Reorg, in particular Determinants as well as some linear algebra from Starlike and Change_Of_Vars
(* Title: HOL/Analysis/Starlike.thy
Author: L C Paulson, University of Cambridge
Author: Robert Himmelmann, TU Muenchen
Author: Bogdan Grechuk, University of Edinburgh
Author: Armin Heller, TU Muenchen
Author: Johannes Hoelzl, TU Muenchen
*)
section \<open>Line Segments\<close>
theory Starlike
imports Convex_Euclidean_Space
begin
subsection \<open>Midpoint\<close>
definition%important midpoint :: "'a::real_vector \<Rightarrow> 'a \<Rightarrow> 'a"
where "midpoint a b = (inverse (2::real)) *\<^sub>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 \<longleftrightarrow> a + b = c + c"
proof -
have "midpoint a b = c \<longleftrightarrow> 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 \<le> b" shows ge_midpoint_1: "a \<le> midpoint a b"
and le_midpoint_1: "midpoint a b \<le> 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 *: "\<And>x y::'a. 2 *\<^sub>R x = - y \<Longrightarrow> norm x = (norm y) / 2"
unfolding equation_minus_iff by auto
have **: "\<And>x y::'a. 2 *\<^sub>R x = y \<Longrightarrow> 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 \<longleftrightarrow> a = b"
"midpoint a b = b \<longleftrightarrow> 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 \<Longrightarrow> midpoint(f a)(f b) = f(midpoint a b)"
by (simp add: linear_iff midpoint_def)
subsection \<open>Line segments\<close>
definition%important closed_segment :: "'a::real_vector \<Rightarrow> 'a \<Rightarrow> 'a set"
where "closed_segment a b = {(1 - u) *\<^sub>R a + u *\<^sub>R b | u::real. 0 \<le> u \<and> u \<le> 1}"
definition%important open_segment :: "'a::real_vector \<Rightarrow> 'a \<Rightarrow> 'a set" where
"open_segment a b \<equiv> closed_segment a b - {a,b}"
lemmas segment = open_segment_def closed_segment_def
lemma in_segment:
"x \<in> closed_segment a b \<longleftrightarrow> (\<exists>u. 0 \<le> u \<and> u \<le> 1 \<and> x = (1 - u) *\<^sub>R a + u *\<^sub>R b)"
"x \<in> open_segment a b \<longleftrightarrow> a \<noteq> b \<and> (\<exists>u. 0 < u \<and> u < 1 \<and> x = (1 - u) *\<^sub>R a + u *\<^sub>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:
"\<lbrakk>linear f; inj f\<rbrakk> \<Longrightarrow> 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 (\<lambda>x. 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 (\<lambda>x. 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:
"\<lbrakk>x \<in> Reals; y \<in> Reals\<rbrakk> \<Longrightarrow> 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:
"\<lbrakk>x \<in> Reals; y \<in> Reals\<rbrakk> \<Longrightarrow> 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') \<in> open_segment (a, a') (b, b')
\<Longrightarrow> (x \<in> open_segment a b \<or> a = b) \<and> (x' \<in> open_segment a' b' \<or> a' = b')"
by (auto simp: in_segment)
lemma closed_segment_PairD:
"(x, x') \<in> closed_segment (a, a') (b, b') \<Longrightarrow> x \<in> closed_segment a b \<and> x' \<in> closed_segment a' b'"
by (auto simp: closed_segment_def)
lemma closed_segment_translation_eq [simp]:
"d + x \<in> closed_segment (d + a) (d + b) \<longleftrightarrow> x \<in> closed_segment a b"
proof -
have *: "\<And>d x a b. x \<in> closed_segment a b \<Longrightarrow> d + x \<in> 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 \<in> open_segment (d + a) (d + b) \<longleftrightarrow> x \<in> open_segment a b"
by (simp add: open_segment_def)
lemma of_real_closed_segment [simp]:
"of_real x \<in> closed_segment (of_real a) (of_real b) \<longleftrightarrow> x \<in> 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 \<in> open_segment (of_real a) (of_real b) \<longleftrightarrow> x \<in> 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 \<longleftrightarrow> (\<forall>a\<in>S. \<forall>b\<in>S. closed_segment a b \<subseteq> S)"
unfolding convex_alt closed_segment_def by auto
lemma closed_segment_in_Reals:
"\<lbrakk>x \<in> closed_segment a b; a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> x \<in> Reals"
by (meson subsetD convex_Reals convex_contains_segment)
lemma open_segment_in_Reals:
"\<lbrakk>x \<in> open_segment a b; a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> x \<in> Reals"
by (metis Diff_iff closed_segment_in_Reals open_segment_def)
lemma closed_segment_subset: "\<lbrakk>x \<in> S; y \<in> S; convex S\<rbrakk> \<Longrightarrow> closed_segment x y \<subseteq> S"
by (simp add: convex_contains_segment)
lemma closed_segment_subset_convex_hull:
"\<lbrakk>x \<in> convex hull S; y \<in> convex hull S\<rbrakk> \<Longrightarrow> closed_segment x y \<subseteq> convex hull S"
using convex_contains_segment by blast
lemma segment_convex_hull:
"closed_segment a b = convex hull {a,b}"
proof -
have *: "\<And>x. {x} \<noteq> {}" 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 \<in> open_segment w z \<Longrightarrow> u \<in> closed_segment w z"
by (auto simp add: closed_segment_def open_segment_def)
lemma segment_open_subset_closed:
"open_segment a b \<subseteq> 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 \<in> closed_segment a b" "b \<in> 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 \<in> X0"
shows "\<forall>\<^sub>F x in at x0 within U. closed_segment x0 x \<subseteq> X0"
proof -
from openE[OF assms]
obtain e where e: "0 < e" "ball x0 e \<subseteq> X0" .
then have "\<forall>\<^sub>F x in at x0 within U. x \<in> ball x0 e"
by (auto simp: dist_commute eventually_at)
then show ?thesis
proof eventually_elim
case (elim x)
have "x0 \<in> ball x0 e" using \<open>e > 0\<close> by simp
from convex_ball[unfolded convex_contains_segment, rule_format, OF this elim]
have "closed_segment x0 x \<subseteq> ball x0 e" .
also note \<open>\<dots> \<subseteq> X0\<close>
finally show ?case .
qed
qed
lemma segment_furthest_le:
fixes a b x y :: "'a::euclidean_space"
assumes "x \<in> closed_segment a b"
shows "norm (y - x) \<le> norm (y - a) \<or> norm (y - x) \<le> norm (y - b)"
proof -
obtain z where "z \<in> {a, b}" "norm (x - y) \<le> 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 \<in> closed_segment a b"
shows "norm (x - a) \<le> norm (b - a)"
proof -
obtain u where "x = (1 - u) *\<^sub>R a + u *\<^sub>R b" "0 \<le> u" "u \<le> 1"
using assms by (auto simp add: closed_segment_def)
then show "norm (x - a) \<le> 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 \<in> closed_segment a b"
shows "norm (x - a) \<le> norm (b - a)" "norm (x - b) \<le> 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 \<union> {a,b}"
using open_segment_def by auto
lemma convex_contains_open_segment:
"convex s \<longleftrightarrow> (\<forall>a\<in>s. \<forall>b\<in>s. open_segment a b \<subseteq> 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 \<le> b then {a .. b} else {b .. a})"
proof -
have "b \<le> a \<Longrightarrow> closed_segment b a = {b .. a}"
and "a \<le> b \<Longrightarrow> 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 \<le> 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 = (\<lambda>x. (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 \<in> closed_segment a b"
shows "dist x a \<le> dist a b \<and> dist x b \<le> dist a b"
proof (intro conjI)
obtain u where u: "0 \<le> u" "u \<le> 1" and x: "x = (1 - u) *\<^sub>R a + u *\<^sub>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 \<open>0 \<le> u\<close> abs_of_nonneg norm_minus_commute norm_scaleR real_vector.scale_right_diff_distrib)
also have "... \<le> dist a b"
by (simp add: mult_left_le_one_le u)
finally show "dist x a \<le> dist a b" .
have "dist x b = norm ((1-u) *\<^sub>R a - (1-u) *\<^sub>R b)"
by (simp add: dist_norm algebra_simps x)
also have "... = (1-u) * dist a b"
proof -
have "norm ((1 - 1 * u) *\<^sub>R (a - b)) = (1 - 1 * u) * norm (a - b)"
using \<open>u \<le> 1\<close> by force
then show ?thesis
by (simp add: dist_norm real_vector.scale_right_diff_distrib)
qed
also have "... \<le> dist a b"
by (simp add: mult_left_le_one_le u)
finally show "dist x b \<le> dist a b" .
qed
lemma dist_in_open_segment:
fixes a :: "'a :: euclidean_space"
assumes "x \<in> open_segment a b"
shows "dist x a < dist a b \<and> dist x b < dist a b"
proof (intro conjI)
obtain u where u: "0 < u" "u < 1" and x: "x = (1 - u) *\<^sub>R a + u *\<^sub>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 \<open>0 < u\<close>)
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 \<open>u < 1\<close>)
finally show "dist x a < dist a b" .
have ab_ne0: "dist a b \<noteq> 0"
using * by fastforce
have "dist x b = norm ((1-u) *\<^sub>R a - (1-u) *\<^sub>R b)"
by (simp add: dist_norm algebra_simps x)
also have "... = (1-u) * dist a b"
proof -
have "norm ((1 - 1 * u) *\<^sub>R (a - b)) = (1 - 1 * u) * norm (a - b)"
using \<open>u < 1\<close> 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 \<open>0 < u\<close> by simp
finally show "dist x b < dist a b" .
qed
lemma dist_decreases_open_segment_0:
fixes x :: "'a :: euclidean_space"
assumes "x \<in> open_segment 0 b"
shows "dist c x < dist c 0 \<or> dist c x < dist c b"
proof (rule ccontr, clarsimp simp: not_less)
obtain u where u: "0 \<noteq> b" "0 < u" "u < 1" and x: "x = u *\<^sub>R b"
using assms by (auto simp: in_segment)
have xb: "x \<bullet> b < b \<bullet> b"
using u x by auto
assume "norm c \<le> dist c x"
then have "c \<bullet> c \<le> (c - x) \<bullet> (c - x)"
by (simp add: dist_norm norm_le)
moreover have "0 < x \<bullet> b"
using u x by auto
ultimately have less: "c \<bullet> b < x \<bullet> b"
by (simp add: x algebra_simps inner_commute u)
assume "dist c b \<le> dist c x"
then have "(c - b) \<bullet> (c - b) \<le> (c - x) \<bullet> (c - x)"
by (simp add: dist_norm norm_le)
then have "(b \<bullet> b) * (1 - u*u) \<le> 2 * (b \<bullet> c) * (1-u)"
by (simp add: x algebra_simps inner_commute)
then have "(1+u) * (b \<bullet> b) * (1-u) \<le> 2 * (b \<bullet> c) * (1-u)"
by (simp add: algebra_simps)
then have "(1+u) * (b \<bullet> b) \<le> 2 * (b \<bullet> c)"
using \<open>u < 1\<close> by auto
with xb have "c \<bullet> b \<ge> x \<bullet> 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 \<in> open_segment a b"
shows "dist c x < dist c a \<or> dist c x < dist c b"
proof -
have *: "x - a \<in> 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 \<in> open_segment a b"
shows "norm (y - x) < norm (y - a) \<or> 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 \<in> closed_segment a b"
shows "dist c x \<le> dist c a \<or> dist c x \<le> dist c b"
apply (cases "x \<in> 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 "\<lbrakk>ball a r \<subseteq> T; T \<subseteq> cball a r\<rbrakk> \<Longrightarrow> 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 \<subseteq> 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 \<noteq> b \<Longrightarrow> a \<notin> 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 "\<lbrakk>closed S; S \<noteq> {}\<rbrakk> \<Longrightarrow> open_segment a (closest_point S a) \<inter> 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 \<noteq> {}"
obtains b where "b \<in> S" "open_segment a b \<inter> S = {}"
by (metis assms segment_to_closest_point closest_point_exists that)
subsubsection\<open>More lemmas, especially for working with the underlying formula\<close>
lemma segment_eq_compose:
fixes a :: "'a :: real_vector"
shows "(\<lambda>u. (1 - u) *\<^sub>R a + u *\<^sub>R b) = (\<lambda>x. a + x) o (\<lambda>u. u *\<^sub>R (b - a))"
by (simp add: o_def algebra_simps)
lemma segment_degen_1:
fixes a :: "'a :: real_vector"
shows "(1 - u) *\<^sub>R a + u *\<^sub>R b = b \<longleftrightarrow> a=b \<or> u=1"
proof -
{ assume "(1 - u) *\<^sub>R a + u *\<^sub>R b = b"
then have "(1 - u) *\<^sub>R a = (1 - u) *\<^sub>R b"
by (simp add: algebra_simps)
then have "a=b \<or> 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) *\<^sub>R a + u *\<^sub>R b = a \<longleftrightarrow> a=b \<or> 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 *\<^sub>R b + v *\<^sub>R a) = (u *\<^sub>R a + v *\<^sub>R b)" "u \<noteq> 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 = (\<lambda>u. (1 - u) *\<^sub>R a + u *\<^sub>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 (\<lambda>u. (1 - u) *\<^sub>R a + u *\<^sub>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 \<in> open_segment a b"
shows "norm (x - a) < norm (b - a)"
proof -
obtain u where "x = (1 - u) *\<^sub>R a + u *\<^sub>R b" "0 < u" "u < 1" "a \<noteq> 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 \<in> 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]:
"closure (open_segment a b) = (if a = b then {} else closed_segment a b)"
for a :: "'a::euclidean_space"
proof (cases "a = b")
case True
then show ?thesis
by simp
next
case False
have "closure ((\<lambda>u. u *\<^sub>R (b - a)) ` {0<..<1}) = (\<lambda>u. u *\<^sub>R (b - a)) ` closure {0<..<1}"
apply (rule closure_injective_linear_image [symmetric])
apply (use False in \<open>auto intro!: injI\<close>)
done
then have "closure
((\<lambda>u. (1 - u) *\<^sub>R a + u *\<^sub>R b) ` {0<..<1}) =
(\<lambda>x. (1 - x) *\<^sub>R a + x *\<^sub>R b) ` closure {0<..<1}"
using closure_translation [of a "((\<lambda>x. x *\<^sub>R b - x *\<^sub>R a) ` {0<..<1})"]
by (simp add: segment_eq_compose field_simps scaleR_diff_left scaleR_diff_right image_image)
then show ?thesis
by (simp add: segment_image_interval closure_greaterThanLessThan [symmetric] del: closure_greaterThanLessThan)
qed
lemma closed_open_segment_iff [simp]:
fixes a :: "'a::euclidean_space" shows "closed(open_segment a b) \<longleftrightarrow> 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) \<longleftrightarrow> 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 ((\<lambda>u. u *\<^sub>R (b - a)) ` {0<..<1})"
by (rule convex_linear_image) auto
then have "convex ((+) a ` (\<lambda>u. u *\<^sub>R (b - a)) ` {0<..<1})"
by (rule convex_translation)
then show ?thesis
by (simp add: image_image open_segment_image_interval segment_eq_compose field_simps scaleR_diff_left scaleR_diff_right)
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 \<Rightarrow> real"
assumes "y \<in> closed_segment (f a) (f b)"
assumes "continuous_on (closed_segment a b) f"
shows "\<exists>x \<in> 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 \<le> b"; cases "f b \<ge> f a") (auto simp: closed_segment_eq_real_ivl continuous_on_minus)
subsection\<open>Starlike sets\<close>
definition%important "starlike S \<longleftrightarrow> (\<exists>a\<in>S. \<forall>x\<in>S. closed_segment a x \<subseteq> S)"
lemma starlike_UNIV [simp]: "starlike UNIV"
by (simp add: starlike_def)
lemma convex_imp_starlike:
"convex S \<Longrightarrow> S \<noteq> {} \<Longrightarrow> 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 \<in> rel_interior S" "b \<in> closure S"
shows "open_segment a b \<subseteq> rel_interior S"
proof
fix x
have [simp]: "(1 - u) *\<^sub>R a + u *\<^sub>R b = b - (1 - u) *\<^sub>R (b - a)" for u
by (simp add: algebra_simps)
assume "x \<in> open_segment a b"
then show "x \<in> 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 \<Union>x \<in> 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 \<subseteq> C" and z: "z \<in> rel_interior C" and dis: "disjnt S (rel_interior C)"
shows "S \<inter> (convex hull (insert z T)) = S \<inter> (convex hull T)" (is "?lhs = ?rhs")
proof
have "T = {} \<Longrightarrow> z \<notin> S"
using dis z by (auto simp add: disjnt_def)
then show "?lhs \<subseteq> ?rhs"
proof (clarsimp simp add: convex_hull_insert_segments)
fix x y
assume "x \<in> S" and y: "y \<in> convex hull T" and "x \<in> closed_segment z y"
have "y \<in> closure C"
by (metis y \<open>convex C\<close> \<open>T \<subseteq> C\<close> closure_subset contra_subsetD convex_hull_eq hull_mono)
moreover have "x \<notin> rel_interior C"
by (meson \<open>x \<in> S\<close> dis disjnt_iff)
moreover have "x \<in> open_segment z y \<union> {z, y}"
using \<open>x \<in> closed_segment z y\<close> closed_segment_eq_open by blast
ultimately show "x \<in> convex hull T"
using rel_interior_closure_convex_segment [OF \<open>convex C\<close> z]
using y z by blast
qed
show "?rhs \<subseteq> ?lhs"
by (meson hull_mono inf_mono subset_insertI subset_refl)
qed
subsection%unimportant\<open>More results about segments\<close>
lemma dist_half_times2:
fixes a :: "'a :: real_normed_vector"
shows "dist ((1 / 2) *\<^sub>R (a + b)) x * 2 = dist (a+b) (2 *\<^sub>R x)"
proof -
have "norm ((1 / 2) *\<^sub>R (a + b) - x) * 2 = norm (2 *\<^sub>R ((1 / 2) *\<^sub>R (a + b) - x))"
by simp
also have "... = norm ((a + b) - 2 *\<^sub>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} \<inter> cball(inverse 2 *\<^sub>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 *: "((\<exists>u v. x = u *\<^sub>R a + v *\<^sub>R b \<and> u + v = 1) \<and>
dist ((1 / 2) *\<^sub>R (a + b)) x * 2 \<le> norm (b - a)) =
(\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and> 0 \<le> u \<and> u \<le> 1)" for x
proof -
have "((\<exists>u v. x = u *\<^sub>R a + v *\<^sub>R b \<and> u + v = 1) \<and>
dist ((1 / 2) *\<^sub>R (a + b)) x * 2 \<le> norm (b - a)) =
((\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b) \<and>
dist ((1 / 2) *\<^sub>R (a + b)) x * 2 \<le> norm (b - a))"
unfolding eq_diff_eq [symmetric] by simp
also have "... = (\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and>
norm ((a+b) - (2 *\<^sub>R x)) \<le> norm (b - a))"
by (simp add: dist_half_times2) (simp add: dist_norm)
also have "... = (\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and>
norm ((a+b) - (2 *\<^sub>R ((1 - u) *\<^sub>R a + u *\<^sub>R b))) \<le> norm (b - a))"
by auto
also have "... = (\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and>
norm ((1 - u * 2) *\<^sub>R (b - a)) \<le> norm (b - a))"
by (simp add: algebra_simps scaleR_2)
also have "... = (\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and>
\<bar>1 - u * 2\<bar> * norm (b - a) \<le> norm (b - a))"
by simp
also have "... = (\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and> \<bar>1 - u * 2\<bar> \<le> 1)"
by (simp add: mult_le_cancel_right2 False)
also have "... = (\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and> 0 \<le> u \<and> u \<le> 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} \<inter> ball(inverse 2 *\<^sub>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 *: "((\<exists>u v. x = u *\<^sub>R a + v *\<^sub>R b \<and> u + v = 1) \<and>
dist ((1 / 2) *\<^sub>R (a + b)) x * 2 < norm (b - a)) =
(\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and> 0 < u \<and> u < 1)" for x
proof -
have "((\<exists>u v. x = u *\<^sub>R a + v *\<^sub>R b \<and> u + v = 1) \<and>
dist ((1 / 2) *\<^sub>R (a + b)) x * 2 < norm (b - a)) =
((\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b) \<and>
dist ((1 / 2) *\<^sub>R (a + b)) x * 2 < norm (b - a))"
unfolding eq_diff_eq [symmetric] by simp
also have "... = (\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and>
norm ((a+b) - (2 *\<^sub>R x)) < norm (b - a))"
by (simp add: dist_half_times2) (simp add: dist_norm)
also have "... = (\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and>
norm ((a+b) - (2 *\<^sub>R ((1 - u) *\<^sub>R a + u *\<^sub>R b))) < norm (b - a))"
by auto
also have "... = (\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and>
norm ((1 - u * 2) *\<^sub>R (b - a)) < norm (b - a))"
by (simp add: algebra_simps scaleR_2)
also have "... = (\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and>
\<bar>1 - u * 2\<bar> * norm (b - a) < norm (b - a))"
by simp
also have "... = (\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and> \<bar>1 - u * 2\<bar> < 1)"
by (simp add: mult_le_cancel_right2 False)
also have "... = (\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and> 0 < u \<and> 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 \<noteq> {}"
by auto
lemma open_segment_eq_empty [simp]: "open_segment a b = {} \<longleftrightarrow> a = b"
proof -
{ assume a1: "open_segment a b = {}"
have "{} \<noteq> {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 \<longleftrightarrow> 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 \<noteq> b"
shows "inj_on (\<lambda>u. (1 - u) *\<^sub>R a + u *\<^sub>R b) I"
proof
fix x y
assume "(1 - x) *\<^sub>R a + x *\<^sub>R b = (1 - y) *\<^sub>R a + y *\<^sub>R b"
then have "x *\<^sub>R (b - a) = y *\<^sub>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) \<longleftrightarrow> 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) \<longleftrightarrow> 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} \<longleftrightarrow> a = c \<and> b = c"
by auto
lemma open_segment_eq_sing: "open_segment a b \<noteq> {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 \<subseteq> closed_segment c d \<longleftrightarrow>
a \<in> closed_segment c d \<and> b \<in> closed_segment c d"
by auto (meson contra_subsetD convex_closed_segment convex_contains_segment)
lemma subset_co_segment:
"closed_segment a b \<subseteq> open_segment c d \<longleftrightarrow>
a \<in> open_segment c d \<and> b \<in> open_segment c d"
using closed_segment_subset by blast
lemma subset_open_segment:
fixes a :: "'a::euclidean_space"
shows "open_segment a b \<subseteq> open_segment c d \<longleftrightarrow>
a = b \<or> a \<in> closed_segment c d \<and> b \<in> 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 \<open>a \<noteq> b\<close> have "c \<noteq> d"
using closed_segment_idem singleton_iff by auto
have "\<exists>uc. (1 - u) *\<^sub>R ((1 - ua) *\<^sub>R c + ua *\<^sub>R d) + u *\<^sub>R ((1 - ub) *\<^sub>R c + ub *\<^sub>R d) =
(1 - uc) *\<^sub>R c + uc *\<^sub>R d \<and> 0 < uc \<and> uc < 1"
if neq: "(1 - ua) *\<^sub>R c + ua *\<^sub>R d \<noteq> (1 - ub) *\<^sub>R c + ub *\<^sub>R d" "c \<noteq> d"
and "a = (1 - ua) *\<^sub>R c + ua *\<^sub>R d" "b = (1 - ub) *\<^sub>R c + ub *\<^sub>R d"
and u: "0 < u" "u < 1" and uab: "0 \<le> ua" "ua \<le> 1" "0 \<le> ub" "ub \<le> 1"
for u ua ub
proof -
have "ua \<noteq> ub"
using neq by auto
moreover have "(u - 1) * ua \<le> 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 "\<not> p \<le> 0" "\<not> q \<le> 0"
using p q not_less by blast+
then show ?thesis
by (metis \<open>ua \<noteq> ub\<close> 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 \<open>ua \<noteq> ub\<close>
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 \<open>a \<noteq> b\<close> \<open>c \<noteq> d\<close> show ?lhs
unfolding open_segment_image_interval closed_segment_def
by (fastforce simp add:)
next
assume lhs: ?lhs
with \<open>a \<noteq> b\<close> have "c \<noteq> d"
by (meson finite_open_segment rev_finite_subset)
have "closure (open_segment a b) \<subseteq> closure (open_segment c d)"
using lhs closure_mono by blast
then have "closed_segment a b \<subseteq> closed_segment c d"
by (simp add: \<open>a \<noteq> b\<close> \<open>c \<noteq> d\<close>)
then show ?rhs
by (force simp: \<open>a \<noteq> b\<close>)
qed
qed
lemma subset_oc_segment:
fixes a :: "'a::euclidean_space"
shows "open_segment a b \<subseteq> closed_segment c d \<longleftrightarrow>
a = b \<or> a \<in> closed_segment c d \<and> b \<in> 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\<open>Betweenness\<close>
definition%important "between = (\<lambda>(a,b) x. x \<in> closed_segment a b)"
lemma betweenI:
assumes "0 \<le> u" "u \<le> 1" "x = (1 - u) *\<^sub>R a + u *\<^sub>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 \<le> u" "u \<le> 1" "x = (1 - u) *\<^sub>R a + u *\<^sub>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 \<noteq> Y"
obtains c where "(X - Y) = c *\<^sub>R (S - Y)"
proof -
from \<open>between (S, T) X\<close> obtain u\<^sub>X where X: "X = u\<^sub>X *\<^sub>R S + (1 - u\<^sub>X) *\<^sub>R T"
by (metis add.commute betweenE eq_diff_eq)
from \<open>between (S, T) Y\<close> obtain u\<^sub>Y where Y: "Y = u\<^sub>Y *\<^sub>R S + (1 - u\<^sub>Y) *\<^sub>R T"
by (metis add.commute betweenE eq_diff_eq)
have "X - Y = (u\<^sub>X - u\<^sub>Y) *\<^sub>R (S - T)"
proof -
from X Y have "X - Y = u\<^sub>X *\<^sub>R S - u\<^sub>Y *\<^sub>R S + ((1 - u\<^sub>X) *\<^sub>R T - (1 - u\<^sub>Y) *\<^sub>R T)" by simp
also have "\<dots> = (u\<^sub>X - u\<^sub>Y) *\<^sub>R S - (u\<^sub>X - u\<^sub>Y) *\<^sub>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\<^sub>Y) *\<^sub>R (S - T)"
by (simp add: real_vector.scale_left_diff_distrib real_vector.scale_right_diff_distrib)
moreover note \<open>S \<noteq> Y\<close>
ultimately have "(X - Y) = ((u\<^sub>X - u\<^sub>Y) / (1 - u\<^sub>Y)) *\<^sub>R (S - Y)" by auto
from this that show thesis by blast
qed
lemma between_mem_segment: "between (a,b) x \<longleftrightarrow> x \<in> closed_segment a b"
unfolding between_def by auto
lemma between: "between (a, b) (x::'a::euclidean_space) \<longleftrightarrow> 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) \<noteq> 0" and Fal2: "norm (a - b) > 0"
by auto
have *: "\<And>u. a - ((1 - u) *\<^sub>R a + u *\<^sub>R b) = u *\<^sub>R (a - b)"
by (auto simp add: algebra_simps)
have "norm (a - x) *\<^sub>R (x - b) = norm (x - b) *\<^sub>R (a - x)" if "x = (1 - u) *\<^sub>R a + u *\<^sub>R b" "0 \<le> u" "u \<le> 1" for u
proof -
have *: "a - x = u *\<^sub>R (a - b)" "x - b = (1 - u) *\<^sub>R (a - b)"
unfolding that(1) by (auto simp add:algebra_simps)
show "norm (a - x) *\<^sub>R (x - b) = norm (x - b) *\<^sub>R (a - x)"
unfolding norm_minus_commute[of x a] * using \<open>0 \<le> u\<close> \<open>u \<le> 1\<close>
by (auto simp add: field_simps)
qed
moreover have "\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and> 0 \<le> u \<and> u \<le> 1" if "dist a b = dist a x + dist x b"
proof -
let ?\<beta> = "norm (a - x) / norm (a - b)"
show "\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and> 0 \<le> u \<and> u \<le> 1"
proof (intro exI conjI)
show "?\<beta> \<le> 1"
using Fal2 unfolding that[unfolded dist_norm] norm_ge_zero by auto
show "x = (1 - ?\<beta>) *\<^sub>R a + (?\<beta>) *\<^sub>R b"
proof (subst euclidean_eq_iff; intro ballI)
fix i :: 'a
assume i: "i \<in> Basis"
have "((1 - ?\<beta>) *\<^sub>R a + (?\<beta>) *\<^sub>R b) \<bullet> i
= ((norm (a - b) - norm (a - x)) * (a \<bullet> i) + norm (a - x) * (b \<bullet> i)) / norm (a - b)"
using Fal by (auto simp add: field_simps inner_simps)
also have "\<dots> = x\<bullet>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 \<bullet> i = ((1 - ?\<beta>) *\<^sub>R a + (?\<beta>) *\<^sub>R b) \<bullet> 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 *: "\<And>x y z. x = (1/2::real) *\<^sub>R z \<Longrightarrow> y = (1/2) *\<^sub>R z \<Longrightarrow> 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 \<longleftrightarrow> x \<in> convex hull {a,b}"
unfolding between_mem_segment segment_convex_hull ..
