Theory HOL-Analysis.Starlike
chapter ‹Unsorted›
theory Starlike
imports
Convex_Euclidean_Space
Line_Segment
begin
lemma affine_hull_closed_segment [simp]:
"affine hull (closed_segment a b) = affine hull {a,b}"
by (simp add: segment_convex_hull)
lemma affine_hull_open_segment [simp]:
fixes a :: "'a::euclidean_space"
shows "affine hull (open_segment a b) = (if a = b then {} else affine hull {a,b})"
by (metis affine_hull_convex_hull affine_hull_empty closure_open_segment closure_same_affine_hull segment_convex_hull)
lemma rel_interior_closure_convex_segment:
fixes S :: "_::euclidean_space set"
assumes "convex S" "a ∈ rel_interior S" "b ∈ closure S"
shows "open_segment a b ⊆ rel_interior S"
proof
fix x
have [simp]: "(1 - u) *⇩R a + u *⇩R b = b - (1 - u) *⇩R (b - a)" for u
by (simp add: algebra_simps)
assume "x ∈ open_segment a b"
then show "x ∈ rel_interior S"
unfolding closed_segment_def open_segment_def using assms
by (auto intro: rel_interior_closure_convex_shrink)
qed
lemma convex_hull_insert_segments:
"convex hull (insert a S) =
(if S = {} then {a} else ⋃x ∈ convex hull S. closed_segment a x)"
by (force simp add: convex_hull_insert_alt in_segment)
lemma Int_convex_hull_insert_rel_exterior:
fixes z :: "'a::euclidean_space"
assumes "convex C" "T ⊆ C" and z: "z ∈ rel_interior C" and dis: "disjnt S (rel_interior C)"
shows "S ∩ (convex hull (insert z T)) = S ∩ (convex hull T)" (is "?lhs = ?rhs")
proof
have *: "T = {} ⟹ z ∉ S"
using dis z by (auto simp add: disjnt_def)
{ fix x y
assume "x ∈ S" and y: "y ∈ convex hull T" and "x ∈ closed_segment z y"
have "y ∈ closure C"
by (metis y ‹convex C› ‹T ⊆ C› closure_subset contra_subsetD convex_hull_eq hull_mono)
moreover have "x ∉ rel_interior C"
by (meson ‹x ∈ S› dis disjnt_iff)
moreover have "x ∈ open_segment z y ∪ {z, y}"
using ‹x ∈ closed_segment z y› closed_segment_eq_open by blast
ultimately have "x ∈ convex hull T"
using rel_interior_closure_convex_segment [OF ‹convex C› z]
using y z by blast
}
with * show "?lhs ⊆ ?rhs"
by (auto simp add: convex_hull_insert_segments)
show "?rhs ⊆ ?lhs"
by (meson hull_mono inf_mono subset_insertI subset_refl)
qed
subsection ‹Shrinking towards the interior of a convex set›
lemma mem_interior_convex_shrink:
fixes S :: "'a::euclidean_space set"
assumes "convex S"
and "c ∈ interior S"
and "x ∈ S"
and "0 < e"
and "e ≤ 1"
shows "x - e *⇩R (x - c) ∈ interior S"
proof -
obtain d where "d > 0" and d: "ball c d ⊆ S"
using assms(2) unfolding mem_interior by auto
show ?thesis
unfolding mem_interior
proof (intro exI subsetI conjI)
fix y
assume "y ∈ ball (x - e *⇩R (x - c)) (e*d)"
then have as: "dist (x - e *⇩R (x - c)) y < e * d"
by simp
have *: "y = (1 - (1 - e)) *⇩R ((1 / e) *⇩R y - ((1 - e) / e) *⇩R x) + (1 - e) *⇩R x"
using ‹e > 0› by (auto simp add: scaleR_left_diff_distrib scaleR_right_diff_distrib)
have "c - ((1 / e) *⇩R y - ((1 - e) / e) *⇩R x) = (1 / e) *⇩R (e *⇩R c - y + (1 - e) *⇩R x)"
using ‹e > 0›
by (auto simp add: euclidean_eq_iff[where 'a='a] field_simps inner_simps)
then have "dist c ((1 / e) *⇩R y - ((1 - e) / e) *⇩R x) = ¦1/e¦ * norm (e *⇩R c - y + (1 - e) *⇩R x)"
by (simp add: dist_norm)
also have "… = ¦1/e¦ * norm (x - e *⇩R (x - c) - y)"
by (auto intro!:arg_cong[where f=norm] simp add: algebra_simps)
also have "… < d"
using as[unfolded dist_norm] and ‹e > 0›
by (auto simp add:pos_divide_less_eq[OF ‹e > 0›] mult.commute)
finally have "(1 - (1 - e)) *⇩R ((1 / e) *⇩R y - ((1 - e) / e) *⇩R x) + (1 - e) *⇩R x ∈ S"
using assms(3-5) d
by (intro convexD_alt [OF ‹convex S›]) (auto intro: convexD_alt [OF ‹convex S›])
with ‹e > 0› show "y ∈ S"
by (auto simp add: scaleR_left_diff_distrib scaleR_right_diff_distrib)
qed (use ‹e>0› ‹d>0› in auto)
qed
lemma mem_interior_closure_convex_shrink:
fixes S :: "'a::euclidean_space set"
assumes "convex S"
and "c ∈ interior S"
and "x ∈ closure S"
and "0 < e"
and "e ≤ 1"
shows "x - e *⇩R (x - c) ∈ interior S"
proof -
obtain d where "d > 0" and d: "ball c d ⊆ S"
using assms(2) unfolding mem_interior by auto
have "∃y∈S. norm (y - x) * (1 - e) < e * d"
proof (cases "x ∈ S")
case True
then show ?thesis
using ‹e > 0› ‹d > 0› by force
next
case False
then have x: "x islimpt S"
using assms(3)[unfolded closure_def] by auto
show ?thesis
proof (cases "e = 1")
case True
obtain y where "y ∈ S" "y ≠ x" "dist y x < 1"
using x[unfolded islimpt_approachable,THEN spec[where x=1]] by auto
then show ?thesis
using True ‹0 < d› by auto
next
case False
then have "0 < e * d / (1 - e)" and *: "1 - e > 0"
using ‹e ≤ 1› ‹e > 0› ‹d > 0› by auto
then obtain y where "y ∈ S" "y ≠ x" "dist y x < e * d / (1 - e)"
using islimpt_approachable x by blast
then have "norm (y - x) * (1 - e) < e * d"
by (metis "*" dist_norm mult_imp_div_pos_le not_less)
then show ?thesis
using ‹y ∈ S› by blast
qed
qed
then obtain y where "y ∈ S" and y: "norm (y - x) * (1 - e) < e * d"
by auto
define z where "z = c + ((1 - e) / e) *⇩R (x - y)"
have *: "x - e *⇩R (x - c) = y - e *⇩R (y - z)"
unfolding z_def using ‹e > 0›
by (auto simp add: scaleR_right_diff_distrib scaleR_right_distrib scaleR_left_diff_distrib)
have "(1 - e) * norm (x - y) / e < d"
using y ‹0 < e› by (simp add: field_simps norm_minus_commute)
then have "z ∈ interior (ball c d)"
using ‹0 < e› ‹e ≤ 1› by (simp add: interior_open[OF open_ball] z_def dist_norm)
then have "z ∈ interior S"
using d interiorI interior_ball by blast
then show ?thesis
unfolding * using mem_interior_convex_shrink ‹y ∈ S› assms by blast
qed
lemma in_interior_closure_convex_segment:
fixes S :: "'a::euclidean_space set"
assumes "convex S" and a: "a ∈ interior S" and b: "b ∈ closure S"
shows "open_segment a b ⊆ interior S"
proof -
{ fix u::real
assume u: "0 < u" "u < 1"
have "(1 - u) *⇩R a + u *⇩R b = b - (1 - u) *⇩R (b - a)"
by (simp add: algebra_simps)
also have "... ∈ interior S" using mem_interior_closure_convex_shrink [OF assms] u
by simp
finally have "(1 - u) *⇩R a + u *⇩R b ∈ interior S" .
}
then show ?thesis
by (clarsimp simp: in_segment)
qed
lemma convex_closure_interior:
fixes S :: "'a::euclidean_space set"
assumes "convex S" and int: "interior S ≠ {}"
shows "closure(interior S) = closure S"
proof -
obtain a where a: "a ∈ interior S"
using int by auto
have "closure S ⊆ closure(interior S)"
proof
fix x
assume x: "x ∈ closure S"
show "x ∈ closure (interior S)"
proof (cases "x=a")
case True
then show ?thesis
using ‹a ∈ interior S› closure_subset by blast
next
case False
{ fix e::real
assume xnotS: "x ∉ interior S" and "0 < e"
have "∃x'∈interior S. x' ≠ x ∧ dist x' x < e"
proof (intro bexI conjI)
show "x - min (e/2 / norm (x - a)) 1 *⇩R (x - a) ≠ x"
using False ‹0 < e› by (auto simp: algebra_simps min_def)
show "dist (x - min (e/2 / norm (x - a)) 1 *⇩R (x - a)) x < e"
using ‹0 < e› by (auto simp: dist_norm min_def)
show "x - min (e/2 / norm (x - a)) 1 *⇩R (x - a) ∈ interior S"
using ‹0 < e› False
by (auto simp add: min_def a intro: mem_interior_closure_convex_shrink [OF ‹convex S› a x])
qed
}
then show ?thesis
by (auto simp add: closure_def islimpt_approachable)
qed
qed
then show ?thesis
by (simp add: closure_mono interior_subset subset_antisym)
qed
lemma openin_subset_relative_interior:
fixes S :: "'a::euclidean_space set"
shows "openin (top_of_set (affine hull T)) S ⟹ (S ⊆ rel_interior T) = (S ⊆ T)"
by (meson order.trans rel_interior_maximal rel_interior_subset)
lemma conic_hull_eq_span_affine_hull:
fixes S :: "'a::euclidean_space set"
assumes "0 ∈ rel_interior S"
shows "conic hull S = span S ∧ conic hull S = affine hull S"
proof -
obtain ε where "ε>0" and ε: "cball 0 ε ∩ affine hull S ⊆ S"
using assms mem_rel_interior_cball by blast
have *: "affine hull S = span S"
by (meson affine_hull_span_0 assms hull_inc mem_rel_interior_cball)
moreover
have "conic hull S ⊆ span S"
by (simp add: hull_minimal span_superset)
moreover
{ fix x
assume "x ∈ affine hull S"
have "x ∈ conic hull S"
proof (cases "x=0")
case True
then show ?thesis
using ‹x ∈ affine hull S› by auto
next
case False
then have "(ε / norm x) *⇩R x ∈ cball 0 ε ∩ affine hull S"
using ‹0 < ε› ‹x ∈ affine hull S› * span_mul by fastforce
then have "(ε / norm x) *⇩R x ∈ S"
by (meson ε subsetD)
then have "∃c xa. x = c *⇩R xa ∧ 0 ≤ c ∧ xa ∈ S"
by (smt (verit, del_insts) ‹0 < ε› divide_nonneg_nonneg eq_vector_fraction_iff norm_eq_zero norm_ge_zero)
then show ?thesis
by (simp add: conic_hull_explicit)
qed
}
then have "affine hull S ⊆ conic hull S"
by auto
ultimately show ?thesis
by blast
qed
lemma conic_hull_eq_span:
fixes S :: "'a::euclidean_space set"
assumes "0 ∈ rel_interior S"
shows "conic hull S = span S"
by (simp add: assms conic_hull_eq_span_affine_hull)
lemma conic_hull_eq_affine_hull:
fixes S :: "'a::euclidean_space set"
assumes "0 ∈ rel_interior S"
shows "conic hull S = affine hull S"
using assms conic_hull_eq_span_affine_hull by blast
lemma conic_hull_eq_span_eq:
fixes S :: "'a::euclidean_space set"
shows "0 ∈ rel_interior(conic hull S) ⟷ conic hull S = span S" (is "?lhs = ?rhs")
proof
show "?lhs ⟹ ?rhs"
by (metis conic_hull_eq_span conic_span hull_hull hull_minimal hull_subset span_eq)
show "?rhs ⟹ ?lhs"
by (metis rel_interior_affine subspace_affine subspace_span)
qed
lemma aff_dim_psubset:
"(affine hull S) ⊂ (affine hull T) ⟹ aff_dim S < aff_dim T"
by (metis aff_dim_affine_hull aff_dim_empty aff_dim_subset affine_affine_hull affine_dim_equal order_less_le)
lemma aff_dim_eq_full_gen:
"S ⊆ T ⟹ (aff_dim S = aff_dim T ⟷ affine hull S = affine hull T)"
by (smt (verit, del_insts) aff_dim_affine_hull2 aff_dim_psubset hull_mono psubsetI)
lemma aff_dim_eq_full:
fixes S :: "'n::euclidean_space set"
shows "aff_dim S = (DIM('n)) ⟷ affine hull S = UNIV"
by (metis aff_dim_UNIV aff_dim_affine_hull affine_hull_UNIV)
lemma closure_convex_Int_superset:
fixes S :: "'a::euclidean_space set"
assumes "convex S" "interior S ≠ {}" "interior S ⊆ closure T"
shows "closure(S ∩ T) = closure S"
proof -
have "closure S ⊆ closure(interior S)"
by (simp add: convex_closure_interior assms)
also have "... ⊆ closure (S ∩ T)"
using interior_subset [of S] assms
by (metis (no_types, lifting) Int_assoc Int_lower2 closure_mono closure_open_Int_superset inf.orderE open_interior)
finally show ?thesis
by (simp add: closure_mono dual_order.antisym)
qed
subsection ‹Some obvious but surprisingly hard simplex lemmas›
lemma simplex:
assumes "finite S"
and "0 ∉ S"
shows "convex hull (insert 0 S) = {y. ∃u. (∀x∈S. 0 ≤ u x) ∧ sum u S ≤ 1 ∧ sum (λx. u x *⇩R x) S = y}"
proof -
{ fix x and u :: "'a ⇒ real"
assume "∀x∈S. 0 ≤ u x" "sum u S ≤ 1"
then have "∃v. 0 ≤ v 0 ∧ (∀x∈S. 0 ≤ v x) ∧ v 0 + sum v S = 1 ∧ (∑x∈S. v x *⇩R x) = (∑x∈S. u x *⇩R x)"
by (rule_tac x="λx. if x = 0 then 1 - sum u S else u x" in exI) (auto simp: sum_delta_notmem assms if_smult)
}
then show ?thesis by (auto simp: convex_hull_finite set_eq_iff assms)
qed
lemma substd_simplex:
assumes d: "d ⊆ Basis"
shows "convex hull (insert 0 d) =
{x. (∀i∈Basis. 0 ≤ x∙i) ∧ (∑i∈d. x∙i) ≤ 1 ∧ (∀i∈Basis. i ∉ d ⟶ x∙i = 0)}"
(is "convex hull (insert 0 ?p) = ?s")
proof -
let ?D = d
have "0 ∉ ?p"
using assms by (auto simp: image_def)
from d have "finite d"
by (blast intro: finite_subset finite_Basis)
show ?thesis
unfolding simplex[OF ‹finite d› ‹0 ∉ ?p›]
proof (intro set_eqI; safe)
fix u :: "'a ⇒ real"
assume as: "∀x∈?D. 0 ≤ u x" "sum u ?D ≤ 1"
let ?x = "(∑x∈?D. u x *⇩R x)"
have ind: "∀i∈Basis. i ∈ d ⟶ u i = ?x ∙ i"
and notind: "(∀i∈Basis. i ∉ d ⟶ ?x ∙ i = 0)"
using substdbasis_expansion_unique[OF assms] by blast+
then have **: "sum u ?D = sum ((∙) ?x) ?D"
using assms by (auto intro!: sum.cong)
show "0 ≤ ?x ∙ i" if "i ∈ Basis" for i
using as(1) ind notind that by fastforce
show "sum ((∙) ?x) ?D ≤ 1"
using "**" as(2) by linarith
show "?x ∙ i = 0" if "i ∈ Basis" "i ∉ d" for i
using notind that by blast
next
fix x
assume "∀i∈Basis. 0 ≤ x ∙ i" "sum ((∙) x) ?D ≤ 1" "(∀i∈Basis. i ∉ d ⟶ x ∙ i = 0)"
with d show "∃u. (∀x∈?D. 0 ≤ u x) ∧ sum u ?D ≤ 1 ∧ (∑x∈?D. u x *⇩R x) = x"
unfolding substdbasis_expansion_unique[OF assms]
by (rule_tac x="inner x" in exI) auto
qed
qed
lemma std_simplex:
"convex hull (insert 0 Basis) =
{x::'a::euclidean_space. (∀i∈Basis. 0 ≤ x∙i) ∧ sum (λi. x∙i) Basis ≤ 1}"
using substd_simplex[of Basis] by auto
lemma interior_std_simplex:
"interior (convex hull (insert 0 Basis)) =
{x::'a::euclidean_space. (∀i∈Basis. 0 < x∙i) ∧ sum (λi. x∙i) Basis < 1}"
unfolding set_eq_iff mem_interior std_simplex
proof (intro allI iffI CollectI; clarify)
fix x :: 'a
fix e
assume "e > 0" and as: "ball x e ⊆ {x. (∀i∈Basis. 0 ≤ x ∙ i) ∧ sum ((∙) x) Basis ≤ 1}"
show "(∀i∈Basis. 0 < x ∙ i) ∧ sum ((∙) x) Basis < 1"
proof safe
fix i :: 'a
assume i: "i ∈ Basis"
then show "0 < x ∙ i"
using as[THEN subsetD[where c="x - (e/2) *⇩R i"]] and ‹e > 0›
by (force simp add: inner_simps)
next
have **: "dist x (x + (e/2) *⇩R (SOME i. i∈Basis)) < e" using ‹e > 0›
unfolding dist_norm
by (auto intro!: mult_strict_left_mono simp: SOME_Basis)
have "⋀i. i ∈ Basis ⟹ (x + (e/2) *⇩R (SOME i. i∈Basis)) ∙ i =
x∙i + (if i = (SOME i. i∈Basis) then e/2 else 0)"
by (auto simp: SOME_Basis inner_Basis inner_simps)
then have *: "sum ((∙) (x + (e/2) *⇩R (SOME i. i∈Basis))) Basis =
sum (λi. x∙i + (if (SOME i. i∈Basis) = i then e/2 else 0)) Basis"
by (auto simp: intro!: sum.cong)
have "sum ((∙) x) Basis < sum ((∙) (x + (e/2) *⇩R (SOME i. i∈Basis))) Basis"
using ‹e > 0› DIM_positive by (auto simp: SOME_Basis sum.distrib *)
also have "… ≤ 1"
using ** as by force
finally show "sum ((∙) x) Basis < 1" by auto
qed
next
fix x :: 'a
assume as: "∀i∈Basis. 0 < x ∙ i" "sum ((∙) x) Basis < 1"
obtain a :: 'b where "a ∈ UNIV" using UNIV_witness ..
let ?d = "(1 - sum ((∙) x) Basis) / real (DIM('a))"
show "∃e>0. ball x e ⊆ {x. (∀i∈Basis. 0 ≤ x ∙ i) ∧ sum ((∙) x) Basis ≤ 1}"
proof (rule_tac x="min (Min (((∙) x) ` Basis)) D" for D in exI, intro conjI subsetI CollectI)
fix y
assume y: "y ∈ ball x (min (Min ((∙) x ` Basis)) ?d)"
have "sum ((∙) y) Basis ≤ sum (λi. x∙i + ?d) Basis"
proof (rule sum_mono)
fix i :: 'a
assume i: "i ∈ Basis"
have "¦y∙i - x∙i¦ ≤ norm (y - x)"
by (metis Basis_le_norm i inner_commute inner_diff_right)
also have "... < ?d"
using y by (simp add: dist_norm norm_minus_commute)
finally have "¦y∙i - x∙i¦ < ?d" .
then show "y ∙ i ≤ x ∙ i + ?d" by auto
qed
also have "… ≤ 1"
unfolding sum.distrib sum_constant
by (auto simp add: Suc_le_eq)
finally show "sum ((∙) y) Basis ≤ 1" .
show "(∀i∈Basis. 0 ≤ y ∙ i)"
proof safe
fix i :: 'a
assume i: "i ∈ Basis"
have "norm (x - y) < Min (((∙) x) ` Basis)"
using y by (auto simp: dist_norm less_eq_real_def)
also have "... ≤ x∙i"
using i by auto
finally have "norm (x - y) < x∙i" .
then show "0 ≤ y∙i"
using Basis_le_norm[OF i, of "x - y"] and as(1)[rule_format, OF i]
by (auto simp: inner_simps)
qed
next
have "Min (((∙) x) ` Basis) > 0"
using as by simp
moreover have "?d > 0"
using as by (auto simp: Suc_le_eq)
ultimately show "0 < min (Min ((∙) x ` Basis)) ((1 - sum ((∙) x) Basis) / real DIM('a))"
by linarith
qed
qed
lemma interior_std_simplex_nonempty:
obtains a :: "'a::euclidean_space" where
"a ∈ interior(convex hull (insert 0 Basis))"
proof -
let ?D = "Basis :: 'a set"
let ?a = "sum (λb::'a. inverse (2 * real DIM('a)) *⇩R b) Basis"
{
fix i :: 'a
assume i: "i ∈ Basis"
have "?a ∙ i = inverse (2 * real DIM('a))"
by (rule trans[of _ "sum (λj. if i = j then inverse (2 * real DIM('a)) else 0) ?D"])
(simp_all add: sum.If_cases i) }
note ** = this
show ?thesis
proof
show "?a ∈ interior(convex hull (insert 0 Basis))"
unfolding interior_std_simplex mem_Collect_eq
proof safe
fix i :: 'a
assume i: "i ∈ Basis"
show "0 < ?a ∙ i"
unfolding **[OF i] by (auto simp add: Suc_le_eq)
next
have "sum ((∙) ?a) ?D = sum (λi. inverse (2 * real DIM('a))) ?D"
by (auto intro: sum.cong)
also have "… < 1"
unfolding sum_constant divide_inverse[symmetric]
by (auto simp add: field_simps)
finally show "sum ((∙) ?a) ?D < 1" by auto
qed
qed
qed
lemma rel_interior_substd_simplex:
assumes D: "D ⊆ Basis"
shows "rel_interior (convex hull (insert 0 D)) =
{x::'a::euclidean_space. (∀i∈D. 0 < x∙i) ∧ (∑i∈D. x∙i) < 1 ∧ (∀i∈Basis. i ∉ D ⟶ x∙i = 0)}"
(is "_ = ?s")
proof -
have "finite D"
using D finite_Basis finite_subset by blast
show ?thesis
proof (cases "D = {}")
case True
then show ?thesis
using rel_interior_sing using euclidean_eq_iff[of _ 0] by auto
next
case False
have h0: "affine hull (convex hull (insert 0 D)) =
{x::'a::euclidean_space. (∀i∈Basis. i ∉ D ⟶ x∙i = 0)}"
using affine_hull_convex_hull affine_hull_substd_basis assms by auto
have aux: "⋀x::'a. ∀i∈Basis. (∀i∈D. 0 ≤ x∙i) ∧ (∀i∈Basis. i ∉ D ⟶ x∙i = 0) ⟶ 0 ≤ x∙i"
by auto
{
fix x :: "'a::euclidean_space"
assume x: "x ∈ rel_interior (convex hull (insert 0 D))"
then obtain e where "e > 0" and
"ball x e ∩ {xa. (∀i∈Basis. i ∉ D ⟶ xa∙i = 0)} ⊆ convex hull (insert 0 D)"
using mem_rel_interior_ball[of x "convex hull (insert 0 D)"] h0 by auto
then have as: "⋀y. ⟦dist x y < e ∧ (∀i∈Basis. i ∉ D ⟶ y∙i = 0)⟧ ⟹
(∀i∈D. 0 ≤ y ∙ i) ∧ sum ((∙) y) D ≤ 1"
using assms by (force simp: substd_simplex)
have x0: "(∀i∈Basis. i ∉ D ⟶ x∙i = 0)"
using x rel_interior_subset substd_simplex[OF assms] by auto
have "(∀i∈D. 0 < x ∙ i) ∧ sum ((∙) x) D < 1 ∧ (∀i∈Basis. i ∉ D ⟶ x∙i = 0)"
proof (intro conjI ballI)
fix i :: 'a
assume "i ∈ D"
then have "∀j∈D. 0 ≤ (x - (e/2) *⇩R i) ∙ j"
using D ‹e > 0› x0
by (intro as[THEN conjunct1]) (force simp: dist_norm inner_simps inner_Basis)
then show "0 < x ∙ i"
using ‹e > 0› ‹i ∈ D› D by (force simp: inner_simps inner_Basis)
next
obtain a where a: "a ∈ D"
using ‹D ≠ {}› by auto
then have **: "dist x (x + (e/2) *⇩R a) < e"
using ‹e > 0› norm_Basis[of a] D by (auto simp: dist_norm)
have "⋀i. i ∈ Basis ⟹ (x + (e/2) *⇩R a) ∙ i = x∙i + (if i = a then e/2 else 0)"
using a D by (auto simp: inner_simps inner_Basis)
then have *: "sum ((∙) (x + (e/2) *⇩R a)) D = sum (λi. x∙i + (if a = i then e/2 else 0)) D"
using D by (intro sum.cong) auto
have "a ∈ Basis"
using ‹a ∈ D› D by auto
then have h1: "(∀i∈Basis. i ∉ D ⟶ (x + (e/2) *⇩R a) ∙ i = 0)"
using x0 D ‹a∈D› by (auto simp add: inner_add_left inner_Basis)
have "sum ((∙) x) D < sum ((∙) (x + (e/2) *⇩R a)) D"
using ‹e > 0› ‹a ∈ D› ‹finite D› by (auto simp add: * sum.distrib)
also have "… ≤ 1"
using ** h1 as[rule_format, of "x + (e/2) *⇩R a"]
by auto
finally show "sum ((∙) x) D < 1" "⋀i. i∈Basis ⟹ i ∉ D ⟶ x∙i = 0"
using x0 by auto
qed
}
moreover
{
fix x :: "'a::euclidean_space"
assume as: "x ∈ ?s"
have "∀i. 0 < x∙i ∨ 0 = x∙i ⟶ 0 ≤ x∙i"
by auto
moreover have "∀i. i ∈ D ∨ i ∉ D" by auto
ultimately
have "∀i. (∀i∈D. 0 < x∙i) ∧ (∀i. i ∉ D ⟶ x∙i = 0) ⟶ 0 ≤ x∙i"
by metis
then have h2: "x ∈ convex hull (insert 0 D)"
using as assms by (force simp add: substd_simplex)
obtain a where a: "a ∈ D"
using ‹D ≠ {}› by auto
define d where "d ≡ (1 - sum ((∙) x) D) / real (card D)"
have "∃e>0. ball x e ∩ {x. ∀i∈Basis. i ∉ D ⟶ x ∙ i = 0} ⊆ convex hull insert 0 D"
unfolding substd_simplex[OF assms]
proof (intro exI; safe)
have "0 < card D" using ‹D ≠ {}› ‹finite D›
by (simp add: card_gt_0_iff)
have "Min (((∙) x) ` D) > 0"
using as ‹D ≠ {}› ‹finite D› by (simp)
moreover have "d > 0"
using as ‹0 < card D› by (auto simp: d_def)
ultimately show "min (Min (((∙) x) ` D)) d > 0"
by auto
fix y :: 'a
assume y2: "∀i∈Basis. i ∉ D ⟶ y∙i = 0"
assume "y ∈ ball x (min (Min ((∙) x ` D)) d)"
then have y: "dist x y < min (Min ((∙) x ` D)) d"
by auto
have "sum ((∙) y) D ≤ sum (λi. x∙i + d) D"
proof (rule sum_mono)
fix i
assume "i ∈ D"
with D have i: "i ∈ Basis"
by auto
have "¦y∙i - x∙i¦ ≤ norm (y - x)"
by (metis i inner_commute inner_diff_right norm_bound_Basis_le order_refl)
also have "... < d"
by (metis dist_norm min_less_iff_conj norm_minus_commute y)
finally have "¦y∙i - x∙i¦ < d" .
then show "y ∙ i ≤ x ∙ i + d" by auto
qed
also have "… ≤ 1"
unfolding sum.distrib sum_constant d_def using ‹0 < card D›
by auto
finally show "sum ((∙) y) D ≤ 1" .
fix i :: 'a
assume i: "i ∈ Basis"
then show "0 ≤ y∙i"
proof (cases "i∈D")
case True
have "norm (x - y) < x∙i"
using y Min_gr_iff[of "(∙) x ` D" "norm (x - y)"] ‹0 < card D› ‹i ∈ D›
by (simp add: dist_norm card_gt_0_iff)
then show "0 ≤ y∙i"
using Basis_le_norm[OF i, of "x - y"] and as(1)[rule_format]
by (auto simp: inner_simps)
qed (use y2 in auto)
qed
then have "x ∈ rel_interior (convex hull (insert 0 D))"
using h0 h2 rel_interior_ball by force
}
ultimately have
"⋀x. x ∈ rel_interior (convex hull insert 0 D) ⟷
x ∈ {x. (∀i∈D. 0 < x ∙ i) ∧ sum ((∙) x) D < 1 ∧ (∀i∈Basis. i ∉ D ⟶ x ∙ i = 0)}"
by blast
then show ?thesis by (rule set_eqI)
qed
qed
lemma rel_interior_substd_simplex_nonempty:
assumes "D ≠ {}"
and "D ⊆ Basis"
obtains a :: "'a::euclidean_space"
where "a ∈ rel_interior (convex hull (insert 0 D))"
proof -
let ?a = "sum (λb::'a::euclidean_space. inverse (2 * real (card D)) *⇩R b) D"
have "finite D"
using assms finite_Basis infinite_super by blast
then have d1: "0 < real (card D)"
using ‹D ≠ {}› by auto
{
fix i
assume "i ∈ D"
have "?a ∙ i = sum (λj. if i = j then inverse (2 * real (card D)) else 0) D"
unfolding inner_sum_left
using ‹i ∈ D› by (auto simp: inner_Basis subsetD[OF assms(2)] intro: sum.cong)
also have "... = inverse (2 * real (card D))"
using ‹i ∈ D› ‹finite D› by auto
finally have "?a ∙ i = inverse (2 * real (card D))" .
