section ‹Line segments, Starlike Sets, etc›
theory Starlike
imports Convex_Euclidean_Space
begin
subsection ‹Midpoint›
definition%important midpoint :: "'a::real_vector ⇒ 'a ⇒ 'a"
where "midpoint a b = (inverse (2::real)) *⇩R (a + b)"
lemma midpoint_idem [simp]: "midpoint x x = x"
unfolding midpoint_def by simp
lemma midpoint_sym: "midpoint a b = midpoint b a"
unfolding midpoint_def by (auto simp add: scaleR_right_distrib)
lemma midpoint_eq_iff: "midpoint a b = c ⟷ a + b = c + c"
proof -
have "midpoint a b = c ⟷ scaleR 2 (midpoint a b) = scaleR 2 c"
by simp
then show ?thesis
unfolding midpoint_def scaleR_2 [symmetric] by simp
qed
lemma
fixes a::real
assumes "a ≤ b" shows ge_midpoint_1: "a ≤ midpoint a b"
and le_midpoint_1: "midpoint a b ≤ b"
by (simp_all add: midpoint_def assms)
lemma dist_midpoint:
fixes a b :: "'a::real_normed_vector" shows
"dist a (midpoint a b) = (dist a b) / 2" (is ?t1)
"dist b (midpoint a b) = (dist a b) / 2" (is ?t2)
"dist (midpoint a b) a = (dist a b) / 2" (is ?t3)
"dist (midpoint a b) b = (dist a b) / 2" (is ?t4)
proof -
have *: "⋀x y::'a. 2 *⇩R x = - y ⟹ norm x = (norm y) / 2"
unfolding equation_minus_iff by auto
have **: "⋀x y::'a. 2 *⇩R x = y ⟹ norm x = (norm y) / 2"
by auto
note scaleR_right_distrib [simp]
show ?t1
unfolding midpoint_def dist_norm
apply (rule **)
apply (simp add: scaleR_right_diff_distrib)
apply (simp add: scaleR_2)
done
show ?t2
unfolding midpoint_def dist_norm
apply (rule *)
apply (simp add: scaleR_right_diff_distrib)
apply (simp add: scaleR_2)
done
show ?t3
unfolding midpoint_def dist_norm
apply (rule *)
apply (simp add: scaleR_right_diff_distrib)
apply (simp add: scaleR_2)
done
show ?t4
unfolding midpoint_def dist_norm
apply (rule **)
apply (simp add: scaleR_right_diff_distrib)
apply (simp add: scaleR_2)
done
qed
lemma midpoint_eq_endpoint [simp]:
"midpoint a b = a ⟷ a = b"
"midpoint a b = b ⟷ a = b"
unfolding midpoint_eq_iff by auto
lemma midpoint_plus_self [simp]: "midpoint a b + midpoint a b = a + b"
using midpoint_eq_iff by metis
lemma midpoint_linear_image:
"linear f ⟹ midpoint(f a)(f b) = f(midpoint a b)"
by (simp add: linear_iff midpoint_def)
subsection ‹Line segments›
definition%important closed_segment :: "'a::real_vector ⇒ 'a ⇒ 'a set"
where "closed_segment a b = {(1 - u) *⇩R a + u *⇩R b | u::real. 0 ≤ u ∧ u ≤ 1}"
definition%important open_segment :: "'a::real_vector ⇒ 'a ⇒ 'a set" where
"open_segment a b ≡ closed_segment a b - {a,b}"
lemmas segment = open_segment_def closed_segment_def
lemma in_segment:
"x ∈ closed_segment a b ⟷ (∃u. 0 ≤ u ∧ u ≤ 1 ∧ x = (1 - u) *⇩R a + u *⇩R b)"
"x ∈ open_segment a b ⟷ a ≠ b ∧ (∃u. 0 < u ∧ u < 1 ∧ x = (1 - u) *⇩R a + u *⇩R b)"
using less_eq_real_def by (auto simp: segment algebra_simps)
lemma closed_segment_linear_image:
"closed_segment (f a) (f b) = f ` (closed_segment a b)" if "linear f"
proof -
interpret linear f by fact
show ?thesis
by (force simp add: in_segment add scale)
qed
lemma open_segment_linear_image:
"⟦linear f; inj f⟧ ⟹ open_segment (f a) (f b) = f ` (open_segment a b)"
by (force simp: open_segment_def closed_segment_linear_image inj_on_def)
lemma closed_segment_translation:
"closed_segment (c + a) (c + b) = image (λx. c + x) (closed_segment a b)"
apply safe
apply (rule_tac x="x-c" in image_eqI)
apply (auto simp: in_segment algebra_simps)
done
lemma open_segment_translation:
"open_segment (c + a) (c + b) = image (λx. c + x) (open_segment a b)"
by (simp add: open_segment_def closed_segment_translation translation_diff)
lemma closed_segment_of_real:
"closed_segment (of_real x) (of_real y) = of_real ` closed_segment x y"
apply (auto simp: image_iff in_segment scaleR_conv_of_real)
apply (rule_tac x="(1-u)*x + u*y" in bexI)
apply (auto simp: in_segment)
done
lemma open_segment_of_real:
"open_segment (of_real x) (of_real y) = of_real ` open_segment x y"
apply (auto simp: image_iff in_segment scaleR_conv_of_real)
apply (rule_tac x="(1-u)*x + u*y" in bexI)
apply (auto simp: in_segment)
done
lemma closed_segment_Reals:
"⟦x ∈ Reals; y ∈ Reals⟧ ⟹ closed_segment x y = of_real ` closed_segment (Re x) (Re y)"
by (metis closed_segment_of_real of_real_Re)
lemma open_segment_Reals:
"⟦x ∈ Reals; y ∈ Reals⟧ ⟹ open_segment x y = of_real ` open_segment (Re x) (Re y)"
by (metis open_segment_of_real of_real_Re)
lemma open_segment_PairD:
"(x, x') ∈ open_segment (a, a') (b, b')
⟹ (x ∈ open_segment a b ∨ a = b) ∧ (x' ∈ open_segment a' b' ∨ a' = b')"
by (auto simp: in_segment)
lemma closed_segment_PairD:
"(x, x') ∈ closed_segment (a, a') (b, b') ⟹ x ∈ closed_segment a b ∧ x' ∈ closed_segment a' b'"
by (auto simp: closed_segment_def)
lemma closed_segment_translation_eq [simp]:
"d + x ∈ closed_segment (d + a) (d + b) ⟷ x ∈ closed_segment a b"
proof -
have *: "⋀d x a b. x ∈ closed_segment a b ⟹ d + x ∈ closed_segment (d + a) (d + b)"
apply (simp add: closed_segment_def)
apply (erule ex_forward)
apply (simp add: algebra_simps)
done
show ?thesis
using * [where d = "-d"] *
by (fastforce simp add:)
qed
lemma open_segment_translation_eq [simp]:
"d + x ∈ open_segment (d + a) (d + b) ⟷ x ∈ open_segment a b"
by (simp add: open_segment_def)
lemma of_real_closed_segment [simp]:
"of_real x ∈ closed_segment (of_real a) (of_real b) ⟷ x ∈ closed_segment a b"
apply (auto simp: in_segment scaleR_conv_of_real elim!: ex_forward)
using of_real_eq_iff by fastforce
lemma of_real_open_segment [simp]:
"of_real x ∈ open_segment (of_real a) (of_real b) ⟷ x ∈ open_segment a b"
apply (auto simp: in_segment scaleR_conv_of_real elim!: ex_forward del: exE)
using of_real_eq_iff by fastforce
lemma convex_contains_segment:
"convex S ⟷ (∀a∈S. ∀b∈S. closed_segment a b ⊆ S)"
unfolding convex_alt closed_segment_def by auto
lemma closed_segment_in_Reals:
"⟦x ∈ closed_segment a b; a ∈ Reals; b ∈ Reals⟧ ⟹ x ∈ Reals"
by (meson subsetD convex_Reals convex_contains_segment)
lemma open_segment_in_Reals:
"⟦x ∈ open_segment a b; a ∈ Reals; b ∈ Reals⟧ ⟹ x ∈ Reals"
by (metis Diff_iff closed_segment_in_Reals open_segment_def)
lemma closed_segment_subset: "⟦x ∈ S; y ∈ S; convex S⟧ ⟹ closed_segment x y ⊆ S"
by (simp add: convex_contains_segment)
lemma closed_segment_subset_convex_hull:
"⟦x ∈ convex hull S; y ∈ convex hull S⟧ ⟹ closed_segment x y ⊆ convex hull S"
using convex_contains_segment by blast
lemma segment_convex_hull:
"closed_segment a b = convex hull {a,b}"
proof -
have *: "⋀x. {x} ≠ {}" by auto
show ?thesis
unfolding segment convex_hull_insert[OF *] convex_hull_singleton
by (safe; rule_tac x="1 - u" in exI; force)
qed
lemma open_closed_segment: "u ∈ open_segment w z ⟹ u ∈ closed_segment w z"
by (auto simp add: closed_segment_def open_segment_def)
lemma segment_open_subset_closed:
"open_segment a b ⊆ closed_segment a b"
by (auto simp: closed_segment_def open_segment_def)
lemma bounded_closed_segment:
fixes a :: "'a::euclidean_space" shows "bounded (closed_segment a b)"
by (simp add: segment_convex_hull compact_convex_hull compact_imp_bounded)
lemma bounded_open_segment:
fixes a :: "'a::euclidean_space" shows "bounded (open_segment a b)"
by (rule bounded_subset [OF bounded_closed_segment segment_open_subset_closed])
lemmas bounded_segment = bounded_closed_segment open_closed_segment
lemma ends_in_segment [iff]: "a ∈ closed_segment a b" "b ∈ closed_segment a b"
unfolding segment_convex_hull
by (auto intro!: hull_subset[unfolded subset_eq, rule_format])
lemma eventually_closed_segment:
fixes x0::"'a::real_normed_vector"
assumes "open X0" "x0 ∈ X0"
shows "∀⇩F x in at x0 within U. closed_segment x0 x ⊆ X0"
proof -
from openE[OF assms]
obtain e where e: "0 < e" "ball x0 e ⊆ X0" .
then have "∀⇩F x in at x0 within U. x ∈ ball x0 e"
by (auto simp: dist_commute eventually_at)
then show ?thesis
proof eventually_elim
case (elim x)
have "x0 ∈ ball x0 e" using ‹e > 0› by simp
from convex_ball[unfolded convex_contains_segment, rule_format, OF this elim]
have "closed_segment x0 x ⊆ ball x0 e" .
also note ‹… ⊆ X0›
finally show ?case .
qed
qed
lemma segment_furthest_le:
fixes a b x y :: "'a::euclidean_space"
assumes "x ∈ closed_segment a b"
shows "norm (y - x) ≤ norm (y - a) ∨ norm (y - x) ≤ norm (y - b)"
proof -
obtain z where "z ∈ {a, b}" "norm (x - y) ≤ norm (z - y)"
using simplex_furthest_le[of "{a, b}" y]
using assms[unfolded segment_convex_hull]
by auto
then show ?thesis
by (auto simp add:norm_minus_commute)
qed
lemma closed_segment_commute: "closed_segment a b = closed_segment b a"
proof -
have "{a, b} = {b, a}" by auto
thus ?thesis
by (simp add: segment_convex_hull)
qed
lemma segment_bound1:
assumes "x ∈ closed_segment a b"
shows "norm (x - a) ≤ norm (b - a)"
proof -
obtain u where "x = (1 - u) *⇩R a + u *⇩R b" "0 ≤ u" "u ≤ 1"
using assms by (auto simp add: closed_segment_def)
then show "norm (x - a) ≤ norm (b - a)"
apply clarify
apply (auto simp: algebra_simps)
apply (simp add: scaleR_diff_right [symmetric] mult_left_le_one_le)
done
qed
lemma segment_bound:
assumes "x ∈ closed_segment a b"
shows "norm (x - a) ≤ norm (b - a)" "norm (x - b) ≤ norm (b - a)"
apply (simp add: assms segment_bound1)
by (metis assms closed_segment_commute dist_commute dist_norm segment_bound1)
lemma open_segment_commute: "open_segment a b = open_segment b a"
proof -
have "{a, b} = {b, a}" by auto
thus ?thesis
by (simp add: closed_segment_commute open_segment_def)
qed
lemma closed_segment_idem [simp]: "closed_segment a a = {a}"
unfolding segment by (auto simp add: algebra_simps)
lemma open_segment_idem [simp]: "open_segment a a = {}"
by (simp add: open_segment_def)
lemma closed_segment_eq_open: "closed_segment a b = open_segment a b ∪ {a,b}"
using open_segment_def by auto
lemma convex_contains_open_segment:
"convex s ⟷ (∀a∈s. ∀b∈s. open_segment a b ⊆ s)"
by (simp add: convex_contains_segment closed_segment_eq_open)
lemma closed_segment_eq_real_ivl:
fixes a b::real
shows "closed_segment a b = (if a ≤ b then {a .. b} else {b .. a})"
proof -
have "b ≤ a ⟹ closed_segment b a = {b .. a}"
and "a ≤ b ⟹ closed_segment a b = {a .. b}"
by (auto simp: convex_hull_eq_real_cbox segment_convex_hull)
thus ?thesis
by (auto simp: closed_segment_commute)
qed
lemma open_segment_eq_real_ivl:
fixes a b::real
shows "open_segment a b = (if a ≤ b then {a<..<b} else {b<..<a})"
by (auto simp: closed_segment_eq_real_ivl open_segment_def split: if_split_asm)
lemma closed_segment_real_eq:
fixes u::real shows "closed_segment u v = (λx. (v - u) * x + u) ` {0..1}"
by (simp add: add.commute [of u] image_affinity_atLeastAtMost [where c=u] closed_segment_eq_real_ivl)
lemma dist_in_closed_segment:
fixes a :: "'a :: euclidean_space"
assumes "x ∈ closed_segment a b"
shows "dist x a ≤ dist a b ∧ dist x b ≤ dist a b"
proof (intro conjI)
obtain u where u: "0 ≤ u" "u ≤ 1" and x: "x = (1 - u) *⇩R a + u *⇩R b"
using assms by (force simp: in_segment algebra_simps)
have "dist x a = u * dist a b"
apply (simp add: dist_norm algebra_simps x)
by (metis ‹0 ≤ u› abs_of_nonneg norm_minus_commute norm_scaleR real_vector.scale_right_diff_distrib)
also have "... ≤ dist a b"
by (simp add: mult_left_le_one_le u)
finally show "dist x a ≤ dist a b" .
have "dist x b = norm ((1-u) *⇩R a - (1-u) *⇩R b)"
by (simp add: dist_norm algebra_simps x)
also have "... = (1-u) * dist a b"
proof -
have "norm ((1 - 1 * u) *⇩R (a - b)) = (1 - 1 * u) * norm (a - b)"
using ‹u ≤ 1› by force
then show ?thesis
by (simp add: dist_norm real_vector.scale_right_diff_distrib)
qed
also have "... ≤ dist a b"
by (simp add: mult_left_le_one_le u)
finally show "dist x b ≤ dist a b" .
qed
lemma dist_in_open_segment:
fixes a :: "'a :: euclidean_space"
assumes "x ∈ open_segment a b"
shows "dist x a < dist a b ∧ dist x b < dist a b"
proof (intro conjI)
obtain u where u: "0 < u" "u < 1" and x: "x = (1 - u) *⇩R a + u *⇩R b"
using assms by (force simp: in_segment algebra_simps)
have "dist x a = u * dist a b"
apply (simp add: dist_norm algebra_simps x)
by (metis abs_of_nonneg less_eq_real_def norm_minus_commute norm_scaleR real_vector.scale_right_diff_distrib ‹0 < u›)
also have *: "... < dist a b"
by (metis (no_types) assms dist_eq_0_iff dist_not_less_zero in_segment(2) linorder_neqE_linordered_idom mult.left_neutral real_mult_less_iff1 ‹u < 1›)
finally show "dist x a < dist a b" .
have ab_ne0: "dist a b ≠ 0"
using * by fastforce
have "dist x b = norm ((1-u) *⇩R a - (1-u) *⇩R b)"
by (simp add: dist_norm algebra_simps x)
also have "... = (1-u) * dist a b"
proof -
have "norm ((1 - 1 * u) *⇩R (a - b)) = (1 - 1 * u) * norm (a - b)"
using ‹u < 1› by force
then show ?thesis
by (simp add: dist_norm real_vector.scale_right_diff_distrib)
qed
also have "... < dist a b"
using ab_ne0 ‹0 < u› by simp
finally show "dist x b < dist a b" .
