(* Title: HOL/Analysis/Finite_Cartesian_Product.thy
Author: Amine Chaieb, University of Cambridge
*)
section \<open>Definition of finite Cartesian product types.\<close>
theory Finite_Cartesian_Product
imports
Euclidean_Space
L2_Norm
"~~/src/HOL/Library/Numeral_Type"
begin
subsection \<open>Finite Cartesian products, with indexing and lambdas.\<close>
typedef ('a, 'b) vec = "UNIV :: (('b::finite) \<Rightarrow> 'a) set"
morphisms vec_nth vec_lambda ..
notation
vec_nth (infixl "$" 90) and
vec_lambda (binder "\<chi>" 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 \<Rightarrow> type \<Rightarrow> type" ("(_ ^/ _)" [15, 16] 15)
parse_translation \<open>
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
\<close>
lemma vec_eq_iff: "(x = y) \<longleftrightarrow> (\<forall>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"
by (simp add: vec_lambda_inverse)
lemma vec_lambda_unique: "(\<forall>i. f$i = g i) \<longleftrightarrow> vec_lambda g = f"
by (auto simp add: vec_eq_iff)
lemma vec_lambda_eta: "(\<chi> i. (g$i)) = g"
by (simp add: vec_eq_iff)
subsection \<open>Group operations and class instances\<close>
instantiation vec :: (zero, finite) zero
begin
definition "0 \<equiv> (\<chi> i. 0)"
instance ..
end
instantiation vec :: (plus, finite) plus
begin
definition "op + \<equiv> (\<lambda> x y. (\<chi> i. x$i + y$i))"
instance ..
end
instantiation vec :: (minus, finite) minus
begin
definition "op - \<equiv> (\<lambda> x y. (\<chi> i. x$i - y$i))"
instance ..
end
instantiation vec :: (uminus, finite) uminus
begin
definition "uminus \<equiv> (\<lambda> x. (\<chi> 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
instance vec :: (semigroup_add, finite) semigroup_add
by standard (simp add: vec_eq_iff add.assoc)
instance vec :: (ab_semigroup_add, finite) ab_semigroup_add
by standard (simp add: vec_eq_iff add.commute)
instance vec :: (monoid_add, finite) monoid_add
by standard (simp_all add: vec_eq_iff)
instance vec :: (comm_monoid_add, finite) comm_monoid_add
by standard (simp add: vec_eq_iff)
instance vec :: (cancel_semigroup_add, finite) cancel_semigroup_add
by standard (simp_all add: vec_eq_iff)
instance vec :: (cancel_ab_semigroup_add, finite) cancel_ab_semigroup_add
by standard (simp_all add: vec_eq_iff diff_diff_eq)
instance vec :: (cancel_comm_monoid_add, finite) cancel_comm_monoid_add ..
