# Theory HOL-Analysis.Product_Vector

```(*  Title:      HOL/Analysis/Product_Vector.thy
Author:     Brian Huffman
Dominique Unruh, University of Tartu
*)

section ‹Cartesian Products as Vector Spaces›

theory Product_Vector
imports
Complex_Main
"HOL-Library.Product_Plus"
begin

lemma Times_eq_image_sum:
fixes S :: "'a :: comm_monoid_add set" and T :: "'b :: comm_monoid_add set"
shows "S × T = {u + v |u v. u ∈ (λx. (x, 0)) ` S ∧ v ∈ Pair 0 ` T}"
by force

subsection ‹Product is a Module›

locale module_prod = module_pair begin

definition scale :: "'a ⇒ 'b × 'c ⇒ 'b × 'c"
where "scale a v = (s1 a (fst v), s2 a (snd v))"

lemma✐‹tag important› scale_prod: "scale x (a, b) = (s1 x a, s2 x b)"
by (auto simp: scale_def)

sublocale✐‹tag important› p: module scale
m1.scale_left_distrib m1.scale_right_distrib m2.scale_left_distrib m2.scale_right_distrib)

lemma subspace_Times: "m1.subspace A ⟹ m2.subspace B ⟹ p.subspace (A × B)"
unfolding m1.subspace_def m2.subspace_def p.subspace_def
by (auto simp: zero_prod_def scale_def)

lemma module_hom_fst: "module_hom scale s1 fst"
by unfold_locales (auto simp: scale_def)

lemma module_hom_snd: "module_hom scale s2 snd"
by unfold_locales (auto simp: scale_def)

end

locale vector_space_prod = vector_space_pair begin

sublocale module_prod s1 s2
rewrites "module_hom = Vector_Spaces.linear"
by unfold_locales (fact module_hom_eq_linear)

sublocale p: vector_space scale by unfold_locales (auto simp: algebra_simps)

lemmas linear_fst = module_hom_fst
and linear_snd = module_hom_snd

end

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

proposition 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

lemma module_prod_scale_eq_scaleR: "module_prod.scale (*⇩R) (*⇩R) = scaleR"
apply (rule ext) apply (rule ext)
apply (subst module_prod.scale_def)
subgoal by unfold_locales

interpretation real_vector?: vector_space_prod "scaleR::_⇒_⇒'a::real_vector" "scaleR::_⇒_⇒'b::real_vector"
rewrites "scale = ((*⇩R)::_⇒_⇒('a × 'b))"
and "module.dependent (*⇩R) = dependent"
and "module.representation (*⇩R) = representation"
and "module.subspace (*⇩R) = subspace"
and "module.span (*⇩R) = span"
and "vector_space.extend_basis (*⇩R) = extend_basis"
and "vector_space.dim (*⇩R) = dim"
and "Vector_Spaces.linear (*⇩R) (*⇩R) = linear"
subgoal by unfold_locales
subgoal by (fact module_prod_scale_eq_scaleR)
unfolding dependent_raw_def representation_raw_def subspace_raw_def span_raw_def
extend_basis_raw_def dim_raw_def linear_def
by (rule refl)+

subsection ‹Product is a Metric Space›

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

instantiation✐‹tag unimportant› 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✐‹tag unimportant› prod :: (uniformity, uniformity) uniformity begin

definition [code del]: ‹(uniformity :: (('a × 'b) × ('a × 'b)) filter) =
filtermap (λ((x1,x2),(y1,y2)). ((x1,y1),(x2,y2))) (uniformity ×⇩F uniformity)›

instance..
