src/HOL/Limits.thy
 author huffman Sun Aug 14 10:25:43 2011 -0700 (2011-08-14) changeset 44205 18da2a87421c parent 44195 f5363511b212 child 44206 5e4a1664106e permissions -rw-r--r--
generalize constant 'lim' and limit uniqueness theorems to class t2_space
 huffman@31349 ` 1` ```(* Title : Limits.thy ``` huffman@31349 ` 2` ``` Author : Brian Huffman ``` huffman@31349 ` 3` ```*) ``` huffman@31349 ` 4` huffman@31349 ` 5` ```header {* Filters and Limits *} ``` huffman@31349 ` 6` huffman@31349 ` 7` ```theory Limits ``` huffman@36822 ` 8` ```imports RealVector ``` huffman@31349 ` 9` ```begin ``` huffman@31349 ` 10` huffman@44081 ` 11` ```subsection {* Filters *} ``` huffman@31392 ` 12` huffman@31392 ` 13` ```text {* ``` huffman@44081 ` 14` ``` This definition also allows non-proper filters. ``` huffman@31392 ` 15` ```*} ``` huffman@31392 ` 16` huffman@36358 ` 17` ```locale is_filter = ``` huffman@44081 ` 18` ``` fixes F :: "('a \ bool) \ bool" ``` huffman@44081 ` 19` ``` assumes True: "F (\x. True)" ``` huffman@44081 ` 20` ``` assumes conj: "F (\x. P x) \ F (\x. Q x) \ F (\x. P x \ Q x)" ``` huffman@44081 ` 21` ``` assumes mono: "\x. P x \ Q x \ F (\x. P x) \ F (\x. Q x)" ``` huffman@36358 ` 22` huffman@44081 ` 23` ```typedef (open) 'a filter = "{F :: ('a \ bool) \ bool. is_filter F}" ``` huffman@31392 ` 24` ```proof ``` huffman@44081 ` 25` ``` show "(\x. True) \ ?filter" by (auto intro: is_filter.intro) ``` huffman@31392 ` 26` ```qed ``` huffman@31349 ` 27` huffman@44195 ` 28` ```lemma is_filter_Rep_filter: "is_filter (Rep_filter F)" ``` huffman@44195 ` 29` ``` using Rep_filter [of F] by simp ``` huffman@31392 ` 30` huffman@44081 ` 31` ```lemma Abs_filter_inverse': ``` huffman@44081 ` 32` ``` assumes "is_filter F" shows "Rep_filter (Abs_filter F) = F" ``` huffman@44081 ` 33` ``` using assms by (simp add: Abs_filter_inverse) ``` huffman@31392 ` 34` huffman@31392 ` 35` huffman@31392 ` 36` ```subsection {* Eventually *} ``` huffman@31349 ` 37` huffman@44081 ` 38` ```definition eventually :: "('a \ bool) \ 'a filter \ bool" ``` huffman@44195 ` 39` ``` where "eventually P F \ Rep_filter F P" ``` huffman@36358 ` 40` huffman@44081 ` 41` ```lemma eventually_Abs_filter: ``` huffman@44081 ` 42` ``` assumes "is_filter F" shows "eventually P (Abs_filter F) = F P" ``` huffman@44081 ` 43` ``` unfolding eventually_def using assms by (simp add: Abs_filter_inverse) ``` huffman@31349 ` 44` huffman@44081 ` 45` ```lemma filter_eq_iff: ``` huffman@44195 ` 46` ``` shows "F = F' \ (\P. eventually P F = eventually P F')" ``` huffman@44081 ` 47` ``` unfolding Rep_filter_inject [symmetric] fun_eq_iff eventually_def .. ``` huffman@36360 ` 48` huffman@44195 ` 49` ```lemma eventually_True [simp]: "eventually (\x. True) F" ``` huffman@44081 ` 50` ``` unfolding eventually_def ``` huffman@44081 ` 51` ``` by (rule is_filter.True [OF is_filter_Rep_filter]) ``` huffman@31349 ` 52` huffman@44195 ` 53` ```lemma always_eventually: "\x. P x \ eventually P F" ``` huffman@36630 ` 54` ```proof - ``` huffman@36630 ` 55` ``` assume "\x. P x" hence "P = (\x. True)" by (simp add: ext) ``` huffman@44195 ` 56` ``` thus "eventually P F" by simp ``` huffman@36630 ` 57` ```qed ``` huffman@36630 ` 58` huffman@31349 ` 59` ```lemma eventually_mono: ``` huffman@44195 ` 60` ``` "(\x. P x \ Q x) \ eventually P F \ eventually Q F" ``` huffman@44081 ` 61` ``` unfolding eventually_def ``` huffman@44081 ` 62` ``` by (rule is_filter.mono [OF is_filter_Rep_filter]) ``` huffman@31349 ` 63` huffman@31349 ` 64` ```lemma eventually_conj: ``` huffman@44195 ` 65` ``` assumes P: "eventually (\x. P x) F" ``` huffman@44195 ` 66` ``` assumes Q: "eventually (\x. Q x) F" ``` huffman@44195 ` 67` ``` shows "eventually (\x. P x \ Q x) F" ``` huffman@44081 ` 68` ``` using assms unfolding eventually_def ``` huffman@44081 ` 69` ``` by (rule is_filter.conj [OF is_filter_Rep_filter]) ``` huffman@31349 ` 70` huffman@31349 ` 71` ```lemma eventually_mp: ``` huffman@44195 ` 72` ``` assumes "eventually (\x. P x \ Q x) F" ``` huffman@44195 ` 73` ``` assumes "eventually (\x. P x) F" ``` huffman@44195 ` 74` ``` shows "eventually (\x. Q x) F" ``` huffman@31349 ` 75` ```proof (rule eventually_mono) ``` huffman@31349 ` 76` ``` show "\x. (P x \ Q x) \ P x \ Q x" by simp ``` huffman@44195 ` 77` ``` show "eventually (\x. (P x \ Q x) \ P x) F" ``` huffman@31349 ` 78` ``` using assms by (rule eventually_conj) ``` huffman@31349 ` 79` ```qed ``` huffman@31349 ` 80` huffman@31349 ` 81` ```lemma eventually_rev_mp: ``` huffman@44195 ` 82` ``` assumes "eventually (\x. P x) F" ``` huffman@44195 ` 83` ``` assumes "eventually (\x. P x \ Q x) F" ``` huffman@44195 ` 84` ``` shows "eventually (\x. Q x) F" ``` huffman@31349 ` 85` ```using assms(2) assms(1) by (rule eventually_mp) ``` huffman@31349 ` 86` huffman@31349 ` 87` ```lemma eventually_conj_iff: ``` huffman@44195 ` 88` ``` "eventually (\x. P x \ Q x) F \ eventually P F \ eventually Q F" ``` huffman@44081 ` 89` ``` by (auto intro: eventually_conj elim: eventually_rev_mp) ``` huffman@31349 ` 90` huffman@31349 ` 91` ```lemma eventually_elim1: ``` huffman@44195 ` 92` ``` assumes "eventually (\i. P i) F" ``` huffman@31349 ` 93` ``` assumes "\i. P i \ Q i" ``` huffman@44195 ` 94` ``` shows "eventually (\i. Q i) F" ``` huffman@44081 ` 95` ``` using assms by (auto elim!: eventually_rev_mp) ``` huffman@31349 ` 96` huffman@31349 ` 97` ```lemma eventually_elim2: ``` huffman@44195 ` 98` ``` assumes "eventually (\i. P i) F" ``` huffman@44195 ` 99` ``` assumes "eventually (\i. Q i) F" ``` huffman@31349 ` 100` ``` assumes "\i. P i \ Q i \ R i" ``` huffman@44195 ` 101` ``` shows "eventually (\i. R i) F" ``` huffman@44081 ` 102` ``` using assms by (auto elim!: eventually_rev_mp) ``` huffman@31349 ` 103` huffman@36360 ` 104` ```subsection {* Finer-than relation *} ``` huffman@36360 ` 105` huffman@44195 ` 106` ```text {* @{term "F \ F'"} means that filter @{term F} is finer than ``` huffman@44195 ` 107` ```filter @{term F'}. *} ``` huffman@36360 ` 108` huffman@44081 ` 109` ```instantiation filter :: (type) complete_lattice ``` huffman@36360 ` 110` ```begin ``` huffman@36360 ` 111` huffman@44081 ` 112` ```definition le_filter_def: ``` huffman@44195 ` 113` ``` "F \ F' \ (\P. eventually P F' \ eventually P F)" ``` huffman@36360 ` 114` huffman@36360 ` 115` ```definition ``` huffman@44195 ` 116` ``` "(F :: 'a