# Theory Product_Vector

theory Product_Vector
imports Inner_Product Product_plus
```(*  Title:      HOL/Library/Product_Vector.thy
Author:     Brian Huffman
*)

section ‹Cartesian Products as Vector Spaces›

theory Product_Vector
imports Inner_Product Product_plus
begin

subsection ‹Product is a real vector space›

instantiation prod :: (real_vector, real_vector) real_vector
begin

definition scaleR_prod_def:
"scaleR r A = (scaleR r (fst A), scaleR r (snd A))"

lemma fst_scaleR [simp]: "fst (scaleR r A) = scaleR r (fst A)"
unfolding scaleR_prod_def by simp

lemma snd_scaleR [simp]: "snd (scaleR r A) = scaleR r (snd A)"
unfolding scaleR_prod_def by simp

lemma scaleR_Pair [simp]: "scaleR r (a, b) = (scaleR r a, scaleR r b)"
unfolding scaleR_prod_def by simp

instance
proof
fix a b :: real and x y :: "'a × 'b"
show "scaleR a (x + y) = scaleR a x + scaleR a y"
show "scaleR (a + b) x = scaleR a x + scaleR b x"
show "scaleR a (scaleR b x) = scaleR (a * b) x"
show "scaleR 1 x = x"
qed

end

subsection ‹Product is a topological space›

instantiation prod :: (topological_space, topological_space) topological_space
begin

definition open_prod_def[code del]:
"open (S :: ('a × 'b) set) ⟷
(∀x∈S. ∃A B. open A ∧ open B ∧ x ∈ A × B ∧ A × B ⊆ S)"

lemma open_prod_elim:
assumes "open S" and "x ∈ S"
obtains A B where "open A" and "open B" and "x ∈ A × B" and "A × B ⊆ S"
using assms unfolding open_prod_def by fast

lemma open_prod_intro:
assumes "⋀x. x ∈ S ⟹ ∃A B. open A ∧ open B ∧ x ∈ A × B ∧ A × B ⊆ S"
shows "open S"
using assms unfolding open_prod_def by fast

instance
proof
show "open (UNIV :: ('a × 'b) set)"
unfolding open_prod_def by auto
next
fix S T :: "('a × 'b) set"
assume "open S" "open T"
show "open (S ∩ T)"
proof (rule open_prod_intro)
fix x assume x: "x ∈ S ∩ T"
from x have "x ∈ S" by simp
obtain Sa Sb where A: "open Sa" "open Sb" "x ∈ Sa × Sb" "Sa × Sb ⊆ S"
using ‹open S› and ‹x ∈ S› by (rule open_prod_elim)
from x have "x ∈ T" by simp
obtain Ta Tb where B: "open Ta" "open Tb" "x ∈ Ta × Tb" "Ta × Tb ⊆ T"
using ‹open T› and ‹x ∈ T› by (rule open_prod_elim)
let ?A = "Sa ∩ Ta" and ?B = "Sb ∩ Tb"
have "open ?A ∧ open ?B ∧ x ∈ ?A × ?B ∧ ?A × ?B ⊆ S ∩ T"
using A B by (auto simp add: open_Int)
thus "∃A B. open A ∧ open B ∧ x ∈ A × B ∧ A × B ⊆ S ∩ T"
by fast
qed
next
fix K :: "('a × 'b) set set"
assume "∀S∈K. open S" thus "open (⋃K)"
unfolding open_prod_def by fast
qed

end

declare [[code abort: "open::('a::topological_space*'b::topological_space) set ⇒ bool"]]

lemma open_Times: "open S ⟹ open T ⟹ open (S × T)"
unfolding open_prod_def by auto

lemma fst_vimage_eq_Times: "fst -` S = S × UNIV"
by auto

lemma snd_vimage_eq_Times: "snd -` S = UNIV × S"
by auto

lemma open_vimage_fst: "open S ⟹ open (fst -` S)"

lemma open_vimage_snd: "open S ⟹ open (snd -` S)"

