(* Title: Convex
ID: $Id:
Author: Robert Himmelmann, TU Muenchen*)
header {* Convex sets, functions and related things. *}
theory Convex_Euclidean_Space
imports Topology_Euclidean_Space
begin
(* ------------------------------------------------------------------------- *)
(* To be moved elsewhere *)
(* ------------------------------------------------------------------------- *)
declare vector_add_ldistrib[simp] vector_ssub_ldistrib[simp] vector_smult_assoc[simp] vector_smult_rneg[simp]
declare vector_sadd_rdistrib[simp] vector_sub_rdistrib[simp]
declare dot_ladd[simp] dot_radd[simp] dot_lsub[simp] dot_rsub[simp]
declare dot_lmult[simp] dot_rmult[simp] dot_lneg[simp] dot_rneg[simp]
declare UNIV_1[simp]
term "(x::real^'n \<Rightarrow> real) 0"
lemma dim1in[intro]:"Suc 0 \<in> {1::nat .. CARD(1)}" by auto
lemmas vector_component_simps = vector_minus_component vector_smult_component vector_add_component vector_less_eq_def Cart_lambda_beta dest_vec1_def basis_component vector_uminus_component
lemmas continuous_intros = continuous_add continuous_vmul continuous_cmul continuous_const continuous_sub continuous_at_id continuous_within_id
lemmas continuous_on_intros = continuous_on_add continuous_on_const continuous_on_id continuous_on_compose continuous_on_cmul continuous_on_neg continuous_on_sub
uniformly_continuous_on_add uniformly_continuous_on_const uniformly_continuous_on_id uniformly_continuous_on_compose uniformly_continuous_on_cmul uniformly_continuous_on_neg uniformly_continuous_on_sub
lemma dest_vec1_simps[simp]: fixes a::"real^1"
shows "a$1 = 0 \<longleftrightarrow> a = 0" (*"a \<le> 1 \<longleftrightarrow> dest_vec1 a \<le> 1" "0 \<le> a \<longleftrightarrow> 0 \<le> dest_vec1 a"*)
"a \<le> b \<longleftrightarrow> dest_vec1 a \<le> dest_vec1 b" "dest_vec1 (1::real^1) = 1"
by(auto simp add:vector_component_simps all_1 Cart_eq)
definition
"is_interval_new (s::((real^'n::finite) set)) \<longleftrightarrow>
(\<forall>a\<in>s. \<forall>b\<in>s. \<forall>x. (\<forall>i. ((a$i \<le> x$i \<and> x$i \<le> b$i) \<or> (b$i \<le> x$i \<and> x$i \<le> a$i))) \<longrightarrow> x \<in> s)"
lemma is_interval_interval_new: "is_interval_new {a .. b}" (is ?th1) "is_interval_new {a<..<b}" (is ?th2) proof -
have *:"\<And>x y z::real. x < y \<Longrightarrow> y < z \<Longrightarrow> x < z" by auto
show ?th1 ?th2 unfolding is_interval_new_def mem_interval Ball_def atLeastAtMost_iff
by(meson real_le_trans le_less_trans less_le_trans *)+ qed
lemma nequals0I:"x\<in>A \<Longrightarrow> A \<noteq> {}" by auto
lemma norm_not_0:"(x::real^'n::finite)\<noteq>0 \<Longrightarrow> norm x \<noteq> 0" by auto
lemma vector_unminus_smult[simp]: "(-1::real) *s x = -x" unfolding pth_3[symmetric] by simp
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_diff1'':assumes "finite A" "a \<in> A"
shows "setsum f (A - {a}) = setsum f A - (f a::'a::ring)" unfolding setsum_diff1'[OF assms] by auto
lemma setsum_delta'': fixes s::"(real^'n) set" assumes "finite s"
shows "(\<Sum>x\<in>s. (if y = x then f x else 0) *s x) = (if y\<in>s then (f y) *s y else 0)"
proof-
have *:"\<And>x y. (if y = x then f x else (0::real)) *s x = (if x=y then (f x) *s x else 0)" by auto
show ?thesis unfolding * using setsum_delta[OF assms, of y "\<lambda>x. f x *s x"] by auto
qed
lemma not_disjointI:"x\<in>A \<Longrightarrow> x\<in>B \<Longrightarrow> A \<inter> B \<noteq> {}" by blast
lemma if_smult:"(if P then x else (y::real)) *s v = (if P then x *s v else y *s v)" by auto
lemma ex_bij_betw_nat_finite_1:
assumes "finite M"
shows "\<exists>h. bij_betw h {1 .. card M} M"
proof-
obtain h where h:"bij_betw h {0..<card M} M" using ex_bij_betw_nat_finite[OF assms] by auto
let ?h = "h \<circ> (\<lambda>i. i - 1)"
have *:"(\<lambda>i. i - 1) ` {1..card M} = {0..<card M}" apply auto unfolding image_iff apply(rule_tac x="Suc x" in bexI) by auto
hence "?h ` {1..card M} = h ` {0..<card M}" unfolding image_compose by auto
hence "?h ` {1..card M} = M" unfolding image_compose using h unfolding * unfolding bij_betw_def by auto
moreover
have "inj_on (\<lambda>i. i - Suc 0) {Suc 0..card M}" unfolding inj_on_def by auto
hence "inj_on ?h {1..card M}" apply(rule_tac comp_inj_on) unfolding * using h[unfolded bij_betw_def] by auto
ultimately show ?thesis apply(rule_tac x="h \<circ> (\<lambda>i. i - 1)" in exI) unfolding o_def and bij_betw_def by auto
qed
lemma finite_subset_image:
assumes "B \<subseteq> f ` A" "finite B"
shows "\<exists>C\<subseteq>A. finite C \<and> B = f ` C"
proof- from assms(1) have "\<forall>x\<in>B. \<exists>y\<in>A. x = f y" by auto
then obtain c where "\<forall>x\<in>B. c x \<in> A \<and> x = f (c x)"
using bchoice[of B "\<lambda>x y. y\<in>A \<and> x = f y"] by auto
thus ?thesis apply(rule_tac x="c ` B" in exI) using assms(2) by auto qed
lemma inj_on_image_eq_iff: assumes "inj_on f (A \<union> B)"
shows "f ` A = f ` B \<longleftrightarrow> A = B"
using assms by(blast dest: inj_onD)
lemma mem_interval_1: fixes x :: "real^1" shows
"(x \<in> {a .. b} \<longleftrightarrow> dest_vec1 a \<le> dest_vec1 x \<and> dest_vec1 x \<le> dest_vec1 b)"
"(x \<in> {a<..<b} \<longleftrightarrow> dest_vec1 a < dest_vec1 x \<and> dest_vec1 x < dest_vec1 b)"
by(simp_all add: Cart_eq vector_less_def vector_less_eq_def dest_vec1_def all_1)
lemma image_smult_interval:"(\<lambda>x. m *s (x::real^'n::finite)) ` {a..b} =
(if {a..b} = {} then {} else if 0 \<le> m then {m *s a..m *s b} else {m *s b..m *s a})"
using image_affinity_interval[of m 0 a b] by auto
lemma dest_vec1_inverval:
"dest_vec1 ` {a .. b} = {dest_vec1 a .. dest_vec1 b}"
"dest_vec1 ` {a<.. b} = {dest_vec1 a<.. dest_vec1 b}"
"dest_vec1 ` {a ..<b} = {dest_vec1 a ..<dest_vec1 b}"
"dest_vec1 ` {a<..<b} = {dest_vec1 a<..<dest_vec1 b}"
apply(rule_tac [!] equalityI)
unfolding subset_eq Ball_def Bex_def mem_interval_1 image_iff
apply(rule_tac [!] allI)apply(rule_tac [!] impI)
apply(rule_tac[2] x="vec1 x" in exI)apply(rule_tac[4] x="vec1 x" in exI)
apply(rule_tac[6] x="vec1 x" in exI)apply(rule_tac[8] x="vec1 x" in exI)
by (auto simp add: vector_less_def vector_less_eq_def all_1 dest_vec1_def
vec1_dest_vec1[unfolded dest_vec1_def One_nat_def])
lemma dest_vec1_setsum: assumes "finite S"
shows " dest_vec1 (setsum f S) = setsum (\<lambda>x. dest_vec1 (f x)) S"
using dest_vec1_sum[OF assms] by auto
lemma dist_triangle_eq:"dist x z = dist x y + dist y z \<longleftrightarrow> norm (x - y) *s (y - z) = norm (y - z) *s (x - y)"
proof- have *:"x - y + (y - z) = x - z" by auto
show ?thesis unfolding dist_def norm_triangle_eq[of "x - y" "y - z", unfolded *]
by(auto simp add:norm_minus_commute) qed
lemma norm_eqI:"x = y \<Longrightarrow> norm x = norm y" by auto
lemma norm_minus_eqI:"(x::real^'n::finite) = - 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 dimindex_ge_1:"CARD(_::finite) \<ge> 1"
using one_le_card_finite by auto
lemma real_dimindex_ge_1:"real (CARD('n::finite)) \<ge> 1"
by(metis dimindex_ge_1 linorder_not_less real_eq_of_nat real_le_trans real_of_nat_1 real_of_nat_le_iff)
lemma real_dimindex_gt_0:"real (CARD('n::finite)) > 0" apply(rule less_le_trans[OF _ real_dimindex_ge_1]) by auto
subsection {* Affine set and affine hull.*}
definition "affine s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u v::real. u + v = 1 \<longrightarrow> (u *s x + v *s y) \<in> s)"
lemma affine_alt: "affine s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u::real. (1 - u) *s x + u *s y \<in> s)"
proof- have *:"\<And>u v ::real. u + v = 1 \<longleftrightarrow> v = 1 - u" by auto
{ fix x y assume "x\<in>s" "y\<in>s"
hence "(\<forall>u v::real. u + v = 1 \<longrightarrow> u *s x + v *s y \<in> s) \<longleftrightarrow> (\<forall>u::real. (1 - u) *s x + u *s y \<in> s)" apply auto
apply(erule_tac[!] x="1 - u" in allE) unfolding * by auto }
thus ?thesis unfolding affine_def by auto qed
lemma affine_empty[intro]: "affine {}"
unfolding affine_def by auto
lemma affine_sing[intro]: "affine {x}"
unfolding affine_alt by (auto simp add: vector_sadd_rdistrib[THEN sym])
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"
proof-
{ fix f assume "f \<subseteq> affine"
hence "affine (\<Inter>f)" using affine_Inter[of f] unfolding subset_eq mem_def by auto }
thus ?thesis using hull_eq[unfolded mem_def, of affine s] by auto
qed
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::"(real^'n) 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) *s 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 *s x) \<in> V"
thus "u *s x + v *s 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: vector_sadd_rdistrib[THEN sym])
next
fix s u assume as:"\<forall>x\<in>V. \<forall>y\<in>V. \<forall>u v. u + v = 1 \<longrightarrow> u *s x + v *s 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 *s 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::"(real^'n) set" and u::"real^'n\<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 *s x + v *s y \<in> V; finite s;
s \<noteq> {}; s \<subseteq> V; setsum u s = 1; n \<equiv> card s \<rbrakk> \<Longrightarrow> (\<Sum>x\<in>s. u x *s x) \<in> V" and
as:"Suc n \<equiv> card s" "2 < card s" "\<forall>x\<in>V. \<forall>y\<in>V. \<forall>u v. u + v = 1 \<longrightarrow> u *s x + v *s 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) *s 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
thus "(\<Sum>x\<in>s. u x *s x) \<in> V" unfolding vector_smult_assoc[THEN sym] and setsum_cmul
apply(subst *) unfolding setsum_clauses(2)[OF *(2)]
using as(3)[THEN bspec[where x=x], THEN bspec[where x="(inverse (1 - u x)) *s (\<Sum>xa\<in>s - {x}. u xa *s xa)"],
THEN spec[where x="u x"], THEN spec[where x="1 - u x"]] and rev_subsetD[OF `x\<in>s` `s\<subseteq>V`] and `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) *s 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 *s 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 *s 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 *s 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 *s 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 *s 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 *s 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 *s 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 *s v) = u *s x + v *s 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 vector_sadd_rdistrib setsum_addf if_smult vector_smult_lzero ** setsum_restrict_set[OF xy, THEN sym]
unfolding vector_smult_assoc[THEN sym] setsum_cmul 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 *s v) s = y}"
unfolding affine_hull_explicit and expand_set_eq 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 *s 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 *s 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 *s v) = x"
thus "\<exists>u. setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *s v) = x" apply(rule_tac x="\<lambda>x. if x\<in>t then u x else 0" in exI)
unfolding if_smult vector_smult_lzero 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:
shows "(\<exists>u::real^'n=>real. setsum u {} = w \<and> setsum (\<lambda>x. u x *s 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 *s x) (insert a s) = y) \<longleftrightarrow>
(\<exists>v u. setsum u s = w - v \<and> setsum (\<lambda>x. u x *s x) s = y - v *s 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 *s 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 *s x) = y - v *s a" by auto
have *:"\<And>x M. (if x = a then v else M) *s x = (if x = a then v *s x else M *s 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 vector_sadd_rdistrib and setsum_addf
unfolding vu and * and pth_4(1)
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 *s x = (if x = a then v *s x else u x *s 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 *s x" "\<lambda>x. if x = a then v *s x else u x *s 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: "affine hull {a,b::real^'n} = {u *s a + v *s 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::real^'n)" by auto
have "?lhs = {y. \<exists>u. setsum u {a, b} = 1 \<and> (\<Sum>v\<in>{a, b}. u v *s 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 *s b = y - v *s 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: "affine hull {a,b,c::real^'n} = { u *s a + v *s b + w *s 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::real^'n)" 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:
"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 *s 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 *s v) = v)"
apply(rule_tac x="x - a" in exI) apply rule defer apply(rule_tac x="(\<lambda>x. x - a) ` (t - {a})" in exI)
apply(rule_tac x="\<lambda>x. u (x + a)" in exI) using as(1)
apply(simp add: setsum_reindex[unfolded inj_on_def] setsum_subtractf setsum_diff1 setsum_vmul[THEN sym])
unfolding as by simp_all qed
lemma affine_hull_insert_span:
assumes "a \<notin> s"
shows "affine hull (insert a s) =
{a + v | v . v \<in> span {x - a | x. x \<in> s}}"
apply(rule, rule affine_hull_insert_subset_span) unfolding subset_eq Ball_def
unfolding affine_hull_explicit and mem_Collect_eq proof(rule,rule,erule exE,erule conjE)
fix y v assume "y = a + v" "v \<in> span {x - a |x. x \<in> s}"
then obtain t u where obt:"finite t" "t \<subseteq> {x - a |x. x \<in> s}" "a + (\<Sum>v\<in>t. u v *s 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) *s (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 *s 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 "{a}", unfolded singleton_iff] and *
by (auto simp add: setsum_subtractf setsum_vmul field_simps) qed
lemma affine_hull_span:
assumes "a \<in> s"
shows "affine hull s = {a + v | v. v \<in> span {x - a | x. x \<in> s - {a}}}"
using affine_hull_insert_span[of a "s - {a}", unfolded insert_Diff[OF assms]] by auto
subsection {* Convexity. *}
definition "convex (s::(real^'n) set) \<longleftrightarrow>
(\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u\<ge>0. \<forall>v\<ge>0. (u + v = 1) \<longrightarrow> (u *s x + v *s y) \<in> s)"
lemma convex_alt: "convex s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u. 0 \<le> u \<and> u \<le> 1 \<longrightarrow> ((1 - u) *s x + u *s y) \<in> s)"
proof- have *:"\<And>u v::real. u + v = 1 \<longleftrightarrow> u = 1 - v" by auto
show ?thesis unfolding convex_def apply auto
apply(erule_tac x=x in ballE) apply(erule_tac x=y in ballE) apply(erule_tac x="1 - u" in allE)
by (auto simp add: *) qed
lemma mem_convex:
assumes "convex s" "a \<in> s" "b \<in> s" "0 \<le> u" "u \<le> 1"
shows "((1 - u) *s a + u *s b) \<in> s"
using assms unfolding convex_alt by auto
lemma convex_empty[intro]: "convex {}"
unfolding convex_def by simp
lemma convex_singleton[intro]: "convex {a}"
unfolding convex_def by (auto simp add:vector_sadd_rdistrib[THEN sym])
lemma convex_UNIV[intro]: "convex UNIV"
unfolding convex_def by auto
lemma convex_Inter: "(\<forall>s\<in>f. convex s) ==> convex(\<Inter> f)"
unfolding convex_def by auto
lemma convex_Int: "convex s \<Longrightarrow> convex t \<Longrightarrow> convex (s \<inter> t)"
unfolding convex_def by auto
lemma convex_halfspace_le: "convex {x. a \<bullet> x \<le> b}"
unfolding convex_def apply auto
unfolding dot_radd dot_rmult by (metis real_convex_bound_le)
lemma convex_halfspace_ge: "convex {x. a \<bullet> x \<ge> b}"
proof- have *:"{x. a \<bullet> x \<ge> b} = {x. -a \<bullet> x \<le> -b}" by auto
show ?thesis apply(unfold *) using convex_halfspace_le[of "-a" "-b"] by auto qed
lemma convex_hyperplane: "convex {x. a \<bullet> x = b}"
proof-
have *:"{x. a \<bullet> x = b} = {x. a \<bullet> x \<le> b} \<inter> {x. a \<bullet> x \<ge> b}" by auto
show ?thesis unfolding * apply(rule convex_Int)
using convex_halfspace_le convex_halfspace_ge by auto
qed
lemma convex_halfspace_lt: "convex {x. a \<bullet> x < b}"
unfolding convex_def by(auto simp add: real_convex_bound_lt dot_radd dot_rmult)
lemma convex_halfspace_gt: "convex {x. a \<bullet> x > b}"
using convex_halfspace_lt[of "-a" "-b"] by(auto simp add: dot_lneg neg_less_iff_less)
lemma convex_positive_orthant: "convex {x::real^'n. (\<forall>i. 0 \<le> x$i)}"
unfolding convex_def apply auto apply(erule_tac x=i in allE)+
apply(rule add_nonneg_nonneg) by(auto simp add: mult_nonneg_nonneg)
subsection {* Explicit expressions for convexity in terms of arbitrary sums. *}
lemma convex: "convex s \<longleftrightarrow>
(\<forall>(k::nat) u x. (\<forall>i. 1\<le>i \<and> i\<le>k \<longrightarrow> 0 \<le> u i \<and> x i \<in>s) \<and> (setsum u {1..k} = 1)
\<longrightarrow> setsum (\<lambda>i. u i *s x i) {1..k} \<in> s)"
unfolding convex_def apply rule apply(rule allI)+ defer apply(rule ballI)+ apply(rule allI)+ proof(rule,rule,rule,rule)
fix x y u v assume as:"\<forall>(k::nat) u x. (\<forall>i. 1 \<le> i \<and> i \<le> k \<longrightarrow> 0 \<le> u i \<and> x i \<in> s) \<and> setsum u {1..k} = 1 \<longrightarrow> (\<Sum>i = 1..k. u i *s x i) \<in> s"
"x \<in> s" "y \<in> s" "0 \<le> u" "0 \<le> v" "u + v = (1::real)"
show "u *s x + v *s y \<in> s" using as(1)[THEN spec[where x=2], THEN spec[where x="\<lambda>n. if n=1 then u else v"], THEN spec[where x="\<lambda>n. if n=1 then x else y"]] and as(2-)
by (auto simp add: setsum_head_Suc)
next
fix k u x assume as:"\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u\<ge>0. \<forall>v\<ge>0. u + v = 1 \<longrightarrow> u *s x + v *s y \<in> s"