filter) < F' \ F \ F' \ \ F' \ F" ``` huffman@36360 ` 117` huffman@36360 ` 118` ```definition ``` huffman@44081 ` 119` ``` "top = Abs_filter (\P. \x. P x)" ``` huffman@36630 ` 120` huffman@36630 ` 121` ```definition ``` huffman@44081 ` 122` ``` "bot = Abs_filter (\P. True)" ``` huffman@36360 ` 123` huffman@36630 ` 124` ```definition ``` huffman@44195 ` 125` ``` "sup F F' = Abs_filter (\P. eventually P F \ eventually P F')" ``` huffman@36630 ` 126` huffman@36630 ` 127` ```definition ``` huffman@44195 ` 128` ``` "inf F F' = Abs_filter ``` huffman@44195 ` 129` ``` (\P. \Q R. eventually Q F \ eventually R F' \ (\x. Q x \ R x \ P x))" ``` huffman@36630 ` 130` huffman@36630 ` 131` ```definition ``` huffman@44195 ` 132` ``` "Sup S = Abs_filter (\P. \F\S. eventually P F)" ``` huffman@36630 ` 133` huffman@36630 ` 134` ```definition ``` huffman@44195 ` 135` ``` "Inf S = Sup {F::'a filter. \F'\S. F \ F'}" ``` huffman@36630 ` 136` huffman@36630 ` 137` ```lemma eventually_top [simp]: "eventually P top \ (\x. P x)" ``` huffman@44081 ` 138` ``` unfolding top_filter_def ``` huffman@44081 ` 139` ``` by (rule eventually_Abs_filter, rule is_filter.intro, auto) ``` huffman@36630 ` 140` huffman@36629 ` 141` ```lemma eventually_bot [simp]: "eventually P bot" ``` huffman@44081 ` 142` ``` unfolding bot_filter_def ``` huffman@44081 ` 143` ``` by (subst eventually_Abs_filter, rule is_filter.intro, auto) ``` huffman@36360 ` 144` huffman@36630 ` 145` ```lemma eventually_sup: ``` huffman@44195 ` 146` ``` "eventually P (sup F F') \ eventually P F \ eventually P F'" ``` huffman@44081 ` 147` ``` unfolding sup_filter_def ``` huffman@44081 ` 148` ``` by (rule eventually_Abs_filter, rule is_filter.intro) ``` huffman@44081 ` 149` ``` (auto elim!: eventually_rev_mp) ``` huffman@36630 ` 150` huffman@36630 ` 151` ```lemma eventually_inf: ``` huffman@44195 ` 152` ``` "eventually P (inf F F') \ ``` huffman@44195 ` 153` ``` (\Q R. eventually Q F \ eventually R F' \ (\x. Q x \ R x \ P x))" ``` huffman@44081 ` 154` ``` unfolding inf_filter_def ``` huffman@44081 ` 155` ``` apply (rule eventually_Abs_filter, rule is_filter.intro) ``` huffman@44081 ` 156` ``` apply (fast intro: eventually_True) ``` huffman@44081 ` 157` ``` apply clarify ``` huffman@44081 ` 158` ``` apply (intro exI conjI) ``` huffman@44081 ` 159` ``` apply (erule (1) eventually_conj) ``` huffman@44081 ` 160` ``` apply (erule (1) eventually_conj) ``` huffman@44081 ` 161` ``` apply simp ``` huffman@44081 ` 162` ``` apply auto ``` huffman@44081 ` 163` ``` done ``` huffman@36630 ` 164` huffman@36630 ` 165` ```lemma eventually_Sup: ``` huffman@44195 ` 166` ``` "eventually P (Sup S) \ (\F\S. eventually P F)" ``` huffman@44081 ` 167` ``` unfolding Sup_filter_def ``` huffman@44081 ` 168` ``` apply (rule eventually_Abs_filter, rule is_filter.intro) ``` huffman@44081 ` 169` ``` apply (auto intro: eventually_conj elim!: eventually_rev_mp) ``` huffman@44081 ` 170` ``` done ``` huffman@36630 ` 171` huffman@36360 ` 172` ```instance proof ``` huffman@44195 ` 173` ``` fix F F' F'' :: "'a filter" and S :: "'a filter set" ``` huffman@44195 ` 174` ``` { show "F < F' \ F \ F' \ \ F' \ F" ``` huffman@44195 ` 175` ``` by (rule less_filter_def) } ``` huffman@44195 ` 176` ``` { show "F \ F" ``` huffman@44195 ` 177` ``` unfolding le_filter_def by simp } ``` huffman@44195 ` 178` ``` { assume "F \ F'" and "F' \ F''" thus "F \ F''" ``` huffman@44195 ` 179` ``` unfolding le_filter_def by simp } ``` huffman@44195 ` 180` ``` { assume "F \ F'" and "F' \ F" thus "F = F'" ``` huffman@44195 ` 181` ``` unfolding le_filter_def filter_eq_iff by fast } ``` huffman@44195 ` 182` ``` { show "F \ top" ``` huffman@44195 ` 183` ``` unfolding le_filter_def eventually_top by (simp add: always_eventually) } ``` huffman@44195 ` 184` ``` { show "bot \ F" ``` huffman@44195 ` 185` ``` unfolding le_filter_def by simp } ``` huffman@44195 ` 186` ``` { show "F \ sup F F'" and "F' \ sup F F'" ``` huffman@44195 ` 187` ``` unfolding le_filter_def eventually_sup by simp_all } ``` huffman@44195 ` 188` ``` { assume "F \ F''" and "F' \ F''" thus "sup F F' \ F''" ``` huffman@44195 ` 189` ``` unfolding le_filter_def eventually_sup by simp } ``` huffman@44195 ` 190` ``` { show "inf F F' \ F" and "inf F F' \ F'" ``` huffman@44195 ` 191` ``` unfolding le_filter_def eventually_inf by (auto intro: eventually_True) } ``` huffman@44195 ` 192` ``` { assume "F \ F'" and "F \ F''" thus "F \ inf F' F''" ``` huffman@44081 ` 193` ``` unfolding le_filter_def eventually_inf ``` huffman@44195 ` 194` ``` by (auto elim!: eventually_mono intro: eventually_conj) } ``` huffman@44195 ` 195` ``` { assume "F \ S" thus "F \ Sup S" ``` huffman@44195 ` 196` ``` unfolding le_filter_def eventually_Sup by simp } ``` huffman@44195 ` 197` ``` { assume "\F. F \ S \ F \ F'" thus "Sup S \ F'" ``` huffman@44195 ` 198` ``` unfolding le_filter_def eventually_Sup by simp } ``` huffman@44195 ` 199` ``` { assume "F'' \ S" thus "Inf S \ F''" ``` huffman@44195 ` 200` ``` unfolding le_filter_def Inf_filter_def eventually_Sup Ball_def by simp } ``` huffman@44195 ` 201` ``` { assume "\F'. F' \ S \ F \ F'" thus "F \ Inf S" ``` huffman@44195 ` 202` ``` unfolding le_filter_def Inf_filter_def eventually_Sup Ball_def by simp } ``` huffman@36360 ` 203` ```qed ``` huffman@36360 ` 204` huffman@36360 ` 205` ```end ``` huffman@36360 ` 206` huffman@44081 ` 207` ```lemma filter_leD: ``` huffman@44195 ` 208` ``` "F \ F' \ eventually P F' \ eventually P F" ``` huffman@44081 ` 209` ``` unfolding le_filter_def by simp ``` huffman@36360 ` 210` huffman@44081 ` 211` ```lemma filter_leI: ``` huffman@44195 ` 212` ``` "(\P. eventually P F' \ eventually P F) \ F \ F'" ``` huffman@44081 ` 213` ``` unfolding le_filter_def by simp ``` huffman@36360 ` 214` huffman@36360 ` 215` ```lemma eventually_False: ``` huffman@44195 ` 216` ``` "eventually (\x. False) F \ F = bot" ``` huffman@44081 ` 217` ``` unfolding filter_eq_iff by (auto elim: eventually_rev_mp) ``` huffman@36360 ` 218` huffman@44081 ` 219` ```subsection {* Map function for filters *} ``` huffman@36654 ` 220` huffman@44081 ` 221` ```definition filtermap :: "('a \ 'b) \ 'a filter \ 'b filter" ``` huffman@44195 ` 222` ``` where "filtermap f F = Abs_filter (\P. eventually (\x. P (f x)) F)" ``` huffman@36654 ` 223` huffman@44081 ` 224` ```lemma eventually_filtermap: ``` huffman@44195 ` 