(* 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 "*" :: (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: expand_prod_eq scaleR_right_distrib)
show "scaleR (a + b) x = scaleR a x + scaleR b x"
by (simp add: expand_prod_eq scaleR_left_distrib)
show "scaleR a (scaleR b x) = scaleR (a * b) x"
by (simp add: expand_prod_eq)
show "scaleR 1 x = x"
by (simp add: expand_prod_eq)
qed
end
subsection {* Product is a topological space *}
instantiation
"*" :: (topological_space, topological_space) topological_space
begin
definition open_prod_def:
"open (S :: ('a \<times> 'b) set) \<longleftrightarrow>
(\<forall>x\<in>S. \<exists>A B. open A \<and> open B \<and> x \<in> A \<times> B \<and> A \<times> B \<subseteq> S)"
lemma open_prod_elim:
assumes "open S" and "x \<in> S"
obtains A B where "open A" and "open B" and "x \<in> A \<times> B" and "A \<times> B \<subseteq> S"
using assms unfolding open_prod_def by fast
lemma open_prod_intro:
assumes "\<And>x. x \<in> S \<Longrightarrow> \<exists>A B. open A \<and> open B \<and> x \<in> A \<times> B \<and> A \<times> B \<subseteq> S"
shows "open S"
using assms unfolding open_prod_def by fast
instance proof
show "open (UNIV :: ('a \<times> 'b) set)"
unfolding open_prod_def by auto
next
fix S T :: "('a \<times> 'b) set"
assume "open S" "open T"
show "open (S \<inter> T)"
proof (rule open_prod_intro)
fix x assume x: "x \<in> S \<inter> T"
from x have "x \<in> S" by simp
obtain Sa Sb where A: "open Sa" "open Sb" "x \<in> Sa \<times> Sb" "Sa \<times> Sb \<subseteq> S"
using `open S` and `x \<in> S` by (rule open_prod_elim)
from x have "x \<in> T" by simp
obtain Ta Tb where B: "open Ta" "open Tb" "x \<in> Ta \<times> Tb" "Ta \<times> Tb \<subseteq> T"
using `open T` and `x \<in> T` by (rule open_prod_elim)
let ?A = "Sa \<inter> Ta" and ?B = "Sb \<inter> Tb"
have "open ?A \<and> open ?B \<and> x \<in> ?A \<times> ?B \<and> ?A \<times> ?B \<subseteq> S \<inter> T"
using A B by (auto simp add: open_Int)
thus "\<exists>A B. open A \<and> open B \<and> x \<in> A \<times> B \<and> A \<times> B \<subseteq> S \<inter> T"
by fast
qed
next
fix K :: "('a \<times> 'b) set set"
assume "\<forall>S\<in>K. open S" thus "open (\<Union>K)"
unfolding open_prod_def by fast
qed
end
lemma open_Times: "open S \<Longrightarrow> open T \<Longrightarrow> open (S \<times> T)"
unfolding open_prod_def by auto
lemma fst_vimage_eq_Times: "fst -` S = S \<times> UNIV"
by auto
lemma snd_vimage_eq_Times: "snd -` S = UNIV \<times> S"
by auto
lemma open_vimage_fst: "open S \<Longrightarrow> open (fst -` S)"
by (simp add: fst_vimage_eq_Times open_Times)
lemma open_vimage_snd: "open S \<Longrightarrow> open (snd -` S)"
by (simp add: snd_vimage_eq_Times open_Times)
lemma closed_vimage_fst: "closed S \<Longrightarrow> closed (fst -` S)"
unfolding closed_open vimage_Compl [symmetric]
by (rule open_vimage_fst)
lemma closed_vimage_snd: "closed S \<Longrightarrow> closed (snd -` S)"
unfolding closed_open vimage_Compl [symmetric]
by (rule open_vimage_snd)
lemma closed_Times: "closed S \<Longrightarrow> closed T \<Longrightarrow> closed (S \<times> T)"
proof -
have "S \<times> T = (fst -` S) \<inter> (snd -` T)" by auto