instance vec :: (group_add, finite) group_add
by standard (simp_all add: vec_eq_iff)
instance vec :: (ab_group_add, finite) ab_group_add
by standard (simp_all add: vec_eq_iff)
subsection \<open>Real vector space\<close>
instantiation vec :: (real_vector, finite) real_vector
begin
definition "scaleR \<equiv> (\<lambda> r x. (\<chi> i. scaleR r (x$i)))"
lemma vector_scaleR_component [simp]: "(scaleR r x)$i = scaleR r (x$i)"
unfolding scaleR_vec_def by simp
instance
by standard (simp_all add: vec_eq_iff scaleR_left_distrib scaleR_right_distrib)
end
subsection \<open>Topological space\<close>
instantiation vec :: (topological_space, finite) topological_space
begin
definition [code del]:
"open (S :: ('a ^ 'b) set) \<longleftrightarrow>
(\<forall>x\<in>S. \<exists>A. (\<forall>i. open (A i) \<and> x$i \<in> A i) \<and>
(\<forall>y. (\<forall>i. y$i \<in> A i) \<longrightarrow> y \<in> 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 \<inter> T)"
unfolding open_vec_def
apply clarify
apply (drule (1) bspec)+
apply (clarify, rename_tac Sa Ta)
apply (rule_tac x="\<lambda>i. Sa i \<inter> Ta i" in exI)
apply (simp add: open_Int)
done
next
fix K :: "('a ^ 'b) set set"
assume "\<forall>S\<in>K. open S" thus "open (\<Union>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: "\<forall>i. open (S i) \<Longrightarrow> open {x. \<forall>i. x $ i \<in> S i}"
unfolding open_vec_def by auto
lemma open_vimage_vec_nth: "open S \<Longrightarrow> open ((\<lambda>x. x $ i) -` S)"
unfolding open_vec_def
apply clarify
apply (rule_tac x="\<lambda>k. if k = i then S else UNIV" in exI, simp)
done
lemma closed_vimage_vec_nth: "closed S \<Longrightarrow> closed ((\<lambda>x. x $ i) -` S)"
unfolding closed_open vimage_Compl [symmetric]
by (rule open_vimage_vec_nth)
lemma closed_vector_box: "\<forall>i. closed (S i) \<Longrightarrow> closed {x. \<forall>i. x $ i \<in> S i}"
proof -
have "{x. \<forall>i. x $ i \<in> S i} = (\<Inter>i. (\<lambda>x. x $ i) -` S i)" by auto
thus "\<forall>i. closed (S i) \<Longrightarrow> closed {x. \<forall>i. x $ i \<in> S i}"
by (simp add: closed_INT closed_vimage_vec_nth)
qed
lemma tendsto_vec_nth [tendsto_intros]:
assumes "((\<lambda>x. f x) \<longlongrightarrow> a) net"
shows "((\<lambda>x. f x $ i) \<longlongrightarrow> a $ i) net"
proof (rule topological_tendstoI)
fix S assume "open S" "a $ i \<in> S"
then have "open ((\<lambda>y. y $ i) -` S)" "a \<in> ((\<lambda>y. y $ i) -` S)"
by (simp_all add: open_vimage_vec_nth)
with assms have "eventually (\<lambda>x. f x \<in> (\<lambda>y. y $ i) -` S) net"
by (rule topological_tendstoD)
then show "eventually (\<lambda>x. f x $ i \<in> S) net"
by simp
qed
lemma isCont_vec_nth [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. f x $ i) a"
unfolding isCont_def by (rule tendsto_vec_nth)
lemma vec_tendstoI:
assumes "\<And>i. ((\<lambda>x. f x $ i) \<longlongrightarrow> a $ i) net"
shows "((\<lambda>x. f x) \<longlongrightarrow> a) net"
proof (rule topological_tendstoI)
fix S assume "open S" and "a \<in> S"
then obtain A where A: "\<And>i. open (A i)" "\<And>i. a $ i \<in> A i"
and S: "\<And>y. \<forall>i. y $ i \<in> A i \<Longrightarrow> y \<in> S"
unfolding open_vec_def by metis
have "\<And>i. eventually (\<lambda>x. f x $ i \<in> A i) net"
using assms A by (rule topological_tendstoD)
hence "eventually (\<lambda>x. \<forall>i. f x $ i \<in> A i) net"
by (rule eventually_all_finite)
thus "eventually (\<lambda>x. f x \<in> S) net"
by (rule eventually_mono, simp add: S)
qed
lemma tendsto_vec_lambda [tendsto_intros]:
assumes "\<And>i. ((\<lambda>x. f x i) \<longlongrightarrow> a i) net"
shows "((\<lambda>x. \<chi> i. f x i) \<longlongrightarrow> (\<chi> i. a i)) net"
using assms by (simp add: vec_tendstoI)
lemma open_image_vec_nth: assumes "open S" shows "open ((\<lambda>x. x $ i) ` S)"
proof (rule openI)
fix a assume "a \<in> (\<lambda>x. x $ i) ` S"
then obtain z where "a = z $ i" and "z \<in> S" ..
