1(* Title: HOL/Topological_Spaces.thy 2 Author: Brian Huffman 3 Author: Johannes H��lzl 4*) 5 6section \<open>Topological Spaces\<close> 7 8theory Topological_Spaces 9 imports Main 10begin 11 12named_theorems continuous_intros "structural introduction rules for continuity" 13 14subsection \<open>Topological space\<close> 15 16class "open" = 17 fixes "open" :: "'a set \<Rightarrow> bool" 18 19class topological_space = "open" + 20 assumes open_UNIV [simp, intro]: "open UNIV" 21 assumes open_Int [intro]: "open S \<Longrightarrow> open T \<Longrightarrow> open (S \<inter> T)" 22 assumes open_Union [intro]: "\<forall>S\<in>K. open S \<Longrightarrow> open (\<Union>K)" 23begin 24 25definition closed :: "'a set \<Rightarrow> bool" 26 where "closed S \<longleftrightarrow> open (- S)" 27 28lemma open_empty [continuous_intros, intro, simp]: "open {}" 29 using open_Union [of "{}"] by simp 30 31lemma open_Un [continuous_intros, intro]: "open S \<Longrightarrow> open T \<Longrightarrow> open (S \<union> T)" 32 using open_Union [of "{S, T}"] by simp 33 34lemma open_UN [continuous_intros, intro]: "\<forall>x\<in>A. open (B x) \<Longrightarrow> open (\<Union>x\<in>A. B x)" 35 using open_Union [of "B ` A"] by simp 36 37lemma open_Inter [continuous_intros, intro]: "finite S \<Longrightarrow> \<forall>T\<in>S. open T \<Longrightarrow> open (\<Inter>S)" 38 by (induct set: finite) auto 39 40lemma open_INT [continuous_intros, intro]: "finite A \<Longrightarrow> \<forall>x\<in>A. open (B x) \<Longrightarrow> open (\<Inter>x\<in>A. B x)" 41 using open_Inter [of "B ` A"] by simp 42 43lemma openI: 44 assumes "\<And>x. x \<in> S \<Longrightarrow> \<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> S" 45 shows "open S" 46proof - 47 have "open (\<Union>{T. open T \<and> T \<subseteq> S})" by auto 48 moreover have "\<Union>{T. open T \<and> T \<subseteq> S} = S" by (auto dest!: assms) 49 ultimately show "open S" by simp 50qed 51 52lemma open_subopen: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> S)" 53by (auto intro: openI) 54 55lemma closed_empty [continuous_intros, intro, simp]: "closed {}" 56 unfolding closed_def by simp 57 58lemma closed_Un [continuous_intros, intro]: "closed S \<Longrightarrow> closed T \<Longrightarrow> closed (S \<union> T)" 59 unfolding closed_def by auto 60 61lemma closed_UNIV [continuous_intros, intro, simp]: "closed UNIV" 62 unfolding closed_def by simp 63 64lemma closed_Int [continuous_intros, intro]: "closed S \<Longrightarrow> closed T \<Longrightarrow> closed (S \<inter> T)" 65 unfolding closed_def by auto 66 67lemma closed_INT [continuous_intros, intro]: "\<forall>x\<in>A. closed (B x) \<Longrightarrow> closed (\<Inter>x\<in>A. B x)" 68 unfolding closed_def by auto 69 70lemma closed_Inter [continuous_intros, intro]: "\<forall>S\<in>K. closed S \<Longrightarrow> closed (\<Inter>K)" 71 unfolding closed_def uminus_Inf by auto 72 73lemma closed_Union [continuous_intros, intro]: "finite S \<Longrightarrow> \<forall>T\<in>S. closed T \<Longrightarrow> closed (\<Union>S)" 74 by (induct set: finite) auto 75 76lemma closed_UN [continuous_intros, intro]: 77 "finite A \<Longrightarrow> \<forall>x\<in>A. closed (B x) \<Longrightarrow> closed (\<Union>x\<in>A. B x)" 78 using closed_Union [of "B ` A"] by simp 79 80lemma open_closed: "open S \<longleftrightarrow> closed (- S)" 81 by (simp add: closed_def) 82 83lemma closed_open: "closed S \<longleftrightarrow> open (- S)" 84 by (rule closed_def) 85 86lemma open_Diff [continuous_intros, intro]: "open S \<Longrightarrow> closed T \<Longrightarrow> open (S - T)" 87 by (simp add: closed_open Diff_eq open_Int) 88 89lemma closed_Diff [continuous_intros, intro]: "closed S \<Longrightarrow> open T \<Longrightarrow> closed (S - T)" 90 by (simp add: open_closed Diff_eq closed_Int) 91 92lemma open_Compl [continuous_intros, intro]: "closed S \<Longrightarrow> open (- S)" 93 by (simp add: closed_open) 94 95lemma closed_Compl [continuous_intros, intro]: "open S \<Longrightarrow> closed (- S)" 96 by (simp add: open_closed) 97 98lemma open_Collect_neg: "closed {x. P x} \<Longrightarrow> open {x. \<not> P x}" 99 unfolding Collect_neg_eq by (rule open_Compl) 100 101lemma open_Collect_conj: 102 assumes "open {x. P x}" "open {x. Q x}" 103 shows "open {x. P x \<and> Q x}" 104 using open_Int[OF assms] by (simp add: Int_def) 105 106lemma open_Collect_disj: 107 assumes "open {x. P x}" "open {x. Q x}" 108 shows "open {x. P x \<or> Q x}" 109 using open_Un[OF assms] by (simp add: Un_def) 110 111lemma open_Collect_ex: "(\<And>i. open {x. P i x}) \<Longrightarrow> open {x. \<exists>i. P i x}" 112 using open_UN[of UNIV "\<lambda>i. {x. P i x}"] unfolding Collect_ex_eq by simp 113 114lemma open_Collect_imp: "closed {x. P x} \<Longrightarrow> open {x. Q x} \<Longrightarrow> open {x. P x \<longrightarrow> Q x}" 115 unfolding imp_conv_disj by (intro open_Collect_disj open_Collect_neg) 116 117lemma open_Collect_const: "open {x. P}" 118 by (cases P) auto 119 120lemma closed_Collect_neg: "open {x. P x} \<Longrightarrow> closed {x. \<not> P x}" 121 unfolding Collect_neg_eq by (rule closed_Compl) 122 123lemma closed_Collect_conj: 124 assumes "closed {x. P x}" "closed {x. Q x}" 125 shows "closed {x. P x \<and> Q x}" 126 using closed_Int[OF assms] by (simp add: Int_def) 127 128lemma closed_Collect_disj: 129 assumes "closed {x. P x}" "closed {x. Q x}" 130 shows "closed {x. P x \<or> Q x}" 131 using closed_Un[OF assms] by (simp add: Un_def) 132 133lemma closed_Collect_all: "(\<And>i. closed {x. P i x}) \<Longrightarrow> closed {x. \<forall>i. P i x}" 134 using closed_INT[of UNIV "\<lambda>i. {x. P i x}"] by (simp add: Collect_all_eq) 135 136lemma closed_Collect_imp: "open {x. P x} \<Longrightarrow> closed {x. Q x} \<Longrightarrow> closed {x. P x \<longrightarrow> Q x}" 137 unfolding imp_conv_disj by (intro closed_Collect_disj closed_Collect_neg) 138 139lemma closed_Collect_const: "closed {x. P}" 140 by (cases P) auto 141 142end 143 144 145subsection \<open>Hausdorff and other separation properties\<close> 146 147class t0_space = topological_space + 148 assumes t0_space: "x \<noteq> y \<Longrightarrow> \<exists>U. open U \<and> \<not> (x \<in> U \<longleftrightarrow> y \<in> U)" 149 150class t1_space = topological_space + 151 assumes t1_space: "x \<noteq> y \<Longrightarrow> \<exists>U. open U \<and> x \<in> U \<and> y \<notin> U" 152 153instance t1_space \<subseteq> t0_space 154 by standard (fast dest: t1_space) 155 156context t1_space begin 157 158lemma separation_t1: "x \<noteq> y \<longleftrightarrow> (\<exists>U. open U \<and> x \<in> U \<and> y \<notin> U)" 159 using t1_space[of x y] by blast 160 161lemma closed_singleton [iff]: "closed {a}" 162proof - 163 let ?T = "\<Union>{S. open S \<and> a \<notin> S}" 164 have "open ?T" 165 by (simp add: open_Union) 166 also have "?T = - {a}" 167 by (auto simp add: set_eq_iff separation_t1) 168 finally show "closed {a}" 169 by (simp only: closed_def) 170qed 171 172lemma closed_insert [continuous_intros, simp]: 173 assumes "closed S" 174 shows "closed (insert a S)" 175proof - 176 from closed_singleton assms have "closed ({a} \<union> S)" 177 by (rule closed_Un) 178 then show "closed (insert a S)" 179 by simp 180qed 181 182lemma finite_imp_closed: "finite S \<Longrightarrow> closed S" 183 by (induct pred: finite) simp_all 184 185end 186 187text \<open>T2 spaces are also known as Hausdorff spaces.\<close> 188 189class t2_space = topological_space + 190 assumes hausdorff: "x \<noteq> y \<Longrightarrow> \<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {}" 191 192instance t2_space \<subseteq> t1_space 193 by standard (fast dest: hausdorff) 194 195lemma (in t2_space) separation_t2: "x \<noteq> y \<longleftrightarrow> (\<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {})" 196 using hausdorff [of x y] by blast 197 198lemma (in t0_space) separation_t0: "x \<noteq> y \<longleftrightarrow> (\<exists>U. open U \<and> \<not> (x \<in> U \<longleftrightarrow> y \<in> U))" 199 using t0_space [of x y] by blast 200 201 202text \<open>A classical separation axiom for topological space, the T3 axiom -- also called regularity: 203if a point is not in a closed set, then there are open sets separating them.\<close> 204 205class t3_space = t2_space + 206 assumes t3_space: "closed S \<Longrightarrow> y \<notin> S \<Longrightarrow> \<exists>U V. open U \<and> open V \<and> y \<in> U \<and> S \<subseteq> V \<and> U \<inter> V = {}" 207 208text \<open>A classical separation axiom for topological space, the T4 axiom -- also called normality: 209if two closed sets are disjoint, then there are open sets separating them.\<close> 210 211class t4_space = t2_space + 212 assumes t4_space: "closed S \<Longrightarrow> closed T \<Longrightarrow> S \<inter> T = {} \<Longrightarrow> \<exists>U V. open U \<and> open V \<and> S \<subseteq> U \<and> T \<subseteq> V \<and> U \<inter> V = {}" 213 214text \<open>T4 is stronger than T3, and weaker than metric.\<close> 215 216instance t4_space \<subseteq> t3_space 217proof 218 fix S and y::'a assume "closed S" "y \<notin> S" 219 then show "\<exists>U V. open U \<and> open V \<and> y \<in> U \<and> S \<subseteq> V \<and> U \<inter> V = {}" 220 using t4_space[of "{y}" S] by auto 221qed 222 223text \<open>A perfect space is a topological space with no isolated points.\<close> 224 225class perfect_space = topological_space + 226 assumes not_open_singleton: "\<not> open {x}" 227 228lemma (in perfect_space) UNIV_not_singleton: "UNIV \<noteq> {x}" 229 for x::'a 230 by (metis (no_types) open_UNIV not_open_singleton) 231 232 233subsection \<open>Generators for toplogies\<close> 234 235inductive generate_topology :: "'a set set \<Rightarrow> 'a set \<Rightarrow> bool" for S :: "'a set set" 236 where 237 UNIV: "generate_topology S UNIV" 238 | Int: "generate_topology S (a \<inter> b)" if "generate_topology S a" and "generate_topology S b" 239 | UN: "generate_topology S (\<Union>K)" if "(\<And>k. k \<in> K \<Longrightarrow> generate_topology S k)" 240 | Basis: "generate_topology S s" if "s \<in> S" 241 242hide_fact (open) UNIV Int UN Basis 243 244lemma generate_topology_Union: 245 "(\<And>k. k \<in> I \<Longrightarrow> generate_topology S (K k)) \<Longrightarrow> generate_topology S (\<Union>k\<in>I. K k)" 246 using generate_topology.UN [of "K ` I"] by auto 247 248lemma topological_space_generate_topology: "class.topological_space (generate_topology S)" 249 by standard (auto intro: generate_topology.intros) 250 251 252subsection \<open>Order topologies\<close> 253 254class order_topology = order + "open" + 255 assumes open_generated_order: "open = generate_topology (range (\<lambda>a. {..< a}) \<union> range (\<lambda>a. {a <..}))" 256begin 257 258subclass topological_space 259 unfolding open_generated_order 260 by (rule topological_space_generate_topology) 261 262lemma open_greaterThan [continuous_intros, simp]: "open {a <..}" 263 unfolding open_generated_order by (auto intro: generate_topology.Basis) 264 265lemma open_lessThan [continuous_intros, simp]: "open {..< a}" 266 unfolding open_generated_order by (auto intro: generate_topology.Basis) 267 268lemma open_greaterThanLessThan [continuous_intros, simp]: "open {a <..< b}" 269 unfolding greaterThanLessThan_eq by (simp add: open_Int) 270 271end 272 273class linorder_topology = linorder + order_topology 274 275lemma closed_atMost [continuous_intros, simp]: "closed {..a}" 276 for a :: "'a::linorder_topology" 277 by (simp add: closed_open) 278 279lemma closed_atLeast [continuous_intros, simp]: "closed {a..}" 280 for a :: "'a::linorder_topology" 281 by (simp add: closed_open) 282 283lemma closed_atLeastAtMost [continuous_intros, simp]: "closed {a..b}" 284 for a b :: "'a::linorder_topology" 285proof - 286 have "{a .. b} = {a ..} \<inter> {.. b}" 287 by auto 288 then show ?thesis 289 by (simp add: closed_Int) 290qed 291 292lemma (in order) less_separate: 293 assumes "x < y" 294 shows "\<exists>a b. x \<in> {..< a} \<and> y \<in> {b <..} \<and> {..< a} \<inter> {b <..} = {}" 295proof (cases "\<exists>z. x < z \<and> z < y") 296 case True 297 then obtain z where "x < z \<and> z < y" .. 298 then have "x \<in> {..< z} \<and> y \<in> {z <..} \<and> {z <..} \<inter> {..< z} = {}" 299 by auto 300 then show ?thesis by blast 301next 302 case False 303 with \<open>x < y\<close> have "x \<in> {..< y}" "y \<in> {x <..}" "{x <..} \<inter> {..< y} = {}" 304 by auto 305 then show ?thesis by blast 306qed 307 308instance linorder_topology \<subseteq> t2_space 309proof 310 fix x y :: 'a 311 show "x \<noteq> y \<Longrightarrow> \<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {}" 312 using less_separate [of x y] less_separate [of y x] 313 by (elim neqE; metis open_lessThan open_greaterThan Int_commute) 314qed 315 316lemma (in linorder_topology) open_right: 317 assumes "open S" "x \<in> S" 318 and gt_ex: "x < y" 319 shows "\<exists>b>x. {x ..< b} \<subseteq> S" 320 using assms unfolding open_generated_order 321proof induct 322 case UNIV 323 then show ?case by blast 324next 325 case (Int A B) 326 then obtain a b where "a > x" "{x ..< a} \<subseteq> A" "b > x" "{x ..< b} \<subseteq> B" 327 by auto 328 then show ?case 329 by (auto intro!: exI[of _ "min a b"]) 330next 331 case UN 332 then show ?case by blast 333next 334 case Basis 335 then show ?case 336 by (fastforce intro: exI[of _ y] gt_ex) 337qed 338 339lemma (in linorder_topology) open_left: 340 assumes "open S" "x \<in> S" 341 and lt_ex: "y < x" 342 shows "\<exists>b<x. {b <.. x} \<subseteq> S" 343 using assms unfolding open_generated_order 344proof induction 345 case UNIV 346 then show ?case by blast 347next 348 case (Int A B) 349 then obtain a b where "a < x" "{a <.. x} \<subseteq> A" "b < x" "{b <.. x} \<subseteq> B" 350 by auto 351 then show ?case 352 by (auto intro!: exI[of _ "max a b"]) 353next 354 case UN 355 then show ?case by blast 356next 357 case Basis 358 then show ?case 359 by (fastforce intro: exI[of _ y] lt_ex) 360qed 361 362 363subsection \<open>Setup some topologies\<close> 364 365subsubsection \<open>Boolean is an order topology\<close> 366 367class discrete_topology = topological_space + 368 assumes open_discrete: "\<And>A. open A" 369 370instance discrete_topology < t2_space 371proof 372 fix x y :: 'a 373 assume "x \<noteq> y" 374 then show "\<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {}" 375 by (intro exI[of _ "{_}"]) (auto intro!: open_discrete) 376qed 377 378instantiation bool :: linorder_topology 379begin 380 381definition open_bool :: "bool set \<Rightarrow> bool" 382 where "open_bool = generate_topology (range (\<lambda>a. {..< a}) \<union> range (\<lambda>a. {a <..}))" 383 384instance 385 by standard (rule open_bool_def) 386 387end 388 389instance bool :: discrete_topology 390proof 391 fix A :: "bool set" 392 have *: "{False <..} = {True}" "{..< True} = {False}" 393 by auto 394 have "A = UNIV \<or> A = {} \<or> A = {False <..} \<or> A = {..< True}" 395 using subset_UNIV[of A] unfolding UNIV_bool * by blast 396 then show "open A" 397 by auto 398qed 399 400instantiation nat :: linorder_topology 401begin 402 403definition open_nat :: "nat set \<Rightarrow> bool" 404 where "open_nat = generate_topology (range (\<lambda>a. {..< a}) \<union> range (\<lambda>a. {a <..}))" 405 406instance 407 by standard (rule open_nat_def) 408 409end 410 411instance nat :: discrete_topology 412proof 413 fix A :: "nat set" 414 have "open {n}" for n :: nat 415 proof (cases n) 416 case 0 417 moreover have "{0} = {..<1::nat}" 418 by auto 419 ultimately show ?thesis 420 by auto 421 next 422 case (Suc n') 423 then have "{n} = {..<Suc n} \<inter> {n' <..}" 424 by auto 425 with Suc show ?thesis 426 by (auto intro: open_lessThan open_greaterThan) 427 qed 428 then have "open (\<Union>a\<in>A. {a})" 429 by (intro open_UN) auto 430 then show "open A" 431 by simp 432qed 433 434instantiation int :: linorder_topology 435begin 436 437definition open_int :: "int set \<Rightarrow> bool" 438 where "open_int = generate_topology (range (\<lambda>a. {..< a}) \<union> range (\<lambda>a. {a <..}))" 439 440instance 441 by standard (rule open_int_def) 442 443end 444 445instance int :: discrete_topology 446proof 447 fix A :: "int set" 448 have "{..<i + 1} \<inter> {i-1 <..} = {i}" for i :: int 449 by auto 450 then have "open {i}" for i :: int 451 using open_Int[OF open_lessThan[of "i + 1"] open_greaterThan[of "i - 1"]] by auto 452 then have "open (\<Union>a\<in>A. {a})" 453 by (intro open_UN) auto 454 then show "open A" 455 by simp 456qed 457 458 459subsubsection \<open>Topological filters\<close> 460 461definition (in topological_space) nhds :: "'a \<Rightarrow> 'a filter" 462 where "nhds a = (INF S\<in>{S. open S \<and> a \<in> S}. principal S)" 463 464definition (in topological_space) at_within :: "'a \<Rightarrow> 'a set \<Rightarrow> 'a filter" 465 ("at (_)/ within (_)" [1000, 60] 60) 466 where "at a within s = inf (nhds a) (principal (s - {a}))" 467 468abbreviation (in topological_space) at :: "'a \<Rightarrow> 'a filter" ("at") 469 where "at x \<equiv> at x within (CONST UNIV)" 470 471abbreviation (in order_topology) at_right :: "'a \<Rightarrow> 'a filter" 472 where "at_right x \<equiv> at x within {x <..}" 473 474abbreviation (in order_topology) at_left :: "'a \<Rightarrow> 'a filter" 475 where "at_left x \<equiv> at x within {..< x}" 476 477lemma (in topological_space) nhds_generated_topology: 478 "open = generate_topology T \<Longrightarrow> nhds x = (INF S\<in>{S\<in>T. x \<in> S}. principal S)" 479 unfolding nhds_def 480proof (safe intro!: antisym INF_greatest) 481 fix S 482 assume "generate_topology T S" "x \<in> S" 483 then show "(INF S\<in>{S \<in> T. x \<in> S}. principal S) \<le> principal S" 484 by induct 485 (auto intro: INF_lower order_trans simp: inf_principal[symmetric] simp del: inf_principal) 486qed (auto intro!: INF_lower intro: generate_topology.intros) 487 488lemma (in topological_space) eventually_nhds: 489 "eventually P (nhds a) \<longleftrightarrow> (\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x))" 490 unfolding nhds_def by (subst eventually_INF_base) (auto simp: eventually_principal) 491 492lemma eventually_eventually: 493 "eventually (\<lambda>y. eventually P (nhds y)) (nhds x) = eventually P (nhds x)" 494 by (auto simp: eventually_nhds) 495 496lemma (in topological_space) eventually_nhds_in_open: 497 "open s \<Longrightarrow> x \<in> s \<Longrightarrow> eventually (\<lambda>y. y \<in> s) (nhds x)" 498 by (subst eventually_nhds) blast 499 500lemma (in topological_space) eventually_nhds_x_imp_x: "eventually P (nhds x) \<Longrightarrow> P x" 501 by (subst (asm) eventually_nhds) blast 502 503lemma (in topological_space) nhds_neq_bot [simp]: "nhds a \<noteq> bot" 504 by (simp add: trivial_limit_def eventually_nhds) 505 506lemma (in t1_space) t1_space_nhds: "x \<noteq> y \<Longrightarrow> (\<forall>\<^sub>F x in nhds x. x \<noteq> y)" 507 by (drule t1_space) (auto simp: eventually_nhds) 508 509lemma (in topological_space) nhds_discrete_open: "open {x} \<Longrightarrow> nhds x = principal {x}" 510 by (auto simp: nhds_def intro!: antisym INF_greatest INF_lower2[of "{x}"]) 511 512lemma (in discrete_topology) nhds_discrete: "nhds x = principal {x}" 513 by (simp add: nhds_discrete_open open_discrete) 514 515lemma (in discrete_topology) at_discrete: "at x within S = bot" 516 unfolding at_within_def nhds_discrete by simp 517 518lemma (in discrete_topology) tendsto_discrete: 519 "filterlim (f :: 'b \<Rightarrow> 'a) (nhds y) F \<longleftrightarrow> eventually (\<lambda>x. f x = y) F" 520 by (auto simp: nhds_discrete filterlim_principal) 521 522lemma (in topological_space) at_within_eq: 523 "at x within s = (INF S\<in>{S. open S \<and> x \<in> S}. principal (S \<inter> s - {x}))" 524 unfolding nhds_def at_within_def 525 by (subst INF_inf_const2[symmetric]) (auto simp: Diff_Int_distrib) 526 527lemma (in topological_space) eventually_at_filter: 528 "eventually P (at a within s) \<longleftrightarrow> eventually (\<lambda>x. x \<noteq> a \<longrightarrow> x \<in> s \<longrightarrow> P x) (nhds a)" 529 by (simp add: at_within_def eventually_inf_principal imp_conjL[symmetric] conj_commute) 530 531lemma (in topological_space) at_le: "s \<subseteq> t \<Longrightarrow> at x within s \<le> at x within t" 532 unfolding at_within_def by (intro inf_mono) auto 533 534lemma (in topological_space) eventually_at_topological: 535 "eventually P (at a within s) \<longleftrightarrow> (\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. x \<noteq> a \<longrightarrow> x \<in> s \<longrightarrow> P x))" 536 by (simp add: eventually_nhds eventually_at_filter) 537 538lemma (in topological_space) at_within_open: "a \<in> S \<Longrightarrow> open S \<Longrightarrow> at a within S = at a" 539 unfolding filter_eq_iff eventually_at_topological by (metis open_Int Int_iff UNIV_I) 540 541lemma (in topological_space) at_within_open_NO_MATCH: 542 "a \<in> s \<Longrightarrow> open s \<Longrightarrow> NO_MATCH UNIV s \<Longrightarrow> at a within s = at a" 543 by (simp only: at_within_open) 544 545lemma (in topological_space) at_within_open_subset: 546 "a \<in> S \<Longrightarrow> open S \<Longrightarrow> S \<subseteq> T \<Longrightarrow> at a within T = at a" 547 by (metis at_le at_within_open dual_order.antisym subset_UNIV) 548 549lemma (in topological_space) at_within_nhd: 550 assumes "x \<in> S" "open S" "T \<inter> S - {x} = U \<inter> S - {x}" 551 shows "at x within T = at x within U" 552 unfolding filter_eq_iff eventually_at_filter 553proof (intro allI eventually_subst) 554 have "eventually (\<lambda>x. x \<in> S) (nhds x)" 555 using \<open>x \<in> S\<close> \<open>open S\<close> by (auto simp: eventually_nhds) 556 then show "\<forall>\<^sub>F n in nhds x. (n \<noteq> x \<longrightarrow> n \<in> T \<longrightarrow> P n) = (n \<noteq> x \<longrightarrow> n \<in> U \<longrightarrow> P n)" for P 557 by eventually_elim (insert \<open>T \<inter> S - {x} = U \<inter> S - {x}\<close>, blast) 558qed 559 560lemma (in topological_space) at_within_empty [simp]: "at a within {} = bot" 561 unfolding at_within_def by simp 562 563lemma (in topological_space) at_within_union: 564 "at x within (S \<union> T) = sup (at x within S) (at x within T)" 565 unfolding filter_eq_iff eventually_sup eventually_at_filter 566 by (auto elim!: eventually_rev_mp) 567 568lemma (in topological_space) at_eq_bot_iff: "at a = bot \<longleftrightarrow> open {a}" 569 unfolding trivial_limit_def eventually_at_topological 570 apply safe 571 apply (case_tac "S = {a}") 572 apply simp 573 apply fast 574 apply fast 575 done 576 577lemma (in perfect_space) at_neq_bot [simp]: "at a \<noteq> bot" 578 by (simp add: at_eq_bot_iff not_open_singleton) 579 580lemma (in order_topology) nhds_order: 581 "nhds x = inf (INF a\<in>{x <..