lemma closed_vimage_fst: "closed S ⟹ closed (fst -` S)"
unfolding closed_open vimage_Compl [symmetric]
by (rule open_vimage_fst)

lemma closed_vimage_snd: "closed S ⟹ closed (snd -` S)"
unfolding closed_open vimage_Compl [symmetric]
by (rule open_vimage_snd)

lemma closed_Times: "closed S ⟹ closed T ⟹ closed (S × T)"
proof -
have "S × T = (fst -` S) ∩ (snd -` T)" by auto
thus "closed S ⟹ closed T ⟹ closed (S × T)"
by (simp add: closed_vimage_fst closed_vimage_snd closed_Int)
qed

lemma subset_fst_imageI: "A × B ⊆ S ⟹ y ∈ B ⟹ A ⊆ fst ` S"
unfolding image_def subset_eq by force

lemma subset_snd_imageI: "A × B ⊆ S ⟹ x ∈ A ⟹ B ⊆ snd ` S"
unfolding image_def subset_eq by force

lemma open_image_fst: assumes "open S" shows "open (fst ` S)"
proof (rule openI)
fix x assume "x ∈ fst ` S"
then obtain y where "(x, y) ∈ S" by auto
then obtain A B where "open A" "open B" "x ∈ A" "y ∈ B" "A × B ⊆ S"
using ‹open S› unfolding open_prod_def by auto
from ‹A × B ⊆ S› ‹y ∈ B› have "A ⊆ fst ` S" by (rule subset_fst_imageI)
with ‹open A› ‹x ∈ A› have "open A ∧ x ∈ A ∧ A ⊆ fst ` S" by simp
then show "∃T. open T ∧ x ∈ T ∧ T ⊆ fst ` S" by - (rule exI)
qed

lemma open_image_snd: assumes "open S" shows "open (snd ` S)"
proof (rule openI)
fix y assume "y ∈ snd ` S"
then obtain x where "(x, y) ∈ S" by auto
then obtain A B where "open A" "open B" "x ∈ A" "y ∈ B" "A × B ⊆ S"
using ‹open S› unfolding open_prod_def by auto
from ‹A × B ⊆ S› ‹x ∈ A› have "B ⊆ snd ` S" by (rule subset_snd_imageI)
with ‹open B› ‹y ∈ B› have "open B ∧ y ∈ B ∧ B ⊆ snd ` S" by simp
then show "∃T. open T ∧ y ∈ T ∧ T ⊆ snd ` S" by - (rule exI)
qed

subsubsection ‹Continuity of operations›

lemma tendsto_fst [tendsto_intros]:
assumes "(f ⤏ a) F"
shows "((λx. fst (f x)) ⤏ fst a) F"
proof (rule topological_tendstoI)
fix S assume "open S" and "fst a ∈ S"
then have "open (fst -` S)" and "a ∈ fst -` S"
with assms have "eventually (λx. f x ∈ fst -` S) F"
by (rule topological_tendstoD)
then show "eventually (λx. fst (f x) ∈ S) F"
by simp
qed

lemma tendsto_snd [tendsto_intros]:
assumes "(f ⤏ a) F"
shows "((λx. snd (f x)) ⤏ snd a) F"
proof (rule topological_tendstoI)
fix S assume "open S" and "snd a ∈ S"
then have "open (snd -` S)" and "a ∈ snd -` S"
with assms have "eventually (λx. f x ∈ snd -` S) F"
by (rule topological_tendstoD)
then show "eventually (λx. snd (f x) ∈ S) F"
by simp
qed

lemma tendsto_Pair [tendsto_intros]:
assumes "(f ⤏ a) F" and "(g ⤏ b) F"
shows "((λx. (f x, g x)) ⤏ (a, b)) F"
proof (rule topological_tendstoI)
fix S assume "open S" and "(a, b) ∈ S"
then obtain A B where "open A" "open B" "a ∈ A" "b ∈ B" "A × B ⊆ S"
unfolding open_prod_def by fast
have "eventually (λx. f x ∈ A) F"
using ‹(f ⤏ a) F› ‹open A› ‹a ∈ A›
by (rule topological_tendstoD)
moreover