lemma between_triv_iff [simp]: "between (a,a) b \<longleftrightarrow> 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 "\<lbrakk>between (b,c) a; between (a,c) b\<rbrakk> \<Longrightarrow> a = b"
by (auto simp: between dist_commute)
lemma between_trans:
fixes a :: "'a :: euclidean_space"
shows "\<lbrakk>between (b,c) a; between (a,c) d\<rbrakk> \<Longrightarrow> 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 \<longleftrightarrow> norm(x - a) *\<^sub>R (b - x) = norm(b - x) *\<^sub>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 \<longleftrightarrow> between (A, Y) X"
using assms by (auto simp add: between)
lemma between_translation [simp]: "between (a + y,a + z) (a + x) \<longleftrightarrow> between (y,z) x"
by (auto simp: between_def)
lemma between_trans_2:
fixes a :: "'a :: euclidean_space"
shows "\<lbrakk>between (b,c) a; between (a,b) d\<rbrakk> \<Longrightarrow> 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 *\<^sub>R v, b *\<^sub>R v) (c *\<^sub>R v) \<longleftrightarrow> v = 0 \<or> 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 \<longleftrightarrow> (a \<le> x \<and> x \<le> b) \<or> (b \<le> x \<and> x \<le> a)"
by (auto simp: between_mem_segment closed_segment_eq_real_ivl)
subsection%unimportant \<open>Shrinking towards the interior of a convex set\<close>
lemma mem_interior_convex_shrink:
fixes S :: "'a::euclidean_space set"
assumes "convex S"
and "c \<in> interior S"
and "x \<in> S"
and "0 < e"
and "e \<le> 1"
shows "x - e *\<^sub>R (x - c) \<in> interior S"
proof -
obtain d where "d > 0" and d: "ball c d \<subseteq> S"
using assms(2) unfolding mem_interior by auto
show ?thesis
unfolding mem_interior
proof (intro exI subsetI conjI)
fix y
assume "y \<in> ball (x - e *\<^sub>R (x - c)) (e*d)"
then have as: "dist (x - e *\<^sub>R (x - c)) y < e * d"
by simp
have *: "y = (1 - (1 - e)) *\<^sub>R ((1 / e) *\<^sub>R y - ((1 - e) / e) *\<^sub>R x) + (1 - e) *\<^sub>R x"
using \<open>e > 0\<close> by (auto simp add: scaleR_left_diff_distrib scaleR_right_diff_distrib)
have "dist c ((1 / e) *\<^sub>R y - ((1 - e) / e) *\<^sub>R x) = \<bar>1/e\<bar> * norm (e *\<^sub>R c - y + (1 - e) *\<^sub>R x)"
unfolding dist_norm
unfolding norm_scaleR[symmetric]
apply (rule arg_cong[where f=norm])
using \<open>e > 0\<close>
by (auto simp add: euclidean_eq_iff[where 'a='a] field_simps inner_simps)
also have "\<dots> = \<bar>1/e\<bar> * norm (x - e *\<^sub>R (x - c) - y)"
by (auto intro!:arg_cong[where f=norm] simp add: algebra_simps)
also have "\<dots> < d"
using as[unfolded dist_norm] and \<open>e > 0\<close>
by (auto simp add:pos_divide_less_eq[OF \<open>e > 0\<close>] mult.commute)
finally show "y \<in> 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 \<open>e>0\<close> \<open>d>0\<close>, auto)
qed
lemma mem_interior_closure_convex_shrink:
fixes S :: "'a::euclidean_space set"
assumes "convex S"
and "c \<in> interior S"
and "x \<in> closure S"
and "0 < e"
and "e \<le> 1"
shows "x - e *\<^sub>R (x - c) \<in> interior S"
proof -
obtain d where "d > 0" and d: "ball c d \<subseteq> S"
using assms(2) unfolding mem_interior by auto
have "\<exists>y\<in>S. norm (y - x) * (1 - e) < e * d"
proof (cases "x \<in> S")
case True
then show ?thesis
using \<open>e > 0\<close> \<open>d > 0\<close>
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 \<in> S" "y \<noteq> 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 \<open>d > 0\<close>
apply auto
done
next
case False
then have "0 < e * d / (1 - e)" and *: "1 - e > 0"
using \<open>e \<le> 1\<close> \<open>e > 0\<close> \<open>d > 0\<close> by auto
then obtain y where "y \<in> S" "y \<noteq> 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 \<in> S" and y: "norm (y - x) * (1 - e) < e * d"
by auto
define z where "z = c + ((1 - e) / e) *\<^sub>R (x - y)"
have *: "x - e *\<^sub>R (x - c) = y - e *\<^sub>R (y - z)"
unfolding z_def using \<open>e > 0\<close>
by (auto simp add: scaleR_right_diff_distrib scaleR_right_distrib scaleR_left_diff_distrib)
have "z \<in> 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 \<open>y \<in> S\<close> assms by blast
qed
lemma in_interior_closure_convex_segment:
fixes S :: "'a::euclidean_space set"
assumes "convex S" and a: "a \<in> interior S" and b: "b \<in> closure S"
shows "open_segment a b \<subseteq> interior S"
proof (clarsimp simp: in_segment)
fix u::real
assume u: "0 < u" "u < 1"
have "(1 - u) *\<^sub>R a + u *\<^sub>R b = b - (1 - u) *\<^sub>R (b - a)"
by (simp add: algebra_simps)
also have "... \<in> interior S" using mem_interior_closure_convex_shrink [OF assms] u
by simp
finally show "(1 - u) *\<^sub>R a + u *\<^sub>R b \<in> interior S" .
qed
lemma closure_open_Int_superset:
assumes "open S" "S \<subseteq> closure T"
shows "closure(S \<inter> T) = closure S"
proof -
have "closure S \<subseteq> closure(S \<inter> 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 \<noteq> {}"
shows "closure(interior S) = closure S"
proof -
obtain a where a: "a \<in> interior S"
using int by auto
have "closure S \<subseteq> closure(interior S)"
proof
fix x
assume x: "x \<in> closure S"
show "x \<in> closure (interior S)"
proof (cases "x=a")
case True
then show ?thesis
using \<open>a \<in> interior S\<close> closure_subset by blast
next
case False
show ?thesis
proof (clarsimp simp add: closure_def islimpt_approachable)
fix e::real
assume xnotS: "x \<notin> interior S" and "0 < e"
show "\<exists>x'\<in>interior S. x' \<noteq> x \<and> dist x' x < e"
proof (intro bexI conjI)
show "x - min (e/2 / norm (x - a)) 1 *\<^sub>R (x - a) \<noteq> x"
using False \<open>0 < e\<close> by (auto simp: algebra_simps min_def)
show "dist (x - min (e/2 / norm (x - a)) 1 *\<^sub>R (x - a)) x < e"
using \<open>0 < e\<close> by (auto simp: dist_norm min_def)
show "x - min (e/2 / norm (x - a)) 1 *\<^sub>R (x - a) \<in> interior S"
apply (clarsimp simp add: min_def a)
apply (rule mem_interior_closure_convex_shrink [OF \<open>convex S\<close> a x])
using \<open>0 < e\<close> 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 \<noteq> {}" "interior S \<subseteq> closure T"
shows "closure(S \<inter> T) = closure S"
proof -
have "closure S \<subseteq> closure(interior S)"
by (simp add: convex_closure_interior assms)
also have "... \<subseteq> closure (S \<inter> 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 \<open>Some obvious but surprisingly hard simplex lemmas\<close>
lemma simplex:
assumes "finite S"
and "0 \<notin> S"
shows "convex hull (insert 0 S) = {y. \<exists>u. (\<forall>x\<in>S. 0 \<le> u x) \<and> sum u S \<le> 1 \<and> sum (\<lambda>x. u x *\<^sub>R x) S = y}"
proof (simp add: convex_hull_finite set_eq_iff assms, safe)
fix x and u :: "'a \<Rightarrow> real"
assume "0 \<le> u 0" "\<forall>x\<in>S. 0 \<le> u x" "u 0 + sum u S = 1"
then show "\<exists>v. (\<forall>x\<in>S. 0 \<le> v x) \<and> sum v S \<le> 1 \<and> (\<Sum>x\<in>S. v x *\<^sub>R x) = (\<Sum>x\<in>S. u x *\<^sub>R x)"
by force
next
fix x and u :: "'a \<Rightarrow> real"
assume "\<forall>x\<in>S. 0 \<le> u x" "sum u S \<le> 1"
then show "\<exists>v. 0 \<le> v 0 \<and> (\<forall>x\<in>S. 0 \<le> v x) \<and> v 0 + sum v S = 1 \<and> (\<Sum>x\<in>S. v x *\<^sub>R x) = (\<Sum>x\<in>S. u x *\<^sub>R x)"
by (rule_tac x="\<lambda>x. 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 \<subseteq> Basis"
shows "convex hull (insert 0 d) =
{x. (\<forall>i\<in>Basis. 0 \<le> x\<bullet>i) \<and> (\<Sum>i\<in>d. x\<bullet>i) \<le> 1 \<and> (\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x\<bullet>i = 0)}"
(is "convex hull (insert 0 ?p) = ?s")
proof -
let ?D = d
have "0 \<notin> ?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 \<open>finite d\<close> \<open>0 \<notin> ?p\<close>]
proof (intro set_eqI; safe)
fix u :: "'a \<Rightarrow> real"
assume as: "\<forall>x\<in>?D. 0 \<le> u x" "sum u ?D \<le> 1"
let ?x = "(\<Sum>x\<in>?D. u x *\<^sub>R x)"
have ind: "\<forall>i\<in>Basis. i \<in> d \<longrightarrow> u i = ?x \<bullet> i"
and notind: "(\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> ?x \<bullet> i = 0)"
using substdbasis_expansion_unique[OF assms] by blast+
then have **: "sum u ?D = sum ((\<bullet>) ?x) ?D"
using assms by (auto intro!: sum.cong)
show "0 \<le> ?x \<bullet> i" if "i \<in> Basis" for i
using as(1) ind notind that by fastforce
show "sum ((\<bullet>) ?x) ?D \<le> 1"
using "**" as(2) by linarith
show "?x \<bullet> i = 0" if "i \<in> Basis" "i \<notin> d" for i
using notind that by blast
next
fix x
assume "\<forall>i\<in>Basis. 0 \<le> x \<bullet> i" "sum ((\<bullet>) x) ?D \<le> 1" "(\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0)"
with d show "\<exists>u. (\<forall>x\<in>?D. 0 \<le> u x) \<and> sum u ?D \<le> 1 \<and> (\<Sum>x\<in>?D. u x *\<^sub>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. (\<forall>i\<in>Basis. 0 \<le> x\<bullet>i) \<and> sum (\<lambda>i. x\<bullet>i) Basis \<le> 1}"
using substd_simplex[of Basis] by auto
lemma interior_std_simplex:
"interior (convex hull (insert 0 Basis)) =
{x::'a::euclidean_space. (\<forall>i\<in>Basis. 0 < x\<bullet>i) \<and> sum (\<lambda>i. x\<bullet>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 \<subseteq> {x. (\<forall>i\<in>Basis. 0 \<le> x \<bullet> i) \<and> sum ((\<bullet>) x) Basis \<le> 1}"
show "(\<forall>i\<in>Basis. 0 < x \<bullet> i) \<and> sum ((\<bullet>) x) Basis < 1"
proof safe
fix i :: 'a
assume i: "i \<in> Basis"
then show "0 < x \<bullet> i"
using as[THEN subsetD[where c="x - (e / 2) *\<^sub>R i"]] and \<open>e > 0\<close>
by (force simp add: inner_simps)
next
have **: "dist x (x + (e / 2) *\<^sub>R (SOME i. i\<in>Basis)) < e" using \<open>e > 0\<close>
unfolding dist_norm
by (auto intro!: mult_strict_left_mono simp: SOME_Basis)
have "\<And>i. i \<in> Basis \<Longrightarrow> (x + (e / 2) *\<^sub>R (SOME i. i\<in>Basis)) \<bullet> i =
x\<bullet>i + (if i = (SOME i. i\<in>Basis) then e/2 else 0)"
by (auto simp: SOME_Basis inner_Basis inner_simps)
then have *: "sum ((\<bullet>) (x + (e / 2) *\<^sub>R (SOME i. i\<in>Basis))) Basis =
sum (\<lambda>i. x\<bullet>i + (if (SOME i. i\<in>Basis) = i then e/2 else 0)) Basis"
by (auto simp: intro!: sum.cong)
have "sum ((\<bullet>) x) Basis < sum ((\<bullet>) (x + (e / 2) *\<^sub>R (SOME i. i\<in>Basis))) Basis"
using \<open>e > 0\<close> DIM_positive by (auto simp: SOME_Basis sum.distrib *)
also have "\<dots> \<le> 1"
using ** as by force
finally show "sum ((\<bullet>) x) Basis < 1" by auto
qed
next
fix x :: 'a
assume as: "\<forall>i\<in>Basis. 0 < x \<bullet> i" "sum ((\<bullet>) x) Basis < 1"
obtain a :: 'b where "a \<in> UNIV" using UNIV_witness ..
let ?d = "(1 - sum ((\<bullet>) x) Basis) / real (DIM('a))"
show "\<exists>e>0. ball x e \<subseteq> {x. (\<forall>i\<in>Basis. 0 \<le> x \<bullet> i) \<and> sum ((\<bullet>) x) Basis \<le> 1}"
proof (rule_tac x="min (Min (((\<bullet>) x) ` Basis)) D" for D in exI, intro conjI subsetI CollectI)
fix y
assume y: "y \<in> ball x (min (Min ((\<bullet>) x ` Basis)) ?d)"
have "sum ((\<bullet>) y) Basis \<le> sum (\<lambda>i. x\<bullet>i + ?d) Basis"
proof (rule sum_mono)
fix i :: 'a
assume i: "i \<in> Basis"
have "\<bar>y\<bullet>i - x\<bullet>i\<bar> \<le> 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 "\<bar>y\<bullet>i - x\<bullet>i\<bar> < ?d" .
then show "y \<bullet> i \<le> x \<bullet> i + ?d" by auto
qed
also have "\<dots> \<le> 1"
unfolding sum.distrib sum_constant
by (auto simp add: Suc_le_eq)
finally show "sum ((\<bullet>) y) Basis \<le> 1" .
show "(\<forall>i\<in>Basis. 0 \<le> y \<bullet> i)"
proof safe
fix i :: 'a
assume i: "i \<in> Basis"
have "norm (x - y) < Min (((\<bullet>) x) ` Basis)"
using y by (auto simp: dist_norm less_eq_real_def)
also have "... \<le> x\<bullet>i"
using i by auto
finally have "norm (x - y) < x\<bullet>i" .
then show "0 \<le> y\<bullet>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 (((\<bullet>) x) ` Basis) > 0"
using as by simp
moreover have "?d > 0"
using as by (auto simp: Suc_le_eq)
ultimately show "0 < min (Min ((\<bullet>) x ` Basis)) ((1 - sum ((\<bullet>) x) Basis) / real DIM('a))"
by linarith
qed
qed
lemma interior_std_simplex_nonempty:
obtains a :: "'a::euclidean_space" where
"a \<in> interior(convex hull (insert 0 Basis))"
proof -
let ?D = "Basis :: 'a set"
let ?a = "sum (\<lambda>b::'a. inverse (2 * real DIM('a)) *\<^sub>R b) Basis"
{
fix i :: 'a
assume i: "i \<in> Basis"
have "?a \<bullet> i = inverse (2 * real DIM('a))"
by (rule trans[of _ "sum (\<lambda>j. 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 \<in> Basis"
show "0 < ?a \<bullet> i"
unfolding **[OF i] by (auto simp add: Suc_le_eq DIM_positive)
next
have "sum ((\<bullet>) ?a) ?D = sum (\<lambda>i. inverse (2 * real DIM('a))) ?D"
apply (rule sum.cong)
apply rule
apply auto
done
also have "\<dots> < 1"
unfolding sum_constant divide_inverse[symmetric]
by (auto simp add: field_simps)
finally show "sum ((\<bullet>) ?a) ?D < 1" by auto
qed
qed
lemma rel_interior_substd_simplex:
assumes D: "D \<subseteq> Basis"
shows "rel_interior (convex hull (insert 0 D)) =
{x::'a::euclidean_space. (\<forall>i\<in>D. 0 < x\<bullet>i) \<and> (\<Sum>i\<in>D. x\<bullet>i) < 1 \<and> (\<forall>i\<in>Basis. i \<notin> D \<longrightarrow> x\<bullet>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. (\<forall>i\<in>Basis. i \<notin> D \<longrightarrow> x\<bullet>i = 0)}"
using affine_hull_convex_hull affine_hull_substd_basis assms by auto
have aux: "\<And>x::'a. \<forall>i\<in>Basis. (\<forall>i\<in>D. 0 \<le> x\<bullet>i) \<and> (\<forall>i\<in>Basis. i \<notin> D \<longrightarrow> x\<bullet>i = 0) \<longrightarrow> 0 \<le> x\<bullet>i"
by auto
{
fix x :: "'a::euclidean_space"
assume x: "x \<in> rel_interior (convex hull (insert 0 ?p))"
then obtain e where "e > 0" and
"ball x e \<inter> {xa. (\<forall>i\<in>Basis. i \<notin> D \<longrightarrow> xa\<bullet>i = 0)} \<subseteq> 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]: "\<And>y. dist x y < e \<and> (\<forall>i\<in>Basis. i \<notin> D \<longrightarrow> y\<bullet>i = 0) \<longrightarrow>
(\<forall>i\<in>D. 0 \<le> y \<bullet> i) \<and> sum ((\<bullet>) y) D \<le> 1"
unfolding ball_def unfolding substd_simplex[OF assms] using assms by auto
have x0: "(\<forall>i\<in>Basis. i \<notin> D \<longrightarrow> x\<bullet>i = 0)"
using x rel_interior_subset substd_simplex[OF assms] by auto
have "(\<forall>i\<in>D. 0 < x \<bullet> i) \<and> sum ((\<bullet>) x) D < 1 \<and> (\<forall>i\<in>Basis. i \<notin> D \<longrightarrow> x\<bullet>i = 0)"
proof (intro conjI ballI)
fix i :: 'a
assume "i \<in> D"
then have "\<forall>j\<in>D. 0 \<le> (x - (e / 2) *\<^sub>R i) \<bullet> j"
apply -
apply (rule as[THEN conjunct1])
using D \<open>e > 0\<close> x0
apply (auto simp: dist_norm inner_simps inner_Basis)
done
then show "0 < x \<bullet> i"
using \<open>e > 0\<close> \<open>i \<in> D\<close> D by (force simp: inner_simps inner_Basis)
next
obtain a where a: "a \<in> D"
using \<open>D \<noteq> {}\<close> by auto
then have **: "dist x (x + (e / 2) *\<^sub>R a) < e"
using \<open>e > 0\<close> norm_Basis[of a] D
unfolding dist_norm
by auto
have "\<And>i. i \<in> Basis \<Longrightarrow> (x + (e / 2) *\<^sub>R a) \<bullet> i = x\<bullet>i + (if i = a then e/2 else 0)"
using a D by (auto simp: inner_simps inner_Basis)
then have *: "sum ((\<bullet>) (x + (e / 2) *\<^sub>R a)) D =
sum (\<lambda>i. x\<bullet>i + (if a = i then e/2 else 0)) D"
using D by (intro sum.cong) auto
have "a \<in> Basis"
using \<open>a \<in> D\<close> D by auto
then have h1: "(\<forall>i\<in>Basis. i \<notin> D \<longrightarrow> (x + (e / 2) *\<^sub>R a) \<bullet> i = 0)"
using x0 D \<open>a\<in>D\<close> by (auto simp add: inner_add_left inner_Basis)
have "sum ((\<bullet>) x) D < sum ((\<bullet>) (x + (e / 2) *\<^sub>R a)) D"
using \<open>e > 0\<close> \<open>a \<in> D\<close> \<open>finite D\<close> by (auto simp add: * sum.distrib)
also have "\<dots> \<le> 1"
using ** h1 as[rule_format, of "x + (e / 2) *\<^sub>R a"]
by auto
finally show "sum ((\<bullet>) x) D < 1" "\<And>i. i\<in>Basis \<Longrightarrow> i \<notin> D \<longrightarrow> x\<bullet>i = 0"
using x0 by auto
qed
}
moreover
{
fix x :: "'a::euclidean_space"
assume as: "x \<in> ?s"
have "\<forall>i. 0 < x\<bullet>i \<or> 0 = x\<bullet>i \<longrightarrow> 0 \<le> x\<bullet>i"
by auto
moreover have "\<forall>i. i \<in> D \<or> i \<notin> D" by auto
ultimately
have "\<forall>i. (\<forall>i\<in>D. 0 < x\<bullet>i) \<and> (\<forall>i. i \<notin> D \<longrightarrow> x\<bullet>i = 0) \<longrightarrow> 0 \<le> x\<bullet>i"
by metis
then have h2: "x \<in> convex hull (insert 0 ?p)"
using as assms
unfolding substd_simplex[OF assms] by fastforce
obtain a where a: "a \<in> D"
using \<open>D \<noteq> {}\<close> by auto
let ?d = "(1 - sum ((\<bullet>) x) D) / real (card D)"
have "0 < card D" using \<open>D \<noteq> {}\<close> \<open>finite D\<close>
by (simp add: card_gt_0_iff)
have "Min (((\<bullet>) x) ` D) > 0"
using as \<open>D \<noteq> {}\<close> \<open>finite D\<close> by (simp add: Min_gr_iff)
moreover have "?d > 0" using as using \<open>0 < card D\<close> by auto
ultimately have h3: "min (Min (((\<bullet>) x) ` D)) ?d > 0"
by auto
have "x \<in> 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 (((\<bullet>) 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 ((\<bullet>) x ` D)) ?d"
assume y2: "\<forall>i\<in>Basis. i \<notin> D \<longrightarrow> y\<bullet>i = 0"
have "sum ((\<bullet>) y) D \<le> sum (\<lambda>i. x\<bullet>i + ?d) D"
proof (rule sum_mono)
fix i
assume "i \<in> D"
with D have i: "i \<in> Basis"
by auto
have "\<bar>y\<bullet>i - x\<bullet>i\<bar> \<le> 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 "\<bar>y\<bullet>i - x\<bullet>i\<bar> < ?d" .
then show "y \<bullet> i \<le> x \<bullet> i + ?d" by auto
qed
also have "\<dots> \<le> 1"
unfolding sum.distrib sum_constant using \<open>0 < card D\<close>
by auto
finally show "sum ((\<bullet>) y) D \<le> 1" .
fix i :: 'a
assume i: "i \<in> Basis"
then show "0 \<le> y\<bullet>i"
proof (cases "i\<in>D")
case True
have "norm (x - y) < x\<bullet>i"
using y[unfolded min_less_iff_conj dist_norm, THEN conjunct1]
using Min_gr_iff[of "(\<bullet>) x ` D" "norm (x - y)"] \<open>0 < card D\<close> \<open>i \<in> D\<close>
by (simp add: card_gt_0_iff)
then show "0 \<le> y\<bullet>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
"\<And>x. x \<in> rel_interior (convex hull insert 0 D) \<longleftrightarrow>
x \<in> {x. (\<forall>i\<in>D. 0 < x \<bullet> i) \<and> sum ((\<bullet>) x) D < 1 \<and> (\<forall>i\<in>Basis. i \<notin> D \<longrightarrow> x \<bullet> i = 0)}"
by blast
then show ?thesis by (rule set_eqI)
qed
qed
lemma rel_interior_substd_simplex_nonempty:
assumes "D \<noteq> {}"
and "D \<subseteq> Basis"
obtains a :: "'a::euclidean_space"
where "a \<in> rel_interior (convex hull (insert 0 D))"
proof -
let ?D = D
let ?a = "sum (\<lambda>b::'a::euclidean_space. inverse (2 * real (card D)) *\<^sub>R b) ?D"
have "finite D"
apply (rule finite_subset)
using assms(2)
apply auto
done
then have d1: "0 < real (card D)"
using \<open>D \<noteq> {}\<close> by auto
{
fix i
assume "i \<in> D"
have "?a \<bullet> i = inverse (2 * real (card D))"
apply (rule trans[of _ "sum (\<lambda>j. if i = j then inverse (2 * real (card D)) else 0) ?D"])
unfolding inner_sum_left
apply (rule sum.cong)
using \<open>i \<in> D\<close> \<open>finite D\<close> sum.delta'[of D i "(\<lambda>k. 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 \<in> D"
have "0 < inverse (2 * real (card D))"
using d1 by auto
also have "\<dots> = ?a \<bullet> i" using **[of i] \<open>i \<in> D\<close>
by auto
finally show "0 < ?a \<bullet> i" by auto
next
have "sum ((\<bullet>) ?a) ?D = sum (\<lambda>i. inverse (2 * real (card D))) ?D"
by (rule sum.cong) (rule refl, rule **)
also have "\<dots> < 1"
unfolding sum_constant divide_real_def[symmetric]
by (auto simp add: field_simps)
finally show "sum ((\<bullet>) ?a) ?D < 1" by auto
next
fix i
assume "i \<in> Basis" and "i \<notin> D"
have "?a \<in> span D"
proof (rule span_sum[of D "(\<lambda>b. b /\<^sub>R (2 * real (card D)))" D])
{
fix x :: "'a::euclidean_space"
assume "x \<in> D"
then have "x \<in> span D"
using span_base[of _ "D"] by auto
then have "x /\<^sub>R (2 * real (card D)) \<in> span D"
using span_mul[of x "D" "(inverse (real (card D)) / 2)"] by auto
}
then show "\<And>x. x\<in>D \<Longrightarrow> x /\<^sub>R (2 * real (card D)) \<in> span D"
by auto
qed
then show "?a \<bullet> i = 0 "
using \<open>i \<notin> D\<close> unfolding span_substd_basis[OF assms(2)] using \<open>i \<in> Basis\<close> by auto
qed
qed
subsection%unimportant \<open>Relative interior of convex set\<close>
lemma rel_interior_convex_nonempty_aux:
fixes S :: "'n::euclidean_space set"
assumes "convex S"
and "0 \<in> S"
shows "rel_interior S \<noteq> {}"
proof (cases "S = {0}")
case True
then show ?thesis using rel_interior_sing by auto
next
case False
obtain B where B: "independent B \<and> B \<le> S \<and> S \<le> span B \<and> card B = dim S"
using basis_exists[of S] by metis
then have "B \<noteq> {}"
using B assms \<open>S \<noteq> {0}\<close> span_empty by auto
have "insert 0 B \<le> span B"
using subspace_span[of B] subspace_0[of "span B"]
span_superset by auto
then have "span (insert 0 B) \<le> span B"
using span_span[of B] span_mono[of "insert 0 B" "span B"] by blast
then have "convex hull insert 0 B \<le> span B"
using convex_hull_subset_span[of "insert 0 B"] by auto
then have "span (convex hull insert 0 B) \<le> 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 \<in> 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 \<Rightarrow> 'n" where
fd: "card d = card B" "linear f" "f ` B = d"
"f ` span B = {x. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = (0::real)} \<and> inj_on f (span B)"
and d: "d \<subseteq> 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 \<noteq> {}"
using fd B \<open>B \<noteq> {}\<close> 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)"] \<open>linear f\<close>
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 \<open>bounded_linear f\<close> fd *
apply auto
done
ultimately have "rel_interior (convex hull insert 0 B) \<noteq> {}"
using rel_interior_substd_simplex_nonempty[OF \<open>d \<noteq> {}\<close> d]
apply auto
apply blast
done
moreover have "convex hull (insert 0 B) \<subseteq> 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 = {} \<longleftrightarrow> S = {}"
proof -
{
assume "S \<noteq> {}"
then obtain a where "a \<in> S" by auto
then have "0 \<in> (+) (-a) ` S"
using assms exI[of "(\<lambda>x. x \<in> S \<and> - a + x = 0)" a] by auto
then have "rel_interior ((+) (-a) ` S) \<noteq> {}"
using rel_interior_convex_nonempty_aux[of "(+) (-a) ` S"]
convex_translation[of S "-a"] assms
by auto
then have "rel_interior S \<noteq> {}"
using rel_interior_translation [of "- a"] by simp
}
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 \<in> 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) \<noteq> {}"
using rel_interior_eq_empty [of "convex hull (insert 0 S)"] by auto
ultimately have "interior (convex hull insert 0 S) \<noteq> {}"
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 \<in> rel_interior S" "y \<in> rel_interior S" "0 \<le> u" "u \<le> 1"
then have "x \<in> S"
using rel_interior_subset by auto
have "x - u *\<^sub>R (x-y) \<in> 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 \<open>x \<in> S\<close> by auto
next
case True
then show ?thesis using assm by auto
qed
then have "(1 - u) *\<^sub>R x + u *\<^sub>R y \<in> 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) \<le> 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 \<in> rel_interior S"
using rel_interior_eq_empty assms by auto
{ fix x
assume x: "x \<in> closure S"
{
assume "x = a"
then have "x \<in> closure (rel_interior S)"
using a unfolding closure_def by auto
}
moreover
{
assume "x \<noteq> a"
{
fix e :: real
assume "e > 0"
define e1 where "e1 = min 1 (e/norm (x - a))"
then have e1: "e1 > 0" "e1 \<le> 1" "e1 * norm (x - a) \<le> e"
using \<open>x \<noteq> a\<close> \<open>e > 0\<close> le_divide_eq[of e1 e "norm (x - a)"]
by simp_all
then have *: "x - e1 *\<^sub>R (x - a) \<in> rel_interior S"
using rel_interior_closure_convex_shrink[of S a x e1] assms x a e1_def
by auto
have "\<exists>y. y \<in> rel_interior S \<and> y \<noteq> x \<and> dist y x \<le> e"
apply (rule_tac x="x - e1 *\<^sub>R (x - a)" in exI)
using * e1 dist_norm[of "x - e1 *\<^sub>R (x - a)" x] \<open>x \<noteq> a\<close>
apply simp
done
}
then have "x islimpt rel_interior S"
unfolding islimpt_approachable_le by auto
then have "x \<in> closure(rel_interior S)"
unfolding closure_def by auto
}
ultimately have "x \<in> 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) *\<^sub>R z = a *\<^sub>R x + b *\<^sub>R y"