}. principal {..< a}) (INF a\<in>{..< x}. principal {a <..})" 582proof - 583 have 1: "{S \<in> range lessThan \<union> range greaterThan. x \<in> S} = 584 (\<lambda>a. {..< a}) ` {x <..} \<union> (\<lambda>a. {a <..}) ` {..< x}" 585 by auto 586 show ?thesis 587 by (simp only: nhds_generated_topology[OF open_generated_order] INF_union 1 INF_image comp_def) 588qed 589 590lemma (in topological_space) filterlim_at_within_If: 591 assumes "filterlim f G (at x within (A \<inter> {x. P x}))" 592 and "filterlim g G (at x within (A \<inter> {x. \<not>P x}))" 593 shows "filterlim (\<lambda>x. if P x then f x else g x) G (at x within A)" 594proof (rule filterlim_If) 595 note assms(1) 596 also have "at x within (A \<inter> {x. P x}) = inf (nhds x) (principal (A \<inter> Collect P - {x}))" 597 by (simp add: at_within_def) 598 also have "A \<inter> Collect P - {x} = (A - {x}) \<inter> Collect P" 599 by blast 600 also have "inf (nhds x) (principal \<dots>) = inf (at x within A) (principal (Collect P))" 601 by (simp add: at_within_def inf_assoc) 602 finally show "filterlim f G (inf (at x within A) (principal (Collect P)))" . 603next 604 note assms(2) 605 also have "at x within (A \<inter> {x. \<not> P x}) = inf (nhds x) (principal (A \<inter> {x. \<not> P x} - {x}))" 606 by (simp add: at_within_def) 607 also have "A \<inter> {x. \<not> P x} - {x} = (A - {x}) \<inter> {x. \<not> P x}" 608 by blast 609 also have "inf (nhds x) (principal \<dots>) = inf (at x within A) (principal {x. \<not> P x})" 610 by (simp add: at_within_def inf_assoc) 611 finally show "filterlim g G (inf (at x within A) (principal {x. \<not> P x}))" . 612qed 613 614lemma (in topological_space) filterlim_at_If: 615 assumes "filterlim f G (at x within {x. P x})" 616 and "filterlim g G (at x within {x. \<not>P x})" 617 shows "filterlim (\<lambda>x. if P x then f x else g x) G (at x)" 618 using assms by (intro filterlim_at_within_If) simp_all 619lemma (in linorder_topology) at_within_order: 620 assumes "UNIV \<noteq> {x}" 621 shows "at x within s = 622 inf (INF a\<in>{x <..}. principal ({..< a} \<inter> s - {x})) 623 (INF a\<in>{..< x}. principal ({a <..} \<inter> s - {x}))" 624proof (cases "{x <..} = {}" "{..< x} = {}" rule: case_split [case_product case_split]) 625 case True_True 626 have "UNIV = {..< x} \<union> {x} \<union> {x <..}" 627 by auto 628 with assms True_True show ?thesis 629 by auto 630qed (auto simp del: inf_principal simp: at_within_def nhds_order Int_Diff 631 inf_principal[symmetric] INF_inf_const2 inf_sup_aci[where 'a="'a filter"]) 632 633lemma (in linorder_topology) at_left_eq: 634 "y < x \<Longrightarrow> at_left x = (INF a\<in>{..< x}. principal {a <..< x})" 635 by (subst at_within_order) 636 (auto simp: greaterThan_Int_greaterThan greaterThanLessThan_eq[symmetric] min.absorb2 INF_constant 637 intro!: INF_lower2 inf_absorb2) 638 639lemma (in linorder_topology) eventually_at_left: 640 "y < x \<Longrightarrow> eventually P (at_left x) \<longleftrightarrow> (\<exists>b<x. \<forall>y>b. y < x \<longrightarrow> P y)" 641 unfolding at_left_eq 642 by (subst eventually_INF_base) (auto simp: eventually_principal Ball_def) 643 644lemma (in linorder_topology) at_right_eq: 645 "x < y \<Longrightarrow> at_right x = (INF a\<in>{x <..}. principal {x <..< a})" 646 by (subst at_within_order) 647 (auto simp: lessThan_Int_lessThan greaterThanLessThan_eq[symmetric] max.absorb2 INF_constant Int_commute 648 intro!: INF_lower2 inf_absorb1) 649 650lemma (in linorder_topology) eventually_at_right: 651 "x < y \<Longrightarrow> eventually P (at_right x) \<longleftrightarrow> (\<exists>b>x. \<forall>y>x. y < b \<longrightarrow> P y)" 652 unfolding at_right_eq 653 by (subst eventually_INF_base) (auto simp: eventually_principal Ball_def) 654 655lemma eventually_at_right_less: "\<forall>\<^sub>F y in at_right (x::'a::{linorder_topology, no_top}). x < y" 656 using gt_ex[of x] eventually_at_right[of x] by auto 657 658lemma trivial_limit_at_right_top: "at_right (top::_::{order_top,linorder_topology}) = bot" 659 by (auto simp: filter_eq_iff eventually_at_topological) 660 661lemma trivial_limit_at_left_bot: "at_left (bot::_::{order_bot,linorder_topology}) = bot" 662 by (auto simp: filter_eq_iff eventually_at_topological) 663 664lemma trivial_limit_at_left_real [simp]: "\<not> trivial_limit (at_left x)" 665 for x :: "'a::{no_bot,dense_order,linorder_topology}" 666 using lt_ex [of x] 667 by safe (auto simp add: trivial_limit_def eventually_at_left dest: dense) 668 669lemma trivial_limit_at_right_real [simp]: "\<not> trivial_limit (at_right x)" 670 for x :: "'a::{no_top,dense_order,linorder_topology}" 671 using gt_ex[of x] 672 by safe (auto simp add: trivial_limit_def eventually_at_right dest: dense) 673 674lemma (in linorder_topology) at_eq_sup_left_right: "at x = sup (at_left x) (at_right x)" 675 by (auto simp: eventually_at_filter filter_eq_iff eventually_sup 676 elim: eventually_elim2 eventually_mono) 677 678lemma (in linorder_topology) eventually_at_split: 679 "eventually P (at x) \<longleftrightarrow> eventually P (at_left x) \<and> eventually P (at_right x)" 680 by (subst at_eq_sup_left_right) (simp add: eventually_sup) 681 682lemma (in order_topology) eventually_at_leftI: 683 assumes "\<And>x. x \<in> {a<..<b} \<Longrightarrow> P x" "a < b" 684 shows "eventually P (at_left b)" 685 using assms unfolding eventually_at_topological by (intro exI[of _ "{a<..}"]) auto 686 687lemma (in order_topology) eventually_at_rightI: 688 assumes "\<And>x. x \<in> {a<..<b} \<Longrightarrow> P x" "a < b" 689 shows "eventually P (at_right a)" 690 using assms unfolding eventually_at_topological by (intro exI[of _ "{..<b}"]) auto 691 692lemma eventually_filtercomap_nhds: 693 "eventually P (filtercomap f (nhds x)) \<longleftrightarrow> (\<exists>S. open S \<and> x \<in> S \<and> (\<forall>x. f x \<in> S \<longrightarrow> P x))" 694 unfolding eventually_filtercomap eventually_nhds by auto 695 696lemma eventually_filtercomap_at_topological: 697 "eventually P (filtercomap f (at A within B)) \<longleftrightarrow> 698 (\<exists>S. open S \<and> A \<in> S \<and> (\<forall>x. f x \<in> S \<inter> B - {A} \<longrightarrow> P x))" (is "?lhs = ?rhs") 699 unfolding at_within_def filtercomap_inf eventually_inf_principal filtercomap_principal 700 eventually_filtercomap_nhds eventually_principal by blast 701 702lemma eventually_at_right_field: 703 "eventually P (at_right x) \<longleftrightarrow> (\<exists>b>x. \<forall>y>x. y < b \<longrightarrow> P y)" 704 for x :: "'a::{linordered_field, linorder_topology}" 705 using linordered_field_no_ub[rule_format, of x] 706 by (auto simp: eventually_at_right) 707 708lemma eventually_at_left_field: 709 "eventually P (at_left x) \<longleftrightarrow> (\<exists>b<x. \<forall>y>b. y < x \<longrightarrow> P y)" 710 for x :: "'a::{linordered_field, linorder_topology}" 711 using linordered_field_no_lb[rule_format, of x] 712 by (auto simp: eventually_at_left) 713 714 715subsubsection \<open>Tendsto\<close> 716 717abbreviation (in topological_space) 718 tendsto :: "('b \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'b filter \<Rightarrow> bool" (infixr "\<longlongrightarrow>" 55) 719 where "(f \<longlongrightarrow> l) F \<equiv> filterlim f (nhds l) F" 720 721definition (in t2_space) Lim :: "'f filter \<Rightarrow> ('f \<Rightarrow> 'a) \<Rightarrow> 'a" 722 where "Lim A f = (THE l. (f \<longlongrightarrow> l) A)" 723 724lemma (in topological_space) tendsto_eq_rhs: "(f \<longlongrightarrow> x) F \<Longrightarrow> x = y \<Longrightarrow> (f \<longlongrightarrow> y) F" 725 by simp 726 727named_theorems tendsto_intros "introduction rules for tendsto" 728setup \<open> 729 Global_Theory.add_thms_dynamic (\<^binding>\<open>tendsto_eq_intros\<close>, 730 fn context => 731 Named_Theorems.get (Context.proof_of context) \<^named_theorems>\<open>tendsto_intros\<close> 732 |> map_filter (try (fn thm => @{thm tendsto_eq_rhs} OF [thm]))) 733\<close> 734 735context topological_space begin 736 737lemma tendsto_def: 738 "(f \<longlongrightarrow> l) F \<longleftrightarrow> (\<forall>S. open S \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) F)" 739 unfolding nhds_def filterlim_INF filterlim_principal by auto 740 741lemma tendsto_cong: "(f \<longlongrightarrow> c) F \<longleftrightarrow> (g \<longlongrightarrow> c) F" if "eventually (\<lambda>x. f x = g x) F" 742 by (rule filterlim_cong [OF refl refl that]) 743 744lemma tendsto_mono: "F \<le> F' \<Longrightarrow> (f \<longlongrightarrow> l) F' \<Longrightarrow> (f \<longlongrightarrow> l) F" 745 unfolding tendsto_def le_filter_def by fast 746 747lemma tendsto_ident_at [tendsto_intros, simp, intro]: "((\<lambda>x. x) \<longlongrightarrow> a) (at a within s)" 748 by (auto simp: tendsto_def eventually_at_topological) 749 750lemma tendsto_const [tendsto_intros, simp, intro]: "((\<lambda>x. k) \<longlongrightarrow> k) F" 751 by (simp add: tendsto_def) 752 753lemma filterlim_at: 754 "(LIM x F. f x :> at b within s) \<longleftrightarrow> eventually (\<lambda>x. f x \<in> s \<and> f x \<noteq> b) F \<and> (f \<longlongrightarrow> b) F" 755 by (simp add: at_within_def filterlim_inf filterlim_principal conj_commute) 756 757lemma (in -) 758 assumes "filterlim f (nhds L) F" 759 shows tendsto_imp_filterlim_at_right: 760 "eventually (\<lambda>x. f x > L) F \<Longrightarrow> filterlim f (at_right L) F" 761 and tendsto_imp_filterlim_at_left: 762 "eventually (\<lambda>x. f x < L) F \<Longrightarrow> filterlim f (at_left L) F" 763 using assms by (auto simp: filterlim_at elim: eventually_mono) 764 765lemma filterlim_at_withinI: 766 assumes "filterlim f (nhds c) F" 767 assumes "eventually (\<lambda>x. f x \<in> A - {c}) F" 768 shows "filterlim f (at c within A) F" 769 using assms by (simp add: filterlim_at) 770 771lemma filterlim_atI: 772 assumes "filterlim f (nhds c) F" 773 assumes "eventually (\<lambda>x. f x \<noteq> c) F" 774 shows "filterlim f (at c) F" 775 using assms by (intro filterlim_at_withinI) simp_all 776 777lemma topological_tendstoI: 778 "(\<And>S. open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) F) \<Longrightarrow> (f \<longlongrightarrow> l) F" 779 by (auto simp: tendsto_def) 780 781lemma topological_tendstoD: 782 "(f \<longlongrightarrow> l) F \<Longrightarrow> open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) F" 783 by (auto simp: tendsto_def) 784 785lemma tendsto_bot [simp]: "(f \<longlongrightarrow> a) bot" 786 by (simp add: tendsto_def) 787 788lemma tendsto_eventually: "eventually (\<lambda>x. f x = l) net \<Longrightarrow> ((\<lambda>x. f x) \<longlongrightarrow> l) net" 789 by (rule topological_tendstoI) (auto elim: eventually_mono) 790 791end 792 793lemma (in topological_space) filterlim_within_subset: 794 "filterlim f l (at x within S) \<Longrightarrow> T \<subseteq> S \<Longrightarrow> filterlim f l (at x within T)" 795 by (blast intro: filterlim_mono at_le) 796 797lemmas tendsto_within_subset = filterlim_within_subset 798 799lemma (in order_topology) order_tendsto_iff: 800 "(f \<longlongrightarrow> x) F \<longleftrightarrow> (\<forall>l<x. eventually (\<lambda>x. l < f x) F) \<and> (\<forall>u>x. eventually (\<lambda>x. f x < u) F)" 801 by (auto simp: nhds_order filterlim_inf filterlim_INF filterlim_principal) 802 803lemma (in order_topology) order_tendstoI: 804 "(\<And>a. a < y \<Longrightarrow> eventually (\<lambda>x. a < f x) F) \<Longrightarrow> (\<And>a. y < a \<Longrightarrow> eventually (\<lambda>x. f x < a) F) \<Longrightarrow> 805 (f \<longlongrightarrow> y) F" 806 by (auto simp: order_tendsto_iff) 807 808lemma (in order_topology) order_tendstoD: 809 assumes "(f \<longlongrightarrow> y) F" 810 shows "a < y \<Longrightarrow> eventually (\<lambda>x. a < f x) F" 811 and "y < a \<Longrightarrow> eventually (\<lambda>x. f x < a) F" 812 using assms by (auto simp: order_tendsto_iff) 813 814lemma (in linorder_topology) tendsto_max[tendsto_intros]: 815 assumes X: "(X \<longlongrightarrow> x) net" 816 and Y: "(Y \<longlongrightarrow> y) net" 817 shows "((\<lambda>x. max (X x) (Y x)) \<longlongrightarrow> max x y) net" 818proof (rule order_tendstoI) 819 fix a 820 assume "a < max x y" 821 then show "eventually (\<lambda>x. a < max (X x) (Y x)) net" 822 using order_tendstoD(1)[OF X, of a] order_tendstoD(1)[OF Y, of a] 823 by (auto simp: less_max_iff_disj elim: eventually_mono) 824next 825 fix a 826 assume "max x y < a" 827 then show "eventually (\<lambda>x. max (X x) (Y x) < a) net" 828 using order_tendstoD(2)[OF X, of a] order_tendstoD(2)[OF Y, of a] 829 by (auto simp: eventually_conj_iff) 830qed 831 832lemma (in linorder_topology) tendsto_min[tendsto_intros]: 833 assumes X: "(X \<longlongrightarrow> x) net" 834 and Y: "(Y \<longlongrightarrow> y) net" 835 shows "((\<lambda>x. min (X x) (Y x)) \<longlongrightarrow> min x y) net" 836proof (rule order_tendstoI) 837 fix a 838 assume "a < min x y" 839 then show "eventually (\<lambda>x. a < min (X x) (Y x)) net" 840 using order_tendstoD(1)[OF X, of a] order_tendstoD(1)[OF Y, of a] 841 by (auto simp: eventually_conj_iff) 842next 843 fix a 844 assume "min x y < a" 845 then show "eventually (\<lambda>x. min (X x) (Y x) < a) net" 846 using order_tendstoD(2)[OF X, of a] order_tendstoD(2)[OF Y, of a] 847 by (auto simp: min_less_iff_disj elim: eventually_mono) 848qed 849 850lemma (in order_topology) 851 assumes "a < b" 852 shows at_within_Icc_at_right: "at a within {a..b} = at_right a" 853 and at_within_Icc_at_left: "at b within {a..b} = at_left b" 854 using order_tendstoD(2)[OF tendsto_ident_at assms, of "{a<..}"] 855 using order_tendstoD(1)[OF tendsto_ident_at assms, of "{..<b}"] 856 by (auto intro!: order_class.antisym filter_leI 857 simp: eventually_at_filter less_le 858 elim: eventually_elim2) 859 860lemma (in order_topology) at_within_Icc_at: "a < x \<Longrightarrow> x < b \<Longrightarrow> at x within {a..b} = at x" 861 by (rule at_within_open_subset[where S="{a<..<b}"]) auto 862 863lemma (in t2_space) tendsto_unique: 864 assumes "F \<noteq> bot" 865 and "(f \<longlongrightarrow> a) F" 866 and "(f \<longlongrightarrow> b) F" 867 shows "a = b" 868proof (rule ccontr) 869 assume "a \<noteq> b" 870 obtain U V where "open U" "open V" "a \<in> U" "b \<in> V" "U \<inter> V = {}" 871 using hausdorff [OF \<open>a \<noteq> b\<close>] by fast 872 have "eventually (\<lambda>x. f x \<in> U) F" 873 using \<open>(f \<longlongrightarrow> a) F\<close> \<open>open U\<close> \<open>a \<in> U\<close> by (rule topological_tendstoD) 874 moreover 875 have "eventually (\<lambda>x. f x \<in> V) F" 876 using \<open>(f \<longlongrightarrow> b) F\<close> \<open>open V\<close> \<open>b \<in> V\<close> by (rule topological_tendstoD) 877 ultimately 878 have "eventually (\<lambda>x. False) F" 879 proof eventually_elim 880 case (elim x) 881 then have "f x \<in> U \<inter> V" by simp 882 with \<open>U \<inter> V = {}\<close> show ?case by simp 883 qed 884 with \<open>\<not> trivial_limit F\<close> show "False" 885 by (simp add: trivial_limit_def) 886qed 887 888lemma (in t2_space) tendsto_const_iff: 889 fixes a b :: 'a 890 assumes "\<not> trivial_limit F" 891 shows "((\<lambda>x. a) \<longlongrightarrow> b) F \<longleftrightarrow> a = b" 892 by (auto intro!: tendsto_unique [OF assms tendsto_const]) 893 894lemma Lim_in_closed_set: 895 assumes "closed S" "eventually (\<lambda>x. f(x) \<in> S) F" "F \<noteq> bot" "(f \<longlongrightarrow> l) F" 896 shows "l \<in> S" 897proof (rule ccontr) 898 assume "l \<notin> S" 899 with \<open>closed S\<close> have "open (- S)" "l \<in> - S" 900 by (simp_all add: open_Compl) 901 with assms(4) have "eventually (\<lambda>x. f x \<in> - S) F" 902 by (rule topological_tendstoD) 903 with assms(2) have "eventually (\<lambda>x. False) F" 904 by (rule eventually_elim2) simp 905 with assms(3) show "False" 906 by (simp add: eventually_False) 907qed 908 909lemma (in t3_space) nhds_closed: 910 assumes "x \<in> A" and "open A" 911 shows "\<exists>A'. x \<in> A' \<and> closed A' \<and> A' \<subseteq> A \<and> eventually (\<lambda>y. y \<in> A') (nhds x)" 912proof - 913 from assms have "\<exists>U V. open U \<and> open V \<and> x \<in> U \<and> - A \<subseteq> V \<and> U \<inter> V = {}" 914 by (intro t3_space) auto 915 then obtain U V where UV: "open U" "open V" "x \<in> U" "-A \<subseteq> V" "U \<inter> V = {}" 916 by auto 917 have "eventually (\<lambda>y. y \<in> U) (nhds x)" 918 using \<open>open U\<close> and \<open>x \<in> U\<close> by (intro eventually_nhds_in_open) 919 hence "eventually (\<lambda>y. y \<in> -V) (nhds x)" 920 by eventually_elim (use UV in auto) 921 with UV show ?thesis by (intro exI[of _ "-V"]) auto 922qed 923 924lemma (in order_topology) increasing_tendsto: 925 assumes bdd: "eventually (\<lambda>n. f n \<le> l) F" 926 and en: "\<And>x. x < l \<Longrightarrow> eventually (\<lambda>n. x < f n) F" 927 shows "(f \<longlongrightarrow> l) F" 928 using assms by (intro order_tendstoI) (auto elim!: eventually_mono) 929 930lemma (in order_topology) decreasing_tendsto: 931 assumes bdd: "eventually (\<lambda>n. l \<le> f n) F" 932 and en: "\<And>x. l < x \<Longrightarrow> eventually (\<lambda>n. f n < x) F" 933 shows "(f \<longlongrightarrow> l) F" 934 using assms by (intro order_tendstoI) (auto elim!: eventually_mono) 935 936lemma (in order_topology) tendsto_sandwich: 937 assumes ev: "eventually (\<lambda>n. f n \<le> g n) net" "eventually (\<lambda>n. g n \<le> h n) net" 938 assumes lim: "(f \<longlongrightarrow> c) net" "(h \<longlongrightarrow> c) net" 939 shows "(g \<longlongrightarrow> c) net" 940proof (rule order_tendstoI) 941 fix a 942 show "a < c \<Longrightarrow> eventually (\<lambda>x. a < g x) net" 943 using order_tendstoD[OF lim(1), of a] ev by (auto elim: eventually_elim2) 944next 945 fix a 946 show "c < a \<Longrightarrow> eventually (\<lambda>x. g x < a) net" 947 using order_tendstoD[OF lim(2), of a] ev by (auto elim: eventually_elim2) 948qed 949 950lemma (in t1_space) limit_frequently_eq: 951 assumes "F \<noteq> bot" 952 and "frequently (\<lambda>x. f x = c) F" 953 and "(f \<longlongrightarrow> d) F" 954 shows "d = c" 955proof (rule ccontr) 956 assume "d \<noteq> c" 957 from t1_space[OF this] obtain U where "open U" "d \<in> U" "c \<notin> U" 958 by blast 959 with assms have "eventually (\<lambda>x. f x \<in> U) F" 960 unfolding tendsto_def by blast 961 then have "eventually (\<lambda>x. f x \<noteq> c) F" 962 by eventually_elim (insert \<open>c \<notin> U\<close>, blast) 963 with assms(2) show False 964 unfolding frequently_def by contradiction 965qed 966 967lemma (in t1_space) tendsto_imp_eventually_ne: 968 assumes "(f \<longlongrightarrow> c) F" "c \<noteq> c'" 969 shows "eventually (\<lambda>z. f z \<noteq> c') F" 970proof (cases "F=bot") 971 case True 972 thus ?thesis by auto 973next 974 case False 975 show ?thesis 976 proof (rule ccontr) 977 assume "\<not> eventually (\<lambda>z. f z \<noteq> c') F" 978 then have "frequently (\<lambda>z. f z = c') F" 979 by (simp add: frequently_def) 980 from limit_frequently_eq[OF False this \<open>(f \<longlongrightarrow> c) F\<close>] and \<open>c \<noteq> c'\<close> show False 981 by contradiction 982 qed 983qed 984 985lemma (in linorder_topology) tendsto_le: 986 assumes F: "\<not> trivial_limit F" 987 and x: "(f \<longlongrightarrow> x) F" 988 and y: "(g \<longlongrightarrow> y) F" 989 and ev: "eventually (\<lambda>x. g x \<le> f x) F" 990 shows "y \<le> x" 991proof (rule ccontr) 992 assume "\<not> y \<le> x" 993 with less_separate[of x y] obtain a b where xy: "x < a" "b < y" "{..<a} \<inter> {b<..