src/HOL/Analysis/Polytope.thy

new material for Analysis

+
−section \<open>Faces, Extreme Points, Polytopes, Polyhedra etc\<close>
+
−
+
−text\<open>Ported from HOL Light by L C Paulson\<close>
+
−
+
−theory Polytope
+
−imports Cartesian_Euclidean_Space
+
−begin
+
−
+
−subsection \<open>Faces of a (usually convex) set\<close>
+
−
+
−definition%important face_of :: "['a::real_vector set, 'a set] \<Rightarrow> bool" (infixr "(face'_of)" 50)
+
−  where
+
−  "T face_of S \<longleftrightarrow>
+
−        T \<subseteq> S \<and> convex T \<and>
+
−        (\<forall>a \<in> S. \<forall>b \<in> S. \<forall>x \<in> T. x \<in> open_segment a b \<longrightarrow> a \<in> T \<and> b \<in> T)"
+
−
+
−lemma face_ofD: "\<lbrakk>T face_of S; x \<in> open_segment a b; a \<in> S; b \<in> S; x \<in> T\<rbrakk> \<Longrightarrow> a \<in> T \<and> b \<in> T"
+
−  unfolding face_of_def by blast
+
−
+
−lemma face_of_translation_eq [simp]:
+
−    "((+) a ` T face_of (+) a ` S) \<longleftrightarrow> T face_of S"
+
−proof -
+
−  have *: "\<And>a T S. T face_of S \<Longrightarrow> ((+) a ` T face_of (+) a ` S)"
+
−    apply (simp add: face_of_def Ball_def, clarify)
+
−    apply (drule open_segment_translation_eq [THEN iffD1])
+
−    using inj_image_mem_iff inj_add_left apply metis
+
−    done
+
−  show ?thesis
+
−    apply (rule iffI)
+
−    apply (force simp: image_comp o_def dest: * [where a = "-a"])
+
−    apply (blast intro: *)
+
−    done
+
−qed
+
−
+
−lemma face_of_linear_image:
+
−  assumes "linear f" "inj f"
+
−    shows "(f ` c face_of f ` S) \<longleftrightarrow> c face_of S"
+
−by (simp add: face_of_def inj_image_subset_iff inj_image_mem_iff open_segment_linear_image assms)
+
−
+
−lemma face_of_refl: "convex S \<Longrightarrow> S face_of S"
+
−  by (auto simp: face_of_def)
+
−
+
−lemma face_of_refl_eq: "S face_of S \<longleftrightarrow> convex S"
+
−  by (auto simp: face_of_def)
+
−
+
−lemma empty_face_of [iff]: "{} face_of S"
+
−  by (simp add: face_of_def)
+
−
+
−lemma face_of_empty [simp]: "S face_of {} \<longleftrightarrow> S = {}"
+
−  by (meson empty_face_of face_of_def subset_empty)
+
−
+
−lemma face_of_trans [trans]: "\<lbrakk>S face_of T; T face_of u\<rbrakk> \<Longrightarrow> S face_of u"
+
−  unfolding face_of_def by (safe; blast)
+
−
+
−lemma face_of_face: "T face_of S \<Longrightarrow> (f face_of T \<longleftrightarrow> f face_of S \<and> f \<subseteq> T)"
+
−  unfolding face_of_def by (safe; blast)
+
−
+
−lemma face_of_subset: "\<lbrakk>F face_of S; F \<subseteq> T; T \<subseteq> S\<rbrakk> \<Longrightarrow> F face_of T"
+
−  unfolding face_of_def by (safe; blast)
+
−
+
−lemma face_of_slice: "\<lbrakk>F face_of S; convex T\<rbrakk> \<Longrightarrow> (F \<inter> T) face_of (S \<inter> T)"
+
−  unfolding face_of_def by (blast intro: convex_Int)
+
−
+
−lemma face_of_Int: "\<lbrakk>t1 face_of S; t2 face_of S\<rbrakk> \<Longrightarrow> (t1 \<inter> t2) face_of S"
+
−  unfolding face_of_def by (blast intro: convex_Int)
+
−
+
−lemma face_of_Inter: "\<lbrakk>A \<noteq> {}; \<And>T. T \<in> A \<Longrightarrow> T face_of S\<rbrakk> \<Longrightarrow> (\<Inter> A) face_of S"
+
−  unfolding face_of_def by (blast intro: convex_Inter)
+
−
+
−lemma face_of_Int_Int: "\<lbrakk>F face_of T; F' face_of t'\<rbrakk> \<Longrightarrow> (F \<inter> F') face_of (T \<inter> t')"
+
−  unfolding face_of_def by (blast intro: convex_Int)
+
−
+
−lemma face_of_imp_subset: "T face_of S \<Longrightarrow> T \<subseteq> S"
+
−  unfolding face_of_def by blast
+
−
+
−proposition face_of_imp_eq_affine_Int:
+
−  fixes S :: "'a::euclidean_space set"
+
−  assumes S: "convex S"  and T: "T face_of S"
+
−  shows "T = (affine hull T) \<inter> S"
+
−proof -
+
−  have "convex T" using T by (simp add: face_of_def)
+
−  have *: False if x: "x \<in> affine hull T" and "x \<in> S" "x \<notin> T" and y: "y \<in> rel_interior T" for x y
+
−  proof -
+
−    obtain e where "e>0" and e: "cball y e \<inter> affine hull T \<subseteq> T"
+
−      using y by (auto simp: rel_interior_cball)
+
−    have "y \<noteq> x" "y \<in> S" "y \<in> T"
+
−      using face_of_imp_subset rel_interior_subset T that by blast+
+
−    then have zne: "\<And>u. \<lbrakk>u \<in> {0<..<1}; (1 - u) *\<^sub>R y + u *\<^sub>R x \<in> T\<rbrakk> \<Longrightarrow>  False"
+
−      using \<open>x \<in> S\<close> \<open>x \<notin> T\<close> \<open>T face_of S\<close> unfolding face_of_def
+
−      apply clarify
+
−      apply (drule_tac x=x in bspec, assumption)
+
−      apply (drule_tac x=y in bspec, assumption)
+
−      apply (subst (asm) open_segment_commute)
+
−      apply (force simp: open_segment_image_interval image_def)
+
−      done
+
−    have in01: "min (1/2) (e / norm (x - y)) \<in> {0<..<1}"
+
−      using \<open>y \<noteq> x\<close> \<open>e > 0\<close> by simp
+
−    show ?thesis
+
−      apply (rule zne [OF in01])
+
−      apply (rule e [THEN subsetD])
+
−      apply (rule IntI)
+
−        using \<open>y \<noteq> x\<close> \<open>e > 0\<close>
+
−        apply (simp add: cball_def dist_norm algebra_simps)
+
−        apply (simp add: Real_Vector_Spaces.scaleR_diff_right [symmetric] norm_minus_commute min_mult_distrib_right)
+
−      apply (rule mem_affine [OF affine_affine_hull _ x])
+
−      using \<open>y \<in> T\<close>  apply (auto simp: hull_inc)
+
−      done
+
−  qed
+
−  show ?thesis
+
−    apply (rule subset_antisym)
+
−    using assms apply (simp add: hull_subset face_of_imp_subset)
+
−    apply (cases "T={}", simp)
+
−    apply (force simp: rel_interior_eq_empty [symmetric] \<open>convex T\<close> intro: *)
+
−    done
+
−qed
+
−
+
−lemma face_of_imp_closed:
+
−     fixes S :: "'a::euclidean_space set"
+
−     assumes "convex S" "closed S" "T face_of S" shows "closed T"
+
−  by (metis affine_affine_hull affine_closed closed_Int face_of_imp_eq_affine_Int assms)
+
−
+
−lemma face_of_Int_supporting_hyperplane_le_strong:
+
−    assumes "convex(S \<inter> {x. a \<bullet> x = b})" and aleb: "\<And>x. x \<in> S \<Longrightarrow> a \<bullet> x \<le> b"
+
−      shows "(S \<inter> {x. a \<bullet> x = b}) face_of S"
+
−proof -
+
−  have *: "a \<bullet> u = a \<bullet> x" if "x \<in> open_segment u v" "u \<in> S" "v \<in> S" and b: "b = a \<bullet> x"
+
−          for u v x
+
−  proof (rule antisym)
+
−    show "a \<bullet> u \<le> a \<bullet> x"
+
−      using aleb \<open>u \<in> S\<close> \<open>b = a \<bullet> x\<close> by blast
+
−  next
+
−    obtain \<xi> where "b = a \<bullet> ((1 - \<xi>) *\<^sub>R u + \<xi> *\<^sub>R v)" "0 < \<xi>" "\<xi> < 1"
+
−      using \<open>b = a \<bullet> x\<close> \<open>x \<in> open_segment u v\<close> in_segment
+
−      by (auto simp: open_segment_image_interval split: if_split_asm)
+
−    then have "b + \<xi> * (a \<bullet> u) \<le> a \<bullet> u + \<xi> * b"
+
−      using aleb [OF \<open>v \<in> S\<close>] by (simp add: algebra_simps)
+
−    then have "(1 - \<xi>) * b \<le> (1 - \<xi>) * (a \<bullet> u)"
+
−      by (simp add: algebra_simps)
+
−    then have "b \<le> a \<bullet> u"
+
−      using \<open>\<xi> < 1\<close> by auto
+
−    with b show "a \<bullet> x \<le> a \<bullet> u" by simp
+
−  qed
+
−  show ?thesis
+
−    apply (simp add: face_of_def assms)
+
−    using "*" open_segment_commute by blast
+
−qed
+
−
+
−lemma face_of_Int_supporting_hyperplane_ge_strong:
+
−   "\<lbrakk>convex(S \<inter> {x. a \<bullet> x = b}); \<And>x. x \<in> S \<Longrightarrow> a \<bullet> x \<ge> b\<rbrakk>
+
−    \<Longrightarrow> (S \<inter> {x. a \<bullet> x = b}) face_of S"
+
−  using face_of_Int_supporting_hyperplane_le_strong [of S "-a" "-b"] by simp
+
−
+
−lemma face_of_Int_supporting_hyperplane_le:
+
−    "\<lbrakk>convex S; \<And>x. x \<in> S \<Longrightarrow> a \<bullet> x \<le> b\<rbrakk> \<Longrightarrow> (S \<inter> {x. a \<bullet> x = b}) face_of S"
+
−  by (simp add: convex_Int convex_hyperplane face_of_Int_supporting_hyperplane_le_strong)
+
−
+
−lemma face_of_Int_supporting_hyperplane_ge:
+
−    "\<lbrakk>convex S; \<And>x. x \<in> S \<Longrightarrow> a \<bullet> x \<ge> b\<rbrakk> \<Longrightarrow> (S \<inter> {x. a \<bullet> x = b}) face_of S"
+
−  by (simp add: convex_Int convex_hyperplane face_of_Int_supporting_hyperplane_ge_strong)
+
−
+
−lemma face_of_imp_convex: "T face_of S \<Longrightarrow> convex T"
+
−  using face_of_def by blast
+
−
+
−lemma face_of_imp_compact:
+
−    fixes S :: "'a::euclidean_space set"
+
−    shows "\<lbrakk>convex S; compact S; T face_of S\<rbrakk> \<Longrightarrow> compact T"
+
−  by (meson bounded_subset compact_eq_bounded_closed face_of_imp_closed face_of_imp_subset)
+
−
+
−lemma face_of_Int_subface:
+
−     "\<lbrakk>A \<inter> B face_of A; A \<inter> B face_of B; C face_of A; D face_of B\<rbrakk>
+
−      \<Longrightarrow> (C \<inter> D) face_of C \<and> (C \<inter> D) face_of D"
+
−  by (meson face_of_Int_Int face_of_face inf_le1 inf_le2)
+
−
+
−lemma subset_of_face_of:
+
−    fixes S :: "'a::real_normed_vector set"
+
−    assumes "T face_of S" "u \<subseteq> S" "T \<inter> (rel_interior u) \<noteq> {}"
+
−      shows "u \<subseteq> T"
+
−proof
+
−  fix c
+
−  assume "c \<in> u"
+
−  obtain b where "b \<in> T" "b \<in> rel_interior u" using assms by auto
+
−  then obtain e where "e>0" "b \<in> u" and e: "cball b e \<inter> affine hull u \<subseteq> u"
+
−    by (auto simp: rel_interior_cball)
+
−  show "c \<in> T"
+
−  proof (cases "b=c")
+
−    case True with \<open>b \<in> T\<close> show ?thesis by blast
+
−  next
+
−    case False
+
−    define d where "d = b + (e / norm(b - c)) *\<^sub>R (b - c)"
+
−    have "d \<in> cball b e \<inter> affine hull u"
+
−      using \<open>e > 0\<close> \<open>b \<in> u\<close> \<open>c \<in> u\<close>
+
−      by (simp add: d_def dist_norm hull_inc mem_affine_3_minus False)
+
−    with e have "d \<in> u" by blast
+
−    have nbc: "norm (b - c) + e > 0" using \<open>e > 0\<close>
+
−      by (metis add.commute le_less_trans less_add_same_cancel2 norm_ge_zero)
+
−    then have [simp]: "d \<noteq> c" using False scaleR_cancel_left [of "1 + (e / norm (b - c))" b c]
+
−      by (simp add: algebra_simps d_def) (simp add: divide_simps)
+
−    have [simp]: "((e - e * e / (e + norm (b - c))) / norm (b - c)) = (e / (e + norm (b - c)))"
+
−      using False nbc
+
−      by (simp add: divide_simps) (simp add: algebra_simps)
+
−    have "b \<in> open_segment d c"
+
−      apply (simp add: open_segment_image_interval)
+
−      apply (simp add: d_def algebra_simps image_def)
+
−      apply (rule_tac x="e / (e + norm (b - c))" in bexI)
+
−      using False nbc \<open>0 < e\<close>
+
−      apply (auto simp: algebra_simps)
+
−      done
+
−    then have "d \<in> T \<and> c \<in> T"
+
−      apply (rule face_ofD [OF \<open>T face_of S\<close>])
+
−      using \<open>d \<in> u\<close>  \<open>c \<in> u\<close> \<open>u \<subseteq> S\<close>  \<open>b \<in> T\<close>  apply auto
+
−      done
+
−    then show ?thesis ..
+
−  qed
+
−qed
+
−
+
−lemma face_of_eq:
+
−    fixes S :: "'a::real_normed_vector set"
+
−    assumes "T face_of S" "u face_of S" "(rel_interior T) \<inter> (rel_interior u) \<noteq> {}"
+
−      shows "T = u"
+
−  apply (rule subset_antisym)
+
−  apply (metis assms disjoint_iff_not_equal face_of_imp_subset rel_interior_subset subsetCE subset_of_face_of)
+
−  by (metis assms disjoint_iff_not_equal face_of_imp_subset rel_interior_subset subset_iff subset_of_face_of)
+
−
+
−lemma face_of_disjoint_rel_interior:
+
−      fixes S :: "'a::real_normed_vector set"
+
−      assumes "T face_of S" "T \<noteq> S"
+
−        shows "T \<inter> rel_interior S = {}"
+
−  by (meson assms subset_of_face_of face_of_imp_subset order_refl subset_antisym)
+
−
+
−lemma face_of_disjoint_interior:
+
−      fixes S :: "'a::real_normed_vector set"
+
−      assumes "T face_of S" "T \<noteq> S"
+
−        shows "T \<inter> interior S = {}"
+
−proof -
+
−  have "T \<inter> interior S \<subseteq> rel_interior S"
+
−    by (meson inf_sup_ord(2) interior_subset_rel_interior order.trans)
+
−  thus ?thesis
+
−    by (metis (no_types) Int_greatest assms face_of_disjoint_rel_interior inf_sup_ord(1) subset_empty)
+
−qed
+
−
+
−lemma face_of_subset_rel_boundary:
+
−  fixes S :: "'a::real_normed_vector set"
+
−  assumes "T face_of S" "T \<noteq> S"
+
−    shows "T \<subseteq> (S - rel_interior S)"
+
−by (meson DiffI assms disjoint_iff_not_equal face_of_disjoint_rel_interior face_of_imp_subset rev_subsetD subsetI)
+
−
+
−lemma face_of_subset_rel_frontier:
+
−    fixes S :: "'a::real_normed_vector set"
+
−    assumes "T face_of S" "T \<noteq> S"
+
−      shows "T \<subseteq> rel_frontier S"
+
−  using assms closure_subset face_of_disjoint_rel_interior face_of_imp_subset rel_frontier_def by fastforce
+
−
+
−lemma face_of_aff_dim_lt:
+
−  fixes S :: "'a::euclidean_space set"
+
−  assumes "convex S" "T face_of S" "T \<noteq> S"
+
−    shows "aff_dim T < aff_dim S"
+
−proof -
+
−  have "aff_dim T \<le> aff_dim S"
+
−    by (simp add: face_of_imp_subset aff_dim_subset assms)
+
−  moreover have "aff_dim T \<noteq> aff_dim S"
+
−  proof (cases "T = {}")
+
−    case True then show ?thesis
+
−      by (metis aff_dim_empty \<open>T \<noteq> S\<close>)
+
−  next case False then show ?thesis
+
−    by (metis Set.set_insert assms convex_rel_frontier_aff_dim dual_order.irrefl face_of_imp_convex face_of_subset_rel_frontier insert_not_empty subsetI)
+
−  qed
+
−  ultimately show ?thesis
+
−    by simp
+
−qed
+
−
+
−lemma subset_of_face_of_affine_hull:
+
−    fixes S :: "'a::euclidean_space set"
+
−  assumes T: "T face_of S" and "convex S" "U \<subseteq> S" and dis: "\<not> disjnt (affine hull T) (rel_interior U)"
+
−  shows "U \<subseteq> T"
+
−  apply (rule subset_of_face_of [OF T \<open>U \<subseteq> S\<close>])
+
−  using face_of_imp_eq_affine_Int [OF \<open>convex S\<close> T]
+
−  using rel_interior_subset [of U] dis
+
−  using \<open>U \<subseteq> S\<close> disjnt_def by fastforce
+
−
+
−lemma affine_hull_face_of_disjoint_rel_interior:
+
−    fixes S :: "'a::euclidean_space set"
+
−  assumes "convex S" "F face_of S" "F \<noteq> S"
+
−  shows "affine hull F \<inter> rel_interior S = {}"
+
−  by (metis assms disjnt_def face_of_imp_subset order_refl subset_antisym subset_of_face_of_affine_hull)
+
−
+
−lemma affine_diff_divide:
+
−    assumes "affine S" "k \<noteq> 0" "k \<noteq> 1" and xy: "x \<in> S" "y /\<^sub>R (1 - k) \<in> S"
+
−      shows "(x - y) /\<^sub>R k \<in> S"
+
−proof -
+
−  have "inverse(k) *\<^sub>R (x - y) = (1 - inverse k) *\<^sub>R inverse(1 - k) *\<^sub>R y + inverse(k) *\<^sub>R x"
+
−    using assms
+
−    by (simp add: algebra_simps) (simp add: scaleR_left_distrib [symmetric] divide_simps)
+
−  then show ?thesis
+
−    using \<open>affine S\<close> xy by (auto simp: affine_alt)
+
−qed
+
−
+
−proposition face_of_convex_hulls:
+
−      assumes S: "finite S" "T \<subseteq> S" and disj: "affine hull T \<inter> convex hull (S - T) = {}"
+
−      shows  "(convex hull T) face_of (convex hull S)"
+
−proof -
+
−  have fin: "finite T" "finite (S - T)" using assms
+
−    by (auto simp: finite_subset)
+
−  have *: "x \<in> convex hull T"
+
−          if x: "x \<in> convex hull S" and y: "y \<in> convex hull S" and w: "w \<in> convex hull T" "w \<in> open_segment x y"
+
−          for x y w
+
−  proof -
+
−    have waff: "w \<in> affine hull T"
+
−      using convex_hull_subset_affine_hull w by blast
+
−    obtain a b where a: "\<And>i. i \<in> S \<Longrightarrow> 0 \<le> a i" and asum: "sum a S = 1" and aeqx: "(\<Sum>i\<in>S. a i *\<^sub>R i) = x"
+
−                 and b: "\<And>i. i \<in> S \<Longrightarrow> 0 \<le> b i" and bsum: "sum b S = 1" and beqy: "(\<Sum>i\<in>S. b i *\<^sub>R i) = y"
+
−      using x y by (auto simp: assms convex_hull_finite)
+
−    obtain u where "(1 - u) *\<^sub>R x + u *\<^sub>R y \<in> convex hull T" "x \<noteq> y" and weq: "w = (1 - u) *\<^sub>R x + u *\<^sub>R y"
+
−               and u01: "0 < u" "u < 1"
+
−      using w by (auto simp: open_segment_image_interval split: if_split_asm)
+
−    define c where "c i = (1 - u) * a i + u * b i" for i
+
−    have cge0: "\<And>i. i \<in> S \<Longrightarrow> 0 \<le> c i"
+
−      using a b u01 by (simp add: c_def)
+
−    have sumc1: "sum c S = 1"
+
−      by (simp add: c_def sum.distrib sum_distrib_left [symmetric] asum bsum)
+
−    have sumci_xy: "(\<Sum>i\<in>S. c i *\<^sub>R i) = (1 - u) *\<^sub>R x + u *\<^sub>R y"
+
−      apply (simp add: c_def sum.distrib scaleR_left_distrib)
+
−      by (simp only: scaleR_scaleR [symmetric] Real_Vector_Spaces.scaleR_right.sum [symmetric] aeqx beqy)
+
−    show ?thesis
+
−    proof (cases "sum c (S - T) = 0")
+
−      case True
+
−      have ci0: "\<And>i. i \<in> (S - T) \<Longrightarrow> c i = 0"
+
−        using True cge0 fin(2) sum_nonneg_eq_0_iff by auto
+
−      have a0: "a i = 0" if "i \<in> (S - T)" for i
+
−        using ci0 [OF that] u01 a [of i] b [of i] that
+
−        by (simp add: c_def Groups.ordered_comm_monoid_add_class.add_nonneg_eq_0_iff)
+
−      have [simp]: "sum a T = 1"
+
−        using assms by (metis sum.mono_neutral_cong_right a0 asum)
+
−      show ?thesis
+
−        apply (simp add: convex_hull_finite \<open>finite T\<close>)
+
−        apply (rule_tac x=a in exI)
+
−        using a0 assms
+
−        apply (auto simp: cge0 a aeqx [symmetric] sum.mono_neutral_right)
+
−        done
+
−    next
+
−      case False
+
−      define k where "k = sum c (S - T)"
+
−      have "k > 0" using False
+
−        unfolding k_def by (metis DiffD1 antisym_conv cge0 sum_nonneg not_less)
+
−      have weq_sumsum: "w = sum (\<lambda>x. c x *\<^sub>R x) T + sum (\<lambda>x. c x *\<^sub>R x) (S - T)"
+
−        by (metis (no_types) add.commute S(1) S(2) sum.subset_diff sumci_xy weq)
+
−      show ?thesis
+
−      proof (cases "k = 1")
+
−        case True
+
−        then have "sum c T = 0"
+
−          by (simp add: S k_def sum_diff sumc1)
+
−        then have [simp]: "sum c (S - T) = 1"
+
−          by (simp add: S sum_diff sumc1)
+
−        have ci0: "\<And>i. i \<in> T \<Longrightarrow> c i = 0"
+
−          by (meson \<open>finite T\<close> \<open>sum c T = 0\<close> \<open>T \<subseteq> S\<close> cge0 sum_nonneg_eq_0_iff subsetCE)
+
−        then have [simp]: "(\<Sum>i\<in>S-T. c i *\<^sub>R i) = w"
+
−          by (simp add: weq_sumsum)
+
−        have "w \<in> convex hull (S - T)"
+
−          apply (simp add: convex_hull_finite fin)
+
−          apply (rule_tac x=c in exI)
+
−          apply (auto simp: cge0 weq True k_def)
+
−          done
+
−        then show ?thesis
+
−          using disj waff by blast
+
−      next
+
−        case False
+
−        then have sumcf: "sum c T = 1 - k"
+
−          by (simp add: S k_def sum_diff sumc1)
+
−        have "(\<Sum>i\<in>T. c i *\<^sub>R i) /\<^sub>R (1 - k) \<in> convex hull T"
+
−          apply (simp add: convex_hull_finite fin)
+
−          apply (rule_tac x="\<lambda>i. inverse (1-k) * c i" in exI)
+
−          apply auto
+
−          apply (metis sumcf cge0 inverse_nonnegative_iff_nonnegative mult_nonneg_nonneg S(2) sum_nonneg subsetCE)
+
−          apply (metis False mult.commute right_inverse right_minus_eq sum_distrib_left sumcf)
+
−          by (metis (mono_tags, lifting) scaleR_right.sum scaleR_scaleR sum.cong)
+
−        with \<open>0 < k\<close>  have "inverse(k) *\<^sub>R (w - sum (\<lambda>i. c i *\<^sub>R i) T) \<in> affine hull T"
+
−          by (simp add: affine_diff_divide [OF affine_affine_hull] False waff convex_hull_subset_affine_hull [THEN subsetD])
+
−        moreover have "inverse(k) *\<^sub>R (w - sum (\<lambda>x. c x *\<^sub>R x) T) \<in> convex hull (S - T)"
+
−          apply (simp add: weq_sumsum convex_hull_finite fin)
+
−          apply (rule_tac x="\<lambda>i. inverse k * c i" in exI)
+
−          using \<open>k > 0\<close> cge0
+
−          apply (auto simp: scaleR_right.sum sum_distrib_left [symmetric] k_def [symmetric])
+
−          done
+
−        ultimately show ?thesis
+
−          using disj by blast
+
−      qed
+
−    qed
+
−  qed
+
−  have [simp]: "convex hull T \<subseteq> convex hull S"
+
−    by (simp add: \<open>T \<subseteq> S\<close> hull_mono)
+
−  show ?thesis
+
−    using open_segment_commute by (auto simp: face_of_def intro: *)
+
−qed
+
−
+
−proposition face_of_convex_hull_insert:
+
−   "\<lbrakk>finite S; a \<notin> affine hull S; T face_of convex hull S\<rbrakk> \<Longrightarrow> T face_of convex hull insert a S"
+
−  apply (rule face_of_trans, blast)
+
−  apply (rule face_of_convex_hulls; force simp: insert_Diff_if)
+
−  done
+
−
+
−proposition face_of_affine_trivial:
+
−    assumes "affine S" "T face_of S"
+
−    shows "T = {} \<or> T = S"
+
−proof (rule ccontr, clarsimp)
+
−  assume "T \<noteq> {}" "T \<noteq> S"
+
−  then obtain a where "a \<in> T" by auto
+
−  then have "a \<in> S"
+
−    using \<open>T face_of S\<close> face_of_imp_subset by blast
+
−  have "S \<subseteq> T"
+
−  proof
+
−    fix b  assume "b \<in> S"
+
−    show "b \<in> T"
+
−    proof (cases "a = b")
+
−      case True with \<open>a \<in> T\<close> show ?thesis by auto
+
−    next
+
−      case False
+
−      then have "a \<in> open_segment (2 *\<^sub>R a - b) b"
+
−        apply (auto simp: open_segment_def closed_segment_def)
+
−        apply (rule_tac x="1/2" in exI)
+
−        apply (simp add: algebra_simps)
+
−        by (simp add: scaleR_2)
+
−      moreover have "2 *\<^sub>R a - b \<in> S"
+
−        by (rule mem_affine [OF \<open>affine S\<close> \<open>a \<in> S\<close> \<open>b \<in> S\<close>, of 2 "-1", simplified])
+
−      moreover note \<open>b \<in> S\<close> \<open>a \<in> T\<close>
+
−      ultimately show ?thesis
+
−        by (rule face_ofD [OF \<open>T face_of S\<close>, THEN conjunct2])
+
−    qed
+
−  qed
+
−  then show False
+
−    using \<open>T \<noteq> S\<close> \<open>T face_of S\<close> face_of_imp_subset by blast
+
−qed
+
−
+
−
+
−lemma face_of_affine_eq:
+
−   "affine S \<Longrightarrow> (T face_of S \<longleftrightarrow> T = {} \<or> T = S)"
+
−using affine_imp_convex face_of_affine_trivial face_of_refl by auto
+
−
+
−
+
−proposition Inter_faces_finite_altbound:
+
−    fixes T :: "'a::euclidean_space set set"
+
−    assumes cfaI: "\<And>c. c \<in> T \<Longrightarrow> c face_of S"
+
−    shows "\<exists>F'. finite F' \<and> F' \<subseteq> T \<and> card F' \<le> DIM('a) + 2 \<and> \<Inter>F' = \<Inter>T"
+
−proof (cases "\<forall>F'. finite F' \<and> F' \<subseteq> T \<and> card F' \<le> DIM('a) + 2 \<longrightarrow> (\<exists>c. c \<in> T \<and> c \<inter> (\<Inter>F') \<subset> (\<Inter>F'))")
+
−  case True
+
−  then obtain c where c:
+
−       "\<And>F'. \<lbrakk>finite F'; F' \<subseteq> T; card F' \<le> DIM('a) + 2\<rbrakk> \<Longrightarrow> c F' \<in> T \<and> c F' \<inter> (\<Inter>F') \<subset> (\<Inter>F')"
+
−    by metis
+
−  define d where "d = rec_nat {c{}} (\<lambda>n r. insert (c r) r)"
+
−  have [simp]: "d 0 = {c {}}"
+
−    by (simp add: d_def)
+
−  have dSuc [simp]: "\<And>n. d (Suc n) = insert (c (d n)) (d n)"
+
−    by (simp add: d_def)
+
−  have dn_notempty: "d n \<noteq> {}" for n
+
−    by (induction n) auto
+
−  have dn_le_Suc: "d n \<subseteq> T \<and> finite(d n) \<and> card(d n) \<le> Suc n" if "n \<le> DIM('a) + 2" for n
+
−  using that
+
−  proof (induction n)
+
−    case 0
+
−    then show ?case by (simp add: c)
+
−  next
+
−    case (Suc n)
+
−    then show ?case by (auto simp: c card_insert_if)
+
−  qed
+
−  have aff_dim_le: "aff_dim(\<Inter>(d n)) \<le> DIM('a) - int n" if "n \<le> DIM('a) + 2" for n
+
−  using that
+
−  proof (induction n)
+
−    case 0
+
−    then show ?case
+
−      by (simp add: aff_dim_le_DIM)
+
−  next
+
−    case (Suc n)
+
−    have fs: "\<Inter>(d (Suc n)) face_of S"
+
−      by (meson Suc.prems cfaI dn_le_Suc dn_notempty face_of_Inter subsetCE)
+
−    have condn: "convex (\<Inter>(d n))"
+
−      using Suc.prems nat_le_linear not_less_eq_eq
+
−      by (blast intro: face_of_imp_convex cfaI convex_Inter dest: dn_le_Suc)
+
−    have fdn: "\<Inter>(d (Suc n)) face_of \<Inter>(d n)"
+
−      by (metis (no_types, lifting) Inter_anti_mono Suc.prems dSuc cfaI dn_le_Suc dn_notempty face_of_Inter face_of_imp_subset face_of_subset subset_iff subset_insertI)
+
−    have ne: "\<Inter>(d (Suc n)) \<noteq> \<Inter>(d n)"
+
−      by (metis (no_types, lifting) Suc.prems Suc_leD c complete_lattice_class.Inf_insert dSuc dn_le_Suc less_irrefl order.trans)
+
−    have *: "\<And>m::int. \<And>d. \<And>d'::int. d < d' \<and> d' \<le> m - n \<Longrightarrow> d \<le> m - of_nat(n+1)"
+
−      by arith
+
−    have "aff_dim (\<Inter>(d (Suc n))) < aff_dim (\<Inter>(d n))"
+
−      by (rule face_of_aff_dim_lt [OF condn fdn ne])
+
−    moreover have "aff_dim (\<Inter>(d n)) \<le> int (DIM('a)) - int n"
+
−      using Suc by auto
+
−    ultimately
+
−    have "aff_dim (\<Inter>(d (Suc n))) \<le> int (DIM('a)) - (n+1)" by arith
+
−    then show ?case by linarith
+
−  qed
+
−  have "aff_dim (\<Inter>(d (DIM('a) + 2))) \<le> -2"
+
−      using aff_dim_le [OF order_refl] by simp
+
−  with aff_dim_geq [of "\<Inter>(d (DIM('a) + 2))"] show ?thesis
+
−    using order.trans by fastforce
+
−next
+
−  case False
+
−  then show ?thesis
+
−    apply simp
+
−    apply (erule ex_forward)
+
−    by blast
+
−qed
+
−
+
−lemma faces_of_translation:
+
−   "{F. F face_of image (\<lambda>x. a + x) S} = image (image (\<lambda>x. a + x)) {F. F face_of S}"
+
−apply (rule subset_antisym, clarify)
+
−apply (auto simp: image_iff)
+
−apply (metis face_of_imp_subset face_of_translation_eq subset_imageE)
+
−done
+
−
+
−proposition face_of_Times:
+
−  assumes "F face_of S" and "F' face_of S'"
+
−    shows "(F \<times> F') face_of (S \<times> S')"
+
−proof -
+
−  have "F \<times> F' \<subseteq> S \<times> S'"
+
−    using assms [unfolded face_of_def] by blast
+
−  moreover
+
−  have "convex (F \<times> F')"
+
−    using assms [unfolded face_of_def] by (blast intro: convex_Times)
+
−  moreover
+
−    have "a \<in> F \<and> a' \<in> F' \<and> b \<in> F \<and> b' \<in> F'"
+
−       if "a \<in> S" "b \<in> S" "a' \<in> S'" "b' \<in> S'" "x \<in> F \<times> F'" "x \<in> open_segment (a,a') (b,b')"
+
−       for a b a' b' x
+
−  proof (cases "b=a \<or> b'=a'")
+
−    case True with that show ?thesis
+
−      using assms
+
−      by (force simp: in_segment dest: face_ofD)
+
−  next
+
−    case False with assms [unfolded face_of_def] that show ?thesis