have "eventually (λx. g x ∈ B) F"
using ‹(g ⤏ b) F› ‹open B› ‹b ∈ B›
by (rule topological_tendstoD)
ultimately
show "eventually (λx. (f x, g x) ∈ S) F"
by (rule eventually_elim2)
(simp add: subsetD [OF ‹A × B ⊆ S›])
qed

lemma continuous_fst[continuous_intros]: "continuous F f ⟹ continuous F (λx. fst (f x))"
unfolding continuous_def by (rule tendsto_fst)

lemma continuous_snd[continuous_intros]: "continuous F f ⟹ continuous F (λx. snd (f x))"
unfolding continuous_def by (rule tendsto_snd)

lemma continuous_Pair[continuous_intros]: "continuous F f ⟹ continuous F g ⟹ continuous F (λx. (f x, g x))"
unfolding continuous_def by (rule tendsto_Pair)

lemma continuous_on_fst[continuous_intros]: "continuous_on s f ⟹ continuous_on s (λx. fst (f x))"
unfolding continuous_on_def by (auto intro: tendsto_fst)

lemma continuous_on_snd[continuous_intros]: "continuous_on s f ⟹ continuous_on s (λx. snd (f x))"
unfolding continuous_on_def by (auto intro: tendsto_snd)

lemma continuous_on_Pair[continuous_intros]: "continuous_on s f ⟹ continuous_on s g ⟹ continuous_on s (λx. (f x, g x))"
unfolding continuous_on_def by (auto intro: tendsto_Pair)

lemma continuous_on_swap[continuous_intros]: "continuous_on A prod.swap"
by (simp add: prod.swap_def continuous_on_fst continuous_on_snd continuous_on_Pair continuous_on_id)

lemma isCont_fst [simp]: "isCont f a ⟹ isCont (λx. fst (f x)) a"
by (fact continuous_fst)

lemma isCont_snd [simp]: "isCont f a ⟹ isCont (λx. snd (f x)) a"
by (fact continuous_snd)

lemma isCont_Pair [simp]: "⟦isCont f a; isCont g a⟧ ⟹ isCont (λx. (f x, g x)) a"
by (fact continuous_Pair)

subsubsection ‹Separation axioms›

instance prod :: (t0_space, t0_space) t0_space
proof
fix x y :: "'a × 'b" assume "x ≠ y"
hence "fst x ≠ fst y ∨ snd x ≠ snd y"
thus "∃U. open U ∧ (x ∈ U) ≠ (y ∈ U)"
by (fast dest: t0_space elim: open_vimage_fst open_vimage_snd)
qed

instance prod :: (t1_space, t1_space) t1_space
proof
fix x y :: "'a × 'b" assume "x ≠ y"
hence "fst x ≠ fst y ∨ snd x ≠ snd y"
thus "∃U. open U ∧ x ∈ U ∧ y ∉ U"
by (fast dest: t1_space elim: open_vimage_fst open_vimage_snd)
qed

instance prod :: (t2_space, t2_space) t2_space
proof
fix x y :: "'a × 'b" assume "x ≠ y"
hence "fst x ≠ fst y ∨ snd x ≠ snd y"
thus "∃U V. open U ∧ open V ∧ x ∈ U ∧ y ∈ V ∧ U ∩ V = {}"
by (fast dest: hausdorff elim: open_vimage_fst open_vimage_snd)
qed

lemma isCont_swap[continuous_intros]: "isCont prod.swap a"
using continuous_on_eq_continuous_within continuous_on_swap by blast

subsection ‹Product is a metric space›

(* TODO: Product of uniform spaces and compatibility with metric_spaces! *)

instantiation prod :: (metric_space, metric_space) dist
begin

definition dist_prod_def[code del]:
"dist x y = sqrt ((dist (fst x) (fst y))⇧2 + (dist (snd x) (snd y))⇧2)"

instance ..
