(* Title: HOL/Analysis/Product_Vector.thy Author: Brian Huffman Dominique Unruh, University of Tartu
*)
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 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" using module_pair_axioms module_prod.scale_def module_prod_def by fastforce
instantiation\<^marker>\<open>tag unimportant\<close> prod :: (uniform_space, uniform_space) uniform_space begin instance proof standard fix U :: \<open>('a \<times> 'b) set\<close> show\<open>open U \<longleftrightarrow> (\<forall>x\<in>U. \<forall>\<^sub>F (x', y) in uniformity. x' = x \<longrightarrow> y \<in> U)\<close> proof (intro iffI ballI) fix x assume\<open>open U\<close> and \<open>x \<in> U\<close> thenobtain A B where\<open>open A\<close> \<open>open B\<close> \<open>x \<in> A\<times>B\<close> \<open>A\<times>B \<subseteq> U\<close> by (metis open_prod_elim)
define UA where\<open>UA = (\<lambda>(x'::'a,y). x' = fst x \<longrightarrow> y \<in> A)\<close> from\<open>open A\<close> \<open>x \<in> A\<times>B\<close> have\<open>eventually UA uniformity\<close> unfolding open_uniformity UA_def by auto
define UB where\<open>UB = (\<lambda>(x'::'b,y). x' = snd x \<longrightarrow> y \<in> B)\<close> from\<open>open A\<close> \<open>open B\<close> \<open>x \<in> A\<times>B\<close> have\<open>eventually UA uniformity\<close> \<open>eventually UB uniformity\<close> unfolding open_uniformity UA_def UB_def by auto thenhave\<open>\<forall>\<^sub>F ((x'1, y1), (x'2, y2)) in uniformity \<times>\<^sub>F uniformity. (x'1,x'2) = x \<longrightarrow> (y1,y2) \<in> U\<close> apply (auto intro!: exI[of _ UA] exI[of _ UB] simp add: eventually_prod_filter) using\<open>A\<times>B \<subseteq> U\<close> by (auto simp: UA_def UB_def) thenshow\<open>\<forall>\<^sub>F (x', y) in uniformity. x' = x \<longrightarrow> y \<in> U\<close> by (simp add: uniformity_prod_def eventually_filtermap case_prod_unfold) next assume asm: \<open>\<forall>x\<in>U. \<forall>\<^sub>F (x', y) in uniformity. x' = x \<longrightarrow> y \<in> U\<close> show\<open>open U\<close> proof (unfold open_prod_def, intro ballI) fix x assume\<open>x \<in> U\<close> with asm have\<open>\<forall>\<^sub>F (x', y) in uniformity. x' = x \<longrightarrow> y \<in> U\<close> by auto thenhave\<open>\<forall>\<^sub>F ((x'1, y1), (x'2, y2)) in uniformity \<times>\<^sub>F uniformity. (x'1,x'2) = x \<longrightarrow> (y1,y2) \<in> U\<close> by (simp add: uniformity_prod_def eventually_filtermap case_prod_unfold) thenobtain UA UB where\<open>eventually UA uniformity\<close> and \<open>eventually UB uniformity\<close> and UA_UB_U: \<open>UA (a1, a2) \<Longrightarrow> UB (b1, b2) \<Longrightarrow> (a1, b1) = x \<Longrightarrow> (a2, b2) \<in> U\<close> for a1 a2 b1 b2 by (force simp: case_prod_beta eventually_prod_filter) have\<open>eventually (\<lambda>a. UA (fst x, a)) (nhds (fst x))\<close> using\<open>eventually UA uniformity\<close> eventually_mono eventually_nhds_uniformity by fastforce thenobtain A where\<open>open A\<close> and A_UA: \<open>A \<subseteq> {a. UA (fst x, a)}\<close> and \<open>fst x \<in> A\<close> by (metis (mono_tags, lifting) eventually_nhds mem_Collect_eq subsetI) have\<open>eventually (\<lambda>b. UB (snd x, b)) (nhds (snd x))\<close> using\<open>eventually UB uniformity\<close> eventually_mono eventually_nhds_uniformity by fastforce thenobtain B where\<open>open B\<close> and B_UB: \<open>B \<subseteq> {b. UB (snd x, b)}\<close> and \<open>snd x \<in> B\<close> by (metis (mono_tags, lifting) eventually_nhds mem_Collect_eq subsetI) have\<open>x \<in> A \<times> B\<close> by (simp add: \<open>fst x \<in> A\<close> \<open>snd x \<in> B\<close> mem_Times_iff) have\<open>A \<times> B \<subseteq> U\<close> using A_UA B_UB UA_UB_U by fastforce show\<open>\<exists>A B. open A \<and> open B \<and> x \<in> A \<times> B \<and> A \<times> B \<subseteq> U\<close> using\<open>A \<times> B \<subseteq> U\<close> \<open>open A\<close> \<open>open B\<close> \<open>x \<in> A \<times> B\<close> by auto qed qed next show\<open>eventually E uniformity \<Longrightarrow> E (x, x)\<close> for E and x :: \<open>'a \<times> 'b\<close> apply (simp add: uniformity_prod_def eventually_filtermap case_prod_unfold eventually_prod_filter) by (metis surj_pair uniformity_refl) next show\<open>eventually E uniformity \<Longrightarrow> \<forall>\<^sub>F (x::'a\<times>'b, y) in uniformity. E (y, x)\<close> for E unfolding uniformity_prod_def eventually_filtermap case_prod_unfold eventually_prod_filter apply clarify
subgoal for Pf Pg apply (rule_tac x=\<open>\<lambda>(x,y). Pf (y,x)\<close> in exI) apply (rule_tac x=\<open>\<lambda>(x,y). Pg (y,x)\<close> in exI) by (auto simp add: uniformity_sym) done next show\<open>\<exists>D. eventually D uniformity \<and> (\<forall>x y z. D (x::'a\<times>'b, y) \<longrightarrow> D (y, z) \<longrightarrow> E (x, z))\<close> if\<open>eventually E uniformity\<close> for E proof - from that obtain EA EB where\<open>eventually EA uniformity\<close> and \<open>eventually EB uniformity\<close> and EA_EB_E: \<open>EA (a1, a2) \<Longrightarrow> EB (b1, b2) \<Longrightarrow> E ((a1, b1), (a2, b2))\<close> for a1 a2 b1 b2 by (auto simp add: uniformity_prod_def eventually_filtermap case_prod_unfold eventually_prod_filter) obtain DA where\<open>eventually DA uniformity\<close> and DA_EA: \<open>DA (x,y) \<Longrightarrow> DA (y,z) \<Longrightarrow> EA (x,z)\<close> for x y z using\<open>eventually EA uniformity\<close> uniformity_transE by blast obtain DB where\<open>eventually DB uniformity\<close> and DB_EB: \<open>DB (x,y) \<Longrightarrow> DB (y,z) \<Longrightarrow> EB (x,z)\<close> for x y z using\<open>eventually EB uniformity\<close> uniformity_transE by blast
define D where\<open>D = (\<lambda>((a1,b1),(a2,b2)). DA (a1,a2) \<and> DB (b1,b2))\<close> have\<open>eventually D uniformity\<close> using\<open>eventually DA uniformity\<close> \<open>eventually DB uniformity\<close> by (auto simp add: uniformity_prod_def eventually_filtermap case_prod_unfold eventually_prod_filter D_def) moreoverhave\<open>D ((a1, b1), (a2, b2)) \<Longrightarrow> D ((a2, b2), (a3, b3)) \<Longrightarrow> E ((a1, b1), (a3, b3))\<close> for a1 b1 a2 b2 a3 b3 using DA_EA DB_EB D_def EA_EB_E by blast ultimatelyshow ?thesis by auto qed qed end
