# Theory Finite_Cartesian_Product

theory Finite_Cartesian_Product
imports Euclidean_Space Numeral_Type
```(*  Title:      HOL/Multivariate_Analysis/Finite_Cartesian_Product.thy
Author:     Amine Chaieb, University of Cambridge
*)

section ‹Definition of finite Cartesian product types.›

theory Finite_Cartesian_Product
imports
Euclidean_Space
L2_Norm
"~~/src/HOL/Library/Numeral_Type"
begin

subsection ‹Finite Cartesian products, with indexing and lambdas.›

typedef ('a, 'b) vec = "UNIV :: (('b::finite) ⇒ 'a) set"
morphisms vec_nth vec_lambda ..

notation
vec_nth (infixl "\$" 90) and
vec_lambda (binder "χ" 10)

(*
Translate "'b ^ 'n" into "'b ^ ('n :: finite)". When 'n has already more than
the finite type class write "vec 'b 'n"
*)

syntax "_finite_vec" :: "type ⇒ type ⇒ type" ("(_ ^/ _)" [15, 16] 15)

parse_translation ‹
let
fun vec t u = Syntax.const @{type_syntax vec} \$ t \$ u;
fun finite_vec_tr [t, u] =
(case Term_Position.strip_positions u of
v as Free (x, _) =>
if Lexicon.is_tid x then
vec t (Syntax.const @{syntax_const "_ofsort"} \$ v \$
Syntax.const @{class_syntax finite})
else vec t u
| _ => vec t u)
in
[(@{syntax_const "_finite_vec"}, K finite_vec_tr)]
end
›

lemma vec_eq_iff: "(x = y) ⟷ (∀i. x\$i = y\$i)"
by (simp add: vec_nth_inject [symmetric] fun_eq_iff)

lemma vec_lambda_beta [simp]: "vec_lambda g \$ i = g i"

lemma vec_lambda_unique: "(∀i. f\$i = g i) ⟷ vec_lambda g = f"

lemma vec_lambda_eta: "(χ i. (g\$i)) = g"

subsection ‹Group operations and class instances›

instantiation vec :: (zero, finite) zero
begin
definition "0 ≡ (χ i. 0)"
instance ..
end

instantiation vec :: (plus, finite) plus
begin
definition "op + ≡ (λ x y. (χ i. x\$i + y\$i))"
instance ..
end

instantiation vec :: (minus, finite) minus
begin
definition "op - ≡ (λ x y. (χ i. x\$i - y\$i))"
instance ..
end

instantiation vec :: (uminus, finite) uminus
begin
definition "uminus ≡ (λ x. (χ i. - (x\$i)))"
instance ..
end

lemma zero_index [simp]: "0 \$ i = 0"
unfolding zero_vec_def by simp

lemma vector_add_component [simp]: "(x + y)\$i = x\$i + y\$i"
unfolding plus_vec_def by simp

lemma vector_minus_component [simp]: "(x - y)\$i = x\$i - y\$i"
unfolding minus_vec_def by simp

lemma vector_uminus_component [simp]: "(- x)\$i = - (x\$i)"
unfolding uminus_vec_def by simp

by standard (simp_all add: vec_eq_iff diff_diff_eq)

subsection ‹Real vector space›

instantiation vec :: (real_vector, finite) real_vector
begin

definition "scaleR ≡ (λ r x. (χ i. scaleR r (x\$i)))"

lemma vector_scaleR_component [simp]: "(scaleR r x)\$i = scaleR r (x\$i)"
unfolding scaleR_vec_def by simp

instance
by standard (simp_all add: vec_eq_iff scaleR_left_distrib scaleR_right_distrib)

end

subsection ‹Topological space›

instantiation vec :: (topological_space, finite) topological_space
begin

definition [code del]:
"open (S :: ('a ^ 'b) set) ⟷
(∀x∈S. ∃A. (∀i. open (A i) ∧ x\$i ∈ A i) ∧
(∀y. (∀i. y\$i ∈ A i) ⟶ y ∈ S))"

instance proof
show "open (UNIV :: ('a ^ 'b) set)"
unfolding open_vec_def by auto
next
fix S T :: "('a ^ 'b) set"
assume "open S" "open T" thus "open (S ∩ T)"
unfolding open_vec_def
apply clarify
apply (drule (1) bspec)+
apply (clarify, rename_tac Sa Ta)