obtains e where "0 < e" "e \<le> 1" "z = y - e *\<^sub>R (y - x)"
proof -
define e where "e = a / (a + b)"
have "z = (1 / (a + b)) *\<^sub>R ((a + b) *\<^sub>R z)"
using assms by (simp add: eq_vector_fraction_iff)
also have "\<dots> = (1 / (a + b)) *\<^sub>R (a *\<^sub>R x + b *\<^sub>R y)"
using assms scaleR_cancel_left[of "1/(a+b)" "(a + b) *\<^sub>R z" "a *\<^sub>R x + b *\<^sub>R y"]
by auto
also have "\<dots> = y - e *\<^sub>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 *\<^sub>R (y-x)"
by auto
moreover have "e > 0" using e_def assms by auto
moreover have "e \<le> 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) \<supseteq> 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 \<in> rel_interior (closure S)"
obtain x where x: "x \<in> rel_interior S"
using \<open>S \<noteq> {}\<close> assms rel_interior_eq_empty by auto
have "z \<in> 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 \<inter> affine hull closure S \<le> 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)) *\<^sub>R (z-x)"
have yball: "y \<in> cball z e"
using mem_cball y_def dist_norm[of z y] e by auto
have "x \<in> affine hull closure S"
using x rel_interior_subset_closure hull_inc[of x "closure S"] by blast
moreover have "z \<in> affine hull closure S"
using z rel_interior_subset hull_subset[of "closure S"] by blast
ultimately have "y \<in> 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 \<in> closure S" using e yball by auto
have "(1 + (e/norm(z-x))) *\<^sub>R z = (e/norm(z-x)) *\<^sub>R x + y"
using y_def by (simp add: algebra_simps)
then obtain e1 where "0 < e1" "e1 \<le> 1" "z = y - e1 *\<^sub>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 \<open>y \<in> closure S\<close>
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 \<longleftrightarrow> 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 \<longleftrightarrow> rel_interior S1 \<le> S2 \<and> S2 \<subseteq> closure S1"
(is "?A \<longleftrightarrow> ?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 \<subseteq> closure S2"
by (metis assms(1) convex_closure_rel_interior closure_mono)
moreover from \<open>?B\<close> have "closure S1 \<supseteq> 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 \<inter> closure S = {} \<longleftrightarrow> A \<inter> 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 *\<^sub>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 \<noteq> {}" and ST: "rel_interior S \<subseteq> T" and TS: "T \<subseteq> closure S"
shows "starlike T"
proof -
have "rel_interior S \<noteq> {}"
by (simp add: assms rel_interior_eq_empty)
then obtain a where a: "a \<in> rel_interior S" by blast
with ST have "a \<in> T" by blast
have *: "\<And>x. x \<in> T \<Longrightarrow> open_segment a x \<subseteq> rel_interior S"
apply (rule rel_interior_closure_convex_segment [OF \<open>convex S\<close> a])
using assms by blast
show ?thesis
unfolding starlike_def
apply (rule bexI [OF _ \<open>a \<in> T\<close>])
apply (simp add: closed_segment_eq_open)
apply (intro conjI ballI a \<open>a \<in> T\<close> rel_interior_closure_convex_segment [OF \<open>convex S\<close> a])
apply (simp add: order_trans [OF * ST])
done
qed
subsection\<open>The relative frontier of a set\<close>
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 = {} \<longleftrightarrow> 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 \<subseteq> 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((\<lambda>x. a + x) ` S) = (\<lambda>x. 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 \<noteq> {} \<Longrightarrow> 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 \<Longrightarrow> rel_frontier S = frontier S"
by (simp add: frontier_def rel_frontier_def rel_interior_interior)
lemma closest_point_in_rel_frontier:
"\<lbrakk>closed S; S \<noteq> {}; x \<in> affine hull S - rel_interior S\<rbrakk>
\<Longrightarrow> closest_point S x \<in> 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 \<Longrightarrow> 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 \<Longrightarrow> 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 \<Longrightarrow> 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 \<Longrightarrow> 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 \<Longrightarrow> 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 "\<lbrakk>convex S; convex T\<rbrakk>
\<Longrightarrow> rel_interior S = rel_interior T \<longleftrightarrow> 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 "\<lbrakk>convex S; convex T\<rbrakk>
\<Longrightarrow> rel_interior S = rel_interior T \<longleftrightarrow>
rel_interior S \<subseteq> T \<and> T \<subseteq> 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 \<noteq> {}"
and "S1 \<le> rel_frontier S2"
shows "aff_dim S1 < aff_dim S2"
proof -
have "S1 \<subseteq> closure S2"
using assms unfolding rel_frontier_def by auto
then have *: "affine hull S1 \<subseteq> affine hull S2"
using hull_mono[of "S1" "closure S2"] closure_same_affine_hull[of S2] by blast
then have "aff_dim S1 \<le> 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 \<noteq> {}"
using aff_dim_empty[of S1] aff_dim_empty[of S2] \<open>S2 \<noteq> {}\<close> by auto
have **: "affine hull S1 = affine hull S2"
apply (rule affine_dim_equal)
using * affine_affine_hull
apply auto
using \<open>S1 \<noteq> {}\<close> 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 \<in> rel_interior S1"
using \<open>S1 \<noteq> {}\<close> rel_interior_eq_empty assms by auto
obtain T where T: "open T" "a \<in> T \<inter> S1" "T \<inter> affine hull S1 \<subseteq> S1"
using mem_rel_interior[of a S1] a by auto
then have "a \<in> T \<inter> closure S2"
using a assms unfolding rel_frontier_def by auto
then obtain b where b: "b \<in> T \<inter> rel_interior S2"
using open_inter_closure_rel_interior[of S2 T] assms T by auto
then have "b \<in> affine hull S1"
using rel_interior_subset hull_subset[of S2] ** by auto
then have "b \<in> 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 \<in> rel_interior S"
shows "\<forall>x\<in>affine hull S. \<exists>m. m > 1 \<and> (\<forall>e. e > 1 \<and> e \<le> m \<longrightarrow> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S)"
proof -
obtain e1 where e1: "e1 > 0 \<and> cball z e1 \<inter> affine hull S \<subseteq> S"
using mem_rel_interior_cball[of z S] assms by auto
{
fix x
assume x: "x \<in> affine hull S"
{
assume "x \<noteq> z"
define m where "m = 1 + e1/norm(x-z)"
hence "m > 1" using e1 \<open>x \<noteq> z\<close> by auto
{
fix e
assume e: "e > 1 \<and> e \<le> m"
have "z \<in> affine hull S"
using assms rel_interior_subset hull_subset[of S] by auto
then have *: "(1 - e)*\<^sub>R x + e *\<^sub>R z \<in> 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 *\<^sub>R x - (x + e *\<^sub>R z)) = norm ((e - 1) *\<^sub>R (x - z))"
by (simp add: algebra_simps)
also have "\<dots> = (e - 1) * norm (x-z)"
using norm_scaleR e by auto
also have "\<dots> \<le> (m - 1) * norm (x - z)"
using e mult_right_mono[of _ _ "norm(x-z)"] by auto
also have "\<dots> = (e1 / norm (x - z)) * norm (x - z)"
using m_def by auto
also have "\<dots> = e1"
using \<open>x \<noteq> z\<close> e1 by simp
finally have **: "norm (z + e *\<^sub>R x - (x + e *\<^sub>R z)) \<le> e1"
by auto
have "(1 - e)*\<^sub>R x+ e *\<^sub>R z \<in> cball z e1"
using m_def **
unfolding cball_def dist_norm
by (auto simp add: algebra_simps)
then have "(1 - e) *\<^sub>R x+ e *\<^sub>R z \<in> S"
using e * e1 by auto
}
then have "\<exists>m. m > 1 \<and> (\<forall>e. e > 1 \<and> e \<le> m \<longrightarrow> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S )"
using \<open>m> 1 \<close> 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 \<and> e \<le> m"
then have "(1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S"
using e1 x \<open>x = z\<close> by (auto simp add: algebra_simps)
then have "(1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S"
using e by auto
}
then have "\<exists>m. m > 1 \<and> (\<forall>e. e > 1 \<and> e \<le> m \<longrightarrow> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S)"
using \<open>m > 1\<close> by auto
}
ultimately have "\<exists>m. m > 1 \<and> (\<forall>e. e > 1 \<and> e \<le> m \<longrightarrow> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> 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 \<in> rel_interior S"
shows "\<forall>x\<in>affine hull S. \<exists>e. e > 1 \<and> (1 - e)*\<^sub>R x + e *\<^sub>R z \<in> 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 \<noteq> {}"
assumes "\<forall>x\<in>S. \<exists>e. e > 1 \<and> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S"
shows "z \<in> rel_interior S"
proof -
obtain x where x: "x \<in> rel_interior S"
using rel_interior_eq_empty assms by auto
then have "x \<in> S"
using rel_interior_subset by auto
then obtain e where e: "e > 1 \<and> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S"
using assms by auto
define y where [abs_def]: "y = (1 - e) *\<^sub>R x + e *\<^sub>R z"
then have "y \<in> S" using e by auto
define e1 where "e1 = 1/e"
then have "0 < e1 \<and> e1 < 1" using e by auto
then have "z =y - (1 - e1) *\<^sub>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"] \<open>0 < e1 \<and> e1 < 1\<close> \<open>y \<in> S\<close> x assms
by auto
qed
lemma convex_rel_interior_iff:
fixes S :: "'n::euclidean_space set"
assumes "convex S"
and "S \<noteq> {}"
shows "z \<in> rel_interior S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e. e > 1 \<and> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> 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 \<noteq> {}"
shows "z \<in> rel_interior S \<longleftrightarrow> (\<forall>x\<in>affine hull S. \<exists>e. e > 1 \<and> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> 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 \<in> interior S \<longleftrightarrow> (\<forall>x. \<exists>e. e > 0 \<and> z + e *\<^sub>R x \<in> S)"
proof (cases "aff_dim S = int DIM('n)")
case False
{ assume "z \<in> interior S"
then have False
using False interior_rel_interior_gen[of S] by auto }
moreover
{ assume r: "\<forall>x. \<exists>e. e > 0 \<and> z + e *\<^sub>R x \<in> S"
{ fix x
obtain e1 where e1: "e1 > 0 \<and> z + e1 *\<^sub>R (x - z) \<in> S"
using r by auto
obtain e2 where e2: "e2 > 0 \<and> z + e2 *\<^sub>R (z - x) \<in> S"
using r by auto
define x1 where [abs_def]: "x1 = z + e1 *\<^sub>R (x - z)"
then have x1: "x1 \<in> affine hull S"
using e1 hull_subset[of S] by auto
define x2 where [abs_def]: "x2 = z + e2 *\<^sub>R (z - x)"
then have x2: "x2 \<in> 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)) *\<^sub>R x1 + (e1/(e1+e2)) *\<^sub>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 \<in> 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) *\<^sub>R (x - z)"
using x1_def x2_def by (auto simp add: algebra_simps)
then have "x = z+(1/(e1+e2)) *\<^sub>R (x1-x2)"
using e1 e2 by simp
then have "x \<in> 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 \<noteq> {}"
using aff_dim_empty[of S] by auto
have *: "affine hull S = UNIV"
using True affine_hull_UNIV by auto
{
assume "z \<in> interior S"
then have "z \<in> rel_interior S"
using True interior_rel_interior_gen[of S] by auto
then have **: "\<forall>x. \<exists>e. e > 1 \<and> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S"
using convex_rel_interior_iff2[of S z] assms \<open>S \<noteq> {}\<close> * by auto
fix x
obtain e1 where e1: "e1 > 1" "(1 - e1) *\<^sub>R (z - x) + e1 *\<^sub>R z \<in> S"
using **[rule_format, of "z-x"] by auto
define e where [abs_def]: "e = e1 - 1"
then have "(1 - e1) *\<^sub>R (z - x) + e1 *\<^sub>R z = z + e *\<^sub>R x"
by (simp add: algebra_simps)
then have "e > 0" "z + e *\<^sub>R x \<in> S"
using e1 e_def by auto
then have "\<exists>e. e > 0 \<and> z + e *\<^sub>R x \<in> S"
by auto
}
moreover
{
assume r: "\<forall>x. \<exists>e. e > 0 \<and> z + e *\<^sub>R x \<in> S"
{
fix x
obtain e1 where e1: "e1 > 0" "z + e1 *\<^sub>R (z - x) \<in> S"
using r[rule_format, of "z-x"] by auto
define e where "e = e1 + 1"
then have "z + e1 *\<^sub>R (z - x) = (1 - e) *\<^sub>R x + e *\<^sub>R z"
by (simp add: algebra_simps)
then have "e > 1" "(1 - e)*\<^sub>R x + e *\<^sub>R z \<in> S"
using e1 e_def by auto
then have "\<exists>e. e > 1 \<and> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S" by auto
}
then have "z \<in> rel_interior S"
using convex_rel_interior_iff2[of S z] assms \<open>S \<noteq> {}\<close> by auto
then have "z \<in> interior S"
using True interior_rel_interior_gen[of S] by auto
}
ultimately show ?thesis by auto
qed
subsubsection%unimportant \<open>Relative interior and closure under common operations\<close>
lemma rel_interior_inter_aux: "\<Inter>{rel_interior S |S. S \<in> I} \<subseteq> \<Inter>I"
proof -
{
fix y
assume "y \<in> \<Inter>{rel_interior S |S. S \<in> I}"
then have y: "\<forall>S \<in> I. y \<in> rel_interior S"
by auto
{
fix S
assume "S \<in> I"
then have "y \<in> S"
using rel_interior_subset y by auto
}
then have "y \<in> \<Inter>I" by auto
}
then show ?thesis by auto
qed
lemma closure_Int: "closure (\<Inter>I) \<le> \<Inter>{closure S |S. S \<in> I}"
proof -
{
fix y
assume "y \<in> \<Inter>I"
then have y: "\<forall>S \<in> I. y \<in> S" by auto
{
fix S
assume "S \<in> I"
then have "y \<in> closure S"
using closure_subset y by auto
}
then have "y \<in> \<Inter>{closure S |S. S \<in> I}"
by auto
}
then have "\<Inter>I \<subseteq> \<Inter>{closure S |S. S \<in> I}"
by auto
moreover have "closed (\<Inter>{closure S |S. S \<in> I})"
unfolding closed_Inter closed_closure by auto
ultimately show ?thesis using closure_hull[of "\<Inter>I"]
hull_minimal[of "\<Inter>I" "\<Inter>{closure S |S. S \<in> I}" "closed"] by auto
qed
lemma convex_closure_rel_interior_inter:
assumes "\<forall>S\<in>I. convex (S :: 'n::euclidean_space set)"
and "\<Inter>{rel_interior S |S. S \<in> I} \<noteq> {}"
shows "\<Inter>{closure S |S. S \<in> I} \<le> closure (\<Inter>{rel_interior S |S. S \<in> I})"
proof -
obtain x where x: "\<forall>S\<in>I. x \<in> rel_interior S"
using assms by auto
{
fix y
assume "y \<in> \<Inter>{closure S |S. S \<in> I}"
then have y: "\<forall>S \<in> I. y \<in> closure S"
by auto
{
assume "y = x"
then have "y \<in> closure (\<Inter>{rel_interior S |S. S \<in> I})"
using x closure_subset[of "\<Inter>{rel_interior S |S. S \<in> I}"] by auto
}
moreover
{
assume "y \<noteq> x"
{ fix e :: real
assume e: "e > 0"
define e1 where "e1 = min 1 (e/norm (y - x))"
then have e1: "e1 > 0" "e1 \<le> 1" "e1 * norm (y - x) \<le> e"
using \<open>y \<noteq> x\<close> \<open>e > 0\<close> le_divide_eq[of e1 e "norm (y - x)"]
by simp_all
define z where "z = y - e1 *\<^sub>R (y - x)"
{
fix S
assume "S \<in> I"
then have "z \<in> 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 \<in> \<Inter>{rel_interior S |S. S \<in> I}"
by auto
have "\<exists>z. z \<in> \<Inter>{rel_interior S |S. S \<in> I} \<and> z \<noteq> y \<and> dist z y \<le> e"
apply (rule_tac x="z" in exI)
using \<open>y \<noteq> x\<close> z_def * e1 e dist_norm[of z y]
apply simp
done
}
then have "y islimpt \<Inter>{rel_interior S |S. S \<in> I}"
unfolding islimpt_approachable_le by blast
then have "y \<in> closure (\<Inter>{rel_interior S |S. S \<in> I})"
unfolding closure_def by auto
}
ultimately have "y \<in> closure (\<Inter>{rel_interior S |S. S \<in> I})"
by auto
}
then show ?thesis by auto
qed
lemma convex_closure_inter:
assumes "\<forall>S\<in>I. convex (S :: 'n::euclidean_space set)"
and "\<Inter>{rel_interior S |S. S \<in> I} \<noteq> {}"
shows "closure (\<Inter>I) = \<Inter>{closure S |S. S \<in> I}"
proof -
have "\<Inter>{closure S |S. S \<in> I} \<le> closure (\<Inter>{rel_interior S |S. S \<in> I})"
using convex_closure_rel_interior_inter assms by auto
moreover
have "closure (\<Inter>{rel_interior S |S. S \<in> I}) \<le> closure (\<Inter>I)"
using rel_interior_inter_aux closure_mono[of "\<Inter>{rel_interior S |S. S \<in> I}" "\<Inter>I"]
by auto
ultimately show ?thesis
using closure_Int[of I] by auto
qed
lemma convex_inter_rel_interior_same_closure:
assumes "\<forall>S\<in>I. convex (S :: 'n::euclidean_space set)"
and "\<Inter>{rel_interior S |S. S \<in> I} \<noteq> {}"
shows "closure (\<Inter>{rel_interior S |S. S \<in> I}) = closure (\<Inter>I)"
proof -
have "\<Inter>{closure S |S. S \<in> I} \<le> closure (\<Inter>{rel_interior S |S. S \<in> I})"
using convex_closure_rel_interior_inter assms by auto
moreover
have "closure (\<Inter>{rel_interior S |S. S \<in> I}) \<le> closure (\<Inter>I)"
using rel_interior_inter_aux closure_mono[of "\<Inter>{rel_interior S |S. S \<in> I}" "\<Inter>I"]
by auto
ultimately show ?thesis
using closure_Int[of I] by auto
qed
lemma convex_rel_interior_inter:
assumes "\<forall>S\<in>I. convex (S :: 'n::euclidean_space set)"
and "\<Inter>{rel_interior S |S. S \<in> I} \<noteq> {}"
shows "rel_interior (\<Inter>I) \<subseteq> \<Inter>{rel_interior S |S. S \<in> I}"
proof -
have "convex (\<Inter>I)"
using assms convex_Inter by auto
moreover
have "convex (\<Inter>{rel_interior S |S. S \<in> I})"
apply (rule convex_Inter)
using assms convex_rel_interior
apply auto
done
ultimately
have "rel_interior (\<Inter>{rel_interior S |S. S \<in> I}) = rel_interior (\<Inter>I)"
using convex_inter_rel_interior_same_closure assms
closure_eq_rel_interior_eq[of "\<Inter>{rel_interior S |S. S \<in> I}" "\<Inter>I"]
by blast
then show ?thesis
using rel_interior_subset[of "\<Inter>{rel_interior S |S. S \<in> I}"] by auto
qed
lemma convex_rel_interior_finite_inter:
assumes "\<forall>S\<in>I. convex (S :: 'n::euclidean_space set)"
and "\<Inter>{rel_interior S |S. S \<in> I} \<noteq> {}"
and "finite I"
shows "rel_interior (\<Inter>I) = \<Inter>{rel_interior S |S. S \<in> I}"
proof -
have "\<Inter>I \<noteq> {}"
using assms rel_interior_inter_aux[of I] by auto
have "convex (\<Inter>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 \<in> \<Inter>{rel_interior S |S. S \<in> I}"
{
fix x
assume x: "x \<in> \<Inter>I"
{
fix S
assume S: "S \<in> I"
then have "z \<in> rel_interior S" "x \<in> S"
using z x by auto
then have "\<exists>m. m > 1 \<and> (\<forall>e. e > 1 \<and> e \<le> m \<longrightarrow> (1 - e)*\<^sub>R x + e *\<^sub>R z \<in> S)"
using convex_rel_interior_if[of S z] S assms hull_subset[of S] by auto
}
then obtain mS where
mS: "\<forall>S\<in>I. mS S > 1 \<and> (\<forall>e. e > 1 \<and> e \<le> mS S \<longrightarrow> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S)" by metis
define e where "e = Min (mS ` I)"
then have "e \<in> mS ` I" using assms \<open>I \<noteq> {}\<close> by simp
then have "e > 1" using mS by auto
moreover have "\<forall>S\<in>I. e \<le> mS S"
using e_def assms by auto
ultimately have "\<exists>e > 1. (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> \<Inter>I"
using mS by auto
}
then have "z \<in> rel_interior (\<Inter>I)"
using convex_rel_interior_iff[of "\<Inter>I" z] \<open>\<Inter>I \<noteq> {}\<close> \<open>convex (\<Inter>I)\<close> 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 \<inter> rel_interior T \<noteq> {}"
shows "closure (S \<inter> T) = closure S \<inter> 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 \<inter> rel_interior T \<noteq> {}"
shows "rel_interior (S \<inter> T) = rel_interior S \<inter> 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 \<inter> T \<noteq> {}"
shows "closure (S \<inter> T) = closure S \<inter> 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 \<longleftrightarrow> closed_segment a b \<subseteq> 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 \<longleftrightarrow> closed_segment a b \<subseteq> 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 \<inter> T \<noteq> {}"
shows "rel_interior (S \<inter> T) = rel_interior S \<inter> 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 \<inter> T \<noteq> {}"
shows "rel_frontier(S \<inter> T) = frontier S \<inter> 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 \<inter> T \<noteq> {}"
shows "rel_interior(S \<inter> T) = interior S \<inter> T"
proof -
obtain a where aS: "a \<in> interior S" and aT:"a \<in> 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 \<inter> T \<noteq> {}"
shows "closure(S \<inter> T) = closure S \<inter> T"
proof
have "closure (S \<inter> T) \<subseteq> closure T"
by (simp add: closure_mono)
also have "... \<subseteq> T"
by (simp add: affine_closed assms)
finally show "closure(S \<inter> T) \<subseteq> closure S \<inter> T"
by (simp add: closure_mono)
next
obtain a where "a \<in> rel_interior S" "a \<in> T"
using assms by auto
then have ssT: "subspace ((\<lambda>x. (-a)+x) ` T)" and "a \<in> S"
using affine_diffs_subspace rel_interior_subset assms by blast+
show "closure S \<inter> T \<subseteq> closure (S \<inter> T)"
proof
fix x assume "x \<in> closure S \<inter> T"
show "x \<in> closure (S \<inter> T)"
proof (cases "x = a")
case True
then show ?thesis
using \<open>a \<in> S\<close> \<open>a \<in> T\<close> closure_subset by fastforce
next
case False
then have "x \<in> closure(open_segment a x)"
by auto
then show ?thesis
using \<open>x \<in> closure S \<inter> T\<close> 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 \<le> closure T"
and "\<not> S \<subseteq> rel_frontier T"
shows "rel_interior S \<subseteq> rel_interior T"
proof -
have *: "S \<inter> closure T = S"
using assms by auto
have "\<not> rel_interior S \<subseteq> 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 \<inter> rel_interior (closure T) \<noteq> {}"
using assms rel_frontier_def[of T] rel_interior_subset convex_rel_interior_closure[of T]
by auto
then have "rel_interior S \<inter> rel_interior T = rel_interior (S \<inter> 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 "\<dots> = rel_interior S"
using * by auto
finally show ?thesis
by auto
qed
lemma rel_interior_convex_linear_image:
fixes f :: "'m::euclidean_space \<Rightarrow> '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) \<subseteq> f ` S"
unfolding image_mono using rel_interior_subset by auto
have "f ` S \<subseteq> f ` (closure S)"
unfolding image_mono using closure_subset by auto
also have "\<dots> = f ` (closure (rel_interior S))"
using convex_closure_rel_interior assms by auto
also have "\<dots> \<subseteq> 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) \<subseteq> f ` rel_interior S"
using rel_interior_subset by auto
moreover
{
fix z
assume "z \<in> f ` rel_interior S"
then obtain z1 where z1: "z1 \<in> rel_interior S" "f z1 = z" by auto
{
fix x
assume "x \<in> f ` S"
then obtain x1 where x1: "x1 \<in> S" "f x1 = x" by auto
then obtain e where e: "e > 1" "(1 - e) *\<^sub>R x1 + e *\<^sub>R z1 \<in> S"
using convex_rel_interior_iff[of S z1] \<open>convex S\<close> x1 z1 by auto
moreover have "f ((1 - e) *\<^sub>R x1 + e *\<^sub>R z1) = (1 - e) *\<^sub>R x + e *\<^sub>R z"
using x1 z1 by (simp add: linear_add linear_scale \<open>linear f\<close>)
ultimately have "(1 - e) *\<^sub>R x + e *\<^sub>R z \<in> f ` S"
using imageI[of "(1 - e) *\<^sub>R x1 + e *\<^sub>R z1" S f] by auto
then have "\<exists>e. e > 1 \<and> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> f ` S"
using e by auto
}
then have "z \<in> rel_interior (f ` S)"
using convex_rel_interior_iff[of "f ` S" z] \<open>convex S\<close> \<open>linear f\<close>
\<open>S \<noteq> {}\<close> 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 \<Rightarrow> 'n::euclidean_space"
assumes "linear f"
and "convex S"
and "f -` (rel_interior S) \<noteq> {}"
shows "rel_interior (f -` S) = f -` (rel_interior S)"
proof -
interpret linear f by fact
have "S \<noteq> {}"
using assms rel_interior_empty by auto
have nonemp: "f -` S \<noteq> {}"
by (metis assms(3) rel_interior_subset subset_empty vimage_mono)
then have "S \<inter> (range f) \<noteq> {}"
by auto
have conv: "convex (f -` S)"
using convex_linear_vimage assms by auto
then have "convex (S \<inter> range f)"
by (simp add: assms(2) convex_Int convex_linear_image linear_axioms)
{
fix z
assume "z \<in> f -` (rel_interior S)"
then have z: "f z \<in> rel_interior S"
by auto
{
fix x
assume "x \<in> f -` S"
then have "f x \<in> S" by auto
then obtain e where e: "e > 1" "(1 - e) *\<^sub>R f x + e *\<^sub>R f z \<in> S"
using convex_rel_interior_iff[of S "f z"] z assms \<open>S \<noteq> {}\<close> by auto
moreover have "(1 - e) *\<^sub>R f x + e *\<^sub>R f z = f ((1 - e) *\<^sub>R x + e *\<^sub>R z)"
using \<open>linear f\<close> by (simp add: linear_iff)
ultimately have "\<exists>e. e > 1 \<and> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> f -` S"
using e by auto
}
then have "z \<in> rel_interior (f -` S)"
using convex_rel_interior_iff[of "f -` S" z] conv nonemp by auto
}
moreover
{
fix z
assume z: "z \<in> rel_interior (f -` S)"
{
fix x
assume "x \<in> S \<inter> range f"
then obtain y where y: "f y = x" "y \<in> f -` S" by auto