qed
lemma dist_decreases_open_segment_0:
fixes x :: "'a :: euclidean_space"
assumes "x ∈ open_segment 0 b"
shows "dist c x < dist c 0 ∨ dist c x < dist c b"
proof (rule ccontr, clarsimp simp: not_less)
obtain u where u: "0 ≠ b" "0 < u" "u < 1" and x: "x = u *⇩R b"
using assms by (auto simp: in_segment)
have xb: "x ∙ b < b ∙ b"
using u x by auto
assume "norm c ≤ dist c x"
then have "c ∙ c ≤ (c - x) ∙ (c - x)"
by (simp add: dist_norm norm_le)
moreover have "0 < x ∙ b"
using u x by auto
ultimately have less: "c ∙ b < x ∙ b"
by (simp add: x algebra_simps inner_commute u)
assume "dist c b ≤ dist c x"
then have "(c - b) ∙ (c - b) ≤ (c - x) ∙ (c - x)"
by (simp add: dist_norm norm_le)
then have "(b ∙ b) * (1 - u*u) ≤ 2 * (b ∙ c) * (1-u)"
by (simp add: x algebra_simps inner_commute)
then have "(1+u) * (b ∙ b) * (1-u) ≤ 2 * (b ∙ c) * (1-u)"
by (simp add: algebra_simps)
then have "(1+u) * (b ∙ b) ≤ 2 * (b ∙ c)"
using ‹u < 1› by auto
with xb have "c ∙ b ≥ x ∙ b"
by (auto simp: x algebra_simps inner_commute)
with less show False by auto
qed
proposition dist_decreases_open_segment:
fixes a :: "'a :: euclidean_space"
assumes "x ∈ open_segment a b"
shows "dist c x < dist c a ∨ dist c x < dist c b"
proof -
have *: "x - a ∈ open_segment 0 (b - a)" using assms
by (metis diff_self open_segment_translation_eq uminus_add_conv_diff)
show ?thesis
using dist_decreases_open_segment_0 [OF *, of "c-a"] assms
by (simp add: dist_norm)
qed
corollary open_segment_furthest_le:
fixes a b x y :: "'a::euclidean_space"
assumes "x ∈ open_segment a b"
shows "norm (y - x) < norm (y - a) ∨ norm (y - x) < norm (y - b)"
by (metis assms dist_decreases_open_segment dist_norm)
corollary dist_decreases_closed_segment:
fixes a :: "'a :: euclidean_space"
assumes "x ∈ closed_segment a b"
shows "dist c x ≤ dist c a ∨ dist c x ≤ dist c b"
apply (cases "x ∈ open_segment a b")
using dist_decreases_open_segment less_eq_real_def apply blast
by (metis DiffI assms empty_iff insertE open_segment_def order_refl)
lemma convex_intermediate_ball:
fixes a :: "'a :: euclidean_space"
shows "⟦ball a r ⊆ T; T ⊆ cball a r⟧ ⟹ convex T"
apply (simp add: convex_contains_open_segment, clarify)
by (metis (no_types, hide_lams) less_le_trans mem_ball mem_cball subsetCE dist_decreases_open_segment)
lemma csegment_midpoint_subset: "closed_segment (midpoint a b) b ⊆ closed_segment a b"
apply (clarsimp simp: midpoint_def in_segment)
apply (rule_tac x="(1 + u) / 2" in exI)
apply (auto simp: algebra_simps add_divide_distrib diff_divide_distrib)
by (metis field_sum_of_halves scaleR_left.add)
lemma notin_segment_midpoint:
fixes a :: "'a :: euclidean_space"
shows "a ≠ b ⟹ a ∉ closed_segment (midpoint a b) b"
by (auto simp: dist_midpoint dest!: dist_in_closed_segment)
lemma segment_to_closest_point:
fixes S :: "'a :: euclidean_space set"
shows "⟦closed S; S ≠ {}⟧ ⟹ open_segment a (closest_point S a) ∩ S = {}"
apply (subst disjoint_iff_not_equal)
apply (clarify dest!: dist_in_open_segment)
by (metis closest_point_le dist_commute le_less_trans less_irrefl)
lemma segment_to_point_exists:
fixes S :: "'a :: euclidean_space set"
assumes "closed S" "S ≠ {}"
obtains b where "b ∈ S" "open_segment a b ∩ S = {}"
by (metis assms segment_to_closest_point closest_point_exists that)
subsubsection‹More lemmas, especially for working with the underlying formula›
lemma segment_eq_compose:
fixes a :: "'a :: real_vector"
shows "(λu. (1 - u) *⇩R a + u *⇩R b) = (λx. a + x) o (λu. u *⇩R (b - a))"
by (simp add: o_def algebra_simps)
lemma segment_degen_1:
fixes a :: "'a :: real_vector"
shows "(1 - u) *⇩R a + u *⇩R b = b ⟷ a=b ∨ u=1"
proof -
{ assume "(1 - u) *⇩R a + u *⇩R b = b"
then have "(1 - u) *⇩R a = (1 - u) *⇩R b"
by (simp add: algebra_simps)
then have "a=b ∨ u=1"
by simp
} then show ?thesis
by (auto simp: algebra_simps)
qed
lemma segment_degen_0:
fixes a :: "'a :: real_vector"
shows "(1 - u) *⇩R a + u *⇩R b = a ⟷ a=b ∨ u=0"
using segment_degen_1 [of "1-u" b a]
by (auto simp: algebra_simps)
lemma add_scaleR_degen:
fixes a b ::"'a::real_vector"
assumes "(u *⇩R b + v *⇩R a) = (u *⇩R a + v *⇩R b)" "u ≠ v"
shows "a=b"
by (metis (no_types, hide_lams) add.commute add_diff_eq diff_add_cancel real_vector.scale_cancel_left real_vector.scale_left_diff_distrib assms)
lemma closed_segment_image_interval:
"closed_segment a b = (λu. (1 - u) *⇩R a + u *⇩R b) ` {0..1}"
by (auto simp: set_eq_iff image_iff closed_segment_def)
lemma open_segment_image_interval:
"open_segment a b = (if a=b then {} else (λu. (1 - u) *⇩R a + u *⇩R b) ` {0<..<1})"
by (auto simp: open_segment_def closed_segment_def segment_degen_0 segment_degen_1)
lemmas segment_image_interval = closed_segment_image_interval open_segment_image_interval
lemma open_segment_bound1:
assumes "x ∈ open_segment a b"
shows "norm (x - a) < norm (b - a)"
proof -
obtain u where "x = (1 - u) *⇩R a + u *⇩R b" "0 < u" "u < 1" "a ≠ b"
using assms by (auto simp add: open_segment_image_interval split: if_split_asm)
then show "norm (x - a) < norm (b - a)"
apply clarify
apply (auto simp: algebra_simps)
apply (simp add: scaleR_diff_right [symmetric])
done
qed
lemma compact_segment [simp]:
fixes a :: "'a::real_normed_vector"
shows "compact (closed_segment a b)"
by (auto simp: segment_image_interval intro!: compact_continuous_image continuous_intros)
lemma closed_segment [simp]:
fixes a :: "'a::real_normed_vector"
shows "closed (closed_segment a b)"
by (simp add: compact_imp_closed)
lemma closure_closed_segment [simp]:
fixes a :: "'a::real_normed_vector"
shows "closure(closed_segment a b) = closed_segment a b"
by simp
lemma open_segment_bound:
assumes "x ∈ open_segment a b"
shows "norm (x - a) < norm (b - a)" "norm (x - b) < norm (b - a)"
apply (simp add: assms open_segment_bound1)
by (metis assms norm_minus_commute open_segment_bound1 open_segment_commute)
lemma closure_open_segment [simp]:
fixes a :: "'a::euclidean_space"
shows "closure(open_segment a b) = (if a = b then {} else closed_segment a b)"
proof -
have "closure ((λu. u *⇩R (b - a)) ` {0<..<1}) = (λu. u *⇩R (b - a)) ` closure {0<..<1}" if "a ≠ b"
apply (rule closure_injective_linear_image [symmetric])
apply (simp add:)
using that by (simp add: inj_on_def)
then show ?thesis
by (simp add: segment_image_interval segment_eq_compose closure_greaterThanLessThan [symmetric]
closure_translation image_comp [symmetric] del: closure_greaterThanLessThan)
qed
lemma closed_open_segment_iff [simp]:
fixes a :: "'a::euclidean_space" shows "closed(open_segment a b) ⟷ a = b"
by (metis open_segment_def DiffE closure_eq closure_open_segment ends_in_segment(1) insert_iff segment_image_interval(2))
lemma compact_open_segment_iff [simp]:
fixes a :: "'a::euclidean_space" shows "compact(open_segment a b) ⟷ a = b"
by (simp add: bounded_open_segment compact_eq_bounded_closed)
lemma convex_closed_segment [iff]: "convex (closed_segment a b)"
unfolding segment_convex_hull by(rule convex_convex_hull)
lemma convex_open_segment [iff]: "convex(open_segment a b)"
proof -
have "convex ((λu. u *⇩R (b-a)) ` {0<..<1})"
by (rule convex_linear_image) auto
then show ?thesis
apply (simp add: open_segment_image_interval segment_eq_compose)
by (metis image_comp convex_translation)
qed
lemmas convex_segment = convex_closed_segment convex_open_segment
lemma connected_segment [iff]:
fixes x :: "'a :: real_normed_vector"
shows "connected (closed_segment x y)"
by (simp add: convex_connected)
lemma is_interval_closed_segment_1[intro, simp]: "is_interval (closed_segment a b)" for a b::real
by (auto simp: is_interval_convex_1)
lemma IVT'_closed_segment_real:
fixes f :: "real ⇒ real"
assumes "y ∈ closed_segment (f a) (f b)"
assumes "continuous_on (closed_segment a b) f"
shows "∃x ∈ closed_segment a b. f x = y"
using IVT'[of f a y b]
IVT'[of "-f" a "-y" b]
IVT'[of f b y a]
IVT'[of "-f" b "-y" a] assms
by (cases "a ≤ b"; cases "f b ≥ f a") (auto simp: closed_segment_eq_real_ivl continuous_on_minus)
subsection‹Starlike sets›
definition%important "starlike S ⟷ (∃a∈S. ∀x∈S. closed_segment a x ⊆ S)"
lemma starlike_UNIV [simp]: "starlike UNIV"
by (simp add: starlike_def)
lemma convex_imp_starlike:
"convex S ⟹ S ≠ {} ⟹ starlike S"
unfolding convex_contains_segment starlike_def by auto
lemma affine_hull_closed_segment [simp]:
"affine hull (closed_segment a b) = affine hull {a,b}"
by (simp add: segment_convex_hull)
lemma affine_hull_open_segment [simp]:
fixes a :: "'a::euclidean_space"
shows "affine hull (open_segment a b) = (if a = b then {} else affine hull {a,b})"
by (metis affine_hull_convex_hull affine_hull_empty closure_open_segment closure_same_affine_hull segment_convex_hull)
lemma rel_interior_closure_convex_segment:
fixes S :: "_::euclidean_space set"
assumes "convex S" "a ∈ 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)
then show "?lhs ⊆ ?rhs"
proof (clarsimp simp add: convex_hull_insert_segments)
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 show "x ∈ convex hull T"
using rel_interior_closure_convex_segment [OF ‹convex C› z]
using y z by blast
qed
show "?rhs ⊆ ?lhs"
by (meson hull_mono inf_mono subset_insertI subset_refl)
qed
subsection%unimportant‹More results about segments›
lemma dist_half_times2:
fixes a :: "'a :: real_normed_vector"
shows "dist ((1 / 2) *⇩R (a + b)) x * 2 = dist (a+b) (2 *⇩R x)"
proof -
have "norm ((1 / 2) *⇩R (a + b) - x) * 2 = norm (2 *⇩R ((1 / 2) *⇩R (a + b) - x))"
by simp
also have "... = norm ((a + b) - 2 *⇩R x)"
by (simp add: real_vector.scale_right_diff_distrib)
finally show ?thesis
by (simp only: dist_norm)
qed
lemma closed_segment_as_ball:
"closed_segment a b = affine hull {a,b} ∩ cball(inverse 2 *⇩R (a + b))(norm(b - a) / 2)"
proof (cases "b = a")
case True then show ?thesis by (auto simp: hull_inc)
next
case False
then have *: "((∃u v. x = u *⇩R a + v *⇩R b ∧ u + v = 1) ∧
dist ((1 / 2) *⇩R (a + b)) x * 2 ≤ norm (b - a)) =
(∃u. x = (1 - u) *⇩R a + u *⇩R b ∧ 0 ≤ u ∧ u ≤ 1)" for x
proof -
have "((∃u v. x = u *⇩R a + v *⇩R b ∧ u + v = 1) ∧
dist ((1 / 2) *⇩R (a + b)) x * 2 ≤ norm (b - a)) =
((∃u. x = (1 - u) *⇩R a + u *⇩R b) ∧
dist ((1 / 2) *⇩R (a + b)) x * 2 ≤ norm (b - a))"
unfolding eq_diff_eq [symmetric] by simp
also have "... = (∃u. x = (1 - u) *⇩R a + u *⇩R b ∧
norm ((a+b) - (2 *⇩R x)) ≤ norm (b - a))"
by (simp add: dist_half_times2) (simp add: dist_norm)
also have "... = (∃u. x = (1 - u) *⇩R a + u *⇩R b ∧
norm ((a+b) - (2 *⇩R ((1 - u) *⇩R a + u *⇩R b))) ≤ norm (b - a))"
by auto
also have "... = (∃u. x = (1 - u) *⇩R a + u *⇩R b ∧
norm ((1 - u * 2) *⇩R (b - a)) ≤ norm (b - a))"
by (simp add: algebra_simps scaleR_2)
also have "... = (∃u. x = (1 - u) *⇩R a + u *⇩R b ∧
¦1 - u * 2¦ * norm (b - a) ≤ norm (b - a))"
by simp
also have "... = (∃u. x = (1 - u) *⇩R a + u *⇩R b ∧ ¦1 - u * 2¦ ≤ 1)"
by (simp add: mult_le_cancel_right2 False)
also have "... = (∃u. x = (1 - u) *⇩R a + u *⇩R b ∧ 0 ≤ u ∧ u ≤ 1)"
by auto
finally show ?thesis .
qed
show ?thesis
by (simp add: affine_hull_2 Set.set_eq_iff closed_segment_def *)
qed
lemma open_segment_as_ball:
"open_segment a b =
affine hull {a,b} ∩ ball(inverse 2 *⇩R (a + b))(norm(b - a) / 2)"
proof (cases "b = a")
case True then show ?thesis by (auto simp: hull_inc)
next
case False
then have *: "((∃u v. x = u *⇩R a + v *⇩R b ∧ u + v = 1) ∧
dist ((1 / 2) *⇩R (a + b)) x * 2 < norm (b - a)) =
(∃u. x = (1 - u) *⇩R a + u *⇩R b ∧ 0 < u ∧ u < 1)" for x
proof -
have "((∃u v. x = u *⇩R a + v *⇩R b ∧ u + v = 1) ∧
dist ((1 / 2) *⇩R (a + b)) x * 2 < norm (b - a)) =
((∃u. x = (1 - u) *⇩R a + u *⇩R b) ∧
dist ((1 / 2) *⇩R (a + b)) x * 2 < norm (b - a))"
unfolding eq_diff_eq [symmetric] by simp
also have "... = (∃u. x = (1 - u) *⇩R a + u *⇩R b ∧
norm ((a+b) - (2 *⇩R x)) < norm (b - a))"
by (simp add: dist_half_times2) (simp add: dist_norm)
also have "... = (∃u. x = (1 - u) *⇩R a + u *⇩R b ∧
norm ((a+b) - (2 *⇩R ((1 - u) *⇩R a + u *⇩R b))) < norm (b - a))"
by auto
also have "... = (∃u. x = (1 - u) *⇩R a + u *⇩R b ∧
norm ((1 - u * 2) *⇩R (b - a)) < norm (b - a))"
by (simp add: algebra_simps scaleR_2)
also have "... = (∃u. x = (1 - u) *⇩R a + u *⇩R b ∧
¦1 - u * 2¦ * norm (b - a) < norm (b - a))"
by simp
also have "... = (∃u. x = (1 - u) *⇩R a + u *⇩R b ∧ ¦1 - u * 2¦ < 1)"
by (simp add: mult_le_cancel_right2 False)
also have "... = (∃u. x = (1 - u) *⇩R a + u *⇩R b ∧ 0 < u ∧ u < 1)"
by auto
finally show ?thesis .
qed
show ?thesis
using False by (force simp: affine_hull_2 Set.set_eq_iff open_segment_image_interval *)
qed
lemmas segment_as_ball = closed_segment_as_ball open_segment_as_ball
lemma closed_segment_neq_empty [simp]: "closed_segment a b ≠ {}"
by auto
lemma open_segment_eq_empty [simp]: "open_segment a b = {} ⟷ a = b"
proof -
{ assume a1: "open_segment a b = {}"
have "{} ≠ {0::real<..<1}"
by simp
then have "a = b"
using a1 open_segment_image_interval by fastforce
} then show ?thesis by auto
qed
lemma open_segment_eq_empty' [simp]: "{} = open_segment a b ⟷ a = b"
using open_segment_eq_empty by blast
lemmas segment_eq_empty = closed_segment_neq_empty open_segment_eq_empty
lemma inj_segment:
fixes a :: "'a :: real_vector"
assumes "a ≠ b"
shows "inj_on (λu. (1 - u) *⇩R a + u *⇩R b) I"
proof
fix x y
assume "(1 - x) *⇩R a + x *⇩R b = (1 - y) *⇩R a + y *⇩R b"
then have "x *⇩R (b - a) = y *⇩R (b - a)"
by (simp add: algebra_simps)
with assms show "x = y"
by (simp add: real_vector.scale_right_imp_eq)
qed
lemma finite_closed_segment [simp]: "finite(closed_segment a b) ⟷ a = b"
apply auto
apply (rule ccontr)
apply (simp add: segment_image_interval)
using infinite_Icc [OF zero_less_one] finite_imageD [OF _ inj_segment] apply blast
done
lemma finite_open_segment [simp]: "finite(open_segment a b) ⟷ a = b"
by (auto simp: open_segment_def)
lemmas finite_segment = finite_closed_segment finite_open_segment
lemma closed_segment_eq_sing: "closed_segment a b = {c} ⟷ a = c ∧ b = c"
by auto
lemma open_segment_eq_sing: "open_segment a b ≠ {c}"
by (metis finite_insert finite_open_segment insert_not_empty open_segment_image_interval)
lemmas segment_eq_sing = closed_segment_eq_sing open_segment_eq_sing
lemma subset_closed_segment:
"closed_segment a b ⊆ closed_segment c d ⟷
a ∈ closed_segment c d ∧ b ∈ closed_segment c d"
by auto (meson contra_subsetD convex_closed_segment convex_contains_segment)
lemma subset_co_segment:
"closed_segment a b ⊆ open_segment c d ⟷
a ∈ open_segment c d ∧ b ∈ open_segment c d"
using closed_segment_subset by blast
lemma subset_open_segment:
fixes a :: "'a::euclidean_space"
shows "open_segment a b ⊆ open_segment c d ⟷
a = b ∨ a ∈ closed_segment c d ∧ b ∈ closed_segment c d"
(is "?lhs = ?rhs")
proof (cases "a = b")
case True then show ?thesis by simp
next
case False show ?thesis
proof
assume rhs: ?rhs
with ‹a ≠ b› have "c ≠ d"
using closed_segment_idem singleton_iff by auto
have "∃uc. (1 - u) *⇩R ((1 - ua) *⇩R c + ua *⇩R d) + u *⇩R ((1 - ub) *⇩R c + ub *⇩R d) =
(1 - uc) *⇩R c + uc *⇩R d ∧ 0 < uc ∧ uc < 1"
if neq: "(1 - ua) *⇩R c + ua *⇩R d ≠ (1 - ub) *⇩R c + ub *⇩R d" "c ≠ d"
and "a = (1 - ua) *⇩R c + ua *⇩R d" "b = (1 - ub) *⇩R c + ub *⇩R d"
and u: "0 < u" "u < 1" and uab: "0 ≤ ua" "ua ≤ 1" "0 ≤ ub" "ub ≤ 1"
for u ua ub
proof -
have "ua ≠ ub"
using neq by auto
moreover have "(u - 1) * ua ≤ 0" using u uab
by (simp add: mult_nonpos_nonneg)
ultimately have lt: "(u - 1) * ua < u * ub" using u uab
by (metis antisym_conv diff_ge_0_iff_ge le_less_trans mult_eq_0_iff mult_le_0_iff not_less)
have "p * ua + q * ub < p+q" if p: "0 < p" and q: "0 < q" for p q
proof -
have "¬ p ≤ 0" "¬ q ≤ 0"
using p q not_less by blast+
then show ?thesis
by (metis ‹ua ≠ ub› add_less_cancel_left add_less_cancel_right add_mono_thms_linordered_field(5)
less_eq_real_def mult_cancel_left1 mult_less_cancel_left2 uab(2) uab(4))
qed
then have "(1 - u) * ua + u * ub < 1" using u ‹ua ≠ ub›
by (metis diff_add_cancel diff_gt_0_iff_gt)
with lt show ?thesis
by (rule_tac x="ua + u*(ub-ua)" in exI) (simp add: algebra_simps)
qed
with rhs ‹a ≠ b› ‹c ≠ d› show ?lhs
unfolding open_segment_image_interval closed_segment_def
by (fastforce simp add:)
next
assume lhs: ?lhs
with ‹a ≠ b› have "c ≠ d"
by (meson finite_open_segment rev_finite_subset)
have "closure (open_segment a b) ⊆ closure (open_segment c d)"
using lhs closure_mono by blast
then have "closed_segment a b ⊆ closed_segment c d"
by (simp add: ‹a ≠ b› ‹c ≠ d›)
then show ?rhs
by (force simp: ‹a ≠ b›)
qed
qed
lemma subset_oc_segment:
fixes a :: "'a::euclidean_space"
shows "open_segment a b ⊆ closed_segment c d ⟷
a = b ∨ a ∈ closed_segment c d ∧ b ∈ closed_segment c d"
apply (simp add: subset_open_segment [symmetric])
apply (rule iffI)
apply (metis closure_closed_segment closure_mono closure_open_segment subset_closed_segment subset_open_segment)
apply (meson dual_order.trans segment_open_subset_closed)
done
lemmas subset_segment = subset_closed_segment subset_co_segment subset_oc_segment subset_open_segment
subsection‹Betweenness›
definition%important "between = (λ(a,b) x. x ∈ closed_segment a b)"
lemma betweenI:
assumes "0 ≤ u" "u ≤ 1" "x = (1 - u) *⇩R a + u *⇩R b"
shows "between (a, b) x"
using assms unfolding between_def closed_segment_def by auto
lemma betweenE:
assumes "between (a, b) x"
obtains u where "0 ≤ u" "u ≤ 1" "x = (1 - u) *⇩R a + u *⇩R b"
using assms unfolding between_def closed_segment_def by auto
lemma between_implies_scaled_diff:
assumes "between (S, T) X" "between (S, T) Y" "S ≠ Y"
obtains c where "(X - Y) = c *⇩R (S - Y)"
proof -
from ‹between (S, T) X› obtain u⇩X where X: "X = u⇩X *⇩R S + (1 - u⇩X) *⇩R T"
by (metis add.commute betweenE eq_diff_eq)
from ‹between (S, T) Y› obtain u⇩Y where Y: "Y = u⇩Y *⇩R S + (1 - u⇩Y) *⇩R T"
by (metis add.commute betweenE eq_diff_eq)
have "X - Y = (u⇩X - u⇩Y) *⇩R (S - T)"
proof -
from X Y have "X - Y = u⇩X *⇩R S - u⇩Y *⇩R S + ((1 - u⇩X) *⇩R T - (1 - u⇩Y) *⇩R T)" by simp
also have "… = (u⇩X - u⇩Y) *⇩R S - (u⇩X - u⇩Y) *⇩R T" by (simp add: scaleR_left.diff)
finally show ?thesis by (simp add: real_vector.scale_right_diff_distrib)
qed
moreover from Y have "S - Y = (1 - u⇩Y) *⇩R (S - T)"
by (simp add: real_vector.scale_left_diff_distrib real_vector.scale_right_diff_distrib)
moreover note ‹S ≠ Y›
ultimately have "(X - Y) = ((u⇩X - u⇩Y) / (1 - u⇩Y)) *⇩R (S - Y)" by auto
from this that show thesis by blast
qed
lemma between_mem_segment: "between (a,b) x ⟷ x ∈ closed_segment a b"
unfolding between_def by auto
lemma between: "between (a, b) (x::'a::euclidean_space) ⟷ dist a b = (dist a x) + (dist x b)"
proof (cases "a = b")
case True
then show ?thesis
by (auto simp add: between_def dist_commute)
next
case False
then have Fal: "norm (a - b) ≠ 0" and Fal2: "norm (a - b) > 0"
by auto
have *: "⋀u. a - ((1 - u) *⇩R a + u *⇩R b) = u *⇩R (a - b)"
by (auto simp add: algebra_simps)
have "norm (a - x) *⇩R (x - b) = norm (x - b) *⇩R (a - x)" if "x = (1 - u) *⇩R a + u *⇩R b" "0 ≤ u" "u ≤ 1" for u
proof -
have *: "a - x = u *⇩R (a - b)" "x - b = (1 - u) *⇩R (a - b)"
unfolding that(1) by (auto simp add:algebra_simps)
show "norm (a - x) *⇩R (x - b) = norm (x - b) *⇩R (a - x)"
unfolding norm_minus_commute[of x a] * using ‹0 ≤ u› ‹u ≤ 1›
by (auto simp add: field_simps)
qed
moreover have "∃u. x = (1 - u) *⇩R a + u *⇩R b ∧ 0 ≤ u ∧ u ≤ 1" if "dist a b = dist a x + dist x b"
proof -
let ?β = "norm (a - x) / norm (a - b)"
show "∃u. x = (1 - u) *⇩R a + u *⇩R b ∧ 0 ≤ u ∧ u ≤ 1"
proof (intro exI conjI)
show "?β ≤ 1"
using Fal2 unfolding that[unfolded dist_norm] norm_ge_zero by auto
show "x = (1 - ?β) *⇩R a + (?β) *⇩R b"
proof (subst euclidean_eq_iff; intro ballI)
fix i :: 'a
assume i: "i ∈ Basis"
have "((1 - ?β) *⇩R a + (?β) *⇩R b) ∙ i
= ((norm (a - b) - norm (a - x)) * (a ∙ i) + norm (a - x) * (b ∙ i)) / norm (a - b)"
using Fal by (auto simp add: field_simps inner_simps)
also have "… = x∙i"
apply (rule divide_eq_imp[OF Fal])
unfolding that[unfolded dist_norm]
using that[unfolded dist_triangle_eq] i
apply (subst (asm) euclidean_eq_iff)
apply (auto simp add: field_simps inner_simps)
done
finally show "x ∙ i = ((1 - ?β) *⇩R a + (?β) *⇩R b) ∙ i"
by auto
qed
qed (use Fal2 in auto)
qed
ultimately show ?thesis
by (force simp add: between_def closed_segment_def dist_triangle_eq)
qed
lemma between_midpoint:
fixes a :: "'a::euclidean_space"
shows "between (a,b) (midpoint a b)" (is ?t1)
and "between (b,a) (midpoint a b)" (is ?t2)
proof -
have *: "⋀x y z. x = (1/2::real) *⇩R z ⟹ y = (1/2) *⇩R z ⟹ norm z = norm x + norm y"
by auto
show ?t1 ?t2
unfolding between midpoint_def dist_norm
by (auto simp add: field_simps inner_simps euclidean_eq_iff[where 'a='a] intro!: *)
qed
lemma between_mem_convex_hull:
"between (a,b) x ⟷ x ∈ convex hull {a,b}"
unfolding between_mem_segment segment_convex_hull ..