+
−      by (blast dest!: open_segment_PairD)
+
−  qed
+
−  ultimately show ?thesis
+
−    unfolding face_of_def by blast
+
−qed
+
−
+
−corollary face_of_Times_decomp:
+
−    fixes S :: "'a::euclidean_space set" and S' :: "'b::euclidean_space set"
+
−    shows "c face_of (S \<times> S') \<longleftrightarrow> (\<exists>F F'. F face_of S \<and> F' face_of S' \<and> c = F \<times> F')"
+
−     (is "?lhs = ?rhs")
+
−proof
+
−  assume c: ?lhs
+
−  show ?rhs
+
−  proof (cases "c = {}")
+
−    case True then show ?thesis by auto
+
−  next
+
−    case False
+
−    have 1: "fst ` c \<subseteq> S" "snd ` c \<subseteq> S'"
+
−      using c face_of_imp_subset by fastforce+
+
−    have "convex c"
+
−      using c by (metis face_of_imp_convex)
+
−    have conv: "convex (fst ` c)" "convex (snd ` c)"
+
−      by (simp_all add: \<open>convex c\<close> convex_linear_image fst_linear snd_linear)
+
−    have fstab: "a \<in> fst ` c \<and> b \<in> fst ` c"
+
−            if "a \<in> S" "b \<in> S" "x \<in> open_segment a b" "(x,x') \<in> c" for a b x x'
+
−    proof -
+
−      have *: "(x,x') \<in> open_segment (a,x') (b,x')"
+
−        using that by (auto simp: in_segment)
+
−      show ?thesis
+
−        using face_ofD [OF c *] that face_of_imp_subset [OF c] by force
+
−    qed
+
−    have fst: "fst ` c face_of S"
+
−      by (force simp: face_of_def 1 conv fstab)
+
−    have sndab: "a' \<in> snd ` c \<and> b' \<in> snd ` c"
+
−            if "a' \<in> S'" "b' \<in> S'" "x' \<in> open_segment a' b'" "(x,x') \<in> c" for a' b' x x'
+
−    proof -
+
−      have *: "(x,x') \<in> open_segment (x,a') (x,b')"
+
−        using that by (auto simp: in_segment)
+
−      show ?thesis
+
−        using face_ofD [OF c *] that face_of_imp_subset [OF c] by force
+
−    qed
+
−    have snd: "snd ` c face_of S'"
+
−      by (force simp: face_of_def 1 conv sndab)
+
−    have cc: "rel_interior c \<subseteq> rel_interior (fst ` c) \<times> rel_interior (snd ` c)"
+
−      by (force simp: face_of_Times rel_interior_Times conv fst snd \<open>convex c\<close> fst_linear snd_linear rel_interior_convex_linear_image [symmetric])
+
−    have "c = fst ` c \<times> snd ` c"
+
−      apply (rule face_of_eq [OF c])
+
−      apply (simp_all add: face_of_Times rel_interior_Times conv fst snd)
+
−      using False rel_interior_eq_empty \<open>convex c\<close> cc
+
−      apply blast
+
−      done
+
−    with fst snd show ?thesis by metis
+
−  qed
+
−next
+
−  assume ?rhs with face_of_Times show ?lhs by auto
+
−qed
+
−
+
−lemma face_of_Times_eq:
+
−    fixes S :: "'a::euclidean_space set" and S' :: "'b::euclidean_space set"
+
−    shows "(F \<times> F') face_of (S \<times> S') \<longleftrightarrow>
+
−           F = {} \<or> F' = {} \<or> F face_of S \<and> F' face_of S'"
+
−by (auto simp: face_of_Times_decomp times_eq_iff)
+
−
+
−lemma hyperplane_face_of_halfspace_le: "{x. a \<bullet> x = b} face_of {x. a \<bullet> x \<le> b}"
+
−proof -
+
−  have "{x. a \<bullet> x \<le> b} \<inter> {x. a \<bullet> x = b} = {x. a \<bullet> x = b}"
+
−    by auto
+
−  with face_of_Int_supporting_hyperplane_le [OF convex_halfspace_le [of a b], of a b]
+
−  show ?thesis by auto
+
−qed
+
−
+
−lemma hyperplane_face_of_halfspace_ge: "{x. a \<bullet> x = b} face_of {x. a \<bullet> x \<ge> b}"
+
−proof -
+
−  have "{x. a \<bullet> x \<ge> b} \<inter> {x. a \<bullet> x = b} = {x. a \<bullet> x = b}"
+
−    by auto
+
−  with face_of_Int_supporting_hyperplane_ge [OF convex_halfspace_ge [of b a], of b a]
+
−  show ?thesis by auto
+
−qed
+
−
+
−lemma face_of_halfspace_le:
+
−  fixes a :: "'n::euclidean_space"
+
−  shows "F face_of {x. a \<bullet> x \<le> b} \<longleftrightarrow>
+
−         F = {} \<or> F = {x. a \<bullet> x = b} \<or> F = {x. a \<bullet> x \<le> b}"
+
−     (is "?lhs = ?rhs")
+
−proof (cases "a = 0")
+
−  case True then show ?thesis
+
−    using face_of_affine_eq affine_UNIV by auto
+
−next
+
−  case False
+
−  then have ine: "interior {x. a \<bullet> x \<le> b} \<noteq> {}"
+
−    using halfspace_eq_empty_lt interior_halfspace_le by blast
+
−  show ?thesis
+
−  proof
+
−    assume L: ?lhs
+
−    have "F \<noteq> {x. a \<bullet> x \<le> b} \<Longrightarrow> F face_of {x. a \<bullet> x = b}"
+
−      using False
+
−      apply (simp add: frontier_halfspace_le [symmetric] rel_frontier_nonempty_interior [OF ine, symmetric])
+
−      apply (rule face_of_subset [OF L])
+
−      apply (simp add: face_of_subset_rel_frontier [OF L])
+
−      apply (force simp: rel_frontier_def closed_halfspace_le)
+
−      done
+
−    with L show ?rhs
+
−      using affine_hyperplane face_of_affine_eq by blast
+
−  next
+
−    assume ?rhs
+
−    then show ?lhs
+
−      by (metis convex_halfspace_le empty_face_of face_of_refl hyperplane_face_of_halfspace_le)
+
−  qed
+
−qed
+
−
+
−lemma face_of_halfspace_ge:
+
−  fixes a :: "'n::euclidean_space"
+
−  shows "F face_of {x. a \<bullet> x \<ge> b} \<longleftrightarrow>
+
−         F = {} \<or> F = {x. a \<bullet> x = b} \<or> F = {x. a \<bullet> x \<ge> b}"
+
−using face_of_halfspace_le [of F "-a" "-b"] by simp
+
−
+
−subsection\<open>Exposed faces\<close>
+
−
+
−text\<open>That is, faces that are intersection with supporting hyperplane\<close>
+
−
+
−definition%important exposed_face_of :: "['a::euclidean_space set, 'a set] \<Rightarrow> bool"
+
−                               (infixr "(exposed'_face'_of)" 50)
+
−  where "T exposed_face_of S \<longleftrightarrow>
+
−         T face_of S \<and> (\<exists>a b. S \<subseteq> {x. a \<bullet> x \<le> b} \<and> T = S \<inter> {x. a \<bullet> x = b})"
+
−
+
−lemma empty_exposed_face_of [iff]: "{} exposed_face_of S"
+
−  apply (simp add: exposed_face_of_def)
+
−  apply (rule_tac x=0 in exI)
+
−  apply (rule_tac x=1 in exI, force)
+
−  done
+
−
+
−lemma exposed_face_of_refl_eq [simp]: "S exposed_face_of S \<longleftrightarrow> convex S"
+
−  apply (simp add: exposed_face_of_def face_of_refl_eq, auto)
+
−  apply (rule_tac x=0 in exI)+
+
−  apply force
+
−  done
+
−
+
−lemma exposed_face_of_refl: "convex S \<Longrightarrow> S exposed_face_of S"
+
−  by simp
+
−
+
−lemma exposed_face_of:
+
−    "T exposed_face_of S \<longleftrightarrow>
+
−     T face_of S \<and>
+
−     (T = {} \<or> T = S \<or>
+
−      (\<exists>a b. a \<noteq> 0 \<and> S \<subseteq> {x. a \<bullet> x \<le> b} \<and> T = S \<inter> {x. a \<bullet> x = b}))"
+
−proof (cases "T = {}")
+
−  case True then show ?thesis
+
−    by simp
+
−next
+
−  case False
+
−  show ?thesis
+
−  proof (cases "T = S")
+
−    case True then show ?thesis
+
−      by (simp add: face_of_refl_eq)
+
−  next
+
−    case False
+
−    with \<open>T \<noteq> {}\<close> show ?thesis
+
−      apply (auto simp: exposed_face_of_def)
+
−      apply (metis inner_zero_left)
+
−      done
+
−  qed
+
−qed
+
−
+
−lemma exposed_face_of_Int_supporting_hyperplane_le:
+
−   "\<lbrakk>convex S; \<And>x. x \<in> S \<Longrightarrow> a \<bullet> x \<le> b\<rbrakk> \<Longrightarrow> (S \<inter> {x. a \<bullet> x = b}) exposed_face_of S"
+
−by (force simp: exposed_face_of_def face_of_Int_supporting_hyperplane_le)
+
−
+
−lemma exposed_face_of_Int_supporting_hyperplane_ge:
+
−   "\<lbrakk>convex S; \<And>x. x \<in> S \<Longrightarrow> a \<bullet> x \<ge> b\<rbrakk> \<Longrightarrow> (S \<inter> {x. a \<bullet> x = b}) exposed_face_of S"
+
−using exposed_face_of_Int_supporting_hyperplane_le [of S "-a" "-b"] by simp
+
−
+
−proposition exposed_face_of_Int:
+
−  assumes "T exposed_face_of S"
+
−      and "u exposed_face_of S"
+
−    shows "(T \<inter> u) exposed_face_of S"
+
−proof -
+
−  obtain a b where T: "S \<inter> {x. a \<bullet> x = b} face_of S"
+
−               and S: "S \<subseteq> {x. a \<bullet> x \<le> b}"
+
−               and teq: "T = S \<inter> {x. a \<bullet> x = b}"
+
−    using assms by (auto simp: exposed_face_of_def)
+
−  obtain a' b' where u: "S \<inter> {x. a' \<bullet> x = b'} face_of S"
+
−                 and s': "S \<subseteq> {x. a' \<bullet> x \<le> b'}"
+
−                 and ueq: "u = S \<inter> {x. a' \<bullet> x = b'}"
+
−    using assms by (auto simp: exposed_face_of_def)
+
−  have tu: "T \<inter> u face_of S"
+
−    using T teq u ueq by (simp add: face_of_Int)
+
−  have ss: "S \<subseteq> {x. (a + a') \<bullet> x \<le> b + b'}"
+
−    using S s' by (force simp: inner_left_distrib)
+
−  show ?thesis
+
−    apply (simp add: exposed_face_of_def tu)
+
−    apply (rule_tac x="a+a'" in exI)
+
−    apply (rule_tac x="b+b'" in exI)
+
−    using S s'
+
−    apply (fastforce simp: ss inner_left_distrib teq ueq)
+
−    done
+
−qed
+
−
+
−proposition exposed_face_of_Inter:
+
−    fixes P :: "'a::euclidean_space set set"
+
−  assumes "P \<noteq> {}"
+
−      and "\<And>T. T \<in> P \<Longrightarrow> T exposed_face_of S"
+
−    shows "\<Inter>P exposed_face_of S"
+
−proof -
+
−  obtain Q where "finite Q" and QsubP: "Q \<subseteq> P" "card Q \<le> DIM('a) + 2" and IntQ: "\<Inter>Q = \<Inter>P"
+
−    using Inter_faces_finite_altbound [of P S] assms [unfolded exposed_face_of]
+
−    by force
+
−  show ?thesis
+
−  proof (cases "Q = {}")
+
−    case True then show ?thesis
+
−      by (metis IntQ Inter_UNIV_conv(2) assms(1) assms(2) ex_in_conv)
+
−  next
+
−    case False
+
−    have "Q \<subseteq> {T. T exposed_face_of S}"
+
−      using QsubP assms by blast
+
−    moreover have "Q \<subseteq> {T. T exposed_face_of S} \<Longrightarrow> \<Inter>Q exposed_face_of S"
+
−      using \<open>finite Q\<close> False
+
−      apply (induction Q rule: finite_induct)
+
−      using exposed_face_of_Int apply fastforce+
+
−      done
+
−    ultimately show ?thesis
+
−      by (simp add: IntQ)
+
−  qed
+
−qed
+
−
+
−proposition exposed_face_of_sums:
+
−  assumes "convex S" and "convex T"
+
−      and "F exposed_face_of {x + y | x y. x \<in> S \<and> y \<in> T}"
+
−          (is "F exposed_face_of ?ST")
+
−  obtains k l
+
−    where "k exposed_face_of S" "l exposed_face_of T"
+
−          "F = {x + y | x y. x \<in> k \<and> y \<in> l}"
+
−proof (cases "F = {}")
+
−  case True then show ?thesis
+
−    using that by blast
+
−next
+
−  case False
+
−  show ?thesis
+
−  proof (cases "F = ?ST")
+
−    case True then show ?thesis
+
−      using assms exposed_face_of_refl_eq that by blast
+
−  next
+
−    case False
+
−    obtain p where "p \<in> F" using \<open>F \<noteq> {}\<close> by blast
+
−    moreover
+
−    obtain u z where T: "?ST \<inter> {x. u \<bullet> x = z} face_of ?ST"
+
−                 and S: "?ST \<subseteq> {x. u \<bullet> x \<le> z}"
+
−                 and feq: "F = ?ST \<inter> {x. u \<bullet> x = z}"
+
−      using assms by (auto simp: exposed_face_of_def)
+
−    ultimately obtain a0 b0
+
−            where p: "p = a0 + b0" and "a0 \<in> S" "b0 \<in> T" and z: "u \<bullet> p = z"
+
−      by auto
+
−    have lez: "u \<bullet> (x + y) \<le> z" if "x \<in> S" "y \<in> T" for x y
+
−      using S that by auto
+
−    have sef: "S \<inter> {x. u \<bullet> x = u \<bullet> a0} exposed_face_of S"
+
−      apply (rule exposed_face_of_Int_supporting_hyperplane_le [OF \<open>convex S\<close>])
+
−      apply (metis p z add_le_cancel_right inner_right_distrib lez [OF _ \<open>b0 \<in> T\<close>])
+
−      done
+
−    have tef: "T \<inter> {x. u \<bullet> x = u \<bullet> b0} exposed_face_of T"
+
−      apply (rule exposed_face_of_Int_supporting_hyperplane_le [OF \<open>convex T\<close>])
+
−      apply (metis p z add.commute add_le_cancel_right inner_right_distrib lez [OF \<open>a0 \<in> S\<close>])
+
−      done
+
−    have "{x + y |x y. x \<in> S \<and> u \<bullet> x = u \<bullet> a0 \<and> y \<in> T \<and> u \<bullet> y = u \<bullet> b0} \<subseteq> F"
+
−      by (auto simp: feq) (metis inner_right_distrib p z)
+
−    moreover have "F \<subseteq> {x + y |x y. x \<in> S \<and> u \<bullet> x = u \<bullet> a0 \<and> y \<in> T \<and> u \<bullet> y = u \<bullet> b0}"
+
−      apply (auto simp: feq)
+
−      apply (rename_tac x y)
+
−      apply (rule_tac x=x in exI)
+
−      apply (rule_tac x=y in exI, simp)
+
−      using z p \<open>a0 \<in> S\<close> \<open>b0 \<in> T\<close>
+
−      apply clarify
+
−      apply (simp add: inner_right_distrib)
+
−      apply (metis add_le_cancel_right antisym lez [unfolded inner_right_distrib] add.commute)
+
−      done
+
−    ultimately have "F = {x + y |x y. x \<in> S \<inter> {x. u \<bullet> x = u \<bullet> a0} \<and> y \<in> T \<inter> {x. u \<bullet> x = u \<bullet> b0}}"
+
−      by blast
+
−    then show ?thesis
+
−      by (rule that [OF sef tef])
+
−  qed
+
−qed
+
−
+
−proposition exposed_face_of_parallel:
+
−   "T exposed_face_of S \<longleftrightarrow>
+
−         T face_of S \<and>
+
−         (\<exists>a b. S \<subseteq> {x. a \<bullet> x \<le> b} \<and> T = S \<inter> {x. a \<bullet> x = b} \<and>
+
−                (T \<noteq> {} \<longrightarrow> T \<noteq> S \<longrightarrow> a \<noteq> 0) \<and>
+
−                (T \<noteq> S \<longrightarrow> (\<forall>w \<in> affine hull S. (w + a) \<in> affine hull S)))"
+
−  (is "?lhs = ?rhs")
+
−proof
+
−  assume ?lhs then show ?rhs
+
−  proof (clarsimp simp: exposed_face_of_def)
+
−    fix a b
+
−    assume faceS: "S \<inter> {x. a \<bullet> x = b} face_of S" and Ssub: "S \<subseteq> {x. a \<bullet> x \<le> b}" 
+
−    show "\<exists>c d. S \<subseteq> {x. c \<bullet> x \<le> d} \<and>
+
−                S \<inter> {x. a \<bullet> x = b} = S \<inter> {x. c \<bullet> x = d} \<and>
+
−                (S \<inter> {x. a \<bullet> x = b} \<noteq> {} \<longrightarrow> S \<inter> {x. a \<bullet> x = b} \<noteq> S \<longrightarrow> c \<noteq> 0) \<and>
+
−                (S \<inter> {x. a \<bullet> x = b} \<noteq> S \<longrightarrow> (\<forall>w \<in> affine hull S. w + c \<in> affine hull S))"
+
−    proof (cases "affine hull S \<inter> {x. -a \<bullet> x \<le> -b} = {} \<or> affine hull S \<subseteq> {x. - a \<bullet> x \<le> - b}")
+
−      case True
+
−      then show ?thesis
+
−      proof
+
−        assume "affine hull S \<inter> {x. - a \<bullet> x \<le> - b} = {}"
+
−       then show ?thesis
+
−         apply (rule_tac x="0" in exI)
+
−         apply (rule_tac x="1" in exI)
+
−         using hull_subset by fastforce
+
−    next
+
−      assume "affine hull S \<subseteq> {x. - a \<bullet> x \<le> - b}"
+
−      then show ?thesis
+
−         apply (rule_tac x="0" in exI)
+
−         apply (rule_tac x="0" in exI)
+
−        using Ssub hull_subset by fastforce
+
−    qed
+
−  next
+
−    case False
+
−    then obtain a' b' where "a' \<noteq> 0" 
+
−      and le: "affine hull S \<inter> {x. a' \<bullet> x \<le> b'} = affine hull S \<inter> {x. - a \<bullet> x \<le> - b}" 
+
−      and eq: "affine hull S \<inter> {x. a' \<bullet> x = b'} = affine hull S \<inter> {x. - a \<bullet> x = - b}" 
+
−      and mem: "\<And>w. w \<in> affine hull S \<Longrightarrow> w + a' \<in> affine hull S"
+
−      using affine_parallel_slice affine_affine_hull by metis 
+
−    show ?thesis
+
−    proof (intro conjI impI allI ballI exI)
+
−      have *: "S \<subseteq> - (affine hull S \<inter> {x. P x}) \<union> affine hull S \<inter> {x. Q x} \<Longrightarrow> S \<subseteq> {x. \<not> P x \<or> Q x}" 
+
−        for P Q 
+
−        using hull_subset by fastforce  
+
−      have "S \<subseteq> {x. \<not> (a' \<bullet> x \<le> b') \<or> a' \<bullet> x = b'}"
+
−        apply (rule *)
+
−        apply (simp only: le eq)
+
−        using Ssub by auto
+
−      then show "S \<subseteq> {x. - a' \<bullet> x \<le> - b'}"
+
−        by auto 
+
−      show "S \<inter> {x. a \<bullet> x = b} = S \<inter> {x. - a' \<bullet> x = - b'}"
+
−        using eq hull_subset [of S affine] by force
+
−      show "\<lbrakk>S \<inter> {x. a \<bullet> x = b} \<noteq> {}; S \<inter> {x. a \<bullet> x = b} \<noteq> S\<rbrakk> \<Longrightarrow> - a' \<noteq> 0"
+
−        using \<open>a' \<noteq> 0\<close> by auto
+
−      show "w + - a' \<in> affine hull S"
+
−        if "S \<inter> {x. a \<bullet> x = b} \<noteq> S" "w \<in> affine hull S" for w
+
−      proof -
+
−        have "w + 1 *\<^sub>R (w - (w + a')) \<in> affine hull S"
+
−          using affine_affine_hull mem mem_affine_3_minus that(2) by blast
+
−        then show ?thesis  by simp
+
−      qed
+
−    qed
+
−  qed
+
−qed
+
−next
+
−  assume ?rhs then show ?lhs
+
−    unfolding exposed_face_of_def by blast
+
−qed
+
−
+
−subsection\<open>Extreme points of a set: its singleton faces\<close>
+
−
+
−definition%important extreme_point_of :: "['a::real_vector, 'a set] \<Rightarrow> bool"
+
−                               (infixr "(extreme'_point'_of)" 50)
+
−  where "x extreme_point_of S \<longleftrightarrow>
+
−         x \<in> S \<and> (\<forall>a \<in> S. \<forall>b \<in> S. x \<notin> open_segment a b)"
+
−
+
−lemma extreme_point_of_stillconvex:
+
−   "convex S \<Longrightarrow> (x extreme_point_of S \<longleftrightarrow> x \<in> S \<and> convex(S - {x}))"
+
−  by (fastforce simp add: convex_contains_segment extreme_point_of_def open_segment_def)
+
−
+
−lemma face_of_singleton:
+
−   "{x} face_of S \<longleftrightarrow> x extreme_point_of S"
+
−by (fastforce simp add: extreme_point_of_def face_of_def)
+
−
+
−lemma extreme_point_not_in_REL_INTERIOR:
+
−    fixes S :: "'a::real_normed_vector set"
+
−    shows "\<lbrakk>x extreme_point_of S; S \<noteq> {x}\<rbrakk> \<Longrightarrow> x \<notin> rel_interior S"
+
−apply (simp add: face_of_singleton [symmetric])
+
−apply (blast dest: face_of_disjoint_rel_interior)
+
−done
+
−
+
−lemma extreme_point_not_in_interior:
+
−    fixes S :: "'a::{real_normed_vector, perfect_space} set"
+
−    shows "x extreme_point_of S \<Longrightarrow> x \<notin> interior S"
+
−apply (case_tac "S = {x}")
+
−apply (simp add: empty_interior_finite)
+
−by (meson contra_subsetD extreme_point_not_in_REL_INTERIOR interior_subset_rel_interior)
+
−
+
−lemma extreme_point_of_face:
+
−     "F face_of S \<Longrightarrow> v extreme_point_of F \<longleftrightarrow> v extreme_point_of S \<and> v \<in> F"
+
−  by (meson empty_subsetI face_of_face face_of_singleton insert_subset)
+
−
+
−lemma extreme_point_of_convex_hull:
+
−   "x extreme_point_of (convex hull S) \<Longrightarrow> x \<in> S"
+
−apply (simp add: extreme_point_of_stillconvex)
+
−using hull_minimal [of S "(convex hull S) - {x}" convex]
+
−using hull_subset [of S convex]
+
−apply blast
+
−done
+
−
+
−proposition extreme_points_of_convex_hull:
+
−   "{x. x extreme_point_of (convex hull S)} \<subseteq> S"
+
−  using extreme_point_of_convex_hull by auto
+
−
+
−lemma extreme_point_of_empty [simp]: "\<not> (x extreme_point_of {})"
+
−  by (simp add: extreme_point_of_def)
+
−
+
−lemma extreme_point_of_singleton [iff]: "x extreme_point_of {a} \<longleftrightarrow> x = a"
+
−  using extreme_point_of_stillconvex by auto
+
−
+
−lemma extreme_point_of_translation_eq:
+
−   "(a + x) extreme_point_of (image (\<lambda>x. a + x) S) \<longleftrightarrow> x extreme_point_of S"
+
−by (auto simp: extreme_point_of_def)
+
−
+
−lemma extreme_points_of_translation:
+
−   "{x. x extreme_point_of (image (\<lambda>x. a + x) S)} =
+
−    (\<lambda>x. a + x) ` {x. x extreme_point_of S}"
+
−  using extreme_point_of_translation_eq
+
−  by auto (metis (no_types, lifting) image_iff mem_Collect_eq minus_add_cancel)
+
−
+
−lemma extreme_points_of_translation_subtract:
+
−   "{x. x extreme_point_of (image (\<lambda>x. x - a) S)} =
+
−    (\<lambda>x. x - a) ` {x. x extreme_point_of S}"
+
−  using extreme_points_of_translation [of "- a" S]
+
−  by simp
+
−
+
−lemma extreme_point_of_Int:
+
−   "\<lbrakk>x extreme_point_of S; x extreme_point_of T\<rbrakk> \<Longrightarrow> x extreme_point_of (S \<inter> T)"
+
−by (simp add: extreme_point_of_def)
+
−
+
−lemma extreme_point_of_Int_supporting_hyperplane_le:
+
−   "\<lbrakk>S \<inter> {x. a \<bullet> x = b} = {c}; \<And>x. x \<in> S \<Longrightarrow> a \<bullet> x \<le> b\<rbrakk> \<Longrightarrow> c extreme_point_of S"
+
−apply (simp add: face_of_singleton [symmetric])
+
−by (metis face_of_Int_supporting_hyperplane_le_strong convex_singleton)
+
−
+
−lemma extreme_point_of_Int_supporting_hyperplane_ge:
+
−   "\<lbrakk>S \<inter> {x. a \<bullet> x = b} = {c}; \<And>x. x \<in> S \<Longrightarrow> a \<bullet> x \<ge> b\<rbrakk> \<Longrightarrow> c extreme_point_of S"
+
−apply (simp add: face_of_singleton [symmetric])
+
−by (metis face_of_Int_supporting_hyperplane_ge_strong convex_singleton)
+
−
+
−lemma exposed_point_of_Int_supporting_hyperplane_le:
+
−   "\<lbrakk>S \<inter> {x. a \<bullet> x = b} = {c}; \<And>x. x \<in> S \<Longrightarrow> a \<bullet> x \<le> b\<rbrakk> \<Longrightarrow> {c} exposed_face_of S"
+
−apply (simp add: exposed_face_of_def face_of_singleton)
+
−apply (force simp: extreme_point_of_Int_supporting_hyperplane_le)
+
−done
+
−
+
−lemma exposed_point_of_Int_supporting_hyperplane_ge:
+
−    "\<lbrakk>S \<inter> {x. a \<bullet> x = b} = {c}; \<And>x. x \<in> S \<Longrightarrow> a \<bullet> x \<ge> b\<rbrakk> \<Longrightarrow> {c} exposed_face_of S"
+
−using exposed_point_of_Int_supporting_hyperplane_le [of S "-a" "-b" c]
+
−by simp
+
−
+
−lemma extreme_point_of_convex_hull_insert:
+
−   "\<lbrakk>finite S; a \<notin> convex hull S\<rbrakk> \<Longrightarrow> a extreme_point_of (convex hull (insert a S))"
+
−apply (case_tac "a \<in> S")
+
−apply (simp add: hull_inc)
+
−using face_of_convex_hulls [of "insert a S" "{a}"]
+
−apply (auto simp: face_of_singleton hull_same)
+
−done
+
−
+
−subsection\<open>Facets\<close>
+
−
+
−definition%important facet_of :: "['a::euclidean_space set, 'a set] \<Rightarrow> bool"
+
−                    (infixr "(facet'_of)" 50)
+
−  where "F facet_of S \<longleftrightarrow> F face_of S \<and> F \<noteq> {} \<and> aff_dim F = aff_dim S - 1"
+
−
+
−lemma facet_of_empty [simp]: "\<not> S facet_of {}"
+
−  by (simp add: facet_of_def)
+
−
+
−lemma facet_of_irrefl [simp]: "\<not> S facet_of S "
+
−  by (simp add: facet_of_def)
+
−
+
−lemma facet_of_imp_face_of: "F facet_of S \<Longrightarrow> F face_of S"
+
−  by (simp add: facet_of_def)
+
−
+
−lemma facet_of_imp_subset: "F facet_of S \<Longrightarrow> F \<subseteq> S"
+
−  by (simp add: face_of_imp_subset facet_of_def)
+
−
+
−lemma hyperplane_facet_of_halfspace_le:
+
−   "a \<noteq> 0 \<Longrightarrow> {x. a \<bullet> x = b} facet_of {x. a \<bullet> x \<le> b}"
+
−unfolding facet_of_def hyperplane_eq_empty
+
−by (auto simp: hyperplane_face_of_halfspace_ge hyperplane_face_of_halfspace_le
+
−           DIM_positive Suc_leI of_nat_diff aff_dim_halfspace_le)
+
−
+
−lemma hyperplane_facet_of_halfspace_ge:
+
−    "a \<noteq> 0 \<Longrightarrow> {x. a \<bullet> x = b} facet_of {x. a \<bullet> x \<ge> b}"
+
−unfolding facet_of_def hyperplane_eq_empty
+
−by (auto simp: hyperplane_face_of_halfspace_le hyperplane_face_of_halfspace_ge
+
−           DIM_positive Suc_leI of_nat_diff aff_dim_halfspace_ge)
+
−
+
−lemma facet_of_halfspace_le:
+
−    "F facet_of {x. a \<bullet> x \<le> b} \<longleftrightarrow> a \<noteq> 0 \<and> F = {x. a \<bullet> x = b}"
+
−    (is "?lhs = ?rhs")
+
−proof
+
−  assume c: ?lhs
+
−  with c facet_of_irrefl show ?rhs
+
−    by (force simp: aff_dim_halfspace_le facet_of_def face_of_halfspace_le cong: conj_cong split: if_split_asm)
+
−next
+
−  assume ?rhs then show ?lhs
+
−    by (simp add: hyperplane_facet_of_halfspace_le)
+
−qed
+
−
+
−lemma facet_of_halfspace_ge:
+
−    "F facet_of {x. a \<bullet> x \<ge> b} \<longleftrightarrow> a \<noteq> 0 \<and> F = {x. a \<bullet> x = b}"
+
−using facet_of_halfspace_le [of F "-a" "-b"] by simp
+
−
+
−subsection \<open>Edges: faces of affine dimension 1\<close> (*FIXME too small subsection, rearrange? *)
+
−
+
−definition%important edge_of :: "['a::euclidean_space set, 'a set] \<Rightarrow> bool"  (infixr "(edge'_of)" 50)
+
−  where "e edge_of S \<longleftrightarrow> e face_of S \<and> aff_dim e = 1"
+
−
+
−lemma edge_of_imp_subset:
+
−   "S edge_of T \<Longrightarrow> S \<subseteq> T"
+
−by (simp add: edge_of_def face_of_imp_subset)
+
−
+
−subsection\<open>Existence of extreme points\<close>
+
−
+
−proposition different_norm_3_collinear_points:
+
−  fixes a :: "'a::euclidean_space"
+
−  assumes "x \<in> open_segment a b" "norm(a) = norm(b)" "norm(x) = norm(b)"
+
−  shows False
+
−proof -
+
−  obtain u where "norm ((1 - u) *\<^sub>R a + u *\<^sub>R b) = norm b"
+
−             and "a \<noteq> b"
+
−             and u01: "0 < u" "u < 1"
+
−    using assms by (auto simp: open_segment_image_interval if_splits)
+
−  then have "(1 - u) *\<^sub>R a \<bullet> (1 - u) *\<^sub>R a + ((1 - u) * 2) *\<^sub>R a \<bullet> u *\<^sub>R b =
+
−             (1 - u * u) *\<^sub>R (a \<bullet> a)"
+
−    using assms by (simp add: norm_eq algebra_simps inner_commute)
+
−  then have "(1 - u) *\<^sub>R ((1 - u) *\<^sub>R a \<bullet> a + (2 * u) *\<^sub>R  a \<bullet> b) =
+
−             (1 - u) *\<^sub>R ((1 + u) *\<^sub>R (a \<bullet> a))"
+
−    by (simp add: algebra_simps)
+
−  then have "(1 - u) *\<^sub>R (a \<bullet> a) + (2 * u) *\<^sub>R (a \<bullet> b) = (1 + u) *\<^sub>R (a \<bullet> a)"
+
−    using u01 by auto
+
−  then have "a \<bullet> b = a \<bullet> a"
+
−    using u01 by (simp add: algebra_simps)
+
−  then have "a = b"
+
−    using \<open>norm(a) = norm(b)\<close> norm_eq vector_eq by fastforce
+
−  then show ?thesis
+
−    using \<open>a \<noteq> b\<close> by force
+
−qed
+
−
+
−proposition extreme_point_exists_convex:
+
−  fixes S :: "'a::euclidean_space set"
+
−  assumes "compact S" "convex S" "S \<noteq> {}"
+
−  obtains x where "x extreme_point_of S"
+
−proof -
+
−  obtain x where "x \<in> S" and xsup: "\<And>y. y \<in> S \<Longrightarrow> norm y \<le> norm x"
+
−    using distance_attains_sup [of S 0] assms by auto
+
−  have False if "a \<in> S" "b \<in> S" and x: "x \<in> open_segment a b" for a b
+
−  proof -
+
−    have noax: "norm a \<le> norm x" and nobx: "norm b \<le> norm x" using xsup that by auto
+
−    have "a \<noteq> b"
+
−      using empty_iff open_segment_idem x by auto
+
−    have *: "(1 - u) * na + u * nb < norm x" if "na < norm x"  "nb \<le> norm x" "0 < u" "u < 1" for na nb u
+
−    proof -
+
−      have "(1 - u) * na + u * nb < (1 - u) * norm x + u * nb"
+
−        by (simp add: that)
+
−      also have "... \<le> (1 - u) * norm x + u * norm x"
+
−        by (simp add: that)
+
−      finally have "(1 - u) * na + u * nb < (1 - u) * norm x + u * norm x" .