apply (rule_tac x="λi. Sa i ∩ Ta i" in exI)
done
next
fix K :: "('a ^ 'b) set set"
assume "∀S∈K. open S" thus "open (⋃K)"
unfolding open_vec_def
apply clarify
apply (drule (1) bspec)
apply (drule (1) bspec)
apply clarify
apply (rule_tac x=A in exI)
apply fast
done
qed

end

lemma open_vector_box: "∀i. open (S i) ⟹ open {x. ∀i. x \$ i ∈ S i}"
unfolding open_vec_def by auto

lemma open_vimage_vec_nth: "open S ⟹ open ((λx. x \$ i) -` S)"
unfolding open_vec_def
apply clarify
apply (rule_tac x="λk. if k = i then S else UNIV" in exI, simp)
done

lemma closed_vimage_vec_nth: "closed S ⟹ closed ((λx. x \$ i) -` S)"
unfolding closed_open vimage_Compl [symmetric]
by (rule open_vimage_vec_nth)

lemma closed_vector_box: "∀i. closed (S i) ⟹ closed {x. ∀i. x \$ i ∈ S i}"
proof -
have "{x. ∀i. x \$ i ∈ S i} = (⋂i. (λx. x \$ i) -` S i)" by auto
thus "∀i. closed (S i) ⟹ closed {x. ∀i. x \$ i ∈ S i}"
qed

lemma tendsto_vec_nth [tendsto_intros]:
assumes "((λx. f x) ⤏ a) net"
shows "((λx. f x \$ i) ⤏ a \$ i) net"
proof (rule topological_tendstoI)
fix S assume "open S" "a \$ i ∈ S"
then have "open ((λy. y \$ i) -` S)" "a ∈ ((λy. y \$ i) -` S)"
with assms have "eventually (λx. f x ∈ (λy. y \$ i) -` S) net"
by (rule topological_tendstoD)
then show "eventually (λx. f x \$ i ∈ S) net"
by simp
qed

lemma isCont_vec_nth [simp]: "isCont f a ⟹ isCont (λx. f x \$ i) a"
unfolding isCont_def by (rule tendsto_vec_nth)

lemma vec_tendstoI:
assumes "⋀i. ((λx. f x \$ i) ⤏ a \$ i) net"
shows "((λx. f x) ⤏ a) net"
proof (rule topological_tendstoI)
fix S assume "open S" and "a ∈ S"
then obtain A where A: "⋀i. open (A i)" "⋀i. a \$ i ∈ A i"
and S: "⋀y. ∀i. y \$ i ∈ A i ⟹ y ∈ S"
unfolding open_vec_def by metis
have "⋀i. eventually (λx. f x \$ i ∈ A i) net"
using assms A by (rule topological_tendstoD)
hence "eventually (λx. ∀i. f x \$ i ∈ A i) net"
by (rule eventually_all_finite)
thus "eventually (λx. f x ∈ S) net"
by (rule eventually_mono, simp add: S)
qed

lemma tendsto_vec_lambda [tendsto_intros]:
assumes "⋀i. ((λx. f x i) ⤏ a i) net"
shows "((λx. χ i. f x i) ⤏ (χ i. a i)) net"
using assms by (simp add: vec_tendstoI)

lemma open_image_vec_nth: assumes "open S" shows "open ((λx. x \$ i) ` S)"
proof (rule openI)
fix a assume "a ∈ (λx. x \$ i) ` S"
then obtain z where "a = z \$ i" and "z ∈ S" ..
then obtain A where A: "∀i. open (A i) ∧ z \$ i ∈ A i"
and S: "∀y. (∀i. y \$ i ∈ A i) ⟶ y ∈ S"
using ‹open S› unfolding open_vec_def by auto
hence "A i ⊆ (λx. x \$ i) ` S"
by (clarsimp, rule_tac x="χ j. if j = i then x else z \$ j" in image_eqI,
simp_all)
hence "open (A i) ∧ a ∈ A i ∧ A i ⊆ (λx. x \$ i) ` S"
using A ‹a = z \$ i› by simp
then show "∃T. open T ∧ a ∈ T ∧ T ⊆ (λx. x \$ i) ` S" by - (rule exI)
qed

instance vec :: (perfect_space, finite) perfect_space
proof
fix x :: "'a ^ 'b" show "¬ open {x}"
proof
assume "open {x}"
hence "∀i. open ((λx. x \$ i) ` {x})" by (fast intro: open_image_vec_nth)
hence "∀i. open {x \$ i}" by simp
thus "False" by (simp add: not_open_singleton)
qed
qed

subsection ‹Metric space›
(* TODO: Product of uniform spaces and compatibility with metric_spaces! *)

instantiation vec :: (metric_space, finite) dist
begin

definition
"dist x y = setL2 (λi. dist (x\$i) (y\$i)) UNIV"

instance ..
