src/HOL/Topological_Spaces.thy
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```     1 (*  Title:      HOL/Topological_Spaces.thy
```
```     2     Author:     Brian Huffman
```
```     3     Author:     Johannes Hölzl
```
```     4 *)
```
```     5
```
```     6 header {* Topological Spaces *}
```
```     7
```
```     8 theory Topological_Spaces
```
```     9 imports Main Conditionally_Complete_Lattices
```
```    10 begin
```
```    11
```
```    12 named_theorems continuous_intros "structural introduction rules for continuity"
```
```    13
```
```    14
```
```    15 subsection {* Topological space *}
```
```    16
```
```    17 class "open" =
```
```    18   fixes "open" :: "'a set \<Rightarrow> bool"
```
```    19
```
```    20 class topological_space = "open" +
```
```    21   assumes open_UNIV [simp, intro]: "open UNIV"
```
```    22   assumes open_Int [intro]: "open S \<Longrightarrow> open T \<Longrightarrow> open (S \<inter> T)"
```
```    23   assumes open_Union [intro]: "\<forall>S\<in>K. open S \<Longrightarrow> open (\<Union> K)"
```
```    24 begin
```
```    25
```
```    26 definition
```
```    27   closed :: "'a set \<Rightarrow> bool" where
```
```    28   "closed S \<longleftrightarrow> open (- S)"
```
```    29
```
```    30 lemma open_empty [continuous_intros, intro, simp]: "open {}"
```
```    31   using open_Union [of "{}"] by simp
```
```    32
```
```    33 lemma open_Un [continuous_intros, intro]: "open S \<Longrightarrow> open T \<Longrightarrow> open (S \<union> T)"
```
```    34   using open_Union [of "{S, T}"] by simp
```
```    35
```
```    36 lemma open_UN [continuous_intros, intro]: "\<forall>x\<in>A. open (B x) \<Longrightarrow> open (\<Union>x\<in>A. B x)"
```
```    37   using open_Union [of "B ` A"] by simp
```
```    38
```
```    39 lemma open_Inter [continuous_intros, intro]: "finite S \<Longrightarrow> \<forall>T\<in>S. open T \<Longrightarrow> open (\<Inter>S)"
```
```    40   by (induct set: finite) auto
```
```    41
```
```    42 lemma open_INT [continuous_intros, intro]: "finite A \<Longrightarrow> \<forall>x\<in>A. open (B x) \<Longrightarrow> open (\<Inter>x\<in>A. B x)"
```
```    43   using open_Inter [of "B ` A"] by simp
```
```    44
```
```    45 lemma openI:
```
```    46   assumes "\<And>x. x \<in> S \<Longrightarrow> \<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> S"
```
```    47   shows "open S"
```
```    48 proof -
```
```    49   have "open (\<Union>{T. open T \<and> T \<subseteq> S})" by auto
```
```    50   moreover have "\<Union>{T. open T \<and> T \<subseteq> S} = S" by (auto dest!: assms)
```
```    51   ultimately show "open S" by simp
```
```    52 qed
```
```    53
```
```    54 lemma closed_empty [continuous_intros, intro, simp]:  "closed {}"
```
```    55   unfolding closed_def by simp
```
```    56
```
```    57 lemma closed_Un [continuous_intros, intro]: "closed S \<Longrightarrow> closed T \<Longrightarrow> closed (S \<union> T)"
```
```    58   unfolding closed_def by auto
```
```    59
```
```    60 lemma closed_UNIV [continuous_intros, intro, simp]: "closed UNIV"
```
```    61   unfolding closed_def by simp
```
```    62
```
```    63 lemma closed_Int [continuous_intros, intro]: "closed S \<Longrightarrow> closed T \<Longrightarrow> closed (S \<inter> T)"
```
```    64   unfolding closed_def by auto
```
```    65
```
```    66 lemma closed_INT [continuous_intros, intro]: "\<forall>x\<in>A. closed (B x) \<Longrightarrow> closed (\<Inter>x\<in>A. B x)"
```
```    67   unfolding closed_def by auto
```
```    68
```
```    69 lemma closed_Inter [continuous_intros, intro]: "\<forall>S\<in>K. closed S \<Longrightarrow> closed (\<Inter> K)"
```
```    70   unfolding closed_def uminus_Inf by auto
```
```    71
```
```    72 lemma closed_Union [continuous_intros, intro]: "finite S \<Longrightarrow> \<forall>T\<in>S. closed T \<Longrightarrow> closed (\<Union>S)"
```
```    73   by (induct set: finite) auto
```
```    74
```
```    75 lemma closed_UN [continuous_intros, intro]: "finite A \<Longrightarrow> \<forall>x\<in>A. closed (B x) \<Longrightarrow> closed (\<Union>x\<in>A. B x)"
```
```    76   using closed_Union [of "B ` A"] by simp
```
```    77
```
```    78 lemma open_closed: "open S \<longleftrightarrow> closed (- S)"
```
```    79   unfolding closed_def by simp
```
```    80
```
```    81 lemma closed_open: "closed S \<longleftrightarrow> open (- S)"
```
```    82   unfolding closed_def by simp
```
```    83
```
```    84 lemma open_Diff [continuous_intros, intro]: "open S \<Longrightarrow> closed T \<Longrightarrow> open (S - T)"
```
```    85   unfolding closed_open Diff_eq by (rule open_Int)
```
```    86
```
```    87 lemma closed_Diff [continuous_intros, intro]: "closed S \<Longrightarrow> open T \<Longrightarrow> closed (S - T)"
```
```    88   unfolding open_closed Diff_eq by (rule closed_Int)
```
```    89
```
```    90 lemma open_Compl [continuous_intros, intro]: "closed S \<Longrightarrow> open (- S)"
```
```    91   unfolding closed_open .
```
```    92
```
```    93 lemma closed_Compl [continuous_intros, intro]: "open S \<Longrightarrow> closed (- S)"
```
```    94   unfolding open_closed .
```
```    95
```
```    96 lemma open_Collect_neg: "closed {x. P x} \<Longrightarrow> open {x. \<not> P x}"
```
```    97   unfolding Collect_neg_eq by (rule open_Compl)
```
```    98
```
```    99 lemma open_Collect_conj: assumes "open {x. P x}" "open {x. Q x}" shows "open {x. P x \<and> Q x}"
```
```   100   using open_Int[OF assms] by (simp add: Int_def)
```
```   101
```
```   102 lemma open_Collect_disj: assumes "open {x. P x}" "open {x. Q x}" shows "open {x. P x \<or> Q x}"
```
```   103   using open_Un[OF assms] by (simp add: Un_def)
```
```   104
```
```   105 lemma open_Collect_ex: "(\<And>i. open {x. P i x}) \<Longrightarrow> open {x. \<exists>i. P i x}"
```
```   106   using open_UN[of UNIV "\<lambda>i. {x. P i x}"] unfolding Collect_ex_eq by simp
```
```   107
```
```   108 lemma open_Collect_imp: "closed {x. P x} \<Longrightarrow> open {x. Q x} \<Longrightarrow> open {x. P x \<longrightarrow> Q x}"
```
```   109   unfolding imp_conv_disj by (intro open_Collect_disj open_Collect_neg)
```
```   110
```
```   111 lemma open_Collect_const: "open {x. P}"
```
```   112   by (cases P) auto
```
```   113
```
```   114 lemma closed_Collect_neg: "open {x. P x} \<Longrightarrow> closed {x. \<not> P x}"
```
```   115   unfolding Collect_neg_eq by (rule closed_Compl)
```
```   116
```
```   117 lemma closed_Collect_conj: assumes "closed {x. P x}" "closed {x. Q x}" shows "closed {x. P x \<and> Q x}"
```
```   118   using closed_Int[OF assms] by (simp add: Int_def)
```
```   119
```
```   120 lemma closed_Collect_disj: assumes "closed {x. P x}" "closed {x. Q x}" shows "closed {x. P x \<or> Q x}"
```
```   121   using closed_Un[OF assms] by (simp add: Un_def)
```
```   122
```
```   123 lemma closed_Collect_all: "(\<And>i. closed {x. P i x}) \<Longrightarrow> closed {x. \<forall>i. P i x}"
```
```   124   using closed_INT[of UNIV "\<lambda>i. {x. P i x}"] unfolding Collect_all_eq by simp
```
```   125
```
```   126 lemma closed_Collect_imp: "open {x. P x} \<Longrightarrow> closed {x. Q x} \<Longrightarrow> closed {x. P x \<longrightarrow> Q x}"
```
```   127   unfolding imp_conv_disj by (intro closed_Collect_disj closed_Collect_neg)
```
```   128
```
```   129 lemma closed_Collect_const: "closed {x. P}"
```
```   130   by (cases P) auto
```
```   131
```
```   132 end
```
```   133
```
```   134 subsection{* Hausdorff and other separation properties *}
```
```   135
```
```   136 class t0_space = topological_space +
```
```   137   assumes t0_space: "x \<noteq> y \<Longrightarrow> \<exists>U. open U \<and> \<not> (x \<in> U \<longleftrightarrow> y \<in> U)"
```
```   138
```
```   139 class t1_space = topological_space +
```
```   140   assumes t1_space: "x \<noteq> y \<Longrightarrow> \<exists>U. open U \<and> x \<in> U \<and> y \<notin> U"
```
```   141
```
```   142 instance t1_space \<subseteq> t0_space
```
```   143 proof qed (fast dest: t1_space)
```
```   144
```
```   145 lemma separation_t1:
```
```   146   fixes x y :: "'a::t1_space"
```
```   147   shows "x \<noteq> y \<longleftrightarrow> (\<exists>U. open U \<and> x \<in> U \<and> y \<notin> U)"
```
```   148   using t1_space[of x y] by blast
```
```   149
```
```   150 lemma closed_singleton:
```
```   151   fixes a :: "'a::t1_space"
```
```   152   shows "closed {a}"
```
```   153 proof -
```
```   154   let ?T = "\<Union>{S. open S \<and> a \<notin> S}"
```
```   155   have "open ?T" by (simp add: open_Union)
```
```   156   also have "?T = - {a}"
```
```   157     by (simp add: set_eq_iff separation_t1, auto)
```
```   158   finally show "closed {a}" unfolding closed_def .
```
```   159 qed
```
```   160
```
```   161 lemma closed_insert [continuous_intros, simp]:
```
```   162   fixes a :: "'a::t1_space"
```
```   163   assumes "closed S" shows "closed (insert a S)"
```
```   164 proof -
```
```   165   from closed_singleton assms
```
```   166   have "closed ({a} \<union> S)" by (rule closed_Un)
```
```   167   thus "closed (insert a S)" by simp
```
```   168 qed
```
```   169
```
```   170 lemma finite_imp_closed:
```
```   171   fixes S :: "'a::t1_space set"
```
```   172   shows "finite S \<Longrightarrow> closed S"
```
```   173 by (induct set: finite, simp_all)
```
```   174
```
```   175 text {* T2 spaces are also known as Hausdorff spaces. *}
```
```   176
```
```   177 class t2_space = topological_space +
```
```   178   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 = {}"
```
```   179
```
```   180 instance t2_space \<subseteq> t1_space
```
```   181 proof qed (fast dest: hausdorff)
```
```   182
```
```   183 lemma separation_t2:
```
```   184   fixes x y :: "'a::t2_space"
```
```   185   shows "x \<noteq> y \<longleftrightarrow> (\<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {})"
```
```   186   using hausdorff[of x y] by blast
```
```   187
```
```   188 lemma separation_t0:
```
```   189   fixes x y :: "'a::t0_space"
```
```   190   shows "x \<noteq> y \<longleftrightarrow> (\<exists>U. open U \<and> ~(x\<in>U \<longleftrightarrow> y\<in>U))"
```
```   191   using t0_space[of x y] by blast
```
```   192
```
```   193 text {* A perfect space is a topological space with no isolated points. *}
```
```   194
```
```   195 class perfect_space = topological_space +
```
```   196   assumes not_open_singleton: "\<not> open {x}"
```
```   197
```
```   198
```
```   199 subsection {* Generators for toplogies *}
```
```   200
```
```   201 inductive generate_topology for S where
```
```   202   UNIV: "generate_topology S UNIV"
```
```   203 | Int: "generate_topology S a \<Longrightarrow> generate_topology S b \<Longrightarrow> generate_topology S (a \<inter> b)"
```
```   204 | UN: "(\<And>k. k \<in> K \<Longrightarrow> generate_topology S k) \<Longrightarrow> generate_topology S (\<Union>K)"
```
```   205 | Basis: "s \<in> S \<Longrightarrow> generate_topology S s"
```
```   206
```
```   207 hide_fact (open) UNIV Int UN Basis
```
```   208
```
```   209 lemma generate_topology_Union:
```
```   210   "(\<And>k. k \<in> I \<Longrightarrow> generate_topology S (K k)) \<Longrightarrow> generate_topology S (\<Union>k\<in>I. K k)"
```
```   211   using generate_topology.UN [of "K ` I"] by auto
```
```   212
```
```   213 lemma topological_space_generate_topology:
```
```   214   "class.topological_space (generate_topology S)"
```
```   215   by default (auto intro: generate_topology.intros)
```
```   216
```
```   217 subsection {* Order topologies *}
```
```   218
```
```   219 class order_topology = order + "open" +
```
```   220   assumes open_generated_order: "open = generate_topology (range (\<lambda>a. {..< a}) \<union> range (\<lambda>a. {a <..}))"
```
```   221 begin
```
```   222
```
```   223 subclass topological_space
```
```   224   unfolding open_generated_order
```
```   225   by (rule topological_space_generate_topology)
```
```   226
```
```   227 lemma open_greaterThan [continuous_intros, simp]: "open {a <..}"
```
```   228   unfolding open_generated_order by (auto intro: generate_topology.Basis)
```
```   229
```
```   230 lemma open_lessThan [continuous_intros, simp]: "open {..< a}"
```
```   231   unfolding open_generated_order by (auto intro: generate_topology.Basis)
```
```   232
```
```   233 lemma open_greaterThanLessThan [continuous_intros, simp]: "open {a <..< b}"
```
```   234    unfolding greaterThanLessThan_eq by (simp add: open_Int)
```
```   235
```
```   236 end
```
```   237
```
```   238 class linorder_topology = linorder + order_topology
```
```   239
```
```   240 lemma closed_atMost [continuous_intros, simp]: "closed {.. a::'a::linorder_topology}"
```
```   241   by (simp add: closed_open)
```
```   242
```
```   243 lemma closed_atLeast [continuous_intros, simp]: "closed {a::'a::linorder_topology ..}"
```
```   244   by (simp add: closed_open)
```
```   245
```
```   246 lemma closed_atLeastAtMost [continuous_intros, simp]: "closed {a::'a::linorder_topology .. b}"
```
```   247 proof -
```
```   248   have "{a .. b} = {a ..} \<inter> {.. b}"
```
```   249     by auto
```
```   250   then show ?thesis
```
```   251     by (simp add: closed_Int)
```
```   252 qed
```
```   253
```
```   254 lemma (in linorder) less_separate:
```
```   255   assumes "x < y"
```
```   256   shows "\<exists>a b. x \<in> {..< a} \<and> y \<in> {b <..} \<and> {..< a} \<inter> {b <..} = {}"
```
```   257 proof (cases "\<exists>z. x < z \<and> z < y")
```
```   258   case True
```
```   259   then obtain z where "x < z \<and> z < y" ..
```
```   260   then have "x \<in> {..< z} \<and> y \<in> {z <..} \<and> {z <..} \<inter> {..< z} = {}"
```
```   261     by auto
```
```   262   then show ?thesis by blast
```
```   263 next
```
```   264   case False
```
```   265   with `x < y` have "x \<in> {..< y} \<and> y \<in> {x <..} \<and> {x <..} \<inter> {..< y} = {}"
```
```   266     by auto
```
```   267   then show ?thesis by blast
```
```   268 qed
```
```   269
```
```   270 instance linorder_topology \<subseteq> t2_space
```
```   271 proof
```
```   272   fix x y :: 'a
```
```   273   from less_separate[of x y] less_separate[of y x]
```
```   274   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 = {}"
```
```   275     by (elim neqE) (metis open_lessThan open_greaterThan Int_commute)+
```
```   276 qed
```
```   277
```
```   278 lemma (in linorder_topology) open_right:
```
```   279   assumes "open S" "x \<in> S" and gt_ex: "x < y" shows "\<exists>b>x. {x ..< b} \<subseteq> S"
```
```   280   using assms unfolding open_generated_order
```
```   281 proof induction
```
```   282   case (Int A B)
```
```   283   then obtain a b where "a > x" "{x ..< a} \<subseteq> A"  "b > x" "{x ..< b} \<subseteq> B" by auto
```
```   284   then show ?case by (auto intro!: exI[of _ "min a b"])
```
```   285 next
```
```   286   case (Basis S) then show ?case by (fastforce intro: exI[of _ y] gt_ex)
```
```   287 qed blast+
```
```   288
```
```   289 lemma (in linorder_topology) open_left:
```
```   290   assumes "open S" "x \<in> S" and lt_ex: "y < x" shows "\<exists>b<x. {b <.. x} \<subseteq> S"
```
```   291   using assms unfolding open_generated_order
```
```   292 proof induction
```
```   293   case (Int A B)
```
```   294   then obtain a b where "a < x" "{a <.. x} \<subseteq> A"  "b < x" "{b <.. x} \<subseteq> B" by auto
```
```   295   then show ?case by (auto intro!: exI[of _ "max a b"])
```
```   296 next
```
```   297   case (Basis S) then show ?case by (fastforce intro: exI[of _ y] lt_ex)
```
```   298 qed blast+
```
```   299
```
```   300 subsection {* Filters *}
```
```   301
```
```   302 text {*
```
```   303   This definition also allows non-proper filters.
```
```   304 *}
```
```   305
```
```   306 locale is_filter =
```
```   307   fixes F :: "('a \<Rightarrow> bool) \<Rightarrow> bool"
```
```   308   assumes True: "F (\<lambda>x. True)"
```
```   309   assumes conj: "F (\<lambda>x. P x) \<Longrightarrow> F (\<lambda>x. Q x) \<Longrightarrow> F (\<lambda>x. P x \<and> Q x)"
```
```   310   assumes mono: "\<forall>x. P x \<longrightarrow> Q x \<Longrightarrow> F (\<lambda>x. P x) \<Longrightarrow> F (\<lambda>x. Q x)"
```
```   311
```
```   312 typedef 'a filter = "{F :: ('a \<Rightarrow> bool) \<Rightarrow> bool. is_filter F}"
```
```   313 proof
```
```   314   show "(\<lambda>x. True) \<in> ?filter" by (auto intro: is_filter.intro)
```
```   315 qed
```
```   316
```
```   317 lemma is_filter_Rep_filter: "is_filter (Rep_filter F)"
```
```   318   using Rep_filter [of F] by simp
```
```   319
```
```   320 lemma Abs_filter_inverse':
```
```   321   assumes "is_filter F" shows "Rep_filter (Abs_filter F) = F"
```
```   322   using assms by (simp add: Abs_filter_inverse)
```
```   323
```
```   324
```
```   325 subsubsection {* Eventually *}
```
```   326
```
```   327 definition eventually :: "('a \<Rightarrow> bool) \<Rightarrow> 'a filter \<Rightarrow> bool"
```
```   328   where "eventually P F \<longleftrightarrow> Rep_filter F P"
```
```   329
```
```   330 lemma eventually_Abs_filter:
```
```   331   assumes "is_filter F" shows "eventually P (Abs_filter F) = F P"
```
```   332   unfolding eventually_def using assms by (simp add: Abs_filter_inverse)
```
```   333
```
```   334 lemma filter_eq_iff:
```
```   335   shows "F = F' \<longleftrightarrow> (\<forall>P. eventually P F = eventually P F')"
```
```   336   unfolding Rep_filter_inject [symmetric] fun_eq_iff eventually_def ..