225` ``` "eventually P (filtermap f F) = eventually (\x. P (f x)) F" ``` huffman@44081 ` 226` ``` unfolding filtermap_def ``` huffman@44081 ` 227` ``` apply (rule eventually_Abs_filter) ``` huffman@44081 ` 228` ``` apply (rule is_filter.intro) ``` huffman@44081 ` 229` ``` apply (auto elim!: eventually_rev_mp) ``` huffman@44081 ` 230` ``` done ``` huffman@36654 ` 231` huffman@44195 ` 232` ```lemma filtermap_ident: "filtermap (\x. x) F = F" ``` huffman@44081 ` 233` ``` by (simp add: filter_eq_iff eventually_filtermap) ``` huffman@36654 ` 234` huffman@44081 ` 235` ```lemma filtermap_filtermap: ``` huffman@44195 ` 236` ``` "filtermap f (filtermap g F) = filtermap (\x. f (g x)) F" ``` huffman@44081 ` 237` ``` by (simp add: filter_eq_iff eventually_filtermap) ``` huffman@36654 ` 238` huffman@44195 ` 239` ```lemma filtermap_mono: "F \ F' \ filtermap f F \ filtermap f F'" ``` huffman@44081 ` 240` ``` unfolding le_filter_def eventually_filtermap by simp ``` huffman@44081 ` 241` huffman@44081 ` 242` ```lemma filtermap_bot [simp]: "filtermap f bot = bot" ``` huffman@44081 ` 243` ``` by (simp add: filter_eq_iff eventually_filtermap) ``` huffman@36654 ` 244` huffman@36654 ` 245` huffman@36662 ` 246` ```subsection {* Sequentially *} ``` huffman@31392 ` 247` huffman@44081 ` 248` ```definition sequentially :: "nat filter" ``` huffman@44081 ` 249` ``` where "sequentially = Abs_filter (\P. \k. \n\k. P n)" ``` huffman@31392 ` 250` huffman@36662 ` 251` ```lemma eventually_sequentially: ``` huffman@36662 ` 252` ``` "eventually P sequentially \ (\N. \n\N. P n)" ``` huffman@36662 ` 253` ```unfolding sequentially_def ``` huffman@44081 ` 254` ```proof (rule eventually_Abs_filter, rule is_filter.intro) ``` huffman@36662 ` 255` ``` fix P Q :: "nat \ bool" ``` huffman@36662 ` 256` ``` assume "\i. \n\i. P n" and "\j. \n\j. Q n" ``` huffman@36662 ` 257` ``` then obtain i j where "\n\i. P n" and "\n\j. Q n" by auto ``` huffman@36662 ` 258` ``` then have "\n\max i j. P n \ Q n" by simp ``` huffman@36662 ` 259` ``` then show "\k. \n\k. P n \ Q n" .. ``` huffman@36662 ` 260` ```qed auto ``` huffman@36662 ` 261` huffman@36662 ` 262` ```lemma sequentially_bot [simp]: "sequentially \ bot" ``` huffman@44081 ` 263` ``` unfolding filter_eq_iff eventually_sequentially by auto ``` huffman@36662 ` 264` huffman@36662 ` 265` ```lemma eventually_False_sequentially [simp]: ``` huffman@36662 ` 266` ``` "\ eventually (\n. False) sequentially" ``` huffman@44081 ` 267` ``` by (simp add: eventually_False) ``` huffman@36662 ` 268` huffman@36662 ` 269` ```lemma le_sequentially: ``` huffman@44195 ` 270` ``` "F \ sequentially \ (\N. eventually (\n. N \ n) F)" ``` huffman@44081 ` 271` ``` unfolding le_filter_def eventually_sequentially ``` huffman@44081 ` 272` ``` by (safe, fast, drule_tac x=N in spec, auto elim: eventually_rev_mp) ``` huffman@36662 ` 273` huffman@36662 ` 274` huffman@44081 ` 275` ```definition trivial_limit :: "'a filter \ bool" ``` huffman@44195 ` 276` ``` where "trivial_limit F \ eventually (\x. False) F" ``` hoelzl@41970 ` 277` huffman@44081 ` 278` ```lemma trivial_limit_sequentially [intro]: "\ trivial_limit sequentially" ``` hoelzl@41970 ` 279` ``` by (auto simp add: trivial_limit_def eventually_sequentially) ``` hoelzl@41970 ` 280` huffman@44081 ` 281` ```subsection {* Standard filters *} ``` huffman@36662 ` 282` huffman@44081 ` 283` ```definition within :: "'a filter \ 'a set \ 'a filter" (infixr "within" 70) ``` huffman@44195 ` 284` ``` where "F within S = Abs_filter (\P. eventually (\x. x \ S \ P x) F)" ``` huffman@31392 ` 285` huffman@44081 ` 286` ```definition nhds :: "'a::topological_space \ 'a filter" ``` huffman@44081 ` 287` ``` where "nhds a = Abs_filter (\P. \S. open S \ a \ S \ (\x\S. P x))" ``` huffman@36654 ` 288` huffman@44081 ` 289` ```definition at :: "'a::topological_space \ 'a filter" ``` huffman@44081 ` 290` ``` where "at a = nhds a within - {a}" ``` huffman@31447 ` 291` huffman@31392 ` 292` ```lemma eventually_within: ``` huffman@44195 ` 293` ``` "eventually P (F within S) = eventually (\x. x \ S \ P x) F" ``` huffman@44081 ` 294` ``` unfolding within_def ``` huffman@44081 ` 295` ``` by (rule eventually_Abs_filter, rule is_filter.intro) ``` huffman@44081 ` 296` ``` (auto elim!: eventually_rev_mp) ``` huffman@31392 ` 297` huffman@44195 ` 298` ```lemma within_UNIV: "F within UNIV = F" ``` huffman@44081 ` 299` ``` unfolding filter_eq_iff eventually_within by simp ``` huffman@36360 ` 300` huffman@36654 ` 301` ```lemma eventually_nhds: ``` huffman@36654 ` 302` ``` "eventually P (nhds a) \ (\S. open S \ a \ S \ (\x\S. P x))" ``` huffman@36654 ` 303` ```unfolding nhds_def ``` huffman@44081 ` 304` ```proof (rule eventually_Abs_filter, rule is_filter.intro) ``` huffman@36654 ` 305` ``` have "open UNIV \ a \ UNIV \ (\x\UNIV. True)" by simp ``` huffman@36654 ` 306` ``` thus "\S. open S \ a \ S \ (\x\S. True)" by - rule ``` huffman@36358 ` 307` ```next ``` huffman@36358 ` 308` ``` fix P Q ``` huffman@36654 ` 309` ``` assume "\S. open S \ a \ S \ (\x\S. P x)" ``` huffman@36654 ` 310` ``` and "\T. open T \ a \ T \ (\x\T. Q x)" ``` huffman@36358 ` 311` ``` then obtain S T where ``` huffman@36654 ` 312` ``` "open S \ a \ S \ (\x\S. P x)" ``` huffman@36654 ` 313` ``` "open T \ a \ T \ (\x\T. Q x)" by auto ``` huffman@36654 ` 314` ``` hence "open (S \ T) \ a \ S \ T \ (\x\(S \ T). P x \ Q x)" ``` huffman@36358 ` 315` ``` by (simp add: open_Int) ``` huffman@36654 ` 316` ``` thus "\S. open S \ a \ S \ (\x\S. P x \ Q x)" by - rule ``` huffman@36358 ` 317` ```qed auto ``` huffman@31447 ` 318` huffman@36656 ` 319` ```lemma eventually_nhds_metric: ``` huffman@36656 ` 320` ``` "eventually P (nhds a) \ (\d>0. \x. dist x a < d \ P x)" ``` huffman@36656 ` 321` ```unfolding eventually_nhds open_dist ``` huffman@31447 ` 322` ```apply safe ``` huffman@31447 ` 323` ```apply fast ``` huffman@31492 ` 324` ```apply (rule_tac x="{x. dist x a < d}" in exI, simp) ``` huffman@31447 ` 325` ```apply clarsimp ``` huffman@31447 ` 326` ```apply (rule_tac x="d - dist x a" in exI, clarsimp) ``` huffman@31447 ` 327` ```apply (simp only: less_diff_eq) ``` huffman@31447 ` 328` ```apply (erule le_less_trans [OF dist_triangle]) ``` huffman@31447 ` 329` ```done ``` huffman@31447 ` 330` huffman@36656 ` 331` ```lemma eventually_at_topological: ``` huffman@36656 ` 332` ``` "eventually P (at a) \ (\S. open S \ a \ S \ (\x\S. x \ a \ P x))" ``` huffman@36656 ` 333` ```unfolding at_def eventually_within eventually_nhds by simp ``` huffman@36656 ` 334` huffman@36656 ` 335` ```lemma eventually_at: ``` huffman@36656 ` 336` ``` fixes a :: "'a::metric_space" ``` huffman@36656 ` 337` ``` shows "eventually P (at a) \ (\d>0. \x. x \ a \ dist x a < d \ P x)" ``` huffman@36656 ` 338` ```unfolding at_def eventually_within eventually_nhds_metric by auto ``` huffman@36656 ` 339` huffman@31392 ` 340` huffman@31355 ` 341` ```subsection {* Boundedness *} ``` huffman@31355 ` 342` huffman@44081 ` 343` ```definition Bfun :: "('a \ 'b::real_normed_vector) \ 'a filter \ bool" ``` huffman@44195 ` 344` ``` where "Bfun f F = (\K>0. eventually (\x. norm (f x) \ K) F)" ``` huffman@31355 ` 345` huffman@31487 ` 346` ```lemma BfunI: ``` huffman@44195 ` 347` ``` assumes K: "eventually (\x. norm (f x) \ K) F" shows "Bfun f F" ``` huffman@31355 ` 348` ```unfolding Bfun_def ``` huffman@31355 ` 349` ```proof (intro exI conjI allI) ``` huffman@31355 ` 350` ``` show "0 < max K 1" by simp ``` huffman@31355 ` 351` ```next ``` huffman@44195 ` 352` ``` show "eventually (\x. norm (f x) \ max K 1) F" ``` huffman@31355 ` 353` ``` using K by (rule eventually_elim1, simp) ``` huffman@31355 ` 354` ```qed ``` huffman@31355 ` 355` huffman@31355 ` 356` ```lemma BfunE: ``` huffman@44195 ` 357` ``` assumes "Bfun f F" ``` huffman@44195 ` 358` ``` obtains B where "0 < B" and "eventually (\x. norm (f x) \ B) F" ``` huffman@31355 ` 359` ```using assms unfolding Bfun_def by fast ``` huffman@31355 ` 360` huffman@31355 ` 361` huffman@31349 ` 362` ```subsection {* Convergence to Zero *} ``` huffman@31349 ` 363` huffman@44081 ` 364` ```definition Zfun :: "('a \ 'b::real_normed_vector) \ 'a filter \ bool" ``` huffman@44195 ` 365` ``` where "Zfun f F = (\r>0. eventually (\x. norm (f x) < r) F)" ``` huffman@31349 ` 366` huffman@31349 ` 367` ```lemma ZfunI: ``` huffman@44195 ` 368` ``` "(\r. 0 < r \ eventually (\x. norm (f x) < r) F) \ Zfun f F" ``` huffman@44081 ` 369` ``` unfolding Zfun_def by simp ``` huffman@31349 ` 370` huffman@31349 ` 371` ```lemma ZfunD: ``` huffman@44195 ` 372` ``` "\Zfun f F; 0 < r\ \ eventually (\x. norm (f x) < r) F" ``` huffman@44081 ` 373` ``` unfolding Zfun_def by simp ``` huffman@31349 ` 374` huffman@31355 ` 375` ```lemma Zfun_ssubst: ``` huffman@44195 ` 376` ``` "eventually (\x. f x = g x) F \ Zfun g F \ Zfun f F" ``` huffman@44081 ` 377` ``` unfolding Zfun_def by (auto elim!: eventually_rev_mp) ``` huffman@31355 ` 378` huffman@44195 ` 379` ```lemma Zfun_zero: "Zfun (\x. 0) F" ``` huffman@44081 ` 380` ``` unfolding Zfun_def by simp ``` huffman@31349 ` 381` huffman@44195 ` 382` ```lemma Zfun_norm_iff: "Zfun (\x. norm (f x)) F = Zfun (\x. f x) F" ``` huffman@44081 ` 383` ``` unfolding Zfun_def by simp ``` huffman@31349 ` 384` huffman@31349 ` 385` ```lemma Zfun_imp_Zfun: ``` huffman@44195 ` 386` ``` assumes f: "Zfun f F" ``` huffman@44195 ` 387` ``` assumes g: "eventually (\x. norm (g x) \ norm (f x) * K) F" ``` huffman@44195 ` 388` ``` shows "Zfun (\x. g x) F" ``` huffman@31349 ` 389` ```proof (cases) ``` huffman@31349 ` 390` ``` assume K: "0 < K" ``` huffman@31349 ` 391` ``` show ?thesis ``` huffman@31349 ` 392` ``` proof (rule ZfunI) ``` huffman@31349 ` 393` ``` fix r::real assume "0 < r" ``` huffman@31349 ` 394` ``` hence "0 < r / K" ``` huffman@31349 ` 395` ``` using K by (rule divide_pos_pos) ``` huffman@44195 ` 396` ``` then have "eventually (\x. norm (f x) < r / K) F" ``` huffman@31487 ` 397` ``` using ZfunD [OF f] by fast ``` huffman@44195 ` 398` ``` with g show "eventually (\x. norm (g x) < r) F" ``` huffman@31355 ` 399` ``` proof (rule eventually_elim2) ``` huffman@31487 ` 400` ``` fix x ``` huffman@31487 ` 401` ``` assume *: "norm (g x) \ norm (f x) * K" ``` huffman@31487 ` 402` ``` assume "norm (f x) < r / K" ``` huffman@31487 ` 403` ``` hence "norm (f x) * K < r" ``` huffman@31349 ` 404` ``` by (simp add: pos_less_divide_eq K) ``` huffman@31487 ` 405` ``` thus "norm (g x) < r" ``` huffman@31355 ` 406` ``` by (simp add: order_le_less_trans [OF *]) ``` huffman@31349 ` 407` ``` qed ``` huffman@31349 ` 408` ``` qed ``` huffman@31349 ` 409` ```next ``` huffman@31349 ` 410` ``` assume "\ 0 < K" ``` huffman@31349 ` 411` ``` hence K: "K \ 0" by (simp only: not_less) ``` huffman@31355 ` 412` ``` show ?thesis ``` huffman@31355 ` 413` ``` proof (rule ZfunI) ``` huffman@31355 ` 414` ``` fix r :: real ``` huffman@31355 ` 415` ``` assume "0 < r" ``` huffman@44195 ` 416` ``` from g show "eventually (\x. norm (g x) < r) F" ``` huffman@31355 ` 417` ``` proof (rule eventually_elim1) ``` huffman@31487 ` 418` ``` fix x ``` huffman@31487 ` 419` ``` assume "norm (g x) \ norm (f x) * K" ``` huffman@31487 ` 420` ``` also have "\ \ norm (f x) * 0" ``` huffman@31355 ` 421` ``` using K norm_ge_zero by (rule mult_left_mono) ``` huffman@31487 ` 422` ``` finally show "norm (g x) < r" ``` huffman@31355 ` 423` ``` using `0 < r` by simp ``` huffman@31355 ` 424` ``` qed ``` huffman@31355 ` 425` ``` qed ``` huffman@31349 ` 426` ```qed ``` huffman@31349 ` 427` huffman@44195 ` 428` ```lemma Zfun_le: "\Zfun g F; \x. norm (f x) \ norm (g x)\ \ Zfun f F" ``` huffman@44081 ` 429` ``` by (erule_tac K="1" in Zfun_imp_Zfun, simp) ``` huffman@31349 ` 430` huffman@31349 ` 431` ```lemma Zfun_add: ``` huffman@44195 ` 432` ``` assumes f: "Zfun f F" and g: "Zfun g F" ``` huffman@44195 ` 433` ``` shows "Zfun (\x. f x + g x) F" ``` huffman@31349 ` 434` ```proof (rule ZfunI) ``` huffman@31349 ` 435` ``` fix r::real assume "0 < r" ``` huffman@31349 ` 436` ``` hence r: "0 < r / 2" by simp ``` huffman@44195 ` 437` ``` have "eventually (\x. norm (f x) < r/2) F" ``` huffman@31487 ` 438` ``` using f r by (rule ZfunD) ``` huffman@31349 ` 439` ``` moreover ``` huffman@44195 ` 440` ``` have "eventually (\x. norm (g x) < r/2) F" ``` huffman@31487 ` 441` ``` using g r by (rule ZfunD) ``` huffman@31349 ` 442` ``` ultimately ``` huffman@44195 ` 443` ``` show "eventually (\x. norm (f x + g x) < r) F" ``` huffman@31349 ` 444` ``` proof (rule eventually_elim2) ``` huffman@31487 ` 445` ``` fix x ``` huffman@31487 ` 446` ``` assume *: "norm (f x) < r/2" "norm (g x) < r/2" ``` huffman@31487 ` 447` ``` have "norm (f x + g x) \ norm (f x) + norm (g x)" ``` huffman@31349 ` 448` ``` by (rule norm_triangle_ineq) ``` huffman@31349 ` 