then obtain e where e: "e > 1" "(1 - e) *\<^sub>R y + e *\<^sub>R z \<in> f -` S"
using convex_rel_interior_iff[of "f -` S" z] z conv by auto
moreover have "(1 - e) *\<^sub>R x + e *\<^sub>R f z = f ((1 - e) *\<^sub>R y + e *\<^sub>R z)"
using \<open>linear f\<close> y by (simp add: linear_iff)
ultimately have "\<exists>e. e > 1 \<and> (1 - e) *\<^sub>R x + e *\<^sub>R f z \<in> S \<inter> range f"
using e by auto
}
then have "f z \<in> rel_interior (S \<inter> range f)"
using \<open>convex (S \<inter> (range f))\<close> \<open>S \<inter> range f \<noteq> {}\<close>
convex_rel_interior_iff[of "S \<inter> (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 \<in> rel_interior S"
using convex_affine_rel_interior_Int[of S "range f"] assms by auto
then have "z \<in> 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 \<times> T) = rel_interior S \<times> rel_interior T"
proof -
{ assume "S = {}"
then have ?thesis
by auto
}
moreover
{ assume "T = {}"
then have ?thesis
by auto
}
moreover
{
assume "S \<noteq> {}" "T \<noteq> {}"
then have ri: "rel_interior S \<noteq> {}" "rel_interior T \<noteq> {}"
using rel_interior_eq_empty assms by auto
then have "fst -` rel_interior S \<noteq> {}"
using fst_vimage_eq_Times[of "rel_interior S"] by auto
then have "rel_interior ((fst :: 'n * 'm \<Rightarrow> 'n) -` S) = fst -` rel_interior S"
using fst_linear \<open>convex S\<close> rel_interior_convex_linear_preimage[of fst S] by auto
then have s: "rel_interior (S \<times> (UNIV :: 'm set)) = rel_interior S \<times> UNIV"
by (simp add: fst_vimage_eq_Times)
from ri have "snd -` rel_interior T \<noteq> {}"
using snd_vimage_eq_Times[of "rel_interior T"] by auto
then have "rel_interior ((snd :: 'n * 'm \<Rightarrow> 'm) -` T) = snd -` rel_interior T"
using snd_linear \<open>convex T\<close> rel_interior_convex_linear_preimage[of snd T] by auto
then have t: "rel_interior ((UNIV :: 'n set) \<times> T) = UNIV \<times> rel_interior T"
by (simp add: snd_vimage_eq_Times)
from s t have *: "rel_interior (S \<times> (UNIV :: 'm set)) \<inter> rel_interior ((UNIV :: 'n set) \<times> T) =
rel_interior S \<times> rel_interior T" by auto
have "S \<times> T = S \<times> (UNIV :: 'm set) \<inter> (UNIV :: 'n set) \<times> T"
by auto
then have "rel_interior (S \<times> T) = rel_interior ((S \<times> (UNIV :: 'm set)) \<inter> ((UNIV :: 'n set) \<times> T))"
by auto
also have "\<dots> = rel_interior (S \<times> (UNIV :: 'm set)) \<inter> rel_interior ((UNIV :: 'n set) \<times> T)"
apply (subst convex_rel_interior_inter_two[of "S \<times> (UNIV :: 'm set)" "(UNIV :: 'n set) \<times> 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 \<noteq> 0"
shows "((*\<^sub>R) c) ` (rel_interior S) = rel_interior (((*\<^sub>R) c) ` S)"
using rel_interior_injective_linear_image[of "((*\<^sub>R) c)" S]
linear_conv_bounded_linear[of "(*\<^sub>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 "((*\<^sub>R) c) ` (rel_interior S) = rel_interior (((*\<^sub>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 (((*\<^sub>R) c) ` S) \<and> rel_open (((*\<^sub>R) c) ` S)"
by (metis assms convex_scaling rel_interior_convex_scaleR rel_open_def)
lemma convex_rel_open_finite_inter:
assumes "\<forall>S\<in>I. convex (S :: 'n::euclidean_space set) \<and> rel_open S"
and "finite I"
shows "convex (\<Inter>I) \<and> rel_open (\<Inter>I)"
proof (cases "\<Inter>{rel_interior S |S. S \<in> I} = {}")
case True
then have "\<Inter>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 (\<Inter>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 \<Rightarrow> 'n::euclidean_space"
assumes "linear f"
and "convex S"
and "rel_open S"
shows "convex (f ` S) \<and> 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 \<Rightarrow> 'n::euclidean_space"
assumes "linear f"
and "convex S"
and "rel_open S"
shows "convex (f -` S) \<and> 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 \<times> 'n::euclidean_space) set"
and f :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space set"
assumes "convex S"
and "f = (\<lambda>y. {z. (y, z) \<in> S})"
shows "(y, z) \<in> rel_interior S \<longleftrightarrow> (y \<in> rel_interior {y. (f y \<noteq> {})} \<and> z \<in> rel_interior (f y))"
proof -
{
fix y
assume "y \<in> {y. f y \<noteq> {}}"
then obtain z where "(y, z) \<in> S"
using assms by auto
then have "\<exists>x. x \<in> S \<and> y = fst x"
apply (rule_tac x="(y, z)" in exI)
apply auto
done
then obtain x where "x \<in> S" "y = fst x"
by blast
then have "y \<in> fst ` S"
unfolding image_def by auto
}
then have "fst ` S = {y. f y \<noteq> {}}"
unfolding fst_def using assms by auto
then have h1: "fst ` rel_interior S = rel_interior {y. f y \<noteq> {}}"
using rel_interior_convex_linear_image[of fst S] assms fst_linear by auto
{
fix y
assume "y \<in> rel_interior {y. f y \<noteq> {}}"
then have "y \<in> fst ` rel_interior S"
using h1 by auto
then have *: "rel_interior S \<inter> fst -` {y} \<noteq> {}"
by auto
moreover have aff: "affine (fst -` {y})"
unfolding affine_alt by (simp add: algebra_simps)
ultimately have **: "rel_interior (S \<inter> fst -` {y}) = rel_interior S \<inter> fst -` {y}"
using convex_affine_rel_interior_Int[of S "fst -` {y}"] assms by auto
have conv: "convex (S \<inter> fst -` {y})"
using convex_Int assms aff affine_imp_convex by auto
{
fix x
assume "x \<in> f y"
then have "(y, x) \<in> S \<inter> (fst -` {y})"
using assms by auto
moreover have "x = snd (y, x)" by auto
ultimately have "x \<in> snd ` (S \<inter> fst -` {y})"
by blast
}
then have "snd ` (S \<inter> fst -` {y}) = f y"
using assms by auto
then have ***: "rel_interior (f y) = snd ` rel_interior (S \<inter> fst -` {y})"
using rel_interior_convex_linear_image[of snd "S \<inter> fst -` {y}"] snd_linear conv
by auto
{
fix z
assume "z \<in> rel_interior (f y)"
then have "z \<in> snd ` rel_interior (S \<inter> fst -` {y})"
using *** by auto
moreover have "{y} = fst ` rel_interior (S \<inter> fst -` {y})"
using * ** rel_interior_subset by auto
ultimately have "(y, z) \<in> rel_interior (S \<inter> fst -` {y})"
by force
then have "(y,z) \<in> rel_interior S"
using ** by auto
}
moreover
{
fix z
assume "(y, z) \<in> rel_interior S"
then have "(y, z) \<in> rel_interior (S \<inter> fst -` {y})"
using ** by auto
then have "z \<in> snd ` rel_interior (S \<inter> fst -` {y})"
by (metis Range_iff snd_eq_Range)
then have "z \<in> rel_interior (f y)"
using *** by auto
}
ultimately have "\<And>z. (y, z) \<in> rel_interior S \<longleftrightarrow> z \<in> rel_interior (f y)"
by auto
}
then have h2: "\<And>y z. y \<in> rel_interior {t. f t \<noteq> {}} \<Longrightarrow>
(y, z) \<in> rel_interior S \<longleftrightarrow> z \<in> rel_interior (f y)"
by auto
{
fix y z
assume asm: "(y, z) \<in> rel_interior S"
then have "y \<in> fst ` rel_interior S"
by (metis Domain_iff fst_eq_Domain)
then have "y \<in> rel_interior {t. f t \<noteq> {}}"
using h1 by auto
then have "y \<in> rel_interior {t. f t \<noteq> {}}" and "(z \<in> 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 \<times> rel_frontier T \<subseteq> rel_frontier (S \<times> T)"
by (force simp: rel_frontier_def rel_interior_Times assms closure_Times)
subsubsection%unimportant \<open>Relative interior of convex cone\<close>
lemma cone_rel_interior:
fixes S :: "'m::euclidean_space set"
assumes "cone S"
shows "cone ({0} \<union> 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 \<in> S \<and> (\<forall>c. c > 0 \<longrightarrow> (*\<^sub>R) c ` S = S)"
using cone_iff[of S] assms by auto
then have *: "0 \<in> ({0} \<union> rel_interior S)"
and "\<forall>c. c > 0 \<longrightarrow> (*\<^sub>R) c ` ({0} \<union> rel_interior S) = ({0} \<union> rel_interior S)"
by (auto simp add: rel_interior_scaleR)
then show ?thesis
using cone_iff[of "{0} \<union> rel_interior S"] by auto
qed
lemma rel_interior_convex_cone_aux:
fixes S :: "'m::euclidean_space set"
assumes "convex S"
shows "(c, x) \<in> rel_interior (cone hull ({(1 :: real)} \<times> S)) \<longleftrightarrow>
c > 0 \<and> x \<in> (((*\<^sub>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 \<in> S" by auto
have conv: "convex ({(1 :: real)} \<times> S)"
using convex_Times[of "{(1 :: real)}" S] assms convex_singleton[of "1 :: real"]
by auto
define f where "f y = {z. (y, z) \<in> cone hull ({1 :: real} \<times> S)}" for y
then have *: "(c, x) \<in> rel_interior (cone hull ({(1 :: real)} \<times> S)) =
(c \<in> rel_interior {y. f y \<noteq> {}} \<and> x \<in> rel_interior (f c))"
apply (subst rel_interior_projection[of "cone hull ({(1 :: real)} \<times> S)" f c x])
using convex_cone_hull[of "{(1 :: real)} \<times> S"] conv
apply auto
done
{
fix y :: real
assume "y \<ge> 0"
then have "y *\<^sub>R (1,s) \<in> cone hull ({1 :: real} \<times> S)"
using cone_hull_expl[of "{(1 :: real)} \<times> S"] \<open>s \<in> S\<close> by auto
then have "f y \<noteq> {}"
using f_def by auto
}
then have "{y. f y \<noteq> {}} = {0..}"
using f_def cone_hull_expl[of "{1 :: real} \<times> S"] by auto
then have **: "rel_interior {y. f y \<noteq> {}} = {0<..}"
using rel_interior_real_semiline by auto
{
fix c :: real
assume "c > 0"
then have "f c = ((*\<^sub>R) c ` S)"
using f_def cone_hull_expl[of "{1 :: real} \<times> S"] by auto
then have "rel_interior (f c) = (*\<^sub>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} \<times> S)) =
{(c, c *\<^sub>R x) | c x. c > 0 \<and> x \<in> rel_interior S}"
(is "?lhs = ?rhs")
proof -
{
fix z
assume "z \<in> ?lhs"
have *: "z = (fst z, snd z)"
by auto
have "z \<in> ?rhs"
using rel_interior_convex_cone_aux[of S "fst z" "snd z"] assms \<open>z \<in> ?lhs\<close>
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 \<in> ?rhs"
then have "z \<in> ?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 "\<forall>i\<in>I. convex (S i) \<and> (S i) \<noteq> {}"
shows "convex hull (\<Union>(S ` I)) =
{sum (\<lambda>i. c i *\<^sub>R s i) I | c s. (\<forall>i\<in>I. c i \<ge> 0) \<and> sum c I = 1 \<and> (\<forall>i\<in>I. s i \<in> S i)}"
(is "?lhs = ?rhs")
proof -
have "?lhs \<supseteq> ?rhs"
proof
fix x
assume "x \<in> ?rhs"
then obtain c s where *: "sum (\<lambda>i. c i *\<^sub>R s i) I = x" "sum c I = 1"
"(\<forall>i\<in>I. c i \<ge> 0) \<and> (\<forall>i\<in>I. s i \<in> S i)" by auto
then have "\<forall>i\<in>I. s i \<in> convex hull (\<Union>(S ` I))"
using hull_subset[of "\<Union>(S ` I)" convex] by auto
then show "x \<in> ?lhs"
unfolding *(1)[symmetric]
apply (subst convex_sum[of I "convex hull \<Union>(S ` I)" c s])
using * assms convex_convex_hull
apply auto
done
qed
{
fix i
assume "i \<in> I"
with assms have "\<exists>p. p \<in> S i" by auto
}
then obtain p where p: "\<forall>i\<in>I. p i \<in> S i" by metis
{
fix i
assume "i \<in> I"
{
fix x
assume "x \<in> S i"
define c where "c j = (if j = i then 1::real else 0)" for j
then have *: "sum c I = 1"
using \<open>finite I\<close> \<open>i \<in> I\<close> sum.delta[of I i "\<lambda>j::'a. 1::real"]
by auto
define s where "s j = (if j = i then x else p j)" for j
then have "\<forall>j. c j *\<^sub>R s j = (if j = i then x else 0)"
using c_def by (auto simp add: algebra_simps)
then have "x = sum (\<lambda>i. c i *\<^sub>R s i) I"
using s_def c_def \<open>finite I\<close> \<open>i \<in> I\<close> sum.delta[of I i "\<lambda>j::'a. x"]
by auto
then have "x \<in> ?rhs"
apply auto
apply (rule_tac x = c in exI)
apply (rule_tac x = s in exI)
using * c_def s_def p \<open>x \<in> S i\<close>
apply auto
done
}
then have "?rhs \<supseteq> S i" by auto
}
then have *: "?rhs \<supseteq> \<Union>(S ` I)" by auto
{
fix u v :: real
assume uv: "u \<ge> 0 \<and> v \<ge> 0 \<and> u + v = 1"
fix x y
assume xy: "x \<in> ?rhs \<and> y \<in> ?rhs"
from xy obtain c s where
xc: "x = sum (\<lambda>i. c i *\<^sub>R s i) I \<and> (\<forall>i\<in>I. c i \<ge> 0) \<and> sum c I = 1 \<and> (\<forall>i\<in>I. s i \<in> S i)"
by auto
from xy obtain d t where
yc: "y = sum (\<lambda>i. d i *\<^sub>R t i) I \<and> (\<forall>i\<in>I. d i \<ge> 0) \<and> sum d I = 1 \<and> (\<forall>i\<in>I. t i \<in> S i)"
by auto
define e where "e i = u * c i + v * d i" for i
have ge0: "\<forall>i\<in>I. e i \<ge> 0"
using e_def xc yc uv by simp
have "sum (\<lambda>i. u * c i) I = u * sum c I"
by (simp add: sum_distrib_left)
moreover have "sum (\<lambda>i. 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) *\<^sub>R s i + (v * d i / e i) *\<^sub>R t i)"
for i
{
fix i
assume i: "i \<in> I"
have "q i \<in> 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: "\<forall>i\<in>I. q i \<in> S i" by auto
{
fix i
assume i: "i \<in> I"
have "(u * c i) *\<^sub>R s i + (v * d i) *\<^sub>R t i = e i *\<^sub>R q i"
proof (cases "e i = 0")
case True
have ge: "u * (c i) \<ge> 0 \<and> v * d i \<ge> 0"
using xc yc uv i by simp
moreover from ge have "u * c i \<le> 0 \<and> v * d i \<le> 0"
using True e_def i by simp
ultimately have "u * c i = 0 \<and> v * d i = 0" by auto
with True show ?thesis by auto
next
case False
then have "(u * (c i)/(e i))*\<^sub>R (s i)+(v * (d i)/(e i))*\<^sub>R (t i) = q i"
using q_def by auto
then have "e i *\<^sub>R ((u * (c i)/(e i))*\<^sub>R (s i)+(v * (d i)/(e i))*\<^sub>R (t i))
= (e i) *\<^sub>R (q i)" by auto
with False show ?thesis by (simp add: algebra_simps)
qed
}
then have *: "\<forall>i\<in>I. (u * c i) *\<^sub>R s i + (v * d i) *\<^sub>R t i = e i *\<^sub>R q i"
by auto
have "u *\<^sub>R x + v *\<^sub>R y = sum (\<lambda>i. (u * c i) *\<^sub>R s i + (v * d i) *\<^sub>R t i) I"
using xc yc by (simp add: algebra_simps scaleR_right.sum sum.distrib)
also have "\<dots> = sum (\<lambda>i. e i *\<^sub>R q i) I"
using * by auto
finally have "u *\<^sub>R x + v *\<^sub>R y = sum (\<lambda>i. (e i) *\<^sub>R (q i)) I"
by auto
then have "u *\<^sub>R x + v *\<^sub>R y \<in> ?rhs"
using ge0 sum1 qs by auto
}
then have "convex ?rhs" unfolding convex_def by auto
then show ?thesis
using \<open>?lhs \<supseteq> ?rhs\<close> * hull_minimal[of "\<Union>(S ` I)" ?rhs convex]
by blast
qed
lemma convex_hull_union_two:
fixes S T :: "'m::euclidean_space set"
assumes "convex S"
and "S \<noteq> {}"
and "convex T"
and "T \<noteq> {}"
shows "convex hull (S \<union> T) =
{u *\<^sub>R s + v *\<^sub>R t | u v s t. u \<ge> 0 \<and> v \<ge> 0 \<and> u + v = 1 \<and> s \<in> S \<and> t \<in> 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 "\<Union>(s ` I) = S \<union> T"
using s_def I_def by auto
then have "convex hull (\<Union>(s ` I)) = convex hull (S \<union> T)"
by auto
moreover have "convex hull \<Union>(s ` I) =
{\<Sum> i\<in>I. c i *\<^sub>R sa i | c sa. (\<forall>i\<in>I. 0 \<le> c i) \<and> sum c I = 1 \<and> (\<forall>i\<in>I. sa i \<in> s i)}"
apply (subst convex_hull_finite_union[of I s])
using assms s_def I_def
apply auto
done
moreover have
"{\<Sum>i\<in>I. c i *\<^sub>R sa i | c sa. (\<forall>i\<in>I. 0 \<le> c i) \<and> sum c I = 1 \<and> (\<forall>i\<in>I. sa i \<in> s i)} \<le> ?rhs"
using s_def I_def by auto
ultimately show "?lhs \<subseteq> ?rhs" by auto
{
fix x
assume "x \<in> ?rhs"
then obtain u v s t where *: "x = u *\<^sub>R s + v *\<^sub>R t \<and> u \<ge> 0 \<and> v \<ge> 0 \<and> u + v = 1 \<and> s \<in> S \<and> t \<in> T"
by auto
then have "x \<in> convex hull {s, t}"
using convex_hull_2[of s t] by auto
then have "x \<in> convex hull (S \<union> T)"
using * hull_mono[of "{s, t}" "S \<union> T"] by auto
}
then show "?lhs \<supseteq> ?rhs" by blast
qed
subsection%unimportant \<open>Convexity on direct sums\<close>
lemma closure_sum:
fixes S T :: "'a::real_normed_vector set"
shows "closure S + closure T \<subseteq> 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 = (\<lambda>(x,y). x + y) ` (rel_interior S \<times> rel_interior T)"
by (simp add: set_plus_image)
also have "\<dots> = (\<lambda>(x,y). x + y) ` rel_interior (S \<times> T)"
using rel_interior_Times assms by auto
also have "\<dots> = rel_interior (S + T)"
using fst_snd_linear convex_Times assms
rel_interior_convex_linear_image[of "(\<lambda>(x,y). x + y)" "S \<times> T"]
by (auto simp add: set_plus_image)
finally show ?thesis ..
qed
lemma rel_interior_sum_gen:
fixes S :: "'a \<Rightarrow> 'n::euclidean_space set"
assumes "\<forall>i\<in>I. convex (S i)"
shows "rel_interior (sum S I) = sum (\<lambda>i. 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 \<times> T) \<and> rel_open (S \<times> 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) \<and> 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 \<noteq> {}"
assumes "\<forall>i\<in>I. convex (S i) \<and> cone (S i) \<and> S i \<noteq> {}"
shows "convex hull (\<Union>(S ` I)) = sum S I"
(is "?lhs = ?rhs")
proof -
{
fix x
assume "x \<in> ?lhs"
then obtain c xs where
x: "x = sum (\<lambda>i. c i *\<^sub>R xs i) I \<and> (\<forall>i\<in>I. c i \<ge> 0) \<and> sum c I = 1 \<and> (\<forall>i\<in>I. xs i \<in> S i)"
using convex_hull_finite_union[of I S] assms by auto
define s where "s i = c i *\<^sub>R xs i" for i
{
fix i
assume "i \<in> I"
then have "s i \<in> S i"
using s_def x assms mem_cone[of "S i" "xs i" "c i"] by auto
}
then have "\<forall>i\<in>I. s i \<in> S i" by auto
moreover have "x = sum s I" using x s_def by auto
ultimately have "x \<in> ?rhs"
using set_sum_alt[of I S] assms by auto
}
moreover
{
fix x
assume "x \<in> ?rhs"
then obtain s where x: "x = sum s I \<and> (\<forall>i\<in>I. s i \<in> S i)"
using set_sum_alt[of I S] assms by auto
define xs where "xs i = of_nat(card I) *\<^sub>R s i" for i
then have "x = sum (\<lambda>i. ((1 :: real) / of_nat(card I)) *\<^sub>R xs i) I"
using x assms by auto
moreover have "\<forall>i\<in>I. xs i \<in> S i"
using x xs_def assms by (simp add: cone_def)
moreover have "\<forall>i\<in>I. (1 :: real) / of_nat (card I) \<ge> 0"
by auto
moreover have "sum (\<lambda>i. (1 :: real) / of_nat (card I)) I = 1"
using assms by auto
ultimately have "x \<in> ?lhs"
apply (subst convex_hull_finite_union[of I S])
using assms
apply blast
using assms
apply blast
apply rule
apply (rule_tac x = "(\<lambda>i. (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 \<noteq> {}"
assumes "convex T"
and "cone T"
and "T \<noteq> {}"
shows "convex hull (S \<union> 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 "\<Union>(A ` I) = S \<union> T"
using A_def I_def by auto
then have "convex hull (\<Union>(A ` I)) = convex hull (S \<union> T)"
by auto
moreover have "convex hull \<Union>(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 \<Rightarrow> 'n::euclidean_space set"
assumes "finite I"
and "\<forall>i\<in>I. convex (S i) \<and> S i \<noteq> {}"
shows "rel_interior (convex hull (\<Union>(S ` I))) =
{sum (\<lambda>i. c i *\<^sub>R s i) I | c s. (\<forall>i\<in>I. c i > 0) \<and> sum c I = 1 \<and>
(\<forall>i\<in>I. s i \<in> 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 (\<Union>(S ` I))"
have "\<forall>i\<in>I. C0 \<ge> S i"
unfolding C0_def using hull_subset[of "\<Union>(S ` I)"] by auto
define K0 where "K0 = cone hull ({1 :: real} \<times> C0)"
define K where "K i = cone hull ({1 :: real} \<times> S i)" for i
have "\<forall>i\<in>I. K i \<noteq> {}"
unfolding K_def using assms
by (simp add: cone_hull_empty_iff[symmetric])
{
fix i
assume "i \<in> 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: "\<forall>i\<in>I. convex (K i)"
by auto
{
fix i
assume "i \<in> I"
then have "K0 \<supseteq> K i"
unfolding K0_def K_def
apply (subst hull_mono)
using \<open>\<forall>i\<in>I. C0 \<ge> S i\<close>
apply auto
done
}
then have "K0 \<supseteq> \<Union>(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 \<supseteq> convex hull (\<Union>(K ` I))"
using hull_minimal[of _ "K0" "convex"] by blast
have "\<forall>i\<in>I. K i \<supseteq> {1 :: real} \<times> S i"
using K_def by (simp add: hull_subset)
then have "\<Union>(K ` I) \<supseteq> {1 :: real} \<times> \<Union>(S ` I)"
by auto
then have "convex hull \<Union>(K ` I) \<supseteq> convex hull ({1 :: real} \<times> \<Union>(S ` I))"
by (simp add: hull_mono)
then have "convex hull \<Union>(K ` I) \<supseteq> {1 :: real} \<times> C0"
unfolding C0_def
using convex_hull_Times[of "{(1 :: real)}" "\<Union>(S ` I)"] convex_hull_singleton
by auto
moreover have "cone (convex hull (\<Union>(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 (\<Union>(K ` I)) \<supseteq> K0"
unfolding K0_def
using hull_minimal[of _ "convex hull (\<Union>(K ` I))" "cone"]
by blast
then have "K0 = convex hull (\<Union>(K ` I))"
using geq by auto
also have "\<dots> = 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 \<open>\<forall>i\<in>I. K i \<noteq> {}\<close> K_def
apply auto
done
finally have "K0 = sum K I" by auto
then have *: "rel_interior K0 = sum (\<lambda>i. (rel_interior (K i))) I"
using rel_interior_sum_gen[of I K] convK by auto
{
fix x
assume "x \<in> ?lhs"
then have "(1::real, x) \<in> 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 \<and> (\<forall>i\<in>I. k i \<in> rel_interior (K i))"
using \<open>finite I\<close> * set_sum_alt[of I "\<lambda>i. rel_interior (K i)"] by auto
{
fix i
assume "i \<in> I"
then have "convex (S i) \<and> k i \<in> rel_interior (cone hull {1} \<times> S i)"
using k K_def assms by auto
then have "\<exists>ci si. k i = (ci, ci *\<^sub>R si) \<and> 0 < ci \<and> si \<in> rel_interior (S i)"
using rel_interior_convex_cone[of "S i"] by auto
}
then obtain c s where
cs: "\<forall>i\<in>I. k i = (c i, c i *\<^sub>R s i) \<and> 0 < c i \<and> s i \<in> rel_interior (S i)"
by metis
then have "x = (\<Sum>i\<in>I. c i *\<^sub>R s i) \<and> sum c I = 1"
using k by (simp add: sum_prod)
then have "x \<in> ?rhs"
using k cs by auto
}
moreover
{
fix x
assume "x \<in> ?rhs"
then obtain c s where cs: "x = sum (\<lambda>i. c i *\<^sub>R s i) I \<and>
(\<forall>i\<in>I. c i > 0) \<and> sum c I = 1 \<and> (\<forall>i\<in>I. s i \<in> rel_interior (S i))"
by auto
define k where "k i = (c i, c i *\<^sub>R s i)" for i
{
fix i assume "i \<in> I"
then have "k i \<in> 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) \<in> rel_interior K0"
using K0_def * set_sum_alt[of I "(\<lambda>i. 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 \<in> ?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 \<Rightarrow> real"
assumes "convex_on I f"
and "x \<in> interior I"
and "y \<in> I"
shows "f y \<ge> f x + Inf ((\<lambda>t. (f x - f t) / (x - t)) ` ({x<..} \<inter> I)) * (y - x)"
(is "_ \<ge> _ + Inf (?F x) * (y - x)")
proof (cases rule: linorder_cases)
assume "x < y"
moreover
have "open (interior I)" by auto
from openE[OF this \<open>x \<in> interior I\<close>]
obtain e where e: "0 < e" "ball x e \<subseteq> interior I" .
moreover define t where "t = min (x + e / 2) ((x + y) / 2)"
ultimately have "x < t" "t < y" "t \<in> ball x e"
by (auto simp: dist_real_def field_simps split: split_min)
with \<open>x \<in> interior I\<close> e interior_subset[of I] have "t \<in> I" "x \<in> I" by auto
have "open (interior I)" by auto
from openE[OF this \<open>x \<in> interior I\<close>]
obtain e where "0 < e" "ball x e \<subseteq> interior I" .
moreover define K where "K = x - e / 2"
with \<open>0 < e\<close> have "K \<in> ball x e" "K < x"
by (auto simp: dist_real_def)
ultimately have "K \<in> I" "K < x" "x \<in> I"
using interior_subset[of I] \<open>x \<in> interior I\<close> by auto
have "Inf (?F x) \<le> (f x - f y) / (x - y)"
proof (intro bdd_belowI cInf_lower2)
show "(f x - f t) / (x - t) \<in> ?F x"
using \<open>t \<in> I\<close> \<open>x < t\<close> by auto
show "(f x - f t) / (x - t) \<le> (f x - f y) / (x - y)"
using \<open>convex_on I f\<close> \<open>x \<in> I\<close> \<open>y \<in> I\<close> \<open>x < t\<close> \<open>t < y\<close>
by (rule convex_on_diff)
next
fix y
assume "y \<in> ?F x"
with order_trans[OF convex_on_diff[OF \<open>convex_on I f\<close> \<open>K \<in> I\<close> _ \<open>K < x\<close> _]]
show "(f K - f x) / (K - x) \<le> y" by auto
qed
then show ?thesis
using \<open>x < y\<close> by (simp add: field_simps)
next
assume "y < x"
moreover
have "open (interior I)" by auto
from openE[OF this \<open>x \<in> interior I\<close>]
obtain e where e: "0 < e" "ball x e \<subseteq> interior I" .
moreover define t where "t = x + e / 2"
ultimately have "x < t" "t \<in> ball x e"
by (auto simp: dist_real_def field_simps)
with \<open>x \<in> interior I\<close> e interior_subset[of I] have "t \<in> I" "x \<in> I" by auto
have "(f x - f y) / (x - y) \<le> Inf (?F x)"
proof (rule cInf_greatest)
have "(f x - f y) / (x - y) = (f y - f x) / (y - x)"
using \<open>y < x\<close> by (auto simp: field_simps)
also
fix z
assume "z \<in> ?F x"
with order_trans[OF convex_on_diff[OF \<open>convex_on I f\<close> \<open>y \<in> I\<close> _ \<open>y < x\<close>]]
have "(f y - f x) / (y - x) \<le> z"
by auto
finally show "(f x - f y) / (x - y) \<le> z" .
next
have "open (interior I)" by auto
from openE[OF this \<open>x \<in> interior I\<close>]
obtain e where e: "0 < e" "ball x e \<subseteq> interior I" .
