Convex hulls: theorems about interior, etc. And a few simple lemmas.
(* Title: HOL/Multivariate_Analysis/Convex_Euclidean_Space.thy
Author: Robert Himmelmann, TU Muenchen
Author: Bogdan Grechuk, University of Edinburgh
*)
section {* Convex sets, functions and related things. *}
theory Convex_Euclidean_Space
imports
Topology_Euclidean_Space
"~~/src/HOL/Library/Convex"
"~~/src/HOL/Library/Set_Algebras"
begin
(* ------------------------------------------------------------------------- *)
(* To be moved elsewhere *)
(* ------------------------------------------------------------------------- *)
lemma linear_scaleR: "linear (\<lambda>x. scaleR c x)"
by (simp add: linear_iff scaleR_add_right)
lemma linear_scaleR_left: "linear (\<lambda>r. scaleR r x)"
by (simp add: linear_iff scaleR_add_left)
lemma injective_scaleR: "c \<noteq> 0 \<Longrightarrow> inj (\<lambda>x::'a::real_vector. scaleR c x)"
by (simp add: inj_on_def)
lemma linear_add_cmul:
assumes "linear f"
shows "f (a *\<^sub>R x + b *\<^sub>R y) = a *\<^sub>R f x + b *\<^sub>R f y"
using linear_add[of f] linear_cmul[of f] assms by simp
lemma mem_convex_alt:
assumes "convex S" "x \<in> S" "y \<in> S" "u \<ge> 0" "v \<ge> 0" "u + v > 0"
shows "((u/(u+v)) *\<^sub>R x + (v/(u+v)) *\<^sub>R y) \<in> S"
apply (rule convexD)
using assms
apply (simp_all add: zero_le_divide_iff add_divide_distrib [symmetric])
done
lemma inj_on_image_mem_iff: "inj_on f B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> f a \<in> f`A \<Longrightarrow> a \<in> B \<Longrightarrow> a \<in> A"
by (blast dest: inj_onD)
lemma independent_injective_on_span_image:
assumes iS: "independent S"
and lf: "linear f"
and fi: "inj_on f (span S)"
shows "independent (f ` S)"
proof -
{
fix a
assume a: "a \<in> S" "f a \<in> span (f ` S - {f a})"
have eq: "f ` S - {f a} = f ` (S - {a})"
using fi a span_inc by (auto simp add: inj_on_def)
from a have "f a \<in> f ` span (S -{a})"
unfolding eq span_linear_image [OF lf, of "S - {a}"] by blast
moreover have "span (S - {a}) \<subseteq> span S"
using span_mono[of "S - {a}" S] by auto
ultimately have "a \<in> span (S - {a})"
using fi a span_inc by (auto simp add: inj_on_def)
with a(1) iS have False
by (simp add: dependent_def)
}
then show ?thesis
unfolding dependent_def by blast
qed
lemma dim_image_eq:
fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
assumes lf: "linear f"
and fi: "inj_on f (span S)"
shows "dim (f ` S) = dim (S::'n::euclidean_space set)"
proof -
obtain B where B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "card B = dim S"
using basis_exists[of S] by auto
then have "span S = span B"
using span_mono[of B S] span_mono[of S "span B"] span_span[of B] by auto
then have "independent (f ` B)"
using independent_injective_on_span_image[of B f] B assms by auto
moreover have "card (f ` B) = card B"
using assms card_image[of f B] subset_inj_on[of f "span S" B] B span_inc by auto
moreover have "(f ` B) \<subseteq> (f ` S)"
using B by auto
ultimately have "dim (f ` S) \<ge> dim S"
using independent_card_le_dim[of "f ` B" "f ` S"] B by auto
then show ?thesis
using dim_image_le[of f S] assms by auto
qed
lemma linear_injective_on_subspace_0:
assumes lf: "linear f"
and "subspace S"
shows "inj_on f S \<longleftrightarrow> (\<forall>x \<in> S. f x = 0 \<longrightarrow> x = 0)"
proof -
have "inj_on f S \<longleftrightarrow> (\<forall>x \<in> S. \<forall>y \<in> S. f x = f y \<longrightarrow> x = y)"
by (simp add: inj_on_def)
also have "\<dots> \<longleftrightarrow> (\<forall>x \<in> S. \<forall>y \<in> S. f x - f y = 0 \<longrightarrow> x - y = 0)"
by simp
also have "\<dots> \<longleftrightarrow> (\<forall>x \<in> S. \<forall>y \<in> S. f (x - y) = 0 \<longrightarrow> x - y = 0)"
by (simp add: linear_sub[OF lf])
also have "\<dots> \<longleftrightarrow> (\<forall>x \<in> S. f x = 0 \<longrightarrow> x = 0)"
using `subspace S` subspace_def[of S] subspace_sub[of S] by auto
finally show ?thesis .
qed
lemma subspace_Inter: "\<forall>s \<in> f. subspace s \<Longrightarrow> subspace (Inter f)"
unfolding subspace_def by auto
lemma span_eq[simp]: "span s = s \<longleftrightarrow> subspace s"
unfolding span_def by (rule hull_eq) (rule subspace_Inter)
lemma substdbasis_expansion_unique:
assumes d: "d \<subseteq> Basis"
shows "(\<Sum>i\<in>d. f i *\<^sub>R i) = (x::'a::euclidean_space) \<longleftrightarrow>
(\<forall>i\<in>Basis. (i \<in> d \<longrightarrow> f i = x \<bullet> i) \<and> (i \<notin> d \<longrightarrow> x \<bullet> i = 0))"
proof -
have *: "\<And>x a b P. x * (if P then a else b) = (if P then x * a else x * b)"
by auto
have **: "finite d"
by (auto intro: finite_subset[OF assms])
have ***: "\<And>i. i \<in> Basis \<Longrightarrow> (\<Sum>i\<in>d. f i *\<^sub>R i) \<bullet> i = (\<Sum>x\<in>d. if x = i then f x else 0)"
using d
by (auto intro!: setsum.cong simp: inner_Basis inner_setsum_left)
show ?thesis
unfolding euclidean_eq_iff[where 'a='a] by (auto simp: setsum.delta[OF **] ***)
qed
lemma independent_substdbasis: "d \<subseteq> Basis \<Longrightarrow> independent d"
by (rule independent_mono[OF independent_Basis])
lemma dim_cball:
assumes "e > 0"
shows "dim (cball (0 :: 'n::euclidean_space) e) = DIM('n)"
proof -
{
fix x :: "'n::euclidean_space"
def y \<equiv> "(e / norm x) *\<^sub>R x"
then have "y \<in> cball 0 e"
using cball_def dist_norm[of 0 y] assms by auto
moreover have *: "x = (norm x / e) *\<^sub>R y"
using y_def assms by simp
moreover from * have "x = (norm x/e) *\<^sub>R y"
by auto
ultimately have "x \<in> span (cball 0 e)"
using span_mul[of y "cball 0 e" "norm x/e"] span_inc[of "cball 0 e"] by auto
}
then have "span (cball 0 e) = (UNIV :: 'n::euclidean_space set)"
by auto
then show ?thesis
using dim_span[of "cball (0 :: 'n::euclidean_space) e"] by (auto simp add: dim_UNIV)
qed
lemma indep_card_eq_dim_span:
fixes B :: "'n::euclidean_space set"
assumes "independent B"
shows "finite B \<and> card B = dim (span B)"
using assms basis_card_eq_dim[of B "span B"] span_inc by auto
lemma setsum_not_0: "setsum f A \<noteq> 0 \<Longrightarrow> \<exists>a \<in> A. f a \<noteq> 0"
by (rule ccontr) auto
lemma translate_inj_on:
fixes A :: "'a::ab_group_add set"
shows "inj_on (\<lambda>x. a + x) A"
unfolding inj_on_def by auto
lemma translation_assoc:
fixes a b :: "'a::ab_group_add"
shows "(\<lambda>x. b + x) ` ((\<lambda>x. a + x) ` S) = (\<lambda>x. (a + b) + x) ` S"
by auto
lemma translation_invert:
fixes a :: "'a::ab_group_add"
assumes "(\<lambda>x. a + x) ` A = (\<lambda>x. a + x) ` B"
shows "A = B"
proof -
have "(\<lambda>x. -a + x) ` ((\<lambda>x. a + x) ` A) = (\<lambda>x. - a + x) ` ((\<lambda>x. a + x) ` B)"
using assms by auto
then show ?thesis
using translation_assoc[of "-a" a A] translation_assoc[of "-a" a B] by auto
qed
lemma translation_galois:
fixes a :: "'a::ab_group_add"
shows "T = ((\<lambda>x. a + x) ` S) \<longleftrightarrow> S = ((\<lambda>x. (- a) + x) ` T)"
using translation_assoc[of "-a" a S]
apply auto
using translation_assoc[of a "-a" T]
apply auto
done
lemma translation_inverse_subset:
assumes "((\<lambda>x. - a + x) ` V) \<le> (S :: 'n::ab_group_add set)"
shows "V \<le> ((\<lambda>x. a + x) ` S)"
proof -
{
fix x
assume "x \<in> V"
then have "x-a \<in> S" using assms by auto
then have "x \<in> {a + v |v. v \<in> S}"
apply auto
apply (rule exI[of _ "x-a"])
apply simp
done
then have "x \<in> ((\<lambda>x. a+x) ` S)" by auto
}
then show ?thesis by auto
qed
lemma basis_to_basis_subspace_isomorphism:
assumes s: "subspace (S:: ('n::euclidean_space) set)"
and t: "subspace (T :: ('m::euclidean_space) set)"
and d: "dim S = dim T"
and B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "card B = dim S"
and C: "C \<subseteq> T" "independent C" "T \<subseteq> span C" "card C = dim T"
shows "\<exists>f. linear f \<and> f ` B = C \<and> f ` S = T \<and> inj_on f S"
proof -
from B independent_bound have fB: "finite B"
by blast
from C independent_bound have fC: "finite C"
by blast
from B(4) C(4) card_le_inj[of B C] d obtain f where
f: "f ` B \<subseteq> C" "inj_on f B" using `finite B` `finite C` by auto
from linear_independent_extend[OF B(2)] obtain g where
g: "linear g" "\<forall>x \<in> B. g x = f x" by blast
from inj_on_iff_eq_card[OF fB, of f] f(2)
have "card (f ` B) = card B" by simp
with B(4) C(4) have ceq: "card (f ` B) = card C" using d
by simp
have "g ` B = f ` B" using g(2)
by (auto simp add: image_iff)
also have "\<dots> = C" using card_subset_eq[OF fC f(1) ceq] .
finally have gBC: "g ` B = C" .
have gi: "inj_on g B" using f(2) g(2)
by (auto simp add: inj_on_def)
note g0 = linear_indep_image_lemma[OF g(1) fB, unfolded gBC, OF C(2) gi]
{
fix x y
assume x: "x \<in> S" and y: "y \<in> S" and gxy: "g x = g y"
from B(3) x y have x': "x \<in> span B" and y': "y \<in> span B"
by blast+
from gxy have th0: "g (x - y) = 0"
by (simp add: linear_sub[OF g(1)])
have th1: "x - y \<in> span B" using x' y'
by (metis span_sub)
have "x = y" using g0[OF th1 th0] by simp
}
then have giS: "inj_on g S" unfolding inj_on_def by blast
from span_subspace[OF B(1,3) s]
have "g ` S = span (g ` B)"
by (simp add: span_linear_image[OF g(1)])
also have "\<dots> = span C"
unfolding gBC ..
also have "\<dots> = T"
using span_subspace[OF C(1,3) t] .
finally have gS: "g ` S = T" .
from g(1) gS giS gBC show ?thesis
by blast
qed
lemma closure_bounded_linear_image:
assumes f: "bounded_linear f"
shows "f ` closure S \<subseteq> closure (f ` S)"
using linear_continuous_on [OF f] closed_closure closure_subset
by (rule image_closure_subset)
lemma closure_linear_image:
fixes f :: "'m::euclidean_space \<Rightarrow> 'n::real_normed_vector"
assumes "linear f"
shows "f ` (closure S) \<le> closure (f ` S)"
using assms unfolding linear_conv_bounded_linear
by (rule closure_bounded_linear_image)
lemma closure_injective_linear_image:
fixes f :: "'n::euclidean_space \<Rightarrow> 'n::euclidean_space"
assumes "linear f" "inj f"
shows "f ` (closure S) = closure (f ` S)"
proof -
obtain f' where f': "linear f' \<and> f \<circ> f' = id \<and> f' \<circ> f = id"
using assms linear_injective_isomorphism[of f] isomorphism_expand by auto
then have "f' ` closure (f ` S) \<le> closure (S)"
using closure_linear_image[of f' "f ` S"] image_comp[of f' f] by auto
then have "f ` f' ` closure (f ` S) \<le> f ` closure S" by auto
then have "closure (f ` S) \<le> f ` closure S"
using image_comp[of f f' "closure (f ` S)"] f' by auto
then show ?thesis using closure_linear_image[of f S] assms by auto
qed
lemma closure_scaleR:
fixes S :: "'a::real_normed_vector set"
shows "(op *\<^sub>R c) ` (closure S) = closure ((op *\<^sub>R c) ` S)"
proof
show "(op *\<^sub>R c) ` (closure S) \<subseteq> closure ((op *\<^sub>R c) ` S)"
using bounded_linear_scaleR_right
by (rule closure_bounded_linear_image)
show "closure ((op *\<^sub>R c) ` S) \<subseteq> (op *\<^sub>R c) ` (closure S)"
by (intro closure_minimal image_mono closure_subset closed_scaling closed_closure)
qed
lemma fst_linear: "linear fst"
unfolding linear_iff by (simp add: algebra_simps)
lemma snd_linear: "linear snd"
unfolding linear_iff by (simp add: algebra_simps)
lemma fst_snd_linear: "linear (\<lambda>(x,y). x + y)"
unfolding linear_iff by (simp add: algebra_simps)
lemma scaleR_2:
fixes x :: "'a::real_vector"
shows "scaleR 2 x = x + x"
unfolding one_add_one [symmetric] scaleR_left_distrib by simp
lemma vector_choose_size:
"0 \<le> c \<Longrightarrow> \<exists>x::'a::euclidean_space. norm x = c"
apply (rule exI [where x="c *\<^sub>R (SOME i. i \<in> Basis)"])
apply (auto simp: SOME_Basis)
done
lemma setsum_delta_notmem:
assumes "x \<notin> s"
shows "setsum (\<lambda>y. if (y = x) then P x else Q y) s = setsum Q s"
and "setsum (\<lambda>y. if (x = y) then P x else Q y) s = setsum Q s"
and "setsum (\<lambda>y. if (y = x) then P y else Q y) s = setsum Q s"
and "setsum (\<lambda>y. if (x = y) then P y else Q y) s = setsum Q s"
apply (rule_tac [!] setsum.cong)
using assms
apply auto
done
lemma setsum_delta'':
fixes s::"'a::real_vector set"
assumes "finite s"
shows "(\<Sum>x\<in>s. (if y = x then f x else 0) *\<^sub>R x) = (if y\<in>s then (f y) *\<^sub>R y else 0)"
proof -
have *: "\<And>x y. (if y = x then f x else (0::real)) *\<^sub>R x = (if x=y then (f x) *\<^sub>R x else 0)"
by auto
show ?thesis
unfolding * using setsum.delta[OF assms, of y "\<lambda>x. f x *\<^sub>R x"] by auto
qed
lemma if_smult: "(if P then x else (y::real)) *\<^sub>R v = (if P then x *\<^sub>R v else y *\<^sub>R v)"
by (fact if_distrib)
lemma dist_triangle_eq:
fixes x y z :: "'a::real_inner"
shows "dist x z = dist x y + dist y z \<longleftrightarrow>
norm (x - y) *\<^sub>R (y - z) = norm (y - z) *\<^sub>R (x - y)"
proof -
have *: "x - y + (y - z) = x - z" by auto
show ?thesis unfolding dist_norm norm_triangle_eq[of "x - y" "y - z", unfolded *]
by (auto simp add:norm_minus_commute)
qed
lemma norm_minus_eqI: "x = - y \<Longrightarrow> norm x = norm y" by auto
lemma Min_grI:
assumes "finite A" "A \<noteq> {}" "\<forall>a\<in>A. x < a"
shows "x < Min A"
unfolding Min_gr_iff[OF assms(1,2)] using assms(3) by auto
lemma norm_lt: "norm x < norm y \<longleftrightarrow> inner x x < inner y y"
unfolding norm_eq_sqrt_inner by simp
lemma norm_le: "norm x \<le> norm y \<longleftrightarrow> inner x x \<le> inner y y"
unfolding norm_eq_sqrt_inner by simp
subsection {* Affine set and affine hull *}
definition affine :: "'a::real_vector set \<Rightarrow> bool"
where "affine s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u v. u + v = 1 \<longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> s)"
lemma affine_alt: "affine s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u::real. (1 - u) *\<^sub>R x + u *\<^sub>R y \<in> s)"
unfolding affine_def by (metis eq_diff_eq')
lemma affine_empty[intro]: "affine {}"
unfolding affine_def by auto
lemma affine_sing[intro]: "affine {x}"
unfolding affine_alt by (auto simp add: scaleR_left_distrib [symmetric])
lemma affine_UNIV[intro]: "affine UNIV"
unfolding affine_def by auto
lemma affine_Inter[intro]: "(\<forall>s\<in>f. affine s) \<Longrightarrow> affine (\<Inter> f)"
unfolding affine_def by auto
lemma affine_Int[intro]: "affine s \<Longrightarrow> affine t \<Longrightarrow> affine (s \<inter> t)"
unfolding affine_def by auto
lemma affine_affine_hull [simp]: "affine(affine hull s)"
unfolding hull_def
using affine_Inter[of "{t. affine t \<and> s \<subseteq> t}"] by auto
lemma affine_hull_eq[simp]: "(affine hull s = s) \<longleftrightarrow> affine s"
by (metis affine_affine_hull hull_same)
subsubsection {* Some explicit formulations (from Lars Schewe) *}
lemma affine:
fixes V::"'a::real_vector set"
shows "affine V \<longleftrightarrow>
(\<forall>s u. finite s \<and> s \<noteq> {} \<and> s \<subseteq> V \<and> setsum u s = 1 \<longrightarrow> (setsum (\<lambda>x. (u x) *\<^sub>R x)) s \<in> V)"
unfolding affine_def
apply rule
apply(rule, rule, rule)
apply(erule conjE)+
defer
apply (rule, rule, rule, rule, rule)
proof -
fix x y u v
assume as: "x \<in> V" "y \<in> V" "u + v = (1::real)"
"\<forall>s u. finite s \<and> s \<noteq> {} \<and> s \<subseteq> V \<and> setsum u s = 1 \<longrightarrow> (\<Sum>x\<in>s. u x *\<^sub>R x) \<in> V"
then show "u *\<^sub>R x + v *\<^sub>R y \<in> V"
apply (cases "x = y")
using as(4)[THEN spec[where x="{x,y}"], THEN spec[where x="\<lambda>w. if w = x then u else v"]]
and as(1-3)
apply (auto simp add: scaleR_left_distrib[symmetric])
done
next
fix s u
assume as: "\<forall>x\<in>V. \<forall>y\<in>V. \<forall>u v. u + v = 1 \<longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> V"
"finite s" "s \<noteq> {}" "s \<subseteq> V" "setsum u s = (1::real)"
def n \<equiv> "card s"
have "card s = 0 \<or> card s = 1 \<or> card s = 2 \<or> card s > 2" by auto
then show "(\<Sum>x\<in>s. u x *\<^sub>R x) \<in> V"
proof (auto simp only: disjE)
assume "card s = 2"
then have "card s = Suc (Suc 0)"
by auto
then obtain a b where "s = {a, b}"
unfolding card_Suc_eq by auto
then show ?thesis
using as(1)[THEN bspec[where x=a], THEN bspec[where x=b]] using as(4,5)
by (auto simp add: setsum_clauses(2))
next
assume "card s > 2"
then show ?thesis using as and n_def
proof (induct n arbitrary: u s)
case 0
then show ?case by auto
next
case (Suc n)
fix s :: "'a set" and u :: "'a \<Rightarrow> real"
assume IA:
"\<And>u s. \<lbrakk>2 < card s; \<forall>x\<in>V. \<forall>y\<in>V. \<forall>u v. u + v = 1 \<longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> V; finite s;
s \<noteq> {}; s \<subseteq> V; setsum u s = 1; n = card s \<rbrakk> \<Longrightarrow> (\<Sum>x\<in>s. u x *\<^sub>R x) \<in> V"
and as:
"Suc n = card s" "2 < card s" "\<forall>x\<in>V. \<forall>y\<in>V. \<forall>u v. u + v = 1 \<longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> V"
"finite s" "s \<noteq> {}" "s \<subseteq> V" "setsum u s = 1"
have "\<exists>x\<in>s. u x \<noteq> 1"
proof (rule ccontr)
assume "\<not> ?thesis"
then have "setsum u s = real_of_nat (card s)"
unfolding card_eq_setsum by auto
then show False
using as(7) and `card s > 2`
by (metis One_nat_def less_Suc0 Zero_not_Suc of_nat_1 of_nat_eq_iff numeral_2_eq_2)
qed
then obtain x where x:"x \<in> s" "u x \<noteq> 1" by auto
have c: "card (s - {x}) = card s - 1"
apply (rule card_Diff_singleton)
using `x\<in>s` as(4)
apply auto
done
have *: "s = insert x (s - {x})" "finite (s - {x})"
using `x\<in>s` and as(4) by auto
have **: "setsum u (s - {x}) = 1 - u x"
using setsum_clauses(2)[OF *(2), of u x, unfolded *(1)[symmetric] as(7)] by auto
have ***: "inverse (1 - u x) * setsum u (s - {x}) = 1"
unfolding ** using `u x \<noteq> 1` by auto
have "(\<Sum>xa\<in>s - {x}. (inverse (1 - u x) * u xa) *\<^sub>R xa) \<in> V"
proof (cases "card (s - {x}) > 2")
case True
then have "s - {x} \<noteq> {}" "card (s - {x}) = n"
unfolding c and as(1)[symmetric]
proof (rule_tac ccontr)
assume "\<not> s - {x} \<noteq> {}"
then have "card (s - {x}) = 0" unfolding card_0_eq[OF *(2)] by simp
then show False using True by auto
qed auto
then show ?thesis
apply (rule_tac IA[of "s - {x}" "\<lambda>y. (inverse (1 - u x) * u y)"])
unfolding setsum_right_distrib[symmetric]
using as and *** and True
apply auto
done
next
case False
then have "card (s - {x}) = Suc (Suc 0)"
using as(2) and c by auto
then obtain a b where "(s - {x}) = {a, b}" "a\<noteq>b"
unfolding card_Suc_eq by auto
then show ?thesis
using as(3)[THEN bspec[where x=a], THEN bspec[where x=b]]
using *** *(2) and `s \<subseteq> V`
unfolding setsum_right_distrib
by (auto simp add: setsum_clauses(2))
qed
then have "u x + (1 - u x) = 1 \<Longrightarrow>
u x *\<^sub>R x + (1 - u x) *\<^sub>R ((\<Sum>xa\<in>s - {x}. u xa *\<^sub>R xa) /\<^sub>R (1 - u x)) \<in> V"
apply -
apply (rule as(3)[rule_format])
unfolding Real_Vector_Spaces.scaleR_right.setsum
using x(1) as(6)
apply auto
done
then show "(\<Sum>x\<in>s. u x *\<^sub>R x) \<in> V"
unfolding scaleR_scaleR[symmetric] and scaleR_right.setsum [symmetric]
apply (subst *)
unfolding setsum_clauses(2)[OF *(2)]
using `u x \<noteq> 1`
apply auto
done
qed
next
assume "card s = 1"
then obtain a where "s={a}"
by (auto simp add: card_Suc_eq)
then show ?thesis
using as(4,5) by simp
qed (insert `s\<noteq>{}` `finite s`, auto)
qed
lemma affine_hull_explicit:
"affine hull p =
{y. \<exists>s u. finite s \<and> s \<noteq> {} \<and> s \<subseteq> p \<and> setsum u s = 1 \<and> setsum (\<lambda>v. (u v) *\<^sub>R v) s = y}"
apply (rule hull_unique)
apply (subst subset_eq)
prefer 3
apply rule
unfolding mem_Collect_eq
apply (erule exE)+
apply (erule conjE)+
prefer 2
apply rule
proof -
fix x
assume "x\<in>p"
then show "\<exists>s u. finite s \<and> s \<noteq> {} \<and> s \<subseteq> p \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = x"
apply (rule_tac x="{x}" in exI)
apply (rule_tac x="\<lambda>x. 1" in exI)
apply auto
done
next
fix t x s u
assume as: "p \<subseteq> t" "affine t" "finite s" "s \<noteq> {}"
"s \<subseteq> p" "setsum u s = 1" "(\<Sum>v\<in>s. u v *\<^sub>R v) = x"
then show "x \<in> t"
using as(2)[unfolded affine, THEN spec[where x=s], THEN spec[where x=u]]
by auto
next
show "affine {y. \<exists>s u. finite s \<and> s \<noteq> {} \<and> s \<subseteq> p \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = y}"
unfolding affine_def
apply (rule, rule, rule, rule, rule)
unfolding mem_Collect_eq
proof -
fix u v :: real
assume uv: "u + v = 1"
fix x
assume "\<exists>s u. finite s \<and> s \<noteq> {} \<and> s \<subseteq> p \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = x"
then obtain sx ux where
x: "finite sx" "sx \<noteq> {}" "sx \<subseteq> p" "setsum ux sx = 1" "(\<Sum>v\<in>sx. ux v *\<^sub>R v) = x"
by auto
fix y
assume "\<exists>s u. finite s \<and> s \<noteq> {} \<and> s \<subseteq> p \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = y"
then obtain sy uy where
y: "finite sy" "sy \<noteq> {}" "sy \<subseteq> p" "setsum uy sy = 1" "(\<Sum>v\<in>sy. uy v *\<^sub>R v) = y" by auto
have xy: "finite (sx \<union> sy)"
using x(1) y(1) by auto
have **: "(sx \<union> sy) \<inter> sx = sx" "(sx \<union> sy) \<inter> sy = sy"
by auto
show "\<exists>s ua. finite s \<and> s \<noteq> {} \<and> s \<subseteq> p \<and>
setsum ua s = 1 \<and> (\<Sum>v\<in>s. ua v *\<^sub>R v) = u *\<^sub>R x + v *\<^sub>R y"
apply (rule_tac x="sx \<union> sy" in exI)
apply (rule_tac x="\<lambda>a. (if a\<in>sx then u * ux a else 0) + (if a\<in>sy then v * uy a else 0)" in exI)
unfolding scaleR_left_distrib setsum.distrib if_smult scaleR_zero_left
** setsum.inter_restrict[OF xy, symmetric]
unfolding scaleR_scaleR[symmetric] Real_Vector_Spaces.scaleR_right.setsum [symmetric]
and setsum_right_distrib[symmetric]
unfolding x y
using x(1-3) y(1-3) uv
apply simp
done
qed
qed
lemma affine_hull_finite:
assumes "finite s"
shows "affine hull s = {y. \<exists>u. setsum u s = 1 \<and> setsum (\<lambda>v. u v *\<^sub>R v) s = y}"
unfolding affine_hull_explicit and set_eq_iff and mem_Collect_eq
apply (rule, rule)
apply (erule exE)+
apply (erule conjE)+
defer
apply (erule exE)
apply (erule conjE)
proof -
fix x u
assume "setsum u s = 1" "(\<Sum>v\<in>s. u v *\<^sub>R v) = x"
then show "\<exists>sa u. finite sa \<and>
\<not> (\<forall>x. (x \<in> sa) = (x \<in> {})) \<and> sa \<subseteq> s \<and> setsum u sa = 1 \<and> (\<Sum>v\<in>sa. u v *\<^sub>R v) = x"
apply (rule_tac x=s in exI, rule_tac x=u in exI)
using assms
apply auto
done
next
fix x t u
assume "t \<subseteq> s"
then have *: "s \<inter> t = t"
by auto
assume "finite t" "\<not> (\<forall>x. (x \<in> t) = (x \<in> {}))" "setsum u t = 1" "(\<Sum>v\<in>t. u v *\<^sub>R v) = x"
then show "\<exists>u. setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = x"
apply (rule_tac x="\<lambda>x. if x\<in>t then u x else 0" in exI)
unfolding if_smult scaleR_zero_left and setsum.inter_restrict[OF assms, symmetric] and *
apply auto
done
qed
subsubsection {* Stepping theorems and hence small special cases *}
lemma affine_hull_empty[simp]: "affine hull {} = {}"
by (rule hull_unique) auto
lemma affine_hull_finite_step:
fixes y :: "'a::real_vector"
shows
"(\<exists>u. setsum u {} = w \<and> setsum (\<lambda>x. u x *\<^sub>R x) {} = y) \<longleftrightarrow> w = 0 \<and> y = 0" (is ?th1)
and
"finite s \<Longrightarrow>
(\<exists>u. setsum u (insert a s) = w \<and> setsum (\<lambda>x. u x *\<^sub>R x) (insert a s) = y) \<longleftrightarrow>
(\<exists>v u. setsum u s = w - v \<and> setsum (\<lambda>x. u x *\<^sub>R x) s = y - v *\<^sub>R a)" (is "_ \<Longrightarrow> ?lhs = ?rhs")
proof -
show ?th1 by simp
assume fin: "finite s"
show "?lhs = ?rhs"
proof
assume ?lhs
then obtain u where u: "setsum u (insert a s) = w \<and> (\<Sum>x\<in>insert a s. u x *\<^sub>R x) = y"
by auto
show ?rhs
proof (cases "a \<in> s")
case True
then have *: "insert a s = s" by auto
show ?thesis
using u[unfolded *]
apply(rule_tac x=0 in exI)
apply auto
done
next
case False
then show ?thesis
apply (rule_tac x="u a" in exI)
using u and fin
apply auto
done
qed
next
assume ?rhs
then obtain v u where vu: "setsum u s = w - v" "(\<Sum>x\<in>s. u x *\<^sub>R x) = y - v *\<^sub>R a"
by auto
have *: "\<And>x M. (if x = a then v else M) *\<^sub>R x = (if x = a then v *\<^sub>R x else M *\<^sub>R x)"
by auto
show ?lhs
proof (cases "a \<in> s")
case True
then show ?thesis
apply (rule_tac x="\<lambda>x. (if x=a then v else 0) + u x" in exI)
unfolding setsum_clauses(2)[OF fin]
apply simp
unfolding scaleR_left_distrib and setsum.distrib
unfolding vu and * and scaleR_zero_left
apply (auto simp add: setsum.delta[OF fin])
done
next
case False
then have **:
"\<And>x. x \<in> s \<Longrightarrow> u x = (if x = a then v else u x)"
"\<And>x. x \<in> s \<Longrightarrow> u x *\<^sub>R x = (if x = a then v *\<^sub>R x else u x *\<^sub>R x)" by auto
from False show ?thesis
apply (rule_tac x="\<lambda>x. if x=a then v else u x" in exI)
unfolding setsum_clauses(2)[OF fin] and * using vu
using setsum.cong [of s _ "\<lambda>x. u x *\<^sub>R x" "\<lambda>x. if x = a then v *\<^sub>R x else u x *\<^sub>R x", OF _ **(2)]
using setsum.cong [of s _ u "\<lambda>x. if x = a then v else u x", OF _ **(1)]
apply auto
done
qed
qed
qed
lemma affine_hull_2:
fixes a b :: "'a::real_vector"
shows "affine hull {a,b} = {u *\<^sub>R a + v *\<^sub>R b| u v. (u + v = 1)}"
(is "?lhs = ?rhs")
proof -
have *:
"\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::real)"
"\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::'a)" by auto
have "?lhs = {y. \<exists>u. setsum u {a, b} = 1 \<and> (\<Sum>v\<in>{a, b}. u v *\<^sub>R v) = y}"
using affine_hull_finite[of "{a,b}"] by auto
also have "\<dots> = {y. \<exists>v u. u b = 1 - v \<and> u b *\<^sub>R b = y - v *\<^sub>R a}"
by (simp add: affine_hull_finite_step(2)[of "{b}" a])
also have "\<dots> = ?rhs" unfolding * by auto
finally show ?thesis by auto
qed
lemma affine_hull_3:
fixes a b c :: "'a::real_vector"
shows "affine hull {a,b,c} = { u *\<^sub>R a + v *\<^sub>R b + w *\<^sub>R c| u v w. u + v + w = 1}"
proof -
have *:
"\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::real)"
"\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::'a)" by auto
show ?thesis
apply (simp add: affine_hull_finite affine_hull_finite_step)
unfolding *
apply auto
apply (rule_tac x=v in exI)
apply (rule_tac x=va in exI)
apply auto
apply (rule_tac x=u in exI)
apply force
done
qed
lemma mem_affine:
assumes "affine S" "x \<in> S" "y \<in> S" "u + v = 1"
shows "u *\<^sub>R x + v *\<^sub>R y \<in> S"
using assms affine_def[of S] by auto
lemma mem_affine_3:
assumes "affine S" "x \<in> S" "y \<in> S" "z \<in> S" "u + v + w = 1"
shows "u *\<^sub>R x + v *\<^sub>R y + w *\<^sub>R z \<in> S"
proof -
have "u *\<^sub>R x + v *\<^sub>R y + w *\<^sub>R z \<in> affine hull {x, y, z}"
using affine_hull_3[of x y z] assms by auto
moreover
have "affine hull {x, y, z} \<subseteq> affine hull S"
using hull_mono[of "{x, y, z}" "S"] assms by auto
moreover
have "affine hull S = S"
using assms affine_hull_eq[of S] by auto
ultimately show ?thesis by auto
qed
lemma mem_affine_3_minus:
assumes "affine S" "x \<in> S" "y \<in> S" "z \<in> S"
shows "x + v *\<^sub>R (y-z) \<in> S"
using mem_affine_3[of S x y z 1 v "-v"] assms
by (simp add: algebra_simps)
corollary mem_affine_3_minus2:
"\<lbrakk>affine S; x \<in> S; y \<in> S; z \<in> S\<rbrakk> \<Longrightarrow> x - v *\<^sub>R (y-z) \<in> S"
by (metis add_uminus_conv_diff mem_affine_3_minus real_vector.scale_minus_left)
subsubsection {* Some relations between affine hull and subspaces *}
lemma affine_hull_insert_subset_span:
"affine hull (insert a s) \<subseteq> {a + v| v . v \<in> span {x - a | x . x \<in> s}}"
unfolding subset_eq Ball_def
unfolding affine_hull_explicit span_explicit mem_Collect_eq
apply (rule, rule)
apply (erule exE)+
apply (erule conjE)+
proof -
fix x t u
assume as: "finite t" "t \<noteq> {}" "t \<subseteq> insert a s" "setsum u t = 1" "(\<Sum>v\<in>t. u v *\<^sub>R v) = x"
have "(\<lambda>x. x - a) ` (t - {a}) \<subseteq> {x - a |x. x \<in> s}"
using as(3) by auto
then show "\<exists>v. x = a + v \<and> (\<exists>S u. finite S \<and> S \<subseteq> {x - a |x. x \<in> s} \<and> (\<Sum>v\<in>S. u v *\<^sub>R v) = v)"
apply (rule_tac x="x - a" in exI)
apply (rule conjI, simp)
apply (rule_tac x="(\<lambda>x. x - a) ` (t - {a})" in exI)
apply (rule_tac x="\<lambda>x. u (x + a)" in exI)
apply (rule conjI) using as(1) apply simp
apply (erule conjI)
using as(1)
apply (simp add: setsum.reindex[unfolded inj_on_def] scaleR_right_diff_distrib
setsum_subtractf scaleR_left.setsum[symmetric] setsum_diff1 scaleR_left_diff_distrib)
unfolding as
apply simp
done
qed
lemma affine_hull_insert_span:
assumes "a \<notin> s"
shows "affine hull (insert a s) = {a + v | v . v \<in> span {x - a | x. x \<in> s}}"
apply (rule, rule affine_hull_insert_subset_span)
unfolding subset_eq Ball_def
unfolding affine_hull_explicit and mem_Collect_eq
proof (rule, rule, erule exE, erule conjE)
fix y v
assume "y = a + v" "v \<in> span {x - a |x. x \<in> s}"
then obtain t u where obt: "finite t" "t \<subseteq> {x - a |x. x \<in> s}" "a + (\<Sum>v\<in>t. u v *\<^sub>R v) = y"
unfolding span_explicit by auto
def f \<equiv> "(\<lambda>x. x + a) ` t"
have f: "finite f" "f \<subseteq> s" "(\<Sum>v\<in>f. u (v - a) *\<^sub>R (v - a)) = y - a"
unfolding f_def using obt by (auto simp add: setsum.reindex[unfolded inj_on_def])
have *: "f \<inter> {a} = {}" "f \<inter> - {a} = f"
using f(2) assms by auto
show "\<exists>sa u. finite sa \<and> sa \<noteq> {} \<and> sa \<subseteq> insert a s \<and> setsum u sa = 1 \<and> (\<Sum>v\<in>sa. u v *\<^sub>R v) = y"
apply (rule_tac x = "insert a f" in exI)
apply (rule_tac x = "\<lambda>x. if x=a then 1 - setsum (\<lambda>x. u (x - a)) f else u (x - a)" in exI)
using assms and f
unfolding setsum_clauses(2)[OF f(1)] and if_smult
unfolding setsum.If_cases[OF f(1), of "\<lambda>x. x = a"]
apply (auto simp add: setsum_subtractf scaleR_left.setsum algebra_simps *)
done
qed
lemma affine_hull_span:
assumes "a \<in> s"
shows "affine hull s = {a + v | v. v \<in> span {x - a | x. x \<in> s - {a}}}"
using affine_hull_insert_span[of a "s - {a}", unfolded insert_Diff[OF assms]] by auto
subsubsection {* Parallel affine sets *}
definition affine_parallel :: "'a::real_vector set \<Rightarrow> 'a::real_vector set \<Rightarrow> bool"
where "affine_parallel S T \<longleftrightarrow> (\<exists>a. T = (\<lambda>x. a + x) ` S)"
lemma affine_parallel_expl_aux:
fixes S T :: "'a::real_vector set"
assumes "\<forall>x. x \<in> S \<longleftrightarrow> a + x \<in> T"
shows "T = (\<lambda>x. a + x) ` S"
proof -
{
fix x
assume "x \<in> T"
then have "( - a) + x \<in> S"
using assms by auto
then have "x \<in> ((\<lambda>x. a + x) ` S)"
using imageI[of "-a+x" S "(\<lambda>x. a+x)"] by auto
}
moreover have "T \<ge> (\<lambda>x. a + x) ` S"
using assms by auto
ultimately show ?thesis by auto
qed
lemma affine_parallel_expl: "affine_parallel S T \<longleftrightarrow> (\<exists>a. \<forall>x. x \<in> S \<longleftrightarrow> a + x \<in> T)"
unfolding affine_parallel_def
using affine_parallel_expl_aux[of S _ T] by auto
lemma affine_parallel_reflex: "affine_parallel S S"
unfolding affine_parallel_def
apply (rule exI[of _ "0"])
apply auto
done
lemma affine_parallel_commut:
assumes "affine_parallel A B"
shows "affine_parallel B A"
proof -
from assms obtain a where B: "B = (\<lambda>x. a + x) ` A"
unfolding affine_parallel_def by auto
have [simp]: "(\<lambda>x. x - a) = plus (- a)" by (simp add: fun_eq_iff)
from B show ?thesis
using translation_galois [of B a A]
unfolding affine_parallel_def by auto
qed
lemma affine_parallel_assoc:
assumes "affine_parallel A B"
and "affine_parallel B C"
shows "affine_parallel A C"
proof -
from assms obtain ab where "B = (\<lambda>x. ab + x) ` A"
unfolding affine_parallel_def by auto
moreover
from assms obtain bc where "C = (\<lambda>x. bc + x) ` B"
unfolding affine_parallel_def by auto
ultimately show ?thesis
using translation_assoc[of bc ab A] unfolding affine_parallel_def by auto
qed
lemma affine_translation_aux:
fixes a :: "'a::real_vector"
assumes "affine ((\<lambda>x. a + x) ` S)"
shows "affine S"
proof -
{
fix x y u v
assume xy: "x \<in> S" "y \<in> S" "(u :: real) + v = 1"
then have "(a + x) \<in> ((\<lambda>x. a + x) ` S)" "(a + y) \<in> ((\<lambda>x. a + x) ` S)"
by auto
then have h1: "u *\<^sub>R (a + x) + v *\<^sub>R (a + y) \<in> (\<lambda>x. a + x) ` S"
using xy assms unfolding affine_def by auto
have "u *\<^sub>R (a + x) + v *\<^sub>R (a + y) = (u + v) *\<^sub>R a + (u *\<^sub>R x + v *\<^sub>R y)"
by (simp add: algebra_simps)
also have "\<dots> = a + (u *\<^sub>R x + v *\<^sub>R y)"
using `u + v = 1` by auto
ultimately have "a + (u *\<^sub>R x + v *\<^sub>R y) \<in> (\<lambda>x. a + x) ` S"
using h1 by auto
then have "u *\<^sub>R x + v *\<^sub>R y : S" by auto
}
then show ?thesis unfolding affine_def by auto
qed
lemma affine_translation:
fixes a :: "'a::real_vector"
shows "affine S \<longleftrightarrow> affine ((\<lambda>x. a + x) ` S)"
proof -
have "affine S \<Longrightarrow> affine ((\<lambda>x. a + x) ` S)"
using affine_translation_aux[of "-a" "((\<lambda>x. a + x) ` S)"]
using translation_assoc[of "-a" a S] by auto
then show ?thesis using affine_translation_aux by auto
qed
lemma parallel_is_affine:
fixes S T :: "'a::real_vector set"
assumes "affine S" "affine_parallel S T"
shows "affine T"
proof -
from assms obtain a where "T = (\<lambda>x. a + x) ` S"
unfolding affine_parallel_def by auto
then show ?thesis
using affine_translation assms by auto
qed
lemma subspace_imp_affine: "subspace s \<Longrightarrow> affine s"
unfolding subspace_def affine_def by auto
subsubsection {* Subspace parallel to an affine set *}
lemma subspace_affine: "subspace S \<longleftrightarrow> affine S \<and> 0 \<in> S"
proof -
have h0: "subspace S \<Longrightarrow> affine S \<and> 0 \<in> S"
using subspace_imp_affine[of S] subspace_0 by auto
{
assume assm: "affine S \<and> 0 \<in> S"
{
fix c :: real
fix x
assume x: "x \<in> S"
have "c *\<^sub>R x = (1-c) *\<^sub>R 0 + c *\<^sub>R x" by auto
moreover
have "(1 - c) *\<^sub>R 0 + c *\<^sub>R x \<in> S"
using affine_alt[of S] assm x by auto
ultimately have "c *\<^sub>R x \<in> S" by auto
}
then have h1: "\<forall>c. \<forall>x \<in> S. c *\<^sub>R x \<in> S" by auto
{
fix x y
assume xy: "x \<in> S" "y \<in> S"
def u == "(1 :: real)/2"
have "(1/2) *\<^sub>R (x+y) = (1/2) *\<^sub>R (x+y)"
by auto
moreover
have "(1/2) *\<^sub>R (x+y)=(1/2) *\<^sub>R x + (1-(1/2)) *\<^sub>R y"
by (simp add: algebra_simps)
moreover
have "(1 - u) *\<^sub>R x + u *\<^sub>R y \<in> S"
using affine_alt[of S] assm xy by auto
ultimately
have "(1/2) *\<^sub>R (x+y) \<in> S"
using u_def by auto
moreover
have "x + y = 2 *\<^sub>R ((1/2) *\<^sub>R (x+y))"
by auto
ultimately
have "x + y \<in> S"
using h1[rule_format, of "(1/2) *\<^sub>R (x+y)" "2"] by auto
}
then have "\<forall>x \<in> S. \<forall>y \<in> S. x + y \<in> S"
by auto
then have "subspace S"
using h1 assm unfolding subspace_def by auto
}
then show ?thesis using h0 by metis
qed
lemma affine_diffs_subspace:
assumes "affine S" "a \<in> S"
shows "subspace ((\<lambda>x. (-a)+x) ` S)"
proof -
have [simp]: "(\<lambda>x. x - a) = plus (- a)" by (simp add: fun_eq_iff)
have "affine ((\<lambda>x. (-a)+x) ` S)"
using affine_translation assms by auto
moreover have "0 : ((\<lambda>x. (-a)+x) ` S)"
using assms exI[of "(\<lambda>x. x\<in>S \<and> -a+x = 0)" a] by auto
ultimately show ?thesis using subspace_affine by auto
qed
lemma parallel_subspace_explicit:
assumes "affine S"
and "a \<in> S"
assumes "L \<equiv> {y. \<exists>x \<in> S. (-a) + x = y}"
shows "subspace L \<and> affine_parallel S L"
proof -
from assms have "L = plus (- a) ` S" by auto
then have par: "affine_parallel S L"
unfolding affine_parallel_def ..