} = {}" 994 by (auto simp: not_le) 995 then have "eventually (\<lambda>x. f x < a) F" "eventually (\<lambda>x. b < g x) F" 996 using x y by (auto intro: order_tendstoD) 997 with ev have "eventually (\<lambda>x. False) F" 998 by eventually_elim (insert xy, fastforce) 999 with F show False 1000 by (simp add: eventually_False) 1001qed 1002 1003lemma (in linorder_topology) tendsto_lowerbound: 1004 assumes x: "(f \<longlongrightarrow> x) F" 1005 and ev: "eventually (\<lambda>i. a \<le> f i) F" 1006 and F: "\<not> trivial_limit F" 1007 shows "a \<le> x" 1008 using F x tendsto_const ev by (rule tendsto_le) 1009 1010lemma (in linorder_topology) tendsto_upperbound: 1011 assumes x: "(f \<longlongrightarrow> x) F" 1012 and ev: "eventually (\<lambda>i. a \<ge> f i) F" 1013 and F: "\<not> trivial_limit F" 1014 shows "a \<ge> x" 1015 by (rule tendsto_le [OF F tendsto_const x ev]) 1016 1017lemma filterlim_at_within_not_equal: 1018 fixes f::"'a \<Rightarrow> 'b::t2_space" 1019 assumes "filterlim f (at a within s) F" 1020 shows "eventually (\<lambda>w. f w\<in>s \<and> f w \<noteq>b) F" 1021proof (cases "a=b") 1022 case True 1023 then show ?thesis using assms by (simp add: filterlim_at) 1024next 1025 case False 1026 from hausdorff[OF this] obtain U V where UV:"open U" "open V" "a \<in> U" "b \<in> V" "U \<inter> V = {}" 1027 by auto 1028 have "(f \<longlongrightarrow> a) F" using assms filterlim_at by auto 1029 then have "\<forall>\<^sub>F x in F. f x \<in> U" using UV unfolding tendsto_def by auto 1030 moreover have "\<forall>\<^sub>F x in F. f x \<in> s \<and> f x\<noteq>a" using assms filterlim_at by auto 1031 ultimately show ?thesis 1032 apply eventually_elim 1033 using UV by auto 1034qed 1035 1036subsubsection \<open>Rules about \<^const>\<open>Lim\<close>\<close> 1037 1038lemma tendsto_Lim: "\<not> trivial_limit net \<Longrightarrow> (f \<longlongrightarrow> l) net \<Longrightarrow> Lim net f = l" 1039 unfolding Lim_def using tendsto_unique [of net f] by auto 1040 1041lemma Lim_ident_at: "\<not> trivial_limit (at x within s) \<Longrightarrow> Lim (at x within s) (\<lambda>x. x) = x" 1042 by (rule tendsto_Lim[OF _ tendsto_ident_at]) auto 1043 1044lemma eventually_Lim_ident_at: 1045 "(\<forall>\<^sub>F y in at x within X. P (Lim (at x within X) (\<lambda>x. x)) y) \<longleftrightarrow> 1046 (\<forall>\<^sub>F y in at x within X. P x y)" for x::"'a::t2_space" 1047 by (cases "at x within X = bot") (auto simp: Lim_ident_at) 1048 1049lemma filterlim_at_bot_at_right: 1050 fixes f :: "'a::linorder_topology \<Rightarrow> 'b::linorder" 1051 assumes mono: "\<And>x y. Q x \<Longrightarrow> Q y \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y" 1052 and bij: "\<And>x. P x \<Longrightarrow> f (g x) = x" "\<And>x. P x \<Longrightarrow> Q (g x)" 1053 and Q: "eventually Q (at_right a)" 1054 and bound: "\<And>b. Q b \<Longrightarrow> a < b" 1055 and P: "eventually P at_bot" 1056 shows "filterlim f at_bot (at_right a)" 1057proof - 1058 from P obtain x where x: "\<And>y. y \<le> x \<Longrightarrow> P y" 1059 unfolding eventually_at_bot_linorder by auto 1060 show ?thesis 1061 proof (intro filterlim_at_bot_le[THEN iffD2] allI impI) 1062 fix z 1063 assume "z \<le> x" 1064 with x have "P z" by auto 1065 have "eventually (\<lambda>x. x \<le> g z) (at_right a)" 1066 using bound[OF bij(2)[OF \<open>P z\<close>]] 1067 unfolding eventually_at_right[OF bound[OF bij(2)[OF \<open>P z\<close>]]] 1068 by (auto intro!: exI[of _ "g z"]) 1069 with Q show "eventually (\<lambda>x. f x \<le> z) (at_right a)" 1070 by eventually_elim (metis bij \<open>P z\<close> mono) 1071 qed 1072qed 1073 1074lemma filterlim_at_top_at_left: 1075 fixes f :: "'a::linorder_topology \<Rightarrow> 'b::linorder" 1076 assumes mono: "\<And>x y. Q x \<Longrightarrow> Q y \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y" 1077 and bij: "\<And>x. P x \<Longrightarrow> f (g x) = x" "\<And>x. P x \<Longrightarrow> Q (g x)" 1078 and Q: "eventually Q (at_left a)" 1079 and bound: "\<And>b. Q b \<Longrightarrow> b < a" 1080 and P: "eventually P at_top" 1081 shows "filterlim f at_top (at_left a)" 1082proof - 1083 from P obtain x where x: "\<And>y. x \<le> y \<Longrightarrow> P y" 1084 unfolding eventually_at_top_linorder by auto 1085 show ?thesis 1086 proof (intro filterlim_at_top_ge[THEN iffD2] allI impI) 1087 fix z 1088 assume "x \<le> z" 1089 with x have "P z" by auto 1090 have "eventually (\<lambda>x. g z \<le> x) (at_left a)" 1091 using bound[OF bij(2)[OF \<open>P z\<close>]] 1092 unfolding eventually_at_left[OF bound[OF bij(2)[OF \<open>P z\<close>]]] 1093 by (auto intro!: exI[of _ "g z"]) 1094 with Q show "eventually (\<lambda>x. z \<le> f x) (at_left a)" 1095 by eventually_elim (metis bij \<open>P z\<close> mono) 1096 qed 1097qed 1098 1099lemma filterlim_split_at: 1100 "filterlim f F (at_left x) \<Longrightarrow> filterlim f F (at_right x) \<Longrightarrow> 1101 filterlim f F (at x)" 1102 for x :: "'a::linorder_topology" 1103 by (subst at_eq_sup_left_right) (rule filterlim_sup) 1104 1105lemma filterlim_at_split: 1106 "filterlim f F (at x) \<longleftrightarrow> filterlim f F (at_left x) \<and> filterlim f F (at_right x)" 1107 for x :: "'a::linorder_topology" 1108 by (subst at_eq_sup_left_right) (simp add: filterlim_def filtermap_sup) 1109 1110lemma eventually_nhds_top: 1111 fixes P :: "'a :: {order_top,linorder_topology} \<Rightarrow> bool" 1112 and b :: 'a 1113 assumes "b < top" 1114 shows "eventually P (nhds top) \<longleftrightarrow> (\<exists>b<top. (\<forall>z. b < z \<longrightarrow> P z))" 1115 unfolding eventually_nhds 1116proof safe 1117 fix S :: "'a set" 1118 assume "open S" "top \<in> S" 1119 note open_left[OF this \<open>b < top\<close>] 1120 moreover assume "\<forall>s\<in>S. P s" 1121 ultimately show "\<exists>b<top. \<forall>z>b. P z" 1122 by (auto simp: subset_eq Ball_def) 1123next 1124 fix b 1125 assume "b < top" "\<forall>z>b. P z" 1126 then show "\<exists>S. open S \<and> top \<in> S \<and> (\<forall>xa\<in>S. P xa)" 1127 by (intro exI[of _ "{b <..}"]) auto 1128qed 1129 1130lemma tendsto_at_within_iff_tendsto_nhds: 1131 "(g \<longlongrightarrow> g l) (at l within S) \<longleftrightarrow> (g \<longlongrightarrow> g l) (inf (nhds l) (principal S))" 1132 unfolding tendsto_def eventually_at_filter eventually_inf_principal 1133 by (intro ext all_cong imp_cong) (auto elim!: eventually_mono) 1134 1135 1136subsection \<open>Limits on sequences\<close> 1137 1138abbreviation (in topological_space) 1139 LIMSEQ :: "[nat \<Rightarrow> 'a, 'a] \<Rightarrow> bool" ("((_)/ \<longlonglongrightarrow> (_))" [60, 60] 60) 1140 where "X \<longlonglongrightarrow> L \<equiv> (X \<longlongrightarrow> L) sequentially" 1141 1142abbreviation (in t2_space) lim :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'a" 1143 where "lim X \<equiv> Lim sequentially X" 1144 1145definition (in topological_space) convergent :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" 1146 where "convergent X = (\<exists>L. X \<longlonglongrightarrow> L)" 1147 1148lemma lim_def: "lim X = (THE L. X \<longlonglongrightarrow> L)" 1149 unfolding Lim_def .. 1150 1151lemma lim_explicit: 1152 "f \<longlonglongrightarrow> f0 \<longleftrightarrow> (\<forall>S. open S \<longrightarrow> f0 \<in> S \<longrightarrow> (\<exists>N. \<forall>n\<ge>N. f n \<in> S))" 1153 unfolding tendsto_def eventually_sequentially by auto 1154 1155 1156subsection \<open>Monotone sequences and subsequences\<close> 1157 1158text \<open> 1159 Definition of monotonicity. 1160 The use of disjunction here complicates proofs considerably. 1161 One alternative is to add a Boolean argument to indicate the direction. 1162 Another is to develop the notions of increasing and decreasing first. 1163\<close> 1164definition monoseq :: "(nat \<Rightarrow> 'a::order) \<Rightarrow> bool" 1165 where "monoseq X \<longleftrightarrow> (\<forall>m. \<forall>n\<ge>m. X m \<le> X n) \<or> (\<forall>m. \<forall>n\<ge>m. X n \<le> X m)" 1166 1167abbreviation incseq :: "(nat \<Rightarrow> 'a::order) \<Rightarrow> bool" 1168 where "incseq X \<equiv> mono X" 1169 1170lemma incseq_def: "incseq X \<longleftrightarrow> (\<forall>m. \<forall>n\<ge>m. X n \<ge> X m)" 1171 unfolding mono_def .. 1172 1173abbreviation decseq :: "(nat \<Rightarrow> 'a::order) \<Rightarrow> bool" 1174 where "decseq X \<equiv> antimono X" 1175 1176lemma decseq_def: "decseq X \<longleftrightarrow> (\<forall>m. \<forall>n\<ge>m. X n \<le> X m)" 1177 unfolding antimono_def .. 1178 1179subsubsection \<open>Definition of subsequence.\<close> 1180 1181(* For compatibility with the old "subseq" *) 1182lemma strict_mono_leD: "strict_mono r \<Longrightarrow> m \<le> n \<Longrightarrow> r m \<le> r n" 1183 by (erule (1) monoD [OF strict_mono_mono]) 1184 1185lemma strict_mono_id: "strict_mono id" 1186 by (simp add: strict_mono_def) 1187 1188lemma incseq_SucI: "(\<And>n. X n \<le> X (Suc n)) \<Longrightarrow> incseq X" 1189 using lift_Suc_mono_le[of X] by (auto simp: incseq_def) 1190 1191lemma incseqD: "incseq f \<Longrightarrow> i \<le> j \<Longrightarrow> f i \<le> f j" 1192 by (auto simp: incseq_def) 1193 1194lemma incseq_SucD: "incseq A \<Longrightarrow> A i \<le> A (Suc i)" 1195 using incseqD[of A i "Suc i"] by auto 1196 1197lemma incseq_Suc_iff: "incseq f \<longleftrightarrow> (\<forall>n. f n \<le> f (Suc n))" 1198 by (auto intro: incseq_SucI dest: incseq_SucD) 1199 1200lemma incseq_const[simp, intro]: "incseq (\<lambda>x. k)" 1201 unfolding incseq_def by auto 1202 1203lemma decseq_SucI: "(\<And>n. X (Suc n) \<le> X n) \<Longrightarrow> decseq X" 1204 using order.lift_Suc_mono_le[OF dual_order, of X] by (auto simp: decseq_def) 1205 1206lemma decseqD: "decseq f \<Longrightarrow> i \<le> j \<Longrightarrow> f j \<le> f i" 1207 by (auto simp: decseq_def) 1208 1209lemma decseq_SucD: "decseq A \<Longrightarrow> A (Suc i) \<le> A i" 1210 using decseqD[of A i "Suc i"] by auto 1211 1212lemma decseq_Suc_iff: "decseq f \<longleftrightarrow> (\<forall>n. f (Suc n) \<le> f n)" 1213 by (auto intro: decseq_SucI dest: decseq_SucD) 1214 1215lemma decseq_const[simp, intro]: "decseq (\<lambda>x. k)" 1216 unfolding decseq_def by auto 1217 1218lemma monoseq_iff: "monoseq X \<longleftrightarrow> incseq X \<or> decseq X" 1219 unfolding monoseq_def incseq_def decseq_def .. 1220 1221lemma monoseq_Suc: "monoseq X \<longleftrightarrow> (\<forall>n. X n \<le> X (Suc n)) \<or> (\<forall>n. X (Suc n) \<le> X n)" 1222 unfolding monoseq_iff incseq_Suc_iff decseq_Suc_iff .. 1223 1224lemma monoI1: "\<forall>m. \<forall>n \<ge> m. X m \<le> X n \<Longrightarrow> monoseq X" 1225 by (simp add: monoseq_def) 1226 1227lemma monoI2: "\<forall>m. \<forall>n \<ge> m. X n \<le> X m \<Longrightarrow> monoseq X" 1228 by (simp add: monoseq_def) 1229 1230lemma mono_SucI1: "\<forall>n. X n \<le> X (Suc n) \<Longrightarrow> monoseq X" 1231 by (simp add: monoseq_Suc) 1232 1233lemma mono_SucI2: "\<forall>n. X (Suc n) \<le> X n \<Longrightarrow> monoseq X" 1234 by (simp add: monoseq_Suc) 1235 1236lemma monoseq_minus: 1237 fixes a :: "nat \<Rightarrow> 'a::ordered_ab_group_add" 1238 assumes "monoseq a" 1239 shows "monoseq (\<lambda> n. - a n)" 1240proof (cases "\<forall>m. \<forall>n \<ge> m. a m \<le> a n") 1241 case True 1242 then have "\<forall>m. \<forall>n \<ge> m. - a n \<le> - a m" by auto 1243 then show ?thesis by (rule monoI2) 1244next 1245 case False 1246 then have "\<forall>m. \<forall>n \<ge> m. - a m \<le> - a n" 1247 using \<open>monoseq a\<close>[unfolded monoseq_def] by auto 1248 then show ?thesis by (rule monoI1) 1249qed 1250 1251 1252subsubsection \<open>Subsequence (alternative definition, (e.g. Hoskins)\<close> 1253 1254lemma strict_mono_Suc_iff: "strict_mono f \<longleftrightarrow> (\<forall>n. f n < f (Suc n))" 1255proof (intro iffI strict_monoI) 1256 assume *: "\<forall>n. f n < f (Suc n)" 1257 fix m n :: nat assume "m < n" 1258 thus "f m < f n" 1259 by (induction rule: less_Suc_induct) (use * in auto) 1260qed (auto simp: strict_mono_def) 1261 1262lemma strict_mono_add: "strict_mono (\<lambda>n::'a::linordered_semidom. n + k)" 1263 by (auto simp: strict_mono_def) 1264 1265text \<open>For any sequence, there is a monotonic subsequence.\<close> 1266lemma seq_monosub: 1267 fixes s :: "nat \<Rightarrow> 'a::linorder" 1268 shows "\<exists>f. strict_mono f \<and> monoseq (\<lambda>n. (s (f n)))" 1269proof (cases "\<forall>n. \<exists>p>n. \<forall>m\<ge>p. s m \<le> s p") 1270 case True 1271 then have "\<exists>f. \<forall>n. (\<forall>m\<ge>f n. s m \<le> s (f n)) \<and> f n < f (Suc n)" 1272 by (intro dependent_nat_choice) (auto simp: conj_commute) 1273 then obtain f :: "nat \<Rightarrow> nat" 1274 where f: "strict_mono f" and mono: "\<And>n m. f n \<le> m \<Longrightarrow> s m \<le> s (f n)" 1275 by (auto simp: strict_mono_Suc_iff) 1276 then have "incseq f" 1277 unfolding strict_mono_Suc_iff incseq_Suc_iff by (auto intro: less_imp_le) 1278 then have "monoseq (\<lambda>n. s (f n))" 1279 by (auto simp add: incseq_def intro!: mono monoI2) 1280 with f show ?thesis 1281 by auto 1282next 1283 case False 1284 then obtain N where N: "p > N \<Longrightarrow> \<exists>m>p. s p < s m" for p 1285 by (force simp: not_le le_less) 1286 have "\<exists>f. \<forall>n. N < f n \<and> f n < f (Suc n) \<and> s (f n) \<le> s (f (Suc n))" 1287 proof (intro dependent_nat_choice) 1288 fix x 1289 assume "N < x" with N[of x] 1290 show "\<exists>y>N. x < y \<and> s x \<le> s y" 1291 by (auto intro: less_trans) 1292 qed auto 1293 then show ?thesis 1294 by (auto simp: monoseq_iff incseq_Suc_iff strict_mono_Suc_iff) 1295qed 1296 1297lemma seq_suble: 1298 assumes sf: "strict_mono (f :: nat \<Rightarrow> nat)" 1299 shows "n \<le> f n" 1300proof (induct n) 1301 case 0 1302 show ?case by simp 1303next 1304 case (Suc n) 1305 with sf [unfolded strict_mono_Suc_iff, rule_format, of n] have "n < f (Suc n)" 1306 by arith 1307 then show ?case by arith 1308qed 1309 1310lemma eventually_subseq: 1311 "strict_mono r \<Longrightarrow> eventually P sequentially \<Longrightarrow> eventually (\<lambda>n. P (r n)) sequentially" 1312 unfolding eventually_sequentially by (metis seq_suble le_trans) 1313 1314lemma not_eventually_sequentiallyD: 1315 assumes "\<not> eventually P sequentially" 1316 shows "\<exists>r::nat\<Rightarrow>nat. strict_mono r \<and> (\<forall>n. \<not> P (r n))" 1317proof - 1318 from assms have "\<forall>n. \<exists>m\<ge>n. \<not> P m" 1319 unfolding eventually_sequentially by (simp add: not_less) 1320 then obtain r where "\<And>n. r n \<ge> n" "\<And>n. \<not> P (r n)" 1321 by (auto simp: choice_iff) 1322 then show ?thesis 1323 by (auto intro!: exI[of _ "\<lambda>n. r (((Suc \<circ> r) ^^ Suc n) 0)"] 1324 simp: less_eq_Suc_le strict_mono_Suc_iff) 1325qed 1326 1327lemma sequentially_offset: 1328 assumes "eventually (\<lambda>i. P i) sequentially" 1329 shows "eventually (\<lambda>i. P (i + k)) sequentially" 1330 using assms by (rule eventually_sequentially_seg [THEN iffD2]) 1331 1332lemma seq_offset_neg: 1333 "(f \<longlongrightarrow> l) sequentially \<Longrightarrow> ((\<lambda>i. f(i - k)) \<longlongrightarrow> l) sequentially" 1334 apply (erule filterlim_compose) 1335 apply (simp add: filterlim_def le_sequentially eventually_filtermap eventually_sequentially, arith) 1336 done 1337 1338lemma filterlim_subseq: "strict_mono f \<Longrightarrow> filterlim f sequentially sequentially" 1339 unfolding filterlim_iff by (metis eventually_subseq) 1340 1341lemma strict_mono_o: "strict_mono r \<Longrightarrow> strict_mono s \<Longrightarrow> strict_mono (r \<circ> s)" 1342 unfolding strict_mono_def by simp 1343 1344lemma strict_mono_compose: "strict_mono r \<Longrightarrow> strict_mono s \<Longrightarrow> strict_mono (\<lambda>x. r (s x))" 1345 using strict_mono_o[of r s] by (simp add: o_def) 1346 1347lemma incseq_imp_monoseq: "incseq X \<Longrightarrow> monoseq X" 1348 by (simp add: incseq_def monoseq_def) 1349 1350lemma decseq_imp_monoseq: "decseq X \<Longrightarrow> monoseq X" 1351 by (simp add: decseq_def monoseq_def) 1352 1353lemma decseq_eq_incseq: "decseq X = incseq (\<lambda>n. - X n)" 1354 for X :: "nat \<Rightarrow> 'a::ordered_ab_group_add" 1355 by (simp add: decseq_def incseq_def) 1356 1357lemma INT_decseq_offset: 1358 assumes "decseq F" 1359 shows "(\<Inter>i. F i) = (\<Inter>i\<in>{n..}. F i)" 1360proof safe 1361 fix x i 1362 assume x: "x \<in> (\<Inter>i\<in>{n..}. F i)" 1363 show "x \<in> F i" 1364 proof cases 1365 from x have "x \<in> F n" by auto 1366 also assume "i \<le> n" with \<open>decseq F\<close> have "F n \<subseteq> F i" 1367 unfolding decseq_def by simp 1368 finally show ?thesis . 1369 qed (insert x, simp) 1370qed auto 1371 1372lemma LIMSEQ_const_iff: "(\<lambda>n. k) \<longlonglongrightarrow> l \<longleftrightarrow> k = l" 1373 for k l :: "'a::t2_space" 1374 using trivial_limit_sequentially by (rule tendsto_const_iff) 1375 1376lemma LIMSEQ_SUP: "incseq X \<Longrightarrow> X \<longlonglongrightarrow> (SUP i. X i :: 'a::{complete_linorder,linorder_topology})" 1377 by (intro increasing_tendsto) 1378 (auto simp: SUP_upper less_SUP_iff incseq_def eventually_sequentially intro: less_le_trans) 1379 1380lemma LIMSEQ_INF: "decseq X \<Longrightarrow> X \<longlonglongrightarrow> (INF i. X i :: 'a::{complete_linorder,linorder_topology})" 1381 by (intro decreasing_tendsto) 1382 (auto simp: INF_lower INF_less_iff decseq_def eventually_sequentially intro: le_less_trans) 1383 1384lemma LIMSEQ_ignore_initial_segment: "f \<longlonglongrightarrow> a \<Longrightarrow> (\<lambda>n. f (n + k)) \<longlonglongrightarrow> a" 1385 unfolding tendsto_def by (subst eventually_sequentially_seg[where k=k]) 1386 1387lemma LIMSEQ_offset: "(\<lambda>n. f (n + k)) \<longlonglongrightarrow> a \<Longrightarrow> f \<longlonglongrightarrow> a" 1388 unfolding tendsto_def 1389 by (subst (asm) eventually_sequentially_seg[where k=k]) 1390 1391lemma LIMSEQ_Suc: "f \<longlonglongrightarrow> l \<Longrightarrow> (\<lambda>n. f (Suc n)) \<longlonglongrightarrow> l" 1392 by (drule LIMSEQ_ignore_initial_segment [where k="Suc 0"]) simp 1393 1394lemma LIMSEQ_imp_Suc: "(\<lambda>n. f (Suc n)) \<longlonglongrightarrow> l \<Longrightarrow> f \<longlonglongrightarrow> l" 1395 by (rule LIMSEQ_offset [where k="Suc 0"]) simp 1396 1397lemma LIMSEQ_Suc_iff: "(\<lambda>n. f (Suc n)) \<longlonglongrightarrow> l = f \<longlonglongrightarrow> l" 1398 by (rule filterlim_sequentially_Suc) 1399 1400lemma LIMSEQ_lessThan_iff_atMost: 1401 shows "(\<lambda>n. f {..<n}) \<longlonglongrightarrow> x \<longleftrightarrow> (\<lambda>n. f {..n}) \<longlonglongrightarrow> x" 1402 apply (subst LIMSEQ_Suc_iff [symmetric]) 1403 apply (simp only: lessThan_Suc_atMost) 1404 done 1405 1406lemma LIMSEQ_unique: "X \<longlonglongrightarrow> a \<Longrightarrow> X \<longlonglongrightarrow> b \<Longrightarrow> a = b" 1407 for a b :: "'a::t2_space" 1408 using trivial_limit_sequentially by (rule tendsto_unique) 1409 1410lemma LIMSEQ_le_const: "X \<longlonglongrightarrow> x \<Longrightarrow> \<exists>N. \<forall>n\<ge>N. a \<le> X n \<Longrightarrow> a \<le> x" 1411 for a x :: "'a::linorder_topology" 1412 by (simp add: eventually_at_top_linorder tendsto_lowerbound) 1413 1414lemma LIMSEQ_le: "X \<longlonglongrightarrow> x \<Longrightarrow> Y \<longlonglongrightarrow> y \<Longrightarrow> \<exists>N. \<forall>n\<ge>N. X n \<le> Y n \<Longrightarrow> x \<le> y" 1415 for x y :: "'a::linorder_topology" 1416 using tendsto_le[of sequentially Y y X x] by (simp add: eventually_sequentially) 1417 1418lemma LIMSEQ_le_const2: "X \<longlonglongrightarrow> x \<Longrightarrow> \<exists>N. \<forall>n\<ge>N. X n \<le> a \<Longrightarrow> x \<le> a" 1419 for a x :: "'a::linorder_topology" 1420 by (rule LIMSEQ_le[of X x "\<lambda>n. a"]) auto 1421 1422lemma Lim_bounded: "f \<longlonglongrightarrow> l \<Longrightarrow> \<forall>n\<ge>M. f n \<le> C \<Longrightarrow> l \<le> C" 1423 for l :: "'a::linorder_topology" 1424 by (intro LIMSEQ_le_const2) auto 1425 1426lemma Lim_bounded2: 1427 fixes f :: "nat \<Rightarrow> 'a::linorder_topology" 1428 assumes lim:"f \<longlonglongrightarrow> l" and ge: "\<forall>n\<ge>N. f n \<ge> C" 1429 shows "l \<ge> C" 1430 using ge 1431 by (intro tendsto_le[OF trivial_limit_sequentially lim tendsto_const]) 1432 (auto simp: eventually_sequentially) 1433 1434lemma lim_mono: 1435 fixes X Y :: "nat \<Rightarrow> 'a::linorder_topology" 1436 assumes "\<And>n. N \<le> n \<Longrightarrow> X n \<le> Y n" 1437 and "X \<longlonglongrightarrow> x" 1438 and "Y \<longlonglongrightarrow> y" 1439 shows "x \<le> y" 1440 using assms(1) by (intro LIMSEQ_le[OF assms(2,3)]) auto 1441 1442lemma Sup_lim: 1443 fixes a :: "'a::{complete_linorder,linorder_topology}" 1444 assumes "\<And>n. b n \<in> s" 1445 and "b \<longlonglongrightarrow> a" 1446 shows "a \<le> Sup s" 1447 by (metis Lim_bounded assms complete_lattice_class.Sup_upper) 1448 1449lemma Inf_lim: 1450 fixes a :: "'a::{complete_linorder,linorder_topology}" 1451 assumes "\<And>n. b n \<in> s" 1452 and "b \<longlonglongrightarrow> a" 1453 shows "Inf s \<le> a" 1454 by (metis Lim_bounded2 assms complete_lattice_class.Inf_lower) 1455 1456lemma SUP_Lim: 1457 fixes X :: "nat \<Rightarrow> 'a::{complete_linorder,linorder_topology}" 1458 assumes inc: "incseq X" 1459 and l: "X \<longlonglongrightarrow> l" 1460 shows "(SUP n. X n) = l" 1461 using LIMSEQ_SUP[OF inc] tendsto_unique[OF trivial_limit_sequentially l] 1462 by simp 1463 1464lemma INF_Lim: 1465 fixes X :: "nat \<Rightarrow> 'a::{complete_linorder,linorder_topology}" 1466 assumes dec: "decseq X" 1467 and l: "X \<longlonglongrightarrow> l" 1468 shows "(INF n. X n) = l" 1469 using LIMSEQ_INF[OF dec] tendsto_unique[OF trivial_limit_sequentially l] 1470 by simp 1471 1472lemma convergentD: "convergent X \<Longrightarrow> \<exists>L. X \<longlonglongrightarrow> L" 1473 by (simp add: convergent_def) 1474 1475lemma convergentI: "X \<longlonglongrightarrow> L \<Longrightarrow> convergent X" 1476 by (auto simp add: convergent_def) 1477 1478lemma convergent_LIMSEQ_iff: "convergent X \<longleftrightarrow> X \<longlonglongrightarrow> lim X" 1479 by (auto intro: theI LIMSEQ_unique simp add: convergent_def lim_def) 1480 1481lemma convergent_const: "convergent (\<lambda>n. c)" 1482 by (rule convergentI) (rule tendsto_const) 1483 1484lemma monoseq_le: 1485 "monoseq a \<Longrightarrow> a \<longlonglongrightarrow> x \<Longrightarrow> 1486 (\<forall>n. a n \<le> x) \<and> (\<forall>m. \<forall>n\<ge>m. a m \<le> a n) \<or> 1487 (\<forall>n. x \<le> a n) \<and> (\<forall>m. \<forall>n\<ge>m. a n \<le> a m)" 1488 for x :: "'a::linorder_topology" 1489 by (metis LIMSEQ_le_const LIMSEQ_le_const2 decseq_def incseq_def monoseq_iff) 1490 1491lemma LIMSEQ_subseq_LIMSEQ: "X \<longlonglongrightarrow> L \<Longrightarrow> strict_mono f \<Longrightarrow> (X \<circ> f) \<longlonglongrightarrow> L" 1492 unfolding comp_def by (rule filterlim_compose [of X, OF _ filterlim_subseq]) 1493 1494lemma convergent_subseq_convergent: "convergent X \<Longrightarrow> strict_mono f \<Longrightarrow> convergent (X \<circ> f)" 1495 by (auto simp: convergent_def intro: LIMSEQ_subseq_LIMSEQ) 1496 1497lemma limI: "X \<longlonglongrightarrow> L \<Longrightarrow> lim X = L" 1498 by (rule tendsto_Lim) (rule trivial_limit_sequentially) 1499 1500lemma lim_le: "convergent f \<Longrightarrow> (\<And>n. f n \<le> x) \<Longrightarrow> lim f \<le> x" 1501 for x :: "'a::linorder_topology" 1502 using LIMSEQ_le_const2[of f "lim f" x] by (simp add: convergent_LIMSEQ_iff) 1503 1504lemma lim_const [simp]: "lim (\<lambda>m. a) = a" 1505 by (simp add: limI) 1506 1507 1508subsubsection \<open>Increasing and Decreasing Series\<close> 1509 1510lemma incseq_le: "incseq X \<Longrightarrow> X \<longlonglongrightarrow> L \<Longrightarrow> X n \<le> L" 1511 for L :: "'a::linorder_topology" 1512 by (metis incseq_def LIMSEQ_le_const) 1513 1514lemma decseq_ge: "decseq X \<Longrightarrow> X \<longlonglongrightarrow> L \<Longrightarrow> L \<le> X n" 1515 for L :: "'a::linorder_topology" 1516 by (metis decseq_def LIMSEQ_le_const2) 