theory Ordered_Euclidean_Space
imports
Topology_Euclidean_Space
"~~/src/HOL/Library/Product_Order"
begin
subsection {* An ordering on euclidean spaces that will allow us to talk about intervals *}
class ordered_euclidean_space = ord + inf + sup + abs + Inf + Sup + euclidean_space +
assumes eucl_le: "x \<le> y \<longleftrightarrow> (\<forall>i\<in>Basis. x \<bullet> i \<le> y \<bullet> i)"
assumes eucl_less_le_not_le: "x < y \<longleftrightarrow> x \<le> y \<and> \<not> y \<le> x"
assumes eucl_inf: "inf x y = (\<Sum>i\<in>Basis. inf (x \<bullet> i) (y \<bullet> i) *\<^sub>R i)"
assumes eucl_sup: "sup x y = (\<Sum>i\<in>Basis. sup (x \<bullet> i) (y \<bullet> i) *\<^sub>R i)"
assumes eucl_Inf: "Inf X = (\<Sum>i\<in>Basis. (INF x:X. x \<bullet> i) *\<^sub>R i)"
assumes eucl_Sup: "Sup X = (\<Sum>i\<in>Basis. (SUP x:X. x \<bullet> i) *\<^sub>R i)"
assumes eucl_abs: "abs x = (\<Sum>i\<in>Basis. abs (x \<bullet> i) *\<^sub>R i)"
begin
subclass order
by default
(auto simp: eucl_le eucl_less_le_not_le intro!: euclidean_eqI antisym intro: order.trans)
subclass ordered_ab_group_add_abs
by default (auto simp: eucl_le inner_add_left eucl_abs abs_leI)
subclass ordered_real_vector
by default (auto simp: eucl_le intro!: mult_left_mono mult_right_mono)
subclass lattice
by default (auto simp: eucl_inf eucl_sup eucl_le)
subclass distrib_lattice
by default (auto simp: eucl_inf eucl_sup sup_inf_distrib1 intro!: euclidean_eqI)
subclass conditionally_complete_lattice
proof
fix z::'a and X::"'a set"
assume "X \<noteq> {}"
hence "\<And>i. (\<lambda>x. x \<bullet> i) ` X \<noteq> {}" by simp
thus "(\<And>x. x \<in> X \<Longrightarrow> z \<le> x) \<Longrightarrow> z \<le> Inf X" "(\<And>x. x \<in> X \<Longrightarrow> x \<le> z) \<Longrightarrow> Sup X \<le> z"
by (auto simp: eucl_Inf eucl_Sup eucl_le Inf_class.INF_def Sup_class.SUP_def
simp del: Inf_class.Inf_image_eq Sup_class.Sup_image_eq
intro!: cInf_greatest cSup_least)
qed (force intro!: cInf_lower cSup_upper
simp: bdd_below_def bdd_above_def preorder_class.bdd_below_def preorder_class.bdd_above_def
eucl_Inf eucl_Sup eucl_le Inf_class.INF_def Sup_class.SUP_def
simp del: Inf_class.Inf_image_eq Sup_class.Sup_image_eq)+
lemma inner_Basis_inf_left: "i \<in> Basis \<Longrightarrow> inf x y \<bullet> i = inf (x \<bullet> i) (y \<bullet> i)"
and inner_Basis_sup_left: "i \<in> Basis \<Longrightarrow> sup x y \<bullet> i = sup (x \<bullet> i) (y \<bullet> i)"
by (simp_all add: eucl_inf eucl_sup inner_setsum_left inner_Basis if_distrib setsum_delta
cong: if_cong)
lemma inner_Basis_INF_left: "i \<in> Basis \<Longrightarrow> (INF x:X. f x) \<bullet> i = (INF x:X. f x \<bullet> i)"
and inner_Basis_SUP_left: "i \<in> Basis \<Longrightarrow> (SUP x:X. f x) \<bullet> i = (SUP x:X. f x \<bullet> i)"
using eucl_Sup [of "f ` X"] eucl_Inf [of "f ` X"] by (simp_all add: comp_def)
lemma abs_inner: "i \<in> Basis \<Longrightarrow> abs x \<bullet> i = abs (x \<bullet> i)"
by (auto simp: eucl_abs)
lemma
abs_scaleR: "\<bar>a *\<^sub>R b\<bar> = \<bar>a\<bar> *\<^sub>R \<bar>b\<bar>"
by (auto simp: eucl_abs abs_mult intro!: euclidean_eqI)
lemma interval_inner_leI:
assumes "x \<in> {a .. b}" "0 \<le> i"
shows "a\<bullet>i \<le> x\<bullet>i" "x\<bullet>i \<le> b\<bullet>i"
using assms
unfolding euclidean_inner[of a i] euclidean_inner[of x i] euclidean_inner[of b i]
by (auto intro!: setsum_mono mult_right_mono simp: eucl_le)
lemma inner_nonneg_nonneg:
shows "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> a \<bullet> b"
using interval_inner_leI[of a 0 a b]
by auto
lemma inner_Basis_mono:
shows "a \<le> b \<Longrightarrow> c \<in> Basis \<Longrightarrow> a \<bullet> c \<le> b \<bullet> c"
by (simp add: eucl_le)
lemma Basis_nonneg[intro, simp]: "i \<in> Basis \<Longrightarrow> 0 \<le> i"
by (auto simp: eucl_le inner_Basis)
lemma Sup_eq_maximum_componentwise:
fixes s::"'a set"
assumes i: "\<And>b. b \<in> Basis \<Longrightarrow> X \<bullet> b = i b \<bullet> b"
assumes sup: "\<And>b x. b \<in> Basis \<Longrightarrow> x \<in> s \<Longrightarrow> x \<bullet> b \<le> X \<bullet> b"
assumes i_s: "\<And>b. b \<in> Basis \<Longrightarrow> (i b \<bullet> b) \<in> (\<lambda>x. x \<bullet> b) ` s"
shows "Sup s = X"
using assms
unfolding eucl_Sup euclidean_representation_setsum
by (auto simp: Sup_class.SUP_def simp del: Sup_class.Sup_image_eq intro!: conditionally_complete_lattice_class.cSup_eq_maximum)
lemma Inf_eq_minimum_componentwise:
assumes i: "\<And>b. b \<in> Basis \<Longrightarrow> X \<bullet> b = i b \<bullet> b"
assumes sup: "\<And>b x. b \<in> Basis \<Longrightarrow> x \<in> s \<Longrightarrow> X \<bullet> b \<le> x \<bullet> b"
