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>\<open>F \<le> F'\<close> means that filter \<^term>\<open>F\<close> is finer than 255filter \<^term>\<open>F'\<close>.\<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 525lemma eventually_INF_finite: 526 assumes "finite A" 527 shows "eventually P (\<Sqinter> x\<in>A. F x) \<longleftrightarrow> 528 (\<exists>Q. (\<forall>x\<in>A. eventually (Q x) (F x)) \<and> (\<forall>y. (\<forall>x\<in>A. Q x y) \<longrightarrow> P y))" 529 using assms 530proof (induction arbitrary: P rule: finite_induct) 531 case (insert a A P) 532 from insert.hyps have [simp]: "x \<noteq> a" if "x \<in> A" for x 533 using that by auto 534 have "eventually P (\<Sqinter> x\<in>insert a A. F x) \<longleftrightarrow> 535 (\<exists>Q R S. eventually Q (F a) \<and> (( (\<forall>x\<in>A. eventually (S x) (F x)) \<and> 536 (\<forall>y. (\<forall>x\<in>A. S x y) \<longrightarrow> R y)) \<and> (\<forall>x. Q x \<and> R x \<longrightarrow> P x)))" 537 unfolding ex_simps by (simp add: eventually_inf insert.IH) 538 also have "\<dots> \<longleftrightarrow> (\<exists>Q. (\<forall>x\<in>insert a A. eventually (Q x) (F x)) \<and> 539 (\<forall>y. (\<forall>x\<in>insert a A. Q x y) \<longrightarrow> P y))" 540 proof (safe, goal_cases) 541 case (1 Q R S) 542 thus ?case using 1 by (intro exI[of _ "S(a := Q)"]) auto 543 next 544 case (2 Q) 545 show ?case 546 by (rule exI[of _ "Q a"], rule exI[of _ "\<lambda>y. \<forall>x\<in>A. Q x y"], 547 rule exI[of _ "Q(a := (\<lambda>_. True))"]) (use 2 in auto) 548 qed 549 finally show ?case . 550qed auto 551 552subsubsection \<open>Map function for filters\<close> 553 554definition filtermap :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a filter \<Rightarrow> 'b filter" 555 where "filtermap f F = Abs_filter (\<lambda>P. eventually (\<lambda>x. P (f x)) F)" 556 557lemma eventually_filtermap: 558 "eventually P (filtermap f F) = eventually (\<lambda>x. P (f x)) F" 559 unfolding filtermap_def 560 apply (rule eventually_Abs_filter) 561 apply (rule is_filter.intro) 562 apply (auto elim!: eventually_rev_mp) 563 done 564 565lemma filtermap_ident: "filtermap (\<lambda>x. x) F = F" 566 by (simp add: filter_eq_iff eventually_filtermap) 567 568lemma filtermap_filtermap: 569 "filtermap f (filtermap g F) = filtermap (\<lambda>x. f (g x)) F" 570 by (simp add: filter_eq_iff eventually_filtermap) 571 572lemma filtermap_mono: "F \<le> F' \<Longrightarrow> filtermap f F \<le> filtermap f F'" 573 unfolding le_filter_def eventually_filtermap by simp 574 575lemma filtermap_bot [simp]: "filtermap f bot = bot" 576 by (simp add: filter_eq_iff eventually_filtermap) 577 578lemma filtermap_bot_iff: "filtermap f F = bot \<longleftrightarrow> F = bot" 579 by (simp add: trivial_limit_def eventually_filtermap) 580 581lemma filtermap_sup: "filtermap f (sup F1 F2) = sup (filtermap f F1) (filtermap f F2)" 582 by (simp add: filter_eq_iff eventually_filtermap eventually_sup) 583 584lemma filtermap_SUP: "filtermap f (\<Squnion>b\<in>B. F b) = (\<Squnion>b\<in>B. filtermap f (F b))" 585 by (simp add: filter_eq_iff eventually_Sup eventually_filtermap) 586 587lemma filtermap_inf: "filtermap f (inf F1 F2) \<le> inf (filtermap f F1) (filtermap f F2)" 588 by (intro inf_greatest filtermap_mono inf_sup_ord) 589 590lemma filtermap_INF: "filtermap f (\<Sqinter>b\<in>B. F b) \<le> (\<Sqinter>b\<in>B. filtermap f (F b))" 591 by (rule INF_greatest, rule filtermap_mono, erule INF_lower) 592 593 594subsubsection \<open>Contravariant map function for filters\<close> 595 596definition filtercomap :: "('a \<Rightarrow> 'b) \<Rightarrow> 'b filter \<Rightarrow> 'a filter" where 597 "filtercomap f F = Abs_filter (\<lambda>P. \<exists>Q. eventually Q F \<and> (\<forall>x. Q (f x) \<longrightarrow> P x))" 598 599lemma eventually_filtercomap: 600 "eventually P (filtercomap f F) \<longleftrightarrow> (\<exists>Q. eventually Q F \<and> (\<forall>x. Q (f x) \<longrightarrow> P x))" 601 unfolding filtercomap_def 602proof (intro eventually_Abs_filter, unfold_locales, goal_cases) 603 case 1 604 show ?case by (auto intro!: exI[of _ "\<lambda>_. True"]) 605next 606 case (2 P Q) 607 from 2(1) guess P' by (elim exE conjE) note P' = this 608 from 2(2) guess Q' by (elim exE conjE) note Q' = this 609 show ?case 610 by (rule exI[of _ "\<lambda>x. P' x \<and> Q' x"]) 611 (insert P' Q', auto intro!: eventually_conj) 612next 613 case (3 P Q) 614 thus ?case by blast 615qed 616 617lemma filtercomap_ident: "filtercomap (\<lambda>x. x) F = F" 618 by (auto simp: filter_eq_iff eventually_filtercomap elim!: eventually_mono) 619 620lemma filtercomap_filtercomap: "filtercomap f (filtercomap g F) = filtercomap (\<lambda>x. g (f x)) F" 621 unfolding filter_eq_iff by (auto simp: eventually_filtercomap) 622 623lemma filtercomap_mono: "F \<le> F' \<Longrightarrow> filtercomap f F \<le> filtercomap f F'" 624 by (auto simp: eventually_filtercomap le_filter_def) 625 626lemma filtercomap_bot [simp]: "filtercomap f bot = bot" 627 by (auto simp: filter_eq_iff eventually_filtercomap) 628 629lemma filtercomap_top [simp]: "filtercomap f top = top" 630 by (auto simp: filter_eq_iff eventually_filtercomap) 631 632lemma filtercomap_inf: "filtercomap f (inf F1 F2) = inf (filtercomap f F1) (filtercomap f F2)" 633 unfolding filter_eq_iff 634proof safe 635 fix P 636 assume "eventually P (filtercomap f (F1 \<sqinter> F2))" 637 then obtain Q R S where *: 638 "eventually Q F1" "eventually R F2" "\<And>x. Q x \<Longrightarrow> R x \<Longrightarrow> S x" "\<And>x. S (f x) \<Longrightarrow> P x" 639 unfolding eventually_filtercomap eventually_inf by blast 640 from * have "eventually (\<lambda>x. Q (f x)) (filtercomap f F1)" 641 "eventually (\<lambda>x. R (f x)) (filtercomap f F2)" 642 by (auto simp: eventually_filtercomap) 643 with * show "eventually P (filtercomap f F1 \<sqinter> filtercomap f F2)" 644 unfolding eventually_inf by blast 645next 646 fix P 647 assume "eventually P (inf (filtercomap f F1) (filtercomap f F2))" 648 then obtain Q Q' R R' where *: 649 "eventually Q F1" "eventually R F2" "\<And>x. Q (f x) \<Longrightarrow> Q' x" "\<And>x. R (f x) \<Longrightarrow> R' x" 650 "\<And>x. Q' x \<Longrightarrow> R' x \<Longrightarrow> P x" 651 unfolding eventually_filtercomap eventually_inf by blast 652 from * have "eventually (\<lambda>x. Q x \<and> R x) (F1 \<sqinter> F2)" by (auto simp: eventually_inf) 653 with * show "eventually P (filtercomap f (F1 \<sqinter> F2))" 654 by (auto simp: eventually_filtercomap) 655qed 656 657lemma filtercomap_sup: "filtercomap f (sup F1 F2) \<ge> sup (filtercomap f F1) (filtercomap f F2)" 658 by (intro sup_least filtercomap_mono inf_sup_ord) 659 660lemma filtercomap_INF: "filtercomap f (\<Sqinter>b\<in>B. F b) = (\<Sqinter>b\<in>B. filtercomap f (F b))" 661proof - 662 have *: "filtercomap f (\<Sqinter>b\<in>B. F b) = (\<Sqinter>b\<in>B. filtercomap f (F b))" if "finite B" for B 663 using that by induction (simp_all add: filtercomap_inf) 664 show ?thesis unfolding filter_eq_iff 665 proof 666 fix P 667 have "eventually P (\<Sqinter>b\<in>B. filtercomap f (F b)) \<longleftrightarrow> 668 (\<exists>X. (X \<subseteq> B \<and> finite X) \<and> eventually P (\<Sqinter>b\<in>X. filtercomap f (F b)))" 669 by (subst eventually_INF) blast 670 also have "\<dots> \<longleftrightarrow> (\<exists>X. (X \<subseteq> B \<and> finite X) \<and> eventually P (filtercomap f (\<Sqinter>b\<in>X. F b)))" 671 by (rule ex_cong) (simp add: *) 672 also have "\<dots> \<longleftrightarrow> eventually P (filtercomap f (\<Sqinter>(F ` B)))" 673 unfolding eventually_filtercomap by (subst eventually_INF) blast 674 finally show "eventually P (filtercomap f (\<Sqinter>(F ` B))) = 675 eventually P (\<Sqinter>b\<in>B. filtercomap f (F b))" .. 