(* Title: HOL/Analysis/Product_Vector.thy
Author: Brian Huffman
*)
section \<open>Cartesian Products as Vector Spaces\<close>
theory Product_Vector
imports
Complex_Main
"HOL-Library.Product_Plus"
begin
lemma Times_eq_image_sum:
fixes S :: "'a :: comm_monoid_add set" and T :: "'b :: comm_monoid_add set"
shows "S \ T = {u + v |u v. u \ (\x. (x, 0)) ` S \ v \ Pair 0 ` T}"
by force
subsection \<open>Product is a Module\<close>
locale module_prod = module_pair begin
definition scale :: "'a \ 'b \ 'c \ 'b \ 'c"
where "scale a v = (s1 a (fst v), s2 a (snd v))"
lemma\<^marker>\<open>tag important\<close> scale_prod: "scale x (a, b) = (s1 x a, s2 x b)"
by (auto simp: scale_def)
sublocale\<^marker>\<open>tag important\<close> p: module scale
proof qed (simp_all add: scale_def
m1.scale_left_distrib m1.scale_right_distrib m2.scale_left_distrib m2.scale_right_distrib)
lemma subspace_Times: "m1.subspace A \ m2.subspace B \ p.subspace (A \ B)"
unfolding m1.subspace_def m2.subspace_def p.subspace_def
by (auto simp: zero_prod_def scale_def)
lemma module_hom_fst: "module_hom scale s1 fst"
by unfold_locales (auto simp: scale_def)
lemma module_hom_snd: "module_hom scale s2 snd"
by unfold_locales (auto simp: scale_def)
end
locale vector_space_prod = vector_space_pair begin
sublocale module_prod s1 s2
rewrites "module_hom = Vector_Spaces.linear"
by unfold_locales (fact module_hom_eq_linear)
sublocale p: vector_space scale by unfold_locales (auto simp: algebra_simps)
lemmas linear_fst = module_hom_fst
and linear_snd = module_hom_snd
end
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
proposition 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: 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
lemma module_prod_scale_eq_scaleR: "module_prod.scale (*\<^sub>R) (*\<^sub>R) = scaleR"
apply (rule ext) apply (rule ext)
apply (subst module_prod.scale_def)
subgoal by unfold_locales
by (simp add: scaleR_prod_def)
interpretation real_vector?: vector_space_prod "scaleR::_\_\'a::real_vector" "scaleR::_\_\'b::real_vector"
rewrites "scale = ((*\<^sub>R)::_\_\('a \ 'b))"
and "module.dependent (*\<^sub>R) = dependent"
and "module.representation (*\<^sub>R) = representation"
and "module.subspace (*\<^sub>R) = subspace"
and "module.span (*\<^sub>R) = span"
and "vector_space.extend_basis (*\<^sub>R) = extend_basis"
and "vector_space.dim (*\<^sub>R) = dim"
and "Vector_Spaces.linear (*\<^sub>R) (*\<^sub>R) = linear"
subgoal by unfold_locales
subgoal by (fact module_prod_scale_eq_scaleR)
unfolding dependent_raw_def representation_raw_def subspace_raw_def span_raw_def
extend_basis_raw_def dim_raw_def linear_def
by (rule refl)+
subsection \<open>Product is a Metric Space\<close>
(* TODO: Product of uniform spaces and compatibility with metric_spaces! *)
instantiation\<^marker>\<open>tag unimportant\<close> 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\<^marker>\<open>tag unimportant\<close> prod :: (metric_space, metric_space) uniformity_dist
begin
definition [code del]:
"(uniformity :: (('a \ 'b) \ ('a \ 'b)) filter) =
(INF e\<in>{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
proposition 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) \ 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 prod_eq_iff 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
fix S :: "('a \ 'b) set"
have *: "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>open S\<close> and \<open>x \<in> S\<close> by (rule open_prod_elim)
obtain r where r: "0 < r" "\y. dist y (fst x) < r \ y \ 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" "\y. dist y (snd x) < s \ y \ 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 \ (\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 \<open>A \<times> B \<subseteq> S\<close> show "y \<in> 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" show "open S"
proof (rule open_prod_intro)
fix x assume "x \ S"
then obtain e where "0 < e" and S: "\y. dist y x < e \ y \ 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 \ A \ B"
unfolding A_def B_def mem_Times_iff
using \<open>0 < r\<close> and \<open>0 < s\<close> by simp
moreover have "A \ B \ S"
proof (clarify)
fix a b assume "a \ A" and "b \ 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) \ S"
by (simp add: S)
qed
ultimately show "\A B. open A \ open B \ x \ A \ B \ A \ B \ S" by fast
qed
qed
show "open S = (\x\S. \\<^sub>F (x', y) in uniformity. x' = x \ y \ 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)\('a*'b) \ real"]]
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 \ Cauchy (\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 (\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: "\m\M. \n\M. dist (X m) (X n) < ?s"
using metric_CauchyD [OF \<open>Cauchy X\<close> \<open>0 < ?s\<close>] ..