assumes i_s: "\<And>b. b \<in> Basis \<Longrightarrow> (i b \<bullet> b) \<in> (\<lambda>x. x \<bullet> b) ` s"
shows "Inf s = X"
using assms
unfolding eucl_Inf euclidean_representation_setsum
by (auto simp: Inf_class.INF_def simp del: Inf_class.Inf_image_eq intro!: conditionally_complete_lattice_class.cInf_eq_minimum)
end
lemma
compact_attains_Inf_componentwise:
fixes b::"'a::ordered_euclidean_space"
assumes "b \<in> Basis" assumes "X \<noteq> {}" "compact X"
obtains x where "x \<in> X" "x \<bullet> b = Inf X \<bullet> b" "\<And>y. y \<in> X \<Longrightarrow> x \<bullet> b \<le> y \<bullet> b"
proof atomize_elim
let ?proj = "(\<lambda>x. x \<bullet> b) ` X"
from assms have "compact ?proj" "?proj \<noteq> {}"
by (auto intro!: compact_continuous_image continuous_on_intros)
from compact_attains_inf[OF this]
obtain s x
where s: "s\<in>(\<lambda>x. x \<bullet> b) ` X" "\<And>t. t\<in>(\<lambda>x. x \<bullet> b) ` X \<Longrightarrow> s \<le> t"
and x: "x \<in> X" "s = x \<bullet> b" "\<And>y. y \<in> X \<Longrightarrow> x \<bullet> b \<le> y \<bullet> b"
by auto
hence "Inf ?proj = x \<bullet> b"
by (auto intro!: conditionally_complete_lattice_class.cInf_eq_minimum simp del: Inf_class.Inf_image_eq)
hence "x \<bullet> b = Inf X \<bullet> b"
by (auto simp: eucl_Inf Inf_class.INF_def inner_setsum_left inner_Basis if_distrib `b \<in> Basis` setsum_delta
simp del: Inf_class.Inf_image_eq
cong: if_cong)
with x show "\<exists>x. x \<in> X \<and> x \<bullet> b = Inf X \<bullet> b \<and> (\<forall>y. y \<in> X \<longrightarrow> x \<bullet> b \<le> y \<bullet> b)" by blast
qed
lemma
compact_attains_Sup_componentwise:
fixes b::"'a::ordered_euclidean_space"
assumes "b \<in> Basis" assumes "X \<noteq> {}" "compact X"
obtains x where "x \<in> X" "x \<bullet> b = Sup X \<bullet> b" "\<And>y. y \<in> X \<Longrightarrow> y \<bullet> b \<le> x \<bullet> b"
proof atomize_elim
let ?proj = "(\<lambda>x. x \<bullet> b) ` X"
from assms have "compact ?proj" "?proj \<noteq> {}"
by (auto intro!: compact_continuous_image continuous_on_intros)
from compact_attains_sup[OF this]
obtain s x
where s: "s\<in>(\<lambda>x. x \<bullet> b) ` X" "\<And>t. t\<in>(\<lambda>x. x \<bullet> b) ` X \<Longrightarrow> t \<le> s"
and x: "x \<in> X" "s = x \<bullet> b" "\<And>y. y \<in> X \<Longrightarrow> y \<bullet> b \<le> x \<bullet> b"
by auto
hence "Sup ?proj = x \<bullet> b"
by (auto intro!: cSup_eq_maximum simp del: Sup_image_eq)
hence "x \<bullet> b = Sup X \<bullet> b"
by (auto simp: eucl_Sup[where 'a='a] SUP_def inner_setsum_left inner_Basis if_distrib `b \<in> Basis` setsum_delta
simp del: Sup_image_eq cong: if_cong)
with x show "\<exists>x. x \<in> X \<and> x \<bullet> b = Sup X \<bullet> b \<and> (\<forall>y. y \<in> X \<longrightarrow> y \<bullet> b \<le> x \<bullet> b)" by blast
qed
lemma (in order) atLeastatMost_empty'[simp]:
"(~ a <= b) \<Longrightarrow> {a..b} = {}"
by (auto)
instance real :: ordered_euclidean_space
by default (auto simp: INF_def SUP_def)
lemma in_Basis_prod_iff:
fixes i::"'a::euclidean_space*'b::euclidean_space"
shows "i \<in> Basis \<longleftrightarrow> fst i = 0 \<and> snd i \<in> Basis \<or> snd i = 0 \<and> fst i \<in> Basis"
by (cases i) (auto simp: Basis_prod_def)
instantiation prod::(abs, abs) abs
begin
definition "abs x = (abs (fst x), abs (snd x))"
instance proof qed
end
instance prod :: (ordered_euclidean_space, ordered_euclidean_space) ordered_euclidean_space
by default
(auto intro!: add_mono simp add: euclidean_representation_setsum' Ball_def inner_prod_def
in_Basis_prod_iff inner_Basis_inf_left inner_Basis_sup_left inner_Basis_INF_left Inf_prod_def
inner_Basis_SUP_left Sup_prod_def less_prod_def less_eq_prod_def eucl_le[where 'a='a]
eucl_le[where 'a='b] abs_prod_def abs_inner)
subsection {* Boxes *}
lemma box:
fixes a :: "'a::euclidean_space"
shows "box a b = {x::'a. \<forall>i\<in>Basis. a\<bullet>i < x\<bullet>i \<and> x\<bullet>i < b\<bullet>i}"
and "cbox a b = {x::'a. \<forall>i\<in>Basis. a\<bullet>i \<le> x\<bullet>i \<and> x\<bullet>i \<le> b\<bullet>i}"
by (auto simp add:set_eq_iff box_def cbox_def)
lemma mem_box:
fixes a :: "'a::euclidean_space"
shows "x \<in> box a b \<longleftrightarrow> (\<forall>i\<in>Basis. a\<bullet>i < x\<bullet>i \<and> x\<bullet>i < b\<bullet>i)"
and "x \<in> cbox a b \<longleftrightarrow> (\<forall>i\<in>Basis. a\<bullet>i \<le> x\<bullet>i \<and> x\<bullet>i \<le> b\<bullet>i)"
using box[of a b]
by auto
lemma box_eq_empty:
fixes a :: "'a::euclidean_space"
shows "(box a b = {} \<longleftrightarrow> (\<exists>i\<in>Basis. b\<bullet>i \<le> a\<bullet>i))" (is ?th1)