```
```   337
```
```   338 lemma eventually_True [simp]: "eventually (\<lambda>x. True) F"
```
```   339   unfolding eventually_def
```
```   340   by (rule is_filter.True [OF is_filter_Rep_filter])
```
```   341
```
```   342 lemma always_eventually: "\<forall>x. P x \<Longrightarrow> eventually P F"
```
```   343 proof -
```
```   344   assume "\<forall>x. P x" hence "P = (\<lambda>x. True)" by (simp add: ext)
```
```   345   thus "eventually P F" by simp
```
```   346 qed
```
```   347
```
```   348 lemma eventually_mono:
```
```   349   "(\<forall>x. P x \<longrightarrow> Q x) \<Longrightarrow> eventually P F \<Longrightarrow> eventually Q F"
```
```   350   unfolding eventually_def
```
```   351   by (rule is_filter.mono [OF is_filter_Rep_filter])
```
```   352
```
```   353 lemma eventually_conj:
```
```   354   assumes P: "eventually (\<lambda>x. P x) F"
```
```   355   assumes Q: "eventually (\<lambda>x. Q x) F"
```
```   356   shows "eventually (\<lambda>x. P x \<and> Q x) F"
```
```   357   using assms unfolding eventually_def
```
```   358   by (rule is_filter.conj [OF is_filter_Rep_filter])
```
```   359
```
```   360 lemma eventually_Ball_finite:
```
```   361   assumes "finite A" and "\<forall>y\<in>A. eventually (\<lambda>x. P x y) net"
```
```   362   shows "eventually (\<lambda>x. \<forall>y\<in>A. P x y) net"
```
```   363 using assms by (induct set: finite, simp, simp add: eventually_conj)
```
```   364
```
```   365 lemma eventually_all_finite:
```
```   366   fixes P :: "'a \<Rightarrow> 'b::finite \<Rightarrow> bool"
```
```   367   assumes "\<And>y. eventually (\<lambda>x. P x y) net"
```
```   368   shows "eventually (\<lambda>x. \<forall>y. P x y) net"
```
```   369 using eventually_Ball_finite [of UNIV P] assms by simp
```
```   370
```
```   371 lemma eventually_mp:
```
```   372   assumes "eventually (\<lambda>x. P x \<longrightarrow> Q x) F"
```
```   373   assumes "eventually (\<lambda>x. P x) F"
```
```   374   shows "eventually (\<lambda>x. Q x) F"
```
```   375 proof (rule eventually_mono)
```
```   376   show "\<forall>x. (P x \<longrightarrow> Q x) \<and> P x \<longrightarrow> Q x" by simp
```
```   377   show "eventually (\<lambda>x. (P x \<longrightarrow> Q x) \<and> P x) F"
```
```   378     using assms by (rule eventually_conj)
```
```   379 qed
```
```   380
```
```   381 lemma eventually_rev_mp:
```
```   382   assumes "eventually (\<lambda>x. P x) F"
```
```   383   assumes "eventually (\<lambda>x. P x \<longrightarrow> Q x) F"
```
```   384   shows "eventually (\<lambda>x. Q x) F"
```
```   385 using assms(2) assms(1) by (rule eventually_mp)
```
```   386
```
```   387 lemma eventually_conj_iff:
```
```   388   "eventually (\<lambda>x. P x \<and> Q x) F \<longleftrightarrow> eventually P F \<and> eventually Q F"
```
```   389   by (auto intro: eventually_conj elim: eventually_rev_mp)
```
```   390
```
```   391 lemma eventually_elim1:
```
```   392   assumes "eventually (\<lambda>i. P i) F"
```
```   393   assumes "\<And>i. P i \<Longrightarrow> Q i"
```
```   394   shows "eventually (\<lambda>i. Q i) F"
```
```   395   using assms by (auto elim!: eventually_rev_mp)
```
```   396
```
```   397 lemma eventually_elim2:
```
```   398   assumes "eventually (\<lambda>i. P i) F"
```
```   399   assumes "eventually (\<lambda>i. Q i) F"
```
```   400   assumes "\<And>i. P i \<Longrightarrow> Q i \<Longrightarrow> R i"
```
```   401   shows "eventually (\<lambda>i. R i) F"
```
```   402   using assms by (auto elim!: eventually_rev_mp)
```
```   403
```
```   404 lemma not_eventually_impI: "eventually P F \<Longrightarrow> \<not> eventually Q F \<Longrightarrow> \<not> eventually (\<lambda>x. P x \<longrightarrow> Q x) F"
```
```   405   by (auto intro: eventually_mp)
```
```   406
```
```   407 lemma not_eventuallyD: "\<not> eventually P F \<Longrightarrow> \<exists>x. \<not> P x"
```
```   408   by (metis always_eventually)
```
```   409
```
```   410 lemma eventually_subst:
```
```   411   assumes "eventually (\<lambda>n. P n = Q n) F"
```
```   412   shows "eventually P F = eventually Q F" (is "?L = ?R")
```
```   413 proof -
```
```   414   from assms have "eventually (\<lambda>x. P x \<longrightarrow> Q x) F"
```
```   415       and "eventually (\<lambda>x. Q x \<longrightarrow> P x) F"
```
```   416     by (auto elim: eventually_elim1)
```
```   417   then show ?thesis by (auto elim: eventually_elim2)
```
```   418 qed
```
```   419
```
```   420 ML {*
```
```   421   fun eventually_elim_tac ctxt thms = SUBGOAL_CASES (fn (_, _, st) =>
```
```   422     let
```
```   423       val thy = Proof_Context.theory_of ctxt
```
```   424       val mp_thms = thms RL [@{thm eventually_rev_mp}]
```
```   425       val raw_elim_thm =
```
```   426         (@{thm allI} RS @{thm always_eventually})
```
```   427         |> fold (fn thm1 => fn thm2 => thm2 RS thm1) mp_thms
```
```   428         |> fold (fn _ => fn thm => @{thm impI} RS thm) thms
```
```   429       val cases_prop = prop_of (raw_elim_thm RS st)
```
```   430       val cases = (Rule_Cases.make_common (thy, cases_prop) [(("elim", []), [])])
```
```   431     in
```
```   432       CASES cases (rtac raw_elim_thm 1)
```
```   433     end) 1
```
```   434 *}
```
```   435
```
```   436 method_setup eventually_elim = {*
```
```   437   Scan.succeed (fn ctxt => METHOD_CASES (eventually_elim_tac ctxt))
```
```   438 *} "elimination of eventually quantifiers"
```
```   439
```
```   440
```
```   441 subsubsection {* Finer-than relation *}
```
```   442
```
```   443 text {* @{term "F \<le> F'"} means that filter @{term F} is finer than
```
```   444 filter @{term F'}. *}
```
```   445
```
```   446 instantiation filter :: (type) complete_lattice
```
```   447 begin
```
```   448
```
```   449 definition le_filter_def:
```
```   450   "F \<le> F' \<longleftrightarrow> (\<forall>P. eventually P F' \<longrightarrow> eventually P F)"
```
```   451
```
```   452 definition
```
```   453   "(F :: 'a filter) < F' \<longleftrightarrow> F \<le> F' \<and> \<not> F' \<le> F"
```
```   454
```
```   455 definition
```
```   456   "top = Abs_filter (\<lambda>P. \<forall>x. P x)"
```
```   457
```
```   458 definition
```
```   459   "bot = Abs_filter (\<lambda>P. True)"
```
```   460
```
```   461 definition
```
```   462   "sup F F' = Abs_filter (\<lambda>P. eventually P F \<and> eventually P F')"
```
```   463
```
```   464 definition
```
```   465   "inf F F' = Abs_filter
```
```   466       (\<lambda>P. \<exists>Q R. eventually Q F \<and> eventually R F' \<and> (\<forall>x. Q x \<and> R x \<longrightarrow> P x))"
```
```   467
```
```   468 definition
```
```   469   "Sup S = Abs_filter (\<lambda>P. \<forall>F\<in>S. eventually P F)"
```
```   470
```
```   471 definition
```
```   472   "Inf S = Sup {F::'a filter. \<forall>F'\<in>S. F \<le> F'}"
```
```   473
```
```   474 lemma eventually_top [simp]: "eventually P top \<longleftrightarrow> (\<forall>x. P x)"
```
```   475   unfolding top_filter_def
```
```   476   by (rule eventually_Abs_filter, rule is_filter.intro, auto)
```
```   477
```
```   478 lemma eventually_bot [simp]: "eventually P bot"
```
```   479   unfolding bot_filter_def
```
```   480   by (subst eventually_Abs_filter, rule is_filter.intro, auto)
```
```   481
```
```   482 lemma eventually_sup:
```
```   483   "eventually P (sup F F') \<longleftrightarrow> eventually P F \<and> eventually P F'"
```
```   484   unfolding sup_filter_def
```
```   485   by (rule eventually_Abs_filter, rule is_filter.intro)
```
```   486      (auto elim!: eventually_rev_mp)
```
```   487
```
```   488 lemma eventually_inf:
```
```   489   "eventually P (inf F F') \<longleftrightarrow>
```
```   490    (\<exists>Q R. eventually Q F \<and> eventually R F' \<and> (\<forall>x. Q x \<and> R x \<longrightarrow> P x))"
```
```   491   unfolding inf_filter_def
```
```   492   apply (rule eventually_Abs_filter, rule is_filter.intro)
```
```   493   apply (fast intro: eventually_True)
```
```   494   apply clarify
```
```   495   apply (intro exI conjI)
```
```   496   apply (erule (1) eventually_conj)
```
```   497   apply (erule (1) eventually_conj)
```
```   498   apply simp
```
```   499   apply auto
```
```   500   done
```
```   501
```
```   502 lemma eventually_Sup:
```
```   503   "eventually P (Sup S) \<longleftrightarrow> (\<forall>F\<in>S. eventually P F)"
```
```   504   unfolding Sup_filter_def
```
```   505   apply (rule eventually_Abs_filter, rule is_filter.intro)
```
```   506   apply (auto intro: eventually_conj elim!: eventually_rev_mp)
```
```   507   done
```
```   508
```
```   509 instance proof
```
```   510   fix F F' F'' :: "'a filter" and S :: "'a filter set"
```
```   511   { show "F < F' \<longleftrightarrow> F \<le> F' \<and> \<not> F' \<le> F"
```
```   512     by (rule less_filter_def) }
```
```   513   { show "F \<le> F"
```
```   514     unfolding le_filter_def by simp }
```
```   515   { assume "F \<le> F'" and "F' \<le> F''" thus "F \<le> F''"
```
```   516     unfolding le_filter_def by simp }
```
```   517   { assume "F \<le> F'" and "F' \<le> F" thus "F = F'"
```
```   518     unfolding le_filter_def filter_eq_iff by fast }
```
```   519   { show "inf F F' \<le> F" and "inf F F' \<le> F'"
```
```   520     unfolding le_filter_def eventually_inf by (auto intro: eventually_True) }
```
```   521   { assume "F \<le> F'" and "F \<le> F''" thus "F \<le> inf F' F''"
```
```   522     unfolding le_filter_def eventually_inf
```
```   523     by (auto elim!: eventually_mono intro: eventually_conj) }
```
```   524   { show "F \<le> sup F F'" and "F' \<le> sup F F'"
```
```   525     unfolding le_filter_def eventually_sup by simp_all }
```
```   526   { assume "F \<le> F''" and "F' \<le> F''" thus "sup F F' \<le> F''"
```
```   527     unfolding le_filter_def eventually_sup by simp }
```
```   528   { assume "F'' \<in> S" thus "Inf S \<le> F''"
```
```   529     unfolding le_filter_def Inf_filter_def eventually_Sup Ball_def by simp }
```
```   530   { assume "\<And>F'. F' \<in> S \<Longrightarrow> F \<le> F'" thus "F \<le> Inf S"
```
```   531     unfolding le_filter_def Inf_filter_def eventually_Sup Ball_def by simp }
```
```   532   { assume "F \<in> S" thus "F \<le> Sup S"
```
```   533     unfolding le_filter_def eventually_Sup by simp }
```
```   534   { assume "\<And>F. F \<in> S \<Longrightarrow> F \<le> F'" thus "Sup S \<le> F'"
```
```   535     unfolding le_filter_def eventually_Sup by simp }
```
```   536   { show "Inf {} = (top::'a filter)"
```
```   537     by (auto simp: top_filter_def Inf_filter_def Sup_filter_def)
```
```   538       (metis (full_types) top_filter_def always_eventually eventually_top) }
```
```   539   { show "Sup {} = (bot::'a filter)"
```
```   540     by (auto simp: bot_filter_def Sup_filter_def) }
```
```   541 qed
```
```   542
```
```   543 end
```
```   544
```
```   545 lemma filter_leD:
```
```   546   "F \<le> F' \<Longrightarrow> eventually P F' \<Longrightarrow> eventually P F"
```
```   547   unfolding le_filter_def by simp
```
```   548
```
```   549 lemma filter_leI:
```
```   550   "(\<And>P. eventually P F' \<Longrightarrow> eventually P F) \<Longrightarrow> F \<le> F'"
```
```   551   unfolding le_filter_def by simp
```
```   552
```
```   553 lemma eventually_False:
```
```   554   "eventually (\<lambda>x. False) F \<longleftrightarrow> F = bot"
```
```   555   unfolding filter_eq_iff by (auto elim: eventually_rev_mp)
```
```   556
```
```   557 abbreviation (input) trivial_limit :: "'a filter \<Rightarrow> bool"
```
```   558   where "trivial_limit F \<equiv> F = bot"
```
```   559
```
```   560 lemma trivial_limit_def: "trivial_limit F \<longleftrightarrow> eventually (\<lambda>x. False) F"
```
```   561   by (rule eventually_False [symmetric])
```
```   562
```
```   563 lemma eventually_const: "\<not> trivial_limit net \<Longrightarrow> eventually (\<lambda>x. P) net \<longleftrightarrow> P"
```
```   564   by (cases P) (simp_all add: eventually_False)
```
```   565
```
```   566 lemma eventually_Inf: "eventually P (Inf B) \<longleftrightarrow> (\<exists>X\<subseteq>B. finite X \<and> eventually P (Inf X))"
```
```   567 proof -
```
```   568   let ?F = "\<lambda>P. \<exists>X\<subseteq>B. finite X \<and> eventually P (Inf X)"
```
```   569
```
```   570   { fix P have "eventually P (Abs_filter ?F) \<longleftrightarrow> ?F P"
```
```   571     proof (rule eventually_Abs_filter is_filter.intro)+
```
```   572       show "?F (\<lambda>x. True)"
```
```   573         by (rule exI[of _ "{}"]) (simp add: le_fun_def)
```
```   574     next
```
```   575       fix P Q
```
```   576       assume "?F P" then guess X ..
```
```   577       moreover
```
```   578       assume "?F Q" then guess Y ..
```
```   579       ultimately show "?F (\<lambda>x. P x \<and> Q x)"
```
```   580         by (intro exI[of _ "X \<union> Y"])
```
```   581            (auto simp: Inf_union_distrib eventually_inf)
```
```   582     next
```
```   583       fix P Q
```
```   584       assume "?F P" then guess X ..
```
```   585       moreover assume "\<forall>x. P x \<longrightarrow> Q x"
```
```   586       ultimately show "?F Q"
```
```   587         by (intro exI[of _ X]) (auto elim: eventually_elim1)
```
```   588     qed }
```
```   589   note eventually_F = this
```
```   590
```
```   591   have "Inf B = Abs_filter ?F"
```
```   592   proof (intro antisym Inf_greatest)
```
```   593     show "Inf B \<le> Abs_filter ?F"
```
```   594       by (auto simp: le_filter_def eventually_F dest: Inf_superset_mono)
```
```   595   next
```
```   596     fix F assume "F \<in> B" then show "Abs_filter ?F \<le> F"
```
```   597       by (auto simp add: le_filter_def eventually_F intro!: exI[of _ "{F}"])
```
```   598   qed
```
```   599   then show ?thesis
```
```   600     by (simp add: eventually_F)
```
```   601 qed
```
```   602
```
```   603 lemma eventually_INF: "eventually P (INF b:B. F b) \<longleftrightarrow> (\<exists>X\<subseteq>B. finite X \<and> eventually P (INF b:X. F b))"
```
```   604   unfolding INF_def[of B] eventually_Inf[of P "F`B"]
```
```   605   by (metis Inf_image_eq finite_imageI image_mono finite_subset_image)
```
```   606
```
```   607 lemma Inf_filter_not_bot:
```
```   608   fixes B :: "'a filter set"
```
```   609   shows "(\<And>X. X \<subseteq> B \<Longrightarrow> finite X \<Longrightarrow> Inf X \<noteq> bot) \<Longrightarrow> Inf B \<noteq> bot"
```
```   610   unfolding trivial_limit_def eventually_Inf[of _ B]
```
```   611     bot_bool_def [symmetric] bot_fun_def [symmetric] bot_unique by simp
```
```   612
```
```   613 lemma INF_filter_not_bot:
```
```   614   fixes F :: "'i \<Rightarrow> 'a filter"
```
```   615   shows "(\<And>X. X \<subseteq> B \<Longrightarrow> finite X \<Longrightarrow> (INF b:X. F b) \<noteq> bot) \<Longrightarrow> (INF b:B. F b) \<noteq> bot"
```
```   616   unfolding trivial_limit_def eventually_INF[of _ B]
```
```   617     bot_bool_def [symmetric] bot_fun_def [symmetric] bot_unique by simp
```
```   618
```
```   619 lemma eventually_Inf_base:
```
```   620   assumes "B \<noteq> {}" and base: "\<And>F G. F \<in> B \<Longrightarrow> G \<in> B \<Longrightarrow> \<exists>x\<in>B. x \<le> inf F G"
```
```   621   shows "eventually P (Inf B) \<longleftrightarrow> (\<exists>b\<in>B. eventually P b)"
```
```   622 proof (subst eventually_Inf, safe)
```
```   623   fix X assume "finite X" "X \<subseteq> B"
```
```   624   then have "\<exists>b\<in>B. \<forall>x\<in>X. b \<le> x"
```
```   625   proof induct
```
```   626     case empty then show ?case
```
```   627       using `B \<noteq> {}` by auto
```
```   628   next
```
```   629     case (insert x X)
```
```   630     then obtain b where "b \<in> B" "\<And>x. x \<in> X \<Longrightarrow> b \<le> x"
```
```   631       by auto
```
```   632     with `insert x X \<subseteq> B` base[of b x] show ?case
```
```   633       by (auto intro: order_trans)
```
```   634   qed
```
```   635   then obtain b where "b \<in> B" "b \<le> Inf X"
```
```   636     by (auto simp: le_Inf_iff)
```
```   637   then show "eventually P (Inf X) \<Longrightarrow> Bex B (eventually P)"
```
```   638     by (intro bexI[of _ b]) (auto simp: le_filter_def)
```
```   639 qed (auto intro!: exI[of _ "{x}" for x])
```
```   640
```
```   641 lemma eventually_INF_base:
```
```   642   "B \<noteq> {} \<Longrightarrow> (\<And>a b. a \<in> B \<Longrightarrow> b \<in> B \<Longrightarrow> \<exists>x\<in>B. F x \<le> inf (F a) (F b)) \<Longrightarrow>
```
```   643     eventually P (INF b:B. F b) \<longleftrightarrow> (\<exists>b\<in>B. eventually P (F b))"
```
```   644   unfolding INF_def by (subst eventually_Inf_base) auto
```
```   645
```
```   646
```
```   647 subsubsection {* Map function for filters *}
```
```   648
```
```   649 definition filtermap :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a filter \<Rightarrow> 'b filter"
```
```   650   where "filtermap f F = Abs_filter (\<lambda>P. eventually (\<lambda>x. P (f x)) F)"
```
```   651
```
```   652 lemma eventually_filtermap:
```
```   653   "eventually P (filtermap f F) = eventually (\<lambda>x. P (f x)) F"
```
```   654   unfolding filtermap_def
```
```   655   apply (rule eventually_Abs_filter)
```
```   656   apply (rule is_filter.intro)
```
```   657   apply (auto elim!: eventually_rev_mp)
```
```   658   done
```
```   659
```
```   660 lemma filtermap_ident: "filtermap (\<lambda>x. x) F = F"
```
```   661   by (simp add: filter_eq_iff eventually_filtermap)
```
```   662
```
```   663 lemma filtermap_filtermap:
```
```   664   "filtermap f (filtermap g F) = filtermap (\<lambda>x. f (g x)) F"
```
```   665   by (simp add: filter_eq_iff eventually_filtermap)
```
```   666
```
```   667 lemma filtermap_mono: "F \<le> F' \<Longrightarrow> filtermap f F \<le> filtermap f F'"
```
```   668   unfolding le_filter_def eventually_filtermap by simp
```
```   669
```
```   670 lemma filtermap_bot [simp]: "filtermap f bot = bot"
```
```   671   by (simp add: filter_eq_iff eventually_filtermap)
```
```   672
```
```   673 lemma filtermap_sup: "filtermap f (sup F1 F2) = sup (filtermap f F1) (filtermap f F2)"
```
```   674   by (auto simp: filter_eq_iff eventually_filtermap eventually_sup)
```
```   675
```
```   676 lemma filtermap_inf: "filtermap f (inf F1 F2) \<le> inf (filtermap f F1) (filtermap f F2)"
```
```   677   by (auto simp: le_filter_def eventually_filtermap eventually_inf)
```
```   678
```
```   679 lemma filtermap_INF: "filtermap f (INF b:B. F b) \<le> (INF b:B. filtermap f (F b))"
```
```   680 proof -
```
```   681   { fix X :: "'c set" assume "finite X"
```
```   682     then have "filtermap f (INFIMUM X F) \<le> (INF b:X. filtermap f (F b))"
```
```   683     proof induct
```
```   684       case (insert x X)
```
```   685       have "filtermap f (INF a:insert x X. F a) \<le> inf (filtermap f (F x)) (filtermap f (INF a:X. F a))"
```
```   686         by (rule order_trans[OF _ filtermap_inf]) simp
```
```   687       also have "\<dots> \<le> inf (filtermap f (F x)) (INF a:X. filtermap f (F a))"
```
```   688         by (intro inf_mono insert order_refl)
```
```   689       finally show ?case
```
```   690         by simp
```
```   691     qed simp }
```
```   692   then show ?thesis
```
```   693     unfolding le_filter_def eventually_filtermap
```
```   694     by (subst (1 2) eventually_INF) auto
```
```   695 qed
```
```   696 subsubsection {* Standard filters *}
```
```   697
```
```   698 definition principal :: "'a set \<Rightarrow> 'a filter" where
```
```   699   "principal S = Abs_filter (\<lambda>P. \<forall>x\<in>S. P x)"
```
```   700
```
```   701 lemma eventually_principal: "eventually P (principal S) \<longleftrightarrow> (\<forall>x\<in>S. P x)"
```
```   702   unfolding principal_def
```
```   703   by (rule eventually_Abs_filter, rule is_filter.intro) auto
```
```   704
```
```   705 lemma eventually_inf_principal: "eventually P (inf F (principal s)) \<longleftrightarrow> eventually (\<lambda>x. x \<in> s \<longrightarrow> P x) F"
```
```   706   unfolding eventually_inf eventually_principal by (auto elim: eventually_elim1)
```
```   707
```
```   708 lemma principal_UNIV[simp]: "principal UNIV = top"
```
```   709   by (auto simp: filter_eq_iff eventually_principal)
```
```   710
```
```   711 lemma principal_empty[simp]: "principal {} = bot"
```
```   712   by (auto simp: filter_eq_iff eventually_principal)
```
```   713
```
```   714 lemma principal_eq_bot_iff: "principal X = bot \<longleftrightarrow> X = {}"
```
```   715   by (auto simp add: filter_eq_iff eventually_principal)
```
```   716
```
```   717 lemma principal_le_iff[iff]: "principal A \<le> principal B \<longleftrightarrow> A \<subseteq> B"
```
```   718   by (auto simp: le_filter_def eventually_principal)
```
```   719
```
```   720 lemma le_principal: "F \<le> principal A \<longleftrightarrow> eventually (\<lambda>x. x \<in> A) F"
```
```   721   unfolding le_filter_def eventually_principal
```
```   722   apply safe
```
```   723   apply (erule_tac x="\<lambda>x. x \<in> A" in allE)
```
```   724   apply (auto elim: eventually_elim1)
```
```   725   done
```
```   726
```
```   727 lemma principal_inject[iff]: "principal A = principal B \<longleftrightarrow> A = B"
```
```   728   unfolding eq_iff by simp
```
```   729
```
```   730 lemma sup_principal[simp]: "sup (principal A) (principal B) = principal (A \<union> B)"
```
```   731   unfolding filter_eq_iff eventually_sup eventually_principal by auto
```
```   732
```
```   733 lemma inf_principal[simp]: "inf (principal A) (principal B) = principal (A \<inter> B)"
```
```   734   unfolding filter_eq_iff eventually_inf eventually_principal
```
```   735   by (auto intro: exI[of _ "\<lambda>x. x \<in> A"] exI[of _ "\<lambda>x. x \<in> B"])
```
```   736
```
```   737 lemma SUP_principal[simp]: "(SUP i : I. principal (A i)) = principal (\<Union>i\<in>I. A i)"
```
```   738   unfolding filter_eq_iff eventually_Sup SUP_def by (auto simp: eventually_principal)
```
```   739
```
```   740 lemma INF_principal_finite: "finite X \<Longrightarrow> (INF x:X. principal (f x)) = principal (\<Inter>x\<in>X. f x)"
```
```   741   by (induct X rule: finite_induct) auto
```
```   742
```
```   743 lemma filtermap_principal[simp]: "filtermap f (principal A) = principal (f ` A)"
```
```   744   unfolding filter_eq_iff eventually_filtermap eventually_principal by simp
```
```   745
```
```   746 subsubsection {* Order filters *}
```
```   747
```
```   748 definition at_top :: "('a::order) filter"
```
```   749   where "at_top = (INF k. principal {k ..})"
```
```   750
```
```   751 lemma at_top_sub: "at_top = (INF k:{c::'a::linorder..}. principal {k ..})"
```
```   752   by (auto intro!: INF_eq max.cobounded1 max.cobounded2 simp: at_top_def)
```
```   753
```
```   754 lemma eventually_at_top_linorder: "eventually P at_top \<longleftrightarrow> (\<exists>N::'a::linorder. \<forall>n\<ge>N. P n)"
```
```   755   unfolding at_top_def
```
```   756   by (subst eventually_INF_base) (auto simp: eventually_principal intro: max.cobounded1 max.cobounded2)
```
```   757
```
```   758 lemma eventually_ge_at_top:
```
```   759   "eventually (\<lambda>x. (c::_::linorder) \<le> x) at_top"
```
```   760   unfolding eventually_at_top_linorder by auto
```
```   761
```
```   762 lemma eventually_at_top_dense: "eventually P at_top \<longleftrightarrow> (\<exists>N::'a::{no_top, linorder}. \<forall>n>N. P n)"
```
```   763 proof -
```
```   764   have "eventually P (INF k. principal {k <..}) \<longleftrightarrow> (\<exists>N::'a. \<forall>n>N. P n)"
```
```   765     by (subst eventually_INF_base) (auto simp: eventually_principal intro: max.cobounded1 max.cobounded2)
```
```   766   also have "(INF k. principal {k::'a <..}) = at_top"
```
```   767     unfolding at_top_def
```
```   768     by (intro INF_eq) (auto intro: less_imp_le simp: Ici_subset_Ioi_iff gt_ex)
```
```   769   finally show ?thesis .