}
note ** = this
show ?thesis
proof
show "?a ∈ rel_interior (convex hull (insert 0 D))"
unfolding rel_interior_substd_simplex[OF assms(2)]
proof safe
fix i
assume "i ∈ D"
have "0 < inverse (2 * real (card D))"
using d1 by auto
also have "… = ?a ∙ i" using **[of i] ‹i ∈ D›
by auto
finally show "0 < ?a ∙ i" by auto
next
have "sum ((∙) ?a) D = sum (λi. inverse (2 * real (card D))) D"
by (rule sum.cong) (rule refl, rule **)
also have "… < 1"
unfolding sum_constant divide_real_def[symmetric]
by (auto simp add: field_simps)
finally show "sum ((∙) ?a) D < 1" by auto
next
fix i
assume "i ∈ Basis" and "i ∉ D"
have "?a ∈ span D"
proof (rule span_sum[of D "(λb. b /⇩R (2 * real (card D)))" D])
{
fix x :: "'a::euclidean_space"
assume "x ∈ D"
then have "x ∈ span D"
using span_base[of _ "D"] by auto
then have "x /⇩R (2 * real (card D)) ∈ span D"
using span_mul[of x "D" "(inverse (real (card D)) / 2)"] by auto
}
then show "⋀x. x∈D ⟹ x /⇩R (2 * real (card D)) ∈ span D"
by auto
qed
then show "?a ∙ i = 0 "
using ‹i ∉ D› unfolding span_substd_basis[OF assms(2)] using ‹i ∈ Basis› by auto
qed
qed
qed
subsection ‹Relative interior of convex set›
lemma rel_interior_convex_nonempty_aux:
fixes S :: "'n::euclidean_space set"
assumes "convex S"
and "0 ∈ S"
shows "rel_interior S ≠ {}"
proof (cases "S = {0}")
case True
then show ?thesis using rel_interior_sing by auto
next
case False
obtain B where B: "independent B ∧ B ≤ S ∧ S ≤ span B ∧ card B = dim S"
using basis_exists[of S] by metis
then have "B ≠ {}"
using B assms ‹S ≠ {0}› span_empty by auto
have "insert 0 B ≤ span B"
using subspace_span[of B] subspace_0[of "span B"]
span_superset by auto
then have "span (insert 0 B) ≤ span B"
using span_span[of B] span_mono[of "insert 0 B" "span B"] by blast
then have "convex hull insert 0 B ≤ span B"
using convex_hull_subset_span[of "insert 0 B"] by auto
then have "span (convex hull insert 0 B) ≤ span B"
using span_span[of B]
span_mono[of "convex hull insert 0 B" "span B"] by blast
then have *: "span (convex hull insert 0 B) = span B"
using span_mono[of B "convex hull insert 0 B"] hull_subset[of "insert 0 B"] by auto
then have "span (convex hull insert 0 B) = span S"
using B span_mono[of B S] span_mono[of S "span B"]
span_span[of B] by auto
moreover have "0 ∈ affine hull (convex hull insert 0 B)"
using hull_subset[of "convex hull insert 0 B"] hull_subset[of "insert 0 B"] by auto
ultimately have **: "affine hull (convex hull insert 0 B) = affine hull S"
using affine_hull_span_0[of "convex hull insert 0 B"] affine_hull_span_0[of "S"]
assms hull_subset[of S]
by auto
obtain d and f :: "'n ⇒ 'n" where
fd: "card d = card B" "linear f" "f ` B = d"
"f ` span B = {x. ∀i∈Basis. i ∉ d ⟶ x ∙ i = (0::real)} ∧ inj_on f (span B)"
and d: "d ⊆ Basis"
using basis_to_substdbasis_subspace_isomorphism[of B,OF _ ] B by auto
then have "bounded_linear f"
using linear_conv_bounded_linear by auto
have "d ≠ {}"
using fd B ‹B ≠ {}› by auto
have "insert 0 d = f ` (insert 0 B)"
using fd linear_0 by auto
then have "(convex hull (insert 0 d)) = f ` (convex hull (insert 0 B))"
using convex_hull_linear_image[of f "(insert 0 d)"]
convex_hull_linear_image[of f "(insert 0 B)"] ‹linear f›
by auto
moreover have "rel_interior (f ` (convex hull insert 0 B)) = f ` rel_interior (convex hull insert 0 B)"
proof (rule rel_interior_injective_on_span_linear_image[OF ‹bounded_linear f›])
show "inj_on f (span (convex hull insert 0 B))"
using fd * by auto
qed
ultimately have "rel_interior (convex hull insert 0 B) ≠ {}"
using rel_interior_substd_simplex_nonempty[OF ‹d ≠ {}› d] by fastforce
moreover have "convex hull (insert 0 B) ⊆ S"
using B assms hull_mono[of "insert 0 B" "S" "convex"] convex_hull_eq by auto
ultimately show ?thesis
using subset_rel_interior[of "convex hull insert 0 B" S] ** by auto
qed
lemma rel_interior_eq_empty:
fixes S :: "'n::euclidean_space set"
assumes "convex S"
shows "rel_interior S = {} ⟷ S = {}"
proof -
{
assume "S ≠ {}"
then obtain a where "a ∈ S" by auto
then have "0 ∈ (+) (-a) ` S"
using assms exI[of "(λx. x ∈ S ∧ - a + x = 0)" a] by auto
then have "rel_interior ((+) (-a) ` S) ≠ {}"
using rel_interior_convex_nonempty_aux[of "(+) (-a) ` S"]
convex_translation[of S "-a"] assms
by auto
then have "rel_interior S ≠ {}"
using rel_interior_translation [of "- a"] by simp
}
then show ?thesis by auto
qed
lemma interior_simplex_nonempty:
fixes S :: "'N :: euclidean_space set"
assumes "independent S" "finite S" "card S = DIM('N)"
obtains a where "a ∈ interior (convex hull (insert 0 S))"
proof -
have "affine hull (insert 0 S) = UNIV"
by (simp add: hull_inc affine_hull_span_0 dim_eq_full[symmetric]
assms(1) assms(3) dim_eq_card_independent)
moreover have "rel_interior (convex hull insert 0 S) ≠ {}"
using rel_interior_eq_empty [of "convex hull (insert 0 S)"] by auto
ultimately have "interior (convex hull insert 0 S) ≠ {}"
by (simp add: rel_interior_interior)
with that show ?thesis
by auto
qed
lemma convex_rel_interior:
fixes S :: "'n::euclidean_space set"
assumes "convex S"
shows "convex (rel_interior S)"
proof -
{
fix x y and u :: real
assume assm: "x ∈ rel_interior S" "y ∈ rel_interior S" "0 ≤ u" "u ≤ 1"
then have "x ∈ S"
using rel_interior_subset by auto
have "x - u *⇩R (x-y) ∈ rel_interior S"
proof (cases "0 = u")
case False
then have "0 < u" using assm by auto
then show ?thesis
using assm rel_interior_convex_shrink[of S y x u] assms ‹x ∈ S› by auto
next
case True
then show ?thesis using assm by auto
qed
then have "(1 - u) *⇩R x + u *⇩R y ∈ rel_interior S"
by (simp add: algebra_simps)
}
then show ?thesis
unfolding convex_alt by auto
qed
lemma convex_closure_rel_interior:
fixes S :: "'n::euclidean_space set"
assumes "convex S"
shows "closure (rel_interior S) = closure S"
proof -
have h1: "closure (rel_interior S) ≤ closure S"
using closure_mono[of "rel_interior S" S] rel_interior_subset[of S] by auto
show ?thesis
proof (cases "S = {}")
case False
then obtain a where a: "a ∈ rel_interior S"
using rel_interior_eq_empty assms by auto
{ fix x
assume x: "x ∈ closure S"
{
assume "x = a"
then have "x ∈ closure (rel_interior S)"
using a unfolding closure_def by auto
}
moreover
{
assume "x ≠ a"
{
fix e :: real
assume "e > 0"
define e1 where "e1 = min 1 (e/norm (x - a))"
then have e1: "e1 > 0" "e1 ≤ 1" "e1 * norm (x - a) ≤ e"
using ‹x ≠ a› ‹e > 0› le_divide_eq[of e1 e "norm (x - a)"]
by simp_all
then have *: "x - e1 *⇩R (x - a) ∈ rel_interior S"
using rel_interior_closure_convex_shrink[of S a x e1] assms x a e1_def
by auto
have "∃y. y ∈ rel_interior S ∧ y ≠ x ∧ dist y x ≤ e"
using "*" ‹x ≠ a› e1 by force
}
then have "x islimpt rel_interior S"
unfolding islimpt_approachable_le by auto
then have "x ∈ closure(rel_interior S)"
unfolding closure_def by auto
}
ultimately have "x ∈ closure(rel_interior S)" by auto
}
then show ?thesis using h1 by auto
qed auto
qed
lemma empty_interior_subset_hyperplane_aux:
fixes S :: "'a::euclidean_space set"
assumes "convex S" "0 ∈ S" and empty_int: "interior S = {}"
shows "∃a b. a≠0 ∧ S ⊆ {x. a ∙ x = b}"
proof -
have False if "⋀a. a = 0 ∨ (∀b. ∃T ∈ S. a ∙ T ≠ b)"
proof -
have rel_int: "rel_interior S ≠ {}"
using assms rel_interior_eq_empty by auto
moreover
have "dim S ≠ dim (UNIV::'a set)"
by (metis aff_dim_zero affine_hull_UNIV ‹0 ∈ S› dim_UNIV empty_int hull_inc rel_int rel_interior_interior)
then obtain a where "a ≠ 0" and a: "span S ⊆ {x. a ∙ x = 0}"
using lowdim_subset_hyperplane
by (metis dim_UNIV dim_subset_UNIV order_less_le)
have "span UNIV = span S"
by (metis span_base span_not_UNIV_orthogonal that)
then have "UNIV ⊆ affine hull S"
by (simp add: ‹0 ∈ S› hull_inc affine_hull_span_0)
ultimately show False
using ‹rel_interior S ≠ {}› empty_int rel_interior_interior by blast
qed
then show ?thesis
by blast
qed
lemma empty_interior_subset_hyperplane:
fixes S :: "'a::euclidean_space set"
assumes "convex S" and int: "interior S = {}"
obtains a b where "a ≠ 0" "S ⊆ {x. a ∙ x = b}"
proof (cases "S = {}")
case True
then show ?thesis
using that by blast
next
case False
then obtain u where "u ∈ S"
by blast
have "∃a b. a ≠ 0 ∧ (λx. x - u) ` S ⊆ {x. a ∙ x = b}"
proof (rule empty_interior_subset_hyperplane_aux)
show "convex ((λx. x - u) ` S)"
using ‹convex S› by force
show "0 ∈ (λx. x - u) ` S"
by (simp add: ‹u ∈ S›)
show "interior ((λx. x - u) ` S) = {}"
by (simp add: int interior_translation_subtract)
qed
then obtain a b where "a ≠ 0" and ab: "(λx. x - u) ` S ⊆ {x. a ∙ x = b}"
by metis
then have "S ⊆ {x. a ∙ x = b + (a ∙ u)}"
using ab by (auto simp: algebra_simps)
then show ?thesis
using ‹a ≠ 0› that by auto
qed
lemma rel_interior_same_affine_hull:
fixes S :: "'n::euclidean_space set"
assumes "convex S"
shows "affine hull (rel_interior S) = affine hull S"
by (metis assms closure_same_affine_hull convex_closure_rel_interior)
lemma rel_interior_aff_dim:
fixes S :: "'n::euclidean_space set"
assumes "convex S"
shows "aff_dim (rel_interior S) = aff_dim S"
by (metis aff_dim_affine_hull2 assms rel_interior_same_affine_hull)
lemma rel_interior_rel_interior:
fixes S :: "'n::euclidean_space set"
assumes "convex S"
shows "rel_interior (rel_interior S) = rel_interior S"
proof -
have "openin (top_of_set (affine hull (rel_interior S))) (rel_interior S)"
using openin_rel_interior[of S] rel_interior_same_affine_hull[of S] assms by auto
then show ?thesis
using rel_interior_def by auto
qed
lemma rel_interior_rel_open:
fixes S :: "'n::euclidean_space set"
assumes "convex S"
shows "rel_open (rel_interior S)"
unfolding rel_open_def using rel_interior_rel_interior assms by auto
lemma convex_rel_interior_closure_aux:
fixes x y z :: "'n::euclidean_space"
assumes "0 < a" "0 < b" "(a + b) *⇩R z = a *⇩R x + b *⇩R y"
obtains e where "0 < e" "e < 1" "z = y - e *⇩R (y - x)"
proof -
define e where "e = a / (a + b)"
have "z = (1 / (a + b)) *⇩R ((a + b) *⇩R z)"
using assms by (simp add: eq_vector_fraction_iff)
also have "… = (1 / (a + b)) *⇩R (a *⇩R x + b *⇩R y)"
using assms scaleR_cancel_left[of "1/(a+b)" "(a + b) *⇩R z" "a *⇩R x + b *⇩R y"]
by auto
also have "… = y - e *⇩R (y-x)"
using e_def assms
by (simp add: divide_simps vector_fraction_eq_iff) (simp add: algebra_simps)
finally have "z = y - e *⇩R (y-x)"
by auto
moreover have "e > 0" "e < 1" using e_def assms by auto
ultimately show ?thesis using that[of e] by auto
qed
lemma convex_rel_interior_closure:
fixes S :: "'n::euclidean_space set"
assumes "convex S"
shows "rel_interior (closure S) = rel_interior S"
proof (cases "S = {}")
case True
then show ?thesis
using assms rel_interior_eq_empty by auto
next
case False
have "rel_interior (closure S) ⊇ rel_interior S"
using subset_rel_interior[of S "closure S"] closure_same_affine_hull closure_subset
by auto
moreover
{
fix z
assume z: "z ∈ rel_interior (closure S)"
obtain x where x: "x ∈ rel_interior S"
using ‹S ≠ {}› assms rel_interior_eq_empty by auto
have "z ∈ rel_interior S"
proof (cases "x = z")
case True
then show ?thesis using x by auto
next
case False
obtain e where e: "e > 0" "cball z e ∩ affine hull closure S ≤ closure S"
using z rel_interior_cball[of "closure S"] by auto
hence *: "0 < e/norm(z-x)" using e False by auto
define y where "y = z + (e/norm(z-x)) *⇩R (z-x)"
have yball: "y ∈ cball z e"
using y_def dist_norm[of z y] e by auto
have "x ∈ affine hull closure S"
using x rel_interior_subset_closure hull_inc[of x "closure S"] by blast
moreover have "z ∈ affine hull closure S"
using z rel_interior_subset hull_subset[of "closure S"] by blast
ultimately have "y ∈ affine hull closure S"
using y_def affine_affine_hull[of "closure S"]
mem_affine_3_minus [of "affine hull closure S" z z x "e/norm(z-x)"] by auto
then have "y ∈ closure S" using e yball by auto
have "(1 + (e/norm(z-x))) *⇩R z = (e/norm(z-x)) *⇩R x + y"
using y_def by (simp add: algebra_simps)
then obtain e1 where "0 < e1" "e1 < 1" "z = y - e1 *⇩R (y - x)"
using * convex_rel_interior_closure_aux[of "e / norm (z - x)" 1 z x y]
by (auto simp add: algebra_simps)
then show ?thesis
using rel_interior_closure_convex_shrink assms x ‹y ∈ closure S›
by fastforce
qed
}
ultimately show ?thesis by auto
qed
lemma convex_interior_closure:
fixes S :: "'n::euclidean_space set"
assumes "convex S"
shows "interior (closure S) = interior S"
using closure_aff_dim[of S] interior_rel_interior_gen[of S]
interior_rel_interior_gen[of "closure S"]
convex_rel_interior_closure[of S] assms
by auto
lemma open_subset_closure_of_interval:
assumes "open U" "is_interval S"
shows "U ⊆ closure S ⟷ U ⊆ interior S"
by (metis assms convex_interior_closure is_interval_convex open_subset_interior)
lemma closure_eq_rel_interior_eq:
fixes S1 S2 :: "'n::euclidean_space set"
assumes "convex S1"
and "convex S2"
shows "closure S1 = closure S2 ⟷ rel_interior S1 = rel_interior S2"
by (metis convex_rel_interior_closure convex_closure_rel_interior assms)
lemma closure_eq_between:
fixes S1 S2 :: "'n::euclidean_space set"
assumes "convex S1"
and "convex S2"
shows "closure S1 = closure S2 ⟷ rel_interior S1 ≤ S2 ∧ S2 ⊆ closure S1"
(is "?A ⟷ ?B")
proof
assume ?A
then show ?B
by (metis assms closure_subset convex_rel_interior_closure rel_interior_subset)
next
assume ?B
then have "closure S1 ⊆ closure S2"
by (metis assms(1) convex_closure_rel_interior closure_mono)
moreover from ‹?B› have "closure S1 ⊇ closure S2"
by (metis closed_closure closure_minimal)
ultimately show ?A ..
qed
lemma open_inter_closure_rel_interior:
fixes S A :: "'n::euclidean_space set"
assumes "convex S"
and "open A"
shows "A ∩ closure S = {} ⟷ A ∩ rel_interior S = {}"
by (metis assms convex_closure_rel_interior open_Int_closure_eq_empty)
lemma rel_interior_open_segment:
fixes a :: "'a :: euclidean_space"
shows "rel_interior(open_segment a b) = open_segment a b"
proof (cases "a = b")
case True then show ?thesis by auto
next
case False then
have "open_segment a b = affine hull {a, b} ∩ ball ((a + b) /⇩R 2) (norm (b - a) / 2)"
by (simp add: open_segment_as_ball)
then show ?thesis
unfolding rel_interior_eq openin_open
by (metis Elementary_Metric_Spaces.open_ball False affine_hull_open_segment)
qed
lemma rel_interior_closed_segment:
fixes a :: "'a :: euclidean_space"
shows "rel_interior(closed_segment a b) =
(if a = b then {a} else open_segment a b)"
proof (cases "a = b")
case True then show ?thesis by auto
next
case False then show ?thesis
by simp
(metis closure_open_segment convex_open_segment convex_rel_interior_closure
rel_interior_open_segment)
qed
lemmas rel_interior_segment = rel_interior_closed_segment rel_interior_open_segment
subsection‹The relative frontier of a set›
definition "rel_frontier S = closure S - rel_interior S"
lemma rel_frontier_empty [simp]: "rel_frontier {} = {}"
by (simp add: rel_frontier_def)
lemma rel_frontier_eq_empty:
fixes S :: "'n::euclidean_space set"
shows "rel_frontier S = {} ⟷ affine S"
unfolding rel_frontier_def
using rel_interior_subset_closure by (auto simp add: rel_interior_eq_closure [symmetric])
lemma rel_frontier_sing [simp]:
fixes a :: "'n::euclidean_space"
shows "rel_frontier {a} = {}"
by (simp add: rel_frontier_def)
lemma rel_frontier_affine_hull:
fixes S :: "'a::euclidean_space set"
shows "rel_frontier S ⊆ affine hull S"
using closure_affine_hull rel_frontier_def by fastforce
lemma rel_frontier_cball [simp]:
fixes a :: "'n::euclidean_space"
shows "rel_frontier(cball a r) = (if r = 0 then {} else sphere a r)"
proof (cases rule: linorder_cases [of r 0])
case less then show ?thesis
by (force simp: sphere_def)
next
case equal then show ?thesis by simp
next
case greater then show ?thesis
by simp (metis centre_in_ball empty_iff frontier_cball frontier_def interior_cball interior_rel_interior_gen rel_frontier_def)
qed
lemma rel_frontier_translation:
fixes a :: "'a::euclidean_space"
shows "rel_frontier((λx. a + x) ` S) = (λx. a + x) ` (rel_frontier S)"
by (simp add: rel_frontier_def translation_diff rel_interior_translation closure_translation)
lemma rel_frontier_nonempty_interior:
fixes S :: "'n::euclidean_space set"
shows "interior S ≠ {} ⟹ rel_frontier S = frontier S"
by (metis frontier_def interior_rel_interior_gen rel_frontier_def)
lemma rel_frontier_frontier:
fixes S :: "'n::euclidean_space set"
shows "affine hull S = UNIV ⟹ rel_frontier S = frontier S"
by (simp add: frontier_def rel_frontier_def rel_interior_interior)
lemma closest_point_in_rel_frontier:
"⟦closed S; S ≠ {}; x ∈ affine hull S - rel_interior S⟧
⟹ closest_point S x ∈ rel_frontier S"
by (simp add: closest_point_in_rel_interior closest_point_in_set rel_frontier_def)
lemma closed_rel_frontier [iff]:
fixes S :: "'n::euclidean_space set"
shows "closed (rel_frontier S)"
proof -
have *: "closedin (top_of_set (affine hull S)) (closure S - rel_interior S)"
by (simp add: closed_subset closedin_diff closure_affine_hull openin_rel_interior)
show ?thesis
proof (rule closedin_closed_trans[of "affine hull S" "rel_frontier S"])
show "closedin (top_of_set (affine hull S)) (rel_frontier S)"
by (simp add: "*" rel_frontier_def)
qed simp
qed
lemma closed_rel_boundary:
fixes S :: "'n::euclidean_space set"
shows "closed S ⟹ closed(S - rel_interior S)"
by (metis closed_rel_frontier closure_closed rel_frontier_def)
lemma compact_rel_boundary:
fixes S :: "'n::euclidean_space set"
shows "compact S ⟹ compact(S - rel_interior S)"
by (metis bounded_diff closed_rel_boundary closure_eq compact_closure compact_imp_closed)
lemma bounded_rel_frontier:
fixes S :: "'n::euclidean_space set"
shows "bounded S ⟹ bounded(rel_frontier S)"
by (simp add: bounded_closure bounded_diff rel_frontier_def)
lemma compact_rel_frontier_bounded:
fixes S :: "'n::euclidean_space set"
shows "bounded S ⟹ compact(rel_frontier S)"
using bounded_rel_frontier closed_rel_frontier compact_eq_bounded_closed by blast
lemma compact_rel_frontier:
fixes S :: "'n::euclidean_space set"
shows "compact S ⟹ compact(rel_frontier S)"
by (meson compact_eq_bounded_closed compact_rel_frontier_bounded)
lemma convex_same_rel_interior_closure:
fixes S :: "'n::euclidean_space set"
shows "⟦convex S; convex T⟧
⟹ rel_interior S = rel_interior T ⟷ closure S = closure T"
by (simp add: closure_eq_rel_interior_eq)
lemma convex_same_rel_interior_closure_straddle:
fixes S :: "'n::euclidean_space set"
shows "⟦convex S; convex T⟧
⟹ rel_interior S = rel_interior T ⟷
rel_interior S ⊆ T ∧ T ⊆ closure S"
by (simp add: closure_eq_between convex_same_rel_interior_closure)
lemma convex_rel_frontier_aff_dim:
fixes S1 S2 :: "'n::euclidean_space set"
assumes "convex S1"
and "convex S2"
and "S2 ≠ {}"
and "S1 ≤ rel_frontier S2"
shows "aff_dim S1 < aff_dim S2"
proof -
have "S1 ⊆ closure S2"
using assms unfolding rel_frontier_def by auto
then have *: "affine hull S1 ⊆ affine hull S2"
using hull_mono[of "S1" "closure S2"] closure_same_affine_hull[of S2] by blast
then have "aff_dim S1 ≤ aff_dim S2"
using * aff_dim_affine_hull[of S1] aff_dim_affine_hull[of S2]
aff_dim_subset[of "affine hull S1" "affine hull S2"]
by auto
moreover
{
assume eq: "aff_dim S1 = aff_dim S2"
then have "S1 ≠ {}"
using aff_dim_empty[of S1] aff_dim_empty[of S2] ‹S2 ≠ {}› by auto
have **: "affine hull S1 = affine hull S2"
by (simp_all add: * eq ‹S1 ≠ {}› affine_dim_equal)
obtain a where a: "a ∈ rel_interior S1"
using ‹S1 ≠ {}› rel_interior_eq_empty assms by auto
obtain T where T: "open T" "a ∈ T ∩ S1" "T ∩ affine hull S1 ⊆ S1"
using mem_rel_interior[of a S1] a by auto
then have "a ∈ T ∩ closure S2"
using a assms unfolding rel_frontier_def by auto
then obtain b where b: "b ∈ T ∩ rel_interior S2"
using open_inter_closure_rel_interior[of S2 T] assms T by auto
then have "b ∈ affine hull S1"
using rel_interior_subset hull_subset[of S2] ** by auto
then have "b ∈ S1"
using T b by auto
then have False
using b assms unfolding rel_frontier_def by auto
}
ultimately show ?thesis
using less_le by auto
qed
lemma convex_rel_interior_if:
fixes S :: "'n::euclidean_space set"
assumes "convex S"
and "z ∈ rel_interior S"
shows "∀x∈affine hull S. ∃m. m > 1 ∧ (∀e. e > 1 ∧ e ≤ m ⟶ (1 - e) *⇩R x + e *⇩R z ∈ S)"
proof -
obtain e1 where e1: "e1 > 0 ∧ cball z e1 ∩ affine hull S ⊆ S"
using mem_rel_interior_cball[of z S] assms by auto
{
fix x
assume x: "x ∈ affine hull S"
{
assume "x ≠ z"
define m where "m = 1 + e1/norm(x-z)"
hence "m > 1" using e1 ‹x ≠ z› by auto
{
fix e
assume e: "e > 1 ∧ e ≤ m"
have "z ∈ affine hull S"
using assms rel_interior_subset hull_subset[of S] by auto
then have *: "(1 - e)*⇩R x + e *⇩R z ∈ affine hull S"
using mem_affine[of "affine hull S" x z "(1-e)" e] affine_affine_hull[of S] x
by auto
have "norm (z + e *⇩R x - (x + e *⇩R z)) = norm ((e - 1) *⇩R (x - z))"
by (simp add: algebra_simps)
also have "… = (e - 1) * norm (x-z)"
using norm_scaleR e by auto
also have "… ≤ (m - 1) * norm (x - z)"
using e mult_right_mono[of _ _ "norm(x-z)"] by auto
also have "… = (e1 / norm (x - z)) * norm (x - z)"
using m_def by auto
also have "… = e1"
using ‹x ≠ z› e1 by simp
finally have **: "norm (z + e *⇩R x - (x + e *⇩R z)) ≤ e1"
by auto
have "(1 - e)*⇩R x+ e *⇩R z ∈ cball z e1"
using m_def **
unfolding cball_def dist_norm
by (auto simp add: algebra_simps)
then have "(1 - e) *⇩R x+ e *⇩R z ∈ S"
using e * e1 by auto
}
then have "∃m. m > 1 ∧ (∀e. e > 1 ∧ e ≤ m ⟶ (1 - e) *⇩R x + e *⇩R z ∈ S )"
using ‹m> 1 › by auto
}
moreover
{
assume "x = z"
define m where "m = 1 + e1"
then have "m > 1"
using e1 by auto
{
fix e
assume e: "e > 1 ∧ e ≤ m"
then have "(1 - e) *⇩R x + e *⇩R z ∈ S"
using e1 x ‹x = z› by (auto simp add: algebra_simps)
then have "(1 - e) *⇩R x + e *⇩R z ∈ S"
using e by auto
}
then have "∃m. m > 1 ∧ (∀e. e > 1 ∧ e ≤ m ⟶ (1 - e) *⇩R x + e *⇩R z ∈ S)"
using ‹m > 1› by auto
}
ultimately have "∃m. m > 1 ∧ (∀e. e > 1 ∧ e ≤ m ⟶ (1 - e) *⇩R x + e *⇩R z ∈ S )"
by blast
}
then show ?thesis by auto
qed
lemma convex_rel_interior_if2:
fixes S :: "'n::euclidean_space set"
assumes "convex S"
assumes "z ∈ rel_interior S"
shows "∀x∈affine hull S. ∃e. e > 1 ∧ (1 - e)*⇩R x + e *⇩R z ∈ S"
using convex_rel_interior_if[of S z] assms by auto
lemma convex_rel_interior_only_if:
fixes S :: "'n::euclidean_space set"
assumes "convex S"
and "S ≠ {}"
assumes "∀x∈S. ∃e. e > 1 ∧ (1 - e) *⇩R x + e *⇩R z ∈ S"
shows "z ∈ rel_interior S"
proof -
obtain x where x: "x ∈ rel_interior S"
using rel_interior_eq_empty assms by auto
then have "x ∈ S"
using rel_interior_subset by auto
then obtain e where e: "e > 1 ∧ (1 - e) *⇩R x + e *⇩R z ∈ S"
using assms by auto
define y where [abs_def]: "y = (1 - e) *⇩R x + e *⇩R z"
then have "y ∈ S" using e by auto
define e1 where "e1 = 1/e"
then have "0 < e1 ∧ e1 < 1" using e by auto
then have "z =y - (1 - e1) *⇩R (y - x)"
using e1_def y_def by (auto simp add: algebra_simps)
then show ?thesis
using rel_interior_convex_shrink[of S x y "1-e1"] ‹0 < e1 ∧ e1 < 1› ‹y ∈ S› x assms
by auto
qed
lemma convex_rel_interior_iff:
fixes S :: "'n::euclidean_space set"
assumes "convex S"
and "S ≠ {}"
shows "z ∈ rel_interior S ⟷ (∀x∈S. ∃e. e > 1 ∧ (1 - e) *⇩R x + e *⇩R z ∈ S)"
using assms hull_subset[of S "affine"]
convex_rel_interior_if[of S z] convex_rel_interior_only_if[of S z]
by auto
lemma convex_rel_interior_iff2:
fixes S :: "'n::euclidean_space set"
assumes "convex S"
and "S ≠ {}"
shows "z ∈ rel_interior S ⟷ (∀x∈affine hull S. ∃e. e > 1 ∧ (1 - e) *⇩R x + e *⇩R z ∈ S)"
using assms hull_subset[of S] convex_rel_interior_if2[of S z] convex_rel_interior_only_if[of S z]
by auto
lemma convex_interior_iff:
fixes S :: "'n::euclidean_space set"
assumes "convex S"
shows "z ∈ interior S ⟷ (∀x. ∃e. e > 0 ∧ z + e *⇩R x ∈ S)"
proof (cases "aff_dim S = int DIM('n)")
case False
{ assume "z ∈ interior S"
then have False
using False interior_rel_interior_gen[of S] by auto }
moreover
{ assume r: "∀x. ∃e. e > 0 ∧ z + e *⇩R x ∈ S"
{ fix x
obtain e1 where e1: "e1 > 0 ∧ z + e1 *⇩R (x - z) ∈ S"
using r by auto
obtain e2 where e2: "e2 > 0 ∧ z + e2 *⇩R (z - x) ∈ S"
using r by auto
define x1 where [abs_def]: "x1 = z + e1 *⇩R (x - z)"
then have x1: "x1 ∈ affine hull S"
using e1 hull_subset[of S] by auto
define x2 where [abs_def]: "x2 = z + e2 *⇩R (z - x)"
then have x2: "x2 ∈ affine hull S"
using e2 hull_subset[of S] by auto
have *: "e1/(e1+e2) + e2/(e1+e2) = 1"
using add_divide_distrib[of e1 e2 "e1+e2"] e1 e2 by simp
then have "z = (e2/(e1+e2)) *⇩R x1 + (e1/(e1+e2)) *⇩R x2"
by (simp add: x1_def x2_def algebra_simps) (simp add: "*" pth_8)
then have z: "z ∈ affine hull S"
using mem_affine[of "affine hull S" x1 x2 "e2/(e1+e2)" "e1/(e1+e2)"]
x1 x2 affine_affine_hull[of S] *
by auto
have "x1 - x2 = (e1 + e2) *⇩R (x - z)"
using x1_def x2_def by (auto simp add: algebra_simps)
then have "x = z+(1/(e1+e2)) *⇩R (x1-x2)"
using e1 e2 by simp
then have "x ∈ affine hull S"
using mem_affine_3_minus[of "affine hull S" z x1 x2 "1/(e1+e2)"]
x1 x2 z affine_affine_hull[of S]
by auto
}
then have "affine hull S = UNIV"
by auto
then have "aff_dim S = int DIM('n)"
using aff_dim_affine_hull[of S] by (simp)
then have False
using False by auto
}
ultimately show ?thesis by auto
next
case True
then have "S ≠ {}"
using aff_dim_empty[of S] by auto
have *: "affine hull S = UNIV"
using True affine_hull_UNIV by auto
{
assume "z ∈ interior S"
then have "z ∈ rel_interior S"
using True interior_rel_interior_gen[of S] by auto
then have **: "∀x. ∃e. e > 1 ∧ (1 - e) *⇩R x + e *⇩R z ∈ S"
using convex_rel_interior_iff2[of S z] assms ‹S ≠ {}› * by auto
fix x
obtain e1 where e1: "e1 > 1" "(1 - e1) *⇩R (z - x) + e1 *⇩R z ∈ S"
using **[rule_format, of "z-x"] by auto
define e where [abs_def]: "e = e1 - 1"
then have "(1 - e1) *⇩R (z - x) + e1 *⇩R z = z + e *⇩R x"
by (simp add: algebra_simps)
then have "e > 0" "z + e *⇩R x ∈ S"
using e1 e_def by auto
then have "∃e. e > 0 ∧ z + e *⇩R x ∈ S"
by auto
}
moreover
{
assume r: "∀x. ∃e. e > 0 ∧ z + e *⇩R x ∈ S"
{
fix x
obtain e1 where e1: "e1 > 0" "z + e1 *⇩R (z - x) ∈ S"
using r[rule_format, of "z-x"] by auto
define e where "e = e1 + 1"
then have "z + e1 *⇩R (z - x) = (1 - e) *⇩R x + e *⇩R z"
by (simp add: algebra_simps)
then have "e > 1" "(1 - e)*⇩R x + e *⇩R z ∈ S"
using e1 e_def by auto
then have "∃e. e > 1 ∧ (1 - e) *⇩R x + e *⇩R z ∈ S" by auto
}
then have "z ∈ rel_interior S"
using convex_rel_interior_iff2[of S z] assms ‹S ≠ {}› by auto
then have "z ∈ interior S"
using True interior_rel_interior_gen[of S] by auto
}
ultimately show ?thesis by auto
qed
subsubsection ‹Relative interior and closure under common operations›
lemma rel_interior_inter_aux: "⋂{rel_interior S |S. S ∈ I} ⊆ ⋂I"
proof -
{ fix y
assume "y ∈ ⋂{rel_interior S |S. S ∈ I}"
then have y: "∀S ∈ I. y ∈ rel_interior S"
by auto
{ fix S
assume "S ∈ I"
then have "y ∈ S"
using rel_interior_subset y by auto
}
then have "y ∈ ⋂I" by auto
}
then show ?thesis by auto
qed
lemma convex_closure_rel_interior_Int:
assumes "⋀S. S∈ℱ ⟹ convex (S :: 'n::euclidean_space set)"
and "⋂(rel_interior ` ℱ) ≠ {}"
shows "⋂(closure ` ℱ) ⊆ closure (⋂(rel_interior ` ℱ))"
proof -
obtain x where x: "∀S∈ℱ. x ∈ rel_interior S"
using assms by auto
show ?thesis
proof
fix y
assume y: "y ∈ ⋂ (closure ` ℱ)"
show "y ∈ closure (⋂(rel_interior ` ℱ))"
proof (cases "y=x")
case True
with closure_subset x show ?thesis
by fastforce
next
case False
show ?thesis
proof (clarsimp simp: closure_approachable_le)
fix ε :: real
assume e: "ε > 0"
define e1 where "e1 = min 1 (ε/norm (y - x))"
then have e1: "e1 > 0" "e1 ≤ 1" "e1 * norm (y - x) ≤ ε"
using ‹y ≠ x› ‹ε > 0› le_divide_eq[of e1 ε "norm (y - x)"]
by simp_all
define z where "z = y - e1 *⇩R (y - x)"
{
fix S
assume "S ∈ ℱ"
then have "z ∈ rel_interior S"
using rel_interior_closure_convex_shrink[of S x y e1] assms x y e1 z_def
by auto
}
then have *: "z ∈ ⋂(rel_interior ` ℱ)"
by auto
show "∃x∈⋂ (rel_interior ` ℱ). dist x y ≤ ε"
using ‹y ≠ x› z_def * e1 e dist_norm[of z y]
by force
qed
qed
qed
qed
lemma closure_Inter_convex:
fixes ℱ :: "'n::euclidean_space set set"
assumes "⋀S. S ∈ ℱ ⟹ convex S" and "⋂(rel_interior ` ℱ) ≠ {}"
shows "closure(⋂ℱ) = ⋂(closure ` ℱ)"
proof -
have "⋂(closure ` ℱ) ≤ closure (⋂(rel_interior ` ℱ))"
by (meson assms convex_closure_rel_interior_Int)
moreover
have "closure (⋂(rel_interior ` ℱ)) ⊆ closure (⋂ℱ)"
using rel_interior_inter_aux closure_mono[of "⋂(rel_interior ` ℱ)" "⋂ℱ"]
by auto
ultimately show ?thesis
using closure_Int[of ℱ] by blast
qed
lemma closure_Inter_convex_open:
"(⋀S::'n::euclidean_space set. S ∈ ℱ ⟹ convex S ∧ open S)
⟹ closure(⋂ℱ) = (if ⋂ℱ = {} then {} else ⋂(closure ` ℱ))"
by (simp add: closure_Inter_convex rel_interior_open)
lemma convex_Inter_rel_interior_same_closure:
fixes ℱ :: "'n::euclidean_space set set"
assumes "⋀S. S ∈ ℱ ⟹ convex S"
and "⋂(rel_interior ` ℱ) ≠ {}"
shows "closure (⋂(rel_interior ` ℱ)) = closure (⋂ℱ)"
proof -
have "⋂(closure ` ℱ) ⊆ closure (⋂(rel_interior ` ℱ))"
by (meson assms convex_closure_rel_interior_Int)
moreover
have "closure (⋂(rel_interior ` ℱ)) ⊆ closure (⋂ℱ)"
by (metis Setcompr_eq_image closure_mono rel_interior_inter_aux)
ultimately show ?thesis
by (simp add: assms closure_Inter_convex)
qed
lemma convex_rel_interior_Inter:
fixes ℱ :: "'n::euclidean_space set set"
assumes "⋀S. S ∈ ℱ ⟹ convex S"
and "⋂(rel_interior ` ℱ) ≠ {}"
shows "rel_interior (⋂ℱ) ⊆ ⋂(rel_interior ` ℱ)"
proof -
have "convex (⋂ℱ)"
using assms convex_Inter by auto
moreover
have "convex (⋂(rel_interior ` ℱ))"
using assms by (metis convex_rel_interior convex_INT)
ultimately
have "rel_interior (⋂(rel_interior ` ℱ)) = rel_interior (⋂ℱ)"
using convex_Inter_rel_interior_same_closure assms
closure_eq_rel_interior_eq[of "⋂(rel_interior ` ℱ)" "⋂ℱ"]
by blast
then show ?thesis
using rel_interior_subset[of "⋂(rel_interior ` ℱ)"] by auto
qed
lemma convex_rel_interior_finite_Inter:
fixes ℱ :: "'n::euclidean_space set set"
assumes "⋀S. S ∈ ℱ ⟹ convex S"
and "⋂(rel_interior ` ℱ) ≠ {}"
and "finite ℱ"
shows "rel_interior (⋂ℱ) = ⋂(rel_interior ` ℱ)"
proof -
have "⋂ℱ ≠ {}"
using assms rel_interior_inter_aux[of ℱ] by auto
have "convex (⋂ℱ)"
using convex_Inter assms by auto
show ?thesis
proof (cases "ℱ = {}")
case True
then show ?thesis
using Inter_empty rel_interior_UNIV by auto
next
case False
{
fix z
assume z: "z ∈ ⋂(rel_interior ` ℱ)"
{
fix x
assume x: "x ∈ ⋂ℱ"
{
fix S
assume S: "S ∈ ℱ"
then have "z ∈ rel_interior S" "x ∈ S"
using z x by auto
then have "∃m. m > 1 ∧ (∀e. e > 1 ∧ e ≤ m ⟶ (1 - e)*⇩R x + e *⇩R z ∈ S)"
using convex_rel_interior_if[of S z] S assms hull_subset[of S] by auto
}
then obtain mS where
mS: "∀S∈ℱ. mS S > 1 ∧ (∀e. e > 1 ∧ e ≤ mS S ⟶ (1 - e) *⇩R x + e *⇩R z ∈ S)" by metis
define e where "e = Min (mS ` ℱ)"
then have "e ∈ mS ` ℱ" using assms ‹ℱ ≠ {}› by simp
then have "e > 1" using mS by auto
moreover have "∀S∈ℱ. e ≤ mS S"
using e_def assms by auto
ultimately have "∃e > 1. (1 - e) *⇩R x + e *⇩R z ∈ ⋂ℱ"
using mS by auto
}
then have "z ∈ rel_interior (⋂ℱ)"
using convex_rel_interior_iff[of "⋂ℱ" z] ‹⋂ℱ ≠ {}› ‹convex (⋂ℱ)› by auto
}
then show ?thesis
using convex_rel_interior_Inter[of ℱ] assms by auto
qed
qed
lemma closure_Int_convex:
fixes S T :: "'n::euclidean_space set"
assumes "convex S"
and "convex T"
assumes "rel_interior S ∩ rel_interior T ≠ {}"
shows "closure (S ∩ T) = closure S ∩ closure T"
using closure_Inter_convex[of "{S,T}"] assms by auto
lemma convex_rel_interior_inter_two:
fixes S T :: "'n::euclidean_space set"
assumes "convex S"