then obtain A where A: "\<forall>i. open (A i) \<and> z $ i \<in> A i"
and S: "\<forall>y. (\<forall>i. y $ i \<in> A i) \<longrightarrow> y \<in> S"
using \<open>open S\<close> unfolding open_vec_def by auto
hence "A i \<subseteq> (\<lambda>x. x $ i) ` S"
by (clarsimp, rule_tac x="\<chi> j. if j = i then x else z $ j" in image_eqI,
simp_all)
hence "open (A i) \<and> a \<in> A i \<and> A i \<subseteq> (\<lambda>x. x $ i) ` S"
using A \<open>a = z $ i\<close> by simp
then show "\<exists>T. open T \<and> a \<in> T \<and> T \<subseteq> (\<lambda>x. x $ i) ` S" by - (rule exI)
qed
instance vec :: (perfect_space, finite) perfect_space
proof
fix x :: "'a ^ 'b" show "\<not> open {x}"
proof
assume "open {x}"
hence "\<forall>i. open ((\<lambda>x. x $ i) ` {x})" by (fast intro: open_image_vec_nth)
hence "\<forall>i. open {x $ i}" by simp
thus "False" by (simp add: not_open_singleton)
qed
qed
subsection \<open>Metric space\<close>
(* TODO: Product of uniform spaces and compatibility with metric_spaces! *)
instantiation vec :: (metric_space, finite) dist
begin
definition
"dist x y = setL2 (\<lambda>i. dist (x$i) (y$i)) UNIV"
instance ..
end
instantiation vec :: (metric_space, finite) uniformity_dist
begin
definition [code del]:
"(uniformity :: (('a, 'b) vec \<times> ('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) \<le> 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 \<longleftrightarrow> x = y"
unfolding dist_vec_def
by (simp add: setL2_eq_0_iff vec_eq_iff)
next
fix x y z :: "'a ^ 'b"
show "dist x y \<le> dist x z + dist y z"
unfolding dist_vec_def
apply (rule order_trans [OF _ setL2_triangle_ineq])
apply (simp add: setL2_mono dist_triangle2)
done
next
fix S :: "('a ^ 'b) set"
have *: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
proof
assume "open S" show "\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S"
proof
fix x assume "x \<in> S"
obtain A where A: "\<forall>i. open (A i)" "\<forall>i. x $ i \<in> A i"
and S: "\<forall>y. (\<forall>i. y $ i \<in> A i) \<longrightarrow> y \<in> S"
using \<open>open S\<close> and \<open>x \<in> S\<close> unfolding open_vec_def by metis
have "\<forall>i\<in>UNIV. \<exists>r>0. \<forall>y. dist y (x $ i) < r \<longrightarrow> y \<in> A i"
using A unfolding open_dist by simp
hence "\<exists>r. \<forall>i\<in>UNIV. 0 < r i \<and> (\<forall>y. dist y (x $ i) < r i \<longrightarrow> y \<in> A i)"
by (rule finite_set_choice [OF finite])
then obtain r where r1: "\<forall>i. 0 < r i"
and r2: "\<forall>i y. dist y (x $ i) < r i \<longrightarrow> y \<in> A i" by fast
have "0 < Min (range r) \<and> (\<forall>y. dist y x < Min (range r) \<longrightarrow> y \<in> S)"
by (simp add: r1 r2 S le_less_trans [OF dist_vec_nth_le])
thus "\<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S" ..
