expand_fun_eq -> ext_iff
expand_set_eq -> set_ext_iff
Naming in line now with multisets
(* Title: HOL/Library/Convex_Euclidean_Space.thy
Author: Robert Himmelmann, TU Muenchen
*)
header {* Convex sets, functions and related things. *}
theory Convex_Euclidean_Space
imports Topology_Euclidean_Space Convex
begin
(* ------------------------------------------------------------------------- *)
(* To be moved elsewhere *)
(* ------------------------------------------------------------------------- *)
lemma basis_0[simp]:"(basis i::'a::euclidean_space) = 0 \<longleftrightarrow> i\<ge>DIM('a)"
using norm_basis[of i, where 'a='a] unfolding norm_eq_zero[where 'a='a,THEN sym] by auto
lemma scaleR_2:
fixes x :: "'a::real_vector"
shows "scaleR 2 x = x + x"
unfolding one_add_one_is_two [symmetric] scaleR_left_distrib by simp
declare euclidean_simps[simp]
lemma vector_choose_size: "0 <= c ==> \<exists>(x::'a::euclidean_space). norm x = c"
apply (rule exI[where x="c *\<^sub>R basis 0 ::'a"]) using DIM_positive[where 'a='a] by auto
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"
"setsum (\<lambda>y. if (x = y) then P x else Q y) s = setsum Q s"
"setsum (\<lambda>y. if (y = x) then P y else Q y) s = setsum Q s"
"setsum (\<lambda>y. if (x = y) then P y else Q y) s = setsum Q s"
apply(rule_tac [!] setsum_cong2) using assms by auto
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 auto
lemma image_smult_interval:"(\<lambda>x. m *\<^sub>R (x::'a::ordered_euclidean_space)) ` {a..b} =
(if {a..b} = {} then {} else if 0 \<le> m then {m *\<^sub>R a..m *\<^sub>R b} else {m *\<^sub>R b..m *\<^sub>R a})"
using image_affinity_interval[of m 0 a b] by auto
lemma dist_triangle_eq:
fixes x y z :: "'a::euclidean_space"
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: "(\<forall>s\<in>f. affine s) \<Longrightarrow> affine (\<Inter> f)"
unfolding affine_def by auto
lemma affine_Int: "affine s \<Longrightarrow> affine t \<Longrightarrow> affine (s \<inter> t)"
unfolding affine_def by auto
lemma affine_affine_hull: "affine(affine hull s)"
unfolding hull_def using affine_Inter[of "{t \<in> affine. s \<subseteq> t}"]
unfolding mem_def by auto
lemma affine_hull_eq[simp]: "(affine hull s = s) \<longleftrightarrow> affine s"
by (metis affine_affine_hull hull_same mem_def)
lemma setsum_restrict_set'': assumes "finite A"
shows "setsum f {x \<in> A. P x} = (\<Sum>x\<in>A. if P x then f x else 0)"
unfolding mem_def[of _ P, symmetric] unfolding setsum_restrict_set'[OF assms] ..
subsection {* 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"
thus "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)
by(auto simp add: scaleR_left_distrib[THEN sym])
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
thus "(\<Sum>x\<in>s. u x *\<^sub>R x) \<in> V" proof(auto simp only: disjE)
assume "card s = 2" hence "card s = Suc (Suc 0)" by auto
then obtain a b where "s = {a, b}" unfolding card_Suc_eq by auto
thus ?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" thus ?thesis using as and n_def proof(induct n arbitrary: u s)
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_tac ccontr)
assume " \<not> (\<exists>x\<in>s. u x \<noteq> 1)" hence "setsum u s = real_of_nat (card s)" unfolding card_eq_setsum by auto
thus False using as(7) and `card s > 2` by (metis Numeral1_eq1_nat less_0_number_of less_int_code(15)
less_nat_number_of not_less_iff_gr_or_eq of_nat_1 of_nat_eq_iff pos2 rel_simps(4)) 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) by auto
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)[THEN sym] 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 hence "s - {x} \<noteq> {}" "card (s - {x}) = n" unfolding c and as(1)[symmetric] proof(rule_tac ccontr)
assume "\<not> s - {x} \<noteq> {}" hence "card (s - {x}) = 0" unfolding card_0_eq[OF *(2)] by simp
thus False using True by auto qed auto
thus ?thesis apply(rule_tac IA[of "s - {x}" "\<lambda>y. (inverse (1 - u x) * u y)"])
unfolding setsum_right_distrib[THEN sym] using as and *** and True by auto
next case False hence "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
thus ?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
hence "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 RealVector.scaleR_right.setsum using x(1) as(6) by auto
thus "(\<Sum>x\<in>s. u x *\<^sub>R x) \<in> V" unfolding scaleR_scaleR[THEN sym] and scaleR_right.setsum [symmetric]
apply(subst *) unfolding setsum_clauses(2)[OF *(2)]
using `u x \<noteq> 1` by auto
qed auto
next assume "card s = 1" then obtain a where "s={a}" by(auto simp add: card_Suc_eq)
thus ?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 and mem_def[of _ affine]
apply (erule exE)+ apply(erule conjE)+ prefer 2 apply rule proof-
fix x assume "x\<in>p" thus "\<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, rule_tac x="\<lambda>x. 1" in exI) by auto
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"
thus "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_addf if_smult scaleR_zero_left ** setsum_restrict_set[OF xy, THEN sym]
unfolding scaleR_scaleR[THEN sym] RealVector.scaleR_right.setsum [symmetric] and setsum_right_distrib[THEN sym]
unfolding x y using x(1-3) y(1-3) uv by simp 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_ext_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"
thus "\<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 by auto
next
fix x t u assume "t \<subseteq> s" hence *:"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"
thus "\<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_restrict_set[OF assms, THEN sym] and * by auto qed
subsection {* Stepping theorems and hence small special cases. *}
lemma affine_hull_empty[simp]: "affine hull {} = {}"
apply(rule hull_unique) unfolding mem_def by 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)
"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 "?as \<Longrightarrow> (?lhs = ?rhs)")
proof-
show ?th1 by simp
assume ?as
{ 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
have ?rhs proof(cases "a\<in>s")
case True hence *:"insert a s = s" by auto
show ?thesis using u[unfolded *] apply(rule_tac x=0 in exI) by auto
next
case False thus ?thesis apply(rule_tac x="u a" in exI) using u and `?as` by auto
qed } moreover
{ 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
have ?lhs proof(cases "a\<in>s")
case True thus ?thesis
apply(rule_tac x="\<lambda>x. (if x=a then v else 0) + u x" in exI)
unfolding setsum_clauses(2)[OF `?as`] apply simp
unfolding scaleR_left_distrib and setsum_addf
unfolding vu and * and scaleR_zero_left
by (auto simp add: setsum_delta[OF `?as`])
next
case False
hence **:"\<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 `?as`] and * using vu
using setsum_cong2[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_cong2[of s u "\<lambda>x. if x = a then v else u x", OF **(1)] by auto
qed }
ultimately show "?lhs = ?rhs" by blast
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}" (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
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) by(auto intro!: exI)
qed
subsection {* Some relations between affine hull and subspaces. *}
lemma affine_hull_insert_subset_span:
fixes a :: "'a::euclidean_space"
shows "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
thus "\<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[THEN sym] setsum_diff1 scaleR_left_diff_distrib)
unfolding as by simp qed
lemma affine_hull_insert_span:
fixes a :: "'a::euclidean_space"
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_cases[OF f(1), of "\<lambda>x. x = a"]
by (auto simp add: setsum_subtractf scaleR_left.setsum algebra_simps *) qed
lemma affine_hull_span:
fixes a :: "'a::euclidean_space"
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
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
subsection {* Conic hull. *}
lemma cone_cone_hull: "cone (cone hull s)"
unfolding hull_def using cone_Inter[of "{t \<in> conic. s \<subseteq> t}"]
by (auto simp add: mem_def)
lemma cone_hull_eq: "(cone hull s = s) \<longleftrightarrow> cone s"
apply(rule hull_eq[unfolded mem_def])
using cone_Inter unfolding subset_eq by (auto simp add: mem_def)
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 by auto
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
thus "\<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, rule_tac x="s - {v}" in exI, rule_tac x="\<lambda>x. - (1 / u v) * u x" in exI)
unfolding scaleR_scaleR[THEN sym] and scaleR_right.setsum [symmetric] unfolding setsum_right_distrib[THEN sym] and setsum_diff1[OF as(1)] using as by auto
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
thus ?rhs apply(rule_tac x="\<lambda>x. if x\<in>t then u x else 0" in exI)
apply auto unfolding * and setsum_restrict_set[OF assms, THEN sym]
unfolding Int_absorb1[OF `t\<subseteq>s`] by auto
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
thus ?lhs unfolding affine_dependent_explicit using assms by auto
qed
subsection {* A general lemma. *}
lemma convex_connected:
fixes s :: "'a::real_normed_vector set"
assumes "convex s" shows "connected s"
proof-
{ fix e1 e2 assume as:"open e1" "open e2" "e1 \<inter> e2 \<inter> s = {}" "s \<subseteq> e1 \<union> e2"
assume "e1 \<inter> s \<noteq> {}" "e2 \<inter> s \<noteq> {}"
then obtain x1 x2 where x1:"x1\<in>e1" "x1\<in>s" and x2:"x2\<in>e2" "x2\<in>s" by auto
hence n:"norm (x1 - x2) > 0" unfolding zero_less_norm_iff using as(3) by auto
{ fix x e::real assume as:"0 \<le> x" "x \<le> 1" "0 < e"
{ fix y have *:"(1 - x) *\<^sub>R x1 + x *\<^sub>R x2 - ((1 - y) *\<^sub>R x1 + y *\<^sub>R x2) = (y - x) *\<^sub>R x1 - (y - x) *\<^sub>R x2"
by (simp add: algebra_simps)
assume "\<bar>y - x\<bar> < e / norm (x1 - x2)"
hence "norm ((1 - x) *\<^sub>R x1 + x *\<^sub>R x2 - ((1 - y) *\<^sub>R x1 + y *\<^sub>R x2)) < e"
unfolding * and scaleR_right_diff_distrib[THEN sym]
unfolding less_divide_eq using n by auto }
hence "\<exists>d>0. \<forall>y. \<bar>y - x\<bar> < d \<longrightarrow> norm ((1 - x) *\<^sub>R x1 + x *\<^sub>R x2 - ((1 - y) *\<^sub>R x1 + y *\<^sub>R x2)) < e"
apply(rule_tac x="e / norm (x1 - x2)" in exI) using as
apply auto unfolding zero_less_divide_iff using n by simp } note * = this
have "\<exists>x\<ge>0. x \<le> 1 \<and> (1 - x) *\<^sub>R x1 + x *\<^sub>R x2 \<notin> e1 \<and> (1 - x) *\<^sub>R x1 + x *\<^sub>R x2 \<notin> e2"
apply(rule connected_real_lemma) apply (simp add: `x1\<in>e1` `x2\<in>e2` dist_commute)+
using * apply(simp add: dist_norm)
using as(1,2)[unfolded open_dist] apply simp
using as(1,2)[unfolded open_dist] apply simp
using assms[unfolded convex_alt, THEN bspec[where x=x1], THEN bspec[where x=x2]] using x1 x2
using as(3) by auto
