src/HOL/Topological_Spaces.thy
 author wenzelm Sun Sep 18 20:33:48 2016 +0200 (2016-09-18) changeset 63915 bab633745c7f parent 63713 009e176e1010 child 63952 354808e9f44b permissions -rw-r--r--
tuned proofs;
```     1 (*  Title:      HOL/Topological_Spaces.thy
```
```     2     Author:     Brian Huffman
```
```     3     Author:     Johannes Hölzl
```
```     4 *)
```
```     5
```
```     6 section \<open>Topological Spaces\<close>
```
```     7
```
```     8 theory Topological_Spaces
```
```     9   imports Main
```
```    10 begin
```
```    11
```
```    12 named_theorems continuous_intros "structural introduction rules for continuity"
```
```    13
```
```    14 subsection \<open>Topological space\<close>
```
```    15
```
```    16 class "open" =
```
```    17   fixes "open" :: "'a set \<Rightarrow> bool"
```
```    18
```
```    19 class 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)"
```
```    23 begin
```
```    24
```
```    25 definition closed :: "'a set \<Rightarrow> bool"
```
```    26   where "closed S \<longleftrightarrow> open (- S)"
```
```    27
```
```    28 lemma open_empty [continuous_intros, intro, simp]: "open {}"
```
```    29   using open_Union [of "{}"] by simp
```
```    30
```
```    31 lemma 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
```
```    34 lemma 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
```
```    37 lemma 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
```
```    40 lemma 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
```
```    43 lemma 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"
```
```    46 proof -
```
```    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
```
```    50 qed
```
```    51
```
```    52 lemma closed_empty [continuous_intros, intro, simp]: "closed {}"
```
```    53   unfolding closed_def by simp
```
```    54
```
```    55 lemma closed_Un [continuous_intros, intro]: "closed S \<Longrightarrow> closed T \<Longrightarrow> closed (S \<union> T)"
```
```    56   unfolding closed_def by auto
```
```    57
```
```    58 lemma closed_UNIV [continuous_intros, intro, simp]: "closed UNIV"
```
```    59   unfolding closed_def by simp
```
```    60
```
```    61 lemma closed_Int [continuous_intros, intro]: "closed S \<Longrightarrow> closed T \<Longrightarrow> closed (S \<inter> T)"
```
```    62   unfolding closed_def by auto
```
```    63
```
```    64 lemma 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
```
```    67 lemma closed_Inter [continuous_intros, intro]: "\<forall>S\<in>K. closed S \<Longrightarrow> closed (\<Inter>K)"
```
```    68   unfolding closed_def uminus_Inf by auto
```
```    69
```
```    70 lemma 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
```
```    73 lemma 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
```
```    77 lemma open_closed: "open S \<longleftrightarrow> closed (- S)"
```
```    78   by (simp add: closed_def)
```
```    79
```
```    80 lemma closed_open: "closed S \<longleftrightarrow> open (- S)"
```
```    81   by (rule closed_def)
```
```    82
```
```    83 lemma 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
```
```    86 lemma 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
```
```    89 lemma open_Compl [continuous_intros, intro]: "closed S \<Longrightarrow> open (- S)"
```
```    90   by (simp add: closed_open)
```
```    91
```
```    92 lemma closed_Compl [continuous_intros, intro]: "open S \<Longrightarrow> closed (- S)"
```
```    93   by (simp add: open_closed)
```
```    94
```
```    95 lemma open_Collect_neg: "closed {x. P x} \<Longrightarrow> open {x. \<not> P x}"
```
```    96   unfolding Collect_neg_eq by (rule open_Compl)
```
```    97
```
```    98 lemma 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
```
```   103 lemma 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
```
```   108 lemma 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
```
```   111 lemma 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
```
```   114 lemma open_Collect_const: "open {x. P}"
```
```   115   by (cases P) auto
```
```   116
```
```   117 lemma closed_Collect_neg: "open {x. P x} \<Longrightarrow> closed {x. \<not> P x}"
```
```   118   unfolding Collect_neg_eq by (rule closed_Compl)
```
```   119
```
```   120 lemma 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
```
```   125 lemma 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
```
```   130 lemma 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
```
```   133 lemma 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
```
```   136 lemma closed_Collect_const: "closed {x. P}"
```
```   137   by (cases P) auto
```
```   138
```
```   139 end
```
```   140
```
```   141
```
```   142 subsection \<open>Hausdorff and other separation properties\<close>
```
```   143
```
```   144 class 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
```
```   147 class 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
```
```   150 instance t1_space \<subseteq> t0_space
```
```   151   by standard (fast dest: t1_space)
```
```   152
```
```   153 lemma separation_t1: "x \<noteq> y \<longleftrightarrow> (\<exists>U. open U \<and> x \<in> U \<and> y \<notin> U)"
```
```   154   for x y :: "'a::t1_space"
```
```   155   using t1_space[of x y] by blast
```
```   156
```
```   157 lemma closed_singleton [iff]: "closed {a}"
```
```   158   for a :: "'a::t1_space"
```
```   159 proof -
```
```   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)
```
```   167 qed
```
```   168
```
```   169 lemma closed_insert [continuous_intros, simp]:
```
```   170   fixes a :: "'a::t1_space"
```
```   171   assumes "closed S"
```
```   172   shows "closed (insert a S)"
```
```   173 proof -
```
```   174   from closed_singleton assms have "closed ({a} \<union> S)"
```
```   175     by (rule closed_Un)
```
```   176   then show "closed (insert a S)"
```
```   177     by simp
```
```   178 qed
```
```   179
```
```   180 lemma finite_imp_closed: "finite S \<Longrightarrow> closed S"
```
```   181   for S :: "'a::t1_space set"
```
```   182   by (induct pred: finite) simp_all
```
```   183
```
```   184
```
```   185 text \<open>T2 spaces are also known as Hausdorff spaces.\<close>
```
```   186
```
```   187 class t2_space = topological_space +
```
```   188   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 = {}"
```
```   189
```
```   190 instance t2_space \<subseteq> t1_space
```
```   191   by standard (fast dest: hausdorff)
```
```   192
```
```   193 lemma 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 = {})"
```
```   194   for x y :: "'a::t2_space"
```
```   195   using hausdorff [of x y] by blast
```
```   196
```
```   197 lemma separation_t0: "x \<noteq> y \<longleftrightarrow> (\<exists>U. open U \<and> \<not> (x \<in> U \<longleftrightarrow> y \<in> U))"
```
```   198   for x y :: "'a::t0_space"
```
```   199   using t0_space [of x y] by blast
```
```   200
```
```   201
```
```   202 text \<open>A perfect space is a topological space with no isolated points.\<close>
```
```   203
```
```   204 class perfect_space = topological_space +
```
```   205   assumes not_open_singleton: "\<not> open {x}"
```
```   206
```
```   207 lemma UNIV_not_singleton: "UNIV \<noteq> {x}"
```
```   208   for x :: "'a::perfect_space"
```
```   209   by (metis open_UNIV not_open_singleton)
```
```   210
```
```   211
```
```   212 subsection \<open>Generators for toplogies\<close>
```
```   213
```
```   214 inductive generate_topology :: "'a set set \<Rightarrow> 'a set \<Rightarrow> bool" for S :: "'a set set"
```
```   215   where
```
```   216     UNIV: "generate_topology S UNIV"
```
```   217   | Int: "generate_topology S (a \<inter> b)" if "generate_topology S a" and "generate_topology S b"
```
```   218   | UN: "generate_topology S (\<Union>K)" if "(\<And>k. k \<in> K \<Longrightarrow> generate_topology S k)"
```
```   219   | Basis: "generate_topology S s" if "s \<in> S"
```
```   220
```
```   221 hide_fact (open) UNIV Int UN Basis
```
```   222
```
```   223 lemma generate_topology_Union:
```
```   224   "(\<And>k. k \<in> I \<Longrightarrow> generate_topology S (K k)) \<Longrightarrow> generate_topology S (\<Union>k\<in>I. K k)"
```
```   225   using generate_topology.UN [of "K ` I"] by auto
```
```   226
```
```   227 lemma topological_space_generate_topology: "class.topological_space (generate_topology S)"
```
```   228   by standard (auto intro: generate_topology.intros)
```
```   229
```
```   230
```
```   231 subsection \<open>Order topologies\<close>
```
```   232
```
```   233 class order_topology = order + "open" +
```
```   234   assumes open_generated_order: "open = generate_topology (range (\<lambda>a. {..< a}) \<union> range (\<lambda>a. {a <..}))"
```
```   235 begin
```
```   236
```
```   237 subclass topological_space
```
```   238   unfolding open_generated_order
```
```   239   by (rule topological_space_generate_topology)
```
```   240
```
```   241 lemma open_greaterThan [continuous_intros, simp]: "open {a <..}"
```
```   242   unfolding open_generated_order by (auto intro: generate_topology.Basis)
```
```   243
```
```   244 lemma open_lessThan [continuous_intros, simp]: "open {..< a}"
```
```   245   unfolding open_generated_order by (auto intro: generate_topology.Basis)
```
```   246
```
```   247 lemma open_greaterThanLessThan [continuous_intros, simp]: "open {a <..< b}"
```
```   248    unfolding greaterThanLessThan_eq by (simp add: open_Int)
```
```   249
```
```   250 end
```
```   251
```
```   252 class linorder_topology = linorder + order_topology
```
```   253
```
```   254 lemma closed_atMost [continuous_intros, simp]: "closed {..a}"
```
```   255   for a :: "'a::linorder_topology"
```
```   256   by (simp add: closed_open)
```
```   257
```
```   258 lemma closed_atLeast [continuous_intros, simp]: "closed {a..}"
```
```   259   for a :: "'a::linorder_topology"
```
```   260   by (simp add: closed_open)
```
```   261
```
```   262 lemma closed_atLeastAtMost [continuous_intros, simp]: "closed {a..b}"
```
```   263   for a b :: "'a::linorder_topology"
```
```   264 proof -
```
```   265   have "{a .. b} = {a ..} \<inter> {.. b}"
```
```   266     by auto
```
```   267   then show ?thesis
```
```   268     by (simp add: closed_Int)
```
```   269 qed
```
```   270
```
```   271 lemma (in linorder) less_separate:
```
```   272   assumes "x < y"
```
```   273   shows "\<exists>a b. x \<in> {..< a} \<and> y \<in> {b <..} \<and> {..< a} \<inter> {b <..} = {}"
```
```   274 proof (cases "\<exists>z. x < z \<and> z < y")
```
```   275   case True
```
```   276   then obtain z where "x < z \<and> z < y" ..
```
```   277   then have "x \<in> {..< z} \<and> y \<in> {z <..} \<and> {z <..} \<inter> {..< z} = {}"
```
```   278     by auto
```
```   279   then show ?thesis by blast
```
```   280 next
```
```   281   case False
```
```   282   with \<open>x < y\<close> have "x \<in> {..< y}" "y \<in> {x <..}" "{x <..} \<inter> {..< y} = {}"
```
```   283     by auto
```
```   284   then show ?thesis by blast
```
```   285 qed
```
```   286
```
```   287 instance linorder_topology \<subseteq> t2_space
```
```   288 proof
```
```   289   fix x y :: 'a
```
```   290   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 = {}"
```
```   291     using less_separate [of x y] less_separate [of y x]
```
```   292     by (elim neqE; metis open_lessThan open_greaterThan Int_commute)
```
```   293 qed
```
```   294
```
```   295 lemma (in linorder_topology) open_right:
```
```   296   assumes "open S" "x \<in> S"
```
```   297     and gt_ex: "x < y"
```
```   298   shows "\<exists>b>x. {x ..< b} \<subseteq> S"
```
```   299   using assms unfolding open_generated_order
```
```   300 proof induct
```
```   301   case UNIV
```
```   302   then show ?case by blast
```
```   303 next
```
```   304   case (Int A B)
```
```   305   then obtain a b where "a > x" "{x ..< a} \<subseteq> A"  "b > x" "{x ..< b} \<subseteq> B"
```
```   306     by auto
```
```   307   then show ?case
```
```   308     by (auto intro!: exI[of _ "min a b"])
```
```   309 next
```
```   310   case UN
```
```   311   then show ?case by blast
```
```   312 next
```
```   313   case Basis
```
```   314   then show ?case
```
```   315     by (fastforce intro: exI[of _ y] gt_ex)
```
```   316 qed
```
```   317
```
```   318 lemma (in linorder_topology) open_left:
```
```   319   assumes "open S" "x \<in> S"
```
```   320     and lt_ex: "y < x"
```
```   321   shows "\<exists>b<x. {b <.. x} \<subseteq> S"
```
```   322   using assms unfolding open_generated_order
```
```   323 proof induction
```
```   324   case UNIV
```
```   325   then show ?case by blast
```
```   326 next
```
```   327   case (Int A B)
```
```   328   then obtain a b where "a < x" "{a <.. x} \<subseteq> A"  "b < x" "{b <.. x} \<subseteq> B"
```
```   329     by auto
```
```   330   then show ?case
```
```   331     by (auto intro!: exI[of _ "max a b"])
```
```   332 next
```
```   333   case UN
```
```   334   then show ?case by blast
```
```   335 next
```
```   336   case Basis
```
```   337   then show ?case
```
```   338     by (fastforce intro: exI[of _ y] lt_ex)
```
```   339 qed
```
```   340
```
```   341
```
```   342 subsection \<open>Setup some topologies\<close>
```
```   343
```
```   344 subsubsection \<open>Boolean is an order topology\<close>
```
```   345
```
```   346 class discrete_topology = topological_space +
```
```   347   assumes open_discrete: "\<And>A. open A"
```
```   348
```
```   349 instance discrete_topology < t2_space
```
```   350 proof
```
```   351   fix x y :: 'a
```
```   352   assume "x \<noteq> y"
```
```   353   then show "\<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {}"
```
```   354     by (intro exI[of _ "{_}"]) (auto intro!: open_discrete)
```
```   355 qed
```
```   356
```
```   357 instantiation bool :: linorder_topology
```
```   358 begin
```
```   359
```
```   360 definition open_bool :: "bool set \<Rightarrow> bool"
```
```   361   where "open_bool = generate_topology (range (\<lambda>a. {..< a}) \<union> range (\<lambda>a. {a <..}))"
```
```   362
```
```   363 instance
```
```   364   by standard (rule open_bool_def)
```
```   365
```
```   366 end
```
```   367
```
```   368 instance bool :: discrete_topology
```
```   369 proof
```
```   370   fix A :: "bool set"
```
```   371   have *: "{False <..} = {True}" "{..< True} = {False}"
```
```   372     by auto
```
```   373   have "A = UNIV \<or> A = {} \<or> A = {False <..} \<or> A = {..< True}"
```
```   374     using subset_UNIV[of A] unfolding UNIV_bool * by blast
```
```   375   then show "open A"
```
```   376     by auto
```
```   377 qed
```
```   378
```
```   379 instantiation nat :: linorder_topology
```
```   380 begin
```
```   381
```
```   382 definition open_nat :: "nat set \<Rightarrow> bool"
```
```   383   where "open_nat = generate_topology (range (\<lambda>a. {..< a}) \<union> range (\<lambda>a. {a <..}))"
```
```   384
```
```   385 instance
```
```   386   by standard (rule open_nat_def)
```
```   387
```
```   388 end
```
```   389
```
```   390 instance nat :: discrete_topology
```
```   391 proof
```
```   392   fix A :: "nat set"
```
```   393   have "open {n}" for n :: nat
```
```   394   proof (cases n)
```
```   395     case 0
```
```   396     moreover have "{0} = {..<1::nat}"
```
```   397       by auto
```
```   398     ultimately show ?thesis
```
```   399        by auto
```
```   400   next
```
```   401     case (Suc n')
```
```   402     then have "{n} = {..<Suc n} \<inter> {n' <..}"
```
```   403       by auto
```
```   404     with Suc show ?thesis
```
```   405       by (auto intro: open_lessThan open_greaterThan)
```
```   406   qed
```
```   407   then have "open (\<Union>a\<in>A. {a})"
```
```   408     by (intro open_UN) auto
```
```   409   then show "open A"
```
```   410     by simp
```
```   411 qed
```
```   412
```
```   413 instantiation int :: linorder_topology
```
```   414 begin
```
```   415
```
```   416 definition open_int :: "int set \<Rightarrow> bool"
```
```   417   where "open_int = generate_topology (range (\<lambda>a. {..< a}) \<union> range (\<lambda>a. {a <..}))"
```
```   418
```
```   419 instance
```
```   420   by standard (rule open_int_def)
```
```   421
```
```   422 end
```
```   423
```
```   424 instance int :: discrete_topology
```
```   425 proof
```
```   426   fix A :: "int set"
```
```   427   have "{..<i + 1} \<inter> {i-1 <..} = {i}" for i :: int
```
```   428     by auto
```
```   429   then have "open {i}" for i :: int
```
```   430     using open_Int[OF open_lessThan[of "i + 1"] open_greaterThan[of "i - 1"]] by auto
```
```   431   then have "open (\<Union>a\<in>A. {a})"
```
```   432     by (intro open_UN) auto
```
```   433   then show "open A"
```
```   434     by simp
```
```   435 qed
```
```   436
```
```   437
```
```   438 subsubsection \<open>Topological filters\<close>
```
```   439
```
```   440 definition (in topological_space) nhds :: "'a \<Rightarrow> 'a filter"
```
```   441   where "nhds a = (INF S:{S. open S \<and> a \<in> S}. principal S)"
```
```   442
```
```   443 definition (in topological_space) at_within :: "'a \<Rightarrow> 'a set \<Rightarrow> 'a filter"
```
```   444     ("at (_)/ within (_)" [1000, 60] 60)
```
```   445   where "at a within s = inf (nhds a) (principal (s - {a}))"
```
```   446
```
```   447 abbreviation (in topological_space) at :: "'a \<Rightarrow> 'a filter"  ("at")
```
```   448   where "at x \<equiv> at x within (CONST UNIV)"
```
```   449
```
```   450 abbreviation (in order_topology) at_right :: "'a \<Rightarrow> 'a filter"
```
```   451   where "at_right x \<equiv> at x within {x <..}"
```
```   452
```
```   453 abbreviation (in order_topology) at_left :: "'a \<Rightarrow> 'a filter"
```
```   454   where "at_left x \<equiv> at x within {..< x}"
```
```   455
```
```   456 lemma (in topological_space) nhds_generated_topology:
```
```   457   "open = generate_topology T \<Longrightarrow> nhds x = (INF S:{S\<in>T. x \<in> S}. principal S)"
```
```   458   unfolding nhds_def
```
```   459 proof (safe intro!: antisym INF_greatest)
```
```   460   fix S
```
```   461   assume "generate_topology T S" "x \<in> S"
```
```   462   then show "(INF S:{S \<in> T. x \<in> S}. principal S) \<le> principal S"
```
```   463     by induct
```
```   464       (auto intro: INF_lower order_trans simp: inf_principal[symmetric] simp del: inf_principal)
```
```   465 qed (auto intro!: INF_lower intro: generate_topology.intros)
```
```   466
```
```   467 lemma (in topological_space) eventually_nhds:
```
```   468   "eventually P (nhds a) \<longleftrightarrow> (\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x))"
```
```   469   unfolding nhds_def by (subst eventually_INF_base) (auto simp: eventually_principal)
```
```   470
```
```   471 lemma (in topological_space) eventually_nhds_in_open:
```
```   472   "open s \<Longrightarrow> x \<in> s \<Longrightarrow> eventually (\<lambda>y. y \<in> s) (nhds x)"
```
```   473   by (subst eventually_nhds) blast
```
```   474
```
```   475 lemma eventually_nhds_x_imp_x: "eventually P (nhds x) \<Longrightarrow> P x"
```
```   476   by (subst (asm) eventually_nhds) blast
```
```   477
```
```   478 lemma nhds_neq_bot [simp]: "nhds a \<noteq> bot"
```
```   479   by (simp add: trivial_limit_def eventually_nhds)
```
```   480
```
```   481 lemma (in t1_space) t1_space_nhds: "x \<noteq> y \<Longrightarrow> (\<forall>\<^sub>F x in nhds x. x \<noteq> y)"
```
```   482   by (drule t1_space) (auto simp: eventually_nhds)
```
```   483
```
```   484 lemma (in topological_space) nhds_discrete_open: "open {x} \<Longrightarrow> nhds x = principal {x}"
```
```   485   by (auto simp: nhds_def intro!: antisym INF_greatest INF_lower2[of "{x}"])
```
```   486
```
```   487 lemma (in discrete_topology) nhds_discrete: "nhds x = principal {x}"
```
```   488   by (simp add: nhds_discrete_open open_discrete)
```
```   489
```
```   490 lemma (in discrete_topology) at_discrete: "at x within S = bot"
```
```   491   unfolding at_within_def nhds_discrete by simp
```
```   492
```
```   493 lemma at_within_eq: "at x within s = (INF S:{S. open S \<and> x \<in> S}. principal (S \<inter> s - {x}))"
```
```   494   unfolding nhds_def at_within_def
```
```   495   by (subst INF_inf_const2[symmetric]) (auto simp: Diff_Int_distrib)
```
```   496
```
```   497 lemma eventually_at_filter:
```
```   498   "eventually P (at a within s) \<longleftrightarrow> eventually (\<lambda>x. x \<noteq> a \<longrightarrow> x \<in> s \<longrightarrow> P x) (nhds a)"
```
```   499   by (simp add: at_within_def eventually_inf_principal imp_conjL[symmetric] conj_commute)
```
```   500
```
```   501 lemma at_le: "s \<subseteq> t \<Longrightarrow> at x within s \<le> at x within t"
```
```   502   unfolding at_within_def by (intro inf_mono) auto
```
```   503
```
```   504 lemma eventually_at_topological:
```
```   505   "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))"
```
```   506   by (simp add: eventually_nhds eventually_at_filter)
```
```   507
```
```   508 lemma at_within_open: "a \<in> S \<Longrightarrow> open S \<Longrightarrow> at a within S = at a"
```
```   509   unfolding filter_eq_iff eventually_at_topological by (metis open_Int Int_iff UNIV_I)
```
```   510
```
```   511 lemma at_within_open_NO_MATCH: "a \<in> s \<Longrightarrow> open s \<Longrightarrow> NO_MATCH UNIV s \<Longrightarrow> at a within s = at a"
```
```   512   by (simp only: at_within_open)
```
```   513
```
```   514 lemma at_within_nhd:
```
```   515   assumes "x \<in> S" "open S" "T \<inter> S - {x} = U \<inter> S - {x}"
```
```   516   shows "at x within T = at x within U"
```
```   517   unfolding filter_eq_iff eventually_at_filter
```
```   518 proof (intro allI eventually_subst)
```
```   519   have "eventually (\<lambda>x. x \<in> S) (nhds x)"
```
```   520     using \<open>x \<in> S\<close> \<open>open S\<close> by (auto simp: eventually_nhds)
```
```   521   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
```
```   522     by eventually_elim (insert \<open>T \<inter> S - {x} = U \<inter> S - {x}\<close>, blast)
```
```   523 qed
```
```   524
```
```   525 lemma at_within_empty [simp]: "at a within {} = bot"
```
```   526   unfolding at_within_def by simp
```
```   527
```
```   528 lemma at_within_union: "at x within (S \<union> T) = sup (at x within S) (at x within T)"
```
```   529   unfolding filter_eq_iff eventually_sup eventually_at_filter
```
```   530   by (auto elim!: eventually_rev_mp)
```
```   531
```
```   532 lemma at_eq_bot_iff: "at a = bot \<longleftrightarrow> open {a}"
```
```   533   unfolding trivial_limit_def eventually_at_topological
```
```   534   apply safe
```
```   535    apply (case_tac "S = {a}")
```
```   536     apply simp
```
```   537    apply fast
```
```   538   apply fast
```
```   539   done
```
```   540
```
```   541 lemma at_neq_bot [simp]: "at a \<noteq> bot"
```
```   542   for a :: "'a::perfect_space"
```
```   543   by (simp add: at_eq_bot_iff not_open_singleton)
```
```   544
```
```   545 lemma (in order_topology) nhds_order:
```
```   546   "nhds x = inf (INF a:{x <..}. principal {..< a}) (INF a:{..< x}. principal {a <..})"
```
```   547 proof -
```
```   548   have 1: "{S \<in> range lessThan \<union> range greaterThan. x \<in> S} =
```
```   549       (\<lambda>a. {..< a}) ` {x <..} \<union> (\<lambda>a. {a <..}) ` {..< x}"
```
```   550     by auto
```
```   551   show ?thesis
```
```   552     by (simp only: nhds_generated_topology[OF open_generated_order] INF_union 1 INF_image comp_def)
```
```   553 qed
```
```   554
```
```   555 lemma filterlim_at_within_If:
```
```   556   assumes "filterlim f G (at x within (A \<inter> {x. P x}))"
```
```   557     and "filterlim g G (at x within (A \<inter> {x. \<not>P x}))"
```
```   558   shows "filterlim (\<lambda>x. if P x then f x else g x) G (at x within A)"
```
```   559 proof (rule filterlim_If)
```
```   560   note assms(1)
```
```   561   also have "at x within (A \<inter> {x. P x}) = inf (nhds x) (principal (A \<inter> Collect P - {x}))"
```
```   562     by (simp add: at_within_def)
```
```   563   also have "A \<inter> Collect P - {x} = (A - {x}) \<inter> Collect P"
```
```   564     by blast
```
```   565   also have "inf (nhds x) (principal \<dots>) = inf (at x within A) (principal (Collect P))"
```
```   566     by (simp add: at_within_def inf_assoc)
```
```   567   finally show "filterlim f G (inf (at x within A) (principal (Collect P)))" .
```
```   568 next
```
```   569   note assms(2)
```
```   570   also have "at x within (A \<inter> {x. \<not> P x}) = inf (nhds x) (principal (A \<inter> {x. \<not> P x} - {x}))"
```
```   571     by (simp add: at_within_def)
```
```   572   also have "A \<inter> {x. \<not> P x} - {x} = (A - {x}) \<inter> {x. \<not> P x}"
```
```   573     by blast
```
```   574   also have "inf (nhds x) (principal \<dots>) = inf (at x within A) (principal {x. \<not> P x})"
```
```   575     by (simp add: at_within_def inf_assoc)
```
```   576   finally show "filterlim g G (inf (at x within A) (principal {x. \<not> P x}))" .