+
−      then show ?thesis
+
−      using scaleR_collapse [symmetric, of "norm x" u] by auto
+
−    qed
+
−    have "norm x < norm x" if "norm a < norm x"
+
−      using x
+
−      apply (clarsimp simp only: open_segment_image_interval \<open>a \<noteq> b\<close> if_False)
+
−      apply (rule norm_triangle_lt)
+
−      apply (simp add: norm_mult)
+
−      using * [of "norm a" "norm b"] nobx that
+
−        apply blast
+
−      done
+
−    moreover have "norm x < norm x" if "norm b < norm x"
+
−      using x
+
−      apply (clarsimp simp only: open_segment_image_interval \<open>a \<noteq> b\<close> if_False)
+
−      apply (rule norm_triangle_lt)
+
−      apply (simp add: norm_mult)
+
−      using * [of "norm b" "norm a" "1-u" for u] noax that
+
−        apply (simp add: add.commute)
+
−      done
+
−    ultimately have "\<not> (norm a < norm x) \<and> \<not> (norm b < norm x)"
+
−      by auto
+
−    then show ?thesis
+
−      using different_norm_3_collinear_points noax nobx that(3) by fastforce
+
−  qed
+
−  then show ?thesis
+
−    apply (rule_tac x=x in that)
+
−    apply (force simp: extreme_point_of_def \<open>x \<in> S\<close>)
+
−    done
+
−qed
+
−
+
−subsection\<open>Krein-Milman, the weaker form\<close>
+
−
+
−proposition Krein_Milman:
+
−  fixes S :: "'a::euclidean_space set"
+
−  assumes "compact S" "convex S"
+
−    shows "S = closure(convex hull {x. x extreme_point_of S})"
+
−proof (cases "S = {}")
+
−  case True then show ?thesis   by simp
+
−next
+
−  case False
+
−  have "closed S"
+
−    by (simp add: \<open>compact S\<close> compact_imp_closed)
+
−  have "closure (convex hull {x. x extreme_point_of S}) \<subseteq> S"
+
−    apply (rule closure_minimal [OF hull_minimal \<open>closed S\<close>])
+
−    using assms
+
−    apply (auto simp: extreme_point_of_def)
+
−    done
+
−  moreover have "u \<in> closure (convex hull {x. x extreme_point_of S})"
+
−                if "u \<in> S" for u
+
−  proof (rule ccontr)
+
−    assume unot: "u \<notin> closure(convex hull {x. x extreme_point_of S})"
+
−    then obtain a b where "a \<bullet> u < b"
+
−          and ab: "\<And>x. x \<in> closure(convex hull {x. x extreme_point_of S}) \<Longrightarrow> b < a \<bullet> x"
+
−      using separating_hyperplane_closed_point [of "closure(convex hull {x. x extreme_point_of S})"]
+
−      by blast
+
−    have "continuous_on S ((\<bullet>) a)"
+
−      by (rule continuous_intros)+
+
−    then obtain m where "m \<in> S" and m: "\<And>y. y \<in> S \<Longrightarrow> a \<bullet> m \<le> a \<bullet> y"
+
−      using continuous_attains_inf [of S "\<lambda>x. a \<bullet> x"] \<open>compact S\<close> \<open>u \<in> S\<close>
+
−      by auto
+
−    define T where "T = S \<inter> {x. a \<bullet> x = a \<bullet> m}"
+
−    have "m \<in> T"
+
−      by (simp add: T_def \<open>m \<in> S\<close>)
+
−    moreover have "compact T"
+
−      by (simp add: T_def compact_Int_closed [OF \<open>compact S\<close> closed_hyperplane])
+
−    moreover have "convex T"
+
−      by (simp add: T_def convex_Int [OF \<open>convex S\<close> convex_hyperplane])
+
−    ultimately obtain v where v: "v extreme_point_of T"
+
−      using extreme_point_exists_convex [of T] by auto
+
−    then have "{v} face_of T"
+
−      by (simp add: face_of_singleton)
+
−    also have "T face_of S"
+
−      by (simp add: T_def m face_of_Int_supporting_hyperplane_ge [OF \<open>convex S\<close>])
+
−    finally have "v extreme_point_of S"
+
−      by (simp add: face_of_singleton)
+
−    then have "b < a \<bullet> v"
+
−      using closure_subset by (simp add: closure_hull hull_inc ab)
+
−    then show False
+
−      using \<open>a \<bullet> u < b\<close> \<open>{v} face_of T\<close> face_of_imp_subset m T_def that by fastforce
+
−  qed
+
−  ultimately show ?thesis
+
−    by blast
+
−qed
+
−
+
−text\<open>Now the sharper form.\<close>
+
−
+
−lemma Krein_Milman_Minkowski_aux:
+
−  fixes S :: "'a::euclidean_space set"
+
−  assumes n: "dim S = n" and S: "compact S" "convex S" "0 \<in> S"
+
−    shows "0 \<in> convex hull {x. x extreme_point_of S}"
+
−using n S
+
−proof (induction n arbitrary: S rule: less_induct)
+
−  case (less n S) show ?case
+
−  proof (cases "0 \<in> rel_interior S")
+
−    case True with Krein_Milman show ?thesis
+
−      by (metis subsetD convex_convex_hull convex_rel_interior_closure less.prems(2) less.prems(3) rel_interior_subset)
+
−  next
+
−    case False
+
−    have "rel_interior S \<noteq> {}"
+
−      by (simp add: rel_interior_convex_nonempty_aux less)
+
−    then obtain c where c: "c \<in> rel_interior S" by blast
+
−    obtain a where "a \<noteq> 0"
+
−              and le_ay: "\<And>y. y \<in> S \<Longrightarrow> a \<bullet> 0 \<le> a \<bullet> y"
+
−              and less_ay: "\<And>y. y \<in> rel_interior S \<Longrightarrow> a \<bullet> 0 < a \<bullet> y"
+
−      by (blast intro: supporting_hyperplane_rel_boundary intro!: less False)
+
−    have face: "S \<inter> {x. a \<bullet> x = 0} face_of S"
+
−      apply (rule face_of_Int_supporting_hyperplane_ge [OF \<open>convex S\<close>])
+
−      using le_ay by auto
+
−    then have co: "compact (S \<inter> {x. a \<bullet> x = 0})" "convex (S \<inter> {x. a \<bullet> x = 0})"
+
−      using less.prems by (blast intro: face_of_imp_compact face_of_imp_convex)+
+
−    have "a \<bullet> y = 0" if "y \<in> span (S \<inter> {x. a \<bullet> x = 0})" for y
+
−    proof -
+
−      have "y \<in> span {x. a \<bullet> x = 0}"
+
−        by (metis inf.cobounded2 span_mono subsetCE that)
+
−      then show ?thesis
+
−        by (blast intro: span_induct [OF _ subspace_hyperplane])
+
−    qed
+
−    then have "dim (S \<inter> {x. a \<bullet> x = 0}) < n"
+
−      by (metis (no_types) less_ay c subsetD dim_eq_span inf.strict_order_iff
+
−           inf_le1 \<open>dim S = n\<close> not_le rel_interior_subset span_0 span_base)
+
−    then have "0 \<in> convex hull {x. x extreme_point_of (S \<inter> {x. a \<bullet> x = 0})}"
+
−      by (rule less.IH) (auto simp: co less.prems)
+
−    then show ?thesis
+
−      by (metis (mono_tags, lifting) Collect_mono_iff \<open>S \<inter> {x. a \<bullet> x = 0} face_of S\<close> extreme_point_of_face hull_mono subset_iff)
+
−  qed
+
−qed
+
−
+
−
+
−theorem Krein_Milman_Minkowski:
+
−  fixes S :: "'a::euclidean_space set"
+
−  assumes "compact S" "convex S"
+
−    shows "S = convex hull {x. x extreme_point_of S}"
+
−proof
+
−  show "S \<subseteq> convex hull {x. x extreme_point_of S}"
+
−  proof
+
−    fix a assume [simp]: "a \<in> S"
+
−    have 1: "compact ((+) (- a) ` S)"
+
−      by (simp add: \<open>compact S\<close> compact_translation_subtract cong: image_cong_simp)
+
−    have 2: "convex ((+) (- a) ` S)"
+
−      by (simp add: \<open>convex S\<close> compact_translation_subtract)
+
−    show a_invex: "a \<in> convex hull {x. x extreme_point_of S}"
+
−      using Krein_Milman_Minkowski_aux [OF refl 1 2]
+
−            convex_hull_translation [of "-a"]
+
−      by (auto simp: extreme_points_of_translation_subtract translation_assoc cong: image_cong_simp)
+
−    qed
+
−next
+
−  show "convex hull {x. x extreme_point_of S} \<subseteq> S"
+
−  proof -
+
−    have "{a. a extreme_point_of S} \<subseteq> S"
+
−      using extreme_point_of_def by blast
+
−    then show ?thesis
+
−      by (simp add: \<open>convex S\<close> hull_minimal)
+
−  qed
+
−qed
+
−
+
−
+
−subsection\<open>Applying it to convex hulls of explicitly indicated finite sets\<close>
+
−
+
−corollary Krein_Milman_polytope:
+
−  fixes S :: "'a::euclidean_space set"
+
−  shows
+
−   "finite S
+
−       \<Longrightarrow> convex hull S =
+
−           convex hull {x. x extreme_point_of (convex hull S)}"
+
−  by (simp add: Krein_Milman_Minkowski finite_imp_compact_convex_hull)
+
−
+
−lemma extreme_points_of_convex_hull_eq:
+
−  fixes S :: "'a::euclidean_space set"
+
−  shows
+
−   "\<lbrakk>compact S; \<And>T. T \<subset> S \<Longrightarrow> convex hull T \<noteq> convex hull S\<rbrakk>
+
−        \<Longrightarrow> {x. x extreme_point_of (convex hull S)} = S"
+
−by (metis (full_types) Krein_Milman_Minkowski compact_convex_hull convex_convex_hull extreme_points_of_convex_hull psubsetI)
+
−
+
−
+
−lemma extreme_point_of_convex_hull_eq:
+
−  fixes S :: "'a::euclidean_space set"
+
−  shows
+
−   "\<lbrakk>compact S; \<And>T. T \<subset> S \<Longrightarrow> convex hull T \<noteq> convex hull S\<rbrakk>
+
−    \<Longrightarrow> (x extreme_point_of (convex hull S) \<longleftrightarrow> x \<in> S)"
+
−using extreme_points_of_convex_hull_eq by auto
+
−
+
−lemma extreme_point_of_convex_hull_convex_independent:
+
−  fixes S :: "'a::euclidean_space set"
+
−  assumes "compact S" and S: "\<And>a. a \<in> S \<Longrightarrow> a \<notin> convex hull (S - {a})"
+
−  shows "(x extreme_point_of (convex hull S) \<longleftrightarrow> x \<in> S)"
+
−proof -
+
−  have "convex hull T \<noteq> convex hull S" if "T \<subset> S" for T
+
−  proof -
+
−    obtain a where  "T \<subseteq> S" "a \<in> S" "a \<notin> T" using \<open>T \<subset> S\<close> by blast
+
−    then show ?thesis
+
−      by (metis (full_types) Diff_eq_empty_iff Diff_insert0 S hull_mono hull_subset insert_Diff_single subsetCE)
+
−  qed
+
−  then show ?thesis
+
−    by (rule extreme_point_of_convex_hull_eq [OF \<open>compact S\<close>])
+
−qed
+
−
+
−lemma extreme_point_of_convex_hull_affine_independent:
+
−  fixes S :: "'a::euclidean_space set"
+
−  shows
+
−   "\<not> affine_dependent S
+
−         \<Longrightarrow> (x extreme_point_of (convex hull S) \<longleftrightarrow> x \<in> S)"
+
−by (metis aff_independent_finite affine_dependent_def affine_hull_convex_hull extreme_point_of_convex_hull_convex_independent finite_imp_compact hull_inc)
+
−
+
−text\<open>Elementary proofs exist, not requiring Euclidean spaces and all this development\<close>
+
−lemma extreme_point_of_convex_hull_2:
+
−  fixes x :: "'a::euclidean_space"
+
−  shows "x extreme_point_of (convex hull {a,b}) \<longleftrightarrow> x = a \<or> x = b"
+
−proof -
+
−  have "x extreme_point_of (convex hull {a,b}) \<longleftrightarrow> x \<in> {a,b}"
+
−    by (intro extreme_point_of_convex_hull_affine_independent affine_independent_2)
+
−  then show ?thesis
+
−    by simp
+
−qed
+
−
+
−lemma extreme_point_of_segment:
+
−  fixes x :: "'a::euclidean_space"
+
−  shows
+
−   "x extreme_point_of closed_segment a b \<longleftrightarrow> x = a \<or> x = b"
+
−by (simp add: extreme_point_of_convex_hull_2 segment_convex_hull)
+
−
+
−lemma face_of_convex_hull_subset:
+
−  fixes S :: "'a::euclidean_space set"
+
−  assumes "compact S" and T: "T face_of (convex hull S)"
+
−  obtains s' where "s' \<subseteq> S" "T = convex hull s'"
+
−apply (rule_tac s' = "{x. x extreme_point_of T}" in that)
+
−using T extreme_point_of_convex_hull extreme_point_of_face apply blast
+
−by (metis (no_types) Krein_Milman_Minkowski assms compact_convex_hull convex_convex_hull face_of_imp_compact face_of_imp_convex)
+
−
+
−
+
−lemma face_of_convex_hull_aux:
+
−  assumes eq: "x *\<^sub>R p = u *\<^sub>R a + v *\<^sub>R b + w *\<^sub>R c"
+
−    and x: "u + v + w = x" "x \<noteq> 0" and S: "affine S" "a \<in> S" "b \<in> S" "c \<in> S"
+
−  shows "p \<in> S"
+
−proof -
+
−  have "p = (u *\<^sub>R a + v *\<^sub>R b + w *\<^sub>R c) /\<^sub>R x"
+
−    by (metis \<open>x \<noteq> 0\<close> eq mult.commute right_inverse scaleR_one scaleR_scaleR)
+
−  moreover have "affine hull {a,b,c} \<subseteq> S"
+
−    by (simp add: S hull_minimal)
+
−  moreover have "(u *\<^sub>R a + v *\<^sub>R b + w *\<^sub>R c) /\<^sub>R x \<in> affine hull {a,b,c}"
+
−    apply (simp add: affine_hull_3)
+
−    apply (rule_tac x="u/x" in exI)
+
−    apply (rule_tac x="v/x" in exI)
+
−    apply (rule_tac x="w/x" in exI)
+
−    using x apply (auto simp: algebra_simps divide_simps)
+
−    done
+
−  ultimately show ?thesis by force
+
−qed
+
−
+
−proposition face_of_convex_hull_insert_eq:
+
−  fixes a :: "'a :: euclidean_space"
+
−  assumes "finite S" and a: "a \<notin> affine hull S"
+
−  shows "(F face_of (convex hull (insert a S)) \<longleftrightarrow>
+
−          F face_of (convex hull S) \<or>
+
−          (\<exists>F'. F' face_of (convex hull S) \<and> F = convex hull (insert a F')))"
+
−         (is "F face_of ?CAS \<longleftrightarrow> _")
+
−proof safe
+
−  assume F: "F face_of ?CAS"
+
−    and *: "\<nexists>F'. F' face_of convex hull S \<and> F = convex hull insert a F'"
+
−  obtain T where T: "T \<subseteq> insert a S" and FeqT: "F = convex hull T"
+
−    by (metis F \<open>finite S\<close> compact_insert finite_imp_compact face_of_convex_hull_subset)
+
−  show "F face_of convex hull S"
+
−  proof (cases "a \<in> T")
+
−    case True
+
−    have "F = convex hull insert a (convex hull T \<inter> convex hull S)"
+
−    proof
+
−      have "T \<subseteq> insert a (convex hull T \<inter> convex hull S)"
+
−        using T hull_subset by fastforce
+
−      then show "F \<subseteq> convex hull insert a (convex hull T \<inter> convex hull S)"
+
−        by (simp add: FeqT hull_mono)
+
−      show "convex hull insert a (convex hull T \<inter> convex hull S) \<subseteq> F"
+
−        apply (rule hull_minimal)
+
−        using True by (auto simp: \<open>F = convex hull T\<close> hull_inc)
+
−    qed
+
−    moreover have "convex hull T \<inter> convex hull S face_of convex hull S"
+
−      by (metis F FeqT convex_convex_hull face_of_slice hull_mono inf.absorb_iff2 subset_insertI)
+
−    ultimately show ?thesis
+
−      using * by force
+
−  next
+
−    case False
+
−    then show ?thesis
+
−      by (metis FeqT F T face_of_subset hull_mono subset_insert subset_insertI)
+
−  qed
+
−next
+
−  assume "F face_of convex hull S"
+
−  show "F face_of ?CAS"
+
−    by (simp add: \<open>F face_of convex hull S\<close> a face_of_convex_hull_insert \<open>finite S\<close>)
+
−next
+
−  fix F
+
−  assume F: "F face_of convex hull S"
+
−  show "convex hull insert a F face_of ?CAS"
+
−  proof (cases "S = {}")
+
−    case True
+
−    then show ?thesis
+
−      using F face_of_affine_eq by auto
+
−  next
+
−    case False
+
−    have anotc: "a \<notin> convex hull S"
+
−      by (metis (no_types) a affine_hull_convex_hull hull_inc)
+
−    show ?thesis
+
−    proof (cases "F = {}")
+
−      case True show ?thesis
+
−        using anotc by (simp add: \<open>F = {}\<close> \<open>finite S\<close> extreme_point_of_convex_hull_insert face_of_singleton)
+
−    next
+
−      case False
+
−      have "convex hull insert a F \<subseteq> ?CAS"
+
−        by (simp add: F a \<open>finite S\<close> convex_hull_subset face_of_convex_hull_insert face_of_imp_subset hull_inc)
+
−      moreover
+
−      have "(\<exists>y v. (1 - ub) *\<^sub>R a + ub *\<^sub>R b = (1 - v) *\<^sub>R a + v *\<^sub>R y \<and>
+
−                   0 \<le> v \<and> v \<le> 1 \<and> y \<in> F) \<and>
+
−            (\<exists>x u. (1 - uc) *\<^sub>R a + uc *\<^sub>R c = (1 - u) *\<^sub>R a + u *\<^sub>R x \<and>
+
−                   0 \<le> u \<and> u \<le> 1 \<and> x \<in> F)"
+
−        if *: "(1 - ux) *\<^sub>R a + ux *\<^sub>R x
+
−               \<in> open_segment ((1 - ub) *\<^sub>R a + ub *\<^sub>R b) ((1 - uc) *\<^sub>R a + uc *\<^sub>R c)"
+
−          and "0 \<le> ub" "ub \<le> 1" "0 \<le> uc" "uc \<le> 1" "0 \<le> ux" "ux \<le> 1"
+
−          and b: "b \<in> convex hull S" and c: "c \<in> convex hull S" and "x \<in> F"
+
−        for b c ub uc ux x
+
−      proof -
+
−        obtain v where ne: "(1 - ub) *\<^sub>R a + ub *\<^sub>R b \<noteq> (1 - uc) *\<^sub>R a + uc *\<^sub>R c"
+
−          and eq: "(1 - ux) *\<^sub>R a + ux *\<^sub>R x =
+
−                    (1 - v) *\<^sub>R ((1 - ub) *\<^sub>R a + ub *\<^sub>R b) + v *\<^sub>R ((1 - uc) *\<^sub>R a + uc *\<^sub>R c)"
+
−          and "0 < v" "v < 1"
+
−          using * by (auto simp: in_segment)
+
−        then have 0: "((1 - ux) - ((1 - v) * (1 - ub) + v * (1 - uc))) *\<^sub>R a +
+
−                      (ux *\<^sub>R x - (((1 - v) * ub) *\<^sub>R b + (v * uc) *\<^sub>R c)) = 0"
+
−          by (auto simp: algebra_simps)
+
−        then have "((1 - ux) - ((1 - v) * (1 - ub) + v * (1 - uc))) *\<^sub>R a =
+
−                   ((1 - v) * ub) *\<^sub>R b + (v * uc) *\<^sub>R c + (-ux) *\<^sub>R x"
+
−          by (auto simp: algebra_simps)
+
−        then have "a \<in> affine hull S" if "1 - ux - ((1 - v) * (1 - ub) + v * (1 - uc)) \<noteq> 0"
+
−          apply (rule face_of_convex_hull_aux)
+
−          using b c that apply (auto simp: algebra_simps)
+
−          using F convex_hull_subset_affine_hull face_of_imp_subset \<open>x \<in> F\<close> apply blast+
+
−          done
+
−        then have "1 - ux - ((1 - v) * (1 - ub) + v * (1 - uc)) = 0"
+
−          using a by blast
+
−        with 0 have equx: "(1 - v) * ub + v * uc = ux"
+
−          and uxx: "ux *\<^sub>R x = (((1 - v) * ub) *\<^sub>R b + (v * uc) *\<^sub>R c)"
+
−          by auto (auto simp: algebra_simps)
+
−        show ?thesis
+
−        proof (cases "uc = 0")
+
−          case True
+
−          then show ?thesis
+
−            using equx 0 \<open>0 \<le> ub\<close> \<open>ub \<le> 1\<close> \<open>v < 1\<close> \<open>x \<in> F\<close>
+
−            apply (auto simp: algebra_simps)
+
−             apply (rule_tac x=x in exI, simp)
+
−             apply (rule_tac x=ub in exI, auto)
+
−             apply (metis add.left_neutral diff_eq_eq less_irrefl mult.commute mult_cancel_right1 real_vector.scale_cancel_left real_vector.scale_left_diff_distrib)
+
−            using \<open>x \<in> F\<close> \<open>uc \<le> 1\<close> apply blast
+
−            done
+
−        next
+
−          case False
+
−          show ?thesis
+
−          proof (cases "ub = 0")
+
−            case True
+
−            then show ?thesis
+
−              using equx 0 \<open>0 \<le> uc\<close> \<open>uc \<le> 1\<close> \<open>0 < v\<close> \<open>x \<in> F\<close> \<open>uc \<noteq> 0\<close> by (force simp: algebra_simps)
+
−          next
+
−            case False
+
−            then have "0 < ub" "0 < uc"
+
−              using \<open>uc \<noteq> 0\<close> \<open>0 \<le> ub\<close> \<open>0 \<le> uc\<close> by auto
+
−            then have "ux \<noteq> 0"
+
−              by (metis \<open>0 < v\<close> \<open>v < 1\<close> diff_ge_0_iff_ge dual_order.strict_implies_order equx leD le_add_same_cancel2 zero_le_mult_iff zero_less_mult_iff)
+
−            have "b \<in> F \<and> c \<in> F"
+
−            proof (cases "b = c")
+
−              case True
+
−              then show ?thesis
+
−                by (metis \<open>ux \<noteq> 0\<close> equx real_vector.scale_cancel_left scaleR_add_left uxx \<open>x \<in> F\<close>)
+
−            next
+
−              case False
+
−              have "x = (((1 - v) * ub) *\<^sub>R b + (v * uc) *\<^sub>R c) /\<^sub>R ux"
+
−                by (metis \<open>ux \<noteq> 0\<close> uxx mult.commute right_inverse scaleR_one scaleR_scaleR)
+
−              also have "... = (1 - v * uc / ux) *\<^sub>R b + (v * uc / ux) *\<^sub>R c"
+
−                using \<open>ux \<noteq> 0\<close> equx apply (auto simp: algebra_simps divide_simps)
+
−                by (metis add.commute add_diff_eq add_divide_distrib diff_add_cancel scaleR_add_left)
+
−              finally have "x = (1 - v * uc / ux) *\<^sub>R b + (v * uc / ux) *\<^sub>R c" .
+
−              then have "x \<in> open_segment b c"
+
−                apply (simp add: in_segment \<open>b \<noteq> c\<close>)
+
−                apply (rule_tac x="(v * uc) / ux" in exI)
+
−                using \<open>0 \<le> ux\<close> \<open>ux \<noteq> 0\<close> \<open>0 < uc\<close> \<open>0 < v\<close> \<open>0 < ub\<close> \<open>v < 1\<close> equx
+
−                apply (force simp: algebra_simps divide_simps)
+
−                done
+
−              then show ?thesis
+
−                by (rule face_ofD [OF F _ b c \<open>x \<in> F\<close>])
+
−            qed
+
−            with \<open>0 \<le> ub\<close> \<open>ub \<le> 1\<close> \<open>0 \<le> uc\<close> \<open>uc \<le> 1\<close> show ?thesis by blast
+
−          qed
+
−        qed
+
−      qed
+
−      moreover have "convex hull F = F"
+
−        by (meson F convex_hull_eq face_of_imp_convex)
+
−      ultimately show ?thesis
+
−        unfolding face_of_def by (fastforce simp: convex_hull_insert_alt \<open>S \<noteq> {}\<close> \<open>F \<noteq> {}\<close>)
+
−    qed
+
−  qed
+
−qed
+
−
+
−lemma face_of_convex_hull_insert2:
+
−  fixes a :: "'a :: euclidean_space"
+
−  assumes S: "finite S" and a: "a \<notin> affine hull S" and F: "F face_of convex hull S"
+
−  shows "convex hull (insert a F) face_of convex hull (insert a S)"
+
−  by (metis F face_of_convex_hull_insert_eq [OF S a])
+
−
+
−proposition face_of_convex_hull_affine_independent:
+
−  fixes S :: "'a::euclidean_space set"
+
−  assumes "\<not> affine_dependent S"
+
−    shows "(T face_of (convex hull S) \<longleftrightarrow> (\<exists>c. c \<subseteq> S \<and> T = convex hull c))"
+
−          (is "?lhs = ?rhs")
+
−proof
+
−  assume ?lhs
+
−  then show ?rhs
+
−    by (meson \<open>T face_of convex hull S\<close> aff_independent_finite assms face_of_convex_hull_subset finite_imp_compact)
+
−next
+
−  assume ?rhs
+
−  then obtain c where "c \<subseteq> S" and T: "T = convex hull c"
+
−    by blast
+
−  have "affine hull c \<inter> affine hull (S - c) = {}"
+
−    apply (rule disjoint_affine_hull [OF assms \<open>c \<subseteq> S\<close>], auto)
+
−    done
+
−  then have "affine hull c \<inter> convex hull (S - c) = {}"
+
−    using convex_hull_subset_affine_hull by fastforce
+
−  then show ?lhs
+
−    by (metis face_of_convex_hulls \<open>c \<subseteq> S\<close> aff_independent_finite assms T)
+
−qed
+
−
+
−lemma facet_of_convex_hull_affine_independent:
+
−  fixes S :: "'a::euclidean_space set"
+
−  assumes "\<not> affine_dependent S"
+
−    shows "T facet_of (convex hull S) \<longleftrightarrow>
+
−           T \<noteq> {} \<and> (\<exists>u. u \<in> S \<and> T = convex hull (S - {u}))"
+
−          (is "?lhs = ?rhs")
+
−proof
+
−  assume ?lhs
+
−  then have "T face_of (convex hull S)" "T \<noteq> {}"
+
−        and afft: "aff_dim T = aff_dim (convex hull S) - 1"
+
−    by (auto simp: facet_of_def)
+
−  then obtain c where "c \<subseteq> S" and c: "T = convex hull c"
+
−    by (auto simp: face_of_convex_hull_affine_independent [OF assms])
+
−  then have affs: "aff_dim S = aff_dim c + 1"
+
−    by (metis aff_dim_convex_hull afft eq_diff_eq)
+
−  have "\<not> affine_dependent c"
+
−    using \<open>c \<subseteq> S\<close> affine_dependent_subset assms by blast
+
−  with affs have "card (S - c) = 1"
+
−    apply (simp add: aff_dim_affine_independent [symmetric] aff_dim_convex_hull)
+
−    by (metis aff_dim_affine_independent aff_independent_finite One_nat_def \<open>c \<subseteq> S\<close> add.commute
+
−                add_diff_cancel_right' assms card_Diff_subset card_mono of_nat_1 of_nat_diff of_nat_eq_iff)
+
−  then obtain u where u: "u \<in> S - c"
+
−    by (metis DiffI \<open>c \<subseteq> S\<close> aff_independent_finite assms cancel_comm_monoid_add_class.diff_cancel
+
−                card_Diff_subset subsetI subset_antisym zero_neq_one)
+
−  then have u: "S = insert u c"
+
−    by (metis Diff_subset \<open>c \<subseteq> S\<close> \<open>card (S - c) = 1\<close> card_1_singletonE double_diff insert_Diff insert_subset singletonD)
+
−  have "T = convex hull (c - {u})"
+
−    by (metis Diff_empty Diff_insert0 \<open>T facet_of convex hull S\<close> c facet_of_irrefl insert_absorb u)
+
−  with \<open>T \<noteq> {}\<close> show ?rhs
+
−    using c u by auto
+
−next
+
−  assume ?rhs
+
−  then obtain u where "T \<noteq> {}" "u \<in> S" and u: "T = convex hull (S - {u})"
+
−    by (force simp: facet_of_def)
+
−  then have "\<not> S \<subseteq> {u}"
+
−    using \<open>T \<noteq> {}\<close> u by auto
+
−  have [simp]: "aff_dim (convex hull (S - {u})) = aff_dim (convex hull S) - 1"
+
−    using assms \<open>u \<in> S\<close>
+
−    apply (simp add: aff_dim_convex_hull affine_dependent_def)
+
−    apply (drule bspec, assumption)
+
−    by (metis add_diff_cancel_right' aff_dim_insert insert_Diff [of u S])
+
−  show ?lhs
+
−    apply (subst u)
+
−    apply (simp add: \<open>\<not> S \<subseteq> {u}\<close> facet_of_def face_of_convex_hull_affine_independent [OF assms], blast)
+
−    done
+
−qed
+
−
+
−lemma facet_of_convex_hull_affine_independent_alt:
+
−  fixes S :: "'a::euclidean_space set"
+
−  shows
+
−   "\<not>affine_dependent S
+
−        \<Longrightarrow> (T facet_of (convex hull S) \<longleftrightarrow>
+
−             2 \<le> card S \<and> (\<exists>u. u \<in> S \<and> T = convex hull (S - {u})))"
+
−apply (simp add: facet_of_convex_hull_affine_independent)
+
−apply (auto simp: Set.subset_singleton_iff)
+
−apply (metis Diff_cancel Int_empty_right Int_insert_right_if1  aff_independent_finite card_eq_0_iff card_insert_if card_mono card_subset_eq convex_hull_eq_empty eq_iff equals0D finite_insert finite_subset inf.absorb_iff2 insert_absorb insert_not_empty  not_less_eq_eq numeral_2_eq_2)
+
−done
+
−
+
−lemma segment_face_of:
+
−  assumes "(closed_segment a b) face_of S"
+
−  shows "a extreme_point_of S" "b extreme_point_of S"
+
−proof -
+
−  have as: "{a} face_of S"
+
−    by (metis (no_types) assms convex_hull_singleton empty_iff extreme_point_of_convex_hull_insert face_of_face face_of_singleton finite.emptyI finite.insertI insert_absorb insert_iff segment_convex_hull)
+
−  moreover have "{b} face_of S"
+
−  proof -
+
−    have "b \<in> convex hull {a} \<or> b extreme_point_of convex hull {b, a}"
+
−      by (meson extreme_point_of_convex_hull_insert finite.emptyI finite.insertI)
+
−    moreover have "closed_segment a b = convex hull {b, a}"
+
−      using closed_segment_commute segment_convex_hull by blast
+
−    ultimately show ?thesis
+
−      by (metis as assms face_of_face convex_hull_singleton empty_iff face_of_singleton insertE)
+
−    qed
+
−  ultimately show "a extreme_point_of S" "b extreme_point_of S"
+
−    using face_of_singleton by blast+
+
−qed
+
−
+
−
+
−proposition Krein_Milman_frontier:
+
−  fixes S :: "'a::euclidean_space set"
+
−  assumes "convex S" "compact S"
+
−    shows "S = convex hull (frontier S)"
+
−          (is "?lhs = ?rhs")
+
−proof
+
−  have "?lhs \<subseteq> convex hull {x. x extreme_point_of S}"
+
−    using Krein_Milman_Minkowski assms by blast
+
−  also have "... \<subseteq> ?rhs"
+
−    apply (rule hull_mono)
+
−    apply (auto simp: frontier_def extreme_point_not_in_interior)
+
−    using closure_subset apply (force simp: extreme_point_of_def)
+
−    done
+
−  finally show "?lhs \<subseteq> ?rhs" .