1517 1518 1519subsection \<open>First countable topologies\<close> 1520 1521class first_countable_topology = topological_space + 1522 assumes first_countable_basis: 1523 "\<exists>A::nat \<Rightarrow> 'a set. (\<forall>i. x \<in> A i \<and> open (A i)) \<and> (\<forall>S. open S \<and> x \<in> S \<longrightarrow> (\<exists>i. A i \<subseteq> S))" 1524 1525lemma (in first_countable_topology) countable_basis_at_decseq: 1526 obtains A :: "nat \<Rightarrow> 'a set" where 1527 "\<And>i. open (A i)" "\<And>i. x \<in> (A i)" 1528 "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> eventually (\<lambda>i. A i \<subseteq> S) sequentially" 1529proof atomize_elim 1530 from first_countable_basis[of x] obtain A :: "nat \<Rightarrow> 'a set" 1531 where nhds: "\<And>i. open (A i)" "\<And>i. x \<in> A i" 1532 and incl: "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> \<exists>i. A i \<subseteq> S" 1533 by auto 1534 define F where "F n = (\<Inter>i\<le>n. A i)" for n 1535 show "\<exists>A. (\<forall>i. open (A i)) \<and> (\<forall>i. x \<in> A i) \<and> 1536 (\<forall>S. open S \<longrightarrow> x \<in> S \<longrightarrow> eventually (\<lambda>i. A i \<subseteq> S) sequentially)" 1537 proof (safe intro!: exI[of _ F]) 1538 fix i 1539 show "open (F i)" 1540 using nhds(1) by (auto simp: F_def) 1541 show "x \<in> F i" 1542 using nhds(2) by (auto simp: F_def) 1543 next 1544 fix S 1545 assume "open S" "x \<in> S" 1546 from incl[OF this] obtain i where "F i \<subseteq> S" 1547 unfolding F_def by auto 1548 moreover have "\<And>j. i \<le> j \<Longrightarrow> F j \<subseteq> F i" 1549 by (simp add: Inf_superset_mono F_def image_mono) 1550 ultimately show "eventually (\<lambda>i. F i \<subseteq> S) sequentially" 1551 by (auto simp: eventually_sequentially) 1552 qed 1553qed 1554 1555lemma (in first_countable_topology) nhds_countable: 1556 obtains X :: "nat \<Rightarrow> 'a set" 1557 where "decseq X" "\<And>n. open (X n)" "\<And>n. x \<in> X n" "nhds x = (INF n. principal (X n))" 1558proof - 1559 from first_countable_basis obtain A :: "nat \<Rightarrow> 'a set" 1560 where *: "\<And>n. x \<in> A n" "\<And>n. open (A n)" "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> \<exists>i. A i \<subseteq> S" 1561 by metis 1562 show thesis 1563 proof 1564 show "decseq (\<lambda>n. \<Inter>i\<le>n. A i)" 1565 by (simp add: antimono_iff_le_Suc atMost_Suc) 1566 show "x \<in> (\<Inter>i\<le>n. A i)" "\<And>n. open (\<Inter>i\<le>n. A i)" for n 1567 using * by auto 1568 show "nhds x = (INF n. principal (\<Inter>i\<le>n. A i))" 1569 using * 1570 unfolding nhds_def 1571 apply - 1572 apply (rule INF_eq) 1573 apply simp_all 1574 apply fastforce 1575 apply (intro exI [of _ "\<Inter>i\<le>n. A i" for n] conjI open_INT) 1576 apply auto 1577 done 1578 qed 1579qed 1580 1581lemma (in first_countable_topology) countable_basis: 1582 obtains A :: "nat \<Rightarrow> 'a set" where 1583 "\<And>i. open (A i)" "\<And>i. x \<in> A i" 1584 "\<And>F. (\<forall>n. F n \<in> A n) \<Longrightarrow> F \<longlonglongrightarrow> x" 1585proof atomize_elim 1586 obtain A :: "nat \<Rightarrow> 'a set" where *: 1587 "\<And>i. open (A i)" 1588 "\<And>i. x \<in> A i" 1589 "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> eventually (\<lambda>i. A i \<subseteq> S) sequentially" 1590 by (rule countable_basis_at_decseq) blast 1591 have "eventually (\<lambda>n. F n \<in> S) sequentially" 1592 if "\<forall>n. F n \<in> A n" "open S" "x \<in> S" for F S 1593 using *(3)[of S] that by (auto elim: eventually_mono simp: subset_eq) 1594 with * show "\<exists>A. (\<forall>i. open (A i)) \<and> (\<forall>i. x \<in> A i) \<and> (\<forall>F. (\<forall>n. F n \<in> A n) \<longrightarrow> F \<longlonglongrightarrow> x)" 1595 by (intro exI[of _ A]) (auto simp: tendsto_def) 1596qed 1597 1598lemma (in first_countable_topology) sequentially_imp_eventually_nhds_within: 1599 assumes "\<forall>f. (\<forall>n. f n \<in> s) \<and> f \<longlonglongrightarrow> a \<longrightarrow> eventually (\<lambda>n. P (f n)) sequentially" 1600 shows "eventually P (inf (nhds a) (principal s))" 1601proof (rule ccontr) 1602 obtain A :: "nat \<Rightarrow> 'a set" where *: 1603 "\<And>i. open (A i)" 1604 "\<And>i. a \<in> A i" 1605 "\<And>F. \<forall>n. F n \<in> A n \<Longrightarrow> F \<longlonglongrightarrow> a" 1606 by (rule countable_basis) blast 1607 assume "\<not> ?thesis" 1608 with * have "\<exists>F. \<forall>n. F n \<in> s \<and> F n \<in> A n \<and> \<not> P (F n)" 1609 unfolding eventually_inf_principal eventually_nhds 1610 by (intro choice) fastforce 1611 then obtain F where F: "\<forall>n. F n \<in> s" and "\<forall>n. F n \<in> A n" and F': "\<forall>n. \<not> P (F n)" 1612 by blast 1613 with * have "F \<longlonglongrightarrow> a" 1614 by auto 1615 then have "eventually (\<lambda>n. P (F n)) sequentially" 1616 using assms F by simp 1617 then show False 1618 by (simp add: F') 1619qed 1620 1621lemma (in first_countable_topology) eventually_nhds_within_iff_sequentially: 1622 "eventually P (inf (nhds a) (principal s)) \<longleftrightarrow> 1623 (\<forall>f. (\<forall>n. f n \<in> s) \<and> f \<longlonglongrightarrow> a \<longrightarrow> eventually (\<lambda>n. P (f n)) sequentially)" 1624proof (safe intro!: sequentially_imp_eventually_nhds_within) 1625 assume "eventually P (inf (nhds a) (principal s))" 1626 then obtain S where "open S" "a \<in> S" "\<forall>x\<in>S. x \<in> s \<longrightarrow> P x" 1627 by (auto simp: eventually_inf_principal eventually_nhds) 1628 moreover 1629 fix f 1630 assume "\<forall>n. f n \<in> s" "f \<longlonglongrightarrow> a" 1631 ultimately show "eventually (\<lambda>n. P (f n)) sequentially" 1632 by (auto dest!: topological_tendstoD elim: eventually_mono) 1633qed 1634 1635lemma (in first_countable_topology) eventually_nhds_iff_sequentially: 1636 "eventually P (nhds a) \<longleftrightarrow> (\<forall>f. f \<longlonglongrightarrow> a \<longrightarrow> eventually (\<lambda>n. P (f n)) sequentially)" 1637 using eventually_nhds_within_iff_sequentially[of P a UNIV] by simp 1638 1639(*Thanks to S��bastien Gou��zel*) 1640lemma Inf_as_limit: 1641 fixes A::"'a::{linorder_topology, first_countable_topology, complete_linorder} set" 1642 assumes "A \<noteq> {}" 1643 shows "\<exists>u. (\<forall>n. u n \<in> A) \<and> u \<longlonglongrightarrow> Inf A" 1644proof (cases "Inf A \<in> A") 1645 case True 1646 show ?thesis 1647 by (rule exI[of _ "\<lambda>n. Inf A"], auto simp add: True) 1648next 1649 case False 1650 obtain y where "y \<in> A" using assms by auto 1651 then have "Inf A < y" using False Inf_lower less_le by auto 1652 obtain F :: "nat \<Rightarrow> 'a set" where F: "\<And>i. open (F i)" "\<And>i. Inf A \<in> F i" 1653 "\<And>u. (\<forall>n. u n \<in> F n) \<Longrightarrow> u \<longlonglongrightarrow> Inf A" 1654 by (metis first_countable_topology_class.countable_basis) 1655 define u where "u = (\<lambda>n. SOME z. z \<in> F n \<and> z \<in> A)" 1656 have "\<exists>z. z \<in> U \<and> z \<in> A" if "Inf A \<in> U" "open U" for U 1657 proof - 1658 obtain b where "b > Inf A" "{Inf A ..<b} \<subseteq> U" 1659 using open_right[OF \<open>open U\<close> \<open>Inf A \<in> U\<close> \<open>Inf A < y\<close>] by auto 1660 obtain z where "z < b" "z \<in> A" 1661 using \<open>Inf A < b\<close> Inf_less_iff by auto 1662 then have "z \<in> {Inf A ..<b}" 1663 by (simp add: Inf_lower) 1664 then show ?thesis using \<open>z \<in> A\<close> \<open>{Inf A ..<b} \<subseteq> U\<close> by auto 1665 qed 1666 then have *: "u n \<in> F n \<and> u n \<in> A" for n 1667 using \<open>Inf A \<in> F n\<close> \<open>open (F n)\<close> unfolding u_def by (metis (no_types, lifting) someI_ex) 1668 then have "u \<longlonglongrightarrow> Inf A" using F(3) by simp 1669 then show ?thesis using * by auto 1670qed 1671 1672lemma tendsto_at_iff_sequentially: 1673 "(f \<longlongrightarrow> a) (at x within s) \<longleftrightarrow> (\<forall>X. (\<forall>i. X i \<in> s - {x}) \<longrightarrow> X \<longlonglongrightarrow> x \<longrightarrow> ((f \<circ> X) \<longlonglongrightarrow> a))" 1674 for f :: "'a::first_countable_topology \<Rightarrow> _" 1675 unfolding filterlim_def[of _ "nhds a"] le_filter_def eventually_filtermap 1676 at_within_def eventually_nhds_within_iff_sequentially comp_def 1677 by metis 1678 1679lemma approx_from_above_dense_linorder: 1680 fixes x::"'a::{dense_linorder, linorder_topology, first_countable_topology}" 1681 assumes "x < y" 1682 shows "\<exists>u. (\<forall>n. u n > x) \<and> (u \<longlonglongrightarrow> x)" 1683proof - 1684 obtain A :: "nat \<Rightarrow> 'a set" where A: "\<And>i. open (A i)" "\<And>i. x \<in> A i" 1685 "\<And>F. (\<forall>n. F n \<in> A n) \<Longrightarrow> F \<longlonglongrightarrow> x" 1686 by (metis first_countable_topology_class.countable_basis) 1687 define u where "u = (\<lambda>n. SOME z. z \<in> A n \<and> z > x)" 1688 have "\<exists>z. z \<in> U \<and> x < z" if "x \<in> U" "open U" for U 1689 using open_right[OF \<open>open U\<close> \<open>x \<in> U\<close> \<open>x < y\<close>] 1690 by (meson atLeastLessThan_iff dense less_imp_le subset_eq) 1691 then have *: "u n \<in> A n \<and> x < u n" for n 1692 using \<open>x \<in> A n\<close> \<open>open (A n)\<close> unfolding u_def by (metis (no_types, lifting) someI_ex) 1693 then have "u \<longlonglongrightarrow> x" using A(3) by simp 1694 then show ?thesis using * by auto 1695qed 1696 1697lemma approx_from_below_dense_linorder: 1698 fixes x::"'a::{dense_linorder, linorder_topology, first_countable_topology}" 1699 assumes "x > y" 1700 shows "\<exists>u. (\<forall>n. u n < x) \<and> (u \<longlonglongrightarrow> x)" 1701proof - 1702 obtain A :: "nat \<Rightarrow> 'a set" where A: "\<And>i. open (A i)" "\<And>i. x \<in> A i" 1703 "\<And>F. (\<forall>n. F n \<in> A n) \<Longrightarrow> F \<longlonglongrightarrow> x" 1704 by (metis first_countable_topology_class.countable_basis) 1705 define u where "u = (\<lambda>n. SOME z. z \<in> A n \<and> z < x)" 1706 have "\<exists>z. z \<in> U \<and> z < x" if "x \<in> U" "open U" for U 1707 using open_left[OF \<open>open U\<close> \<open>x \<in> U\<close> \<open>x > y\<close>] 1708 by (meson dense greaterThanAtMost_iff less_imp_le subset_eq) 1709 then have *: "u n \<in> A n \<and> u n < x" for n 1710 using \<open>x \<in> A n\<close> \<open>open (A n)\<close> unfolding u_def by (metis (no_types, lifting) someI_ex) 1711 then have "u \<longlonglongrightarrow> x" using A(3) by simp 1712 then show ?thesis using * by auto 1713qed 1714 1715 1716subsection \<open>Function limit at a point\<close> 1717 1718abbreviation LIM :: "('a::topological_space \<Rightarrow> 'b::topological_space) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool" 1719 ("((_)/ \<midarrow>(_)/\<rightarrow> (_))" [60, 0, 60] 60) 1720 where "f \<midarrow>a\<rightarrow> L \<equiv> (f \<longlongrightarrow> L) (at a)" 1721 1722lemma tendsto_within_open: "a \<in> S \<Longrightarrow> open S \<Longrightarrow> (f \<longlongrightarrow> l) (at a within S) \<longleftrightarrow> (f \<midarrow>a\<rightarrow> l)" 1723 by (simp add: tendsto_def at_within_open[where S = S]) 1724 1725lemma tendsto_within_open_NO_MATCH: 1726 "a \<in> S \<Longrightarrow> NO_MATCH UNIV S \<Longrightarrow> open S \<Longrightarrow> (f \<longlongrightarrow> l)(at a within S) \<longleftrightarrow> (f \<longlongrightarrow> l)(at a)" 1727 for f :: "'a::topological_space \<Rightarrow> 'b::topological_space" 1728 using tendsto_within_open by blast 1729 1730lemma LIM_const_not_eq[tendsto_intros]: "k \<noteq> L \<Longrightarrow> \<not> (\<lambda>x. k) \<midarrow>a\<rightarrow> L" 1731 for a :: "'a::perfect_space" and k L :: "'b::t2_space" 1732 by (simp add: tendsto_const_iff) 1733 1734lemmas LIM_not_zero = LIM_const_not_eq [where L = 0] 1735 1736lemma LIM_const_eq: "(\<lambda>x. k) \<midarrow>a\<rightarrow> L \<Longrightarrow> k = L" 1737 for a :: "'a::perfect_space" and k L :: "'b::t2_space" 1738 by (simp add: tendsto_const_iff) 1739 1740lemma LIM_unique: "f \<midarrow>a\<rightarrow> L \<Longrightarrow> f \<midarrow>a\<rightarrow> M \<Longrightarrow> L = M" 1741 for a :: "'a::perfect_space" and L M :: "'b::t2_space" 1742 using at_neq_bot by (rule tendsto_unique) 1743 1744 1745text \<open>Limits are equal for functions equal except at limit point.\<close> 1746lemma LIM_equal: "\<forall>x. x \<noteq> a \<longrightarrow> f x = g x \<Longrightarrow> (f \<midarrow>a\<rightarrow> l) \<longleftrightarrow> (g \<midarrow>a\<rightarrow> l)" 1747 by (simp add: tendsto_def eventually_at_topological) 1748 1749lemma LIM_cong: "a = b \<Longrightarrow> (\<And>x. x \<noteq> b \<Longrightarrow> f x = g x) \<Longrightarrow> l = m \<Longrightarrow> (f \<midarrow>a\<rightarrow> l) \<longleftrightarrow> (g \<midarrow>b\<rightarrow> m)" 1750 by (simp add: LIM_equal) 1751 1752lemma tendsto_cong_limit: "(f \<longlongrightarrow> l) F \<Longrightarrow> k = l \<Longrightarrow> (f \<longlongrightarrow> k) F" 1753 by simp 1754 1755lemma tendsto_at_iff_tendsto_nhds: "g \<midarrow>l\<rightarrow> g l \<longleftrightarrow> (g \<longlongrightarrow> g l) (nhds l)" 1756 unfolding tendsto_def eventually_at_filter 1757 by (intro ext all_cong imp_cong) (auto elim!: eventually_mono) 1758 1759lemma tendsto_compose: "g \<midarrow>l\<rightarrow> g l \<Longrightarrow> (f \<longlongrightarrow> l) F \<Longrightarrow> ((\<lambda>x. g (f x)) \<longlongrightarrow> g l) F" 1760 unfolding tendsto_at_iff_tendsto_nhds by (rule filterlim_compose[of g]) 1761 1762lemma tendsto_compose_eventually: 1763 "g \<midarrow>l\<rightarrow> m \<Longrightarrow> (f \<longlongrightarrow> l) F \<Longrightarrow> eventually (\<lambda>x. f x \<noteq> l) F \<Longrightarrow> ((\<lambda>x. g (f x)) \<longlongrightarrow> m) F" 1764 by (rule filterlim_compose[of g _ "at l"]) (auto simp add: filterlim_at) 1765 1766lemma LIM_compose_eventually: 1767 assumes "f \<midarrow>a\<rightarrow> b" 1768 and "g \<midarrow>b\<rightarrow> c" 1769 and "eventually (\<lambda>x. f x \<noteq> b) (at a)" 1770 shows "(\<lambda>x. g (f x)) \<midarrow>a\<rightarrow> c" 1771 using assms(2,1,3) by (rule tendsto_compose_eventually) 1772 1773lemma tendsto_compose_filtermap: "((g \<circ> f) \<longlongrightarrow> T) F \<longleftrightarrow> (g \<longlongrightarrow> T) (filtermap f F)" 1774 by (simp add: filterlim_def filtermap_filtermap comp_def) 1775 1776lemma tendsto_compose_at: 1777 assumes f: "(f \<longlongrightarrow> y) F" and g: "(g \<longlongrightarrow> z) (at y)" and fg: "eventually (\<lambda>w. f w = y \<longrightarrow> g y = z) F" 1778 shows "((g \<circ> f) \<longlongrightarrow> z) F" 1779proof - 1780 have "(\<forall>\<^sub>F a in F. f a \<noteq> y) \<or> g y = z" 1781 using fg by force 1782 moreover have "(g \<longlongrightarrow> z) (filtermap f F) \<or> \<not> (\<forall>\<^sub>F a in F. f a \<noteq> y)" 1783 by (metis (no_types) filterlim_atI filterlim_def tendsto_mono f g) 1784 ultimately show ?thesis 1785 by (metis (no_types) f filterlim_compose filterlim_filtermap g tendsto_at_iff_tendsto_nhds tendsto_compose_filtermap) 1786qed 1787 1788 1789subsubsection \<open>Relation of \<open>LIM\<close> and \<open>LIMSEQ\<close>\<close> 1790 1791lemma (in first_countable_topology) sequentially_imp_eventually_within: 1792 "(\<forall>f. (\<forall>n. f n \<in> s \<and> f n \<noteq> a) \<and> f \<longlonglongrightarrow> a \<longrightarrow> eventually (\<lambda>n. P (f n)) sequentially) \<Longrightarrow> 1793 eventually P (at a within s)" 1794 unfolding at_within_def 1795 by (intro sequentially_imp_eventually_nhds_within) auto 1796 1797lemma (in first_countable_topology) sequentially_imp_eventually_at: 1798 "(\<forall>f. (\<forall>n. f n \<noteq> a) \<and> f \<longlonglongrightarrow> a \<longrightarrow> eventually (\<lambda>n. P (f n)) sequentially) \<Longrightarrow> eventually P (at a)" 1799 using sequentially_imp_eventually_within [where s=UNIV] by simp 1800 1801lemma LIMSEQ_SEQ_conv1: 1802 fixes f :: "'a::topological_space \<Rightarrow> 'b::topological_space" 1803 assumes f: "f \<midarrow>a\<rightarrow> l" 1804 shows "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S \<longlonglongrightarrow> a \<longrightarrow> (\<lambda>n. f (S n)) \<longlonglongrightarrow> l" 1805 using tendsto_compose_eventually [OF f, where F=sequentially] by simp 1806 1807lemma LIMSEQ_SEQ_conv2: 1808 fixes f :: "'a::first_countable_topology \<Rightarrow> 'b::topological_space" 1809 assumes "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S \<longlonglongrightarrow> a \<longrightarrow> (\<lambda>n. f (S n)) \<longlonglongrightarrow> l" 1810 shows "f \<midarrow>a\<rightarrow> l" 1811 using assms unfolding tendsto_def [where l=l] by (simp add: sequentially_imp_eventually_at) 1812 1813lemma LIMSEQ_SEQ_conv: "(\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S \<longlonglongrightarrow> a \<longrightarrow> (\<lambda>n. X (S n)) \<longlonglongrightarrow> L) \<longleftrightarrow> X \<midarrow>a\<rightarrow> L" 1814 for a :: "'a::first_countable_topology" and L :: "'b::topological_space" 1815 using LIMSEQ_SEQ_conv2 LIMSEQ_SEQ_conv1 .. 1816 1817lemma sequentially_imp_eventually_at_left: 1818 fixes a :: "'a::{linorder_topology,first_countable_topology}" 1819 assumes b[simp]: "b < a" 1820 and *: "\<And>f. (\<And>n. b < f n) \<Longrightarrow> (\<And>n. f n < a) \<Longrightarrow> incseq f \<Longrightarrow> f \<longlonglongrightarrow> a \<Longrightarrow> 1821 eventually (\<lambda>n. P (f n)) sequentially" 1822 shows "eventually P (at_left a)" 1823proof (safe intro!: sequentially_imp_eventually_within) 1824 fix X 1825 assume X: "\<forall>n. X n \<in> {..< a} \<and> X n \<noteq> a" "X \<longlonglongrightarrow> a" 1826 show "eventually (\<lambda>n. P (X n)) sequentially" 1827 proof (rule ccontr) 1828 assume neg: "\<not> ?thesis" 1829 have "\<exists>s. \<forall>n. (\<not> P (X (s n)) \<and> b < X (s n)) \<and> (X (s n) \<le> X (s (Suc n)) \<and> Suc (s n) \<le> s (Suc n))" 1830 (is "\<exists>s. ?P s") 1831 proof (rule dependent_nat_choice) 1832 have "\<not> eventually (\<lambda>n. b < X n \<longrightarrow> P (X n)) sequentially" 1833 by (intro not_eventually_impI neg order_tendstoD(1) [OF X(2) b]) 1834 then show "\<exists>x. \<not> P (X x) \<and> b < X x" 1835 by (auto dest!: not_eventuallyD) 1836 next 1837 fix x n 1838 have "\<not> eventually (\<lambda>n. Suc x \<le> n \<longrightarrow> b < X n \<longrightarrow> X x < X n \<longrightarrow> P (X n)) sequentially" 1839 using X 1840 by (intro not_eventually_impI order_tendstoD(1)[OF X(2)] eventually_ge_at_top neg) auto 1841 then show "\<exists>n. (\<not> P (X n) \<and> b < X n) \<and> (X x \<le> X n \<and> Suc x \<le> n)" 1842 by (auto dest!: not_eventuallyD) 1843 qed 1844 then obtain s where "?P s" .. 1845 with X have "b < X (s n)" 1846 and "X (s n) < a" 1847 and "incseq (\<lambda>n. X (s n))" 1848 and "(\<lambda>n. X (s n)) \<longlonglongrightarrow> a" 1849 and "\<not> P (X (s n))" 1850 for n 1851 by (auto simp: strict_mono_Suc_iff Suc_le_eq incseq_Suc_iff 1852 intro!: LIMSEQ_subseq_LIMSEQ[OF \<open>X \<longlonglongrightarrow> a\<close>, unfolded comp_def]) 1853 from *[OF this(1,2,3,4)] this(5) show False 1854 by auto 1855 qed 1856qed 1857 1858lemma tendsto_at_left_sequentially: 1859 fixes a b :: "'b::{linorder_topology,first_countable_topology}" 1860 assumes "b < a" 1861 assumes *: "\<And>S. (\<And>n. S n < a) \<Longrightarrow> (\<And>n. b < S n) \<Longrightarrow> incseq S \<Longrightarrow> S \<longlonglongrightarrow> a \<Longrightarrow> 1862 (\<lambda>n. X (S n)) \<longlonglongrightarrow> L" 1863 shows "(X \<longlongrightarrow> L) (at_left a)" 1864 using assms by (simp add: tendsto_def [where l=L] sequentially_imp_eventually_at_left) 1865 1866lemma sequentially_imp_eventually_at_right: 1867 fixes a b :: "'a::{linorder_topology,first_countable_topology}" 1868 assumes b[simp]: "a < b" 1869 assumes *: "\<And>f. (\<And>n. a < f n) \<Longrightarrow> (\<And>n. f n < b) \<Longrightarrow> decseq f \<Longrightarrow> f \<longlonglongrightarrow> a \<Longrightarrow> 1870 eventually (\<lambda>n. P (f n)) sequentially" 1871 shows "eventually P (at_right a)" 1872proof (safe intro!: sequentially_imp_eventually_within) 1873 fix X 1874 assume X: "\<forall>n. X n \<in> {a <..} \<and> X n \<noteq> a" "X \<longlonglongrightarrow> a" 1875 show "eventually (\<lambda>n. P (X n)) sequentially" 1876 proof (rule ccontr) 1877 assume neg: "\<not> ?thesis" 1878 have "\<exists>s. \<forall>n. (\<not> P (X (s n)) \<and> X (s n) < b) \<and> (X (s (Suc n)) \<le> X (s n) \<and> Suc (s n) \<le> s (Suc n))" 1879 (is "\<exists>s. ?P s") 1880 proof (rule dependent_nat_choice) 1881 have "\<not> eventually (\<lambda>n. X n < b \<longrightarrow> P (X n)) sequentially" 1882 by (intro not_eventually_impI neg order_tendstoD(2) [OF X(2) b]) 1883 then show "\<exists>x. \<not> P (X x) \<and> X x < b" 1884 by (auto dest!: not_eventuallyD) 1885 next 1886 fix x n 1887 have "\<not> eventually (\<lambda>n. Suc x \<le> n \<longrightarrow> X n < b \<longrightarrow> X n < X x \<longrightarrow> P (X n)) sequentially" 1888 using X 1889 by (intro not_eventually_impI order_tendstoD(2)[OF X(2)] eventually_ge_at_top neg) auto 1890 then show "\<exists>n. (\<not> P (X n) \<and> X n < b) \<and> (X n \<le> X x \<and> Suc x \<le> n)" 1891 by (auto dest!: not_eventuallyD) 1892 qed 1893 then obtain s where "?P s" .. 