```
```   577 qed
```
```   578
```
```   579 lemma filterlim_at_If:
```
```   580   assumes "filterlim f G (at x within {x. P x})"
```
```   581     and "filterlim g G (at x within {x. \<not>P x})"
```
```   582   shows "filterlim (\<lambda>x. if P x then f x else g x) G (at x)"
```
```   583   using assms by (intro filterlim_at_within_If) simp_all
```
```   584
```
```   585 lemma (in linorder_topology) at_within_order:
```
```   586   assumes "UNIV \<noteq> {x}"
```
```   587   shows "at x within s =
```
```   588     inf (INF a:{x <..}. principal ({..< a} \<inter> s - {x}))
```
```   589         (INF a:{..< x}. principal ({a <..} \<inter> s - {x}))"
```
```   590 proof (cases "{x <..} = {}" "{..< x} = {}" rule: case_split [case_product case_split])
```
```   591   case True_True
```
```   592   have "UNIV = {..< x} \<union> {x} \<union> {x <..}"
```
```   593     by auto
```
```   594   with assms True_True show ?thesis
```
```   595     by auto
```
```   596 qed (auto simp del: inf_principal simp: at_within_def nhds_order Int_Diff
```
```   597       inf_principal[symmetric] INF_inf_const2 inf_sup_aci[where 'a="'a filter"])
```
```   598
```
```   599 lemma (in linorder_topology) at_left_eq:
```
```   600   "y < x \<Longrightarrow> at_left x = (INF a:{..< x}. principal {a <..< x})"
```
```   601   by (subst at_within_order)
```
```   602      (auto simp: greaterThan_Int_greaterThan greaterThanLessThan_eq[symmetric] min.absorb2 INF_constant
```
```   603            intro!: INF_lower2 inf_absorb2)
```
```   604
```
```   605 lemma (in linorder_topology) eventually_at_left:
```
```   606   "y < x \<Longrightarrow> eventually P (at_left x) \<longleftrightarrow> (\<exists>b<x. \<forall>y>b. y < x \<longrightarrow> P y)"
```
```   607   unfolding at_left_eq
```
```   608   by (subst eventually_INF_base) (auto simp: eventually_principal Ball_def)
```
```   609
```
```   610 lemma (in linorder_topology) at_right_eq:
```
```   611   "x < y \<Longrightarrow> at_right x = (INF a:{x <..}. principal {x <..< a})"
```
```   612   by (subst at_within_order)
```
```   613      (auto simp: lessThan_Int_lessThan greaterThanLessThan_eq[symmetric] max.absorb2 INF_constant Int_commute
```
```   614            intro!: INF_lower2 inf_absorb1)
```
```   615
```
```   616 lemma (in linorder_topology) eventually_at_right:
```
```   617   "x < y \<Longrightarrow> eventually P (at_right x) \<longleftrightarrow> (\<exists>b>x. \<forall>y>x. y < b \<longrightarrow> P y)"
```
```   618   unfolding at_right_eq
```
```   619   by (subst eventually_INF_base) (auto simp: eventually_principal Ball_def)
```
```   620
```
```   621 lemma eventually_at_right_less: "\<forall>\<^sub>F y in at_right (x::'a::{linorder_topology, no_top}). x < y"
```
```   622   using gt_ex[of x] eventually_at_right[of x] by auto
```
```   623
```
```   624 lemma trivial_limit_at_right_top: "at_right (top::_::{order_top,linorder_topology}) = bot"
```
```   625   by (auto simp: filter_eq_iff eventually_at_topological)
```
```   626
```
```   627 lemma trivial_limit_at_left_bot: "at_left (bot::_::{order_bot,linorder_topology}) = bot"
```
```   628   by (auto simp: filter_eq_iff eventually_at_topological)
```
```   629
```
```   630 lemma trivial_limit_at_left_real [simp]: "\<not> trivial_limit (at_left x)"
```
```   631   for x :: "'a::{no_bot,dense_order,linorder_topology}"
```
```   632   using lt_ex [of x]
```
```   633   by safe (auto simp add: trivial_limit_def eventually_at_left dest: dense)
```
```   634
```
```   635 lemma trivial_limit_at_right_real [simp]: "\<not> trivial_limit (at_right x)"
```
```   636   for x :: "'a::{no_top,dense_order,linorder_topology}"
```
```   637   using gt_ex[of x]
```
```   638   by safe (auto simp add: trivial_limit_def eventually_at_right dest: dense)
```
```   639
```
```   640 lemma at_eq_sup_left_right: "at x = sup (at_left x) (at_right x)"
```
```   641   for x :: "'a::linorder_topology"
```
```   642   by (auto simp: eventually_at_filter filter_eq_iff eventually_sup
```
```   643       elim: eventually_elim2 eventually_mono)
```
```   644
```
```   645 lemma eventually_at_split:
```
```   646   "eventually P (at x) \<longleftrightarrow> eventually P (at_left x) \<and> eventually P (at_right x)"
```
```   647   for x :: "'a::linorder_topology"
```
```   648   by (subst at_eq_sup_left_right) (simp add: eventually_sup)
```
```   649
```
```   650 lemma eventually_at_leftI:
```
```   651   assumes "\<And>x. x \<in> {a<..<b} \<Longrightarrow> P x" "a < b"
```
```   652   shows   "eventually P (at_left b)"
```
```   653   using assms unfolding eventually_at_topological by (intro exI[of _ "{a<..}"]) auto
```
```   654
```
```   655 lemma eventually_at_rightI:
```
```   656   assumes "\<And>x. x \<in> {a<..<b} \<Longrightarrow> P x" "a < b"
```
```   657   shows   "eventually P (at_right a)"
```
```   658   using assms unfolding eventually_at_topological by (intro exI[of _ "{..<b}"]) auto
```
```   659
```
```   660
```
```   661 subsubsection \<open>Tendsto\<close>
```
```   662
```
```   663 abbreviation (in topological_space)
```
```   664   tendsto :: "('b \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'b filter \<Rightarrow> bool"  (infixr "\<longlongrightarrow>" 55)
```
```   665   where "(f \<longlongrightarrow> l) F \<equiv> filterlim f (nhds l) F"
```
```   666
```
```   667 definition (in t2_space) Lim :: "'f filter \<Rightarrow> ('f \<Rightarrow> 'a) \<Rightarrow> 'a"
```
```   668   where "Lim A f = (THE l. (f \<longlongrightarrow> l) A)"
```
```   669
```
```   670 lemma tendsto_eq_rhs: "(f \<longlongrightarrow> x) F \<Longrightarrow> x = y \<Longrightarrow> (f \<longlongrightarrow> y) F"
```
```   671   by simp
```
```   672
```
```   673 named_theorems tendsto_intros "introduction rules for tendsto"
```
```   674 setup \<open>
```
```   675   Global_Theory.add_thms_dynamic (@{binding tendsto_eq_intros},
```
```   676     fn context =>
```
```   677       Named_Theorems.get (Context.proof_of context) @{named_theorems tendsto_intros}
```
```   678       |> map_filter (try (fn thm => @{thm tendsto_eq_rhs} OF [thm])))
```
```   679 \<close>
```
```   680
```
```   681 lemma (in topological_space) tendsto_def:
```
```   682    "(f \<longlongrightarrow> l) F \<longleftrightarrow> (\<forall>S. open S \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) F)"
```
```   683    unfolding nhds_def filterlim_INF filterlim_principal by auto
```
```   684
```
```   685 lemma tendsto_cong: "(f \<longlongrightarrow> c) F \<longleftrightarrow> (g \<longlongrightarrow> c) F" if "eventually (\<lambda>x. f x = g x) F"
```
```   686   by (rule filterlim_cong [OF refl refl that])
```
```   687
```
```   688 lemma tendsto_mono: "F \<le> F' \<Longrightarrow> (f \<longlongrightarrow> l) F' \<Longrightarrow> (f \<longlongrightarrow> l) F"
```
```   689   unfolding tendsto_def le_filter_def by fast
```
```   690
```
```   691 lemma tendsto_within_subset: "(f \<longlongrightarrow> l) (at x within S) \<Longrightarrow> T \<subseteq> S \<Longrightarrow> (f \<longlongrightarrow> l) (at x within T)"
```
```   692   by (blast intro: tendsto_mono at_le)
```
```   693
```
```   694 lemma filterlim_at:
```
```   695   "(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"
```
```   696   by (simp add: at_within_def filterlim_inf filterlim_principal conj_commute)
```
```   697
```
```   698 lemma filterlim_at_withinI:
```
```   699   assumes "filterlim f (nhds c) F"
```
```   700   assumes "eventually (\<lambda>x. f x \<in> A - {c}) F"
```
```   701   shows   "filterlim f (at c within A) F"
```
```   702   using assms by (simp add: filterlim_at)
```
```   703
```
```   704 lemma filterlim_atI:
```
```   705   assumes "filterlim f (nhds c) F"
```
```   706   assumes "eventually (\<lambda>x. f x \<noteq> c) F"
```
```   707   shows   "filterlim f (at c) F"
```
```   708   using assms by (intro filterlim_at_withinI) simp_all
```
```   709
```
```   710 lemma (in topological_space) topological_tendstoI:
```
```   711   "(\<And>S. open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) F) \<Longrightarrow> (f \<longlongrightarrow> l) F"
```
```   712   by (auto simp: tendsto_def)
```
```   713
```
```   714 lemma (in topological_space) topological_tendstoD:
```
```   715   "(f \<longlongrightarrow> l) F \<Longrightarrow> open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) F"
```
```   716   by (auto simp: tendsto_def)
```
```   717
```
```   718 lemma (in order_topology) order_tendsto_iff:
```
```   719   "(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)"
```
```   720   by (auto simp: nhds_order filterlim_inf filterlim_INF filterlim_principal)
```
```   721
```
```   722 lemma (in order_topology) order_tendstoI:
```
```   723   "(\<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>
```
```   724     (f \<longlongrightarrow> y) F"
```
```   725   by (auto simp: order_tendsto_iff)
```
```   726
```
```   727 lemma (in order_topology) order_tendstoD:
```
```   728   assumes "(f \<longlongrightarrow> y) F"
```
```   729   shows "a < y \<Longrightarrow> eventually (\<lambda>x. a < f x) F"
```
```   730     and "y < a \<Longrightarrow> eventually (\<lambda>x. f x < a) F"
```
```   731   using assms by (auto simp: order_tendsto_iff)
```
```   732
```
```   733 lemma tendsto_bot [simp]: "(f \<longlongrightarrow> a) bot"
```
```   734   by (simp add: tendsto_def)
```
```   735
```
```   736 lemma (in linorder_topology) tendsto_max:
```
```   737   assumes X: "(X \<longlongrightarrow> x) net"
```
```   738     and Y: "(Y \<longlongrightarrow> y) net"
```
```   739   shows "((\<lambda>x. max (X x) (Y x)) \<longlongrightarrow> max x y) net"
```
```   740 proof (rule order_tendstoI)
```
```   741   fix a
```
```   742   assume "a < max x y"
```
```   743   then show "eventually (\<lambda>x. a < max (X x) (Y x)) net"
```
```   744     using order_tendstoD(1)[OF X, of a] order_tendstoD(1)[OF Y, of a]
```
```   745     by (auto simp: less_max_iff_disj elim: eventually_mono)
```
```   746 next
```
```   747   fix a
```
```   748   assume "max x y < a"
```
```   749   then show "eventually (\<lambda>x. max (X x) (Y x) < a) net"
```
```   750     using order_tendstoD(2)[OF X, of a] order_tendstoD(2)[OF Y, of a]
```
```   751     by (auto simp: eventually_conj_iff)
```
```   752 qed
```
```   753
```
```   754 lemma (in linorder_topology) tendsto_min:
```
```   755   assumes X: "(X \<longlongrightarrow> x) net"
```
```   756     and Y: "(Y \<longlongrightarrow> y) net"
```
```   757   shows "((\<lambda>x. min (X x) (Y x)) \<longlongrightarrow> min x y) net"
```
```   758 proof (rule order_tendstoI)
```
```   759   fix a
```
```   760   assume "a < min x y"
```
```   761   then show "eventually (\<lambda>x. a < min (X x) (Y x)) net"
```
```   762     using order_tendstoD(1)[OF X, of a] order_tendstoD(1)[OF Y, of a]
```
```   763     by (auto simp: eventually_conj_iff)
```
```   764 next
```
```   765   fix a
```
```   766   assume "min x y < a"
```
```   767   then show "eventually (\<lambda>x. min (X x) (Y x) < a) net"
```
```   768     using order_tendstoD(2)[OF X, of a] order_tendstoD(2)[OF Y, of a]
```
```   769     by (auto simp: min_less_iff_disj elim: eventually_mono)
```
```   770 qed
```
```   771
```
```   772 lemma tendsto_ident_at [tendsto_intros, simp, intro]: "((\<lambda>x. x) \<longlongrightarrow> a) (at a within s)"
```
```   773   by (auto simp: tendsto_def eventually_at_topological)
```
```   774
```
```   775 lemma (in topological_space) tendsto_const [tendsto_intros, simp, intro]: "((\<lambda>x. k) \<longlongrightarrow> k) F"
```
```   776   by (simp add: tendsto_def)
```
```   777
```
```   778 lemma (in t2_space) tendsto_unique:
```
```   779   assumes "F \<noteq> bot"
```
```   780     and "(f \<longlongrightarrow> a) F"
```
```   781     and "(f \<longlongrightarrow> b) F"
```
```   782   shows "a = b"
```
```   783 proof (rule ccontr)
```
```   784   assume "a \<noteq> b"
```
```   785   obtain U V where "open U" "open V" "a \<in> U" "b \<in> V" "U \<inter> V = {}"
```
```   786     using hausdorff [OF \<open>a \<noteq> b\<close>] by fast
```
```   787   have "eventually (\<lambda>x. f x \<in> U) F"
```
```   788     using \<open>(f \<longlongrightarrow> a) F\<close> \<open>open U\<close> \<open>a \<in> U\<close> by (rule topological_tendstoD)
```
```   789   moreover
```
```   790   have "eventually (\<lambda>x. f x \<in> V) F"
```
```   791     using \<open>(f \<longlongrightarrow> b) F\<close> \<open>open V\<close> \<open>b \<in> V\<close> by (rule topological_tendstoD)
```
```   792   ultimately
```
```   793   have "eventually (\<lambda>x. False) F"
```
```   794   proof eventually_elim
```
```   795     case (elim x)
```
```   796     then have "f x \<in> U \<inter> V" by simp
```
```   797     with \<open>U \<inter> V = {}\<close> show ?case by simp
```
```   798   qed
```
```   799   with \<open>\<not> trivial_limit F\<close> show "False"
```
```   800     by (simp add: trivial_limit_def)
```
```   801 qed
```
```   802
```
```   803 lemma (in t2_space) tendsto_const_iff:
```
```   804   fixes a b :: 'a
```
```   805   assumes "\<not> trivial_limit F"
```
```   806   shows "((\<lambda>x. a) \<longlongrightarrow> b) F \<longleftrightarrow> a = b"
```
```   807   by (auto intro!: tendsto_unique [OF assms tendsto_const])
```
```   808
```
```   809 lemma increasing_tendsto:
```
```   810   fixes f :: "_ \<Rightarrow> 'a::order_topology"
```
```   811   assumes bdd: "eventually (\<lambda>n. f n \<le> l) F"
```
```   812     and en: "\<And>x. x < l \<Longrightarrow> eventually (\<lambda>n. x < f n) F"
```
```   813   shows "(f \<longlongrightarrow> l) F"
```
```   814   using assms by (intro order_tendstoI) (auto elim!: eventually_mono)
```
```   815
```
```   816 lemma decreasing_tendsto:
```
```   817   fixes f :: "_ \<Rightarrow> 'a::order_topology"
```
```   818   assumes bdd: "eventually (\<lambda>n. l \<le> f n) F"
```
```   819     and en: "\<And>x. l < x \<Longrightarrow> eventually (\<lambda>n. f n < x) F"
```
```   820   shows "(f \<longlongrightarrow> l) F"
```
```   821   using assms by (intro order_tendstoI) (auto elim!: eventually_mono)
```
```   822
```
```   823 lemma tendsto_sandwich:
```
```   824   fixes f g h :: "'a \<Rightarrow> 'b::order_topology"
```
```   825   assumes ev: "eventually (\<lambda>n. f n \<le> g n) net" "eventually (\<lambda>n. g n \<le> h n) net"
```
```   826   assumes lim: "(f \<longlongrightarrow> c) net" "(h \<longlongrightarrow> c) net"
```
```   827   shows "(g \<longlongrightarrow> c) net"
```
```   828 proof (rule order_tendstoI)
```
```   829   fix a
```
```   830   show "a < c \<Longrightarrow> eventually (\<lambda>x. a < g x) net"
```
```   831     using order_tendstoD[OF lim(1), of a] ev by (auto elim: eventually_elim2)
```
```   832 next
```
```   833   fix a
```
```   834   show "c < a \<Longrightarrow> eventually (\<lambda>x. g x < a) net"
```
```   835     using order_tendstoD[OF lim(2), of a] ev by (auto elim: eventually_elim2)
```
```   836 qed
```
```   837
```
```   838 lemma limit_frequently_eq:
```
```   839   fixes c d :: "'a::t1_space"
```
```   840   assumes "F \<noteq> bot"
```
```   841     and "frequently (\<lambda>x. f x = c) F"
```
```   842     and "(f \<longlongrightarrow> d) F"
```
```   843   shows "d = c"
```
```   844 proof (rule ccontr)
```
```   845   assume "d \<noteq> c"
```
```   846   from t1_space[OF this] obtain U where "open U" "d \<in> U" "c \<notin> U"
```
```   847     by blast
```
```   848   with assms have "eventually (\<lambda>x. f x \<in> U) F"
```
```   849     unfolding tendsto_def by blast
```
```   850   then have "eventually (\<lambda>x. f x \<noteq> c) F"
```
```   851     by eventually_elim (insert \<open>c \<notin> U\<close>, blast)
```
```   852   with assms(2) show False
```
```   853     unfolding frequently_def by contradiction
```
```   854 qed
```
```   855
```
```   856 lemma tendsto_imp_eventually_ne:
```
```   857   fixes c :: "'a::t1_space"
```
```   858   assumes "F \<noteq> bot" "(f \<longlongrightarrow> c) F" "c \<noteq> c'"
```
```   859   shows "eventually (\<lambda>z. f z \<noteq> c') F"
```
```   860 proof (rule ccontr)
```
```   861   assume "\<not> eventually (\<lambda>z. f z \<noteq> c') F"
```
```   862   then have "frequently (\<lambda>z. f z = c') F"
```
```   863     by (simp add: frequently_def)
```
```   864   from limit_frequently_eq[OF assms(1) this assms(2)] and assms(3) show False
```
```   865     by contradiction
```
```   866 qed
```
```   867
```
```   868 lemma tendsto_le:
```
```   869   fixes f g :: "'a \<Rightarrow> 'b::linorder_topology"
```
```   870   assumes F: "\<not> trivial_limit F"
```
```   871     and x: "(f \<longlongrightarrow> x) F"
```
```   872     and y: "(g \<longlongrightarrow> y) F"
```
```   873     and ev: "eventually (\<lambda>x. g x \<le> f x) F"
```
```   874   shows "y \<le> x"
```
```   875 proof (rule ccontr)
```
```   876   assume "\<not> y \<le> x"
```
```   877   with less_separate[of x y] obtain a b where xy: "x < a" "b < y" "{..<a} \<inter> {b<..} = {}"
```
```   878     by (auto simp: not_le)
```
```   879   then have "eventually (\<lambda>x. f x < a) F" "eventually (\<lambda>x. b < g x) F"
```
```   880     using x y by (auto intro: order_tendstoD)
```
```   881   with ev have "eventually (\<lambda>x. False) F"
```
```   882     by eventually_elim (insert xy, fastforce)
```
```   883   with F show False
```
```   884     by (simp add: eventually_False)
```
```   885 qed
```
```   886
```
```   887 lemma tendsto_le_const:
```
```   888   fixes f :: "'a \<Rightarrow> 'b::linorder_topology"
```
```   889   assumes F: "\<not> trivial_limit F"
```
```   890     and x: "(f \<longlongrightarrow> x) F"
```
```   891     and ev: "eventually (\<lambda>i. a \<le> f i) F"
```
```   892   shows "a \<le> x"
```
```   893   using F x tendsto_const ev by (rule tendsto_le)
```
```   894
```
```   895 lemma tendsto_ge_const:
```
```   896   fixes f :: "'a \<Rightarrow> 'b::linorder_topology"
```
```   897   assumes F: "\<not> trivial_limit F"
```
```   898     and x: "(f \<longlongrightarrow> x) F"
```
```   899     and ev: "eventually (\<lambda>i. a \<ge> f i) F"
```
```   900   shows "a \<ge> x"
```
```   901   by (rule tendsto_le [OF F tendsto_const x ev])
```
```   902
```
```   903
```
```   904 subsubsection \<open>Rules about @{const Lim}\<close>
```
```   905
```
```   906 lemma tendsto_Lim: "\<not> trivial_limit net \<Longrightarrow> (f \<longlongrightarrow> l) net \<Longrightarrow> Lim net f = l"
```
```   907   unfolding Lim_def using tendsto_unique [of net f] by auto
```
```   908
```
```   909 lemma Lim_ident_at: "\<not> trivial_limit (at x within s) \<Longrightarrow> Lim (at x within s) (\<lambda>x. x) = x"
```
```   910   by (rule tendsto_Lim[OF _ tendsto_ident_at]) auto
```
```   911
```
```   912 lemma filterlim_at_bot_at_right:
```
```   913   fixes f :: "'a::linorder_topology \<Rightarrow> 'b::linorder"
```
```   914   assumes mono: "\<And>x y. Q x \<Longrightarrow> Q y \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y"
```
```   915     and bij: "\<And>x. P x \<Longrightarrow> f (g x) = x" "\<And>x. P x \<Longrightarrow> Q (g x)"
```
```   916     and Q: "eventually Q (at_right a)"
```
```   917     and bound: "\<And>b. Q b \<Longrightarrow> a < b"
```
```   918     and P: "eventually P at_bot"
```
```   919   shows "filterlim f at_bot (at_right a)"
```
```   920 proof -
```
```   921   from P obtain x where x: "\<And>y. y \<le> x \<Longrightarrow> P y"
```
```   922     unfolding eventually_at_bot_linorder by auto
```
```   923   show ?thesis
```
```   924   proof (intro filterlim_at_bot_le[THEN iffD2] allI impI)
```
```   925     fix z
```
```   926     assume "z \<le> x"
```
```   927     with x have "P z" by auto
```
```   928     have "eventually (\<lambda>x. x \<le> g z) (at_right a)"
```
```   929       using bound[OF bij(2)[OF \<open>P z\<close>]]
```
```   930       unfolding eventually_at_right[OF bound[OF bij(2)[OF \<open>P z\<close>]]]
```
```   931       by (auto intro!: exI[of _ "g z"])
```
```   932     with Q show "eventually (\<lambda>x. f x \<le> z) (at_right a)"
```
```   933       by eventually_elim (metis bij \<open>P z\<close> mono)
```
```   934   qed
```
```   935 qed
```
```   936
```
```   937 lemma filterlim_at_top_at_left:
```
```   938   fixes f :: "'a::linorder_topology \<Rightarrow> 'b::linorder"
```
```   939   assumes mono: "\<And>x y. Q x \<Longrightarrow> Q y \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y"
```
```   940     and bij: "\<And>x. P x \<Longrightarrow> f (g x) = x" "\<And>x. P x \<Longrightarrow> Q (g x)"
```
```   941     and Q: "eventually Q (at_left a)"
```
```   942     and bound: "\<And>b. Q b \<Longrightarrow> b < a"
```
```   943     and P: "eventually P at_top"
```
```   944   shows "filterlim f at_top (at_left a)"
```
```   945 proof -
```
```   946   from P obtain x where x: "\<And>y. x \<le> y \<Longrightarrow> P y"
```
```   947     unfolding eventually_at_top_linorder by auto
```
```   948   show ?thesis
```
```   949   proof (intro filterlim_at_top_ge[THEN iffD2] allI impI)
```
```   950     fix z
```
```   951     assume "x \<le> z"
```
```   952     with x have "P z" by auto
```
```   953     have "eventually (\<lambda>x. g z \<le> x) (at_left a)"
```
```   954       using bound[OF bij(2)[OF \<open>P z\<close>]]
```
```   955       unfolding eventually_at_left[OF bound[OF bij(2)[OF \<open>P z\<close>]]]
```
```   956       by (auto intro!: exI[of _ "g z"])
```
```   957     with Q show "eventually (\<lambda>x. z \<le> f x) (at_left a)"
```
```   958       by eventually_elim (metis bij \<open>P z\<close> mono)
```
```   959   qed
```
```   960 qed
```
```   961
```
```   962 lemma filterlim_split_at:
```
```   963   "filterlim f F (at_left x) \<Longrightarrow> filterlim f F (at_right x) \<Longrightarrow>
```
```   964     filterlim f F (at x)"
```
```   965   for x :: "'a::linorder_topology"
```
```   966   by (subst at_eq_sup_left_right) (rule filterlim_sup)
```
```   967
```
```   968 lemma filterlim_at_split:
```
```   969   "filterlim f F (at x) \<longleftrightarrow> filterlim f F (at_left x) \<and> filterlim f F (at_right x)"
```
```   970   for x :: "'a::linorder_topology"
```
```   971   by (subst at_eq_sup_left_right) (simp add: filterlim_def filtermap_sup)
```
```   972
```
```   973 lemma eventually_nhds_top:
```
```   974   fixes P :: "'a :: {order_top,linorder_topology} \<Rightarrow> bool"
```
```   975     and b :: 'a
```
```   976   assumes "b < top"
```
```   977   shows "eventually P (nhds top) \<longleftrightarrow> (\<exists>b<top. (\<forall>z. b < z \<longrightarrow> P z))"
```
```   978   unfolding eventually_nhds
```
```   979 proof safe
```
```   980   fix S :: "'a set"
```
```   981   assume "open S" "top \<in> S"
```
```   982   note open_left[OF this \<open>b < top\<close>]
```
```   983   moreover assume "\<forall>s\<in>S. P s"
```
```   984   ultimately show "\<exists>b<top. \<forall>z>b. P z"
```
```   985     by (auto simp: subset_eq Ball_def)
```
```   986 next
```
```   987   fix b
```
```   988   assume "b < top" "\<forall>z>b. P z"
```
```   989   then show "\<exists>S. open S \<and> top \<in> S \<and> (\<forall>xa\<in>S. P xa)"
```
```   990     by (intro exI[of _ "{b <..}"]) auto
```
```   991 qed
```
```   992
```
```   993 lemma tendsto_at_within_iff_tendsto_nhds:
```
```   994   "(g \<longlongrightarrow> g l) (at l within S) \<longleftrightarrow> (g \<longlongrightarrow> g l) (inf (nhds l) (principal S))"
```
```   995   unfolding tendsto_def eventually_at_filter eventually_inf_principal
```
```   996   by (intro ext all_cong imp_cong) (auto elim!: eventually_mono)
```
```   997
```
```   998
```
```   999 subsection \<open>Limits on sequences\<close>
```
```  1000
```
```  1001 abbreviation (in topological_space)
```
```  1002   LIMSEQ :: "[nat \<Rightarrow> 'a, 'a] \<Rightarrow> bool"  ("((_)/ \<longlonglongrightarrow> (_))" [60, 60] 60)
```
```  1003   where "X \<longlonglongrightarrow> L \<equiv> (X \<longlongrightarrow> L) sequentially"
```
```  1004
```
```  1005 abbreviation (in t2_space) lim :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'a"
```
```  1006   where "lim X \<equiv> Lim sequentially X"
```
```  1007
```
```  1008 definition (in topological_space) convergent :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"
```
```  1009   where "convergent X = (\<exists>L. X \<longlonglongrightarrow> L)"
```
```  1010
```
```  1011 lemma lim_def: "lim X = (THE L. X \<longlonglongrightarrow> L)"
```
```  1012   unfolding Lim_def ..