end

subsubsection ‹Uniform spaces›

instantiation✐‹tag unimportant› prod :: (uniform_space, uniform_space) uniform_space
begin
instance
proof standard
fix U :: ‹('a × 'b) set›
show ‹open U ⟷ (∀x∈U. ∀⇩F (x', y) in uniformity. x' = x ⟶ y ∈ U)›
proof (intro iffI ballI)
fix x assume ‹open U› and ‹x ∈ U›
then obtain A B where ‹open A› ‹open B› ‹x ∈ A×B› ‹A×B ⊆ U›
by (metis open_prod_elim)
define UA where ‹UA = (λ(x'::'a,y). x' = fst x ⟶ y ∈ A)›
from ‹open A› ‹x ∈ A×B›
have ‹eventually UA uniformity›
unfolding open_uniformity UA_def by auto
define UB where ‹UB = (λ(x'::'b,y). x' = snd x ⟶ y ∈ B)›
from ‹open A› ‹open B› ‹x ∈ A×B›
have ‹eventually UA uniformity› ‹eventually UB uniformity›
unfolding open_uniformity UA_def UB_def by auto
then have ‹∀⇩F ((x'1, y1), (x'2, y2)) in uniformity ×⇩F uniformity. (x'1,x'2) = x ⟶ (y1,y2) ∈ U›
apply (auto intro!: exI[of _ UA] exI[of _ UB] simp add: eventually_prod_filter)
using ‹A×B ⊆ U› by (auto simp: UA_def UB_def)
then show ‹∀⇩F (x', y) in uniformity. x' = x ⟶ y ∈ U›
by (simp add: uniformity_prod_def eventually_filtermap case_prod_unfold)
next
assume asm: ‹∀x∈U. ∀⇩F (x', y) in uniformity. x' = x ⟶ y ∈ U›
show ‹open U›
proof (unfold open_prod_def, intro ballI)
fix x assume ‹x ∈ U›
with asm have ‹∀⇩F (x', y) in uniformity. x' = x ⟶ y ∈ U›
by auto
then have ‹∀⇩F ((x'1, y1), (x'2, y2)) in uniformity ×⇩F uniformity. (x'1,x'2) = x ⟶ (y1,y2) ∈ U›
by (simp add: uniformity_prod_def eventually_filtermap case_prod_unfold)
then obtain UA UB where ‹eventually UA uniformity› and ‹eventually UB uniformity›
and UA_UB_U: ‹UA (a1, a2) ⟹ UB (b1, b2) ⟹ (a1, b1) = x ⟹ (a2, b2) ∈ U› for a1 a2 b1 b2
apply atomize_elim by (simp add: case_prod_beta eventually_prod_filter)
have ‹eventually (λa. UA (fst x, a)) (nhds (fst x))›
using ‹eventually UA uniformity› eventually_mono eventually_nhds_uniformity by fastforce
then obtain A where ‹open A› and A_UA: ‹A ⊆ {a. UA (fst x, a)}› and ‹fst x ∈ A›
by (metis (mono_tags, lifting) eventually_nhds mem_Collect_eq subsetI)
have ‹eventually (λb. UB (snd x, b)) (nhds (snd x))›
using ‹eventually UB uniformity› eventually_mono eventually_nhds_uniformity by fastforce
then obtain B where ‹open B› and B_UB: ‹B ⊆ {b. UB (snd x, b)}› and ‹snd x ∈ B›
by (metis (mono_tags, lifting) eventually_nhds mem_Collect_eq subsetI)
have ‹x ∈ A × B›
by (simp add: ‹fst x ∈ A› ‹snd x ∈ B› mem_Times_iff)
have ‹A × B ⊆ U›
using A_UA B_UB UA_UB_U by fastforce
show ‹∃A B. open A ∧ open B ∧ x ∈ A × B ∧ A × B ⊆ U›
using ‹A × B ⊆ U› ‹open A› ‹open B› ‹x ∈ A × B› by auto
qed
qed
next
show ‹eventually E uniformity ⟹ E (x, x)› for E and x :: ‹'a × 'b›
apply (simp add: uniformity_prod_def eventually_filtermap case_prod_unfold eventually_prod_filter)
by (metis surj_pair uniformity_refl)
next
show ‹eventually E uniformity ⟹ ∀⇩F (x::'a×'b, y) in uniformity. E (y, x)› for E
apply (simp only: uniformity_prod_def eventually_filtermap case_prod_unfold eventually_prod_filter)
