src/HOL/Library/Product_Vector.thy
 author huffman Thu Jun 11 16:26:06 2009 -0700 (2009-06-11) changeset 31568 963caf6fa234 parent 31565 da5a5589418e child 31587 a7e187205fc0 permissions -rw-r--r--
 huffman@30019 ` 1` ```(* Title: HOL/Library/Product_Vector.thy ``` huffman@30019 ` 2` ``` Author: Brian Huffman ``` huffman@30019 ` 3` ```*) ``` huffman@30019 ` 4` huffman@30019 ` 5` ```header {* Cartesian Products as Vector Spaces *} ``` huffman@30019 ` 6` huffman@30019 ` 7` ```theory Product_Vector ``` huffman@30019 ` 8` ```imports Inner_Product Product_plus ``` huffman@30019 ` 9` ```begin ``` huffman@30019 ` 10` huffman@30019 ` 11` ```subsection {* Product is a real vector space *} ``` huffman@30019 ` 12` huffman@30019 ` 13` ```instantiation "*" :: (real_vector, real_vector) real_vector ``` huffman@30019 ` 14` ```begin ``` huffman@30019 ` 15` huffman@30019 ` 16` ```definition scaleR_prod_def: ``` huffman@30019 ` 17` ``` "scaleR r A = (scaleR r (fst A), scaleR r (snd A))" ``` huffman@30019 ` 18` huffman@30019 ` 19` ```lemma fst_scaleR [simp]: "fst (scaleR r A) = scaleR r (fst A)" ``` huffman@30019 ` 20` ``` unfolding scaleR_prod_def by simp ``` huffman@30019 ` 21` huffman@30019 ` 22` ```lemma snd_scaleR [simp]: "snd (scaleR r A) = scaleR r (snd A)" ``` huffman@30019 ` 23` ``` unfolding scaleR_prod_def by simp ``` huffman@30019 ` 24` huffman@30019 ` 25` ```lemma scaleR_Pair [simp]: "scaleR r (a, b) = (scaleR r a, scaleR r b)" ``` huffman@30019 ` 26` ``` unfolding scaleR_prod_def by simp ``` huffman@30019 ` 27` huffman@30019 ` 28` ```instance proof ``` huffman@30019 ` 29` ``` fix a b :: real and x y :: "'a \ 'b" ``` huffman@30019 ` 30` ``` show "scaleR a (x + y) = scaleR a x + scaleR a y" ``` huffman@30019 ` 31` ``` by (simp add: expand_prod_eq scaleR_right_distrib) ``` huffman@30019 ` 32` ``` show "scaleR (a + b) x = scaleR a x + scaleR b x" ``` huffman@30019 ` 33` ``` by (simp add: expand_prod_eq scaleR_left_distrib) ``` huffman@30019 ` 34` ``` show "scaleR a (scaleR b x) = scaleR (a * b) x" ``` huffman@30019 ` 35` ``` by (simp add: expand_prod_eq) ``` huffman@30019 ` 36` ``` show "scaleR 1 x = x" ``` huffman@30019 ` 37` ``` by (simp add: expand_prod_eq) ``` huffman@30019 ` 38` ```qed ``` huffman@30019 ` 39` huffman@30019 ` 40` ```end ``` huffman@30019 ` 41` huffman@31415 ` 42` ```subsection {* Product is a topological space *} ``` huffman@31415 ` 43` huffman@31415 ` 44` ```instantiation ``` huffman@31415 ` 45` ``` "*" :: (topological_space, topological_space) topological_space ``` huffman@31415 ` 46` ```begin ``` huffman@31415 ` 47` huffman@31492 ` 48` ```definition open_prod_def: ``` huffman@31492 ` 49` ``` "open (S :: ('a \ 'b) set) \ ``` huffman@31492 ` 50` ``` (\x\S. \A B. open A \ open B \ x \ A \ B \ A \ B \ S)" ``` huffman@31415 ` 51` huffman@31415 ` 52` ```instance proof ``` huffman@31492 ` 53` ``` show "open (UNIV :: ('a \ 'b) set)" ``` huffman@31492 ` 54` ``` unfolding open_prod_def by auto ``` huffman@31415 ` 55` ```next ``` huffman@31415 ` 56` ``` fix S T :: "('a \ 'b) set" ``` huffman@31492 ` 57` ``` assume "open S" "open T" thus "open (S \ T)" ``` huffman@31492 ` 58` ``` unfolding open_prod_def ``` huffman@31415 ` 59` ``` apply clarify ``` huffman@31415 ` 60` ``` apply (drule (1) bspec)+ ``` huffman@31415 ` 61` ``` apply (clarify, rename_tac Sa Ta Sb Tb) ``` huffman@31492 ` 62` ``` apply (rule_tac x="Sa \ Ta" in exI) ``` huffman@31492 ` 63` ``` apply (rule_tac x="Sb \ Tb" in exI) ``` huffman@31492 ` 64` ``` apply (simp add: open_Int) ``` huffman@31415 ` 65` ``` apply fast ``` huffman@31415 ` 66` ``` done ``` huffman@31415 ` 67` ```next ``` huffman@31492 ` 68` ``` fix K :: "('a \ 'b) set set" ``` huffman@31492 ` 69` ``` assume "\S\K. open S" thus "open (\K)" ``` huffman@31492 ` 70` ``` unfolding open_prod_def by fast ``` huffman@31415 ` 71` ```qed ``` huffman@31415 ` 72` huffman@31415 ` 73` ```end ``` huffman@31415 ` 74` huffman@31562 ` 75` ```lemma open_Times: "open S \ open T \ open (S \ T)" ``` huffman@31562 ` 76` ```unfolding open_prod_def by auto ``` huffman@31562 ` 77` huffman@31562 ` 78` ```lemma fst_vimage_eq_Times: "fst -` S = S \ UNIV" ``` huffman@31562 ` 79` ```by auto ``` huffman@31562 ` 80` huffman@31562 ` 81` ```lemma snd_vimage_eq_Times: "snd -` S = UNIV \ S" ``` huffman@31562 ` 82` ```by auto ``` huffman@31562 ` 83` huffman@31562 ` 84` ```lemma open_vimage_fst: "open S \ open (fst -` S)" ``` huffman@31562 ` 85` ```by (simp add: fst_vimage_eq_Times open_Times) ``` huffman@31562 ` 86` huffman@31562 ` 87` ```lemma open_vimage_snd: "open S \ open (snd -` S)" ``` huffman@31562 ` 88` ```by (simp add: snd_vimage_eq_Times open_Times) ``` huffman@31562 ` 89` huffman@31568 ` 90` ```lemma closed_vimage_fst: "closed S \ closed (fst -` S)" ``` huffman@31568 ` 91` ```unfolding closed_open vimage_Compl [symmetric] ``` huffman@31568 ` 92` ```by (rule open_vimage_fst) ``` huffman@31568 ` 93` huffman@31568 ` 94` ```lemma closed_vimage_snd: "closed S \ closed (snd -` S)" ``` huffman@31568 ` 95` ```unfolding closed_open vimage_Compl [symmetric] ``` huffman@31568 ` 96` ```by (rule open_vimage_snd) ``` huffman@31568 ` 97` huffman@31568 ` 98` ```lemma closed_Times: "closed S \ closed T \ closed (S \ T)" ``` huffman@31568 ` 99` ```proof - ``` huffman@31568 ` 100` ``` have "S \ T = (fst -` S) \ (snd -` T)" by auto ``` huffman@31568 ` 101` ``` thus "closed S \ closed T \ closed (S \ T)" ``` huffman@31568 ` 102` ``` by (simp add: closed_vimage_fst closed_vimage_snd closed_Int) ``` huffman@31568 ` 103` ```qed ``` huffman@31568 ` 104` huffman@31568 ` 105` huffman@31339 ` 106` ```subsection {* Product is a metric space *} ``` huffman@31339 ` 107` huffman@31339 ` 108` ```instantiation ``` huffman@31339 ` 109` ``` "*" :: (metric_space, metric_space) metric_space ``` huffman@31339 ` 110` ```begin ``` huffman@31339 ` 111` huffman@31339 ` 112` ```definition dist_prod_def: ``` huffman@31339 ` 113` ``` "dist (x::'a \ 'b) y = sqrt ((dist (fst x) (fst y))\ + (dist (snd x) (snd y))\)" ``` huffman@31339 ` 114` huffman@31339 ` 115` ```lemma dist_Pair_Pair: "dist (a, b) (c, d) = sqrt ((dist a c)\ + (dist b d)\)" ``` huffman@31339 ` 116` ``` unfolding dist_prod_def by simp ``` huffman@31339 ` 117` huffman@31339 ` 118` ```instance proof ``` huffman@31339 ` 119` ``` fix x y :: "'a \ 'b" ``` huffman@31339 ` 120` ``` show "dist x y = 0 \ x = y" ``` huffman@31563 ` 121` ``` unfolding dist_prod_def expand_prod_eq by simp ``` huffman@31339 ` 122` ```next ``` huffman@31339 ` 123` ``` fix x y z :: "'a \ 'b" ``` huffman@31339 ` 124` ``` show "dist x y \ dist x z + dist y z" ``` huffman@31339 ` 125` ``` unfolding dist_prod_def ``` huffman@31563 ` 126` ``` by (intro order_trans [OF _ real_sqrt_sum_squares_triangle_ineq] ``` huffman@31563 ` 127` ``` real_sqrt_le_mono