```
```   770 qed
```
```   771
```
```   772 lemma eventually_gt_at_top:
```
```   773   "eventually (\<lambda>x. (c::_::unbounded_dense_linorder) < x) at_top"
```
```   774   unfolding eventually_at_top_dense by auto
```
```   775
```
```   776 definition at_bot :: "('a::order) filter"
```
```   777   where "at_bot = (INF k. principal {.. k})"
```
```   778
```
```   779 lemma at_bot_sub: "at_bot = (INF k:{.. c::'a::linorder}. principal {.. k})"
```
```   780   by (auto intro!: INF_eq min.cobounded1 min.cobounded2 simp: at_bot_def)
```
```   781
```
```   782 lemma eventually_at_bot_linorder:
```
```   783   fixes P :: "'a::linorder \<Rightarrow> bool" shows "eventually P at_bot \<longleftrightarrow> (\<exists>N. \<forall>n\<le>N. P n)"
```
```   784   unfolding at_bot_def
```
```   785   by (subst eventually_INF_base) (auto simp: eventually_principal intro: min.cobounded1 min.cobounded2)
```
```   786
```
```   787 lemma eventually_le_at_bot:
```
```   788   "eventually (\<lambda>x. x \<le> (c::_::linorder)) at_bot"
```
```   789   unfolding eventually_at_bot_linorder by auto
```
```   790
```
```   791 lemma eventually_at_bot_dense: "eventually P at_bot \<longleftrightarrow> (\<exists>N::'a::{no_bot, linorder}. \<forall>n<N. P n)"
```
```   792 proof -
```
```   793   have "eventually P (INF k. principal {..< k}) \<longleftrightarrow> (\<exists>N::'a. \<forall>n<N. P n)"
```
```   794     by (subst eventually_INF_base) (auto simp: eventually_principal intro: min.cobounded1 min.cobounded2)
```
```   795   also have "(INF k. principal {..< k::'a}) = at_bot"
```
```   796     unfolding at_bot_def
```
```   797     by (intro INF_eq) (auto intro: less_imp_le simp: Iic_subset_Iio_iff lt_ex)
```
```   798   finally show ?thesis .
```
```   799 qed
```
```   800
```
```   801 lemma eventually_gt_at_bot:
```
```   802   "eventually (\<lambda>x. x < (c::_::unbounded_dense_linorder)) at_bot"
```
```   803   unfolding eventually_at_bot_dense by auto
```
```   804
```
```   805 lemma trivial_limit_at_bot_linorder: "\<not> trivial_limit (at_bot ::('a::linorder) filter)"
```
```   806   unfolding trivial_limit_def
```
```   807   by (metis eventually_at_bot_linorder order_refl)
```
```   808
```
```   809 lemma trivial_limit_at_top_linorder: "\<not> trivial_limit (at_top ::('a::linorder) filter)"
```
```   810   unfolding trivial_limit_def
```
```   811   by (metis eventually_at_top_linorder order_refl)
```
```   812
```
```   813 subsection {* Sequentially *}
```
```   814
```
```   815 abbreviation sequentially :: "nat filter"
```
```   816   where "sequentially \<equiv> at_top"
```
```   817
```
```   818 lemma eventually_sequentially:
```
```   819   "eventually P sequentially \<longleftrightarrow> (\<exists>N. \<forall>n\<ge>N. P n)"
```
```   820   by (rule eventually_at_top_linorder)
```
```   821
```
```   822 lemma sequentially_bot [simp, intro]: "sequentially \<noteq> bot"
```
```   823   unfolding filter_eq_iff eventually_sequentially by auto
```
```   824
```
```   825 lemmas trivial_limit_sequentially = sequentially_bot
```
```   826
```
```   827 lemma eventually_False_sequentially [simp]:
```
```   828   "\<not> eventually (\<lambda>n. False) sequentially"
```
```   829   by (simp add: eventually_False)
```
```   830
```
```   831 lemma le_sequentially:
```
```   832   "F \<le> sequentially \<longleftrightarrow> (\<forall>N. eventually (\<lambda>n. N \<le> n) F)"
```
```   833   by (simp add: at_top_def le_INF_iff le_principal)
```
```   834
```
```   835 lemma eventually_sequentiallyI:
```
```   836   assumes "\<And>x. c \<le> x \<Longrightarrow> P x"
```
```   837   shows "eventually P sequentially"
```
```   838 using assms by (auto simp: eventually_sequentially)
```
```   839
```
```   840 lemma eventually_sequentially_seg:
```
```   841   "eventually (\<lambda>n. P (n + k)) sequentially \<longleftrightarrow> eventually P sequentially"
```
```   842   unfolding eventually_sequentially
```
```   843   apply safe
```
```   844    apply (rule_tac x="N + k" in exI)
```
```   845    apply rule
```
```   846    apply (erule_tac x="n - k" in allE)
```
```   847    apply auto []
```
```   848   apply (rule_tac x=N in exI)
```
```   849   apply auto []
```
```   850   done
```
```   851
```
```   852 subsubsection {* Topological filters *}
```
```   853
```
```   854 definition (in topological_space) nhds :: "'a \<Rightarrow> 'a filter"
```
```   855   where "nhds a = (INF S:{S. open S \<and> a \<in> S}. principal S)"
```
```   856
```
```   857 definition (in topological_space) at_within :: "'a \<Rightarrow> 'a set \<Rightarrow> 'a filter" ("at (_) within (_)" [1000, 60] 60)
```
```   858   where "at a within s = inf (nhds a) (principal (s - {a}))"
```
```   859
```
```   860 abbreviation (in topological_space) at :: "'a \<Rightarrow> 'a filter" ("at") where
```
```   861   "at x \<equiv> at x within (CONST UNIV)"
```
```   862
```
```   863 abbreviation (in order_topology) at_right :: "'a \<Rightarrow> 'a filter" where
```
```   864   "at_right x \<equiv> at x within {x <..}"
```
```   865
```
```   866 abbreviation (in order_topology) at_left :: "'a \<Rightarrow> 'a filter" where
```
```   867   "at_left x \<equiv> at x within {..< x}"
```
```   868
```
```   869 lemma (in topological_space) nhds_generated_topology:
```
```   870   "open = generate_topology T \<Longrightarrow> nhds x = (INF S:{S\<in>T. x \<in> S}. principal S)"
```
```   871   unfolding nhds_def
```
```   872 proof (safe intro!: antisym INF_greatest)
```
```   873   fix S assume "generate_topology T S" "x \<in> S"
```
```   874   then show "(INF S:{S \<in> T. x \<in> S}. principal S) \<le> principal S"
```
```   875     by induction
```
```   876        (auto intro: INF_lower order_trans simp add: inf_principal[symmetric] simp del: inf_principal)
```
```   877 qed (auto intro!: INF_lower intro: generate_topology.intros)
```
```   878
```
```   879 lemma (in topological_space) eventually_nhds:
```
```   880   "eventually P (nhds a) \<longleftrightarrow> (\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x))"
```
```   881   unfolding nhds_def by (subst eventually_INF_base) (auto simp: eventually_principal)
```
```   882
```
```   883 lemma nhds_neq_bot [simp]: "nhds a \<noteq> bot"
```
```   884   unfolding trivial_limit_def eventually_nhds by simp
```
```   885
```
```   886 lemma at_within_eq: "at x within s = (INF S:{S. open S \<and> x \<in> S}. principal (S \<inter> s - {x}))"
```
```   887   unfolding nhds_def at_within_def by (subst INF_inf_const2[symmetric]) (auto simp add: Diff_Int_distrib)
```
```   888
```
```   889 lemma eventually_at_filter:
```
```   890   "eventually P (at a within s) \<longleftrightarrow> eventually (\<lambda>x. x \<noteq> a \<longrightarrow> x \<in> s \<longrightarrow> P x) (nhds a)"
```
```   891   unfolding at_within_def eventually_inf_principal by (simp add: imp_conjL[symmetric] conj_commute)
```
```   892
```
```   893 lemma at_le: "s \<subseteq> t \<Longrightarrow> at x within s \<le> at x within t"
```
```   894   unfolding at_within_def by (intro inf_mono) auto
```
```   895
```
```   896 lemma eventually_at_topological:
```
```   897   "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))"
```
```   898   unfolding eventually_nhds eventually_at_filter by simp
```
```   899
```
```   900 lemma at_within_open: "a \<in> S \<Longrightarrow> open S \<Longrightarrow> at a within S = at a"
```
```   901   unfolding filter_eq_iff eventually_at_topological by (metis open_Int Int_iff UNIV_I)
```
```   902
```
```   903 lemma at_within_empty [simp]: "at a within {} = bot"
```
```   904   unfolding at_within_def by simp
```
```   905
```
```   906 lemma at_within_union: "at x within (S \<union> T) = sup (at x within S) (at x within T)"
```
```   907   unfolding filter_eq_iff eventually_sup eventually_at_filter
```
```   908   by (auto elim!: eventually_rev_mp)
```
```   909
```
```   910 lemma at_eq_bot_iff: "at a = bot \<longleftrightarrow> open {a}"
```
```   911   unfolding trivial_limit_def eventually_at_topological
```
```   912   by (safe, case_tac "S = {a}", simp, fast, fast)
```
```   913
```
```   914 lemma at_neq_bot [simp]: "at (a::'a::perfect_space) \<noteq> bot"
```
```   915   by (simp add: at_eq_bot_iff not_open_singleton)
```
```   916
```
```   917 lemma (in order_topology) nhds_order: "nhds x =
```
```   918   inf (INF a:{x <..}. principal {..< a}) (INF a:{..< x}. principal {a <..})"
```
```   919 proof -
```
```   920   have 1: "{S \<in> range lessThan \<union> range greaterThan. x \<in> S} =
```
```   921       (\<lambda>a. {..< a}) ` {x <..} \<union> (\<lambda>a. {a <..}) ` {..< x}"
```
```   922     by auto
```
```   923   show ?thesis
```
```   924     unfolding nhds_generated_topology[OF open_generated_order] INF_union 1 INF_image comp_def ..
```
```   925 qed
```
```   926
```
```   927 lemma (in linorder_topology) at_within_order: "UNIV \<noteq> {x} \<Longrightarrow>
```
```   928   at x within s = inf (INF a:{x <..}. principal ({..< a} \<inter> s - {x}))
```
```   929                       (INF a:{..< x}. principal ({a <..} \<inter> s - {x}))"
```
```   930 proof (cases "{x <..} = {}" "{..< x} = {}" rule: case_split[case_product case_split])
```
```   931   assume "UNIV \<noteq> {x}" "{x<..} = {}" "{..< x} = {}"
```
```   932   moreover have "UNIV = {..< x} \<union> {x} \<union> {x <..}"
```
```   933     by auto
```
```   934   ultimately show ?thesis
```
```   935     by auto
```
```   936 qed (auto simp: at_within_def nhds_order Int_Diff inf_principal[symmetric] INF_inf_const2
```
```   937                 inf_sup_aci[where 'a="'a filter"]
```
```   938           simp del: inf_principal)
```
```   939
```
```   940 lemma (in linorder_topology) at_left_eq:
```
```   941   "y < x \<Longrightarrow> at_left x = (INF a:{..< x}. principal {a <..< x})"
```
```   942   by (subst at_within_order)
```
```   943      (auto simp: greaterThan_Int_greaterThan greaterThanLessThan_eq[symmetric] min.absorb2 INF_constant
```
```   944            intro!: INF_lower2 inf_absorb2)
```
```   945
```
```   946 lemma (in linorder_topology) eventually_at_left:
```
```   947   "y < x \<Longrightarrow> eventually P (at_left x) \<longleftrightarrow> (\<exists>b<x. \<forall>y>b. y < x \<longrightarrow> P y)"
```
```   948   unfolding at_left_eq by (subst eventually_INF_base) (auto simp: eventually_principal Ball_def)
```
```   949
```
```   950 lemma (in linorder_topology) at_right_eq:
```
```   951   "x < y \<Longrightarrow> at_right x = (INF a:{x <..}. principal {x <..< a})"
```
```   952   by (subst at_within_order)
```
```   953      (auto simp: lessThan_Int_lessThan greaterThanLessThan_eq[symmetric] max.absorb2 INF_constant Int_commute
```
```   954            intro!: INF_lower2 inf_absorb1)
```
```   955
```
```   956 lemma (in linorder_topology) eventually_at_right:
```
```   957   "x < y \<Longrightarrow> eventually P (at_right x) \<longleftrightarrow> (\<exists>b>x. \<forall>y>x. y < b \<longrightarrow> P y)"
```
```   958   unfolding at_right_eq by (subst eventually_INF_base) (auto simp: eventually_principal Ball_def)
```
```   959
```
```   960 lemma trivial_limit_at_right_top: "at_right (top::_::{order_top, linorder_topology}) = bot"
```
```   961   unfolding filter_eq_iff eventually_at_topological by auto
```
```   962
```
```   963 lemma trivial_limit_at_left_bot: "at_left (bot::_::{order_bot, linorder_topology}) = bot"
```
```   964   unfolding filter_eq_iff eventually_at_topological by auto
```
```   965
```
```   966 lemma trivial_limit_at_left_real [simp]:
```
```   967   "\<not> trivial_limit (at_left (x::'a::{no_bot, dense_order, linorder_topology}))"
```
```   968   using lt_ex[of x]
```
```   969   by safe (auto simp add: trivial_limit_def eventually_at_left dest: dense)
```
```   970
```
```   971 lemma trivial_limit_at_right_real [simp]:
```
```   972   "\<not> trivial_limit (at_right (x::'a::{no_top, dense_order, linorder_topology}))"
```
```   973   using gt_ex[of x]
```
```   974   by safe (auto simp add: trivial_limit_def eventually_at_right dest: dense)
```
```   975
```
```   976 lemma at_eq_sup_left_right: "at (x::'a::linorder_topology) = sup (at_left x) (at_right x)"
```
```   977   by (auto simp: eventually_at_filter filter_eq_iff eventually_sup
```
```   978            elim: eventually_elim2 eventually_elim1)
```
```   979
```
```   980 lemma eventually_at_split:
```
```   981   "eventually P (at (x::'a::linorder_topology)) \<longleftrightarrow> eventually P (at_left x) \<and> eventually P (at_right x)"
```
```   982   by (subst at_eq_sup_left_right) (simp add: eventually_sup)
```
```   983
```
```   984 subsection {* Limits *}
```
```   985
```
```   986 definition filterlim :: "('a \<Rightarrow> 'b) \<Rightarrow> 'b filter \<Rightarrow> 'a filter \<Rightarrow> bool" where
```
```   987   "filterlim f F2 F1 \<longleftrightarrow> filtermap f F1 \<le> F2"
```
```   988
```
```   989 syntax
```
```   990   "_LIM" :: "pttrns \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> bool" ("(3LIM (_)/ (_)./ (_) :> (_))" [1000, 10, 0, 10] 10)
```
```   991
```
```   992 translations
```
```   993   "LIM x F1. f :> F2"   == "CONST filterlim (%x. f) F2 F1"
```
```   994
```
```   995 lemma filterlim_iff:
```
```   996   "(LIM x F1. f x :> F2) \<longleftrightarrow> (\<forall>P. eventually P F2 \<longrightarrow> eventually (\<lambda>x. P (f x)) F1)"
```
```   997   unfolding filterlim_def le_filter_def eventually_filtermap ..
```
```   998
```
```   999 lemma filterlim_compose:
```
```  1000   "filterlim g F3 F2 \<Longrightarrow> filterlim f F2 F1 \<Longrightarrow> filterlim (\<lambda>x. g (f x)) F3 F1"
```
```  1001   unfolding filterlim_def filtermap_filtermap[symmetric] by (metis filtermap_mono order_trans)
```
```  1002
```
```  1003 lemma filterlim_mono:
```
```  1004   "filterlim f F2 F1 \<Longrightarrow> F2 \<le> F2' \<Longrightarrow> F1' \<le> F1 \<Longrightarrow> filterlim f F2' F1'"
```
```  1005   unfolding filterlim_def by (metis filtermap_mono order_trans)
```
```  1006
```
```  1007 lemma filterlim_ident: "LIM x F. x :> F"
```
```  1008   by (simp add: filterlim_def filtermap_ident)
```
```  1009
```
```  1010 lemma filterlim_cong:
```
```  1011   "F1 = F1' \<Longrightarrow> F2 = F2' \<Longrightarrow> eventually (\<lambda>x. f x = g x) F2 \<Longrightarrow> filterlim f F1 F2 = filterlim g F1' F2'"
```
```  1012   by (auto simp: filterlim_def le_filter_def eventually_filtermap elim: eventually_elim2)
```
```  1013
```
```  1014 lemma filterlim_mono_eventually:
```
```  1015   assumes "filterlim f F G" and ord: "F \<le> F'" "G' \<le> G"
```
```  1016   assumes eq: "eventually (\<lambda>x. f x = f' x) G'"
```
```  1017   shows "filterlim f' F' G'"
```
```  1018   apply (rule filterlim_cong[OF refl refl eq, THEN iffD1])
```
```  1019   apply (rule filterlim_mono[OF _ ord])
```
```  1020   apply fact
```
```  1021   done
```
```  1022
```
```  1023 lemma filtermap_mono_strong: "inj f \<Longrightarrow> filtermap f F \<le> filtermap f G \<longleftrightarrow> F \<le> G"
```
```  1024   apply (auto intro!: filtermap_mono) []
```
```  1025   apply (auto simp: le_filter_def eventually_filtermap)
```
```  1026   apply (erule_tac x="\<lambda>x. P (inv f x)" in allE)
```
```  1027   apply auto
```
```  1028   done
```
```  1029
```
```  1030 lemma filtermap_eq_strong: "inj f \<Longrightarrow> filtermap f F = filtermap f G \<longleftrightarrow> F = G"
```
```  1031   by (simp add: filtermap_mono_strong eq_iff)
```
```  1032
```
```  1033 lemma filterlim_principal:
```
```  1034   "(LIM x F. f x :> principal S) \<longleftrightarrow> (eventually (\<lambda>x. f x \<in> S) F)"
```
```  1035   unfolding filterlim_def eventually_filtermap le_principal ..
```
```  1036
```
```  1037 lemma filterlim_inf:
```
```  1038   "(LIM x F1. f x :> inf F2 F3) \<longleftrightarrow> ((LIM x F1. f x :> F2) \<and> (LIM x F1. f x :> F3))"
```
```  1039   unfolding filterlim_def by simp
```
```  1040
```
```  1041 lemma filterlim_INF:
```
```  1042   "(LIM x F. f x :> (INF b:B. G b)) \<longleftrightarrow> (\<forall>b\<in>B. LIM x F. f x :> G b)"
```
```  1043   unfolding filterlim_def le_INF_iff ..
