# Theory Polytope

theory Polytope
imports Cartesian_Euclidean_Space
```section ‹Faces, Extreme Points, Polytopes, Polyhedra etc›

text‹Ported from HOL Light by L C Paulson›

theory Polytope
imports Cartesian_Euclidean_Space
begin

subsection ‹Faces of a (usually convex) set›

definition face_of :: "['a::real_vector set, 'a set] ⇒ bool" (infixr "(face'_of)" 50)
where
"T face_of S ⟷
T ⊆ S ∧ convex T ∧
(∀a ∈ S. ∀b ∈ S. ∀x ∈ T. x ∈ open_segment a b ⟶ a ∈ T ∧ b ∈ T)"

lemma face_ofD: "⟦T face_of S; x ∈ open_segment a b; a ∈ S; b ∈ S; x ∈ T⟧ ⟹ a ∈ T ∧ b ∈ T"
unfolding face_of_def by blast

lemma face_of_translation_eq [simp]:
"((+) a ` T face_of (+) a ` S) ⟷ T face_of S"
proof -
have *: "⋀a T S. T face_of S ⟹ ((+) a ` T face_of (+) a ` S)"
apply (simp add: face_of_def Ball_def, clarify)
apply (drule open_segment_translation_eq [THEN iffD1])
done
show ?thesis
apply (rule iffI)
apply (force simp: image_comp o_def dest: * [where a = "-a"])
apply (blast intro: *)
done
qed

lemma face_of_linear_image:
assumes "linear f" "inj f"
shows "(f ` c face_of f ` S) ⟷ c face_of S"
by (simp add: face_of_def inj_image_subset_iff inj_image_mem_iff open_segment_linear_image assms)

lemma face_of_refl: "convex S ⟹ S face_of S"
by (auto simp: face_of_def)

lemma face_of_refl_eq: "S face_of S ⟷ convex S"
by (auto simp: face_of_def)

lemma empty_face_of [iff]: "{} face_of S"

lemma face_of_empty [simp]: "S face_of {} ⟷ S = {}"
by (meson empty_face_of face_of_def subset_empty)

lemma face_of_trans [trans]: "⟦S face_of T; T face_of u⟧ ⟹ S face_of u"
unfolding face_of_def by (safe; blast)

lemma face_of_face: "T face_of S ⟹ (f face_of T ⟷ f face_of S ∧ f ⊆ T)"
unfolding face_of_def by (safe; blast)

lemma face_of_subset: "⟦F face_of S; F ⊆ T; T ⊆ S⟧ ⟹ F face_of T"
unfolding face_of_def by (safe; blast)

lemma face_of_slice: "⟦F face_of S; convex T⟧ ⟹ (F ∩ T) face_of (S ∩ T)"
unfolding face_of_def by (blast intro: convex_Int)

lemma face_of_Int: "⟦t1 face_of S; t2 face_of S⟧ ⟹ (t1 ∩ t2) face_of S"
unfolding face_of_def by (blast intro: convex_Int)

lemma face_of_Inter: "⟦A ≠ {}; ⋀T. T ∈ A ⟹ T face_of S⟧ ⟹ (⋂ A) face_of S"
unfolding face_of_def by (blast intro: convex_Inter)

lemma face_of_Int_Int: "⟦F face_of T; F' face_of t'⟧ ⟹ (F ∩ F') face_of (T ∩ t')"
unfolding face_of_def by (blast intro: convex_Int)

lemma face_of_imp_subset: "T face_of S ⟹ T ⊆ S"
unfolding face_of_def by blast

lemma 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) ∩ S"
proof -
have "convex T" using T by (simp add: face_of_def)
have *: False if x: "x ∈ affine hull T" and "x ∈ S" "x ∉ T" and y: "y ∈ rel_interior T" for x y
proof -
obtain e where "e>0" and e: "cball y e ∩ affine hull T ⊆ T"
using y by (auto simp: rel_interior_cball)
have "y ≠ x" "y ∈ S" "y ∈ T"
using face_of_imp_subset rel_interior_subset T that by blast+
then have zne: "⋀u. ⟦u ∈ {0<..<1}; (1 - u) *⇩R y + u *⇩R x ∈ T⟧ ⟹  False"
using ‹x ∈ S› ‹x ∉ T› ‹T face_of S› 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)) ∈ {0<..<1}"
using ‹y ≠ x› ‹e > 0› by simp
show ?thesis
apply (rule zne [OF in01])
apply (rule e [THEN subsetD])
apply (rule IntI)
using ‹y ≠ x› ‹e > 0›
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 ‹y ∈ T›  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] ‹convex T› intro: *)
done
qed

lemma face_of_imp_closed:
fixes S :: "'a::euclidean_space set"
assumes "convex S" "closed S" "T face_of S" shows "closed T"
by (metis affine_affine_hull affine_closed closed_Int face_of_imp_eq_affine_Int assms)

lemma face_of_Int_supporting_hyperplane_le_strong:
assumes "convex(S ∩ {x. a ∙ x = b})" and aleb: "⋀x. x ∈ S ⟹ a ∙ x ≤ b"
shows "(S ∩ {x. a ∙ x = b}) face_of S"
proof -
have *: "a ∙ u = a ∙ x" if "x ∈ open_segment u v" "u ∈ S" "v ∈ S" and b: "b = a ∙ x"
for u v x
proof (rule antisym)
show "a ∙ u ≤ a ∙ x"
using aleb ‹u ∈ S› ‹b = a ∙ x› by blast
next
obtain ξ where "b = a ∙ ((1 - ξ) *⇩R u + ξ *⇩R v)" "0 < ξ" "ξ < 1"
using ‹b = a ∙ x› ‹x ∈ open_segment u v› in_segment
by (auto simp: open_segment_image_interval split: if_split_asm)
then have "b + ξ * (a ∙ u) ≤ a ∙ u + ξ * b"
using aleb [OF ‹v ∈ S›] by (simp add: algebra_simps)
then have "(1 - ξ) * b ≤ (1 - ξ) * (a ∙ u)"
then have "b ≤ a ∙ u"
using ‹ξ < 1› by auto
with b show "a ∙ x ≤ a ∙ u" by simp
qed
show ?thesis
using "*" open_segment_commute by blast
qed

lemma face_of_Int_supporting_hyperplane_ge_strong:
"⟦convex(S ∩ {x. a ∙ x = b}); ⋀x. x ∈ S ⟹ a ∙ x ≥ b⟧
⟹ (S ∩ {x. a ∙ x = b}) face_of S"
using face_of_Int_supporting_hyperplane_le_strong [of S "-a" "-b"] by simp

lemma face_of_Int_supporting_hyperplane_le:
"⟦convex S; ⋀x. x ∈ S ⟹ a ∙ x ≤ b⟧ ⟹ (S ∩ {x. a ∙ x = b}) face_of S"
by (simp add: convex_Int convex_hyperplane face_of_Int_supporting_hyperplane_le_strong)

lemma face_of_Int_supporting_hyperplane_ge:
"⟦convex S; ⋀x. x ∈ S ⟹ a ∙ x ≥ b⟧ ⟹ (S ∩ {x. a ∙ x = b}) face_of S"
by (simp add: convex_Int convex_hyperplane face_of_Int_supporting_hyperplane_ge_strong)

lemma face_of_imp_convex: "T face_of S ⟹ convex T"
using face_of_def by blast

lemma face_of_imp_compact:
fixes S :: "'a::euclidean_space set"
shows "⟦convex S; compact S; T face_of S⟧ ⟹ compact T"
by (meson bounded_subset compact_eq_bounded_closed face_of_imp_closed face_of_imp_subset)

lemma face_of_Int_subface:
"⟦A ∩ B face_of A; A ∩ B face_of B; C face_of A; D face_of B⟧
⟹ (C ∩ D) face_of C ∧ (C ∩ D) face_of D"
by (meson face_of_Int_Int face_of_face inf_le1 inf_le2)

lemma subset_of_face_of:
fixes S :: "'a::real_normed_vector set"
assumes "T face_of S" "u ⊆ S" "T ∩ (rel_interior u) ≠ {}"
shows "u ⊆ T"
proof
fix c
assume "c ∈ u"
obtain b where "b ∈ T" "b ∈ rel_interior u" using assms by auto
then obtain e where "e>0" "b ∈ u" and e: "cball b e ∩ affine hull u ⊆ u"
by (auto simp: rel_interior_cball)
show "c ∈ T"
proof (cases "b=c")
case True with ‹b ∈ T› show ?thesis by blast
next
case False
define d where "d = b + (e / norm(b - c)) *⇩R (b - c)"
have "d ∈ cball b e ∩ affine hull u"
using ‹e > 0› ‹b ∈ u› ‹c ∈ u›
by (simp add: d_def dist_norm hull_inc mem_affine_3_minus False)
with e have "d ∈ u" by blast
have nbc: "norm (b - c) + e > 0" using ‹e > 0›
then have [simp]: "d ≠ c" using False scaleR_cancel_left [of "1 + (e / norm (b - c))" b c]
have [simp]: "((e - e * e / (e + norm (b - c))) / norm (b - c)) = (e / (e + norm (b - c)))"
using False nbc
have "b ∈ open_segment d c"
apply (simp add: d_def algebra_simps image_def)
apply (rule_tac x="e / (e + norm (b - c))" in bexI)
using False nbc ‹0 < e›
apply (auto simp: algebra_simps)
done
then have "d ∈ T ∧ c ∈ T"
apply (rule face_ofD [OF ‹T face_of S›])
using ‹d ∈ u›  ‹c ∈ u› ‹u ⊆ S›  ‹b ∈ T›  apply auto
done
then show ?thesis ..
qed
qed

lemma face_of_eq:
fixes S :: "'a::real_normed_vector set"
assumes "T face_of S" "u face_of S" "(rel_interior T) ∩ (rel_interior u) ≠ {}"
shows "T = u"
apply (rule subset_antisym)
apply (metis assms disjoint_iff_not_equal face_of_imp_subset rel_interior_subset subsetCE subset_of_face_of)
by (metis assms disjoint_iff_not_equal face_of_imp_subset rel_interior_subset subset_iff subset_of_face_of)

lemma face_of_disjoint_rel_interior:
fixes S :: "'a::real_normed_vector set"
assumes "T face_of S" "T ≠ S"
shows "T ∩ rel_interior S = {}"
by (meson assms subset_of_face_of face_of_imp_subset order_refl subset_antisym)

lemma face_of_disjoint_interior:
fixes S :: "'a::real_normed_vector set"
assumes "T face_of S" "T ≠ S"
shows "T ∩ interior S = {}"
proof -
have "T ∩ interior S ⊆ rel_interior S"
by (meson inf_sup_ord(2) interior_subset_rel_interior order.trans)
thus ?thesis
by (metis (no_types) Int_greatest assms face_of_disjoint_rel_interior inf_sup_ord(1) subset_empty)
qed

lemma face_of_subset_rel_boundary:
fixes S :: "'a::real_normed_vector set"
assumes "T face_of S" "T ≠ S"
shows "T ⊆ (S - rel_interior S)"
by (meson DiffI assms disjoint_iff_not_equal face_of_disjoint_rel_interior face_of_imp_subset rev_subsetD subsetI)

lemma face_of_subset_rel_frontier:
fixes S :: "'a::real_normed_vector set"
assumes "T face_of S" "T ≠ S"
shows "T ⊆ rel_frontier S"
using assms closure_subset face_of_disjoint_rel_interior face_of_imp_subset rel_frontier_def by fastforce

lemma face_of_aff_dim_lt:
fixes S :: "'a::euclidean_space set"
assumes "convex S" "T face_of S" "T ≠ S"
shows "aff_dim T < aff_dim S"
proof -
have "aff_dim T ≤ aff_dim S"
by (simp add: face_of_imp_subset aff_dim_subset assms)
moreover have "aff_dim T ≠ aff_dim S"
proof (cases "T = {}")
case True then show ?thesis
by (metis aff_dim_empty ‹T ≠ S›)
next case False then show ?thesis
by (metis Set.set_insert assms convex_rel_frontier_aff_dim dual_order.irrefl face_of_imp_convex face_of_subset_rel_frontier insert_not_empty subsetI)
qed
ultimately show ?thesis
by simp
qed

lemma subset_of_face_of_affine_hull:
fixes S :: "'a::euclidean_space set"
assumes T: "T face_of S" and "convex S" "U ⊆ S" and dis: "~disjnt (affine hull T) (rel_interior U)"
shows "U ⊆ T"
apply (rule subset_of_face_of [OF T ‹U ⊆ S›])
using face_of_imp_eq_affine_Int [OF ‹convex S› T]
using rel_interior_subset [of U] dis
using ‹U ⊆ S› disjnt_def by fastforce

lemma affine_hull_face_of_disjoint_rel_interior:
fixes S :: "'a::euclidean_space set"
assumes "convex S" "F face_of S" "F ≠ S"
shows "affine hull F ∩ rel_interior S = {}"
by (metis assms disjnt_def face_of_imp_subset order_refl subset_antisym subset_of_face_of_affine_hull)

lemma affine_diff_divide:
assumes "affine S" "k ≠ 0" "k ≠ 1" and xy: "x ∈ S" "y /⇩R (1 - k) ∈ S"
shows "(x - y) /⇩R k ∈ S"
proof -
have "inverse(k) *⇩R (x - y) = (1 - inverse k) *⇩R inverse(1 - k) *⇩R y + inverse(k) *⇩R x"
using assms
then show ?thesis
using ‹affine S› xy by (auto simp: affine_alt)
qed

lemma face_of_convex_hulls:
assumes S: "finite S" "T ⊆ S" and disj: "affine hull T ∩ convex hull (S - T) = {}"
shows  "(convex hull T) face_of (convex hull S)"
proof -
have fin: "finite T" "finite (S - T)" using assms
by (auto simp: finite_subset)
have *: "x ∈ convex hull T"
if x: "x ∈ convex hull S" and y: "y ∈ convex hull S" and w: "w ∈ convex hull T" "w ∈ open_segment x y"
for x y w
proof -
have waff: "w ∈ affine hull T"
using convex_hull_subset_affine_hull w by blast
obtain a b where a: "⋀i. i ∈ S ⟹ 0 ≤ a i" and asum: "sum a S = 1" and aeqx: "(∑i∈S. a i *⇩R i) = x"
and b: "⋀i. i ∈ S ⟹ 0 ≤ b i" and bsum: "sum b S = 1" and beqy: "(∑i∈S. b i *⇩R i) = y"
using x y by (auto simp: assms convex_hull_finite)
obtain u where "(1 - u) *⇩R x + u *⇩R y ∈ convex hull T" "x ≠ y" and weq: "w = (1 - u) *⇩R x + u *⇩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: "⋀i. i ∈ S ⟹ 0 ≤ 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: "(∑i∈S. c i *⇩R i) = (1 - u) *⇩R x + u *⇩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: "⋀i. i ∈ (S - T) ⟹ c i = 0"
using True cge0 fin(2) sum_nonneg_eq_0_iff by auto
have a0: "a i = 0" if "i ∈ (S - T)" for i
using ci0 [OF that] u01 a [of i] b [of i] that
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 ‹finite T›)
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 (λx. c x *⇩R x) T + sum (λx. c x *⇩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: "⋀i. i ∈ T ⟹ c i = 0"
by (meson ‹finite T› ‹sum c T = 0› ‹T ⊆ S› cge0 sum_nonneg_eq_0_iff subsetCE)
then have [simp]: "(∑i∈S-T. c i *⇩R i) = w"
have "w ∈ convex hull (S - T)"
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 "(∑i∈T. c i *⇩R i) /⇩R (1 - k) ∈ convex hull T"
apply (rule_tac x="λi. 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 ‹0 < k›  have "inverse(k) *⇩R (w - sum (λi. c i *⇩R i) T) ∈ 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) *⇩R (w - sum (λx. c x *⇩R x) T) ∈ convex hull (S - T)"
apply (simp add: weq_sumsum convex_hull_finite fin)
apply (rule_tac x="λi. inverse k * c i" in exI)
using ‹k > 0› 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 ⊆ convex hull S"
by (simp add: ‹T ⊆ S› hull_mono)
show ?thesis
using open_segment_commute by (auto simp: face_of_def intro: *)
qed

proposition face_of_convex_hull_insert:
"⟦finite S; a ∉ affine hull S; T face_of convex hull S⟧ ⟹ T face_of convex hull insert a S"
apply (rule face_of_trans, blast)
apply (rule face_of_convex_hulls; force simp: insert_Diff_if)
done

proposition face_of_affine_trivial:
assumes "affine S" "T face_of S"
shows "T = {} ∨ T = S"
proof (rule ccontr, clarsimp)
assume "T ≠ {}" "T ≠ S"
then obtain a where "a ∈ T" by auto
then have "a ∈ S"
using ‹T face_of S› face_of_imp_subset by blast
have "S ⊆ T"
proof
fix b  assume "b ∈ S"
show "b ∈ T"
proof (cases "a = b")
case True with ‹a ∈ T› show ?thesis by auto
next
case False
then have "a ∈ open_segment (2 *⇩R a - b) b"
apply (auto simp: open_segment_def closed_segment_def)
apply (rule_tac x="1/2" in exI)
moreover have "2 *⇩R a - b ∈ S"
by (rule mem_affine [OF ‹affine S› ‹a ∈ S› ‹b ∈ S›, of 2 "-1", simplified])
moreover note ‹b ∈ S› ‹a ∈ T›
ultimately show ?thesis
by (rule face_ofD [OF ‹T face_of S›, THEN conjunct2])
qed
qed
then show False
using ‹T ≠ S› ‹T face_of S› face_of_imp_subset by blast
qed

lemma face_of_affine_eq:
"affine S ⟹ (T face_of S ⟷ T = {} ∨ T = S)"
using affine_imp_convex face_of_affine_trivial face_of_refl by auto

lemma Inter_faces_finite_altbound:
fixes T :: "'a::euclidean_space set set"
assumes cfaI: "⋀c. c ∈ T ⟹ c face_of S"
shows "∃F'. finite F' ∧ F' ⊆ T ∧ card F' ≤ DIM('a) + 2 ∧ ⋂F' = ⋂T"
proof (cases "∀F'. finite F' ∧ F' ⊆ T ∧ card F' ≤ DIM('a) + 2 ⟶ (∃c. c ∈ T ∧ c ∩ (⋂F') ⊂ (⋂F'))")
case True
then obtain c where c:
"⋀F'. ⟦finite F'; F' ⊆ T; card F' ≤ DIM('a) + 2⟧ ⟹ c F' ∈ T ∧ c F' ∩ (⋂F') ⊂ (⋂F')"
by metis
define d where "d = rec_nat {c{}} (λn r. insert (c r) r)"
have [simp]: "d 0 = {c {}}"
have dSuc [simp]: "⋀n. d (Suc n) = insert (c (d n)) (d n)"
have dn_notempty: "d n ≠ {}" for n
by (induction n) auto
have dn_le_Suc: "d n ⊆ T ∧ finite(d n) ∧ card(d n) ≤ Suc n" if "n ≤ 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(⋂(d n)) ≤ DIM('a) - int n" if "n ≤ DIM('a) + 2" for n
using that
proof (induction n)
case 0
then show ?case
next
case (Suc n)
have fs: "⋂d (Suc n) face_of S"
by (meson Suc.prems cfaI dn_le_Suc dn_notempty face_of_Inter subsetCE)
have condn: "convex (⋂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: "⋂d (Suc n) face_of ⋂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: "⋂d (Suc n) ≠ ⋂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 *: "⋀m::int. ⋀d. ⋀d'::int. d < d' ∧ d' ≤ m - n ⟹ d ≤ m - of_nat(n+1)"