```
```  1013
```
```  1014
```
```  1015 subsubsection \<open>Monotone sequences and subsequences\<close>
```
```  1016
```
```  1017 text \<open>
```
```  1018   Definition of monotonicity.
```
```  1019   The use of disjunction here complicates proofs considerably.
```
```  1020   One alternative is to add a Boolean argument to indicate the direction.
```
```  1021   Another is to develop the notions of increasing and decreasing first.
```
```  1022 \<close>
```
```  1023 definition monoseq :: "(nat \<Rightarrow> 'a::order) \<Rightarrow> bool"
```
```  1024   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)"
```
```  1025
```
```  1026 abbreviation incseq :: "(nat \<Rightarrow> 'a::order) \<Rightarrow> bool"
```
```  1027   where "incseq X \<equiv> mono X"
```
```  1028
```
```  1029 lemma incseq_def: "incseq X \<longleftrightarrow> (\<forall>m. \<forall>n\<ge>m. X n \<ge> X m)"
```
```  1030   unfolding mono_def ..
```
```  1031
```
```  1032 abbreviation decseq :: "(nat \<Rightarrow> 'a::order) \<Rightarrow> bool"
```
```  1033   where "decseq X \<equiv> antimono X"
```
```  1034
```
```  1035 lemma decseq_def: "decseq X \<longleftrightarrow> (\<forall>m. \<forall>n\<ge>m. X n \<le> X m)"
```
```  1036   unfolding antimono_def ..
```
```  1037
```
```  1038 text \<open>Definition of subsequence.\<close>
```
```  1039 definition subseq :: "(nat \<Rightarrow> nat) \<Rightarrow> bool"
```
```  1040   where "subseq f \<longleftrightarrow> (\<forall>m. \<forall>n>m. f m < f n)"
```
```  1041
```
```  1042 lemma incseq_SucI: "(\<And>n. X n \<le> X (Suc n)) \<Longrightarrow> incseq X"
```
```  1043   using lift_Suc_mono_le[of X] by (auto simp: incseq_def)
```
```  1044
```
```  1045 lemma incseqD: "incseq f \<Longrightarrow> i \<le> j \<Longrightarrow> f i \<le> f j"
```
```  1046   by (auto simp: incseq_def)
```
```  1047
```
```  1048 lemma incseq_SucD: "incseq A \<Longrightarrow> A i \<le> A (Suc i)"
```
```  1049   using incseqD[of A i "Suc i"] by auto
```
```  1050
```
```  1051 lemma incseq_Suc_iff: "incseq f \<longleftrightarrow> (\<forall>n. f n \<le> f (Suc n))"
```
```  1052   by (auto intro: incseq_SucI dest: incseq_SucD)
```
```  1053
```
```  1054 lemma incseq_const[simp, intro]: "incseq (\<lambda>x. k)"
```
```  1055   unfolding incseq_def by auto
```
```  1056
```
```  1057 lemma decseq_SucI: "(\<And>n. X (Suc n) \<le> X n) \<Longrightarrow> decseq X"
```
```  1058   using order.lift_Suc_mono_le[OF dual_order, of X] by (auto simp: decseq_def)
```
```  1059
```
```  1060 lemma decseqD: "decseq f \<Longrightarrow> i \<le> j \<Longrightarrow> f j \<le> f i"
```
```  1061   by (auto simp: decseq_def)
```
```  1062
```
```  1063 lemma decseq_SucD: "decseq A \<Longrightarrow> A (Suc i) \<le> A i"
```
```  1064   using decseqD[of A i "Suc i"] by auto
```
```  1065
```
```  1066 lemma decseq_Suc_iff: "decseq f \<longleftrightarrow> (\<forall>n. f (Suc n) \<le> f n)"
```
```  1067   by (auto intro: decseq_SucI dest: decseq_SucD)
```
```  1068
```
```  1069 lemma decseq_const[simp, intro]: "decseq (\<lambda>x. k)"
```
```  1070   unfolding decseq_def by auto
```
```  1071
```
```  1072 lemma monoseq_iff: "monoseq X \<longleftrightarrow> incseq X \<or> decseq X"
```
```  1073   unfolding monoseq_def incseq_def decseq_def ..
```
```  1074
```
```  1075 lemma monoseq_Suc: "monoseq X \<longleftrightarrow> (\<forall>n. X n \<le> X (Suc n)) \<or> (\<forall>n. X (Suc n) \<le> X n)"
```
```  1076   unfolding monoseq_iff incseq_Suc_iff decseq_Suc_iff ..
```
```  1077
```
```  1078 lemma monoI1: "\<forall>m. \<forall>n \<ge> m. X m \<le> X n \<Longrightarrow> monoseq X"
```
```  1079   by (simp add: monoseq_def)
```
```  1080
```
```  1081 lemma monoI2: "\<forall>m. \<forall>n \<ge> m. X n \<le> X m \<Longrightarrow> monoseq X"
```
```  1082   by (simp add: monoseq_def)
```
```  1083
```
```  1084 lemma mono_SucI1: "\<forall>n. X n \<le> X (Suc n) \<Longrightarrow> monoseq X"
```
```  1085   by (simp add: monoseq_Suc)
```
```  1086
```
```  1087 lemma mono_SucI2: "\<forall>n. X (Suc n) \<le> X n \<Longrightarrow> monoseq X"
```
```  1088   by (simp add: monoseq_Suc)
```
```  1089
```
```  1090 lemma monoseq_minus:
```
```  1091   fixes a :: "nat \<Rightarrow> 'a::ordered_ab_group_add"
```
```  1092   assumes "monoseq a"
```
```  1093   shows "monoseq (\<lambda> n. - a n)"
```
```  1094 proof (cases "\<forall>m. \<forall>n \<ge> m. a m \<le> a n")
```
```  1095   case True
```
```  1096   then have "\<forall>m. \<forall>n \<ge> m. - a n \<le> - a m" by auto
```
```  1097   then show ?thesis by (rule monoI2)
```
```  1098 next
```
```  1099   case False
```
```  1100   then have "\<forall>m. \<forall>n \<ge> m. - a m \<le> - a n"
```
```  1101     using \<open>monoseq a\<close>[unfolded monoseq_def] by auto
```
```  1102   then show ?thesis by (rule monoI1)
```
```  1103 qed
```
```  1104
```
```  1105
```
```  1106 text \<open>Subsequence (alternative definition, (e.g. Hoskins)\<close>
```
```  1107
```
```  1108 lemma subseq_Suc_iff: "subseq f \<longleftrightarrow> (\<forall>n. f n < f (Suc n))"
```
```  1109   apply (simp add: subseq_def)
```
```  1110   apply (auto dest!: less_imp_Suc_add)
```
```  1111   apply (induct_tac k)
```
```  1112    apply (auto intro: less_trans)
```
```  1113   done
```
```  1114
```
```  1115 lemma subseq_add: "subseq (\<lambda>n. n + k)"
```
```  1116   by (auto simp: subseq_Suc_iff)
```
```  1117
```
```  1118 text \<open>For any sequence, there is a monotonic subsequence.\<close>
```
```  1119 lemma seq_monosub:
```
```  1120   fixes s :: "nat \<Rightarrow> 'a::linorder"
```
```  1121   shows "\<exists>f. subseq f \<and> monoseq (\<lambda>n. (s (f n)))"
```
```  1122 proof (cases "\<forall>n. \<exists>p>n. \<forall>m\<ge>p. s m \<le> s p")
```
```  1123   case True
```
```  1124   then have "\<exists>f. \<forall>n. (\<forall>m\<ge>f n. s m \<le> s (f n)) \<and> f n < f (Suc n)"
```
```  1125     by (intro dependent_nat_choice) (auto simp: conj_commute)
```
```  1126   then obtain f where f: "subseq f" and mono: "\<And>n m. f n \<le> m \<Longrightarrow> s m \<le> s (f n)"
```
```  1127     by (auto simp: subseq_Suc_iff)
```
```  1128   then have "incseq f"
```
```  1129     unfolding subseq_Suc_iff incseq_Suc_iff by (auto intro: less_imp_le)
```
```  1130   then have "monoseq (\<lambda>n. s (f n))"
```
```  1131     by (auto simp add: incseq_def intro!: mono monoI2)
```
```  1132   with f show ?thesis
```
```  1133     by auto
```
```  1134 next
```
```  1135   case False
```
```  1136   then obtain N where N: "p > N \<Longrightarrow> \<exists>m>p. s p < s m" for p
```
```  1137     by (force simp: not_le le_less)
```
```  1138   have "\<exists>f. \<forall>n. N < f n \<and> f n < f (Suc n) \<and> s (f n) \<le> s (f (Suc n))"
```
```  1139   proof (intro dependent_nat_choice)
```
```  1140     fix x
```
```  1141     assume "N < x" with N[of x]
```
```  1142     show "\<exists>y>N. x < y \<and> s x \<le> s y"
```
```  1143       by (auto intro: less_trans)
```
```  1144   qed auto
```
```  1145   then show ?thesis
```
```  1146     by (auto simp: monoseq_iff incseq_Suc_iff subseq_Suc_iff)
```
```  1147 qed
```
```  1148
```
```  1149 lemma seq_suble:
```
```  1150   assumes sf: "subseq f"
```
```  1151   shows "n \<le> f n"
```
```  1152 proof (induct n)
```
```  1153   case 0
```
```  1154   show ?case by simp
```
```  1155 next
```
```  1156   case (Suc n)
```
```  1157   with sf [unfolded subseq_Suc_iff, rule_format, of n] have "n < f (Suc n)"
```
```  1158      by arith
```
```  1159   then show ?case by arith
```
```  1160 qed
```
```  1161
```
```  1162 lemma eventually_subseq:
```
```  1163   "subseq r \<Longrightarrow> eventually P sequentially \<Longrightarrow> eventually (\<lambda>n. P (r n)) sequentially"
```
```  1164   unfolding eventually_sequentially by (metis seq_suble le_trans)
```
```  1165
```
```  1166 lemma not_eventually_sequentiallyD:
```
```  1167   assumes "\<not> eventually P sequentially"
```
```  1168   shows "\<exists>r. subseq r \<and> (\<forall>n. \<not> P (r n))"
```
```  1169 proof -
```
```  1170   from assms have "\<forall>n. \<exists>m\<ge>n. \<not> P m"
```
```  1171     unfolding eventually_sequentially by (simp add: not_less)
```
```  1172   then obtain r where "\<And>n. r n \<ge> n" "\<And>n. \<not> P (r n)"
```
```  1173     by (auto simp: choice_iff)
```
```  1174   then show ?thesis
```
```  1175     by (auto intro!: exI[of _ "\<lambda>n. r (((Suc \<circ> r) ^^ Suc n) 0)"]
```
```  1176              simp: less_eq_Suc_le subseq_Suc_iff)
```
```  1177 qed
```
```  1178
```
```  1179 lemma filterlim_subseq: "subseq f \<Longrightarrow> filterlim f sequentially sequentially"
```
```  1180   unfolding filterlim_iff by (metis eventually_subseq)
```
```  1181
```
```  1182 lemma subseq_o: "subseq r \<Longrightarrow> subseq s \<Longrightarrow> subseq (r \<circ> s)"
```
```  1183   unfolding subseq_def by simp
```
```  1184
```
```  1185 lemma subseq_mono: "subseq r \<Longrightarrow> m < n \<Longrightarrow> r m < r n"
```
```  1186   by (auto simp: subseq_def)
```
```  1187
```
```  1188 lemma subseq_imp_inj_on: "subseq g \<Longrightarrow> inj_on g A"
```
```  1189 proof (rule inj_onI)
```
```  1190   assume g: "subseq g"
```
```  1191   fix x y
```
```  1192   assume "g x = g y"
```
```  1193   with subseq_mono[OF g, of x y] subseq_mono[OF g, of y x] show "x = y"
```
```  1194     by (cases x y rule: linorder_cases) simp_all
```
```  1195 qed
```
```  1196
```
```  1197 lemma subseq_strict_mono: "subseq g \<Longrightarrow> strict_mono g"
```
```  1198   by (intro strict_monoI subseq_mono[of g])
```
```  1199
```
```  1200 lemma incseq_imp_monoseq:  "incseq X \<Longrightarrow> monoseq X"
```
```  1201   by (simp add: incseq_def monoseq_def)
```
```  1202
```
```  1203 lemma decseq_imp_monoseq:  "decseq X \<Longrightarrow> monoseq X"
```
```  1204   by (simp add: decseq_def monoseq_def)
```
```  1205
```
```  1206 lemma decseq_eq_incseq: "decseq X = incseq (\<lambda>n. - X n)"
```
```  1207   for X :: "nat \<Rightarrow> 'a::ordered_ab_group_add"
```
```  1208   by (simp add: decseq_def incseq_def)
```
```  1209
```
```  1210 lemma INT_decseq_offset:
```
```  1211   assumes "decseq F"
```
```  1212   shows "(\<Inter>i. F i) = (\<Inter>i\<in>{n..}. F i)"
```
```  1213 proof safe
```
```  1214   fix x i
```
```  1215   assume x: "x \<in> (\<Inter>i\<in>{n..}. F i)"
```
```  1216   show "x \<in> F i"
```
```  1217   proof cases
```
```  1218     from x have "x \<in> F n" by auto
```
```  1219     also assume "i \<le> n" with \<open>decseq F\<close> have "F n \<subseteq> F i"
```
```  1220       unfolding decseq_def by simp
```
```  1221     finally show ?thesis .
```
```  1222   qed (insert x, simp)
```
```  1223 qed auto
```
```  1224
```
```  1225 lemma LIMSEQ_const_iff: "(\<lambda>n. k) \<longlonglongrightarrow> l \<longleftrightarrow> k = l"
```
```  1226   for k l :: "'a::t2_space"
```
```  1227   using trivial_limit_sequentially by (rule tendsto_const_iff)
```
```  1228
```
```  1229 lemma LIMSEQ_SUP: "incseq X \<Longrightarrow> X \<longlonglongrightarrow> (SUP i. X i :: 'a::{complete_linorder,linorder_topology})"
```
```  1230   by (intro increasing_tendsto)
```
```  1231     (auto simp: SUP_upper less_SUP_iff incseq_def eventually_sequentially intro: less_le_trans)
```
```  1232
```
```  1233 lemma LIMSEQ_INF: "decseq X \<Longrightarrow> X \<longlonglongrightarrow> (INF i. X i :: 'a::{complete_linorder,linorder_topology})"
```
```  1234   by (intro decreasing_tendsto)
```
```  1235     (auto simp: INF_lower INF_less_iff decseq_def eventually_sequentially intro: le_less_trans)
```
```  1236
```
```  1237 lemma LIMSEQ_ignore_initial_segment: "f \<longlonglongrightarrow> a \<Longrightarrow> (\<lambda>n. f (n + k)) \<longlonglongrightarrow> a"
```
```  1238   unfolding tendsto_def by (subst eventually_sequentially_seg[where k=k])
```
```  1239
```
```  1240 lemma LIMSEQ_offset: "(\<lambda>n. f (n + k)) \<longlonglongrightarrow> a \<Longrightarrow> f \<longlonglongrightarrow> a"
```
```  1241   unfolding tendsto_def
```
```  1242   by (subst (asm) eventually_sequentially_seg[where k=k])
```
```  1243
```
```  1244 lemma LIMSEQ_Suc: "f \<longlonglongrightarrow> l \<Longrightarrow> (\<lambda>n. f (Suc n)) \<longlonglongrightarrow> l"
```
```  1245   by (drule LIMSEQ_ignore_initial_segment [where k="Suc 0"]) simp
```
```  1246
```
```  1247 lemma LIMSEQ_imp_Suc: "(\<lambda>n. f (Suc n)) \<longlonglongrightarrow> l \<Longrightarrow> f \<longlonglongrightarrow> l"
```
```  1248   by (rule LIMSEQ_offset [where k="Suc 0"]) simp
```
```  1249
```
```  1250 lemma LIMSEQ_Suc_iff: "(\<lambda>n. f (Suc n)) \<longlonglongrightarrow> l = f \<longlonglongrightarrow> l"
```
```  1251   by (blast intro: LIMSEQ_imp_Suc LIMSEQ_Suc)
```
```  1252
```
```  1253 lemma LIMSEQ_unique: "X \<longlonglongrightarrow> a \<Longrightarrow> X \<longlonglongrightarrow> b \<Longrightarrow> a = b"
```
```  1254   for a b :: "'a::t2_space"
```
```  1255   using trivial_limit_sequentially by (rule tendsto_unique)
```
```  1256
```
```  1257 lemma LIMSEQ_le_const: "X \<longlonglongrightarrow> x \<Longrightarrow> \<exists>N. \<forall>n\<ge>N. a \<le> X n \<Longrightarrow> a \<le> x"
```
```  1258   for a x :: "'a::linorder_topology"
```
```  1259   using tendsto_le_const[of sequentially X x a] by (simp add: eventually_sequentially)
```
```  1260
```
```  1261 lemma 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"
```
```  1262   for x y :: "'a::linorder_topology"
```
```  1263   using tendsto_le[of sequentially Y y X x] by (simp add: eventually_sequentially)
```
```  1264
```
```  1265 lemma LIMSEQ_le_const2: "X \<longlonglongrightarrow> x \<Longrightarrow> \<exists>N. \<forall>n\<ge>N. X n \<le> a \<Longrightarrow> x \<le> a"
```
```  1266   for a x :: "'a::linorder_topology"
```
```  1267   by (rule LIMSEQ_le[of X x "\<lambda>n. a"]) auto
```
```  1268
```
```  1269 lemma convergentD: "convergent X \<Longrightarrow> \<exists>L. X \<longlonglongrightarrow> L"
```
```  1270   by (simp add: convergent_def)
```
```  1271
```
```  1272 lemma convergentI: "X \<longlonglongrightarrow> L \<Longrightarrow> convergent X"
```
```  1273   by (auto simp add: convergent_def)
```
```  1274
```
```  1275 lemma convergent_LIMSEQ_iff: "convergent X \<longleftrightarrow> X \<longlonglongrightarrow> lim X"
```
```  1276   by (auto intro: theI LIMSEQ_unique simp add: convergent_def lim_def)
```
```  1277
```
```  1278 lemma convergent_const: "convergent (\<lambda>n. c)"
```
```  1279   by (rule convergentI) (rule tendsto_const)
```
```  1280
```
```  1281 lemma monoseq_le:
```
```  1282   "monoseq a \<Longrightarrow> a \<longlonglongrightarrow> x \<Longrightarrow>
```
```  1283     (\<forall>n. a n \<le> x) \<and> (\<forall>m. \<forall>n\<ge>m. a m \<le> a n) \<or>
```
```  1284     (\<forall>n. x \<le> a n) \<and> (\<forall>m. \<forall>n\<ge>m. a n \<le> a m)"
```
```  1285   for x :: "'a::linorder_topology"
```
```  1286   by (metis LIMSEQ_le_const LIMSEQ_le_const2 decseq_def incseq_def monoseq_iff)
```
```  1287
```
```  1288 lemma LIMSEQ_subseq_LIMSEQ: "X \<longlonglongrightarrow> L \<Longrightarrow> subseq f \<Longrightarrow> (X \<circ> f) \<longlonglongrightarrow> L"
```
```  1289   unfolding comp_def by (rule filterlim_compose [of X, OF _ filterlim_subseq])
```
```  1290
```
```  1291 lemma convergent_subseq_convergent: "convergent X \<Longrightarrow> subseq f \<Longrightarrow> convergent (X \<circ> f)"
```
```  1292   by (auto simp: convergent_def intro: LIMSEQ_subseq_LIMSEQ)
```
```  1293
```
```  1294 lemma limI: "X \<longlonglongrightarrow> L \<Longrightarrow> lim X = L"
```
```  1295   by (rule tendsto_Lim) (rule trivial_limit_sequentially)
```
```  1296
```
```  1297 lemma lim_le: "convergent f \<Longrightarrow> (\<And>n. f n \<le> x) \<Longrightarrow> lim f \<le> x"
```
```  1298   for x :: "'a::linorder_topology"
```
```  1299   using LIMSEQ_le_const2[of f "lim f" x] by (simp add: convergent_LIMSEQ_iff)
```
```  1300
```
```  1301 lemma lim_const [simp]: "lim (\<lambda>m. a) = a"
```
```  1302   by (simp add: limI)
```
```  1303
```
```  1304
```
```  1305 subsubsection \<open>Increasing and Decreasing Series\<close>
```
```  1306
```
```  1307 lemma incseq_le: "incseq X \<Longrightarrow> X \<longlonglongrightarrow> L \<Longrightarrow> X n \<le> L"
```
```  1308   for L :: "'a::linorder_topology"
```
```  1309   by (metis incseq_def LIMSEQ_le_const)
```
```  1310
```
```  1311 lemma decseq_le: "decseq X \<Longrightarrow> X \<longlonglongrightarrow> L \<Longrightarrow> L \<le> X n"
```
```  1312   for L :: "'a::linorder_topology"
```
```  1313   by (metis decseq_def LIMSEQ_le_const2)
```
```  1314
```
```  1315
```
```  1316 subsection \<open>First countable topologies\<close>
```
```  1317
```
```  1318 class first_countable_topology = topological_space +
```
```  1319   assumes first_countable_basis:
```
```  1320     "\<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))"
```
```  1321
```
```  1322 lemma (in first_countable_topology) countable_basis_at_decseq:
```
```  1323   obtains A :: "nat \<Rightarrow> 'a set" where
```
```  1324     "\<And>i. open (A i)" "\<And>i. x \<in> (A i)"
```
```  1325     "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> eventually (\<lambda>i. A i \<subseteq> S) sequentially"
```
```  1326 proof atomize_elim
```
```  1327   from first_countable_basis[of x] obtain A :: "nat \<Rightarrow> 'a set"
```
```  1328     where nhds: "\<And>i. open (A i)" "\<And>i. x \<in> A i"
```
```  1329       and incl: "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> \<exists>i. A i \<subseteq> S"
```
```  1330     by auto
```
```  1331   define F where "F n = (\<Inter>i\<le>n. A i)" for n
```
```  1332   show "\<exists>A. (\<forall>i. open (A i)) \<and> (\<forall>i. x \<in> A i) \<and>
```
```  1333     (\<forall>S. open S \<longrightarrow> x \<in> S \<longrightarrow> eventually (\<lambda>i. A i \<subseteq> S) sequentially)"
```
```  1334   proof (safe intro!: exI[of _ F])
```
```  1335     fix i
```
```  1336     show "open (F i)"
```
```  1337       using nhds(1) by (auto simp: F_def)
```
```  1338     show "x \<in> F i"
```
```  1339       using nhds(2) by (auto simp: F_def)
```
```  1340   next
```
```  1341     fix S
```
```  1342     assume "open S" "x \<in> S"
```
```  1343     from incl[OF this] obtain i where "F i \<subseteq> S"
```
```  1344       unfolding F_def by auto
```
```  1345     moreover have "\<And>j. i \<le> j \<Longrightarrow> F j \<subseteq> F i"
```
```  1346       by (simp add: Inf_superset_mono F_def image_mono)
```
```  1347     ultimately show "eventually (\<lambda>i. F i \<subseteq> S) sequentially"
```
```  1348       by (auto simp: eventually_sequentially)
```
```  1349   qed
```
```  1350 qed
```
```  1351
```
```  1352 lemma (in first_countable_topology) nhds_countable:
```
```  1353   obtains X :: "nat \<Rightarrow> 'a set"
```
```  1354   where "decseq X" "\<And>n. open (X n)" "\<And>n. x \<in> X n" "nhds x = (INF n. principal (X n))"
```
```  1355 proof -
```
```  1356   from first_countable_basis obtain A :: "nat \<Rightarrow> 'a set"
```