then obtain x where "x\<ge>0" "x\<le>1" "(1 - x) *\<^sub>R x1 + x *\<^sub>R x2 \<notin> e1" "(1 - x) *\<^sub>R x1 + x *\<^sub>R x2 \<notin> e2" by auto
hence False using as(4)
using assms[unfolded convex_alt, THEN bspec[where x=x1], THEN bspec[where x=x2]]
using x1(2) x2(2) by auto }
thus ?thesis unfolding connected_def by auto
qed
subsection {* One rather trivial consequence. *}
lemma connected_UNIV[intro]: "connected (UNIV :: 'a::real_normed_vector set)"
by(simp add: convex_connected convex_UNIV)
subsection {* Balls, being convex, are connected. *}
lemma convex_box: fixes a::"'a::euclidean_space"
assumes "\<And>i. i<DIM('a) \<Longrightarrow> convex {x. P i x}"
shows "convex {x. \<forall>i<DIM('a). P i (x$$i)}"
using assms unfolding convex_def by(auto simp add:euclidean_simps)
lemma convex_positive_orthant: "convex {x::'a::euclidean_space. (\<forall>i<DIM('a). 0 \<le> x$$i)}"
by (rule convex_box) (simp add: atLeast_def[symmetric] convex_real_interval)
lemma convex_local_global_minimum:
fixes s :: "'a::real_normed_vector set"
assumes "0<e" "convex_on s f" "ball x e \<subseteq> s" "\<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> (\<forall>y\<in>s. f x \<le> f y)"
then obtain y where "y\<in>s" and y:"f x > f y" by auto
hence xy:"0 < dist x y" by (auto simp add: dist_nz[THEN sym])
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"] using xy `e>0` and divide_pos_pos[of e "dist x y"] by auto
hence "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[THEN sym]
using u unfolding pos_less_divide_eq[OF xy] by auto
hence "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_distance[of "ball x e" x, unfolded convex_on_def, THEN bspec[where x=y], THEN bspec[where x=z]] by auto
thus "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(auto simp add: convex_def Ball_def)
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_distance[of "cball x e" x, unfolded convex_on_def, THEN bspec[where x=y], THEN bspec[where x=z]] by auto
thus "dist x (u *\<^sub>R y + v *\<^sub>R z) \<le> e" using convex_bound_le[OF yz uv] by auto
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\<in>convex. s\<subseteq>t}"]
unfolding mem_def by auto
lemma convex_hull_eq: "convex hull s = s \<longleftrightarrow> convex s"
by (metis convex_convex_hull hull_same mem_def)
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[unfolded mem_def, of convex, OF convex_cball]
unfolding subset_eq mem_cball dist_norm using B by auto 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
subsection {* Stepping theorems for convex hulls of finite sets. *}
lemma convex_hull_empty[simp]: "convex hull {} = {}"
apply(rule hull_unique) unfolding mem_def by auto
lemma convex_hull_singleton[simp]: "convex hull {a} = {a}"
apply(rule hull_unique) unfolding mem_def by 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 "?xyz = ?hull")
apply(rule,rule hull_minimal,rule) unfolding mem_def[of _ convex] and insert_iff prefer 3 apply rule proof-
fix x assume x:"x = a \<or> x \<in> s"
thus "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] by auto
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
thus "x\<in> convex hull insert a s" unfolding obt(5) using convex_convex_hull[of "insert a s", unfolded convex_def]
apply(erule_tac x=a in ballE) apply(erule_tac x=b in ballE) apply(erule_tac x=u in allE) using obt by auto
next
show "convex ?hull" unfolding convex_def apply(rule,rule,rule,rule,rule,rule,rule) proof-
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")
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)
case True hence **:"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+
hence "u * u1 + v * u2 = 1" using as(3) obt1(3) obt2(3) by auto
thus ?thesis unfolding obt1(5) obt2(5) * using assms hull_subset[of s convex] by(auto simp add: ** scaleR_right_distrib)
next
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
case False have "0 \<le> u * v1 + v * v2" "0 \<le> u * v1" "0 \<le> u * v1 + v * v2" "0 \<le> v * v2" apply -
apply(rule add_nonneg_nonneg) prefer 4 apply(rule add_nonneg_nonneg) apply(rule_tac [!] mult_nonneg_nonneg)
using as(1,2) obt1(1,2) obt2(1,2) by auto
thus ?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 convex_convex_hull[of s, unfolded convex_def, rule_format]) using obt1(4) obt2(4)
unfolding add_divide_distrib[THEN sym] and real_0_le_divide_iff
by (auto simp add: scaleR_left_distrib scaleR_right_distrib)
qed note * = this
have u1:"u1 \<le> 1" unfolding obt1(3)[THEN sym] and not_le using obt1(2) by auto
have u2:"u2 \<le> 1" unfolding obt2(3)[THEN sym] 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) by auto
also have "\<dots> \<le> 1" unfolding mult.add_right[THEN sym] 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) * by(auto intro!: mult_nonneg_nonneg add_nonneg_nonneg simp add: algebra_simps)
qed
qed
subsection {* 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) unfolding mem_def[of _ convex] apply(rule) defer
apply(subst convex_def) apply(rule,rule,rule,rule,rule,rule,rule)
proof-
fix x assume "x\<in>s"
thus "x \<in> ?hull" unfolding mem_Collect_eq apply(rule_tac x=1 in exI, rule_tac x="\<lambda>x. 1" in exI) by auto
next
fix t assume as:"s \<subseteq> t" "convex t"
show "?hull \<subseteq> t" apply(rule) unfolding mem_Collect_eq apply(erule exE | erule 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)[THEN sym] apply(rule as(2)[unfolded convex, rule_format])
using assm(1,2) as(1) by auto qed
next
fix x y u v assume uv:"0\<le>u" "0\<le>v" "u+v=(1::real)" and 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) by(auto simp add: not_le)
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, 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_cases[OF finite_atLeastAtMost[of 1 "k1 + k2"]] and setsum_reindex[OF inj] and o_def Collect_mem_eq
unfolding scaleR_scaleR[THEN sym] scaleR_right.setsum [symmetric] setsum_right_distrib[THEN sym] 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 thus ?thesis using mult_nonneg_nonneg[of u "u1 i"] and uv(1) x(1)[THEN bspec[where x=i]] by auto
next def j \<equiv> "i - k1"
case False with i have "j \<in> {1..k2}" unfolding j_def by auto
thus ?thesis unfolding j_def[symmetric] using False
using mult_nonneg_nonneg[of v "u2 j"] and uv(2) y(1)[THEN bspec[where x=j]] by auto 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: mem_def[of _ convex] convex_def[of ?set])
fix x assume "x\<in>s" thus " \<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] by auto
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"
hence "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, metis add_nonneg_nonneg mult_nonneg_nonneg uv(1) uv(2)) }
moreover have "(\<Sum>x\<in>s. u * ux x + v * uy x) = 1"
unfolding setsum_addf and setsum_right_distrib[THEN sym] 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_addf and scaleR_scaleR[THEN sym] 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) by auto
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)"
thus "(\<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
subsection {* 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}"
hence "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) by auto }
moreover
have "(\<Sum>v\<in>y ` {1..k}. setsum u {i \<in> {1..k}. y i = v}) = 1"
unfolding setsum_image_gen[OF fin, THEN sym] 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, THEN sym]
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) by auto
hence "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}"
hence "f i \<in> s" apply(subst f(2)[THEN sym]) by auto
hence "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
hence "{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) by auto
hence "card {x. Suc 0 \<le> x \<and> x \<le> card s \<and> f x = y} = 1" by auto
hence "(\<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) }
hence "(\<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_cong2[of 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_cong2 [of 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) by fastsimp
hence "y \<in> ?lhs" unfolding convex_hull_indexed by auto }
ultimately show ?thesis unfolding set_ext_iff by blast
qed
subsection {* 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" hence *:"insert a s = s" by auto
assume ?lhs thus ?rhs unfolding * apply(rule_tac x=0 in exI) by auto
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" thus ?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` by auto
next
assume "a\<in>s" hence *:"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_addf 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` by auto
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" "(\<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_cong2) defer apply(rule_tac setsum_cong2) using `a\<notin>s` by auto
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] by auto
qed
subsection {* 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) by simp 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 unfolding Collect_def
proof(rule ext) 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) = (\<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) by (auto simp add: algebra_simps) 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"
"\<And>x y z ::_::euclidean_space. 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 Collect_def and convex_hull_finite_step[OF fin(2)] and *
unfolding convex_hull_finite_step[OF fin(3)] apply(rule ext) 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) by simp 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) by (simp add: algebra_simps) qed
subsection {* Relations among closure notions and corresponding hulls. *}
text {* TODO: Generalize linear algebra concepts defined in @{text
Euclidean_Space.thy} so that we can generalize these lemmas. *}
lemma subspace_imp_affine:
fixes s :: "(_::euclidean_space) set" shows "subspace s \<Longrightarrow> affine s"
unfolding subspace_def affine_def by auto
lemma affine_imp_convex: "affine s \<Longrightarrow> convex s"
unfolding affine_def convex_def by auto
lemma subspace_imp_convex:
fixes s :: "(_::euclidean_space) set" shows "subspace s \<Longrightarrow> convex s"
using subspace_imp_affine affine_imp_convex by auto
lemma affine_hull_subset_span:
fixes s :: "(_::euclidean_space) set" shows "(affine hull s) \<subseteq> (span s)"
by (metis hull_minimal mem_def span_inc subspace_imp_affine subspace_span)
lemma convex_hull_subset_span:
fixes s :: "(_::euclidean_space) set" shows "(convex hull s) \<subseteq> (span s)"
by (metis hull_minimal mem_def 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 mem_def)
lemma affine_dependent_imp_dependent:
fixes s :: "(_::euclidean_space) set" shows "affine_dependent s \<Longrightarrow> dependent s"