show "(\<forall>i::nat. 1 \<le> i \<and> i \<le> k \<longrightarrow> 0 \<le> u i \<and> x i \<in> s) \<and> setsum u {1..k} = 1 \<longrightarrow> (\<Sum>i = 1..k. u i *s x i) \<in> s" apply(rule,erule conjE) proof(induct k arbitrary: u)
case (Suc k) show ?case proof(cases "u (Suc k) = 1")
case True hence "(\<Sum>i = Suc 0..k. u i *s x i) = 0" apply(rule_tac setsum_0') apply(rule ccontr) unfolding ball_simps apply(erule bexE) proof-
fix i assume i:"i \<in> {Suc 0..k}" "u i *s x i \<noteq> 0"
hence ui:"u i \<noteq> 0" by auto
hence "setsum (\<lambda>k. if k=i then u i else 0) {1 .. k} \<le> setsum u {1 .. k}" apply(rule_tac setsum_mono) using Suc(2) by auto
hence "setsum u {1 .. k} \<ge> u i" using i(1) by(auto simp add: setsum_delta)
hence "setsum u {1 .. k} > 0" using ui apply(rule_tac less_le_trans[of _ "u i"]) using Suc(2)[THEN spec[where x=i]] and i(1) by auto
thus False using Suc(3) unfolding setsum_cl_ivl_Suc and True by simp qed
thus ?thesis unfolding setsum_cl_ivl_Suc using True and Suc(2) by auto
next
have *:"setsum u {1..k} = 1 - u (Suc k)" using Suc(3)[unfolded setsum_cl_ivl_Suc] by auto
have **:"u (Suc k) \<le> 1" apply(rule ccontr) unfolding not_le using Suc(3) using setsum_nonneg[of "{1..k}" u] using Suc(2) by auto
have ***:"\<And>i k. (u i / (1 - u (Suc k))) *s x i = (inverse (1 - u (Suc k))) *s (u i *s x i)" unfolding real_divide_def by auto
case False hence nn:"1 - u (Suc k) \<noteq> 0" by auto
have "(\<Sum>i = 1..k. (u i / (1 - u (Suc k))) *s x i) \<in> s" apply(rule Suc(1)) unfolding setsum_divide_distrib[THEN sym] and *
apply(rule_tac allI) apply(rule,rule) apply(rule divide_nonneg_pos) using nn Suc(2) ** by auto
hence "(1 - u (Suc k)) *s (\<Sum>i = 1..k. (u i / (1 - u (Suc k))) *s x i) + u (Suc k) *s x (Suc k) \<in> s"
apply(rule as[THEN bspec, THEN bspec, THEN spec, THEN mp, THEN spec, THEN mp, THEN mp]) using Suc(2)[THEN spec[where x="Suc k"]] and ** by auto
thus ?thesis unfolding setsum_cl_ivl_Suc and *** and setsum_cmul using nn by auto qed qed auto qed
lemma convex_explicit: "convex (s::(real^'n) set) \<longleftrightarrow>
(\<forall>t u. finite t \<and> t \<subseteq> s \<and> (\<forall>x\<in>t. 0 \<le> u x) \<and> setsum u t = 1 \<longrightarrow> setsum (\<lambda>x. u x *s x) t \<in> s)"
unfolding convex_def apply(rule,rule,rule) apply(subst imp_conjL,rule) defer apply(rule,rule,rule,rule,rule,rule,rule) proof-
fix x y u v assume as:"\<forall>t u. finite t \<and> t \<subseteq> s \<and> (\<forall>x\<in>t. 0 \<le> u x) \<and> setsum u t = 1 \<longrightarrow> (\<Sum>x\<in>t. u x *s x) \<in> s" "x \<in> s" "y \<in> s" "0 \<le> u" "0 \<le> v" "u + v = (1::real)"
show "u *s x + v *s y \<in> s" proof(cases "x=y")
case True show ?thesis unfolding True and vector_sadd_rdistrib[THEN sym] using as(3,6) by auto next
case False thus ?thesis using as(1)[THEN spec[where x="{x,y}"], THEN spec[where x="\<lambda>z. if z=x then u else v"]] and as(2-) by auto qed
next
fix t u assume asm:"\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u\<ge>0. \<forall>v\<ge>0. u + v = 1 \<longrightarrow> u *s x + v *s y \<in> s" "finite (t::(real^'n) set)"
(*"finite t" "t \<subseteq> s" "\<forall>x\<in>t. (0::real) \<le> u x" "setsum u t = 1"*)
from this(2) have "\<forall>u. t \<subseteq> s \<and> (\<forall>x\<in>t. 0 \<le> u x) \<and> setsum u t = 1 \<longrightarrow> (\<Sum>x\<in>t. u x *s x) \<in> s" apply(induct_tac t rule:finite_induct)
prefer 3 apply (rule,rule) apply(erule conjE)+ proof-
fix x f u assume ind:"\<forall>u. f \<subseteq> s \<and> (\<forall>x\<in>f. 0 \<le> u x) \<and> setsum u f = 1 \<longrightarrow> (\<Sum>x\<in>f. u x *s x) \<in> s"
assume as:"finite f" "x \<notin> f" "insert x f \<subseteq> s" "\<forall>x\<in>insert x f. 0 \<le> u x" "setsum u (insert x f) = (1::real)"
show "(\<Sum>x\<in>insert x f. u x *s x) \<in> s" proof(cases "u x = 1")
case True hence "setsum (\<lambda>x. u x *s x) f = 0" apply(rule_tac setsum_0') apply(rule ccontr) unfolding ball_simps apply(erule bexE) proof-
fix y assume y:"y \<in> f" "u y *s y \<noteq> 0"
hence uy:"u y \<noteq> 0" by auto
hence "setsum (\<lambda>k. if k=y then u y else 0) f \<le> setsum u f" apply(rule_tac setsum_mono) using as(4) by auto
hence "setsum u f \<ge> u y" using y(1) and as(1) by(auto simp add: setsum_delta)
hence "setsum u f > 0" using uy apply(rule_tac less_le_trans[of _ "u y"]) using as(4) and y(1) by auto
thus False using as(2,5) unfolding setsum_clauses(2)[OF as(1)] and True by auto qed
thus ?thesis unfolding setsum_clauses(2)[OF as(1)] using as(2,3) unfolding True by auto
next
have *:"setsum u f = setsum u (insert x f) - u x" using as(2) unfolding setsum_clauses(2)[OF as(1)] by auto
have **:"u x \<le> 1" apply(rule ccontr) unfolding not_le using as(5)[unfolded setsum_clauses(2)[OF as(1)]] and as(2)
using setsum_nonneg[of f u] and as(4) by auto
case False hence "inverse (1 - u x) *s (\<Sum>x\<in>f. u x *s x) \<in> s" unfolding setsum_cmul[THEN sym] and vector_smult_assoc
apply(rule_tac ind[THEN spec, THEN mp]) apply rule defer apply rule apply rule apply(rule mult_nonneg_nonneg)
unfolding setsum_right_distrib[THEN sym] and * using as and ** by auto
hence "u x *s x + (1 - u x) *s ((inverse (1 - u x)) *s setsum (\<lambda>x. u x *s x) f) \<in>s"
apply(rule_tac asm(1)[THEN bspec, THEN bspec, THEN spec, THEN mp, THEN spec, THEN mp, THEN mp]) using as and ** False by auto
thus ?thesis unfolding setsum_clauses(2)[OF as(1)] using as(2) and False by auto qed
qed auto thus "t \<subseteq> s \<and> (\<forall>x\<in>t. 0 \<le> u x) \<and> setsum u t = 1 \<longrightarrow> (\<Sum>x\<in>t. u x *s x) \<in> s" by auto
qed
lemma convex_finite: assumes "finite s"
shows "convex s \<longleftrightarrow> (\<forall>u. (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1
\<longrightarrow> setsum (\<lambda>x. u x *s x) s \<in> s)"
unfolding convex_explicit apply(rule, rule, rule) defer apply(rule,rule,rule)apply(erule conjE)+ proof-
fix t u assume as:"\<forall>u. (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<longrightarrow> (\<Sum>x\<in>s. u x *s x) \<in> s" " finite t" "t \<subseteq> s" "\<forall>x\<in>t. 0 \<le> u x" "setsum u t = (1::real)"
have *:"s \<inter> t = t" using as(3) by auto
show "(\<Sum>x\<in>t. u x *s x) \<in> s" using as(1)[THEN spec[where x="\<lambda>x. if x\<in>t then u x else 0"]]
unfolding if_smult and setsum_cases[OF assms] and * using as(2-) by auto
qed (erule_tac x=s in allE, erule_tac x=u in allE, auto)
subsection {* Cones. *}
definition "cone (s::(real^'n) set) \<longleftrightarrow> (\<forall>x\<in>s. \<forall>c\<ge>0. (c *s 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 (s::(real^'n) set) \<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 *s 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 *s 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 *s 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 *s 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 *s 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 vector_smult_assoc[THEN sym] and setsum_cmul unfolding setsum_right_distrib[THEN sym] and setsum_diff1''[OF as(1,5)] using as by auto
qed
lemma affine_dependent_explicit_finite:
assumes "finite (s::(real^'n) set)"
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 *s v) s = 0)"
(is "?lhs = ?rhs")
proof
have *:"\<And>vt u v. (if vt then u v else 0) *s v = (if vt then (u v) *s v else (0::real^'n))" 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 *s 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_absorb2[OF `t\<subseteq>s`, unfolded Int_commute] 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 *s v) = 0" by auto
thus ?lhs unfolding affine_dependent_explicit using assms by auto
qed
subsection {* A general lemma. *}
lemma convex_connected:
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) *s x1 + x *s x2 - ((1 - y) *s x1 + y *s x2) = (y - x) *s x1 - (y - x) *s x2"
by(simp add: ring_simps vector_sadd_rdistrib vector_sub_rdistrib)
assume "\<bar>y - x\<bar> < e / norm (x1 - x2)"
hence "norm ((1 - x) *s x1 + x *s x2 - ((1 - y) *s x1 + y *s x2)) < e"
unfolding * and vector_ssub_ldistrib[THEN sym] and norm_mul
unfolding less_divide_eq using n by auto }
hence "\<exists>d>0. \<forall>y. \<bar>y - x\<bar> < d \<longrightarrow> norm ((1 - x) *s x1 + x *s x2 - ((1 - y) *s x1 + y *s 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) *s x1 + x *s x2 \<notin> e1 \<and> (1 - x) *s x1 + x *s x2 \<notin> e2"
apply(rule connected_real_lemma) apply (simp add: `x1\<in>e1` `x2\<in>e2` dist_sym)+
using * apply(simp add: dist_def)
using as(1,2)[unfolded open_def] apply simp
using as(1,2)[unfolded open_def] 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) *s x1 + x *s x2 \<notin> e1" "(1 - x) *s x1 + x *s 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: "connected UNIV"
by(simp add: convex_connected convex_UNIV)
subsection {* Convex functions into the reals. *}
definition "convex_on s (f::real^'n \<Rightarrow> real) =
(\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u\<ge>0. \<forall>v\<ge>0. u + v = 1 \<longrightarrow> f (u *s x + v *s y) \<le> u * f x + v * f y)"
lemma convex_on_subset: "convex_on t f \<Longrightarrow> s \<subseteq> t \<Longrightarrow> convex_on s f"
unfolding convex_on_def by auto
lemma convex_add:
assumes "convex_on s f" "convex_on s g"
shows "convex_on s (\<lambda>x. f x + g x)"
proof-
{ fix x y assume "x\<in>s" "y\<in>s" moreover
fix u v ::real assume "0 \<le> u" "0 \<le> v" "u + v = 1"
ultimately have "f (u *s x + v *s y) + g (u *s x + v *s y) \<le> (u * f x + v * f y) + (u * g x + v * g y)"
using assms(1)[unfolded convex_on_def, THEN bspec[where x=x], THEN bspec[where x=y], THEN spec[where x=u]]
using assms(2)[unfolded convex_on_def, THEN bspec[where x=x], THEN bspec[where x=y], THEN spec[where x=u]]
apply - apply(rule add_mono) by auto
hence "f (u *s x + v *s y) + g (u *s x + v *s y) \<le> u * (f x + g x) + v * (f y + g y)" by (simp add: ring_simps) }
thus ?thesis unfolding convex_on_def by auto
qed
lemma convex_cmul:
assumes "0 \<le> (c::real)" "convex_on s f"
shows "convex_on s (\<lambda>x. c * f x)"
proof-
have *:"\<And>u c fx v fy ::real. u * (c * fx) + v * (c * fy) = c * (u * fx + v * fy)" by (simp add: ring_simps)
show ?thesis using assms(2) and mult_mono1[OF _ assms(1)] unfolding convex_on_def and * by auto
qed
lemma convex_lower:
assumes "convex_on s f" "x\<in>s" "y \<in> s" "0 \<le> u" "0 \<le> v" "u + v = 1"
shows "f (u *s x + v *s y) \<le> max (f x) (f y)"
proof-
let ?m = "max (f x) (f y)"
have "u * f x + v * f y \<le> u * max (f x) (f y) + v * max (f x) (f y)" apply(rule add_mono)
using assms(4,5) by(auto simp add: mult_mono1)
also have "\<dots> = max (f x) (f y)" using assms(6) unfolding distrib[THEN sym] by auto
finally show ?thesis using assms(1)[unfolded convex_on_def, THEN bspec[where x=x], THEN bspec[where x=y], THEN spec[where x=u]]
using assms(2-6) by auto
qed
lemma convex_local_global_minimum:
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) *s x + u *s 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) *s x + u *s y) = u *s (x - y)" by (simp add: vector_ssub_ldistrib vector_sub_rdistrib)
have "(1 - u) *s x + u *s y \<in> ball x e" unfolding mem_ball dist_def unfolding * and norm_mul and abs_of_pos[OF `0<u`] unfolding dist_def[THEN sym]
using u unfolding pos_less_divide_eq[OF xy] by auto
hence "f x \<le> f ((1 - u) *s x + u *s 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_distance: "convex_on s (\<lambda>x. dist a x)"
proof(auto simp add: convex_on_def dist_def)
fix x y assume "x\<in>s" "y\<in>s"
fix u v ::real assume "0 \<le> u" "0 \<le> v" "u + v = 1"
have "a = u *s a + v *s a" unfolding vector_sadd_rdistrib[THEN sym] and `u+v=1` by simp
hence *:"a - (u *s x + v *s y) = (u *s (a - x)) + (v *s (a - y))" by auto
show "norm (a - (u *s x + v *s y)) \<le> u * norm (a - x) + v * norm (a - y)"
unfolding * using norm_triangle_ineq[of "u *s (a - x)" "v *s (a - y)"] unfolding norm_mul
using `0 \<le> u` `0 \<le> v` by auto
qed
subsection {* Arithmetic operations on sets preserve convexity. *}
lemma convex_scaling: "convex s \<Longrightarrow> convex ((\<lambda>x. c *s x) ` s)"
unfolding convex_def and image_iff apply auto
apply (rule_tac x="u *s x+v *s y" in bexI) by (auto simp add: field_simps)
lemma convex_negations: "convex s \<Longrightarrow> convex ((\<lambda>x. -x)` s)"
unfolding convex_def and image_iff apply auto
apply (rule_tac x="u *s x+v *s y" in bexI) by auto
lemma convex_sums:
assumes "convex s" "convex t"
shows "convex {x + y| x y. x \<in> s \<and> y \<in> t}"
proof(auto simp add: convex_def image_iff)
fix xa xb ya yb assume xy:"xa\<in>s" "xb\<in>s" "ya\<in>t" "yb\<in>t"
fix u v ::real assume uv:"0 \<le> u" "0 \<le> v" "u + v = 1"
show "\<exists>x y. u *s xa + u *s ya + (v *s xb + v *s yb) = x + y \<and> x \<in> s \<and> y \<in> t"
apply(rule_tac x="u *s xa + v *s xb" in exI) apply(rule_tac x="u *s ya + v *s yb" in exI)
using assms(1)[unfolded convex_def, THEN bspec[where x=xa], THEN bspec[where x=xb]]
using assms(2)[unfolded convex_def, THEN bspec[where x=ya], THEN bspec[where x=yb]]
using uv xy by auto
qed
lemma convex_differences:
assumes "convex s" "convex t"
shows "convex {x - y| x y. x \<in> s \<and> y \<in> t}"
proof-
have "{x - y| x y. x \<in> s \<and> y \<in> t} = {x + y |x y. x \<in> s \<and> y \<in> uminus ` t}" unfolding image_iff apply auto
apply(rule_tac x=xa in exI) apply(rule_tac x="-y" in exI) apply simp
apply(rule_tac x=xa in exI) apply(rule_tac x=xb in exI) by simp
thus ?thesis using convex_sums[OF assms(1) convex_negations[OF assms(2)]] by auto
qed
lemma convex_translation: assumes "convex s" shows "convex ((\<lambda>x. a + x) ` s)"
proof- have "{a + y |y. y \<in> s} = (\<lambda>x. a + x) ` s" by auto
thus ?thesis using convex_sums[OF convex_singleton[of a] assms] by auto qed
lemma convex_affinity: assumes "convex (s::(real^'n) set)" shows "convex ((\<lambda>x. a + c *s x) ` s)"
proof- have "(\<lambda>x. a + c *s x) ` s = op + a ` op *s c ` s" by auto
thus ?thesis using convex_translation[OF convex_scaling[OF assms], of a c] by auto qed
lemma convex_linear_image: assumes c:"convex s" and l:"linear f" shows "convex(f ` s)"
proof(auto simp add: convex_def)
fix x y assume xy:"x \<in> s" "y \<in> s"
fix u v ::real assume uv:"0 \<le> u" "0 \<le> v" "u + v = 1"
show "u *s f x + v *s f y \<in> f ` s" unfolding image_iff
apply(rule_tac x="u *s x + v *s y" in bexI)
unfolding linear_add[OF l] linear_cmul[OF l]
using c[unfolded convex_def] xy uv by auto
qed
subsection {* Balls, being convex, are connected. *}
lemma convex_ball: "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 *s y + v *s 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 *s y + v *s z) < e" using real_convex_bound_lt[OF yz uv] by auto
qed
lemma convex_cball: "convex(cball x e)"
proof(auto simp add: convex_def Ball_def mem_cball)
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 *s y + v *s 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 *s y + v *s z) \<le> e" using real_convex_bound_le[OF yz uv] by auto
qed
lemma connected_ball: "connected(ball x e)"
using convex_connected convex_ball by auto
lemma connected_cball: "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" apply(rule hull_eq[unfolded mem_def])
using convex_Inter[unfolded Ball_def mem_def] by auto
lemma bounded_convex_hull: 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_def 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_def using B by auto qed
lemma finite_imp_bounded_convex_hull:
"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:
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 *s a + v *s 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 *s a + v *s 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 *s a + v1 *s 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 *s a + v2 *s b2" by auto
have *:"\<And>x s1 s2. x - s1 *s x - s2 *s x = ((1::real) - (s1 + s2)) *s x" by auto
have "\<exists>b \<in> convex hull s. u *s x + v *s y = (u * u1) *s a + (v * u2) *s a + (b - (u * u1) *s b - (v * u2) *s b)"
proof(cases "u * v1 + v * v2 = 0")
have *:"\<And>x s1 s2. x - s1 *s x - s2 *s x = ((1::real) - (s1 + s2)) *s x" by auto
case True hence **:"u * v1 = 0" "v * v2 = 0" apply- apply(rule_tac [!] ccontr)
using mult_nonneg_nonneg[OF `u\<ge>0` `v1\<ge>0`] mult_nonneg_nonneg[OF `v\<ge>0` `v2\<ge>0`] by auto
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: **)
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)) *s b1 + ((v * v2) / (u * v1 + v * v2)) *s 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
qed note * = this
have u1:"u1 \<le> 1" apply(rule ccontr) unfolding obt1(3)[THEN sym] and not_le using obt1(2) by auto
have u2:"u2 \<le> 1" apply(rule ccontr) 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 *s x + v *s 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:field_simps)
qed
qed
subsection {* Explicit expression for convex hull. *}
lemma convex_hull_indexed:
"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 *s 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 *s 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 *s 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 *s x2 i) = y" by auto
have *:"\<And>P x1 x2 s1 s2 i.(if P i then s1 else s2) *s (if P i then x1 else x2) = (if P i then s1 *s x1 else s2 *s 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 *s x + v *s 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