apply (erule exE, erule exE, rename_tac Pf Pg)
apply (rule_tac x=‹λ(x,y). Pf (y,x)› in exI)
apply (rule_tac x=‹λ(x,y). Pg (y,x)› in exI)
next
show ‹∃D. eventually D uniformity ∧ (∀x y z. D (x::'a×'b, y) ⟶ D (y, z) ⟶ E (x, z))›
if ‹eventually E uniformity› for E
proof -
from that
obtain EA EB where ‹eventually EA uniformity› and ‹eventually EB uniformity›
and EA_EB_E: ‹EA (a1, a2) ⟹ EB (b1, b2) ⟹ E ((a1, b1), (a2, b2))› for a1 a2 b1 b2
by (auto simp add: uniformity_prod_def eventually_filtermap case_prod_unfold eventually_prod_filter)
obtain DA where ‹eventually DA uniformity› and DA_EA: ‹DA (x,y) ⟹ DA (y,z) ⟹ EA (x,z)› for x y z
using ‹eventually EA uniformity› uniformity_transE by blast
obtain DB where ‹eventually DB uniformity› and DB_EB: ‹DB (x,y) ⟹ DB (y,z) ⟹ EB (x,z)› for x y z
using ‹eventually EB uniformity› uniformity_transE by blast
define D where ‹D = (λ((a1,b1),(a2,b2)). DA (a1,a2) ∧ DB (b1,b2))›
have ‹eventually D uniformity›
using ‹eventually DA uniformity› ‹eventually DB uniformity›
by (auto simp add: uniformity_prod_def eventually_filtermap case_prod_unfold eventually_prod_filter D_def)
moreover have ‹D ((a1, b1), (a2, b2)) ⟹ D ((a2, b2), (a3, b3)) ⟹ E ((a1, b1), (a3, b3))› for a1 b1 a2 b2 a3 b3
using DA_EA DB_EB D_def EA_EB_E by blast
ultimately show ?thesis
by auto
qed
qed
end

lemma (in uniform_space) nhds_eq_comap_uniformity: "nhds x = filtercomap (λy. (x, y)) uniformity"
proof -
have *: "eventually P (filtercomap (λy. (x, y)) F) ⟷
eventually (λz. fst z = x ⟶ P (snd z)) F" for P :: "'a ⇒ bool" and F
unfolding eventually_filtercomap
by (smt (verit) eventually_elim2 fst_conv prod.collapse snd_conv)
thus ?thesis
unfolding filter_eq_iff
by (subst *) (auto simp: eventually_nhds_uniformity case_prod_unfold)
qed

lemma uniformity_of_uniform_continuous_invariant:
fixes f :: "'a :: uniform_space ⇒ 'a ⇒ 'a"
assumes "filterlim (λ((a,b),(c,d)). (f a c, f b d)) uniformity (uniformity ×⇩F uniformity)"
assumes "eventually P uniformity"
obtains Q where "eventually Q uniformity" "⋀a b c. Q (a, b) ⟹ P (f a c, f b c)"
using eventually_compose_filterlim[OF assms(2,1)] uniformity_refl
by (fastforce simp: case_prod_unfold eventually_filtercomap eventually_prod_same)

"filterlim (λ((a,b), (c,d)). (a + c, b + d)) uniformity (uniformity ×⇩F uniformity)"

"uniformly_continuous_on UNIV (λ(x :: 'a :: uniform_topological_monoid_add,y). x + y)"
by (simp add: uniformly_continuous_on_uniformity case_prod_unfold uniformity_prod_def filterlim_filtermap)

lemma filterlim_fst: "filterlim fst F (F ×⇩F G)"

lemma filterlim_snd: "filterlim snd G (F ×⇩F G)"

assumes uniformly_continuous_uminus': "filterlim (λ(a, b). (-a, -b)) uniformity uniformity"
begin

lemma uniformly_continuous_minus':
"filterlim (λ((a,b), (c,d)). (a - c, b - d)) uniformity (uniformity ×⇩F uniformity)"
proof -
have "filterlim ((λ((a,b), (c,d)). (a + c, b + d)) ∘ (λ((a,b), (c,d)). ((a, b), (-c, -d))))