then have "x + e / 2 \<in> ball x e"
by (auto simp: dist_real_def)
with e interior_subset[of I] have "x + e / 2 \<in> {x<..} \<inter> I"
by auto
then show "?F x \<noteq> {}"
by blast
qed
then show ?thesis
using \<open>y < x\<close> by (simp add: field_simps)
qed simp
subsection%unimportant\<open>Explicit formulas for interior and relative interior of convex hull\<close>
lemma at_within_cbox_finite:
assumes "x \<in> box a b" "x \<notin> 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 \<open>finite S\<close> 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 "\<not> affine_dependent s" "t \<subseteq> s"
shows "convex hull t = affine hull t \<inter> 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" "\<forall>x\<in>s. 0 \<le> v x" "sum v s = 1"
"(\<Sum>x\<in>s. v x *\<^sub>R x) = (\<Sum>v\<in>t. u v *\<^sub>R v)" "x \<in> t"
then have s: "s = (s - t) \<union> t" \<comment> \<open>split into separate cases\<close>
using assms by auto
have [simp]: "(\<Sum>x\<in>t. v x *\<^sub>R x) + (\<Sum>x\<in>s - t. v x *\<^sub>R x) = (\<Sum>x\<in>t. u x *\<^sub>R x)"
"sum v t + sum v (s - t) = 1"
using uv fin s
by (auto simp: sum.union_disjoint [symmetric] Un_commute)
have "(\<Sum>x\<in>s. if x \<in> t then v x - u x else v x) = 0"
"(\<Sum>x\<in>s. (if x \<in> t then v x - u x else v x) *\<^sub>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 \<subseteq> affine hull t"
using convex_hull_subset_affine_hull by blast
moreover have "convex hull t \<subseteq> convex hull s"
using assms hull_mono by blast
moreover have "affine hull t \<inter> convex hull s \<subseteq> 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="\<lambda>x. if x \<in> 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 "\<not> 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 \<notin> 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: "\<not> 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 \<le> 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 \<le> 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
\<Longrightarrow> {y. \<exists>u. (\<forall>x \<in> s. 0 < u x \<and> u x < 1) \<and> sum u s = 1 \<and> sum (\<lambda>x. u x *\<^sub>R x) s = y}
\<subseteq> rel_interior (convex hull s)"
by (force simp add: rel_interior_convex_hull_union [where S="\<lambda>x. {x}" and I=s, simplified])
lemma explicit_subset_rel_interior_convex_hull_minimal:
fixes s :: "'a::euclidean_space set"
shows "finite s
\<Longrightarrow> {y. \<exists>u. (\<forall>x \<in> s. 0 < u x) \<and> sum u s = 1 \<and> sum (\<lambda>x. u x *\<^sub>R x) s = y}
\<subseteq> rel_interior (convex hull s)"
by (force simp add: rel_interior_convex_hull_union [where S="\<lambda>x. {x}" and I=s, simplified])
lemma rel_interior_convex_hull_explicit:
fixes s :: "'a::euclidean_space set"
assumes "\<not> affine_dependent s"
shows "rel_interior(convex hull s) =
{y. \<exists>u. (\<forall>x \<in> s. 0 < u x) \<and> sum u s = 1 \<and> sum (\<lambda>x. u x *\<^sub>R x) s = y}"
(is "?lhs = ?rhs")
proof
show "?rhs \<le> ?lhs"
by (simp add: aff_independent_finite explicit_subset_rel_interior_convex_hull_minimal assms)
next
show "?lhs \<le> ?rhs"
proof (cases "\<exists>a. s = {a}")
case True then show "?lhs \<le> ?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 \<in> s" "b \<in> s" "a \<noteq> b"
then have s: "s = (s - {a,b}) \<union> {a,b}" \<comment> \<open>split into separate cases\<close>
by auto
have "(\<Sum>x\<in>s. if x = a then - d else if x = b then d else 0) = 0"
"(\<Sum>x\<in>s. (if x = a then - d else if x = b then d else 0) *\<^sub>R x) = d *\<^sub>R b - d *\<^sub>R a"
using ab fs
by (subst s, subst sum.union_disjoint, auto)+
} note [simp] = this
{ fix y
assume y: "y \<in> convex hull s" "y \<notin> ?rhs"
{ fix u T a
assume ua: "\<forall>x\<in>s. 0 \<le> u x" "sum u s = 1" "\<not> 0 < u a" "a \<in> s"
and yT: "y = (\<Sum>x\<in>s. u x *\<^sub>R x)" "y \<in> T" "open T"
and sb: "T \<inter> affine hull s \<subseteq> {w. \<exists>u. (\<forall>x\<in>s. 0 \<le> u x) \<and> sum u s = 1 \<and> (\<Sum>x\<in>s. u x *\<^sub>R x) = w}"
have ua0: "u a = 0"
using ua by auto
obtain b where b: "b\<in>s" "a \<noteq> b"
using ua False by auto
obtain e where e: "0 < e" "ball (\<Sum>x\<in>s. u x *\<^sub>R x) e \<subseteq> T"
using yT by (auto elim: openE)
with b obtain d where d: "0 < d" "norm(d *\<^sub>R (a-b)) < e"
by (auto intro: that [of "e / 2 / norm(a-b)"])
have "(\<Sum>x\<in>s. u x *\<^sub>R x) \<in> affine hull s"
using yT y by (metis affine_hull_convex_hull hull_redundant_eq)
then have "(\<Sum>x\<in>s. u x *\<^sub>R x) - d *\<^sub>R (a - b) \<in> affine hull s"
using ua b by (auto simp: hull_inc intro: mem_affine_3_minus2)
then have "y - d *\<^sub>R (a - b) \<in> T \<inter> affine hull s"
using d e yT by auto
then obtain v where "\<forall>x\<in>s. 0 \<le> v x"
"sum v s = 1"
"(\<Sum>x\<in>s. v x *\<^sub>R x) = (\<Sum>x\<in>s. u x *\<^sub>R x) - d *\<^sub>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="\<lambda>x. (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 \<notin> 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 \<le> ?rhs"
by blast
qed
qed
lemma interior_convex_hull_explicit_minimal:
fixes s :: "'a::euclidean_space set"
shows
"\<not> affine_dependent s
==> interior(convex hull s) =
(if card(s) \<le> DIM('a) then {}
else {y. \<exists>u. (\<forall>x \<in> s. 0 < u x) \<and> sum u s = 1 \<and> (\<Sum>x\<in>s. u x *\<^sub>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_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 "\<not> affine_dependent s"
shows
"interior(convex hull s) =
(if card(s) \<le> DIM('a) then {}
else {y. \<exists>u. (\<forall>x \<in> s. 0 < u x \<and> u x < 1) \<and> sum u s = 1 \<and> (\<Sum>x\<in>s. u x *\<^sub>R x) = y})"
proof -
{ fix u :: "'a \<Rightarrow> real" and a
assume "card Basis < card s" and u: "\<And>x. x\<in>s \<Longrightarrow> 0 < u x" "sum u s = 1" and a: "a \<in> s"
then have cs: "Suc 0 < card s"
by (metis DIM_positive less_trans_Suc)
obtain b where b: "b \<in> s" "a \<noteq> b"
proof (cases "s \<le> {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 \<le> sum u {a,b}"
using a b by simp
also have "... \<le> 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 \<open>b \<in> s\<close> 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 \<le> DIM('a)"
shows "interior(closed_segment a b) = {}"
using assms unfolding segment_convex_hull
proof -
have "card {a, b} \<le> 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 \<le> DIM('a) then {} else open_segment a b)"
proof (simp add: not_le, intro conjI impI)
assume "2 \<le> 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 \<le> 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 \<longleftrightarrow> {a,b} = {c,d}"
proof
assume abcd: "closed_segment a b = closed_segment c d"
show "{a,b} = {c,d}"
proof (cases "a=b \<or> c=d")
case True with abcd show ?thesis by force
next
case False
then have neq: "a \<noteq> b \<and> c \<noteq> 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 \<in> {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 \<in> {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 \<noteq> 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 \<noteq> 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 \<longleftrightarrow> a = b \<and> c = d \<or> {a,b} = {c,d}"
(is "?lhs = ?rhs")
proof
assume abcd: ?lhs
show ?rhs
proof (cases "a=b \<or> c=d")
case True with abcd show ?thesis
using finite_open_segment by fastforce
next
case False
then have a2: "a \<noteq> b \<and> c \<noteq> 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\<open>Similar results for closure and (relative or absolute) frontier\<close>
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 "\<not> affine_dependent s"
shows "rel_frontier(convex hull s) =
{y. \<exists>u. (\<forall>x \<in> s. 0 \<le> u x) \<and> (\<exists>x \<in> s. u x = 0) \<and> sum u s = 1 \<and> sum (\<lambda>x. u x *\<^sub>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 = "\<lambda>x. 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 "\<not> affine_dependent s"
shows "frontier(convex hull s) =
{y. \<exists>u. (\<forall>x \<in> s. 0 \<le> u x) \<and> (DIM ('a) < card s \<longrightarrow> (\<exists>x \<in> s. u x = 0)) \<and>
sum u s = 1 \<and> sum (\<lambda>x. u x *\<^sub>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 \<le> 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 "\<not> affine_dependent s"
shows "rel_frontier(convex hull s) = \<Union>{convex hull (s - {x}) |x. x \<in> s}"
proof -
have fs: "finite s"
using assms by (simp add: aff_independent_finite)
{ fix u a
have "\<forall>x\<in>s. 0 \<le> u x \<Longrightarrow> a \<in> s \<Longrightarrow> u a = 0 \<Longrightarrow> sum u s = 1 \<Longrightarrow>
\<exists>x v. x \<in> s \<and>
(\<forall>x\<in>s - {x}. 0 \<le> v x) \<and>
sum v (s - {x}) = 1 \<and> (\<Sum>x\<in>s - {x}. v x *\<^sub>R x) = (\<Sum>x\<in>s. u x *\<^sub>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 \<in> s \<Longrightarrow> \<forall>x\<in>s - {a}. 0 \<le> u x \<Longrightarrow> sum u (s - {a}) = 1 \<Longrightarrow>
\<exists>v. (\<forall>x\<in>s. 0 \<le> v x) \<and>
(\<exists>x\<in>s. v x = 0) \<and> sum v s = 1 \<and> (\<Sum>x\<in>s. v x *\<^sub>R x) = (\<Sum>x\<in>s - {a}. u x *\<^sub>R x)"
apply (rule_tac x="\<lambda>x. 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 "\<not> affine_dependent s"
shows "frontier(convex hull s) =
(if card s \<le> 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 "\<not> affine_dependent s"
shows "frontier(convex hull s) =
(if card s \<le> DIM ('a) then convex hull s else \<Union>{convex hull (s - {x}) |x. x \<in> 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 \<le> Suc (DIM ('a))" "x \<in> s"
shows "x \<in> 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 "\<exists>y \<in> s. y \<noteq> x"
by (cases "s \<le> {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 \<le> Suc (DIM ('a))" "x \<in> s"
shows "x \<notin> 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) = {} \<longleftrightarrow> affine_dependent s"
proof -
{ fix a b
assume ab: "a \<in> interior (convex hull s)" "b \<in> s" "b \<in> 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 \<open>Coplanarity, and collinearity in terms of affine hull\<close>
definition%important coplanar where
"coplanar s \<equiv> \<exists>u v w. s \<subseteq> affine hull {u,v,w}"
lemma collinear_affine_hull:
"collinear s \<longleftrightarrow> (\<exists>u v. s \<subseteq> affine hull {u,v})"
proof (cases "s={}")
case True then show ?thesis
by simp
next
case False
then obtain x where x: "x \<in> s" by auto
{ fix u
assume *: "\<And>x y. \<lbrakk>x\<in>s; y\<in>s\<rbrakk> \<Longrightarrow> \<exists>c. x - y = c *\<^sub>R u"
have "\<exists>u v. s \<subseteq> {a *\<^sub>R u + b *\<^sub>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 \<subseteq> {a *\<^sub>R u + b *\<^sub>R v |a b. a + b = 1}"
have "x\<in>s \<Longrightarrow> y\<in>s \<Longrightarrow> \<exists>c. x - y = c *\<^sub>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} \<longleftrightarrow> between (b,c) a \<or> between (c,a) b \<or> between (a,b) c"
(is "?lhs = ?rhs")
proof
assume ?lhs
then obtain u v where uv: "\<And>x. x \<in> {a, b, c} \<Longrightarrow> \<exists>c. x = u + c *\<^sub>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 \<Rightarrow> real"
assumes "continuous_on (closed_segment a b) f"
shows "closed_segment (f a) (f b) \<subseteq> 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 \<Rightarrow> 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) \<subseteq> f ` closed_segment a b"
by (metis subset_continuous_image_segment_1 contf)
show "f ` closed_segment a b \<subseteq> 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 \<noteq> f b"
using inj_onD injf by fastforce
moreover
have "f a \<notin> open_segment (f c) (f b)" if c: "c \<in> closed_segment a b" for c
proof (clarsimp simp add: open_segment_def)
assume fa: "f a \<in> closed_segment (f c) (f b)"
moreover have "closed_segment (f c) (f b) \<subseteq> 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 \<in> f ` closed_segment c b"
by blast
then have a: "a \<in> 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 \<subseteq> 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 \<notin> open_segment (f a) (f c)" if c: "c \<in> closed_segment a b" for c
proof (clarsimp simp add: open_segment_def fnot eq_commute)
assume fb: "f b \<in> closed_segment (f a) (f c)"
moreover have "closed_segment (f a) (f c) \<subseteq> 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 \<in> f ` closed_segment a c"
by blast
then have b: "b \<in> 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 \<subseteq> 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 \<Rightarrow> 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 \<le> 2"
shows "collinear s"
proof -
have "card s = 0 \<or> card s = 1 \<or> 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 \<le> 3"
shows "coplanar s"
proof -
have "card s \<le> 2 \<or> 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) \<longleftrightarrow> 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) \<longleftrightarrow> 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 \<Rightarrow> 'b::real_normed_vector"
assumes "coplanar s" "linear f" shows "coplanar(f ` s)"
proof -
{ fix u v w
assume "s \<subseteq> affine hull {u, v, w}"
then have "f ` s \<subseteq> f ` (affine hull {u, v, w})"
by (simp add: image_mono)
then have "f ` s \<subseteq> 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 \<Longrightarrow> coplanar ((\<lambda>x. 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((\<lambda>x. a + x) ` s) \<longleftrightarrow> coplanar s"
by (metis (no_types) coplanar_translation_imp translation_galois)
lemma coplanar_linear_image_eq:
fixes f :: "'a::euclidean_space \<Rightarrow> '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 \<circ> f = id"
using linear_injective_left_inverse [OF assms]
by blast
assume "coplanar (f ` s)"
then obtain u v w where "f ` s \<subseteq> affine hull {u, v, w}"
by (auto simp: coplanar_def)
then have "g ` f ` s \<subseteq> g ` (affine hull {u, v, w})"
by blast
then have "s \<subseteq> 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] \<open>linear g\<close> linear_conv_bounded_linear
by fastforce
qed
(*The HOL Light proof is simply
MATCH_ACCEPT_TAC(LINEAR_INVARIANT_RULE COPLANAR_LINEAR_IMAGE));;
*)
lemma coplanar_subset: "\<lbrakk>coplanar t; s \<subseteq> t\<rbrakk> \<Longrightarrow> coplanar s"
by (meson coplanar_def order_trans)
lemma affine_hull_3_imp_collinear: "c \<in> affine hull {a,b} \<Longrightarrow> collinear {a,b,c}"
by (metis collinear_2 collinear_affine_hull_collinear hull_redundant insert_commute)
lemma collinear_3_imp_in_affine_hull: "\<lbrakk>collinear {a,b,c}; a \<noteq> b\<rbrakk> \<Longrightarrow> c \<in> 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 \<noteq> b"
shows "collinear {a,b,c} \<longleftrightarrow> c \<in> 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} \<longleftrightarrow> a = b \<or> a = c \<or> b = c \<or> 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} \<Longrightarrow> collinear{a,b,c}"
by (simp add: collinear_3_eq_affine_dependent)
lemma collinear_3: "NO_MATCH 0 x \<Longrightarrow> collinear {x,y,z} \<longleftrightarrow> collinear {0, x-y, z-y}"
by (auto simp add: collinear_def)
lemma collinear_3_expand:
"collinear{a,b,c} \<longleftrightarrow> a = c \<or> (\<exists>u. b = u *\<^sub>R a + (1 - u) *\<^sub>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 "... \<longleftrightarrow> (a = c \<or> b = c \<or> (\<exists>ca. b - c = ca *\<^sub>R (a - c)))"
by (simp add: collinear_lemma)
also have "... \<longleftrightarrow> a = c \<or> (\<exists>u. b = u *\<^sub>R a + (1 - u) *\<^sub>R c)"
by (cases "a = c \<or> b = c") (auto simp: algebra_simps)
finally show ?thesis .
qed
lemma collinear_aff_dim: "collinear S \<longleftrightarrow> aff_dim S \<le> 1"
proof
assume "collinear S"
then obtain u and v :: "'a" where "aff_dim S \<le> aff_dim {u,v}"
by (metis \<open>collinear S\<close> aff_dim_affine_hull aff_dim_subset collinear_affine_hull)
then show "aff_dim S \<le> 1"
using order_trans by fastforce
next
assume "aff_dim S \<le> 1"
then have le1: "aff_dim (affine hull S) \<le> 1"
by simp
obtain B where "B \<subseteq> S" and B: "\<not> affine_dependent B" "affine hull S = affine hull B"
using affine_basis_exists [of S] by auto
then have "finite B" "card B \<le> 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 \<open>affine hull S = affine hull B\<close> 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 \<noteq> c"
shows "b = midpoint a c \<longleftrightarrow> collinear{a,b,c} \<and> dist a b = dist b c"
proof -
have *: "a - (u *\<^sub>R a + (1 - u) *\<^sub>R c) = (1 - u) *\<^sub>R (a - c)"
"u *\<^sub>R a + (1 - u) *\<^sub>R c - c = u *\<^sub>R (a - c)"
"\<bar>1 - u\<bar> = \<bar>u\<bar> \<longleftrightarrow> u = 1/2" for u::real
by (auto simp: algebra_simps)
have "b = midpoint a c \<Longrightarrow> collinear{a,b,c} "
using collinear_midpoint by blast
moreover have "collinear{a,b,c} \<Longrightarrow> b = midpoint a c \<longleftrightarrow> 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 \<or> x = b \<or> 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 \<longleftrightarrow> between (a,c) b \<and> 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 \<noteq> b"
shows "collinear(insert a (insert b S)) \<longleftrightarrow> (\<forall>x \<in> S. collinear{a,b,x})"
(is "?lhs = ?rhs")
proof safe
fix x
assume ?lhs and "x \<in> S"
then show "collinear {a, b, x}"
using collinear_subset by force
next
assume ?rhs
then have "\<forall>x \<in> S. collinear{a,x,b}"
by (simp add: insert_commute)
then have *: "\<exists>u. x = u *\<^sub>R a + (1 - u) *\<^sub>R b" if "x \<in> (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 \<noteq> b"
shows "collinear {a,b,c,d} \<longleftrightarrow> collinear{a,b,c} \<and> 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 \<noteq> 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 = {} \<longleftrightarrow> S = {}"
using affine_hull_nonempty by blast
lemma affine_hull_2_alt:
fixes a b :: "'a::real_vector"
shows "affine hull {a,b} = range (\<lambda>u. a + u *\<^sub>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 "\<lbrakk>\<not> collinear{a,b,c}; DIM('a) = 2\<rbrakk>
\<Longrightarrow> interior(convex hull {a,b,c}) =
{v. \<exists>x y z. 0 < x \<and> 0 < y \<and> 0 < z \<and> x + y + z = 1 \<and>
x *\<^sub>R a + y *\<^sub>R b + z *\<^sub>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="\<lambda>r. (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%unimportant\<open>Basic lemmas about hyperplanes and halfspaces\<close>
lemma halfspace_Int_eq:
"{x. a \<bullet> x \<le> b} \<inter> {x. b \<le> a \<bullet> x} = {x. a \<bullet> x = b}"
"{x. b \<le> a \<bullet> x} \<inter> {x. a \<bullet> x \<le> b} = {x. a \<bullet> x = b}"
by auto
lemma hyperplane_eq_Ex:
assumes "a \<noteq> 0" obtains x where "a \<bullet> x = b"
by (rule_tac x = "(b / (a \<bullet> a)) *\<^sub>R a" in that) (simp add: assms)
lemma hyperplane_eq_empty:
"{x. a \<bullet> x = b} = {} \<longleftrightarrow> a = 0 \<and> b \<noteq> 0"
using hyperplane_eq_Ex apply auto[1]
using inner_zero_right by blast
lemma hyperplane_eq_UNIV:
"{x. a \<bullet> x = b} = UNIV \<longleftrightarrow> a = 0 \<and> b = 0"
proof -
have "UNIV \<subseteq> {x. a \<bullet> x = b} \<Longrightarrow> a = 0 \<and> b = 0"
apply (drule_tac c = "((b+1) / (a \<bullet> a)) *\<^sub>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 \<bullet> x < b} = {} \<longleftrightarrow> a = 0 \<and> b \<le> 0"
proof -
have "{x. a \<bullet> x < b} \<subseteq> {} \<Longrightarrow> a = 0 \<and> b \<le> 0"
apply (rule ccontr)
apply (drule_tac c = "((b-1) / (a \<bullet> a)) *\<^sub>R a" in subsetD)
apply force+
done
then show ?thesis by force
qed
lemma halfspace_eq_empty_gt:
"{x. a \<bullet> x > b} = {} \<longleftrightarrow> a = 0 \<and> b \<ge> 0"
using halfspace_eq_empty_lt [of "-a" "-b"]
by simp
lemma halfspace_eq_empty_le:
"{x. a \<bullet> x \<le> b} = {} \<longleftrightarrow> a = 0 \<and> b < 0"
proof -
have "{x. a \<bullet> x \<le> b} \<subseteq> {} \<Longrightarrow> a = 0 \<and> b < 0"
apply (rule ccontr)
apply (drule_tac c = "((b-1) / (a \<bullet> a)) *\<^sub>R a" in subsetD)
apply force+
done
then show ?thesis by force
qed
lemma halfspace_eq_empty_ge:
"{x. a \<bullet> x \<ge> b} = {} \<longleftrightarrow> a = 0 \<and> b > 0"
using halfspace_eq_empty_le [of "-a" "-b"]
by simp
subsection%unimportant\<open>Use set distance for an easy proof of separation properties\<close>
proposition%unimportant separation_closures:
fixes S :: "'a::euclidean_space set"
assumes "S \<inter> closure T = {}" "T \<inter> closure S = {}"
obtains U V where "U \<inter> V = {}" "open U" "open V" "S \<subseteq> U" "T \<subseteq> V"
proof (cases "S = {} \<or> T = {}")
case True with that show ?thesis by auto
next
case False
define f where "f \<equiv> \<lambda>x. 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} \<inter> {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 \<subseteq> {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 \<subseteq> {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 \<inter> T = {}"
obtains U V where "open U" "open V" "S \<subseteq> U" "T \<subseteq> V" "U \<inter> 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 \<inter> T = {}"
obtains S' T' where "openin (subtopology euclidean U) S'"
"openin (subtopology euclidean U) T'"
"S \<subseteq> S'" "T \<subseteq> T'" "S' \<inter> T' = {}"
proof (cases "S = {} \<or> T = {}")
case True with that show ?thesis
using UT US by (blast dest: closedin_subset)
next
case False
define f where "f \<equiv> \<lambda>x. 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 \<inter> f -` {0<..})" and T' = "(U \<inter> f -` {..<0})" in that)
show "(U \<inter> f -` {0<..}) \<inter> (U \<inter> f -` {..<0}) = {}"
by auto
show "openin (subtopology euclidean U) (U \<inter> f -` {0<..})"
by (rule continuous_openin_preimage [where T=UNIV]) (simp_all add: contf)
next
show "openin (subtopology euclidean U) (U \<inter> f -` {..<0})"
by (rule continuous_openin_preimage [where T=UNIV]) (simp_all add: contf)
next
have "S \<subseteq> U" "T \<subseteq> U"
using closedin_imp_subset assms by blast+
then show "S \<subseteq> U \<inter> f -` {0<..}" "T \<subseteq> U \<inter> 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 \<inter> T = {}"
obtains U V where "open U" "compact(closure U)" "open V" "S \<subseteq> U" "T \<subseteq> V" "U \<inter> 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 \<subseteq> ball 0 r"
by (auto dest!: bounded_subset_ballD)
have **: "closed (T \<union> - ball 0 r)" "S \<inter> (T \<union> - ball 0 r) = {}"
using assms r by blast+
then show ?thesis
apply (rule separation_normal [OF \<open>closed S\<close>])
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\<open>Connectedness of the intersection of a chain\<close>
proposition connected_chain:
fixes \<F> :: "'a :: euclidean_space set set"
assumes cc: "\<And>S. S \<in> \<F> \<Longrightarrow> compact S \<and> connected S"
and linear: "\<And>S T. S \<in> \<F> \<and> T \<in> \<F> \<Longrightarrow> S \<subseteq> T \<or> T \<subseteq> S"
shows "connected(\<Inter>\<F>)"
proof (cases "\<F> = {}")
case True then show ?thesis
by auto
next
case False
then have cf: "compact(\<Inter>\<F>)"
by (simp add: cc compact_Inter)
have False if AB: "closed A" "closed B" "A \<inter> B = {}"
and ABeq: "A \<union> B = \<Inter>\<F>" and "A \<noteq> {}" "B \<noteq> {}" for A B
proof -
obtain U V where "open U" "open V" "A \<subseteq> U" "B \<subseteq> V" "U \<inter> V = {}"
using separation_normal [OF AB] by metis
obtain K where "K \<in> \<F>" "compact K"
using cc False by blast
then obtain N where "open N" and "K \<subseteq> N"
by blast
let ?\<C> = "insert (U \<union> V) ((\<lambda>S. N - S) ` \<F>)"
obtain \<D> where "\<D> \<subseteq> ?\<C>" "finite \<D>" "K \<subseteq> \<Union>\<D>"
proof (rule compactE [OF \<open>compact K\<close>])
show "K \<subseteq> \<Union>insert (U \<union> V) ((-) N ` \<F>)"
using \<open>K \<subseteq> N\<close> ABeq \<open>A \<subseteq> U\<close> \<open>B \<subseteq> V\<close> by auto
show "\<And>B. B \<in> insert (U \<union> V) ((-) N ` \<F>) \<Longrightarrow> open B"
by (auto simp: \<open>open U\<close> \<open>open V\<close> open_Un \<open>open N\<close> cc compact_imp_closed open_Diff)
qed
then have "finite(\<D> - {U \<union> V})"
by blast
moreover have "\<D> - {U \<union> V} \<subseteq> (\<lambda>S. N - S) ` \<F>"
using \<open>\<D> \<subseteq> ?\<C>\<close> by blast
ultimately obtain \<G> where "\<G> \<subseteq> \<F>" "finite \<G>" and Deq: "\<D> - {U \<union> V} = (\<lambda>S. N-S) ` \<G>"
using finite_subset_image by metis
obtain J where "J \<in> \<F>" and J: "(\<Union>S\<in>\<G>. N - S) \<subseteq> N - J"
proof (cases "\<G> = {}")
case True
with \<open>\<F> \<noteq> {}\<close> that show ?thesis
by auto
next
case False
have "\<And>S T. \<lbrakk>S \<in> \<G>; T \<in> \<G>\<rbrakk> \<Longrightarrow> S \<subseteq> T \<or> T \<subseteq> S"
by (meson \<open>\<G> \<subseteq> \<F>\<close> in_mono local.linear)
with \<open>finite \<G>\<close> \<open>\<G> \<noteq> {}\<close>
have "\<exists>J \<in> \<G>. (\<Union>S\<in>\<G>. N - S) \<subseteq> 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 "\<And>S T. \<lbrakk>S \<in> \<H>; T \<in> \<H>\<rbrakk> \<Longrightarrow> S \<subseteq> T \<or> T \<subseteq> S"
by (simp add: insert.prems)
with insert.IH False obtain J where "J \<in> \<H>" and J: "(\<Union>Y\<in>\<H>. N - Y) \<subseteq> N - J"
by metis
have "N - J \<subseteq> N - X \<or> N - X \<subseteq> N - J"
by (meson Diff_mono \<open>J \<in> \<H>\<close> insert.prems(2) insert_iff order_refl)
then show ?thesis
proof
assume "N - J \<subseteq> N - X" with J show ?thesis
by auto
next
assume "N - X \<subseteq> N - J"
with J have "N - X \<union> \<Union> ((-) N ` \<H>) \<subseteq> N - J"
by auto
with \<open>J \<in> \<H>\<close> show ?thesis
by blast
qed
qed
qed simp
with \<open>\<G> \<subseteq> \<F>\<close> show ?thesis by (blast intro: that)
qed
have "K \<subseteq> \<Union>(insert (U \<union> V) (\<D> - {U \<union> V}))"
using \<open>K \<subseteq> \<Union>\<D>\<close> by auto
also have "... \<subseteq> (U \<union> V) \<union> (N - J)"
by (metis (no_types, hide_lams) Deq Un_subset_iff Un_upper2 J Union_insert order_trans sup_ge1)
finally have "J \<inter> K \<subseteq> U \<union> V"
by blast
moreover have "connected(J \<inter> K)"
by (metis Int_absorb1 \<open>J \<in> \<F>\<close> \<open>K \<in> \<F>\<close> cc inf.orderE local.linear)
moreover have "U \<inter> (J \<inter> K) \<noteq> {}"
using ABeq \<open>J \<in> \<F>\<close> \<open>K \<in> \<F>\<close> \<open>A \<noteq> {}\<close> \<open>A \<subseteq> U\<close> by blast
moreover have "V \<inter> (J \<inter> K) \<noteq> {}"
using ABeq \<open>J \<in> \<F>\<close> \<open>K \<in> \<F>\<close> \<open>B \<noteq> {}\<close> \<open>B \<subseteq> V\<close> by blast
ultimately show False
using connectedD [of "J \<inter> K" U V] \<open>open U\<close> \<open>open V\<close> \<open>U \<inter> V = {}\<close> 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 \<in> \<F>" "compact X"
and cc: "\<And>T. T \<in> \<F> \<Longrightarrow> closed T \<and> connected T"
and linear: "\<And>S T. S \<in> \<F> \<and> T \<in> \<F> \<Longrightarrow> S \<subseteq> T \<or> T \<subseteq> S"