unfolding affine_dependent_def dependent_def
using affine_hull_subset_span by auto
lemma dependent_imp_affine_dependent:
fixes s :: "(_::euclidean_space) set"
assumes "dependent {x - a| x . x \<in> s}" "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
hence "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_cong2) using `a\<notin>s` `t\<subseteq>s` by auto
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 * by auto
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 by auto
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_cong2) using `a\<notin>s` `t\<subseteq>s` by auto
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_addf scaleR_right_distrib)
hence "(\<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) by auto
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
hence "2 *\<^sub>R x \<in>s" "2 *\<^sub>R y \<in> s" unfolding cone_def by auto
hence "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) by auto }
thus ?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 by auto
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`, THEN sym])
apply(rule dependent_imp_affine_dependent)
apply(rule dependent_biggerset) by auto qed
lemma affine_dependent_biggerset_general:
assumes "finite (s::('a::euclidean_space) set)" "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 by auto
have "dim {x - a |x. x \<in> s - {a}} \<le> dim s"
apply(rule subset_le_dim) unfolding subset_eq
using `a\<in>s` by (auto simp add:span_superset span_sub)
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`, THEN sym])
apply(rule dependent_imp_affine_dependent) apply(rule dependent_biggerset_general) unfolding ** by auto 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_ext_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
hence "\<exists>n\<le>N. (\<forall>k<n. \<not> ?P k) \<and> ?P n" apply(rule_tac ex_least_nat_le) by auto
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"
hence "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"
hence "setsum w (s - {v}) \<ge> 0" apply(rule_tac setsum_nonneg) by auto
hence "setsum w s > 0" unfolding setsum_diff1'[OF obt(1) `v\<in>s`]
using as[THEN bspec[where x=v]] and `v\<in>s` using `w v \<noteq> 0` by auto
thus False using wv(1) by auto
qed hence "i\<noteq>{}" unfolding i_def by auto
hence "t \<ge> 0" using Min_ge_iff[of i 0 ] and obt(1) unfolding t_def i_def
using obt(4)[unfolded le_less] apply auto unfolding divide_le_0_iff by auto
have t:"\<forall>v\<in>s. u v + t * w v \<ge> 0" proof
fix v assume "v\<in>s" hence 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 apply(rule_tac add_nonneg_nonneg)
using v apply simp apply(rule mult_nonneg_nonneg) using `t\<ge>0` by auto next
case True hence "t \<le> u v / (- w v)" using `v\<in>s`
unfolding t_def i_def apply(rule_tac Min_le) using obt(1) by auto
thus ?thesis unfolding real_0_le_add_iff
using pos_le_divide_eq[OF True[unfolded neg_0_less_iff_less[THEN sym]]] 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
hence 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_diff1'[OF obt(1) `a\<in>s`] by auto
have "(\<Sum>v\<in>s. u v + t * w v) = 1"
unfolding setsum_addf wv(1) setsum_right_distrib[THEN sym] 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_addf obt(6) scaleR_scaleR[THEN sym] 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
by (auto simp add: * scaleR_left_distrib)
thus False using smallest[THEN spec[where x="n - 1"]] by auto qed
thus "\<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_ext_iff apply(rule, 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
thus "\<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]] by auto
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
thus "x \<in> convex hull p" using hull_mono[OF `s\<subseteq>p`] by auto
qed
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(10)
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>{}" unfolding i_def 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` by auto
next fix y assume "y \<in> cball a (Min i)"
hence y:"norm (a - y) \<le> Min i" unfolding dist_norm[THEN sym] by auto
{ fix x assume "x\<in>t"
hence "Min i \<le> b x" unfolding i_def apply(rule_tac Min_le) using obt(1) by auto
hence "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_addf setsum_subtractf scaleR_left.setsum[THEN sym] 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) by auto
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_ext) unfolding image_iff mem_Collect_eq
apply rule apply auto
apply (rule_tac x=u in rev_bexI, simp)
apply (erule rev_bexI, erule rev_bexI, simp)
by auto
have "continuous_on ({0..1} \<times> s \<times> t)
(\<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)
thus ?thesis unfolding *
apply (rule compact_continuous_image)
apply (intro compact_Times compact_interval assms)
done
qed
lemma compact_convex_hull: fixes s::"('a::euclidean_space) set"
assumes "compact s" shows "compact(convex hull s)"
proof(cases "s={}")
case True thus ?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"))
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
case 0 thus ?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_ext_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 thus ?thesis using t(4) unfolding card_0_eq[OF t(1)] by simp
next
case False hence "card t = Suc 0" using t(3) `n=0` by auto
then obtain a where "t = {a}" unfolding card_Suc_eq by auto
thus ?thesis using t(2,4) by simp
qed
next
fix x assume "x\<in>s"
thus "\<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 by auto
qed thus ?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_ext_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"] by auto
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) by (auto simp add: card_insert_if)
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
let ?P = "\<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)"
show ?P proof(cases "card t = Suc n")
case False hence "card t \<le> n" using t(3) by auto
thus ?P 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
by(auto intro!: exI[where x=t])
next
case True then obtain a u where au:"t = insert a u" "a\<notin>u" apply(drule_tac card_eq_SucD) by auto
show ?P proof(cases "u={}")
case True hence "x=a" using t(4)[unfolded au] by auto
show ?P unfolding `x=a` apply(rule_tac x=a in exI, rule_tac x=a in exI, rule_tac x=1 in exI)
using t and `n\<noteq>0` unfolding au by(auto intro!: exI[where x="{a}"])
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 ?P apply(rule_tac x=a in exI, rule_tac x=b in exI, rule_tac x=vx in exI)
using obt and t(1-3) unfolding au and * using card_insert_disjoint[OF _ au(2)]
by(auto intro!: exI[where x=u])
qed
qed
qed
thus ?thesis using compact_convex_combinations[OF assms Suc] by simp
qed
qed
qed
lemma finite_imp_compact_convex_hull:
fixes s :: "('a::euclidean_space) set"
shows "finite s \<Longrightarrow> compact(convex hull s)"
by (metis compact_convex_hull finite_imp_compact)
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 hence "0 < inner d d + (inner a d * 2 - inner b d * 2)"
apply(rule_tac add_pos_pos) using assms by auto
thus ?thesis apply(rule_tac disjI2) unfolding dist_norm and norm_eq_sqrt_inner and real_sqrt_less_iff
by (simp add: algebra_simps inner_commute)
next
case False hence "0 < inner d d + (inner b d * 2 - inner a d * 2)"
apply(rule_tac add_pos_nonneg) using assms by auto
thus ?thesis apply(rule_tac disjI1) unfolding dist_norm and norm_eq_sqrt_inner and real_sqrt_less_iff
by (simp add: algebra_simps inner_commute)
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))"
proof(induct_tac rule: finite_induct[of s])
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
thus ?thesis apply(rule_tac x=z in bexI) unfolding convex_hull_insert[OF False] by auto
next
case False show ?thesis using obt(3) proof(cases "u=0", case_tac[!] "v=0")
assume "u=0" "v\<noteq>0" hence "y = b" using obt by auto
thus ?thesis using False and obt(4) by auto
next
assume "u\<noteq>0" "v=0" hence "y = x" using obt by auto
thus ?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(rule ccontr)
assume "\<not> x\<noteq>b" hence "y=b" unfolding obt(5)
using obt(3) by(auto simp add: scaleR_left_distrib[THEN sym])
thus False using obt(4) and False by simp qed
hence *:"w *\<^sub>R (x - b) \<noteq> 0" using w(1) by auto
show ?thesis using dist_increases_online[OF *, of a y]
proof(erule_tac disjE)
assume "dist a y < dist a (y + w *\<^sub>R (x - b))"
hence "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) by auto
ultimately show ?thesis by auto
next
assume "dist a y < dist a (y - w *\<^sub>R (x - b))"
hence "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) by auto
ultimately show ?thesis by auto
qed
qed auto
qed
qed auto
qed (auto simp add: assms)
lemma simplex_furthest_le:
fixes s :: "('a::euclidean_space) set"
assumes "finite s" "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
thus ?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
thus ?thesis using x(2)[THEN bspec[where x=y]] by auto
qed auto
qed
lemma simplex_furthest_le_exists:
fixes s :: "('a::euclidean_space) 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::euclidean_space) set"
assumes "finite s" "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
thus ?thesis proof(cases "u\<notin>s \<or> v\<notin>s", erule_tac 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
thus ?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
thus ?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::euclidean_space) 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" "s \<noteq> {}"
shows "closest_point s a \<in> s" "\<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] by auto
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 by auto
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
thus ?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" "v \<le> inner y z / inner z z"
thus "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(rule divide_pos_pos, 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> inner (a - x) (y - x) \<le> 0"
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
thus 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) by auto
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] by fastsimp
lemma closest_point_lipschitz:
assumes "convex s" "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"
"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)] by auto
thus ?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" "closed s" "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" "closed s" "s \<noteq> {}"
shows "continuous_on t (closest_point s)"
by(metis continuous_at_imp_continuous_on continuous_at_closest_point[OF assms])
