moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
(* Title: HOL/Analysis/Product_Vector.thy
Author: Brian Huffman
*)
section \<open>Cartesian Products as Vector Spaces\<close>
theory Product_Vector
imports
Inner_Product
"HOL-Library.Product_Plus"
begin
subsection \<open>Product is a real vector space\<close>
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 \<times> '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 \<open>Product is a metric space\<close>
(* 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))\<^sup>2 + (dist (snd x) (snd y))\<^sup>2)"
instance ..
end
instantiation prod :: (metric_space, metric_space) uniformity_dist
begin
definition [code del]:
"(uniformity :: (('a \<times> 'b) \<times> ('a \<times> '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 \<times> '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)\<^sup>2 + (dist b d)\<^sup>2)"
unfolding dist_prod_def by simp
lemma dist_fst_le: "dist (fst x) (fst y) \<le> dist x y"
unfolding dist_prod_def by (rule real_sqrt_sum_squares_ge1)
lemma dist_snd_le: "dist (snd x) (snd y) \<le> dist x y"
unfolding dist_prod_def by (rule real_sqrt_sum_squares_ge2)
instance
proof
fix x y :: "'a \<times> 'b"
show "dist x y = 0 \<longleftrightarrow> x = y"
unfolding dist_prod_def prod_eq_iff by simp
next
fix x y z :: "'a \<times> 'b"
show "dist x y \<le> 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 \<times> '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 B where "open A" "open B" "x \<in> A \<times> B" "A \<times> B \<subseteq> S"
using \<open>open S\<close> and \<open>x \<in> S\<close> by (rule open_prod_elim)
obtain r where r: "0 < r" "\<forall>y. dist y (fst x) < r \<longrightarrow> y \<in> A"
using \<open>open A\<close> and \<open>x \<in> A \<times> B\<close> unfolding open_dist by auto
obtain s where s: "0 < s" "\<forall>y. dist y (snd x) < s \<longrightarrow> y \<in> B"
using \<open>open B\<close> and \<open>x \<in> A \<times> B\<close> unfolding open_dist by auto
let ?e = "min r s"
have "0 < ?e \<and> (\<forall>y. dist y x < ?e \<longrightarrow> y \<in> 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 \<in> A" and "snd y \<in> B"
by (simp_all add: r(2) s(2))
hence "y \<in> A \<times> B" by (induct y, simp)
with \<open>A \<times> B \<subseteq> S\<close> show "y \<in> S" ..
qed
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 (rule open_prod_intro)
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 "r = e / sqrt 2"
define s where "s = e / sqrt 2"
from \<open>0 < e\<close> have "0 < r" and "0 < s"
unfolding r_def s_def by simp_all
from \<open>0 < e\<close> have "e = sqrt (r\<^sup>2 + s\<^sup>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 \<in> A \<times> B"
unfolding A_def B_def mem_Times_iff
using \<open>0 < r\<close> and \<open>0 < s\<close> by simp
moreover have "A \<times> B \<subseteq> S"
proof (clarify)
fix a b assume "a \<in> A" and "b \<in> 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 \<open>e = sqrt (r\<^sup>2 + s\<^sup>2)\<close>
by (simp add: add_strict_mono power_strict_mono)
thus "(a, b) \<in> S"
by (simp add: S)
qed
ultimately show "\<exists>A B. open A \<and> open B \<and> x \<in> A \<times> B \<and> A \<times> B \<subseteq> S" by fast
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_prod_def dist_commute)
qed
end
declare [[code abort: "dist::('a::metric_space*'b::metric_space)\<Rightarrow>('a*'b) \<Rightarrow> real"]]
lemma Cauchy_fst: "Cauchy X \<Longrightarrow> Cauchy (\<lambda>n. fst (X n))"
unfolding Cauchy_def by (fast elim: le_less_trans [OF dist_fst_le])
lemma Cauchy_snd: "Cauchy X \<Longrightarrow> Cauchy (\<lambda>n. 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 (\<lambda>n. (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: "\<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < ?s"
using metric_CauchyD [OF \<open>Cauchy X\<close> \<open>0 < ?s\<close>] ..
obtain N where N: "\<forall>m\<ge>N. \<forall>n\<ge>N. dist (Y m) (Y n) < ?s"
using metric_CauchyD [OF \<open>Cauchy Y\<close> \<open>0 < ?s\<close>] ..
have "\<forall>m\<ge>max M N. \<forall>n\<ge>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 "\<exists>n0. \<forall>m\<ge>n0. \<forall>n\<ge>n0. dist (X m, Y m) (X n, Y n) < r" ..