unfolding vector_smult_assoc[THEN sym] setsum_cmul 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:
assumes "finite (s::(real^'n)set)"
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 *s 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 *s 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) *s x) = u *s (\<Sum>x\<in>s. ux x *s x) + v *s (\<Sum>x\<in>s. uy x *s x)"
unfolding vector_sadd_rdistrib and setsum_addf and vector_smult_assoc[THEN sym] and setsum_cmul 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 *s x) = u *s (\<Sum>x\<in>s. ux x *s x) + v *s (\<Sum>x\<in>s. uy x *s 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 *s 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 convex_hull_explicit:
"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 *s 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 *s 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} *s v) = x"
using setsum_image_gen[OF fin, of "\<lambda>i. u i *s y i" y, THEN sym]
unfolding setsum_vmul[OF fin'] 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 *s 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 *s 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) *s f x) = u y *s y" by auto }
hence "(\<Sum>x = 1..card s. u (f x)) = 1" "(\<Sum>i = 1..card s. u (f i) *s f i) = y"
unfolding setsum_image_gen[OF *(1), of "\<lambda>x. u (f x) *s 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) *s f x)" "\<lambda>v. u v *s 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 *s 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 expand_set_eq by blast
qed
subsection {* A stepping theorem for that expansion. *}
lemma convex_hull_finite_step:
assumes "finite (s::(real^'n) set)"
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 *s 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 *s x) s = y - v *s 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 *s 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 *s x) = y - v *s a" by auto
show ?lhs apply(rule_tac x="\<lambda>x. (if a = x then v else 0) + u x" in exI) unfolding vector_sadd_rdistrib 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 *s x) = y - v *s 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) *s x) = (\<Sum>x\<in>s. u x *s 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 *s a + v *s 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 *s (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 *s a + u *s b \<and> 0 \<le> v \<and> 0 \<le> u \<and> v + u = 1) = (\<exists>u. x = a + u *s (b - a) \<and> 0 \<le> u \<and> u \<le> 1)"
unfolding * apply auto apply(rule_tac[!] x=u in exI) by auto qed
lemma convex_hull_3:
"convex hull {a::real^'n,b,c} = { u *s a + v *s b + w *s 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 ::real^'n. x + y + z = 1 \<longleftrightarrow> x = 1 - y - z" by (auto simp add: ring_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 *s (b - a) + v *s (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
apply(rule_tac x=u in exI) apply(rule_tac x=v in exI) by simp qed
subsection {* Relations among closure notions and corresponding hulls. *}
lemma subspace_imp_affine: "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: "subspace s \<Longrightarrow> convex s"
using subspace_imp_affine affine_imp_convex by auto
lemma affine_hull_subset_span: "(affine hull s) \<subseteq> (span s)"
unfolding span_def apply(rule hull_antimono) unfolding subset_eq Ball_def mem_def
using subspace_imp_affine by auto
lemma convex_hull_subset_span: "(convex hull s) \<subseteq> (span s)"
unfolding span_def apply(rule hull_antimono) unfolding subset_eq Ball_def mem_def
using subspace_imp_convex by auto
lemma convex_hull_subset_affine_hull: "(convex hull s) \<subseteq> (affine hull s)"
unfolding span_def apply(rule hull_antimono) unfolding subset_eq Ball_def mem_def
using affine_imp_convex by auto
lemma affine_dependent_imp_dependent: "affine_dependent s \<Longrightarrow> dependent s"
unfolding affine_dependent_def dependent_def
using affine_hull_subset_span by auto
lemma dependent_imp_affine_dependent:
assumes "dependent {x - a| x . x \<in> s}" "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 *s 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) *s x) = (\<Sum>x\<in>t. Q x *s x)"
apply(rule setsum_cong2) using `a\<notin>s` `t\<subseteq>s` by auto
have "(\<Sum>x\<in>t. u (x - a)) *s a = (\<Sum>v\<in>t. u (v - a) *s v)"
unfolding setsum_vmul[OF fin(1)] unfolding t_def and setsum_reindex[OF inj] and o_def
using obt(5) by (auto simp add: setsum_addf)
hence "(\<Sum>v\<in>insert a t. (if v = a then - (\<Sum>x\<in>t. u (x - a)) else u (v - a)) *s v) = 0"
unfolding setsum_clauses(2)[OF fin] using `a\<notin>s` `t\<subseteq>s` by (auto simp add: * vector_smult_lneg)
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 *s x) \<in> s)" (is "?lhs = ?rhs")
proof-
{ fix x y assume "x\<in>s" "y\<in>s" and ?lhs
hence "2 *s x \<in>s" "2 *s 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*s x" in ballE) apply(erule_tac x="2*s 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::"(real^'n::finite) set"
assumes "finite s" "card s \<ge> CARD('n) + 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> > CARD('n)" 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::(real^'n::finite) 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::"(real^'n::finite) set"
shows "convex hull p = {y. \<exists>s u. finite s \<and> s \<subseteq> p \<and> card s \<le> CARD('n) + 1 \<and>
(\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<and> setsum (\<lambda>v. u v *s v) s = y}"
unfolding convex_hull_explicit expand_set_eq 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 *s 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 *s 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 *s v) = y" by auto
have "card s \<le> CARD('n) + 1" proof(rule ccontr, simp only: not_le)
assume "CARD('n) + 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 *s 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)::'a::ring)" 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 *s v + (t * w v) *s v) - (u a *s a + (t * w a) *s a) = y"
unfolding setsum_addf obt(6) vector_smult_assoc[THEN sym] setsum_cmul wv(4)
by (metis diff_0_right a(2) pth_5 pth_8 pth_d vector_mul_eq_0)
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: *)
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> CARD('n) + 1
\<and> (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *s v) = y" using obt by auto
qed auto
lemma caratheodory:
"convex hull p = {x::real^'n::finite. \<exists>s. finite s \<and> s \<subseteq> p \<and>
card s \<le> CARD('n) + 1 \<and> x \<in> convex hull s}"
unfolding expand_set_eq 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> CARD('n) + 1"
"\<forall>x\<in>s. 0 \<le> u x" "setsum u s = 1" "(\<Sum>v\<in>s. u v *s v) = x"unfolding convex_hull_caratheodory by auto
thus "\<exists>s. finite s \<and> s \<subseteq> p \<and> card s \<le> CARD('n) + 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> CARD('n) + 1 \<and> x \<in> convex hull s"
then obtain s where "finite s" "s \<subseteq> p" "card s \<le> CARD('n) + 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:
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 *s 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 *s 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 *s 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_def[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_def 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 auto }
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)) *s v) = y"
unfolding setsum_reindex[OF *] o_def using obt(4,5)
by (simp add: setsum_addf setsum_subtractf setsum_vmul[OF obt(1), THEN sym])
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 *s 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:
assumes "compact s" "compact t"
shows "compact { (1 - u) *s x + u *s y | x y u. 0 \<le> u \<and> u \<le> 1 \<and> x \<in> s \<and> y \<in> t}"
proof-
let ?X = "{ pastecart u w | u w. u \<in> {vec1 0 .. vec1 1} \<and> w \<in> { pastecart x y |x y. x \<in> s \<and> y \<in> t} }"
let ?h = "(\<lambda>z. (1 - dest_vec1(fstcart z)) *s fstcart(sndcart z) + dest_vec1(fstcart z) *s sndcart(sndcart z))"
have *:"{ (1 - u) *s x + u *s 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 unfolding mem_interval_1 vec1_dest_vec1
apply rule apply auto apply(rule_tac x="pastecart (vec1 u) (pastecart xa y)" in exI) apply simp
apply(rule_tac x="vec1 u" in exI) apply(rule_tac x="pastecart xa y" in exI) by auto
{ fix u::"real^1" fix x y assume as:"0 \<le> dest_vec1 u" "dest_vec1 u \<le> 1" "x \<in> s" "y \<in> t"
hence "continuous (at (pastecart u (pastecart x y)))
(\<lambda>z. fstcart (sndcart z) - dest_vec1 (fstcart z) *s fstcart (sndcart z) +
dest_vec1 (fstcart z) *s sndcart (sndcart z))"
apply (auto intro!: continuous_add continuous_sub continuous_mul simp add: o_def vec1_dest_vec1)
using linear_continuous_at linear_fstcart linear_sndcart linear_sndcart
using linear_compose[unfolded o_def] by auto }
hence "continuous_on {pastecart u w |u w. u \<in> {vec1 0..vec1 1} \<and> w \<in> {pastecart x y |x y. x \<in> s \<and> y \<in> t}}
(\<lambda>z. (1 - dest_vec1 (fstcart z)) *s fstcart (sndcart z) + dest_vec1 (fstcart z) *s sndcart (sndcart z))"
apply(rule_tac continuous_at_imp_continuous_on) unfolding mem_Collect_eq
unfolding mem_interval_1 vec1_dest_vec1 by auto
thus ?thesis unfolding * apply(rule compact_continuous_image)
defer apply(rule compact_pastecart) defer apply(rule compact_pastecart)
using compact_interval assms by auto
qed
lemma compact_convex_hull: fixes s::"(real^'n::finite) 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 "CARD('n) + 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 simp add: convex_hull_empty)
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 expand_set_eq 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 add: convex_hull_empty)
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 add: convex_hull_singleton)
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) *s x + u *s 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 expand_set_eq and mem_Collect_eq proof(rule,rule)
fix x assume "\<exists>u v c. x = (1 - c) *s u + c *s 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) *s u + c *s 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) *s u + c *s 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) *s u + c *s 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}"] simp add: convex_hull_singleton)
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 *s a + vx *s 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:
"finite s \<Longrightarrow> compact(convex hull s)"
apply(drule finite_imp_compact, drule compact_convex_hull) by assumption
subsection {* Extremal points of a simplex are some vertices. *}
lemma dist_increases_online: assumes "d \<noteq> 0"
shows "dist a (b + d) > dist a b \<or> dist a (b - d) > dist a b"
proof(cases "a \<bullet> d - b \<bullet> d > 0")
case True hence "0 < d \<bullet> d + (a \<bullet> d * 2 - b \<bullet> d * 2)"
apply(rule_tac add_pos_pos) using assms by auto
thus ?thesis apply(rule_tac disjI2) unfolding dist_def and real_vector_norm_def and real_sqrt_less_iff
by(simp add: dot_rsub dot_radd dot_lsub dot_ladd dot_sym field_simps)
next
case False hence "0 < d \<bullet> d + (b \<bullet> d * 2 - a \<bullet> d * 2)"
apply(rule_tac add_pos_nonneg) using assms by auto
thus ?thesis apply(rule_tac disjI1) unfolding dist_def and real_vector_norm_def and real_sqrt_less_iff
by(simp add: dot_rsub dot_radd dot_lsub dot_ladd dot_sym field_simps)
qed
lemma norm_increases_online:
"(d::real^'n::finite) \<noteq> 0 \<Longrightarrow> norm(a + d) > norm a \<or> norm(a - d) > norm a"
using dist_increases_online[of d a 0] unfolding dist_def by auto
lemma simplex_furthest_lt:
fixes s::"(real^'n::finite) 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 *s x + v *s 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: vector_sadd_rdistrib[THEN sym])
thus False using obt(4) and False by simp qed
hence *:"w *s (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 *s (x - b))"
hence "norm (y - a) < norm ((u + w) *s x + (v - w) *s b - a)"
unfolding dist_sym[of a] unfolding dist_def obt(5) by (simp add: ring_simps)
moreover have "(u + w) *s x + (v - w) *s 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 *s (x - b))"
hence "norm (y - a) < norm ((u - w) *s x + (v + w) *s b - a)"
unfolding dist_sym[of a] unfolding dist_def obt(5) by (simp add: ring_simps)
moreover have "(u - w) *s x + (v + w) *s 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:
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_sym[of a] unfolding dist_def 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:
"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:
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:
"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 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: fixes y::"real^'n::finite"
assumes "y \<bullet> z > 0"
shows "\<exists>u>0. \<forall>v>0. v \<le> u \<longrightarrow> norm(v *s z - y) < norm y"
proof- have z:"z \<bullet> z > 0" unfolding dot_pos_lt using assms by auto
thus ?thesis using assms apply(rule_tac x="(y \<bullet> z) / (z \<bullet> z)" in exI) apply(rule) defer proof(rule+)
fix v assume "0<v" "v \<le> y \<bullet> z / (z \<bullet> z)"
thus "norm (v *s z - y) < norm y" unfolding norm_lt using z and assms
by (simp add: field_simps dot_sym mult_strict_left_mono[OF _ `0<v`])
qed(rule divide_pos_pos, auto) qed
lemma closer_point_lemma:
assumes "(y - x) \<bullet> (z - x) > 0"
shows "\<exists>u>0. u \<le> 1 \<and> dist (x + u *s (z - x)) y < dist x y"
proof- obtain u where "u>0" and u:"\<forall>v>0. v \<le> u \<longrightarrow> norm (v *s (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_def 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 "(a - x) \<bullet> (y - x) \<le> 0"
proof(rule ccontr) assume "\<not> (a - x) \<bullet> (y - x) \<le> 0"
then obtain u where u:"u>0" "u\<le>1" "dist (x + u *s (y - x)) a < dist x a" using closer_point_lemma[of a x y] by auto
let ?z = "(1 - u) *s x + u *s 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_sym field_simps) qed
lemma any_closest_point_unique:
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
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 "(a - closest_point s a) \<bullet> (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 "(x - closest_point s x) \<bullet> (closest_point s y - closest_point s x) \<le> 0"
"(y - closest_point s y) \<bullet> (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_def and norm_le
using dot_pos_le[of "(x - closest_point s x) - (y - closest_point s y)"]
by (auto simp add: dot_sym dot_ladd dot_radd) 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)"
apply(rule continuous_at_imp_continuous_on) using continuous_at_closest_point[OF assms] by auto
subsection {* Various point-to-set separating/supporting hyperplane theorems. *}
lemma supporting_hyperplane_closed_point:
assumes "convex s" "closed s" "s \<noteq> {}" "z \<notin> s"
shows "\<exists>a b. \<exists>y\<in>s. a \<bullet> z < b \<and> (a \<bullet> y = b) \<and> (\<forall>x\<in>s. a \<bullet> 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="(y - z) \<bullet> 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 "(y - z) \<bullet> z < (y - z) \<bullet> y" apply(subst diff_less_iff(1)[THEN sym])
unfolding dot_rsub[THEN sym] and dot_pos_lt 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) *s y + u *s x)"
using assms(1)[unfolded convex_alt] and y and `x\<in>s` and `y\<in>s` by auto
assume "\<not> (y - z) \<bullet> y \<le> (y - z) \<bullet> x" then obtain v where
"v>0" "v\<le>1" "dist (y + v *s (x - y)) z < dist y z" using closer_point_lemma[of z y x] by auto
thus False using *[THEN spec[where x=v]] by(auto simp add: dist_sym field_simps)
qed auto
qed
lemma separating_hyperplane_closed_point:
assumes "convex s" "closed s" "z \<notin> s"
shows "\<exists>a b. a \<bullet> z < b \<and> (\<forall>x\<in>s. a \<bullet> 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 _ dot_pos_le[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="(y - z) \<bullet> z + (norm(y - z))\<twosuperior> / 2" in exI)
apply rule defer apply rule proof-
fix x assume "x\<in>s"
have "\<not> 0 < (z - y) \<bullet> (x - y)" apply(rule_tac notI) proof(drule closer_point_lemma)
assume "\<exists>u>0. u \<le> 1 \<and> dist (y + u *s (x - y)) z < dist y z"
then obtain u where "u>0" "u\<le>1" "dist (y + u *s (x - y)) z < dist y z" by auto
thus False using y[THEN bspec[where x="y + u *s (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_sym field_simps) qed
moreover have "0 < norm (y - z) ^ 2" using `y\<in>s` `z\<notin>s` by auto
hence "0 < (y - z) \<bullet> (y - z)" unfolding norm_pow_2 by simp
ultimately show "(y - z) \<bullet> z + (norm (y - z))\<twosuperior> / 2 < (y - z) \<bullet> x"
unfolding norm_pow_2 and dlo_simps(3) by (auto simp add: field_simps dot_sym)
qed(insert `y\<in>s` `z\<notin>s`, auto)
qed
lemma separating_hyperplane_closed_0:
assumes "convex (s::(real^'n::finite) set)" "closed s" "0 \<notin> s"
shows "\<exists>a b. a \<noteq> 0 \<and> 0 < b \<and> (\<forall>x\<in>s. a \<bullet> x > b)"
proof(cases "s={}") guess a using UNIV_witness[where 'a='n] ..