then have "affine L" using assms parallel_is_affine by auto
moreover have "0 \<in> L"
using assms by auto
ultimately show ?thesis
using subspace_affine par by auto
qed
lemma parallel_subspace_aux:
assumes "subspace A"
and "subspace B"
and "affine_parallel A B"
shows "A \<supseteq> B"
proof -
from assms obtain a where a: "\<forall>x. x \<in> A \<longleftrightarrow> a + x \<in> B"
using affine_parallel_expl[of A B] by auto
then have "-a \<in> A"
using assms subspace_0[of B] by auto
then have "a \<in> A"
using assms subspace_neg[of A "-a"] by auto
then show ?thesis
using assms a unfolding subspace_def by auto
qed
lemma parallel_subspace:
assumes "subspace A"
and "subspace B"
and "affine_parallel A B"
shows "A = B"
proof
show "A \<supseteq> B"
using assms parallel_subspace_aux by auto
show "A \<subseteq> B"
using assms parallel_subspace_aux[of B A] affine_parallel_commut by auto
qed
lemma affine_parallel_subspace:
assumes "affine S" "S \<noteq> {}"
shows "\<exists>!L. subspace L \<and> affine_parallel S L"
proof -
have ex: "\<exists>L. subspace L \<and> affine_parallel S L"
using assms parallel_subspace_explicit by auto
{
fix L1 L2
assume ass: "subspace L1 \<and> affine_parallel S L1" "subspace L2 \<and> affine_parallel S L2"
then have "affine_parallel L1 L2"
using affine_parallel_commut[of S L1] affine_parallel_assoc[of L1 S L2] by auto
then have "L1 = L2"
using ass parallel_subspace by auto
}
then show ?thesis using ex by auto
qed
subsection {* Cones *}
definition cone :: "'a::real_vector set \<Rightarrow> bool"
where "cone s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>c\<ge>0. c *\<^sub>R x \<in> s)"
lemma cone_empty[intro, simp]: "cone {}"
unfolding cone_def by auto
lemma cone_univ[intro, simp]: "cone UNIV"
unfolding cone_def by auto
lemma cone_Inter[intro]: "\<forall>s\<in>f. cone s \<Longrightarrow> cone (\<Inter>f)"
unfolding cone_def by auto
subsubsection {* Conic hull *}
lemma cone_cone_hull: "cone (cone hull s)"
unfolding hull_def by auto
lemma cone_hull_eq: "cone hull s = s \<longleftrightarrow> cone s"
apply (rule hull_eq)
using cone_Inter
unfolding subset_eq
apply auto
done
lemma mem_cone:
assumes "cone S" "x \<in> S" "c \<ge> 0"
shows "c *\<^sub>R x : S"
using assms cone_def[of S] by auto
lemma cone_contains_0:
assumes "cone S"
shows "S \<noteq> {} \<longleftrightarrow> 0 \<in> S"
proof -
{
assume "S \<noteq> {}"
then obtain a where "a \<in> S" by auto
then have "0 \<in> S"
using assms mem_cone[of S a 0] by auto
}
then show ?thesis by auto
qed
lemma cone_0: "cone {0}"
unfolding cone_def by auto
lemma cone_Union[intro]: "(\<forall>s\<in>f. cone s) \<longrightarrow> cone (Union f)"
unfolding cone_def by blast
lemma cone_iff:
assumes "S \<noteq> {}"
shows "cone S \<longleftrightarrow> 0 \<in> S \<and> (\<forall>c. c > 0 \<longrightarrow> (op *\<^sub>R c) ` S = S)"
proof -
{
assume "cone S"
{
fix c :: real
assume "c > 0"
{
fix x
assume "x \<in> S"
then have "x \<in> (op *\<^sub>R c) ` S"
unfolding image_def
using `cone S` `c>0` mem_cone[of S x "1/c"]
exI[of "(\<lambda>t. t \<in> S \<and> x = c *\<^sub>R t)" "(1 / c) *\<^sub>R x"]
by auto
}
moreover
{
fix x
assume "x \<in> (op *\<^sub>R c) ` S"
then have "x \<in> S"
using `cone S` `c > 0`
unfolding cone_def image_def `c > 0` by auto
}
ultimately have "(op *\<^sub>R c) ` S = S" by auto
}
then have "0 \<in> S \<and> (\<forall>c. c > 0 \<longrightarrow> (op *\<^sub>R c) ` S = S)"
using `cone S` cone_contains_0[of S] assms by auto
}
moreover
{
assume a: "0 \<in> S \<and> (\<forall>c. c > 0 \<longrightarrow> (op *\<^sub>R c) ` S = S)"
{
fix x
assume "x \<in> S"
fix c1 :: real
assume "c1 \<ge> 0"
then have "c1 = 0 \<or> c1 > 0" by auto
then have "c1 *\<^sub>R x \<in> S" using a `x \<in> S` by auto
}
then have "cone S" unfolding cone_def by auto
}
ultimately show ?thesis by blast
qed
lemma cone_hull_empty: "cone hull {} = {}"
by (metis cone_empty cone_hull_eq)
lemma cone_hull_empty_iff: "S = {} \<longleftrightarrow> cone hull S = {}"
by (metis bot_least cone_hull_empty hull_subset xtrans(5))
lemma cone_hull_contains_0: "S \<noteq> {} \<longleftrightarrow> 0 \<in> cone hull S"
using cone_cone_hull[of S] cone_contains_0[of "cone hull S"] cone_hull_empty_iff[of S]
by auto
lemma mem_cone_hull:
assumes "x \<in> S" "c \<ge> 0"
shows "c *\<^sub>R x \<in> cone hull S"
by (metis assms cone_cone_hull hull_inc mem_cone)
lemma cone_hull_expl: "cone hull S = {c *\<^sub>R x | c x. c \<ge> 0 \<and> x \<in> S}"
(is "?lhs = ?rhs")
proof -
{
fix x
assume "x \<in> ?rhs"
then obtain cx :: real and xx where x: "x = cx *\<^sub>R xx" "cx \<ge> 0" "xx \<in> S"
by auto
fix c :: real
assume c: "c \<ge> 0"
then have "c *\<^sub>R x = (c * cx) *\<^sub>R xx"
using x by (simp add: algebra_simps)
moreover
have "c * cx \<ge> 0" using c x by auto
ultimately
have "c *\<^sub>R x \<in> ?rhs" using x by auto
}
then have "cone ?rhs"
unfolding cone_def by auto
then have "?rhs \<in> Collect cone"
unfolding mem_Collect_eq by auto
{
fix x
assume "x \<in> S"
then have "1 *\<^sub>R x \<in> ?rhs"
apply auto
apply (rule_tac x = 1 in exI)
apply auto
done
then have "x \<in> ?rhs" by auto
}
then have "S \<subseteq> ?rhs" by auto
then have "?lhs \<subseteq> ?rhs"
using `?rhs \<in> Collect cone` hull_minimal[of S "?rhs" "cone"] by auto
moreover
{
fix x
assume "x \<in> ?rhs"
then obtain cx :: real and xx where x: "x = cx *\<^sub>R xx" "cx \<ge> 0" "xx \<in> S"
by auto
then have "xx \<in> cone hull S"
using hull_subset[of S] by auto
then have "x \<in> ?lhs"
using x cone_cone_hull[of S] cone_def[of "cone hull S"] by auto
}
ultimately show ?thesis by auto
qed
lemma cone_closure:
fixes S :: "'a::real_normed_vector set"
assumes "cone S"
shows "cone (closure S)"
proof (cases "S = {}")
case True
then show ?thesis by auto
next
case False
then have "0 \<in> S \<and> (\<forall>c. c > 0 \<longrightarrow> op *\<^sub>R c ` S = S)"
using cone_iff[of S] assms by auto
then have "0 \<in> closure S \<and> (\<forall>c. c > 0 \<longrightarrow> op *\<^sub>R c ` closure S = closure S)"
using closure_subset by (auto simp add: closure_scaleR)
then show ?thesis
using cone_iff[of "closure S"] by auto
qed
subsection {* Affine dependence and consequential theorems (from Lars Schewe) *}
definition affine_dependent :: "'a::real_vector set \<Rightarrow> bool"
where "affine_dependent s \<longleftrightarrow> (\<exists>x\<in>s. x \<in> affine hull (s - {x}))"
lemma affine_dependent_explicit:
"affine_dependent p \<longleftrightarrow>
(\<exists>s u. finite s \<and> s \<subseteq> p \<and> setsum u s = 0 \<and>
(\<exists>v\<in>s. u v \<noteq> 0) \<and> setsum (\<lambda>v. u v *\<^sub>R v) s = 0)"
unfolding affine_dependent_def affine_hull_explicit mem_Collect_eq
apply rule
apply (erule bexE, erule exE, erule exE)
apply (erule conjE)+
defer
apply (erule exE, erule exE)
apply (erule conjE)+
apply (erule bexE)
proof -
fix x s u
assume as: "x \<in> p" "finite s" "s \<noteq> {}" "s \<subseteq> p - {x}" "setsum u s = 1" "(\<Sum>v\<in>s. u v *\<^sub>R v) = x"
have "x \<notin> s" using as(1,4) by auto
show "\<exists>s u. finite s \<and> s \<subseteq> p \<and> setsum u s = 0 \<and> (\<exists>v\<in>s. u v \<noteq> 0) \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = 0"
apply (rule_tac x="insert x s" in exI, rule_tac x="\<lambda>v. if v = x then - 1 else u v" in exI)
unfolding if_smult and setsum_clauses(2)[OF as(2)] and setsum_delta_notmem[OF `x\<notin>s`] and as
using as
apply auto
done
next
fix s u v
assume as: "finite s" "s \<subseteq> p" "setsum u s = 0" "(\<Sum>v\<in>s. u v *\<^sub>R v) = 0" "v \<in> s" "u v \<noteq> 0"
have "s \<noteq> {v}"
using as(3,6) by auto
then show "\<exists>x\<in>p. \<exists>s u. finite s \<and> s \<noteq> {} \<and> s \<subseteq> p - {x} \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = x"
apply (rule_tac x=v in bexI)
apply (rule_tac x="s - {v}" in exI)
apply (rule_tac x="\<lambda>x. - (1 / u v) * u x" in exI)
unfolding scaleR_scaleR[symmetric] and scaleR_right.setsum [symmetric]
unfolding setsum_right_distrib[symmetric] and setsum_diff1[OF as(1)]
using as
apply auto
done
qed
lemma affine_dependent_explicit_finite:
fixes s :: "'a::real_vector set"
assumes "finite s"
shows "affine_dependent s \<longleftrightarrow>
(\<exists>u. setsum u s = 0 \<and> (\<exists>v\<in>s. u v \<noteq> 0) \<and> setsum (\<lambda>v. u v *\<^sub>R v) s = 0)"
(is "?lhs = ?rhs")
proof
have *: "\<And>vt u v. (if vt then u v else 0) *\<^sub>R v = (if vt then (u v) *\<^sub>R v else 0::'a)"
by auto
assume ?lhs
then obtain t u v where
"finite t" "t \<subseteq> s" "setsum u t = 0" "v\<in>t" "u v \<noteq> 0" "(\<Sum>v\<in>t. u v *\<^sub>R v) = 0"
unfolding affine_dependent_explicit by auto
then show ?rhs
apply (rule_tac x="\<lambda>x. if x\<in>t then u x else 0" in exI)
apply auto unfolding * and setsum.inter_restrict[OF assms, symmetric]
unfolding Int_absorb1[OF `t\<subseteq>s`]
apply auto
done
next
assume ?rhs
then obtain u v where "setsum u s = 0" "v\<in>s" "u v \<noteq> 0" "(\<Sum>v\<in>s. u v *\<^sub>R v) = 0"
by auto
then show ?lhs unfolding affine_dependent_explicit
using assms by auto
qed
subsection {* Connectedness of convex sets *}
lemma connectedD:
"connected S \<Longrightarrow> open A \<Longrightarrow> open B \<Longrightarrow> S \<subseteq> A \<union> B \<Longrightarrow> A \<inter> B \<inter> S = {} \<Longrightarrow> A \<inter> S = {} \<or> B \<inter> S = {}"
by (metis connected_def)
lemma convex_connected:
fixes s :: "'a::real_normed_vector set"
assumes "convex s"
shows "connected s"
proof (rule connectedI)
fix A B
assume "open A" "open B" "A \<inter> B \<inter> s = {}" "s \<subseteq> A \<union> B"
moreover
assume "A \<inter> s \<noteq> {}" "B \<inter> s \<noteq> {}"
then obtain a b where a: "a \<in> A" "a \<in> s" and b: "b \<in> B" "b \<in> s" by auto
def f \<equiv> "\<lambda>u. u *\<^sub>R a + (1 - u) *\<^sub>R b"
then have "continuous_on {0 .. 1} f"
by (auto intro!: continuous_intros)
then have "connected (f ` {0 .. 1})"
by (auto intro!: connected_continuous_image)
note connectedD[OF this, of A B]
moreover have "a \<in> A \<inter> f ` {0 .. 1}"
using a by (auto intro!: image_eqI[of _ _ 1] simp: f_def)
moreover have "b \<in> B \<inter> f ` {0 .. 1}"
using b by (auto intro!: image_eqI[of _ _ 0] simp: f_def)
moreover have "f ` {0 .. 1} \<subseteq> s"
using `convex s` a b unfolding convex_def f_def by auto
ultimately show False by auto
qed
text {* One rather trivial consequence. *}
lemma connected_UNIV[intro]: "connected (UNIV :: 'a::real_normed_vector set)"
by(simp add: convex_connected convex_UNIV)
text {* Balls, being convex, are connected. *}
lemma convex_prod:
assumes "\<And>i. i \<in> Basis \<Longrightarrow> convex {x. P i x}"
shows "convex {x. \<forall>i\<in>Basis. P i (x\<bullet>i)}"
using assms unfolding convex_def
by (auto simp: inner_add_left)
lemma convex_positive_orthant: "convex {x::'a::euclidean_space. (\<forall>i\<in>Basis. 0 \<le> x\<bullet>i)}"
by (rule convex_prod) (simp add: atLeast_def[symmetric] convex_real_interval)
lemma convex_local_global_minimum:
fixes s :: "'a::real_normed_vector set"
assumes "e > 0"
and "convex_on s f"
and "ball x e \<subseteq> s"
and "\<forall>y\<in>ball x e. f x \<le> f y"
shows "\<forall>y\<in>s. f x \<le> f y"
proof (rule ccontr)
have "x \<in> s" using assms(1,3) by auto
assume "\<not> ?thesis"
then obtain y where "y\<in>s" and y: "f x > f y" by auto
then have xy: "0 < dist x y"
by (auto simp add: dist_nz[symmetric])
then obtain u where "0 < u" "u \<le> 1" and u: "u < e / dist x y"
using real_lbound_gt_zero[of 1 "e / dist x y"] xy `e>0` by auto
then have "f ((1-u) *\<^sub>R x + u *\<^sub>R y) \<le> (1-u) * f x + u * f y"
using `x\<in>s` `y\<in>s`
using assms(2)[unfolded convex_on_def,
THEN bspec[where x=x], THEN bspec[where x=y], THEN spec[where x="1-u"]]
by auto
moreover
have *: "x - ((1 - u) *\<^sub>R x + u *\<^sub>R y) = u *\<^sub>R (x - y)"
by (simp add: algebra_simps)
have "(1 - u) *\<^sub>R x + u *\<^sub>R y \<in> ball x e"
unfolding mem_ball dist_norm
unfolding * and norm_scaleR and abs_of_pos[OF `0<u`]
unfolding dist_norm[symmetric]
using u
unfolding pos_less_divide_eq[OF xy]
by auto
then have "f x \<le> f ((1 - u) *\<^sub>R x + u *\<^sub>R y)"
using assms(4) by auto
ultimately show False
using mult_strict_left_mono[OF y `u>0`]
unfolding left_diff_distrib
by auto
qed
lemma convex_ball:
fixes x :: "'a::real_normed_vector"
shows "convex (ball x e)"
proof (auto simp add: convex_def)
fix y z
assume yz: "dist x y < e" "dist x z < e"
fix u v :: real
assume uv: "0 \<le> u" "0 \<le> v" "u + v = 1"
have "dist x (u *\<^sub>R y + v *\<^sub>R z) \<le> u * dist x y + v * dist x z"
using uv yz
using convex_on_dist [of "ball x e" x, unfolded convex_on_def,
THEN bspec[where x=y], THEN bspec[where x=z]]
by auto
then show "dist x (u *\<^sub>R y + v *\<^sub>R z) < e"
using convex_bound_lt[OF yz uv] by auto
qed
lemma convex_cball:
fixes x :: "'a::real_normed_vector"
shows "convex (cball x e)"
proof -
{
fix y z
assume yz: "dist x y \<le> e" "dist x z \<le> e"
fix u v :: real
assume uv: "0 \<le> u" "0 \<le> v" "u + v = 1"
have "dist x (u *\<^sub>R y + v *\<^sub>R z) \<le> u * dist x y + v * dist x z"
using uv yz
using convex_on_dist [of "cball x e" x, unfolded convex_on_def,
THEN bspec[where x=y], THEN bspec[where x=z]]
by auto
then have "dist x (u *\<^sub>R y + v *\<^sub>R z) \<le> e"
using convex_bound_le[OF yz uv] by auto
}
then show ?thesis by (auto simp add: convex_def Ball_def)
qed
lemma connected_ball:
fixes x :: "'a::real_normed_vector"
shows "connected (ball x e)"
using convex_connected convex_ball by auto
lemma connected_cball:
fixes x :: "'a::real_normed_vector"
shows "connected (cball x e)"
using convex_connected convex_cball by auto
subsection {* Convex hull *}
lemma convex_convex_hull: "convex (convex hull s)"
unfolding hull_def
using convex_Inter[of "{t. convex t \<and> s \<subseteq> t}"]
by auto
lemma convex_hull_eq: "convex hull s = s \<longleftrightarrow> convex s"
by (metis convex_convex_hull hull_same)
lemma bounded_convex_hull:
fixes s :: "'a::real_normed_vector set"
assumes "bounded s"
shows "bounded (convex hull s)"
proof -
from assms obtain B where B: "\<forall>x\<in>s. norm x \<le> B"
unfolding bounded_iff by auto
show ?thesis
apply (rule bounded_subset[OF bounded_cball, of _ 0 B])
unfolding subset_hull[of convex, OF convex_cball]
unfolding subset_eq mem_cball dist_norm using B
apply auto
done
qed
lemma finite_imp_bounded_convex_hull:
fixes s :: "'a::real_normed_vector set"
shows "finite s \<Longrightarrow> bounded (convex hull s)"
using bounded_convex_hull finite_imp_bounded
by auto
subsubsection {* Convex hull is "preserved" by a linear function *}
lemma convex_hull_linear_image:
assumes f: "linear f"
shows "f ` (convex hull s) = convex hull (f ` s)"
proof
show "convex hull (f ` s) \<subseteq> f ` (convex hull s)"
by (intro hull_minimal image_mono hull_subset convex_linear_image assms convex_convex_hull)
show "f ` (convex hull s) \<subseteq> convex hull (f ` s)"
proof (unfold image_subset_iff_subset_vimage, rule hull_minimal)
show "s \<subseteq> f -` (convex hull (f ` s))"
by (fast intro: hull_inc)
show "convex (f -` (convex hull (f ` s)))"
by (intro convex_linear_vimage [OF f] convex_convex_hull)
qed
qed
lemma in_convex_hull_linear_image:
assumes "linear f"
and "x \<in> convex hull s"
shows "f x \<in> convex hull (f ` s)"
using convex_hull_linear_image[OF assms(1)] assms(2) by auto
lemma convex_hull_Times:
"convex hull (s \<times> t) = (convex hull s) \<times> (convex hull t)"
proof
show "convex hull (s \<times> t) \<subseteq> (convex hull s) \<times> (convex hull t)"
by (intro hull_minimal Sigma_mono hull_subset convex_Times convex_convex_hull)
have "\<forall>x\<in>convex hull s. \<forall>y\<in>convex hull t. (x, y) \<in> convex hull (s \<times> t)"
proof (intro hull_induct)
fix x y assume "x \<in> s" and "y \<in> t"
then show "(x, y) \<in> convex hull (s \<times> t)"
by (simp add: hull_inc)
next
fix x let ?S = "((\<lambda>y. (0, y)) -` (\<lambda>p. (- x, 0) + p) ` (convex hull s \<times> t))"
have "convex ?S"
by (intro convex_linear_vimage convex_translation convex_convex_hull,
simp add: linear_iff)
also have "?S = {y. (x, y) \<in> convex hull (s \<times> t)}"
by (auto simp add: image_def Bex_def)
finally show "convex {y. (x, y) \<in> convex hull (s \<times> t)}" .
next
show "convex {x. \<forall>y\<in>convex hull t. (x, y) \<in> convex hull (s \<times> t)}"
proof (unfold Collect_ball_eq, rule convex_INT [rule_format])
fix y let ?S = "((\<lambda>x. (x, 0)) -` (\<lambda>p. (0, - y) + p) ` (convex hull s \<times> t))"
have "convex ?S"
by (intro convex_linear_vimage convex_translation convex_convex_hull,
simp add: linear_iff)
also have "?S = {x. (x, y) \<in> convex hull (s \<times> t)}"
by (auto simp add: image_def Bex_def)
finally show "convex {x. (x, y) \<in> convex hull (s \<times> t)}" .
qed
qed
then show "(convex hull s) \<times> (convex hull t) \<subseteq> convex hull (s \<times> t)"
unfolding subset_eq split_paired_Ball_Sigma .
qed
subsubsection {* Stepping theorems for convex hulls of finite sets *}
lemma convex_hull_empty[simp]: "convex hull {} = {}"
by (rule hull_unique) auto
lemma convex_hull_singleton[simp]: "convex hull {a} = {a}"
by (rule hull_unique) auto
lemma convex_hull_insert:
fixes s :: "'a::real_vector set"
assumes "s \<noteq> {}"
shows "convex hull (insert a s) =
{x. \<exists>u\<ge>0. \<exists>v\<ge>0. \<exists>b. (u + v = 1) \<and> b \<in> (convex hull s) \<and> (x = u *\<^sub>R a + v *\<^sub>R b)}"
(is "_ = ?hull")
apply (rule, rule hull_minimal, rule)
unfolding insert_iff
prefer 3
apply rule
proof -
fix x
assume x: "x = a \<or> x \<in> s"
then show "x \<in> ?hull"
apply rule
unfolding mem_Collect_eq
apply (rule_tac x=1 in exI)
defer
apply (rule_tac x=0 in exI)
using assms hull_subset[of s convex]
apply auto
done
next
fix x
assume "x \<in> ?hull"
then obtain u v b where obt: "u\<ge>0" "v\<ge>0" "u + v = 1" "b \<in> convex hull s" "x = u *\<^sub>R a + v *\<^sub>R b"
by auto
have "a \<in> convex hull insert a s" "b \<in> convex hull insert a s"
using hull_mono[of s "insert a s" convex] hull_mono[of "{a}" "insert a s" convex] and obt(4)
by auto
then show "x \<in> convex hull insert a s"
unfolding obt(5) using obt(1-3)
by (rule convexD [OF convex_convex_hull])
next
show "convex ?hull"
proof (rule convexI)
fix x y u v
assume as: "(0::real) \<le> u" "0 \<le> v" "u + v = 1" "x\<in>?hull" "y\<in>?hull"
from as(4) obtain u1 v1 b1 where
obt1: "u1\<ge>0" "v1\<ge>0" "u1 + v1 = 1" "b1 \<in> convex hull s" "x = u1 *\<^sub>R a + v1 *\<^sub>R b1"
by auto
from as(5) obtain u2 v2 b2 where
obt2: "u2\<ge>0" "v2\<ge>0" "u2 + v2 = 1" "b2 \<in> convex hull s" "y = u2 *\<^sub>R a + v2 *\<^sub>R b2"
by auto
have *: "\<And>(x::'a) s1 s2. x - s1 *\<^sub>R x - s2 *\<^sub>R x = ((1::real) - (s1 + s2)) *\<^sub>R x"
by (auto simp add: algebra_simps)
have **: "\<exists>b \<in> convex hull s. u *\<^sub>R x + v *\<^sub>R y =
(u * u1) *\<^sub>R a + (v * u2) *\<^sub>R a + (b - (u * u1) *\<^sub>R b - (v * u2) *\<^sub>R b)"
proof (cases "u * v1 + v * v2 = 0")
case True
have *: "\<And>(x::'a) s1 s2. x - s1 *\<^sub>R x - s2 *\<^sub>R x = ((1::real) - (s1 + s2)) *\<^sub>R x"
by (auto simp add: algebra_simps)
from True have ***: "u * v1 = 0" "v * v2 = 0"
using mult_nonneg_nonneg[OF `u\<ge>0` `v1\<ge>0`] mult_nonneg_nonneg[OF `v\<ge>0` `v2\<ge>0`]
by arith+
then have "u * u1 + v * u2 = 1"
using as(3) obt1(3) obt2(3) by auto
then show ?thesis
unfolding obt1(5) obt2(5) *
using assms hull_subset[of s convex]
by (auto simp add: *** scaleR_right_distrib)
next
case False
have "1 - (u * u1 + v * u2) = (u + v) - (u * u1 + v * u2)"
using as(3) obt1(3) obt2(3) by (auto simp add: field_simps)
also have "\<dots> = u * (v1 + u1 - u1) + v * (v2 + u2 - u2)"
using as(3) obt1(3) obt2(3) by (auto simp add: field_simps)
also have "\<dots> = u * v1 + v * v2"
by simp
finally have **:"1 - (u * u1 + v * u2) = u * v1 + v * v2" by auto
have "0 \<le> u * v1 + v * v2" "0 \<le> u * v1" "0 \<le> u * v1 + v * v2" "0 \<le> v * v2"
using as(1,2) obt1(1,2) obt2(1,2) by auto
then show ?thesis
unfolding obt1(5) obt2(5)
unfolding * and **
using False
apply (rule_tac
x = "((u * v1) / (u * v1 + v * v2)) *\<^sub>R b1 + ((v * v2) / (u * v1 + v * v2)) *\<^sub>R b2" in bexI)
defer
apply (rule convexD [OF convex_convex_hull])
using obt1(4) obt2(4)
unfolding add_divide_distrib[symmetric] and zero_le_divide_iff
apply (auto simp add: scaleR_left_distrib scaleR_right_distrib)
done
qed
have u1: "u1 \<le> 1"
unfolding obt1(3)[symmetric] and not_le using obt1(2) by auto
have u2: "u2 \<le> 1"
unfolding obt2(3)[symmetric] and not_le using obt2(2) by auto
have "u1 * u + u2 * v \<le> max u1 u2 * u + max u1 u2 * v"
apply (rule add_mono)
apply (rule_tac [!] mult_right_mono)
using as(1,2) obt1(1,2) obt2(1,2)
apply auto
done
also have "\<dots> \<le> 1"
unfolding distrib_left[symmetric] and as(3) using u1 u2 by auto
finally show "u *\<^sub>R x + v *\<^sub>R y \<in> ?hull"
unfolding mem_Collect_eq
apply (rule_tac x="u * u1 + v * u2" in exI)
apply (rule conjI)
defer
apply (rule_tac x="1 - u * u1 - v * u2" in exI)
unfolding Bex_def
using as(1,2) obt1(1,2) obt2(1,2) **
apply (auto simp add: algebra_simps)
done
qed
qed
subsubsection {* Explicit expression for convex hull *}
lemma convex_hull_indexed:
fixes s :: "'a::real_vector set"
shows "convex hull s =
{y. \<exists>k u x.
(\<forall>i\<in>{1::nat .. k}. 0 \<le> u i \<and> x i \<in> s) \<and>
(setsum u {1..k} = 1) \<and> (setsum (\<lambda>i. u i *\<^sub>R x i) {1..k} = y)}"
(is "?xyz = ?hull")
apply (rule hull_unique)
apply rule
defer
apply (rule convexI)
proof -
fix x
assume "x\<in>s"
then show "x \<in> ?hull"
unfolding mem_Collect_eq
apply (rule_tac x=1 in exI, rule_tac x="\<lambda>x. 1" in exI)
apply auto
done
next
fix t
assume as: "s \<subseteq> t" "convex t"
show "?hull \<subseteq> t"
apply rule
unfolding mem_Collect_eq
apply (elim exE conjE)
proof -
fix x k u y
assume assm:
"\<forall>i\<in>{1::nat..k}. 0 \<le> u i \<and> y i \<in> s"
"setsum u {1..k} = 1" "(\<Sum>i = 1..k. u i *\<^sub>R y i) = x"
show "x\<in>t"
unfolding assm(3) [symmetric]
apply (rule as(2)[unfolded convex, rule_format])
using assm(1,2) as(1) apply auto
done
qed
next
fix x y u v
assume uv: "0 \<le> u" "0 \<le> v" "u + v = (1::real)"
assume xy: "x \<in> ?hull" "y \<in> ?hull"
from xy obtain k1 u1 x1 where
x: "\<forall>i\<in>{1::nat..k1}. 0\<le>u1 i \<and> x1 i \<in> s" "setsum u1 {Suc 0..k1} = 1" "(\<Sum>i = Suc 0..k1. u1 i *\<^sub>R x1 i) = x"
by auto
from xy obtain k2 u2 x2 where
y: "\<forall>i\<in>{1::nat..k2}. 0\<le>u2 i \<and> x2 i \<in> s" "setsum u2 {Suc 0..k2} = 1" "(\<Sum>i = Suc 0..k2. u2 i *\<^sub>R x2 i) = y"
by auto
have *: "\<And>P (x1::'a) x2 s1 s2 i.