449` ``` also have "\ < r/2 + r/2" ``` huffman@31349 ` 450` ``` using * by (rule add_strict_mono) ``` huffman@31487 ` 451` ``` finally show "norm (f x + g x) < r" ``` huffman@31349 ` 452` ``` by simp ``` huffman@31349 ` 453` ``` qed ``` huffman@31349 ` 454` ```qed ``` huffman@31349 ` 455` huffman@44195 ` 456` ```lemma Zfun_minus: "Zfun f F \ Zfun (\x. - f x) F" ``` huffman@44081 ` 457` ``` unfolding Zfun_def by simp ``` huffman@31349 ` 458` huffman@44195 ` 459` ```lemma Zfun_diff: "\Zfun f F; Zfun g F\ \ Zfun (\x. f x - g x) F" ``` huffman@44081 ` 460` ``` by (simp only: diff_minus Zfun_add Zfun_minus) ``` huffman@31349 ` 461` huffman@31349 ` 462` ```lemma (in bounded_linear) Zfun: ``` huffman@44195 ` 463` ``` assumes g: "Zfun g F" ``` huffman@44195 ` 464` ``` shows "Zfun (\x. f (g x)) F" ``` huffman@31349 ` 465` ```proof - ``` huffman@31349 ` 466` ``` obtain K where "\x. norm (f x) \ norm x * K" ``` huffman@31349 ` 467` ``` using bounded by fast ``` huffman@44195 ` 468` ``` then have "eventually (\x. norm (f (g x)) \ norm (g x) * K) F" ``` huffman@31355 ` 469` ``` by simp ``` huffman@31487 ` 470` ``` with g show ?thesis ``` huffman@31349 ` 471` ``` by (rule Zfun_imp_Zfun) ``` huffman@31349 ` 472` ```qed ``` huffman@31349 ` 473` huffman@31349 ` 474` ```lemma (in bounded_bilinear) Zfun: ``` huffman@44195 ` 475` ``` assumes f: "Zfun f F" ``` huffman@44195 ` 476` ``` assumes g: "Zfun g F" ``` huffman@44195 ` 477` ``` shows "Zfun (\x. f x ** g x) F" ``` huffman@31349 ` 478` ```proof (rule ZfunI) ``` huffman@31349 ` 479` ``` fix r::real assume r: "0 < r" ``` huffman@31349 ` 480` ``` obtain K where K: "0 < K" ``` huffman@31349 ` 481` ``` and norm_le: "\x y. norm (x ** y) \ norm x * norm y * K" ``` huffman@31349 ` 482` ``` using pos_bounded by fast ``` huffman@31349 ` 483` ``` from K have K': "0 < inverse K" ``` huffman@31349 ` 484` ``` by (rule positive_imp_inverse_positive) ``` huffman@44195 ` 485` ``` have "eventually (\x. norm (f x) < r) F" ``` huffman@31487 ` 486` ``` using f r by (rule ZfunD) ``` huffman@31349 ` 487` ``` moreover ``` huffman@44195 ` 488` ``` have "eventually (\x. norm (g x) < inverse K) F" ``` huffman@31487 ` 489` ``` using g K' by (rule ZfunD) ``` huffman@31349 ` 490` ``` ultimately ``` huffman@44195 ` 491` ``` show "eventually (\x. norm (f x ** g x) < r) F" ``` huffman@31349 ` 492` ``` proof (rule eventually_elim2) ``` huffman@31487 ` 493` ``` fix x ``` huffman@31487 ` 494` ``` assume *: "norm (f x) < r" "norm (g x) < inverse K" ``` huffman@31487 ` 495` ``` have "norm (f x ** g x) \ norm (f x) * norm (g x) * K" ``` huffman@31349 ` 496` ``` by (rule norm_le) ``` huffman@31487 ` 497` ``` also have "norm (f x) * norm (g x) * K < r * inverse K * K" ``` huffman@31349 ` 498` ``` by (intro mult_strict_right_mono mult_strict_mono' norm_ge_zero * K) ``` huffman@31349 ` 499` ``` also from K have "r * inverse K * K = r" ``` huffman@31349 ` 500` ``` by simp ``` huffman@31487 ` 501` ``` finally show "norm (f x ** g x) < r" . ``` huffman@31349 ` 502` ``` qed ``` huffman@31349 ` 503` ```qed ``` huffman@31349 ` 504` huffman@31349 ` 505` ```lemma (in bounded_bilinear) Zfun_left: ``` huffman@44195 ` 506` ``` "Zfun f F \ Zfun (\x. f x ** a) F" ``` huffman@44081 ` 507` ``` by (rule bounded_linear_left [THEN bounded_linear.Zfun]) ``` huffman@31349 ` 508` huffman@31349 ` 509` ```lemma (in bounded_bilinear) Zfun_right: ``` huffman@44195 ` 510` ``` "Zfun f F \ Zfun (\x. a ** f x) F" ``` huffman@44081 ` 511` ``` by (rule bounded_linear_right [THEN bounded_linear.Zfun]) ``` huffman@31349 ` 512` huffman@31349 ` 513` ```lemmas Zfun_mult = mult.Zfun ``` huffman@31349 ` 514` ```lemmas Zfun_mult_right = mult.Zfun_right ``` huffman@31349 ` 515` ```lemmas Zfun_mult_left = mult.Zfun_left ``` huffman@31349 ` 516` huffman@31349 ` 517` wenzelm@31902 ` 518` ```subsection {* Limits *} ``` huffman@31349 ` 519` huffman@44081 ` 520` ```definition tendsto :: "('a \ 'b::topological_space) \ 'b \ 'a filter \ bool" ``` haftmann@37767 ` 521` ``` (infixr "--->" 55) where ``` huffman@44195 ` 522` ``` "(f ---> l) F \ (\S. open S \ l \ S \ eventually (\x. f x \ S) F)" ``` huffman@31349 ` 523` wenzelm@31902 ` 524` ```ML {* ``` wenzelm@31902 ` 525` ```structure Tendsto_Intros = Named_Thms ``` wenzelm@31902 ` 526` ```( ``` wenzelm@31902 ` 527` ``` val name = "tendsto_intros" ``` wenzelm@31902 ` 528` ``` val description = "introduction rules for tendsto" ``` wenzelm@31902 ` 529` ```) ``` huffman@31565 ` 530` ```*} ``` huffman@31565 ` 531` wenzelm@31902 ` 532` ```setup Tendsto_Intros.setup ``` huffman@31565 ` 533` huffman@44195 ` 534` ```lemma tendsto_mono: "F \ F' \ (f ---> l) F' \ (f ---> l) F" ``` huffman@44081 ` 535` ``` unfolding tendsto_def le_filter_def by fast ``` huffman@36656 ` 536` huffman@31488 ` 537` ```lemma topological_tendstoI: ``` huffman@44195 ` 538` ``` "(\S. open S \ l \ S \ eventually (\x. f x \ S) F) ``` huffman@44195 ` 539` ``` \ (f ---> l) F" ``` huffman@31349 ` 540` ``` unfolding tendsto_def by auto ``` huffman@31349 ` 541` huffman@31488 ` 542` ```lemma topological_tendstoD: ``` huffman@44195 ` 543` ``` "(f ---> l) F \ open S \ l \ S \ eventually (\x. f x \ S) F" ``` huffman@31488 ` 544` ``` unfolding tendsto_def by auto ``` huffman@31488 ` 545` huffman@31488 ` 546` ```lemma tendstoI: ``` huffman@44195 ` 547` ``` assumes "\e. 0 < e \ eventually (\x. dist (f x) l < e) F" ``` huffman@44195 ` 548` ``` shows "(f ---> l) F" ``` huffman@44081 ` 549` ``` apply (rule topological_tendstoI) ``` huffman@44081 ` 550` ``` apply (simp add: open_dist) ``` huffman@44081 ` 551` ``` apply (drule (1) bspec, clarify) ``` huffman@44081 ` 552` ``` apply (drule assms) ``` huffman@44081 ` 553` ``` apply (erule eventually_elim1, simp) ``` huffman@44081 ` 554` ``` done ``` huffman@31488 ` 555` huffman@31349 ` 556` ```lemma tendstoD: ``` huffman@44195 ` 557` ``` "(f ---> l) F \ 0 < e \ eventually (\x. dist (f x) l < e) F" ``` huffman@44081 ` 558` ``` apply (drule_tac S="{x. dist x l < e}" in topological_tendstoD) ``` huffman@44081 ` 559` ``` apply (clarsimp simp add: open_dist) ``` huffman@44081 ` 560` ``` apply (rule_tac x="e - dist x l" in exI, clarsimp) ``` huffman@44081 ` 561` ``` apply (simp only: less_diff_eq) ``` huffman@44081 ` 562` ``` apply (erule le_less_trans [OF dist_triangle]) ``` huffman@44081 ` 563` ``` apply simp ``` huffman@44081 ` 564` ``` apply simp ``` huffman@44081 ` 565` ``` done ``` huffman@31488 ` 566` huffman@31488 ` 567` ```lemma tendsto_iff: ``` huffman@44195 ` 568` ``` "(f ---> l) F \ (\e>0. eventually (\x. dist (f x) l < e) F)" ``` huffman@44081 ` 569` ``` using tendstoI tendstoD by fast ``` huffman@31349 ` 570` huffman@44195 ` 571` ```lemma tendsto_Zfun_iff: "(f ---> a) F = Zfun (\x. f x - a) F" ``` huffman@44081 ` 572` ``` by (simp only: tendsto_iff Zfun_def dist_norm) ``` huffman@31349 ` 573` huffman@31565 ` 574` ```lemma tendsto_ident_at [tendsto_intros]: "((\x. x) ---> a) (at a)" ``` huffman@44081 ` 575` ``` unfolding tendsto_def eventually_at_topological by auto ``` huffman@31565 ` 576` huffman@31565 ` 577` ```lemma tendsto_ident_at_within [tendsto_intros]: ``` huffman@36655 ` 578` ``` "((\x. x) ---> a) (at a within S)" ``` huffman@44081 ` 579` ``` unfolding tendsto_def eventually_within eventually_at_topological by auto ``` huffman@31565 ` 580` huffman@44195 ` 581` ```lemma tendsto_const [tendsto_intros]: "((\x. k) ---> k) F" ``` huffman@44081 ` 582` ``` by (simp add: tendsto_def) ``` huffman@31349 ` 583` huffman@44205 ` 584` ```lemma tendsto_unique: ``` huffman@44205 ` 585` ``` fixes f :: "'a \ 'b::t2_space" ``` huffman@44205 ` 586` ``` assumes "\ trivial_limit F" and "(f ---> a) F" and "(f ---> b) F" ``` huffman@44205 ` 587` ``` shows "a = b" ``` huffman@44205 ` 588` ```proof (rule ccontr) ``` huffman@44205 ` 589` ``` assume "a \ b" ``` huffman@44205 ` 590` ``` obtain U V where "open U" "open V" "a \ U" "b \ V" "U \ V = {}" ``` huffman@44205 ` 591` ``` using hausdorff [OF `a \ b`] by fast ``` huffman@44205 ` 592` ``` have "eventually (\x. f x \ U) F" ``` huffman@44205 ` 593` ``` using `(f ---> a) F` `open U` `a \ U` by (rule topological_tendstoD) ``` huffman@44205 ` 594` ``` moreover ``` huffman@44205 ` 595` ``` have "eventually (\x. f x \ V) F" ``` huffman@44205 ` 596` ``` using `(f ---> b) F` `open V` `b \ V` by (rule topological_tendstoD) ``` huffman@44205 ` 597` ``` ultimately ``` huffman@44205 ` 598` ``` have "eventually (\x. False) F" ``` huffman@44205 ` 599` ``` proof (rule eventually_elim2) ``` huffman@44205 ` 600` ``` fix x ``` huffman@44205 ` 601` ``` assume "f x \ U" "f x \ V" ``` huffman@44205 ` 602` ``` hence "f x \ U \ V" by simp ``` huffman@44205 ` 603` ``` with `U \ V = {}` show "False" by simp ``` huffman@44205 ` 604` ``` qed ``` huffman@44205 ` 605` ``` with `\ trivial_limit F` show "False" ``` huffman@44205 ` 606` ``` by (simp add: trivial_limit_def) ``` huffman@44205 ` 607` ```qed ``` huffman@44205 ` 608` huffman@36662 ` 609` ```lemma tendsto_const_iff: ``` huffman@44205 ` 610` ``` fixes a b :: "'a::t2_space" ``` huffman@44205 ` 611` ``` assumes "\ trivial_limit F" shows "((\x. a) ---> b) F \ a = b" ``` huffman@44205 ` 612` ``` by (safe intro!: tendsto_const tendsto_unique [OF assms tendsto_const]) ``` huffman@44205 ` 613` huffman@44205 ` 614` ```subsubsection {* Distance and norms *} ``` huffman@36662 ` 615` huffman@31565 ` 616` ```lemma tendsto_dist [tendsto_intros]: ``` huffman@44195 ` 617` ``` assumes f: "(f ---> l) F" and g: "(g ---> m) F" ``` huffman@44195 ` 618` ``` shows "((\x. dist (f x) (g x)) ---> dist l m) F" ``` huffman@31565 ` 619` ```proof (rule tendstoI) ``` huffman@31565 ` 620` ``` fix e :: real assume "0 < e" ``` huffman@31565 ` 621` ``` hence e2: "0 < e/2" by simp ``` huffman@31565 ` 622` ``` from tendstoD [OF f e2] tendstoD [OF g e2] ``` huffman@44195 ` 623` ``` show "eventually (\x. dist (dist (f x) (g x)) (dist l m) < e) F" ``` huffman@31565 ` 624` ``` proof (rule eventually_elim2) ``` huffman@31565 ` 625` ``` fix x assume "dist (f x) l < e/2" "dist (g x) m < e/2" ``` huffman@31565 ` 626` ``` then show "dist (dist (f x) (g x)) (dist l m) < e" ``` huffman@31565 ` 627` ``` unfolding dist_real_def ``` huffman@31565 ` 628` ``` using dist_triangle2 [of "f x" "g x" "l"] ``` huffman@31565 ` 629` ``` using dist_triangle2 [of "g x" "l" "m"] ``` huffman@31565 ` 630` ``` using dist_triangle3 [of "l" "m" "f x"] ``` huffman@31565 ` 631` ``` using dist_triangle [of "f x" "m" "g x"] ``` huffman@31565 ` 632` ``` by arith ``` huffman@31565 ` 633` ``` qed ``` huffman@31565 ` 634` ```qed ``` huffman@31565 ` 635` huffman@36662 ` 636` ```lemma norm_conv_dist: "norm x = dist x 0" ``` huffman@44081 ` 637` ``` unfolding dist_norm by simp ``` huffman@36662 ` 638` huffman@31565 ` 639` ```lemma tendsto_norm [tendsto_intros]: ``` huffman@44195 ` 640` ``` "(f ---> a) F \ ((\x. norm (f x)) ---> norm a) F" ``` huffman@44081 ` 641` ``` unfolding norm_conv_dist by (intro tendsto_intros) ``` huffman@36662 ` 642` huffman@36662 ` 643` ```lemma tendsto_norm_zero: ``` huffman@44195 ` 644` ``` "(f ---> 0) F \ ((\x. norm (f x)) ---> 0) F" ``` huffman@44081 ` 645` ``` by (drule tendsto_norm, simp) ``` huffman@36662 ` 646` huffman@36662 ` 647` ```lemma tendsto_norm_zero_cancel: ``` huffman@44195 ` 648` ``` "((\x. norm (f x)) ---> 0) F \ (f ---> 0) F" ``` huffman@44081 ` 649` ``` unfolding tendsto_iff dist_norm by simp ``` huffman@36662 ` 650` huffman@36662 ` 651` ```lemma tendsto_norm_zero_iff: ``` huffman@44195 ` 652` ``` "((\x. norm (f x)) ---> 0) F \ (f ---> 0) F" ``` huffman@44081 ` 653` ``` unfolding tendsto_iff dist_norm by simp ``` huffman@31349 ` 654` huffman@44194 ` 655` ```lemma tendsto_rabs [tendsto_intros]: ``` huffman@44195 ` 656` ``` "(f ---> (l::real)) F \ ((\x. \f x\) ---> \l\) F" ``` huffman@44194 ` 657` ``` by (fold real_norm_def, rule tendsto_norm) ``` huffman@44194 ` 658` huffman@44194 ` 659` ```lemma tendsto_rabs_zero: ``` huffman@44195 ` 660` ``` "(f ---> (0::real)) F \ ((\x. \f x\) ---> 0) F" ``` huffman@44194 ` 661` ``` by (fold real_norm_def, rule tendsto_norm_zero) ``` huffman@44194 ` 662` huffman@44194 ` 663` ```lemma tendsto_rabs_zero_cancel: ``` huffman@44195 ` 664` ``` "((\x. \f x\) ---> (0::real)) F \ (f ---> 0) F" ``` huffman@44194 ` 665` ``` by (fold real_norm_def, rule tendsto_norm_zero_cancel) ``` huffman@44194 ` 666` huffman@44194 ` 667` ```lemma tendsto_rabs_zero_iff: ``` huffman@44195 ` 668` ``` "((\x. \f x\) ---> (0::real)) F \ (f ---> 0) F" ``` huffman@44194 ` 669` ``` by (fold real_norm_def, rule tendsto_norm_zero_iff) ``` huffman@44194 ` 670` huffman@44194 ` 671` ```subsubsection {* Addition and subtraction *} ``` huffman@44194 ` 672` huffman@31565 ` 673` ```lemma tendsto_add [tendsto_intros]: ``` huffman@31349 ` 