1(* Title: HOL/Filter.thy 2 Author: Brian Huffman 3 Author: Johannes H��lzl 4*) 5 6section \<open>Filters on predicates\<close> 7 8theory Filter 9imports Set_Interval Lifting_Set 10begin 11 12subsection \<open>Filters\<close> 13 14text \<open> 15 This definition also allows non-proper filters. 16\<close> 17 18locale is_filter = 19 fixes F :: "('a \<Rightarrow> bool) \<Rightarrow> bool" 20 assumes True: "F (\<lambda>x. True)" 21 assumes conj: "F (\<lambda>x. P x) \<Longrightarrow> F (\<lambda>x. Q x) \<Longrightarrow> F (\<lambda>x. P x \<and> Q x)" 22 assumes mono: "\<forall>x. P x \<longrightarrow> Q x \<Longrightarrow> F (\<lambda>x. P x) \<Longrightarrow> F (\<lambda>x. Q x)" 23 24typedef 'a filter = "{F :: ('a \<Rightarrow> bool) \<Rightarrow> bool. is_filter F}" 25proof 26 show "(\<lambda>x. True) \<in> ?filter" by (auto intro: is_filter.intro) 27qed 28 29lemma is_filter_Rep_filter: "is_filter (Rep_filter F)" 30 using Rep_filter [of F] by simp 31 32lemma Abs_filter_inverse': 33 assumes "is_filter F" shows "Rep_filter (Abs_filter F) = F" 34 using assms by (simp add: Abs_filter_inverse) 35 36 37subsubsection \<open>Eventually\<close> 38 39definition eventually :: "('a \<Rightarrow> bool) \<Rightarrow> 'a filter \<Rightarrow> bool" 40 where "eventually P F \<longleftrightarrow> Rep_filter F P" 41 42syntax 43 "_eventually" :: "pttrn => 'a filter => bool => bool" ("(3\<forall>\<^sub>F _ in _./ _)" [0, 0, 10] 10) 44translations 45 "\<forall>\<^sub>Fx in F. P" == "CONST eventually (\<lambda>x. P) F" 46 47lemma eventually_Abs_filter: 48 assumes "is_filter F" shows "eventually P (Abs_filter F) = F P" 49 unfolding eventually_def using assms by (simp add: Abs_filter_inverse) 50 51lemma filter_eq_iff: 52 shows "F = F' \<longleftrightarrow> (\<forall>P. eventually P F = eventually P F')" 53 unfolding Rep_filter_inject [symmetric] fun_eq_iff eventually_def .. 54 55lemma eventually_True [simp]: "eventually (\<lambda>x. True) F" 56 unfolding eventually_def 57 by (rule is_filter.True [OF is_filter_Rep_filter]) 58 59lemma always_eventually: "\<forall>x. P x \<Longrightarrow> eventually P F" 60proof - 61 assume "\<forall>x. P x" hence "P = (\<lambda>x. True)" by (simp add: ext) 62 thus "eventually P F" by simp 63qed 64 65lemma eventuallyI: "(\<And>x. P x) \<Longrightarrow> eventually P F" 66 by (auto intro: always_eventually) 67 68lemma eventually_mono: 69 "\<lbrakk>eventually P F; \<And>x. P x \<Longrightarrow> Q x\<rbrakk> \<Longrightarrow> eventually Q F" 70 unfolding eventually_def 71 by (blast intro: is_filter.mono [OF is_filter_Rep_filter]) 72 73lemma eventually_conj: 74 assumes P: "eventually (\<lambda>x. P x) F" 75 assumes Q: "eventually (\<lambda>x. Q x) F" 76 shows "eventually (\<lambda>x. P x \<and> Q x) F" 77 using assms unfolding eventually_def 78 by (rule is_filter.conj [OF is_filter_Rep_filter]) 79 80lemma eventually_mp: 81 assumes "eventually (\<lambda>x. P x \<longrightarrow> Q x) F" 82 assumes "eventually (\<lambda>x. P x) F" 83 shows "eventually (\<lambda>x. Q x) F" 84proof - 85 have "eventually (\<lambda>x. (P x \<longrightarrow> Q x) \<and> P x) F" 86 using assms by (rule eventually_conj) 87 then show ?thesis 88 by (blast intro: eventually_mono) 89qed 90 91lemma eventually_rev_mp: 92 assumes "eventually (\<lambda>x. P x) F" 93 assumes "eventually (\<lambda>x. P x \<longrightarrow> Q x) F" 94 shows "eventually (\<lambda>x. Q x) F" 95using assms(2) assms(1) by (rule eventually_mp) 96 97lemma eventually_conj_iff: 98 "eventually (\<lambda>x. P x \<and> Q x) F \<longleftrightarrow> eventually P F \<and> eventually Q F" 99 by (auto intro: eventually_conj elim: eventually_rev_mp) 100 101lemma eventually_elim2: 102 assumes "eventually (\<lambda>i. P i) F" 103 assumes "eventually (\<lambda>i. Q i) F" 104 assumes "\<And>i. P i \<Longrightarrow> Q i \<Longrightarrow> R i" 105 shows "eventually (\<lambda>i. R i) F" 106 using assms by (auto elim!: eventually_rev_mp) 107 108lemma eventually_ball_finite_distrib: 109 "finite A \<Longrightarrow> (eventually (\<lambda>x. \<forall>y\<in>A. P x y) net) \<longleftrightarrow> (\<forall>y\<in>A. eventually (\<lambda>x. P x y) net)" 110 by (induction A rule: finite_induct) (auto simp: eventually_conj_iff) 111 112lemma eventually_ball_finite: 113 "finite A \<Longrightarrow> \<forall>y\<in>A. eventually (\<lambda>x. P x y) net \<Longrightarrow> eventually (\<lambda>x. \<forall>y\<in>A. P x y) net" 114 by (auto simp: eventually_ball_finite_distrib) 115 116lemma eventually_all_finite: 117 fixes P :: "'a \<Rightarrow> 'b::finite \<Rightarrow> bool" 118 assumes "\<And>y. eventually (\<lambda>x. P x y) net" 119 shows "eventually (\<lambda>x. \<forall>y. P x y) net" 120using eventually_ball_finite [of UNIV P] assms by simp 121 122lemma eventually_ex: "(\<forall>\<^sub>Fx in F. \<exists>y. P x y) \<longleftrightarrow> (\<exists>Y. \<forall>\<^sub>Fx in F. P x (Y x))" 123proof 124 assume "\<forall>\<^sub>Fx in F. \<exists>y. P x y" 125 then have "\<forall>\<^sub>Fx in F. P x (SOME y. P x y)" 126 by (auto intro: someI_ex eventually_mono) 127 then show "\<exists>Y. \<forall>\<^sub>Fx in F. P x (Y x)" 128 by auto 129qed (auto intro: eventually_mono) 130 131lemma not_eventually_impI: "eventually P F \<Longrightarrow> \<not> eventually Q F \<Longrightarrow> \<not> eventually (\<lambda>x. P x \<longrightarrow> Q x) F" 132 by (auto intro: eventually_mp) 133 134lemma not_eventuallyD: "\<not> eventually P F \<Longrightarrow> \<exists>x. \<not> P x" 135 by (metis always_eventually) 136 137lemma eventually_subst: 138 assumes "eventually (\<lambda>n. P n = Q n) F" 139 shows "eventually P F = eventually Q F" (is "?L = ?R") 140proof - 141 from assms have "eventually (\<lambda>x. P x \<longrightarrow> Q x) F" 142 and "eventually (\<lambda>x. Q x \<longrightarrow> P x) F" 143 by (auto elim: eventually_mono) 144 then show ?thesis by (auto elim: eventually_elim2) 145qed 146 147subsection \<open> Frequently as dual to eventually \<close> 148 149definition frequently :: "('a \<Rightarrow> bool) \<Rightarrow> 'a filter \<Rightarrow> bool" 150 where "frequently P F \<longleftrightarrow> \<not> eventually (\<lambda>x. \<not> P x) F" 151 152syntax 153 "_frequently" :: "pttrn \<Rightarrow> 'a filter \<Rightarrow> bool \<Rightarrow> bool" ("(3\<exists>\<^sub>F _ in _./ _)" [0, 0, 10] 10) 154translations 155 "\<exists>\<^sub>Fx in F. P" == "CONST frequently (\<lambda>x. P) F" 156 157lemma not_frequently_False [simp]: "\<not> (\<exists>\<^sub>Fx in F. False)" 158 by (simp add: frequently_def) 159 160lemma frequently_ex: "\<exists>\<^sub>Fx in F. P x \<Longrightarrow> \<exists>x. P x" 161 by (auto simp: frequently_def dest: not_eventuallyD) 162 163lemma frequentlyE: assumes "frequently P F" obtains x where "P x" 164 using frequently_ex[OF assms] by auto 165 166lemma frequently_mp: 167 assumes ev: "\<forall>\<^sub>Fx in F. P x \<longrightarrow> Q x" and P: "\<exists>\<^sub>Fx in F. P x" shows "\<exists>\<^sub>Fx in F. Q x" 168proof - 169 from ev have "eventually (\<lambda>x. \<not> Q x \<longrightarrow> \<not> P x) F" 170 by (rule eventually_rev_mp) (auto intro!: always_eventually) 171 from eventually_mp[OF this] P show ?thesis 172 by (auto simp: frequently_def) 173qed 174 175lemma frequently_rev_mp: 176 assumes "\<exists>\<^sub>Fx in F. P x" 177 assumes "\<forall>\<^sub>Fx in F. P x \<longrightarrow> Q x" 178 shows "\<exists>\<^sub>Fx in F. Q x" 179using assms(2) assms(1) by (rule frequently_mp) 180 181lemma frequently_mono: "(\<forall>x. P x \<longrightarrow> Q x) \<Longrightarrow> frequently P F \<Longrightarrow> frequently Q F" 182 using frequently_mp[of P Q] by (simp add: always_eventually) 183 184lemma frequently_elim1: "\<exists>\<^sub>Fx in F. P x \<Longrightarrow> (\<And>i. P i \<Longrightarrow> Q i) \<Longrightarrow> \<exists>\<^sub>Fx in F. Q x" 185 by (metis frequently_mono) 186 187lemma frequently_disj_iff: "(\<exists>\<^sub>Fx in F. P x \<or> Q x) \<longleftrightarrow> (\<exists>\<^sub>Fx in F. P x) \<or> (\<exists>\<^sub>Fx in F. Q x)" 188 by (simp add: frequently_def eventually_conj_iff) 189 190lemma frequently_disj: "\<exists>\<^sub>Fx in F. P x \<Longrightarrow> \<exists>\<^sub>Fx in F. Q x \<Longrightarrow> \<exists>\<^sub>Fx in F. P x \<or> Q x" 191 by (simp add: frequently_disj_iff) 192 193lemma frequently_bex_finite_distrib: 194 assumes "finite A" shows "(\<exists>\<^sub>Fx in F. \<exists>y\<in>A. P x y) \<longleftrightarrow> (\<exists>y\<in>A. \<exists>\<^sub>Fx in F. P x y)" 195 using assms by induction (auto simp: frequently_disj_iff) 196 197lemma frequently_bex_finite: "finite A \<Longrightarrow> \<exists>\<^sub>Fx in F. \<exists>y\<in>A. P x y \<Longrightarrow> \<exists>y\<in>A. \<exists>\<^sub>Fx in F. P x y" 198 by (simp add: frequently_bex_finite_distrib) 199 200lemma frequently_all: "(\<exists>\<^sub>Fx in F. \<forall>y. P x y) \<longleftrightarrow> (\<forall>Y. \<exists>\<^sub>Fx in F. P x (Y x))" 201 using eventually_ex[of "\<lambda>x y. \<not> P x y" F] by (simp add: frequently_def) 202 203lemma 204 shows not_eventually: "\<not> eventually P F \<longleftrightarrow> (\<exists>\<^sub>Fx in F. \<not> P x)" 205 and not_frequently: "\<not> frequently P F \<longleftrightarrow> (\<forall>\<^sub>Fx in F. \<not> P x)" 206 by (auto simp: frequently_def) 207 208lemma frequently_imp_iff: 209 "(\<exists>\<^sub>Fx in F. P x \<longrightarrow> Q x) \<longleftrightarrow> (eventually P F \<longrightarrow> frequently Q F)" 210 unfolding imp_conv_disj frequently_disj_iff not_eventually[symmetric] .. 211 212lemma eventually_frequently_const_simps: 213 "(\<exists>\<^sub>Fx in F. P x \<and> C) \<longleftrightarrow> (\<exists>\<^sub>Fx in F. P x) \<and> C" 214 "(\<exists>\<^sub>Fx in F. C \<and> P x) \<longleftrightarrow> C \<and> (\<exists>\<^sub>Fx in F. P x)" 215 "(\<forall>\<^sub>Fx in F. P x \<or> C) \<longleftrightarrow> (\<forall>\<^sub>Fx in F. P x) \<or> C" 216 "(\<forall>\<^sub>Fx in F. C \<or> P x) \<longleftrightarrow> C \<or> (\<forall>\<^sub>Fx in F. P x)" 217 "(\<forall>\<^sub>Fx in F. P x \<longrightarrow> C) \<longleftrightarrow> ((\<exists>\<^sub>Fx in F. P x) \<longrightarrow> C)" 218 "(\<forall>\<^sub>Fx in F. C \<longrightarrow> P x) \<longleftrightarrow> (C \<longrightarrow> (\<forall>\<^sub>Fx in F. P x))" 219 by (cases C; simp add: not_frequently)+ 220 221lemmas eventually_frequently_simps = 222 eventually_frequently_const_simps 223 not_eventually 224 eventually_conj_iff 225 eventually_ball_finite_distrib 226 eventually_ex 227 not_frequently 228 frequently_disj_iff 229 frequently_bex_finite_distrib 230 frequently_all 231 frequently_imp_iff 232 233ML \<open> 234 fun eventually_elim_tac facts = 235 CONTEXT_SUBGOAL (fn (goal, i) => fn (ctxt, st) => 236 let 237 val mp_facts = facts RL @{thms eventually_rev_mp} 238 val rule = 239 @{thm eventuallyI} 240 |> fold (fn mp_fact => fn th => th RS mp_fact) mp_facts 241 |> funpow (length facts) (fn th => @{thm impI} RS th) 242 val cases_prop = 243 Thm.prop_of (Rule_Cases.internalize_params (rule RS Goal.init (Thm.cterm_of ctxt goal))) 244 val cases = Rule_Cases.make_common ctxt cases_prop [(("elim", []), [])] 245 in CONTEXT_CASES cases (resolve_tac ctxt [rule] i) (ctxt, st) end) 246\<close> 247 248method_setup eventually_elim = \<open> 249 Scan.succeed (fn _ => CONTEXT_METHOD (fn facts => eventually_elim_tac facts 1)) 250\<close> "elimination of eventually quantifiers" 251 252subsubsection \<open>Finer-than relation\<close> 253 254text \<open>@{term "F \<le> F'"} means that filter @{term F} is finer than 255filter @{term F'}.