diff --git a/src/HOL/Analysis/Elementary_Topology.thy b/src/HOL/Analysis/Elementary_Topology.thy
--- a/src/HOL/Analysis/Elementary_Topology.thy
+++ b/src/HOL/Analysis/Elementary_Topology.thy
@@ -1,2660 +1,2693 @@
 (*  Author:     L C Paulson, University of Cambridge
     Author:     Amine Chaieb, University of Cambridge
     Author:     Robert Himmelmann, TU Muenchen
     Author:     Brian Huffman, Portland State University
 *)
 
 chapter \<open>Topology\<close>
 
 theory Elementary_Topology
 imports
   "HOL-Library.Set_Idioms"
   "HOL-Library.Disjoint_Sets"
   Product_Vector
 begin
 
 
 section \<open>Elementary Topology\<close>
 
 
 subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Affine transformations of intervals\<close>
 
 lemma real_affinity_le: "0 < m \<Longrightarrow> m * x + c \<le> y \<longleftrightarrow> x \<le> inverse m * y + - (c / m)"
   for m :: "'a::linordered_field"
   by (simp add: field_simps)
 
 lemma real_le_affinity: "0 < m \<Longrightarrow> y \<le> m * x + c \<longleftrightarrow> inverse m * y + - (c / m) \<le> x"
   for m :: "'a::linordered_field"
   by (simp add: field_simps)
 
 lemma real_affinity_lt: "0 < m \<Longrightarrow> m * x + c < y \<longleftrightarrow> x < inverse m * y + - (c / m)"
   for m :: "'a::linordered_field"
   by (simp add: field_simps)
 
 lemma real_lt_affinity: "0 < m \<Longrightarrow> y < m * x + c \<longleftrightarrow> inverse m * y + - (c / m) < x"
   for m :: "'a::linordered_field"
   by (simp add: field_simps)
 
 lemma real_affinity_eq: "m \<noteq> 0 \<Longrightarrow> m * x + c = y \<longleftrightarrow> x = inverse m * y + - (c / m)"
   for m :: "'a::linordered_field"
   by (simp add: field_simps)
 
 lemma real_eq_affinity: "m \<noteq> 0 \<Longrightarrow> y = m * x + c  \<longleftrightarrow> inverse m * y + - (c / m) = x"
   for m :: "'a::linordered_field"
   by (simp add: field_simps)
 
 
 subsection \<open>Topological Basis\<close>
 
 context topological_space
 begin
 
 definition\<^marker>\<open>tag important\<close> "topological_basis B \<longleftrightarrow>
   (\<forall>b\<in>B. open b) \<and> (\<forall>x. open x \<longrightarrow> (\<exists>B'. B' \<subseteq> B \<and> \<Union>B' = x))"
 
 lemma topological_basis:
   "topological_basis B \<longleftrightarrow> (\<forall>x. open x \<longleftrightarrow> (\<exists>B'. B' \<subseteq> B \<and> \<Union>B' = x))"
   unfolding topological_basis_def
   apply safe
      apply fastforce
     apply fastforce
    apply (erule_tac x=x in allE, simp)
    apply (rule_tac x="{x}" in exI, auto)
   done
 
 lemma topological_basis_iff:
   assumes "\<And>B'. B' \<in> B \<Longrightarrow> open B'"
   shows "topological_basis B \<longleftrightarrow> (\<forall>O'. open O' \<longrightarrow> (\<forall>x\<in>O'. \<exists>B'\<in>B. x \<in> B' \<and> B' \<subseteq> O'))"
     (is "_ \<longleftrightarrow> ?rhs")
 proof safe
   fix O' and x::'a
   assume H: "topological_basis B" "open O'" "x \<in> O'"
   then have "(\<exists>B'\<subseteq>B. \<Union>B' = O')" by (simp add: topological_basis_def)
   then obtain B' where "B' \<subseteq> B" "O' = \<Union>B'" by auto
   then show "\<exists>B'\<in>B. x \<in> B' \<and> B' \<subseteq> O'" using H by auto
 next
   assume H: ?rhs
   show "topological_basis B"
     using assms unfolding topological_basis_def
   proof safe
     fix O' :: "'a set"
     assume "open O'"
     with H obtain f where "\<forall>x\<in>O'. f x \<in> B \<and> x \<in> f x \<and> f x \<subseteq> O'"
       by (force intro: bchoice simp: Bex_def)
     then show "\<exists>B'\<subseteq>B. \<Union>B' = O'"
       by (auto intro: exI[where x="{f x |x. x \<in> O'}"])
   qed
 qed
 
 lemma topological_basisI:
   assumes "\<And>B'. B' \<in> B \<Longrightarrow> open B'"
     and "\<And>O' x. open O' \<Longrightarrow> x \<in> O' \<Longrightarrow> \<exists>B'\<in>B. x \<in> B' \<and> B' \<subseteq> O'"
   shows "topological_basis B"
   using assms by (subst topological_basis_iff) auto
 
 lemma topological_basisE:
   fixes O'
   assumes "topological_basis B"
     and "open O'"
     and "x \<in> O'"
   obtains B' where "B' \<in> B" "x \<in> B'" "B' \<subseteq> O'"
 proof atomize_elim
   from assms have "\<And>B'. B'\<in>B \<Longrightarrow> open B'"
     by (simp add: topological_basis_def)
   with topological_basis_iff assms
   show  "\<exists>B'. B' \<in> B \<and> x \<in> B' \<and> B' \<subseteq> O'"
     using assms by (simp add: Bex_def)
 qed
 
 lemma topological_basis_open:
   assumes "topological_basis B"
     and "X \<in> B"
   shows "open X"
   using assms by (simp add: topological_basis_def)
 
 lemma topological_basis_imp_subbasis:
   assumes B: "topological_basis B"
   shows "open = generate_topology B"
 proof (intro ext iffI)
   fix S :: "'a set"
   assume "open S"
   with B obtain B' where "B' \<subseteq> B" "S = \<Union>B'"
     unfolding topological_basis_def by blast
   then show "generate_topology B S"
     by (auto intro: generate_topology.intros dest: topological_basis_open)
 next
   fix S :: "'a set"
   assume "generate_topology B S"
   then show "open S"
     by induct (auto dest: topological_basis_open[OF B])
 qed
 
 lemma basis_dense:
   fixes B :: "'a set set"
     and f :: "'a set \<Rightarrow> 'a"
   assumes "topological_basis B"
     and choosefrom_basis: "\<And>B'. B' \<noteq> {} \<Longrightarrow> f B' \<in> B'"
   shows "\<forall>X. open X \<longrightarrow> X \<noteq> {} \<longrightarrow> (\<exists>B' \<in> B. f B' \<in> X)"
 proof (intro allI impI)
   fix X :: "'a set"
   assume "open X" and "X \<noteq> {}"
   from topological_basisE[OF \<open>topological_basis B\<close> \<open>open X\<close> choosefrom_basis[OF \<open>X \<noteq> {}\<close>]]
   obtain B' where "B' \<in> B" "f X \<in> B'" "B' \<subseteq> X" .
   then show "\<exists>B'\<in>B. f B' \<in> X"
     by (auto intro!: choosefrom_basis)
 qed
 
 end
 
 lemma topological_basis_prod:
   assumes A: "topological_basis A"
     and B: "topological_basis B"
   shows "topological_basis ((\<lambda>(a, b). a \<times> b) ` (A \<times> B))"
   unfolding topological_basis_def
 proof (safe, simp_all del: ex_simps add: subset_image_iff ex_simps(1)[symmetric])
   fix S :: "('a \<times> 'b) set"
   assume "open S"
   then show "\<exists>X\<subseteq>A \<times> B. (\<Union>(a,b)\<in>X. a \<times> b) = S"
   proof (safe intro!: exI[of _ "{x\<in>A \<times> B. fst x \<times> snd x \<subseteq> S}"])
     fix x y
     assume "(x, y) \<in> S"
     from open_prod_elim[OF \<open>open S\<close> this]
     obtain a b where a: "open a""x \<in> a" and b: "open b" "y \<in> b" and "a \<times> b \<subseteq> S"
       by (metis mem_Sigma_iff)
     moreover
     from A a obtain A0 where "A0 \<in> A" "x \<in> A0" "A0 \<subseteq> a"
       by (rule topological_basisE)
     moreover
     from B b obtain B0 where "B0 \<in> B" "y \<in> B0" "B0 \<subseteq> b"
       by (rule topological_basisE)
     ultimately show "(x, y) \<in> (\<Union>(a, b)\<in>{X \<in> A \<times> B. fst X \<times> snd X \<subseteq> S}. a \<times> b)"
       by (intro UN_I[of "(A0, B0)"]) auto
   qed auto
 qed (metis A B topological_basis_open open_Times)
 
 
 subsection \<open>Countable Basis\<close>
 
 locale\<^marker>\<open>tag important\<close> countable_basis = topological_space p for p::"'a set \<Rightarrow> bool" +
   fixes B :: "'a set set"
   assumes is_basis: "topological_basis B"
     and countable_basis: "countable B"
 begin
 
 lemma open_countable_basis_ex:
   assumes "p X"
   shows "\<exists>B' \<subseteq> B. X = \<Union>B'"
   using assms countable_basis is_basis
   unfolding topological_basis_def by blast
 
 lemma open_countable_basisE:
   assumes "p X"
   obtains B' where "B' \<subseteq> B" "X = \<Union>B'"
   using assms open_countable_basis_ex
   by atomize_elim simp
 
 lemma countable_dense_exists:
   "\<exists>D::'a set. countable D \<and> (\<forall>X. p X \<longrightarrow> X \<noteq> {} \<longrightarrow> (\<exists>d \<in> D. d \<in> X))"
 proof -
   let ?f = "(\<lambda>B'. SOME x. x \<in> B')"
   have "countable (?f ` B)" using countable_basis by simp
   with basis_dense[OF is_basis, of ?f] show ?thesis
     by (intro exI[where x="?f ` B"]) (metis (mono_tags) all_not_in_conv imageI someI)
 qed
 
 lemma countable_dense_setE:
   obtains D :: "'a set"
   where "countable D" "\<And>X. p X \<Longrightarrow> X \<noteq> {} \<Longrightarrow> \<exists>d \<in> D. d \<in> X"
   using countable_dense_exists by blast
 
 end
 
 lemma countable_basis_openI: "countable_basis open B"
   if "countable B" "topological_basis B"
   using that
   by unfold_locales
     (simp_all add: topological_basis topological_space.topological_basis topological_space_axioms)
 
 lemma (in first_countable_topology) first_countable_basisE:
   fixes x :: 'a
   obtains \<A> where "countable \<A>" "\<And>A. A \<in> \<A> \<Longrightarrow> x \<in> A" "\<And>A. A \<in> \<A> \<Longrightarrow> open A"
     "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> (\<exists>A\<in>\<A>. A \<subseteq> S)"
 proof -
   obtain \<A> where \<A>: "(\<forall>i::nat. x \<in> \<A> i \<and> open (\<A> i))" "(\<forall>S. open S \<and> x \<in> S \<longrightarrow> (\<exists>i. \<A> i \<subseteq> S))"
     using first_countable_basis[of x] by metis
   show thesis
   proof 
     show "countable (range \<A>)"
       by simp
   qed (use \<A> in auto)
 qed
 
 lemma (in first_countable_topology) first_countable_basis_Int_stableE:
   obtains \<A> where "countable \<A>" "\<And>A. A \<in> \<A> \<Longrightarrow> x \<in> A" "\<And>A. A \<in> \<A> \<Longrightarrow> open A"
     "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> (\<exists>A\<in>\<A>. A \<subseteq> S)"
     "\<And>A B. A \<in> \<A> \<Longrightarrow> B \<in> \<A> \<Longrightarrow> A \<inter> B \<in> \<A>"
 proof atomize_elim
   obtain \<B> where \<B>:
     "countable \<B>"
     "\<And>B. B \<in> \<B> \<Longrightarrow> x \<in> B"
     "\<And>B. B \<in> \<B> \<Longrightarrow> open B"
     "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> \<exists>B\<in>\<B>. B \<subseteq> S"
     by (rule first_countable_basisE) blast
   define \<A> where [abs_def]:
     "\<A> = (\<lambda>N. \<Inter>((\<lambda>n. from_nat_into \<B> n) ` N)) ` (Collect finite::nat set set)"
   then show "\<exists>\<A>. countable \<A> \<and> (\<forall>A. A \<in> \<A> \<longrightarrow> x \<in> A) \<and> (\<forall>A. A \<in> \<A> \<longrightarrow> open A) \<and>
         (\<forall>S. open S \<longrightarrow> x \<in> S \<longrightarrow> (\<exists>A\<in>\<A>. A \<subseteq> S)) \<and> (\<forall>A B. A \<in> \<A> \<longrightarrow> B \<in> \<A> \<longrightarrow> A \<inter> B \<in> \<A>)"
   proof (safe intro!: exI[where x=\<A>])
     show "countable \<A>"
       unfolding \<A>_def by (intro countable_image countable_Collect_finite)
     fix A
     assume "A \<in> \<A>"
     then show "x \<in> A" "open A"
       using \<B>(4)[OF open_UNIV] by (auto simp: \<A>_def intro: \<B> from_nat_into)
   next
     let ?int = "\<lambda>N. \<Inter>(from_nat_into \<B> ` N)"
     fix A B
     assume "A \<in> \<A>" "B \<in> \<A>"
     then obtain N M where "A = ?int N" "B = ?int M" "finite (N \<union> M)"
       by (auto simp: \<A>_def)
     then show "A \<inter> B \<in> \<A>"
       by (auto simp: \<A>_def intro!: image_eqI[where x="N \<union> M"])
   next
     fix S
     assume "open S" "x \<in> S"
     then obtain a where a: "a\<in>\<B>" "a \<subseteq> S" using \<B> by blast
     then show "\<exists>a\<in>\<A>. a \<subseteq> S" using a \<B>
       by (intro bexI[where x=a]) (auto simp: \<A>_def intro: image_eqI[where x="{to_nat_on \<B> a}"])
   qed
 qed
 
 lemma (in topological_space) first_countableI:
   assumes "countable \<A>"
     and 1: "\<And>A. A \<in> \<A> \<Longrightarrow> x \<in> A" "\<And>A. A \<in> \<A> \<Longrightarrow> open A"
     and 2: "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> \<exists>A\<in>\<A>. A \<subseteq> S"
   shows "\<exists>\<A>::nat \<Rightarrow> 'a set. (\<forall>i. x \<in> \<A> i \<and> open (\<A> i)) \<and> (\<forall>S. open S \<and> x \<in> S \<longrightarrow> (\<exists>i. \<A> i \<subseteq> S))"
 proof (safe intro!: exI[of _ "from_nat_into \<A>"])
   fix i
   have "\<A> \<noteq> {}" using 2[of UNIV] by auto
   show "x \<in> from_nat_into \<A> i" "open (from_nat_into \<A> i)"
     using range_from_nat_into_subset[OF \<open>\<A> \<noteq> {}\<close>] 1 by auto
 next
   fix S
   assume "open S" "x\<in>S" from 2[OF this]
   show "\<exists>i. from_nat_into \<A> i \<subseteq> S"
     using subset_range_from_nat_into[OF \<open>countable \<A>\<close>] by auto
 qed
 
 instance prod :: (first_countable_topology, first_countable_topology) first_countable_topology
 proof
   fix x :: "'a \<times> 'b"
   obtain \<A> where \<A>:
       "countable \<A>"
       "\<And>a. a \<in> \<A> \<Longrightarrow> fst x \<in> a"
       "\<And>a. a \<in> \<A> \<Longrightarrow> open a"
       "\<And>S. open S \<Longrightarrow> fst x \<in> S \<Longrightarrow> \<exists>a\<in>\<A>. a \<subseteq> S"
     by (rule first_countable_basisE[of "fst x"]) blast
   obtain B where B:
       "countable B"
       "\<And>a. a \<in> B \<Longrightarrow> snd x \<in> a"
       "\<And>a. a \<in> B \<Longrightarrow> open a"
       "\<And>S. open S \<Longrightarrow> snd x \<in> S \<Longrightarrow> \<exists>a\<in>B. a \<subseteq> S"
     by (rule first_countable_basisE[of "snd x"]) blast
   show "\<exists>\<A>::nat \<Rightarrow> ('a \<times> 'b) set.
     (\<forall>i. x \<in> \<A> i \<and> open (\<A> i)) \<and> (\<forall>S. open S \<and> x \<in> S \<longrightarrow> (\<exists>i. \<A> i \<subseteq> S))"
   proof (rule first_countableI[of "(\<lambda>(a, b). a \<times> b) ` (\<A> \<times> B)"], safe)
     fix a b
     assume x: "a \<in> \<A>" "b \<in> B"
     show "x \<in> a \<times> b" 
       by (simp add: \<A>(2) B(2) mem_Times_iff x)
     show "open (a \<times> b)"
       by (simp add: \<A>(3) B(3) open_Times x)
   next
     fix S
     assume "open S" "x \<in> S"
     then obtain a' b' where a'b': "open a'" "open b'" "x \<in> a' \<times> b'" "a' \<times> b' \<subseteq> S"
       by (rule open_prod_elim)
     moreover
     from a'b' \<A>(4)[of a'] B(4)[of b']
     obtain a b where "a \<in> \<A>" "a \<subseteq> a'" "b \<in> B" "b \<subseteq> b'"
       by auto
     ultimately
     show "\<exists>a\<in>(\<lambda>(a, b). a \<times> b) ` (\<A> \<times> B). a \<subseteq> S"
       by (auto intro!: bexI[of _ "a \<times> b"] bexI[of _ a] bexI[of _ b])
   qed (simp add: \<A> B)
 qed
 
 class second_countable_topology = topological_space +
   assumes ex_countable_subbasis:
     "\<exists>B::'a set set. countable B \<and> open = generate_topology B"
 begin
 
 lemma ex_countable_basis: "\<exists>B::'a set set. countable B \<and> topological_basis B"
 proof -
   from ex_countable_subbasis obtain B where B: "countable B" "open = generate_topology B"
     by blast
   let ?B = "Inter ` {b. finite b \<and> b \<subseteq> B }"
 
