Theory Ordered_Euclidean_Space
section ‹Ordered Euclidean Space›
theory Ordered_Euclidean_Space
imports
Convex_Euclidean_Space Abstract_Limits
"HOL-Library.Product_Order"
begin
text ‹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 ≤ y ⟷ (∀i∈Basis. x ∙ i ≤ y ∙ i)"
assumes eucl_less_le_not_le: "x < y ⟷ x ≤ y ∧ ¬ y ≤ x"
assumes eucl_inf: "inf x y = (∑i∈Basis. inf (x ∙ i) (y ∙ i) *⇩R i)"
assumes eucl_sup: "sup x y = (∑i∈Basis. sup (x ∙ i) (y ∙ i) *⇩R i)"
assumes eucl_Inf: "Inf X = (∑i∈Basis. (INF x∈X. x ∙ i) *⇩R i)"
assumes eucl_Sup: "Sup X = (∑i∈Basis. (SUP x∈X. x ∙ i) *⇩R i)"
assumes eucl_abs: "¦x¦ = (∑i∈Basis. ¦x ∙ i¦ *⇩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 ≠ {}"
hence "⋀i. (λx. x ∙ i) ` X ≠ {}" by simp
thus "(⋀x. x ∈ X ⟹ z ≤ x) ⟹ z ≤ Inf X" "(⋀x. x ∈ X ⟹ x ≤ z) ⟹ Sup X ≤ 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 ∈ Basis ⟹ inf x y ∙ i = inf (x ∙ i) (y ∙ i)"
and inner_Basis_sup_left: "i ∈ Basis ⟹ sup x y ∙ i = sup (x ∙ i) (y ∙ i)"
by (simp_all add: eucl_inf eucl_sup inner_sum_left inner_Basis if_distrib
cong: if_cong)
lemma inner_Basis_INF_left: "i ∈ Basis ⟹ (INF x∈X. f x) ∙ i = (INF x∈X. f x ∙ i)"
and inner_Basis_SUP_left: "i ∈ Basis ⟹ (SUP x∈X. f x) ∙ i = (SUP x∈X. f x ∙ i)"
using eucl_Sup [of "f ` X"] eucl_Inf [of "f ` X"] by (simp_all add: image_comp)
lemma abs_inner: "i ∈ Basis ⟹ ¦x¦ ∙ i = ¦x ∙ i¦"
by (auto simp: eucl_abs)
lemma
abs_scaleR: "¦a *⇩R b¦ = ¦a¦ *⇩R ¦b¦"
by (auto simp: eucl_abs abs_mult intro!: euclidean_eqI)
lemma interval_inner_leI:
assumes "x ∈ {a .. b}" "0 ≤ i"
shows "a∙i ≤ x∙i" "x∙i ≤ b∙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 ≤ a ⟹ 0 ≤ b ⟹ 0 ≤ a ∙ b"
using interval_inner_leI[of a 0 a b]
by auto
lemma inner_Basis_mono:
shows "a ≤ b ⟹ c ∈ Basis ⟹ a ∙ c ≤ b ∙ c"
by (simp add: eucl_le)
lemma Basis_nonneg[intro, simp]: "i ∈ Basis ⟹ 0 ≤ i"
by (auto simp: eucl_le inner_Basis)
lemma Sup_eq_maximum_componentwise:
fixes s::"'a set"
assumes i: "⋀b. b ∈ Basis ⟹ X ∙ b = i b ∙ b"
assumes sup: "⋀b x. b ∈ Basis ⟹ x ∈ s ⟹ x ∙ b ≤ X ∙ b"
assumes i_s: "⋀b. b ∈ Basis ⟹ (i b ∙ b) ∈ (λx. x ∙ 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: "⋀b. b ∈ Basis ⟹ X ∙ b = i b ∙ b"
assumes sup: "⋀b x. b ∈ Basis ⟹ x ∈ s ⟹ X ∙ b ≤ x ∙ b"
assumes i_s: "⋀b. b ∈ Basis ⟹ (i b ∙ b) ∈ (λx. x ∙ 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
proposition compact_attains_Inf_componentwise:
fixes b::"'a::ordered_euclidean_space"
assumes "b ∈ Basis" assumes "X ≠ {}" "compact X"
obtains x where "x ∈ X" "x ∙ b = Inf X ∙ b" "⋀y. y ∈ X ⟹ x ∙ b ≤ y ∙ b"
proof atomize_elim
let ?proj = "(λx. x ∙ b) ` X"
from assms have "compact ?proj" "?proj ≠ {}"
by (auto intro!: compact_continuous_image continuous_intros)