lemma between_triv_iff [simp]: "between (a,a) b ⟷ a=b"
by (auto simp: between_def)
lemma between_triv1 [simp]: "between (a,b) a"
by (auto simp: between_def)
lemma between_triv2 [simp]: "between (a,b) b"
by (auto simp: between_def)
lemma between_commute:
"between (a,b) = between (b,a)"
by (auto simp: between_def closed_segment_commute)
lemma between_antisym:
fixes a :: "'a :: euclidean_space"
shows "⟦between (b,c) a; between (a,c) b⟧ ⟹ a = b"
by (auto simp: between dist_commute)
lemma between_trans:
fixes a :: "'a :: euclidean_space"
shows "⟦between (b,c) a; between (a,c) d⟧ ⟹ between (b,c) d"
using dist_triangle2 [of b c d] dist_triangle3 [of b d a]
by (auto simp: between dist_commute)
lemma between_norm:
fixes a :: "'a :: euclidean_space"
shows "between (a,b) x ⟷ norm(x - a) *⇩R (b - x) = norm(b - x) *⇩R (x - a)"
by (auto simp: between dist_triangle_eq norm_minus_commute algebra_simps)
lemma between_swap:
fixes A B X Y :: "'a::euclidean_space"
assumes "between (A, B) X"
assumes "between (A, B) Y"
shows "between (X, B) Y ⟷ between (A, Y) X"
using assms by (auto simp add: between)
lemma between_translation [simp]: "between (a + y,a + z) (a + x) ⟷ between (y,z) x"
by (auto simp: between_def)
lemma between_trans_2:
fixes a :: "'a :: euclidean_space"
shows "⟦between (b,c) a; between (a,b) d⟧ ⟹ between (c,d) a"
by (metis between_commute between_swap between_trans)
lemma between_scaleR_lift [simp]:
fixes v :: "'a::euclidean_space"
shows "between (a *⇩R v, b *⇩R v) (c *⇩R v) ⟷ v = 0 ∨ between (a, b) c"
by (simp add: between dist_norm scaleR_left_diff_distrib [symmetric] distrib_right [symmetric])
lemma between_1:
fixes x::real
shows "between (a,b) x ⟷ (a ≤ x ∧ x ≤ b) ∨ (b ≤ x ∧ x ≤ a)"
by (auto simp: between_mem_segment closed_segment_eq_real_ivl)
subsection%unimportant ‹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 "dist c ((1 / e) *⇩R y - ((1 - e) / e) *⇩R x) = ¦1/e¦ * norm (e *⇩R c - y + (1 - e) *⇩R x)"
unfolding dist_norm
unfolding norm_scaleR[symmetric]
apply (rule arg_cong[where f=norm])
using ‹e > 0›
by (auto simp add: euclidean_eq_iff[where 'a='a] field_simps inner_simps)
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 show "y ∈ S"
apply (subst *)
apply (rule assms(1)[unfolded convex_alt,rule_format])
apply (rule d[unfolded subset_eq,rule_format])
unfolding mem_ball
using assms(3-5)
apply auto
done
qed (insert ‹e>0› ‹d>0›, 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›
apply (rule_tac bexI[where x=x])
apply (auto)
done
next
case False
then have x: "x islimpt S"
using assms(3)[unfolded closure_def] by auto
show ?thesis
proof (cases "e = 1")
case True
obtain y where "y ∈ S" "y ≠ x" "dist y x < 1"
using x[unfolded islimpt_approachable,THEN spec[where x=1]] by auto
then show ?thesis
apply (rule_tac x=y in bexI)
unfolding True
using ‹d > 0›
apply auto
done
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 x[unfolded islimpt_approachable,THEN spec[where x="e*d / (1 - e)"]] by auto
then show ?thesis
apply (rule_tac x=y in bexI)
unfolding dist_norm
using pos_less_divide_eq[OF *]
apply auto
done
qed
qed
then obtain y where "y ∈ 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 "z ∈ interior S"
apply (rule interior_mono[OF d,unfolded subset_eq,rule_format])
unfolding interior_open[OF open_ball] mem_ball z_def dist_norm using y and assms(4,5)
apply (auto simp add:field_simps norm_minus_commute)
done
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 (clarsimp simp: in_segment)
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 show "(1 - u) *⇩R a + u *⇩R b ∈ interior S" .
qed
lemma closure_open_Int_superset:
assumes "open S" "S ⊆ closure T"
shows "closure(S ∩ T) = closure S"
proof -
have "closure S ⊆ closure(S ∩ T)"
by (metis assms closed_closure closure_minimal inf.orderE open_Int_closure_subset)
then show ?thesis
by (simp add: closure_mono dual_order.antisym)
qed
lemma convex_closure_interior:
fixes S :: "'a::euclidean_space set"
assumes "convex S" and int: "interior S ≠ {}"
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
show ?thesis
proof (clarsimp simp add: closure_def islimpt_approachable)
fix e::real
assume xnotS: "x ∉ interior S" and "0 < e"
show "∃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"
apply (clarsimp simp add: min_def a)
apply (rule mem_interior_closure_convex_shrink [OF ‹convex S› a x])
using ‹0 < e› False apply (auto simp: divide_simps)
done
qed
qed
qed
qed
then show ?thesis
by (simp add: closure_mono interior_subset subset_antisym)
qed
lemma closure_convex_Int_superset:
fixes S :: "'a::euclidean_space set"
assumes "convex S" "interior S ≠ {}" "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%unimportant ‹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 (simp add: convex_hull_finite set_eq_iff assms, safe)
fix x and u :: "'a ⇒ real"
assume "0 ≤ u 0" "∀x∈S. 0 ≤ u x" "u 0 + sum u S = 1"
then show "∃v. (∀x∈S. 0 ≤ v x) ∧ sum v S ≤ 1 ∧ (∑x∈S. v x *⇩R x) = (∑x∈S. u x *⇩R x)"
by force
next
fix x and u :: "'a ⇒ real"
assume "∀x∈S. 0 ≤ u x" "sum u S ≤ 1"
then show "∃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)
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) < MINIMUM Basis ((∙) x)"
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
apply (rule that[of ?a])
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 DIM_positive)
next
have "sum ((∙) ?a) ?D = sum (λi. inverse (2 * real DIM('a))) ?D"
apply (rule sum.cong)
apply rule
apply auto
done
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
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 "rel_interior (convex hull (insert 0 ?p)) = ?s")
proof -
have "finite D"
using D finite_Basis finite_subset by blast
show ?thesis
proof (cases "D = {}")
case True
then show ?thesis
using rel_interior_sing using euclidean_eq_iff[of _ 0] by auto
next
case False
have h0: "affine hull (convex hull (insert 0 ?p)) =
{x::'a::euclidean_space. (∀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 ?p))"
then obtain e where "e > 0" and
"ball x e ∩ {xa. (∀i∈Basis. i ∉ D ⟶ xa∙i = 0)} ⊆ convex hull (insert 0 ?p)"
using mem_rel_interior_ball[of x "convex hull (insert 0 ?p)"] h0 by auto
then have as [rule_format]: "⋀y. dist x y < e ∧ (∀i∈Basis. i ∉ D ⟶ y∙i = 0) ⟶
(∀i∈D. 0 ≤ y ∙ i) ∧ sum ((∙) y) D ≤ 1"
unfolding ball_def unfolding substd_simplex[OF assms] using assms by auto
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"
apply -
apply (rule as[THEN conjunct1])
using D ‹e > 0› x0
apply (auto simp: dist_norm inner_simps inner_Basis)
done
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
unfolding dist_norm
by auto
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 ?p)"
using as assms
unfolding substd_simplex[OF assms] by fastforce
obtain a where a: "a ∈ D"
using ‹D ≠ {}› by auto
let ?d = "(1 - sum ((∙) x) D) / real (card D)"
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 add: Min_gr_iff)
moreover have "?d > 0" using as using ‹0 < card D› by auto
ultimately have h3: "min (Min (((∙) x) ` D)) ?d > 0"
by auto
have "x ∈ rel_interior (convex hull (insert 0 ?p))"
unfolding rel_interior_ball mem_Collect_eq h0
apply (rule,rule h2)
unfolding substd_simplex[OF assms]
apply (rule_tac x="min (Min (((∙) x) ` D)) ?d" in exI)
apply (rule, rule h3)
apply safe
unfolding mem_ball
proof -
fix y :: 'a
assume y: "dist x y < min (Min ((∙) x ` D)) ?d"
assume y2: "∀i∈Basis. i ∉ D ⟶ y∙i = 0"
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 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[unfolded min_less_iff_conj dist_norm, THEN conjunct1]
using Min_gr_iff[of "(∙) x ` D" "norm (x - y)"] ‹0 < card D› ‹i ∈ D›
by (simp add: 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 (insert y2, auto)
qed
}
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 ?D = D
let ?a = "sum (λb::'a::euclidean_space. inverse (2 * real (card D)) *⇩R b) ?D"
have "finite D"
apply (rule finite_subset)
using assms(2)
apply auto
done
then have d1: "0 < real (card D)"
using ‹D ≠ {}› by auto
{
fix i
assume "i ∈ D"
have "?a ∙ i = inverse (2 * real (card D))"
apply (rule trans[of _ "sum (λj. if i = j then inverse (2 * real (card D)) else 0) ?D"])
unfolding inner_sum_left
apply (rule sum.cong)
using ‹i ∈ D› ‹finite D› sum.delta'[of D i "(λk. inverse (2 * real (card D)))"]
d1 assms(2)
by (auto simp: inner_Basis set_rev_mp[OF _ assms(2)])
}
note ** = this
show ?thesis
apply (rule that[of ?a])
unfolding rel_interior_substd_simplex[OF assms(2)] mem_Collect_eq
proof safe
fix i
assume "i ∈ 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
subsection%unimportant ‹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)"
apply (rule rel_interior_injective_on_span_linear_image[of f "(convex hull insert 0 B)"])
using ‹bounded_linear f› fd *
apply auto
done
ultimately have "rel_interior (convex hull insert 0 B) ≠ {}"
using rel_interior_substd_simplex_nonempty[OF ‹d ≠ {}› d]
apply auto
apply blast
done
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 by auto
}
then show ?thesis
using rel_interior_empty 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"
apply (rule_tac x="x - e1 *⇩R (x - a)" in exI)
using * e1 dist_norm[of "x - e1 *⇩R (x - a)" x] ‹x ≠ a›
apply simp
done
}
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
next
case True
then have "rel_interior S = {}"
using rel_interior_empty by auto
then have "closure (rel_interior S) = {}"
using closure_empty by auto
with True show ?thesis by auto
qed
qed
lemma rel_interior_same_affine_hull:
fixes S :: "'n::euclidean_space set"
assumes "convex S"
shows "affine hull (rel_interior S) = affine hull S"
by (metis assms closure_same_affine_hull convex_closure_rel_interior)
lemma rel_interior_aff_dim:
fixes S :: "'n::euclidean_space set"
assumes "convex S"
shows "aff_dim (rel_interior S) = aff_dim S"
by (metis aff_dim_affine_hull2 assms rel_interior_same_affine_hull)
lemma rel_interior_rel_interior:
fixes S :: "'n::euclidean_space set"
assumes "convex S"
shows "rel_interior (rel_interior S) = rel_interior S"
proof -
have "openin (subtopology euclidean (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
apply (simp add: algebra_simps)
using scaleR_left_distrib[of "a/(a+b)" "b/(a+b)" y] assms add_divide_distrib[of a b "a+b"]
apply auto
done
finally have "z = y - e *⇩R (y-x)"
by auto
moreover have "e > 0" using e_def assms by auto
moreover have "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 mem_cball 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 auto
qed
}
ultimately show ?thesis by auto
qed
lemma convex_interior_closure:
fixes S :: "'n::euclidean_space set"
assumes "convex S"
shows "interior (closure S) = interior S"
using closure_aff_dim[of S] interior_rel_interior_gen[of S]
interior_rel_interior_gen[of "closure S"]
convex_rel_interior_closure[of S] assms
by auto
lemma closure_eq_rel_interior_eq:
fixes S1 S2 :: "'n::euclidean_space set"
assumes "convex S1"
and "convex S2"
shows "closure S1 = closure S2 ⟷ 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 show ?thesis
apply (simp add: rel_interior_eq openin_open)
apply (rule_tac x="ball (inverse 2 *⇩R (a + b)) (norm(b - a) / 2)" in exI)
apply (simp add: open_segment_as_ball)
done
qed
lemma rel_interior_closed_segment:
fixes a :: "'a :: euclidean_space"
shows "rel_interior(closed_segment a b) =
(if a = b then {a} else open_segment a b)"
proof (cases "a = b")
case True then show ?thesis by auto
next
case False then show ?thesis
by simp
(metis closure_open_segment convex_open_segment convex_rel_interior_closure
rel_interior_open_segment)
qed
lemmas rel_interior_segment = rel_interior_closed_segment rel_interior_open_segment
lemma starlike_convex_tweak_boundary_points:
fixes S :: "'a::euclidean_space set"
assumes "convex S" "S ≠ {}" and ST: "rel_interior S ⊆ T" and TS: "T ⊆ closure S"
shows "starlike T"
proof -
have "rel_interior S ≠ {}"
by (simp add: assms rel_interior_eq_empty)
then obtain a where a: "a ∈ rel_interior S" by blast
with ST have "a ∈ T" by blast
have *: "⋀x. x ∈ T ⟹ open_segment a x ⊆ rel_interior S"
apply (rule rel_interior_closure_convex_segment [OF ‹convex S› a])
using assms by blast
show ?thesis
unfolding starlike_def
apply (rule bexI [OF _ ‹a ∈ T›])
apply (simp add: closed_segment_eq_open)
apply (intro conjI ballI a ‹a ∈ T› rel_interior_closure_convex_segment [OF ‹convex S› a])
apply (simp add: order_trans [OF * ST])
done
qed
subsection‹The relative frontier of a set›
definition%important "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
apply simp
by (metis centre_in_ball empty_iff frontier_cball frontier_def interior_cball interior_rel_interior_gen rel_frontier_def)
qed
lemma rel_frontier_translation:
fixes a :: "'a::euclidean_space"
shows "rel_frontier((λx. a + x) ` S) = (λx. a + x) ` (rel_frontier S)"
by (simp add: rel_frontier_def translation_diff rel_interior_translation closure_translation)
lemma closed_affine_hull [iff]:
fixes S :: "'n::euclidean_space set"
shows "closed (affine hull S)"
by (metis affine_affine_hull affine_closed)
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 (subtopology euclidean (affine hull S)) (closure S - rel_interior S)"
by (simp add: closed_subset closedin_diff closure_affine_hull openin_rel_interior)
show ?thesis
apply (rule closedin_closed_trans[of "affine hull S" "rel_frontier S"])
unfolding rel_frontier_def
using * closed_affine_hull
apply auto
done
qed
lemma closed_rel_boundary:
fixes S :: "'n::euclidean_space set"
shows "closed S ⟹ 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"
apply (rule affine_dim_equal)
using * affine_affine_hull
apply auto
using ‹S1 ≠ {}› hull_subset[of S1]
apply auto
using eq aff_dim_affine_hull[of S1] aff_dim_affine_hull[of S2]
apply auto
done
obtain a where a: "a ∈ 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"
using x1_def x2_def
apply (auto simp add: algebra_simps)
using scaleR_left_distrib[of "e1/(e1+e2)" "e2/(e1+e2)" z]
apply auto
done
then have z: "z ∈ 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 add: aff_dim_UNIV)
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%unimportant ‹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 closure_Int: "closure (⋂I) ≤ ⋂{closure S |S. S ∈ I}"
proof -
{
fix y
assume "y ∈ ⋂I"
then have y: "∀S ∈ I. y ∈ S" by auto
{
fix S
assume "S ∈ I"
then have "y ∈ closure S"
using closure_subset y by auto
}
then have "y ∈ ⋂{closure S |S. S ∈ I}"
by auto