   show ?thesis
   proof (intro exI conjI)
     show "countable ?B"
       by (intro countable_image countable_Collect_finite_subset B)
     {
       fix S
       assume "open S"
       then have "\<exists>B'\<subseteq>{b. finite b \<and> b \<subseteq> B}. (\<Union>b\<in>B'. \<Inter>b) = S"
         unfolding B
       proof induct
         case UNIV
         show ?case by (intro exI[of _ "{{}}"]) simp
       next
         case (Int a b)
         then obtain x y where x: "a = \<Union>(Inter ` x)" "\<And>i. i \<in> x \<Longrightarrow> finite i \<and> i \<subseteq> B"
           and y: "b = \<Union>(Inter ` y)" "\<And>i. i \<in> y \<Longrightarrow> finite i \<and> i \<subseteq> B"
           by blast
         show ?case
           unfolding x y Int_UN_distrib2
           by (intro exI[of _ "{i \<union> j| i j.  i \<in> x \<and> j \<in> y}"]) (auto dest: x(2) y(2))
       next
         case (UN K)
         then have "\<forall>k\<in>K. \<exists>B'\<subseteq>{b. finite b \<and> b \<subseteq> B}. \<Union> (Inter ` B') = k" by auto
         then obtain k where
             "\<forall>ka\<in>K. k ka \<subseteq> {b. finite b \<and> b \<subseteq> B} \<and> \<Union>(Inter ` (k ka)) = ka"
           unfolding bchoice_iff ..
         then show "\<exists>B'\<subseteq>{b. finite b \<and> b \<subseteq> B}. \<Union> (Inter ` B') = \<Union>K"
           by (intro exI[of _ "\<Union>(k ` K)"]) auto
       next
         case (Basis S)
         then show ?case
           by (intro exI[of _ "{{S}}"]) auto
       qed
       then have "(\<exists>B'\<subseteq>Inter ` {b. finite b \<and> b \<subseteq> B}. \<Union>B' = S)"
         unfolding subset_image_iff by blast }
     then show "topological_basis ?B"
       unfolding topological_basis_def
       by (safe intro!: open_Inter)
          (simp_all add: B generate_topology.Basis subset_eq)
   qed
 qed
 
 
 end
 
 lemma univ_second_countable:
   obtains \<B> :: "'a::second_countable_topology set set"
   where "countable \<B>" "\<And>C. C \<in> \<B> \<Longrightarrow> open C"
        "\<And>S. open S \<Longrightarrow> \<exists>U. U \<subseteq> \<B> \<and> S = \<Union>U"
 by (metis ex_countable_basis topological_basis_def)
 
 proposition Lindelof:
   fixes \<F> :: "'a::second_countable_topology set set"
   assumes \<F>: "\<And>S. S \<in> \<F> \<Longrightarrow> open S"
   obtains \<F>' where "\<F>' \<subseteq> \<F>" "countable \<F>'" "\<Union>\<F>' = \<Union>\<F>"
 proof -
   obtain \<B> :: "'a set set"
     where "countable \<B>" "\<And>C. C \<in> \<B> \<Longrightarrow> open C"
       and \<B>: "\<And>S. open S \<Longrightarrow> \<exists>U. U \<subseteq> \<B> \<and> S = \<Union>U"
     using univ_second_countable by blast
   define \<D> where "\<D> \<equiv> {S. S \<in> \<B> \<and> (\<exists>U. U \<in> \<F> \<and> S \<subseteq> U)}"
   have "countable \<D>"
     apply (rule countable_subset [OF _ \<open>countable \<B>\<close>])
     apply (force simp: \<D>_def)
     done
   have "\<And>S. \<exists>U. S \<in> \<D> \<longrightarrow> U \<in> \<F> \<and> S \<subseteq> U"
     by (simp add: \<D>_def)
   then obtain G where G: "\<And>S. S \<in> \<D> \<longrightarrow> G S \<in> \<F> \<and> S \<subseteq> G S"
     by metis
   have "\<Union>\<F> \<subseteq> \<Union>\<D>"
     unfolding \<D>_def by (blast dest: \<F> \<B>)
   moreover have "\<Union>\<D> \<subseteq> \<Union>\<F>"
     using \<D>_def by blast
   ultimately have eq1: "\<Union>\<F> = \<Union>\<D>" ..
   have eq2: "\<Union>\<D> = \<Union> (G ` \<D>)"
     using G eq1 by auto
   show ?thesis
     apply (rule_tac \<F>' = "G ` \<D>" in that)
     using G \<open>countable \<D>\<close>
     by (auto simp: eq1 eq2)
 qed
 
 lemma countable_disjoint_open_subsets:
   fixes \<F> :: "'a::second_countable_topology set set"
   assumes "\<And>S. S \<in> \<F> \<Longrightarrow> open S" and pw: "pairwise disjnt \<F>"
     shows "countable \<F>"
 proof -
   obtain \<F>' where "\<F>' \<subseteq> \<F>" "countable \<F>'" "\<Union>\<F>' = \<Union>\<F>"
     by (meson assms Lindelof)
   with pw have "\<F> \<subseteq> insert {} \<F>'"
     by (fastforce simp add: pairwise_def disjnt_iff)
   then show ?thesis
     by (simp add: \<open>countable \<F>'\<close> countable_subset)
 qed
 
 sublocale second_countable_topology <
   countable_basis "open" "SOME B. countable B \<and> topological_basis B"
   using someI_ex[OF ex_countable_basis]
   by unfold_locales safe
 
 
 instance prod :: (second_countable_topology, second_countable_topology) second_countable_topology
 proof
   obtain A :: "'a set set" where "countable A" "topological_basis A"
     using ex_countable_basis by auto
   moreover
   obtain B :: "'b set set" where "countable B" "topological_basis B"
     using ex_countable_basis by auto
   ultimately show "\<exists>B::('a \<times> 'b) set set. countable B \<and> open = generate_topology B"
     by (auto intro!: exI[of _ "(\<lambda>(a, b). a \<times> b) ` (A \<times> B)"] topological_basis_prod
       topological_basis_imp_subbasis)
 qed
 
 instance second_countable_topology \<subseteq> first_countable_topology
 proof
   fix x :: 'a
   define B :: "'a set set" where "B = (SOME B. countable B \<and> topological_basis B)"
   then have B: "countable B" "topological_basis B"
     using countable_basis is_basis
     by (auto simp: countable_basis is_basis)
   then show "\<exists>A::nat \<Rightarrow> 'a set.
     (\<forall>i. x \<in> A i \<and> open (A i)) \<and> (\<forall>S. open S \<and> x \<in> S \<longrightarrow> (\<exists>i. A i \<subseteq> S))"
     by (intro first_countableI[of "{b\<in>B. x \<in> b}"])
        (fastforce simp: topological_space_class.topological_basis_def)+
 qed
 
 instance nat :: second_countable_topology
 proof
   show "\<exists>B::nat set set. countable B \<and> open = generate_topology B"
     by (intro exI[of _ "range lessThan \<union> range greaterThan"]) (auto simp: open_nat_def)
 qed
 
 lemma countable_separating_set_linorder1:
   shows "\<exists>B::('a::{linorder_topology, second_countable_topology} set). countable B \<and> (\<forall>x y. x < y \<longrightarrow> (\<exists>b \<in> B. x < b \<and> b \<le> y))"
 proof -
   obtain A::"'a set set" where "countable A" "topological_basis A" using ex_countable_basis by auto
   define B1 where "B1 = {(LEAST x. x \<in> U)| U. U \<in> A}"
   then have "countable B1" using \<open>countable A\<close> by (simp add: Setcompr_eq_image)
   define B2 where "B2 = {(SOME x. x \<in> U)| U. U \<in> A}"
   then have "countable B2" using \<open>countable A\<close> by (simp add: Setcompr_eq_image)
   have "\<exists>b \<in> B1 \<union> B2. x < b \<and> b \<le> y" if "x < y" for x y
   proof (cases)
     assume "\<exists>z. x < z \<and> z < y"
     then obtain z where z: "x < z \<and> z < y" by auto
     define U where "U = {x<..<y}"
     then have "open U" by simp
     moreover have "z \<in> U" using z U_def by simp
     ultimately obtain V where "V \<in> A" "z \<in> V" "V \<subseteq> U" using topological_basisE[OF \<open>topological_basis A\<close>] by auto
     define w where "w = (SOME x. x \<in> V)"
     then have "w \<in> V" using \<open>z \<in> V\<close> by (metis someI2)
     then have "x < w \<and> w \<le> y" using \<open>w \<in> V\<close> \<open>V \<subseteq> U\<close> U_def by fastforce
     moreover have "w \<in> B1 \<union> B2" using w_def B2_def \<open>V \<in> A\<close> by auto
     ultimately show ?thesis by auto
   next
     assume "\<not>(\<exists>z. x < z \<and> z < y)"
     then have *: "\<And>z. z > x \<Longrightarrow> z \<ge> y" by auto
     define U where "U = {x<..}"
     then have "open U" by simp
     moreover have "y \<in> U" using \<open>x < y\<close> U_def by simp
     ultimately obtain "V" where "V \<in> A" "y \<in> V" "V \<subseteq> U" using topological_basisE[OF \<open>topological_basis A\<close>] by auto
     have "U = {y..}" unfolding U_def using * \<open>x < y\<close> by auto
     then have "V \<subseteq> {y..}" using \<open>V \<subseteq> U\<close> by simp
     then have "(LEAST w. w \<in> V) = y" using \<open>y \<in> V\<close> by (meson Least_equality atLeast_iff subsetCE)
     then have "y \<in> B1 \<union> B2" using \<open>V \<in> A\<close> B1_def by auto
     moreover have "x < y \<and> y \<le> y" using \<open>x < y\<close> by simp
     ultimately show ?thesis by auto
   qed
   moreover have "countable (B1 \<union> B2)" using \<open>countable B1\<close> \<open>countable B2\<close> by simp
   ultimately show ?thesis by auto
 qed
 
 lemma countable_separating_set_linorder2:
   shows "\<exists>B::('a::{linorder_topology, second_countable_topology} set). countable B \<and> (\<forall>x y. x < y \<longrightarrow> (\<exists>b \<in> B. x \<le> b \<and> b < y))"
 proof -
   obtain A::"'a set set" where "countable A" "topological_basis A" using ex_countable_basis by auto
   define B1 where "B1 = {(GREATEST x. x \<in> U) | U. U \<in> A}"
   then have "countable B1" using \<open>countable A\<close> by (simp add: Setcompr_eq_image)
   define B2 where "B2 = {(SOME x. x \<in> U)| U. U \<in> A}"
   then have "countable B2" using \<open>countable A\<close> by (simp add: Setcompr_eq_image)
   have "\<exists>b \<in> B1 \<union> B2. x \<le> b \<and> b < y" if "x < y" for x y
   proof (cases)
     assume "\<exists>z. x < z \<and> z < y"
     then obtain z where z: "x < z \<and> z < y" by auto
     define U where "U = {x<..<y}"
     then have "open U" by simp
     moreover have "z \<in> U" using z U_def by simp
     ultimately obtain "V" where "V \<in> A" "z \<in> V" "V \<subseteq> U" using topological_basisE[OF \<open>topological_basis A\<close>] by auto
     define w where "w = (SOME x. x \<in> V)"
     then have "w \<in> V" using \<open>z \<in> V\<close> by (metis someI2)
     then have "x \<le> w \<and> w < y" using \<open>w \<in> V\<close> \<open>V \<subseteq> U\<close> U_def by fastforce
     moreover have "w \<in> B1 \<union> B2" using w_def B2_def \<open>V \<in> A\<close> by auto
     ultimately show ?thesis by auto
   next
     assume "\<not>(\<exists>z. x < z \<and> z < y)"
     then have *: "\<And>z. z < y \<Longrightarrow> z \<le> x" using leI by blast
     define U where "U = {..<y}"
     then have "open U" by simp
     moreover have "x \<in> U" using \<open>x < y\<close> U_def by simp
     ultimately obtain "V" where "V \<in> A" "x \<in> V" "V \<subseteq> U" using topological_basisE[OF \<open>topological_basis A\<close>] by auto
     have "U = {..x}" unfolding U_def using * \<open>x < y\<close> by auto
     then have "V \<subseteq> {..x}" using \<open>V \<subseteq> U\<close> by simp
     then have "(GREATEST x. x \<in> V) = x" using \<open>x \<in> V\<close> by (meson Greatest_equality atMost_iff subsetCE)
     then have "x \<in> B1 \<union> B2" using \<open>V \<in> A\<close> B1_def by auto
     moreover have "x \<le> x \<and> x < y" using \<open>x < y\<close> by simp
     ultimately show ?thesis by auto
   qed
   moreover have "countable (B1 \<union> B2)" using \<open>countable B1\<close> \<open>countable B2\<close> by simp
   ultimately show ?thesis by auto
 qed
 
 lemma countable_separating_set_dense_linorder:
   shows "\<exists>B::('a::{linorder_topology, dense_linorder, second_countable_topology} set). countable B \<and> (\<forall>x y. x < y \<longrightarrow> (\<exists>b \<in> B. x < b \<and> b < y))"
 proof -
   obtain B::"'a set" where B: "countable B" "\<And>x y. x < y \<Longrightarrow> (\<exists>b \<in> B. x < b \<and> b \<le> y)"
     using countable_separating_set_linorder1 by auto
   have "\<exists>b \<in> B. x < b \<and> b < y" if "x < y" for x y
   proof -
     obtain z where "x < z" "z < y" using \<open>x < y\<close> dense by blast
     then obtain b where "b \<in> B" "x < b \<and> b \<le> z" using B(2) by auto
     then have "x < b \<and> b < y" using \<open>z < y\<close> by auto
     then show ?thesis using \<open>b \<in> B\<close> by auto
   qed
   then show ?thesis using B(1) by auto
 qed
 
 
 subsection \<open>Polish spaces\<close>
 
 text \<open>Textbooks define Polish spaces as completely metrizable.
   We assume the topology to be complete for a given metric.\<close>
 
 class polish_space = complete_space + second_countable_topology
 
 
 subsection \<open>Limit Points\<close>
 
 definition\<^marker>\<open>tag important\<close> (in topological_space) islimpt:: "'a \<Rightarrow> 'a set \<Rightarrow> bool"  (infixr "islimpt" 60)
   where "x islimpt S \<longleftrightarrow> (\<forall>T. x\<in>T \<longrightarrow> open T \<longrightarrow> (\<exists>y\<in>S. y\<in>T \<and> y\<noteq>x))"
 
 lemma islimptI:
   assumes "\<And>T. x \<in> T \<Longrightarrow> open T \<Longrightarrow> \<exists>y\<in>S. y \<in> T \<and> y \<noteq> x"
   shows "x islimpt S"
   using assms unfolding islimpt_def by auto
 
 lemma islimptE:
   assumes "x islimpt S" and "x \<in> T" and "open T"
   obtains y where "y \<in> S" and "y \<in> T" and "y \<noteq> x"
   using assms unfolding islimpt_def by auto
 
 lemma islimpt_iff_eventually: "x islimpt S \<longleftrightarrow> \<not> eventually (\<lambda>y. y \<notin> S) (at x)"
   unfolding islimpt_def eventually_at_topological by auto
 
 lemma islimpt_subset: "x islimpt S \<Longrightarrow> S \<subseteq> T \<Longrightarrow> x islimpt T"
   unfolding islimpt_def by fast
 
 lemma islimpt_UNIV_iff: "x islimpt UNIV \<longleftrightarrow> \<not> open {x}"
   unfolding islimpt_def by (safe, fast, case_tac "T = {x}", fast, fast)
 
 lemma islimpt_punctured: "x islimpt S = x islimpt (S-{x})"
   unfolding islimpt_def by blast
 
 text \<open>A perfect space has no isolated points.\<close>
 
 lemma islimpt_UNIV [simp, intro]: "x islimpt UNIV"
   for x :: "'a::perfect_space"
   unfolding islimpt_UNIV_iff by (rule not_open_singleton)
 
 lemma closed_limpt: "closed S \<longleftrightarrow> (\<forall>x. x islimpt S \<longrightarrow> x \<in> S)"
   unfolding closed_def
   apply (subst open_subopen)
   apply (simp add: islimpt_def subset_eq)
   apply (metis ComplE ComplI)
   done
 
 lemma islimpt_EMPTY[simp]: "\<not> x islimpt {}"
   by (auto simp: islimpt_def)
 
 lemma islimpt_Un: "x islimpt (S \<union> T) \<longleftrightarrow> x islimpt S \<or> x islimpt T"
   by (simp add: islimpt_iff_eventually eventually_conj_iff)
 
 
 lemma islimpt_insert:
   fixes x :: "'a::t1_space"
   shows "x islimpt (insert a s) \<longleftrightarrow> x islimpt s"
 proof
   assume *: "x islimpt (insert a s)"
   show "x islimpt s"
   proof (rule islimptI)
     fix t
     assume t: "x \<in> t" "open t"
     show "\<exists>y\<in>s. y \<in> t \<and> y \<noteq> x"
     proof (cases "x = a")
       case True
       obtain y where "y \<in> insert a s" "y \<in> t" "y \<noteq> x"
         using * t by (rule islimptE)
       with \<open>x = a\<close> show ?thesis by auto
     next
       case False
       with t have t': "x \<in> t - {a}" "open (t - {a})"
         by (simp_all add: open_Diff)
       obtain y where "y \<in> insert a s" "y \<in> t - {a}" "y \<noteq> x"
         using * t' by (rule islimptE)
       then show ?thesis by auto
     qed
   qed
 next
   assume "x islimpt s"
   then show "x islimpt (insert a s)"
     by (rule islimpt_subset) auto
 qed
 
 lemma islimpt_finite:
   fixes x :: "'a::t1_space"
   shows "finite s \<Longrightarrow> \<not> x islimpt s"
   by (induct set: finite) (simp_all add: islimpt_insert)
 
 lemma islimpt_Un_finite:
   fixes x :: "'a::t1_space"
   shows "finite s \<Longrightarrow> x islimpt (s \<union> t) \<longleftrightarrow> x islimpt t"
   by (simp add: islimpt_Un islimpt_finite)
 
 lemma islimpt_eq_acc_point:
   fixes l :: "'a :: t1_space"
   shows "l islimpt S \<longleftrightarrow> (\<forall>U. l\<in>U \<longrightarrow> open U \<longrightarrow> infinite (U \<inter> S))"
 proof (safe intro!: islimptI)
   fix U
   assume "l islimpt S" "l \<in> U" "open U" "finite (U \<inter> S)"
   then have "l islimpt S" "l \<in> (U - (U \<inter> S - {l}))" "open (U - (U \<inter> S - {l}))"
     by (auto intro: finite_imp_closed)
   then show False
     by (rule islimptE) auto
 next
   fix T
   assume *: "\<forall>U. l\<in>U \<longrightarrow> open U \<longrightarrow> infinite (U \<inter> S)" "l \<in> T" "open T"
   then have "infinite (T \<inter> S - {l})"
     by auto
   then have "\<exists>x. x \<in> (T \<inter> S - {l})"
     unfolding ex_in_conv by (intro notI) simp
   then show "\<exists>y\<in>S. y \<in> T \<and> y \<noteq> l"
     by auto
 qed
 
