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