and "convex T"
and "rel_interior S ∩ rel_interior T ≠ {}"
shows "rel_interior (S ∩ T) = rel_interior S ∩ rel_interior T"
using convex_rel_interior_finite_Inter[of "{S,T}"] assms by auto
lemma convex_affine_closure_Int:
fixes S T :: "'n::euclidean_space set"
assumes "convex S"
and "affine T"
and "rel_interior S ∩ T ≠ {}"
shows "closure (S ∩ T) = closure S ∩ T"
by (metis affine_imp_convex assms closure_Int_convex rel_interior_affine rel_interior_eq_closure)
lemma connected_component_1_gen:
fixes S :: "'a :: euclidean_space set"
assumes "DIM('a) = 1"
shows "connected_component S a b ⟷ closed_segment a b ⊆ S"
unfolding connected_component_def
by (metis (no_types, lifting) assms subsetD subsetI convex_contains_segment convex_segment(1)
ends_in_segment connected_convex_1_gen)
lemma connected_component_1:
fixes S :: "real set"
shows "connected_component S a b ⟷ closed_segment a b ⊆ S"
by (simp add: connected_component_1_gen)
lemma convex_affine_rel_interior_Int:
fixes S T :: "'n::euclidean_space set"
assumes "convex S"
and "affine T"
and "rel_interior S ∩ T ≠ {}"
shows "rel_interior (S ∩ T) = rel_interior S ∩ T"
by (simp add: affine_imp_convex assms convex_rel_interior_inter_two rel_interior_affine)
lemma convex_affine_rel_frontier_Int:
fixes S T :: "'n::euclidean_space set"
assumes "convex S"
and "affine T"
and "interior S ∩ T ≠ {}"
shows "rel_frontier(S ∩ T) = frontier S ∩ T"
using assms
unfolding rel_frontier_def frontier_def
using convex_affine_closure_Int convex_affine_rel_interior_Int rel_interior_nonempty_interior by fastforce
lemma rel_interior_convex_Int_affine:
fixes S :: "'a::euclidean_space set"
assumes "convex S" "affine T" "interior S ∩ T ≠ {}"
shows "rel_interior(S ∩ T) = interior S ∩ T"
by (metis Int_emptyI assms convex_affine_rel_interior_Int empty_iff interior_rel_interior_gen)
lemma subset_rel_interior_convex:
fixes S T :: "'n::euclidean_space set"
assumes "convex S"
and "convex T"
and "S ≤ closure T"
and "¬ S ⊆ rel_frontier T"
shows "rel_interior S ⊆ rel_interior T"
proof -
have *: "S ∩ closure T = S"
using assms by auto
have "¬ rel_interior S ⊆ rel_frontier T"
using closure_mono[of "rel_interior S" "rel_frontier T"] closed_rel_frontier[of T]
closure_closed[of S] convex_closure_rel_interior[of S] closure_subset[of S] assms
by auto
then have "rel_interior S ∩ rel_interior (closure T) ≠ {}"
using assms rel_frontier_def[of T] rel_interior_subset convex_rel_interior_closure[of T]
by auto
then have "rel_interior S ∩ rel_interior T = rel_interior (S ∩ closure T)"
using assms convex_closure convex_rel_interior_inter_two[of S "closure T"]
convex_rel_interior_closure[of T]
by auto
also have "… = rel_interior S"
using * by auto
finally show ?thesis
by auto
qed
lemma rel_interior_convex_linear_image:
fixes f :: "'m::euclidean_space ⇒ 'n::euclidean_space"
assumes "linear f"
and "convex S"
shows "f ` (rel_interior S) = rel_interior (f ` S)"
proof (cases "S = {}")
case True
then show ?thesis
using assms by auto
next
case False
interpret linear f by fact
have *: "f ` (rel_interior S) ⊆ f ` S"
unfolding image_mono using rel_interior_subset by auto
have "f ` S ⊆ f ` (closure S)"
unfolding image_mono using closure_subset by auto
also have "… = f ` (closure (rel_interior S))"
using convex_closure_rel_interior assms by auto
also have "… ⊆ closure (f ` (rel_interior S))"
using closure_linear_image_subset assms by auto
finally have "closure (f ` S) = closure (f ` rel_interior S)"
using closure_mono[of "f ` S" "closure (f ` rel_interior S)"] closure_closure
closure_mono[of "f ` rel_interior S" "f ` S"] *
by auto
then have "rel_interior (f ` S) = rel_interior (f ` rel_interior S)"
using assms convex_rel_interior
linear_conv_bounded_linear[of f] convex_linear_image[of _ S]
convex_linear_image[of _ "rel_interior S"]
closure_eq_rel_interior_eq[of "f ` S" "f ` rel_interior S"]
by auto
then have "rel_interior (f ` S) ⊆ f ` rel_interior S"
using rel_interior_subset by auto
moreover
{
fix z
assume "z ∈ f ` rel_interior S"
then obtain z1 where z1: "z1 ∈ rel_interior S" "f z1 = z" by auto
{
fix x
assume "x ∈ f ` S"
then obtain x1 where x1: "x1 ∈ S" "f x1 = x" by auto
then obtain e where e: "e > 1" "(1 - e) *⇩R x1 + e *⇩R z1 ∈ S"
using convex_rel_interior_iff[of S z1] ‹convex S› x1 z1 by auto
moreover have "f ((1 - e) *⇩R x1 + e *⇩R z1) = (1 - e) *⇩R x + e *⇩R z"
using x1 z1 by (simp add: linear_add linear_scale ‹linear f›)
ultimately have "(1 - e) *⇩R x + e *⇩R z ∈ f ` S"
using imageI[of "(1 - e) *⇩R x1 + e *⇩R z1" S f] by auto
then have "∃e. e > 1 ∧ (1 - e) *⇩R x + e *⇩R z ∈ f ` S"
using e by auto
}
then have "z ∈ rel_interior (f ` S)"
using convex_rel_interior_iff[of "f ` S" z] ‹convex S› ‹linear f›
‹S ≠ {}› convex_linear_image[of f S] linear_conv_bounded_linear[of f]
by auto
}
ultimately show ?thesis by auto
qed
lemma rel_interior_convex_linear_preimage:
fixes f :: "'m::euclidean_space ⇒ 'n::euclidean_space"
assumes "linear f"
and "convex S"
and "f -` (rel_interior S) ≠ {}"
shows "rel_interior (f -` S) = f -` (rel_interior S)"
proof -
interpret linear f by fact
have "S ≠ {}"
using assms by auto
have nonemp: "f -` S ≠ {}"
by (metis assms(3) rel_interior_subset subset_empty vimage_mono)
then have "S ∩ (range f) ≠ {}"
by auto
have conv: "convex (f -` S)"
using convex_linear_vimage assms by auto
then have "convex (S ∩ range f)"
by (simp add: assms(2) convex_Int convex_linear_image linear_axioms)
{
fix z
assume "z ∈ f -` (rel_interior S)"
then have z: "f z ∈ rel_interior S"
by auto
{
fix x
assume "x ∈ f -` S"
then have "f x ∈ S" by auto
then obtain e where e: "e > 1" "(1 - e) *⇩R f x + e *⇩R f z ∈ S"
using convex_rel_interior_iff[of S "f z"] z assms ‹S ≠ {}› by auto
moreover have "(1 - e) *⇩R f x + e *⇩R f z = f ((1 - e) *⇩R x + e *⇩R z)"
using ‹linear f› by (simp add: linear_iff)
ultimately have "∃e. e > 1 ∧ (1 - e) *⇩R x + e *⇩R z ∈ f -` S"
using e by auto
}
then have "z ∈ rel_interior (f -` S)"
using convex_rel_interior_iff[of "f -` S" z] conv nonemp by auto
}
moreover
{
fix z
assume z: "z ∈ rel_interior (f -` S)"
{
fix x
assume "x ∈ S ∩ range f"
then obtain y where y: "f y = x" "y ∈ f -` S" by auto
then obtain e where e: "e > 1" "(1 - e) *⇩R y + e *⇩R z ∈ f -` S"
using convex_rel_interior_iff[of "f -` S" z] z conv by auto
moreover have "(1 - e) *⇩R x + e *⇩R f z = f ((1 - e) *⇩R y + e *⇩R z)"
using ‹linear f› y by (simp add: linear_iff)
ultimately have "∃e. e > 1 ∧ (1 - e) *⇩R x + e *⇩R f z ∈ S ∩ range f"
using e by auto
}
then have "f z ∈ rel_interior (S ∩ range f)"
using ‹convex (S ∩ (range f))› ‹S ∩ range f ≠ {}›
convex_rel_interior_iff[of "S ∩ (range f)" "f z"]
by auto
moreover have "affine (range f)"
by (simp add: linear_axioms linear_subspace_image subspace_imp_affine)
ultimately have "f z ∈ rel_interior S"
using convex_affine_rel_interior_Int[of S "range f"] assms by auto
then have "z ∈ f -` (rel_interior S)"
by auto
}
ultimately show ?thesis by auto
qed
lemma rel_interior_Times:
fixes S :: "'n::euclidean_space set"
and T :: "'m::euclidean_space set"
assumes "convex S"
and "convex T"
shows "rel_interior (S × T) = rel_interior S × rel_interior T"
proof (cases "S = {} ∨ T = {}")
case True
then show ?thesis
by auto
next
case False
then have "S ≠ {}" "T ≠ {}"
by auto
then have ri: "rel_interior S ≠ {}" "rel_interior T ≠ {}"
using rel_interior_eq_empty assms by auto
then have "fst -` rel_interior S ≠ {}"
using fst_vimage_eq_Times[of "rel_interior S"] by auto
then have "rel_interior ((fst :: 'n * 'm ⇒ 'n) -` S) = fst -` rel_interior S"
using linear_fst ‹convex S› rel_interior_convex_linear_preimage[of fst S] by auto
then have s: "rel_interior (S × (UNIV :: 'm set)) = rel_interior S × UNIV"
by (simp add: fst_vimage_eq_Times)
from ri have "snd -` rel_interior T ≠ {}"
using snd_vimage_eq_Times[of "rel_interior T"] by auto
then have "rel_interior ((snd :: 'n * 'm ⇒ 'm) -` T) = snd -` rel_interior T"
using linear_snd ‹convex T› rel_interior_convex_linear_preimage[of snd T] by auto
then have t: "rel_interior ((UNIV :: 'n set) × T) = UNIV × rel_interior T"
by (simp add: snd_vimage_eq_Times)
from s t have *: "rel_interior (S × (UNIV :: 'm set)) ∩ rel_interior ((UNIV :: 'n set) × T) =
rel_interior S × rel_interior T" by auto
have "S × T = S × (UNIV :: 'm set) ∩ (UNIV :: 'n set) × T"
by auto
then have "rel_interior (S × T) = rel_interior ((S × (UNIV :: 'm set)) ∩ ((UNIV :: 'n set) × T))"
by auto
also have "… = rel_interior (S × (UNIV :: 'm set)) ∩ rel_interior ((UNIV :: 'n set) × T)"
using * ri assms convex_Times
by (subst convex_rel_interior_inter_two) auto
finally show ?thesis using * by auto
qed
lemma rel_interior_scaleR:
fixes S :: "'n::euclidean_space set"
assumes "c ≠ 0"
shows "((*⇩R) c) ` (rel_interior S) = rel_interior (((*⇩R) c) ` S)"
using rel_interior_injective_linear_image[of "((*⇩R) c)" S]
linear_conv_bounded_linear[of "(*⇩R) c"] linear_scaleR injective_scaleR[of c] assms
by auto
lemma rel_interior_convex_scaleR:
fixes S :: "'n::euclidean_space set"
assumes "convex S"
shows "((*⇩R) c) ` (rel_interior S) = rel_interior (((*⇩R) c) ` S)"
by (metis assms linear_scaleR rel_interior_convex_linear_image)
lemma convex_rel_open_scaleR:
fixes S :: "'n::euclidean_space set"
assumes "convex S"
and "rel_open S"
shows "convex (((*⇩R) c) ` S) ∧ rel_open (((*⇩R) c) ` S)"
by (metis assms convex_scaling rel_interior_convex_scaleR rel_open_def)
lemma convex_rel_open_finite_Inter:
fixes ℱ :: "'n::euclidean_space set set"
assumes "⋀S. S ∈ ℱ ⟹ convex S ∧ rel_open S"
and "finite ℱ"
shows "convex (⋂ℱ) ∧ rel_open (⋂ℱ)"
proof (cases "⋂{rel_interior S |S. S ∈ ℱ} = {}")
case True
then have "⋂ℱ = {}"
using assms unfolding rel_open_def by auto
then show ?thesis
unfolding rel_open_def by auto
next
case False
then have "rel_open (⋂ℱ)"
using assms convex_rel_interior_finite_Inter[of ℱ] by (force simp: rel_open_def)
then show ?thesis
using convex_Inter assms by auto
qed
lemma convex_rel_open_linear_image:
fixes f :: "'m::euclidean_space ⇒ 'n::euclidean_space"
assumes "linear f"
and "convex S"
and "rel_open S"
shows "convex (f ` S) ∧ rel_open (f ` S)"
by (metis assms convex_linear_image rel_interior_convex_linear_image rel_open_def)
lemma convex_rel_open_linear_preimage:
fixes f :: "'m::euclidean_space ⇒ 'n::euclidean_space"
assumes "linear f"
and "convex S"
and "rel_open S"
shows "convex (f -` S) ∧ rel_open (f -` S)"
proof (cases "f -` (rel_interior S) = {}")
case True
then have "f -` S = {}"
using assms unfolding rel_open_def by auto
then show ?thesis
unfolding rel_open_def by auto
next
case False
then have "rel_open (f -` S)"
using assms unfolding rel_open_def
using rel_interior_convex_linear_preimage[of f S]
by auto
then show ?thesis
using convex_linear_vimage assms
by auto
qed
lemma rel_interior_projection:
fixes S :: "('m::euclidean_space × 'n::euclidean_space) set"
and f :: "'m::euclidean_space ⇒ 'n::euclidean_space set"
assumes "convex S"
and "f = (λy. {z. (y, z) ∈ S})"
shows "(y, z) ∈ rel_interior S ⟷ (y ∈ rel_interior {y. (f y ≠ {})} ∧ z ∈ rel_interior (f y))"
proof -
{
fix y
assume "y ∈ {y. f y ≠ {}}"
then obtain z where "(y, z) ∈ S"
using assms by auto
then have "∃x. x ∈ S ∧ y = fst x"
by auto
then obtain x where "x ∈ S" "y = fst x"
by blast
then have "y ∈ fst ` S"
unfolding image_def by auto
}
then have "fst ` S = {y. f y ≠ {}}"
unfolding fst_def using assms by auto
then have h1: "fst ` rel_interior S = rel_interior {y. f y ≠ {}}"
using rel_interior_convex_linear_image[of fst S] assms linear_fst by auto
{
fix y
assume "y ∈ rel_interior {y. f y ≠ {}}"
then have "y ∈ fst ` rel_interior S"
using h1 by auto
then have *: "rel_interior S ∩ fst -` {y} ≠ {}"
by auto
moreover have aff: "affine (fst -` {y})"
unfolding affine_alt by (simp add: algebra_simps)
ultimately have **: "rel_interior (S ∩ fst -` {y}) = rel_interior S ∩ fst -` {y}"
using convex_affine_rel_interior_Int[of S "fst -` {y}"] assms by auto
have conv: "convex (S ∩ fst -` {y})"
using convex_Int assms aff affine_imp_convex by auto
{
fix x
assume "x ∈ f y"
then have "(y, x) ∈ S ∩ (fst -` {y})"
using assms by auto
moreover have "x = snd (y, x)" by auto
ultimately have "x ∈ snd ` (S ∩ fst -` {y})"
by blast
}
then have "snd ` (S ∩ fst -` {y}) = f y"
using assms by auto
then have ***: "rel_interior (f y) = snd ` rel_interior (S ∩ fst -` {y})"
using rel_interior_convex_linear_image[of snd "S ∩ fst -` {y}"] linear_snd conv
by auto
{
fix z
assume "z ∈ rel_interior (f y)"
then have "z ∈ snd ` rel_interior (S ∩ fst -` {y})"
using *** by auto
moreover have "{y} = fst ` rel_interior (S ∩ fst -` {y})"
using * ** rel_interior_subset by auto
ultimately have "(y, z) ∈ rel_interior (S ∩ fst -` {y})"
by force
then have "(y,z) ∈ rel_interior S"
using ** by auto
}
moreover
{
fix z
assume "(y, z) ∈ rel_interior S"
then have "(y, z) ∈ rel_interior (S ∩ fst -` {y})"
using ** by auto
then have "z ∈ snd ` rel_interior (S ∩ fst -` {y})"
by (metis Range_iff snd_eq_Range)
then have "z ∈ rel_interior (f y)"
using *** by auto
}
ultimately have "⋀z. (y, z) ∈ rel_interior S ⟷ z ∈ rel_interior (f y)"
by auto
}
then have h2: "⋀y z. y ∈ rel_interior {t. f t ≠ {}} ⟹
(y, z) ∈ rel_interior S ⟷ z ∈ rel_interior (f y)"
by auto
{
fix y z
assume asm: "(y, z) ∈ rel_interior S"
then have "y ∈ fst ` rel_interior S"
by (metis Domain_iff fst_eq_Domain)
then have "y ∈ rel_interior {t. f t ≠ {}}"
using h1 by auto
then have "y ∈ rel_interior {t. f t ≠ {}}" and "(z ∈ rel_interior (f y))"
using h2 asm by auto
}
then show ?thesis using h2 by blast
qed
lemma rel_frontier_Times:
fixes S :: "'n::euclidean_space set"
and T :: "'m::euclidean_space set"
assumes "convex S"
and "convex T"
shows "rel_frontier S × rel_frontier T ⊆ rel_frontier (S × T)"
by (force simp: rel_frontier_def rel_interior_Times assms closure_Times)
subsubsection ‹Relative interior of convex cone›
lemma cone_rel_interior:
fixes S :: "'m::euclidean_space set"
assumes "cone S"
shows "cone ({0} ∪ rel_interior S)"
proof (cases "S = {}")
case True
then show ?thesis
by (simp add: cone_0)
next
case False
then have *: "0 ∈ S ∧ (∀c. c > 0 ⟶ (*⇩R) c ` S = S)"
using cone_iff[of S] assms by auto
then have *: "0 ∈ ({0} ∪ rel_interior S)"
and "∀c. c > 0 ⟶ (*⇩R) c ` ({0} ∪ rel_interior S) = ({0} ∪ rel_interior S)"
by (auto simp add: rel_interior_scaleR)
then show ?thesis
using cone_iff[of "{0} ∪ rel_interior S"] by auto
qed
lemma rel_interior_convex_cone_aux:
fixes S :: "'m::euclidean_space set"
assumes "convex S"
shows "(c, x) ∈ rel_interior (cone hull ({(1 :: real)} × S)) ⟷
c > 0 ∧ x ∈ (((*⇩R) c) ` (rel_interior S))"
proof (cases "S = {}")
case True
then show ?thesis
by (simp add: cone_hull_empty)
next
case False
then obtain s where "s ∈ S" by auto
have conv: "convex ({(1 :: real)} × S)"
using convex_Times[of "{(1 :: real)}" S] assms convex_singleton[of "1 :: real"]
by auto
define f where "f y = {z. (y, z) ∈ cone hull ({1 :: real} × S)}" for y
then have *: "(c, x) ∈ rel_interior (cone hull ({(1 :: real)} × S)) =
(c ∈ rel_interior {y. f y ≠ {}} ∧ x ∈ rel_interior (f c))"
using convex_cone_hull[of "{(1 :: real)} × S"] conv
by (subst rel_interior_projection) auto
{
fix y :: real
assume "y ≥ 0"
then have "y *⇩R (1,s) ∈ cone hull ({1 :: real} × S)"
using cone_hull_expl[of "{(1 :: real)} × S"] ‹s ∈ S› by auto
then have "f y ≠ {}"
using f_def by auto
}
then have "{y. f y ≠ {}} = {0..}"
using f_def cone_hull_expl[of "{1 :: real} × S"] by auto
then have **: "rel_interior {y. f y ≠ {}} = {0<..}"
using rel_interior_real_semiline by auto
{
fix c :: real
assume "c > 0"
then have "f c = ((*⇩R) c ` S)"
using f_def cone_hull_expl[of "{1 :: real} × S"] by auto
then have "rel_interior (f c) = (*⇩R) c ` rel_interior S"
using rel_interior_convex_scaleR[of S c] assms by auto
}
then show ?thesis using * ** by auto
qed
lemma rel_interior_convex_cone:
fixes S :: "'m::euclidean_space set"
assumes "convex S"
shows "rel_interior (cone hull ({1 :: real} × S)) =
{(c, c *⇩R x) | c x. c > 0 ∧ x ∈ rel_interior S}"
(is "?lhs = ?rhs")
proof -
{
fix z
assume "z ∈ ?lhs"
have *: "z = (fst z, snd z)"
by auto
then have "z ∈ ?rhs"
using rel_interior_convex_cone_aux[of S "fst z" "snd z"] assms ‹z ∈ ?lhs› by fastforce
}
moreover
{
fix z
assume "z ∈ ?rhs"
then have "z ∈ ?lhs"
using rel_interior_convex_cone_aux[of S "fst z" "snd z"] assms
by auto
}
ultimately show ?thesis by blast
qed
lemma convex_hull_finite_union:
assumes "finite I"
assumes "∀i∈I. convex (S i) ∧ (S i) ≠ {}"
shows "convex hull (⋃(S ` I)) =
{sum (λi. c i *⇩R s i) I | c s. (∀i∈I. c i ≥ 0) ∧ sum c I = 1 ∧ (∀i∈I. s i ∈ S i)}"
(is "?lhs = ?rhs")
proof -
have "?lhs ⊇ ?rhs"
proof
fix x
assume "x ∈ ?rhs"
then obtain c s where *: "sum (λi. c i *⇩R s i) I = x" "sum c I = 1"
"(∀i∈I. c i ≥ 0) ∧ (∀i∈I. s i ∈ S i)" by auto
then have "∀i∈I. s i ∈ convex hull (⋃(S ` I))"
using hull_subset[of "⋃(S ` I)" convex] by auto
then show "x ∈ ?lhs"
unfolding *(1)[symmetric]
using * assms convex_convex_hull
by (subst convex_sum) auto
qed
{
fix i
assume "i ∈ I"
with assms have "∃p. p ∈ S i" by auto
}
then obtain p where p: "∀i∈I. p i ∈ S i" by metis
{
fix i
assume "i ∈ I"
{
fix x
assume "x ∈ S i"
define c where "c j = (if j = i then 1::real else 0)" for j
then have *: "sum c I = 1"
using ‹finite I› ‹i ∈ I› sum.delta[of I i "λj::'a. 1::real"]
by auto
define s where "s j = (if j = i then x else p j)" for j
then have "∀j. c j *⇩R s j = (if j = i then x else 0)"
using c_def by (auto simp add: algebra_simps)
then have "x = sum (λi. c i *⇩R s i) I"
using s_def c_def ‹finite I› ‹i ∈ I› sum.delta[of I i "λj::'a. x"]
by auto
moreover have "(∀i∈I. 0 ≤ c i) ∧ sum c I = 1 ∧ (∀i∈I. s i ∈ S i)"
using * c_def s_def p ‹x ∈ S i› by auto
ultimately have "x ∈ ?rhs"
by force
}
then have "?rhs ⊇ S i" by auto
}
then have *: "?rhs ⊇ ⋃(S ` I)" by auto
{
fix u v :: real
assume uv: "u ≥ 0 ∧ v ≥ 0 ∧ u + v = 1"
fix x y
assume xy: "x ∈ ?rhs ∧ y ∈ ?rhs"
from xy obtain c s where
xc: "x = sum (λi. c i *⇩R s i) I ∧ (∀i∈I. c i ≥ 0) ∧ sum c I = 1 ∧ (∀i∈I. s i ∈ S i)"
by auto
from xy obtain d t where
yc: "y = sum (λi. d i *⇩R t i) I ∧ (∀i∈I. d i ≥ 0) ∧ sum d I = 1 ∧ (∀i∈I. t i ∈ S i)"
by auto
define e where "e i = u * c i + v * d i" for i
have ge0: "∀i∈I. e i ≥ 0"
using e_def xc yc uv by simp
have "sum (λi. u * c i) I = u * sum c I"
by (simp add: sum_distrib_left)
moreover have "sum (λi. v * d i) I = v * sum d I"
by (simp add: sum_distrib_left)
ultimately have sum1: "sum e I = 1"
using e_def xc yc uv by (simp add: sum.distrib)
define q where "q i = (if e i = 0 then p i else (u * c i / e i) *⇩R s i + (v * d i / e i) *⇩R t i)"
for i
{
fix i
assume i: "i ∈ I"
have "q i ∈ S i"
proof (cases "e i = 0")
case True
then show ?thesis using i p q_def by auto
next
case False
then show ?thesis
using mem_convex_alt[of "S i" "s i" "t i" "u * (c i)" "v * (d i)"]
mult_nonneg_nonneg[of u "c i"] mult_nonneg_nonneg[of v "d i"]
assms q_def e_def i False xc yc uv
by (auto simp del: mult_nonneg_nonneg)
qed
}
then have qs: "∀i∈I. q i ∈ S i" by auto
{
fix i
assume i: "i ∈ I"
have "(u * c i) *⇩R s i + (v * d i) *⇩R t i = e i *⇩R q i"
proof (cases "e i = 0")
case True
have ge: "u * (c i) ≥ 0 ∧ v * d i ≥ 0"
using xc yc uv i by simp
moreover from ge have "u * c i ≤ 0 ∧ v * d i ≤ 0"
using True e_def i by simp
ultimately have "u * c i = 0 ∧ v * d i = 0" by auto
with True show ?thesis by auto
next
case False
then have "(u * (c i)/(e i))*⇩R (s i)+(v * (d i)/(e i))*⇩R (t i) = q i"
using q_def by auto
then have "e i *⇩R ((u * (c i)/(e i))*⇩R (s i)+(v * (d i)/(e i))*⇩R (t i))
= (e i) *⇩R (q i)" by auto
with False show ?thesis by (simp add: algebra_simps)
qed
}
then have *: "∀i∈I. (u * c i) *⇩R s i + (v * d i) *⇩R t i = e i *⇩R q i"
by auto
have "u *⇩R x + v *⇩R y = sum (λi. (u * c i) *⇩R s i + (v * d i) *⇩R t i) I"
using xc yc by (simp add: algebra_simps scaleR_right.sum sum.distrib)
also have "… = sum (λi. e i *⇩R q i) I"
using * by auto
finally have "u *⇩R x + v *⇩R y = sum (λi. (e i) *⇩R (q i)) I"
by auto
then have "u *⇩R x + v *⇩R y ∈ ?rhs"
using ge0 sum1 qs by auto
}
then have "convex ?rhs" unfolding convex_def by auto
then show ?thesis
using ‹?lhs ⊇ ?rhs› * hull_minimal[of "⋃(S ` I)" ?rhs convex]
by blast
qed
lemma convex_hull_union_two:
fixes S T :: "'m::euclidean_space set"
assumes "convex S"
and "S ≠ {}"
and "convex T"
and "T ≠ {}"
shows "convex hull (S ∪ T) =
{u *⇩R s + v *⇩R t | u v s t. u ≥ 0 ∧ v ≥ 0 ∧ u + v = 1 ∧ s ∈ S ∧ t ∈ T}"
(is "?lhs = ?rhs")
proof
define I :: "nat set" where "I = {1, 2}"
define s where "s i = (if i = (1::nat) then S else T)" for i
have "⋃(s ` I) = S ∪ T"
using s_def I_def by auto
then have "convex hull (⋃(s ` I)) = convex hull (S ∪ T)"
by auto
moreover have "convex hull ⋃(s ` I) =
{∑ i∈I. c i *⇩R sa i | c sa. (∀i∈I. 0 ≤ c i) ∧ sum c I = 1 ∧ (∀i∈I. sa i ∈ s i)}"
using assms s_def I_def
by (subst convex_hull_finite_union) auto
moreover have
"{∑i∈I. c i *⇩R sa i | c sa. (∀i∈I. 0 ≤ c i) ∧ sum c I = 1 ∧ (∀i∈I. sa i ∈ s i)} ≤ ?rhs"
using s_def I_def by auto
ultimately show "?lhs ⊆ ?rhs" by auto
{
fix x
assume "x ∈ ?rhs"
then obtain u v s t where *: "x = u *⇩R s + v *⇩R t ∧ u ≥ 0 ∧ v ≥ 0 ∧ u + v = 1 ∧ s ∈ S ∧ t ∈ T"
by auto
then have "x ∈ convex hull {s, t}"
using convex_hull_2[of s t] by auto
then have "x ∈ convex hull (S ∪ T)"
using * hull_mono[of "{s, t}" "S ∪ T"] by auto
}
then show "?lhs ⊇ ?rhs" by blast
qed
proposition ray_to_rel_frontier:
fixes a :: "'a::real_inner"
assumes "bounded S"
and a: "a ∈ rel_interior S"
and aff: "(a + l) ∈ affine hull S"
and "l ≠ 0"
obtains d where "0 < d" "(a + d *⇩R l) ∈ rel_frontier S"
"⋀e. ⟦0 ≤ e; e < d⟧ ⟹ (a + e *⇩R l) ∈ rel_interior S"
proof -
have aaff: "a ∈ affine hull S"
by (meson a hull_subset rel_interior_subset rev_subsetD)
let ?D = "{d. 0 < d ∧ a + d *⇩R l ∉ rel_interior S}"
obtain B where "B > 0" and B: "S ⊆ ball a B"
using bounded_subset_ballD [OF ‹bounded S›] by blast
have "a + (B / norm l) *⇩R l ∉ ball a B"
by (simp add: dist_norm ‹l ≠ 0›)
with B have "a + (B / norm l) *⇩R l ∉ rel_interior S"
using rel_interior_subset subsetCE by blast
with ‹B > 0› ‹l ≠ 0› have nonMT: "?D ≠ {}"
using divide_pos_pos zero_less_norm_iff by fastforce
have bdd: "bdd_below ?D"
by (metis (no_types, lifting) bdd_belowI le_less mem_Collect_eq)
have relin_Ex: "⋀x. x ∈ rel_interior S ⟹
∃e>0. ∀x'∈affine hull S. dist x' x < e ⟶ x' ∈ rel_interior S"
using openin_rel_interior [of S] by (simp add: openin_euclidean_subtopology_iff)
define d where "d = Inf ?D"
obtain ε where "0 < ε" and ε: "⋀η. ⟦0 ≤ η; η < ε⟧ ⟹ (a + η *⇩R l) ∈ rel_interior S"
proof -
obtain e where "e>0"
and e: "⋀x'. x' ∈ affine hull S ⟹ dist x' a < e ⟹ x' ∈ rel_interior S"
using relin_Ex a by blast
show thesis
proof (rule_tac ε = "e / norm l" in that)
show "0 < e / norm l" by (simp add: ‹0 < e› ‹l ≠ 0›)
next
show "a + η *⇩R l ∈ rel_interior S" if "0 ≤ η" "η < e / norm l" for η
proof (rule e)
show "a + η *⇩R l ∈ affine hull S"
by (metis (no_types) add_diff_cancel_left' aff affine_affine_hull mem_affine_3_minus aaff)
show "dist (a + η *⇩R l) a < e"
using that by (simp add: ‹l ≠ 0› dist_norm pos_less_divide_eq)
qed
qed
qed
have inint: "⋀e. ⟦0 ≤ e; e < d⟧ ⟹ a + e *⇩R l ∈ rel_interior S"
unfolding d_def using cInf_lower [OF _ bdd]
by (metis (no_types, lifting) a add.right_neutral le_less mem_Collect_eq not_less real_vector.scale_zero_left)
have "ε ≤ d"
unfolding d_def
using ε dual_order.strict_implies_order le_less_linear
by (blast intro: cInf_greatest [OF nonMT])
with ‹0 < ε› have "0 < d" by simp
have "a + d *⇩R l ∉ rel_interior S"
proof
assume adl: "a + d *⇩R l ∈ rel_interior S"
obtain e where "e > 0"
and e: "⋀x'. x' ∈ affine hull S ⟹ dist x' (a + d *⇩R l) < e ⟹ x' ∈ rel_interior S"
using relin_Ex adl by blast
have "d + e / norm l ≤ x"
if "0 < x" and nonrel: "a + x *⇩R l ∉ rel_interior S" for x
proof (cases "x < d")
case True with inint nonrel ‹0 < x›
show ?thesis by auto
next
case False
then have dle: "x < d + e / norm l ⟹ dist (a + x *⇩R l) (a + d *⇩R l) < e"
by (simp add: field_simps ‹l ≠ 0›)
have ain: "a + x *⇩R l ∈ affine hull S"
by (metis add_diff_cancel_left' aff affine_affine_hull mem_affine_3_minus aaff)
show ?thesis
using e [OF ain] nonrel dle by force
qed
then
have "d + e / norm l ≤ Inf {d. 0 < d ∧ a + d *⇩R l ∉ rel_interior S}"
by (force simp add: intro: cInf_greatest [OF nonMT])
then show False
using ‹0 < e› ‹l ≠ 0› by (simp add: d_def [symmetric] field_simps)
qed
moreover
have "∃y∈S. dist y (a + d *⇩R l) < η" if "0 < η" for η::real
proof -
have 1: "a + (d - min d (η / 2 / norm l)) *⇩R l ∈ S"
proof (rule subsetD [OF rel_interior_subset inint])
show "d - min d (η / 2 / norm l) < d"
using ‹l ≠ 0› ‹0 < d› ‹0 < η› by auto
qed auto
have "norm l * min d (η / (norm l * 2)) ≤ norm l * (η / (norm l * 2))"
by (metis min_def mult_left_mono norm_ge_zero order_refl)
also have "... < η"
using ‹l ≠ 0› ‹0 < η› by (simp add: field_simps)
finally have 2: "norm l * min d (η / (norm l * 2)) < η" .
show ?thesis
using 1 2 ‹0 < d› ‹0 < η›
by (rule_tac x="a + (d - min d (η / 2 / norm l)) *⇩R l" in bexI) (auto simp: algebra_simps)
qed
then have "a + d *⇩R l ∈ closure S"
by (auto simp: closure_approachable)
ultimately have infront: "a + d *⇩R l ∈ rel_frontier S"
by (simp add: rel_frontier_def)
show ?thesis
by (rule that [OF ‹0 < d› infront inint])
qed
corollary ray_to_frontier:
fixes a :: "'a::euclidean_space"
assumes "bounded S"
and a: "a ∈ interior S"
and "l ≠ 0"
obtains d where "0 < d" "(a + d *⇩R l) ∈ frontier S"
"⋀e. ⟦0 ≤ e; e < d⟧ ⟹ (a + e *⇩R l) ∈ interior S"
proof -
have §: "interior S = rel_interior S"
using a rel_interior_nonempty_interior by auto
then have "a ∈ rel_interior S"
using a by simp
moreover have "a + l ∈ affine hull S"
using a affine_hull_nonempty_interior by blast
ultimately show thesis
by (metis § ‹bounded S› ‹l ≠ 0› frontier_def ray_to_rel_frontier rel_frontier_def that)
qed
lemma segment_to_rel_frontier_aux:
fixes x :: "'a::euclidean_space"
assumes "convex S" "bounded S" and x: "x ∈ rel_interior S" and y: "y ∈ S" and xy: "x ≠ y"
obtains z where "z ∈ rel_frontier S" "y ∈ closed_segment x z"
"open_segment x z ⊆ rel_interior S"
proof -
have "x + (y - x) ∈ affine hull S"
using hull_inc [OF y] by auto
then obtain d where "0 < d" and df: "(x + d *⇩R (y-x)) ∈ rel_frontier S"
and di: "⋀e. ⟦0 ≤ e; e < d⟧ ⟹ (x + e *⇩R (y-x)) ∈ rel_interior S"
by (rule ray_to_rel_frontier [OF ‹bounded S› x]) (use xy in auto)
show ?thesis
proof
show "x + d *⇩R (y - x) ∈ rel_frontier S"
by (simp add: df)
next
have "open_segment x y ⊆ rel_interior S"
using rel_interior_closure_convex_segment [OF ‹convex S› x] closure_subset y by blast
moreover have "x + d *⇩R (y - x) ∈ open_segment x y" if "d < 1"
using xy ‹0 < d› that by (force simp: in_segment algebra_simps)
ultimately have "1 ≤ d"
using df rel_frontier_def by fastforce
moreover have "x = (1 / d) *⇩R x + ((d - 1) / d) *⇩R x"
by (metis ‹0 < d› add.commute add_divide_distrib diff_add_cancel divide_self_if less_irrefl scaleR_add_left scaleR_one)
ultimately show "y ∈ closed_segment x (x + d *⇩R (y - x))"
unfolding in_segment
by (rule_tac x="1/d" in exI) (auto simp: algebra_simps)
next
show "open_segment x (x + d *⇩R (y - x)) ⊆ rel_interior S"
proof (rule rel_interior_closure_convex_segment [OF ‹convex S› x])
show "x + d *⇩R (y - x) ∈ closure S"
using df rel_frontier_def by auto
qed
qed
qed
lemma segment_to_rel_frontier:
fixes x :: "'a::euclidean_space"
assumes S: "convex S" "bounded S" and x: "x ∈ rel_interior S"
and y: "y ∈ S" and xy: "¬(x = y ∧ S = {x})"
obtains z where "z ∈ rel_frontier S" "y ∈ closed_segment x z"
"open_segment x z ⊆ rel_interior S"
proof (cases "x=y")
case True
with xy have "S ≠ {x}"
by blast
with True show ?thesis
by (metis Set.set_insert all_not_in_conv ends_in_segment(1) insert_iff segment_to_rel_frontier_aux[OF S x] that y)
next
case False
then show ?thesis
using segment_to_rel_frontier_aux [OF S x y] that by blast
qed
proposition rel_frontier_not_sing:
fixes a :: "'a::euclidean_space"
assumes "bounded S"
shows "rel_frontier S ≠ {a}"
proof (cases "S = {}")
case True then show ?thesis by simp
next
case False
then obtain z where "z ∈ S"
by blast
then show ?thesis
proof (cases "S = {z}")
case True then show ?thesis by simp
next
case False
then obtain w where "w ∈ S" "w ≠ z"
using ‹z ∈ S› by blast
show ?thesis
proof
assume "rel_frontier S = {a}"
then consider "w ∉ rel_frontier S" | "z ∉ rel_frontier S"
using ‹w ≠ z› by auto
then show False
proof cases
case 1
then have w: "w ∈ rel_interior S"
using ‹w ∈ S› closure_subset rel_frontier_def by fastforce
have "w + (w - z) ∈ affine hull S"
by (metis ‹w ∈ S› ‹z ∈ S› affine_affine_hull hull_inc mem_affine_3_minus scaleR_one)
then obtain e where "0 < e" "(w + e *⇩R (w - z)) ∈ rel_frontier S"
using ‹w ≠ z› ‹z ∈ S› by (metis assms ray_to_rel_frontier right_minus_eq w)
moreover obtain d where "0 < d" "(w + d *⇩R (z - w)) ∈ rel_frontier S"
using ray_to_rel_frontier [OF ‹bounded S› w, of "1 *⇩R (z - w)"] ‹w ≠ z› ‹z ∈ S›
by (metis add.commute add.right_neutral diff_add_cancel hull_inc scaleR_one)
ultimately have "d *⇩R (z - w) = e *⇩R (w - z)"
using ‹rel_frontier S = {a}› by force
moreover have "e ≠ -d "
using ‹0 < e› ‹0 < d› by force
ultimately show False
by (metis (no_types, lifting) ‹w ≠ z› eq_iff_diff_eq_0 minus_diff_eq real_vector.scale_cancel_right real_vector.scale_minus_right scaleR_left.minus)
next
case 2
then have z: "z ∈ rel_interior S"
using ‹z ∈ S› closure_subset rel_frontier_def by fastforce
have "z + (z - w) ∈ affine hull S"
by (metis ‹z ∈ S› ‹w ∈ S› affine_affine_hull hull_inc mem_affine_3_minus scaleR_one)
then obtain e where "0 < e" "(z + e *⇩R (z - w)) ∈ rel_frontier S"
using ‹w ≠ z› ‹w ∈ S› by (metis assms ray_to_rel_frontier right_minus_eq z)
moreover obtain d where "0 < d" "(z + d *⇩R (w - z)) ∈ rel_frontier S"
using ray_to_rel_frontier [OF ‹bounded S› z, of "1 *⇩R (w - z)"] ‹w ≠ z› ‹w ∈ S›
by (metis add.commute add.right_neutral diff_add_cancel hull_inc scaleR_one)
ultimately have "d *⇩R (w - z) = e *⇩R (z - w)"
using ‹rel_frontier S = {a}› by force
moreover have "e ≠ -d "
using ‹0 < e› ‹0 < d› by force
ultimately show False
by (metis (no_types, lifting) ‹w ≠ z› eq_iff_diff_eq_0 minus_diff_eq real_vector.scale_cancel_right real_vector.scale_minus_right scaleR_left.minus)
qed
qed
qed
qed
subsection ‹Convexity on direct sums›
lemma closure_sum:
fixes S T :: "'a::real_normed_vector set"
shows "closure S + closure T ⊆ closure (S + T)"
unfolding set_plus_image closure_Times [symmetric] split_def
by (intro closure_bounded_linear_image_subset bounded_linear_add
bounded_linear_fst bounded_linear_snd)
lemma fst_snd_linear: "linear (λ(x,y). x + y)"
unfolding linear_iff by (simp add: algebra_simps)
lemma rel_interior_sum:
fixes S T :: "'n::euclidean_space set"
assumes "convex S"
and "convex T"
shows "rel_interior (S + T) = rel_interior S + rel_interior T"
proof -
have "rel_interior S + rel_interior T = (λ(x,y). x + y) ` (rel_interior S × rel_interior T)"
by (simp add: set_plus_image)
also have "… = (λ(x,y). x + y) ` rel_interior (S × T)"
using rel_interior_Times assms by auto
also have "… = rel_interior (S + T)"
using fst_snd_linear convex_Times assms
rel_interior_convex_linear_image[of "(λ(x,y). x + y)" "S × T"]
by (auto simp add: set_plus_image)
finally show ?thesis ..
