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%important \<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%unimportant 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%unimportant 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%unimportant 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%unimportant face_of_refl: "convex S \<Longrightarrow> S face_of S"
by (auto simp: face_of_def)
lemma%unimportant face_of_refl_eq: "S face_of S \<longleftrightarrow> convex S"
by (auto simp: face_of_def)
lemma%unimportant empty_face_of [iff]: "{} face_of S"
by (simp add: face_of_def)
lemma%unimportant face_of_empty [simp]: "S face_of {} \<longleftrightarrow> S = {}"
by (meson empty_face_of face_of_def subset_empty)
lemma%unimportant 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%unimportant 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%unimportant 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%unimportant 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%unimportant 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%unimportant 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%unimportant 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%unimportant face_of_imp_subset: "T face_of S \<Longrightarrow> T \<subseteq> S"
unfolding face_of_def by blast
lemma%important 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%unimportant -
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%unimportant 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%important 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%unimportant -
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%unimportant 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%unimportant 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%unimportant 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%unimportant face_of_imp_convex: "T face_of S \<Longrightarrow> convex T"
using face_of_def by blast
lemma%unimportant 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%unimportant 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%unimportant 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%unimportant 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%unimportant 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%unimportant 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%unimportant 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%unimportant 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%unimportant 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%unimportant 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%unimportant 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%unimportant 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
lemma%important 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%unimportant -
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%important 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%important face_of_affine_trivial:
assumes "affine S" "T face_of S"
shows "T = {} \<or> T = S"
proof%unimportant (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%unimportant 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
lemma%important 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%unimportant (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%unimportant 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%important face_of_Times:
assumes "F face_of S" and "F' face_of S'"
shows "(F \<times> F') face_of (S \<times> S')"
proof%unimportant -
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%important 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%unimportant
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%unimportant 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%unimportant 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%unimportant 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%important 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%unimportant (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%unimportant 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%important\<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%unimportant 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%unimportant 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%unimportant exposed_face_of_refl: "convex S \<Longrightarrow> S exposed_face_of S"
by simp
lemma%unimportant 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%unimportant 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%unimportant 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%important 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%unimportant -
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%important 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%unimportant -
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%important 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%unimportant (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
lemma%important 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%unimportant
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%important\<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%unimportant 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%unimportant 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%unimportant 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%important 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%unimportant 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%unimportant 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
lemma%important extreme_points_of_convex_hull:
"{x. x extreme_point_of (convex hull S)} \<subseteq> S"
using%unimportant extreme_point_of_convex_hull by auto
lemma%unimportant extreme_point_of_empty [simp]: "\<not> (x extreme_point_of {})"
by (simp add: extreme_point_of_def)
lemma%unimportant extreme_point_of_singleton [iff]: "x extreme_point_of {a} \<longleftrightarrow> x = a"
using extreme_point_of_stillconvex by auto
lemma%unimportant 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%important 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%unimportant extreme_point_of_translation_eq
by%unimportant auto (metis (no_types, lifting) image_iff mem_Collect_eq minus_add_cancel)
lemma%unimportant 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%important 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%unimportant 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%unimportant 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%unimportant 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%unimportant 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%important\<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%unimportant facet_of_empty [simp]: "\<not> S facet_of {}"
by (simp add: facet_of_def)
lemma%unimportant facet_of_irrefl [simp]: "\<not> S facet_of S "
by (simp add: facet_of_def)
lemma%unimportant facet_of_imp_face_of: "F facet_of S \<Longrightarrow> F face_of S"
by (simp add: facet_of_def)
lemma%unimportant facet_of_imp_subset: "F facet_of S \<Longrightarrow> F \<subseteq> S"
by (simp add: face_of_imp_subset facet_of_def)
lemma%unimportant 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%unimportant 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%important 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%unimportant
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%unimportant 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%important \<open>Edges: faces of affine dimension 1\<close>
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%unimportant edge_of_imp_subset:
"S edge_of T \<Longrightarrow> S \<subseteq> T"
by (simp add: edge_of_def face_of_imp_subset)
subsection%important\<open>Existence of extreme points\<close>
lemma%important 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%unimportant -
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%important 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%unimportant -
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%important\<open>Krein-Milman, the weaker form\<close>
proposition%important 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%unimportant (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%important 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%unimportant (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%important 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%unimportant
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)
have 2: "convex ((+) (- a) ` S)"
by (simp add: \<open>convex S\<close> convex_translation)
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 translation_assoc)
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%important\<open>Applying it to convex hulls of explicitly indicated finite sets\<close>
lemma%important 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%unimportant (simp add: Krein_Milman_Minkowski finite_imp_compact_convex_hull)
lemma%unimportant 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%unimportant 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%important 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%unimportant -