qed
subsection \<open>Product is a complete metric space\<close>
instance prod :: (complete_space, complete_space) complete_space
proof
fix X :: "nat \<Rightarrow> 'a \<times> 'b" assume "Cauchy X"
have 1: "(\<lambda>n. fst (X n)) \<longlonglongrightarrow> lim (\<lambda>n. fst (X n))"
using Cauchy_fst [OF \<open>Cauchy X\<close>]
by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
have 2: "(\<lambda>n. snd (X n)) \<longlonglongrightarrow> lim (\<lambda>n. snd (X n))"
using Cauchy_snd [OF \<open>Cauchy X\<close>]
by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
have "X \<longlonglongrightarrow> (lim (\<lambda>n. fst (X n)), lim (\<lambda>n. snd (X n)))"
using tendsto_Pair [OF 1 2] by simp
then show "convergent X"
by (rule convergentI)
qed
subsection \<open>Product is a normed vector space\<close>
instantiation prod :: (real_normed_vector, real_normed_vector) real_normed_vector
begin
definition norm_prod_def[code del]:
"norm x = sqrt ((norm (fst x))\<^sup>2 + (norm (snd x))\<^sup>2)"
definition sgn_prod_def:
"sgn (x::'a \<times> 'b) = scaleR (inverse (norm x)) x"
lemma norm_Pair: "norm (a, b) = sqrt ((norm a)\<^sup>2 + (norm b)\<^sup>2)"
unfolding norm_prod_def by simp
instance
proof
fix r :: real and x y :: "'a \<times> 'b"
show "norm x = 0 \<longleftrightarrow> x = 0"
unfolding norm_prod_def
by (simp add: prod_eq_iff)
show "norm (x + y) \<le> 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) = \<bar>r\<bar> * 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
declare [[code abort: "norm::('a::real_normed_vector*'b::real_normed_vector) \<Rightarrow> real"]]
instance prod :: (banach, banach) banach ..
subsubsection \<open>Pair operations are linear\<close>
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)
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 (\<lambda>x. (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 *\<^sub>R x), g (r *\<^sub>R x)) = r *\<^sub>R (f x, g x)"
by (simp add: f.scaleR g.scaleR)
obtain Kf where "0 < Kf" and norm_f: "\<And>x. norm (f x) \<le> norm x * Kf"
using f.pos_bounded by fast
obtain Kg where "0 < Kg" and norm_g: "\<And>x. norm (g x) \<le> norm x * Kg"
using g.pos_bounded by fast
have "\<forall>x. norm (f x, g x) \<le> 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 "\<exists>K. \<forall>x. norm (f x, g x) \<le> norm x * K" ..
qed
subsubsection \<open>Frechet derivatives involving pairs\<close>
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 "((\<lambda>x. (f x, g x)) has_derivative (\<lambda>h. (f' h, g' h))) (at x within s)"
proof (rule has_derivativeI_sandwich[of 1])
show "bounded_linear (\<lambda>h. (f' h, g' h))"
using f g by (intro bounded_linear_Pair has_derivative_bounded_linear)
let ?Rf = "\<lambda>y. f y - f x - f' (y - x)"
let ?Rg = "\<lambda>y. g y - g x - g' (y - x)"
let ?R = "\<lambda>y. ((f y, g y) - (f x, g x) - (f' (y - x), g' (y - x)))"
show "((\<lambda>y. norm (?Rf y) / norm (y - x) + norm (?Rg y) / norm (y - x)) \<longlongrightarrow> 0) (at x within s)"
using f g by (intro tendsto_add_zero) (auto simp: has_derivative_iff_norm)
fix y :: 'a assume "y \<noteq> x"
show "norm (?R y) / norm (y - x) \<le> 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
lemma differentiable_Pair [simp, derivative_intros]:
"f differentiable at x within s \<Longrightarrow> g differentiable at x within s \<Longrightarrow>
(\<lambda>x. (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]:
"((\<lambda>p. f (fst p) (snd p)) has_derivative f') F \<Longrightarrow> ((\<lambda>(a, b). f a b) has_derivative f') F"
unfolding split_beta' .
subsubsection \<open>Vector derivatives involving pairs\<close>
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 "((\<lambda>x. (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)
subsection \<open>Product is an inner product space\<close>
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 \<times> '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 \<le> inner x x"
unfolding inner_prod_def
by (intro add_nonneg_nonneg inner_ge_zero)
show "inner x x = 0 \<longleftrightarrow> 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
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)"
by (simp add: norm_Pair)
lemma norm_fst_le: "norm x \<le> norm (x,y)"
by (metis dist_fst_le fst_conv fst_zero norm_conv_dist)
lemma norm_snd_le: "norm y \<le> norm (x,y)"
by (metis dist_snd_le snd_conv snd_zero norm_conv_dist)
lemma norm_Pair_le:
shows "norm (x, y) \<le> norm x + norm y"
unfolding norm_Pair
by (metis norm_ge_zero sqrt_sum_squares_le_sum)
end