add_mono power_mono dist_triangle2 zero_le_dist) ``` huffman@31415 ` 128` ```next ``` huffman@31415 ` 129` ``` (* FIXME: long proof! *) ``` huffman@31415 ` 130` ``` (* Maybe it would be easier to define topological spaces *) ``` huffman@31415 ` 131` ``` (* in terms of neighborhoods instead of open sets? *) ``` huffman@31492 ` 132` ``` fix S :: "('a \ 'b) set" ``` huffman@31492 ` 133` ``` show "open S \ (\x\S. \e>0. \y. dist y x < e \ y \ S)" ``` huffman@31563 ` 134` ``` proof ``` huffman@31563 ` 135` ``` assume "open S" thus "\x\S. \e>0. \y. dist y x < e \ y \ S" ``` huffman@31492 ` 136` ``` unfolding open_prod_def open_dist ``` huffman@31492 ` 137` ``` apply safe ``` huffman@31415 ` 138` ``` apply (drule (1) bspec) ``` huffman@31415 ` 139` ``` apply clarify ``` huffman@31415 ` 140` ``` apply (drule (1) bspec)+ ``` huffman@31415 ` 141` ``` apply (clarify, rename_tac r s) ``` huffman@31415 ` 142` ``` apply (rule_tac x="min r s" in exI, simp) ``` huffman@31415 ` 143` ``` apply (clarify, rename_tac c d) ``` huffman@31415 ` 144` ``` apply (erule subsetD) ``` huffman@31415 ` 145` ``` apply (simp add: dist_Pair_Pair) ``` huffman@31415 ` 146` ``` apply (rule conjI) ``` huffman@31415 ` 147` ``` apply (drule spec, erule mp) ``` huffman@31415 ` 148` ``` apply (erule le_less_trans [OF real_sqrt_sum_squares_ge1]) ``` huffman@31415 ` 149` ``` apply (drule spec, erule mp) ``` huffman@31415 ` 150` ``` apply (erule le_less_trans [OF real_sqrt_sum_squares_ge2]) ``` huffman@31563 ` 151` ``` done ``` huffman@31563 ` 152` ``` next ``` huffman@31563 ` 153` ``` assume "\x\S. \e>0. \y. dist y x < e \ y \ S" thus "open S" ``` huffman@31563 ` 154` ``` unfolding open_prod_def open_dist ``` huffman@31563 ` 155` ``` apply safe ``` huffman@31415 ` 156` ``` apply (drule (1) bspec) ``` huffman@31415 ` 157` ``` apply clarify ``` huffman@31415 ` 158` ``` apply (subgoal_tac "\r>0. \s>0. e = sqrt (r\ + s\)") ``` huffman@31415 ` 159` ``` apply clarify ``` huffman@31492 ` 160` ``` apply (rule_tac x="{y. dist y a < r}" in exI) ``` huffman@31492 ` 161` ``` apply (rule_tac x="{y. dist y b < s}" in exI) ``` huffman@31492 ` 162` ``` apply (rule conjI) ``` huffman@31415 ` 163` ``` apply clarify ``` huffman@31415 ` 164` ``` apply (rule_tac x="r - dist x a" in exI, rule conjI, simp) ``` huffman@31415 ` 165` ``` apply clarify ``` huffman@31563 ` 166` ``` apply (simp add: less_diff_eq) ``` huffman@31563 ` 167` ``` apply (erule le_less_trans [OF dist_triangle]) ``` huffman@31492 ` 168` ``` apply (rule conjI) ``` huffman@31415 ` 169` ``` apply clarify ``` huffman@31415 ` 170` ``` apply (rule_tac x="s - dist x b" in exI, rule conjI, simp) ``` huffman@31415 ` 171` ``` apply clarify ``` huffman@31563 ` 172` ``` apply (simp add: less_diff_eq) ``` huffman@31563 ` 173` ``` apply (erule le_less_trans [OF dist_triangle]) ``` huffman@31415 ` 174` ``` apply (rule conjI) ``` huffman@31415 ` 175` ``` apply simp ``` huffman@31415 ` 176` ``` apply (clarify, rename_tac c d) ``` huffman@31415 ` 177` ``` apply (drule spec, erule mp) ``` huffman@31415 ` 178` ``` apply (simp add: dist_Pair_Pair add_strict_mono power_strict_mono) ``` huffman@31415 ` 179` ``` apply (rule_tac x="e / sqrt 2" in exI, simp add: divide_pos_pos) ``` huffman@31415 ` 180` ``` apply (rule_tac x="e / sqrt 2" in exI, simp add: divide_pos_pos) ``` huffman@31415 ` 181` ``` apply (simp add: power_divide) ``` huffman@31415 ` 182` ``` done ``` huffman@31563 ` 183` ``` qed ``` huffman@31339 ` 184` ```qed ``` huffman@31339 ` 185` huffman@31339 ` 186` ```end ``` huffman@31339 ` 187` huffman@31405 ` 188` ```subsection {* Continuity of operations *} ``` huffman@31405 ` 189` huffman@31405 ` 190` ```lemma dist_fst_le: "dist (fst x) (fst y) \ dist x y" ``` huffman@31405 ` 191` ```unfolding dist_prod_def by simp ``` huffman@31405 ` 192` huffman@31405 ` 193` ```lemma dist_snd_le: "dist (snd x) (snd y) \ dist x y" ``` huffman@31405 ` 194` ```unfolding dist_prod_def by simp ``` huffman@31405 ` 195` huffman@31565 ` 196` ```lemma tendsto_fst [tendsto_intros]: ``` huffman@31491 ` 197` ``` assumes "(f ---> a) net" ``` huffman@31491 ` 198` ``` shows "((\x. fst (f x)) ---> fst a) net" ``` huffman@31491 ` 199` ```proof (rule topological_tendstoI) ``` huffman@31492 ` 200` ``` fix S assume "open S" "fst a \ S" ``` huffman@31492 ` 201` ``` then have "open (fst -` S)" "a \ fst -` S" ``` huffman@31492 ` 202` ``` unfolding open_prod_def ``` huffman@31491 ` 203` ``` apply simp_all ``` huffman@31491 ` 204` ``` apply clarify ``` huffman@31492 ` 205` ``` apply (rule exI, erule conjI) ``` huffman@31492 ` 206` ``` apply (rule exI, rule conjI [OF open_UNIV]) ``` huffman@31491 ` 207` ``` apply auto ``` huffman@31491 ` 208` ``` done ``` huffman@31491 ` 209` ``` with assms have "eventually (\x. f x \ fst -` S) net" ``` huffman@31491 ` 210` ``` by (rule topological_tendstoD) ``` huffman@31491 ` 211` ``` then show "eventually (\x. fst (f x) \ S) net" ``` huffman@31491 ` 212` ``` by simp ``` huffman@31405 ` 213` ```qed ``` huffman@31405 ` 214` huffman@31565 ` 215` ```lemma tendsto_snd [tendsto_intros]: ``` huffman@31491 ` 216` ``` assumes "(f ---> a) net" ``` huffman@31491 ` 217` ``` shows "((\x. snd (f x)) ---> snd a) net" ``` huffman@31491 ` 218` ```proof (rule topological_tendstoI) ``` huffman@31492 ` 219` ``` fix S assume "open S" "snd a \ S" ``` huffman@31492 ` 220` ``` then have "open (snd -` S)" "a \ snd -` S" ``` huffman@31492 ` 221` ``` unfolding open_prod_def ``` huffman@31491 ` 222` ``` apply simp_all ``` huffman@31491 ` 223` ``` apply clarify ``` huffman@31492 ` 224` ``` apply (rule exI, rule conjI [OF open_UNIV]) ``` huffman@31492 ` 225` ``` apply (rule exI, erule conjI) ``` huffman@31491 ` 226` ``` apply auto ``` huffman@31491 ` 227` ``` done ``` huffman@31491 ` 228` ``` with assms have "eventually (\x. f x \ snd -` S) net" ``` huffman@31491 ` 229` ``` by (rule topological_tendstoD) ``` huffman@31491 ` 230` ``` then show "eventually (\x. snd (f x) \ S) net" ``` huffman@31491 ` 231` ``` by simp ``` huffman@31405 ` 232` ```qed ``` huffman@31405 ` 233` huffman@31565 ` 234` ```lemma tendsto_Pair [tendsto_intros]: ``` huffman@31491 ` 235` ``` assumes "(f ---> a) net" and "(g ---> b) net" ``` huffman@31491 ` 236` ``` shows "((\x. (f x, g x)) ---> (a, b)) net" ``` huffman@31491 ` 237` ```proof (rule topological_tendstoI) ``` huffman@31492 ` 238` ``` fix S assume "open S" "(a, b) \ S" ``` huffman@31492 ` 239` ``` then obtain A B where "open A" "open B" "a \ A" "b \ B" "A \ B \ S" ``` huffman@31492 ` 240` ``` unfolding open_prod_def by auto ``` huffman@31491 ` 241` ``` have "eventually (\x. f x \ A) net" ``` huffman@31492 ` 242` ``` using `(f ---> a) net` `open A` `a \ A` ``` huffman@31491 ` 243` ``` by (rule topological_tendstoD) ``` huffman@31405 ` 