src/HOL/Library/Product_Vector.thy
 author wenzelm Thu, 26 Mar 2009 20:08:55 +0100 changeset 30729 461ee3e49ad3 parent 30019 a2f19e0a28b2 child 31290 f41c023d90bc permissions -rw-r--r--
interpretation/interpret: prefixes are mandatory by default;
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(*  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"
show "scaleR (a + b) x = scaleR a x + scaleR b x"
show "scaleR a (scaleR b x) = scaleR (a * b) x"
show "scaleR 1 x = x"
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
show "norm (x + y) \<le> norm x + norm y"
unfolding norm_prod_def
apply (rule order_trans [OF _ real_sqrt_sum_squares_triangle_ineq])
done
show "norm (scaleR r x) = \<bar>r\<bar> * norm x"
unfolding norm_prod_def
done
show "sgn x = scaleR (inverse (norm x)) x"
by (rule sgn_prod_def)
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
show "inner (x + y) z = inner x z + inner y z"
unfolding inner_prod_def
show "inner (scaleR r x) y = r * inner x y"
unfolding inner_prod_def
show "0 \<le> inner x x"
unfolding inner_prod_def
show "inner x x = 0 \<longleftrightarrow> x = 0"
unfolding inner_prod_def expand_prod_eq
show "norm x = sqrt (inner x x)"
unfolding norm_prod_def inner_prod_def
qed

end

subsection {* Pair operations are linear and continuous *}

interpretation fst: bounded_linear fst
apply (unfold_locales)
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_scaleR)
apply (rule_tac x="1" in exI, simp add: norm_Pair)
done

text {* TODO: move to NthRoot *}
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: mult_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)"
show "(f (r *\<^sub>R x), g (r *\<^sub>R x)) = r *\<^sub>R (f x, g x)"
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 (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 next three proofs are nearly identical to each other.
Is there a good way to factor out the common parts?
*}

lemma LIMSEQ_Pair:
assumes "X ----> a" and "Y ----> b"
shows "(\<lambda>n. (X n, Y n)) ----> (a, b)"
proof (rule LIMSEQ_I)
fix r :: real assume "0 < r"
then have "0 < r / sqrt 2" (is "0 < ?s")
obtain M where M: "\<forall>n\<ge>M. norm (X n - a) < ?s"
using LIMSEQ_D [OF `X ----> a` `0 < ?s`] ..
obtain N where N: "\<forall>n\<ge>N. norm (Y n - b) < ?s"
using LIMSEQ_D [OF `Y ----> b` `0 < ?s`] ..
have "\<forall>n\<ge>max M N. norm ((X n, Y n) - (a, b)) < r"
using M N by (simp add: real_sqrt_sum_squares_less norm_Pair)
then show "\<exists>n0. \<forall>n\<ge>n0. norm ((X n, Y n) - (a, b)) < r" ..
qed

lemma Cauchy_Pair:
assumes "Cauchy X" and "Cauchy Y"
shows "Cauchy (\<lambda>n. (X n, Y n))"
proof (rule CauchyI)
fix r :: real assume "0 < r"
then have "0 < r / sqrt 2" (is "0 < ?s")
obtain M where M: "\<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < ?s"
using CauchyD [OF `Cauchy X` `0 < ?s`] ..
obtain N where N: "\<forall>m\<ge>N. \<forall>n\<ge>N. norm (Y m - Y n) < ?s"
using CauchyD [OF `Cauchy Y` `0 < ?s`] ..
have "\<forall>m\<ge>max M N. \<forall>n\<ge>max M N. norm ((X m, Y m) - (X n, Y n)) < r"
using M N by (simp add: real_sqrt_sum_squares_less norm_Pair)
then show "\<exists>n0. \<forall>m\<ge>n0. \<forall>n\<ge>n0. norm ((X m, Y m) - (X n, Y n)) < r" ..
qed

lemma LIM_Pair:
assumes "f -- x --> a" and "g -- x --> b"
shows "(\<lambda>x. (f x, g x)) -- x --> (a, b)"
proof (rule LIM_I)
fix r :: real assume "0 < r"
then have "0 < r / sqrt 2" (is "0 < ?e")
obtain s where s: "0 < s"
"\<forall>z. z \<noteq> x \<and> norm (z - x) < s \<longrightarrow> norm (f z - a) < ?e"
using LIM_D [OF `f -- x --> a` `0 < ?e`] by fast
obtain t where t: "0 < t"
"\<forall>z. z \<noteq> x \<and> norm (z - x) < t \<longrightarrow> norm (g z - b) < ?e"
using LIM_D [OF `g -- x --> b` `0 < ?e`] by fast
have "0 < min s t \<and>
(\<forall>z. z \<noteq> x \<and> norm (z - x) < min s t \<longrightarrow> norm ((f z, g z) - (a, b)) < r)"
using s t by (simp add: real_sqrt_sum_squares_less norm_Pair)
then show
"\<exists>s>0. \<forall>z. z \<noteq> x \<and> norm (z - x) < s \<longrightarrow> norm ((f z, g z) - (a, b)) < 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`]
have 2: "(\<lambda>n. snd (X n)) ----> lim (\<lambda>n. snd (X n))"
using snd.Cauchy [OF `Cauchy X`]
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 (rule real_LIM_sandwich_zero)
apply (rule FDERIV_D [OF f])
apply (rule FDERIV_D [OF g])
apply (rename_tac h)