(if P i then s1 else s2) *\<^sub>R (if P i then x1 else x2) = (if P i then s1 *\<^sub>R x1 else s2 *\<^sub>R x2)"
"{1..k1 + k2} \<inter> {1..k1} = {1..k1}" "{1..k1 + k2} \<inter> - {1..k1} = (\<lambda>i. i + k1) ` {1..k2}"
prefer 3
apply (rule, rule)
unfolding image_iff
apply (rule_tac x = "x - k1" in bexI)
apply (auto simp add: not_le)
done
have inj: "inj_on (\<lambda>i. i + k1) {1..k2}"
unfolding inj_on_def by auto
show "u *\<^sub>R x + v *\<^sub>R y \<in> ?hull"
apply rule
apply (rule_tac x="k1 + k2" in exI)
apply (rule_tac x="\<lambda>i. if i \<in> {1..k1} then u * u1 i else v * u2 (i - k1)" in exI)
apply (rule_tac x="\<lambda>i. if i \<in> {1..k1} then x1 i else x2 (i - k1)" in exI)
apply (rule, rule)
defer
apply rule
unfolding * and setsum.If_cases[OF finite_atLeastAtMost[of 1 "k1 + k2"]] and
setsum.reindex[OF inj] and o_def Collect_mem_eq
unfolding scaleR_scaleR[symmetric] scaleR_right.setsum [symmetric] setsum_right_distrib[symmetric]
proof -
fix i
assume i: "i \<in> {1..k1+k2}"
show "0 \<le> (if i \<in> {1..k1} then u * u1 i else v * u2 (i - k1)) \<and>
(if i \<in> {1..k1} then x1 i else x2 (i - k1)) \<in> s"
proof (cases "i\<in>{1..k1}")
case True
then show ?thesis
using uv(1) x(1)[THEN bspec[where x=i]] by auto
next
case False
def j \<equiv> "i - k1"
from i False have "j \<in> {1..k2}"
unfolding j_def by auto
then show ?thesis
using False uv(2) y(1)[THEN bspec[where x=j]]
by (auto simp: j_def[symmetric])
qed
qed (auto simp add: not_le x(2,3) y(2,3) uv(3))
qed
lemma convex_hull_finite:
fixes s :: "'a::real_vector set"
assumes "finite s"
shows "convex hull s = {y. \<exists>u. (\<forall>x\<in>s. 0 \<le> u x) \<and>
setsum u s = 1 \<and> setsum (\<lambda>x. u x *\<^sub>R x) s = y}"
(is "?HULL = ?set")
proof (rule hull_unique, auto simp add: convex_def[of ?set])
fix x
assume "x \<in> s"
then show "\<exists>u. (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<and> (\<Sum>x\<in>s. u x *\<^sub>R x) = x"
apply (rule_tac x="\<lambda>y. if x=y then 1 else 0" in exI)
apply auto
unfolding setsum.delta'[OF assms] and setsum_delta''[OF assms]
apply auto
done
next
fix u v :: real
assume uv: "0 \<le> u" "0 \<le> v" "u + v = 1"
fix ux assume ux: "\<forall>x\<in>s. 0 \<le> ux x" "setsum ux s = (1::real)"
fix uy assume uy: "\<forall>x\<in>s. 0 \<le> uy x" "setsum uy s = (1::real)"
{
fix x
assume "x\<in>s"
then have "0 \<le> u * ux x + v * uy x"
using ux(1)[THEN bspec[where x=x]] uy(1)[THEN bspec[where x=x]] and uv(1,2)
by auto
}
moreover
have "(\<Sum>x\<in>s. u * ux x + v * uy x) = 1"
unfolding setsum.distrib and setsum_right_distrib[symmetric] and ux(2) uy(2)
using uv(3) by auto
moreover
have "(\<Sum>x\<in>s. (u * ux x + v * uy x) *\<^sub>R x) = u *\<^sub>R (\<Sum>x\<in>s. ux x *\<^sub>R x) + v *\<^sub>R (\<Sum>x\<in>s. uy x *\<^sub>R x)"
unfolding scaleR_left_distrib and setsum.distrib and scaleR_scaleR[symmetric]
and scaleR_right.setsum [symmetric]
by auto
ultimately
show "\<exists>uc. (\<forall>x\<in>s. 0 \<le> uc x) \<and> setsum uc s = 1 \<and>
(\<Sum>x\<in>s. uc x *\<^sub>R x) = u *\<^sub>R (\<Sum>x\<in>s. ux x *\<^sub>R x) + v *\<^sub>R (\<Sum>x\<in>s. uy x *\<^sub>R x)"
apply (rule_tac x="\<lambda>x. u * ux x + v * uy x" in exI)
apply auto
done
next
fix t
assume t: "s \<subseteq> t" "convex t"
fix u
assume u: "\<forall>x\<in>s. 0 \<le> u x" "setsum u s = (1::real)"
then show "(\<Sum>x\<in>s. u x *\<^sub>R x) \<in> t"
using t(2)[unfolded convex_explicit, THEN spec[where x=s], THEN spec[where x=u]]
using assms and t(1) by auto
qed
subsubsection {* Another formulation from Lars Schewe *}
lemma setsum_constant_scaleR:
fixes y :: "'a::real_vector"
shows "(\<Sum>x\<in>A. y) = of_nat (card A) *\<^sub>R y"
apply (cases "finite A")
apply (induct set: finite)
apply (simp_all add: algebra_simps)
done
lemma convex_hull_explicit:
fixes p :: "'a::real_vector set"
shows "convex hull p =
{y. \<exists>s u. finite s \<and> s \<subseteq> p \<and> (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<and> setsum (\<lambda>v. u v *\<^sub>R v) s = y}"
(is "?lhs = ?rhs")
proof -
{
fix x
assume "x\<in>?lhs"
then obtain k u y where
obt: "\<forall>i\<in>{1::nat..k}. 0 \<le> u i \<and> y i \<in> p" "setsum u {1..k} = 1" "(\<Sum>i = 1..k. u i *\<^sub>R y i) = x"
unfolding convex_hull_indexed by auto
have fin: "finite {1..k}" by auto
have fin': "\<And>v. finite {i \<in> {1..k}. y i = v}" by auto
{
fix j
assume "j\<in>{1..k}"
then have "y j \<in> p" "0 \<le> setsum u {i. Suc 0 \<le> i \<and> i \<le> k \<and> y i = y j}"
using obt(1)[THEN bspec[where x=j]] and obt(2)
apply simp
apply (rule setsum_nonneg)
using obt(1)
apply auto
done
}
moreover
have "(\<Sum>v\<in>y ` {1..k}. setsum u {i \<in> {1..k}. y i = v}) = 1"
unfolding setsum_image_gen[OF fin, symmetric] using obt(2) by auto
moreover have "(\<Sum>v\<in>y ` {1..k}. setsum u {i \<in> {1..k}. y i = v} *\<^sub>R v) = x"
using setsum_image_gen[OF fin, of "\<lambda>i. u i *\<^sub>R y i" y, symmetric]
unfolding scaleR_left.setsum using obt(3) by auto
ultimately
have "\<exists>s u. finite s \<and> s \<subseteq> p \<and> (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = x"
apply (rule_tac x="y ` {1..k}" in exI)
apply (rule_tac x="\<lambda>v. setsum u {i\<in>{1..k}. y i = v}" in exI)
apply auto
done
then have "x\<in>?rhs" by auto
}
moreover
{
fix y
assume "y\<in>?rhs"
then obtain s u where
obt: "finite s" "s \<subseteq> p" "\<forall>x\<in>s. 0 \<le> u x" "setsum u s = 1" "(\<Sum>v\<in>s. u v *\<^sub>R v) = y"
by auto
obtain f where f: "inj_on f {1..card s}" "f ` {1..card s} = s"
using ex_bij_betw_nat_finite_1[OF obt(1)] unfolding bij_betw_def by auto
{
fix i :: nat
assume "i\<in>{1..card s}"
then have "f i \<in> s"
apply (subst f(2)[symmetric])
apply auto
done
then have "0 \<le> u (f i)" "f i \<in> p" using obt(2,3) by auto
}
moreover have *: "finite {1..card s}" by auto
{
fix y
assume "y\<in>s"
then obtain i where "i\<in>{1..card s}" "f i = y"
using f using image_iff[of y f "{1..card s}"]
by auto
then have "{x. Suc 0 \<le> x \<and> x \<le> card s \<and> f x = y} = {i}"
apply auto
using f(1)[unfolded inj_on_def]
apply(erule_tac x=x in ballE)
apply auto
done
then have "card {x. Suc 0 \<le> x \<and> x \<le> card s \<and> f x = y} = 1" by auto
then have "(\<Sum>x\<in>{x \<in> {1..card s}. f x = y}. u (f x)) = u y"
"(\<Sum>x\<in>{x \<in> {1..card s}. f x = y}. u (f x) *\<^sub>R f x) = u y *\<^sub>R y"
by (auto simp add: setsum_constant_scaleR)
}
then have "(\<Sum>x = 1..card s. u (f x)) = 1" "(\<Sum>i = 1..card s. u (f i) *\<^sub>R f i) = y"
unfolding setsum_image_gen[OF *(1), of "\<lambda>x. u (f x) *\<^sub>R f x" f]
and setsum_image_gen[OF *(1), of "\<lambda>x. u (f x)" f]
unfolding f
using setsum.cong [of s s "\<lambda>y. (\<Sum>x\<in>{x \<in> {1..card s}. f x = y}. u (f x) *\<^sub>R f x)" "\<lambda>v. u v *\<^sub>R v"]
using setsum.cong [of s s "\<lambda>y. (\<Sum>x\<in>{x \<in> {1..card s}. f x = y}. u (f x))" u]
unfolding obt(4,5)
by auto
ultimately
have "\<exists>k u x. (\<forall>i\<in>{1..k}. 0 \<le> u i \<and> x i \<in> p) \<and> setsum u {1..k} = 1 \<and>
(\<Sum>i::nat = 1..k. u i *\<^sub>R x i) = y"
apply (rule_tac x="card s" in exI)
apply (rule_tac x="u \<circ> f" in exI)
apply (rule_tac x=f in exI)
apply fastforce
done
then have "y \<in> ?lhs"
unfolding convex_hull_indexed by auto
}
ultimately show ?thesis
unfolding set_eq_iff by blast
qed
subsubsection {* A stepping theorem for that expansion *}
lemma convex_hull_finite_step:
fixes s :: "'a::real_vector set"
assumes "finite s"
shows
"(\<exists>u. (\<forall>x\<in>insert a s. 0 \<le> u x) \<and> setsum u (insert a s) = w \<and> setsum (\<lambda>x. u x *\<^sub>R x) (insert a s) = y)
\<longleftrightarrow> (\<exists>v\<ge>0. \<exists>u. (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = w - v \<and> setsum (\<lambda>x. u x *\<^sub>R x) s = y - v *\<^sub>R a)"
(is "?lhs = ?rhs")
proof (rule, case_tac[!] "a\<in>s")
assume "a \<in> s"
then have *: "insert a s = s" by auto
assume ?lhs
then show ?rhs
unfolding *
apply (rule_tac x=0 in exI)
apply auto
done
next
assume ?lhs
then obtain u where
u: "\<forall>x\<in>insert a s. 0 \<le> u x" "setsum u (insert a s) = w" "(\<Sum>x\<in>insert a s. u x *\<^sub>R x) = y"
by auto
assume "a \<notin> s"
then show ?rhs
apply (rule_tac x="u a" in exI)
using u(1)[THEN bspec[where x=a]]
apply simp
apply (rule_tac x=u in exI)
using u[unfolded setsum_clauses(2)[OF assms]] and `a\<notin>s`
apply auto
done
next
assume "a \<in> s"
then have *: "insert a s = s" by auto
have fin: "finite (insert a s)" using assms by auto
assume ?rhs
then obtain v u where uv: "v\<ge>0" "\<forall>x\<in>s. 0 \<le> u x" "setsum u s = w - v" "(\<Sum>x\<in>s. u x *\<^sub>R x) = y - v *\<^sub>R a"
by auto
show ?lhs
apply (rule_tac x = "\<lambda>x. (if a = x then v else 0) + u x" in exI)
unfolding scaleR_left_distrib and setsum.distrib and setsum_delta''[OF fin] and setsum.delta'[OF fin]
unfolding setsum_clauses(2)[OF assms]
using uv and uv(2)[THEN bspec[where x=a]] and `a\<in>s`
apply auto
done
next
assume ?rhs
then obtain v u where
uv: "v\<ge>0" "\<forall>x\<in>s. 0 \<le> u x" "setsum u s = w - v" "(\<Sum>x\<in>s. u x *\<^sub>R x) = y - v *\<^sub>R a"
by auto
moreover
assume "a \<notin> s"
moreover
have "(\<Sum>x\<in>s. if a = x then v else u x) = setsum u s"
and "(\<Sum>x\<in>s. (if a = x then v else u x) *\<^sub>R x) = (\<Sum>x\<in>s. u x *\<^sub>R x)"
apply (rule_tac setsum.cong) apply rule
defer
apply (rule_tac setsum.cong) apply rule
using `a \<notin> s`
apply auto
done
ultimately show ?lhs
apply (rule_tac x="\<lambda>x. if a = x then v else u x" in exI)
unfolding setsum_clauses(2)[OF assms]
apply auto
done
qed
subsubsection {* Hence some special cases *}
lemma convex_hull_2:
"convex hull {a,b} = {u *\<^sub>R a + v *\<^sub>R b | u v. 0 \<le> u \<and> 0 \<le> v \<and> u + v = 1}"
proof -
have *: "\<And>u. (\<forall>x\<in>{a, b}. 0 \<le> u x) \<longleftrightarrow> 0 \<le> u a \<and> 0 \<le> u b"
by auto
have **: "finite {b}" by auto
show ?thesis
apply (simp add: convex_hull_finite)
unfolding convex_hull_finite_step[OF **, of a 1, unfolded * conj_assoc]
apply auto
apply (rule_tac x=v in exI)
apply (rule_tac x="1 - v" in exI)
apply simp
apply (rule_tac x=u in exI)
apply simp
apply (rule_tac x="\<lambda>x. v" in exI)
apply simp
done
qed
lemma convex_hull_2_alt: "convex hull {a,b} = {a + u *\<^sub>R (b - a) | u. 0 \<le> u \<and> u \<le> 1}"
unfolding convex_hull_2
proof (rule Collect_cong)
have *: "\<And>x y ::real. x + y = 1 \<longleftrightarrow> x = 1 - y"
by auto
fix x
show "(\<exists>v u. x = v *\<^sub>R a + u *\<^sub>R b \<and> 0 \<le> v \<and> 0 \<le> u \<and> v + u = 1) \<longleftrightarrow>
(\<exists>u. x = a + u *\<^sub>R (b - a) \<and> 0 \<le> u \<and> u \<le> 1)"
unfolding *
apply auto
apply (rule_tac[!] x=u in exI)
apply (auto simp add: algebra_simps)
done
qed
lemma convex_hull_3:
"convex hull {a,b,c} = { u *\<^sub>R a + v *\<^sub>R b + w *\<^sub>R c | u v w. 0 \<le> u \<and> 0 \<le> v \<and> 0 \<le> w \<and> u + v + w = 1}"
proof -
have fin: "finite {a,b,c}" "finite {b,c}" "finite {c}"
by auto
have *: "\<And>x y z ::real. x + y + z = 1 \<longleftrightarrow> x = 1 - y - z"
by (auto simp add: field_simps)
show ?thesis
unfolding convex_hull_finite[OF fin(1)] and convex_hull_finite_step[OF fin(2)] and *
unfolding convex_hull_finite_step[OF fin(3)]
apply (rule Collect_cong)
apply simp
apply auto
apply (rule_tac x=va in exI)
apply (rule_tac x="u c" in exI)
apply simp
apply (rule_tac x="1 - v - w" in exI)
apply simp
apply (rule_tac x=v in exI)
apply simp
apply (rule_tac x="\<lambda>x. w" in exI)
apply simp
done
qed
lemma convex_hull_3_alt:
"convex hull {a,b,c} = {a + u *\<^sub>R (b - a) + v *\<^sub>R (c - a) | u v. 0 \<le> u \<and> 0 \<le> v \<and> u + v \<le> 1}"
proof -
have *: "\<And>x y z ::real. x + y + z = 1 \<longleftrightarrow> x = 1 - y - z"
by auto
show ?thesis
unfolding convex_hull_3
apply (auto simp add: *)
apply (rule_tac x=v in exI)
apply (rule_tac x=w in exI)
apply (simp add: algebra_simps)
apply (rule_tac x=u in exI)
apply (rule_tac x=v in exI)
apply (simp add: algebra_simps)
done
qed
subsection {* Relations among closure notions and corresponding hulls *}
lemma affine_imp_convex: "affine s \<Longrightarrow> convex s"
unfolding affine_def convex_def by auto
lemma subspace_imp_convex: "subspace s \<Longrightarrow> convex s"
using subspace_imp_affine affine_imp_convex by auto
lemma affine_hull_subset_span: "(affine hull s) \<subseteq> (span s)"
by (metis hull_minimal span_inc subspace_imp_affine subspace_span)
lemma convex_hull_subset_span: "(convex hull s) \<subseteq> (span s)"
by (metis hull_minimal span_inc subspace_imp_convex subspace_span)
lemma convex_hull_subset_affine_hull: "(convex hull s) \<subseteq> (affine hull s)"
by (metis affine_affine_hull affine_imp_convex hull_minimal hull_subset)
lemma affine_dependent_imp_dependent: "affine_dependent s \<Longrightarrow> dependent s"
unfolding affine_dependent_def dependent_def
using affine_hull_subset_span by auto
lemma dependent_imp_affine_dependent:
assumes "dependent {x - a| x . x \<in> s}"
and "a \<notin> s"
shows "affine_dependent (insert a s)"
proof -
from assms(1)[unfolded dependent_explicit] obtain S u v
where obt: "finite S" "S \<subseteq> {x - a |x. x \<in> s}" "v\<in>S" "u v \<noteq> 0" "(\<Sum>v\<in>S. u v *\<^sub>R v) = 0"
by auto
def t \<equiv> "(\<lambda>x. x + a) ` S"
have inj: "inj_on (\<lambda>x. x + a) S"
unfolding inj_on_def by auto
have "0 \<notin> S"
using obt(2) assms(2) unfolding subset_eq by auto
have fin: "finite t" and "t \<subseteq> s"
unfolding t_def using obt(1,2) by auto
then have "finite (insert a t)" and "insert a t \<subseteq> insert a s"
by auto
moreover have *: "\<And>P Q. (\<Sum>x\<in>t. (if x = a then P x else Q x)) = (\<Sum>x\<in>t. Q x)"
apply (rule setsum.cong)
using `a\<notin>s` `t\<subseteq>s`
apply auto
done
have "(\<Sum>x\<in>insert a t. if x = a then - (\<Sum>x\<in>t. u (x - a)) else u (x - a)) = 0"
unfolding setsum_clauses(2)[OF fin]
using `a\<notin>s` `t\<subseteq>s`
apply auto
unfolding *
apply auto
done
moreover have "\<exists>v\<in>insert a t. (if v = a then - (\<Sum>x\<in>t. u (x - a)) else u (v - a)) \<noteq> 0"
apply (rule_tac x="v + a" in bexI)
using obt(3,4) and `0\<notin>S`
unfolding t_def
apply auto
done
moreover have *: "\<And>P Q. (\<Sum>x\<in>t. (if x = a then P x else Q x) *\<^sub>R x) = (\<Sum>x\<in>t. Q x *\<^sub>R x)"
apply (rule setsum.cong)
using `a\<notin>s` `t\<subseteq>s`
apply auto
done
have "(\<Sum>x\<in>t. u (x - a)) *\<^sub>R a = (\<Sum>v\<in>t. u (v - a) *\<^sub>R v)"
unfolding scaleR_left.setsum
unfolding t_def and setsum.reindex[OF inj] and o_def
using obt(5)
by (auto simp add: setsum.distrib scaleR_right_distrib)
then have "(\<Sum>v\<in>insert a t. (if v = a then - (\<Sum>x\<in>t. u (x - a)) else u (v - a)) *\<^sub>R v) = 0"
unfolding setsum_clauses(2)[OF fin]
using `a\<notin>s` `t\<subseteq>s`
by (auto simp add: *)
ultimately show ?thesis
unfolding affine_dependent_explicit
apply (rule_tac x="insert a t" in exI)
apply auto
done
qed
lemma convex_cone:
"convex s \<and> cone s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. (x + y) \<in> s) \<and> (\<forall>x\<in>s. \<forall>c\<ge>0. (c *\<^sub>R x) \<in> s)"
(is "?lhs = ?rhs")
proof -
{
fix x y
assume "x\<in>s" "y\<in>s" and ?lhs
then have "2 *\<^sub>R x \<in>s" "2 *\<^sub>R y \<in> s"
unfolding cone_def by auto
then have "x + y \<in> s"
using `?lhs`[unfolded convex_def, THEN conjunct1]
apply (erule_tac x="2*\<^sub>R x" in ballE)
apply (erule_tac x="2*\<^sub>R y" in ballE)
apply (erule_tac x="1/2" in allE)
apply simp
apply (erule_tac x="1/2" in allE)
apply auto
done
}
then show ?thesis
unfolding convex_def cone_def by blast
qed
lemma affine_dependent_biggerset:
fixes s :: "'a::euclidean_space set"
assumes "finite s" "card s \<ge> DIM('a) + 2"
shows "affine_dependent s"
proof -
have "s \<noteq> {}" using assms by auto
then obtain a where "a\<in>s" by auto
have *: "{x - a |x. x \<in> s - {a}} = (\<lambda>x. x - a) ` (s - {a})"
by auto
have "card {x - a |x. x \<in> s - {a}} = card (s - {a})"
unfolding *
apply (rule card_image)
unfolding inj_on_def
apply auto
done
also have "\<dots> > DIM('a)" using assms(2)
unfolding card_Diff_singleton[OF assms(1) `a\<in>s`] by auto
finally show ?thesis
apply (subst insert_Diff[OF `a\<in>s`, symmetric])
apply (rule dependent_imp_affine_dependent)
apply (rule dependent_biggerset)
apply auto
done
qed
lemma affine_dependent_biggerset_general:
assumes "finite (s :: 'a::euclidean_space set)"
and "card s \<ge> dim s + 2"
shows "affine_dependent s"
proof -
from assms(2) have "s \<noteq> {}" by auto
then obtain a where "a\<in>s" by auto
have *: "{x - a |x. x \<in> s - {a}} = (\<lambda>x. x - a) ` (s - {a})"
by auto
have **: "card {x - a |x. x \<in> s - {a}} = card (s - {a})"
unfolding *
apply (rule card_image)
unfolding inj_on_def
apply auto
done
have "dim {x - a |x. x \<in> s - {a}} \<le> dim s"
apply (rule subset_le_dim)
unfolding subset_eq
using `a\<in>s`
apply (auto simp add:span_superset span_sub)
done
also have "\<dots> < dim s + 1" by auto
also have "\<dots> \<le> card (s - {a})"
using assms
using card_Diff_singleton[OF assms(1) `a\<in>s`]
by auto
finally show ?thesis
apply (subst insert_Diff[OF `a\<in>s`, symmetric])
apply (rule dependent_imp_affine_dependent)
apply (rule dependent_biggerset_general)
unfolding **
apply auto
done
qed
subsection {* Caratheodory's theorem. *}
lemma convex_hull_caratheodory:
fixes p :: "('a::euclidean_space) set"
shows "convex hull p =
{y. \<exists>s u. finite s \<and> s \<subseteq> p \<and> card s \<le> DIM('a) + 1 \<and>
(\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<and> setsum (\<lambda>v. u v *\<^sub>R v) s = y}"
unfolding convex_hull_explicit set_eq_iff mem_Collect_eq
proof (rule, rule)
fix y
let ?P = "\<lambda>n. \<exists>s u. finite s \<and> card s = n \<and> s \<subseteq> p \<and> (\<forall>x\<in>s. 0 \<le> u x) \<and>
setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = y"
assume "\<exists>s u. finite s \<and> s \<subseteq> p \<and> (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = y"
then obtain N where "?P N" by auto
then have "\<exists>n\<le>N. (\<forall>k<n. \<not> ?P k) \<and> ?P n"
apply (rule_tac ex_least_nat_le)
apply auto
done
then obtain n where "?P n" and smallest: "\<forall>k<n. \<not> ?P k"
by blast
then obtain s u where obt: "finite s" "card s = n" "s\<subseteq>p" "\<forall>x\<in>s. 0 \<le> u x"
"setsum u s = 1" "(\<Sum>v\<in>s. u v *\<^sub>R v) = y" by auto
have "card s \<le> DIM('a) + 1"
proof (rule ccontr, simp only: not_le)
assume "DIM('a) + 1 < card s"
then have "affine_dependent s"
using affine_dependent_biggerset[OF obt(1)] by auto
then obtain w v where wv: "setsum w s = 0" "v\<in>s" "w v \<noteq> 0" "(\<Sum>v\<in>s. w v *\<^sub>R v) = 0"
using affine_dependent_explicit_finite[OF obt(1)] by auto
def i \<equiv> "(\<lambda>v. (u v) / (- w v)) ` {v\<in>s. w v < 0}"
def t \<equiv> "Min i"
have "\<exists>x\<in>s. w x < 0"
proof (rule ccontr, simp add: not_less)
assume as:"\<forall>x\<in>s. 0 \<le> w x"
then have "setsum w (s - {v}) \<ge> 0"
apply (rule_tac setsum_nonneg)
apply auto
done
then have "setsum w s > 0"
unfolding setsum.remove[OF obt(1) `v\<in>s`]
using as[THEN bspec[where x=v]] and `v\<in>s`
using `w v \<noteq> 0`
by auto
then show False using wv(1) by auto
qed
then have "i \<noteq> {}" unfolding i_def by auto
then have "t \<ge> 0"
using Min_ge_iff[of i 0 ] and obt(1)
unfolding t_def i_def
using obt(4)[unfolded le_less]
by (auto simp: divide_le_0_iff)
have t: "\<forall>v\<in>s. u v + t * w v \<ge> 0"
proof
fix v
assume "v \<in> s"
then have v: "0 \<le> u v"
using obt(4)[THEN bspec[where x=v]] by auto
show "0 \<le> u v + t * w v"
proof (cases "w v < 0")
case False
thus ?thesis using v `t\<ge>0` by auto
next
case True
then have "t \<le> u v / (- w v)"
using `v\<in>s`
unfolding t_def i_def
apply (rule_tac Min_le)
using obt(1)
apply auto
done
then show ?thesis
unfolding real_0_le_add_iff
using pos_le_divide_eq[OF True[unfolded neg_0_less_iff_less[symmetric]]]
by auto
qed
qed
obtain a where "a \<in> s" and "t = (\<lambda>v. (u v) / (- w v)) a" and "w a < 0"
using Min_in[OF _ `i\<noteq>{}`] and obt(1) unfolding i_def t_def by auto
then have a: "a \<in> s" "u a + t * w a = 0" by auto
have *: "\<And>f. setsum f (s - {a}) = setsum f s - ((f a)::'b::ab_group_add)"
unfolding setsum.remove[OF obt(1) `a\<in>s`] by auto
have "(\<Sum>v\<in>s. u v + t * w v) = 1"
unfolding setsum.distrib wv(1) setsum_right_distrib[symmetric] obt(5) by auto
moreover have "(\<Sum>v\<in>s. u v *\<^sub>R v + (t * w v) *\<^sub>R v) - (u a *\<^sub>R a + (t * w a) *\<^sub>R a) = y"
unfolding setsum.distrib obt(6) scaleR_scaleR[symmetric] scaleR_right.setsum [symmetric] wv(4)
using a(2) [THEN eq_neg_iff_add_eq_0 [THEN iffD2]] by simp
ultimately have "?P (n - 1)"
apply (rule_tac x="(s - {a})" in exI)
apply (rule_tac x="\<lambda>v. u v + t * w v" in exI)
using obt(1-3) and t and a
apply (auto simp add: * scaleR_left_distrib)
done
then show False
using smallest[THEN spec[where x="n - 1"]] by auto
qed
then show "\<exists>s u. finite s \<and> s \<subseteq> p \<and> card s \<le> DIM('a) + 1 \<and>
(\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = y"
using obt by auto
qed auto
lemma caratheodory:
"convex hull p =
{x::'a::euclidean_space. \<exists>s. finite s \<and> s \<subseteq> p \<and>
card s \<le> DIM('a) + 1 \<and> x \<in> convex hull s}"
unfolding set_eq_iff
apply rule
apply rule
unfolding mem_Collect_eq
proof -
fix x
assume "x \<in> convex hull p"
then obtain s u where "finite s" "s \<subseteq> p" "card s \<le> DIM('a) + 1"
"\<forall>x\<in>s. 0 \<le> u x" "setsum u s = 1" "(\<Sum>v\<in>s. u v *\<^sub>R v) = x"
unfolding convex_hull_caratheodory by auto
then show "\<exists>s. finite s \<and> s \<subseteq> p \<and> card s \<le> DIM('a) + 1 \<and> x \<in> convex hull s"
apply (rule_tac x=s in exI)
using hull_subset[of s convex]
using convex_convex_hull[unfolded convex_explicit, of s,
THEN spec[where x=s], THEN spec[where x=u]]
apply auto
done
next
fix x
assume "\<exists>s. finite s \<and> s \<subseteq> p \<and> card s \<le> DIM('a) + 1 \<and> x \<in> convex hull s"
then obtain s where "finite s" "s \<subseteq> p" "card s \<le> DIM('a) + 1" "x \<in> convex hull s"
by auto
then show "x \<in> convex hull p"
using hull_mono[OF `s\<subseteq>p`] by auto
qed
subsection {* Some Properties of Affine Dependent Sets *}
lemma affine_independent_empty: "\<not> affine_dependent {}"
by (simp add: affine_dependent_def)
lemma affine_independent_sing: "\<not> affine_dependent {a}"
by (simp add: affine_dependent_def)
lemma affine_hull_translation: "affine hull ((\<lambda>x. a + x) ` S) = (\<lambda>x. a + x) ` (affine hull S)"
proof -
have "affine ((\<lambda>x. a + x) ` (affine hull S))"
using affine_translation affine_affine_hull by blast
moreover have "(\<lambda>x. a + x) ` S \<subseteq> (\<lambda>x. a + x) ` (affine hull S)"
using hull_subset[of S] by auto
ultimately have h1: "affine hull ((\<lambda>x. a + x) ` S) \<subseteq> (\<lambda>x. a + x) ` (affine hull S)"
by (metis hull_minimal)
have "affine((\<lambda>x. -a + x) ` (affine hull ((\<lambda>x. a + x) ` S)))"
using affine_translation affine_affine_hull by blast
moreover have "(\<lambda>x. -a + x) ` (\<lambda>x. a + x) ` S \<subseteq> (\<lambda>x. -a + x) ` (affine hull ((\<lambda>x. a + x) ` S))"
using hull_subset[of "(\<lambda>x. a + x) ` S"] by auto
moreover have "S = (\<lambda>x. -a + x) ` (\<lambda>x. a + x) ` S"
using translation_assoc[of "-a" a] by auto
ultimately have "(\<lambda>x. -a + x) ` (affine hull ((\<lambda>x. a + x) ` S)) >= (affine hull S)"
by (metis hull_minimal)
then have "affine hull ((\<lambda>x. a + x) ` S) >= (\<lambda>x. a + x) ` (affine hull S)"
by auto
then show ?thesis using h1 by auto
qed
lemma affine_dependent_translation:
assumes "affine_dependent S"
shows "affine_dependent ((\<lambda>x. a + x) ` S)"
proof -
obtain x where x: "x \<in> S \<and> x \<in> affine hull (S - {x})"
using assms affine_dependent_def by auto
have "op + a ` (S - {x}) = op + a ` S - {a + x}"
by auto
then have "a + x \<in> affine hull ((\<lambda>x. a + x) ` S - {a + x})"
using affine_hull_translation[of a "S - {x}"] x by auto
moreover have "a + x \<in> (\<lambda>x. a + x) ` S"
using x by auto
ultimately show ?thesis
unfolding affine_dependent_def by auto
qed
lemma affine_dependent_translation_eq:
"affine_dependent S \<longleftrightarrow> affine_dependent ((\<lambda>x. a + x) ` S)"
proof -
{
assume "affine_dependent ((\<lambda>x. a + x) ` S)"
then have "affine_dependent S"
using affine_dependent_translation[of "((\<lambda>x. a + x) ` S)" "-a"] translation_assoc[of "-a" a]
by auto
}
then show ?thesis
using affine_dependent_translation by auto
qed
lemma affine_hull_0_dependent:
assumes "0 \<in> affine hull S"
shows "dependent S"
proof -
obtain s u where s_u: "finite s \<and> s \<noteq> {} \<and> s \<subseteq> S \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = 0"
using assms affine_hull_explicit[of S] by auto
then have "\<exists>v\<in>s. u v \<noteq> 0"
using setsum_not_0[of "u" "s"] by auto
then have "finite s \<and> s \<subseteq> S \<and> (\<exists>v\<in>s. u v \<noteq> 0 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = 0)"
using s_u by auto
then show ?thesis
unfolding dependent_explicit[of S] by auto
qed
lemma affine_dependent_imp_dependent2:
assumes "affine_dependent (insert 0 S)"
shows "dependent S"
proof -
obtain x where x: "x \<in> insert 0 S \<and> x \<in> affine hull (insert 0 S - {x})"
using affine_dependent_def[of "(insert 0 S)"] assms by blast
then have "x \<in> span (insert 0 S - {x})"
using affine_hull_subset_span by auto
moreover have "span (insert 0 S - {x}) = span (S - {x})"
using insert_Diff_if[of "0" S "{x}"] span_insert_0[of "S-{x}"] by auto
ultimately have "x \<in> span (S - {x})" by auto
then have "x \<noteq> 0 \<Longrightarrow> dependent S"
using x dependent_def by auto
moreover
{
assume "x = 0"
then have "0 \<in> affine hull S"
using x hull_mono[of "S - {0}" S] by auto
then have "dependent S"
using affine_hull_0_dependent by auto
}
ultimately show ?thesis by auto
qed
lemma affine_dependent_iff_dependent:
assumes "a \<notin> S"
shows "affine_dependent (insert a S) \<longleftrightarrow> dependent ((\<lambda>x. -a + x) ` S)"
proof -
have "(op + (- a) ` S) = {x - a| x . x : S}" by auto
then show ?thesis
using affine_dependent_translation_eq[of "(insert a S)" "-a"]
affine_dependent_imp_dependent2 assms
dependent_imp_affine_dependent[of a S]
by (auto simp del: uminus_add_conv_diff)
qed
lemma affine_dependent_iff_dependent2:
assumes "a \<in> S"
shows "affine_dependent S \<longleftrightarrow> dependent ((\<lambda>x. -a + x) ` (S-{a}))"
proof -
have "insert a (S - {a}) = S"
using assms by auto
then show ?thesis
using assms affine_dependent_iff_dependent[of a "S-{a}"] by auto
qed
lemma affine_hull_insert_span_gen:
"affine hull (insert a s) = (\<lambda>x. a + x) ` span ((\<lambda>x. - a + x) ` s)"
proof -
have h1: "{x - a |x. x \<in> s} = ((\<lambda>x. -a+x) ` s)"
by auto
{
assume "a \<notin> s"
then have ?thesis
using affine_hull_insert_span[of a s] h1 by auto
}
moreover
{
assume a1: "a \<in> s"
have "\<exists>x. x \<in> s \<and> -a+x=0"
apply (rule exI[of _ a])
using a1
apply auto
done
then have "insert 0 ((\<lambda>x. -a+x) ` (s - {a})) = (\<lambda>x. -a+x) ` s"
by auto
then have "span ((\<lambda>x. -a+x) ` (s - {a}))=span ((\<lambda>x. -a+x) ` s)"
using span_insert_0[of "op + (- a) ` (s - {a})"] by (auto simp del: uminus_add_conv_diff)
moreover have "{x - a |x. x \<in> (s - {a})} = ((\<lambda>x. -a+x) ` (s - {a}))"
by auto
moreover have "insert a (s - {a}) = insert a s"
using assms by auto
ultimately have ?thesis
using assms affine_hull_insert_span[of "a" "s-{a}"] by auto
}
ultimately show ?thesis by auto
qed
lemma affine_hull_span2:
assumes "a \<in> s"
shows "affine hull s = (\<lambda>x. a+x) ` span ((\<lambda>x. -a+x) ` (s-{a}))"
using affine_hull_insert_span_gen[of a "s - {a}", unfolded insert_Diff[OF assms]]
by auto
lemma affine_hull_span_gen:
assumes "a \<in> affine hull s"
shows "affine hull s = (\<lambda>x. a+x) ` span ((\<lambda>x. -a+x) ` s)"
proof -
have "affine hull (insert a s) = affine hull s"
using hull_redundant[of a affine s] assms by auto
then show ?thesis
using affine_hull_insert_span_gen[of a "s"] by auto
qed
lemma affine_hull_span_0:
assumes "0 \<in> affine hull S"
shows "affine hull S = span S"
using affine_hull_span_gen[of "0" S] assms by auto
lemma extend_to_affine_basis:
fixes S V :: "'n::euclidean_space set"
assumes "\<not> affine_dependent S" "S \<subseteq> V" "S \<noteq> {}"
shows "\<exists>T. \<not> affine_dependent T \<and> S \<subseteq> T \<and> T \<subseteq> V \<and> affine hull T = affine hull V"
proof -
obtain a where a: "a \<in> S"
using assms by auto
then have h0: "independent ((\<lambda>x. -a + x) ` (S-{a}))"
using affine_dependent_iff_dependent2 assms by auto
then obtain B where B:
"(\<lambda>x. -a+x) ` (S - {a}) \<subseteq> B \<and> B \<subseteq> (\<lambda>x. -a+x) ` V \<and> independent B \<and> (\<lambda>x. -a+x) ` V \<subseteq> span B"
using maximal_independent_subset_extend[of "(\<lambda>x. -a+x) ` (S-{a})" "(\<lambda>x. -a + x) ` V"] assms
by blast
def T \<equiv> "(\<lambda>x. a+x) ` insert 0 B"
then have "T = insert a ((\<lambda>x. a+x) ` B)"
by auto
then have "affine hull T = (\<lambda>x. a+x) ` span B"
using affine_hull_insert_span_gen[of a "((\<lambda>x. a+x) ` B)"] translation_assoc[of "-a" a B]
by auto
then have "V \<subseteq> affine hull T"
using B assms translation_inverse_subset[of a V "span B"]
by auto
moreover have "T \<subseteq> V"
using T_def B a assms by auto
ultimately have "affine hull T = affine hull V"
by (metis Int_absorb1 Int_absorb2 hull_hull hull_mono)
moreover have "S \<subseteq> T"
using T_def B translation_inverse_subset[of a "S-{a}" B]
by auto
moreover have "\<not> affine_dependent T"
using T_def affine_dependent_translation_eq[of "insert 0 B"]
affine_dependent_imp_dependent2 B
by auto
ultimately show ?thesis using `T \<subseteq> V` by auto
qed
lemma affine_basis_exists:
fixes V :: "'n::euclidean_space set"
shows "\<exists>B. B \<subseteq> V \<and> \<not> affine_dependent B \<and> affine hull V = affine hull B"
proof (cases "V = {}")
case True
then show ?thesis
using affine_independent_empty by auto
next
case False
then obtain x where "x \<in> V" by auto
then show ?thesis
using affine_dependent_def[of "{x}"] extend_to_affine_basis[of "{x}" V]
by auto
qed
subsection {* Affine Dimension of a Set *}
definition "aff_dim V =
(SOME d :: int.