qed
next
assume *: "\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S" show "open S"
proof (unfold open_vec_def, rule)
fix x assume "x \<in> S"
then obtain e where "0 < e" and S: "\<forall>y. dist y x < e \<longrightarrow> y \<in> S"
using * by fast
define r where [abs_def]: "r i = e / sqrt (of_nat CARD('b))" for i :: 'b
from \<open>0 < e\<close> have r: "\<forall>i. 0 < r i"
unfolding r_def by simp_all
from \<open>0 < e\<close> have e: "e = setL2 r UNIV"
unfolding r_def by (simp add: setL2_constant)
define A where "A i = {y. dist (x $ i) y < r i}" for i
have "\<forall>i. open (A i) \<and> x $ i \<in> A i"
unfolding A_def by (simp add: open_ball r)
moreover have "\<forall>y. (\<forall>i. y $ i \<in> A i) \<longrightarrow> y \<in> S"
by (simp add: A_def S dist_vec_def e setL2_strict_mono dist_commute)
ultimately show "\<exists>A. (\<forall>i. open (A i) \<and> x $ i \<in> A i) \<and>
(\<forall>y. (\<forall>i. y $ i \<in> A i) \<longrightarrow> y \<in> S)" by metis
qed
qed
show "open S = (\<forall>x\<in>S. \<forall>\<^sub>F (x', y) in uniformity. x' = x \<longrightarrow> y \<in> S)"
unfolding * eventually_uniformity_metric
by (simp del: split_paired_All add: dist_vec_def dist_commute)
qed
end
lemma Cauchy_vec_nth:
"Cauchy (\<lambda>n. X n) \<Longrightarrow> Cauchy (\<lambda>n. X n $ i)"
unfolding Cauchy_def by (fast intro: le_less_trans [OF dist_vec_nth_le])
lemma vec_CauchyI:
fixes X :: "nat \<Rightarrow> 'a::metric_space ^ 'n"
assumes X: "\<And>i. Cauchy (\<lambda>n. X n $ i)"
shows "Cauchy (\<lambda>n. X n)"
proof (rule metric_CauchyI)
fix r :: real assume "0 < r"
hence "0 < r / of_nat CARD('n)" (is "0 < ?s") by simp
define N where "N i = (LEAST N. \<forall>m\<ge>N. \<forall>n\<ge>N. dist (X m $ i) (X n $ i) < ?s)" for i
define M where "M = Max (range N)"
have "\<And>i. \<exists>N. \<forall>m\<ge>N. \<forall>n\<ge>N. dist (X m $ i) (X n $ i) < ?s"
using X \<open>0 < ?s\<close> by (rule metric_CauchyD)
hence "\<And>i. \<forall>m\<ge>N i. \<forall>n\<ge>N i. dist (X m $ i) (X n $ i) < ?s"
unfolding N_def by (rule LeastI_ex)
hence M: "\<And>i. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m $ i) (X n $ i) < ?s"
unfolding M_def by simp
{
fix m n :: nat
assume "M \<le> m" "M \<le> n"
have "dist (X m) (X n) = setL2 (\<lambda>i. dist (X m $ i) (X n $ i)) UNIV"
unfolding dist_vec_def ..
also have "\<dots> \<le> sum (\<lambda>i. dist (X m $ i) (X n $ i)) UNIV"
by (rule setL2_le_sum [OF zero_le_dist])
also have "\<dots> < sum (\<lambda>i::'n. ?s) UNIV"
by (rule sum_strict_mono, simp_all add: M \<open>M \<le> m\<close> \<open>M \<le> n\<close>)
also have "\<dots> = r"
by simp
finally have "dist (X m) (X n) < r" .
}
hence "\<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < r"
by simp
then show "\<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < r" ..