from compact_attains_inf[OF this]
obtain s x
where s: "s∈(λx. x ∙ b) ` X" "⋀t. t∈(λx. x ∙ b) ` X ⟹ s ≤ t"
and x: "x ∈ X" "s = x ∙ b" "⋀y. y ∈ X ⟹ x ∙ b ≤ y ∙ b"
by auto
hence "Inf ?proj = x ∙ b"
by (auto intro!: conditionally_complete_lattice_class.cInf_eq_minimum)
hence "x ∙ b = Inf X ∙ b"
by (auto simp: eucl_Inf inner_sum_left inner_Basis if_distrib ‹b ∈ Basis›
cong: if_cong)
with x show "∃x. x ∈ X ∧ x ∙ b = Inf X ∙ b ∧ (∀y. y ∈ X ⟶ x ∙ b ≤ y ∙ b)" by blast
qed
proposition
compact_attains_Sup_componentwise:
fixes b::"'a::ordered_euclidean_space"
assumes "b ∈ Basis" assumes "X ≠ {}" "compact X"
obtains x where "x ∈ X" "x ∙ b = Sup X ∙ b" "⋀y. y ∈ X ⟹ y ∙ b ≤ x ∙ b"
proof atomize_elim
let ?proj = "(λx. x ∙ b) ` X"
from assms have "compact ?proj" "?proj ≠ {}"
by (auto intro!: compact_continuous_image continuous_intros)
from compact_attains_sup[OF this]
obtain s x
where s: "s∈(λx. x ∙ b) ` X" "⋀t. t∈(λx. x ∙ b) ` X ⟹ t ≤ s"
and x: "x ∈ X" "s = x ∙ b" "⋀y. y ∈ X ⟹ y ∙ b ≤ x ∙ b"
by auto
hence "Sup ?proj = x ∙ b"
by (auto intro!: cSup_eq_maximum)
hence "x ∙ b = Sup X ∙ b"
by (auto simp: eucl_Sup[where 'a='a] inner_sum_left inner_Basis if_distrib ‹b ∈ Basis›
cong: if_cong)
with x show "∃x. x ∈ X ∧ x ∙ b = Sup X ∙ b ∧ (∀y. y ∈ X ⟶ y ∙ b ≤ x ∙ b)" by blast
qed
lemma tendsto_sup[tendsto_intros]:
fixes X :: "'a ⇒ 'b::ordered_euclidean_space"
assumes "(X ⤏ x) net" "(Y ⤏ y) net"
shows "((λi. sup (X i) (Y i)) ⤏ sup x y) net"
unfolding sup_max eucl_sup by (intro assms tendsto_intros)
lemma tendsto_inf[tendsto_intros]:
fixes X :: "'a ⇒ 'b::ordered_euclidean_space"
assumes "(X ⤏ x) net" "(Y ⤏ y) net"
shows "((λi. inf (X i) (Y i)) ⤏ inf x y) net"
unfolding inf_min eucl_inf by (intro assms tendsto_intros)
lemma tendsto_Inf[tendsto_intros]:
fixes f :: "'a ⇒ 'b ⇒ 'c::ordered_euclidean_space"
assumes "finite K" "⋀i. i ∈ K ⟹ ((λx. f x i) ⤏ l i) F"
shows "((λx. Inf (f x ` K)) ⤏ Inf (l ` K)) F"
using assms
by (induction K rule: finite_induct) (auto simp: cInf_insert_If tendsto_inf)
lemma tendsto_Sup[tendsto_intros]:
fixes f :: "'a ⇒ 'b ⇒ 'c::ordered_euclidean_space"
assumes "finite K" "⋀i. i ∈ K ⟹ ((λx. f x i) ⤏ l i) F"
shows "((λx. Sup (f x ` K)) ⤏ Sup (l ` K)) F"
using assms
by (induction K rule: finite_induct) (auto simp: cSup_insert_If tendsto_sup)
lemma continuous_map_Inf [continuous_intros]:
fixes f :: "'a ⇒ 'b ⇒ 'c::ordered_euclidean_space"
assumes "finite K" "⋀i. i ∈ K ⟹ continuous_map X euclidean (λx. f x i)"
shows "continuous_map X euclidean (λx. INF i∈K. f x i)"
using assms by (simp add: continuous_map_atin tendsto_Inf)
lemma continuous_map_Sup [continuous_intros]:
fixes f :: "'a ⇒ 'b ⇒ 'c::ordered_euclidean_space"
assumes "finite K" "⋀i. i ∈ K ⟹ continuous_map X euclidean (λx. f x i)"