case True have "norm ((basis a)::real^'n::finite) = 1"
using norm_basis and dimindex_ge_1 by auto
thus ?thesis apply(rule_tac x="basis a" in exI, rule_tac x=1 in exI) using True by auto
next case False thus ?thesis using False using separating_hyperplane_closed_point[OF assms]
apply - apply(erule exE)+ unfolding dot_rzero 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::(real^'n::finite) set)" "closed s" "convex t" "compact t" "t \<noteq> {}" "s \<inter> t = {}"
shows "\<exists>a b. (\<forall>x\<in>s. a \<bullet> x < b) \<and> (\<forall>x\<in>t. a \<bullet> 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::"real^'n" 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:"a \<bullet> z < b" "\<forall>x\<in>t. b < a \<bullet> 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 < a \<bullet> 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 + a \<bullet> y < a \<bullet> x" apply- apply(rule,rule) apply(erule_tac x="x - y" in ballE) by auto
def k \<equiv> "rsup ((\<lambda>x. a \<bullet> 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 dot_lneg and neg_less_iff_less proof-
from ab have "((\<lambda>x. a \<bullet> x) ` t) *<= (a \<bullet> y - b)"
apply(erule_tac x=y in ballE) apply(rule setleI) using `y\<in>s` by auto
hence k:"isLub UNIV ((\<lambda>x. a \<bullet> x) ` t) k" unfolding k_def apply(rule_tac rsup) using assms(5) by auto
fix x assume "x\<in>t" thus "a \<bullet> x < (k + b / 2)" using `0<b` and isLubD2[OF k, of "a \<bullet> x"] by auto
next
fix x assume "x\<in>s"
hence "k \<le> a \<bullet> x - b" unfolding k_def apply(rule_tac rsup_le) using assms(5)
unfolding setle_def
using ab[THEN bspec[where x=x]] by auto
thus "k + b / 2 < a \<bullet> x" using `0 < b` by auto
qed
qed
lemma separating_hyperplane_compact_closed:
assumes "convex s" "compact s" "s \<noteq> {}" "convex t" "closed t" "s \<inter> t = {}"
shows "\<exists>a b. (\<forall>x\<in>s. a \<bullet> x < b) \<and> (\<forall>x\<in>t. a \<bullet> x > b)"
proof- obtain a b where "(\<forall>x\<in>t. a \<bullet> x < b) \<and> (\<forall>x\<in>s. b < a \<bullet> 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::real^'n::finite) \<notin> s"
shows "\<exists>a. a \<noteq> 0 \<and> (\<forall>x\<in>s. 0 \<le> a \<bullet> x)"
proof- let ?k = "\<lambda>c. {x::real^'n. 0 \<le> c \<bullet> 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] by auto
then obtain a b where ab:"a \<noteq> 0" "0 < b" "\<forall>x\<in>convex hull c. b < a \<bullet> 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> y \<bullet> x)" apply(rule_tac x="inverse(norm a) *s a" in exI)
using hull_subset[of c convex] unfolding subset_eq and dot_rmult
apply- apply rule defer apply rule apply(rule mult_nonneg_nonneg)
by(auto simp add: dot_sym elim!: ballE)
thus "frontier (cball 0 1) \<inter> \<Inter>f \<noteq> {}" unfolding c(1) frontier_cball dist_def 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_def by auto
thus ?thesis apply(rule_tac x=x in exI) by(auto simp add: dot_sym) qed
lemma separating_hyperplane_sets:
assumes "convex s" "convex (t::(real^'n::finite) set)" "s \<noteq> {}" "t \<noteq> {}" "s \<inter> t = {}"
shows "\<exists>a b. a \<noteq> 0 \<and> (\<forall>x\<in>s. a \<bullet> x \<le> b) \<and> (\<forall>x\<in>t. a \<bullet> 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> a \<bullet> x" using assms(3-5) by auto
hence "\<forall>x\<in>t. \<forall>y\<in>s. a \<bullet> y \<le> a \<bullet> x" apply- apply(rule, rule) apply(erule_tac x="x - y" in ballE) by auto
thus ?thesis apply(rule_tac x=a in exI, rule_tac x="rsup ((\<lambda>x. a \<bullet> x) ` s)" in exI) using `a\<noteq>0`
apply(rule) apply(rule,rule) apply(rule rsup[THEN isLubD2]) prefer 4 apply(rule,rule rsup_le) unfolding setle_def
prefer 4 using assms(3-5) by blast+ qed
subsection {* More convexity generalities. *}
lemma convex_closure: 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 *s xb n + v *s 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: 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) *s x + u *s 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) *s x + u *s y) (min d e)"
hence "(1- u) *s (z - u *s (y - x)) + u *s (z + (1 - u) *s (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_def by(auto simp add: ring_simps)
thus "z \<in> s" using u by (auto simp add: ring_simps) qed(insert u ed(3-4), auto) qed
lemma convex_hull_eq_empty: "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)"
apply(rule hull_minimal, rule image_mono, rule hull_subset) unfolding mem_def
using convex_translation[OF convex_convex_hull, of a s] by assumption
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 *s x) ` s)) \<subseteq> (\<lambda>x. c *s x) ` (convex hull s)"
apply(rule hull_minimal, rule image_mono, rule hull_subset)
unfolding mem_def by(rule convex_scaling, rule convex_convex_hull)
lemma convex_hull_scaling:
"convex hull ((\<lambda>x. c *s x) ` s) = (\<lambda>x. c *s 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 vector_smult_assoc by(auto simp add:image_constant_conv convex_hull_eq_empty)
lemma convex_hull_affinity:
"convex hull ((\<lambda>x. a + c *s x) ` s) = (\<lambda>x. a + c *s x) ` (convex hull s)"
unfolding image_image[THEN sym] convex_hull_scaling convex_hull_translation ..
subsection {* Convex set as intersection of halfspaces. *}
lemma convex_halfspace_intersection:
assumes "closed s" "convex s"
shows "s = \<Inter> {h. s \<subseteq> h \<and> (\<exists>a b. h = {x. a \<bullet> 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. a \<bullet> x \<le> b}) \<longrightarrow> x \<in> xa"
hence "\<forall>a b. s \<subseteq> {x. a \<bullet> x \<le> b} \<longrightarrow> x \<in> {x. a \<bullet> 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 *s 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 vector_smult_lzero
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::real^'n)"
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 *s 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})) *s setsum (\<lambda>x. u x *s 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 real_less_def 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 *s x) = (\<Sum>x\<in>c. u x *s 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 *s x) = - (\<Sum>x\<in>{x \<in> c. 0 > u x}. u x *s 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 vector_smult_assoc[THEN sym] setsum_cmul and z_def
by(auto simp add: setsum_negf vector_smult_lneg 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 vector_smult_assoc[THEN sym] setsum_cmul and z_def using *
by(auto simp add: setsum_negf vector_smult_lneg 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::"(real^'n::finite) set set"
assumes "f hassize n" "n \<ge> CARD('n) + 1"
"\<forall>s\<in>f. convex s" "\<forall>t\<subseteq>f. card t = CARD('n) + 1 \<longrightarrow> \<Inter> t \<noteq> {}"
shows "\<Inter> f \<noteq> {}"
using assms unfolding hassize_def apply(erule_tac conjE) proof(induct n arbitrary: f)
case (Suc n)
show "\<Inter> f \<noteq> {}" apply(cases "n = CARD('n)") apply(rule Suc(4)[rule_format])
unfolding card_Diff_singleton_if[OF Suc(5)] and Suc(6) proof-
assume ng:"n \<noteq> CARD('n)" 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 Suc(5)] and Suc(6)
defer apply(rule Suc(3)[rule_format]) defer apply(rule Suc(4)[rule_format]) using Suc(2,5) 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 Suc(6) using Suc(2,5) 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_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(3) 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(insert dimindex_ge_1, auto) qed(auto)
lemma helly: fixes f::"(real^'n::finite) set set"
assumes "finite f" "card f \<ge> CARD('n) + 1" "\<forall>s\<in>f. convex s"
"\<forall>t\<subseteq>f. card t = CARD('n) + 1 \<longrightarrow> \<Inter> t \<noteq> {}"
shows "\<Inter> f \<noteq>{}"
apply(rule helly_induct) unfolding hassize_def using assms by auto
subsection {* Convex hull is "preserved" by a linear function. *}
lemma convex_hull_linear_image:
assumes "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- show "convex {x. f x \<in> convex hull f ` s}"
unfolding convex_def by(auto simp add: linear_cmul[OF assms] linear_add[OF assms]
convex_convex_hull[unfolded convex_def, rule_format]) next
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: linear_cmul[OF assms, THEN sym] linear_add[OF assms, THEN sym])
qed auto
lemma in_convex_hull_linear_image:
assumes "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:
assumes "compact s" "0 \<in> s" "x \<noteq> 0"
obtains u where "0 \<le> u" "(u *s x) \<in> frontier s" "\<forall>v>u. (v *s 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 *s x)}"
have A:"?A = (\<lambda>u. dest_vec1 u *s x) ` {0 .. vec1 (b / norm x)}"
unfolding image_image[of "\<lambda>u. u *s x" "\<lambda>x. dest_vec1 x", THEN sym]
unfolding dest_vec1_inverval vec1_dest_vec1 by auto
have "compact ?A" unfolding A apply(rule compact_continuous_image, rule continuous_at_imp_continuous_on)
apply(rule, rule continuous_vmul) unfolding o_def vec1_dest_vec1 apply(rule continuous_at_id) by(rule compact_interval)
moreover have "{y. \<exists>u\<ge>0. u \<le> b / norm x \<and> y = u *s x} \<inter> s \<noteq> {}" apply(rule not_disjointI[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 *s 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 *s 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`] and norm_mul by auto
hence "norm (v *s 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) unfolding norm_mul using `u\<ge>0` `norm x > 0` `v>u` by(auto simp add:field_simps)
} note u_max = this
have "u *s x \<in> frontier s" unfolding frontier_straddle apply(rule,rule,rule) apply(rule_tac x="u *s x" in bexI) unfolding obt(3)[THEN sym]
prefer 3 apply(rule_tac x="(u + (e / 2) / norm x) *s x" in exI) apply(rule, rule) proof-
fix e assume "0 < e" and as:"(u + e / 2 / norm x) *s 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_def obt(3))
thus ?thesis apply(rule_tac that[of u]) apply(rule obt(1), assumption)
apply(rule,rule,rule ccontr) apply(rule u_max) by auto qed
lemma starlike_compact_projective:
assumes "compact s" "cball (0::real^'n::finite) 1 \<subseteq> s "
"\<forall>x\<in>s. \<forall>u. 0 \<le> u \<and> u < 1 \<longrightarrow> (u *s x) \<in> (s - frontier s )"
shows "s homeomorphic (cball (0::real^'n::finite) 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::real^'n. inverse (norm x) *s 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) unfolding o_def
apply(rule continuous_at_inv[unfolded o_def]) unfolding continuous_at_vec1_range[unfolded o_def]
apply(rule,rule) apply(rule_tac x=e in exI) apply(rule,assumption,rule,rule)
proof- fix e x y assume "0 < e" "norm (y - x::real^'n) < e"
thus "\<bar>norm y - norm x\<bar> < e" using norm_triangle_ineq3[of y x] by auto
qed(auto intro!:continuous_at_id)
def sphere \<equiv> "{x::real^'n. norm x = 1}"
have pi:"\<And>x. x \<noteq> 0 \<Longrightarrow> pi x \<in> sphere" "\<And>x u. u>0 \<Longrightarrow> pi (u *s 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 *s 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 *s x \<in> frontier s" "\<forall>w>v. w *s 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 *s x"], THEN spec[where x="inverse v"]]
using v and x and fs unfolding inverse_less_1_iff by auto qed
show "u *s 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 *s 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 *s x \<in> frontier s" "\<forall>v>u. v *s 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 *s 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 *s ((inverse (norm y)) *s y)" unfolding as(3)[unfolded pi_def, THEN sym] by auto
from nor have y:"y = norm y *s ((inverse (norm x)) *s 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::real^'n) 1"
have "norm x *s 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_def)
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 *s 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 *s (?a *s surf (pi x)) = x" apply(rule_tac vector_mul_lcancel_imp[OF invn]) unfolding pi_def 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) *s surf (pi x))"
unfolding norm_mul 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) *s 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))) *s x) \<le> 1" unfolding dist_def
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))) *s x" in bexI)
apply(subst injpi[THEN sym]) unfolding norm_mul 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 *s 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::"real^'n" assume as:"x \<in> cball 0 1"
thus "continuous (at x) (\<lambda>x. norm x *s surf (pi x))" proof(cases "x=0")
case False thus ?thesis apply(rule_tac continuous_mul, rule_tac continuous_at_vec1_norm[THEN spec])
using cont_surfpi unfolding continuous_on_eq_continuous_at[OF open_delete[OF open_UNIV]] o_def by auto
next guess a using UNIV_witness[where 'a = 'n] ..