by arith
have "aff_dim (⋂d (Suc n)) < aff_dim (⋂d n)"
by (rule face_of_aff_dim_lt [OF condn fdn ne])
moreover have "aff_dim (⋂d n) ≤ int (DIM('a)) - int n"
using Suc by auto
ultimately
have "aff_dim (⋂d (Suc n)) ≤ int (DIM('a)) - (n+1)" by arith
then show ?case by linarith
qed
have "aff_dim (⋂d (DIM('a) + 2)) ≤ -2"
using aff_dim_le [OF order_refl] by simp
with aff_dim_geq [of "⋂d (DIM('a) + 2)"] show ?thesis
using order.trans by fastforce
next
case False
then show ?thesis
apply simp
apply (erule ex_forward)
by blast
qed

lemma faces_of_translation:
"{F. F face_of image (λx. a + x) S} = image (image (λx. a + x)) {F. F face_of S}"
apply (rule subset_antisym, clarify)
apply (auto simp: image_iff)
apply (metis face_of_imp_subset face_of_translation_eq subset_imageE)
done

proposition face_of_Times:
assumes "F face_of S" and "F' face_of S'"
shows "(F × F') face_of (S × S')"
proof -
have "F × F' ⊆ S × S'"
using assms [unfolded face_of_def] by blast
moreover
have "convex (F × F')"
using assms [unfolded face_of_def] by (blast intro: convex_Times)
moreover
have "a ∈ F ∧ a' ∈ F' ∧ b ∈ F ∧ b' ∈ F'"
if "a ∈ S" "b ∈ S" "a' ∈ S'" "b' ∈ S'" "x ∈ F × F'" "x ∈ open_segment (a,a') (b,b')"
for a b a' b' x
proof (cases "b=a ∨ b'=a'")
case True with that show ?thesis
using assms
by (force simp: in_segment dest: face_ofD)
next
case False with assms [unfolded face_of_def] that show ?thesis
by (blast dest!: open_segment_PairD)
qed
ultimately show ?thesis
unfolding face_of_def by blast
qed

corollary face_of_Times_decomp:
fixes S :: "'a::euclidean_space set" and S' :: "'b::euclidean_space set"
shows "c face_of (S × S') ⟷ (∃F F'. F face_of S ∧ F' face_of S' ∧ c = F × F')"
(is "?lhs = ?rhs")
proof
assume c: ?lhs
show ?rhs
proof (cases "c = {}")
case True then show ?thesis by auto
next
case False
have 1: "fst ` c ⊆ S" "snd ` c ⊆ 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: ‹convex c› convex_linear_image fst_linear snd_linear)
have fstab: "a ∈ fst ` c ∧ b ∈ fst ` c"
if "a ∈ S" "b ∈ S" "x ∈ open_segment a b" "(x,x') ∈ c" for a b x x'
proof -
have *: "(x,x') ∈ 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' ∈ snd ` c ∧ b' ∈ snd ` c"
if "a' ∈ S'" "b' ∈ S'" "x' ∈ open_segment a' b'" "(x,x') ∈ c" for a' b' x x'
proof -
have *: "(x,x') ∈ 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 ⊆ rel_interior (fst ` c) × rel_interior (snd ` c)"
by (force simp: face_of_Times rel_interior_Times conv fst snd ‹convex c› fst_linear snd_linear rel_interior_convex_linear_image [symmetric])
have "c = fst ` c × 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 ‹convex c› cc
apply blast
done
with fst snd show ?thesis by metis
qed
next
assume ?rhs with face_of_Times show ?lhs by auto
qed

lemma face_of_Times_eq:
fixes S :: "'a::euclidean_space set" and S' :: "'b::euclidean_space set"
shows "(F × F') face_of (S × S') ⟷
F = {} ∨ F' = {} ∨ F face_of S ∧ F' face_of S'"
by (auto simp: face_of_Times_decomp times_eq_iff)

lemma hyperplane_face_of_halfspace_le: "{x. a ∙ x = b} face_of {x. a ∙ x ≤ b}"
proof -
have "{x. a ∙ x ≤ b} ∩ {x. a ∙ x = b} = {x. a ∙ x = b}"
by auto
with face_of_Int_supporting_hyperplane_le [OF convex_halfspace_le [of a b], of a b]
show ?thesis by auto
qed

lemma hyperplane_face_of_halfspace_ge: "{x. a ∙ x = b} face_of {x. a ∙ x ≥ b}"
proof -
have "{x. a ∙ x ≥ b} ∩ {x. a ∙ x = b} = {x. a ∙ x = b}"
by auto
with face_of_Int_supporting_hyperplane_ge [OF convex_halfspace_ge [of b a], of b a]
show ?thesis by auto
qed

lemma face_of_halfspace_le:
fixes a :: "'n::euclidean_space"
shows "F face_of {x. a ∙ x ≤ b} ⟷
F = {} ∨ F = {x. a ∙ x = b} ∨ F = {x. a ∙ x ≤ b}"
(is "?lhs = ?rhs")
proof (cases "a = 0")
case True then show ?thesis
using face_of_affine_eq affine_UNIV by auto
next
case False
then have ine: "interior {x. a ∙ x ≤ b} ≠ {}"
using halfspace_eq_empty_lt interior_halfspace_le by blast
show ?thesis
proof
assume L: ?lhs
have "F ≠ {x. a ∙ x ≤ b} ⟹ F face_of {x. a ∙ x = b}"
using False
apply (simp add: frontier_halfspace_le [symmetric] rel_frontier_nonempty_interior [OF ine, symmetric])
apply (rule face_of_subset [OF L])
apply (simp add: face_of_subset_rel_frontier [OF L])
apply (force simp: rel_frontier_def closed_halfspace_le)
done
with L show ?rhs
using affine_hyperplane face_of_affine_eq by blast
next
assume ?rhs
then show ?lhs
by (metis convex_halfspace_le empty_face_of face_of_refl hyperplane_face_of_halfspace_le)
qed
qed

lemma face_of_halfspace_ge:
fixes a :: "'n::euclidean_space"
shows "F face_of {x. a ∙ x ≥ b} ⟷
F = {} ∨ F = {x. a ∙ x = b} ∨ F = {x. a ∙ x ≥ b}"
using face_of_halfspace_le [of F "-a" "-b"] by simp

subsection‹Exposed faces›

text‹That is, faces that are intersection with supporting hyperplane›

definition exposed_face_of :: "['a::euclidean_space set, 'a set] ⇒ bool"
(infixr "(exposed'_face'_of)" 50)
where "T exposed_face_of S ⟷
T face_of S ∧ (∃a b. S ⊆ {x. a ∙ x ≤ b} ∧ T = S ∩ {x. a ∙ x = b})"

lemma empty_exposed_face_of [iff]: "{} exposed_face_of S"
apply (rule_tac x=0 in exI)
apply (rule_tac x=1 in exI, force)
done

lemma exposed_face_of_refl_eq [simp]: "S exposed_face_of S ⟷ convex S"
apply (simp add: exposed_face_of_def face_of_refl_eq, auto)
apply (rule_tac x=0 in exI)+
apply force
done

lemma exposed_face_of_refl: "convex S ⟹ S exposed_face_of S"
by simp

lemma exposed_face_of:
"T exposed_face_of S ⟷
T face_of S ∧
(T = {} ∨ T = S ∨
(∃a b. a ≠ 0 ∧ S ⊆ {x. a ∙ x ≤ b} ∧ T = S ∩ {x. a ∙ 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
next
case False
with ‹T ≠ {}› show ?thesis
apply (auto simp: exposed_face_of_def)
apply (metis inner_zero_left)
done
qed
qed

lemma exposed_face_of_Int_supporting_hyperplane_le:
"⟦convex S; ⋀x. x ∈ S ⟹ a ∙ x ≤ b⟧ ⟹ (S ∩ {x. a ∙ x = b}) exposed_face_of S"
by (force simp: exposed_face_of_def face_of_Int_supporting_hyperplane_le)

lemma exposed_face_of_Int_supporting_hyperplane_ge:
"⟦convex S; ⋀x. x ∈ S ⟹ a ∙ x ≥ b⟧ ⟹ (S ∩ {x. a ∙ x = b}) exposed_face_of S"
using exposed_face_of_Int_supporting_hyperplane_le [of S "-a" "-b"] by simp

proposition exposed_face_of_Int:
assumes "T exposed_face_of S"
and "u exposed_face_of S"
shows "(T ∩ u) exposed_face_of S"
proof -
obtain a b where T: "S ∩ {x. a ∙ x = b} face_of S"
and S: "S ⊆ {x. a ∙ x ≤ b}"
and teq: "T = S ∩ {x. a ∙ x = b}"
using assms by (auto simp: exposed_face_of_def)
obtain a' b' where u: "S ∩ {x. a' ∙ x = b'} face_of S"
and s': "S ⊆ {x. a' ∙ x ≤ b'}"
and ueq: "u = S ∩ {x. a' ∙ x = b'}"
using assms by (auto simp: exposed_face_of_def)
have tu: "T ∩ u face_of S"
using T teq u ueq by (simp add: face_of_Int)
have ss: "S ⊆ {x. (a + a') ∙ x ≤ b + b'}"
using S s' by (force simp: inner_left_distrib)
show ?thesis
apply (rule_tac x="a+a'" in exI)
apply (rule_tac x="b+b'" in exI)
using S s'
apply (fastforce simp: ss inner_left_distrib teq ueq)
done
qed

proposition exposed_face_of_Inter:
fixes P :: "'a::euclidean_space set set"
assumes "P ≠ {}"
and "⋀T. T ∈ P ⟹ T exposed_face_of S"
shows "⋂P exposed_face_of S"
proof -
obtain Q where "finite Q" and QsubP: "Q ⊆ P" "card Q ≤ DIM('a) + 2" and IntQ: "⋂Q = ⋂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 ⊆ {T. T exposed_face_of S}"
using QsubP assms by blast
moreover have "Q ⊆ {T. T exposed_face_of S} ⟹ ⋂Q exposed_face_of S"
using ‹finite Q› False
apply (induction Q rule: finite_induct)
using exposed_face_of_Int apply fastforce+
done
ultimately show ?thesis
qed
qed

proposition exposed_face_of_sums:
assumes "convex S" and "convex T"
and "F exposed_face_of {x + y | x y. x ∈ S ∧ y ∈ 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 ∈ k ∧ y ∈ l}"
proof (cases "F = {}")
case True then show ?thesis
using that by blast
next
case False
show ?thesis
proof (cases "F = ?ST")
case True then show ?thesis
using assms exposed_face_of_refl_eq that by blast
next
case False
obtain p where "p ∈ F" using ‹F ≠ {}› by blast
moreover
obtain u z where T: "?ST ∩ {x. u ∙ x = z} face_of ?ST"
and S: "?ST ⊆ {x. u ∙ x ≤ z}"
and feq: "F = ?ST ∩ {x. u ∙ x = z}"
using assms by (auto simp: exposed_face_of_def)
ultimately obtain a0 b0
where p: "p = a0 + b0" and "a0 ∈ S" "b0 ∈ T" and z: "u ∙ p = z"
by auto
have lez: "u ∙ (x + y) ≤ z" if "x ∈ S" "y ∈ T" for x y
using S that by auto
have sef: "S ∩ {x. u ∙ x = u ∙ a0} exposed_face_of S"
apply (rule exposed_face_of_Int_supporting_hyperplane_le [OF ‹convex S›])
apply (metis p z add_le_cancel_right inner_right_distrib lez [OF _ ‹b0 ∈ T›])
done
have tef: "T ∩ {x. u ∙ x = u ∙ b0} exposed_face_of T"
apply (rule exposed_face_of_Int_supporting_hyperplane_le [OF ‹convex T›])
apply (metis p z add.commute add_le_cancel_right inner_right_distrib lez [OF ‹a0 ∈ S›])
done
have "{x + y |x y. x ∈ S ∧ u ∙ x = u ∙ a0 ∧ y ∈ T ∧ u ∙ y = u ∙ b0} ⊆ F"
by (auto simp: feq) (metis inner_right_distrib p z)
moreover have "F ⊆ {x + y |x y. x ∈ S ∧ u ∙ x = u ∙ a0 ∧ y ∈ T ∧ u ∙ y = u ∙ 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 ‹a0 ∈ S› ‹b0 ∈ T›
apply clarify
done
ultimately have "F = {x + y |x y. x ∈ S ∩ {x. u ∙ x = u ∙ a0} ∧ y ∈ T ∩ {x. u ∙ x = u ∙ b0}}"
by blast
then show ?thesis
by (rule that [OF sef tef])
qed
qed

lemma exposed_face_of_parallel:
"T exposed_face_of S ⟷
T face_of S ∧
(∃a b. S ⊆ {x. a ∙ x ≤ b} ∧ T = S ∩ {x. a ∙ x = b} ∧
(T ≠ {} ⟶ T ≠ S ⟶ a ≠ 0) ∧
(T ≠ S ⟶ (∀w ∈ affine hull S. (w + a) ∈ affine hull S)))"
(is "?lhs = ?rhs")
proof
assume ?lhs then show ?rhs
proof (clarsimp simp: exposed_face_of_def)
fix a b
assume faceS: "S ∩ {x. a ∙ x = b} face_of S" and Ssub: "S ⊆ {x. a ∙ x ≤ b}"
show "∃c d. S ⊆ {x. c ∙ x ≤ d} ∧
S ∩ {x. a ∙ x = b} = S ∩ {x. c ∙ x = d} ∧
(S ∩ {x. a ∙ x = b} ≠ {} ⟶ S ∩ {x. a ∙ x = b} ≠ S ⟶ c ≠ 0) ∧
(S ∩ {x. a ∙ x = b} ≠ S ⟶ (∀w ∈ affine hull S. w + c ∈ affine hull S))"
proof (cases "affine hull S ∩ {x. -a ∙ x ≤ -b} = {} ∨ affine hull S ⊆ {x. - a ∙ x ≤ - b}")
case True
then show ?thesis
proof
assume "affine hull S ∩ {x. - a ∙ x ≤ - 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 ⊆ {x. - a ∙ x ≤ - 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' ≠ 0"
and le: "affine hull S ∩ {x. a' ∙ x ≤ b'} = affine hull S ∩ {x. - a ∙ x ≤ - b}"
and eq: "affine hull S ∩ {x. a' ∙ x = b'} = affine hull S ∩ {x. - a ∙ x = - b}"
and mem: "⋀w. w ∈ affine hull S ⟹ w + a' ∈ affine hull S"
using affine_parallel_slice affine_affine_hull by metis
show ?thesis
proof (intro conjI impI allI ballI exI)
have *: "S ⊆ - (affine hull S ∩ {x. P x}) ∪ affine hull S ∩ {x. Q x} ⟹ S ⊆ {x. ~P x ∨ Q x}"
for P Q
using hull_subset by fastforce
have "S ⊆ {x. ~ (a' ∙ x ≤ b') ∨ a' ∙ x = b'}"
apply (rule *)
apply (simp only: le eq)
using Ssub by auto
then show "S ⊆ {x. - a' ∙ x ≤ - b'}"
by auto
show "S ∩ {x. a ∙ x = b} = S ∩ {x. - a' ∙ x = - b'}"
using eq hull_subset [of S affine] by force
show "⟦S ∩ {x. a ∙ x = b} ≠ {}; S ∩ {x. a ∙ x = b} ≠ S⟧ ⟹ - a' ≠ 0"
using ‹a' ≠ 0› by auto
show "w + - a' ∈ affine hull S"
if "S ∩ {x. a ∙ x = b} ≠ S" "w ∈ affine hull S" for w
proof -
have "w + 1 *⇩R (w - (w + a')) ∈ 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‹Extreme points of a set: its singleton faces›

definition extreme_point_of :: "['a::real_vector, 'a set] ⇒ bool"
(infixr "(extreme'_point'_of)" 50)
where "x extreme_point_of S ⟷
x ∈ S ∧ (∀a ∈ S. ∀b ∈ S. x ∉ open_segment a b)"

lemma extreme_point_of_stillconvex:
"convex S ⟹ (x extreme_point_of S ⟷ x ∈ S ∧ convex(S - {x}))"
by (fastforce simp add: convex_contains_segment extreme_point_of_def open_segment_def)

lemma face_of_singleton:
"{x} face_of S ⟷ x extreme_point_of S"
by (fastforce simp add: extreme_point_of_def face_of_def)

lemma extreme_point_not_in_REL_INTERIOR:
fixes S :: "'a::real_normed_vector set"
shows "⟦x extreme_point_of S; S ≠ {x}⟧ ⟹ x ∉ rel_interior S"
apply (blast dest: face_of_disjoint_rel_interior)
done

lemma extreme_point_not_in_interior:
fixes S :: "'a::{real_normed_vector, perfect_space} set"
shows "x extreme_point_of S ⟹ x ∉ interior S"
apply (case_tac "S = {x}")
by (meson contra_subsetD extreme_point_not_in_REL_INTERIOR interior_subset_rel_interior)

lemma extreme_point_of_face:
"F face_of S ⟹ v extreme_point_of F ⟷ v extreme_point_of S ∧ v ∈ F"
by (meson empty_subsetI face_of_face face_of_singleton insert_subset)

lemma extreme_point_of_convex_hull:
"x extreme_point_of (convex hull S) ⟹ x ∈ S"
using hull_minimal [of S "(convex hull S) - {x}" convex]
using hull_subset [of S convex]
apply blast
done

lemma extreme_points_of_convex_hull:
"{x. x extreme_point_of (convex hull S)} ⊆ S"
using extreme_point_of_convex_hull by auto

lemma extreme_point_of_empty [simp]: "~ (x extreme_point_of {})"

lemma extreme_point_of_singleton [iff]: "x extreme_point_of {a} ⟷ x = a"
using extreme_point_of_stillconvex by auto

lemma extreme_point_of_translation_eq:
"(a + x) extreme_point_of (image (λx. a + x) S) ⟷ x extreme_point_of S"
by (auto simp: extreme_point_of_def)

lemma extreme_points_of_translation:
"{x. x extreme_point_of (image (λx. a + x) S)} =
(λx. a + x) ` {x. x extreme_point_of S}"
using extreme_point_of_translation_eq
by auto (metis (no_types, lifting) image_iff mem_Collect_eq minus_add_cancel)

lemma extreme_point_of_Int:
"⟦x extreme_point_of S; x extreme_point_of T⟧ ⟹ x extreme_point_of (S ∩ T)"

lemma extreme_point_of_Int_supporting_hyperplane_le:
"⟦S ∩ {x. a ∙ x = b} = {c}; ⋀x. x ∈ S ⟹ a ∙ x ≤ b⟧ ⟹ c extreme_point_of S"
by (metis face_of_Int_supporting_hyperplane_le_strong convex_singleton)

lemma extreme_point_of_Int_supporting_hyperplane_ge:
"⟦S ∩ {x. a ∙ x = b} = {c}; ⋀x. x ∈ S ⟹ a ∙ x ≥ b⟧ ⟹ c extreme_point_of S"
by (metis face_of_Int_supporting_hyperplane_ge_strong convex_singleton)

lemma exposed_point_of_Int_supporting_hyperplane_le:
"⟦S ∩ {x. a ∙ x = b} = {c}; ⋀x. x ∈ S ⟹ a ∙ x ≤ b⟧ ⟹ {c} exposed_face_of S"
apply (force simp: extreme_point_of_Int_supporting_hyperplane_le)
done

lemma exposed_point_of_Int_supporting_hyperplane_ge:
"⟦S ∩ {x. a ∙ x = b} = {c}; ⋀x. x ∈ S ⟹ a ∙ x ≥ b⟧ ⟹ {c} exposed_face_of S"
using exposed_point_of_Int_supporting_hyperplane_le [of S "-a" "-b" c]
by simp

lemma extreme_point_of_convex_hull_insert:
"⟦finite S; a ∉ convex hull S⟧ ⟹ a extreme_point_of (convex hull (insert a S))"
apply (case_tac "a ∈ S")
using face_of_convex_hulls [of "insert a S" "{a}"]