lemma (in uniform_space) nhds_eq_comap_uniformity: "nhds x = filtercomap (\y. (x, y)) uniformity" proof - have *: "eventually P (filtercomap (\y. (x, y)) F) \
eventually (\<lambda>z. fst z = x \<longrightarrow> P (snd z)) F" for P :: "'a \<Rightarrow> bool" and F unfolding eventually_filtercomap by (smt (verit, best) eventually_mono split_pairs2) thus ?thesis unfolding filter_eq_iff * by (auto simp: eventually_nhds_uniformity case_prod_unfold) qed
lemma uniformity_of_uniform_continuous_invariant: fixes f :: "'a :: uniform_space \ 'a \ 'a" assumes"filterlim (\((a,b),(c,d)). (f a c, f b d)) uniformity (uniformity \\<^sub>F uniformity)" assumes"eventually P uniformity" obtains Q where"eventually Q uniformity""\a b c. Q (a, b) \ P (f a c, f b c)" using eventually_compose_filterlim[OF assms(2,1)] uniformity_refl by (fastforce simp: case_prod_unfold eventually_filtercomap eventually_prod_same)
class uniform_topological_monoid_add = topological_monoid_add + uniform_space + assumes uniformly_continuous_add': "filterlim (\((a,b), (c,d)). (a + c, b + d)) uniformity (uniformity \\<^sub>F uniformity)"
lemma uniformly_continuous_add: "uniformly_continuous_on UNIV (\(x :: 'a :: uniform_topological_monoid_add,y). x + y)" using uniformly_continuous_add'[where ?'a = 'a] by (simp add: uniformly_continuous_on_uniformity case_prod_unfold uniformity_prod_def filterlim_filtermap)
lemma filterlim_fst: "filterlim fst F (F \\<^sub>F G)" by (simp add: filterlim_def filtermap_fst_prod_filter)
lemma filterlim_snd: "filterlim snd G (F \\<^sub>F G)" by (simp add: filterlim_def filtermap_snd_prod_filter)
class uniform_topological_group_add = topological_group_add + uniform_topological_monoid_add + assumes uniformly_continuous_uminus': "filterlim (\(a, b). (-a, -b)) uniformity uniformity" begin
lemma uniformly_continuous_minus': "filterlim (\((a,b), (c,d)). (a - c, b - d)) uniformity (uniformity \\<^sub>F uniformity)" proof - have"filterlim ((\((a,b), (c,d)). (a + c, b + d)) \ (\((a,b), (c,d)). ((a, b), (-c, -d))))
uniformity (uniformity \<times>\<^sub>F uniformity)" unfolding o_def using uniformly_continuous_uminus' by (intro filterlim_compose[OF uniformly_continuous_add'])
(auto simp: case_prod_unfold intro!: filterlim_Pair
filterlim_fst filterlim_compose[OF _ filterlim_snd]) thus ?thesis by (simp add: o_def case_prod_unfold) qed
end
lemma uniformly_continuous_uminus: "uniformly_continuous_on UNIV (\x :: 'a :: uniform_topological_group_add. -x)" using uniformly_continuous_uminus'[where ?'a = 'a] by (simp add: uniformly_continuous_on_uniformity)
lemma uniformly_continuous_minus: "uniformly_continuous_on UNIV (\(x :: 'a :: uniform_topological_group_add,y). x - y)" using uniformly_continuous_minus'[where ?'a = 'a] by (simp add: uniformly_continuous_on_uniformity case_prod_unfold uniformity_prod_def filterlim_filtermap)