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%unimportant 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%unimportant 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%unimportant 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%unimportant 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%important 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%unimportant -
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%important 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%unimportant 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%unimportant 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%important 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%unimportant
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%unimportant 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%unimportant 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%unimportant 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
lemma%important Krein_Milman_frontier:
fixes S :: "'a::euclidean_space set"
assumes "convex S" "compact S"
shows "S = convex hull (frontier S)"
(is "?lhs = ?rhs")
proof%unimportant
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%important\<open>Polytopes\<close>
definition%important polytope where
"polytope S \<equiv> \<exists>v. finite v \<and> S = convex hull v"
lemma%unimportant 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%unimportant 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%unimportant polytope_empty: "polytope {}"
using convex_hull_empty polytope_def by blast
lemma%unimportant polytope_convex_hull: "finite S \<Longrightarrow> polytope(convex hull S)"
using polytope_def by auto
lemma%unimportant 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%unimportant 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%unimportant 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%unimportant finite_polytope_facets:
assumes "polytope S"
shows "finite {T. T facet_of S}"
by (simp add: assms facet_of_def finite_polytope_faces)
lemma%unimportant polytope_scaling:
assumes "polytope S" shows "polytope (image (\<lambda>x. c *\<^sub>R x) S)"
by (simp add: assms polytope_linear_image)
lemma%unimportant 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%unimportant polytope_imp_convex: "polytope S \<Longrightarrow> convex S"
by (metis convex_convex_hull polytope_def)
lemma%unimportant 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%unimportant 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%unimportant polytope_interval: "polytope(cbox a b)"
unfolding polytope_def by (meson closed_interval_as_convex_hull)
lemma%unimportant polytope_sing: "polytope {a}"
using polytope_def by force
lemma%unimportant 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)
lemma%important 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%unimportant -
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%important\<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%unimportant 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%unimportant polyhedron_UNIV [iff]: "polyhedron UNIV"
unfolding polyhedron_def
by (rule_tac x="{}" in exI) auto
lemma%unimportant 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%unimportant 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%unimportant 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%unimportant 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%important polyhedron_hyperplane:
fixes a :: "'a :: euclidean_space"
shows "polyhedron {x. a \<bullet> x = b}"
proof%unimportant -
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%unimportant 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%unimportant 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%unimportant 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%unimportant polyhedron_affine_hull:
fixes S :: "'a :: euclidean_space set"
shows "polyhedron(affine hull S)"
by (simp add: affine_imp_polyhedron)
subsection%important\<open>Canonical polyhedron representation making facial structure explicit\<close>
lemma%important 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%unimportant
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%important 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%unimportant -
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%important 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%unimportant
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%important 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%unimportant
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%unimportant 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%important 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%unimportant (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
define inff where "inff =
(INF j\<in>F - {h}.
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)"
have "0 < inff"
apply (simp add: inff_def)
apply (rule finite_imp_less_Inf)
using \<open>finite F\<close> apply blast
using h' apply blast
apply simp
using awlt apply (force simp: divide_simps)
done
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%important 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%unimportant -
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%important 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%unimportant -
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%important 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%unimportant -
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%important 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%unimportant -
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%unimportant 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%unimportant 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%unimportant 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%unimportant 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%unimportant 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%unimportant 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%unimportant 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%important 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%unimportant -
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%unimportant 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%unimportant 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%unimportant 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%important 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%unimportant
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%important 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%unimportant
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%unimportant 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%unimportant 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%important\<open>Relation between polytopes and polyhedra\<close>
lemma%important polytope_eq_bounded_polyhedron:
fixes S :: "'a :: euclidean_space set"
shows "polytope S \<longleftrightarrow> polyhedron S \<and> bounded S"
(is "?lhs = ?rhs")
proof%unimportant
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%unimportant 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%important polytope_Int_polyhedron:
fixes S :: "'a :: euclidean_space set"
shows "\<lbrakk>polytope S; polyhedron T\<rbrakk> \<Longrightarrow> polytope(S \<inter> T)"
by%unimportant (simp add: bounded_Int polytope_eq_bounded_polyhedron)
lemma%important polyhedron_Int_polytope:
fixes S :: "'a :: euclidean_space set"
shows "\<lbrakk>polyhedron S; polytope T\<rbrakk> \<Longrightarrow> polytope(S \<inter> T)"
by%unimportant (simp add: bounded_Int polytope_eq_bounded_polyhedron)
lemma%important polytope_imp_polyhedron:
fixes S :: "'a :: euclidean_space set"
shows "polytope S \<Longrightarrow> polyhedron S"
by%unimportant (simp add: polytope_eq_bounded_polyhedron)
lemma%unimportant 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%unimportant polyhedron_interval [iff]: "polyhedron(cbox a b)"
by (metis polytope_imp_polyhedron polytope_interval)
lemma%unimportant 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%important\<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 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%important \<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%important\<open>Refining a cell complex to a simplicial complex\<close>
lemma%important 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%unimportant
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%important 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%unimportant (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%important 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%unimportant (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%important 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%unimportant (blast intro: simplicial_subdivision_of_cell_complex_lowdim [OF assms aff_dim_le_DIM])
corollary%important 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%unimportant -
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%important\<open>Some results on cell division with full-dimensional cells only\<close>
lemma%important 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%unimportant -
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%important 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%unimportant -
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