\<close> 256 257instantiation filter :: (type) complete_lattice 258begin 259 260definition le_filter_def: 261 "F \<le> F' \<longleftrightarrow> (\<forall>P. eventually P F' \<longrightarrow> eventually P F)" 262 263definition 264 "(F :: 'a filter) < F' \<longleftrightarrow> F \<le> F' \<and> \<not> F' \<le> F" 265 266definition 267 "top = Abs_filter (\<lambda>P. \<forall>x. P x)" 268 269definition 270 "bot = Abs_filter (\<lambda>P. True)" 271 272definition 273 "sup F F' = Abs_filter (\<lambda>P. eventually P F \<and> eventually P F')" 274 275definition 276 "inf F F' = Abs_filter 277 (\<lambda>P. \<exists>Q R. eventually Q F \<and> eventually R F' \<and> (\<forall>x. Q x \<and> R x \<longrightarrow> P x))" 278 279definition 280 "Sup S = Abs_filter (\<lambda>P. \<forall>F\<in>S. eventually P F)" 281 282definition 283 "Inf S = Sup {F::'a filter. \<forall>F'\<in>S. F \<le> F'}" 284 285lemma eventually_top [simp]: "eventually P top \<longleftrightarrow> (\<forall>x. P x)" 286 unfolding top_filter_def 287 by (rule eventually_Abs_filter, rule is_filter.intro, auto) 288 289lemma eventually_bot [simp]: "eventually P bot" 290 unfolding bot_filter_def 291 by (subst eventually_Abs_filter, rule is_filter.intro, auto) 292 293lemma eventually_sup: 294 "eventually P (sup F F') \<longleftrightarrow> eventually P F \<and> eventually P F'" 295 unfolding sup_filter_def 296 by (rule eventually_Abs_filter, rule is_filter.intro) 297 (auto elim!: eventually_rev_mp) 298 299lemma eventually_inf: 300 "eventually P (inf F F') \<longleftrightarrow> 301 (\<exists>Q R. eventually Q F \<and> eventually R F' \<and> (\<forall>x. Q x \<and> R x \<longrightarrow> P x))" 302 unfolding inf_filter_def 303 apply (rule eventually_Abs_filter, rule is_filter.intro) 304 apply (fast intro: eventually_True) 305 apply clarify 306 apply (intro exI conjI) 307 apply (erule (1) eventually_conj) 308 apply (erule (1) eventually_conj) 309 apply simp 310 apply auto 311 done 312 313lemma eventually_Sup: 314 "eventually P (Sup S) \<longleftrightarrow> (\<forall>F\<in>S. eventually P F)" 315 unfolding Sup_filter_def 316 apply (rule eventually_Abs_filter, rule is_filter.intro) 317 apply (auto intro: eventually_conj elim!: eventually_rev_mp) 318 done 319 320instance proof 321 fix F F' F'' :: "'a filter" and S :: "'a filter set" 322 { show "F < F' \<longleftrightarrow> F \<le> F' \<and> \<not> F' \<le> F" 323 by (rule less_filter_def) } 324 { show "F \<le> F" 325 unfolding le_filter_def by simp } 326 { assume "F \<le> F'" and "F' \<le> F''" thus "F \<le> F''" 327 unfolding le_filter_def by simp } 328 { assume "F \<le> F'" and "F' \<le> F" thus "F = F'" 329 unfolding le_filter_def filter_eq_iff by fast } 330 { show "inf F F' \<le> F" and "inf F F' \<le> F'" 331 unfolding le_filter_def eventually_inf by (auto intro: eventually_True) } 332 { assume "F \<le> F'" and "F \<le> F''" thus "F \<le> inf F' F''" 333 unfolding le_filter_def eventually_inf 334 by (auto intro: eventually_mono [OF eventually_conj]) } 335 { show "F \<le> sup F F'" and "F' \<le> sup F F'" 336 unfolding le_filter_def eventually_sup by simp_all } 337 { assume "F \<le> F''" and "F' \<le> F''" thus "sup F F' \<le> F''" 338 unfolding le_filter_def eventually_sup by simp } 339 { assume "F'' \<in> S" thus "Inf S \<le> F''" 340 unfolding le_filter_def Inf_filter_def eventually_Sup Ball_def by simp } 341 { assume "\<And>F'. F' \<in> S \<Longrightarrow> F \<le> F'" thus "F \<le> Inf S" 342 unfolding le_filter_def Inf_filter_def eventually_Sup Ball_def by simp } 343 { assume "F \<in> S" thus "F \<le> Sup S" 344 unfolding le_filter_def eventually_Sup by simp } 345 { assume "\<And>F. F \<in> S \<Longrightarrow> F \<le> F'" thus "Sup S \<le> F'" 346 unfolding le_filter_def eventually_Sup by simp } 347 { show "Inf {} = (top::'a filter)" 348 by (auto simp: top_filter_def Inf_filter_def Sup_filter_def) 349 (metis (full_types) top_filter_def always_eventually eventually_top) } 350 { show "Sup {} = (bot::'a filter)" 351 by (auto simp: bot_filter_def Sup_filter_def) } 352qed 353 354end 355 356instance filter :: (type) distrib_lattice 357proof 358 fix F G H :: "'a filter" 359 show "sup F (inf G H) = inf (sup F G) (sup F H)" 360 proof (rule order.antisym) 361 show "inf (sup F G) (sup F H) \<le> sup F (inf G H)" 362 unfolding le_filter_def eventually_sup 363 proof safe 364 fix P assume 1: "eventually P F" and 2: "eventually P (inf G H)" 365 from 2 obtain Q R 366 where QR: "eventually Q G" "eventually R H" "\<And>x. Q x \<Longrightarrow> R x \<Longrightarrow> P x" 367 by (auto simp: eventually_inf) 368 define Q' where "Q' = (\<lambda>x. Q x \<or> P x)" 369 define R' where "R' = (\<lambda>x. R x \<or> P x)" 370 from 1 have "eventually Q' F" 371 by (elim eventually_mono) (auto simp: Q'_def) 372 moreover from 1 have "eventually R' F" 373 by (elim eventually_mono) (auto simp: R'_def) 374 moreover from QR(1) have "eventually Q' G" 375 by (elim eventually_mono) (auto simp: Q'_def) 376 moreover from QR(2) have "eventually R' H" 377 by (elim eventually_mono)(auto simp: R'_def) 378 moreover from QR have "P x" if "Q' x" "R' x" for x 379 using that by (auto simp: Q'_def R'_def) 380 ultimately show "eventually P (inf (sup F G) (sup F H))" 381 by (auto simp: eventually_inf eventually_sup) 382 qed 383 qed (auto intro: inf.coboundedI1 inf.coboundedI2) 384qed 385 386 387lemma filter_leD: 388 "F \<le> F' \<Longrightarrow> eventually P F' \<Longrightarrow> eventually P F" 389 unfolding le_filter_def by simp 390 391lemma filter_leI: 392 "(\<And>P. eventually P F' \<Longrightarrow> eventually P F) \<Longrightarrow> F \<le> F'" 393 unfolding le_filter_def by simp 394 395lemma eventually_False: 396 "eventually (\<lambda>x. False) F \<longleftrightarrow> F = bot" 397 unfolding filter_eq_iff by (auto elim: eventually_rev_mp) 398 399lemma eventually_frequently: "F \<noteq> bot \<Longrightarrow> eventually P F \<Longrightarrow> frequently P F" 400 using eventually_conj[of P F "\<lambda>x. \<not> P x"] 401 by (auto simp add: frequently_def eventually_False) 402 403lemma eventually_frequentlyE: 404 assumes "eventually P F" 405 assumes "eventually (\<lambda>x. \<not> P x \<or> Q x) F" "F\<noteq>bot" 406 shows "frequently Q F" 407proof - 408 have "eventually Q F" 409 using eventually_conj[OF assms(1,2),simplified] by (auto elim:eventually_mono) 410 then show ?thesis using eventually_frequently[OF \<open>F\<noteq>bot\<close>] by auto 411qed 412 413lemma eventually_const_iff: "eventually (\<lambda>x. P) F \<longleftrightarrow> P \<or> F = bot" 414 by (cases P) (auto simp: eventually_False) 415 416lemma eventually_const[simp]: "F \<noteq> bot \<Longrightarrow> eventually (\<lambda>x. P) F \<longleftrightarrow> P" 417 by (simp add: eventually_const_iff) 418 419lemma frequently_const_iff: "frequently (\<lambda>x. P) F \<longleftrightarrow> P \<and> F \<noteq> bot" 420 by (simp add: frequently_def eventually_const_iff) 421 422lemma frequently_const[simp]: "F \<noteq> bot \<Longrightarrow> frequently (\<lambda>x. P) F \<longleftrightarrow> P" 423 by (simp add: frequently_const_iff) 424 425lemma eventually_happens: "eventually P net \<Longrightarrow> net = bot \<or> (\<exists>x. P x)" 426 by (metis frequentlyE eventually_frequently) 427 428lemma eventually_happens': 429 assumes "F \<noteq> bot" "eventually P F" 430 shows "\<exists>x. P x" 431 using assms eventually_frequently frequentlyE by blast 432 433abbreviation (input) trivial_limit :: "'a filter \<Rightarrow> bool" 434 where "trivial_limit F \<equiv> F = bot" 435 436lemma trivial_limit_def: "trivial_limit F \<longleftrightarrow> eventually (\<lambda>x. False) F" 437 by (rule eventually_False [symmetric]) 438 439lemma False_imp_not_eventually: "(\<forall>x. \<not> P x ) \<Longrightarrow> \<not> trivial_limit net \<Longrightarrow> \<not> eventually (\<lambda>x. P x) net" 440 by (simp add: eventually_False) 441 442lemma eventually_Inf: "eventually P (Inf B) \<longleftrightarrow> (\<exists>X\<subseteq>B. finite X \<and> eventually P (Inf X))" 443proof - 444 let ?F = "\<lambda>P. \<exists>X\<subseteq>B. finite X \<and> eventually P (Inf X)" 445 446 { fix P have "eventually P (Abs_filter ?F) \<longleftrightarrow> ?F P" 447 proof (rule eventually_Abs_filter is_filter.intro)+ 448 show "?F (\<lambda>x. True)" 449 by (rule exI[of _ "{}"]) (simp add: le_fun_def) 450 next 451 fix P Q 452 assume "?F P" then guess X .. 453 moreover 454 assume "?F Q" then guess Y .. 455 ultimately show "?F (\<lambda>x. P x \<and> Q x)" 456 by (intro exI[of _ "X \<union> Y"]) 457 (auto simp: Inf_union_distrib eventually_inf) 458 next 459 fix P Q 460 assume "?F P" then guess X .. 461 moreover assume "\<forall>x. P x \<longrightarrow> Q x" 462 ultimately show "?F Q" 463 by (intro exI[of _ X]) (auto elim: eventually_mono) 464 qed } 465 note eventually_F = this 466 467 have "Inf B = Abs_filter ?F" 468 proof (intro antisym Inf_greatest) 469 show "Inf B \<le> Abs_filter ?F" 470 by (auto simp: le_filter_def eventually_F dest: Inf_superset_mono) 471 next 472 fix F assume "F \<in> B" then show "Abs_filter ?F \<le> F" 473 by (auto simp add: le_filter_def eventually_F intro!: exI[of _ "{F}"]) 474 qed 475 then show ?thesis 476 by (simp add: eventually_F) 477qed 478 479lemma eventually_INF: "eventually P (\<Sqinter>b\<in>B. F b) \<longleftrightarrow> (\<exists>X\<subseteq>B. finite X \<and> eventually P (\<Sqinter>b\<in>X. F b))" 480 unfolding eventually_Inf [of P "F`B"] 481 by (metis finite_imageI image_mono finite_subset_image) 482 483lemma Inf_filter_not_bot: 484 fixes B :: "'a filter set" 485 shows "(\<And>X. X \<subseteq> B \<Longrightarrow> finite X \<Longrightarrow> Inf X \<noteq> bot) \<Longrightarrow> Inf B \<noteq> bot" 486 unfolding trivial_limit_def eventually_Inf[of _ B] 487 bot_bool_def [symmetric] bot_fun_def [symmetric] bot_unique by simp 488 489lemma INF_filter_not_bot: 490 fixes F :: "'i \<Rightarrow> 'a filter" 491 shows "(\<And>X. X \<subseteq> B \<Longrightarrow> finite X \<Longrightarrow> (\<Sqinter>b\<in>X. F b) \<noteq> bot) \<Longrightarrow> (\<Sqinter>b\<in>B. F b) \<noteq> bot" 492 unfolding trivial_limit_def eventually_INF [of _ _ B] 493 bot_bool_def [symmetric] bot_fun_def [symmetric] bot_unique by simp 494 495lemma eventually_Inf_base: 496 assumes "B \<noteq> {}" and base: "\<And>F G. F \<in> B \<Longrightarrow> G \<in> B \<Longrightarrow> \<exists>x\<in>B. x \<le> inf F G" 497 shows "eventually P (Inf B) \<longleftrightarrow> (\<exists>b\<in>B. eventually P b)" 498proof (subst eventually_Inf, safe) 499 fix X assume "finite X" "X \<subseteq> B" 500 then have "\<exists>b\<in>B. \<forall>x\<in>X. b \<le> x" 501 proof induct 502 case empty then show ?case 503 using \<open>B \<noteq> {}\<close> by auto 504 next 505 case (insert x X) 506 then obtain b where "b \<in> B" "\<And>x. x \<in> X \<Longrightarrow> b \<le> x" 507 by auto 508 with \<open>insert x X \<subseteq> B\<close> base[of b x] show ?case 509 by (auto intro: order_trans) 510 qed 511 then obtain b where "b \<in> B" "b \<le> Inf X" 512 by (auto simp: le_Inf_iff) 513 then show "eventually P (Inf X) \<Longrightarrow> Bex B (eventually P)" 514 by (intro bexI[of _ b]) (auto simp: le_filter_def) 515qed (auto intro!: exI[of _ "{x}" for x]) 516 517lemma eventually_INF_base: 518 "B \<noteq> {} \<Longrightarrow> (\<And>a b. a \<in> B \<Longrightarrow> b \<in> B \<Longrightarrow> \<exists>x\<in>B. F x \<le> inf (F a) (F b)) \<Longrightarrow> 519 eventually P (\<Sqinter>b\<in>B. F b) \<longleftrightarrow> (\<exists>b\<in>B. eventually P (F b))" 520 by (subst eventually_Inf_base) auto 521 522lemma eventually_INF1: "i \<in> I \<Longrightarrow> eventually P (F i) \<Longrightarrow> eventually P (\<Sqinter>i\<in>I. F i)" 523 using filter_leD[OF INF_lower] . 