```
```  1044
```
```  1045 lemma filterlim_INF_INF:
```
```  1046   "(\<And>m. m \<in> J \<Longrightarrow> \<exists>i\<in>I. filtermap f (F i) \<le> G m) \<Longrightarrow> LIM x (INF i:I. F i). f x :> (INF j:J. G j)"
```
```  1047   unfolding filterlim_def by (rule order_trans[OF filtermap_INF INF_mono])
```
```  1048
```
```  1049 lemma filterlim_base:
```
```  1050   "(\<And>m x. m \<in> J \<Longrightarrow> i m \<in> I) \<Longrightarrow> (\<And>m x. m \<in> J \<Longrightarrow> x \<in> F (i m) \<Longrightarrow> f x \<in> G m) \<Longrightarrow>
```
```  1051     LIM x (INF i:I. principal (F i)). f x :> (INF j:J. principal (G j))"
```
```  1052   by (force intro!: filterlim_INF_INF simp: image_subset_iff)
```
```  1053
```
```  1054 lemma filterlim_base_iff:
```
```  1055   assumes "I \<noteq> {}" and chain: "\<And>i j. i \<in> I \<Longrightarrow> j \<in> I \<Longrightarrow> F i \<subseteq> F j \<or> F j \<subseteq> F i"
```
```  1056   shows "(LIM x (INF i:I. principal (F i)). f x :> INF j:J. principal (G j)) \<longleftrightarrow>
```
```  1057     (\<forall>j\<in>J. \<exists>i\<in>I. \<forall>x\<in>F i. f x \<in> G j)"
```
```  1058   unfolding filterlim_INF filterlim_principal
```
```  1059 proof (subst eventually_INF_base)
```
```  1060   fix i j assume "i \<in> I" "j \<in> I"
```
```  1061   with chain[OF this] show "\<exists>x\<in>I. principal (F x) \<le> inf (principal (F i)) (principal (F j))"
```
```  1062     by auto
```
```  1063 qed (auto simp: eventually_principal `I \<noteq> {}`)
```
```  1064
```
```  1065 lemma filterlim_filtermap: "filterlim f F1 (filtermap g F2) = filterlim (\<lambda>x. f (g x)) F1 F2"
```
```  1066   unfolding filterlim_def filtermap_filtermap ..
```
```  1067
```
```  1068 lemma filterlim_sup:
```
```  1069   "filterlim f F F1 \<Longrightarrow> filterlim f F F2 \<Longrightarrow> filterlim f F (sup F1 F2)"
```
```  1070   unfolding filterlim_def filtermap_sup by auto
```
```  1071
```
```  1072 lemma eventually_sequentially_Suc: "eventually (\<lambda>i. P (Suc i)) sequentially \<longleftrightarrow> eventually P sequentially"
```
```  1073   unfolding eventually_sequentially by (metis Suc_le_D Suc_le_mono le_Suc_eq)
```
```  1074
```
```  1075 lemma filterlim_sequentially_Suc:
```
```  1076   "(LIM x sequentially. f (Suc x) :> F) \<longleftrightarrow> (LIM x sequentially. f x :> F)"
```
```  1077   unfolding filterlim_iff by (subst eventually_sequentially_Suc) simp
```
```  1078
```
```  1079 lemma filterlim_Suc: "filterlim Suc sequentially sequentially"
```
```  1080   by (simp add: filterlim_iff eventually_sequentially) (metis le_Suc_eq)
```
```  1081
```
```  1082 subsubsection {* Tendsto *}
```
```  1083
```
```  1084 abbreviation (in topological_space)
```
```  1085   tendsto :: "('b \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'b filter \<Rightarrow> bool" (infixr "--->" 55) where
```
```  1086   "(f ---> l) F \<equiv> filterlim f (nhds l) F"
```
```  1087
```
```  1088 definition (in t2_space) Lim :: "'f filter \<Rightarrow> ('f \<Rightarrow> 'a) \<Rightarrow> 'a" where
```
```  1089   "Lim A f = (THE l. (f ---> l) A)"
```
```  1090
```
```  1091 lemma tendsto_eq_rhs: "(f ---> x) F \<Longrightarrow> x = y \<Longrightarrow> (f ---> y) F"
```
```  1092   by simp
```
```  1093
```
```  1094 named_theorems tendsto_intros "introduction rules for tendsto"
```
```  1095 setup {*
```
```  1096   Global_Theory.add_thms_dynamic (@{binding tendsto_eq_intros},
```
```  1097     fn context =>
```
```  1098       Named_Theorems.get (Context.proof_of context) @{named_theorems tendsto_intros}
```
```  1099       |> map_filter (try (fn thm => @{thm tendsto_eq_rhs} OF [thm])))
```
```  1100 *}
```
```  1101
```
```  1102 lemma (in topological_space) tendsto_def:
```
```  1103    "(f ---> l) F \<longleftrightarrow> (\<forall>S. open S \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) F)"
```
```  1104    unfolding nhds_def filterlim_INF filterlim_principal by auto
```
```  1105
```
```  1106 lemma tendsto_mono: "F \<le> F' \<Longrightarrow> (f ---> l) F' \<Longrightarrow> (f ---> l) F"
```
```  1107   unfolding tendsto_def le_filter_def by fast
```
```  1108
```
```  1109 lemma tendsto_within_subset: "(f ---> l) (at x within S) \<Longrightarrow> T \<subseteq> S \<Longrightarrow> (f ---> l) (at x within T)"
```
```  1110   by (blast intro: tendsto_mono at_le)
```
```  1111
```
```  1112 lemma filterlim_at:
```
```  1113   "(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 ---> b) F)"
```
```  1114   by (simp add: at_within_def filterlim_inf filterlim_principal conj_commute)
```
```  1115
```
```  1116 lemma (in topological_space) topological_tendstoI:
```
```  1117   "(\<And>S. open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) F) \<Longrightarrow> (f ---> l) F"
```
```  1118   unfolding tendsto_def by auto
```
```  1119
```
```  1120 lemma (in topological_space) topological_tendstoD:
```
```  1121   "(f ---> l) F \<Longrightarrow> open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) F"
```
```  1122   unfolding tendsto_def by auto
```
```  1123
```
```  1124 lemma (in order_topology) order_tendsto_iff:
```
```  1125   "(f ---> x) F \<longleftrightarrow> (\<forall>l<x. eventually (\<lambda>x. l < f x) F) \<and> (\<forall>u>x. eventually (\<lambda>x. f x < u) F)"
```
```  1126   unfolding nhds_order filterlim_inf filterlim_INF filterlim_principal by auto
```
```  1127
```
```  1128 lemma (in order_topology) order_tendstoI:
```
```  1129   "(\<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>
```
```  1130     (f ---> y) F"
```
```  1131   unfolding order_tendsto_iff by auto
```
```  1132
```
```  1133 lemma (in order_topology) order_tendstoD:
```
```  1134   assumes "(f ---> y) F"
```
```  1135   shows "a < y \<Longrightarrow> eventually (\<lambda>x. a < f x) F"
```
```  1136     and "y < a \<Longrightarrow> eventually (\<lambda>x. f x < a) F"
```
```  1137   using assms unfolding order_tendsto_iff by auto
```
```  1138
```
```  1139 lemma tendsto_bot [simp]: "(f ---> a) bot"
```
```  1140   unfolding tendsto_def by simp
```
```  1141
```
```  1142 lemma (in linorder_topology) tendsto_max:
```
```  1143   assumes X: "(X ---> x) net"
```
```  1144   assumes Y: "(Y ---> y) net"
```
```  1145   shows "((\<lambda>x. max (X x) (Y x)) ---> max x y) net"
```
```  1146 proof (rule order_tendstoI)
```
```  1147   fix a assume "a < max x y"
```
```  1148   then show "eventually (\<lambda>x. a < max (X x) (Y x)) net"
```
```  1149     using order_tendstoD(1)[OF X, of a] order_tendstoD(1)[OF Y, of a]
```
```  1150     by (auto simp: less_max_iff_disj elim: eventually_elim1)
```
```  1151 next
```
```  1152   fix a assume "max x y < a"
```
```  1153   then show "eventually (\<lambda>x. max (X x) (Y x) < a) net"
```
```  1154     using order_tendstoD(2)[OF X, of a] order_tendstoD(2)[OF Y, of a]
```
```  1155     by (auto simp: eventually_conj_iff)
```
```  1156 qed
```
```  1157
```
```  1158 lemma (in linorder_topology) tendsto_min:
```
```  1159   assumes X: "(X ---> x) net"
```
```  1160   assumes Y: "(Y ---> y) net"
```
```  1161   shows "((\<lambda>x. min (X x) (Y x)) ---> min x y) net"
```
```  1162 proof (rule order_tendstoI)
```
```  1163   fix a assume "a < min x y"
```
```  1164   then show "eventually (\<lambda>x. a < min (X x) (Y x)) net"
```
```  1165     using order_tendstoD(1)[OF X, of a] order_tendstoD(1)[OF Y, of a]
```
```  1166     by (auto simp: eventually_conj_iff)
```
```  1167 next
```
```  1168   fix a assume "min x y < a"
```
```  1169   then show "eventually (\<lambda>x. min (X x) (Y x) < a) net"
```
```  1170     using order_tendstoD(2)[OF X, of a] order_tendstoD(2)[OF Y, of a]
```
```  1171     by (auto simp: min_less_iff_disj elim: eventually_elim1)
```
```  1172 qed
```
```  1173
```
```  1174 lemma tendsto_ident_at [tendsto_intros]: "((\<lambda>x. x) ---> a) (at a within s)"
```
```  1175   unfolding tendsto_def eventually_at_topological by auto
```
```  1176
```
```  1177 lemma (in topological_space) tendsto_const [tendsto_intros]: "((\<lambda>x. k) ---> k) F"
```
```  1178   by (simp add: tendsto_def)
```
```  1179
```
```  1180 lemma (in t2_space) tendsto_unique:
```
```  1181   assumes "F \<noteq> bot" and "(f ---> a) F" and "(f ---> b) F"
```
```  1182   shows "a = b"
```
```  1183 proof (rule ccontr)
```
```  1184   assume "a \<noteq> b"
```
```  1185   obtain U V where "open U" "open V" "a \<in> U" "b \<in> V" "U \<inter> V = {}"
```
```  1186     using hausdorff [OF `a \<noteq> b`] by fast
```
```  1187   have "eventually (\<lambda>x. f x \<in> U) F"
```
```  1188     using `(f ---> a) F` `open U` `a \<in> U` by (rule topological_tendstoD)
```
```  1189   moreover
```
```  1190   have "eventually (\<lambda>x. f x \<in> V) F"
```
```  1191     using `(f ---> b) F` `open V` `b \<in> V` by (rule topological_tendstoD)
```
```  1192   ultimately
```
```  1193   have "eventually (\<lambda>x. False) F"
```
```  1194   proof eventually_elim
```
```  1195     case (elim x)
```
```  1196     hence "f x \<in> U \<inter> V" by simp
```
```  1197     with `U \<inter> V = {}` show ?case by simp
```
```  1198   qed
```
```  1199   with `\<not> trivial_limit F` show "False"
```
```  1200     by (simp add: trivial_limit_def)
```
```  1201 qed
```
```  1202
```
```  1203 lemma (in t2_space) tendsto_const_iff:
```
```  1204   assumes "\<not> trivial_limit F" shows "((\<lambda>x. a :: 'a) ---> b) F \<longleftrightarrow> a = b"
```
```  1205   by (safe intro!: tendsto_const tendsto_unique [OF assms tendsto_const])
```
```  1206
```
```  1207 lemma increasing_tendsto:
```
```  1208   fixes f :: "_ \<Rightarrow> 'a::order_topology"
```
```  1209   assumes bdd: "eventually (\<lambda>n. f n \<le> l) F"
```
```  1210       and en: "\<And>x. x < l \<Longrightarrow> eventually (\<lambda>n. x < f n) F"
```
```  1211   shows "(f ---> l) F"
```
```  1212   using assms by (intro order_tendstoI) (auto elim!: eventually_elim1)
```
```  1213
```
```  1214 lemma decreasing_tendsto:
```
```  1215   fixes f :: "_ \<Rightarrow> 'a::order_topology"
```
```  1216   assumes bdd: "eventually (\<lambda>n. l \<le> f n) F"
```
```  1217       and en: "\<And>x. l < x \<Longrightarrow> eventually (\<lambda>n. f n < x) F"
```
```  1218   shows "(f ---> l) F"
```
```  1219   using assms by (intro order_tendstoI) (auto elim!: eventually_elim1)
```
```  1220
```
```  1221 lemma tendsto_sandwich:
```
```  1222   fixes f g h :: "'a \<Rightarrow> 'b::order_topology"
```
```  1223   assumes ev: "eventually (\<lambda>n. f n \<le> g n) net" "eventually (\<lambda>n. g n \<le> h n) net"
```
```  1224   assumes lim: "(f ---> c) net" "(h ---> c) net"
```
```  1225   shows "(g ---> c) net"
```
```  1226 proof (rule order_tendstoI)
```
```  1227   fix a show "a < c \<Longrightarrow> eventually (\<lambda>x. a < g x) net"
```
```  1228     using order_tendstoD[OF lim(1), of a] ev by (auto elim: eventually_elim2)
```
```  1229 next
```
```  1230   fix a show "c < a \<Longrightarrow> eventually (\<lambda>x. g x < a) net"
```
```  1231     using order_tendstoD[OF lim(2), of a] ev by (auto elim: eventually_elim2)
```
```  1232 qed
```
```  1233
```
```  1234 lemma tendsto_le:
```
```  1235   fixes f g :: "'a \<Rightarrow> 'b::linorder_topology"
```
```  1236   assumes F: "\<not> trivial_limit F"
```
```  1237   assumes x: "(f ---> x) F" and y: "(g ---> y) F"
```
```  1238   assumes ev: "eventually (\<lambda>x. g x \<le> f x) F"
```
```  1239   shows "y \<le> x"
```
```  1240 proof (rule ccontr)
```
```  1241   assume "\<not> y \<le> x"
```
```  1242   with less_separate[of x y] obtain a b where xy: "x < a" "b < y" "{..<a} \<inter> {b<..} = {}"
```
```  1243     by (auto simp: not_le)
```
```  1244   then have "eventually (\<lambda>x. f x < a) F" "eventually (\<lambda>x. b < g x) F"
```
```  1245     using x y by (auto intro: order_tendstoD)
```
```  1246   with ev have "eventually (\<lambda>x. False) F"
```
```  1247     by eventually_elim (insert xy, fastforce)
```
```  1248   with F show False
```
```  1249     by (simp add: eventually_False)
```
```  1250 qed
```
```  1251
```
```  1252 lemma tendsto_le_const:
```
```  1253   fixes f :: "'a \<Rightarrow> 'b::linorder_topology"
```
```  1254   assumes F: "\<not> trivial_limit F"
```
```  1255   assumes x: "(f ---> x) F" and a: "eventually (\<lambda>i. a \<le> f i) F"
```
```  1256   shows "a \<le> x"
```
```  1257   using F x tendsto_const a by (rule tendsto_le)
```
```  1258
```
```  1259 lemma tendsto_ge_const:
```
```  1260   fixes f :: "'a \<Rightarrow> 'b::linorder_topology"
```
```  1261   assumes F: "\<not> trivial_limit F"
```
```  1262   assumes x: "(f ---> x) F" and a: "eventually (\<lambda>i. a \<ge> f i) F"
```
```  1263   shows "a \<ge> x"
```
```  1264   by (rule tendsto_le [OF F tendsto_const x a])
```
```  1265
```
```  1266 subsubsection {* Rules about @{const Lim} *}
```
```  1267
```
```  1268 lemma tendsto_Lim:
```
```  1269   "\<not>(trivial_limit net) \<Longrightarrow> (f ---> l) net \<Longrightarrow> Lim net f = l"
```
```  1270   unfolding Lim_def using tendsto_unique[of net f] by auto
```
```  1271
```
```  1272 lemma Lim_ident_at: "\<not> trivial_limit (at x within s) \<Longrightarrow> Lim (at x within s) (\<lambda>x. x) = x"
```
```  1273   by (rule tendsto_Lim[OF _ tendsto_ident_at]) auto
```
```  1274
```
```  1275 subsection {* Limits to @{const at_top} and @{const at_bot} *}
```
```  1276
```
```  1277 lemma filterlim_at_top:
```
```  1278   fixes f :: "'a \<Rightarrow> ('b::linorder)"
```
```  1279   shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. Z \<le> f x) F)"
```
```  1280   by (auto simp: filterlim_iff eventually_at_top_linorder elim!: eventually_elim1)
```
```  1281
```
```  1282 lemma filterlim_at_top_mono:
```
```  1283   "LIM x F. f x :> at_top \<Longrightarrow> eventually (\<lambda>x. f x \<le> (g x::'a::linorder)) F \<Longrightarrow>
```
```  1284     LIM x F. g x :> at_top"
```
```  1285   by (auto simp: filterlim_at_top elim: eventually_elim2 intro: order_trans)
```
```  1286
```
```  1287 lemma filterlim_at_top_dense:
```
```  1288   fixes f :: "'a \<Rightarrow> ('b::unbounded_dense_linorder)"
```
```  1289   shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. Z < f x) F)"
```
```  1290   by (metis eventually_elim1[of _ F] eventually_gt_at_top order_less_imp_le
```
```  1291             filterlim_at_top[of f F] filterlim_iff[of f at_top F])
```
```  1292
```
```  1293 lemma filterlim_at_top_ge:
```
```  1294   fixes f :: "'a \<Rightarrow> ('b::linorder)" and c :: "'b"
```
```  1295   shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z\<ge>c. eventually (\<lambda>x. Z \<le> f x) F)"
```
```  1296   unfolding at_top_sub[of c] filterlim_INF by (auto simp add: filterlim_principal)
```
```  1297
```
```  1298 lemma filterlim_at_top_at_top:
```
```  1299   fixes f :: "'a::linorder \<Rightarrow> 'b::linorder"
```
```  1300   assumes mono: "\<And>x y. Q x \<Longrightarrow> Q y \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y"
```
```  1301   assumes bij: "\<And>x. P x \<Longrightarrow> f (g x) = x" "\<And>x. P x \<Longrightarrow> Q (g x)"
```
```  1302   assumes Q: "eventually Q at_top"
```
```  1303   assumes P: "eventually P at_top"
```
```  1304   shows "filterlim f at_top at_top"
```
```  1305 proof -
```
```  1306   from P obtain x where x: "\<And>y. x \<le> y \<Longrightarrow> P y"
```
```  1307     unfolding eventually_at_top_linorder by auto
```
```  1308   show ?thesis
```
```  1309   proof (intro filterlim_at_top_ge[THEN iffD2] allI impI)
```
```  1310     fix z assume "x \<le> z"
```
```  1311     with x have "P z" by auto
```
```  1312     have "eventually (\<lambda>x. g z \<le> x) at_top"
```
```  1313       by (rule eventually_ge_at_top)
```
```  1314     with Q show "eventually (\<lambda>x. z \<le> f x) at_top"
```
```  1315       by eventually_elim (metis mono bij `P z`)
```
```  1316   qed
```
```  1317 qed
```
```  1318
```
```  1319 lemma filterlim_at_top_gt:
```
```  1320   fixes f :: "'a \<Rightarrow> ('b::unbounded_dense_linorder)" and c :: "'b"
```
```  1321   shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z>c. eventually (\<lambda>x. Z \<le> f x) F)"
```
```  1322   by (metis filterlim_at_top order_less_le_trans gt_ex filterlim_at_top_ge)
```
```  1323
```
```  1324 lemma filterlim_at_bot:
```
```  1325   fixes f :: "'a \<Rightarrow> ('b::linorder)"
```
```  1326   shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. f x \<le> Z) F)"
```
```  1327   by (auto simp: filterlim_iff eventually_at_bot_linorder elim!: eventually_elim1)
```
```  1328
```
```  1329 lemma filterlim_at_bot_dense:
```
```  1330   fixes f :: "'a \<Rightarrow> ('b::{dense_linorder, no_bot})"
```
```  1331   shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. f x < Z) F)"
```
```  1332 proof (auto simp add: filterlim_at_bot[of f F])
```
```  1333   fix Z :: 'b
```
```  1334   from lt_ex [of Z] obtain Z' where 1: "Z' < Z" ..