obtain B where B:"\<forall>x\<in>s. norm x \<le> B" using compact_imp_bounded[OF assms(1)] unfolding bounded_def by auto
hence "B > 0" using assms(2) unfolding subset_eq apply(erule_tac x="basis a" in ballE) defer apply(erule_tac x="basis a" in ballE)
unfolding Ball_def mem_cball dist_def 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_0 vector_smult_lzero dist_def diff_0_right norm_mul abs_norm_cancel proof-
fix e and x::"real^'n" 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 *s 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 *s surf (pi x) = norm y *s 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::"(real^'n::finite) set"
assumes "convex s" "compact s" "cball 0 1 \<subseteq> s"
shows "s homeomorphic (cball (0::real^'n) 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 *s x \<in> interior s" unfolding interior_def mem_Collect_eq
apply(rule_tac x="ball (u *s 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 *s x) y < 1 - u"
hence "inverse (1 - u) *s (y - u *s x) \<in> s"
using assms(3) apply(erule_tac subsetD) unfolding mem_cball dist_sym dist_def
unfolding group_add_class.diff_0 group_add_class.diff_0_right norm_minus_cancel norm_mul
apply (rule mult_left_le_imp_le[of "1 - u"])
unfolding class_semiring.mul_a using `u<1` by auto
thus "y \<in> s" using assms(1)[unfolded convex_def, rule_format, of "inverse(1 - u) *s (y - u *s x)" x "1 - u" u]
using as unfolding vector_smult_assoc by auto qed auto
thus "u *s x \<in> s - frontier s" using frontier_def and interior_subset by auto qed
lemma homeomorphic_convex_compact_cball: fixes e::real and s::"(real^'n::finite) set"
assumes "convex s" "compact s" "interior s \<noteq> {}" "0 < e"
shows "s homeomorphic (cball (b::real^'n::finite) 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::real^'n"
have "cball ?n 1 \<subseteq> (\<lambda>x. inverse d *s (x - a)) ` s"
apply(rule, rule_tac x="d *s x + a" in image_eqI) defer
apply(rule d[unfolded subset_eq, rule_format]) using `d>0` unfolding mem_cball dist_def
by(auto simp add: mult_right_le_one_le)
hence "(\<lambda>x. inverse d *s (x - a)) ` s homeomorphic cball ?n 1"
using homeomorphic_convex_compact_lemma[of "(\<lambda>x. ?d *s -a + ?d *s x) ` s", OF convex_affinity compact_affinity]
using assms(1,2) by(auto simp add: uminus_add_conv_diff)
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 *s -a"]])
using `d>0` `e>0` by(auto simp add: uminus_add_conv_diff) qed
lemma homeomorphic_convex_compact: fixes s::"(real^'n::finite) set" and t::"(real^'n) 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::real^'n \<Rightarrow> real) = {xy. fstcart xy \<in> s \<and> f(fstcart xy) \<le> dest_vec1 (sndcart xy)}"
lemma mem_epigraph: "(pastecart x (vec1 y)) \<in> epigraph s f \<longleftrightarrow> x \<in> s \<and> f x \<le> y" unfolding epigraph_def by auto
lemma convex_epigraph:
"convex(epigraph s f) \<longleftrightarrow> convex_on s f \<and> convex s"
unfolding convex_def convex_on_def unfolding Ball_def forall_pastecart epigraph_def
unfolding mem_Collect_eq fstcart_pastecart sndcart_pastecart sndcart_add sndcart_cmul fstcart_add fstcart_cmul
unfolding Ball_def[symmetric] unfolding dest_vec1_add dest_vec1_cmul
apply(subst forall_dest_vec1[THEN sym])+ by(meson real_le_refl real_le_trans add_mono mult_left_mono)
lemma convex_epigraphI: assumes "convex_on s f" "convex s"
shows "convex(epigraph s f)" using assms unfolding convex_epigraph by auto
lemma convex_epigraph_convex: "convex s \<Longrightarrow> (convex_on s f \<longleftrightarrow> convex(epigraph s f))"
using convex_epigraph by auto
subsection {* Use this to derive general bound property of convex function. *}
lemma forall_of_pastecart:
"(\<forall>p. P (\<lambda>x. fstcart (p x)) (\<lambda>x. sndcart (p x))) \<longleftrightarrow> (\<forall>x y. P x y)" apply meson
apply(erule_tac x="\<lambda>a. pastecart (x a) (y a)" in allE) unfolding o_def by auto
lemma forall_of_pastecart':
"(\<forall>p. P (fstcart p) (sndcart p)) \<longleftrightarrow> (\<forall>x y. P x y)" apply meson
apply(erule_tac x="pastecart x y" in allE) unfolding o_def by auto
lemma forall_of_dest_vec1: "(\<forall>v. P (\<lambda>x. dest_vec1 (v x))) \<longleftrightarrow> (\<forall>x. P x)"
apply rule apply rule apply(erule_tac x="(vec1 \<circ> x)" in allE) unfolding o_def vec1_dest_vec1 by auto
lemma forall_of_dest_vec1': "(\<forall>v. P (dest_vec1 v)) \<longleftrightarrow> (\<forall>x. P x)"
apply rule apply rule apply(erule_tac x="(vec1 x)" in allE) defer apply rule
apply(erule_tac x="dest_vec1 v" in allE) unfolding o_def vec1_dest_vec1 by auto
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 *s 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 sndcart_setsum[OF finite_atLeastAtMost] fstcart_setsum[OF finite_atLeastAtMost] dest_vec1_setsum[OF finite_atLeastAtMost]
unfolding fstcart_pastecart sndcart_pastecart sndcart_add sndcart_cmul fstcart_add fstcart_cmul
unfolding dest_vec1_add dest_vec1_cmul apply(subst forall_of_pastecart)+ apply(subst forall_of_dest_vec1)+ apply rule
using assms[unfolded convex] apply simp apply(rule,rule,rule)
apply(erule_tac x=k in allE, erule_tac x=u in allE, erule_tac x=x in allE) apply rule apply rule apply rule defer
apply(rule_tac j="\<Sum>i = 1..k. u i * f (x i)" in real_le_trans)
defer apply(rule setsum_mono) apply(erule conjE)+ apply(erule_tac x=i in allE)apply(rule mult_left_mono)
using assms[unfolded convex] by auto
subsection {* Convexity of general and special intervals. *}
lemma is_interval_convex: assumes "is_interval_new s" shows "convex s"
unfolding convex_def apply(rule,rule,rule,rule,rule,rule,rule) proof-
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: ring_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 *s x + v *s y \<in> s" apply- apply(rule assms[unfolded is_interval_new_def, rule_format, OF as(1,2)])
using as(3-) dimindex_ge_1 apply- by(auto simp add: vector_component) qed
lemma is_interval_connected: "is_interval_new s \<Longrightarrow> connected s"
using is_interval_convex convex_connected by auto
lemma convex_interval: "convex {a .. b}" "convex {a<..<b::real^'n::finite}"
apply(rule_tac[!] is_interval_convex) using is_interval_interval_new by auto
subsection {* On real^1, is_interval, convex and connected are all equivalent. *}
lemma is_interval_1:
"is_interval_new 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_new_def dest_vec1_def forall_1 by auto
lemma is_interval_connected_1: "is_interval_new 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. basis 1 \<bullet> z < dest_vec1 x} " and ?halfr = "{z. basis 1 \<bullet> 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: basis_component field_simps dest_vec1_eq[unfolded dest_vec1_def One_nat_def] dest_vec1_def) }
moreover have "a\<in>?halfl" "b\<in>?halfr" using * by (auto simp add: basis_component field_simps dest_vec1_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) apply(rule, rule, rule ccontr)
by(auto simp add: basis_component field_simps) qed
lemma is_interval_convex_1:
"is_interval_new s \<longleftrightarrow> convex (s::(real^1) set)"
using is_interval_convex convex_connected is_interval_connected_1 by auto
lemma convex_connected_1:
"connected s \<longleftrightarrow> convex (s::(real^1) set)"
using is_interval_convex convex_connected is_interval_connected_1 by auto
subsection {* Another intermediate value theorem formulation. *}
lemma ivt_increasing_component_on_1: fixes f::"real^1 \<Rightarrow> real^'n::finite"
assumes "dest_vec1 a \<le> dest_vec1 b" "continuous_on {a .. b} f" "(f a)$k \<le> y" "y \<le> (f b)$k"
shows "\<exists>x\<in>{a..b}. (f x)$k = y"
proof- have "f a \<in> f ` {a..b}" "f b \<in> f ` {a..b}" apply(rule_tac[!] imageI)
using assms(1) by(auto simp add: vector_less_eq_def dest_vec1_def)
thus ?thesis using connected_ivt_component[of "f ` {a..b}" "f a" "f b" k y]
using connected_continuous_image[OF assms(2) convex_connected[OF convex_interval(1)]]
using assms by(auto intro!: imageI) qed
lemma ivt_increasing_component_1: fixes f::"real^1 \<Rightarrow> real^'n::finite"
assumes "dest_vec1 a \<le> dest_vec1 b"
"\<forall>x\<in>{a .. b}. continuous (at x) f" "f a$k \<le> y" "y \<le> f b$k"
shows "\<exists>x\<in>{a..b}. (f x)$k = y"
apply(rule ivt_increasing_component_on_1) using assms using continuous_at_imp_continuous_on by auto
lemma ivt_decreasing_component_on_1: fixes f::"real^1 \<Rightarrow> real^'n::finite"
assumes "dest_vec1 a \<le> dest_vec1 b" "continuous_on {a .. b} f" "(f b)$k \<le> y" "y \<le> (f a)$k"
shows "\<exists>x\<in>{a..b}. (f x)$k = y"
apply(subst neg_equal_iff_equal[THEN sym]) unfolding vector_uminus_component[THEN sym]
apply(rule ivt_increasing_component_on_1) using assms using continuous_on_neg
by(auto simp add:vector_uminus_component)
lemma ivt_decreasing_component_1: fixes f::"real^1 \<Rightarrow> real^'n::finite"
assumes "dest_vec1 a \<le> dest_vec1 b" "\<forall>x\<in>{a .. b}. continuous (at x) f" "f b$k \<le> y" "y \<le> f a$k"
shows "\<exists>x\<in>{a..b}. (f x)$k = y"
apply(rule ivt_decreasing_component_on_1) using assms using continuous_at_imp_continuous_on by auto
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 *s 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::real^'n::finite .. 1} = convex hull {x. \<forall>i. (x$i = 0) \<or> (x$i = 1)}" (is "?int = convex hull ?points")
proof- have 01:"{0,1} \<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::real^'n .. 1}" "n \<le> CARD('n)" "card {i. x$i \<noteq> 0} \<le> n"
hence "x\<in>convex hull ?points" proof(induct n arbitrary: x)
case 0 hence "x = 0" apply(subst Cart_eq) apply rule by auto
thus "x\<in>convex hull ?points" using 01 by auto
next
case (Suc n) show "x\<in>convex hull ?points" proof(cases "{i. x$i \<noteq> 0} = {}")
case True hence "x = 0" unfolding Cart_eq by auto
thus "x\<in>convex hull ?points" using 01 by auto
next
case False def xi \<equiv> "Min ((\<lambda>i. x$i) ` {i. x$i \<noteq> 0})"
have "xi \<in> (\<lambda>i. x$i) ` {i. x$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" by auto
have i:"\<And>j. 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 simp add: Cart_lambda_beta)
show ?thesis proof(cases "x$i=1")
case True have "\<forall>j\<in>{i. x$i \<noteq> 0}. x$j = 1" apply(rule, rule ccontr) unfolding mem_Collect_eq proof-
fix j assume "x $ j \<noteq> 0" "x $ j \<noteq> 1"
hence j:"x$j \<in> {0<..<1}" using Suc(2) by(auto simp add: vector_less_eq_def elim!:allE[where x=j])
hence "x$j \<in> op $ x ` {i. x $ i \<noteq> 0}" 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: vector_less_eq_def elim!:ballE[where x=j]) qed
thus "x\<in>convex hull ?points" apply(rule_tac hull_subset[unfolded subset_eq, rule_format])
by(auto simp add: Cart_lambda_beta)
next let ?y = "\<lambda>j. if x$j = 0 then 0 else (x$j - x$i) / (1 - x$i)"
case False hence *:"x = x$i *s (\<chi> j. if x$j = 0 then 0 else 1) + (1 - x$i) *s (\<chi> j. ?y j)" unfolding Cart_eq
by(auto simp add: Cart_lambda_beta vector_add_component vector_smult_component vector_minus_component field_simps)
{ fix j have "x$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] by(auto simp add:field_simps Cart_lambda_beta)
hence "0 \<le> ?y j \<and> ?y j \<le> 1" by auto }
moreover have "i\<in>{j. x$j \<noteq> 0} - {j. ((\<chi> j. ?y j)::real^'n) $ j \<noteq> 0}" using i01 by(auto simp add: Cart_lambda_beta)
hence "{j. x$j \<noteq> 0} \<noteq> {j. ((\<chi> j. ?y j)::real^'n::finite) $ j \<noteq> 0}" by auto
hence **:"{j. ((\<chi> j. ?y j)::real^'n::finite) $ j \<noteq> 0} \<subset> {j. x$j \<noteq> 0}" apply - apply rule by(auto simp add: Cart_lambda_beta)
have "card {j. ((\<chi> j. ?y j)::real^'n) $ 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 simp add: Cart_lambda_beta)
qed qed qed } note * = this
show ?thesis apply rule defer apply(rule hull_minimal) unfolding subset_eq prefer 3 apply rule
apply(rule_tac n2="CARD('n)" in *) prefer 3 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: vector_less_eq_def 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 .. 1::real^'n::finite} = convex hull s"
apply(rule that[of "{x::real^'n::finite. \<forall>i. x$i=0 \<or> x$i=1}"])
apply(rule finite_subset[of _ "(\<lambda>s. (\<chi> i. if i\<in>s then 1::real else 0)::real^'n::finite) ` UNIV"])
prefer 3 apply(rule unit_interval_convex_hull) apply rule unfolding mem_Collect_eq proof-
fix x::"real^'n" assume as:"\<forall>i. x $ i = 0 \<or> x $ i = 1"
show "x \<in> (\<lambda>s. \<chi> i. if i \<in> s then 1 else 0) ` UNIV" apply(rule image_eqI[where x="{i. x$i = 1}"])
unfolding Cart_eq using as by(auto simp add:Cart_lambda_beta) qed auto
subsection {* Hence any cube (could do any nonempty interval). *}
lemma cube_convex_hull:
assumes "0 < d" obtains s::"(real^'n::finite) set" where "finite s" "{x - (\<chi> i. d) .. x + (\<chi> i. d)} = convex hull s" proof-
let ?d = "(\<chi> i. d)::real^'n"
have *:"{x - ?d .. x + ?d} = (\<lambda>y. x - ?d + (2 * d) *s y) ` {0 .. 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::'n have "x $ i \<le> d + y $ i" "y $ i \<le> d + x $ i" using as[unfolded mem_interval, THEN spec[where x=i]]
by(auto simp add: vector_component)
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 right_inverse)
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) *s (y - (x - ?d)) \<in> {0..1}" unfolding mem_interval using assms
by(auto simp add: Cart_eq vector_component_simps field_simps)
thus "\<exists>z\<in>{0..1}. y = x - ?d + (2 * d) *s z" apply- apply(rule_tac x="inverse (2 * d) *s (y - (x - ?d))" in bexI)
using assms by(auto simp add: Cart_eq vector_less_eq_def Cart_lambda_beta)
next
fix y z assume as:"z\<in>{0..1}" "y = x - ?d + (2*d) *s z"
have "\<And>i. 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 simp add: vector_component_simps Cart_eq)
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: vector_component_simps Cart_eq) qed
obtain s where "finite s" "{0..1::real^'n} = convex hull s" using unit_cube_convex_hull by auto
thus ?thesis apply(rule_tac that[of "(\<lambda>y. x - ?d + (2 * d) *s y)` s"]) unfolding * and convex_hull_affinity by auto qed
subsection {* Bounded convex function on open set is continuous. *}
lemma convex_on_bounded_continuous:
assumes "open s" "convex_on s f" "\<forall>x\<in>s. abs(f x) \<le> b"
shows "continuous_on s (vec1 o f)"
apply(rule continuous_at_imp_continuous_on) unfolding continuous_at_vec1_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_def
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 *s (y - x)"
have "w\<in>s" unfolding w_def apply(rule k[unfolded subset_eq,rule_format]) unfolding mem_cball dist_def
unfolding t_def using `k>0` by(auto simp add: norm_mul simp del: vector_ssub_ldistrib)
have "(1 / t) *s x + - x + ((t - 1) / t) *s x = (1 / t - 1 + (t - 1) / t) *s x" by auto
also have "\<dots> = 0" using `t>0` by(auto simp add:field_simps simp del:vector_sadd_rdistrib)
finally have w:"(1 / t) *s w + ((t - 1) / t) *s x = y" unfolding w_def using False and `t>0` by auto
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 *s (y - x)"
have "w\<in>s" unfolding w_def apply(rule k[unfolded subset_eq,rule_format]) unfolding mem_cball dist_def
unfolding t_def using `k>0` by(auto simp add: norm_mul simp del: vector_ssub_ldistrib)
have "(1 / (1 + t)) *s x + (t / (1 + t)) *s x = (1 / (1 + t) + t / (1 + t)) *s x" by auto
also have "\<dots>=x" using `t>0` by (auto simp add:field_simps simp del:vector_sadd_rdistrib)
finally have w:"(1 / (1+t)) *s w + (t / (1 + t)) *s y = x" unfolding w_def using False and `t>0` by auto
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 real_divide_def 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:
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 *s x - y"
have *:"x - (2 *s x - y) = y - x" by vector
have z:"z\<in>cball x e" using y unfolding z_def mem_cball dist_def * by(auto simp add: norm_minus_commute)
have "(1 / 2) *s y + (1 / 2) *s z = x" unfolding z_def by auto
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