\<exists>B. affine hull B = affine hull V \<and> \<not> affine_dependent B \<and> of_nat (card B) = d + 1)"
lemma aff_dim_basis_exists:
fixes V :: "('n::euclidean_space) set"
shows "\<exists>B. affine hull B = affine hull V \<and> \<not> affine_dependent B \<and> of_nat (card B) = aff_dim V + 1"
proof -
obtain B where "\<not> affine_dependent B \<and> affine hull B = affine hull V"
using affine_basis_exists[of V] by auto
then show ?thesis
unfolding aff_dim_def
some_eq_ex[of "\<lambda>d. \<exists>B. affine hull B = affine hull V \<and> \<not> affine_dependent B \<and> of_nat (card B) = d + 1"]
apply auto
apply (rule exI[of _ "int (card B) - (1 :: int)"])
apply (rule exI[of _ "B"])
apply auto
done
qed
lemma affine_hull_nonempty: "S \<noteq> {} \<longleftrightarrow> affine hull S \<noteq> {}"
proof -
have "S = {} \<Longrightarrow> affine hull S = {}"
using affine_hull_empty by auto
moreover have "affine hull S = {} \<Longrightarrow> S = {}"
unfolding hull_def by auto
ultimately show ?thesis by blast
qed
lemma aff_dim_parallel_subspace_aux:
fixes B :: "'n::euclidean_space set"
assumes "\<not> affine_dependent B" "a \<in> B"
shows "finite B \<and> ((card B) - 1 = dim (span ((\<lambda>x. -a+x) ` (B-{a}))))"
proof -
have "independent ((\<lambda>x. -a + x) ` (B-{a}))"
using affine_dependent_iff_dependent2 assms by auto
then have fin: "dim (span ((\<lambda>x. -a+x) ` (B-{a}))) = card ((\<lambda>x. -a + x) ` (B-{a}))"
"finite ((\<lambda>x. -a + x) ` (B - {a}))"
using indep_card_eq_dim_span[of "(\<lambda>x. -a+x) ` (B-{a})"] by auto
show ?thesis
proof (cases "(\<lambda>x. -a + x) ` (B - {a}) = {}")
case True
have "B = insert a ((\<lambda>x. a + x) ` (\<lambda>x. -a + x) ` (B - {a}))"
using translation_assoc[of "a" "-a" "(B - {a})"] assms by auto
then have "B = {a}" using True by auto
then show ?thesis using assms fin by auto
next
case False
then have "card ((\<lambda>x. -a + x) ` (B - {a})) > 0"
using fin by auto
moreover have h1: "card ((\<lambda>x. -a + x) ` (B-{a})) = card (B-{a})"
apply (rule card_image)
using translate_inj_on
apply (auto simp del: uminus_add_conv_diff)
done
ultimately have "card (B-{a}) > 0" by auto
then have *: "finite (B - {a})"
using card_gt_0_iff[of "(B - {a})"] by auto
then have "card (B - {a}) = card B - 1"
using card_Diff_singleton assms by auto
with * show ?thesis using fin h1 by auto
qed
qed
lemma aff_dim_parallel_subspace:
fixes V L :: "'n::euclidean_space set"
assumes "V \<noteq> {}"
and "subspace L"
and "affine_parallel (affine hull V) L"
shows "aff_dim V = int (dim L)"
proof -
obtain B where
B: "affine hull B = affine hull V \<and> \<not> affine_dependent B \<and> int (card B) = aff_dim V + 1"
using aff_dim_basis_exists by auto
then have "B \<noteq> {}"
using assms B affine_hull_nonempty[of V] affine_hull_nonempty[of B]
by auto
then obtain a where a: "a \<in> B" by auto
def Lb \<equiv> "span ((\<lambda>x. -a+x) ` (B-{a}))"
moreover have "affine_parallel (affine hull B) Lb"
using Lb_def B assms affine_hull_span2[of a B] a
affine_parallel_commut[of "Lb" "(affine hull B)"]
unfolding affine_parallel_def
by auto
moreover have "subspace Lb"
using Lb_def subspace_span by auto
moreover have "affine hull B \<noteq> {}"
using assms B affine_hull_nonempty[of V] by auto
ultimately have "L = Lb"
using assms affine_parallel_subspace[of "affine hull B"] affine_affine_hull[of B] B
by auto
then have "dim L = dim Lb"
by auto
moreover have "card B - 1 = dim Lb" and "finite B"
using Lb_def aff_dim_parallel_subspace_aux a B by auto
ultimately show ?thesis
using B `B \<noteq> {}` card_gt_0_iff[of B] by auto
qed
lemma aff_independent_finite:
fixes B :: "'n::euclidean_space set"
assumes "\<not> affine_dependent B"
shows "finite B"
proof -
{
assume "B \<noteq> {}"
then obtain a where "a \<in> B" by auto
then have ?thesis
using aff_dim_parallel_subspace_aux assms by auto
}
then show ?thesis by auto
qed
lemma independent_finite:
fixes B :: "'n::euclidean_space set"
assumes "independent B"
shows "finite B"
using affine_dependent_imp_dependent[of B] aff_independent_finite[of B] assms
by auto
lemma subspace_dim_equal:
assumes "subspace (S :: ('n::euclidean_space) set)"
and "subspace T"
and "S \<subseteq> T"
and "dim S \<ge> dim T"
shows "S = T"
proof -
obtain B where B: "B \<le> S" "independent B \<and> S \<subseteq> span B" "card B = dim S"
using basis_exists[of S] by auto
then have "span B \<subseteq> S"
using span_mono[of B S] span_eq[of S] assms by metis
then have "span B = S"
using B by auto
have "dim S = dim T"
using assms dim_subset[of S T] by auto
then have "T \<subseteq> span B"
using card_eq_dim[of B T] B independent_finite assms by auto
then show ?thesis
using assms `span B = S` by auto
qed
lemma span_substd_basis:
assumes d: "d \<subseteq> Basis"
shows "span d = {x. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x\<bullet>i = 0}"
(is "_ = ?B")
proof -
have "d \<subseteq> ?B"
using d by (auto simp: inner_Basis)
moreover have s: "subspace ?B"
using subspace_substandard[of "\<lambda>i. i \<notin> d"] .
ultimately have "span d \<subseteq> ?B"
using span_mono[of d "?B"] span_eq[of "?B"] by blast
moreover have *: "card d \<le> dim (span d)"
using independent_card_le_dim[of d "span d"] independent_substdbasis[OF assms] span_inc[of d]
by auto
moreover from * have "dim ?B \<le> dim (span d)"
using dim_substandard[OF assms] by auto
ultimately show ?thesis
using s subspace_dim_equal[of "span d" "?B"] subspace_span[of d] by auto
qed
lemma basis_to_substdbasis_subspace_isomorphism:
fixes B :: "'a::euclidean_space set"
assumes "independent B"
shows "\<exists>f d::'a set. card d = card B \<and> linear f \<and> f ` B = d \<and>
f ` span B = {x. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0} \<and> inj_on f (span B) \<and> d \<subseteq> Basis"
proof -
have B: "card B = dim B"
using dim_unique[of B B "card B"] assms span_inc[of B] by auto
have "dim B \<le> card (Basis :: 'a set)"
using dim_subset_UNIV[of B] by simp
from ex_card[OF this] obtain d :: "'a set" where d: "d \<subseteq> Basis" and t: "card d = dim B"
by auto
let ?t = "{x::'a::euclidean_space. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x\<bullet>i = 0}"
have "\<exists>f. linear f \<and> f ` B = d \<and> f ` span B = ?t \<and> inj_on f (span B)"
apply (rule basis_to_basis_subspace_isomorphism[of "span B" ?t B "d"])
apply (rule subspace_span)
apply (rule subspace_substandard)
defer
apply (rule span_inc)
apply (rule assms)
defer
unfolding dim_span[of B]
apply(rule B)
unfolding span_substd_basis[OF d, symmetric]
apply (rule span_inc)
apply (rule independent_substdbasis[OF d])
apply rule
apply assumption
unfolding t[symmetric] span_substd_basis[OF d] dim_substandard[OF d]
apply auto
done
with t `card B = dim B` d show ?thesis by auto
qed
lemma aff_dim_empty:
fixes S :: "'n::euclidean_space set"
shows "S = {} \<longleftrightarrow> aff_dim S = -1"
proof -
obtain B where *: "affine hull B = affine hull S"
and "\<not> affine_dependent B"
and "int (card B) = aff_dim S + 1"
using aff_dim_basis_exists by auto
moreover
from * have "S = {} \<longleftrightarrow> B = {}"
using affine_hull_nonempty[of B] affine_hull_nonempty[of S] by auto
ultimately show ?thesis
using aff_independent_finite[of B] card_gt_0_iff[of B] by auto
qed
lemma aff_dim_affine_hull: "aff_dim (affine hull S) = aff_dim S"
unfolding aff_dim_def using hull_hull[of _ S] by auto
lemma aff_dim_affine_hull2:
assumes "affine hull S = affine hull T"
shows "aff_dim S = aff_dim T"
unfolding aff_dim_def using assms by auto
lemma aff_dim_unique:
fixes B V :: "'n::euclidean_space set"
assumes "affine hull B = affine hull V \<and> \<not> affine_dependent B"
shows "of_nat (card B) = aff_dim V + 1"
proof (cases "B = {}")
case True
then have "V = {}"
using affine_hull_nonempty[of V] affine_hull_nonempty[of B] assms
by auto
then have "aff_dim V = (-1::int)"
using aff_dim_empty by auto
then show ?thesis
using `B = {}` by auto
next
case False
then obtain a where a: "a \<in> B" by auto
def Lb \<equiv> "span ((\<lambda>x. -a+x) ` (B-{a}))"
have "affine_parallel (affine hull B) Lb"
using Lb_def affine_hull_span2[of a B] a
affine_parallel_commut[of "Lb" "(affine hull B)"]
unfolding affine_parallel_def by auto
moreover have "subspace Lb"
using Lb_def subspace_span by auto
ultimately have "aff_dim B = int(dim Lb)"
using aff_dim_parallel_subspace[of B Lb] `B \<noteq> {}` by auto
moreover have "(card B) - 1 = dim Lb" "finite B"
using Lb_def aff_dim_parallel_subspace_aux a assms by auto
ultimately have "of_nat (card B) = aff_dim B + 1"
using `B \<noteq> {}` card_gt_0_iff[of B] by auto
then show ?thesis
using aff_dim_affine_hull2 assms by auto
qed
lemma aff_dim_affine_independent:
fixes B :: "'n::euclidean_space set"
assumes "\<not> affine_dependent B"
shows "of_nat (card B) = aff_dim B + 1"
using aff_dim_unique[of B B] assms by auto
lemma aff_dim_sing:
fixes a :: "'n::euclidean_space"
shows "aff_dim {a} = 0"
using aff_dim_affine_independent[of "{a}"] affine_independent_sing by auto
lemma aff_dim_inner_basis_exists:
fixes V :: "('n::euclidean_space) set"
shows "\<exists>B. B \<subseteq> V \<and> affine hull B = affine hull V \<and>
\<not> affine_dependent B \<and> of_nat (card B) = aff_dim V + 1"
proof -
obtain B where B: "\<not> affine_dependent B" "B \<subseteq> V" "affine hull B = affine hull V"
using affine_basis_exists[of V] by auto
then have "of_nat(card B) = aff_dim V+1" using aff_dim_unique by auto
with B show ?thesis by auto
qed
lemma aff_dim_le_card:
fixes V :: "'n::euclidean_space set"
assumes "finite V"
shows "aff_dim V \<le> of_nat (card V) - 1"
proof -
obtain B where B: "B \<subseteq> V" "of_nat (card B) = aff_dim V + 1"
using aff_dim_inner_basis_exists[of V] by auto
then have "card B \<le> card V"
using assms card_mono by auto
with B show ?thesis by auto
qed
lemma aff_dim_parallel_eq:
fixes S T :: "'n::euclidean_space set"
assumes "affine_parallel (affine hull S) (affine hull T)"
shows "aff_dim S = aff_dim T"
proof -
{
assume "T \<noteq> {}" "S \<noteq> {}"
then obtain L where L: "subspace L \<and> affine_parallel (affine hull T) L"
using affine_parallel_subspace[of "affine hull T"]
affine_affine_hull[of T] affine_hull_nonempty
by auto
then have "aff_dim T = int (dim L)"
using aff_dim_parallel_subspace `T \<noteq> {}` by auto
moreover have *: "subspace L \<and> affine_parallel (affine hull S) L"
using L affine_parallel_assoc[of "affine hull S" "affine hull T" L] assms by auto
moreover from * have "aff_dim S = int (dim L)"
using aff_dim_parallel_subspace `S \<noteq> {}` by auto
ultimately have ?thesis by auto
}
moreover
{
assume "S = {}"
then have "S = {}" and "T = {}"
using assms affine_hull_nonempty
unfolding affine_parallel_def
by auto
then have ?thesis using aff_dim_empty by auto
}
moreover
{
assume "T = {}"
then have "S = {}" and "T = {}"
using assms affine_hull_nonempty
unfolding affine_parallel_def
by auto
then have ?thesis
using aff_dim_empty by auto
}
ultimately show ?thesis by blast
qed
lemma aff_dim_translation_eq:
fixes a :: "'n::euclidean_space"
shows "aff_dim ((\<lambda>x. a + x) ` S) = aff_dim S"
proof -
have "affine_parallel (affine hull S) (affine hull ((\<lambda>x. a + x) ` S))"
unfolding affine_parallel_def
apply (rule exI[of _ "a"])
using affine_hull_translation[of a S]
apply auto
done
then show ?thesis
using aff_dim_parallel_eq[of S "(\<lambda>x. a + x) ` S"] by auto
qed
lemma aff_dim_affine:
fixes S L :: "'n::euclidean_space set"
assumes "S \<noteq> {}"
and "affine S"
and "subspace L"
and "affine_parallel S L"
shows "aff_dim S = int (dim L)"
proof -
have *: "affine hull S = S"
using assms affine_hull_eq[of S] by auto
then have "affine_parallel (affine hull S) L"
using assms by (simp add: *)
then show ?thesis
using assms aff_dim_parallel_subspace[of S L] by blast
qed
lemma dim_affine_hull:
fixes S :: "'n::euclidean_space set"
shows "dim (affine hull S) = dim S"
proof -
have "dim (affine hull S) \<ge> dim S"
using dim_subset by auto
moreover have "dim (span S) \<ge> dim (affine hull S)"
using dim_subset affine_hull_subset_span by blast
moreover have "dim (span S) = dim S"
using dim_span by auto
ultimately show ?thesis by auto
qed
lemma aff_dim_subspace:
fixes S :: "'n::euclidean_space set"
assumes "S \<noteq> {}"
and "subspace S"
shows "aff_dim S = int (dim S)"
using aff_dim_affine[of S S] assms subspace_imp_affine[of S] affine_parallel_reflex[of S]
by auto
lemma aff_dim_zero:
fixes S :: "'n::euclidean_space set"
assumes "0 \<in> affine hull S"
shows "aff_dim S = int (dim S)"
proof -
have "subspace (affine hull S)"
using subspace_affine[of "affine hull S"] affine_affine_hull assms
by auto
then have "aff_dim (affine hull S) = int (dim (affine hull S))"
using assms aff_dim_subspace[of "affine hull S"] by auto
then show ?thesis
using aff_dim_affine_hull[of S] dim_affine_hull[of S]
by auto
qed
lemma aff_dim_univ: "aff_dim (UNIV :: 'n::euclidean_space set) = int(DIM('n))"
using aff_dim_subspace[of "(UNIV :: 'n::euclidean_space set)"]
dim_UNIV[where 'a="'n::euclidean_space"]
by auto
lemma aff_dim_geq:
fixes V :: "'n::euclidean_space set"
shows "aff_dim V \<ge> -1"
proof -
obtain B where "affine hull B = affine hull V"
and "\<not> affine_dependent B"
and "int (card B) = aff_dim V + 1"
using aff_dim_basis_exists by auto
then show ?thesis by auto
qed
lemma independent_card_le_aff_dim:
fixes B :: "'n::euclidean_space set"
assumes "B \<subseteq> V"
assumes "\<not> affine_dependent B"
shows "int (card B) \<le> aff_dim V + 1"
proof (cases "B = {}")
case True
then have "-1 \<le> aff_dim V"
using aff_dim_geq by auto
with True show ?thesis by auto
next
case False
then obtain T where T: "\<not> affine_dependent T \<and> B \<subseteq> T \<and> T \<subseteq> V \<and> affine hull T = affine hull V"
using assms extend_to_affine_basis[of B V] by auto
then have "of_nat (card T) = aff_dim V + 1"
using aff_dim_unique by auto
then show ?thesis
using T card_mono[of T B] aff_independent_finite[of T] by auto
qed
lemma aff_dim_subset:
fixes S T :: "'n::euclidean_space set"
assumes "S \<subseteq> T"
shows "aff_dim S \<le> aff_dim T"
proof -
obtain B where B: "\<not> affine_dependent B" "B \<subseteq> S" "affine hull B = affine hull S"
"of_nat (card B) = aff_dim S + 1"
using aff_dim_inner_basis_exists[of S] by auto
then have "int (card B) \<le> aff_dim T + 1"
using assms independent_card_le_aff_dim[of B T] by auto
with B show ?thesis by auto
qed
lemma aff_dim_subset_univ:
fixes S :: "'n::euclidean_space set"
shows "aff_dim S \<le> int (DIM('n))"
proof -
have "aff_dim (UNIV :: 'n::euclidean_space set) = int(DIM('n))"
using aff_dim_univ by auto
then show "aff_dim (S:: 'n::euclidean_space set) \<le> int(DIM('n))"
using assms aff_dim_subset[of S "(UNIV :: ('n::euclidean_space) set)"] subset_UNIV by auto
qed
lemma affine_dim_equal:
fixes S :: "'n::euclidean_space set"
assumes "affine S" "affine T" "S \<noteq> {}" "S \<subseteq> T" "aff_dim S = aff_dim T"
shows "S = T"
proof -
obtain a where "a \<in> S" using assms by auto
then have "a \<in> T" using assms by auto
def LS \<equiv> "{y. \<exists>x \<in> S. (-a) + x = y}"
then have ls: "subspace LS" "affine_parallel S LS"
using assms parallel_subspace_explicit[of S a LS] `a \<in> S` by auto
then have h1: "int(dim LS) = aff_dim S"
using assms aff_dim_affine[of S LS] by auto
have "T \<noteq> {}" using assms by auto
def LT \<equiv> "{y. \<exists>x \<in> T. (-a) + x = y}"
then have lt: "subspace LT \<and> affine_parallel T LT"
using assms parallel_subspace_explicit[of T a LT] `a \<in> T` by auto
then have "int(dim LT) = aff_dim T"
using assms aff_dim_affine[of T LT] `T \<noteq> {}` by auto
then have "dim LS = dim LT"
using h1 assms by auto
moreover have "LS \<le> LT"
using LS_def LT_def assms by auto
ultimately have "LS = LT"
using subspace_dim_equal[of LS LT] ls lt by auto
moreover have "S = {x. \<exists>y \<in> LS. a+y=x}"
using LS_def by auto
moreover have "T = {x. \<exists>y \<in> LT. a+y=x}"
using LT_def by auto
ultimately show ?thesis by auto
qed
lemma affine_hull_univ:
fixes S :: "'n::euclidean_space set"
assumes "aff_dim S = int(DIM('n))"
shows "affine hull S = (UNIV :: ('n::euclidean_space) set)"
proof -
have "S \<noteq> {}"
using assms aff_dim_empty[of S] by auto
have h0: "S \<subseteq> affine hull S"
using hull_subset[of S _] by auto
have h1: "aff_dim (UNIV :: ('n::euclidean_space) set) = aff_dim S"
using aff_dim_univ assms by auto
then have h2: "aff_dim (affine hull S) \<le> aff_dim (UNIV :: ('n::euclidean_space) set)"
using aff_dim_subset_univ[of "affine hull S"] assms h0 by auto
have h3: "aff_dim S \<le> aff_dim (affine hull S)"
using h0 aff_dim_subset[of S "affine hull S"] assms by auto
then have h4: "aff_dim (affine hull S) = aff_dim (UNIV :: ('n::euclidean_space) set)"
using h0 h1 h2 by auto
then show ?thesis
using affine_dim_equal[of "affine hull S" "(UNIV :: ('n::euclidean_space) set)"]
affine_affine_hull[of S] affine_UNIV assms h4 h0 `S \<noteq> {}`
by auto
qed
lemma aff_dim_convex_hull:
fixes S :: "'n::euclidean_space set"
shows "aff_dim (convex hull S) = aff_dim S"
using aff_dim_affine_hull[of S] convex_hull_subset_affine_hull[of S]
hull_subset[of S "convex"] aff_dim_subset[of S "convex hull S"]
aff_dim_subset[of "convex hull S" "affine hull S"]
by auto
lemma aff_dim_cball:
fixes a :: "'n::euclidean_space"
assumes "e > 0"
shows "aff_dim (cball a e) = int (DIM('n))"
proof -
have "(\<lambda>x. a + x) ` (cball 0 e) \<subseteq> cball a e"
unfolding cball_def dist_norm by auto
then have "aff_dim (cball (0 :: 'n::euclidean_space) e) \<le> aff_dim (cball a e)"
using aff_dim_translation_eq[of a "cball 0 e"]
aff_dim_subset[of "op + a ` cball 0 e" "cball a e"]
by auto
moreover have "aff_dim (cball (0 :: 'n::euclidean_space) e) = int (DIM('n))"
using hull_inc[of "(0 :: 'n::euclidean_space)" "cball 0 e"]
centre_in_cball[of "(0 :: 'n::euclidean_space)"] assms
by (simp add: dim_cball[of e] aff_dim_zero[of "cball 0 e"])
ultimately show ?thesis
using aff_dim_subset_univ[of "cball a e"] by auto
qed
lemma aff_dim_open:
fixes S :: "'n::euclidean_space set"
assumes "open S"
and "S \<noteq> {}"
shows "aff_dim S = int (DIM('n))"
proof -
obtain x where "x \<in> S"
using assms by auto
then obtain e where e: "e > 0" "cball x e \<subseteq> S"
using open_contains_cball[of S] assms by auto
then have "aff_dim (cball x e) \<le> aff_dim S"
using aff_dim_subset by auto
with e show ?thesis
using aff_dim_cball[of e x] aff_dim_subset_univ[of S] by auto
qed
lemma low_dim_interior:
fixes S :: "'n::euclidean_space set"
assumes "\<not> aff_dim S = int (DIM('n))"
shows "interior S = {}"
proof -
have "aff_dim(interior S) \<le> aff_dim S"
using interior_subset aff_dim_subset[of "interior S" S] by auto
then show ?thesis
using aff_dim_open[of "interior S"] aff_dim_subset_univ[of S] assms by auto
qed
corollary empty_interior_lowdim:
fixes S :: "'n::euclidean_space set"
shows "dim S < DIM ('n) \<Longrightarrow> interior S = {}"
by (metis low_dim_interior affine_hull_univ dim_affine_hull less_not_refl dim_UNIV)
subsection {* Relative interior of a set *}
definition "rel_interior S =
{x. \<exists>T. openin (subtopology euclidean (affine hull S)) T \<and> x \<in> T \<and> T \<subseteq> S}"
lemma rel_interior:
"rel_interior S = {x \<in> S. \<exists>T. open T \<and> x \<in> T \<and> T \<inter> affine hull S \<subseteq> S}"
unfolding rel_interior_def[of S] openin_open[of "affine hull S"]
apply auto
proof -
fix x T
assume *: "x \<in> S" "open T" "x \<in> T" "T \<inter> affine hull S \<subseteq> S"
then have **: "x \<in> T \<inter> affine hull S"
using hull_inc by auto
show "\<exists>Tb. (\<exists>Ta. open Ta \<and> Tb = affine hull S \<inter> Ta) \<and> x \<in> Tb \<and> Tb \<subseteq> S"
apply (rule_tac x = "T \<inter> (affine hull S)" in exI)
using * **
apply auto
done
qed
lemma mem_rel_interior: "x \<in> rel_interior S \<longleftrightarrow> (\<exists>T. open T \<and> x \<in> T \<inter> S \<and> T \<inter> affine hull S \<subseteq> S)"
by (auto simp add: rel_interior)
lemma mem_rel_interior_ball:
"x \<in> rel_interior S \<longleftrightarrow> x \<in> S \<and> (\<exists>e. e > 0 \<and> ball x e \<inter> affine hull S \<subseteq> S)"
apply (simp add: rel_interior, safe)
apply (force simp add: open_contains_ball)
apply (rule_tac x = "ball x e" in exI)
apply simp
done
lemma rel_interior_ball:
"rel_interior S = {x \<in> S. \<exists>e. e > 0 \<and> ball x e \<inter> affine hull S \<subseteq> S}"
using mem_rel_interior_ball [of _ S] by auto
lemma mem_rel_interior_cball:
"x \<in> rel_interior S \<longleftrightarrow> x \<in> S \<and> (\<exists>e. e > 0 \<and> cball x e \<inter> affine hull S \<subseteq> S)"
apply (simp add: rel_interior, safe)
apply (force simp add: open_contains_cball)
apply (rule_tac x = "ball x e" in exI)
apply (simp add: subset_trans [OF ball_subset_cball])
apply auto
done
lemma rel_interior_cball:
"rel_interior S = {x \<in> S. \<exists>e. e > 0 \<and> cball x e \<inter> affine hull S \<subseteq> S}"
using mem_rel_interior_cball [of _ S] by auto
lemma rel_interior_empty [simp]: "rel_interior {} = {}"
by (auto simp add: rel_interior_def)
lemma affine_hull_sing [simp]: "affine hull {a :: 'n::euclidean_space} = {a}"
by (metis affine_hull_eq affine_sing)
lemma rel_interior_sing [simp]: "rel_interior {a :: 'n::euclidean_space} = {a}"
unfolding rel_interior_ball affine_hull_sing
apply auto
apply (rule_tac x = "1 :: real" in exI)
apply simp
done
lemma subset_rel_interior:
fixes S T :: "'n::euclidean_space set"
assumes "S \<subseteq> T"
and "affine hull S = affine hull T"
shows "rel_interior S \<subseteq> rel_interior T"
using assms by (auto simp add: rel_interior_def)
lemma rel_interior_subset: "rel_interior S \<subseteq> S"
by (auto simp add: rel_interior_def)
lemma rel_interior_subset_closure: "rel_interior S \<subseteq> closure S"
using rel_interior_subset by (auto simp add: closure_def)
lemma interior_subset_rel_interior: "interior S \<subseteq> rel_interior S"
by (auto simp add: rel_interior interior_def)
lemma interior_rel_interior:
fixes S :: "'n::euclidean_space set"
assumes "aff_dim S = int(DIM('n))"
shows "rel_interior S = interior S"
proof -
have "affine hull S = UNIV"
using assms affine_hull_univ[of S] by auto
then show ?thesis
unfolding rel_interior interior_def by auto
qed
lemma rel_interior_interior:
fixes S :: "'n::euclidean_space set"
assumes "affine hull S = UNIV"
shows "rel_interior S = interior S"
using assms unfolding rel_interior interior_def by auto
lemma rel_interior_open:
fixes S :: "'n::euclidean_space set"
assumes "open S"
shows "rel_interior S = S"
by (metis assms interior_eq interior_subset_rel_interior rel_interior_subset set_eq_subset)
lemma interior_rel_interior_gen:
fixes S :: "'n::euclidean_space set"
shows "interior S = (if aff_dim S = int(DIM('n)) then rel_interior S else {})"
by (metis interior_rel_interior low_dim_interior)
lemma rel_interior_univ:
fixes S :: "'n::euclidean_space set"
shows "rel_interior (affine hull S) = affine hull S"
proof -
have *: "rel_interior (affine hull S) \<subseteq> affine hull S"
using rel_interior_subset by auto
{
fix x
assume x: "x \<in> affine hull S"
def e \<equiv> "1::real"
then have "e > 0" "ball x e \<inter> affine hull (affine hull S) \<subseteq> affine hull S"
using hull_hull[of _ S] by auto
then have "x \<in> rel_interior (affine hull S)"
using x rel_interior_ball[of "affine hull S"] by auto
}
then show ?thesis using * by auto
qed
lemma rel_interior_univ2: "rel_interior (UNIV :: ('n::euclidean_space) set) = UNIV"
by (metis open_UNIV rel_interior_open)
lemma rel_interior_convex_shrink:
fixes S :: "'a::euclidean_space set"
assumes "convex S"
and "c \<in> rel_interior S"
and "x \<in> S"
and "0 < e"
and "e \<le> 1"
shows "x - e *\<^sub>R (x - c) \<in> rel_interior S"
proof -
obtain d where "d > 0" and d: "ball c d \<inter> affine hull S \<subseteq> S"
using assms(2) unfolding mem_rel_interior_ball by auto
{
fix y
assume as: "dist (x - e *\<^sub>R (x - c)) y < e * d" "y \<in> affine hull S"
have *: "y = (1 - (1 - e)) *\<^sub>R ((1 / e) *\<^sub>R y - ((1 - e) / e) *\<^sub>R x) + (1 - e) *\<^sub>R x"
using `e > 0` by (auto simp add: scaleR_left_diff_distrib scaleR_right_diff_distrib)
have "x \<in> affine hull S"
using assms hull_subset[of S] by auto
moreover have "1 / e + - ((1 - e) / e) = 1"
using `e > 0` left_diff_distrib[of "1" "(1-e)" "1/e"] by auto
ultimately have **: "(1 / e) *\<^sub>R y - ((1 - e) / e) *\<^sub>R x \<in> affine hull S"
using as affine_affine_hull[of S] mem_affine[of "affine hull S" y x "(1 / e)" "-((1 - e) / e)"]
by (simp add: algebra_simps)
have "dist c ((1 / e) *\<^sub>R y - ((1 - e) / e) *\<^sub>R x) = abs(1/e) * norm (e *\<^sub>R c - y + (1 - e) *\<^sub>R x)"
unfolding dist_norm norm_scaleR[symmetric]
apply (rule arg_cong[where f=norm])
using `e > 0`
apply (auto simp add: euclidean_eq_iff[where 'a='a] field_simps inner_simps)
done
also have "\<dots> = abs (1/e) * 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 `e > 0`
by (auto simp add:pos_divide_less_eq[OF `e > 0`] mult.commute)
finally have "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
}
then have "ball (x - e *\<^sub>R (x - c)) (e*d) \<inter> affine hull S \<subseteq> S"
by auto
moreover have "e * d > 0"
using `e > 0` `d > 0` by simp
moreover have c: "c \<in> S"
using assms rel_interior_subset by auto
moreover from c have "x - e *\<^sub>R (x - c) \<in> S"
using mem_convex[of S x c e]
apply (simp add: algebra_simps)
using assms
apply auto
done
ultimately show ?thesis
using mem_rel_interior_ball[of "x - e *\<^sub>R (x - c)" S] `e > 0` by auto
qed
lemma interior_real_semiline:
fixes a :: real
shows "interior {a..} = {a<..}"
proof -
{
fix y
assume "a < y"
then have "y \<in> interior {a..}"
apply (simp add: mem_interior)
apply (rule_tac x="(y-a)" in exI)
apply (auto simp add: dist_norm)
done
}
moreover
{
fix y
assume "y \<in> interior {a..}"
then obtain e where e: "e > 0" "cball y e \<subseteq> {a..}"
using mem_interior_cball[of y "{a..}"] by auto
moreover from e have "y - e \<in> cball y e"
by (auto simp add: cball_def dist_norm)
ultimately have "a \<le> y - e" by blast
then have "a < y" using e by auto
}
ultimately show ?thesis by auto
qed
lemma rel_interior_real_box:
fixes a b :: real
assumes "a < b"
shows "rel_interior {a .. b} = {a <..< b}"
proof -
have "box a b \<noteq> {}"
using assms
unfolding set_eq_iff
by (auto intro!: exI[of _ "(a + b) / 2"] simp: box_def)
then show ?thesis
using interior_rel_interior_gen[of "cbox a b", symmetric]
by (simp split: split_if_asm del: box_real add: box_real[symmetric] interior_cbox)
qed
lemma rel_interior_real_semiline:
fixes a :: real
shows "rel_interior {a..} = {a<..}"
proof -
have *: "{a<..} \<noteq> {}"
unfolding set_eq_iff by (auto intro!: exI[of _ "a + 1"])
then show ?thesis using interior_real_semiline interior_rel_interior_gen[of "{a..}"]
by (auto split: split_if_asm)
qed
subsubsection {* Relative open sets *}
definition "rel_open S \<longleftrightarrow> rel_interior S = S"
lemma rel_open: "rel_open S \<longleftrightarrow> openin (subtopology euclidean (affine hull S)) S"
unfolding rel_open_def rel_interior_def
apply auto
using openin_subopen[of "subtopology euclidean (affine hull S)" S]
apply auto
done
lemma opein_rel_interior: "openin (subtopology euclidean (affine hull S)) (rel_interior S)"
apply (simp add: rel_interior_def)
apply (subst openin_subopen)
apply blast
done
lemma affine_rel_open:
fixes S :: "'n::euclidean_space set"
assumes "affine S"
shows "rel_open S"
unfolding rel_open_def
using assms rel_interior_univ[of S] affine_hull_eq[of S]
by metis
lemma affine_closed:
fixes S :: "'n::euclidean_space set"
assumes "affine S"
shows "closed S"
proof -
{
assume "S \<noteq> {}"
then obtain L where L: "subspace L" "affine_parallel S L"
using assms affine_parallel_subspace[of S] by auto
then obtain a where a: "S = (op + a ` L)"
using affine_parallel_def[of L S] affine_parallel_commut by auto
from L have "closed L" using closed_subspace by auto
then have "closed S"
using closed_translation a by auto
}
then show ?thesis by auto
qed
lemma closure_affine_hull:
fixes S :: "'n::euclidean_space set"
shows "closure S \<subseteq> affine hull S"
by (intro closure_minimal hull_subset affine_closed affine_affine_hull)
lemma closure_same_affine_hull:
fixes S :: "'n::euclidean_space set"
shows "affine hull (closure S) = affine hull S"
proof -
have "affine hull (closure S) \<subseteq> affine hull S"
using hull_mono[of "closure S" "affine hull S" "affine"]
closure_affine_hull[of S] hull_hull[of "affine" S]
by auto
moreover have "affine hull (closure S) \<supseteq> affine hull S"
using hull_mono[of "S" "closure S" "affine"] closure_subset by auto
ultimately show ?thesis by auto
qed
lemma closure_aff_dim:
fixes S :: "'n::euclidean_space set"
shows "aff_dim (closure S) = aff_dim S"
proof -
have "aff_dim S \<le> aff_dim (closure S)"
using aff_dim_subset closure_subset by auto
moreover have "aff_dim (closure S) \<le> aff_dim (affine hull S)"
using aff_dim_subset closure_affine_hull by auto
moreover have "aff_dim (affine hull S) = aff_dim S"
using aff_dim_affine_hull by auto
ultimately show ?thesis by auto
qed
lemma rel_interior_closure_convex_shrink:
fixes S :: "_::euclidean_space set"
assumes "convex S"
and "c \<in> rel_interior S"
and "x \<in> closure S"
and "e > 0"
and "e \<le> 1"
shows "x - e *\<^sub>R (x - c) \<in> rel_interior S"
proof -
obtain d where "d > 0" and d: "ball c d \<inter> affine hull S \<subseteq> S"
using assms(2) unfolding mem_rel_interior_ball 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 `e > 0` `d > 0`
apply (rule_tac bexI[where x=x])
apply (auto)
done
next
case False
then have x: "x islimpt S"
using assms(3)[unfolded closure_def] by auto
show ?thesis
proof (cases "e = 1")
case True
obtain y where "y \<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 `d > 0`
apply auto
done
next
case False
then have "0 < e * d / (1 - e)" and *: "1 - e > 0"
using `e \<le> 1` `e > 0` `d > 0` 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
def z \<equiv> "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 `e > 0`
by (auto simp add: scaleR_right_diff_distrib scaleR_right_distrib scaleR_left_diff_distrib)
have zball: "z \<in> ball c d"
using mem_ball z_def dist_norm[of c]
using y and assms(4,5)
by (auto simp add:field_simps norm_minus_commute)
have "x \<in> affine hull S"
using closure_affine_hull assms by auto
moreover have "y \<in> affine hull S"
using `y \<in> S` hull_subset[of S] by auto
moreover have "c \<in> affine hull S"
using assms rel_interior_subset hull_subset[of S] by auto
ultimately have "z \<in> affine hull S"
using z_def affine_affine_hull[of S]
mem_affine_3_minus [of "affine hull S" c x y "(1 - e) / e"]
assms
by (auto simp add: field_simps)
then have "z \<in> S" using d zball by auto
obtain d1 where "d1 > 0" and d1: "ball z d1 \<le> ball c d"
using zball open_ball[of c d] openE[of "ball c d" z] by auto
then have "ball z d1 \<inter> affine hull S \<subseteq> ball c d \<inter> affine hull S"
by auto
then have "ball z d1 \<inter> affine hull S \<subseteq> S"
using d by auto
then have "z \<in> rel_interior S"
using mem_rel_interior_ball using `d1 > 0` `z \<in> S` by auto
then have "y - e *\<^sub>R (y - z) \<in> rel_interior S"
using rel_interior_convex_shrink[of S z y e] assms `y \<in> S` by auto
then show ?thesis using * by auto
qed
subsubsection{* Relative interior preserves under linear transformations *}
lemma rel_interior_translation_aux:
fixes a :: "'n::euclidean_space"
shows "((\<lambda>x. a + x) ` rel_interior S) \<subseteq> rel_interior ((\<lambda>x. a + x) ` S)"
proof -
{
fix x
assume x: "x \<in> rel_interior S"
then obtain T where "open T" "x \<in> T \<inter> S" "T \<inter> affine hull S \<subseteq> S"
using mem_rel_interior[of x S] by auto
then have "open ((\<lambda>x. a + x) ` T)"
and "a + x \<in> ((\<lambda>x. a + x) ` T) \<inter> ((\<lambda>x. a + x) ` S)"
and "((\<lambda>x. a + x) ` T) \<inter> affine hull ((\<lambda>x. a + x) ` S) \<subseteq> (\<lambda>x. a + x) ` S"
using affine_hull_translation[of a S] open_translation[of T a] x by auto
then have "a + x \<in> rel_interior ((\<lambda>x. a + x) ` S)"
using mem_rel_interior[of "a+x" "((\<lambda>x. a + x) ` S)"] by auto
}
then show ?thesis by auto
qed
lemma rel_interior_translation:
fixes a :: "'n::euclidean_space"
shows "rel_interior ((\<lambda>x. a + x) ` S) = (\<lambda>x. a + x) ` rel_interior S"
proof -
have "(\<lambda>x. (-a) + x) ` rel_interior ((\<lambda>x. a + x) ` S) \<subseteq> rel_interior S"
using rel_interior_translation_aux[of "-a" "(\<lambda>x. a + x) ` S"]
translation_assoc[of "-a" "a"]
by auto
then have "((\<lambda>x. a + x) ` rel_interior S) \<supseteq> rel_interior ((\<lambda>x. a + x) ` S)"
using translation_inverse_subset[of a "rel_interior (op + a ` S)" "rel_interior S"]
by auto
then show ?thesis
using rel_interior_translation_aux[of a S] by auto
qed
lemma affine_hull_linear_image:
assumes "bounded_linear f"
shows "f ` (affine hull s) = affine hull f ` s"
apply rule
unfolding subset_eq ball_simps
apply (rule_tac[!] hull_induct, rule hull_inc)
prefer 3
apply (erule imageE)
apply (rule_tac x=xa in image_eqI)
apply assumption
apply (rule hull_subset[unfolded subset_eq, rule_format])
apply assumption
proof -
interpret f: bounded_linear f by fact
show "affine {x. f x \<in> affine hull f ` s}"
unfolding affine_def
by (auto simp add: f.scaleR f.add affine_affine_hull[unfolded affine_def, rule_format])
show "affine {x. x \<in> f ` (affine hull s)}"
using affine_affine_hull[unfolded affine_def, of s]
unfolding affine_def by (auto simp add: f.scaleR [symmetric] f.add [symmetric])
qed auto
lemma rel_interior_injective_on_span_linear_image:
fixes f :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space"
and S :: "'m::euclidean_space set"
assumes "bounded_linear f"
and "inj_on f (span S)"
shows "rel_interior (f ` S) = f ` (rel_interior S)"
proof -
{
fix z
assume z: "z \<in> rel_interior (f ` S)"
then have "z \<in> f ` S"
using rel_interior_subset[of "f ` S"] by auto
then obtain x where x: "x \<in> S" "f x = z" by auto
obtain e2 where e2: "e2 > 0" "cball z e2 \<inter> affine hull (f ` S) \<subseteq> (f ` S)"
using z rel_interior_cball[of "f ` S"] by auto
obtain K where K: "K > 0" "\<And>x. norm (f x) \<le> norm x * K"
using assms Real_Vector_Spaces.bounded_linear.pos_bounded[of f] by auto
def e1 \<equiv> "1 / K"
then have e1: "e1 > 0" "\<And>x. e1 * norm (f x) \<le> norm x"
using K pos_le_divide_eq[of e1] by auto
def e \<equiv> "e1 * e2"
then have "e > 0" using e1 e2 by auto
{
fix y
assume y: "y \<in> cball x e \<inter> affine hull S"
then have h1: "f y \<in> affine hull (f ` S)"
using affine_hull_linear_image[of f S] assms by auto
from y have "norm (x-y) \<le> e1 * e2"
using cball_def[of x e] dist_norm[of x y] e_def by auto
moreover have "f x - f y = f (x - y)"
using assms linear_sub[of f x y] linear_conv_bounded_linear[of f] by auto
moreover have "e1 * norm (f (x-y)) \<le> norm (x - y)"
using e1 by auto
ultimately have "e1 * norm ((f x)-(f y)) \<le> e1 * e2"
by auto
then have "f y \<in> cball z e2"
using cball_def[of "f x" e2] dist_norm[of "f x" "f y"] e1 x by auto
then have "f y \<in> f ` S"
using y e2 h1 by auto
then have "y \<in> S"
using assms y hull_subset[of S] affine_hull_subset_span
inj_on_image_mem_iff[of f "span S" S y]
by auto
}
then have "z \<in> f ` (rel_interior S)"
using mem_rel_interior_cball[of x S] `e > 0` x by auto
}
moreover
{
fix x
assume x: "x \<in> rel_interior S"
then obtain e2 where e2: "e2 > 0" "cball x e2 \<inter> affine hull S \<subseteq> S"
using rel_interior_cball[of S] by auto
have "x \<in> S" using x rel_interior_subset by auto
then have *: "f x \<in> f ` S" by auto
have "\<forall>x\<in>span S. f x = 0 \<longrightarrow> x = 0"
using assms subspace_span linear_conv_bounded_linear[of f]
linear_injective_on_subspace_0[of f "span S"]
by auto
then obtain e1 where e1: "e1 > 0" "\<forall>x \<in> span S. e1 * norm x \<le> norm (f x)"
using assms injective_imp_isometric[of "span S" f]
subspace_span[of S] closed_subspace[of "span S"]
by auto
def e \<equiv> "e1 * e2"
hence "e > 0" using e1 e2 by auto
{
fix y
assume y: "y \<in> cball (f x) e \<inter> affine hull (f ` S)"
then have "y \<in> f ` (affine hull S)"
using affine_hull_linear_image[of f S] assms by auto
then obtain xy where xy: "xy \<in> affine hull S" "f xy = y" by auto
with y have "norm (f x - f xy) \<le> e1 * e2"
using cball_def[of "f x" e] dist_norm[of "f x" y] e_def by auto
moreover have "f x - f xy = f (x - xy)"
using assms linear_sub[of f x xy] linear_conv_bounded_linear[of f] by auto
moreover have *: "x - xy \<in> span S"
using subspace_sub[of "span S" x xy] subspace_span `x \<in> S` xy
affine_hull_subset_span[of S] span_inc
by auto
moreover from * have "e1 * norm (x - xy) \<le> norm (f (x - xy))"
using e1 by auto
ultimately have "e1 * norm (x - xy) \<le> e1 * e2"
by auto
then have "xy \<in> cball x e2"
using cball_def[of x e2] dist_norm[of x xy] e1 by auto
then have "y \<in> f ` S"
using xy e2 by auto
}
then have "f x \<in> rel_interior (f ` S)"
using mem_rel_interior_cball[of "(f x)" "(f ` S)"] * `e > 0` by auto
}
ultimately show ?thesis by auto
qed
lemma rel_interior_injective_linear_image:
fixes f :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space"
assumes "bounded_linear f"
and "inj f"
shows "rel_interior (f ` S) = f ` (rel_interior S)"
using assms rel_interior_injective_on_span_linear_image[of f S]
subset_inj_on[of f "UNIV" "span S"]
by auto
subsection{* Some Properties of subset of standard basis *}
lemma affine_hull_substd_basis:
assumes "d \<subseteq> Basis"
shows "affine hull (insert 0 d) = {x::'a::euclidean_space. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x\<bullet>i = 0}"
(is "affine hull (insert 0 ?A) = ?B")
proof -
have *: "\<And>A. op + (0\<Colon>'a) ` A = A" "\<And>A. op + (- (0\<Colon>'a)) ` A = A"
by auto
show ?thesis
unfolding affine_hull_insert_span_gen span_substd_basis[OF assms,symmetric] * ..