thus "closed S \<Longrightarrow> closed T \<Longrightarrow> closed (S \<times> T)"
by (simp add: closed_vimage_fst closed_vimage_snd closed_Int)
qed
lemma openI: (* TODO: move *)
assumes "\<And>x. x \<in> S \<Longrightarrow> \<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> S"
shows "open S"
proof -
have "open (\<Union>{T. open T \<and> T \<subseteq> S})" by auto
moreover have "\<Union>{T. open T \<and> T \<subseteq> S} = S" by (auto dest!: assms)
ultimately show "open S" by simp
qed
lemma subset_fst_imageI: "A \<times> B \<subseteq> S \<Longrightarrow> y \<in> B \<Longrightarrow> A \<subseteq> fst ` S"
unfolding image_def subset_eq by force
lemma subset_snd_imageI: "A \<times> B \<subseteq> S \<Longrightarrow> x \<in> A \<Longrightarrow> B \<subseteq> 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 \<in> fst ` S"
then obtain y where "(x, y) \<in> S" by auto
then obtain A B where "open A" "open B" "x \<in> A" "y \<in> B" "A \<times> B \<subseteq> S"
using `open S` unfolding open_prod_def by auto
from `A \<times> B \<subseteq> S` `y \<in> B` have "A \<subseteq> fst ` S" by (rule subset_fst_imageI)
with `open A` `x \<in> A` have "open A \<and> x \<in> A \<and> A \<subseteq> fst ` S" by simp
then show "\<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> fst ` S" by - (rule exI)
qed
lemma open_image_snd: assumes "open S" shows "open (snd ` S)"
proof (rule openI)
fix y assume "y \<in> snd ` S"
then obtain x where "(x, y) \<in> S" by auto
then obtain A B where "open A" "open B" "x \<in> A" "y \<in> B" "A \<times> B \<subseteq> S"
using `open S` unfolding open_prod_def by auto
from `A \<times> B \<subseteq> S` `x \<in> A` have "B \<subseteq> snd ` S" by (rule subset_snd_imageI)
with `open B` `y \<in> B` have "open B \<and> y \<in> B \<and> B \<subseteq> snd ` S" by simp
then show "\<exists>T. open T \<and> y \<in> T \<and> T \<subseteq> snd ` S" by - (rule exI)
qed
subsection {* Product is a metric space *}
instantiation
"*" :: (metric_space, metric_space) metric_space
begin
definition dist_prod_def:
"dist (x::'a \<times> 'b) y = sqrt ((dist (fst x) (fst y))\<twosuperior> + (dist (snd x) (snd y))\<twosuperior>)"
lemma dist_Pair_Pair: "dist (a, b) (c, d) = sqrt ((dist a c)\<twosuperior> + (dist b d)\<twosuperior>)"
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 expand_prod_eq 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
(* FIXME: long proof! *)
(* Maybe it would be easier to define topological spaces *)
(* in terms of neighborhoods instead of open sets? *)
fix S :: "('a \<times> 'b) set"
show "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 S` and `x \<in> S` by (rule open_prod_elim)
obtain r where r: "0 < r" "\<forall>y. dist y (fst x) < r \<longrightarrow> y \<in> A"
using `open A` and `x \<in> A \<times> B` unfolding open_dist by auto
obtain s where s: "0 < s" "\<forall>y. dist y (snd x) < s \<longrightarrow> y \<in> B"
using `open B` and `x \<in> A \<times> B` 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 `A \<times> B \<subseteq> S` 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" thus "open S"
unfolding open_prod_def open_dist
apply safe
apply (drule (1) bspec)