uniformity (uniformity ×⇩F uniformity)"
unfolding o_def using uniformly_continuous_uminus'
(auto simp: case_prod_unfold intro!: filterlim_Pair
filterlim_fst filterlim_compose[OF _ filterlim_snd])
thus ?thesis
qed

end

lemma uniformly_continuous_uminus:
"uniformly_continuous_on UNIV (λx :: 'a :: uniform_topological_group_add. -x)"
using uniformly_continuous_uminus'[where ?'a = 'a]

lemma uniformly_continuous_minus:
"uniformly_continuous_on UNIV (λ(x :: 'a :: uniform_topological_group_add,y). x - y)"
using uniformly_continuous_minus'[where ?'a = 'a]
by (simp add: uniformly_continuous_on_uniformity case_prod_unfold uniformity_prod_def filterlim_filtermap)

proof
show "filterlim (λ((a::'a,b), (c,d)). (a + c, b + d)) uniformity (uniformity ×⇩F uniformity)"
unfolding filterlim_def le_filter_def eventually_filtermap case_prod_unfold
proof safe
fix P :: "'a × 'a ⇒ bool"
assume "eventually P uniformity"
then obtain ε where ε: "ε > 0" "⋀x y. dist x y < ε ⟹ P (x, y)"
by (auto simp: eventually_uniformity_metric)
define Q where "Q = (λ(x::'a,y). dist x y < ε / 2)"
have Q: "eventually Q uniformity"
unfolding eventually_uniformity_metric Q_def using ‹ε > 0›
by (meson case_prodI divide_pos_pos zero_less_numeral)
have "P (a + c, b + d)" if "Q (a, b)" "Q (c, d)" for a b c d
proof -
have "dist (a + c) (b + d) ≤ dist a b + dist c d"
also have "… < ε"
using that by (auto simp: Q_def)
finally show ?thesis
by (intro ε)
qed
thus "∀⇩F x in uniformity ×⇩F uniformity. P (fst (fst x) + fst (snd x), snd (fst x) + snd (snd x))"
unfolding eventually_prod_filter by (intro exI[of _ Q] conjI Q) auto
qed
next
show "filterlim (λ((a::'a), b). (-a, -b)) uniformity uniformity"
unfolding filterlim_def le_filter_def eventually_filtermap
proof safe
fix P :: "'a × 'a ⇒ bool"
assume "eventually P uniformity"
then obtain ε where ε: "ε > 0" "⋀x y. dist x y < ε ⟹ P (x, y)"
by (auto simp: eventually_uniformity_metric)
show "∀⇩F x in uniformity. P (case x of (a, b) ⇒ (- a, - b))"
unfolding eventually_uniformity_metric
by (intro exI[of _ ε]) (auto intro!: ε simp: dist_norm norm_minus_commute)
qed
qed

lemma cauchy_seq_finset_iff_vanishing:
"uniformity = filtercomap (λ(x,y). y - x :: 'a :: uniform_topological_group_add) (nhds 0)"
proof -
have "filtercomap (λx. (0, case x of (x, y) ⇒ y - (x :: 'a))) uniformity ≤ uniformity"
by (metis diff_self eq_diff_eq)
moreover
have "uniformity ≤ filtercomap (λx. (0, case x of (x, y) ⇒ y - (x :: 'a))) uniformity"
using uniformity_of_uniform_continuous_invariant[OF uniformly_continuous_minus']
by (metis (mono_tags) diff_self eventually_mono surjective_pairing)
ultimately show ?thesis
qed

subsubsection ‹Metric spaces›

instantiation✐‹tag unimportant› prod :: (metric_space, metric_space) uniformity_dist begin
instance
proof
show ‹uniformity = (INF e∈{0 <..}. principal {(x::'a×'b, y). dist x y < e})›
proof (subst filter_eq_iff, intro allI iffI)
fix P :: ‹('a × 'b) × ('a × 'b) ⇒ bool›

have 1: ‹∃e∈{0<..}.