244` ``` moreover ``` huffman@31491 ` 245` ``` have "eventually (\x. g x \ B) net" ``` huffman@31492 ` 246` ``` using `(g ---> b) net` `open B` `b \ B` ``` huffman@31491 ` 247` ``` by (rule topological_tendstoD) ``` huffman@31405 ` 248` ``` ultimately ``` huffman@31491 ` 249` ``` show "eventually (\x. (f x, g x) \ S) net" ``` huffman@31405 ` 250` ``` by (rule eventually_elim2) ``` huffman@31491 ` 251` ``` (simp add: subsetD [OF `A \ B \ S`]) ``` huffman@31405 ` 252` ```qed ``` huffman@31405 ` 253` huffman@31405 ` 254` ```lemma LIMSEQ_fst: "(X ----> a) \ (\n. fst (X n)) ----> fst a" ``` huffman@31405 ` 255` ```unfolding LIMSEQ_conv_tendsto by (rule tendsto_fst) ``` huffman@31405 ` 256` huffman@31405 ` 257` ```lemma LIMSEQ_snd: "(X ----> a) \ (\n. snd (X n)) ----> snd a" ``` huffman@31405 ` 258` ```unfolding LIMSEQ_conv_tendsto by (rule tendsto_snd) ``` huffman@31405 ` 259` huffman@31405 ` 260` ```lemma LIMSEQ_Pair: ``` huffman@31405 ` 261` ``` assumes "X ----> a" and "Y ----> b" ``` huffman@31405 ` 262` ``` shows "(\n. (X n, Y n)) ----> (a, b)" ``` huffman@31405 ` 263` ```using assms unfolding LIMSEQ_conv_tendsto ``` huffman@31405 ` 264` ```by (rule tendsto_Pair) ``` huffman@31405 ` 265` huffman@31405 ` 266` ```lemma LIM_fst: "f -- x --> a \ (\x. fst (f x)) -- x --> fst a" ``` huffman@31405 ` 267` ```unfolding LIM_conv_tendsto by (rule tendsto_fst) ``` huffman@31405 ` 268` huffman@31405 ` 269` ```lemma LIM_snd: "f -- x --> a \ (\x. snd (f x)) -- x --> snd a" ``` huffman@31405 ` 270` ```unfolding LIM_conv_tendsto by (rule tendsto_snd) ``` huffman@31405 ` 271` huffman@31405 ` 272` ```lemma LIM_Pair: ``` huffman@31405 ` 273` ``` assumes "f -- x --> a" and "g -- x --> b" ``` huffman@31405 ` 274` ``` shows "(\x. (f x, g x)) -- x --> (a, b)" ``` huffman@31405 ` 275` ```using assms unfolding LIM_conv_tendsto ``` huffman@31405 ` 276` ```by (rule tendsto_Pair) ``` huffman@31405 ` 277` huffman@31405 ` 278` ```lemma Cauchy_fst: "Cauchy X \ Cauchy (\n. fst (X n))" ``` huffman@31405 ` 279` ```unfolding Cauchy_def by (fast elim: le_less_trans [OF dist_fst_le]) ``` huffman@31405 ` 280` huffman@31405 ` 281` ```lemma Cauchy_snd: "Cauchy X \ Cauchy (\n. snd (X n))" ``` huffman@31405 ` 282` ```unfolding Cauchy_def by (fast elim: le_less_trans [OF dist_snd_le]) ``` huffman@31405 ` 283` huffman@31405 ` 284` ```lemma Cauchy_Pair: ``` huffman@31405 ` 285` ``` assumes "Cauchy X" and "Cauchy Y" ``` huffman@31405 ` 286` ``` shows "Cauchy (\n. (X n, Y n))" ``` huffman@31405 ` 287` ```proof (rule metric_CauchyI) ``` huffman@31405 ` 288` ``` fix r :: real assume "0 < r" ``` huffman@31405 ` 289` ``` then have "0 < r / sqrt 2" (is "0 < ?s") ``` huffman@31405 ` 290` ``` by (simp add: divide_pos_pos) ``` huffman@31405 ` 291` ``` obtain M where M: "\m\M. \n\M. dist (X m) (X n) < ?s" ``` huffman@31405 ` 292` ``` using metric_CauchyD [OF `Cauchy X` `0 < ?s`] .. ``` huffman@31405 ` 293` ``` obtain N where N: "\m\N. \n\N. dist (Y m) (Y n) < ?s" ``` huffman@31405 ` 294` ``` using metric_CauchyD [OF `Cauchy Y` `0 < ?s`] .. ``` huffman@31405 ` 295` ``` have "\m\max M N. \n\max M N. dist (X m, Y m) (X n, Y n) < r" ``` huffman@31405 ` 296` ``` using M N by (simp add: real_sqrt_sum_squares_less dist_Pair_Pair) ``` huffman@31405 ` 297` ``` then show "\n0. \m\n0. \n\n0. dist (X m, Y m) (X n, Y n) < r" .. ``` huffman@31405 ` 298` ```qed ``` huffman@31405 ` 299` huffman@31405 ` 300` ```lemma isCont_Pair [simp]: ``` huffman@31405 ` 301` ``` "\isCont f x; isCont g x\ \ isCont (\x. (f x, g x)) x" ``` huffman@31405 ` 302` ``` unfolding isCont_def by (rule LIM_Pair) ``` huffman@31405 ` 303` huffman@31405 ` 304` ```subsection {* Product is a complete metric space *} ``` huffman@31405 ` 305` huffman@31405 ` 306` ```instance "*" :: (complete_space, complete_space) complete_space ``` huffman@31405 ` 307` ```proof ``` huffman@31405 ` 308` ``` fix X :: "nat \ 'a \ 'b" assume "Cauchy X" ``` huffman@31405 ` 309` ``` have 1: "(\n. fst (X n)) ----> lim (\n. fst (X n))" ``` huffman@31405 ` 310` ``` using Cauchy_fst [OF `Cauchy X`] ``` huffman@31405 ` 311` ``` by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff) ``` huffman@31405 ` 312` ``` have 2: "(\n. snd (X n)) ----> lim (\n. snd (X n))" ``` huffman@31405 ` 313` ``` using Cauchy_snd [OF `Cauchy X`] ``` huffman@31405 ` 314` ``` by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff) ``` huffman@31405 ` 315` ``` have "X ----> (lim (\n. fst (X n)), lim (\n. snd (X n)))" ``` huffman@31405 ` 316` ``` using LIMSEQ_Pair [OF 1 2] by simp ``` huffman@31405 ` 317` ``` then show "convergent X" ``` huffman@31405 ` 318` ``` by (rule convergentI) ``` huffman@31405 ` 319` ```qed ``` huffman@31405 ` 320` huffman@30019 ` 321` ```subsection {* Product is a normed vector space *} ``` huffman@30019 ` 322` huffman@30019 ` 323` ```instantiation ``` huffman@30019 ` 324` ``` "*" :: (real_normed_vector, real_normed_vector) real_normed_vector ``` huffman@30019 ` 325` ```begin ``` huffman@30019 ` 326` huffman@30019 ` 327` ```definition norm_prod_def: ``` huffman@30019 ` 328` ``` "norm x = sqrt ((norm (fst x))\ + (norm (snd x))\)" ``` huffman@30019 ` 329` huffman@30019 ` 330` ```definition sgn_prod_def: ``` huffman@30019 ` 331` ``` "sgn (x::'a \ 'b) = scaleR (inverse (norm x)) x" ``` huffman@30019 ` 332` huffman@30019 ` 333` ```lemma norm_Pair: "norm (a, b) = sqrt ((norm a)\ + (norm b)\)" ``` huffman@30019 ` 334` ``` unfolding norm_prod_def by simp ``` huffman@30019 ` 335` huffman@30019 ` 336` ```instance proof ``` huffman@30019 ` 337` ``` fix r :: real and x y :: "'a \ 'b" ``` huffman@30019 ` 338` ``` show "0 \ norm x" ``` huffman@30019 ` 339` ``` unfolding norm_prod_def by simp ``` huffman@30019 ` 340` ``` show "norm x = 0 \ x = 0" ``` huffman@30019 ` 341` ``` unfolding norm_prod_def ``` huffman@30019 ` 342` ``` by (simp add: expand_prod_eq) ``` huffman@30019 ` 343` ``` show "norm (x + y) \ norm x + norm y" ``` huffman@30019 ` 344` ``` unfolding norm_prod_def ``` huffman@30019 ` 345` ``` apply (rule order_trans [OF _ real_sqrt_sum_squares_triangle_ineq]) ``` huffman@30019 ` 346` ``` apply (simp add: add_mono power_mono norm_triangle_ineq) ``` huffman@30019 ` 347` ``` done ``` huffman@30019 ` 348` ``` show "norm (scaleR r x) = \r\ * norm x" ``` huffman@30019 ` 349` ``` unfolding norm_prod_def ``` huffman@30019 ` 350` ``` apply (simp add: norm_scaleR power_mult_distrib) ``` huffman@30019 ` 351` ``` apply (simp add: right_distrib [symmetric]) ``` huffman@30019 ` 352` ``` apply (simp add: real_sqrt_mult_distrib) ``` huffman@30019 ` 353` ``` done ``` huffman@30019 ` 354` ``` show "sgn x = scaleR (inverse (norm x)) x" ``` huffman@30019 ` 355` ``` by (rule sgn_prod_def) ``` huffman@31290 ` 356` ``` show "dist x y = norm (x - y)" ``` huffman@31339 ` 357` ``` unfolding