end

instantiation vec :: (metric_space, finite) uniformity_dist
begin

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

instance
by standard (rule uniformity_vec_def)
end

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

instantiation vec :: (metric_space, finite) metric_space
begin

lemma dist_vec_nth_le: "dist (x \$ i) (y \$ i) ≤ dist x y"
unfolding dist_vec_def by (rule member_le_setL2) simp_all

instance proof
fix x y :: "'a ^ 'b"
show "dist x y = 0 ⟷ x = y"
unfolding dist_vec_def
next
fix x y z :: "'a ^ 'b"
show "dist x y ≤ dist x z + dist y z"
unfolding dist_vec_def
apply (rule order_trans [OF _ setL2_triangle_ineq])
done
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 where A: "∀i. open (A i)" "∀i. x \$ i ∈ A i"
and S: "∀y. (∀i. y \$ i ∈ A i) ⟶ y ∈ S"
using ‹open S› and ‹x ∈ S› unfolding open_vec_def by metis
have "∀i∈UNIV. ∃r>0. ∀y. dist y (x \$ i) < r ⟶ y ∈ A i"
using A unfolding open_dist by simp
hence "∃r. ∀i∈UNIV. 0 < r i ∧ (∀y. dist y (x \$ i) < r i ⟶ y ∈ A i)"
by (rule finite_set_choice [OF finite])
then obtain r where r1: "∀i. 0 < r i"
and r2: "∀i y. dist y (x \$ i) < r i ⟶ y ∈ A i" by fast
have "0 < Min (range r) ∧ (∀y. dist y x < Min (range r) ⟶ y ∈ S)"
by (simp add: r1 r2 S le_less_trans [OF dist_vec_nth_le])
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 (unfold open_vec_def, rule)
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 ≡ "λi::'b. e / sqrt (of_nat CARD('b))"
from ‹0 < e› have r: "∀i. 0 < r i"
unfolding r_def by simp_all
from ‹0 < e› have e: "e = setL2 r UNIV"
unfolding r_def by (simp add: setL2_constant)
def A ≡ "λi. {y. dist (x \$ i) y < r i}"
have "∀i. open (A i) ∧ x \$ i ∈ A i"
unfolding A_def by (simp add: open_ball r)
moreover have "∀y. (∀i. y \$ i ∈ A i) ⟶ y ∈ S"
by (simp add: A_def S dist_vec_def e setL2_strict_mono dist_commute)
ultimately show "∃A. (∀i. open (A i) ∧ x \$ i ∈ A i) ∧
(∀y. (∀i. y \$ i ∈ A i) ⟶ y ∈ S)" by metis
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_vec_def dist_commute)
qed

end

lemma Cauchy_vec_nth:
"Cauchy (λn. X n) ⟹ Cauchy (λn. X n \$ i)"
unfolding Cauchy_def by (fast intro: le_less_trans [OF dist_vec_nth_le])

lemma vec_CauchyI:
fixes X :: "nat ⇒ 'a::metric_space ^ 'n"
assumes X: "⋀i. Cauchy (λn. X n \$ i)"
shows "Cauchy (λn. X n)"
proof (rule metric_CauchyI)
fix r :: real assume "0 < r"
hence "0 < r / of_nat CARD('n)" (is "0 < ?s") by simp
def N ≡ "λi. LEAST N. ∀m≥N. ∀n≥N. dist (X m \$ i) (X n \$ i) < ?s"
def M ≡ "Max (range N)"
have "⋀i. ∃N. ∀m≥N. ∀n≥N. dist (X m \$ i) (X n \$ i) < ?s"
using X ‹0 < ?s› by (rule metric_CauchyD)
hence "⋀i. ∀m≥N i. ∀n≥N i. dist (X m \$ i) (X n \$ i) < ?s"
unfolding N_def by (rule LeastI_ex)
hence M: "⋀i. ∀m≥M. ∀n≥M. dist (X m \$ i) (X n \$ i) < ?s"
unfolding M_def by simp
{
fix m n :: nat
assume "M ≤ m" "M ≤ n"
have "dist (X m) (X n) = setL2 (λi. dist (X m \$ i) (X n \$ i)) UNIV"
unfolding dist_vec_def ..
also have "… ≤ setsum (λi. dist (X m \$ i) (X n \$ i)) UNIV"
by (rule setL2_le_setsum [OF zero_le_dist])
also have "… < setsum (λi::'n. ?s) UNIV"
by (rule setsum_strict_mono, simp_all add: M ‹M ≤ m› ‹M ≤ n›)
also have "… = r"
by simp
finally have "dist (X m) (X n) < r" .