676 qed 677qed 678 679lemma filtercomap_SUP: 680 "filtercomap f (\<Squnion>b\<in>B. F b) \<ge> (\<Squnion>b\<in>B. filtercomap f (F b))" 681 by (intro SUP_least filtercomap_mono SUP_upper) 682 683lemma filtermap_le_iff_le_filtercomap: "filtermap f F \<le> G \<longleftrightarrow> F \<le> filtercomap f G" 684 unfolding le_filter_def eventually_filtermap eventually_filtercomap 685 using eventually_mono by auto 686 687lemma filtercomap_neq_bot: 688 assumes "\<And>P. eventually P F \<Longrightarrow> \<exists>x. P (f x)" 689 shows "filtercomap f F \<noteq> bot" 690 using assms by (auto simp: trivial_limit_def eventually_filtercomap) 691 692lemma filtercomap_neq_bot_surj: 693 assumes "F \<noteq> bot" and "surj f" 694 shows "filtercomap f F \<noteq> bot" 695proof (rule filtercomap_neq_bot) 696 fix P assume *: "eventually P F" 697 show "\<exists>x. P (f x)" 698 proof (rule ccontr) 699 assume **: "\<not>(\<exists>x. P (f x))" 700 from * have "eventually (\<lambda>_. False) F" 701 proof eventually_elim 702 case (elim x) 703 from \<open>surj f\<close> obtain y where "x = f y" by auto 704 with elim and ** show False by auto 705 qed 706 with assms show False by (simp add: trivial_limit_def) 707 qed 708qed 709 710lemma eventually_filtercomapI [intro]: 711 assumes "eventually P F" 712 shows "eventually (\<lambda>x. P (f x)) (filtercomap f F)" 713 using assms by (auto simp: eventually_filtercomap) 714 715lemma filtermap_filtercomap: "filtermap f (filtercomap f F) \<le> F" 716 by (auto simp: le_filter_def eventually_filtermap eventually_filtercomap) 717 718lemma filtercomap_filtermap: "filtercomap f (filtermap f F) \<ge> F" 719 unfolding le_filter_def eventually_filtermap eventually_filtercomap 720 by (auto elim!: eventually_mono) 721 722 723subsubsection \<open>Standard filters\<close> 724 725definition principal :: "'a set \<Rightarrow> 'a filter" where 726 "principal S = Abs_filter (\<lambda>P. \<forall>x\<in>S. P x)" 727 728lemma eventually_principal: "eventually P (principal S) \<longleftrightarrow> (\<forall>x\<in>S. P x)" 729 unfolding principal_def 730 by (rule eventually_Abs_filter, rule is_filter.intro) auto 731 732lemma eventually_inf_principal: "eventually P (inf F (principal s)) \<longleftrightarrow> eventually (\<lambda>x. x \<in> s \<longrightarrow> P x) F" 733 unfolding eventually_inf eventually_principal by (auto elim: eventually_mono) 734 735lemma principal_UNIV[simp]: "principal UNIV = top" 736 by (auto simp: filter_eq_iff eventually_principal) 737 738lemma principal_empty[simp]: "principal {} = bot" 739 by (auto simp: filter_eq_iff eventually_principal) 740 741lemma principal_eq_bot_iff: "principal X = bot \<longleftrightarrow> X = {}" 742 by (auto simp add: filter_eq_iff eventually_principal) 743 744lemma principal_le_iff[iff]: "principal A \<le> principal B \<longleftrightarrow> A \<subseteq> B" 745 by (auto simp: le_filter_def eventually_principal) 746 747lemma le_principal: "F \<le> principal A \<longleftrightarrow> eventually (\<lambda>x. x \<in> A) F" 748 unfolding le_filter_def eventually_principal 749 apply safe 750 apply (erule_tac x="\<lambda>x. x \<in> A" in allE) 751 apply (auto elim: eventually_mono) 752 done 753 754lemma principal_inject[iff]: "principal A = principal B \<longleftrightarrow> A = B" 755 unfolding eq_iff by simp 756 757lemma sup_principal[simp]: "sup (principal A) (principal B) = principal (A \<union> B)" 758 unfolding filter_eq_iff eventually_sup eventually_principal by auto 759 760lemma inf_principal[simp]: "inf (principal A) (principal B) = principal (A \<inter> B)" 761 unfolding filter_eq_iff eventually_inf eventually_principal 762 by (auto intro: exI[of _ "\<lambda>x. x \<in> A"] exI[of _ "\<lambda>x. x \<in> B"]) 763 764lemma SUP_principal[simp]: "(\<Squnion>i\<in>I. principal (A i)) = principal (\<Union>i\<in>I. A i)" 765 unfolding filter_eq_iff eventually_Sup by (auto simp: eventually_principal) 766 767lemma INF_principal_finite: "finite X \<Longrightarrow> (\<Sqinter>x\<in>X. principal (f x)) = principal (\<Inter>x\<in>X. f x)" 768 by (induct X rule: finite_induct) auto 769 770lemma filtermap_principal[simp]: "filtermap f (principal A) = principal (f ` A)" 771 unfolding filter_eq_iff eventually_filtermap eventually_principal by simp 772 773lemma filtercomap_principal[simp]: "filtercomap f (principal A) = principal (f -` A)" 774 unfolding filter_eq_iff eventually_filtercomap eventually_principal by fast 775 776subsubsection \<open>Order filters\<close> 777 778definition at_top :: "('a::order) filter" 779 where "at_top = (\<Sqinter>k. principal {k ..})" 780 781lemma at_top_sub: "at_top = (\<Sqinter>k\<in>{c::'a::linorder..}. principal {k ..})" 782 by (auto intro!: INF_eq max.cobounded1 max.cobounded2 simp: at_top_def) 783 784lemma eventually_at_top_linorder: "eventually P at_top \<longleftrightarrow> (\<exists>N::'a::linorder. \<forall>n\<ge>N. P n)" 785 unfolding at_top_def 786 by (subst eventually_INF_base) (auto simp: eventually_principal intro: max.cobounded1 max.cobounded2) 787 788lemma eventually_filtercomap_at_top_linorder: 789 "eventually P (filtercomap f at_top) \<longleftrightarrow> (\<exists>N::'a::linorder. \<forall>x. f x \<ge> N \<longrightarrow> P x)" 790 by (auto simp: eventually_filtercomap eventually_at_top_linorder) 791 792lemma eventually_at_top_linorderI: 793 fixes c::"'a::linorder" 794 assumes "\<And>x. c \<le> x \<Longrightarrow> P x" 795 shows "eventually P at_top" 796 using assms by (auto simp: eventually_at_top_linorder) 797 798lemma eventually_ge_at_top [simp]: 799 "eventually (\<lambda>x. (c::_::linorder) \<le> x) at_top" 800 unfolding eventually_at_top_linorder by auto 801 802lemma eventually_at_top_dense: "eventually P at_top \<longleftrightarrow> (\<exists>N::'a::{no_top, linorder}. \<forall>n>N. P n)" 803proof - 804 have "eventually P (\<Sqinter>k. principal {k <..}) \<longleftrightarrow> (\<exists>N::'a. \<forall>n>N. P n)" 805 by (subst eventually_INF_base) (auto simp: eventually_principal intro: max.cobounded1 max.cobounded2) 806 also have "(\<Sqinter>k. principal {k::'a <..}) = at_top" 807 unfolding at_top_def 808 by (intro INF_eq) (auto intro: less_imp_le simp: Ici_subset_Ioi_iff gt_ex) 809 finally show ?thesis . 810qed 811 812lemma eventually_filtercomap_at_top_dense: 813 "eventually P (filtercomap f at_top) \<longleftrightarrow> (\<exists>N::'a::{no_top, linorder}. \<forall>x. f x > N \<longrightarrow> P x)" 814 by (auto simp: eventually_filtercomap eventually_at_top_dense) 815 816lemma eventually_at_top_not_equal [simp]: "eventually (\<lambda>x::'a::{no_top, linorder}. x \<noteq> c) at_top" 817 unfolding eventually_at_top_dense by auto 818 819lemma eventually_gt_at_top [simp]: "eventually (\<lambda>x. (c::_::{no_top, linorder}) < x) at_top" 820 unfolding eventually_at_top_dense by auto 821 822lemma eventually_all_ge_at_top: 823 assumes "eventually P (at_top :: ('a :: linorder) filter)" 824 shows "eventually (\<lambda>x. \<forall>y\<ge>x. P y) at_top" 825proof - 826 from assms obtain x where "\<And>y. y \<ge> x \<Longrightarrow> P y" by (auto simp: eventually_at_top_linorder) 827 hence "\<forall>z\<ge>y. P z" if "y \<ge> x" for y using that by simp 828 thus ?thesis by (auto simp: eventually_at_top_linorder) 829qed 830 831definition at_bot :: "('a::order) filter" 832 where "at_bot = (\<Sqinter>k. principal {.. k})" 833 834lemma at_bot_sub: "at_bot = (\<Sqinter>k\<in>{.. c::'a::linorder}. principal {.. k})" 835 by (auto intro!: INF_eq min.cobounded1 min.cobounded2 simp: at_bot_def) 836 837lemma eventually_at_bot_linorder: 838 fixes P :: "'a::linorder \<Rightarrow> bool" shows "eventually P at_bot \<longleftrightarrow> (\<exists>N. \<forall>n\<le>N. P n)" 839 unfolding at_bot_def 840 by (subst eventually_INF_base) (auto simp: eventually_principal intro: min.cobounded1 min.cobounded2) 841 842lemma eventually_filtercomap_at_bot_linorder: 843 "eventually P (filtercomap f at_bot) \<longleftrightarrow> (\<exists>N::'a::linorder. \<forall>x. f x \<le> N \<longrightarrow> P x)" 844 by (auto simp: eventually_filtercomap eventually_at_bot_linorder) 845 846lemma eventually_le_at_bot [simp]: 847 "eventually (\<lambda>x. x \<le> (c::_::linorder)) at_bot" 848 unfolding eventually_at_bot_linorder by auto 849 850lemma eventually_at_bot_dense: "eventually P at_bot \<longleftrightarrow> (\<exists>N::'a::{no_bot, linorder}. \<forall>n<N. P n)" 851proof - 852 have "eventually P (\<Sqinter>k. principal {..< k}) \<longleftrightarrow> (\<exists>N::'a. \<forall>n<N. P n)" 853 by (subst eventually_INF_base) (auto simp: eventually_principal intro: min.cobounded1 min.cobounded2) 854 also have "(\<Sqinter>k. principal {..< k::'a}) = at_bot" 855 unfolding at_bot_def 856 by (intro INF_eq) (auto intro: less_imp_le simp: Iic_subset_Iio_iff lt_ex) 857 finally show ?thesis . 858qed 859 860lemma eventually_filtercomap_at_bot_dense: 861 "eventually P (filtercomap f at_bot) \<longleftrightarrow> (\<exists>N::'a::{no_bot, linorder}. \<forall>x. f x < N \<longrightarrow> P x)" 862 by (auto simp: eventually_filtercomap eventually_at_bot_dense) 863 864lemma eventually_at_bot_not_equal [simp]: "eventually (\<lambda>x::'a::{no_bot, linorder}. x \<noteq> c) at_bot" 865 unfolding eventually_at_bot_dense by auto 866 867lemma eventually_gt_at_bot [simp]: 868 "eventually (\<lambda>x. x < (c::_::unbounded_dense_linorder)) at_bot" 869 unfolding eventually_at_bot_dense by auto 870 871lemma trivial_limit_at_bot_linorder [simp]: "\<not> trivial_limit (at_bot ::('a::linorder) filter)" 872 unfolding trivial_limit_def 873 by (metis eventually_at_bot_linorder order_refl) 874 875lemma trivial_limit_at_top_linorder [simp]: "\<not> trivial_limit (at_top ::('a::linorder) filter)" 876 unfolding trivial_limit_def 877 by (metis eventually_at_top_linorder order_refl) 878 879subsection \<open>Sequentially\<close> 880 881abbreviation sequentially :: "nat filter" 882 where "sequentially \<equiv> at_top" 883 884lemma eventually_sequentially: 885 "eventually P sequentially \<longleftrightarrow> (\<exists>N. \<forall>n\<ge>N. P n)" 886 by (rule eventually_at_top_linorder) 887 888lemma sequentially_bot [simp, intro]: "sequentially \<noteq> bot" 889 unfolding filter_eq_iff eventually_sequentially by auto 890 891lemmas trivial_limit_sequentially = sequentially_bot 892 893lemma eventually_False_sequentially [simp]: 894 "\<not> eventually (\<lambda>n. False) sequentially" 