lemma real_normed_vector_is_uniform_topological_group_add [Pure.intro]: "OFCLASS('a :: real_normed_vector, uniform_topological_group_add_class)" proof show"filterlim (\((a::'a,b), (c,d)). (a + c, b + d)) uniformity (uniformity \\<^sub>F uniformity)" unfolding filterlim_def le_filter_def eventually_filtermap case_prod_unfold proof safe fix P :: "'a \ 'a \ bool" assume"eventually P uniformity" thenobtain\<epsilon> where \<epsilon>: "\<epsilon> > 0" "\<And>x y. dist x y < \<epsilon> \<Longrightarrow> P (x, y)" by (auto simp: eventually_uniformity_metric)
define Q where"Q = (\(x::'a,y). dist x y < \ / 2)" have Q: "eventually Q uniformity" unfolding eventually_uniformity_metric Q_def using\<open>\<epsilon> > 0\<close> by (meson case_prodI divide_pos_pos zero_less_numeral) have"P (a + c, b + d)"if"Q (a, b)""Q (c, d)"for a b c d proof - have"dist (a + c) (b + d) \ dist a b + dist c d" by (simp add: dist_norm norm_diff_triangle_ineq) alsohave"\ < \" using that by (auto simp: Q_def) finallyshow ?thesis by (intro \<epsilon>) qed thus"\\<^sub>F x in uniformity \\<^sub>F uniformity. P (fst (fst x) + fst (snd x), snd (fst x) + snd (snd x))" unfolding eventually_prod_filter by (intro exI[of _ Q] conjI Q) auto qed next show"filterlim (\((a::'a), b). (-a, -b)) uniformity uniformity" unfolding filterlim_def le_filter_def eventually_filtermap proof safe fix P :: "'a \ 'a \ bool" assume"eventually P uniformity" thenobtain\<epsilon> where \<epsilon>: "\<epsilon> > 0" "\<And>x y. dist x y < \<epsilon> \<Longrightarrow> P (x, y)" by (auto simp: eventually_uniformity_metric) show"\\<^sub>F x in uniformity. P (case x of (a, b) \ (- a, - b))" unfolding eventually_uniformity_metric by (intro exI[of _ \<epsilon>]) (auto intro!: \<epsilon> simp: dist_norm norm_minus_commute) qed qed
instance real :: uniform_topological_group_add .. instance complex :: uniform_topological_group_add ..
lemma cauchy_seq_finset_iff_vanishing: "uniformity = filtercomap (\(x,y). y - x :: 'a :: uniform_topological_group_add) (nhds 0)" proof - have"filtercomap (\x. (0, case x of (x, y) \ y - (x :: 'a))) uniformity \ uniformity" apply (simp add: le_filter_def eventually_filtercomap) using uniformity_of_uniform_continuous_invariant[OF uniformly_continuous_add'] by (metis diff_self eq_diff_eq) moreover have"uniformity \ filtercomap (\x. (0, case x of (x, y) \ y - (x :: 'a))) uniformity" apply (simp add: le_filter_def eventually_filtercomap) using uniformity_of_uniform_continuous_invariant[OF uniformly_continuous_minus'] by (metis (mono_tags) diff_self eventually_mono surjective_pairing) ultimatelyshow ?thesis by (simp add: nhds_eq_comap_uniformity filtercomap_filtercomap) qed
subsubsection \<open>Metric spaces\<close>
instantiation\<^marker>\<open>tag unimportant\<close> prod :: (metric_space, metric_space) uniformity_dist begin instance proof show\<open>uniformity = (INF e\<in>{0 <..}. principal {(x::'a\<times>'b, y). dist x y < e})\<close> proof (subst filter_eq_iff, intro allI iffI) fix P :: \<open>('a \<times> 'b) \<times> ('a \<times> 'b) \<Rightarrow> bool\<close>
have 1: \<open>\<exists>e\<in>{0<..}.
