\<close> adjoint_adjoint adjoint_linear assms surj_adjoint_iff_inj)
 next
   assume "surj f"
   then have "range f = (adjoint f -` {0})\<^sup>\<bottom>"
     by (simp add: adjoint_adjoint adjoint_linear assms ker_orthogonal_comp_adjoint)
   then have "{0} = adjoint f -` {0}"
     using \<open>surj f\<close> adjoint_adjoint adjoint_linear assms ker_orthogonal_comp_adjoint by force
   then show "inj (adjoint f)"
     by (simp add: \<open>surj f\<close> adjoint_adjoint adjoint_linear assms linear_surj_adj_imp_inj)
 qed
 
 lemma linear_singular_into_hyperplane:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'n"
   assumes "linear f"
   shows "\<not> inj f \<longleftrightarrow> (\<exists>a. a \<noteq> 0 \<and> (\<forall>x. a \<bullet> f x = 0))" (is "_ = ?rhs")
 proof
   assume "\<not>inj f"
   then show ?rhs
     using all_zero_iff
     by (metis (no_types, hide_lams) adjoint_clauses(2) adjoint_linear assms
         linear_injective_0 linear_injective_imp_surjective linear_surj_adj_imp_inj)
 next
   assume ?rhs
   then show "\<not>inj f"
     by (metis assms linear_injective_isomorphism all_zero_iff)
 qed
 
 lemma linear_singular_image_hyperplane:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'n"
   assumes "linear f" "\<not>inj f"
   obtains a where "a \<noteq> 0" "\<And>S. f ` S \<subseteq> {x. a \<bullet> x = 0}"
   using assms by (fastforce simp add: linear_singular_into_hyperplane)
 
 end