apply clarify
apply (subgoal_tac "\<exists>r>0. \<exists>s>0. e = sqrt (r\<twosuperior> + s\<twosuperior>)")
apply clarify
apply (rule_tac x="{y. dist y a < r}" in exI)
apply (rule_tac x="{y. dist y b < s}" in exI)
apply (rule conjI)
apply clarify
apply (rule_tac x="r - dist x a" in exI, rule conjI, simp)
apply clarify
apply (simp add: less_diff_eq)
apply (erule le_less_trans [OF dist_triangle])
apply (rule conjI)
apply clarify
apply (rule_tac x="s - dist x b" in exI, rule conjI, simp)
apply clarify
apply (simp add: less_diff_eq)
apply (erule le_less_trans [OF dist_triangle])
apply (rule conjI)
apply simp
apply (clarify, rename_tac c d)
apply (drule spec, erule mp)
apply (simp add: dist_Pair_Pair add_strict_mono power_strict_mono)
apply (rule_tac x="e / sqrt 2" in exI, simp add: divide_pos_pos)
apply (rule_tac x="e / sqrt 2" in exI, simp add: divide_pos_pos)
apply (simp add: power_divide)
done
qed
qed
end
subsection {* Continuity of operations *}
lemma tendsto_fst [tendsto_intros]:
assumes "(f ---> a) net"
shows "((\<lambda>x. fst (f x)) ---> fst a) net"
proof (rule topological_tendstoI)
fix S assume "open S" "fst a \<in> S"
then have "open (fst -` S)" "a \<in> fst -` S"
unfolding open_prod_def
apply simp_all
apply clarify
apply (rule exI, erule conjI)
apply (rule exI, rule conjI [OF open_UNIV])
apply auto
done
with assms have "eventually (\<lambda>x. f x \<in> fst -` S) net"
by (rule topological_tendstoD)
then show "eventually (\<lambda>x. fst (f x) \<in> S) net"
by simp
qed
lemma tendsto_snd [tendsto_intros]:
assumes "(f ---> a) net"
shows "((\<lambda>x. snd (f x)) ---> snd a) net"
proof (rule topological_tendstoI)
fix S assume "open S" "snd a \<in> S"
then have "open (snd -` S)" "a \<in> snd -` S"
unfolding open_prod_def
apply simp_all
apply clarify
apply (rule exI, rule conjI [OF open_UNIV])
apply (rule exI, erule conjI)
apply auto
done
with assms have "eventually (\<lambda>x. f x \<in> snd -` S) net"
by (rule topological_tendstoD)
then show "eventually (\<lambda>x. snd (f x) \<in> S) net"
by simp
qed
lemma tendsto_Pair [tendsto_intros]:
assumes "(f ---> a) net" and "(g ---> b) net"
shows "((\<lambda>x. (f x, g x)) ---> (a, b)) net"
proof (rule topological_tendstoI)
fix S assume "open S" "(a, b) \<in> S"
then obtain A B where "open A" "open B" "a \<in> A" "b \<in> B" "A \<times> B \<subseteq> S"
unfolding open_prod_def by auto
have "eventually (\<lambda>x. f x \<in> A) net"
using `(f ---> a) net` `open A` `a \<in> A`
by (rule topological_tendstoD)
moreover
have "eventually (\<lambda>x. g x \<in> B) net"
using `(g ---> b) net` `open B` `b \<in> B`
by (rule topological_tendstoD)
ultimately
show "eventually (\<lambda>x. (f x, g x) \<in> S) net"
by (rule eventually_elim2)
(simp add: subsetD [OF `A \<times> B \<subseteq> S`])
qed
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"
then have "0 < r / sqrt 2" (is "0 < ?s")
by (simp add: divide_pos_pos)
obtain M where M: "\<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < ?s"
using metric_CauchyD [OF `Cauchy X` `0 < ?s`] ..
obtain N where N: "\<forall>m\<ge>N. \<forall>n\<ge>N. dist (Y m) (Y n) < ?s"
using metric_CauchyD [OF `Cauchy Y` `0 < ?s`] ..