qed
lemma affine_hull_convex_hull [simp]: "affine hull (convex hull S) = affine hull S"
by (metis Int_absorb1 Int_absorb2 convex_hull_subset_affine_hull hull_hull hull_mono hull_subset)
subsection {* Openness and compactness are preserved by convex hull operation. *}
lemma open_convex_hull[intro]:
fixes s :: "'a::real_normed_vector set"
assumes "open s"
shows "open (convex hull s)"
unfolding open_contains_cball convex_hull_explicit
unfolding mem_Collect_eq ball_simps(8)
proof (rule, rule)
fix a
assume "\<exists>sa u. finite sa \<and> sa \<subseteq> s \<and> (\<forall>x\<in>sa. 0 \<le> u x) \<and> setsum u sa = 1 \<and> (\<Sum>v\<in>sa. u v *\<^sub>R v) = a"
then obtain t u where obt: "finite t" "t\<subseteq>s" "\<forall>x\<in>t. 0 \<le> u x" "setsum u t = 1" "(\<Sum>v\<in>t. u v *\<^sub>R v) = a"
by auto
from assms[unfolded open_contains_cball] obtain b
where b: "\<forall>x\<in>s. 0 < b x \<and> cball x (b x) \<subseteq> s"
using bchoice[of s "\<lambda>x e. e > 0 \<and> cball x e \<subseteq> s"] by auto
have "b ` t \<noteq> {}"
using obt by auto
def i \<equiv> "b ` t"
show "\<exists>e > 0.
cball a e \<subseteq> {y. \<exists>sa u. finite sa \<and> sa \<subseteq> s \<and> (\<forall>x\<in>sa. 0 \<le> u x) \<and> setsum u sa = 1 \<and> (\<Sum>v\<in>sa. u v *\<^sub>R v) = y}"
apply (rule_tac x = "Min i" in exI)
unfolding subset_eq
apply rule
defer
apply rule
unfolding mem_Collect_eq
proof -
show "0 < Min i"
unfolding i_def and Min_gr_iff[OF finite_imageI[OF obt(1)] `b \` t\<noteq>{}`]
using b
apply simp
apply rule
apply (erule_tac x=x in ballE)
using `t\<subseteq>s`
apply auto
done
next
fix y
assume "y \<in> cball a (Min i)"
then have y: "norm (a - y) \<le> Min i"
unfolding dist_norm[symmetric] by auto
{
fix x
assume "x \<in> t"
then have "Min i \<le> b x"
unfolding i_def
apply (rule_tac Min_le)
using obt(1)
apply auto
done
then have "x + (y - a) \<in> cball x (b x)"
using y unfolding mem_cball dist_norm by auto
moreover from `x\<in>t` have "x \<in> s"
using obt(2) by auto
ultimately have "x + (y - a) \<in> s"
using y and b[THEN bspec[where x=x]] unfolding subset_eq by fast
}
moreover
have *: "inj_on (\<lambda>v. v + (y - a)) t"
unfolding inj_on_def by auto
have "(\<Sum>v\<in>(\<lambda>v. v + (y - a)) ` t. u (v - (y - a))) = 1"
unfolding setsum.reindex[OF *] o_def using obt(4) by auto
moreover have "(\<Sum>v\<in>(\<lambda>v. v + (y - a)) ` t. u (v - (y - a)) *\<^sub>R v) = y"
unfolding setsum.reindex[OF *] o_def using obt(4,5)
by (simp add: setsum.distrib setsum_subtractf scaleR_left.setsum[symmetric] scaleR_right_distrib)
ultimately
show "\<exists>sa u. finite sa \<and> (\<forall>x\<in>sa. x \<in> s) \<and> (\<forall>x\<in>sa. 0 \<le> u x) \<and> setsum u sa = 1 \<and> (\<Sum>v\<in>sa. u v *\<^sub>R v) = y"
apply (rule_tac x="(\<lambda>v. v + (y - a)) ` t" in exI)
apply (rule_tac x="\<lambda>v. u (v - (y - a))" in exI)
using obt(1, 3)
apply auto
done
qed
qed
lemma compact_convex_combinations:
fixes s t :: "'a::real_normed_vector set"
assumes "compact s" "compact t"
shows "compact { (1 - u) *\<^sub>R x + u *\<^sub>R y | x y u. 0 \<le> u \<and> u \<le> 1 \<and> x \<in> s \<and> y \<in> t}"
proof -
let ?X = "{0..1} \<times> s \<times> t"
let ?h = "(\<lambda>z. (1 - fst z) *\<^sub>R fst (snd z) + fst z *\<^sub>R snd (snd z))"
have *: "{ (1 - u) *\<^sub>R x + u *\<^sub>R y | x y u. 0 \<le> u \<and> u \<le> 1 \<and> x \<in> s \<and> y \<in> t} = ?h ` ?X"
apply (rule set_eqI)
unfolding image_iff mem_Collect_eq
apply rule
apply auto
apply (rule_tac x=u in rev_bexI)
apply simp
apply (erule rev_bexI)
apply (erule rev_bexI)
apply simp
apply auto
done
have "continuous_on ?X (\<lambda>z. (1 - fst z) *\<^sub>R fst (snd z) + fst z *\<^sub>R snd (snd z))"
unfolding continuous_on by (rule ballI) (intro tendsto_intros)
then show ?thesis
unfolding *
apply (rule compact_continuous_image)
apply (intro compact_Times compact_Icc assms)
done
qed
lemma finite_imp_compact_convex_hull:
fixes s :: "'a::real_normed_vector set"
assumes "finite s"
shows "compact (convex hull s)"
proof (cases "s = {}")
case True
then show ?thesis by simp
next
case False
with assms show ?thesis
proof (induct rule: finite_ne_induct)
case (singleton x)
show ?case by simp
next
case (insert x A)
let ?f = "\<lambda>(u, y::'a). u *\<^sub>R x + (1 - u) *\<^sub>R y"
let ?T = "{0..1::real} \<times> (convex hull A)"
have "continuous_on ?T ?f"
unfolding split_def continuous_on by (intro ballI tendsto_intros)
moreover have "compact ?T"
by (intro compact_Times compact_Icc insert)
ultimately have "compact (?f ` ?T)"
by (rule compact_continuous_image)
also have "?f ` ?T = convex hull (insert x A)"
unfolding convex_hull_insert [OF `A \<noteq> {}`]
apply safe
apply (rule_tac x=a in exI, simp)
apply (rule_tac x="1 - a" in exI, simp)
apply fast
apply (rule_tac x="(u, b)" in image_eqI, simp_all)
done
finally show "compact (convex hull (insert x A))" .
qed
qed
lemma compact_convex_hull:
fixes s :: "'a::euclidean_space set"
assumes "compact s"
shows "compact (convex hull s)"
proof (cases "s = {}")
case True
then show ?thesis using compact_empty by simp
next
case False
then obtain w where "w \<in> s" by auto
show ?thesis
unfolding caratheodory[of s]
proof (induct ("DIM('a) + 1"))
case 0
have *: "{x.\<exists>sa. finite sa \<and> sa \<subseteq> s \<and> card sa \<le> 0 \<and> x \<in> convex hull sa} = {}"
using compact_empty by auto
from 0 show ?case unfolding * by simp
next
case (Suc n)
show ?case
proof (cases "n = 0")
case True
have "{x. \<exists>t. finite t \<and> t \<subseteq> s \<and> card t \<le> Suc n \<and> x \<in> convex hull t} = s"
unfolding set_eq_iff and mem_Collect_eq
proof (rule, rule)
fix x
assume "\<exists>t. finite t \<and> t \<subseteq> s \<and> card t \<le> Suc n \<and> x \<in> convex hull t"
then obtain t where t: "finite t" "t \<subseteq> s" "card t \<le> Suc n" "x \<in> convex hull t"
by auto
show "x \<in> s"
proof (cases "card t = 0")
case True
then show ?thesis
using t(4) unfolding card_0_eq[OF t(1)] by simp
next
case False
then have "card t = Suc 0" using t(3) `n=0` by auto
then obtain a where "t = {a}" unfolding card_Suc_eq by auto
then show ?thesis using t(2,4) by simp
qed
next
fix x assume "x\<in>s"
then show "\<exists>t. finite t \<and> t \<subseteq> s \<and> card t \<le> Suc n \<and> x \<in> convex hull t"
apply (rule_tac x="{x}" in exI)
unfolding convex_hull_singleton
apply auto
done
qed
then show ?thesis using assms by simp
next
case False
have "{x. \<exists>t. finite t \<and> t \<subseteq> s \<and> card t \<le> Suc n \<and> x \<in> convex hull t} =
{(1 - u) *\<^sub>R x + u *\<^sub>R y | x y u.
0 \<le> u \<and> u \<le> 1 \<and> x \<in> s \<and> y \<in> {x. \<exists>t. finite t \<and> t \<subseteq> s \<and> card t \<le> n \<and> x \<in> convex hull t}}"
unfolding set_eq_iff and mem_Collect_eq
proof (rule, rule)
fix x
assume "\<exists>u v c. x = (1 - c) *\<^sub>R u + c *\<^sub>R v \<and>
0 \<le> c \<and> c \<le> 1 \<and> u \<in> s \<and> (\<exists>t. finite t \<and> t \<subseteq> s \<and> card t \<le> n \<and> v \<in> convex hull t)"
then obtain u v c t where obt: "x = (1 - c) *\<^sub>R u + c *\<^sub>R v"
"0 \<le> c \<and> c \<le> 1" "u \<in> s" "finite t" "t \<subseteq> s" "card t \<le> n" "v \<in> convex hull t"
by auto
moreover have "(1 - c) *\<^sub>R u + c *\<^sub>R v \<in> convex hull insert u t"
apply (rule mem_convex)
using obt(2) and convex_convex_hull and hull_subset[of "insert u t" convex]
using obt(7) and hull_mono[of t "insert u t"]
apply auto
done
ultimately show "\<exists>t. finite t \<and> t \<subseteq> s \<and> card t \<le> Suc n \<and> x \<in> convex hull t"
apply (rule_tac x="insert u t" in exI)
apply (auto simp add: card_insert_if)
done
next
fix x
assume "\<exists>t. finite t \<and> t \<subseteq> s \<and> card t \<le> Suc n \<and> x \<in> convex hull t"
then obtain t where t: "finite t" "t \<subseteq> s" "card t \<le> Suc n" "x \<in> convex hull t"
by auto
show "\<exists>u v c. x = (1 - c) *\<^sub>R u + c *\<^sub>R v \<and>
0 \<le> c \<and> c \<le> 1 \<and> u \<in> s \<and> (\<exists>t. finite t \<and> t \<subseteq> s \<and> card t \<le> n \<and> v \<in> convex hull t)"
proof (cases "card t = Suc n")
case False
then have "card t \<le> n" using t(3) by auto
then show ?thesis
apply (rule_tac x=w in exI, rule_tac x=x in exI, rule_tac x=1 in exI)
using `w\<in>s` and t
apply (auto intro!: exI[where x=t])
done
next
case True
then obtain a u where au: "t = insert a u" "a\<notin>u"
apply (drule_tac card_eq_SucD)
apply auto
done
show ?thesis
proof (cases "u = {}")
case True
then have "x = a" using t(4)[unfolded au] by auto
show ?thesis unfolding `x = a`
apply (rule_tac x=a in exI)
apply (rule_tac x=a in exI)
apply (rule_tac x=1 in exI)
using t and `n \<noteq> 0`
unfolding au
apply (auto intro!: exI[where x="{a}"])
done
next
case False
obtain ux vx b where obt: "ux\<ge>0" "vx\<ge>0" "ux + vx = 1"
"b \<in> convex hull u" "x = ux *\<^sub>R a + vx *\<^sub>R b"
using t(4)[unfolded au convex_hull_insert[OF False]]
by auto
have *: "1 - vx = ux" using obt(3) by auto
show ?thesis
apply (rule_tac x=a in exI)
apply (rule_tac x=b in exI)
apply (rule_tac x=vx in exI)
using obt and t(1-3)
unfolding au and * using card_insert_disjoint[OF _ au(2)]
apply (auto intro!: exI[where x=u])
done
qed
qed
qed
then show ?thesis
using compact_convex_combinations[OF assms Suc] by simp
qed
qed
qed
subsection {* Extremal points of a simplex are some vertices. *}
lemma dist_increases_online:
fixes a b d :: "'a::real_inner"
assumes "d \<noteq> 0"
shows "dist a (b + d) > dist a b \<or> dist a (b - d) > dist a b"
proof (cases "inner a d - inner b d > 0")
case True
then have "0 < inner d d + (inner a d * 2 - inner b d * 2)"
apply (rule_tac add_pos_pos)
using assms
apply auto
done
then show ?thesis
apply (rule_tac disjI2)
unfolding dist_norm and norm_eq_sqrt_inner and real_sqrt_less_iff
apply (simp add: algebra_simps inner_commute)
done
next
case False
then have "0 < inner d d + (inner b d * 2 - inner a d * 2)"
apply (rule_tac add_pos_nonneg)
using assms
apply auto
done
then show ?thesis
apply (rule_tac disjI1)
unfolding dist_norm and norm_eq_sqrt_inner and real_sqrt_less_iff
apply (simp add: algebra_simps inner_commute)
done
qed
lemma norm_increases_online:
fixes d :: "'a::real_inner"
shows "d \<noteq> 0 \<Longrightarrow> norm (a + d) > norm a \<or> norm(a - d) > norm a"
using dist_increases_online[of d a 0] unfolding dist_norm by auto
lemma simplex_furthest_lt:
fixes s :: "'a::real_inner set"
assumes "finite s"
shows "\<forall>x \<in> convex hull s. x \<notin> s \<longrightarrow> (\<exists>y \<in> convex hull s. norm (x - a) < norm(y - a))"
using assms
proof induct
fix x s
assume as: "finite s" "x\<notin>s" "\<forall>x\<in>convex hull s. x \<notin> s \<longrightarrow> (\<exists>y\<in>convex hull s. norm (x - a) < norm (y - a))"
show "\<forall>xa\<in>convex hull insert x s. xa \<notin> insert x s \<longrightarrow>
(\<exists>y\<in>convex hull insert x s. norm (xa - a) < norm (y - a))"
proof (rule, rule, cases "s = {}")
case False
fix y
assume y: "y \<in> convex hull insert x s" "y \<notin> insert x s"
obtain u v b where obt: "u\<ge>0" "v\<ge>0" "u + v = 1" "b \<in> convex hull s" "y = u *\<^sub>R x + v *\<^sub>R b"
using y(1)[unfolded convex_hull_insert[OF False]] by auto
show "\<exists>z\<in>convex hull insert x s. norm (y - a) < norm (z - a)"
proof (cases "y \<in> convex hull s")
case True
then obtain z where "z \<in> convex hull s" "norm (y - a) < norm (z - a)"
using as(3)[THEN bspec[where x=y]] and y(2) by auto
then show ?thesis
apply (rule_tac x=z in bexI)
unfolding convex_hull_insert[OF False]
apply auto
done
next
case False
show ?thesis
using obt(3)
proof (cases "u = 0", case_tac[!] "v = 0")
assume "u = 0" "v \<noteq> 0"
then have "y = b" using obt by auto
then show ?thesis using False and obt(4) by auto
next
assume "u \<noteq> 0" "v = 0"
then have "y = x" using obt by auto
then show ?thesis using y(2) by auto
next
assume "u \<noteq> 0" "v \<noteq> 0"
then obtain w where w: "w>0" "w<u" "w<v"
using real_lbound_gt_zero[of u v] and obt(1,2) by auto
have "x \<noteq> b"
proof
assume "x = b"
then have "y = b" unfolding obt(5)
using obt(3) by (auto simp add: scaleR_left_distrib[symmetric])
then show False using obt(4) and False by simp
qed
then have *: "w *\<^sub>R (x - b) \<noteq> 0" using w(1) by auto
show ?thesis
using dist_increases_online[OF *, of a y]
proof (elim disjE)
assume "dist a y < dist a (y + w *\<^sub>R (x - b))"
then have "norm (y - a) < norm ((u + w) *\<^sub>R x + (v - w) *\<^sub>R b - a)"
unfolding dist_commute[of a]
unfolding dist_norm obt(5)
by (simp add: algebra_simps)
moreover have "(u + w) *\<^sub>R x + (v - w) *\<^sub>R b \<in> convex hull insert x s"
unfolding convex_hull_insert[OF `s\<noteq>{}`] and mem_Collect_eq
apply (rule_tac x="u + w" in exI)
apply rule
defer
apply (rule_tac x="v - w" in exI)
using `u \<ge> 0` and w and obt(3,4)
apply auto
done
ultimately show ?thesis by auto
next
assume "dist a y < dist a (y - w *\<^sub>R (x - b))"
then have "norm (y - a) < norm ((u - w) *\<^sub>R x + (v + w) *\<^sub>R b - a)"
unfolding dist_commute[of a]
unfolding dist_norm obt(5)
by (simp add: algebra_simps)
moreover have "(u - w) *\<^sub>R x + (v + w) *\<^sub>R b \<in> convex hull insert x s"
unfolding convex_hull_insert[OF `s\<noteq>{}`] and mem_Collect_eq
apply (rule_tac x="u - w" in exI)
apply rule
defer
apply (rule_tac x="v + w" in exI)
using `u \<ge> 0` and w and obt(3,4)
apply auto
done
ultimately show ?thesis by auto
qed
qed auto
qed
qed auto
qed (auto simp add: assms)
lemma simplex_furthest_le:
fixes s :: "'a::real_inner set"
assumes "finite s"
and "s \<noteq> {}"
shows "\<exists>y\<in>s. \<forall>x\<in> convex hull s. norm (x - a) \<le> norm (y - a)"
proof -
have "convex hull s \<noteq> {}"
using hull_subset[of s convex] and assms(2) by auto
then obtain x where x: "x \<in> convex hull s" "\<forall>y\<in>convex hull s. norm (y - a) \<le> norm (x - a)"
using distance_attains_sup[OF finite_imp_compact_convex_hull[OF assms(1)], of a]
unfolding dist_commute[of a]
unfolding dist_norm
by auto
show ?thesis
proof (cases "x \<in> s")
case False
then obtain y where "y \<in> convex hull s" "norm (x - a) < norm (y - a)"
using simplex_furthest_lt[OF assms(1), THEN bspec[where x=x]] and x(1)
by auto
then show ?thesis
using x(2)[THEN bspec[where x=y]] by auto
next
case True
with x show ?thesis by auto
qed
qed
lemma simplex_furthest_le_exists:
fixes s :: "('a::real_inner) set"
shows "finite s \<Longrightarrow> \<forall>x\<in>(convex hull s). \<exists>y\<in>s. norm (x - a) \<le> norm (y - a)"
using simplex_furthest_le[of s] by (cases "s = {}") auto
lemma simplex_extremal_le:
fixes s :: "'a::real_inner set"
assumes "finite s"
and "s \<noteq> {}"
shows "\<exists>u\<in>s. \<exists>v\<in>s. \<forall>x\<in>convex hull s. \<forall>y \<in> convex hull s. norm (x - y) \<le> norm (u - v)"
proof -
have "convex hull s \<noteq> {}"
using hull_subset[of s convex] and assms(2) by auto
then obtain u v where obt: "u \<in> convex hull s" "v \<in> convex hull s"
"\<forall>x\<in>convex hull s. \<forall>y\<in>convex hull s. norm (x - y) \<le> norm (u - v)"
using compact_sup_maxdistance[OF finite_imp_compact_convex_hull[OF assms(1)]]
by (auto simp: dist_norm)
then show ?thesis
proof (cases "u\<notin>s \<or> v\<notin>s", elim disjE)
assume "u \<notin> s"
then obtain y where "y \<in> convex hull s" "norm (u - v) < norm (y - v)"
using simplex_furthest_lt[OF assms(1), THEN bspec[where x=u]] and obt(1)
by auto
then show ?thesis
using obt(3)[THEN bspec[where x=y], THEN bspec[where x=v]] and obt(2)
by auto
next
assume "v \<notin> s"
then obtain y where "y \<in> convex hull s" "norm (v - u) < norm (y - u)"
using simplex_furthest_lt[OF assms(1), THEN bspec[where x=v]] and obt(2)
by auto
then show ?thesis
using obt(3)[THEN bspec[where x=u], THEN bspec[where x=y]] and obt(1)
by (auto simp add: norm_minus_commute)
qed auto
qed
lemma simplex_extremal_le_exists:
fixes s :: "'a::real_inner set"
shows "finite s \<Longrightarrow> x \<in> convex hull s \<Longrightarrow> y \<in> convex hull s \<Longrightarrow>
\<exists>u\<in>s. \<exists>v\<in>s. norm (x - y) \<le> norm (u - v)"
using convex_hull_empty simplex_extremal_le[of s]
by(cases "s = {}") auto
subsection {* Closest point of a convex set is unique, with a continuous projection. *}
definition closest_point :: "'a::{real_inner,heine_borel} set \<Rightarrow> 'a \<Rightarrow> 'a"
where "closest_point s a = (SOME x. x \<in> s \<and> (\<forall>y\<in>s. dist a x \<le> dist a y))"
lemma closest_point_exists:
assumes "closed s"
and "s \<noteq> {}"
shows "closest_point s a \<in> s"
and "\<forall>y\<in>s. dist a (closest_point s a) \<le> dist a y"
unfolding closest_point_def
apply(rule_tac[!] someI2_ex)
using distance_attains_inf[OF assms(1,2), of a]
apply auto
done
lemma closest_point_in_set: "closed s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> closest_point s a \<in> s"
by (meson closest_point_exists)
lemma closest_point_le: "closed s \<Longrightarrow> x \<in> s \<Longrightarrow> dist a (closest_point s a) \<le> dist a x"
using closest_point_exists[of s] by auto
lemma closest_point_self:
assumes "x \<in> s"
shows "closest_point s x = x"
unfolding closest_point_def
apply (rule some1_equality, rule ex1I[of _ x])
using assms
apply auto
done
lemma closest_point_refl: "closed s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> closest_point s x = x \<longleftrightarrow> x \<in> s"
using closest_point_in_set[of s x] closest_point_self[of x s]
by auto
lemma closer_points_lemma:
assumes "inner y z > 0"
shows "\<exists>u>0. \<forall>v>0. v \<le> u \<longrightarrow> norm(v *\<^sub>R z - y) < norm y"
proof -
have z: "inner z z > 0"
unfolding inner_gt_zero_iff using assms by auto
then show ?thesis
using assms
apply (rule_tac x = "inner y z / inner z z" in exI)
apply rule
defer
proof rule+
fix v
assume "0 < v" and "v \<le> inner y z / inner z z"
then show "norm (v *\<^sub>R z - y) < norm y"
unfolding norm_lt using z and assms
by (simp add: field_simps inner_diff inner_commute mult_strict_left_mono[OF _ `0<v`])
qed auto
qed
lemma closer_point_lemma:
assumes "inner (y - x) (z - x) > 0"
shows "\<exists>u>0. u \<le> 1 \<and> dist (x + u *\<^sub>R (z - x)) y < dist x y"
proof -
obtain u where "u > 0"
and u: "\<forall>v>0. v \<le> u \<longrightarrow> norm (v *\<^sub>R (z - x) - (y - x)) < norm (y - x)"
using closer_points_lemma[OF assms] by auto
show ?thesis
apply (rule_tac x="min u 1" in exI)
using u[THEN spec[where x="min u 1"]] and `u > 0`
unfolding dist_norm by (auto simp add: norm_minus_commute field_simps)
qed
lemma any_closest_point_dot:
assumes "convex s" "closed s" "x \<in> s" "y \<in> s" "\<forall>z\<in>s. dist a x \<le> dist a z"
shows "inner (a - x) (y - x) \<le> 0"
proof (rule ccontr)
assume "\<not> ?thesis"
then obtain u where u: "u>0" "u\<le>1" "dist (x + u *\<^sub>R (y - x)) a < dist x a"
using closer_point_lemma[of a x y] by auto
let ?z = "(1 - u) *\<^sub>R x + u *\<^sub>R y"
have "?z \<in> s"
using mem_convex[OF assms(1,3,4), of u] using u by auto
then show False
using assms(5)[THEN bspec[where x="?z"]] and u(3)
by (auto simp add: dist_commute algebra_simps)
qed
lemma any_closest_point_unique:
fixes x :: "'a::real_inner"
assumes "convex s" "closed s" "x \<in> s" "y \<in> s"
"\<forall>z\<in>s. dist a x \<le> dist a z" "\<forall>z\<in>s. dist a y \<le> dist a z"
shows "x = y"
using any_closest_point_dot[OF assms(1-4,5)] and any_closest_point_dot[OF assms(1-2,4,3,6)]
unfolding norm_pths(1) and norm_le_square
by (auto simp add: algebra_simps)
lemma closest_point_unique:
assumes "convex s" "closed s" "x \<in> s" "\<forall>z\<in>s. dist a x \<le> dist a z"
shows "x = closest_point s a"
using any_closest_point_unique[OF assms(1-3) _ assms(4), of "closest_point s a"]
using closest_point_exists[OF assms(2)] and assms(3) by auto
lemma closest_point_dot:
assumes "convex s" "closed s" "x \<in> s"
shows "inner (a - closest_point s a) (x - closest_point s a) \<le> 0"
apply (rule any_closest_point_dot[OF assms(1,2) _ assms(3)])
using closest_point_exists[OF assms(2)] and assms(3)
apply auto
done
lemma closest_point_lt:
assumes "convex s" "closed s" "x \<in> s" "x \<noteq> closest_point s a"
shows "dist a (closest_point s a) < dist a x"
apply (rule ccontr)
apply (rule_tac notE[OF assms(4)])
apply (rule closest_point_unique[OF assms(1-3), of a])
using closest_point_le[OF assms(2), of _ a]
apply fastforce
done
lemma closest_point_lipschitz:
assumes "convex s"
and "closed s" "s \<noteq> {}"
shows "dist (closest_point s x) (closest_point s y) \<le> dist x y"
proof -
have "inner (x - closest_point s x) (closest_point s y - closest_point s x) \<le> 0"
and "inner (y - closest_point s y) (closest_point s x - closest_point s y) \<le> 0"
apply (rule_tac[!] any_closest_point_dot[OF assms(1-2)])
using closest_point_exists[OF assms(2-3)]
apply auto
done
then show ?thesis unfolding dist_norm and norm_le
using inner_ge_zero[of "(x - closest_point s x) - (y - closest_point s y)"]
by (simp add: inner_add inner_diff inner_commute)
qed
lemma continuous_at_closest_point:
assumes "convex s"
and "closed s"
and "s \<noteq> {}"
shows "continuous (at x) (closest_point s)"
unfolding continuous_at_eps_delta
using le_less_trans[OF closest_point_lipschitz[OF assms]] by auto
lemma continuous_on_closest_point:
assumes "convex s"
and "closed s"
and "s \<noteq> {}"
shows "continuous_on t (closest_point s)"
by (metis continuous_at_imp_continuous_on continuous_at_closest_point[OF assms])
subsubsection {* Various point-to-set separating/supporting hyperplane theorems. *}
lemma supporting_hyperplane_closed_point:
fixes z :: "'a::{real_inner,heine_borel}"
assumes "convex s"
and "closed s"
and "s \<noteq> {}"
and "z \<notin> s"
shows "\<exists>a b. \<exists>y\<in>s. inner a z < b \<and> inner a y = b \<and> (\<forall>x\<in>s. inner a x \<ge> b)"
proof -
from distance_attains_inf[OF assms(2-3)]
obtain y where "y \<in> s" and y: "\<forall>x\<in>s. dist z y \<le> dist z x"
by auto
show ?thesis
apply (rule_tac x="y - z" in exI)
apply (rule_tac x="inner (y - z) y" in exI)
apply (rule_tac x=y in bexI)
apply rule
defer
apply rule
defer
apply rule
apply (rule ccontr)
using `y \<in> s`
proof -
show "inner (y - z) z < inner (y - z) y"
apply (subst diff_less_iff(1)[symmetric])
unfolding inner_diff_right[symmetric] and inner_gt_zero_iff
using `y\<in>s` `z\<notin>s`
apply auto
done
next
fix x
assume "x \<in> s"
have *: "\<forall>u. 0 \<le> u \<and> u \<le> 1 \<longrightarrow> dist z y \<le> dist z ((1 - u) *\<^sub>R y + u *\<^sub>R x)"
using assms(1)[unfolded convex_alt] and y and `x\<in>s` and `y\<in>s` by auto
assume "\<not> inner (y - z) y \<le> inner (y - z) x"
then obtain v where "v > 0" "v \<le> 1" "dist (y + v *\<^sub>R (x - y)) z < dist y z"
using closer_point_lemma[of z y x] by (auto simp add: inner_diff)
then show False
using *[THEN spec[where x=v]] by (auto simp add: dist_commute algebra_simps)
qed auto
qed
lemma separating_hyperplane_closed_point:
fixes z :: "'a::{real_inner,heine_borel}"
assumes "convex s"
and "closed s"
and "z \<notin> s"
shows "\<exists>a b. inner a z < b \<and> (\<forall>x\<in>s. inner a x > b)"
proof (cases "s = {}")
case True
then show ?thesis
apply (rule_tac x="-z" in exI)
apply (rule_tac x=1 in exI)
using less_le_trans[OF _ inner_ge_zero[of z]]
apply auto
done
next
case False
obtain y where "y \<in> s" and y: "\<forall>x\<in>s. dist z y \<le> dist z x"
using distance_attains_inf[OF assms(2) False] by auto
show ?thesis
apply (rule_tac x="y - z" in exI)
apply (rule_tac x="inner (y - z) z + (norm (y - z))\<^sup>2 / 2" in exI)
apply rule
defer
apply rule
proof -
fix x
assume "x \<in> s"
have "\<not> 0 < inner (z - y) (x - y)"
apply (rule notI)
apply (drule closer_point_lemma)
proof -
assume "\<exists>u>0. u \<le> 1 \<and> dist (y + u *\<^sub>R (x - y)) z < dist y z"
then obtain u where "u > 0" "u \<le> 1" "dist (y + u *\<^sub>R (x - y)) z < dist y z"
by auto
then show False using y[THEN bspec[where x="y + u *\<^sub>R (x - y)"]]
using assms(1)[unfolded convex_alt, THEN bspec[where x=y]]
using `x\<in>s` `y\<in>s` by (auto simp add: dist_commute algebra_simps)
qed
moreover have "0 < (norm (y - z))\<^sup>2"
using `y\<in>s` `z\<notin>s` by auto
then have "0 < inner (y - z) (y - z)"
unfolding power2_norm_eq_inner by simp
ultimately show "inner (y - z) z + (norm (y - z))\<^sup>2 / 2 < inner (y - z) x"
unfolding power2_norm_eq_inner and not_less
by (auto simp add: field_simps inner_commute inner_diff)
qed (insert `y\<in>s` `z\<notin>s`, auto)
qed
lemma separating_hyperplane_closed_0:
assumes "convex (s::('a::euclidean_space) set)"
and "closed s"
and "0 \<notin> s"
shows "\<exists>a b. a \<noteq> 0 \<and> 0 < b \<and> (\<forall>x\<in>s. inner a x > b)"
proof (cases "s = {}")
case True
have "norm ((SOME i. i\<in>Basis)::'a) = 1" "(SOME i. i\<in>Basis) \<noteq> (0::'a)"
defer
apply (subst norm_le_zero_iff[symmetric])
apply (auto simp: SOME_Basis)
done
then show ?thesis
apply (rule_tac x="SOME i. i\<in>Basis" in exI)
apply (rule_tac x=1 in exI)
using True using DIM_positive[where 'a='a]
apply auto
done
next
case False
then show ?thesis
using False using separating_hyperplane_closed_point[OF assms]
apply (elim exE)
unfolding inner_zero_right
apply (rule_tac x=a in exI)
apply (rule_tac x=b in exI)
apply auto
done
qed
subsubsection {* Now set-to-set for closed/compact sets *}
lemma separating_hyperplane_closed_compact:
fixes s :: "'a::euclidean_space set"
assumes "convex s"
and "closed s"
and "convex t"
and "compact t"
and "t \<noteq> {}"
and "s \<inter> t = {}"
shows "\<exists>a b. (\<forall>x\<in>s. inner a x < b) \<and> (\<forall>x\<in>t. inner a x > b)"
proof (cases "s = {}")
case True
obtain b where b: "b > 0" "\<forall>x\<in>t. norm x \<le> b"
using compact_imp_bounded[OF assms(4)] unfolding bounded_pos by auto
obtain z :: 'a where z: "norm z = b + 1"
using vector_choose_size[of "b + 1"] and b(1) by auto
then have "z \<notin> t" using b(2)[THEN bspec[where x=z]] by auto
then obtain a b where ab: "inner a z < b" "\<forall>x\<in>t. b < inner a x"
using separating_hyperplane_closed_point[OF assms(3) compact_imp_closed[OF assms(4)], of z]
by auto
then show ?thesis
using True by auto
next
case False
then obtain y where "y \<in> s" by auto
obtain a b where "0 < b" "\<forall>x\<in>{x - y |x y. x \<in> s \<and> y \<in> t}. b < inner a x"
using separating_hyperplane_closed_point[OF convex_differences[OF assms(1,3)], of 0]
using closed_compact_differences[OF assms(2,4)]
using assms(6) by auto blast
then have ab: "\<forall>x\<in>s. \<forall>y\<in>t. b + inner a y < inner a x"
apply -
apply rule
apply rule
apply (erule_tac x="x - y" in ballE)
apply (auto simp add: inner_diff)
done
def k \<equiv> "SUP x:t. a \<bullet> x"
show ?thesis
apply (rule_tac x="-a" in exI)
apply (rule_tac x="-(k + b / 2)" in exI)
apply (intro conjI ballI)
unfolding inner_minus_left and neg_less_iff_less
proof -
fix x assume "x \<in> t"
then have "inner a x - b / 2 < k"
unfolding k_def
proof (subst less_cSUP_iff)
show "t \<noteq> {}" by fact
show "bdd_above (op \<bullet> a ` t)"
using ab[rule_format, of y] `y \<in> s`