```
```  1335   assume "\<forall>Z. eventually (\<lambda>x. f x \<le> Z) F"
```
```  1336   hence "eventually (\<lambda>x. f x \<le> Z') F" by auto
```
```  1337   thus "eventually (\<lambda>x. f x < Z) F"
```
```  1338     apply (rule eventually_mono[rotated])
```
```  1339     using 1 by auto
```
```  1340   next
```
```  1341     fix Z :: 'b
```
```  1342     show "\<forall>Z. eventually (\<lambda>x. f x < Z) F \<Longrightarrow> eventually (\<lambda>x. f x \<le> Z) F"
```
```  1343       by (drule spec [of _ Z], erule eventually_mono[rotated], auto simp add: less_imp_le)
```
```  1344 qed
```
```  1345
```
```  1346 lemma filterlim_at_bot_le:
```
```  1347   fixes f :: "'a \<Rightarrow> ('b::linorder)" and c :: "'b"
```
```  1348   shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z\<le>c. eventually (\<lambda>x. Z \<ge> f x) F)"
```
```  1349   unfolding filterlim_at_bot
```
```  1350 proof safe
```
```  1351   fix Z assume *: "\<forall>Z\<le>c. eventually (\<lambda>x. Z \<ge> f x) F"
```
```  1352   with *[THEN spec, of "min Z c"] show "eventually (\<lambda>x. Z \<ge> f x) F"
```
```  1353     by (auto elim!: eventually_elim1)
```
```  1354 qed simp
```
```  1355
```
```  1356 lemma filterlim_at_bot_lt:
```
```  1357   fixes f :: "'a \<Rightarrow> ('b::unbounded_dense_linorder)" and c :: "'b"
```
```  1358   shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z<c. eventually (\<lambda>x. Z \<ge> f x) F)"
```
```  1359   by (metis filterlim_at_bot filterlim_at_bot_le lt_ex order_le_less_trans)
```
```  1360
```
```  1361 lemma filterlim_at_bot_at_right:
```
```  1362   fixes f :: "'a::linorder_topology \<Rightarrow> 'b::linorder"
```
```  1363   assumes mono: "\<And>x y. Q x \<Longrightarrow> Q y \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y"
```
```  1364   assumes bij: "\<And>x. P x \<Longrightarrow> f (g x) = x" "\<And>x. P x \<Longrightarrow> Q (g x)"
```
```  1365   assumes Q: "eventually Q (at_right a)" and bound: "\<And>b. Q b \<Longrightarrow> a < b"
```
```  1366   assumes P: "eventually P at_bot"
```
```  1367   shows "filterlim f at_bot (at_right a)"
```
```  1368 proof -
```
```  1369   from P obtain x where x: "\<And>y. y \<le> x \<Longrightarrow> P y"
```
```  1370     unfolding eventually_at_bot_linorder by auto
```
```  1371   show ?thesis
```
```  1372   proof (intro filterlim_at_bot_le[THEN iffD2] allI impI)
```
```  1373     fix z assume "z \<le> x"
```
```  1374     with x have "P z" by auto
```
```  1375     have "eventually (\<lambda>x. x \<le> g z) (at_right a)"
```
```  1376       using bound[OF bij(2)[OF `P z`]]
```
```  1377       unfolding eventually_at_right[OF bound[OF bij(2)[OF `P z`]]] by (auto intro!: exI[of _ "g z"])
```
```  1378     with Q show "eventually (\<lambda>x. f x \<le> z) (at_right a)"
```
```  1379       by eventually_elim (metis bij `P z` mono)
```
```  1380   qed
```
```  1381 qed
```
```  1382
```
```  1383 lemma filterlim_at_top_at_left:
```
```  1384   fixes f :: "'a::linorder_topology \<Rightarrow> 'b::linorder"
```
```  1385   assumes mono: "\<And>x y. Q x \<Longrightarrow> Q y \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y"
```
```  1386   assumes bij: "\<And>x. P x \<Longrightarrow> f (g x) = x" "\<And>x. P x \<Longrightarrow> Q (g x)"
```
```  1387   assumes Q: "eventually Q (at_left a)" and bound: "\<And>b. Q b \<Longrightarrow> b < a"
```
```  1388   assumes P: "eventually P at_top"
```
```  1389   shows "filterlim f at_top (at_left a)"
```
```  1390 proof -
```
```  1391   from P obtain x where x: "\<And>y. x \<le> y \<Longrightarrow> P y"
```
```  1392     unfolding eventually_at_top_linorder by auto
```
```  1393   show ?thesis
```
```  1394   proof (intro filterlim_at_top_ge[THEN iffD2] allI impI)
```
```  1395     fix z assume "x \<le> z"
```
```  1396     with x have "P z" by auto
```
```  1397     have "eventually (\<lambda>x. g z \<le> x) (at_left a)"
```
```  1398       using bound[OF bij(2)[OF `P z`]]
```
```  1399       unfolding eventually_at_left[OF bound[OF bij(2)[OF `P z`]]] by (auto intro!: exI[of _ "g z"])
```
```  1400     with Q show "eventually (\<lambda>x. z \<le> f x) (at_left a)"
```
```  1401       by eventually_elim (metis bij `P z` mono)
```
```  1402   qed
```
```  1403 qed
```
```  1404
```
```  1405 lemma filterlim_split_at:
```
```  1406   "filterlim f F (at_left x) \<Longrightarrow> filterlim f F (at_right x) \<Longrightarrow> filterlim f F (at (x::'a::linorder_topology))"
```
```  1407   by (subst at_eq_sup_left_right) (rule filterlim_sup)
```
```  1408
```
```  1409 lemma filterlim_at_split:
```
```  1410   "filterlim f F (at (x::'a::linorder_topology)) \<longleftrightarrow> filterlim f F (at_left x) \<and> filterlim f F (at_right x)"
```
```  1411   by (subst at_eq_sup_left_right) (simp add: filterlim_def filtermap_sup)
```
```  1412
```
```  1413 lemma eventually_nhds_top:
```
```  1414   fixes P :: "'a :: {order_top, linorder_topology} \<Rightarrow> bool"
```
```  1415   assumes "(b::'a) < top"
```
```  1416   shows "eventually P (nhds top) \<longleftrightarrow> (\<exists>b<top. (\<forall>z. b < z \<longrightarrow> P z))"
```
```  1417   unfolding eventually_nhds
```
```  1418 proof safe
```
```  1419   fix S :: "'a set" assume "open S" "top \<in> S"
```
```  1420   note open_left[OF this `b < top`]
```
```  1421   moreover assume "\<forall>s\<in>S. P s"
```
```  1422   ultimately show "\<exists>b<top. \<forall>z>b. P z"
```
```  1423     by (auto simp: subset_eq Ball_def)
```
```  1424 next
```
```  1425   fix b assume "b < top" "\<forall>z>b. P z"
```
```  1426   then show "\<exists>S. open S \<and> top \<in> S \<and> (\<forall>xa\<in>S. P xa)"
```
```  1427     by (intro exI[of _ "{b <..}"]) auto
```
```  1428 qed
```
```  1429
```
```  1430 lemma tendsto_at_within_iff_tendsto_nhds:
```
```  1431   "(g ---> g l) (at l within S) \<longleftrightarrow> (g ---> g l) (inf (nhds l) (principal S))"
```
```  1432   unfolding tendsto_def eventually_at_filter eventually_inf_principal
```
```  1433   by (intro ext all_cong imp_cong) (auto elim!: eventually_elim1)
```
```  1434
```
```  1435 subsection {* Limits on sequences *}
```
```  1436
```
```  1437 abbreviation (in topological_space)
```
```  1438   LIMSEQ :: "[nat \<Rightarrow> 'a, 'a] \<Rightarrow> bool"
```
```  1439     ("((_)/ ----> (_))" [60, 60] 60) where
```
```  1440   "X ----> L \<equiv> (X ---> L) sequentially"
```
```  1441
```
```  1442 abbreviation (in t2_space) lim :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'a" where
```
```  1443   "lim X \<equiv> Lim sequentially X"
```
```  1444
```
```  1445 definition (in topological_space) convergent :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" where
```
```  1446   "convergent X = (\<exists>L. X ----> L)"
```
```  1447
```
```  1448 lemma lim_def: "lim X = (THE L. X ----> L)"
```
```  1449   unfolding Lim_def ..
```
```  1450
```
```  1451 subsubsection {* Monotone sequences and subsequences *}
```
```  1452
```
```  1453 definition
```
```  1454   monoseq :: "(nat \<Rightarrow> 'a::order) \<Rightarrow> bool" where
```
```  1455     --{*Definition of monotonicity.
```
```  1456         The use of disjunction here complicates proofs considerably.
```
```  1457         One alternative is to add a Boolean argument to indicate the direction.
```
```  1458         Another is to develop the notions of increasing and decreasing first.*}
```
```  1459   "monoseq X = ((\<forall>m. \<forall>n\<ge>m. X m \<le> X n) \<or> (\<forall>m. \<forall>n\<ge>m. X n \<le> X m))"
```
```  1460
```
```  1461 abbreviation incseq :: "(nat \<Rightarrow> 'a::order) \<Rightarrow> bool" where
```
```  1462   "incseq X \<equiv> mono X"
```
```  1463
```
```  1464 lemma incseq_def: "incseq X \<longleftrightarrow> (\<forall>m. \<forall>n\<ge>m. X n \<ge> X m)"
```
```  1465   unfolding mono_def ..
```
```  1466
```
```  1467 abbreviation decseq :: "(nat \<Rightarrow> 'a::order) \<Rightarrow> bool" where
```
```  1468   "decseq X \<equiv> antimono X"
```
```  1469
```
```  1470 lemma decseq_def: "decseq X \<longleftrightarrow> (\<forall>m. \<forall>n\<ge>m. X n \<le> X m)"
```
```  1471   unfolding antimono_def ..
```
```  1472
```
```  1473 definition
```
```  1474   subseq :: "(nat \<Rightarrow> nat) \<Rightarrow> bool" where
```
```  1475     --{*Definition of subsequence*}
```
```  1476   "subseq f \<longleftrightarrow> (\<forall>m. \<forall>n>m. f m < f n)"
```
```  1477
```
```  1478 lemma incseq_SucI:
```
```  1479   "(\<And>n. X n \<le> X (Suc n)) \<Longrightarrow> incseq X"
```
```  1480   using lift_Suc_mono_le[of X]
```
```  1481   by (auto simp: incseq_def)
```
```  1482
```
```  1483 lemma incseqD: "\<And>i j. incseq f \<Longrightarrow> i \<le> j \<Longrightarrow> f i \<le> f j"
```
```  1484   by (auto simp: incseq_def)
```
```  1485
```
```  1486 lemma incseq_SucD: "incseq A \<Longrightarrow> A i \<le> A (Suc i)"
```
```  1487   using incseqD[of A i "Suc i"] by auto
```
```  1488
```
```  1489 lemma incseq_Suc_iff: "incseq f \<longleftrightarrow> (\<forall>n. f n \<le> f (Suc n))"
```
```  1490   by (auto intro: incseq_SucI dest: incseq_SucD)
```
```  1491
```
```  1492 lemma incseq_const[simp, intro]: "incseq (\<lambda>x. k)"
```
```  1493   unfolding incseq_def by auto
```
```  1494
```
```  1495 lemma decseq_SucI:
```
```  1496   "(\<And>n. X (Suc n) \<le> X n) \<Longrightarrow> decseq X"
```
```  1497   using order.lift_Suc_mono_le[OF dual_order, of X]
```
```  1498   by (auto simp: decseq_def)
```
```  1499
```
```  1500 lemma decseqD: "\<And>i j. decseq f \<Longrightarrow> i \<le> j \<Longrightarrow> f j \<le> f i"
```
```  1501   by (auto simp: decseq_def)
```
```  1502
```
```  1503 lemma decseq_SucD: "decseq A \<Longrightarrow> A (Suc i) \<le> A i"
```
```  1504   using decseqD[of A i "Suc i"] by auto
```
```  1505
```
```  1506 lemma decseq_Suc_iff: "decseq f \<longleftrightarrow> (\<forall>n. f (Suc n) \<le> f n)"
```
```  1507   by (auto intro: decseq_SucI dest: decseq_SucD)
```
```  1508
```
```  1509 lemma decseq_const[simp, intro]: "decseq (\<lambda>x. k)"
```
```  1510   unfolding decseq_def by auto
```
```  1511
```
```  1512 lemma monoseq_iff: "monoseq X \<longleftrightarrow> incseq X \<or> decseq X"
```
```  1513   unfolding monoseq_def incseq_def decseq_def ..
```
```  1514
```
```  1515 lemma monoseq_Suc:
```
```  1516   "monoseq X \<longleftrightarrow> (\<forall>n. X n \<le> X (Suc n)) \<or> (\<forall>n. X (Suc n) \<le> X n)"
```
```  1517   unfolding monoseq_iff incseq_Suc_iff decseq_Suc_iff ..
```
```  1518
```
```  1519 lemma monoI1: "\<forall>m. \<forall> n \<ge> m. X m \<le> X n ==> monoseq X"
```
```  1520 by (simp add: monoseq_def)
```
```  1521
```
```  1522 lemma monoI2: "\<forall>m. \<forall> n \<ge> m. X n \<le> X m ==> monoseq X"
```
```  1523 by (simp add: monoseq_def)
```
```  1524
```
```  1525 lemma mono_SucI1: "\<forall>n. X n \<le> X (Suc n) ==> monoseq X"
```
```  1526 by (simp add: monoseq_Suc)
```
```  1527
```
```  1528 lemma mono_SucI2: "\<forall>n. X (Suc n) \<le> X n ==> monoseq X"
```
```  1529 by (simp add: monoseq_Suc)
```
```  1530
```
```  1531 lemma monoseq_minus:
```
```  1532   fixes a :: "nat \<Rightarrow> 'a::ordered_ab_group_add"
```
```  1533   assumes "monoseq a"
```
```  1534   shows "monoseq (\<lambda> n. - a n)"
```
```  1535 proof (cases "\<forall> m. \<forall> n \<ge> m. a m \<le> a n")
```
```  1536   case True
```
```  1537   hence "\<forall> m. \<forall> n \<ge> m. - a n \<le> - a m" by auto
```
```  1538   thus ?thesis by (rule monoI2)
```
```  1539 next
```
```  1540   case False
```
```  1541   hence "\<forall> m. \<forall> n \<ge> m. - a m \<le> - a n" using `monoseq a`[unfolded monoseq_def] by auto
```
```  1542   thus ?thesis by (rule monoI1)
```
```  1543 qed
```
```  1544
```
```  1545 text{*Subsequence (alternative definition, (e.g. Hoskins)*}
```
```  1546
```
```  1547 lemma subseq_Suc_iff: "subseq f = (\<forall>n. (f n) < (f (Suc n)))"
```
```  1548 apply (simp add: subseq_def)
```
```  1549 apply (auto dest!: less_imp_Suc_add)
```
```  1550 apply (induct_tac k)
```
```  1551 apply (auto intro: less_trans)
```
```  1552 done
```
```  1553
```
```  1554 text{* for any sequence, there is a monotonic subsequence *}
```
```  1555 lemma seq_monosub:
```
```  1556   fixes s :: "nat => 'a::linorder"
```
```  1557   shows "\<exists>f. subseq f \<and> monoseq (\<lambda>n. (s (f n)))"
```
```  1558 proof cases
```
```  1559   assume "\<forall>n. \<exists>p>n. \<forall>m\<ge>p. s m \<le> s p"
```
```  1560   then have "\<exists>f. \<forall>n. (\<forall>m\<ge>f n. s m \<le> s (f n)) \<and> f n < f (Suc n)"
```
```  1561     by (intro dependent_nat_choice) (auto simp: conj_commute)
```
```  1562   then obtain f where "subseq f" and mono: "\<And>n m. f n \<le> m \<Longrightarrow> s m \<le> s (f n)"
```
```  1563     by (auto simp: subseq_Suc_iff)
```
```  1564   moreover
```
```  1565   then have "incseq f"
```
```  1566     unfolding subseq_Suc_iff incseq_Suc_iff by (auto intro: less_imp_le)
```
```  1567   then have "monoseq (\<lambda>n. s (f n))"
```
```  1568     by (auto simp add: incseq_def intro!: mono monoI2)
```
```  1569   ultimately show ?thesis
```
```  1570     by auto
```
```  1571 next
```
```  1572   assume "\<not> (\<forall>n. \<exists>p>n. (\<forall>m\<ge>p. s m \<le> s p))"
```
```  1573   then obtain N where N: "\<And>p. p > N \<Longrightarrow> \<exists>m>p. s p < s m" by (force simp: not_le le_less)
```
```  1574   have "\<exists>f. \<forall>n. N < f n \<and> f n < f (Suc n) \<and> s (f n) \<le> s (f (Suc n))"
```
```  1575   proof (intro dependent_nat_choice)
```
```  1576     fix x assume "N < x" with N[of x] show "\<exists>y>N. x < y \<and> s x \<le> s y"
```
```  1577       by (auto intro: less_trans)
```
```  1578   qed auto
```
```  1579   then show ?thesis
```
```  1580     by (auto simp: monoseq_iff incseq_Suc_iff subseq_Suc_iff)
```
```  1581 qed
```
```  1582
```
```  1583 lemma seq_suble: assumes sf: "subseq f" shows "n \<le> f n"
```
```  1584 proof(induct n)
```
```  1585   case 0 thus ?case by simp
```
```  1586 next
```
```  1587   case (Suc n)
```
```  1588   from sf[unfolded subseq_Suc_iff, rule_format, of n] Suc.hyps
```
```  1589   have "n < f (Suc n)" by arith
```
```  1590   thus ?case by arith
```
```  1591 qed
```
```  1592
```
```  1593 lemma eventually_subseq:
```
```  1594   "subseq r \<Longrightarrow> eventually P sequentially \<Longrightarrow> eventually (\<lambda>n. P (r n)) sequentially"
```
```  1595   unfolding eventually_sequentially by (metis seq_suble le_trans)
```
```  1596
```
```  1597 lemma not_eventually_sequentiallyD:
```
```  1598   assumes P: "\<not> eventually P sequentially"
```
```  1599   shows "\<exists>r. subseq r \<and> (\<forall>n. \<not> P (r n))"
```
```  1600 proof -
```
```  1601   from P have "\<forall>n. \<exists>m\<ge>n. \<not> P m"
```
```  1602     unfolding eventually_sequentially by (simp add: not_less)
```
```  1603   then obtain r where "\<And>n. r n \<ge> n" "\<And>n. \<not> P (r n)"
```
```  1604     by (auto simp: choice_iff)
```
```  1605   then show ?thesis
```
```  1606     by (auto intro!: exI[of _ "\<lambda>n. r (((Suc \<circ> r) ^^ Suc n) 0)"]
```
```  1607              simp: less_eq_Suc_le subseq_Suc_iff)
```
```  1608 qed
```
```  1609
```
```  1610 lemma filterlim_subseq: "subseq f \<Longrightarrow> filterlim f sequentially sequentially"
```
```  1611   unfolding filterlim_iff by (metis eventually_subseq)
```
```  1612
```
```  1613 lemma subseq_o: "subseq r \<Longrightarrow> subseq s \<Longrightarrow> subseq (r \<circ> s)"
```
```  1614   unfolding subseq_def by simp
```
```  1615
```
```  1616 lemma subseq_mono: assumes "subseq r" "m < n" shows "r m < r n"
```
```  1617   using assms by (auto simp: subseq_def)
```
```  1618
```
```  1619 lemma incseq_imp_monoseq:  "incseq X \<Longrightarrow> monoseq X"
```
```  1620   by (simp add: incseq_def monoseq_def)
```
```  1621
```
```  1622 lemma decseq_imp_monoseq:  "decseq X \<Longrightarrow> monoseq X"
```
```  1623   by (simp add: decseq_def monoseq_def)
```
```  1624
```
```  1625 lemma decseq_eq_incseq:
```
```  1626   fixes X :: "nat \<Rightarrow> 'a::ordered_ab_group_add" shows "decseq X = incseq (\<lambda>n. - X n)"
```
```  1627   by (simp add: decseq_def incseq_def)
```
```  1628
```
```  1629 lemma INT_decseq_offset:
```
```  1630   assumes "decseq F"
```
```  1631   shows "(\<Inter>i. F i) = (\<Inter>i\<in>{n..}. F i)"
```
```  1632 proof safe
```
```  1633   fix x i assume x: "x \<in> (\<Inter>i\<in>{n..}. F i)"
```
```  1634   show "x \<in> F i"
```
```  1635   proof cases
```
```  1636     from x have "x \<in> F n" by auto
```
```  1637     also assume "i \<le> n" with `decseq F` have "F n \<subseteq> F i"
```
```  1638       unfolding decseq_def by simp
```
```  1639     finally show ?thesis .