and "(cbox a b = {} \<longleftrightarrow> (\<exists>i\<in>Basis. b\<bullet>i < a\<bullet>i))" (is ?th2)
proof -
{
fix i x
assume i: "i\<in>Basis" and as:"b\<bullet>i \<le> a\<bullet>i" and x:"x\<in>box a b"
then have "a \<bullet> i < x \<bullet> i \<and> x \<bullet> i < b \<bullet> i"
unfolding mem_box by (auto simp: box_def)
then have "a\<bullet>i < b\<bullet>i" by auto
then have False using as by auto
}
moreover
{
assume as: "\<forall>i\<in>Basis. \<not> (b\<bullet>i \<le> a\<bullet>i)"
let ?x = "(1/2) *\<^sub>R (a + b)"
{
fix i :: 'a
assume i: "i \<in> Basis"
have "a\<bullet>i < b\<bullet>i"
using as[THEN bspec[where x=i]] i by auto
then have "a\<bullet>i < ((1/2) *\<^sub>R (a+b)) \<bullet> i" "((1/2) *\<^sub>R (a+b)) \<bullet> i < b\<bullet>i"
by (auto simp: inner_add_left)
}
then have "box a b \<noteq> {}"
using mem_box(1)[of "?x" a b] by auto
}
ultimately show ?th1 by blast
{
fix i x
assume i: "i \<in> Basis" and as:"b\<bullet>i < a\<bullet>i" and x:"x\<in>cbox a b"
then have "a \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> b \<bullet> i"
unfolding mem_box by auto
then have "a\<bullet>i \<le> b\<bullet>i" by auto
then have False using as by auto
}
moreover
{
assume as:"\<forall>i\<in>Basis. \<not> (b\<bullet>i < a\<bullet>i)"
let ?x = "(1/2) *\<^sub>R (a + b)"
{
fix i :: 'a
assume i:"i \<in> Basis"
have "a\<bullet>i \<le> b\<bullet>i"
using as[THEN bspec[where x=i]] i by auto
then have "a\<bullet>i \<le> ((1/2) *\<^sub>R (a+b)) \<bullet> i" "((1/2) *\<^sub>R (a+b)) \<bullet> i \<le> b\<bullet>i"
by (auto simp: inner_add_left)
}
then have "cbox a b \<noteq> {}"
using mem_box(2)[of "?x" a b] by auto
}
ultimately show ?th2 by blast
qed
lemma box_ne_empty:
fixes a :: "'a::euclidean_space"
shows "cbox a b \<noteq> {} \<longleftrightarrow> (\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i)"
and "box a b \<noteq> {} \<longleftrightarrow> (\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i)"
unfolding box_eq_empty[of a b] by fastforce+
lemma
fixes a :: "'a::euclidean_space"
shows cbox_sing: "cbox a a = {a}"
and box_sing: "box a a = {}"
unfolding set_eq_iff mem_box eq_iff [symmetric]
by (auto intro!: euclidean_eqI[where 'a='a])
(metis all_not_in_conv nonempty_Basis)
lemma subset_box_imp:
fixes a :: "'a::euclidean_space"
shows "(\<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i) \<Longrightarrow> cbox c d \<subseteq> cbox a b"
and "(\<forall>i\<in>Basis. a\<bullet>i < c\<bullet>i \<and> d\<bullet>i < b\<bullet>i) \<Longrightarrow> cbox c d \<subseteq> box a b"
and "(\<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i) \<Longrightarrow> box c d \<subseteq> cbox a b"
and "(\<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i) \<Longrightarrow> box c d \<subseteq> box a b"
unfolding subset_eq[unfolded Ball_def] unfolding mem_box
by (best intro: order_trans less_le_trans le_less_trans less_imp_le)+
lemma box_subset_cbox:
fixes a :: "'a::euclidean_space"
shows "box a b \<subseteq> cbox a b"
unfolding subset_eq [unfolded Ball_def] mem_box
by (fast intro: less_imp_le)
lemma subset_box:
fixes a :: "'a::euclidean_space"
shows "cbox c d \<subseteq> cbox a b \<longleftrightarrow> (\<forall>i\<in>Basis. c\<bullet>i \<le> d\<bullet>i) --> (\<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i)" (is ?th1)
and "cbox c d \<subseteq> box a b \<longleftrightarrow> (\<forall>i\<in>Basis. c\<bullet>i \<le> d\<bullet>i) --> (\<forall>i\<in>Basis. a\<bullet>i < c\<bullet>i \<and> d\<bullet>i < b\<bullet>i)" (is ?th2)
and "box c d \<subseteq> cbox a b \<longleftrightarrow> (\<forall>i\<in>Basis. c\<bullet>i < d\<bullet>i) --> (\<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i)" (is ?th3)
and "box c d \<subseteq> box a b \<longleftrightarrow> (\<forall>i\<in>Basis. c\<bullet>i < d\<bullet>i) --> (\<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i)" (is ?th4)
proof -
show ?th1
unfolding subset_eq and Ball_def and mem_box
by (auto intro: order_trans)
show ?th2
unfolding subset_eq and Ball_def and mem_box
by (auto intro: le_less_trans less_le_trans order_trans less_imp_le)
{
assume as: "box c d \<subseteq> cbox a b" "\<forall>i\<in>Basis. c\<bullet>i < d\<bullet>i"
then have "box c d \<noteq> {}"
unfolding box_eq_empty by auto
fix i :: 'a
assume i: "i \<in> Basis"
(** TODO combine the following two parts as done in the HOL_light version. **)
{
let ?x = "(\<Sum>j\<in>Basis. (if j=i then ((min (a\<bullet>j) (d\<bullet>j))+c\<bullet>j)/2 else (c\<bullet>j+d\<bullet>j)/2) *\<^sub>R j)::'a"
assume as2: "a\<bullet>i > c\<bullet>i"
{
fix j :: 'a
assume j: "j \<in> Basis"