```  1357     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"
```
```  1358     by metis
```
```  1359   show thesis
```
```  1360   proof
```
```  1361     show "decseq (\<lambda>n. \<Inter>i\<le>n. A i)"
```
```  1362       by (simp add: antimono_iff_le_Suc atMost_Suc)
```
```  1363     show "x \<in> (\<Inter>i\<le>n. A i)" "\<And>n. open (\<Inter>i\<le>n. A i)" for n
```
```  1364       using * by auto
```
```  1365     show "nhds x = (INF n. principal (\<Inter>i\<le>n. A i))"
```
```  1366       using *
```
```  1367       unfolding nhds_def
```
```  1368       apply -
```
```  1369       apply (rule INF_eq)
```
```  1370        apply simp_all
```
```  1371        apply fastforce
```
```  1372       apply (intro exI [of _ "\<Inter>i\<le>n. A i" for n] conjI open_INT)
```
```  1373          apply auto
```
```  1374       done
```
```  1375   qed
```
```  1376 qed
```
```  1377
```
```  1378 lemma (in first_countable_topology) countable_basis:
```
```  1379   obtains A :: "nat \<Rightarrow> 'a set" where
```
```  1380     "\<And>i. open (A i)" "\<And>i. x \<in> A i"
```
```  1381     "\<And>F. (\<forall>n. F n \<in> A n) \<Longrightarrow> F \<longlonglongrightarrow> x"
```
```  1382 proof atomize_elim
```
```  1383   obtain A :: "nat \<Rightarrow> 'a set" where *:
```
```  1384     "\<And>i. open (A i)"
```
```  1385     "\<And>i. x \<in> A i"
```
```  1386     "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> eventually (\<lambda>i. A i \<subseteq> S) sequentially"
```
```  1387     by (rule countable_basis_at_decseq) blast
```
```  1388   have "eventually (\<lambda>n. F n \<in> S) sequentially"
```
```  1389     if "\<forall>n. F n \<in> A n" "open S" "x \<in> S" for F S
```
```  1390     using *(3)[of S] that by (auto elim: eventually_mono simp: subset_eq)
```
```  1391   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)"
```
```  1392     by (intro exI[of _ A]) (auto simp: tendsto_def)
```
```  1393 qed
```
```  1394
```
```  1395 lemma (in first_countable_topology) sequentially_imp_eventually_nhds_within:
```
```  1396   assumes "\<forall>f. (\<forall>n. f n \<in> s) \<and> f \<longlonglongrightarrow> a \<longrightarrow> eventually (\<lambda>n. P (f n)) sequentially"
```
```  1397   shows "eventually P (inf (nhds a) (principal s))"
```
```  1398 proof (rule ccontr)
```
```  1399   obtain A :: "nat \<Rightarrow> 'a set" where *:
```
```  1400     "\<And>i. open (A i)"
```
```  1401     "\<And>i. a \<in> A i"
```
```  1402     "\<And>F. \<forall>n. F n \<in> A n \<Longrightarrow> F \<longlonglongrightarrow> a"
```
```  1403     by (rule countable_basis) blast
```
```  1404   assume "\<not> ?thesis"
```
```  1405   with * have "\<exists>F. \<forall>n. F n \<in> s \<and> F n \<in> A n \<and> \<not> P (F n)"
```
```  1406     unfolding eventually_inf_principal eventually_nhds
```
```  1407     by (intro choice) fastforce
```
```  1408   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)"
```
```  1409     by blast
```
```  1410   with * have "F \<longlonglongrightarrow> a"
```
```  1411     by auto
```
```  1412   then have "eventually (\<lambda>n. P (F n)) sequentially"
```
```  1413     using assms F by simp
```
```  1414   then show False
```
```  1415     by (simp add: F')
```
```  1416 qed
```
```  1417
```
```  1418 lemma (in first_countable_topology) eventually_nhds_within_iff_sequentially:
```
```  1419   "eventually P (inf (nhds a) (principal s)) \<longleftrightarrow>
```
```  1420     (\<forall>f. (\<forall>n. f n \<in> s) \<and> f \<longlonglongrightarrow> a \<longrightarrow> eventually (\<lambda>n. P (f n)) sequentially)"
```
```  1421 proof (safe intro!: sequentially_imp_eventually_nhds_within)
```
```  1422   assume "eventually P (inf (nhds a) (principal s))"
```
```  1423   then obtain S where "open S" "a \<in> S" "\<forall>x\<in>S. x \<in> s \<longrightarrow> P x"
```
```  1424     by (auto simp: eventually_inf_principal eventually_nhds)
```
```  1425   moreover
```
```  1426   fix f
```
```  1427   assume "\<forall>n. f n \<in> s" "f \<longlonglongrightarrow> a"
```
```  1428   ultimately show "eventually (\<lambda>n. P (f n)) sequentially"
```
```  1429     by (auto dest!: topological_tendstoD elim: eventually_mono)
```
```  1430 qed
```
```  1431
```
```  1432 lemma (in first_countable_topology) eventually_nhds_iff_sequentially:
```
```  1433   "eventually P (nhds a) \<longleftrightarrow> (\<forall>f. f \<longlonglongrightarrow> a \<longrightarrow> eventually (\<lambda>n. P (f n)) sequentially)"
```
```  1434   using eventually_nhds_within_iff_sequentially[of P a UNIV] by simp
```
```  1435
```
```  1436 lemma tendsto_at_iff_sequentially:
```
```  1437   "(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))"
```
```  1438   for f :: "'a::first_countable_topology \<Rightarrow> _"
```
```  1439   unfolding filterlim_def[of _ "nhds a"] le_filter_def eventually_filtermap
```
```  1440     at_within_def eventually_nhds_within_iff_sequentially comp_def
```
```  1441   by metis
```
```  1442
```
```  1443
```
```  1444 subsection \<open>Function limit at a point\<close>
```
```  1445
```
```  1446 abbreviation LIM :: "('a::topological_space \<Rightarrow> 'b::topological_space) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool"
```
```  1447     ("((_)/ \<midarrow>(_)/\<rightarrow> (_))" [60, 0, 60] 60)
```
```  1448   where "f \<midarrow>a\<rightarrow> L \<equiv> (f \<longlongrightarrow> L) (at a)"
```
```  1449
```
```  1450 lemma tendsto_within_open: "a \<in> S \<Longrightarrow> open S \<Longrightarrow> (f \<longlongrightarrow> l) (at a within S) \<longleftrightarrow> (f \<midarrow>a\<rightarrow> l)"
```
```  1451   by (simp add: tendsto_def at_within_open[where S = S])
```
```  1452
```
```  1453 lemma tendsto_within_open_NO_MATCH:
```
```  1454   "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)"
```
```  1455   for f :: "'a::topological_space \<Rightarrow> 'b::topological_space"
```
```  1456   using tendsto_within_open by blast
```
```  1457
```
```  1458 lemma LIM_const_not_eq[tendsto_intros]: "k \<noteq> L \<Longrightarrow> \<not> (\<lambda>x. k) \<midarrow>a\<rightarrow> L"
```
```  1459   for a :: "'a::perfect_space" and k L :: "'b::t2_space"
```
```  1460   by (simp add: tendsto_const_iff)
```
```  1461
```
```  1462 lemmas LIM_not_zero = LIM_const_not_eq [where L = 0]
```
```  1463
```
```  1464 lemma LIM_const_eq: "(\<lambda>x. k) \<midarrow>a\<rightarrow> L \<Longrightarrow> k = L"
```
```  1465   for a :: "'a::perfect_space" and k L :: "'b::t2_space"
```
```  1466   by (simp add: tendsto_const_iff)
```
```  1467
```
```  1468 lemma LIM_unique: "f \<midarrow>a\<rightarrow> L \<Longrightarrow> f \<midarrow>a\<rightarrow> M \<Longrightarrow> L = M"
```
```  1469   for a :: "'a::perfect_space" and L M :: "'b::t2_space"
```
```  1470   using at_neq_bot by (rule tendsto_unique)
```
```  1471
```
```  1472
```
```  1473 text \<open>Limits are equal for functions equal except at limit point.\<close>
```
```  1474 lemma LIM_equal: "\<forall>x. x \<noteq> a \<longrightarrow> f x = g x \<Longrightarrow> (f \<midarrow>a\<rightarrow> l) \<longleftrightarrow> (g \<midarrow>a\<rightarrow> l)"
```
```  1475   by (simp add: tendsto_def eventually_at_topological)
```
```  1476
```
```  1477 lemma 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)"
```
```  1478   by (simp add: LIM_equal)
```
```  1479
```
```  1480 lemma LIM_cong_limit: "f \<midarrow>x\<rightarrow> L \<Longrightarrow> K = L \<Longrightarrow> f \<midarrow>x\<rightarrow> K"
```
```  1481   by simp
```
```  1482
```
```  1483 lemma tendsto_at_iff_tendsto_nhds: "g \<midarrow>l\<rightarrow> g l \<longleftrightarrow> (g \<longlongrightarrow> g l) (nhds l)"
```
```  1484   unfolding tendsto_def eventually_at_filter
```
```  1485   by (intro ext all_cong imp_cong) (auto elim!: eventually_mono)
```
```  1486
```
```  1487 lemma tendsto_compose: "g \<midarrow>l\<rightarrow> g l \<Longrightarrow> (f \<longlongrightarrow> l) F \<Longrightarrow> ((\<lambda>x. g (f x)) \<longlongrightarrow> g l) F"
```
```  1488   unfolding tendsto_at_iff_tendsto_nhds by (rule filterlim_compose[of g])
```
```  1489
```
```  1490 lemma LIM_o: "g \<midarrow>l\<rightarrow> g l \<Longrightarrow> f \<midarrow>a\<rightarrow> l \<Longrightarrow> (g \<circ> f) \<midarrow>a\<rightarrow> g l"
```
```  1491   unfolding o_def by (rule tendsto_compose)
```
```  1492
```
```  1493 lemma tendsto_compose_eventually:
```
```  1494   "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"
```
```  1495   by (rule filterlim_compose[of g _ "at l"]) (auto simp add: filterlim_at)
```
```  1496
```
```  1497 lemma LIM_compose_eventually:
```
```  1498   assumes "f \<midarrow>a\<rightarrow> b"
```
```  1499     and "g \<midarrow>b\<rightarrow> c"
```
```  1500     and "eventually (\<lambda>x. f x \<noteq> b) (at a)"
```
```  1501   shows "(\<lambda>x. g (f x)) \<midarrow>a\<rightarrow> c"
```
```  1502   using assms(2,1,3) by (rule tendsto_compose_eventually)
```
```  1503
```
```  1504 lemma tendsto_compose_filtermap: "((g \<circ> f) \<longlongrightarrow> T) F \<longleftrightarrow> (g \<longlongrightarrow> T) (filtermap f F)"
```
```  1505   by (simp add: filterlim_def filtermap_filtermap comp_def)
```
```  1506
```
```  1507
```
```  1508 subsubsection \<open>Relation of \<open>LIM\<close> and \<open>LIMSEQ\<close>\<close>
```
```  1509
```
```  1510 lemma (in first_countable_topology) sequentially_imp_eventually_within:
```
```  1511   "(\<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>
```
```  1512     eventually P (at a within s)"
```
```  1513   unfolding at_within_def
```
```  1514   by (intro sequentially_imp_eventually_nhds_within) auto
```
```  1515
```
```  1516 lemma (in first_countable_topology) sequentially_imp_eventually_at:
```
```  1517   "(\<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)"
```
```  1518   using sequentially_imp_eventually_within [where s=UNIV] by simp
```
```  1519
```
```  1520 lemma LIMSEQ_SEQ_conv1:
```
```  1521   fixes f :: "'a::topological_space \<Rightarrow> 'b::topological_space"
```
```  1522   assumes f: "f \<midarrow>a\<rightarrow> l"
```
```  1523   shows "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S \<longlonglongrightarrow> a \<longrightarrow> (\<lambda>n. f (S n)) \<longlonglongrightarrow> l"
```
```  1524   using tendsto_compose_eventually [OF f, where F=sequentially] by simp
```
```  1525
```
```  1526 lemma LIMSEQ_SEQ_conv2:
```
```  1527   fixes f :: "'a::first_countable_topology \<Rightarrow> 'b::topological_space"
```
```  1528   assumes "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S \<longlonglongrightarrow> a \<longrightarrow> (\<lambda>n. f (S n)) \<longlonglongrightarrow> l"
```
```  1529   shows "f \<midarrow>a\<rightarrow> l"
```
```  1530   using assms unfolding tendsto_def [where l=l] by (simp add: sequentially_imp_eventually_at)
```
```  1531
```
```  1532 lemma 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"
```
```  1533   for a :: "'a::first_countable_topology" and L :: "'b::topological_space"
```
```  1534   using LIMSEQ_SEQ_conv2 LIMSEQ_SEQ_conv1 ..
```
```  1535
```
```  1536 lemma sequentially_imp_eventually_at_left:
```
```  1537   fixes a :: "'a::{linorder_topology,first_countable_topology}"
```
```  1538   assumes b[simp]: "b < a"
```
```  1539     and *: "\<And>f. (\<And>n. b < f n) \<Longrightarrow> (\<And>n. f n < a) \<Longrightarrow> incseq f \<Longrightarrow> f \<longlonglongrightarrow> a \<Longrightarrow>
```
```  1540       eventually (\<lambda>n. P (f n)) sequentially"
```
```  1541   shows "eventually P (at_left a)"
```
```  1542 proof (safe intro!: sequentially_imp_eventually_within)
```
```  1543   fix X
```
```  1544   assume X: "\<forall>n. X n \<in> {..< a} \<and> X n \<noteq> a" "X \<longlonglongrightarrow> a"
```
```  1545   show "eventually (\<lambda>n. P (X n)) sequentially"
```
```  1546   proof (rule ccontr)
```
```  1547     assume neg: "\<not> ?thesis"
```
```  1548     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))"
```
```  1549       (is "\<exists>s. ?P s")
```
```  1550     proof (rule dependent_nat_choice)
```
```  1551       have "\<not> eventually (\<lambda>n. b < X n \<longrightarrow> P (X n)) sequentially"
```
```  1552         by (intro not_eventually_impI neg order_tendstoD(1) [OF X(2) b])
```
```  1553       then show "\<exists>x. \<not> P (X x) \<and> b < X x"
```
```  1554         by (auto dest!: not_eventuallyD)
```
```  1555     next
```
```  1556       fix x n
```
```  1557       have "\<not> eventually (\<lambda>n. Suc x \<le> n \<longrightarrow> b < X n \<longrightarrow> X x < X n \<longrightarrow> P (X n)) sequentially"
```
```  1558         using X
```
```  1559         by (intro not_eventually_impI order_tendstoD(1)[OF X(2)] eventually_ge_at_top neg) auto
```
```  1560       then show "\<exists>n. (\<not> P (X n) \<and> b < X n) \<and> (X x \<le> X n \<and> Suc x \<le> n)"
```
```  1561         by (auto dest!: not_eventuallyD)
```
```  1562     qed
```
```  1563     then obtain s where "?P s" ..
```
```  1564     with X have "b < X (s n)"
```
```  1565       and "X (s n) < a"
```
```  1566       and "incseq (\<lambda>n. X (s n))"
```
```  1567       and "(\<lambda>n. X (s n)) \<longlonglongrightarrow> a"
```
```  1568       and "\<not> P (X (s n))"
```
```  1569       for n
```
```  1570       by (auto simp: subseq_Suc_iff Suc_le_eq incseq_Suc_iff
```
```  1571           intro!: LIMSEQ_subseq_LIMSEQ[OF \<open>X \<longlonglongrightarrow> a\<close>, unfolded comp_def])
```
```  1572     from *[OF this(1,2,3,4)] this(5) show False
```
```  1573       by auto
```
```  1574   qed
```
```  1575 qed
```
```  1576
```
```  1577 lemma tendsto_at_left_sequentially:
```
```  1578   fixes a b :: "'b::{linorder_topology,first_countable_topology}"
```
```  1579   assumes "b < a"
```
```  1580   assumes *: "\<And>S. (\<And>n. S n < a) \<Longrightarrow> (\<And>n. b < S n) \<Longrightarrow> incseq S \<Longrightarrow> S \<longlonglongrightarrow> a \<Longrightarrow>
```
```  1581     (\<lambda>n. X (S n)) \<longlonglongrightarrow> L"
```
```  1582   shows "(X \<longlongrightarrow> L) (at_left a)"
```
```  1583   using assms by (simp add: tendsto_def [where l=L] sequentially_imp_eventually_at_left)
```
```  1584
```
```  1585 lemma sequentially_imp_eventually_at_right:
```
```  1586   fixes a b :: "'a::{linorder_topology,first_countable_topology}"
```
```  1587   assumes b[simp]: "a < b"
```
```  1588   assumes *: "\<And>f. (\<And>n. a < f n) \<Longrightarrow> (\<And>n. f n < b) \<Longrightarrow> decseq f \<Longrightarrow> f \<longlonglongrightarrow> a \<Longrightarrow>
```
```  1589     eventually (\<lambda>n. P (f n)) sequentially"
```
```  1590   shows "eventually P (at_right a)"
```
```  1591 proof (safe intro!: sequentially_imp_eventually_within)
```
```  1592   fix X
```
```  1593   assume X: "\<forall>n. X n \<in> {a <..} \<and> X n \<noteq> a" "X \<longlonglongrightarrow> a"
```
```  1594   show "eventually (\<lambda>n. P (X n)) sequentially"
```
```  1595   proof (rule ccontr)
```
```  1596     assume neg: "\<not> ?thesis"
```
```  1597     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))"
```
```  1598       (is "\<exists>s. ?P s")
```
```  1599     proof (rule dependent_nat_choice)
```
```  1600       have "\<not> eventually (\<lambda>n. X n < b \<longrightarrow> P (X n)) sequentially"
```
```  1601         by (intro not_eventually_impI neg order_tendstoD(2) [OF X(2) b])
```
```  1602       then show "\<exists>x. \<not> P (X x) \<and> X x < b"
```
```  1603         by (auto dest!: not_eventuallyD)
```
```  1604     next
```
```  1605       fix x n
```
```  1606       have "\<not> eventually (\<lambda>n. Suc x \<le> n \<longrightarrow> X n < b \<longrightarrow> X n < X x \<longrightarrow> P (X n)) sequentially"
```
```  1607         using X
```
```  1608         by (intro not_eventually_impI order_tendstoD(2)[OF X(2)] eventually_ge_at_top neg) auto
```
```  1609       then show "\<exists>n. (\<not> P (X n) \<and> X n < b) \<and> (X n \<le> X x \<and> Suc x \<le> n)"
```
```  1610         by (auto dest!: not_eventuallyD)
```
```  1611     qed
```
```  1612     then obtain s where "?P s" ..
```
```  1613     with X have "a < X (s n)"
```
```  1614       and "X (s n) < b"
```
```  1615       and "decseq (\<lambda>n. X (s n))"
```
```  1616       and "(\<lambda>n. X (s n)) \<longlonglongrightarrow> a"
```
```  1617       and "\<not> P (X (s n))"
```
```  1618       for n
```
```  1619       by (auto simp: subseq_Suc_iff Suc_le_eq decseq_Suc_iff
```
```  1620           intro!: LIMSEQ_subseq_LIMSEQ[OF \<open>X \<longlonglongrightarrow> a\<close>, unfolded comp_def])
```
```  1621     from *[OF this(1,2,3,4)] this(5) show False
```
```  1622       by auto
```
```  1623   qed
```
```  1624 qed
```
```  1625
```
```  1626 lemma tendsto_at_right_sequentially:
```
```  1627   fixes a :: "_ :: {linorder_topology, first_countable_topology}"
```
```  1628   assumes "a < b"
```
```  1629     and *: "\<And>S. (\<And>n. a < S n) \<Longrightarrow> (\<And>n. S n < b) \<Longrightarrow> decseq S \<Longrightarrow> S \<longlonglongrightarrow> a \<Longrightarrow>
```
```  1630       (\<lambda>n. X (S n)) \<longlonglongrightarrow> L"
```
```  1631   shows "(X \<longlongrightarrow> L) (at_right a)"
```
```  1632   using assms by (simp add: tendsto_def [where l=L] sequentially_imp_eventually_at_right)
```
```  1633
```
```  1634
```
```  1635 subsection \<open>Continuity\<close>
```
```  1636
```
```  1637 subsubsection \<open>Continuity on a set\<close>
```
```  1638
```
```  1639 definition continuous_on :: "'a set \<Rightarrow> ('a::topological_space \<Rightarrow> 'b::topological_space) \<Rightarrow> bool"
```
```  1640   where "continuous_on s f \<longleftrightarrow> (\<forall>x\<in>s. (f \<longlongrightarrow> f x) (at x within s))"
```
```  1641
```
```  1642 lemma continuous_on_cong [cong]:
```
```  1643   "s = t \<Longrightarrow> (\<And>x. x \<in> t \<Longrightarrow> f x = g x) \<Longrightarrow> continuous_on s f \<longleftrightarrow> continuous_on t g"
```
```  1644   unfolding continuous_on_def
```
```  1645   by (intro ball_cong filterlim_cong) (auto simp: eventually_at_filter)
```
```  1646
```
```  1647 lemma continuous_on_topological:
```
```  1648   "continuous_on s f \<longleftrightarrow>
```
```  1649     (\<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)))"
```
```  1650   unfolding continuous_on_def tendsto_def eventually_at_topological by metis
```
```  1651
```
```  1652 lemma continuous_on_open_invariant:
```
```  1653   "continuous_on s f \<longleftrightarrow> (\<forall>B. open B \<longrightarrow> (\<exists>A. open A \<and> A \<inter> s = f -` B \<inter> s))"
```
```  1654 proof safe
```
```  1655   fix B :: "'b set"
```
```  1656   assume "continuous_on s f" "open B"
```
```  1657   then have "\<forall>x\<in>f -` B \<inter> s. (\<exists>A. open A \<and> x \<in> A \<and> s \<inter> A \<subseteq> f -` B)"
```
```  1658     by (auto simp: continuous_on_topological subset_eq Ball_def imp_conjL)
```
```  1659   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"
```
```  1660     unfolding bchoice_iff ..