shows "connected(\<Inter>\<F>)"
proof -
have "\<Inter>\<F> = (\<Inter>T\<in>\<F>. X \<inter> T)"
using X by blast
moreover have "connected (\<Inter>T\<in>\<F>. X \<inter> T)"
proof (rule connected_chain)
show "\<And>T. T \<in> (\<inter>) X ` \<F> \<Longrightarrow> compact T \<and> connected T"
using cc X by auto (metis inf.absorb2 inf.orderE local.linear)
show "\<And>S T. S \<in> (\<inter>) X ` \<F> \<and> T \<in> (\<inter>) X ` \<F> \<Longrightarrow> S \<subseteq> T \<or> T \<subseteq> S"
using local.linear by blast
qed
ultimately show ?thesis
by metis
qed
lemma connected_nest:
fixes S :: "'a::linorder \<Rightarrow> 'b::euclidean_space set"
assumes S: "\<And>n. compact(S n)" "\<And>n. connected(S n)"
and nest: "\<And>m n. m \<le> n \<Longrightarrow> S n \<subseteq> S m"
shows "connected(\<Inter> (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 \<Rightarrow> 'b::euclidean_space set"
assumes S: "\<And>n. closed(S n)" "\<And>n. connected(S n)" "compact(S k)"
and nest: "\<And>m n. m \<le> n \<Longrightarrow> S n \<subseteq> S m"
shows "connected(\<Inter> (range S))"
apply (rule connected_chain_gen [of "S k"])
using S apply auto
by (meson le_cases nest subsetCE)
subsection\<open>Proper maps, including projections out of compact sets\<close>
lemma finite_indexed_bound:
assumes A: "finite A" "\<And>x. x \<in> A \<Longrightarrow> \<exists>n::'a::linorder. P x n"
shows "\<exists>m. \<forall>x \<in> A. \<exists>k\<le>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 "\<forall>x \<in> A. \<exists>k\<le>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 \<Rightarrow> 'b::heine_borel"
assumes "closedin (subtopology euclidean S) K"
and com: "\<And>U. \<lbrakk>U \<subseteq> T; compact U\<rbrakk> \<Longrightarrow> compact (S \<inter> f -` U)"
and "f ` S \<subseteq> T"
shows "closedin (subtopology euclidean T) (f ` K)"
proof -
have "K \<subseteq> S"
using assms closedin_imp_subset by metis
obtain C where "closed C" and Keq: "K = S \<inter> C"
using assms by (auto simp: closedin_closed)
have *: "y \<in> f ` K" if "y \<in> T" and y: "y islimpt f ` K" for y
proof -
obtain h where "\<forall>n. (\<exists>x\<in>K. h n = f x) \<and> h n \<noteq> y" "inj h" and hlim: "(h \<longlongrightarrow> y) sequentially"
using \<open>y \<in> T\<close> y by (force simp: limpt_sequential_inj)
then obtain X where X: "\<And>n. X n \<in> K \<and> h n = f (X n) \<and> h n \<noteq> y"
by metis
then have fX: "\<And>n. f (X n) = h n"
by metis
have "compact (C \<inter> (S \<inter> f -` insert y (range (\<lambda>i. f(X(n + i))))))" for n
apply (rule closed_Int_compact [OF \<open>closed C\<close>])
apply (rule com)
using X \<open>K \<subseteq> S\<close> \<open>f ` S \<subseteq> T\<close> \<open>y \<in> T\<close> 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 \<in> K. f a \<in> insert y (range (\<lambda>i. f(X(n + i))))}" for n
by (simp add: Keq Int_def conj_commute)
have ne: "\<Inter>\<F> \<noteq> {}"
if "finite \<F>"
and \<F>: "\<And>t. t \<in> \<F> \<Longrightarrow>
(\<exists>n. t = {a \<in> K. f a \<in> insert y (range (\<lambda>i. f (X (n + i))))})"
for \<F>
proof -
obtain m where m: "\<And>t. t \<in> \<F> \<Longrightarrow> \<exists>k\<le>m. t = {a \<in> K. f a \<in> insert y (range (\<lambda>i. f (X (k + i))))}"
apply (rule exE)
apply (rule finite_indexed_bound [OF \<open>finite \<F>\<close> \<F>], assumption, force)
done
have "X m \<in> \<Inter>\<F>"
using X le_Suc_ex by (fastforce dest: m)
then show ?thesis by blast
qed
have "\<Inter>{{a. a \<in> K \<and> f a \<in> insert y (range (\<lambda>i. f(X(n + i))))} |n. n \<in> UNIV}
\<noteq> {}"
apply (rule compact_fip_Heine_Borel)
using comf apply force
using ne apply (simp add: subset_iff del: insert_iff)
done
then have "\<exists>x. x \<in> (\<Inter>n. {a \<in> K. f a \<in> insert y (range (\<lambda>i. f (X (n + i))))})"
by blast
then show ?thesis
apply (simp add: image_iff fX)
by (metis \<open>inj h\<close> 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 \<Rightarrow> 'b::heine_borel"
assumes f: "inj_on f S"
shows "continuous_on S f \<longleftrightarrow> (\<forall>T. compact T \<and> T \<subseteq> S \<longrightarrow> 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: "\<And>x. x \<in> S \<Longrightarrow> g (f x) = x"
by (metis inv_into_f_f f)
then have *: "(S \<inter> f -` U) = g ` U" if "U \<subseteq> f ` S" for U
using that by fastforce
have gfim: "g ` f ` S \<subseteq> S" using gf by auto
have **: "compact (f ` S \<inter> g -` C)" if C: "C \<subseteq> S" "compact C" for C
proof -
obtain h where "h C \<in> C \<and> h C \<notin> S \<or> compact (f ` C)"
by (force simp: C RHS)
moreover have "f ` C = (f ` S \<inter> 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\<open>Trivial fact: convexity equals connectedness for collinear sets\<close>
lemma convex_connected_collinear:
fixes S :: "'a::euclidean_space set"
assumes "collinear S"
shows "convex S \<longleftrightarrow> 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 \<in> S" by auto
have "collinear (affine hull S)"
by (simp add: assms collinear_affine_hull_collinear)
then obtain z where "z \<noteq> 0" "\<And>x. x \<in> affine hull S \<Longrightarrow> \<exists>c. x - a = c *\<^sub>R z"
by (meson \<open>a \<in> S\<close> collinear hull_inc)
then obtain f where f: "\<And>x. x \<in> affine hull S \<Longrightarrow> x - a = f x *\<^sub>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) *\<^sub>R z" if x: "x \<in> affine hull S" and y: "y \<in> affine hull S" for x y
proof -
have "f x *\<^sub>R z = x - a"
by (simp add: f hull_inc x)
moreover have "f y *\<^sub>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 \<open>z \<noteq> 0\<close> 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 \<in> S" "y \<in> S" "z \<in> closed_segment x y"
have False if "z \<notin> S"
proof -
have "f ` (closed_segment x y) = closed_segment (f x) (f y)"
apply (rule continuous_injective_image_segment_1)
apply (meson \<open>x \<in> S\<close> \<open>y \<in> S\<close> convex_affine_hull convex_contains_segment hull_inc continuous_on_subset [OF cont_f])
by (meson \<open>x \<in> S\<close> \<open>y \<in> S\<close> convex_affine_hull convex_contains_segment hull_inc inj_on_subset [OF inj_f])
then have fz: "f z \<in> closed_segment (f x) (f y)"
using \<open>z \<in> closed_segment x y\<close> by blast
have "z \<in> affine hull S"
by (meson \<open>x \<in> S\<close> \<open>y \<in> S\<close> \<open>z \<in> closed_segment x y\<close> convex_affine_hull convex_contains_segment hull_inc subset_eq)
then have fz_notin: "f z \<notin> f ` S"
using hull_subset inj_f inj_onD that by fastforce
moreover have "{..<f z} \<inter> f ` S \<noteq> {}" "{f z<..} \<inter> f ` S \<noteq> {}"
proof -
have "{..<f z} \<inter> f ` {x,y} \<noteq> {}" "{f z<..} \<inter> f ` {x,y} \<noteq> {}"
using fz fz_notin \<open>x \<in> S\<close> \<open>y \<in> S\<close>
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} \<inter> f ` S \<noteq> {}" "{f z<..} \<inter> f ` S \<noteq> {}"
using \<open>x \<in> S\<close> \<open>y \<in> S\<close> by blast+
qed
ultimately show False
using connectedD [OF conn_fS, of "{..<f z}" "{f z<..}"] by force
qed
then show "z \<in> S" by meson
qed
qed
qed
lemma compact_convex_collinear_segment_alt:
fixes S :: "'a::euclidean_space set"
assumes "S \<noteq> {}" "compact S" "connected S" "collinear S"
obtains a b where "S = closed_segment a b"
proof -
obtain \<xi> where "\<xi> \<in> S" using \<open>S \<noteq> {}\<close> by auto
have "collinear (affine hull S)"
by (simp add: assms collinear_affine_hull_collinear)
then obtain z where "z \<noteq> 0" "\<And>x. x \<in> affine hull S \<Longrightarrow> \<exists>c. x - \<xi> = c *\<^sub>R z"
by (meson \<open>\<xi> \<in> S\<close> collinear hull_inc)
then obtain f where f: "\<And>x. x \<in> affine hull S \<Longrightarrow> x - \<xi> = f x *\<^sub>R z"
by metis
let ?g = "\<lambda>r. r *\<^sub>R z + \<xi>"
have gf: "?g (f x) = x" if "x \<in> 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) *\<^sub>R z" if x: "x \<in> affine hull S" and y: "y \<in> affine hull S" for x y
proof -
have "f x *\<^sub>R z = x - \<xi>"
by (simp add: f hull_inc x)
moreover have "f y *\<^sub>R z = y - \<xi>"
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 \<open>z \<noteq> 0\<close> 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 \<open>connected S\<close> connected_continuous_image continuous_on_subset hull_subset)
moreover have "compact (f ` S)"
by (meson \<open>compact S\<close> 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 \<open>S \<noteq> {}\<close> closed_segment_eq_real_ivl by auto
obtain a b where "a \<in> S" "f a = x" "b \<in> 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 \<open>a \<in> S\<close> \<open>b \<in> S\<close> convex_affine_hull convex_contains_segment hull_inc continuous_on_subset [OF cont_f])
by (meson \<open>a \<in> S\<close> \<open>b \<in> S\<close> 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: \<open>f a = x\<close> \<open>f b = y\<close> fS_eq)
then have "?g ` f ` (closed_segment a b) = ?g ` f ` S"
by simp
moreover have "(\<lambda>x. f x *\<^sub>R z + \<xi>) ` closed_segment a b = closed_segment a b"
apply safe
apply (metis (mono_tags, hide_lams) \<open>a \<in> S\<close> \<open>b \<in> S\<close> convex_affine_hull convex_contains_segment gf hull_inc subsetCE)
by (metis (mono_tags, lifting) \<open>a \<in> S\<close> \<open>b \<in> S\<close> 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 \<noteq> {}" "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 \<Rightarrow> 'b::euclidean_space"
assumes contf: "continuous_on S f" and imf: "f ` S \<subseteq> T" and "compact S"
"closedin (subtopology euclidean T) K"
shows "compact (S \<inter> f -` K)"
by (rule closedin_compact [OF \<open>compact S\<close>] continuous_closedin_preimage_gen assms)+
lemma proper_map_fst:
assumes "compact T" "K \<subseteq> S" "compact K"
shows "compact (S \<times> T \<inter> fst -` K)"
proof -
have "(S \<times> T \<inter> fst -` K) = K \<times> 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 \<times> T)) c"
shows "closedin (subtopology euclidean S) (fst ` c)"
proof -
have *: "fst ` (S \<times> T) \<subseteq> 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 \<subseteq> T" "compact K"
shows "compact (S \<times> T \<inter> snd -` K)"
proof -
have "(S \<times> T \<inter> snd -` K) = S \<times> 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 \<times> T)) c"
shows "closedin (subtopology euclidean T) (snd ` c)"
proof -
have *: "snd ` (S \<times> T) \<subseteq> 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 \<times> T)) U"
shows "closedin (subtopology euclidean T) {y. \<exists>x. x \<in> S \<and> (x, y) \<in> U}"
proof -
have "U \<subseteq> S \<times> T"
by (metis clo closedin_imp_subset)
then have "{y. \<exists>x. x \<in> S \<and> (x, y) \<in> 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. \<exists>x. x \<in> S \<and> (x, y) \<in> T}"
proof -
have *: "{y. \<exists>x. x \<in> S \<and> Pair x y \<in> T} =
{y. \<exists>x. x \<in> S \<and> Pair x y \<in> ((S \<times> UNIV) \<inter> T)}"
by auto
show ?thesis
apply (subst *)
apply (rule closedin_closed_trans [OF _ closed_UNIV])
apply (rule closedin_compact_projection [OF \<open>compact S\<close>])
by (simp add: clo closedin_closed_Int)
qed
subsubsection%unimportant\<open>Representing affine hull as a finite intersection of hyperplanes\<close>
proposition%unimportant affine_hull_convex_Int_nonempty_interior:
fixes S :: "'a::real_normed_vector set"
assumes "convex S" "S \<inter> interior T \<noteq> {}"
shows "affine hull (S \<inter> T) = affine hull S"
proof
show "affine hull (S \<inter> T) \<subseteq> affine hull S"
by (simp add: hull_mono)
next
obtain a where "a \<in> S" "a \<in> T" and at: "a \<in> interior T"
using assms interior_subset by blast
then obtain e where "e > 0" and e: "cball a e \<subseteq> T"
using mem_interior_cball by blast
have *: "x \<in> (+) a ` span ((\<lambda>x. x - a) ` (S \<inter> T))" if "x \<in> 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 \<open>e > 0\<close> False by (auto simp: k_def)
then have xa: "(x-a) = inverse k *\<^sub>R k *\<^sub>R (x-a)"
by simp
have "e / norm (x - a) \<ge> k"
using k_def by linarith
then have "a + k *\<^sub>R (x - a) \<in> cball a e"
using \<open>0 < k\<close> False by (simp add: dist_norm field_simps)
then have T: "a + k *\<^sub>R (x - a) \<in> T"
using e by blast
have S: "a + k *\<^sub>R (x - a) \<in> S"
using k \<open>a \<in> S\<close> convexD [OF \<open>convex S\<close> \<open>a \<in> S\<close> \<open>x \<in> S\<close>, of "1-k" k]
by (simp add: algebra_simps)
have "inverse k *\<^sub>R k *\<^sub>R (x-a) \<in> span ((\<lambda>x. x - a) ` (S \<inter> T))"
apply (rule span_mul)
apply (rule span_base)
apply (rule image_eqI [where x = "a + k *\<^sub>R (x - a)"])
apply (auto simp: S T)
done
with xa image_iff show ?thesis by fastforce
qed
show "affine hull S \<subseteq> affine hull (S \<inter> T)"
apply (simp add: subset_hull)
apply (simp add: \<open>a \<in> S\<close> \<open>a \<in> T\<close> 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 \<inter> T \<noteq> {}"
shows "affine hull (S \<inter> 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 \<inter> interior T \<noteq> {}"
shows "affine hull (S \<inter> 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 \<inter> T \<noteq> {}"
shows "affine hull (S \<inter> 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 \<inter> T \<noteq> {}"
shows "affine hull (S \<inter> 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 \<noteq> {}"
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 \<noteq> {}"
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 "\<lbrakk>convex S; S \<inter> interior T \<noteq> {}\<rbrakk> \<Longrightarrow> aff_dim(S \<inter> 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 "\<lbrakk>convex S; open T; S \<inter> T \<noteq> {}\<rbrakk> \<Longrightarrow> aff_dim(S \<inter> 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 \<subset> 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 \<noteq> {}"
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 \<bullet> x < r} = (if a = 0 \<and> r \<le> 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 \<bullet> x \<le> r} = (if a = 0 \<and> 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 \<bullet> x < r} = UNIV"
using affine_hull_halfspace_lt closure_same_affine_hull by fastforce
moreover have "{x. a \<bullet> x < r} \<subseteq> {x. a \<bullet> x \<le> 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 \<bullet> x > r} = (if a = 0 \<and> r \<ge> 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 \<bullet> x \<ge> r} = (if a = 0 \<and> 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 \<bullet> x < r} =
(if a = 0 \<and> r \<le> 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 \<bullet> x \<le> r} =
(if a = 0 \<and> 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 \<bullet> x \<le> r}) = DIM('a)" if "(a = 0 \<longrightarrow> r \<ge> 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 \<bullet> x > r} =
(if a = 0 \<and> r \<ge> 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 \<bullet> x \<ge> r} =
(if a = 0 \<and> r > 0 then -1 else DIM('a))"
using aff_dim_halfspace_le [of "-a" "-r"] by simp
proposition aff_dim_eq_hyperplane:
fixes S :: "'a::euclidean_space set"
shows "aff_dim S = DIM('a) - 1 \<longleftrightarrow> (\<exists>a b. a \<noteq> 0 \<and> affine hull S = {x. a \<bullet> 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 \<in> S" by blast
show ?thesis
proof (cases "c = 0")
case True show ?thesis
using span_zero [of S]
apply (simp add: aff_dim_eq_dim [of c] affine_hull_span_gen [of c] \<open>c \<in> S\<close> hull_inc dim_eq_hyperplane
del: One_nat_def)
apply (auto simp add: \<open>c = 0\<close>)
done
next
case False
have xc_im: "x \<in> (+) c ` {y. a \<bullet> y = 0}" if "a \<bullet> x = a \<bullet> c" for a x
proof -
have "\<exists>y. a \<bullet> y = 0 \<and> c + y = x"
by (metis that add.commute diff_add_cancel inner_commute inner_diff_left right_minus_eq)
then show "x \<in> (+) c ` {y. a \<bullet> y = 0}"
by blast
qed
have 2: "span ((\<lambda>x. x - c) ` S) = {x. a \<bullet> x = 0}"
if "(+) c ` span ((\<lambda>x. x - c) ` S) = {x. a \<bullet> x = b}" for a b
proof -
have "b = a \<bullet> c"
using span_0 that by fastforce
with that have "(+) c ` span ((\<lambda>x. x - c) ` S) = {x. a \<bullet> x = a \<bullet> c}"
by simp
then have "span ((\<lambda>x. x - c) ` S) = (\<lambda>x. x - c) ` {x. a \<bullet> x = a \<bullet> c}"
by (metis (no_types) image_cong translation_galois uminus_add_conv_diff)
also have "... = {x. a \<bullet> 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] \<open>c \<in> S\<close> hull_inc dim_eq_hyperplane
del: One_nat_def cong: image_cong_simp, 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 \<noteq> 0 \<Longrightarrow> aff_dim {x. a \<bullet> x = r} = DIM('a) - 1"
by (metis aff_dim_eq_hyperplane affine_hull_eq affine_hyperplane)
subsection%unimportant\<open>Some stepping theorems\<close>
lemma aff_dim_insert:
fixes a :: "'a::euclidean_space"
shows "aff_dim (insert a S) = (if a \<in> 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 \<notin> 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
cong: image_cong_simp)
qed
lemma affine_dependent_choose:
fixes a :: "'a :: euclidean_space"
assumes "\<not>(affine_dependent S)"
shows "affine_dependent(insert a S) \<longleftrightarrow> a \<notin> S \<and> a \<in> affine hull S"
(is "?lhs = ?rhs")
proof safe
assume "affine_dependent (insert a S)" and "a \<in> S"
then show "False"
using \<open>a \<in> S\<close> assms insert_absorb by fastforce
next
assume lhs: "affine_dependent (insert a S)"
then have "a \<notin> 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) \<noteq> int (card S)"
using \<open>finite S\<close> affine_independent_iff_card \<open>a \<notin> S\<close> lhs by fastforce
ultimately show "a \<in> affine hull S"
by (metis aff_dim_affine_independent aff_dim_insert assms)
next
assume "a \<notin> S" and "a \<in> affine hull S"
show "affine_dependent (insert a S)"
by (simp add: \<open>a \<in> affine hull S\<close> \<open>a \<notin> S\<close> affine_dependent_def)
qed
lemma affine_independent_insert:
fixes a :: "'a :: euclidean_space"
shows "\<lbrakk>\<not> affine_dependent S; a \<notin> affine hull S\<rbrakk> \<Longrightarrow> \<not> 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 \<longleftrightarrow> S = {0}"
proof -
have "False" if "bounded S" "x \<in> S" "x \<noteq> 0" for x
proof -
obtain B where B: "\<And>y. y \<in> S \<Longrightarrow> norm y < B" "B > 0"
using \<open>bounded S\<close> by (force simp: bounded_pos_less)
have "(B / norm x) *\<^sub>R x \<in> S"
using assms subspace_mul \<open>x \<in> S\<close> by auto
moreover have "norm ((B / norm x) *\<^sub>R x) = B"
using that B by (simp add: algebra_simps)
ultimately show False using B by force
qed
then have "bounded S \<Longrightarrow> 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 \<longleftrightarrow> S = {} \<or> (\<exists>a. S = {a})"
proof (cases "S = {}")
case True then show ?thesis
by simp
next
case False
then obtain b where "b \<in> S" by blast
with False assms show ?thesis
apply safe
using affine_diffs_subspace [OF assms \<open>b \<in> S\<close>]
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 \<longleftrightarrow> aff_dim S \<le> 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 \<noteq> 0"
shows "bounded {x. a \<bullet> x = 0} \<longleftrightarrow> DIM('a) = 1"
proof
assume "bounded {x. a \<bullet> x = 0}"
then have "aff_dim {x. a \<bullet> x = 0} \<le> 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 \<bullet> 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 \<bullet> x = r} \<longleftrightarrow> (if a = 0 then r \<noteq> 0 else DIM('a) = 1)"
proof (simp add: bounded_hyperplane_eq_trivial_0, clarify)
assume "r \<noteq> 0" "a \<noteq> 0"
have "aff_dim {x. y \<bullet> x = 0} = aff_dim {x. a \<bullet> x = r}" if "y \<noteq> 0" for y::'a
by (metis that \<open>a \<noteq> 0\<close> aff_dim_hyperplane)
then show "bounded {x. a \<bullet> x = r} = (DIM('a) = Suc 0)"
by (metis One_nat_def \<open>a \<noteq> 0\<close> affine_bounded_eq_lowdim affine_hyperplane bounded_hyperplane_eq_trivial_0)
qed
subsection%unimportant\<open>General case without assuming closure and getting non-strict separation\<close>
proposition%unimportant separating_hyperplane_closed_point_inset:
fixes S :: "'a::euclidean_space set"
assumes "convex S" "closed S" "S \<noteq> {}" "z \<notin> S"
obtains a b where "a \<in> S" "(a - z) \<bullet> z < b" "\<And>x. x \<in> S \<Longrightarrow> b < (a - z) \<bullet> x"
proof -
obtain y where "y \<in> S" and y: "\<And>u. u \<in> S \<Longrightarrow> dist z y \<le> dist z u"
using distance_attains_inf [of S z] assms by auto
then have *: "(y - z) \<bullet> z < (y - z) \<bullet> z + (norm (y - z))\<^sup>2 / 2"
using \<open>y \<in> S\<close> \<open>z \<notin> S\<close> by auto
show ?thesis
proof (rule that [OF \<open>y \<in> S\<close> *])
fix x
assume "x \<in> S"
have yz: "0 < (y - z) \<bullet> (y - z)"
using \<open>y \<in> S\<close> \<open>z \<notin> S\<close> by auto
{ assume 0: "0 < ((z - y) \<bullet> (x - y))"
with any_closest_point_dot [OF \<open>convex S\<close> \<open>closed S\<close>]
have False
using y \<open>x \<in> S\<close> \<open>y \<in> S\<close> not_less by blast
}
then have "0 \<le> ((y - z) \<bullet> (x - y))"
by (force simp: not_less inner_diff_left)
with yz have "0 < 2 * ((y - z) \<bullet> (x - y)) + (y - z) \<bullet> (y - z)"
by (simp add: algebra_simps)
then show "(y - z) \<bullet> z + (norm (y - z))\<^sup>2 / 2 < (y - z) \<bullet> 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 \<noteq> {}" "0 \<notin> S"
obtains a b where "a \<in> S" "a \<noteq> 0" "0 < b" "\<And>x. x \<in> S \<Longrightarrow> a \<bullet> x > b"
using separating_hyperplane_closed_point_inset [OF assms]
by simp (metis \<open>0 \<notin> S\<close>)
proposition%unimportant separating_hyperplane_set_0_inspan:
fixes S :: "'a::euclidean_space set"
assumes "convex S" "S \<noteq> {}" "0 \<notin> S"
obtains a where "a \<in> span S" "a \<noteq> 0" "\<And>x. x \<in> S \<Longrightarrow> 0 \<le> a \<bullet> x"
proof -
define k where [abs_def]: "k c = {x. 0 \<le> c \<bullet> x}" for c :: 'a
have *: "span S \<inter> frontier (cball 0 1) \<inter> \<Inter>f' \<noteq> {}"
if f': "finite f'" "f' \<subseteq> k ` S" for f'
proof -
obtain C where "C \<subseteq> S" "finite C" and C: "f' = k ` C"
using finite_subset_image [OF f'] by blast
obtain a where "a \<in> S" "a \<noteq> 0"
using \<open>S \<noteq> {}\<close> \<open>0 \<notin> S\<close> ex_in_conv by blast
then have "norm (a /\<^sub>R (norm a)) = 1"
by simp
moreover have "a /\<^sub>R (norm a) \<in> span S"
by (simp add: \<open>a \<in> S\<close> span_scale span_base)
ultimately have ass: "a /\<^sub>R (norm a) \<in> span S \<inter> 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 \<open>finite C\<close> compact_eq_bounded_closed finite_imp_compact_convex_hull by auto
moreover have "convex hull C \<noteq> {}"
by (simp add: False)
moreover have "0 \<notin> convex hull C"
by (metis \<open>C \<subseteq> S\<close> \<open>convex S\<close> \<open>0 \<notin> S\<close> convex_hull_subset hull_same insert_absorb insert_subset)
ultimately obtain a b
where "a \<in> convex hull C" "a \<noteq> 0" "0 < b"
and ab: "\<And>x. x \<in> convex hull C \<Longrightarrow> a \<bullet> x > b"
using separating_hyperplane_closed_0_inset by blast
then have "a \<in> S"
by (metis \<open>C \<subseteq> S\<close> assms(1) subsetCE subset_hull)
moreover have "norm (a /\<^sub>R (norm a)) = 1"
using \<open>a \<noteq> 0\<close> by simp
moreover have "a /\<^sub>R (norm a) \<in> span S"
by (simp add: \<open>a \<in> S\<close> span_scale span_base)
ultimately have ass: "a /\<^sub>R (norm a) \<in> span S \<inter> sphere 0 1"
by simp
have aa: "a /\<^sub>R (norm a) \<in> (\<Inter>c\<in>C. {x. 0 \<le> c \<bullet> x})"
apply (clarsimp simp add: divide_simps)
using ab \<open>0 < b\<close>
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 \<inter> frontier(cball 0 1)) \<inter> (\<Inter> (k ` S)) \<noteq> {}"
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 \<noteq> {}" and zno: "z \<notin> S"
obtains a b where "(z + a) \<in> affine hull (insert z S)"
and "a \<noteq> 0" and "a \<bullet> z \<le> b"
and "\<And>x. x \<in> S \<Longrightarrow> a \<bullet> x \<ge> b"
proof -
from separating_hyperplane_set_0_inspan [of "image (\<lambda>x. -z + x) S"]
have "convex ((+) (- z) ` S)"
using \<open>convex S\<close> by simp
moreover have "(+) (- z) ` S \<noteq> {}"
by (simp add: \<open>S \<noteq> {}\<close>)
moreover have "0 \<notin> (+) (- z) ` S"
using zno by auto
ultimately obtain a where "a \<in> span ((+) (- z) ` S)" "a \<noteq> 0"
and a: "\<And>x. x \<in> ((+) (- z) ` S) \<Longrightarrow> 0 \<le> a \<bullet> x"
using separating_hyperplane_set_0_inspan [of "image (\<lambda>x. -z + x) S"]
by blast
then have szx: "\<And>x. x \<in> S \<Longrightarrow> a \<bullet> z \<le> a \<bullet> 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 \<bullet> z" in that, simp_all)
using \<open>a \<in> span ((+) (- z) ` S)\<close> affine_hull_insert_span_gen apply blast
apply (simp_all add: \<open>a \<noteq> 0\<close> szx)
done
qed
proposition%unimportant supporting_hyperplane_rel_boundary:
fixes S :: "'a::euclidean_space set"
assumes "convex S" "x \<in> S" and xno: "x \<notin> rel_interior S"
obtains a where "a \<noteq> 0"
and "\<And>y. y \<in> S \<Longrightarrow> a \<bullet> x \<le> a \<bullet> y"
and "\<And>y. y \<in> rel_interior S \<Longrightarrow> a \<bullet> x < a \<bullet> y"
proof -
obtain a b where aff: "(x + a) \<in> affine hull (insert x (rel_interior S))"
and "a \<noteq> 0" and "a \<bullet> x \<le> b"
and ageb: "\<And>u. u \<in> (rel_interior S) \<Longrightarrow> a \<bullet> u \<ge> 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 \<bullet> x \<le> a \<bullet> y" if "y \<in> S" for y
proof -
have con: "continuous_on (closure (rel_interior S)) ((\<bullet>) a)"
by (rule continuous_intros continuous_on_subset | blast)+
have y: "y \<in> closure (rel_interior S)"
using \<open>convex S\<close> closure_def convex_closure_rel_interior \<open>y \<in> S\<close>
by fastforce
show ?thesis
using continuous_ge_on_closure [OF con y] ageb \<open>a \<bullet> x \<le> b\<close>
by fastforce
qed
have 3: "a \<bullet> x < a \<bullet> y" if "y \<in> rel_interior S" for y
proof -
obtain e where "0 < e" "y \<in> S" and e: "cball y e \<inter> affine hull S \<subseteq> S"
using \<open>y \<in> rel_interior S\<close> by (force simp: rel_interior_cball)
define y' where "y' = y - (e / norm a) *\<^sub>R ((x + a) - x)"
have "y' \<in> cball y e"
unfolding y'_def using \<open>0 < e\<close> by force
moreover have "y' \<in> affine hull S"
unfolding y'_def
by (metis \<open>x \<in> S\<close> \<open>y \<in> S\<close> \<open>convex S\<close> aff affine_affine_hull hull_redundant
rel_interior_same_affine_hull hull_inc mem_affine_3_minus2)
ultimately have "y' \<in> S"
using e by auto
have "a \<bullet> x \<le> a \<bullet> y"
using le_ay \<open>a \<noteq> 0\<close> \<open>y \<in> S\<close> by blast
moreover have "a \<bullet> x \<noteq> a \<bullet> y"
using le_ay [OF \<open>y' \<in> S\<close>] \<open>a \<noteq> 0\<close>
apply (simp add: y'_def inner_diff dot_square_norm power2_eq_square)
by (metis \<open>0 < e\<close> 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 \<open>a \<noteq> 0\<close> le_ay 3])