then have "c \<bullet> j < ?x \<bullet> j \<and> ?x \<bullet> j < d \<bullet> j"
apply (cases "j = i")
using as(2)[THEN bspec[where x=j]] i
apply (auto simp add: as2)
done
}
then have "?x\<in>box c d"
using i unfolding mem_box by auto
moreover
have "?x \<notin> cbox a b"
unfolding mem_box
apply auto
apply (rule_tac x=i in bexI)
using as(2)[THEN bspec[where x=i]] and as2 i
apply auto
done
ultimately have False using as by auto
}
then have "a\<bullet>i \<le> c\<bullet>i" by (rule ccontr) auto
moreover
{
let ?x = "(\<Sum>j\<in>Basis. (if j=i then ((max (b\<bullet>j) (c\<bullet>j))+d\<bullet>j)/2 else (c\<bullet>j+d\<bullet>j)/2) *\<^sub>R j)::'a"
assume as2: "b\<bullet>i < d\<bullet>i"
{
fix j :: 'a
assume "j\<in>Basis"
then have "d \<bullet> j > ?x \<bullet> j \<and> ?x \<bullet> j > c \<bullet> j"
apply (cases "j = i")
using as(2)[THEN bspec[where x=j]]
apply (auto simp add: as2)
done
}
then have "?x\<in>box c d"
unfolding mem_box by auto
moreover
have "?x\<notin>cbox a b"
unfolding mem_box
apply auto
apply (rule_tac x=i in bexI)
using as(2)[THEN bspec[where x=i]] and as2 using i
apply auto
done
ultimately have False using as by auto
}
then have "b\<bullet>i \<ge> d\<bullet>i" by (rule ccontr) auto
ultimately
have "a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i" by auto
} note part1 = this
show ?th3
unfolding subset_eq and Ball_def and mem_box
apply (rule, rule, rule, rule)
apply (rule part1)
unfolding subset_eq and Ball_def and mem_box
prefer 4
apply auto
apply (erule_tac x=xa in allE, erule_tac x=xa in allE, fastforce)+
done
{
assume as: "box c d \<subseteq> box a b" "\<forall>i\<in>Basis. c\<bullet>i < d\<bullet>i"
fix i :: 'a
assume i:"i\<in>Basis"
from as(1) have "box c d \<subseteq> cbox a b"
using box_subset_cbox[of a b] by auto
then have "a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i"
using part1 and as(2) using i by auto
} note * = this
show ?th4
unfolding subset_eq and Ball_def and mem_box
apply (rule, rule, rule, rule)
apply (rule *)
unfolding subset_eq and Ball_def and mem_box
prefer 4
apply auto
apply (erule_tac x=xa in allE, simp)+
done
qed
lemma inter_interval:
fixes a :: "'a::euclidean_space"
shows "cbox a b \<inter> cbox c d =
cbox (\<Sum>i\<in>Basis. max (a\<bullet>i) (c\<bullet>i) *\<^sub>R i) (\<Sum>i\<in>Basis. min (b\<bullet>i) (d\<bullet>i) *\<^sub>R i)"
unfolding set_eq_iff and Int_iff and mem_box
by auto
lemma disjoint_interval:
fixes a::"'a::euclidean_space"
shows "cbox a b \<inter> cbox c d = {} \<longleftrightarrow> (\<exists>i\<in>Basis. (b\<bullet>i < a\<bullet>i \<or> d\<bullet>i < c\<bullet>i \<or> b\<bullet>i < c\<bullet>i \<or> d\<bullet>i < a\<bullet>i))" (is ?th1)
and "cbox a b \<inter> box c d = {} \<longleftrightarrow> (\<exists>i\<in>Basis. (b\<bullet>i < a\<bullet>i \<or> d\<bullet>i \<le> c\<bullet>i \<or> b\<bullet>i \<le> c\<bullet>i \<or> d\<bullet>i \<le> a\<bullet>i))" (is ?th2)
and "box a b \<inter> cbox c d = {} \<longleftrightarrow> (\<exists>i\<in>Basis. (b\<bullet>i \<le> a\<bullet>i \<or> d\<bullet>i < c\<bullet>i \<or> b\<bullet>i \<le> c\<bullet>i \<or> d\<bullet>i \<le> a\<bullet>i))" (is ?th3)
and "box a b \<inter> box c d = {} \<longleftrightarrow> (\<exists>i\<in>Basis. (b\<bullet>i \<le> a\<bullet>i \<or> d\<bullet>i \<le> c\<bullet>i \<or> b\<bullet>i \<le> c\<bullet>i \<or> d\<bullet>i \<le> a\<bullet>i))" (is ?th4)
proof -
let ?z = "(\<Sum>i\<in>Basis. (((max (a\<bullet>i) (c\<bullet>i)) + (min (b\<bullet>i) (d\<bullet>i))) / 2) *\<^sub>R i)::'a"
have **: "\<And>P Q. (\<And>i :: 'a. i \<in> Basis \<Longrightarrow> Q ?z i \<Longrightarrow> P i) \<Longrightarrow>
(\<And>i x :: 'a. i \<in> Basis \<Longrightarrow> P i \<Longrightarrow> Q x i) \<Longrightarrow> (\<forall>x. \<exists>i\<in>Basis. Q x i) \<longleftrightarrow> (\<exists>i\<in>Basis. P i)"
by blast
note * = set_eq_iff Int_iff empty_iff mem_box ball_conj_distrib[symmetric] eq_False ball_simps(10)
show ?th1 unfolding * by (intro **) auto
show ?th2 unfolding * by (intro **) auto
show ?th3 unfolding * by (intro **) auto
show ?th4 unfolding * by (intro **) auto
qed
(* Moved box_subset_cbox a bit upwards *)
lemma open_box[intro]:
fixes a b :: "'a::euclidean_space"
shows "open (box a b)"
proof -
have "open (\<Inter>i\<in>Basis. (\<lambda>x. x\<bullet>i) -` {a\<bullet>i<..<b\<bullet>i})"
by (intro open_INT finite_lessThan ballI continuous_open_vimage allI
linear_continuous_at open_real_greaterThanLessThan finite_Basis bounded_linear_inner_left)
also have "(\<Inter>i\<in>Basis. (\<lambda>x. x\<bullet>i) -` {a\<bullet>i<..<b\<bullet>i}) = box a b"
by (auto simp add: box)
finally show "open (box a b)" .