apply (auto simp: face_of_singleton hull_same)
done

subsection‹Facets›

definition facet_of :: "['a::euclidean_space set, 'a set] ⇒ bool"
(infixr "(facet'_of)" 50)
where "F facet_of S ⟷ F face_of S ∧ F ≠ {} ∧ aff_dim F = aff_dim S - 1"

lemma facet_of_empty [simp]: "~ S facet_of {}"

lemma facet_of_irrefl [simp]: "~ S facet_of S "

lemma facet_of_imp_face_of: "F facet_of S ⟹ F face_of S"

lemma facet_of_imp_subset: "F facet_of S ⟹ F ⊆ S"

lemma hyperplane_facet_of_halfspace_le:
"a ≠ 0 ⟹ {x. a ∙ x = b} facet_of {x. a ∙ x ≤ b}"
unfolding facet_of_def hyperplane_eq_empty
by (auto simp: hyperplane_face_of_halfspace_ge hyperplane_face_of_halfspace_le
DIM_positive Suc_leI of_nat_diff aff_dim_halfspace_le)

lemma hyperplane_facet_of_halfspace_ge:
"a ≠ 0 ⟹ {x. a ∙ x = b} facet_of {x. a ∙ x ≥ b}"
unfolding facet_of_def hyperplane_eq_empty
by (auto simp: hyperplane_face_of_halfspace_le hyperplane_face_of_halfspace_ge
DIM_positive Suc_leI of_nat_diff aff_dim_halfspace_ge)

lemma facet_of_halfspace_le:
"F facet_of {x. a ∙ x ≤ b} ⟷ a ≠ 0 ∧ F = {x. a ∙ x = b}"
(is "?lhs = ?rhs")
proof
assume c: ?lhs
with c facet_of_irrefl show ?rhs
by (force simp: aff_dim_halfspace_le facet_of_def face_of_halfspace_le cong: conj_cong split: if_split_asm)
next
assume ?rhs then show ?lhs
qed

lemma facet_of_halfspace_ge:
"F facet_of {x. a ∙ x ≥ b} ⟷ a ≠ 0 ∧ F = {x. a ∙ x = b}"
using facet_of_halfspace_le [of F "-a" "-b"] by simp

subsection ‹Edges: faces of affine dimension 1›

definition edge_of :: "['a::euclidean_space set, 'a set] ⇒ bool"  (infixr "(edge'_of)" 50)
where "e edge_of S ⟷ e face_of S ∧ aff_dim e = 1"

lemma edge_of_imp_subset:
"S edge_of T ⟹ S ⊆ T"

subsection‹Existence of extreme points›

lemma different_norm_3_collinear_points:
fixes a :: "'a::euclidean_space"
assumes "x ∈ open_segment a b" "norm(a) = norm(b)" "norm(x) = norm(b)"
shows False
proof -
obtain u where "norm ((1 - u) *⇩R a + u *⇩R b) = norm b"
and "a ≠ b"
and u01: "0 < u" "u < 1"
using assms by (auto simp: open_segment_image_interval if_splits)
then have "(1 - u) *⇩R a ∙ (1 - u) *⇩R a + ((1 - u) * 2) *⇩R a ∙ u *⇩R b =
(1 - u * u) *⇩R (a ∙ a)"
using assms by (simp add: norm_eq algebra_simps inner_commute)
then have "(1 - u) *⇩R ((1 - u) *⇩R a ∙ a + (2 * u) *⇩R  a ∙ b) =
(1 - u) *⇩R ((1 + u) *⇩R (a ∙ a))"
then have "(1 - u) *⇩R (a ∙ a) + (2 * u) *⇩R (a ∙ b) = (1 + u) *⇩R (a ∙ a)"
using u01 by auto
then have "a ∙ b = a ∙ a"
using u01 by (simp add: algebra_simps)
then have "a = b"
using ‹norm(a) = norm(b)› norm_eq vector_eq by fastforce
then show ?thesis
using ‹a ≠ b› by force
qed

proposition extreme_point_exists_convex:
fixes S :: "'a::euclidean_space set"
assumes "compact S" "convex S" "S ≠ {}"
obtains x where "x extreme_point_of S"
proof -
obtain x where "x ∈ S" and xsup: "⋀y. y ∈ S ⟹ norm y ≤ norm x"
using distance_attains_sup [of S 0] assms by auto
have False if "a ∈ S" "b ∈ S" and x: "x ∈ open_segment a b" for a b
proof -
have noax: "norm a ≤ norm x" and nobx: "norm b ≤ norm x" using xsup that by auto
have "a ≠ b"
using empty_iff open_segment_idem x by auto
have *: "(1 - u) * na + u * nb < norm x" if "na < norm x"  "nb ≤ norm x" "0 < u" "u < 1" for na nb u
proof -
have "(1 - u) * na + u * nb < (1 - u) * norm x + u * nb"
also have "... ≤ (1 - u) * norm x + u * norm x"
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 ‹a ≠ b› if_False)
apply (rule norm_triangle_lt)
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 ‹a ≠ b› if_False)
apply (rule norm_triangle_lt)
using * [of "norm b" "norm a" "1-u" for u] noax that
done
ultimately have "~ (norm a < norm x) ∧ ~ (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 ‹x ∈ S›)
done
qed

subsection‹Krein-Milman, the weaker form›

proposition Krein_Milman:
fixes S :: "'a::euclidean_space set"
assumes "compact S" "convex S"
shows "S = closure(convex hull {x. x extreme_point_of S})"
proof (cases "S = {}")
case True then show ?thesis   by simp
next
case False
have "closed S"
by (simp add: ‹compact S› compact_imp_closed)
have "closure (convex hull {x. x extreme_point_of S}) ⊆ S"
apply (rule closure_minimal [OF hull_minimal ‹closed S›])
using assms
apply (auto simp: extreme_point_of_def)
done
moreover have "u ∈ closure (convex hull {x. x extreme_point_of S})"
if "u ∈ S" for u
proof (rule ccontr)
assume unot: "u ∉ closure(convex hull {x. x extreme_point_of S})"
then obtain a b where "a ∙ u < b"
and ab: "⋀x. x ∈ closure(convex hull {x. x extreme_point_of S}) ⟹ b < a ∙ x"
using separating_hyperplane_closed_point [of "closure(convex hull {x. x extreme_point_of S})"]
by blast
have "continuous_on S ((∙) a)"
by (rule continuous_intros)+
then obtain m where "m ∈ S" and m: "⋀y. y ∈ S ⟹ a ∙ m ≤ a ∙ y"
using continuous_attains_inf [of S "λx. a ∙ x"] ‹compact S› ‹u ∈ S›
by auto
define T where "T = S ∩ {x. a ∙ x = a ∙ m}"
have "m ∈ T"
by (simp add: T_def ‹m ∈ S›)
moreover have "compact T"
by (simp add: T_def compact_Int_closed [OF ‹compact S› closed_hyperplane])
moreover have "convex T"
by (simp add: T_def convex_Int [OF ‹convex S› 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"
also have "T face_of S"
by (simp add: T_def m face_of_Int_supporting_hyperplane_ge [OF ‹convex S›])
finally have "v extreme_point_of S"
then have "b < a ∙ v"
using closure_subset by (simp add: closure_hull hull_inc ab)
then show False
using ‹a ∙ u < b› ‹{v} face_of T› face_of_imp_subset m T_def that by fastforce
qed
ultimately show ?thesis
by blast
qed

text‹Now the sharper form.›

lemma Krein_Milman_Minkowski_aux:
fixes S :: "'a::euclidean_space set"
assumes n: "dim S = n" and S: "compact S" "convex S" "0 ∈ S"
shows "0 ∈ convex hull {x. x extreme_point_of S}"
using n S
proof (induction n arbitrary: S rule: less_induct)
case (less n S) show ?case
proof (cases "0 ∈ 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 ≠ {}"
then obtain c where c: "c ∈ rel_interior S" by blast
obtain a where "a ≠ 0"
and le_ay: "⋀y. y ∈ S ⟹ a ∙ 0 ≤ a ∙ y"
and less_ay: "⋀y. y ∈ rel_interior S ⟹ a ∙ 0 < a ∙ y"
by (blast intro: supporting_hyperplane_rel_boundary intro!: less False)
have face: "S ∩ {x. a ∙ x = 0} face_of S"
apply (rule face_of_Int_supporting_hyperplane_ge [OF ‹convex S›])
using le_ay by auto
then have co: "compact (S ∩ {x. a ∙ x = 0})" "convex (S ∩ {x. a ∙ x = 0})"
using less.prems by (blast intro: face_of_imp_compact face_of_imp_convex)+
have "a ∙ y = 0" if "y ∈ span (S ∩ {x. a ∙ x = 0})" for y
proof -
have "y ∈ span {x. a ∙ 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 ∩ {x. a ∙ x = 0}) < n"
by (metis (no_types) less_ay c subsetD dim_eq_span inf.strict_order_iff
inf_le1 ‹dim S = n› not_le rel_interior_subset span_0 span_base)
then have "0 ∈ convex hull {x. x extreme_point_of (S ∩ {x. a ∙ x = 0})}"
by (rule less.IH) (auto simp: co less.prems)
then show ?thesis
by (metis (mono_tags, lifting) Collect_mono_iff ‹S ∩ {x. a ∙ x = 0} face_of S› extreme_point_of_face hull_mono subset_iff)
qed
qed

theorem Krein_Milman_Minkowski:
fixes S :: "'a::euclidean_space set"
assumes "compact S" "convex S"
shows "S = convex hull {x. x extreme_point_of S}"
proof
show "S ⊆ convex hull {x. x extreme_point_of S}"
proof
fix a assume [simp]: "a ∈ S"
have 1: "compact ((+) (- a) ` S)"
by (simp add: ‹compact S› compact_translation)
have 2: "convex ((+) (- a) ` S)"
by (simp add: ‹convex S› convex_translation)
show a_invex: "a ∈ 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} ⊆ S"
proof -
have "{a. a extreme_point_of S} ⊆ S"
using extreme_point_of_def by blast
then show ?thesis
by (simp add: ‹convex S› hull_minimal)
qed
qed

subsection‹Applying it to convex hulls of explicitly indicated finite sets›

lemma Krein_Milman_polytope:
fixes S :: "'a::euclidean_space set"
shows
"finite S
⟹ convex hull S =
convex hull {x. x extreme_point_of (convex hull S)}"

lemma extreme_points_of_convex_hull_eq:
fixes S :: "'a::euclidean_space set"
shows
"⟦compact S; ⋀T. T ⊂ S ⟹ convex hull T ≠ convex hull S⟧
⟹ {x. x extreme_point_of (convex hull S)} = S"
by (metis (full_types) Krein_Milman_Minkowski compact_convex_hull convex_convex_hull extreme_points_of_convex_hull psubsetI)

lemma extreme_point_of_convex_hull_eq:
fixes S :: "'a::euclidean_space set"
shows
"⟦compact S; ⋀T. T ⊂ S ⟹ convex hull T ≠ convex hull S⟧
⟹ (x extreme_point_of (convex hull S) ⟷ x ∈ S)"
using extreme_points_of_convex_hull_eq by auto

lemma extreme_point_of_convex_hull_convex_independent:
fixes S :: "'a::euclidean_space set"
assumes "compact S" and S: "⋀a. a ∈ S ⟹ a ∉ convex hull (S - {a})"
shows "(x extreme_point_of (convex hull S) ⟷ x ∈ S)"
proof -
have "convex hull T ≠ convex hull S" if "T ⊂ S" for T
proof -
obtain a where  "T ⊆ S" "a ∈ S" "a ∉ T" using ‹T ⊂ S› 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 ‹compact S›])
qed

lemma extreme_point_of_convex_hull_affine_independent:
fixes S :: "'a::euclidean_space set"
shows
"~ affine_dependent S
⟹ (x extreme_point_of (convex hull S) ⟷ x ∈ 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‹Elementary proofs exist, not requiring Euclidean spaces and all this development›
lemma extreme_point_of_convex_hull_2:
fixes x :: "'a::euclidean_space"
shows "x extreme_point_of (convex hull {a,b}) ⟷ x = a ∨ x = b"
proof -
have "x extreme_point_of (convex hull {a,b}) ⟷ x ∈ {a,b}"
by (intro extreme_point_of_convex_hull_affine_independent affine_independent_2)
then show ?thesis
by simp
qed

lemma extreme_point_of_segment:
fixes x :: "'a::euclidean_space"
shows
"x extreme_point_of closed_segment a b ⟷ x = a ∨ x = b"

lemma face_of_convex_hull_subset:
fixes S :: "'a::euclidean_space set"
assumes "compact S" and T: "T face_of (convex hull S)"
obtains s' where "s' ⊆ S" "T = convex hull s'"
apply (rule_tac s' = "{x. x extreme_point_of T}" in that)
using T extreme_point_of_convex_hull extreme_point_of_face apply blast
by (metis (no_types) Krein_Milman_Minkowski assms compact_convex_hull convex_convex_hull face_of_imp_compact face_of_imp_convex)

lemma face_of_convex_hull_aux:
assumes eq: "x *⇩R p = u *⇩R a + v *⇩R b + w *⇩R c"
and x: "u + v + w = x" "x ≠ 0" and S: "affine S" "a ∈ S" "b ∈ S" "c ∈ S"
shows "p ∈ S"
proof -
have "p = (u *⇩R a + v *⇩R b + w *⇩R c) /⇩R x"
by (metis ‹x ≠ 0› eq mult.commute right_inverse scaleR_one scaleR_scaleR)
moreover have "affine hull {a,b,c} ⊆ S"
moreover have "(u *⇩R a + v *⇩R b + w *⇩R c) /⇩R x ∈ affine hull {a,b,c}"
apply (rule_tac x="u/x" in exI)
apply (rule_tac x="v/x" in exI)
apply (rule_tac x="w/x" in exI)
using x apply (auto simp: algebra_simps divide_simps)
done
ultimately show ?thesis by force
qed

proposition face_of_convex_hull_insert_eq:
fixes a :: "'a :: euclidean_space"
assumes "finite S" and a: "a ∉ affine hull S"
shows "(F face_of (convex hull (insert a S)) ⟷
F face_of (convex hull S) ∨
(∃F'. F' face_of (convex hull S) ∧ F = convex hull (insert a F')))"
(is "F face_of ?CAS ⟷ _")
proof safe
assume F: "F face_of ?CAS"
and *: "∄F'. F' face_of convex hull S ∧ F = convex hull insert a F'"
obtain T where T: "T ⊆ insert a S" and FeqT: "F = convex hull T"
by (metis F ‹finite S› compact_insert finite_imp_compact face_of_convex_hull_subset)
show "F face_of convex hull S"
proof (cases "a ∈ T")
case True
have "F = convex hull insert a (convex hull T ∩ convex hull S)"
proof
have "T ⊆ insert a (convex hull T ∩ convex hull S)"
using T hull_subset by fastforce
then show "F ⊆ convex hull insert a (convex hull T ∩ convex hull S)"
show "convex hull insert a (convex hull T ∩ convex hull S) ⊆ F"
apply (rule hull_minimal)
using True by (auto simp: ‹F = convex hull T› hull_inc)
qed
moreover have "convex hull T ∩ 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: ‹F face_of convex hull S› a face_of_convex_hull_insert ‹finite S›)
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 ∉ 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: ‹F = {}› ‹finite S› extreme_point_of_convex_hull_insert face_of_singleton)
next
case False
have "convex hull insert a F ⊆ ?CAS"
by (simp add: F a ‹finite S› convex_hull_subset face_of_convex_hull_insert face_of_imp_subset hull_inc)
moreover
have "(∃y v. (1 - ub) *⇩R a + ub *⇩R b = (1 - v) *⇩R a + v *⇩R y ∧
0 ≤ v ∧ v ≤ 1 ∧ y ∈ F) ∧
(∃x u. (1 - uc) *⇩R a + uc *⇩R c = (1 - u) *⇩R a + u *⇩R x ∧
0 ≤ u ∧ u ≤ 1 ∧ x ∈ F)"
if *: "(1 - ux) *⇩R a + ux *⇩R x
∈ open_segment ((1 - ub) *⇩R a + ub *⇩R b) ((1 - uc) *⇩R a + uc *⇩R c)"
and "0 ≤ ub" "ub ≤ 1" "0 ≤ uc" "uc ≤ 1" "0 ≤ ux" "ux ≤ 1"
and b: "b ∈ convex hull S" and c: "c ∈ convex hull S" and "x ∈ F"
for b c ub uc ux x
proof -
obtain v where ne: "(1 - ub) *⇩R a + ub *⇩R b ≠ (1 - uc) *⇩R a + uc *⇩R c"
and eq: "(1 - ux) *⇩R a + ux *⇩R x =
(1 - v) *⇩R ((1 - ub) *⇩R a + ub *⇩R b) + v *⇩R ((1 - uc) *⇩R a + uc *⇩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))) *⇩R a +
(ux *⇩R x - (((1 - v) * ub) *⇩R b + (v * uc) *⇩R c)) = 0"
by (auto simp: algebra_simps)
then have "((1 - ux) - ((1 - v) * (1 - ub) + v * (1 - uc))) *⇩R a =
((1 - v) * ub) *⇩R b + (v * uc) *⇩R c + (-ux) *⇩R x"
by (auto simp: algebra_simps)
then have "a ∈ affine hull S" if "1 - ux - ((1 - v) * (1 - ub) + v * (1 - uc)) ≠ 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 ‹x ∈ F› 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 *⇩R x = (((1 - v) * ub) *⇩R b + (v * uc) *⇩R c)"
by auto (auto simp: algebra_simps)
show ?thesis
proof (cases "uc = 0")
case True
then show ?thesis
using equx 0 ‹0 ≤ ub› ‹ub ≤ 1› ‹v < 1› ‹x ∈ F›
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 ‹x ∈ F› ‹uc ≤ 1› apply blast
done
next
case False
show ?thesis
proof (cases "ub = 0")
case True
then show ?thesis
using equx 0 ‹0 ≤ uc› ‹uc ≤ 1› ‹0 < v› ‹x ∈ F› ‹uc ≠ 0› by (force simp: algebra_simps)
next
case False
then have "0 < ub" "0 < uc"
using ‹uc ≠ 0› ‹0 ≤ ub› ‹0 ≤ uc› by auto
then have "ux ≠ 0"
by (metis ‹0 < v› ‹v < 1› 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 ∈ F ∧ c ∈ F"
proof (cases "b = c")
case True
then show ?thesis
by (metis ‹ux ≠ 0› equx real_vector.scale_cancel_left scaleR_add_left uxx ‹x ∈ F›)
next
case False
have "x = (((1 - v) * ub) *⇩R b + (v * uc) *⇩R c) /⇩R ux"
by (metis ‹ux ≠ 0› uxx mult.commute right_inverse scaleR_one scaleR_scaleR)
also have "... = (1 - v * uc / ux) *⇩R b + (v * uc / ux) *⇩R c"
using ‹ux ≠ 0› equx apply (auto simp: algebra_simps divide_simps)
finally have "x = (1 - v * uc / ux) *⇩R b + (v * uc / ux) *⇩R c" .