}
then have "⋂I ⊆ ⋂{closure S |S. S ∈ I}"
by auto
moreover have "closed (⋂{closure S |S. S ∈ I})"
unfolding closed_Inter closed_closure by auto
ultimately show ?thesis using closure_hull[of "⋂I"]
hull_minimal[of "⋂I" "⋂{closure S |S. S ∈ I}" "closed"] by auto
qed
lemma convex_closure_rel_interior_inter:
assumes "∀S∈I. convex (S :: 'n::euclidean_space set)"
and "⋂{rel_interior S |S. S ∈ I} ≠ {}"
shows "⋂{closure S |S. S ∈ I} ≤ closure (⋂{rel_interior S |S. S ∈ I})"
proof -
obtain x where x: "∀S∈I. x ∈ rel_interior S"
using assms by auto
{
fix y
assume "y ∈ ⋂{closure S |S. S ∈ I}"
then have y: "∀S ∈ I. y ∈ closure S"
by auto
{
assume "y = x"
then have "y ∈ closure (⋂{rel_interior S |S. S ∈ I})"
using x closure_subset[of "⋂{rel_interior S |S. S ∈ I}"] by auto
}
moreover
{
assume "y ≠ x"
{ fix e :: real
assume e: "e > 0"
define e1 where "e1 = min 1 (e/norm (y - x))"
then have e1: "e1 > 0" "e1 ≤ 1" "e1 * norm (y - x) ≤ e"
using ‹y ≠ x› ‹e > 0› le_divide_eq[of e1 e "norm (y - x)"]
by simp_all
define z where "z = y - e1 *⇩R (y - x)"
{
fix S
assume "S ∈ I"
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 S |S. S ∈ I}"
by auto
have "∃z. z ∈ ⋂{rel_interior S |S. S ∈ I} ∧ z ≠ y ∧ dist z y ≤ e"
apply (rule_tac x="z" in exI)
using ‹y ≠ x› z_def * e1 e dist_norm[of z y]
apply simp
done
}
then have "y islimpt ⋂{rel_interior S |S. S ∈ I}"
unfolding islimpt_approachable_le by blast
then have "y ∈ closure (⋂{rel_interior S |S. S ∈ I})"
unfolding closure_def by auto
}
ultimately have "y ∈ closure (⋂{rel_interior S |S. S ∈ I})"
by auto
}
then show ?thesis by auto
qed
lemma convex_closure_inter:
assumes "∀S∈I. convex (S :: 'n::euclidean_space set)"
and "⋂{rel_interior S |S. S ∈ I} ≠ {}"
shows "closure (⋂I) = ⋂{closure S |S. S ∈ I}"
proof -
have "⋂{closure S |S. S ∈ I} ≤ closure (⋂{rel_interior S |S. S ∈ I})"
using convex_closure_rel_interior_inter assms by auto
moreover
have "closure (⋂{rel_interior S |S. S ∈ I}) ≤ closure (⋂I)"
using rel_interior_inter_aux closure_mono[of "⋂{rel_interior S |S. S ∈ I}" "⋂I"]
by auto
ultimately show ?thesis
using closure_Int[of I] by auto
qed
lemma convex_inter_rel_interior_same_closure:
assumes "∀S∈I. convex (S :: 'n::euclidean_space set)"
and "⋂{rel_interior S |S. S ∈ I} ≠ {}"
shows "closure (⋂{rel_interior S |S. S ∈ I}) = closure (⋂I)"
proof -
have "⋂{closure S |S. S ∈ I} ≤ closure (⋂{rel_interior S |S. S ∈ I})"
using convex_closure_rel_interior_inter assms by auto
moreover
have "closure (⋂{rel_interior S |S. S ∈ I}) ≤ closure (⋂I)"
using rel_interior_inter_aux closure_mono[of "⋂{rel_interior S |S. S ∈ I}" "⋂I"]
by auto
ultimately show ?thesis
using closure_Int[of I] by auto
qed
lemma convex_rel_interior_inter:
assumes "∀S∈I. convex (S :: 'n::euclidean_space set)"
and "⋂{rel_interior S |S. S ∈ I} ≠ {}"
shows "rel_interior (⋂I) ⊆ ⋂{rel_interior S |S. S ∈ I}"
proof -
have "convex (⋂I)"
using assms convex_Inter by auto
moreover
have "convex (⋂{rel_interior S |S. S ∈ I})"
apply (rule convex_Inter)
using assms convex_rel_interior
apply auto
done
ultimately
have "rel_interior (⋂{rel_interior S |S. S ∈ I}) = rel_interior (⋂I)"
using convex_inter_rel_interior_same_closure assms
closure_eq_rel_interior_eq[of "⋂{rel_interior S |S. S ∈ I}" "⋂I"]
by blast
then show ?thesis
using rel_interior_subset[of "⋂{rel_interior S |S. S ∈ I}"] by auto
qed
lemma convex_rel_interior_finite_inter:
assumes "∀S∈I. convex (S :: 'n::euclidean_space set)"
and "⋂{rel_interior S |S. S ∈ I} ≠ {}"
and "finite I"
shows "rel_interior (⋂I) = ⋂{rel_interior S |S. S ∈ I}"
proof -
have "⋂I ≠ {}"
using assms rel_interior_inter_aux[of I] by auto
have "convex (⋂I)"
using convex_Inter assms by auto
show ?thesis
proof (cases "I = {}")
case True
then show ?thesis
using Inter_empty rel_interior_UNIV by auto
next
case False
{
fix z
assume z: "z ∈ ⋂{rel_interior S |S. S ∈ I}"
{
fix x
assume x: "x ∈ ⋂I"
{
fix S
assume S: "S ∈ I"
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∈I. 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 ` I)"
then have "e ∈ mS ` I" using assms ‹I ≠ {}› by simp
then have "e > 1" using mS by auto
moreover have "∀S∈I. e ≤ mS S"
using e_def assms by auto
ultimately have "∃e > 1. (1 - e) *⇩R x + e *⇩R z ∈ ⋂I"
using mS by auto
}
then have "z ∈ rel_interior (⋂I)"
using convex_rel_interior_iff[of "⋂I" z] ‹⋂I ≠ {}› ‹convex (⋂I)› by auto
}
then show ?thesis
using convex_rel_interior_inter[of I] assms by auto
qed
qed
lemma convex_closure_inter_two:
fixes S T :: "'n::euclidean_space set"
assumes "convex S"
and "convex T"
assumes "rel_interior S ∩ rel_interior T ≠ {}"
shows "closure (S ∩ T) = closure S ∩ closure T"
using convex_closure_inter[of "{S,T}"] assms by auto
lemma convex_rel_interior_inter_two:
fixes S T :: "'n::euclidean_space set"
assumes "convex S"
and "convex T"
and "rel_interior S ∩ 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"
proof -
have "affine hull T = T"
using assms by auto
then have "rel_interior T = T"
using rel_interior_affine_hull[of T] by metis
moreover have "closure T = T"
using assms affine_closed[of T] by auto
ultimately show ?thesis
using convex_closure_inter_two[of S T] assms affine_imp_convex by auto
qed
lemma connected_component_1_gen:
fixes S :: "'a :: euclidean_space set"
assumes "DIM('a) = 1"
shows "connected_component S a b ⟷ 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"
proof -
have "affine hull T = T"
using assms by auto
then have "rel_interior T = T"
using rel_interior_affine_hull[of T] by metis
moreover have "closure T = T"
using assms affine_closed[of T] by auto
ultimately show ?thesis
using convex_rel_interior_inter_two[of S T] assms affine_imp_convex by auto
qed
lemma convex_affine_rel_frontier_Int:
fixes S T :: "'n::euclidean_space set"
assumes "convex S"
and "affine T"
and "interior S ∩ T ≠ {}"
shows "rel_frontier(S ∩ T) = frontier S ∩ T"
using assms
apply (simp add: rel_frontier_def convex_affine_closure_Int frontier_def)
by (metis Diff_Int_distrib2 Int_emptyI convex_affine_closure_Int convex_affine_rel_interior_Int empty_iff interior_rel_interior_gen)
lemma rel_interior_convex_Int_affine:
fixes S :: "'a::euclidean_space set"
assumes "convex S" "affine T" "interior S ∩ T ≠ {}"
shows "rel_interior(S ∩ T) = interior S ∩ T"
proof -
obtain a where aS: "a ∈ interior S" and aT:"a ∈ T"
using assms by force
have "rel_interior S = interior S"
by (metis (no_types) aS affine_hull_nonempty_interior equals0D rel_interior_interior)
then show ?thesis
by (metis (no_types) affine_imp_convex assms convex_rel_interior_inter_two hull_same rel_interior_affine_hull)
qed
lemma closure_convex_Int_affine:
fixes S :: "'a::euclidean_space set"
assumes "convex S" "affine T" "rel_interior S ∩ T ≠ {}"
shows "closure(S ∩ T) = closure S ∩ T"
proof
have "closure (S ∩ T) ⊆ closure T"
by (simp add: closure_mono)
also have "... ⊆ T"
by (simp add: affine_closed assms)
finally show "closure(S ∩ T) ⊆ closure S ∩ T"
by (simp add: closure_mono)
next
obtain a where "a ∈ rel_interior S" "a ∈ T"
using assms by auto
then have ssT: "subspace ((λx. (-a)+x) ` T)" and "a ∈ S"
using affine_diffs_subspace rel_interior_subset assms by blast+
show "closure S ∩ T ⊆ closure (S ∩ T)"
proof
fix x assume "x ∈ closure S ∩ T"
show "x ∈ closure (S ∩ T)"
proof (cases "x = a")
case True
then show ?thesis
using ‹a ∈ S› ‹a ∈ T› closure_subset by fastforce
next
case False
then have "x ∈ closure(open_segment a x)"
by auto
then show ?thesis
using ‹x ∈ closure S ∩ T› assms convex_affine_closure_Int by blast
qed
qed
qed
lemma subset_rel_interior_convex:
fixes S T :: "'n::euclidean_space set"
assumes "convex S"
and "convex T"
and "S ≤ 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 rel_interior_empty rel_interior_eq_empty 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 rel_interior_empty 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 -
{ assume "S = {}"
then have ?thesis
by auto
}
moreover
{ assume "T = {}"
then have ?thesis
by auto
}
moreover
{
assume "S ≠ {}" "T ≠ {}"
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 fst_linear ‹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 snd_linear ‹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)"
apply (subst convex_rel_interior_inter_two[of "S × (UNIV :: 'm set)" "(UNIV :: 'n set) × T"])
using * ri assms convex_Times
apply auto
done
finally have ?thesis using * by auto
}
ultimately show ?thesis by blast
qed
lemma rel_interior_scaleR:
fixes S :: "'n::euclidean_space set"
assumes "c ≠ 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:
assumes "∀S∈I. convex (S :: 'n::euclidean_space set) ∧ rel_open S"
and "finite I"
shows "convex (⋂I) ∧ rel_open (⋂I)"
proof (cases "⋂{rel_interior S |S. S ∈ I} = {}")
case True
then have "⋂I = {}"
using assms unfolding rel_open_def by auto
then show ?thesis
unfolding rel_open_def using rel_interior_empty by auto
next
case False
then have "rel_open (⋂I)"
using assms unfolding rel_open_def
using convex_rel_interior_finite_inter[of I]
by auto
then show ?thesis
using convex_Inter assms by auto
qed
lemma convex_rel_open_linear_image:
fixes f :: "'m::euclidean_space ⇒ '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 using rel_interior_empty 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"
apply (rule_tac x="(y, z)" in exI)
apply auto
done
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 fst_linear 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}"] snd_linear 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%unimportant ‹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: rel_interior_empty 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: rel_interior_empty 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))"
apply (subst rel_interior_projection[of "cone hull ({(1 :: real)} × S)" f c x])
using convex_cone_hull[of "{(1 :: real)} × S"] conv
apply auto
done
{
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
have "z ∈ ?rhs"
using rel_interior_convex_cone_aux[of S "fst z" "snd z"] assms ‹z ∈ ?lhs›
apply auto
apply (rule_tac x = "fst z" in exI)
apply (rule_tac x = x in exI)
using *
apply auto
done
}
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]
apply (subst convex_sum[of I "convex hull ⋃(S ` I)" c s])
using * assms convex_convex_hull
apply auto
done
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
then have "x ∈ ?rhs"
apply auto
apply (rule_tac x = c in exI)
apply (rule_tac x = s in exI)
using * c_def s_def p ‹x ∈ S i›
apply auto
done
}
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)}"
apply (subst convex_hull_finite_union[of I s])
using assms s_def I_def
apply auto
done
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
subsection%unimportant ‹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 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. convex (S i)"
shows "rel_interior (sum S I) = sum (λi. rel_interior (S i)) I"
apply (subst sum_set_cond_linear[of convex])
using rel_interior_sum rel_interior_sing[of "0"] assms
apply (auto simp add: convex_set_plus)
done
lemma convex_rel_open_direct_sum:
fixes S T :: "'n::euclidean_space set"
assumes "convex S"
and "rel_open S"
and "convex T"
and "rel_open T"
shows "convex (S × 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. 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
{
fix i
assume "i ∈ I"
then have "s i ∈ S i"
using s_def x assms mem_cone[of "S i" "xs i" "c i"] by auto
}
then have "∀i∈I. s i ∈ S i" by auto
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"
apply (subst convex_hull_finite_union[of I S])
using assms
apply blast
using assms
apply blast
apply rule
apply (rule_tac x = "(λi. (1 :: real) / of_nat (card I))" in exI)
apply auto
done
}
ultimately show ?thesis by auto
qed
lemma convex_hull_union_cones_two:
fixes S T :: "'m::euclidean_space set"
assumes "convex S"
and "cone S"
and "S ≠ {}"
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"
apply (subst convex_hull_finite_union_cones[of I A])
using assms A_def I_def
apply auto
done
moreover have "sum A I = S + T"
using A_def I_def
unfolding set_plus_def
apply auto
unfolding set_plus_def
apply auto
done
ultimately show ?thesis by auto
qed
lemma rel_interior_convex_hull_union:
fixes S :: "'a ⇒ '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 rel_interior_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])
{
fix i
assume "i ∈ I"
then have "convex (K i)"
unfolding K_def
apply (subst convex_cone_hull)
apply (subst convex_Times)
using assms
apply auto
done
}
then have convK: "∀i∈I. convex (K i)"
by auto
{
fix i
assume "i ∈ I"
then have "K0 ⊇ K i"
unfolding K0_def K_def
apply (subst hull_mono)
using ‹∀i∈I. C0 ≥ S i›
apply auto
done
}
then have "K0 ⊇ ⋃(K ` I)" by auto
moreover have "convex K0"
unfolding K0_def
apply (subst convex_cone_hull)
apply (subst convex_Times)
unfolding C0_def
using convex_convex_hull
apply auto
done
ultimately have geq: "K0 ⊇ 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)))"
apply (subst cone_convex_hull)
using cone_Union[of "K ` I"]
apply auto
unfolding K_def
using cone_cone_hull
apply auto
done
ultimately have "convex hull (⋃(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"
apply (subst convex_hull_finite_union_cones[of I K])
using assms
apply blast
using False
apply blast
unfolding K_def
apply rule
apply (subst convex_cone_hull)
apply (subst convex_Times)
using assms cone_cone_hull ‹∀i∈I. K i ≠ {}› K_def
apply auto
done
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::real, x) ∈ rel_interior K0"
using K0_def * set_sum_alt[of I "(λi. rel_interior (K i))"] assms k_def cs
apply auto
apply (rule_tac x = k in exI)
apply (simp add: sum_prod)
done
then have "x ∈ ?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
have "open (interior I)" by auto
from openE[OF this ‹x ∈ interior I›]
obtain e where "0 < e" "ball x e ⊆ interior I" .
moreover define K where "K = x - e / 2"
with ‹0 < e› have "K ∈ ball x e" "K < x"
by (auto simp: dist_real_def)
ultimately have "K ∈ I" "K < x" "x ∈ I"
using interior_subset[of I] ‹x ∈ interior I› by auto
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_diff)
next
fix y
assume "y ∈ ?F x"
with order_trans[OF convex_on_diff[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_diff[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 "open (interior I)" by auto
from openE[OF this ‹x ∈ interior I›]
obtain e where e: "0 < e" "ball x e ⊆ interior I" .