qed
lemma closed_cbox[intro]:
fixes a b :: "'a::euclidean_space"
shows "closed (cbox a b)"
proof -
have "closed (\<Inter>i\<in>Basis. (\<lambda>x. x\<bullet>i) -` {a\<bullet>i .. b\<bullet>i})"
by (intro closed_INT ballI continuous_closed_vimage allI
linear_continuous_at closed_real_atLeastAtMost finite_Basis bounded_linear_inner_left)
also have "(\<Inter>i\<in>Basis. (\<lambda>x. x\<bullet>i) -` {a\<bullet>i .. b\<bullet>i}) = cbox a b"
by (auto simp add: cbox_def)
finally show "closed (cbox a b)" .
qed
lemma interior_cbox [intro]:
fixes a b :: "'a::euclidean_space"
shows "interior (cbox a b) = box a b" (is "?L = ?R")
proof(rule subset_antisym)
show "?R \<subseteq> ?L"
using box_subset_cbox open_box
by (rule interior_maximal)
{
fix x
assume "x \<in> interior (cbox a b)"
then obtain s where s: "open s" "x \<in> s" "s \<subseteq> cbox a b" ..
then obtain e where "e>0" and e:"\<forall>x'. dist x' x < e \<longrightarrow> x' \<in> cbox a b"
unfolding open_dist and subset_eq by auto
{
fix i :: 'a
assume i: "i \<in> Basis"
have "dist (x - (e / 2) *\<^sub>R i) x < e"
and "dist (x + (e / 2) *\<^sub>R i) x < e"
unfolding dist_norm
apply auto
unfolding norm_minus_cancel
using norm_Basis[OF i] `e>0`
apply auto
done
then have "a \<bullet> i \<le> (x - (e / 2) *\<^sub>R i) \<bullet> i" and "(x + (e / 2) *\<^sub>R i) \<bullet> i \<le> b \<bullet> i"
using e[THEN spec[where x="x - (e/2) *\<^sub>R i"]]
and e[THEN spec[where x="x + (e/2) *\<^sub>R i"]]
unfolding mem_box
using i
by blast+
then have "a \<bullet> i < x \<bullet> i" and "x \<bullet> i < b \<bullet> i"
using `e>0` i
by (auto simp: inner_diff_left inner_Basis inner_add_left)
}
then have "x \<in> box a b"
unfolding mem_box by auto
}
then show "?L \<subseteq> ?R" ..
qed
lemma bounded_cbox:
fixes a :: "'a::euclidean_space"
shows "bounded (cbox a b)"
proof -
let ?b = "\<Sum>i\<in>Basis. \<bar>a\<bullet>i\<bar> + \<bar>b\<bullet>i\<bar>"
{
fix x :: "'a"
assume x: "\<forall>i\<in>Basis. a \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> b \<bullet> i"
{
fix i :: 'a
assume "i \<in> Basis"
then have "\<bar>x\<bullet>i\<bar> \<le> \<bar>a\<bullet>i\<bar> + \<bar>b\<bullet>i\<bar>"
using x[THEN bspec[where x=i]] by auto
}
then have "(\<Sum>i\<in>Basis. \<bar>x \<bullet> i\<bar>) \<le> ?b"
apply -
apply (rule setsum_mono)
apply auto
done
then have "norm x \<le> ?b"
using norm_le_l1[of x] by auto
}
then show ?thesis
unfolding box and bounded_iff by auto
qed
lemma bounded_interval:
fixes a :: "'a::euclidean_space"
shows "bounded (cbox a b) \<and> bounded (box a b)"
using bounded_cbox[of a b]
using box_subset_cbox[of a b]
using bounded_subset[of "cbox a b" "box a b"]
by simp
lemma not_interval_univ:
fixes a :: "'a::euclidean_space"
shows "cbox a b \<noteq> UNIV \<and> box a b \<noteq> UNIV"
using bounded_interval[of a b] by auto
lemma compact_cbox:
fixes a :: "'a::euclidean_space"
shows "compact (cbox a b)"
using bounded_closed_imp_seq_compact[of "cbox a b"] using bounded_interval[of a b]
by (auto simp: compact_eq_seq_compact_metric)
lemma open_interval_midpoint:
fixes a :: "'a::euclidean_space"
assumes "box a b \<noteq> {}"
shows "((1/2) *\<^sub>R (a + b)) \<in> box a b"
proof -
{
fix i :: 'a
assume "i \<in> Basis"
then have "a \<bullet> i < ((1 / 2) *\<^sub>R (a + b)) \<bullet> i \<and> ((1 / 2) *\<^sub>R (a + b)) \<bullet> i < b \<bullet> i"
using assms[unfolded box_ne_empty, THEN bspec[where x=i]] by (auto simp: inner_add_left)
}
then show ?thesis unfolding mem_box by auto
qed
lemma open_closed_interval_convex:
fixes x :: "'a::euclidean_space"
assumes x: "x \<in> box a b"
and y: "y \<in> cbox a b"
and e: "0 < e" "e \<le> 1"
shows "(e *\<^sub>R x + (1 - e) *\<^sub>R y) \<in> box a b"
proof -
{
fix i :: 'a
assume i: "i \<in> Basis"
have "a \<bullet> i = e * (a \<bullet> i) + (1 - e) * (a \<bullet> i)"
unfolding left_diff_distrib by simp
also have "\<dots> < e * (x \<bullet> i) + (1 - e) * (y \<bullet> i)"
apply (rule add_less_le_mono)
using e unfolding mult_less_cancel_left and mult_le_cancel_left
apply simp_all
using x unfolding mem_box using i
apply simp
using y unfolding mem_box using i
apply simp
done
finally have "a \<bullet> i < (e *\<^sub>R x + (1 - e) *\<^sub>R y) \<bullet> i"
unfolding inner_simps by auto
moreover
{
have "b \<bullet> i = e * (b\<bullet>i) + (1 - e) * (b\<bullet>i)"
unfolding left_diff_distrib by simp
also have "\<dots> > e * (x \<bullet> i) + (1 - e) * (y \<bullet> i)"
apply (rule add_less_le_mono)