end

instantiation prod :: (metric_space, metric_space) uniformity_dist
begin

definition [code del]:
"(uniformity :: (('a × 'b) × ('a × 'b)) filter) =
(INF e:{0 <..}. principal {(x, y). dist x y < e})"

instance
by standard (rule uniformity_prod_def)
end

declare uniformity_Abort[where 'a="'a :: metric_space × 'b :: metric_space", code]

instantiation prod :: (metric_space, metric_space) metric_space
begin

lemma dist_Pair_Pair: "dist (a, b) (c, d) = sqrt ((dist a c)⇧2 + (dist b d)⇧2)"
unfolding dist_prod_def by simp

lemma dist_fst_le: "dist (fst x) (fst y) ≤ dist x y"
unfolding dist_prod_def by (rule real_sqrt_sum_squares_ge1)

lemma dist_snd_le: "dist (snd x) (snd y) ≤ dist x y"
unfolding dist_prod_def by (rule real_sqrt_sum_squares_ge2)

instance
proof
fix x y :: "'a × 'b"
show "dist x y = 0 ⟷ x = y"
unfolding dist_prod_def prod_eq_iff by simp
next
fix x y z :: "'a × 'b"
show "dist x y ≤ dist x z + dist y z"
unfolding dist_prod_def
by (intro order_trans [OF _ real_sqrt_sum_squares_triangle_ineq]
next
fix S :: "('a × 'b) set"
have *: "open S ⟷ (∀x∈S. ∃e>0. ∀y. dist y x < e ⟶ y ∈ S)"
proof
assume "open S" show "∀x∈S. ∃e>0. ∀y. dist y x < e ⟶ y ∈ S"
proof
fix x assume "x ∈ S"
obtain A B where "open A" "open B" "x ∈ A × B" "A × B ⊆ S"
using ‹open S› and ‹x ∈ S› by (rule open_prod_elim)
obtain r where r: "0 < r" "∀y. dist y (fst x) < r ⟶ y ∈ A"
using ‹open A› and ‹x ∈ A × B› unfolding open_dist by auto
obtain s where s: "0 < s" "∀y. dist y (snd x) < s ⟶ y ∈ B"
using ‹open B› and ‹x ∈ A × B› unfolding open_dist by auto
let ?e = "min r s"
have "0 < ?e ∧ (∀y. dist y x < ?e ⟶ y ∈ S)"
proof (intro allI impI conjI)
show "0 < min r s" by (simp add: r(1) s(1))
next
fix y assume "dist y x < min r s"
hence "dist y x < r" and "dist y x < s"
by simp_all
hence "dist (fst y) (fst x) < r" and "dist (snd y) (snd x) < s"
by (auto intro: le_less_trans dist_fst_le dist_snd_le)
hence "fst y ∈ A" and "snd y ∈ B"
hence "y ∈ A × B" by (induct y, simp)
with ‹A × B ⊆ S› show "y ∈ S" ..
qed
thus "∃e>0. ∀y. dist y x < e ⟶ y ∈ S" ..
qed
next
assume *: "∀x∈S. ∃e>0. ∀y. dist y x < e ⟶ y ∈ S" show "open S"
proof (rule open_prod_intro)
fix x assume "x ∈ S"
then obtain e where "0 < e" and S: "∀y. dist y x < e ⟶ y ∈ S"
using * by fast
def r ≡ "e / sqrt 2" and s ≡ "e / sqrt 2"
from ‹0 < e› have "0 < r" and "0 < s"
unfolding r_def s_def by simp_all
from ‹0 < e› have "e = sqrt (r⇧2 + s⇧2)"
unfolding r_def s_def by (simp add: power_divide)
def A ≡ "{y. dist (fst x) y < r}" and B ≡ "{y. dist (snd x) y < s}"
have "open A" and "open B"
unfolding A_def B_def by (simp_all add: open_ball)
moreover have "x ∈ A × B"
unfolding A_def B_def mem_Times_iff
using ‹0 < r› and ‹0 < s› by simp
moreover have "A × B ⊆ S"
proof (clarify)
fix a b assume "a ∈ A" and "b ∈ B"
hence "dist a (fst x) < r" and "dist b (snd x) < s"
unfolding A_def B_def by (simp_all add: dist_commute)
hence "dist (a, b) x < e"
unfolding dist_prod_def ‹e = sqrt (r⇧2 + s⇧2)›
thus "(a, b) ∈ S"
qed
ultimately show "∃A B. open A ∧ open B ∧ x ∈ A × B ∧ A × B ⊆ S" by fast
qed
qed
show "open S = (∀x∈S. ∀⇩F (x', y) in uniformity. x' = x ⟶ y ∈ S)"
unfolding * eventually_uniformity_metric
by (simp del: split_paired_All add: dist_prod_def dist_commute)
qed

end

declare [[code abort: "dist::('a::metric_space*'b::metric_space)⇒('a*'b) ⇒ real"]]

lemma Cauchy_fst: "Cauchy X ⟹ Cauchy (λn. fst (X n))"
unfolding Cauchy_def by (fast elim: le_less_trans [OF dist_fst_le])

lemma Cauchy_snd: "Cauchy X ⟹ Cauchy (λn. snd (X n))"
unfolding Cauchy_def by (fast elim: le_less_trans [OF dist_snd_le])

lemma Cauchy_Pair:
assumes "Cauchy X" and "Cauchy Y"
shows "Cauchy (λn. (X n, Y n))"
proof (rule metric_CauchyI)
fix r :: real assume "0 < r"
hence "0 < r / sqrt 2" (is "0 < ?s") by simp
obtain M where M: "∀m≥M. ∀n≥M. dist (X m) (X n) < ?s"
using metric_CauchyD [OF ‹Cauchy X› ‹0 < ?s›] ..