{(x,y). dist x y < e} \<subseteq> {(x,y). dist x y < a} \<and>
{(x,y). dist x y < e} \<subseteq> {(x,y). dist x y < b}\<close> if \<open>a>0\<close> \<open>b>0\<close> for a b using that by (auto intro: bexI[of _ \<open>min a b\<close>]) have 2: \<open>mono (\<lambda>P. eventually (\<lambda>x. P (Q x)) F)\<close> for F :: \<open>'z filter\<close> and Q :: \<open>'z \<Rightarrow> 'y\<close> unfolding mono_def using eventually_mono le_funD by fastforce have\<open>\<forall>\<^sub>F ((x1::'a,y1),(x2::'b,y2)) in uniformity \<times>\<^sub>F uniformity. dist x1 y1 < e/2 \<and> dist x2 y2 < e/2\<close> if \<open>e>0\<close> for e by (auto intro!: eventually_prodI exI[of _ \<open>e/2\<close>] simp: case_prod_unfold eventually_uniformity_metric that) thenhave 3: \<open>\<forall>\<^sub>F ((x1::'a,y1),(x2::'b,y2)) in uniformity \<times>\<^sub>F uniformity. dist (x1,x2) (y1,y2) < e\<close> if \<open>e>0\<close> for e apply (rule eventually_rev_mp) by (auto intro!: that eventuallyI simp: case_prod_unfold dist_prod_def sqrt_sum_squares_half_less) show\<open>eventually P (INF e\<in>{0<..}. principal {(x, y). dist x y < e}) \<Longrightarrow> eventually P uniformity\<close> apply (subst (asm) eventually_INF_base) using 1 3 apply (auto simp: uniformity_prod_def case_prod_unfold eventually_filtermap 2 eventually_principal) by (smt (verit, best) eventually_mono) next fix P :: \<open>('a \<times> 'b) \<times> ('a \<times> 'b) \<Rightarrow> bool\<close> assume\<open>eventually P uniformity\<close> thenobtain P1 P2 where\<open>eventually P1 uniformity\<close> \<open>eventually P2 uniformity\<close> and P1P2P: \<open>P1 (x1, y1) \<Longrightarrow> P2 (x2, y2) \<Longrightarrow> P ((x1, x2), (y1, y2))\<close> for x1 y1 x2 y2 by (auto simp: eventually_filtermap case_prod_beta eventually_prod_filter uniformity_prod_def) from\<open>eventually P1 uniformity\<close> obtain e1 where \<open>e1>0\<close> and e1P1: \<open>dist x y < e1 \<Longrightarrow> P1 (x,y)\<close> for x y using eventually_uniformity_metric by blast from\<open>eventually P2 uniformity\<close> obtain e2 where \<open>e2>0\<close> and e2P2: \<open>dist x y < e2 \<Longrightarrow> P2 (x,y)\<close> for x y using eventually_uniformity_metric by blast
define e where\<open>e = min e1 e2\<close> have\<open>e > 0\<close> using\<open>0 < e1\<close> \<open>0 < e2\<close> e_def by auto have\<open>dist (x1,x2) (y1,y2) < e \<Longrightarrow> dist x1 y1 < e1\<close> for x1 y1 :: 'a and x2 y2 :: 'b unfolding dist_prod_def e_def by (metis real_sqrt_sum_squares_ge1 fst_conv min_less_iff_conj order_le_less_trans snd_conv) moreoverhave\<open>dist (x1,x2) (y1,y2) < e \<Longrightarrow> dist x2 y2 < e2\<close> for x1 y1 :: 'a and x2 y2 :: 'b unfolding dist_prod_def e_def using real_sqrt_sum_squares_ge1 [of "dist x1 y1""dist x2 y2"] by (metis min_less_iff_conj order_le_less_trans real_sqrt_sum_squares_ge2 snd_conv) ultimatelyhave *: \<open>dist (x1,x2) (y1,y2) < e \<Longrightarrow> P ((x1, x2), (y1, y2))\<close> for x1 y1 x2 y2 using e1P1 e2P2 P1P2P by auto
show\<open>eventually P (INF e\<in>{0<..}. principal {(x, y). dist x y < e})\<close> using\<open>e > 0\<close> * by (auto simp: eventually_principal intro: eventually_INF1) qed qed end
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 (meson add_mono dist_triangle2 order_trans power_mono real_sqrt_le_iff