895 by (simp add: eventually_False) 896 897lemma le_sequentially: 898 "F \<le> sequentially \<longleftrightarrow> (\<forall>N. eventually (\<lambda>n. N \<le> n) F)" 899 by (simp add: at_top_def le_INF_iff le_principal) 900 901lemma eventually_sequentiallyI [intro?]: 902 assumes "\<And>x. c \<le> x \<Longrightarrow> P x" 903 shows "eventually P sequentially" 904using assms by (auto simp: eventually_sequentially) 905 906lemma eventually_sequentially_Suc [simp]: "eventually (\<lambda>i. P (Suc i)) sequentially \<longleftrightarrow> eventually P sequentially" 907 unfolding eventually_sequentially by (metis Suc_le_D Suc_le_mono le_Suc_eq) 908 909lemma eventually_sequentially_seg [simp]: "eventually (\<lambda>n. P (n + k)) sequentially \<longleftrightarrow> eventually P sequentially" 910 using eventually_sequentially_Suc[of "\<lambda>n. P (n + k)" for k] by (induction k) auto 911 912lemma filtermap_sequentually_ne_bot: "filtermap f sequentially \<noteq> bot" 913 by (simp add: filtermap_bot_iff) 914 915subsection \<open>Increasing finite subsets\<close> 916 917definition finite_subsets_at_top where 918 "finite_subsets_at_top A = (\<Sqinter> X\<in>{X. finite X \<and> X \<subseteq> A}. principal {Y. finite Y \<and> X \<subseteq> Y \<and> Y \<subseteq> A})" 919 920lemma eventually_finite_subsets_at_top: 921 "eventually P (finite_subsets_at_top A) \<longleftrightarrow> 922 (\<exists>X. finite X \<and> X \<subseteq> A \<and> (\<forall>Y. finite Y \<and> X \<subseteq> Y \<and> Y \<subseteq> A \<longrightarrow> P Y))" 923 unfolding finite_subsets_at_top_def 924proof (subst eventually_INF_base, goal_cases) 925 show "{X. finite X \<and> X \<subseteq> A} \<noteq> {}" by auto 926next 927 case (2 B C) 928 thus ?case by (intro bexI[of _ "B \<union> C"]) auto 929qed (simp_all add: eventually_principal) 930 931lemma eventually_finite_subsets_at_top_weakI [intro]: 932 assumes "\<And>X. finite X \<Longrightarrow> X \<subseteq> A \<Longrightarrow> P X" 933 shows "eventually P (finite_subsets_at_top A)" 934proof - 935 have "eventually (\<lambda>X. finite X \<and> X \<subseteq> A) (finite_subsets_at_top A)" 936 by (auto simp: eventually_finite_subsets_at_top) 937 thus ?thesis by eventually_elim (use assms in auto) 938qed 939 940lemma finite_subsets_at_top_neq_bot [simp]: "finite_subsets_at_top A \<noteq> bot" 941proof - 942 have "\<not>eventually (\<lambda>x. False) (finite_subsets_at_top A)" 943 by (auto simp: eventually_finite_subsets_at_top) 944 thus ?thesis by auto 945qed 946 947lemma filtermap_image_finite_subsets_at_top: 948 assumes "inj_on f A" 949 shows "filtermap ((`) f) (finite_subsets_at_top A) = finite_subsets_at_top (f ` A)" 950 unfolding filter_eq_iff eventually_filtermap 951proof (safe, goal_cases) 952 case (1 P) 953 then obtain X where X: "finite X" "X \<subseteq> A" "\<And>Y. finite Y \<Longrightarrow> X \<subseteq> Y \<Longrightarrow> Y \<subseteq> A \<Longrightarrow> P (f ` Y)" 954 unfolding eventually_finite_subsets_at_top by force 955 show ?case unfolding eventually_finite_subsets_at_top eventually_filtermap 956 proof (rule exI[of _ "f ` X"], intro conjI allI impI, goal_cases) 957 case (3 Y) 958 with assms and X(1,2) have "P (f ` (f -` Y \<inter> A))" using X(1,2) 959 by (intro X(3) finite_vimage_IntI) auto 960 also have "f ` (f -` Y \<inter> A) = Y" using assms 3 by blast 961 finally show ?case . 962 qed (insert assms X(1,2), auto intro!: finite_vimage_IntI) 963next 964 case (2 P) 965 then obtain X where X: "finite X" "X \<subseteq> f ` A" "\<And>Y. finite Y \<Longrightarrow> X \<subseteq> Y \<Longrightarrow> Y \<subseteq> f ` A \<Longrightarrow> P Y" 966 unfolding eventually_finite_subsets_at_top by force 967 show ?case unfolding eventually_finite_subsets_at_top eventually_filtermap 968 proof (rule exI[of _ "f -` X \<inter> A"], intro conjI allI impI, goal_cases) 969 case (3 Y) 970 with X(1,2) and assms show ?case by (intro X(3)) force+ 971 qed (insert assms X(1), auto intro!: finite_vimage_IntI) 972qed 973 974lemma eventually_finite_subsets_at_top_finite: 975 assumes "finite A" 976 shows "eventually P (finite_subsets_at_top A) \<longleftrightarrow> P A" 977 unfolding eventually_finite_subsets_at_top using assms by force 978 979lemma finite_subsets_at_top_finite: "finite A \<Longrightarrow> finite_subsets_at_top A = principal {A}" 980 by (auto simp: filter_eq_iff eventually_finite_subsets_at_top_finite eventually_principal) 981 982 983subsection \<open>The cofinite filter\<close> 984 985definition "cofinite = Abs_filter (\<lambda>P. finite {x. \<not> P x})" 986 987abbreviation Inf_many :: "('a \<Rightarrow> bool) \<Rightarrow> bool" (binder "\<exists>\<^sub>\<infinity>" 10) 988 where "Inf_many P \<equiv> frequently P cofinite" 989 990abbreviation Alm_all :: "('a \<Rightarrow> bool) \<Rightarrow> bool" (binder "\<forall>\<^sub>\<infinity>" 10) 991 where "Alm_all P \<equiv> eventually P cofinite" 992 993notation (ASCII) 994 Inf_many (binder "INFM " 10) and 995 Alm_all (binder "MOST " 10) 996 997lemma eventually_cofinite: "eventually P cofinite \<longleftrightarrow> finite {x. \<not> P x}" 998 unfolding cofinite_def 999proof (rule eventually_Abs_filter, rule is_filter.intro) 1000 fix P Q :: "'a \<Rightarrow> bool" assume "finite {x. \<not> P x}" "finite {x. \<not> Q x}" 1001 from finite_UnI[OF this] show "finite {x. \<not> (P x \<and> Q x)}" 1002 by (rule rev_finite_subset) auto 1003next 1004 fix P Q :: "'a \<Rightarrow> bool" assume P: "finite {x. \<not> P x}" and *: "\<forall>x. P x \<longrightarrow> Q x" 1005 from * show "finite {x. \<not> Q x}" 1006 by (intro finite_subset[OF _ P]) auto 1007qed simp 1008 1009lemma frequently_cofinite: "frequently P cofinite \<longleftrightarrow> \<not> finite {x. P x}" 1010 by (simp add: frequently_def eventually_cofinite) 1011 1012lemma cofinite_bot[simp]: "cofinite = (bot::'a filter) \<longleftrightarrow> finite (UNIV :: 'a set)" 1013 unfolding trivial_limit_def eventually_cofinite by simp 1014 1015lemma cofinite_eq_sequentially: "cofinite = sequentially" 1016 unfolding filter_eq_iff eventually_sequentially eventually_cofinite 1017proof safe 1018 fix P :: "nat \<Rightarrow> bool" assume [simp]: "finite {x. \<not> P x}" 1019 show "\<exists>N. \<forall>n\<ge>N. P n" 1020 proof cases 1021 assume "{x. \<not> P x} \<noteq> {}" then show ?thesis 1022 by (intro exI[of _ "Suc (Max {x. \<not> P x})"]) (auto simp: Suc_le_eq) 1023 qed auto 1024next 1025 fix P :: "nat \<Rightarrow> bool" and N :: nat assume "\<forall>n\<ge>N. P n" 1026 then have "{x. \<not> P x} \<subseteq> {..< N}" 1027 by (auto simp: not_le) 1028 then show "finite {x. \<not> P x}" 1029 by (blast intro: finite_subset) 1030qed 1031 1032subsubsection \<open>Product of filters\<close> 1033 1034definition prod_filter :: "'a filter \<Rightarrow> 'b filter \<Rightarrow> ('a \<times> 'b) filter" (infixr "\<times>\<^sub>F" 80) where 1035 "prod_filter F G = 1036 (\<Sqinter>(P, Q)\<in>{(P, Q). eventually P F \<and> eventually Q G}. principal {(x, y). P x \<and> Q y})" 1037 1038lemma eventually_prod_filter: "eventually P (F \<times>\<^sub>F G) \<longleftrightarrow> 1039 (\<exists>Pf Pg. eventually Pf F \<and> eventually Pg G \<and> (\<forall>x y. Pf x \<longrightarrow> Pg y \<longrightarrow> P (x, y)))" 1040 unfolding prod_filter_def 1041proof (subst eventually_INF_base, goal_cases) 1042 case 2 1043 moreover have "eventually Pf F \<Longrightarrow> eventually Qf F \<Longrightarrow> eventually Pg G \<Longrightarrow> eventually Qg G \<Longrightarrow> 1044 \<exists>P Q. eventually P F \<and> eventually Q G \<and> 1045 Collect P \<times> Collect Q \<subseteq> Collect Pf \<times> Collect Pg \<inter> Collect Qf \<times> Collect Qg" for Pf Pg Qf Qg 1046 by (intro conjI exI[of _ "inf Pf Qf"] exI[of _ "inf Pg Qg"]) 1047 (auto simp: inf_fun_def eventually_conj) 1048 ultimately show ?case 1049 by auto 1050qed (auto simp: eventually_principal intro: eventually_True) 1051 1052lemma eventually_prod1: 1053 assumes "B \<noteq> bot" 1054 shows "(\<forall>\<^sub>F (x, y) in A \<times>\<^sub>F B. P x) \<longleftrightarrow> (\<forall>\<^sub>F x in A. P x)" 1055 unfolding eventually_prod_filter 1056proof safe 1057 fix R Q 1058 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" 1059 with \<open>B \<noteq> bot\<close> obtain y where "Q y" by (auto dest: eventually_happens) 1060 with * show "eventually P A" 1061 by (force elim: eventually_mono) 1062next 1063 assume "eventually P A" 1064 then show "\<exists>Pf Pg. eventually Pf A \<and> eventually Pg B \<and> (\<forall>x y. Pf x \<longrightarrow> Pg y \<longrightarrow> P x)" 1065 by (intro exI[of _ P] exI[of _ "\<lambda>x. True"]) auto 1066qed 1067 1068lemma eventually_prod2: 1069 assumes "A \<noteq> bot" 1070 shows "(\<forall>\<^sub>F (x, y) in A \<times>\<^sub>F B. P y) \<longleftrightarrow> (\<forall>\<^sub>F y in B. P y)" 1071 unfolding eventually_prod_filter 1072proof safe 1073 fix R Q 1074 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" 1075 with \<open>A \<noteq> bot\<close> obtain x where "R x" by (auto dest: eventually_happens) 1076 with * show "eventually