1894 with X have "a < X (s n)" 1895 and "X (s n) < b" 1896 and "decseq (\<lambda>n. X (s n))" 1897 and "(\<lambda>n. X (s n)) \<longlonglongrightarrow> a" 1898 and "\<not> P (X (s n))" 1899 for n 1900 by (auto simp: strict_mono_Suc_iff Suc_le_eq decseq_Suc_iff 1901 intro!: LIMSEQ_subseq_LIMSEQ[OF \<open>X \<longlonglongrightarrow> a\<close>, unfolded comp_def]) 1902 from *[OF this(1,2,3,4)] this(5) show False 1903 by auto 1904 qed 1905qed 1906 1907lemma tendsto_at_right_sequentially: 1908 fixes a :: "_ :: {linorder_topology, first_countable_topology}" 1909 assumes "a < b" 1910 and *: "\<And>S. (\<And>n. a < S n) \<Longrightarrow> (\<And>n. S n < b) \<Longrightarrow> decseq S \<Longrightarrow> S \<longlonglongrightarrow> a \<Longrightarrow> 1911 (\<lambda>n. X (S n)) \<longlonglongrightarrow> L" 1912 shows "(X \<longlongrightarrow> L) (at_right a)" 1913 using assms by (simp add: tendsto_def [where l=L] sequentially_imp_eventually_at_right) 1914 1915 1916subsection \<open>Continuity\<close> 1917 1918subsubsection \<open>Continuity on a set\<close> 1919 1920definition continuous_on :: "'a set \<Rightarrow> ('a::topological_space \<Rightarrow> 'b::topological_space) \<Rightarrow> bool" 1921 where "continuous_on s f \<longleftrightarrow> (\<forall>x\<in>s. (f \<longlongrightarrow> f x) (at x within s))" 1922 1923lemma continuous_on_cong [cong]: 1924 "s = t \<Longrightarrow> (\<And>x. x \<in> t \<Longrightarrow> f x = g x) \<Longrightarrow> continuous_on s f \<longleftrightarrow> continuous_on t g" 1925 unfolding continuous_on_def 1926 by (intro ball_cong filterlim_cong) (auto simp: eventually_at_filter) 1927 1928lemma continuous_on_cong_simp: 1929 "s = t \<Longrightarrow> (\<And>x. x \<in> t =simp=> f x = g x) \<Longrightarrow> continuous_on s f \<longleftrightarrow> continuous_on t g" 1930 unfolding simp_implies_def by (rule continuous_on_cong) 1931 1932lemma continuous_on_topological: 1933 "continuous_on s f \<longleftrightarrow> 1934 (\<forall>x\<in>s. \<forall>B. open B \<longrightarrow> f x \<in> B \<longrightarrow> (\<exists>A. open A \<and> x \<in> A \<and> (\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B)))" 1935 unfolding continuous_on_def tendsto_def eventually_at_topological by metis 1936 1937lemma continuous_on_open_invariant: 1938 "continuous_on s f \<longleftrightarrow> (\<forall>B. open B \<longrightarrow> (\<exists>A. open A \<and> A \<inter> s = f -` B \<inter> s))" 1939proof safe 1940 fix B :: "'b set" 1941 assume "continuous_on s f" "open B" 1942 then have "\<forall>x\<in>f -` B \<inter> s. (\<exists>A. open A \<and> x \<in> A \<and> s \<inter> A \<subseteq> f -` B)" 1943 by (auto simp: continuous_on_topological subset_eq Ball_def imp_conjL) 1944 then obtain A where "\<forall>x\<in>f -` B \<inter> s. open (A x) \<and> x \<in> A x \<and> s \<inter> A x \<subseteq> f -` B" 1945 unfolding bchoice_iff .. 1946 then show "\<exists>A. open A \<and> A \<inter> s = f -` B \<inter> s" 1947 by (intro exI[of _ "\<Union>x\<in>f -` B \<inter> s. A x"]) auto 1948next 1949 assume B: "\<forall>B. open B \<longrightarrow> (\<exists>A. open A \<and> A \<inter> s = f -` B \<inter> s)" 1950 show "continuous_on s f" 1951 unfolding continuous_on_topological 1952 proof safe 1953 fix x B 1954 assume "x \<in> s" "open B" "f x \<in> B" 1955 with B obtain A where A: "open A" "A \<inter> s = f -` B \<inter> s" 1956 by auto 1957 with \<open>x \<in> s\<close> \<open>f x \<in> B\<close> show "\<exists>A. open A \<and> x \<in> A \<and> (\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B)" 1958 by (intro exI[of _ A]) auto 1959 qed 1960qed 1961 1962lemma continuous_on_open_vimage: 1963 "open s \<Longrightarrow> continuous_on s f \<longleftrightarrow> (\<forall>B. open B \<longrightarrow> open (f -` B \<inter> s))" 1964 unfolding continuous_on_open_invariant 1965 by (metis open_Int Int_absorb Int_commute[of s] Int_assoc[of _ _ s]) 1966 1967corollary continuous_imp_open_vimage: 1968 assumes "continuous_on s f" "open s" "open B" "f -` B \<subseteq> s" 1969 shows "open (f -` B)" 1970 by (metis assms continuous_on_open_vimage le_iff_inf) 1971 1972corollary open_vimage[continuous_intros]: 1973 assumes "open s" 1974 and "continuous_on UNIV f" 1975 shows "open (f -` s)" 1976 using assms by (simp add: continuous_on_open_vimage [OF open_UNIV]) 1977 1978lemma continuous_on_closed_invariant: 1979 "continuous_on s f \<longleftrightarrow> (\<forall>B. closed B \<longrightarrow> (\<exists>A. closed A \<and> A \<inter> s = f -` B \<inter> s))" 1980proof - 1981 have *: "(\<And>A. P A \<longleftrightarrow> Q (- A)) \<Longrightarrow> (\<forall>A. P A) \<longleftrightarrow> (\<forall>A. Q A)" 1982 for P Q :: "'b set \<Rightarrow> bool" 1983 by (metis double_compl) 1984 show ?thesis 1985 unfolding continuous_on_open_invariant 1986 by (intro *) (auto simp: open_closed[symmetric]) 1987qed 1988 1989lemma continuous_on_closed_vimage: 1990 "closed s \<Longrightarrow> continuous_on s f \<longleftrightarrow> (\<forall>B. closed B \<longrightarrow> closed (f -` B \<inter> s))" 1991 unfolding continuous_on_closed_invariant 1992 by (metis closed_Int Int_absorb Int_commute[of s] Int_assoc[of _ _ s]) 1993 1994corollary closed_vimage_Int[continuous_intros]: 1995 assumes "closed s" 1996 and "continuous_on t f" 1997 and t: "closed t" 1998 shows "closed (f -` s \<inter> t)" 1999 using assms by (simp add: continuous_on_closed_vimage [OF t]) 2000 2001corollary closed_vimage[continuous_intros]: 2002 assumes "closed s" 2003 and "continuous_on UNIV f" 2004 shows "closed (f -` s)" 2005 using closed_vimage_Int [OF assms] by simp 2006 2007lemma continuous_on_empty [simp]: "continuous_on {} f" 2008 by (simp add: continuous_on_def) 2009 2010lemma continuous_on_sing [simp]: "continuous_on {x} f" 2011 by (simp add: continuous_on_def at_within_def) 2012 2013lemma continuous_on_open_Union: 2014 "(\<And>s. s \<in> S \<Longrightarrow> open s) \<Longrightarrow> (\<And>s. s \<in> S \<Longrightarrow> continuous_on s f) \<Longrightarrow> continuous_on (\<Union>S) f" 2015 unfolding continuous_on_def 2016 by safe (metis open_Union at_within_open UnionI) 2017 2018lemma continuous_on_open_UN: 2019 "(\<And>s. s \<in> S \<Longrightarrow> open (A s)) \<Longrightarrow> (\<And>s. s \<in> S \<Longrightarrow> continuous_on (A s) f) \<Longrightarrow> 2020 continuous_on (\<Union>s\<in>S. A s) f" 2021 by (rule continuous_on_open_Union) auto 2022 2023lemma continuous_on_open_Un: 2024 "open s \<Longrightarrow> open t \<Longrightarrow> continuous_on s f \<Longrightarrow> continuous_on t f \<Longrightarrow> continuous_on (s \<union> t) f" 2025 using continuous_on_open_Union [of "{s,t}"] by auto 2026 2027lemma continuous_on_closed_Un: 2028 "closed s \<Longrightarrow> closed t \<Longrightarrow> continuous_on s f \<Longrightarrow> continuous_on t f \<Longrightarrow> continuous_on (s \<union> t) f" 2029 by (auto simp add: continuous_on_closed_vimage closed_Un Int_Un_distrib) 2030 2031lemma continuous_on_closed_Union: 2032 assumes "finite I" 2033 "\<And>i. i \<in> I \<Longrightarrow> closed (U i)" 2034 "\<And>i. i \<in> I \<Longrightarrow> continuous_on (U i) f" 2035 shows "continuous_on (\<Union> i \<in> I. U i) f" 2036 using assms 2037 by (induction I) (auto intro!: continuous_on_closed_Un) 2038 2039lemma continuous_on_If: 2040 assumes closed: "closed s" "closed t" 2041 and cont: "continuous_on s f" "continuous_on t g" 2042 and P: "\<And>x. x \<in> s \<Longrightarrow> \<not> P x \<Longrightarrow> f x = g x" "\<And>x. x \<in> t \<Longrightarrow> P x \<Longrightarrow> f x = g x" 2043 shows "continuous_on (s \<union> t) (\<lambda>x. if P x then f x else g x)" 2044 (is "continuous_on _ ?h") 2045proof- 2046 from P have "\<forall>x\<in>s. f x = ?h x" "\<forall>x\<in>t. g x = ?h x" 2047 by auto 2048 with cont have "continuous_on s ?h" "continuous_on t ?h" 2049 by simp_all 2050 with closed show ?thesis 2051 by (rule continuous_on_closed_Un) 2052qed 2053 2054lemma continuous_on_cases: 2055 "closed s \<Longrightarrow> closed t \<Longrightarrow> continuous_on s f \<Longrightarrow> continuous_on t g \<Longrightarrow> 2056 \<forall>x. (x\<in>s \<and> \<not> P x) \<or> (x \<in> t \<and> P x) \<longrightarrow> f x = g x \<Longrightarrow> 2057 continuous_on (s \<union> t) (\<lambda>x. if P x then f x else g x)" 2058 by (rule continuous_on_If) auto 2059 2060lemma continuous_on_id[continuous_intros,simp]: "continuous_on s (\<lambda>x. x)" 2061 unfolding continuous_on_def by fast 2062 2063lemma continuous_on_id'[continuous_intros,simp]: "continuous_on s id" 2064 unfolding continuous_on_def id_def by fast 2065 2066lemma continuous_on_const[continuous_intros,simp]: "continuous_on s (\<lambda>x. c)" 2067 unfolding continuous_on_def by auto 2068 2069lemma continuous_on_subset: "continuous_on s f \<Longrightarrow> t \<subseteq> s \<Longrightarrow> continuous_on t f" 2070 unfolding continuous_on_def 2071 by (metis subset_eq tendsto_within_subset) 2072 2073lemma continuous_on_compose[continuous_intros]: 2074 "continuous_on s f \<Longrightarrow> continuous_on (f ` s) g \<Longrightarrow> continuous_on s (g \<circ> f)" 2075 unfolding continuous_on_topological by simp metis 2076 2077lemma continuous_on_compose2: 2078 "continuous_on t g \<Longrightarrow> continuous_on s f \<Longrightarrow> f ` s \<subseteq> t \<Longrightarrow> continuous_on s (\<lambda>x. g (f x))" 2079 using continuous_on_compose[of s f g] continuous_on_subset by (force simp add: comp_def) 2080 2081lemma continuous_on_generate_topology: 2082 assumes *: "open = generate_topology X" 2083 and **: "\<And>B. B \<in> X \<Longrightarrow> \<exists>C. open C \<and> C \<inter> A = f -` B \<inter> A" 2084 shows "continuous_on A f" 2085 unfolding continuous_on_open_invariant 2086proof safe 2087 fix B :: "'a set" 2088 assume "open B" 2089 then show "\<exists>C. open C \<and> C \<inter> A = f -` B \<inter> A" 2090 unfolding * 2091 proof induct 2092 case (UN K) 2093 then obtain C where "\<And>k. k \<in> K \<Longrightarrow> open (C k)" "\<And>k. k \<in> K \<Longrightarrow> C k \<inter> A = f -` k \<inter> A" 2094 by metis 2095 then show ?case 2096 by (intro exI[of _ "\<Union>k\<in>K. C k"]) blast 2097 qed (auto intro: **) 2098qed 2099 2100lemma continuous_onI_mono: 2101 fixes f :: "'a::linorder_topology \<Rightarrow> 'b::{dense_order,linorder_topology}" 2102 assumes "open (f`A)" 2103 and mono: "\<And>x y. x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y" 2104 shows "continuous_on A f" 2105proof (rule continuous_on_generate_topology[OF open_generated_order], safe) 2106 have monoD: "\<And>x y. x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> f x < f y \<Longrightarrow> x < y" 2107 by (auto simp: not_le[symmetric] mono) 2108 have "\<exists>x. x \<in> A \<and> f x < b \<and> a < x" if a: "a \<in> A" and fa: "f a < b" for a b 2109 proof - 2110 obtain y where "f a < y" "{f a ..< y} \<subseteq> f`A" 2111 using open_right[OF \<open>open (f`A)\<close>, of "f a" b] a fa 2112 by auto 2113 obtain z where z: "f a < z" "z < min b y" 2114 using dense[of "f a" "min b y"] \<open>f a < y\<close> \<open>f a < b\<close> by auto 2115 then obtain c where "z = f c" "c \<in> A" 2116 using \<open>{f a ..< y} \<subseteq> f`A\<close>[THEN subsetD, of z] by (auto simp: less_imp_le) 2117 with a z show ?thesis 2118 by (auto intro!: exI[of _ c] simp: monoD) 2119 qed 2120 then show "\<exists>C. open C \<and> C \<inter> A = f -` {..<b} \<inter> A" for b 2121 by (intro exI[of _ "(\<Union>x\<in>{x\<in>A. f x < b}. {..< x})"]) 2122 (auto intro: le_less_trans[OF mono] less_imp_le) 2123 2124 have "\<exists>x. x \<in> A \<and> b < f x \<and> x < a" if a: "a \<in> A" and fa: "b < f a" for a b 2125 proof - 2126 note a fa 2127 moreover 2128 obtain y where "y < f a" "{y <.. f a} \<subseteq> f`A" 2129 using open_left[OF \<open>open (f`A)\<close>, of "f a" b] a fa 2130 by auto 2131 then obtain z where z: "max b y < z" "z < f a" 2132 using dense[of "max b y" "f a"] \<open>y < f a\<close> \<open>b < f a\<close> by auto 2133 then obtain c where "z = f c" "c \<in> A" 2134 using \<open>{y <.. f a} \<subseteq> f`A\<close>[THEN subsetD, of z] by (auto simp: less_imp_le) 2135 with a z show ?thesis 2136 by (auto intro!: exI[of _ c] simp: monoD) 2137 qed 2138 then show "\<exists>C. open C \<and> C \<inter> A = f -` {b <..} \<inter> A" for b 2139 by (intro exI[of _ "(\<Union>x\<in>{x\<in>A. b < f x}. {x <..})"]) 2140 (auto intro: less_le_trans[OF _ mono] less_imp_le) 2141qed 2142 2143lemma continuous_on_IccI: 2144 "\<lbrakk>(f \<longlongrightarrow> f a) (at_right a); 2145 (f \<longlongrightarrow> f b) (at_left b); 2146 (\<And>x. a < x \<Longrightarrow> x < b \<Longrightarrow> f \<midarrow>x\<rightarrow> f x); a < b\<rbrakk> \<Longrightarrow> 2147 continuous_on {a .. b} f" 2148 for a::"'a::linorder_topology" 2149 using at_within_open[of _ "{a<..<b}"] 2150 by (auto simp: continuous_on_def at_within_Icc_at_right at_within_Icc_at_left le_less 2151 at_within_Icc_at) 2152 2153lemma 2154 fixes a b::"'a::linorder_topology" 2155 assumes "continuous_on {a .. b} f" "a < b" 2156 shows continuous_on_Icc_at_rightD: "(f \<longlongrightarrow> f a) (at_right a)" 2157 and continuous_on_Icc_at_leftD: "(f \<longlongrightarrow> f b) (at_left b)" 2158 using assms 2159 by (auto simp: at_within_Icc_at_right at_within_Icc_at_left continuous_on_def 2160 dest: bspec[where x=a] bspec[where x=b]) 2161 2162lemma continuous_on_discrete [simp]: 2163 "continuous_on A (f :: 'a :: discrete_topology \<Rightarrow> _)" 2164 by (auto simp: continuous_on_def at_discrete) 2165 2166subsubsection \<open>Continuity at a point\<close> 2167 2168definition continuous :: "'a::t2_space filter \<Rightarrow> ('a \<Rightarrow> 'b::topological_space) \<Rightarrow> bool" 2169 where "continuous F f \<longleftrightarrow> (f \<longlongrightarrow> f (Lim F (\<lambda>x. x))) F" 2170 2171lemma continuous_bot[continuous_intros, simp]: "continuous bot f" 2172 unfolding continuous_def by auto 2173 2174lemma continuous_trivial_limit: "trivial_limit net \<Longrightarrow> continuous net f" 2175 by simp 2176 2177lemma continuous_within: "continuous (at x within s) f \<longleftrightarrow> (f \<longlongrightarrow> f x) (at x within s)" 2178 by (cases "trivial_limit (at x within s)") (auto simp add: Lim_ident_at continuous_def) 2179 2180lemma continuous_within_topological: 2181 "continuous (at x within s) f \<longleftrightarrow> 2182 (\<forall>B. open B \<longrightarrow> f x \<in> B \<longrightarrow> (\<exists>A. open A \<and> x \<in> A \<and> (\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B)))" 2183 unfolding continuous_within tendsto_def eventually_at_topological by metis 2184 2185lemma continuous_within_compose[continuous_intros]: 2186 "continuous (at x within s) f \<Longrightarrow> continuous (at (f x) within f ` s) g \<Longrightarrow> 2187 continuous (at x within s) (g \<circ> f)" 2188 by (simp add: continuous_within_topological) metis 2189 2190lemma continuous_within_compose2: 2191 "continuous (at x within s) f \<Longrightarrow> continuous (at (f x) within f ` s) g \<Longrightarrow> 2192 continuous (at x within s) (\<lambda>x. g (f x))" 2193 using continuous_within_compose[of x s f g] by (simp add: comp_def) 2194 2195lemma continuous_at: "continuous (at x) f \<longleftrightarrow> f \<midarrow>x\<rightarrow> f x" 2196 using continuous_within[of x UNIV f] by simp 2197 2198lemma continuous_ident[continuous_intros, simp]: "continuous (at x within S) (\<lambda>x. x)" 2199 unfolding continuous_within by (rule tendsto_ident_at) 2200 2201lemma continuous_id[continuous_intros, simp]: "continuous (at x within S) id" 2202 by (simp add: id_def) 2203 2204lemma continuous_const[continuous_intros, simp]: "continuous F (\<lambda>x. c)" 2205 unfolding continuous_def by (rule tendsto_const) 2206 2207lemma continuous_on_eq_continuous_within: 2208 "continuous_on s f \<longleftrightarrow> (\<forall>x\<in>s. continuous (at x within s) f)" 2209 unfolding continuous_on_def continuous_within .. 2210 2211lemma continuous_discrete [simp]: 2212 "continuous (at x within A) (f :: 'a :: discrete_topology \<Rightarrow> _)" 2213 by (auto simp: continuous_def at_discrete) 2214 2215abbreviation isCont :: "('a::t2_space \<Rightarrow> 'b::topological_space) \<Rightarrow> 'a \<Rightarrow> bool" 2216 where "isCont f a \<equiv> continuous (at a) f" 2217 2218lemma isCont_def: "isCont f a \<longleftrightarrow> f \<midarrow>a\<rightarrow> f a" 2219 by (rule continuous_at) 2220 2221lemma isContD: "isCont f x \<Longrightarrow> f \<midarrow>x\<rightarrow> f x" 2222 by (simp add: isCont_def) 2223 2224lemma isCont_cong: 2225 assumes "eventually (\<lambda>x. f x = g x) (nhds x)" 2226 shows "isCont f x \<longleftrightarrow> isCont g x" 2227proof - 2228 from assms have [simp]: "f x = g x" 2229 by (rule eventually_nhds_x_imp_x) 2230 from assms have "eventually (\<lambda>x. f x = g x) (at x)" 2231 by (auto simp: eventually_at_filter elim!: eventually_mono) 2232 with assms have "isCont f x \<longleftrightarrow> isCont g x" unfolding isCont_def 2233 by (intro filterlim_cong) (auto elim!: eventually_mono) 2234 with assms show ?thesis by simp 2235qed 2236 2237lemma continuous_at_imp_continuous_at_within: "isCont f x \<Longrightarrow> continuous (at x within s) f" 2238 by (auto intro: tendsto_mono at_le simp: continuous_at continuous_within) 2239 2240lemma continuous_on_eq_continuous_at: "open s \<Longrightarrow> continuous_on s f \<longleftrightarrow> (\<forall>x\<in>s. isCont f x)" 2241 by (simp add: continuous_on_def continuous_at at_within_open[of _ s]) 2242 2243lemma continuous_within_open: "a \<in> A \<Longrightarrow> open A \<Longrightarrow> continuous (at a within A) f \<longleftrightarrow> isCont f a" 2244 by (simp add: at_within_open_NO_MATCH) 2245 2246lemma continuous_at_imp_continuous_on: "\<forall>x\<in>s. isCont f x \<Longrightarrow> continuous_on s f" 2247 by (auto intro: continuous_at_imp_continuous_at_within simp: continuous_on_eq_continuous_within) 2248 2249lemma isCont_o2: "isCont f a \<Longrightarrow> isCont g (f a) \<Longrightarrow> isCont (\<lambda>x. g (f x)) a" 2250 unfolding isCont_def by (rule tendsto_compose) 2251 2252lemma continuous_at_compose[continuous_intros]: "isCont f a \<Longrightarrow> isCont g (f a) \<Longrightarrow> isCont (g \<circ> f) a" 2253 unfolding o_def by (rule isCont_o2) 2254 2255lemma isCont_tendsto_compose: "isCont g l \<Longrightarrow> (f \<longlongrightarrow> l) F \<Longrightarrow> ((\<lambda>x. g (f x)) \<longlongrightarrow> g l) F" 2256 unfolding isCont_def by (rule tendsto_compose) 2257 2258lemma continuous_on_tendsto_compose: 2259 assumes f_cont: "continuous_on s f" 2260 and g: "(g \<longlongrightarrow> l) F" 2261 and l: "l \<in> s" 2262 and ev: "\<forall>\<^sub>Fx in F. g x \<in> s" 2263 shows "((\<lambda>x. f (g x)) \<longlongrightarrow> f l) F" 2264proof - 2265 from f_cont l have f: "(f \<longlongrightarrow> f l) (at l within s)" 2266 by (simp add: continuous_on_def) 2267 have i: "((\<lambda>x. if g x = l then f l else f (g x)) \<longlongrightarrow> f l) F" 2268 by (rule filterlim_If) 2269 (auto intro!: filterlim_compose[OF f] eventually_conj tendsto_mono[OF _ g] 2270 simp: filterlim_at eventually_inf_principal eventually_mono[OF ev]) 2271 show ?thesis 2272 by (rule filterlim_cong[THEN iffD1[OF _ i]]) auto 2273qed 2274 2275lemma continuous_within_compose3: 2276 "isCont g (f x) \<Longrightarrow> continuous (at x within s) f \<Longrightarrow> continuous (at x within s) (\<lambda>x. g (f x))" 2277 using continuous_at_imp_continuous_at_within continuous_within_compose2 by blast 2278 2279lemma filtermap_nhds_open_map: 2280 assumes cont: "isCont f a" 2281 and open_map: "\<And>S. open S \<Longrightarrow> open (f`S)" 2282 shows "filtermap f (nhds a) = nhds (f a)" 2283 unfolding filter_eq_iff 2284proof safe 2285 fix P 2286 assume "eventually P (filtermap f (nhds a))" 2287 then obtain S where "open S" "a \<in> S" "\<forall>x\<in>S. P (f x)" 2288 by (auto simp: eventually_filtermap eventually_nhds) 2289 then show "eventually P (nhds (f a))" 2290 unfolding eventually_nhds by (intro exI[of _ "f`S"]) (auto intro!: open_map) 2291qed (metis filterlim_iff tendsto_at_iff_tendsto_nhds isCont_def eventually_filtermap cont) 2292 2293lemma continuous_at_split: 2294 "continuous (at x) f \<longleftrightarrow> continuous (at_left x) f \<and> continuous (at_right x) f" 2295 for x :: "'a::linorder_topology" 2296 by (simp add: continuous_within filterlim_at_split) 2297 2298lemma continuous_on_max [continuous_intros]: 2299 fixes f g :: "'a::topological_space \<Rightarrow> 'b::linorder_topology" 2300 shows "continuous_on A f \<Longrightarrow> continuous_on A g \<Longrightarrow> continuous_on A (\<lambda>x. max (f x) (g x))" 2301 by (auto simp: continuous_on_def intro!: tendsto_max) 2302 2303lemma continuous_on_min [continuous_intros]: 2304 fixes f g :: "'a::topological_space \<Rightarrow> 'b::linorder_topology" 2305 shows "continuous_on A f \<Longrightarrow> continuous_on A g \<Longrightarrow> continuous_on A (\<lambda>x. min (f x) (g x))" 2306 by (auto simp: continuous_on_def intro!: tendsto_min) 2307 2308lemma continuous_max [continuous_intros]: 2309 fixes f :: "'a::t2_space \<Rightarrow> 'b::linorder_topology" 2310 shows "\<lbrakk>continuous F f; continuous F g\<rbrakk> \<Longrightarrow> continuous F (\<lambda>x. (max (f x) (g x)))" 2311 by (simp add: tendsto_max continuous_def) 2312 2313lemma continuous_min [continuous_intros]: 2314 fixes f :: "'a::t2_space \<Rightarrow> 'b::linorder_topology" 2315 shows "\<lbrakk>continuous F f; continuous F g\<rbrakk> \<Longrightarrow> continuous F (\<lambda>x. (min (f x) (g x)))" 2316 by (simp add: tendsto_min continuous_def) 2317 2318text \<open> 2319 The following open/closed Collect lemmas are ported from 2320 S��bastien Gou��zel's \<open>Ergodic_Theory\<close>. 2321\<close> 2322lemma open_Collect_neq: 2323 fixes f g :: "'a::topological_space \<Rightarrow> 'b::t2_space" 2324 assumes f: "continuous_on UNIV f" and g: "continuous_on UNIV g" 2325 shows "open {x. f x \<noteq> g x}" 2326proof (rule openI) 2327 fix t 2328 assume "t \<in> {x. f x \<noteq> g x}" 2329 then obtain U V where *: "open U" "open V" "f t \<in> U" "g t \<in> V" "U \<inter> V = {}" 2330 by (auto simp add: separation_t2) 2331 with open_vimage[OF \<open>open U\<close> f] open_vimage[OF \<open>open V\<close> g] 2332 show "\<exists>T. open T \<and> t \<in> T \<and> T \<subseteq> {x. f x \<noteq> g x}" 2333 by (intro exI[of _ "f -` U \<inter> g -` V"]) auto 2334qed 2335 2336lemma closed_Collect_eq: 2337 fixes f g :: "'a::topological_space \<Rightarrow> 'b::t2_space" 2338 assumes f: "continuous_on UNIV f" and g: "continuous_on UNIV g" 2339 shows "closed {x. f x = g x}" 2340 using open_Collect_neq[OF f g] by (simp add: closed_def Collect_neg_eq) 2341 2342lemma open_Collect_less: 2343 fixes f g :: "'a::topological_space \<Rightarrow> 'b::linorder_topology" 2344 assumes f: "continuous_on UNIV f" and g: "continuous_on UNIV g" 2345 shows "open {x. f x < g x}" 2346proof (rule openI) 2347 fix t 2348 assume t: "t \<in> {x. f x < g x}" 2349 show "\<exists>T. open T \<and> t \<in> T \<and> T \<subseteq> {x. f x < g x}" 2350 proof (cases "\<exists>z. f t < z \<and> z < g t") 2351 case True 2352 then obtain z where "f t < z \<and> z < g t" by blast 2353 then show ?thesis 2354 using open_vimage[OF _ f, of "{..