 lemma acc_point_range_imp_convergent_subsequence:
   fixes l :: "'a :: first_countable_topology"
   assumes l: "\<forall>U. l\<in>U \<longrightarrow> open U \<longrightarrow> infinite (U \<inter> range f)"
   shows "\<exists>r::nat\<Rightarrow>nat. strict_mono r \<and> (f \<circ> r) \<longlonglongrightarrow> l"
 proof -
   from countable_basis_at_decseq[of l]
   obtain A where A:
       "\<And>i. open (A i)"
       "\<And>i. l \<in> A i"
       "\<And>S. open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>i. A i \<subseteq> S) sequentially"
     by blast
   define s where "s n i = (SOME j. i < j \<and> f j \<in> A (Suc n))" for n i
   {
     fix n i
     have "infinite (A (Suc n) \<inter> range f - f`{.. i})"
       using l A by auto
     then have "\<exists>x. x \<in> A (Suc n) \<inter> range f - f`{.. i}"
       unfolding ex_in_conv by (intro notI) simp
     then have "\<exists>j. f j \<in> A (Suc n) \<and> j \<notin> {.. i}"
       by auto
     then have "\<exists>a. i < a \<and> f a \<in> A (Suc n)"
       by (auto simp: not_le)
     then have "i < s n i" "f (s n i) \<in> A (Suc n)"
       unfolding s_def by (auto intro: someI2_ex)
   }
   note s = this
   define r where "r = rec_nat (s 0 0) s"
   have "strict_mono r"
     by (auto simp: r_def s strict_mono_Suc_iff)
   moreover
   have "(\<lambda>n. f (r n)) \<longlonglongrightarrow> l"
   proof (rule topological_tendstoI)
     fix S
     assume "open S" "l \<in> S"
     with A(3) have "eventually (\<lambda>i. A i \<subseteq> S) sequentially"
       by auto
     moreover
     {
       fix i
       assume "Suc 0 \<le> i"
       then have "f (r i) \<in> A i"
         by (cases i) (simp_all add: r_def s)
     }
     then have "eventually (\<lambda>i. f (r i) \<in> A i) sequentially"
       by (auto simp: eventually_sequentially)
     ultimately show "eventually (\<lambda>i. f (r i) \<in> S) sequentially"
       by eventually_elim auto
   qed
   ultimately show "\<exists>r::nat\<Rightarrow>nat. strict_mono r \<and> (f \<circ> r) \<longlonglongrightarrow> l"
     by (auto simp: convergent_def comp_def)
 qed
 
 lemma islimpt_range_imp_convergent_subsequence:
   fixes l :: "'a :: {t1_space, first_countable_topology}"
   assumes l: "l islimpt (range f)"
   shows "\<exists>r::nat\<Rightarrow>nat. strict_mono r \<and> (f \<circ> r) \<longlonglongrightarrow> l"
   using l unfolding islimpt_eq_acc_point
   by (rule acc_point_range_imp_convergent_subsequence)
 
 lemma sequence_unique_limpt:
   fixes f :: "nat \<Rightarrow> 'a::t2_space"
   assumes "(f \<longlongrightarrow> l) sequentially"
     and "l' islimpt (range f)"
   shows "l' = l"
 proof (rule ccontr)
   assume "l' \<noteq> l"
   obtain s t where "open s" "open t" "l' \<in> s" "l \<in> t" "s \<inter> t = {}"
     using hausdorff [OF \<open>l' \<noteq> l\<close>] by auto
   have "eventually (\<lambda>n. f n \<in> t) sequentially"
     using assms(1) \<open>open t\<close> \<open>l \<in> t\<close> by (rule topological_tendstoD)
   then obtain N where "\<forall>n\<ge>N. f n \<in> t"
     unfolding eventually_sequentially by auto
 
   have "UNIV = {..<N} \<union> {N..}"
     by auto
   then have "l' islimpt (f ` ({..<N} \<union> {N..}))"
     using assms(2) by simp
   then have "l' islimpt (f ` {..<N} \<union> f ` {N..})"
     by (simp add: image_Un)
   then have "l' islimpt (f ` {N..})"
     by (simp add: islimpt_Un_finite)
   then obtain y where "y \<in> f ` {N..}" "y \<in> s" "y \<noteq> l'"
     using \<open>l' \<in> s\<close> \<open>open s\<close> by (rule islimptE)
   then obtain n where "N \<le> n" "f n \<in> s" "f n \<noteq> l'"
     by auto
   with \<open>\<forall>n\<ge>N. f n \<in> t\<close> have "f n \<in> s \<inter> t"
     by simp
   with \<open>s \<inter> t = {}\<close> show False
     by simp
 qed
 
 
 subsection \<open>Interior of a Set\<close>
 
 definition\<^marker>\<open>tag important\<close> interior :: "('a::topological_space) set \<Rightarrow> 'a set" where
 "interior S = \<Union>{T. open T \<and> T \<subseteq> S}"
 
 lemma interiorI [intro?]:
   assumes "open T" and "x \<in> T" and "T \<subseteq> S"
   shows "x \<in> interior S"
   using assms unfolding interior_def by fast
 
 lemma interiorE [elim?]:
   assumes "x \<in> interior S"
   obtains T where "open T" and "x \<in> T" and "T \<subseteq> S"
   using assms unfolding interior_def by fast
 
 lemma open_interior [simp, intro]: "open (interior S)"
   by (simp add: interior_def open_Union)
 
 lemma interior_subset: "interior S \<subseteq> S"
   by (auto simp: interior_def)
 
 lemma interior_maximal: "T \<subseteq> S \<Longrightarrow> open T \<Longrightarrow> T \<subseteq> interior S"
   by (auto simp: interior_def)
 
 lemma interior_open: "open S \<Longrightarrow> interior S = S"
   by (intro equalityI interior_subset interior_maximal subset_refl)
 
 lemma interior_eq: "interior S = S \<longleftrightarrow> open S"
   by (metis open_interior interior_open)
 
 lemma open_subset_interior: "open S \<Longrightarrow> S \<subseteq> interior T \<longleftrightarrow> S \<subseteq> T"
   by (metis interior_maximal interior_subset subset_trans)
 
 lemma interior_empty [simp]: "interior {} = {}"
   using open_empty by (rule interior_open)
 
 lemma interior_UNIV [simp]: "interior UNIV = UNIV"
   using open_UNIV by (rule interior_open)
 
 lemma interior_interior [simp]: "interior (interior S) = interior S"
   using open_interior by (rule interior_open)
 
 lemma interior_mono: "S \<subseteq> T \<Longrightarrow> interior S \<subseteq> interior T"
   by (auto simp: interior_def)
 
 lemma interior_unique:
   assumes "T \<subseteq> S" and "open T"
   assumes "\<And>T'. T' \<subseteq> S \<Longrightarrow> open T' \<Longrightarrow> T' \<subseteq> T"
   shows "interior S = T"
   by (intro equalityI assms interior_subset open_interior interior_maximal)
 
 lemma interior_singleton [simp]: "interior {a} = {}"
   for a :: "'a::perfect_space"
   apply (rule interior_unique, simp_all)
   using not_open_singleton subset_singletonD
   apply fastforce
   done
 
 lemma interior_Int [simp]: "interior (S \<inter> T) = interior S \<inter> interior T"
   by (intro equalityI Int_mono Int_greatest interior_mono Int_lower1
     Int_lower2 interior_maximal interior_subset open_Int open_interior)
 
 lemma eventually_nhds_in_nhd: "x \<in> interior s \<Longrightarrow> eventually (\<lambda>y. y \<in> s) (nhds x)"
   using interior_subset[of s] by (subst eventually_nhds) blast
 
 lemma interior_limit_point [intro]:
   fixes x :: "'a::perfect_space"
   assumes x: "x \<in> interior S"
   shows "x islimpt S"
   using x islimpt_UNIV [of x]
   unfolding interior_def islimpt_def
   apply (clarsimp, rename_tac T T')
   apply (drule_tac x="T \<inter> T'" in spec)
   apply (auto simp: open_Int)
   done
 
 lemma interior_closed_Un_empty_interior:
   assumes cS: "closed S"
     and iT: "interior T = {}"
   shows "interior (S \<union> T) = interior S"
 proof
   show "interior S \<subseteq> interior (S \<union> T)"
     by (rule interior_mono) (rule Un_upper1)
   show "interior (S \<union> T) \<subseteq> interior S"
   proof
     fix x
     assume "x \<in> interior (S \<union> T)"
     then obtain R where "open R" "x \<in> R" "R \<subseteq> S \<union> T" ..
     show "x \<in> interior S"
     proof (rule ccontr)
       assume "x \<notin> interior S"
       with \<open>x \<in> R\<close> \<open>open R\<close> obtain y where "y \<in> R - S"
         unfolding interior_def by fast
       from \<open>open R\<close> \<open>closed S\<close> have "open (R - S)"
         by (rule open_Diff)
       from \<open>R \<subseteq> S \<union> T\<close> have "R - S \<subseteq> T"
         by fast
       from \<open>y \<in> R - S\<close> \<open>open (R - S)\<close> \<open>R - S \<subseteq> T\<close> \<open>interior T = {}\<close> show False
         unfolding interior_def by fast
     qed
   qed
 qed
 
 lemma interior_Times: "interior (A \<times> B) = interior A \<times> interior B"
 proof (rule interior_unique)
   show "interior A \<times> interior B \<subseteq> A \<times> B"
     by (intro Sigma_mono interior_subset)
   show "open (interior A \<times> interior B)"
     by (intro open_Times open_interior)
   fix T
   assume "T \<subseteq> A \<times> B" and "open T"
   then show "T \<subseteq> interior A \<times> interior B"
   proof safe
     fix x y
     assume "(x, y) \<in> T"
     then obtain C D where "open C" "open D" "C \<times> D \<subseteq> T" "x \<in> C" "y \<in> D"
       using \<open>open T\<close> unfolding open_prod_def by fast
     then have "open C" "open D" "C \<subseteq> A" "D \<subseteq> B" "x \<in> C" "y \<in> D"
       using \<open>T \<subseteq> A \<times> B\<close> by auto
     then show "x \<in> interior A" and "y \<in> interior B"
       by (auto intro: interiorI)
   qed
 qed
 
 lemma interior_Ici:
   fixes x :: "'a :: {dense_linorder,linorder_topology}"
   assumes "b < x"
   shows "interior {x ..} = {x <..}"
 proof (rule interior_unique)
   fix T
   assume "T \<subseteq> {x ..}" "open T"
   moreover have "x \<notin> T"
   proof
     assume "x \<in> T"
     obtain y where "y < x" "{y <.. x} \<subseteq> T"
       using open_left[OF \<open>open T\<close> \<open>x \<in> T\<close> \<open>b < x\<close>] by auto
     with dense[OF \<open>y < x\<close>] obtain z where "z \<in> T" "z < x"
       by (auto simp: subset_eq Ball_def)
     with \<open>T \<subseteq> {x ..}\<close> show False by auto
   qed
   ultimately show "T \<subseteq> {x <..}"
     by (auto simp: subset_eq less_le)
 qed auto
 
 lemma interior_Iic:
   fixes x :: "'a ::{dense_linorder,linorder_topology}"
   assumes "x < b"
   shows "interior {.. x} = {..< x}"
 proof (rule interior_unique)
   fix T
   assume "T \<subseteq> {.. x}" "open T"
   moreover have "x \<notin> T"
   proof
     assume "x \<in> T"
     obtain y where "x < y" "{x ..< y} \<subseteq> T"
       using open_right[OF \<open>open T\<close> \<open>x \<in> T\<close> \<open>x < b\<close>] by auto
     with dense[OF \<open>x < y\<close>] obtain z where "z \<in> T" "x < z"
       by (auto simp: subset_eq Ball_def less_le)
     with \<open>T \<subseteq> {.. x}\<close> show False by auto
   qed
   ultimately show "T \<subseteq> {..< x}"
     by (auto simp: subset_eq less_le)
 qed auto
 
 lemma countable_disjoint_nonempty_interior_subsets:
   fixes \<F> :: "'a::second_countable_topology set set"
   assumes pw: "pairwise disjnt \<F>" and int: "\<And>S. \<lbrakk>S \<in> \<F>; interior S = {}\<rbrakk> \<Longrightarrow> S = {}"
   shows "countable \<F>"
 proof (rule countable_image_inj_on)
   have "disjoint (interior ` \<F>)"
     using pw by (simp add: disjoint_image_subset interior_subset)
   then show "countable (interior ` \<F>)"
     by (auto intro: countable_disjoint_open_subsets)
   show "inj_on interior \<F>"
     using pw apply (clarsimp simp: inj_on_def pairwise_def)
     apply (metis disjnt_def disjnt_subset1 inf.orderE int interior_subset)
     done
 qed
 
 subsection \<open>Closure of a Set\<close>
 
 definition\<^marker>\<open>tag important\<close> closure :: "('a::topological_space) set \<Rightarrow> 'a set" where
 "closure S = S \<union> {x . x islimpt S}"
 
 lemma interior_closure: "interior S = - (closure (- S))"
   by (auto simp: interior_def closure_def islimpt_def)
 
 lemma closure_interior: "closure S = - interior (- S)"
   by (simp add: interior_closure)
 
 lemma closed_closure[simp, intro]: "closed (closure S)"
   by (simp add: closure_interior closed_Compl)
 
 lemma closure_subset: "S \<subseteq> closure S"
   by (simp add: closure_def)
 
 lemma closure_hull: "closure S = closed hull S"
   by (auto simp: hull_def closure_interior interior_def)
 
 lemma closure_eq: "closure S = S \<longleftrightarrow> closed S"
   unfolding closure_hull using closed_Inter by (rule hull_eq)
 
 lemma closure_closed [simp]: "closed S \<Longrightarrow> closure S = S"
   by (simp only: closure_eq)
 
 lemma closure_closure [simp]: "closure (closure S) = closure S"
   unfolding closure_hull by (rule hull_hull)
 
 lemma closure_mono: "S \<subseteq> T \<Longrightarrow> closure S \<subseteq> closure T"
   unfolding closure_hull by (rule hull_mono)
 
 lemma closure_minimal: "S \<subseteq> T \<Longrightarrow> closed T \<Longrightarrow> closure S \<subseteq> T"
   unfolding closure_hull by (rule hull_minimal)
 
 lemma closure_unique:
   assumes "S \<subseteq> T"
     and "closed T"
     and "\<And>T'. S \<subseteq> T' \<Longrightarrow> closed T' \<Longrightarrow> T \<subseteq> T'"
   shows "closure S = T"
   using assms unfolding closure_hull by (rule hull_unique)
 
 lemma closure_empty [simp]: "closure {} = {}"
   using closed_empty by (rule closure_closed)
 
 lemma closure_UNIV [simp]: "closure UNIV = UNIV"
   using closed_UNIV by (rule closure_closed)
 
 lemma closure_Un [simp]: "closure (S \<union> T) = closure S \<union> closure T"
   by (simp add: closure_interior)
 
 lemma closure_eq_empty [iff]: "closure S = {} \<longleftrightarrow> S = {}"
   using closure_empty closure_subset[of S] by blast
 
 lemma closure_subset_eq: "closure S \<subseteq> S \<longleftrightarrow> closed S"
   using closure_eq[of S] closure_subset[of S] by simp
 
 lemma open_Int_closure_eq_empty: "open S \<Longrightarrow> (S \<inter> closure T) = {} \<longleftrightarrow> S \<inter> T = {}"
   using open_subset_interior[of S "- T"]
   using interior_subset[of "- T"]
   by (auto simp: closure_interior)
 
 lemma open_Int_closure_subset: "open S \<Longrightarrow> S \<inter> closure T \<subseteq> closure (S \<inter> T)"
 proof
   fix x
   assume *: "open S" "x \<in> S \<inter> closure T"
   have "x islimpt (S \<inter> T)" if **: "x islimpt T"
   proof (rule islimptI)
     fix A
     assume "x \<in> A" "open A"
     with * have "x \<in> A \<inter> S" "open (A \<inter> S)"
       by (simp_all add: open_Int)
     with ** obtain y where "y \<in> T" "y \<in> A \<inter> S" "y \<noteq> x"
       by (rule islimptE)
     then have "y \<in> S \<inter> T" "y \<in> A \<and> y \<noteq> x"
       by simp_all
     then show "\<exists>y\<in>(S \<inter> T). y \<in> A \<and> y \<noteq> x" ..
   qed
   with * show "x \<in> closure (S \<inter> T)"
     unfolding closure_def by blast
 qed
 
 lemma closure_complement: "closure (- S) = - interior S"
   by (simp add: closure_interior)
 
 lemma interior_complement: "interior (- S) = - closure S"
   by (simp add: closure_interior)
 
 lemma interior_diff: "interior(S - T) = interior S - closure T"
   by (simp add: Diff_eq interior_complement)
 
 lemma closure_Times: "closure (A \<times> B) = closure A \<times> closure B"
 proof (rule closure_unique)
   show "A \<times> B \<subseteq> closure A \<times> closure B"
     by (intro Sigma_mono closure_subset)
   show "closed (closure A \<times> closure B)"
     by (intro closed_Times closed_closure)
   fix T
   assume "A \<times> B \<subseteq> T" and "closed T"
   then show "closure A \<times> closure B \<subseteq> T"
     apply (simp add: closed_def open_prod_def, clarify)
     apply (rule ccontr)
     apply (drule_tac x="(a, b)" in bspec, simp, clarify, rename_tac C D)
     apply (simp add: closure_interior interior_def)
     apply (drule_tac x=C in spec)
     apply (drule_tac x=D in spec, auto)
     done
 qed
 