{(x,y). dist x y < e} ⊆ {(x,y). dist x y < a} ∧
{(x,y). dist x y < e} ⊆ {(x,y). dist x y < b}› if ‹a>0› ‹b>0› for a b
apply (rule bexI[of _ ‹min a b›])
using that by auto
have 2: ‹mono (λP. eventually (λx. P (Q x)) F)› for F :: ‹'z filter› and Q :: ‹'z ⇒ 'y›
unfolding mono_def using eventually_mono le_funD by fastforce
have ‹∀⇩F ((x1::'a,y1),(x2::'b,y2)) in uniformity ×⇩F uniformity. dist x1 y1 < e/2 ∧ dist x2 y2 < e/2› if ‹e>0› for e
by (auto intro!: eventually_prodI exI[of _ ‹e/2›] simp: case_prod_unfold eventually_uniformity_metric that)
then have 3: ‹∀⇩F ((x1::'a,y1),(x2::'b,y2)) in uniformity ×⇩F uniformity. dist (x1,x2) (y1,y2) < e› if ‹e>0› for e
apply (rule eventually_rev_mp)
by (auto intro!: that eventuallyI simp: case_prod_unfold dist_prod_def sqrt_sum_squares_half_less)
show ‹eventually P (INF e∈{0<..}. principal {(x, y). dist x y < e}) ⟹ eventually P uniformity›
apply (subst (asm) eventually_INF_base)
using 1 3 apply (auto simp: uniformity_prod_def case_prod_unfold eventually_filtermap 2 eventually_principal)
by (smt (verit, best) eventually_mono)
next
fix P :: ‹('a × 'b) × ('a × 'b) ⇒ bool›
assume ‹eventually P uniformity›
then obtain P1 P2 where ‹eventually P1 uniformity› ‹eventually P2 uniformity›
and P1P2P: ‹P1 (x1, y1) ⟹ P2 (x2, y2) ⟹ P ((x1, x2), (y1, y2))› for x1 y1 x2 y2
by (auto simp: eventually_filtermap case_prod_beta eventually_prod_filter uniformity_prod_def)
from ‹eventually P1 uniformity› obtain e1 where ‹e1>0› and e1P1: ‹dist x y < e1 ⟹ P1 (x,y)› for x y
using eventually_uniformity_metric by blast
from ‹eventually P2 uniformity› obtain e2 where ‹e2>0› and e2P2: ‹dist x y < e2 ⟹ P2 (x,y)› for x y
using eventually_uniformity_metric by blast
define e where ‹e = min e1 e2›
have ‹e > 0›
using ‹0 < e1› ‹0 < e2› e_def by auto
have ‹dist (x1,x2) (y1,y2) < e ⟹ dist x1 y1 < e1› for x1 y1 :: 'a and x2 y2 :: 'b
unfolding dist_prod_def e_def apply auto
by (smt (verit, best) real_sqrt_sum_squares_ge1)
moreover have ‹dist (x1,x2) (y1,y2) < e ⟹ dist x2 y2 < e2› for x1 y1 :: 'a and x2 y2 :: 'b
unfolding dist_prod_def e_def apply auto
by (smt (verit, best) real_sqrt_sum_squares_ge1)
ultimately have *: ‹dist (x1,x2) (y1,y2) < e ⟹ P ((x1, x2), (y1, y2))› for x1 y1 x2 y2
using e1P1 e2P2 P1P2P by auto

show ‹eventually P (INF e∈{0<..}. principal {(x, y). dist x y < e})›
apply (rule eventually_INF1[where i=e])
using ‹e > 0› * by (auto simp: eventually_principal)
qed
qed
end

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

instantiation prod :: (metric_space, metric_space) metric_space
begin

proposition 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
define r where "r = e / sqrt 2"
define s where "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)
define A where "A = {y. dist (fst x) y < r}"
define B where "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
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 :: 'a::metric_space × 'b::metric_space))"