qed
lemma rel_interior_sum_gen:
fixes S :: "'a ⇒ 'n::euclidean_space set"
assumes "⋀i. i∈I ⟹ convex (S i)"
shows "rel_interior (sum S I) = sum (λi. rel_interior (S i)) I"
using rel_interior_sum rel_interior_sing[of "0"] assms
by (subst sum_set_cond_linear[of convex], auto simp add: convex_set_plus)
lemma convex_rel_open_direct_sum:
fixes S T :: "'n::euclidean_space set"
assumes "convex S"
and "rel_open S"
and "convex T"
and "rel_open T"
shows "convex (S × T) ∧ rel_open (S × T)"
by (metis assms convex_Times rel_interior_Times rel_open_def)
lemma convex_rel_open_sum:
fixes S T :: "'n::euclidean_space set"
assumes "convex S"
and "rel_open S"
and "convex T"
and "rel_open T"
shows "convex (S + T) ∧ rel_open (S + T)"
by (metis assms convex_set_plus rel_interior_sum rel_open_def)
lemma convex_hull_finite_union_cones:
assumes "finite I"
and "I ≠ {}"
assumes "⋀i. i∈I ⟹ convex (S i) ∧ cone (S i) ∧ S i ≠ {}"
shows "convex hull (⋃(S ` I)) = sum S I"
(is "?lhs = ?rhs")
proof -
{
fix x
assume "x ∈ ?lhs"
then obtain c xs where
x: "x = sum (λi. c i *⇩R xs i) I ∧ (∀i∈I. c i ≥ 0) ∧ sum c I = 1 ∧ (∀i∈I. xs i ∈ S i)"
using convex_hull_finite_union[of I S] assms by auto
define s where "s i = c i *⇩R xs i" for i
have "∀i∈I. s i ∈ S i"
using s_def x assms by (simp add: mem_cone)
moreover have "x = sum s I" using x s_def by auto
ultimately have "x ∈ ?rhs"
using set_sum_alt[of I S] assms by auto
}
moreover
{
fix x
assume "x ∈ ?rhs"
then obtain s where x: "x = sum s I ∧ (∀i∈I. s i ∈ S i)"
using set_sum_alt[of I S] assms by auto
define xs where "xs i = of_nat(card I) *⇩R s i" for i
then have "x = sum (λi. ((1 :: real) / of_nat(card I)) *⇩R xs i) I"
using x assms by auto
moreover have "∀i∈I. xs i ∈ S i"
using x xs_def assms by (simp add: cone_def)
moreover have "∀i∈I. (1 :: real) / of_nat (card I) ≥ 0"
by auto
moreover have "sum (λi. (1 :: real) / of_nat (card I)) I = 1"
using assms by auto
ultimately have "x ∈ ?lhs"
using assms
apply (simp add: convex_hull_finite_union[of I S])
by (rule_tac x = "(λi. 1 / (card I))" in exI) auto
}
ultimately show ?thesis by auto
qed
lemma convex_hull_union_cones_two:
fixes S T :: "'m::euclidean_space set"
assumes "convex S"
and "cone S"
and "S ≠ {}"
assumes "convex T"
and "cone T"
and "T ≠ {}"
shows "convex hull (S ∪ T) = S + T"
proof -
define I :: "nat set" where "I = {1, 2}"
define A where "A i = (if i = (1::nat) then S else T)" for i
have "⋃(A ` I) = S ∪ T"
using A_def I_def by auto
then have "convex hull (⋃(A ` I)) = convex hull (S ∪ T)"
by auto
moreover have "convex hull ⋃(A ` I) = sum A I"
using A_def I_def
by (metis assms convex_hull_finite_union_cones empty_iff finite.emptyI finite.insertI insertI1)
moreover have "sum A I = S + T"
using A_def I_def by (force simp add: set_plus_def)
ultimately show ?thesis by auto
qed
lemma rel_interior_convex_hull_union:
fixes S :: "'a ⇒ 'n::euclidean_space set"
assumes "finite I"
and "∀i∈I. convex (S i) ∧ S i ≠ {}"
shows "rel_interior (convex hull (⋃(S ` I))) =
{sum (λi. c i *⇩R s i) I | c s. (∀i∈I. c i > 0) ∧ sum c I = 1 ∧
(∀i∈I. s i ∈ rel_interior(S i))}"
(is "?lhs = ?rhs")
proof (cases "I = {}")
case True
then show ?thesis
using convex_hull_empty by auto
next
case False
define C0 where "C0 = convex hull (⋃(S ` I))"
have "∀i∈I. C0 ≥ S i"
unfolding C0_def using hull_subset[of "⋃(S ` I)"] by auto
define K0 where "K0 = cone hull ({1 :: real} × C0)"
define K where "K i = cone hull ({1 :: real} × S i)" for i
have "∀i∈I. K i ≠ {}"
unfolding K_def using assms
by (simp add: cone_hull_empty_iff[symmetric])
have convK: "∀i∈I. convex (K i)"
unfolding K_def
by (simp add: assms(2) convex_Times convex_cone_hull)
have "K0 ⊇ K i" if "i ∈ I" for i
unfolding K0_def K_def
by (simp add: Sigma_mono ‹∀i∈I. S i ⊆ C0› hull_mono that)
then have "K0 ⊇ ⋃(K ` I)" by auto
moreover have "convex K0"
unfolding K0_def by (simp add: C0_def convex_Times convex_cone_hull)
ultimately have geq: "K0 ⊇ convex hull (⋃(K ` I))"
using hull_minimal[of _ "K0" "convex"] by blast
have "∀i∈I. K i ⊇ {1 :: real} × S i"
using K_def by (simp add: hull_subset)
then have "⋃(K ` I) ⊇ {1 :: real} × ⋃(S ` I)"
by auto
then have "convex hull ⋃(K ` I) ⊇ convex hull ({1 :: real} × ⋃(S ` I))"
by (simp add: hull_mono)
then have "convex hull ⋃(K ` I) ⊇ {1 :: real} × C0"
unfolding C0_def
using convex_hull_Times[of "{(1 :: real)}" "⋃(S ` I)"] convex_hull_singleton
by auto
moreover have "cone (convex hull (⋃(K ` I)))"
by (simp add: K_def cone_Union cone_cone_hull cone_convex_hull)
ultimately have "convex hull (⋃(K ` I)) ⊇ K0"
unfolding K0_def
using hull_minimal[of _ "convex hull (⋃(K ` I))" "cone"]
by blast
then have "K0 = convex hull (⋃(K ` I))"
using geq by auto
also have "… = sum K I"
using assms False ‹∀i∈I. K i ≠ {}› cone_hull_eq convK
by (intro convex_hull_finite_union_cones; fastforce simp: K_def)
finally have "K0 = sum K I" by auto
then have *: "rel_interior K0 = sum (λi. (rel_interior (K i))) I"
using rel_interior_sum_gen[of I K] convK by auto
{
fix x
assume "x ∈ ?lhs"
then have "(1::real, x) ∈ rel_interior K0"
using K0_def C0_def rel_interior_convex_cone_aux[of C0 "1::real" x] convex_convex_hull
by auto
then obtain k where k: "(1::real, x) = sum k I ∧ (∀i∈I. k i ∈ rel_interior (K i))"
using ‹finite I› * set_sum_alt[of I "λi. rel_interior (K i)"] by auto
{
fix i
assume "i ∈ I"
then have "convex (S i) ∧ k i ∈ rel_interior (cone hull {1} × S i)"
using k K_def assms by auto
then have "∃ci si. k i = (ci, ci *⇩R si) ∧ 0 < ci ∧ si ∈ rel_interior (S i)"
using rel_interior_convex_cone[of "S i"] by auto
}
then obtain c s where cs: "∀i∈I. k i = (c i, c i *⇩R s i) ∧ 0 < c i ∧ s i ∈ rel_interior (S i)"
by metis
then have "x = (∑i∈I. c i *⇩R s i) ∧ sum c I = 1"
using k by (simp add: sum_prod)
then have "x ∈ ?rhs"
using k cs by auto
}
moreover
{
fix x
assume "x ∈ ?rhs"
then obtain c s where cs: "x = sum (λi. c i *⇩R s i) I ∧
(∀i∈I. c i > 0) ∧ sum c I = 1 ∧ (∀i∈I. s i ∈ rel_interior (S i))"
by auto
define k where "k i = (c i, c i *⇩R s i)" for i
{
fix i assume "i ∈ I"
then have "k i ∈ rel_interior (K i)"
using k_def K_def assms cs rel_interior_convex_cone[of "S i"]
by auto
}
then have "(1, x) ∈ rel_interior K0"
using * set_sum_alt[of I "(λi. rel_interior (K i))"] assms cs
by (simp add: k_def) (metis (mono_tags, lifting) sum_prod)
then have "x ∈ ?lhs"
using K0_def C0_def rel_interior_convex_cone_aux[of C0 1 x]
by auto
}
ultimately show ?thesis by blast
qed
lemma convex_le_Inf_differential:
fixes f :: "real ⇒ real"
assumes "convex_on I f"
and "x ∈ interior I"
and "y ∈ I"
shows "f y ≥ f x + Inf ((λt. (f x - f t) / (x - t)) ` ({x<..} ∩ I)) * (y - x)"
(is "_ ≥ _ + Inf (?F x) * (y - x)")
proof (cases rule: linorder_cases)
assume "x < y"
moreover
have "open (interior I)" by auto
from openE[OF this ‹x ∈ interior I›]
obtain e where e: "0 < e" "ball x e ⊆ interior I" .
moreover define t where "t = min (x + e / 2) ((x + y) / 2)"
ultimately have "x < t" "t < y" "t ∈ ball x e"
by (auto simp: dist_real_def field_simps split: split_min)
with ‹x ∈ interior I› e interior_subset[of I] have "t ∈ I" "x ∈ I" by auto
define K where "K = x - e / 2"
with ‹0 < e› have "K ∈ ball x e" "K < x"
by (auto simp: dist_real_def)
then have "K ∈ I"
using ‹interior I ⊆ I› e(2) by blast
have "Inf (?F x) ≤ (f x - f y) / (x - y)"
proof (intro bdd_belowI cInf_lower2)
show "(f x - f t) / (x - t) ∈ ?F x"
using ‹t ∈ I› ‹x < t› by auto
show "(f x - f t) / (x - t) ≤ (f x - f y) / (x - y)"
using ‹convex_on I f› ‹x ∈ I› ‹y ∈ I› ‹x < t› ‹t < y›
by (rule convex_on_slope_le)
next
fix y
assume "y ∈ ?F x"
with order_trans[OF convex_on_slope_le[OF ‹convex_on I f› ‹K ∈ I› _ ‹K < x› _]]
show "(f K - f x) / (K - x) ≤ y" by auto
qed
then show ?thesis
using ‹x < y› by (simp add: field_simps)
next
assume "y < x"
moreover
have "open (interior I)" by auto
from openE[OF this ‹x ∈ interior I›]
obtain e where e: "0 < e" "ball x e ⊆ interior I" .
moreover define t where "t = x + e / 2"
ultimately have "x < t" "t ∈ ball x e"
by (auto simp: dist_real_def field_simps)
with ‹x ∈ interior I› e interior_subset[of I] have "t ∈ I" "x ∈ I" by auto
have "(f x - f y) / (x - y) ≤ Inf (?F x)"
proof (rule cInf_greatest)
have "(f x - f y) / (x - y) = (f y - f x) / (y - x)"
using ‹y < x› by (auto simp: field_simps)
also
fix z
assume "z ∈ ?F x"
with order_trans[OF convex_on_slope_le[OF ‹convex_on I f› ‹y ∈ I› _ ‹y < x›]]
have "(f y - f x) / (y - x) ≤ z"
by auto
finally show "(f x - f y) / (x - y) ≤ z" .
next
have "x + e / 2 ∈ ball x e"
using e by (auto simp: dist_real_def)
with e interior_subset[of I] have "x + e / 2 ∈ {x<..} ∩ I"
by auto
then show "?F x ≠ {}"
by blast
qed
then show ?thesis
using ‹y < x› by (simp add: field_simps)
qed simp
subsection‹Explicit formulas for interior and relative interior of convex hull›
lemma at_within_cbox_finite:
assumes "x ∈ box a b" "x ∉ S" "finite S"
shows "(at x within cbox a b - S) = at x"
proof -
have "interior (cbox a b - S) = box a b - S"
using ‹finite S› by (simp add: interior_diff finite_imp_closed)
then show ?thesis
using at_within_interior assms by fastforce
qed
lemma affine_independent_convex_affine_hull:
fixes S :: "'a::euclidean_space set"
assumes "¬ affine_dependent S" "T ⊆ S"
shows "convex hull T = affine hull T ∩ convex hull S"
proof -
have fin: "finite S" "finite T" using assms aff_independent_finite finite_subset by auto
have "convex hull T ⊆ affine hull T"
using convex_hull_subset_affine_hull by blast
moreover have "convex hull T ⊆ convex hull S"
using assms hull_mono by blast
moreover have "affine hull T ∩ convex hull S ⊆ convex hull T"
proof -
have 0: "⋀u. sum u S = 0 ⟹ (∀v∈S. u v = 0) ∨ (∑v∈S. u v *⇩R v) ≠ 0"
using affine_dependent_explicit_finite assms(1) fin(1) by auto
show ?thesis
proof (clarsimp simp add: affine_hull_finite fin)
fix u
assume S: "(∑v∈T. u v *⇩R v) ∈ convex hull S"
and T1: "sum u T = 1"
then obtain v where v: "∀x∈S. 0 ≤ v x" "sum v S = 1" "(∑x∈S. v x *⇩R x) = (∑v∈T. u v *⇩R v)"
by (auto simp add: convex_hull_finite fin)
{ fix x
assume"x ∈ T"
then have S: "S = (S - T) ∪ T"
using assms by auto
have [simp]: "(∑x∈T. v x *⇩R x) + (∑x∈S - T. v x *⇩R x) = (∑x∈T. u x *⇩R x)"
"sum v T + sum v (S - T) = 1"
using v fin S
by (auto simp: sum.union_disjoint [symmetric] Un_commute)
have "(∑x∈S. if x ∈ T then v x - u x else v x) = 0"
"(∑x∈S. (if x ∈ T then v x - u x else v x) *⇩R x) = 0"
using v fin T1
by (subst S, subst sum.union_disjoint, auto simp: algebra_simps sum_subtractf)+
} note [simp] = this
have "(∀x∈T. 0 ≤ u x)"
using 0 [of "λx. if x ∈ T then v x - u x else v x"] ‹T ⊆ S› v(1) by fastforce
then show "(∑v∈T. u v *⇩R v) ∈ convex hull T"
using 0 [of "λx. if x ∈ T then v x - u x else v x"] ‹T ⊆ S› T1
by (fastforce simp add: convex_hull_finite fin)
qed
qed
ultimately show ?thesis
by blast
qed
lemma affine_independent_span_eq:
fixes S :: "'a::euclidean_space set"
assumes "¬ affine_dependent S" "card S = Suc (DIM ('a))"
shows "affine hull S = UNIV"
proof (cases "S = {}")
case True then show ?thesis
using assms by simp
next
case False
then obtain a T where T: "a ∉ T" "S = insert a T"
by blast
then have fin: "finite T" using assms
by (metis finite_insert aff_independent_finite)
have "UNIV ⊆ (+) a ` span ((λx. x - a) ` T)"
proof (intro card_ge_dim_independent Fun.vimage_subsetD)
show "independent ((λx. x - a) ` T)"
using T affine_dependent_iff_dependent assms(1) by auto
show "dim ((+) a -` UNIV) ≤ card ((λx. x - a) ` T)"
using assms T fin by (auto simp: card_image inj_on_def)
qed (use surj_plus in auto)
then show ?thesis
using T(2) affine_hull_insert_span_gen equalityI by fastforce
qed
lemma affine_independent_span_gt:
fixes S :: "'a::euclidean_space set"
assumes ind: "¬ affine_dependent S" and dim: "DIM ('a) < card S"
shows "affine hull S = UNIV"
proof (intro affine_independent_span_eq [OF ind] antisym)
show "card S ≤ Suc DIM('a)"
using aff_independent_finite affine_dependent_biggerset ind by fastforce
show "Suc DIM('a) ≤ card S"
using Suc_leI dim by blast
qed
lemma empty_interior_affine_hull:
fixes S :: "'a::euclidean_space set"
assumes "finite S" and dim: "card S ≤ DIM ('a)"
shows "interior(affine hull S) = {}"
using assms
proof (induct S rule: finite_induct)
case (insert x S)
then have "dim (span ((λy. y - x) ` S)) < DIM('a)"
by (auto simp: Suc_le_lessD card_image_le dual_order.trans intro!: dim_le_card'[THEN le_less_trans])
then show ?case
by (simp add: empty_interior_lowdim affine_hull_insert_span_gen interior_translation)
qed auto
lemma empty_interior_convex_hull:
fixes S :: "'a::euclidean_space set"
assumes "finite S" and dim: "card S ≤ DIM ('a)"
shows "interior(convex hull S) = {}"
by (metis Diff_empty Diff_eq_empty_iff convex_hull_subset_affine_hull
interior_mono empty_interior_affine_hull [OF assms])
lemma explicit_subset_rel_interior_convex_hull:
fixes S :: "'a::euclidean_space set"
shows "finite S
⟹ {y. ∃u. (∀x ∈ S. 0 < u x ∧ u x < 1) ∧ sum u S = 1 ∧ sum (λx. u x *⇩R x) S = y}
⊆ rel_interior (convex hull S)"
by (force simp add: rel_interior_convex_hull_union [where S="λx. {x}" and I=S, simplified])
lemma explicit_subset_rel_interior_convex_hull_minimal:
fixes S :: "'a::euclidean_space set"
shows "finite S
⟹ {y. ∃u. (∀x ∈ S. 0 < u x) ∧ sum u S = 1 ∧ sum (λx. u x *⇩R x) S = y}
⊆ rel_interior (convex hull S)"
by (force simp add: rel_interior_convex_hull_union [where S="λx. {x}" and I=S, simplified])
lemma rel_interior_convex_hull_explicit:
fixes S :: "'a::euclidean_space set"
assumes "¬ affine_dependent S"
shows "rel_interior(convex hull S) =
{y. ∃u. (∀x ∈ S. 0 < u x) ∧ sum u S = 1 ∧ sum (λx. u x *⇩R x) S = y}"
(is "?lhs = ?rhs")
proof
show "?rhs ≤ ?lhs"
by (simp add: aff_independent_finite explicit_subset_rel_interior_convex_hull_minimal assms)
next
show "?lhs ≤ ?rhs"
proof (cases "∃a. S = {a}")
case True then show "?lhs ≤ ?rhs"
by force
next
case False
have fs: "finite S"
using assms by (simp add: aff_independent_finite)
{ fix a b and d::real
assume ab: "a ∈ S" "b ∈ S" "a ≠ b"
then have S: "S = (S - {a,b}) ∪ {a,b}"
by auto
have "(∑x∈S. if x = a then - d else if x = b then d else 0) = 0"
"(∑x∈S. (if x = a then - d else if x = b then d else 0) *⇩R x) = d *⇩R b - d *⇩R a"
using ab fs
by (subst S, subst sum.union_disjoint, auto)+
} note [simp] = this
{ fix y
assume y: "y ∈ convex hull S" "y ∉ ?rhs"
have *: False if
ua: "∀x∈S. 0 ≤ u x" "sum u S = 1" "¬ 0 < u a" "a ∈ S"
and yT: "y = (∑x∈S. u x *⇩R x)" "y ∈ T" "open T"
and sb: "T ∩ affine hull S ⊆ {w. ∃u. (∀x∈S. 0 ≤ u x) ∧ sum u S = 1 ∧ (∑x∈S. u x *⇩R x) = w}"
for u T a
proof -
have ua0: "u a = 0"
using ua by auto
obtain b where b: "b∈S" "a ≠ b"
using ua False by auto
obtain e where e: "0 < e" "ball (∑x∈S. u x *⇩R x) e ⊆ T"
using yT by (auto elim: openE)
with b obtain d where d: "0 < d" "norm(d *⇩R (a-b)) < e"
by (auto intro: that [of "e / 2 / norm(a-b)"])
have "(∑x∈S. u x *⇩R x) ∈ affine hull S"
using yT y by (metis affine_hull_convex_hull hull_redundant_eq)
then have "(∑x∈S. u x *⇩R x) - d *⇩R (a - b) ∈ affine hull S"
using ua b by (auto simp: hull_inc intro: mem_affine_3_minus2)
then have "y - d *⇩R (a - b) ∈ T ∩ affine hull S"
using d e yT by auto
then obtain v where v: "∀x∈S. 0 ≤ v x"
"sum v S = 1"
"(∑x∈S. v x *⇩R x) = (∑x∈S. u x *⇩R x) - d *⇩R (a - b)"
using subsetD [OF sb] yT
by auto
have aff: "⋀u. sum u S = 0 ⟹ (∀v∈S. u v = 0) ∨ (∑v∈S. u v *⇩R v) ≠ 0"
using assms by (simp add: affine_dependent_explicit_finite fs)
show False
using ua b d v aff [of "λx. (v x - u x) - (if x = a then -d else if x = b then d else 0)"]
by (auto simp: algebra_simps sum_subtractf sum.distrib)
qed
have "y ∉ rel_interior (convex hull S)"
using y convex_hull_finite [OF fs] *
apply simp
by (metis (no_types, lifting) IntD1 affine_hull_convex_hull mem_rel_interior)
} with rel_interior_subset show "?lhs ≤ ?rhs"
by blast
qed
qed
lemma interior_convex_hull_explicit_minimal:
fixes S :: "'a::euclidean_space set"
assumes "¬ affine_dependent S"
shows
"interior(convex hull S) =
(if card(S) ≤ DIM('a) then {}
else {y. ∃u. (∀x ∈ S. 0 < u x) ∧ sum u S = 1 ∧ (∑x∈S. u x *⇩R x) = y})"
(is "_ = (if _ then _ else ?rhs)")
proof -
{ assume S: "¬ card S ≤ DIM('a)"
have "interior (convex hull S) = rel_interior(convex hull S)"
using assms S by (simp add: affine_independent_span_gt rel_interior_interior)
then have "interior(convex hull S) = ?rhs"
by (simp add: assms S rel_interior_convex_hull_explicit)
}
then show ?thesis
by (auto simp: aff_independent_finite empty_interior_convex_hull assms)
qed
lemma interior_convex_hull_explicit:
fixes S :: "'a::euclidean_space set"
assumes "¬ affine_dependent S"
shows
"interior(convex hull S) =
(if card(S) ≤ DIM('a) then {}
else {y. ∃u. (∀x ∈ S. 0 < u x ∧ u x < 1) ∧ sum u S = 1 ∧ (∑x∈S. u x *⇩R x) = y})"
proof -
{ fix u :: "'a ⇒ real" and a
assume "card Basis < card S" and u: "⋀x. x∈S ⟹ 0 < u x" "sum u S = 1" and a: "a ∈ S"
then have cs: "Suc 0 < card S"
by (metis DIM_positive less_trans_Suc)
obtain b where b: "b ∈ S" "a ≠ b"
proof (cases "S ≤ {a}")
case True
then show thesis
using cs subset_singletonD by fastforce
qed blast
have "u a + u b ≤ sum u {a,b}"
using a b by simp
also have "... ≤ sum u S"
using a b u
by (intro Groups_Big.sum_mono2) (auto simp: less_imp_le aff_independent_finite assms)
finally have "u a < 1"
using ‹b ∈ S› u by fastforce
} note [simp] = this
show ?thesis
using assms by (force simp add: not_le interior_convex_hull_explicit_minimal)
qed
lemma interior_closed_segment_ge2:
fixes a :: "'a::euclidean_space"
assumes "2 ≤ DIM('a)"
shows "interior(closed_segment a b) = {}"
using assms unfolding segment_convex_hull
proof -
have "card {a, b} ≤ DIM('a)"
using assms
by (simp add: card_insert_if linear not_less_eq_eq numeral_2_eq_2)
then show "interior (convex hull {a, b}) = {}"
by (metis empty_interior_convex_hull finite.insertI finite.emptyI)
qed
lemma interior_open_segment:
fixes a :: "'a::euclidean_space"
shows "interior(open_segment a b) =
(if 2 ≤ DIM('a) then {} else open_segment a b)"
proof (cases "2 ≤ DIM('a)")
case True
then have "interior (open_segment a b) = {}"
using interior_closed_segment_ge2 interior_mono segment_open_subset_closed by blast
with True show ?thesis
by auto
next
case ge2: False
have "interior (open_segment a b) = open_segment a b"
proof (cases "a = b")
case True then show ?thesis by auto
next
case False
with ge2 have "affine hull (open_segment a b) = UNIV"
by (simp add: False affine_independent_span_gt)
then show "interior (open_segment a b) = open_segment a b"
using rel_interior_interior rel_interior_open_segment by blast
qed
with ge2 show ?thesis
by auto
qed
lemma interior_closed_segment:
fixes a :: "'a::euclidean_space"
shows "interior(closed_segment a b) =
(if 2 ≤ DIM('a) then {} else open_segment a b)"
proof (cases "a = b")
case True then show ?thesis by simp
next
case False
then have "closure (open_segment a b) = closed_segment a b"
by simp
then show ?thesis
by (metis (no_types) convex_interior_closure convex_open_segment interior_open_segment)
qed
lemmas interior_segment = interior_closed_segment interior_open_segment
lemma closed_segment_eq [simp]:
fixes a :: "'a::euclidean_space"
shows "closed_segment a b = closed_segment c d ⟷ {a,b} = {c,d}"
proof
assume abcd: "closed_segment a b = closed_segment c d"
show "{a,b} = {c,d}"
proof (cases "a=b ∨ c=d")
case True with abcd show ?thesis by force
next
case False
then have neq: "a ≠ b ∧ c ≠ d" by force
have *: "closed_segment c d - {a, b} = rel_interior (closed_segment c d)"
using neq abcd by (metis (no_types) open_segment_def rel_interior_closed_segment)
have "b ∈ {c, d}"
proof -
have "insert b (closed_segment c d) = closed_segment c d"
using abcd by blast
then show ?thesis
by (metis DiffD2 Diff_insert2 False * insertI1 insert_Diff_if open_segment_def rel_interior_closed_segment)
qed
moreover have "a ∈ {c, d}"
by (metis Diff_iff False * abcd ends_in_segment(1) insertI1 open_segment_def rel_interior_closed_segment)
ultimately show "{a, b} = {c, d}"
using neq by fastforce
qed
next
assume "{a,b} = {c,d}"
then show "closed_segment a b = closed_segment c d"
by (simp add: segment_convex_hull)
qed
lemma closed_open_segment_eq [simp]:
fixes a :: "'a::euclidean_space"
shows "closed_segment a b ≠ open_segment c d"
by (metis DiffE closed_segment_neq_empty closure_closed_segment closure_open_segment ends_in_segment(1) insertI1 open_segment_def)
lemma open_closed_segment_eq [simp]:
fixes a :: "'a::euclidean_space"
shows "open_segment a b ≠ closed_segment c d"
using closed_open_segment_eq by blast
lemma open_segment_eq [simp]:
fixes a :: "'a::euclidean_space"
shows "open_segment a b = open_segment c d ⟷ a = b ∧ c = d ∨ {a,b} = {c,d}"
(is "?lhs = ?rhs")
proof
assume abcd: ?lhs
show ?rhs
proof (cases "a=b ∨ c=d")
case True with abcd show ?thesis
using finite_open_segment by fastforce
next
case False
then have a2: "a ≠ b ∧ c ≠ d" by force
with abcd show ?rhs
unfolding open_segment_def
by (metis (no_types) abcd closed_segment_eq closure_open_segment)
qed
next
assume ?rhs
then show ?lhs
by (metis Diff_cancel convex_hull_singleton insert_absorb2 open_segment_def segment_convex_hull)
qed
subsection‹Similar results for closure and (relative or absolute) frontier›
lemma closure_convex_hull [simp]:
fixes S :: "'a::euclidean_space set"
shows "compact S ==> closure(convex hull S) = convex hull S"
by (simp add: compact_imp_closed compact_convex_hull)
lemma rel_frontier_convex_hull_explicit:
fixes S :: "'a::euclidean_space set"
assumes "¬ affine_dependent S"
shows "rel_frontier(convex hull S) =
{y. ∃u. (∀x ∈ S. 0 ≤ u x) ∧ (∃x ∈ S. u x = 0) ∧ sum u S = 1 ∧ sum (λx. u x *⇩R x) S = y}"
proof -
have fs: "finite S"
using assms by (simp add: aff_independent_finite)
have "⋀u y v.
⟦y ∈ S; u y = 0; sum u S = 1; ∀x∈S. 0 < v x;
sum v S = 1; (∑x∈S. v x *⇩R x) = (∑x∈S. u x *⇩R x)⟧
⟹ ∃u. sum u S = 0 ∧ (∃v∈S. u v ≠ 0) ∧ (∑v∈S. u v *⇩R v) = 0"
apply (rule_tac x = "λx. u x - v x" in exI)
apply (force simp: sum_subtractf scaleR_diff_left)
done
then show ?thesis
using fs assms
apply (simp add: rel_frontier_def finite_imp_compact rel_interior_convex_hull_explicit)
apply (auto simp: convex_hull_finite)
apply (metis less_eq_real_def)
by (simp add: affine_dependent_explicit_finite)
qed
lemma frontier_convex_hull_explicit:
fixes S :: "'a::euclidean_space set"
assumes "¬ affine_dependent S"
shows "frontier(convex hull S) =
{y. ∃u. (∀x ∈ S. 0 ≤ u x) ∧ (DIM ('a) < card S ⟶ (∃x ∈ S. u x = 0)) ∧
sum u S = 1 ∧ sum (λx. u x *⇩R x) S = y}"
proof -
have fs: "finite S"
using assms by (simp add: aff_independent_finite)
show ?thesis
proof (cases "DIM ('a) < card S")
case True
with assms fs show ?thesis
by (simp add: rel_frontier_def frontier_def rel_frontier_convex_hull_explicit [symmetric]
interior_convex_hull_explicit_minimal rel_interior_convex_hull_explicit)
next
case False
then have "card S ≤ DIM ('a)"
by linarith
then show ?thesis
using assms fs
apply (simp add: frontier_def interior_convex_hull_explicit finite_imp_compact)
apply (simp add: convex_hull_finite)
done
qed
qed
lemma rel_frontier_convex_hull_cases:
fixes S :: "'a::euclidean_space set"
assumes "¬ affine_dependent S"
shows "rel_frontier(convex hull S) = ⋃{convex hull (S - {x}) |x. x ∈ S}"
proof -
have fs: "finite S"
using assms by (simp add: aff_independent_finite)
{ fix u a
have "∀x∈S. 0 ≤ u x ⟹ a ∈ S ⟹ u a = 0 ⟹ sum u S = 1 ⟹
∃x v. x ∈ S ∧
(∀x∈S - {x}. 0 ≤ v x) ∧
sum v (S - {x}) = 1 ∧ (∑x∈S - {x}. v x *⇩R x) = (∑x∈S. u x *⇩R x)"
apply (rule_tac x=a in exI)
apply (rule_tac x=u in exI)
apply (simp add: Groups_Big.sum_diff1 fs)
done }
moreover
{ fix a u
have "a ∈ S ⟹ ∀x∈S - {a}. 0 ≤ u x ⟹ sum u (S - {a}) = 1 ⟹
∃v. (∀x∈S. 0 ≤ v x) ∧
(∃x∈S. v x = 0) ∧ sum v S = 1 ∧ (∑x∈S. v x *⇩R x) = (∑x∈S - {a}. u x *⇩R x)"
apply (rule_tac x="λx. if x = a then 0 else u x" in exI)
apply (auto simp: sum.If_cases Diff_eq if_smult fs)
done }
ultimately show ?thesis
using assms
apply (simp add: rel_frontier_convex_hull_explicit)
apply (auto simp add: convex_hull_finite fs Union_SetCompr_eq)
done
qed
lemma frontier_convex_hull_eq_rel_frontier:
fixes S :: "'a::euclidean_space set"
assumes "¬ affine_dependent S"
shows "frontier(convex hull S) =
(if card S ≤ DIM ('a) then convex hull S else rel_frontier(convex hull S))"
using assms
unfolding rel_frontier_def frontier_def
by (simp add: affine_independent_span_gt rel_interior_interior
finite_imp_compact empty_interior_convex_hull aff_independent_finite)
lemma frontier_convex_hull_cases:
fixes S :: "'a::euclidean_space set"
assumes "¬ affine_dependent S"
shows "frontier(convex hull S) =
(if card S ≤ DIM ('a) then convex hull S else ⋃{convex hull (S - {x}) |x. x ∈ S})"
by (simp add: assms frontier_convex_hull_eq_rel_frontier rel_frontier_convex_hull_cases)
lemma in_frontier_convex_hull:
fixes S :: "'a::euclidean_space set"
assumes "finite S" "card S ≤ Suc (DIM ('a))" "x ∈ S"
shows "x ∈ frontier(convex hull S)"
proof (cases "affine_dependent S")
case True
with assms obtain y where "y ∈ S" and y: "y ∈ affine hull (S - {y})"
by (auto simp: affine_dependent_def)
moreover have "x ∈ closure (convex hull S)"
by (meson closure_subset hull_inc subset_eq ‹x ∈ S›)
moreover have "x ∉ interior (convex hull S)"
using assms
by (metis Suc_mono affine_hull_convex_hull affine_hull_nonempty_interior ‹y ∈ S› y card.remove empty_iff empty_interior_affine_hull finite_Diff hull_redundant insert_Diff interior_UNIV not_less)
ultimately show ?thesis
unfolding frontier_def by blast
next
case False
{ assume "card S = Suc (card Basis)"
then have cs: "Suc 0 < card S"
by (simp)
with subset_singletonD have "∃y ∈ S. y ≠ x"
by (cases "S ≤ {x}") fastforce+
} note [dest!] = this
show ?thesis using assms
unfolding frontier_convex_hull_cases [OF False] Union_SetCompr_eq
by (auto simp: le_Suc_eq hull_inc)
qed
lemma not_in_interior_convex_hull:
fixes S :: "'a::euclidean_space set"
assumes "finite S" "card S ≤ Suc (DIM ('a))" "x ∈ S"
shows "x ∉ interior(convex hull S)"
using in_frontier_convex_hull [OF assms]
by (metis Diff_iff frontier_def)
lemma interior_convex_hull_eq_empty:
fixes S :: "'a::euclidean_space set"
assumes "card S = Suc (DIM ('a))"
shows "interior(convex hull S) = {} ⟷ affine_dependent S"
proof
show "affine_dependent S ⟹ interior (convex hull S) = {}"
proof (clarsimp simp: affine_dependent_def)
fix a b
assume "b ∈ S" "b ∈ affine hull (S - {b})"
then have "interior(affine hull S) = {}" using assms
by (metis DIM_positive One_nat_def Suc_mono card.remove card.infinite empty_interior_affine_hull eq_iff hull_redundant insert_Diff not_less zero_le_one)
then show "interior (convex hull S) = {}"
using affine_hull_nonempty_interior by fastforce
qed
next
show "interior (convex hull S) = {} ⟹ affine_dependent S"
by (metis affine_hull_convex_hull affine_hull_empty affine_independent_span_eq assms convex_convex_hull empty_not_UNIV rel_interior_eq_empty rel_interior_interior)
qed
subsection ‹Coplanarity, and collinearity in terms of affine hull›
definition coplanar where
"coplanar S ≡ ∃u v w. S ⊆ affine hull {u,v,w}"
lemma collinear_affine_hull:
"collinear S ⟷ (∃u v. S ⊆ affine hull {u,v})"
proof (cases "S={}")
case True then show ?thesis
by simp
next
case False
then obtain x where x: "x ∈ S" by auto
{ fix u
assume *: "⋀x y. ⟦x∈S; y∈S⟧ ⟹ ∃c. x - y = c *⇩R u"
have "⋀y c. x - y = c *⇩R u ⟹ ∃a b. y = a *⇩R x + b *⇩R (x + u) ∧ a + b = 1"
by (rule_tac x="1+c" in exI, rule_tac x="-c" in exI, simp add: algebra_simps)
then have "∃u v. S ⊆ {a *⇩R u + b *⇩R v |a b. a + b = 1}"
using * [OF x] by (rule_tac x=x in exI, rule_tac x="x+u" in exI, force)
} moreover
{ fix u v x y
assume *: "S ⊆ {a *⇩R u + b *⇩R v |a b. a + b = 1}"
have "∃c. x - y = c *⇩R (v-u)" if "x∈S" "y∈S"
proof -
obtain a r where "a + r = 1" "x = a *⇩R u + r *⇩R v"
using "*" ‹x ∈ S› by blast
moreover
obtain b s where "b + s = 1" "y = b *⇩R u + s *⇩R v"
using "*" ‹y ∈ S› by blast
ultimately have "x - y = (r-s) *⇩R (v-u)"
by (simp add: algebra_simps) (metis scaleR_left.add)
then show ?thesis
by blast
qed
} ultimately
show ?thesis
unfolding collinear_def affine_hull_2
by blast
qed
lemma collinear_closed_segment [simp]: "collinear (closed_segment a b)"
by (metis affine_hull_convex_hull collinear_affine_hull hull_subset segment_convex_hull)
lemma collinear_open_segment [simp]: "collinear (open_segment a b)"
unfolding open_segment_def
by (metis convex_hull_subset_affine_hull segment_convex_hull dual_order.trans
convex_hull_subset_affine_hull Diff_subset collinear_affine_hull)
lemma collinear_between_cases:
fixes c :: "'a::euclidean_space"
shows "collinear {a,b,c} ⟷ between (b,c) a ∨ between (c,a) b ∨ between (a,b) c"
(is "?lhs = ?rhs")
proof
assume ?lhs
then obtain u v where uv: "⋀x. x ∈ {a, b, c} ⟹ ∃c. x = u + c *⇩R v"
by (auto simp: collinear_alt)
show ?rhs
using uv [of a] uv [of b] uv [of c] by (auto simp: between_1)
next
assume ?rhs
then show ?lhs
unfolding between_mem_convex_hull
by (metis (no_types, opaque_lifting) collinear_closed_segment collinear_subset hull_redundant hull_subset insert_commute segment_convex_hull)
qed
lemma subset_continuous_image_segment_1:
fixes f :: "'a::euclidean_space ⇒ real"
assumes "continuous_on (closed_segment a b) f"
shows "closed_segment (f a) (f b) ⊆ image f (closed_segment a b)"
by (metis connected_segment convex_contains_segment ends_in_segment imageI
is_interval_connected_1 is_interval_convex connected_continuous_image [OF assms])
lemma continuous_injective_image_segment_1:
fixes f :: "'a::euclidean_space ⇒ real"
assumes contf: "continuous_on (closed_segment a b) f"
and injf: "inj_on f (closed_segment a b)"
shows "f ` (closed_segment a b) = closed_segment (f a) (f b)"
proof
show "closed_segment (f a) (f b) ⊆ f ` closed_segment a b"
by (metis subset_continuous_image_segment_1 contf)
show "f ` closed_segment a b ⊆ closed_segment (f a) (f b)"
proof (cases "a = b")
case True
then show ?thesis by auto
next
case False
then have fnot: "f a ≠ f b"
using inj_onD injf by fastforce
moreover
have "f a ∉ open_segment (f c) (f b)" if c: "c ∈ closed_segment a b" for c
proof (clarsimp simp add: open_segment_def)
assume fa: "f a ∈ closed_segment (f c) (f b)"
moreover have "closed_segment (f c) (f b) ⊆ f ` closed_segment c b"
by (meson closed_segment_subset contf continuous_on_subset convex_closed_segment ends_in_segment(2) subset_continuous_image_segment_1 that)
ultimately have "f a ∈ f ` closed_segment c b"
by blast
then have a: "a ∈ closed_segment c b"
by (meson ends_in_segment inj_on_image_mem_iff injf subset_closed_segment that)
have cb: "closed_segment c b ⊆ closed_segment a b"
by (simp add: closed_segment_subset that)
show "f a = f c"
proof (rule between_antisym)
show "between (f c, f b) (f a)"
by (simp add: between_mem_segment fa)
show "between (f a, f b) (f c)"
by (metis a cb between_antisym between_mem_segment between_triv1 subset_iff)
qed
qed
moreover
have "f b ∉ open_segment (f a) (f c)" if c: "c ∈ closed_segment a b" for c
proof (clarsimp simp add: open_segment_def fnot eq_commute)
assume fb: "f b ∈ closed_segment (f a) (f c)"
moreover have "closed_segment (f a) (f c) ⊆ f ` closed_segment a c"
by (meson contf continuous_on_subset ends_in_segment(1) subset_closed_segment subset_continuous_image_segment_1 that)
ultimately have "f b ∈ f ` closed_segment a c"
by blast
then have b: "b ∈ closed_segment a c"
by (meson ends_in_segment inj_on_image_mem_iff injf subset_closed_segment that)
have ca: "closed_segment a c ⊆ closed_segment a b"
by (simp add: closed_segment_subset that)
show "f b = f c"
proof (rule between_antisym)
show "between (f c, f a) (f b)"
by (simp add: between_commute between_mem_segment fb)
show "between (f b, f a) (f c)"
by (metis b between_antisym between_commute between_mem_segment between_triv2 that)
qed
qed
ultimately show ?thesis
by (force simp: closed_segment_eq_real_ivl open_segment_eq_real_ivl split: if_split_asm)
qed
qed
lemma continuous_injective_image_open_segment_1:
fixes f :: "'a::euclidean_space ⇒ real"
assumes contf: "continuous_on (closed_segment a b) f"
and injf: "inj_on f (closed_segment a b)"
shows "f ` (open_segment a b) = open_segment (f a) (f b)"
proof -
have "f ` (open_segment a b) = f ` (closed_segment a b) - {f a, f b}"
by (metis (no_types, opaque_lifting) empty_subsetI ends_in_segment image_insert image_is_empty inj_on_image_set_diff injf insert_subset open_segment_def segment_open_subset_closed)
also have "... = open_segment (f a) (f b)"
using continuous_injective_image_segment_1 [OF assms]
by (simp add: open_segment_def inj_on_image_set_diff [OF injf])
finally show ?thesis .