qed
lemma supporting_hyperplane_relative_frontier:
fixes S :: "'a::euclidean_space set"
assumes "convex S" "x \<in> closure S" "x \<notin> rel_interior S"
obtains a where "a \<noteq> 0"
and "\<And>y. y \<in> closure S \<Longrightarrow> a \<bullet> x \<le> a \<bullet> y"
and "\<And>y. y \<in> rel_interior S \<Longrightarrow> a \<bullet> x < a \<bullet> y"
using supporting_hyperplane_rel_boundary [of "closure S" x]
by (metis assms convex_closure convex_rel_interior_closure)
subsection%unimportant\<open> Some results on decomposing convex hulls: intersections, simplicial subdivision\<close>
lemma
fixes s :: "'a::euclidean_space set"
assumes "\<not> affine_dependent(s \<union> t)"
shows convex_hull_Int_subset: "convex hull s \<inter> convex hull t \<subseteq> convex hull (s \<inter> t)" (is ?C)
and affine_hull_Int_subset: "affine hull s \<inter> affine hull t \<subseteq> affine hull (s \<inter> t)" (is ?A)
proof -
have [simp]: "finite s" "finite t"
using aff_independent_finite assms by blast+
have "sum u (s \<inter> t) = 1 \<and>
(\<Sum>v\<in>s \<inter> t. u v *\<^sub>R v) = (\<Sum>v\<in>s. u v *\<^sub>R v)"
if [simp]: "sum u s = 1"
"sum v t = 1"
and eq: "(\<Sum>x\<in>t. v x *\<^sub>R x) = (\<Sum>x\<in>s. u x *\<^sub>R x)" for u v
proof -
define f where "f x = (if x \<in> s then u x else 0) - (if x \<in> t then v x else 0)" for x
have "sum f (s \<union> t) = 0"
apply (simp add: f_def sum_Un sum_subtractf)
apply (simp add: sum.inter_restrict [symmetric] Int_commute)
done
moreover have "(\<Sum>x\<in>(s \<union> t). f x *\<^sub>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 "\<And>v. v \<in> s \<union> t \<Longrightarrow> f v = 0"
using aff_independent_finite assms unfolding affine_dependent_explicit
by blast
then have u [simp]: "\<And>x. x \<in> s \<Longrightarrow> u x = (if x \<in> t then v x else 0)"
by (simp add: f_def) presburger
have "sum u (s \<inter> t) = sum u s"
by (simp add: sum.inter_restrict)
then have "sum u (s \<inter> t) = 1"
using that by linarith
moreover have "(\<Sum>v\<in>s \<inter> t. u v *\<^sub>R v) = (\<Sum>v\<in>s. u v *\<^sub>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%unimportant affine_hull_Int:
fixes s :: "'a::euclidean_space set"
assumes "\<not> affine_dependent(s \<union> t)"
shows "affine hull (s \<inter> t) = affine hull s \<inter> affine hull t"
apply (rule subset_antisym)
apply (simp add: hull_mono)
by (simp add: affine_hull_Int_subset assms)
proposition%unimportant convex_hull_Int:
fixes s :: "'a::euclidean_space set"
assumes "\<not> affine_dependent(s \<union> t)"
shows "convex hull (s \<inter> t) = convex hull s \<inter> convex hull t"
apply (rule subset_antisym)
apply (simp add: hull_mono)
by (simp add: convex_hull_Int_subset assms)
proposition%unimportant
fixes s :: "'a::euclidean_space set set"
assumes "\<not> affine_dependent (\<Union>s)"
shows affine_hull_Inter: "affine hull (\<Inter>s) = (\<Inter>t\<in>s. affine hull t)" (is "?A")
and convex_hull_Inter: "convex hull (\<Inter>s) = (\<Inter>t\<in>s. convex hull t)" (is "?C")
proof -
have "finite s"
using aff_independent_finite assms finite_UnionD by blast
then have "?A \<and> ?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]: "\<not> affine_dependent (t \<union> \<Inter>F)"
by (auto intro: affine_dependent_subset)
have [simp]: "\<not> affine_dependent (\<Union>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%unimportant in_convex_hull_exchange_unique:
fixes S :: "'a::euclidean_space set"
assumes naff: "\<not> affine_dependent S" and a: "a \<in> convex hull S"
and S: "T \<subseteq> S" "T' \<subseteq> S"
and x: "x \<in> convex hull (insert a T)"
and x': "x \<in> convex hull (insert a T')"
shows "x \<in> convex hull (insert a (T \<inter> T'))"
proof (cases "a \<in> S")
case True
then have "\<not> affine_dependent (insert a T \<union> insert a T')"
using affine_dependent_subset assms by auto
then have "x \<in> convex hull (insert a T \<inter> insert a T')"
by (metis IntI convex_hull_Int x x')
then show ?thesis
by simp
next
case False
then have anot: "a \<notin> T" "a \<notin> T'"
using assms by auto
have [simp]: "finite S"
by (simp add: aff_independent_finite assms)
then obtain b where b0: "\<And>s. s \<in> S \<Longrightarrow> 0 \<le> b s"
and b1: "sum b S = 1" and aeq: "a = (\<Sum>s\<in>S. b s *\<^sub>R s)"
using a by (auto simp: convex_hull_finite)
have fin [simp]: "finite T" "finite T'"
using assms infinite_super \<open>finite S\<close> by blast+
then obtain c c' where c0: "\<And>t. t \<in> insert a T \<Longrightarrow> 0 \<le> c t"
and c1: "sum c (insert a T) = 1"
and xeq: "x = (\<Sum>t \<in> insert a T. c t *\<^sub>R t)"
and c'0: "\<And>t. t \<in> insert a T' \<Longrightarrow> 0 \<le> c' t"
and c'1: "sum c' (insert a T') = 1"
and x'eq: "x = (\<Sum>t \<in> insert a T'. c' t *\<^sub>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: "(\<Sum>t \<in> T. c t *\<^sub>R t) = x - c a *\<^sub>R a"
by simp_all
have wsumT': "(\<Sum>t \<in> T'. c' t *\<^sub>R t) = x - c' a *\<^sub>R a"
using x'eq fin anot by simp
define cc where "cc \<equiv> \<lambda>x. if x \<in> T then c x else 0"
define cc' where "cc' \<equiv> \<lambda>x. if x \<in> T' then c' x else 0"
define dd where "dd \<equiv> \<lambda>x. 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: "(\<Sum>t \<in> S. cc t *\<^sub>R t) = x - c a *\<^sub>R a" "(\<Sum>t \<in> S. cc' t *\<^sub>R t) = x - c' a *\<^sub>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 "(\<Sum>v\<in>S. (b v * x) *\<^sub>R v) = x *\<^sub>R (\<Sum>v\<in>S. b v *\<^sub>R v)" for x
by (simp add: pth_5 real_vector.scale_sum_right mult.commute)
then have *: "(\<Sum>v\<in>S. (b v * x) *\<^sub>R v) = x *\<^sub>R a" for x
using aeq by blast
have "(\<Sum>v \<in> S. dd v *\<^sub>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 \<in> S" for v
using naff that \<open>finite S\<close> 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 \<le> c a" | "c a \<le> c' a" by linarith
then show ?thesis
proof cases
case 1
then have "sum cc S \<le> sum cc' S"
by (simp add: sumSS')
then have le: "cc x \<le> cc' x" if "x \<in> 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 \<in> S" "x \<notin> T \<inter> T'" for x
using le [OF \<open>x \<in> S\<close>] 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 "\<forall>x\<in>T \<inter> T'. 0 \<le> (cc(a := c a)) x"
by (simp add: c0 cc_def)
show "0 \<le> (cc(a := c a)) a"
by (simp add: c0)
have "sum (cc(a := c a)) (insert a (T \<inter> T')) = c a + sum (cc(a := c a)) (T \<inter> T')"
by (simp add: anot)
also have "... = c a + sum (cc(a := c a)) S"
apply simp
apply (rule sum.mono_neutral_left)
using \<open>T \<subseteq> S\<close> apply (auto simp: \<open>a \<notin> S\<close> cc0)
done
also have "... = c a + (1 - c a)"
by (metis \<open>a \<notin> S\<close> fun_upd_other sum.cong sumSS')
finally show "sum (cc(a := c a)) (insert a (T \<inter> T')) = 1"
by simp
have "(\<Sum>x\<in>insert a (T \<inter> T'). (cc(a := c a)) x *\<^sub>R x) = c a *\<^sub>R a + (\<Sum>x \<in> T \<inter> T'. (cc(a := c a)) x *\<^sub>R x)"
by (simp add: anot)
also have "... = c a *\<^sub>R a + (\<Sum>x \<in> S. (cc(a := c a)) x *\<^sub>R x)"
apply simp
apply (rule sum.mono_neutral_left)
using \<open>T \<subseteq> S\<close> apply (auto simp: \<open>a \<notin> S\<close> cc0)
done
also have "... = c a *\<^sub>R a + x - c a *\<^sub>R a"
by (simp add: wsumSS \<open>a \<notin> S\<close> if_smult sum_delta_notmem)
finally show "(\<Sum>x\<in>insert a (T \<inter> T'). (cc(a := c a)) x *\<^sub>R x) = x"
by simp
qed
next
case 2
then have "sum cc' S \<le> sum cc S"
by (simp add: sumSS')
then have le: "cc' x \<le> cc x" if "x \<in> 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 \<in> S" "x \<notin> T \<inter> T'" for x
using le [OF \<open>x \<in> S\<close>] 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 "\<forall>x\<in>T \<inter> T'. 0 \<le> (cc'(a := c' a)) x"
by (simp add: c'0 cc'_def)
show "0 \<le> (cc'(a := c' a)) a"
by (simp add: c'0)
have "sum (cc'(a := c' a)) (insert a (T \<inter> T')) = c' a + sum (cc'(a := c' a)) (T \<inter> T')"
by (simp add: anot)
also have "... = c' a + sum (cc'(a := c' a)) S"
apply simp
apply (rule sum.mono_neutral_left)
using \<open>T \<subseteq> S\<close> apply (auto simp: \<open>a \<notin> S\<close> cc0)
done
also have "... = c' a + (1 - c' a)"
by (metis \<open>a \<notin> S\<close> fun_upd_other sum.cong sumSS')
finally show "sum (cc'(a := c' a)) (insert a (T \<inter> T')) = 1"
by simp
have "(\<Sum>x\<in>insert a (T \<inter> T'). (cc'(a := c' a)) x *\<^sub>R x) = c' a *\<^sub>R a + (\<Sum>x \<in> T \<inter> T'. (cc'(a := c' a)) x *\<^sub>R x)"
by (simp add: anot)
also have "... = c' a *\<^sub>R a + (\<Sum>x \<in> S. (cc'(a := c' a)) x *\<^sub>R x)"
apply simp
apply (rule sum.mono_neutral_left)
using \<open>T \<subseteq> S\<close> apply (auto simp: \<open>a \<notin> S\<close> cc0)
done
also have "... = c a *\<^sub>R a + x - c a *\<^sub>R a"
by (simp add: wsumSS \<open>a \<notin> S\<close> if_smult sum_delta_notmem)
finally show "(\<Sum>x\<in>insert a (T \<inter> T'). (cc'(a := c' a)) x *\<^sub>R x) = x"
by simp
qed
qed
qed
corollary%unimportant convex_hull_exchange_Int:
fixes a :: "'a::euclidean_space"
assumes "\<not> affine_dependent S" "a \<in> convex hull S" "T \<subseteq> S" "T' \<subseteq> S"
shows "(convex hull (insert a T)) \<inter> (convex hull (insert a T')) =
convex hull (insert a (T \<inter> 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 \<in> closed_segment a c \<or> \<not> collinear{a,b,c}"
shows "closed_segment a b \<inter> 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 \<in> closed_segment a c"
moreover have "\<not> 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: "\<not> collinear {a, b, c}"
have False if "closed_segment a b \<inter> closed_segment b c \<noteq> {b}"
proof -
have "b \<in> closed_segment a b" and "b \<in> closed_segment b c"
by auto
with that obtain d where "b \<noteq> d" "d \<in> closed_segment a b" "d \<in> 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 _ _ \<open>b \<noteq> d\<close>])
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 = \<Inter>f"
"\<And>h. h \<in> f \<Longrightarrow> \<exists>a b. a \<noteq> 0 \<and> h = {x. a \<bullet> x = b}"
proof -
obtain b where "b \<subseteq> s"
and indb: "\<not> affine_dependent b"
and eq: "affine hull s = affine hull b"
using affine_basis_exists by blast
obtain c where indc: "\<not> affine_dependent c" and "b \<subseteq> 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 \<le> card c"
using \<open>b \<subseteq> c\<close> infinite_super by (auto simp: card_mono)
have imeq: "(\<lambda>x. affine hull x) ` ((\<lambda>a. c - {a}) ` (c - b)) = ((\<lambda>a. affine hull (c - {a})) ` (c - b))"
by blast
have card1: "card ((\<lambda>a. 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: \<open>\<not> affine_dependent c\<close> \<open>b \<subseteq> c\<close> 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: "\<not> affine_dependent (\<Union>a\<in>c - b. c - {a})"
by (rule affine_independent_subset [OF indc]) auto
have affeq: "affine hull s = (\<Inter>x\<in>(\<lambda>a. c - {a}) ` (c - b). affine hull x)"
using \<open>b \<subseteq> c\<close> 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 \<in> 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 = "(\<lambda>x. affine hull x) ` ((\<lambda>a. c - {a}) ` (c - b))" in that)
using \<open>finite c\<close> 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 \<in> S" and "w \<in> S"
and "a \<bullet> w \<noteq> 0"
shows "{x + y| x y. x \<in> S \<and> a \<bullet> y = 0} = UNIV"
proof -
have "subspace S"
by (simp add: assms subspace_affine)
have span1: "span {y. a \<bullet> y = 0} \<subseteq> span {x + y |x y. x \<in> S \<and> a \<bullet> y = 0}"
apply (rule span_mono)
using \<open>0 \<in> S\<close> add.left_neutral by force
have "w \<notin> span {y. a \<bullet> y = 0}"
using \<open>a \<bullet> w \<noteq> 0\<close> span_induct subspace_hyperplane by auto
moreover have "w \<in> span {x + y |x y. x \<in> S \<and> a \<bullet> y = 0}"
using \<open>w \<in> S\<close>
by (metis (mono_tags, lifting) inner_zero_right mem_Collect_eq pth_d span_base)
ultimately have span2: "span {y. a \<bullet> y = 0} \<noteq> span {x + y |x y. x \<in> S \<and> a \<bullet> y = 0}"
by blast
have "a \<noteq> 0" using assms inner_zero_left by blast
then have "DIM('a) - 1 = dim {y. a \<bullet> y = 0}"
by (simp add: dim_hyperplane)
also have "... < dim {x + y |x y. x \<in> S \<and> a \<bullet> y = 0}"
using span1 span2 by (blast intro: dim_psubset)
finally have DIM_lt: "DIM('a) - 1 < dim {x + y |x y. x \<in> S \<and> a \<bullet> y = 0}" .
have subs: "subspace {x + y| x y. x \<in> S \<and> a \<bullet> y = 0}"
using subspace_sums [OF \<open>subspace S\<close> subspace_hyperplane] by simp
moreover have "span {x + y| x y. x \<in> S \<and> a \<bullet> 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%unimportant affine_hyperplane_sums_eq_UNIV:
fixes S :: "'a :: euclidean_space set"
assumes "affine S"
and "S \<inter> {v. a \<bullet> v = b} \<noteq> {}"
and "S - {v. a \<bullet> v = b} \<noteq> {}"
shows "{x + y| x y. x \<in> S \<and> a \<bullet> 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 \<in> S" and c: "a \<bullet> c = b"
using assms by force
with affine_diffs_subspace [OF \<open>affine S\<close>]
have "subspace ((+) (- c) ` S)" by blast
then have aff: "affine ((+) (- c) ` S)"
by (simp add: subspace_imp_affine)
have 0: "0 \<in> (+) (- c) ` S"
by (simp add: \<open>c \<in> S\<close>)
obtain d where "d \<in> S" and "a \<bullet> d \<noteq> b" and dc: "d-c \<in> (+) (- c) ` S"
using assms by auto
then have adc: "a \<bullet> (d - c) \<noteq> 0"
by (simp add: c inner_diff_right)
let ?U = "(+) (c+c) ` {x + y |x y. x \<in> (+) (- c) ` S \<and> a \<bullet> y = 0}"
have "u + v \<in> (+) (c + c) ` {x + v |x v. x \<in> (+) (- c) ` S \<and> a \<bullet> v = 0}"
if "u \<in> S" "b = a \<bullet> 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 "\<lbrakk>a \<bullet> v = 0; u \<in> S\<rbrakk>
\<Longrightarrow> \<exists>x ya. v + (u + c) = x + ya \<and> x \<in> S \<and> a \<bullet> ya = b" for v u
by (metis add.left_commute c inner_right_distrib pth_d)
ultimately have "{x + y |x y. x \<in> S \<and> a \<bullet> y = b} = ?U"
by (fastforce simp: algebra_simps)
also have "... = range ((+) (c + c))"
by (simp only: affine_hyperplane_sums_eq_UNIV_0 [OF aff 0 dc adc])
also have "... = UNIV"
by simp
finally show ?thesis .
qed
lemma aff_dim_sums_Int_0:
assumes "affine S"
and "affine T"
and "0 \<in> S" "0 \<in> T"
shows "aff_dim {x + y| x y. x \<in> S \<and> y \<in> T} = (aff_dim S + aff_dim T) - aff_dim(S \<inter> T)"
proof -
have "0 \<in> {x + y |x y. x \<in> S \<and> y \<in> T}"
using assms by force
then have 0: "0 \<in> affine hull {x + y |x y. x \<in> S \<and> y \<in> 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 \<inter> T \<noteq> {}"
shows "aff_dim {x + y| x y. x \<in> S \<and> y \<in> T} = (aff_dim S + aff_dim T) - aff_dim(S \<inter> T)"
proof -
obtain a where a: "a \<in> S" "a \<in> T" using assms by force
have aff: "affine ((+) (-a) ` S)" "affine ((+) (-a) ` T)"
using affine_translation [symmetric, of "- a"] assms by (simp_all cong: image_cong_simp)
have zero: "0 \<in> ((+) (-a) ` S)" "0 \<in> ((+) (-a) ` T)"
using a assms by auto
have "{x + y |x y. x \<in> (+) (- a) ` S \<and> y \<in> (+) (- a) ` T} =
(+) (- 2 *\<^sub>R a) ` {x + y| x y. x \<in> S \<and> y \<in> T}"
by (force simp: algebra_simps scaleR_2)
moreover have "(+) (- a) ` S \<inter> (+) (- a) ` T = (+) (- a) ` (S \<inter> T)"
by auto
ultimately show ?thesis
using aff_dim_sums_Int_0 [OF aff zero] aff_dim_translation_eq
by (metis (lifting))
qed
lemma aff_dim_affine_Int_hyperplane:
fixes a :: "'a::euclidean_space"
assumes "affine S"
shows "aff_dim(S \<inter> {x. a \<bullet> x = b}) =
(if S \<inter> {v. a \<bullet> v = b} = {} then - 1
else if S \<subseteq> {v. a \<bullet> 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 \<inter> {x. a \<bullet> x = b}) = aff_dim S - 1"
if "x \<in> S" "a \<bullet> x \<noteq> b" and non: "S \<inter> {v. a \<bullet> v = b} \<noteq> {}" for x
proof -
have [simp]: "{x + y| x y. x \<in> S \<and> a \<bullet> 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) \<longleftrightarrow> (affine hull S \<noteq> UNIV)"
by (metis (no_types) aff_dim_affine_hull aff_dim_le_DIM aff_dim_UNIV affine_hull_UNIV less_le)
lemma aff_dim_openin:
fixes S :: "'a::euclidean_space set"
assumes ope: "openin (subtopology euclidean T) S" and "affine T" "S \<noteq> {}"
shows "aff_dim S = aff_dim T"
proof -
show ?thesis
proof (rule order_antisym)
show "aff_dim S \<le> aff_dim T"
by (blast intro: aff_dim_subset [OF openin_imp_subset] ope)
next
obtain a where "a \<in> S"
using \<open>S \<noteq> {}\<close> by blast
have "S \<subseteq> T"
using ope openin_imp_subset by auto
then have "a \<in> T"
using \<open>a \<in> S\<close> by auto
then have subT': "subspace ((\<lambda>x. - a + x) ` T)"
using affine_diffs_subspace \<open>affine T\<close> by auto
then obtain B where Bsub: "B \<subseteq> ((\<lambda>x. - a + x) ` T)" and po: "pairwise orthogonal B"
and eq1: "\<And>x. x \<in> B \<Longrightarrow> norm x = 1" and "independent B"
and cardB: "card B = dim ((\<lambda>x. - a + x) ` T)"
and spanB: "span B = ((\<lambda>x. - a + x) ` T)"
by (rule orthonormal_basis_subspace) auto
obtain e where "0 < e" and e: "cball a e \<inter> T \<subseteq> S"
by (meson \<open>a \<in> S\<close> openin_contains_cball ope)
have "aff_dim T = aff_dim ((\<lambda>x. - a + x) ` T)"
by (metis aff_dim_translation_eq)
also have "... = dim ((\<lambda>x. - a + x) ` T)"
using aff_dim_subspace subT' by blast
also have "... = card B"
by (simp add: cardB)
also have "... = card ((\<lambda>x. e *\<^sub>R x) ` B)"
using \<open>0 < e\<close> by (force simp: inj_on_def card_image)
also have "... \<le> dim ((\<lambda>x. - a + x) ` S)"
proof (simp, rule independent_card_le_dim)
have e': "cball 0 e \<inter> (\<lambda>x. x - a) ` T \<subseteq> (\<lambda>x. x - a) ` S"
using e by (auto simp: dist_norm norm_minus_commute subset_eq)
have "(\<lambda>x. e *\<^sub>R x) ` B \<subseteq> cball 0 e \<inter> (\<lambda>x. x - a) ` T"
using Bsub \<open>0 < e\<close> eq1 subT' \<open>a \<in> T\<close> by (auto simp: subspace_def)
then show "(\<lambda>x. e *\<^sub>R x) ` B \<subseteq> (\<lambda>x. x - a) ` S"
using e' by blast
show "independent ((\<lambda>x. e *\<^sub>R x) ` B)"
using linear_scale_self \<open>independent B\<close>
apply (rule linear_independent_injective_image)
using \<open>0 < e\<close> inj_on_def by fastforce
qed
also have "... = aff_dim S"
using \<open>a \<in> S\<close> aff_dim_eq_dim hull_inc by (force cong: image_cong_simp)
finally show "aff_dim T \<le> 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 \<noteq> {}"
shows "dim S = dim T"
proof (rule order_antisym)
show "dim S \<le> 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 \<open>subspace T\<close> \<open>S \<noteq> {}\<close> ope subspace_affine)
also have "... \<le> dim S"
by (metis aff_dim_subset aff_dim_subspace dim_span span_superset
subspace_span)
finally show "dim T \<le> dim S" by simp
qed
subsection\<open>Lower-dimensional affine subsets are nowhere dense\<close>
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 \<subseteq> S" for U
proof -
have "span U \<subset> span S"
by (metis neq_iff psubsetI span_eq_dim span_mono that)
then obtain a where "a \<noteq> 0" "a \<in> span S" and a: "\<And>y. y \<in> span U \<Longrightarrow> orthogonal a y"
using orthogonal_to_subspace_exists_gen by metis
show ?thesis
proof
have "closed S"
by (simp add: \<open>subspace S\<close> closed_subspace)
then show "closure (S - U) \<subseteq> S"
by (simp add: closure_minimal)
show "S \<subseteq> closure (S - U)"
proof (clarsimp simp: closure_approachable)
fix x and e::real
assume "x \<in> S" "0 < e"
show "\<exists>y\<in>S - U. dist y x < e"
proof (cases "x \<in> U")
case True
let ?y = "x + (e/2 / norm a) *\<^sub>R a"
show ?thesis
proof
show "dist ?y x < e"
using \<open>0 < e\<close> by (simp add: dist_norm)
next
have "?y \<in> S"
by (metis \<open>a \<in> span S\<close> \<open>x \<in> S\<close> assms(2) span_eq_iff subspace_add subspace_scale)
moreover have "?y \<notin> U"
proof -
have "e/2 / norm a \<noteq> 0"
using \<open>0 < e\<close> \<open>a \<noteq> 0\<close> by auto
then show ?thesis
by (metis True \<open>a \<noteq> 0\<close> a orthogonal_scaleR orthogonal_self real_vector.scale_eq_0_iff span_add_eq span_base)
qed
ultimately show "?y \<in> S - U" by blast
qed
next
case False
with \<open>0 < e\<close> \<open>x \<in> S\<close> show ?thesis by force
qed
qed
qed
qed
moreover have "S - S \<inter> T = S-T"
by blast
moreover have "dim (S \<inter> 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%unimportant 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 \<inter> T = {}")
case True
then show ?thesis
by (metis Diff_triv affine_hull_eq \<open>affine S\<close> closure_same_affine_hull closure_subset hull_subset subset_antisym)
next
case False
then obtain z where z: "z \<in> S \<inter> T" by blast
then have "subspace ((+) (- z) ` S)"
by (meson IntD1 affine_diffs_subspace \<open>affine S\<close>)
moreover have "int (dim ((+) (- z) ` T)) < int (dim ((+) (- z) ` S))"
thm aff_dim_eq_dim
using z less by (simp add: aff_dim_eq_dim_subtract [of z] hull_inc cong: image_cong_simp)
ultimately have "closure(((+) (- z) ` S) - ((+) (- z) ` T)) = ((+) (- z) ` S)"
by (simp add: dense_complement_subspace)
then show ?thesis
by (metis closure_translation translation_diff translation_invert)
qed
corollary%unimportant 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 \<subseteq> affine hull S"
by blast
then have "closure (S \<inter> closure (affine hull S - T)) = closure (S \<inter> (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%unimportant 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) \<subseteq> closure S"
by (simp add: 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 \<open>convex S\<close> rel_interior_rel_open rel_open by blast
then show "closure S \<subseteq> closure (S - T)"
by (metis Diff_mono \<open>convex S\<close> closure_mono convex_closure_rel_interior order_refl rel_interior_subset)
qed
corollary%unimportant 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\<open>Parallel slices, etc\<close>
text\<open> 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.\<close>
proposition%unimportant affine_parallel_slice:
fixes S :: "'a :: euclidean_space set"
assumes "affine S"
and "S \<inter> {x. a \<bullet> x \<le> b} \<noteq> {}"
and "\<not> (S \<subseteq> {x. a \<bullet> x \<le> b})"
obtains a' b' where "a' \<noteq> 0"
"S \<inter> {x. a' \<bullet> x \<le> b'} = S \<inter> {x. a \<bullet> x \<le> b}"
"S \<inter> {x. a' \<bullet> x = b'} = S \<inter> {x. a \<bullet> x = b}"
"\<And>w. w \<in> S \<Longrightarrow> (w + a') \<in> S"
proof (cases "S \<inter> {x. a \<bullet> x = b} = {}")
case True
then obtain u v where "u \<in> S" "v \<in> S" "a \<bullet> u \<le> b" "a \<bullet> v > b"
using assms by (auto simp: not_le)
define \<eta> where "\<eta> = u + ((b - a \<bullet> u) / (a \<bullet> v - a \<bullet> u)) *\<^sub>R (v - u)"
have "\<eta> \<in> S"
by (simp add: \<eta>_def \<open>u \<in> S\<close> \<open>v \<in> S\<close> \<open>affine S\<close> mem_affine_3_minus)
moreover have "a \<bullet> \<eta> = b"
using \<open>a \<bullet> u \<le> b\<close> \<open>b < a \<bullet> v\<close>
by (simp add: \<eta>_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 \<in> S" and z: "a \<bullet> z = b"
using assms by auto
with affine_diffs_subspace [OF \<open>affine S\<close>]
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' \<in> span ((+) (- z) ` S)" and a: "a = a' + a''"
and "\<And>w. w \<in> span ((+) (- z) ` S) \<Longrightarrow> orthogonal a'' w"
using orthogonal_subspace_decomp_exists [of "(+) (- z) ` S" "a"] by metis
then have "\<And>w. w \<in> S \<Longrightarrow> a'' \<bullet> (w-z) = 0"
by (simp add: span_base orthogonal_def)
then have a'': "\<And>w. w \<in> S \<Longrightarrow> a'' \<bullet> w = (a - a') \<bullet> z"
by (simp add: a inner_diff_right)
then have ba'': "\<And>w. w \<in> S \<Longrightarrow> a'' \<bullet> w = b - a' \<bullet> z"
by (simp add: inner_diff_left z)
have "\<And>w. w \<in> (+) (- z) ` S \<Longrightarrow> (w + a') \<in> (+) (- z) ` S"
by (metis subspace_add a' span_eq_iff sub)
then have Sclo: "\<And>w. w \<in> S \<Longrightarrow> (w + a') \<in> 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' \<bullet> z" in that)
apply (auto simp: a ba'' inner_left_distrib False Sclo)
done
qed
qed
lemma diffs_affine_hull_span:
assumes "a \<in> S"
shows "{x - a |x. x \<in> affine hull S} = span {x - a |x. x \<in> S}"
proof -
have *: "((\<lambda>x. x - a) ` (S - {a})) = {x. x + a \<in> 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 \<in> S"
shows "aff_dim S = dim {x - a |x. x \<in> S}"
proof -
obtain B where aff: "affine hull B = affine hull S"
and ind: "\<not> affine_dependent B"
and card: "of_nat (card B) = aff_dim S + 1"
using aff_dim_basis_exists by blast
then have "B \<noteq> {}" using assms
by (metis affine_hull_eq_empty ex_in_conv)
then obtain c where "c \<in> B" by auto
then have "c \<in> S"
by (metis aff affine_hull_eq \<open>affine S\<close> hull_inc)
have xy: "x - c = y - a \<longleftrightarrow> y = x + 1 *\<^sub>R (a - c)" for x y c and a::'a
by (auto simp: algebra_simps)
have *: "{x - c |x. x \<in> S} = {x - a |x. x \<in> S}"
apply safe
apply (simp_all only: xy)
using mem_affine_3_minus [OF \<open>affine S\<close>] \<open>a \<in> S\<close> \<open>c \<in> S\<close> apply blast+
done
have affS: "affine hull S = S"
by (simp add: \<open>affine S\<close>)
have "aff_dim S = of_nat (card B) - 1"
using card by simp
also have "... = dim {x - c |x. x \<in> B}"
by (simp add: affine_independent_card_dim_diffs [OF ind \<open>c \<in> B\<close>])
also have "... = dim {x - c | x. x \<in> affine hull B}"
by (simp add: diffs_affine_hull_span \<open>c \<in> B\<close> dim_span)
also have "... = dim {x - a |x. x \<in> S}"
by (simp add: affS aff *)
finally show ?thesis .
qed
lemma aff_dim_linear_image_le:
assumes "linear f"
shows "aff_dim(f ` S) \<le> aff_dim S"
proof -
have "aff_dim (f ` T) \<le> 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 \<in> T" by auto
have 1: "((\<lambda>x. x - f a) ` f ` T) = {x - f a |x. x \<in> f ` T}"
by auto
have 2: "{x - f a| x. x \<in> f ` T} = f ` {x - a| x. x \<in> T}"
by (force simp: linear_diff [OF assms])
have "aff_dim (f ` T) = int (dim {x - f a |x. x \<in> f ` T})"
by (simp add: \<open>a \<in> T\<close> hull_inc aff_dim_eq_dim [of "f a"] 1 cong: image_cong_simp)
also have "... = int (dim (f ` {x - a| x. x \<in> T}))"
by (force simp: linear_diff [OF assms] 2)
also have "... \<le> int (dim {x - a| x. x \<in> T})"
by (simp add: dim_image_le [OF assms])
also have "... \<le> aff_dim T"
by (simp add: aff_dim_dim_affine_diffs [symmetric] \<open>a \<in> T\<close> \<open>affine T\<close>)
finally show ?thesis .
qed
then
have "aff_dim (f ` (affine hull S)) \<le> 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) \<le> aff_dim S"
by (simp add: aff_dim_linear_image_le assms(1))
next
obtain g where "linear g" "g \<circ> f = id"
using assms(1) assms(2) linear_injective_left_inverse by blast
then have "aff_dim S \<le> aff_dim(g ` f ` S)"
by (simp add: image_comp)
also have "... \<le> aff_dim (f ` S)"
by (simp add: \<open>linear g\<close> aff_dim_linear_image_le)
finally show "aff_dim S \<le> aff_dim (f ` S)" .