then have "x ∈ open_segment b c"
apply (simp add: in_segment ‹b ≠ c›)
apply (rule_tac x="(v * uc) / ux" in exI)
using ‹0 ≤ ux› ‹ux ≠ 0› ‹0 < uc› ‹0 < v› ‹0 < ub› ‹v < 1› equx
apply (force simp: algebra_simps divide_simps)
done
then show ?thesis
by (rule face_ofD [OF F _ b c ‹x ∈ F›])
qed
with ‹0 ≤ ub› ‹ub ≤ 1› ‹0 ≤ uc› ‹uc ≤ 1› 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 ‹S ≠ {}› ‹F ≠ {}›)
qed
qed
qed

lemma face_of_convex_hull_insert2:
fixes a :: "'a :: euclidean_space"
assumes S: "finite S" and a: "a ∉ affine hull S" and F: "F face_of convex hull S"
shows "convex hull (insert a F) face_of convex hull (insert a S)"
by (metis F face_of_convex_hull_insert_eq [OF S a])

proposition face_of_convex_hull_affine_independent:
fixes S :: "'a::euclidean_space set"
assumes "~ affine_dependent S"
shows "(T face_of (convex hull S) ⟷ (∃c. c ⊆ S ∧ T = convex hull c))"
(is "?lhs = ?rhs")
proof
assume ?lhs
then show ?rhs
by (meson ‹T face_of convex hull S› aff_independent_finite assms face_of_convex_hull_subset finite_imp_compact)
next
assume ?rhs
then obtain c where "c ⊆ S" and T: "T = convex hull c"
by blast
have "affine hull c ∩ affine hull (S - c) = {}"
apply (rule disjoint_affine_hull [OF assms ‹c ⊆ S›], auto)
done
then have "affine hull c ∩ convex hull (S - c) = {}"
using convex_hull_subset_affine_hull by fastforce
then show ?lhs
by (metis face_of_convex_hulls ‹c ⊆ S› aff_independent_finite assms T)
qed

lemma facet_of_convex_hull_affine_independent:
fixes S :: "'a::euclidean_space set"
assumes "~ affine_dependent S"
shows "T facet_of (convex hull S) ⟷
T ≠ {} ∧ (∃u. u ∈ S ∧ T = convex hull (S - {u}))"
(is "?lhs = ?rhs")
proof
assume ?lhs
then have "T face_of (convex hull S)" "T ≠ {}"
and afft: "aff_dim T = aff_dim (convex hull S) - 1"
by (auto simp: facet_of_def)
then obtain c where "c ⊆ 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 "~ affine_dependent c"
using ‹c ⊆ S› 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 ‹c ⊆ S› 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 ∈ S - c"
by (metis DiffI ‹c ⊆ S› 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 ‹c ⊆ S› ‹card (S - c) = 1› card_1_singletonE double_diff insert_Diff insert_subset singletonD)
have "T = convex hull (c - {u})"
by (metis Diff_empty Diff_insert0 ‹T facet_of convex hull S› c facet_of_irrefl insert_absorb u)
with ‹T ≠ {}› show ?rhs
using c u by auto
next
assume ?rhs
then obtain u where "T ≠ {}" "u ∈ S" and u: "T = convex hull (S - {u})"
by (force simp: facet_of_def)
then have "¬ S ⊆ {u}"
using ‹T ≠ {}› u by auto
have [simp]: "aff_dim (convex hull (S - {u})) = aff_dim (convex hull S) - 1"
using assms ‹u ∈ S›
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: ‹¬ S ⊆ {u}› facet_of_def face_of_convex_hull_affine_independent [OF assms], blast)
done
qed

lemma facet_of_convex_hull_affine_independent_alt:
fixes S :: "'a::euclidean_space set"
shows
"~affine_dependent S
⟹ (T facet_of (convex hull S) ⟷
2 ≤ card S ∧ (∃u. u ∈ S ∧ T = convex hull (S - {u})))"
apply (auto simp: Set.subset_singleton_iff)
apply (metis Diff_cancel Int_empty_right Int_insert_right_if1  aff_independent_finite card_eq_0_iff card_insert_if card_mono card_subset_eq convex_hull_eq_empty eq_iff equals0D finite_insert finite_subset inf.absorb_iff2 insert_absorb insert_not_empty  not_less_eq_eq numeral_2_eq_2)
done

lemma segment_face_of:
assumes "(closed_segment a b) face_of S"
shows "a extreme_point_of S" "b extreme_point_of S"
proof -
have as: "{a} face_of S"
by (metis (no_types) assms convex_hull_singleton empty_iff extreme_point_of_convex_hull_insert face_of_face face_of_singleton finite.emptyI finite.insertI insert_absorb insert_iff segment_convex_hull)
moreover have "{b} face_of S"
proof -
have "b ∈ convex hull {a} ∨ 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 Krein_Milman_frontier:
fixes S :: "'a::euclidean_space set"
assumes "convex S" "compact S"
shows "S = convex hull (frontier S)"
(is "?lhs = ?rhs")
proof
have "?lhs ⊆ convex hull {x. x extreme_point_of S}"
using Krein_Milman_Minkowski assms by blast
also have "... ⊆ ?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 ⊆ ?rhs" .
next
have "?rhs ⊆ convex hull S"
by (metis Diff_subset ‹compact S› closure_closed compact_eq_bounded_closed frontier_def hull_mono)
also have "... ⊆ ?lhs"
by (simp add: ‹convex S› hull_same)
finally show "?rhs ⊆ ?lhs" .
qed

subsection‹Polytopes›

definition polytope where
"polytope S ≡ ∃v. finite v ∧ S = convex hull v"

lemma polytope_translation_eq: "polytope (image (λx. a + x) S) ⟷ polytope S"
apply (metis convex_hull_translation finite_imageI translation_galois)
by (metis convex_hull_translation finite_imageI)

lemma polytope_linear_image: "⟦linear f; polytope p⟧ ⟹ polytope(image f p)"
unfolding polytope_def using convex_hull_linear_image by blast

lemma polytope_empty: "polytope {}"
using convex_hull_empty polytope_def by blast

lemma polytope_convex_hull: "finite S ⟹ polytope(convex hull S)"
using polytope_def by auto

lemma polytope_Times: "⟦polytope S; polytope T⟧ ⟹ polytope(S × T)"
unfolding polytope_def
by (metis finite_cartesian_product convex_hull_Times)

lemma face_of_polytope_polytope:
fixes S :: "'a::euclidean_space set"
shows "⟦polytope S; F face_of S⟧ ⟹ polytope F"
unfolding polytope_def
by (meson face_of_convex_hull_subset finite_imp_compact finite_subset)

lemma finite_polytope_faces:
fixes S :: "'a::euclidean_space set"
assumes "polytope S"
shows "finite {F. F face_of S}"
proof -
obtain v where "finite v" "S = convex hull v"
using assms polytope_def by auto
have "finite ((hull) convex ` {T. T ⊆ v})"
moreover have "{F. F face_of S} ⊆ ((hull) convex ` {T. T ⊆ v})"
by (metis (no_types, lifting) ‹finite v› ‹S = convex hull v› face_of_convex_hull_subset finite_imp_compact image_eqI mem_Collect_eq subsetI)
ultimately show ?thesis
by (blast intro: finite_subset)
qed

lemma finite_polytope_facets:
assumes "polytope S"
shows "finite {T. T facet_of S}"
by (simp add: assms facet_of_def finite_polytope_faces)

lemma polytope_scaling:
assumes "polytope S"  shows "polytope (image (λx. c *⇩R x) S)"

lemma polytope_imp_compact:
fixes S :: "'a::real_normed_vector set"
shows "polytope S ⟹ compact S"
by (metis finite_imp_compact_convex_hull polytope_def)

lemma polytope_imp_convex: "polytope S ⟹ convex S"
by (metis convex_convex_hull polytope_def)

lemma polytope_imp_closed:
fixes S :: "'a::real_normed_vector set"
shows "polytope S ⟹ closed S"

lemma polytope_imp_bounded:
fixes S :: "'a::real_normed_vector set"
shows "polytope S ⟹ bounded S"

lemma polytope_interval: "polytope(cbox a b)"
unfolding polytope_def by (meson closed_interval_as_convex_hull)

lemma polytope_sing: "polytope {a}"
using polytope_def by force

lemma face_of_polytope_insert:
"⟦polytope S; a ∉ affine hull S; F face_of S⟧ ⟹ 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 face_of_polytope_insert2:
fixes a :: "'a :: euclidean_space"
assumes "polytope S" "a ∉ affine hull S" "F face_of S"
shows "convex hull (insert a F) face_of convex hull (insert a S)"
proof -
obtain V where "finite V" "S = convex hull V"
using assms by (auto simp: polytope_def)
then have "convex hull (insert a F) face_of convex hull (insert a V)"
using affine_hull_convex_hull assms face_of_convex_hull_insert2 by blast
then show ?thesis
by (metis ‹S = convex hull V› hull_insert)
qed

subsection‹Polyhedra›

definition polyhedron where
"polyhedron S ≡
∃F. finite F ∧
S = ⋂ F ∧
(∀h ∈ F. ∃a b. a ≠ 0 ∧ h = {x. a ∙ x ≤ b})"

lemma polyhedron_Int [intro,simp]:
"⟦polyhedron S; polyhedron T⟧ ⟹ polyhedron (S ∩ T)"
apply (rename_tac F G)
apply (rule_tac x="F ∪ G" in exI, auto)
done

lemma polyhedron_UNIV [iff]: "polyhedron UNIV"
unfolding polyhedron_def
by (rule_tac x="{}" in exI) auto

lemma polyhedron_Inter [intro,simp]:
"⟦finite F; ⋀S. S ∈ F ⟹ polyhedron S⟧ ⟹ polyhedron(⋂F)"
by (induction F rule: finite_induct) auto

lemma polyhedron_empty [iff]: "polyhedron ({} :: 'a :: euclidean_space set)"
proof -
have "∃a. a ≠ 0 ∧
(∃b. {x. (SOME i. i ∈ Basis) ∙ x ≤ - 1} = {x. a ∙ x ≤ b})"
by (rule_tac x="(SOME i. i ∈ Basis)" in exI) (force simp: SOME_Basis nonzero_Basis)
moreover have "∃a b. a ≠ 0 ∧
{x. - (SOME i. i ∈ Basis) ∙ x ≤ - 1} = {x. a ∙ x ≤ b}"
apply (rule_tac x="-(SOME i. i ∈ Basis)" in exI)
apply (rule_tac x="-1" in exI)
done
ultimately show ?thesis
unfolding polyhedron_def
apply (rule_tac x="{{x. (SOME i. i ∈ Basis) ∙ x ≤ -1},
{x. -(SOME i. i ∈ Basis) ∙ x ≤ -1}}" in exI)
apply force
done
qed

lemma polyhedron_halfspace_le:
fixes a :: "'a :: euclidean_space"
shows "polyhedron {x. a ∙ x ≤ 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 ∙ x ≤ b}}" in exI) auto
qed

lemma polyhedron_halfspace_ge:
fixes a :: "'a :: euclidean_space"
shows "polyhedron {x. a ∙ x ≥ b}"
using polyhedron_halfspace_le [of "-a" "-b"] by simp

lemma polyhedron_hyperplane:
fixes a :: "'a :: euclidean_space"
shows "polyhedron {x. a ∙ x = b}"
proof -
have "{x. a ∙ x = b} = {x. a ∙ x ≤ b} ∩ {x. a ∙ x ≥ b}"
by force
then show ?thesis
qed

lemma affine_imp_polyhedron:
fixes S :: "'a :: euclidean_space set"
shows "affine S ⟹ polyhedron S"
by (metis affine_hull_eq polyhedron_Inter polyhedron_hyperplane affine_hull_finite_intersection_hyperplanes [of S])

lemma polyhedron_imp_closed:
fixes S :: "'a :: euclidean_space set"
shows "polyhedron S ⟹ closed S"
using closed_halfspace_le by fastforce

lemma polyhedron_imp_convex:
fixes S :: "'a :: euclidean_space set"
shows "polyhedron S ⟹ convex S"
using convex_Inter convex_halfspace_le by fastforce

lemma polyhedron_affine_hull:
fixes S :: "'a :: euclidean_space set"
shows "polyhedron(affine hull S)"

subsection‹Canonical polyhedron representation making facial structure explicit›

lemma polyhedron_Int_affine:
fixes S :: "'a :: euclidean_space set"
shows "polyhedron S ⟷
(∃F. finite F ∧ S = (affine hull S) ∩ ⋂F ∧
(∀h ∈ F. ∃a b. a ≠ 0 ∧ h = {x. a ∙ x ≤ b}))"
(is "?lhs = ?rhs")
proof
assume ?lhs then show ?rhs
apply (erule ex_forward)
using hull_subset apply force
done
next
assume ?rhs then show ?lhs
apply clarify
apply (erule ssubst)
apply (force intro: polyhedron_affine_hull polyhedron_halfspace_le)
done
qed

proposition rel_interior_polyhedron_explicit:
assumes "finite F"
and seq: "S = affine hull S ∩ ⋂F"
and faceq: "⋀h. h ∈ F ⟹ a h ≠ 0 ∧ h = {x. a h ∙ x ≤ b h}"
and psub: "⋀F'. F' ⊂ F ⟹ S ⊂ affine hull S ∩ ⋂F'"
shows "rel_interior S = {x ∈ S. ∀h ∈ F. a h ∙ x < b h}"
proof -
have rels: "⋀x. x ∈ rel_interior S ⟹ x ∈ S"
by (meson IntE mem_rel_interior)
moreover have "a i ∙ x < b i" if x: "x ∈ rel_interior S" and "i ∈ F" for x i
proof -
have fif: "F - {i} ⊂ F"
using ‹i ∈ F› Diff_insert_absorb Diff_subset set_insert psubsetI by blast
then have "S ⊂ affine hull S ∩ ⋂(F - {i})"
by (rule psub)
then obtain z where ssub: "S ⊆ ⋂(F - {i})" and zint: "z ∈ ⋂(F - {i})"
and "z ∉ S" and zaff: "z ∈ affine hull S"
by auto
have "z ≠ x"
using ‹z ∉ S› rels x by blast
have "z ∉ affine hull S ∩ ⋂F"
using ‹z ∉ S› seq by auto
then have aiz: "a i ∙ z > b i"
using faceq zint zaff by fastforce
obtain e where "e > 0" "x ∈ S" and e: "ball x e ∩ affine hull S ⊆ S"
using x by (auto simp: mem_rel_interior_ball)
then have ins: "⋀y. ⟦norm (x - y) < e; y ∈ affine hull S⟧ ⟹ y ∈ S"
by (metis IntI subsetD dist_norm mem_ball)
define ξ where "ξ = min (1/2) (e / 2 / norm(z - x))"
have "norm (ξ *⇩R x - ξ *⇩R z) = norm (ξ *⇩R (x - z))"
by (simp add: ξ_def algebra_simps norm_mult)
also have "... = ξ * norm (x - z)"
using ‹e > 0› by (simp add: ξ_def)
also have "... < e"
using ‹z ≠ x› ‹e > 0› by (simp add: ξ_def min_def divide_simps norm_minus_commute)
finally have lte: "norm (ξ *⇩R x - ξ *⇩R z) < e" .
have ξ_aff: "ξ *⇩R z + (1 - ξ) *⇩R x ∈ affine hull S"
have "ξ *⇩R z + (1 - ξ) *⇩R x ∈ S"
apply (rule ins [OF _ ξ_aff])
done
then obtain l where l: "0 < l" "l < 1" and ls: "(l *⇩R z + (1 - l) *⇩R x) ∈ S"
apply (rule_tac l = ξ in that)
using ‹e > 0› ‹z ≠ x›  apply (auto simp: ξ_def)
done
then have i: "l *⇩R z + (1 - l) *⇩R x ∈ i"
using seq ‹i ∈ F› by auto
have "b i * l + (a i ∙ x) * (1 - l) < a i ∙ (l *⇩R z + (1 - l) *⇩R x)"
using l by (simp add: algebra_simps aiz)
also have "… ≤ b i" using i l
using faceq mem_Collect_eq ‹i ∈ F› by blast
finally have "(a i ∙ x) * (1 - l) < b i * (1 - l)"
with l show ?thesis
by simp
qed
moreover have "x ∈ rel_interior S"
if "x ∈ S" and less: "⋀h. h ∈ F ⟹ a h ∙ x < b h" for x
proof -
have 1: "⋀h. h ∈ F ⟹ x ∈ interior h"
by (metis interior_halfspace_le mem_Collect_eq less faceq)
have 2: "⋀y. ⟦∀h∈F. y ∈ interior h; y ∈ affine hull S⟧ ⟹ y ∈ S"
by (metis IntI Inter_iff contra_subsetD interior_subset seq)
show ?thesis
apply (simp add: rel_interior ‹x ∈ S›)
apply (rule_tac x="⋂h∈F. interior h" in exI)
apply (auto simp: ‹finite F› open_INT 1 2)
done
qed
ultimately show ?thesis by blast
qed

lemma polyhedron_Int_affine_parallel:
fixes S :: "'a :: euclidean_space set"
shows "polyhedron S ⟷
(∃F. finite F ∧
S = (affine hull S) ∩ (⋂F) ∧
(∀h ∈ F. ∃a b. a ≠ 0 ∧ h = {x. a ∙ x ≤ b} ∧
(∀x ∈ affine hull S. (x + a) ∈ affine hull S)))"
(is "?lhs = ?rhs")
proof
assume ?lhs
then obtain F where "finite F" and seq: "S = (affine hull S) ∩ ⋂F"
and faces: "⋀h. h ∈ F ⟹ ∃a b. a ≠ 0 ∧ h = {x. a ∙ x ≤ b}"
then obtain a b where ab: "⋀h. h ∈ F ⟹ a h ≠ 0 ∧ h = {x. a h ∙ x ≤ b h}"
by metis
show ?rhs
proof -
have "∃a' b'. a' ≠ 0 ∧
affine hull S ∩ {x. a' ∙ x ≤ b'} = affine hull S ∩ h ∧
(∀w ∈ affine hull S. (w + a') ∈ affine hull S)"
if "h ∈ F" "~(affine hull S ⊆ h)" for h
proof -
have "a h ≠ 0" and "h = {x. a h ∙ x ≤ b h}" "h ∩ ⋂F = ⋂F"
using ‹h ∈ F› ab by auto
then have "(affine hull S) ∩ {x. a h ∙ x ≤ b h} ≠ {}"
by (metis (no_types) affine_hull_eq_empty inf.absorb_iff2 inf_assoc inf_bot_right inf_commute seq that(2))
moreover have "~ (affine hull S ⊆ {x. a h ∙ x ≤ b h})"
using ‹h = {x. a h ∙ x ≤ b h}› that(2) by blast
ultimately show ?thesis
using affine_parallel_slice [of "affine hull S"]
by (metis ‹h = {x. a h ∙ x ≤ b h}› affine_affine_hull)
qed
then obtain a b
where ab: "⋀h. ⟦h ∈ F; ~ (affine hull S ⊆ h)⟧
⟹ a h ≠ 0 ∧
affine hull S ∩ {x. a h ∙ x ≤ b h} = affine hull S ∩ h ∧
(∀w ∈ affine hull S. (w + a h) ∈ affine hull S)"
by metis
have seq2: "S = affine hull S ∩ (⋂h∈{h ∈ F. ¬ affine hull S ⊆ h}. {x. a h ∙ x ≤ b h})"
by (subst seq) (auto simp: ab INT_extend_simps)
show ?thesis
apply (rule_tac x="(λh. {x. a h ∙ x ≤ b h}) ` {h. h ∈ F ∧ ~(affine hull S ⊆ h)}" in exI)
apply (intro conjI seq2)
using ‹finite F› apply force
using ab apply blast
done
qed
next
assume ?rhs then show ?lhs
by metis
qed

proposition polyhedron_Int_affine_parallel_minimal:
fixes S :: "'a :: euclidean_space set"
shows "polyhedron S ⟷
(∃F. finite F ∧
S = (affine hull S) ∩ (⋂F) ∧
(∀h ∈ F. ∃a b. a ≠ 0 ∧ h = {x. a ∙ x ≤ b} ∧
(∀x ∈ affine hull S. (x + a) ∈ affine hull S)) ∧
(∀F'. F' ⊂ F ⟶ S ⊂ (affine hull S) ∩ (⋂F')))"
(is "?lhs = ?rhs")
proof
assume ?lhs
then obtain f0
where f0: "finite f0"
"S = (affine hull S) ∩ (⋂f0)"
(is "?P f0")
"∀h ∈ f0. ∃a b. a ≠ 0 ∧ h = {x. a ∙ x ≤ b} ∧
(∀x ∈ affine hull S. (x + a) ∈ affine hull S)"
(is "?Q f0")
by (force simp: polyhedron_Int_affine_parallel)
define n where "n = (LEAST n. ∃F. card F = n ∧ finite F ∧ ?P F ∧ ?Q F)"
have nf: "∃F. card F = n ∧ finite F ∧ ?P F ∧ ?Q F"
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 "~ (finite g ∧ ?P g ∧ ?Q g)" if "card g < n" for g
using that by (auto simp: n_def dest!: not_less_Least)
then have *: "~ (?P g ∧ ?Q g)" if "g ⊂ F" for g
using that ‹finite F› psubset_card_mono ‹card F = n›
by (metis finite_Int inf.strict_order_iff)
have 1: "⋀F'. F' ⊂ F ⟹ S ⊆ affine hull S ∩ ⋂F'"
by (subst seq) blast
have 2: "⋀F'. F' ⊂ F ⟹ S ≠ affine hull S ∩ ⋂F'"
apply (frule *)
by (metis aff subsetCE subset_iff_psubset_eq)
show ?rhs
by (metis ‹finite F› seq aff psubsetI 1 2)
next
assume ?rhs then show ?lhs
by (auto simp: polyhedron_Int_affine_parallel)
qed

lemma polyhedron_Int_affine_minimal:
fixes S :: "'a :: euclidean_space set"
shows "polyhedron S ⟷
(∃F. finite F ∧ S = (affine hull S) ∩ ⋂F ∧
(∀h ∈ F. ∃a b. a ≠ 0 ∧ h = {x. a ∙ x ≤ b}) ∧
(∀F'. F' ⊂ F ⟶ S ⊂ (affine hull S) ∩ ⋂F'))"
apply (rule iffI)
apply (force simp: polyhedron_Int_affine_parallel_minimal elim!: ex_forward)
apply (auto simp: polyhedron_Int_affine elim!: ex_forward)
done

proposition facet_of_polyhedron_explicit:
assumes "finite F"
and seq: "S = affine hull S ∩ ⋂F"
and faceq: "⋀h. h ∈ F ⟹ a h ≠ 0 ∧ h = {x. a h ∙ x ≤ b h}"
and psub: "⋀F'. F' ⊂ F ⟹ S ⊂ affine hull S ∩ ⋂F'"
shows "c facet_of S ⟷ (∃h. h ∈ F ∧ c = S ∩ {x. a h ∙ x = b h})"
proof (cases "S = {}")
case True with psub show ?thesis by force
next
case False
have "polyhedron S"
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 ≠ {}" by blast
then obtain x where "x ∈ rel_interior S" by auto
then obtain T where "open T" "x ∈ T" "x ∈ S" "T ∩ affine hull S ⊆ S"
by (force simp: mem_rel_interior)
then have xaff: "x ∈ affine hull S" and xint: "x ∈ ⋂F"
using seq hull_inc by auto
have "rel_interior S = {x ∈ S. ∀h∈F. a h ∙ x < b h}"
by (rule rel_interior_polyhedron_explicit [OF ‹finite F› seq faceq psub])
with ‹x ∈ rel_interior S›
have [simp]: "⋀h. h∈F ⟹ a h ∙ x < b h" by blast
have *: "(S ∩ {x. a h ∙ x = b h}) facet_of S" if "h ∈ F" for h
proof -
have "S ⊂ affine hull S ∩ ⋂(F - {h})"
using psub that by (metis Diff_disjoint Diff_subset insert_disjoint(2) psubsetI)
then obtain z where zaff: "z ∈ affine hull S" and zint: "z ∈ ⋂(F - {h})" and "z ∉ S"
by force
then have "z ≠ x" "z ∉ h" using seq ‹x ∈ S› by auto
have "x ∈ h" using that xint by auto
then have able: "a h ∙ x ≤ b h"
using faceq that by blast
also have "... < a h ∙ z" using ‹z ∉ h› faceq [OF that] xint by auto
finally have xltz: "a h ∙ x < a h ∙ z" .