+
−next
+
−  have "?rhs \<subseteq> convex hull S"
+
−    by (metis Diff_subset \<open>compact S\<close> closure_closed compact_eq_bounded_closed frontier_def hull_mono)
+
−  also have "... \<subseteq> ?lhs"
+
−    by (simp add: \<open>convex S\<close> hull_same)
+
−  finally show "?rhs \<subseteq> ?lhs" .
+
−qed
+
−
+
−subsection\<open>Polytopes\<close>
+
−
+
−definition%important polytope where
+
− "polytope S \<equiv> \<exists>v. finite v \<and> S = convex hull v"
+
−
+
−lemma polytope_translation_eq: "polytope (image (\<lambda>x. a + x) S) \<longleftrightarrow> polytope S"
+
−apply (simp add: polytope_def, safe)
+
−apply (metis convex_hull_translation finite_imageI translation_galois)
+
−by (metis convex_hull_translation finite_imageI)
+
−
+
−lemma polytope_linear_image: "\<lbrakk>linear f; polytope p\<rbrakk> \<Longrightarrow> polytope(image f p)"
+
−  unfolding polytope_def using convex_hull_linear_image by blast
+
−
+
−lemma polytope_empty: "polytope {}"
+
−  using convex_hull_empty polytope_def by blast
+
−
+
−lemma polytope_convex_hull: "finite S \<Longrightarrow> polytope(convex hull S)"
+
−  using polytope_def by auto
+
−
+
−lemma polytope_Times: "\<lbrakk>polytope S; polytope T\<rbrakk> \<Longrightarrow> polytope(S \<times> T)"
+
−  unfolding polytope_def
+
−  by (metis finite_cartesian_product convex_hull_Times)
+
−
+
−lemma face_of_polytope_polytope:
+
−  fixes S :: "'a::euclidean_space set"
+
−  shows "\<lbrakk>polytope S; F face_of S\<rbrakk> \<Longrightarrow> polytope F"
+
−unfolding polytope_def
+
−by (meson face_of_convex_hull_subset finite_imp_compact finite_subset)
+
−
+
−lemma finite_polytope_faces:
+
−  fixes S :: "'a::euclidean_space set"
+
−  assumes "polytope S"
+
−  shows "finite {F. F face_of S}"
+
−proof -
+
−  obtain v where "finite v" "S = convex hull v"
+
−    using assms polytope_def by auto
+
−  have "finite ((hull) convex ` {T. T \<subseteq> v})"
+
−    by (simp add: \<open>finite v\<close>)
+
−  moreover have "{F. F face_of S} \<subseteq> ((hull) convex ` {T. T \<subseteq> v})"
+
−    by (metis (no_types, lifting) \<open>finite v\<close> \<open>S = convex hull v\<close> face_of_convex_hull_subset finite_imp_compact image_eqI mem_Collect_eq subsetI)
+
−  ultimately show ?thesis
+
−    by (blast intro: finite_subset)
+
−qed
+
−
+
−lemma finite_polytope_facets:
+
−  assumes "polytope S"
+
−  shows "finite {T. T facet_of S}"
+
−by (simp add: assms facet_of_def finite_polytope_faces)
+
−
+
−lemma polytope_scaling:
+
−  assumes "polytope S"  shows "polytope (image (\<lambda>x. c *\<^sub>R x) S)"
+
−by (simp add: assms polytope_linear_image)
+
−
+
−lemma polytope_imp_compact:
+
−  fixes S :: "'a::real_normed_vector set"
+
−  shows "polytope S \<Longrightarrow> compact S"
+
−by (metis finite_imp_compact_convex_hull polytope_def)
+
−
+
−lemma polytope_imp_convex: "polytope S \<Longrightarrow> convex S"
+
−  by (metis convex_convex_hull polytope_def)
+
−
+
−lemma polytope_imp_closed:
+
−  fixes S :: "'a::real_normed_vector set"
+
−  shows "polytope S \<Longrightarrow> closed S"
+
−by (simp add: compact_imp_closed polytope_imp_compact)
+
−
+
−lemma polytope_imp_bounded:
+
−  fixes S :: "'a::real_normed_vector set"
+
−  shows "polytope S \<Longrightarrow> bounded S"
+
−by (simp add: compact_imp_bounded polytope_imp_compact)
+
−
+
−lemma polytope_interval: "polytope(cbox a b)"
+
−  unfolding polytope_def by (meson closed_interval_as_convex_hull)
+
−
+
−lemma polytope_sing: "polytope {a}"
+
−  using polytope_def by force
+
−
+
−lemma face_of_polytope_insert:
+
−     "\<lbrakk>polytope S; a \<notin> affine hull S; F face_of S\<rbrakk> \<Longrightarrow> F face_of convex hull (insert a S)"
+
−  by (metis (no_types, lifting) affine_hull_convex_hull face_of_convex_hull_insert hull_insert polytope_def)
+
−
+
−proposition face_of_polytope_insert2:
+
−  fixes a :: "'a :: euclidean_space"
+
−  assumes "polytope S" "a \<notin> affine hull S" "F face_of S"
+
−  shows "convex hull (insert a F) face_of convex hull (insert a S)"
+
−proof -
+
−  obtain V where "finite V" "S = convex hull V"
+
−    using assms by (auto simp: polytope_def)
+
−  then have "convex hull (insert a F) face_of convex hull (insert a V)"
+
−    using affine_hull_convex_hull assms face_of_convex_hull_insert2 by blast
+
−  then show ?thesis
+
−    by (metis \<open>S = convex hull V\<close> hull_insert)
+
−qed
+
−
+
−
+
−subsection\<open>Polyhedra\<close>
+
−
+
−definition%important polyhedron where
+
− "polyhedron S \<equiv>
+
−        \<exists>F. finite F \<and>
+
−            S = \<Inter> F \<and>
+
−            (\<forall>h \<in> F. \<exists>a b. a \<noteq> 0 \<and> h = {x. a \<bullet> x \<le> b})"
+
−
+
−lemma polyhedron_Int [intro,simp]:
+
−   "\<lbrakk>polyhedron S; polyhedron T\<rbrakk> \<Longrightarrow> polyhedron (S \<inter> T)"
+
−  apply (simp add: polyhedron_def, clarify)
+
−  apply (rename_tac F G)
+
−  apply (rule_tac x="F \<union> G" in exI, auto)
+
−  done
+
−
+
−lemma polyhedron_UNIV [iff]: "polyhedron UNIV"
+
−  unfolding polyhedron_def
+
−  by (rule_tac x="{}" in exI) auto
+
−
+
−lemma polyhedron_Inter [intro,simp]:
+
−   "\<lbrakk>finite F; \<And>S. S \<in> F \<Longrightarrow> polyhedron S\<rbrakk> \<Longrightarrow> polyhedron(\<Inter>F)"
+
−by (induction F rule: finite_induct) auto
+
−
+
−
+
−lemma polyhedron_empty [iff]: "polyhedron ({} :: 'a :: euclidean_space set)"
+
−proof -
+
−  have "\<exists>a. a \<noteq> 0 \<and>
+
−             (\<exists>b. {x. (SOME i. i \<in> Basis) \<bullet> x \<le> - 1} = {x. a \<bullet> x \<le> b})"
+
−    by (rule_tac x="(SOME i. i \<in> Basis)" in exI) (force simp: SOME_Basis nonzero_Basis)
+
−  moreover have "\<exists>a b. a \<noteq> 0 \<and>
+
−                       {x. - (SOME i. i \<in> Basis) \<bullet> x \<le> - 1} = {x. a \<bullet> x \<le> b}"
+
−      apply (rule_tac x="-(SOME i. i \<in> Basis)" in exI)
+
−      apply (rule_tac x="-1" in exI)
+
−      apply (simp add: SOME_Basis nonzero_Basis)
+
−      done
+
−  ultimately show ?thesis
+
−    unfolding polyhedron_def
+
−    apply (rule_tac x="{{x. (SOME i. i \<in> Basis) \<bullet> x \<le> -1},
+
−                        {x. -(SOME i. i \<in> Basis) \<bullet> x \<le> -1}}" in exI)
+
−    apply force
+
−    done
+
−qed
+
−
+
−lemma polyhedron_halfspace_le:
+
−  fixes a :: "'a :: euclidean_space"
+
−  shows "polyhedron {x. a \<bullet> x \<le> b}"
+
−proof (cases "a = 0")
+
−  case True then show ?thesis by auto
+
−next
+
−  case False
+
−  then show ?thesis
+
−    unfolding polyhedron_def
+
−    by (rule_tac x="{{x. a \<bullet> x \<le> b}}" in exI) auto
+
−qed
+
−
+
−lemma polyhedron_halfspace_ge:
+
−  fixes a :: "'a :: euclidean_space"
+
−  shows "polyhedron {x. a \<bullet> x \<ge> b}"
+
−using polyhedron_halfspace_le [of "-a" "-b"] by simp
+
−
+
−lemma polyhedron_hyperplane:
+
−  fixes a :: "'a :: euclidean_space"
+
−  shows "polyhedron {x. a \<bullet> x = b}"
+
−proof -
+
−  have "{x. a \<bullet> x = b} = {x. a \<bullet> x \<le> b} \<inter> {x. a \<bullet> x \<ge> b}"
+
−    by force
+
−  then show ?thesis
+
−    by (simp add: polyhedron_halfspace_ge polyhedron_halfspace_le)
+
−qed
+
−
+
−lemma affine_imp_polyhedron:
+
−  fixes S :: "'a :: euclidean_space set"
+
−  shows "affine S \<Longrightarrow> polyhedron S"
+
−by (metis affine_hull_eq polyhedron_Inter polyhedron_hyperplane affine_hull_finite_intersection_hyperplanes [of S])
+
−
+
−lemma polyhedron_imp_closed:
+
−  fixes S :: "'a :: euclidean_space set"
+
−  shows "polyhedron S \<Longrightarrow> closed S"
+
−apply (simp add: polyhedron_def)
+
−using closed_halfspace_le by fastforce
+
−
+
−lemma polyhedron_imp_convex:
+
−  fixes S :: "'a :: euclidean_space set"
+
−  shows "polyhedron S \<Longrightarrow> convex S"
+
−apply (simp add: polyhedron_def)
+
−using convex_Inter convex_halfspace_le by fastforce
+
−
+
−lemma polyhedron_affine_hull:
+
−  fixes S :: "'a :: euclidean_space set"
+
−  shows "polyhedron(affine hull S)"
+
−by (simp add: affine_imp_polyhedron)
+
−
+
−
+
−subsection\<open>Canonical polyhedron representation making facial structure explicit\<close>
+
−
+
−proposition polyhedron_Int_affine:
+
−  fixes S :: "'a :: euclidean_space set"
+
−  shows "polyhedron S \<longleftrightarrow>
+
−           (\<exists>F. finite F \<and> S = (affine hull S) \<inter> \<Inter>F \<and>
+
−                (\<forall>h \<in> F. \<exists>a b. a \<noteq> 0 \<and> h = {x. a \<bullet> x \<le> b}))"
+
−        (is "?lhs = ?rhs")
+
−proof
+
−  assume ?lhs then show ?rhs
+
−    apply (simp add: polyhedron_def)
+
−    apply (erule ex_forward)
+
−    using hull_subset apply force
+
−    done
+
−next
+
−  assume ?rhs then show ?lhs
+
−    apply clarify
+
−    apply (erule ssubst)
+
−    apply (force intro: polyhedron_affine_hull polyhedron_halfspace_le)
+
−    done
+
−qed
+
−
+
−proposition rel_interior_polyhedron_explicit:
+
−  assumes "finite F"
+
−      and seq: "S = affine hull S \<inter> \<Inter>F"
+
−      and faceq: "\<And>h. h \<in> F \<Longrightarrow> a h \<noteq> 0 \<and> h = {x. a h \<bullet> x \<le> b h}"
+
−      and psub: "\<And>F'. F' \<subset> F \<Longrightarrow> S \<subset> affine hull S \<inter> \<Inter>F'"
+
−    shows "rel_interior S = {x \<in> S. \<forall>h \<in> F. a h \<bullet> x < b h}"
+
−proof -
+
−  have rels: "\<And>x. x \<in> rel_interior S \<Longrightarrow> x \<in> S"
+
−    by (meson IntE mem_rel_interior)
+
−  moreover have "a i \<bullet> x < b i" if x: "x \<in> rel_interior S" and "i \<in> F" for x i
+
−  proof -
+
−    have fif: "F - {i} \<subset> F"
+
−      using \<open>i \<in> F\<close> Diff_insert_absorb Diff_subset set_insert psubsetI by blast
+
−    then have "S \<subset> affine hull S \<inter> \<Inter>(F - {i})"
+
−      by (rule psub)
+
−    then obtain z where ssub: "S \<subseteq> \<Inter>(F - {i})" and zint: "z \<in> \<Inter>(F - {i})"
+
−                    and "z \<notin> S" and zaff: "z \<in> affine hull S"
+
−      by auto
+
−    have "z \<noteq> x"
+
−      using \<open>z \<notin> S\<close> rels x by blast
+
−    have "z \<notin> affine hull S \<inter> \<Inter>F"
+
−      using \<open>z \<notin> S\<close> seq by auto
+
−    then have aiz: "a i \<bullet> z > b i"
+
−      using faceq zint zaff by fastforce
+
−    obtain e where "e > 0" "x \<in> S" and e: "ball x e \<inter> affine hull S \<subseteq> S"
+
−      using x by (auto simp: mem_rel_interior_ball)
+
−    then have ins: "\<And>y. \<lbrakk>norm (x - y) < e; y \<in> affine hull S\<rbrakk> \<Longrightarrow> y \<in> S"
+
−      by (metis IntI subsetD dist_norm mem_ball)
+
−    define \<xi> where "\<xi> = min (1/2) (e / 2 / norm(z - x))"
+
−    have "norm (\<xi> *\<^sub>R x - \<xi> *\<^sub>R z) = norm (\<xi> *\<^sub>R (x - z))"
+
−      by (simp add: \<xi>_def algebra_simps norm_mult)
+
−    also have "... = \<xi> * norm (x - z)"
+
−      using \<open>e > 0\<close> by (simp add: \<xi>_def)
+
−    also have "... < e"
+
−      using \<open>z \<noteq> x\<close> \<open>e > 0\<close> by (simp add: \<xi>_def min_def divide_simps norm_minus_commute)
+
−    finally have lte: "norm (\<xi> *\<^sub>R x - \<xi> *\<^sub>R z) < e" .
+
−    have \<xi>_aff: "\<xi> *\<^sub>R z + (1 - \<xi>) *\<^sub>R x \<in> affine hull S"
+
−      by (metis \<open>x \<in> S\<close> add.commute affine_affine_hull diff_add_cancel hull_inc mem_affine zaff)
+
−    have "\<xi> *\<^sub>R z + (1 - \<xi>) *\<^sub>R x \<in> S"
+
−      apply (rule ins [OF _ \<xi>_aff])
+
−      apply (simp add: algebra_simps lte)
+
−      done
+
−    then obtain l where l: "0 < l" "l < 1" and ls: "(l *\<^sub>R z + (1 - l) *\<^sub>R x) \<in> S"
+
−      apply (rule_tac l = \<xi> in that)
+
−      using \<open>e > 0\<close> \<open>z \<noteq> x\<close>  apply (auto simp: \<xi>_def)
+
−      done
+
−    then have i: "l *\<^sub>R z + (1 - l) *\<^sub>R x \<in> i"
+
−      using seq \<open>i \<in> F\<close> by auto
+
−    have "b i * l + (a i \<bullet> x) * (1 - l) < a i \<bullet> (l *\<^sub>R z + (1 - l) *\<^sub>R x)"
+
−      using l by (simp add: algebra_simps aiz)
+
−    also have "\<dots> \<le> b i" using i l
+
−      using faceq mem_Collect_eq \<open>i \<in> F\<close> by blast
+
−    finally have "(a i \<bullet> x) * (1 - l) < b i * (1 - l)"
+
−      by (simp add: algebra_simps)
+
−    with l show ?thesis
+
−      by simp
+
−  qed
+
−  moreover have "x \<in> rel_interior S"
+
−           if "x \<in> S" and less: "\<And>h. h \<in> F \<Longrightarrow> a h \<bullet> x < b h" for x
+
−  proof -
+
−    have 1: "\<And>h. h \<in> F \<Longrightarrow> x \<in> interior h"
+
−      by (metis interior_halfspace_le mem_Collect_eq less faceq)
+
−    have 2: "\<And>y. \<lbrakk>\<forall>h\<in>F. y \<in> interior h; y \<in> affine hull S\<rbrakk> \<Longrightarrow> y \<in> S"
+
−      by (metis IntI Inter_iff contra_subsetD interior_subset seq)
+
−    show ?thesis
+
−      apply (simp add: rel_interior \<open>x \<in> S\<close>)
+
−      apply (rule_tac x="\<Inter>h\<in>F. interior h" in exI)
+
−      apply (auto simp: \<open>finite F\<close> open_INT 1 2)
+
−      done
+
−  qed
+
−  ultimately show ?thesis by blast
+
−qed
+
−
+
−
+
−lemma polyhedron_Int_affine_parallel:
+
−  fixes S :: "'a :: euclidean_space set"
+
−  shows "polyhedron S \<longleftrightarrow>
+
−         (\<exists>F. finite F \<and>
+
−              S = (affine hull S) \<inter> (\<Inter>F) \<and>
+
−              (\<forall>h \<in> F. \<exists>a b. a \<noteq> 0 \<and> h = {x. a \<bullet> x \<le> b} \<and>
+
−                             (\<forall>x \<in> affine hull S. (x + a) \<in> affine hull S)))"
+
−    (is "?lhs = ?rhs")
+
−proof
+
−  assume ?lhs
+
−  then obtain F where "finite F" and seq: "S = (affine hull S) \<inter> \<Inter>F"
+
−                  and faces: "\<And>h. h \<in> F \<Longrightarrow> \<exists>a b. a \<noteq> 0 \<and> h = {x. a \<bullet> x \<le> b}"
+
−    by (fastforce simp add: polyhedron_Int_affine)
+
−  then obtain a b where ab: "\<And>h. h \<in> F \<Longrightarrow> a h \<noteq> 0 \<and> h = {x. a h \<bullet> x \<le> b h}"
+
−    by metis
+
−  show ?rhs
+
−  proof -
+
−    have "\<exists>a' b'. a' \<noteq> 0 \<and>
+
−                  affine hull S \<inter> {x. a' \<bullet> x \<le> b'} = affine hull S \<inter> h \<and>
+
−                  (\<forall>w \<in> affine hull S. (w + a') \<in> affine hull S)"
+
−        if "h \<in> F" "\<not>(affine hull S \<subseteq> h)" for h
+
−    proof -
+
−      have "a h \<noteq> 0" and "h = {x. a h \<bullet> x \<le> b h}" "h \<inter> \<Inter>F = \<Inter>F"
+
−        using \<open>h \<in> F\<close> ab by auto
+
−      then have "(affine hull S) \<inter> {x. a h \<bullet> x \<le> b h} \<noteq> {}"
+
−        by (metis (no_types) affine_hull_eq_empty inf.absorb_iff2 inf_assoc inf_bot_right inf_commute seq that(2))
+
−      moreover have "\<not> (affine hull S \<subseteq> {x. a h \<bullet> x \<le> b h})"
+
−        using \<open>h = {x. a h \<bullet> x \<le> b h}\<close> that(2) by blast
+
−      ultimately show ?thesis
+
−        using affine_parallel_slice [of "affine hull S"]
+
−        by (metis \<open>h = {x. a h \<bullet> x \<le> b h}\<close> affine_affine_hull)
+
−    qed
+
−    then obtain a b
+
−         where ab: "\<And>h. \<lbrakk>h \<in> F; \<not> (affine hull S \<subseteq> h)\<rbrakk>
+
−             \<Longrightarrow> a h \<noteq> 0 \<and>
+
−                  affine hull S \<inter> {x. a h \<bullet> x \<le> b h} = affine hull S \<inter> h \<and>
+
−                  (\<forall>w \<in> affine hull S. (w + a h) \<in> affine hull S)"
+
−      by metis
+
−    have seq2: "S = affine hull S \<inter> (\<Inter>h\<in>{h \<in> F. \<not> affine hull S \<subseteq> h}. {x. a h \<bullet> x \<le> b h})"
+
−      by (subst seq) (auto simp: ab INT_extend_simps)
+
−    show ?thesis
+
−      apply (rule_tac x="(\<lambda>h. {x. a h \<bullet> x \<le> b h}) ` {h. h \<in> F \<and> \<not>(affine hull S \<subseteq> h)}" in exI)
+
−      apply (intro conjI seq2)
+
−        using \<open>finite F\<close> apply force
+
−       using ab apply blast
+
−       done
+
−  qed
+
−next
+
−  assume ?rhs then show ?lhs
+
−    apply (simp add: polyhedron_Int_affine)
+
−    by metis
+
−qed
+
−
+
−
+
−proposition polyhedron_Int_affine_parallel_minimal:
+
−  fixes S :: "'a :: euclidean_space set"
+
−  shows "polyhedron S \<longleftrightarrow>
+
−         (\<exists>F. finite F \<and>
+
−              S = (affine hull S) \<inter> (\<Inter>F) \<and>
+
−              (\<forall>h \<in> F. \<exists>a b. a \<noteq> 0 \<and> h = {x. a \<bullet> x \<le> b} \<and>
+
−                             (\<forall>x \<in> affine hull S. (x + a) \<in> affine hull S)) \<and>
+
−              (\<forall>F'. F' \<subset> F \<longrightarrow> S \<subset> (affine hull S) \<inter> (\<Inter>F')))"
+
−    (is "?lhs = ?rhs")
+
−proof
+
−  assume ?lhs
+
−  then obtain f0
+
−           where f0: "finite f0"
+
−                 "S = (affine hull S) \<inter> (\<Inter>f0)"
+
−                   (is "?P f0")
+
−                 "\<forall>h \<in> f0. \<exists>a b. a \<noteq> 0 \<and> h = {x. a \<bullet> x \<le> b} \<and>
+
−                             (\<forall>x \<in> affine hull S. (x + a) \<in> affine hull S)"
+
−                   (is "?Q f0")
+
−    by (force simp: polyhedron_Int_affine_parallel)
+
−  define n where "n = (LEAST n. \<exists>F. card F = n \<and> finite F \<and> ?P F \<and> ?Q F)"
+
−  have nf: "\<exists>F. card F = n \<and> finite F \<and> ?P F \<and> ?Q F"
+
−    apply (simp add: n_def)
+
−    apply (rule LeastI [where k = "card f0"])
+
−    using f0 apply auto
+
−    done
+
−  then obtain F where F: "card F = n" "finite F" and seq: "?P F" and aff: "?Q F"
+
−    by blast
+
−  then have "\<not> (finite g \<and> ?P g \<and> ?Q g)" if "card g < n" for g
+
−    using that by (auto simp: n_def dest!: not_less_Least)
+
−  then have *: "\<not> (?P g \<and> ?Q g)" if "g \<subset> F" for g
+
−    using that \<open>finite F\<close> psubset_card_mono \<open>card F = n\<close>
+
−    by (metis finite_Int inf.strict_order_iff)
+
−  have 1: "\<And>F'. F' \<subset> F \<Longrightarrow> S \<subseteq> affine hull S \<inter> \<Inter>F'"
+
−    by (subst seq) blast
+
−  have 2: "\<And>F'. F' \<subset> F \<Longrightarrow> S \<noteq> affine hull S \<inter> \<Inter>F'"
+
−    apply (frule *)
+
−    by (metis aff subsetCE subset_iff_psubset_eq)
+
−  show ?rhs
+
−    by (metis \<open>finite F\<close> seq aff psubsetI 1 2)
+
−next
+
−  assume ?rhs then show ?lhs
+
−    by (auto simp: polyhedron_Int_affine_parallel)
+
−qed
+
−
+
−
+
−lemma polyhedron_Int_affine_minimal:
+
−  fixes S :: "'a :: euclidean_space set"
+
−  shows "polyhedron S \<longleftrightarrow>
+
−         (\<exists>F. finite F \<and> S = (affine hull S) \<inter> \<Inter>F \<and>
+
−              (\<forall>h \<in> F. \<exists>a b. a \<noteq> 0 \<and> h = {x. a \<bullet> x \<le> b}) \<and>
+
−              (\<forall>F'. F' \<subset> F \<longrightarrow> S \<subset> (affine hull S) \<inter> \<Inter>F'))"
+
−apply (rule iffI)
+
− apply (force simp: polyhedron_Int_affine_parallel_minimal elim!: ex_forward)
+
−apply (auto simp: polyhedron_Int_affine elim!: ex_forward)
+
−done
+
−
+
−proposition facet_of_polyhedron_explicit:
+
−  assumes "finite F"
+
−      and seq: "S = affine hull S \<inter> \<Inter>F"
+
−      and faceq: "\<And>h. h \<in> F \<Longrightarrow> a h \<noteq> 0 \<and> h = {x. a h \<bullet> x \<le> b h}"
+
−      and psub: "\<And>F'. F' \<subset> F \<Longrightarrow> S \<subset> affine hull S \<inter> \<Inter>F'"
+
−    shows "c facet_of S \<longleftrightarrow> (\<exists>h. h \<in> F \<and> c = S \<inter> {x. a h \<bullet> x = b h})"
+
−proof (cases "S = {}")
+
−  case True with psub show ?thesis by force
+
−next
+
−  case False
+
−  have "polyhedron S"
+
−    apply (simp add: polyhedron_Int_affine)
+
−    apply (rule_tac x=F in exI)
+
−    using assms  apply force
+
−    done
+
−  then have "convex S"
+
−    by (rule polyhedron_imp_convex)
+
−  with False rel_interior_eq_empty have "rel_interior S \<noteq> {}" by blast
+
−  then obtain x where "x \<in> rel_interior S" by auto
+
−  then obtain T where "open T" "x \<in> T" "x \<in> S" "T \<inter> affine hull S \<subseteq> S"
+
−    by (force simp: mem_rel_interior)
+
−  then have xaff: "x \<in> affine hull S" and xint: "x \<in> \<Inter>F"
+
−    using seq hull_inc by auto
+
−  have "rel_interior S = {x \<in> S. \<forall>h\<in>F. a h \<bullet> x < b h}"
+
−    by (rule rel_interior_polyhedron_explicit [OF \<open>finite F\<close> seq faceq psub])
+
−  with \<open>x \<in> rel_interior S\<close>
+
−  have [simp]: "\<And>h. h\<in>F \<Longrightarrow> a h \<bullet> x < b h" by blast
+
−  have *: "(S \<inter> {x. a h \<bullet> x = b h}) facet_of S" if "h \<in> F" for h
+
−  proof -
+
−    have "S \<subset> affine hull S \<inter> \<Inter>(F - {h})"
+
−      using psub that by (metis Diff_disjoint Diff_subset insert_disjoint(2) psubsetI)
+
−    then obtain z where zaff: "z \<in> affine hull S" and zint: "z \<in> \<Inter>(F - {h})" and "z \<notin> S"
+
−      by force
+
−    then have "z \<noteq> x" "z \<notin> h" using seq \<open>x \<in> S\<close> by auto
+
−    have "x \<in> h" using that xint by auto
+
−    then have able: "a h \<bullet> x \<le> b h"
+
−      using faceq that by blast
+
−    also have "... < a h \<bullet> z" using \<open>z \<notin> h\<close> faceq [OF that] xint by auto
+
−    finally have xltz: "a h \<bullet> x < a h \<bullet> z" .