subsection {* Various point-to-set separating/supporting hyperplane theorems. *}
lemma supporting_hyperplane_closed_point:
fixes z :: "'a::{real_inner,heine_borel}"
assumes "convex s" "closed s" "s \<noteq> {}" "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, rule_tac x="inner (y - z) y" in exI, rule_tac x=y in bexI)
apply rule defer apply rule defer apply(rule, rule ccontr) using `y\<in>s` proof-
show "inner (y - z) z < inner (y - z) y" apply(subst diff_less_iff(1)[THEN sym])
unfolding inner_diff_right[THEN sym] and inner_gt_zero_iff using `y\<in>s` `z\<notin>s` by auto
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] apply - by (auto simp add: inner_diff)
thus 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" "closed s" "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 thus ?thesis apply(rule_tac x="-z" in exI, rule_tac x=1 in exI)
using less_le_trans[OF _ inner_ge_zero[of z]] by auto
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, rule_tac x="inner (y - z) z + (norm(y - z))\<twosuperior> / 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_tac notI) proof(drule closer_point_lemma)
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
thus 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) ^ 2" using `y\<in>s` `z\<notin>s` by auto
hence "0 < inner (y - z) (y - z)" unfolding power2_norm_eq_inner by simp
ultimately show "inner (y - z) z + (norm (y - z))\<twosuperior> / 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)" "closed s" "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 ((basis 0)::'a) = 1" by auto
hence "norm ((basis 0)::'a) = 1" "basis 0 \<noteq> (0::'a)" defer
apply(subst norm_le_zero_iff[THEN sym]) by auto
thus ?thesis apply(rule_tac x="basis 0" in exI, rule_tac x=1 in exI)
using True using DIM_positive[where 'a='a] by auto
next case False thus ?thesis using False using separating_hyperplane_closed_point[OF assms]
apply - apply(erule exE)+ unfolding inner.zero_right apply(rule_tac x=a in exI, rule_tac x=b in exI) by auto qed
subsection {* Now set-to-set for closed/compact sets. *}
lemma separating_hyperplane_closed_compact:
assumes "convex (s::('a::euclidean_space) set)" "closed s" "convex t" "compact t" "t \<noteq> {}" "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
hence "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
thus ?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)
hence ab:"\<forall>x\<in>s. \<forall>y\<in>t. b + inner a y < inner a x" apply- apply(rule,rule) apply(erule_tac x="x - y" in ballE) by (auto simp add: inner_diff)
def k \<equiv> "Sup ((\<lambda>x. inner a x) ` t)"
show ?thesis apply(rule_tac x="-a" in exI, rule_tac x="-(k + b / 2)" in exI)
apply(rule,rule) defer apply(rule) unfolding inner_minus_left and neg_less_iff_less proof-
from ab have "((\<lambda>x. inner a x) ` t) *<= (inner a y - b)"
apply(erule_tac x=y in ballE) apply(rule setleI) using `y\<in>s` by auto
hence k:"isLub UNIV ((\<lambda>x. inner a x) ` t) k" unfolding k_def apply(rule_tac Sup) using assms(5) by auto
fix x assume "x\<in>t" thus "inner a x < (k + b / 2)" using `0<b` and isLubD2[OF k, of "inner a x"] by auto
next
fix x assume "x\<in>s"
hence "k \<le> inner a x - b" unfolding k_def apply(rule_tac Sup_least) using assms(5)
using ab[THEN bspec[where x=x]] by auto
thus "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" "compact s" "s \<noteq> {}" "convex t" "closed t" "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
thus ?thesis apply(rule_tac x="-a" in exI, rule_tac x="-b" in exI) by auto qed
subsection {* 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,rule,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[unfolded mem_def, of convex, OF assms(1), THEN sym, of c] by auto
hence "\<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
apply- apply rule defer apply rule apply(rule mult_nonneg_nonneg)
by(auto simp add: inner_commute elim!: ballE)
thus "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
thus ?thesis apply(rule_tac x=x in exI) by(auto simp add: inner_commute) qed
lemma separating_hyperplane_sets:
assumes "convex s" "convex (t::('a::euclidean_space) set)" "s \<noteq> {}" "t \<noteq> {}" "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
hence "\<forall>x\<in>t. \<forall>y\<in>s. inner a y \<le> inner a x"
by (force simp add: inner_diff)
thus ?thesis
apply(rule_tac x=a in exI, rule_tac x="Sup ((\<lambda>x. inner a x) ` s)" in exI) using `a\<noteq>0`
apply auto
apply (rule Sup[THEN isLubD2])
prefer 4
apply (rule Sup_least)
using assms(3-5) apply (auto simp add: setle_def)
apply metis
done
qed
subsection {* More convexity generalities. *}
lemma convex_closure:
fixes s :: "'a::real_normed_vector set"
assumes "convex s" shows "convex(closure s)"
unfolding convex_def Ball_def closure_sequential
apply(rule,rule,rule,rule,rule,rule,rule,rule,rule) apply(erule_tac exE)+
apply(rule_tac x="\<lambda>n. u *\<^sub>R xb n + v *\<^sub>R xc n" in exI) apply(rule,rule)
apply(rule assms[unfolded convex_def, rule_format]) prefer 6
apply(rule Lim_add) apply(rule_tac [1-2] Lim_cmul) by auto
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(erule exE | erule 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)"
hence "(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 by(auto simp add: algebra_simps)
thus "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_translation_lemma:
"convex hull ((\<lambda>x. a + x) ` s) \<subseteq> (\<lambda>x. a + x) ` (convex hull s)"
by (metis convex_convex_hull convex_translation hull_minimal hull_subset image_mono mem_def)
lemma convex_hull_bilemma: fixes neg
assumes "(\<forall>s a. (convex hull (up a s)) \<subseteq> up a (convex hull s))"
shows "(\<forall>s. up a (up (neg a) s) = s) \<and> (\<forall>s. up (neg a) (up a s) = s) \<and> (\<forall>s t a. s \<subseteq> t \<longrightarrow> up a s \<subseteq> up a t)
\<Longrightarrow> \<forall>s. (convex hull (up a s)) = up a (convex hull s)"
using assms by(metis subset_antisym)
lemma convex_hull_translation:
"convex hull ((\<lambda>x. a + x) ` s) = (\<lambda>x. a + x) ` (convex hull s)"
apply(rule convex_hull_bilemma[rule_format, of _ _ "\<lambda>a. -a"], rule convex_hull_translation_lemma) unfolding image_image by auto
lemma convex_hull_scaling_lemma:
"(convex hull ((\<lambda>x. c *\<^sub>R x) ` s)) \<subseteq> (\<lambda>x. c *\<^sub>R x) ` (convex hull s)"
by (metis convex_convex_hull convex_scaling hull_subset mem_def subset_hull subset_image_iff)
lemma convex_hull_scaling:
"convex hull ((\<lambda>x. c *\<^sub>R x) ` s) = (\<lambda>x. c *\<^sub>R x) ` (convex hull s)"
apply(cases "c=0") defer apply(rule convex_hull_bilemma[rule_format, of _ _ inverse]) apply(rule convex_hull_scaling_lemma)
unfolding image_image scaleR_scaleR by(auto simp add:image_constant_conv)
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[THEN sym] convex_hull_scaling convex_hull_translation)
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_ext, 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"
hence "\<forall>a b. s \<subseteq> {x. inner a x \<le> b} \<longrightarrow> x \<in> {x. inner a x \<le> b}" by blast
thus "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, erule_tac x="-b" in allE) by auto
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] guess s .. then guess u ..
thus ?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_restrict_set[OF assms(1), THEN sym] by(auto simp add: Int_absorb1) qed
lemma radon_s_lemma:
assumes "finite s" "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[THEN sym] and setsum_restrict_set''[OF assms(1)] and setsum_addf[THEN sym] and *
using assms(2) by assumption qed
lemma radon_v_lemma:
assumes "finite s" "setsum f s = 0" "\<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_restrict_set''[OF assms(1)] and setsum_addf[THEN sym] and *
using assms(2) by assumption 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 hence "u v < 0" by auto
thus ?thesis proof(cases "\<exists>w\<in>{x \<in> c. 0 < u x}. u w > 0")
case True thus ?thesis using setsum_nonneg_eq_0_iff[of _ u, OF fin(1)] by auto
next
case False hence "setsum u c \<le> setsum (\<lambda>x. if x=v then u v else 0) c" apply(rule_tac setsum_mono) by auto
thus ?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)
hence *:"setsum u {x\<in>c. u x > 0} > 0" unfolding less_le apply(rule_tac conjI, rule_tac setsum_nonneg) by auto
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_zero_left) by auto
hence "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_Un_zero[OF fin, THEN sym])
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 * by auto
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, rule_tac x="\<lambda>y. inverse (setsum u {x\<in>c. u x > 0}) * - u y" in exI)
using assms(1) unfolding scaleR_scaleR[THEN sym] scaleR_right.setsum [symmetric] and z_def
by(auto simp add: setsum_negf mult_right.setsum[THEN sym])
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 * by auto
hence "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, rule_tac x="\<lambda>y. inverse (setsum u {x\<in>c. u x > 0}) * u y" in exI)
using assms(1) unfolding scaleR_scaleR[THEN sym] scaleR_right.setsum [symmetric] and z_def using *
by(auto simp add: setsum_negf mult_right.setsum[THEN sym])
ultimately show ?thesis apply(rule_tac x="{v\<in>c. u v \<le> 0}" in exI, rule_tac x="{v\<in>c. u v > 0}" in exI) by auto
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] guess s .. then guess u ..
hence *:"finite s" "affine_dependent s" and s:"s \<subseteq> c" unfolding affine_dependent_explicit by auto
from radon_partition[OF *] guess m .. then guess p ..
thus ?thesis apply(rule_tac that[of p m]) using s by auto qed
subsection {* Helly's theorem. *}
lemma helly_induct: fixes f::"('a::euclidean_space) set set"
assumes "card f = n" "n \<ge> DIM('a) + 1"
"\<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 (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)" hence "\<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` by auto
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
hence *:"\<Inter> f = \<Inter> (f - {s}) \<inter> \<Inter> (f - {t})" by auto
show ?thesis unfolding * unfolding ex_in_conv[THEN sym] apply(rule_tac x="X s" in exI)
apply(rule, rule X[rule_format]) using X st by auto
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
hence "f \<union> (g \<union> h) = f" by auto
hence 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[THEN sym] 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 mem_def 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 guess y unfolding image_iff ..
thus "x\<in>\<Inter>h" using X[THEN bspec[where x=y]] using * f by auto next
fix x assume "x\<in>X ` h" then guess y unfolding image_iff ..