unfolding Cauchy_def by (fast elim: le_less_trans [OF dist_fst_le])

lemma Cauchy_snd: "Cauchy X ⟹ Cauchy (λn. snd (X n :: 'a::metric_space × 'b::metric_space))"
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 :: 'a::metric_space, Y n :: 'a::metric_space))"
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

text ‹Analogue to @{thm [source] uniformly_continuous_on_def} for two-argument functions.›
lemma uniformly_continuous_on_prod_metric:
fixes f :: ‹('a::metric_space × 'b::metric_space) ⇒ 'c::metric_space›
shows ‹uniformly_continuous_on (S×T) f ⟷ (∀e>0. ∃d>0. ∀x∈S. ∀y∈S. ∀x'∈T. ∀y'∈T. dist x y < d ⟶ dist x' y' < d ⟶ dist (f (x, x')) (f (y, y')) < e)›
proof (unfold uniformly_continuous_on_def, intro iffI impI allI)
fix e :: real
assume ‹e > 0› and ‹∀e>0. ∃d>0. ∀x∈S. ∀y∈S. ∀x'∈T. ∀y'∈T. dist x y < d ⟶ dist x' y' < d ⟶ dist (f (x, x')) (f (y, y')) < e›
then obtain d where ‹d > 0›
and d: ‹∀x∈S. ∀y∈S. ∀x'∈T. ∀y'∈T. dist x y < d ⟶ dist x' y' < d ⟶ dist (f (x, x')) (f (y, y')) < e›
by auto
show ‹∃d>0. ∀x∈S×T. ∀y∈S×T. dist y x < d ⟶ dist (f y) (f x) < e›
apply (rule exI[of _ d])
using ‹d>0› d[rule_format] apply auto
by (smt (verit, del_insts) dist_fst_le dist_snd_le fst_conv snd_conv)
next
fix e :: real
assume ‹e > 0› and ‹∀e>0. ∃d>0. ∀x∈S×T. ∀x'∈S×T. dist x' x < d ⟶ dist (f x') (f x) < e›
then obtain d where ‹d > 0› and d: ‹∀x∈S×T. ∀x'∈S×T. dist x' x < d ⟶ dist (f x') (f x) < e›
by auto
show ‹∃d>0. ∀x∈S. ∀y∈S. ∀x'∈T. ∀y'∈T. dist x y < d ⟶ dist x' y' < d ⟶ dist (f (x, x')) (f (y, y')) < e›
proof (intro exI conjI impI ballI)
from ‹d > 0› show ‹d / 2 > 0› by auto
fix x y x' y'
assume [simp]: ‹x ∈ S› ‹y ∈ S› ‹x' ∈ T› ‹y' ∈ T›
assume ‹dist x y < d / 2› and ‹dist x' y' < d / 2›
then have ‹dist (x, x') (y, y') < d›
with d show ‹dist (f (x, x')) (f (y, y')) < e›
by auto
qed
qed

text ‹Analogue to @{thm [source] isUCont_def} for two-argument functions.›
lemma isUCont_prod_metric:
fixes f :: ‹('a::metric_space × 'b::metric_space) ⇒ 'c::metric_space›
shows ‹isUCont f ⟷ (∀e>0. ∃d>0. ∀x. ∀y. ∀x'. ∀y'. dist x y < d ⟶ dist x' y' < d ⟶ dist (f (x, x')) (f (y, y')) < e)›
using uniformly_continuous_on_prod_metric[of UNIV UNIV]
by auto

text ‹This logically belong with the real vector spaces but we only have the necessary lemmas now.›
lemma isUCont_plus[simp]:
shows ‹isUCont (λ(x::'a::real_normed_vector,y). x+y)›
proof (rule isUCont_prod_metric[THEN iffD2], intro allI impI, simp)
fix e :: real assume ‹0 < e›
show ‹∃d>0. ∀x y :: 'a. dist x y < d ⟶ (∀x' y'. dist x' y' < d ⟶ dist (x + x') (y + y') < e)›
apply (rule exI[of _ ‹e/2›])
using ‹0 < e› apply auto
qed

subsection ‹Product is a Complete Metric Space›

instance✐‹tag important› 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"

proposition 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✐‹tag important› prod :: (banach, banach) banach ..

