theory Ordered_Euclidean_Space
imports
Cartesian_Euclidean_Space
"HOL-Library.Product_Order"
begin
subsection \<open>An ordering on euclidean spaces that will allow us to talk about intervals\<close>
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: "\<bar>x\<bar> = (\<Sum>i\<in>Basis. \<bar>x \<bullet> i\<bar> *\<^sub>R i)"
begin
subclass order
by standard
(auto simp: eucl_le eucl_less_le_not_le intro!: euclidean_eqI antisym intro: order.trans)
subclass ordered_ab_group_add_abs
by standard (auto simp: eucl_le inner_add_left eucl_abs abs_leI)
subclass ordered_real_vector
by standard (auto simp: eucl_le intro!: mult_left_mono mult_right_mono)
subclass lattice
by standard (auto simp: eucl_inf eucl_sup eucl_le)
subclass distrib_lattice
by standard (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
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)+
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_sum_left inner_Basis if_distrib comm_monoid_add_class.sum.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> \<bar>x\<bar> \<bullet> i = \<bar>x \<bullet> i\<bar>"
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!: ordered_comm_monoid_add_class.sum_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_sum
by (auto 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_sum
by (auto 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_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)
hence "x \<bullet> b = Inf X \<bullet> b"
by (auto simp: eucl_Inf inner_sum_left inner_Basis if_distrib \<open>b \<in> Basis\<close> sum.delta
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_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)
hence "x \<bullet> b = Sup X \<bullet> b"
by (auto simp: eucl_Sup[where 'a='a] inner_sum_left inner_Basis if_distrib \<open>b \<in> Basis\<close> sum.delta
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 standard auto
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 "\<bar>x\<bar> = (\<bar>fst x\<bar>, \<bar>snd x\<bar>)"
instance ..
end
instance prod :: (ordered_euclidean_space, ordered_euclidean_space) ordered_euclidean_space
by standard
(auto intro!: add_mono simp add: euclidean_representation_sum' 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)
text\<open>Instantiation for intervals on \<open>ordered_euclidean_space\<close>\<close>
lemma
fixes a :: "'a::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 vec_nth_real_1_iff_cbox [simp]:
fixes a b :: real
shows "(\<lambda>x::real^1. x $ 1) ` S = {a..b} \<longleftrightarrow> S = cbox (vec a) (vec b)"
by (metis interval_cbox vec_nth_1_iff_cbox)
lemma closed_eucl_atLeastAtMost[simp, intro]:
fixes a :: "'a::ordered_euclidean_space"
shows "closed {a..b}"
by (simp add: cbox_interval[symmetric] closed_cbox)
lemma closed_eucl_atMost[simp, intro]:
fixes a :: "'a::ordered_euclidean_space"
shows "closed {..a}"
by (simp add: closed_interval_left eucl_le_atMost[symmetric])
lemma closed_eucl_atLeast[simp, intro]:
fixes a :: "'a::ordered_euclidean_space"
shows "closed {a..}"
by (simp add: closed_interval_right eucl_le_atLeast[symmetric])
lemma bounded_closed_interval [simp]:
fixes a :: "'a::ordered_euclidean_space"
shows "bounded {a .. b}"
using bounded_cbox[of a b]
by (metis interval_cbox)
lemma convex_closed_interval [simp]:
fixes a :: "'a::ordered_euclidean_space"
shows "convex {a .. b}"
using convex_box[of a b]
by (metis interval_cbox)
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)
lemma [simp]:
fixes a b::"'a::ordered_euclidean_space" and r s::real
shows is_interval_io: "is_interval {..<r}"
and is_interval_ic: "is_interval {..a}"
and is_interval_oi: "is_interval {r<..}"
and is_interval_ci: "is_interval {a..}"
and is_interval_oo: "is_interval {r<..<s}"
and is_interval_oc: "is_interval {r<..s}"
and is_interval_co: "is_interval {r..<s}"
and is_interval_cc: "is_interval {b..a}"
by (force simp: is_interval_def eucl_le[where 'a='a])+
lemma is_interval_real_ereal_oo: "is_interval (real_of_ereal ` {N<..<M::ereal})"
by (auto simp: real_atLeastGreaterThan_eq)
lemma compact_interval [simp]:
fixes a b::"'a::ordered_euclidean_space"
shows "compact {a .. b}"
by (metis compact_cbox interval_cbox)
lemma homeomorphic_closed_intervals:
fixes a :: "'a::euclidean_space" and b and c :: "'a::euclidean_space" and d
assumes "box a b \<noteq> {}" and "box c d \<noteq> {}"
shows "(cbox a b) homeomorphic (cbox c d)"
apply (rule homeomorphic_convex_compact)
using assms apply auto
done
lemma homeomorphic_closed_intervals_real:
fixes a::real and b and c::real and d
assumes "a<b" and "c<d"
shows "{a..b} homeomorphic {c..d}"
by (metis assms compact_interval continuous_on_id convex_real_interval(5) emptyE homeomorphic_convex_compact interior_atLeastAtMost_real mvt)
no_notation
eucl_less (infix "<e" 50)
lemma One_nonneg: "0 \<le> (\<Sum>Basis::'a::ordered_euclidean_space)"
by (auto intro: sum_nonneg)
lemma
fixes a b::"'a::ordered_euclidean_space"
shows bdd_above_cbox[intro, simp]: "bdd_above (cbox a b)"
and bdd_below_cbox[intro, simp]: "bdd_below (cbox a b)"
and bdd_above_box[intro, simp]: "bdd_above (box a b)"
and bdd_below_box[intro, simp]: "bdd_below (box a b)"
unfolding atomize_conj
by (metis bdd_above_Icc bdd_above_mono bdd_below_Icc bdd_below_mono bounded_box
bounded_subset_cbox_symmetric interval_cbox)
instantiation vec :: (ordered_euclidean_space, finite) ordered_euclidean_space
begin
definition "inf x y = (\<chi> i. inf (x $ i) (y $ i))"
definition "sup x y = (\<chi> i. sup (x $ i) (y $ i))"
definition "Inf X = (\<chi> i. (INF x:X. x $ i))"
definition "Sup X = (\<chi> i. (SUP x:X. x $ i))"
definition "\<bar>x\<bar> = (\<chi> i. \<bar>x $ i\<bar>)"
instance
apply standard
unfolding euclidean_representation_sum'
apply (auto simp: less_eq_vec_def inf_vec_def sup_vec_def Inf_vec_def Sup_vec_def inner_axis
Basis_vec_def inner_Basis_inf_left inner_Basis_sup_left inner_Basis_INF_left
inner_Basis_SUP_left eucl_le[where 'a='a] less_le_not_le abs_vec_def abs_inner)
done
end
lemma ANR_interval [iff]:
fixes a :: "'a::ordered_euclidean_space"
shows "ANR{a..b}"
by (simp add: interval_cbox)
lemma ENR_interval [iff]:
fixes a :: "'a::ordered_euclidean_space"
shows "ENR{a..b}"
by (auto simp: interval_cbox)
end