thus "x\<in>\<Inter>g" using X[THEN bspec[where x=y]] using * f by auto
qed(auto)
thus ?thesis unfolding f using mp(3)[unfolded gh] by blast qed
qed(auto) qed(auto)
lemma helly: fixes f::"('a::euclidean_space) set set"
assumes "card f \<ge> DIM('a) + 1" "\<forall>s\<in>f. convex s"
"\<forall>t\<subseteq>f. card t = DIM('a) + 1 \<longrightarrow> \<Inter> t \<noteq> {}"
shows "\<Inter> f \<noteq>{}"
apply(rule helly_induct) using assms by auto
subsection {* Convex hull is "preserved" by a linear function. *}
lemma convex_hull_linear_image:
assumes "bounded_linear f"
shows "f ` (convex hull s) = convex 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 "convex {x. f x \<in> convex hull f ` s}"
unfolding convex_def by(auto simp add: f.scaleR f.add convex_convex_hull[unfolded convex_def, rule_format]) next
interpret f: bounded_linear f by fact
show "convex {x. x \<in> f ` (convex hull s)}" using convex_convex_hull[unfolded convex_def, of s]
unfolding convex_def by (auto simp add: f.scaleR [symmetric] f.add [symmetric])
qed auto
lemma in_convex_hull_linear_image:
assumes "bounded_linear f" "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
subsection {* Homeomorphism of all convex compact sets with nonempty interior. *}
lemma compact_frontier_line_lemma:
fixes s :: "('a::euclidean_space) set"
assumes "compact s" "0 \<in> s" "x \<noteq> 0"
obtains u where "0 \<le> u" "(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, rule continuous_at_imp_continuous_on)
apply(rule, rule continuous_vmul)
apply(rule continuous_at_id) by(rule compact_interval)
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) by(auto intro!: divide_nonneg_pos)
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[THEN sym]] by auto
{ fix v assume as:"v > u" "v *\<^sub>R x \<in> s"
hence "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
hence "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` by(auto simp add:field_simps)
hence 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)[THEN sym]
prefer 3 apply(rule_tac x="(u + (e / 2) / norm x) *\<^sub>R x" in exI) apply(rule, rule) proof-
fix e assume "0 < e" and as:"(u + e / 2 / norm x) *\<^sub>R x \<in> s"
hence "u + e / 2 / norm x > u" using`norm x > 0` by(auto simp del:zero_less_norm_iff intro!: divide_pos_pos)
thus False using u_max[OF _ as] by auto
qed(insert `y\<in>s`, auto simp add: dist_norm scaleR_left_distrib obt(3))
thus ?thesis by(metis that[of u] u_max obt(1))
qed
lemma starlike_compact_projective:
assumes "compact s" "cball (0::'a::euclidean_space) 1 \<subseteq> s "
"\<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)] by simp
def pi \<equiv> "\<lambda>x::'a. inverse (norm x) *\<^sub>R x"
have "0 \<notin> frontier s" unfolding frontier_straddle apply(rule ccontr) unfolding not_not apply(erule_tac x=1 in allE)
using assms(2)[unfolded subset_eq Ball_def mem_cball] by auto
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 (rule continuous_mul)
apply (rule continuous_at_inv[unfolded o_def])
apply (rule continuous_at_norm)
apply simp
apply (rule continuous_at_id)
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 u assume x:"x\<in>frontier s" and "(0::real)\<le>u"
hence "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" thus False using v(3)[THEN spec[where x=1]] using x and fs by auto next
assume "v>1" thus 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" thus "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 by auto 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_ext,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
thus "x \<in> sphere" using pi(1)[of y] and `0 \<notin> frontier s` by auto
next fix x assume "x\<in>sphere" hence "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
thus "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` by auto
next fix x y assume as:"x \<in> frontier s" "y \<in> frontier s" "pi x = pi y"
hence 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, THEN sym] 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)" "0 \<le> norm x * inverse (norm y)"
unfolding divide_inverse[THEN sym] apply(rule_tac[!] divide_nonneg_pos) using nor by auto
hence "norm x = norm y" apply(rule_tac 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 by(auto simp add:field_simps divide_le_eq_1 divide_inverse[THEN sym])
thus "x = y" apply(subst injpi[THEN sym]) using as(3) by auto
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, rule contpi)
apply(rule continuous_on_subset[of sphere], rule surf(6)) using pi(1) by auto
{ 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 hence "pi x \<in> sphere" "norm x < 1" using pi(1)[of x] as by(auto simp add: dist_norm)
thus ?thesis apply(rule_tac assms(3)[rule_format, THEN DiffD1])
apply(rule_tac fs[unfolded subset_eq, rule_format])
unfolding surf(5)[THEN sym] by auto
next case True thus ?thesis apply rule defer unfolding pi_def apply(rule fs[unfolded subset_eq, rule_format])
unfolding surf(5)[unfolded sphere_def, THEN sym] using `0\<in>s` by auto qed } note hom = this
{ fix x assume "x\<in>s"
hence "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) by auto
next let ?a = "inverse (norm (surf (pi x)))"
case False hence invn:"inverse (norm x) \<noteq> 0" by auto
from False have pix:"pi x\<in>sphere" using pi(1) by auto
hence "pi (surf (pi x)) = pi x" apply(rule_tac surf(4)[rule_format]) by assumption
hence **:"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 by auto
hence *:"?a * norm x > 0" and"?a > 0" "?a \<noteq> 0" using surf(5) `0\<notin>frontier s` apply -
apply(rule_tac mult_pos_pos) using False[unfolded zero_less_norm_iff[THEN sym]] by auto
have "norm (surf (pi x)) \<noteq> 0" using ** False by auto
hence "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
hence "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[THEN sym]) unfolding abs_mult abs_norm_cancel abs_of_pos[OF `?a > 0`]
unfolding pi(2)[OF `?a > 0`] by auto
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_ext, rule, erule imageE, drule hom)
prefer 4 apply(rule continuous_at_imp_continuous_on, rule) apply(rule_tac [3] hom2) proof-
fix x::"'a" assume as:"x \<in> cball 0 1"
thus "continuous (at x) (\<lambda>x. norm x *\<^sub>R surf (pi x))" proof(cases "x=0")
case False thus ?thesis apply(rule_tac continuous_mul, rule_tac continuous_at_norm)
using cont_surfpi unfolding continuous_on_eq_continuous_at[OF open_delete[OF open_UNIV]] o_def by auto
next obtain B where B:"\<forall>x\<in>s. norm x \<le> B" using compact_imp_bounded[OF assms(1)] unfolding bounded_iff by auto
hence "B > 0" using assms(2) unfolding subset_eq apply(erule_tac x="basis 0" in ballE) defer
apply(erule_tac x="basis 0" in ballE)
unfolding Ball_def mem_cball dist_norm using DIM_positive[where 'a='a]
by(auto simp add:norm_basis[unfolded One_nat_def])
case True 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" "0<e"
hence "surf (pi x) \<in> frontier s" using pi(1)[of x] unfolding surf(5)[THEN sym] by auto
hence "norm (surf (pi x)) \<le> B" using B fs by auto
hence "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` by auto
also have "\<dots> = e" using `B>0` by auto
finally show "norm x * norm (surf (pi x)) < e" by assumption
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" hence "pi x \<in> sphere" using pi(1) by auto
hence "surf (pi x) \<in> frontier s" using surf(5) by auto
thus 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)"
thus "x=y" proof(cases "x=0 \<or> y=0")
case True thus ?thesis using as by(auto elim: surf_0) next
case False
hence "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" "compact s" "cball 0 1 \<subseteq> s"
shows "s homeomorphic (cball (0::'a) 1)"
apply(rule starlike_compact_projective[OF assms(2-3)]) proof(rule,rule,rule,erule conjE)
fix x u assume as:"x \<in> s" "0 \<le> u" "u < (1::real)"
hence "u *\<^sub>R x \<in> interior s" unfolding interior_def mem_Collect_eq
apply(rule_tac x="ball (u *\<^sub>R x) (1 - u)" in exI) apply(rule, rule open_ball)
unfolding centre_in_ball apply rule defer apply(rule) unfolding mem_ball proof-
fix y assume "dist (u *\<^sub>R x) y < 1 - u"
hence "inverse (1 - u) *\<^sub>R (y - u *\<^sub>R x) \<in> s"
using assms(3) apply(erule_tac subsetD) unfolding mem_cball dist_commute dist_norm
unfolding group_add_class.diff_0 group_add_class.diff_0_right norm_minus_cancel norm_scaleR
apply (rule mult_left_le_imp_le[of "1 - u"])
unfolding mult_assoc[symmetric] using `u<1` by auto
thus "y \<in> s" using assms(1)[unfolded convex_def, rule_format, of "inverse(1 - u) *\<^sub>R (y - u *\<^sub>R x)" x "1 - u" u]
using as unfolding scaleR_scaleR by auto qed auto
thus "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" "compact s" "interior s \<noteq> {}" "0 < e"
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, 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
by(auto simp add: mult_right_le_one_le)
hence "(\<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: uminus_add_conv_diff scaleR_right_diff_distrib)
thus ?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` by(auto simp add: uminus_add_conv_diff scaleR_right_diff_distrib) qed