real_sqrt_sum_squares_triangle_ineq 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" thenobtain 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) moreoverhave"x \ A \ B" unfolding A_def B_def mem_Times_iff using\<open>0 < r\<close> and \<open>0 < s\<close> by simp moreoverhave"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 ultimatelyshow"\A B. open A \ open B \ x \ A \ B \ A \ B \ S" by fast qed qed qed
lemma Cauchy_fst: "Cauchy X \ Cauchy (\n. fst (X n :: 'a::metric_space \ 'b::metric_space))" unfolding Cauchy_def by (fast elim: le_less_trans [OF dist_fst_le])
lemma Cauchy_snd: "Cauchy X \ Cauchy (\n. snd (X n :: 'a::metric_space \ 'b::metric_space))" 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 :: 'a::metric_space, Y n :: 'a::metric_space))" 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) thenshow"\n0. \m\n0. \n\n0. dist (X m, Y m) (X n, Y n) < r" .. qed
text\<open>Analogue to @{thm [source] uniformly_continuous_on_def} for two-argument functions.\<close> lemma uniformly_continuous_on_prod_metric: fixes f :: \<open>('a::metric_space \<times> 'b::metric_space) \<Rightarrow> 'c::metric_space\<close> shows\<open>uniformly_continuous_on (S\<times>T) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0. \<forall>x\<in>S. \<forall>y\<in>S. \<forall>x'\<in>T. \<forall>y'\<in>T. dist x y < d \<longrightarrow> dist x' y' < d \<longrightarrow> dist (f (x, x')) (f (y, y')) < e)\<close> proof (unfold uniformly_continuous_on_def, intro iffI impI allI) fix e :: real assume\<open>e > 0\<close> and \<open>\<forall>e>0. \<exists>d>0. \<forall>x\<in>S. \<forall>y\<in>S. \<forall>x'\<in>T. \<forall>y'\<in>T. dist x y < d \<longrightarrow> dist x' y' < d \<longrightarrow> dist (f (x, x')) (f (y, y')) < e\<close> thenobtain d where\<open>d > 0\<close> and d: \<open>\<forall>x\<in>S. \<forall>y\<in>S. \<forall>x'\<in>T. \<forall>y'\<in>T. dist x y < d \<longrightarrow> dist x' y' < d \<longrightarrow> dist (f (x, x')) (f (y, y')) < e\<close> by auto show\<open>\<exists>d>0. \<forall>x\<in>S\<times>T. \<forall>y\<in>S\<times>T. dist y x < d \<longrightarrow> dist (f y) (f x) < e\<close> apply (rule exI[of _ d]) by (metis SigmaE \<open>0 < d\<close> d dist_fst_le dist_snd_le fst_eqD order_le_less_trans
snd_conv) next fix e :: real assume\<open>e > 0\<close> and \<open>\<forall>e>0. \<exists>d>0. \<forall>x\<in>S\<times>T. \<forall>x'\<in>S\<times>T. dist x' x < d \<longrightarrow> dist (f x') (f x) < e\<close> thenobtain d where\<open>d > 0\<close> and d: \<open>\<forall>x\<in>S\<times>T. \<forall>x'\<in>S\<times>T. dist x' x < d \<longrightarrow> dist (f x') (f x) < e\<close> by auto show\<open>\<exists>d>0. \<forall>x\<in>S. \<forall>y\<in>S. \<forall>x'\<in>T. \<forall>y'\<in>T. dist x y < d \<longrightarrow> dist x' y' < d \<longrightarrow> dist (f (x, x')) (f (y, y')) < e\<close> proof (intro exI conjI impI ballI) from\<open>d > 0\<close> show \<open>d / 2 > 0\<close> by auto fix x y x' y' assume [simp]: \<open>x \<in> S\<close> \<open>y \<in> S\<close> \<open>x' \<in> T\<close> \<open>y' \<in> T\<close> assume\<open>dist x y < d / 2\<close> and \<open>dist x' y' < d / 2\<close> thenhave\<open>dist (x, x') (y, y') < d\<close> by (simp add: dist_Pair_Pair sqrt_sum_squares_half_less) with d show\<open>dist (f (x, x')) (f (y, y')) < e\<close> by auto qed qed
text\<open>Analogue to @{thm [source] isUCont_def} for two-argument functions.\<close> lemma isUCont_prod_metric: fixes f :: \<open>('a::metric_space \<times> 'b::metric_space) \<Rightarrow> 'c::metric_space\<close> shows\<open>isUCont f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0. \<forall>x. \<forall>y. \<forall>x'. \<forall>y'. dist x y < d \<longrightarrow> dist x' y' < d \<longrightarrow> dist (f (x, x')) (f (y, y')) < e)\<close> using uniformly_continuous_on_prod_metric[of UNIV UNIV] by auto