```
```  1661   then show "\<exists>A. open A \<and> A \<inter> s = f -` B \<inter> s"
```
```  1662     by (intro exI[of _ "\<Union>x\<in>f -` B \<inter> s. A x"]) auto
```
```  1663 next
```
```  1664   assume B: "\<forall>B. open B \<longrightarrow> (\<exists>A. open A \<and> A \<inter> s = f -` B \<inter> s)"
```
```  1665   show "continuous_on s f"
```
```  1666     unfolding continuous_on_topological
```
```  1667   proof safe
```
```  1668     fix x B
```
```  1669     assume "x \<in> s" "open B" "f x \<in> B"
```
```  1670     with B obtain A where A: "open A" "A \<inter> s = f -` B \<inter> s"
```
```  1671       by auto
```
```  1672     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)"
```
```  1673       by (intro exI[of _ A]) auto
```
```  1674   qed
```
```  1675 qed
```
```  1676
```
```  1677 lemma continuous_on_open_vimage:
```
```  1678   "open s \<Longrightarrow> continuous_on s f \<longleftrightarrow> (\<forall>B. open B \<longrightarrow> open (f -` B \<inter> s))"
```
```  1679   unfolding continuous_on_open_invariant
```
```  1680   by (metis open_Int Int_absorb Int_commute[of s] Int_assoc[of _ _ s])
```
```  1681
```
```  1682 corollary continuous_imp_open_vimage:
```
```  1683   assumes "continuous_on s f" "open s" "open B" "f -` B \<subseteq> s"
```
```  1684   shows "open (f -` B)"
```
```  1685   by (metis assms continuous_on_open_vimage le_iff_inf)
```
```  1686
```
```  1687 corollary open_vimage[continuous_intros]:
```
```  1688   assumes "open s"
```
```  1689     and "continuous_on UNIV f"
```
```  1690   shows "open (f -` s)"
```
```  1691   using assms by (simp add: continuous_on_open_vimage [OF open_UNIV])
```
```  1692
```
```  1693 lemma continuous_on_closed_invariant:
```
```  1694   "continuous_on s f \<longleftrightarrow> (\<forall>B. closed B \<longrightarrow> (\<exists>A. closed A \<and> A \<inter> s = f -` B \<inter> s))"
```
```  1695 proof -
```
```  1696   have *: "(\<And>A. P A \<longleftrightarrow> Q (- A)) \<Longrightarrow> (\<forall>A. P A) \<longleftrightarrow> (\<forall>A. Q A)"
```
```  1697     for P Q :: "'b set \<Rightarrow> bool"
```
```  1698     by (metis double_compl)
```
```  1699   show ?thesis
```
```  1700     unfolding continuous_on_open_invariant
```
```  1701     by (intro *) (auto simp: open_closed[symmetric])
```
```  1702 qed
```
```  1703
```
```  1704 lemma continuous_on_closed_vimage:
```
```  1705   "closed s \<Longrightarrow> continuous_on s f \<longleftrightarrow> (\<forall>B. closed B \<longrightarrow> closed (f -` B \<inter> s))"
```
```  1706   unfolding continuous_on_closed_invariant
```
```  1707   by (metis closed_Int Int_absorb Int_commute[of s] Int_assoc[of _ _ s])
```
```  1708
```
```  1709 corollary closed_vimage_Int[continuous_intros]:
```
```  1710   assumes "closed s"
```
```  1711     and "continuous_on t f"
```
```  1712     and t: "closed t"
```
```  1713   shows "closed (f -` s \<inter> t)"
```
```  1714   using assms by (simp add: continuous_on_closed_vimage [OF t])
```
```  1715
```
```  1716 corollary closed_vimage[continuous_intros]:
```
```  1717   assumes "closed s"
```
```  1718     and "continuous_on UNIV f"
```
```  1719   shows "closed (f -` s)"
```
```  1720   using closed_vimage_Int [OF assms] by simp
```
```  1721
```
```  1722 lemma continuous_on_empty [simp]: "continuous_on {} f"
```
```  1723   by (simp add: continuous_on_def)
```
```  1724
```
```  1725 lemma continuous_on_sing [simp]: "continuous_on {x} f"
```
```  1726   by (simp add: continuous_on_def at_within_def)
```
```  1727
```
```  1728 lemma continuous_on_open_Union:
```
```  1729   "(\<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"
```
```  1730   unfolding continuous_on_def
```
```  1731   by safe (metis open_Union at_within_open UnionI)
```
```  1732
```
```  1733 lemma continuous_on_open_UN:
```
```  1734   "(\<And>s. s \<in> S \<Longrightarrow> open (A s)) \<Longrightarrow> (\<And>s. s \<in> S \<Longrightarrow> continuous_on (A s) f) \<Longrightarrow>
```
```  1735     continuous_on (\<Union>s\<in>S. A s) f"
```
```  1736   by (rule continuous_on_open_Union) auto
```
```  1737
```
```  1738 lemma continuous_on_open_Un:
```
```  1739   "open s \<Longrightarrow> open t \<Longrightarrow> continuous_on s f \<Longrightarrow> continuous_on t f \<Longrightarrow> continuous_on (s \<union> t) f"
```
```  1740   using continuous_on_open_Union [of "{s,t}"] by auto
```
```  1741
```
```  1742 lemma continuous_on_closed_Un:
```
```  1743   "closed s \<Longrightarrow> closed t \<Longrightarrow> continuous_on s f \<Longrightarrow> continuous_on t f \<Longrightarrow> continuous_on (s \<union> t) f"
```
```  1744   by (auto simp add: continuous_on_closed_vimage closed_Un Int_Un_distrib)
```
```  1745
```
```  1746 lemma continuous_on_If:
```
```  1747   assumes closed: "closed s" "closed t"
```
```  1748     and cont: "continuous_on s f" "continuous_on t g"
```
```  1749     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"
```
```  1750   shows "continuous_on (s \<union> t) (\<lambda>x. if P x then f x else g x)"
```
```  1751     (is "continuous_on _ ?h")
```
```  1752 proof-
```
```  1753   from P have "\<forall>x\<in>s. f x = ?h x" "\<forall>x\<in>t. g x = ?h x"
```
```  1754     by auto
```
```  1755   with cont have "continuous_on s ?h" "continuous_on t ?h"
```
```  1756     by simp_all
```
```  1757   with closed show ?thesis
```
```  1758     by (rule continuous_on_closed_Un)
```
```  1759 qed
```
```  1760
```
```  1761 lemma continuous_on_id[continuous_intros]: "continuous_on s (\<lambda>x. x)"
```
```  1762   unfolding continuous_on_def by fast
```
```  1763
```
```  1764 lemma continuous_on_id'[continuous_intros]: "continuous_on s id"
```
```  1765   unfolding continuous_on_def id_def by fast
```
```  1766
```
```  1767 lemma continuous_on_const[continuous_intros]: "continuous_on s (\<lambda>x. c)"
```
```  1768   unfolding continuous_on_def by auto
```
```  1769
```
```  1770 lemma continuous_on_subset: "continuous_on s f \<Longrightarrow> t \<subseteq> s \<Longrightarrow> continuous_on t f"
```
```  1771   unfolding continuous_on_def by (metis subset_eq tendsto_within_subset)
```
```  1772
```
```  1773 lemma continuous_on_compose[continuous_intros]:
```
```  1774   "continuous_on s f \<Longrightarrow> continuous_on (f ` s) g \<Longrightarrow> continuous_on s (g \<circ> f)"
```
```  1775   unfolding continuous_on_topological by simp metis
```
```  1776
```
```  1777 lemma continuous_on_compose2:
```
```  1778   "continuous_on t g \<Longrightarrow> continuous_on s f \<Longrightarrow> f ` s \<subseteq> t \<Longrightarrow> continuous_on s (\<lambda>x. g (f x))"
```
```  1779   using continuous_on_compose[of s f g] continuous_on_subset by (force simp add: comp_def)
```
```  1780
```
```  1781 lemma continuous_on_generate_topology:
```
```  1782   assumes *: "open = generate_topology X"
```
```  1783     and **: "\<And>B. B \<in> X \<Longrightarrow> \<exists>C. open C \<and> C \<inter> A = f -` B \<inter> A"
```
```  1784   shows "continuous_on A f"
```
```  1785   unfolding continuous_on_open_invariant
```
```  1786 proof safe
```
```  1787   fix B :: "'a set"
```
```  1788   assume "open B"
```
```  1789   then show "\<exists>C. open C \<and> C \<inter> A = f -` B \<inter> A"
```
```  1790     unfolding *
```
```  1791   proof induct
```
```  1792     case (UN K)
```
```  1793     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"
```
```  1794       by metis
```
```  1795     then show ?case
```
```  1796       by (intro exI[of _ "\<Union>k\<in>K. C k"]) blast
```
```  1797   qed (auto intro: **)
```
```  1798 qed
```
```  1799
```
```  1800 lemma continuous_onI_mono:
```
```  1801   fixes f :: "'a::linorder_topology \<Rightarrow> 'b::{dense_order,linorder_topology}"
```
```  1802   assumes "open (f`A)"
```
```  1803     and mono: "\<And>x y. x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y"
```
```  1804   shows "continuous_on A f"
```
```  1805 proof (rule continuous_on_generate_topology[OF open_generated_order], safe)
```
```  1806   have monoD: "\<And>x y. x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> f x < f y \<Longrightarrow> x < y"
```
```  1807     by (auto simp: not_le[symmetric] mono)
```
```  1808   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
```
```  1809   proof -
```
```  1810     obtain y where "f a < y" "{f a ..< y} \<subseteq> f`A"
```
```  1811       using open_right[OF \<open>open (f`A)\<close>, of "f a" b] a fa
```
```  1812       by auto
```
```  1813     obtain z where z: "f a < z" "z < min b y"
```
```  1814       using dense[of "f a" "min b y"] \<open>f a < y\<close> \<open>f a < b\<close> by auto
```
```  1815     then obtain c where "z = f c" "c \<in> A"
```
```  1816       using \<open>{f a ..< y} \<subseteq> f`A\<close>[THEN subsetD, of z] by (auto simp: less_imp_le)
```
```  1817     with a z show ?thesis
```
```  1818       by (auto intro!: exI[of _ c] simp: monoD)
```
```  1819   qed
```
```  1820   then show "\<exists>C. open C \<and> C \<inter> A = f -` {..<b} \<inter> A" for b
```
```  1821     by (intro exI[of _ "(\<Union>x\<in>{x\<in>A. f x < b}. {..< x})"])
```
```  1822        (auto intro: le_less_trans[OF mono] less_imp_le)
```
```  1823
```
```  1824   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
```
```  1825   proof -
```
```  1826     note a fa
```
```  1827     moreover
```
```  1828     obtain y where "y < f a" "{y <.. f a} \<subseteq> f`A"
```
```  1829       using open_left[OF \<open>open (f`A)\<close>, of "f a" b]  a fa
```
```  1830       by auto
```
```  1831     then obtain z where z: "max b y < z" "z < f a"
```
```  1832       using dense[of "max b y" "f a"] \<open>y < f a\<close> \<open>b < f a\<close> by auto
```
```  1833     then obtain c where "z = f c" "c \<in> A"
```
```  1834       using \<open>{y <.. f a} \<subseteq> f`A\<close>[THEN subsetD, of z] by (auto simp: less_imp_le)
```
```  1835     with a z show ?thesis
```
```  1836       by (auto intro!: exI[of _ c] simp: monoD)
```
```  1837   qed
```
```  1838   then show "\<exists>C. open C \<and> C \<inter> A = f -` {b <..} \<inter> A" for b
```
```  1839     by (intro exI[of _ "(\<Union>x\<in>{x\<in>A. b < f x}. {x <..})"])
```
```  1840        (auto intro: less_le_trans[OF _ mono] less_imp_le)
```
```  1841 qed
```
```  1842
```
```  1843
```
```  1844 subsubsection \<open>Continuity at a point\<close>
```
```  1845
```
```  1846 definition continuous :: "'a::t2_space filter \<Rightarrow> ('a \<Rightarrow> 'b::topological_space) \<Rightarrow> bool"
```
```  1847   where "continuous F f \<longleftrightarrow> (f \<longlongrightarrow> f (Lim F (\<lambda>x. x))) F"
```
```  1848
```
```  1849 lemma continuous_bot[continuous_intros, simp]: "continuous bot f"
```
```  1850   unfolding continuous_def by auto
```
```  1851
```
```  1852 lemma continuous_trivial_limit: "trivial_limit net \<Longrightarrow> continuous net f"
```
```  1853   by simp
```
```  1854
```
```  1855 lemma continuous_within: "continuous (at x within s) f \<longleftrightarrow> (f \<longlongrightarrow> f x) (at x within s)"
```
```  1856   by (cases "trivial_limit (at x within s)") (auto simp add: Lim_ident_at continuous_def)
```
```  1857
```
```  1858 lemma continuous_within_topological:
```
```  1859   "continuous (at x within s) f \<longleftrightarrow>
```
```  1860     (\<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)))"
```
```  1861   unfolding continuous_within tendsto_def eventually_at_topological by metis
```
```  1862
```
```  1863 lemma continuous_within_compose[continuous_intros]:
```
```  1864   "continuous (at x within s) f \<Longrightarrow> continuous (at (f x) within f ` s) g \<Longrightarrow>
```
```  1865     continuous (at x within s) (g \<circ> f)"
```
```  1866   by (simp add: continuous_within_topological) metis
```
```  1867
```
```  1868 lemma continuous_within_compose2:
```
```  1869   "continuous (at x within s) f \<Longrightarrow> continuous (at (f x) within f ` s) g \<Longrightarrow>
```
```  1870     continuous (at x within s) (\<lambda>x. g (f x))"
```
```  1871   using continuous_within_compose[of x s f g] by (simp add: comp_def)
```
```  1872
```
```  1873 lemma continuous_at: "continuous (at x) f \<longleftrightarrow> f \<midarrow>x\<rightarrow> f x"
```
```  1874   using continuous_within[of x UNIV f] by simp
```
```  1875
```
```  1876 lemma continuous_ident[continuous_intros, simp]: "continuous (at x within S) (\<lambda>x. x)"
```
```  1877   unfolding continuous_within by (rule tendsto_ident_at)
```
```  1878
```
```  1879 lemma continuous_const[continuous_intros, simp]: "continuous F (\<lambda>x. c)"
```
```  1880   unfolding continuous_def by (rule tendsto_const)
```
```  1881
```
```  1882 lemma continuous_on_eq_continuous_within:
```
```  1883   "continuous_on s f \<longleftrightarrow> (\<forall>x\<in>s. continuous (at x within s) f)"
```
```  1884   unfolding continuous_on_def continuous_within ..
```
```  1885
```
```  1886 abbreviation isCont :: "('a::t2_space \<Rightarrow> 'b::topological_space) \<Rightarrow> 'a \<Rightarrow> bool"
```
```  1887   where "isCont f a \<equiv> continuous (at a) f"
```
```  1888
```
```  1889 lemma isCont_def: "isCont f a \<longleftrightarrow> f \<midarrow>a\<rightarrow> f a"
```
```  1890   by (rule continuous_at)
```
```  1891
```
```  1892 lemma isCont_cong:
```
```  1893   assumes "eventually (\<lambda>x. f x = g x) (nhds x)"
```
```  1894   shows "isCont f x \<longleftrightarrow> isCont g x"
```
```  1895 proof -
```
```  1896   from assms have [simp]: "f x = g x"
```
```  1897     by (rule eventually_nhds_x_imp_x)
```
```  1898   from assms have "eventually (\<lambda>x. f x = g x) (at x)"
```
```  1899     by (auto simp: eventually_at_filter elim!: eventually_mono)
```
```  1900   with assms have "isCont f x \<longleftrightarrow> isCont g x" unfolding isCont_def
```
```  1901     by (intro filterlim_cong) (auto elim!: eventually_mono)
```
```  1902   with assms show ?thesis by simp
```
```  1903 qed
```
```  1904
```
```  1905 lemma continuous_at_imp_continuous_at_within: "isCont f x \<Longrightarrow> continuous (at x within s) f"
```
```  1906   by (auto intro: tendsto_mono at_le simp: continuous_at continuous_within)
```
```  1907
```
```  1908 lemma continuous_on_eq_continuous_at: "open s \<Longrightarrow> continuous_on s f \<longleftrightarrow> (\<forall>x\<in>s. isCont f x)"
```
```  1909   by (simp add: continuous_on_def continuous_at at_within_open[of _ s])
```
```  1910
```
```  1911 lemma continuous_within_open: "a \<in> A \<Longrightarrow> open A \<Longrightarrow> continuous (at a within A) f \<longleftrightarrow> isCont f a"
```
```  1912   by (simp add: at_within_open_NO_MATCH)
```
```  1913
```
```  1914 lemma continuous_at_imp_continuous_on: "\<forall>x\<in>s. isCont f x \<Longrightarrow> continuous_on s f"
```
```  1915   by (auto intro: continuous_at_imp_continuous_at_within simp: continuous_on_eq_continuous_within)
```
```  1916
```
```  1917 lemma isCont_o2: "isCont f a \<Longrightarrow> isCont g (f a) \<Longrightarrow> isCont (\<lambda>x. g (f x)) a"
```
```  1918   unfolding isCont_def by (rule tendsto_compose)
```
```  1919
```
```  1920 lemma isCont_o[continuous_intros]: "isCont f a \<Longrightarrow> isCont g (f a) \<Longrightarrow> isCont (g \<circ> f) a"
```
```  1921   unfolding o_def by (rule isCont_o2)
```
```  1922
```
```  1923 lemma isCont_tendsto_compose: "isCont g l \<Longrightarrow> (f \<longlongrightarrow> l) F \<Longrightarrow> ((\<lambda>x. g (f x)) \<longlongrightarrow> g l) F"
```
```  1924   unfolding isCont_def by (rule tendsto_compose)
```
```  1925
```
```  1926 lemma continuous_on_tendsto_compose:
```
```  1927   assumes f_cont: "continuous_on s f"
```
```  1928     and g: "(g \<longlongrightarrow> l) F"
```
```  1929     and l: "l \<in> s"
```
```  1930     and ev: "\<forall>\<^sub>Fx in F. g x \<in> s"
```
```  1931   shows "((\<lambda>x. f (g x)) \<longlongrightarrow> f l) F"
```
```  1932 proof -
```
```  1933   from f_cont l have f: "(f \<longlongrightarrow> f l) (at l within s)"
```
```  1934     by (simp add: continuous_on_def)
```
```  1935   have i: "((\<lambda>x. if g x = l then f l else f (g x)) \<longlongrightarrow> f l) F"
```
```  1936     by (rule filterlim_If)
```
```  1937        (auto intro!: filterlim_compose[OF f] eventually_conj tendsto_mono[OF _ g]
```
```  1938              simp: filterlim_at eventually_inf_principal eventually_mono[OF ev])
```
```  1939   show ?thesis
```
```  1940     by (rule filterlim_cong[THEN iffD1[OF _ i]]) auto
```
```  1941 qed
```
```  1942
```
```  1943 lemma continuous_within_compose3:
```
```  1944   "isCont g (f x) \<Longrightarrow> continuous (at x within s) f \<Longrightarrow> continuous (at x within s) (\<lambda>x. g (f x))"
```
```  1945   using continuous_at_imp_continuous_at_within continuous_within_compose2 by blast
```
```  1946
```
```  1947 lemma filtermap_nhds_open_map:
```
```  1948   assumes cont: "isCont f a"
```
```  1949     and open_map: "\<And>S. open S \<Longrightarrow> open (f`S)"
```
```  1950   shows "filtermap f (nhds a) = nhds (f a)"
```
```  1951   unfolding filter_eq_iff
```
```  1952 proof safe
```
```  1953   fix P
```
```  1954   assume "eventually P (filtermap f (nhds a))"
```
```  1955   then obtain S where "open S" "a \<in> S" "\<forall>x\<in>S. P (f x)"
```
```  1956     by (auto simp: eventually_filtermap eventually_nhds)
```
```  1957   then show "eventually P (nhds (f a))"
```
```  1958     unfolding eventually_nhds by (intro exI[of _ "f`S"]) (auto intro!: open_map)
```
```  1959 qed (metis filterlim_iff tendsto_at_iff_tendsto_nhds isCont_def eventually_filtermap cont)
```
```  1960
```
```  1961 lemma continuous_at_split:
```
```  1962   "continuous (at x) f \<longleftrightarrow> continuous (at_left x) f \<and> continuous (at_right x) f"
```
```  1963   for x :: "'a::linorder_topology"
```
```  1964   by (simp add: continuous_within filterlim_at_split)
```
```  1965
```
```  1966 text \<open>
```
```  1967   The following open/closed Collect lemmas are ported from
```
```  1968   Sébastien Gouëzel's \<open>Ergodic_Theory\<close>.
```
```  1969 \<close>
```
```  1970 lemma open_Collect_neq:
```
```  1971   fixes f g :: "'a::topological_space \<Rightarrow> 'b::t2_space"
```
```  1972   assumes f: "continuous_on UNIV f" and g: "continuous_on UNIV g"
```
```  1973   shows "open {x. f x \<noteq> g x}"
```
```  1974 proof (rule openI)
```
```  1975   fix t
```
```  1976   assume "t \<in> {x. f x \<noteq> g x}"
```
```  1977   then obtain U V where *: "open U" "open V" "f t \<in> U" "g t \<in> V" "U \<inter> V = {}"
```
```  1978     by (auto simp add: separation_t2)
```
```  1979   with open_vimage[OF \<open>open U\<close> f] open_vimage[OF \<open>open V\<close> g]
```
```  1980   show "\<exists>T. open T \<and> t \<in> T \<and> T \<subseteq> {x. f x \<noteq> g x}"
```
```  1981     by (intro exI[of _ "f -` U \<inter> g -` V"]) auto
```
```  1982 qed
```
```  1983
```
```  1984 lemma closed_Collect_eq:
```
```  1985   fixes f g :: "'a::topological_space \<Rightarrow> 'b::t2_space"
```
```  1986   assumes f: "continuous_on UNIV f" and g: "continuous_on UNIV g"
```
```  1987   shows "closed {x. f x = g x}"
```
```  1988   using open_Collect_neq[OF f g] by (simp add: closed_def Collect_neg_eq)
```
```  1989
```
```  1990 lemma open_Collect_less:
```
```  1991   fixes f g :: "'a::topological_space \<Rightarrow> 'b::linorder_topology"
```
```  1992   assumes f: "continuous_on UNIV f" and g: "continuous_on UNIV g"
```
```  1993   shows "open {x. f x < g x}"
```
```  1994 proof (rule openI)
```
```  1995   fix t
```
```  1996   assume t: "t \<in> {x. f x < g x}"
```
```  1997   show "\<exists>T. open T \<and> t \<in> T \<and> T \<subseteq> {x. f x < g x}"
```
```  1998   proof (cases "\<exists>z. f t < z \<and> z < g t")
```
```  1999     case True
```
```  2000     then obtain z where "f t < z \<and> z < g t" by blast
```
```  2001     then show ?thesis
```
```  2002       using open_vimage[OF _ f, of "{..< z}"] open_vimage[OF _ g, of "{z <..}"]
```
```  2003       by (intro exI[of _ "f -` {..<z} \<inter> g -` {z<..}"]) auto
```
```  2004   next
```
```  2005     case False
```
```  2006     then have *: "{g t ..} = {f t <..}" "{..< g t} = {.. f t}"
```
```  2007       using t by (auto intro: leI)
```
```  2008     show ?thesis
```
```  2009       using open_vimage[OF _ f, of "{..< g t}"] open_vimage[OF _ g, of "{f t <..}"] t
```
```  2010       apply (intro exI[of _ "f -` {..< g t} \<inter> g -` {f t<..}"])
```
```  2011       apply (simp add: open_Int)
```
```  2012       apply (auto simp add: *)
```
```  2013       done
```
```  2014   qed
```
```  2015 qed
```
```  2016
```
```  2017 lemma closed_Collect_le:
```
```  2018   fixes f g :: "'a :: topological_space \<Rightarrow> 'b::linorder_topology"
```
```  2019   assumes f: "continuous_on UNIV f"
```
```  2020     and g: "continuous_on UNIV g"
```
```  2021   shows "closed {x. f x \<le> g x}"
```
```  2022   using open_Collect_less [OF g f]
```
```  2023   by (simp add: closed_def Collect_neg_eq[symmetric] not_le)
```
```  2024
```
```  2025
```
```  2026 subsubsection \<open>Open-cover compactness\<close>
```
```  2027
```
```  2028 context topological_space
```
```  2029 begin
```
```  2030
```
```  2031 definition compact :: "'a set \<Rightarrow> bool"
```
```  2032   where compact_eq_heine_borel:  (* This name is used for backwards compatibility *)
```
```  2033     "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))"
```
```  2034
```
```  2035 lemma compactI:
```
```  2036   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'"
```
```  2037   shows "compact s"
```
```  2038   unfolding compact_eq_heine_borel using assms by metis
```
```  2039
```
```  2040 lemma compact_empty[simp]: "compact {}"
```
```  2041   by (auto intro!: compactI)
```
```  2042
```
```  2043 lemma compactE:
```
```  2044   assumes "compact s"
```
```  2045     and "\<forall>t\<in>C. open t"
```
```  2046     and "s \<subseteq> \<Union>C"
```
```  2047   obtains C' where "C' \<subseteq> C" and "finite C'" and "s \<subseteq> \<Union>C'"
```
```  2048   using assms unfolding compact_eq_heine_borel by metis
```
```  2049
```
```  2050 lemma compactE_image:
```
```  2051   assumes "compact s"
```
```  2052     and "\<forall>t\<in>C. open (f t)"
```
```  2053     and "s \<subseteq> (\<Union>c\<in>C. f c)"
```
```  2054   obtains C' where "C' \<subseteq> C" and "finite C'" and "s \<subseteq> (\<Union>c\<in>C'. f c)"
```
```  2055   using assms unfolding ball_simps [symmetric]
```
```  2056   by (metis (lifting) finite_subset_image compact_eq_heine_borel[of s])
```
```  2057
```
```  2058 lemma compact_Int_closed [intro]:
```
```  2059   assumes "compact s"
```
```  2060     and "closed t"
```
```  2061   shows "compact (s \<inter> t)"
```
```  2062 proof (rule compactI)
```
```  2063   fix C
```
```  2064   assume C: "\<forall>c\<in>C. open c"
```
```  2065   assume cover: "s \<inter> t \<subseteq> \<Union>C"
```
```  2066   from C \<open>closed t\<close> have "\<forall>c\<in>C \<union> {- t}. open c"
```
```  2067     by auto
```
```  2068   moreover from cover have "s \<subseteq> \<Union>(C \<union> {- t})"
```
```  2069     by auto
```
```  2070   ultimately have "\<exists>D\<subseteq>C \<union> {- t}. finite D \<and> s \<subseteq> \<Union>D"
```
```  2071     using \<open>compact s\<close> unfolding compact_eq_heine_borel by auto
```
```  2072   then obtain D where "D \<subseteq> C \<union> {- t} \<and> finite D \<and> s \<subseteq> \<Union>D" ..