lemma homeomorphic_convex_compact: fixes s::"('a::euclidean_space) set" and t::"('a) set"
assumes "convex s" "compact s" "interior s \<noteq> {}"
"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
(** This might break sooner or later. In fact it did already once. **)
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,erule_tac x="f x" in allE) apply safe
apply(erule_tac x=xa in allE,erule_tac x="f xa" in allE) prefer 3
apply(rule_tac y="u * f a + v * f aa" in order_trans) defer by(auto intro!:mult_left_mono add_mono)
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)
subsection {* 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] by auto
subsection {* Convexity of general and special intervals. *}
lemma convexI: (* TODO: move to Library/Convex.thy *)
assumes "\<And>x y u v. \<lbrakk>x \<in> s; y \<in> s; 0 \<le> u; 0 \<le> v; u + v = 1\<rbrakk> \<Longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> s"
shows "convex s"
using assms unfolding convex_def by fast
lemma is_interval_convex:
fixes s :: "('a::euclidean_space) set"
assumes "is_interval s" shows "convex s"
proof (rule convexI)
fix x y u v assume as:"x \<in> s" "y \<in> s" "0 \<le> u" "0 \<le> v" "u + v = (1::real)"
hence *:"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"
hence "u * a < (1 - v) * b" unfolding not_le using as(4) by(auto simp add: field_simps)
hence "a < b" unfolding * using as(4) *(2) apply(rule_tac mult_left_less_imp_less[of "1 - v"]) by(auto simp add: field_simps)
hence "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"
hence "v * b > (1 - u) * a" unfolding not_le using as(4) by(auto simp add: field_simps)
hence "a < b" unfolding * using as(4) apply(rule_tac mult_left_less_imp_less) by(auto simp add: field_simps)
hence "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] by(auto simp add:euclidean_simps) 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_interval: "convex {a .. b}" "convex {a<..<b::'a::ordered_euclidean_space}"
apply(rule_tac[!] is_interval_convex) using is_interval_interval by auto
(* FIXME: rewrite these lemmas without using vec1
subsection {* On @{text "real^1"}, @{text "is_interval"}, @{text "convex"} and @{text "connected"} are all equivalent. *}
lemma is_interval_1:
"is_interval s \<longleftrightarrow> (\<forall>a\<in>s. \<forall>b\<in>s. \<forall> x. dest_vec1 a \<le> dest_vec1 x \<and> dest_vec1 x \<le> dest_vec1 b \<longrightarrow> x \<in> s)"
unfolding is_interval_def forall_1 by auto
lemma is_interval_connected_1: "is_interval s \<longleftrightarrow> connected (s::(real^1) set)"
apply(rule, rule is_interval_connected, assumption) unfolding is_interval_1
apply(rule,rule,rule,rule,erule conjE,rule ccontr) proof-
fix a b x assume as:"connected s" "a \<in> s" "b \<in> s" "dest_vec1 a \<le> dest_vec1 x" "dest_vec1 x \<le> dest_vec1 b" "x\<notin>s"
hence *:"dest_vec1 a < dest_vec1 x" "dest_vec1 x < dest_vec1 b" apply(rule_tac [!] ccontr) unfolding not_less by auto
let ?halfl = "{z. inner (basis 1) z < dest_vec1 x} " and ?halfr = "{z. inner (basis 1) z > dest_vec1 x} "
{ fix y assume "y \<in> s" have "y \<in> ?halfr \<union> ?halfl" apply(rule ccontr)
using as(6) `y\<in>s` by (auto simp add: inner_vector_def) }
moreover have "a\<in>?halfl" "b\<in>?halfr" using * by (auto simp add: inner_vector_def)
hence "?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, rule_tac x="?halfr" in exI)
apply(rule, rule open_halfspace_lt, rule, rule open_halfspace_gt)
by(auto simp add: field_simps) qed
lemma is_interval_convex_1:
"is_interval s \<longleftrightarrow> convex (s::(real^1) set)"
by(metis is_interval_convex convex_connected is_interval_connected_1)
lemma convex_connected_1:
"connected s \<longleftrightarrow> convex (s::(real^1) set)"
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" "continuous_on {a .. b} f" "(f a)$$k \<le> y" "y \<le> (f b)$$k"
shows "\<exists>x\<in>{a..b}. (f x)$$k = y"
proof- have "f a \<in> f ` {a..b}" "f b \<in> f ` {a..b}" apply(rule_tac[!] imageI)
using assms(1) by auto
thus ?thesis using connected_ivt_component[of "f ` {a..b}" "f a" "f b" k y]
using connected_continuous_image[OF assms(2) convex_connected[OF convex_real_interval(5)]]
using assms by(auto intro!: imageI) qed
lemma ivt_increasing_component_1: fixes f::"real \<Rightarrow> 'a::euclidean_space"
shows "a \<le> b \<Longrightarrow> \<forall>x\<in>{a .. b}. continuous (at x) f
\<Longrightarrow> f a$$k \<le> y \<Longrightarrow> y \<le> f b$$k \<Longrightarrow> \<exists>x\<in>{a..b}. (f x)$$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" "continuous_on {a .. b} f" "(f b)$$k \<le> y" "y \<le> (f a)$$k"
shows "\<exists>x\<in>{a..b}. (f x)$$k = y"
apply(subst neg_equal_iff_equal[THEN sym])
using ivt_increasing_component_on_1[of a b "\<lambda>x. - f x" k "- y"] using assms using continuous_on_neg
by (auto simp add:euclidean_simps)
lemma ivt_decreasing_component_1: fixes f::"real \<Rightarrow> 'a::euclidean_space"
shows "a \<le> b \<Longrightarrow> \<forall>x\<in>{a .. b}. continuous (at x) f
\<Longrightarrow> f b$$k \<le> y \<Longrightarrow> y \<le> f a$$k \<Longrightarrow> \<exists>x\<in>{a..b}. (f x)$$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" "\<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[THEN sym] obt(2) mult_1 apply(drule_tac meta_mp) apply(rule mult_left_mono)
using assms(2) obt(1) by auto
thus "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 unit_interval_convex_hull:
"{0::'a::ordered_euclidean_space .. (\<chi>\<chi> i. 1)} = convex hull {x. \<forall>i<DIM('a). (x$$i = 0) \<or> (x$$i = 1)}"
(is "?int = convex hull ?points")
proof- have 01:"{0,(\<chi>\<chi> i. 1)} \<subseteq> convex hull ?points" apply rule apply(rule_tac hull_subset[unfolded subset_eq, rule_format]) by auto
{ fix n x assume "x\<in>{0::'a::ordered_euclidean_space .. \<chi>\<chi> i. 1}" "n \<le> DIM('a)" "card {i. i<DIM('a) \<and> x$$i \<noteq> 0} \<le> n"
hence "x\<in>convex hull ?points" proof(induct n arbitrary: x)
case 0 hence "x = 0" apply(subst euclidean_eq) apply rule by auto
thus "x\<in>convex hull ?points" using 01 by auto
next
case (Suc n) show "x\<in>convex hull ?points" proof(cases "{i. i<DIM('a) \<and> x$$i \<noteq> 0} = {}")
case True hence "x = 0" apply(subst euclidean_eq) by auto
thus "x\<in>convex hull ?points" using 01 by auto
next
case False def xi \<equiv> "Min ((\<lambda>i. x$$i) ` {i. i<DIM('a) \<and> x$$i \<noteq> 0})"
have "xi \<in> (\<lambda>i. x$$i) ` {i. i<DIM('a) \<and> x$$i \<noteq> 0}" unfolding xi_def apply(rule Min_in) using False by auto
then obtain i where i':"x$$i = xi" "x$$i \<noteq> 0" "i<DIM('a)" by auto
have i:"\<And>j. j<DIM('a) \<Longrightarrow> x$$j > 0 \<Longrightarrow> x$$i \<le> x$$j"
unfolding i'(1) xi_def apply(rule_tac Min_le) unfolding image_iff
defer apply(rule_tac x=j in bexI) using i' by auto
have i01:"x$$i \<le> 1" "x$$i > 0" using Suc(2)[unfolded mem_interval,rule_format,of i]
using i'(2-) `x$$i \<noteq> 0` by auto
show ?thesis proof(cases "x$$i=1")
case True have "\<forall>j\<in>{i. i<DIM('a) \<and> x$$i \<noteq> 0}. x$$j = 1" apply(rule, rule ccontr) unfolding mem_Collect_eq
proof(erule conjE) fix j assume as:"x $$ j \<noteq> 0" "x $$ j \<noteq> 1" "j<DIM('a)"
hence j:"x$$j \<in> {0<..<1}" using Suc(2) by(auto simp add: eucl_le[where 'a='a] elim!:allE[where x=j])
hence "x$$j \<in> op $$ x ` {i. i<DIM('a) \<and> x $$ i \<noteq> 0}" using as(3) by auto
hence "x$$j \<ge> x$$i" unfolding i'(1) xi_def apply(rule_tac Min_le) by auto
thus False using True Suc(2) j by(auto simp add: elim!:ballE[where x=j]) qed
thus "x\<in>convex hull ?points" apply(rule_tac hull_subset[unfolded subset_eq, rule_format])
by auto
next let ?y = "\<lambda>j. if x$$j = 0 then 0 else (x$$j - x$$i) / (1 - x$$i)"
case False hence *:"x = x$$i *\<^sub>R (\<chi>\<chi> j. if x$$j = 0 then 0 else 1) + (1 - x$$i) *\<^sub>R (\<chi>\<chi> j. ?y j)"
apply(subst euclidean_eq) by(auto simp add: field_simps euclidean_simps)
{ fix j assume j:"j<DIM('a)"
have "x$$j \<noteq> 0 \<Longrightarrow> 0 \<le> (x $$ j - x $$ i) / (1 - x $$ i)" "(x $$ j - x $$ i) / (1 - x $$ i) \<le> 1"
apply(rule_tac divide_nonneg_pos) using i(1)[of j] using False i01
using Suc(2)[unfolded mem_interval, rule_format, of j] using j
by(auto simp add:field_simps euclidean_simps)
hence "0 \<le> ?y j \<and> ?y j \<le> 1" by auto }
moreover have "i\<in>{j. j<DIM('a) \<and> x$$j \<noteq> 0} - {j. j<DIM('a) \<and> ((\<chi>\<chi> j. ?y j)::'a) $$ j \<noteq> 0}"
using i01 using i'(3) by auto
hence "{j. j<DIM('a) \<and> x$$j \<noteq> 0} \<noteq> {j. j<DIM('a) \<and> ((\<chi>\<chi> j. ?y j)::'a) $$ j \<noteq> 0}" using i'(3) by blast
hence **:"{j. j<DIM('a) \<and> ((\<chi>\<chi> j. ?y j)::'a) $$ j \<noteq> 0} \<subset> {j. j<DIM('a) \<and> x$$j \<noteq> 0}" apply - apply rule
by( auto simp add:euclidean_simps)
have "card {j. j<DIM('a) \<and> ((\<chi>\<chi> j. ?y j)::'a) $$ j \<noteq> 0} \<le> n"
using less_le_trans[OF psubset_card_mono[OF _ **] Suc(4)] by auto
ultimately show ?thesis apply(subst *) apply(rule convex_convex_hull[unfolded convex_def, rule_format])
