(* 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 \ '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 \ 'b) set) \
(\x\S. \A B. open A \ open B \ x \ A \ B \ A \ B \ S)"
lemma open_prod_elim:
assumes "open S" and "x \ S"
obtains A B where "open A" and "open B" and "x \ A \ B" and "A \ B \ S"
using assms unfolding open_prod_def by fast
lemma open_prod_intro:
assumes "\x. x \ S \ \A B. open A \ open B \ x \ A \ B \ A \ B \ S"
shows "open S"
using assms unfolding open_prod_def by fast
instance proof
show "open (UNIV :: ('a \ 'b) set)"
unfolding open_prod_def by auto
next
fix S T :: "('a \ 'b) set"
assume "open S" "open T"
show "open (S \ T)"
proof (rule open_prod_intro)
fix x assume x: "x \ S \ T"
from x have "x \ S" by simp
obtain Sa Sb where A: "open Sa" "open Sb" "x \ Sa \ Sb" "Sa \ Sb \ S"
using `open S` and `x \ S` by (rule open_prod_elim)
from x have "x \ T" by simp
obtain Ta Tb where B: "open Ta" "open Tb" "x \ Ta \ Tb" "Ta \ Tb \ T"
using `open T` and `x \ T` by (rule open_prod_elim)
let ?A = "Sa \ Ta" and ?B = "Sb \ Tb"
have "open ?A \ open ?B \ x \ ?A \ ?B \ ?A \ ?B \ S \ T"
using A B by (auto simp add: open_Int)
thus "\A B. open A \ open B \ x \ A \ B \ A \ B \ S \ T"
by fast
qed
next
fix K :: "('a \ 'b) set set"
assume "\S\K. open S" thus "open (\K)"
unfolding open_prod_def by fast
qed
end
lemma open_Times: "open S \ open T \ open (S \ T)"
unfolding open_prod_def by auto
lemma fst_vimage_eq_Times: "fst -` S = S \ UNIV"
by auto
lemma snd_vimage_eq_Times: "snd -` S = UNIV \ S"
by auto
lemma open_vimage_fst: "open S \ open (fst -` S)"
by (simp add: fst_vimage_eq_Times open_Times)
lemma open_vimage_snd: "open S \ open (snd -` S)"
by (simp add: snd_vimage_eq_Times open_Times)
lemma closed_vimage_fst: "closed S \ closed (fst -` S)"
unfolding closed_open vimage_Compl [symmetric]
by (rule open_vimage_fst)
lemma closed_vimage_snd: "closed S \ closed (snd -` S)"
unfolding closed_open vimage_Compl [symmetric]
by (rule open_vimage_snd)
lemma closed_Times: "closed S \ closed T \ closed (S \ T)"
proof -
have "S \ T = (fst -` S) \ (snd -` T)" by auto
thus "closed S \ closed T \ closed (S \ T)"
by (simp add: closed_vimage_fst closed_vimage_snd closed_Int)
qed
lemma openI: (* TODO: move *)
assumes "\x. x \ S \ \T. open T \ x \ T \ T \ S"
shows "open S"
proof -
have "open (\{T. open T \ T \ S})" by auto
moreover have "\{T. open T \ T \ S} = S" by (auto dest!: assms)
ultimately show "open S" by simp
qed
lemma subset_fst_imageI: "A \ B \ S \ y \ B \ A \ fst ` S"
unfolding image_def subset_eq by force
lemma subset_snd_imageI: "A \ B \ S \ x \ A \ B \ 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 \ fst ` S"
then obtain y where "(x, y) \ S" by auto
then obtain A B where "open A" "open B" "x \ A" "y \ B" "A \ B \ S"
using `open S` unfolding open_prod_def by auto
from `A \ B \ S` `y \ B` have "A \ fst ` S" by (rule subset_fst_imageI)
with `open A` `x \ A` have "open A \ x \ A \ A \ fst ` S" by simp
then show "\T. open T \ x \ T \ T \ fst ` S" by - (rule exI)
qed
lemma open_image_snd: assumes "open S" shows "open (snd ` S)"
proof (rule openI)
fix y assume "y \ snd ` S"
then obtain x where "(x, y) \ S" by auto
then obtain A B where "open A" "open B" "x \ A" "y \ B" "A \ B \ S"
using `open S` unfolding open_prod_def by auto
from `A \ B \ S` `x \ A` have "B \ snd ` S" by (rule subset_snd_imageI)
with `open B` `y \ B` have "open B \ y \ B \ B \ snd ` S" by simp
then show "\T. open T \ y \ T \ T \ 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 \ 'b) y = sqrt ((dist (fst x) (fst y))\ + (dist (snd x) (snd y))\)"
lemma dist_Pair_Pair: "dist (a, b) (c, d) = sqrt ((dist a c)\ + (dist b d)\)"
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 expand_prod_eq 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]
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 \ 'b) set"
show "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"
by (simp_all add: r(2) s(2))
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" thus "open S"
unfolding open_prod_def open_dist
apply safe
apply (drule (1) bspec)
apply clarify
apply (subgoal_tac "\r>0. \s>0. e = sqrt (r\ + s\)")
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 "((\x. fst (f x)) ---> fst a) net"
proof (rule topological_tendstoI)
fix S assume "open S" "fst a \ S"
then have "open (fst -` S)" "a \ 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 (\x. f x \ fst -` S) net"
by (rule topological_tendstoD)
then show "eventually (\x. fst (f x) \ S) net"
by simp
qed
lemma tendsto_snd [tendsto_intros]:
assumes "(f ---> a) net"
shows "((\x. snd (f x)) ---> snd a) net"
proof (rule topological_tendstoI)
fix S assume "open S" "snd a \ S"
then have "open (snd -` S)" "a \ 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 (\x. f x \ snd -` S) net"
by (rule topological_tendstoD)
then show "eventually (\x. snd (f x) \ S) net"
by simp
qed
lemma tendsto_Pair [tendsto_intros]:
assumes "(f ---> a) net" and "(g ---> b) net"
shows "((\x. (f x, g x)) ---> (a, b)) net"
proof (rule topological_tendstoI)
fix S assume "open S" "(a, b) \ S"
then obtain A B where "open A" "open B" "a \ A" "b \ B" "A \ B \ S"
unfolding open_prod_def by auto
have "eventually (\x. f x \ A) net"
using `(f ---> a) net` `open A` `a \ A`
by (rule topological_tendstoD)
moreover
have "eventually (\x. g x \ B) net"
using `(g ---> b) net` `open B` `b \ B`
by (rule topological_tendstoD)
ultimately
show "eventually (\x. (f x, g x) \ S) net"
by (rule eventually_elim2)
(simp add: subsetD [OF `A \ B \ S`])
qed
lemma LIM_fst: "f -- x --> a \ (\x. fst (f x)) -- x --> fst a"
unfolding LIM_conv_tendsto by (rule tendsto_fst)
lemma LIM_snd: "f -- x --> a \ (\x. snd (f x)) -- x --> snd a"
unfolding LIM_conv_tendsto by (rule tendsto_snd)
lemma LIM_Pair:
assumes "f -- x --> a" and "g -- x --> b"
shows "(\x. (f x, g x)) -- x --> (a, b)"
using assms unfolding LIM_conv_tendsto
by (rule tendsto_Pair)
lemma Cauchy_fst: "Cauchy X \ Cauchy (\n. fst (X n))"
unfolding Cauchy_def by (fast elim: le_less_trans [OF dist_fst_le])
lemma Cauchy_snd: "Cauchy X \