+lemma closure_open_Int_superset:
+  assumes "open S" "S \<subseteq> closure T"
+  shows "closure(S \<inter> T) = closure S"
+proof -
+  have "closure S \<subseteq> closure(S \<inter> T)"
+    by (metis assms closed_closure closure_minimal inf.orderE open_Int_closure_subset)
+  then show ?thesis
+    by (simp add: closure_mono dual_order.antisym)
+qed
+
+lemma closure_Int: "closure (\<Inter>I) \<le> \<Inter>{closure S |S. S \<in> I}"
+proof -
+  {
+    fix y
+    assume "y \<in> \<Inter>I"
+    then have y: "\<forall>S \<in> I. y \<in> S" by auto
+    {
+      fix S
+      assume "S \<in> I"
+      then have "y \<in> closure S"
+        using closure_subset y by auto
+    }
+    then have "y \<in> \<Inter>{closure S |S. S \<in> I}"
+      by auto
+  }
+  then have "\<Inter>I \<subseteq> \<Inter>{closure S |S. S \<in> I}"
+    by auto
+  moreover have "closed (\<Inter>{closure S |S. S \<in> I})"
+    unfolding closed_Inter closed_closure by auto
+  ultimately show ?thesis using closure_hull[of "\<Inter>I"]
+    hull_minimal[of "\<Inter>I" "\<Inter>{closure S |S. S \<in> I}" "closed"] by auto
+qed
+
 lemma islimpt_in_closure: "(x islimpt S) = (x\<in>closure(S-{x}))"
   unfolding closure_def using islimpt_punctured by blast
 
 lemma connected_imp_connected_closure: "connected S \<Longrightarrow> connected (closure S)"
   by (rule connectedI) (meson closure_subset open_Int open_Int_closure_eq_empty subset_trans connectedD)
 
 lemma bdd_below_closure:
   fixes A :: "real set"
   assumes "bdd_below A"
   shows "bdd_below (closure A)"
 proof -
   from assms obtain m where "\<And>x. x \<in> A \<Longrightarrow> m \<le> x"
     by (auto simp: bdd_below_def)
   then have "A \<subseteq> {m..}" by auto
   then have "closure A \<subseteq> {m..}"
     using closed_real_atLeast by (rule closure_minimal)
   then show ?thesis
     by (auto simp: bdd_below_def)
 qed
 
 
 subsection \<open>Frontier (also known as boundary)\<close>
 
 definition\<^marker>\<open>tag important\<close> frontier :: "('a::topological_space) set \<Rightarrow> 'a set" where
 "frontier S = closure S - interior S"
 
 lemma frontier_closed [iff]: "closed (frontier S)"
   by (simp add: frontier_def closed_Diff)
 
 lemma frontier_closures: "frontier S = closure S \<inter> closure (- S)"
   by (auto simp: frontier_def interior_closure)
 
 lemma frontier_Int: "frontier(S \<inter> T) = closure(S \<inter> T) \<inter> (frontier S \<union> frontier T)"
 proof -
   have "closure (S \<inter> T) \<subseteq> closure S" "closure (S \<inter> T) \<subseteq> closure T"
     by (simp_all add: closure_mono)
   then show ?thesis
     by (auto simp: frontier_closures)
 qed
 
 lemma frontier_Int_subset: "frontier(S \<inter> T) \<subseteq> frontier S \<union> frontier T"
   by (auto simp: frontier_Int)
 
 lemma frontier_Int_closed:
   assumes "closed S" "closed T"
   shows "frontier(S \<inter> T) = (frontier S \<inter> T) \<union> (S \<inter> frontier T)"
 proof -
   have "closure (S \<inter> T) = T \<inter> S"
     using assms by (simp add: Int_commute closed_Int)
   moreover have "T \<inter> (closure S \<inter> closure (- S)) = frontier S \<inter> T"
     by (simp add: Int_commute frontier_closures)
   ultimately show ?thesis
     by (simp add: Int_Un_distrib Int_assoc Int_left_commute assms frontier_closures)
 qed
 
 lemma frontier_subset_closed: "closed S \<Longrightarrow> frontier S \<subseteq> S"
   by (metis frontier_def closure_closed Diff_subset)
 
 lemma frontier_empty [simp]: "frontier {} = {}"
   by (simp add: frontier_def)
 
 lemma frontier_subset_eq: "frontier S \<subseteq> S \<longleftrightarrow> closed S"
 proof -
   {
     assume "frontier S \<subseteq> S"
     then have "closure S \<subseteq> S"
       using interior_subset unfolding frontier_def by auto
     then have "closed S"
       using closure_subset_eq by auto
   }
   then show ?thesis using frontier_subset_closed[of S] ..
 qed
 
 lemma frontier_complement [simp]: "frontier (- S) = frontier S"
   by (auto simp: frontier_def closure_complement interior_complement)
 
 lemma frontier_Un_subset: "frontier(S \<union> T) \<subseteq> frontier S \<union> frontier T"
   by (metis compl_sup frontier_Int_subset frontier_complement)
 
 lemma frontier_disjoint_eq: "frontier S \<inter> S = {} \<longleftrightarrow> open S"
   using frontier_complement frontier_subset_eq[of "- S"]
   unfolding open_closed by auto
 
 lemma frontier_UNIV [simp]: "frontier UNIV = {}"
   using frontier_complement frontier_empty by fastforce
 
 lemma frontier_interiors: "frontier s = - interior(s) - interior(-s)"
   by (simp add: Int_commute frontier_def interior_closure)
 
 lemma frontier_interior_subset: "frontier(interior S) \<subseteq> frontier S"
   by (simp add: Diff_mono frontier_interiors interior_mono interior_subset)
 
 lemma closure_Un_frontier: "closure S = S \<union> frontier S"
 proof -
   have "S \<union> interior S = S"
     using interior_subset by auto
   then show ?thesis
     using closure_subset by (auto simp: frontier_def)
 qed
 
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>Filters and the ``eventually true'' quantifier\<close>
 
 text \<open>Identify Trivial limits, where we can't approach arbitrarily closely.\<close>
 
 lemma trivial_limit_within: "trivial_limit (at a within S) \<longleftrightarrow> \<not> a islimpt S"
 proof
   assume "trivial_limit (at a within S)"
   then show "\<not> a islimpt S"
     unfolding trivial_limit_def
     unfolding eventually_at_topological
     unfolding islimpt_def
     apply (clarsimp simp add: set_eq_iff)
     apply (rename_tac T, rule_tac x=T in exI)
     apply (clarsimp, drule_tac x=y in bspec, simp_all)
     done
 next
   assume "\<not> a islimpt S"
   then show "trivial_limit (at a within S)"
     unfolding trivial_limit_def eventually_at_topological islimpt_def
     by metis
 qed
 
 lemma trivial_limit_at_iff: "trivial_limit (at a) \<longleftrightarrow> \<not> a islimpt UNIV"
   using trivial_limit_within [of a UNIV] by simp
 
 lemma trivial_limit_at: "\<not> trivial_limit (at a)"
   for a :: "'a::perfect_space"
   by (rule at_neq_bot)
 
 lemma not_trivial_limit_within: "\<not> trivial_limit (at x within S) = (x \<in> closure (S - {x}))"
   using islimpt_in_closure by (metis trivial_limit_within)
 
 lemma not_in_closure_trivial_limitI:
   "x \<notin> closure s \<Longrightarrow> trivial_limit (at x within s)"
   using not_trivial_limit_within[of x s]
   by safe (metis Diff_empty Diff_insert0 closure_subset contra_subsetD)
 
 lemma filterlim_at_within_closure_implies_filterlim: "filterlim f l (at x within s)"
   if "x \<in> closure s \<Longrightarrow> filterlim f l (at x within s)"
   by (metis bot.extremum filterlim_filtercomap filterlim_mono not_in_closure_trivial_limitI that)
 
 lemma at_within_eq_bot_iff: "at c within A = bot \<longleftrightarrow> c \<notin> closure (A - {c})"
   using not_trivial_limit_within[of c A] by blast
 
 text \<open>Some property holds "sufficiently close" to the limit point.\<close>
 
 lemma trivial_limit_eventually: "trivial_limit net \<Longrightarrow> eventually P net"
   by simp
 
 lemma trivial_limit_eq: "trivial_limit net \<longleftrightarrow> (\<forall>P. eventually P net)"
   by (simp add: filter_eq_iff)
 
 lemma Lim_topological:
   "(f \<longlongrightarrow> l) net \<longleftrightarrow>
     trivial_limit net \<or> (\<forall>S. open S \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) net)"
   unfolding tendsto_def trivial_limit_eq by auto
 
 lemma eventually_within_Un:
   "eventually P (at x within (s \<union> t)) \<longleftrightarrow>
     eventually P (at x within s) \<and> eventually P (at x within t)"
   unfolding eventually_at_filter
   by (auto elim!: eventually_rev_mp)
 
 lemma Lim_within_union:
  "(f \<longlongrightarrow> l) (at x within (s \<union> t)) \<longleftrightarrow>
   (f \<longlongrightarrow> l) (at x within s) \<and> (f \<longlongrightarrow> l) (at x within t)"
   unfolding tendsto_def
   by (auto simp: eventually_within_Un)
 
 
 subsection \<open>Limits\<close>
 
 text \<open>The expected monotonicity property.\<close>
 
 lemma Lim_Un:
   assumes "(f \<longlongrightarrow> l) (at x within S)" "(f \<longlongrightarrow> l) (at x within T)"
   shows "(f \<longlongrightarrow> l) (at x within (S \<union> T))"
   using assms unfolding at_within_union by (rule filterlim_sup)
 
 lemma Lim_Un_univ:
   "(f \<longlongrightarrow> l) (at x within S) \<Longrightarrow> (f \<longlongrightarrow> l) (at x within T) \<Longrightarrow>
     S \<union> T = UNIV \<Longrightarrow> (f \<longlongrightarrow> l) (at x)"
   by (metis Lim_Un)
 
 text \<open>Interrelations between restricted and unrestricted limits.\<close>
 
 lemma Lim_at_imp_Lim_at_within: "(f \<longlongrightarrow> l) (at x) \<Longrightarrow> (f \<longlongrightarrow> l) (at x within S)"
   by (metis order_refl filterlim_mono subset_UNIV at_le)
 
 lemma eventually_within_interior:
   assumes "x \<in> interior S"
   shows "eventually P (at x within S) \<longleftrightarrow> eventually P (at x)"
   (is "?lhs = ?rhs")
 proof
   from assms obtain T where T: "open T" "x \<in> T" "T \<subseteq> S" ..
   {
     assume ?lhs
     then obtain A where "open A" and "x \<in> A" and "\<forall>y\<in>A. y \<noteq> x \<longrightarrow> y \<in> S \<longrightarrow> P y"
       by (auto simp: eventually_at_topological)
     with T have "open (A \<inter> T)" and "x \<in> A \<inter> T" and "\<forall>y \<in> A \<inter> T. y \<noteq> x \<longrightarrow> P y"
       by auto
     then show ?rhs
       by (auto simp: eventually_at_topological)
   next
     assume ?rhs
     then show ?lhs
       by (auto elim: eventually_mono simp: eventually_at_filter)
   }
 qed
 
 lemma at_within_interior: "x \<in> interior S \<Longrightarrow> at x within S = at x"
   unfolding filter_eq_iff by (intro allI eventually_within_interior)
 
 lemma Lim_within_LIMSEQ:
   fixes a :: "'a::first_countable_topology"
   assumes "\<forall>S. (\<forall>n. S n \<noteq> a \<and> S n \<in> T) \<and> S \<longlonglongrightarrow> a \<longrightarrow> (\<lambda>n. X (S n)) \<longlonglongrightarrow> L"
   shows "(X \<longlongrightarrow> L) (at a within T)"
   using assms unfolding tendsto_def [where l=L]
   by (simp add: sequentially_imp_eventually_within)
 
 lemma Lim_right_bound:
   fixes f :: "'a :: {linorder_topology, conditionally_complete_linorder, no_top} \<Rightarrow>
     'b::{linorder_topology, conditionally_complete_linorder}"
   assumes mono: "\<And>a b. a \<in> I \<Longrightarrow> b \<in> I \<Longrightarrow> x < a \<Longrightarrow> a \<le> b \<Longrightarrow> f a \<le> f b"
     and bnd: "\<And>a. a \<in> I \<Longrightarrow> x < a \<Longrightarrow> K \<le> f a"
   shows "(f \<longlongrightarrow> Inf (f ` ({x<..} \<inter> I))) (at x within ({x<..} \<inter> I))"
 proof (cases "{x<..} \<inter> I = {}")
   case True
   then show ?thesis by simp
 next
   case False
   show ?thesis
   proof (rule order_tendstoI)
     fix a
     assume a: "a < Inf (f ` ({x<..} \<inter> I))"
     {
       fix y
       assume "y \<in> {x<..} \<inter> I"
       with False bnd have "Inf (f ` ({x<..} \<inter> I)) \<le> f y"
         by (auto intro!: cInf_lower bdd_belowI2)
       with a have "a < f y"
         by (blast intro: less_le_trans)
     }
     then show "eventually (\<lambda>x. a < f x) (at x within ({x<..} \<inter> I))"
       by (auto simp: eventually_at_filter intro: exI[of _ 1] zero_less_one)
   next
     fix a
     assume "Inf (f ` ({x<..} \<inter> I)) < a"
     from cInf_lessD[OF _ this] False obtain y where y: "x < y" "y \<in> I" "f y < a"
       by auto
     then have "eventually (\<lambda>x. x \<in> I \<longrightarrow> f x < a) (at_right x)"
       unfolding eventually_at_right[OF \<open>x < y\<close>] by (metis less_imp_le le_less_trans mono)
     then show "eventually (\<lambda>x. f x < a) (at x within ({x<..} \<inter> I))"
       unfolding eventually_at_filter by eventually_elim simp
   qed
 qed
 
 (*could prove directly from islimpt_sequential_inj, but only for metric spaces*)
 lemma islimpt_sequential:
   fixes x :: "'a::first_countable_topology"
   shows "x islimpt S \<longleftrightarrow> (\<exists>f. (\<forall>n::nat. f n \<in> S - {x}) \<and> (f \<longlongrightarrow> x) sequentially)"
     (is "?lhs = ?rhs")
 proof
   assume ?lhs
   from countable_basis_at_decseq[of x] obtain A where A:
       "\<And>i. open (A i)"
       "\<And>i. x \<in> A i"
       "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> eventually (\<lambda>i. A i \<subseteq> S) sequentially"
     by blast
   define f where "f n = (SOME y. y \<in> S \<and> y \<in> A n \<and> x \<noteq> y)" for n
   {
     fix n
     from \<open>?lhs\<close> have "\<exists>y. y \<in> S \<and> y \<in> A n \<and> x \<noteq> y"
       unfolding islimpt_def using A(1,2)[of n] by auto
     then have "f n \<in> S \<and> f n \<in> A n \<and> x \<noteq> f n"
       unfolding f_def by (rule someI_ex)
     then have "f n \<in> S" "f n \<in> A n" "x \<noteq> f n" by auto
   }
   then have "\<forall>n. f n \<in> S - {x}" by auto
   moreover have "(\<lambda>n. f n) \<longlonglongrightarrow> x"
   proof (rule topological_tendstoI)
     fix S
     assume "open S" "x \<in> S"
     from A(3)[OF this] \<open>\<And>n. f n \<in> A n\<close>
     show "eventually (\<lambda>x. f x \<in> S) sequentially"
       by (auto elim!: eventually_mono)
   qed
   ultimately show ?rhs by fast
 next
   assume ?rhs
   then obtain f :: "nat \<Rightarrow> 'a" where f: "\<And>n. f n \<in> S - {x}" and lim: "f \<longlonglongrightarrow> x"
     by auto
   show ?lhs
     unfolding islimpt_def
   proof safe
     fix T
     assume "open T" "x \<in> T"
     from lim[THEN topological_tendstoD, OF this] f
     show "\<exists>y\<in>S. y \<in> T \<and> y \<noteq> x"
       unfolding eventually_sequentially by auto
   qed
 qed
 
 text\<open>These are special for limits out of the same topological space.\<close>
 
 lemma Lim_within_id: "(id \<longlongrightarrow> a) (at a within s)"
   unfolding id_def by (rule tendsto_ident_at)
 
 lemma Lim_at_id: "(id \<longlongrightarrow> a) (at a)"
   unfolding id_def by (rule tendsto_ident_at)
 
 text\<open>It's also sometimes useful to extract the limit point from the filter.\<close>
 
 abbreviation netlimit :: "'a::t2_space filter \<Rightarrow> 'a"
   where "netlimit F \<equiv> Lim F (\<lambda>x. x)"
 
 lemma netlimit_at [simp]:
   fixes a :: "'a::{perfect_space,t2_space}"
   shows "netlimit (at a) = a"
   using Lim_ident_at [of a UNIV] by simp
 
 lemma lim_within_interior:
   "x \<in> interior S \<Longrightarrow> (f \<longlongrightarrow> l) (at x within S) \<longleftrightarrow> (f \<longlongrightarrow> l) (at x)"
   by (metis at_within_interior)
 
 lemma netlimit_within_interior:
   fixes x :: "'a::{t2_space,perfect_space}"
   assumes "x \<in> interior S"
   shows "netlimit (at x within S) = x"
   using assms by (metis at_within_interior netlimit_at)
 
 text\<open>Useful lemmas on closure and set of possible sequential limits.\<close>
 
 lemma closure_sequential:
   fixes l :: "'a::first_countable_topology"
   shows "l \<in> closure S \<longleftrightarrow> (\<exists>x. (\<forall>n. x n \<in> S) \<and> (x \<longlongrightarrow> l) sequentially)"
   (is "?lhs = ?rhs")
 proof
   assume "?lhs"
   moreover
   {
     assume "l \<in> S"
     then have "?rhs" using tendsto_const[of l sequentially] by auto
   }
   moreover
   {
     assume "l islimpt S"
     then have "?rhs" unfolding islimpt_sequential by auto
   }
   ultimately show "?rhs"
     unfolding closure_def by auto
 next
   assume "?rhs"
   then show "?lhs" unfolding closure_def islimpt_sequential by auto
 qed
 
 lemma closed_sequential_limits:
   fixes S :: "'a::first_countable_topology set"
   shows "closed S \<longleftrightarrow> (\<forall>x l. (\<forall>n. x n \<in> S) \<and> (x \<longlongrightarrow> l) sequentially \<longrightarrow> l \<in> S)"
 by (metis closure_sequential closure_subset_eq subset_iff)
 
 lemma tendsto_If_within_closures:
   assumes f: "x \<in> s \<union> (closure s \<inter> closure t) \<Longrightarrow>
       (f \<longlongrightarrow> l x) (at x within s \<union> (closure s \<inter> closure t))"
   assumes g: "x \<in> t \<union> (closure s \<inter> closure t) \<Longrightarrow>
       (g \<longlongrightarrow> l x) (at x within t \<union> (closure s \<inter> closure t))"
   assumes "x \<in> s \<union> t"
   shows "((\<lambda>x. if x \<in> s then f x else g x) \<longlongrightarrow> l x) (at x within s \<union> t)"
 proof -
   have *: "(s \<union> t) \<inter> {x. x \<in> s} = s" "(s \<union> t) \<inter> {x. x \<notin> s} = t - s"
     by auto
   have "(f \<longlongrightarrow> l x) (at x within s)"
     by (rule filterlim_at_within_closure_implies_filterlim)
        (use \<open>x \<in> _\<close> in \<open>auto simp: inf_commute closure_def intro: tendsto_within_subset[OF f]\<close>)
   moreover
   have "(g \<longlongrightarrow> l x) (at x within t - s)"
     by (rule filterlim_at_within_closure_implies_filterlim)
       (use \<open>x \<in> _\<close> in
         \<open>auto intro!: tendsto_within_subset[OF g] simp: closure_def intro: islimpt_subset\<close>)
   ultimately show ?thesis
     by (intro filterlim_at_within_If) (simp_all only: *)
 qed
 
 
 subsection \<open>Compactness\<close>
 
 lemma brouwer_compactness_lemma:
   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
   assumes "compact s"
     and "continuous_on s f"
     and "\<not> (\<exists>x\<in>s. f x = 0)"
   obtains d where "0 < d" and "\<forall>x\<in>s. d \<le> norm (f x)"
 proof (cases "s = {}")
   case True
   show thesis
     by (rule that [of 1]) (auto simp: True)
 next
   case False
   have "continuous_on s (norm \<circ> f)"
     by (rule continuous_intros continuous_on_norm assms(2))+
   with False obtain x where x: "x \<in> s" "\<forall>y\<in>s. (norm \<circ> f) x \<le> (norm \<circ> f) y"
     using continuous_attains_inf[OF assms(1), of "norm \<circ> f"]
     unfolding o_def
     by auto
   have "(norm \<circ> f) x > 0"
     using assms(3) and x(1)
     by auto
   then show ?thesis
     by (rule that) (insert x(2), auto simp: o_def)
 qed
 