apply(rule_tac hull_subset[unfolded subset_eq, rule_format]) defer apply(rule Suc(1))
unfolding mem_interval using i01 Suc(3) by auto
qed qed qed } note * = this
have **:"DIM('a) = card {..<DIM('a)}" by auto
show ?thesis apply rule defer apply(rule hull_minimal) unfolding subset_eq prefer 3 apply rule
apply(rule_tac n2="DIM('a)" in *) prefer 3 apply(subst(2) **)
apply(rule card_mono) using 01 and convex_interval(1) prefer 5 apply - apply rule
unfolding mem_interval apply rule unfolding mem_Collect_eq apply(erule_tac x=i in allE)
by(auto simp add: mem_def[of _ convex]) qed
subsection {* And this is a finite set of vertices. *}
lemma unit_cube_convex_hull: obtains s where "finite s" "{0 .. (\<chi>\<chi> i. 1)::'a::ordered_euclidean_space} = convex hull s"
apply(rule that[of "{x::'a. \<forall>i<DIM('a). x$$i=0 \<or> x$$i=1}"])
apply(rule finite_subset[of _ "(\<lambda>s. (\<chi>\<chi> i. if i\<in>s then 1::real else 0)::'a) ` Pow {..<DIM('a)}"])
prefer 3 apply(rule unit_interval_convex_hull) apply rule unfolding mem_Collect_eq proof-
fix x::"'a" assume as:"\<forall>i<DIM('a). x $$ i = 0 \<or> x $$ i = 1"
show "x \<in> (\<lambda>s. \<chi>\<chi> i. if i \<in> s then 1 else 0) ` Pow {..<DIM('a)}"
apply(rule image_eqI[where x="{i. i<DIM('a) \<and> x$$i = 1}"])
using as apply(subst euclidean_eq) by auto qed auto
subsection {* Hence any cube (could do any nonempty interval). *}
lemma cube_convex_hull:
assumes "0 < d" obtains s::"('a::ordered_euclidean_space) set" where
"finite s" "{x - (\<chi>\<chi> i. d) .. x + (\<chi>\<chi> i. d)} = convex hull s" proof-
let ?d = "(\<chi>\<chi> i. d)::'a"
have *:"{x - ?d .. x + ?d} = (\<lambda>y. x - ?d + (2 * d) *\<^sub>R y) ` {0 .. \<chi>\<chi> i. 1}" apply(rule set_ext, rule)
unfolding image_iff defer apply(erule bexE) proof-
fix y assume as:"y\<in>{x - ?d .. x + ?d}"
{ fix i assume i:"i<DIM('a)" have "x $$ i \<le> d + y $$ i" "y $$ i \<le> d + x $$ i"
using as[unfolded mem_interval, THEN spec[where x=i]] i
by(auto simp add:euclidean_simps)
hence "1 \<ge> inverse d * (x $$ i - y $$ i)" "1 \<ge> inverse d * (y $$ i - x $$ i)"
apply(rule_tac[!] mult_left_le_imp_le[OF _ assms]) unfolding mult_assoc[THEN sym]
using assms by(auto simp add: field_simps)
hence "inverse d * (x $$ i * 2) \<le> 2 + inverse d * (y $$ i * 2)"
"inverse d * (y $$ i * 2) \<le> 2 + inverse d * (x $$ i * 2)" by(auto simp add:field_simps) }
hence "inverse (2 * d) *\<^sub>R (y - (x - ?d)) \<in> {0..\<chi>\<chi> i.1}" unfolding mem_interval using assms
by(auto simp add: euclidean_simps field_simps)
thus "\<exists>z\<in>{0..\<chi>\<chi> i.1}. y = x - ?d + (2 * d) *\<^sub>R z" apply- apply(rule_tac x="inverse (2 * d) *\<^sub>R (y - (x - ?d))" in bexI)
using assms by auto
next
fix y z assume as:"z\<in>{0..\<chi>\<chi> i.1}" "y = x - ?d + (2*d) *\<^sub>R z"
have "\<And>i. i<DIM('a) \<Longrightarrow> 0 \<le> d * z $$ i \<and> d * z $$ i \<le> d"
using assms as(1)[unfolded mem_interval] apply(erule_tac x=i in allE)
apply rule apply(rule mult_nonneg_nonneg) prefer 3 apply(rule mult_right_le_one_le)
using assms by auto
thus "y \<in> {x - ?d..x + ?d}" unfolding as(2) mem_interval apply- apply rule using as(1)[unfolded mem_interval]
apply(erule_tac x=i in allE) using assms by(auto simp add: euclidean_simps) qed
obtain s where "finite s" "{0::'a..\<chi>\<chi> i.1} = convex hull s" using unit_cube_convex_hull by auto
thus ?thesis apply(rule_tac that[of "(\<lambda>y. x - ?d + (2 * d) *\<^sub>R y)` s"]) unfolding * and convex_hull_affinity by auto 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" "convex_on s f" "\<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 e assume "x\<in>s" "(0::real) < e"
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]) by auto
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` by auto
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)
hence "(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)
hence 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` by auto
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)
hence *:"(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` unfolding divide_inverse by (auto simp add:field_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 add:field_simps intro!:mult_pos_pos) 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" "\<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)
thus "\<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"
hence "dist x y < 0" using False unfolding mem_cball not_le by (auto simp del: dist_not_less_zero)
thus "\<bar>f y\<bar> \<le> b + 2 * \<bar>f x\<bar>" using zero_le_dist[of x y] by auto qed
subsection {* Hence a convex function on an open set is continuous. *}
lemma convex_on_continuous:
assumes "open (s::('a::ordered_euclidean_space) set)" "convex_on s f"
(* FIXME: generalize to euclidean_space *)
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(rule_tac divide_pos_pos, auto)
let ?d = "(\<chi>\<chi> i. d)::'a"
obtain c where c:"finite c" "{x - ?d..x + ?d} = convex hull c" using cube_convex_hull[OF `d>0`, of x] by auto
have "x\<in>{x - ?d..x + ?d}" using `d>0` unfolding mem_interval by(auto simp add:euclidean_simps)
hence "c\<noteq>{}" using c by auto
def k \<equiv> "Max (f ` c)"
have "convex_on {x - ?d..x + ?d} f" apply(rule convex_on_subset[OF assms(2)])
apply(rule subset_trans[OF _ e(1)]) unfolding subset_eq mem_cball proof
fix z assume z:"z\<in>{x - ?d..x + ?d}"
have e:"e = setsum (\<lambda>i. d) {..<DIM('a)}" unfolding setsum_constant d_def using dimge1
unfolding real_eq_of_nat by auto
show "dist x z \<le> e" unfolding dist_norm e apply(rule_tac order_trans[OF norm_le_l1], rule setsum_mono)
using z[unfolded mem_interval] apply(erule_tac x=i in allE) by(auto simp add:euclidean_simps) qed
hence k:"\<forall>y\<in>{x - ?d..x + ?d}. f y \<le> k" unfolding c(2) apply(rule_tac convex_on_convex_hull_bound) apply assumption
unfolding k_def apply(rule, rule Max_ge) using c(1) by auto
have "d \<le> e" unfolding d_def apply(rule mult_imp_div_pos_le) using `e>0` dimge1 unfolding mult_le_cancel_left1 by auto
hence dsube:"cball x d \<subseteq> cball x e" unfolding subset_eq Ball_def mem_cball by auto
have conv:"convex_on (cball x d) f" apply(rule convex_on_subset, rule convex_on_subset[OF assms(2)]) apply(rule e(1)) using dsube by auto
hence "\<forall>y\<in>cball x d. abs (f y) \<le> k + 2 * abs (f x)" apply(rule_tac convex_bounds_lemma) apply assumption proof
fix y assume y:"y\<in>cball x d"
{ fix i assume "i<DIM('a)" hence "x $$ i - d \<le> y $$ i" "y $$ i \<le> x $$ i + d"
using order_trans[OF component_le_norm y[unfolded mem_cball dist_norm], of i] by(auto simp add:euclidean_simps) }
thus "f y \<le> k" apply(rule_tac k[rule_format]) unfolding mem_cball mem_interval dist_norm
by(auto simp add:euclidean_simps) qed
hence "continuous_on (ball x d) f" apply(rule_tac convex_on_bounded_continuous)
apply(rule open_ball, rule convex_on_subset[OF conv], rule ball_subset_cball)
apply force
done
thus "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). closed_segment a b)"
lemmas segment = open_segment_def closed_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[THEN sym] 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
thus ?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 **)
by (simp add: scaleR_right_diff_distrib, simp add: scaleR_2)
show ?t2 unfolding midpoint_def dist_norm apply (rule *)
by (simp add: scaleR_right_diff_distrib, simp add: scaleR_2)
show ?t3 unfolding midpoint_def dist_norm apply (rule *)
by (simp add: scaleR_right_diff_distrib, simp add: scaleR_2)
show ?t4 unfolding midpoint_def dist_norm apply (rule **)
by (simp add: scaleR_right_diff_distrib, simp add: scaleR_2)
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_ext)
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(erule exE, (erule conjE)?)+ apply(rule_tac x="1 - u" in exI) unfolding ** by auto 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]) by auto
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
thus ?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 between_mem_segment: "between (a,b) x \<longleftrightarrow> x \<in> closed_segment a b"
unfolding between_def mem_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 thus ?thesis unfolding between_def split_conv mem_def[of x, symmetric]
by(auto simp add:segment_refl dist_commute) next
case False hence 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 mem_def[of x, symmetric] closed_segment_def mem_Collect_eq
apply rule apply(erule exE, (erule 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"
hence *:"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" unfolding divide_le_eq_1_pos[OF Fal2]
unfolding as[unfolded dist_norm] norm_ge_zero by auto
thus "\<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) apply rule defer apply(rule, rule divide_nonneg_pos) prefer 4
proof(rule,rule) fix i assume i:"i<DIM('a)"
have "((1 - norm (a - x) / norm (a - b)) *\<^sub>R a + (norm (a - x) / norm (a - b)) *\<^sub>R b) $$ i =
((norm (a - b) - norm (a - x)) * (a $$ i) + norm (a - x) * (b $$ i)) / norm (a - b)"
using Fal by(auto simp add: field_simps euclidean_simps)