```
```  2073   then show "\<exists>D\<subseteq>C. finite D \<and> s \<inter> t \<subseteq> \<Union>D"
```
```  2074     by (intro exI[of _ "D - {-t}"]) auto
```
```  2075 qed
```
```  2076
```
```  2077 lemma inj_setminus: "inj_on uminus (A::'a set set)"
```
```  2078   by (auto simp: inj_on_def)
```
```  2079
```
```  2080
```
```  2081 subsection \<open>Finite intersection property\<close>
```
```  2082
```
```  2083 lemma compact_fip:
```
```  2084   "compact U \<longleftrightarrow>
```
```  2085     (\<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> {})"
```
```  2086   (is "_ \<longleftrightarrow> ?R")
```
```  2087 proof (safe intro!: compact_eq_heine_borel[THEN iffD2])
```
```  2088   fix A
```
```  2089   assume "compact U"
```
```  2090   assume A: "\<forall>a\<in>A. closed a" "U \<inter> \<Inter>A = {}"
```
```  2091   assume fin: "\<forall>B \<subseteq> A. finite B \<longrightarrow> U \<inter> \<Inter>B \<noteq> {}"
```
```  2092   from A have "(\<forall>a\<in>uminus`A. open a) \<and> U \<subseteq> \<Union>(uminus`A)"
```
```  2093     by auto
```
```  2094   with \<open>compact U\<close> obtain B where "B \<subseteq> A" "finite (uminus`B)" "U \<subseteq> \<Union>(uminus`B)"
```
```  2095     unfolding compact_eq_heine_borel by (metis subset_image_iff)
```
```  2096   with fin[THEN spec, of B] show False
```
```  2097     by (auto dest: finite_imageD intro: inj_setminus)
```
```  2098 next
```
```  2099   fix A
```
```  2100   assume ?R
```
```  2101   assume "\<forall>a\<in>A. open a" "U \<subseteq> \<Union>A"
```
```  2102   then have "U \<inter> \<Inter>(uminus`A) = {}" "\<forall>a\<in>uminus`A. closed a"
```
```  2103     by auto
```
```  2104   with \<open>?R\<close> obtain B where "B \<subseteq> A" "finite (uminus`B)" "U \<inter> \<Inter>(uminus`B) = {}"
```
```  2105     by (metis subset_image_iff)
```
```  2106   then show "\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T"
```
```  2107     by (auto intro!: exI[of _ B] inj_setminus dest: finite_imageD)
```
```  2108 qed
```
```  2109
```
```  2110 lemma compact_imp_fip:
```
```  2111   assumes "compact S"
```
```  2112     and "\<And>T. T \<in> F \<Longrightarrow> closed T"
```
```  2113     and "\<And>F'. finite F' \<Longrightarrow> F' \<subseteq> F \<Longrightarrow> S \<inter> (\<Inter>F') \<noteq> {}"
```
```  2114   shows "S \<inter> (\<Inter>F) \<noteq> {}"
```
```  2115   using assms unfolding compact_fip by auto
```
```  2116
```
```  2117 lemma compact_imp_fip_image:
```
```  2118   assumes "compact s"
```
```  2119     and P: "\<And>i. i \<in> I \<Longrightarrow> closed (f i)"
```
```  2120     and Q: "\<And>I'. finite I' \<Longrightarrow> I' \<subseteq> I \<Longrightarrow> (s \<inter> (\<Inter>i\<in>I'. f i) \<noteq> {})"
```
```  2121   shows "s \<inter> (\<Inter>i\<in>I. f i) \<noteq> {}"
```
```  2122 proof -
```
```  2123   note \<open>compact s\<close>
```
```  2124   moreover from P have "\<forall>i \<in> f ` I. closed i"
```
```  2125     by blast
```
```  2126   moreover have "\<forall>A. finite A \<and> A \<subseteq> f ` I \<longrightarrow> (s \<inter> (\<Inter>A) \<noteq> {})"
```
```  2127     apply rule
```
```  2128     apply rule
```
```  2129     apply (erule conjE)
```
```  2130   proof -
```
```  2131     fix A :: "'a set set"
```
```  2132     assume "finite A" and "A \<subseteq> f ` I"
```
```  2133     then obtain B where "B \<subseteq> I" and "finite B" and "A = f ` B"
```
```  2134       using finite_subset_image [of A f I] by blast
```
```  2135     with Q [of B] show "s \<inter> \<Inter>A \<noteq> {}"
```
```  2136       by simp
```
```  2137   qed
```
```  2138   ultimately have "s \<inter> (\<Inter>(f ` I)) \<noteq> {}"
```
```  2139     by (metis compact_imp_fip)
```
```  2140   then show ?thesis by simp
```
```  2141 qed
```
```  2142
```
```  2143 end
```
```  2144
```
```  2145 lemma (in t2_space) compact_imp_closed:
```
```  2146   assumes "compact s"
```
```  2147   shows "closed s"
```
```  2148   unfolding closed_def
```
```  2149 proof (rule openI)
```
```  2150   fix y
```
```  2151   assume "y \<in> - s"
```
```  2152   let ?C = "\<Union>x\<in>s. {u. open u \<and> x \<in> u \<and> eventually (\<lambda>y. y \<notin> u) (nhds y)}"
```
```  2153   note \<open>compact s\<close>
```
```  2154   moreover have "\<forall>u\<in>?C. open u" by simp
```
```  2155   moreover have "s \<subseteq> \<Union>?C"
```
```  2156   proof
```
```  2157     fix x
```
```  2158     assume "x \<in> s"
```
```  2159     with \<open>y \<in> - s\<close> have "x \<noteq> y" by clarsimp
```
```  2160     then have "\<exists>u v. open u \<and> open v \<and> x \<in> u \<and> y \<in> v \<and> u \<inter> v = {}"
```
```  2161       by (rule hausdorff)
```
```  2162     with \<open>x \<in> s\<close> show "x \<in> \<Union>?C"
```
```  2163       unfolding eventually_nhds by auto
```
```  2164   qed
```
```  2165   ultimately obtain D where "D \<subseteq> ?C" and "finite D" and "s \<subseteq> \<Union>D"
```
```  2166     by (rule compactE)
```
```  2167   from \<open>D \<subseteq> ?C\<close> have "\<forall>x\<in>D. eventually (\<lambda>y. y \<notin> x) (nhds y)"
```
```  2168     by auto
```
```  2169   with \<open>finite D\<close> have "eventually (\<lambda>y. y \<notin> \<Union>D) (nhds y)"
```
```  2170     by (simp add: eventually_ball_finite)
```
```  2171   with \<open>s \<subseteq> \<Union>D\<close> have "eventually (\<lambda>y. y \<notin> s) (nhds y)"
```
```  2172     by (auto elim!: eventually_mono)
```
```  2173   then show "\<exists>t. open t \<and> y \<in> t \<and> t \<subseteq> - s"
```
```  2174     by (simp add: eventually_nhds subset_eq)
```
```  2175 qed
```
```  2176
```
```  2177 lemma compact_continuous_image:
```
```  2178   assumes f: "continuous_on s f"
```
```  2179     and s: "compact s"
```
```  2180   shows "compact (f ` s)"
```
```  2181 proof (rule compactI)
```
```  2182   fix C
```
```  2183   assume "\<forall>c\<in>C. open c" and cover: "f`s \<subseteq> \<Union>C"
```
```  2184   with f have "\<forall>c\<in>C. \<exists>A. open A \<and> A \<inter> s = f -` c \<inter> s"
```
```  2185     unfolding continuous_on_open_invariant by blast
```
```  2186   then obtain A where A: "\<forall>c\<in>C. open (A c) \<and> A c \<inter> s = f -` c \<inter> s"
```
```  2187     unfolding bchoice_iff ..
```
```  2188   with cover have "\<forall>c\<in>C. open (A c)" "s \<subseteq> (\<Union>c\<in>C. A c)"
```
```  2189     by (fastforce simp add: subset_eq set_eq_iff)+
```
```  2190   from compactE_image[OF s this] obtain D where "D \<subseteq> C" "finite D" "s \<subseteq> (\<Union>c\<in>D. A c)" .
```
```  2191   with A show "\<exists>D \<subseteq> C. finite D \<and> f`s \<subseteq> \<Union>D"
```
```  2192     by (intro exI[of _ D]) (fastforce simp add: subset_eq set_eq_iff)+
```
```  2193 qed
```
```  2194
```
```  2195 lemma continuous_on_inv:
```
```  2196   fixes f :: "'a::topological_space \<Rightarrow> 'b::t2_space"
```
```  2197   assumes "continuous_on s f"
```
```  2198     and "compact s"
```
```  2199     and "\<forall>x\<in>s. g (f x) = x"
```
```  2200   shows "continuous_on (f ` s) g"
```
```  2201   unfolding continuous_on_topological
```
```  2202 proof (clarsimp simp add: assms(3))
```
```  2203   fix x :: 'a and B :: "'a set"
```
```  2204   assume "x \<in> s" and "open B" and "x \<in> B"
```
```  2205   have 1: "\<forall>x\<in>s. f x \<in> f ` (s - B) \<longleftrightarrow> x \<in> s - B"
```
```  2206     using assms(3) by (auto, metis)
```
```  2207   have "continuous_on (s - B) f"
```
```  2208     using \<open>continuous_on s f\<close> Diff_subset
```
```  2209     by (rule continuous_on_subset)
```
```  2210   moreover have "compact (s - B)"
```
```  2211     using \<open>open B\<close> and \<open>compact s\<close>
```
```  2212     unfolding Diff_eq by (intro compact_Int_closed closed_Compl)
```
```  2213   ultimately have "compact (f ` (s - B))"
```
```  2214     by (rule compact_continuous_image)
```
```  2215   then have "closed (f ` (s - B))"
```
```  2216     by (rule compact_imp_closed)
```
```  2217   then have "open (- f ` (s - B))"
```
```  2218     by (rule open_Compl)
```
```  2219   moreover have "f x \<in> - f ` (s - B)"
```
```  2220     using \<open>x \<in> s\<close> and \<open>x \<in> B\<close> by (simp add: 1)
```
```  2221   moreover have "\<forall>y\<in>s. f y \<in> - f ` (s - B) \<longrightarrow> y \<in> B"
```
```  2222     by (simp add: 1)
```
```  2223   ultimately show "\<exists>A. open A \<and> f x \<in> A \<and> (\<forall>y\<in>s. f y \<in> A \<longrightarrow> y \<in> B)"
```
```  2224     by fast
```
```  2225 qed
```
```  2226
```
```  2227 lemma continuous_on_inv_into:
```
```  2228   fixes f :: "'a::topological_space \<Rightarrow> 'b::t2_space"
```
```  2229   assumes s: "continuous_on s f" "compact s"
```
```  2230     and f: "inj_on f s"
```
```  2231   shows "continuous_on (f ` s) (the_inv_into s f)"
```
```  2232   by (rule continuous_on_inv[OF s]) (auto simp: the_inv_into_f_f[OF f])
```
```  2233
```
```  2234 lemma (in linorder_topology) compact_attains_sup:
```
```  2235   assumes "compact S" "S \<noteq> {}"
```
```  2236   shows "\<exists>s\<in>S. \<forall>t\<in>S. t \<le> s"
```
```  2237 proof (rule classical)
```
```  2238   assume "\<not> (\<exists>s\<in>S. \<forall>t\<in>S. t \<le> s)"
```
```  2239   then obtain t where t: "\<forall>s\<in>S. t s \<in> S" and "\<forall>s\<in>S. s < t s"
```
```  2240     by (metis not_le)
```
```  2241   then have "\<forall>s\<in>S. open {..< t s}" "S \<subseteq> (\<Union>s\<in>S. {..< t s})"
```
```  2242     by auto
```
```  2243   with \<open>compact S\<close> obtain C where "C \<subseteq> S" "finite C" and C: "S \<subseteq> (\<Union>s\<in>C. {..< t s})"
```
```  2244     by (erule compactE_image)
```
```  2245   with \<open>S \<noteq> {}\<close> have Max: "Max (t`C) \<in> t`C" and "\<forall>s\<in>t`C. s \<le> Max (t`C)"
```
```  2246     by (auto intro!: Max_in)
```
```  2247   with C have "S \<subseteq> {..< Max (t`C)}"
```
```  2248     by (auto intro: less_le_trans simp: subset_eq)
```
```  2249   with t Max \<open>C \<subseteq> S\<close> show ?thesis
```
```  2250     by fastforce
```
```  2251 qed
```
```  2252
```
```  2253 lemma (in linorder_topology) compact_attains_inf:
```
```  2254   assumes "compact S" "S \<noteq> {}"
```
```  2255   shows "\<exists>s\<in>S. \<forall>t\<in>S. s \<le> t"
```
```  2256 proof (rule classical)
```
```  2257   assume "\<not> (\<exists>s\<in>S. \<forall>t\<in>S. s \<le> t)"
```
```  2258   then obtain t where t: "\<forall>s\<in>S. t s \<in> S" and "\<forall>s\<in>S. t s < s"
```
```  2259     by (metis not_le)
```
```  2260   then have "\<forall>s\<in>S. open {t s <..}" "S \<subseteq> (\<Union>s\<in>S. {t s <..})"
```
```  2261     by auto
```
```  2262   with \<open>compact S\<close> obtain C where "C \<subseteq> S" "finite C" and C: "S \<subseteq> (\<Union>s\<in>C. {t s <..})"
```
```  2263     by (erule compactE_image)
```
```  2264   with \<open>S \<noteq> {}\<close> have Min: "Min (t`C) \<in> t`C" and "\<forall>s\<in>t`C. Min (t`C) \<le> s"
```
```  2265     by (auto intro!: Min_in)
```
```  2266   with C have "S \<subseteq> {Min (t`C) <..}"
```
```  2267     by (auto intro: le_less_trans simp: subset_eq)
```
```  2268   with t Min \<open>C \<subseteq> S\<close> show ?thesis
```
```  2269     by fastforce
```
```  2270 qed
```
```  2271
```
```  2272 lemma continuous_attains_sup:
```
```  2273   fixes f :: "'a::topological_space \<Rightarrow> 'b::linorder_topology"
```
```  2274   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)"
```
```  2275   using compact_attains_sup[of "f ` s"] compact_continuous_image[of s f] by auto
```
```  2276
```
```  2277 lemma continuous_attains_inf:
```
```  2278   fixes f :: "'a::topological_space \<Rightarrow> 'b::linorder_topology"
```
```  2279   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)"
```
```  2280   using compact_attains_inf[of "f ` s"] compact_continuous_image[of s f] by auto
```
```  2281
```
```  2282
```
```  2283 subsection \<open>Connectedness\<close>
```
```  2284
```
```  2285 context topological_space
```
```  2286 begin
```
```  2287
```
```  2288 definition "connected S \<longleftrightarrow>
```
```  2289   \<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> {})"
```
```  2290
```
```  2291 lemma connectedI:
```
```  2292   "(\<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)
```
```  2293   \<Longrightarrow> connected U"
```
```  2294   by (auto simp: connected_def)
```
```  2295
```
```  2296 lemma connected_empty [simp]: "connected {}"
```
```  2297   by (auto intro!: connectedI)
```
```  2298
```
```  2299 lemma connected_sing [simp]: "connected {x}"
```
```  2300   by (auto intro!: connectedI)
```
```  2301
```
```  2302 lemma connectedD:
```
```  2303   "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 = {}"
```
```  2304   by (auto simp: connected_def)
```
```  2305
```
```  2306 end
```
```  2307
```
```  2308 lemma connected_closed:
```
```  2309   "connected s \<longleftrightarrow>
```
```  2310     \<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> {})"
```
```  2311   apply (simp add: connected_def del: ex_simps, safe)
```
```  2312    apply (drule_tac x="-A" in spec)
```
```  2313    apply (drule_tac x="-B" in spec)
```
```  2314    apply (fastforce simp add: closed_def [symmetric])
```
```  2315   apply (drule_tac x="-A" in spec)
```
```  2316   apply (drule_tac x="-B" in spec)
```
```  2317   apply (fastforce simp add: open_closed [symmetric])
```
```  2318   done
```
```  2319
```
```  2320 lemma connected_closedD:
```
```  2321   "\<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 = {}"
```
```  2322   by (simp add: connected_closed)
```
```  2323
```
```  2324 lemma connected_Union:
```
```  2325   assumes cs: "\<And>s. s \<in> S \<Longrightarrow> connected s"
```
```  2326     and ne: "\<Inter>S \<noteq> {}"
```
```  2327   shows "connected(\<Union>S)"
```
```  2328 proof (rule connectedI)
```
```  2329   fix A B
```
```  2330   assume A: "open A" and B: "open B" and Alap: "A \<inter> \<Union>S \<noteq> {}" and Blap: "B \<inter> \<Union>S \<noteq> {}"
```
```  2331     and disj: "A \<inter> B \<inter> \<Union>S = {}" and cover: "\<Union>S \<subseteq> A \<union> B"
```
```  2332   have disjs:"\<And>s. s \<in> S \<Longrightarrow> A \<inter> B \<inter> s = {}"
```
```  2333     using disj by auto
```
```  2334   obtain sa where sa: "sa \<in> S" "A \<inter> sa \<noteq> {}"
```
```  2335     using Alap by auto
```
```  2336   obtain sb where sb: "sb \<in> S" "B \<inter> sb \<noteq> {}"
```
```  2337     using Blap by auto
```
```  2338   obtain x where x: "\<And>s. s \<in> S \<Longrightarrow> x \<in> s"
```
```  2339     using ne by auto
```
```  2340   then have "x \<in> \<Union>S"
```
```  2341     using \<open>sa \<in> S\<close> by blast
```
```  2342   then have "x \<in> A \<or> x \<in> B"
```
```  2343     using cover by auto
```
```  2344   then show False
```
```  2345     using cs [unfolded connected_def]
```
```  2346     by (metis A B IntI Sup_upper sa sb disjs x cover empty_iff subset_trans)
```
```  2347 qed
```
```  2348
```
```  2349 lemma connected_Un: "connected s \<Longrightarrow> connected t \<Longrightarrow> s \<inter> t \<noteq> {} \<Longrightarrow> connected (s \<union> t)"
```
```  2350   using connected_Union [of "{s,t}"] by auto
```
```  2351
```
```  2352 lemma connected_diff_open_from_closed:
```
```  2353   assumes st: "s \<subseteq> t"
```
```  2354     and tu: "t \<subseteq> u"
```
```  2355     and s: "open s"
```
```  2356     and t: "closed t"
```
```  2357     and u: "connected u"
```
```  2358     and ts: "connected (t - s)"
```
```  2359   shows "connected(u - s)"
```
```  2360 proof (rule connectedI)
```
```  2361   fix A B
```
```  2362   assume AB: "open A" "open B" "A \<inter> (u - s) \<noteq> {}" "B \<inter> (u - s) \<noteq> {}"
```
```  2363     and disj: "A \<inter> B \<inter> (u - s) = {}"
```
```  2364     and cover: "u - s \<subseteq> A \<union> B"
```
```  2365   then consider "A \<inter> (t - s) = {}" | "B \<inter> (t - s) = {}"
```
```  2366     using st ts tu connectedD [of "t-s" "A" "B"] by auto
```
```  2367   then show False
```
```  2368   proof cases
```
```  2369     case 1
```
```  2370     then have "(A - t) \<inter> (B \<union> s) \<inter> u = {}"
```
```  2371       using disj st by auto
```
```  2372     moreover have "u \<subseteq> (A - t) \<union> (B \<union> s)"
```
```  2373       using 1 cover by auto
```
```  2374     ultimately show False
```
```  2375       using connectedD [of u "A - t" "B \<union> s"] AB s t 1 u by auto
```
```  2376   next
```
```  2377     case 2
```
```  2378     then have "(A \<union> s) \<inter> (B - t) \<inter> u = {}"
```
```  2379       using disj st by auto
```
```  2380     moreover have "u \<subseteq> (A \<union> s) \<union> (B - t)"
```
```  2381       using 2 cover by auto
```
```  2382     ultimately show False
```
```  2383       using connectedD [of u "A \<union> s" "B - t"] AB s t 2 u by auto
```
```  2384   qed
```
```  2385 qed
```
```  2386
```
```  2387 lemma connected_iff_const:
```
```  2388   fixes S :: "'a::topological_space set"
```
```  2389   shows "connected S \<longleftrightarrow> (\<forall>P::'a \<Rightarrow> bool. continuous_on S P \<longrightarrow> (\<exists>c. \<forall>s\<in>S. P s = c))"
```
```  2390 proof safe
```
```  2391   fix P :: "'a \<Rightarrow> bool"
```
```  2392   assume "connected S" "continuous_on S P"
```
```  2393   then have "\<And>b. \<exists>A. open A \<and> A \<inter> S = P -` {b} \<inter> S"
```
```  2394     unfolding continuous_on_open_invariant by (simp add: open_discrete)
```
```  2395   from this[of True] this[of False]
```
```  2396   obtain t f where "open t" "open f" and *: "f \<inter> S = P -` {False} \<inter> S" "t \<inter> S = P -` {True} \<inter> S"
```
```  2397     by meson
```
```  2398   then have "t \<inter> S = {} \<or> f \<inter> S = {}"
```
```  2399     by (intro connectedD[OF \<open>connected S\<close>])  auto
```
```  2400   then show "\<exists>c. \<forall>s\<in>S. P s = c"
```
```  2401   proof (rule disjE)
```
```  2402     assume "t \<inter> S = {}"
```
```  2403     then show ?thesis
```
```  2404       unfolding * by (intro exI[of _ False]) auto
```
```  2405   next
```
```  2406     assume "f \<inter> S = {}"
```
```  2407     then show ?thesis
```
```  2408       unfolding * by (intro exI[of _ True]) auto
```
```  2409   qed
```
```  2410 next
```
```  2411   assume P: "\<forall>P::'a \<Rightarrow> bool. continuous_on S P \<longrightarrow> (\<exists>c. \<forall>s\<in>S. P s = c)"
```
```  2412   show "connected S"
```
```  2413   proof (rule connectedI)
```
```  2414     fix A B
```
```  2415     assume *: "open A" "open B" "A \<inter> S \<noteq> {}" "B \<inter> S \<noteq> {}" "A \<inter> B \<inter> S = {}" "S \<subseteq> A \<union> B"
```
```  2416     have "continuous_on S (\<lambda>x. x \<in> A)"
```
```  2417       unfolding continuous_on_open_invariant
```
```  2418     proof safe
```
```  2419       fix C :: "bool set"
```
```  2420       have "C = UNIV \<or> C = {True} \<or> C = {False} \<or> C = {}"
```
```  2421         using subset_UNIV[of C] unfolding UNIV_bool by auto
```
```  2422       with * show "\<exists>T. open T \<and> T \<inter> S = (\<lambda>x. x \<in> A) -` C \<inter> S"
```
```  2423         by (intro exI[of _ "(if True \<in> C then A else {}) \<union> (if False \<in> C then B else {})"]) auto
```
```  2424     qed
```
```  2425     from P[rule_format, OF this] obtain c where "\<And>s. s \<in> S \<Longrightarrow> (s \<in> A) = c"
```
```  2426       by blast
```
```  2427     with * show False
```
```  2428       by (cases c) auto
```
```  2429   qed
```
```  2430 qed
```
```  2431
```
```  2432 lemma connectedD_const: "connected S \<Longrightarrow> continuous_on S P \<Longrightarrow> \<exists>c. \<forall>s\<in>S. P s = c"
```
```  2433   for P :: "'a::topological_space \<Rightarrow> bool"
```
```  2434   by (auto simp: connected_iff_const)
```
```  2435
```
```  2436 lemma connectedI_const:
```
```  2437   "(\<And>P::'a::topological_space \<Rightarrow> bool. continuous_on S P \<Longrightarrow> \<exists>c. \<forall>s\<in>S. P s = c) \<Longrightarrow> connected S"
```
```  2438   by (auto simp: connected_iff_const)
```
```  2439
```
```  2440 lemma connected_local_const:
```
```  2441   assumes "connected A" "a \<in> A" "b \<in> A"
```
```  2442     and *: "\<forall>a\<in>A. eventually (\<lambda>b. f a = f b) (at a within A)"
```
```  2443   shows "f a = f b"
```
```  2444 proof -
```
```  2445   obtain S where S: "\<And>a. a \<in> A \<Longrightarrow> a \<in> S a" "\<And>a. a \<in> A \<Longrightarrow> open (S a)"
```
```  2446     "\<And>a x. a \<in> A \<Longrightarrow> x \<in> S a \<Longrightarrow> x \<in> A \<Longrightarrow> f a = f x"
```
```  2447     using * unfolding eventually_at_topological by metis
```
```  2448   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"
```
```  2449   have "?P \<inter> A = {} \<or> ?N \<inter> A = {}"
```
```  2450     using \<open>connected A\<close> S \<open>a\<in>A\<close>
```
```  2451     by (intro connectedD) (auto, metis)
```
```  2452   then show "f a = f b"
```
```  2453   proof
```
```  2454     assume "?N \<inter> A = {}"
```
```  2455     then have "\<forall>x\<in>A. f a = f x"
```
```  2456       using S(1) by auto
```
```  2457     with \<open>b\<in>A\<close> show ?thesis by auto
```
```  2458   next
```
```  2459     assume "?P \<inter> A = {}" then show ?thesis
```
```  2460       using \<open>a \<in> A\<close> S(1)[of a] by auto
```
```  2461   qed
```
```  2462 qed
```
```  2463
```
```  2464 lemma (in linorder_topology) connectedD_interval:
```
```  2465   assumes "connected U"
```
```  2466     and xy: "x \<in> U" "y \<in> U"
```
```  2467     and "x \<le> z" "z \<le> y"
```
```  2468   shows "z \<in> U"
```
```  2469 proof -
```
```  2470   have eq: "{..<z} \<union> {z<..} = - {z}"
```
```  2471     by auto
```
```  2472   have "\<not> connected U" if "z \<notin> U" "x < z" "z < y"
```
```  2473     using xy that
```
```  2474     apply (simp only: connected_def simp_thms)
```
```  2475     apply (rule_tac exI[of _ "{..< z}"])
```
```  2476     apply (rule_tac exI[of _ "{z <..}"])
```
```  2477     apply (auto simp add: eq)
```
```  2478     done
```
```  2479   with assms show "z \<in> U"
```
```  2480     by (metis less_le)
```
```  2481 qed
```
```  2482
```
```  2483 lemma connected_continuous_image:
```
```  2484   assumes *: "continuous_on s f"
```
```  2485     and "connected s"
```
```  2486   shows "connected (f ` s)"
```
```  2487 proof (rule connectedI_const)
```
```  2488   fix P :: "'b \<Rightarrow> bool"
```
```  2489   assume "continuous_on (f ` s) P"
```
```  2490   then have "continuous_on s (P \<circ> f)"
```
```  2491     by (rule continuous_on_compose[OF *])
```
```  2492   from connectedD_const[OF \<open>connected s\<close> this] show "\<exists>c. \<forall>s\<in>f ` s. P s = c"
```
```  2493     by auto
```
```  2494 qed
```
```  2495
```
```  2496
```
```  2497 section \<open>Linear Continuum Topologies\<close>
```
```  2498
```
```  2499 class linear_continuum_topology = linorder_topology + linear_continuum
```
```  2500 begin
```
```  2501
```
```  2502 lemma Inf_notin_open:
```
```  2503   assumes A: "open A"
```
```  2504     and bnd: "\<forall>a\<in>A. x < a"
```
```  2505   shows "Inf A \<notin> A"
```
```  2506 proof
```
```  2507   assume "Inf A \<in> A"
```
```  2508   then obtain b where "b < Inf A" "{b <.. Inf A} \<subseteq> A"
```
```  2509     using open_left[of A "Inf A" x] assms by auto
```
```  2510   with dense[of b "Inf A"] obtain c where "c < Inf A" "c \<in> A"
```
```  2511     by (auto simp: subset_eq)
```
```  2512   then show False
```
```  2513     using cInf_lower[OF \<open>c \<in> A\<close>] bnd
```
```  2514     by (metis not_le less_imp_le bdd_belowI)
```
```  2515 qed
```
```  2516
```
```  2517 lemma Sup_notin_open:
```
```  2518   assumes A: "open A"
```
```  2519     and bnd: "\<forall>a\<in>A. a < x"
```
```  2520   shows "Sup A \<notin> A"
```
```  2521 proof
```
```  2522   assume "Sup A \<in> A"
```
```  2523   with assms obtain b where "Sup A < b" "{Sup A ..< b} \<subseteq> A"
```
```  2524     using open_right[of A "Sup A" x] by auto
```
```  2525   with dense[of "Sup A" b] obtain c where "Sup A < c" "c \<in> A"
```
```  2526     by (auto simp: subset_eq)
```
```  2527   then show False
```
```  2528     using cSup_upper[OF \<open>c \<in> A\<close>] bnd
```
```  2529     by (metis less_imp_le not_le bdd_aboveI)
```
```  2530 qed
```
```  2531
```
```  2532 end
```
```  2533
```
```  2534 instance linear_continuum_topology \<subseteq> perfect_space
```
```  2535 proof
```
```  2536   fix x :: 'a
```
```  2537   obtain y where "x < y \<or> y < x"
```
```  2538     using ex_gt_or_lt [of x] ..