obtain N where N: "∀m≥N. ∀n≥N. dist (Y m) (Y n) < ?s"
using metric_CauchyD [OF ‹Cauchy Y› ‹0 < ?s›] ..
have "∀m≥max M N. ∀n≥max M N. dist (X m, Y m) (X n, Y n) < r"
using M N by (simp add: real_sqrt_sum_squares_less dist_Pair_Pair)
then show "∃n0. ∀m≥n0. ∀n≥n0. dist (X m, Y m) (X n, Y n) < r" ..
qed

subsection ‹Product is a complete metric space›

instance prod :: (complete_space, complete_space) complete_space
proof
fix X :: "nat ⇒ 'a × 'b" assume "Cauchy X"
have 1: "(λn. fst (X n)) ⇢ lim (λn. fst (X n))"
using Cauchy_fst [OF ‹Cauchy X›]
have 2: "(λn. snd (X n)) ⇢ lim (λn. snd (X n))"
using Cauchy_snd [OF ‹Cauchy X›]
have "X ⇢ (lim (λn. fst (X n)), lim (λn. snd (X n)))"
using tendsto_Pair [OF 1 2] by simp
then show "convergent X"
by (rule convergentI)
qed

subsection ‹Product is a normed vector space›

instantiation prod :: (real_normed_vector, real_normed_vector) real_normed_vector
begin

definition norm_prod_def[code del]:
"norm x = sqrt ((norm (fst x))⇧2 + (norm (snd x))⇧2)"

definition sgn_prod_def:
"sgn (x::'a × 'b) = scaleR (inverse (norm x)) x"

lemma norm_Pair: "norm (a, b) = sqrt ((norm a)⇧2 + (norm b)⇧2)"
unfolding norm_prod_def by simp

instance
proof
fix r :: real and x y :: "'a × 'b"
show "norm x = 0 ⟷ x = 0"
unfolding norm_prod_def
show "norm (x + y) ≤ norm x + norm y"
unfolding norm_prod_def
apply (rule order_trans [OF _ real_sqrt_sum_squares_triangle_ineq])
done
show "norm (scaleR r x) = ¦r¦ * norm x"
unfolding norm_prod_def
done
show "sgn x = scaleR (inverse (norm x)) x"
by (rule sgn_prod_def)
show "dist x y = norm (x - y)"
unfolding dist_prod_def norm_prod_def
qed

end

declare [[code abort: "norm::('a::real_normed_vector*'b::real_normed_vector) ⇒ real"]]

instance prod :: (banach, banach) banach ..