P B" 1077 by (force elim: eventually_mono) 1078next 1079 assume "eventually P B" 1080 then show "\<exists>Pf Pg. eventually Pf A \<and> eventually Pg B \<and> (\<forall>x y. Pf x \<longrightarrow> Pg y \<longrightarrow> P y)" 1081 by (intro exI[of _ P] exI[of _ "\<lambda>x. True"]) auto 1082qed 1083 1084lemma INF_filter_bot_base: 1085 fixes F :: "'a \<Rightarrow> 'b filter" 1086 assumes *: "\<And>i j. i \<in> I \<Longrightarrow> j \<in> I \<Longrightarrow> \<exists>k\<in>I. F k \<le> F i \<sqinter> F j" 1087 shows "(\<Sqinter>i\<in>I. F i) = bot \<longleftrightarrow> (\<exists>i\<in>I. F i = bot)" 1088proof (cases "\<exists>i\<in>I. F i = bot") 1089 case True 1090 then have "(\<Sqinter>i\<in>I. F i) \<le> bot" 1091 by (auto intro: INF_lower2) 1092 with True show ?thesis 1093 by (auto simp: bot_unique) 1094next 1095 case False 1096 moreover have "(\<Sqinter>i\<in>I. F i) \<noteq> bot" 1097 proof (cases "I = {}") 1098 case True 1099 then show ?thesis 1100 by (auto simp add: filter_eq_iff) 1101 next 1102 case False': False 1103 show ?thesis 1104 proof (rule INF_filter_not_bot) 1105 fix J 1106 assume "finite J" "J \<subseteq> I" 1107 then have "\<exists>k\<in>I. F k \<le> (\<Sqinter>i\<in>J. F i)" 1108 proof (induct J) 1109 case empty 1110 then show ?case 1111 using \<open>I \<noteq> {}\<close> by auto 1112 next 1113 case (insert i J) 1114 then obtain k where "k \<in> I" "F k \<le> (\<Sqinter>i\<in>J. F i)" by auto 1115 with insert *[of i k] show ?case 1116 by auto 1117 qed 1118 with False show "(\<Sqinter>i\<in>J. F i) \<noteq> \<bottom>" 1119 by (auto simp: bot_unique) 1120 qed 1121 qed 1122 ultimately show ?thesis 1123 by auto 1124qed 1125 1126lemma Collect_empty_eq_bot: "Collect P = {} \<longleftrightarrow> P = \<bottom>" 1127 by auto 1128 1129lemma prod_filter_eq_bot: "A \<times>\<^sub>F B = bot \<longleftrightarrow> A = bot \<or> B = bot" 1130 unfolding trivial_limit_def 1131proof 1132 assume "\<forall>\<^sub>F x in A \<times>\<^sub>F B. False" 1133 then obtain Pf Pg 1134 where Pf: "eventually (\<lambda>x. Pf x) A" and Pg: "eventually (\<lambda>y. Pg y) B" 1135 and *: "\<forall>x y. Pf x \<longrightarrow> Pg y \<longrightarrow> False" 1136 unfolding eventually_prod_filter by fast 1137 from * have "(\<forall>x. \<not> Pf x) \<or> (\<forall>y. \<not> Pg y)" by fast 1138 with Pf Pg show "(\<forall>\<^sub>F x in A. False) \<or> (\<forall>\<^sub>F x in B. False)" by auto 1139next 1140 assume "(\<forall>\<^sub>F x in A. False) \<or> (\<forall>\<^sub>F x in B. False)" 1141 then show "\<forall>\<^sub>F x in A \<times>\<^sub>F B. False" 1142 unfolding eventually_prod_filter by (force intro: eventually_True) 1143qed 1144 1145lemma prod_filter_mono: "F \<le> F' \<Longrightarrow> G \<le> G' \<Longrightarrow> F \<times>\<^sub>F G \<le> F' \<times>\<^sub>F G'" 1146 by (auto simp: le_filter_def eventually_prod_filter) 1147 1148lemma prod_filter_mono_iff: 1149 assumes nAB: "A \<noteq> bot" "B \<noteq> bot" 1150 shows "A \<times>\<^sub>F B \<le> C \<times>\<^sub>F D \<longleftrightarrow> A \<le> C \<and> B \<le> D" 1151proof safe 1152 assume *: "A \<times>\<^sub>F B \<le> C \<times>\<^sub>F D" 1153 with assms have "A \<times>\<^sub>F B \<noteq> bot" 1154 by (auto simp: bot_unique prod_filter_eq_bot) 1155 with * have "C \<times>\<^sub>F D \<noteq> bot" 1156 by (auto simp: bot_unique) 1157 then have nCD: "C \<noteq> bot" "D \<noteq> bot" 1158 by (auto simp: prod_filter_eq_bot) 1159 1160 show "A \<le> C" 1161 proof (rule filter_leI) 1162 fix P assume "eventually P C" with *[THEN filter_leD, of "\<lambda>(x, y). P x"] show "eventually P A" 1163 using nAB nCD by (simp add: eventually_prod1 eventually_prod2) 1164 qed 1165 1166 show "B \<le> D" 1167 proof (rule filter_leI) 1168 fix P assume "eventually P D" with *[THEN filter_leD, of "\<lambda>(x, y). P y"] show "eventually P B" 1169 using nAB nCD by (simp add: eventually_prod1 eventually_prod2) 1170 qed 1171qed (intro prod_filter_mono) 1172 1173lemma eventually_prod_same: "eventually P (F \<times>\<^sub>F F) \<longleftrightarrow> 1174 (\<exists>Q. eventually Q F \<and> (\<forall>x y. Q x \<longrightarrow> Q y \<longrightarrow> P (x, y)))" 1175 unfolding eventually_prod_filter 1176 apply safe 1177 apply (rule_tac x="inf Pf Pg" in exI) 1178 apply (auto simp: inf_fun_def intro!: eventually_conj) 1179 done 1180 1181lemma eventually_prod_sequentially: 1182 "eventually P (sequentially \<times>\<^sub>F sequentially) \<longleftrightarrow> (\<exists>N. \<forall>m \<ge> N. \<forall>n \<ge> N. P (n, m))" 1183 unfolding eventually_prod_same eventually_sequentially by auto 1184 1185lemma principal_prod_principal: "principal A \<times>\<^sub>F principal B = principal (A \<times> B)" 1186 unfolding filter_eq_iff eventually_prod_filter eventually_principal 1187 by (fast intro: exI[of _ "\<lambda>x. x \<in> A"] exI[of _ "\<lambda>x. x \<in> B"]) 1188 1189lemma le_prod_filterI: 1190 "filtermap fst F \<le> A \<Longrightarrow> filtermap snd F \<le> B \<Longrightarrow> F \<le> A \<times>\<^sub>F B" 1191 unfolding le_filter_def eventually_filtermap eventually_prod_filter 1192 by (force elim: eventually_elim2) 1193 1194lemma filtermap_fst_prod_filter: "filtermap fst (A \<times>\<^sub>F B) \<le> A" 1195 unfolding le_filter_def eventually_filtermap eventually_prod_filter 1196 by (force intro: eventually_True) 1197 1198lemma filtermap_snd_prod_filter: "filtermap snd (A \<times>\<^sub>F B) \<le> B" 1199 unfolding le_filter_def eventually_filtermap eventually_prod_filter 1200 by (force intro: eventually_True) 1201 1202lemma prod_filter_INF: 1203 assumes "I \<noteq> {}" and "J \<noteq> {}" 1204 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)" 1205proof (rule antisym) 1206 from \<open>I \<noteq> {}\<close> obtain i where "i \<in> I" by auto 1207 from \<open>J \<noteq> {}\<close> obtain j where "j \<in> J" by auto 1208 1209 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)" 1210 by (fast intro: le_prod_filterI INF_greatest INF_lower2 1211 order_trans[OF filtermap_INF] \<open>i \<in> I\<close> \<open>j \<in> J\<close> 1212 filtermap_fst_prod_filter filtermap_snd_prod_filter) 1213 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)" 1214 by (intro INF_greatest prod_filter_mono INF_lower) 1215qed 1216 1217lemma filtermap_Pair: "filtermap (\<lambda>x. (f x, g x)) F \<le> filtermap f F \<times>\<^sub>F filtermap g F" 1218 by (rule le_prod_filterI, simp_all add: filtermap_filtermap) 1219 1220lemma 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)" 1221 unfolding eventually_prod_filter by auto 1222 1223lemma 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)" 1224 using prod_filter_INF[of I "{B}" A "\<lambda>x. x"] by simp 1225 1226lemma 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)" 1227 using prod_filter_INF[of "{A}" J "\<lambda>x. x" B] by simp 1228 1229lemma prod_filtermap1: "prod_filter (filtermap f F) G = filtermap (apfst f) (prod_filter F G)" 1230 apply(clarsimp simp add: filter_eq_iff eventually_filtermap eventually_prod_filter; safe) 1231 subgoal by auto 1232 subgoal for P Q R by(rule exI[where x="\<lambda>y. \<exists>x. y = f x \<and> Q x"])(auto intro: eventually_mono) 1233 done 1234 1235lemma prod_filtermap2: "prod_filter F (filtermap g G) = filtermap (apsnd g) (prod_filter F G)" 1236 apply(clarsimp simp add: filter_eq_iff eventually_filtermap eventually_prod_filter; safe) 1237 subgoal by auto 1238 subgoal for P Q R by(auto intro: exI[where x="\<lambda>y. \<exists>x. y = g x \<and> R x"] eventually_mono) 1239 done 1240 1241lemma prod_filter_assoc: 1242 "prod_filter (prod_filter F G) H = filtermap (\<lambda>(x, y, z). ((x, y), z)) (prod_filter F (prod_filter G H))" 1243 apply(clarsimp simp add: filter_eq_iff eventually_filtermap eventually_prod_filter; safe) 1244 subgoal for P Q R S T by(auto 4 4 intro: exI[where x="\<lambda>(a, b). T a \<and> S b"]) 1245 subgoal for P Q R S T by(auto 4 3 intro: exI[where x="\<lambda>(a, b). Q a \<and> S b"]) 1246 done 1247 1248lemma prod_filter_principal_singleton: "prod_filter (principal {x}) F = filtermap (Pair x) F" 1249 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"]) 1250 1251lemma prod_filter_principal_singleton2: "prod_filter F (principal {x}) = filtermap (\<lambda>a. (a, x)) F" 1252 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"]) 1253 1254lemma prod_filter_commute: "prod_filter F G = filtermap prod.swap (prod_filter G F)" 1255 by(auto simp add: filter_eq_iff eventually_prod_filter eventually_filtermap) 1256 1257subsection \<open>Limits\<close> 1258 1259definition filterlim :: "('a \<Rightarrow> 'b) \<Rightarrow> 'b filter \<Rightarrow> 'a filter \<Rightarrow> bool" where 1260 "filterlim f F2 F1 \<longleftrightarrow> filtermap f F1 \<le> F2" 1261 1262syntax 1263 "_LIM" :: "pttrns \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> bool" ("(3LIM (_)/ (_)./ (_) :> (_))" [1000, 10, 0, 10] 10) 1264 1265translations 1266 "LIM x F1. f :> F2" == "CONST filterlim (\<lambda>x. f) F2 F1" 1267 1268lemma filterlim_top [simp]: "filterlim f top F" 1269 by (simp add: filterlim_def) 1270 1271lemma filterlim_iff: 1272 "(LIM x F1. f x :> F2) \<longleftrightarrow> (\<forall>P. eventually P F2 \<longrightarrow> eventually (\<lambda>x. P (f x)) F1)" 1273 unfolding filterlim_def le_filter_def eventually_filtermap .. 