qed
lemma collinear_imp_coplanar:
"collinear s ==> coplanar s"
by (metis collinear_affine_hull coplanar_def insert_absorb2)
lemma collinear_small:
assumes "finite s" "card s ≤ 2"
shows "collinear s"
proof -
have "card s = 0 ∨ card s = 1 ∨ card s = 2"
using assms by linarith
then show ?thesis using assms
using card_eq_SucD numeral_2_eq_2 by (force simp: card_1_singleton_iff)
qed
lemma coplanar_small:
assumes "finite s" "card s ≤ 3"
shows "coplanar s"
proof -
consider "card s ≤ 2" | "card s = Suc (Suc (Suc 0))"
using assms by linarith
then show ?thesis
proof cases
case 1
then show ?thesis
by (simp add: ‹finite s› collinear_imp_coplanar collinear_small)
next
case 2
then show ?thesis
using hull_subset [of "{_,_,_}"]
by (fastforce simp: coplanar_def dest!: card_eq_SucD)
qed
qed
lemma coplanar_empty: "coplanar {}"
by (simp add: coplanar_small)
lemma coplanar_sing: "coplanar {a}"
by (simp add: coplanar_small)
lemma coplanar_2: "coplanar {a,b}"
by (auto simp: card_insert_if coplanar_small)
lemma coplanar_3: "coplanar {a,b,c}"
by (auto simp: card_insert_if coplanar_small)
lemma collinear_affine_hull_collinear: "collinear(affine hull s) ⟷ collinear s"
unfolding collinear_affine_hull
by (metis affine_affine_hull subset_hull hull_hull hull_mono)
lemma coplanar_affine_hull_coplanar: "coplanar(affine hull s) ⟷ coplanar s"
unfolding coplanar_def
by (metis affine_affine_hull subset_hull hull_hull hull_mono)
lemma coplanar_linear_image:
fixes f :: "'a::euclidean_space ⇒ 'b::real_normed_vector"
assumes "coplanar S" "linear f" shows "coplanar(f ` S)"
proof -
{ fix u v w
assume "S ⊆ affine hull {u, v, w}"
then have "f ` S ⊆ f ` (affine hull {u, v, w})"
by (simp add: image_mono)
then have "f ` S ⊆ affine hull (f ` {u, v, w})"
by (metis assms(2) linear_conv_bounded_linear affine_hull_linear_image)
} then
show ?thesis
by auto (meson assms(1) coplanar_def)
qed
lemma coplanar_translation_imp:
assumes "coplanar S" shows "coplanar ((λx. a + x) ` S)"
proof -
obtain u v w where "S ⊆ affine hull {u,v,w}"
by (meson assms coplanar_def)
then have "(+) a ` S ⊆ affine hull {u + a, v + a, w + a}"
using affine_hull_translation [of a "{u,v,w}" for u v w]
by (force simp: add.commute)
then show ?thesis
unfolding coplanar_def by blast
qed
lemma coplanar_translation_eq: "coplanar((λx. a + x) ` S) ⟷ coplanar S"
by (metis (no_types) coplanar_translation_imp translation_galois)
lemma coplanar_linear_image_eq:
fixes f :: "'a::euclidean_space ⇒ 'b::euclidean_space"
assumes "linear f" "inj f" shows "coplanar(f ` S) = coplanar S"
proof
assume "coplanar S"
then show "coplanar (f ` S)"
using assms(1) coplanar_linear_image by blast
next
obtain g where g: "linear g" "g ∘ f = id"
using linear_injective_left_inverse [OF assms]
by blast
assume "coplanar (f ` S)"
then show "coplanar S"
by (metis coplanar_linear_image g(1) g(2) id_apply image_comp image_id)
qed
lemma coplanar_subset: "⟦coplanar t; S ⊆ t⟧ ⟹ coplanar S"
by (meson coplanar_def order_trans)
lemma affine_hull_3_imp_collinear: "c ∈ affine hull {a,b} ⟹ collinear {a,b,c}"
by (metis collinear_2 collinear_affine_hull_collinear hull_redundant insert_commute)
lemma collinear_3_imp_in_affine_hull:
assumes "collinear {a,b,c}" "a ≠ b" shows "c ∈ affine hull {a,b}"
proof -
obtain u x y where "b - a = y *⇩R u" "c - a = x *⇩R u"
using assms unfolding collinear_def by auto
with ‹a ≠ b› have "∃v. c = (1 - x / y) *⇩R a + v *⇩R b ∧ 1 - x / y + v = 1"
by (simp add: algebra_simps)
then show ?thesis
by (simp add: hull_inc mem_affine)
qed
lemma collinear_3_affine_hull:
assumes "a ≠ b"
shows "collinear {a,b,c} ⟷ c ∈ affine hull {a,b}"
using affine_hull_3_imp_collinear assms collinear_3_imp_in_affine_hull by blast
lemma collinear_3_eq_affine_dependent:
"collinear{a,b,c} ⟷ a = b ∨ a = c ∨ b = c ∨ affine_dependent {a,b,c}"
proof (cases "a = b ∨ a = c ∨ b = c")
case True
then show ?thesis
by (auto simp: insert_commute)
next
case False
then have "collinear{a,b,c}" if "affine_dependent {a,b,c}"
using that unfolding affine_dependent_def
by (auto simp: insert_Diff_if; metis affine_hull_3_imp_collinear insert_commute)
moreover
have "affine_dependent {a,b,c}" if "collinear{a,b,c}"
using False that by (auto simp: affine_dependent_def collinear_3_affine_hull insert_Diff_if)
ultimately
show ?thesis
using False by blast
qed
lemma affine_dependent_imp_collinear_3:
"affine_dependent {a,b,c} ⟹ collinear{a,b,c}"
by (simp add: collinear_3_eq_affine_dependent)
lemma collinear_3: "NO_MATCH 0 x ⟹ collinear {x,y,z} ⟷ collinear {0, x-y, z-y}"
by (auto simp add: collinear_def)
lemma collinear_3_expand:
"collinear{a,b,c} ⟷ a = c ∨ (∃u. b = u *⇩R a + (1 - u) *⇩R c)"
proof -
have "collinear{a,b,c} = collinear{a,c,b}"
by (simp add: insert_commute)
also have "... = collinear {0, a - c, b - c}"
by (simp add: collinear_3)
also have "... ⟷ (a = c ∨ b = c ∨ (∃ca. b - c = ca *⇩R (a - c)))"
by (simp add: collinear_lemma)
also have "... ⟷ a = c ∨ (∃u. b = u *⇩R a + (1 - u) *⇩R c)"
by (cases "a = c ∨ b = c") (auto simp: algebra_simps)
finally show ?thesis .
qed
lemma collinear_aff_dim: "collinear S ⟷ aff_dim S ≤ 1"
proof
assume "collinear S"
then obtain u and v :: "'a" where "aff_dim S ≤ aff_dim {u,v}"
by (metis ‹collinear S› aff_dim_affine_hull aff_dim_subset collinear_affine_hull)
then show "aff_dim S ≤ 1"
using order_trans by fastforce
next
assume "aff_dim S ≤ 1"
then have le1: "aff_dim (affine hull S) ≤ 1"
by simp
obtain B where "B ⊆ S" and B: "¬ affine_dependent B" "affine hull S = affine hull B"
using affine_basis_exists [of S] by auto
then have "finite B" "card B ≤ 2"
using B le1 by (auto simp: affine_independent_iff_card)
then have "collinear B"
by (rule collinear_small)
then show "collinear S"
by (metis ‹affine hull S = affine hull B› collinear_affine_hull_collinear)
qed
lemma collinear_midpoint: "collinear{a, midpoint a b, b}"
proof -
have §: "⟦a ≠ midpoint a b; b - midpoint a b ≠ - 1 *⇩R (a - midpoint a b)⟧ ⟹ b = midpoint a b"
by (simp add: algebra_simps)
show ?thesis
by (auto simp: collinear_3 collinear_lemma intro: §)
qed
lemma midpoint_collinear:
fixes a b c :: "'a::real_normed_vector"
assumes "a ≠ c"
shows "b = midpoint a c ⟷ collinear{a,b,c} ∧ dist a b = dist b c"
proof -
have *: "a - (u *⇩R a + (1 - u) *⇩R c) = (1 - u) *⇩R (a - c)"
"u *⇩R a + (1 - u) *⇩R c - c = u *⇩R (a - c)"
"¦1 - u¦ = ¦u¦ ⟷ u = 1/2" for u::real
by (auto simp: algebra_simps)
have "b = midpoint a c ⟹ collinear{a,b,c}"
using collinear_midpoint by blast
moreover have "b = midpoint a c ⟷ dist a b = dist b c" if "collinear{a,b,c}"
proof -
consider "a = c" | u where "b = u *⇩R a + (1 - u) *⇩R c"
using ‹collinear {a,b,c}› unfolding collinear_3_expand by blast
then show ?thesis
proof cases
case 2
with assms have "dist a b = dist b c ⟹ b = midpoint a c"
by (simp add: dist_norm * midpoint_def scaleR_add_right del: divide_const_simps)
then show ?thesis
by (auto simp: dist_midpoint)
qed (use assms in auto)
qed
ultimately show ?thesis by blast
qed
lemma between_imp_collinear:
fixes x :: "'a :: euclidean_space"
assumes "between (a,b) x"
shows "collinear {a,x,b}"
proof (cases "x = a ∨ x = b ∨ a = b")
case True with assms show ?thesis
by (auto simp: dist_commute)
next
case False
then have False if "⋀c. b - x ≠ c *⇩R (a - x)"
using that [of "-(norm(b - x) / norm(x - a))"] assms
by (simp add: between_norm vector_add_divide_simps flip: real_vector.scale_minus_right)
then show ?thesis
by (auto simp: collinear_3 collinear_lemma)
qed
lemma midpoint_between:
fixes a b :: "'a::euclidean_space"
shows "b = midpoint a c ⟷ between (a,c) b ∧ dist a b = dist b c"
proof (cases "a = c")
case False
show ?thesis
using False between_imp_collinear between_midpoint(1) midpoint_collinear by blast
qed (auto simp: dist_commute)
lemma collinear_triples:
assumes "a ≠ b"
shows "collinear(insert a (insert b S)) ⟷ (∀x ∈ S. collinear{a,b,x})"
(is "?lhs = ?rhs")
proof safe
fix x
assume ?lhs and "x ∈ S"
then show "collinear {a, b, x}"
using collinear_subset by force
next
assume ?rhs
then have "∀x ∈ S. collinear{a,x,b}"
by (simp add: insert_commute)
then have *: "∃u. x = u *⇩R a + (1 - u) *⇩R b" if "x ∈ insert a (insert b S)" for x
using that assms collinear_3_expand by fastforce+
have "∃c. x - y = c *⇩R (b - a)"
if x: "x ∈ insert a (insert b S)" and y: "y ∈ insert a (insert b S)" for x y
proof -
obtain u v where "x = u *⇩R a + (1 - u) *⇩R b" "y = v *⇩R a + (1 - v) *⇩R b"
using "*" x y by presburger
then have "x - y = (v - u) *⇩R (b - a)"
by (simp add: scale_left_diff_distrib scale_right_diff_distrib)
then show ?thesis ..
qed
then show ?lhs
unfolding collinear_def by metis
qed
lemma collinear_4_3:
assumes "a ≠ b"
shows "collinear {a,b,c,d} ⟷ collinear{a,b,c} ∧ collinear{a,b,d}"
using collinear_triples [OF assms, of "{c,d}"] by (force simp:)
lemma collinear_3_trans:
assumes "collinear{a,b,c}" "collinear{b,c,d}" "b ≠ c"
shows "collinear{a,b,d}"
proof -
have "collinear{b,c,a,d}"
by (metis (full_types) assms collinear_4_3 insert_commute)
then show ?thesis
by (simp add: collinear_subset)
qed
lemma affine_hull_2_alt:
fixes a b :: "'a::real_vector"
shows "affine hull {a,b} = range (λu. a + u *⇩R (b - a))"
proof -
have 1: "u *⇩R a + v *⇩R b = a + v *⇩R (b - a)" if "u + v = 1" for u v
using that
by (simp add: algebra_simps flip: scaleR_add_left)
have 2: "a + u *⇩R (b - a) = (1 - u) *⇩R a + u *⇩R b" for u
by (auto simp: algebra_simps)
show ?thesis
by (force simp add: affine_hull_2 dest: 1 intro!: 2)
qed
lemma interior_convex_hull_3_minimal:
fixes a :: "'a::euclidean_space"
assumes "¬ collinear{a,b,c}" and 2: "DIM('a) = 2"
shows "interior(convex hull {a,b,c}) =
{v. ∃x y z. 0 < x ∧ 0 < y ∧ 0 < z ∧ x + y + z = 1 ∧ x *⇩R a + y *⇩R b + z *⇩R c = v}"
(is "?lhs = ?rhs")
proof
have abc: "a ≠ b" "a ≠ c" "b ≠ c" "¬ affine_dependent {a, b, c}"
using assms by (auto simp: collinear_3_eq_affine_dependent)
with 2 show "?lhs ⊆ ?rhs"
by (fastforce simp add: interior_convex_hull_explicit_minimal)
show "?rhs ⊆ ?lhs"
using abc 2
apply (clarsimp simp add: interior_convex_hull_explicit_minimal)
subgoal for x y z
by (rule_tac x="λr. (if r=a then x else if r=b then y else if r=c then z else 0)" in exI) auto
done
qed
subsection‹Basic lemmas about hyperplanes and halfspaces›
lemma halfspace_Int_eq:
"{x. a ∙ x ≤ b} ∩ {x. b ≤ a ∙ x} = {x. a ∙ x = b}"
"{x. b ≤ a ∙ x} ∩ {x. a ∙ x ≤ b} = {x. a ∙ x = b}"
by auto
lemma hyperplane_eq_Ex:
assumes "a ≠ 0" obtains x where "a ∙ x = b"
by (rule_tac x = "(b / (a ∙ a)) *⇩R a" in that) (simp add: assms)
lemma hyperplane_eq_empty:
"{x. a ∙ x = b} = {} ⟷ a = 0 ∧ b ≠ 0"
using hyperplane_eq_Ex
by (metis (mono_tags, lifting) empty_Collect_eq inner_zero_left)
lemma hyperplane_eq_UNIV:
"{x. a ∙ x = b} = UNIV ⟷ a = 0 ∧ b = 0"
proof -
have "a = 0 ∧ b = 0" if "UNIV ⊆ {x. a ∙ x = b}"
using subsetD [OF that, where c = "((b+1) / (a ∙ a)) *⇩R a"]
by (simp add: field_split_simps split: if_split_asm)
then show ?thesis by force
qed
lemma halfspace_eq_empty_lt:
"{x. a ∙ x < b} = {} ⟷ a = 0 ∧ b ≤ 0"
proof -
have "a = 0 ∧ b ≤ 0" if "{x. a ∙ x < b} ⊆ {}"
using subsetD [OF that, where c = "((b-1) / (a ∙ a)) *⇩R a"]
by (force simp add: field_split_simps split: if_split_asm)
then show ?thesis by force
qed
lemma halfspace_eq_empty_gt:
"{x. a ∙ x > b} = {} ⟷ a = 0 ∧ b ≥ 0"
using halfspace_eq_empty_lt [of "-a" "-b"]
by simp
lemma halfspace_eq_empty_le:
"{x. a ∙ x ≤ b} = {} ⟷ a = 0 ∧ b < 0"
proof -
have "a = 0 ∧ b < 0" if "{x. a ∙ x ≤ b} ⊆ {}"
using subsetD [OF that, where c = "((b-1) / (a ∙ a)) *⇩R a"]
by (force simp add: field_split_simps split: if_split_asm)
then show ?thesis by force
qed
lemma halfspace_eq_empty_ge:
"{x. a ∙ x ≥ b} = {} ⟷ a = 0 ∧ b > 0"
using halfspace_eq_empty_le [of "-a" "-b"] by simp
subsection‹Use set distance for an easy proof of separation properties›
proposition separation_closures:
fixes S :: "'a::euclidean_space set"
assumes "S ∩ closure T = {}" "T ∩ closure S = {}"
obtains U V where "U ∩ V = {}" "open U" "open V" "S ⊆ U" "T ⊆ V"
proof (cases "S = {} ∨ T = {}")
case True with that show ?thesis by auto
next
case False
define f where "f ≡ λx. setdist {x} T - setdist {x} S"
have contf: "continuous_on UNIV f"
unfolding f_def by (intro continuous_intros continuous_on_setdist)
show ?thesis
proof (rule_tac U = "{x. f x > 0}" and V = "{x. f x < 0}" in that)
show "{x. 0 < f x} ∩ {x. f x < 0} = {}"
by auto
show "open {x. 0 < f x}"
by (simp add: open_Collect_less contf)
show "open {x. f x < 0}"
by (simp add: open_Collect_less contf)
have "⋀x. x ∈ S ⟹ setdist {x} T ≠ 0" "⋀x. x ∈ T ⟹ setdist {x} S ≠ 0"
by (meson False assms disjoint_iff setdist_eq_0_sing_1)+
then show "S ⊆ {x. 0 < f x}" "T ⊆ {x. f x < 0}"
using less_eq_real_def by (fastforce simp add: f_def setdist_sing_in_set)+
qed
qed
lemma separation_normal:
fixes S :: "'a::euclidean_space set"
assumes "closed S" "closed T" "S ∩ T = {}"
obtains U V where "open U" "open V" "S ⊆ U" "T ⊆ V" "U ∩ V = {}"
using separation_closures [of S T]
by (metis assms closure_closed disjnt_def inf_commute)
lemma separation_normal_local:
fixes S :: "'a::euclidean_space set"
assumes US: "closedin (top_of_set U) S"
and UT: "closedin (top_of_set U) T"
and "S ∩ T = {}"
obtains S' T' where "openin (top_of_set U) S'"
"openin (top_of_set U) T'"
"S ⊆ S'" "T ⊆ T'" "S' ∩ T' = {}"
proof (cases "S = {} ∨ T = {}")
case True with that show ?thesis
using UT US by (blast dest: closedin_subset)
next
case False
define f where "f ≡ λx. setdist {x} T - setdist {x} S"
have contf: "continuous_on U f"
unfolding f_def by (intro continuous_intros)
show ?thesis
proof (rule_tac S' = "(U ∩ f -` {0<..})" and T' = "(U ∩ f -` {..<0})" in that)
show "(U ∩ f -` {0<..}) ∩ (U ∩ f -` {..<0}) = {}"
by auto
show "openin (top_of_set U) (U ∩ f -` {0<..})"
by (rule continuous_openin_preimage [where T=UNIV]) (simp_all add: contf)
next
show "openin (top_of_set U) (U ∩ f -` {..<0})"
by (rule continuous_openin_preimage [where T=UNIV]) (simp_all add: contf)
next
have "S ⊆ U" "T ⊆ U"
using closedin_imp_subset assms by blast+
then show "S ⊆ U ∩ f -` {0<..}" "T ⊆ U ∩ f -` {..<0}"
using assms False by (force simp add: f_def setdist_sing_in_set intro!: setdist_gt_0_closedin)+
qed
qed
lemma separation_normal_compact:
fixes S :: "'a::euclidean_space set"
assumes "compact S" "closed T" "S ∩ T = {}"
obtains U V where "open U" "compact(closure U)" "open V" "S ⊆ U" "T ⊆ V" "U ∩ V = {}"
proof -
have "closed S" "bounded S"
using assms by (auto simp: compact_eq_bounded_closed)
then obtain r where "r>0" and r: "S ⊆ ball 0 r"
by (auto dest!: bounded_subset_ballD)
have **: "closed (T ∪ - ball 0 r)" "S ∩ (T ∪ - ball 0 r) = {}"
using assms r by blast+
then obtain U V where UV: "open U" "open V" "S ⊆ U" "T ∪ - ball 0 r ⊆ V" "U ∩ V = {}"
by (meson ‹closed S› separation_normal)
then have "compact(closure U)"
by (meson bounded_ball bounded_subset compact_closure compl_le_swap2 disjoint_eq_subset_Compl le_sup_iff)
with UV show thesis
using that by auto
qed
subsection‹Connectedness of the intersection of a chain›
proposition connected_chain:
fixes ℱ :: "'a :: euclidean_space set set"
assumes cc: "⋀S. S ∈ ℱ ⟹ compact S ∧ connected S"
and linear: "⋀S T. S ∈ ℱ ∧ T ∈ ℱ ⟹ S ⊆ T ∨ T ⊆ S"
shows "connected(⋂ℱ)"
proof (cases "ℱ = {}")
case True then show ?thesis
by auto
next
case False
then have cf: "compact(⋂ℱ)"
by (simp add: cc compact_Inter)
have False if AB: "closed A" "closed B" "A ∩ B = {}"
and ABeq: "A ∪ B = ⋂ℱ" and "A ≠ {}" "B ≠ {}" for A B
proof -
obtain U V where "open U" "open V" "A ⊆ U" "B ⊆ V" "U ∩ V = {}"
using separation_normal [OF AB] by metis
obtain K where "K ∈ ℱ" "compact K"
using cc False by blast
then obtain N where "open N" and "K ⊆ N"
by blast
let ?𝒞 = "insert (U ∪ V) ((λS. N - S) ` ℱ)"
obtain 𝒟 where "𝒟 ⊆ ?𝒞" "finite 𝒟" "K ⊆ ⋃𝒟"
proof (rule compactE [OF ‹compact K›])
show "K ⊆ ⋃(insert (U ∪ V) ((-) N ` ℱ))"
using ‹K ⊆ N› ABeq ‹A ⊆ U› ‹B ⊆ V› by auto
show "⋀B. B ∈ insert (U ∪ V) ((-) N ` ℱ) ⟹ open B"
by (auto simp: ‹open U› ‹open V› open_Un ‹open N› cc compact_imp_closed open_Diff)
qed
then have "finite(𝒟 - {U ∪ V})"
by blast
moreover have "𝒟 - {U ∪ V} ⊆ (λS. N - S) ` ℱ"
using ‹𝒟 ⊆ ?𝒞› by blast
ultimately obtain 𝒢 where "𝒢 ⊆ ℱ" "finite 𝒢" and Deq: "𝒟 - {U ∪ V} = (λS. N-S) ` 𝒢"
using finite_subset_image by metis
obtain J where "J ∈ ℱ" and J: "(⋃S∈𝒢. N - S) ⊆ N - J"
proof (cases "𝒢 = {}")
case True
with ‹ℱ ≠ {}› that show ?thesis
by auto
next
case False
have "⋀S T. ⟦S ∈ 𝒢; T ∈ 𝒢⟧ ⟹ S ⊆ T ∨ T ⊆ S"
by (meson ‹𝒢 ⊆ ℱ› in_mono local.linear)
with ‹finite 𝒢› ‹𝒢 ≠ {}›
have "∃J ∈ 𝒢. (⋃S∈𝒢. N - S) ⊆ N - J"
proof induction
case (insert X ℋ)
show ?case
proof (cases "ℋ = {}")
case True then show ?thesis by auto
next
case False
then have "⋀S T. ⟦S ∈ ℋ; T ∈ ℋ⟧ ⟹ S ⊆ T ∨ T ⊆ S"
by (simp add: insert.prems)
with insert.IH False obtain J where "J ∈ ℋ" and J: "(⋃Y∈ℋ. N - Y) ⊆ N - J"
by metis
have "N - J ⊆ N - X ∨ N - X ⊆ N - J"
by (meson Diff_mono ‹J ∈ ℋ› insert.prems(2) insert_iff order_refl)
then show ?thesis
proof
assume "N - J ⊆ N - X" with J show ?thesis
by auto
next
assume "N - X ⊆ N - J"
with J have "N - X ∪ ⋃ ((-) N ` ℋ) ⊆ N - J"
by auto
with ‹J ∈ ℋ› show ?thesis
by blast
qed
qed
qed simp
with ‹𝒢 ⊆ ℱ› show ?thesis by (blast intro: that)
qed
have "K ⊆ ⋃(insert (U ∪ V) (𝒟 - {U ∪ V}))"
using ‹K ⊆ ⋃𝒟› by auto
also have "... ⊆ (U ∪ V) ∪ (N - J)"
by (metis (no_types, opaque_lifting) Deq Un_subset_iff Un_upper2 J Union_insert order_trans sup_ge1)
finally have "J ∩ K ⊆ U ∪ V"
by blast
moreover have "connected(J ∩ K)"
by (metis Int_absorb1 ‹J ∈ ℱ› ‹K ∈ ℱ› cc inf.orderE local.linear)
moreover have "U ∩ (J ∩ K) ≠ {}"
using ABeq ‹J ∈ ℱ› ‹K ∈ ℱ› ‹A ≠ {}› ‹A ⊆ U› by blast
moreover have "V ∩ (J ∩ K) ≠ {}"
using ABeq ‹J ∈ ℱ› ‹K ∈ ℱ› ‹B ≠ {}› ‹B ⊆ V› by blast
ultimately show False
using connectedD [of "J ∩ K" U V] ‹open U› ‹open V› ‹U ∩ V = {}› by auto
qed
with cf show ?thesis
by (auto simp: connected_closed_set compact_imp_closed)
qed
lemma connected_chain_gen:
fixes ℱ :: "'a :: euclidean_space set set"
assumes X: "X ∈ ℱ" "compact X"
and cc: "⋀T. T ∈ ℱ ⟹ closed T ∧ connected T"
and linear: "⋀S T. S ∈ ℱ ∧ T ∈ ℱ ⟹ S ⊆ T ∨ T ⊆ S"
shows "connected(⋂ℱ)"
proof -
have "⋂ℱ = (⋂T∈ℱ. X ∩ T)"
using X by blast
moreover have "connected (⋂T∈ℱ. X ∩ T)"
proof (rule connected_chain)
show "⋀T. T ∈ (∩) X ` ℱ ⟹ compact T ∧ connected T"
using cc X by auto (metis inf.absorb2 inf.orderE local.linear)
show "⋀S T. S ∈ (∩) X ` ℱ ∧ T ∈ (∩) X ` ℱ ⟹ S ⊆ T ∨ T ⊆ S"
using local.linear by blast
qed
ultimately show ?thesis
by metis
qed
lemma connected_nest:
fixes S :: "'a::linorder ⇒ 'b::euclidean_space set"
assumes S: "⋀n. compact(S n)" "⋀n. connected(S n)"
and nest: "⋀m n. m ≤ n ⟹ S n ⊆ S m"
shows "connected(⋂ (range S))"
proof (rule connected_chain)
show "⋀A T. A ∈ range S ∧ T ∈ range S ⟹ A ⊆ T ∨ T ⊆ A"
by (metis image_iff le_cases nest)
qed (use S in blast)
lemma connected_nest_gen:
fixes S :: "'a::linorder ⇒ 'b::euclidean_space set"
assumes S: "⋀n. closed(S n)" "⋀n. connected(S n)" "compact(S k)"
and nest: "⋀m n. m ≤ n ⟹ S n ⊆ S m"
shows "connected(⋂ (range S))"
proof (rule connected_chain_gen [of "S k"])
show "⋀A T. A ∈ range S ∧ T ∈ range S ⟹ A ⊆ T ∨ T ⊆ A"
by (metis imageE le_cases nest)
qed (use S in auto)
subsection‹Proper maps, including projections out of compact sets›
lemma finite_indexed_bound:
assumes A: "finite A" "⋀x. x ∈ A ⟹ ∃n::'a::linorder. P x n"
shows "∃m. ∀x ∈ A. ∃k≤m. P x k"
using A
proof (induction A)
case empty then show ?case by force
next
case (insert a A)
then obtain m n where "∀x ∈ A. ∃k≤m. P x k" "P a n"
by force
then show ?case
by (metis dual_order.trans insert_iff le_cases)
qed
proposition proper_map:
fixes f :: "'a::heine_borel ⇒ 'b::heine_borel"
assumes "closedin (top_of_set S) K"
and com: "⋀U. ⟦U ⊆ T; compact U⟧ ⟹ compact (S ∩ f -` U)"
and "f ` S ⊆ T"
shows "closedin (top_of_set T) (f ` K)"
proof -
have "K ⊆ S"
using assms closedin_imp_subset by metis
obtain C where "closed C" and Keq: "K = S ∩ C"
using assms by (auto simp: closedin_closed)
have *: "y ∈ f ` K" if "y ∈ T" and y: "y islimpt f ` K" for y
proof -
obtain h where "∀n. (∃x∈K. h n = f x) ∧ h n ≠ y" "inj h" and hlim: "(h ⤏ y) sequentially"
using ‹y ∈ T› y by (force simp: limpt_sequential_inj)
then obtain X where X: "⋀n. X n ∈ K ∧ h n = f (X n) ∧ h n ≠ y"
by metis
then have fX: "⋀n. f (X n) = h n"
by metis
define Ψ where "Ψ ≡ λn. {a ∈ K. f a ∈ insert y (range (λi. f (X (n + i))))}"
have "compact (C ∩ (S ∩ f -` insert y (range (λi. f(X(n + i))))))" for n
proof (intro closed_Int_compact [OF ‹closed C› com] compact_sequence_with_limit)
show "insert y (range (λi. f (X (n + i)))) ⊆ T"
using X ‹K ⊆ S› ‹f ` S ⊆ T› ‹y ∈ T› by blast
show "(λi. f (X (n + i))) ⇢ y"
by (simp add: fX add.commute [of n] LIMSEQ_ignore_initial_segment [OF hlim])
qed
then have comf: "compact (Ψ n)" for n
by (simp add: Keq Int_def Ψ_def conj_commute)
have ne: "⋂ℱ ≠ {}"
if "finite ℱ"
and ℱ: "⋀t. t ∈ ℱ ⟹ (∃n. t = Ψ n)"
for ℱ
proof -
obtain m where m: "⋀t. t ∈ ℱ ⟹ ∃k≤m. t = Ψ k"
by (rule exE [OF finite_indexed_bound [OF ‹finite ℱ› ℱ]], force+)
have "X m ∈ ⋂ℱ"
using X le_Suc_ex by (fastforce simp: Ψ_def dest: m)
then show ?thesis by blast
qed
have "(⋂n. Ψ n) ≠ {}"
proof (rule compact_fip_Heine_Borel)
show "⋀ℱ'. ⟦finite ℱ'; ℱ' ⊆ range Ψ⟧ ⟹ ⋂ ℱ' ≠ {}"
by (meson ne rangeE subset_eq)
qed (use comf in blast)
then obtain x where "x ∈ K" "⋀n. (f x = y ∨ (∃u. f x = h (n + u)))"
by (force simp add: Ψ_def fX)
then show ?thesis
unfolding image_iff by (metis ‹inj h› le_add1 not_less_eq_eq rangeI range_ex1_eq)
qed
with assms closedin_subset show ?thesis
by (force simp: closedin_limpt)
qed
subsection ‹Closure of conic hulls›
proposition closedin_conic_hull:
fixes S :: "'a::euclidean_space set"
assumes "compact T" "0 ∉ T" "T ⊆ S"
shows "closedin (top_of_set (conic hull S)) (conic hull T)"
proof -
have **: "compact ({0..} × T ∩ (λz. fst z *⇩R snd z) -` K)" (is "compact ?L")
if "K ⊆ (λz. (fst z) *⇩R snd z) ` ({0..} × S)" "compact K" for K
proof -
obtain r where "r > 0" and r: "⋀x. x ∈ K ⟹ norm x ≤ r"
by (metis ‹compact K› bounded_normE compact_imp_bounded)
show ?thesis
unfolding compact_eq_bounded_closed
proof
have "bounded ({0..r / setdist{0}T} × T)"
by (simp add: assms(1) bounded_Times compact_imp_bounded)
moreover have "?L ⊆ ({0..r / setdist{0}T} × T)"
proof clarsimp
fix a b
assume "a *⇩R b ∈ K" and "b ∈ T" and "0 ≤ a"
have "setdist {0} T ≠ 0"
using ‹b ∈ T› assms compact_imp_closed setdist_eq_0_closed by auto
then have T0: "setdist {0} T > 0"
using less_eq_real_def by fastforce
then have "a * setdist {0} T ≤ r"
by (smt (verit, ccfv_SIG) ‹0 ≤ a› ‹a *⇩R b ∈ K› ‹b ∈ T› dist_0_norm mult_mono' norm_scaleR r setdist_le_dist singletonI)
with T0 ‹r>0› show "a ≤ r / setdist {0} T"
by (simp add: divide_simps)
qed
ultimately show "bounded ?L"
by (meson bounded_subset)
show "closed ?L"
proof (rule continuous_closed_preimage)
show "continuous_on ({0..} × T) (λz. fst z *⇩R snd z)"
by (intro continuous_intros)
show "closed ({0::real..} × T)"
by (simp add: assms(1) closed_Times compact_imp_closed)
show "closed K"
by (simp add: compact_imp_closed that(2))
qed
qed
qed
show ?thesis
unfolding conic_hull_as_image
proof (rule proper_map)
show "compact ({0..} × T ∩ (λz. fst z *⇩R snd z) -` K)" (is "compact ?L")
if "K ⊆ (λz. (fst z) *⇩R snd z) ` ({0..} × S)" "compact K" for K
proof -
obtain r where "r > 0" and r: "⋀x. x ∈ K ⟹ norm x ≤ r"
by (metis ‹compact K› bounded_normE compact_imp_bounded)
show ?thesis
unfolding compact_eq_bounded_closed
proof
have "bounded ({0..r / setdist{0}T} × T)"
by (simp add: assms(1) bounded_Times compact_imp_bounded)
moreover have "?L ⊆ ({0..r / setdist{0}T} × T)"
proof clarsimp
fix a b
assume "a *⇩R b ∈ K" and "b ∈ T" and "0 ≤ a"
have "setdist {0} T ≠ 0"
using ‹b ∈ T› assms compact_imp_closed setdist_eq_0_closed by auto
then have T0: "setdist {0} T > 0"
using less_eq_real_def by fastforce
then have "a * setdist {0} T ≤ r"
by (smt (verit, ccfv_SIG) ‹0 ≤ a› ‹a *⇩R b ∈ K› ‹b ∈ T› dist_0_norm mult_mono' norm_scaleR r setdist_le_dist singletonI)
with T0 ‹r>0› show "a ≤ r / setdist {0} T"
by (simp add: divide_simps)
qed
ultimately show "bounded ?L"
by (meson bounded_subset)
show "closed ?L"
proof (rule continuous_closed_preimage)
show "continuous_on ({0..} × T) (λz. fst z *⇩R snd z)"
by (intro continuous_intros)
show "closed ({0::real..} × T)"
by (simp add: assms(1) closed_Times compact_imp_closed)
show "closed K"
by (simp add: compact_imp_closed that(2))
qed
qed
qed
show "(λz. fst z *⇩R snd z) ` ({0::real..} × T) ⊆ (λz. fst z *⇩R snd z) ` ({0..} × S)"
using ‹T ⊆ S› by force
qed auto
qed
lemma closed_conic_hull:
fixes S :: "'a::euclidean_space set"
assumes "0 ∈ rel_interior S ∨ compact S ∧ 0 ∉ S"
shows "closed(conic hull S)"
using assms
proof
assume "0 ∈ rel_interior S"
then show "closed (conic hull S)"
by (simp add: conic_hull_eq_span)
next
assume "compact S ∧ 0 ∉ S"
then have "closedin (top_of_set UNIV) (conic hull S)"
using closedin_conic_hull by force
then show "closed (conic hull S)"
by simp
qed
lemma conic_closure:
fixes S :: "'a::euclidean_space set"
shows "conic S ⟹ conic(closure S)"
by (meson Convex.cone_def cone_closure conic_def)
lemma closure_conic_hull:
fixes S :: "'a::euclidean_space set"
assumes "0 ∈ rel_interior S ∨ bounded S ∧ ~(0 ∈ closure S)"
shows "closure(conic hull S) = conic hull (closure S)"
using assms
proof
assume "0 ∈ rel_interior S"
then show "closure (conic hull S) = conic hull closure S"
by (metis closed_affine_hull closure_closed closure_same_affine_hull closure_subset conic_hull_eq_affine_hull subsetD subset_rel_interior)
next
have "⋀x. x ∈ conic hull closure S ⟹ x ∈ closure (conic hull S)"
by (metis (no_types, opaque_lifting) closure_mono conic_closure conic_conic_hull subset_eq subset_hull)
moreover
assume "bounded S ∧ 0 ∉ closure S"
then have "⋀x. x ∈ closure (conic hull S) ⟹ x ∈ conic hull closure S"
by (metis closed_conic_hull closure_Un_frontier closure_closed closure_mono compact_closure hull_Un_subset le_sup_iff subsetD)
ultimately show "closure (conic hull S) = conic hull closure S"
by blast
qed
lemma compact_continuous_image_eq:
fixes f :: "'a::heine_borel ⇒ 'b::heine_borel"
assumes f: "inj_on f S"
shows "continuous_on S f ⟷ (∀T. compact T ∧ T ⊆ S ⟶ compact(f ` T))"
(is "?lhs = ?rhs")
proof
assume ?lhs then show ?rhs
by (metis continuous_on_subset compact_continuous_image)
next
assume RHS: ?rhs
obtain g where gf: "⋀x. x ∈ S ⟹ g (f x) = x"
by (metis inv_into_f_f f)
then have *: "(S ∩ f -` U) = g ` U" if "U ⊆ f ` S" for U
using that by fastforce
have gfim: "g ` f ` S ⊆ S" using gf by auto
have **: "compact (f ` S ∩ g -` C)" if C: "C ⊆ S" "compact C" for C
proof -
obtain h where "h C ∈ C ∧ h C ∉ S ∨ compact (f ` C)"
by (force simp: C RHS)
moreover have "f ` C = (f ` S ∩ g -` C)"
using C gf by auto
ultimately show ?thesis
using C by auto
qed
show ?lhs
using proper_map [OF _ _ gfim] **
by (simp add: continuous_on_closed * closedin_imp_subset)
qed
subsection‹Trivial fact: convexity equals connectedness for collinear sets›
lemma convex_connected_collinear:
fixes S :: "'a::euclidean_space set"
assumes "collinear S"
shows "convex S ⟷ connected S"
proof
assume "convex S"
then show "connected S"
using convex_connected by blast
next
assume S: "connected S"
show "convex S"
proof (cases "S = {}")
case True
then show ?thesis by simp
next
case False
then obtain a where "a ∈ S" by auto
have "collinear (affine hull S)"
by (simp add: assms collinear_affine_hull_collinear)