obtain N where N: "\m\N. \n\N. dist (Y m) (Y n) < ?s"
using metric_CauchyD [OF \<open>Cauchy Y\<close> \<open>0 < ?s\<close>] ..
have "\m\max M N. \n\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 "\n0. \m\n0. \n\n0. dist (X m, Y m) (X n, Y n) < r" ..
qed
subsection \<open>Product is a Complete Metric Space\<close>
instance\<^marker>\<open>tag important\<close> prod :: (complete_space, complete_space) complete_space
proof
fix X :: "nat \ 'a \ 'b" assume "Cauchy X"
have 1: "(\n. fst (X n)) \ lim (\n. fst (X n))"
using Cauchy_fst [OF \<open>Cauchy X\<close>]
by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
have 2: "(\n. snd (X n)) \ lim (\n. snd (X n))"
using Cauchy_snd [OF \<open>Cauchy X\<close>]
by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
have "X \ (lim (\n. fst (X n)), lim (\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 \ 'b) = scaleR (inverse (norm x)) x"
proposition 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 \ 'b"
show "norm x = 0 \ x = 0"
unfolding norm_prod_def
by (simp add: prod_eq_iff)
show "norm (x + y) \ 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) = \r\ * norm x"
unfolding norm_prod_def
apply (simp add: power_mult_distrib)
apply (simp add: distrib_left [symmetric])
apply (simp add: real_sqrt_mult)
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) \ real"]]
instance\<^marker>\<open>tag important\<close> prod :: (banach, banach) banach ..
subsubsection\<^marker>\<open>tag unimportant\<close> \<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 (\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.scale g.scale)
obtain Kf where "0 < Kf" and norm_f: "\x. norm (f x) \ norm x * Kf"
using f.pos_bounded by fast
obtain Kg where "0 < Kg" and norm_g: "\x. norm (g x) \ norm x * Kg"
using g.pos_bounded by fast
have "\x. norm (f x, g x) \ 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 "\K. \x. norm (f x, g x) \ norm x * K" ..
qed
subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Frechet derivatives involving pairs\<close>
text\<^marker>\<open>tag important\<close> \<open>%whitespace\<close>
proposition 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 "((\x. (f x, g x)) has_derivative (\h. (f' h, g' h))) (at x within s)"
proof (rule has_derivativeI_sandwich[of 1])
show "bounded_linear (\h. (f' h, g' h))"
using f g by (intro bounded_linear_Pair has_derivative_bounded_linear)
let ?Rf = "\y. f y - f x - f' (y - x)"
let ?Rg = "\y. g y - g x - g' (y - x)"
let ?R = "\y. ((f y, g y) - (f x, g x) - (f' (y - x), g' (y - x)))"
show "((\y. norm (?Rf y) / norm (y - x) + norm (?Rg y) / norm (y - x)) \ 0) (at x within s)"
using f g by (intro tendsto_add_zero) (auto simp: has_derivative_iff_norm)
fix y :: 'a assume "y \ x"
show "norm (?R y) / norm (y - x) \ 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 \ g differentiable at x within s \
(\<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]:
"((\p. f (fst p) (snd p)) has_derivative f') F \ ((\(a, b). f a b) has_derivative f') F"
unfolding split_beta' .
subsubsection\<^marker>\<open>tag unimportant\<close> \<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 "((\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)
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 \ norm (x,y)"
by (metis dist_fst_le fst_conv fst_zero norm_conv_dist)
lemma norm_snd_le: "norm y \ norm (x,y)"
by (metis dist_snd_le snd_conv snd_zero norm_conv_dist)
lemma norm_Pair_le:
shows "norm (x, y) \ norm x + norm y"
unfolding norm_Pair
by (metis norm_ge_zero sqrt_sum_squares_le_sum)
lemma (in vector_space_prod) span_Times_sing1: "p.span ({0} \ B) = {0} \ vs2.span B"
apply (rule p.span_unique)
subgoal by (auto intro!: vs1.span_base vs2.span_base)
subgoal using vs1.subspace_single_0 vs2.subspace_span by (rule subspace_Times)
subgoal for T
proof safe
fix b
assume subset_T: "{0} \ B \ T" and subspace: "p.subspace T" and b_span: "b \ vs2.span B"
then obtain t r where b: "b = (\a\t. r a *b a)" and t: "finite t" "t \ B"
by (auto simp: vs2.span_explicit)
have "(0, b) = (\b\t. scale (r b) (0, b))"
unfolding b scale_prod sum_prod
by simp
also have "\ \ T"
using \<open>t \<subseteq> B\<close> subset_T
by (auto intro!: p.subspace_sum p.subspace_scale subspace)
finally show "(0, b) \ T" .