524 525subsubsection \<open>Map function for filters\<close> 526 527definition filtermap :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a filter \<Rightarrow> 'b filter" 528 where "filtermap f F = Abs_filter (\<lambda>P. eventually (\<lambda>x. P (f x)) F)" 529 530lemma eventually_filtermap: 531 "eventually P (filtermap f F) = eventually (\<lambda>x. P (f x)) F" 532 unfolding filtermap_def 533 apply (rule eventually_Abs_filter) 534 apply (rule is_filter.intro) 535 apply (auto elim!: eventually_rev_mp) 536 done 537 538lemma filtermap_ident: "filtermap (\<lambda>x. x) F = F" 539 by (simp add: filter_eq_iff eventually_filtermap) 540 541lemma filtermap_filtermap: 542 "filtermap f (filtermap g F) = filtermap (\<lambda>x. f (g x)) F" 543 by (simp add: filter_eq_iff eventually_filtermap) 544 545lemma filtermap_mono: "F \<le> F' \<Longrightarrow> filtermap f F \<le> filtermap f F'" 546 unfolding le_filter_def eventually_filtermap by simp 547 548lemma filtermap_bot [simp]: "filtermap f bot = bot" 549 by (simp add: filter_eq_iff eventually_filtermap) 550 551lemma filtermap_bot_iff: "filtermap f F = bot \<longleftrightarrow> F = bot" 552 by (simp add: trivial_limit_def eventually_filtermap) 553 554lemma filtermap_sup: "filtermap f (sup F1 F2) = sup (filtermap f F1) (filtermap f F2)" 555 by (simp add: filter_eq_iff eventually_filtermap eventually_sup) 556 557lemma filtermap_SUP: "filtermap f (\<Squnion>b\<in>B. F b) = (\<Squnion>b\<in>B. filtermap f (F b))" 558 by (simp add: filter_eq_iff eventually_Sup eventually_filtermap) 559 560lemma filtermap_inf: "filtermap f (inf F1 F2) \<le> inf (filtermap f F1) (filtermap f F2)" 561 by (intro inf_greatest filtermap_mono inf_sup_ord) 562 563lemma filtermap_INF: "filtermap f (\<Sqinter>b\<in>B. F b) \<le> (\<Sqinter>b\<in>B. filtermap f (F b))" 564 by (rule INF_greatest, rule filtermap_mono, erule INF_lower) 565 566 567subsubsection \<open>Contravariant map function for filters\<close> 568 569definition filtercomap :: "('a \<Rightarrow> 'b) \<Rightarrow> 'b filter \<Rightarrow> 'a filter" where 570 "filtercomap f F = Abs_filter (\<lambda>P. \<exists>Q. eventually Q F \<and> (\<forall>x. Q (f x) \<longrightarrow> P x))" 571 572lemma eventually_filtercomap: 573 "eventually P (filtercomap f F) \<longleftrightarrow> (\<exists>Q. eventually Q F \<and> (\<forall>x. Q (f x) \<longrightarrow> P x))" 574 unfolding filtercomap_def 575proof (intro eventually_Abs_filter, unfold_locales, goal_cases) 576 case 1 577 show ?case by (auto intro!: exI[of _ "\<lambda>_. True"]) 578next 579 case (2 P Q) 580 from 2(1) guess P' by (elim exE conjE) note P' = this 581 from 2(2) guess Q' by (elim exE conjE) note Q' = this 582 show ?case 583 by (rule exI[of _ "\<lambda>x. P' x \<and> Q' x"]) 584 (insert P' Q', auto intro!: eventually_conj) 585next 586 case (3 P Q) 587 thus ?case by blast 588qed 589 590lemma filtercomap_ident: "filtercomap (\<lambda>x. x) F = F" 591 by (auto simp: filter_eq_iff eventually_filtercomap elim!: eventually_mono) 592 593lemma filtercomap_filtercomap: "filtercomap f (filtercomap g F) = filtercomap (\<lambda>x. g (f x)) F" 594 unfolding filter_eq_iff by (auto simp: eventually_filtercomap) 595 596lemma filtercomap_mono: "F \<le> F' \<Longrightarrow> filtercomap f F \<le> filtercomap f F'" 597 by (auto simp: eventually_filtercomap le_filter_def) 598 599lemma filtercomap_bot [simp]: "filtercomap f bot = bot" 600 by (auto simp: filter_eq_iff eventually_filtercomap) 601 602lemma filtercomap_top [simp]: "filtercomap f top = top" 603 by (auto simp: filter_eq_iff eventually_filtercomap) 604 605lemma filtercomap_inf: "filtercomap f (inf F1 F2) = inf (filtercomap f F1) (filtercomap f F2)" 606 unfolding filter_eq_iff 607proof safe 608 fix P 609 assume "eventually P (filtercomap f (F1 \<sqinter> F2))" 610 then obtain Q R S where *: 611 "eventually Q F1" "eventually R F2" "\<And>x. Q x \<Longrightarrow> R x \<Longrightarrow> S x" "\<And>x. S (f x) \<Longrightarrow> P x" 612 unfolding eventually_filtercomap eventually_inf by blast 613 from * have "eventually (\<lambda>x. Q (f x)) (filtercomap f F1)" 614 "eventually (\<lambda>x. R (f x)) (filtercomap f F2)" 615 by (auto simp: eventually_filtercomap) 616 with * show "eventually P (filtercomap f F1 \<sqinter> filtercomap f F2)" 617 unfolding eventually_inf by blast 618next 619 fix P 620 assume "eventually P (inf (filtercomap f F1) (filtercomap f F2))" 621 then obtain Q Q' R R' where *: 622 "eventually Q F1" "eventually R F2" "\<And>x. Q (f x) \<Longrightarrow> Q' x" "\<And>x. R (f x) \<Longrightarrow> R' x" 623 "\<And>x. Q' x \<Longrightarrow> R' x \<Longrightarrow> P x" 624 unfolding eventually_filtercomap eventually_inf by blast 625 from * have "eventually (\<lambda>x. Q x \<and> R x) (F1 \<sqinter> F2)" by (auto simp: eventually_inf) 626 with * show "eventually P (filtercomap f (F1 \<sqinter> F2))" 627 by (auto simp: eventually_filtercomap) 628qed 629 630lemma filtercomap_sup: "filtercomap f (sup F1 F2) \<ge> sup (filtercomap f F1) (filtercomap f F2)" 631 by (intro sup_least filtercomap_mono inf_sup_ord) 632 633lemma filtercomap_INF: "filtercomap f (\<Sqinter>b\<in>B. F b) = (\<Sqinter>b\<in>B. filtercomap f (F b))" 634proof - 635 have *: "filtercomap f (\<Sqinter>b\<in>B. F b) = (\<Sqinter>b\<in>B. filtercomap f (F b))" if "finite B" for B 636 using that by induction (simp_all add: filtercomap_inf) 637 show ?thesis unfolding filter_eq_iff 638 proof 639 fix P 640 have "eventually P (\<Sqinter>b\<in>B. filtercomap f (F b)) \<longleftrightarrow> 641 (\<exists>X. (X \<subseteq> B \<and> finite X) \<and> eventually P (\<Sqinter>b\<in>X. filtercomap f (F b)))" 642 by (subst eventually_INF) blast 643 also have "\<dots> \<longleftrightarrow> (\<exists>X. (X \<subseteq> B \<and> finite X) \<and> eventually P (filtercomap f (\<Sqinter>b\<in>X. F b)))" 644 by (rule ex_cong) (simp add: *) 645 also have "\<dots> \<longleftrightarrow> eventually P (filtercomap f (INFIMUM B F))" 646 unfolding eventually_filtercomap by (subst eventually_INF) blast 647 finally show "eventually P (filtercomap f (INFIMUM B F)) = 648 eventually P (\<Sqinter>b\<in>B. filtercomap f (F b))" .. 649 qed 650qed 651 652lemma filtercomap_SUP: 653 "filtercomap f (\<Squnion>b\<in>B. F b) \<ge> (\<Squnion>b\<in>B. filtercomap f (F b))" 654 by (intro SUP_least filtercomap_mono SUP_upper) 655 656lemma eventually_filtercomapI [intro]: 657 assumes "eventually P F" 658 shows "eventually (\<lambda>x. P (f x)) (filtercomap f F)" 659 using assms by (auto simp: eventually_filtercomap) 660 661lemma filtermap_filtercomap: "filtermap f (filtercomap f F) \<le> F" 662 by (auto simp: le_filter_def eventually_filtermap eventually_filtercomap) 663 664lemma filtercomap_filtermap: "filtercomap f (filtermap f F) \<ge> F" 665 unfolding le_filter_def eventually_filtermap eventually_filtercomap 666 by (auto elim!: eventually_mono) 667 668 669subsubsection \<open>Standard filters\<close> 670 671definition principal :: "'a set \<Rightarrow> 'a filter" where 672 "principal S = Abs_filter (\<lambda>P. \<forall>x\<in>S. P x)" 673 674lemma eventually_principal: "eventually P (principal S) \<longleftrightarrow> (\<forall>x\<in>S. P x)" 675 unfolding principal_def 676 by (rule eventually_Abs_filter, rule is_filter.intro) auto 677 678lemma eventually_inf_principal: "eventually P (inf F (principal s)) \<longleftrightarrow> eventually (\<lambda>x. x \<in> s \<longrightarrow> P x) F" 679 unfolding eventually_inf eventually_principal by (auto elim: eventually_mono) 680 681lemma principal_UNIV[simp]: "principal UNIV = top" 682 by (auto simp: filter_eq_iff eventually_principal) 683 684lemma principal_empty[simp]: "principal {} = bot" 685 by (auto simp: filter_eq_iff eventually_principal) 686 687lemma principal_eq_bot_iff: "principal X = bot \<longleftrightarrow> X = {}" 688 by (auto simp add: filter_eq_iff eventually_principal) 689 690lemma principal_le_iff[iff]: "principal A \<le> principal B \<longleftrightarrow> A \<subseteq> B" 691 by (auto simp: le_filter_def eventually_principal) 692 693lemma le_principal: "F \<le> principal A \<longleftrightarrow> eventually (\<lambda>x. x \<in> A) F" 694 unfolding le_filter_def eventually_principal 695 apply safe 696 apply (erule_tac x="\<lambda>x. x \<in> A" in allE) 697 apply (auto elim: eventually_mono) 698 done 699 700lemma principal_inject[iff]: "principal A = principal B \<longleftrightarrow> A = B" 701 unfolding eq_iff by simp 702 703lemma sup_principal[simp]: "sup (principal A) (principal B) = principal (A \<union> B)" 704 unfolding filter_eq_iff eventually_sup eventually_principal by auto 705 706lemma inf_principal[simp]: "inf (principal A) (principal B) = principal (A \<inter> B)" 707 unfolding filter_eq_iff eventually_inf eventually_principal 708 by (auto intro: exI[of _ "\<lambda>x. x \<in> A"] exI[of _ "\<lambda>x. x \<in> B"]) 709 710lemma SUP_principal[simp]: "(\<Squnion>i\<in>I. principal (A i)) = principal (\<Union>i\<in>I. A i)" 711 unfolding filter_eq_iff eventually_Sup by (auto simp: eventually_principal) 712 713lemma INF_principal_finite: "finite X \<Longrightarrow> (\<Sqinter>x\<in>X. principal (f x)) = principal (\<Inter>x\<in>X. f x)" 714 by (induct X rule: finite_induct) auto 715 716lemma filtermap_principal[simp]: "filtermap f (principal A) = principal (f ` A)" 717 unfolding filter_eq_iff eventually_filtermap eventually_principal by simp 718 719lemma filtercomap_principal[simp]: "filtercomap f (principal A) = principal (f -` A)" 720 unfolding filter_eq_iff eventually_filtercomap eventually_principal by fast 721 722subsubsection \<open>Order filters\<close> 723 724definition at_top :: "('a::order) filter" 725 where "at_top = (\<Sqinter>k. principal {k ..})" 726 727lemma at_top_sub: "at_top = (\<Sqinter>k\<in>{c::'a::linorder..}. principal {k ..})" 728 by (auto intro!: INF_eq max.cobounded1 max.cobounded2 simp: at_top_def) 729 730lemma eventually_at_top_linorder: "eventually P at_top \<longleftrightarrow> (\<exists>N::'a::linorder. \<forall>n\<ge>N. P n)" 731 unfolding at_top_def 732 by (subst eventually_INF_base) (auto simp: eventually_principal intro: max.cobounded1 max.cobounded2) 733 734lemma eventually_filtercomap_at_top_linorder: 735 "eventually P (filtercomap f at_top) \<longleftrightarrow> (\<exists>N::'a::linorder. \<forall>x. f x \<ge> N \<longrightarrow> P x)" 736 by (auto simp: eventually_filtercomap eventually_at_top_linorder) 737 738lemma eventually_at_top_linorderI: 739 fixes c::"'a::linorder" 740 assumes "\<And>x. c \<le> x \<Longrightarrow> P x" 741 shows "eventually P at_top" 742 using assms by (auto simp: eventually_at_top_linorder) 743 744lemma eventually_ge_at_top [simp]: 745 "eventually (\<lambda>x. (c::_::linorder) \<le> x) at_top" 746 unfolding eventually_at_top_linorder by auto 747 748lemma eventually_at_top_dense: "eventually P at_top \<longleftrightarrow> (\<exists>N::'a::{no_top, linorder}. \<forall>n>N. P n)" 749proof - 750 have "eventually P (\<Sqinter>k. principal {k <..