 subsubsection \<open>Bolzano-Weierstrass property\<close>
 
 proposition Heine_Borel_imp_Bolzano_Weierstrass:
   assumes "compact s"
     and "infinite t"
     and "t \<subseteq> s"
   shows "\<exists>x \<in> s. x islimpt t"
 proof (rule ccontr)
   assume "\<not> (\<exists>x \<in> s. x islimpt t)"
   then obtain f where f: "\<forall>x\<in>s. x \<in> f x \<and> open (f x) \<and> (\<forall>y\<in>t. y \<in> f x \<longrightarrow> y = x)"
     unfolding islimpt_def
     using bchoice[of s "\<lambda> x T. x \<in> T \<and> open T \<and> (\<forall>y\<in>t. y \<in> T \<longrightarrow> y = x)"]
     by auto
   obtain g where g: "g \<subseteq> {t. \<exists>x. x \<in> s \<and> t = f x}" "finite g" "s \<subseteq> \<Union>g"
     using assms(1)[unfolded compact_eq_Heine_Borel, THEN spec[where x="{t. \<exists>x. x\<in>s \<and> t = f x}"]]
     using f by auto
   from g(1,3) have g':"\<forall>x\<in>g. \<exists>xa \<in> s. x = f xa"
     by auto
   {
     fix x y
     assume "x \<in> t" "y \<in> t" "f x = f y"
     then have "x \<in> f x"  "y \<in> f x \<longrightarrow> y = x"
       using f[THEN bspec[where x=x]] and \<open>t \<subseteq> s\<close> by auto
     then have "x = y"
       using \<open>f x = f y\<close> and f[THEN bspec[where x=y]] and \<open>y \<in> t\<close> and \<open>t \<subseteq> s\<close>
       by auto
   }
   then have "inj_on f t"
     unfolding inj_on_def by simp
   then have "infinite (f ` t)"
     using assms(2) using finite_imageD by auto
   moreover
   {
     fix x
     assume "x \<in> t" "f x \<notin> g"
     from g(3) assms(3) \<open>x \<in> t\<close> obtain h where "h \<in> g" and "x \<in> h"
       by auto
     then obtain y where "y \<in> s" "h = f y"
       using g'[THEN bspec[where x=h]] by auto
     then have "y = x"
       using f[THEN bspec[where x=y]] and \<open>x\<in>t\<close> and \<open>x\<in>h\<close>[unfolded \<open>h = f y\<close>]
       by auto
     then have False
       using \<open>f x \<notin> g\<close> \<open>h \<in> g\<close> unfolding \<open>h = f y\<close>
       by auto
   }
   then have "f ` t \<subseteq> g" by auto
   ultimately show False
     using g(2) using finite_subset by auto
 qed
 
 lemma sequence_infinite_lemma:
   fixes f :: "nat \<Rightarrow> 'a::t1_space"
   assumes "\<forall>n. f n \<noteq> l"
     and "(f \<longlongrightarrow> l) sequentially"
   shows "infinite (range f)"
 proof
   assume "finite (range f)"
   then have "closed (range f)"
     by (rule finite_imp_closed)
   then have "open (- range f)"
     by (rule open_Compl)
   from assms(1) have "l \<in> - range f"
     by auto
   from assms(2) have "eventually (\<lambda>n. f n \<in> - range f) sequentially"
     using \<open>open (- range f)\<close> \<open>l \<in> - range f\<close>
     by (rule topological_tendstoD)
   then show False
     unfolding eventually_sequentially
     by auto
 qed
 
 lemma Bolzano_Weierstrass_imp_closed:
   fixes s :: "'a::{first_countable_topology,t2_space} set"
   assumes "\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t)"
   shows "closed s"
 proof -
   {
     fix x l
     assume as: "\<forall>n::nat. x n \<in> s" "(x \<longlongrightarrow> l) sequentially"
     then have "l \<in> s"
     proof (cases "\<forall>n. x n \<noteq> l")
       case False
       then show "l\<in>s" using as(1) by auto
     next
       case True note cas = this
       with as(2) have "infinite (range x)"
         using sequence_infinite_lemma[of x l] by auto
       then obtain l' where "l'\<in>s" "l' islimpt (range x)"
         using assms[THEN spec[where x="range x"]] as(1) by auto
       then show "l\<in>s" using sequence_unique_limpt[of x l l']
         using as cas by auto
     qed
   }
   then show ?thesis
     unfolding closed_sequential_limits by fast
 qed
 
 lemma closure_insert:
   fixes x :: "'a::t1_space"
   shows "closure (insert x s) = insert x (closure s)"
   apply (rule closure_unique)
   apply (rule insert_mono [OF closure_subset])
   apply (rule closed_insert [OF closed_closure])
   apply (simp add: closure_minimal)
   done
 
 
 text\<open>In particular, some common special cases.\<close>
 
 lemma compact_Un [intro]:
   assumes "compact s"
     and "compact t"
   shows " compact (s \<union> t)"
 proof (rule compactI)
   fix f
   assume *: "Ball f open" "s \<union> t \<subseteq> \<Union>f"
   from * \<open>compact s\<close> obtain s' where "s' \<subseteq> f \<and> finite s' \<and> s \<subseteq> \<Union>s'"
     unfolding compact_eq_Heine_Borel by (auto elim!: allE[of _ f])
   moreover
   from * \<open>compact t\<close> obtain t' where "t' \<subseteq> f \<and> finite t' \<and> t \<subseteq> \<Union>t'"
     unfolding compact_eq_Heine_Borel by (auto elim!: allE[of _ f])
   ultimately show "\<exists>f'\<subseteq>f. finite f' \<and> s \<union> t \<subseteq> \<Union>f'"
     by (auto intro!: exI[of _ "s' \<union> t'"])
 qed
 
 lemma compact_Union [intro]: "finite S \<Longrightarrow> (\<And>T. T \<in> S \<Longrightarrow> compact T) \<Longrightarrow> compact (\<Union>S)"
   by (induct set: finite) auto
 
 lemma compact_UN [intro]:
   "finite A \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> compact (B x)) \<Longrightarrow> compact (\<Union>x\<in>A. B x)"
   by (rule compact_Union) auto
 
 lemma closed_Int_compact [intro]:
   assumes "closed s"
     and "compact t"
   shows "compact (s \<inter> t)"
   using compact_Int_closed [of t s] assms
   by (simp add: Int_commute)
 
 lemma compact_Int [intro]:
   fixes s t :: "'a :: t2_space set"
   assumes "compact s"
     and "compact t"
   shows "compact (s \<inter> t)"
   using assms by (intro compact_Int_closed compact_imp_closed)
 
 lemma compact_sing [simp]: "compact {a}"
   unfolding compact_eq_Heine_Borel by auto
 
 lemma compact_insert [simp]:
   assumes "compact s"
   shows "compact (insert x s)"
 proof -
   have "compact ({x} \<union> s)"
     using compact_sing assms by (rule compact_Un)
   then show ?thesis by simp
 qed
 
 lemma finite_imp_compact: "finite s \<Longrightarrow> compact s"
   by (induct set: finite) simp_all
 
 lemma open_delete:
   fixes s :: "'a::t1_space set"
   shows "open s \<Longrightarrow> open (s - {x})"
   by (simp add: open_Diff)
 
 
 text\<open>Compactness expressed with filters\<close>
 
 lemma closure_iff_nhds_not_empty:
   "x \<in> closure X \<longleftrightarrow> (\<forall>A. \<forall>S\<subseteq>A. open S \<longrightarrow> x \<in> S \<longrightarrow> X \<inter> A \<noteq> {})"
 proof safe
   assume x: "x \<in> closure X"
   fix S A
   assume "open S" "x \<in> S" "X \<inter> A = {}" "S \<subseteq> A"
   then have "x \<notin> closure (-S)"
     by (auto simp: closure_complement subset_eq[symmetric] intro: interiorI)
   with x have "x \<in> closure X - closure (-S)"
     by auto
   also have "\<dots> \<subseteq> closure (X \<inter> S)"
     using \<open>open S\<close> open_Int_closure_subset[of S X] by (simp add: closed_Compl ac_simps)
   finally have "X \<inter> S \<noteq> {}" by auto
   then show False using \<open>X \<inter> A = {}\<close> \<open>S \<subseteq> A\<close> by auto
 next
   assume "\<forall>A S. S \<subseteq> A \<longrightarrow> open S \<longrightarrow> x \<in> S \<longrightarrow> X \<inter> A \<noteq> {}"
   from this[THEN spec, of "- X", THEN spec, of "- closure X"]
   show "x \<in> closure X"
     by (simp add: closure_subset open_Compl)
 qed
 
 lemma compact_filter:
   "compact U \<longleftrightarrow> (\<forall>F. F \<noteq> bot \<longrightarrow> eventually (\<lambda>x. x \<in> U) F \<longrightarrow> (\<exists>x\<in>U. inf (nhds x) F \<noteq> bot))"
 proof (intro allI iffI impI compact_fip[THEN iffD2] notI)
   fix F
   assume "compact U"
   assume F: "F \<noteq> bot" "eventually (\<lambda>x. x \<in> U) F"
   then have "U \<noteq> {}"
     by (auto simp: eventually_False)
 
   define Z where "Z = closure ` {A. eventually (\<lambda>x. x \<in> A) F}"
   then have "\<forall>z\<in>Z. closed z"
     by auto
   moreover
   have ev_Z: "\<And>z. z \<in> Z \<Longrightarrow> eventually (\<lambda>x. x \<in> z) F"
     unfolding Z_def by (auto elim: eventually_mono intro: subsetD[OF closure_subset])
   have "(\<forall>B \<subseteq> Z. finite B \<longrightarrow> U \<inter> \<Inter>B \<noteq> {})"
   proof (intro allI impI)
     fix B assume "finite B" "B \<subseteq> Z"
     with \<open>finite B\<close> ev_Z F(2) have "eventually (\<lambda>x. x \<in> U \<inter> (\<Inter>B)) F"
       by (auto simp: eventually_ball_finite_distrib eventually_conj_iff)
     with F show "U \<inter> \<Inter>B \<noteq> {}"
       by (intro notI) (simp add: eventually_False)
   qed
   ultimately have "U \<inter> \<Inter>Z \<noteq> {}"
     using \<open>compact U\<close> unfolding compact_fip by blast
   then obtain x where "x \<in> U" and x: "\<And>z. z \<in> Z \<Longrightarrow> x \<in> z"
     by auto
 
   have "\<And>P. eventually P (inf (nhds x) F) \<Longrightarrow> P \<noteq> bot"
     unfolding eventually_inf eventually_nhds
   proof safe
     fix P Q R S
     assume "eventually R F" "open S" "x \<in> S"
     with open_Int_closure_eq_empty[of S "{x. R x}"] x[of "closure {x. R x}"]
     have "S \<inter> {x. R x} \<noteq> {}" by (auto simp: Z_def)
     moreover assume "Ball S Q" "\<forall>x. Q x \<and> R x \<longrightarrow> bot x"
     ultimately show False by (auto simp: set_eq_iff)
   qed
   with \<open>x \<in> U\<close> show "\<exists>x\<in>U. inf (nhds x) F \<noteq> bot"
     by (metis eventually_bot)
 next
   fix A
   assume A: "\<forall>a\<in>A. closed a" "\<forall>B\<subseteq>A. finite B \<longrightarrow> U \<inter> \<Inter>B \<noteq> {}" "U \<inter> \<Inter>A = {}"
   define F where "F = (INF a\<in>insert U A. principal a)"
   have "F \<noteq> bot"
     unfolding F_def
   proof (rule INF_filter_not_bot)
     fix X
     assume X: "X \<subseteq> insert U A" "finite X"
     with A(2)[THEN spec, of "X - {U}"] have "U \<inter> \<Inter>(X - {U}) \<noteq> {}"
       by auto
     with X show "(INF a\<in>X. principal a) \<noteq> bot"
       by (auto simp: INF_principal_finite principal_eq_bot_iff)
   qed
   moreover
   have "F \<le> principal U"
     unfolding F_def by auto
   then have "eventually (\<lambda>x. x \<in> U) F"
     by (auto simp: le_filter_def eventually_principal)
   moreover
   assume "\<forall>F. F \<noteq> bot \<longrightarrow> eventually (\<lambda>x. x \<in> U) F \<longrightarrow> (\<exists>x\<in>U. inf (nhds x) F \<noteq> bot)"
   ultimately obtain x where "x \<in> U" and x: "inf (nhds x) F \<noteq> bot"
     by auto
 
   { fix V assume "V \<in> A"
     then have "F \<le> principal V"
       unfolding F_def by (intro INF_lower2[of V]) auto
     then have V: "eventually (\<lambda>x. x \<in> V) F"
       by (auto simp: le_filter_def eventually_principal)
     have "x \<in> closure V"
       unfolding closure_iff_nhds_not_empty
     proof (intro impI allI)
       fix S A
       assume "open S" "x \<in> S" "S \<subseteq> A"
       then have "eventually (\<lambda>x. x \<in> A) (nhds x)"
         by (auto simp: eventually_nhds)
       with V have "eventually (\<lambda>x. x \<in> V \<inter> A) (inf (nhds x) F)"
         by (auto simp: eventually_inf)
       with x show "V \<inter> A \<noteq> {}"
         by (auto simp del: Int_iff simp add: trivial_limit_def)
     qed
     then have "x \<in> V"
       using \<open>V \<in> A\<close> A(1) by simp
   }
   with \<open>x\<in>U\<close> have "x \<in> U \<inter> \<Inter>A" by auto
   with \<open>U \<inter> \<Inter>A = {}\<close> show False by auto
 qed
 
 definition\<^marker>\<open>tag important\<close> countably_compact :: "('a::topological_space) set \<Rightarrow> bool" where
 "countably_compact U \<longleftrightarrow>
   (\<forall>A. countable A \<longrightarrow> (\<forall>a\<in>A. open a) \<longrightarrow> U \<subseteq> \<Union>A
      \<longrightarrow> (\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T))"
 
 lemma countably_compactE:
   assumes "countably_compact s" and "\<forall>t\<in>C. open t" and "s \<subseteq> \<Union>C" "countable C"
   obtains C' where "C' \<subseteq> C" and "finite C'" and "s \<subseteq> \<Union>C'"
   using assms unfolding countably_compact_def by metis
 
 lemma countably_compactI:
   assumes "\<And>C. \<forall>t\<in>C. open t \<Longrightarrow> s \<subseteq> \<Union>C \<Longrightarrow> countable C \<Longrightarrow> (\<exists>C'\<subseteq>C. finite C' \<and> s \<subseteq> \<Union>C')"
   shows "countably_compact s"
   using assms unfolding countably_compact_def by metis
 
 lemma compact_imp_countably_compact: "compact U \<Longrightarrow> countably_compact U"
   by (auto simp: compact_eq_Heine_Borel countably_compact_def)
 
 lemma countably_compact_imp_compact:
   assumes "countably_compact U"
     and ccover: "countable B" "\<forall>b\<in>B. open b"
     and basis: "\<And>T x. open T \<Longrightarrow> x \<in> T \<Longrightarrow> x \<in> U \<Longrightarrow> \<exists>b\<in>B. x \<in> b \<and> b \<inter> U \<subseteq> T"
   shows "compact U"
   using \<open>countably_compact U\<close>
   unfolding compact_eq_Heine_Borel countably_compact_def
 proof safe
   fix A
   assume A: "\<forall>a\<in>A. open a" "U \<subseteq> \<Union>A"
   assume *: "\<forall>A. countable A \<longrightarrow> (\<forall>a\<in>A. open a) \<longrightarrow> U \<subseteq> \<Union>A \<longrightarrow> (\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T)"
   moreover define C where "C = {b\<in>B. \<exists>a\<in>A. b \<inter> U \<subseteq> a}"
   ultimately have "countable C" "\<forall>a\<in>C. open a"
     unfolding C_def using ccover by auto
   moreover
   have "\<Union>A \<inter> U \<subseteq> \<Union>C"
   proof safe
     fix x a
     assume "x \<in> U" "x \<in> a" "a \<in> A"
     with basis[of a x] A obtain b where "b \<in> B" "x \<in> b" "b \<inter> U \<subseteq> a"
       by blast
     with \<open>a \<in> A\<close> show "x \<in> \<Union>C"
       unfolding C_def by auto
   qed
   then have "U \<subseteq> \<Union>C" using \<open>U \<subseteq> \<Union>A\<close> by auto
   ultimately obtain T where T: "T\<subseteq>C" "finite T" "U \<subseteq> \<Union>T"
     using * by metis
   then have "\<forall>t\<in>T. \<exists>a\<in>A. t \<inter> U \<subseteq> a"
     by (auto simp: C_def)
   then obtain f where "\<forall>t\<in>T. f t \<in> A \<and> t \<inter> U \<subseteq> f t"
     unfolding bchoice_iff Bex_def ..
   with T show "\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T"
     unfolding C_def by (intro exI[of _ "f`T"]) fastforce
 qed
 
 proposition countably_compact_imp_compact_second_countable:
   "countably_compact U \<Longrightarrow> compact (U :: 'a :: second_countable_topology set)"
 proof (rule countably_compact_imp_compact)
   fix T and x :: 'a
   assume "open T" "x \<in> T"
   from topological_basisE[OF is_basis this] obtain b where
     "b \<in> (SOME B. countable B \<and> topological_basis B)" "x \<in> b" "b \<subseteq> T" .
   then show "\<exists>b\<in>SOME B. countable B \<and> topological_basis B. x \<in> b \<and> b \<inter> U \<subseteq> T"
     by blast
 qed (insert countable_basis topological_basis_open[OF is_basis], auto)
 
 lemma countably_compact_eq_compact:
   "countably_compact U \<longleftrightarrow> compact (U :: 'a :: second_countable_topology set)"
   using countably_compact_imp_compact_second_countable compact_imp_countably_compact by blast
 
 subsubsection\<open>Sequential compactness\<close>
 
 definition\<^marker>\<open>tag important\<close> seq_compact :: "'a::topological_space set \<Rightarrow> bool" where
 "seq_compact S \<longleftrightarrow>
   (\<forall>f. (\<forall>n. f n \<in> S)
     \<longrightarrow> (\<exists>l\<in>S. \<exists>r::nat\<Rightarrow>nat. strict_mono r \<and> ((f \<circ> r) \<longlongrightarrow> l) sequentially))"
 
 lemma seq_compactI:
   assumes "\<And>f. \<forall>n. f n \<in> S \<Longrightarrow> \<exists>l\<in>S. \<exists>r::nat\<Rightarrow>nat. strict_mono r \<and> ((f \<circ> r) \<longlongrightarrow> l) sequentially"
   shows "seq_compact S"
   unfolding seq_compact_def using assms by fast
 
 lemma seq_compactE:
   assumes "seq_compact S" "\<forall>n. f n \<in> S"
   obtains l r where "l \<in> S" "strict_mono (r :: nat \<Rightarrow> nat)" "((f \<circ> r) \<longlongrightarrow> l) sequentially"
   using assms unfolding seq_compact_def by fast
 
 lemma closed_sequentially: (* TODO: move upwards *)
   assumes "closed s" and "\<forall>n. f n \<in> s" and "f \<longlonglongrightarrow> l"
   shows "l \<in> s"
 proof (rule ccontr)
   assume "l \<notin> s"
   with \<open>closed s\<close> and \<open>f \<longlonglongrightarrow> l\<close> have "eventually (\<lambda>n. f n \<in> - s) sequentially"
     by (fast intro: topological_tendstoD)
   with \<open>\<forall>n. f n \<in> s\<close> show "False"
     by simp
 qed
 