qed
lemma choose_affine_subset:
assumes "affine S" "-1 \<le> d" and dle: "d \<le> aff_dim S"
obtains T where "affine T" "T \<subseteq> S" "aff_dim T = d"
proof (cases "d = -1 \<or> 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 \<in> S" "0 \<le> d" by auto
with assms have ss: "subspace ((+) (- a) ` S)"
by (simp add: affine_diffs_subspace_subtract cong: image_cong_simp)
have "nat d \<le> 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 \<subseteq> 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 \<subseteq> S"
proof -
have "T \<subseteq> (+) (- a) ` S"
by (metis (no_types) span_eq_iff Tsb ss)
then show "(+) a ` T \<subseteq> S"
using add_ac by auto
qed
moreover have "aff_dim ((+) a ` T) = d"
by (simp add: aff_dim_subspace Tdim \<open>0 \<le> d\<close> \<open>subspace T\<close> aff_dim_translation_eq)
ultimately show ?thesis
by (rule that)
qed
subsection\<open>Paracompactness\<close>
proposition paracompact:
fixes S :: "'a :: euclidean_space set"
assumes "S \<subseteq> \<Union>\<C>" and opC: "\<And>T. T \<in> \<C> \<Longrightarrow> open T"
obtains \<C>' where "S \<subseteq> \<Union> \<C>'"
and "\<And>U. U \<in> \<C>' \<Longrightarrow> open U \<and> (\<exists>T. T \<in> \<C> \<and> U \<subseteq> T)"
and "\<And>x. x \<in> S
\<Longrightarrow> \<exists>V. open V \<and> x \<in> V \<and>
finite {U. U \<in> \<C>' \<and> (U \<inter> V \<noteq> {})}"
proof (cases "S = {}")
case True with that show ?thesis by blast
next
case False
have "\<exists>T U. x \<in> U \<and> open U \<and> closure U \<subseteq> T \<and> T \<in> \<C>" if "x \<in> S" for x
proof -
obtain T where "x \<in> T" "T \<in> \<C>" "open T"
using assms \<open>x \<in> S\<close> by blast
then obtain e where "e > 0" "cball x e \<subseteq> 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 \<open>T \<in> \<C>\<close>
apply (simp add: closure_minimal)
done
qed
then obtain F G where Gin: "x \<in> G x" and oG: "open (G x)"
and clos: "closure (G x) \<subseteq> F x" and Fin: "F x \<in> \<C>"
if "x \<in> S" for x
by metis
then obtain \<F> where "\<F> \<subseteq> G ` S" "countable \<F>" "\<Union>\<F> = \<Union>(G ` S)"
using Lindelof [of "G ` S"] by (metis image_iff)
then obtain K where K: "K \<subseteq> S" "countable K" and eq: "\<Union>(G ` K) = \<Union>(G ` S)"
by (metis countable_subset_image)
with False Gin have "K \<noteq> {}" by force
then obtain a :: "nat \<Rightarrow> 'a" where "range a = K"
by (metis range_from_nat_into \<open>countable K\<close>)
then have odif: "\<And>n. open (F (a n) - \<Union>{closure (G (a m)) |m. m < n})"
using \<open>K \<subseteq> S\<close> Fin opC by (fastforce simp add:)
let ?C = "range (\<lambda>n. F(a n) - \<Union>{closure(G(a m)) |m. m < n})"
have enum_S: "\<exists>n. x \<in> F(a n) \<and> x \<in> G(a n)" if "x \<in> S" for x
proof -
have "\<exists>y \<in> K. x \<in> G y" using eq that Gin by fastforce
then show ?thesis
using clos K \<open>range a = K\<close> closure_subset by blast
qed
have 1: "S \<subseteq> Union ?C"
proof
fix x assume "x \<in> S"
define n where "n \<equiv> LEAST n. x \<in> F(a n)"
have n: "x \<in> F(a n)"
using enum_S [OF \<open>x \<in> S\<close>] by (force simp: n_def intro: LeastI)
have notn: "x \<notin> F(a m)" if "m < n" for m
using that not_less_Least by (force simp: n_def)
then have "x \<notin> \<Union>{closure (G (a m)) |m. m < n}"
using n \<open>K \<subseteq> S\<close> \<open>range a = K\<close> clos notn by fastforce
with n show "x \<in> Union ?C"
by blast
qed
have 3: "\<exists>V. open V \<and> x \<in> V \<and> finite {U. U \<in> ?C \<and> (U \<inter> V \<noteq> {})}" if "x \<in> S" for x
proof -
obtain n where n: "x \<in> F(a n)" "x \<in> G(a n)"
using \<open>x \<in> S\<close> enum_S by auto
have "{U \<in> ?C. U \<inter> G (a n) \<noteq> {}} \<subseteq> (\<lambda>n. F(a n) - \<Union>{closure(G(a m)) |m. m < n}) ` atMost n"
proof clarsimp
fix k assume "(F (a k) - \<Union>{closure (G (a m)) |m. m < k}) \<inter> G (a n) \<noteq> {}"
then have "k \<le> n"
by auto (metis closure_subset not_le subsetCE)
then show "F (a k) - \<Union>{closure (G (a m)) |m. m < k}
\<in> (\<lambda>n. F (a n) - \<Union>{closure (G (a m)) |m. m < n}) ` {..n}"
by force
qed
moreover have "finite ((\<lambda>n. F(a n) - \<Union>{closure(G(a m)) |m. m < n}) ` atMost n)"
by force
ultimately have *: "finite {U \<in> ?C. U \<inter> G (a n) \<noteq> {}}"
using finite_subset by blast
show ?thesis
apply (rule_tac x="G (a n)" in exI)
apply (intro conjI oG n *)
using \<open>K \<subseteq> S\<close> \<open>range a = K\<close> apply blast
done
qed
show ?thesis
apply (rule that [OF 1 _ 3])
using Fin \<open>K \<subseteq> S\<close> \<open>range a = K\<close> apply (auto simp: odif)
done
qed
corollary paracompact_closedin:
fixes S :: "'a :: euclidean_space set"
assumes cin: "closedin (subtopology euclidean U) S"
and oin: "\<And>T. T \<in> \<C> \<Longrightarrow> openin (subtopology euclidean U) T"
and "S \<subseteq> \<Union>\<C>"
obtains \<C>' where "S \<subseteq> \<Union> \<C>'"
and "\<And>V. V \<in> \<C>' \<Longrightarrow> openin (subtopology euclidean U) V \<and> (\<exists>T. T \<in> \<C> \<and> V \<subseteq> T)"
and "\<And>x. x \<in> U
\<Longrightarrow> \<exists>V. openin (subtopology euclidean U) V \<and> x \<in> V \<and>
finite {X. X \<in> \<C>' \<and> (X \<inter> V \<noteq> {})}"
proof -
have "\<exists>Z. open Z \<and> (T = U \<inter> Z)" if "T \<in> \<C>" for T
using oin [OF that] by (auto simp: openin_open)
then obtain F where opF: "open (F T)" and intF: "U \<inter> F T = T" if "T \<in> \<C>" for T
by metis
obtain K where K: "closed K" "U \<inter> K = S"
using cin by (auto simp: closedin_closed)
have 1: "U \<subseteq> \<Union>insert (- K) (F ` \<C>)"
by clarsimp (metis Int_iff Union_iff \<open>U \<inter> K = S\<close> \<open>S \<subseteq> \<Union>\<C>\<close> subsetD intF)
have 2: "\<And>T. T \<in> insert (- K) (F ` \<C>) \<Longrightarrow> open T"
using \<open>closed K\<close> by (auto simp: opF)
obtain \<D> where "U \<subseteq> \<Union>\<D>"
and D1: "\<And>U. U \<in> \<D> \<Longrightarrow> open U \<and> (\<exists>T. T \<in> insert (- K) (F ` \<C>) \<and> U \<subseteq> T)"
and D2: "\<And>x. x \<in> U \<Longrightarrow> \<exists>V. open V \<and> x \<in> V \<and> finite {U \<in> \<D>. U \<inter> V \<noteq> {}}"
using paracompact [OF 1 2] by auto
let ?C = "{U \<inter> V |V. V \<in> \<D> \<and> (V \<inter> K \<noteq> {})}"
show ?thesis
proof (rule_tac \<C>' = "{U \<inter> V |V. V \<in> \<D> \<and> (V \<inter> K \<noteq> {})}" in that)
show "S \<subseteq> \<Union>?C"
using \<open>U \<inter> K = S\<close> \<open>U \<subseteq> \<Union>\<D>\<close> K by (blast dest!: subsetD)
show "\<And>V. V \<in> ?C \<Longrightarrow> openin (subtopology euclidean U) V \<and> (\<exists>T. T \<in> \<C> \<and> V \<subseteq> T)"
using D1 intF by fastforce
have *: "{X. (\<exists>V. X = U \<inter> V \<and> V \<in> \<D> \<and> V \<inter> K \<noteq> {}) \<and> X \<inter> (U \<inter> V) \<noteq> {}} \<subseteq>
(\<lambda>x. U \<inter> x) ` {U \<in> \<D>. U \<inter> V \<noteq> {}}" for V
by blast
show "\<exists>V. openin (subtopology euclidean U) V \<and> x \<in> V \<and> finite {X \<in> ?C. X \<inter> V \<noteq> {}}"
if "x \<in> U" for x
using D2 [OF that]
apply clarify
apply (rule_tac x="U \<inter> V" in exI)
apply (auto intro: that finite_subset [OF *])
done
qed
qed
corollary%unimportant paracompact_closed:
fixes S :: "'a :: euclidean_space set"
assumes "closed S"
and opC: "\<And>T. T \<in> \<C> \<Longrightarrow> open T"
and "S \<subseteq> \<Union>\<C>"
obtains \<C>' where "S \<subseteq> \<Union>\<C>'"
and "\<And>U. U \<in> \<C>' \<Longrightarrow> open U \<and> (\<exists>T. T \<in> \<C> \<and> U \<subseteq> T)"
and "\<And>x. \<exists>V. open V \<and> x \<in> V \<and>
finite {U. U \<in> \<C>' \<and> (U \<inter> V \<noteq> {})}"
using paracompact_closedin [of UNIV S \<C>] assms by auto
subsection%unimportant\<open>Closed-graph characterization of continuity\<close>
lemma continuous_closed_graph_gen:
fixes T :: "'b::real_normed_vector set"
assumes contf: "continuous_on S f" and fim: "f ` S \<subseteq> T"
shows "closedin (subtopology euclidean (S \<times> T)) ((\<lambda>x. Pair x (f x)) ` S)"
proof -
have eq: "((\<lambda>x. Pair x (f x)) ` S) =(S \<times> T \<inter> (\<lambda>z. (f \<circ> 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 \<Rightarrow> 'b::euclidean_space"
assumes "compact T" and fim: "f ` S \<subseteq> T"
shows "continuous_on S f \<longleftrightarrow>
closedin (subtopology euclidean (S \<times> T)) ((\<lambda>x. 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 \<inter> f -` U) = fst ` (((\<lambda>x. Pair x (f x)) ` S) \<inter> (S \<times> U))"
by (force simp: image_iff)
show "closedin (subtopology euclidean S) (S \<inter> f -` U)"
by (simp add: U closedin_Int closedin_Times closed_map_fst [OF \<open>compact T\<close>] that eq)
qed
with continuous_closed_graph_gen assms show ?thesis by blast
qed
lemma continuous_closed_graph:
fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
assumes "closed S" and contf: "continuous_on S f"
shows "closed ((\<lambda>x. Pair x (f x)) ` S)"
apply (rule closedin_closed_trans)
apply (rule continuous_closed_graph_gen [OF contf subset_UNIV])
by (simp add: \<open>closed S\<close> closed_Times)
lemma continuous_from_closed_graph:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
assumes "compact T" and fim: "f ` S \<subseteq> T" and clo: "closed ((\<lambda>x. Pair x (f x)) ` S)"
shows "continuous_on S f"
using fim clo
by (auto intro: closed_subset simp: continuous_closed_graph_eq [OF \<open>compact T\<close> fim])
lemma continuous_on_Un_local_open:
assumes opS: "openin (subtopology euclidean (S \<union> T)) S"
and opT: "openin (subtopology euclidean (S \<union> T)) T"
and contf: "continuous_on S f" and contg: "continuous_on T f"
shows "continuous_on (S \<union> T) f"
using pasting_lemma [of "{S,T}" "S \<union> T" "\<lambda>i. i" "\<lambda>i. f" f] contf contg opS opT by blast
lemma continuous_on_cases_local_open:
assumes opS: "openin (subtopology euclidean (S \<union> T)) S"
and opT: "openin (subtopology euclidean (S \<union> T)) T"
and contf: "continuous_on S f" and contg: "continuous_on T g"
and fg: "\<And>x. x \<in> S \<and> \<not>P x \<or> x \<in> T \<and> P x \<Longrightarrow> f x = g x"
shows "continuous_on (S \<union> T) (\<lambda>x. if P x then f x else g x)"
proof -
have "\<And>x. x \<in> S \<Longrightarrow> (if P x then f x else g x) = f x" "\<And>x. x \<in> T \<Longrightarrow> (if P x then f x else g x) = g x"
by (simp_all add: fg)
then have "continuous_on S (\<lambda>x. if P x then f x else g x)" "continuous_on T (\<lambda>x. 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\<open>The union of two collinear segments is another segment\<close>
proposition%unimportant in_convex_hull_exchange:
fixes a :: "'a::euclidean_space"
assumes a: "a \<in> convex hull S" and xS: "x \<in> convex hull S"
obtains b where "b \<in> S" "x \<in> convex hull (insert a (S - {b}))"
proof (cases "a \<in> S")
case True
with xS insert_Diff that show ?thesis by fastforce
next
case False
show ?thesis
proof (cases "finite S \<and> card S \<le> Suc (DIM('a))")
case True
then obtain u where u0: "\<And>i. i \<in> S \<Longrightarrow> 0 \<le> u i" and u1: "sum u S = 1"
and ua: "(\<Sum>i\<in>S. u i *\<^sub>R i) = a"
using a by (auto simp: convex_hull_finite)
obtain v where v0: "\<And>i. i \<in> S \<Longrightarrow> 0 \<le> v i" and v1: "sum v S = 1"
and vx: "(\<Sum>i\<in>S. v i *\<^sub>R i) = x"
using True xS by (auto simp: convex_hull_finite)
show ?thesis
proof (cases "\<exists>b. b \<in> S \<and> v b = 0")
case True
then obtain b where b: "b \<in> S" "v b = 0"
by blast
show ?thesis
proof
have fin: "finite (insert a (S - {b}))"
using sum.infinite v1 by fastforce
show "x \<in> convex hull insert a (S - {b})"
unfolding convex_hull_finite [OF fin] mem_Collect_eq
proof (intro conjI exI ballI)
have "(\<Sum>x \<in> insert a (S - {b}). if x = a then 0 else v x) =
(\<Sum>x \<in> S - {b}. if x = a then 0 else v x)"
apply (rule sum.mono_neutral_right)
using fin by auto
also have "... = (\<Sum>x \<in> S - {b}. v x)"
using b False by (auto intro!: sum.cong split: if_split_asm)
also have "... = (\<Sum>x\<in>S. v x)"
by (metis \<open>v b = 0\<close> diff_zero sum.infinite sum_diff1 u1 zero_neq_one)
finally show "(\<Sum>x\<in>insert a (S - {b}). if x = a then 0 else v x) = 1"
by (simp add: v1)
show "\<And>x. x \<in> insert a (S - {b}) \<Longrightarrow> 0 \<le> (if x = a then 0 else v x)"
by (auto simp: v0)
have "(\<Sum>x \<in> insert a (S - {b}). (if x = a then 0 else v x) *\<^sub>R x) =
(\<Sum>x \<in> S - {b}. (if x = a then 0 else v x) *\<^sub>R x)"
apply (rule sum.mono_neutral_right)
using fin by auto
also have "... = (\<Sum>x \<in> S - {b}. v x *\<^sub>R x)"
using b False by (auto intro!: sum.cong split: if_split_asm)
also have "... = (\<Sum>x\<in>S. v x *\<^sub>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 "(\<Sum>x\<in>insert a (S - {b}). (if x = a then 0 else v x) *\<^sub>R x) = x"
by (simp add: vx)
qed
qed (rule \<open>b \<in> S\<close>)
next
case False
have le_Max: "u i / v i \<le> Max ((\<lambda>i. u i / v i) ` S)" if "i \<in> S" for i
by (simp add: True that)
have "Max ((\<lambda>i. u i / v i) ` S) \<in> (\<lambda>i. u i / v i) ` S"
using True v1 by (auto intro: Max_in)
then obtain b where "b \<in> S" and beq: "Max ((\<lambda>b. u b / v b) ` S) = u b / v b"
by blast
then have "0 \<noteq> 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 \<open>b \<in> S\<close> u0 v0 by force+
have fin: "finite (insert a (S - {b}))"
using sum.infinite v1 by fastforce
show ?thesis
proof
show "x \<in> convex hull insert a (S - {b})"
unfolding convex_hull_finite [OF fin] mem_Collect_eq
proof (intro conjI exI ballI)
have "(\<Sum>x \<in> 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 + (\<Sum>x \<in> S - {b}. v x - (v b / u b) * u x)"
using \<open>a \<notin> S\<close> \<open>b \<in> S\<close> True apply simp
apply (rule sum.cong, auto)
done
also have "... = v b / u b + (\<Sum>x \<in> S - {b}. v x) - (v b / u b) * (\<Sum>x \<in> S - {b}. u x)"
by (simp add: Groups_Big.sum_subtractf sum_distrib_left)
also have "... = (\<Sum>x\<in>S. v x)"
using \<open>0 < u b\<close> True by (simp add: Groups_Big.sum_diff1 u1 field_simps)
finally show "sum (\<lambda>x. 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 \<le> (if i = a then v b / u b else v i - v b / u b * u i)"
if "i \<in> insert a (S - {b})" for i
using \<open>0 < u b\<close> \<open>0 < v b\<close> v0 [of i] le_Max [of i] beq that False
by (auto simp: field_simps split: if_split_asm)
have "(\<Sum>x\<in>insert a (S - {b}). (if x=a then v b / u b else v x - v b / u b * u x) *\<^sub>R x) =
(v b / u b) *\<^sub>R a + (\<Sum>x\<in>S - {b}. (v x - v b / u b * u x) *\<^sub>R x)"
using \<open>a \<notin> S\<close> \<open>b \<in> S\<close> True apply simp
apply (rule sum.cong, auto)
done
also have "... = (v b / u b) *\<^sub>R a + (\<Sum>x \<in> S - {b}. v x *\<^sub>R x) - (v b / u b) *\<^sub>R (\<Sum>x \<in> S - {b}. u x *\<^sub>R x)"
by (simp add: Groups_Big.sum_subtractf scaleR_left_diff_distrib sum_distrib_left scale_sum_right)
also have "... = (\<Sum>x\<in>S. v x *\<^sub>R x)"
using \<open>0 < u b\<close> True by (simp add: ua vx Groups_Big.sum_diff1 algebra_simps)
finally
show "(\<Sum>x\<in>insert a (S - {b}). (if x=a then v b / u b else v x - v b / u b * u x) *\<^sub>R x) = x"
by (simp add: vx)
qed
qed (rule \<open>b \<in> S\<close>)
qed
next
case False
obtain T where "finite T" "T \<subseteq> S" and caT: "card T \<le> Suc (DIM('a))" and xT: "x \<in> convex hull T"
using xS by (auto simp: caratheodory [of S])
with False obtain b where b: "b \<in> S" "b \<notin> T"
by (metis antisym subsetI)
show ?thesis
proof
show "x \<in> convex hull insert a (S - {b})"
using \<open>T \<subseteq> S\<close> b by (blast intro: subsetD [OF hull_mono xT])
qed (rule \<open>b \<in> S\<close>)
qed
qed
lemma convex_hull_exchange_Union:
fixes a :: "'a::euclidean_space"
assumes "a \<in> convex hull S"
shows "convex hull S = (\<Union>b \<in> S. convex hull (insert a (S - {b})))" (is "?lhs = ?rhs")
proof
show "?lhs \<subseteq> ?rhs"
by (blast intro: in_convex_hull_exchange [OF assms])
show "?rhs \<subseteq> ?lhs"
proof clarify
fix x b
assume"b \<in> S" "x \<in> convex hull insert a (S - {b})"
then show "x \<in> convex hull S" if "b \<in> 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 \<in> closed_segment a c"
shows "closed_segment a b \<union> 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} \<union> convex hull {b, c} = (\<Union>ba\<in>{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 \<in> open_segment a c"
shows "open_segment a b \<union> {b} \<union> open_segment b c = open_segment a c"
proof -
have b: "b \<in> closed_segment a c"
by (simp add: assms open_closed_segment)
have *: "open_segment a c \<subseteq> insert b (open_segment a b \<union> open_segment b c)"
if "{b,c,a} \<union> open_segment a b \<union> open_segment b c = {c,a} \<union> open_segment a c"
proof -
have "insert a (insert c (insert b (open_segment a b \<union> 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\<open>Covering an open set by a countable chain of compact sets\<close>
proposition open_Union_compact_subsets:
fixes S :: "'a::euclidean_space set"
assumes "open S"
obtains C where "\<And>n. compact(C n)" "\<And>n. C n \<subseteq> S"
"\<And>n. C n \<subseteq> interior(C(Suc n))"
"\<Union>(range C) = S"
"\<And>K. \<lbrakk>compact K; K \<subseteq> S\<rbrakk> \<Longrightarrow> \<exists>N. \<forall>n\<ge>N. K \<subseteq> (C n)"
proof (cases "S = {}")
case True
then show ?thesis
by (rule_tac C = "\<lambda>n. {}" in that) auto
next
case False
then obtain a where "a \<in> S"
by auto
let ?C = "\<lambda>n. cball a (real n) - (\<Union>x \<in> -S. \<Union>e \<in> ball 0 (1 / real(Suc n)). {x + e})"
have "\<exists>N. \<forall>n\<ge>N. K \<subseteq> (f n)"
if "\<And>n. compact(f n)" and sub_int: "\<And>n. f n \<subseteq> interior (f(Suc n))"
and eq: "\<Union>(range f) = S" and "compact K" "K \<subseteq> S" for f K
proof -
have *: "\<forall>n. f n \<subseteq> (\<Union>n. interior (f n))"
by (meson Sup_upper2 UNIV_I \<open>\<And>n. f n \<subseteq> interior (f (Suc n))\<close> image_iff)
have mono: "\<And>m n. m \<le> n \<Longrightarrow>f m \<subseteq> f n"
by (meson dual_order.trans interior_subset lift_Suc_mono_le sub_int)
obtain I where "finite I" and I: "K \<subseteq> (\<Union>i\<in>I. interior (f i))"
proof (rule compactE_image [OF \<open>compact K\<close>])
show "K \<subseteq> (\<Union>n. interior (f n))"
using \<open>K \<subseteq> S\<close> \<open>\<Union>(f ` UNIV) = S\<close> * by blast
qed auto
{ fix n
assume n: "Max I \<le> n"
have "(\<Union>i\<in>I. interior (f i)) \<subseteq> f n"
by (rule UN_least) (meson dual_order.trans interior_subset mono I Max_ge [OF \<open>finite I\<close>] n)
then have "K \<subseteq> f n"
using I by auto
}
then show ?thesis
by blast
qed
moreover have "\<exists>f. (\<forall>n. compact(f n)) \<and> (\<forall>n. (f n) \<subseteq> S) \<and> (\<forall>n. (f n) \<subseteq> interior(f(Suc n))) \<and>
((\<Union>(range f) = S))"
proof (intro exI conjI allI)
show "\<And>n. compact (?C n)"
by (auto simp: compact_diff open_sums)
show "\<And>n. ?C n \<subseteq> S"
by auto
show "?C n \<subseteq> interior (?C (Suc n))" for n
proof (simp add: interior_diff, rule Diff_mono)
show "cball a (real n) \<subseteq> ball a (1 + real n)"
by (simp add: cball_subset_ball_iff)
have cl: "closed (\<Union>x\<in>- S. \<Union>e\<in>cball 0 (1 / (2 + real n)). {x + e})"
using assms by (auto intro: closed_compact_sums)
have "closure (\<Union>x\<in>- S. \<Union>y\<in>ball 0 (1 / (2 + real n)). {x + y})
\<subseteq> (\<Union>x \<in> -S. \<Union>e \<in> cball 0 (1 / (2 + real n)). {x + e})"
by (intro closure_minimal UN_mono ball_subset_cball order_refl cl)
also have "... \<subseteq> (\<Union>x \<in> -S. \<Union>y\<in>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 (\<Union>x\<in>- S. \<Union>y\<in>ball 0 (1 / (2 + real n)). {x + y})
\<subseteq> (\<Union>x \<in> -S. \<Union>y\<in>ball 0 (1 / (1 + real n)). {x + y})" .
qed
have "S \<subseteq> \<Union> (range ?C)"
proof
fix x
assume x: "x \<in> S"
then obtain e where "e > 0" and e: "ball x e \<subseteq> 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) \<le> real N2"
by (meson real_arch_simple)
have N12: "inverse((N1 + N2) + 1) \<le> inverse(N1)"
using \<open>N1 > 0\<close> by (auto simp: divide_simps)
have "x \<noteq> y + z" if "y \<notin> 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: \<open>0 < e\<close>)
then have "1 / (1 + (real N1 + real N2)) < e"
using N1 \<open>e > 0\<close>
by (metis divide_less_eq less_trans mult.commute of_nat_add of_nat_less_0_iff of_nat_Suc)
then have "x - z \<in> ball x e"
using that by simp
then have "x - z \<in> S"
using e by blast
with that show ?thesis
by auto
qed
with N2 show "x \<in> \<Union> (range ?C)"
by (rule_tac a = "N1+N2" in UN_I) (auto simp: dist_norm norm_minus_commute)
qed
then show "\<Union> (range ?C) = S" by auto
qed
ultimately show ?thesis
using that by metis
qed
subsection\<open>Orthogonal complement\<close>
definition%important orthogonal_comp ("_\<^sup>\<bottom>" [80] 80)
where "orthogonal_comp W \<equiv> {x. \<forall>y \<in> W. orthogonal y x}"
proposition subspace_orthogonal_comp: "subspace (W\<^sup>\<bottom>)"
unfolding subspace_def orthogonal_comp_def orthogonal_def
by (auto simp: inner_right_distrib)
lemma orthogonal_comp_anti_mono:
assumes "A \<subseteq> B"
shows "B\<^sup>\<bottom> \<subseteq> A\<^sup>\<bottom>"
proof
fix x assume x: "x \<in> B\<^sup>\<bottom>"
show "x \<in> 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}\<^sup>\<bottom> = UNIV"
by (auto simp: orthogonal_comp_def orthogonal_def)
lemma orthogonal_comp_UNIV [simp]: "UNIV\<^sup>\<bottom> = {0}"
unfolding orthogonal_comp_def orthogonal_def
by auto (use inner_eq_zero_iff in blast)
lemma orthogonal_comp_subset: "U \<subseteq> U\<^sup>\<bottom>\<^sup>\<bottom>"
by (auto simp: orthogonal_comp_def orthogonal_def inner_commute)
lemma subspace_sum_minimal:
assumes "S \<subseteq> U" "T \<subseteq> U" "subspace U"
shows "S + T \<subseteq> U"
proof
fix x
assume "x \<in> S + T"
then obtain xs xt where "xs \<in> S" "xt \<in> T" "x = xs+xt"
by (meson set_plus_elim)
then show "x \<in> U"
by (meson assms subsetCE subspace_add)
qed
proposition subspace_sum_orthogonal_comp:
fixes U :: "'a :: euclidean_space set"
assumes "subspace U"
shows "U + U\<^sup>\<bottom> = UNIV"
proof -
obtain B where "B \<subseteq> U"
and ortho: "pairwise orthogonal B" "\<And>x. x \<in> B \<Longrightarrow> 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 *: "\<forall>x\<in>B. \<forall>y\<in>B. x \<bullet> y = (if x=y then 1 else 0)"
using ortho norm_eq_1 by (auto simp: orthogonal_def pairwise_def)
{ fix v
let ?u = "\<Sum>b\<in>B. (v \<bullet> b) *\<^sub>R b"
have "v = ?u + (v - ?u)"
by simp
moreover have "?u \<in> U"
by (metis (no_types, lifting) \<open>span B = U\<close> assms subspace_sum span_base span_mul)
moreover have "(v - ?u) \<in> U\<^sup>\<bottom>"
proof (clarsimp simp: orthogonal_comp_def orthogonal_def)
fix y
assume "y \<in> U"
with \<open>span B = U\<close> span_finite [OF \<open>finite B\<close>]
obtain u where u: "y = (\<Sum>b\<in>B. u b *\<^sub>R b)"
by auto
have "b \<bullet> (v - ?u) = 0" if "b \<in> B" for b
using that \<open>finite B\<close>
by (simp add: * algebra_simps inner_sum_right if_distrib [of "(*)v" for v] inner_commute cong: if_cong)
then show "y \<bullet> (v - ?u) = 0"
by (simp add: u inner_sum_left)
qed
ultimately have "v \<in> U + U\<^sup>\<bottom>"
using set_plus_intro by fastforce
} then show ?thesis
by auto
qed
lemma orthogonal_Int_0:
assumes "subspace U"
shows "U \<inter> U\<^sup>\<bottom> = {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\<^sup>\<bottom>\<^sup>\<bottom> = U"
proof
have ssU': "subspace (U\<^sup>\<bottom>)"
by (simp add: subspace_orthogonal_comp)
have "u \<in> U" if "u \<in> U\<^sup>\<bottom>\<^sup>\<bottom>" for u
proof -
obtain v w where "u = v+w" "v \<in> U" "w \<in> U\<^sup>\<bottom>"
using subspace_sum_orthogonal_comp [OF assms] set_plus_elim by blast
then have "u-v \<in> U\<^sup>\<bottom>"
by simp
moreover have "v \<in> U\<^sup>\<bottom>\<^sup>\<bottom>"
using \<open>v \<in> U\<close> orthogonal_comp_subset by blast
then have "u-v \<in> U\<^sup>\<bottom>\<^sup>\<bottom>"
by (simp add: subspace_diff subspace_orthogonal_comp that)
ultimately have "u-v = 0"
using orthogonal_Int_0 ssU' by blast
with \<open>v \<in> U\<close> show ?thesis
by auto
qed
then show "U\<^sup>\<bottom>\<^sup>\<bottom> \<subseteq> U"
by auto
qed (use orthogonal_comp_subset in auto)
lemma ker_orthogonal_comp_adjoint:
fixes f :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space"
assumes "linear f"
shows "f -` {0} = (range (adjoint f))\<^sup>\<bottom>"
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%unimportant \<open>A non-injective linear function maps into a hyperplane.\<close>
lemma linear_surj_adj_imp_inj:
fixes f :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space"
assumes "linear f" "surj (adjoint f)"
shows "inj f"
proof -
have "\<exists>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
(*http://mathonline.wikidot.com/injectivity-and-surjectivity-of-the-adjoint-of-a-linear-map*)
lemma surj_adjoint_iff_inj [simp]:
fixes f :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space"
assumes "linear f"
shows "surj (adjoint f) \<longleftrightarrow> 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 \<open>inj f\<close> linear_0 linear_injective_0 by fastforce
moreover have "f -` {0} = range (adjoint f)\<^sup>\<bottom>"
by (intro ker_orthogonal_comp_adjoint assms)
ultimately have "range (adjoint f)\<^sup>\<bottom>\<^sup>\<bottom> = UNIV"
by (metis orthogonal_comp_null)
then show "surj (adjoint f)"
using adjoint_linear \<open>linear f\<close>
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 \<Rightarrow> 'n::euclidean_space"
assumes "linear f"
shows "inj (adjoint f) \<longleftrightarrow> surj f"
proof
assume "inj (adjoint f)"
have "(adjoint f) -` {0} = {0}"
by (metis \<open>inj (adjoint f)\<close> adjoint_linear assms surj_adjoint_iff_inj ker_orthogonal_comp_adjoint orthogonal_comp_UNIV)
then have "(range(f))\<^sup>\<bottom> = {0}"
by (metis (no_types, hide_lams) adjoint_adjoint adjoint_linear assms ker_orthogonal_comp_adjoint set_zero)
then show "surj f"
by (metis \<open>inj (adjoint f)\<close> adjoint_adjoint adjoint_linear assms surj_adjoint_iff_inj)
next
assume "surj f"
then have "range f = (adjoint f -` {0})\<^sup>\<bottom>"
by (simp add: adjoint_adjoint adjoint_linear assms ker_orthogonal_comp_adjoint)
then have "{0} = adjoint f -` {0}"
using \<open>surj f\<close> adjoint_adjoint adjoint_linear assms ker_orthogonal_comp_adjoint by force
then show "inj (adjoint f)"
by (simp add: \<open>surj f\<close> adjoint_adjoint adjoint_linear assms linear_surj_adj_imp_inj)
qed
lemma linear_singular_into_hyperplane:
fixes f :: "'n::euclidean_space \<Rightarrow> 'n"
assumes "linear f"
shows "\<not> inj f \<longleftrightarrow> (\<exists>a. a \<noteq> 0 \<and> (\<forall>x. a \<bullet> f x = 0))" (is "_ = ?rhs")
proof
assume "\<not>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 "\<not>inj f"
by (metis assms linear_injective_isomorphism all_zero_iff)
qed
lemma linear_singular_image_hyperplane:
fixes f :: "'n::euclidean_space \<Rightarrow> 'n"
assumes "linear f" "\<not>inj f"
obtains a where "a \<noteq> 0" "\<And>S. f ` S \<subseteq> {x. a \<bullet> x = 0}"
using assms by (fastforce simp add: linear_singular_into_hyperplane)
end