then obtain z where "z ≠ 0" "⋀x. x ∈ affine hull S ⟹ ∃c. x - a = c *⇩R z"
by (meson ‹a ∈ S› collinear hull_inc)
then obtain f where f: "⋀x. x ∈ affine hull S ⟹ x - a = f x *⇩R z"
by metis
then have inj_f: "inj_on f (affine hull S)"
by (metis diff_add_cancel inj_onI)
have diff: "x - y = (f x - f y) *⇩R z" if x: "x ∈ affine hull S" and y: "y ∈ affine hull S" for x y
proof -
have "f x *⇩R z = x - a"
by (simp add: f hull_inc x)
moreover have "f y *⇩R z = y - a"
by (simp add: f hull_inc y)
ultimately show ?thesis
by (simp add: scaleR_left.diff)
qed
have cont_f: "continuous_on (affine hull S) f"
proof (clarsimp simp: dist_norm continuous_on_iff diff)
show "⋀x e. 0 < e ⟹ ∃d>0. ∀y ∈ affine hull S. ¦f y - f x¦ * norm z < d ⟶ ¦f y - f x¦ < e"
by (metis ‹z ≠ 0› mult_pos_pos mult_less_cancel_right_pos zero_less_norm_iff)
qed
then have conn_fS: "connected (f ` S)"
by (meson S connected_continuous_image continuous_on_subset hull_subset)
show ?thesis
proof (clarsimp simp: convex_contains_segment)
fix x y z
assume "x ∈ S" "y ∈ S" "z ∈ closed_segment x y"
have False if "z ∉ S"
proof -
have "f ` (closed_segment x y) = closed_segment (f x) (f y)"
proof (rule continuous_injective_image_segment_1)
show "continuous_on (closed_segment x y) f"
by (meson ‹x ∈ S› ‹y ∈ S› convex_affine_hull convex_contains_segment hull_inc continuous_on_subset [OF cont_f])
show "inj_on f (closed_segment x y)"
by (meson ‹x ∈ S› ‹y ∈ S› convex_affine_hull convex_contains_segment hull_inc inj_on_subset [OF inj_f])
qed
then have fz: "f z ∈ closed_segment (f x) (f y)"
using ‹z ∈ closed_segment x y› by blast
have "z ∈ affine hull S"
by (meson ‹x ∈ S› ‹y ∈ S› ‹z ∈ closed_segment x y› convex_affine_hull convex_contains_segment hull_inc subset_eq)
then have fz_notin: "f z ∉ f ` S"
using hull_subset inj_f inj_onD that by fastforce
moreover have "{..<f z} ∩ f ` S ≠ {}" "{f z<..} ∩ f ` S ≠ {}"
proof -
consider "f x ≤ f z ∧ f z ≤ f y" | "f y ≤ f z ∧ f z ≤ f x"
using fz
by (auto simp add: closed_segment_eq_real_ivl split: if_split_asm)
then have "{..<f z} ∩ f ` {x,y} ≠ {} ∧ {f z<..} ∩ f ` {x,y} ≠ {}"
by cases (use fz_notin ‹x ∈ S› ‹y ∈ S› in ‹auto simp: image_iff›)
then show "{..<f z} ∩ f ` S ≠ {}" "{f z<..} ∩ f ` S ≠ {}"
using ‹x ∈ S› ‹y ∈ S› by blast+
qed
ultimately show False
using connectedD [OF conn_fS, of "{..<f z}" "{f z<..}"] by force
qed
then show "z ∈ S" by meson
qed
qed
qed
lemma compact_convex_collinear_segment_alt:
fixes S :: "'a::euclidean_space set"
assumes "S ≠ {}" "compact S" "connected S" "collinear S"
obtains a b where "S = closed_segment a b"
proof -
obtain ξ where "ξ ∈ S" using ‹S ≠ {}› by auto
have "collinear (affine hull S)"
by (simp add: assms collinear_affine_hull_collinear)
then obtain z where "z ≠ 0" "⋀x. x ∈ affine hull S ⟹ ∃c. x - ξ = c *⇩R z"
by (meson ‹ξ ∈ S› collinear hull_inc)
then obtain f where f: "⋀x. x ∈ affine hull S ⟹ x - ξ = f x *⇩R z"
by metis
let ?g = "λr. r *⇩R z + ξ"
have gf: "?g (f x) = x" if "x ∈ affine hull S" for x
by (metis diff_add_cancel f that)
then have inj_f: "inj_on f (affine hull S)"
by (metis inj_onI)
have diff: "x - y = (f x - f y) *⇩R z" if x: "x ∈ affine hull S" and y: "y ∈ affine hull S" for x y
proof -
have "f x *⇩R z = x - ξ"
by (simp add: f hull_inc x)
moreover have "f y *⇩R z = y - ξ"
by (simp add: f hull_inc y)
ultimately show ?thesis
by (simp add: scaleR_left.diff)
qed
have cont_f: "continuous_on (affine hull S) f"
proof (clarsimp simp: dist_norm continuous_on_iff diff)
show "⋀x e. 0 < e ⟹ ∃d>0. ∀y ∈ affine hull S. ¦f y - f x¦ * norm z < d ⟶ ¦f y - f x¦ < e"
by (metis ‹z ≠ 0› mult_pos_pos mult_less_cancel_right_pos zero_less_norm_iff)
qed
then have "connected (f ` S)"
by (meson ‹connected S› connected_continuous_image continuous_on_subset hull_subset)
moreover have "compact (f ` S)"
by (meson ‹compact S› compact_continuous_image_eq cont_f hull_subset inj_f)
ultimately obtain x y where "f ` S = {x..y}"
by (meson connected_compact_interval_1)
then have fS_eq: "f ` S = closed_segment x y"
using ‹S ≠ {}› closed_segment_eq_real_ivl by auto
obtain a b where "a ∈ S" "f a = x" "b ∈ S" "f b = y"
by (metis (full_types) ends_in_segment fS_eq imageE)
have "f ` (closed_segment a b) = closed_segment (f a) (f b)"
proof (rule continuous_injective_image_segment_1)
show "continuous_on (closed_segment a b) f"
by (meson ‹a ∈ S› ‹b ∈ S› convex_affine_hull convex_contains_segment hull_inc continuous_on_subset [OF cont_f])
show "inj_on f (closed_segment a b)"
by (meson ‹a ∈ S› ‹b ∈ S› convex_affine_hull convex_contains_segment hull_inc inj_on_subset [OF inj_f])
qed
then have "f ` (closed_segment a b) = f ` S"
by (simp add: ‹f a = x› ‹f b = y› fS_eq)
then have "?g ` f ` (closed_segment a b) = ?g ` f ` S"
by simp
moreover have "(λx. f x *⇩R z + ξ) ` closed_segment a b = closed_segment a b"
unfolding image_def using ‹a ∈ S› ‹b ∈ S›
by (safe; metis (mono_tags, lifting) convex_affine_hull convex_contains_segment gf hull_subset subsetCE)
ultimately have "closed_segment a b = S"
using gf by (simp add: image_comp o_def hull_inc cong: image_cong)
then show ?thesis
using that by blast
qed
lemma compact_convex_collinear_segment:
fixes S :: "'a::euclidean_space set"
assumes "S ≠ {}" "compact S" "convex S" "collinear S"
obtains a b where "S = closed_segment a b"
using assms convex_connected_collinear compact_convex_collinear_segment_alt by blast
lemma proper_map_from_compact:
fixes f :: "'a::euclidean_space ⇒ 'b::euclidean_space"
assumes contf: "continuous_on S f" and imf: "f ∈ S → T" and "compact S"
"closedin (top_of_set T) K"
shows "compact (S ∩ f -` K)"
by (rule closedin_compact [OF ‹compact S›] continuous_closedin_preimage_gen assms)+
lemma proper_map_fst:
assumes "compact T" "K ⊆ S" "compact K"
shows "compact (S × T ∩ fst -` K)"
proof -
have "(S × T ∩ fst -` K) = K × T"
using assms by auto
then show ?thesis by (simp add: assms compact_Times)
qed
lemma closed_map_fst:
fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
assumes "compact T" "closedin (top_of_set (S × T)) c"
shows "closedin (top_of_set S) (fst ` c)"
proof -
have *: "fst ` (S × T) ⊆ S"
by auto
show ?thesis
using proper_map [OF _ _ *] by (simp add: proper_map_fst assms)
qed
lemma proper_map_snd:
assumes "compact S" "K ⊆ T" "compact K"
shows "compact (S × T ∩ snd -` K)"
proof -
have "(S × T ∩ snd -` K) = S × K"
using assms by auto
then show ?thesis by (simp add: assms compact_Times)
qed
lemma closed_map_snd:
fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
assumes "compact S" "closedin (top_of_set (S × T)) c"
shows "closedin (top_of_set T) (snd ` c)"
proof -
have *: "snd ` (S × T) ⊆ T"
by auto
show ?thesis
using proper_map [OF _ _ *] by (simp add: proper_map_snd assms)
qed
lemma closedin_compact_projection:
fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
assumes "compact S" and clo: "closedin (top_of_set (S × T)) U"
shows "closedin (top_of_set T) {y. ∃x. x ∈ S ∧ (x, y) ∈ U}"
proof -
have "U ⊆ S × T"
by (metis clo closedin_imp_subset)
then have "{y. ∃x. x ∈ S ∧ (x, y) ∈ U} = snd ` U"
by force
moreover have "closedin (top_of_set T) (snd ` U)"
by (rule closed_map_snd [OF assms])
ultimately show ?thesis
by simp
qed
lemma closed_compact_projection:
fixes S :: "'a::euclidean_space set"
and T :: "('a * 'b::euclidean_space) set"
assumes "compact S" and clo: "closed T"
shows "closed {y. ∃x. x ∈ S ∧ (x, y) ∈ T}"
proof -
have *: "{y. ∃x. x ∈ S ∧ Pair x y ∈ T} = {y. ∃x. x ∈ S ∧ Pair x y ∈ ((S × UNIV) ∩ T)}"
by auto
show ?thesis
unfolding *
by (intro clo closedin_closed_Int closedin_closed_trans [OF _ closed_UNIV] closedin_compact_projection [OF ‹compact S›])
qed
subsubsection‹Representing affine hull as a finite intersection of hyperplanes›
proposition affine_hull_convex_Int_nonempty_interior:
fixes S :: "'a::real_normed_vector set"
assumes "convex S" "S ∩ interior T ≠ {}"
shows "affine hull (S ∩ T) = affine hull S"
proof
show "affine hull (S ∩ T) ⊆ affine hull S"
by (simp add: hull_mono)
next
obtain a where "a ∈ S" "a ∈ T" and at: "a ∈ interior T"
using assms interior_subset by blast
then obtain e where "e > 0" and e: "cball a e ⊆ T"
using mem_interior_cball by blast
have *: "x ∈ (+) a ` span ((λx. x - a) ` (S ∩ T))" if "x ∈ S" for x
proof (cases "x = a")
case True with that span_0 eq_add_iff image_def mem_Collect_eq show ?thesis
by blast
next
case False
define k where "k = min (1/2) (e / norm (x-a))"
have k: "0 < k" "k < 1"
using ‹e > 0› False by (auto simp: k_def)
then have xa: "(x-a) = inverse k *⇩R k *⇩R (x-a)"
by simp
have "e / norm (x - a) ≥ k"
using k_def by linarith
then have "a + k *⇩R (x - a) ∈ cball a e"
using ‹0 < k› False
by (simp add: dist_norm) (simp add: field_simps)
then have T: "a + k *⇩R (x - a) ∈ T"
using e by blast
have S: "a + k *⇩R (x - a) ∈ S"
using k ‹a ∈ S› convexD [OF ‹convex S› ‹a ∈ S› ‹x ∈ S›, of "1-k" k]
by (simp add: algebra_simps)
have "inverse k *⇩R k *⇩R (x-a) ∈ span ((λx. x - a) ` (S ∩ T))"
by (intro span_mul [OF span_base] image_eqI [where x = "a + k *⇩R (x - a)"]) (auto simp: S T)
with xa image_iff show ?thesis by fastforce
qed
have "S ⊆ affine hull (S ∩ T)"
by (force simp: * ‹a ∈ S› ‹a ∈ T› hull_inc affine_hull_span_gen [of a])
then show "affine hull S ⊆ affine hull (S ∩ T)"
by (simp add: subset_hull)
qed
corollary affine_hull_convex_Int_open:
fixes S :: "'a::real_normed_vector set"
assumes "convex S" "open T" "S ∩ T ≠ {}"
shows "affine hull (S ∩ T) = affine hull S"
using affine_hull_convex_Int_nonempty_interior assms interior_eq by blast
corollary affine_hull_affine_Int_nonempty_interior:
fixes S :: "'a::real_normed_vector set"
assumes "affine S" "S ∩ interior T ≠ {}"
shows "affine hull (S ∩ T) = affine hull S"
by (simp add: affine_hull_convex_Int_nonempty_interior affine_imp_convex assms)
corollary affine_hull_affine_Int_open:
fixes S :: "'a::real_normed_vector set"
assumes "affine S" "open T" "S ∩ T ≠ {}"
shows "affine hull (S ∩ T) = affine hull S"
by (simp add: affine_hull_convex_Int_open affine_imp_convex assms)
corollary affine_hull_convex_Int_openin:
fixes S :: "'a::real_normed_vector set"
assumes "convex S" "openin (top_of_set (affine hull S)) T" "S ∩ T ≠ {}"
shows "affine hull (S ∩ T) = affine hull S"
using assms unfolding openin_open
by (metis affine_hull_convex_Int_open hull_subset inf.orderE inf_assoc)
corollary affine_hull_openin:
fixes S :: "'a::real_normed_vector set"
assumes "openin (top_of_set (affine hull T)) S" "S ≠ {}"
shows "affine hull S = affine hull T"
using assms unfolding openin_open
by (metis affine_affine_hull affine_hull_affine_Int_open hull_hull)
corollary affine_hull_open:
fixes S :: "'a::real_normed_vector set"
assumes "open S" "S ≠ {}"
shows "affine hull S = UNIV"
by (metis affine_hull_convex_Int_nonempty_interior assms convex_UNIV hull_UNIV inf_top.left_neutral interior_open)
lemma aff_dim_convex_Int_nonempty_interior:
fixes S :: "'a::euclidean_space set"
shows "⟦convex S; S ∩ interior T ≠ {}⟧ ⟹ aff_dim(S ∩ T) = aff_dim S"
using aff_dim_affine_hull2 affine_hull_convex_Int_nonempty_interior by blast
lemma aff_dim_convex_Int_open:
fixes S :: "'a::euclidean_space set"
shows "⟦convex S; open T; S ∩ T ≠ {}⟧ ⟹ aff_dim(S ∩ T) = aff_dim S"
using aff_dim_convex_Int_nonempty_interior interior_eq by blast
lemma affine_hull_Diff:
fixes S:: "'a::real_normed_vector set"
assumes ope: "openin (top_of_set (affine hull S)) S" and "finite F" "F ⊂ S"
shows "affine hull (S - F) = affine hull S"
proof -
have clo: "closedin (top_of_set S) F"
using assms finite_imp_closedin by auto
moreover have "S - F ≠ {}"
using assms by auto
ultimately show ?thesis
by (metis ope closedin_def topspace_euclidean_subtopology affine_hull_openin openin_trans)
qed
lemma affine_hull_halfspace_lt:
fixes a :: "'a::euclidean_space"
shows "affine hull {x. a ∙ x < r} = (if a = 0 ∧ r ≤ 0 then {} else UNIV)"
using halfspace_eq_empty_lt [of a r]
by (simp add: open_halfspace_lt affine_hull_open)
lemma affine_hull_halfspace_le:
fixes a :: "'a::euclidean_space"
shows "affine hull {x. a ∙ x ≤ r} = (if a = 0 ∧ r < 0 then {} else UNIV)"
proof (cases "a = 0")
case True then show ?thesis by simp
next
case False
then have "affine hull closure {x. a ∙ x < r} = UNIV"
using affine_hull_halfspace_lt closure_same_affine_hull by fastforce
moreover have "{x. a ∙ x < r} ⊆ {x. a ∙ x ≤ r}"
by (simp add: Collect_mono)
ultimately show ?thesis using False antisym_conv hull_mono top_greatest
by (metis affine_hull_halfspace_lt)
qed
lemma affine_hull_halfspace_gt:
fixes a :: "'a::euclidean_space"
shows "affine hull {x. a ∙ x > r} = (if a = 0 ∧ r ≥ 0 then {} else UNIV)"
using halfspace_eq_empty_gt [of r a]
by (simp add: open_halfspace_gt affine_hull_open)
lemma affine_hull_halfspace_ge:
fixes a :: "'a::euclidean_space"
shows "affine hull {x. a ∙ x ≥ r} = (if a = 0 ∧ r > 0 then {} else UNIV)"
using affine_hull_halfspace_le [of "-a" "-r"] by simp
lemma aff_dim_halfspace_lt:
fixes a :: "'a::euclidean_space"
shows "aff_dim {x. a ∙ x < r} =
(if a = 0 ∧ r ≤ 0 then -1 else DIM('a))"
by simp (metis aff_dim_open halfspace_eq_empty_lt open_halfspace_lt)
lemma aff_dim_halfspace_le:
fixes a :: "'a::euclidean_space"
shows "aff_dim {x. a ∙ x ≤ r} =
(if a = 0 ∧ r < 0 then -1 else DIM('a))"
proof -
have "int (DIM('a)) = aff_dim (UNIV::'a set)"
by (simp)
then have "aff_dim (affine hull {x. a ∙ x ≤ r}) = DIM('a)" if "(a = 0 ⟶ r ≥ 0)"
using that by (simp add: affine_hull_halfspace_le not_less)
then show ?thesis
by (force)
qed
lemma aff_dim_halfspace_gt:
fixes a :: "'a::euclidean_space"
shows "aff_dim {x. a ∙ x > r} =
(if a = 0 ∧ r ≥ 0 then -1 else DIM('a))"
by simp (metis aff_dim_open halfspace_eq_empty_gt open_halfspace_gt)
lemma aff_dim_halfspace_ge:
fixes a :: "'a::euclidean_space"
shows "aff_dim {x. a ∙ x ≥ r} =
(if a = 0 ∧ r > 0 then -1 else DIM('a))"
using aff_dim_halfspace_le [of "-a" "-r"] by simp
proposition aff_dim_eq_hyperplane:
fixes S :: "'a::euclidean_space set"
shows "aff_dim S = DIM('a) - 1 ⟷ (∃a b. a ≠ 0 ∧ affine hull S = {x. a ∙ x = b})"
(is "?lhs = ?rhs")
proof (cases "S = {}")
case True then show ?thesis
by (auto simp: dest: hyperplane_eq_Ex)
next
case False
then obtain c where "c ∈ S" by blast
show ?thesis
proof (cases "c = 0")
case True
have "?lhs ⟷ (∃a. a ≠ 0 ∧ span ((λx. x - c) ` S) = {x. a ∙ x = 0})"
by (simp add: aff_dim_eq_dim [of c] ‹c ∈ S› hull_inc dim_eq_hyperplane del: One_nat_def)
also have "... ⟷ ?rhs"
using span_zero [of S] True ‹c ∈ S› affine_hull_span_0 hull_inc
by (fastforce simp add: affine_hull_span_gen [of c] ‹c = 0›)
finally show ?thesis .
next
case False
have xc_im: "x ∈ (+) c ` {y. a ∙ y = 0}" if "a ∙ x = a ∙ c" for a x
proof -
have "∃y. a ∙ y = 0 ∧ c + y = x"
by (metis that add.commute diff_add_cancel inner_commute inner_diff_left right_minus_eq)
then show "x ∈ (+) c ` {y. a ∙ y = 0}"
by blast
qed
have 2: "span ((λx. x - c) ` S) = {x. a ∙ x = 0}"
if "(+) c ` span ((λx. x - c) ` S) = {x. a ∙ x = b}" for a b
proof -
have "b = a ∙ c"
using span_0 that by fastforce
with that have "(+) c ` span ((λx. x - c) ` S) = {x. a ∙ x = a ∙ c}"
by simp
then have "span ((λx. x - c) ` S) = (λx. x - c) ` {x. a ∙ x = a ∙ c}"
by (metis (no_types) image_cong translation_galois uminus_add_conv_diff)
also have "... = {x. a ∙ x = 0}"
by (force simp: inner_distrib inner_diff_right
intro: image_eqI [where x="x+c" for x])
finally show ?thesis .
qed
have "?lhs = (∃a. a ≠ 0 ∧ span ((λx. x - c) ` S) = {x. a ∙ x = 0})"
by (simp add: aff_dim_eq_dim [of c] ‹c ∈ S› hull_inc dim_eq_hyperplane del: One_nat_def)
also have "... = ?rhs"
by (fastforce simp add: affine_hull_span_gen [of c] ‹c ∈ S› hull_inc inner_distrib intro: xc_im intro!: 2)
finally show ?thesis .
qed
qed
corollary aff_dim_hyperplane [simp]:
fixes a :: "'a::euclidean_space"
shows "a ≠ 0 ⟹ aff_dim {x. a ∙ x = r} = DIM('a) - 1"
by (metis aff_dim_eq_hyperplane affine_hull_eq affine_hyperplane)
subsection‹Some stepping theorems›
lemma aff_dim_insert:
fixes a :: "'a::euclidean_space"
shows "aff_dim (insert a S) = (if a ∈ affine hull S then aff_dim S else aff_dim S + 1)"
proof (cases "S = {}")
case True then show ?thesis
by simp
next
case False
then obtain x s' where S: "S = insert x s'" "x ∉ s'"
by (meson Set.set_insert all_not_in_conv)
show ?thesis using S
by (force simp add: affine_hull_insert_span_gen span_zero insert_commute [of a] aff_dim_eq_dim [of x] dim_insert)
qed
lemma affine_dependent_choose:
fixes a :: "'a :: euclidean_space"
assumes "¬(affine_dependent S)"
shows "affine_dependent(insert a S) ⟷ a ∉ S ∧ a ∈ affine hull S"
(is "?lhs = ?rhs")
proof safe
assume "affine_dependent (insert a S)" and "a ∈ S"
then show "False"
using ‹a ∈ S› assms insert_absorb by fastforce
next
assume lhs: "affine_dependent (insert a S)"
then have "a ∉ S"
by (metis (no_types) assms insert_absorb)
moreover have "finite S"
using affine_independent_iff_card assms by blast
moreover have "aff_dim (insert a S) ≠ int (card S)"
using ‹finite S› affine_independent_iff_card ‹a ∉ S› lhs by fastforce
ultimately show "a ∈ affine hull S"
by (metis aff_dim_affine_independent aff_dim_insert assms)
next
assume "a ∉ S" and "a ∈ affine hull S"
show "affine_dependent (insert a S)"
by (simp add: ‹a ∈ affine hull S› ‹a ∉ S› affine_dependent_def)
qed
lemma affine_independent_insert:
fixes a :: "'a :: euclidean_space"
shows "⟦¬ affine_dependent S; a ∉ affine hull S⟧ ⟹ ¬ affine_dependent(insert a S)"
by (simp add: affine_dependent_choose)
lemma subspace_bounded_eq_trivial:
fixes S :: "'a::real_normed_vector set"
assumes "subspace S"
shows "bounded S ⟷ S = {0}"
proof -
have "False" if "bounded S" "x ∈ S" "x ≠ 0" for x
proof -
obtain B where B: "⋀y. y ∈ S ⟹ norm y < B" "B > 0"
using ‹bounded S› by (force simp: bounded_pos_less)
have "(B / norm x) *⇩R x ∈ S"
using assms subspace_mul ‹x ∈ S› by auto
moreover have "norm ((B / norm x) *⇩R x) = B"
using that B by (simp add: algebra_simps)
ultimately show False using B by force
qed
then have "bounded S ⟹ S = {0}"
using assms subspace_0 by fastforce
then show ?thesis
by blast
qed
lemma affine_bounded_eq_trivial:
fixes S :: "'a::real_normed_vector set"
assumes "affine S"
shows "bounded S ⟷ S = {} ∨ (∃a. S = {a})"
proof (cases "S = {}")
case True then show ?thesis
by simp
next
case False
then obtain b where "b ∈ S" by blast
with False assms
have "bounded S ⟹ S = {b}"
using affine_diffs_subspace [OF assms ‹b ∈ S›]
by (metis (no_types, lifting) ab_group_add_class.ab_left_minus bounded_translation image_empty image_insert subspace_bounded_eq_trivial translation_invert)
then show ?thesis by auto
qed
lemma affine_bounded_eq_lowdim:
fixes S :: "'a::euclidean_space set"
assumes "affine S"
shows "bounded S ⟷ aff_dim S ≤ 0"
proof
show "aff_dim S ≤ 0 ⟹ bounded S"
by (metis aff_dim_sing aff_dim_subset affine_dim_equal affine_sing all_not_in_conv assms bounded_empty bounded_insert dual_order.antisym empty_subsetI insert_subset)
qed (use affine_bounded_eq_trivial assms in fastforce)
lemma bounded_hyperplane_eq_trivial_0:
fixes a :: "'a::euclidean_space"
assumes "a ≠ 0"
shows "bounded {x. a ∙ x = 0} ⟷ DIM('a) = 1"
proof
assume "bounded {x. a ∙ x = 0}"
then have 0: "aff_dim {x. a ∙ x = 0} ≤ 0"
by (simp add: affine_bounded_eq_lowdim affine_hyperplane)
with assms 0
have "int DIM('a) = 1"
using order_neq_le_trans by fastforce
then show "DIM('a) = 1" by auto
next
assume "DIM('a) = 1"
then show "bounded {x. a ∙ x = 0}"
by (simp add: affine_bounded_eq_lowdim affine_hyperplane assms)
qed
lemma bounded_hyperplane_eq_trivial:
fixes a :: "'a::euclidean_space"
shows "bounded {x. a ∙ x = r} ⟷ (if a = 0 then r ≠ 0 else DIM('a) = 1)"
proof -
{ assume "r ≠ 0" "a ≠ 0"
have "aff_dim {x. y ∙ x = 0} = aff_dim {x. a ∙ x = r}" if "y ≠ 0" for y::'a
by (metis that ‹a ≠ 0› aff_dim_hyperplane)
then have "bounded {x. a ∙ x = r} = (DIM('a) = Suc 0)"
by (metis One_nat_def ‹a ≠ 0› affine_bounded_eq_lowdim affine_hyperplane bounded_hyperplane_eq_trivial_0)
}
then show ?thesis
by (auto simp: bounded_hyperplane_eq_trivial_0)
qed
subsection‹General case without assuming closure and getting non-strict separation›
proposition separating_hyperplane_closed_point_inset:
fixes S :: "'a::euclidean_space set"
assumes "convex S" "closed S" "S ≠ {}" "z ∉ S"
obtains a b where "a ∈ S" "(a - z) ∙ z < b" "⋀x. x ∈ S ⟹ b < (a - z) ∙ x"
proof -
obtain y where "y ∈ S" and y: "⋀u. u ∈ S ⟹ dist z y ≤ dist z u"
using distance_attains_inf [of S z] assms by auto
then have *: "(y - z) ∙ z < (y - z) ∙ z + (norm (y - z))⇧2 / 2"
using ‹y ∈ S› ‹z ∉ S› by auto
show ?thesis
proof (rule that [OF ‹y ∈ S› *])
fix x
assume "x ∈ S"
have yz: "0 < (y - z) ∙ (y - z)"
using ‹y ∈ S› ‹z ∉ S› by auto
{ assume 0: "0 < ((z - y) ∙ (x - y))"
with any_closest_point_dot [OF ‹convex S› ‹closed S›]
have False
using y ‹x ∈ S› ‹y ∈ S› not_less by blast
}
then have "0 ≤ ((y - z) ∙ (x - y))"
by (force simp: not_less inner_diff_left)
with yz have "0 < 2 * ((y - z) ∙ (x - y)) + (y - z) ∙ (y - z)"
by (simp add: algebra_simps)
then show "(y - z) ∙ z + (norm (y - z))⇧2 / 2 < (y - z) ∙ x"
by (simp add: field_simps inner_diff_left inner_diff_right dot_square_norm [symmetric])
qed
qed
lemma separating_hyperplane_closed_0_inset:
fixes S :: "'a::euclidean_space set"
assumes "convex S" "closed S" "S ≠ {}" "0 ∉ S"
obtains a b where "a ∈ S" "a ≠ 0" "0 < b" "⋀x. x ∈ S ⟹ a ∙ x > b"
using separating_hyperplane_closed_point_inset [OF assms] by simp (metis ‹0 ∉ S›)
proposition separating_hyperplane_set_0_inspan:
fixes S :: "'a::euclidean_space set"
assumes "convex S" "S ≠ {}" "0 ∉ S"
obtains a where "a ∈ span S" "a ≠ 0" "⋀x. x ∈ S ⟹ 0 ≤ a ∙ x"
proof -
define k where [abs_def]: "k c = {x. 0 ≤ c ∙ x}" for c :: 'a
have "span S ∩ frontier (cball 0 1) ∩ ⋂f' ≠ {}"
if f': "finite f'" "f' ⊆ k ` S" for f'
proof -
obtain C where "C ⊆ S" "finite C" and C: "f' = k ` C"
using finite_subset_image [OF f'] by blast
obtain a where "a ∈ S" "a ≠ 0"
using ‹S ≠ {}› ‹0 ∉ S› ex_in_conv by blast
then have "norm (a /⇩R (norm a)) = 1"
by simp
moreover have "a /⇩R (norm a) ∈ span S"
by (simp add: ‹a ∈ S› span_scale span_base)
ultimately have ass: "a /⇩R (norm a) ∈ span S ∩ sphere 0 1"
by simp
show ?thesis
proof (cases "C = {}")
case True with C ass show ?thesis
by auto
next
case False
have "closed (convex hull C)"
using ‹finite C› compact_eq_bounded_closed finite_imp_compact_convex_hull by auto
moreover have "convex hull C ≠ {}"
by (simp add: False)
moreover have "0 ∉ convex hull C"
by (metis ‹C ⊆ S› ‹convex S› ‹0 ∉ S› convex_hull_subset hull_same insert_absorb insert_subset)
ultimately obtain a b
where "a ∈ convex hull C" "a ≠ 0" "0 < b"
and ab: "⋀x. x ∈ convex hull C ⟹ a ∙ x > b"
using separating_hyperplane_closed_0_inset by blast
then have "a ∈ S"
by (metis ‹C ⊆ S› assms(1) subsetCE subset_hull)
moreover have "norm (a /⇩R (norm a)) = 1"
using ‹a ≠ 0› by simp
moreover have "a /⇩R (norm a) ∈ span S"
by (simp add: ‹a ∈ S› span_scale span_base)
ultimately have ass: "a /⇩R (norm a) ∈ span S ∩ sphere 0 1"
by simp
have "⋀x. ⟦a ≠ 0; x ∈ C⟧ ⟹ 0 ≤ x ∙ a"
using ab ‹0 < b› by (metis hull_inc inner_commute less_eq_real_def less_trans)
then have aa: "a /⇩R (norm a) ∈ (⋂c∈C. {x. 0 ≤ c ∙ x})"
by (auto simp add: field_split_simps)
show ?thesis
unfolding C k_def
using ass aa Int_iff empty_iff by force
qed
qed
moreover have "⋀T. T ∈ k ` S ⟹ closed T"
using closed_halfspace_ge k_def by blast
ultimately have "(span S ∩ frontier(cball 0 1)) ∩ (⋂ (k ` S)) ≠ {}"
by (metis compact_imp_fip closed_Int_compact closed_span compact_cball compact_frontier)
then show ?thesis
unfolding set_eq_iff k_def
by simp (metis inner_commute norm_eq_zero that zero_neq_one)
qed
lemma separating_hyperplane_set_point_inaff:
fixes S :: "'a::euclidean_space set"
assumes "convex S" "S ≠ {}" and zno: "z ∉ S"
obtains a b where "(z + a) ∈ affine hull (insert z S)"
and "a ≠ 0" and "a ∙ z ≤ b"
and "⋀x. x ∈ S ⟹ a ∙ x ≥ b"
proof -
from separating_hyperplane_set_0_inspan [of "image (λx. -z + x) S"]
have "convex ((+) (- z) ` S)"
using ‹convex S› by simp
moreover have "(+) (- z) ` S ≠ {}"
by (simp add: ‹S ≠ {}›)
moreover have "0 ∉ (+) (- z) ` S"
using zno by auto
ultimately obtain a where "a ∈ span ((+) (- z) ` S)" "a ≠ 0"
and a: "⋀x. x ∈ ((+) (- z) ` S) ⟹ 0 ≤ a ∙ x"
using separating_hyperplane_set_0_inspan [of "image (λx. -z + x) S"]
by blast
then have szx: "⋀x. x ∈ S ⟹ a ∙ z ≤ a ∙ x"
by (metis (no_types, lifting) imageI inner_minus_right inner_right_distrib minus_add neg_le_0_iff_le neg_le_iff_le real_add_le_0_iff)
moreover
have "z + a ∈ affine hull insert z S"
using ‹a ∈ span ((+) (- z) ` S)› affine_hull_insert_span_gen by blast
ultimately show ?thesis
using ‹a ≠ 0› szx that by auto
qed
proposition supporting_hyperplane_rel_boundary:
fixes S :: "'a::euclidean_space set"
assumes "convex S" "x ∈ S" and xno: "x ∉ rel_interior S"
obtains a where "a ≠ 0"
and "⋀y. y ∈ S ⟹ a ∙ x ≤ a ∙ y"
and "⋀y. y ∈ rel_interior S ⟹ a ∙ x < a ∙ y"
proof -
obtain a b where aff: "(x + a) ∈ affine hull (insert x (rel_interior S))"
and "a ≠ 0" and "a ∙ x ≤ b"
and ageb: "⋀u. u ∈ (rel_interior S) ⟹ a ∙ u ≥ b"
using separating_hyperplane_set_point_inaff [of "rel_interior S" x] assms
by (auto simp: rel_interior_eq_empty convex_rel_interior)
have le_ay: "a ∙ x ≤ a ∙ y" if "y ∈ S" for y
proof -
have con: "continuous_on (closure (rel_interior S)) ((∙) a)"
by (rule continuous_intros continuous_on_subset | blast)+
have y: "y ∈ closure (rel_interior S)"
using ‹convex S› closure_def convex_closure_rel_interior ‹y ∈ S›
by fastforce
show ?thesis
using continuous_ge_on_closure [OF con y] ageb ‹a ∙ x ≤ b›
by fastforce
qed
have 3: "a ∙ x < a ∙ y" if "y ∈ rel_interior S" for y
proof -
obtain e where "0 < e" "y ∈ S" and e: "cball y e ∩ affine hull S ⊆ S"
using ‹y ∈ rel_interior S› by (force simp: rel_interior_cball)
define y' where "y' = y - (e / norm a) *⇩R ((x + a) - x)"
have "y' ∈ cball y e"
unfolding y'_def using ‹0 < e› by force
moreover have "y' ∈ affine hull S"
unfolding y'_def
by (metis ‹x ∈ S› ‹y ∈ S› ‹convex S› aff affine_affine_hull hull_redundant
rel_interior_same_affine_hull hull_inc mem_affine_3_minus2)
ultimately have "y' ∈ S"
using e by auto
have "a ∙ x ≤ a ∙ y"
using le_ay ‹a ≠ 0› ‹y ∈ S› by blast
moreover have "a ∙ x ≠ a ∙ y"
using le_ay [OF ‹y' ∈ S›] ‹a ≠ 0› ‹0 < e› not_le
by (fastforce simp add: y'_def inner_diff dot_square_norm power2_eq_square)
ultimately show ?thesis by force
qed
show ?thesis
by (rule that [OF ‹a ≠ 0› le_ay 3])
qed
lemma supporting_hyperplane_relative_frontier:
fixes S :: "'a::euclidean_space set"
assumes "convex S" "x ∈ closure S" "x ∉ rel_interior S"
obtains a where "a ≠ 0"
and "⋀y. y ∈ closure S ⟹ a ∙ x ≤ a ∙ y"
and "⋀y. y ∈ rel_interior S ⟹ a ∙ x < a ∙ y"
using supporting_hyperplane_rel_boundary [of "closure S" x]
by (metis assms convex_closure convex_rel_interior_closure)
subsection‹ Some results on decomposing convex hulls: intersections, simplicial subdivision›
lemma
fixes S :: "'a::euclidean_space set"
assumes "¬ affine_dependent(S ∪ T)"
shows convex_hull_Int_subset: "convex hull S ∩ convex hull T ⊆ convex hull (S ∩ T)" (is ?C)
and affine_hull_Int_subset: "affine hull S ∩ affine hull T ⊆ affine hull (S ∩ T)" (is ?A)
proof -
have [simp]: "finite S" "finite T"
using aff_independent_finite assms by blast+
have "sum u (S ∩ T) = 1 ∧
(∑v∈S ∩ T. u v *⇩R v) = (∑v∈S. u v *⇩R v)"
if [simp]: "sum u S = 1"
"sum v T = 1"
and eq: "(∑x∈T. v x *⇩R x) = (∑x∈S. u x *⇩R x)" for u v
proof -