 lemma seq_compact_Int_closed:
   assumes "seq_compact s" and "closed t"
   shows "seq_compact (s \<inter> t)"
 proof (rule seq_compactI)
   fix f assume "\<forall>n::nat. f n \<in> s \<inter> t"
   hence "\<forall>n. f n \<in> s" and "\<forall>n. f n \<in> t"
     by simp_all
   from \<open>seq_compact s\<close> and \<open>\<forall>n. f n \<in> s\<close>
   obtain l r where "l \<in> s" and r: "strict_mono r" and l: "(f \<circ> r) \<longlonglongrightarrow> l"
     by (rule seq_compactE)
   from \<open>\<forall>n. f n \<in> t\<close> have "\<forall>n. (f \<circ> r) n \<in> t"
     by simp
   from \<open>closed t\<close> and this and l have "l \<in> t"
     by (rule closed_sequentially)
   with \<open>l \<in> s\<close> and r and l show "\<exists>l\<in>s \<inter> t. \<exists>r. strict_mono r \<and> (f \<circ> r) \<longlonglongrightarrow> l"
     by fast
 qed
 
 lemma seq_compact_closed_subset:
   assumes "closed s" and "s \<subseteq> t" and "seq_compact t"
   shows "seq_compact s"
   using assms seq_compact_Int_closed [of t s] by (simp add: Int_absorb1)
 
 lemma seq_compact_imp_countably_compact:
   fixes U :: "'a :: first_countable_topology set"
   assumes "seq_compact U"
   shows "countably_compact U"
 proof (safe intro!: countably_compactI)
   fix A
   assume A: "\<forall>a\<in>A. open a" "U \<subseteq> \<Union>A" "countable A"
   have subseq: "\<And>X. range X \<subseteq> U \<Longrightarrow> \<exists>r x. x \<in> U \<and> strict_mono (r :: nat \<Rightarrow> nat) \<and> (X \<circ> r) \<longlonglongrightarrow> x"
     using \<open>seq_compact U\<close> by (fastforce simp: seq_compact_def subset_eq)
   show "\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T"
   proof cases
     assume "finite A"
     with A show ?thesis by auto
   next
     assume "infinite A"
     then have "A \<noteq> {}" by auto
     show ?thesis
     proof (rule ccontr)
       assume "\<not> (\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T)"
       then have "\<forall>T. \<exists>x. T \<subseteq> A \<and> finite T \<longrightarrow> (x \<in> U - \<Union>T)"
         by auto
       then obtain X' where T: "\<And>T. T \<subseteq> A \<Longrightarrow> finite T \<Longrightarrow> X' T \<in> U - \<Union>T"
         by metis
       define X where "X n = X' (from_nat_into A ` {.. n})" for n
       have X: "\<And>n. X n \<in> U - (\<Union>i\<le>n. from_nat_into A i)"
         using \<open>A \<noteq> {}\<close> unfolding X_def by (intro T) (auto intro: from_nat_into)
       then have "range X \<subseteq> U"
         by auto
       with subseq[of X] obtain r x where "x \<in> U" and r: "strict_mono r" "(X \<circ> r) \<longlonglongrightarrow> x"
         by auto
       from \<open>x\<in>U\<close> \<open>U \<subseteq> \<Union>A\<close> from_nat_into_surj[OF \<open>countable A\<close>]
       obtain n where "x \<in> from_nat_into A n" by auto
       with r(2) A(1) from_nat_into[OF \<open>A \<noteq> {}\<close>, of n]
       have "eventually (\<lambda>i. X (r i) \<in> from_nat_into A n) sequentially"
         unfolding tendsto_def by (auto simp: comp_def)
       then obtain N where "\<And>i. N \<le> i \<Longrightarrow> X (r i) \<in> from_nat_into A n"
         by (auto simp: eventually_sequentially)
       moreover from X have "\<And>i. n \<le> r i \<Longrightarrow> X (r i) \<notin> from_nat_into A n"
         by auto
       moreover from \<open>strict_mono r\<close>[THEN seq_suble, of "max n N"] have "\<exists>i. n \<le> r i \<and> N \<le> i"
         by (auto intro!: exI[of _ "max n N"])
       ultimately show False
         by auto
     qed
   qed
 qed
 
 lemma compact_imp_seq_compact:
   fixes U :: "'a :: first_countable_topology set"
   assumes "compact U"
   shows "seq_compact U"
   unfolding seq_compact_def
 proof safe
   fix X :: "nat \<Rightarrow> 'a"
   assume "\<forall>n. X n \<in> U"
   then have "eventually (\<lambda>x. x \<in> U) (filtermap X sequentially)"
     by (auto simp: eventually_filtermap)
   moreover
   have "filtermap X sequentially \<noteq> bot"
     by (simp add: trivial_limit_def eventually_filtermap)
   ultimately
   obtain x where "x \<in> U" and x: "inf (nhds x) (filtermap X sequentially) \<noteq> bot" (is "?F \<noteq> _")
     using \<open>compact U\<close> by (auto simp: compact_filter)
 
   from countable_basis_at_decseq[of x]
   obtain A where A:
       "\<And>i. open (A i)"
       "\<And>i. x \<in> A i"
       "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> eventually (\<lambda>i. A i \<subseteq> S) sequentially"
     by blast
   define s where "s n i = (SOME j. i < j \<and> X j \<in> A (Suc n))" for n i
   {
     fix n i
     have "\<exists>a. i < a \<and> X a \<in> A (Suc n)"
     proof (rule ccontr)
       assume "\<not> (\<exists>a>i. X a \<in> A (Suc n))"
       then have "\<And>a. Suc i \<le> a \<Longrightarrow> X a \<notin> A (Suc n)"
         by auto
       then have "eventually (\<lambda>x. x \<notin> A (Suc n)) (filtermap X sequentially)"
         by (auto simp: eventually_filtermap eventually_sequentially)
       moreover have "eventually (\<lambda>x. x \<in> A (Suc n)) (nhds x)"
         using A(1,2)[of "Suc n"] by (auto simp: eventually_nhds)
       ultimately have "eventually (\<lambda>x. False) ?F"
         by (auto simp: eventually_inf)
       with x show False
         by (simp add: eventually_False)
     qed
     then have "i < s n i" "X (s n i) \<in> A (Suc n)"
       unfolding s_def by (auto intro: someI2_ex)
   }
   note s = this
   define r where "r = rec_nat (s 0 0) s"
   have "strict_mono r"
     by (auto simp: r_def s strict_mono_Suc_iff)
   moreover
   have "(\<lambda>n. X (r n)) \<longlonglongrightarrow> x"
   proof (rule topological_tendstoI)
     fix S
     assume "open S" "x \<in> S"
     with A(3) have "eventually (\<lambda>i. A i \<subseteq> S) sequentially"
       by auto
     moreover
     {
       fix i
       assume "Suc 0 \<le> i"
       then have "X (r i) \<in> A i"
         by (cases i) (simp_all add: r_def s)
     }
     then have "eventually (\<lambda>i. X (r i) \<in> A i) sequentially"
       by (auto simp: eventually_sequentially)
     ultimately show "eventually (\<lambda>i. X (r i) \<in> S) sequentially"
       by eventually_elim auto
   qed
   ultimately show "\<exists>x \<in> U. \<exists>r. strict_mono r \<and> (X \<circ> r) \<longlonglongrightarrow> x"
     using \<open>x \<in> U\<close> by (auto simp: convergent_def comp_def)
 qed
 
 lemma countably_compact_imp_acc_point:
   assumes "countably_compact s"
     and "countable t"
     and "infinite t"
     and "t \<subseteq> s"
   shows "\<exists>x\<in>s. \<forall>U. x\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> t)"
 proof (rule ccontr)
   define C where "C = (\<lambda>F. interior (F \<union> (- t))) ` {F. finite F \<and> F \<subseteq> t }"
   note \<open>countably_compact s\<close>
   moreover have "\<forall>t\<in>C. open t"
     by (auto simp: C_def)
   moreover
   assume "\<not> (\<exists>x\<in>s. \<forall>U. x\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> t))"
   then have s: "\<And>x. x \<in> s \<Longrightarrow> \<exists>U. x\<in>U \<and> open U \<and> finite (U \<inter> t)" by metis
   have "s \<subseteq> \<Union>C"
     using \<open>t \<subseteq> s\<close>
     unfolding C_def
     apply (safe dest!: s)
     apply (rule_tac a="U \<inter> t" in UN_I)
     apply (auto intro!: interiorI simp add: finite_subset)
     done
   moreover
   from \<open>countable t\<close> have "countable C"
     unfolding C_def by (auto intro: countable_Collect_finite_subset)
   ultimately
   obtain D where "D \<subseteq> C" "finite D" "s \<subseteq> \<Union>D"
     by (rule countably_compactE)
   then obtain E where E: "E \<subseteq> {F. finite F \<and> F \<subseteq> t }" "finite E"
     and s: "s \<subseteq> (\<Union>F\<in>E. interior (F \<union> (- t)))"
     by (metis (lifting) finite_subset_image C_def)
   from s \<open>t \<subseteq> s\<close> have "t \<subseteq> \<Union>E"
     using interior_subset by blast
   moreover have "finite (\<Union>E)"
     using E by auto
   ultimately show False using \<open>infinite t\<close>
     by (auto simp: finite_subset)
 qed
 
 lemma countable_acc_point_imp_seq_compact:
   fixes s :: "'a::first_countable_topology set"
   assumes "\<forall>t. infinite t \<and> countable t \<and> t \<subseteq> s \<longrightarrow>
     (\<exists>x\<in>s. \<forall>U. x\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> t))"
   shows "seq_compact s"
 proof -
   {
     fix f :: "nat \<Rightarrow> 'a"
     assume f: "\<forall>n. f n \<in> s"
     have "\<exists>l\<in>s. \<exists>r. strict_mono r \<and> ((f \<circ> r) \<longlongrightarrow> l) sequentially"
     proof (cases "finite (range f)")
       case True
       obtain l where "infinite {n. f n = f l}"
         using pigeonhole_infinite[OF _ True] by auto
       then obtain r :: "nat \<Rightarrow> nat" where "strict_mono  r" and fr: "\<forall>n. f (r n) = f l"
         using infinite_enumerate by blast
       then have "strict_mono r \<and> (f \<circ> r) \<longlonglongrightarrow> f l"
         by (simp add: fr o_def)
       with f show "\<exists>l\<in>s. \<exists>r. strict_mono  r \<and> (f \<circ> r) \<longlonglongrightarrow> l"
         by auto
     next
       case False
       with f assms have "\<exists>x\<in>s. \<forall>U. x\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> range f)"
         by auto
       then obtain l where "l \<in> s" "\<forall>U. l\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> range f)" ..
       from this(2) have "\<exists>r. strict_mono r \<and> ((f \<circ> r) \<longlongrightarrow> l) sequentially"
         using acc_point_range_imp_convergent_subsequence[of l f] by auto
       with \<open>l \<in> s\<close> show "\<exists>l\<in>s. \<exists>r. strict_mono r \<and> ((f \<circ> r) \<longlongrightarrow> l) sequentially" ..
     qed
   }
   then show ?thesis
     unfolding seq_compact_def by auto
 qed
 
 lemma seq_compact_eq_countably_compact:
   fixes U :: "'a :: first_countable_topology set"
   shows "seq_compact U \<longleftrightarrow> countably_compact U"
   using
     countable_acc_point_imp_seq_compact
     countably_compact_imp_acc_point
     seq_compact_imp_countably_compact
   by metis
 
 lemma seq_compact_eq_acc_point:
   fixes s :: "'a :: first_countable_topology set"
   shows "seq_compact s \<longleftrightarrow>
     (\<forall>t. infinite t \<and> countable t \<and> t \<subseteq> s --> (\<exists>x\<in>s. \<forall>U. x\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> t)))"
   using
     countable_acc_point_imp_seq_compact[of s]
     countably_compact_imp_acc_point[of s]
     seq_compact_imp_countably_compact[of s]
   by metis
 
 lemma seq_compact_eq_compact:
   fixes U :: "'a :: second_countable_topology set"
   shows "seq_compact U \<longleftrightarrow> compact U"
   using seq_compact_eq_countably_compact countably_compact_eq_compact by blast
 
 proposition Bolzano_Weierstrass_imp_seq_compact:
   fixes s :: "'a::{t1_space, first_countable_topology} set"
   shows "\<forall>t. infinite t \<and> t \<subseteq> s \<longrightarrow> (\<exists>x \<in> s. x islimpt t) \<Longrightarrow> seq_compact s"
   by (rule countable_acc_point_imp_seq_compact) (metis islimpt_eq_acc_point)
 
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>Cartesian products\<close>
 
 lemma seq_compact_Times: "seq_compact s \<Longrightarrow> seq_compact t \<Longrightarrow> seq_compact (s \<times> t)"
   unfolding seq_compact_def
   apply clarify
   apply (drule_tac x="fst \<circ> f" in spec)
   apply (drule mp, simp add: mem_Times_iff)
   apply (clarify, rename_tac l1 r1)
   apply (drule_tac x="snd \<circ> f \<circ> r1" in spec)
   apply (drule mp, simp add: mem_Times_iff)
   apply (clarify, rename_tac l2 r2)
   apply (rule_tac x="(l1, l2)" in rev_bexI, simp)
   apply (rule_tac x="r1 \<circ> r2" in exI)
   apply (rule conjI, simp add: strict_mono_def)
   apply (drule_tac f=r2 in LIMSEQ_subseq_LIMSEQ, assumption)
   apply (drule (1) tendsto_Pair) back
   apply (simp add: o_def)
   done
 
 lemma compact_Times:
   assumes "compact s" "compact t"
   shows "compact (s \<times> t)"
 proof (rule compactI)
   fix C
   assume C: "\<forall>t\<in>C. open t" "s \<times> t \<subseteq> \<Union>C"
   have "\<forall>x\<in>s. \<exists>a. open a \<and> x \<in> a \<and> (\<exists>d\<subseteq>C. finite d \<and> a \<times> t \<subseteq> \<Union>d)"
   proof
     fix x
     assume "x \<in> s"
     have "\<forall>y\<in>t. \<exists>a b c. c \<in> C \<and> open a \<and> open b \<and> x \<in> a \<and> y \<in> b \<and> a \<times> b \<subseteq> c" (is "\<forall>y\<in>t. ?P y")
     proof
       fix y
       assume "y \<in> t"
       with \<open>x \<in> s\<close> C obtain c where "c \<in> C" "(x, y) \<in> c" "open c" by auto
       then show "?P y" by (auto elim!: open_prod_elim)
     qed
     then obtain a b c where b: "\<And>y. y \<in> t \<Longrightarrow> open (b y)"
       and c: "\<And>y. y \<in> t \<Longrightarrow> c y \<in> C \<and> open (a y) \<and> open (b y) \<and> x \<in> a y \<and> y \<in> b y \<and> a y \<times> b y \<subseteq> c y"
       by metis
     then have "\<forall>y\<in>t. open (b y)" "t \<subseteq> (\<Union>y\<in>t. b y)" by auto
     with compactE_image[OF \<open>compact t\<close>] obtain D where D: "D \<subseteq> t" "finite D" "t \<subseteq> (\<Union>y\<in>D. b y)"
       by metis
     moreover from D c have "(\<Inter>y\<in>D. a y) \<times> t \<subseteq> (\<Union>y\<in>D. c y)"
       by (fastforce simp: subset_eq)
     ultimately show "\<exists>a. open a \<and> x \<in> a \<and> (\<exists>d\<subseteq>C. finite d \<and> a \<times> t \<subseteq> \<Union>d)"
       using c by (intro exI[of _ "c`D"] exI[of _ "\<Inter>(a`D)"] conjI) (auto intro!: open_INT)
   qed
   then obtain a d where a: "\<And>x. x\<in>s \<Longrightarrow> open (a x)" "s \<subseteq> (\<Union>x\<in>s. a x)"
     and d: "\<And>x. x \<in> s \<Longrightarrow> d x \<subseteq> C \<and> finite (d x) \<and> a x \<times> t \<subseteq> \<Union>(d x)"
     unfolding subset_eq UN_iff by metis
   moreover
   from compactE_image[OF \<open>compact s\<close> a]
   obtain e where e: "e \<subseteq> s" "finite e" and s: "s \<subseteq> (\<Union>x\<in>e. a x)"
     by auto
   moreover
   {
     from s have "s \<times> t \<subseteq> (\<Union>x\<in>e. a x \<times> t)"
       by auto
     also have "\<dots> \<subseteq> (\<Union>x\<in>e. \<Union>(d x))"
       using d \<open>e \<subseteq> s\<close> by (intro UN_mono) auto
     finally have "s \<times> t \<subseteq> (\<Union>x\<in>e. \<Union>(d x))" .
   }
   ultimately show "\<exists>C'\<subseteq>C. finite C' \<and> s \<times> t \<subseteq> \<Union>C'"
     by (intro exI[of _ "(\<Union>x\<in>e. d x)"]) (auto simp: subset_eq)
 qed
 
 
 lemma tube_lemma:
   assumes "compact K"
   assumes "open W"
   assumes "{x0} \<times> K \<subseteq> W"
   shows "\<exists>X0. x0 \<in> X0 \<and> open X0 \<and> X0 \<times> K \<subseteq> W"
 proof -
   {
     fix y assume "y \<in> K"
     then have "(x0, y) \<in> W" using assms by auto
     with \<open>open W\<close>
     have "\<exists>X0 Y. open X0 \<and> open Y \<and> x0 \<in> X0 \<and> y \<in> Y \<and> X0 \<times> Y \<subseteq> W"
       by (rule open_prod_elim) blast
   }
   then obtain X0 Y where
     *: "\<forall>y \<in> K. open (X0 y) \<and> open (Y y) \<and> x0 \<in> X0 y \<and> y \<in> Y y \<and> X0 y \<times> Y y \<subseteq> W"
     by metis
   from * have "\<forall>t\<in>Y ` K. open t" "K \<subseteq> \<Union>(Y ` K)" by auto
   with \<open>compact K\<close> obtain CC where CC: "CC \<subseteq> Y ` K" "finite CC" "K \<subseteq> \<Union>CC"
     by (meson compactE)
   then obtain c where c: "\<And>C. C \<in> CC \<Longrightarrow> c C \<in> K \<and> C = Y (c C)"
     by (force intro!: choice)
   with * CC show ?thesis
     by (force intro!: exI[where x="\<Inter>C\<in>CC. X0 (c C)"]) (* SLOW *)
 qed
 
 lemma continuous_on_prod_compactE:
   fixes fx::"'a::topological_space \<times> 'b::topological_space \<Rightarrow> 'c::metric_space"
     and e::real
   assumes cont_fx: "continuous_on (U \<times> C) fx"
   assumes "compact C"
   assumes [intro]: "x0 \<in> U"
   notes [continuous_intros] = continuous_on_compose2[OF cont_fx]
   assumes "e > 0"
   obtains X0 where "x0 \<in> X0" "open X0"
     "\<forall>x\<in>X0 \<inter> U. \<forall>t \<in> C. dist (fx (x, t)) (fx (x0, t)) \<le> e"
 proof -
   define psi where "psi = (\<lambda>(x, t). dist (fx (x, t)) (fx (x0, t)))"
   define W0 where "W0 = {(x, t) \<in> U \<times> C. psi (x, t) < e}"
   have W0_eq: "W0 = psi -` {..<e} \<inter> U \<times> C"
     by (auto simp: vimage_def W0_def)
   have "open {..<e}" by simp
   have "continuous_on (U \<times> C) psi"
     by (auto intro!: continuous_intros simp: psi_def split_beta')
   from this[unfolded continuous_on_open_invariant, rule_format, OF \<open>open {..<e}\<close>]
   obtain W where W: "open W" "W \<inter> U \<times> C = W0 \<inter> U \<times> C"
     unfolding W0_eq by blast
   have "{x0} \<times> C \<subseteq> W \<inter> U \<times> C"
     unfolding W
     by (auto simp: W0_def psi_def \<open>0 < e\<close>)
   then have "{x0} \<times> C \<subseteq> W" by blast
   from tube_lemma[OF \<open>compact C\<close> \<open>open W\<close> this]
   obtain X0 where X0: "x0 \<in> X0" "open X0" "X0 \<times> C \<subseteq> W"
     by blast
 