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
lemma isCont_Pair [simp]:
"\<lbrakk>isCont f x; isCont g x\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. (f x, g x)) x"
unfolding isCont_def by (rule tendsto_Pair)
subsection {* Product is a complete metric space *}
instance "*" :: (complete_space, complete_space) complete_space
proof
fix X :: "nat \<Rightarrow> 'a \<times> 'b" assume "Cauchy X"
have 1: "(\<lambda>n. fst (X n)) ----> lim (\<lambda>n. fst (X n))"
using Cauchy_fst [OF `Cauchy X`]
by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
have 2: "(\<lambda>n. snd (X n)) ----> lim (\<lambda>n. snd (X n))"
using Cauchy_snd [OF `Cauchy X`]
by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
have "X ----> (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 {* Product is a normed vector space *}
instantiation
"*" :: (real_normed_vector, real_normed_vector) real_normed_vector
begin
definition norm_prod_def:
"norm x = sqrt ((norm (fst x))\<twosuperior> + (norm (snd x))\<twosuperior>)"
definition sgn_prod_def:
"sgn (x::'a \<times> 'b) = scaleR (inverse (norm x)) x"
lemma norm_Pair: "norm (a, b) = sqrt ((norm a)\<twosuperior> + (norm b)\<twosuperior>)"
unfolding norm_prod_def by simp
instance proof
fix r :: real and x y :: "'a \<times> 'b"
show "0 \<le> norm x"
unfolding norm_prod_def by simp
show "norm x = 0 \<longleftrightarrow> x = 0"
unfolding norm_prod_def
by (simp add: expand_prod_eq)
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: right_distrib [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 "*" :: (banach, banach) banach ..
subsection {* Product is an inner product space *}
instantiation "*" :: (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: right_distrib)
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 expand_prod_eq
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
subsection {* Pair operations are linear *}
interpretation fst: bounded_linear fst
apply (unfold_locales)
apply (rule fst_add)
apply (rule fst_scaleR)
apply (rule_tac x="1" in exI, simp add: norm_Pair)
done
interpretation snd: bounded_linear snd
apply (unfold_locales)
apply (rule snd_add)
apply (rule snd_scaleR)
apply (rule_tac x="1" in exI, simp add: norm_Pair)
done
text {* TODO: move to NthRoot *}
lemma sqrt_add_le_add_sqrt:
assumes x: "0 \<le> x" and y: "0 \<le> y"
shows "sqrt (x + y) \<le> sqrt x + sqrt y"
apply (rule power2_le_imp_le)
apply (simp add: real_sum_squared_expand add_nonneg_nonneg x y)
apply (simp add: mult_nonneg_nonneg x y)
apply (simp add: add_nonneg_nonneg x y)
done
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: right_distrib)
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
subsection {* Frechet derivatives involving pairs *}
lemma FDERIV_Pair:
assumes f: "FDERIV f x :> f'" and g: "FDERIV g x :> g'"
shows "FDERIV (\<lambda>x. (f x, g x)) x :> (\<lambda>h. (f' h, g' h))"
apply (rule FDERIV_I)
apply (rule bounded_linear_Pair)
apply (rule FDERIV_bounded_linear [OF f])
apply (rule FDERIV_bounded_linear [OF g])
apply (simp add: norm_Pair)
apply (rule real_LIM_sandwich_zero)
apply (rule LIM_add_zero)
apply (rule FDERIV_D [OF f])
apply (rule FDERIV_D [OF g])
apply (rename_tac h)
apply (simp add: divide_nonneg_pos)
apply (rename_tac h)
apply (subst add_divide_distrib [symmetric])
apply (rule divide_right_mono [OF _ norm_ge_zero])
apply (rule order_trans [OF sqrt_add_le_add_sqrt])
apply simp
apply simp
apply simp
done
end