+
−    define l where "l = (b h - a h \<bullet> x) / (a h \<bullet> z - a h \<bullet> x)"
+
−    define w where "w = (1 - l) *\<^sub>R x + l *\<^sub>R z"
+
−    have "0 < l" "l < 1"
+
−      using able xltz \<open>b h < a h \<bullet> z\<close> \<open>h \<in> F\<close>
+
−      by (auto simp: l_def divide_simps)
+
−    have awlt: "a i \<bullet> w < b i" if "i \<in> F" "i \<noteq> h" for i
+
−    proof -
+
−      have "(1 - l) * (a i \<bullet> x) < (1 - l) * b i"
+
−        by (simp add: \<open>l < 1\<close> \<open>i \<in> F\<close>)
+
−      moreover have "l * (a i \<bullet> z) \<le> l * b i"
+
−        apply (rule mult_left_mono)
+
−        apply (metis Diff_insert_absorb Inter_iff Set.set_insert \<open>h \<in> F\<close> faceq insertE mem_Collect_eq that zint)
+
−        using \<open>0 < l\<close>
+
−        apply simp
+
−        done
+
−      ultimately show ?thesis by (simp add: w_def algebra_simps)
+
−    qed
+
−    have weq: "a h \<bullet> w = b h"
+
−      using xltz unfolding w_def l_def
+
−      by (simp add: algebra_simps) (simp add: field_simps)
+
−    have "w \<in> affine hull S"
+
−      by (simp add: w_def mem_affine xaff zaff)
+
−    moreover have "w \<in> \<Inter>F"
+
−      using \<open>a h \<bullet> w = b h\<close> awlt faceq less_eq_real_def by blast
+
−    ultimately have "w \<in> S"
+
−      using seq by blast
+
−    with weq have "S \<inter> {x. a h \<bullet> x = b h} \<noteq> {}" by blast
+
−    moreover have "S \<inter> {x. a h \<bullet> x = b h} face_of S"
+
−      apply (rule face_of_Int_supporting_hyperplane_le)
+
−      apply (rule \<open>convex S\<close>)
+
−      apply (subst (asm) seq)
+
−      using faceq that apply fastforce
+
−      done
+
−    moreover have "affine hull (S \<inter> {x. a h \<bullet> x = b h}) =
+
−                   (affine hull S) \<inter> {x. a h \<bullet> x = b h}"
+
−    proof
+
−      show "affine hull (S \<inter> {x. a h \<bullet> x = b h}) \<subseteq> affine hull S \<inter> {x. a h \<bullet> x = b h}"
+
−        apply (intro Int_greatest hull_mono Int_lower1)
+
−        apply (metis affine_hull_eq affine_hyperplane hull_mono inf_le2)
+
−        done
+
−    next
+
−      show "affine hull S \<inter> {x. a h \<bullet> x = b h} \<subseteq> affine hull (S \<inter> {x. a h \<bullet> x = b h})"
+
−      proof
+
−        fix y
+
−        assume yaff: "y \<in> affine hull S \<inter> {y. a h \<bullet> y = b h}"
+
−        obtain T where "0 < T"
+
−                 and T: "\<And>j. \<lbrakk>j \<in> F; j \<noteq> h\<rbrakk> \<Longrightarrow> T * (a j \<bullet> y - a j \<bullet> w) \<le> b j - a j \<bullet> w"
+
−        proof (cases "F - {h} = {}")
+
−          case True then show ?thesis
+
−            by (rule_tac T=1 in that) auto
+
−        next
+
−          case False
+
−          then obtain h' where h': "h' \<in> F - {h}" by auto
+
−          let ?body = "(\<lambda>j. if 0 < a j \<bullet> y - a j \<bullet> w
+
−              then (b j - a j \<bullet> w) / (a j \<bullet> y - a j \<bullet> w)
+
−              else 1) ` (F - {h})"
+
−          define inff where "inff = Inf ?body"
+
−          from \<open>finite F\<close> have "finite ?body"
+
−            by blast
+
−          moreover from h' have "?body \<noteq> {}"
+
−            by blast
+
−          moreover have "j > 0" if "j \<in> ?body" for j
+
−          proof -
+
−            from that obtain x where "x \<in> F" and "x \<noteq> h" and *: "j =
+
−              (if 0 < a x \<bullet> y - a x \<bullet> w
+
−                then (b x - a x \<bullet> w) / (a x \<bullet> y - a x \<bullet> w) else 1)"
+
−              by blast
+
−            with awlt [of x] have "a x \<bullet> w < b x"
+
−              by simp
+
−            with * show ?thesis
+
−              by simp
+
−          qed
+
−          ultimately have "0 < inff"
+
−            by (simp_all add: finite_less_Inf_iff inff_def)
+
−          moreover have "inff * (a j \<bullet> y - a j \<bullet> w) \<le> b j - a j \<bullet> w"
+
−                        if "j \<in> F" "j \<noteq> h" for j
+
−          proof (cases "a j \<bullet> w < a j \<bullet> y")
+
−            case True
+
−            then have "inff \<le> (b j - a j \<bullet> w) / (a j \<bullet> y - a j \<bullet> w)"
+
−              apply (simp add: inff_def)
+
−              apply (rule cInf_le_finite)
+
−              using \<open>finite F\<close> apply blast
+
−              apply (simp add: that split: if_split_asm)
+
−              done
+
−            then show ?thesis
+
−              using \<open>0 < inff\<close> awlt [OF that] mult_strict_left_mono
+
−              by (fastforce simp add: algebra_simps divide_simps split: if_split_asm)
+
−          next
+
−            case False
+
−            with \<open>0 < inff\<close> have "inff * (a j \<bullet> y - a j \<bullet> w) \<le> 0"
+
−              by (simp add: mult_le_0_iff)
+
−            also have "... < b j - a j \<bullet> w"
+
−              by (simp add: awlt that)
+
−            finally show ?thesis by simp
+
−          qed
+
−          ultimately show ?thesis
+
−            by (blast intro: that)
+
−        qed
+
−        define c where "c = (1 - T) *\<^sub>R w + T *\<^sub>R y"
+
−        have "(1 - T) *\<^sub>R w + T *\<^sub>R y \<in> j" if "j \<in> F" for j
+
−        proof (cases "j = h")
+
−          case True
+
−          have "(1 - T) *\<^sub>R w + T *\<^sub>R y \<in> {x. a h \<bullet> x \<le> b h}"
+
−            using weq yaff by (auto simp: algebra_simps)
+
−          with True faceq [OF that] show ?thesis by metis
+
−        next
+
−          case False
+
−          with T that have "(1 - T) *\<^sub>R w + T *\<^sub>R y \<in> {x. a j \<bullet> x \<le> b j}"
+
−            by (simp add: algebra_simps)
+
−          with faceq [OF that] show ?thesis by simp
+
−        qed
+
−        moreover have "(1 - T) *\<^sub>R w + T *\<^sub>R y \<in> affine hull S"
+
−          apply (rule affine_affine_hull [simplified affine_alt, rule_format])
+
−          apply (simp add: \<open>w \<in> affine hull S\<close>)
+
−          using yaff apply blast
+
−          done
+
−        ultimately have "c \<in> S"
+
−          using seq by (force simp: c_def)
+
−        moreover have "a h \<bullet> c = b h"
+
−          using yaff by (force simp: c_def algebra_simps weq)
+
−        ultimately have caff: "c \<in> affine hull (S \<inter> {y. a h \<bullet> y = b h})"
+
−          by (simp add: hull_inc)
+
−        have waff: "w \<in> affine hull (S \<inter> {y. a h \<bullet> y = b h})"
+
−          using \<open>w \<in> S\<close> weq by (blast intro: hull_inc)
+
−        have yeq: "y = (1 - inverse T) *\<^sub>R w + c /\<^sub>R T"
+
−          using \<open>0 < T\<close> by (simp add: c_def algebra_simps)
+
−        show "y \<in> affine hull (S \<inter> {y. a h \<bullet> y = b h})"
+
−          by (metis yeq affine_affine_hull [simplified affine_alt, rule_format, OF waff caff])
+
−      qed
+
−    qed
+
−    ultimately show ?thesis
+
−      apply (simp add: facet_of_def)
+
−      apply (subst aff_dim_affine_hull [symmetric])
+
−      using  \<open>b h < a h \<bullet> z\<close> zaff
+
−      apply (force simp: aff_dim_affine_Int_hyperplane)
+
−      done
+
−  qed
+
−  show ?thesis
+
−  proof
+
−    show "\<exists>h. h \<in> F \<and> c = S \<inter> {x. a h \<bullet> x = b h} \<Longrightarrow> c facet_of S"
+
−      using * by blast
+
−  next
+
−    assume "c facet_of S"
+
−    then have "c face_of S" "convex c" "c \<noteq> {}" and affc: "aff_dim c = aff_dim S - 1"
+
−      by (auto simp: facet_of_def face_of_imp_convex)
+
−    then obtain x where x: "x \<in> rel_interior c"
+
−      by (force simp: rel_interior_eq_empty)
+
−    then have "x \<in> c"
+
−      by (meson subsetD rel_interior_subset)
+
−    then have "x \<in> S"
+
−      using \<open>c facet_of S\<close> facet_of_imp_subset by blast
+
−    have rels: "rel_interior S = {x \<in> S. \<forall>h\<in>F. a h \<bullet> x < b h}"
+
−      by (rule rel_interior_polyhedron_explicit [OF assms])
+
−    have "c \<noteq> S"
+
−      using \<open>c facet_of S\<close> facet_of_irrefl by blast
+
−    then have "x \<notin> rel_interior S"
+
−      by (metis IntI empty_iff \<open>x \<in> c\<close> \<open>c \<noteq> S\<close> \<open>c face_of S\<close> face_of_disjoint_rel_interior)
+
−    with rels \<open>x \<in> S\<close> obtain i where "i \<in> F" and i: "a i \<bullet> x \<ge> b i"
+
−      by force
+
−    have "x \<in> {u. a i \<bullet> u \<le> b i}"
+
−      by (metis IntD2 InterE \<open>i \<in> F\<close> \<open>x \<in> S\<close> faceq seq)
+
−    then have "a i \<bullet> x \<le> b i" by simp
+
−    then have "a i \<bullet> x = b i" using i by auto
+
−    have "c \<subseteq> S \<inter> {x. a i \<bullet> x = b i}"
+
−      apply (rule subset_of_face_of [of _ S])
+
−        apply (simp add: "*" \<open>i \<in> F\<close> facet_of_imp_face_of)
+
−       apply (simp add: \<open>c face_of S\<close> face_of_imp_subset)
+
−      using \<open>a i \<bullet> x = b i\<close> \<open>x \<in> S\<close> x by blast
+
−    then have cface: "c face_of (S \<inter> {x. a i \<bullet> x = b i})"
+
−      by (meson \<open>c face_of S\<close> face_of_subset inf_le1)
+
−    have con: "convex (S \<inter> {x. a i \<bullet> x = b i})"
+
−      by (simp add: \<open>convex S\<close> convex_Int convex_hyperplane)
+
−    show "\<exists>h. h \<in> F \<and> c = S \<inter> {x. a h \<bullet> x = b h}"
+
−      apply (rule_tac x=i in exI)
+
−      apply (simp add: \<open>i \<in> F\<close>)
+
−      by (metis (no_types) * \<open>i \<in> F\<close> affc facet_of_def less_irrefl face_of_aff_dim_lt [OF con cface])
+
−  qed
+
−qed
+
−
+
−
+
−lemma face_of_polyhedron_subset_explicit:
+
−  fixes S :: "'a :: euclidean_space set"
+
−  assumes "finite F"
+
−      and seq: "S = affine hull S \<inter> \<Inter>F"
+
−      and faceq: "\<And>h. h \<in> F \<Longrightarrow> a h \<noteq> 0 \<and> h = {x. a h \<bullet> x \<le> b h}"
+
−      and psub: "\<And>F'. F' \<subset> F \<Longrightarrow> S \<subset> affine hull S \<inter> \<Inter>F'"
+
−      and c: "c face_of S" and "c \<noteq> {}" "c \<noteq> S"
+
−   obtains h where "h \<in> F" "c \<subseteq> S \<inter> {x. a h \<bullet> x = b h}"
+
−proof -
+
−  have "c \<subseteq> S" using \<open>c face_of S\<close>
+
−    by (simp add: face_of_imp_subset)
+
−  have "polyhedron S"
+
−    apply (simp add: polyhedron_Int_affine)
+
−    by (metis \<open>finite F\<close> faceq seq)
+
−  then have "convex S"
+
−    by (simp add: polyhedron_imp_convex)
+
−  then have *: "(S \<inter> {x. a h \<bullet> x = b h}) face_of S" if "h \<in> F" for h
+
−    apply (rule face_of_Int_supporting_hyperplane_le)
+
−    using faceq seq that by fastforce
+
−  have "rel_interior c \<noteq> {}"
+
−    using c \<open>c \<noteq> {}\<close> face_of_imp_convex rel_interior_eq_empty by blast
+
−  then obtain x where "x \<in> rel_interior c" by auto
+
−  have rels: "rel_interior S = {x \<in> S. \<forall>h\<in>F. a h \<bullet> x < b h}"
+
−    by (rule rel_interior_polyhedron_explicit [OF \<open>finite F\<close> seq faceq psub])
+
−  then have xnot: "x \<notin> rel_interior S"
+
−    by (metis IntI \<open>x \<in> rel_interior c\<close> c \<open>c \<noteq> S\<close> contra_subsetD empty_iff face_of_disjoint_rel_interior rel_interior_subset)
+
−  then have "x \<in> S"
+
−    using \<open>c \<subseteq> S\<close> \<open>x \<in> rel_interior c\<close> rel_interior_subset by auto
+
−  then have xint: "x \<in> \<Inter>F"
+
−    using seq by blast
+
−  have "F \<noteq> {}" using assms
+
−    by (metis affine_Int affine_Inter affine_affine_hull ex_in_conv face_of_affine_trivial)
+
−  then obtain i where "i \<in> F" "\<not> (a i \<bullet> x < b i)"
+
−    using \<open>x \<in> S\<close> rels xnot by auto
+
−  with xint have "a i \<bullet> x = b i"
+
−    by (metis eq_iff mem_Collect_eq not_le Inter_iff faceq)
+
−  have face: "S \<inter> {x. a i \<bullet> x = b i} face_of S"
+
−    by (simp add: "*" \<open>i \<in> F\<close>)
+
−  show ?thesis
+
−    apply (rule_tac h = i in that)
+
−     apply (rule \<open>i \<in> F\<close>)
+
−    apply (rule subset_of_face_of [OF face \<open>c \<subseteq> S\<close>])
+
−    using \<open>a i \<bullet> x = b i\<close> \<open>x \<in> rel_interior c\<close> \<open>x \<in> S\<close> apply blast
+
−    done
+
−qed
+
−
+
−text\<open>Initial part of proof duplicates that above\<close>
+
−proposition face_of_polyhedron_explicit:
+
−  fixes S :: "'a :: euclidean_space set"
+
−  assumes "finite F"
+
−      and seq: "S = affine hull S \<inter> \<Inter>F"
+
−      and faceq: "\<And>h. h \<in> F \<Longrightarrow> a h \<noteq> 0 \<and> h = {x. a h \<bullet> x \<le> b h}"
+
−      and psub: "\<And>F'. F' \<subset> F \<Longrightarrow> S \<subset> affine hull S \<inter> \<Inter>F'"
+
−      and c: "c face_of S" and "c \<noteq> {}" "c \<noteq> S"
+
−    shows "c = \<Inter>{S \<inter> {x. a h \<bullet> x = b h} | h. h \<in> F \<and> c \<subseteq> S \<inter> {x. a h \<bullet> x = b h}}"
+
−proof -
+
−  let ?ab = "\<lambda>h. {x. a h \<bullet> x = b h}"
+
−  have "c \<subseteq> S" using \<open>c face_of S\<close>
+
−    by (simp add: face_of_imp_subset)
+
−  have "polyhedron S"
+
−    apply (simp add: polyhedron_Int_affine)
+
−    by (metis \<open>finite F\<close> faceq seq)
+
−  then have "convex S"
+
−    by (simp add: polyhedron_imp_convex)
+
−  then have *: "(S \<inter> ?ab h) face_of S" if "h \<in> F" for h
+
−    apply (rule face_of_Int_supporting_hyperplane_le)
+
−    using faceq seq that by fastforce
+
−  have "rel_interior c \<noteq> {}"
+
−    using c \<open>c \<noteq> {}\<close> face_of_imp_convex rel_interior_eq_empty by blast
+
−  then obtain z where z: "z \<in> rel_interior c" by auto
+
−  have rels: "rel_interior S = {z \<in> S. \<forall>h\<in>F. a h \<bullet> z < b h}"
+
−    by (rule rel_interior_polyhedron_explicit [OF \<open>finite F\<close> seq faceq psub])
+
−  then have xnot: "z \<notin> rel_interior S"
+
−    by (metis IntI \<open>z \<in> rel_interior c\<close> c \<open>c \<noteq> S\<close> contra_subsetD empty_iff face_of_disjoint_rel_interior rel_interior_subset)
+
−  then have "z \<in> S"
+
−    using \<open>c \<subseteq> S\<close> \<open>z \<in> rel_interior c\<close> rel_interior_subset by auto
+
−  with seq have xint: "z \<in> \<Inter>F" by blast
+
−  have "open (\<Inter>h\<in>{h \<in> F. a h \<bullet> z < b h}. {w. a h \<bullet> w < b h})"
+
−    by (auto simp: \<open>finite F\<close> open_halfspace_lt open_INT)
+
−  then obtain e where "0 < e"
+
−                 "ball z e \<subseteq> (\<Inter>h\<in>{h \<in> F. a h \<bullet> z < b h}. {w. a h \<bullet> w < b h})"
+
−    by (auto intro: openE [of _ z])
+
−  then have e: "\<And>h. \<lbrakk>h \<in> F; a h \<bullet> z < b h\<rbrakk> \<Longrightarrow> ball z e \<subseteq> {w. a h \<bullet> w < b h}"
+
−    by blast
+
−  have "c \<subseteq> (S \<inter> ?ab h) \<longleftrightarrow> z \<in> S \<inter> ?ab h" if "h \<in> F" for h
+
−  proof
+
−    show "z \<in> S \<inter> ?ab h \<Longrightarrow> c \<subseteq> S \<inter> ?ab h"
+
−      apply (rule subset_of_face_of [of _ S])
+
−      using that \<open>c \<subseteq> S\<close> \<open>z \<in> rel_interior c\<close>
+
−      using facet_of_polyhedron_explicit [OF \<open>finite F\<close> seq faceq psub]
+
−            unfolding facet_of_def
+
−      apply auto
+
−      done
+
−  next
+
−    show "c \<subseteq> S \<inter> ?ab h \<Longrightarrow> z \<in> S \<inter> ?ab h"
+
−      using \<open>z \<in> rel_interior c\<close> rel_interior_subset by force
+
−  qed
+
−  then have **: "{S \<inter> ?ab h | h. h \<in> F \<and> c \<subseteq> S \<and> c \<subseteq> ?ab h} =
+
−                 {S \<inter> ?ab h |h. h \<in> F \<and> z \<in> S \<inter> ?ab h}"
+
−    by blast
+
−  have bsub: "ball z e \<inter> affine hull \<Inter>{S \<inter> ?ab h |h. h \<in> F \<and> a h \<bullet> z = b h}
+
−             \<subseteq> affine hull S \<inter> \<Inter>F \<inter> \<Inter>{?ab h |h. h \<in> F \<and> a h \<bullet> z = b h}"
+
−            if "i \<in> F" and i: "a i \<bullet> z = b i" for i
+
−  proof -
+
−    have sub: "ball z e \<inter> \<Inter>{?ab h |h. h \<in> F \<and> a h \<bullet> z = b h} \<subseteq> j"
+
−             if "j \<in> F" for j
+
−    proof -
+
−      have "a j \<bullet> z \<le> b j" using faceq that xint by auto
+
−      then consider "a j \<bullet> z < b j" | "a j \<bullet> z = b j" by linarith
+
−      then have "\<exists>G. G \<in> {?ab h |h. h \<in> F \<and> a h \<bullet> z = b h} \<and> ball z e \<inter> G \<subseteq> j"
+
−      proof cases
+
−        assume "a j \<bullet> z < b j"
+
−        then have "ball z e \<inter> {x. a i \<bullet> x = b i} \<subseteq> j"
+
−          using e [OF \<open>j \<in> F\<close>] faceq that
+
−          by (fastforce simp: ball_def)
+
−        then show ?thesis
+
−          by (rule_tac x="{x. a i \<bullet> x = b i}" in exI) (force simp: \<open>i \<in> F\<close> i)
+
−      next
+
−        assume eq: "a j \<bullet> z = b j"
+
−        with faceq that show ?thesis
+
−          by (rule_tac x="{x. a j \<bullet> x = b j}" in exI) (fastforce simp add: \<open>j \<in> F\<close>)
+
−      qed
+
−      then show ?thesis  by blast
+
−    qed
+
−    have 1: "affine hull \<Inter>{S \<inter> ?ab h |h. h \<in> F \<and> a h \<bullet> z = b h} \<subseteq> affine hull S"
+
−      apply (rule hull_mono)
+
−      using that \<open>z \<in> S\<close> by auto
+
−    have 2: "affine hull \<Inter>{S \<inter> ?ab h |h. h \<in> F \<and> a h \<bullet> z = b h}
+
−          \<subseteq> \<Inter>{?ab h |h. h \<in> F \<and> a h \<bullet> z = b h}"
+
−      by (rule hull_minimal) (auto intro: affine_hyperplane)
+
−    have 3: "ball z e \<inter> \<Inter>{?ab h |h. h \<in> F \<and> a h \<bullet> z = b h} \<subseteq> \<Inter>F"
+
−      by (iprover intro: sub Inter_greatest)
+
−    have *: "\<lbrakk>A \<subseteq> (B :: 'a set); A \<subseteq> C; E \<inter> C \<subseteq> D\<rbrakk> \<Longrightarrow> E \<inter> A \<subseteq> (B \<inter> D) \<inter> C"
+
−             for A B C D E  by blast
+
−    show ?thesis by (intro * 1 2 3)
+
−  qed
+
−  have "\<exists>h. h \<in> F \<and> c \<subseteq> ?ab h"
+
−    apply (rule face_of_polyhedron_subset_explicit [OF \<open>finite F\<close> seq faceq psub])
+
−    using assms by auto
+
−  then have fac: "\<Inter>{S \<inter> ?ab h |h. h \<in> F \<and> c \<subseteq> S \<inter> ?ab h} face_of S"
+
−    using * by (force simp: \<open>c \<subseteq> S\<close> intro: face_of_Inter)
+
−  have red:
+
−     "(\<And>a. P a \<Longrightarrow> T \<subseteq> S \<inter> \<Inter>{F x |x. P x}) \<Longrightarrow> T \<subseteq> \<Inter>{S \<inter> F x |x. P x}"
+
−     for P T F   by blast
+
−  have "ball z e \<inter> affine hull \<Inter>{S \<inter> ?ab h |h. h \<in> F \<and> a h \<bullet> z = b h}
+
−        \<subseteq> \<Inter>{S \<inter> ?ab h |h. h \<in> F \<and> a h \<bullet> z = b h}"
+
−    apply (rule red)
+
−    apply (metis seq bsub)
+
−    done
+
−  with \<open>0 < e\<close> have zinrel: "z \<in> rel_interior
+
−                    (\<Inter>{S \<inter> ?ab h |h. h \<in> F \<and> z \<in> S \<and> a h \<bullet> z = b h})"
+
−    by (auto simp: mem_rel_interior_ball \<open>z \<in> S\<close>)
+
−  show ?thesis
+
−    apply (rule face_of_eq [OF c fac])
+
−    using z zinrel apply (force simp: **)
+
−    done
+
−qed
+
−
+
−
+
−subsection\<open>More general corollaries from the explicit representation\<close>
+
−
+
−corollary facet_of_polyhedron:
+
−  assumes "polyhedron S" and "c facet_of S"
+
−  obtains a b where "a \<noteq> 0" "S \<subseteq> {x. a \<bullet> x \<le> b}" "c = S \<inter> {x. a \<bullet> x = b}"
+
−proof -
+
−  obtain F where "finite F" and seq: "S = affine hull S \<inter> \<Inter>F"
+
−             and faces: "\<And>h. h \<in> F \<Longrightarrow> \<exists>a b. a \<noteq> 0 \<and> h = {x. a \<bullet> x \<le> b}"
+
−             and min: "\<And>F'. F' \<subset> F \<Longrightarrow> S \<subset> (affine hull S) \<inter> \<Inter>F'"
+
−    using assms by (simp add: polyhedron_Int_affine_minimal) meson
+
−  then obtain a b where ab: "\<And>h. h \<in> F \<Longrightarrow> a h \<noteq> 0 \<and> h = {x. a h \<bullet> x \<le> b h}"
+
−    by metis
+
−  obtain i where "i \<in> F" and c: "c = S \<inter> {x. a i \<bullet> x = b i}"
+
−    using facet_of_polyhedron_explicit [OF \<open>finite F\<close> seq ab min] assms
+
−    by force
+
−  moreover have ssub: "S \<subseteq> {x. a i \<bullet> x \<le> b i}"
+
−     apply (subst seq)
+
−     using \<open>i \<in> F\<close> ab by auto
+
−  ultimately show ?thesis
+
−    by (rule_tac a = "a i" and b = "b i" in that) (simp_all add: ab)
+
−qed
+
−
+
−corollary face_of_polyhedron:
+
−  assumes "polyhedron S" and "c face_of S" and "c \<noteq> {}" and "c \<noteq> S"
+
−    shows "c = \<Inter>{F. F facet_of S \<and> c \<subseteq> F}"
+
−proof -
+
−  obtain F where "finite F" and seq: "S = affine hull S \<inter> \<Inter>F"
+
−             and faces: "\<And>h. h \<in> F \<Longrightarrow> \<exists>a b. a \<noteq> 0 \<and> h = {x. a \<bullet> x \<le> b}"
+
−             and min: "\<And>F'. F' \<subset> F \<Longrightarrow> S \<subset> (affine hull S) \<inter> \<Inter>F'"
+
−    using assms by (simp add: polyhedron_Int_affine_minimal) meson
+
−  then obtain a b where ab: "\<And>h. h \<in> F \<Longrightarrow> a h \<noteq> 0 \<and> h = {x. a h \<bullet> x \<le> b h}"
+
−    by metis
+
−  show ?thesis
+
−    apply (subst face_of_polyhedron_explicit [OF \<open>finite F\<close> seq ab min])
+
−    apply (auto simp: assms facet_of_polyhedron_explicit [OF \<open>finite F\<close> seq ab min] cong: Collect_cong)
+
−    done
+
−qed
+
−
+
−lemma face_of_polyhedron_subset_facet:
+
−  assumes "polyhedron S" and "c face_of S" and "c \<noteq> {}" and "c \<noteq> S"
+
−  obtains F where "F facet_of S" "c \<subseteq> F"
+
−using face_of_polyhedron assms
+
−by (metis (no_types, lifting) Inf_greatest antisym_conv face_of_imp_subset mem_Collect_eq)
+
−
+
−
+
−lemma exposed_face_of_polyhedron:
+
−  assumes "polyhedron S"
+
−    shows "F exposed_face_of S \<longleftrightarrow> F face_of S"
+
−proof
+
−  show "F exposed_face_of S \<Longrightarrow> F face_of S"
+
−    by (simp add: exposed_face_of_def)
+
−next
+
−  assume "F face_of S"
+
−  show "F exposed_face_of S"
+
−  proof (cases "F = {} \<or> F = S")
+
−    case True then show ?thesis
+
−      using \<open>F face_of S\<close> exposed_face_of by blast
+
−  next
+
−    case False
+
−    then have "{g. g facet_of S \<and> F \<subseteq> g} \<noteq> {}"
+
−      by (metis Collect_empty_eq_bot \<open>F face_of S\<close> assms empty_def face_of_polyhedron_subset_facet)
+
−    moreover have "\<And>T. \<lbrakk>T facet_of S; F \<subseteq> T\<rbrakk> \<Longrightarrow> T exposed_face_of S"
+
−      by (metis assms exposed_face_of facet_of_imp_face_of facet_of_polyhedron)
+
−    ultimately have "\<Inter>{fa.
+
−       fa facet_of S \<and> F \<subseteq> fa} exposed_face_of S"
+
−      by (metis (no_types, lifting) mem_Collect_eq exposed_face_of_Inter)
+
−    then show ?thesis
+
−      using False
+
−      apply (subst face_of_polyhedron [OF assms \<open>F face_of S\<close>], auto)
+
−      done
+
−  qed
+
−qed
+
−
+
−lemma face_of_polyhedron_polyhedron:
+
−  fixes S :: "'a :: euclidean_space set"
+
−  assumes "polyhedron S" "c face_of S" shows "polyhedron c"
+
−by (metis assms face_of_imp_eq_affine_Int polyhedron_Int polyhedron_affine_hull polyhedron_imp_convex)
+
−
+
−lemma finite_polyhedron_faces:
+
−  fixes S :: "'a :: euclidean_space set"
+
−  assumes "polyhedron S"
+
−    shows "finite {F. F face_of S}"
+
−proof -
+
−  obtain F where "finite F" and seq: "S = affine hull S \<inter> \<Inter>F"
+
−             and faces: "\<And>h. h \<in> F \<Longrightarrow> \<exists>a b. a \<noteq> 0 \<and> h = {x. a \<bullet> x \<le> b}"
+
−             and min:   "\<And>F'. F' \<subset> F \<Longrightarrow> S \<subset> (affine hull S) \<inter> \<Inter>F'"
+
−    using assms by (simp add: polyhedron_Int_affine_minimal) meson
+
−  then obtain a b where ab: "\<And>h. h \<in> F \<Longrightarrow> a h \<noteq> 0 \<and> h = {x. a h \<bullet> x \<le> b h}"
+
−    by metis
+
−  have "finite {\<Inter>{S \<inter> {x. a h \<bullet> x = b h} |h. h \<in> F'}| F'. F' \<in> Pow F}"
+
−    by (simp add: \<open>finite F\<close>)
+
−  moreover have "{F. F face_of S} - {{}, S} \<subseteq> {\<Inter>{S \<inter> {x. a h \<bullet> x = b h} |h. h \<in> F'}| F'. F' \<in> Pow F}"
+
−    apply clarify
+
−    apply (rename_tac c)
+
−    apply (drule face_of_polyhedron_explicit [OF \<open>finite F\<close> seq ab min, simplified], simp_all)
+
−    apply (erule ssubst)
+
−    apply (rule_tac x="{h \<in> F. c \<subseteq> S \<inter> {x. a h \<bullet> x = b h}}" in exI, auto)
+
−    done
+
−  ultimately show ?thesis
+
−    by (meson finite.emptyI finite.insertI finite_Diff2 finite_subset)
+
−qed
+
−
+
−lemma finite_polyhedron_exposed_faces:
+
−   "polyhedron S \<Longrightarrow> finite {F. F exposed_face_of S}"
+
−using exposed_face_of_polyhedron finite_polyhedron_faces by fastforce
+
−
+
−lemma finite_polyhedron_extreme_points:
+
−  fixes S :: "'a :: euclidean_space set"
+
−  shows "polyhedron S \<Longrightarrow> finite {v. v extreme_point_of S}"
+
−apply (simp add: face_of_singleton [symmetric])
+
−apply (rule finite_subset [OF _ finite_vimageI [OF finite_polyhedron_faces]], auto)
+
−done
+
−
+
−lemma finite_polyhedron_facets:
+
−  fixes S :: "'a :: euclidean_space set"
+
−  shows "polyhedron S \<Longrightarrow> finite {F. F facet_of S}"
+
−unfolding facet_of_def
+
−by (blast intro: finite_subset [OF _ finite_polyhedron_faces])
+
−
+
−
+
−proposition rel_interior_of_polyhedron:
+
−  fixes S :: "'a :: euclidean_space set"
+
−  assumes "polyhedron S"
+
−    shows "rel_interior S = S - \<Union>{F. F facet_of S}"
+
−proof -
+
−  obtain F where "finite F" and seq: "S = affine hull S \<inter> \<Inter>F"
+
−             and faces: "\<And>h. h \<in> F \<Longrightarrow> \<exists>a b. a \<noteq> 0 \<and> h = {x. a \<bullet> x \<le> b}"
+
−             and min: "\<And>F'. F' \<subset> F \<Longrightarrow> S \<subset> (affine hull S) \<inter> \<Inter>F'"
+
−    using assms by (simp add: polyhedron_Int_affine_minimal) meson
+
−  then obtain a b where ab: "\<And>h. h \<in> F \<Longrightarrow> a h \<noteq> 0 \<and> h = {x. a h \<bullet> x \<le> b h}"
+
−    by metis
+
−  have facet: "(c facet_of S) \<longleftrightarrow> (\<exists>h. h \<in> F \<and> c = S \<inter> {x. a h \<bullet> x = b h})" for c
+
−    by (rule facet_of_polyhedron_explicit [OF \<open>finite F\<close> seq ab min])
+
−  have rel: "rel_interior S = {x \<in> S. \<forall>h\<in>F. a h \<bullet> x < b h}"
+
−    by (rule rel_interior_polyhedron_explicit [OF \<open>finite F\<close> seq ab min])
+
−  have "a h \<bullet> x < b h" if "x \<in> S" "h \<in> F" and xnot: "x \<notin> \<Union>{F. F facet_of S}" for x h
+
−  proof -
+
−    have "x \<in> \<Inter>F" using seq that by force
+
−    with \<open>h \<in> F\<close> ab have "a h \<bullet> x \<le> b h" by auto
+
−    then consider "a h \<bullet> x < b h" | "a h \<bullet> x = b h" by linarith
+
−    then show ?thesis
+
−    proof cases
+
−      case 1 then show ?thesis .
+
−    next
+
−      case 2
+
−      have "Collect ((\<in>) x) \<notin> Collect ((\<in>) (\<Union>{A. A facet_of S}))"
+
−        using xnot by fastforce
+
−      then have "F \<notin> Collect ((\<in>) h)"
+
−        using 2 \<open>x \<in> S\<close> facet by blast
+
−      with \<open>h \<in> F\<close> have "\<Inter>F \<subseteq> S \<inter> {x. a h \<bullet> x = b h}" by blast
+
−      with 2 that \<open>x \<in> \<Inter>F\<close> show ?thesis
+
−        apply simp
+
−        apply (drule_tac x="\<Inter>F" in spec)
+
−        apply (simp add: facet)
+
−        apply (drule_tac x=h in spec)
+
−        using seq by auto
+
−      qed
+
−  qed
+
−  moreover have "\<exists>h\<in>F. a h \<bullet> x \<ge> b h" if "x \<in> \<Union>{F. F facet_of S}" for x
+
−    using that by (force simp: facet)
+
−  ultimately show ?thesis
+
−    by (force simp: rel)
+
−qed
+
−
+
−lemma rel_boundary_of_polyhedron:
+
−  fixes S :: "'a :: euclidean_space set"
+
−  assumes "polyhedron S"
+
−    shows "S - rel_interior S = \<Union> {F. F facet_of S}"
+
−using facet_of_imp_subset by (fastforce simp add: rel_interior_of_polyhedron assms)
+
−
+
−lemma rel_frontier_of_polyhedron:
+
−  fixes S :: "'a :: euclidean_space set"
+
−  assumes "polyhedron S"
+
−    shows "rel_frontier S = \<Union> {F. F facet_of S}"
+
−by (simp add: assms rel_frontier_def polyhedron_imp_closed rel_boundary_of_polyhedron)
+
−
+
−lemma rel_frontier_of_polyhedron_alt:
+
−  fixes S :: "'a :: euclidean_space set"
+
−  assumes "polyhedron S"
+
−    shows "rel_frontier S = \<Union> {F. F face_of S \<and> (F \<noteq> S)}"
+
−apply (rule subset_antisym)
+
−  apply (force simp: rel_frontier_of_polyhedron facet_of_def assms)
+
−using face_of_subset_rel_frontier by fastforce
+
−
+
−
+
−text\<open>A characterization of polyhedra as having finitely many faces\<close>
+
−
+
−proposition polyhedron_eq_finite_exposed_faces:
+
−  fixes S :: "'a :: euclidean_space set"
+
−  shows "polyhedron S \<longleftrightarrow> closed S \<and> convex S \<and> finite {F. F exposed_face_of S}"
+
−         (is "?lhs = ?rhs")
+
−proof
+
−  assume ?lhs
+
−  then show ?rhs
+
−    by (auto simp: polyhedron_imp_closed polyhedron_imp_convex finite_polyhedron_exposed_faces)
+
−next
+
−  assume ?rhs
+
−  then have "closed S" "convex S" and fin: "finite {F. F exposed_face_of S}" by auto
+
−  show ?lhs
+
−  proof (cases "S = {}")
+
−    case True then show ?thesis by auto
+
−  next
+
−    case False
+
−    define F where "F = {h. h exposed_face_of S \<and> h \<noteq> {} \<and> h \<noteq> S}"
+
−    have "finite F" by (simp add: fin F_def)
+
−    have hface: "h face_of S"
+
−      and "\<exists>a b. a \<noteq> 0 \<and> S \<subseteq> {x. a \<bullet> x \<le> b} \<and> h = S \<inter> {x. a \<bullet> x = b}"
+
−      if "h \<in> F" for h
+
−      using exposed_face_of F_def that by simp_all auto
+
−    then obtain a b where ab:
+
−      "\<And>h. h \<in> F \<Longrightarrow> a h \<noteq> 0 \<and> S \<subseteq> {x. a h \<bullet> x \<le> b h} \<and> h = S \<inter> {x. a h \<bullet> x = b h}"
+
−      by metis
+
−    have *: "False"
+
−      if paff: "p \<in> affine hull S" and "p \<notin> S"
+
−      and pint: "p \<in> \<Inter>{{x. a h \<bullet> x \<le> b h} |h. h \<in> F}" for p
+
−    proof -
+
−      have "rel_interior S \<noteq> {}"
+
−        by (simp add: \<open>S \<noteq> {}\<close> \<open>convex S\<close> rel_interior_eq_empty)
+
−      then obtain c where c: "c \<in> rel_interior S" by auto
+
−      with rel_interior_subset have "c \<in> S"  by blast
+
−      have ccp: "closed_segment c p \<subseteq> affine hull S"
+
−        by (meson affine_affine_hull affine_imp_convex c closed_segment_subset hull_subset paff rel_interior_subset subsetCE)
+
−      obtain x where xcl: "x \<in> closed_segment c p" and "x \<in> S" and xnot: "x \<notin> rel_interior S"
+
−        using connected_openin [of "closed_segment c p"]
+
−        apply simp
+
−        apply (drule_tac x="closed_segment c p \<inter> rel_interior S" in spec)
+
−        apply (erule impE)
+
−         apply (force simp: openin_rel_interior openin_Int intro: openin_subtopology_Int_subset [OF _ ccp])
+
−        apply (drule_tac x="closed_segment c p \<inter> (- S)" in spec)
+
−        using rel_interior_subset \<open>closed S\<close> c \<open>p \<notin> S\<close> apply blast
+
−        done
+
−      then obtain \<mu> where "0 \<le> \<mu>" "\<mu> \<le> 1" and xeq: "x = (1 - \<mu>) *\<^sub>R c + \<mu> *\<^sub>R p"
+
−        by (auto simp: in_segment)
+
−      show False
+
−      proof (cases "\<mu>=0 \<or> \<mu>=1")
+
−        case True with xeq c xnot \<open>x \<in> S\<close> \<open>p \<notin> S\<close>
+
−        show False by auto
+
−      next
+
−        case False
+
−        then have xos: "x \<in> open_segment c p"
+
−          using \<open>x \<in> S\<close> c open_segment_def that(2) xcl xnot by auto
+
−        have xclo: "x \<in> closure S"
+
−          using \<open>x \<in> S\<close> closure_subset by blast
+
−        obtain d where "d \<noteq> 0"
+
−              and dle: "\<And>y. y \<in> closure S \<Longrightarrow> d \<bullet> x \<le> d \<bullet> y"
+
−              and dless: "\<And>y. y \<in> rel_interior S \<Longrightarrow> d \<bullet> x < d \<bullet> y"
+
−          by (metis supporting_hyperplane_relative_frontier [OF \<open>convex S\<close> xclo xnot])
+
−        have sex: "S \<inter> {y. d \<bullet> y = d \<bullet> x} exposed_face_of S"
+
−          by (simp add: \<open>closed S\<close> dle exposed_face_of_Int_supporting_hyperplane_ge [OF \<open>convex S\<close>])
+
−        have sne: "S \<inter> {y. d \<bullet> y = d \<bullet> x} \<noteq> {}"
+
−          using \<open>x \<in> S\<close> by blast
+
−        have sns: "S \<inter> {y. d \<bullet> y = d \<bullet> x} \<noteq> S"
+
−          by (metis (mono_tags) Int_Collect c subsetD dless not_le order_refl rel_interior_subset)
+
−        obtain h where "h \<in> F" "x \<in> h"
+
−          apply (rule_tac h="S \<inter> {y. d \<bullet> y = d \<bullet> x}" in that)
+
−          apply (simp_all add: F_def sex sne sns \<open>x \<in> S\<close>)
+
−          done
+
−        have abface: "{y. a h \<bullet> y = b h} face_of {y. a h \<bullet> y \<le> b h}"
+
−          using hyperplane_face_of_halfspace_le by blast
+
−        then have "c \<in> h"
+
−          using face_ofD [OF abface xos] \<open>c \<in> S\<close> \<open>h \<in> F\<close> ab pint \<open>x \<in> h\<close> by blast
+
−        with c have "h \<inter> rel_interior S \<noteq> {}" by blast
+
−        then show False
+
−          using \<open>h \<in> F\<close> F_def face_of_disjoint_rel_interior hface by auto
+
−      qed
+
−    qed
+
−    have "S \<subseteq> affine hull S \<inter> \<Inter>{{x. a h \<bullet> x \<le> b h} |h. h \<in> F}"
+
−      using ab by (auto simp: hull_subset)
+
−    moreover have "affine hull S \<inter> \<Inter>{{x. a h \<bullet> x \<le> b h} |h. h \<in> F} \<subseteq> S"
+
−      using * by blast
+
−    ultimately have "S = affine hull S \<inter> \<Inter> {{x. a h \<bullet> x \<le> b h} |h. h \<in> F}" ..