define f where "f x = (if x ∈ S then u x else 0) - (if x ∈ T then v x else 0)" for x
have "sum f (S ∪ T) = 0"
by (simp add: f_def sum_Un sum_subtractf flip: sum.inter_restrict)
moreover have "(∑x∈(S ∪ T). f x *⇩R x) = 0"
by (simp add: eq f_def sum_Un scaleR_left_diff_distrib sum_subtractf if_smult flip: sum.inter_restrict cong: if_cong)
ultimately have "⋀v. v ∈ S ∪ T ⟹ f v = 0"
using aff_independent_finite assms unfolding affine_dependent_explicit
by blast
then have u [simp]: "⋀x. x ∈ S ⟹ u x = (if x ∈ T then v x else 0)"
by (simp add: f_def) presburger
have "sum u (S ∩ T) = sum u S"
by (simp add: sum.inter_restrict)
then have "sum u (S ∩ T) = 1"
using that by linarith
moreover have "(∑v∈S ∩ T. u v *⇩R v) = (∑v∈S. u v *⇩R v)"
by (auto simp: if_smult sum.inter_restrict intro: sum.cong)
ultimately show ?thesis
by force
qed
then show ?A ?C
by (auto simp: convex_hull_finite affine_hull_finite)
qed
proposition affine_hull_Int:
fixes S :: "'a::euclidean_space set"
assumes "¬ affine_dependent(S ∪ T)"
shows "affine hull (S ∩ T) = affine hull S ∩ affine hull T"
by (simp add: affine_hull_Int_subset assms hull_mono subset_antisym)
proposition convex_hull_Int:
fixes S :: "'a::euclidean_space set"
assumes "¬ affine_dependent(S ∪ T)"
shows "convex hull (S ∩ T) = convex hull S ∩ convex hull T"
by (simp add: convex_hull_Int_subset assms hull_mono subset_antisym)
proposition
fixes S :: "'a::euclidean_space set set"
assumes "¬ affine_dependent (⋃S)"
shows affine_hull_Inter: "affine hull (⋂S) = (⋂T∈S. affine hull T)" (is "?A")
and convex_hull_Inter: "convex hull (⋂S) = (⋂T∈S. convex hull T)" (is "?C")
proof -
have "finite S"
using aff_independent_finite assms finite_UnionD by blast
then have "?A ∧ ?C" using assms
proof (induction S rule: finite_induct)
case empty then show ?case by auto
next
case (insert T F)
then show ?case
proof (cases "F={}")
case True then show ?thesis by simp
next
case False
with "insert.prems" have [simp]: "¬ affine_dependent (T ∪ ⋂F)"
by (auto intro: affine_dependent_subset)
have [simp]: "¬ affine_dependent (⋃F)"
using affine_independent_subset insert.prems by fastforce
show ?thesis
by (simp add: affine_hull_Int convex_hull_Int insert.IH)
qed
qed
then show "?A" "?C"
by auto
qed
proposition in_convex_hull_exchange_unique:
fixes S :: "'a::euclidean_space set"
assumes naff: "¬ affine_dependent S" and a: "a ∈ convex hull S"
and S: "T ⊆ S" "T' ⊆ S"
and x: "x ∈ convex hull (insert a T)"
and x': "x ∈ convex hull (insert a T')"
shows "x ∈ convex hull (insert a (T ∩ T'))"
proof (cases "a ∈ S")
case True
then have "¬ affine_dependent (insert a T ∪ insert a T')"
using affine_dependent_subset assms by auto
then have "x ∈ convex hull (insert a T ∩ insert a T')"
by (metis IntI convex_hull_Int x x')
then show ?thesis
by simp
next
case False
then have anot: "a ∉ T" "a ∉ T'"
using assms by auto
have [simp]: "finite S"
by (simp add: aff_independent_finite assms)
then obtain b where b0: "⋀s. s ∈ S ⟹ 0 ≤ b s"
and b1: "sum b S = 1" and aeq: "a = (∑s∈S. b s *⇩R s)"
using a by (auto simp: convex_hull_finite)
have fin [simp]: "finite T" "finite T'"
using assms infinite_super ‹finite S› by blast+
then obtain c c' where c0: "⋀t. t ∈ insert a T ⟹ 0 ≤ c t"
and c1: "sum c (insert a T) = 1"
and xeq: "x = (∑t ∈ insert a T. c t *⇩R t)"
and c'0: "⋀t. t ∈ insert a T' ⟹ 0 ≤ c' t"
and c'1: "sum c' (insert a T') = 1"
and x'eq: "x = (∑t ∈ insert a T'. c' t *⇩R t)"
using x x' by (auto simp: convex_hull_finite)
with fin anot
have sumTT': "sum c T = 1 - c a" "sum c' T' = 1 - c' a"
and wsumT: "(∑t ∈ T. c t *⇩R t) = x - c a *⇩R a"
by simp_all
have wsumT': "(∑t ∈ T'. c' t *⇩R t) = x - c' a *⇩R a"
using x'eq fin anot by simp
define cc where "cc ≡ λx. if x ∈ T then c x else 0"
define cc' where "cc' ≡ λx. if x ∈ T' then c' x else 0"
define dd where "dd ≡ λx. cc x - cc' x + (c a - c' a) * b x"
have sumSS': "sum cc S = 1 - c a" "sum cc' S = 1 - c' a"
unfolding cc_def cc'_def using S
by (simp_all add: Int_absorb1 Int_absorb2 sum_subtractf sum.inter_restrict [symmetric] sumTT')
have wsumSS: "(∑t ∈ S. cc t *⇩R t) = x - c a *⇩R a" "(∑t ∈ S. cc' t *⇩R t) = x - c' a *⇩R a"
unfolding cc_def cc'_def using S
by (simp_all add: Int_absorb1 Int_absorb2 if_smult sum.inter_restrict [symmetric] wsumT wsumT' cong: if_cong)
have sum_dd0: "sum dd S = 0"
unfolding dd_def using S
by (simp add: sumSS' comm_monoid_add_class.sum.distrib sum_subtractf
algebra_simps sum_distrib_right [symmetric] b1)
have "(∑v∈S. (b v * x) *⇩R v) = x *⇩R (∑v∈S. b v *⇩R v)" for x
by (simp add: pth_5 real_vector.scale_sum_right mult.commute)
then have *: "(∑v∈S. (b v * x) *⇩R v) = x *⇩R a" for x
using aeq by blast
have "(∑v ∈ S. dd v *⇩R v) = 0"
unfolding dd_def using S
by (simp add: * wsumSS sum.distrib sum_subtractf algebra_simps)
then have dd0: "dd v = 0" if "v ∈ S" for v
using naff [unfolded affine_dependent_explicit not_ex, rule_format, of S dd]
using that sum_dd0 by force
consider "c' a ≤ c a" | "c a ≤ c' a" by linarith
then show ?thesis
proof cases
case 1
then have "sum cc S ≤ sum cc' S"
by (simp add: sumSS')
then have le: "cc x ≤ cc' x" if "x ∈ S" for x
using dd0 [OF that] 1 b0 mult_left_mono that
by (fastforce simp add: dd_def algebra_simps)
have cc0: "cc x = 0" if "x ∈ S" "x ∉ T ∩ T'" for x
using le [OF ‹x ∈ S›] that c0
by (force simp: cc_def cc'_def split: if_split_asm)
have ge0: "∀x∈T ∩ T'. 0 ≤ (cc(a := c a)) x"
by (simp add: c0 cc_def)
have "sum (cc(a := c a)) (insert a (T ∩ T')) = c a + sum (cc(a := c a)) (T ∩ T')"
by (simp add: anot)
also have "... = c a + sum (cc(a := c a)) S"
using ‹T ⊆ S› False cc0 cc_def ‹a ∉ S› by (fastforce intro!: sum.mono_neutral_left split: if_split_asm)
also have "... = c a + (1 - c a)"
by (metis ‹a ∉ S› fun_upd_other sum.cong sumSS'(1))
finally have 1: "sum (cc(a := c a)) (insert a (T ∩ T')) = 1"
by simp
have "(∑x∈insert a (T ∩ T'). (cc(a := c a)) x *⇩R x) = c a *⇩R a + (∑x ∈ T ∩ T'. (cc(a := c a)) x *⇩R x)"
by (simp add: anot)
also have "... = c a *⇩R a + (∑x ∈ S. (cc(a := c a)) x *⇩R x)"
using ‹T ⊆ S› False cc0 by (fastforce intro!: sum.mono_neutral_left split: if_split_asm)
also have "... = c a *⇩R a + x - c a *⇩R a"
by (simp add: wsumSS ‹a ∉ S› if_smult sum_delta_notmem)
finally have self: "(∑x∈insert a (T ∩ T'). (cc(a := c a)) x *⇩R x) = x"
by simp
show ?thesis
by (force simp: convex_hull_finite c0 intro!: ge0 1 self exI [where x = "cc(a := c a)"])
next
case 2
then have "sum cc' S ≤ sum cc S"
by (simp add: sumSS')
then have le: "cc' x ≤ cc x" if "x ∈ S" for x
using dd0 [OF that] 2 b0 mult_left_mono that
by (fastforce simp add: dd_def algebra_simps)
have cc0: "cc' x = 0" if "x ∈ S" "x ∉ T ∩ T'" for x
using le [OF ‹x ∈ S›] that c'0
by (force simp: cc_def cc'_def split: if_split_asm)
have ge0: "∀x∈T ∩ T'. 0 ≤ (cc'(a := c' a)) x"
by (simp add: c'0 cc'_def)
have "sum (cc'(a := c' a)) (insert a (T ∩ T')) = c' a + sum (cc'(a := c' a)) (T ∩ T')"
by (simp add: anot)
also have "... = c' a + sum (cc'(a := c' a)) S"
using ‹T ⊆ S› False cc0 by (fastforce intro!: sum.mono_neutral_left split: if_split_asm)
also have "... = c' a + (1 - c' a)"
by (metis ‹a ∉ S› fun_upd_other sum.cong sumSS')
finally have 1: "sum (cc'(a := c' a)) (insert a (T ∩ T')) = 1"
by simp
have "(∑x∈insert a (T ∩ T'). (cc'(a := c' a)) x *⇩R x) = c' a *⇩R a + (∑x ∈ T ∩ T'. (cc'(a := c' a)) x *⇩R x)"
by (simp add: anot)
also have "... = c' a *⇩R a + (∑x ∈ S. (cc'(a := c' a)) x *⇩R x)"
using ‹T ⊆ S› False cc0 by (fastforce intro!: sum.mono_neutral_left split: if_split_asm)
also have "... = c a *⇩R a + x - c a *⇩R a"
by (simp add: wsumSS ‹a ∉ S› if_smult sum_delta_notmem)
finally have self: "(∑x∈insert a (T ∩ T'). (cc'(a := c' a)) x *⇩R x) = x"
by simp
show ?thesis
by (force simp: convex_hull_finite c'0 intro!: ge0 1 self exI [where x = "cc'(a := c' a)"])
qed
qed
corollary convex_hull_exchange_Int:
fixes a :: "'a::euclidean_space"
assumes "¬ affine_dependent S" "a ∈ convex hull S" "T ⊆ S" "T' ⊆ S"
shows "(convex hull (insert a T)) ∩ (convex hull (insert a T')) =
convex hull (insert a (T ∩ T'))" (is "?lhs = ?rhs")
proof (rule subset_antisym)
show "?lhs ⊆ ?rhs"
using in_convex_hull_exchange_unique assms by blast
show "?rhs ⊆ ?lhs"
by (metis hull_mono inf_le1 inf_le2 insert_inter_insert le_inf_iff)
qed
lemma Int_closed_segment:
fixes b :: "'a::euclidean_space"
assumes "b ∈ closed_segment a c ∨ ¬ collinear{a,b,c}"
shows "closed_segment a b ∩ closed_segment b c = {b}"
proof (cases "c = a")
case True
then show ?thesis
using assms collinear_3_eq_affine_dependent by fastforce
next
case False
from assms show ?thesis
proof
assume "b ∈ closed_segment a c"
moreover have "¬ affine_dependent {a, c}"
by (simp)
ultimately show ?thesis
using False convex_hull_exchange_Int [of "{a,c}" b "{a}" "{c}"]
by (simp add: segment_convex_hull insert_commute)
next
assume ncoll: "¬ collinear {a, b, c}"
have False if "closed_segment a b ∩ closed_segment b c ≠ {b}"
proof -
have "b ∈ closed_segment a b" and "b ∈ closed_segment b c"
by auto
with that obtain d where "b ≠ d" "d ∈ closed_segment a b" "d ∈ closed_segment b c"
by force
then have d: "collinear {a, d, b}" "collinear {b, d, c}"
by (auto simp: between_mem_segment between_imp_collinear)
have "collinear {a, b, c}"
by (metis (full_types) ‹b ≠ d› collinear_3_trans d insert_commute)
with ncoll show False ..
qed
then show ?thesis
by blast
qed
qed
lemma affine_hull_finite_intersection_hyperplanes:
fixes S :: "'a::euclidean_space set"
obtains ℱ where
"finite ℱ"
"of_nat (card ℱ) + aff_dim S = DIM('a)"
"affine hull S = ⋂ℱ"
"⋀h. h ∈ ℱ ⟹ ∃a b. a ≠ 0 ∧ h = {x. a ∙ x = b}"
proof -
obtain b where "b ⊆ S"
and indb: "¬ affine_dependent b"
and eq: "affine hull S = affine hull b"
using affine_basis_exists by blast
obtain c where indc: "¬ affine_dependent c" and "b ⊆ c"
and affc: "affine hull c = UNIV"
by (metis extend_to_affine_basis affine_UNIV hull_same indb subset_UNIV)
then have "finite c"
by (simp add: aff_independent_finite)
then have fbc: "finite b" "card b ≤ card c"
using ‹b ⊆ c› infinite_super by (auto simp: card_mono)
have imeq: "(λx. affine hull x) ` ((λa. c - {a}) ` (c - b)) = ((λa. affine hull (c - {a})) ` (c - b))"
by blast
have card_cb: "(card (c - b)) + aff_dim S = DIM('a)"
proof -
have aff: "aff_dim (UNIV::'a set) = aff_dim c"
by (metis aff_dim_affine_hull affc)
have "aff_dim b = aff_dim S"
by (metis (no_types) aff_dim_affine_hull eq)
then have "int (card b) = 1 + aff_dim S"
by (simp add: aff_dim_affine_independent indb)
then show ?thesis
using fbc aff
by (simp add: ‹¬ affine_dependent c› ‹b ⊆ c› aff_dim_affine_independent card_Diff_subset of_nat_diff)
qed
show ?thesis
proof (cases "c = b")
case True show ?thesis
proof
show "int (card {}) + aff_dim S = int DIM('a)"
using True card_cb by auto
show "affine hull S = ⋂ {}"
using True affc eq by blast
qed auto
next
case False
have ind: "¬ affine_dependent (⋃a∈c - b. c - {a})"
by (rule affine_independent_subset [OF indc]) auto
have *: "1 + aff_dim (c - {t}) = int (DIM('a))" if t: "t ∈ c" for t
proof -
have "insert t c = c"
using t by blast
then show ?thesis
by (metis (full_types) add.commute aff_dim_affine_hull aff_dim_insert aff_dim_UNIV affc affine_dependent_def indc insert_Diff_single t)
qed
let ?ℱ = "(λx. affine hull x) ` ((λa. c - {a}) ` (c - b))"
show ?thesis
proof
have "card ((λa. affine hull (c - {a})) ` (c - b)) = card (c - b)"
proof (rule card_image)
show "inj_on (λa. affine hull (c - {a})) (c - b)"
unfolding inj_on_def
by (metis Diff_eq_empty_iff Diff_iff indc affine_dependent_def hull_subset insert_iff)
qed
then show "int (card ?ℱ) + aff_dim S = int DIM('a)"
by (simp add: imeq card_cb)
show "affine hull S = ⋂ ?ℱ"
by (metis Diff_eq_empty_iff INT_simps(4) UN_singleton order_refl ‹b ⊆ c› False eq double_diff affine_hull_Inter [OF ind])
have "⋀a. ⟦a ∈ c; a ∉ b⟧ ⟹ aff_dim (c - {a}) = int (DIM('a) - Suc 0)"
by (metis "*" DIM_ge_Suc0 One_nat_def add_diff_cancel_left' int_ops(2) of_nat_diff)
then show "⋀h. h ∈ ?ℱ ⟹ ∃a b. a ≠ 0 ∧ h = {x. a ∙ x = b}"
by (auto simp only: One_nat_def aff_dim_eq_hyperplane [symmetric])
qed (use ‹finite c› in auto)
qed
qed
lemma affine_hyperplane_sums_eq_UNIV_0:
fixes S :: "'a :: euclidean_space set"
assumes "affine S"
and "0 ∈ S" and "w ∈ S"
and "a ∙ w ≠ 0"
shows "{x + y| x y. x ∈ S ∧ a ∙ y = 0} = UNIV"
proof -
have "subspace S"
by (simp add: assms subspace_affine)
have span1: "span {y. a ∙ y = 0} ⊆ span {x + y |x y. x ∈ S ∧ a ∙ y = 0}"
using ‹0 ∈ S› add.left_neutral by (intro span_mono) force
have "w ∉ span {y. a ∙ y = 0}"
using ‹a ∙ w ≠ 0› span_induct subspace_hyperplane by auto
moreover have "w ∈ span {x + y |x y. x ∈ S ∧ a ∙ y = 0}"
using ‹w ∈ S›
by (metis (mono_tags, lifting) inner_zero_right mem_Collect_eq pth_d span_base)
ultimately have span2: "span {y. a ∙ y = 0} ≠ span {x + y |x y. x ∈ S ∧ a ∙ y = 0}"
by blast
have "a ≠ 0" using assms inner_zero_left by blast
then have "DIM('a) - 1 = dim {y. a ∙ y = 0}"
by (simp add: dim_hyperplane)
also have "... < dim {x + y |x y. x ∈ S ∧ a ∙ y = 0}"
using span1 span2 by (blast intro: dim_psubset)
finally have "DIM('a) - 1 < dim {x + y |x y. x ∈ S ∧ a ∙ y = 0}" .
then have DD: "dim {x + y |x y. x ∈ S ∧ a ∙ y = 0} = DIM('a)"
using antisym dim_subset_UNIV lowdim_subset_hyperplane not_le by fastforce
have subs: "subspace {x + y| x y. x ∈ S ∧ a ∙ y = 0}"
using subspace_sums [OF ‹subspace S› subspace_hyperplane] by simp
moreover have "span {x + y| x y. x ∈ S ∧ a ∙ y = 0} = UNIV"
using DD dim_eq_full by blast
ultimately show ?thesis
by (simp add: subs) (metis (lifting) span_eq_iff subs)
qed
proposition affine_hyperplane_sums_eq_UNIV:
fixes S :: "'a :: euclidean_space set"
assumes "affine S"
and "S ∩ {v. a ∙ v = b} ≠ {}"
and "S - {v. a ∙ v = b} ≠ {}"
shows "{x + y| x y. x ∈ S ∧ a ∙ y = b} = UNIV"
proof (cases "a = 0")
case True with assms show ?thesis
by (auto simp: if_splits)
next
case False
obtain c where "c ∈ S" and c: "a ∙ c = b"
using assms by force
with affine_diffs_subspace [OF ‹affine S›]
have "subspace ((+) (- c) ` S)" by blast
then have aff: "affine ((+) (- c) ` S)"
by (simp add: subspace_imp_affine)
have 0: "0 ∈ (+) (- c) ` S"
by (simp add: ‹c ∈ S›)
obtain d where "d ∈ S" and "a ∙ d ≠ b" and dc: "d-c ∈ (+) (- c) ` S"
using assms by auto
then have adc: "a ∙ (d - c) ≠ 0"
by (simp add: c inner_diff_right)
define U where "U ≡ {x + y |x y. x ∈ (+) (- c) ` S ∧ a ∙ y = 0}"
have "u + v ∈ (+) (c+c) ` U"
if "u ∈ S" "b = a ∙ v" for u v
proof
show "u + v = c + c + (u + v - c - c)"
by (simp add: algebra_simps)
have "∃x y. u + v - c - c = x + y ∧ (∃xa∈S. x = xa - c) ∧ a ∙ y = 0"
proof (intro exI conjI)
show "u + v - c - c = (u-c) + (v-c)" "a ∙ (v - c) = 0"
by (simp_all add: algebra_simps c that)
qed (use that in auto)
then show "u + v - c - c ∈ U"
by (auto simp: U_def image_def)
qed
moreover have "⟦a ∙ v = 0; u ∈ S⟧
⟹ ∃x ya. v + (u + c) = x + ya ∧ x ∈ S ∧ a ∙ ya = b" for v u
by (metis add.left_commute c inner_right_distrib pth_d)
ultimately have "{x + y |x y. x ∈ S ∧ a ∙ y = b} = (+) (c+c) ` U"
by (fastforce simp: algebra_simps U_def)
also have "... = range ((+) (c + c))"
by (simp only: U_def affine_hyperplane_sums_eq_UNIV_0 [OF aff 0 dc adc])
also have "... = UNIV"
by simp
finally show ?thesis .
qed
lemma aff_dim_sums_Int_0:
assumes "affine S"
and "affine T"
and "0 ∈ S" "0 ∈ T"
shows "aff_dim {x + y| x y. x ∈ S ∧ y ∈ T} = (aff_dim S + aff_dim T) - aff_dim(S ∩ T)"
proof -
have "0 ∈ {x + y |x y. x ∈ S ∧ y ∈ T}"
using assms by force
then have 0: "0 ∈ affine hull {x + y |x y. x ∈ S ∧ y ∈ T}"
by (metis (lifting) hull_inc)
have sub: "subspace S" "subspace T"
using assms by (auto simp: subspace_affine)
show ?thesis
using dim_sums_Int [OF sub] by (simp add: aff_dim_zero assms 0 hull_inc)
qed
proposition aff_dim_sums_Int:
assumes "affine S"
and "affine T"
and "S ∩ T ≠ {}"
shows "aff_dim {x + y| x y. x ∈ S ∧ y ∈ T} = (aff_dim S + aff_dim T) - aff_dim(S ∩ T)"
proof -
obtain a where a: "a ∈ S" "a ∈ T" using assms by force
have aff: "affine ((+) (-a) ` S)" "affine ((+) (-a) ` T)"
using affine_translation [symmetric, of "- a"] assms by (simp_all cong: image_cong_simp)
have zero: "0 ∈ ((+) (-a) ` S)" "0 ∈ ((+) (-a) ` T)"
using a assms by auto
have "{x + y |x y. x ∈ (+) (- a) ` S ∧ y ∈ (+) (- a) ` T} =
(+) (- 2 *⇩R a) ` {x + y| x y. x ∈ S ∧ y ∈ T}"
by (force simp: algebra_simps scaleR_2)
moreover have "(+) (- a) ` S ∩ (+) (- a) ` T = (+) (- a) ` (S ∩ T)"
by auto
ultimately show ?thesis
using aff_dim_sums_Int_0 [OF aff zero] aff_dim_translation_eq
by (metis (lifting))
qed
lemma aff_dim_affine_Int_hyperplane:
fixes a :: "'a::euclidean_space"
assumes "affine S"
shows "aff_dim(S ∩ {x. a ∙ x = b}) =
(if S ∩ {v. a ∙ v = b} = {} then - 1
else if S ⊆ {v. a ∙ v = b} then aff_dim S
else aff_dim S - 1)"
proof (cases "a = 0")
case True with assms show ?thesis
by auto
next
case False
then have "aff_dim (S ∩ {x. a ∙ x = b}) = aff_dim S - 1"
if "x ∈ S" "a ∙ x ≠ b" and non: "S ∩ {v. a ∙ v = b} ≠ {}" for x
proof -
have [simp]: "{x + y| x y. x ∈ S ∧ a ∙ y = b} = UNIV"
using affine_hyperplane_sums_eq_UNIV [OF assms non] that by blast
show ?thesis
using aff_dim_sums_Int [OF assms affine_hyperplane non]
by (simp add: of_nat_diff False)
qed
then show ?thesis
by (metis (mono_tags, lifting) inf.orderE aff_dim_empty_eq mem_Collect_eq subsetI)
qed
lemma aff_dim_lt_full:
fixes S :: "'a::euclidean_space set"
shows "aff_dim S < DIM('a) ⟷ (affine hull S ≠ UNIV)"
by (metis (no_types) aff_dim_affine_hull aff_dim_le_DIM aff_dim_UNIV affine_hull_UNIV less_le)
lemma aff_dim_openin:
fixes S :: "'a::euclidean_space set"
assumes ope: "openin (top_of_set T) S" and "affine T" "S ≠ {}"
shows "aff_dim S = aff_dim T"
proof -
show ?thesis
proof (rule order_antisym)
show "aff_dim S ≤ aff_dim T"
by (blast intro: aff_dim_subset [OF openin_imp_subset] ope)
next
obtain a where "a ∈ S"
using ‹S ≠ {}› by blast
have "S ⊆ T"
using ope openin_imp_subset by auto
then have "a ∈ T"
using ‹a ∈ S› by auto
then have subT': "subspace ((λx. - a + x) ` T)"
using affine_diffs_subspace ‹affine T› by auto
then obtain B where Bsub: "B ⊆ ((λx. - a + x) ` T)" and po: "pairwise orthogonal B"
and eq1: "⋀x. x ∈ B ⟹ norm x = 1" and "independent B"
and cardB: "card B = dim ((λx. - a + x) ` T)"
and spanB: "span B = ((λx. - a + x) ` T)"
by (rule orthonormal_basis_subspace) auto
obtain e where "0 < e" and e: "cball a e ∩ T ⊆ S"
by (meson ‹a ∈ S› openin_contains_cball ope)
have "aff_dim T = aff_dim ((λx. - a + x) ` T)"
by (metis aff_dim_translation_eq)
also have "... = dim ((λx. - a + x) ` T)"
using aff_dim_subspace subT' by blast
also have "... = card B"
by (simp add: cardB)
also have "... = card ((λx. e *⇩R x) ` B)"
using ‹0 < e› by (force simp: inj_on_def card_image)
also have "... ≤ dim ((λx. - a + x) ` S)"
proof -
have e': "cball 0 e ∩ (λx. x - a) ` T ⊆ (λx. x - a) ` S"
using e by (auto simp: dist_norm norm_minus_commute subset_eq)
have "(λx. e *⇩R x) ` B ⊆ cball 0 e ∩ (λx. x - a) ` T"
using Bsub ‹0 < e› eq1 subT' ‹a ∈ T› by (auto simp: subspace_def)
then have "(λx. e *⇩R x) ` B ⊆ (λx. x - a) ` S"
using e' by blast
moreover
have "inj_on ((*⇩R) e) (span B)"
using ‹0 < e› inj_on_def by fastforce
then have "independent ((λx. e *⇩R x) ` B)"
using linear_scale_self ‹independent B› linear_dependent_inj_imageD by blast
ultimately show ?thesis
by (auto simp: intro!: independent_card_le_dim)
qed
also have "... = aff_dim S"
using ‹a ∈ S› aff_dim_eq_dim hull_inc by (force cong: image_cong_simp)
finally show "aff_dim T ≤ aff_dim S" .
qed
qed
lemma dim_openin:
fixes S :: "'a::euclidean_space set"
assumes ope: "openin (top_of_set T) S" and "subspace T" "S ≠ {}"
shows "dim S = dim T"
proof (rule order_antisym)
show "dim S ≤ dim T"
by (metis ope dim_subset openin_subset topspace_euclidean_subtopology)
next
have "dim T = aff_dim S"
using aff_dim_openin
by (metis aff_dim_subspace ‹subspace T› ‹S ≠ {}› ope subspace_affine)
also have "... ≤ dim S"
by (metis aff_dim_subset aff_dim_subspace dim_span span_superset
subspace_span)
finally show "dim T ≤ dim S" by simp
qed
subsection‹Lower-dimensional affine subsets are nowhere dense›
proposition dense_complement_subspace:
fixes S :: "'a :: euclidean_space set"
assumes dim_less: "dim T < dim S" and "subspace S" shows "closure(S - T) = S"
proof -
have "closure(S - U) = S" if "dim U < dim S" "U ⊆ S" for U
proof -
have "span U ⊂ span S"
by (metis neq_iff psubsetI span_eq_dim span_mono that)
then obtain a where "a ≠ 0" "a ∈ span S" and a: "⋀y. y ∈ span U ⟹ orthogonal a y"
using orthogonal_to_subspace_exists_gen by metis
show ?thesis
proof
have "closed S"
by (simp add: ‹subspace S› closed_subspace)
then show "closure (S - U) ⊆ S"
by (simp add: closure_minimal)
show "S ⊆ closure (S - U)"
proof (clarsimp simp: closure_approachable)
fix x and e::real
assume "x ∈ S" "0 < e"
show "∃y∈S - U. dist y x < e"
proof (cases "x ∈ U")
case True
let ?y = "x + (e/2 / norm a) *⇩R a"
show ?thesis
proof
show "dist ?y x < e"
using ‹0 < e› by (simp add: dist_norm)
next
have "?y ∈ S"
by (metis ‹a ∈ span S› ‹x ∈ S› assms(2) span_eq_iff subspace_add subspace_scale)
moreover have "?y ∉ U"
proof -
have "e/2 / norm a ≠ 0"
using ‹0 < e› ‹a ≠ 0› by auto
then show ?thesis
by (metis True ‹a ≠ 0› a orthogonal_scaleR orthogonal_self real_vector.scale_eq_0_iff span_add_eq span_base)
qed
ultimately show "?y ∈ S - U" by blast
qed
next
case False
with ‹0 < e› ‹x ∈ S› show ?thesis by force
qed
qed
qed
qed
moreover have "S - S ∩ T = S-T"
by blast
moreover have "dim (S ∩ T) < dim S"
by (metis dim_less dim_subset inf.cobounded2 inf.orderE inf.strict_boundedE not_le)
ultimately show ?thesis
by force
qed
corollary dense_complement_affine:
fixes S :: "'a :: euclidean_space set"
assumes less: "aff_dim T < aff_dim S" and "affine S" shows "closure(S - T) = S"
proof (cases "S ∩ T = {}")
case True
then show ?thesis
by (metis Diff_triv affine_hull_eq ‹affine S› closure_same_affine_hull closure_subset hull_subset subset_antisym)
next
case False
then obtain z where z: "z ∈ S ∩ T" by blast
then have "subspace ((+) (- z) ` S)"
by (meson IntD1 affine_diffs_subspace ‹affine S›)
moreover have "int (dim ((+) (- z) ` T)) < int (dim ((+) (- z) ` S))"
thm aff_dim_eq_dim
using z less by (simp add: aff_dim_eq_dim_subtract [of z] hull_inc cong: image_cong_simp)
ultimately have "closure(((+) (- z) ` S) - ((+) (- z) ` T)) = ((+) (- z) ` S)"
by (simp add: dense_complement_subspace)
then show ?thesis
by (metis closure_translation translation_diff translation_invert)
qed
corollary dense_complement_openin_affine_hull:
fixes S :: "'a :: euclidean_space set"
assumes less: "aff_dim T < aff_dim S"
and ope: "openin (top_of_set (affine hull S)) S"
shows "closure(S - T) = closure S"
proof -
have "affine hull S - T ⊆ affine hull S"
by blast
then have "closure (S ∩ closure (affine hull S - T)) = closure (S ∩ (affine hull S - T))"
by (rule closure_openin_Int_closure [OF ope])
then show ?thesis
by (metis Int_Diff aff_dim_affine_hull affine_affine_hull dense_complement_affine hull_subset inf.orderE less)
qed
corollary dense_complement_convex:
fixes S :: "'a :: euclidean_space set"
assumes "aff_dim T < aff_dim S" "convex S"
shows "closure(S - T) = closure S"
proof
show "closure (S - T) ⊆ closure S"
by (simp add: closure_mono)
have "closure (rel_interior S - T) = closure (rel_interior S)"
by (simp add: assms dense_complement_openin_affine_hull openin_rel_interior rel_interior_aff_dim rel_interior_same_affine_hull)
then show "closure S ⊆ closure (S - T)"
by (metis Diff_mono ‹convex S› closure_mono convex_closure_rel_interior order_refl rel_interior_subset)
qed
corollary dense_complement_convex_closed:
fixes S :: "'a :: euclidean_space set"
assumes "aff_dim T < aff_dim S" "convex S" "closed S"
shows "closure(S - T) = S"
by (simp add: assms dense_complement_convex)
subsection‹Parallel slices, etc›
text‹ If we take a slice out of a set, we can do it perpendicularly,
with the normal vector to the slice parallel to the affine hull.›
proposition affine_parallel_slice:
fixes S :: "'a :: euclidean_space set"
assumes "affine S"
and "S ∩ {x. a ∙ x ≤ b} ≠ {}"
and "¬ (S ⊆ {x. a ∙ x ≤ b})"
obtains a' b' where "a' ≠ 0"
"S ∩ {x. a' ∙ x ≤ b'} = S ∩ {x. a ∙ x ≤ b}"
"S ∩ {x. a' ∙ x = b'} = S ∩ {x. a ∙ x = b}"
"⋀w. w ∈ S ⟹ (w + a') ∈ S"
proof (cases "S ∩ {x. a ∙ x = b} = {}")
case True
then obtain u v where "u ∈ S" "v ∈ S" "a ∙ u ≤ b" "a ∙ v > b"
using assms by (auto simp: not_le)
define η where "η = u + ((b - a ∙ u) / (a ∙ v - a ∙ u)) *⇩R (v - u)"
have "η ∈ S"
by (simp add: η_def ‹u ∈ S› ‹v ∈ S› ‹affine S› mem_affine_3_minus)
moreover have "a ∙ η = b"
using ‹a ∙ u ≤ b› ‹b < a ∙ v›
by (simp add: η_def algebra_simps) (simp add: field_simps)
ultimately have False
using True by force
then show ?thesis ..
next
case False
then obtain z where "z ∈ S" and z: "a ∙ z = b"
using assms by auto
with affine_diffs_subspace [OF ‹affine S›]
have sub: "subspace ((+) (- z) ` S)" by blast
then have aff: "affine ((+) (- z) ` S)" and span: "span ((+) (- z) ` S) = ((+) (- z) ` S)"
by (auto simp: subspace_imp_affine)
obtain a' a'' where a': "a' ∈ span ((+) (- z) ` S)" and a: "a = a' + a''"
and "⋀w. w ∈ span ((+) (- z) ` S) ⟹ orthogonal a'' w"
using orthogonal_subspace_decomp_exists [of "(+) (- z) ` S" "a"] by metis
then have "⋀w. w ∈ S ⟹ a'' ∙ (w-z) = 0"
by (simp add: span_base orthogonal_def)
then have a'': "⋀w. w ∈ S ⟹ a'' ∙ w = (a - a') ∙ z"
by (simp add: a inner_diff_right)
then have ba'': "⋀w. w ∈ S ⟹ a'' ∙ w = b - a' ∙ z"
by (simp add: inner_diff_left z)
show ?thesis
proof (cases "a' = 0")
case True
with a assms True a'' diff_zero less_irrefl show ?thesis
by auto
next
case False
show ?thesis
proof
show "S ∩ {x. a' ∙ x ≤ a' ∙ z} = S ∩ {x. a ∙ x ≤ b}"
"S ∩ {x. a' ∙ x = a' ∙ z} = S ∩ {x. a ∙ x = b}"
by (auto simp: a ba'' inner_left_distrib)
have "⋀w. w ∈ (+) (- z) ` S ⟹ (w + a') ∈ (+) (- z) ` S"
by (metis subspace_add a' span_eq_iff sub)
then show "⋀w. w ∈ S ⟹ (w + a') ∈ S"
by fastforce
qed (use False in auto)
qed
qed
lemma diffs_affine_hull_span:
assumes "a ∈ S"
shows "(λx. x - a) ` (affine hull S) = span ((λx. x - a) ` S)"
proof -
have *: "((λx. x - a) ` (S - {a})) = ((λx. x - a) ` S) - {0}"
by (auto simp: algebra_simps)
show ?thesis
by (auto simp add: algebra_simps affine_hull_span2 [OF assms] *)
qed
lemma aff_dim_dim_affine_diffs:
fixes S :: "'a :: euclidean_space set"
assumes "affine S" "a ∈ S"
shows "aff_dim S = dim ((λx. x - a) ` S)"
proof -
obtain B where aff: "affine hull B = affine hull S"
and ind: "¬ affine_dependent B"
and card: "of_nat (card B) = aff_dim S + 1"
using aff_dim_basis_exists by blast
then have "B ≠ {}" using assms
by (metis affine_hull_eq_empty ex_in_conv)
then obtain c where "c ∈ B" by auto
then have "c ∈ S"
by (metis aff affine_hull_eq ‹affine S› hull_inc)
have xy: "x - c = y - a ⟷ y = x + 1 *⇩R (a - c)" for x y c and a::'a
by (auto simp: algebra_simps)
have *: "(λx. x - c) ` S = (λx. x - a) ` S"
using assms ‹c ∈ S›
by (auto simp: image_iff xy; metis mem_affine_3_minus pth_1)
have affS: "affine hull S = S"
by (simp add: ‹affine S›)
have "aff_dim S = of_nat (card B) - 1"
using card by simp
also have "... = dim ((λx. x - c) ` B)"
using affine_independent_card_dim_diffs [OF ind ‹c ∈ B›]
by (simp add: affine_independent_card_dim_diffs [OF ind ‹c ∈ B›])
also have "... = dim ((λx. x - c) ` (affine hull B))"
by (simp add: diffs_affine_hull_span ‹c ∈ B›)
also have "... = dim ((λx. x - a) ` S)"
by (simp add: affS aff *)
finally show ?thesis .