}) \<longleftrightarrow> (\<exists>N::'a. \<forall>n>N. P n)" 751 by (subst eventually_INF_base) (auto simp: eventually_principal intro: max.cobounded1 max.cobounded2) 752 also have "(\<Sqinter>k. principal {k::'a <..}) = at_top" 753 unfolding at_top_def 754 by (intro INF_eq) (auto intro: less_imp_le simp: Ici_subset_Ioi_iff gt_ex) 755 finally show ?thesis . 756qed 757 758lemma eventually_filtercomap_at_top_dense: 759 "eventually P (filtercomap f at_top) \<longleftrightarrow> (\<exists>N::'a::{no_top, linorder}. \<forall>x. f x > N \<longrightarrow> P x)" 760 by (auto simp: eventually_filtercomap eventually_at_top_dense) 761 762lemma eventually_at_top_not_equal [simp]: "eventually (\<lambda>x::'a::{no_top, linorder}. x \<noteq> c) at_top" 763 unfolding eventually_at_top_dense by auto 764 765lemma eventually_gt_at_top [simp]: "eventually (\<lambda>x. (c::_::{no_top, linorder}) < x) at_top" 766 unfolding eventually_at_top_dense by auto 767 768lemma eventually_all_ge_at_top: 769 assumes "eventually P (at_top :: ('a :: linorder) filter)" 770 shows "eventually (\<lambda>x. \<forall>y\<ge>x. P y) at_top" 771proof - 772 from assms obtain x where "\<And>y. y \<ge> x \<Longrightarrow> P y" by (auto simp: eventually_at_top_linorder) 773 hence "\<forall>z\<ge>y. P z" if "y \<ge> x" for y using that by simp 774 thus ?thesis by (auto simp: eventually_at_top_linorder) 775qed 776 777definition at_bot :: "('a::order) filter" 778 where "at_bot = (\<Sqinter>k. principal {.. k})" 779 780lemma at_bot_sub: "at_bot = (\<Sqinter>k\<in>{.. c::'a::linorder}. principal {.. k})" 781 by (auto intro!: INF_eq min.cobounded1 min.cobounded2 simp: at_bot_def) 782 783lemma eventually_at_bot_linorder: 784 fixes P :: "'a::linorder \<Rightarrow> bool" shows "eventually P at_bot \<longleftrightarrow> (\<exists>N. \<forall>n\<le>N. P n)" 785 unfolding at_bot_def 786 by (subst eventually_INF_base) (auto simp: eventually_principal intro: min.cobounded1 min.cobounded2) 787 788lemma eventually_filtercomap_at_bot_linorder: 789 "eventually P (filtercomap f at_bot) \<longleftrightarrow> (\<exists>N::'a::linorder. \<forall>x. f x \<le> N \<longrightarrow> P x)" 790 by (auto simp: eventually_filtercomap eventually_at_bot_linorder) 791 792lemma eventually_le_at_bot [simp]: 793 "eventually (\<lambda>x. x \<le> (c::_::linorder)) at_bot" 794 unfolding eventually_at_bot_linorder by auto 795 796lemma eventually_at_bot_dense: "eventually P at_bot \<longleftrightarrow> (\<exists>N::'a::{no_bot, linorder}. \<forall>n<N. P n)" 797proof - 798 have "eventually P (\<Sqinter>k. principal {..< k}) \<longleftrightarrow> (\<exists>N::'a. \<forall>n<N. P n)" 799 by (subst eventually_INF_base) (auto simp: eventually_principal intro: min.cobounded1 min.cobounded2) 800 also have "(\<Sqinter>k. principal {..< k::'a}) = at_bot" 801 unfolding at_bot_def 802 by (intro INF_eq) (auto intro: less_imp_le simp: Iic_subset_Iio_iff lt_ex) 803 finally show ?thesis . 804qed 805 806lemma eventually_filtercomap_at_bot_dense: 807 "eventually P (filtercomap f at_bot) \<longleftrightarrow> (\<exists>N::'a::{no_bot, linorder}. \<forall>x. f x < N \<longrightarrow> P x)" 808 by (auto simp: eventually_filtercomap eventually_at_bot_dense) 809 810lemma eventually_at_bot_not_equal [simp]: "eventually (\<lambda>x::'a::{no_bot, linorder}. x \<noteq> c) at_bot" 811 unfolding eventually_at_bot_dense by auto 812 813lemma eventually_gt_at_bot [simp]: 814 "eventually (\<lambda>x. x < (c::_::unbounded_dense_linorder)) at_bot" 815 unfolding eventually_at_bot_dense by auto 816 817lemma trivial_limit_at_bot_linorder [simp]: "\<not> trivial_limit (at_bot ::('a::linorder) filter)" 818 unfolding trivial_limit_def 819 by (metis eventually_at_bot_linorder order_refl) 820 821lemma trivial_limit_at_top_linorder [simp]: "\<not> trivial_limit (at_top ::('a::linorder) filter)" 822 unfolding trivial_limit_def 823 by (metis eventually_at_top_linorder order_refl) 824 825subsection \<open>Sequentially\<close> 826 827abbreviation sequentially :: "nat filter" 828 where "sequentially \<equiv> at_top" 829 830lemma eventually_sequentially: 831 "eventually P sequentially \<longleftrightarrow> (\<exists>N. \<forall>n\<ge>N. P n)" 832 by (rule eventually_at_top_linorder) 833 834lemma sequentially_bot [simp, intro]: "sequentially \<noteq> bot" 835 unfolding filter_eq_iff eventually_sequentially by auto 836 837lemmas trivial_limit_sequentially = sequentially_bot 838 839lemma eventually_False_sequentially [simp]: 840 "\<not> eventually (\<lambda>n. False) sequentially" 841 by (simp add: eventually_False) 842 843lemma le_sequentially: 844 "F \<le> sequentially \<longleftrightarrow> (\<forall>N. eventually (\<lambda>n. N \<le> n) F)" 845 by (simp add: at_top_def le_INF_iff le_principal) 846 847lemma eventually_sequentiallyI [intro?]: 848 assumes "\<And>x. c \<le> x \<Longrightarrow> P x" 849 shows "eventually P sequentially" 850using assms by (auto simp: eventually_sequentially) 851 852lemma eventually_sequentially_Suc [simp]: "eventually (\<lambda>i. P (Suc i)) sequentially \<longleftrightarrow> eventually P sequentially" 853 unfolding eventually_sequentially by (metis Suc_le_D Suc_le_mono le_Suc_eq) 854 855lemma eventually_sequentially_seg [simp]: "eventually (\<lambda>n. P (n + k)) sequentially \<longleftrightarrow> eventually P sequentially" 856 using eventually_sequentially_Suc[of "\<lambda>n. P (n + k)" for k] by (induction k) auto 857 858lemma filtermap_sequentually_ne_bot: "filtermap f sequentially \<noteq> bot" 859 by (simp add: filtermap_bot_iff) 860 861subsection \<open>The cofinite filter\<close> 862 863definition "cofinite = Abs_filter (\<lambda>P. finite {x. \<not> P x})" 864 865abbreviation Inf_many :: "('a \<Rightarrow> bool) \<Rightarrow> bool" (binder "\<exists>\<^sub>\<infinity>" 10) 866 where "Inf_many P \<equiv> frequently P cofinite" 867 868abbreviation Alm_all :: "('a \<Rightarrow> bool) \<Rightarrow> bool" (binder "\<forall>\<^sub>\<infinity>" 10) 869 where "Alm_all P \<equiv> eventually P cofinite" 870 871notation (ASCII) 872 Inf_many (binder "INFM " 10) and 873 Alm_all (binder "MOST " 10) 874 875lemma eventually_cofinite: "eventually P cofinite \<longleftrightarrow> finite {x. \<not> P x}" 876 unfolding cofinite_def 877proof (rule eventually_Abs_filter, rule is_filter.intro) 878 fix P Q :: "'a \<Rightarrow> bool" assume "finite {x. \<not> P x}" "finite {x. \<not> Q x}" 879 from finite_UnI[OF this] show "finite {x. \<not> (P x \<and> Q x)}" 880 by (rule rev_finite_subset) auto 881next 882 fix P Q :: "'a \<Rightarrow> bool" assume P: "finite {x. \<not> P x}" and *: "\<forall>x. P x \<longrightarrow> Q x" 883 from * show "finite {x. \<not> Q x}" 884 by (intro finite_subset[OF _ P]) auto 885qed simp 886 887lemma frequently_cofinite: "frequently P cofinite \<longleftrightarrow> \<not> finite {x. P x}" 888 by (simp add: frequently_def eventually_cofinite) 889 890lemma cofinite_bot[simp]: "cofinite = (bot::'a filter) \<longleftrightarrow> finite (UNIV :: 'a set)" 891 unfolding trivial_limit_def eventually_cofinite by simp 892 893lemma cofinite_eq_sequentially: "cofinite = sequentially" 894 unfolding filter_eq_iff eventually_sequentially eventually_cofinite 895proof safe 896 fix P :: "nat \<Rightarrow> bool" assume [simp]: "finite {x. \<not> P x}" 897 show "\<exists>N. \<forall>n\<ge>N. P n" 898 proof cases 899 assume "{x. \<not> P x} \<noteq> {}" then show ?thesis 900 by (intro exI[of _ "Suc (Max {x. \<not> P x})"]) (auto simp: Suc_le_eq) 901 qed auto 902next 903 fix P :: "nat \<Rightarrow> bool" and N :: nat assume "\<forall>n\<ge>N. P n" 904 then have "{x. \<not> P x} \<subseteq> {..< N}" 905 by (auto simp: not_le) 906 then show "finite {x. \<not> P x}" 907 by (blast intro: finite_subset) 908qed 909 910subsubsection \<open>Product of filters\<close> 911 912definition prod_filter :: "'a filter \<Rightarrow> 'b filter \<Rightarrow> ('a \<times> 'b) filter" (infixr "\<times>\<^sub>F" 80) where 913 "prod_filter F G = 914 (\<Sqinter>(P, Q)\<in>{(P, Q). eventually P F \<and> eventually Q G}. principal {(x, y). P x \<and> Q y})" 915 916lemma eventually_prod_filter: "eventually P (F \<times>\<^sub>F G) \<longleftrightarrow> 917 (\<exists>Pf Pg. eventually Pf F \<and> eventually Pg G \<and> (\<forall>x y. Pf x \<longrightarrow> Pg y \<longrightarrow> P (x, y)))" 918 unfolding prod_filter_def 919proof (subst eventually_INF_base, goal_cases) 920 case 2 921 moreover have "eventually Pf F \<Longrightarrow> eventually Qf F \<Longrightarrow> eventually Pg G \<Longrightarrow> eventually Qg G \<Longrightarrow> 922 \<exists>P Q. eventually P F \<and> eventually Q G \<and> 923 Collect P \<times> Collect Q \<subseteq> Collect Pf \<times> Collect Pg \<inter> Collect Qf \<times> Collect Qg" for Pf Pg Qf Qg 924 by (intro conjI exI[of _ "inf Pf Qf"] exI[of _ "inf Pg Qg"]) 925 (auto simp: inf_fun_def eventually_conj) 926 ultimately show ?case 927 by auto 928qed (auto simp: eventually_principal intro: eventually_True) 929 930lemma eventually_prod1: 931 assumes "B \<noteq> bot" 932 shows "(\<forall>\<^sub>F (x, y) in A \<times>\<^sub>F B. P x) \<longleftrightarrow> (\<forall>\<^sub>F x in A. P x)" 933 unfolding eventually_prod_filter 934proof safe 935 fix R Q 936 assume *: "\<forall>\<^sub>F x in A. R x" "\<forall>\<^sub>F x in B. Q x" "\<forall>x y. R x \<longrightarrow> Q y \<longrightarrow> P x" 937 with \<open>B \<noteq> bot\<close> obtain y where "Q y" by (auto dest: eventually_happens) 938 with * show "eventually P A" 939 by (force elim: eventually_mono) 940next 941 assume "eventually P A" 942 then show "\<exists>Pf Pg. eventually Pf A \<and> eventually Pg B \<and> (\<forall>x y. Pf x \<longrightarrow> Pg y \<longrightarrow> P x)" 943 by (intro exI[of _ P] exI[of _ "\<lambda>x. True"]) auto 944qed 945 946lemma eventually_prod2: 947 assumes "A \<noteq> bot" 948 shows "(\<forall>\<^sub>F (x, y) in A \<times>\<^sub>F B. P y) \<longleftrightarrow> (\<forall>\<^sub>F y in B. P y)" 949 unfolding eventually_prod_filter 950proof safe 951 fix R Q 952 assume *: "\<forall>\<^sub>F x in A. R x" "\<forall>\<^sub>F x in B. Q x" "\<forall>x y. R x \<longrightarrow> Q y \<longrightarrow> P y" 953 with \<open>A \<noteq> bot\<close> obtain x where "R x" by (auto dest: eventually_happens) 954 with * show "eventually P B" 955 by (force elim: eventually_mono) 956next 957 assume "eventually P B" 958 then show "\<exists>Pf Pg. eventually Pf A \<and> eventually Pg B \<and> (\<forall>x y. Pf x \<longrightarrow> Pg y \<longrightarrow> P y)" 959 by (intro exI[of _ P] exI[of _ "\<lambda>x. True"]) auto 960qed 961 962lemma INF_filter_bot_base: 963 fixes F :: "'a \<Rightarrow> 'b filter" 964 assumes *: "\<And>i j. i \<in> I \<Longrightarrow> j \<in> I \<Longrightarrow> \<exists>k\<in>I. F k \<le> F i \<sqinter> F j" 965 shows "(\<Sqinter>i\<in>I. F i) = bot \<longleftrightarrow> (\<exists>i\<in>I. F i = bot)" 966proof (cases "\<exists>i\<in>I. F i = bot") 967 case True 968 then have "(\<Sqinter>i\<in>I. F i) \<le> bot" 969 by (auto intro: INF_lower2) 970 with True show ?thesis 971 by (auto simp: bot_unique) 972next 973 case False 974 moreover have "(\<Sqinter>i\<in>I. F i) \<noteq> bot" 975 proof (cases "I = {}") 976 case True 977 then show ?thesis 978 by (auto simp add: filter_eq_iff) 979 next 980 case False': False 981 show ?thesis 982 proof (rule INF_filter_not_bot) 983 fix J 984 assume "finite J" "J \<subseteq> I" 985 then have "\<exists>k\<in>I. F k \<le> (\<Sqinter>i\<in>J. F i)" 986 proof (induct J) 987 case empty 988 then show ?case 989 using \<open>I \<noteq> {}\<close> by auto 990 next 991 case (insert i J) 992 then obtain k where "k \<in> I" "F k \<le> (\<Sqinter>i\<in>J. F i)" by auto 993 with insert *[of i k] show ?case 994 by auto 995 qed 996 with False show "(\<Sqinter>i\<in>J. F i) \<noteq> \<bottom>" 997 by (auto simp: bot_unique) 998 qed 999 qed 1000 ultimately show ?thesis 1001 by auto 1002qed 1003 1004lemma Collect_empty_eq_bot: "Collect P = {} \<longleftrightarrow> P = \<bottom>" 1005 by auto 1006 1007lemma prod_filter_eq_bot: "A \<times>\<^sub>F B = bot \<longleftrightarrow> A = bot \<or> B = bot" 1008 unfolding trivial_limit_def 1009proof 1010 assume "\<forall>\<^sub>F x in A \<times>\<^sub>F B. False" 1011 then obtain Pf Pg 1012 where Pf: "eventually (\<lambda>x. Pf x) A" and Pg: "eventually (\<lambda>y. Pg y) B" 1013 and *: "\<forall>x y. Pf x \<longrightarrow> Pg y \<longrightarrow> False" 1014 unfolding eventually_prod_filter by fast 1015 from * have "(\<forall>x. \<not> Pf x) \<or> (\<forall>y. \<not> Pg y)" by fast 1016 with Pf Pg show "(\<forall>\<^sub>F x in A. False) \<or> (\<forall>\<^sub>F x in B. False)" by auto 1017next 1018 assume "(\<forall>\<^sub>F x in A. False) \<or> (\<forall>\<^sub>F x in B. False)" 1019 then show "\<forall>\<^sub>F x in A \<times>\<^sub>F B. False" 1020 unfolding eventually_prod_filter by (force intro: eventually_True) 1021qed 1022 1023lemma prod_filter_mono: "F \<le> F' \<Longrightarrow> G \<le> G' \<Longrightarrow> F \<times>\<^sub>F G \<le> F' \<times>\<^sub>F G'" 1024 by (auto simp: le_filter_def eventually_prod_filter) 1025 1026lemma prod_filter_mono_iff: 1027 assumes nAB: "A \<noteq> bot" "B \<noteq> bot" 1028 shows "A \<times>\<^sub>F B \<le> C \<times>\<^sub>F D \<longleftrightarrow> A \<le> C \<and> B \<le> D" 1029proof safe 1030 assume *: "A \<times>\<^sub>F B \<le> C \<times>\<^sub>F D" 1031 with assms have "A \<times>\<^sub>F B \<noteq> bot" 1032 by (auto simp: bot_unique prod_filter_eq_bot) 1033 with * have "C \<times>\<^sub>F D \<noteq> bot" 1034 by (auto simp: bot_unique) 1035 then have nCD: "C \<noteq> bot" "D \<noteq> bot" 1036 by (auto simp: prod_filter_eq_bot) 1037 1038 show "A \<le> C" 1039 proof (rule filter_leI) 1040 fix P assume "eventually P C" with *[THEN filter_leD, of "\<lambda>(x, y). P x"] show "eventually P A" 1041 using nAB nCD by (simp add: eventually_prod1 eventually_prod2) 1042 qed 1043 1044 show "B \<le> D" 1045 proof (rule filter_leI) 1046 fix P assume "eventually P D" with *[THEN filter_leD, of "\<lambda>(x, y). P y"] show "eventually P B" 1047 using nAB nCD by (simp add: eventually_prod1 eventually_prod2) 1048 qed 1049qed (intro prod_filter_mono) 1050 1051lemma eventually_prod_same: "eventually P (F \<times>\<^sub>F F) \<longleftrightarrow> 1052 (\<exists>Q. eventually Q F \<and> (\<forall>x y. Q x \<longrightarrow> Q y \<longrightarrow> P (x, y)))" 1053 unfolding eventually_prod_filter 1054 apply safe 1055 apply (rule_tac x="inf Pf Pg" in exI) 1056 apply (auto simp: inf_fun_def intro!: eventually_conj) 1057 done 1058 1059lemma eventually_prod_sequentially: 1060 "eventually P (sequentially \<times>\<^sub>F sequentially) \<longleftrightarrow> (\<exists>N. \<forall>m \<ge> N. \<forall>n \<ge> N. P (n, m))" 1061 unfolding eventually_prod_same eventually_sequentially by auto 1062 1063lemma principal_prod_principal: "principal A \<times>\<^sub>F principal B = principal (A \<times> B)" 1064 unfolding filter_eq_iff eventually_prod_filter eventually_principal 1065 by (fast intro: exI[of _ "\<lambda>x. x \<in> A"] exI[of _ "\<lambda>x. x \<in> B"]) 1066 1067lemma le_prod_filterI: 1068 "filtermap fst F \<le> A \<Longrightarrow> filtermap snd F \<le> B \<Longrightarrow> F \<le> A \<times>\<^sub>F B" 1069 unfolding le_filter_def eventually_filtermap eventually_prod_filter 1070 by (force elim: eventually_elim2) 1071 1072lemma filtermap_fst_prod_filter: "filtermap fst (A \<times>\<^sub>F B) \<le> A" 1073 unfolding le_filter_def eventually_filtermap eventually_prod_filter 1074 by (force intro: eventually_True) 1075 1076lemma filtermap_snd_prod_filter: "filtermap snd (A \<times>\<^sub>F B) \<le> B" 1077 unfolding le_filter_def eventually_filtermap eventually_prod_filter 1078 by (force intro: eventually_True) 1079 1080lemma prod_filter_INF: 1081 assumes "I \<noteq> {}" and "J \<noteq> {}" 1082 shows "(\<Sqinter>i\<in>I. A i) \<times>\<^sub>F (\<Sqinter>j\<in>J. B j) = (\<Sqinter>i\<in>I. \<Sqinter>j\<in>J. A i \<times>\<^sub>F B j)" 1083proof (rule antisym) 1084 from \<open>I \<noteq> {}\<close> obtain i where "i \<in> I" by auto 1085 from \<open>J \<noteq> {}\<close> obtain j where "j \<in> J" by auto 1086 1087 show "(\<Sqinter>i\<in>I. \<Sqinter>j\<in>J. A i \<times>\<^sub>F B j) \<le> (\<Sqinter>i\<in>I. A i) \<times>\<^sub>F (\<Sqinter>j\<in>J. B j)" 1088 by (fast intro: le_prod_filterI INF_greatest INF_lower2 1089 order_trans[OF filtermap_INF] `i \<in> I` `j \<in> J` 1090 filtermap_fst_prod_filter filtermap_snd_prod_filter) 1091 show "(\<Sqinter>i\<in>I. A i) \<times>\<^sub>F (\<Sqinter>j\<in>J. B j) \<le> (\<Sqinter>i\<in>I. \<Sqinter>j\<in>J. A i \<times>\<^sub>F B j)" 1092 by (intro INF_greatest prod_filter_mono INF_lower) 1093qed 1094 1095lemma filtermap_Pair: "filtermap (\<lambda>x. (f x, g x)) F \<le> filtermap f F \<times>\<^sub>F filtermap g F" 1096 by (rule le_prod_filterI, simp_all add: filtermap_filtermap) 1097 1098lemma eventually_prodI: "eventually P F \<Longrightarrow> eventually Q G \<Longrightarrow> eventually (\<lambda>x. P (fst x) \<and> Q (snd x)) (F \<times>\<^sub>F G)" 1099 unfolding eventually_prod_filter by auto 1100 1101lemma prod_filter_INF1: "I \<noteq> {} \<Longrightarrow> (\<Sqinter>i\<in>I. A i) \<times>\<^sub>F B = (\<Sqinter>i\<in>I. A i \<times>\<^sub>F B)" 1102 using prod_filter_INF[of I "{B}" A "\<lambda>x. x"] by simp 1103 1104lemma prod_filter_INF2: "J \<noteq> {} \<Longrightarrow> A \<times>\<^sub>F (\<Sqinter>i\<in>J. B i) = (\<Sqinter>i\<in>J. A \<times>\<^sub>F B i)" 1105 using prod_filter_INF[of "{A}" J "\<lambda>x. x" B] by simp 1106 1107lemma prod_filtermap1: "prod_filter (filtermap f F) G = filtermap (apfst f) (prod_filter F G)" 1108 apply(clarsimp simp add: filter_eq_iff eventually_filtermap eventually_prod_filter; safe) 1109 subgoal by auto 1110 subgoal for P Q R by(rule exI[where x="\<lambda>y. \<exists>x. y = f x \<and> Q x"])(auto intro: eventually_mono) 1111 done 1112 1113lemma prod_filtermap2: "prod_filter F (filtermap g G) = filtermap (apsnd g) (prod_filter F G)" 1114 apply(clarsimp simp add: filter_eq_iff eventually_filtermap eventually_prod_filter; safe) 1115 subgoal by auto 1116 subgoal for P Q R by(auto intro: exI[where x="\<lambda>y. \<exists>x. y = g x \<and> R x"] eventually_mono) 1117 done 1118 1119lemma prod_filter_assoc: 1120 "prod_filter (prod_filter F G) H = filtermap (\<lambda>(x, y, z). ((x, y), z)) (prod_filter F (prod_filter G H))" 1121 apply(clarsimp simp add: filter_eq_iff eventually_filtermap eventually_prod_filter; safe) 1122 subgoal for P Q R S T by(auto 4 4 intro: exI[where x="\<lambda>(a, b). T a \<and> S b"]) 1123 subgoal for P Q R S T by(auto 4 3 intro: exI[where x="\<lambda>(a, b). Q a \<and> S b"]) 1124 done 1125 1126lemma prod_filter_principal_singleton: "prod_filter (principal {x}) F = filtermap (Pair x) F" 1127 by(fastforce simp add: filter_eq_iff eventually_prod_filter eventually_principal eventually_filtermap elim: eventually_mono intro: exI[where x="\<lambda>a. a = x"]) 1128 1129lemma prod_filter_principal_singleton2: "prod_filter F (principal {x}) = filtermap (\<lambda>a. (a, x)) F" 1130 by(fastforce simp add: filter_eq_iff eventually_prod_filter eventually_principal eventually_filtermap elim: eventually_mono intro: exI[where x="\<lambda>a. a = x"]) 1131 1132lemma prod_filter_commute: "prod_filter F G = filtermap prod.swap (prod_filter G F)" 1133 by(auto simp add: filter_eq_iff eventually_prod_filter eventually_filtermap) 1134 1135subsection \<open>Limits\<close> 1136 1137definition filterlim :: "('a \<Rightarrow> 'b) \<Rightarrow> 'b filter \<Rightarrow> 'a filter \<Rightarrow> bool" where 1138 "filterlim f F2 F1 \<longleftrightarrow> filtermap f F1 \<le> F2" 1139 1140syntax 1141 "_LIM" :: "pttrns \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> bool" ("(3LIM (_)/ (_)./ (_) :> (_))" [1000, 10, 0, 10] 10) 1142 1143translations 1144 "LIM x F1. f :> F2" == "CONST filterlim (\<lambda>x. f) F2 F1" 1145 1146lemma filterlim_top [simp]: "filterlim f top F" 1147 by (simp add: filterlim_def) 1148 1149lemma filterlim_iff: 1150 "(LIM x F1. f x :> F2) \<longleftrightarrow> (\<forall>P. eventually P F2 \<longrightarrow> eventually (\<lambda>x. P (f x)) F1)" 1151 unfolding filterlim_def le_filter_def eventually_filtermap .. 1152 1153lemma filterlim_compose: 1154 "filterlim g F3 F2 \<Longrightarrow> filterlim f F2 F1 \<Longrightarrow> filterlim (\<lambda>x. g (f x)) F3 F1" 1155 unfolding filterlim_def filtermap_filtermap[symmetric] by (metis filtermap_mono order_trans) 1156 1157lemma filterlim_mono: 1158 "filterlim f F2 F1 \<Longrightarrow> F2 \<le> F2' \<Longrightarrow> F1' \<le> F1 \<Longrightarrow> filterlim f F2' F1'" 1159 unfolding filterlim_def by (metis filtermap_mono order_trans) 1160 1161lemma filterlim_ident: "LIM x F. x :> F" 1162 by (simp add: filterlim_def filtermap_ident) 1163 1164lemma filterlim_cong: 1165 "F1 = F1' \<Longrightarrow> F2 = F2' \<Longrightarrow> eventually (\<lambda>x. f x = g x) F2 \<Longrightarrow> filterlim f F1 F2 = filterlim g F1' F2'" 1166 by (auto simp: filterlim_def le_filter_def eventually_filtermap elim: eventually_elim2) 1167 1168lemma filterlim_mono_eventually: 1169 assumes "filterlim f F G" and ord: "F \<le> F'" "G' \<le> G" 1170 assumes eq: "eventually (\<lambda>x. f x = f' x) G'" 1171 shows "filterlim f' F' G'" 1172 apply (rule filterlim_cong[OF refl refl eq, THEN iffD1]) 1173 apply (rule filterlim_mono[OF _ ord]) 1174 apply fact 1175 done 1176 1177lemma filtermap_mono_strong: "inj f \<Longrightarrow> filtermap f F \<le> filtermap f G \<longleftrightarrow> F \<le> G" 1178 apply (safe intro!: filtermap_mono) 1179 apply (auto simp: le_filter_def eventually_filtermap) 1180 apply (erule_tac x="\<lambda>x. P (inv f x)" in allE) 1181 apply auto 1182 done 1183 1184lemma eventually_compose_filterlim: 1185 assumes "eventually P F" "filterlim f F G" 1186 shows "eventually (\<lambda>x. P (f x)) G" 1187 using assms by (simp add: filterlim_iff) 1188 1189lemma filtermap_eq_strong: "inj f \<Longrightarrow> filtermap f F = filtermap f G \<longleftrightarrow> F = G" 1190 by (simp add: filtermap_mono_strong eq_iff) 1191 1192lemma filtermap_fun_inverse: 1193 assumes g: "filterlim g F G" 1194 assumes f: "filterlim f G F" 1195 assumes ev: "eventually (\<lambda>x. f (g x) = x) G" 1196 shows "filtermap f F = G" 1197proof (rule antisym) 1198 show "filtermap f F \<le> G" 1199 using f unfolding filterlim_def . 