+
−    then show ?thesis
+
−      apply (rule ssubst)
+
−      apply (force intro: polyhedron_affine_hull polyhedron_halfspace_le simp: \<open>finite F\<close>)
+
−      done
+
−  qed
+
−qed
+
−
+
−corollary polyhedron_eq_finite_faces:
+
−  fixes S :: "'a :: euclidean_space set"
+
−  shows "polyhedron S \<longleftrightarrow> closed S \<and> convex S \<and> finite {F. F face_of S}"
+
−         (is "?lhs = ?rhs")
+
−proof
+
−  assume ?lhs
+
−  then show ?rhs
+
−    by (simp add: finite_polyhedron_faces polyhedron_imp_closed polyhedron_imp_convex)
+
−next
+
−  assume ?rhs
+
−  then show ?lhs
+
−    by (force simp: polyhedron_eq_finite_exposed_faces exposed_face_of intro: finite_subset)
+
−qed
+
−
+
−lemma polyhedron_linear_image_eq:
+
−  fixes h :: "'a :: euclidean_space \<Rightarrow> 'b :: euclidean_space"
+
−  assumes "linear h" "bij h"
+
−    shows "polyhedron (h ` S) \<longleftrightarrow> polyhedron S"
+
−proof -
+
−  have *: "{f. P f} = (image h) ` {f. P (h ` f)}" for P
+
−    apply safe
+
−    apply (rule_tac x="inv h ` x" in image_eqI)
+
−    apply (auto simp: \<open>bij h\<close> bij_is_surj image_f_inv_f)
+
−    done
+
−  have "inj h" using bij_is_inj assms by blast
+
−  then have injim: "inj_on ((`) h) A" for A
+
−    by (simp add: inj_on_def inj_image_eq_iff)
+
−  show ?thesis
+
−    using \<open>linear h\<close> \<open>inj h\<close>
+
−    apply (simp add: polyhedron_eq_finite_faces closed_injective_linear_image_eq)
+
−    apply (simp add: * face_of_linear_image [of h _ S, symmetric] finite_image_iff injim)
+
−    done
+
−qed
+
−
+
−lemma polyhedron_negations:
+
−  fixes S :: "'a :: euclidean_space set"
+
−  shows   "polyhedron S \<Longrightarrow> polyhedron(image uminus S)"
+
−  by (subst polyhedron_linear_image_eq)
+
−    (auto simp: bij_uminus intro!: linear_uminus)
+
−
+
−subsection\<open>Relation between polytopes and polyhedra\<close>
+
−
+
−proposition polytope_eq_bounded_polyhedron:
+
−  fixes S :: "'a :: euclidean_space set"
+
−  shows "polytope S \<longleftrightarrow> polyhedron S \<and> bounded S"
+
−         (is "?lhs = ?rhs")
+
−proof
+
−  assume ?lhs
+
−  then show ?rhs
+
−    by (simp add: finite_polytope_faces polyhedron_eq_finite_faces
+
−                  polytope_imp_closed polytope_imp_convex polytope_imp_bounded)
+
−next
+
−  assume ?rhs then show ?lhs
+
−    unfolding polytope_def
+
−    apply (rule_tac x="{v. v extreme_point_of S}" in exI)
+
−    apply (simp add: finite_polyhedron_extreme_points Krein_Milman_Minkowski compact_eq_bounded_closed polyhedron_imp_closed polyhedron_imp_convex)
+
−    done
+
−qed
+
−
+
−lemma polytope_Int:
+
−  fixes S :: "'a :: euclidean_space set"
+
−  shows "\<lbrakk>polytope S; polytope T\<rbrakk> \<Longrightarrow> polytope(S \<inter> T)"
+
−by (simp add: polytope_eq_bounded_polyhedron bounded_Int)
+
−
+
−
+
−lemma polytope_Int_polyhedron:
+
−  fixes S :: "'a :: euclidean_space set"
+
−  shows "\<lbrakk>polytope S; polyhedron T\<rbrakk> \<Longrightarrow> polytope(S \<inter> T)"
+
−  by (simp add: bounded_Int polytope_eq_bounded_polyhedron)
+
−
+
−lemma polyhedron_Int_polytope:
+
−  fixes S :: "'a :: euclidean_space set"
+
−  shows "\<lbrakk>polyhedron S; polytope T\<rbrakk> \<Longrightarrow> polytope(S \<inter> T)"
+
−  by (simp add: bounded_Int polytope_eq_bounded_polyhedron)
+
−
+
−lemma polytope_imp_polyhedron:
+
−  fixes S :: "'a :: euclidean_space set"
+
−  shows "polytope S \<Longrightarrow> polyhedron S"
+
−  by (simp add: polytope_eq_bounded_polyhedron)
+
−
+
−lemma polytope_facet_exists:
+
−  fixes p :: "'a :: euclidean_space set"
+
−  assumes "polytope p" "0 < aff_dim p"
+
−  obtains F where "F facet_of p"
+
−proof (cases "p = {}")
+
−  case True with assms show ?thesis by auto
+
−next
+
−  case False
+
−  then obtain v where "v extreme_point_of p"
+
−    using extreme_point_exists_convex
+
−    by (blast intro: \<open>polytope p\<close> polytope_imp_compact polytope_imp_convex)
+
−  then
+
−  show ?thesis
+
−    by (metis face_of_polyhedron_subset_facet polytope_imp_polyhedron aff_dim_sing
+
−       all_not_in_conv assms face_of_singleton less_irrefl singletonI that)
+
−qed
+
−
+
−lemma polyhedron_interval [iff]: "polyhedron(cbox a b)"
+
−by (metis polytope_imp_polyhedron polytope_interval)
+
−
+
−lemma polyhedron_convex_hull:
+
−  fixes S :: "'a :: euclidean_space set"
+
−  shows "finite S \<Longrightarrow> polyhedron(convex hull S)"
+
−by (simp add: polytope_convex_hull polytope_imp_polyhedron)
+
−
+
−
+
−subsection\<open>Relative and absolute frontier of a polytope\<close>
+
−
+
−lemma rel_boundary_of_convex_hull:
+
−    fixes S :: "'a::euclidean_space set"
+
−    assumes "\<not> affine_dependent S"
+
−      shows "(convex hull S) - rel_interior(convex hull S) = (\<Union>a\<in>S. convex hull (S - {a}))"
+
−proof -
+
−  have "finite S" by (metis assms aff_independent_finite)
+
−  then consider "card S = 0" | "card S = 1" | "2 \<le> card S" by arith
+
−  then show ?thesis
+
−  proof cases
+
−    case 1 then have "S = {}" by (simp add: \<open>finite S\<close>)
+
−    then show ?thesis by simp
+
−  next
+
−    case 2 show ?thesis
+
−      by (auto intro: card_1_singletonE [OF \<open>card S = 1\<close>])
+
−  next
+
−    case 3
+
−    with assms show ?thesis
+
−      by (auto simp: polyhedron_convex_hull rel_boundary_of_polyhedron facet_of_convex_hull_affine_independent_alt \<open>finite S\<close>)
+
−  qed
+
−qed
+
−
+
−proposition frontier_of_convex_hull:
+
−    fixes S :: "'a::euclidean_space set"
+
−    assumes "card S = Suc (DIM('a))"
+
−      shows "frontier(convex hull S) = \<Union> {convex hull (S - {a}) | a. a \<in> S}"
+
−proof (cases "affine_dependent S")
+
−  case True
+
−    have [iff]: "finite S"
+
−      using assms using card_infinite by force
+
−    then have ccs: "closed (convex hull S)"
+
−      by (simp add: compact_imp_closed finite_imp_compact_convex_hull)
+
−    { fix x T
+
−      assume "finite T" "T \<subseteq> S" "int (card T) \<le> aff_dim S + 1" "x \<in> convex hull T"
+
−      then have "S \<noteq> T"
+
−        using True \<open>finite S\<close> aff_dim_le_card affine_independent_iff_card by fastforce
+
−      then obtain a where "a \<in> S" "a \<notin> T"
+
−        using \<open>T \<subseteq> S\<close> by blast
+
−      then have "x \<in> (\<Union>a\<in>S. convex hull (S - {a}))"
+
−        using True affine_independent_iff_card [of S]
+
−        apply simp
+
−        apply (metis (no_types, hide_lams) Diff_eq_empty_iff Diff_insert0 \<open>a \<notin> T\<close> \<open>T \<subseteq> S\<close> \<open>x \<in> convex hull T\<close>  hull_mono insert_Diff_single   subsetCE)
+
−        done
+
−    } note * = this
+
−    have 1: "convex hull S \<subseteq> (\<Union> a\<in>S. convex hull (S - {a}))"
+
−      apply (subst caratheodory_aff_dim)
+
−      apply (blast intro: *)
+
−      done
+
−    have 2: "\<Union>((\<lambda>a. convex hull (S - {a})) ` S) \<subseteq> convex hull S"
+
−      by (rule Union_least) (metis (no_types, lifting)  Diff_subset hull_mono imageE)
+
−    show ?thesis using True
+
−      apply (simp add: segment_convex_hull frontier_def)
+
−      using interior_convex_hull_eq_empty [OF assms]
+
−      apply (simp add: closure_closed [OF ccs])
+
−      apply (rule subset_antisym)
+
−      using 1 apply blast
+
−      using 2 apply blast
+
−      done
+
−next
+
−  case False
+
−  then have "frontier (convex hull S) = (convex hull S) - rel_interior(convex hull S)"
+
−    apply (simp add: rel_boundary_of_convex_hull [symmetric] frontier_def)
+
−    by (metis aff_independent_finite assms closure_convex_hull finite_imp_compact_convex_hull hull_hull interior_convex_hull_eq_empty rel_interior_nonempty_interior)
+
−  also have "... = \<Union>{convex hull (S - {a}) |a. a \<in> S}"
+
−  proof -
+
−    have "convex hull S - rel_interior (convex hull S) = rel_frontier (convex hull S)"
+
−      by (simp add: False aff_independent_finite polyhedron_convex_hull rel_boundary_of_polyhedron rel_frontier_of_polyhedron)
+
−    then show ?thesis
+
−      by (simp add: False rel_frontier_convex_hull_cases)
+
−  qed
+
−  finally show ?thesis .
+
−qed
+
−
+
−subsection\<open>Special case of a triangle\<close>
+
−
+
−proposition frontier_of_triangle:
+
−    fixes a :: "'a::euclidean_space"
+
−    assumes "DIM('a) = 2"
+
−    shows "frontier(convex hull {a,b,c}) = closed_segment a b \<union> closed_segment b c \<union> closed_segment c a"
+
−          (is "?lhs = ?rhs")
+
−proof (cases "b = a \<or> c = a \<or> c = b")
+
−  case True then show ?thesis
+
−    by (auto simp: assms segment_convex_hull frontier_def empty_interior_convex_hull insert_commute card_insert_le_m1 hull_inc insert_absorb)
+
−next
+
−  case False then have [simp]: "card {a, b, c} = Suc (DIM('a))"
+
−    by (simp add: card_insert Set.insert_Diff_if assms)
+
−  show ?thesis
+
−  proof
+
−    show "?lhs \<subseteq> ?rhs"
+
−      using False
+
−      by (force simp: segment_convex_hull frontier_of_convex_hull insert_Diff_if insert_commute split: if_split_asm)
+
−    show "?rhs \<subseteq> ?lhs"
+
−      using False
+
−      apply (simp add: frontier_of_convex_hull segment_convex_hull)
+
−      apply (intro conjI subsetI)
+
−        apply (rule_tac X="convex hull {a,b}" in UnionI; force simp: Set.insert_Diff_if)
+
−       apply (rule_tac X="convex hull {b,c}" in UnionI; force)
+
−      apply (rule_tac X="convex hull {a,c}" in UnionI; force simp: insert_commute Set.insert_Diff_if)
+
−      done
+
−  qed
+
−qed
+
−
+
−corollary inside_of_triangle:
+
−    fixes a :: "'a::euclidean_space"
+
−    assumes "DIM('a) = 2"
+
−    shows "inside (closed_segment a b \<union> closed_segment b c \<union> closed_segment c a) = interior(convex hull {a,b,c})"
+
−by (metis assms frontier_of_triangle bounded_empty bounded_insert convex_convex_hull inside_frontier_eq_interior bounded_convex_hull)
+
−
+
−corollary interior_of_triangle:
+
−    fixes a :: "'a::euclidean_space"
+
−    assumes "DIM('a) = 2"
+
−    shows "interior(convex hull {a,b,c}) =
+
−           convex hull {a,b,c} - (closed_segment a b \<union> closed_segment b c \<union> closed_segment c a)"
+
−  using interior_subset
+
−  by (force simp: frontier_of_triangle [OF assms, symmetric] frontier_def Diff_Diff_Int)
+
−
+
−subsection\<open>Subdividing a cell complex\<close>
+
−
+
−lemma subdivide_interval:
+
−  fixes x::real
+
−  assumes "a < \<bar>x - y\<bar>" "0 < a"
+
−  obtains n where "n \<in> \<int>" "x < n * a \<and> n * a < y \<or> y <  n * a \<and> n * a < x"
+
−proof -
+
−  consider "a + x < y" | "a + y < x"
+
−    using assms by linarith
+
−  then show ?thesis
+
−  proof cases
+
−    case 1
+
−    let ?n = "of_int (floor (x/a)) + 1"
+
−    have x: "x < ?n * a"
+
−      by (meson \<open>0 < a\<close> divide_less_eq floor_eq_iff)
+
−    have "?n * a \<le> a + x"
+
−      apply (simp add: algebra_simps)
+
−      by (metis \<open>0 < a\<close> floor_correct less_irrefl nonzero_mult_div_cancel_left real_mult_le_cancel_iff2 times_divide_eq_right)
+
−    also have "... < y"
+
−      by (rule 1)
+
−    finally have "?n * a < y" .
+
−    with x show ?thesis
+
−      using Ints_1 Ints_add Ints_of_int that by blast
+
−  next
+
−    case 2
+
−    let ?n = "of_int (floor (y/a)) + 1"
+
−    have y: "y < ?n * a"
+
−      by (meson \<open>0 < a\<close> divide_less_eq floor_eq_iff)
+
−    have "?n * a \<le> a + y"
+
−      apply (simp add: algebra_simps)
+
−      by (metis \<open>0 < a\<close> floor_correct less_irrefl nonzero_mult_div_cancel_left real_mult_le_cancel_iff2 times_divide_eq_right)
+
−    also have "... < x"
+
−      by (rule 2)
+
−    finally have "?n * a < x" .
+
−    then show ?thesis
+
−      using Ints_1 Ints_add Ints_of_int that y by blast
+
−  qed
+
−qed
+
−
+
−lemma cell_subdivision_lemma:
+
−  assumes "finite \<F>"
+
−      and "\<And>X. X \<in> \<F> \<Longrightarrow> polytope X"
+
−      and "\<And>X. X \<in> \<F> \<Longrightarrow> aff_dim X \<le> d"
+
−      and "\<And>X Y. \<lbrakk>X \<in> \<F>; Y \<in> \<F>\<rbrakk> \<Longrightarrow> (X \<inter> Y) face_of X \<and> (X \<inter> Y) face_of Y"
+
−      and "finite I"
+
−    shows "\<exists>\<G>. \<Union>\<G> = \<Union>\<F> \<and>
+
−                 finite \<G> \<and>
+
−                 (\<forall>C \<in> \<G>. \<exists>D. D \<in> \<F> \<and> C \<subseteq> D) \<and>
+
−                 (\<forall>C \<in> \<F>. \<forall>x \<in> C. \<exists>D. D \<in> \<G> \<and> x \<in> D \<and> D \<subseteq> C) \<and>
+
−                 (\<forall>X \<in> \<G>. polytope X) \<and>
+
−                 (\<forall>X \<in> \<G>. aff_dim X \<le> d) \<and>
+
−                 (\<forall>X \<in> \<G>. \<forall>Y \<in> \<G>. X \<inter> Y face_of X \<and> X \<inter> Y face_of Y) \<and>
+
−                 (\<forall>X \<in> \<G>. \<forall>x \<in> X. \<forall>y \<in> X. \<forall>a b.
+
−                          (a,b) \<in> I \<longrightarrow> a \<bullet> x \<le> b \<and> a \<bullet> y \<le> b \<or>
+
−                                        a \<bullet> x \<ge> b \<and> a \<bullet> y \<ge> b)"
+
−  using \<open>finite I\<close>
+
−proof induction
+
−  case empty
+
−  then show ?case
+
−    by (rule_tac x="\<F>" in exI) (auto simp: assms)
+
−next
+
−  case (insert ab I)
+
−  then obtain \<G> where eq: "\<Union>\<G> = \<Union>\<F>" and "finite \<G>"
+
−                   and sub1: "\<And>C. C \<in> \<G> \<Longrightarrow> \<exists>D. D \<in> \<F> \<and> C \<subseteq> D"
+
−                   and sub2: "\<And>C x. C \<in> \<F> \<and> x \<in> C \<Longrightarrow> \<exists>D. D \<in> \<G> \<and> x \<in> D \<and> D \<subseteq> C"
+
−                   and poly: "\<And>X. X \<in> \<G> \<Longrightarrow> polytope X"
+
−                   and aff: "\<And>X. X \<in> \<G> \<Longrightarrow> aff_dim X \<le> d"
+
−                   and face: "\<And>X Y. \<lbrakk>X \<in> \<G>; Y \<in> \<G>\<rbrakk> \<Longrightarrow> X \<inter> Y face_of X \<and> X \<inter> Y face_of Y"
+
−                   and I: "\<And>X x y a b.  \<lbrakk>X \<in> \<G>; x \<in> X; y \<in> X; (a,b) \<in> I\<rbrakk> \<Longrightarrow>
+
−                                    a \<bullet> x \<le> b \<and> a \<bullet> y \<le> b \<or> a \<bullet> x \<ge> b \<and> a \<bullet> y \<ge> b"
+
−    by (auto simp: that)
+
−  obtain a b where "ab = (a,b)"
+
−    by fastforce
+
−  let ?\<G> = "(\<lambda>X. X \<inter> {x. a \<bullet> x \<le> b}) ` \<G> \<union> (\<lambda>X. X \<inter> {x. a \<bullet> x \<ge> b}) ` \<G>"
+
−  have eqInt: "(S \<inter> Collect P) \<inter> (T \<inter> Collect Q) = (S \<inter> T) \<inter> (Collect P \<inter> Collect Q)" for S T::"'a set" and P Q
+
−    by blast
+
−  show ?case
+
−  proof (intro conjI exI)
+
−    show "\<Union>?\<G> = \<Union>\<F>"
+
−      by (force simp: eq [symmetric])
+
−    show "finite ?\<G>"
+
−      using \<open>finite \<G>\<close> by force
+
−    show "\<forall>X \<in> ?\<G>. polytope X"
+
−      by (force simp: poly polytope_Int_polyhedron polyhedron_halfspace_le polyhedron_halfspace_ge)
+
−    show "\<forall>X \<in> ?\<G>. aff_dim X \<le> d"
+
−      by (auto; metis order_trans aff aff_dim_subset inf_le1)
+
−    show "\<forall>X \<in> ?\<G>. \<forall>x \<in> X. \<forall>y \<in> X. \<forall>a b.
+
−                          (a,b) \<in> insert ab I \<longrightarrow> a \<bullet> x \<le> b \<and> a \<bullet> y \<le> b \<or>
+
−                                                  a \<bullet> x \<ge> b \<and> a \<bullet> y \<ge> b"
+
−      using \<open>ab = (a, b)\<close> I by fastforce
+
−    show "\<forall>X \<in> ?\<G>. \<forall>Y \<in> ?\<G>. X \<inter> Y face_of X \<and> X \<inter> Y face_of Y"
+
−      by (auto simp: eqInt halfspace_Int_eq face_of_Int_Int face face_of_halfspace_le face_of_halfspace_ge)
+
−    show "\<forall>C \<in> ?\<G>. \<exists>D. D \<in> \<F> \<and> C \<subseteq> D"
+
−      using sub1 by force
+
−    show "\<forall>C\<in>\<F>. \<forall>x\<in>C. \<exists>D. D \<in> ?\<G> \<and> x \<in> D \<and> D \<subseteq> C"
+
−    proof (intro ballI)
+
−      fix C z
+
−      assume "C \<in> \<F>" "z \<in> C"
+
−      with sub2 obtain D where D: "D \<in> \<G>" "z \<in> D" "D \<subseteq> C" by blast
+
−      have "D \<in> \<G> \<and> z \<in> D \<inter> {x. a \<bullet> x \<le> b} \<and> D \<inter> {x. a \<bullet> x \<le> b} \<subseteq> C \<or>
+
−            D \<in> \<G> \<and> z \<in> D \<inter> {x. a \<bullet> x \<ge> b} \<and> D \<inter> {x. a \<bullet> x \<ge> b} \<subseteq> C"
+
−        using linorder_class.linear [of "a \<bullet> z" b] D by blast
+
−      then show "\<exists>D. D \<in> ?\<G> \<and> z \<in> D \<and> D \<subseteq> C"
+
−        by blast
+
−    qed
+
−  qed
+
−qed
+
−
+
−
+
−proposition cell_complex_subdivision_exists:
+
−  fixes \<F> :: "'a::euclidean_space set set"
+
−  assumes "0 < e" "finite \<F>"
+
−      and poly: "\<And>X. X \<in> \<F> \<Longrightarrow> polytope X"
+
−      and aff: "\<And>X. X \<in> \<F> \<Longrightarrow> aff_dim X \<le> d"
+
−      and face: "\<And>X Y. \<lbrakk>X \<in> \<F>; Y \<in> \<F>\<rbrakk> \<Longrightarrow> X \<inter> Y face_of X \<and> X \<inter> Y face_of Y"
+
−  obtains "\<F>'" where "finite \<F>'" "\<Union>\<F>' = \<Union>\<F>" "\<And>X. X \<in> \<F>' \<Longrightarrow> diameter X < e"
+
−                "\<And>X. X \<in> \<F>' \<Longrightarrow> polytope X" "\<And>X. X \<in> \<F>' \<Longrightarrow> aff_dim X \<le> d"
+
−                "\<And>X Y. \<lbrakk>X \<in> \<F>'; Y \<in> \<F>'\<rbrakk> \<Longrightarrow> X \<inter> Y face_of X \<and> X \<inter> Y face_of Y"
+
−                "\<And>C. C \<in> \<F>' \<Longrightarrow> \<exists>D. D \<in> \<F> \<and> C \<subseteq> D"
+
−                "\<And>C x. C \<in> \<F> \<and> x \<in> C \<Longrightarrow> \<exists>D. D \<in> \<F>' \<and> x \<in> D \<and> D \<subseteq> C"
+
−proof -
+
−  have "bounded(\<Union>\<F>)"
+
−    by (simp add: \<open>finite \<F>\<close> poly bounded_Union polytope_imp_bounded)
+
−  then obtain B where "B > 0" and B: "\<And>x. x \<in> \<Union>\<F> \<Longrightarrow> norm x < B"
+
−    by (meson bounded_pos_less)
+
−  define C where "C \<equiv> {z \<in> \<int>. \<bar>z * e / 2 / real DIM('a)\<bar> \<le> B}"
+
−  define I where "I \<equiv> \<Union>i \<in> Basis. \<Union>j \<in> C. { (i::'a, j * e / 2 / DIM('a)) }"
+
−  have "finite C"
+
−    using finite_int_segment [of "-B / (e / 2 / DIM('a))" "B / (e / 2 / DIM('a))"]
+
−    apply (simp add: C_def)
+
−    apply (erule rev_finite_subset)
+
−    using \<open>0 < e\<close>
+
−    apply (auto simp: divide_simps)
+
−    done
+
−  then have "finite I"
+
−    by (simp add: I_def)
+
−  obtain \<F>' where eq: "\<Union>\<F>' = \<Union>\<F>" and "finite \<F>'"
+
−              and poly: "\<And>X. X \<in> \<F>' \<Longrightarrow> polytope X"
+
−              and aff: "\<And>X. X \<in> \<F>' \<Longrightarrow> aff_dim X \<le> d"
+
−              and face: "\<And>X Y. \<lbrakk>X \<in> \<F>'; Y \<in> \<F>'\<rbrakk> \<Longrightarrow> X \<inter> Y face_of X \<and> X \<inter> Y face_of Y"
+
−              and I: "\<And>X x y a b.  \<lbrakk>X \<in> \<F>'; x \<in> X; y \<in> X; (a,b) \<in> I\<rbrakk> \<Longrightarrow>
+
−                                     a \<bullet> x \<le> b \<and> a \<bullet> y \<le> b \<or> a \<bullet> x \<ge> b \<and> a \<bullet> y \<ge> b"
+
−              and sub1: "\<And>C. C \<in> \<F>' \<Longrightarrow> \<exists>D. D \<in> \<F> \<and> C \<subseteq> D"
+
−              and sub2: "\<And>C x. C \<in> \<F> \<and> x \<in> C \<Longrightarrow> \<exists>D. D \<in> \<F>' \<and> x \<in> D \<and> D \<subseteq> C"
+
−    apply (rule exE [OF cell_subdivision_lemma])
+
−    using assms \<open>finite I\<close> apply auto
+
−    done
+
−  show ?thesis
+
−  proof (rule_tac \<F>'="\<F>'" in that)
+
−    show "diameter X < e" if "X \<in> \<F>'" for X
+
−    proof -
+
−      have "diameter X \<le> e/2"
+
−      proof (rule diameter_le)
+
−        show "norm (x - y) \<le> e / 2" if "x \<in> X" "y \<in> X" for x y
+
−        proof -
+
−          have "norm x < B" "norm y < B"
+
−            using B \<open>X \<in> \<F>'\<close> eq that by fastforce+
+
−          have "norm (x - y) \<le> (\<Sum>b\<in>Basis. \<bar>(x-y) \<bullet> b\<bar>)"
+
−            by (rule norm_le_l1)
+
−          also have "... \<le> of_nat (DIM('a)) * (e / 2 / DIM('a))"
+
−          proof (rule sum_bounded_above)
+
−            fix i::'a
+
−            assume "i \<in> Basis"
+
−            then have I': "\<And>z b. \<lbrakk>z \<in> C; b = z * e / (2 * real DIM('a))\<rbrakk> \<Longrightarrow> i \<bullet> x \<le> b \<and> i \<bullet> y \<le> b \<or> i \<bullet> x \<ge> b \<and> i \<bullet> y \<ge> b"
+
−              using I \<open>X \<in> \<F>'\<close> that
+
−              by (fastforce simp: I_def)
+
−            show "\<bar>(x - y) \<bullet> i\<bar> \<le> e / 2 / real DIM('a)"
+
−            proof (rule ccontr)
+
−              assume "\<not> \<bar>(x - y) \<bullet> i\<bar> \<le> e / 2 / real DIM('a)"
+
−              then have xyi: "\<bar>i \<bullet> x - i \<bullet> y\<bar> > e / 2 / real DIM('a)"
+
−                by (simp add: inner_commute inner_diff_right)
+
−              obtain n where "n \<in> \<int>" and n: "i \<bullet> x < n * (e / 2 / real DIM('a)) \<and> n * (e / 2 / real DIM('a)) < i \<bullet> y \<or> i \<bullet> y < n * (e / 2 / real DIM('a)) \<and> n * (e / 2 / real DIM('a)) < i \<bullet> x"
+
−                using subdivide_interval [OF xyi] DIM_positive \<open>0 < e\<close>
+
−                by (auto simp: zero_less_divide_iff)
+
−              have "\<bar>i \<bullet> x\<bar> < B"
+
−                by (metis \<open>i \<in> Basis\<close> \<open>norm x < B\<close> inner_commute norm_bound_Basis_lt)
+
−              have "\<bar>i \<bullet> y\<bar> < B"
+
−                by (metis \<open>i \<in> Basis\<close> \<open>norm y < B\<close> inner_commute norm_bound_Basis_lt)
+
−              have *: "\<bar>n * e\<bar> \<le> B * (2 * real DIM('a))"
+
−                      if "\<bar>ix\<bar> < B" "\<bar>iy\<bar> < B"
+
−                         and ix: "ix * (2 * real DIM('a)) < n * e"
+
−                         and iy: "n * e < iy * (2 * real DIM('a))" for ix iy
+
−              proof (rule abs_leI)
+
−                have "iy * (2 * real DIM('a)) \<le> B * (2 * real DIM('a))"
+
−                  by (rule mult_right_mono) (use \<open>\<bar>iy\<bar> < B\<close> in linarith)+
+
−                then show "n * e \<le> B * (2 * real DIM('a))"
+
−                  using iy by linarith
+
−              next
+
−                have "- ix * (2 * real DIM('a)) \<le> B * (2 * real DIM('a))"
+
−                  by (rule mult_right_mono) (use \<open>\<bar>ix\<bar> < B\<close> in linarith)+
+
−                then show "- (n * e) \<le> B * (2 * real DIM('a))"
+
−                  using ix by linarith
+
−              qed
+
−              have "n \<in> C"
+
−                using \<open>n \<in> \<int>\<close> n  by (auto simp: C_def divide_simps intro: * \<open>\<bar>i \<bullet> x\<bar> < B\<close> \<open>\<bar>i \<bullet> y\<bar> < B\<close>)
+
−              show False
+
−                using  I' [OF \<open>n \<in> C\<close> refl] n  by auto
+
−            qed
+
−          qed
+
−          also have "... = e / 2"
+
−            by simp
+
−          finally show ?thesis .