using e unfolding mult_less_cancel_left and mult_le_cancel_left
apply simp_all
using x
unfolding mem_box
using i
apply simp
using y
unfolding mem_box
using i
apply simp
done
finally have "(e *\<^sub>R x + (1 - e) *\<^sub>R y) \<bullet> i < b \<bullet> i"
unfolding inner_simps by auto
}
ultimately have "a \<bullet> i < (e *\<^sub>R x + (1 - e) *\<^sub>R y) \<bullet> i \<and> (e *\<^sub>R x + (1 - e) *\<^sub>R y) \<bullet> i < b \<bullet> i"
by auto
}
then show ?thesis
unfolding mem_box by auto
qed
notation
eucl_less (infix "<e" 50)
lemma closure_open_interval:
fixes a :: "'a::euclidean_space"
assumes "box a b \<noteq> {}"
shows "closure (box a b) = cbox a b"
proof -
have ab: "a <e b"
using assms by (simp add: eucl_less_def box_ne_empty)
let ?c = "(1 / 2) *\<^sub>R (a + b)"
{
fix x
assume as:"x \<in> cbox a b"
def f \<equiv> "\<lambda>n::nat. x + (inverse (real n + 1)) *\<^sub>R (?c - x)"
{
fix n
assume fn: "f n <e b \<longrightarrow> a <e f n \<longrightarrow> f n = x" and xc: "x \<noteq> ?c"
have *: "0 < inverse (real n + 1)" "inverse (real n + 1) \<le> 1"
unfolding inverse_le_1_iff by auto
have "(inverse (real n + 1)) *\<^sub>R ((1 / 2) *\<^sub>R (a + b)) + (1 - inverse (real n + 1)) *\<^sub>R x =
x + (inverse (real n + 1)) *\<^sub>R (((1 / 2) *\<^sub>R (a + b)) - x)"
by (auto simp add: algebra_simps)
then have "f n <e b" and "a <e f n"
using open_closed_interval_convex[OF open_interval_midpoint[OF assms] as *]
unfolding f_def by (auto simp: box eucl_less_def)
then have False
using fn unfolding f_def using xc by auto
}
moreover
{
assume "\<not> (f ---> x) sequentially"
{
fix e :: real
assume "e > 0"
then have "\<exists>N::nat. inverse (real (N + 1)) < e"
using real_arch_inv[of e]
apply (auto simp add: Suc_pred')
apply (rule_tac x="n - 1" in exI)
apply auto
done
then obtain N :: nat where "inverse (real (N + 1)) < e"
by auto
then have "\<forall>n\<ge>N. inverse (real n + 1) < e"
apply auto
apply (metis Suc_le_mono le_SucE less_imp_inverse_less nat_le_real_less order_less_trans
real_of_nat_Suc real_of_nat_Suc_gt_zero)
done
then have "\<exists>N::nat. \<forall>n\<ge>N. inverse (real n + 1) < e" by auto
}
then have "((\<lambda>n. inverse (real n + 1)) ---> 0) sequentially"
unfolding LIMSEQ_def by(auto simp add: dist_norm)
then have "(f ---> x) sequentially"
unfolding f_def
using tendsto_add[OF tendsto_const, of "\<lambda>n::nat. (inverse (real n + 1)) *\<^sub>R ((1 / 2) *\<^sub>R (a + b) - x)" 0 sequentially x]
using tendsto_scaleR [OF _ tendsto_const, of "\<lambda>n::nat. inverse (real n + 1)" 0 sequentially "((1 / 2) *\<^sub>R (a + b) - x)"]
by auto
}
ultimately have "x \<in> closure (box a b)"
using as and open_interval_midpoint[OF assms]
unfolding closure_def
unfolding islimpt_sequential
by (cases "x=?c") (auto simp: in_box_eucl_less)
}
then show ?thesis
using closure_minimal[OF box_subset_cbox, of a b] by blast
qed
lemma bounded_subset_open_interval_symmetric:
fixes s::"('a::euclidean_space) set"
assumes "bounded s"
shows "\<exists>a. s \<subseteq> box (-a) a"
proof -
obtain b where "b>0" and b: "\<forall>x\<in>s. norm x \<le> b"
using assms[unfolded bounded_pos] by auto
def a \<equiv> "(\<Sum>i\<in>Basis. (b + 1) *\<^sub>R i)::'a"
{
fix x
assume "x \<in> s"
fix i :: 'a
assume i: "i \<in> Basis"
then have "(-a)\<bullet>i < x\<bullet>i" and "x\<bullet>i < a\<bullet>i"
using b[THEN bspec[where x=x], OF `x\<in>s`]
using Basis_le_norm[OF i, of x]
unfolding inner_simps and a_def
by auto
}
then show ?thesis
by (auto intro: exI[where x=a] simp add: box)
qed
lemma bounded_subset_open_interval:
fixes s :: "('a::euclidean_space) set"
shows "bounded s \<Longrightarrow> (\<exists>a b. s \<subseteq> box a b)"
by (auto dest!: bounded_subset_open_interval_symmetric)
lemma bounded_subset_cbox_symmetric:
fixes s :: "('a::euclidean_space) set"
assumes "bounded s"
shows "\<exists>a. s \<subseteq> cbox (-a) a"
proof -
obtain a where "s \<subseteq> box (-a) a"
using bounded_subset_open_interval_symmetric[OF assms] by auto
then show ?thesis
using box_subset_cbox[of "-a" a] by auto
qed
lemma bounded_subset_cbox:
fixes s :: "('a::euclidean_space) set"
shows "bounded s \<Longrightarrow> \<exists>a b. s \<subseteq> cbox a b"
using bounded_subset_cbox_symmetric[of s] by auto
lemma frontier_closed_interval:
fixes a b :: "'a::euclidean_space"
shows "frontier (cbox a b) = cbox a b - box a b"
unfolding frontier_def unfolding interior_cbox and closure_closed[OF closed_cbox] ..