   have "\<forall>x\<in>X0 \<inter> U. \<forall>t \<in> C. dist (fx (x, t)) (fx (x0, t)) \<le> e"
   proof safe
     fix x assume x: "x \<in> X0" "x \<in> U"
     fix t assume t: "t \<in> C"
     have "dist (fx (x, t)) (fx (x0, t)) = psi (x, t)"
       by (auto simp: psi_def)
     also
     {
       have "(x, t) \<in> X0 \<times> C"
         using t x
         by auto
       also note \<open>\<dots> \<subseteq> W\<close>
       finally have "(x, t) \<in> W" .
       with t x have "(x, t) \<in> W \<inter> U \<times> C"
         by blast
       also note \<open>W \<inter> U \<times> C = W0 \<inter> U \<times> C\<close>
       finally  have "psi (x, t) < e"
         by (auto simp: W0_def)
     }
     finally show "dist (fx (x, t)) (fx (x0, t)) \<le> e" by simp
   qed
   from X0(1,2) this show ?thesis ..
 qed
 
 
 subsection \<open>Continuity\<close>
 
 lemma continuous_at_imp_continuous_within:
   "continuous (at x) f \<Longrightarrow> continuous (at x within s) f"
   unfolding continuous_within continuous_at using Lim_at_imp_Lim_at_within by auto
 
 lemma Lim_trivial_limit: "trivial_limit net \<Longrightarrow> (f \<longlongrightarrow> l) net"
   by simp
 
 lemmas continuous_on = continuous_on_def \<comment> \<open>legacy theorem name\<close>
 
 lemma continuous_within_subset:
   "continuous (at x within s) f \<Longrightarrow> t \<subseteq> s \<Longrightarrow> continuous (at x within t) f"
   unfolding continuous_within by(metis tendsto_within_subset)
 
 lemma continuous_on_interior:
   "continuous_on s f \<Longrightarrow> x \<in> interior s \<Longrightarrow> continuous (at x) f"
   by (metis continuous_on_eq_continuous_at continuous_on_subset interiorE)
 
 lemma continuous_on_eq:
   "\<lbrakk>continuous_on s f; \<And>x. x \<in> s \<Longrightarrow> f x = g x\<rbrakk> \<Longrightarrow> continuous_on s g"
   unfolding continuous_on_def tendsto_def eventually_at_topological
   by simp
 
 text \<open>Characterization of various kinds of continuity in terms of sequences.\<close>
 
 lemma continuous_within_sequentiallyI:
   fixes f :: "'a::{first_countable_topology, t2_space} \<Rightarrow> 'b::topological_space"
   assumes "\<And>u::nat \<Rightarrow> 'a. u \<longlonglongrightarrow> a \<Longrightarrow> (\<forall>n. u n \<in> s) \<Longrightarrow> (\<lambda>n. f (u n)) \<longlonglongrightarrow> f a"
   shows "continuous (at a within s) f"
   using assms unfolding continuous_within tendsto_def[where l = "f a"]
   by (auto intro!: sequentially_imp_eventually_within)
 
 lemma continuous_within_tendsto_compose:
   fixes f::"'a::t2_space \<Rightarrow> 'b::topological_space"
   assumes "continuous (at a within s) f"
           "eventually (\<lambda>n. x n \<in> s) F"
           "(x \<longlongrightarrow> a) F "
   shows "((\<lambda>n. f (x n)) \<longlongrightarrow> f a) F"
 proof -
   have *: "filterlim x (inf (nhds a) (principal s)) F"
     using assms(2) assms(3) unfolding at_within_def filterlim_inf by (auto simp: filterlim_principal eventually_mono)
   show ?thesis
     by (auto simp: assms(1) continuous_within[symmetric] tendsto_at_within_iff_tendsto_nhds[symmetric] intro!: filterlim_compose[OF _ *])
 qed
 
 lemma continuous_within_tendsto_compose':
   fixes f::"'a::t2_space \<Rightarrow> 'b::topological_space"
   assumes "continuous (at a within s) f"
     "\<And>n. x n \<in> s"
     "(x \<longlongrightarrow> a) F "
   shows "((\<lambda>n. f (x n)) \<longlongrightarrow> f a) F"
   by (auto intro!: continuous_within_tendsto_compose[OF assms(1)] simp add: assms)
 
 lemma continuous_within_sequentially:
   fixes f :: "'a::{first_countable_topology, t2_space} \<Rightarrow> 'b::topological_space"
   shows "continuous (at a within s) f \<longleftrightarrow>
     (\<forall>x. (\<forall>n::nat. x n \<in> s) \<and> (x \<longlongrightarrow> a) sequentially
          \<longrightarrow> ((f \<circ> x) \<longlongrightarrow> f a) sequentially)"
   using continuous_within_tendsto_compose'[of a s f _ sequentially]
     continuous_within_sequentiallyI[of a s f]
   by (auto simp: o_def)
 
 lemma continuous_at_sequentiallyI:
   fixes f :: "'a::{first_countable_topology, t2_space} \<Rightarrow> 'b::topological_space"
   assumes "\<And>u. u \<longlonglongrightarrow> a \<Longrightarrow> (\<lambda>n. f (u n)) \<longlonglongrightarrow> f a"
   shows "continuous (at a) f"
   using continuous_within_sequentiallyI[of a UNIV f] assms by auto
 
 lemma continuous_at_sequentially:
   fixes f :: "'a::metric_space \<Rightarrow> 'b::topological_space"
   shows "continuous (at a) f \<longleftrightarrow>
     (\<forall>x. (x \<longlongrightarrow> a) sequentially --> ((f \<circ> x) \<longlongrightarrow> f a) sequentially)"
   using continuous_within_sequentially[of a UNIV f] by simp
 
 lemma continuous_on_sequentiallyI:
   fixes f :: "'a::{first_countable_topology, t2_space} \<Rightarrow> 'b::topological_space"
   assumes "\<And>u a. (\<forall>n. u n \<in> s) \<Longrightarrow> a \<in> s \<Longrightarrow> u \<longlonglongrightarrow> a \<Longrightarrow> (\<lambda>n. f (u n)) \<longlonglongrightarrow> f a"
   shows "continuous_on s f"
   using assms unfolding continuous_on_eq_continuous_within
   using continuous_within_sequentiallyI[of _ s f] by auto
 
 lemma continuous_on_sequentially:
   fixes f :: "'a::{first_countable_topology, t2_space} \<Rightarrow> 'b::topological_space"
   shows "continuous_on s f \<longleftrightarrow>
     (\<forall>x. \<forall>a \<in> s. (\<forall>n. x(n) \<in> s) \<and> (x \<longlongrightarrow> a) sequentially
       --> ((f \<circ> x) \<longlongrightarrow> f a) sequentially)"
     (is "?lhs = ?rhs")
 proof
   assume ?rhs
   then show ?lhs
     using continuous_within_sequentially[of _ s f]
     unfolding continuous_on_eq_continuous_within
     by auto
 next
   assume ?lhs
   then show ?rhs
     unfolding continuous_on_eq_continuous_within
     using continuous_within_sequentially[of _ s f]
     by auto
 qed
 
 text \<open>Continuity in terms of open preimages.\<close>
 
 lemma continuous_at_open:
   "continuous (at x) f \<longleftrightarrow> (\<forall>t. open t \<and> f x \<in> t --> (\<exists>s. open s \<and> x \<in> s \<and> (\<forall>x' \<in> s. (f x') \<in> t)))"
   unfolding continuous_within_topological [of x UNIV f]
   unfolding imp_conjL
   by (intro all_cong imp_cong ex_cong conj_cong refl) auto
 
 lemma continuous_imp_tendsto:
   assumes "continuous (at x0) f"
     and "x \<longlonglongrightarrow> x0"
   shows "(f \<circ> x) \<longlonglongrightarrow> (f x0)"
 proof (rule topological_tendstoI)
   fix S
   assume "open S" "f x0 \<in> S"
   then obtain T where T_def: "open T" "x0 \<in> T" "\<forall>x\<in>T. f x \<in> S"
      using assms continuous_at_open by metis
   then have "eventually (\<lambda>n. x n \<in> T) sequentially"
     using assms T_def by (auto simp: tendsto_def)
   then show "eventually (\<lambda>n. (f \<circ> x) n \<in> S) sequentially"
     using T_def by (auto elim!: eventually_mono)
 qed
 
 subsection \<open>Homeomorphisms\<close>
 
 definition\<^marker>\<open>tag important\<close> "homeomorphism s t f g \<longleftrightarrow>
   (\<forall>x\<in>s. (g(f x) = x)) \<and> (f ` s = t) \<and> continuous_on s f \<and>
   (\<forall>y\<in>t. (f(g y) = y)) \<and> (g ` t = s) \<and> continuous_on t g"
 
 lemma homeomorphismI [intro?]:
   assumes "continuous_on S f" "continuous_on T g"
           "f ` S \<subseteq> T" "g ` T \<subseteq> S" "\<And>x. x \<in> S \<Longrightarrow> g(f x) = x" "\<And>y. y \<in> T \<Longrightarrow> f(g y) = y"
     shows "homeomorphism S T f g"
   using assms by (force simp: homeomorphism_def)
 
 lemma homeomorphism_translation:
   fixes a :: "'a :: real_normed_vector"
   shows "homeomorphism ((+) a ` S) S ((+) (- a)) ((+) a)"
 unfolding homeomorphism_def by (auto simp: algebra_simps continuous_intros)
 
 lemma homeomorphism_ident: "homeomorphism T T (\<lambda>a. a) (\<lambda>a. a)"
   by (rule homeomorphismI) auto
 
 lemma homeomorphism_compose:
   assumes "homeomorphism S T f g" "homeomorphism T U h k"
     shows "homeomorphism S U (h o f) (g o k)"
   using assms
   unfolding homeomorphism_def
   by (intro conjI ballI continuous_on_compose) (auto simp: image_iff)
 
 lemma homeomorphism_cong:
   "homeomorphism X' Y' f' g'"
     if "homeomorphism X Y f g" "X' = X" "Y' = Y" "\<And>x. x \<in> X \<Longrightarrow> f' x = f x" "\<And>y. y \<in> Y \<Longrightarrow> g' y = g y"
   using that by (auto simp add: homeomorphism_def)
 
 lemma homeomorphism_empty [simp]:
   "homeomorphism {} {} f g"
   unfolding homeomorphism_def by auto
 
 lemma homeomorphism_symD: "homeomorphism S t f g \<Longrightarrow> homeomorphism t S g f"
   by (simp add: homeomorphism_def)
 
 lemma homeomorphism_sym: "homeomorphism S t f g = homeomorphism t S g f"
   by (force simp: homeomorphism_def)
 
 definition\<^marker>\<open>tag important\<close> homeomorphic :: "'a::topological_space set \<Rightarrow> 'b::topological_space set \<Rightarrow> bool"
     (infixr "homeomorphic" 60)
   where "s homeomorphic t \<equiv> (\<exists>f g. homeomorphism s t f g)"
 
 lemma homeomorphic_empty [iff]:
      "S homeomorphic {} \<longleftrightarrow> S = {}" "{} homeomorphic S \<longleftrightarrow> S = {}"
   by (auto simp: homeomorphic_def homeomorphism_def)
 
 lemma homeomorphic_refl: "s homeomorphic s"
   unfolding homeomorphic_def homeomorphism_def
   using continuous_on_id
   apply (rule_tac x = "(\<lambda>x. x)" in exI)
   apply (rule_tac x = "(\<lambda>x. x)" in exI)
   apply blast
   done
 
 lemma homeomorphic_sym: "s homeomorphic t \<longleftrightarrow> t homeomorphic s"
   unfolding homeomorphic_def homeomorphism_def
   by blast
 
 lemma homeomorphic_trans [trans]:
   assumes "S homeomorphic T"
       and "T homeomorphic U"
     shows "S homeomorphic U"
   using assms
   unfolding homeomorphic_def
 by (metis homeomorphism_compose)
 
 lemma homeomorphic_minimal:
   "s homeomorphic t \<longleftrightarrow>
     (\<exists>f g. (\<forall>x\<in>s. f(x) \<in> t \<and> (g(f(x)) = x)) \<and>
            (\<forall>y\<in>t. g(y) \<in> s \<and> (f(g(y)) = y)) \<and>
            continuous_on s f \<and> continuous_on t g)"
    (is "?lhs = ?rhs")
 proof
   assume ?lhs
   then show ?rhs
     by (fastforce simp: homeomorphic_def homeomorphism_def)
 next
   assume ?rhs
   then show ?lhs
     apply clarify
     unfolding homeomorphic_def homeomorphism_def
     by (metis equalityI image_subset_iff subsetI)
  qed
 
 lemma homeomorphicI [intro?]:
    "\<lbrakk>f ` S = T; g ` T = S;
      continuous_on S f; continuous_on T g;
      \<And>x. x \<in> S \<Longrightarrow> g(f(x)) = x;
      \<And>y. y \<in> T \<Longrightarrow> f(g(y)) = y\<rbrakk> \<Longrightarrow> S homeomorphic T"
 unfolding homeomorphic_def homeomorphism_def by metis
 
 lemma homeomorphism_of_subsets:
    "\<lbrakk>homeomorphism S T f g; S' \<subseteq> S; T'' \<subseteq> T; f ` S' = T'\<rbrakk>
     \<Longrightarrow> homeomorphism S' T' f g"
 apply (auto simp: homeomorphism_def elim!: continuous_on_subset)
 by (metis subsetD imageI)
 
 lemma homeomorphism_apply1: "\<lbrakk>homeomorphism S T f g; x \<in> S\<rbrakk> \<Longrightarrow> g(f x) = x"
   by (simp add: homeomorphism_def)
 
 lemma homeomorphism_apply2: "\<lbrakk>homeomorphism S T f g; x \<in> T\<rbrakk> \<Longrightarrow> f(g x) = x"
   by (simp add: homeomorphism_def)
 
 lemma homeomorphism_image1: "homeomorphism S T f g \<Longrightarrow> f ` S = T"
   by (simp add: homeomorphism_def)
 
 lemma homeomorphism_image2: "homeomorphism S T f g \<Longrightarrow> g ` T = S"
   by (simp add: homeomorphism_def)
 
 lemma homeomorphism_cont1: "homeomorphism S T f g \<Longrightarrow> continuous_on S f"
   by (simp add: homeomorphism_def)
 
 lemma homeomorphism_cont2: "homeomorphism S T f g \<Longrightarrow> continuous_on T g"
   by (simp add: homeomorphism_def)
 
 lemma continuous_on_no_limpt:
    "(\<And>x. \<not> x islimpt S) \<Longrightarrow> continuous_on S f"
   unfolding continuous_on_def
   by (metis UNIV_I empty_iff eventually_at_topological islimptE open_UNIV tendsto_def trivial_limit_within)
 
 lemma continuous_on_finite:
   fixes S :: "'a::t1_space set"
   shows "finite S \<Longrightarrow> continuous_on S f"
 by (metis continuous_on_no_limpt islimpt_finite)
 
 lemma homeomorphic_finite:
   fixes S :: "'a::t1_space set" and T :: "'b::t1_space set"
   assumes "finite T"
   shows "S homeomorphic T \<longleftrightarrow> finite S \<and> finite T \<and> card S = card T" (is "?lhs = ?rhs")
 proof
   assume "S homeomorphic T"
   with assms show ?rhs
     apply (auto simp: homeomorphic_def homeomorphism_def)
      apply (metis finite_imageI)
     by (metis card_image_le finite_imageI le_antisym)
 next
   assume R: ?rhs
   with finite_same_card_bij obtain h where "bij_betw h S T"
     by auto
   with R show ?lhs
     apply (auto simp: homeomorphic_def homeomorphism_def continuous_on_finite)
     apply (rule_tac x=h in exI)
     apply (rule_tac x="inv_into S h" in exI)
     apply (auto simp:  bij_betw_inv_into_left bij_betw_inv_into_right bij_betw_imp_surj_on inv_into_into bij_betwE)
     apply (metis bij_betw_def bij_betw_inv_into)
     done
 qed
 
 text \<open>Relatively weak hypotheses if a set is compact.\<close>
 
 lemma homeomorphism_compact:
   fixes f :: "'a::topological_space \<Rightarrow> 'b::t2_space"
   assumes "compact s" "continuous_on s f"  "f ` s = t"  "inj_on f s"
   shows "\<exists>g. homeomorphism s t f g"
 proof -
   define g where "g x = (SOME y. y\<in>s \<and> f y = x)" for x
   have g: "\<forall>x\<in>s. g (f x) = x"
     using assms(3) assms(4)[unfolded inj_on_def] unfolding g_def by auto
   {
     fix y
     assume "y \<in> t"
     then obtain x where x:"f x = y" "x\<in>s"
       using assms(3) by auto
     then have "g (f x) = x" using g by auto
     then have "f (g y) = y" unfolding x(1)[symmetric] by auto
   }
   then have g':"\<forall>x\<in>t. f (g x) = x" by auto
   moreover
   {
     fix x
     have "x\<in>s \<Longrightarrow> x \<in> g ` t"
       using g[THEN bspec[where x=x]]
       unfolding image_iff
       using assms(3)
       by (auto intro!: bexI[where x="f x"])
     moreover
     {
       assume "x\<in>g ` t"
       then obtain y where y:"y\<in>t" "g y = x" by auto
       then obtain x' where x':"x'\<in>s" "f x' = y"
         using assms(3) by auto
       then have "x \<in> s"
         unfolding g_def
         using someI2[of "\<lambda>b. b\<in>s \<and> f b = y" x' "\<lambda>x. x\<in>s"]
         unfolding y(2)[symmetric] and g_def
         by auto
     }
     ultimately have "x\<in>s \<longleftrightarrow> x \<in> g ` t" ..
   }
   then have "g ` t = s" by auto
   ultimately show ?thesis
     unfolding homeomorphism_def homeomorphic_def
     apply (rule_tac x=g in exI)
     using g and assms(3) and continuous_on_inv[OF assms(2,1), of g, unfolded assms(3)] and assms(2)
     apply auto
     done
 qed
 
 lemma homeomorphic_compact:
   fixes f :: "'a::topological_space \<Rightarrow> 'b::t2_space"
   shows "compact s \<Longrightarrow> continuous_on s f \<Longrightarrow> (f ` s = t) \<Longrightarrow> inj_on f s \<Longrightarrow> s homeomorphic t"
   unfolding homeomorphic_def by (metis homeomorphism_compact)
 
 text\<open>Preservation of topological properties.\<close>
 
 lemma homeomorphic_compactness: "s homeomorphic t \<Longrightarrow> (compact s \<longleftrightarrow> compact t)"
   unfolding homeomorphic_def homeomorphism_def
   by (metis compact_continuous_image)
 