shows "continuous_map X euclidean (λx. SUP i∈K. f x i)"
using assms by (simp add: continuous_map_atin tendsto_Sup)
lemma tendsto_componentwise_max:
assumes f: "(f ⤏ l) F" and g: "(g ⤏ m) F"
shows "((λx. (∑i∈Basis. max (f x ∙ i) (g x ∙ i) *⇩R i)) ⤏ (∑i∈Basis. max (l ∙ i) (m ∙ i) *⇩R i)) F"
by (intro tendsto_intros assms)
lemma tendsto_componentwise_min:
assumes f: "(f ⤏ l) F" and g: "(g ⤏ m) F"
shows "((λx. (∑i∈Basis. min (f x ∙ i) (g x ∙ i) *⇩R i)) ⤏ (∑i∈Basis. min (l ∙ i) (m ∙ i) *⇩R i)) F"
by (intro tendsto_intros assms)
instance real :: ordered_euclidean_space
by standard auto
lemma in_Basis_prod_iff:
fixes i::"'a::euclidean_space*'b::euclidean_space"
shows "i ∈ Basis ⟷ fst i = 0 ∧ snd i ∈ Basis ∨ snd i = 0 ∧ fst i ∈ Basis"
by (cases i) (auto simp: Basis_prod_def)
instantiation prod :: (abs, abs) abs
begin
definition "¦x¦ = (¦fst x¦, ¦snd x¦)"
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‹Instantiation for intervals on ‹ordered_euclidean_space››
proposition
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. ∀i∈Basis. x ∙ i <= a ∙ i} = {..a}"
and eucl_le_atLeast: "{x. ∀i∈Basis. a ∙ i <= x ∙ i} = {a..}"
by (auto simp: eucl_le[where 'a='a] eucl_less_def box_def cbox_def)
lemma sums_vec_nth :
assumes "f sums a"
shows "(λx. f x $ i) sums a $ i"
using assms unfolding sums_def
by (auto dest: tendsto_vec_nth [where i=i])
lemma summable_vec_nth :
assumes "summable f"
shows "summable (λx. f x $ i)"
using assms unfolding summable_def
by (blast intro: sums_vec_nth)
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:"(λx. m *⇩R (x::_::ordered_euclidean_space)) ` {a .. b} =
(if {a .. b} = {} then {} else if 0 ≤ m then {m *⇩R a .. m *⇩R b} else {m *⇩R b .. m *⇩R a})"
using image_smult_cbox[of m a b]
by (simp add: cbox_interval)
lemma [simp]:
fixes a b::"'a::ordered_euclidean_space"
shows is_interval_ic: "is_interval {..a}"
and is_interval_ci: "is_interval {a..}"
and is_interval_cc: "is_interval {b..a}"
by (force simp: is_interval_def eucl_le[where 'a='a])+
lemma connected_interval [simp]:
fixes a b::"'a::ordered_euclidean_space"
shows "connected {a..b}"
using is_interval_cc is_interval_connected by blast
lemma compact_interval [simp]:
fixes a b::"'a::ordered_euclidean_space"
shows "compact {a .. b}"
by (metis compact_cbox interval_cbox)
no_notation eucl_less (infix ‹<e› 50)
lemma One_nonneg: "0 ≤ (∑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 = (χ i. inf (x $ i) (y $ i))"
definition "sup x y = (χ i. sup (x $ i) (y $ i))"
definition "Inf X = (χ i. (INF x∈X. x $ i))"
definition "Sup X = (χ i. (SUP x∈X. x $ i))"
definition "¦x¦ = (χ i. ¦x $ i¦)"
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
end