lemma frontier_open_interval:
fixes a b :: "'a::euclidean_space"
shows "frontier (box a b) = (if box a b = {} then {} else cbox a b - box a b)"
proof (cases "box a b = {}")
case True
then show ?thesis
using frontier_empty by auto
next
case False
then show ?thesis
unfolding frontier_def and closure_open_interval[OF False] and interior_open[OF open_box]
by auto
qed
lemma inter_interval_mixed_eq_empty:
fixes a :: "'a::euclidean_space"
assumes "box c d \<noteq> {}"
shows "box a b \<inter> cbox c d = {} \<longleftrightarrow> box a b \<inter> box c d = {}"
unfolding closure_open_interval[OF assms, symmetric]
unfolding open_inter_closure_eq_empty[OF open_box] ..
lemma diameter_closed_interval:
fixes a b::"'a::ordered_euclidean_space"
shows "a \<le> b \<Longrightarrow> diameter {a..b} = dist a b"
by (force simp add: diameter_def SUP_def simp del: Sup_image_eq intro!: cSup_eq_maximum setL2_mono
simp: euclidean_dist_l2[where 'a='a] eucl_le[where 'a='a] dist_norm)
text {* Intervals in general, including infinite and mixtures of open and closed. *}
definition "is_interval (s::('a::euclidean_space) set) \<longleftrightarrow>
(\<forall>a\<in>s. \<forall>b\<in>s. \<forall>x. (\<forall>i\<in>Basis. ((a\<bullet>i \<le> x\<bullet>i \<and> x\<bullet>i \<le> b\<bullet>i) \<or> (b\<bullet>i \<le> x\<bullet>i \<and> x\<bullet>i \<le> a\<bullet>i))) \<longrightarrow> x \<in> s)"
lemma is_interval_interval: "is_interval (cbox a (b::'a::euclidean_space))" (is ?th1)
"is_interval (box a b)" (is ?th2) proof -
show ?th1 ?th2 unfolding is_interval_def mem_box Ball_def atLeastAtMost_iff
by(meson order_trans le_less_trans less_le_trans less_trans)+ qed
lemma is_interval_empty:
"is_interval {}"
unfolding is_interval_def
by simp
lemma is_interval_univ:
"is_interval UNIV"
unfolding is_interval_def
by simp
lemma mem_is_intervalI:
assumes "is_interval s"
assumes "a \<in> s" "b \<in> s"
assumes "\<And>i. i \<in> Basis \<Longrightarrow> a \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> b \<bullet> i \<or> b \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> a \<bullet> i"
shows "x \<in> s"
by (rule assms(1)[simplified is_interval_def, rule_format, OF assms(2,3,4)])
lemma interval_subst:
fixes S::"'a::ordered_euclidean_space set"
assumes "is_interval S"
assumes "x \<in> S" "y j \<in> S"
assumes "j \<in> Basis"
shows "(\<Sum>i\<in>Basis. (if i = j then y i \<bullet> i else x \<bullet> i) *\<^sub>R i) \<in> S"
by (rule mem_is_intervalI[OF assms(1,2)]) (auto simp: assms)
lemma mem_box_componentwiseI:
fixes S::"'a::ordered_euclidean_space set"
assumes "is_interval S"
assumes "\<And>i. i \<in> Basis \<Longrightarrow> x \<bullet> i \<in> ((\<lambda>x. x \<bullet> i) ` S)"
shows "x \<in> S"
proof -
from assms have "\<forall>i \<in> Basis. \<exists>s \<in> S. x \<bullet> i = s \<bullet> i"
by auto
with finite_Basis obtain s and bs::"'a list" where
s: "\<And>i. i \<in> Basis \<Longrightarrow> x \<bullet> i = s i \<bullet> i" "\<And>i. i \<in> Basis \<Longrightarrow> s i \<in> S" and
bs: "set bs = Basis" "distinct bs"
by (metis finite_distinct_list)
from nonempty_Basis s obtain j where j: "j \<in> Basis" "s j \<in> S" by blast
def y \<equiv> "rec_list
(s j)
(\<lambda>j _ Y. (\<Sum>i\<in>Basis. (if i = j then s i \<bullet> i else Y \<bullet> i) *\<^sub>R i))"
have "x = (\<Sum>i\<in>Basis. (if i \<in> set bs then s i \<bullet> i else s j \<bullet> i) *\<^sub>R i)"
using bs by (auto simp add: s(1)[symmetric] euclidean_representation)
also have [symmetric]: "y bs = \<dots>"
using bs(2) bs(1)[THEN equalityD1]
by (induct bs) (auto simp: y_def euclidean_representation intro!: euclidean_eqI[where 'a='a])
also have "y bs \<in> S"
using bs(1)[THEN equalityD1]
apply (induct bs)
apply (auto simp: y_def j)
apply (rule interval_subst[OF assms(1)])
apply (auto simp: s)
done
finally show ?thesis .