define l where "l = (b h - a h ∙ x) / (a h ∙ z - a h ∙ x)"
define w where "w = (1 - l) *⇩R x + l *⇩R z"
have "0 < l" "l < 1"
using able xltz ‹b h < a h ∙ z› ‹h ∈ F›
by (auto simp: l_def divide_simps)
have awlt: "a i ∙ w < b i" if "i ∈ F" "i ≠ h" for i
proof -
have "(1 - l) * (a i ∙ x) < (1 - l) * b i"
by (simp add: ‹l < 1› ‹i ∈ F›)
moreover have "l * (a i ∙ z) ≤ l * b i"
apply (rule mult_left_mono)
apply (metis Diff_insert_absorb Inter_iff Set.set_insert ‹h ∈ F› faceq insertE mem_Collect_eq that zint)
using ‹0 < l›
apply simp
done
ultimately show ?thesis by (simp add: w_def algebra_simps)
qed
have weq: "a h ∙ w = b h"
using xltz unfolding w_def l_def
have "w ∈ affine hull S"
by (simp add: w_def mem_affine xaff zaff)
moreover have "w ∈ ⋂F"
using ‹a h ∙ w = b h› awlt faceq less_eq_real_def by blast
ultimately have "w ∈ S"
using seq by blast
with weq have "S ∩ {x. a h ∙ x = b h} ≠ {}" by blast
moreover have "S ∩ {x. a h ∙ x = b h} face_of S"
apply (rule face_of_Int_supporting_hyperplane_le)
apply (rule ‹convex S›)
apply (subst (asm) seq)
using faceq that apply fastforce
done
moreover have "affine hull (S ∩ {x. a h ∙ x = b h}) =
(affine hull S) ∩ {x. a h ∙ x = b h}"
proof
show "affine hull (S ∩ {x. a h ∙ x = b h}) ⊆ affine hull S ∩ {x. a h ∙ 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 ∩ {x. a h ∙ x = b h} ⊆ affine hull (S ∩ {x. a h ∙ x = b h})"
proof
fix y
assume yaff: "y ∈ affine hull S ∩ {y. a h ∙ y = b h}"
obtain T where "0 < T"
and T: "⋀j. ⟦j ∈ F; j ≠ h⟧ ⟹ T * (a j ∙ y - a j ∙ w) ≤ b j - a j ∙ 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' ∈ F - {h}" by auto
define inff where "inff =
(INF j:F - {h}.
if 0 < a j ∙ y - a j ∙ w
then (b j - a j ∙ w) / (a j ∙ y - a j ∙ w)
else 1)"
have "0 < inff"
apply (rule finite_imp_less_Inf)
using ‹finite F› apply blast
using h' apply blast
apply simp
using awlt apply (force simp: divide_simps)
done
moreover have "inff * (a j ∙ y - a j ∙ w) ≤ b j - a j ∙ w"
if "j ∈ F" "j ≠ h" for j
proof (cases "a j ∙ w < a j ∙ y")
case True
then have "inff ≤ (b j - a j ∙ w) / (a j ∙ y - a j ∙ w)"
apply (rule cInf_le_finite)
using ‹finite F› apply blast
apply (simp add: that split: if_split_asm)
done
then show ?thesis
using ‹0 < inff› awlt [OF that] mult_strict_left_mono
by (fastforce simp add: algebra_simps divide_simps split: if_split_asm)
next
case False
with ‹0 < inff› have "inff * (a j ∙ y - a j ∙ w) ≤ 0"
also have "... < b j - a j ∙ w"
finally show ?thesis by simp
qed
ultimately show ?thesis
by (blast intro: that)
qed
define c where "c = (1 - T) *⇩R w + T *⇩R y"
have "(1 - T) *⇩R w + T *⇩R y ∈ j" if "j ∈ F" for j
proof (cases "j = h")
case True
have "(1 - T) *⇩R w + T *⇩R y ∈ {x. a h ∙ x ≤ 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) *⇩R w + T *⇩R y ∈ {x. a j ∙ x ≤ b j}"
with faceq [OF that] show ?thesis by simp
qed
moreover have "(1 - T) *⇩R w + T *⇩R y ∈ affine hull S"
apply (rule affine_affine_hull [simplified affine_alt, rule_format])
apply (simp add: ‹w ∈ affine hull S›)
using yaff apply blast
done
ultimately have "c ∈ S"
using seq by (force simp: c_def)
moreover have "a h ∙ c = b h"
using yaff by (force simp: c_def algebra_simps weq)
ultimately have caff: "c ∈ affine hull (S ∩ {y. a h ∙ y = b h})"
have waff: "w ∈ affine hull (S ∩ {y. a h ∙ y = b h})"
using ‹w ∈ S› weq by (blast intro: hull_inc)
have yeq: "y = (1 - inverse T) *⇩R w + c /⇩R T"
using ‹0 < T› by (simp add: c_def algebra_simps)
show "y ∈ affine hull (S ∩ {y. a h ∙ y = b h})"
by (metis yeq affine_affine_hull [simplified affine_alt, rule_format, OF waff caff])
qed
qed
ultimately show ?thesis
apply (subst aff_dim_affine_hull [symmetric])
using  ‹b h < a h ∙ z› zaff
apply (force simp: aff_dim_affine_Int_hyperplane)
done
qed
show ?thesis
proof
show "∃h. h ∈ F ∧ c = S ∩ {x. a h ∙ x = b h} ⟹ c facet_of S"
using * by blast
next
assume "c facet_of S"
then have "c face_of S" "convex c" "c ≠ {}" 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 ∈ rel_interior c"
by (force simp: rel_interior_eq_empty)
then have "x ∈ c"
by (meson subsetD rel_interior_subset)
then have "x ∈ S"
using ‹c facet_of S› facet_of_imp_subset by blast
have rels: "rel_interior S = {x ∈ S. ∀h∈F. a h ∙ x < b h}"
by (rule rel_interior_polyhedron_explicit [OF assms])
have "c ≠ S"
using ‹c facet_of S› facet_of_irrefl by blast
then have "x ∉ rel_interior S"
by (metis IntI empty_iff ‹x ∈ c› ‹c ≠ S› ‹c face_of S› face_of_disjoint_rel_interior)
with rels ‹x ∈ S› obtain i where "i ∈ F" and i: "a i ∙ x ≥ b i"
by force
have "x ∈ {u. a i ∙ u ≤ b i}"
by (metis IntD2 InterE ‹i ∈ F› ‹x ∈ S› faceq seq)
then have "a i ∙ x ≤ b i" by simp
then have "a i ∙ x = b i" using i by auto
have "c ⊆ S ∩ {x. a i ∙ x = b i}"
apply (rule subset_of_face_of [of _ S])
apply (simp add: "*" ‹i ∈ F› facet_of_imp_face_of)
apply (simp add: ‹c face_of S› face_of_imp_subset)
using ‹a i ∙ x = b i› ‹x ∈ S› x by blast
then have cface: "c face_of (S ∩ {x. a i ∙ x = b i})"
by (meson ‹c face_of S› face_of_subset inf_le1)
have con: "convex (S ∩ {x. a i ∙ x = b i})"
by (simp add: ‹convex S› convex_Int convex_hyperplane)
show "∃h. h ∈ F ∧ c = S ∩ {x. a h ∙ x = b h}"
apply (rule_tac x=i in exI)
apply (simp add: ‹i ∈ F›)
by (metis (no_types) * ‹i ∈ F› affc facet_of_def less_irrefl face_of_aff_dim_lt [OF con cface])
qed
qed

lemma face_of_polyhedron_subset_explicit:
fixes S :: "'a :: euclidean_space set"
assumes "finite F"
and seq: "S = affine hull S ∩ ⋂F"
and faceq: "⋀h. h ∈ F ⟹ a h ≠ 0 ∧ h = {x. a h ∙ x ≤ b h}"
and psub: "⋀F'. F' ⊂ F ⟹ S ⊂ affine hull S ∩ ⋂F'"
and c: "c face_of S" and "c ≠ {}" "c ≠ S"
obtains h where "h ∈ F" "c ⊆ S ∩ {x. a h ∙ x = b h}"
proof -
have "c ⊆ S" using ‹c face_of S›
have "polyhedron S"
by (metis ‹finite F› faceq seq)
then have "convex S"
then have *: "(S ∩ {x. a h ∙ x = b h}) face_of S" if "h ∈ F" for h
apply (rule face_of_Int_supporting_hyperplane_le)
using faceq seq that by fastforce
have "rel_interior c ≠ {}"
using c ‹c ≠ {}› face_of_imp_convex rel_interior_eq_empty by blast
then obtain x where "x ∈ rel_interior c" by auto
have rels: "rel_interior S = {x ∈ S. ∀h∈F. a h ∙ x < b h}"
by (rule rel_interior_polyhedron_explicit [OF ‹finite F› seq faceq psub])
then have xnot: "x ∉ rel_interior S"
by (metis IntI ‹x ∈ rel_interior c› c ‹c ≠ S› contra_subsetD empty_iff face_of_disjoint_rel_interior rel_interior_subset)
then have "x ∈ S"
using ‹c ⊆ S› ‹x ∈ rel_interior c› rel_interior_subset by auto
then have xint: "x ∈ ⋂F"
using seq by blast
have "F ≠ {}" using assms
by (metis affine_Int affine_Inter affine_affine_hull ex_in_conv face_of_affine_trivial)
then obtain i where "i ∈ F" "~ (a i ∙ x < b i)"
using ‹x ∈ S› rels xnot by auto
with xint have "a i ∙ x = b i"
by (metis eq_iff mem_Collect_eq not_le Inter_iff faceq)
have face: "S ∩ {x. a i ∙ x = b i} face_of S"
by (simp add: "*" ‹i ∈ F›)
show ?thesis
apply (rule_tac h = i in that)
apply (rule ‹i ∈ F›)
apply (rule subset_of_face_of [OF face ‹c ⊆ S›])
using ‹a i ∙ x = b i› ‹x ∈ rel_interior c› ‹x ∈ S› apply blast
done
qed

text‹Initial part of proof duplicates that above›
proposition face_of_polyhedron_explicit:
fixes S :: "'a :: euclidean_space set"
assumes "finite F"
and seq: "S = affine hull S ∩ ⋂F"
and faceq: "⋀h. h ∈ F ⟹ a h ≠ 0 ∧ h = {x. a h ∙ x ≤ b h}"
and psub: "⋀F'. F' ⊂ F ⟹ S ⊂ affine hull S ∩ ⋂F'"
and c: "c face_of S" and "c ≠ {}" "c ≠ S"
shows "c = ⋂{S ∩ {x. a h ∙ x = b h} | h. h ∈ F ∧ c ⊆ S ∩ {x. a h ∙ x = b h}}"
proof -
let ?ab = "λh. {x. a h ∙ x = b h}"
have "c ⊆ S" using ‹c face_of S›
have "polyhedron S"
by (metis ‹finite F› faceq seq)
then have "convex S"
then have *: "(S ∩ ?ab h) face_of S" if "h ∈ F" for h
apply (rule face_of_Int_supporting_hyperplane_le)
using faceq seq that by fastforce
have "rel_interior c ≠ {}"
using c ‹c ≠ {}› face_of_imp_convex rel_interior_eq_empty by blast
then obtain z where z: "z ∈ rel_interior c" by auto
have rels: "rel_interior S = {z ∈ S. ∀h∈F. a h ∙ z < b h}"
by (rule rel_interior_polyhedron_explicit [OF ‹finite F› seq faceq psub])
then have xnot: "z ∉ rel_interior S"
by (metis IntI ‹z ∈ rel_interior c› c ‹c ≠ S› contra_subsetD empty_iff face_of_disjoint_rel_interior rel_interior_subset)
then have "z ∈ S"
using ‹c ⊆ S› ‹z ∈ rel_interior c› rel_interior_subset by auto
with seq have xint: "z ∈ ⋂F" by blast
have "open (⋂h∈{h ∈ F. a h ∙ z < b h}. {w. a h ∙ w < b h})"
by (auto simp: ‹finite F› open_halfspace_lt open_INT)
then obtain e where "0 < e"
"ball z e ⊆ (⋂h∈{h ∈ F. a h ∙ z < b h}. {w. a h ∙ w < b h})"
by (auto intro: openE [of _ z])
then have e: "⋀h. ⟦h ∈ F; a h ∙ z < b h⟧ ⟹ ball z e ⊆ {w. a h ∙ w < b h}"
by blast
have "c ⊆ (S ∩ ?ab h) ⟷ z ∈ S ∩ ?ab h" if "h ∈ F" for h
proof
show "z ∈ S ∩ ?ab h ⟹ c ⊆ S ∩ ?ab h"
apply (rule subset_of_face_of [of _ S])
using that ‹c ⊆ S› ‹z ∈ rel_interior c›
using facet_of_polyhedron_explicit [OF ‹finite F› seq faceq psub]
unfolding facet_of_def
apply auto
done
next
show "c ⊆ S ∩ ?ab h ⟹ z ∈ S ∩ ?ab h"
using ‹z ∈ rel_interior c› rel_interior_subset by force
qed
then have **: "{S ∩ ?ab h | h. h ∈ F ∧ c ⊆ S ∧ c ⊆ ?ab h} =
{S ∩ ?ab h |h. h ∈ F ∧ z ∈ S ∩ ?ab h}"
by blast
have bsub: "ball z e ∩ affine hull ⋂{S ∩ ?ab h |h. h ∈ F ∧ a h ∙ z = b h}
⊆ affine hull S ∩ ⋂F ∩ ⋂{?ab h |h. h ∈ F ∧ a h ∙ z = b h}"
if "i ∈ F" and i: "a i ∙ z = b i" for i
proof -
have sub: "ball z e ∩ ⋂{?ab h |h. h ∈ F ∧ a h ∙ z = b h} ⊆ j"
if "j ∈ F" for j
proof -
have "a j ∙ z ≤ b j" using faceq that xint by auto
then consider "a j ∙ z < b j" | "a j ∙ z = b j" by linarith
then have "∃G. G ∈ {?ab h |h. h ∈ F ∧ a h ∙ z = b h} ∧ ball z e ∩ G ⊆ j"
proof cases
assume "a j ∙ z < b j"
then have "ball z e ∩ {x. a i ∙ x = b i} ⊆ j"
using e [OF ‹j ∈ F›] faceq that
by (fastforce simp: ball_def)
then show ?thesis
by (rule_tac x="{x. a i ∙ x = b i}" in exI) (force simp: ‹i ∈ F› i)
next
assume eq: "a j ∙ z = b j"
with faceq that show ?thesis
by (rule_tac x="{x. a j ∙ x = b j}" in exI) (fastforce simp add: ‹j ∈ F›)
qed
then show ?thesis  by blast
qed
have 1: "affine hull ⋂{S ∩ ?ab h |h. h ∈ F ∧ a h ∙ z = b h} ⊆ affine hull S"
apply (rule hull_mono)
using that ‹z ∈ S› by auto
have 2: "affine hull ⋂{S ∩ ?ab h |h. h ∈ F ∧ a h ∙ z = b h}
⊆ ⋂{?ab h |h. h ∈ F ∧ a h ∙ z = b h}"
by (rule hull_minimal) (auto intro: affine_hyperplane)
have 3: "ball z e ∩ ⋂{?ab h |h. h ∈ F ∧ a h ∙ z = b h} ⊆ ⋂F"
by (iprover intro: sub Inter_greatest)
have *: "⟦A ⊆ (B :: 'a set); A ⊆ C; E ∩ C ⊆ D⟧ ⟹ E ∩ A ⊆ (B ∩ D) ∩ C"
for A B C D E  by blast
show ?thesis by (intro * 1 2 3)
qed
have "∃h. h ∈ F ∧ c ⊆ ?ab h"
apply (rule face_of_polyhedron_subset_explicit [OF ‹finite F› seq faceq psub])
using assms by auto
then have fac: "⋂{S ∩ ?ab h |h. h ∈ F ∧ c ⊆ S ∩ ?ab h} face_of S"
using * by (force simp: ‹c ⊆ S› intro: face_of_Inter)
have red:
"(⋀a. P a ⟹ T ⊆ S ∩ ⋂{F x |x. P x}) ⟹ T ⊆ ⋂{S ∩ F x |x. P x}"
for P T F   by blast
have "ball z e ∩ affine hull ⋂{S ∩ ?ab h |h. h ∈ F ∧ a h ∙ z = b h}
⊆ ⋂{S ∩ ?ab h |h. h ∈ F ∧ a h ∙ z = b h}"
apply (rule red)
apply (metis seq bsub)
done
with ‹0 < e› have zinrel: "z ∈ rel_interior
(⋂{S ∩ ?ab h |h. h ∈ F ∧ z ∈ S ∧ a h ∙ z = b h})"
by (auto simp: mem_rel_interior_ball ‹z ∈ S›)
show ?thesis
apply (rule face_of_eq [OF c fac])
using z zinrel apply (force simp: **)
done
qed

subsection‹More general corollaries from the explicit representation›

corollary facet_of_polyhedron:
assumes "polyhedron S" and "c facet_of S"
obtains a b where "a ≠ 0" "S ⊆ {x. a ∙ x ≤ b}" "c = S ∩ {x. a ∙ x = b}"
proof -
obtain F where "finite F" and seq: "S = affine hull S ∩ ⋂F"
and faces: "⋀h. h ∈ F ⟹ ∃a b. a ≠ 0 ∧ h = {x. a ∙ x ≤ b}"
and min: "⋀F'. F' ⊂ F ⟹ S ⊂ (affine hull S) ∩ ⋂F'"
using assms by (simp add: polyhedron_Int_affine_minimal) meson
then obtain a b where ab: "⋀h. h ∈ F ⟹ a h ≠ 0 ∧ h = {x. a h ∙ x ≤ b h}"
by metis
obtain i where "i ∈ F" and c: "c = S ∩ {x. a i ∙ x = b i}"
using facet_of_polyhedron_explicit [OF ‹finite F› seq ab min] assms
by force
moreover have ssub: "S ⊆ {x. a i ∙ x ≤ b i}"
apply (subst seq)
using ‹i ∈ F› ab by auto
ultimately show ?thesis
by (rule_tac a = "a i" and b = "b i" in that) (simp_all add: ab)
qed

corollary face_of_polyhedron:
assumes "polyhedron S" and "c face_of S" and "c ≠ {}" and "c ≠ S"
shows "c = ⋂{F. F facet_of S ∧ c ⊆ F}"
proof -
obtain F where "finite F" and seq: "S = affine hull S ∩ ⋂F"
and faces: "⋀h. h ∈ F ⟹ ∃a b. a ≠ 0 ∧ h = {x. a ∙ x ≤ b}"
and min: "⋀F'. F' ⊂ F ⟹ S ⊂ (affine hull S) ∩ ⋂F'"
using assms by (simp add: polyhedron_Int_affine_minimal) meson
then obtain a b where ab: "⋀h. h ∈ F ⟹ a h ≠ 0 ∧ h = {x. a h ∙ x ≤ b h}"
by metis
show ?thesis
apply (subst face_of_polyhedron_explicit [OF ‹finite F› seq ab min])
apply (auto simp: assms facet_of_polyhedron_explicit [OF ‹finite F› seq ab min] cong: Collect_cong)
done
qed

lemma face_of_polyhedron_subset_facet:
assumes "polyhedron S" and "c face_of S" and "c ≠ {}" and "c ≠ S"
obtains F where "F facet_of S" "c ⊆ F"
using face_of_polyhedron assms
by (metis (no_types, lifting) Inf_greatest antisym_conv face_of_imp_subset mem_Collect_eq)

lemma exposed_face_of_polyhedron:
assumes "polyhedron S"
shows "F exposed_face_of S ⟷ F face_of S"
proof
show "F exposed_face_of S ⟹ F face_of S"
next
assume "F face_of S"
show "F exposed_face_of S"
proof (cases "F = {} ∨ F = S")
case True then show ?thesis
using ‹F face_of S› exposed_face_of by blast
next
case False
then have "{g. g facet_of S ∧ F ⊆ g} ≠ {}"
by (metis Collect_empty_eq_bot ‹F face_of S› assms empty_def face_of_polyhedron_subset_facet)
moreover have "⋀T. ⟦T facet_of S; F ⊆ T⟧ ⟹ T exposed_face_of S"
by (metis assms exposed_face_of facet_of_imp_face_of facet_of_polyhedron)
ultimately have "⋂{fa.