+
−        qed
+
−      qed (use \<open>0 < e\<close> in force)
+
−      also have "... < e"
+
−        by (simp add: \<open>0 < e\<close>)
+
−      finally show ?thesis .
+
−    qed
+
−  qed (auto simp: eq poly aff face sub1 sub2 \<open>finite \<F>'\<close>)
+
−qed
+
−
+
−
+
−subsection\<open>Simplexes\<close>
+
−
+
−text\<open>The notion of n-simplex for integer \<^term>\<open>n \<ge> -1\<close>\<close>
+
−
+
−definition%important simplex :: "int \<Rightarrow> 'a::euclidean_space set \<Rightarrow> bool" (infix "simplex" 50)
+
−  where "n simplex S \<equiv> \<exists>C. \<not> affine_dependent C \<and> int(card C) = n + 1 \<and> S = convex hull C"
+
−
+
−lemma simplex:
+
−    "n simplex S \<longleftrightarrow> (\<exists>C. finite C \<and>
+
−                       \<not> affine_dependent C \<and>
+
−                       int(card C) = n + 1 \<and>
+
−                       S = convex hull C)"
+
−  by (auto simp add: simplex_def intro: aff_independent_finite)
+
−
+
−lemma simplex_convex_hull:
+
−   "\<not> affine_dependent C \<and> int(card C) = n + 1 \<Longrightarrow> n simplex (convex hull C)"
+
−  by (auto simp add: simplex_def)
+
−
+
−lemma convex_simplex: "n simplex S \<Longrightarrow> convex S"
+
−  by (metis convex_convex_hull simplex_def)
+
−
+
−lemma compact_simplex: "n simplex S \<Longrightarrow> compact S"
+
−  unfolding simplex
+
−  using finite_imp_compact_convex_hull by blast
+
−
+
−lemma closed_simplex: "n simplex S \<Longrightarrow> closed S"
+
−  by (simp add: compact_imp_closed compact_simplex)
+
−
+
−lemma simplex_imp_polytope:
+
−   "n simplex S \<Longrightarrow> polytope S"
+
−  unfolding simplex_def polytope_def
+
−  using aff_independent_finite by blast
+
−
+
−lemma simplex_imp_polyhedron:
+
−   "n simplex S \<Longrightarrow> polyhedron S"
+
−  by (simp add: polytope_imp_polyhedron simplex_imp_polytope)
+
−
+
−lemma simplex_dim_ge: "n simplex S \<Longrightarrow> -1 \<le> n"
+
−  by (metis (no_types, hide_lams) aff_dim_geq affine_independent_iff_card diff_add_cancel diff_diff_eq2 simplex_def)
+
−
+
−lemma simplex_empty [simp]: "n simplex {} \<longleftrightarrow> n = -1"
+
−proof
+
−  assume "n simplex {}"
+
−  then show "n = -1"
+
−    unfolding simplex by (metis card_empty convex_hull_eq_empty diff_0 diff_eq_eq of_nat_0)
+
−next
+
−  assume "n = -1" then show "n simplex {}"
+
−    by (fastforce simp: simplex)
+
−qed
+
−
+
−lemma simplex_minus_1 [simp]: "-1 simplex S \<longleftrightarrow> S = {}"
+
−  by (metis simplex cancel_comm_monoid_add_class.diff_cancel card_0_eq diff_minus_eq_add of_nat_eq_0_iff simplex_empty)
+
−
+
−
+
−lemma aff_dim_simplex:
+
−   "n simplex S \<Longrightarrow> aff_dim S = n"
+
−  by (metis simplex add.commute add_diff_cancel_left' aff_dim_convex_hull affine_independent_iff_card)
+
−
+
−lemma zero_simplex_sing: "0 simplex {a}"
+
−  apply (simp add: simplex_def)
+
−  by (metis affine_independent_1 card_empty card_insert_disjoint convex_hull_singleton empty_iff finite.emptyI)
+
−
+
−lemma simplex_sing [simp]: "n simplex {a} \<longleftrightarrow> n = 0"
+
−  using aff_dim_simplex aff_dim_sing zero_simplex_sing by blast
+
−
+
−lemma simplex_zero: "0 simplex S \<longleftrightarrow> (\<exists>a. S = {a})"
+
−apply (auto simp: )
+
−  using aff_dim_eq_0 aff_dim_simplex by blast
+
−
+
−lemma one_simplex_segment: "a \<noteq> b \<Longrightarrow> 1 simplex closed_segment a b"
+
−  apply (simp add: simplex_def)
+
−  apply (rule_tac x="{a,b}" in exI)
+
−  apply (auto simp: segment_convex_hull)
+
−  done
+
−
+
−lemma simplex_segment_cases:
+
−   "(if a = b then 0 else 1) simplex closed_segment a b"
+
−  by (auto simp: one_simplex_segment)
+
−
+
−lemma simplex_segment:
+
−   "\<exists>n. n simplex closed_segment a b"
+
−  using simplex_segment_cases by metis
+
−
+
−lemma polytope_lowdim_imp_simplex:
+
−  assumes "polytope P" "aff_dim P \<le> 1"
+
−  obtains n where "n simplex P"
+
−proof (cases "P = {}")
+
−  case True
+
−  then show ?thesis
+
−    by (simp add: that)
+
−next
+
−  case False
+
−  then show ?thesis
+
−    by (metis assms compact_convex_collinear_segment collinear_aff_dim polytope_imp_compact polytope_imp_convex simplex_segment_cases that)
+
−qed
+
−
+
−lemma simplex_insert_dimplus1:
+
−  fixes n::int
+
−  assumes "n simplex S" and a: "a \<notin> affine hull S"
+
−  shows "(n+1) simplex (convex hull (insert a S))"
+
−proof -
+
−  obtain C where C: "finite C" "\<not> affine_dependent C" "int(card C) = n+1" and S: "S = convex hull C"
+
−    using assms unfolding simplex by force
+
−  show ?thesis
+
−    unfolding simplex
+
−  proof (intro exI conjI)
+
−      have "aff_dim S = n"
+
−        using aff_dim_simplex assms(1) by blast
+
−      moreover have "a \<notin> affine hull C"
+
−        using S a affine_hull_convex_hull by blast
+
−      moreover have "a \<notin> C"
+
−          using S a hull_inc by fastforce
+
−      ultimately show "\<not> affine_dependent (insert a C)"
+
−        by (simp add: C S aff_dim_convex_hull aff_dim_insert affine_independent_iff_card)
+
−  next
+
−    have "a \<notin> C"
+
−      using S a hull_inc by fastforce
+
−    then show "int (card (insert a C)) = n + 1 + 1"
+
−      by (simp add: C)
+
−  next
+
−    show "convex hull insert a S = convex hull (insert a C)"
+
−      by (simp add: S convex_hull_insert_segments)
+
−  qed (use C in auto)
+
−qed
+
−
+
−subsection \<open>Simplicial complexes and triangulations\<close>
+
−
+
−definition%important simplicial_complex where
+
− "simplicial_complex \<C> \<equiv>
+
−        finite \<C> \<and>
+
−        (\<forall>S \<in> \<C>. \<exists>n. n simplex S) \<and>
+
−        (\<forall>F S. S \<in> \<C> \<and> F face_of S \<longrightarrow> F \<in> \<C>) \<and>
+
−        (\<forall>S S'. S \<in> \<C> \<and> S' \<in> \<C>
+
−                \<longrightarrow> (S \<inter> S') face_of S \<and> (S \<inter> S') face_of S')"
+
−
+
−definition%important triangulation where
+
− "triangulation \<T> \<equiv>
+
−        finite \<T> \<and>
+
−        (\<forall>T \<in> \<T>. \<exists>n. n simplex T) \<and>
+
−        (\<forall>T T'. T \<in> \<T> \<and> T' \<in> \<T>
+
−                \<longrightarrow> (T \<inter> T') face_of T \<and> (T \<inter> T') face_of T')"
+
−
+
−
+
−subsection\<open>Refining a cell complex to a simplicial complex\<close>
+
−
+
−proposition convex_hull_insert_Int_eq:
+
−  fixes z :: "'a :: euclidean_space"
+
−  assumes z: "z \<in> rel_interior S"
+
−      and T: "T \<subseteq> rel_frontier S"
+
−      and U: "U \<subseteq> rel_frontier S"
+
−      and "convex S" "convex T" "convex U"
+
−  shows "convex hull (insert z T) \<inter> convex hull (insert z U) = convex hull (insert z (T \<inter> U))"
+
−    (is "?lhs = ?rhs")
+
−proof
+
−  show "?lhs \<subseteq> ?rhs"
+
−  proof (cases "T={} \<or> U={}")
+
−    case True then show ?thesis by auto
+
−  next
+
−    case False
+
−    then have "T \<noteq> {}" "U \<noteq> {}" by auto
+
−    have TU: "convex (T \<inter> U)"
+
−      by (simp add: \<open>convex T\<close> \<open>convex U\<close> convex_Int)
+
−    have "(\<Union>x\<in>T. closed_segment z x) \<inter> (\<Union>x\<in>U. closed_segment z x)
+
−          \<subseteq> (if T \<inter> U = {} then {z} else \<Union>((closed_segment z) ` (T \<inter> U)))" (is "_ \<subseteq> ?IF")
+
−    proof clarify
+
−      fix x t u
+
−      assume xt: "x \<in> closed_segment z t"
+
−        and xu: "x \<in> closed_segment z u"
+
−        and "t \<in> T" "u \<in> U"
+
−      then have ne: "t \<noteq> z" "u \<noteq> z"
+
−        using T U z unfolding rel_frontier_def by blast+
+
−      show "x \<in> ?IF"
+
−      proof (cases "x = z")
+
−        case True then show ?thesis by auto
+
−      next
+
−        case False
+
−        have t: "t \<in> closure S"
+
−          using T \<open>t \<in> T\<close> rel_frontier_def by auto
+
−        have u: "u \<in> closure S"
+
−          using U \<open>u \<in> U\<close> rel_frontier_def by auto
+
−        show ?thesis
+
−        proof (cases "t = u")
+
−          case True
+
−          then show ?thesis
+
−            using \<open>t \<in> T\<close> \<open>u \<in> U\<close> xt by auto
+
−        next
+
−          case False
+
−          have tnot: "t \<notin> closed_segment u z"
+
−          proof -
+
−            have "t \<in> closure S - rel_interior S"
+
−              using T \<open>t \<in> T\<close> rel_frontier_def by blast
+
−            then have "t \<notin> open_segment z u"
+
−              by (meson DiffD2 rel_interior_closure_convex_segment [OF \<open>convex S\<close> z u] subsetD)
+
−            then show ?thesis
+
−              by (simp add: \<open>t \<noteq> u\<close> \<open>t \<noteq> z\<close> open_segment_commute open_segment_def)
+
−          qed
+
−          moreover have "u \<notin> closed_segment z t"
+
−            using rel_interior_closure_convex_segment [OF \<open>convex S\<close> z t] \<open>u \<in> U\<close> \<open>u \<noteq> z\<close>
+
−              U [unfolded rel_frontier_def] tnot
+
−            by (auto simp: closed_segment_eq_open)
+
−          ultimately
+
−          have "\<not>(between (t,u) z | between (u,z) t | between (z,t) u)" if "x \<noteq> z"
+
−            using that xt xu
+
−            apply (simp add: between_mem_segment [symmetric])
+
−            by (metis between_commute between_trans_2 between_antisym)
+
−          then have "\<not> collinear {t, z, u}" if "x \<noteq> z"
+
−            by (auto simp: that collinear_between_cases between_commute)
+
−          moreover have "collinear {t, z, x}"
+
−            by (metis closed_segment_commute collinear_2 collinear_closed_segment collinear_triples ends_in_segment(1) insert_absorb insert_absorb2 xt)
+
−          moreover have "collinear {z, x, u}"
+
−            by (metis closed_segment_commute collinear_2 collinear_closed_segment collinear_triples ends_in_segment(1) insert_absorb insert_absorb2 xu)
+
−          ultimately have False
+
−            using collinear_3_trans [of t z x u] \<open>x \<noteq> z\<close> by blast
+
−          then show ?thesis by metis
+
−        qed
+
−      qed
+
−    qed
+
−    then show ?thesis
+
−      using False \<open>convex T\<close> \<open>convex U\<close> TU
+
−      by (simp add: convex_hull_insert_segments hull_same split: if_split_asm)
+
−  qed
+
−  show "?rhs \<subseteq> ?lhs"
+
−    by (metis inf_greatest hull_mono inf.cobounded1 inf.cobounded2 insert_mono)
+
−qed
+
−
+
−lemma simplicial_subdivision_aux:
+
−  assumes "finite \<M>"
+
−      and "\<And>C. C \<in> \<M> \<Longrightarrow> polytope C"
+
−      and "\<And>C. C \<in> \<M> \<Longrightarrow> aff_dim C \<le> of_nat n"
+
−      and "\<And>C F. \<lbrakk>C \<in> \<M>; F face_of C\<rbrakk> \<Longrightarrow> F \<in> \<M>"
+
−      and "\<And>C1 C2. \<lbrakk>C1 \<in> \<M>; C2 \<in> \<M>\<rbrakk> \<Longrightarrow> C1 \<inter> C2 face_of C1 \<and> C1 \<inter> C2 face_of C2"
+
−    shows "\<exists>\<T>. simplicial_complex \<T> \<and>
+
−                (\<forall>K \<in> \<T>. aff_dim K \<le> of_nat n) \<and>
+
−                \<Union>\<T> = \<Union>\<M> \<and>
+
−                (\<forall>C \<in> \<M>. \<exists>F. finite F \<and> F \<subseteq> \<T> \<and> C = \<Union>F) \<and>
+
−                (\<forall>K \<in> \<T>. \<exists>C. C \<in> \<M> \<and> K \<subseteq> C)"
+
−  using assms
+
−proof (induction n arbitrary: \<M> rule: less_induct)
+
−  case (less n)
+
−  then have poly\<M>: "\<And>C. C \<in> \<M> \<Longrightarrow> polytope C"
+
−      and aff\<M>:    "\<And>C. C \<in> \<M> \<Longrightarrow> aff_dim C \<le> of_nat n"
+
−      and face\<M>:   "\<And>C F. \<lbrakk>C \<in> \<M>; F face_of C\<rbrakk> \<Longrightarrow> F \<in> \<M>"
+
−      and intface\<M>: "\<And>C1 C2. \<lbrakk>C1 \<in> \<M>; C2 \<in> \<M>\<rbrakk> \<Longrightarrow> C1 \<inter> C2 face_of C1 \<and> C1 \<inter> C2 face_of C2"
+
−    by metis+
+
−  show ?case
+
−  proof (cases "n \<le> 1")
+
−    case True
+
−    have "\<And>s. \<lbrakk>n \<le> 1; s \<in> \<M>\<rbrakk> \<Longrightarrow> \<exists>m. m simplex s"
+
−      using poly\<M> aff\<M> by (force intro: polytope_lowdim_imp_simplex)
+
−    then show ?thesis
+
−      unfolding simplicial_complex_def
+
−      apply (rule_tac x="\<M>" in exI)
+
−      using True by (auto simp: less.prems)
+
−  next
+
−    case False
+
−    define \<S> where "\<S> \<equiv> {C \<in> \<M>. aff_dim C < n}"
+
−    have "finite \<S>" "\<And>C. C \<in> \<S> \<Longrightarrow> polytope C" "\<And>C. C \<in> \<S> \<Longrightarrow> aff_dim C \<le> int (n - 1)"
+
−         "\<And>C F. \<lbrakk>C \<in> \<S>; F face_of C\<rbrakk> \<Longrightarrow> F \<in> \<S>"
+
−         "\<And>C1 C2. \<lbrakk>C1 \<in> \<S>; C2 \<in> \<S>\<rbrakk>  \<Longrightarrow> C1 \<inter> C2 face_of C1 \<and> C1 \<inter> C2 face_of C2"
+
−      using less.prems
+
−      apply (auto simp: \<S>_def)
+
−      by (metis aff_dim_subset face_of_imp_subset less_le not_le)
+
−    with less.IH [of "n-1" \<S>] False
+
−    obtain \<U> where "simplicial_complex \<U>"
+
−           and aff_dim\<U>: "\<And>K. K \<in> \<U> \<Longrightarrow> aff_dim K \<le> int (n - 1)"
+
−           and        "\<Union>\<U> = \<Union>\<S>"
+
−           and fin\<U>:  "\<And>C. C \<in> \<S> \<Longrightarrow> \<exists>F. finite F \<and> F \<subseteq> \<U> \<and> C = \<Union>F"
+
−           and C\<U>:    "\<And>K. K \<in> \<U> \<Longrightarrow> \<exists>C. C \<in> \<S> \<and> K \<subseteq> C"
+
−      by auto
+
−    then have "finite \<U>"
+
−         and simpl\<U>: "\<And>S. S \<in> \<U> \<Longrightarrow> \<exists>n. n simplex S"
+
−         and face\<U>:  "\<And>F S. \<lbrakk>S \<in> \<U>; F face_of S\<rbrakk> \<Longrightarrow> F \<in> \<U>"
+
−         and faceI\<U>: "\<And>S S'. \<lbrakk>S \<in> \<U>; S' \<in> \<U>\<rbrakk> \<Longrightarrow> (S \<inter> S') face_of S \<and> (S \<inter> S') face_of S'"
+
−      by (auto simp: simplicial_complex_def)
+
−    define \<N> where "\<N> \<equiv> {C \<in> \<M>. aff_dim C = n}"
+
−    have "finite \<N>"
+
−      by (simp add: \<N>_def less.prems(1))
+
−    have poly\<N>: "\<And>C. C \<in> \<N> \<Longrightarrow> polytope C"
+
−      and convex\<N>: "\<And>C. C \<in> \<N> \<Longrightarrow> convex C"
+
−      and closed\<N>: "\<And>C. C \<in> \<N> \<Longrightarrow> closed C"
+
−      by (auto simp: \<N>_def poly\<M> polytope_imp_convex polytope_imp_closed)
+
−    have in_rel_interior: "(SOME z. z \<in> rel_interior C) \<in> rel_interior C" if "C \<in> \<N>" for C
+
−        using that poly\<M> polytope_imp_convex rel_interior_aff_dim some_in_eq by (fastforce simp: \<N>_def)
+
−    have *: "\<exists>T. \<not> affine_dependent T \<and> card T \<le> n \<and> aff_dim K < n \<and> K = convex hull T"
+
−      if "K \<in> \<U>" for K
+
−    proof -
+
−      obtain r where r: "r simplex K"
+
−        using \<open>K \<in> \<U>\<close> simpl\<U> by blast
+
−      have "r = aff_dim K"
+
−        using \<open>r simplex K\<close> aff_dim_simplex by blast
+
−      with r
+
−      show ?thesis
+
−        unfolding simplex_def
+
−        using False \<open>\<And>K. K \<in> \<U> \<Longrightarrow> aff_dim K \<le> int (n - 1)\<close> that by fastforce
+
−    qed
+
−    have ahK_C_disjoint: "affine hull K \<inter> rel_interior C = {}"
+
−      if "C \<in> \<N>" "K \<in> \<U>" "K \<subseteq> rel_frontier C" for C K
+
−    proof -
+
−      have "convex C" "closed C"
+
−        by (auto simp: convex\<N> closed\<N> \<open>C \<in> \<N>\<close>)
+
−      obtain F where F: "F face_of C" and "F \<noteq> C" "K \<subseteq> F"
+
−      proof -
+
−        obtain L where "L \<in> \<S>" "K \<subseteq> L"
+
−          using \<open>K \<in> \<U>\<close> C\<U> by blast
+
−        have "K \<le> rel_frontier C"
+
−          by (simp add: \<open>K \<subseteq> rel_frontier C\<close>)
+
−        also have "... \<le> C"
+
−          by (simp add: \<open>closed C\<close> rel_frontier_def subset_iff)
+
−        finally have "K \<subseteq> C" .
+
−        have "L \<inter> C face_of C"
+
−          using \<N>_def \<S>_def \<open>C \<in> \<N>\<close> \<open>L \<in> \<S>\<close> intface\<M> by auto
+
−        moreover have "L \<inter> C \<noteq> C"
+
−          using \<open>C \<in> \<N>\<close> \<open>L \<in> \<S>\<close>
+
−          apply (clarsimp simp: \<N>_def \<S>_def)
+
−          by (metis aff_dim_subset inf_le1 not_le)
+
−        moreover have "K \<subseteq> L \<inter> C"
+
−          using \<open>C \<in> \<N>\<close> \<open>L \<in> \<S>\<close> \<open>K \<subseteq> C\<close> \<open>K \<subseteq> L\<close>
+
−          by (auto simp: \<N>_def \<S>_def)
+
−        ultimately show ?thesis using that by metis
+
−      qed
+
−      have "affine hull F \<inter> rel_interior C = {}"
+
−        by (rule affine_hull_face_of_disjoint_rel_interior [OF \<open>convex C\<close> F \<open>F \<noteq> C\<close>])
+
−      with hull_mono [OF \<open>K \<subseteq> F\<close>]
+
−      show "affine hull K \<inter> rel_interior C = {}"
+
−        by fastforce
+
−    qed
+
−    let ?\<T> = "(\<Union>C \<in> \<N>. \<Union>K \<in> \<U> \<inter> Pow (rel_frontier C).
+
−                     {convex hull (insert (SOME z. z \<in> rel_interior C) K)})"
+
−    have "\<exists>\<T>. simplicial_complex \<T> \<and>
+
−              (\<forall>K \<in> \<T>. aff_dim K \<le> of_nat n) \<and>
+
−              (\<forall>C \<in> \<M>. \<exists>F. F \<subseteq> \<T> \<and> C = \<Union>F) \<and>
+
−              (\<forall>K \<in> \<T>. \<exists>C. C \<in> \<M> \<and> K \<subseteq> C)"
+
−    proof (rule exI, intro conjI ballI)
+
−      show "simplicial_complex (\<U> \<union> ?\<T>)"
+
−        unfolding simplicial_complex_def
+
−      proof (intro conjI impI ballI allI)
+
−        show "finite (\<U> \<union> ?\<T>)"
+
−          using \<open>finite \<U>\<close> \<open>finite \<N>\<close> by simp
+
−        show "\<exists>n. n simplex S" if "S \<in> \<U> \<union> ?\<T>" for S
+
−          using that ahK_C_disjoint in_rel_interior simpl\<U> simplex_insert_dimplus1 by fastforce
+
−        show "F \<in> \<U> \<union> ?\<T>" if S: "S \<in> \<U> \<union> ?\<T> \<and> F face_of S" for F S
+
−        proof -
+
−          have "F \<in> \<U>" if "S \<in> \<U>"
+
−            using S face\<U> that by blast
+
−          moreover have "F \<in> \<U> \<union> ?\<T>"
+
−            if "F face_of S" "C \<in> \<N>" "K \<in> \<U>" and "K \<subseteq> rel_frontier C"
+
−              and S: "S = convex hull insert (SOME z. z \<in> rel_interior C) K" for C K
+
−          proof -
+
−            let ?z = "SOME z. z \<in> rel_interior C"
+
−            have "?z \<in> rel_interior C"
+
−              by (simp add: in_rel_interior \<open>C \<in> \<N>\<close>)
+
−            moreover
+
−            obtain I where "\<not> affine_dependent I" "card I \<le> n" "aff_dim K < int n" "K = convex hull I"
+
−              using * [OF \<open>K \<in> \<U>\<close>] by auto
+
−            ultimately have "?z \<notin> affine hull I"
+
−              using ahK_C_disjoint affine_hull_convex_hull that by blast
+
−            have "compact I" "finite I"
+
−              by (auto simp: \<open>\<not> affine_dependent I\<close> aff_independent_finite finite_imp_compact)
+
−            moreover have "F face_of convex hull insert ?z I"
+
−              by (metis S \<open>F face_of S\<close> \<open>K = convex hull I\<close> convex_hull_eq_empty convex_hull_insert_segments hull_hull)
+
−            ultimately obtain J where "J \<subseteq> insert ?z I" "F = convex hull J"
+
−              using face_of_convex_hull_subset [of "insert ?z I" F] by auto
+
−            show ?thesis
+
−            proof (cases "?z \<in> J")
+
−              case True
+
−              have "F \<in> (\<Union>K\<in>\<U> \<inter> Pow (rel_frontier C). {convex hull insert ?z K})"
+
−              proof
+
−                have "convex hull (J - {?z}) face_of K"
+
−                  by (metis True \<open>J \<subseteq> insert ?z I\<close> \<open>K = convex hull I\<close> \<open>\<not> affine_dependent I\<close> face_of_convex_hull_affine_independent subset_insert_iff)
+
−                then have "convex hull (J - {?z}) \<in> \<U>"
+
−                  by (rule face\<U> [OF \<open>K \<in> \<U>\<close>])
+
−                moreover
+
−                have "\<And>x. x \<in> convex hull (J - {?z}) \<Longrightarrow> x \<in> rel_frontier C"
+
−                  by (metis True \<open>J \<subseteq> insert ?z I\<close> \<open>K = convex hull I\<close> subsetD hull_mono subset_insert_iff that(4))
+
−                ultimately show "convex hull (J - {?z}) \<in> \<U> \<inter> Pow (rel_frontier C)" by auto
+
−                let ?F = "convex hull insert ?z (convex hull (J - {?z}))"
+
−                have "F \<subseteq> ?F"
+
−                  apply (clarsimp simp: \<open>F = convex hull J\<close>)
+
−                  by (metis True subsetD hull_mono hull_subset subset_insert_iff)
+
−                moreover have "?F \<subseteq> F"
+
−                  apply (clarsimp simp: \<open>F = convex hull J\<close>)
+
−                  by (metis (no_types, lifting) True convex_hull_eq_empty convex_hull_insert_segments hull_hull insert_Diff)
+
−                ultimately
+
−                show "F \<in> {?F}" by auto
+
−              qed
+
−              with \<open>C\<in>\<N>\<close> show ?thesis by auto
+
−            next
+
−              case False
+
−              then have "F \<in> \<U>"
+
−                using face_of_convex_hull_affine_independent [OF \<open>\<not> affine_dependent I\<close>]
+
−                by (metis Int_absorb2 Int_insert_right_if0 \<open>F = convex hull J\<close> \<open>J \<subseteq> insert ?z I\<close> \<open>K = convex hull I\<close> face\<U> inf_le2 \<open>K \<in> \<U>\<close>)
+
−              then show "F \<in> \<U> \<union> ?\<T>"
+
−                by blast
+
−            qed
+
−          qed
+
−          ultimately show ?thesis
+
−            using that by auto
+
−        qed
+
−        have "(S \<inter> S' face_of S) \<and> (S \<inter> S' face_of S')"
+
−          if "S \<in> \<U> \<union> ?\<T>" "S' \<in> \<U> \<union> ?\<T>" for S S'
+
−        proof -
+
−          have symmy: "\<lbrakk>\<And>X Y. R X Y \<Longrightarrow> R Y X;
+
−                        \<And>X Y. \<lbrakk>X \<in> \<U>; Y \<in> \<U>\<rbrakk> \<Longrightarrow> R X Y;
+
−                        \<And>X Y. \<lbrakk>X \<in> \<U>; Y \<in> ?\<T>\<rbrakk> \<Longrightarrow> R X Y;
+
−                        \<And>X Y. \<lbrakk>X \<in> ?\<T>; Y \<in> ?\<T>\<rbrakk> \<Longrightarrow> R X Y\<rbrakk> \<Longrightarrow> R S S'" for R
+
−            using that by (metis (no_types, lifting) Un_iff)
+
−          show ?thesis
+
−          proof (rule symmy)
+
−            show "Y \<inter> X face_of Y \<and> Y \<inter> X face_of X"
+
−              if "X \<inter> Y face_of X \<and> X \<inter> Y face_of Y" for X Y :: "'a set"
+
−              by (simp add: inf_commute that)
+
−          next
+
−            show "X \<inter> Y face_of X \<and> X \<inter> Y face_of Y"
+
−              if "X \<in> \<U>" and "Y \<in> \<U>" for X Y
+
−              by (simp add: faceI\<U> that)
+
−          next
+
−            show "X \<inter> Y face_of X \<and> X \<inter> Y face_of Y"
+
−              if XY: "X \<in> \<U>" "Y \<in> ?\<T>" for X Y
+
−            proof -
+
−              obtain C K
+
−                where "C \<in> \<N>" "K \<in> \<U>" "K \<subseteq> rel_frontier C"
+
−                and Y: "Y = convex hull insert (SOME z. z \<in> rel_interior C) K"
+
−                using XY by blast
+
−              have "convex C"
+
−                by (simp add: \<open>C \<in> \<N>\<close> convex\<N>)
+
−              have "K \<subseteq> C"
+
−                by (metis DiffE \<open>C \<in> \<N>\<close> \<open>K \<subseteq> rel_frontier C\<close> closed\<N> closure_closed rel_frontier_def subset_iff)
+
−              let ?z = "(SOME z. z \<in> rel_interior C)"
+
−              have z: "?z \<in> rel_interior C"
+
−                using \<open>C \<in> \<N>\<close> in_rel_interior by blast
+
−              obtain D where "D \<in> \<S>" "X \<subseteq> D"
+
−                using C\<U> \<open>X \<in> \<U>\<close> by blast
+
−              have "D \<inter> rel_interior C = (C \<inter> D) \<inter> rel_interior C"
+
−                using rel_interior_subset by blast
+
−              also have "(C \<inter> D) \<inter> rel_interior C = {}"
+
−              proof (rule face_of_disjoint_rel_interior)
+
−                show "C \<inter> D face_of C"
+
−                  using \<N>_def \<S>_def \<open>C \<in> \<N>\<close> \<open>D \<in> \<S>\<close> intface\<M> by blast
+
−                show "C \<inter> D \<noteq> C"
+
−                  by (metis (mono_tags, lifting) Int_lower2 \<N>_def \<S>_def \<open>C \<in> \<N>\<close> \<open>D \<in> \<S>\<close> aff_dim_subset mem_Collect_eq not_le)
+
−              qed
+
−              finally have DC: "D \<inter> rel_interior C = {}" .
+
−              have eq: "X \<inter> convex hull (insert ?z K) = X \<inter> convex hull K"
+
−                apply (rule Int_convex_hull_insert_rel_exterior [OF \<open>convex C\<close> \<open>K \<subseteq> C\<close> z])
+
−                using DC by (meson \<open>X \<subseteq> D\<close> disjnt_def disjnt_subset1)
+
−              obtain I where I: "\<not> affine_dependent I"
+
−                         and Keq: "K = convex hull I" and [simp]: "convex hull K = K"
+
−                using "*" \<open>K \<in> \<U>\<close> by force
+
−              then have "?z \<notin> affine hull I"
+
−                using ahK_C_disjoint \<open>C \<in> \<N>\<close> \<open>K \<in> \<U>\<close> \<open>K \<subseteq> rel_frontier C\<close> affine_hull_convex_hull z by blast
+
−              have "X \<inter> K face_of K"
+
−                by (simp add: \<open>K \<in> \<U>\<close> faceI\<U> \<open>X \<in> \<U>\<close>)
+
−              also have "... face_of convex hull insert ?z K"
+
−                by (metis I Keq \<open>?z \<notin> affine hull I\<close> aff_independent_finite convex_convex_hull face_of_convex_hull_insert face_of_refl hull_insert)
+
−              finally have "X \<inter> K face_of convex hull insert ?z K" .