1200 have "G = filtermap f (filtermap g G)" 1201 using ev by (auto elim: eventually_elim2 simp: filter_eq_iff eventually_filtermap) 1202 also have "\<dots> \<le> filtermap f F" 1203 using g by (intro filtermap_mono) (simp add: filterlim_def) 1204 finally show "G \<le> filtermap f F" . 1205qed 1206 1207lemma filterlim_principal: 1208 "(LIM x F. f x :> principal S) \<longleftrightarrow> (eventually (\<lambda>x. f x \<in> S) F)" 1209 unfolding filterlim_def eventually_filtermap le_principal .. 1210 1211lemma filterlim_inf: 1212 "(LIM x F1. f x :> inf F2 F3) \<longleftrightarrow> ((LIM x F1. f x :> F2) \<and> (LIM x F1. f x :> F3))" 1213 unfolding filterlim_def by simp 1214 1215lemma filterlim_INF: 1216 "(LIM x F. f x :> (\<Sqinter>b\<in>B. G b)) \<longleftrightarrow> (\<forall>b\<in>B. LIM x F. f x :> G b)" 1217 unfolding filterlim_def le_INF_iff .. 1218 1219lemma filterlim_INF_INF: 1220 "(\<And>m. m \<in> J \<Longrightarrow> \<exists>i\<in>I. filtermap f (F i) \<le> G m) \<Longrightarrow> LIM x (\<Sqinter>i\<in>I. F i). f x :> (\<Sqinter>j\<in>J. G j)" 1221 unfolding filterlim_def by (rule order_trans[OF filtermap_INF INF_mono]) 1222 1223lemma filterlim_base: 1224 "(\<And>m x. m \<in> J \<Longrightarrow> i m \<in> I) \<Longrightarrow> (\<And>m x. m \<in> J \<Longrightarrow> x \<in> F (i m) \<Longrightarrow> f x \<in> G m) \<Longrightarrow> 1225 LIM x (\<Sqinter>i\<in>I. principal (F i)). f x :> (\<Sqinter>j\<in>J. principal (G j))" 1226 by (force intro!: filterlim_INF_INF simp: image_subset_iff) 1227 1228lemma filterlim_base_iff: 1229 assumes "I \<noteq> {}" and chain: "\<And>i j. i \<in> I \<Longrightarrow> j \<in> I \<Longrightarrow> F i \<subseteq> F j \<or> F j \<subseteq> F i" 1230 shows "(LIM x (\<Sqinter>i\<in>I. principal (F i)). f x :> \<Sqinter>j\<in>J. principal (G j)) \<longleftrightarrow> 1231 (\<forall>j\<in>J. \<exists>i\<in>I. \<forall>x\<in>F i. f x \<in> G j)" 1232 unfolding filterlim_INF filterlim_principal 1233proof (subst eventually_INF_base) 1234 fix i j assume "i \<in> I" "j \<in> I" 1235 with chain[OF this] show "\<exists>x\<in>I. principal (F x) \<le> inf (principal (F i)) (principal (F j))" 1236 by auto 1237qed (auto simp: eventually_principal \<open>I \<noteq> {}\<close>) 1238 1239lemma filterlim_filtermap: "filterlim f F1 (filtermap g F2) = filterlim (\<lambda>x. f (g x)) F1 F2" 1240 unfolding filterlim_def filtermap_filtermap .. 1241 1242lemma filterlim_sup: 1243 "filterlim f F F1 \<Longrightarrow> filterlim f F F2 \<Longrightarrow> filterlim f F (sup F1 F2)" 1244 unfolding filterlim_def filtermap_sup by auto 1245 1246lemma filterlim_sequentially_Suc: 1247 "(LIM x sequentially. f (Suc x) :> F) \<longleftrightarrow> (LIM x sequentially. f x :> F)" 1248 unfolding filterlim_iff by (subst eventually_sequentially_Suc) simp 1249 1250lemma filterlim_Suc: "filterlim Suc sequentially sequentially" 1251 by (simp add: filterlim_iff eventually_sequentially) 1252 1253lemma filterlim_If: 1254 "LIM x inf F (principal {x. P x}). f x :> G \<Longrightarrow> 1255 LIM x inf F (principal {x. \<not> P x}). g x :> G \<Longrightarrow> 1256 LIM x F. if P x then f x else g x :> G" 1257 unfolding filterlim_iff eventually_inf_principal by (auto simp: eventually_conj_iff) 1258 1259lemma filterlim_Pair: 1260 "LIM x F. f x :> G \<Longrightarrow> LIM x F. g x :> H \<Longrightarrow> LIM x F. (f x, g x) :> G \<times>\<^sub>F H" 1261 unfolding filterlim_def 1262 by (rule order_trans[OF filtermap_Pair prod_filter_mono]) 1263 1264subsection \<open>Limits to @{const at_top} and @{const at_bot}\<close> 1265 1266lemma filterlim_at_top: 1267 fixes f :: "'a \<Rightarrow> ('b::linorder)" 1268 shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. Z \<le> f x) F)" 1269 by (auto simp: filterlim_iff eventually_at_top_linorder elim!: eventually_mono) 1270 1271lemma filterlim_at_top_mono: 1272 "LIM x F. f x :> at_top \<Longrightarrow> eventually (\<lambda>x. f x \<le> (g x::'a::linorder)) F \<Longrightarrow> 1273 LIM x F. g x :> at_top" 1274 by (auto simp: filterlim_at_top elim: eventually_elim2 intro: order_trans) 1275 1276lemma filterlim_at_top_dense: 1277 fixes f :: "'a \<Rightarrow> ('b::unbounded_dense_linorder)" 1278 shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. Z < f x) F)" 1279 by (metis eventually_mono[of _ F] eventually_gt_at_top order_less_imp_le 1280 filterlim_at_top[of f F] filterlim_iff[of f at_top F]) 1281 1282lemma filterlim_at_top_ge: 1283 fixes f :: "'a \<Rightarrow> ('b::linorder)" and c :: "'b" 1284 shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z\<ge>c. eventually (\<lambda>x. Z \<le> f x) F)" 1285 unfolding at_top_sub[of c] filterlim_INF by (auto simp add: filterlim_principal) 1286 1287lemma filterlim_at_top_at_top: 1288 fixes f :: "'a::linorder \<Rightarrow> 'b::linorder" 1289 assumes mono: "\<And>x y. Q x \<Longrightarrow> Q y \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y" 1290 assumes bij: "\<And>x. P x \<Longrightarrow> f (g x) = x" "\<And>x. P x \<Longrightarrow> Q (g x)" 1291 assumes Q: "eventually Q at_top" 1292 assumes P: "eventually P at_top" 1293 shows "filterlim f at_top at_top" 1294proof - 1295 from P obtain x where x: "\<And>y. x \<le> y \<Longrightarrow> P y" 1296 unfolding eventually_at_top_linorder by auto 1297 show ?thesis 1298 proof (intro filterlim_at_top_ge[THEN iffD2] allI impI) 1299 fix z assume "x \<le> z" 1300 with x have "P z" by auto 1301 have "eventually (\<lambda>x. g z \<le> x) at_top" 1302 by (rule eventually_ge_at_top) 1303 with Q show "eventually (\<lambda>x. z \<le> f x) at_top" 1304 by eventually_elim (metis mono bij \<open>P z\<close>) 1305 qed 1306qed 1307 1308lemma filterlim_at_top_gt: 1309 fixes f :: "'a \<Rightarrow> ('b::unbounded_dense_linorder)" and c :: "'b" 1310 shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z>c. eventually (\<lambda>x. Z \<le> f x) F)" 1311 by (metis filterlim_at_top order_less_le_trans gt_ex filterlim_at_top_ge) 1312 1313lemma filterlim_at_bot: 1314 fixes f :: "'a \<Rightarrow> ('b::linorder)" 1315 shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. f x \<le> Z) F)" 1316 by (auto simp: filterlim_iff eventually_at_bot_linorder elim!: eventually_mono) 1317 1318lemma filterlim_at_bot_dense: 1319 fixes f :: "'a \<Rightarrow> ('b::{dense_linorder, no_bot})" 1320 shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. f x < Z) F)" 1321proof (auto simp add: filterlim_at_bot[of f F]) 1322 fix Z :: 'b 1323 from lt_ex [of Z] obtain Z' where 1: "Z' < Z" .. 1324 assume "\<forall>Z. eventually (\<lambda>x. f x \<le> Z) F" 1325 hence "eventually (\<lambda>x. f x \<le> Z') F" by auto 1326 thus "eventually (\<lambda>x. f x < Z) F" 1327 apply (rule eventually_mono) 1328 using 1 by auto 1329 next 1330 fix Z :: 'b 1331 show "\<forall>Z. eventually (\<lambda>x. f x < Z) F \<Longrightarrow> eventually (\<lambda>x. f x \<le> Z) F" 1332 by (drule spec [of _ Z], erule eventually_mono, auto simp add: less_imp_le) 1333qed 1334 1335lemma filterlim_at_bot_le: 1336 fixes f :: "'a \<Rightarrow> ('b::linorder)" and c :: "'b" 1337 shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z\<le>c. eventually (\<lambda>x. Z \<ge> f x) F)" 1338 unfolding filterlim_at_bot 1339proof safe 1340 fix Z assume *: "\<forall>Z\<le>c. eventually (\<lambda>x. Z \<ge> f x) F" 1341 with *[THEN spec, of "min Z c"] show "eventually (\<lambda>x. Z \<ge> f x) F" 1342 by (auto elim!: eventually_mono) 1343qed simp 1344 1345lemma filterlim_at_bot_lt: 1346 fixes f :: "'a \<Rightarrow> ('b::unbounded_dense_linorder)" and c :: "'b" 1347 shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z<c. eventually (\<lambda>x. Z \<ge> f x) F)" 1348 by (metis filterlim_at_bot filterlim_at_bot_le lt_ex order_le_less_trans) 1349 1350lemma filterlim_filtercomap [intro]: "filterlim f F (filtercomap f F)" 1351 unfolding filterlim_def by (rule filtermap_filtercomap) 1352 1353 1354subsection \<open>Setup @{typ "'a filter"} for lifting and transfer\<close> 1355 1356lemma filtermap_id [simp, id_simps]: "filtermap id = id" 1357by(simp add: fun_eq_iff id_def filtermap_ident) 1358 1359lemma filtermap_id' [simp]: "filtermap (\<lambda>x. x) = (\<lambda>F. F)" 1360using filtermap_id unfolding id_def . 1361 1362context includes lifting_syntax 1363begin 1364 1365definition map_filter_on :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a filter \<Rightarrow> 'b filter" where 1366 "map_filter_on X f F = Abs_filter (\<lambda>P. eventually (\<lambda>x. P (f x) \<and> x \<in> X) F)" 1367 1368lemma is_filter_map_filter_on: 1369 "is_filter (\<lambda>P. \<forall>\<^sub>F x in F. P (f x) \<and> x \<in> X) \<longleftrightarrow> eventually (\<lambda>x. x \<in> X) F" 1370proof(rule iffI; unfold_locales) 1371 show "\<forall>\<^sub>F x in F. True \<and> x \<in> X" if "eventually (\<lambda>x. x \<in> X) F" using that by simp 1372 show "\<forall>\<^sub>F x in F. (P (f x) \<and> Q (f x)) \<and> x \<in> X" if "\<forall>\<^sub>F x in F. P (f x) \<and> x \<in> X" "\<forall>\<^sub>F x in F. Q (f x) \<and> x \<in> X" for P Q 1373 using eventually_conj[OF that] by(auto simp add: conj_ac cong: conj_cong) 1374 show "\<forall>\<^sub>F x in F. Q (f x) \<and> x \<in> X" if "\<forall>x. P x \<longrightarrow> Q x" "\<forall>\<^sub>F x in F. P (f x) \<and> x \<in> X" for P Q 1375 using that(2) by(rule eventually_mono)(use that(1) in auto) 1376 show "eventually (\<lambda>x. x \<in> X) F" if "is_filter (\<lambda>P. \<forall>\<^sub>F x in F. P (f x) \<and> x \<in> X)" 1377 using is_filter.True[OF that] by simp 1378qed 1379 1380lemma eventually_map_filter_on: "eventually P (map_filter_on X f F) = (\<forall>\<^sub>F x in F. P (f x) \<and> x \<in> X)" 1381 if "eventually (\<lambda>x. x \<in> X) F" 1382 by(simp add: is_filter_map_filter_on map_filter_on_def eventually_Abs_filter that) 1383 1384lemma map_filter_on_UNIV: "map_filter_on UNIV = filtermap" 1385 by(simp add: map_filter_on_def filtermap_def fun_eq_iff) 1386 1387lemma map_filter_on_comp: "map_filter_on X f (map_filter_on Y g F) = map_filter_on Y (f \<circ> g) F" 1388 if "g ` Y \<subseteq> X" and "eventually (\<lambda>x. x \<in> Y) F" 1389 unfolding map_filter_on_def using that(1) 1390 by(auto simp add: eventually_Abs_filter that(2) is_filter_map_filter_on intro!: arg_cong[where f=Abs_filter] arg_cong2[where f=eventually]) 1391 1392inductive rel_filter :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a filter \<Rightarrow> 'b filter \<Rightarrow> bool" for R F G where 1393 "rel_filter R F G" if "eventually (case_prod R) Z" "map_filter_on {(x, y). R x y} fst Z = F" "map_filter_on {(x, y). R x y} snd Z = G" 1394 1395lemma rel_filter_eq [relator_eq]: "rel_filter (=) = (=)" 1396proof(intro ext iffI)+ 1397 show "F = G" if "rel_filter (=) F G" for F G using that 1398 by cases(clarsimp simp add: filter_eq_iff eventually_map_filter_on split_def cong: rev_conj_cong) 1399 show "rel_filter (=) F G" if "F = G" for F G unfolding \<open>F = G\<close> 1400 proof 1401 let ?Z = "map_filter_on