by (intro bdd_aboveI2[where M="inner a y - b"]) (auto simp: field_simps intro: less_imp_le)
qed (auto intro!: bexI[of _ x] `0<b`)
then show "inner a x < k + b / 2"
by auto
next
fix x
assume "x \<in> s"
then have "k \<le> inner a x - b"
unfolding k_def
apply (rule_tac cSUP_least)
using assms(5)
using ab[THEN bspec[where x=x]]
apply auto
done
then show "k + b / 2 < inner a x"
using `0 < b` by auto
qed
qed
lemma separating_hyperplane_compact_closed:
fixes s :: "'a::euclidean_space set"
assumes "convex s"
and "compact s"
and "s \<noteq> {}"
and "convex t"
and "closed t"
and "s \<inter> t = {}"
shows "\<exists>a b. (\<forall>x\<in>s. inner a x < b) \<and> (\<forall>x\<in>t. inner a x > b)"
proof -
obtain a b where "(\<forall>x\<in>t. inner a x < b) \<and> (\<forall>x\<in>s. b < inner a x)"
using separating_hyperplane_closed_compact[OF assms(4-5,1-2,3)] and assms(6)
by auto
then show ?thesis
apply (rule_tac x="-a" in exI)
apply (rule_tac x="-b" in exI)
apply auto
done
qed
subsubsection {* General case without assuming closure and getting non-strict separation *}
lemma separating_hyperplane_set_0:
assumes "convex s" "(0::'a::euclidean_space) \<notin> s"
shows "\<exists>a. a \<noteq> 0 \<and> (\<forall>x\<in>s. 0 \<le> inner a x)"
proof -
let ?k = "\<lambda>c. {x::'a. 0 \<le> inner c x}"
have "frontier (cball 0 1) \<inter> (\<Inter> (?k ` s)) \<noteq> {}"
apply (rule compact_imp_fip)
apply (rule compact_frontier[OF compact_cball])
defer
apply rule
apply rule
apply (erule conjE)
proof -
fix f
assume as: "f \<subseteq> ?k ` s" "finite f"
obtain c where c: "f = ?k ` c" "c \<subseteq> s" "finite c"
using finite_subset_image[OF as(2,1)] by auto
then obtain a b where ab: "a \<noteq> 0" "0 < b" "\<forall>x\<in>convex hull c. b < inner a x"
using separating_hyperplane_closed_0[OF convex_convex_hull, of c]
using finite_imp_compact_convex_hull[OF c(3), THEN compact_imp_closed] and assms(2)
using subset_hull[of convex, OF assms(1), symmetric, of c]
by auto
then have "\<exists>x. norm x = 1 \<and> (\<forall>y\<in>c. 0 \<le> inner y x)"
apply (rule_tac x = "inverse(norm a) *\<^sub>R a" in exI)
using hull_subset[of c convex]
unfolding subset_eq and inner_scaleR
by (auto simp add: inner_commute del: ballE elim!: ballE)
then show "frontier (cball 0 1) \<inter> \<Inter>f \<noteq> {}"
unfolding c(1) frontier_cball dist_norm by auto
qed (insert closed_halfspace_ge, auto)
then obtain x where "norm x = 1" "\<forall>y\<in>s. x\<in>?k y"
unfolding frontier_cball dist_norm by auto
then show ?thesis
apply (rule_tac x=x in exI)
apply (auto simp add: inner_commute)
done
qed
lemma separating_hyperplane_sets:
fixes s t :: "'a::euclidean_space set"
assumes "convex s"
and "convex t"
and "s \<noteq> {}"
and "t \<noteq> {}"
and "s \<inter> t = {}"
shows "\<exists>a b. a \<noteq> 0 \<and> (\<forall>x\<in>s. inner a x \<le> b) \<and> (\<forall>x\<in>t. inner a x \<ge> b)"
proof -
from separating_hyperplane_set_0[OF convex_differences[OF assms(2,1)]]
obtain a where "a \<noteq> 0" "\<forall>x\<in>{x - y |x y. x \<in> t \<and> y \<in> s}. 0 \<le> inner a x"
using assms(3-5) by auto
then have *: "\<And>x y. x \<in> t \<Longrightarrow> y \<in> s \<Longrightarrow> inner a y \<le> inner a x"
by (force simp add: inner_diff)
then have bdd: "bdd_above ((op \<bullet> a)`s)"
using `t \<noteq> {}` by (auto intro: bdd_aboveI2[OF *])
show ?thesis
using `a\<noteq>0`
by (intro exI[of _ a] exI[of _ "SUP x:s. a \<bullet> x"])
(auto intro!: cSUP_upper bdd cSUP_least `a \<noteq> 0` `s \<noteq> {}` *)
qed
subsection {* More convexity generalities *}
lemma convex_closure:
fixes s :: "'a::real_normed_vector set"
assumes "convex s"
shows "convex (closure s)"
apply (rule convexI)
apply (unfold closure_sequential, elim exE)
apply (rule_tac x="\<lambda>n. u *\<^sub>R xa n + v *\<^sub>R xb n" in exI)
apply (rule,rule)
apply (rule convexD [OF assms])
apply (auto del: tendsto_const intro!: tendsto_intros)
done
lemma convex_interior:
fixes s :: "'a::real_normed_vector set"
assumes "convex s"
shows "convex (interior s)"
unfolding convex_alt Ball_def mem_interior
apply (rule,rule,rule,rule,rule,rule)
apply (elim exE conjE)
proof -
fix x y u
assume u: "0 \<le> u" "u \<le> (1::real)"
fix e d
assume ed: "ball x e \<subseteq> s" "ball y d \<subseteq> s" "0<d" "0<e"
show "\<exists>e>0. ball ((1 - u) *\<^sub>R x + u *\<^sub>R y) e \<subseteq> s"
apply (rule_tac x="min d e" in exI)
apply rule
unfolding subset_eq
defer
apply rule
proof -
fix z
assume "z \<in> ball ((1 - u) *\<^sub>R x + u *\<^sub>R y) (min d e)"
then have "(1- u) *\<^sub>R (z - u *\<^sub>R (y - x)) + u *\<^sub>R (z + (1 - u) *\<^sub>R (y - x)) \<in> s"
apply (rule_tac assms[unfolded convex_alt, rule_format])
using ed(1,2) and u
unfolding subset_eq mem_ball Ball_def dist_norm
apply (auto simp add: algebra_simps)
done
then show "z \<in> s"
using u by (auto simp add: algebra_simps)
qed(insert u ed(3-4), auto)
qed
lemma convex_hull_eq_empty[simp]: "convex hull s = {} \<longleftrightarrow> s = {}"
using hull_subset[of s convex] convex_hull_empty by auto
subsection {* Moving and scaling convex hulls. *}
lemma convex_hull_set_plus:
"convex hull (s + t) = convex hull s + convex hull t"
unfolding set_plus_image
apply (subst convex_hull_linear_image [symmetric])
apply (simp add: linear_iff scaleR_right_distrib)
apply (simp add: convex_hull_Times)
done
lemma translation_eq_singleton_plus: "(\<lambda>x. a + x) ` t = {a} + t"
unfolding set_plus_def by auto
lemma convex_hull_translation:
"convex hull ((\<lambda>x. a + x) ` s) = (\<lambda>x. a + x) ` (convex hull s)"
unfolding translation_eq_singleton_plus
by (simp only: convex_hull_set_plus convex_hull_singleton)
lemma convex_hull_scaling:
"convex hull ((\<lambda>x. c *\<^sub>R x) ` s) = (\<lambda>x. c *\<^sub>R x) ` (convex hull s)"
using linear_scaleR by (rule convex_hull_linear_image [symmetric])
lemma convex_hull_affinity:
"convex hull ((\<lambda>x. a + c *\<^sub>R x) ` s) = (\<lambda>x. a + c *\<^sub>R x) ` (convex hull s)"
by(simp only: image_image[symmetric] convex_hull_scaling convex_hull_translation)
subsection {* Convexity of cone hulls *}
lemma convex_cone_hull:
assumes "convex S"
shows "convex (cone hull S)"
proof (rule convexI)
fix x y
assume xy: "x \<in> cone hull S" "y \<in> cone hull S"
then have "S \<noteq> {}"
using cone_hull_empty_iff[of S] by auto
fix u v :: real
assume uv: "u \<ge> 0" "v \<ge> 0" "u + v = 1"
then have *: "u *\<^sub>R x \<in> cone hull S" "v *\<^sub>R y \<in> cone hull S"
using cone_cone_hull[of S] xy cone_def[of "cone hull S"] by auto
from * obtain cx :: real and xx where x: "u *\<^sub>R x = cx *\<^sub>R xx" "cx \<ge> 0" "xx \<in> S"
using cone_hull_expl[of S] by auto
from * obtain cy :: real and yy where y: "v *\<^sub>R y = cy *\<^sub>R yy" "cy \<ge> 0" "yy \<in> S"
using cone_hull_expl[of S] by auto
{
assume "cx + cy \<le> 0"
then have "u *\<^sub>R x = 0" and "v *\<^sub>R y = 0"
using x y by auto
then have "u *\<^sub>R x + v *\<^sub>R y = 0"
by auto
then have "u *\<^sub>R x + v *\<^sub>R y \<in> cone hull S"
using cone_hull_contains_0[of S] `S \<noteq> {}` by auto
}
moreover
{
assume "cx + cy > 0"
then have "(cx / (cx + cy)) *\<^sub>R xx + (cy / (cx + cy)) *\<^sub>R yy \<in> S"
using assms mem_convex_alt[of S xx yy cx cy] x y by auto
then have "cx *\<^sub>R xx + cy *\<^sub>R yy \<in> cone hull S"
using mem_cone_hull[of "(cx/(cx+cy)) *\<^sub>R xx + (cy/(cx+cy)) *\<^sub>R yy" S "cx+cy"] `cx+cy>0`
by (auto simp add: scaleR_right_distrib)
then have "u *\<^sub>R x + v *\<^sub>R y \<in> cone hull S"
using x y by auto
}
moreover have "cx + cy \<le> 0 \<or> cx + cy > 0" by auto
ultimately show "u *\<^sub>R x + v *\<^sub>R y \<in> cone hull S" by blast
qed
lemma cone_convex_hull:
assumes "cone S"
shows "cone (convex hull S)"
proof (cases "S = {}")
case True
then show ?thesis by auto
next
case False
then have *: "0 \<in> S \<and> (\<forall>c. c > 0 \<longrightarrow> op *\<^sub>R c ` S = S)"
using cone_iff[of S] assms by auto
{
fix c :: real
assume "c > 0"
then have "op *\<^sub>R c ` (convex hull S) = convex hull (op *\<^sub>R c ` S)"
using convex_hull_scaling[of _ S] by auto
also have "\<dots> = convex hull S"
using * `c > 0` by auto
finally have "op *\<^sub>R c ` (convex hull S) = convex hull S"
by auto
}
then have "0 \<in> convex hull S" "\<And>c. c > 0 \<Longrightarrow> (op *\<^sub>R c ` (convex hull S)) = (convex hull S)"
using * hull_subset[of S convex] by auto
then show ?thesis
using `S \<noteq> {}` cone_iff[of "convex hull S"] by auto
qed
subsection {* Convex set as intersection of halfspaces *}
lemma convex_halfspace_intersection:
fixes s :: "('a::euclidean_space) set"
assumes "closed s" "convex s"
shows "s = \<Inter> {h. s \<subseteq> h \<and> (\<exists>a b. h = {x. inner a x \<le> b})}"
apply (rule set_eqI)
apply rule
unfolding Inter_iff Ball_def mem_Collect_eq
apply (rule,rule,erule conjE)
proof -
fix x
assume "\<forall>xa. s \<subseteq> xa \<and> (\<exists>a b. xa = {x. inner a x \<le> b}) \<longrightarrow> x \<in> xa"
then have "\<forall>a b. s \<subseteq> {x. inner a x \<le> b} \<longrightarrow> x \<in> {x. inner a x \<le> b}"
by blast
then show "x \<in> s"
apply (rule_tac ccontr)
apply (drule separating_hyperplane_closed_point[OF assms(2,1)])
apply (erule exE)+
apply (erule_tac x="-a" in allE)
apply (erule_tac x="-b" in allE)
apply auto
done
qed auto
subsection {* Radon's theorem (from Lars Schewe) *}
lemma radon_ex_lemma:
assumes "finite c" "affine_dependent c"
shows "\<exists>u. setsum u c = 0 \<and> (\<exists>v\<in>c. u v \<noteq> 0) \<and> setsum (\<lambda>v. u v *\<^sub>R v) c = 0"
proof -
from assms(2)[unfolded affine_dependent_explicit]
obtain s u where
"finite s" "s \<subseteq> c" "setsum u s = 0" "\<exists>v\<in>s. u v \<noteq> 0" "(\<Sum>v\<in>s. u v *\<^sub>R v) = 0"
by blast
then show ?thesis
apply (rule_tac x="\<lambda>v. if v\<in>s then u v else 0" in exI)
unfolding if_smult scaleR_zero_left and setsum.inter_restrict[OF assms(1), symmetric]
apply (auto simp add: Int_absorb1)
done
qed
lemma radon_s_lemma:
assumes "finite s"
and "setsum f s = (0::real)"
shows "setsum f {x\<in>s. 0 < f x} = - setsum f {x\<in>s. f x < 0}"
proof -
have *: "\<And>x. (if f x < 0 then f x else 0) + (if 0 < f x then f x else 0) = f x"
by auto
show ?thesis
unfolding real_add_eq_0_iff[symmetric] and setsum.inter_filter[OF assms(1)]
and setsum.distrib[symmetric] and *
using assms(2)
apply assumption
done
qed
lemma radon_v_lemma:
assumes "finite s"
and "setsum f s = 0"
and "\<forall>x. g x = (0::real) \<longrightarrow> f x = (0::'a::euclidean_space)"
shows "(setsum f {x\<in>s. 0 < g x}) = - setsum f {x\<in>s. g x < 0}"
proof -
have *: "\<And>x. (if 0 < g x then f x else 0) + (if g x < 0 then f x else 0) = f x"
using assms(3) by auto
show ?thesis
unfolding eq_neg_iff_add_eq_0 and setsum.inter_filter[OF assms(1)]
and setsum.distrib[symmetric] and *
using assms(2)
apply assumption
done
qed
lemma radon_partition:
assumes "finite c" "affine_dependent c"
shows "\<exists>m p. m \<inter> p = {} \<and> m \<union> p = c \<and> (convex hull m) \<inter> (convex hull p) \<noteq> {}"
proof -
obtain u v where uv: "setsum u c = 0" "v\<in>c" "u v \<noteq> 0" "(\<Sum>v\<in>c. u v *\<^sub>R v) = 0"
using radon_ex_lemma[OF assms] by auto
have fin: "finite {x \<in> c. 0 < u x}" "finite {x \<in> c. 0 > u x}"
using assms(1) by auto
def z \<equiv> "inverse (setsum u {x\<in>c. u x > 0}) *\<^sub>R setsum (\<lambda>x. u x *\<^sub>R x) {x\<in>c. u x > 0}"
have "setsum u {x \<in> c. 0 < u x} \<noteq> 0"
proof (cases "u v \<ge> 0")
case False
then have "u v < 0" by auto
then show ?thesis
proof (cases "\<exists>w\<in>{x \<in> c. 0 < u x}. u w > 0")
case True
then show ?thesis
using setsum_nonneg_eq_0_iff[of _ u, OF fin(1)] by auto
next
case False
then have "setsum u c \<le> setsum (\<lambda>x. if x=v then u v else 0) c"
apply (rule_tac setsum_mono)
apply auto
done
then show ?thesis
unfolding setsum.delta[OF assms(1)] using uv(2) and `u v < 0` and uv(1) by auto
qed
qed (insert setsum_nonneg_eq_0_iff[of _ u, OF fin(1)] uv(2-3), auto)
then have *: "setsum u {x\<in>c. u x > 0} > 0"
unfolding less_le
apply (rule_tac conjI)
apply (rule_tac setsum_nonneg)
apply auto
done
moreover have "setsum u ({x \<in> c. 0 < u x} \<union> {x \<in> c. u x < 0}) = setsum u c"
"(\<Sum>x\<in>{x \<in> c. 0 < u x} \<union> {x \<in> c. u x < 0}. u x *\<^sub>R x) = (\<Sum>x\<in>c. u x *\<^sub>R x)"
using assms(1)
apply (rule_tac[!] setsum.mono_neutral_left)
apply auto
done
then have "setsum u {x \<in> c. 0 < u x} = - setsum u {x \<in> c. 0 > u x}"
"(\<Sum>x\<in>{x \<in> c. 0 < u x}. u x *\<^sub>R x) = - (\<Sum>x\<in>{x \<in> c. 0 > u x}. u x *\<^sub>R x)"
unfolding eq_neg_iff_add_eq_0
using uv(1,4)
by (auto simp add: setsum.union_inter_neutral[OF fin, symmetric])
moreover have "\<forall>x\<in>{v \<in> c. u v < 0}. 0 \<le> inverse (setsum u {x \<in> c. 0 < u x}) * - u x"
apply rule
apply (rule mult_nonneg_nonneg)
using *
apply auto
done
ultimately have "z \<in> convex hull {v \<in> c. u v \<le> 0}"
unfolding convex_hull_explicit mem_Collect_eq
apply (rule_tac x="{v \<in> c. u v < 0}" in exI)
apply (rule_tac x="\<lambda>y. inverse (setsum u {x\<in>c. u x > 0}) * - u y" in exI)
using assms(1) unfolding scaleR_scaleR[symmetric] scaleR_right.setsum [symmetric] and z_def
apply (auto simp add: setsum_negf setsum_right_distrib[symmetric])
done
moreover have "\<forall>x\<in>{v \<in> c. 0 < u v}. 0 \<le> inverse (setsum u {x \<in> c. 0 < u x}) * u x"
apply rule
apply (rule mult_nonneg_nonneg)
using *
apply auto
done
then have "z \<in> convex hull {v \<in> c. u v > 0}"
unfolding convex_hull_explicit mem_Collect_eq
apply (rule_tac x="{v \<in> c. 0 < u v}" in exI)
apply (rule_tac x="\<lambda>y. inverse (setsum u {x\<in>c. u x > 0}) * u y" in exI)
using assms(1)
unfolding scaleR_scaleR[symmetric] scaleR_right.setsum [symmetric] and z_def
using *
apply (auto simp add: setsum_negf setsum_right_distrib[symmetric])
done
ultimately show ?thesis
apply (rule_tac x="{v\<in>c. u v \<le> 0}" in exI)
apply (rule_tac x="{v\<in>c. u v > 0}" in exI)
apply auto
done
qed
lemma radon:
assumes "affine_dependent c"
obtains m p where "m \<subseteq> c" "p \<subseteq> c" "m \<inter> p = {}" "(convex hull m) \<inter> (convex hull p) \<noteq> {}"
proof -
from assms[unfolded affine_dependent_explicit]
obtain s u where
"finite s" "s \<subseteq> c" "setsum u s = 0" "\<exists>v\<in>s. u v \<noteq> 0" "(\<Sum>v\<in>s. u v *\<^sub>R v) = 0"
by blast
then have *: "finite s" "affine_dependent s" and s: "s \<subseteq> c"
unfolding affine_dependent_explicit by auto
from radon_partition[OF *]
obtain m p where "m \<inter> p = {}" "m \<union> p = s" "convex hull m \<inter> convex hull p \<noteq> {}"
by blast
then show ?thesis
apply (rule_tac that[of p m])
using s
apply auto
done
qed
subsection {* Helly's theorem *}
lemma helly_induct:
fixes f :: "'a::euclidean_space set set"
assumes "card f = n"
and "n \<ge> DIM('a) + 1"
and "\<forall>s\<in>f. convex s" "\<forall>t\<subseteq>f. card t = DIM('a) + 1 \<longrightarrow> \<Inter> t \<noteq> {}"
shows "\<Inter>f \<noteq> {}"
using assms
proof (induct n arbitrary: f)
case 0
then show ?case by auto
next
case (Suc n)
have "finite f"
using `card f = Suc n` by (auto intro: card_ge_0_finite)
show "\<Inter>f \<noteq> {}"
apply (cases "n = DIM('a)")
apply (rule Suc(5)[rule_format])
unfolding `card f = Suc n`
proof -
assume ng: "n \<noteq> DIM('a)"
then have "\<exists>X. \<forall>s\<in>f. X s \<in> \<Inter>(f - {s})"
apply (rule_tac bchoice)
unfolding ex_in_conv
apply (rule, rule Suc(1)[rule_format])
unfolding card_Diff_singleton_if[OF `finite f`] `card f = Suc n`
defer
defer
apply (rule Suc(4)[rule_format])
defer
apply (rule Suc(5)[rule_format])
using Suc(3) `finite f`
apply auto
done
then obtain X where X: "\<forall>s\<in>f. X s \<in> \<Inter>(f - {s})" by auto
show ?thesis
proof (cases "inj_on X f")
case False
then obtain s t where st: "s\<noteq>t" "s\<in>f" "t\<in>f" "X s = X t"
unfolding inj_on_def by auto
then have *: "\<Inter>f = \<Inter>(f - {s}) \<inter> \<Inter>(f - {t})" by auto
show ?thesis
unfolding *
unfolding ex_in_conv[symmetric]
apply (rule_tac x="X s" in exI)
apply rule
apply (rule X[rule_format])
using X st
apply auto
done
next
case True
then obtain m p where mp: "m \<inter> p = {}" "m \<union> p = X ` f" "convex hull m \<inter> convex hull p \<noteq> {}"
using radon_partition[of "X ` f"] and affine_dependent_biggerset[of "X ` f"]
unfolding card_image[OF True] and `card f = Suc n`
using Suc(3) `finite f` and ng
by auto
have "m \<subseteq> X ` f" "p \<subseteq> X ` f"
using mp(2) by auto
then obtain g h where gh:"m = X ` g" "p = X ` h" "g \<subseteq> f" "h \<subseteq> f"
unfolding subset_image_iff by auto
then have "f \<union> (g \<union> h) = f" by auto
then have f: "f = g \<union> h"
using inj_on_Un_image_eq_iff[of X f "g \<union> h"] and True
unfolding mp(2)[unfolded image_Un[symmetric] gh]
by auto
have *: "g \<inter> h = {}"
using mp(1)
unfolding gh
using inj_on_image_Int[OF True gh(3,4)]
by auto
have "convex hull (X ` h) \<subseteq> \<Inter>g" "convex hull (X ` g) \<subseteq> \<Inter>h"
apply (rule_tac [!] hull_minimal)
using Suc gh(3-4)
unfolding subset_eq
apply (rule_tac [2] convex_Inter, rule_tac [4] convex_Inter)
apply rule
prefer 3
apply rule
proof -
fix x
assume "x \<in> X ` g"
then obtain y where "y \<in> g" "x = X y"
unfolding image_iff ..
then show "x \<in> \<Inter>h"
using X[THEN bspec[where x=y]] using * f by auto
next
fix x
assume "x \<in> X ` h"
then obtain y where "y \<in> h" "x = X y"
unfolding image_iff ..
then show "x \<in> \<Inter>g"
using X[THEN bspec[where x=y]] using * f by auto
qed auto
then show ?thesis
unfolding f using mp(3)[unfolded gh] by blast
qed
qed auto
qed
lemma helly:
fixes f :: "'a::euclidean_space set set"
assumes "card f \<ge> DIM('a) + 1" "\<forall>s\<in>f. convex s"
and "\<forall>t\<subseteq>f. card t = DIM('a) + 1 \<longrightarrow> \<Inter> t \<noteq> {}"
shows "\<Inter>f \<noteq> {}"
apply (rule helly_induct)
using assms
apply auto
done
subsection {* Homeomorphism of all convex compact sets with nonempty interior *}
lemma compact_frontier_line_lemma:
fixes s :: "'a::euclidean_space set"
assumes "compact s"
and "0 \<in> s"
and "x \<noteq> 0"
obtains u where "0 \<le> u" and "(u *\<^sub>R x) \<in> frontier s" "\<forall>v>u. (v *\<^sub>R x) \<notin> s"
proof -
obtain b where b: "b > 0" "\<forall>x\<in>s. norm x \<le> b"
using compact_imp_bounded[OF assms(1), unfolded bounded_pos] by auto
let ?A = "{y. \<exists>u. 0 \<le> u \<and> u \<le> b / norm(x) \<and> (y = u *\<^sub>R x)}"
have A: "?A = (\<lambda>u. u *\<^sub>R x) ` {0 .. b / norm x}"
by auto
have *: "\<And>x A B. x\<in>A \<Longrightarrow> x\<in>B \<Longrightarrow> A\<inter>B \<noteq> {}" by blast
have "compact ?A"
unfolding A
apply (rule compact_continuous_image)
apply (rule continuous_at_imp_continuous_on)
apply rule
apply (intro continuous_intros)
apply (rule compact_Icc)
done
moreover have "{y. \<exists>u\<ge>0. u \<le> b / norm x \<and> y = u *\<^sub>R x} \<inter> s \<noteq> {}"
apply(rule *[OF _ assms(2)])
unfolding mem_Collect_eq
using `b > 0` assms(3)
apply auto
done
ultimately obtain u y where obt: "u\<ge>0" "u \<le> b / norm x" "y = u *\<^sub>R x"
"y \<in> ?A" "y \<in> s" "\<forall>z\<in>?A \<inter> s. dist 0 z \<le> dist 0 y"
using distance_attains_sup[OF compact_inter[OF _ assms(1), of ?A], of 0]
by auto
have "norm x > 0"
using assms(3)[unfolded zero_less_norm_iff[symmetric]] by auto
{
fix v
assume as: "v > u" "v *\<^sub>R x \<in> s"
then have "v \<le> b / norm x"
using b(2)[rule_format, OF as(2)]
using `u\<ge>0`
unfolding pos_le_divide_eq[OF `norm x > 0`]
by auto
then have "norm (v *\<^sub>R x) \<le> norm y"
apply (rule_tac obt(6)[rule_format, unfolded dist_0_norm])
apply (rule IntI)
defer
apply (rule as(2))
unfolding mem_Collect_eq
apply (rule_tac x=v in exI)
using as(1) `u\<ge>0`
apply (auto simp add: field_simps)
done
then have False
unfolding obt(3) using `u\<ge>0` `norm x > 0` `v > u`
by (auto simp add:field_simps)
} note u_max = this
have "u *\<^sub>R x \<in> frontier s"
unfolding frontier_straddle
apply (rule,rule,rule)
apply (rule_tac x="u *\<^sub>R x" in bexI)
unfolding obt(3)[symmetric]
prefer 3
apply (rule_tac x="(u + (e / 2) / norm x) *\<^sub>R x" in exI)
apply (rule, rule)
proof -
fix e
assume "e > 0" and as: "(u + e / 2 / norm x) *\<^sub>R x \<in> s"
then have "u + e / 2 / norm x > u"
using `norm x > 0` by (auto simp del:zero_less_norm_iff)
then show False using u_max[OF _ as] by auto
qed (insert `y\<in>s`, auto simp add: dist_norm scaleR_left_distrib obt(3))
then show ?thesis by(metis that[of u] u_max obt(1))
qed
lemma starlike_compact_projective:
assumes "compact s"
and "cball (0::'a::euclidean_space) 1 \<subseteq> s "
and "\<forall>x\<in>s. \<forall>u. 0 \<le> u \<and> u < 1 \<longrightarrow> u *\<^sub>R x \<in> s - frontier s"
shows "s homeomorphic (cball (0::'a::euclidean_space) 1)"
proof -
have fs: "frontier s \<subseteq> s"
apply (rule frontier_subset_closed)
using compact_imp_closed[OF assms(1)]
apply simp
done
def pi \<equiv> "\<lambda>x::'a. inverse (norm x) *\<^sub>R x"
have "0 \<notin> frontier s"
unfolding frontier_straddle
apply (rule notI)
apply (erule_tac x=1 in allE)
using assms(2)[unfolded subset_eq Ball_def mem_cball]
apply auto
done
have injpi: "\<And>x y. pi x = pi y \<and> norm x = norm y \<longleftrightarrow> x = y"
unfolding pi_def by auto
have contpi: "continuous_on (UNIV - {0}) pi"
apply (rule continuous_at_imp_continuous_on)
apply rule unfolding pi_def
apply (intro continuous_intros)
apply simp
done
def sphere \<equiv> "{x::'a. norm x = 1}"
have pi: "\<And>x. x \<noteq> 0 \<Longrightarrow> pi x \<in> sphere" "\<And>x u. u>0 \<Longrightarrow> pi (u *\<^sub>R x) = pi x"
unfolding pi_def sphere_def by auto
have "0 \<in> s"
using assms(2) and centre_in_cball[of 0 1] by auto
have front_smul: "\<forall>x\<in>frontier s. \<forall>u\<ge>0. u *\<^sub>R x \<in> s \<longleftrightarrow> u \<le> 1"
proof (rule,rule,rule)
fix x and u :: real
assume x: "x \<in> frontier s" and "0 \<le> u"
then have "x \<noteq> 0"
using `0 \<notin> frontier s` by auto
obtain v where v: "0 \<le> v" "v *\<^sub>R x \<in> frontier s" "\<forall>w>v. w *\<^sub>R x \<notin> s"
using compact_frontier_line_lemma[OF assms(1) `0\<in>s` `x\<noteq>0`] by auto
have "v = 1"
apply (rule ccontr)
unfolding neq_iff
apply (erule disjE)
proof -
assume "v < 1"
then show False
using v(3)[THEN spec[where x=1]] using x and fs by auto
next
assume "v > 1"
then show False
using assms(3)[THEN bspec[where x="v *\<^sub>R x"], THEN spec[where x="inverse v"]]
using v and x and fs
unfolding inverse_less_1_iff by auto
qed
show "u *\<^sub>R x \<in> s \<longleftrightarrow> u \<le> 1"
apply rule
using v(3)[unfolded `v=1`, THEN spec[where x=u]]
proof -
assume "u \<le> 1"
then show "u *\<^sub>R x \<in> s"
apply (cases "u = 1")
using assms(3)[THEN bspec[where x=x], THEN spec[where x=u]]
using `0\<le>u` and x and fs
apply auto
done
qed auto
qed
have "\<exists>surf. homeomorphism (frontier s) sphere pi surf"
apply (rule homeomorphism_compact)
apply (rule compact_frontier[OF assms(1)])
apply (rule continuous_on_subset[OF contpi])
defer
apply (rule set_eqI)
apply rule
unfolding inj_on_def
prefer 3
apply(rule,rule,rule)
proof -
fix x
assume "x \<in> pi ` frontier s"
then obtain y where "y \<in> frontier s" "x = pi y" by auto
then show "x \<in> sphere"
using pi(1)[of y] and `0 \<notin> frontier s` by auto
next
fix x
assume "x \<in> sphere"
then have "norm x = 1" "x \<noteq> 0"
unfolding sphere_def by auto
then obtain u where "0 \<le> u" "u *\<^sub>R x \<in> frontier s" "\<forall>v>u. v *\<^sub>R x \<notin> s"
using compact_frontier_line_lemma[OF assms(1) `0\<in>s`, of x] by auto
then show "x \<in> pi ` frontier s"
unfolding image_iff le_less pi_def
apply (rule_tac x="u *\<^sub>R x" in bexI)
using `norm x = 1` `0 \<notin> frontier s`
apply auto
done
next
fix x y
assume as: "x \<in> frontier s" "y \<in> frontier s" "pi x = pi y"
then have xys: "x \<in> s" "y \<in> s"
using fs by auto
from as(1,2) have nor: "norm x \<noteq> 0" "norm y \<noteq> 0"
using `0\<notin>frontier s` by auto
from nor have x: "x = norm x *\<^sub>R ((inverse (norm y)) *\<^sub>R y)"
unfolding as(3)[unfolded pi_def, symmetric] by auto
from nor have y: "y = norm y *\<^sub>R ((inverse (norm x)) *\<^sub>R x)"
unfolding as(3)[unfolded pi_def] by auto
have "0 \<le> norm y * inverse (norm x)" and "0 \<le> norm x * inverse (norm y)"
using nor
apply auto
done
then have "norm x = norm y"
apply -
apply (rule ccontr)
unfolding neq_iff
using x y and front_smul[THEN bspec, OF as(1), THEN spec[where x="norm y * (inverse (norm x))"]]
using front_smul[THEN bspec, OF as(2), THEN spec[where x="norm x * (inverse (norm y))"]]
using xys nor
apply (auto simp add: field_simps)
done
then show "x = y"
apply (subst injpi[symmetric])
using as(3)
apply auto
done
qed (insert `0 \<notin> frontier s`, auto)
then obtain surf where
surf: "\<forall>x\<in>frontier s. surf (pi x) = x" "pi ` frontier s = sphere" "continuous_on (frontier s) pi"
"\<forall>y\<in>sphere. pi (surf y) = y" "surf ` sphere = frontier s" "continuous_on sphere surf"
unfolding homeomorphism_def by auto
have cont_surfpi: "continuous_on (UNIV - {0}) (surf \<circ> pi)"
apply (rule continuous_on_compose)
apply (rule contpi)
apply (rule continuous_on_subset[of sphere])
apply (rule surf(6))
using pi(1)
apply auto
done
{
fix x
assume as: "x \<in> cball (0::'a) 1"
have "norm x *\<^sub>R surf (pi x) \<in> s"
proof (cases "x=0 \<or> norm x = 1")
case False
then have "pi x \<in> sphere" "norm x < 1"
using pi(1)[of x] as by(auto simp add: dist_norm)
then show ?thesis
apply (rule_tac assms(3)[rule_format, THEN DiffD1])
apply (rule_tac fs[unfolded subset_eq, rule_format])
unfolding surf(5)[symmetric]
apply auto
done
next
case True
then show ?thesis
apply rule
defer
unfolding pi_def
apply (rule fs[unfolded subset_eq, rule_format])
unfolding surf(5)[unfolded sphere_def, symmetric]
using `0\<in>s`
apply auto
done
qed
} note hom = this
{
fix x
assume "x \<in> s"
then have "x \<in> (\<lambda>x. norm x *\<^sub>R surf (pi x)) ` cball 0 1"
proof (cases "x = 0")
case True
show ?thesis
unfolding image_iff True
apply (rule_tac x=0 in bexI)
apply auto
done
next
let ?a = "inverse (norm (surf (pi x)))"
case False
then have invn: "inverse (norm x) \<noteq> 0" by auto
from False have pix: "pi x\<in>sphere" using pi(1) by auto
then have "pi (surf (pi x)) = pi x"
apply (rule_tac surf(4)[rule_format])
apply assumption
done
then have **: "norm x *\<^sub>R (?a *\<^sub>R surf (pi x)) = x"
apply (rule_tac scaleR_left_imp_eq[OF invn])
unfolding pi_def
using invn
apply auto
done
then have *: "?a * norm x > 0" and "?a > 0" "?a \<noteq> 0"
using surf(5) `0\<notin>frontier s`
apply -
apply (rule mult_pos_pos)
using False[unfolded zero_less_norm_iff[symmetric]]
apply auto
done
have "norm (surf (pi x)) \<noteq> 0"
using ** False by auto
then have "norm x = norm ((?a * norm x) *\<^sub>R surf (pi x))"
unfolding norm_scaleR abs_mult abs_norm_cancel abs_of_pos[OF `?a > 0`] by auto
moreover have "pi x = pi ((inverse (norm (surf (pi x))) * norm x) *\<^sub>R surf (pi x))"
unfolding pi(2)[OF *] surf(4)[rule_format, OF pix] ..