fa facet_of S ∧ F ⊆ 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 ‹F face_of S›], auto)
done
qed
qed

lemma face_of_polyhedron_polyhedron:
fixes S :: "'a :: euclidean_space set"
assumes "polyhedron S" "c face_of S" shows "polyhedron c"
by (metis assms face_of_imp_eq_affine_Int polyhedron_Int polyhedron_affine_hull polyhedron_imp_convex)

lemma finite_polyhedron_faces:
fixes S :: "'a :: euclidean_space set"
assumes "polyhedron S"
shows "finite {F. F face_of S}"
proof -
obtain F where "finite F" and seq: "S = affine hull S ∩ ⋂F"
and faces: "⋀h. h ∈ F ⟹ ∃a b. a ≠ 0 ∧ h = {x. a ∙ x ≤ b}"
and min:   "⋀F'. F' ⊂ F ⟹ S ⊂ (affine hull S) ∩ ⋂F'"
using assms by (simp add: polyhedron_Int_affine_minimal) meson
then obtain a b where ab: "⋀h. h ∈ F ⟹ a h ≠ 0 ∧ h = {x. a h ∙ x ≤ b h}"
by metis
have "finite {⋂{S ∩ {x. a h ∙ x = b h} |h. h ∈ F'}| F'. F' ∈ Pow F}"
moreover have "{F. F face_of S} - {{}, S} ⊆ {⋂{S ∩ {x. a h ∙ x = b h} |h. h ∈ F'}| F'. F' ∈ Pow F}"
apply clarify
apply (rename_tac c)
apply (drule face_of_polyhedron_explicit [OF ‹finite F› seq ab min, simplified], simp_all)
apply (erule ssubst)
apply (rule_tac x="{h ∈ F. c ⊆ S ∩ {x. a h ∙ x = b h}}" in exI, auto)
done
ultimately show ?thesis
by (meson finite.emptyI finite.insertI finite_Diff2 finite_subset)
qed

lemma finite_polyhedron_exposed_faces:
"polyhedron S ⟹ finite {F. F exposed_face_of S}"
using exposed_face_of_polyhedron finite_polyhedron_faces by fastforce

lemma finite_polyhedron_extreme_points:
fixes S :: "'a :: euclidean_space set"
shows "polyhedron S ⟹ finite {v. v extreme_point_of S}"
apply (rule finite_subset [OF _ finite_vimageI [OF finite_polyhedron_faces]], auto)
done

lemma finite_polyhedron_facets:
fixes S :: "'a :: euclidean_space set"
shows "polyhedron S ⟹ finite {F. F facet_of S}"
unfolding facet_of_def
by (blast intro: finite_subset [OF _ finite_polyhedron_faces])

proposition rel_interior_of_polyhedron:
fixes S :: "'a :: euclidean_space set"
assumes "polyhedron S"
shows "rel_interior S = S - ⋃{F. F facet_of S}"
proof -
obtain F where "finite F" and seq: "S = affine hull S ∩ ⋂F"
and faces: "⋀h. h ∈ F ⟹ ∃a b. a ≠ 0 ∧ h = {x. a ∙ x ≤ b}"
and min: "⋀F'. F' ⊂ F ⟹ S ⊂ (affine hull S) ∩ ⋂F'"
using assms by (simp add: polyhedron_Int_affine_minimal) meson
then obtain a b where ab: "⋀h. h ∈ F ⟹ a h ≠ 0 ∧ h = {x. a h ∙ x ≤ b h}"
by metis
have facet: "(c facet_of S) ⟷ (∃h. h ∈ F ∧ c = S ∩ {x. a h ∙ x = b h})" for c
by (rule facet_of_polyhedron_explicit [OF ‹finite F› seq ab min])
have rel: "rel_interior S = {x ∈ S. ∀h∈F. a h ∙ x < b h}"
by (rule rel_interior_polyhedron_explicit [OF ‹finite F› seq ab min])
have "a h ∙ x < b h" if "x ∈ S" "h ∈ F" and xnot: "x ∉ ⋃{F. F facet_of S}" for x h
proof -
have "x ∈ ⋂F" using seq that by force
with ‹h ∈ F› ab have "a h ∙ x ≤ b h" by auto
then consider "a h ∙ x < b h" | "a h ∙ x = b h" by linarith
then show ?thesis
proof cases
case 1 then show ?thesis .
next
case 2
have "Collect ((∈) x) ∉ Collect ((∈) (⋃{A. A facet_of S}))"
using xnot by fastforce
then have "F ∉ Collect ((∈) h)"
using 2 ‹x ∈ S› facet by blast
with ‹h ∈ F› have "⋂F ⊆ S ∩ {x. a h ∙ x = b h}" by blast
with 2 that ‹x ∈ ⋂F› show ?thesis
apply simp
apply (drule_tac x="⋂F" in spec)
apply (drule_tac x=h in spec)
using seq by auto
qed
qed
moreover have "∃h∈F. a h ∙ x ≥ b h" if "x ∈ ⋃{F. F facet_of S}" for x
using that by (force simp: facet)
ultimately show ?thesis
by (force simp: rel)
qed

lemma rel_boundary_of_polyhedron:
fixes S :: "'a :: euclidean_space set"
assumes "polyhedron S"
shows "S - rel_interior S = ⋃ {F. F facet_of S}"
using facet_of_imp_subset by (fastforce simp add: rel_interior_of_polyhedron assms)

lemma rel_frontier_of_polyhedron:
fixes S :: "'a :: euclidean_space set"
assumes "polyhedron S"
shows "rel_frontier S = ⋃ {F. F facet_of S}"
by (simp add: assms rel_frontier_def polyhedron_imp_closed rel_boundary_of_polyhedron)

lemma rel_frontier_of_polyhedron_alt:
fixes S :: "'a :: euclidean_space set"
assumes "polyhedron S"
shows "rel_frontier S = ⋃ {F. F face_of S ∧ (F ≠ 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‹A characterization of polyhedra as having finitely many faces›

proposition polyhedron_eq_finite_exposed_faces:
fixes S :: "'a :: euclidean_space set"
shows "polyhedron S ⟷ closed S ∧ convex S ∧ finite {F. F exposed_face_of S}"
(is "?lhs = ?rhs")
proof
assume ?lhs
then show ?rhs
by (auto simp: polyhedron_imp_closed polyhedron_imp_convex finite_polyhedron_exposed_faces)
next
assume ?rhs
then have "closed S" "convex S" and fin: "finite {F. F exposed_face_of S}" by auto
show ?lhs
proof (cases "S = {}")
case True then show ?thesis by auto
next
case False
define F where "F = {h. h exposed_face_of S ∧ h ≠ {} ∧ h ≠ S}"
have "finite F" by (simp add: fin F_def)
have hface: "h face_of S"
and "∃a b. a ≠ 0 ∧ S ⊆ {x. a ∙ x ≤ b} ∧ h = S ∩ {x. a ∙ x = b}"
if "h ∈ F" for h
using exposed_face_of F_def that by simp_all auto
then obtain a b where ab:
"⋀h. h ∈ F ⟹ a h ≠ 0 ∧ S ⊆ {x. a h ∙ x ≤ b h} ∧ h = S ∩ {x. a h ∙ x = b h}"
by metis
have *: "False"
if paff: "p ∈ affine hull S" and "p ∉ S"
and pint: "p ∈ ⋂{{x. a h ∙ x ≤ b h} |h. h ∈ F}" for p
proof -
have "rel_interior S ≠ {}"
by (simp add: ‹S ≠ {}› ‹convex S› rel_interior_eq_empty)
then obtain c where c: "c ∈ rel_interior S" by auto
with rel_interior_subset have "c ∈ S"  by blast
have ccp: "closed_segment c p ⊆ 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 ∈ closed_segment c p" and "x ∈ S" and xnot: "x ∉ rel_interior S"
using connected_openin [of "closed_segment c p"]
apply simp
apply (drule_tac x="closed_segment c p ∩ 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 ∩ (- S)" in spec)
using rel_interior_subset ‹closed S› c ‹p ∉ S› apply blast
done
then obtain μ where "0 ≤ μ" "μ ≤ 1" and xeq: "x = (1 - μ) *⇩R c + μ *⇩R p"
by (auto simp: in_segment)
show False
proof (cases "μ=0 ∨ μ=1")
case True with xeq c xnot ‹x ∈ S› ‹p ∉ S›
show False by auto
next
case False
then have xos: "x ∈ open_segment c p"
using ‹x ∈ S› c open_segment_def that(2) xcl xnot by auto
have xclo: "x ∈ closure S"
using ‹x ∈ S› closure_subset by blast
obtain d where "d ≠ 0"
and dle: "⋀y. y ∈ closure S ⟹ d ∙ x ≤ d ∙ y"
and dless: "⋀y. y ∈ rel_interior S ⟹ d ∙ x < d ∙ y"
by (metis supporting_hyperplane_relative_frontier [OF ‹convex S› xclo xnot])
have sex: "S ∩ {y. d ∙ y = d ∙ x} exposed_face_of S"
by (simp add: ‹closed S› dle exposed_face_of_Int_supporting_hyperplane_ge [OF ‹convex S›])
have sne: "S ∩ {y. d ∙ y = d ∙ x} ≠ {}"
using ‹x ∈ S› by blast
have sns: "S ∩ {y. d ∙ y = d ∙ x} ≠ S"
by (metis (mono_tags) Int_Collect c subsetD dless not_le order_refl rel_interior_subset)
obtain h where "h ∈ F" "x ∈ h"
apply (rule_tac h="S ∩ {y. d ∙ y = d ∙ x}" in that)
apply (simp_all add: F_def sex sne sns ‹x ∈ S›)
done
have abface: "{y. a h ∙ y = b h} face_of {y. a h ∙ y ≤ b h}"
using hyperplane_face_of_halfspace_le by blast
then have "c ∈ h"
using face_ofD [OF abface xos] ‹c ∈ S› ‹h ∈ F› ab pint ‹x ∈ h› by blast
with c have "h ∩ rel_interior S ≠ {}" by blast
then show False
using ‹h ∈ F› F_def face_of_disjoint_rel_interior hface by auto
qed
qed
have "S ⊆ affine hull S ∩ ⋂{{x. a h ∙ x ≤ b h} |h. h ∈ F}"
using ab by (auto simp: hull_subset)
moreover have "affine hull S ∩ ⋂{{x. a h ∙ x ≤ b h} |h. h ∈ F} ⊆ S"
using * by blast
ultimately have "S = affine hull S ∩ ⋂ {{x. a h ∙ x ≤ b h} |h. h ∈ F}" ..
then show ?thesis
apply (rule ssubst)
apply (force intro: polyhedron_affine_hull polyhedron_halfspace_le simp: ‹finite F›)
done
qed
qed

corollary polyhedron_eq_finite_faces:
fixes S :: "'a :: euclidean_space set"
shows "polyhedron S ⟷ closed S ∧ convex S ∧ finite {F. F face_of S}"
(is "?lhs = ?rhs")
proof
assume ?lhs
then show ?rhs
by (simp add: finite_polyhedron_faces polyhedron_imp_closed polyhedron_imp_convex)
next
assume ?rhs
then show ?lhs
by (force simp: polyhedron_eq_finite_exposed_faces exposed_face_of intro: finite_subset)
qed

lemma polyhedron_linear_image_eq:
fixes h :: "'a :: euclidean_space ⇒ 'b :: euclidean_space"
assumes "linear h" "bij h"
shows "polyhedron (h ` S) ⟷ 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: ‹bij h› 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
show ?thesis
using ‹linear h› ‹inj h›
apply (simp add: * face_of_linear_image [of h _ S, symmetric] finite_image_iff injim)
done
qed

lemma polyhedron_negations:
fixes S :: "'a :: euclidean_space set"
shows   "polyhedron S ⟹ polyhedron(image uminus S)"
by (subst polyhedron_linear_image_eq)
(auto simp: bij_uminus intro!: linear_uminus)

subsection‹Relation between polytopes and polyhedra›

lemma polytope_eq_bounded_polyhedron:
fixes S :: "'a :: euclidean_space set"
shows "polytope S ⟷ polyhedron S ∧ bounded S"
(is "?lhs = ?rhs")
proof
assume ?lhs
then show ?rhs
polytope_imp_closed polytope_imp_convex polytope_imp_bounded)
next
assume ?rhs then show ?lhs
unfolding polytope_def
apply (rule_tac x="{v. v extreme_point_of S}" in exI)
apply (simp add: finite_polyhedron_extreme_points Krein_Milman_Minkowski compact_eq_bounded_closed polyhedron_imp_closed polyhedron_imp_convex)
done
qed

lemma polytope_Int:
fixes S :: "'a :: euclidean_space set"
shows "⟦polytope S; polytope T⟧ ⟹ polytope(S ∩ T)"

lemma polytope_Int_polyhedron:
fixes S :: "'a :: euclidean_space set"
shows "⟦polytope S; polyhedron T⟧ ⟹ polytope(S ∩ T)"

lemma polyhedron_Int_polytope:
fixes S :: "'a :: euclidean_space set"
shows "⟦polyhedron S; polytope T⟧ ⟹ polytope(S ∩ T)"

lemma polytope_imp_polyhedron:
fixes S :: "'a :: euclidean_space set"
shows "polytope S ⟹ polyhedron S"

lemma polytope_facet_exists:
fixes p :: "'a :: euclidean_space set"
assumes "polytope p" "0 < aff_dim p"
obtains F where "F facet_of p"
proof (cases "p = {}")
case True with assms show ?thesis by auto
next
case False
then obtain v where "v extreme_point_of p"
using extreme_point_exists_convex
by (blast intro: ‹polytope p› polytope_imp_compact polytope_imp_convex)
then
show ?thesis
by (metis face_of_polyhedron_subset_facet polytope_imp_polyhedron aff_dim_sing
all_not_in_conv assms face_of_singleton less_irrefl singletonI that)
qed

lemma polyhedron_interval [iff]: "polyhedron(cbox a b)"
by (metis polytope_imp_polyhedron polytope_interval)

lemma polyhedron_convex_hull:
fixes S :: "'a :: euclidean_space set"
shows "finite S ⟹ polyhedron(convex hull S)"

subsection‹Relative and absolute frontier of a polytope›

lemma rel_boundary_of_convex_hull:
fixes S :: "'a::euclidean_space set"
assumes "~ affine_dependent S"
shows "(convex hull S) - rel_interior(convex hull S) = (⋃a∈S. convex hull (S - {a}))"
proof -
have "finite S" by (metis assms aff_independent_finite)
then consider "card S = 0" | "card S = 1" | "2 ≤ card S" by arith
then show ?thesis
proof cases
case 1 then have "S = {}" by (simp add: ‹finite S›)
then show ?thesis by simp
next
case 2 show ?thesis
by (auto intro: card_1_singletonE [OF ‹card S = 1›])
next
case 3
with assms show ?thesis
by (auto simp: polyhedron_convex_hull rel_boundary_of_polyhedron facet_of_convex_hull_affine_independent_alt ‹finite S›)
qed
qed

proposition frontier_of_convex_hull:
fixes S :: "'a::euclidean_space set"
assumes "card S = Suc (DIM('a))"
shows "frontier(convex hull S) = ⋃ {convex hull (S - {a}) | a. a ∈ 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)"
{ fix x T
assume "finite T" "T ⊆ S" "int (card T) ≤ aff_dim S + 1" "x ∈ convex hull T"
then have "S ≠ T"
using True ‹finite S› aff_dim_le_card affine_independent_iff_card by fastforce
then obtain a where "a ∈ S" "a ∉ T"
using ‹T ⊆ S› by blast
then have "x ∈ (⋃a∈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 ‹a ∉ T› ‹T ⊆ S› ‹x ∈ convex hull T›  hull_mono insert_Diff_single   subsetCE)
done
} note * = this
have 1: "convex hull S ⊆ (⋃ a∈S. convex hull (S - {a}))"
apply (subst caratheodory_aff_dim)
apply (blast intro: *)
done
have 2: "⋃((λa. convex hull (S - {a})) ` S) ⊆ convex hull S"
by (rule Union_least) (metis (no_types, lifting)  Diff_subset hull_mono imageE)
show ?thesis using True
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 "... = ⋃{convex hull (S - {a}) |a. a ∈ 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
qed
finally show ?thesis .
qed

subsection‹Special case of a triangle›

proposition frontier_of_triangle:
fixes a :: "'a::euclidean_space"
assumes "DIM('a) = 2"
shows "frontier(convex hull {a,b,c}) = closed_segment a b ∪ closed_segment b c ∪ closed_segment c a"
(is "?lhs = ?rhs")
proof (cases "b = a ∨ c = a ∨ 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 ⊆ ?rhs"
using False
by (force simp: segment_convex_hull frontier_of_convex_hull insert_Diff_if insert_commute split: if_split_asm)
show "?rhs ⊆ ?lhs"
using False
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 ∪ closed_segment b c ∪ 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 ∪ closed_segment b c ∪ closed_segment c a)"
using interior_subset
by (force simp: frontier_of_triangle [OF assms, symmetric] frontier_def Diff_Diff_Int)

subsection‹Subdividing a cell complex›

lemma subdivide_interval:
fixes x::real
assumes "a < ¦x - y¦" "0 < a"
obtains n where "n ∈ ℤ" "x < n * a ∧ n * a < y ∨ y <  n * a ∧ 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 ‹0 < a› divide_less_eq floor_eq_iff)
have "?n * a ≤ a + x"
by (metis ‹0 < a› 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 ‹0 < a› divide_less_eq floor_eq_iff)
have "?n * a ≤ a + y"
by (metis ‹0 < a› 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 "⋀X. X ∈ ℱ ⟹ polytope X"
and "⋀X. X ∈ ℱ ⟹ aff_dim X ≤ d"
and "⋀X Y. ⟦X ∈ ℱ; Y ∈ ℱ⟧ ⟹ (X ∩ Y) face_of X ∧ (X ∩ Y) face_of Y"
and "finite I"
shows "∃𝒢. ⋃𝒢 = ⋃ℱ ∧
finite 𝒢 ∧
(∀C ∈ 𝒢. ∃D. D ∈ ℱ ∧ C ⊆ D) ∧
(∀C ∈ ℱ. ∀x ∈ C. ∃D. D ∈ 𝒢 ∧ x ∈ D ∧ D ⊆ C) ∧
(∀X ∈ 𝒢. polytope X) ∧
(∀X ∈ 𝒢. aff_dim X ≤ d) ∧
(∀X ∈ 𝒢. ∀Y ∈ 𝒢. X ∩ Y face_of X ∧ X ∩ Y face_of Y) ∧
(∀X ∈ 𝒢. ∀x ∈ X. ∀y ∈ X. ∀a b.