}
hence "∀m≥M. ∀n≥M. dist (X m) (X n) < r"
by simp
then show "∃M. ∀m≥M. ∀n≥M. dist (X m) (X n) < r" ..
qed

instance vec :: (complete_space, finite) complete_space
proof
fix X :: "nat ⇒ 'a ^ 'b" assume "Cauchy X"
have "⋀i. (λn. X n \$ i) ⇢ lim (λn. X n \$ i)"
using Cauchy_vec_nth [OF ‹Cauchy X›]
hence "X ⇢ vec_lambda (λi. lim (λn. X n \$ i))"
then show "convergent X"
by (rule convergentI)
qed

subsection ‹Normed vector space›

instantiation vec :: (real_normed_vector, finite) real_normed_vector
begin

definition "norm x = setL2 (λi. norm (x\$i)) UNIV"

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

instance proof
fix a :: real and x y :: "'a ^ 'b"
show "norm x = 0 ⟷ x = 0"
unfolding norm_vec_def
show "norm (x + y) ≤ norm x + norm y"
unfolding norm_vec_def
apply (rule order_trans [OF _ setL2_triangle_ineq])
done
show "norm (scaleR a x) = ¦a¦ * norm x"
unfolding norm_vec_def
show "sgn x = scaleR (inverse (norm x)) x"
by (rule sgn_vec_def)
show "dist x y = norm (x - y)"
unfolding dist_vec_def norm_vec_def
qed

end

lemma norm_nth_le: "norm (x \$ i) ≤ norm x"
unfolding norm_vec_def
by (rule member_le_setL2) simp_all

lemma bounded_linear_vec_nth: "bounded_linear (λx. x \$ i)"
apply standard
apply (rule vector_scaleR_component)
apply (rule_tac x="1" in exI, simp add: norm_nth_le)
done

instance vec :: (banach, finite) banach ..

subsection ‹Inner product space›

instantiation vec :: (real_inner, finite) real_inner
begin

definition "inner x y = setsum (λi. inner (x\$i) (y\$i)) UNIV"

instance proof
fix r :: real and x y z :: "'a ^ 'b"
show "inner x y = inner y x"
unfolding inner_vec_def
show "inner (x + y) z = inner x z + inner y z"
unfolding inner_vec_def
show "inner (scaleR r x) y = r * inner x y"
unfolding inner_vec_def
show "0 ≤ inner x x"
unfolding inner_vec_def
show "inner x x = 0 ⟷ x = 0"
unfolding inner_vec_def
show "norm x = sqrt (inner x x)"
unfolding inner_vec_def norm_vec_def setL2_def
qed

end

subsection ‹Euclidean space›

text ‹Vectors pointing along a single axis.›

definition "axis k x = (χ i. if i = k then x else 0)"

lemma axis_nth [simp]: "axis i x \$ i = x"
unfolding axis_def by simp

lemma axis_eq_axis: "axis i x = axis j y ⟷ x = y ∧ i = j ∨ x = 0 ∧ y = 0"
unfolding axis_def vec_eq_iff by auto

lemma inner_axis_axis:
"inner (axis i x) (axis j y) = (if i = j then inner x y else 0)"
unfolding inner_vec_def
apply (cases "i = j")
apply clarsimp
apply (subst setsum.remove [of _ j], simp_all)
apply (rule setsum.neutral, simp add: axis_def)
apply (rule setsum.neutral, simp add: axis_def)
done

lemma setsum_single:
assumes "finite A" and "k ∈ A" and "f k = y"
assumes "⋀i. i ∈ A ⟹ i ≠ k ⟹ f i = 0"
shows "(∑i∈A. f i) = y"
apply (subst setsum.remove [OF assms(1,2)])
done

lemma inner_axis: "inner x (axis i y) = inner (x \$ i) y"
unfolding inner_vec_def
apply (rule_tac k=i in setsum_single)
apply simp_all
done

instantiation vec :: (euclidean_space, finite) euclidean_space
begin

definition "Basis = (⋃i. ⋃u∈Basis. {axis i u})"

instance proof
show "(Basis :: ('a ^ 'b) set) ≠ {}"
unfolding Basis_vec_def by simp
next
show "finite (Basis :: ('a ^ 'b) set)"
unfolding Basis_vec_def by simp
next
fix u v :: "'a ^ 'b"
assume "u ∈ Basis" and "v ∈ Basis"
thus "inner u v = (if u = v then 1 else 0)"
unfolding Basis_vec_def
by (auto simp add: inner_axis_axis axis_eq_axis inner_Basis)
next
fix x :: "'a ^ 'b"
show "(∀u∈Basis. inner x u = 0) ⟷ x = 0"
unfolding Basis_vec_def
by (simp add: inner_axis euclidean_all_zero_iff vec_eq_iff)
qed

lemma DIM_cart[simp]: "DIM('a^'b) = CARD('b) * DIM('a)"