1274 1275lemma filterlim_compose: 1276 "filterlim g F3 F2 \<Longrightarrow> filterlim f F2 F1 \<Longrightarrow> filterlim (\<lambda>x. g (f x)) F3 F1" 1277 unfolding filterlim_def filtermap_filtermap[symmetric] by (metis filtermap_mono order_trans) 1278 1279lemma filterlim_mono: 1280 "filterlim f F2 F1 \<Longrightarrow> F2 \<le> F2' \<Longrightarrow> F1' \<le> F1 \<Longrightarrow> filterlim f F2' F1'" 1281 unfolding filterlim_def by (metis filtermap_mono order_trans) 1282 1283lemma filterlim_ident: "LIM x F. x :> F" 1284 by (simp add: filterlim_def filtermap_ident) 1285 1286lemma filterlim_cong: 1287 "F1 = F1' \<Longrightarrow> F2 = F2' \<Longrightarrow> eventually (\<lambda>x. f x = g x) F2 \<Longrightarrow> filterlim f F1 F2 = filterlim g F1' F2'" 1288 by (auto simp: filterlim_def le_filter_def eventually_filtermap elim: eventually_elim2) 1289 1290lemma filterlim_mono_eventually: 1291 assumes "filterlim f F G" and ord: "F \<le> F'" "G' \<le> G" 1292 assumes eq: "eventually (\<lambda>x. f x = f' x) G'" 1293 shows "filterlim f' F' G'" 1294 apply (rule filterlim_cong[OF refl refl eq, THEN iffD1]) 1295 apply (rule filterlim_mono[OF _ ord]) 1296 apply fact 1297 done 1298 1299lemma filtermap_mono_strong: "inj f \<Longrightarrow> filtermap f F \<le> filtermap f G \<longleftrightarrow> F \<le> G" 1300 apply (safe intro!: filtermap_mono) 1301 apply (auto simp: le_filter_def eventually_filtermap) 1302 apply (erule_tac x="\<lambda>x. P (inv f x)" in allE) 1303 apply auto 1304 done 1305 1306lemma eventually_compose_filterlim: 1307 assumes "eventually P F" "filterlim f F G" 1308 shows "eventually (\<lambda>x. P (f x)) G" 1309 using assms by (simp add: filterlim_iff) 1310 1311lemma filtermap_eq_strong: "inj f \<Longrightarrow> filtermap f F = filtermap f G \<longleftrightarrow> F = G" 1312 by (simp add: filtermap_mono_strong eq_iff) 1313 1314lemma filtermap_fun_inverse: 1315 assumes g: "filterlim g F G" 1316 assumes f: "filterlim f G F" 1317 assumes ev: "eventually (\<lambda>x. f (g x) = x) G" 1318 shows "filtermap f F = G" 1319proof (rule antisym) 1320 show "filtermap f F \<le> G" 1321 using f unfolding filterlim_def . 1322 have "G = filtermap f (filtermap g G)" 1323 using ev by (auto elim: eventually_elim2 simp: filter_eq_iff eventually_filtermap) 1324 also have "\<dots> \<le> filtermap f F" 1325 using g by (intro filtermap_mono) (simp add: filterlim_def) 1326 finally show "G \<le> filtermap f F" . 1327qed 1328 1329lemma filterlim_principal: 1330 "(LIM x F. f x :> principal S) \<longleftrightarrow> (eventually (\<lambda>x. f x \<in> S) F)" 1331 unfolding filterlim_def eventually_filtermap le_principal .. 1332 1333lemma filterlim_filtercomap [intro]: "filterlim f F (filtercomap f F)" 1334 unfolding filterlim_def by (rule filtermap_filtercomap) 1335 1336lemma filterlim_inf: 1337 "(LIM x F1. f x :> inf F2 F3) \<longleftrightarrow> ((LIM x F1. f x :> F2) \<and> (LIM x F1. f x :> F3))" 1338 unfolding filterlim_def by simp 1339 1340lemma filterlim_INF: 1341 "(LIM x F. f x :> (\<Sqinter>b\<in>B. G b)) \<longleftrightarrow> (\<forall>b\<in>B. LIM x F. f x :> G b)" 1342 unfolding filterlim_def le_INF_iff .. 1343 1344lemma filterlim_INF_INF: 1345 "(\<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)" 1346 unfolding filterlim_def by (rule order_trans[OF filtermap_INF INF_mono]) 1347 1348lemma filterlim_INF': "x \<in> A \<Longrightarrow> filterlim f F (G x) \<Longrightarrow> filterlim f F (\<Sqinter> x\<in>A. G x)" 1349 unfolding filterlim_def by (rule order.trans[OF filtermap_mono[OF INF_lower]]) 1350 1351lemma filterlim_filtercomap_iff: "filterlim f (filtercomap g G) F \<longleftrightarrow> filterlim (g \<circ> f) G F" 1352 by (simp add: filterlim_def filtermap_le_iff_le_filtercomap filtercomap_filtercomap o_def) 1353 1354lemma filterlim_iff_le_filtercomap: "filterlim f F G \<longleftrightarrow> G \<le> filtercomap f F" 1355 by (simp add: filterlim_def filtermap_le_iff_le_filtercomap) 1356 1357lemma filterlim_base: 1358 "(\<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> 1359 LIM x (\<Sqinter>i\<in>I. principal (F i)). f x :> (\<Sqinter>j\<in>J. principal (G j))" 1360 by (force intro!: filterlim_INF_INF simp: image_subset_iff) 1361 1362lemma filterlim_base_iff: 1363 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" 1364 shows "(LIM x (\<Sqinter>i\<in>I. principal (F i)). f x :> \<Sqinter>j\<in>J. principal (G j)) \<longleftrightarrow> 1365 (\<forall>j\<in>J. \<exists>i\<in>I. \<forall>x\<in>F i. f x \<in> G j)" 1366 unfolding filterlim_INF filterlim_principal 1367proof (subst eventually_INF_base) 1368 fix i j assume "i \<in> I" "j \<in> I" 1369 with chain[OF this] show "\<exists>x\<in>I. principal (F x) \<le> inf (principal (F i)) (principal (F j))" 1370 by auto 1371qed (auto simp: eventually_principal \<open>I \<noteq> {}\<close>) 1372 1373lemma filterlim_filtermap: "filterlim f F1 (filtermap g F2) = filterlim (\<lambda>x. f (g x)) F1 F2" 1374 unfolding filterlim_def filtermap_filtermap .. 1375 1376lemma filterlim_sup: 1377 "filterlim f F F1 \<Longrightarrow> filterlim f F F2 \<Longrightarrow> filterlim f F (sup F1 F2)" 1378 unfolding filterlim_def filtermap_sup by auto 1379 1380lemma filterlim_sequentially_Suc: 1381 "(LIM x sequentially. f (Suc x) :> F) \<longleftrightarrow> (LIM x sequentially. f x :> F)" 1382 unfolding filterlim_iff by (subst eventually_sequentially_Suc) simp 1383 1384lemma filterlim_Suc: "filterlim Suc sequentially sequentially" 1385 by (simp add: filterlim_iff eventually_sequentially) 1386 1387lemma filterlim_If: 1388 "LIM x inf F (principal {x. P x}). f x :> G \<Longrightarrow> 1389 LIM x inf F (principal {x. \<not> P x}). g x :> G \<Longrightarrow> 1390 LIM x F. if P x then f x else g x :> G" 1391 unfolding filterlim_iff eventually_inf_principal by (auto simp: eventually_conj_iff) 1392 1393lemma filterlim_Pair: 1394 "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" 1395 unfolding filterlim_def 1396 by (rule order_trans[OF filtermap_Pair prod_filter_mono]) 1397 1398subsection \<open>Limits to \<^const>\<open>at_top\<close> and \<^const>\<open>at_bot\<close>\<close> 1399 1400lemma filterlim_at_top: 1401 fixes f :: "'a \<Rightarrow> ('b::linorder)" 1402 shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. Z \<le> f x) F)" 1403 by (auto simp: filterlim_iff eventually_at_top_linorder elim!: eventually_mono) 1404 1405lemma filterlim_at_top_mono: 1406 "LIM x F. f x :> at_top \<Longrightarrow> eventually (\<lambda>x. f x \<le> (g x::'a::linorder)) F \<Longrightarrow> 1407 LIM x F. g x :> at_top" 1408 by (auto simp: filterlim_at_top elim: eventually_elim2 intro: order_trans) 1409 1410lemma filterlim_at_top_dense: 1411 fixes f :: "'a \<Rightarrow> ('b::unbounded_dense_linorder)" 1412 shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. Z < f x) F)" 1413 by (metis eventually_mono[of _ F] eventually_gt_at_top order_less_imp_le 1414 filterlim_at_top[of f F] filterlim_iff[of f at_top F]) 1415 1416lemma filterlim_at_top_ge: 1417 fixes f :: "'a \<Rightarrow> ('b::linorder)" and c :: "'b" 1418 shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z\<ge>c. eventually (\<lambda>x. Z \<le> f x) F)" 1419 unfolding at_top_sub[of c] filterlim_INF by (auto simp add: filterlim_principal) 1420 1421lemma filterlim_at_top_at_top: 1422 fixes f :: "'a::linorder \<Rightarrow> 'b::linorder" 1423 assumes mono: "\<And>x y. Q x \<Longrightarrow> Q y \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y" 1424 assumes bij: "\<And>x. P x \<Longrightarrow> f (g x) = x" "\<And>x. P x \<Longrightarrow> Q (g x)" 1425 assumes Q: "eventually Q at_top" 1426 assumes P: "eventually P at_top" 1427 shows "filterlim f at_top at_top" 1428proof - 1429 from P obtain x where x: "\<And>y. x \<le> y \<Longrightarrow> P y" 1430 unfolding eventually_at_top_linorder by auto 1431 show ?thesis 1432 proof (intro filterlim_at_top_ge[THEN iffD2] allI impI) 1433 fix z assume "x \<le> z" 1434 with x have "P z" by auto 1435 have "eventually (\<lambda>x. g z \<le> x) at_top" 1436 by (rule eventually_ge_at_top) 1437 with Q show "eventually (\<lambda>x. z \<le> f x) at_top" 1438 by eventually_elim (metis mono bij \<open>P z\<close>) 1439 qed 1440qed 1441 1442lemma filterlim_at_top_gt: 1443 fixes f :: "'a \<Rightarrow> ('b::unbounded_dense_linorder)" and c :: "'b" 1444 shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z>c. eventually (\<lambda>x. Z \<le> f x) F)" 1445 by (metis filterlim_at_top order_less_le_trans gt_ex filterlim_at_top_ge) 1446 1447lemma filterlim_at_bot: 1448 fixes f :: "'a \<Rightarrow> ('b::linorder)" 1449 shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. f x \<le> Z) F)" 1450 by (auto simp: filterlim_iff eventually_at_bot_linorder elim!: eventually_mono) 1451 1452lemma filterlim_at_bot_dense: 1453 fixes f :: "'a \<Rightarrow> ('b::{dense_linorder, no_bot})" 1454 shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. f x < Z) F)" 1455proof (auto simp add: filterlim_at_bot[of f F]) 1456 fix Z :: 'b 1457 from lt_ex [of Z] obtain Z' where 1: "Z' < Z" .. 