thus "\<bar>f y\<bar> \<le> b + 2 * \<bar>f x\<bar>" using dist_pos_le[of x y] by auto qed
subsection {* Hence a convex function on an open set is continuous. *}
lemma convex_on_continuous:
assumes "open (s::(real^'n::finite) set)" "convex_on s f"
shows "continuous_on s (vec1 \<circ> f)"
unfolding continuous_on_eq_continuous_at[OF assms(1)] proof
note dimge1 = dimindex_ge_1[where 'a='n]
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 CARD('n)"
have "0 < d" unfolding d_def using `e>0` dimge1 by(rule_tac divide_pos_pos, auto)
let ?d = "(\<chi> i. d)::real^'n"
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:vector_component_simps)
hence "c\<noteq>{}" apply(rule_tac ccontr) using c by(auto simp add:convex_hull_empty)
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) (UNIV::'n set)" unfolding setsum_constant d_def using dimge1
by (metis card_enum field_simps d_def not_one_le_zero of_nat_le_iff real_eq_of_nat real_of_nat_1)
show "dist x z \<le> e" unfolding dist_def 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:field_simps vector_component_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 using real_dimindex_ge_1 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::'n have "x $ i - d \<le> y $ i" "y $ i \<le> x $ i + d"
using order_trans[OF component_le_norm y[unfolded mem_cball dist_def], of i] by(auto simp add: vector_component) }
thus "f y \<le> k" apply(rule_tac k[rule_format]) unfolding mem_cball mem_interval dist_def
by(auto simp add: vector_component_simps) qed
hence "continuous_on (ball x d) (vec1 \<circ> f)" apply(rule_tac convex_on_bounded_continuous)
apply(rule open_ball, rule convex_on_subset[OF conv], rule ball_subset_cball) by auto
thus "continuous (at x) (vec1 \<circ> 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 b = (inverse (2::real)) *s (a + b)"
definition "open_segment a b = {(1 - u) *s a + u *s b | u::real. 0 < u \<and> u < 1}"
definition "closed_segment a b = {(1 - u) *s a + u *s 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 vector_add_ldistrib unfolding vector_sadd_rdistrib[THEN sym] by auto
lemma midpoint_sym: "midpoint a b = midpoint b a" unfolding midpoint_def by auto
lemma dist_midpoint:
"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::real^'n::finite. 2 *s x = - y \<Longrightarrow> norm x = (norm y) / 2" unfolding equation_minus_iff by auto
have **:"\<And>x y::real^'n::finite. 2 *s x = y \<Longrightarrow> norm x = (norm y) / 2" by auto
show ?t1 unfolding midpoint_def dist_def apply (rule **) by(auto,vector)
show ?t2 unfolding midpoint_def dist_def apply (rule *) by(auto,vector)
show ?t3 unfolding midpoint_def dist_def apply (rule *) by(auto,vector)
show ?t4 unfolding midpoint_def dist_def apply (rule **) by(auto,vector) qed
lemma midpoint_eq_endpoint:
"midpoint a b = a \<longleftrightarrow> a = (b::real^'n::finite)"
"midpoint a b = b \<longleftrightarrow> a = b"
unfolding dist_eq_0[THEN sym] dist_midpoint 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:
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:
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
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::real^'n::finite) \<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_sym) next
case False hence Fal:"norm (a - b) \<noteq> 0" and Fal2: "norm (a - b) > 0" by auto
have *:"\<And>u. a - ((1 - u) *s a + u *s b) = u *s (a - b)" by auto
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) *s a + u *s b" "0 \<le> u" "u \<le> 1"
hence *:"a - x = u *s (a - b)" "x - b = (1 - u) *s (a - b)"
unfolding as(1) by(auto simp add:field_simps)
show "norm (a - x) *s (x - b) = norm (x - b) *s (a - x)"
unfolding norm_minus_commute[of x a] * norm_mul Cart_eq using as(2,3)
by(auto simp add: vector_component_simps 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_def] norm_ge_zero by auto
thus "\<exists>u. x = (1 - u) *s a + u *s b \<and> 0 \<le> u \<and> u \<le> 1" apply(rule_tac x="dist a x / dist a b" in exI)
unfolding dist_def Cart_eq apply- apply rule defer apply(rule, rule divide_nonneg_pos) prefer 4 proof rule
fix i::'n have "((1 - norm (a - x) / norm (a - b)) *s a + (norm (a - x) / norm (a - b)) *s b) $ i =
((norm (a - b) - norm (a - x)) * (a $ i) + norm (a - x) * (b $ i)) / norm (a - b)"
using Fal by(auto simp add:vector_component_simps field_simps)
also have "\<dots> = x$i" apply(rule divide_eq_imp[OF Fal])
unfolding as[unfolded dist_def] using as[unfolded dist_triangle_eq Cart_eq,rule_format, of i]
by(auto simp add:field_simps vector_component_simps)
finally show "x $ i = ((1 - norm (a - x) / norm (a - b)) *s a + (norm (a - x) / norm (a - b)) *s b) $ i" by auto
qed(insert Fal2, auto) qed qed
lemma between_midpoint:
"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) *s z \<Longrightarrow> y = (1/2) *s z \<Longrightarrow> norm z = norm x + norm y" by auto
show ?t1 ?t2 unfolding between midpoint_def dist_def apply(rule_tac[!] *)
by(auto simp add:field_simps Cart_eq vector_component_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:
assumes "convex s" "c \<in> interior s" "x \<in> s" "0 < e" "e \<le> 1"
shows "x - e *s (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 *s (x - c)) y < e * d"
have *:"y = (1 - (1 - e)) *s ((1 / e) *s y - ((1 - e) / e) *s x) + (1 - e) *s x" using `e>0` by auto
have "dist c ((1 / e) *s y - ((1 - e) / e) *s x) = abs(1/e) * norm (e *s c - y + (1 - e) *s x)"
unfolding dist_def unfolding norm_mul[THEN sym] apply(rule norm_eqI) using `e>0`
by(auto simp add:vector_component_simps Cart_eq field_simps)
also have "\<dots> = abs(1/e) * norm (x - e *s (x - c) - y)" by(auto intro!:norm_eqI)
also have "\<dots> < d" using as[unfolded dist_def] and `e>0`
by(auto simp add:pos_divide_less_eq[OF `e>0`] real_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:
assumes "convex s" "c \<in> interior s" "x \<in> closure s" "0 < e" "e \<le> 1"
shows "x - e *s (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_def 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) *s (x - y)"
have *:"x - e *s (x - c) = y - e *s (y - z)" unfolding z_def using `e>0` by auto
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_def using y and assms(4,5)
by(auto simp del:vector_ssub_ldistrib 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 *s 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\<in>UNIV}) =
{x::real^'n::finite . (\<forall>i. 0 \<le> x$i) \<and> setsum (\<lambda>i. x$i) UNIV \<le> 1 }" (is "convex hull (insert 0 ?p) = ?s")
proof- let ?D = "UNIV::'n set"
have "0\<notin>?p" by(auto simp add: basis_nonzero)
have "{(basis i)::real^'n |i. i \<in> ?D} = basis ` ?D" by auto
note sumbas = this setsum_reindex[OF basis_inj, unfolded o_def]
show ?thesis unfolding simplex[OF finite_stdbasis `0\<notin>?p`] apply(rule set_ext) unfolding mem_Collect_eq apply rule
apply(erule exE, (erule conjE)+) apply(erule_tac[2] conjE)+ proof-
fix x::"real^'n" and u assume as: "\<forall>x\<in>{basis i |i. i \<in>?D}. 0 \<le> u x" "setsum u {basis i |i. i \<in> ?D} \<le> 1" "(\<Sum>x\<in>{basis i |i. i \<in>?D}. u x *s x) = x"
have *:"\<forall>i. u (basis i) = x$i" using as(3) unfolding sumbas and basis_expansion_unique by auto
hence **:"setsum u {basis i |i. i \<in> ?D} = setsum (op $ x) ?D" unfolding sumbas by(rule_tac setsum_cong, auto)
show " (\<forall>i. 0 \<le> x $ i) \<and> setsum (op $ x) ?D \<le> 1" apply - proof(rule,rule)
fix i::'n show "0 \<le> x$i" unfolding *[rule_format,of i,THEN sym] apply(rule_tac as(1)[rule_format]) by auto
qed(insert as(2)[unfolded **], auto)
next fix x::"real^'n" assume as:"\<forall>i. 0 \<le> x $ i" "setsum (op $ x) ?D \<le> 1"
show "\<exists>u. (\<forall>x\<in>{basis i |i. i \<in> ?D}. 0 \<le> u x) \<and> setsum u {basis i |i. i \<in> ?D} \<le> 1 \<and> (\<Sum>x\<in>{basis i |i. i \<in> ?D}. u x *s x) = x"
apply(rule_tac x="\<lambda>y. y \<bullet> x" in exI) apply(rule,rule) unfolding mem_Collect_eq apply(erule exE) using as(1) apply(erule_tac x=i in allE)
unfolding sumbas using as(2) and basis_expansion_unique by(auto simp add:dot_basis) qed qed
lemma interior_std_simplex:
"interior (convex hull (insert 0 { basis i| i. i\<in>UNIV})) =
{x::real^'n::finite. (\<forall>i. 0 < x$i) \<and> setsum (\<lambda>i. x$i) UNIV < 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::"real^'n" and e assume "0<e" and as:"\<forall>xa. dist x xa < e \<longrightarrow> (\<forall>x. 0 \<le> xa $ x) \<and> setsum (op $ xa) UNIV \<le> 1"
show "(\<forall>xa. 0 < x $ xa) \<and> setsum (op $ x) UNIV < 1" apply(rule,rule) proof-
fix i::'n show "0 < x $ i" using as[THEN spec[where x="x - (e / 2) *s basis i"]] and `e>0`
unfolding dist_def by(auto simp add: norm_basis vector_component_simps basis_component elim:allE[where x=i])
next guess a using UNIV_witness[where 'a='n] ..
have **:"dist x (x + (e / 2) *s basis a) < e" using `e>0` and norm_basis[of a]
unfolding dist_def by(auto simp add: vector_component_simps basis_component intro!: mult_strict_left_mono_comm)
have "\<And>i. (x + (e / 2) *s basis a) $ i = x$i + (if i = a then e/2 else 0)" by(auto simp add:vector_component_simps)
hence *:"setsum (op $ (x + (e / 2) *s basis a)) UNIV = setsum (\<lambda>i. x$i + (if a = i then e/2 else 0)) UNIV" by(rule_tac setsum_cong, auto)
have "setsum (op $ x) UNIV < setsum (op $ (x + (e / 2) *s basis a)) UNIV" unfolding * setsum_addf
using `0<e` dimindex_ge_1 by(auto simp add: setsum_delta')
also have "\<dots> \<le> 1" using ** apply(drule_tac as[rule_format]) by auto
finally show "setsum (op $ x) UNIV < 1" by auto qed
next
fix x::"real^'n::finite" assume as:"\<forall>i. 0 < x $ i" "setsum (op $ x) UNIV < 1"
guess a using UNIV_witness[where 'a='b] ..
let ?d = "(1 - setsum (op $ x) UNIV) / real (CARD('n))"
have "Min ((op $ x) ` UNIV) > 0" apply(rule Min_grI) using as(1) dimindex_ge_1 by auto
moreover have"?d > 0" apply(rule divide_pos_pos) using as(2) using dimindex_ge_1 by(auto simp add: Suc_le_eq)
ultimately show "\<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> (\<forall>i. 0 \<le> y $ i) \<and> setsum (op $ y) UNIV \<le> 1"
apply(rule_tac x="min (Min ((op $ x) ` UNIV)) ?D" in exI) apply rule defer apply(rule,rule) proof-
fix y assume y:"dist x y < min (Min (op $ x ` UNIV)) ?d"
have "setsum (op $ y) UNIV \<le> setsum (\<lambda>i. x$i + ?d) UNIV" proof(rule setsum_mono)
fix i::'n have "abs (y$i - x$i) < ?d" apply(rule le_less_trans) using component_le_norm[of "y - x" i]
using y[unfolded min_less_iff_conj dist_def, THEN conjunct2] by(auto simp add:vector_component_simps 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 using dimindex_ge_1 by(auto simp add: Suc_le_eq)
finally show "(\<forall>i. 0 \<le> y $ i) \<and> setsum (op $ y) UNIV \<le> 1" apply- proof(rule,rule)
fix i::'n have "norm (x - y) < x$i" using y[unfolded min_less_iff_conj dist_def, THEN conjunct1]
using Min_gr_iff[of "op $ x ` dimset x"] dimindex_ge_1 by auto
thus "0 \<le> y$i" using component_le_norm[of "x - y" i] and as(1)[rule_format, of i] by(auto simp add: vector_component_simps)
qed auto qed auto qed
lemma interior_std_simplex_nonempty: obtains a::"real^'n::finite" where
"a \<in> interior(convex hull (insert 0 {basis i | i . i \<in> UNIV}))" proof-
let ?D = "UNIV::'n set" let ?a = "setsum (\<lambda>b. inverse (2 * real CARD('n)) *s b) {(basis i) | i. i \<in> ?D}"
have *:"{basis i | i. i \<in> ?D} = basis ` ?D" by auto
{ fix i have "?a $ i = inverse (2 * real CARD('n))"
unfolding setsum_component vector_smult_component and * and setsum_reindex[OF basis_inj] and o_def
apply(rule trans[of _ "setsum (\<lambda>j. if i = j then inverse (2 * real CARD('n)) else 0) ?D"]) apply(rule setsum_cong2)
unfolding setsum_delta'[OF finite_UNIV[where 'a='n]] and real_dimindex_ge_1[where 'n='n] by(auto simp add: basis_component[of i]) }
note ** = this
show ?thesis apply(rule that[of ?a]) unfolding interior_std_simplex mem_Collect_eq proof(rule,rule)
fix i::'n show "0 < ?a $ i" unfolding ** using dimindex_ge_1 by(auto simp add: Suc_le_eq) next
have "setsum (op $ ?a) ?D = setsum (\<lambda>i. inverse (2 * real CARD('n))) ?D" by(rule setsum_cong2, rule **)
also have "\<dots> < 1" unfolding setsum_constant card_enum real_eq_of_nat real_divide_def[THEN sym] by (auto simp add:field_simps)
finally show "setsum (op $ ?a) ?D < 1" by auto qed qed
subsection {* Paths. *}
definition "path (g::real^1 \<Rightarrow> real^'n::finite) \<longleftrightarrow> continuous_on {0 .. 1} g"
definition "pathstart (g::real^1 \<Rightarrow> real^'n) = g 0"
definition "pathfinish (g::real^1 \<Rightarrow> real^'n) = g 1"
definition "path_image (g::real^1 \<Rightarrow> real^'n) = g ` {0 .. 1}"
definition "reversepath (g::real^1 \<Rightarrow> real^'n) = (\<lambda>x. g(1 - x))"
definition joinpaths:: "(real^1 \<Rightarrow> real^'n) \<Rightarrow> (real^1 \<Rightarrow> real^'n) \<Rightarrow> (real^1 \<Rightarrow> real^'n)" (infixr "+++" 75)
where "joinpaths g1 g2 = (\<lambda>x. if dest_vec1 x \<le> ((1 / 2)::real) then g1 (2 *s x) else g2(2 *s x - 1))"
definition "simple_path (g::real^1 \<Rightarrow> real^'n) \<longleftrightarrow>
(\<forall>x\<in>{0..1}. \<forall>y\<in>{0..1}. g x = g y \<longrightarrow> x = y \<or> x = 0 \<and> y = 1 \<or> x = 1 \<and> y = 0)"
definition "injective_path (g::real^1 \<Rightarrow> real^'n) \<longleftrightarrow>
(\<forall>x\<in>{0..1}. \<forall>y\<in>{0..1}. g x = g y \<longrightarrow> x = y)"
subsection {* Some lemmas about these concepts. *}
lemma injective_imp_simple_path:
"injective_path g \<Longrightarrow> simple_path g"
unfolding injective_path_def simple_path_def by auto
lemma path_image_nonempty: "path_image g \<noteq> {}"
unfolding path_image_def image_is_empty interval_eq_empty by auto
lemma pathstart_in_path_image[intro]: "(pathstart g) \<in> path_image g"
unfolding pathstart_def path_image_def apply(rule imageI)
unfolding mem_interval_1 vec_1[THEN sym] dest_vec1_vec by auto
lemma pathfinish_in_path_image[intro]: "(pathfinish g) \<in> path_image g"
unfolding pathfinish_def path_image_def apply(rule imageI)
unfolding mem_interval_1 vec_1[THEN sym] dest_vec1_vec by auto
lemma connected_path_image[intro]: "path g \<Longrightarrow> connected(path_image g)"
unfolding path_def path_image_def apply(rule connected_continuous_image, assumption)
by(rule convex_connected, rule convex_interval)
lemma compact_path_image[intro]: "path g \<Longrightarrow> compact(path_image g)"
unfolding path_def path_image_def apply(rule compact_continuous_image, assumption)
by(rule compact_interval)
lemma reversepath_reversepath[simp]: "reversepath(reversepath g) = g"
unfolding reversepath_def by auto
lemma pathstart_reversepath[simp]: "pathstart(reversepath g) = pathfinish g"
unfolding pathstart_def reversepath_def pathfinish_def by auto
lemma pathfinish_reversepath[simp]: "pathfinish(reversepath g) = pathstart g"
unfolding pathstart_def reversepath_def pathfinish_def by auto
lemma pathstart_join[simp]: "pathstart(g1 +++ g2) = pathstart g1"
unfolding pathstart_def joinpaths_def pathfinish_def by auto
lemma pathfinish_join[simp]:"pathfinish(g1 +++ g2) = pathfinish g2" proof-
have "2 *s 1 - 1 = (1::real^1)" unfolding Cart_eq by(auto simp add:vector_component_simps)
thus ?thesis unfolding pathstart_def joinpaths_def pathfinish_def
unfolding vec_1[THEN sym] dest_vec1_vec by auto qed
lemma path_image_reversepath[simp]: "path_image(reversepath g) = path_image g" proof-
have *:"\<And>g. path_image(reversepath g) \<subseteq> path_image g"
unfolding path_image_def subset_eq reversepath_def Ball_def image_iff apply(rule,rule,erule bexE)
apply(rule_tac x="1 - xa" in bexI) by(auto simp add:vector_less_eq_def vector_component_simps elim!:ballE)
show ?thesis using *[of g] *[of "reversepath g"] unfolding reversepath_reversepath by auto qed
lemma path_reversepath[simp]: "path(reversepath g) \<longleftrightarrow> path g" proof-
have *:"\<And>g. path g \<Longrightarrow> path(reversepath g)" unfolding path_def reversepath_def
apply(rule continuous_on_compose[unfolded o_def, of _ "\<lambda>x. 1 - x"])
apply(rule continuous_on_sub, rule continuous_on_const, rule continuous_on_id)