(a,b) ∈ I ⟶ a ∙ x ≤ b ∧ a ∙ y ≤ b ∨
a ∙ x ≥ b ∧ a ∙ y ≥ b)"
using ‹finite I›
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: "⋃𝒢 = ⋃ℱ" and "finite 𝒢"
and sub1: "⋀C. C ∈ 𝒢 ⟹ ∃D. D ∈ ℱ ∧ C ⊆ D"
and sub2: "⋀C x. C ∈ ℱ ∧ x ∈ C ⟹ ∃D. D ∈ 𝒢 ∧ x ∈ D ∧ D ⊆ C"
and poly: "⋀X. X ∈ 𝒢 ⟹ polytope X"
and aff: "⋀X. X ∈ 𝒢 ⟹ aff_dim X ≤ d"
and face: "⋀X Y. ⟦X ∈ 𝒢; Y ∈ 𝒢⟧ ⟹ X ∩ Y face_of X ∧ X ∩ Y face_of Y"
and I: "⋀X x y a b.  ⟦X ∈ 𝒢; x ∈ X; y ∈ X; (a,b) ∈ I⟧ ⟹
a ∙ x ≤ b ∧ a ∙ y ≤ b ∨ a ∙ x ≥ b ∧ a ∙ y ≥ b"
by (auto simp: that)
obtain a b where "ab = (a,b)"
by fastforce
let ?𝒢 = "(λX. X ∩ {x. a ∙ x ≤ b}) ` 𝒢 ∪ (λX. X ∩ {x. a ∙ x ≥ b}) ` 𝒢"
have eqInt: "(S ∩ Collect P) ∩ (T ∩ Collect Q) = (S ∩ T) ∩ (Collect P ∩ Collect Q)" for S T::"'a set" and P Q
by blast
show ?case
proof (intro conjI exI)
show "⋃?𝒢 = ⋃ℱ"
by (force simp: eq [symmetric])
show "finite ?𝒢"
using ‹finite 𝒢› by force
show "∀X ∈ ?𝒢. polytope X"
by (force simp: poly polytope_Int_polyhedron polyhedron_halfspace_le polyhedron_halfspace_ge)
show "∀X ∈ ?𝒢. aff_dim X ≤ d"
by (auto; metis order_trans aff aff_dim_subset inf_le1)
show "∀X ∈ ?𝒢. ∀x ∈ X. ∀y ∈ X. ∀a b.
(a,b) ∈ insert ab I ⟶ a ∙ x ≤ b ∧ a ∙ y ≤ b ∨
a ∙ x ≥ b ∧ a ∙ y ≥ b"
using ‹ab = (a, b)› I by fastforce
show "∀X ∈ ?𝒢. ∀Y ∈ ?𝒢. X ∩ Y face_of X ∧ X ∩ 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 "∀C ∈ ?𝒢. ∃D. D ∈ ℱ ∧ C ⊆ D"
using sub1 by force
show "∀C∈ℱ. ∀x∈C. ∃D. D ∈ ?𝒢 ∧ x ∈ D ∧ D ⊆ C"
proof (intro ballI)
fix C z
assume "C ∈ ℱ" "z ∈ C"
with sub2 obtain D where D: "D ∈ 𝒢" "z ∈ D" "D ⊆ C" by blast
have "D ∈ 𝒢 ∧ z ∈ D ∩ {x. a ∙ x ≤ b} ∧ D ∩ {x. a ∙ x ≤ b} ⊆ C ∨
D ∈ 𝒢 ∧ z ∈ D ∩ {x. a ∙ x ≥ b} ∧ D ∩ {x. a ∙ x ≥ b} ⊆ C"
using linorder_class.linear [of "a ∙ z" b] D by blast
then show "∃D. D ∈ ?𝒢 ∧ z ∈ D ∧ D ⊆ 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: "⋀X. X ∈ ℱ ⟹ polytope X"
and aff: "⋀X. X ∈ ℱ ⟹ aff_dim X ≤ d"
and face: "⋀X Y. ⟦X ∈ ℱ; Y ∈ ℱ⟧ ⟹ X ∩ Y face_of X ∧ X ∩ Y face_of Y"
obtains "ℱ'" where "finite ℱ'" "⋃ℱ' = ⋃ℱ" "⋀X. X ∈ ℱ' ⟹ diameter X < e"
"⋀X. X ∈ ℱ' ⟹ polytope X" "⋀X. X ∈ ℱ' ⟹ aff_dim X ≤ d"
"⋀X Y. ⟦X ∈ ℱ'; Y ∈ ℱ'⟧ ⟹ X ∩ Y face_of X ∧ X ∩ Y face_of Y"
"⋀C. C ∈ ℱ' ⟹ ∃D. D ∈ ℱ ∧ C ⊆ D"
"⋀C x. C ∈ ℱ ∧ x ∈ C ⟹ ∃D. D ∈ ℱ' ∧ x ∈ D ∧ D ⊆ C"
proof -
have "bounded(⋃ℱ)"
by (simp add: ‹finite ℱ› poly bounded_Union polytope_imp_bounded)
then obtain B where "B > 0" and B: "⋀x. x ∈ ⋃ℱ ⟹ norm x < B"
by (meson bounded_pos_less)
define C where "C ≡ {z ∈ ℤ. ¦z * e / 2 / real DIM('a)¦ ≤ B}"
define I where "I ≡ ⋃i ∈ Basis. ⋃j ∈ 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 (erule rev_finite_subset)
using ‹0 < e›
apply (auto simp: divide_simps)
done
then have "finite I"
obtain ℱ' where eq: "⋃ℱ' = ⋃ℱ" and "finite ℱ'"
and poly: "⋀X. X ∈ ℱ' ⟹ polytope X"
and aff: "⋀X. X ∈ ℱ' ⟹ aff_dim X ≤ d"
and face: "⋀X Y. ⟦X ∈ ℱ'; Y ∈ ℱ'⟧ ⟹ X ∩ Y face_of X ∧ X ∩ Y face_of Y"
and I: "⋀X x y a b.  ⟦X ∈ ℱ'; x ∈ X; y ∈ X; (a,b) ∈ I⟧ ⟹
a ∙ x ≤ b ∧ a ∙ y ≤ b ∨ a ∙ x ≥ b ∧ a ∙ y ≥ b"
and sub1: "⋀C. C ∈ ℱ' ⟹ ∃D. D ∈ ℱ ∧ C ⊆ D"
and sub2: "⋀C x. C ∈ ℱ ∧ x ∈ C ⟹ ∃D. D ∈ ℱ' ∧ x ∈ D ∧ D ⊆ C"
apply (rule exE [OF cell_subdivision_lemma])
using assms ‹finite I› apply auto
done
show ?thesis
proof (rule_tac ℱ'="ℱ'" in that)
show "diameter X < e" if "X ∈ ℱ'" for X
proof -
have "diameter X ≤ e/2"
proof (rule diameter_le)
show "norm (x - y) ≤ e / 2" if "x ∈ X" "y ∈ X" for x y
proof -
have "norm x < B" "norm y < B"
using B ‹X ∈ ℱ'› eq that by fastforce+
have "norm (x - y) ≤ (∑b∈Basis. ¦(x-y) ∙ b¦)"
by (rule norm_le_l1)
also have "... ≤ of_nat (DIM('a)) * (e / 2 / DIM('a))"
proof (rule sum_bounded_above)
fix i::'a
assume "i ∈ Basis"
then have I': "⋀z b. ⟦z ∈ C; b = z * e / (2 * real DIM('a))⟧ ⟹ i ∙ x ≤ b ∧ i ∙ y ≤ b ∨ i ∙ x ≥ b ∧ i ∙ y ≥ b"
using I ‹X ∈ ℱ'› that
by (fastforce simp: I_def)
show "¦(x - y) ∙ i¦ ≤ e / 2 / real DIM('a)"
proof (rule ccontr)
assume "¬ ¦(x - y) ∙ i¦ ≤ e / 2 / real DIM('a)"
then have xyi: "¦i ∙ x - i ∙ y¦ > e / 2 / real DIM('a)"
obtain n where "n ∈ ℤ" and n: "i ∙ x < n * (e / 2 / real DIM('a)) ∧ n * (e / 2 / real DIM('a)) < i ∙ y ∨ i ∙ y < n * (e / 2 / real DIM('a)) ∧ n * (e / 2 / real DIM('a)) < i ∙ x"
using subdivide_interval [OF xyi] DIM_positive ‹0 < e›
by (auto simp: zero_less_divide_iff)
have "¦i ∙ x¦ < B"
by (metis ‹i ∈ Basis› ‹norm x < B› inner_commute norm_bound_Basis_lt)
have "¦i ∙ y¦ < B"
by (metis ‹i ∈ Basis› ‹norm y < B› inner_commute norm_bound_Basis_lt)
have *: "¦n * e¦ ≤ B * (2 * real DIM('a))"
if "¦ix¦ < B" "¦iy¦ < 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)) ≤ B * (2 * real DIM('a))"
by (rule mult_right_mono) (use ‹¦iy¦ < B› in linarith)+
then show "n * e ≤ B * (2 * real DIM('a))"
using iy by linarith
next
have "- ix * (2 * real DIM('a)) ≤ B * (2 * real DIM('a))"
by (rule mult_right_mono) (use ‹¦ix¦ < B› in linarith)+
then show "- (n * e) ≤ B * (2 * real DIM('a))"
using ix by linarith
qed
have "n ∈ C"
using ‹n ∈ ℤ› n  by (auto simp: C_def divide_simps intro: * ‹¦i ∙ x¦ < B› ‹¦i ∙ y¦ < B›)
show False
using  I' [OF ‹n ∈ C› refl] n  by auto
qed
qed
also have "... = e / 2"
by simp
finally show ?thesis .
qed
qed (use ‹0 < e› in force)
also have "... < e"
by (simp add: ‹0 < e›)
finally show ?thesis .
qed
qed (auto simp: eq poly aff face sub1 sub2 ‹finite ℱ'›)
qed

subsection‹Simplexes›

text‹The notion of n-simplex for integer @{term"n ≥ -1"}›
definition simplex :: "int ⇒ 'a::euclidean_space set ⇒ bool" (infix "simplex" 50)
where "n simplex S ≡ ∃C. ~(affine_dependent C) ∧ int(card C) = n + 1 ∧ S = convex hull C"

lemma simplex:
"n simplex S ⟷ (∃C. finite C ∧
~(affine_dependent C) ∧
int(card C) = n + 1 ∧
S = convex hull C)"
by (auto simp add: simplex_def intro: aff_independent_finite)

lemma simplex_convex_hull:
"~affine_dependent C ∧ int(card C) = n + 1 ⟹ n simplex (convex hull C)"

lemma convex_simplex: "n simplex S ⟹ convex S"
by (metis convex_convex_hull simplex_def)

lemma compact_simplex: "n simplex S ⟹ compact S"
unfolding simplex
using finite_imp_compact_convex_hull by blast

lemma closed_simplex: "n simplex S ⟹ closed S"

lemma simplex_imp_polytope:
"n simplex S ⟹ polytope S"
unfolding simplex_def polytope_def
using aff_independent_finite by blast

lemma simplex_imp_polyhedron:
"n simplex S ⟹ polyhedron S"

lemma simplex_dim_ge: "n simplex S ⟹ -1 ≤ 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 {} ⟷ 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 ⟷ S = {}"

lemma aff_dim_simplex:
"n simplex S ⟹ aff_dim S = n"

lemma zero_simplex_sing: "0 simplex {a}"
by (metis affine_independent_1 card_empty card_insert_disjoint convex_hull_singleton empty_iff finite.emptyI)

lemma simplex_sing [simp]: "n simplex {a} ⟷ n = 0"
using aff_dim_simplex aff_dim_sing zero_simplex_sing by blast

lemma simplex_zero: "0 simplex S ⟷ (∃a. S = {a})"
apply (auto simp: )
using aff_dim_eq_0 aff_dim_simplex by blast

lemma one_simplex_segment: "a ≠ b ⟹ 1 simplex closed_segment a b"
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:
"∃n. n simplex closed_segment a b"
using simplex_segment_cases by metis

lemma polytope_lowdim_imp_simplex:
assumes "polytope P" "aff_dim P ≤ 1"
obtains n where "n simplex P"
proof (cases "P = {}")
case True
then show ?thesis
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 ∉ affine hull S"
shows "(n+1) simplex (convex hull (insert a S))"
proof -
obtain C where C: "finite C" "~(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 ∉ affine hull C"
using S a affine_hull_convex_hull by blast
moreover have "a ∉ C"
using S a hull_inc by fastforce
ultimately show "¬ affine_dependent (insert a C)"
by (simp add: C S aff_dim_convex_hull aff_dim_insert affine_independent_iff_card)
next
have "a ∉ C"
using S a hull_inc by fastforce
then show "int (card (insert a C)) = n + 1 + 1"
next
show "convex hull insert a S = convex hull (insert a C)"
qed (use C in auto)
qed

subsection‹Simplicial complexes and triangulations›

definition simplicial_complex where
"simplicial_complex 𝒞 ≡
finite 𝒞 ∧
(∀S ∈ 𝒞. ∃n. n simplex S) ∧
(∀F S. S ∈ 𝒞 ∧ F face_of S ⟶ F ∈ 𝒞) ∧
(∀S S'. S ∈ 𝒞 ∧ S' ∈ 𝒞
⟶ (S ∩ S') face_of S ∧ (S ∩ S') face_of S')"

definition triangulation where
"triangulation 𝒯 ≡
finite 𝒯 ∧
(∀T ∈ 𝒯. ∃n. n simplex T) ∧
(∀T T'. T ∈ 𝒯 ∧ T' ∈ 𝒯
⟶ (T ∩ T') face_of T ∧ (T ∩ T') face_of T')"

subsection‹Refining a cell complex to a simplicial complex›

lemma convex_hull_insert_Int_eq:
fixes z :: "'a :: euclidean_space"
assumes z: "z ∈ rel_interior S"
and T: "T ⊆ rel_frontier S"
and U: "U ⊆ rel_frontier S"
and "convex S" "convex T" "convex U"
shows "convex hull (insert z T) ∩ convex hull (insert z U) = convex hull (insert z (T ∩ U))"
(is "?lhs = ?rhs")
proof
show "?lhs ⊆ ?rhs"
proof (cases "T={} ∨ U={}")
case True then show ?thesis by auto
next
case False
then have "T ≠ {}" "U ≠ {}" by auto
have TU: "convex (T ∩ U)"
by (simp add: ‹convex T› ‹convex U› convex_Int)
have "(⋃x∈T. closed_segment z x) ∩ (⋃x∈U. closed_segment z x)
⊆ (if T ∩ U = {} then {z} else UNION (T ∩ U) (closed_segment z))" (is "_ ⊆ ?IF")
proof clarify
fix x t u
assume xt: "x ∈ closed_segment z t"
and xu: "x ∈ closed_segment z u"
and "t ∈ T" "u ∈ U"
then have ne: "t ≠ z" "u ≠ z"
using T U z unfolding rel_frontier_def by blast+
show "x ∈ ?IF"
proof (cases "x = z")
case True then show ?thesis by auto
next
case False
have t: "t ∈ closure S"
using T ‹t ∈ T› rel_frontier_def by auto
have u: "u ∈ closure S"
using U ‹u ∈ U› rel_frontier_def by auto
show ?thesis
proof (cases "t = u")
case True
then show ?thesis
using ‹t ∈ T› ‹u ∈ U› xt by auto
next
case False
have tnot: "t ∉ closed_segment u z"
proof -
have "t ∈ closure S - rel_interior S"
using T ‹t ∈ T› rel_frontier_def by blast
then have "t ∉ open_segment z u"
by (meson DiffD2 rel_interior_closure_convex_segment [OF ‹convex S› z u] subsetD)
then show ?thesis
by (simp add: ‹t ≠ u› ‹t ≠ z› open_segment_commute open_segment_def)
qed
moreover have "u ∉ closed_segment z t"
using rel_interior_closure_convex_segment [OF ‹convex S› z t] ‹u ∈ U› ‹u ≠ z›
U [unfolded rel_frontier_def] tnot
by (auto simp: closed_segment_eq_open)
ultimately
have "~(between (t,u) z | between (u,z) t | between (z,t) u)" if "x ≠ z"
using that xt xu
by (metis between_commute between_trans_2 between_antisym)
then have "~ collinear {t, z, u}" if "x ≠ 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] ‹x ≠ z› by blast
then show ?thesis by metis
qed
qed
qed
then show ?thesis
using False ‹convex T› ‹convex U› TU
by (simp add: convex_hull_insert_segments hull_same split: if_split_asm)
qed
show "?rhs ⊆ ?lhs"
by (metis inf_greatest hull_mono inf.cobounded1 inf.cobounded2 insert_mono)
qed

lemma simplicial_subdivision_aux:
assumes "finite ℳ"
and "⋀C. C ∈ ℳ ⟹ polytope C"
and "⋀C. C ∈ ℳ ⟹ aff_dim C ≤ of_nat n"
and "⋀C F. ⟦C ∈ ℳ; F face_of C⟧ ⟹ F ∈ ℳ"
and "⋀C1 C2. ⟦C1 ∈ ℳ; C2 ∈ ℳ⟧ ⟹ C1 ∩ C2 face_of C1 ∧ C1 ∩ C2 face_of C2"
shows "∃𝒯. simplicial_complex 𝒯 ∧
(∀K ∈ 𝒯. aff_dim K ≤ of_nat n) ∧
⋃𝒯 = ⋃ℳ ∧
(∀C ∈ ℳ. ∃F. finite F ∧ F ⊆ 𝒯 ∧ C = ⋃F) ∧
(∀K ∈ 𝒯. ∃C. C ∈ ℳ ∧ K ⊆ C)"
using assms
proof (induction n arbitrary: ℳ rule: less_induct)
case (less n)
then have polyℳ: "⋀C. C ∈ ℳ ⟹ polytope C"
and affℳ:    "⋀C. C ∈ ℳ ⟹ aff_dim C ≤ of_nat n"
and faceℳ:   "⋀C F. ⟦C ∈ ℳ; F face_of C⟧ ⟹ F ∈ ℳ"
and intfaceℳ: "⋀C1 C2. ⟦C1 ∈ ℳ; C2 ∈ ℳ⟧ ⟹ C1 ∩ C2 face_of C1 ∧ C1 ∩ C2 face_of C2"
by metis+
show ?case
proof (cases "n ≤ 1")
case True
have "⋀s. ⟦n ≤ 1; s ∈ ℳ⟧ ⟹ ∃m. m simplex s"
using polyℳ affℳ 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 ≡ {C ∈ ℳ. aff_dim C < n}"
have "finite 𝒮" "⋀C. C ∈ 𝒮 ⟹ polytope C" "⋀C. C ∈ 𝒮 ⟹ aff_dim C ≤ int (n - 1)"
"⋀C F. ⟦C ∈ 𝒮; F face_of C⟧ ⟹ F ∈ 𝒮"
"⋀C1 C2. ⟦C1 ∈ 𝒮; C2 ∈ 𝒮⟧  ⟹ C1 ∩ C2 face_of C1 ∧ C1 ∩ 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_dim𝒰: "⋀K. K ∈ 𝒰 ⟹ aff_dim K ≤ int (n - 1)"
and        "⋃𝒰 = ⋃𝒮"
and fin𝒰:  "⋀C. C ∈ 𝒮 ⟹ ∃F. finite F ∧ F ⊆ 𝒰 ∧ C = ⋃F"
and C𝒰:    "⋀K. K ∈ 𝒰 ⟹ ∃C. C ∈ 𝒮 ∧ K ⊆ C"
by auto
then have "finite 𝒰"
and simpl𝒰: "⋀S. S ∈ 𝒰 ⟹ ∃n. n simplex S"
and face𝒰:  "⋀F S. ⟦S ∈ 𝒰; F face_of S⟧ ⟹ F ∈ 𝒰"
and faceI𝒰: "⋀S S'. ⟦S ∈ 𝒰; S' ∈ 𝒰⟧ ⟹ (S ∩ S') face_of S ∧ (S ∩ S') face_of S'"
by (auto simp: simplicial_complex_def)
define 𝒩 where "𝒩 ≡ {C ∈ ℳ. aff_dim C = n}"
have "finite 𝒩"
have poly𝒩: "⋀C. C ∈ 𝒩 ⟹ polytope C"
and convex𝒩: "⋀C. C ∈ 𝒩 ⟹ convex C"
and closed𝒩: "⋀C. C ∈ 𝒩 ⟹ closed C"
by (auto simp: 𝒩_def polyℳ polytope_imp_convex polytope_imp_closed)
have in_rel_interior: "(SOME z. z ∈ rel_interior C) ∈ rel_interior C" if "C ∈ 𝒩" for C
using that polyℳ polytope_imp_convex rel_interior_aff_dim some_in_eq by (fastforce simp: 𝒩_def)
have *: "∃T. ~affine_dependent T ∧ card T ≤ n ∧ aff_dim K < n ∧ K = convex hull T"
if "K ∈ 𝒰" for K
proof -
obtain r where r: "r simplex K"
using ‹K ∈ 𝒰› simpl𝒰 by blast
have "r = aff_dim K"
using ‹r simplex K› aff_dim_simplex by blast
with r
show ?thesis
unfolding simplex_def
using False ‹⋀K. K ∈ 𝒰 ⟹ aff_dim K ≤ int (n - 1)› that by fastforce
qed
have ahK_C_disjoint: "affine hull K ∩ rel_interior C = {}"
if "C ∈ 𝒩" "K ∈ 𝒰" "K ⊆ rel_frontier C" for C K
proof -
have "convex C" "closed C"
by (auto simp: convex𝒩 closed𝒩 ‹C ∈ 𝒩›)
obtain F where F: "F face_of C" and "F ≠ C" "K ⊆ F"
proof -
obtain L where "L ∈ 𝒮" "K ⊆ L"
using ‹K ∈ 𝒰› C𝒰 by blast
have "K ≤ rel_frontier C"
by (simp add: ‹K ⊆ rel_frontier C›)
also have "... ≤ C"
by (simp add: ‹closed C› rel_frontier_def subset_iff)
finally have "K ⊆ C" .