1458 assume "\<forall>Z. eventually (\<lambda>x. f x \<le> Z) F" 1459 hence "eventually (\<lambda>x. f x \<le> Z') F" by auto 1460 thus "eventually (\<lambda>x. f x < Z) F" 1461 apply (rule eventually_mono) 1462 using 1 by auto 1463 next 1464 fix Z :: 'b 1465 show "\<forall>Z. eventually (\<lambda>x. f x < Z) F \<Longrightarrow> eventually (\<lambda>x. f x \<le> Z) F" 1466 by (drule spec [of _ Z], erule eventually_mono, auto simp add: less_imp_le) 1467qed 1468 1469lemma filterlim_at_bot_le: 1470 fixes f :: "'a \<Rightarrow> ('b::linorder)" and c :: "'b" 1471 shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z\<le>c. eventually (\<lambda>x. Z \<ge> f x) F)" 1472 unfolding filterlim_at_bot 1473proof safe 1474 fix Z assume *: "\<forall>Z\<le>c. eventually (\<lambda>x. Z \<ge> f x) F" 1475 with *[THEN spec, of "min Z c"] show "eventually (\<lambda>x. Z \<ge> f x) F" 1476 by (auto elim!: eventually_mono) 1477qed simp 1478 1479lemma filterlim_at_bot_lt: 1480 fixes f :: "'a \<Rightarrow> ('b::unbounded_dense_linorder)" and c :: "'b" 1481 shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z<c. eventually (\<lambda>x. Z \<ge> f x) F)" 1482 by (metis filterlim_at_bot filterlim_at_bot_le lt_ex order_le_less_trans) 1483 1484lemma filterlim_finite_subsets_at_top: 1485 "filterlim f (finite_subsets_at_top A) F \<longleftrightarrow> 1486 (\<forall>X. finite X \<and> X \<subseteq> A \<longrightarrow> eventually (\<lambda>y. finite (f y) \<and> X \<subseteq> f y \<and> f y \<subseteq> A) F)" 1487 (is "?lhs = ?rhs") 1488proof 1489 assume ?lhs 1490 thus ?rhs 1491 proof (safe, goal_cases) 1492 case (1 X) 1493 hence *: "(\<forall>\<^sub>F x in F. P (f x))" if "eventually P (finite_subsets_at_top A)" for P 1494 using that by (auto simp: filterlim_def le_filter_def eventually_filtermap) 1495 have "\<forall>\<^sub>F Y in finite_subsets_at_top A. finite Y \<and> X \<subseteq> Y \<and> Y \<subseteq> A" 1496 using 1 unfolding eventually_finite_subsets_at_top by force 1497 thus ?case by (intro *) auto 1498 qed 1499next 1500 assume rhs: ?rhs 1501 show ?lhs unfolding filterlim_def le_filter_def eventually_finite_subsets_at_top 1502 proof (safe, goal_cases) 1503 case (1 P X) 1504 with rhs have "\<forall>\<^sub>F y in F. finite (f y) \<and> X \<subseteq> f y \<and> f y \<subseteq> A" by auto 1505 thus "eventually P (filtermap f F)" unfolding eventually_filtermap 1506 by eventually_elim (insert 1, auto) 1507 qed 1508qed 1509 1510lemma filterlim_atMost_at_top: 1511 "filterlim (\<lambda>n. {..n}) (finite_subsets_at_top (UNIV :: nat set)) at_top" 1512 unfolding filterlim_finite_subsets_at_top 1513proof (safe, goal_cases) 1514 case (1 X) 1515 then obtain n where n: "X \<subseteq> {..n}" by (auto simp: finite_nat_set_iff_bounded_le) 1516 show ?case using eventually_ge_at_top[of n] 1517 by eventually_elim (insert n, auto) 1518qed 1519 1520lemma filterlim_lessThan_at_top: 1521 "filterlim (\<lambda>n. {..<n}) (finite_subsets_at_top (UNIV :: nat set)) at_top" 1522 unfolding filterlim_finite_subsets_at_top 1523proof (safe, goal_cases) 1524 case (1 X) 1525 then obtain n where n: "X \<subseteq> {..<n}" by (auto simp: finite_nat_set_iff_bounded) 1526 show ?case using eventually_ge_at_top[of n] 1527 by eventually_elim (insert n, auto) 1528qed 1529 1530subsection \<open>Setup \<^typ>\<open>'a filter\<close> for lifting and transfer\<close> 1531 1532lemma filtermap_id [simp, id_simps]: "filtermap id = id" 1533by(simp add: fun_eq_iff id_def filtermap_ident) 1534 1535lemma filtermap_id' [simp]: "filtermap (\<lambda>x. x) = (\<lambda>F. F)" 1536using filtermap_id unfolding id_def . 1537 1538context includes lifting_syntax 1539begin 1540 1541definition map_filter_on :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a filter \<Rightarrow> 'b filter" where 1542 "map_filter_on X f F = Abs_filter (\<lambda>P. eventually (\<lambda>x. P (f x) \<and> x \<in> X) F)" 1543 1544lemma is_filter_map_filter_on: 1545 "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" 1546proof(rule iffI; unfold_locales) 1547 show "\<forall>\<^sub>F x in F. True \<and> x \<in> X" if "eventually (\<lambda>x. x \<in> X) F" using that by simp 1548 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 1549 using eventually_conj[OF that] by(auto simp add: conj_ac cong: conj_cong) 1550 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 1551 using that(2) by(rule eventually_mono)(use that(1) in auto) 1552 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)" 1553 using is_filter.True[OF that] by simp 1554qed 1555 1556lemma 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)" 1557 if "eventually (\<lambda>x. x \<in> X) F" 1558 by(simp add: is_filter_map_filter_on map_filter_on_def eventually_Abs_filter that) 1559 1560lemma map_filter_on_UNIV: "map_filter_on UNIV = filtermap" 1561 by(simp add: map_filter_on_def filtermap_def fun_eq_iff) 1562 1563lemma map_filter_on_comp: "map_filter_on X f (map_filter_on Y g F) = map_filter_on Y (f \<circ> g) F" 1564 if "g ` Y \<subseteq> X" and "eventually (\<lambda>x. x \<in> Y) F" 1565 unfolding map_filter_on_def using that(1) 1566 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]) 1567 1568inductive rel_filter :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a filter \<Rightarrow> 'b filter \<Rightarrow> bool" for R F G where 1569 "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" 1570 1571lemma rel_filter_eq [relator_eq]: "rel_filter (=) = (=)" 1572proof(intro ext iffI)+ 1573 show "F = G" if "rel_filter (=) F G" for F G using that 1574 by cases(clarsimp simp add: filter_eq_iff eventually_map_filter_on split_def cong: rev_conj_cong) 1575 show "rel_filter (=) F G" if "F = G" for F G unfolding \<open>F = G\<close> 1576 proof 1577 let ?Z = "map_filter_on UNIV (\<lambda>x. (x, x)) G" 1578 have [simp]: "range (\<lambda>x. (x, x)) \<subseteq> {(x, y). x = y}" by auto 1579 show "map_filter_on {(x, y). x = y} fst ?Z = G" and "map_filter_on {(x, y). x = y} snd ?Z = G" 1580 by(simp_all add: map_filter_on_comp)(simp_all add: map_filter_on_UNIV o_def) 1581 show "\<forall>\<^sub>F (x, y) in ?Z. x = y" by(simp add: eventually_map_filter_on) 1582 qed 1583qed 1584 1585lemma rel_filter_mono [relator_mono]: "rel_filter A \<le> rel_filter B" if le: "A \<le> B" 1586proof(clarify elim!: rel_filter.cases) 1587 show "rel_filter B (map_filter_on {(x, y). A x y} fst Z) (map_filter_on {(x, y). A x y} snd Z)" 1588 (is "rel_filter _ ?X ?Y") if "\<forall>\<^sub>F (x, y) in Z. A x y" for Z 1589 proof 1590 let ?Z = "map_filter_on {(x, y). A x y} id Z" 1591 show "\<forall>\<^sub>F (x, y) in ?Z. B x y" using le that 1592 by(simp add: eventually_map_filter_on le_fun_def split_def conj_commute cong: conj_cong) 1593 have [simp]: "{(x, y). A x y} \<subseteq> {(x, y). B x y}" using le by auto 1594 show "map_filter_on {(x, y). B x y} fst ?Z = ?X" "map_filter_on {(x, y). B x y} snd ?Z = ?Y" 1595 using le that by(simp_all add: le_fun_def map_filter_on_comp) 1596 qed 1597qed 1598 1599lemma rel_filter_conversep: "rel_filter A\<inverse>\<inverse> = (rel_filter A)\<inverse>\<inverse>" 1600proof(safe intro!: ext elim!: rel_filter.cases) 1601 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)" 1602 (is "rel_filter _ ?X ?Y") if "\<forall>\<^sub>F (x, y) in Z. A\<inverse>\<inverse> x y" for A Z 1603 proof 1604 let ?Z = "map_filter_on {(x, y). A y x} prod.swap Z" 1605 show "\<forall>\<^sub>F (x, y) in ?Z. A x y" using that by(simp add: eventually_map_filter_on) 1606 have [simp]: "prod.swap ` {(x, y). A y x} \<subseteq> {(x, y). A x y}" by auto 1607 show "map_filter_on {(x, y). A x y} fst ?Z = ?X" "map_filter_on {(x, y). A x y} snd ?Z = ?Y" 1608 using that by(simp_all add: map_filter_on_comp o_def) 1609 qed 1610 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)" 1611 if "\<forall>\<^sub>F (x, y) in Z. A x y" for Z using *[of "A\<inverse>\<inverse>" Z] that by simp 1612qed 1613 1614lemma rel_filter_distr [relator_distr]: 1615 "rel_filter A OO rel_filter B = rel_filter (A OO B)" 1616proof(safe intro!: ext elim!: rel_filter.cases) 1617 let ?AB = "{(x, y). (A OO B) x y}" 1618 show "(rel_filter A OO rel_filter B) 1619 (map_filter_on {(x, y). (A OO B) x y} fst Z) (map_filter_on {(x, y). (A OO B) x y} snd Z)" 1620 (is "(_ OO _) ?F ?H") if "\<forall>\<^sub>F (x, y) in Z. (A OO B) x y" for Z 1621 proof 1622 let ?G = "map_filter_on ?AB (\<lambda>(x, y). SOME z. A x z \<and> B z y) Z" 1623 show "rel_filter A ?F ?G" 1624 proof 1625 let ?Z = "map_filter_on ?AB (\<lambda>(x, y). (x, SOME z. A x z \<and> B z y)) Z" 1626 show "\<forall>\<^sub>F (x, y) in ?Z. A x y" using that 1627 by(auto simp add: eventually_map_filter_on split_def elim!: eventually_mono intro: someI2) 1628 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) 1629 show "map_filter_on {(x, y). A x y} fst ?Z = ?F" "map_filter_on {(x, y). A x y} snd ?Z = ?G" 1630 using that by(simp_all add: map_filter_on_comp split_def o_def) 1631 qed 1632 show "rel_filter B ?G ?H" 1633 proof 1634 let ?Z = "map_filter_on ?AB (\<lambda>(x, y). (SOME z. A x z \<and> B z y, y)) Z" 1635 show "\<forall>\<^sub>F (x, y) in ?Z. B x y" using that 1636 by(auto simp add: eventually_map_filter_on split_def elim!: eventually_mono intro: someI2) 1637 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) 1638 show "map_filter_on {(x, y). B x y} fst ?Z = ?G" "map_filter_on {(x, y). B x y} snd ?Z = ?H" 1639 using that by(simp_all add: map_filter_on_comp split_def o_def) 1640 qed 1641 qed 1642 1643 fix F G 1644 assume F: "\<forall>\<^sub>F (x, y) in F. A x y" and G: "\<forall>\<^sub>F (x, y) in G. B x y" 1645 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") 1646 let ?X = "map_filter_on {(x, y). A x y} fst F" 1647 and ?Z = "(map_filter_on {(x, y). B x y} snd G)" 1648 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)}" 1649 if P: "eventually P F" and Q: "eventually Q G" for P Q 1650 proof - 1651 let ?P = "\<lambda>(x, y). P (x, y) \<and> A x y" and ?Q = "\<lambda>(y, z). Q (y, z) \<and> B y z" 1652 define P' where "P' \<equiv> \<lambda>(x, y). ?P (x, y) \<and> (\<exists>z. ?Q (y, z))" 1653 define Q' where "Q' \<equiv> \<lambda>(y, z). ?Q (y, z) \<and> (\<exists>x. ?P (x, y))" 1654 have "P' \<le> P" "Q' \<le> Q" "{y. \<exists>x. P' (x, y)} = {y. \<exists>z. Q' (y, z)}" 1655 by(auto simp add: P'_def Q'_def) 1656 moreover 1657 from P Q F G have P': "eventually ?P F" and Q': "eventually ?Q G" 1658 by(simp_all add: eventually_conj_iff split_def) 1659 from P' F have "\<forall>\<^sub>F y in ?Y1. \<exists>x. P (x, y) \<and> A x y" 1660 by(auto simp add: eventually_map_filter_on elim!: eventually_mono) 1661 from this[folded eq] obtain Q'' where Q'': "eventually Q'' G" 1662 and Q''P: "{y. \<exists>z. Q'' (y, z)} \<subseteq> {y. \<exists>x. ?P (x, y)}" 1663 using G by(fastforce simp add: eventually_map_filter_on) 1664 have "eventually (inf Q'' ?Q) G" using Q'' Q' by(auto intro: eventually_conj simp add: inf_fun_def) 1665 then have "eventually Q' G" using Q''P by(auto elim!: eventually_mono simp add: Q'_def) 1666 moreover 1667 from Q' G have "\<forall>\<^sub>F y in ?Y2. \<exists>z. Q (y, z) \<and> B y z" 1668 by(auto simp add: eventually_map_filter_on elim!: eventually_mono) 1669 from this[unfolded eq] obtain P'' where P'': "eventually P'' F" 1670 and P''Q: "{y. \<exists>x. P'' (x, y)} \<subseteq> {y. \<exists>z. ?Q (y, z)}" 1671 using F by(fastforce simp add: eventually_map_filter_on) 1672 have "eventually (inf P'' ?P) F" using P'' P' by(auto intro: eventually_conj simp add: inf_fun_def) 1673 then have "eventually P' F" using P''Q by(auto elim!: eventually_mono simp add: P'_def) 1674 ultimately show ?thesis by blast 1675 qed 1676 1677 show "rel_filter (A OO B) ?X ?Z" 1678 proof 1679 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" 1680 have Y: "is_filter ?Y" 1681 proof 1682 show "?Y (\<lambda>_. True)" by(auto simp add: le_fun_def intro: eventually_True) 1683 show "?Y (\<lambda>x. P x \<and> Q x)" if "?Y P" "?Y Q" for P Q using that 1684 apply clarify 1685 apply(intro exI conjI; (elim eventually_rev_mp; fold imp_conjL; intro always_eventually allI; rule imp_refl)?) 1686 apply auto 1687 done 1688 show "?Y Q" if "?Y P" "\<forall>x. P x \<longrightarrow> Q x" for P Q using that by blast 1689 qed 1690 define Y where "Y = Abs_filter ?Y" 1691 have eventually_Y: "eventually P Y \<longleftrightarrow> ?Y P" for P 1692 using eventually_Abs_filter[OF Y, of P] by(simp add: Y_def) 1693 show YY: "\<forall>\<^sub>F (x, y) in Y. (A OO B) x y" using F G 1694 by(auto simp add: eventually_Y eventually_map_filter_on eventually_conj_iff intro!: eventually_True) 1695 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 1696 proof 1697 show ?lhs if ?rhs using G F that 1698 by(auto 4 3 intro: exI[where x="\<lambda>_. True"] simp add: eventually_map_filter_on split_def) 1699 assume ?lhs 1700 then obtain X Z where "\<forall>\<^sub>F (x, y) in F. X x \<and> A x y" 1701 and "\<forall>\<^sub>F (x, y) in G. Z y \<and> B x y" 1702 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)" 1703 using F G by(auto simp add: eventually_map_filter_on split_def) 1704 from step[OF this(1, 2)] this(3) 1705 show ?rhs by(clarsimp elim!: eventually_rev_mp simp add: le_fun_def)(fastforce intro: always_eventually) 1706 qed 1707 then show "map_filter_on ?AB fst Y = ?X" 1708 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) 1709 1710 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 1711 proof 1712 show ?lhs if ?rhs using G F that 1713 by(auto 4 3 intro: exI[where x="\<lambda>_. True"] simp add: eventually_map_filter_on split_def) 1714 assume ?lhs 1715 then obtain X Z where "\<forall>\<^sub>F (x, y) in F. X x \<and> A x y" 1716 and "\<forall>\<^sub>F (x, y) in G. Z y \<and> B x y" 1717 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)" 1718 using F G by(auto simp add: eventually_map_filter_on split_def) 1719 from step[OF this(1, 2)] this(3) 1720 show ?rhs by(clarsimp elim!: eventually_rev_mp simp add: le_fun_def)(fastforce intro: always_eventually) 1721 qed 1722 then show "map_filter_on ?AB snd Y = ?Z" 1723 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) 1724 qed 1725qed 1726 1727lemma filtermap_parametric: "((A ===> B) ===> rel_filter A ===> rel_filter B) filtermap filtermap" 1728proof(intro rel_funI; erule rel_filter.cases; hypsubst) 1729 fix f g Z 1730 assume fg: "(A ===> B) f g" and Z: "\<forall>\<^sub>F (x, y) in Z. A x y" 1731 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)" 1732 (is "rel_filter _ ?F ?G") 1733 proof 1734 let ?Z = "map_filter_on {(x, y). A x y} (map_prod f g) Z" 1735 show "\<forall>\<^sub>F (x, y) in ?Z. B x y" using fg Z 1736 by(auto simp add: eventually_map_filter_on split_def elim!: eventually_mono rel_funD) 1737 have [simp]: "map_prod f g ` {p. A (fst p) (snd p)} \<subseteq> {p. B (fst p) (snd p)}" 1738 using fg by(auto dest: rel_funD) 1739 show "map_filter_on {(x, y). B x y} fst ?Z = ?F" "map_filter_on {(x, y). B x y} snd ?Z = ?G" 1740 using Z by(auto simp add: map_filter_on_comp split_def) 1741 qed 1742 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))" 1743 using Z by(simp add: map_filter_on_UNIV[symmetric] map_filter_on_comp) 1744qed 1745 1746lemma rel_filter_Grp: "rel_filter (Grp UNIV f) = Grp UNIV (filtermap f)" 1747proof((intro antisym predicate2I; (elim GrpE; hypsubst)?), rule GrpI[OF _ UNIV_I]) 1748 fix F G 1749 assume "rel_filter (Grp UNIV f) F G" 1750 hence "rel_filter (=) (filtermap f F) (filtermap id G)" 1751 by(rule filtermap_parametric[THEN rel_funD, THEN rel_funD, rotated])(simp add: Grp_def rel_fun_def) 1752 thus "filtermap f F = G" by(simp add: rel_filter_eq) 1753next 1754 fix F :: "'a filter" 1755 have "rel_filter (=) F F" by(simp add: rel_filter_eq) 1756 hence "rel_filter (Grp UNIV f) (filtermap id F) (filtermap f F)" 1757 by(rule filtermap_parametric[THEN rel_funD, THEN rel_funD, rotated])(simp add: Grp_def rel_fun_def) 1758 thus "rel_filter (Grp UNIV f) F (filtermap f F)" by simp 1759qed 1760 1761lemma Quotient_filter [quot_map]: 1762 "Quotient R Abs Rep T \<Longrightarrow> Quotient (rel_filter R) (filtermap Abs) (filtermap Rep) (rel_filter T)" 1763 unfolding Quotient_alt_def5 rel_filter_eq[symmetric] rel_filter_Grp[symmetric] 1764 by(simp add: rel_filter_distr[symmetric] rel_filter_conversep[symmetric] rel_filter_mono) 1765 1766lemma left_total_rel_filter [transfer_rule]: "left_total A \<Longrightarrow> left_total (rel_filter A)" 1767unfolding left_total_alt_def rel_filter_eq[symmetric] rel_filter_conversep[symmetric] rel_filter_distr 1768by(rule rel_filter_mono) 1769 1770lemma right_total_rel_filter [transfer_rule]: "right_total A \<Longrightarrow> right_total (rel_filter A)" 1771using left_total_rel_filter[of "A\<inverse>\<inverse>"] by(simp add: rel_filter_conversep) 1772 1773lemma bi_total_rel_filter [transfer_rule]: "bi_total A \<Longrightarrow> bi_total (rel_filter A)" 1774unfolding bi_total_alt_def by(simp add: left_total_rel_filter right_total_rel_filter) 1775 1776lemma left_unique_rel_filter [transfer_rule]: "left_unique A \<Longrightarrow> left_unique (rel_filter A)" 1777unfolding left_unique_alt_def rel_filter_eq[symmetric] rel_filter_conversep[symmetric] rel_filter_distr 1778by(rule rel_filter_mono) 1779 1780lemma right_unique_rel_filter [transfer_rule]: 1781 "right_unique A \<Longrightarrow> right_unique (rel_filter A)" 1782using left_unique_rel_filter[of "A\<inverse>\<inverse>"] by(simp add: rel_filter_conversep) 1783 1784lemma bi_unique_rel_filter [transfer_rule]: "bi_unique A \<Longrightarrow> bi_unique (rel_filter A)" 1785by(simp add: bi_unique_alt_def left_unique_rel_filter right_unique_rel_filter) 1786 1787lemma eventually_parametric [transfer_rule]: 1788 "((A ===> (=)) ===> rel_filter A ===> (=)) eventually eventually" 1789by(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) 1790 1791lemma frequently_parametric [transfer_rule]: "((A ===> (=)) ===> rel_filter A ===> (=)) frequently frequently" 1792 unfolding frequently_def[abs_def] by transfer_prover 