apply(rule continuous_on_subset[of "{0..1}"], assumption)
by (auto, auto simp add:vector_less_eq_def vector_component_simps elim!:ballE)
show ?thesis using *[of g] *[of "reversepath g"] unfolding reversepath_reversepath by auto qed
lemmas reversepath_simps = path_reversepath path_image_reversepath pathstart_reversepath pathfinish_reversepath
lemma path_join[simp]: assumes "pathfinish g1 = pathstart g2" shows "path (g1 +++ g2) \<longleftrightarrow> path g1 \<and> path g2"
unfolding path_def pathfinish_def pathstart_def apply rule defer apply(erule conjE) proof-
assume as:"continuous_on {0..1} (g1 +++ g2)"
have *:"g1 = (\<lambda>x. g1 (2 *s x)) \<circ> (\<lambda>x. (1/2) *s x)"
"g2 = (\<lambda>x. g2 (2 *s x - 1)) \<circ> (\<lambda>x. (1/2) *s (x + 1))" unfolding o_def by auto
have "op *s (1 / 2) ` {0::real^1..1} \<subseteq> {0..1}" "(\<lambda>x. (1 / 2) *s (x + 1)) ` {(0::real^1)..1} \<subseteq> {0..1}"
unfolding image_smult_interval by (auto, auto simp add:vector_less_eq_def vector_component_simps elim!:ballE)
thus "continuous_on {0..1} g1 \<and> continuous_on {0..1} g2" apply -apply rule
apply(subst *) defer apply(subst *) apply (rule_tac[!] continuous_on_compose)
apply (rule continuous_on_cmul, rule continuous_on_add, rule continuous_on_id, rule continuous_on_const) defer
apply (rule continuous_on_cmul, rule continuous_on_id) apply(rule_tac[!] continuous_on_eq[of _ "g1 +++ g2"]) defer prefer 3
apply(rule_tac[1-2] continuous_on_subset[of "{0 .. 1}"]) apply(rule as, assumption, rule as, assumption)
apply(rule) defer apply rule proof-
fix x assume "x \<in> op *s (1 / 2) ` {0::real^1..1}"
hence "dest_vec1 x \<le> 1 / 2" unfolding image_iff by(auto simp add: vector_component_simps)
thus "(g1 +++ g2) x = g1 (2 *s x)" unfolding joinpaths_def by auto next
fix x assume "x \<in> (\<lambda>x. (1 / 2) *s (x + 1)) ` {0::real^1..1}"
hence "dest_vec1 x \<ge> 1 / 2" unfolding image_iff by(auto simp add: vector_component_simps)
thus "(g1 +++ g2) x = g2 (2 *s x - 1)" proof(cases "dest_vec1 x = 1 / 2")
case True hence "x = (1/2) *s 1" unfolding Cart_eq by(auto simp add: forall_1 vector_component_simps)
thus ?thesis unfolding joinpaths_def using assms[unfolded pathstart_def pathfinish_def] by auto
qed (auto simp add:le_less joinpaths_def) qed
next assume as:"continuous_on {0..1} g1" "continuous_on {0..1} g2"
have *:"{0 .. 1::real^1} = {0.. (1/2)*s 1} \<union> {(1/2) *s 1 .. 1}" by(auto simp add: vector_component_simps)
have **:"op *s 2 ` {0..(1 / 2) *s 1} = {0..1::real^1}" apply(rule set_ext, rule) unfolding image_iff
defer apply(rule_tac x="(1/2)*s x" in bexI) by(auto simp add: vector_component_simps)
have ***:"(\<lambda>x. 2 *s x - 1) ` {(1 / 2) *s 1..1} = {0..1::real^1}"
unfolding image_affinity_interval[of _ "- 1", unfolded diff_def[symmetric]] and interval_eq_empty_1
by(auto simp add: vector_component_simps)
have ****:"\<And>x::real^1. x $ 1 * 2 = 1 \<longleftrightarrow> x = (1/2) *s 1" unfolding Cart_eq by(auto simp add: forall_1 vector_component_simps)
show "continuous_on {0..1} (g1 +++ g2)" unfolding * apply(rule continuous_on_union) apply(rule closed_interval)+ proof-
show "continuous_on {0..(1 / 2) *s 1} (g1 +++ g2)" apply(rule continuous_on_eq[of _ "\<lambda>x. g1 (2 *s x)"]) defer
unfolding o_def[THEN sym] apply(rule continuous_on_compose) apply(rule continuous_on_cmul, rule continuous_on_id)
unfolding ** apply(rule as(1)) unfolding joinpaths_def by(auto simp add: vector_component_simps) next
show "continuous_on {(1/2)*s1..1} (g1 +++ g2)" apply(rule continuous_on_eq[of _ "g2 \<circ> (\<lambda>x. 2 *s x - 1)"]) defer
apply(rule continuous_on_compose) apply(rule continuous_on_sub, rule continuous_on_cmul, rule continuous_on_id, rule continuous_on_const)
unfolding *** o_def joinpaths_def apply(rule as(2)) using assms[unfolded pathstart_def pathfinish_def]
by(auto simp add: vector_component_simps ****) qed qed
lemma path_image_join_subset: "path_image(g1 +++ g2) \<subseteq> (path_image g1 \<union> path_image g2)" proof
fix x assume "x \<in> path_image (g1 +++ g2)"
then obtain y where y:"y\<in>{0..1}" "x = (if dest_vec1 y \<le> 1 / 2 then g1 (2 *s y) else g2 (2 *s y - 1))"
unfolding path_image_def image_iff joinpaths_def by auto
thus "x \<in> path_image g1 \<union> path_image g2" apply(cases "dest_vec1 y \<le> 1/2")
apply(rule_tac UnI1) defer apply(rule_tac UnI2) unfolding y(2) path_image_def using y(1)
by(auto intro!: imageI simp add: vector_component_simps) qed
lemma subset_path_image_join:
assumes "path_image g1 \<subseteq> s" "path_image g2 \<subseteq> s" shows "path_image(g1 +++ g2) \<subseteq> s"
using path_image_join_subset[of g1 g2] and assms by auto
lemma path_image_join:
assumes "path g1" "path g2" "pathfinish g1 = pathstart g2"
shows "path_image(g1 +++ g2) = (path_image g1) \<union> (path_image g2)"
apply(rule, rule path_image_join_subset, rule) unfolding Un_iff proof(erule disjE)
fix x assume "x \<in> path_image g1"
then obtain y where y:"y\<in>{0..1}" "x = g1 y" unfolding path_image_def image_iff by auto
thus "x \<in> path_image (g1 +++ g2)" unfolding joinpaths_def path_image_def image_iff
apply(rule_tac x="(1/2) *s y" in bexI) by(auto simp add: vector_component_simps) next
fix x assume "x \<in> path_image g2"
then obtain y where y:"y\<in>{0..1}" "x = g2 y" unfolding path_image_def image_iff by auto
moreover have *:"y $ 1 = 0 \<Longrightarrow> y = 0" unfolding Cart_eq by auto
ultimately show "x \<in> path_image (g1 +++ g2)" unfolding joinpaths_def path_image_def image_iff
apply(rule_tac x="(1/2) *s (y + 1)" in bexI) using assms(3)[unfolded pathfinish_def pathstart_def]
by(auto simp add: vector_component_simps) qed
lemma not_in_path_image_join:
assumes "x \<notin> path_image g1" "x \<notin> path_image g2" shows "x \<notin> path_image(g1 +++ g2)"
using assms and path_image_join_subset[of g1 g2] by auto
lemma simple_path_reversepath: assumes "simple_path g" shows "simple_path (reversepath g)"
using assms unfolding simple_path_def reversepath_def apply- apply(rule ballI)+
apply(erule_tac x="1-x" in ballE, erule_tac x="1-y" in ballE)
unfolding mem_interval_1 by(auto simp add:vector_component_simps)
lemma simple_path_join_loop:
assumes "injective_path g1" "injective_path g2" "pathfinish g2 = pathstart g1"
"(path_image g1 \<inter> path_image g2) \<subseteq> {pathstart g1,pathstart g2}"
shows "simple_path(g1 +++ g2)"
unfolding simple_path_def proof((rule ballI)+, rule impI) let ?g = "g1 +++ g2"
note inj = assms(1,2)[unfolded injective_path_def, rule_format]
fix x y::"real^1" assume xy:"x \<in> {0..1}" "y \<in> {0..1}" "?g x = ?g y"
show "x = y \<or> x = 0 \<and> y = 1 \<or> x = 1 \<and> y = 0" proof(case_tac "x$1 \<le> 1/2",case_tac[!] "y$1 \<le> 1/2", unfold not_le)
assume as:"x $ 1 \<le> 1 / 2" "y $ 1 \<le> 1 / 2"
hence "g1 (2 *s x) = g1 (2 *s y)" using xy(3) unfolding joinpaths_def dest_vec1_def by auto
moreover have "2 *s x \<in> {0..1}" "2 *s y \<in> {0..1}" using xy(1,2) as
unfolding mem_interval_1 dest_vec1_def by(auto simp add:vector_component_simps)
ultimately show ?thesis using inj(1)[of "2*s x" "2*s y"] by auto
next assume as:"x $ 1 > 1 / 2" "y $ 1 > 1 / 2"
hence "g2 (2 *s x - 1) = g2 (2 *s y - 1)" using xy(3) unfolding joinpaths_def dest_vec1_def by auto
moreover have "2 *s x - 1 \<in> {0..1}" "2 *s y - 1 \<in> {0..1}" using xy(1,2) as
unfolding mem_interval_1 dest_vec1_def by(auto simp add:vector_component_simps)
ultimately show ?thesis using inj(2)[of "2*s x - 1" "2*s y - 1"] by auto
next assume as:"x $ 1 \<le> 1 / 2" "y $ 1 > 1 / 2"
hence "?g x \<in> path_image g1" "?g y \<in> path_image g2" unfolding path_image_def joinpaths_def
using xy(1,2)[unfolded mem_interval_1] by(auto simp add:vector_component_simps intro!: imageI)
moreover have "?g y \<noteq> pathstart g2" using as(2) unfolding pathstart_def joinpaths_def
using inj(2)[of "2 *s y - 1" 0] and xy(2)[unfolded mem_interval_1]
apply(rule_tac ccontr) by(auto simp add:vector_component_simps field_simps Cart_eq)
ultimately have *:"?g x = pathstart g1" using assms(4) unfolding xy(3) by auto
hence "x = 0" unfolding pathstart_def joinpaths_def using as(1) and xy(1)[unfolded mem_interval_1]
using inj(1)[of "2 *s x" 0] by(auto simp add:vector_component_simps)
moreover have "y = 1" using * unfolding xy(3) assms(3)[THEN sym]
unfolding joinpaths_def pathfinish_def using as(2) and xy(2)[unfolded mem_interval_1]
using inj(2)[of "2 *s y - 1" 1] by (auto simp add:vector_component_simps Cart_eq)
ultimately show ?thesis by auto
next assume as:"x $ 1 > 1 / 2" "y $ 1 \<le> 1 / 2"
hence "?g x \<in> path_image g2" "?g y \<in> path_image g1" unfolding path_image_def joinpaths_def
using xy(1,2)[unfolded mem_interval_1] by(auto simp add:vector_component_simps intro!: imageI)
moreover have "?g x \<noteq> pathstart g2" using as(1) unfolding pathstart_def joinpaths_def
using inj(2)[of "2 *s x - 1" 0] and xy(1)[unfolded mem_interval_1]
apply(rule_tac ccontr) by(auto simp add:vector_component_simps field_simps Cart_eq)
ultimately have *:"?g y = pathstart g1" using assms(4) unfolding xy(3) by auto
hence "y = 0" unfolding pathstart_def joinpaths_def using as(2) and xy(2)[unfolded mem_interval_1]
using inj(1)[of "2 *s y" 0] by(auto simp add:vector_component_simps)
moreover have "x = 1" using * unfolding xy(3)[THEN sym] assms(3)[THEN sym]
unfolding joinpaths_def pathfinish_def using as(1) and xy(1)[unfolded mem_interval_1]
using inj(2)[of "2 *s x - 1" 1] by(auto simp add:vector_component_simps Cart_eq)
ultimately show ?thesis by auto qed qed
lemma injective_path_join:
assumes "injective_path g1" "injective_path g2" "pathfinish g1 = pathstart g2"
"(path_image g1 \<inter> path_image g2) \<subseteq> {pathstart g2}"
shows "injective_path(g1 +++ g2)"
unfolding injective_path_def proof(rule,rule,rule) let ?g = "g1 +++ g2"
note inj = assms(1,2)[unfolded injective_path_def, rule_format]
fix x y assume xy:"x \<in> {0..1}" "y \<in> {0..1}" "(g1 +++ g2) x = (g1 +++ g2) y"
show "x = y" proof(cases "x$1 \<le> 1/2", case_tac[!] "y$1 \<le> 1/2", unfold not_le)
assume "x $ 1 \<le> 1 / 2" "y $ 1 \<le> 1 / 2" thus ?thesis using inj(1)[of "2*s x" "2*s y"] and xy
unfolding mem_interval_1 joinpaths_def by(auto simp add:vector_component_simps)
next assume "x $ 1 > 1 / 2" "y $ 1 > 1 / 2" thus ?thesis using inj(2)[of "2*s x - 1" "2*s y - 1"] and xy
unfolding mem_interval_1 joinpaths_def by(auto simp add:vector_component_simps)
next assume as:"x $ 1 \<le> 1 / 2" "y $ 1 > 1 / 2"
hence "?g x \<in> path_image g1" "?g y \<in> path_image g2" unfolding path_image_def joinpaths_def
using xy(1,2)[unfolded mem_interval_1] by(auto simp add:vector_component_simps intro!: imageI)
hence "?g x = pathfinish g1" "?g y = pathstart g2" using assms(4) unfolding assms(3) xy(3) by auto
thus ?thesis using as and inj(1)[of "2 *s x" 1] inj(2)[of "2 *s y - 1" 0] and xy(1,2)
unfolding pathstart_def pathfinish_def joinpaths_def mem_interval_1
by(auto simp add:vector_component_simps Cart_eq forall_1)
next assume as:"x $ 1 > 1 / 2" "y $ 1 \<le> 1 / 2"
hence "?g x \<in> path_image g2" "?g y \<in> path_image g1" unfolding path_image_def joinpaths_def
using xy(1,2)[unfolded mem_interval_1] by(auto simp add:vector_component_simps intro!: imageI)
hence "?g x = pathstart g2" "?g y = pathfinish g1" using assms(4) unfolding assms(3) xy(3) by auto
thus ?thesis using as and inj(2)[of "2 *s x - 1" 0] inj(1)[of "2 *s y" 1] and xy(1,2)
unfolding pathstart_def pathfinish_def joinpaths_def mem_interval_1
by(auto simp add:vector_component_simps forall_1 Cart_eq) qed qed
lemmas join_paths_simps = path_join path_image_join pathstart_join pathfinish_join
subsection {* Reparametrizing a closed curve to start at some chosen point. *}
definition "shiftpath a (f::real^1 \<Rightarrow> real^'n) =
(\<lambda>x. if dest_vec1 (a + x) \<le> 1 then f(a + x) else f(a + x - 1))"
lemma pathstart_shiftpath: "a \<le> 1 \<Longrightarrow> pathstart(shiftpath a g) = g a"
unfolding pathstart_def shiftpath_def by auto
(** move this **)
declare forall_1[simp] ex_1[simp]
lemma pathfinish_shiftpath: assumes "0 \<le> a" "pathfinish g = pathstart g"
shows "pathfinish(shiftpath a g) = g a"
using assms unfolding pathstart_def pathfinish_def shiftpath_def
by(auto simp add: vector_component_simps)
lemma endpoints_shiftpath:
assumes "pathfinish g = pathstart g" "a \<in> {0 .. 1}"
shows "pathfinish(shiftpath a g) = g a" "pathstart(shiftpath a g) = g a"
using assms by(auto intro!:pathfinish_shiftpath pathstart_shiftpath)
lemma closed_shiftpath:
assumes "pathfinish g = pathstart g" "a \<in> {0..1}"
shows "pathfinish(shiftpath a g) = pathstart(shiftpath a g)"
using endpoints_shiftpath[OF assms] by auto
lemma path_shiftpath:
assumes "path g" "pathfinish g = pathstart g" "a \<in> {0..1}"
shows "path(shiftpath a g)" proof-
have *:"{0 .. 1} = {0 .. 1-a} \<union> {1-a .. 1}" using assms(3) by(auto simp add: vector_component_simps)
have **:"\<And>x. x + a = 1 \<Longrightarrow> g (x + a - 1) = g (x + a)"
using assms(2)[unfolded pathfinish_def pathstart_def] by auto
show ?thesis unfolding path_def shiftpath_def * apply(rule continuous_on_union)
apply(rule closed_interval)+ apply(rule continuous_on_eq[of _ "g \<circ> (\<lambda>x. a + x)"]) prefer 3
apply(rule continuous_on_eq[of _ "g \<circ> (\<lambda>x. a - 1 + x)"]) defer prefer 3
apply(rule continuous_on_intros)+ prefer 2 apply(rule continuous_on_intros)+
apply(rule_tac[1-2] continuous_on_subset[OF assms(1)[unfolded path_def]])
using assms(3) and ** by(auto simp add:vector_component_simps field_simps Cart_eq) qed
lemma shiftpath_shiftpath: assumes "pathfinish g = pathstart g" "a \<in> {0..1}" "x \<in> {0..1}"
shows "shiftpath (1 - a) (shiftpath a g) x = g x"
using assms unfolding pathfinish_def pathstart_def shiftpath_def
by(auto simp add: vector_component_simps)
lemma path_image_shiftpath:
assumes "a \<in> {0..1}" "pathfinish g = pathstart g"
shows "path_image(shiftpath a g) = path_image g" proof-
{ fix x assume as:"g 1 = g 0" "x \<in> {0..1::real^1}" " \<forall>y\<in>{0..1} \<inter> {x. \<not> a $ 1 + x $ 1 \<le> 1}. g x \<noteq> g (a + y - 1)"
hence "\<exists>y\<in>{0..1} \<inter> {x. a $ 1 + x $ 1 \<le> 1}. g x = g (a + y)" proof(cases "a \<le> x")
case False thus ?thesis apply(rule_tac x="1 + x - a" in bexI)
using as(1,2) and as(3)[THEN bspec[where x="1 + x - a"]] and assms(1)
by(auto simp add:vector_component_simps field_simps atomize_not) next
case True thus ?thesis using as(1-2) and assms(1) apply(rule_tac x="x - a" in bexI)
by(auto simp add:vector_component_simps field_simps) qed }
thus ?thesis using assms unfolding shiftpath_def path_image_def pathfinish_def pathstart_def
by(auto simp add:vector_component_simps image_iff) qed
subsection {* Special case of straight-line paths. *}
definition "linepath a b = (\<lambda>x. (1 - dest_vec1 x) *s a + dest_vec1 x *s b)"
lemma pathstart_linepath[simp]: "pathstart(linepath a b) = a"
unfolding pathstart_def linepath_def by auto
lemma pathfinish_linepath[simp]: "pathfinish(linepath a b) = b"
unfolding pathfinish_def linepath_def by auto
lemma continuous_linepath_at[intro]: "continuous (at x) (linepath a b)"
unfolding linepath_def by(auto simp add: vec1_dest_vec1 o_def intro!: continuous_intros)
lemma continuous_on_linepath[intro]: "continuous_on s (linepath a b)"
using continuous_linepath_at by(auto intro!: continuous_at_imp_continuous_on)
lemma path_linepath[intro]: "path(linepath a b)"
unfolding path_def by(rule continuous_on_linepath)
lemma path_image_linepath[simp]: "path_image(linepath a b) = (closed_segment a b)"
unfolding path_image_def segment linepath_def apply (rule set_ext, rule) defer
unfolding mem_Collect_eq image_iff apply(erule exE) apply(rule_tac x="u *s 1" in bexI)
by(auto simp add:vector_component_simps)
lemma reversepath_linepath[simp]: "reversepath(linepath a b) = linepath b a"
unfolding reversepath_def linepath_def by(rule ext, auto simp add:vector_component_simps)
lemma injective_path_linepath: assumes "a \<noteq> b" shows "injective_path(linepath a b)" proof-