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>On Linorder Topologies\<close>
 
 lemma islimpt_greaterThanLessThan1:
   fixes a b::"'a::{linorder_topology, dense_order}"
   assumes "a < b"
   shows  "a islimpt {a<..<b}"
 proof (rule islimptI)
   fix T
   assume "open T" "a \<in> T"
   from open_right[OF this \<open>a < b\<close>]
   obtain c where c: "a < c" "{a..<c} \<subseteq> T" by auto
   with assms dense[of a "min c b"]
   show "\<exists>y\<in>{a<..<b}. y \<in> T \<and> y \<noteq> a"
     by (metis atLeastLessThan_iff greaterThanLessThan_iff min_less_iff_conj
       not_le order.strict_implies_order subset_eq)
 qed
 
 lemma islimpt_greaterThanLessThan2:
   fixes a b::"'a::{linorder_topology, dense_order}"
   assumes "a < b"
   shows  "b islimpt {a<..<b}"
 proof (rule islimptI)
   fix T
   assume "open T" "b \<in> T"
   from open_left[OF this \<open>a < b\<close>]
   obtain c where c: "c < b" "{c<..b} \<subseteq> T" by auto
   with assms dense[of "max a c" b]
   show "\<exists>y\<in>{a<..<b}. y \<in> T \<and> y \<noteq> b"
     by (metis greaterThanAtMost_iff greaterThanLessThan_iff max_less_iff_conj
       not_le order.strict_implies_order subset_eq)
 qed
 
 lemma closure_greaterThanLessThan[simp]:
   fixes a b::"'a::{linorder_topology, dense_order}"
   shows "a < b \<Longrightarrow> closure {a <..< b} = {a .. b}" (is "_ \<Longrightarrow> ?l = ?r")
 proof
   have "?l \<subseteq> closure ?r"
     by (rule closure_mono) auto
   thus "closure {a<..<b} \<subseteq> {a..b}" by simp
 qed (auto simp: closure_def order.order_iff_strict islimpt_greaterThanLessThan1
   islimpt_greaterThanLessThan2)
 
 lemma closure_greaterThan[simp]:
   fixes a b::"'a::{no_top, linorder_topology, dense_order}"
   shows "closure {a<..} = {a..}"
 proof -
   from gt_ex obtain b where "a < b" by auto
   hence "{a<..} = {a<..<b} \<union> {b..}" by auto
   also have "closure \<dots> = {a..}" using \<open>a < b\<close> unfolding closure_Un
     by auto
   finally show ?thesis .
 qed
 
 lemma closure_lessThan[simp]:
   fixes b::"'a::{no_bot, linorder_topology, dense_order}"
   shows "closure {..<b} = {..b}"
 proof -
   from lt_ex obtain a where "a < b" by auto
   hence "{..<b} = {a<..<b} \<union> {..a}" by auto
   also have "closure \<dots> = {..b}" using \<open>a < b\<close> unfolding closure_Un
     by auto
   finally show ?thesis .
 qed
 
 lemma closure_atLeastLessThan[simp]:
   fixes a b::"'a::{linorder_topology, dense_order}"
   assumes "a < b"
   shows "closure {a ..< b} = {a .. b}"
 proof -
   from assms have "{a ..< b} = {a} \<union> {a <..< b}" by auto
   also have "closure \<dots> = {a .. b}" unfolding closure_Un
     by (auto simp: assms less_imp_le)
   finally show ?thesis .
 qed
 
 lemma closure_greaterThanAtMost[simp]:
   fixes a b::"'a::{linorder_topology, dense_order}"
   assumes "a < b"
   shows "closure {a <.. b} = {a .. b}"
 proof -
   from assms have "{a <.. b} = {b} \<union> {a <..< b}" by auto
   also have "closure \<dots> = {a .. b}" unfolding closure_Un
     by (auto simp: assms less_imp_le)
   finally show ?thesis .
 qed
 
 end
\ No newline at end of file
diff --git a/src/HOL/Analysis/Starlike.thy b/src/HOL/Analysis/Starlike.thy
--- a/src/HOL/Analysis/Starlike.thy
+++ b/src/HOL/Analysis/Starlike.thy
@@ -1,6917 +1,6884 @@
 (* 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
     Abstract_Limits
     Line_Segment
 begin
 
 subsection\<open>Starlike sets\<close>
 
 definition\<^marker>\<open>tag important\<close> "starlike S \<longleftrightarrow> (\<exists>a\<in>S. \<forall>x\<in>S. closed_segment a x \<subseteq> S)"
 
 lemma starlike_UNIV [simp]: "starlike UNIV"
   by (simp add: starlike_def)
 
 lemma convex_imp_starlike:
   "convex S \<Longrightarrow> S \<noteq> {} \<Longrightarrow> starlike S"
   unfolding convex_contains_segment starlike_def by auto
 
 
 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>More results about segments\<close>
 
 lemma dist_half_times2:
   fixes a :: "'a :: real_normed_vector"
   shows "dist ((1 / 2) *\<^sub>R (a + b)) x * 2 = dist (a+b) (2 *\<^sub>R x)"
 proof -
   have "norm ((1 / 2) *\<^sub>R (a + b) - x) * 2 = norm (2 *\<^sub>R ((1 / 2) *\<^sub>R (a + b) - x))"
     by simp
   also have "... = norm ((a + b) - 2 *\<^sub>R x)"
     by (simp add: real_vector.scale_right_diff_distrib)
   finally show ?thesis
     by (simp only: dist_norm)
 qed
 
 lemma closed_segment_as_ball:
     "closed_segment a b = affine hull {a,b} \<inter> cball(inverse 2 *\<^sub>R (a + b))(norm(b - a) / 2)"
 proof (cases "b = a")
   case True then show ?thesis by (auto simp: hull_inc)
 next
   case False
   then have *: "((\<exists>u v. x = u *\<^sub>R a + v *\<^sub>R b \<and> u + v = 1) \<and>
                   dist ((1 / 2) *\<^sub>R (a + b)) x * 2 \<le> norm (b - a)) =
                  (\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and> 0 \<le> u \<and> u \<le> 1)" for x
   proof -
     have "((\<exists>u v. x = u *\<^sub>R a + v *\<^sub>R b \<and> u + v = 1) \<and>
                   dist ((1 / 2) *\<^sub>R (a + b)) x * 2 \<le> norm (b - a)) =
           ((\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b) \<and>
                   dist ((1 / 2) *\<^sub>R (a + b)) x * 2 \<le> norm (b - a))"
       unfolding eq_diff_eq [symmetric] by simp
     also have "... = (\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and>
                           norm ((a+b) - (2 *\<^sub>R x)) \<le> norm (b - a))"
       by (simp add: dist_half_times2) (simp add: dist_norm)
     also have "... = (\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and>
             norm ((a+b) - (2 *\<^sub>R ((1 - u) *\<^sub>R a + u *\<^sub>R b))) \<le> norm (b - a))"
       by auto
     also have "... = (\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and>
                 norm ((1 - u * 2) *\<^sub>R (b - a)) \<le> norm (b - a))"
       by (simp add: algebra_simps scaleR_2)
     also have "... = (\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and>
                           \<bar>1 - u * 2\<bar> * norm (b - a) \<le> norm (b - a))"
       by simp
     also have "... = (\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and> \<bar>1 - u * 2\<bar> \<le> 1)"
       by (simp add: mult_le_cancel_right2 False)
     also have "... = (\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and> 0 \<le> u \<and> u \<le> 1)"
       by auto
     finally show ?thesis .
   qed
   show ?thesis
     by (simp add: affine_hull_2 Set.set_eq_iff closed_segment_def *)
 qed
 
 lemma open_segment_as_ball:
     "open_segment a b =
      affine hull {a,b} \<inter> ball(inverse 2 *\<^sub>R (a + b))(norm(b - a) / 2)"
 proof (cases "b = a")
   case True then show ?thesis by (auto simp: hull_inc)
 next
   case False
   then have *: "((\<exists>u v. x = u *\<^sub>R a + v *\<^sub>R b \<and> u + v = 1) \<and>
                   dist ((1 / 2) *\<^sub>R (a + b)) x * 2 < norm (b - a)) =
                  (\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and> 0 < u \<and> u < 1)" for x
   proof -
     have "((\<exists>u v. x = u *\<^sub>R a + v *\<^sub>R b \<and> u + v = 1) \<and>
                   dist ((1 / 2) *\<^sub>R (a + b)) x * 2 < norm (b - a)) =
           ((\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b) \<and>
                   dist ((1 / 2) *\<^sub>R (a + b)) x * 2 < norm (b - a))"
       unfolding eq_diff_eq [symmetric] by simp
     also have "... = (\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and>
                           norm ((a+b) - (2 *\<^sub>R x)) < norm (b - a))"
       by (simp add: dist_half_times2) (simp add: dist_norm)
     also have "... = (\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and>
             norm ((a+b) - (2 *\<^sub>R ((1 - u) *\<^sub>R a + u *\<^sub>R b))) < norm (b - a))"
       by auto
     also have "... = (\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and>
                 norm ((1 - u * 2) *\<^sub>R (b - a)) < norm (b - a))"
       by (simp add: algebra_simps scaleR_2)
     also have "... = (\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and>
                           \<bar>1 - u * 2\<bar> * norm (b - a) < norm (b - a))"
       by simp
     also have "... = (\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and> \<bar>1 - u * 2\<bar> < 1)"
       by (simp add: mult_le_cancel_right2 False)
     also have "... = (\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and> 0 < u \<and> u < 1)"
       by auto
     finally show ?thesis .
   qed
   show ?thesis
     using False by (force simp: affine_hull_2 Set.set_eq_iff open_segment_image_interval *)
 qed
 
 lemmas segment_as_ball = closed_segment_as_ball open_segment_as_ball
 
 lemma closed_segment_neq_empty [simp]: "closed_segment a b \<noteq> {}"
   by auto
 
 lemma open_segment_eq_empty [simp]: "open_segment a b = {} \<longleftrightarrow> a = b"
 proof -
   { assume a1: "open_segment a b = {}"
     have "{} \<noteq> {0::real<..<1}"
       by simp
     then have "a = b"
       using a1 open_segment_image_interval by fastforce
   } then show ?thesis by auto
 qed
 
 lemma open_segment_eq_empty' [simp]: "{} = open_segment a b \<longleftrightarrow> a = b"
   using open_segment_eq_empty by blast
 
 lemmas segment_eq_empty = closed_segment_neq_empty open_segment_eq_empty
 
 lemma inj_segment:
   fixes a :: "'a :: real_vector"
   assumes "a \<noteq> b"
     shows "inj_on (\<lambda>u. (1 - u) *\<^sub>R a + u *\<^sub>R b) I"
 proof
   fix x y
   assume "(1 - x) *\<^sub>R a + x *\<^sub>R b = (1 - y) *\<^sub>R a + y *\<^sub>R b"
   then have "x *\<^sub>R (b - a) = y *\<^sub>R (b - a)"
     by (simp add: algebra_simps)
   with assms show "x = y"
     by (simp add: real_vector.scale_right_imp_eq)
 qed
 
 lemma finite_closed_segment [simp]: "finite(closed_segment a b) \<longleftrightarrow> a = b"
   apply auto
   apply (rule ccontr)
   apply (simp add: segment_image_interval)
   using infinite_Icc [OF zero_less_one] finite_imageD [OF _ inj_segment] apply blast
   done
 
 lemma finite_open_segment [simp]: "finite(open_segment a b) \<longleftrightarrow> a = b"
   by (auto simp: open_segment_def)
 
 lemmas finite_segment = finite_closed_segment finite_open_segment
 
 lemma closed_segment_eq_sing: "closed_segment a b = {c} \<longleftrightarrow> a = c \<and> b = c"
   by auto
 
 lemma open_segment_eq_sing: "open_segment a b \<noteq> {c}"
   by (metis finite_insert finite_open_segment insert_not_empty open_segment_image_interval)
 
 lemmas segment_eq_sing = closed_segment_eq_sing open_segment_eq_sing
 
 lemma subset_closed_segment:
     "closed_segment a b \<subseteq> closed_segment c d \<longleftrightarrow>
      a \<in> closed_segment c d \<and> b \<in> closed_segment c d"
   by auto (meson contra_subsetD convex_closed_segment convex_contains_segment)
 
 lemma subset_co_segment:
     "closed_segment a b \<subseteq> open_segment c d \<longleftrightarrow>
      a \<in> open_segment c d \<and> b \<in> open_segment c d"
 using closed_segment_subset by blast
 
 lemma subset_open_segment:
   fixes a :: "'a::euclidean_space"
   shows "open_segment a b \<subseteq> open_segment c d \<longleftrightarrow>
          a = b \<or> a \<in> closed_segment c d \<and> b \<in> closed_segment c d"
         (is "?lhs = ?rhs")
 proof (cases "a = b")
   case True then show ?thesis by simp
 next
   case False show ?thesis
   proof
     assume rhs: ?rhs
     with \<open>a \<noteq> b\<close> have "c \<noteq> d"
       using closed_segment_idem singleton_iff by auto
     have "\<exists>uc. (1 - u) *\<^sub>R ((1 - ua) *\<^sub>R c + ua *\<^sub>R d) + u *\<^sub>R ((1 - ub) *\<^sub>R c + ub *\<^sub>R d) =
                (1 - uc) *\<^sub>R c + uc *\<^sub>R d \<and> 0 < uc \<and> uc < 1"
         if neq: "(1 - ua) *\<^sub>R c + ua *\<^sub>R d \<noteq> (1 - ub) *\<^sub>R c + ub *\<^sub>R d" "c \<noteq> d"
            and "a = (1 - ua) *\<^sub>R c + ua *\<^sub>R d" "b = (1 - ub) *\<^sub>R c + ub *\<^sub>R d"
            and u: "0 < u" "u < 1" and uab: "0 \<le> ua" "ua \<le> 1" "0 \<le> ub" "ub \<le> 1"
         for u ua ub
     proof -
       have "ua \<noteq> ub"
         using neq by auto
       moreover have "(u - 1) * ua \<le> 0" using u uab
         by (simp add: mult_nonpos_nonneg)
       ultimately have lt: "(u - 1) * ua < u * ub" using u uab
         by (metis antisym_conv diff_ge_0_iff_ge le_less_trans mult_eq_0_iff mult_le_0_iff not_less)
       have "p * ua + q * ub < p+q" if p: "0 < p" and  q: "0 < q" for p q
       proof -
         have "\<not> p \<le> 0" "\<not> q \<le> 0"
           using p q not_less by blast+
         then show ?thesis
           by (metis \<open>ua \<noteq> ub\<close> add_less_cancel_left add_less_cancel_right add_mono_thms_linordered_field(5)
                     less_eq_real_def mult_cancel_left1 mult_less_cancel_left2 uab(2) uab(4))
       qed
       then have "(1 - u) * ua + u * ub < 1" using u \<open>ua \<noteq> ub\<close>
         by (metis diff_add_cancel diff_gt_0_iff_gt)
       with lt show ?thesis
         by (rule_tac x="ua + u*(ub-ua)" in exI) (simp add: algebra_simps)
     qed
     with rhs \<open>a \<noteq> b\<close> \<open>c \<noteq> d\<close> show ?lhs
       unfolding open_segment_image_interval closed_segment_def
       by (fastforce simp add:)
   next
     assume lhs: ?lhs
     with \<open>a \<noteq> b\<close> have "c \<noteq> d"
       by (meson finite_open_segment rev_finite_subset)
     have "closure (open_segment a b) \<subseteq> closure (open_segment c d)"
       using lhs closure_mono by blast
     then have "closed_segment a b \<subseteq> closed_segment c d"
       by (simp add: \<open>a \<noteq> b\<close> \<open>c \<noteq> d\<close>)
     then show ?rhs
       by (force simp: \<open>a \<noteq> b\<close>)
   qed
 qed
 
 lemma subset_oc_segment:
   fixes a :: "'a::euclidean_space"
   shows "open_segment a b \<subseteq> closed_segment c d \<longleftrightarrow>
          a = b \<or> a \<in> closed_segment c d \<and> b \<in> closed_segment c d"
 apply (simp add: subset_open_segment [symmetric])
 apply (rule iffI)
  apply (metis closure_closed_segment closure_mono closure_open_segment subset_closed_segment subset_open_segment)
 apply (meson dual_order.trans segment_open_subset_closed)
 done
 
 lemmas subset_segment = subset_closed_segment subset_co_segment subset_oc_segment subset_open_segment
 
 
 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 closure_open_Int_superset:
-  assumes "open S" "S \<subseteq> closure T"
-  shows "closure(S \<inter> T) = closure S"
-proof -
-  have "closure S \<subseteq> closure(S \<inter> T)"
-    by (metis assms closed_closure closure_minimal inf.orderE open_Int_closure_subset)
-  then show ?thesis
-    by (simp add: closure_mono dual_order.antisym)
-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
 
 lemma starlike_convex_tweak_boundary_points:
   fixes S :: "'a::euclidean_space set"
   assumes "convex S" "S \<noteq> {}" and ST: "rel_interior S \<subseteq> T" and TS: "T \<subseteq> closure S"
   shows "starlike T"
 proof -
   have "rel_interior S \<noteq> {}"
     by (simp add: assms rel_interior_eq_empty)
   then obtain a where a: "a \<in> rel_interior S"  by blast
   with ST have "a \<in> T"  by blast
   have *: "\<And>x. x \<in> T \<Longrightarrow> open_segment a x \<subseteq> rel_interior S"
     apply (rule rel_interior_closure_convex_segment [OF \<open>convex S\<close> a])
     using assms by blast
   show ?thesis
     unfolding starlike_def
     apply (rule bexI [OF _ \<open>a \<in> T\<close>])
     apply (simp add: closed_segment_eq_open)
     apply (intro conjI ballI a \<open>a \<in> T\<close> rel_interior_closure_convex_segment [OF \<open>convex S\<close> a])
     apply (simp add: order_trans [OF * ST])
     done
 qed
 
 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 closure_Int: "closure (\<Inter>I) \<le> \<Inter>{closure S |S. S \<in> I}"
-proof -
-  {
-    fix y
-    assume "y \<in> \<Inter>I"
-    then have y: "\<forall>S \<in> I. y \<in> S" by auto
-    {
-      fix S
-      assume "S \<in> I"
-      then have "y \<in> closure S"
-        using closure_subset y by auto
-    }
-    then have "y \<in> \<Inter>{closure S |S. S \<in> I}"
-      by auto
-  }
-  then have "\<Inter>I \<subseteq> \<Inter>{closure S |S. S \<in> I}"
-    by auto
-  moreover have "closed (\<Inter>{closure S |S. S \<in> I})"
-    unfolding closed_Inter closed_closure by auto
-  ultimately show ?thesis using closure_hull[of "\<Inter>I"]
-    hull_minimal[of "\<Inter>I" "\<Inter>{closure S |S. S \<in> I}" "closed"] 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
     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
     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
   {
     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
       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_eq_empty [simp]: "affine hull S = {} \<longleftrightarrow> S = {}"
   using affine_hull_nonempty by blast
 
 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
 
 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
 
 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