subsubsection✐‹tag unimportant› ‹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✐‹tag unimportant› ‹Frechet derivatives involving pairs›

text✐‹tag important› ‹%whitespace›
proposition 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

lemma differentiable_Pair [simp, derivative_intros]:
"f differentiable at x within s ⟹ g differentiable at x within s ⟹
(λx. (f x, g x)) differentiable at x within s"
unfolding differentiable_def by (blast intro: has_derivative_Pair)

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' .

subsubsection✐‹tag unimportant› ‹Vector derivatives involving pairs›

lemma has_vector_derivative_Pair[derivative_intros]:
assumes "(f has_vector_derivative f') (at x within s)"
"(g has_vector_derivative g') (at x within s)"
shows "((λx. (f x, g x)) has_vector_derivative (f', g')) (at x within s)"
using assms
by (auto simp: has_vector_derivative_def intro!: derivative_eq_intros)

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)"

lemma norm_fst_le: "norm x ≤ norm (x,y)"
by (metis dist_fst_le fst_conv fst_zero norm_conv_dist)

lemma norm_snd_le: "norm y ≤ norm (x,y)"
by (metis dist_snd_le snd_conv snd_zero norm_conv_dist)

lemma norm_Pair_le:
shows "norm (x, y) ≤ norm x + norm y"
unfolding norm_Pair
by (metis norm_ge_zero sqrt_sum_squares_le_sum)

lemma (in vector_space_prod) span_Times_sing1: "p.span ({0} × B) = {0} × vs2.span B"
apply (rule p.span_unique)
subgoal by (auto intro!: vs1.span_base vs2.span_base)
subgoal using vs1.subspace_single_0 vs2.subspace_span by (rule subspace_Times)
subgoal for T
proof safe
fix b
assume subset_T: "{0} × B ⊆ T" and subspace: "p.subspace T" and b_span: "b ∈ vs2.span B"
then obtain t r where b: "b = (∑a∈t. r a *b a)" and t: "finite t" "t ⊆ B"
by (auto simp: vs2.span_explicit)
have "(0, b) = (∑b∈t. scale (r b) (0, b))"
unfolding b scale_prod sum_prod
by simp
also have "… ∈ T"
using ‹t ⊆ B› subset_T
by (auto intro!: p.subspace_sum p.subspace_scale subspace)
finally show "(0, b) ∈ T" .
qed
done

lemma (in vector_space_prod) span_Times_sing2: "p.span (A × {0}) = vs1.span A × {0}"
apply (rule p.span_unique)
subgoal by (auto intro!: vs1.span_base vs2.span_base)
subgoal using vs1.subspace_span vs2.subspace_single_0 by (rule subspace_Times)
subgoal for T
proof safe
fix a
assume subset_T: "A × {0} ⊆ T" and subspace: "p.subspace T" and a_span: "a ∈ vs1.span A"
then obtain t r where a: "a = (∑a∈t. r a *a a)" and t: "finite t" "t ⊆ A"
by (auto simp: vs1.span_explicit)
have "(a, 0) = (∑a∈t. scale (r a) (a, 0))"
unfolding a scale_prod sum_prod
by simp
also have "… ∈ T"
using ‹t ⊆ A› subset_T
by (auto intro!: p.subspace_sum p.subspace_scale subspace)
finally show "(a, 0) ∈ T" .