```
```  1640   qed (insert x, simp)
```
```  1641 qed auto
```
```  1642
```
```  1643 lemma LIMSEQ_const_iff:
```
```  1644   fixes k l :: "'a::t2_space"
```
```  1645   shows "(\<lambda>n. k) ----> l \<longleftrightarrow> k = l"
```
```  1646   using trivial_limit_sequentially by (rule tendsto_const_iff)
```
```  1647
```
```  1648 lemma LIMSEQ_SUP:
```
```  1649   "incseq X \<Longrightarrow> X ----> (SUP i. X i :: 'a :: {complete_linorder, linorder_topology})"
```
```  1650   by (intro increasing_tendsto)
```
```  1651      (auto simp: SUP_upper less_SUP_iff incseq_def eventually_sequentially intro: less_le_trans)
```
```  1652
```
```  1653 lemma LIMSEQ_INF:
```
```  1654   "decseq X \<Longrightarrow> X ----> (INF i. X i :: 'a :: {complete_linorder, linorder_topology})"
```
```  1655   by (intro decreasing_tendsto)
```
```  1656      (auto simp: INF_lower INF_less_iff decseq_def eventually_sequentially intro: le_less_trans)
```
```  1657
```
```  1658 lemma LIMSEQ_ignore_initial_segment:
```
```  1659   "f ----> a \<Longrightarrow> (\<lambda>n. f (n + k)) ----> a"
```
```  1660   unfolding tendsto_def
```
```  1661   by (subst eventually_sequentially_seg[where k=k])
```
```  1662
```
```  1663 lemma LIMSEQ_offset:
```
```  1664   "(\<lambda>n. f (n + k)) ----> a \<Longrightarrow> f ----> a"
```
```  1665   unfolding tendsto_def
```
```  1666   by (subst (asm) eventually_sequentially_seg[where k=k])
```
```  1667
```
```  1668 lemma LIMSEQ_Suc: "f ----> l \<Longrightarrow> (\<lambda>n. f (Suc n)) ----> l"
```
```  1669 by (drule_tac k="Suc 0" in LIMSEQ_ignore_initial_segment, simp)
```
```  1670
```
```  1671 lemma LIMSEQ_imp_Suc: "(\<lambda>n. f (Suc n)) ----> l \<Longrightarrow> f ----> l"
```
```  1672 by (rule_tac k="Suc 0" in LIMSEQ_offset, simp)
```
```  1673
```
```  1674 lemma LIMSEQ_Suc_iff: "(\<lambda>n. f (Suc n)) ----> l = f ----> l"
```
```  1675 by (blast intro: LIMSEQ_imp_Suc LIMSEQ_Suc)
```
```  1676
```
```  1677 lemma LIMSEQ_unique:
```
```  1678   fixes a b :: "'a::t2_space"
```
```  1679   shows "\<lbrakk>X ----> a; X ----> b\<rbrakk> \<Longrightarrow> a = b"
```
```  1680   using trivial_limit_sequentially by (rule tendsto_unique)
```
```  1681
```
```  1682 lemma LIMSEQ_le_const:
```
```  1683   "\<lbrakk>X ----> (x::'a::linorder_topology); \<exists>N. \<forall>n\<ge>N. a \<le> X n\<rbrakk> \<Longrightarrow> a \<le> x"
```
```  1684   using tendsto_le_const[of sequentially X x a] by (simp add: eventually_sequentially)
```
```  1685
```
```  1686 lemma LIMSEQ_le:
```
```  1687   "\<lbrakk>X ----> x; Y ----> y; \<exists>N. \<forall>n\<ge>N. X n \<le> Y n\<rbrakk> \<Longrightarrow> x \<le> (y::'a::linorder_topology)"
```
```  1688   using tendsto_le[of sequentially Y y X x] by (simp add: eventually_sequentially)
```
```  1689
```
```  1690 lemma LIMSEQ_le_const2:
```
```  1691   "\<lbrakk>X ----> (x::'a::linorder_topology); \<exists>N. \<forall>n\<ge>N. X n \<le> a\<rbrakk> \<Longrightarrow> x \<le> a"
```
```  1692   by (rule LIMSEQ_le[of X x "\<lambda>n. a"]) (auto simp: tendsto_const)
```
```  1693
```
```  1694 lemma convergentD: "convergent X ==> \<exists>L. (X ----> L)"
```
```  1695 by (simp add: convergent_def)
```
```  1696
```
```  1697 lemma convergentI: "(X ----> L) ==> convergent X"
```
```  1698 by (auto simp add: convergent_def)
```
```  1699
```
```  1700 lemma convergent_LIMSEQ_iff: "convergent X = (X ----> lim X)"
```
```  1701 by (auto intro: theI LIMSEQ_unique simp add: convergent_def lim_def)
```
```  1702
```
```  1703 lemma convergent_const: "convergent (\<lambda>n. c)"
```
```  1704   by (rule convergentI, rule tendsto_const)
```
```  1705
```
```  1706 lemma monoseq_le:
```
```  1707   "monoseq a \<Longrightarrow> a ----> (x::'a::linorder_topology) \<Longrightarrow>
```
```  1708     ((\<forall> n. a n \<le> x) \<and> (\<forall>m. \<forall>n\<ge>m. a m \<le> a n)) \<or> ((\<forall> n. x \<le> a n) \<and> (\<forall>m. \<forall>n\<ge>m. a n \<le> a m))"
```
```  1709   by (metis LIMSEQ_le_const LIMSEQ_le_const2 decseq_def incseq_def monoseq_iff)
```
```  1710
```
```  1711 lemma LIMSEQ_subseq_LIMSEQ:
```
```  1712   "\<lbrakk> X ----> L; subseq f \<rbrakk> \<Longrightarrow> (X o f) ----> L"
```
```  1713   unfolding comp_def by (rule filterlim_compose[of X, OF _ filterlim_subseq])
```
```  1714
```
```  1715 lemma convergent_subseq_convergent:
```
```  1716   "\<lbrakk>convergent X; subseq f\<rbrakk> \<Longrightarrow> convergent (X o f)"
```
```  1717   unfolding convergent_def by (auto intro: LIMSEQ_subseq_LIMSEQ)
```
```  1718
```
```  1719 lemma limI: "X ----> L ==> lim X = L"
```
```  1720   by (rule tendsto_Lim) (rule trivial_limit_sequentially)
```
```  1721
```
```  1722 lemma lim_le: "convergent f \<Longrightarrow> (\<And>n. f n \<le> (x::'a::linorder_topology)) \<Longrightarrow> lim f \<le> x"
```
```  1723   using LIMSEQ_le_const2[of f "lim f" x] by (simp add: convergent_LIMSEQ_iff)
```
```  1724
```
```  1725 subsubsection{*Increasing and Decreasing Series*}
```
```  1726
```
```  1727 lemma incseq_le: "incseq X \<Longrightarrow> X ----> L \<Longrightarrow> X n \<le> (L::'a::linorder_topology)"
```
```  1728   by (metis incseq_def LIMSEQ_le_const)
```
```  1729
```
```  1730 lemma decseq_le: "decseq X \<Longrightarrow> X ----> L \<Longrightarrow> (L::'a::linorder_topology) \<le> X n"
```
```  1731   by (metis decseq_def LIMSEQ_le_const2)
```
```  1732
```
```  1733 subsection {* First countable topologies *}
```
```  1734
```
```  1735 class first_countable_topology = topological_space +
```
```  1736   assumes first_countable_basis:
```
```  1737     "\<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))"
```
```  1738
```
```  1739 lemma (in first_countable_topology) countable_basis_at_decseq:
```
```  1740   obtains A :: "nat \<Rightarrow> 'a set" where
```
```  1741     "\<And>i. open (A i)" "\<And>i. x \<in> (A i)"
```
```  1742     "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> eventually (\<lambda>i. A i \<subseteq> S) sequentially"
```
```  1743 proof atomize_elim
```
```  1744   from first_countable_basis[of x] obtain A :: "nat \<Rightarrow> 'a set" where
```
```  1745     nhds: "\<And>i. open (A i)" "\<And>i. x \<in> A i"
```
```  1746     and incl: "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> \<exists>i. A i \<subseteq> S"  by auto
```
```  1747   def F \<equiv> "\<lambda>n. \<Inter>i\<le>n. A i"
```
```  1748   show "\<exists>A. (\<forall>i. open (A i)) \<and> (\<forall>i. x \<in> A i) \<and>
```
```  1749       (\<forall>S. open S \<longrightarrow> x \<in> S \<longrightarrow> eventually (\<lambda>i. A i \<subseteq> S) sequentially)"
```
```  1750   proof (safe intro!: exI[of _ F])
```
```  1751     fix i
```
```  1752     show "open (F i)" using nhds(1) by (auto simp: F_def)
```
```  1753     show "x \<in> F i" using nhds(2) by (auto simp: F_def)
```
```  1754   next
```
```  1755     fix S assume "open S" "x \<in> S"
```
```  1756     from incl[OF this] obtain i where "F i \<subseteq> S" unfolding F_def by auto
```
```  1757     moreover have "\<And>j. i \<le> j \<Longrightarrow> F j \<subseteq> F i"
```
```  1758       by (auto simp: F_def)
```
```  1759     ultimately show "eventually (\<lambda>i. F i \<subseteq> S) sequentially"
```
```  1760       by (auto simp: eventually_sequentially)
```
```  1761   qed
```
```  1762 qed
```
```  1763
```
```  1764 lemma (in first_countable_topology) nhds_countable:
```
```  1765   obtains X :: "nat \<Rightarrow> 'a set"
```
```  1766   where "decseq X" "\<And>n. open (X n)" "\<And>n. x \<in> X n" "nhds x = (INF n. principal (X n))"
```
```  1767 proof -
```
```  1768   from first_countable_basis obtain A :: "nat \<Rightarrow> 'a set"
```
```  1769     where A: "\<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"
```
```  1770     by metis
```
```  1771   show thesis
```
```  1772   proof
```
```  1773     show "decseq (\<lambda>n. \<Inter>i\<le>n. A i)"
```
```  1774       by (auto simp: decseq_def)
```
```  1775     show "\<And>n. x \<in> (\<Inter>i\<le>n. A i)" "\<And>n. open (\<Inter>i\<le>n. A i)"
```
```  1776       using A by auto
```
```  1777     show "nhds x = (INF n. principal (\<Inter> i\<le>n. A i))"
```
```  1778       using A unfolding nhds_def
```
```  1779       apply (intro INF_eq)
```
```  1780       apply simp_all
```
```  1781       apply force
```
```  1782       apply (intro exI[of _ "\<Inter> i\<le>n. A i" for n] conjI open_INT)
```
```  1783       apply auto
```
```  1784       done
```
```  1785   qed
```
```  1786 qed
```
```  1787
```
```  1788 lemma (in first_countable_topology) countable_basis:
```
```  1789   obtains A :: "nat \<Rightarrow> 'a set" where
```
```  1790     "\<And>i. open (A i)" "\<And>i. x \<in> A i"
```
```  1791     "\<And>F. (\<forall>n. F n \<in> A n) \<Longrightarrow> F ----> x"
```
```  1792 proof atomize_elim
```
```  1793   obtain A :: "nat \<Rightarrow> 'a set" where A:
```
```  1794     "\<And>i. open (A i)"
```
```  1795     "\<And>i. x \<in> A i"
```
```  1796     "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> eventually (\<lambda>i. A i \<subseteq> S) sequentially"
```
```  1797     by (rule countable_basis_at_decseq) blast
```
```  1798   {
```
```  1799     fix F S assume "\<forall>n. F n \<in> A n" "open S" "x \<in> S"
```
```  1800     with A(3)[of S] have "eventually (\<lambda>n. F n \<in> S) sequentially"
```
```  1801       by (auto elim: eventually_elim1 simp: subset_eq)
```
```  1802   }
```
```  1803   with A 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 ----> x)"
```
```  1804     by (intro exI[of _ A]) (auto simp: tendsto_def)
```
```  1805 qed
```
```  1806
```
```  1807 lemma (in first_countable_topology) sequentially_imp_eventually_nhds_within:
```
```  1808   assumes "\<forall>f. (\<forall>n. f n \<in> s) \<and> f ----> a \<longrightarrow> eventually (\<lambda>n. P (f n)) sequentially"
```
```  1809   shows "eventually P (inf (nhds a) (principal s))"
```
```  1810 proof (rule ccontr)
```
```  1811   obtain A :: "nat \<Rightarrow> 'a set" where A:
```
```  1812     "\<And>i. open (A i)"
```
```  1813     "\<And>i. a \<in> A i"
```
```  1814     "\<And>F. \<forall>n. F n \<in> A n \<Longrightarrow> F ----> a"
```
```  1815     by (rule countable_basis) blast
```
```  1816   assume "\<not> ?thesis"
```
```  1817   with A have P: "\<exists>F. \<forall>n. F n \<in> s \<and> F n \<in> A n \<and> \<not> P (F n)"
```
```  1818     unfolding eventually_inf_principal eventually_nhds by (intro choice) fastforce
```
```  1819   then obtain F where F0: "\<forall>n. F n \<in> s" and F2: "\<forall>n. F n \<in> A n" and F3: "\<forall>n. \<not> P (F n)"
```
```  1820     by blast
```
```  1821   with A have "F ----> a" by auto
```
```  1822   hence "eventually (\<lambda>n. P (F n)) sequentially"
```
```  1823     using assms F0 by simp
```
```  1824   thus "False" by (simp add: F3)
```
```  1825 qed
```
```  1826
```
```  1827 lemma (in first_countable_topology) eventually_nhds_within_iff_sequentially:
```
```  1828   "eventually P (inf (nhds a) (principal s)) \<longleftrightarrow>
```
```  1829     (\<forall>f. (\<forall>n. f n \<in> s) \<and> f ----> a \<longrightarrow> eventually (\<lambda>n. P (f n)) sequentially)"
```
```  1830 proof (safe intro!: sequentially_imp_eventually_nhds_within)
```
```  1831   assume "eventually P (inf (nhds a) (principal s))"
```
```  1832   then obtain S where "open S" "a \<in> S" "\<forall>x\<in>S. x \<in> s \<longrightarrow> P x"
```
```  1833     by (auto simp: eventually_inf_principal eventually_nhds)
```
```  1834   moreover fix f assume "\<forall>n. f n \<in> s" "f ----> a"
```
```  1835   ultimately show "eventually (\<lambda>n. P (f n)) sequentially"
```
```  1836     by (auto dest!: topological_tendstoD elim: eventually_elim1)
```
```  1837 qed
```
```  1838
```
```  1839 lemma (in first_countable_topology) eventually_nhds_iff_sequentially:
```
```  1840   "eventually P (nhds a) \<longleftrightarrow> (\<forall>f. f ----> a \<longrightarrow> eventually (\<lambda>n. P (f n)) sequentially)"
```
```  1841   using eventually_nhds_within_iff_sequentially[of P a UNIV] by simp
```
```  1842
```
```  1843 lemma tendsto_at_iff_sequentially:
```
```  1844   fixes f :: "'a :: first_countable_topology \<Rightarrow> _"
```
```  1845   shows "(f ---> a) (at x within s) \<longleftrightarrow> (\<forall>X. (\<forall>i. X i \<in> s - {x}) \<longrightarrow> X ----> x \<longrightarrow> ((f \<circ> X) ----> a))"
```
```  1846   unfolding filterlim_def[of _ "nhds a"] le_filter_def eventually_filtermap at_within_def eventually_nhds_within_iff_sequentially comp_def
```
```  1847   by metis
```
```  1848
```
```  1849 subsection {* Function limit at a point *}
```
```  1850
```
```  1851 abbreviation
```
```  1852   LIM :: "('a::topological_space \<Rightarrow> 'b::topological_space) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool"
```
```  1853         ("((_)/ -- (_)/ --> (_))" [60, 0, 60] 60) where
```
```  1854   "f -- a --> L \<equiv> (f ---> L) (at a)"
```
```  1855
```
```  1856 lemma tendsto_within_open: "a \<in> S \<Longrightarrow> open S \<Longrightarrow> (f ---> l) (at a within S) \<longleftrightarrow> (f -- a --> l)"
```
```  1857   unfolding tendsto_def by (simp add: at_within_open[where S=S])
```
```  1858
```
```  1859 lemma LIM_const_not_eq[tendsto_intros]:
```
```  1860   fixes a :: "'a::perfect_space"
```
```  1861   fixes k L :: "'b::t2_space"
```
```  1862   shows "k \<noteq> L \<Longrightarrow> \<not> (\<lambda>x. k) -- a --> L"
```
```  1863   by (simp add: tendsto_const_iff)
```
```  1864
```
```  1865 lemmas LIM_not_zero = LIM_const_not_eq [where L = 0]
```
```  1866
```
```  1867 lemma LIM_const_eq:
```
```  1868   fixes a :: "'a::perfect_space"
```
```  1869   fixes k L :: "'b::t2_space"
```
```  1870   shows "(\<lambda>x. k) -- a --> L \<Longrightarrow> k = L"
```
```  1871   by (simp add: tendsto_const_iff)
```
```  1872
```
```  1873 lemma LIM_unique:
```
```  1874   fixes a :: "'a::perfect_space" and L M :: "'b::t2_space"
```
```  1875   shows "f -- a --> L \<Longrightarrow> f -- a --> M \<Longrightarrow> L = M"
```
```  1876   using at_neq_bot by (rule tendsto_unique)
```
```  1877
```
```  1878 text {* Limits are equal for functions equal except at limit point *}
```
```  1879
```
```  1880 lemma LIM_equal: "\<forall>x. x \<noteq> a --> (f x = g x) \<Longrightarrow> (f -- a --> l) \<longleftrightarrow> (g -- a --> l)"
```
```  1881   unfolding tendsto_def eventually_at_topological by simp
```
```  1882
```
```  1883 lemma LIM_cong: "a = b \<Longrightarrow> (\<And>x. x \<noteq> b \<Longrightarrow> f x = g x) \<Longrightarrow> l = m \<Longrightarrow> (f -- a --> l) \<longleftrightarrow> (g -- b --> m)"
```
```  1884   by (simp add: LIM_equal)
```
```  1885
```
```  1886 lemma LIM_cong_limit: "f -- x --> L \<Longrightarrow> K = L \<Longrightarrow> f -- x --> K"
```
```  1887   by simp
```
```  1888
```
```  1889 lemma tendsto_at_iff_tendsto_nhds:
```
```  1890   "g -- l --> g l \<longleftrightarrow> (g ---> g l) (nhds l)"
```
```  1891   unfolding tendsto_def eventually_at_filter
```
```  1892   by (intro ext all_cong imp_cong) (auto elim!: eventually_elim1)
```
```  1893
```
```  1894 lemma tendsto_compose:
```
```  1895   "g -- l --> g l \<Longrightarrow> (f ---> l) F \<Longrightarrow> ((\<lambda>x. g (f x)) ---> g l) F"
```
```  1896   unfolding tendsto_at_iff_tendsto_nhds by (rule filterlim_compose[of g])
```
```  1897
```
```  1898 lemma LIM_o: "\<lbrakk>g -- l --> g l; f -- a --> l\<rbrakk> \<Longrightarrow> (g \<circ> f) -- a --> g l"
```
```  1899   unfolding o_def by (rule tendsto_compose)
```
```  1900
```
```  1901 lemma tendsto_compose_eventually:
```
```  1902   "g -- l --> m \<Longrightarrow> (f ---> l) F \<Longrightarrow> eventually (\<lambda>x. f x \<noteq> l) F \<Longrightarrow> ((\<lambda>x. g (f x)) ---> m) F"
```
```  1903   by (rule filterlim_compose[of g _ "at l"]) (auto simp add: filterlim_at)
```
```  1904
```
```  1905 lemma LIM_compose_eventually:
```
```  1906   assumes f: "f -- a --> b"
```
```  1907   assumes g: "g -- b --> c"
```
```  1908   assumes inj: "eventually (\<lambda>x. f x \<noteq> b) (at a)"
```
```  1909   shows "(\<lambda>x. g (f x)) -- a --> c"
```
```  1910   using g f inj by (rule tendsto_compose_eventually)
```
```  1911
```
```  1912 lemma tendsto_compose_filtermap: "((g \<circ> f) ---> T) F \<longleftrightarrow> (g ---> T) (filtermap f F)"
```
```  1913   by (simp add: filterlim_def filtermap_filtermap comp_def)
```
```  1914
```
```  1915 subsubsection {* Relation of LIM and LIMSEQ *}
```
```  1916
```
```  1917 lemma (in first_countable_topology) sequentially_imp_eventually_within:
```
```  1918   "(\<forall>f. (\<forall>n. f n \<in> s \<and> f n \<noteq> a) \<and> f ----> a \<longrightarrow> eventually (\<lambda>n. P (f n)) sequentially) \<Longrightarrow>
```
```  1919     eventually P (at a within s)"
```
```  1920   unfolding at_within_def
```
```  1921   by (intro sequentially_imp_eventually_nhds_within) auto
```
```  1922
```
```  1923 lemma (in first_countable_topology) sequentially_imp_eventually_at:
```
```  1924   "(\<forall>f. (\<forall>n. f n \<noteq> a) \<and> f ----> a \<longrightarrow> eventually (\<lambda>n. P (f n)) sequentially) \<Longrightarrow> eventually P (at a)"
```
```  1925   using assms sequentially_imp_eventually_within [where s=UNIV] by simp
```
```  1926
```
```  1927 lemma LIMSEQ_SEQ_conv1:
```
```  1928   fixes f :: "'a::topological_space \<Rightarrow> 'b::topological_space"
```
```  1929   assumes f: "f -- a --> l"
```
```  1930   shows "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. f (S n)) ----> l"
```
```  1931   using tendsto_compose_eventually [OF f, where F=sequentially] by simp
```
```  1932
```
```  1933 lemma LIMSEQ_SEQ_conv2:
```
```  1934   fixes f :: "'a::first_countable_topology \<Rightarrow> 'b::topological_space"
```
```  1935   assumes "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. f (S n)) ----> l"
```
```  1936   shows "f -- a --> l"
```
```  1937   using assms unfolding tendsto_def [where l=l] by (simp add: sequentially_imp_eventually_at)
```
```  1938
```
```  1939 lemma LIMSEQ_SEQ_conv:
```
```  1940   "(\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> (a::'a::first_countable_topology) \<longrightarrow> (\<lambda>n. X (S n)) ----> L) =
```
```  1941    (X -- a --> (L::'b::topological_space))"
```
```  1942   using LIMSEQ_SEQ_conv2 LIMSEQ_SEQ_conv1 ..
```
```  1943
```
```  1944 lemma sequentially_imp_eventually_at_left:
```
```  1945   fixes a :: "'a :: {dense_linorder, linorder_topology, first_countable_topology}"
```
```  1946   assumes b[simp]: "b < a"
```
```  1947   assumes *: "\<And>f. (\<And>n. b < f n) \<Longrightarrow> (\<And>n. f n < a) \<Longrightarrow> incseq f \<Longrightarrow> f ----> a \<Longrightarrow> eventually (\<lambda>n. P (f n)) sequentially"
```
```  1948   shows "eventually P (at_left a)"
```
```  1949 proof (safe intro!: sequentially_imp_eventually_within)
```
```  1950   fix X assume X: "\<forall>n. X n \<in> {..< a} \<and> X n \<noteq> a" "X ----> a"
```
```  1951   show "eventually (\<lambda>n. P (X n)) sequentially"
```
```  1952   proof (rule ccontr)
```
```  1953     assume neg: "\<not> eventually (\<lambda>n. P (X n)) sequentially"
```
```  1954     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))"
```
```  1955     proof (rule dependent_nat_choice)
```
```  1956       have "\<not> eventually (\<lambda>n. b < X n \<longrightarrow> P (X n)) sequentially"
```
```  1957         by (intro not_eventually_impI neg order_tendstoD(1) [OF X(2) b])
```
```  1958       then show "\<exists>x. \<not> P (X x) \<and> b < X x"
```
```  1959         by (auto dest!: not_eventuallyD)
```
```  1960     next
```
```  1961       fix x n
```
```  1962       have "\<not> eventually (\<lambda>n. Suc x \<le> n \<longrightarrow> b < X n \<longrightarrow> X x < X n \<longrightarrow> P (X n)) sequentially"
```
```  1963         using X by (intro not_eventually_impI order_tendstoD(1)[OF X(2)] eventually_ge_at_top neg) auto
```
```  1964       then show "\<exists>n. (\<not> P (X n) \<and> b < X n) \<and> (X x \<le> X n \<and> Suc x \<le> n)"
```
```  1965         by (auto dest!: not_eventuallyD)
```
```  1966     qed
```
```  1967     then guess s ..
```
```  1968     then have "\<And>n. b < X (s n)" "\<And>n. X (s n) < a" "incseq (\<lambda>n. X (s n))" "(\<lambda>n. X (s n)) ----> a" "\<And>n. \<not> P (X (s n))"
```
```  1969       using X by (auto simp: subseq_Suc_iff Suc_le_eq incseq_Suc_iff intro!: LIMSEQ_subseq_LIMSEQ[OF `X ----> a`, unfolded comp_def])
```
```  1970     from *[OF this(1,2,3,4)] this(5) show False by auto
```
```  1971   qed
```
```  1972 qed
```
```  1973
```
```  1974 lemma tendsto_at_left_sequentially:
```
```  1975   fixes a :: "_ :: {dense_linorder, linorder_topology, first_countable_topology}"
```
```  1976   assumes "b < a"
```
```  1977   assumes *: "\<And>S. (\<And>n. S n < a) \<Longrightarrow> (\<And>n. b < S n) \<Longrightarrow> incseq S \<Longrightarrow> S ----> a \<Longrightarrow> (\<lambda>n. X (S n)) ----> L"
```
```  1978   shows "(X ---> L) (at_left a)"
```
```  1979   using assms unfolding tendsto_def [where l=L]
```
```  1980   by (simp add: sequentially_imp_eventually_at_left)
```
```  1981
```
```  1982 lemma sequentially_imp_eventually_at_right:
```
```  1983   fixes a :: "'a :: {dense_linorder, linorder_topology, first_countable_topology}"
```
```  1984   assumes b[simp]: "a < b"
```
```  1985   assumes *: "\<And>f. (\<And>n. a < f n) \<Longrightarrow> (\<And>n. f n < b) \<Longrightarrow> decseq f \<Longrightarrow> f ----> a \<Longrightarrow> eventually (\<lambda>n. P (f n)) sequentially"
```
```  1986   shows "eventually P (at_right a)"
```
```  1987 proof (safe intro!: sequentially_imp_eventually_within)
```
```  1988   fix X assume X: "\<forall>n. X n \<in> {a <..} \<and> X n \<noteq> a" "X ----> a"
```
```  1989   show "eventually (\<lambda>n. P (X n)) sequentially"
```
```  1990   proof (rule ccontr)
```
```  1991     assume neg: "\<not> eventually (\<lambda>n. P (X n)) sequentially"
```
```  1992     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))"
```
```  1993     proof (rule dependent_nat_choice)
```
```  1994       have "\<not> eventually (\<lambda>n. X n < b \<longrightarrow> P (X n)) sequentially"
```
```  1995         by (intro not_eventually_impI neg order_tendstoD(2) [OF X(2) b])
```
```  1996       then show "\<exists>x. \<not> P (X x) \<and> X x < b"
```
```  1997         by (auto dest!: not_eventuallyD)
```
```  1998     next
```
```  1999       fix x n
```
```  2000       have "\<not> eventually (\<lambda>n. Suc x \<le> n \<longrightarrow> X n < b \<longrightarrow> X n < X x \<longrightarrow> P (X n)) sequentially"
```
```  2001         using X by (intro not_eventually_impI order_tendstoD(2)[OF X(2)] eventually_ge_at_top neg) auto
```
```  2002       then show "\<exists>n. (\<not> P (X n) \<and> X n < b) \<and> (X n \<le> X x \<and> Suc x \<le> n)"
```
```  2003         by (auto dest!: not_eventuallyD)
```
```  2004     qed
```
```  2005     then guess s ..