qed
instance vec :: (complete_space, finite) complete_space
proof
fix X :: "nat \<Rightarrow> 'a ^ 'b" assume "Cauchy X"
have "\<And>i. (\<lambda>n. X n $ i) \<longlonglongrightarrow> lim (\<lambda>n. X n $ i)"
using Cauchy_vec_nth [OF \<open>Cauchy X\<close>]
by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
hence "X \<longlonglongrightarrow> vec_lambda (\<lambda>i. lim (\<lambda>n. X n $ i))"
by (simp add: vec_tendstoI)
then show "convergent X"
by (rule convergentI)
qed
subsection \<open>Normed vector space\<close>
instantiation vec :: (real_normed_vector, finite) real_normed_vector
begin
definition "norm x = setL2 (\<lambda>i. 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 \<longleftrightarrow> x = 0"
unfolding norm_vec_def
by (simp add: setL2_eq_0_iff vec_eq_iff)
show "norm (x + y) \<le> norm x + norm y"
unfolding norm_vec_def
apply (rule order_trans [OF _ setL2_triangle_ineq])
apply (simp add: setL2_mono norm_triangle_ineq)
done
show "norm (scaleR a x) = \<bar>a\<bar> * norm x"
unfolding norm_vec_def
by (simp add: setL2_right_distrib)
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
by (simp add: dist_norm)
qed
end
lemma norm_nth_le: "norm (x $ i) \<le> norm x"
unfolding norm_vec_def
by (rule member_le_setL2) simp_all
lemma bounded_linear_vec_nth: "bounded_linear (\<lambda>x. x $ i)"
apply standard
apply (rule vector_add_component)
apply (rule vector_scaleR_component)
apply (rule_tac x="1" in exI, simp add: norm_nth_le)
done
instance vec :: (banach, finite) banach ..
subsection \<open>Inner product space\<close>
instantiation vec :: (real_inner, finite) real_inner
begin
definition "inner x y = sum (\<lambda>i. 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
by (simp add: inner_commute)
show "inner (x + y) z = inner x z + inner y z"
unfolding inner_vec_def
by (simp add: inner_add_left sum.distrib)
show "inner (scaleR r x) y = r * inner x y"
unfolding inner_vec_def
by (simp add: sum_distrib_left)
show "0 \<le> inner x x"
unfolding inner_vec_def
by (simp add: sum_nonneg)
show "inner x x = 0 \<longleftrightarrow> x = 0"
unfolding inner_vec_def
by (simp add: vec_eq_iff sum_nonneg_eq_0_iff)
show "norm x = sqrt (inner x x)"
unfolding inner_vec_def norm_vec_def setL2_def
by (simp add: power2_norm_eq_inner)
qed
end
subsection \<open>Euclidean space\<close>
text \<open>Vectors pointing along a single axis.\<close>
definition "axis k x = (\<chi> 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 \<longleftrightarrow> x = y \<and> i = j \<or> x = 0 \<and> 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 sum.remove [of _ j], simp_all)
apply (rule sum.neutral, simp add: axis_def)
apply (rule sum.neutral, simp add: axis_def)
done
lemma sum_single:
assumes "finite A" and "k \<in> A" and "f k = y"
assumes "\<And>i. i \<in> A \<Longrightarrow> i \<noteq> k \<Longrightarrow> f i = 0"
shows "(\<Sum>i\<in>A. f i) = y"
apply (subst sum.remove [OF assms(1,2)])
apply (simp add: sum.neutral assms(3,4))
done
lemma inner_axis: "inner x (axis i y) = inner (x $ i) y"
unfolding inner_vec_def
apply (rule_tac k=i in sum_single)
apply simp_all
apply (simp add: axis_def)
done
instantiation vec :: (euclidean_space, finite) euclidean_space
begin
definition "Basis = (\<Union>i. \<Union>u\<in>Basis. {axis i u})"
instance proof
show "(Basis :: ('a ^ 'b) set) \<noteq> {}"
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 \<in> Basis" and "v \<in> 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 "(\<forall>u\<in>Basis. inner x u = 0) \<longleftrightarrow> 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)"
apply (simp add: Basis_vec_def)
apply (subst card_UN_disjoint)
apply simp
apply simp
apply (auto simp: axis_eq_axis) [1]
apply (subst card_UN_disjoint)
apply (auto simp: axis_eq_axis)
done
end
lemma cart_eq_inner_axis: "a $ i = inner a (axis i 1)"
by (simp add: inner_axis)
lemma axis_in_Basis: "a \<in> Basis \<Longrightarrow> axis i a \<in> Basis"
by (auto simp add: Basis_vec_def axis_eq_axis)
end