qed
done

subsection ‹Product is Finite Dimensional›

lemma (in finite_dimensional_vector_space) zero_not_in_Basis[simp]: "0 ∉ Basis"
using dependent_zero local.independent_Basis by blast

locale finite_dimensional_vector_space_prod = vector_space_prod + finite_dimensional_vector_space_pair begin

definition "Basis_pair = B1 × {0} ∪ {0} × B2"

sublocale p: finite_dimensional_vector_space scale Basis_pair
proof unfold_locales
show "finite Basis_pair"
by (auto intro!: finite_cartesian_product vs1.finite_Basis vs2.finite_Basis simp: Basis_pair_def)
show "p.independent Basis_pair"
unfolding p.dependent_def Basis_pair_def
proof safe
fix a
assume a: "a ∈ B1"
assume "(a, 0) ∈ p.span (B1 × {0} ∪ {0} × B2 - {(a, 0)})"
also have "B1 × {0} ∪ {0} × B2 - {(a, 0)} = (B1 - {a}) × {0} ∪ {0} × B2"
by auto
finally show False
using a vs1.dependent_def vs1.independent_Basis
by (auto simp: p.span_Un span_Times_sing1 span_Times_sing2)
next
fix b
assume b: "b ∈ B2"
assume "(0, b) ∈ p.span (B1 × {0} ∪ {0} × B2 - {(0, b)})"
also have "(B1 × {0} ∪ {0} × B2 - {(0, b)}) = B1 × {0} ∪ {0} × (B2 - {b})"
by auto
finally show False
using b vs2.dependent_def vs2.independent_Basis
by (auto simp: p.span_Un span_Times_sing1 span_Times_sing2)
qed
show "p.span Basis_pair = UNIV"
by (auto simp: p.span_Un span_Times_sing2 span_Times_sing1 vs1.span_Basis vs2.span_Basis
Basis_pair_def)
qed

proposition dim_Times:
assumes "vs1.subspace S" "vs2.subspace T"
shows "p.dim(S × T) = vs1.dim S + vs2.dim T"
proof -
interpret p1: Vector_Spaces.linear s1 scale "(λx. (x, 0))"
by unfold_locales (auto simp: scale_def)
interpret pair1: finite_dimensional_vector_space_pair "(*a)" B1 scale Basis_pair
by unfold_locales
interpret p2: Vector_Spaces.linear s2 scale "(λx. (0, x))"
by unfold_locales (auto simp: scale_def)
interpret pair2: finite_dimensional_vector_space_pair "(*b)" B2 scale Basis_pair
by unfold_locales
have ss: "p.subspace ((λx. (x, 0)) ` S)" "p.subspace (Pair 0 ` T)"
by (rule p1.subspace_image p2.subspace_image assms)+
have "p.dim(S × T) = p.dim({u + v |u v. u ∈ (λx. (x, 0)) ` S ∧ v ∈ Pair 0 ` T})"
moreover have "p.dim ((λx. (x, 0::'c)) ` S) = vs1.dim S" "p.dim (Pair (0::'b) ` T) = vs2.dim T"
by (simp_all add: inj_on_def p1.linear_axioms pair1.dim_image_eq p2.linear_axioms pair2.dim_image_eq)
moreover have "p.dim ((λx. (x, 0)) ` S ∩ Pair 0 ` T) = 0"
by (subst p.dim_eq_0) auto
ultimately show ?thesis
using p.dim_sums_Int [OF ss] by linarith
qed

lemma dimension_pair: "p.dimension = vs1.dimension + vs2.dimension"
using dim_Times[OF vs1.subspace_UNIV vs2.subspace_UNIV]
by (auto simp: p.dimension_def vs1.dimension_def vs2.dimension_def)

end

end
```