```
```  2006     then have "\<And>n. a < X (s n)" "\<And>n. X (s n) < b" "decseq (\<lambda>n. X (s n))" "(\<lambda>n. X (s n)) ----> a" "\<And>n. \<not> P (X (s n))"
```
```  2007       using X by (auto simp: subseq_Suc_iff Suc_le_eq decseq_Suc_iff intro!: LIMSEQ_subseq_LIMSEQ[OF `X ----> a`, unfolded comp_def])
```
```  2008     from *[OF this(1,2,3,4)] this(5) show False by auto
```
```  2009   qed
```
```  2010 qed
```
```  2011
```
```  2012 lemma tendsto_at_right_sequentially:
```
```  2013   fixes a :: "_ :: {dense_linorder, linorder_topology, first_countable_topology}"
```
```  2014   assumes "a < b"
```
```  2015   assumes *: "\<And>S. (\<And>n. a < S n) \<Longrightarrow> (\<And>n. S n < b) \<Longrightarrow> decseq S \<Longrightarrow> S ----> a \<Longrightarrow> (\<lambda>n. X (S n)) ----> L"
```
```  2016   shows "(X ---> L) (at_right a)"
```
```  2017   using assms unfolding tendsto_def [where l=L]
```
```  2018   by (simp add: sequentially_imp_eventually_at_right)
```
```  2019
```
```  2020 subsection {* Continuity *}
```
```  2021
```
```  2022 subsubsection {* Continuity on a set *}
```
```  2023
```
```  2024 definition continuous_on :: "'a set \<Rightarrow> ('a :: topological_space \<Rightarrow> 'b :: topological_space) \<Rightarrow> bool" where
```
```  2025   "continuous_on s f \<longleftrightarrow> (\<forall>x\<in>s. (f ---> f x) (at x within s))"
```
```  2026
```
```  2027 lemma continuous_on_cong [cong]:
```
```  2028   "s = t \<Longrightarrow> (\<And>x. x \<in> t \<Longrightarrow> f x = g x) \<Longrightarrow> continuous_on s f \<longleftrightarrow> continuous_on t g"
```
```  2029   unfolding continuous_on_def by (intro ball_cong filterlim_cong) (auto simp: eventually_at_filter)
```
```  2030
```
```  2031 lemma continuous_on_topological:
```
```  2032   "continuous_on s f \<longleftrightarrow>
```
```  2033     (\<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)))"
```
```  2034   unfolding continuous_on_def tendsto_def eventually_at_topological by metis
```
```  2035
```
```  2036 lemma continuous_on_open_invariant:
```
```  2037   "continuous_on s f \<longleftrightarrow> (\<forall>B. open B \<longrightarrow> (\<exists>A. open A \<and> A \<inter> s = f -` B \<inter> s))"
```
```  2038 proof safe
```
```  2039   fix B :: "'b set" assume "continuous_on s f" "open B"
```
```  2040   then have "\<forall>x\<in>f -` B \<inter> s. (\<exists>A. open A \<and> x \<in> A \<and> s \<inter> A \<subseteq> f -` B)"
```
```  2041     by (auto simp: continuous_on_topological subset_eq Ball_def imp_conjL)
```
```  2042   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"
```
```  2043     unfolding bchoice_iff ..
```
```  2044   then show "\<exists>A. open A \<and> A \<inter> s = f -` B \<inter> s"
```
```  2045     by (intro exI[of _ "\<Union>x\<in>f -` B \<inter> s. A x"]) auto
```
```  2046 next
```
```  2047   assume B: "\<forall>B. open B \<longrightarrow> (\<exists>A. open A \<and> A \<inter> s = f -` B \<inter> s)"
```
```  2048   show "continuous_on s f"
```
```  2049     unfolding continuous_on_topological
```
```  2050   proof safe
```
```  2051     fix x B assume "x \<in> s" "open B" "f x \<in> B"
```
```  2052     with B obtain A where A: "open A" "A \<inter> s = f -` B \<inter> s" by auto
```
```  2053     with `x \<in> s` `f x \<in> B` show "\<exists>A. open A \<and> x \<in> A \<and> (\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B)"
```
```  2054       by (intro exI[of _ A]) auto
```
```  2055   qed
```
```  2056 qed
```
```  2057
```
```  2058 lemma continuous_on_open_vimage:
```
```  2059   "open s \<Longrightarrow> continuous_on s f \<longleftrightarrow> (\<forall>B. open B \<longrightarrow> open (f -` B \<inter> s))"
```
```  2060   unfolding continuous_on_open_invariant
```
```  2061   by (metis open_Int Int_absorb Int_commute[of s] Int_assoc[of _ _ s])
```
```  2062
```
```  2063 corollary continuous_imp_open_vimage:
```
```  2064   assumes "continuous_on s f" "open s" "open B" "f -` B \<subseteq> s"
```
```  2065     shows "open (f -` B)"
```
```  2066 by (metis assms continuous_on_open_vimage le_iff_inf)
```
```  2067
```
```  2068 corollary open_vimage[continuous_intros]:
```
```  2069   assumes "open s" and "continuous_on UNIV f"
```
```  2070   shows "open (f -` s)"
```
```  2071   using assms unfolding continuous_on_open_vimage [OF open_UNIV]
```
```  2072   by simp
```
```  2073
```
```  2074 lemma continuous_on_closed_invariant:
```
```  2075   "continuous_on s f \<longleftrightarrow> (\<forall>B. closed B \<longrightarrow> (\<exists>A. closed A \<and> A \<inter> s = f -` B \<inter> s))"
```
```  2076 proof -
```
```  2077   have *: "\<And>P Q::'b set\<Rightarrow>bool. (\<And>A. P A \<longleftrightarrow> Q (- A)) \<Longrightarrow> (\<forall>A. P A) \<longleftrightarrow> (\<forall>A. Q A)"
```
```  2078     by (metis double_compl)
```
```  2079   show ?thesis
```
```  2080     unfolding continuous_on_open_invariant by (intro *) (auto simp: open_closed[symmetric])
```
```  2081 qed
```
```  2082
```
```  2083 lemma continuous_on_closed_vimage:
```
```  2084   "closed s \<Longrightarrow> continuous_on s f \<longleftrightarrow> (\<forall>B. closed B \<longrightarrow> closed (f -` B \<inter> s))"
```
```  2085   unfolding continuous_on_closed_invariant
```
```  2086   by (metis closed_Int Int_absorb Int_commute[of s] Int_assoc[of _ _ s])
```
```  2087
```
```  2088 corollary closed_vimage[continuous_intros]:
```
```  2089   assumes "closed s" and "continuous_on UNIV f"
```
```  2090   shows "closed (f -` s)"
```
```  2091   using assms unfolding continuous_on_closed_vimage [OF closed_UNIV]
```
```  2092   by simp
```
```  2093
```
```  2094 lemma continuous_on_open_Union:
```
```  2095   "(\<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"
```
```  2096   unfolding continuous_on_def by safe (metis open_Union at_within_open UnionI)
```
```  2097
```
```  2098 lemma continuous_on_open_UN:
```
```  2099   "(\<And>s. s \<in> S \<Longrightarrow> open (A s)) \<Longrightarrow> (\<And>s. s \<in> S \<Longrightarrow> continuous_on (A s) f) \<Longrightarrow> continuous_on (\<Union>s\<in>S. A s) f"
```
```  2100   unfolding Union_image_eq[symmetric] by (rule continuous_on_open_Union) auto
```
```  2101
```
```  2102 lemma continuous_on_closed_Un:
```
```  2103   "closed s \<Longrightarrow> closed t \<Longrightarrow> continuous_on s f \<Longrightarrow> continuous_on t f \<Longrightarrow> continuous_on (s \<union> t) f"
```
```  2104   by (auto simp add: continuous_on_closed_vimage closed_Un Int_Un_distrib)
```
```  2105
```
```  2106 lemma continuous_on_If:
```
```  2107   assumes closed: "closed s" "closed t" and cont: "continuous_on s f" "continuous_on t g"
```
```  2108     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"
```
```  2109   shows "continuous_on (s \<union> t) (\<lambda>x. if P x then f x else g x)" (is "continuous_on _ ?h")
```
```  2110 proof-
```
```  2111   from P have "\<forall>x\<in>s. f x = ?h x" "\<forall>x\<in>t. g x = ?h x"
```
```  2112     by auto
```
```  2113   with cont have "continuous_on s ?h" "continuous_on t ?h"
```
```  2114     by simp_all
```
```  2115   with closed show ?thesis
```
```  2116     by (rule continuous_on_closed_Un)
```
```  2117 qed
```
```  2118
```
```  2119 lemma continuous_on_id[continuous_intros]: "continuous_on s (\<lambda>x. x)"
```
```  2120   unfolding continuous_on_def by (fast intro: tendsto_ident_at)
```
```  2121
```
```  2122 lemma continuous_on_const[continuous_intros]: "continuous_on s (\<lambda>x. c)"
```
```  2123   unfolding continuous_on_def by (auto intro: tendsto_const)
```
```  2124
```
```  2125 lemma continuous_on_compose[continuous_intros]:
```
```  2126   "continuous_on s f \<Longrightarrow> continuous_on (f ` s) g \<Longrightarrow> continuous_on s (g o f)"
```
```  2127   unfolding continuous_on_topological by simp metis
```
```  2128
```
```  2129 lemma continuous_on_compose2:
```
```  2130   "continuous_on t g \<Longrightarrow> continuous_on s f \<Longrightarrow> t = f ` s \<Longrightarrow> continuous_on s (\<lambda>x. g (f x))"
```
```  2131   using continuous_on_compose[of s f g] by (simp add: comp_def)
```
```  2132
```
```  2133 subsubsection {* Continuity at a point *}
```
```  2134
```
```  2135 definition continuous :: "'a::t2_space filter \<Rightarrow> ('a \<Rightarrow> 'b::topological_space) \<Rightarrow> bool" where
```
```  2136   "continuous F f \<longleftrightarrow> (f ---> f (Lim F (\<lambda>x. x))) F"
```
```  2137
```
```  2138 lemma continuous_bot[continuous_intros, simp]: "continuous bot f"
```
```  2139   unfolding continuous_def by auto
```
```  2140
```
```  2141 lemma continuous_trivial_limit: "trivial_limit net \<Longrightarrow> continuous net f"
```
```  2142   by simp
```
```  2143
```
```  2144 lemma continuous_within: "continuous (at x within s) f \<longleftrightarrow> (f ---> f x) (at x within s)"
```
```  2145   by (cases "trivial_limit (at x within s)") (auto simp add: Lim_ident_at continuous_def)
```
```  2146
```
```  2147 lemma continuous_within_topological:
```
```  2148   "continuous (at x within s) f \<longleftrightarrow>
```
```  2149     (\<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)))"
```
```  2150   unfolding continuous_within tendsto_def eventually_at_topological by metis
```
```  2151
```
```  2152 lemma continuous_within_compose[continuous_intros]:
```
```  2153   "continuous (at x within s) f \<Longrightarrow> continuous (at (f x) within f ` s) g \<Longrightarrow>
```
```  2154   continuous (at x within s) (g o f)"
```
```  2155   by (simp add: continuous_within_topological) metis
```
```  2156
```
```  2157 lemma continuous_within_compose2:
```
```  2158   "continuous (at x within s) f \<Longrightarrow> continuous (at (f x) within f ` s) g \<Longrightarrow>
```
```  2159   continuous (at x within s) (\<lambda>x. g (f x))"
```
```  2160   using continuous_within_compose[of x s f g] by (simp add: comp_def)
```
```  2161
```
```  2162 lemma continuous_at: "continuous (at x) f \<longleftrightarrow> f -- x --> f x"
```
```  2163   using continuous_within[of x UNIV f] by simp
```
```  2164
```
```  2165 lemma continuous_ident[continuous_intros, simp]: "continuous (at x within S) (\<lambda>x. x)"
```
```  2166   unfolding continuous_within by (rule tendsto_ident_at)
```
```  2167
```
```  2168 lemma continuous_const[continuous_intros, simp]: "continuous F (\<lambda>x. c)"
```
```  2169   unfolding continuous_def by (rule tendsto_const)
```
```  2170
```
```  2171 lemma continuous_on_eq_continuous_within:
```
```  2172   "continuous_on s f \<longleftrightarrow> (\<forall>x\<in>s. continuous (at x within s) f)"
```
```  2173   unfolding continuous_on_def continuous_within ..
```
```  2174
```
```  2175 abbreviation isCont :: "('a::t2_space \<Rightarrow> 'b::topological_space) \<Rightarrow> 'a \<Rightarrow> bool" where
```
```  2176   "isCont f a \<equiv> continuous (at a) f"
```
```  2177
```
```  2178 lemma isCont_def: "isCont f a \<longleftrightarrow> f -- a --> f a"
```
```  2179   by (rule continuous_at)
```
```  2180
```
```  2181 lemma continuous_at_within: "isCont f x \<Longrightarrow> continuous (at x within s) f"
```
```  2182   by (auto intro: tendsto_mono at_le simp: continuous_at continuous_within)
```
```  2183
```
```  2184 lemma continuous_on_eq_continuous_at: "open s \<Longrightarrow> continuous_on s f \<longleftrightarrow> (\<forall>x\<in>s. isCont f x)"
```
```  2185   by (simp add: continuous_on_def continuous_at at_within_open[of _ s])
```
```  2186
```
```  2187 lemma continuous_on_subset: "continuous_on s f \<Longrightarrow> t \<subseteq> s \<Longrightarrow> continuous_on t f"
```
```  2188   unfolding continuous_on_def by (metis subset_eq tendsto_within_subset)
```
```  2189
```
```  2190 lemma continuous_at_imp_continuous_on: "\<forall>x\<in>s. isCont f x \<Longrightarrow> continuous_on s f"
```
```  2191   by (auto intro: continuous_at_within simp: continuous_on_eq_continuous_within)
```
```  2192
```
```  2193 lemma isContI_continuous: "continuous (at x within UNIV) f \<Longrightarrow> isCont f x"
```
```  2194   by simp
```
```  2195
```
```  2196 lemma isCont_ident[continuous_intros, simp]: "isCont (\<lambda>x. x) a"
```
```  2197   using continuous_ident by (rule isContI_continuous)
```
```  2198
```
```  2199 lemmas isCont_const = continuous_const
```
```  2200
```
```  2201 lemma isCont_o2: "isCont f a \<Longrightarrow> isCont g (f a) \<Longrightarrow> isCont (\<lambda>x. g (f x)) a"
```
```  2202   unfolding isCont_def by (rule tendsto_compose)
```
```  2203
```
```  2204 lemma isCont_o[continuous_intros]: "isCont f a \<Longrightarrow> isCont g (f a) \<Longrightarrow> isCont (g \<circ> f) a"
```
```  2205   unfolding o_def by (rule isCont_o2)
```
```  2206
```
```  2207 lemma isCont_tendsto_compose: "isCont g l \<Longrightarrow> (f ---> l) F \<Longrightarrow> ((\<lambda>x. g (f x)) ---> g l) F"
```
```  2208   unfolding isCont_def by (rule tendsto_compose)
```
```  2209
```
```  2210 lemma continuous_within_compose3:
```
```  2211   "isCont g (f x) \<Longrightarrow> continuous (at x within s) f \<Longrightarrow> continuous (at x within s) (\<lambda>x. g (f x))"
```
```  2212   using continuous_within_compose2[of x s f g] by (simp add: continuous_at_within)
```
```  2213
```
```  2214 lemma filtermap_nhds_open_map:
```
```  2215   assumes cont: "isCont f a" and open_map: "\<And>S. open S \<Longrightarrow> open (f`S)"
```
```  2216   shows "filtermap f (nhds a) = nhds (f a)"
```
```  2217   unfolding filter_eq_iff
```
```  2218 proof safe
```
```  2219   fix P assume "eventually P (filtermap f (nhds a))"
```
```  2220   then guess S unfolding eventually_filtermap eventually_nhds ..
```
```  2221   then show "eventually P (nhds (f a))"
```
```  2222     unfolding eventually_nhds by (intro exI[of _ "f`S"]) (auto intro!: open_map)
```
```  2223 qed (metis filterlim_iff tendsto_at_iff_tendsto_nhds isCont_def eventually_filtermap cont)
```
```  2224
```
```  2225 lemma continuous_at_split:
```
```  2226   "continuous (at (x::'a::linorder_topology)) f = (continuous (at_left x) f \<and> continuous (at_right x) f)"
```
```  2227   by (simp add: continuous_within filterlim_at_split)
```
```  2228
```
```  2229 subsubsection{* Open-cover compactness *}
```
```  2230
```
```  2231 context topological_space
```
```  2232 begin
```
```  2233
```
```  2234 definition compact :: "'a set \<Rightarrow> bool" where
```
```  2235   compact_eq_heine_borel: -- "This name is used for backwards compatibility"
```
```  2236     "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))"
```
```  2237
```
```  2238 lemma compactI:
```
```  2239   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'"
```
```  2240   shows "compact s"
```
```  2241   unfolding compact_eq_heine_borel using assms by metis
```
```  2242
```
```  2243 lemma compact_empty[simp]: "compact {}"
```
```  2244   by (auto intro!: compactI)
```
```  2245
```
```  2246 lemma compactE:
```
```  2247   assumes "compact s" and "\<forall>t\<in>C. open t" and "s \<subseteq> \<Union>C"
```
```  2248   obtains C' where "C' \<subseteq> C" and "finite C'" and "s \<subseteq> \<Union>C'"
```
```  2249   using assms unfolding compact_eq_heine_borel by metis
```
```  2250
```
```  2251 lemma compactE_image:
```
```  2252   assumes "compact s" and "\<forall>t\<in>C. open (f t)" and "s \<subseteq> (\<Union>c\<in>C. f c)"
```
```  2253   obtains C' where "C' \<subseteq> C" and "finite C'" and "s \<subseteq> (\<Union>c\<in>C'. f c)"
```
```  2254   using assms unfolding ball_simps[symmetric] SUP_def
```
```  2255   by (metis (lifting) finite_subset_image compact_eq_heine_borel[of s])
```
```  2256
```
```  2257 lemma compact_inter_closed [intro]:
```
```  2258   assumes "compact s" and "closed t"
```
```  2259   shows "compact (s \<inter> t)"
```
```  2260 proof (rule compactI)
```
```  2261   fix C assume C: "\<forall>c\<in>C. open c" and cover: "s \<inter> t \<subseteq> \<Union>C"
```
```  2262   from C `closed t` have "\<forall>c\<in>C \<union> {-t}. open c" by auto
```
```  2263   moreover from cover have "s \<subseteq> \<Union>(C \<union> {-t})" by auto
```
```  2264   ultimately have "\<exists>D\<subseteq>C \<union> {-t}. finite D \<and> s \<subseteq> \<Union>D"
```
```  2265     using `compact s` unfolding compact_eq_heine_borel by auto
```
```  2266   then obtain D where "D \<subseteq> C \<union> {- t} \<and> finite D \<and> s \<subseteq> \<Union>D" ..