```
```  2539   with Inf_notin_open[of "{x}" y] Sup_notin_open[of "{x}" y] show "\<not> open {x}"
```
```  2540     by auto
```
```  2541 qed
```
```  2542
```
```  2543 lemma connectedI_interval:
```
```  2544   fixes U :: "'a :: linear_continuum_topology set"
```
```  2545   assumes *: "\<And>x y z. x \<in> U \<Longrightarrow> y \<in> U \<Longrightarrow> x \<le> z \<Longrightarrow> z \<le> y \<Longrightarrow> z \<in> U"
```
```  2546   shows "connected U"
```
```  2547 proof (rule connectedI)
```
```  2548   {
```
```  2549     fix A B
```
```  2550     assume "open A" "open B" "A \<inter> B \<inter> U = {}" "U \<subseteq> A \<union> B"
```
```  2551     fix x y
```
```  2552     assume "x < y" "x \<in> A" "y \<in> B" "x \<in> U" "y \<in> U"
```
```  2553
```
```  2554     let ?z = "Inf (B \<inter> {x <..})"
```
```  2555
```
```  2556     have "x \<le> ?z" "?z \<le> y"
```
```  2557       using \<open>y \<in> B\<close> \<open>x < y\<close> by (auto intro: cInf_lower cInf_greatest)
```
```  2558     with \<open>x \<in> U\<close> \<open>y \<in> U\<close> have "?z \<in> U"
```
```  2559       by (rule *)
```
```  2560     moreover have "?z \<notin> B \<inter> {x <..}"
```
```  2561       using \<open>open B\<close> by (intro Inf_notin_open) auto
```
```  2562     ultimately have "?z \<in> A"
```
```  2563       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
```
```  2564     have "\<exists>b\<in>B. b \<in> A \<and> b \<in> U" if "?z < y"
```
```  2565     proof -
```
```  2566       obtain a where "?z < a" "{?z ..< a} \<subseteq> A"
```
```  2567         using open_right[OF \<open>open A\<close> \<open>?z \<in> A\<close> \<open>?z < y\<close>] by auto
```
```  2568       moreover obtain b where "b \<in> B" "x < b" "b < min a y"
```
```  2569         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>
```
```  2570         by auto
```
```  2571       moreover have "?z \<le> b"
```
```  2572         using \<open>b \<in> B\<close> \<open>x < b\<close>
```
```  2573         by (intro cInf_lower) auto
```
```  2574       moreover have "b \<in> U"
```
```  2575         using \<open>x \<le> ?z\<close> \<open>?z \<le> b\<close> \<open>b < min a y\<close>
```
```  2576         by (intro *[OF \<open>x \<in> U\<close> \<open>y \<in> U\<close>]) (auto simp: less_imp_le)
```
```  2577       ultimately show ?thesis
```
```  2578         by (intro bexI[of _ b]) auto
```
```  2579     qed
```
```  2580     then have False
```
```  2581       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>
```
```  2582       unfolding le_less by blast
```
```  2583   }
```
```  2584   note not_disjoint = this
```
```  2585
```
```  2586   fix A B assume AB: "open A" "open B" "U \<subseteq> A \<union> B" "A \<inter> B \<inter> U = {}"
```
```  2587   moreover assume "A \<inter> U \<noteq> {}" then obtain x where x: "x \<in> U" "x \<in> A" by auto
```
```  2588   moreover assume "B \<inter> U \<noteq> {}" then obtain y where y: "y \<in> U" "y \<in> B" by auto
```
```  2589   moreover note not_disjoint[of B A y x] not_disjoint[of A B x y]
```
```  2590   ultimately show False
```
```  2591     by (cases x y rule: linorder_cases) auto
```
```  2592 qed
```
```  2593
```
```  2594 lemma 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)"
```
```  2595   for U :: "'a::linear_continuum_topology set"
```
```  2596   by (auto intro: connectedI_interval dest: connectedD_interval)
```
```  2597
```
```  2598 lemma connected_UNIV[simp]: "connected (UNIV::'a::linear_continuum_topology set)"
```
```  2599   by (simp add: connected_iff_interval)
```
```  2600
```
```  2601 lemma connected_Ioi[simp]: "connected {a<..}"
```
```  2602   for a :: "'a::linear_continuum_topology"
```
```  2603   by (auto simp: connected_iff_interval)
```
```  2604
```
```  2605 lemma connected_Ici[simp]: "connected {a..}"
```
```  2606   for a :: "'a::linear_continuum_topology"
```
```  2607   by (auto simp: connected_iff_interval)
```
```  2608
```
```  2609 lemma connected_Iio[simp]: "connected {..<a}"
```
```  2610   for a :: "'a::linear_continuum_topology"
```
```  2611   by (auto simp: connected_iff_interval)
```
```  2612
```
```  2613 lemma connected_Iic[simp]: "connected {..a}"
```
```  2614   for a :: "'a::linear_continuum_topology"
```
```  2615   by (auto simp: connected_iff_interval)
```
```  2616
```
```  2617 lemma connected_Ioo[simp]: "connected {a<..<b}"
```
```  2618   for a b :: "'a::linear_continuum_topology"
```
```  2619   unfolding connected_iff_interval by auto
```
```  2620
```
```  2621 lemma connected_Ioc[simp]: "connected {a<..b}"
```
```  2622   for a b :: "'a::linear_continuum_topology"
```
```  2623   by (auto simp: connected_iff_interval)
```
```  2624
```
```  2625 lemma connected_Ico[simp]: "connected {a..<b}"
```
```  2626   for a b :: "'a::linear_continuum_topology"
```
```  2627   by (auto simp: connected_iff_interval)
```
```  2628
```
```  2629 lemma connected_Icc[simp]: "connected {a..b}"
```
```  2630   for a b :: "'a::linear_continuum_topology"
```
```  2631   by (auto simp: connected_iff_interval)
```
```  2632
```
```  2633 lemma connected_contains_Ioo:
```
```  2634   fixes A :: "'a :: linorder_topology set"
```
```  2635   assumes "connected A" "a \<in> A" "b \<in> A" shows "{a <..< b} \<subseteq> A"
```
```  2636   using connectedD_interval[OF assms] by (simp add: subset_eq Ball_def less_imp_le)
```
```  2637
```
```  2638 lemma connected_contains_Icc:
```
```  2639   fixes A :: "'a::linorder_topology set"
```
```  2640   assumes "connected A" "a \<in> A" "b \<in> A"
```
```  2641   shows "{a..b} \<subseteq> A"
```
```  2642 proof
```
```  2643   fix x assume "x \<in> {a..b}"
```
```  2644   then have "x = a \<or> x = b \<or> x \<in> {a<..<b}"
```
```  2645     by auto
```
```  2646   then show "x \<in> A"
```
```  2647     using assms connected_contains_Ioo[of A a b] by auto
```
```  2648 qed
```
```  2649
```
```  2650
```
```  2651 subsection \<open>Intermediate Value Theorem\<close>
```
```  2652
```
```  2653 lemma IVT':
```
```  2654   fixes f :: "'a::linear_continuum_topology \<Rightarrow> 'b::linorder_topology"
```
```  2655   assumes y: "f a \<le> y" "y \<le> f b" "a \<le> b"
```
```  2656     and *: "continuous_on {a .. b} f"
```
```  2657   shows "\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = y"
```
```  2658 proof -
```
```  2659   have "connected {a..b}"
```
```  2660     unfolding connected_iff_interval by auto
```
```  2661   from connected_continuous_image[OF * this, THEN connectedD_interval, of "f a" "f b" y] y
```
```  2662   show ?thesis
```
```  2663     by (auto simp add: atLeastAtMost_def atLeast_def atMost_def)
```
```  2664 qed
```
```  2665
```
```  2666 lemma IVT2':
```
```  2667   fixes f :: "'a :: linear_continuum_topology \<Rightarrow> 'b :: linorder_topology"
```
```  2668   assumes y: "f b \<le> y" "y \<le> f a" "a \<le> b"
```
```  2669     and *: "continuous_on {a .. b} f"
```
```  2670   shows "\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = y"
```
```  2671 proof -
```
```  2672   have "connected {a..b}"
```
```  2673     unfolding connected_iff_interval by auto
```
```  2674   from connected_continuous_image[OF * this, THEN connectedD_interval, of "f b" "f a" y] y
```
```  2675   show ?thesis
```
```  2676     by (auto simp add: atLeastAtMost_def atLeast_def atMost_def)
```
```  2677 qed
```
```  2678
```
```  2679 lemma IVT:
```
```  2680   fixes f :: "'a::linear_continuum_topology \<Rightarrow> 'b::linorder_topology"
```
```  2681   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>
```
```  2682     \<exists>x. a \<le> x \<and> x \<le> b \<and> f x = y"
```
```  2683   by (rule IVT') (auto intro: continuous_at_imp_continuous_on)
```
```  2684
```
```  2685 lemma IVT2:
```
```  2686   fixes f :: "'a::linear_continuum_topology \<Rightarrow> 'b::linorder_topology"
```
```  2687   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>
```
```  2688     \<exists>x. a \<le> x \<and> x \<le> b \<and> f x = y"
```
```  2689   by (rule IVT2') (auto intro: continuous_at_imp_continuous_on)
```
```  2690
```
```  2691 lemma continuous_inj_imp_mono:
```
```  2692   fixes f :: "'a::linear_continuum_topology \<Rightarrow> 'b::linorder_topology"
```
```  2693   assumes x: "a < x" "x < b"
```
```  2694     and cont: "continuous_on {a..b} f"
```
```  2695     and inj: "inj_on f {a..b}"
```
```  2696   shows "(f a < f x \<and> f x < f b) \<or> (f b < f x \<and> f x < f a)"
```
```  2697 proof -
```
```  2698   note I = inj_on_eq_iff[OF inj]
```
```  2699   {
```
```  2700     assume "f x < f a" "f x < f b"
```
```  2701     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"
```
```  2702       using IVT'[of f x "min (f a) (f b)" b] IVT2'[of f x "min (f a) (f b)" a] x
```
```  2703       by (auto simp: continuous_on_subset[OF cont] less_imp_le)
```
```  2704     with x I have False by auto
```
```  2705   }
```
```  2706   moreover
```
```  2707   {
```
```  2708     assume "f a < f x" "f b < f x"
```
```  2709     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"
```
```  2710       using IVT'[of f a "max (f a) (f b)" x] IVT2'[of f b "max (f a) (f b)" x] x
```
```  2711       by (auto simp: continuous_on_subset[OF cont] less_imp_le)
```
```  2712     with x I have False by auto
```
```  2713   }
```
```  2714   ultimately show ?thesis
```
```  2715     using I[of a x] I[of x b] x less_trans[OF x]
```
```  2716     by (auto simp add: le_less less_imp_neq neq_iff)
```
```  2717 qed
```
```  2718
```
```  2719 lemma continuous_at_Sup_mono:
```
```  2720   fixes f :: "'a::{linorder_topology,conditionally_complete_linorder} \<Rightarrow>
```
```  2721     'b::{linorder_topology,conditionally_complete_linorder}"
```
```  2722   assumes "mono f"
```
```  2723     and cont: "continuous (at_left (Sup S)) f"
```
```  2724     and S: "S \<noteq> {}" "bdd_above S"
```
```  2725   shows "f (Sup S) = (SUP s:S. f s)"
```
```  2726 proof (rule antisym)
```
```  2727   have f: "(f \<longlongrightarrow> f (Sup S)) (at_left (Sup S))"
```
```  2728     using cont unfolding continuous_within .
```
```  2729   show "f (Sup S) \<le> (SUP s:S. f s)"
```
```  2730   proof cases
```
```  2731     assume "Sup S \<in> S"
```
```  2732     then show ?thesis
```
```  2733       by (rule cSUP_upper) (auto intro: bdd_above_image_mono S \<open>mono f\<close>)
```
```  2734   next
```
```  2735     assume "Sup S \<notin> S"
```
```  2736     from \<open>S \<noteq> {}\<close> obtain s where "s \<in> S"
```
```  2737       by auto
```
```  2738     with \<open>Sup S \<notin> S\<close> S have "s < Sup S"
```
```  2739       unfolding less_le by (blast intro: cSup_upper)
```
```  2740     show ?thesis
```
```  2741     proof (rule ccontr)
```
```  2742       assume "\<not> ?thesis"
```
```  2743       with order_tendstoD(1)[OF f, of "SUP s:S. f s"] obtain b where "b < Sup S"
```
```  2744         and *: "\<And>y. b < y \<Longrightarrow> y < Sup S \<Longrightarrow> (SUP s:S. f s) < f y"
```
```  2745         by (auto simp: not_le eventually_at_left[OF \<open>s < Sup S\<close>])
```
```  2746       with \<open>S \<noteq> {}\<close> obtain c where "c \<in> S" "b < c"
```
```  2747         using less_cSupD[of S b] by auto
```
```  2748       with \<open>Sup S \<notin> S\<close> S have "c < Sup S"
```
```  2749         unfolding less_le by (blast intro: cSup_upper)
```
```  2750       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]]
```
```  2751       show False
```
```  2752         by (auto simp: assms)
```
```  2753     qed
```
```  2754   qed
```
```  2755 qed (intro cSUP_least \<open>mono f\<close>[THEN monoD] cSup_upper S)
```
```  2756
```
```  2757 lemma continuous_at_Sup_antimono:
```
```  2758   fixes f :: "'a::{linorder_topology,conditionally_complete_linorder} \<Rightarrow>
```
```  2759     'b::{linorder_topology,conditionally_complete_linorder}"
```
```  2760   assumes "antimono f"
```
```  2761     and cont: "continuous (at_left (Sup S)) f"
```
```  2762     and S: "S \<noteq> {}" "bdd_above S"
```
```  2763   shows "f (Sup S) = (INF s:S. f s)"
```
```  2764 proof (rule antisym)
```
```  2765   have f: "(f \<longlongrightarrow> f (Sup S)) (at_left (Sup S))"
```
```  2766     using cont unfolding continuous_within .
```
```  2767   show "(INF s:S. f s) \<le> f (Sup S)"
```
```  2768   proof cases
```
```  2769     assume "Sup S \<in> S"
```
```  2770     then show ?thesis
```
```  2771       by (intro cINF_lower) (auto intro: bdd_below_image_antimono S \<open>antimono f\<close>)
```
```  2772   next
```
```  2773     assume "Sup S \<notin> S"
```
```  2774     from \<open>S \<noteq> {}\<close> obtain s where "s \<in> S"
```
```  2775       by auto
```
```  2776     with \<open>Sup S \<notin> S\<close> S have "s < Sup S"
```
```  2777       unfolding less_le by (blast intro: cSup_upper)
```
```  2778     show ?thesis
```
```  2779     proof (rule ccontr)
```
```  2780       assume "\<not> ?thesis"
```
```  2781       with order_tendstoD(2)[OF f, of "INF s:S. f s"] obtain b where "b < Sup S"
```
```  2782         and *: "\<And>y. b < y \<Longrightarrow> y < Sup S \<Longrightarrow> f y < (INF s:S. f s)"
```
```  2783         by (auto simp: not_le eventually_at_left[OF \<open>s < Sup S\<close>])
```
```  2784       with \<open>S \<noteq> {}\<close> obtain c where "c \<in> S" "b < c"
```
```  2785         using less_cSupD[of S b] by auto
```
```  2786       with \<open>Sup S \<notin> S\<close> S have "c < Sup S"
```
```  2787         unfolding less_le by (blast intro: cSup_upper)
```
```  2788       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>
```
```  2789       show False
```
```  2790         by (auto simp: assms)
```
```  2791     qed
```
```  2792   qed
```
```  2793 qed (intro cINF_greatest \<open>antimono f\<close>[THEN antimonoD] cSup_upper S)
```
```  2794
```
```  2795 lemma continuous_at_Inf_mono:
```
```  2796   fixes f :: "'a::{linorder_topology,conditionally_complete_linorder} \<Rightarrow>
```
```  2797     'b::{linorder_topology,conditionally_complete_linorder}"
```
```  2798   assumes "mono f"
```
```  2799     and cont: "continuous (at_right (Inf S)) f"
```
```  2800     and S: "S \<noteq> {}" "bdd_below S"
```
```  2801   shows "f (Inf S) = (INF s:S. f s)"
```
```  2802 proof (rule antisym)
```
```  2803   have f: "(f \<longlongrightarrow> f (Inf S)) (at_right (Inf S))"
```
```  2804     using cont unfolding continuous_within .
```
```  2805   show "(INF s:S. f s) \<le> f (Inf S)"
```
```  2806   proof cases
```
```  2807     assume "Inf S \<in> S"
```
```  2808     then show ?thesis
```
```  2809       by (rule cINF_lower[rotated]) (auto intro: bdd_below_image_mono S \<open>mono f\<close>)
```
```  2810   next
```
```  2811     assume "Inf S \<notin> S"
```
```  2812     from \<open>S \<noteq> {}\<close> obtain s where "s \<in> S"
```
```  2813       by auto
```
```  2814     with \<open>Inf S \<notin> S\<close> S have "Inf S < s"
```
```  2815       unfolding less_le by (blast intro: cInf_lower)
```
```  2816     show ?thesis
```
```  2817     proof (rule ccontr)
```
```  2818       assume "\<not> ?thesis"
```
```  2819       with order_tendstoD(2)[OF f, of "INF s:S. f s"] obtain b where "Inf S < b"
```
```  2820         and *: "\<And>y. Inf S < y \<Longrightarrow> y < b \<Longrightarrow> f y < (INF s:S. f s)"
```
```  2821         by (auto simp: not_le eventually_at_right[OF \<open>Inf S < s\<close>])
```
```  2822       with \<open>S \<noteq> {}\<close> obtain c where "c \<in> S" "c < b"
```
```  2823         using cInf_lessD[of S b] by auto
```
```  2824       with \<open>Inf S \<notin> S\<close> S have "Inf S < c"
```
```  2825         unfolding less_le by (blast intro: cInf_lower)
```
```  2826       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>]
```
```  2827       show False
```
```  2828         by (auto simp: assms)
```
```  2829     qed
```
```  2830   qed
```
```  2831 qed (intro cINF_greatest \<open>mono f\<close>[THEN monoD] cInf_lower \<open>bdd_below S\<close> \<open>S \<noteq> {}\<close>)
```
```  2832
```
```  2833 lemma continuous_at_Inf_antimono:
```
```  2834   fixes f :: "'a::{linorder_topology,conditionally_complete_linorder} \<Rightarrow>
```
```  2835     'b::{linorder_topology,conditionally_complete_linorder}"
```
```  2836   assumes "antimono f"
```
```  2837     and cont: "continuous (at_right (Inf S)) f"
```
```  2838     and S: "S \<noteq> {}" "bdd_below S"
```
```  2839   shows "f (Inf S) = (SUP s:S. f s)"
```
```  2840 proof (rule antisym)
```
```  2841   have f: "(f \<longlongrightarrow> f (Inf S)) (at_right (Inf S))"
```
```  2842     using cont unfolding continuous_within .