qed
lemma aff_dim_linear_image_le:
assumes "linear f"
shows "aff_dim(f ` S) ≤ aff_dim S"
proof -
have "aff_dim (f ` T) ≤ aff_dim T" if "affine T" for T
proof (cases "T = {}")
case True then show ?thesis by (simp add: aff_dim_geq)
next
case False
then obtain a where "a ∈ T" by auto
have 1: "((λx. x - f a) ` f ` T) = {x - f a |x. x ∈ f ` T}"
by auto
have 2: "{x - f a| x. x ∈ f ` T} = f ` ((λx. x - a) ` T)"
by (force simp: linear_diff [OF assms])
have "aff_dim (f ` T) = int (dim {x - f a |x. x ∈ f ` T})"
by (simp add: ‹a ∈ T› hull_inc aff_dim_eq_dim [of "f a"] 1 cong: image_cong_simp)
also have "... = int (dim (f ` ((λx. x - a) ` T)))"
by (force simp: linear_diff [OF assms] 2)
also have "... ≤ int (dim ((λx. x - a) ` T))"
by (simp add: dim_image_le [OF assms])
also have "... ≤ aff_dim T"
by (simp add: aff_dim_dim_affine_diffs [symmetric] ‹a ∈ T› ‹affine T›)
finally show ?thesis .
qed
then
have "aff_dim (f ` (affine hull S)) ≤ aff_dim (affine hull S)"
using affine_affine_hull [of S] by blast
then show ?thesis
using affine_hull_linear_image assms linear_conv_bounded_linear by fastforce
qed
lemma aff_dim_injective_linear_image [simp]:
assumes "linear f" "inj f"
shows "aff_dim (f ` S) = aff_dim S"
proof (rule antisym)
show "aff_dim (f ` S) ≤ aff_dim S"
by (simp add: aff_dim_linear_image_le assms(1))
next
obtain g where "linear g" "g ∘ f = id"
using assms(1) assms(2) linear_injective_left_inverse by blast
then have "aff_dim S ≤ aff_dim(g ` f ` S)"
by (simp add: image_comp)
also have "... ≤ aff_dim (f ` S)"
by (simp add: ‹linear g› aff_dim_linear_image_le)
finally show "aff_dim S ≤ aff_dim (f ` S)" .
qed
lemma choose_affine_subset:
assumes "affine S" "-1 ≤ d" and dle: "d ≤ aff_dim S"
obtains T where "affine T" "T ⊆ S" "aff_dim T = d"
proof (cases "d = -1 ∨ S={}")
case True with assms show ?thesis
by (metis aff_dim_empty affine_empty bot.extremum that eq_iff)
next
case False
with assms obtain a where "a ∈ S" "0 ≤ d" by auto
with assms have ss: "subspace ((+) (- a) ` S)"
by (simp add: affine_diffs_subspace_subtract cong: image_cong_simp)
have "nat d ≤ dim ((+) (- a) ` S)"
by (metis aff_dim_subspace aff_dim_translation_eq dle nat_int nat_mono ss)
then obtain T where "subspace T" and Tsb: "T ⊆ span ((+) (- a) ` S)"
and Tdim: "dim T = nat d"
using choose_subspace_of_subspace [of "nat d" "(+) (- a) ` S"] by blast
then have "affine T"
using subspace_affine by blast
then have "affine ((+) a ` T)"
by (metis affine_hull_eq affine_hull_translation)
moreover have "(+) a ` T ⊆ S"
proof -
have "T ⊆ (+) (- a) ` S"
by (metis (no_types) span_eq_iff Tsb ss)
then show "(+) a ` T ⊆ S"
using add_ac by auto
qed
moreover have "aff_dim ((+) a ` T) = d"
by (simp add: aff_dim_subspace Tdim ‹0 ≤ d› ‹subspace T› aff_dim_translation_eq)
ultimately show ?thesis
by (rule that)
qed
subsection‹Paracompactness›
proposition paracompact:
fixes S :: "'a :: {metric_space,second_countable_topology} set"
assumes "S ⊆ ⋃𝒞" and opC: "⋀T. T ∈ 𝒞 ⟹ open T"
obtains 𝒞' where "S ⊆ ⋃ 𝒞'"
and "⋀U. U ∈ 𝒞' ⟹ open U ∧ (∃T. T ∈ 𝒞 ∧ U ⊆ T)"
and "⋀x. x ∈ S
⟹ ∃V. open V ∧ x ∈ V ∧ finite {U. U ∈ 𝒞' ∧ (U ∩ V ≠ {})}"
proof (cases "S = {}")
case True with that show ?thesis by blast
next
case False
have "∃T U. x ∈ U ∧ open U ∧ closure U ⊆ T ∧ T ∈ 𝒞" if "x ∈ S" for x
proof -
obtain T where "x ∈ T" "T ∈ 𝒞" "open T"
using assms ‹x ∈ S› by blast
then obtain e where "e > 0" "cball x e ⊆ T"
by (force simp: open_contains_cball)
then show ?thesis
by (meson open_ball ‹T ∈ 𝒞› ball_subset_cball centre_in_ball closed_cball closure_minimal dual_order.trans)
qed
then obtain F G where Gin: "x ∈ G x" and oG: "open (G x)"
and clos: "closure (G x) ⊆ F x" and Fin: "F x ∈ 𝒞"
if "x ∈ S" for x
by metis
then obtain ℱ where "ℱ ⊆ G ` S" "countable ℱ" "⋃ℱ = ⋃(G ` S)"
using Lindelof [of "G ` S"] by (metis image_iff)
then obtain K where K: "K ⊆ S" "countable K" and eq: "⋃(G ` K) = ⋃(G ` S)"
by (metis countable_subset_image)
with False Gin have "K ≠ {}" by force
then obtain a :: "nat ⇒ 'a" where "range a = K"
by (metis range_from_nat_into ‹countable K›)
then have odif: "⋀n. open (F (a n) - ⋃{closure (G (a m)) |m. m < n})"
using ‹K ⊆ S› Fin opC by (fastforce simp add:)
let ?C = "range (λn. F(a n) - ⋃{closure(G(a m)) |m. m < n})"
have enum_S: "∃n. x ∈ F(a n) ∧ x ∈ G(a n)" if "x ∈ S" for x
proof -
have "∃y ∈ K. x ∈ G y" using eq that Gin by fastforce
then show ?thesis
using clos K ‹range a = K› closure_subset by blast
qed
show ?thesis
proof
show "S ⊆ Union ?C"
proof
fix x assume "x ∈ S"
define n where "n ≡ LEAST n. x ∈ F(a n)"
have n: "x ∈ F(a n)"
using enum_S [OF ‹x ∈ S›] by (force simp: n_def intro: LeastI)
have notn: "x ∉ F(a m)" if "m < n" for m
using that not_less_Least by (force simp: n_def)
then have "x ∉ ⋃{closure (G (a m)) |m. m < n}"
using n ‹K ⊆ S› ‹range a = K› clos notn by fastforce
with n show "x ∈ Union ?C"
by blast
qed
show "⋀U. U ∈ ?C ⟹ open U ∧ (∃T. T ∈ 𝒞 ∧ U ⊆ T)"
using Fin ‹K ⊆ S› ‹range a = K› by (auto simp: odif)
show "∃V. open V ∧ x ∈ V ∧ finite {U. U ∈ ?C ∧ (U ∩ V ≠ {})}" if "x ∈ S" for x
proof -
obtain n where n: "x ∈ F(a n)" "x ∈ G(a n)"
using ‹x ∈ S› enum_S by auto
have "{U ∈ ?C. U ∩ G (a n) ≠ {}} ⊆ (λn. F(a n) - ⋃{closure(G(a m)) |m. m < n}) ` atMost n"
proof clarsimp
fix k assume "(F (a k) - ⋃{closure (G (a m)) |m. m < k}) ∩ G (a n) ≠ {}"
then have "k ≤ n"
by auto (metis closure_subset not_le subsetCE)
then show "F (a k) - ⋃{closure (G (a m)) |m. m < k}
∈ (λn. F (a n) - ⋃{closure (G (a m)) |m. m < n}) ` {..n}"
by force
qed
moreover have "finite ((λn. F(a n) - ⋃{closure(G(a m)) |m. m < n}) ` atMost n)"
by force
ultimately have *: "finite {U ∈ ?C. U ∩ G (a n) ≠ {}}"
using finite_subset by blast
have "a n ∈ S"
using ‹K ⊆ S› ‹range a = K› by blast
then show ?thesis
by (blast intro: oG n *)
qed
qed
qed
corollary paracompact_closedin:
fixes S :: "'a :: {metric_space,second_countable_topology} set"
assumes cin: "closedin (top_of_set U) S"
and oin: "⋀T. T ∈ 𝒞 ⟹ openin (top_of_set U) T"
and "S ⊆ ⋃𝒞"
obtains 𝒞' where "S ⊆ ⋃ 𝒞'"
and "⋀V. V ∈ 𝒞' ⟹ openin (top_of_set U) V ∧ (∃T. T ∈ 𝒞 ∧ V ⊆ T)"
and "⋀x. x ∈ U
⟹ ∃V. openin (top_of_set U) V ∧ x ∈ V ∧
finite {X. X ∈ 𝒞' ∧ (X ∩ V ≠ {})}"
proof -
have "∃Z. open Z ∧ (T = U ∩ Z)" if "T ∈ 𝒞" for T
using oin [OF that] by (auto simp: openin_open)
then obtain F where opF: "open (F T)" and intF: "U ∩ F T = T" if "T ∈ 𝒞" for T
by metis
obtain K where K: "closed K" "U ∩ K = S"
using cin by (auto simp: closedin_closed)
have 1: "U ⊆ ⋃(insert (- K) (F ` 𝒞))"
by clarsimp (metis Int_iff Union_iff ‹U ∩ K = S› ‹S ⊆ ⋃𝒞› subsetD intF)
have 2: "⋀T. T ∈ insert (- K) (F ` 𝒞) ⟹ open T"
using ‹closed K› by (auto simp: opF)
obtain 𝒟 where "U ⊆ ⋃𝒟"
and D1: "⋀U. U ∈ 𝒟 ⟹ open U ∧ (∃T. T ∈ insert (- K) (F ` 𝒞) ∧ U ⊆ T)"
and D2: "⋀x. x ∈ U ⟹ ∃V. open V ∧ x ∈ V ∧ finite {U ∈ 𝒟. U ∩ V ≠ {}}"
by (blast intro: paracompact [OF 1 2])
let ?C = "{U ∩ V |V. V ∈ 𝒟 ∧ (V ∩ K ≠ {})}"
show ?thesis
proof (rule_tac 𝒞' = "{U ∩ V |V. V ∈ 𝒟 ∧ (V ∩ K ≠ {})}" in that)
show "S ⊆ ⋃?C"
using ‹U ∩ K = S› ‹U ⊆ ⋃𝒟› K by (blast dest!: subsetD)
show "⋀V. V ∈ ?C ⟹ openin (top_of_set U) V ∧ (∃T. T ∈ 𝒞 ∧ V ⊆ T)"
using D1 intF by fastforce
have *: "{X. (∃V. X = U ∩ V ∧ V ∈ 𝒟 ∧ V ∩ K ≠ {}) ∧ X ∩ (U ∩ V) ≠ {}} ⊆
(λx. U ∩ x) ` {U ∈ 𝒟. U ∩ V ≠ {}}" for V
by blast
show "∃V. openin (top_of_set U) V ∧ x ∈ V ∧ finite {X ∈ ?C. X ∩ V ≠ {}}"
if "x ∈ U" for x
proof -
from D2 [OF that] obtain V where "open V" "x ∈ V" "finite {U ∈ 𝒟. U ∩ V ≠ {}}"
by auto
with * show ?thesis
by (rule_tac x="U ∩ V" in exI) (auto intro: that finite_subset [OF *])
qed
qed
qed
corollary paracompact_closed:
fixes S :: "'a :: {metric_space,second_countable_topology} set"
assumes "closed S"
and opC: "⋀T. T ∈ 𝒞 ⟹ open T"
and "S ⊆ ⋃𝒞"
obtains 𝒞' where "S ⊆ ⋃𝒞'"
and "⋀U. U ∈ 𝒞' ⟹ open U ∧ (∃T. T ∈ 𝒞 ∧ U ⊆ T)"
and "⋀x. ∃V. open V ∧ x ∈ V ∧
finite {U. U ∈ 𝒞' ∧ (U ∩ V ≠ {})}"
by (rule paracompact_closedin [of UNIV S 𝒞]) (auto simp: assms)
subsection‹Closed-graph characterization of continuity›
lemma continuous_closed_graph_gen:
fixes T :: "'b::real_normed_vector set"
assumes contf: "continuous_on S f" and fim: "f ` S ⊆ T"
shows "closedin (top_of_set (S × T)) ((λx. Pair x (f x)) ` S)"
proof -
have eq: "((λx. Pair x (f x)) ` S) = (S × T ∩ (λz. (f ∘ fst)z - snd z) -` {0})"
using fim by auto
show ?thesis
unfolding eq
by (intro continuous_intros continuous_closedin_preimage continuous_on_subset [OF contf]) auto
qed
lemma continuous_closed_graph_eq:
fixes f :: "'a::euclidean_space ⇒ 'b::euclidean_space"
assumes "compact T" and fim: "f ∈ S → T"
shows "continuous_on S f ⟷
closedin (top_of_set (S × T)) ((λx. Pair x (f x)) ` S)"
(is "?lhs = ?rhs")
proof -
have "?lhs" if ?rhs
proof (clarsimp simp add: continuous_on_closed_gen [OF fim])
fix U
assume U: "closedin (top_of_set T) U"
have eq: "(S ∩ f -` U) = fst ` (((λx. Pair x (f x)) ` S) ∩ (S × U))"
by (force simp: image_iff)
show "closedin (top_of_set S) (S ∩ f -` U)"
by (simp add: U closedin_Int closedin_Times closed_map_fst [OF ‹compact T›] that eq)
qed
with continuous_closed_graph_gen assms show ?thesis by blast
qed
lemma continuous_closed_graph:
fixes f :: "'a::topological_space ⇒ 'b::real_normed_vector"
assumes "closed S" and contf: "continuous_on S f"
shows "closed ((λx. Pair x (f x)) ` S)"
proof (rule closedin_closed_trans)
show "closedin (top_of_set (S × UNIV)) ((λx. (x, f x)) ` S)"
by (rule continuous_closed_graph_gen [OF contf subset_UNIV])
qed (simp add: ‹closed S› closed_Times)
lemma continuous_from_closed_graph:
fixes f :: "'a::euclidean_space ⇒ 'b::euclidean_space"
assumes "compact T" and fim: "f ∈ S → T" and clo: "closed ((λx. Pair x (f x)) ` S)"
shows "continuous_on S f"
using fim clo
by (auto intro: closed_subset simp: continuous_closed_graph_eq [OF ‹compact T› fim])
lemma continuous_on_Un_local_open:
assumes opS: "openin (top_of_set (S ∪ T)) S"
and opT: "openin (top_of_set (S ∪ T)) T"
and contf: "continuous_on S f" and contg: "continuous_on T f"
shows "continuous_on (S ∪ T) f"
using pasting_lemma [of "{S,T}" "top_of_set (S ∪ T)" id euclidean "λi. f" f] contf contg opS opT
by (simp add: subtopology_subtopology) (metis inf.absorb2 openin_imp_subset)
lemma continuous_on_cases_local_open:
assumes opS: "openin (top_of_set (S ∪ T)) S"
and opT: "openin (top_of_set (S ∪ T)) T"
and contf: "continuous_on S f" and contg: "continuous_on T g"
and fg: "⋀x. x ∈ S ∧ ¬P x ∨ x ∈ T ∧ P x ⟹ f x = g x"
shows "continuous_on (S ∪ T) (λx. if P x then f x else g x)"
proof -
have "⋀x. x ∈ S ⟹ (if P x then f x else g x) = f x" "⋀x. x ∈ T ⟹ (if P x then f x else g x) = g x"
by (simp_all add: fg)
then have "continuous_on S (λx. if P x then f x else g x)" "continuous_on T (λx. if P x then f x else g x)"
by (simp_all add: contf contg cong: continuous_on_cong)
then show ?thesis
by (rule continuous_on_Un_local_open [OF opS opT])
qed
subsection‹The union of two collinear segments is another segment›
proposition in_convex_hull_exchange:
fixes a :: "'a::euclidean_space"
assumes a: "a ∈ convex hull S" and xS: "x ∈ convex hull S"
obtains b where "b ∈ S" "x ∈ convex hull (insert a (S - {b}))"
proof (cases "a ∈ S")
case True
with xS insert_Diff that show ?thesis by fastforce
next
case False
show ?thesis
proof (cases "finite S ∧ card S ≤ Suc (DIM('a))")
case True
then obtain u where u0: "⋀i. i ∈ S ⟹ 0 ≤ u i" and u1: "sum u S = 1"
and ua: "(∑i∈S. u i *⇩R i) = a"
using a by (auto simp: convex_hull_finite)
obtain v where v0: "⋀i. i ∈ S ⟹ 0 ≤ v i" and v1: "sum v S = 1"
and vx: "(∑i∈S. v i *⇩R i) = x"
using True xS by (auto simp: convex_hull_finite)
show ?thesis
proof (cases "∃b. b ∈ S ∧ v b = 0")
case True
then obtain b where b: "b ∈ S" "v b = 0"
by blast
show ?thesis
proof
have fin: "finite (insert a (S - {b}))"
using sum.infinite v1 by fastforce
show "x ∈ convex hull insert a (S - {b})"
unfolding convex_hull_finite [OF fin] mem_Collect_eq
proof (intro conjI exI ballI)
have "(∑x ∈ insert a (S - {b}). if x = a then 0 else v x) =
(∑x ∈ S - {b}. if x = a then 0 else v x)"
using fin by (force intro: sum.mono_neutral_right)
also have "... = (∑x ∈ S - {b}. v x)"
using b False by (auto intro!: sum.cong split: if_split_asm)
also have "... = (∑x∈S. v x)"
by (metis ‹v b = 0› diff_zero sum.infinite sum_diff1 u1 zero_neq_one)
finally show "(∑x∈insert a (S - {b}). if x = a then 0 else v x) = 1"
by (simp add: v1)
show "⋀x. x ∈ insert a (S - {b}) ⟹ 0 ≤ (if x = a then 0 else v x)"
by (auto simp: v0)
have "(∑x ∈ insert a (S - {b}). (if x = a then 0 else v x) *⇩R x) =
(∑x ∈ S - {b}. (if x = a then 0 else v x) *⇩R x)"
using fin by (force intro: sum.mono_neutral_right)
also have "... = (∑x ∈ S - {b}. v x *⇩R x)"
using b False by (auto intro!: sum.cong split: if_split_asm)
also have "... = (∑x∈S. v x *⇩R x)"
by (metis (no_types, lifting) b(2) diff_zero fin finite.emptyI finite_Diff2 finite_insert scale_eq_0_iff sum_diff1)
finally show "(∑x∈insert a (S - {b}). (if x = a then 0 else v x) *⇩R x) = x"
by (simp add: vx)
qed
qed (rule ‹b ∈ S›)
next
case False
have le_Max: "u i / v i ≤ Max ((λi. u i / v i) ` S)" if "i ∈ S" for i
by (simp add: True that)
have "Max ((λi. u i / v i) ` S) ∈ (λi. u i / v i) ` S"
using True v1 by (auto intro: Max_in)
then obtain b where "b ∈ S" and beq: "Max ((λb. u b / v b) ` S) = u b / v b"
by blast
then have "0 ≠ u b / v b"
using le_Max beq divide_le_0_iff le_numeral_extra(2) sum_nonpos u1
by (metis False eq_iff v0)
then have "0 < u b" "0 < v b"
using False ‹b ∈ S› u0 v0 by force+
have fin: "finite (insert a (S - {b}))"
using sum.infinite v1 by fastforce
show ?thesis
proof
show "x ∈ convex hull insert a (S - {b})"
unfolding convex_hull_finite [OF fin] mem_Collect_eq
proof (intro conjI exI ballI)
have "(∑x ∈ insert a (S - {b}). if x=a then v b / u b else v x - (v b / u b) * u x) =
v b / u b + (∑x ∈ S - {b}. v x - (v b / u b) * u x)"
using ‹a ∉ S› ‹b ∈ S› True
by (auto intro!: sum.cong split: if_split_asm)
also have "... = v b / u b + (∑x ∈ S - {b}. v x) - (v b / u b) * (∑x ∈ S - {b}. u x)"
by (simp add: Groups_Big.sum_subtractf sum_distrib_left)
also have "... = (∑x∈S. v x)"
using ‹0 < u b› True by (simp add: Groups_Big.sum_diff1 u1 field_simps)
finally show "sum (λx. if x=a then v b / u b else v x - (v b / u b) * u x) (insert a (S - {b})) = 1"
by (simp add: v1)
show "0 ≤ (if i = a then v b / u b else v i - v b / u b * u i)"
if "i ∈ insert a (S - {b})" for i
using ‹0 < u b› ‹0 < v b› v0 [of i] le_Max [of i] beq that False
by (auto simp: field_simps split: if_split_asm)
have "(∑x∈insert a (S - {b}). (if x=a then v b / u b else v x - v b / u b * u x) *⇩R x) =
(v b / u b) *⇩R a + (∑x∈S - {b}. (v x - v b / u b * u x) *⇩R x)"
using ‹a ∉ S› ‹b ∈ S› True by (auto intro!: sum.cong split: if_split_asm)
also have "... = (v b / u b) *⇩R a + (∑x ∈ S - {b}. v x *⇩R x) - (v b / u b) *⇩R (∑x ∈ S - {b}. u x *⇩R x)"
by (simp add: Groups_Big.sum_subtractf scaleR_left_diff_distrib sum_distrib_left scale_sum_right)
also have "... = (∑x∈S. v x *⇩R x)"
using ‹0 < u b› True by (simp add: ua vx Groups_Big.sum_diff1 algebra_simps)
finally
show "(∑x∈insert a (S - {b}). (if x=a then v b / u b else v x - v b / u b * u x) *⇩R x) = x"
by (simp add: vx)
qed
qed (rule ‹b ∈ S›)
qed
next
case False
obtain T where "finite T" "T ⊆ S" and caT: "card T ≤ Suc (DIM('a))" and xT: "x ∈ convex hull T"
using xS by (auto simp: caratheodory [of S])
with False obtain b where b: "b ∈ S" "b ∉ T"
by (metis antisym subsetI)
show ?thesis
proof
show "x ∈ convex hull insert a (S - {b})"
using ‹T ⊆ S› b by (blast intro: subsetD [OF hull_mono xT])
qed (rule ‹b ∈ S›)
qed
qed
lemma convex_hull_exchange_Union:
fixes a :: "'a::euclidean_space"
assumes "a ∈ convex hull S"
shows "convex hull S = (⋃b ∈ S. convex hull (insert a (S - {b})))" (is "?lhs = ?rhs")
proof
show "?lhs ⊆ ?rhs"
by (blast intro: in_convex_hull_exchange [OF assms])
show "?rhs ⊆ ?lhs"
proof clarify
fix x b
assume"b ∈ S" "x ∈ convex hull insert a (S - {b})"
then show "x ∈ convex hull S" if "b ∈ S"
by (metis (no_types) that assms order_refl hull_mono hull_redundant insert_Diff_single insert_subset subsetCE)
qed
qed
lemma Un_closed_segment:
fixes a :: "'a::euclidean_space"
assumes "b ∈ closed_segment a c"
shows "closed_segment a b ∪ closed_segment b c = closed_segment a c"
proof (cases "c = a")
case True
with assms show ?thesis by simp
next
case False
with assms have "convex hull {a, b} ∪ convex hull {b, c} = (⋃ba∈{a, c}. convex hull insert b ({a, c} - {ba}))"
by (auto simp: insert_Diff_if insert_commute)
then show ?thesis
using convex_hull_exchange_Union
by (metis assms segment_convex_hull)
qed
lemma Un_open_segment:
fixes a :: "'a::euclidean_space"
assumes "b ∈ open_segment a c"
shows "open_segment a b ∪ {b} ∪ open_segment b c = open_segment a c" (is "?lhs = ?rhs")
proof -
have b: "b ∈ closed_segment a c"
by (simp add: assms open_closed_segment)
have *: "?rhs ⊆ insert b (open_segment a b ∪ open_segment b c)"
if "{b,c,a} ∪ open_segment a b ∪ open_segment b c = {c,a} ∪ ?rhs"
proof -
have "insert a (insert c (insert b (open_segment a b ∪ open_segment b c))) = insert a (insert c (?rhs))"
using that by (simp add: insert_commute)
then show ?thesis
by (metis (no_types) Diff_cancel Diff_eq_empty_iff Diff_insert2 open_segment_def)
qed
show ?thesis
proof
show "?lhs ⊆ ?rhs"
by (simp add: assms b subset_open_segment)
show "?rhs ⊆ ?lhs"
using Un_closed_segment [OF b] *
by (simp add: closed_segment_eq_open insert_commute)
qed
qed
subsection‹Covering an open set by a countable chain of compact sets›
proposition open_Union_compact_subsets:
fixes S :: "'a::euclidean_space set"
assumes "open S"
obtains C where "⋀n. compact(C n)" "⋀n. C n ⊆ S"
"⋀n. C n ⊆ interior(C(Suc n))"
"⋃(range C) = S"
"⋀K. ⟦compact K; K ⊆ S⟧ ⟹ ∃N. ∀n≥N. K ⊆ (C n)"
proof (cases "S = {}")
case True
then show ?thesis
by (rule_tac C = "λn. {}" in that) auto
next
case False
then obtain a where "a ∈ S"
by auto
let ?C = "λn. cball a (real n) - (⋃x ∈ -S. ⋃e ∈ ball 0 (1 / real(Suc n)). {x + e})"
have "∃N. ∀n≥N. K ⊆ (f n)"
if "⋀n. compact(f n)" and sub_int: "⋀n. f n ⊆ interior (f(Suc n))"
and eq: "⋃(range f) = S" and "compact K" "K ⊆ S" for f K
proof -
have *: "∀n. f n ⊆ (⋃n. interior (f n))"
by (meson Sup_upper2 UNIV_I ‹⋀n. f n ⊆ interior (f (Suc n))› image_iff)
have mono: "⋀m n. m ≤ n ⟹f m ⊆ f n"
by (meson dual_order.trans interior_subset lift_Suc_mono_le sub_int)
obtain I where "finite I" and I: "K ⊆ (⋃i∈I. interior (f i))"
proof (rule compactE_image [OF ‹compact K›])
show "K ⊆ (⋃n. interior (f n))"
using ‹K ⊆ S› ‹⋃(f ` UNIV) = S› * by blast
qed auto
{ fix n
assume n: "Max I ≤ n"
have "(⋃i∈I. interior (f i)) ⊆ f n"
by (rule UN_least) (meson dual_order.trans interior_subset mono I Max_ge [OF ‹finite I›] n)
then have "K ⊆ f n"
using I by auto
}
then show ?thesis
by blast
qed
moreover have "∃f. (∀n. compact(f n)) ∧ (∀n. (f n) ⊆ S) ∧ (∀n. (f n) ⊆ interior(f(Suc n))) ∧
((⋃(range f) = S))"
proof (intro exI conjI allI)
show "⋀n. compact (?C n)"
by (auto simp: compact_diff open_sums)
show "⋀n. ?C n ⊆ S"
by auto
show "?C n ⊆ interior (?C (Suc n))" for n
proof (simp add: interior_diff, rule Diff_mono)
show "cball a (real n) ⊆ ball a (1 + real n)"
by (simp add: cball_subset_ball_iff)
have cl: "closed (⋃x∈- S. ⋃e∈cball 0 (1 / (2 + real n)). {x + e})"
using assms by (auto intro: closed_compact_sums)
have "closure (⋃x∈- S. ⋃y∈ball 0 (1 / (2 + real n)). {x + y})
⊆ (⋃x ∈ -S. ⋃e ∈ cball 0 (1 / (2 + real n)). {x + e})"
by (intro closure_minimal UN_mono ball_subset_cball order_refl cl)
also have "... ⊆ (⋃x ∈ -S. ⋃y∈ball 0 (1 / (1 + real n)). {x + y})"
by (simp add: cball_subset_ball_iff field_split_simps UN_mono)
finally show "closure (⋃x∈- S. ⋃y∈ball 0 (1 / (2 + real n)). {x + y})
⊆ (⋃x ∈ -S. ⋃y∈ball 0 (1 / (1 + real n)). {x + y})" .
qed
have "S ⊆ ⋃ (range ?C)"
proof
fix x
assume x: "x ∈ S"
then obtain e where "e > 0" and e: "ball x e ⊆ S"
using assms open_contains_ball by blast
then obtain N1 where "N1 > 0" and N1: "real N1 > 1/e"
using reals_Archimedean2
by (metis divide_less_0_iff less_eq_real_def neq0_conv not_le of_nat_0 of_nat_1 of_nat_less_0_iff)
obtain N2 where N2: "norm(x - a) ≤ real N2"
by (meson real_arch_simple)
have N12: "inverse((N1 + N2) + 1) ≤ inverse(N1)"
using ‹N1 > 0› by (auto simp: field_split_simps)
have "x ≠ y + z" if "y ∉ S" "norm z < 1 / (1 + (real N1 + real N2))" for y z
proof -
have "e * real N1 < e * (1 + (real N1 + real N2))"
by (simp add: ‹0 < e›)
then have "1 / (1 + (real N1 + real N2)) < e"
using N1 ‹e > 0›
by (metis divide_less_eq less_trans mult.commute of_nat_add of_nat_less_0_iff of_nat_Suc)
then have "x - z ∈ ball x e"
using that by simp
then have "x - z ∈ S"
using e by blast
with that show ?thesis
by auto
qed
with N2 show "x ∈ ⋃ (range ?C)"
by (rule_tac a = "N1+N2" in UN_I) (auto simp: dist_norm norm_minus_commute)
qed
then show "⋃ (range ?C) = S" by auto
qed
ultimately show ?thesis
using that by metis
qed
subsection‹Orthogonal complement›
definition orthogonal_comp (‹(‹open_block notation=‹postfix ⊥››_⇧⊥)› [80] 80)
where "orthogonal_comp W ≡ {x. ∀y ∈ W. orthogonal y x}"
proposition subspace_orthogonal_comp: "subspace (W⇧⊥)"
unfolding subspace_def orthogonal_comp_def orthogonal_def
by (auto simp: inner_right_distrib)
lemma orthogonal_comp_anti_mono:
assumes "A ⊆ B"
shows "B⇧⊥ ⊆ A⇧⊥"
proof
fix x assume x: "x ∈ B⇧⊥"
show "x ∈ orthogonal_comp A" using x unfolding orthogonal_comp_def
by (simp add: orthogonal_def, metis assms in_mono)
qed
lemma orthogonal_comp_null [simp]: "{0}⇧⊥ = UNIV"
by (auto simp: orthogonal_comp_def orthogonal_def)
lemma orthogonal_comp_UNIV [simp]: "UNIV⇧⊥ = {0}"
unfolding orthogonal_comp_def orthogonal_def
by auto (use inner_eq_zero_iff in blast)
lemma orthogonal_comp_subset: "U ⊆ U⇧⊥⇧⊥"
by (auto simp: orthogonal_comp_def orthogonal_def inner_commute)
lemma subspace_sum_minimal:
assumes "S ⊆ U" "T ⊆ U" "subspace U"
shows "S + T ⊆ U"
proof
fix x
assume "x ∈ S + T"
then obtain xs xt where "xs ∈ S" "xt ∈ T" "x = xs+xt"
by (meson set_plus_elim)
then show "x ∈ U"
by (meson assms subsetCE subspace_add)
qed
proposition subspace_sum_orthogonal_comp:
fixes U :: "'a :: euclidean_space set"
assumes "subspace U"
shows "U + U⇧⊥ = UNIV"
proof -
obtain B where "B ⊆ U"
and ortho: "pairwise orthogonal B" "⋀x. x ∈ B ⟹ norm x = 1"
and "independent B" "card B = dim U" "span B = U"
using orthonormal_basis_subspace [OF assms] by metis
then have "finite B"
by (simp add: indep_card_eq_dim_span)
have *: "∀x∈B. ∀y∈B. x ∙ y = (if x=y then 1 else 0)"
using ortho norm_eq_1 by (auto simp: orthogonal_def pairwise_def)
{ fix v
let ?u = "∑b∈B. (v ∙ b) *⇩R b"
have "v = ?u + (v - ?u)"
by simp
moreover have "?u ∈ U"
by (metis (no_types, lifting) ‹span B = U› assms subspace_sum span_base span_mul)
moreover have "(v - ?u) ∈ U⇧⊥"
proof (clarsimp simp: orthogonal_comp_def orthogonal_def)
fix y
assume "y ∈ U"
with ‹span B = U› span_finite [OF ‹finite B›]
obtain u where u: "y = (∑b∈B. u b *⇩R b)"
by auto
have "b ∙ (v - ?u) = 0" if "b ∈ B" for b
using that ‹finite B›
by (simp add: * algebra_simps inner_sum_right if_distrib [of "(*)v" for v] inner_commute cong: if_cong)
then show "y ∙ (v - ?u) = 0"
by (simp add: u inner_sum_left)
qed
ultimately have "v ∈ U + U⇧⊥"
using set_plus_intro by fastforce
} then show ?thesis
by auto
qed
lemma orthogonal_Int_0:
assumes "subspace U"
shows "U ∩ U⇧⊥ = {0}"
using orthogonal_comp_def orthogonal_self
by (force simp: assms subspace_0 subspace_orthogonal_comp)
lemma orthogonal_comp_self:
fixes U :: "'a :: euclidean_space set"
assumes "subspace U"
shows "U⇧⊥⇧⊥ = U"
proof
have ssU': "subspace (U⇧⊥)"
by (simp add: subspace_orthogonal_comp)
have "u ∈ U" if "u ∈ U⇧⊥⇧⊥" for u
proof -
obtain v w where "u = v+w" "v ∈ U" "w ∈ U⇧⊥"
using subspace_sum_orthogonal_comp [OF assms] set_plus_elim by blast
then have "u-v ∈ U⇧⊥"
by simp
moreover have "v ∈ U⇧⊥⇧⊥"
using ‹v ∈ U› orthogonal_comp_subset by blast
then have "u-v ∈ U⇧⊥⇧⊥"
by (simp add: subspace_diff subspace_orthogonal_comp that)
ultimately have "u-v = 0"
using orthogonal_Int_0 ssU' by blast
with ‹v ∈ U› show ?thesis
by auto
qed
then show "U⇧⊥⇧⊥ ⊆ U"
by auto
qed (use orthogonal_comp_subset in auto)
lemma ker_orthogonal_comp_adjoint:
fixes f :: "'m::euclidean_space ⇒ 'n::euclidean_space"
assumes "linear f"
shows "f -` {0} = (range (adjoint f))⇧⊥"
proof -
have "⋀x. ⟦∀y. y ∙ f x = 0⟧ ⟹ f x = 0"
using assms inner_commute all_zero_iff by metis
then show ?thesis
using assms
by (auto simp: orthogonal_comp_def orthogonal_def adjoint_works inner_commute)
qed
subsection ‹A non-injective linear function maps into a hyperplane.›
lemma linear_surj_adj_imp_inj:
fixes f :: "'m::euclidean_space ⇒ 'n::euclidean_space"
assumes "linear f" "surj (adjoint f)"
shows "inj f"
proof -
have "∃x. y = adjoint f x" for y
using assms by (simp add: surjD)
then show "inj f"
using assms unfolding inj_on_def image_def
by (metis (no_types) adjoint_works euclidean_eqI)
qed
lemma surj_adjoint_iff_inj [simp]:
fixes f :: "'m::euclidean_space ⇒ 'n::euclidean_space"
assumes "linear f"
shows "surj (adjoint f) ⟷ inj f"
proof
assume "surj (adjoint f)"
then show "inj f"
by (simp add: assms linear_surj_adj_imp_inj)
next
assume "inj f"
have "f -` {0} = {0}"
using assms ‹inj f› linear_0 linear_injective_0 by fastforce
moreover have "f -` {0} = range (adjoint f)⇧⊥"
by (intro ker_orthogonal_comp_adjoint assms)
ultimately have "range (adjoint f)⇧⊥⇧⊥ = UNIV"
by (metis orthogonal_comp_null)
then show "surj (adjoint f)"
using adjoint_linear ‹linear f›
by (subst (asm) orthogonal_comp_self)
(simp add: adjoint_linear linear_subspace_image)
qed
lemma inj_adjoint_iff_surj [simp]:
fixes f :: "'m::euclidean_space ⇒ 'n::euclidean_space"
assumes "linear f"
shows "inj (adjoint f) ⟷ surj f"
proof
assume "inj (adjoint f)"
have "(adjoint f) -` {0} = {0}"
by (metis ‹inj (adjoint f)› adjoint_linear assms surj_adjoint_iff_inj ker_orthogonal_comp_adjoint orthogonal_comp_UNIV)
then have "(range(f))⇧⊥ = {0}"
by (metis (no_types, opaque_lifting) adjoint_adjoint adjoint_linear assms ker_orthogonal_comp_adjoint set_zero)
then show "surj f"
by (metis ‹inj (adjoint f)› adjoint_adjoint adjoint_linear assms surj_adjoint_iff_inj)
next
assume "surj f"
then have "range f = (adjoint f -` {0})⇧⊥"
by (simp add: adjoint_adjoint adjoint_linear assms ker_orthogonal_comp_adjoint)
then have "{0} = adjoint f -` {0}"
using ‹surj f› adjoint_adjoint adjoint_linear assms ker_orthogonal_comp_adjoint by force
then show "inj (adjoint f)"
by (simp add: ‹surj f› adjoint_adjoint adjoint_linear assms linear_surj_adj_imp_inj)
qed
lemma linear_singular_into_hyperplane:
fixes f :: "'n::euclidean_space ⇒ 'n"
assumes "linear f"
shows "¬ inj f ⟷ (∃a. a ≠ 0 ∧ (∀x. a ∙ f x = 0))" (is "_ = ?rhs")
proof
assume "¬inj f"
then show ?rhs
using all_zero_iff
by (metis (no_types, opaque_lifting) adjoint_clauses(2) adjoint_linear assms
linear_injective_0 linear_injective_imp_surjective linear_surj_adj_imp_inj)
next
assume ?rhs
then show "¬inj f"
by (metis assms linear_injective_isomorphism all_zero_iff)
qed
lemma linear_singular_image_hyperplane:
fixes f :: "'n::euclidean_space ⇒ 'n"
assumes "linear f" "¬inj f"
obtains a where "a ≠ 0" "⋀S. f ` S ⊆ {x. a ∙ x = 0}"
using assms by (fastforce simp add: linear_singular_into_hyperplane)
end