< z}"] open_vimage[OF _ g, of "{z <..}"] 2355 by (intro exI[of _ "f -` {..<z} \<inter> g -` {z<..}"]) auto 2356 next 2357 case False 2358 then have *: "{g t ..} = {f t <..}" "{..< g t} = {.. f t}" 2359 using t by (auto intro: leI) 2360 show ?thesis 2361 using open_vimage[OF _ f, of "{..< g t}"] open_vimage[OF _ g, of "{f t <..}"] t 2362 apply (intro exI[of _ "f -` {..< g t} \<inter> g -` {f t<..}"]) 2363 apply (simp add: open_Int) 2364 apply (auto simp add: *) 2365 done 2366 qed 2367qed 2368 2369lemma closed_Collect_le: 2370 fixes f g :: "'a :: topological_space \<Rightarrow> 'b::linorder_topology" 2371 assumes f: "continuous_on UNIV f" 2372 and g: "continuous_on UNIV g" 2373 shows "closed {x. f x \<le> g x}" 2374 using open_Collect_less [OF g f] 2375 by (simp add: closed_def Collect_neg_eq[symmetric] not_le) 2376 2377 2378subsubsection \<open>Open-cover compactness\<close> 2379 2380context topological_space 2381begin 2382 2383definition compact :: "'a set \<Rightarrow> bool" where 2384compact_eq_Heine_Borel: (* This name is used for backwards compatibility *) 2385 "compact S \<longleftrightarrow> (\<forall>C. (\<forall>c\<in>C. open c) \<and> S \<subseteq> \<Union>C \<longrightarrow> (\<exists>D\<subseteq>C. finite D \<and> S \<subseteq> \<Union>D))" 2386 2387lemma compactI: 2388 assumes "\<And>C. \<forall>t\<in>C. open t \<Longrightarrow> s \<subseteq> \<Union>C \<Longrightarrow> \<exists>C'. C' \<subseteq> C \<and> finite C' \<and> s \<subseteq> \<Union>C'" 2389 shows "compact s" 2390 unfolding compact_eq_Heine_Borel using assms by metis 2391 2392lemma compact_empty[simp]: "compact {}" 2393 by (auto intro!: compactI) 2394 2395lemma compactE: (*related to COMPACT_IMP_HEINE_BOREL in HOL Light*) 2396 assumes "compact S" "S \<subseteq> \<Union>\<T>" "\<And>B. B \<in> \<T> \<Longrightarrow> open B" 2397 obtains \<T>' where "\<T>' \<subseteq> \<T>" "finite \<T>'" "S \<subseteq> \<Union>\<T>'" 2398 by (meson assms compact_eq_Heine_Borel) 2399 2400lemma compactE_image: 2401 assumes "compact S" 2402 and opn: "\<And>T. T \<in> C \<Longrightarrow> open (f T)" 2403 and S: "S \<subseteq> (\<Union>c\<in>C. f c)" 2404 obtains C' where "C' \<subseteq> C" and "finite C'" and "S \<subseteq> (\<Union>c\<in>C'. f c)" 2405 apply (rule compactE[OF \<open>compact S\<close> S]) 2406 using opn apply force 2407 by (metis finite_subset_image) 2408 2409lemma compact_Int_closed [intro]: 2410 assumes "compact S" 2411 and "closed T" 2412 shows "compact (S \<inter> T)" 2413proof (rule compactI) 2414 fix C 2415 assume C: "\<forall>c\<in>C. open c" 2416 assume cover: "S \<inter> T \<subseteq> \<Union>C" 2417 from C \<open>closed T\<close> have "\<forall>c\<in>C \<union> {- T}. open c" 2418 by auto 2419 moreover from cover have "S \<subseteq> \<Union>(C \<union> {- T})" 2420 by auto 2421 ultimately have "\<exists>D\<subseteq>C \<union> {- T}. finite D \<and> S \<subseteq> \<Union>D" 2422 using \<open>compact S\<close> unfolding compact_eq_Heine_Borel by auto 2423 then obtain D where "D \<subseteq> C \<union> {- T} \<and> finite D \<and> S \<subseteq> \<Union>D" .. 2424 then show "\<exists>D\<subseteq>C. finite D \<and> S \<inter> T \<subseteq> \<Union>D" 2425 by (intro exI[of _ "D - {-T}"]) auto 2426qed 2427 2428lemma compact_diff: "\<lbrakk>compact S; open T\<rbrakk> \<Longrightarrow> compact(S - T)" 2429 by (simp add: Diff_eq compact_Int_closed open_closed) 2430 2431lemma inj_setminus: "inj_on uminus (A::'a set set)" 2432 by (auto simp: inj_on_def) 2433 2434 2435subsection \<open>Finite intersection property\<close> 2436 2437lemma compact_fip: 2438 "compact U \<longleftrightarrow> 2439 (\<forall>A. (\<forall>a\<in>A. closed a) \<longrightarrow> (\<forall>B \<subseteq> A. finite B \<longrightarrow> U \<inter> \<Inter>B \<noteq> {}) \<longrightarrow> U \<inter> \<Inter>A \<noteq> {})" 2440 (is "_ \<longleftrightarrow> ?R") 2441proof (safe intro!: compact_eq_Heine_Borel[THEN iffD2]) 2442 fix A 2443 assume "compact U" 2444 assume A: "\<forall>a\<in>A. closed a" "U \<inter> \<Inter>A = {}" 2445 assume fin: "\<forall>B \<subseteq> A. finite B \<longrightarrow> U \<inter> \<Inter>B \<noteq> {}" 2446 from A have "(\<forall>a\<in>uminus`A. open a) \<and> U \<subseteq> \<Union>(uminus`A)" 2447 by auto 2448 with \<open>compact U\<close> obtain B where "B \<subseteq> A" "finite (uminus`B)" "U \<subseteq> \<Union>(uminus`B)" 2449 unfolding compact_eq_Heine_Borel by (metis subset_image_iff) 2450 with fin[THEN spec, of B] show False 2451 by (auto dest: finite_imageD intro: inj_setminus) 2452next 2453 fix A 2454 assume ?R 2455 assume "\<forall>a\<in>A. open a" "U \<subseteq> \<Union>A" 2456 then have "U \<inter> \<Inter>(uminus`A) = {}" "\<forall>a\<in>uminus`A. closed a" 2457 by auto 2458 with \<open>?R\<close> obtain B where "B \<subseteq> A" "finite (uminus`B)" "U \<inter> \<Inter>(uminus`B) = {}" 2459 by (metis subset_image_iff) 2460 then show "\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T" 2461 by (auto intro!: exI[of _ B] inj_setminus dest: finite_imageD) 2462qed 2463 2464lemma compact_imp_fip: 2465 assumes "compact S" 2466 and "\<And>T. T \<in> F \<Longrightarrow> closed T" 2467 and "\<And>F'. finite F' \<Longrightarrow> F' \<subseteq> F \<Longrightarrow> S \<inter> (\<Inter>F') \<noteq> {}" 2468 shows "S \<inter> (\<Inter>F) \<noteq> {}" 2469 using assms unfolding compact_fip by auto 2470 2471lemma compact_imp_fip_image: 2472 assumes "compact s" 2473 and P: "\<And>i. i \<in> I \<Longrightarrow> closed (f i)" 2474 and Q: "\<And>I'. finite I' \<Longrightarrow> I' \<subseteq> I \<Longrightarrow> (s \<inter> (\<Inter>i\<in>I'. f i) \<noteq> {})" 2475 shows "s \<inter> (\<Inter>i\<in>I. f i) \<noteq> {}" 2476proof - 2477 note \<open>compact s\<close> 2478 moreover from P have "\<forall>i \<in> f ` I. closed i" 2479 by blast 2480 moreover have "\<forall>A. finite A \<and> A \<subseteq> f ` I \<longrightarrow> (s \<inter> (\<Inter>A) \<noteq> {})" 2481 apply rule 2482 apply rule 2483 apply (erule conjE) 2484 proof - 2485 fix A :: "'a set set" 2486 assume "finite A" and "A \<subseteq> f ` I" 2487 then obtain B where "B \<subseteq> I" and "finite B" and "A = f ` B" 2488 using finite_subset_image [of A f I] by blast 2489 with Q [of B] show "s \<inter> \<Inter>A \<noteq> {}" 2490 by simp 2491 qed 2492 ultimately have "s \<inter> (\<Inter>(f ` I)) \<noteq> {}" 2493 by (metis compact_imp_fip) 2494 then show ?thesis by simp 2495qed 2496 2497end 2498 2499lemma (in t2_space) compact_imp_closed: 2500 assumes "compact s" 2501 shows "closed s" 2502 unfolding closed_def 2503proof (rule openI) 2504 fix y 2505 assume "y \<in> - s" 2506 let ?C = "\<Union>x\<in>s. {u. open u \<and> x \<in> u \<and> eventually (\<lambda>y. y \<notin> u) (nhds y)}" 2507 have "s \<subseteq> \<Union>?C" 2508 proof 2509 fix x 2510 assume "x \<in> s" 2511 with \<open>y \<in> - s\<close> have "x \<noteq> y" by clarsimp 2512 then have "\<exists>u v. open u \<and> open v \<and> x \<in> u \<and> y \<in> v \<and> u \<inter> v = {}" 2513 by (rule hausdorff) 2514 with \<open>x \<in> s\<close> show "x \<in> \<Union>?C" 2515 unfolding eventually_nhds by auto 2516 qed 2517 then obtain D where "D \<subseteq> ?C" and "finite D" and "s \<subseteq> \<Union>D" 2518 by (rule compactE [OF \<open>compact s\<close>]) auto 2519 from \<open>D \<subseteq> ?C\<close> have "\<forall>x\<in>D. eventually (\<lambda>y. y \<notin> x) (nhds y)" 2520 by auto 2521 with \<open>finite D\<close> have "eventually (\<lambda>y. y \<notin> \<Union>D) (nhds y)" 2522 by (simp add: eventually_ball_finite) 2523 with \<open>s \<subseteq> \<Union>D\<close> have "eventually (\<lambda>y. y \<notin> s) (nhds y)" 2524 by (auto elim!: eventually_mono) 2525 then show "\<exists>t. open t \<and> y \<in> t \<and> t \<subseteq> - s" 2526 by (simp add: eventually_nhds subset_eq) 2527qed 2528 2529lemma compact_continuous_image: 2530 assumes f: "continuous_on s f" 2531 and s: "compact s" 2532 shows "compact (f ` s)" 2533proof (rule compactI) 2534 fix C 2535 assume "\<forall>c\<in>C. open c" and cover: "f`s \<subseteq> \<Union>C" 2536 with f have "\<forall>c\<in>C. \<exists>A. open A \<and> A \<inter> s = f -` c \<inter> s" 2537 unfolding continuous_on_open_invariant by blast 2538 then obtain A where A: "\<forall>c\<in>C. open (A c) \<and> A c \<inter> s = f -` c \<inter> s" 2539 unfolding bchoice_iff .. 2540 with cover have "\<And>c. c \<in> C \<Longrightarrow> open (A c)" "s \<subseteq> (\<Union>c\<in>C. A c)" 2541 by (fastforce simp add: subset_eq set_eq_iff)+ 2542 from compactE_image[OF s this] obtain D where "D \<subseteq> C" "finite D" "s \<subseteq> (\<Union>c\<in>D. A c)" . 2543 with A show "\<exists>D \<subseteq> C. finite D \<and> f`s \<subseteq> \<Union>D" 2544 by (intro exI[of _ D]) (fastforce simp add: subset_eq set_eq_iff)+ 2545qed 2546 2547lemma continuous_on_inv: 2548 fixes f :: "'a::topological_space \<Rightarrow> 'b::t2_space" 2549 assumes "continuous_on s f" 2550 and "compact s" 2551 and "\<forall>x\<in>s. g (f x) = x" 2552 shows "continuous_on (f ` s) g" 2553 unfolding continuous_on_topological 2554proof (clarsimp simp add: assms(3)) 2555 fix x :: 'a and B :: "'a set" 2556 assume "x \<in> s" and "open B" and "x \<in> B" 2557 have 1: "\<forall>x\<in>s. f x \<in> f ` (s - B) \<longleftrightarrow> x \<in> s - B" 2558 using assms(3) by (auto, metis) 2559 have "continuous_on (s - B) f" 2560 using \<open>continuous_on s f\<close> Diff_subset 2561 by (rule continuous_on_subset) 2562 moreover have "compact (s - B)" 2563 using \<open>open B\<close> and \<open>compact s\<close> 2564 unfolding Diff_eq by (intro compact_Int_closed closed_Compl) 2565 ultimately have "compact (f ` (s - B))" 2566 by (rule compact_continuous_image) 2567 then have "closed (f ` (s - B))" 2568 by (rule compact_imp_closed) 2569 then have "open (- f ` (s - B))" 2570 by (rule open_Compl) 2571 moreover have "f x \<in> - f ` (s - B)" 2572 using \<open>x \<in> s\<close> and \<open>x \<in> B\<close> by (simp add: 1) 2573 moreover have "\<forall>y\<in>s. f y \<in> - f ` (s - B) \<longrightarrow> y \<in> B" 2574 by (simp add: 1) 2575 ultimately show "\<exists>A. open A \<and> f x \<in> A \<and> (\<forall>y\<in>s. f y \<in> A \<longrightarrow> y \<in> B)" 2576 by fast 2577qed 2578 2579lemma continuous_on_inv_into: 2580 fixes f :: "'a::topological_space \<Rightarrow> 'b::t2_space" 2581 assumes s: "continuous_on s f" "compact s" 2582 and f: "inj_on f s" 2583 shows "continuous_on (f ` s) (the_inv_into s f)" 2584 by (rule continuous_on_inv[OF s]) (auto simp: the_inv_into_f_f[OF f]) 2585 2586lemma (in linorder_topology) compact_attains_sup: 2587 assumes "compact S" "S \<noteq> {}" 2588 shows "\<exists>s\<in>S. \<forall>t\<in>S. t \<le> s" 2589proof (rule classical) 2590 assume "\<not> (\<exists>s\<in>S. \<forall>t\<in>S. t \<le> s)" 2591 then obtain t where t: "\<forall>s\<in>S. t s \<in> S" and "\<forall>s\<in>S. s < t s" 2592 by (metis not_le) 2593 then have "\<And>s. s\<in>S \<Longrightarrow> open {..< t s}" "S \<subseteq> (\<Union>s\<in>S. {..< t s})" 2594 by auto 2595 with \<open>compact S\<close> obtain C where "C \<subseteq> S" "finite C" and C: "S \<subseteq> (\<Union>s\<in>C. {..< t s})" 2596 by (metis compactE_image) 2597 with \<open>S \<noteq> {}\<close> have Max: "Max (t`C) \<in> t`C" and "\<forall>s\<in>t`C. s \<le> Max (t`C)" 2598 by (auto intro!: Max_in) 2599 with C have "S \<subseteq> {..< Max (t`C)}" 2600 by (auto intro: less_le_trans simp: subset_eq) 2601 with t Max \<open>C \<subseteq> S\<close> show ?thesis 2602 by fastforce 2603qed 2604 2605lemma (in linorder_topology) compact_attains_inf: 2606 assumes "compact S" "S \<noteq> {}" 2607 shows "\<exists>s\<in>S. \<forall>t\<in>S. s \<le> t" 2608proof (rule classical) 2609 assume "\<not> (\<exists>s\<in>S. \<forall>t\<in>S. s \<le> t)" 2610 then obtain t where t: "\<forall>s\<in>S. t s \<in> S" and "\<forall>s\<in>S. t s < s" 2611 by (metis not_le) 2612 then have "\<And>s. s\<in>S \<Longrightarrow> open {t s <..}" "S \<subseteq> (\<Union>s\<in>S. {t s <..})" 2613 by auto 2614 with \<open>compact S\<close> obtain C where "C \<subseteq> S" "finite C" and C: "S \<subseteq> (\<Union>s\<in>C. {t s <..})" 2615 by (metis compactE_image) 2616 with \<open>S \<noteq> {}\<close> have Min: "Min (t`C) \<in> t`C" and "\<forall>s\<in>t`C. Min (t`C) \<le> s" 2617 by (auto intro!: Min_in) 2618 with C have "S \<subseteq> {Min (t`C) <..}" 2619 by (auto intro: le_less_trans simp: subset_eq) 2620 with t Min \<open>C \<subseteq> S\<close> show ?thesis 2621 by fastforce 2622qed 2623 2624lemma continuous_attains_sup: 2625 fixes f :: "'a::topological_space \<Rightarrow> 'b::linorder_topology" 2626 shows "compact s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> continuous_on s f \<Longrightarrow> (\<exists>x\<in>s. \<forall>y\<in>s. f y \<le> f x)" 2627 using compact_attains_sup[of "f ` s"] compact_continuous_image[of s f] by auto 2628 2629lemma continuous_attains_inf: 2630 fixes f :: "'a::topological_space \<Rightarrow> 'b::linorder_topology" 2631 shows "compact s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> continuous_on s f \<Longrightarrow> (\<exists>x\<in>s. \<forall>y\<in>s. f x \<le> f y)" 2632 using compact_attains_inf[of "f ` s"] compact_continuous_image[of s f] by auto 2633 2634 2635subsection \<open>Connectedness\<close> 2636 2637context topological_space 2638begin 2639 2640definition "connected S \<longleftrightarrow> 2641 \<not> (\<exists>A B. open A \<and> open B \<and> S \<subseteq> A \<union> B \<and> A \<inter> B \<inter> S = {} \<and> A \<inter> S \<noteq> {} \<and> B \<inter> S \<noteq> {})" 2642 2643lemma connectedI: 2644 "(\<And>A B. open A \<Longrightarrow> open B \<Longrightarrow> A \<inter> U \<noteq> {} \<Longrightarrow> B \<inter> U \<noteq> {} \<Longrightarrow> A \<inter> B \<inter> U = {} \<Longrightarrow> U \<subseteq> A \<union> B \<Longrightarrow> False) 2645 \<Longrightarrow> connected U" 2646 by (auto simp: connected_def) 2647 2648lemma connected_empty [simp]: "connected {}" 2649 by (auto intro!: connectedI) 2650 2651lemma connected_sing [simp]: "connected {x}" 2652 by (auto intro!: connectedI) 2653 2654lemma connectedD: 2655 "connected A \<Longrightarrow> open U \<Longrightarrow> open V \<Longrightarrow> U \<inter> V \<inter> A = {} \<Longrightarrow> A \<subseteq> U \<union> V \<Longrightarrow> U \<inter> A = {} \<or> V \<inter> A = {}" 2656 by (auto simp: connected_def) 2657 2658end 2659 2660lemma connected_closed: 2661 "connected s \<longleftrightarrow> 2662 \<not> (\<exists>A B. closed A \<and> closed B \<and> s \<subseteq> A \<union> B \<and> A \<inter> B \<inter> s = {} \<and> A \<inter> s \<noteq> {} \<and> B \<inter> s \<noteq> {})" 2663 apply (simp add: connected_def del: ex_simps, safe) 2664 apply (drule_tac x="-A" in spec) 2665 apply (drule_tac x="-B" in spec) 2666 apply (fastforce simp add: closed_def [symmetric]) 2667 apply (drule_tac x="-A" in spec) 2668 apply (drule_tac x="-B" in spec) 2669 apply (fastforce simp add: open_closed [symmetric]) 2670 done 2671 2672lemma connected_closedD: 2673 "\<lbrakk>connected s; A \<inter> B \<inter> s = {}; s \<subseteq> A \<union> B; closed A; closed B\<rbrakk> \<Longrightarrow> A \<inter> s = {} \<or> B \<inter> s = {}" 2674 by (simp add: connected_closed) 2675 2676lemma connected_Union: 2677 assumes cs: "\<And>s. s \<in> S \<Longrightarrow> connected s" 2678 and ne: "\<Inter>S \<noteq> {}" 2679 shows "connected(\<Union>S)" 2680proof (rule connectedI) 2681 fix A B 2682 assume A: "open A" and B: "open B" and Alap: "A \<inter> \<Union>S \<noteq> {}" and Blap: "B \<inter> \<Union>S \<noteq> {}" 2683 and disj: "A \<inter> B \<inter> \<Union>S = {}" and cover: "\<Union>S \<subseteq> A \<union> B" 2684 have disjs:"\<And>s. s \<in> S \<Longrightarrow> A \<inter> B \<inter> s = {}" 2685 using disj by auto 2686 obtain sa where sa: "sa \<in> S" "A \<inter> sa \<noteq> {}" 2687 using Alap by auto 2688 obtain sb where sb: "sb \<in> S" "B \<inter> sb \<noteq> {}" 2689 using Blap by auto 2690 obtain x where x: "\<And>s. s \<in> S \<Longrightarrow> x \<in> s" 2691 using ne by auto 2692 then have "x \<in> \<Union>S" 2693 using \<open>sa \<in> S\<close> by blast 2694 then have "x \<in> A \<or> x \<in> B" 2695 using cover by auto 2696 then show False 2697 using cs [unfolded connected_def] 2698 by (metis A B IntI Sup_upper sa sb disjs x cover empty_iff subset_trans) 2699qed 2700 2701lemma connected_Un: "connected s \<Longrightarrow> connected t \<Longrightarrow> s \<inter> t \<noteq> {} \<Longrightarrow> connected (s \<union> t)" 2702 using connected_Union [of "{s,t}"] by auto 2703 2704lemma connected_diff_open_from_closed: 2705 assumes st: "s \<subseteq> t" 2706 and tu: "t \<subseteq> u" 2707 and s: "open s" 2708 and t: "closed t" 2709 and u: "connected u" 2710 and ts: "connected (t - s)" 2711 shows "connected(u - s)" 2712proof (rule connectedI) 2713 fix A B 2714 assume AB: "open A" "open B" "A \<inter> (u - s) \<noteq> {}" "B \<inter> (u - s) \<noteq> {}" 2715 and disj: "A \<inter> B \<inter> (u - s) = {}" 2716 and cover: "u - s \<subseteq> A \<union> B" 2717 then consider "A \<inter> (t - s) = {}" | "B \<inter> (t - s) = {}" 2718 using st ts tu connectedD [of "t-s" "A" "B"] by auto 2719 then show False 2720 proof cases 2721 case 1 2722 then have "(A - t) \<inter> (B \<union> s) \<inter> u = {}" 2723 using disj st by auto 2724 moreover have "u \<subseteq> (A - t) \<union> (B \<union> s)" 2725 using 1 cover by auto 2726 ultimately show False 2727 using connectedD [of u "A - t" "B \<union> s"] AB s t 1 u by auto 2728 next 2729 case 2 2730 then have "(A \<union> s) \<inter> (B - t) \<inter> u = {}" 2731 using disj st by auto 2732 moreover have "u \<subseteq> (A \<union> s) \<union> (B - t)" 2733 using 2 cover by auto 2734 ultimately show False 2735 using connectedD [of u "A \<union> s" "B - t"] AB s t 2 u by auto 2736 qed 2737qed 2738 2739lemma connected_iff_const: 2740 fixes S :: "'a::topological_space set" 2741 shows "connected S \<longleftrightarrow> (\<forall>P::'a \<Rightarrow> bool. continuous_on S P \<longrightarrow> (\<exists>c. \<forall>s\<in>S. P s = c))" 2742proof safe 2743 fix P :: "'a \<Rightarrow> bool" 2744 assume "connected S" "continuous_on S P" 2745 then have "\<And>b. \<exists>A. open A \<and> A \<inter> S = P -` {b} \<inter> S" 2746 unfolding continuous_on_open_invariant by (simp add: open_discrete) 2747 from this[of True] this[of False] 2748 obtain t f where "open t" "open f" and *: "f \<inter> S = P -` {False} \<inter> S" "t \<inter> S = P -` {True} \<inter> S" 2749 by meson 2750 then have "t \<inter> S = {} \<or> f \<inter> S = {}" 2751 by (intro connectedD[OF \<open>connected S\<close>]) auto 2752 then show "\<exists>c. \<forall>s\<in>S. P s = c" 2753 proof (rule disjE) 2754 assume "t \<inter> S = {}" 2755 then show ?thesis 2756 unfolding * by (intro exI[of _ False]) auto 2757 next 2758 assume "f \<inter> S = {}" 2759 then show ?thesis 2760 unfolding * by (intro exI[of _ True]) auto 2761 qed 2762next 2763 assume P: "\<forall>P::'a \<Rightarrow> bool. continuous_on S P \<longrightarrow> (\<exists>c. \<forall>s\<in>S. P s = c)" 2764 show "connected S" 2765 proof (rule connectedI) 2766 fix A B 2767 assume *: "open A" "open B" "A \<inter> S \<noteq> {}" "B \<inter> S \<noteq> {}" "A \<inter> B \<inter> S = {}" "S \<subseteq> A \<union> B" 2768 have "continuous_on S (\<lambda>x. x \<in> A)" 2769 unfolding continuous_on_open_invariant 2770 proof safe 2771 fix C :: "bool set" 2772 have "C = UNIV \<or> C = {True} \<or> C = {False} \<or> C = {}" 2773 using subset_UNIV[of C] unfolding UNIV_bool by auto 2774 with * show "\<exists>T. open T \<and> T \<inter> S = (\<lambda>x. x \<in> A) -` C \<inter> S" 2775 by (intro exI[of _ "(if True \<in> C then A else {}) \<union> (if False \<in> C then B else {})"]) auto 2776 qed 2777 from P[rule_format, OF this] obtain c where "\<And>s. s \<in> S \<Longrightarrow> (s \<in> A) = c" 2778 by blast 2779 with * show False 2780 by (cases c) auto 2781 qed 2782qed 2783 2784lemma connectedD_const: "connected S \<Longrightarrow> continuous_on S P \<Longrightarrow> \<exists>c. \<forall>s\<in>S. P s = c" 2785 for P :: "'a::topological_space \<Rightarrow> bool" 2786 by (auto simp: connected_iff_const) 2787 2788lemma connectedI_const: 2789 "(\<And>P::'a::topological_space \<Rightarrow> bool. continuous_on S P \<Longrightarrow> \<exists>c. \<forall>s\<in>S. P s = c) \<Longrightarrow> connected S" 2790 by (auto simp: connected_iff_const) 2791 2792lemma connected_local_const: 2793 assumes "connected A" "a \<in> A" "b \<in> A" 2794 and *: "\<forall>a\<in>A. eventually (\<lambda>b. f a = f b) (at a within A)" 2795 shows "f a = f b" 2796proof - 2797 obtain S where S: "\<And>a. a \<in> A \<Longrightarrow> a \<in> S a" "\<And>a. a \<in> A \<Longrightarrow> open (S a)" 2798 "\<And>a x. a \<in> A \<Longrightarrow> x \<in> S a \<Longrightarrow> x \<in> A \<Longrightarrow> f a = f x" 2799 using * unfolding eventually_at_topological by metis 2800 let ?P = "\<Union>b\<in>{b\<in>A. f a = f b}. S b" and ?N = "\<Union>b\<in>{b\<in>A. f a \<noteq> f b}. S b" 2801 have "?P \<inter> A = {} \<or> ?N \<inter> A = {}" 2802 using \<open>connected A\<close> S \<open>a\<in>A\<close> 2803 by (intro connectedD) (auto, metis) 2804 then show "f a = f b" 2805 proof 2806 assume "?N \<inter> A = {}" 2807 then have "\<forall>x\<in>A. f a = f x" 2808 using S(1) by auto 2809 with \<open>b\<in>A\<close> show ?thesis by auto 2810 next 2811 assume "?P \<inter> A = {}" then show ?thesis 2812 using \<open>a \<in> A\<close> S(1)[of a] by auto 2813 qed 2814qed 2815 2816lemma (in linorder_topology) connectedD_interval: 2817 assumes "connected U" 2818 and xy: "x \<in> U" "y \<in> U" 2819 and "x \<le> z" "z \<le> y" 2820 shows "z \<in> U" 2821proof - 2822 have eq: "{..