{ obtain i where i:"a$i \<noteq> b$i" using assms[unfolded Cart_eq] by auto
fix x y::"real^1" assume "x $ 1 *s b + y $ 1 *s a = x $ 1 *s a + y $ 1 *s b"
hence "x$1 * (b$i - a$i) = y$1 * (b$i - a$i)" unfolding Cart_eq by(auto simp add:field_simps vector_component_simps)
hence "x = y" unfolding mult_cancel_right Cart_eq using i(1) by(auto simp add:field_simps) }
thus ?thesis unfolding injective_path_def linepath_def by(auto simp add:vector_component_simps field_simps) qed
lemma simple_path_linepath[intro]: "a \<noteq> b \<Longrightarrow> simple_path(linepath a b)" by(auto intro!: injective_imp_simple_path injective_path_linepath)
subsection {* Bounding a point away from a path. *}
lemma not_on_path_ball: assumes "path g" "z \<notin> path_image g"
shows "\<exists>e>0. ball z e \<inter> (path_image g) = {}" proof-
obtain a where "a\<in>path_image g" "\<forall>y\<in>path_image g. dist z a \<le> dist z y"
using distance_attains_inf[OF _ path_image_nonempty, of g z]
using compact_path_image[THEN compact_imp_closed, OF assms(1)] by auto
thus ?thesis apply(rule_tac x="dist z a" in exI) using assms(2) by(auto intro!: dist_pos_lt) qed
lemma not_on_path_cball: assumes "path g" "z \<notin> path_image g"
shows "\<exists>e>0. cball z e \<inter> (path_image g) = {}" proof-
obtain e where "ball z e \<inter> path_image g = {}" "e>0" using not_on_path_ball[OF assms] by auto
moreover have "cball z (e/2) \<subseteq> ball z e" using `e>0` by auto
ultimately show ?thesis apply(rule_tac x="e/2" in exI) by auto qed
subsection {* Path component, considered as a "joinability" relation (from Tom Hales). *}
definition "path_component s x y \<longleftrightarrow> (\<exists>g. path g \<and> path_image g \<subseteq> s \<and> pathstart g = x \<and> pathfinish g = y)"
lemmas path_defs = path_def pathstart_def pathfinish_def path_image_def path_component_def
lemma path_component_mem: assumes "path_component s x y" shows "x \<in> s" "y \<in> s"
using assms unfolding path_defs by auto
lemma path_component_refl: assumes "x \<in> s" shows "path_component s x x"
unfolding path_defs apply(rule_tac x="\<lambda>u. x" in exI) using assms
by(auto intro!:continuous_on_intros)
lemma path_component_refl_eq: "path_component s x x \<longleftrightarrow> x \<in> s"
by(auto intro!: path_component_mem path_component_refl)
lemma path_component_sym: "path_component s x y \<Longrightarrow> path_component s y x"
using assms unfolding path_component_def apply(erule exE) apply(rule_tac x="reversepath g" in exI)
by(auto simp add: reversepath_simps)
lemma path_component_trans: assumes "path_component s x y" "path_component s y z" shows "path_component s x z"
using assms unfolding path_component_def apply- apply(erule exE)+ apply(rule_tac x="g +++ ga" in exI) by(auto simp add: path_image_join)
lemma path_component_of_subset: "s \<subseteq> t \<Longrightarrow> path_component s x y \<Longrightarrow> path_component t x y"
unfolding path_component_def by auto
subsection {* Can also consider it as a set, as the name suggests. *}
lemma path_component_set: "path_component s x = { y. (\<exists>g. path g \<and> path_image g \<subseteq> s \<and> pathstart g = x \<and> pathfinish g = y )}"
apply(rule set_ext) unfolding mem_Collect_eq unfolding mem_def path_component_def by auto
lemma mem_path_component_set:"x \<in> path_component s y \<longleftrightarrow> path_component s y x" unfolding mem_def by auto
lemma path_component_subset: "(path_component s x) \<subseteq> s"
apply(rule, rule path_component_mem(2)) by(auto simp add:mem_def)
lemma path_component_eq_empty: "path_component s x = {} \<longleftrightarrow> x \<notin> s"
apply rule apply(drule equals0D[of _ x]) defer apply(rule equals0I) unfolding mem_path_component_set
apply(drule path_component_mem(1)) using path_component_refl by auto
subsection {* Path connectedness of a space. *}
definition "path_connected s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. \<exists>g. path g \<and> (path_image g) \<subseteq> s \<and> pathstart g = x \<and> pathfinish g = y)"
lemma path_connected_component: "path_connected s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. path_component s x y)"
unfolding path_connected_def path_component_def by auto
lemma path_connected_component_set: "path_connected s \<longleftrightarrow> (\<forall>x\<in>s. path_component s x = s)"
unfolding path_connected_component apply(rule, rule, rule, rule path_component_subset)
unfolding subset_eq mem_path_component_set Ball_def mem_def by auto
subsection {* Some useful lemmas about path-connectedness. *}
lemma convex_imp_path_connected: assumes "convex s" shows "path_connected s"
unfolding path_connected_def apply(rule,rule,rule_tac x="linepath x y" in exI)
unfolding path_image_linepath using assms[unfolded convex_contains_segment] by auto
lemma path_connected_imp_connected: assumes "path_connected s" shows "connected s"
unfolding connected_def not_ex apply(rule,rule,rule ccontr) unfolding not_not apply(erule conjE)+ proof-
fix e1 e2 assume as:"open e1" "open e2" "s \<subseteq> e1 \<union> e2" "e1 \<inter> e2 \<inter> s = {}" "e1 \<inter> s \<noteq> {}" "e2 \<inter> s \<noteq> {}"
then obtain x1 x2 where obt:"x1\<in>e1\<inter>s" "x2\<in>e2\<inter>s" by auto
then obtain g where g:"path g" "path_image g \<subseteq> s" "pathstart g = x1" "pathfinish g = x2"
using assms[unfolded path_connected_def,rule_format,of x1 x2] by auto
have *:"connected {0..1::real^1}" by(auto intro!: convex_connected convex_interval)
have "{0..1} \<subseteq> {x \<in> {0..1}. g x \<in> e1} \<union> {x \<in> {0..1}. g x \<in> e2}" using as(3) g(2)[unfolded path_defs] by blast
moreover have "{x \<in> {0..1}. g x \<in> e1} \<inter> {x \<in> {0..1}. g x \<in> e2} = {}" using as(4) g(2)[unfolded path_defs] unfolding subset_eq by auto
moreover have "{x \<in> {0..1}. g x \<in> e1} \<noteq> {} \<and> {x \<in> {0..1}. g x \<in> e2} \<noteq> {}" using g(3,4)[unfolded path_defs] using obt by(auto intro!: exI)
ultimately show False using *[unfolded connected_local not_ex,rule_format, of "{x\<in>{0..1}. g x \<in> e1}" "{x\<in>{0..1}. g x \<in> e2}"]
using continuous_open_in_preimage[OF g(1)[unfolded path_def] as(1)]
using continuous_open_in_preimage[OF g(1)[unfolded path_def] as(2)] by auto qed
lemma open_path_component: assumes "open s" shows "open(path_component s x)"
unfolding open_contains_ball proof
fix y assume as:"y \<in> path_component s x"
hence "y\<in>s" apply- apply(rule path_component_mem(2)) unfolding mem_def by auto
then obtain e where e:"e>0" "ball y e \<subseteq> s" using assms[unfolded open_contains_ball] by auto
show "\<exists>e>0. ball y e \<subseteq> path_component s x" apply(rule_tac x=e in exI) apply(rule,rule `e>0`,rule) unfolding mem_ball mem_path_component_set proof-
fix z assume "dist y z < e" thus "path_component s x z" apply(rule_tac path_component_trans[of _ _ y]) defer
apply(rule path_component_of_subset[OF e(2)]) apply(rule convex_imp_path_connected[OF convex_ball, unfolded path_connected_component, rule_format]) using `e>0`
using as[unfolded mem_def] by auto qed qed
lemma open_non_path_component: assumes "open s" shows "open(s - path_component s x)" unfolding open_contains_ball proof
fix y assume as:"y\<in>s - path_component s x"
then obtain e where e:"e>0" "ball y e \<subseteq> s" using assms[unfolded open_contains_ball] by auto
show "\<exists>e>0. ball y e \<subseteq> s - path_component s x" apply(rule_tac x=e in exI) apply(rule,rule `e>0`,rule,rule) defer proof(rule ccontr)
fix z assume "z\<in>ball y e" "\<not> z \<notin> path_component s x"
hence "y \<in> path_component s x" unfolding not_not mem_path_component_set using `e>0`
apply- apply(rule path_component_trans,assumption) apply(rule path_component_of_subset[OF e(2)])
apply(rule convex_imp_path_connected[OF convex_ball, unfolded path_connected_component, rule_format]) by auto
thus False using as by auto qed(insert e(2), auto) qed
lemma connected_open_path_connected: assumes "open s" "connected s" shows "path_connected s"
unfolding path_connected_component_set proof(rule,rule,rule path_component_subset, rule)
fix x y assume "x \<in> s" "y \<in> s" show "y \<in> path_component s x" proof(rule ccontr)
assume "y \<notin> path_component s x" moreover
have "path_component s x \<inter> s \<noteq> {}" using `x\<in>s` path_component_eq_empty path_component_subset[of s x] by auto
ultimately show False using `y\<in>s` open_non_path_component[OF assms(1)] open_path_component[OF assms(1)]
using assms(2)[unfolded connected_def not_ex, rule_format, of"path_component s x" "s - path_component s x"] by auto
qed qed
lemma path_connected_continuous_image:
assumes "continuous_on s f" "path_connected s" shows "path_connected (f ` s)"
unfolding path_connected_def proof(rule,rule)
fix x' y' assume "x' \<in> f ` s" "y' \<in> f ` s"
then obtain x y where xy:"x\<in>s" "y\<in>s" "x' = f x" "y' = f y" by auto
guess g using assms(2)[unfolded path_connected_def,rule_format,OF xy(1,2)] ..
thus "\<exists>g. path g \<and> path_image g \<subseteq> f ` s \<and> pathstart g = x' \<and> pathfinish g = y'"
unfolding xy apply(rule_tac x="f \<circ> g" in exI) unfolding path_defs
using assms(1) by(auto intro!: continuous_on_compose continuous_on_subset[of _ _ "g ` {0..1}"]) qed
lemma homeomorphic_path_connectedness:
"s homeomorphic t \<Longrightarrow> (path_connected s \<longleftrightarrow> path_connected t)"
unfolding homeomorphic_def homeomorphism_def apply(erule exE|erule conjE)+ apply rule
apply(drule_tac f=f in path_connected_continuous_image) prefer 3
apply(drule_tac f=g in path_connected_continuous_image) by auto
lemma path_connected_empty: "path_connected {}"
unfolding path_connected_def by auto
lemma path_connected_singleton: "path_connected {a}"
unfolding path_connected_def apply(rule,rule)
apply(rule_tac x="linepath a a" in exI) by(auto simp add:segment)
lemma path_connected_Un: assumes "path_connected s" "path_connected t" "s \<inter> t \<noteq> {}"
shows "path_connected (s \<union> t)" unfolding path_connected_component proof(rule,rule)
fix x y assume as:"x \<in> s \<union> t" "y \<in> s \<union> t"
from assms(3) obtain z where "z \<in> s \<inter> t" by auto
thus "path_component (s \<union> t) x y" using as using assms(1-2)[unfolded path_connected_component] apply-
apply(erule_tac[!] UnE)+ apply(rule_tac[2-3] path_component_trans[of _ _ z])
by(auto simp add:path_component_of_subset [OF Un_upper1] path_component_of_subset[OF Un_upper2]) qed
subsection {* sphere is path-connected. *}
lemma path_connected_punctured_universe:
assumes "2 \<le> CARD('n::finite)" shows "path_connected((UNIV::(real^'n::finite) set) - {a})" proof-
obtain \<psi> where \<psi>:"bij_betw \<psi> {1..CARD('n)} (UNIV::'n set)" using ex_bij_betw_nat_finite_1[OF finite_UNIV] by auto
let ?U = "UNIV::(real^'n) set" let ?u = "?U - {0}"
let ?basis = "\<lambda>k. basis (\<psi> k)"
let ?A = "\<lambda>k. {x::real^'n. \<exists>i\<in>{1..k}. (basis (\<psi> i)) \<bullet> x \<noteq> 0}"
have "\<forall>k\<in>{2..CARD('n)}. path_connected (?A k)" proof
have *:"\<And>k. ?A (Suc k) = {x. ?basis (Suc k) \<bullet> x < 0} \<union> {x. ?basis (Suc k) \<bullet> x > 0} \<union> ?A k" apply(rule set_ext,rule) defer
apply(erule UnE)+ unfolding mem_Collect_eq apply(rule_tac[1-2] x="Suc k" in bexI)
by(auto elim!: ballE simp add: not_less le_Suc_eq)
fix k assume "k \<in> {2..CARD('n)}" thus "path_connected (?A k)" proof(induct k)
case (Suc k) show ?case proof(cases "k = 1")
case False from Suc have d:"k \<in> {1..CARD('n)}" "Suc k \<in> {1..CARD('n)}" by auto
hence "\<psi> k \<noteq> \<psi> (Suc k)" using \<psi>[unfolded bij_betw_def inj_on_def, THEN conjunct1, THEN bspec[where x=k]] by auto
hence **:"?basis k + ?basis (Suc k) \<in> {x. 0 < ?basis (Suc k) \<bullet> x} \<inter> (?A k)"
"?basis k - ?basis (Suc k) \<in> {x. 0 > ?basis (Suc k) \<bullet> x} \<inter> ({x. 0 < ?basis (Suc k) \<bullet> x} \<union> (?A k))" using d
by(auto simp add: dot_basis vector_component_simps intro!:bexI[where x=k])
show ?thesis unfolding * Un_assoc apply(rule path_connected_Un) defer apply(rule path_connected_Un)
prefer 5 apply(rule_tac[1-2] convex_imp_path_connected, rule convex_halfspace_lt, rule convex_halfspace_gt)
apply(rule Suc(1)) apply(rule_tac[2-3] ccontr) using d ** False by auto
next case True hence d:"1\<in>{1..CARD('n)}" "2\<in>{1..CARD('n)}" using Suc(2) by auto
have ***:"Suc 1 = 2" by auto
have **:"\<And>s t P Q. s \<union> t \<union> {x. P x \<or> Q x} = (s \<union> {x. P x}) \<union> (t \<union> {x. Q x})" by auto
have "\<psi> 2 \<noteq> \<psi> (Suc 0)" apply(rule ccontr) using \<psi>[unfolded bij_betw_def inj_on_def, THEN conjunct1, THEN bspec[where x=2]] using assms by auto
thus ?thesis unfolding * True unfolding ** neq_iff bex_disj_distrib apply -
apply(rule path_connected_Un, rule_tac[1-2] path_connected_Un) defer 3 apply(rule_tac[1-4] convex_imp_path_connected)
apply(rule_tac[5] x=" ?basis 1 + ?basis 2" in nequals0I)
apply(rule_tac[6] x="-?basis 1 + ?basis 2" in nequals0I)
apply(rule_tac[7] x="-?basis 1 - ?basis 2" in nequals0I)
using d unfolding *** by(auto intro!: convex_halfspace_gt convex_halfspace_lt, auto simp add:vector_component_simps dot_basis)
qed qed auto qed note lem = this
have ***:"\<And>x::real^'n. (\<exists>i\<in>{1..CARD('n)}. basis (\<psi> i) \<bullet> x \<noteq> 0) \<longleftrightarrow> (\<exists>i. basis i \<bullet> x \<noteq> 0)"
apply rule apply(erule bexE) apply(rule_tac x="\<psi> i" in exI) defer apply(erule exE) proof-
fix x::"real^'n" and i assume as:"basis i \<bullet> x \<noteq> 0"
have "i\<in>\<psi> ` {1..CARD('n)}" using \<psi>[unfolded bij_betw_def, THEN conjunct2] by auto
then obtain j where "j\<in>{1..CARD('n)}" "\<psi> j = i" by auto
thus "\<exists>i\<in>{1..CARD('n)}. basis (\<psi> i) \<bullet> x \<noteq> 0" apply(rule_tac x=j in bexI) using as by auto qed auto
have *:"?U - {a} = (\<lambda>x. x + a) ` {x. x \<noteq> 0}" apply(rule set_ext) unfolding image_iff
apply rule apply(rule_tac x="x - a" in bexI) by auto
have **:"\<And>x::real^'n. x\<noteq>0 \<longleftrightarrow> (\<exists>i. basis i \<bullet> x \<noteq> 0)" unfolding Cart_eq by(auto simp add: dot_basis)
show ?thesis unfolding * apply(rule path_connected_continuous_image) apply(rule continuous_on_intros)+
unfolding ** apply(rule lem[THEN bspec[where x="CARD('n)"], unfolded ***]) using assms by auto qed
lemma path_connected_sphere: assumes "2 \<le> CARD('n::finite)" shows "path_connected {x::real^'n::finite. norm(x - a) = r}" proof(cases "r\<le>0")
case True thus ?thesis proof(cases "r=0")
case False hence "{x::real^'n. norm(x - a) = r} = {}" using True by auto
thus ?thesis using path_connected_empty by auto
qed(auto intro!:path_connected_singleton) next
case False hence *:"{x::real^'n. norm(x - a) = r} = (\<lambda>x. a + r *s x) ` {x. norm x = 1}" unfolding not_le apply -apply(rule set_ext,rule)
unfolding image_iff apply(rule_tac x="(1/r) *s (x - a)" in bexI) unfolding mem_Collect_eq norm_mul by auto
have ***:"\<And>xa. (if xa = 0 then 0 else 1) \<noteq> 1 \<Longrightarrow> xa = 0" apply(rule ccontr) by auto
have **:"{x::real^'n. norm x = 1} = (\<lambda>x. (1/norm x) *s x) ` (UNIV - {0})" apply(rule set_ext,rule)
unfolding image_iff apply(rule_tac x=x in bexI) unfolding mem_Collect_eq norm_mul by(auto intro!: ***)
have "continuous_on (UNIV - {0}) (vec1 \<circ> (\<lambda>x::real^'n. 1 / norm x))" unfolding o_def continuous_on_eq_continuous_within
apply(rule, rule continuous_at_within_inv[unfolded o_def inverse_eq_divide]) apply(rule continuous_at_within)
apply(rule continuous_at_vec1_norm[unfolded o_def,THEN spec]) by auto
thus ?thesis unfolding * ** using path_connected_punctured_universe[OF assms]
by(auto intro!: path_connected_continuous_image continuous_on_intros continuous_on_mul) qed
lemma connected_sphere: "2 \<le> CARD('n) \<Longrightarrow> connected {x::real^'n::finite. norm(x - a) = r}"
using path_connected_sphere path_connected_imp_connected by auto
(** In continuous_at_vec1_norm : Use \<And> instead of \<forall>. **)
end