```
```  2843   show "f (Inf S) \<le> (SUP s:S. f s)"
```
```  2844   proof cases
```
```  2845     assume "Inf S \<in> S"
```
```  2846     then show ?thesis
```
```  2847       by (rule cSUP_upper) (auto intro: bdd_above_image_antimono S \<open>antimono f\<close>)
```
```  2848   next
```
```  2849     assume "Inf S \<notin> S"
```
```  2850     from \<open>S \<noteq> {}\<close> obtain s where "s \<in> S"
```
```  2851       by auto
```
```  2852     with \<open>Inf S \<notin> S\<close> S have "Inf S < s"
```
```  2853       unfolding less_le by (blast intro: cInf_lower)
```
```  2854     show ?thesis
```
```  2855     proof (rule ccontr)
```
```  2856       assume "\<not> ?thesis"
```
```  2857       with order_tendstoD(1)[OF f, of "SUP s:S. f s"] obtain b where "Inf S < b"
```
```  2858         and *: "\<And>y. Inf S < y \<Longrightarrow> y < b \<Longrightarrow> (SUP s:S. f s) < f y"
```
```  2859         by (auto simp: not_le eventually_at_right[OF \<open>Inf S < s\<close>])
```
```  2860       with \<open>S \<noteq> {}\<close> obtain c where "c \<in> S" "c < b"
```
```  2861         using cInf_lessD[of S b] by auto
```
```  2862       with \<open>Inf S \<notin> S\<close> S have "Inf S < c"
```
```  2863         unfolding less_le by (blast intro: cInf_lower)
```
```  2864       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]]
```
```  2865       show False
```
```  2866         by (auto simp: assms)
```
```  2867     qed
```
```  2868   qed
```
```  2869 qed (intro cSUP_least \<open>antimono f\<close>[THEN antimonoD] cInf_lower S)
```
```  2870
```
```  2871
```
```  2872 subsection \<open>Uniform spaces\<close>
```
```  2873
```
```  2874 class uniformity =
```
```  2875   fixes uniformity :: "('a \<times> 'a) filter"
```
```  2876 begin
```
```  2877
```
```  2878 abbreviation uniformity_on :: "'a set \<Rightarrow> ('a \<times> 'a) filter"
```
```  2879   where "uniformity_on s \<equiv> inf uniformity (principal (s\<times>s))"
```
```  2880
```
```  2881 end
```
```  2882
```
```  2883 lemma uniformity_Abort:
```
```  2884   "uniformity =
```
```  2885     Filter.abstract_filter (\<lambda>u. Code.abort (STR ''uniformity is not executable'') (\<lambda>u. uniformity))"
```
```  2886   by simp
```
```  2887
```
```  2888 class open_uniformity = "open" + uniformity +
```
```  2889   assumes open_uniformity:
```
```  2890     "\<And>U. open U \<longleftrightarrow> (\<forall>x\<in>U. eventually (\<lambda>(x', y). x' = x \<longrightarrow> y \<in> U) uniformity)"
```
```  2891
```
```  2892 class uniform_space = open_uniformity +
```
```  2893   assumes uniformity_refl: "eventually E uniformity \<Longrightarrow> E (x, x)"
```
```  2894     and uniformity_sym: "eventually E uniformity \<Longrightarrow> eventually (\<lambda>(x, y). E (y, x)) uniformity"
```
```  2895     and uniformity_trans:
```
```  2896       "eventually E uniformity \<Longrightarrow>
```
```  2897         \<exists>D. eventually D uniformity \<and> (\<forall>x y z. D (x, y) \<longrightarrow> D (y, z) \<longrightarrow> E (x, z))"
```
```  2898 begin
```
```  2899
```
```  2900 subclass topological_space
```
```  2901   by standard (force elim: eventually_mono eventually_elim2 simp: split_beta' open_uniformity)+
```
```  2902
```
```  2903 lemma uniformity_bot: "uniformity \<noteq> bot"
```
```  2904   using uniformity_refl by auto
```
```  2905
```
```  2906 lemma uniformity_trans':
```
```  2907   "eventually E uniformity \<Longrightarrow>
```
```  2908     eventually (\<lambda>((x, y), (y', z)). y = y' \<longrightarrow> E (x, z)) (uniformity \<times>\<^sub>F uniformity)"
```
```  2909   by (drule uniformity_trans) (auto simp add: eventually_prod_same)
```
```  2910
```
```  2911 lemma uniformity_transE:
```
```  2912   assumes "eventually E uniformity"
```
```  2913   obtains D where "eventually D uniformity" "\<And>x y z. D (x, y) \<Longrightarrow> D (y, z) \<Longrightarrow> E (x, z)"
```
```  2914   using uniformity_trans [OF assms] by auto
```
```  2915
```
```  2916 lemma eventually_nhds_uniformity:
```
```  2917   "eventually P (nhds x) \<longleftrightarrow> eventually (\<lambda>(x', y). x' = x \<longrightarrow> P y) uniformity"
```
```  2918   (is "_ \<longleftrightarrow> ?N P x")
```
```  2919   unfolding eventually_nhds
```
```  2920 proof safe
```
```  2921   assume *: "?N P x"
```
```  2922   have "?N (?N P) x" if "?N P x" for x
```
```  2923   proof -
```
```  2924     from that obtain D where ev: "eventually D uniformity"
```
```  2925       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
```
```  2926       by (rule uniformity_transE) simp
```
```  2927     from ev show ?thesis
```
```  2928       by eventually_elim (insert ev D, force elim: eventually_mono split: prod.split)
```
```  2929   qed
```
```  2930   then have "open {x. ?N P x}"
```
```  2931     by (simp add: open_uniformity)
```
```  2932   then show "\<exists>S. open S \<and> x \<in> S \<and> (\<forall>x\<in>S. P x)"
```
```  2933     by (intro exI[of _ "{x. ?N P x}"]) (auto dest: uniformity_refl simp: *)
```
```  2934 qed (force simp add: open_uniformity elim: eventually_mono)
```
```  2935
```
```  2936
```
```  2937 subsubsection \<open>Totally bounded sets\<close>
```
```  2938
```
```  2939 definition totally_bounded :: "'a set \<Rightarrow> bool"
```
```  2940   where "totally_bounded S \<longleftrightarrow>
```
```  2941     (\<forall>E. eventually E uniformity \<longrightarrow> (\<exists>X. finite X \<and> (\<forall>s\<in>S. \<exists>x\<in>X. E (x, s))))"
```
```  2942
```
```  2943 lemma totally_bounded_empty[iff]: "totally_bounded {}"
```
```  2944   by (auto simp add: totally_bounded_def)
```
```  2945
```
```  2946 lemma totally_bounded_subset: "totally_bounded S \<Longrightarrow> T \<subseteq> S \<Longrightarrow> totally_bounded T"
```
```  2947   by (fastforce simp add: totally_bounded_def)
```
```  2948
```
```  2949 lemma totally_bounded_Union[intro]:
```
```  2950   assumes M: "finite M" "\<And>S. S \<in> M \<Longrightarrow> totally_bounded S"
```
```  2951   shows "totally_bounded (\<Union>M)"
```
```  2952   unfolding totally_bounded_def
```
```  2953 proof safe
```
```  2954   fix E
```
```  2955   assume "eventually E uniformity"
```
```  2956   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))"
```
```  2957     by (metis totally_bounded_def)
```
```  2958   with \<open>finite M\<close> show "\<exists>X. finite X \<and> (\<forall>s\<in>\<Union>M. \<exists>x\<in>X. E (x, s))"
```
```  2959     by (intro exI[of _ "\<Union>S\<in>M. X S"]) force
```
```  2960 qed
```
```  2961
```
```  2962
```
```  2963 subsubsection \<open>Cauchy filter\<close>
```
```  2964
```
```  2965 definition cauchy_filter :: "'a filter \<Rightarrow> bool"
```
```  2966   where "cauchy_filter F \<longleftrightarrow> F \<times>\<^sub>F F \<le> uniformity"
```
```  2967
```
```  2968 definition Cauchy :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"
```
```  2969   where Cauchy_uniform: "Cauchy X = cauchy_filter (filtermap X sequentially)"
```
```  2970
```
```  2971 lemma Cauchy_uniform_iff:
```
```  2972   "Cauchy X \<longleftrightarrow> (\<forall>P. eventually P uniformity \<longrightarrow> (\<exists>N. \<forall>n\<ge>N. \<forall>m\<ge>N. P (X n, X m)))"
```
```  2973   unfolding Cauchy_uniform cauchy_filter_def le_filter_def eventually_prod_same
```
```  2974     eventually_filtermap eventually_sequentially
```
```  2975 proof safe
```
```  2976   let ?U = "\<lambda>P. eventually P uniformity"
```
```  2977   {
```
```  2978     fix P
```
```  2979     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)))"
```
```  2980     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)"
```
```  2981       by metis
```
```  2982     then show "\<exists>N. \<forall>n\<ge>N. \<forall>m\<ge>N. P (X n, X m)"
```
```  2983       by blast
```
```  2984   next
```
```  2985     fix P
```
```  2986     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))"
```
```  2987     then obtain Q where "?U Q" and Q: "\<And>x y z. Q (x, y) \<Longrightarrow> Q (y, z) \<Longrightarrow> P (x, z)"
```
```  2988       by (auto elim: uniformity_transE)
```
```  2989     then have "?U (\<lambda>x. Q x \<and> (\<lambda>(x, y). Q (y, x)) x)"
```
```  2990       unfolding eventually_conj_iff by (simp add: uniformity_sym)
```
```  2991     from P[rule_format, OF this]
```
```  2992     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)"
```
```  2993       by auto
```
```  2994     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))"
```
```  2995     proof (safe intro!: exI[of _ "\<lambda>x. \<forall>n\<ge>N. Q (x, X n) \<and> Q (X n, x)"] exI[of _ N] N)
```
```  2996       fix x y
```
```  2997       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)"
```
```  2998       then have "Q (x, X N)" "Q (X N, y)" by auto
```
```  2999       then show "P (x, y)"
```
```  3000         by (rule Q)
```
```  3001     qed
```
```  3002   }
```
```  3003 qed
```
```  3004
```
```  3005 lemma nhds_imp_cauchy_filter:
```
```  3006   assumes *: "F \<le> nhds x"
```
```  3007   shows "cauchy_filter F"
```
```  3008 proof -
```
```  3009   have "F \<times>\<^sub>F F \<le> nhds x \<times>\<^sub>F nhds x"
```
```  3010     by (intro prod_filter_mono *)
```
```  3011   also have "\<dots> \<le> uniformity"
```
```  3012     unfolding le_filter_def eventually_nhds_uniformity eventually_prod_same
```
```  3013   proof safe
```
```  3014     fix P
```
```  3015     assume "eventually P uniformity"
```
```  3016     then obtain Ql where ev: "eventually Ql uniformity"
```
```  3017       and "Ql (x, y) \<Longrightarrow> Ql (y, z) \<Longrightarrow> P (x, z)" for x y z
```
```  3018       by (rule uniformity_transE) simp
```
```  3019     with ev[THEN uniformity_sym]
```
```  3020     show "\<exists>Q. eventually (\<lambda>(x', y). x' = x \<longrightarrow> Q y) uniformity \<and>
```
```  3021         (\<forall>x y. Q x \<longrightarrow> Q y \<longrightarrow> P (x, y))"
```
```  3022       by (rule_tac exI[of _ "\<lambda>y. Ql (y, x) \<and> Ql (x, y)"]) (fastforce elim: eventually_elim2)
```
```  3023   qed
```
```  3024   finally show ?thesis
```
```  3025     by (simp add: cauchy_filter_def)
```
```  3026 qed
```
```  3027
```
```  3028 lemma LIMSEQ_imp_Cauchy: "X \<longlonglongrightarrow> x \<Longrightarrow> Cauchy X"
```
```  3029   unfolding Cauchy_uniform filterlim_def by (intro nhds_imp_cauchy_filter)
```
```  3030
```
```  3031 lemma Cauchy_subseq_Cauchy:
```
```  3032   assumes "Cauchy X" "subseq f"
```
```  3033   shows "Cauchy (X \<circ> f)"
```
```  3034   unfolding Cauchy_uniform comp_def filtermap_filtermap[symmetric] cauchy_filter_def
```
```  3035   by (rule order_trans[OF _ \<open>Cauchy X\<close>[unfolded Cauchy_uniform cauchy_filter_def]])
```
```  3036      (intro prod_filter_mono filtermap_mono filterlim_subseq[OF \<open>subseq f\<close>, unfolded filterlim_def])
```
```  3037
```
```  3038 lemma convergent_Cauchy: "convergent X \<Longrightarrow> Cauchy X"
```
```  3039   unfolding convergent_def by (erule exE, erule LIMSEQ_imp_Cauchy)
```
```  3040
```
```  3041 definition complete :: "'a set \<Rightarrow> bool"
```
```  3042   where complete_uniform: "complete S \<longleftrightarrow>
```
```  3043     (\<forall>F \<le> principal S. F \<noteq> bot \<longrightarrow> cauchy_filter F \<longrightarrow> (\<exists>x\<in>S. F \<le> nhds x))"
```
```  3044
```
```  3045 end
```
```  3046
```
```  3047
```
```  3048 subsubsection \<open>Uniformly continuous functions\<close>
```
```  3049
```
```  3050 definition uniformly_continuous_on :: "'a set \<Rightarrow> ('a::uniform_space \<Rightarrow> 'b::uniform_space) \<Rightarrow> bool"
```
```  3051   where uniformly_continuous_on_uniformity: "uniformly_continuous_on s f \<longleftrightarrow>
```
```  3052     (LIM (x, y) (uniformity_on s). (f x, f y) :> uniformity)"
```
```  3053
```
```  3054 lemma uniformly_continuous_onD:
```
```  3055   "uniformly_continuous_on s f \<Longrightarrow> eventually E uniformity \<Longrightarrow>
```
```  3056     eventually (\<lambda>(x, y). x \<in> s \<longrightarrow> y \<in> s \<longrightarrow> E (f x, f y)) uniformity"
```
```  3057   by (simp add: uniformly_continuous_on_uniformity filterlim_iff
```
```  3058       eventually_inf_principal split_beta' mem_Times_iff imp_conjL)
```
```  3059
```
```  3060 lemma uniformly_continuous_on_const[continuous_intros]: "uniformly_continuous_on s (\<lambda>x. c)"
```
```  3061   by (auto simp: uniformly_continuous_on_uniformity filterlim_iff uniformity_refl)
```
```  3062
```
```  3063 lemma uniformly_continuous_on_id[continuous_intros]: "uniformly_continuous_on s (\<lambda>x. x)"
```
```  3064   by (auto simp: uniformly_continuous_on_uniformity filterlim_def)
```
```  3065
```
```  3066 lemma uniformly_continuous_on_compose[continuous_intros]:
```
```  3067   "uniformly_continuous_on s g \<Longrightarrow> uniformly_continuous_on (g`s) f \<Longrightarrow>
```
```  3068     uniformly_continuous_on s (\<lambda>x. f (g x))"
```
```  3069   using filterlim_compose[of "\<lambda>(x, y). (f x, f y)" uniformity
```
```  3070       "uniformity_on (g`s)"  "\<lambda>(x, y). (g x, g y)" "uniformity_on s"]
```
```  3071   by (simp add: split_beta' uniformly_continuous_on_uniformity
```
```  3072       filterlim_inf filterlim_principal eventually_inf_principal mem_Times_iff)
```
```  3073
```
```  3074 lemma uniformly_continuous_imp_continuous:
```
```  3075   assumes f: "uniformly_continuous_on s f"
```
```  3076   shows "continuous_on s f"
```
```  3077   by (auto simp: filterlim_iff eventually_at_filter eventually_nhds_uniformity continuous_on_def
```
```  3078            elim: eventually_mono dest!: uniformly_continuous_onD[OF f])
```
```  3079
```
```  3080
```
```  3081 section \<open>Product Topology\<close>
```
```  3082
```
```  3083 subsection \<open>Product is a topological space\<close>
```
```  3084
```
```  3085 instantiation prod :: (topological_space, topological_space) topological_space
```
```  3086 begin
```
```  3087
```
```  3088 definition open_prod_def[code del]:
```
```  3089   "open (S :: ('a \<times> 'b) set) \<longleftrightarrow>
```
```  3090     (\<forall>x\<in>S. \<exists>A B. open A \<and> open B \<and> x \<in> A \<times> B \<and> A \<times> B \<subseteq> S)"
```
```  3091
```
```  3092 lemma open_prod_elim:
```
```  3093   assumes "open S" and "x \<in> S"
```
```  3094   obtains A B where "open A" and "open B" and "x \<in> A \<times> B" and "A \<times> B \<subseteq> S"
```
```  3095   using assms unfolding open_prod_def by fast
```
```  3096
```
```  3097 lemma open_prod_intro:
```
```  3098   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"
```
```  3099   shows "open S"
```
```  3100   using assms unfolding open_prod_def by fast
```
```  3101
```
```  3102 instance
```
```  3103 proof
```
```  3104   show "open (UNIV :: ('a \<times> 'b) set)"
```
```  3105     unfolding open_prod_def by auto
```
```  3106 next
```
```  3107   fix S T :: "('a \<times> 'b) set"
```
```  3108   assume "open S" "open T"
```
```  3109   show "open (S \<inter> T)"
```
```  3110   proof (rule open_prod_intro)
```
```  3111     fix x
```
```  3112     assume x: "x \<in> S \<inter> T"
```
```  3113     from x have "x \<in> S" by simp
```
```  3114     obtain Sa Sb where A: "open Sa" "open Sb" "x \<in> Sa \<times> Sb" "Sa \<times> Sb \<subseteq> S"
```
```  3115       using \<open>open S\<close> and \<open>x \<in> S\<close> by (rule open_prod_elim)
```
```  3116     from x have "x \<in> T" by simp
```
```  3117     obtain Ta Tb where B: "open Ta" "open Tb" "x \<in> Ta \<times> Tb" "Ta \<times> Tb \<subseteq> T"
```
```  3118       using \<open>open T\<close> and \<open>x \<in> T\<close> by (rule open_prod_elim)
```
```  3119     let ?A = "Sa \<inter> Ta" and ?B = "Sb \<inter> Tb"
```
```  3120     have "open ?A \<and> open ?B \<and> x \<in> ?A \<times> ?B \<and> ?A \<times> ?B \<subseteq> S \<inter> T"
```
```  3121       using A B by (auto simp add: open_Int)
```
```  3122     then show "\<exists>A B. open A \<and> open B \<and> x \<in> A \<times> B \<and> A \<times> B \<subseteq> S \<inter> T"
```
```  3123       by fast
```
```  3124   qed
```
```  3125 next
```
```  3126   fix K :: "('a \<times> 'b) set set"
```
```  3127   assume "\<forall>S\<in>K. open S"
```
```  3128   then show "open (\<Union>K)"
```
```  3129     unfolding open_prod_def by fast
```
```  3130 qed
```
```  3131
```
```  3132 end
```
```  3133
```
```  3134 declare [[code abort: "open :: ('a::topological_space \<times> 'b::topological_space) set \<Rightarrow> bool"]]
```
```  3135
```
```  3136 lemma open_Times: "open S \<Longrightarrow> open T \<Longrightarrow> open (S \<times> T)"
```
```  3137   unfolding open_prod_def by auto
```
```  3138
```
```  3139 lemma fst_vimage_eq_Times: "fst -` S = S \<times> UNIV"
```
```  3140   by auto
```
```  3141
```
```  3142 lemma snd_vimage_eq_Times: "snd -` S = UNIV \<times> S"
```
```  3143   by auto
```
```  3144
```
```  3145 lemma open_vimage_fst: "open S \<Longrightarrow> open (fst -` S)"
```
```  3146   by (simp add: fst_vimage_eq_Times open_Times)
```
```  3147
```
```  3148 lemma open_vimage_snd: "open S \<Longrightarrow> open (snd -` S)"
```
```  3149   by (simp add: snd_vimage_eq_Times open_Times)
```
```  3150
```
```  3151 lemma closed_vimage_fst: "closed S \<Longrightarrow> closed (fst -` S)"
```
```  3152   unfolding closed_open vimage_Compl [symmetric]
```
```  3153   by (rule open_vimage_fst)
```
```  3154
```
```  3155 lemma closed_vimage_snd: "closed S \<Longrightarrow> closed (snd -` S)"
```
```  3156   unfolding closed_open vimage_Compl [symmetric]
```
```  3157   by (rule open_vimage_snd)
```
```  3158
```
```  3159 lemma closed_Times: "closed S \<Longrightarrow> closed T \<Longrightarrow> closed (S \<times> T)"
```
```  3160 proof -
```
```  3161   have "S \<times> T = (fst -` S) \<inter> (snd -` T)"
```
```  3162     by auto
```
```  3163   then show "closed S \<Longrightarrow> closed T \<Longrightarrow> closed (S \<times> T)"
```
```  3164     by (simp add: closed_vimage_fst closed_vimage_snd closed_Int)
```
```  3165 qed
```
```  3166
```
```  3167 lemma subset_fst_imageI: "A \<times> B \<subseteq> S \<Longrightarrow> y \<in> B \<Longrightarrow> A \<subseteq> fst ` S"
```
```  3168   unfolding image_def subset_eq by force
```
```  3169
```
```  3170 lemma subset_snd_imageI: "A \<times> B \<subseteq> S \<Longrightarrow> x \<in> A \<Longrightarrow> B \<subseteq> snd ` S"
```
```  3171   unfolding image_def subset_eq by force
```
```  3172
```
```  3173 lemma open_image_fst:
```
```  3174   assumes "open S"
```
```  3175   shows "open (fst ` S)"
```
```  3176 proof (rule openI)
```
```  3177   fix x
```
```  3178   assume "x \<in> fst ` S"
```
```  3179   then obtain y where "(x, y) \<in> S"
```
```  3180     by auto
```
```  3181   then obtain A B where "open A" "open B" "x \<in> A" "y \<in> B" "A \<times> B \<subseteq> S"
```
```  3182     using \<open>open S\<close> unfolding open_prod_def by auto
```
```  3183   from \<open>A \<times> B \<subseteq> S\<close> \<open>y \<in> B\<close> have "A \<subseteq> fst ` S"
```
```  3184     by (rule subset_fst_imageI)
```
```  3185   with \<open>open A\<close> \<open>x \<in> A\<close> have "open A \<and> x \<in> A \<and> A \<subseteq> fst ` S"
```
```  3186     by simp
```
```  3187   then show "\<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> fst ` S" ..
```
```  3188 qed
```
```  3189
```
```  3190 lemma open_image_snd:
```
```  3191   assumes "open S"
```
```  3192   shows "open (snd ` S)"
```
```  3193 proof (rule openI)
```
```  3194   fix y
```
```  3195   assume "y \<in> snd ` S"
```
```  3196   then obtain x where "(x, y) \<in> S"
```
```  3197     by auto
```
```  3198   then obtain A B where "open A" "open B" "x \<in> A" "y \<in> B" "A \<times> B \<subseteq> S"
```
```  3199     using \<open>open S\<close> unfolding open_prod_def by auto
```
```  3200   from \<open>A \<times> B \<subseteq> S\<close> \<open>x \<in> A\<close> have "B \<subseteq> snd ` S"
```
```  3201     by (rule subset_snd_imageI)
```
```  3202   with \<open>open B\<close> \<open>y \<in> B\<close> have "open B \<and> y \<in> B \<and> B \<subseteq> snd ` S"
```
```  3203     by simp
```
```  3204   then show "\<exists>T. open T \<and> y \<in> T \<and> T \<subseteq> snd ` S" ..
```
```  3205 qed
```
```  3206
```
```  3207 lemma nhds_prod: "nhds (a, b) = nhds a \<times>\<^sub>F nhds b"
```
```  3208   unfolding nhds_def
```
```  3209 proof (subst prod_filter_INF, auto intro!: antisym INF_greatest simp: principal_prod_principal)
```
```  3210   fix S T
```
```  3211   assume "open S" "a \<in> S" "open T" "b \<in> T"
```
```  3212   then show "(INF x : {S. open S \<and> (a, b) \<in> S}. principal x) \<le> principal (S \<times> T)"
```
```  3213     by (intro INF_lower) (auto intro!: open_Times)
```
```  3214 next
```
```  3215   fix S'
```
```  3216   assume "open S'" "(a, b) \<in> S'"
```
```  3217   then obtain S T where "open S" "a \<in> S" "open T" "b \<in> T" "S \<times> T \<subseteq> S'"
```
```  3218     by (auto elim: open_prod_elim)
```
```  3219   then show "(INF x : {S. open S \<and> a \<in> S}. INF y : {S. open S \<and> b \<in> S}.
```
```  3220       principal (x \<times> y)) \<le> principal S'"
```
```  3221     by (auto intro!: INF_lower2)
```
```  3222 qed
```
```  3223
```
```  3224
```
```  3225 subsubsection \<open>Continuity of operations\<close>
```
```  3226
```
```  3227 lemma tendsto_fst [tendsto_intros]:
```
```  3228   assumes "(f \<longlongrightarrow> a) F"
```
```  3229   shows "((\<lambda>x. fst (f x)) \<longlongrightarrow> fst a) F"
```
```  3230 proof (rule topological_tendstoI)
```
```  3231   fix S
```
```  3232   assume "open S" and "fst a \<in> S"
```
```  3233   then have "open (fst -` S)" and "a \<in> fst -` S"
```
```  3234     by (simp_all add: open_vimage_fst)
```
```  3235   with assms have "eventually (\<lambda>x. f x \<in> fst -` S) F"
```
```  3236     by (rule topological_tendstoD)
```
```  3237   then show "eventually (\<lambda>x. fst (f x) \<in> S) F"
```
```  3238     by simp
```
```  3239 qed
```
```  3240
```
```  3241 lemma tendsto_snd [tendsto_intros]:
```
```  3242   assumes "(f \<longlongrightarrow> a) F"
```
```  3243   shows "((\<lambda>x. snd (f x)) \<longlongrightarrow> snd a) F"
```
```  3244 proof (rule topological_tendstoI)
```
```  3245   fix S
```
```  3246   assume "open S" and "snd a \<in> S"
```
```  3247   then have "open (snd -` S)" and "a \<in> snd -` S"
```
```  3248     by (simp_all add: open_vimage_snd)
```
```  3249   with assms have "eventually (\<lambda>x. f x \<in> snd -` S) F"
```
```  3250     by (rule topological_tendstoD)
```
```  3251   then show "eventually (\<lambda>x. snd (f x) \<in> S) F"
```
```  3252     by simp
```
```  3253 qed
```
```  3254
```
```  3255 lemma tendsto_Pair [tendsto_intros]:
```
```  3256   assumes "(f \<longlongrightarrow> a) F" and "(g \<longlongrightarrow> b) F"
```
```  3257   shows "((\<lambda>x. (f x, g x)) \<longlongrightarrow> (a, b)) F"
```
```  3258 proof (rule topological_tendstoI)
```
```  3259   fix S
```
```  3260   assume "open S" and "(a, b) \<in> S"
```
```  3261   then obtain A B where "open A" "open B" "a \<in> A" "b \<in> B" "A \<times> B \<subseteq> S"
```
```  3262     unfolding open_prod_def by fast
```
```  3263   have "eventually (\<lambda>x. f x \<in> A) F"
```
```  3264     using \<open>(f \<longlongrightarrow> a) F\<close> \<open>open A\<close> \<open>a \<in> A\<close>
```
```  3265     by (rule topological_tendstoD)
```
```  3266   moreover
```
```  3267   have "eventually (\<lambda>x. g x \<in> B) F"
```
```  3268     using \<open>(g \<longlongrightarrow> b) F\<close> \<open>open B\<close> \<open>b \<in> B\<close>
```
```  3269     by (rule topological_tendstoD)
```
```  3270   ultimately
```
```  3271   show "eventually (\<lambda>x. (f x, g x) \<in> S) F"
```
```  3272     by (rule eventually_elim2)
```
```  3273        (simp add: subsetD [OF \<open>A \<times> B \<subseteq> S\<close>])
```
```  3274 qed
```
```  3275
```
```  3276 lemma continuous_fst[continuous_intros]: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. fst (f x))"
```
```  3277   unfolding continuous_def by (rule tendsto_fst)
```
```  3278
```
```  3279 lemma continuous_snd[continuous_intros]: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. snd (f x))"
```
```  3280   unfolding continuous_def by (rule tendsto_snd)
```
```  3281
```
```  3282 lemma continuous_Pair[continuous_intros]:
```
```  3283   "continuous F f \<Longrightarrow> continuous F g \<Longrightarrow> continuous F (\<lambda>x. (f x, g x))"
```
```  3284   unfolding continuous_def by (rule tendsto_Pair)
```
```  3285
```
```  3286 lemma continuous_on_fst[continuous_intros]:
```
```  3287   "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. fst (f x))"
```
```  3288   unfolding continuous_on_def by (auto intro: tendsto_fst)
```
```  3289
```
```  3290 lemma continuous_on_snd[continuous_intros]:
```
```  3291   "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. snd (f x))"
```
```  3292   unfolding continuous_on_def by (auto intro: tendsto_snd)
```
```  3293
```
```  3294 lemma continuous_on_Pair[continuous_intros]:
```
```  3295   "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. (f x, g x))"
```
```  3296   unfolding continuous_on_def by (auto intro: tendsto_Pair)
```
```  3297
```
```  3298 lemma continuous_on_swap[continuous_intros]: "continuous_on A prod.swap"
```
```  3299   by (simp add: prod.swap_def continuous_on_fst continuous_on_snd
```
```  3300       continuous_on_Pair continuous_on_id)
```
```  3301
```
```  3302 lemma continuous_on_swap_args:
```
```  3303   assumes "continuous_on (A\<times>B) (\<lambda>(x,y). d x y)"
```
```  3304     shows "continuous_on (B\<times>A) (\<lambda>(x,y). d y x)"
```
```  3305 proof -
```
```  3306   have "(\<lambda>(x,y). d y x) = (\<lambda>(x,y). d x y) \<circ> prod.swap"
```
```  3307     by force
```
```  3308   then show ?thesis
```
```  3309     apply (rule ssubst)
```
```  3310     apply (rule continuous_on_compose)
```
```  3311      apply (force intro: continuous_on_subset [OF continuous_on_swap])
```
```  3312     apply (force intro: continuous_on_subset [OF assms])
```
```  3313     done
```
```  3314 qed
```
```  3315
```
```  3316 lemma isCont_fst [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. fst (f x)) a"
```
```  3317   by (fact continuous_fst)
```
```  3318
```
```  3319 lemma isCont_snd [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. snd (f x)) a"
```
```  3320   by (fact continuous_snd)
```
```  3321
```
```  3322 lemma isCont_Pair [simp]: "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. (f x, g x)) a"
```
```  3323   by (fact continuous_Pair)
```
```  3324
```
```  3325
```
```  3326 subsubsection \<open>Separation axioms\<close>
```
```  3327
```
```  3328 instance prod :: (t0_space, t0_space) t0_space
```
```  3329 proof
```
```  3330   fix x y :: "'a \<times> 'b"
```
```  3331   assume "x \<noteq> y"
```
```  3332   then have "fst x \<noteq> fst y \<or> snd x \<noteq> snd y"
```
```  3333     by (simp add: prod_eq_iff)
```
```  3334   then show "\<exists>U. open U \<and> (x \<in> U) \<noteq> (y \<in> U)"
```
```  3335     by (fast dest: t0_space elim: open_vimage_fst open_vimage_snd)
```
```  3336 qed
```
```  3337
```
```  3338 instance prod :: (t1_space, t1_space) t1_space
```
```  3339 proof
```
```  3340   fix x y :: "'a \<times> 'b"
```
```  3341   assume "x \<noteq> y"
```
```  3342   then have "fst x \<noteq> fst y \<or> snd x \<noteq> snd y"
```
```  3343     by (simp add: prod_eq_iff)
```
```  3344   then show "\<exists>U. open U \<and> x \<in> U \<and> y \<notin> U"
```
```  3345     by (fast dest: t1_space elim: open_vimage_fst open_vimage_snd)
```
```  3346 qed
```
```  3347
```
```  3348 instance prod :: (t2_space, t2_space) t2_space
```
```  3349 proof
```
```  3350   fix x y :: "'a \<times> 'b"
```
```  3351   assume "x \<noteq> y"
```
```  3352   then have "fst x \<noteq> fst y \<or> snd x \<noteq> snd y"
```
```  3353     by (simp add: prod_eq_iff)
```
```  3354   then show "\<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {}"
```
```  3355     by (fast dest: hausdorff elim: open_vimage_fst open_vimage_snd)
```
```  3356 qed
```
```  3357
```
```  3358 lemma isCont_swap[continuous_intros]: "isCont prod.swap a"
```
```  3359   using continuous_on_eq_continuous_within continuous_on_swap by blast
```
```  3360
```
```  3361 end
```