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theory Product_Vector(* 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 normed vector space *} instantiation "*" :: (real_normed_vector, real_normed_vector) real_normed_vector begin definition norm_prod_def: "norm x = sqrt ((norm (fst x))² + (norm (snd x))²)" definition sgn_prod_def: "sgn (x::'a × 'b) = scaleR (inverse (norm x)) x" lemma norm_Pair: "norm (a, b) = sqrt ((norm a)² + (norm b)²)" unfolding norm_prod_def by simp instance proof fix r :: real and x y :: "'a × 'b" show "0 ≤ norm x" unfolding norm_prod_def by simp show "norm x = 0 <-> x = 0" unfolding norm_prod_def by (simp add: expand_prod_eq) 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: 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) 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 ≤ inner x x" unfolding inner_prod_def by (intro add_nonneg_nonneg inner_ge_zero) show "inner x x = 0 <-> 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 ≤ x" and y: "0 ≤ y" shows "sqrt (x + y) ≤ 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 (λx. (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 *R x), g (r *R x)) = r *R (f x, g x)" by (simp add: f.scaleR g.scaleR) 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: right_distrib) apply (rule add_mono [OF norm_f norm_g]) done then show "∃K. ∀x. norm (f x, g x) ≤ 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 "(λn. (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") by (simp add: divide_pos_pos) obtain M where M: "∀n≥M. norm (X n - a) < ?s" using LIMSEQ_D [OF `X ----> a` `0 < ?s`] .. obtain N where N: "∀n≥N. norm (Y n - b) < ?s" using LIMSEQ_D [OF `Y ----> b` `0 < ?s`] .. have "∀n≥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 "∃n0. ∀n≥n0. norm ((X n, Y n) - (a, b)) < r" .. qed lemma Cauchy_Pair: assumes "Cauchy X" and "Cauchy Y" shows "Cauchy (λn. (X n, Y n))" proof (rule 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: "∀m≥M. ∀n≥M. norm (X m - X n) < ?s" using CauchyD [OF `Cauchy X` `0 < ?s`] .. obtain N where N: "∀m≥N. ∀n≥N. norm (Y m - Y n) < ?s" using CauchyD [OF `Cauchy Y` `0 < ?s`] .. have "∀m≥max M N. ∀n≥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 "∃n0. ∀m≥n0. ∀n≥n0. norm ((X m, Y m) - (X n, Y n)) < r" .. qed lemma LIM_Pair: assumes "f -- x --> a" and "g -- x --> b" shows "(λx. (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") by (simp add: divide_pos_pos) obtain s where s: "0 < s" "∀z. z ≠ x ∧ norm (z - x) < s --> norm (f z - a) < ?e" using LIM_D [OF `f -- x --> a` `0 < ?e`] by fast obtain t where t: "0 < t" "∀z. z ≠ x ∧ norm (z - x) < t --> norm (g z - b) < ?e" using LIM_D [OF `g -- x --> b` `0 < ?e`] by fast have "0 < min s t ∧ (∀z. z ≠ x ∧ norm (z - x) < min s t --> norm ((f z, g z) - (a, b)) < r)" using s t by (simp add: real_sqrt_sum_squares_less norm_Pair) then show "∃s>0. ∀z. z ≠ x ∧ norm (z - x) < s --> norm ((f z, g z) - (a, b)) < r" .. qed lemma isCont_Pair [simp]: "[|isCont f x; isCont g x|] ==> isCont (λx. (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 => 'a × 'b" assume "Cauchy X" have 1: "(λn. fst (X n)) ----> lim (λn. fst (X n))" using fst.Cauchy [OF `Cauchy X`] by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff) have 2: "(λn. snd (X n)) ----> lim (λn. snd (X n))" using snd.Cauchy [OF `Cauchy X`] by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff) have "X ----> (lim (λn. fst (X n)), lim (λn. 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 (λx. (f x, g x)) x :> (λh. (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