subsubsection ‹Pair operations are linear›

lemma bounded_linear_fst: "bounded_linear fst"
by (rule bounded_linear_intro [where K=1], simp add: norm_prod_def)

lemma bounded_linear_snd: "bounded_linear snd"
by (rule bounded_linear_intro [where K=1], simp add: norm_prod_def)

lemmas bounded_linear_fst_comp = bounded_linear_fst[THEN bounded_linear_compose]

lemmas bounded_linear_snd_comp = bounded_linear_snd[THEN bounded_linear_compose]

lemma bounded_linear_Pair:
assumes f: "bounded_linear f"
assumes g: "bounded_linear g"
shows "bounded_linear (λx. (f x, g x))"
proof
interpret f: bounded_linear f by fact
interpret g: bounded_linear g by fact
fix x y and r :: real
show "(f (x + y), g (x + y)) = (f x, g x) + (f y, g y)"
show "(f (r *⇩R x), g (r *⇩R x)) = r *⇩R (f x, g x)"
obtain Kf where "0 < Kf" and norm_f: "⋀x. norm (f x) ≤ norm x * Kf"
using f.pos_bounded by fast
obtain Kg where "0 < Kg" and norm_g: "⋀x. norm (g x) ≤ norm x * Kg"
using g.pos_bounded by fast
have "∀x. norm (f x, g x) ≤ norm x * (Kf + Kg)"
apply (rule allI)
apply (rule add_mono [OF norm_f norm_g])
done
then show "∃K. ∀x. norm (f x, g x) ≤ norm x * K" ..
qed

subsubsection ‹Frechet derivatives involving pairs›

lemma has_derivative_Pair [derivative_intros]:
assumes f: "(f has_derivative f') (at x within s)" and g: "(g has_derivative g') (at x within s)"
shows "((λx. (f x, g x)) has_derivative (λh. (f' h, g' h))) (at x within s)"
proof (rule has_derivativeI_sandwich[of 1])
show "bounded_linear (λh. (f' h, g' h))"
using f g by (intro bounded_linear_Pair has_derivative_bounded_linear)
let ?Rf = "λy. f y - f x - f' (y - x)"
let ?Rg = "λy. g y - g x - g' (y - x)"
let ?R = "λy. ((f y, g y) - (f x, g x) - (f' (y - x), g' (y - x)))"

show "((λy. norm (?Rf y) / norm (y - x) + norm (?Rg y) / norm (y - x)) ⤏ 0) (at x within s)"
using f g by (intro tendsto_add_zero) (auto simp: has_derivative_iff_norm)

fix y :: 'a assume "y ≠ x"
show "norm (?R y) / norm (y - x) ≤ norm (?Rf y) / norm (y - x) + norm (?Rg y) / norm (y - x)"
qed simp

lemmas has_derivative_fst [derivative_intros] = bounded_linear.has_derivative [OF bounded_linear_fst]
lemmas has_derivative_snd [derivative_intros] = bounded_linear.has_derivative [OF bounded_linear_snd]

lemma has_derivative_split [derivative_intros]:
"((λp. f (fst p) (snd p)) has_derivative f') F ⟹ ((λ(a, b). f a b) has_derivative f') F"
unfolding split_beta' .

subsection ‹Product is an inner product space›

instantiation prod :: (real_inner, real_inner) real_inner
begin

definition inner_prod_def:
"inner x y = inner (fst x) (fst y) + inner (snd x) (snd y)"

lemma inner_Pair [simp]: "inner (a, b) (c, d) = inner a c + inner b d"
unfolding inner_prod_def by simp

instance
proof
fix r :: real
fix x y z :: "'a::real_inner × 'b::real_inner"
show "inner x y = inner y x"
unfolding inner_prod_def
show "inner (x + y) z = inner x z + inner y z"
unfolding inner_prod_def
show "inner (scaleR r x) y = r * inner x y"
unfolding inner_prod_def
show "0 ≤ inner x x"
unfolding inner_prod_def
show "inner x x = 0 ⟷ x = 0"
unfolding inner_prod_def prod_eq_iff
show "norm x = sqrt (inner x x)"
unfolding norm_prod_def inner_prod_def
qed

end

lemma inner_Pair_0: "inner x (0, b) = inner (snd x) b" "inner x (a, 0) = inner (fst x) a"
by (cases x, simp)+

lemma
fixes x :: "'a::real_normed_vector"
shows norm_Pair1 [simp]: "norm (0,x) = norm x"
and norm_Pair2 [simp]: "norm (x,0) = norm x"
by (auto simp: norm_Pair)

lemma norm_commute: "norm (x,y) = norm (y,x)"