Theory Product_Vector

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


header {* 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"
by (simp add: prod_eq_iff scaleR_right_distrib)
show "scaleR (a + b) x = scaleR a x + scaleR b x"
by (simp add: prod_eq_iff scaleR_left_distrib)
show "scaleR a (scaleR b x) = scaleR (a * b) x"
by (simp add: prod_eq_iff)
show "scaleR 1 x = x"
by (simp add: prod_eq_iff)
qed

end

subsection {* Product is a topological space *}

instantiation prod :: (topological_space, topological_space) topological_space
begin

definition open_prod_def:
"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 (\<Union>K)"
unfolding open_prod_def by fast
qed

end

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)"
by (simp add: fst_vimage_eq_Times open_Times)

lemma open_vimage_snd: "open S ==> open (snd -` S)"
by (simp add: snd_vimage_eq_Times open_Times)

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"
by (simp_all add: open_vimage_fst)
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"
by (simp_all add: open_vimage_snd)
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_on_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_on_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_on_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 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 *}

lemma mem_Times_iff: "x ∈ A × B <-> fst x ∈ A ∧ snd x ∈ B"
by (induct x) simp (* TODO: move elsewhere *)

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"
by (simp add: prod_eq_iff)
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"
by (simp add: prod_eq_iff)
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"
by (simp add: prod_eq_iff)
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

subsection {* Product is a metric space *}

instantiation prod :: (metric_space, metric_space) metric_space
begin

definition dist_prod_def:
"dist x y = sqrt ((dist (fst x) (fst y))2 + (dist (snd x) (snd y))2)"

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]
real_sqrt_le_mono add_mono power_mono dist_triangle2 zero_le_dist)
next
fix S :: "('a × 'b) set"
show "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"
by (simp_all add: r(2) s(2))
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 add: divide_pos_pos)
from `0 < e` have "e = sqrt (r2 + s2)"
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 (r2 + s2)`
by (simp add: add_strict_mono power_strict_mono)
thus "(a, b) ∈ S"
by (simp add: S)
qed
ultimately show "∃A B. open A ∧ open B ∧ x ∈ A × B ∧ A × B ⊆ S" by fast
qed
qed
qed

end

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"
then have "0 < r / sqrt 2" (is "0 < ?s")
by (simp add: divide_pos_pos)
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`]
by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
have 2: "(λn. snd (X n)) ----> lim (λn. snd (X n))"
using Cauchy_snd [OF `Cauchy X`]
by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
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:
"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
by (simp add: prod_eq_iff)
show "norm (x + y) ≤ norm x + norm y"
unfolding norm_prod_def
apply (rule order_trans [OF _ real_sqrt_sum_squares_triangle_ineq])
apply (simp add: add_mono power_mono norm_triangle_ineq)
done
show "norm (scaleR r x) = ¦r¦ * norm x"
unfolding norm_prod_def
apply (simp add: power_mult_distrib)
apply (simp add: distrib_left [symmetric])
apply (simp add: real_sqrt_mult_distrib)
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
by (simp add: dist_norm)
qed

end

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

subsubsection {* Pair operations are linear *}

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

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

text {* TODO: move to NthRoot *}
lemma sqrt_add_le_add_sqrt:
assumes x: "0 ≤ x" and y: "0 ≤ y"
shows "sqrt (x + y) ≤ sqrt x + sqrt y"
apply (rule power2_le_imp_le)
apply (simp add: power2_sum x y)
apply (simp add: mult_nonneg_nonneg x y)
apply (simp add: x y)
done

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)"
by (simp add: f.add g.add)
show "(f (r *R x), g (r *R x)) = r *R (f x, g x)"
by (simp add: f.scaleR g.scaleR)
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 (simp add: norm_Pair)
apply (rule order_trans [OF sqrt_add_le_add_sqrt], simp, simp)
apply (simp add: distrib_left)
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 FDERIV_Pair [FDERIV_intros]:
assumes f: "FDERIV f x : s :> f'" and g: "FDERIV g x : s :> g'"
shows "FDERIV (λx. (f x, g x)) x : s :> (λh. (f' h, g' h))"
proof (rule FDERIV_I_sandwich[of 1])
show "bounded_linear (λh. (f' h, g' h))"
using f g by (intro bounded_linear_Pair FDERIV_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: FDERIV_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)"
unfolding add_divide_distrib [symmetric]
by (simp add: norm_Pair divide_right_mono order_trans [OF sqrt_add_le_add_sqrt])
qed simp

lemmas FDERIV_fst [FDERIV_intros] = bounded_linear.FDERIV [OF bounded_linear_fst]
lemmas FDERIV_snd [FDERIV_intros] = bounded_linear.FDERIV [OF bounded_linear_snd]

lemma FDERIV_split [FDERIV_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
by (simp add: inner_commute)
show "inner (x + y) z = inner x z + inner y z"
unfolding inner_prod_def
by (simp add: inner_add_left)
show "inner (scaleR r x) y = r * inner x y"
unfolding inner_prod_def
by (simp add: distrib_left)
show "0 ≤ inner x x"
unfolding inner_prod_def
by (intro add_nonneg_nonneg inner_ge_zero)
show "inner x x = 0 <-> x = 0"
unfolding inner_prod_def prod_eq_iff
by (simp add: add_nonneg_eq_0_iff)
show "norm x = sqrt (inner x x)"
unfolding norm_prod_def inner_prod_def
by (simp add: power2_norm_eq_inner)
qed

end

end