then have "x + e / 2 ∈ ball x 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%unimportant‹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
{ fix u v x
assume uv: "sum u t = 1" "∀x∈s. 0 ≤ v x" "sum v s = 1"
"(∑x∈s. v x *⇩R x) = (∑v∈t. u v *⇩R v)" "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 uv 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 uv fin
by (subst s, subst sum.union_disjoint, auto simp: algebra_simps sum_subtractf)+
} note [simp] = this
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"
using assms
apply (simp add: convex_hull_finite affine_hull_finite fin affine_dependent_explicit)
apply (drule_tac x=s in spec)
apply (auto simp: fin)
apply (rule_tac x=u in exI)
apply (rename_tac v)
apply (drule_tac x="λx. if x ∈ t then v x - u x else v x" in spec)
apply (force)+
done
ultimately show ?thesis
by blast
qed
lemma affine_independent_span_eq:
fixes s :: "'a::euclidean_space set"
assumes "~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)
show ?thesis
using assms t fin
apply (simp add: affine_dependent_iff_dependent affine_hull_insert_span_gen)
apply (rule subset_antisym)
apply force
apply (rule Fun.vimage_subsetD)
apply (metis add.commute diff_add_cancel surj_def)
apply (rule card_ge_dim_independent)
apply (auto simp: card_image inj_on_def dim_subset_UNIV)
done
qed
lemma affine_independent_span_gt:
fixes s :: "'a::euclidean_space set"
assumes ind: "~ affine_dependent s" and dim: "DIM ('a) < card s"
shows "affine hull s = UNIV"
apply (rule affine_independent_span_eq [OF ind])
apply (rule antisym)
using assms
apply auto
apply (metis add_2_eq_Suc' not_less_eq_eq affine_dependent_biggerset aff_independent_finite)
done
lemma empty_interior_affine_hull:
fixes s :: "'a::euclidean_space set"
assumes "finite s" and dim: "card s ≤ DIM ('a)"
shows "interior(affine hull s) = {}"
using assms
apply (induct s rule: finite_induct)
apply (simp_all add: affine_dependent_iff_dependent affine_hull_insert_span_gen interior_translation)
apply (rule empty_interior_lowdim)
by (auto simp: Suc_le_lessD card_image_le dual_order.trans intro!: dim_le_card'[THEN le_less_trans])
lemma empty_interior_convex_hull:
fixes s :: "'a::euclidean_space set"
assumes "finite s" and dim: "card s ≤ 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"
{ fix u T a
assume 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}"
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 "∀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
then have False
using assms
apply (simp add: affine_dependent_explicit_finite fs)
apply (drule_tac x="λx. (v x - u x) - (if x = a then -d else if x = b then d else 0)" in spec)
using ua b d
apply (auto simp: algebra_simps sum_subtractf sum.distrib)
done
} note * = this
have "y ∉ rel_interior (convex hull s)"
using y
apply (simp add: mem_rel_interior affine_hull_convex_hull)
apply (auto simp: convex_hull_finite [OF fs])
apply (drule_tac x=u in spec)
apply (auto intro: *)
done
} with rel_interior_subset show "?lhs ≤ ?rhs"
by blast
qed
qed
lemma interior_convex_hull_explicit_minimal:
fixes s :: "'a::euclidean_space set"
shows
"~ affine_dependent s
==> 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})"
apply (simp add: aff_independent_finite empty_interior_convex_hull, clarify)
apply (rule trans [of _ "rel_interior(convex hull s)"])
apply (simp add: affine_hull_convex_hull affine_independent_span_gt rel_interior_interior)
by (simp add: rel_interior_convex_hull_explicit)
lemma interior_convex_hull_explicit:
fixes s :: "'a::euclidean_space set"
assumes "~ 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
next
case False
then show thesis
by (blast intro: that)
qed
have "u a + u b ≤ sum u {a,b}"
using a b by simp
also have "... ≤ sum u s"
apply (rule Groups_Big.sum_mono2)
using a b u
apply (auto simp: less_imp_le aff_independent_finite assms)
done
finally have "u a < 1"
using ‹b ∈ s› u by fastforce
} note [simp] = this
show ?thesis
using assms
apply (auto simp: interior_convex_hull_explicit_minimal)
apply (rule_tac x=u in exI)
apply (auto simp: not_le)
done
qed
lemma interior_closed_segment_ge2:
fixes a :: "'a::euclidean_space"
assumes "2 ≤ 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 (simp add: not_le, intro conjI impI)
assume "2 ≤ DIM('a)"
then show "interior (open_segment a b) = {}"
apply (simp add: segment_convex_hull open_segment_def)
apply (metis Diff_subset interior_mono segment_convex_hull subset_empty interior_closed_segment_ge2)
done
next
assume le2: "DIM('a) < 2"
show "interior (open_segment a b) = open_segment a b"
proof (cases "a = b")
case True then show ?thesis by auto
next
case False
with le2 have "affine hull (open_segment a b) = UNIV"
apply simp
apply (rule affine_independent_span_gt)
apply (simp_all add: affine_dependent_def insert_Diff_if)
done
then show "interior (open_segment a b) = open_segment a b"
using rel_interior_interior rel_interior_open_segment by blast
qed
qed
lemma interior_closed_segment:
fixes a :: "'a::euclidean_space"
shows "interior(closed_segment a b) =
(if 2 ≤ 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%unimportant‹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)
show ?thesis
apply (simp add: rel_frontier_def finite_imp_compact rel_interior_convex_hull_explicit assms fs)
apply (auto simp: convex_hull_finite fs)
apply (drule_tac x=u in spec)
apply (rule_tac x=u in exI)
apply force
apply (rename_tac v)
apply (rule notE [OF assms])
apply (simp add: affine_dependent_explicit)
apply (rule_tac x=s in exI)
apply (auto simp: fs)
apply (rule_tac x = "λx. u x - v x" in exI)
apply (force simp: sum_subtractf scaleR_diff_left)
done
qed
lemma frontier_convex_hull_explicit:
fixes s :: "'a::euclidean_space set"
assumes "~ 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 (simp add: convex_hull_finite fs Union_SetCompr_eq, auto)
done
qed
lemma frontier_convex_hull_eq_rel_frontier:
fixes s :: "'a::euclidean_space set"
assumes "~ 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 show ?thesis
apply (auto simp: affine_dependent_def frontier_def finite_imp_compact hull_inc)
by (metis card.insert_remove convex_hull_subset_affine_hull empty_interior_affine_hull finite_Diff hull_redundant insert_Diff insert_Diff_single insert_not_empty interior_mono not_less_eq_eq subset_empty)
next
case False
{ assume "card s = Suc (card Basis)"
then have cs: "Suc 0 < card s"
by (simp add: DIM_positive)
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 -
{ fix a b
assume ab: "a ∈ interior (convex hull s)" "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 have False using ab
by (metis convex_hull_subset_affine_hull equals0D interior_mono subset_eq)
} then
show ?thesis
using assms
apply auto
apply (metis UNIV_I affine_hull_convex_hull affine_hull_empty affine_independent_span_eq convex_convex_hull empty_iff rel_interior_interior rel_interior_same_affine_hull)
apply (auto simp: affine_dependent_def)
done
qed
subsection ‹Coplanarity, and collinearity in terms of affine hull›
definition%important 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 "∃u v. s ⊆ {a *⇩R u + b *⇩R v |a b. a + b = 1}"
apply (rule_tac x=x in exI)
apply (rule_tac x="x+u" in exI, clarify)
apply (erule exE [OF * [OF x]])
apply (rename_tac c)
apply (rule_tac x="1+c" in exI)
apply (rule_tac x="-c" in exI)
apply (simp add: algebra_simps)
done
} moreover
{ fix u v x y
assume *: "s ⊆ {a *⇩R u + b *⇩R v |a b. a + b = 1}"
have "x∈s ⟹ y∈s ⟹ ∃c. x - y = c *⇩R (v-u)"
apply (drule subsetD [OF *])+
apply simp
apply clarify
apply (rename_tac r1 r2)
apply (rule_tac x="r1-r2" in exI)
apply (simp add: algebra_simps)
apply (metis scaleR_left.add)
done
} ultimately
show ?thesis
unfolding collinear_def affine_hull_2
by blast
qed
lemma collinear_closed_segment [simp]: "collinear (closed_segment a b)"
by (metis affine_hull_convex_hull collinear_affine_hull hull_subset segment_convex_hull)
lemma collinear_open_segment [simp]: "collinear (open_segment a b)"
unfolding open_segment_def
by (metis convex_hull_subset_affine_hull segment_convex_hull dual_order.trans
convex_hull_subset_affine_hull Diff_subset collinear_affine_hull)
lemma collinear_between_cases:
fixes c :: "'a::euclidean_space"
shows "collinear {a,b,c} ⟷ 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, hide_lams) collinear_closed_segment collinear_subset hull_redundant hull_subset insert_commute segment_convex_hull)
qed
lemma subset_continuous_image_segment_1:
fixes f :: "'a::euclidean_space ⇒ 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_alt 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_alt 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, hide_lams) empty_subsetI ends_in_segment image_insert image_is_empty inj_on_image_set_diff injf insert_subset open_segment_def segment_open_subset_closed)
also have "... = open_segment (f a) (f b)"
using continuous_injective_image_segment_1 [OF assms]
by (simp add: open_segment_def inj_on_image_set_diff [OF injf])
finally show ?thesis .
qed
lemma collinear_imp_coplanar:
"collinear s ==> coplanar s"
by (metis collinear_affine_hull coplanar_def insert_absorb2)
lemma collinear_small:
assumes "finite s" "card s ≤ 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
by auto (metis collinear_2 numeral_2_eq_2)
qed
lemma coplanar_small:
assumes "finite s" "card s ≤ 3"
shows "coplanar s"
proof -
have "card s ≤ 2 ∨ card s = Suc (Suc (Suc 0))"
using assms by linarith
then show ?thesis using assms
apply safe
apply (simp add: collinear_small collinear_imp_coplanar)
apply (safe dest!: card_eq_SucD)
apply (auto simp: coplanar_def)
apply (metis hull_subset insert_subset)
done
qed
lemma coplanar_empty: "coplanar {}"
by (simp add: coplanar_small)
lemma coplanar_sing: "coplanar {a}"
by (simp add: coplanar_small)
lemma coplanar_2: "coplanar {a,b}"
by (auto simp: card_insert_if coplanar_small)
lemma coplanar_3: "coplanar {a,b,c}"
by (auto simp: card_insert_if coplanar_small)
lemma collinear_affine_hull_collinear: "collinear(affine hull s) ⟷ 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: "coplanar s ⟹ coplanar ((λx. a + x) ` s)"
unfolding coplanar_def
apply clarify
apply (rule_tac x="u+a" in exI)
apply (rule_tac x="v+a" in exI)
apply (rule_tac x="w+a" in exI)
using affine_hull_translation [of a "{u,v,w}" for u v w]
apply (force simp: add.commute)
done
lemma coplanar_translation_eq: "coplanar((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)"
unfolding coplanar_def
using affine_hull_linear_image [of f "{u,v,w}" for u v w] assms
by (meson coplanar_def coplanar_linear_image)
next
obtain g where g: "linear g" "g ∘ f = id"
using linear_injective_left_inverse [OF assms]
by blast
assume "coplanar (f ` s)"
then obtain u v w where "f ` s ⊆ affine hull {u, v, w}"
by (auto simp: coplanar_def)
then have "g ` f ` s ⊆ g ` (affine hull {u, v, w})"
by blast
then have "s ⊆ g ` (affine hull {u, v, w})"
using g by (simp add: Fun.image_comp)
then show "coplanar s"
unfolding coplanar_def
using affine_hull_linear_image [of g "{u,v,w}" for u v w] ‹linear g› linear_conv_bounded_linear
by fastforce
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: "⟦collinear {a,b,c}; a ≠ b⟧ ⟹ c ∈ affine hull {a,b}"
unfolding collinear_def
apply clarify
apply (frule_tac x=b in bspec, blast, drule_tac x=a in bspec, blast, erule exE)
apply (drule_tac x=c in bspec, blast, drule_tac x=a in bspec, blast, erule exE)
apply (rename_tac y x)
apply (simp add: affine_hull_2)
apply (rule_tac x="1 - x/y" in exI)
apply (simp add: algebra_simps)
done
lemma collinear_3_affine_hull:
assumes "a ≠ 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}"
apply (case_tac "a=b", simp)
apply (case_tac "a=c")
apply (simp add: insert_commute)
apply (case_tac "b=c")
apply (simp add: insert_commute)
apply (auto simp: affine_dependent_def collinear_3_affine_hull insert_Diff_if)
apply (metis collinear_3_affine_hull insert_commute)+
done
lemma affine_dependent_imp_collinear_3:
"affine_dependent {a,b,c} ⟹ 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}"
apply (auto simp: collinear_3 collinear_lemma)
apply (drule_tac x="-1" in spec)
apply (simp add: algebra_simps)
done
lemma midpoint_collinear:
fixes a b c :: "'a::real_normed_vector"
assumes "a ≠ 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 "collinear{a,b,c} ⟹ b = midpoint a c ⟷ dist a b = dist b c"
apply (auto simp: collinear_3_expand assms dist_midpoint)
apply (simp add: dist_norm * assms midpoint_def del: divide_const_simps)
apply (simp add: algebra_simps)
done
ultimately show ?thesis by blast
qed
lemma between_imp_collinear:
fixes x :: "'a :: euclidean_space"
assumes "between (a,b) x"
shows "collinear {a,x,b}"
proof (cases "x = a ∨ x = b ∨ a = b")
case True with assms show ?thesis
by (auto simp: dist_commute)
next
case False with assms show ?thesis
apply (auto simp: collinear_3 collinear_lemma between_norm)
apply (drule_tac x="-(norm(b - x) / norm(x - a))" in spec)
apply (simp add: vector_add_divide_simps eq_vector_fraction_iff real_vector.scale_minus_right [symmetric])
done
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 True then show ?thesis
by (auto simp: dist_commute)
next
case False
show ?thesis
apply (rule iffI)
apply (simp add: between_midpoint(1) dist_midpoint)
using False between_imp_collinear midpoint_collinear by blast
qed
lemma collinear_triples:
assumes "a ≠ 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+
show ?lhs
unfolding collinear_def
apply (rule_tac x="b-a" in exI)
apply (clarify dest!: *)
by (metis (no_types, hide_lams) add.commute diff_add_cancel diff_diff_eq2 real_vector.scale_right_diff_distrib scaleR_left.diff)
qed
lemma collinear_4_3:
assumes "a ≠ 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_eq_empty [simp]: "affine hull S = {} ⟷ S = {}"
using affine_hull_nonempty by blast
lemma affine_hull_2_alt:
fixes a b :: "'a::real_vector"
shows "affine hull {a,b} = range (λu. a + u *⇩R (b - a))"
apply (simp add: affine_hull_2, safe)
apply (rule_tac x=v in image_eqI)
apply (simp add: algebra_simps)
apply (metis scaleR_add_left scaleR_one, simp)
apply (rule_tac x="1-u" in exI)
apply (simp add: algebra_simps)
done
lemma interior_convex_hull_3_minimal:
fixes a :: "'a::euclidean_space"
shows "⟦~ collinear{a,b,c}; DIM('a) = 2⟧
⟹ 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}"
apply (simp add: collinear_3_eq_affine_dependent interior_convex_hull_explicit_minimal, safe)
apply (rule_tac x="u a" in exI, simp)
apply (rule_tac x="u b" in exI, simp)
apply (rule_tac x="u c" in exI, simp)
apply (rename_tac uu x y z)
apply (rule_tac x="λr. (if r=a then x else if r=b then y else if r=c then z else 0)" in exI)
apply simp
done
subsection‹The infimum of the distance between two sets›
definition%important setdist :: "'a::metric_space set ⇒ 'a set ⇒ real" where
"setdist s t ≡
(if s = {} ∨ t = {} then 0
else Inf {dist x y| x y. x ∈ s ∧ y ∈ t})"
lemma setdist_empty1 [simp]: "setdist {} t = 0"
by (simp add: setdist_def)
lemma setdist_empty2 [simp]: "setdist t {} = 0"
by (simp add: setdist_def)
lemma setdist_pos_le [simp]: "0 ≤ setdist s t"
by (auto simp: setdist_def ex_in_conv [symmetric] intro: cInf_greatest)
lemma le_setdistI:
assumes "s ≠ {}" "t ≠ {}" "⋀x y. ⟦x ∈ s; y ∈ t⟧ ⟹ d ≤ dist x y"
shows "d ≤ setdist s t"
using assms
by (auto simp: setdist_def Set.ex_in_conv [symmetric] intro: cInf_greatest)
lemma setdist_le_dist: "⟦x ∈ s; y ∈ t⟧ ⟹ setdist s t ≤ dist x y"
unfolding setdist_def
by (auto intro!: bdd_belowI [where m=0] cInf_lower)
lemma le_setdist_iff:
"d ≤ setdist s t ⟷
(∀x ∈ s. ∀y ∈ t. d ≤ dist x y) ∧ (s = {} ∨ t = {} ⟶ d ≤ 0)"
apply (cases "s = {} ∨ t = {}")
apply (force simp add: setdist_def)
apply (intro iffI conjI)
using setdist_le_dist apply fastforce
apply (auto simp: intro: le_setdistI)
done
lemma setdist_ltE:
assumes "setdist s t < b" "s ≠ {}" "t ≠ {}"
obtains x y where "x ∈ s" "y ∈ t" "dist x y < b"
using assms
by (auto simp: not_le [symmetric] le_setdist_iff)
lemma setdist_refl: "setdist s s = 0"
apply (cases "s = {}")
apply (force simp add: setdist_def)