<z} \<union> {z<..} = - {z}" 2823 by auto 2824 have "\<not> connected U" if "z \<notin> U" "x < z" "z < y" 2825 using xy that 2826 apply (simp only: connected_def simp_thms) 2827 apply (rule_tac exI[of _ "{..< z}"]) 2828 apply (rule_tac exI[of _ "{z <..}"]) 2829 apply (auto simp add: eq) 2830 done 2831 with assms show "z \<in> U" 2832 by (metis less_le) 2833qed 2834 2835lemma (in linorder_topology) not_in_connected_cases: 2836 assumes conn: "connected S" 2837 assumes nbdd: "x \<notin> S" 2838 assumes ne: "S \<noteq> {}" 2839 obtains "bdd_above S" "\<And>y. y \<in> S \<Longrightarrow> x \<ge> y" | "bdd_below S" "\<And>y. y \<in> S \<Longrightarrow> x \<le> y" 2840proof - 2841 obtain s where "s \<in> S" using ne by blast 2842 { 2843 assume "s \<le> x" 2844 have "False" if "x \<le> y" "y \<in> S" for y 2845 using connectedD_interval[OF conn \<open>s \<in> S\<close> \<open>y \<in> S\<close> \<open>s \<le> x\<close> \<open>x \<le> y\<close>] \<open>x \<notin> S\<close> 2846 by simp 2847 then have wit: "y \<in> S \<Longrightarrow> x \<ge> y" for y 2848 using le_cases by blast 2849 then have "bdd_above S" 2850 by (rule local.bdd_aboveI) 2851 note this wit 2852 } moreover { 2853 assume "x \<le> s" 2854 have "False" if "x \<ge> y" "y \<in> S" for y 2855 using connectedD_interval[OF conn \<open>y \<in> S\<close> \<open>s \<in> S\<close> \<open>x \<ge> y\<close> \<open>s \<ge> x\<close> ] \<open>x \<notin> S\<close> 2856 by simp 2857 then have wit: "y \<in> S \<Longrightarrow> x \<le> y" for y 2858 using le_cases by blast 2859 then have "bdd_below S" 2860 by (rule bdd_belowI) 2861 note this wit 2862 } ultimately show ?thesis 2863 by (meson le_cases that) 2864qed 2865 2866lemma connected_continuous_image: 2867 assumes *: "continuous_on s f" 2868 and "connected s" 2869 shows "connected (f ` s)" 2870proof (rule connectedI_const) 2871 fix P :: "'b \<Rightarrow> bool" 2872 assume "continuous_on (f ` s) P" 2873 then have "continuous_on s (P \<circ> f)" 2874 by (rule continuous_on_compose[OF *]) 2875 from connectedD_const[OF \<open>connected s\<close> this] show "\<exists>c. \<forall>s\<in>f ` s. P s = c" 2876 by auto 2877qed 2878 2879 2880section \<open>Linear Continuum Topologies\<close> 2881 2882class linear_continuum_topology = linorder_topology + linear_continuum 2883begin 2884 2885lemma Inf_notin_open: 2886 assumes A: "open A" 2887 and bnd: "\<forall>a\<in>A. x < a" 2888 shows "Inf A \<notin> A" 2889proof 2890 assume "Inf A \<in> A" 2891 then obtain b where "b < Inf A" "{b <.. Inf A} \<subseteq> A" 2892 using open_left[of A "Inf A" x] assms by auto 2893 with dense[of b "Inf A"] obtain c where "c < Inf A" "c \<in> A" 2894 by (auto simp: subset_eq) 2895 then show False 2896 using cInf_lower[OF \<open>c \<in> A\<close>] bnd 2897 by (metis not_le less_imp_le bdd_belowI) 2898qed 2899 2900lemma Sup_notin_open: 2901 assumes A: "open A" 2902 and bnd: "\<forall>a\<in>A. a < x" 2903 shows "Sup A \<notin> A" 2904proof 2905 assume "Sup A \<in> A" 2906 with assms obtain b where "Sup A < b" "{Sup A ..< b} \<subseteq> A" 2907 using open_right[of A "Sup A" x] by auto 2908 with dense[of "Sup A" b] obtain c where "Sup A < c" "c \<in> A" 2909 by (auto simp: subset_eq) 2910 then show False 2911 using cSup_upper[OF \<open>c \<in> A\<close>] bnd 2912 by (metis less_imp_le not_le bdd_aboveI) 2913qed 2914 2915end 2916 2917instance linear_continuum_topology \<subseteq> perfect_space 2918proof 2919 fix x :: 'a 2920 obtain y where "x < y \<or> y < x" 2921 using ex_gt_or_lt [of x] .. 2922 with Inf_notin_open[of "{x}" y] Sup_notin_open[of "{x}" y] show "\<not> open {x}" 2923 by auto 2924qed 2925 2926lemma connectedI_interval: 2927 fixes U :: "'a :: linear_continuum_topology set" 2928 assumes *: "\<And>x y z. x \<in> U \<Longrightarrow> y \<in> U \<Longrightarrow> x \<le> z \<Longrightarrow> z \<le> y \<Longrightarrow> z \<in> U" 2929 shows "connected U" 2930proof (rule connectedI) 2931 { 2932 fix A B 2933 assume "open A" "open B" "A \<inter> B \<inter> U = {}" "U \<subseteq> A \<union> B" 2934 fix x y 2935 assume "x < y" "x \<in> A" "y \<in> B" "x \<in> U" "y \<in> U" 2936 2937 let ?z = "Inf (B \<inter> {x <..})" 2938 2939 have "x \<le> ?z" "?z \<le> y" 2940 using \<open>y \<in> B\<close> \<open>x < y\<close> by (auto intro: cInf_lower cInf_greatest) 2941 with \<open>x \<in> U\<close> \<open>y \<in> U\<close> have "?z \<in> U" 2942 by (rule *) 2943 moreover have "?z \<notin> B \<inter> {x <..}" 2944 using \<open>open B\<close> by (intro Inf_notin_open) auto 2945 ultimately have "?z \<in> A" 2946 using \<open>x \<le> ?z\<close> \<open>A \<inter> B \<inter> U = {}\<close> \<open>x \<in> A\<close> \<open>U \<subseteq> A \<union> B\<close> by auto 2947 have "\<exists>b\<in>B. b \<in> A \<and> b \<in> U" if "?z < y" 2948 proof - 2949 obtain a where "?z < a" "{?z ..< a} \<subseteq> A" 2950 using open_right[OF \<open>open A\<close> \<open>?z \<in> A\<close> \<open>?z < y\<close>] by auto 2951 moreover obtain b where "b \<in> B" "x < b" "b < min a y" 2952 using cInf_less_iff[of "B \<inter> {x <..}" "min a y"] \<open>?z < a\<close> \<open>?z < y\<close> \<open>x < y\<close> \<open>y \<in> B\<close> 2953 by auto 2954 moreover have "?z \<le> b" 2955 using \<open>b \<in> B\<close> \<open>x < b\<close> 2956 by (intro cInf_lower) auto 2957 moreover have "b \<in> U" 2958 using \<open>x \<le> ?z\<close> \<open>?z \<le> b\<close> \<open>b < min a y\<close> 2959 by (intro *[OF \<open>x \<in> U\<close> \<open>y \<in> U\<close>]) (auto simp: less_imp_le) 2960 ultimately show ?thesis 2961 by (intro bexI[of _ b]) auto 2962 qed 2963 then have False 2964 using \<open>?z \<le> y\<close> \<open>?z \<in> A\<close> \<open>y \<in> B\<close> \<open>y \<in> U\<close> \<open>A \<inter> B \<inter> U = {}\<close> 2965 unfolding le_less by blast 2966 } 2967 note not_disjoint = this 2968 2969 fix A B assume AB: "open A" "open B" "U \<subseteq> A \<union> B" "A \<inter> B \<inter> U = {}" 2970 moreover assume "A \<inter> U \<noteq> {}" then obtain x where x: "x \<in> U" "x \<in> A" by auto 2971 moreover assume "B \<inter> U \<noteq> {}" then obtain y where y: "y \<in> U" "y \<in> B" by auto 2972 moreover note not_disjoint[of B A y x] not_disjoint[of A B x y] 2973 ultimately show False 2974 by (cases x y rule: linorder_cases) auto 2975qed 2976 2977lemma connected_iff_interval: "connected U \<longleftrightarrow> (\<forall>x\<in>U. \<forall>y\<in>U. \<forall>z. x \<le> z \<longrightarrow> z \<le> y \<longrightarrow> z \<in> U)" 2978 for U :: "'a::linear_continuum_topology set" 2979 by (auto intro: connectedI_interval dest: connectedD_interval) 2980 2981lemma connected_UNIV[simp]: "connected (UNIV::'a::linear_continuum_topology set)" 2982 by (simp add: connected_iff_interval) 2983 2984lemma connected_Ioi[simp]: "connected {a<..}" 2985 for a :: "'a::linear_continuum_topology" 2986 by (auto simp: connected_iff_interval) 2987 2988lemma connected_Ici[simp]: "connected {a..}" 2989 for a :: "'a::linear_continuum_topology" 2990 by (auto simp: connected_iff_interval) 2991 2992lemma connected_Iio[simp]: "connected {..<a}" 2993 for a :: "'a::linear_continuum_topology" 2994 by (auto simp: connected_iff_interval) 2995 2996lemma connected_Iic[simp]: "connected {..a}" 2997 for a :: "'a::linear_continuum_topology" 2998 by (auto simp: connected_iff_interval) 2999 3000lemma connected_Ioo[simp]: "connected {a<..<b}" 3001 for a b :: "'a::linear_continuum_topology" 3002 unfolding connected_iff_interval by auto 3003 3004lemma connected_Ioc[simp]: "connected {a<..b}" 3005 for a b :: "'a::linear_continuum_topology" 3006 by (auto simp: connected_iff_interval) 3007 3008lemma connected_Ico[simp]: "connected {a..<b}" 3009 for a b :: "'a::linear_continuum_topology" 3010 by (auto simp: connected_iff_interval) 3011 3012lemma connected_Icc[simp]: "connected {a..b}" 3013 for a b :: "'a::linear_continuum_topology" 3014 by (auto simp: connected_iff_interval) 3015 3016lemma connected_contains_Ioo: 3017 fixes A :: "'a :: linorder_topology set" 3018 assumes "connected A" "a \<in> A" "b \<in> A" shows "{a <..< b} \<subseteq> A" 3019 using connectedD_interval[OF assms] by (simp add: subset_eq Ball_def less_imp_le) 3020 3021lemma connected_contains_Icc: 3022 fixes A :: "'a::linorder_topology set" 3023 assumes "connected A" "a \<in> A" "b \<in> A" 3024 shows "{a..b} \<subseteq> A" 3025proof 3026 fix x assume "x \<in> {a..b}" 3027 then have "x = a \<or> x = b \<or> x \<in> {a<..<b}" 3028 by auto 3029 then show "x \<in> A" 3030 using assms connected_contains_Ioo[of A a b] by auto 3031qed 3032 3033 3034subsection \<open>Intermediate Value Theorem\<close> 3035 3036lemma IVT': 3037 fixes f :: "'a::linear_continuum_topology \<Rightarrow> 'b::linorder_topology" 3038 assumes y: "f a \<le> y" "y \<le> f b" "a \<le> b" 3039 and *: "continuous_on {a .. b} f" 3040 shows "\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = y" 3041proof - 3042 have "connected {a..b}" 3043 unfolding connected_iff_interval by auto 3044 from connected_continuous_image[OF * this, THEN connectedD_interval, of "f a" "f b" y] y 3045 show ?thesis 3046 by (auto simp add: atLeastAtMost_def atLeast_def atMost_def) 3047qed 3048 3049lemma IVT2': 3050 fixes f :: "'a :: linear_continuum_topology \<Rightarrow> 'b :: linorder_topology" 3051 assumes y: "f b \<le> y" "y \<le> f a" "a \<le> b" 3052 and *: "continuous_on {a .. b} f" 3053 shows "\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = y" 3054proof - 3055 have "connected {a..b}" 3056 unfolding connected_iff_interval by auto 3057 from connected_continuous_image[OF * this, THEN connectedD_interval, of "f b" "f a" y] y 3058 show ?thesis 3059 by (auto simp add: atLeastAtMost_def atLeast_def atMost_def) 3060qed 3061 3062lemma IVT: 3063 fixes f :: "'a::linear_continuum_topology \<Rightarrow> 'b::linorder_topology" 3064 shows "f a \<le> y \<Longrightarrow> y \<le> f b \<Longrightarrow> a \<le> b \<Longrightarrow> (\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x) \<Longrightarrow> 3065 \<exists>x. a \<le> x \<and> x \<le> b \<and> f x = y" 3066 by (rule IVT') (auto intro: continuous_at_imp_continuous_on) 3067 3068lemma IVT2: 3069 fixes f :: "'a::linear_continuum_topology \<Rightarrow> 'b::linorder_topology" 3070 shows "f b \<le> y \<Longrightarrow> y \<le> f a \<Longrightarrow> a \<le> b \<Longrightarrow> (\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x) \<Longrightarrow> 3071 \<exists>x. a \<le> x \<and> x \<le> b \<and> f x = y" 3072 by (rule IVT2') (auto intro: continuous_at_imp_continuous_on) 3073 3074lemma continuous_inj_imp_mono: 3075 fixes f :: "'a::linear_continuum_topology \<Rightarrow> 'b::linorder_topology" 3076 assumes x: "a < x" "x < b" 3077 and cont: "continuous_on {a..b} f" 3078 and inj: "inj_on f {a..b}" 3079 shows "(f a < f x \<and> f x < f b) \<or> (f b < f x \<and> f x < f a)" 3080proof - 3081 note I = inj_on_eq_iff[OF inj] 3082 { 3083 assume "f x < f a" "f x < f b" 3084 then obtain s t where "x \<le> s" "s \<le> b" "a \<le> t" "t \<le> x" "f s = f t" "f x < f s" 3085 using IVT'[of f x "min (f a) (f b)" b] IVT2'[of f x "min (f a) (f b)" a] x 3086 by (auto simp: continuous_on_subset[OF cont] less_imp_le) 3087 with x I have False by auto 3088 } 3089 moreover 3090 { 3091 assume "f a < f x" "f b < f x" 3092 then obtain s t where "x \<le> s" "s \<le> b" "a \<le> t" "t \<le> x" "f s = f t" "f s < f x" 3093 using IVT'[of f a "max (f a) (f b)" x] IVT2'[of f b "max (f a) (f b)" x] x 3094 by (auto simp: continuous_on_subset[OF cont] less_imp_le) 3095 with x I have False by auto 3096 } 3097 ultimately show ?thesis 3098 using I[of a x] I[of x b] x less_trans[OF x] 3099 by (auto simp add: le_less less_imp_neq neq_iff) 3100qed 3101 3102lemma continuous_at_Sup_mono: 3103 fixes f :: "'a::{linorder_topology,conditionally_complete_linorder} \<Rightarrow> 3104 'b::{linorder_topology,conditionally_complete_linorder}" 3105 assumes "mono f" 3106 and cont: "continuous (at_left (Sup S)) f" 3107 and S: "S \<noteq> {}" "bdd_above S" 3108 shows "f (Sup S) = (SUP s\<in>S. f s)" 3109proof (rule antisym) 3110 have f: "(f \<longlongrightarrow> f (Sup S)) (at_left (Sup S))" 3111 using cont unfolding continuous_within . 3112 show "f (Sup S) \<le> (SUP s\<in>S. f s)" 3113 proof cases 3114 assume "Sup S \<in> S" 3115 then show ?thesis 3116 by (rule cSUP_upper) (auto intro: bdd_above_image_mono S \<open>mono f\<close>) 3117 next 3118 assume "Sup S \<notin> S" 3119 from \<open>S \<noteq> {}\<close> obtain s where "s \<in> S" 3120 by auto 3121 with \<open>Sup S \<notin> S\<close> S have "s < Sup S" 3122 unfolding less_le by (blast intro: cSup_upper) 3123 show ?thesis 3124 proof (rule ccontr) 3125 assume "\<not> ?thesis" 3126 with order_tendstoD(1)[OF f, of "SUP s\<in>S. f s"] obtain b where "b < Sup S" 3127 and *: "\<And>y. b < y \<Longrightarrow> y < Sup S \<Longrightarrow> (SUP s\<in>S. f s) < f y" 3128 by (auto simp: not_le eventually_at_left[OF \<open>s < Sup S\<close>]) 3129 with \<open>S \<noteq> {}\<close> obtain c where "c \<in> S" "b < c" 3130 using less_cSupD[of S b] by auto 3131 with \<open>Sup S \<notin> S\<close> S have "c < Sup S" 3132 unfolding less_le by (blast intro: cSup_upper) 3133 from *[OF \<open>b < c\<close> \<open>c < Sup S\<close>] cSUP_upper[OF \<open>c \<in> S\<close> bdd_above_image_mono[of f]] 3134 show False 3135 by (auto simp: assms) 3136 qed 3137 qed 3138qed (intro cSUP_least \<open>mono f\<close>[THEN monoD] cSup_upper S) 3139 3140lemma continuous_at_Sup_antimono: 3141 fixes f :: "'a::{linorder_topology,conditionally_complete_linorder} \<Rightarrow> 3142 'b::{linorder_topology,conditionally_complete_linorder}" 3143 assumes "antimono f" 3144 and cont: "continuous (at_left (Sup S)) f" 3145 and S: "S \<noteq> {}" "bdd_above S" 3146 shows "f (Sup S) = (INF s\<in>S. f s)" 3147proof (rule antisym) 3148 have f: "(f \<longlongrightarrow> f (Sup S)) (at_left (Sup S))" 3149 using cont unfolding continuous_within . 3150 show "(INF s\<in>S. f s) \<le> f (Sup S)" 3151 proof cases 3152 assume "Sup S \<in> S" 3153 then show ?thesis 3154 by (intro cINF_lower) (auto intro: bdd_below_image_antimono S \<open>antimono f\<close>) 3155 next 3156 assume "Sup S \<notin> S" 3157 from \<open>S \<noteq> {}\<close> obtain s where "s \<in> S" 3158 by auto 3159 with \<open>Sup S \<notin> S\<close> S have "s < Sup S" 3160 unfolding less_le by (blast intro: cSup_upper) 3161 show ?thesis 3162 proof (rule ccontr) 3163 assume "\<not> ?thesis" 3164 with order_tendstoD(2)[OF f, of "INF s\<in>S. f s"] obtain b where "b < Sup S" 3165 and *: "\<And>y. b < y \<Longrightarrow> y < Sup S \<Longrightarrow> f y < (INF s\<in>S. f s)" 3166 by (auto simp: not_le eventually_at_left[OF \<open>s < Sup S\<close>]) 3167 with \<open>S \<noteq> {}\<close> obtain c where "c \<in> S" "b < c" 3168 using less_cSupD[of S b] by auto 3169 with \<open>Sup S \<notin> S\<close> S have "c < Sup S" 3170 unfolding less_le by (blast intro: cSup_upper) 3171 from *[OF \<open>b < c\<close> \<open>c < Sup S\<close>] cINF_lower[OF bdd_below_image_antimono, of f S c] \<open>c \<in> S\<close> 3172 show False 3173 by (auto simp: assms) 3174 qed 3175 qed 3176qed (intro cINF_greatest \<open>antimono f\<close>[THEN antimonoD] cSup_upper S) 3177 3178lemma continuous_at_Inf_mono: 3179 fixes f :: "'a::{linorder_topology,conditionally_complete_linorder} \<Rightarrow> 3180 'b::{linorder_topology,conditionally_complete_linorder}" 3181 assumes "mono f" 3182 and cont: "continuous (at_right (Inf S)) f" 3183 and S: "S \<noteq> {}" "bdd_below S" 3184 shows "f (Inf S) = (INF s\<in>S. f s)" 3185proof (rule antisym) 3186 have f: "(f \<longlongrightarrow> f (Inf S)) (at_right (Inf S))" 3187 using cont unfolding continuous_within . 3188 show "(INF s\<in>S. f s) \<le> f (Inf S)" 3189 proof cases 3190 assume "Inf S \<in> S" 3191 then show ?thesis 3192 by (rule cINF_lower[rotated]) (auto intro: bdd_below_image_mono S \<open>mono f\<close>) 3193 next 3194 assume "Inf S \<notin> S" 3195 from \<open>S \<noteq> {}\<close> obtain s where "s \<in> S" 3196 by auto 3197 with \<open>Inf S \<notin> S\<close> S have "Inf S < s" 3198 unfolding less_le by (blast intro: cInf_lower) 3199 show ?thesis 3200 proof (rule ccontr) 3201 assume "\<not> ?thesis" 3202 with order_tendstoD(2)[OF f, of "INF s\<in>S. f s"] obtain b where "Inf S < b" 3203 and *: "\<And>y. Inf S < y \<Longrightarrow> y < b \<Longrightarrow> f y < (INF s\<in>S. f s)" 3204 by (auto simp: not_le eventually_at_right[OF \<open>Inf S < s\<close>]) 3205 with \<open>S \<noteq> {}\<close> obtain c where "c \<in> S" "c < b" 3206 using cInf_lessD[of S b] by auto 3207 with \<open>Inf S \<notin> S\<close> S have "Inf S < c" 3208 unfolding less_le by (blast intro: cInf_lower) 3209 from *[OF \<open>Inf S < c\<close> \<open>c < b\<close>] cINF_lower[OF bdd_below_image_mono[of f] \<open>c \<in> S\<close>] 3210 show False 3211 by (auto simp: assms) 3212 qed 3213 qed 3214qed (intro cINF_greatest \<open>mono f\<close>[THEN monoD] cInf_lower \<open>bdd_below S\<close> \<open>S \<noteq> {}\<close>) 3215 3216lemma continuous_at_Inf_antimono: 3217 fixes f :: "'a::{linorder_topology,conditionally_complete_linorder} \<Rightarrow> 3218 'b::{linorder_topology,conditionally_complete_linorder}" 3219 assumes "antimono f" 3220 and cont: "continuous (at_right (Inf S)) f" 3221 and S: "S \<noteq> {}" "bdd_below S" 3222 shows "f (Inf S) = (SUP s\<in>S. f s)" 3223proof (rule antisym) 3224 have f: "(f \<longlongrightarrow> f (Inf S)) (at_right (Inf S))" 3225 using cont unfolding continuous_within . 3226 show "f (Inf S) \<le> (SUP s\<in>S. f s)" 3227 proof cases 3228 assume "Inf S \<in> S" 3229 then show ?thesis 3230 by (rule cSUP_upper) (auto intro: bdd_above_image_antimono S \<open>antimono f\<close>) 3231 next 3232 assume "Inf S \<notin> S" 3233 from \<open>S \<noteq> {}\<close> obtain s where "s \<in> S" 3234 by auto 3235 with \<open>Inf S \<notin> S\<close> S have "Inf S < s" 3236 unfolding less_le by (blast intro: cInf_lower) 3237 show ?thesis 3238 proof (rule ccontr) 3239 assume "\<not> ?thesis" 3240 with order_tendstoD(1)[OF f, of "SUP s\<in>S. f s"] obtain b where "Inf S < b" 3241 and *: "\<And>y. Inf S < y \<Longrightarrow> y < b \<Longrightarrow> (SUP s\<in>S. f s) < f y" 3242 by (auto simp: not_le eventually_at_right[OF \<open>Inf S < s\<close>]) 3243 with \<open>S \<noteq> {}\<close> obtain c where "c \<in> S" "c < b" 3244 using cInf_lessD[of S b] by auto 3245 with \<open>Inf S \<notin> S\<close> S have "Inf S < c" 3246 unfolding less_le by (blast intro: cInf_lower) 3247 from *[OF \<open>Inf S < c\<close> \<open>c < b\<close>] cSUP_upper[OF \<open>c \<in> S\<close> bdd_above_image_antimono[of f]] 3248 show False 3249 by (auto simp: assms) 3250 qed 3251 qed 3252qed (intro cSUP_least \<open>antimono f\<close>[THEN antimonoD] cInf_lower S) 3253 3254 3255subsection \<open>Uniform spaces\<close> 3256 3257class uniformity = 3258 fixes uniformity :: "('a \<times> 'a) filter" 3259begin 3260 3261abbreviation uniformity_on :: "'a set \<Rightarrow> ('a \<times> 'a) filter" 3262 where "uniformity_on s \<equiv> inf uniformity (principal (s\<times>s))" 3263 3264end 3265 3266lemma uniformity_Abort: 3267 "uniformity = 3268 Filter.abstract_filter (\<lambda>u. Code.abort (STR ''uniformity is not executable'') (\<lambda>u. uniformity))" 3269 by simp 3270 3271class open_uniformity = "open" + uniformity + 3272 assumes open_uniformity: 3273 "\<And>U. open U \<longleftrightarrow> (\<forall>x\<in>U. eventually (\<lambda>(x', y). x' = x \<longrightarrow> y \<in> U) uniformity)" 3274begin 3275 3276subclass topological_space 3277 by standard (force elim: eventually_mono eventually_elim2 simp: split_beta' open_uniformity)+ 3278 3279end 3280 3281class uniform_space = open_uniformity + 3282 assumes uniformity_refl: "eventually E uniformity \<Longrightarrow> E (x, x)" 3283 and uniformity_sym: "eventually E uniformity \<Longrightarrow> eventually (\<lambda>(x, y). E (y, x)) uniformity" 3284 and uniformity_trans: 3285 "eventually E uniformity \<Longrightarrow> 3286 \<exists>D. eventually D uniformity \<and> (\<forall>x y z. D (x, y) \<longrightarrow> D (y, z) \<longrightarrow> E (x, z))" 3287begin 3288 3289lemma uniformity_bot: "uniformity \<noteq> bot" 3290 using uniformity_refl by auto 3291 3292lemma uniformity_trans': 3293 "eventually E uniformity \<Longrightarrow> 3294 eventually (\<lambda>((x, y), (y', z)). y = y' \<longrightarrow> E (x, z)) (uniformity \<times>\<^sub>F uniformity)" 3295 by (drule uniformity_trans) (auto simp add: eventually_prod_same) 3296 3297lemma uniformity_transE: 3298 assumes "eventually E uniformity" 3299 obtains D where "eventually D uniformity" "\<And>x y z. D (x, y) \<Longrightarrow> D (y, z) \<Longrightarrow> E (x, z)" 3300 using uniformity_trans [OF assms] by auto 3301 3302lemma eventually_nhds_uniformity: 3303 "eventually P (nhds x) \<longleftrightarrow> eventually (\<lambda>(x', y). x' = x \<longrightarrow> P y) uniformity" 3304 (is "_ \<longleftrightarrow> ?N P x") 3305 unfolding eventually_nhds 3306proof safe 3307 assume *: "?N P x" 3308 have "?N (?N P) x" if "?N P x" for x 3309 proof - 3310 from that obtain D where ev: "eventually D uniformity" 3311 and D: "D (a, b) \<Longrightarrow> D (b, c) \<Longrightarrow> case (a, c) of (x', y) \<Rightarrow> x' = x \<longrightarrow> P y" for a b c 3312 by (rule uniformity_transE) simp 3313 from ev show ?thesis 3314 by eventually_elim (insert ev D, force elim: eventually_mono split: prod.split) 3315 qed 3316 then have "open {x. ?N P x}" 3317 by (simp add: open_uniformity) 3318 then show "\<exists>S. open S \<and> x \<in> S \<and> (\<forall>x\<in>S. P x)" 3319 by (intro exI[of _ "{x. ?N P x}"]) (auto dest: uniformity_refl simp: *) 3320qed (force simp add: open_uniformity elim: eventually_mono) 3321 3322 3323subsubsection \<open>Totally bounded sets\<close> 3324 3325definition totally_bounded :: "'a set \<Rightarrow> bool" 3326 where "totally_bounded S \<longleftrightarrow> 3327 (\<forall>E. eventually E uniformity \<longrightarrow> (\<exists>X. finite X \<and> (\<forall>s\<in>S. \<exists>x\<in>X. E (x, s))))" 3328 3329lemma totally_bounded_empty[iff]: "totally_bounded {}" 3330 by (auto simp add: totally_bounded_def) 3331 3332lemma totally_bounded_subset: "totally_bounded S \<Longrightarrow> T \<subseteq> S \<Longrightarrow> totally_bounded T" 3333 by (fastforce simp add: totally_bounded_def) 3334 3335lemma totally_bounded_Union[intro]: 3336 assumes M: "finite M" "\<And>S. S \<in> M \<Longrightarrow> totally_bounded S" 3337 shows "totally_bounded (\<Union>M)" 3338 unfolding totally_bounded_def 3339proof safe 3340 fix E 3341 assume "eventually E uniformity" 3342 with M obtain X where "\<forall>S\<in>M. finite (X S) \<and> (\<forall>s\<in>S. \<exists>x\<in>X S. E (x, s))" 3343 by (metis totally_bounded_def) 3344 with \<open>finite M\<close> show "\<exists>X. finite X \<and> (\<forall>s\<in>\<Union>M. \<exists>x\<in>X. E (x, s))" 3345 by (intro exI[of _ "\<Union>S\<in>M. X S"]) force 3346qed 3347 3348 3349subsubsection \<open>Cauchy filter\<close> 3350 3351definition cauchy_filter :: "'a filter \<Rightarrow> bool" 3352 where "cauchy_filter F \<longleftrightarrow> F \<times>\<^sub>F F \<le> uniformity" 3353 3354definition Cauchy :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" 3355 where Cauchy_uniform: "Cauchy X = cauchy_filter (filtermap X sequentially)" 3356 3357lemma Cauchy_uniform_iff: 3358 "Cauchy X \<longleftrightarrow> (\<forall>P. eventually P uniformity \<longrightarrow> (\<exists>N. \<forall>n\<ge>N. \<forall>m\<ge>N. P (X n, X m)))" 3359 unfolding Cauchy_uniform cauchy_filter_def le_filter_def eventually_prod_same 3360 eventually_filtermap eventually_sequentially 3361proof safe 3362 let ?U = "\<lambda>P. eventually P uniformity" 3363 { 3364 fix P 3365 assume "?U P" "\<forall>P. ?U P \<longrightarrow> (\<exists>Q. (\<exists>N. \<forall>n\<ge>N. Q (X n)) \<and> (\<forall>x y. Q x \<longrightarrow> Q y \<longrightarrow> P (x, y)))" 3366 then obtain Q N where "\<And>n. n \<ge> N \<Longrightarrow> Q (X n)" "\<And>x y. Q x \<Longrightarrow> Q y \<Longrightarrow> P (x, y)" 3367 by metis 3368 then show "\<exists>N. \<forall>n\<ge>N. \<forall>m\<ge>N. P (X n, X m)" 3369 by blast 3370 next 3371 fix P 3372 assume "?U P" and P: "\<forall>P. ?U P \<longrightarrow> (\<exists>N. \<forall>n\<ge>N. \<forall>m\<ge>N. P (X n, X m))" 3373 then obtain Q where "?U Q" and Q: "\<And>x y z. Q (x, y) \<Longrightarrow> Q (y, z) \<Longrightarrow> P (x, z)" 3374 by (auto elim: uniformity_transE) 3375 then have "?U (\<lambda>x. Q x \<and> (\<lambda>(x, y). Q (y, x)) x)" 3376 unfolding eventually_conj_iff by (simp add: uniformity_sym) 3377 from P[rule_format, OF this] 3378 obtain N where N: "\<And>n m. n \<ge> N \<Longrightarrow> m \<ge> N \<Longrightarrow> Q (X n, X m) \<and> Q (X m, X n)" 3379 by auto 3380 show "\<exists>Q. (\<exists>N. \<forall>n\<ge>N. Q (X n)) \<and> (\<forall>x y. Q x \<longrightarrow> Q y \<longrightarrow> P (x, y))" 3381 proof (safe intro!: exI[of _ "\<lambda>x. \<forall>n\<ge>N. Q (x, X n) \<and> Q (X n, x)"] exI[of _ N] N) 3382 fix x y 3383 assume "\<forall>n\<ge>N. Q (x, X n) \<and> Q (X n, x)" "\<forall>n\<ge>N. Q (y, X n) \<and> Q (X n, y)" 3384 then have "Q (x, X N)" "Q (X N, y)" by auto 3385 then show "P (x, y)" 3386 by (rule Q) 3387 qed 3388 } 3389qed 3390 3391lemma nhds_imp_cauchy_filter: 3392 assumes *: "F \<le> nhds x" 3393 shows "cauchy_filter F" 3394proof - 3395 have "F \<times>\<^sub>F F \<le> nhds x \<times>\<^sub>F nhds x" 3396 by (intro prod_filter_mono *) 3397 also have "\<dots> \<le> uniformity" 3398 unfolding le_filter_def eventually_nhds_uniformity eventually_prod_same 3399 proof safe 3400 fix P 3401 assume "eventually P uniformity" 3402 then obtain Ql where ev: "eventually Ql uniformity" 3403 and "Ql (x, y) \<Longrightarrow> Ql (y, z) \<Longrightarrow> P (x, z)" for x y z 3404 by (rule uniformity_transE) simp 3405 with ev[THEN uniformity_sym] 3406 show "\<exists>Q. eventually (\<lambda>(x', y). x' = x \<longrightarrow> Q y) uniformity \<and> 3407 (\<forall>x y. Q x \<longrightarrow> Q y \<longrightarrow> P (x, y))" 3408 by (rule_tac exI[of _ "\<lambda>y. Ql (y, x) \<and> Ql (x, y)"]) (fastforce elim: eventually_elim2) 3409 qed 3410 finally show ?thesis 3411 by (simp add: cauchy_filter_def) 3412qed 3413 3414lemma LIMSEQ_imp_Cauchy: "X \<longlonglongrightarrow> x \<Longrightarrow> Cauchy X" 3415 unfolding Cauchy_uniform filterlim_def by (intro nhds_imp_cauchy_filter) 3416 3417lemma Cauchy_subseq_Cauchy: 3418 assumes "Cauchy X" "strict_mono f" 3419 shows "Cauchy (X \<circ> f)" 3420 unfolding Cauchy_uniform comp_def filtermap_filtermap[symmetric] cauchy_filter_def 3421 by (rule order_trans[OF _ \<open>Cauchy X\<close>[unfolded Cauchy_uniform cauchy_filter_def]]) 3422 (intro prod_filter_mono filtermap_mono filterlim_subseq[OF \<open>strict_mono f\<close>, unfolded filterlim_def]) 3423 3424lemma convergent_Cauchy: "convergent X \<Longrightarrow> Cauchy X" 3425 unfolding convergent_def by (erule exE, erule LIMSEQ_imp_Cauchy) 3426 3427definition complete :: "'a set \<Rightarrow> bool" 3428 where complete_uniform: "complete S \<longleftrightarrow> 3429 (\<forall>F \<le> principal S. F \<noteq> bot \<longrightarrow> cauchy_filter F \<longrightarrow> (\<exists>x\<in>S. F \<le> nhds x))" 3430 3431end 3432 3433 3434subsubsection \<open>Uniformly continuous functions\<close> 3435 3436definition uniformly_continuous_on :: "'a set \<Rightarrow> ('a::uniform_space \<Rightarrow> 'b::uniform_space) \<Rightarrow> bool" 3437 where uniformly_continuous_on_uniformity: "uniformly_continuous_on s f \<longleftrightarrow> 3438 (LIM (x, y) (uniformity_on s). (f x, f y) :> uniformity)" 3439 3440lemma uniformly_continuous_onD: 3441 "uniformly_continuous_on s f \<Longrightarrow> eventually E uniformity \<Longrightarrow> 3442 eventually (\<lambda>(x, y). x \<in> s \<longrightarrow> y \<in> s \<longrightarrow> E (f x, f y)) uniformity" 3443 by (simp add: uniformly_continuous_on_uniformity filterlim_iff 3444 eventually_inf_principal split_beta' mem_Times_iff imp_conjL) 3445 3446lemma uniformly_continuous_on_const[continuous_intros]: "uniformly_continuous_on s (\<lambda>x. c)" 3447 by (auto simp: uniformly_continuous_on_uniformity filterlim_iff uniformity_refl) 3448 3449lemma uniformly_continuous_on_id[continuous_intros]: "uniformly_continuous_on s (\<lambda>x. x)" 3450 by (auto simp: uniformly_continuous_on_uniformity filterlim_def) 3451 3452lemma uniformly_continuous_on_compose[continuous_intros]: 3453 "uniformly_continuous_on s g \<Longrightarrow> uniformly_continuous_on (g`s) f \<Longrightarrow> 3454 uniformly_continuous_on s (\<lambda>x. f (g x))" 3455 using filterlim_compose[of "\<lambda>(x, y). (f x, f y)" uniformity 3456 "uniformity_on (g`s)" "\<lambda>(x, y). (g x, g y)" "uniformity_on s"] 3457 by (simp add: split_beta' uniformly_continuous_on_uniformity 3458 filterlim_inf filterlim_principal eventually_inf_principal mem_Times_iff) 3459 3460lemma uniformly_continuous_imp_continuous: 3461 assumes f: "uniformly_continuous_on s f" 3462 shows "continuous_on s f" 3463 by (auto simp: filterlim_iff eventually_at_filter eventually_nhds_uniformity continuous_on_def 3464 elim: eventually_mono dest!: uniformly_continuous_onD[OF f]) 3465 3466 3467section \<open>Product Topology\<close> 3468 3469subsection \<open>Product is a topological space\<close> 3470 3471instantiation prod :: (topological_space, topological_space) topological_space 3472begin 3473 3474definition open_prod_def[code del]: 3475 "open (S :: ('a \<times> 'b) set) \<longleftrightarrow> 3476 (\<forall>x\<in>S. \<exists>A B. open A \<and> open B \<and> x \<in> A \<times> B \<and> A \<times> B \<subseteq> S)" 3477 3478lemma open_prod_elim: 3479 assumes "open S" and "x \<in> S" 3480 obtains A B where "open A" and "open B" and "x \<in> A \<times> B" and "A \<times> B \<subseteq> S" 3481 using assms unfolding open_prod_def by fast 3482 3483lemma open_prod_intro: 3484 assumes "\<And>x. x \<in> S \<Longrightarrow> \<exists>A B. open A \<and> open B \<and> x \<in> A \<times> B \<and> A \<times> B \<subseteq> S" 3485 shows "open S" 3486 using assms unfolding open_prod_def by fast 3487 3488instance 3489proof 3490 show "open (UNIV :: ('a \<times> 'b) set)" 3491 unfolding open_prod_def by auto 3492next 3493 fix S T :: "('a \<times> 'b) set" 3494 assume "open S" "open T" 3495 show "open (S \<inter> T)" 3496 proof (rule open_prod_intro) 3497 fix x 3498 assume x: "x \<in> S \<inter> T" 3499 from x have "x \<in> S" by simp 3500 obtain Sa Sb where A: "open Sa" "open Sb" "x \<in> Sa \<times> Sb" "Sa \<times> Sb \<subseteq> S" 3501 using \<open>open S\<close> and \<open>x \<in> S\<close> by (rule open_prod_elim) 3502 from x have "x \<in> T" by simp 3503 obtain Ta Tb where B: "open Ta" "open Tb" "x \<in> Ta \<times> Tb" "Ta \<times> Tb \<subseteq> T" 3504 using \<open>open T\<close> and \<open>x \<in> T\<close> by (rule open_prod_elim) 3505 let ?A = "Sa \<inter> Ta" and ?B = "Sb \<inter> Tb" 3506 have "open ?A \<and> open ?B \<and> x \<in> ?A \<times> ?B \<and> ?A \<times> ?B \<subseteq> S \<inter> T" 3507 using A B by (auto simp add: open_Int) 3508 then show "\<exists>A B. open A \<and> open B \<and> x \<in> A \<times> B \<and> A \<times> B \<subseteq> S \<inter> T" 3509 by fast 3510 qed 3511next 3512 fix K :: "('a \<times> 'b) set set" 3513 assume "\<forall>S\<in>K. open S" 3514 then show "open (\<Union>K)" 3515 unfolding open_prod_def by fast 3516qed 3517 3518end 3519 3520declare [[code abort: "open :: ('a::topological_space \<times> 'b::topological_space) set \<Rightarrow> bool"]] 3521 3522lemma open_Times: "open S \<Longrightarrow> open T \<Longrightarrow> open (S \<times> T)" 3523 unfolding open_prod_def by auto 3524 3525lemma fst_vimage_eq_Times: "fst -` S = S \<times> UNIV" 3526 by auto 3527 3528lemma snd_vimage_eq_Times: "snd -` S = UNIV \<times> S" 3529 by auto 3530 3531lemma open_vimage_fst: "open S \<Longrightarrow> open (fst -` S)" 3532 by (simp add: fst_vimage_eq_Times open_Times) 3533 3534lemma open_vimage_snd: "open S \<Longrightarrow> open (snd -` S)" 3535 by (simp add: snd_vimage_eq_Times open_Times) 3536 3537lemma closed_vimage_fst: "closed S \<Longrightarrow> closed (fst -` S)" 3538 unfolding closed_open vimage_Compl [symmetric] 3539 by (rule open_vimage_fst) 3540 3541lemma closed_vimage_snd: "closed S \<Longrightarrow> closed (snd -` S)" 3542 unfolding closed_open vimage_Compl [symmetric] 3543 by (rule open_vimage_snd) 3544 3545lemma closed_Times: "closed S \<Longrightarrow> closed T \<Longrightarrow> closed (S \<times> T)" 3546proof - 3547 have "S \<times> T = (fst -` S) \<inter> (snd -` T)" 3548 by auto 3549 then show "closed S \<Longrightarrow> closed T \<Longrightarrow> closed (S \<times> T)" 3550 by (simp add: closed_vimage_fst closed_vimage_snd closed_Int) 3551qed 3552 3553lemma subset_fst_imageI: "A \<times> B \<subseteq> S \<Longrightarrow> y \<in> B \<Longrightarrow> A \<subseteq> fst ` S" 3554 unfolding image_def subset_eq by force 3555 3556lemma subset_snd_imageI: "A \<times> B \<subseteq> S \<Longrightarrow> x \<in> A \<Longrightarrow> B \<subseteq> snd ` S" 3557 unfolding image_def subset_eq by force 3558 3559lemma open_image_fst: 3560 assumes "open S" 3561 shows "open (fst ` S)" 3562proof (rule openI) 3563 fix x 3564 assume "x \<in> fst ` S" 3565 then obtain y where "(x, y) \<in> S" 3566 by auto 3567 then obtain A B where "open A" "open B" "x \<in> A" "y \<in> B" "A \<times> B \<subseteq> S" 3568 using \<open>open S\<close> unfolding open_prod_def by auto 3569 from \<open>A \<times> B \<subseteq> S\<close> \<open>y \<in> B\<close> have "A \<subseteq> fst ` S" 3570 by (rule subset_fst_imageI) 3571 with \<open>open A\<close> \<open>x \<in> A\<close> have "open A \<and> x \<in> A \<and> A \<subseteq> fst ` S" 3572 by simp 3573 then show "\<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> fst ` S" .. 3574qed 3575 3576lemma open_image_snd: 3577 assumes "open S" 3578 shows "open (snd ` S)" 3579proof (rule openI) 3580 fix y 3581 assume "y \<in> snd ` S" 3582 then obtain x where "(x, y) \<in> S" 3583 by auto 3584 then obtain A B where "open A" "open B" "x \<in> A" "y \<in> B" "A \<times> B \<subseteq> S" 3585 using \<open>open S\<close> unfolding open_prod_def by auto 3586 from \<open>A \<times> B \<subseteq> S\<close> \<open>x \<in> A\<close> have "B \<subseteq> snd ` S" 3587 by (rule subset_snd_imageI) 3588 with \<open>open B\<close> \<open>y \<in> B\<close> have "open B \<and> y \<in> B \<and> B \<subseteq> snd ` S" 3589 by simp 3590 then show "\<exists>T. open T \<and> y \<in> T \<and> T \<subseteq> snd ` S" .. 3591qed 3592 3593lemma nhds_prod: "nhds (a, b) = nhds a \<times>\<^sub>F nhds b" 3594 unfolding nhds_def 3595proof (subst prod_filter_INF, auto intro!: antisym INF_greatest simp: principal_prod_principal) 3596 fix S T 3597 assume "open S" "a \<in> S" "open T" "b \<in> T" 3598 then show "(INF x \<in> {S. open S \<and> (a, b) \<in> S}. principal x) \<le> principal (S \<times> T)" 3599 by (intro INF_lower) (auto intro!: open_Times) 3600next 3601 fix S' 3602 assume "open S'" "(a, b) \<in> S'" 3603 then obtain S T where "open S" "a \<in> S" "open T" "b \<in> T" "S \<times> T \<subseteq> S'" 3604 by (auto elim: open_prod_elim) 3605 then show "(INF x \<in> {S. open S \<and> a \<in> S}. INF y \<in> {S. open S \<and> b \<in> S}. 3606 principal (x \<times> y)) \<le> principal S'" 3607 by (auto intro!: INF_lower2) 3608qed 3609 3610 3611subsubsection \<open>Continuity of operations\<close> 3612 3613lemma tendsto_fst [tendsto_intros]: 3614 assumes "(f \<longlongrightarrow> a) F" 3615 shows "((\<lambda>x. fst (f x)) \<longlongrightarrow> fst a) F" 3616proof (rule topological_tendstoI) 3617 fix S 3618 assume "open S" and "fst a \<in> S" 3619 then have "open (fst -` S)" and "a \<in> fst -` S" 3620 by (simp_all add: open_vimage_fst) 3621 with assms have "eventually (\<lambda>x. f x \<in> fst -` S) F" 3622 by (rule topological_tendstoD) 3623 then show "eventually (\<lambda>x. fst (f x) \<in> S) F" 3624 by simp 3625qed 3626 3627lemma tendsto_snd [tendsto_intros]: 3628 assumes "(f \<longlongrightarrow> a) F" 3629 shows "((\<lambda>x. snd (f x)) \<longlongrightarrow> snd a) F" 3630proof (rule topological_tendstoI) 3631 fix S 3632 assume "open S" and "snd a \<in> S" 3633 then have "open (snd -` S)" and "a \<in> snd -` S" 3634 by (simp_all add: open_vimage_snd) 3635 with assms have "eventually (\<lambda>x. f x \<in> snd -` S) F" 3636 by (rule topological_tendstoD) 3637 then show "eventually (\<lambda>x. snd (f x) \<in> S) F" 3638 by simp 3639qed 3640 3641lemma tendsto_Pair [tendsto_intros]: 3642 assumes "(f \<longlongrightarrow> a) F" and "(g \<longlongrightarrow> b) F" 3643 shows "((\<lambda>x. (f x, g x)) \<longlongrightarrow> (a, b)) F" 3644 unfolding nhds_prod using assms by (rule filterlim_Pair) 3645 3646lemma continuous_fst[continuous_intros]: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. fst (f x))" 3647 unfolding continuous_def by (rule tendsto_fst) 3648 3649lemma continuous_snd[continuous_intros]: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. snd (f x))" 3650 unfolding continuous_def by (rule tendsto_snd) 3651 3652lemma continuous_Pair[continuous_intros]: 3653 "continuous F f \<Longrightarrow> continuous F g \<Longrightarrow> continuous F (\<lambda>x. (f x, g x))" 3654 unfolding continuous_def by (rule tendsto_Pair) 3655 3656lemma continuous_on_fst[continuous_intros]: 3657 "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. fst (f x))" 3658 unfolding continuous_on_def by (auto intro: tendsto_fst) 3659 3660lemma continuous_on_snd[continuous_intros]: 3661 "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. snd (f x))" 3662 unfolding continuous_on_def by (auto intro: tendsto_snd) 3663 3664lemma continuous_on_Pair[continuous_intros]: 3665 "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. (f x, g x))" 3666 unfolding continuous_on_def by (auto intro: tendsto_Pair) 3667 3668lemma continuous_on_swap[continuous_intros]: "continuous_on A prod.swap" 3669 by (simp add: prod.swap_def continuous_on_fst continuous_on_snd 3670 continuous_on_Pair continuous_on_id) 3671 3672lemma continuous_on_swap_args: 3673 assumes "continuous_on (A\<times>B) (\<lambda>(x,y). d x y)" 3674 shows "continuous_on (B\<times>A) (\<lambda>(x,y). d y x)" 3675proof - 3676 have "(\<lambda>(x,y). d y x) = (\<lambda>(x,y). d x y) \<circ> prod.swap" 3677 by force 3678 then show ?thesis 3679 by (metis assms continuous_on_compose continuous_on_swap product_swap) 3680qed 3681 3682lemma isCont_fst [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. fst (f x)) a" 3683 by (fact continuous_fst) 3684 3685lemma isCont_snd [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. snd (f x)) a" 3686 by (fact continuous_snd) 3687 3688lemma isCont_Pair [simp]: "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. (f x, g x)) a" 3689 by (fact continuous_Pair) 3690 3691lemma continuous_on_compose_Pair: 3692 assumes f: "continuous_on (Sigma A B) (\<lambda>(a, b). f a b)" 3693 assumes g: "continuous_on C g" 3694 assumes h: "continuous_on C h" 3695 assumes subset: "\<And>c. c \<in> C \<Longrightarrow> g c \<in> A" "\<And>c. c \<in> C \<Longrightarrow> h c \<in> B (g c)" 3696 shows "continuous_on C (\<lambda>c. f (g c) (h c))" 3697 using continuous_on_compose2[OF f continuous_on_Pair[OF g h]] subset 3698 by auto 3699 3700 3701subsubsection \<open>Connectedness of products\<close> 3702 3703proposition connected_Times: 3704 assumes S: "connected S" and T: "connected T" 3705 shows "connected (S \<times> T)" 3706proof (rule connectedI_const) 3707 fix P::"'a \<times> 'b \<Rightarrow> bool" 3708 assume P[THEN continuous_on_compose2, continuous_intros]: "continuous_on (S \<times> T) P" 3709 have "continuous_on S (\<lambda>s. P (s, t))" if "t \<in> T" for t 3710 by (auto intro!: continuous_intros that) 3711 from connectedD_const[OF S this] 3712 obtain c1 where c1: "\<And>s t. t \<in> T \<Longrightarrow> s \<in> S \<Longrightarrow> P (s, t) = c1 t" 3713 by metis 3714 moreover 3715 have "continuous_on T (\<lambda>t. P (s, t))" if "s \<in> S" for s 3716 by (auto intro!: continuous_intros that) 3717 from connectedD_const[OF T this] 3718 obtain c2 where "\<And>s t. t \<in> T \<Longrightarrow> s \<in> S \<Longrightarrow> P (s, t) = c2 s" 3719 by metis 3720 ultimately show "\<exists>c. \<forall>s\<in>S \<times> T. P s = c" 3721 by auto 3722qed 3723 3724corollary connected_Times_eq [simp]: 3725 "connected (S \<times> T) \<longleftrightarrow> S = {} \<or> T = {} \<or> connected S \<and> connected T" (is "?lhs = ?rhs") 3726proof 3727 assume L: ?lhs 3728 show ?rhs 3729 proof cases 3730 assume "S \<noteq> {} \<and> T \<noteq> {}" 3731 moreover 3732 have "connected (fst ` (S \<times> T))" "connected (snd ` (S \<times> T))" 3733 using continuous_on_fst continuous_on_snd continuous_on_id 3734 by (blast intro: connected_continuous_image [OF _ L])+ 3735 ultimately show ?thesis 3736 by auto 3737 qed auto 3738qed (auto simp: connected_Times) 3739 3740 3741subsubsection \<open>Separation axioms\<close> 3742 3743instance prod :: (t0_space, t0_space) t0_space 3744proof 3745 fix x y :: "'a \<times> 'b" 3746 assume "x \<noteq> y" 3747 then have "fst x \<noteq> fst y \<or> snd x \<noteq> snd y" 3748 by (simp add: prod_eq_iff) 3749 then show "\<exists>U. open U \<and> (x \<in> U) \<noteq> (y \<in> U)" 3750 by (fast dest: t0_space elim: open_vimage_fst open_vimage_snd) 3751qed 3752 3753instance prod :: (t1_space, t1_space) t1_space 3754proof 3755 fix x y :: "'a \<times> 'b" 3756 assume "x \<noteq> y" 3757 then have "fst x \<noteq> fst y \<or> snd x \<noteq> snd y" 3758 by (simp add: prod_eq_iff) 3759 then show "\<exists>U. open U \<and> x \<in> U \<and> y \<notin> U" 3760 by (fast dest: t1_space elim: open_vimage_fst open_vimage_snd) 3761qed 3762 3763instance prod :: (t2_space, t2_space) t2_space 3764proof 3765 fix x y :: "'a \<times> 'b" 3766 assume "x \<noteq> y" 3767 then have "fst x \<noteq> fst y \<or> snd x \<noteq> snd y" 3768 by (simp add: prod_eq_iff) 3769 then show "\<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {}" 3770 by (fast dest: hausdorff elim: open_vimage_fst open_vimage_snd) 3771qed 3772 3773lemma isCont_swap[continuous_intros]: "isCont prod.swap a" 3774 using continuous_on_eq_continuous_within continuous_on_swap by blast 3775 3776lemma open_diagonal_complement: 3777 "open {(x,y) |x y. x \<noteq> (y::('a::t2_space))}" 3778proof - 3779 have "open {(x, y). x \<noteq> (y::'a)}" 3780 unfolding split_def by (intro open_Collect_neq continuous_intros) 3781 also have "{(x, y). x \<noteq> (y::'a)} = {(x, y) |x y. x \<noteq> (y::'a)}" 3782 by auto 3783 finally show ?thesis . 3784qed 3785 3786lemma closed_diagonal: 3787 "closed {y. \<exists> x::('a::t2_space). y = (x,x)}" 3788proof - 3789 have "{y. \<exists> x::'a. y = (x,x)} = UNIV - {(x,y) | x y. x \<noteq> y}" by auto 3790 then show ?thesis using open_diagonal_complement closed_Diff by auto 3791qed 3792 3793lemma open_superdiagonal: 3794 "open {(x,y) | x y. x > (y::'a::{linorder_topology})}" 3795proof - 3796 have "open {(x, y). x > (y::'a)}" 3797 unfolding split_def by (intro open_Collect_less continuous_intros) 3798 also have "{(x, y). x > (y::'a)} = {(x, y) |x y. x > (y::'a)}" 3799 by auto 3800 finally show ?thesis . 3801qed 3802 3803lemma closed_subdiagonal: 3804 "closed {(x,y) | x y. x \<le> (y::'a::{linorder_topology})}" 3805proof - 3806 have "{(x,y) | x y. x \<le> (y::'a)} = UNIV - {(x,y) | x y. x > (y::'a)}" by auto 3807 then show ?thesis using open_superdiagonal closed_Diff by auto 3808qed 3809 3810lemma open_subdiagonal: 3811 "open {(x,y) | x y. x < (y::'a::{linorder_topology})}" 3812proof - 3813 have "open {(x, y). x < (y::'a)}" 3814 unfolding split_def by (intro open_Collect_less continuous_intros) 3815 also have "{(x, y). x < (y::'a)} = {(x, y) |x y. x < (y::'a)}" 3816 by auto 3817 finally show ?thesis . 3818qed 3819 3820lemma closed_superdiagonal: 3821 "closed {(x,y) | x y. x \<ge> (y::('a::{linorder_topology}))}" 3822proof - 3823 have "{(x,y) | x y. x \<ge> (y::'a)} = UNIV - {(x,y) | x y. x < y}" by auto 3824 then show ?thesis using open_subdiagonal closed_Diff by auto 3825qed 3826 3827end 3828