qed
done
lemma (in vector_space_prod) span_Times_sing2: "p.span (A \ {0}) = vs1.span A \ {0}"
apply (rule p.span_unique)
subgoal by (auto intro!: vs1.span_base vs2.span_base)
subgoal using vs1.subspace_span vs2.subspace_single_0 by (rule subspace_Times)
subgoal for T
proof safe
fix a
assume subset_T: "A \ {0} \ T" and subspace: "p.subspace T" and a_span: "a \ vs1.span A"
then obtain t r where a: "a = (\a\t. r a *a a)" and t: "finite t" "t \ A"
by (auto simp: vs1.span_explicit)
have "(a, 0) = (\a\t. scale (r a) (a, 0))"
unfolding a scale_prod sum_prod
by simp
also have "\ \ T"
using \<open>t \<subseteq> A\<close> subset_T
by (auto intro!: p.subspace_sum p.subspace_scale subspace)
finally show "(a, 0) \ T" .
qed
done
subsection \<open>Product is Finite Dimensional\<close>
lemma (in finite_dimensional_vector_space) zero_not_in_Basis[simp]: "0 \ Basis"
using dependent_zero local.independent_Basis by blast
locale finite_dimensional_vector_space_prod = vector_space_prod + finite_dimensional_vector_space_pair begin
definition "Basis_pair = B1 \ {0} \ {0} \ B2"
sublocale p: finite_dimensional_vector_space scale Basis_pair
proof unfold_locales
show "finite Basis_pair"
by (auto intro!: finite_cartesian_product vs1.finite_Basis vs2.finite_Basis simp: Basis_pair_def)
show "p.independent Basis_pair"
unfolding p.dependent_def Basis_pair_def
proof safe
fix a
assume a: "a \ B1"
assume "(a, 0) \ p.span (B1 \ {0} \ {0} \ B2 - {(a, 0)})"
also have "B1 \ {0} \ {0} \ B2 - {(a, 0)} = (B1 - {a}) \ {0} \ {0} \ B2"
by auto
finally show False
using a vs1.dependent_def vs1.independent_Basis
by (auto simp: p.span_Un span_Times_sing1 span_Times_sing2)
next
fix b
assume b: "b \ B2"
assume "(0, b) \ p.span (B1 \ {0} \ {0} \ B2 - {(0, b)})"
also have "(B1 \ {0} \ {0} \ B2 - {(0, b)}) = B1 \ {0} \ {0} \ (B2 - {b})"
by auto
finally show False
using b vs2.dependent_def vs2.independent_Basis
by (auto simp: p.span_Un span_Times_sing1 span_Times_sing2)
qed
show "p.span Basis_pair = UNIV"
by (auto simp: p.span_Un span_Times_sing2 span_Times_sing1 vs1.span_Basis vs2.span_Basis
Basis_pair_def)
qed
proposition dim_Times:
assumes "vs1.subspace S" "vs2.subspace T"
shows "p.dim(S \ T) = vs1.dim S + vs2.dim T"
proof -
interpret p1: Vector_Spaces.linear s1 scale "(\x. (x, 0))"
by unfold_locales (auto simp: scale_def)
interpret pair1: finite_dimensional_vector_space_pair "(*a)" B1 scale Basis_pair
by unfold_locales
interpret p2: Vector_Spaces.linear s2 scale "(\x. (0, x))"
by unfold_locales (auto simp: scale_def)
interpret pair2: finite_dimensional_vector_space_pair "(*b)" B2 scale Basis_pair
by unfold_locales
have ss: "p.subspace ((\x. (x, 0)) ` S)" "p.subspace (Pair 0 ` T)"
by (rule p1.subspace_image p2.subspace_image assms)+
have "p.dim(S \ T) = p.dim({u + v |u v. u \ (\x. (x, 0)) ` S \ v \ Pair 0 ` T})"
by (simp add: Times_eq_image_sum)
moreover have "p.dim ((\x. (x, 0::'c)) ` S) = vs1.dim S" "p.dim (Pair (0::'b) ` T) = vs2.dim T"
by (simp_all add: inj_on_def p1.linear_axioms pair1.dim_image_eq p2.linear_axioms pair2.dim_image_eq)
moreover have "p.dim ((\x. (x, 0)) ` S \ Pair 0 ` T) = 0"
by (subst p.dim_eq_0) auto
ultimately show ?thesis
using p.dim_sums_Int [OF ss] by linarith
qed
lemma dimension_pair: "p.dimension = vs1.dimension + vs2.dimension"
using dim_Times[OF vs1.subspace_UNIV vs2.subspace_UNIV]
by (auto simp: p.dimension_def vs1.dimension_def vs2.dimension_def)
end
end
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