also have "\<dots> = x$$i" apply(rule divide_eq_imp[OF Fal])
unfolding as[unfolded dist_norm] using as[unfolded dist_triangle_eq] apply-
apply(subst (asm) euclidean_eq) using i apply(erule_tac x=i in allE) by(auto simp add:field_simps euclidean_simps)
finally show "x $$ i = ((1 - norm (a - x) / norm (a - b)) *\<^sub>R a + (norm (a - x) / norm (a - b)) *\<^sub>R b) $$ 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)
"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[where 'a='a]
by(auto simp add:field_simps euclidean_simps) 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" "c \<in> interior s" "x \<in> s" "0 < e" "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[THEN sym] apply(rule arg_cong[where f=norm]) using `e>0`
by(auto simp add: euclidean_simps euclidean_eq[where 'a='a] field_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) by auto
qed(rule mult_pos_pos, insert `e>0` `d>0`, auto) qed
lemma mem_interior_closure_convex_shrink:
fixes s :: "('a::euclidean_space) set"
assumes "convex s" "c \<in> interior s" "x \<in> closure s" "0 < e" "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 thus ?thesis using `e>0` `d>0` by(rule_tac bexI[where x=x], auto intro!: mult_pos_pos) next
case False hence 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
thus ?thesis apply(rule_tac x=y in bexI) unfolding True using `d>0` by auto next
case False hence "0 < e * d / (1 - e)" and *:"1 - e > 0"
using `e\<le>1` `e>0` `d>0` by(auto intro!:mult_pos_pos divide_pos_pos)
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
thus ?thesis apply(rule_tac x=y in bexI) unfolding dist_norm using pos_less_divide_eq[OF *] by auto 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 subset_interior[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)
by(auto simp add:field_simps norm_minus_commute)
thus ?thesis unfolding * apply - apply(rule mem_interior_convex_shrink)
using assms(1,4-5) `y\<in>s` by auto qed
subsection {* Some obvious but surprisingly hard simplex lemmas. *}
lemma simplex:
assumes "finite s" "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_ext, 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)] by auto
lemma std_simplex:
"convex hull (insert 0 { basis i | i. i<DIM('a)}) =
{x::'a::euclidean_space . (\<forall>i<DIM('a). 0 \<le> x$$i) \<and> setsum (\<lambda>i. x$$i) {..<DIM('a)} \<le> 1 }"
(is "convex hull (insert 0 ?p) = ?s")
proof- let ?D = "{..<DIM('a)}"
have *:"finite ?p" "0\<notin>?p" by auto
have "{(basis i)::'a |i. i<DIM('a)} = basis ` ?D" by auto
note sumbas = this setsum_reindex[OF basis_inj, unfolded o_def]
show ?thesis unfolding simplex[OF *] apply(rule set_ext) unfolding mem_Collect_eq apply rule
apply(erule exE, (erule conjE)+) apply(erule_tac[2] conjE)+ proof-
fix x::"'a" and u assume as: "\<forall>x\<in>{basis i |i. i<DIM('a)}. 0 \<le> u x"
"setsum u {basis i |i. i<DIM('a)} \<le> 1" "(\<Sum>x\<in>{basis i |i. i<DIM('a)}. u x *\<^sub>R x) = x"
have *:"\<forall>i<DIM('a). u (basis i) = x$$i"
proof safe case goal1
have "x$$i = (\<Sum>j<DIM('a). (if j = i then u (basis j) else 0))"
unfolding as(3)[THEN sym] euclidean_component.setsum unfolding sumbas
apply(rule setsum_cong2) by(auto simp add: basis_component)
also have "... = u (basis i)" apply(subst setsum_delta) using goal1 by auto
finally show ?case by auto
qed
hence **:"setsum u {basis i |i. i<DIM('a)} = setsum (op $$ x) ?D" unfolding sumbas
apply-apply(rule setsum_cong2) by auto
show "(\<forall>i<DIM('a). 0 \<le> x $$ i) \<and> setsum (op $$ x) ?D \<le> 1" apply - proof(rule,rule,rule)
fix i assume i:"i<DIM('a)" show "0 \<le> x$$i" unfolding *[rule_format,OF i,THEN sym]
apply(rule_tac as(1)[rule_format]) using i by auto
qed(insert as(2)[unfolded **], auto)
next fix x::"'a" assume as:"\<forall>i<DIM('a). 0 \<le> x $$ i" "setsum (op $$ x) ?D \<le> 1"
show "\<exists>u. (\<forall>x\<in>{basis i |i. i<DIM('a)}. 0 \<le> u x) \<and>
setsum u {basis i |i. i<DIM('a)} \<le> 1 \<and> (\<Sum>x\<in>{basis i |i. i<DIM('a)}. u x *\<^sub>R x) = x"
apply(rule_tac x="\<lambda>y. inner y x" in exI) apply safe using as(1)
proof- show "(\<Sum>y\<in>{basis i |i. i < DIM('a)}. y \<bullet> x) \<le> 1" unfolding sumbas
using as(2) unfolding euclidean_component_def[THEN sym] .
show "(\<Sum>xa\<in>{basis i |i. i < DIM('a)}. (xa \<bullet> x) *\<^sub>R xa) = x" unfolding sumbas
apply(subst (7) euclidean_representation) apply(rule setsum_cong2)
unfolding euclidean_component_def by auto
qed (auto simp add:euclidean_component_def)
qed qed
lemma interior_std_simplex:
"interior (convex hull (insert 0 { basis i| i. i<DIM('a)})) =
{x::'a::euclidean_space. (\<forall>i<DIM('a). 0 < x$$i) \<and> setsum (\<lambda>i. x$$i) {..<DIM('a)} < 1 }"
apply(rule set_ext) unfolding mem_interior std_simplex unfolding subset_eq mem_Collect_eq Ball_def mem_ball
unfolding Ball_def[symmetric] apply rule apply(erule exE, (erule conjE)+) defer apply(erule conjE) proof-
fix x::"'a" and e assume "0<e" and as:"\<forall>xa. dist x xa < e \<longrightarrow> (\<forall>x<DIM('a). 0 \<le> xa $$ x) \<and> setsum (op $$ xa) {..<DIM('a)} \<le> 1"
show "(\<forall>xa<DIM('a). 0 < x $$ xa) \<and> setsum (op $$ x) {..<DIM('a)} < 1" apply(safe) proof-
fix i assume i:"i<DIM('a)" thus "0 < x $$ i" using as[THEN spec[where x="x - (e / 2) *\<^sub>R basis i"]] and `e>0`
unfolding dist_norm by(auto simp add: inner_simps euclidean_component_def dot_basis elim!:allE[where x=i])
next have **:"dist x (x + (e / 2) *\<^sub>R basis 0) < e" using `e>0`
unfolding dist_norm by(auto intro!: mult_strict_left_mono)
have "\<And>i. i<DIM('a) \<Longrightarrow> (x + (e / 2) *\<^sub>R basis 0) $$ i = x$$i + (if i = 0 then e/2 else 0)"
unfolding euclidean_component_def by(auto simp add:inner_simps dot_basis)
hence *:"setsum (op $$ (x + (e / 2) *\<^sub>R basis 0)) {..<DIM('a)} = setsum (\<lambda>i. x$$i + (if 0 = i then e/2 else 0)) {..<DIM('a)}"
apply(rule_tac setsum_cong) by auto
have "setsum (op $$ x) {..<DIM('a)} < setsum (op $$ (x + (e / 2) *\<^sub>R basis 0)) {..<DIM('a)}" unfolding * setsum_addf
using `0<e` DIM_positive[where 'a='a] apply(subst setsum_delta') by auto
also have "\<dots> \<le> 1" using ** apply(drule_tac as[rule_format]) by auto
finally show "setsum (op $$ x) {..<DIM('a)} < 1" by auto qed
next fix x::"'a" assume as:"\<forall>i<DIM('a). 0 < x $$ i" "setsum (op $$ x) {..<DIM('a)} < 1"
guess a using UNIV_witness[where 'a='b] ..
let ?d = "(1 - setsum (op $$ x) {..<DIM('a)}) / real (DIM('a))"
have "Min ((op $$ x) ` {..<DIM('a)}) > 0" apply(rule Min_grI) using as(1) by auto
moreover have"?d > 0" apply(rule divide_pos_pos) using as(2) by(auto simp add: Suc_le_eq)
ultimately show "\<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> (\<forall>i<DIM('a). 0 \<le> y $$ i) \<and> setsum (op $$ y) {..<DIM('a)} \<le> 1"
apply(rule_tac x="min (Min ((op $$ x) ` {..<DIM('a)})) ?D" in exI) apply rule defer apply(rule,rule) proof-
fix y assume y:"dist x y < min (Min (op $$ x ` {..<DIM('a)})) ?d"
have "setsum (op $$ y) {..<DIM('a)} \<le> setsum (\<lambda>i. x$$i + ?d) {..<DIM('a)}" proof(rule setsum_mono)
fix i assume "i\<in>{..<DIM('a)}" hence "abs (y$$i - x$$i) < ?d" apply-apply(rule le_less_trans)
using component_le_norm[of "y - x" i]
using y[unfolded min_less_iff_conj dist_norm, THEN conjunct2] by(auto simp add: norm_minus_commute)
thus "y $$ i \<le> x $$ i + ?d" by auto qed
also have "\<dots> \<le> 1" unfolding setsum_addf setsum_constant card_enum real_eq_of_nat by(auto simp add: Suc_le_eq)
finally show "(\<forall>i<DIM('a). 0 \<le> y $$ i) \<and> setsum (op $$ y) {..<DIM('a)} \<le> 1"
proof safe fix i assume i:"i<DIM('a)"
have "norm (x - y) < x$$i" apply(rule less_le_trans)
apply(rule y[unfolded min_less_iff_conj dist_norm, THEN conjunct1]) using i by auto
thus "0 \<le> y$$i" using component_le_norm[of "x - y" i] and as(1)[rule_format, of i] by auto
qed qed auto qed
lemma interior_std_simplex_nonempty: obtains a::"'a::euclidean_space" where
"a \<in> interior(convex hull (insert 0 {basis i | i . i<DIM('a)}))" proof-
let ?D = "{..<DIM('a)}" let ?a = "setsum (\<lambda>b::'a. inverse (2 * real DIM('a)) *\<^sub>R b) {(basis i) | i. i<DIM('a)}"
have *:"{basis i :: 'a | i. i<DIM('a)} = basis ` ?D" by auto
{ fix i assume i:"i<DIM('a)" have "?a $$ i = inverse (2 * real DIM('a))"
unfolding euclidean_component.setsum * and setsum_reindex[OF basis_inj] and o_def
apply(rule trans[of _ "setsum (\<lambda>j. if i = j then inverse (2 * real DIM('a)) else 0) ?D"]) apply(rule setsum_cong2)
defer apply(subst setsum_delta') unfolding euclidean_component_def using i by(auto simp add:dot_basis) }
note ** = this
show ?thesis apply(rule that[of ?a]) unfolding interior_std_simplex mem_Collect_eq proof safe
fix i assume i:"i<DIM('a)" show "0 < ?a $$ i" unfolding **[OF i] by(auto simp add: Suc_le_eq)
next have "setsum (op $$ ?a) ?D = setsum (\<lambda>i. inverse (2 * real DIM('a))) ?D" apply(rule setsum_cong2, rule **) by auto
also have "\<dots> < 1" unfolding setsum_constant card_enum real_eq_of_nat divide_inverse[THEN sym] by (auto simp add:field_simps)
finally show "setsum (op $$ ?a) ?D < 1" by auto qed qed
end