apply (rule antisym [OF _ setdist_pos_le])
apply (metis all_not_in_conv dist_self setdist_le_dist)
done
lemma setdist_sym: "setdist s t = setdist t s"
by (force simp: setdist_def dist_commute intro!: arg_cong [where f=Inf])
lemma setdist_triangle: "setdist s t ≤ setdist s {a} + setdist {a} t"
proof (cases "s = {} ∨ t = {}")
case True then show ?thesis
using setdist_pos_le by fastforce
next
case False
have "⋀x. x ∈ s ⟹ setdist s t - dist x a ≤ setdist {a} t"
apply (rule le_setdistI, blast)
using False apply (fastforce intro: le_setdistI)
apply (simp add: algebra_simps)
apply (metis dist_commute dist_triangle3 order_trans [OF setdist_le_dist])
done
then have "setdist s t - setdist {a} t ≤ setdist s {a}"
using False by (fastforce intro: le_setdistI)
then show ?thesis
by (simp add: algebra_simps)
qed
lemma setdist_singletons [simp]: "setdist {x} {y} = dist x y"
by (simp add: setdist_def)
lemma setdist_Lipschitz: "¦setdist {x} s - setdist {y} s¦ ≤ dist x y"
apply (subst setdist_singletons [symmetric])
by (metis abs_diff_le_iff diff_le_eq setdist_triangle setdist_sym)
lemma continuous_at_setdist [continuous_intros]: "continuous (at x) (λy. (setdist {y} s))"
by (force simp: continuous_at_eps_delta dist_real_def intro: le_less_trans [OF setdist_Lipschitz])
lemma continuous_on_setdist [continuous_intros]: "continuous_on t (λy. (setdist {y} s))"
by (metis continuous_at_setdist continuous_at_imp_continuous_on)
lemma uniformly_continuous_on_setdist: "uniformly_continuous_on t (λy. (setdist {y} s))"
by (force simp: uniformly_continuous_on_def dist_real_def intro: le_less_trans [OF setdist_Lipschitz])
lemma setdist_subset_right: "⟦t ≠ {}; t ⊆ u⟧ ⟹ setdist s u ≤ setdist s t"
apply (cases "s = {} ∨ u = {}", force)
apply (auto simp: setdist_def intro!: bdd_belowI [where m=0] cInf_superset_mono)
done
lemma setdist_subset_left: "⟦s ≠ {}; s ⊆ t⟧ ⟹ setdist t u ≤ setdist s u"
by (metis setdist_subset_right setdist_sym)
lemma setdist_closure_1 [simp]: "setdist (closure s) t = setdist s t"
proof (cases "s = {} ∨ t = {}")
case True then show ?thesis by force
next
case False
{ fix y
assume "y ∈ t"
have "continuous_on (closure s) (λa. dist a y)"
by (auto simp: continuous_intros dist_norm)
then have *: "⋀x. x ∈ closure s ⟹ setdist s t ≤ dist x y"
apply (rule continuous_ge_on_closure)
apply assumption
apply (blast intro: setdist_le_dist ‹y ∈ t› )
done
} note * = this
show ?thesis
apply (rule antisym)
using False closure_subset apply (blast intro: setdist_subset_left)
using False *
apply (force simp add: closure_eq_empty intro!: le_setdistI)
done
qed
lemma setdist_closure_2 [simp]: "setdist t (closure s) = setdist t s"
by (metis setdist_closure_1 setdist_sym)
lemma setdist_compact_closed:
fixes S :: "'a::euclidean_space set"
assumes S: "compact S" and T: "closed T"
and "S ≠ {}" "T ≠ {}"
shows "∃x ∈ S. ∃y ∈ T. dist x y = setdist S T"
proof -
have "(⋃x∈ S. ⋃y ∈ T. {x - y}) ≠ {}"
using assms by blast
then have "∃x ∈ S. ∃y ∈ T. dist x y ≤ setdist S T"
apply (rule distance_attains_inf [where a=0, OF compact_closed_differences [OF S T]])
apply (simp add: dist_norm le_setdist_iff)
apply blast
done
then show ?thesis
by (blast intro!: antisym [OF _ setdist_le_dist] )
qed
lemma setdist_closed_compact:
fixes S :: "'a::euclidean_space set"
assumes S: "closed S" and T: "compact T"
and "S ≠ {}" "T ≠ {}"
shows "∃x ∈ S. ∃y ∈ T. dist x y = setdist S T"
using setdist_compact_closed [OF T S ‹T ≠ {}› ‹S ≠ {}›]
by (metis dist_commute setdist_sym)
lemma setdist_eq_0I: "⟦x ∈ S; x ∈ T⟧ ⟹ setdist S T = 0"
by (metis antisym dist_self setdist_le_dist setdist_pos_le)
lemma setdist_eq_0_compact_closed:
fixes S :: "'a::euclidean_space set"
assumes S: "compact S" and T: "closed T"
shows "setdist S T = 0 ⟷ S = {} ∨ T = {} ∨ S ∩ T ≠ {}"
apply (cases "S = {} ∨ T = {}", force)
using setdist_compact_closed [OF S T]
apply (force intro: setdist_eq_0I )
done
corollary setdist_gt_0_compact_closed:
fixes S :: "'a::euclidean_space set"
assumes S: "compact S" and T: "closed T"
shows "setdist S T > 0 ⟷ (S ≠ {} ∧ T ≠ {} ∧ S ∩ T = {})"
using setdist_pos_le [of S T] setdist_eq_0_compact_closed [OF assms]
by linarith
lemma setdist_eq_0_closed_compact:
fixes S :: "'a::euclidean_space set"
assumes S: "closed S" and T: "compact T"
shows "setdist S T = 0 ⟷ S = {} ∨ T = {} ∨ S ∩ T ≠ {}"
using setdist_eq_0_compact_closed [OF T S]
by (metis Int_commute setdist_sym)
lemma setdist_eq_0_bounded:
fixes S :: "'a::euclidean_space set"
assumes "bounded S ∨ bounded T"
shows "setdist S T = 0 ⟷ S = {} ∨ T = {} ∨ closure S ∩ closure T ≠ {}"
apply (cases "S = {} ∨ T = {}", force)
using setdist_eq_0_compact_closed [of "closure S" "closure T"]
setdist_eq_0_closed_compact [of "closure S" "closure T"] assms
apply (force simp add: bounded_closure compact_eq_bounded_closed)
done
lemma setdist_unique:
"⟦a ∈ S; b ∈ T; ⋀x y. x ∈ S ∧ y ∈ T ==> dist a b ≤ dist x y⟧
⟹ setdist S T = dist a b"
by (force simp add: setdist_le_dist le_setdist_iff intro: antisym)
lemma setdist_closest_point:
"⟦closed S; S ≠ {}⟧ ⟹ setdist {a} S = dist a (closest_point S a)"
apply (rule setdist_unique)
using closest_point_le
apply (auto simp: closest_point_in_set)
done
lemma setdist_eq_0_sing_1:
fixes S :: "'a::euclidean_space set"
shows "setdist {x} S = 0 ⟷ S = {} ∨ x ∈ closure S"
by (auto simp: setdist_eq_0_bounded)
lemma setdist_eq_0_sing_2:
fixes S :: "'a::euclidean_space set"
shows "setdist S {x} = 0 ⟷ S = {} ∨ x ∈ closure S"
by (auto simp: setdist_eq_0_bounded)
lemma setdist_neq_0_sing_1:
fixes S :: "'a::euclidean_space set"
shows "⟦setdist {x} S = a; a ≠ 0⟧ ⟹ S ≠ {} ∧ x ∉ closure S"
by (auto simp: setdist_eq_0_sing_1)
lemma setdist_neq_0_sing_2:
fixes S :: "'a::euclidean_space set"
shows "⟦setdist S {x} = a; a ≠ 0⟧ ⟹ S ≠ {} ∧ x ∉ closure S"
by (auto simp: setdist_eq_0_sing_2)
lemma setdist_sing_in_set:
fixes S :: "'a::euclidean_space set"
shows "x ∈ S ⟹ setdist {x} S = 0"
using closure_subset by (auto simp: setdist_eq_0_sing_1)
lemma setdist_le_sing: "x ∈ S ==> setdist S T ≤ setdist {x} T"
using setdist_subset_left by auto
lemma setdist_eq_0_closed:
fixes S :: "'a::euclidean_space set"
shows "closed S ⟹ (setdist {x} S = 0 ⟷ S = {} ∨ x ∈ S)"
by (simp add: setdist_eq_0_sing_1)
lemma setdist_eq_0_closedin:
fixes S :: "'a::euclidean_space set"
shows "⟦closedin (subtopology euclidean U) S; x ∈ U⟧
⟹ (setdist {x} S = 0 ⟷ S = {} ∨ x ∈ S)"
by (auto simp: closedin_limpt setdist_eq_0_sing_1 closure_def)
lemma setdist_gt_0_closedin:
fixes S :: "'a::euclidean_space set"
shows "⟦closedin (subtopology euclidean U) S; x ∈ U; S ≠ {}; x ∉ S⟧
⟹ setdist {x} S > 0"
using less_eq_real_def setdist_eq_0_closedin by fastforce
subsection%unimportant‹Basic lemmas about hyperplanes and halfspaces›
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 apply auto[1]
using inner_zero_right by blast
lemma hyperplane_eq_UNIV:
"{x. a ∙ x = b} = UNIV ⟷ a = 0 ∧ b = 0"
proof -
have "UNIV ⊆ {x. a ∙ x = b} ⟹ a = 0 ∧ b = 0"
apply (drule_tac c = "((b+1) / (a ∙ a)) *⇩R a" in subsetD)
apply simp_all
by (metis add_cancel_right_right zero_neq_one)
then show ?thesis by force
qed
lemma halfspace_eq_empty_lt:
"{x. a ∙ x < b} = {} ⟷ a = 0 ∧ b ≤ 0"
proof -
have "{x. a ∙ x < b} ⊆ {} ⟹ a = 0 ∧ b ≤ 0"
apply (rule ccontr)
apply (drule_tac c = "((b-1) / (a ∙ a)) *⇩R a" in subsetD)
apply force+
done
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 "{x. a ∙ x ≤ b} ⊆ {} ⟹ a = 0 ∧ b < 0"
apply (rule ccontr)
apply (drule_tac c = "((b-1) / (a ∙ a)) *⇩R a" in subsetD)
apply force+
done
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%unimportant‹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 continuous_on_const)
show "open {x. f x < 0}"
by (simp add: open_Collect_less contf continuous_on_const)
show "S ⊆ {x. 0 < f x}"
apply (clarsimp simp add: f_def setdist_sing_in_set)
using assms
by (metis False IntI empty_iff le_less setdist_eq_0_sing_2 setdist_pos_le setdist_sym)
show "T ⊆ {x. f x < 0}"
apply (clarsimp simp add: f_def setdist_sing_in_set)
using assms
by (metis False IntI empty_iff le_less setdist_eq_0_sing_2 setdist_pos_le setdist_sym)
qed
qed
lemma separation_normal:
fixes S :: "'a::euclidean_space set"
assumes "closed S" "closed T" "S ∩ 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 (subtopology euclidean U) S"
and UT: "closedin (subtopology euclidean U) T"
and "S ∩ T = {}"
obtains S' T' where "openin (subtopology euclidean U) S'"
"openin (subtopology euclidean 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 (subtopology euclidean U) (U ∩ f -` {0<..})"
by (rule continuous_openin_preimage [where T=UNIV]) (simp_all add: contf)
next
show "openin (subtopology euclidean 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 show ?thesis
apply (rule separation_normal [OF ‹closed S›])
apply (rule_tac U=U and V=V in that)
by auto (meson bounded_ball bounded_subset compl_le_swap2 disjoint_eq_subset_Compl)
qed
subsection‹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 ∪ UNION ℋ ((-) N) ⊆ 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, hide_lams) 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))"
apply (rule connected_chain)
using S apply blast
by (metis image_iff le_cases nest)
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))"
apply (rule connected_chain_gen [of "S k"])
using S apply auto
by (meson le_cases nest subsetCE)
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
apply (rule_tac x="max m n" in exI, safe)
using max.cobounded2 apply blast
by (meson le_max_iff_disj)
qed
proposition proper_map:
fixes f :: "'a::heine_borel ⇒ 'b::heine_borel"
assumes "closedin (subtopology euclidean S) K"
and com: "⋀U. ⟦U ⊆ T; compact U⟧ ⟹ compact (S ∩ f -` U)"
and "f ` S ⊆ T"
shows "closedin (subtopology euclidean 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
have "compact (C ∩ (S ∩ f -` insert y (range (λi. f(X(n + i))))))" for n
apply (rule closed_Int_compact [OF ‹closed C›])
apply (rule com)
using X ‹K ⊆ S› ‹f ` S ⊆ T› ‹y ∈ T› apply blast
apply (rule compact_sequence_with_limit)
apply (simp add: fX add.commute [of n] LIMSEQ_ignore_initial_segment [OF hlim])
done
then have comf: "compact {a ∈ K. f a ∈ insert y (range (λi. f(X(n + i))))}" for n
by (simp add: Keq Int_def conj_commute)
have ne: "⋂ℱ ≠ {}"
if "finite ℱ"
and ℱ: "⋀t. t ∈ ℱ ⟹
(∃n. t = {a ∈ K. f a ∈ insert y (range (λi. f (X (n + i))))})"
for ℱ
proof -
obtain m where m: "⋀t. t ∈ ℱ ⟹ ∃k≤m. t = {a ∈ K. f a ∈ insert y (range (λi. f (X (k + i))))}"
apply (rule exE)
apply (rule finite_indexed_bound [OF ‹finite ℱ› ℱ], assumption, force)
done
have "X m ∈ ⋂ℱ"
using X le_Suc_ex by (fastforce dest: m)
then show ?thesis by blast
qed
have "⋂{{a. a ∈ K ∧ f a ∈ insert y (range (λi. f(X(n + i))))} |n. n ∈ UNIV}
≠ {}"
apply (rule compact_fip_heine_borel)
using comf apply force
using ne apply (simp add: subset_iff del: insert_iff)
done
then have "∃x. x ∈ (⋂n. {a ∈ K. f a ∈ insert y (range (λi. f (X (n + i))))})"
by blast
then show ?thesis
apply (simp add: image_iff fX)
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
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%unimportant‹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"
apply (clarsimp simp: dist_norm continuous_on_iff diff)
by (metis ‹z ≠ 0› mult.commute mult_less_cancel_left_pos norm_minus_commute real_norm_def zero_less_mult_iff zero_less_norm_iff)
then have conn_fS: "connected (f ` S)"
by (meson S connected_continuous_image continuous_on_subset hull_subset)
show ?thesis
proof (clarsimp simp: convex_contains_segment)
fix x y z
assume "x ∈ 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)"
apply (rule continuous_injective_image_segment_1)
apply (meson ‹x ∈ S› ‹y ∈ S› convex_affine_hull convex_contains_segment hull_inc continuous_on_subset [OF cont_f])
by (meson ‹x ∈ S› ‹y ∈ S› convex_affine_hull convex_contains_segment hull_inc inj_on_subset [OF inj_f])
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 -
have "{..<f z} ∩ f ` {x,y} ≠ {}" "{f z<..} ∩ f ` {x,y} ≠ {}"
using fz fz_notin ‹x ∈ S› ‹y ∈ S›
apply (auto simp: closed_segment_eq_real_ivl split: if_split_asm)
apply (metis image_eqI less_eq_real_def)+
done
then show "{..<f z} ∩ 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"
apply (clarsimp simp: dist_norm continuous_on_iff diff)
by (metis ‹z ≠ 0› mult.commute mult_less_cancel_left_pos norm_minus_commute real_norm_def zero_less_mult_iff zero_less_norm_iff)
then have "connected (f ` S)"
by (meson ‹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)"
apply (rule continuous_injective_image_segment_1)
apply (meson ‹a ∈ S› ‹b ∈ S› convex_affine_hull convex_contains_segment hull_inc continuous_on_subset [OF cont_f])
by (meson ‹a ∈ S› ‹b ∈ S› convex_affine_hull convex_contains_segment hull_inc inj_on_subset [OF inj_f])
then have "f ` (closed_segment a b) = f ` S"
by (simp add: ‹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"
apply safe
apply (metis (mono_tags, hide_lams) ‹a ∈ S› ‹b ∈ S› convex_affine_hull convex_contains_segment gf hull_inc subsetCE)
by (metis (mono_tags, lifting) ‹a ∈ S› ‹b ∈ S› convex_affine_hull convex_contains_segment gf hull_subset image_iff subsetCE)
ultimately have "closed_segment a b = S"
using gf by (simp add: image_comp o_def hull_inc cong: image_cong)
then show ?thesis
using that by blast
qed
lemma compact_convex_collinear_segment:
fixes S :: "'a::euclidean_space set"
assumes "S ≠ {}" "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 (subtopology euclidean 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 (subtopology euclidean (S × T)) c"
shows "closedin (subtopology euclidean 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 (subtopology euclidean (S × T)) c"
shows "closedin (subtopology euclidean 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 (subtopology euclidean (S × T)) U"
shows "closedin (subtopology euclidean 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 (subtopology euclidean 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
apply (subst *)
apply (rule closedin_closed_trans [OF _ closed_UNIV])
apply (rule closedin_compact_projection [OF ‹compact S›])
by (simp add: clo closedin_closed_Int)
qed
subsubsection%unimportant‹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 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))"
apply (rule span_mul)
apply (rule span_base)
apply (rule image_eqI [where x = "a + k *⇩R (x - a)"])
apply (auto simp: S T)
done
with xa image_iff show ?thesis by fastforce
qed
show "affine hull S ⊆ affine hull (S ∩ T)"
apply (simp add: subset_hull)
apply (simp add: ‹a ∈ S› ‹a ∈ T› hull_inc affine_hull_span_gen [of a])
apply (force simp: *)
done
qed
corollary affine_hull_convex_Int_open:
fixes S :: "'a::real_normed_vector set"
assumes "convex S" "open T" "S ∩ 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 (subtopology euclidean (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 (subtopology euclidean (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 (subtopology euclidean (affine hull S)) S" and "finite F" "F ⊂ S"
shows "affine hull (S - F) = affine hull S"
proof -
have clo: "closedin (subtopology euclidean 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 add: aff_dim_UNIV)
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 simp: aff_dim_affine_hull)
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
subsection%unimportant‹Properties of special hyperplanes›
lemma subspace_hyperplane: "subspace {x. a ∙ x = 0}"
by (simp add: subspace_def inner_right_distrib)
lemma subspace_hyperplane2: "subspace {x. x ∙ a = 0}"
by (simp add: inner_commute inner_right_distrib subspace_def)
lemma special_hyperplane_span:
fixes S :: "'n::euclidean_space set"
assumes "k ∈ Basis"
shows "{x. k ∙ x = 0} = span (Basis - {k})"
proof -
have *: "x ∈ span (Basis - {k})" if "k ∙ x = 0" for x
proof -
have "x = (∑b∈Basis. (x ∙ b) *⇩R b)"
by (simp add: euclidean_representation)
also have "... = (∑b ∈ Basis - {k}. (x ∙ b) *⇩R b)"
by (auto simp: sum.remove [of _ k] inner_commute assms that)
finally have "x = (∑b∈Basis - {k}. (x ∙ b) *⇩R b)" .