text\<open>This logically belong with the real vector spaces but we only have the necessary lemmas now.\<close> lemma isUCont_plus[simp]: shows\<open>isUCont (\<lambda>(x::'a::real_normed_vector,y). x+y)\<close> proof (rule isUCont_prod_metric[THEN iffD2], intro allI impI, simp) fix e :: real assume\<open>0 < e\<close> show\<open>\<exists>d>0. \<forall>x y :: 'a. dist x y < d \<longrightarrow> (\<forall>x' y'. dist x' y' < d \<longrightarrow> dist (x + x') (y + y') < e)\<close> apply (rule exI[of _ \<open>e/2\<close>]) using\<open>0 < e\<close> by (smt (verit) dist_add_cancel dist_add_cancel2 dist_commute
dist_triangle_lt field_sum_of_halves) 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 thenshow"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
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. sqrt ((norm (f x))\<^sup>2) + sqrt ((norm (g x))\<^sup>2) \ norm x * (Kf + Kg)" by (simp add: add_mono distrib_left norm_f norm_g) thenhave"\x. norm (f x, g x) \ norm x * (Kf + Kg)" by (smt (verit) norm_ge_zero norm_prod_def prod.sel sqrt_add_le_add_sqrt zero_le_power) thenshow"\K. \x. norm (f x, g x) \ norm x * K" .. qed
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)
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' .
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_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" proof (rule p.span_unique) show"{0} \ B \ {0} \ vs2.span B" by (auto intro!: vs1.span_base vs2.span_base) show"p.subspace ({0} \ vs2.span B)" using vs1.subspace_single_0 vs2.subspace_span by (rule subspace_Times) fix T :: "('b \ 'c) set" assume subset_T: "{0} \ B \ T" and subspace: "p.subspace T" show"{0} \ vs2.span B \ T" proof clarify fix b assume b_span: "b \ vs2.span B" thenobtain 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 alsohave"\ \ T" using\<open>t \<subseteq> B\<close> subset_T by (auto intro!: p.subspace_sum p.subspace_scale subspace) finallyshow"(0, b) \ T" . qed qed
lemma (in vector_space_prod) span_Times_sing2: "p.span (A \ {0}) = vs1.span A \ {0}" proof (rule p.span_unique) show"A \ {0} \ vs1.span A \ {0}" by (auto intro!: vs1.span_base vs2.span_base) show"p.subspace (vs1.span A \ {0})" using vs1.subspace_span vs2.subspace_single_0 by (rule subspace_Times) fix T :: "('b \ 'c) set" assume subset_T: "A \ {0} \ T" and subspace: "p.subspace T" show"vs1.span A \ {0} \ T" proof clarify fix a assume a_span: "a \ vs1.span A" thenobtain 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 alsohave"\ \ T" using\<open>t \<subseteq> A\<close> subset_T by (auto intro!: p.subspace_sum p.subspace_scale subspace) finallyshow"(a, 0) \ T" . qed qed
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)})" alsohave"B1 \ {0} \ {0} \ B2 - {(a, 0)} = (B1 - {a}) \ {0} \ {0} \ B2" by auto finallyshow 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)})" alsohave"(B1 \ {0} \ {0} \ B2 - {(0, b)}) = B1 \ {0} \ {0} \ (B2 - {b})" by auto finallyshow 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) moreoverhave"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) moreoverhave"p.dim ((\x. (x, 0)) ` S \ Pair 0 ` T) = 0" by (subst p.dim_eq_0) auto ultimatelyshow ?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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