moreover have "surf (pi x) \<in> frontier s"
using surf(5) pix by auto
then have "dist 0 (inverse (norm (surf (pi x))) *\<^sub>R x) \<le> 1"
unfolding dist_norm
using ** and *
using front_smul[THEN bspec[where x="surf (pi x)"], THEN spec[where x="norm x * ?a"]]
using False `x\<in>s`
by (auto simp add: field_simps)
ultimately show ?thesis
unfolding image_iff
apply (rule_tac x="inverse (norm (surf(pi x))) *\<^sub>R x" in bexI)
apply (subst injpi[symmetric])
unfolding abs_mult abs_norm_cancel abs_of_pos[OF `?a > 0`]
unfolding pi(2)[OF `?a > 0`]
apply auto
done
qed
} note hom2 = this
show ?thesis
apply (subst homeomorphic_sym)
apply (rule homeomorphic_compact[where f="\<lambda>x. norm x *\<^sub>R surf (pi x)"])
apply (rule compact_cball)
defer
apply (rule set_eqI)
apply rule
apply (erule imageE)
apply (drule hom)
prefer 4
apply (rule continuous_at_imp_continuous_on)
apply rule
apply (rule_tac [3] hom2)
proof -
fix x :: 'a
assume as: "x \<in> cball 0 1"
then show "continuous (at x) (\<lambda>x. norm x *\<^sub>R surf (pi x))"
proof (cases "x = 0")
case False
then show ?thesis
apply (intro continuous_intros)
using cont_surfpi
unfolding continuous_on_eq_continuous_at[OF open_delete[OF open_UNIV]] o_def
apply auto
done
next
case True
obtain B where B: "\<forall>x\<in>s. norm x \<le> B"
using compact_imp_bounded[OF assms(1)] unfolding bounded_iff by auto
then have "B > 0"
using assms(2)
unfolding subset_eq
apply (erule_tac x="SOME i. i\<in>Basis" in ballE)
defer
apply (erule_tac x="SOME i. i\<in>Basis" in ballE)
unfolding Ball_def mem_cball dist_norm
using DIM_positive[where 'a='a]
apply (auto simp: SOME_Basis)
done
show ?thesis
unfolding True continuous_at Lim_at
apply(rule,rule)
apply(rule_tac x="e / B" in exI)
apply rule
apply (rule divide_pos_pos)
prefer 3
apply(rule,rule,erule conjE)
unfolding norm_zero scaleR_zero_left dist_norm diff_0_right norm_scaleR abs_norm_cancel
proof -
fix e and x :: 'a
assume as: "norm x < e / B" "0 < norm x" "e > 0"
then have "surf (pi x) \<in> frontier s"
using pi(1)[of x] unfolding surf(5)[symmetric] by auto
then have "norm (surf (pi x)) \<le> B"
using B fs by auto
then have "norm x * norm (surf (pi x)) \<le> norm x * B"
using as(2) by auto
also have "\<dots> < e / B * B"
apply (rule mult_strict_right_mono)
using as(1) `B>0`
apply auto
done
also have "\<dots> = e" using `B > 0` by auto
finally show "norm x * norm (surf (pi x)) < e" .
qed (insert `B>0`, auto)
qed
next
{
fix x
assume as: "surf (pi x) = 0"
have "x = 0"
proof (rule ccontr)
assume "x \<noteq> 0"
then have "pi x \<in> sphere"
using pi(1) by auto
then have "surf (pi x) \<in> frontier s"
using surf(5) by auto
then show False
using `0\<notin>frontier s` unfolding as by simp
qed
} note surf_0 = this
show "inj_on (\<lambda>x. norm x *\<^sub>R surf (pi x)) (cball 0 1)"
unfolding inj_on_def
proof (rule,rule,rule)
fix x y
assume as: "x \<in> cball 0 1" "y \<in> cball 0 1" "norm x *\<^sub>R surf (pi x) = norm y *\<^sub>R surf (pi y)"
then show "x = y"
proof (cases "x=0 \<or> y=0")
case True
then show ?thesis
using as by (auto elim: surf_0)
next
case False
then have "pi (surf (pi x)) = pi (surf (pi y))"
using as(3)
using pi(2)[of "norm x" "surf (pi x)"] pi(2)[of "norm y" "surf (pi y)"]
by auto
moreover have "pi x \<in> sphere" "pi y \<in> sphere"
using pi(1) False by auto
ultimately have *: "pi x = pi y"
using surf(4)[THEN bspec[where x="pi x"]] surf(4)[THEN bspec[where x="pi y"]]
by auto
moreover have "norm x = norm y"
using as(3)[unfolded *] using False
by (auto dest:surf_0)
ultimately show ?thesis
using injpi by auto
qed
qed
qed auto
qed
lemma homeomorphic_convex_compact_lemma:
fixes s :: "'a::euclidean_space set"
assumes "convex s"
and "compact s"
and "cball 0 1 \<subseteq> s"
shows "s homeomorphic (cball (0::'a) 1)"
proof (rule starlike_compact_projective[OF assms(2-3)], clarify)
fix x u
assume "x \<in> s" and "0 \<le> u" and "u < (1::real)"
have "open (ball (u *\<^sub>R x) (1 - u))"
by (rule open_ball)
moreover have "u *\<^sub>R x \<in> ball (u *\<^sub>R x) (1 - u)"
unfolding centre_in_ball using `u < 1` by simp
moreover have "ball (u *\<^sub>R x) (1 - u) \<subseteq> s"
proof
fix y
assume "y \<in> ball (u *\<^sub>R x) (1 - u)"
then have "dist (u *\<^sub>R x) y < 1 - u"
unfolding mem_ball .
with `u < 1` have "inverse (1 - u) *\<^sub>R (y - u *\<^sub>R x) \<in> cball 0 1"
by (simp add: dist_norm inverse_eq_divide norm_minus_commute)
with assms(3) have "inverse (1 - u) *\<^sub>R (y - u *\<^sub>R x) \<in> s" ..
with assms(1) have "(1 - u) *\<^sub>R ((y - u *\<^sub>R x) /\<^sub>R (1 - u)) + u *\<^sub>R x \<in> s"
using `x \<in> s` `0 \<le> u` `u < 1` [THEN less_imp_le] by (rule mem_convex)
then show "y \<in> s" using `u < 1`
by simp
qed
ultimately have "u *\<^sub>R x \<in> interior s" ..
then show "u *\<^sub>R x \<in> s - frontier s"
using frontier_def and interior_subset by auto
qed
lemma homeomorphic_convex_compact_cball:
fixes e :: real
and s :: "'a::euclidean_space set"
assumes "convex s"
and "compact s"
and "interior s \<noteq> {}"
and "e > 0"
shows "s homeomorphic (cball (b::'a) e)"
proof -
obtain a where "a \<in> interior s"
using assms(3) by auto
then obtain d where "d > 0" and d: "cball a d \<subseteq> s"
unfolding mem_interior_cball by auto
let ?d = "inverse d" and ?n = "0::'a"
have "cball ?n 1 \<subseteq> (\<lambda>x. inverse d *\<^sub>R (x - a)) ` s"
apply rule
apply (rule_tac x="d *\<^sub>R x + a" in image_eqI)
defer
apply (rule d[unfolded subset_eq, rule_format])
using `d > 0`
unfolding mem_cball dist_norm
apply (auto simp add: mult_right_le_one_le)
done
then have "(\<lambda>x. inverse d *\<^sub>R (x - a)) ` s homeomorphic cball ?n 1"
using homeomorphic_convex_compact_lemma[of "(\<lambda>x. ?d *\<^sub>R -a + ?d *\<^sub>R x) ` s",
OF convex_affinity compact_affinity]
using assms(1,2)
by (auto simp add: scaleR_right_diff_distrib)
then show ?thesis
apply (rule_tac homeomorphic_trans[OF _ homeomorphic_balls(2)[of 1 _ ?n]])
apply (rule homeomorphic_trans[OF homeomorphic_affinity[of "?d" s "?d *\<^sub>R -a"]])
using `d>0` `e>0`
apply (auto simp add: scaleR_right_diff_distrib)
done
qed
lemma homeomorphic_convex_compact:
fixes s :: "'a::euclidean_space set"
and t :: "'a set"
assumes "convex s" "compact s" "interior s \<noteq> {}"
and "convex t" "compact t" "interior t \<noteq> {}"
shows "s homeomorphic t"
using assms
by (meson zero_less_one homeomorphic_trans homeomorphic_convex_compact_cball homeomorphic_sym)
subsection {* Epigraphs of convex functions *}
definition "epigraph s (f :: _ \<Rightarrow> real) = {xy. fst xy \<in> s \<and> f (fst xy) \<le> snd xy}"
lemma mem_epigraph: "(x, y) \<in> epigraph s f \<longleftrightarrow> x \<in> s \<and> f x \<le> y"
unfolding epigraph_def by auto
lemma convex_epigraph: "convex (epigraph s f) \<longleftrightarrow> convex_on s f \<and> convex s"
unfolding convex_def convex_on_def
unfolding Ball_def split_paired_All epigraph_def
unfolding mem_Collect_eq fst_conv snd_conv fst_add snd_add fst_scaleR snd_scaleR Ball_def[symmetric]
apply safe
defer
apply (erule_tac x=x in allE)
apply (erule_tac x="f x" in allE)
apply safe
apply (erule_tac x=xa in allE)
apply (erule_tac x="f xa" in allE)
prefer 3
apply (rule_tac y="u * f a + v * f aa" in order_trans)
defer
apply (auto intro!:mult_left_mono add_mono)
done
lemma convex_epigraphI: "convex_on s f \<Longrightarrow> convex s \<Longrightarrow> convex (epigraph s f)"
unfolding convex_epigraph by auto
lemma convex_epigraph_convex: "convex s \<Longrightarrow> convex_on s f \<longleftrightarrow> convex(epigraph s f)"
by (simp add: convex_epigraph)
subsubsection {* Use this to derive general bound property of convex function *}
lemma convex_on:
assumes "convex s"
shows "convex_on s f \<longleftrightarrow>
(\<forall>k u x. (\<forall>i\<in>{1..k::nat}. 0 \<le> u i \<and> x i \<in> s) \<and> setsum u {1..k} = 1 \<longrightarrow>
f (setsum (\<lambda>i. u i *\<^sub>R x i) {1..k} ) \<le> setsum (\<lambda>i. u i * f(x i)) {1..k})"
unfolding convex_epigraph_convex[OF assms] convex epigraph_def Ball_def mem_Collect_eq
unfolding fst_setsum snd_setsum fst_scaleR snd_scaleR
apply safe
apply (drule_tac x=k in spec)
apply (drule_tac x=u in spec)
apply (drule_tac x="\<lambda>i. (x i, f (x i))" in spec)
apply simp
using assms[unfolded convex]
apply simp
apply (rule_tac y="\<Sum>i = 1..k. u i * f (fst (x i))" in order_trans)
defer
apply (rule setsum_mono)
apply (erule_tac x=i in allE)
unfolding real_scaleR_def
apply (rule mult_left_mono)
using assms[unfolded convex]
apply auto
done
subsection {* Convexity of general and special intervals *}
lemma is_interval_convex:
fixes s :: "'a::euclidean_space set"
assumes "is_interval s"
shows "convex s"
proof (rule convexI)
fix x y and u v :: real
assume as: "x \<in> s" "y \<in> s" "0 \<le> u" "0 \<le> v" "u + v = 1"
then have *: "u = 1 - v" "1 - v \<ge> 0" and **: "v = 1 - u" "1 - u \<ge> 0"
by auto
{
fix a b
assume "\<not> b \<le> u * a + v * b"
then have "u * a < (1 - v) * b"
unfolding not_le using as(4) by (auto simp add: field_simps)
then have "a < b"
unfolding * using as(4) *(2)
apply (rule_tac mult_left_less_imp_less[of "1 - v"])
apply (auto simp add: field_simps)
done
then have "a \<le> u * a + v * b"
unfolding * using as(4)
by (auto simp add: field_simps intro!:mult_right_mono)
}
moreover
{
fix a b
assume "\<not> u * a + v * b \<le> a"
then have "v * b > (1 - u) * a"
unfolding not_le using as(4) by (auto simp add: field_simps)
then have "a < b"
unfolding * using as(4)
apply (rule_tac mult_left_less_imp_less)
apply (auto simp add: field_simps)
done
then have "u * a + v * b \<le> b"
unfolding **
using **(2) as(3)
by (auto simp add: field_simps intro!:mult_right_mono)
}
ultimately show "u *\<^sub>R x + v *\<^sub>R y \<in> s"
apply -
apply (rule assms[unfolded is_interval_def, rule_format, OF as(1,2)])
using as(3-) DIM_positive[where 'a='a]
apply (auto simp: inner_simps)
done
qed
lemma is_interval_connected:
fixes s :: "'a::euclidean_space set"
shows "is_interval s \<Longrightarrow> connected s"
using is_interval_convex convex_connected by auto
lemma convex_box: "convex (cbox a b)" "convex (box a (b::'a::euclidean_space))"
apply (rule_tac[!] is_interval_convex)+
using is_interval_box is_interval_cbox
apply auto
done
subsection {* On @{text "real"}, @{text "is_interval"}, @{text "convex"} and @{text "connected"} are all equivalent. *}
lemma is_interval_1:
"is_interval (s::real set) \<longleftrightarrow> (\<forall>a\<in>s. \<forall>b\<in>s. \<forall> x. a \<le> x \<and> x \<le> b \<longrightarrow> x \<in> s)"
unfolding is_interval_def by auto
lemma is_interval_connected_1:
fixes s :: "real set"
shows "is_interval s \<longleftrightarrow> connected s"
apply rule
apply (rule is_interval_connected, assumption)
unfolding is_interval_1
apply rule
apply rule
apply rule
apply rule
apply (erule conjE)
apply (rule ccontr)
proof -
fix a b x
assume as: "connected s" "a \<in> s" "b \<in> s" "a \<le> x" "x \<le> b" "x \<notin> s"
then have *: "a < x" "x < b"
unfolding not_le [symmetric] by auto
let ?halfl = "{..<x} "
let ?halfr = "{x<..}"
{
fix y
assume "y \<in> s"
with `x \<notin> s` have "x \<noteq> y" by auto
then have "y \<in> ?halfr \<union> ?halfl" by auto
}
moreover have "a \<in> ?halfl" "b \<in> ?halfr" using * by auto
then have "?halfl \<inter> s \<noteq> {}" "?halfr \<inter> s \<noteq> {}"
using as(2-3) by auto
ultimately show False
apply (rule_tac notE[OF as(1)[unfolded connected_def]])
apply (rule_tac x = ?halfl in exI)
apply (rule_tac x = ?halfr in exI)
apply rule
apply (rule open_lessThan)
apply rule
apply (rule open_greaterThan)
apply auto
done
qed
lemma is_interval_convex_1:
fixes s :: "real set"
shows "is_interval s \<longleftrightarrow> convex s"
by (metis is_interval_convex convex_connected is_interval_connected_1)
lemma convex_connected_1:
fixes s :: "real set"
shows "connected s \<longleftrightarrow> convex s"
by (metis is_interval_convex convex_connected is_interval_connected_1)
subsection {* Another intermediate value theorem formulation *}
lemma ivt_increasing_component_on_1:
fixes f :: "real \<Rightarrow> 'a::euclidean_space"
assumes "a \<le> b"
and "continuous_on (cbox a b) f"
and "(f a)\<bullet>k \<le> y" "y \<le> (f b)\<bullet>k"
shows "\<exists>x\<in>cbox a b. (f x)\<bullet>k = y"
proof -
have "f a \<in> f ` cbox a b" "f b \<in> f ` cbox a b"
apply (rule_tac[!] imageI)
using assms(1)
apply auto
done
then show ?thesis
using connected_ivt_component[of "f ` cbox a b" "f a" "f b" k y]
using connected_continuous_image[OF assms(2) convex_connected[OF convex_box(1)]]
using assms
by auto
qed
lemma ivt_increasing_component_1:
fixes f :: "real \<Rightarrow> 'a::euclidean_space"
shows "a \<le> b \<Longrightarrow> \<forall>x\<in>cbox a b. continuous (at x) f \<Longrightarrow>
f a\<bullet>k \<le> y \<Longrightarrow> y \<le> f b\<bullet>k \<Longrightarrow> \<exists>x\<in>cbox a b. (f x)\<bullet>k = y"
by (rule ivt_increasing_component_on_1) (auto simp add: continuous_at_imp_continuous_on)
lemma ivt_decreasing_component_on_1:
fixes f :: "real \<Rightarrow> 'a::euclidean_space"
assumes "a \<le> b"
and "continuous_on (cbox a b) f"
and "(f b)\<bullet>k \<le> y"
and "y \<le> (f a)\<bullet>k"
shows "\<exists>x\<in>cbox a b. (f x)\<bullet>k = y"
apply (subst neg_equal_iff_equal[symmetric])
using ivt_increasing_component_on_1[of a b "\<lambda>x. - f x" k "- y"]
using assms using continuous_on_minus
apply auto
done
lemma ivt_decreasing_component_1:
fixes f :: "real \<Rightarrow> 'a::euclidean_space"
shows "a \<le> b \<Longrightarrow> \<forall>x\<in>cbox a b. continuous (at x) f \<Longrightarrow>
f b\<bullet>k \<le> y \<Longrightarrow> y \<le> f a\<bullet>k \<Longrightarrow> \<exists>x\<in>cbox a b. (f x)\<bullet>k = y"
by (rule ivt_decreasing_component_on_1) (auto simp: continuous_at_imp_continuous_on)
subsection {* A bound within a convex hull, and so an interval *}
lemma convex_on_convex_hull_bound:
assumes "convex_on (convex hull s) f"
and "\<forall>x\<in>s. f x \<le> b"
shows "\<forall>x\<in> convex hull s. f x \<le> b"
proof
fix x
assume "x \<in> convex hull s"
then obtain k u v where
obt: "\<forall>i\<in>{1..k::nat}. 0 \<le> u i \<and> v i \<in> s" "setsum u {1..k} = 1" "(\<Sum>i = 1..k. u i *\<^sub>R v i) = x"
unfolding convex_hull_indexed mem_Collect_eq by auto
have "(\<Sum>i = 1..k. u i * f (v i)) \<le> b"
using setsum_mono[of "{1..k}" "\<lambda>i. u i * f (v i)" "\<lambda>i. u i * b"]
unfolding setsum_left_distrib[symmetric] obt(2) mult_1
apply (drule_tac meta_mp)
apply (rule mult_left_mono)
using assms(2) obt(1)
apply auto
done
then show "f x \<le> b"
using assms(1)[unfolded convex_on[OF convex_convex_hull], rule_format, of k u v]
unfolding obt(2-3)
using obt(1) and hull_subset[unfolded subset_eq, rule_format, of _ s]
by auto
qed
lemma inner_setsum_Basis[simp]: "i \<in> Basis \<Longrightarrow> (\<Sum>Basis) \<bullet> i = 1"
by (simp add: inner_setsum_left setsum.If_cases inner_Basis)
lemma convex_set_plus:
assumes "convex s" and "convex t" shows "convex (s + t)"
proof -
have "convex {x + y |x y. x \<in> s \<and> y \<in> t}"
using assms by (rule convex_sums)
moreover have "{x + y |x y. x \<in> s \<and> y \<in> t} = s + t"
unfolding set_plus_def by auto
finally show "convex (s + t)" .
qed
lemma convex_set_setsum:
assumes "\<And>i. i \<in> A \<Longrightarrow> convex (B i)"
shows "convex (\<Sum>i\<in>A. B i)"
proof (cases "finite A")
case True then show ?thesis using assms
by induct (auto simp: convex_set_plus)
qed auto
lemma finite_set_setsum:
assumes "finite A" and "\<forall>i\<in>A. finite (B i)" shows "finite (\<Sum>i\<in>A. B i)"
using assms by (induct set: finite, simp, simp add: finite_set_plus)
lemma set_setsum_eq:
"finite A \<Longrightarrow> (\<Sum>i\<in>A. B i) = {\<Sum>i\<in>A. f i |f. \<forall>i\<in>A. f i \<in> B i}"
apply (induct set: finite)
apply simp
apply simp
apply (safe elim!: set_plus_elim)
apply (rule_tac x="fun_upd f x a" in exI)
apply simp
apply (rule_tac f="\<lambda>x. a + x" in arg_cong)
apply (rule setsum.cong [OF refl])
apply clarsimp
apply fast
done
lemma box_eq_set_setsum_Basis:
shows "{x. \<forall>i\<in>Basis. x\<bullet>i \<in> B i} = (\<Sum>i\<in>Basis. image (\<lambda>x. x *\<^sub>R i) (B i))"
apply (subst set_setsum_eq [OF finite_Basis])
apply safe
apply (fast intro: euclidean_representation [symmetric])
apply (subst inner_setsum_left)
apply (subgoal_tac "(\<Sum>x\<in>Basis. f x \<bullet> i) = f i \<bullet> i")
apply (drule (1) bspec)
apply clarsimp
apply (frule setsum.remove [OF finite_Basis])
apply (erule trans)
apply simp
apply (rule setsum.neutral)
apply clarsimp
apply (frule_tac x=i in bspec, assumption)
apply (drule_tac x=x in bspec, assumption)
apply clarsimp
apply (cut_tac u=x and v=i in inner_Basis, assumption+)
apply (rule ccontr)
apply simp
done
lemma convex_hull_set_setsum:
"convex hull (\<Sum>i\<in>A. B i) = (\<Sum>i\<in>A. convex hull (B i))"
proof (cases "finite A")
assume "finite A" then show ?thesis
by (induct set: finite, simp, simp add: convex_hull_set_plus)
qed simp
lemma convex_hull_eq_real_cbox:
fixes x y :: real assumes "x \<le> y"
shows "convex hull {x, y} = cbox x y"
proof (rule hull_unique)
show "{x, y} \<subseteq> cbox x y" using `x \<le> y` by auto
show "convex (cbox x y)"
by (rule convex_box)
next
fix s assume "{x, y} \<subseteq> s" and "convex s"
then show "cbox x y \<subseteq> s"
unfolding is_interval_convex_1 [symmetric] is_interval_def Basis_real_def
by - (clarify, simp (no_asm_use), fast)
qed
lemma unit_interval_convex_hull:
"cbox (0::'a::euclidean_space) One = convex hull {x. \<forall>i\<in>Basis. (x\<bullet>i = 0) \<or> (x\<bullet>i = 1)}"
(is "?int = convex hull ?points")
proof -
have One[simp]: "\<And>i. i \<in> Basis \<Longrightarrow> One \<bullet> i = 1"
by (simp add: One_def inner_setsum_left setsum.If_cases inner_Basis)
have "?int = {x. \<forall>i\<in>Basis. x \<bullet> i \<in> cbox 0 1}"
by (auto simp: cbox_def)
also have "\<dots> = (\<Sum>i\<in>Basis. (\<lambda>x. x *\<^sub>R i) ` cbox 0 1)"
by (simp only: box_eq_set_setsum_Basis)
also have "\<dots> = (\<Sum>i\<in>Basis. (\<lambda>x. x *\<^sub>R i) ` (convex hull {0, 1}))"
by (simp only: convex_hull_eq_real_cbox zero_le_one)
also have "\<dots> = (\<Sum>i\<in>Basis. convex hull ((\<lambda>x. x *\<^sub>R i) ` {0, 1}))"
by (simp only: convex_hull_linear_image linear_scaleR_left)
also have "\<dots> = convex hull (\<Sum>i\<in>Basis. (\<lambda>x. x *\<^sub>R i) ` {0, 1})"
by (simp only: convex_hull_set_setsum)
also have "\<dots> = convex hull {x. \<forall>i\<in>Basis. x\<bullet>i \<in> {0, 1}}"
by (simp only: box_eq_set_setsum_Basis)
also have "convex hull {x. \<forall>i\<in>Basis. x\<bullet>i \<in> {0, 1}} = convex hull ?points"
by simp
finally show ?thesis .
qed
text {* And this is a finite set of vertices. *}
lemma unit_cube_convex_hull:
obtains s :: "'a::euclidean_space set"
where "finite s" and "cbox 0 (\<Sum>Basis) = convex hull s"
apply (rule that[of "{x::'a. \<forall>i\<in>Basis. x\<bullet>i=0 \<or> x\<bullet>i=1}"])
apply (rule finite_subset[of _ "(\<lambda>s. (\<Sum>i\<in>Basis. (if i\<in>s then 1 else 0) *\<^sub>R i)::'a) ` Pow Basis"])
prefer 3
apply (rule unit_interval_convex_hull)
apply rule
unfolding mem_Collect_eq
proof -
fix x :: 'a
assume as: "\<forall>i\<in>Basis. x \<bullet> i = 0 \<or> x \<bullet> i = 1"
show "x \<in> (\<lambda>s. \<Sum>i\<in>Basis. (if i\<in>s then 1 else 0) *\<^sub>R i) ` Pow Basis"
apply (rule image_eqI[where x="{i. i\<in>Basis \<and> x\<bullet>i = 1}"])
using as
apply (subst euclidean_eq_iff)
apply auto
done
qed auto
text {* Hence any cube (could do any nonempty interval). *}
lemma cube_convex_hull:
assumes "d > 0"
obtains s :: "'a::euclidean_space set" where
"finite s" and "cbox (x - (\<Sum>i\<in>Basis. d*\<^sub>Ri)) (x + (\<Sum>i\<in>Basis. d*\<^sub>Ri)) = convex hull s"
proof -
let ?d = "(\<Sum>i\<in>Basis. d*\<^sub>Ri)::'a"
have *: "cbox (x - ?d) (x + ?d) = (\<lambda>y. x - ?d + (2 * d) *\<^sub>R y) ` cbox 0 (\<Sum>Basis)"
apply (rule set_eqI, rule)
unfolding image_iff
defer
apply (erule bexE)
proof -
fix y
assume as: "y\<in>cbox (x - ?d) (x + ?d)"
then have "inverse (2 * d) *\<^sub>R (y - (x - ?d)) \<in> cbox 0 (\<Sum>Basis)"
using assms by (simp add: mem_box field_simps inner_simps)
with `0 < d` show "\<exists>z\<in>cbox 0 (\<Sum>Basis). y = x - ?d + (2 * d) *\<^sub>R z"
by (intro bexI[of _ "inverse (2 * d) *\<^sub>R (y - (x - ?d))"]) auto
next
fix y z
assume as: "z\<in>cbox 0 (\<Sum>Basis)" "y = x - ?d + (2*d) *\<^sub>R z"
have "\<And>i. i\<in>Basis \<Longrightarrow> 0 \<le> d * (z \<bullet> i) \<and> d * (z \<bullet> i) \<le> d"
using assms as(1)[unfolded mem_box]
apply (erule_tac x=i in ballE)
apply rule
prefer 2
apply (rule mult_right_le_one_le)
using assms
apply auto
done
then show "y \<in> cbox (x - ?d) (x + ?d)"
unfolding as(2) mem_box
apply -
apply rule
using as(1)[unfolded mem_box]
apply (erule_tac x=i in ballE)
using assms
apply (auto simp: inner_simps)
done
qed
obtain s where "finite s" "cbox 0 (\<Sum>Basis::'a) = convex hull s"
using unit_cube_convex_hull by auto
then show ?thesis
apply (rule_tac that[of "(\<lambda>y. x - ?d + (2 * d) *\<^sub>R y)` s"])
unfolding * and convex_hull_affinity
apply auto
done
qed
subsection {* Bounded convex function on open set is continuous *}
lemma convex_on_bounded_continuous:
fixes s :: "('a::real_normed_vector) set"
assumes "open s"
and "convex_on s f"
and "\<forall>x\<in>s. abs(f x) \<le> b"
shows "continuous_on s f"
apply (rule continuous_at_imp_continuous_on)
unfolding continuous_at_real_range
proof (rule,rule,rule)
fix x and e :: real
assume "x \<in> s" "e > 0"
def B \<equiv> "abs b + 1"
have B: "0 < B" "\<And>x. x\<in>s \<Longrightarrow> abs (f x) \<le> B"
unfolding B_def
defer
apply (drule assms(3)[rule_format])
apply auto
done
obtain k where "k > 0" and k: "cball x k \<subseteq> s"
using assms(1)[unfolded open_contains_cball, THEN bspec[where x=x]]
using `x\<in>s` by auto
show "\<exists>d>0. \<forall>x'. norm (x' - x) < d \<longrightarrow> \<bar>f x' - f x\<bar> < e"
apply (rule_tac x="min (k / 2) (e / (2 * B) * k)" in exI)
apply rule
defer
proof (rule, rule)
fix y
assume as: "norm (y - x) < min (k / 2) (e / (2 * B) * k)"
show "\<bar>f y - f x\<bar> < e"
proof (cases "y = x")
case False
def t \<equiv> "k / norm (y - x)"
have "2 < t" "0<t"
unfolding t_def using as False and `k>0`
by (auto simp add:field_simps)
have "y \<in> s"
apply (rule k[unfolded subset_eq,rule_format])
unfolding mem_cball dist_norm
apply (rule order_trans[of _ "2 * norm (x - y)"])
using as
by (auto simp add: field_simps norm_minus_commute)
{
def w \<equiv> "x + t *\<^sub>R (y - x)"
have "w \<in> s"
unfolding w_def
apply (rule k[unfolded subset_eq,rule_format])
unfolding mem_cball dist_norm
unfolding t_def
using `k>0`
apply auto
done
have "(1 / t) *\<^sub>R x + - x + ((t - 1) / t) *\<^sub>R x = (1 / t - 1 + (t - 1) / t) *\<^sub>R x"
by (auto simp add: algebra_simps)
also have "\<dots> = 0"
using `t > 0` by (auto simp add:field_simps)
finally have w: "(1 / t) *\<^sub>R w + ((t - 1) / t) *\<^sub>R x = y"
unfolding w_def using False and `t > 0`
by (auto simp add: algebra_simps)
have "2 * B < e * t"
unfolding t_def using `0 < e` `0 < k` `B > 0` and as and False
by (auto simp add:field_simps)
then have "(f w - f x) / t < e"
using B(2)[OF `w\<in>s`] and B(2)[OF `x\<in>s`]
using `t > 0` by (auto simp add:field_simps)
then have th1: "f y - f x < e"
apply -
apply (rule le_less_trans)
defer
apply assumption
using assms(2)[unfolded convex_on_def,rule_format,of w x "1/t" "(t - 1)/t", unfolded w]
using `0 < t` `2 < t` and `x \<in> s` `w \<in> s`
by (auto simp add:field_simps)
}
moreover
{
def w \<equiv> "x - t *\<^sub>R (y - x)"
have "w \<in> s"
unfolding w_def
apply (rule k[unfolded subset_eq,rule_format])
unfolding mem_cball dist_norm
unfolding t_def
using `k > 0`
apply auto
done
have "(1 / (1 + t)) *\<^sub>R x + (t / (1 + t)) *\<^sub>R x = (1 / (1 + t) + t / (1 + t)) *\<^sub>R x"
by (auto simp add: algebra_simps)
also have "\<dots> = x"
using `t > 0` by (auto simp add:field_simps)
finally have w: "(1 / (1+t)) *\<^sub>R w + (t / (1 + t)) *\<^sub>R y = x"
unfolding w_def using False and `t > 0`
by (auto simp add: algebra_simps)
have "2 * B < e * t"
unfolding t_def
using `0 < e` `0 < k` `B > 0` and as and False
by (auto simp add:field_simps)
then have *: "(f w - f y) / t < e"
using B(2)[OF `w\<in>s`] and B(2)[OF `y\<in>s`]
using `t > 0`
by (auto simp add:field_simps)
have "f x \<le> 1 / (1 + t) * f w + (t / (1 + t)) * f y"
using assms(2)[unfolded convex_on_def,rule_format,of w y "1/(1+t)" "t / (1+t)",unfolded w]
using `0 < t` `2 < t` and `y \<in> s` `w \<in> s`
by (auto simp add:field_simps)
also have "\<dots> = (f w + t * f y) / (1 + t)"
using `t > 0` by (auto simp add: divide_simps)
also have "\<dots> < e + f y"
using `t > 0` * `e > 0` by (auto simp add: field_simps)
finally have "f x - f y < e" by auto
}
ultimately show ?thesis by auto
qed (insert `0<e`, auto)
qed (insert `0<e` `0<k` `0<B`, auto simp: field_simps)
qed
subsection {* Upper bound on a ball implies upper and lower bounds *}
lemma convex_bounds_lemma:
fixes x :: "'a::real_normed_vector"
assumes "convex_on (cball x e) f"
and "\<forall>y \<in> cball x e. f y \<le> b"
shows "\<forall>y \<in> cball x e. abs (f y) \<le> b + 2 * abs (f x)"
apply rule
proof (cases "0 \<le> e")
case True
fix y
assume y: "y \<in> cball x e"
def z \<equiv> "2 *\<^sub>R x - y"
have *: "x - (2 *\<^sub>R x - y) = y - x"
by (simp add: scaleR_2)
have z: "z \<in> cball x e"
using y unfolding z_def mem_cball dist_norm * by (auto simp add: norm_minus_commute)
have "(1 / 2) *\<^sub>R y + (1 / 2) *\<^sub>R z = x"
unfolding z_def by (auto simp add: algebra_simps)
then show "\<bar>f y\<bar> \<le> b + 2 * \<bar>f x\<bar>"
using assms(1)[unfolded convex_on_def,rule_format, OF y z, of "1/2" "1/2"]
using assms(2)[rule_format,OF y] assms(2)[rule_format,OF z]
by (auto simp add:field_simps)
next
case False
fix y
assume "y \<in> cball x e"
then have "dist x y < 0"
using False unfolding mem_cball not_le by (auto simp del: dist_not_less_zero)
then show "\<bar>f y\<bar> \<le> b + 2 * \<bar>f x\<bar>"
using zero_le_dist[of x y] by auto
qed
subsubsection {* Hence a convex function on an open set is continuous *}
lemma real_of_nat_ge_one_iff: "1 \<le> real (n::nat) \<longleftrightarrow> 1 \<le> n"
by auto
lemma convex_on_continuous:
assumes "open (s::('a::euclidean_space) set)" "convex_on s f"
shows "continuous_on s f"
unfolding continuous_on_eq_continuous_at[OF assms(1)]
proof
note dimge1 = DIM_positive[where 'a='a]
fix x
assume "x \<in> s"
then obtain e where e: "cball x e \<subseteq> s" "e > 0"
using assms(1) unfolding open_contains_cball by auto
def d \<equiv> "e / real DIM('a)"
have "0 < d"
unfolding d_def using `e > 0` dimge1 by auto
let ?d = "(\<Sum>i\<in>Basis. d *\<^sub>R i)::'a"
obtain c
where c: "finite c"
and c1: "convex hull c \<subseteq> cball x e"
and c2: "cball x d \<subseteq> convex hull c"
proof
def c \<equiv> "\<Sum>i\<in>Basis. (\<lambda>a. a *\<^sub>R i) ` {x\<bullet>i - d, x\<bullet>i + d}"
show "finite c"
unfolding c_def by (simp add: finite_set_setsum)
have 1: "convex hull c = {a. \<forall>i\<in>Basis. a \<bullet> i \<in> cbox (x \<bullet> i - d) (x \<bullet> i + d)}"
unfolding box_eq_set_setsum_Basis
unfolding c_def convex_hull_set_setsum
apply (subst convex_hull_linear_image [symmetric])
apply (simp add: linear_iff scaleR_add_left)
apply (rule setsum.cong [OF refl])
apply (rule image_cong [OF _ refl])
apply (rule convex_hull_eq_real_cbox)
apply (cut_tac `0 < d`, simp)
done
then have 2: "convex hull c = {a. \<forall>i\<in>Basis. a \<bullet> i \<in> cball (x \<bullet> i) d}"
by (simp add: dist_norm abs_le_iff algebra_simps)
show "cball x d \<subseteq> convex hull c"
unfolding 2
apply clarsimp
apply (simp only: dist_norm)
apply (subst inner_diff_left [symmetric])
apply simp
apply (erule (1) order_trans [OF Basis_le_norm])
done
have e': "e = (\<Sum>(i::'a)\<in>Basis. d)"
by (simp add: d_def real_of_nat_def DIM_positive)
show "convex hull c \<subseteq> cball x e"
unfolding 2
apply clarsimp
apply (subst euclidean_dist_l2)
apply (rule order_trans [OF setL2_le_setsum])
apply (rule zero_le_dist)
unfolding e'
apply (rule setsum_mono)
apply simp
done
qed
def k \<equiv> "Max (f ` c)"
have "convex_on (convex hull c) f"
apply(rule convex_on_subset[OF assms(2)])
apply(rule subset_trans[OF _ e(1)])
apply(rule c1)
done
then have k: "\<forall>y\<in>convex hull c. f y \<le> k"
apply (rule_tac convex_on_convex_hull_bound)
apply assumption
unfolding k_def
apply (rule, rule Max_ge)
using c(1)
apply auto
done
have "d \<le> e"
unfolding d_def
apply (rule mult_imp_div_pos_le)
using `e > 0`
unfolding mult_le_cancel_left1
apply (auto simp: real_of_nat_ge_one_iff Suc_le_eq DIM_positive)
done
then have dsube: "cball x d \<subseteq> cball x e"
by (rule subset_cball)
have conv: "convex_on (cball x d) f"
apply (rule convex_on_subset)
apply (rule convex_on_subset[OF assms(2)])
apply (rule e(1))
apply (rule dsube)
done
then have "\<forall>y\<in>cball x d. abs (f y) \<le> k + 2 * abs (f x)"
apply (rule convex_bounds_lemma)
apply (rule ballI)
apply (rule k [rule_format])
apply (erule rev_subsetD)
apply (rule c2)
done
then have "continuous_on (ball x d) f"
apply (rule_tac convex_on_bounded_continuous)
apply (rule open_ball, rule convex_on_subset[OF conv])
apply (rule ball_subset_cball)
apply force
done
then show "continuous (at x) f"
unfolding continuous_on_eq_continuous_at[OF open_ball]
using `d > 0` by auto
qed
subsection {* Line segments, Starlike Sets, etc. *}
(* Use the same overloading tricks as for intervals, so that
segment[a,b] is closed and segment(a,b) is open relative to affine hull. *)
definition midpoint :: "'a::real_vector \<Rightarrow> 'a \<Rightarrow> 'a"
where "midpoint a b = (inverse (2::real)) *\<^sub>R (a + b)"
definition open_segment :: "'a::real_vector \<Rightarrow> 'a \<Rightarrow> 'a set"
where "open_segment a b = {(1 - u) *\<^sub>R a + u *\<^sub>R b | u::real. 0 < u \<and> u < 1}"
definition 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 "between = (\<lambda>(a,b) x. x \<in> closed_segment a b)"
lemmas segment = open_segment_def closed_segment_def
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)
definition "starlike s \<longleftrightarrow> (\<exists>a\<in>s. \<forall>x\<in>s. closed_segment a x \<subseteq> s)"
lemma midpoint_refl: "midpoint x x = x"
unfolding midpoint_def
unfolding scaleR_right_distrib
unfolding scaleR_left_distrib[symmetric]
by auto
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 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:
"midpoint a b = a \<longleftrightarrow> a = b"
"midpoint a b = b \<longleftrightarrow> a = b"
unfolding midpoint_eq_iff by auto
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 convex_imp_starlike:
"convex s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> starlike s"
unfolding convex_contains_segment starlike_def by auto
lemma segment_convex_hull:
"closed_segment a b = convex hull {a,b}"
proof -
have *: "\<And>x. {x} \<noteq> {}" by auto
have **: "\<And>u v. u + v = 1 \<longleftrightarrow> u = 1 - (v::real)" by auto
show ?thesis
unfolding segment convex_hull_insert[OF *] convex_hull_singleton
apply (rule set_eqI)
unfolding mem_Collect_eq
apply (rule, erule exE)
apply (rule_tac x="1 - u" in exI)
apply rule
defer
apply (rule_tac x=u in exI)
defer
apply (elim exE conjE)
apply (rule_tac x="1 - u" in exI)
unfolding **
apply auto
done
qed
lemma convex_segment: "convex (closed_segment a b)"
unfolding segment_convex_hull by(rule convex_convex_hull)
lemma ends_in_segment: "a \<in> closed_segment a b" "b \<in> closed_segment a b"
unfolding segment_convex_hull
apply (rule_tac[!] hull_subset[unfolded subset_eq, rule_format])
apply auto
done
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 segment_bound:
fixes x a b :: "'a::euclidean_space"
assumes "x \<in> closed_segment a b"
shows "norm (x - a) \<le> norm (b - a)" "norm (x - b) \<le> norm (b - a)"
using segment_furthest_le[OF assms, of a]
using segment_furthest_le[OF assms, of b]
by (auto simp add:norm_minus_commute)
lemma segment_refl: "closed_segment a a = {a}"
unfolding segment by (auto simp add: algebra_simps)
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 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 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
unfolding between_def split_conv
by (auto simp add:segment_refl 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)
show ?thesis
unfolding between_def split_conv closed_segment_def mem_Collect_eq
apply rule
apply (elim exE conjE)
apply (subst dist_triangle_eq)
proof -
fix u
assume as: "x = (1 - u) *\<^sub>R a + u *\<^sub>R b" "0 \<le> u" "u \<le> 1"
then have *: "a - x = u *\<^sub>R (a - b)" "x - b = (1 - u) *\<^sub>R (a - b)"
unfolding as(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 as(2,3)
by (auto simp add: field_simps)
next
assume as: "dist a b = dist a x + dist x b"
have "norm (a - x) / norm (a - b) \<le> 1"
using Fal2 unfolding as[unfolded dist_norm] norm_ge_zero by auto
then show "\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and> 0 \<le> u \<and> u \<le> 1"
apply (rule_tac x="dist a x / dist a b" in exI)
unfolding dist_norm
apply (subst euclidean_eq_iff)
apply rule
defer
apply rule
prefer 3
apply rule
proof -
fix i :: 'a
assume i: "i \<in> Basis"
have "((1 - norm (a - x) / norm (a - b)) *\<^sub>R a + (norm (a - x) / norm (a - b)) *\<^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 as[unfolded dist_norm]
using as[unfolded dist_triangle_eq]
apply -
apply (subst (asm) euclidean_eq_iff)
using i
apply (erule_tac x=i in ballE)
apply (auto simp add: field_simps inner_simps)
done
finally show "x \<bullet> i =
((1 - norm (a - x) / norm (a - b)) *\<^sub>R a + (norm (a - x) / norm (a - b)) *\<^sub>R b) \<bullet> i"
by auto
qed (insert Fal2, auto)
qed
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
apply(rule_tac[!] *)
unfolding euclidean_eq_iff[where 'a='a]
apply (auto simp add: field_simps inner_simps)
done
qed
lemma between_mem_convex_hull:
"between (a,b) x \<longleftrightarrow> x \<in> convex hull {a,b}"
unfolding between_mem_segment segment_convex_hull ..