then show ?thesis
by (simp add: span_finite)
qed
show ?thesis
apply (rule span_subspace [symmetric])
using assms
apply (auto simp: inner_not_same_Basis intro: * subspace_hyperplane)
done
qed
lemma dim_special_hyperplane:
fixes k :: "'n::euclidean_space"
shows "k ∈ Basis ⟹ dim {x. k ∙ x = 0} = DIM('n) - 1"
apply (simp add: special_hyperplane_span)
apply (rule dim_unique [OF subset_refl])
apply (auto simp: Diff_subset independent_substdbasis)
apply (metis member_remove remove_def span_base)
done
proposition dim_hyperplane:
fixes a :: "'a::euclidean_space"
assumes "a ≠ 0"
shows "dim {x. a ∙ x = 0} = DIM('a) - 1"
proof -
have span0: "span {x. a ∙ x = 0} = {x. a ∙ x = 0}"
by (rule span_unique) (auto simp: subspace_hyperplane)
then obtain B where "independent B"
and Bsub: "B ⊆ {x. a ∙ x = 0}"
and subspB: "{x. a ∙ x = 0} ⊆ span B"
and card0: "(card B = dim {x. a ∙ x = 0})"
and ortho: "pairwise orthogonal B"
using orthogonal_basis_exists by metis
with assms have "a ∉ span B"
by (metis (mono_tags, lifting) span_eq inner_eq_zero_iff mem_Collect_eq span0)
then have ind: "independent (insert a B)"
by (simp add: ‹independent B› independent_insert)
have "finite B"
using ‹independent B› independent_bound by blast
have "UNIV ⊆ span (insert a B)"
proof fix y::'a
obtain r z where z: "y = r *⇩R a + z" "a ∙ z = 0"
apply (rule_tac r="(a ∙ y) / (a ∙ a)" and z = "y - ((a ∙ y) / (a ∙ a)) *⇩R a" in that)
using assms
by (auto simp: algebra_simps)
show "y ∈ span (insert a B)"
by (metis (mono_tags, lifting) z Bsub span_eq_iff
add_diff_cancel_left' mem_Collect_eq span0 span_breakdown_eq span_subspace subspB)
qed
then have dima: "DIM('a) = dim(insert a B)"
by (metis independent_Basis span_Basis dim_eq_card top.extremum_uniqueI)
then show ?thesis
by (metis (mono_tags, lifting) Bsub Diff_insert_absorb ‹a ∉ span B› ind card0
card_Diff_singleton dim_span indep_card_eq_dim_span insertI1 subsetCE
subspB)
qed
lemma lowdim_eq_hyperplane:
fixes S :: "'a::euclidean_space set"
assumes "dim S = DIM('a) - 1"
obtains a where "a ≠ 0" and "span S = {x. a ∙ x = 0}"
proof -
have dimS: "dim S < DIM('a)"
by (simp add: assms)
then obtain b where b: "b ≠ 0" "span S ⊆ {a. b ∙ a = 0}"
using lowdim_subset_hyperplane [of S] by fastforce
show ?thesis
apply (rule that[OF b(1)])
apply (rule subspace_dim_equal)
by (auto simp: assms b dim_hyperplane dim_span subspace_hyperplane
subspace_span)
qed
lemma dim_eq_hyperplane:
fixes S :: "'n::euclidean_space set"
shows "dim S = DIM('n) - 1 ⟷ (∃a. a ≠ 0 ∧ span S = {x. a ∙ x = 0})"
by (metis One_nat_def dim_hyperplane dim_span lowdim_eq_hyperplane)
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})"
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 show ?thesis
apply (simp add: aff_dim_eq_dim [of c] affine_hull_span_gen [of c] ‹c ∈ S› hull_inc dim_eq_hyperplane
del: One_nat_def)
apply (rule ex_cong)
apply (metis (mono_tags) span_0 ‹c = 0› image_add_0 inner_zero_right mem_Collect_eq)
done
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
show ?thesis
apply (simp add: aff_dim_eq_dim [of c] affine_hull_span_gen [of c] ‹c ∈ S› hull_inc dim_eq_hyperplane
del: One_nat_def, safe)
apply (fastforce simp add: inner_distrib intro: xc_im)
apply (force simp: intro!: 2)
done
qed
qed
corollary aff_dim_hyperplane [simp]:
fixes a :: "'a::euclidean_space"
shows "a ≠ 0 ⟹ aff_dim {x. a ∙ x = r} = DIM('a) - 1"
by (metis aff_dim_eq_hyperplane affine_hull_eq affine_hyperplane)
subsection%unimportant‹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
apply (simp add: hull_redundant cong: aff_dim_affine_hull2)
apply (simp add: affine_hull_insert_span_gen hull_inc)
by (force simp add:span_zero insert_commute [of a] hull_inc aff_dim_eq_dim [of x] dim_insert)
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 show ?thesis
apply safe
using affine_diffs_subspace [OF assms ‹b ∈ S›]
apply (metis (no_types, lifting) subspace_bounded_eq_trivial ab_left_minus bounded_translation
image_empty image_insert translation_invert)
apply force
done
qed
lemma affine_bounded_eq_lowdim:
fixes S :: "'a::euclidean_space set"
assumes "affine S"
shows "bounded S ⟷ aff_dim S ≤ 0"
apply safe
using affine_bounded_eq_trivial assms apply fastforce
by (metis aff_dim_sing aff_dim_subset affine_dim_equal affine_sing all_not_in_conv assms bounded_empty bounded_insert dual_order.antisym empty_subsetI insert_subset)
lemma bounded_hyperplane_eq_trivial_0:
fixes a :: "'a::euclidean_space"
assumes "a ≠ 0"
shows "bounded {x. a ∙ x = 0} ⟷ DIM('a) = 1"
proof
assume "bounded {x. a ∙ x = 0}"
then have "aff_dim {x. a ∙ x = 0} ≤ 0"
by (simp add: affine_bounded_eq_lowdim affine_hyperplane)
with assms show "DIM('a) = 1"
by (simp add: le_Suc_eq aff_dim_hyperplane)
next
assume "DIM('a) = 1"
then show "bounded {x. a ∙ x = 0}"
by (simp add: aff_dim_hyperplane 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 (simp add: bounded_hyperplane_eq_trivial_0, clarify)
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 show "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)
qed
subsection%unimportant‹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 aa: "a /⇩R (norm a) ∈ (⋂c∈C. {x. 0 ≤ c ∙ x})"
apply (clarsimp simp add: divide_simps)
using ab ‹0 < b›
by (metis hull_inc inner_commute less_eq_real_def less_trans)
show ?thesis
apply (simp add: C k_def)
using ass aa Int_iff empty_iff by blast
qed
qed
have "(span S ∩ frontier(cball 0 1)) ∩ (⋂ (k ` S)) ≠ {}"
apply (rule compact_imp_fip)
apply (blast intro: compact_cball)
using closed_halfspace_ge k_def apply blast
apply (metis *)
done
then show ?thesis
unfolding set_eq_iff k_def
by simp (metis inner_commute norm_eq_zero that zero_neq_one)
qed
lemma separating_hyperplane_set_point_inaff:
fixes S :: "'a::euclidean_space set"
assumes "convex S" "S ≠ {}" 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)"
by (simp add: ‹convex S›)
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)
show ?thesis
apply (rule_tac a=a and b = "a ∙ z" in that, simp_all)
using ‹a ∈ span ((+) (- z) ` S)› affine_hull_insert_span_gen apply blast
apply (simp_all add: ‹a ≠ 0› szx)
done
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›
apply (simp add: y'_def inner_diff dot_square_norm power2_eq_square)
by (metis ‹0 < e› add_le_same_cancel1 inner_commute inner_real_def inner_zero_left le_diff_eq norm_le_zero_iff real_mult_le_cancel_iff2)
ultimately show ?thesis by force
qed
show ?thesis
by (rule that [OF ‹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%unimportant‹ 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"
apply (simp add: f_def sum_Un sum_subtractf)
apply (simp add: sum.inter_restrict [symmetric] Int_commute)
done
moreover have "(∑x∈(s ∪ t). f x *⇩R x) = 0"
apply (simp add: f_def sum_Un scaleR_left_diff_distrib sum_subtractf)
apply (simp add: if_smult sum.inter_restrict [symmetric] Int_commute eq
cong del: if_weak_cong)
done
ultimately have "⋀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"
apply (rule subset_antisym)
apply (simp add: hull_mono)
by (simp add: affine_hull_Int_subset assms)
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"
apply (rule subset_antisym)
apply (simp add: hull_mono)
by (simp add: convex_hull_Int_subset assms)
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 that ‹finite S› sum_dd0 unfolding affine_dependent_explicit
apply (simp only: not_ex)
apply (drule_tac x=S in spec)
apply (drule_tac x=dd in spec, simp)
done
consider "c' a ≤ 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)
show ?thesis
proof (simp add: convex_hull_finite, intro exI conjI)
show "∀x∈T ∩ T'. 0 ≤ (cc(a := c a)) x"
by (simp add: c0 cc_def)
show "0 ≤ (cc(a := c a)) a"
by (simp add: c0)
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"
apply simp
apply (rule sum.mono_neutral_left)
using ‹T ⊆ S› apply (auto simp: ‹a ∉ S› cc0)
done
also have "... = c a + (1 - c a)"
by (metis ‹a ∉ S› fun_upd_other sum.cong sumSS')
finally show "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)"
apply simp
apply (rule sum.mono_neutral_left)
using ‹T ⊆ S› apply (auto simp: ‹a ∉ S› cc0)
done
also have "... = c a *⇩R a + x - c a *⇩R a"
by (simp add: wsumSS ‹a ∉ S› if_smult sum_delta_notmem)
finally show "(∑x∈insert a (T ∩ T'). (cc(a := c a)) x *⇩R x) = x"
by simp
qed
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)
show ?thesis
proof (simp add: convex_hull_finite, intro exI conjI)
show "∀x∈T ∩ T'. 0 ≤ (cc'(a := c' a)) x"
by (simp add: c'0 cc'_def)
show "0 ≤ (cc'(a := c' a)) a"
by (simp add: c'0)
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"
apply simp
apply (rule sum.mono_neutral_left)
using ‹T ⊆ S› apply (auto simp: ‹a ∉ S› cc0)
done
also have "... = c' a + (1 - c' a)"
by (metis ‹a ∉ S› fun_upd_other sum.cong sumSS')
finally show "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)"
apply simp
apply (rule sum.mono_neutral_left)
using ‹T ⊆ S› apply (auto simp: ‹a ∉ S› cc0)
done
also have "... = c a *⇩R a + x - c a *⇩R a"
by (simp add: wsumSS ‹a ∉ S› if_smult sum_delta_notmem)
finally show "(∑x∈insert a (T ∩ T'). (cc'(a := c' a)) x *⇩R x) = x"
by simp
qed
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'))"
apply (rule subset_antisym)
using in_convex_hull_exchange_unique assms apply blast
by (metis hull_mono inf_le1 inf_le2 insert_inter_insert le_inf_iff)
lemma Int_closed_segment:
fixes b :: "'a::euclidean_space"
assumes "b ∈ 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 add: affine_independent_2)
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}"
apply (rule collinear_3_trans [OF _ _ ‹b ≠ d›])
using d by (auto simp: insert_commute)
with ncoll show False ..
qed
then show ?thesis
by blast
qed
qed
lemma affine_hull_finite_intersection_hyperplanes:
fixes s :: "'a::euclidean_space set"
obtains f where
"finite f"
"of_nat (card f) + aff_dim s = DIM('a)"
"affine hull s = ⋂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 card1: "card ((λa. affine hull (c - {a})) ` (c - b)) = card (c - b)"
apply (rule card_image [OF inj_onI])
by (metis Diff_eq_empty_iff Diff_iff indc affine_dependent_def hull_subset insert_iff)
have card2: "(card (c - b)) + aff_dim s = DIM('a)"
proof -
have aff: "aff_dim (UNIV::'a set) = aff_dim c"
by (metis aff_dim_affine_hull affc)
have "aff_dim b = aff_dim s"
by (metis (no_types) aff_dim_affine_hull eq)
then have "int (card b) = 1 + aff_dim s"
by (simp add: aff_dim_affine_independent indb)
then show ?thesis
using fbc aff
by (simp add: ‹¬ affine_dependent c› ‹b ⊆ c› aff_dim_affine_independent aff_dim_UNIV card_Diff_subset of_nat_diff)
qed
show ?thesis
proof (cases "c = b")
case True show ?thesis
apply (rule_tac f="{}" in that)
using True affc
apply (simp_all add: eq [symmetric])
by (metis aff_dim_UNIV aff_dim_affine_hull)
next
case False
have ind: "¬ affine_dependent (⋃a∈c - b. c - {a})"
by (rule affine_independent_subset [OF indc]) auto
have affeq: "affine hull s = (⋂x∈(λa. c - {a}) ` (c - b). affine hull x)"
using ‹b ⊆ c› False
apply (subst affine_hull_Inter [OF ind, symmetric])
apply (simp add: eq double_diff)
done
have *: "1 + aff_dim (c - {t}) = int (DIM('a))"
if t: "t ∈ c" for t
proof -
have "insert t c = c"
using t by blast
then show ?thesis
by (metis (full_types) add.commute aff_dim_affine_hull aff_dim_insert aff_dim_UNIV affc affine_dependent_def indc insert_Diff_single t)
qed
show ?thesis
apply (rule_tac f = "(λx. affine hull x) ` ((λa. c - {a}) ` (c - b))" in that)
using ‹finite c› apply blast
apply (simp add: imeq card1 card2)
apply (simp add: affeq, clarify)
apply (metis DIM_positive One_nat_def Suc_leI add_diff_cancel_left' of_nat_1 aff_dim_eq_hyperplane of_nat_diff *)
done
qed
qed
lemma affine_hyperplane_sums_eq_UNIV_0:
fixes S :: "'a :: euclidean_space set"
assumes "affine S"
and "0 ∈ 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}"
apply (rule span_mono)
using ‹0 ∈ S› add.left_neutral by 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_lt: "DIM('a) - 1 < dim {x + y |x y. x ∈ S ∧ a ∙ y = 0}" .
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"
apply (rule dim_eq_full [THEN iffD1])
apply (rule antisym [OF dim_subset_UNIV])
using DIM_lt apply simp
done
ultimately show ?thesis
by (simp add: subs) (metis (lifting) span_eq_iff subs)
qed
proposition 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)
let ?U = "(+) (c+c) ` {x + y |x y. x ∈ (+) (- c) ` S ∧ a ∙ y = 0}"
have "u + v ∈ (+) (c + c) ` {x + v |x v. x ∈ (+) (- c) ` S ∧ a ∙ v = 0}"
if "u ∈ S" "b = a ∙ v" for u v
apply (rule_tac x="u+v-c-c" in image_eqI)
apply (simp_all add: algebra_simps)
apply (rule_tac x="u-c" in exI)
apply (rule_tac x="v-c" in exI)
apply (simp add: algebra_simps that c)
done
moreover have "⟦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} = ?U"
by (fastforce simp: algebra_simps)
also have "... = (+) (c+c) ` UNIV"
by (simp add: affine_hyperplane_sums_eq_UNIV_0 [OF aff 0 dc adc])
also have "... = UNIV"
by (simp add: translation_UNIV)
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 assms by (auto simp: affine_translation [symmetric])
have zero: "0 ∈ ((+) (-a) ` S)" "0 ∈ ((+) (-a) ` T)"
using a assms by auto
have [simp]: "{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)
have [simp]: "(+) (- a) ` S ∩ (+) (- a) ` T = (+) (- a) ` (S ∩ T)"
by auto
show ?thesis
using aff_dim_sums_Int_0 [OF aff zero]
by (auto simp: aff_dim_translation_eq)
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)
subsection‹ Orthogonal bases, Gram-Schmidt process, and related theorems›
lemma pairwise_orthogonal_independent:
assumes "pairwise orthogonal S" and "0 ∉ S"
shows "independent S"
proof -
have 0: "⋀x y. ⟦x ≠ y; x ∈ S; y ∈ S⟧ ⟹ x ∙ y = 0"
using assms by (simp add: pairwise_def orthogonal_def)
have "False" if "a ∈ S" and a: "a ∈ span (S - {a})" for a
proof -
obtain T U where "T ⊆ S - {a}" "a = (∑v∈T. U v *⇩R v)"
using a by (force simp: span_explicit)
then have "a ∙ a = a ∙ (∑v∈T. U v *⇩R v)"
by simp
also have "... = 0"
apply (simp add: inner_sum_right)
apply (rule comm_monoid_add_class.sum.neutral)
by (metis "0" DiffE ‹T ⊆ S - {a}› mult_not_zero singletonI subsetCE ‹a ∈ S›)
finally show ?thesis
using ‹0 ∉ S› ‹a ∈ S› by auto
qed
then show ?thesis
by (force simp: dependent_def)
qed
lemma pairwise_orthogonal_imp_finite:
fixes S :: "'a::euclidean_space set"
assumes "pairwise orthogonal S"
shows "finite S"
proof -
have "independent (S - {0})"
apply (rule pairwise_orthogonal_independent)
apply (metis Diff_iff assms pairwise_def)
by blast
then show ?thesis
by (meson independent_imp_finite infinite_remove)
qed
lemma subspace_orthogonal_to_vector: "subspace {y. orthogonal x y}"
by (simp add: subspace_def orthogonal_clauses)
lemma subspace_orthogonal_to_vectors: "subspace {y. ∀x ∈ S. orthogonal x y}"
by (simp add: subspace_def orthogonal_clauses)
lemma orthogonal_to_span:
assumes a: "a ∈ span S" and x: "⋀y. y ∈ S ⟹ orthogonal x y"
shows "orthogonal x a"
by (metis a orthogonal_clauses(1,2,4)
span_induct_alt x)
proposition Gram_Schmidt_step:
fixes S :: "'a::euclidean_space set"
assumes S: "pairwise orthogonal S" and x: "x ∈ span S"
shows "orthogonal x (a - (∑b∈S. (b ∙ a / (b ∙ b)) *⇩R b))"
proof -
have "finite S"
by (simp add: S pairwise_orthogonal_imp_finite)
have "orthogonal (a - (∑b∈S. (b ∙ a / (b ∙ b)) *⇩R b)) x"
if "x ∈ S" for x
proof -
have "a ∙ x = (∑y∈S. if y = x then y ∙ a else 0)"
by (simp add: ‹finite S› inner_commute sum.delta that)
also have "... = (∑b∈S. b ∙ a * (b ∙ x) / (b ∙ b))"
apply (rule sum.cong [OF refl], simp)
by (meson S orthogonal_def pairwise_def that)
finally show ?thesis
by (simp add: orthogonal_def algebra_simps inner_sum_left)
qed
then show ?thesis
using orthogonal_to_span orthogonal_commute x by blast
qed
lemma orthogonal_extension_aux:
fixes S :: "'a::euclidean_space set"
assumes "finite T" "finite S" "pairwise orthogonal S"
shows "∃U. pairwise orthogonal (S ∪ U) ∧ span (S ∪ U) = span (S ∪ T)"
using assms
proof (induction arbitrary: S)
case empty then show ?case
by simp (metis sup_bot_right)
next
case (insert a T)
have 0: "⋀x y. ⟦x ≠ y; x ∈ S; y ∈ S⟧ ⟹ x ∙ y = 0"
using insert by (simp add: pairwise_def orthogonal_def)
define a' where "a' = a - (∑b∈S. (b ∙ a / (b ∙ b)) *⇩R b)"
obtain U where orthU: "pairwise orthogonal (S ∪ insert a' U)"
and spanU: "span (insert a' S ∪ U) = span (insert a' S ∪ T)"
by (rule exE [OF insert.IH [of "insert a' S"]])
(auto simp: Gram_Schmidt_step a'_def insert.prems orthogonal_commute
pairwise_orthogonal_insert span_clauses)
have orthS: "⋀x. x ∈ S ⟹ a' ∙ x = 0"
apply (simp add: a'_def)
using Gram_Schmidt_step [OF ‹pairwise orthogonal S›]
apply (force simp: orthogonal_def inner_commute span_superset [THEN subsetD])
done
have "span (S ∪ insert a' U) = span (insert a' (S ∪ T))"
using spanU by simp
also have "... = span (insert a (S ∪ T))"
apply (rule eq_span_insert_eq)
apply (simp add: a'_def span_neg span_sum span_base span_mul)
done
also have "... = span (S ∪ insert a T)"
by simp
finally show ?case
by (rule_tac x="insert a' U" in exI) (use orthU in auto)
qed
proposition orthogonal_extension:
fixes S :: "'a::euclidean_space set"
assumes S: "pairwise orthogonal S"
obtains U where "pairwise orthogonal (S ∪ U)" "span (S ∪ U) = span (S ∪ T)"
proof -
obtain B where "finite B" "span B = span T"
using basis_subspace_exists [of "span T"] subspace_span by metis
with orthogonal_extension_aux [of B S]
obtain U where "pairwise orthogonal (S ∪ U)" "span (S ∪ U) = span (S ∪ B)"
using assms pairwise_orthogonal_imp_finite by auto
with ‹span B = span T› show ?thesis
by (rule_tac U=U in that) (auto simp: span_Un)
qed
corollary orthogonal_extension_strong:
fixes S :: "'a::euclidean_space set"
assumes S: "pairwise orthogonal S"
obtains U where "U ∩ (insert 0 S) = {}" "pairwise orthogonal (S ∪ U)"
"span (S ∪ U) = span (S ∪ T)"
proof -
obtain U where "pairwise orthogonal (S ∪ U)" "span (S ∪ U) = span (S ∪ T)"
using orthogonal_extension assms by blast
then show ?thesis
apply (rule_tac U = "U - (insert 0 S)" in that)
apply blast
apply (force simp: pairwise_def)
apply (metis Un_Diff_cancel Un_insert_left span_redundant span_zero)
done
qed
subsection‹Decomposing a vector into parts in orthogonal subspaces›
text‹existence of orthonormal basis for a subspace.›
lemma orthogonal_spanningset_subspace:
fixes S :: "'a :: euclidean_space set"
assumes "subspace S"
obtains B where "B ⊆ S" "pairwise orthogonal B" "span B = S"
proof -
obtain B where "B ⊆ S" "independent B" "S ⊆ span B" "card B = dim S"
using basis_exists by blast
with orthogonal_extension [of "{}" B]
show ?thesis
by (metis Un_empty_left assms pairwise_empty span_superset span_subspace that)
qed
lemma orthogonal_basis_subspace:
fixes S :: "'a :: euclidean_space set"
assumes "subspace S"
obtains B where "0 ∉ B" "B ⊆ S" "pairwise orthogonal B" "independent B"
"card B = dim S" "span B = S"
proof -
obtain B where "B ⊆ S" "pairwise orthogonal B" "span B = S"
using assms orthogonal_spanningset_subspace by blast
then show ?thesis
apply (rule_tac B = "B - {0}" in that)
apply (auto simp: indep_card_eq_dim_span pairwise_subset Diff_subset pairwise_orthogonal_independent elim: pairwise_subset)
done
qed
proposition orthonormal_basis_subspace:
fixes S :: "'a :: euclidean_space set"
assumes "subspace S"
obtains B where "B ⊆ S" "pairwise orthogonal B"
and "⋀x. x ∈ B ⟹ norm x = 1"
and "independent B" "card B = dim S" "span B = S"
proof -
obtain B where "0 ∉ B" "B ⊆ S"
and orth: "pairwise orthogonal B"
and "independent B" "card B = dim S" "span B = S"
by (blast intro: orthogonal_basis_subspace [OF assms])
have 1: "(λx. x /⇩R norm x) ` B ⊆ S"
using ‹span B = S› span_superset span_mul by fastforce
have 2: "pairwise orthogonal ((λx. x /⇩R norm x) ` B)"
using orth by (force simp: pairwise_def orthogonal_clauses)
have 3: "⋀x. x ∈ (λx. x /⇩R norm x) ` B ⟹ norm x = 1"
by (metis (no_types, lifting) ‹0 ∉ B› image_iff norm_sgn sgn_div_norm)
have 4: "independent ((λx. x /⇩R norm x) ` B)"
by (metis "2" "3" norm_zero pairwise_orthogonal_independent zero_neq_one)
have "inj_on (λx. x /⇩R norm x) B"
proof
fix x y
assume "x ∈ B" "y ∈ B" "x /⇩R norm x = y /⇩R norm y"
moreover have "⋀i. i ∈ B ⟹ norm (i /⇩R norm i) = 1"
using 3 by blast
ultimately show "x = y"
by (metis norm_eq_1 orth orthogonal_clauses(7) orthogonal_commute orthogonal_def pairwise_def zero_neq_one)
qed
then have 5: "card ((λx. x /⇩R norm x) ` B) = dim S"
by (metis ‹card B = dim S› card_image)
have 6: "span ((λx. x /⇩R norm x) ` B) = S"
by (metis "1" "4" "5" assms card_eq_dim independent_finite span_subspace)
show ?thesis
by (rule that [OF 1 2 3 4 5 6])
qed
proposition orthogonal_to_subspace_exists_gen:
fixes S :: "'a :: euclidean_space set"
assumes "span S ⊂ span T"
obtains x where "x ≠ 0" "x ∈ span T" "⋀y. y ∈ span S ⟹ orthogonal x y"
proof -
obtain B where "B ⊆ span S" and orthB: "pairwise orthogonal B"
and "⋀x. x ∈ B ⟹ norm x = 1"
and "independent B" "card B = dim S" "span B = span S"
by (rule orthonormal_basis_subspace [of "span S", OF subspace_span])
(auto simp: dim_span)
with assms obtain u where spanBT: "span B ⊆ span T" and "u ∉ span B" "u ∈ span T"
by auto
obtain C where orthBC: "pairwise orthogonal (B ∪ C)" and spanBC: "span (B ∪ C) = span (B ∪ {u})"
by (blast intro: orthogonal_extension [OF orthB])
show thesis
proof (cases "C ⊆ insert 0 B")
case True
then have "C ⊆ span B"
using span_eq
by (metis span_insert_0 subset_trans)
moreover have "u ∈ span (B ∪ C)"
using ‹span (B ∪ C) = span (B ∪ {u})› span_superset by force
ultimately show ?thesis
using True ‹u ∉ span B›
by (metis Un_insert_left span_insert_0 sup.orderE)
next
case False
then obtain x where "x ∈ C" "x ≠ 0" "x ∉ B"
by blast
then have "x ∈ span T"
by (metis (no_types, lifting) Un_insert_right Un_upper2 ‹u ∈ span T› spanBT spanBC
‹u ∈ span T› insert_subset span_superset span_mono
span_span subsetCE subset_trans sup_bot.comm_neutral)
moreover have "orthogonal x y" if "y ∈ span B" for y
using that
proof (rule span_induct)
show "subspace {a. orthogonal x a}"
by (simp add: subspace_orthogonal_to_vector)
show "⋀b. b ∈ B ⟹ orthogonal x b"
by (metis Un_iff ‹x ∈ C› ‹x ∉ B› orthBC pairwise_def)
qed
ultimately show ?thesis
using ‹x ≠ 0› that ‹span B = span S› by auto
qed
qed
corollary orthogonal_to_subspace_exists:
fixes S :: "'a :: euclidean_space set"
assumes "dim S < DIM('a)"
obtains x where "x ≠ 0" "⋀y. y ∈ span S ⟹ orthogonal x y"
proof -
have "span S ⊂ UNIV"
by (metis (mono_tags) UNIV_I assms inner_eq_zero_iff less_le lowdim_subset_hyperplane
mem_Collect_eq top.extremum_strict top.not_eq_extremum)
with orthogonal_to_subspace_exists_gen [of S UNIV] that show ?thesis
by (auto simp: span_UNIV)
qed
corollary orthogonal_to_vector_exists:
fixes x :: "'a :: euclidean_space"
assumes "2 ≤ DIM('a)"
obtains y where "y ≠ 0" "orthogonal x y"
proof -
have "dim {x} < DIM('a)"
using assms by auto
then show thesis
by (rule orthogonal_to_subspace_exists) (simp add: orthogonal_commute span_base that)
qed
proposition orthogonal_subspace_decomp_exists:
fixes S :: "'a :: euclidean_space set"
obtains y z where "y ∈ span S" "⋀w. w ∈ span S ⟹ orthogonal z w" "x = y + z"
proof -
obtain T where "0 ∉ T" "T ⊆ span S" "pairwise orthogonal T" "independent T"
"card T = dim (span S)" "span T = span S"
using orthogonal_basis_subspace subspace_span by blast
let ?a = "∑b∈T. (b ∙ x / (b ∙ b)) *⇩R b"
have orth: "orthogonal (x - ?a) w" if "w ∈ span S" for w
by (simp add: Gram_Schmidt_step ‹pairwise orthogonal T› ‹span T = span S›
orthogonal_commute that)
show ?thesis
apply (rule_tac y = "?a" and z = "x - ?a" in that)
apply (meson ‹T ⊆ span S› span_scale span_sum subsetCE)
apply (fact orth, simp)
done
qed
lemma orthogonal_subspace_decomp_unique:
fixes S :: "'a :: euclidean_space set"
assumes "x + y = x' + y'"
and ST: "x ∈ span S" "x' ∈ span S" "y ∈ span T" "y' ∈ span T"
and orth: "⋀a b. ⟦a ∈ S; b ∈ T⟧ ⟹ orthogonal a b"
shows "x = x' ∧ y = y'"
proof -
have "x + y - y' = x'"
by (simp add: assms)
moreover have "⋀a b. ⟦a ∈ span S; b ∈ span T⟧ ⟹ orthogonal a b"
by (meson orth orthogonal_commute orthogonal_to_span)
ultimately have "0 = x' - x"
by (metis (full_types) add_diff_cancel_left' ST diff_right_commute orthogonal_clauses(10) orthogonal_clauses(5) orthogonal_self)
with assms show ?thesis by auto
qed
lemma vector_in_orthogonal_spanningset:
fixes a :: "'a::euclidean_space"
obtains S where "a ∈ S" "pairwise orthogonal S" "span S = UNIV"
by (metis UNIV_I Un_iff empty_iff insert_subset orthogonal_extension pairwise_def
pairwise_orthogonal_insert span_UNIV subsetI subset_antisym)
lemma vector_in_orthogonal_basis:
fixes a :: "'a::euclidean_space"
assumes "a ≠ 0"
obtains S where "a ∈ S" "0 ∉ S" "pairwise orthogonal S" "independent S" "finite S"
"span S = UNIV" "card S = DIM('a)"
proof -
obtain S where S: "a ∈ S" "pairwise orthogonal S" "span S = UNIV"
using vector_in_orthogonal_spanningset .