qed
lemma eucl_less_eq_halfspaces:
fixes a :: "'a\<Colon>euclidean_space"
shows "{x. x <e a} = (\<Inter>i\<in>Basis. {x. x \<bullet> i < a \<bullet> i})"
"{x. a <e x} = (\<Inter>i\<in>Basis. {x. a \<bullet> i < x \<bullet> i})"
by (auto simp: eucl_less_def)
lemma eucl_le_eq_halfspaces:
fixes a :: "'a\<Colon>euclidean_space"
shows "{x. \<forall>i\<in>Basis. x \<bullet> i \<le> a \<bullet> i} = (\<Inter>i\<in>Basis. {x. x \<bullet> i \<le> a \<bullet> i})"
"{x. \<forall>i\<in>Basis. a \<bullet> i \<le> x \<bullet> i} = (\<Inter>i\<in>Basis. {x. a \<bullet> i \<le> x \<bullet> i})"
by auto
lemma open_Collect_eucl_less[simp, intro]:
fixes a :: "'a\<Colon>euclidean_space"
shows "open {x. x <e a}"
"open {x. a <e x}"
by (auto simp: eucl_less_eq_halfspaces open_halfspace_component_lt open_halfspace_component_gt)
lemma closed_Collect_eucl_le[simp, intro]:
fixes a :: "'a\<Colon>euclidean_space"
shows "closed {x. \<forall>i\<in>Basis. a \<bullet> i \<le> x \<bullet> i}"
"closed {x. \<forall>i\<in>Basis. x \<bullet> i \<le> a \<bullet> i}"
unfolding eucl_le_eq_halfspaces
by (simp_all add: closed_INT closed_Collect_le)
lemma image_affinity_cbox: fixes m::real
fixes a b c :: "'a::euclidean_space"
shows "(\<lambda>x. m *\<^sub>R x + c) ` cbox a b =
(if cbox a b = {} then {}
else (if 0 \<le> m then cbox (m *\<^sub>R a + c) (m *\<^sub>R b + c)
else cbox (m *\<^sub>R b + c) (m *\<^sub>R a + c)))"
proof (cases "m = 0")
case True
{
fix x
assume "\<forall>i\<in>Basis. x \<bullet> i \<le> c \<bullet> i" "\<forall>i\<in>Basis. c \<bullet> i \<le> x \<bullet> i"
then have "x = c"
apply -
apply (subst euclidean_eq_iff)
apply (auto intro: order_antisym)
done
}
moreover have "c \<in> cbox (m *\<^sub>R a + c) (m *\<^sub>R b + c)"
unfolding True by (auto simp add: cbox_sing)
ultimately show ?thesis using True by (auto simp: cbox_def)
next
case False
{
fix y
assume "\<forall>i\<in>Basis. a \<bullet> i \<le> y \<bullet> i" "\<forall>i\<in>Basis. y \<bullet> i \<le> b \<bullet> i" "m > 0"
then have "\<forall>i\<in>Basis. (m *\<^sub>R a + c) \<bullet> i \<le> (m *\<^sub>R y + c) \<bullet> i" and "\<forall>i\<in>Basis. (m *\<^sub>R y + c) \<bullet> i \<le> (m *\<^sub>R b + c) \<bullet> i"
by (auto simp: inner_distrib)
}
moreover
{
fix y
assume "\<forall>i\<in>Basis. a \<bullet> i \<le> y \<bullet> i" "\<forall>i\<in>Basis. y \<bullet> i \<le> b \<bullet> i" "m < 0"
then have "\<forall>i\<in>Basis. (m *\<^sub>R b + c) \<bullet> i \<le> (m *\<^sub>R y + c) \<bullet> i" and "\<forall>i\<in>Basis. (m *\<^sub>R y + c) \<bullet> i \<le> (m *\<^sub>R a + c) \<bullet> i"
by (auto simp add: mult_left_mono_neg inner_distrib)
}
moreover
{
fix y
assume "m > 0" and "\<forall>i\<in>Basis. (m *\<^sub>R a + c) \<bullet> i \<le> y \<bullet> i" and "\<forall>i\<in>Basis. y \<bullet> i \<le> (m *\<^sub>R b + c) \<bullet> i"
then have "y \<in> (\<lambda>x. m *\<^sub>R x + c) ` cbox a b"
unfolding image_iff Bex_def mem_box
apply (intro exI[where x="(1 / m) *\<^sub>R (y - c)"])
apply (auto simp add: pos_le_divide_eq pos_divide_le_eq mult_commute diff_le_iff inner_distrib inner_diff_left)
done
}
moreover
{
fix y
assume "\<forall>i\<in>Basis. (m *\<^sub>R b + c) \<bullet> i \<le> y \<bullet> i" "\<forall>i\<in>Basis. y \<bullet> i \<le> (m *\<^sub>R a + c) \<bullet> i" "m < 0"
then have "y \<in> (\<lambda>x. m *\<^sub>R x + c) ` cbox a b"
unfolding image_iff Bex_def mem_box
apply (intro exI[where x="(1 / m) *\<^sub>R (y - c)"])
apply (auto simp add: neg_le_divide_eq neg_divide_le_eq mult_commute diff_le_iff inner_distrib inner_diff_left)
done
}
ultimately show ?thesis using False by (auto simp: cbox_def)
qed
lemma image_smult_cbox:"(\<lambda>x. m *\<^sub>R (x::_::euclidean_space)) ` cbox a b =
(if cbox a b = {} then {} else if 0 \<le> m then cbox (m *\<^sub>R a) (m *\<^sub>R b) else cbox (m *\<^sub>R b) (m *\<^sub>R a))"
using image_affinity_cbox[of m 0 a b] by auto
text{* Instantiation for intervals on @{text ordered_euclidean_space} *}
lemma
fixes a :: "'a\<Colon>ordered_euclidean_space"
shows cbox_interval: "cbox a b = {a..b}"
and interval_cbox: "{a..b} = cbox a b"
and eucl_le_atMost: "{x. \<forall>i\<in>Basis. x \<bullet> i <= a \<bullet> i} = {..a}"
and eucl_le_atLeast: "{x. \<forall>i\<in>Basis. a \<bullet> i <= x \<bullet> i} = {a..}"
by (auto simp: eucl_le[where 'a='a] eucl_less_def box_def cbox_def)
lemma closed_eucl_atLeastAtMost[simp, intro]:
fixes a :: "'a\<Colon>ordered_euclidean_space"
shows "closed {a..b}"
by (simp add: cbox_interval[symmetric] closed_cbox)
lemma closed_eucl_atMost[simp, intro]:
fixes a :: "'a\<Colon>ordered_euclidean_space"
shows "closed {..a}"
by (simp add: eucl_le_atMost[symmetric])
lemma closed_eucl_atLeast[simp, intro]:
fixes a :: "'a\<Colon>ordered_euclidean_space"
shows "closed {a..}"
by (simp add: eucl_le_atLeast[symmetric])
lemma image_affinity_interval: fixes m::real
fixes a b c :: "'a::ordered_euclidean_space"
shows "(\<lambda>x. m *\<^sub>R x + c) ` {a..b} =
(if {a..b} = {} then {}
else (if 0 \<le> m then {m *\<^sub>R a + c .. m *\<^sub>R b + c}
else {m *\<^sub>R b + c .. m *\<^sub>R a + c}))"
using image_affinity_cbox[of m c a b]
by (simp add: cbox_interval)
lemma image_smult_interval:"(\<lambda>x. m *\<^sub>R (x::_::ordered_euclidean_space)) ` {a .. b} =
(if {a .. b} = {} then {} else if 0 \<le> m then {m *\<^sub>R a .. m *\<^sub>R b} else {m *\<^sub>R b .. m *\<^sub>R a})"
using image_smult_cbox[of m a b]
by (simp add: cbox_interval)
no_notation
eucl_less (infix "<e" 50)
end