have "L ∩ C face_of C"
using 𝒩_def 𝒮_def ‹C ∈ 𝒩› ‹L ∈ 𝒮› intfaceℳ by auto
moreover have "L ∩ C ≠ C"
using ‹C ∈ 𝒩› ‹L ∈ 𝒮›
apply (clarsimp simp: 𝒩_def 𝒮_def)
by (metis aff_dim_subset inf_le1 not_le)
moreover have "K ⊆ L ∩ C"
using ‹C ∈ 𝒩› ‹L ∈ 𝒮› ‹K ⊆ C› ‹K ⊆ L›
by (auto simp: 𝒩_def 𝒮_def)
ultimately show ?thesis using that by metis
qed
have "affine hull F ∩ rel_interior C = {}"
by (rule affine_hull_face_of_disjoint_rel_interior [OF ‹convex C› F ‹F ≠ C›])
with hull_mono [OF ‹K ⊆ F›]
show "affine hull K ∩ rel_interior C = {}"
by fastforce
qed
let ?𝒯 = "(⋃C ∈ 𝒩. ⋃K ∈ 𝒰 ∩ Pow (rel_frontier C).
{convex hull (insert (SOME z. z ∈ rel_interior C) K)})"
have "∃𝒯. simplicial_complex 𝒯 ∧
(∀K ∈ 𝒯. aff_dim K ≤ of_nat n) ∧
(∀C ∈ ℳ. ∃F. F ⊆ 𝒯 ∧ C = ⋃F) ∧
(∀K ∈ 𝒯. ∃C. C ∈ ℳ ∧ K ⊆ C)"
proof (rule exI, intro conjI ballI)
show "simplicial_complex (𝒰 ∪ ?𝒯)"
unfolding simplicial_complex_def
proof (intro conjI impI ballI allI)
show "finite (𝒰 ∪ ?𝒯)"
using ‹finite 𝒰› ‹finite 𝒩› by simp
show "∃n. n simplex S" if "S ∈ 𝒰 ∪ ?𝒯" for S
using that ahK_C_disjoint in_rel_interior simpl𝒰 simplex_insert_dimplus1 by fastforce
show "F ∈ 𝒰 ∪ ?𝒯" if S: "S ∈ 𝒰 ∪ ?𝒯 ∧ F face_of S" for F S
proof -
have "F ∈ 𝒰" if "S ∈ 𝒰"
using S face𝒰 that by blast
moreover have "F ∈ 𝒰 ∪ ?𝒯"
if "F face_of S" "C ∈ 𝒩" "K ∈ 𝒰" and "K ⊆ rel_frontier C"
and S: "S = convex hull insert (SOME z. z ∈ rel_interior C) K" for C K
proof -
let ?z = "SOME z. z ∈ rel_interior C"
have "?z ∈ rel_interior C"
by (simp add: in_rel_interior ‹C ∈ 𝒩›)
moreover
obtain I where "¬ affine_dependent I" "card I ≤ n" "aff_dim K < int n" "K = convex hull I"
using * [OF ‹K ∈ 𝒰›] by auto
ultimately have "?z ∉ affine hull I"
using ahK_C_disjoint affine_hull_convex_hull that by blast
have "compact I" "finite I"
by (auto simp: ‹¬ affine_dependent I› aff_independent_finite finite_imp_compact)
moreover have "F face_of convex hull insert ?z I"
by (metis S ‹F face_of S› ‹K = convex hull I› convex_hull_eq_empty convex_hull_insert_segments hull_hull)
ultimately obtain J where "J ⊆ 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 ∈ J")
case True
have "F ∈ (⋃K∈𝒰 ∩ Pow (rel_frontier C). {convex hull insert ?z K})"
proof
have "convex hull (J - {?z}) face_of K"
by (metis True ‹J ⊆ insert ?z I› ‹K = convex hull I› ‹¬ affine_dependent I› face_of_convex_hull_affine_independent subset_insert_iff)
then have "convex hull (J - {?z}) ∈ 𝒰"
by (rule face𝒰 [OF ‹K ∈ 𝒰›])
moreover
have "⋀x. x ∈ convex hull (J - {?z}) ⟹ x ∈ rel_frontier C"
by (metis True ‹J ⊆ insert ?z I› ‹K = convex hull I› subsetD hull_mono subset_insert_iff that(4))
ultimately show "convex hull (J - {?z}) ∈ 𝒰 ∩ Pow (rel_frontier C)" by auto
let ?F = "convex hull insert ?z (convex hull (J - {?z}))"
have "F ⊆ ?F"
apply (clarsimp simp: ‹F = convex hull J›)
by (metis True subsetD hull_mono hull_subset subset_insert_iff)
moreover have "?F ⊆ F"
apply (clarsimp simp: ‹F = convex hull J›)
by (metis (no_types, lifting) True convex_hull_eq_empty convex_hull_insert_segments hull_hull insert_Diff)
ultimately
show "F ∈ {?F}" by auto
qed
with ‹C∈𝒩› show ?thesis by auto
next
case False
then have "F ∈ 𝒰"
using face_of_convex_hull_affine_independent [OF ‹¬ affine_dependent I›]
by (metis Int_absorb2 Int_insert_right_if0 ‹F = convex hull J› ‹J ⊆ insert ?z I› ‹K = convex hull I› face𝒰 inf_le2 ‹K ∈ 𝒰›)
then show "F ∈ 𝒰 ∪ ?𝒯"
by blast
qed
qed
ultimately show ?thesis
using that by auto
qed
have "(S ∩ S' face_of S) ∧ (S ∩ S' face_of S')"
if "S ∈ 𝒰 ∪ ?𝒯" "S' ∈ 𝒰 ∪ ?𝒯" for S S'
proof -
have symmy: "⟦⋀X Y. R X Y ⟹ R Y X;
⋀X Y. ⟦X ∈ 𝒰; Y ∈ 𝒰⟧ ⟹ R X Y;
⋀X Y. ⟦X ∈ 𝒰; Y ∈ ?𝒯⟧ ⟹ R X Y;
⋀X Y. ⟦X ∈ ?𝒯; Y ∈ ?𝒯⟧ ⟹ R X Y⟧ ⟹ R S S'" for R
using that by (metis (no_types, lifting) Un_iff)
show ?thesis
proof (rule symmy)
show "Y ∩ X face_of Y ∧ Y ∩ X face_of X"
if "X ∩ Y face_of X ∧ X ∩ Y face_of Y" for X Y :: "'a set"
next
show "X ∩ Y face_of X ∧ X ∩ Y face_of Y"
if "X ∈ 𝒰" and "Y ∈ 𝒰" for X Y
next
show "X ∩ Y face_of X ∧ X ∩ Y face_of Y"
if XY: "X ∈ 𝒰" "Y ∈ ?𝒯" for X Y
proof -
obtain C K
where "C ∈ 𝒩" "K ∈ 𝒰" "K ⊆ rel_frontier C"
and Y: "Y = convex hull insert (SOME z. z ∈ rel_interior C) K"
using XY by blast
have "convex C"
by (simp add: ‹C ∈ 𝒩› convex𝒩)
have "K ⊆ C"
by (metis DiffE ‹C ∈ 𝒩› ‹K ⊆ rel_frontier C› closed𝒩 closure_closed rel_frontier_def subset_iff)
let ?z = "(SOME z. z ∈ rel_interior C)"
have z: "?z ∈ rel_interior C"
using ‹C ∈ 𝒩› in_rel_interior by blast
obtain D where "D ∈ 𝒮" "X ⊆ D"
using C𝒰 ‹X ∈ 𝒰› by blast
have "D ∩ rel_interior C = (C ∩ D) ∩ rel_interior C"
using rel_interior_subset by blast
also have "(C ∩ D) ∩ rel_interior C = {}"
proof (rule face_of_disjoint_rel_interior)
show "C ∩ D face_of C"
using 𝒩_def 𝒮_def ‹C ∈ 𝒩› ‹D ∈ 𝒮› intfaceℳ by blast
show "C ∩ D ≠ C"
by (metis (mono_tags, lifting) Int_lower2 𝒩_def 𝒮_def ‹C ∈ 𝒩› ‹D ∈ 𝒮› aff_dim_subset mem_Collect_eq not_le)
qed
finally have DC: "D ∩ rel_interior C = {}" .
have eq: "X ∩ convex hull (insert ?z K) = X ∩ convex hull K"
apply (rule Int_convex_hull_insert_rel_exterior [OF ‹convex C› ‹K ⊆ C› z])
using DC by (meson ‹X ⊆ D› disjnt_def disjnt_subset1)
obtain I where I: "¬ affine_dependent I"
and Keq: "K = convex hull I" and [simp]: "convex hull K = K"
using "*" ‹K ∈ 𝒰› by force
then have "?z ∉ affine hull I"
using ahK_C_disjoint ‹C ∈ 𝒩› ‹K ∈ 𝒰› ‹K ⊆ rel_frontier C› affine_hull_convex_hull z by blast
have "X ∩ K face_of K"
by (simp add: ‹K ∈ 𝒰› faceI𝒰 ‹X ∈ 𝒰›)
also have "... face_of convex hull insert ?z K"
by (metis I Keq ‹?z ∉ affine hull I› aff_independent_finite convex_convex_hull face_of_convex_hull_insert face_of_refl hull_insert)
finally have "X ∩ K face_of convex hull insert ?z K" .
then show ?thesis
using "*" ‹K ∈ 𝒰› faceI𝒰 that(1) by (fastforce simp add: Y eq)
qed
next
show "X ∩ Y face_of X ∧ X ∩ Y face_of Y"
if XY: "X ∈ ?𝒯" "Y ∈ ?𝒯" for X Y
proof -
obtain C K D L
where "C ∈ 𝒩" "K ∈ 𝒰" "K ⊆ rel_frontier C"
and X: "X = convex hull insert (SOME z. z ∈ rel_interior C) K"
and "D ∈ 𝒩" "L ∈ 𝒰" "L ⊆ rel_frontier D"
and Y: "Y = convex hull insert (SOME z. z ∈ rel_interior D) L"
using XY by blast
let ?z = "(SOME z. z ∈ rel_interior C)"
have z: "?z ∈ rel_interior C"
using ‹C ∈ 𝒩› in_rel_interior by blast
have "convex C"
by (simp add: ‹C ∈ 𝒩› convex𝒩)
have "convex K"
using "*" ‹K ∈ 𝒰› by blast
have "convex L"
by (meson ‹L ∈ 𝒰› convex_simplex simpl𝒰)
show ?thesis
proof (cases "D=C")
case True
then have "L ⊆ rel_frontier C"
using ‹L ⊆ rel_frontier D› by auto
show ?thesis
apply (simp add: X Y True)
apply (simp add: convex_hull_insert_Int_eq [OF z] ‹K ⊆ rel_frontier C› ‹L ⊆ rel_frontier C› ‹convex C› ‹convex K› ‹convex L›)
using face_of_polytope_insert2
by (metis "*" IntI ‹C ∈ 𝒩› ‹K ∈ 𝒰› ‹L ∈ 𝒰›‹K ⊆ rel_frontier C› ‹L ⊆ rel_frontier C› aff_independent_finite ahK_C_disjoint empty_iff faceI𝒰 polytope_convex_hull z)
next
case False
have "convex D"
by (simp add: ‹D ∈ 𝒩› convex𝒩)
have "K ⊆ C"
by (metis DiffE ‹C ∈ 𝒩› ‹K ⊆ rel_frontier C› closed𝒩 closure_closed rel_frontier_def subset_eq)
have "L ⊆ D"
by (metis DiffE ‹D ∈ 𝒩› ‹L ⊆ rel_frontier D› closed𝒩 closure_closed rel_frontier_def subset_eq)
let ?w = "(SOME w. w ∈ rel_interior D)"
have w: "?w ∈ rel_interior D"
using ‹D ∈ 𝒩› in_rel_interior by blast
have "C ∩ rel_interior D = (D ∩ C) ∩ rel_interior D"
using rel_interior_subset by blast
also have "(D ∩ C) ∩ rel_interior D = {}"
proof (rule face_of_disjoint_rel_interior)
show "D ∩ C face_of D"
using 𝒩_def ‹C ∈ 𝒩› ‹D ∈ 𝒩› intfaceℳ by blast
have "D ∈ ℳ ∧ aff_dim D = int n"
using 𝒩_def ‹D ∈ 𝒩› by blast
moreover have "C ∈ ℳ ∧ aff_dim C = int n"
using 𝒩_def ‹C ∈ 𝒩› by blast
ultimately show "D ∩ C ≠ D"
by (metis False face_of_aff_dim_lt inf.idem inf_le1 intfaceℳ not_le polyℳ polytope_imp_convex)
qed
finally have CD: "C ∩ (rel_interior D) = {}" .
have zKC: "(convex hull insert ?z K) ⊆ C"
by (metis DiffE ‹C ∈ 𝒩› ‹K ⊆ rel_frontier C› closed𝒩 closure_closed convex𝒩 hull_minimal insert_subset rel_frontier_def rel_interior_subset subset_iff z)
have eq: "convex hull (insert ?z K) ∩ convex hull (insert ?w L) =
convex hull (insert ?z K) ∩ convex hull L"
apply (rule Int_convex_hull_insert_rel_exterior [OF ‹convex D› ‹L ⊆ D› w])
using zKC CD apply (force simp: disjnt_def)
done
have ch_id: "convex hull K = K" "convex hull L = L"
using "*" ‹K ∈ 𝒰› ‹L ∈ 𝒰› hull_same by auto
have "convex C"
by (simp add: ‹C ∈ 𝒩› convex𝒩)
have "convex hull (insert ?z K) ∩ L = L ∩ convex hull (insert ?z K)"
by blast
also have "... = convex hull K ∩ L"
proof (subst Int_convex_hull_insert_rel_exterior [OF ‹convex C› ‹K ⊆ C› z])
have "(C ∩ D) ∩ rel_interior C = {}"
proof (rule face_of_disjoint_rel_interior)
show "C ∩ D face_of C"
using 𝒩_def ‹C ∈ 𝒩› ‹D ∈ 𝒩› intfaceℳ by blast
have "D ∈ ℳ" "aff_dim D = int n"
using 𝒩_def ‹D ∈ 𝒩› by fastforce+
moreover have "C ∈ ℳ" "aff_dim C = int n"
using 𝒩_def ‹C ∈ 𝒩› by fastforce+
ultimately have "aff_dim D + - 1 * aff_dim C ≤ 0"
by fastforce
then have "¬ C face_of D"
using False ‹convex D› face_of_aff_dim_lt by fastforce
show "C ∩ D ≠ C"
using ‹C ∈ ℳ› ‹D ∈ ℳ› ‹¬ C face_of D› intfaceℳ by fastforce
qed
then have "D ∩ 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 ‹L ⊆ D› disjnt_def disjnt_subset1)
next
show "L ∩ convex hull K = convex hull K ∩ L"
by force
qed
finally have chKL: "convex hull (insert ?z K) ∩ L = convex hull K ∩ L" .
have "convex hull insert ?z K ∩ convex hull L face_of K"
by (simp add: ‹K ∈ 𝒰› ‹L ∈ 𝒰› ch_id chKL faceI𝒰)
also have "... face_of convex hull insert ?z K"
proof -
obtain I where I: "¬ affine_dependent I" "K = convex hull I"
using * [OF ‹K ∈ 𝒰›] by auto
then have "⋀a. a ∉ rel_interior C ∨ a ∉ affine hull I"
using ahK_C_disjoint ‹C ∈ 𝒩› ‹K ∈ 𝒰› ‹K ⊆ rel_frontier C› 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 ∩ convex hull L face_of convex hull insert ?z K" .
have "convex hull insert ?z K ∩ convex hull L face_of L"
by (simp add: ‹K ∈ 𝒰› ‹L ∈ 𝒰› ch_id chKL faceI𝒰)
also have "... face_of convex hull insert ?w L"
proof -
obtain I where I: "¬ affine_dependent I" "L = convex hull I"
using * [OF ‹L ∈ 𝒰›] by auto
then have "⋀a. a ∉ rel_interior D ∨ a ∉ affine hull I"
using ‹D ∈ 𝒩› ‹L ∈ 𝒰› ‹L ⊆ rel_frontier D› 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 ∩ 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```