674` ``` fixes a b :: "'a::real_normed_vector" ``` huffman@44195 ` 675` ``` shows "\(f ---> a) F; (g ---> b) F\ \ ((\x. f x + g x) ---> a + b) F" ``` huffman@44081 ` 676` ``` by (simp only: tendsto_Zfun_iff add_diff_add Zfun_add) ``` huffman@31349 ` 677` huffman@44194 ` 678` ```lemma tendsto_add_zero: ``` huffman@44194 ` 679` ``` fixes f g :: "'a::type \ 'b::real_normed_vector" ``` huffman@44195 ` 680` ``` shows "\(f ---> 0) F; (g ---> 0) F\ \ ((\x. f x + g x) ---> 0) F" ``` huffman@44194 ` 681` ``` by (drule (1) tendsto_add, simp) ``` huffman@44194 ` 682` huffman@31565 ` 683` ```lemma tendsto_minus [tendsto_intros]: ``` huffman@31349 ` 684` ``` fixes a :: "'a::real_normed_vector" ``` huffman@44195 ` 685` ``` shows "(f ---> a) F \ ((\x. - f x) ---> - a) F" ``` huffman@44081 ` 686` ``` by (simp only: tendsto_Zfun_iff minus_diff_minus Zfun_minus) ``` huffman@31349 ` 687` huffman@31349 ` 688` ```lemma tendsto_minus_cancel: ``` huffman@31349 ` 689` ``` fixes a :: "'a::real_normed_vector" ``` huffman@44195 ` 690` ``` shows "((\x. - f x) ---> - a) F \ (f ---> a) F" ``` huffman@44081 ` 691` ``` by (drule tendsto_minus, simp) ``` huffman@31349 ` 692` huffman@31565 ` 693` ```lemma tendsto_diff [tendsto_intros]: ``` huffman@31349 ` 694` ``` fixes a b :: "'a::real_normed_vector" ``` huffman@44195 ` 695` ``` shows "\(f ---> a) F; (g ---> b) F\ \ ((\x. f x - g x) ---> a - b) F" ``` huffman@44081 ` 696` ``` by (simp add: diff_minus tendsto_add tendsto_minus) ``` huffman@31349 ` 697` huffman@31588 ` 698` ```lemma tendsto_setsum [tendsto_intros]: ``` huffman@31588 ` 699` ``` fixes f :: "'a \ 'b \ 'c::real_normed_vector" ``` huffman@44195 ` 700` ``` assumes "\i. i \ S \ (f i ---> a i) F" ``` huffman@44195 ` 701` ``` shows "((\x. \i\S. f i x) ---> (\i\S. a i)) F" ``` huffman@31588 ` 702` ```proof (cases "finite S") ``` huffman@31588 ` 703` ``` assume "finite S" thus ?thesis using assms ``` huffman@44194 ` 704` ``` by (induct, simp add: tendsto_const, simp add: tendsto_add) ``` huffman@31588 ` 705` ```next ``` huffman@31588 ` 706` ``` assume "\ finite S" thus ?thesis ``` huffman@31588 ` 707` ``` by (simp add: tendsto_const) ``` huffman@31588 ` 708` ```qed ``` huffman@31588 ` 709` huffman@44194 ` 710` ```subsubsection {* Linear operators and multiplication *} ``` huffman@44194 ` 711` huffman@31565 ` 712` ```lemma (in bounded_linear) tendsto [tendsto_intros]: ``` huffman@44195 ` 713` ``` "(g ---> a) F \ ((\x. f (g x)) ---> f a) F" ``` huffman@44081 ` 714` ``` by (simp only: tendsto_Zfun_iff diff [symmetric] Zfun) ``` huffman@31349 ` 715` huffman@44194 ` 716` ```lemma (in bounded_linear) tendsto_zero: ``` huffman@44195 ` 717` ``` "(g ---> 0) F \ ((\x. f (g x)) ---> 0) F" ``` huffman@44194 ` 718` ``` by (drule tendsto, simp only: zero) ``` huffman@44194 ` 719` huffman@31565 ` 720` ```lemma (in bounded_bilinear) tendsto [tendsto_intros]: ``` huffman@44195 ` 721` ``` "\(f ---> a) F; (g ---> b) F\ \ ((\x. f x ** g x) ---> a ** b) F" ``` huffman@44081 ` 722` ``` by (simp only: tendsto_Zfun_iff prod_diff_prod ``` huffman@44081 ` 723` ``` Zfun_add Zfun Zfun_left Zfun_right) ``` huffman@31349 ` 724` huffman@44194 ` 725` ```lemma (in bounded_bilinear) tendsto_zero: ``` huffman@44195 ` 726` ``` assumes f: "(f ---> 0) F" ``` huffman@44195 ` 727` ``` assumes g: "(g ---> 0) F" ``` huffman@44195 ` 728` ``` shows "((\x. f x ** g x) ---> 0) F" ``` huffman@44194 ` 729` ``` using tendsto [OF f g] by (simp add: zero_left) ``` huffman@31355 ` 730` huffman@44194 ` 731` ```lemma (in bounded_bilinear) tendsto_left_zero: ``` huffman@44195 ` 732` ``` "(f ---> 0) F \ ((\x. f x ** c) ---> 0) F" ``` huffman@44194 ` 733` ``` by (rule bounded_linear.tendsto_zero [OF bounded_linear_left]) ``` huffman@44194 ` 734` huffman@44194 ` 735` ```lemma (in bounded_bilinear) tendsto_right_zero: ``` huffman@44195 ` 736` ``` "(f ---> 0) F \ ((\x. c ** f x) ---> 0) F" ``` huffman@44194 ` 737` ``` by (rule bounded_linear.tendsto_zero [OF bounded_linear_right]) ``` huffman@44194 ` 738` huffman@44194 ` 739` ```lemmas tendsto_mult = mult.tendsto ``` huffman@44194 ` 740` huffman@44194 ` 741` ```lemma tendsto_power [tendsto_intros]: ``` huffman@44194 ` 742` ``` fixes f :: "'a \ 'b::{power,real_normed_algebra}" ``` huffman@44195 ` 743` ``` shows "(f ---> a) F \ ((\x. f x ^ n) ---> a ^ n) F" ``` huffman@44194 ` 744` ``` by (induct n) (simp_all add: tendsto_const tendsto_mult) ``` huffman@44194 ` 745` huffman@44194 ` 746` ```lemma tendsto_setprod [tendsto_intros]: ``` huffman@44194 ` 747` ``` fixes f :: "'a \ 'b \ 'c::{real_normed_algebra,comm_ring_1}" ``` huffman@44195 ` 748` ``` assumes "\i. i \ S \ (f i ---> L i) F" ``` huffman@44195 ` 749` ``` shows "((\x. \i\S. f i x) ---> (\i\S. L i)) F" ``` huffman@44194 ` 750` ```proof (cases "finite S") ``` huffman@44194 ` 751` ``` assume "finite S" thus ?thesis using assms ``` huffman@44194 ` 752` ``` by (induct, simp add: tendsto_const, simp add: tendsto_mult) ``` huffman@44194 ` 753` ```next ``` huffman@44194 ` 754` ``` assume "\ finite S" thus ?thesis ``` huffman@44194 ` 755` ``` by (simp add: tendsto_const) ``` huffman@44194 ` 756` ```qed ``` huffman@44194 ` 757` huffman@44194 ` 758` ```subsubsection {* Inverse and division *} ``` huffman@31355 ` 759` huffman@31355 ` 760` ```lemma (in bounded_bilinear) Zfun_prod_Bfun: ``` huffman@44195 ` 761` ``` assumes f: "Zfun f F" ``` huffman@44195 ` 762` ``` assumes g: "Bfun g F" ``` huffman@44195 ` 763` ``` shows "Zfun (\x. f x ** g x) F" ``` huffman@31355 ` 764` ```proof - ``` huffman@31355 ` 765` ``` obtain K where K: "0 \ K" ``` huffman@31355 ` 766` ``` and norm_le: "\x y. norm (x ** y) \ norm x * norm y * K" ``` huffman@31355 ` 767` ``` using nonneg_bounded by fast ``` huffman@31355 ` 768` ``` obtain B where B: "0 < B" ``` huffman@44195 ` 769` ``` and norm_g: "eventually (\x. norm (g x) \ B) F" ``` huffman@31487 ` 770` ``` using g by (rule BfunE) ``` huffman@44195 ` 771` ``` have "eventually (\x. norm (f x ** g x) \ norm (f x) * (B * K)) F" ``` huffman@31487 ` 772` ``` using norm_g proof (rule eventually_elim1) ``` huffman@31487 ` 773` ``` fix x ``` huffman@31487 ` 774` ``` assume *: "norm (g x) \ B" ``` huffman@31487 ` 775` ``` have "norm (f x ** g x) \ norm (f x) * norm (g x) * K" ``` huffman@31355 ` 776` ``` by (rule norm_le) ``` huffman@31487 ` 777` ``` also have "\ \ norm (f x) * B * K" ``` huffman@31487 ` 778` ``` by (intro mult_mono' order_refl norm_g norm_ge_zero ``` huffman@31355 ` 779` ``` mult_nonneg_nonneg K *) ``` huffman@31487 ` 780` ``` also have "\ = norm (f x) * (B * K)" ``` huffman@31355 ` 781` ``` by (rule mult_assoc) ``` huffman@31487 ` 782` ``` finally show "norm (f x ** g x) \ norm (f x) * (B * K)" . ``` huffman@31355 ` 783` ``` qed ``` huffman@31487 ` 784` ``` with f show ?thesis ``` huffman@31487 ` 785` ``` by (rule Zfun_imp_Zfun) ``` huffman@31355 ` 786` ```qed ``` huffman@31355 ` 787` huffman@31355 ` 788` ```lemma (in bounded_bilinear) flip: ``` huffman@31355 ` 789` ``` "bounded_bilinear (\x y. y ** x)" ``` huffman@44081 ` 790` ``` apply default ``` huffman@44081 ` 791` ``` apply (rule add_right) ``` huffman@44081 ` 792` ``` apply (rule add_left) ``` huffman@44081 ` 793` ``` apply (rule scaleR_right) ``` huffman@44081 ` 794` ``` apply (rule scaleR_left) ``` huffman@44081 ` 795` ``` apply (subst mult_commute) ``` huffman@44081 ` 796` ``` using bounded by fast ``` huffman@31355 ` 797` huffman@31355 ` 798` ```lemma (in bounded_bilinear) Bfun_prod_Zfun: ``` huffman@44195 ` 799` ``` assumes f: "Bfun f F" ``` huffman@44195 ` 800` ``` assumes g: "Zfun g F" ``` huffman@44195 ` 801` ``` shows "Zfun (\x. f x ** g x) F" ``` huffman@44081 ` 802` ``` using flip g f by (rule bounded_bilinear.Zfun_prod_Bfun) ``` huffman@31355 ` 803` huffman@31355 ` 804` ```lemma Bfun_inverse_lemma: ``` huffman@31355 ` 805` ``` fixes x :: "'a::real_normed_div_algebra" ``` huffman@31355 ` 806` ``` shows "\r \ norm x; 0 < r\ \ norm (inverse x) \ inverse r" ``` huffman@44081 ` 807` ``` apply (subst nonzero_norm_inverse, clarsimp) ``` huffman@44081 ` 808` ``` apply (erule (1) le_imp_inverse_le) ``` huffman@44081 ` 809` ``` done ``` huffman@31355 ` 810` huffman@31355 ` 811` ```lemma Bfun_inverse: ``` huffman@31355 ` 812` ``` fixes a :: "'a::real_normed_div_algebra" ``` huffman@44195 ` 813` ``` assumes f: "(f ---> a) F" ``` huffman@31355 ` 814` ``` assumes a: "a \ 0" ``` huffman@44195 ` 815` ``` shows "Bfun (\x. inverse (f x)) F" ``` huffman@31355 ` 816` ```proof - ``` huffman@31355 ` 817` ``` from a have "0 < norm a" by simp ``` huffman@31355 ` 818` ``` hence "\r>0. r < norm a" by (rule dense) ``` huffman@31355 ` 819` ``` then obtain r where r1: "0 < r" and r2: "r < norm a" by fast ``` huffman@44195 ` 820` ``` have "eventually (\x. dist (f x) a < r) F" ``` huffman@31487 ` 821` ``` using tendstoD [OF f r1] by fast ``` huffman@44195 ` 822` ``` hence "eventually (\x. norm (inverse (f x)) \ inverse (norm a - r)) F" ``` huffman@31355 ` 823` ``` proof (rule eventually_elim1) ``` huffman@31487 ` 824` ``` fix x ``` huffman@31487 ` 825` ``` assume "dist (f x) a < r" ``` huffman@31487 ` 826` ``` hence 1: "norm (f x - a) < r" ``` huffman@31355 ` 827` ``` by (simp add: dist_norm) ``` huffman@31487 ` 828` ``` hence 2: "f x \ 0" using r2 by auto ``` huffman@31487 ` 829` ``` hence "norm (inverse (f x)) = inverse (norm (f x))" ``` huffman@31355 ` 830` ``` by (rule nonzero_norm_inverse) ``` huffman@31355 ` 831` ``` also have "\ \ inverse (norm a - r)" ``` huffman@31355 ` 832` ``` proof (rule le_imp_inverse_le) ``` huffman@31355 ` 833` ``` show "0 < norm a - r" using r2 by simp ``` huffman@31355 ` 834` ``` next ``` huffman@31487 ` 835` ``` have "norm a - norm (f x) \ norm (a - f x)" ``` huffman@31355 ` 836` ``` by (rule norm_triangle_ineq2) ``` huffman@31487 ` 837` ``` also have "\ = norm (f x - a)" ``` huffman@31355 ` 838` ``` by (rule norm_minus_commute) ``` huffman@31355 ` 839` ``` also have "\ < r" using 1 . ``` huffman@31487 ` 840` ``` finally show "norm a - r \ norm (f x)" by simp ``` huffman@31355 ` 841` ``` qed ``` huffman@31487 ` 842` ``` finally show "norm (inverse (f x)) \ inverse (norm a - r)" . ``` huffman@31355 ` 843` ``` qed ``` huffman@31355 ` 844` ``` thus ?thesis by (rule BfunI) ``` huffman@31355 ` 845` ```qed ``` huffman@31355 ` 846` huffman@31355 ` 847` ```lemma tendsto_inverse_lemma: ``` huffman@31355 ` 848` ``` fixes a :: "'a::real_normed_div_algebra" ``` huffman@44195 ` 849` ``` shows "\(f ---> a) F; a \ 0; eventually (\x. f x \ 0) F\ ``` huffman@44195 ` 850` ``` \ ((\x. inverse (f x)) ---> inverse a) F" ``` huffman@44081 ` 851` ``` apply (subst tendsto_Zfun_iff) ``` huffman@44081 ` 852` ``` apply (rule Zfun_ssubst) ``` huffman@44081 ` 853` ``` apply (erule eventually_elim1) ``` huffman@44081 ` 854` ``` apply (erule (1) inverse_diff_inverse) ``` huffman@44081 ` 855` ``` apply (rule Zfun_minus) ``` huffman@44081 ` 856` ``` apply (rule Zfun_mult_left) ``` huffman@44081 ` 857` ``` apply (rule mult.Bfun_prod_Zfun) ``` huffman@44081 ` 858` ``` apply (erule (1) Bfun_inverse) ``` huffman@44081 ` 859` ``` apply (simp add: tendsto_Zfun_iff) ``` huffman@44081 ` 860` ``` done ``` huffman@31355 ` 861` huffman@31565 ` 862` ```lemma tendsto_inverse [tendsto_intros]: ``` huffman@31355 ` 863` ``` fixes a :: "'a::real_normed_div_algebra" ``` huffman@44195 ` 864` ``` assumes f: "(f ---> a) F" ``` huffman@31355 ` 865` ``` assumes a: "a \ 0" ``` huffman@44195 ` 866` ``` shows "((\x. inverse (f x)) ---> inverse a) F" ``` huffman@31355 ` 867` ```proof - ``` huffman@31355 ` 868` ``` from a have "0 < norm a" by simp ``` huffman@44195 ` 869` ``` with f have "eventually (\x. dist (f x) a < norm a) F" ``` huffman@31355 ` 870` ``` by (rule tendstoD) ``` huffman@44195 ` 871` ``` then have "eventually (\x. f x \ 0) F" ``` huffman@31355 ` 872` ``` unfolding dist_norm by (auto elim!: eventually_elim1) ``` huffman@31487 ` 873` ``` with f a show ?thesis ``` huffman@31355 ` 874` ``` by (rule tendsto_inverse_lemma) ``` huffman@31355 ` 875` ```qed ``` huffman@31355 ` 876` huffman@31565 ` 877` ```lemma tendsto_divide [tendsto_intros]: ``` huffman@31355 ` 878` ``` fixes a b :: "'a::real_normed_field" ``` huffman@44195 ` 879` ``` shows "\(f ---> a) F; (g ---> b) F; b \ 0\ ``` huffman@44195 ` 880` ``` \ ((\x. f x / g x) ---> a / b) F" ``` huffman@44081 ` 881` ``` by (simp add: mult.tendsto tendsto_inverse divide_inverse) ``` huffman@31355 ` 882` huffman@44194 ` 883` ```lemma tendsto_sgn [tendsto_intros]: ``` huffman@44194 ` 884` ``` fixes l :: "'a::real_normed_vector" ``` huffman@44195 ` 885` ``` shows "\(f ---> l) F; l \ 0\ \ ((\x. sgn (f x)) ---> sgn l) F" ``` huffman@44194 ` 886` ``` unfolding sgn_div_norm by (simp add: tendsto_intros) ``` huffman@44194 ` 887` huffman@31349 ` 888` ```end ```