dist_prod_def norm_prod_def ``` huffman@31339 ` 358` ``` by (simp add: dist_norm) ``` huffman@30019 ` 359` ```qed ``` huffman@30019 ` 360` huffman@30019 ` 361` ```end ``` huffman@30019 ` 362` huffman@31405 ` 363` ```instance "*" :: (banach, banach) banach .. ``` huffman@31405 ` 364` huffman@30019 ` 365` ```subsection {* Product is an inner product space *} ``` huffman@30019 ` 366` huffman@30019 ` 367` ```instantiation "*" :: (real_inner, real_inner) real_inner ``` huffman@30019 ` 368` ```begin ``` huffman@30019 ` 369` huffman@30019 ` 370` ```definition inner_prod_def: ``` huffman@30019 ` 371` ``` "inner x y = inner (fst x) (fst y) + inner (snd x) (snd y)" ``` huffman@30019 ` 372` huffman@30019 ` 373` ```lemma inner_Pair [simp]: "inner (a, b) (c, d) = inner a c + inner b d" ``` huffman@30019 ` 374` ``` unfolding inner_prod_def by simp ``` huffman@30019 ` 375` huffman@30019 ` 376` ```instance proof ``` huffman@30019 ` 377` ``` fix r :: real ``` huffman@30019 ` 378` ``` fix x y z :: "'a::real_inner * 'b::real_inner" ``` huffman@30019 ` 379` ``` show "inner x y = inner y x" ``` huffman@30019 ` 380` ``` unfolding inner_prod_def ``` huffman@30019 ` 381` ``` by (simp add: inner_commute) ``` huffman@30019 ` 382` ``` show "inner (x + y) z = inner x z + inner y z" ``` huffman@30019 ` 383` ``` unfolding inner_prod_def ``` huffman@30019 ` 384` ``` by (simp add: inner_left_distrib) ``` huffman@30019 ` 385` ``` show "inner (scaleR r x) y = r * inner x y" ``` huffman@30019 ` 386` ``` unfolding inner_prod_def ``` huffman@30019 ` 387` ``` by (simp add: inner_scaleR_left right_distrib) ``` huffman@30019 ` 388` ``` show "0 \ inner x x" ``` huffman@30019 ` 389` ``` unfolding inner_prod_def ``` huffman@30019 ` 390` ``` by (intro add_nonneg_nonneg inner_ge_zero) ``` huffman@30019 ` 391` ``` show "inner x x = 0 \ x = 0" ``` huffman@30019 ` 392` ``` unfolding inner_prod_def expand_prod_eq ``` huffman@30019 ` 393` ``` by (simp add: add_nonneg_eq_0_iff) ``` huffman@30019 ` 394` ``` show "norm x = sqrt (inner x x)" ``` huffman@30019 ` 395` ``` unfolding norm_prod_def inner_prod_def ``` huffman@30019 ` 396` ``` by (simp add: power2_norm_eq_inner) ``` huffman@30019 ` 397` ```qed ``` huffman@30019 ` 398` huffman@30019 ` 399` ```end ``` huffman@30019 ` 400` huffman@31405 ` 401` ```subsection {* Pair operations are linear *} ``` huffman@30019 ` 402` wenzelm@30729 ` 403` ```interpretation fst: bounded_linear fst ``` huffman@30019 ` 404` ``` apply (unfold_locales) ``` huffman@30019 ` 405` ``` apply (rule fst_add) ``` huffman@30019 ` 406` ``` apply (rule fst_scaleR) ``` huffman@30019 ` 407` ``` apply (rule_tac x="1" in exI, simp add: norm_Pair) ``` huffman@30019 ` 408` ``` done ``` huffman@30019 ` 409` wenzelm@30729 ` 410` ```interpretation snd: bounded_linear snd ``` huffman@30019 ` 411` ``` apply (unfold_locales) ``` huffman@30019 ` 412` ``` apply (rule snd_add) ``` huffman@30019 ` 413` ``` apply (rule snd_scaleR) ``` huffman@30019 ` 414` ``` apply (rule_tac x="1" in exI, simp add: norm_Pair) ``` huffman@30019 ` 415` ``` done ``` huffman@30019 ` 416` huffman@30019 ` 417` ```text {* TODO: move to NthRoot *} ``` huffman@30019 ` 418` ```lemma sqrt_add_le_add_sqrt: ``` huffman@30019 ` 419` ``` assumes x: "0 \ x" and y: "0 \ y" ``` huffman@30019 ` 420` ``` shows "sqrt (x + y) \ sqrt x + sqrt y" ``` huffman@30019 ` 421` ```apply (rule power2_le_imp_le) ``` huffman@30019 ` 422` ```apply (simp add: real_sum_squared_expand add_nonneg_nonneg x y) ``` huffman@30019 ` 423` ```apply (simp add: mult_nonneg_nonneg x y) ``` huffman@30019 ` 424` ```apply (simp add: add_nonneg_nonneg x y) ``` huffman@30019 ` 425` ```done ``` huffman@30019 ` 426` huffman@30019 ` 427` ```lemma bounded_linear_Pair: ``` huffman@30019 ` 428` ``` assumes f: "bounded_linear f" ``` huffman@30019 ` 429` ``` assumes g: "bounded_linear g" ``` huffman@30019 ` 430` ``` shows "bounded_linear (\x. (f x, g x))" ``` huffman@30019 ` 431` ```proof ``` huffman@30019 ` 432` ``` interpret f: bounded_linear f by fact ``` huffman@30019 ` 433` ``` interpret g: bounded_linear g by fact ``` huffman@30019 ` 434` ``` fix x y and r :: real ``` huffman@30019 ` 435` ``` show "(f (x + y), g (x + y)) = (f x, g x) + (f y, g y)" ``` huffman@30019 ` 436` ``` by (simp add: f.add g.add) ``` huffman@30019 ` 437` ``` show "(f (r *\<^sub>R x), g (r *\<^sub>R x)) = r *\<^sub>R (f x, g x)" ``` huffman@30019 ` 438` ``` by (simp add: f.scaleR g.scaleR) ``` huffman@30019 ` 439` ``` obtain Kf where "0 < Kf" and norm_f: "\x. norm (f x) \ norm x * Kf" ``` huffman@30019 ` 440` ``` using f.pos_bounded by fast ``` huffman@30019 ` 441` ``` obtain Kg where "0 < Kg" and norm_g: "\x. norm (g x) \ norm x * Kg" ``` huffman@30019 ` 442` ``` using g.pos_bounded by fast ``` huffman@30019 ` 443` ``` have "\x. norm (f x, g x) \ norm x * (Kf + Kg)" ``` huffman@30019 ` 444` ``` apply (rule allI) ``` huffman@30019 ` 445` ``` apply (simp add: norm_Pair) ``` huffman@30019 ` 446` ``` apply (rule order_trans [OF sqrt_add_le_add_sqrt], simp, simp) ``` huffman@30019 ` 447` ``` apply (simp add: right_distrib) ``` huffman@30019 ` 448` ``` apply (rule add_mono [OF norm_f norm_g]) ``` huffman@30019 ` 449` ``` done ``` huffman@30019 ` 450` ``` then show "\K. \x. norm (f x, g x) \ norm x * K" .. ``` huffman@30019 ` 451` ```qed ``` huffman@30019 ` 452` huffman@30019 ` 453` ```subsection {* Frechet derivatives involving pairs *} ``` huffman@30019 ` 454` huffman@30019 ` 455` ```lemma FDERIV_Pair: ``` huffman@30019 ` 456` ``` assumes f: "FDERIV f x :> f'" and g: "FDERIV g x :> g'" ``` huffman@30019 ` 457` ``` shows "FDERIV (\x. (f x, g x)) x :> (\h. (f' h, g' h))" ``` huffman@30019 ` 458` ```apply (rule FDERIV_I) ``` huffman@30019 ` 459` ```apply (rule bounded_linear_Pair) ``` huffman@30019 ` 460` ```apply (rule FDERIV_bounded_linear [OF f]) ``` huffman@30019 ` 461` ```apply (rule FDERIV_bounded_linear [OF g]) ``` huffman@30019 ` 462` ```apply (simp add: norm_Pair) ``` huffman@30019 ` 463` ```apply (rule real_LIM_sandwich_zero) ``` huffman@30019 ` 464` ```apply (rule LIM_add_zero) ``` huffman@30019 ` 465` ```apply (rule FDERIV_D [OF f]) ``` huffman@30019 ` 466` ```apply (rule FDERIV_D [OF g]) ``` huffman@30019 ` 467` ```apply (rename_tac h) ``` huffman@30019 ` 468` ```apply (simp add: divide_nonneg_pos) ``` huffman@30019 ` 469` ```apply (rename_tac h) ``` huffman@30019 ` 470` ```apply (subst add_divide_distrib [symmetric]) ``` huffman@30019 ` 471` ```apply (rule divide_right_mono [OF _ norm_ge_zero]) ``` huffman@30019 ` 472` ```apply (rule order_trans [OF sqrt_add_le_add_sqrt]) ``` huffman@30019 ` 473` ```apply simp ``` huffman@30019 ` 474` ```apply simp ``` huffman@30019 ` 475` ```apply simp ``` huffman@30019 ` 476` ```done ``` huffman@30019 ` 477` huffman@30019 ` 478` ```end ```