show thesis
proof
show "pairwise orthogonal (S - {0})"
using pairwise_mono S(2) by blast
show "independent (S - {0})"
by (simp add: ‹pairwise orthogonal (S - {0})› pairwise_orthogonal_independent)
show "finite (S - {0})"
using ‹independent (S - {0})› independent_finite by blast
show "card (S - {0}) = DIM('a)"
using span_delete_0 [of S] S
by (simp add: ‹independent (S - {0})› indep_card_eq_dim_span dim_UNIV)
qed (use S ‹a ≠ 0› in auto)
qed
lemma vector_in_orthonormal_basis:
fixes a :: "'a::euclidean_space"
assumes "norm a = 1"
obtains S where "a ∈ S" "pairwise orthogonal S" "⋀x. x ∈ S ⟹ norm x = 1"
"independent S" "card S = DIM('a)" "span S = UNIV"
proof -
have "a ≠ 0"
using assms by auto
then obtain S where "a ∈ S" "0 ∉ S" "finite S"
and S: "pairwise orthogonal S" "independent S" "span S = UNIV" "card S = DIM('a)"
by (metis vector_in_orthogonal_basis)
let ?S = "(λx. x /⇩R norm x) ` S"
show thesis
proof
show "a ∈ ?S"
using ‹a ∈ S› assms image_iff by fastforce
next
show "pairwise orthogonal ?S"
using ‹pairwise orthogonal S› by (auto simp: pairwise_def orthogonal_def)
show "⋀x. x ∈ (λx. x /⇩R norm x) ` S ⟹ norm x = 1"
using ‹0 ∉ S› by (auto simp: divide_simps)
then show "independent ?S"
by (metis ‹pairwise orthogonal ((λx. x /⇩R norm x) ` S)› norm_zero pairwise_orthogonal_independent zero_neq_one)
have "inj_on (λx. x /⇩R norm x) S"
unfolding inj_on_def
by (metis (full_types) S(1) ‹0 ∉ S› inverse_nonzero_iff_nonzero norm_eq_zero orthogonal_scaleR orthogonal_self pairwise_def)
then show "card ?S = DIM('a)"
by (simp add: card_image S)
show "span ?S = UNIV"
by (metis (no_types) ‹0 ∉ S› ‹finite S› ‹span S = UNIV›
field_class.field_inverse_zero inverse_inverse_eq less_irrefl span_image_scale
zero_less_norm_iff)
qed
qed
proposition dim_orthogonal_sum:
fixes A :: "'a::euclidean_space set"
assumes "⋀x y. ⟦x ∈ A; y ∈ B⟧ ⟹ x ∙ y = 0"
shows "dim(A ∪ B) = dim A + dim B"
proof -
have 1: "⋀x y. ⟦x ∈ span A; y ∈ B⟧ ⟹ x ∙ y = 0"
by (erule span_induct [OF _ subspace_hyperplane2]; simp add: assms)
have "⋀x y. ⟦x ∈ span A; y ∈ span B⟧ ⟹ x ∙ y = 0"
using 1 by (simp add: span_induct [OF _ subspace_hyperplane])
then have 0: "⋀x y. ⟦x ∈ span A; y ∈ span B⟧ ⟹ x ∙ y = 0"
by simp
have "dim(A ∪ B) = dim (span (A ∪ B))"
by (simp add: dim_span)
also have "span (A ∪ B) = ((λ(a, b). a + b) ` (span A × span B))"
by (auto simp add: span_Un image_def)
also have "dim … = dim {x + y |x y. x ∈ span A ∧ y ∈ span B}"
by (auto intro!: arg_cong [where f=dim])
also have "... = dim {x + y |x y. x ∈ span A ∧ y ∈ span B} + dim(span A ∩ span B)"
by (auto simp: dest: 0)
also have "... = dim (span A) + dim (span B)"
by (rule dim_sums_Int) (auto simp: subspace_span)
also have "... = dim A + dim B"
by (simp add: dim_span)
finally show ?thesis .
qed
lemma dim_subspace_orthogonal_to_vectors:
fixes A :: "'a::euclidean_space set"
assumes "subspace A" "subspace B" "A ⊆ B"
shows "dim {y ∈ B. ∀x ∈ A. orthogonal x y} + dim A = dim B"
proof -
have "dim (span ({y ∈ B. ∀x∈A. orthogonal x y} ∪ A)) = dim (span B)"
proof (rule arg_cong [where f=dim, OF subset_antisym])
show "span ({y ∈ B. ∀x∈A. orthogonal x y} ∪ A) ⊆ span B"
by (simp add: ‹A ⊆ B› Collect_restrict span_mono)
next
have *: "x ∈ span ({y ∈ B. ∀x∈A. orthogonal x y} ∪ A)"
if "x ∈ B" for x
proof -
obtain y z where "x = y + z" "y ∈ span A" and orth: "⋀w. w ∈ span A ⟹ orthogonal z w"
using orthogonal_subspace_decomp_exists [of A x] that by auto
have "y ∈ span B"
using ‹y ∈ span A› assms(3) span_mono by blast
then have "z ∈ {a ∈ B. ∀x. x ∈ A ⟶ orthogonal x a}"
apply simp
using ‹x = y + z› assms(1) assms(2) orth orthogonal_commute span_add_eq
span_eq_iff that by blast
then have z: "z ∈ span {y ∈ B. ∀x∈A. orthogonal x y}"
by (meson span_superset subset_iff)
then show ?thesis
apply (auto simp: span_Un image_def ‹x = y + z› ‹y ∈ span A›)
using ‹y ∈ span A› add.commute by blast
qed
show "span B ⊆ span ({y ∈ B. ∀x∈A. orthogonal x y} ∪ A)"
by (rule span_minimal)
(auto intro: * span_minimal simp: subspace_span)
qed
then show ?thesis
by (metis (no_types, lifting) dim_orthogonal_sum dim_span mem_Collect_eq
orthogonal_commute orthogonal_def)
qed
lemma aff_dim_openin:
fixes S :: "'a::euclidean_space set"
assumes ope: "openin (subtopology euclidean 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 (simp, rule independent_card_le_dim)
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 show "(λx. e *⇩R x) ` B ⊆ (λx. x - a) ` S"
using e' by blast
show "independent ((λx. e *⇩R x) ` B)"
using linear_scale_self ‹independent B›
apply (rule linear_independent_injective_image)
using ‹0 < e› inj_on_def by fastforce
qed
also have "... = aff_dim S"
using ‹a ∈ S› aff_dim_eq_dim hull_inc by force
finally show "aff_dim T ≤ aff_dim S" .
qed
qed
lemma dim_openin:
fixes S :: "'a::euclidean_space set"
assumes ope: "openin (subtopology euclidean 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: Diff_subset 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))"
using z less by (simp add: aff_dim_eq_dim [symmetric] hull_inc)
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 (subtopology euclidean (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: Diff_subset closure_mono)
have "closure (rel_interior S - T) = closure (rel_interior S)"
apply (rule dense_complement_openin_affine_hull)
apply (simp add: assms rel_interior_aff_dim)
using ‹convex S› rel_interior_rel_open rel_open by blast
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%unimportant‹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: imageI orthogonal_def span)
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)
have "⋀w. w ∈ (+) (- z) ` S ⟹ (w + a') ∈ (+) (- z) ` S"
by (metis subspace_add a' span_eq_iff sub)
then have Sclo: "⋀w. w ∈ S ⟹ (w + a') ∈ S"
by fastforce
show ?thesis
proof (cases "a' = 0")
case True
with a assms True a'' diff_zero less_irrefl show ?thesis
by auto
next
case False
show ?thesis
apply (rule_tac a' = "a'" and b' = "a' ∙ z" in that)
apply (auto simp: a ba'' inner_left_distrib False Sclo)
done
qed
qed
lemma diffs_affine_hull_span:
assumes "a ∈ S"
shows "{x - a |x. x ∈ affine hull S} = span {x - a |x. x ∈ S}"
proof -
have *: "((λx. x - a) ` (S - {a})) = {x. x + a ∈ S} - {0}"
by (auto simp: algebra_simps)
show ?thesis
apply (simp add: affine_hull_span2 [OF assms] *)
apply (auto simp: algebra_simps)
done
qed
lemma aff_dim_dim_affine_diffs:
fixes S :: "'a :: euclidean_space set"
assumes "affine S" "a ∈ S"
shows "aff_dim S = dim {x - a |x. x ∈ 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 *: "{x - c |x. x ∈ S} = {x - a |x. x ∈ S}"
apply safe
apply (simp_all only: xy)
using mem_affine_3_minus [OF ‹affine S›] ‹a ∈ S› ‹c ∈ S› apply blast+
done
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 {x - c |x. x ∈ B}"
by (simp add: affine_independent_card_dim_diffs [OF ind ‹c ∈ B›])
also have "... = dim {x - c | x. x ∈ affine hull B}"
by (simp add: diffs_affine_hull_span ‹c ∈ B› dim_span)
also have "... = dim {x - a |x. x ∈ 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 ` {x - a| x. x ∈ 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)
also have "... = int (dim (f ` {x - a| x. x ∈ T}))"
by (force simp: linear_diff [OF assms] 2)
also have "... ≤ int (dim {x - a| x. x ∈ 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)
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‹Several Variants of Paracompactness›
proposition paracompact:
fixes S :: "'a :: euclidean_space 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
apply (rule_tac x = T in exI)
apply (rule_tac x = "ball x e" in exI)
using ‹T ∈ 𝒞›
apply (simp add: closure_minimal)
done
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 = UNION S G"
using Lindelof [of "G ` S"] by (metis image_iff)
then obtain K where K: "K ⊆ S" "countable K" and eq: "UNION K G = UNION S G"
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
have 1: "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
have 3: "∃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
show ?thesis
apply (rule_tac x="G (a n)" in exI)
apply (intro conjI oG n *)
using ‹K ⊆ S› ‹range a = K› apply blast
done
qed
show ?thesis
apply (rule that [OF 1 _ 3])
using Fin ‹K ⊆ S› ‹range a = K› apply (auto simp: odif)
done
qed
corollary paracompact_closedin:
fixes S :: "'a :: euclidean_space set"
assumes cin: "closedin (subtopology euclidean U) S"
and oin: "⋀T. T ∈ 𝒞 ⟹ openin (subtopology euclidean U) T"
and "S ⊆ ⋃𝒞"
obtains 𝒞' where "S ⊆ ⋃ 𝒞'"
and "⋀V. V ∈ 𝒞' ⟹ openin (subtopology euclidean U) V ∧ (∃T. T ∈ 𝒞 ∧ V ⊆ T)"
and "⋀x. x ∈ U
⟹ ∃V. openin (subtopology euclidean 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 ≠ {}}"
using paracompact [OF 1 2] by auto
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 (subtopology euclidean 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 (subtopology euclidean U) V ∧ x ∈ V ∧ finite {X ∈ ?C. X ∩ V ≠ {}}"
if "x ∈ U" for x
using D2 [OF that]
apply clarify
apply (rule_tac x="U ∩ V" in exI)
apply (auto intro: that finite_subset [OF *])
done
qed
qed
corollary paracompact_closed:
fixes S :: "'a :: euclidean_space 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 ≠ {})}"
using paracompact_closedin [of UNIV S 𝒞] assms by auto
subsection%unimportant‹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 (subtopology euclidean (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
apply (subst eq)
apply (intro continuous_intros continuous_closedin_preimage continuous_on_subset [OF contf])
by auto
qed
lemma continuous_closed_graph_eq:
fixes f :: "'a::euclidean_space ⇒ 'b::euclidean_space"
assumes "compact T" and fim: "f ` S ⊆ T"
shows "continuous_on S f ⟷
closedin (subtopology euclidean (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 (subtopology euclidean T) U"
have eq: "(S ∩ f -` U) = fst ` (((λx. Pair x (f x)) ` S) ∩ (S × U))"
by (force simp: image_iff)
show "closedin (subtopology euclidean 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)"
apply (rule closedin_closed_trans)
apply (rule continuous_closed_graph_gen [OF contf subset_UNIV])
by (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 (subtopology euclidean (S ∪ T)) S"
and opT: "openin (subtopology euclidean (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}" "S ∪ T" "λi. i" "λi. f" f] contf contg opS opT by blast
lemma continuous_on_cases_local_open:
assumes opS: "openin (subtopology euclidean (S ∪ T)) S"
and opT: "openin (subtopology euclidean (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%unimportant‹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)"
apply (rule sum.mono_neutral_right)
using fin by auto
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)"
apply (rule sum.mono_neutral_right)
using fin by auto
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 apply simp
apply (rule sum.cong, auto)
done
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 apply simp
apply (rule sum.cong, auto)
done
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"
proof -
have b: "b ∈ closed_segment a c"
by (simp add: assms open_closed_segment)
have *: "open_segment a c ⊆ insert b (open_segment a b ∪ open_segment b c)"
if "{b,c,a} ∪ open_segment a b ∪ open_segment b c = {c,a} ∪ open_segment a c"
proof -
have "insert a (insert c (insert b (open_segment a b ∪ open_segment b c))) = insert a (insert c (open_segment a c))"
using that by (simp add: insert_commute)
then show ?thesis
by (metis (no_types) Diff_cancel Diff_eq_empty_iff Diff_insert2 open_segment_def)
qed
show ?thesis
using Un_closed_segment [OF b]
apply (simp add: closed_segment_eq_open)
apply (rule equalityI)
using assms
apply (simp add: b subset_open_segment)
using * by (simp add: insert_commute)
qed
subsection‹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› ‹UNION UNIV f = 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})"
apply (intro UN_mono order_refl)
apply (simp add: cball_subset_ball_iff divide_simps)
done
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 ⊆ UNION UNIV ?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: divide_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 ∈ UNION UNIV ?C"
by (rule_tac a = "N1+N2" in UN_I) (auto simp: dist_norm norm_minus_commute)
qed
then show "UNION UNIV ?C = S" by auto
qed
ultimately show ?thesis
using that by metis
qed
subsection{*Orthogonal complement*}
definition orthogonal_comp ("_⇧⊥" [80] 80)
where "orthogonal_comp W ≡ {x. ∀y ∈ W. orthogonal y x}"
lemma 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
lemma 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))⇧⊥"
apply (auto simp: orthogonal_comp_def orthogonal_def)
apply (simp add: adjoint_works assms(1) inner_commute)
by (metis adjoint_works all_zero_iff assms(1) inner_commute)
subsection‹ 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, hide_lams) 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
proposition 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, hide_lams) adjoint_clauses(2) adjoint_linear assms
linear_injective_0 linear_injective_imp_surjective linear_surj_adj_imp_inj)
next
assume ?rhs
then show "¬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