+
−              then show ?thesis
+
−                using "*" \<open>K \<in> \<U>\<close> faceI\<U> that(1) by (fastforce simp add: Y eq)
+
−            qed
+
−          next
+
−            show "X \<inter> Y face_of X \<and> X \<inter> Y face_of Y"
+
−              if XY: "X \<in> ?\<T>" "Y \<in> ?\<T>" for X Y
+
−            proof -
+
−              obtain C K D L
+
−                where "C \<in> \<N>" "K \<in> \<U>" "K \<subseteq> rel_frontier C"
+
−                and X: "X = convex hull insert (SOME z. z \<in> rel_interior C) K"
+
−                and "D \<in> \<N>" "L \<in> \<U>" "L \<subseteq> rel_frontier D"
+
−                and Y: "Y = convex hull insert (SOME z. z \<in> rel_interior D) L"
+
−                using XY by blast
+
−              let ?z = "(SOME z. z \<in> rel_interior C)"
+
−              have z: "?z \<in> rel_interior C"
+
−                using \<open>C \<in> \<N>\<close> in_rel_interior by blast
+
−              have "convex C"
+
−                by (simp add: \<open>C \<in> \<N>\<close> convex\<N>)
+
−              have "convex K"
+
−                using "*" \<open>K \<in> \<U>\<close> by blast
+
−              have "convex L"
+
−                by (meson \<open>L \<in> \<U>\<close> convex_simplex simpl\<U>)
+
−              show ?thesis
+
−              proof (cases "D=C")
+
−                case True
+
−                then have "L \<subseteq> rel_frontier C"
+
−                  using \<open>L \<subseteq> rel_frontier D\<close> by auto
+
−                show ?thesis
+
−                  apply (simp add: X Y True)
+
−                  apply (simp add: convex_hull_insert_Int_eq [OF z] \<open>K \<subseteq> rel_frontier C\<close> \<open>L \<subseteq> rel_frontier C\<close> \<open>convex C\<close> \<open>convex K\<close> \<open>convex L\<close>)
+
−                  using face_of_polytope_insert2
+
−                  by (metis "*" IntI \<open>C \<in> \<N>\<close> \<open>K \<in> \<U>\<close> \<open>L \<in> \<U>\<close>\<open>K \<subseteq> rel_frontier C\<close> \<open>L \<subseteq> rel_frontier C\<close> aff_independent_finite ahK_C_disjoint empty_iff faceI\<U> polytope_convex_hull z)
+
−              next
+
−                case False
+
−                have "convex D"
+
−                  by (simp add: \<open>D \<in> \<N>\<close> convex\<N>)
+
−                have "K \<subseteq> C"
+
−                  by (metis DiffE \<open>C \<in> \<N>\<close> \<open>K \<subseteq> rel_frontier C\<close> closed\<N> closure_closed rel_frontier_def subset_eq)
+
−                have "L \<subseteq> D"
+
−                  by (metis DiffE \<open>D \<in> \<N>\<close> \<open>L \<subseteq> rel_frontier D\<close> closed\<N> closure_closed rel_frontier_def subset_eq)
+
−                let ?w = "(SOME w. w \<in> rel_interior D)"
+
−                have w: "?w \<in> rel_interior D"
+
−                  using \<open>D \<in> \<N>\<close> in_rel_interior by blast
+
−                have "C \<inter> rel_interior D = (D \<inter> C) \<inter> rel_interior D"
+
−                  using rel_interior_subset by blast
+
−                also have "(D \<inter> C) \<inter> rel_interior D = {}"
+
−                proof (rule face_of_disjoint_rel_interior)
+
−                  show "D \<inter> C face_of D"
+
−                    using \<N>_def \<open>C \<in> \<N>\<close> \<open>D \<in> \<N>\<close> intface\<M> by blast
+
−                  have "D \<in> \<M> \<and> aff_dim D = int n"
+
−                    using \<N>_def \<open>D \<in> \<N>\<close> by blast
+
−                  moreover have "C \<in> \<M> \<and> aff_dim C = int n"
+
−                    using \<N>_def \<open>C \<in> \<N>\<close> by blast
+
−                  ultimately show "D \<inter> C \<noteq> D"
+
−                    by (metis False face_of_aff_dim_lt inf.idem inf_le1 intface\<M> not_le poly\<M> polytope_imp_convex)
+
−                qed
+
−                finally have CD: "C \<inter> (rel_interior D) = {}" .
+
−                have zKC: "(convex hull insert ?z K) \<subseteq> C"
+
−                  by (metis DiffE \<open>C \<in> \<N>\<close> \<open>K \<subseteq> rel_frontier C\<close> closed\<N> closure_closed convex\<N> hull_minimal insert_subset rel_frontier_def rel_interior_subset subset_iff z)
+
−                have eq: "convex hull (insert ?z K) \<inter> convex hull (insert ?w L) =
+
−                          convex hull (insert ?z K) \<inter> convex hull L"
+
−                  apply (rule Int_convex_hull_insert_rel_exterior [OF \<open>convex D\<close> \<open>L \<subseteq> D\<close> w])
+
−                  using zKC CD apply (force simp: disjnt_def)
+
−                  done
+
−                have ch_id: "convex hull K = K" "convex hull L = L"
+
−                  using "*" \<open>K \<in> \<U>\<close> \<open>L \<in> \<U>\<close> hull_same by auto
+
−                have "convex C"
+
−                  by (simp add: \<open>C \<in> \<N>\<close> convex\<N>)
+
−                have "convex hull (insert ?z K) \<inter> L = L \<inter> convex hull (insert ?z K)"
+
−                  by blast
+
−                also have "... = convex hull K \<inter> L"
+
−                proof (subst Int_convex_hull_insert_rel_exterior [OF \<open>convex C\<close> \<open>K \<subseteq> C\<close> z])
+
−                  have "(C \<inter> D) \<inter> rel_interior C = {}"
+
−                  proof (rule face_of_disjoint_rel_interior)
+
−                    show "C \<inter> D face_of C"
+
−                      using \<N>_def \<open>C \<in> \<N>\<close> \<open>D \<in> \<N>\<close> intface\<M> by blast
+
−                    have "D \<in> \<M>" "aff_dim D = int n"
+
−                      using \<N>_def \<open>D \<in> \<N>\<close> by fastforce+
+
−                    moreover have "C \<in> \<M>" "aff_dim C = int n"
+
−                      using \<N>_def \<open>C \<in> \<N>\<close> by fastforce+
+
−                    ultimately have "aff_dim D + - 1 * aff_dim C \<le> 0"
+
−                      by fastforce
+
−                    then have "\<not> C face_of D"
+
−                      using False \<open>convex D\<close> face_of_aff_dim_lt by fastforce
+
−                    show "C \<inter> D \<noteq> C"
+
−                      using \<open>C \<in> \<M>\<close> \<open>D \<in> \<M>\<close> \<open>\<not> C face_of D\<close> intface\<M> by fastforce
+
−                  qed
+
−                  then have "D \<inter> rel_interior C = {}"
+
−                    by (metis inf.absorb_iff2 inf_assoc inf_sup_aci(1) rel_interior_subset)
+
−                  then show "disjnt L (rel_interior C)"
+
−                    by (meson \<open>L \<subseteq> D\<close> disjnt_def disjnt_subset1)
+
−                next
+
−                  show "L \<inter> convex hull K = convex hull K \<inter> L"
+
−                    by force
+
−                qed
+
−                finally have chKL: "convex hull (insert ?z K) \<inter> L = convex hull K \<inter> L" .
+
−                have "convex hull insert ?z K \<inter> convex hull L face_of K"
+
−                  by (simp add: \<open>K \<in> \<U>\<close> \<open>L \<in> \<U>\<close> ch_id chKL faceI\<U>)
+
−                also have "... face_of convex hull insert ?z K"
+
−                proof -
+
−                  obtain I where I: "\<not> affine_dependent I" "K = convex hull I"
+
−                    using * [OF \<open>K \<in> \<U>\<close>] by auto
+
−                  then have "\<And>a. a \<notin> rel_interior C \<or> a \<notin> affine hull I"
+
−                    using ahK_C_disjoint \<open>C \<in> \<N>\<close> \<open>K \<in> \<U>\<close> \<open>K \<subseteq> rel_frontier C\<close> affine_hull_convex_hull by blast
+
−                  then show ?thesis
+
−                    by (metis I affine_independent_insert face_of_convex_hull_affine_independent hull_insert subset_insertI z)
+
−                qed
+
−                finally have 1: "convex hull insert ?z K \<inter> convex hull L face_of convex hull insert ?z K" .
+
−                have "convex hull insert ?z K \<inter> convex hull L face_of L"
+
−                  by (simp add: \<open>K \<in> \<U>\<close> \<open>L \<in> \<U>\<close> ch_id chKL faceI\<U>)
+
−                also have "... face_of convex hull insert ?w L"
+
−                proof -
+
−                  obtain I where I: "\<not> affine_dependent I" "L = convex hull I"
+
−                    using * [OF \<open>L \<in> \<U>\<close>] by auto
+
−                  then have "\<And>a. a \<notin> rel_interior D \<or> a \<notin> affine hull I"
+
−                    using \<open>D \<in> \<N>\<close> \<open>L \<in> \<U>\<close> \<open>L \<subseteq> rel_frontier D\<close> affine_hull_convex_hull ahK_C_disjoint by blast
+
−                  then show ?thesis
+
−                    by (metis I aff_independent_finite convex_convex_hull face_of_convex_hull_insert face_of_refl hull_insert w)
+
−                qed
+
−                finally have 2: "convex hull insert ?z K \<inter> convex hull L face_of convex hull insert ?w L" .
+
−                show ?thesis
+
−                  by (simp add: X Y eq 1 2)
+
−              qed
+
−            qed
+
−          qed
+
−        qed
+
−        then
+
−        show "S \<inter> S' face_of S" "S \<inter> S' face_of S'" if "S \<in> \<U> \<union> ?\<T> \<and> S' \<in> \<U> \<union> ?\<T>" for S S'
+
−          using that by auto
+
−      qed
+
−      show "\<exists>F \<subseteq> \<U> \<union> ?\<T>. C = \<Union>F" if "C \<in> \<M>" for C
+
−      proof (cases "C \<in> \<S>")
+
−        case True
+
−        then show ?thesis
+
−          by (meson UnCI fin\<U> subsetD subsetI)
+
−      next
+
−        case False
+
−        then have "C \<in> \<N>"
+
−          by (simp add: \<N>_def \<S>_def aff\<M> less_le that)
+
−        let ?z = "SOME z. z \<in> rel_interior C"
+
−        have z: "?z \<in> rel_interior C"
+
−          using \<open>C \<in> \<N>\<close> in_rel_interior by blast
+
−        let ?F = "\<Union>K \<in> \<U> \<inter> Pow (rel_frontier C). {convex hull (insert ?z K)}"
+
−        have "?F \<subseteq> ?\<T>"
+
−          using \<open>C \<in> \<N>\<close> by blast
+
−        moreover have "C \<subseteq> \<Union>?F"
+
−        proof
+
−          fix x
+
−          assume "x \<in> C"
+
−          have "convex C"
+
−            using \<open>C \<in> \<N>\<close> convex\<N> by blast
+
−          have "bounded C"
+
−            using \<open>C \<in> \<N>\<close> by (simp add: poly\<M> polytope_imp_bounded that)
+
−          have "polytope C"
+
−            using \<open>C \<in> \<N>\<close> poly\<N> by auto
+
−          have "\<not> (?z = x \<and> C = {?z})"
+
−            using \<open>C \<in> \<N>\<close> aff_dim_sing [of ?z] \<open>\<not> n \<le> 1\<close> by (force simp: \<N>_def)
+
−          then obtain y where y: "y \<in> rel_frontier C" and xzy: "x \<in> closed_segment ?z y"
+
−            and sub: "open_segment ?z y \<subseteq> rel_interior C"
+
−            by (blast intro: segment_to_rel_frontier [OF \<open>convex C\<close> \<open>bounded C\<close> z \<open>x \<in> C\<close>])
+
−          then obtain F where "y \<in> F" "F face_of C" "F \<noteq> C"
+
−            by (auto simp: rel_frontier_of_polyhedron_alt [OF polytope_imp_polyhedron [OF \<open>polytope C\<close>]])
+
−          then obtain \<G> where "finite \<G>" "\<G> \<subseteq> \<U>" "F = \<Union>\<G>"
+
−            by (metis (mono_tags, lifting) \<S>_def \<open>C \<in> \<M>\<close> \<open>convex C\<close> aff\<M> face\<M> face_of_aff_dim_lt fin\<U> le_less_trans mem_Collect_eq not_less)
+
−          then obtain K where "y \<in> K" "K \<in> \<G>"
+
−            using \<open>y \<in> F\<close> by blast
+
−          moreover have x: "x \<in> convex hull {?z,y}"
+
−            using segment_convex_hull xzy by auto
+
−          moreover have "convex hull {?z,y} \<subseteq> convex hull insert ?z K"
+
−            by (metis (full_types) \<open>y \<in> K\<close> hull_mono empty_subsetI insertCI insert_subset)
+
−          moreover have "K \<in> \<U>"
+
−            using \<open>K \<in> \<G>\<close> \<open>\<G> \<subseteq> \<U>\<close> by blast
+
−          moreover have "K \<subseteq> rel_frontier C"
+
−            using \<open>F = \<Union>\<G>\<close> \<open>F \<noteq> C\<close> \<open>F face_of C\<close> \<open>K \<in> \<G>\<close> face_of_subset_rel_frontier by fastforce
+
−          ultimately show "x \<in> \<Union>?F"
+
−            by force
+
−        qed
+
−        moreover
+
−        have "convex hull insert (SOME z. z \<in> rel_interior C) K \<subseteq> C"
+
−          if "K \<in> \<U>" "K \<subseteq> rel_frontier C" for K
+
−        proof (rule hull_minimal)
+
−          show "insert (SOME z. z \<in> rel_interior C) K \<subseteq> C"
+
−            using that \<open>C \<in> \<N>\<close> in_rel_interior rel_interior_subset
+
−            by (force simp: closure_eq rel_frontier_def closed\<N>)
+
−          show "convex C"
+
−            by (simp add: \<open>C \<in> \<N>\<close> convex\<N>)
+
−        qed
+
−        then have "\<Union>?F \<subseteq> C"
+
−          by auto
+
−        ultimately show ?thesis
+
−          by blast
+
−      qed
+
−
+
−      have "(\<exists>C. C \<in> \<M> \<and> L \<subseteq> C) \<and> aff_dim L \<le> int n"  if "L \<in> \<U> \<union> ?\<T>" for L
+
−        using that
+
−      proof
+
−        assume "L \<in> \<U>"
+
−        then show ?thesis
+
−          using C\<U> \<S>_def "*" by fastforce
+
−      next
+
−        assume "L \<in> ?\<T>"
+
−        then obtain C K where "C \<in> \<N>"
+
−          and L: "L = convex hull insert (SOME z. z \<in> rel_interior C) K"
+
−          and K: "K \<in> \<U>" "K \<subseteq> rel_frontier C"
+
−          by auto
+
−        then have "convex hull C = C"
+
−          by (meson convex\<N> convex_hull_eq)
+
−        then have "convex C"
+
−          by (metis (no_types) convex_convex_hull)
+
−        have "rel_frontier C \<subseteq> C"
+
−          by (metis DiffE closed\<N> \<open>C \<in> \<N>\<close> closure_closed rel_frontier_def subsetI)
+
−        have "K \<subseteq> C"
+
−          using K \<open>rel_frontier C \<subseteq> C\<close> by blast
+
−        have "C \<in> \<M>"
+
−          using \<N>_def \<open>C \<in> \<N>\<close> by auto
+
−        moreover have "L \<subseteq> C"
+
−          using K L \<open>C \<in> \<N>\<close>
+
−          by (metis \<open>K \<subseteq> C\<close> \<open>convex hull C = C\<close> contra_subsetD hull_mono in_rel_interior insert_subset rel_interior_subset)
+
−        ultimately show ?thesis
+
−          using \<open>rel_frontier C \<subseteq> C\<close> \<open>L \<subseteq> C\<close> aff\<M> aff_dim_subset \<open>C \<in> \<M>\<close> dual_order.trans by blast
+
−      qed
+
−      then show "\<exists>C. C \<in> \<M> \<and> L \<subseteq> C" "aff_dim L \<le> int n" if "L \<in> \<U> \<union> ?\<T>" for L
+
−        using that by auto
+
−    qed
+
−    then show ?thesis
+
−      apply (rule ex_forward, safe)
+
−        apply (meson Union_iff subsetCE, fastforce)
+
−      by (meson infinite_super simplicial_complex_def)
+
−  qed
+
−qed
+
−
+
−
+
−lemma simplicial_subdivision_of_cell_complex_lowdim:
+
−  assumes "finite \<M>"
+
−      and poly: "\<And>C. C \<in> \<M> \<Longrightarrow> polytope C"
+
−      and face: "\<And>C1 C2. \<lbrakk>C1 \<in> \<M>; C2 \<in> \<M>\<rbrakk> \<Longrightarrow> C1 \<inter> C2 face_of C1 \<and> C1 \<inter> C2 face_of C2"
+
−      and aff: "\<And>C. C \<in> \<M> \<Longrightarrow> aff_dim C \<le> d"
+
−  obtains \<T> where "simplicial_complex \<T>" "\<And>K. K \<in> \<T> \<Longrightarrow> aff_dim K \<le> d"
+
−                  "\<Union>\<T> = \<Union>\<M>"
+
−                  "\<And>C. C \<in> \<M> \<Longrightarrow> \<exists>F. finite F \<and> F \<subseteq> \<T> \<and> C = \<Union>F"
+
−                  "\<And>K. K \<in> \<T> \<Longrightarrow> \<exists>C. C \<in> \<M> \<and> K \<subseteq> C"
+
−proof (cases "d \<ge> 0")
+
−  case True
+
−  then obtain n where n: "d = of_nat n"
+
−    using zero_le_imp_eq_int by blast
+
−  have "\<exists>\<T>. simplicial_complex \<T> \<and>
+
−            (\<forall>K\<in>\<T>. aff_dim K \<le> int n) \<and>
+
−            \<Union>\<T> = \<Union>(\<Union>C\<in>\<M>. {F. F face_of C}) \<and>
+
−            (\<forall>C\<in>\<Union>C\<in>\<M>. {F. F face_of C}.
+
−                \<exists>F. finite F \<and> F \<subseteq> \<T> \<and> C = \<Union>F) \<and>
+
−            (\<forall>K\<in>\<T>. \<exists>C. C \<in> (\<Union>C\<in>\<M>. {F. F face_of C}) \<and> K \<subseteq> C)"
+
−  proof (rule simplicial_subdivision_aux)
+
−    show "finite (\<Union>C\<in>\<M>. {F. F face_of C})"
+
−      using \<open>finite \<M>\<close> poly polyhedron_eq_finite_faces polytope_imp_polyhedron by fastforce
+
−    show "polytope F" if "F \<in> (\<Union>C\<in>\<M>. {F. F face_of C})" for F
+
−      using poly that face_of_polytope_polytope by blast
+
−    show "aff_dim F \<le> int n" if "F \<in> (\<Union>C\<in>\<M>. {F. F face_of C})" for F
+
−      using that
+
−      by clarify (metis n aff_dim_subset aff face_of_imp_subset order_trans)
+
−    show "F \<in> (\<Union>C\<in>\<M>. {F. F face_of C})"
+
−      if "G \<in> (\<Union>C\<in>\<M>. {F. F face_of C})" and "F face_of G" for F G
+
−      using that face_of_trans by blast
+
−  next
+
−    show "F1 \<inter> F2 face_of F1 \<and> F1 \<inter> F2 face_of F2"
+
−      if "F1 \<in> (\<Union>C\<in>\<M>. {F. F face_of C})" and "F2 \<in> (\<Union>C\<in>\<M>. {F. F face_of C})" for F1 F2
+
−      using that
+
−      by safe (meson face face_of_Int_subface)+
+
−  qed
+
−  moreover
+
−  have "\<Union>(\<Union>C\<in>\<M>. {F. F face_of C}) = \<Union>\<M>"
+
−    using face_of_imp_subset face by blast
+
−  ultimately show ?thesis
+
−    apply clarify
+
−    apply (rule that, assumption+)
+
−       using n apply blast
+
−      apply (simp_all add: poly face_of_refl polytope_imp_convex)
+
−    using face_of_imp_subset by fastforce
+
−next
+
−  case False
+
−  then have m1: "\<And>C. C \<in> \<M> \<Longrightarrow> aff_dim C = -1"
+
−    by (metis aff aff_dim_empty_eq aff_dim_negative_iff dual_order.trans not_less)
+
−  then have face\<M>: "\<And>F S. \<lbrakk>S \<in> \<M>; F face_of S\<rbrakk> \<Longrightarrow> F \<in> \<M>"
+
−    by (metis aff_dim_empty face_of_empty)
+
−  show ?thesis
+
−  proof
+
−    have "\<And>S. S \<in> \<M> \<Longrightarrow> \<exists>n. n simplex S"
+
−      by (metis (no_types) m1 aff_dim_empty simplex_minus_1)
+
−    then show "simplicial_complex \<M>"
+
−      by (auto simp: simplicial_complex_def \<open>finite \<M>\<close> face intro: face\<M>)
+
−    show "aff_dim K \<le> d" if "K \<in> \<M>" for K
+
−      by (simp add: that aff)
+
−    show "\<exists>F. finite F \<and> F \<subseteq> \<M> \<and> C = \<Union>F" if "C \<in> \<M>" for C
+
−      using \<open>C \<in> \<M>\<close> equals0I by auto
+
−    show "\<exists>C. C \<in> \<M> \<and> K \<subseteq> C" if "K \<in> \<M>" for K
+
−      using \<open>K \<in> \<M>\<close> by blast
+
−  qed auto
+
−qed
+
−
+
−proposition simplicial_subdivision_of_cell_complex:
+
−  assumes "finite \<M>"
+
−      and poly: "\<And>C. C \<in> \<M> \<Longrightarrow> polytope C"
+
−      and face: "\<And>C1 C2. \<lbrakk>C1 \<in> \<M>; C2 \<in> \<M>\<rbrakk> \<Longrightarrow> C1 \<inter> C2 face_of C1 \<and> C1 \<inter> C2 face_of C2"
+
−  obtains \<T> where "simplicial_complex \<T>"
+
−                  "\<Union>\<T> = \<Union>\<M>"
+
−                  "\<And>C. C \<in> \<M> \<Longrightarrow> \<exists>F. finite F \<and> F \<subseteq> \<T> \<and> C = \<Union>F"
+
−                  "\<And>K. K \<in> \<T> \<Longrightarrow> \<exists>C. C \<in> \<M> \<and> K \<subseteq> C"
+
−  by (blast intro: simplicial_subdivision_of_cell_complex_lowdim [OF assms aff_dim_le_DIM])
+
−
+
−corollary fine_simplicial_subdivision_of_cell_complex:
+
−  assumes "0 < e" "finite \<M>"
+
−      and poly: "\<And>C. C \<in> \<M> \<Longrightarrow> polytope C"
+
−      and face: "\<And>C1 C2. \<lbrakk>C1 \<in> \<M>; C2 \<in> \<M>\<rbrakk> \<Longrightarrow> C1 \<inter> C2 face_of C1 \<and> C1 \<inter> C2 face_of C2"
+
−  obtains \<T> where "simplicial_complex \<T>"
+
−                  "\<And>K. K \<in> \<T> \<Longrightarrow> diameter K < e"
+
−                  "\<Union>\<T> = \<Union>\<M>"
+
−                  "\<And>C. C \<in> \<M> \<Longrightarrow> \<exists>F. finite F \<and> F \<subseteq> \<T> \<and> C = \<Union>F"
+
−                  "\<And>K. K \<in> \<T> \<Longrightarrow> \<exists>C. C \<in> \<M> \<and> K \<subseteq> C"
+
−proof -
+
−  obtain \<N> where \<N>: "finite \<N>" "\<Union>\<N> = \<Union>\<M>" 
+
−              and diapoly: "\<And>X. X \<in> \<N> \<Longrightarrow> diameter X < e" "\<And>X. X \<in> \<N> \<Longrightarrow> polytope X"
+
−               and      "\<And>X Y. \<lbrakk>X \<in> \<N>; Y \<in> \<N>\<rbrakk> \<Longrightarrow> X \<inter> Y face_of X \<and> X \<inter> Y face_of Y"
+
−               and \<N>covers: "\<And>C x. C \<in> \<M> \<and> x \<in> C \<Longrightarrow> \<exists>D. D \<in> \<N> \<and> x \<in> D \<and> D \<subseteq> C"
+
−               and \<N>covered: "\<And>C. C \<in> \<N> \<Longrightarrow> \<exists>D. D \<in> \<M> \<and> C \<subseteq> D"
+
−    by (blast intro: cell_complex_subdivision_exists [OF \<open>0 < e\<close> \<open>finite \<M>\<close> poly aff_dim_le_DIM face])
+
−  then obtain \<T> where \<T>: "simplicial_complex \<T>" "\<Union>\<T> = \<Union>\<N>"
+
−                   and \<T>covers: "\<And>C. C \<in> \<N> \<Longrightarrow> \<exists>F. finite F \<and> F \<subseteq> \<T> \<and> C = \<Union>F"
+
−                   and \<T>covered: "\<And>K. K \<in> \<T> \<Longrightarrow> \<exists>C. C \<in> \<N> \<and> K \<subseteq> C"
+
−    using simplicial_subdivision_of_cell_complex [OF \<open>finite \<N>\<close>] by metis
+
−  show ?thesis
+
−  proof
+
−    show "simplicial_complex \<T>"
+
−      by (rule \<T>)
+
−    show "diameter K < e" if "K \<in> \<T>" for K
+
−      by (metis le_less_trans diapoly \<T>covered diameter_subset polytope_imp_bounded that)
+
−    show "\<Union>\<T> = \<Union>\<M>"
+
−      by (simp add: \<N>(2) \<open>\<Union>\<T> = \<Union>\<N>\<close>)
+
−    show "\<exists>F. finite F \<and> F \<subseteq> \<T> \<and> C = \<Union>F" if "C \<in> \<M>" for C
+
−    proof -
+
−      { fix x
+
−        assume "x \<in> C"
+
−        then obtain D where "D \<in> \<T>" "x \<in> D" "D \<subseteq> C"
+
−          using \<N>covers \<open>C \<in> \<M>\<close> \<T>covers by force
+
−        then have "\<exists>X\<in>\<T> \<inter> Pow C. x \<in> X"
+
−          using \<open>D \<in> \<T>\<close> \<open>D \<subseteq> C\<close> \<open>x \<in> D\<close> by blast
+
−      }
+
−      moreover
+
−      have "finite (\<T> \<inter> Pow C)"
+
−        using \<open>simplicial_complex \<T>\<close> simplicial_complex_def by auto
+
−      ultimately show ?thesis
+
−        by (rule_tac x="(\<T> \<inter> Pow C)" in exI) auto
+
−    qed
+
−    show "\<exists>C. C \<in> \<M> \<and> K \<subseteq> C" if "K \<in> \<T>" for K
+
−      by (meson \<N>covered \<T>covered order_trans that)
+
−  qed
+
−qed
+
−
+
−subsection\<open>Some results on cell division with full-dimensional cells only\<close>
+
−
+
−lemma convex_Union_fulldim_cells:
+
−  assumes "finite \<S>" and clo: "\<And>C. C \<in> \<S> \<Longrightarrow> closed C" and con: "\<And>C. C \<in> \<S> \<Longrightarrow> convex C"
+
−      and eq: "\<Union>\<S> = U"and  "convex U"
+
− shows "\<Union>{C \<in> \<S>. aff_dim C = aff_dim U} = U"  (is "?lhs = U")
+
−proof -
+
−  have "closed U"
+
−    using \<open>finite \<S>\<close> clo eq by blast
+
−  have "?lhs \<subseteq> U"
+
−    using eq by blast
+
−  moreover have "U \<subseteq> ?lhs"
+
−  proof (cases "\<forall>C \<in> \<S>. aff_dim C = aff_dim U")
+
−    case True
+
−    then show ?thesis
+
−      using eq by blast
+
−  next
+
−    case False
+
−    have "closed ?lhs"
+
−      by (simp add: \<open>finite \<S>\<close> clo closed_Union)
+
−    moreover have "U \<subseteq> closure ?lhs"
+
−    proof -
+
−      have "U \<subseteq> closure(\<Inter>{U - C |C. C \<in> \<S> \<and> aff_dim C < aff_dim U})"
+
−      proof (rule Baire [OF \<open>closed U\<close>])
+
−        show "countable {U - C |C. C \<in> \<S> \<and> aff_dim C < aff_dim U}"
+
−          using \<open>finite \<S>\<close> uncountable_infinite by fastforce
+
−        have "\<And>C. C \<in> \<S> \<Longrightarrow> openin (subtopology euclidean U) (U-C)"
+
−          by (metis Sup_upper clo closed_limpt closedin_limpt eq openin_diff openin_subtopology_self)
+
−        then show "openin (subtopology euclidean U) T \<and> U \<subseteq> closure T"
+
−          if "T \<in> {U - C |C. C \<in> \<S> \<and> aff_dim C < aff_dim U}" for T
+
−          using that dense_complement_convex_closed \<open>closed U\<close> \<open>convex U\<close> by auto
+
−      qed
+
−      also have "... \<subseteq> closure ?lhs"
+
−      proof -
+
−        obtain C where "C \<in> \<S>" "aff_dim C < aff_dim U"
+
−          by (metis False Sup_upper aff_dim_subset eq eq_iff not_le)
+
−        have "\<exists>X. X \<in> \<S> \<and> aff_dim X = aff_dim U \<and> x \<in> X"
+
−          if "\<And>V. (\<exists>C. V = U - C \<and> C \<in> \<S> \<and> aff_dim C < aff_dim U) \<Longrightarrow> x \<in> V" for x
+
−        proof -
+
−          have "x \<in> U \<and> x \<in> \<Union>\<S>"
+
−            using \<open>C \<in> \<S>\<close> \<open>aff_dim C < aff_dim U\<close> eq that by blast
+
−          then show ?thesis
+
−            by (metis Diff_iff Sup_upper Union_iff aff_dim_subset dual_order.order_iff_strict eq that)
+
−        qed
+
−        then show ?thesis
+
−          by (auto intro!: closure_mono)
+
−      qed
+
−      finally show ?thesis .
+
−    qed
+
−    ultimately show ?thesis
+
−      using closure_subset_eq by blast
+
−  qed
+
−  ultimately show ?thesis by blast
+
−qed
+
−
+
−proposition fine_triangular_subdivision_of_cell_complex:
+
−  assumes "0 < e" "finite \<M>"
+
−      and poly: "\<And>C. C \<in> \<M> \<Longrightarrow> polytope C"
+
−      and aff: "\<And>C. C \<in> \<M> \<Longrightarrow> aff_dim C = d"
+
−      and face: "\<And>C1 C2. \<lbrakk>C1 \<in> \<M>; C2 \<in> \<M>\<rbrakk> \<Longrightarrow> C1 \<inter> C2 face_of C1 \<and> C1 \<inter> C2 face_of C2"
+
−  obtains \<T> where "triangulation \<T>" "\<And>k. k \<in> \<T> \<Longrightarrow> diameter k < e"
+
−                 "\<And>k. k \<in> \<T> \<Longrightarrow> aff_dim k = d" "\<Union>\<T> = \<Union>\<M>"
+
−                 "\<And>C. C \<in> \<M> \<Longrightarrow> \<exists>f. finite f \<and> f \<subseteq> \<T> \<and> C = \<Union>f"
+
−                 "\<And>k. k \<in> \<T> \<Longrightarrow> \<exists>C. C \<in> \<M> \<and> k \<subseteq> C"
+
−proof -
+
−  obtain \<T> where "simplicial_complex \<T>"
+
−             and dia\<T>: "\<And>K. K \<in> \<T> \<Longrightarrow> diameter K < e"
+
−             and "\<Union>\<T> = \<Union>\<M>"
+
−             and in\<M>: "\<And>C. C \<in> \<M> \<Longrightarrow> \<exists>F. finite F \<and> F \<subseteq> \<T> \<and> C = \<Union>F"
+
−             and in\<T>: "\<And>K. K \<in> \<T> \<Longrightarrow> \<exists>C. C \<in> \<M> \<and> K \<subseteq> C"
+
−    by (blast intro: fine_simplicial_subdivision_of_cell_complex [OF \<open>e > 0\<close> \<open>finite \<M>\<close> poly face])
+
−  let ?\<T> = "{K \<in> \<T>. aff_dim K = d}"
+
−  show thesis
+
−  proof
+
−    show "triangulation ?\<T>"
+
−      using \<open>simplicial_complex \<T>\<close> by (auto simp: triangulation_def simplicial_complex_def)
+
−    show "diameter L < e" if "L \<in> {K \<in> \<T>. aff_dim K = d}" for L
+
−      using that by (auto simp: dia\<T>)
+
−    show "aff_dim L = d" if "L \<in> {K \<in> \<T>. aff_dim K = d}" for L
+
−      using that by auto
+
−    show "\<exists>F. finite F \<and> F \<subseteq> {K \<in> \<T>. aff_dim K = d} \<and> C = \<Union>F" if "C \<in> \<M>" for C
+
−    proof -
+
−      obtain F where "finite F" "F \<subseteq> \<T>" "C = \<Union>F"
+
−        using in\<M> [OF \<open>C \<in> \<M>\<close>] by auto
+
−      show ?thesis
+
−      proof (intro exI conjI)
+
−        show "finite {K \<in> F. aff_dim K = d}"
+
−          by (simp add: \<open>finite F\<close>)
+
−        show "{K \<in> F. aff_dim K = d} \<subseteq> {K \<in> \<T>. aff_dim K = d}"
+
−          using \<open>F \<subseteq> \<T>\<close> by blast
+
−        have "d = aff_dim C"
+
−          by (simp add: aff that)
+
−        moreover have "\<And>K. K \<in> F \<Longrightarrow> closed K \<and> convex K"
+
−          using \<open>simplicial_complex \<T>\<close> \<open>F \<subseteq> \<T>\<close>
+
−          unfolding simplicial_complex_def by (metis subsetCE \<open>F \<subseteq> \<T>\<close> closed_simplex convex_simplex)
+
−        moreover have "convex (\<Union>F)"
+
−          using \<open>C = \<Union>F\<close> poly polytope_imp_convex that by blast
+
−        ultimately show "C = \<Union>{K \<in> F. aff_dim K = d}"
+
−          by (simp add: convex_Union_fulldim_cells \<open>C = \<Union>F\<close> \<open>finite F\<close>)
+
−      qed
+
−    qed
+
−    then show "\<Union>{K \<in> \<T>. aff_dim K = d} = \<Union>\<M>"
+
−      by auto (meson in\<T> subsetCE)
+
−    show "\<exists>C. C \<in> \<M> \<and> L \<subseteq> C"
+
−      if "L \<in> {K \<in> \<T>. aff_dim K = d}" for L
+
−      using that by (auto simp: in\<T>)
+
−  qed
+
−qed
+
−
+
−end