1793 1794lemma is_filter_parametric [transfer_rule]: 1795 assumes [transfer_rule]: "bi_total A" 1796 assumes [transfer_rule]: "bi_unique A" 1797 shows "(((A ===> (=)) ===> (=)) ===> (=)) is_filter is_filter" 1798 unfolding is_filter_def by transfer_prover 1799 1800lemma top_filter_parametric [transfer_rule]: "rel_filter A top top" if "bi_total A" 1801proof 1802 let ?Z = "principal {(x, y). A x y}" 1803 show "\<forall>\<^sub>F (x, y) in ?Z. A x y" by(simp add: eventually_principal) 1804 show "map_filter_on {(x, y). A x y} fst ?Z = top" "map_filter_on {(x, y). A x y} snd ?Z = top" 1805 using that by(auto simp add: filter_eq_iff eventually_map_filter_on eventually_principal bi_total_def) 1806qed 1807 1808lemma bot_filter_parametric [transfer_rule]: "rel_filter A bot bot" 1809proof 1810 show "\<forall>\<^sub>F (x, y) in bot. A x y" by simp 1811 show "map_filter_on {(x, y). A x y} fst bot = bot" "map_filter_on {(x, y). A x y} snd bot = bot" 1812 by(simp_all add: filter_eq_iff eventually_map_filter_on) 1813qed 1814 1815lemma principal_parametric [transfer_rule]: "(rel_set A ===> rel_filter A) principal principal" 1816proof(rule rel_funI rel_filter.intros)+ 1817 fix S S' 1818 assume *: "rel_set A S S'" 1819 define SS' where "SS' = S \<times> S' \<inter> {(x, y). A x y}" 1820 have SS': "SS' \<subseteq> {(x, y). A x y}" and [simp]: "S = fst ` SS'" "S' = snd ` SS'" 1821 using * by(auto 4 3 dest: rel_setD1 rel_setD2 intro: rev_image_eqI simp add: SS'_def) 1822 let ?Z = "principal SS'" 1823 show "\<forall>\<^sub>F (x, y) in ?Z. A x y" using SS' by(auto simp add: eventually_principal) 1824 then show "map_filter_on {(x, y). A x y} fst ?Z = principal S" 1825 and "map_filter_on {(x, y). A x y} snd ?Z = principal S'" 1826 by(auto simp add: filter_eq_iff eventually_map_filter_on eventually_principal) 1827qed 1828 1829lemma sup_filter_parametric [transfer_rule]: 1830 "(rel_filter A ===> rel_filter A ===> rel_filter A) sup sup" 1831proof(intro rel_funI; elim rel_filter.cases; hypsubst) 1832 show "rel_filter A 1833 (map_filter_on {(x, y). A x y} fst FG \<squnion> map_filter_on {(x, y). A x y} fst FG') 1834 (map_filter_on {(x, y). A x y} snd FG \<squnion> map_filter_on {(x, y). A x y} snd FG')" 1835 (is "rel_filter _ (sup ?F ?G) (sup ?F' ?G')") 1836 if "\<forall>\<^sub>F (x, y) in FG. A x y" "\<forall>\<^sub>F (x, y) in FG'. A x y" for FG FG' 1837 proof 1838 let ?Z = "sup FG FG'" 1839 show "\<forall>\<^sub>F (x, y) in ?Z. A x y" by(simp add: eventually_sup that) 1840 then show "map_filter_on {(x, y). A x y} fst ?Z = sup ?F ?G" 1841 and "map_filter_on {(x, y). A x y} snd ?Z = sup ?F' ?G'" 1842 by(simp_all add: filter_eq_iff eventually_map_filter_on eventually_sup) 1843 qed 1844qed 1845 1846lemma Sup_filter_parametric [transfer_rule]: "(rel_set (rel_filter A) ===> rel_filter A) Sup Sup" 1847proof(rule rel_funI) 1848 fix S S' 1849 define SS' where "SS' = S \<times> S' \<inter> {(F, G). rel_filter A F G}" 1850 assume "rel_set (rel_filter A) S S'" 1851 then have SS': "SS' \<subseteq> {(F, G). rel_filter A F G}" and [simp]: "S = fst ` SS'" "S' = snd ` SS'" 1852 by(auto 4 3 dest: rel_setD1 rel_setD2 intro: rev_image_eqI simp add: SS'_def) 1853 from SS' obtain Z where Z: "\<And>F G. (F, G) \<in> SS' \<Longrightarrow> 1854 (\<forall>\<^sub>F (x, y) in Z F G. A x y) \<and> 1855 id F = map_filter_on {(x, y). A x y} fst (Z F G) \<and> 1856 id G = map_filter_on {(x, y). A x y} snd (Z F G)" 1857 unfolding rel_filter.simps by atomize_elim((rule choice allI)+; auto) 1858 have id: "eventually P F = eventually P (id F)" "eventually Q G = eventually Q (id G)" 1859 if "(F, G) \<in> SS'" for P Q F G by simp_all 1860 show "rel_filter A (Sup S) (Sup S')" 1861 proof 1862 let ?Z = "\<Squnion>(F, G)\<in>SS'. Z F G" 1863 show *: "\<forall>\<^sub>F (x, y) in ?Z. A x y" using Z by(auto simp add: eventually_Sup) 1864 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'" 1865 unfolding filter_eq_iff 1866 by(auto 4 4 simp add: id eventually_Sup eventually_map_filter_on *[simplified eventually_Sup] simp del: id_apply dest: Z) 1867 qed 1868qed 1869 1870context 1871 fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool" 1872 assumes [transfer_rule]: "bi_unique A" 1873begin 1874 1875lemma le_filter_parametric [transfer_rule]: 1876 "(rel_filter A ===> rel_filter A ===> (=)) (\<le>) (\<le>)" 1877unfolding le_filter_def[abs_def] by transfer_prover 1878 1879lemma less_filter_parametric [transfer_rule]: 1880 "(rel_filter A ===> rel_filter A ===> (=)) (<) (<)" 1881unfolding less_filter_def[abs_def] by transfer_prover 1882 1883context 1884 assumes [transfer_rule]: "bi_total A" 1885begin 1886 1887lemma Inf_filter_parametric [transfer_rule]: 1888 "(rel_set (rel_filter A) ===> rel_filter A) Inf Inf" 1889unfolding Inf_filter_def[abs_def] by transfer_prover 1890 1891lemma inf_filter_parametric [transfer_rule]: 1892 "(rel_filter A ===> rel_filter A ===> rel_filter A) inf inf" 1893proof(intro rel_funI)+ 1894 fix F F' G G' 1895 assume [transfer_rule]: "rel_filter A F F'" "rel_filter A G G'" 1896 have "rel_filter A (Inf {F, G}) (Inf {F', G'})" by transfer_prover 1897 thus "rel_filter A (inf F G) (inf F' G')" by simp 1898qed 1899 1900end 1901 1902end 1903 1904end 1905 1906context 1907 includes lifting_syntax 1908begin 1909 1910lemma prod_filter_parametric [transfer_rule]: 1911 "(rel_filter R ===> rel_filter S ===> rel_filter (rel_prod R S)) prod_filter prod_filter" 1912proof(intro rel_funI; elim rel_filter.cases; hypsubst) 1913 fix F G 1914 assume F: "\<forall>\<^sub>F (x, y) in F. R x y" and G: "\<forall>\<^sub>F (x, y) in G. S x y" 1915 show "rel_filter (rel_prod R S) 1916 (map_filter_on {(x, y). R x y} fst F \<times>\<^sub>F map_filter_on {(x, y). S x y} fst G) 1917 (map_filter_on {(x, y). R x y} snd F \<times>\<^sub>F map_filter_on {(x, y). S x y} snd G)" 1918 (is "rel_filter ?RS ?F ?G") 1919 proof 1920 let ?Z = "filtermap (\<lambda>((a, b), (a', b')). ((a, a'), (b, b'))) (prod_filter F G)" 1921 show *: "\<forall>\<^sub>F (x, y) in ?Z. rel_prod R S x y" using F G 1922 by(auto simp add: eventually_filtermap split_beta eventually_prod_filter) 1923 show "map_filter_on {(x, y). ?RS x y} fst ?Z = ?F" 1924 using F G 1925 apply(clarsimp simp add: filter_eq_iff eventually_map_filter_on *) 1926 apply(simp add: eventually_filtermap split_beta eventually_prod_filter) 1927 apply(subst eventually_map_filter_on; simp)+ 1928 apply(rule iffI; clarsimp) 1929 subgoal for P P' P'' 1930 apply(rule exI[where x="\<lambda>a. \<exists>b. P' (a, b) \<and> R a b"]; rule conjI) 1931 subgoal by(fastforce elim: eventually_rev_mp eventually_mono) 1932 subgoal 1933 by(rule exI[where x="\<lambda>a. \<exists>b. P'' (a, b) \<and> S a b"])(fastforce elim: eventually_rev_mp eventually_mono) 1934 done 1935 subgoal by fastforce 1936 done 1937 show "map_filter_on {(x, y). ?RS x y} snd ?Z = ?G" 1938 using F G 1939 apply(clarsimp simp add: filter_eq_iff eventually_map_filter_on *) 1940 apply(simp add: eventually_filtermap split_beta eventually_prod_filter) 1941 apply(subst eventually_map_filter_on; simp)+ 1942 apply(rule iffI; clarsimp) 1943 subgoal for P P' P'' 1944 apply(rule exI[where x="\<lambda>b. \<exists>a. P' (a, b) \<and> R a b"]; rule conjI) 1945 subgoal by(fastforce elim: eventually_rev_mp eventually_mono) 1946 subgoal 1947 by(rule exI[where x="\<lambda>b. \<exists>a. P'' (a, b) \<and> S a b"])(fastforce elim: eventually_rev_mp eventually_mono) 1948 done 1949 subgoal by fastforce 1950 done 1951 qed 1952qed 1953 1954end 1955 1956 1957text \<open>Code generation for filters\<close> 1958 1959definition abstract_filter :: "(unit \<Rightarrow> 'a filter) \<Rightarrow> 'a filter" 1960 where [simp]: "abstract_filter f = f ()" 1961 1962code_datatype principal abstract_filter 1963 1964hide_const (open) abstract_filter 1965 1966declare [[code drop: filterlim prod_filter filtermap eventually 1967 "inf :: _ filter \<Rightarrow> _" "sup :: _ filter \<Rightarrow> _" "less_eq :: _ filter \<Rightarrow> _" 1968 Abs_filter]] 1969 1970declare filterlim_principal [code] 1971declare principal_prod_principal [code] 1972declare filtermap_principal [code] 1973declare filtercomap_principal [code] 1974declare eventually_principal [code] 1975declare inf_principal [code] 1976declare sup_principal [code] 1977declare principal_le_iff [code] 1978 1979lemma Rep_filter_iff_eventually [simp, code]: 1980 "Rep_filter F P \<longleftrightarrow> eventually P F" 1981 by (simp add: eventually_def) 1982 1983lemma bot_eq_principal_empty [code]: 1984 "bot = principal {}" 1985 by simp 1986 1987lemma top_eq_principal_UNIV [code]: 1988 "top = principal UNIV" 1989 by simp 1990 1991instantiation filter :: (equal) equal 1992begin 1993 1994definition equal_filter :: "'a filter \<Rightarrow> 'a filter \<Rightarrow> bool" 1995 where "equal_filter F F' \<longleftrightarrow> F = F'" 1996 1997lemma equal_filter [code]: 1998 "HOL.equal (principal A) (principal B) \<longleftrightarrow> A = B" 1999 by (simp add: equal_filter_def) 2000 2001instance 2002 by standard (simp add: equal_filter_def) 2003 2004end 2005 2006end 2007