UNIV (\<lambda>x. (x, x)) G" 1402 have [simp]: "range (\<lambda>x. (x, x)) \<subseteq> {(x, y). x = y}" by auto 1403 show "map_filter_on {(x, y). x = y} fst ?Z = G" and "map_filter_on {(x, y). x = y} snd ?Z = G" 1404 by(simp_all add: map_filter_on_comp)(simp_all add: map_filter_on_UNIV o_def) 1405 show "\<forall>\<^sub>F (x, y) in ?Z. x = y" by(simp add: eventually_map_filter_on) 1406 qed 1407qed 1408 1409lemma rel_filter_mono [relator_mono]: "rel_filter A \<le> rel_filter B" if le: "A \<le> B" 1410proof(clarify elim!: rel_filter.cases) 1411 show "rel_filter B (map_filter_on {(x, y). A x y} fst Z) (map_filter_on {(x, y). A x y} snd Z)" 1412 (is "rel_filter _ ?X ?Y") if "\<forall>\<^sub>F (x, y) in Z. A x y" for Z 1413 proof 1414 let ?Z = "map_filter_on {(x, y). A x y} id Z" 1415 show "\<forall>\<^sub>F (x, y) in ?Z. B x y" using le that 1416 by(simp add: eventually_map_filter_on le_fun_def split_def conj_commute cong: conj_cong) 1417 have [simp]: "{(x, y). A x y} \<subseteq> {(x, y). B x y}" using le by auto 1418 show "map_filter_on {(x, y). B x y} fst ?Z = ?X" "map_filter_on {(x, y). B x y} snd ?Z = ?Y" 1419 using le that by(simp_all add: le_fun_def map_filter_on_comp) 1420 qed 1421qed 1422 1423lemma rel_filter_conversep: "rel_filter A\<inverse>\<inverse> = (rel_filter A)\<inverse>\<inverse>" 1424proof(safe intro!: ext elim!: rel_filter.cases) 1425 show *: "rel_filter A (map_filter_on {(x, y). A\<inverse>\<inverse> x y} snd Z) (map_filter_on {(x, y). A\<inverse>\<inverse> x y} fst Z)" 1426 (is "rel_filter _ ?X ?Y") if "\<forall>\<^sub>F (x, y) in Z. A\<inverse>\<inverse> x y" for A Z 1427 proof 1428 let ?Z = "map_filter_on {(x, y). A y x} prod.swap Z" 1429 show "\<forall>\<^sub>F (x, y) in ?Z. A x y" using that by(simp add: eventually_map_filter_on) 1430 have [simp]: "prod.swap ` {(x, y). A y x} \<subseteq> {(x, y). A x y}" by auto 1431 show "map_filter_on {(x, y). A x y} fst ?Z = ?X" "map_filter_on {(x, y). A x y} snd ?Z = ?Y" 1432 using that by(simp_all add: map_filter_on_comp o_def) 1433 qed 1434 show "rel_filter A\<inverse>\<inverse> (map_filter_on {(x, y). A x y} snd Z) (map_filter_on {(x, y). A x y} fst Z)" 1435 if "\<forall>\<^sub>F (x, y) in Z. A x y" for Z using *[of "A\<inverse>\<inverse>" Z] that by simp 1436qed 1437 1438lemma rel_filter_distr [relator_distr]: 1439 "rel_filter A OO rel_filter B = rel_filter (A OO B)" 1440proof(safe intro!: ext elim!: rel_filter.cases) 1441 let ?AB = "{(x, y). (A OO B) x y}" 1442 show "(rel_filter A OO rel_filter B) 1443 (map_filter_on {(x, y). (A OO B) x y} fst Z) (map_filter_on {(x, y). (A OO B) x y} snd Z)" 1444 (is "(_ OO _) ?F ?H") if "\<forall>\<^sub>F (x, y) in Z. (A OO B) x y" for Z 1445 proof 1446 let ?G = "map_filter_on ?AB (\<lambda>(x, y). SOME z. A x z \<and> B z y) Z" 1447 show "rel_filter A ?F ?G" 1448 proof 1449 let ?Z = "map_filter_on ?AB (\<lambda>(x, y). (x, SOME z. A x z \<and> B z y)) Z" 1450 show "\<forall>\<^sub>F (x, y) in ?Z. A x y" using that 1451 by(auto simp add: eventually_map_filter_on split_def elim!: eventually_mono intro: someI2) 1452 have [simp]: "(\<lambda>p. (fst p, SOME z. A (fst p) z \<and> B z (snd p))) ` {p. (A OO B) (fst p) (snd p)} \<subseteq> {p. A (fst p) (snd p)}" by(auto intro: someI2) 1453 show "map_filter_on {(x, y). A x y} fst ?Z = ?F" "map_filter_on {(x, y). A x y} snd ?Z = ?G" 1454 using that by(simp_all add: map_filter_on_comp split_def o_def) 1455 qed 1456 show "rel_filter B ?G ?H" 1457 proof 1458 let ?Z = "map_filter_on ?AB (\<lambda>(x, y). (SOME z. A x z \<and> B z y, y)) Z" 1459 show "\<forall>\<^sub>F (x, y) in ?Z. B x y" using that 1460 by(auto simp add: eventually_map_filter_on split_def elim!: eventually_mono intro: someI2) 1461 have [simp]: "(\<lambda>p. (SOME z. A (fst p) z \<and> B z (snd p), snd p)) ` {p. (A OO B) (fst p) (snd p)} \<subseteq> {p. B (fst p) (snd p)}" by(auto intro: someI2) 1462 show "map_filter_on {(x, y). B x y} fst ?Z = ?G" "map_filter_on {(x, y). B x y} snd ?Z = ?H" 1463 using that by(simp_all add: map_filter_on_comp split_def o_def) 1464 qed 1465 qed 1466 1467 fix F G 1468 assume F: "\<forall>\<^sub>F (x, y) in F. A x y" and G: "\<forall>\<^sub>F (x, y) in G. B x y" 1469 and eq: "map_filter_on {(x, y). B x y} fst G = map_filter_on {(x, y). A x y} snd F" (is "?Y2 = ?Y1") 1470 let ?X = "map_filter_on {(x, y). A x y} fst F" 1471 and ?Z = "(map_filter_on {(x, y). B x y} snd G)" 1472 have step: "\<exists>P'\<le>P. \<exists>Q' \<le> Q. eventually P' F \<and> eventually Q' G \<and> {y. \<exists>x. P' (x, y)} = {y. \<exists>z. Q' (y, z)}" 1473 if P: "eventually P F" and Q: "eventually Q G" for P Q 1474 proof - 1475 let ?P = "\<lambda>(x, y). P (x, y) \<and> A x y" and ?Q = "\<lambda>(y, z). Q (y, z) \<and> B y z" 1476 define P' where "P' \<equiv> \<lambda>(x, y). ?P (x, y) \<and> (\<exists>z. ?Q (y, z))" 1477 define Q' where "Q' \<equiv> \<lambda>(y, z). ?Q (y, z) \<and> (\<exists>x. ?P (x, y))" 1478 have "P' \<le> P" "Q' \<le> Q" "{y. \<exists>x. P' (x, y)} = {y. \<exists>z. Q' (y, z)}" 1479 by(auto simp add: P'_def Q'_def) 1480 moreover 1481 from P Q F G have P': "eventually ?P F" and Q': "eventually ?Q G" 1482 by(simp_all add: eventually_conj_iff split_def) 1483 from P' F have "\<forall>\<^sub>F y in ?Y1. \<exists>x. P (x, y) \<and> A x y" 1484 by(auto simp add: eventually_map_filter_on elim!: eventually_mono) 1485 from this[folded eq] obtain Q'' where Q'': "eventually Q'' G" 1486 and Q''P: "{y. \<exists>z. Q'' (y, z)} \<subseteq> {y. \<exists>x. ?P (x, y)}" 1487 using G by(fastforce simp add: eventually_map_filter_on) 1488 have "eventually (inf Q'' ?Q) G" using Q'' Q' by(auto intro: eventually_conj simp add: inf_fun_def) 1489 then have "eventually Q' G" using Q''P by(auto elim!: eventually_mono simp add: Q'_def) 1490 moreover 1491 from Q' G have "\<forall>\<^sub>F y in ?Y2. \<exists>z. Q (y, z) \<and> B y z" 1492 by(auto simp add: eventually_map_filter_on elim!: eventually_mono) 1493 from this[unfolded eq] obtain P'' where P'': "eventually P'' F" 1494 and P''Q: "{y. \<exists>x. P'' (x, y)} \<subseteq> {y. \<exists>z. ?Q (y, z)}" 1495 using F by(fastforce simp add: eventually_map_filter_on) 1496 have "eventually (inf P'' ?P) F" using P'' P' by(auto intro: eventually_conj simp add: inf_fun_def) 1497 then have "eventually P' F" using P''Q by(auto elim!: eventually_mono simp add: P'_def) 1498 ultimately show ?thesis by blast 1499 qed 1500 1501 show "rel_filter (A OO B) ?X ?Z" 1502 proof 1503 let ?Y = "\<lambda>Y. \<exists>X Z. eventually X ?X \<and> eventually Z ?Z \<and> (\<lambda>(x, z). X x \<and> Z z \<and> (A OO B) x z) \<le> Y" 1504 have Y: "is_filter ?Y" 1505 proof 1506 show "?Y (\<lambda>_. True)" by(auto simp add: le_fun_def intro: eventually_True) 1507 show "?Y (\<lambda>x. P x \<and> Q x)" if "?Y P" "?Y Q" for P Q using that 1508 apply clarify 1509 apply(intro exI conjI; (elim eventually_rev_mp; fold imp_conjL; intro always_eventually allI; rule imp_refl)?) 1510 apply auto 1511 done 1512 show "?Y Q" if "?Y P" "\<forall>x. P x \<longrightarrow> Q x" for P Q using that by blast 1513 qed 1514 define Y where "Y = Abs_filter ?Y" 1515 have eventually_Y: "eventually P Y \<longleftrightarrow> ?Y P" for P 1516 using eventually_Abs_filter[OF Y, of P] by(simp add: Y_def) 1517 show YY: "\<forall>\<^sub>F (x, y) in Y. (A OO B) x y" using F G 1518 by(auto simp add: eventually_Y eventually_map_filter_on eventually_conj_iff intro!: eventually_True) 1519 have "?Y (\<lambda>(x, z). P x \<and> (A OO B) x z) \<longleftrightarrow> (\<forall>\<^sub>F (x, y) in F. P x \<and> A x y)" (is "?lhs = ?rhs") for P 1520 proof 1521 show ?lhs if ?rhs using G F that 1522 by(auto 4 3 intro: exI[where x="\<lambda>_. True"] simp add: eventually_map_filter_on split_def) 1523 assume ?lhs 1524 then obtain X Z where "\<forall>\<^sub>F (x, y) in F. X x \<and> A x y" 1525 and "\<forall>\<^sub>F (x, y) in G. Z y \<and> B x y" 1526 and "(\<lambda>(x, z). X x \<and> Z z \<and> (A OO B) x z) \<le> (\<lambda>(x, z). P x \<and> (A OO B) x z)" 1527 using F G by(auto simp add: eventually_map_filter_on split_def) 1528 from step[OF this(1, 2)] this(3) 1529 show ?rhs by(clarsimp elim!: eventually_rev_mp simp add: le_fun_def)(fastforce intro: always_eventually) 1530 qed 1531 then show "map_filter_on ?AB fst Y = ?X" 1532 by(simp add: filter_eq_iff YY eventually_map_filter_on)(simp add: eventually_Y eventually_map_filter_on F G; simp add: split_def) 1533 1534 have "?Y (\<lambda>(x, z). P z \<and> (A OO B) x z) \<longleftrightarrow> (\<forall>\<^sub>F (x, y) in G. P y \<and> B x y)" (is "?lhs = ?rhs") for P 1535 proof 1536 show ?lhs if ?rhs using G F that 1537 by(auto 4 3 intro: exI[where x="\<lambda>_. True"] simp add: eventually_map_filter_on split_def) 1538 assume ?lhs 1539 then obtain X Z where "\<forall>\<^sub>F (x, y) in F. X x \<and> A x y" 1540 and "\<forall>\<^sub>F (x, y) in G. Z y \<and> B x y" 1541 and "(\<lambda>(x, z). X x \<and> Z z \<and> (A OO B) x z) \<le> (\<lambda>(x, z). P z \<and> (A OO B) x z)" 1542 using F G by(auto simp add: eventually_map_filter_on split_def) 1543 from step[OF this(1, 2)] this(3) 1544 show ?rhs by(clarsimp elim!: eventually_rev_mp simp add: le_fun_def)(fastforce intro: always_eventually) 1545 qed 1546 then show "map_filter_on ?AB snd Y = ?Z" 1547 by(simp add: filter_eq_iff YY eventually_map_filter_on)(simp add: eventually_Y eventually_map_filter_on F G; simp add: split_def) 1548 qed 1549qed 1550 1551lemma filtermap_parametric: "((A ===> B) ===> rel_filter A ===> rel_filter B) filtermap filtermap" 1552proof(intro rel_funI; erule rel_filter.cases; hypsubst) 1553 fix f g Z 1554 assume fg: "(A ===> B) f g" and Z: "\<forall>\<^sub>F (x, y) in Z. A x y" 1555 have "rel_filter B (map_filter_on {(x, y). A x y} (f \<circ> fst) Z) (map_filter_on {(x, y). A x y} (g \<circ> snd) Z)" 1556 (is "rel_filter _ ?F ?G") 1557 proof 1558 let ?Z = "map_filter_on {(x, y). A x y} (map_prod f g) Z" 1559 show "\<forall>\<^sub>F (x, y) in ?Z. B x y" using fg Z 1560 by(auto simp add: eventually_map_filter_on split_def elim!: eventually_mono rel_funD) 1561 have [simp]: "map_prod f g ` {p. A (fst p) (snd p)} \<subseteq> {p. B (fst p) (snd p)}" 1562 using fg by(auto dest: rel_funD) 1563 show "map_filter_on {(x, y). B x y} fst ?Z = ?F" "map_filter_on {(x, y). B x y} snd ?Z = ?G" 1564 using Z by(auto simp add: map_filter_on_comp split_def) 1565 qed 1566 thus "rel_filter B (filtermap f (map_filter_on {(x, y). A x y} fst Z)) (filtermap g (map_filter_on {(x, y). A x y} snd Z))" 1567 using Z by(simp add: map_filter_on_UNIV[symmetric] map_filter_on_comp) 1568qed 1569 1570lemma rel_filter_Grp: "rel_filter (Grp UNIV f) = Grp UNIV (filtermap f)" 1571proof((intro antisym predicate2I; (elim GrpE; hypsubst)?), rule GrpI[OF _ UNIV_I]) 1572 fix F G 1573 assume "rel_filter (Grp UNIV f) F G" 1574 hence "rel_filter (=) (filtermap f F) (filtermap id G)" 1575 by(rule filtermap_parametric[THEN rel_funD, THEN rel_funD, rotated])(simp add: Grp_def rel_fun_def) 1576 thus "filtermap f F = G" by(simp add: rel_filter_eq) 1577next 1578 fix F :: "'a filter" 1579 have "rel_filter (=) F F" by(simp add: rel_filter_eq) 1580 hence "rel_filter (Grp UNIV f) (filtermap id F) (filtermap f F)" 1581 by(rule filtermap_parametric[THEN rel_funD, THEN rel_funD, rotated])(simp add: Grp_def rel_fun_def) 1582 thus "rel_filter (Grp UNIV f) F (filtermap f F)" by simp 1583qed 1584 1585lemma Quotient_filter [quot_map]: 1586 "Quotient R Abs Rep T \<Longrightarrow> Quotient (rel_filter R) (filtermap Abs) (filtermap Rep) (rel_filter T)" 1587 unfolding Quotient_alt_def5 rel_filter_eq[symmetric] rel_filter_Grp[symmetric] 1588 by(simp add: rel_filter_distr[symmetric] rel_filter_conversep[symmetric] rel_filter_mono) 1589 1590lemma left_total_rel_filter [transfer_rule]: "left_total A \<Longrightarrow> left_total (rel_filter A)" 1591unfolding left_total_alt_def rel_filter_eq[symmetric] rel_filter_conversep[symmetric] rel_filter_distr 1592by(rule rel_filter_mono) 1593 1594lemma right_total_rel_filter [transfer_rule]: "right_total A \<Longrightarrow> right_total (rel_filter A)" 1595using left_total_rel_filter[of "A\<inverse>\<inverse>"] by(simp add: rel_filter_conversep) 1596 1597lemma bi_total_rel_filter [transfer_rule]: "bi_total A \<Longrightarrow> bi_total (rel_filter A)" 1598unfolding bi_total_alt_def by(simp add: left_total_rel_filter right_total_rel_filter) 1599 1600lemma left_unique_rel_filter [transfer_rule]: "left_unique A \<Longrightarrow> left_unique (rel_filter A)" 1601unfolding left_unique_alt_def rel_filter_eq[symmetric] rel_filter_conversep[symmetric] rel_filter_distr 1602by(rule rel_filter_mono) 1603 1604lemma right_unique_rel_filter [transfer_rule]: 1605 "right_unique A \<Longrightarrow> right_unique (rel_filter A)" 1606using left_unique_rel_filter[of "A\<inverse>\<inverse>"] by(simp add: rel_filter_conversep) 1607 1608lemma bi_unique_rel_filter [transfer_rule]: "bi_unique A \<Longrightarrow> bi_unique (rel_filter A)" 1609by(simp add: bi_unique_alt_def left_unique_rel_filter right_unique_rel_filter) 1610 1611lemma eventually_parametric [transfer_rule]: 1612 "((A ===> (=)) ===> rel_filter A ===> (=)) eventually eventually" 1613by(auto 4 4 intro!: rel_funI elim!: rel_filter.cases simp add: eventually_map_filter_on dest: rel_funD intro: always_eventually elim!: eventually_rev_mp) 1614 1615lemma frequently_parametric [transfer_rule]: "((A ===> (=)) ===> rel_filter A ===> (=)) frequently frequently" 1616 unfolding frequently_def[abs_def] by transfer_prover 1617 1618lemma is_filter_parametric [transfer_rule]: 1619 assumes [transfer_rule]: "bi_total A" 1620 assumes [transfer_rule]: "bi_unique A" 1621 shows "(((A ===> (=)) ===> (=)) ===> (=)) is_filter is_filter" 1622 unfolding is_filter_def by transfer_prover 1623 1624lemma top_filter_parametric [transfer_rule]: "rel_filter A top top" if "bi_total A" 1625proof 1626 let ?Z = "principal {(x, y). A x y}" 1627 show "\<forall>\<^sub>F (x, y) in ?Z. A x y" by(simp add: eventually_principal) 1628 show "map_filter_on {(x, y). A x y} fst ?Z = top" "map_filter_on {(x, y). A x y} snd ?Z = top" 1629 using that by(auto simp add: filter_eq_iff eventually_map_filter_on eventually_principal bi_total_def) 1630qed 1631 1632lemma bot_filter_parametric [transfer_rule]: "rel_filter A bot bot" 1633proof 1634 show "\<forall>\<^sub>F (x, y) in bot. A x y" by simp 1635 show "map_filter_on {(x, y). A x y} fst bot = bot" "map_filter_on {(x, y). A x y} snd bot = bot" 1636 by(simp_all add: filter_eq_iff eventually_map_filter_on) 1637qed 1638 1639lemma principal_parametric [transfer_rule]: "(rel_set A ===> rel_filter A) principal principal" 1640proof(rule rel_funI rel_filter.intros)+ 1641 fix S S' 1642 assume *: "rel_set A S S'" 1643 define SS' where "SS' = S \<times> S' \<inter> {(x, y). A x y}" 1644 have SS': "SS' \<subseteq> {(x, y). A x y}" and [simp]: "S = fst ` SS'" "S' = snd ` SS'" 1645 using * by(auto 4 3 dest: rel_setD1 rel_setD2 intro: rev_image_eqI simp add: SS'_def) 1646 let ?Z = "principal SS'" 1647 show "\<forall>\<^sub>F (x, y) in ?Z. A x y" using SS' by(auto simp add: eventually_principal) 1648 then show "map_filter_on {(x, y). A x y} fst ?Z = principal S" 1649 and "map_filter_on {(x, y). A x y} snd ?Z = principal S'" 1650 by(auto simp add: filter_eq_iff eventually_map_filter_on eventually_principal) 1651qed 1652 1653lemma sup_filter_parametric [transfer_rule]: 1654 "(rel_filter A ===> rel_filter A ===> rel_filter A) sup sup" 1655proof(intro rel_funI; elim rel_filter.cases; hypsubst) 1656 show "rel_filter A 1657 (map_filter_on {(x, y). A x y} fst FG \<squnion> map_filter_on {(x, y). A x y} fst FG') 1658 (map_filter_on {(x, y). A x y} snd FG \<squnion> map_filter_on {(x, y). A x y} snd FG')" 1659 (is "rel_filter _ (sup ?F ?G) (sup ?F' ?G')") 1660 if "\<forall>\<^sub>F (x, y) in FG. A x y" "\<forall>\<^sub>F (x, y) in FG'. A x y" for FG FG' 1661 proof 1662 let ?Z = "sup FG FG'" 1663 show "\<forall>\<^sub>F (x, y) in ?Z. A x y" by(simp add: eventually_sup that) 1664 then show "map_filter_on {(x, y). A x y} fst ?Z = sup ?F ?G" 1665 and "map_filter_on {(x, y). A x y} snd ?Z = sup ?F' ?G'" 1666 by(simp_all add: filter_eq_iff eventually_map_filter_on eventually_sup) 1667 qed 1668qed 1669 1670lemma Sup_filter_parametric [transfer_rule]: "(rel_set (rel_filter A) ===> rel_filter A) Sup Sup" 1671proof(rule rel_funI) 1672 fix S S' 1673 define SS' where "SS' = S \<times> S' \<inter> {(F, G). rel_filter A F G}" 1674 assume "rel_set (rel_filter A) S S'" 1675 then have SS': "SS' \<subseteq> {(F, G). rel_filter A F G}" and [simp]: "S = fst ` SS'" "S' = snd ` SS'" 1676 by(auto 4 3 dest: rel_setD1 rel_setD2 intro: rev_image_eqI simp add: SS'_def) 1677 from SS' obtain Z where Z: "\<And>F G. (F, G) \<in> SS' \<Longrightarrow> 1678 (\<forall>\<^sub>F (x, y) in Z F G. A x y) \<and> 1679 id F = map_filter_on {(x, y). A x y} fst (Z F G) \<and> 1680 id G = map_filter_on {(x, y). A x y} snd (Z F G)" 1681 unfolding rel_filter.simps by atomize_elim((rule choice allI)+; auto) 1682 have id: "eventually P F = eventually P (id F)" "eventually Q G = eventually Q (id G)" 1683 if "(F, G) \<in> SS'" for P Q F G by simp_all 1684 show "rel_filter A (Sup S) (Sup S')" 1685 proof 1686 let ?Z = "SUP (F, G):SS'. Z F G" 1687 show *: "\<forall>\<^sub>F (x, y) in ?Z. A x y" using Z by(auto simp add: eventually_Sup) 1688 show "map_filter_on {(x, y). A x y} fst ?Z = Sup S" "map_filter_on {(x, y). A x y} snd ?Z = Sup S'" 1689 unfolding filter_eq_iff 1690 by(auto 4 4 simp add: id eventually_Sup eventually_map_filter_on *[simplified eventually_Sup] simp del: id_apply dest: Z) 1691 qed 1692qed 1693 1694context 1695 fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool" 1696 assumes [transfer_rule]: "bi_unique A" 1697begin 1698 1699lemma le_filter_parametric [transfer_rule]: 1700 "(rel_filter A ===> rel_filter A ===> (=)) (\<le>) (\<le>)" 1701unfolding le_filter_def[abs_def] by transfer_prover 1702 1703lemma less_filter_parametric [transfer_rule]: 1704 "(rel_filter A ===> rel_filter A ===> (=)) (<) (<)" 1705unfolding less_filter_def[abs_def] by transfer_prover 1706 1707context 1708 assumes [transfer_rule]: "bi_total A" 1709begin 1710 1711lemma Inf_filter_parametric [transfer_rule]: 1712 "(rel_set (rel_filter A) ===> rel_filter A) Inf Inf" 1713unfolding Inf_filter_def[abs_def] by transfer_prover 1714 1715lemma inf_filter_parametric [transfer_rule]: 1716 "(rel_filter A ===> rel_filter A ===> rel_filter A) inf inf" 1717proof(intro rel_funI)+ 1718 fix F F' G G' 1719 assume [transfer_rule]: "rel_filter A F F'" "rel_filter A G G'" 1720 have "rel_filter A (Inf {F, G}) (Inf {F', G'})" by transfer_prover 1721 thus "rel_filter A (inf F G) (inf F' G')" by simp 1722qed 1723 1724end 1725 1726end 1727 1728end 1729 1730lemma prod_filter_parametric [transfer_rule]: includes lifting_syntax shows 1731 "(rel_filter R ===> rel_filter S ===> rel_filter (rel_prod R S)) prod_filter prod_filter" 1732proof(intro rel_funI; elim rel_filter.cases; hypsubst) 1733 fix F G 1734 assume F: "\<forall>\<^sub>F (x, y) in F. R x y" and G: "\<forall>\<^sub>F (x, y) in G. S x y" 1735 show "rel_filter (rel_prod R S) 1736 (map_filter_on {(x, y). R x y} fst F \<times>\<^sub>F map_filter_on {(x, y). S x y} fst G) 1737 (map_filter_on {(x, y). R x y} snd F \<times>\<^sub>F map_filter_on {(x, y). S x y} snd G)" 1738 (is "rel_filter ?RS ?F ?G") 1739 proof 1740 let ?Z = "filtermap (\<lambda>((a, b), (a', b')). ((a, a'), (b, b'))) (prod_filter F G)" 1741 show *: "\<forall>\<^sub>F (x, y) in ?Z. rel_prod R S x y" using F G 1742 by(auto simp add: eventually_filtermap split_beta eventually_prod_filter) 1743 show "map_filter_on {(x, y). ?RS x y} fst ?Z = ?F" 1744 using F G 1745 apply(clarsimp simp add: filter_eq_iff eventually_map_filter_on *) 1746 apply(simp add: eventually_filtermap split_beta eventually_prod_filter) 1747 apply(subst eventually_map_filter_on; simp)+ 1748 apply(rule iffI; clarsimp) 1749 subgoal for P P' P'' 1750 apply(rule exI[where x="\<lambda>a. \<exists>b. P' (a, b) \<and> R a b"]; rule conjI) 1751 subgoal by(fastforce elim: eventually_rev_mp eventually_mono) 1752 subgoal 1753 by(rule exI[where x="\<lambda>a. \<exists>b. P'' (a, b) \<and> S a b"])(fastforce elim: eventually_rev_mp eventually_mono) 1754 done 1755 subgoal by fastforce 1756 done 1757 show "map_filter_on {(x, y). ?RS x y} snd ?Z = ?G" 1758 using F G 1759 apply(clarsimp simp add: filter_eq_iff eventually_map_filter_on *) 1760 apply(simp add: eventually_filtermap split_beta eventually_prod_filter) 1761 apply(subst eventually_map_filter_on; simp)+ 1762 apply(rule iffI; clarsimp) 1763 subgoal for P P' P'' 1764 apply(rule exI[where x="\<lambda>b. \<exists>a. P' (a, b) \<and> R a b"]; rule conjI) 1765 subgoal by(fastforce elim: eventually_rev_mp eventually_mono) 1766 subgoal 1767 by(rule exI[where x="\<lambda>b. \<exists>a. P'' (a, b) \<and> S a b"])(fastforce elim: eventually_rev_mp eventually_mono) 1768 done 1769 subgoal by fastforce 1770 done 1771 qed 1772qed 1773 1774text \<open>Code generation for filters\<close> 1775 1776definition abstract_filter :: "(unit \<Rightarrow> 'a filter) \<Rightarrow> 'a filter" 1777 where [simp]: "abstract_filter f = f ()" 1778 1779code_datatype principal abstract_filter 1780 1781hide_const (open) abstract_filter 1782 1783declare [[code drop: filterlim prod_filter filtermap eventually 1784 "inf :: _ filter \<Rightarrow> _" "sup :: _ filter \<Rightarrow> _" "less_eq :: _ filter \<Rightarrow> _" 1785 Abs_filter]] 1786 1787declare filterlim_principal [code] 1788declare principal_prod_principal [code] 1789declare filtermap_principal [code] 1790declare filtercomap_principal [code] 1791declare eventually_principal [code] 1792declare inf_principal [code] 1793declare sup_principal [code] 1794declare principal_le_iff [code] 1795 1796lemma Rep_filter_iff_eventually [simp, code]: 1797 "Rep_filter F P \<longleftrightarrow> eventually P F" 1798 by (simp add: eventually_def) 1799 1800lemma bot_eq_principal_empty [code]: 1801 "bot = principal {}" 1802 by simp 1803 1804lemma top_eq_principal_UNIV [code]: 1805 "top = principal UNIV" 1806 by simp 1807 1808instantiation filter :: (equal) equal 1809begin 1810 1811definition equal_filter :: "'a filter \<Rightarrow> 'a filter \<Rightarrow> bool" 1812 where "equal_filter F F' \<longleftrightarrow> F = F'" 1813 1814lemma equal_filter [code]: 1815 "HOL.equal (principal A) (principal B) \<longleftrightarrow> A = B" 1816 by (simp add: equal_filter_def) 1817 1818instance 1819 by standard (simp add: equal_filter_def) 1820 1821end 1822 1823end 1824