(* 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 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
instance proof
fix x y :: "'a \<times> 'b"
show "dist x y = 0 \<longleftrightarrow> x = y"
unfolding dist_prod_def
by (simp add: expand_prod_eq)
next
fix x y z :: "'a \<times> 'b"
show "dist x y \<le> dist x z + dist y z"
unfolding dist_prod_def
apply (rule order_trans [OF _ real_sqrt_sum_squares_triangle_ineq])
apply (rule real_sqrt_le_mono)
apply (rule order_trans [OF add_mono])
apply (rule power_mono [OF dist_triangle2 [of _ _ "fst z"] zero_le_dist])
apply (rule power_mono [OF dist_triangle2 [of _ _ "snd z"] zero_le_dist])
apply (simp only: real_sum_squared_expand)
done
qed
end
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: norm_scaleR 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
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_left_distrib)
show "inner (scaleR r x) y = r * inner x y"
unfolding inner_prod_def
by (simp add: inner_scaleR_left 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 and continuous *}
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
text {*
TODO: The "tendsto" notion generalizes LIM and LIMSEQ.
But the Cauchy proof still requires a lot of duplication.
Is there a good way to factor out the common parts?
*}
lemma tendsto_Pair:
assumes "tendsto X a net" and "tendsto Y b net"
shows "tendsto (\<lambda>i. (X i, Y i)) (a, b) net"
proof (rule tendstoI)
fix r :: real assume "0 < r"
then have "0 < r / sqrt 2" (is "0 < ?s")
by (simp add: divide_pos_pos)
have "eventually (\<lambda>i. dist (X i) a < ?s) net"
using `tendsto X a net` `0 < ?s` by (rule tendstoD)
moreover
have "eventually (\<lambda>i. dist (Y i) b < ?s) net"
using `tendsto Y b net` `0 < ?s` by (rule tendstoD)
ultimately
show "eventually (\<lambda>i. dist (X i, Y i) (a, b) < r) net"
by (rule eventually_elim2)
(simp add: real_sqrt_sum_squares_less dist_Pair_Pair)
qed
lemma LIMSEQ_Pair:
assumes "X ----> a" and "Y ----> b"
shows "(\<lambda>n. (X n, Y n)) ----> (a, b)"
using assms unfolding LIMSEQ_conv_tendsto
by (rule tendsto_Pair)
lemma LIM_Pair:
assumes "f -- x --> a" and "g -- x --> b"
shows "(\<lambda>x. (f x, g x)) -- x --> (a, b)"
using assms unfolding LIM_conv_tendsto
by (rule tendsto_Pair)
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 LIM_Pair)
subsection {* Product is a complete vector space *}
instance "*" :: (banach, banach) banach
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 fst.Cauchy [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 snd.Cauchy [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 LIMSEQ_Pair [OF 1 2] by simp
then show "convergent X"
by (rule convergentI)
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