subsection {* Shrinking towards the interior of a convex set *}
lemma mem_interior_convex_shrink:
fixes s :: "'a::euclidean_space set"
assumes "convex s"
and "c \<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
apply (rule_tac x="e*d" in exI)
apply rule
defer
unfolding subset_eq Ball_def mem_ball
proof (rule, rule)
fix y
assume as: "dist (x - e *\<^sub>R (x - c)) y < e * d"
have *: "y = (1 - (1 - e)) *\<^sub>R ((1 / e) *\<^sub>R y - ((1 - e) / e) *\<^sub>R x) + (1 - e) *\<^sub>R x"
using `e > 0` 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) = abs(1/e) * 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 `e > 0`
by (auto simp add: euclidean_eq_iff[where 'a='a] field_simps inner_simps)
also have "\<dots> = abs (1/e) * 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 `e > 0`
by (auto simp add:pos_divide_less_eq[OF `e > 0`] 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 `e>0` `d>0`, 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 `e > 0` `d > 0`
apply (rule_tac bexI[where x=x])
apply (auto)
done
next
case False
then have x: "x islimpt s"
using assms(3)[unfolded closure_def] by auto
show ?thesis
proof (cases "e = 1")
case True
obtain y where "y \<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 `d > 0`
apply auto
done
next
case False
then have "0 < e * d / (1 - e)" and *: "1 - e > 0"
using `e \<le> 1` `e > 0` `d > 0` 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
def z \<equiv> "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 `e > 0`
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 *
apply -
apply (rule mem_interior_convex_shrink)
using assms(1,4-5) `y\<in>s`
apply auto
done
qed
subsection {* Some obvious but surprisingly hard simplex lemmas *}
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> setsum u s \<le> 1 \<and> setsum (\<lambda>x. u x *\<^sub>R x) s = y)}"
unfolding convex_hull_finite[OF finite.insertI[OF assms(1)]]
apply (rule set_eqI, rule)
unfolding mem_Collect_eq
apply (erule_tac[!] exE)
apply (erule_tac[!] conjE)+
unfolding setsum_clauses(2)[OF assms(1)]
apply (rule_tac x=u in exI)
defer
apply (rule_tac x="\<lambda>x. if x = 0 then 1 - setsum u s else u x" in exI)
using assms(2)
unfolding if_smult and setsum_delta_notmem[OF assms(2)]
apply auto
done
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 `finite d` `0 \<notin> ?p`]
apply (rule set_eqI)
unfolding mem_Collect_eq
apply rule
apply (elim exE conjE)
apply (erule_tac[2] conjE)+
proof -
fix x :: "'a::euclidean_space"
fix u
assume as: "\<forall>x\<in>?D. 0 \<le> u x" "setsum u ?D \<le> 1" "(\<Sum>x\<in>?D. u x *\<^sub>R x) = x"
have *: "\<forall>i\<in>Basis. i:d \<longrightarrow> u i = x\<bullet>i"
and "(\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0)"
using as(3)
unfolding substdbasis_expansion_unique[OF assms]
by auto
then have **: "setsum u ?D = setsum (op \<bullet> x) ?D"
apply -
apply (rule setsum.cong)
using assms
apply auto
done
have "(\<forall>i\<in>Basis. 0 \<le> x\<bullet>i) \<and> setsum (op \<bullet> x) ?D \<le> 1"
proof (rule,rule)
fix i :: 'a
assume i: "i \<in> Basis"
have "i \<in> d \<Longrightarrow> 0 \<le> x\<bullet>i"
unfolding *[rule_format,OF i,symmetric]
apply (rule_tac as(1)[rule_format])
apply auto
done
moreover have "i \<notin> d \<Longrightarrow> 0 \<le> x\<bullet>i"
using `(\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0)`[rule_format, OF i] by auto
ultimately show "0 \<le> x\<bullet>i" by auto
qed (insert as(2)[unfolded **], auto)
then show "(\<forall>i\<in>Basis. 0 \<le> x\<bullet>i) \<and> setsum (op \<bullet> x) ?D \<le> 1 \<and> (\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0)"
using `(\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0)` by auto
next
fix x :: "'a::euclidean_space"
assume as: "\<forall>i\<in>Basis. 0 \<le> x \<bullet> i" "setsum (op \<bullet> x) ?D \<le> 1" "(\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0)"
show "\<exists>u. (\<forall>x\<in>?D. 0 \<le> u x) \<and> setsum u ?D \<le> 1 \<and> (\<Sum>x\<in>?D. u x *\<^sub>R x) = x"
using as d
unfolding substdbasis_expansion_unique[OF assms]
apply (rule_tac x="inner x" in exI)
apply auto
done
qed
qed
lemma std_simplex:
"convex hull (insert 0 Basis) =
{x::'a::euclidean_space. (\<forall>i\<in>Basis. 0 \<le> x\<bullet>i) \<and> setsum (\<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> setsum (\<lambda>i. x\<bullet>i) Basis < 1}"
apply (rule set_eqI)
unfolding mem_interior std_simplex
unfolding subset_eq mem_Collect_eq Ball_def mem_ball
unfolding Ball_def[symmetric]
apply rule
apply (elim exE conjE)
defer
apply (erule conjE)
proof -
fix x :: 'a
fix e
assume "e > 0" and as: "\<forall>xa. dist x xa < e \<longrightarrow> (\<forall>x\<in>Basis. 0 \<le> xa \<bullet> x) \<and> setsum (op \<bullet> xa) Basis \<le> 1"
show "(\<forall>xa\<in>Basis. 0 < x \<bullet> xa) \<and> setsum (op \<bullet> x) Basis < 1"
apply safe
proof -
fix i :: 'a
assume i: "i \<in> Basis"
then show "0 < x \<bullet> i"
using as[THEN spec[where x="x - (e / 2) *\<^sub>R i"]] and `e > 0`
unfolding dist_norm
by (auto elim!: ballE[where x=i] simp: inner_simps)
next
have **: "dist x (x + (e / 2) *\<^sub>R (SOME i. i\<in>Basis)) < e" using `e > 0`
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 *: "setsum (op \<bullet> (x + (e / 2) *\<^sub>R (SOME i. i\<in>Basis))) Basis =
setsum (\<lambda>i. x\<bullet>i + (if (SOME i. i\<in>Basis) = i then e/2 else 0)) Basis"
apply (rule_tac setsum.cong)
apply auto
done
have "setsum (op \<bullet> x) Basis < setsum (op \<bullet> (x + (e / 2) *\<^sub>R (SOME i. i\<in>Basis))) Basis"
unfolding * setsum.distrib
using `e > 0` DIM_positive[where 'a='a]
apply (subst setsum.delta')
apply (auto simp: SOME_Basis)
done
also have "\<dots> \<le> 1"
using **
apply (drule_tac as[rule_format])
apply auto
done
finally show "setsum (op \<bullet> x) Basis < 1" by auto
qed
next
fix x :: 'a
assume as: "\<forall>i\<in>Basis. 0 < x \<bullet> i" "setsum (op \<bullet> x) Basis < 1"
obtain a :: 'b where "a \<in> UNIV" using UNIV_witness ..
let ?d = "(1 - setsum (op \<bullet> x) Basis) / real (DIM('a))"
have "Min ((op \<bullet> x) ` Basis) > 0"
apply (rule Min_grI)
using as(1)
apply auto
done
moreover have "?d > 0"
using as(2) by (auto simp: Suc_le_eq DIM_positive)
ultimately show "\<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> (\<forall>i\<in>Basis. 0 \<le> y \<bullet> i) \<and> setsum (op \<bullet> y) Basis \<le> 1"
apply (rule_tac x="min (Min ((op \<bullet> x) ` Basis)) D" for D in exI)
apply rule
defer
apply (rule, rule)
proof -
fix y
assume y: "dist x y < min (Min (op \<bullet> x ` Basis)) ?d"
have "setsum (op \<bullet> y) Basis \<le> setsum (\<lambda>i. x\<bullet>i + ?d) Basis"
proof (rule setsum_mono)
fix i :: 'a
assume i: "i \<in> Basis"
then have "abs (y\<bullet>i - x\<bullet>i) < ?d"
apply -
apply (rule le_less_trans)
using Basis_le_norm[OF i, of "y - x"]
using y[unfolded min_less_iff_conj dist_norm, THEN conjunct2]
apply (auto simp add: norm_minus_commute inner_diff_left)
done
then show "y \<bullet> i \<le> x \<bullet> i + ?d" by auto
qed
also have "\<dots> \<le> 1"
unfolding setsum.distrib setsum_constant real_eq_of_nat
by (auto simp add: Suc_le_eq)
finally show "(\<forall>i\<in>Basis. 0 \<le> y \<bullet> i) \<and> setsum (op \<bullet> y) Basis \<le> 1"
proof safe
fix i :: 'a
assume i: "i \<in> Basis"
have "norm (x - y) < x\<bullet>i"
apply (rule less_le_trans)
apply (rule y[unfolded min_less_iff_conj dist_norm, THEN conjunct1])
using i
apply auto
done
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
qed auto
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 = "setsum (\<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 _ "setsum (\<lambda>j. if i = j then inverse (2 * real DIM('a)) else 0) ?D"])
(simp_all add: setsum.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 "setsum (op \<bullet> ?a) ?D = setsum (\<lambda>i. inverse (2 * real DIM('a))) ?D"
apply (rule setsum.cong)
apply rule
apply auto
done
also have "\<dots> < 1"
unfolding setsum_constant real_eq_of_nat divide_inverse[symmetric]
by (auto simp add: field_simps)
finally show "setsum (op \<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"
apply (rule finite_subset)
using assms
apply auto
done
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 e0: "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: "\<forall>xa. dist x xa < e \<and> (\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> xa\<bullet>i = 0) \<longrightarrow>
(\<forall>i\<in>d. 0 \<le> xa \<bullet> i) \<and> setsum (op \<bullet> xa) 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> setsum (op \<bullet> x) d < 1 \<and> (\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x\<bullet>i = 0)"
apply rule
apply rule
proof -
fix i :: 'a
assume "i \<in> d"
then have "\<forall>ia\<in>d. 0 \<le> (x - (e / 2) *\<^sub>R i) \<bullet> ia"
apply -
apply (rule as[rule_format,THEN conjunct1])
unfolding dist_norm
using d `e > 0` x0
apply (auto simp: inner_simps inner_Basis)
done
then show "0 < x \<bullet> i"
apply (erule_tac x=i in ballE)
using `e > 0` `i \<in> d` d
apply (auto simp: inner_simps inner_Basis)
done
next
obtain a where a: "a \<in> d"
using `d \<noteq> {}` by auto
then have **: "dist x (x + (e / 2) *\<^sub>R a) < e"
using `e > 0` 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 *: "setsum (op \<bullet> (x + (e / 2) *\<^sub>R a)) d =
setsum (\<lambda>i. x\<bullet>i + (if a = i then e/2 else 0)) d"
using d by (intro setsum.cong) auto
have "a \<in> Basis"
using `a \<in> d` 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 `a\<in>d` by (auto simp add: inner_add_left inner_Basis)
have "setsum (op \<bullet> x) d < setsum (op \<bullet> (x + (e / 2) *\<^sub>R a)) d"
unfolding * setsum.distrib
using `e > 0` `a \<in> d`
using `finite d`
by (auto simp add: setsum.delta')
also have "\<dots> \<le> 1"
using ** h1 as[rule_format, of "x + (e / 2) *\<^sub>R a"]
by auto
finally show "setsum (op \<bullet> x) d < 1 \<and> (\<forall>i\<in>Basis. 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 `d \<noteq> {}` by auto
let ?d = "(1 - setsum (op \<bullet> x) d) / real (card d)"
have "0 < card d" using `d \<noteq> {}` `finite d`
by (simp add: card_gt_0_iff)
have "Min ((op \<bullet> x) ` d) > 0"
using as `d \<noteq> {}` `finite d` by (simp add: Min_grI)
moreover have "?d > 0" using as using `0 < card d` by auto
ultimately have h3: "min (Min ((op \<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 ((op \<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 (op \<bullet> x ` d)) ?d"
assume y2: "\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> y\<bullet>i = 0"
have "setsum (op \<bullet> y) d \<le> setsum (\<lambda>i. x\<bullet>i + ?d) d"
proof (rule setsum_mono)
fix i
assume "i \<in> d"
with d have i: "i \<in> Basis"
by auto
have "abs (y\<bullet>i - x\<bullet>i) < ?d"
apply (rule le_less_trans)
using Basis_le_norm[OF i, of "y - x"]
using y[unfolded min_less_iff_conj dist_norm, THEN conjunct2]
apply (auto simp add: norm_minus_commute inner_simps)
done
then show "y \<bullet> i \<le> x \<bullet> i + ?d" by auto
qed
also have "\<dots> \<le> 1"
unfolding setsum.distrib setsum_constant real_eq_of_nat
using `0 < card d`
by auto
finally show "setsum (op \<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 "op \<bullet> x ` d" "norm (x - y)"] `0 < card d` `i:d`
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> setsum (op \<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 = "setsum (\<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 `d \<noteq> {}` by auto
{
fix i
assume "i \<in> d"
have "?a \<bullet> i = inverse (2 * real (card d))"
apply (rule trans[of _ "setsum (\<lambda>j. if i = j then inverse (2 * real (card d)) else 0) ?D"])
unfolding inner_setsum_left
apply (rule setsum.cong)
using `i \<in> d` `finite d` setsum.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] `i \<in> d`
by auto
finally show "0 < ?a \<bullet> i" by auto
next
have "setsum (op \<bullet> ?a) ?D = setsum (\<lambda>i. inverse (2 * real (card d))) ?D"
by (rule setsum.cong) (rule refl, rule **)
also have "\<dots> < 1"
unfolding setsum_constant real_eq_of_nat divide_real_def[symmetric]
by (auto simp add: field_simps)
finally show "setsum (op \<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_setsum[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_superset[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 "\<forall>x\<in>d. x /\<^sub>R (2 * real (card d)) \<in> span d"
by auto
qed
then show "?a \<bullet> i = 0 "
using `i \<notin> d` unfolding span_substd_basis[OF assms(2)] using `i \<in> Basis` by auto
qed
qed
subsection {* Relative interior of convex set *}
lemma rel_interior_convex_nonempty_aux:
fixes S :: "'n::euclidean_space set"
assumes "convex S"
and "0 \<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 auto
then have "B \<noteq> {}"
using B assms `S \<noteq> {0}` span_empty by auto
have "insert 0 B \<le> span B"
using subspace_span[of B] subspace_0[of "span B"] span_inc 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 `B \<noteq> {}` by auto
have "insert 0 d = f ` (insert 0 B)"
using fd linear_0 by auto
then have "(convex hull (insert 0 d)) = f ` (convex hull (insert 0 B))"
using convex_hull_linear_image[of f "(insert 0 d)"]
convex_hull_linear_image[of f "(insert 0 B)"] `linear f`
by auto
moreover have "rel_interior (f ` (convex hull insert 0 B)) =
f ` rel_interior (convex hull insert 0 B)"
apply (rule rel_interior_injective_on_span_linear_image[of f "(convex hull insert 0 B)"])
using `bounded_linear f` fd *
apply auto
done
ultimately have "rel_interior (convex hull insert 0 B) \<noteq> {}"
using rel_interior_substd_simplex_nonempty[OF `d \<noteq> {}` 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_convex_nonempty:
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> op + (-a) ` S"
using assms exI[of "(\<lambda>x. x \<in> S \<and> - a + x = 0)" a] by auto
then have "rel_interior (op + (-a) ` S) \<noteq> {}"
using rel_interior_convex_nonempty_aux[of "op + (-a) ` S"]
convex_translation[of S "-a"] assms
by auto
then have "rel_interior S \<noteq> {}"
using rel_interior_translation by auto
}
then show ?thesis
using rel_interior_empty 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 `x \<in> S` 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_convex_nonempty 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"
def e1 \<equiv> "min 1 (e/norm (x - a))"
then have e1: "e1 > 0" "e1 \<le> 1" "e1 * norm (x - a) \<le> e"
using `x \<noteq> a` `e > 0` le_divide_eq[of e1 e "norm (x - a)"]
by simp_all
then have *: "x - e1 *\<^sub>R (x - a) : rel_interior S"
using rel_interior_closure_convex_shrink[of S a x e1] assms x a e1_def
by auto
have "\<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] `x \<noteq> a`
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 opein_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 -
def e \<equiv> "a / (a + b)"
have "z = (1 / (a + b)) *\<^sub>R ((a + b) *\<^sub>R z)"
apply auto
using assms
apply simp
done
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_convex_nonempty 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 `S \<noteq> {}` assms rel_interior_convex_nonempty 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
def y \<equiv> "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 auto
moreover have "z \<in> affine hull closure S"
using z rel_interior_subset hull_subset[of "closure S"] by auto
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 `y \<in> closure S`
by auto
qed
}
ultimately show ?thesis by auto
qed
lemma convex_interior_closure:
fixes S :: "'n::euclidean_space set"
assumes "convex S"
shows "interior (closure S) = interior S"
using closure_aff_dim[of S] interior_rel_interior_gen[of S]
interior_rel_interior_gen[of "closure S"]
convex_rel_interior_closure[of S] assms
by auto
lemma closure_eq_rel_interior_eq:
fixes S1 S2 :: "'n::euclidean_space set"
assumes "convex S1"
and "convex S2"
shows "closure S1 = closure S2 \<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 `?B` 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_inter_closure_eq_empty)
definition "rel_frontier S = closure S - rel_interior S"
lemma closed_affine_hull:
fixes S :: "'n::euclidean_space set"
shows "closed (affine hull S)"
by (metis affine_affine_hull affine_closed)
lemma closed_rel_frontier:
fixes S :: "'n::euclidean_space set"
shows "closed (rel_frontier S)"
proof -
have *: "closedin (subtopology euclidean (affine hull S)) (closure S - rel_interior S)"
apply (rule closedin_diff[of "subtopology euclidean (affine hull S)""closure S" "rel_interior S"])
using closed_closedin_trans[of "affine hull S" "closure S"] closed_affine_hull[of S]
closure_affine_hull[of S] opein_rel_interior[of S]
apply auto
done
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 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 auto
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] `S2 \<noteq> {}` by auto
have **: "affine hull S1 = affine hull S2"
apply (rule affine_dim_equal)
using * affine_affine_hull
apply auto
using `S1 \<noteq> {}` 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 `S1 \<noteq> {}` rel_interior_convex_nonempty 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"
def m \<equiv> "1 + e1/norm(x-z)"
hence "m > 1" using e1 `x \<noteq> z` 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 `x \<noteq> z` 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 `m> 1 ` by auto
}
moreover
{
assume "x = z"
def m \<equiv> "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 `x = z` 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 `m > 1` 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 auto
}
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_convex_nonempty 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
def y \<equiv> "(1 - e) *\<^sub>R x + e *\<^sub>R z"
then have "y \<in> S" using e by auto
def e1 \<equiv> "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"] `0 < e1 \<and> e1 < 1` `y \<in> S` 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
def x1 \<equiv> "z + e1 *\<^sub>R (x - z)"
then have x1: "x1 \<in> affine hull S"
using e1 hull_subset[of S] by auto
def x2 \<equiv> "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 `S \<noteq> {}` * 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
def e \<equiv> "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
def e \<equiv> "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 `S \<noteq> {}` 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 {* Relative interior and closure under common operations *}
lemma rel_interior_inter_aux: "\<Inter>{rel_interior S |S. S : I} \<subseteq> \<Inter>I"
proof -
{
fix y
assume "y \<in> \<Inter>{rel_interior S |S. S : 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_inter: "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"
def e1 \<equiv> "min 1 (e/norm (y - x))"
then have e1: "e1 > 0" "e1 \<le> 1" "e1 * norm (y - x) \<le> e"
using `y \<noteq> x` `e > 0` le_divide_eq[of e1 e "norm (y - x)"]
by simp_all
def z \<equiv> "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 `y \<noteq> x` 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_inter[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_inter[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_univ2 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
def e \<equiv> "Min (mS ` I)"
then have "e \<in> mS ` I" using assms `I \<noteq> {}` 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] `\<Inter>I \<noteq> {}` `convex (\<Inter>I)` by auto
}
then show ?thesis
using convex_rel_interior_inter[of I] assms by auto
qed
qed
lemma convex_closure_inter_two:
fixes S T :: "'n::euclidean_space set"
assumes "convex S"
and "convex T"
assumes "rel_interior S \<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_inter:
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_univ[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 convex_affine_rel_interior_inter:
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_univ[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 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_convex_nonempty by auto
next
case False
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 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 : S"
using convex_rel_interior_iff[of S z1] `convex S` 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 `linear f` by (simp add: linear_add_cmul)
ultimately have "(1 - e) *\<^sub>R x + e *\<^sub>R z : 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 : f ` S"
using e by auto
}
then have "z \<in> rel_interior (f ` S)"
using convex_rel_interior_iff[of "f ` S" z] `convex S`
`linear f` `S \<noteq> {}` 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 -
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 (metis assms(1) assms(2) convex_Int subspace_UNIV subspace_imp_convex subspace_linear_image)
{
fix z
assume "z \<in> f -` (rel_interior S)"
then have z: "f z : 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 `S \<noteq> {}` 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 `linear f` 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 `linear f` 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 `convex (S \<inter> (range f))` `S \<inter> range f \<noteq> {}`
convex_rel_interior_iff[of "S \<inter> (range f)" "f z"]
by auto
moreover have "affine (range f)"
by (metis assms(1) subspace_UNIV subspace_imp_affine subspace_linear_image)
ultimately have "f z \<in> rel_interior S"
using convex_affine_rel_interior_inter[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_direct_sum:
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_convex_nonempty 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 `convex S` 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 `convex T` 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 "(op *\<^sub>R c) ` (rel_interior S) = rel_interior ((op *\<^sub>R c) ` S)"
using rel_interior_injective_linear_image[of "(op *\<^sub>R c)" S]
linear_conv_bounded_linear[of "op *\<^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 "(op *\<^sub>R c) ` (rel_interior S) = rel_interior ((op *\<^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 ((op *\<^sub>R c) ` S) \<and> rel_open ((op *\<^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_inter[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 : rel_interior (f y))"
using h2 asm by auto
}
then show ?thesis using h2 by blast
qed
subsubsection {* Relative interior of convex cone *}
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> op *\<^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> op *\<^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> ((op *\<^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
def f \<equiv> "\<lambda>y. {z. (y, z) \<in> cone hull ({1 :: real} \<times> S)}"
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"] `s \<in> S` 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 = (op *\<^sub>R c ` S)"
using f_def cone_hull_expl[of "{1 :: real} \<times> S"] by auto
then have "rel_interior (f c) = op *\<^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 `z \<in> ?lhs`
apply auto
apply (rule_tac x = "fst z" in exI)
apply (rule_tac x = x in exI)
using *
apply auto
done
}
moreover
{
fix z
assume "z \<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)) =
{setsum (\<lambda>i. c i *\<^sub>R s i) I | c s. (\<forall>i\<in>I. c i \<ge> 0) \<and> setsum 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 : ?rhs"
then obtain c s where *: "setsum (\<lambda>i. c i *\<^sub>R s i) I = x" "setsum 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_setsum[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"
def c \<equiv> "\<lambda>j. if j = i then 1::real else 0"
then have *: "setsum c I = 1"
using `finite I` `i \<in> I` setsum.delta[of I i "\<lambda>j::'a. 1::real"]
by auto
def s \<equiv> "\<lambda>j. if j = i then x else p 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 = setsum (\<lambda>i. c i *\<^sub>R s i) I"
using s_def c_def `finite I` `i \<in> I` setsum.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 `x \<in> S i`
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 = setsum (\<lambda>i. c i *\<^sub>R s i) I \<and> (\<forall>i\<in>I. c i \<ge> 0) \<and> setsum c I = 1 \<and> (\<forall>i\<in>I. s i \<in> S i)"
by auto
from xy obtain d t where
yc: "y = setsum (\<lambda>i. d i *\<^sub>R t i) I \<and> (\<forall>i\<in>I. d i \<ge> 0) \<and> setsum d I = 1 \<and> (\<forall>i\<in>I. t i \<in> S i)"
by auto
def e \<equiv> "\<lambda>i. u * c i + v * d i"
have ge0: "\<forall>i\<in>I. e i \<ge> 0"
using e_def xc yc uv by simp
have "setsum (\<lambda>i. u * c i) I = u * setsum c I"
by (simp add: setsum_right_distrib)
moreover have "setsum (\<lambda>i. v * d i) I = v * setsum d I"
by (simp add: setsum_right_distrib)
ultimately have sum1: "setsum e I = 1"
using e_def xc yc uv by (simp add: setsum.distrib)
def q \<equiv> "\<lambda>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"
{
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 = setsum (\<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.setsum setsum.distrib)
also have "\<dots> = setsum (\<lambda>i. e i *\<^sub>R q i) I"
using * by auto
finally have "u *\<^sub>R x + v *\<^sub>R y = setsum (\<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 `?lhs \<supseteq> ?rhs` * 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
def I \<equiv> "{1::nat, 2}"
def s \<equiv> "\<lambda>i. if i = (1::nat) then S else T"
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> setsum 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> setsum 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 {* Convexity on direct sums *}
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 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_direct_sum 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 (setsum S I) = setsum (\<lambda>i. rel_interior (S i)) I"
apply (subst setsum_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_direct_sum 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)) = setsum S I"
(is "?lhs = ?rhs")
proof -
{
fix x
assume "x \<in> ?lhs"
then obtain c xs where
x: "x = setsum (\<lambda>i. c i *\<^sub>R xs i) I \<and> (\<forall>i\<in>I. c i \<ge> 0) \<and> setsum c I = 1 \<and> (\<forall>i\<in>I. xs i \<in> S i)"
using convex_hull_finite_union[of I S] assms by auto
def s \<equiv> "\<lambda>i. c i *\<^sub>R xs 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 = setsum s I" using x s_def by auto
ultimately have "x \<in> ?rhs"
using set_setsum_alt[of I S] assms by auto
}
moreover
{
fix x
assume "x \<in> ?rhs"
then obtain s where x: "x = setsum s I \<and> (\<forall>i\<in>I. s i \<in> S i)"
using set_setsum_alt[of I S] assms by auto
def xs \<equiv> "\<lambda>i. of_nat(card I) *\<^sub>R s i"
then have "x = setsum (\<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 "setsum (\<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 -
def I \<equiv> "{1::nat, 2}"
def A \<equiv> "(\<lambda>i. if i = (1::nat) then S else T)"
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) = setsum A I"
apply (subst convex_hull_finite_union_cones[of I A])
using assms A_def I_def
apply auto
done
moreover have "setsum 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))) =
{setsum (\<lambda>i. c i *\<^sub>R s i) I | c s. (\<forall>i\<in>I. c i > 0) \<and> setsum 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
def C0 \<equiv> "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
def K0 \<equiv> "cone hull ({1 :: real} \<times> C0)"
def K \<equiv> "\<lambda>i. cone hull ({1 :: real} \<times> S 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 `\<forall>i\<in>I. C0 \<ge> S i`
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> = setsum 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 `\<forall>i\<in>I. K i \<noteq> {}` K_def
apply auto
done
finally have "K0 = setsum K I" by auto
then have *: "rel_interior K0 = setsum (\<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) = setsum k I \<and> (\<forall>i\<in>I. k i \<in> rel_interior (K i))"
using `finite I` * set_setsum_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> setsum c I = 1"
using k by (simp add: setsum_prod)
then have "x \<in> ?rhs"
using k
apply auto
apply (rule_tac x = c in exI)
apply (rule_tac x = s in exI)
using cs
apply auto
done
}
moreover
{
fix x
assume "x \<in> ?rhs"
then obtain c s where cs: "x = setsum (\<lambda>i. c i *\<^sub>R s i) I \<and>
(\<forall>i\<in>I. c i > 0) \<and> setsum c I = 1 \<and> (\<forall>i\<in>I. s i \<in> rel_interior (S i))"
by auto
def k \<equiv> "\<lambda>i. (c i, c i *\<^sub>R s i)"
{
fix i assume "i: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_setsum_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: setsum_prod)
done
then have "x \<in> ?lhs"
using K0_def C0_def rel_interior_convex_cone_aux[of C0 1 x]
by (auto simp add: convex_convex_hull)
}
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 `x \<in> interior I`]
obtain e where e: "0 < e" "ball x e \<subseteq> interior I" .
moreover def t \<equiv> "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 `x \<in> interior I` e interior_subset[of I] have "t \<in> I" "x \<in> I" by auto
have "open (interior I)" by auto
from openE[OF this `x \<in> interior I`]
obtain e where "0 < e" "ball x e \<subseteq> interior I" .
moreover def K \<equiv> "x - e / 2"
with `0 < e` 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] `x \<in> interior I` 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 `t \<in> I` `x < t` by auto
show "(f x - f t) / (x - t) \<le> (f x - f y) / (x - y)"
using `convex_on I f` `x \<in> I` `y \<in> I` `x < t` `t < y`
by (rule convex_on_diff)
next
fix y
assume "y \<in> ?F x"
with order_trans[OF convex_on_diff[OF `convex_on I f` `K \<in> I` _ `K < x` _]]
show "(f K - f x) / (K - x) \<le> y" by auto
qed
then show ?thesis
using `x < y` by (simp add: field_simps)
next
assume "y < x"
moreover
have "open (interior I)" by auto
from openE[OF this `x \<in> interior I`]
obtain e where e: "0 < e" "ball x e \<subseteq> interior I" .
moreover def t \<equiv> "x + e / 2"
ultimately have "x < t" "t \<in> ball x e"
by (auto simp: dist_real_def field_simps)
with `x \<in> interior I` 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 `y < x` by (auto simp: field_simps)
also
fix z
assume "z \<in> ?F x"
with order_trans[OF convex_on_diff[OF `convex_on I f` `y \<in> I` _ `y < x`]]
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 `x \<in> interior I`]
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 `y < x` by (simp add: field_simps)
qed simp
subsection{* Explicit formulas for interior and relative interior of convex hull*}
lemma affine_independent_convex_affine_hull:
fixes s :: "'a::euclidean_space set"
assumes "~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: "setsum u t = 1" "\<forall>x\<in>s. 0 \<le> v x" "setsum 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" --{*split into separate cases*}
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)"
"setsum v t + setsum v (s - t) = 1"
using uv fin s
by (auto simp: setsum.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 setsum.union_disjoint, auto simp: algebra_simps setsum_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 "~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: "~ 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)
apply (simp add: affine_dependent_iff_dependent affine_hull_insert_span_gen)
apply (metis Suc_le_lessD not_less order_trans card_image_le finite_imageI dim_le_card)
done
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> setsum u s = 1 \<and> setsum (\<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> setsum u s = 1 \<and> setsum (\<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 "~ affine_dependent s"
shows "rel_interior(convex hull s) =
{y. \<exists>u. (\<forall>x \<in> s. 0 < u x) \<and> setsum u s = 1 \<and> setsum (\<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}" --{*split into separate cases*}
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 setsum.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" "setsum 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> setsum 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"
"setsum 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 setsum_subtractf setsum.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
"~ 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> setsum 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_hull_convex_hull affine_independent_span_gt rel_interior_interior)
by (simp add: rel_interior_convex_hull_explicit)
lemma interior_convex_hull_explicit:
fixes s :: "'a::euclidean_space set"
assumes "~ affine_dependent s"
shows
"interior(convex hull s) =
(if card(s) \<le> DIM('a) then {}
else {y. \<exists>u. (\<forall>x \<in> s. 0 < u x \<and> u x < 1) \<and> setsum 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" "setsum 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> setsum u {a,b}"
using a b by simp
also have "... \<le> setsum u s"
apply (rule Groups_Big.setsum_mono2)
using a b u
apply (auto simp: less_imp_le aff_independent_finite assms)
done
finally have "u a < 1"
using `b \<in> s` u by fastforce
} note [simp] = this
show ?thesis
using assms
apply (auto simp: interior_convex_hull_explicit_minimal)
apply (rule_tac x=u in exI)
apply (auto simp: not_le)
done
qed
subsection{* Similar results for closure and (relative or absolute) frontier.*}
lemma closure_convex_hull [simp]:
fixes s :: "'a::euclidean_space set"
shows "compact s ==> closure(convex hull s) = convex hull s"
by (simp add: compact_imp_closed compact_convex_hull)
lemma rel_frontier_convex_hull_explicit:
fixes s :: "'a::euclidean_space set"
assumes "~ affine_dependent s"
shows "rel_frontier(convex hull s) =
{y. \<exists>u. (\<forall>x \<in> s. 0 \<le> u x) \<and> (\<exists>x \<in> s. u x = 0) \<and> setsum u s = 1 \<and> setsum (\<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: setsum_subtractf scaleR_diff_left)
done
qed
lemma frontier_convex_hull_explicit:
fixes s :: "'a::euclidean_space set"
assumes "~ affine_dependent s"
shows "frontier(convex hull s) =
{y. \<exists>u. (\<forall>x \<in> s. 0 \<le> u x) \<and> (DIM ('a) < card s \<longrightarrow> (\<exists>x \<in> s. u x = 0)) \<and>
setsum u s = 1 \<and> setsum (\<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 "~ 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> setsum u s = 1 \<Longrightarrow>
\<exists>x v. x \<in> s \<and>
(\<forall>x\<in>s - {x}. 0 \<le> v x) \<and>
setsum 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.setsum_diff1 fs)
done }
moreover
{ fix a u
have "a \<in> s \<Longrightarrow> \<forall>x\<in>s - {a}. 0 \<le> u x \<Longrightarrow> setsum u (s - {a}) = 1 \<Longrightarrow>
\<exists>v. (\<forall>x\<in>s. 0 \<le> v x) \<and>
(\<exists>x\<in>s. v x = 0) \<and> setsum 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: setsum.If_cases Diff_eq if_smult fs)
done }
ultimately show ?thesis
using assms
apply (simp add: rel_frontier_convex_hull_explicit)
apply (simp add: convex_hull_finite fs Union_SetCompr_eq, auto)
done
qed
lemma frontier_convex_hull_eq_rel_frontier:
fixes s :: "'a::euclidean_space set"
assumes "~ affine_dependent s"
shows "frontier(convex hull s) =
(if card s \<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 "~ 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))" "x \<in> s"
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
end