```
```  2267   then show "\<exists>D\<subseteq>C. finite D \<and> s \<inter> t \<subseteq> \<Union>D"
```
```  2268     by (intro exI[of _ "D - {-t}"]) auto
```
```  2269 qed
```
```  2270
```
```  2271 lemma inj_setminus: "inj_on uminus (A::'a set set)"
```
```  2272   by (auto simp: inj_on_def)
```
```  2273
```
```  2274 lemma compact_fip:
```
```  2275   "compact U \<longleftrightarrow>
```
```  2276     (\<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> {})"
```
```  2277   (is "_ \<longleftrightarrow> ?R")
```
```  2278 proof (safe intro!: compact_eq_heine_borel[THEN iffD2])
```
```  2279   fix A
```
```  2280   assume "compact U"
```
```  2281     and A: "\<forall>a\<in>A. closed a" "U \<inter> \<Inter>A = {}"
```
```  2282     and fi: "\<forall>B \<subseteq> A. finite B \<longrightarrow> U \<inter> \<Inter>B \<noteq> {}"
```
```  2283   from A have "(\<forall>a\<in>uminus`A. open a) \<and> U \<subseteq> \<Union>(uminus`A)"
```
```  2284     by auto
```
```  2285   with `compact U` obtain B where "B \<subseteq> A" "finite (uminus`B)" "U \<subseteq> \<Union>(uminus`B)"
```
```  2286     unfolding compact_eq_heine_borel by (metis subset_image_iff)
```
```  2287   with fi[THEN spec, of B] show False
```
```  2288     by (auto dest: finite_imageD intro: inj_setminus)
```
```  2289 next
```
```  2290   fix A
```
```  2291   assume ?R
```
```  2292   assume "\<forall>a\<in>A. open a" "U \<subseteq> \<Union>A"
```
```  2293   then have "U \<inter> \<Inter>(uminus`A) = {}" "\<forall>a\<in>uminus`A. closed a"
```
```  2294     by auto
```
```  2295   with `?R` obtain B where "B \<subseteq> A" "finite (uminus`B)" "U \<inter> \<Inter>(uminus`B) = {}"
```
```  2296     by (metis subset_image_iff)
```
```  2297   then show "\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T"
```
```  2298     by  (auto intro!: exI[of _ B] inj_setminus dest: finite_imageD)
```
```  2299 qed
```
```  2300
```
```  2301 lemma compact_imp_fip:
```
```  2302   "compact s \<Longrightarrow> \<forall>t \<in> f. closed t \<Longrightarrow> \<forall>f'. finite f' \<and> f' \<subseteq> f \<longrightarrow> (s \<inter> (\<Inter> f') \<noteq> {}) \<Longrightarrow>
```
```  2303     s \<inter> (\<Inter> f) \<noteq> {}"
```
```  2304   unfolding compact_fip by auto
```
```  2305
```
```  2306 lemma compact_imp_fip_image:
```
```  2307   assumes "compact s"
```
```  2308     and P: "\<And>i. i \<in> I \<Longrightarrow> closed (f i)"
```
```  2309     and Q: "\<And>I'. finite I' \<Longrightarrow> I' \<subseteq> I \<Longrightarrow> (s \<inter> (\<Inter>i\<in>I'. f i) \<noteq> {})"
```
```  2310   shows "s \<inter> (\<Inter>i\<in>I. f i) \<noteq> {}"
```
```  2311 proof -
```
```  2312   note `compact s`
```
```  2313   moreover from P have "\<forall>i \<in> f ` I. closed i" by blast
```
```  2314   moreover have "\<forall>A. finite A \<and> A \<subseteq> f ` I \<longrightarrow> (s \<inter> (\<Inter>A) \<noteq> {})"
```
```  2315   proof (rule, rule, erule conjE)
```
```  2316     fix A :: "'a set set"
```
```  2317     assume "finite A"
```
```  2318     moreover assume "A \<subseteq> f ` I"
```
```  2319     ultimately obtain B where "B \<subseteq> I" and "finite B" and "A = f ` B"
```
```  2320       using finite_subset_image [of A f I] by blast
```
```  2321     with Q [of B] show "s \<inter> \<Inter>A \<noteq> {}" by simp
```
```  2322   qed
```
```  2323   ultimately have "s \<inter> (\<Inter>(f ` I)) \<noteq> {}" by (rule compact_imp_fip)
```
```  2324   then show ?thesis by simp
```
```  2325 qed
```
```  2326
```
```  2327 end
```
```  2328
```
```  2329 lemma (in t2_space) compact_imp_closed:
```
```  2330   assumes "compact s" shows "closed s"
```
```  2331 unfolding closed_def
```
```  2332 proof (rule openI)
```
```  2333   fix y assume "y \<in> - s"
```
```  2334   let ?C = "\<Union>x\<in>s. {u. open u \<and> x \<in> u \<and> eventually (\<lambda>y. y \<notin> u) (nhds y)}"
```
```  2335   note `compact s`
```
```  2336   moreover have "\<forall>u\<in>?C. open u" by simp
```
```  2337   moreover have "s \<subseteq> \<Union>?C"
```
```  2338   proof
```
```  2339     fix x assume "x \<in> s"
```
```  2340     with `y \<in> - s` have "x \<noteq> y" by clarsimp
```
```  2341     hence "\<exists>u v. open u \<and> open v \<and> x \<in> u \<and> y \<in> v \<and> u \<inter> v = {}"
```
```  2342       by (rule hausdorff)
```
```  2343     with `x \<in> s` show "x \<in> \<Union>?C"
```
```  2344       unfolding eventually_nhds by auto
```
```  2345   qed
```
```  2346   ultimately obtain D where "D \<subseteq> ?C" and "finite D" and "s \<subseteq> \<Union>D"
```
```  2347     by (rule compactE)
```
```  2348   from `D \<subseteq> ?C` have "\<forall>x\<in>D. eventually (\<lambda>y. y \<notin> x) (nhds y)" by auto
```
```  2349   with `finite D` have "eventually (\<lambda>y. y \<notin> \<Union>D) (nhds y)"
```
```  2350     by (simp add: eventually_Ball_finite)
```
```  2351   with `s \<subseteq> \<Union>D` have "eventually (\<lambda>y. y \<notin> s) (nhds y)"
```
```  2352     by (auto elim!: eventually_mono [rotated])
```
```  2353   thus "\<exists>t. open t \<and> y \<in> t \<and> t \<subseteq> - s"
```
```  2354     by (simp add: eventually_nhds subset_eq)
```
```  2355 qed
```
```  2356
```
```  2357 lemma compact_continuous_image:
```
```  2358   assumes f: "continuous_on s f" and s: "compact s"
```
```  2359   shows "compact (f ` s)"
```
```  2360 proof (rule compactI)
```
```  2361   fix C assume "\<forall>c\<in>C. open c" and cover: "f`s \<subseteq> \<Union>C"
```
```  2362   with f have "\<forall>c\<in>C. \<exists>A. open A \<and> A \<inter> s = f -` c \<inter> s"
```
```  2363     unfolding continuous_on_open_invariant by blast
```
```  2364   then obtain A where A: "\<forall>c\<in>C. open (A c) \<and> A c \<inter> s = f -` c \<inter> s"
```
```  2365     unfolding bchoice_iff ..
```
```  2366   with cover have "\<forall>c\<in>C. open (A c)" "s \<subseteq> (\<Union>c\<in>C. A c)"
```
```  2367     by (fastforce simp add: subset_eq set_eq_iff)+
```
```  2368   from compactE_image[OF s this] obtain D where "D \<subseteq> C" "finite D" "s \<subseteq> (\<Union>c\<in>D. A c)" .
```
```  2369   with A show "\<exists>D \<subseteq> C. finite D \<and> f`s \<subseteq> \<Union>D"
```
```  2370     by (intro exI[of _ D]) (fastforce simp add: subset_eq set_eq_iff)+
```
```  2371 qed
```
```  2372
```
```  2373 lemma continuous_on_inv:
```
```  2374   fixes f :: "'a::topological_space \<Rightarrow> 'b::t2_space"
```
```  2375   assumes "continuous_on s f"  "compact s"  "\<forall>x\<in>s. g (f x) = x"
```
```  2376   shows "continuous_on (f ` s) g"
```
```  2377 unfolding continuous_on_topological
```
```  2378 proof (clarsimp simp add: assms(3))
```
```  2379   fix x :: 'a and B :: "'a set"
```
```  2380   assume "x \<in> s" and "open B" and "x \<in> B"
```
```  2381   have 1: "\<forall>x\<in>s. f x \<in> f ` (s - B) \<longleftrightarrow> x \<in> s - B"
```
```  2382     using assms(3) by (auto, metis)
```
```  2383   have "continuous_on (s - B) f"
```
```  2384     using `continuous_on s f` Diff_subset
```
```  2385     by (rule continuous_on_subset)
```
```  2386   moreover have "compact (s - B)"
```
```  2387     using `open B` and `compact s`
```
```  2388     unfolding Diff_eq by (intro compact_inter_closed closed_Compl)
```
```  2389   ultimately have "compact (f ` (s - B))"
```
```  2390     by (rule compact_continuous_image)
```
```  2391   hence "closed (f ` (s - B))"
```
```  2392     by (rule compact_imp_closed)
```
```  2393   hence "open (- f ` (s - B))"
```
```  2394     by (rule open_Compl)
```
```  2395   moreover have "f x \<in> - f ` (s - B)"
```
```  2396     using `x \<in> s` and `x \<in> B` by (simp add: 1)
```
```  2397   moreover have "\<forall>y\<in>s. f y \<in> - f ` (s - B) \<longrightarrow> y \<in> B"
```
```  2398     by (simp add: 1)
```
```  2399   ultimately show "\<exists>A. open A \<and> f x \<in> A \<and> (\<forall>y\<in>s. f y \<in> A \<longrightarrow> y \<in> B)"
```
```  2400     by fast
```
```  2401 qed
```
```  2402
```
```  2403 lemma continuous_on_inv_into:
```
```  2404   fixes f :: "'a::topological_space \<Rightarrow> 'b::t2_space"
```
```  2405   assumes s: "continuous_on s f" "compact s" and f: "inj_on f s"
```
```  2406   shows "continuous_on (f ` s) (the_inv_into s f)"
```
```  2407   by (rule continuous_on_inv[OF s]) (auto simp: the_inv_into_f_f[OF f])
```
```  2408
```
```  2409 lemma (in linorder_topology) compact_attains_sup:
```
```  2410   assumes "compact S" "S \<noteq> {}"
```
```  2411   shows "\<exists>s\<in>S. \<forall>t\<in>S. t \<le> s"
```
```  2412 proof (rule classical)
```
```  2413   assume "\<not> (\<exists>s\<in>S. \<forall>t\<in>S. t \<le> s)"
```
```  2414   then obtain t where t: "\<forall>s\<in>S. t s \<in> S" and "\<forall>s\<in>S. s < t s"
```
```  2415     by (metis not_le)
```
```  2416   then have "\<forall>s\<in>S. open {..< t s}" "S \<subseteq> (\<Union>s\<in>S. {..< t s})"
```
```  2417     by auto
```
```  2418   with `compact S` obtain C where "C \<subseteq> S" "finite C" and C: "S \<subseteq> (\<Union>s\<in>C. {..< t s})"
```
```  2419     by (erule compactE_image)
```
```  2420   with `S \<noteq> {}` have Max: "Max (t`C) \<in> t`C" and "\<forall>s\<in>t`C. s \<le> Max (t`C)"
```
```  2421     by (auto intro!: Max_in)
```
```  2422   with C have "S \<subseteq> {..< Max (t`C)}"
```
```  2423     by (auto intro: less_le_trans simp: subset_eq)
```
```  2424   with t Max `C \<subseteq> S` show ?thesis
```
```  2425     by fastforce
```
```  2426 qed
```
```  2427
```
```  2428 lemma (in linorder_topology) compact_attains_inf:
```
```  2429   assumes "compact S" "S \<noteq> {}"
```
```  2430   shows "\<exists>s\<in>S. \<forall>t\<in>S. s \<le> t"
```
```  2431 proof (rule classical)
```
```  2432   assume "\<not> (\<exists>s\<in>S. \<forall>t\<in>S. s \<le> t)"
```
```  2433   then obtain t where t: "\<forall>s\<in>S. t s \<in> S" and "\<forall>s\<in>S. t s < s"
```
```  2434     by (metis not_le)
```
```  2435   then have "\<forall>s\<in>S. open {t s <..}" "S \<subseteq> (\<Union>s\<in>S. {t s <..})"
```
```  2436     by auto
```
```  2437   with `compact S` obtain C where "C \<subseteq> S" "finite C" and C: "S \<subseteq> (\<Union>s\<in>C. {t s <..})"
```
```  2438     by (erule compactE_image)
```
```  2439   with `S \<noteq> {}` have Min: "Min (t`C) \<in> t`C" and "\<forall>s\<in>t`C. Min (t`C) \<le> s"
```
```  2440     by (auto intro!: Min_in)
```
```  2441   with C have "S \<subseteq> {Min (t`C) <..}"
```
```  2442     by (auto intro: le_less_trans simp: subset_eq)
```
```  2443   with t Min `C \<subseteq> S` show ?thesis
```
```  2444     by fastforce
```
```  2445 qed
```
```  2446
```
```  2447 lemma continuous_attains_sup:
```
```  2448   fixes f :: "'a::topological_space \<Rightarrow> 'b::linorder_topology"
```
```  2449   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)"
```
```  2450   using compact_attains_sup[of "f ` s"] compact_continuous_image[of s f] by auto
```
```  2451
```
```  2452 lemma continuous_attains_inf:
```
```  2453   fixes f :: "'a::topological_space \<Rightarrow> 'b::linorder_topology"
```
```  2454   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)"
```
```  2455   using compact_attains_inf[of "f ` s"] compact_continuous_image[of s f] by auto
```
```  2456
```
```  2457 subsection {* Connectedness *}
```
```  2458
```
```  2459 context topological_space
```
```  2460 begin
```
```  2461
```
```  2462 definition "connected S \<longleftrightarrow>
```
```  2463   \<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> {})"
```
```  2464
```
```  2465 lemma connectedI:
```
```  2466   "(\<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)
```
```  2467   \<Longrightarrow> connected U"
```
```  2468   by (auto simp: connected_def)
```
```  2469
```
```  2470 lemma connected_empty[simp]: "connected {}"
```
```  2471   by (auto intro!: connectedI)
```
```  2472
```
```  2473 lemma connectedD:
```
```  2474   "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 = {}"
```
```  2475   by (auto simp: connected_def)
```
```  2476
```
```  2477 end
```
```  2478
```
```  2479 lemma connected_local_const:
```
```  2480   assumes "connected A" "a \<in> A" "b \<in> A"
```
```  2481   assumes *: "\<forall>a\<in>A. eventually (\<lambda>b. f a = f b) (at a within A)"
```
```  2482   shows "f a = f b"
```
```  2483 proof -
```
```  2484   obtain S where S: "\<And>a. a \<in> A \<Longrightarrow> a \<in> S a" "\<And>a. a \<in> A \<Longrightarrow> open (S a)"
```
```  2485     "\<And>a x. a \<in> A \<Longrightarrow> x \<in> S a \<Longrightarrow> x \<in> A \<Longrightarrow> f a = f x"
```
```  2486     using * unfolding eventually_at_topological by metis
```
```  2487
```
```  2488   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"
```
```  2489   have "?P \<inter> A = {} \<or> ?N \<inter> A = {}"
```
```  2490     using `connected A` S `a\<in>A`
```
```  2491     by (intro connectedD) (auto, metis)
```
```  2492   then show "f a = f b"
```
```  2493   proof
```
```  2494     assume "?N \<inter> A = {}"
```
```  2495     then have "\<forall>x\<in>A. f a = f x"
```
```  2496       using S(1) by auto
```
```  2497     with `b\<in>A` show ?thesis by auto
```
```  2498   next
```
```  2499     assume "?P \<inter> A = {}" then show ?thesis
```
```  2500       using `a \<in> A` S(1)[of a] by auto
```
```  2501   qed
```
```  2502 qed
```
```  2503
```
```  2504 lemma (in linorder_topology) connectedD_interval:
```
```  2505   assumes "connected U" and xy: "x \<in> U" "y \<in> U" and "x \<le> z" "z \<le> y"
```
```  2506   shows "z \<in> U"
```
```  2507 proof -
```
```  2508   have eq: "{..<z} \<union> {z<..} = - {z}"
```
```  2509     by auto
```
```  2510   { assume "z \<notin> U" "x < z" "z < y"
```
```  2511     with xy have "\<not> connected U"
```
```  2512       unfolding connected_def simp_thms
```
```  2513       apply (rule_tac exI[of _ "{..< z}"])
```
```  2514       apply (rule_tac exI[of _ "{z <..}"])
```
```  2515       apply (auto simp add: eq)
```
```  2516       done }
```
```  2517   with assms show "z \<in> U"
```
```  2518     by (metis less_le)
```
```  2519 qed
```
```  2520
```
```  2521 lemma connected_continuous_image:
```
```  2522   assumes *: "continuous_on s f"
```
```  2523   assumes "connected s"
```
```  2524   shows "connected (f ` s)"
```
```  2525 proof (rule connectedI)
```
```  2526   fix A B assume A: "open A" "A \<inter> f ` s \<noteq> {}" and B: "open B" "B \<inter> f ` s \<noteq> {}" and
```
```  2527     AB: "A \<inter> B \<inter> f ` s = {}" "f ` s \<subseteq> A \<union> B"
```
```  2528   obtain A' where A': "open A'" "f -` A \<inter> s = A' \<inter> s"
```
```  2529     using * `open A` unfolding continuous_on_open_invariant by metis
```
```  2530   obtain B' where B': "open B'" "f -` B \<inter> s = B' \<inter> s"
```
```  2531     using * `open B` unfolding continuous_on_open_invariant by metis
```
```  2532
```
```  2533   have "\<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> {}"
```
```  2534   proof (rule exI[of _ A'], rule exI[of _ B'], intro conjI)
```
```  2535     have "s \<subseteq> (f -` A \<inter> s) \<union> (f -` B \<inter> s)" using AB by auto
```
```  2536     then show "s \<subseteq> A' \<union> B'" using A' B' by auto
```
```  2537   next
```
```  2538     have "(f -` A \<inter> s) \<inter> (f -` B \<inter> s) = {}" using AB by auto
```
```  2539     then show "A' \<inter> B' \<inter> s = {}" using A' B' by auto
```
```  2540   qed (insert A' B' A B, auto)
```
```  2541   with `connected s` show False
```
```  2542     unfolding connected_def by blast
```
```  2543 qed
```
```  2544
```
```  2545
```
```  2546 section {* Connectedness *}
```
```  2547
```
```  2548 class linear_continuum_topology = linorder_topology + linear_continuum
```
```  2549 begin
```
```  2550
```
```  2551 lemma Inf_notin_open:
```
```  2552   assumes A: "open A" and bnd: "\<forall>a\<in>A. x < a"
```
```  2553   shows "Inf A \<notin> A"
```
```  2554 proof
```
```  2555   assume "Inf A \<in> A"
```
```  2556   then obtain b where "b < Inf A" "{b <.. Inf A} \<subseteq> A"
```
```  2557     using open_left[of A "Inf A" x] assms by auto
```
```  2558   with dense[of b "Inf A"] obtain c where "c < Inf A" "c \<in> A"
```
```  2559     by (auto simp: subset_eq)
```
```  2560   then show False
```
```  2561     using cInf_lower[OF `c \<in> A`] bnd by (metis not_le less_imp_le bdd_belowI)
```
```  2562 qed
```
```  2563
```
```  2564 lemma Sup_notin_open:
```
```  2565   assumes A: "open A" and bnd: "\<forall>a\<in>A. a < x"
```
```  2566   shows "Sup A \<notin> A"
```
```  2567 proof
```
```  2568   assume "Sup A \<in> A"
```
```  2569   then obtain b where "Sup A < b" "{Sup A ..< b} \<subseteq> A"
```
```  2570     using open_right[of A "Sup A" x] assms by auto
```
```  2571   with dense[of "Sup A" b] obtain c where "Sup A < c" "c \<in> A"
```
```  2572     by (auto simp: subset_eq)
```
```  2573   then show False
```
```  2574     using cSup_upper[OF `c \<in> A`] bnd by (metis less_imp_le not_le bdd_aboveI)
```
```  2575 qed
```
```  2576
```
```  2577 end
```
```  2578
```
```  2579 instance linear_continuum_topology \<subseteq> perfect_space
```
```  2580 proof
```
```  2581   fix x :: 'a
```
```  2582   obtain y where "x < y \<or> y < x"
```
```  2583     using ex_gt_or_lt [of x] ..
```
```  2584   with Inf_notin_open[of "{x}" y] Sup_notin_open[of "{x}" y]
```
```  2585   show "\<not> open {x}"
```
```  2586     by auto
```
```  2587 qed
```
```  2588
```
```  2589 lemma connectedI_interval:
```
```  2590   fixes U :: "'a :: linear_continuum_topology set"
```
```  2591   assumes *: "\<And>x y z. x \<in> U \<Longrightarrow> y \<in> U \<Longrightarrow> x \<le> z \<Longrightarrow> z \<le> y \<Longrightarrow> z \<in> U"
```
```  2592   shows "connected U"
```
```  2593 proof (rule connectedI)
```
```  2594   { fix A B assume "open A" "open B" "A \<inter> B \<inter> U = {}" "U \<subseteq> A \<union> B"
```
```  2595     fix x y assume "x < y" "x \<in> A" "y \<in> B" "x \<in> U" "y \<in> U"
```
```  2596
```
```  2597     let ?z = "Inf (B \<inter> {x <..})"
```
```  2598
```
```  2599     have "x \<le> ?z" "?z \<le> y"
```
```  2600       using `y \<in> B` `x < y` by (auto intro: cInf_lower cInf_greatest)
```
```  2601     with `x \<in> U` `y \<in> U` have "?z \<in> U"
```
```  2602       by (rule *)
```
```  2603     moreover have "?z \<notin> B \<inter> {x <..}"
```
```  2604       using `open B` by (intro Inf_notin_open) auto
```
```  2605     ultimately have "?z \<in> A"
```
```  2606       using `x \<le> ?z` `A \<inter> B \<inter> U = {}` `x \<in> A` `U \<subseteq> A \<union> B` by auto
```
```  2607
```
```  2608     { assume "?z < y"
```
```  2609       obtain a where "?z < a" "{?z ..< a} \<subseteq> A"
```
```  2610         using open_right[OF `open A` `?z \<in> A` `?z < y`] by auto
```
```  2611       moreover obtain b where "b \<in> B" "x < b" "b < min a y"
```
```  2612         using cInf_less_iff[of "B \<inter> {x <..}" "min a y"] `?z < a` `?z < y` `x < y` `y \<in> B`
```
```  2613         by (auto intro: less_imp_le)
```
```  2614       moreover have "?z \<le> b"
```
```  2615         using `b \<in> B` `x < b`
```
```  2616         by (intro cInf_lower) auto
```
```  2617       moreover have "b \<in> U"
```
```  2618         using `x \<le> ?z` `?z \<le> b` `b < min a y`
```
```  2619         by (intro *[OF `x \<in> U` `y \<in> U`]) (auto simp: less_imp_le)
```
```  2620       ultimately have "\<exists>b\<in>B. b \<in> A \<and> b \<in> U"
```
```  2621         by (intro bexI[of _ b]) auto }
```
```  2622     then have False
```
```  2623       using `?z \<le> y` `?z \<in> A` `y \<in> B` `y \<in> U` `A \<inter> B \<inter> U = {}` unfolding le_less by blast }
```
```  2624   note not_disjoint = this
```
```  2625
```
```  2626   fix A B assume AB: "open A" "open B" "U \<subseteq> A \<union> B" "A \<inter> B \<inter> U = {}"
```
```  2627   moreover assume "A \<inter> U \<noteq> {}" then obtain x where x: "x \<in> U" "x \<in> A" by auto
```
```  2628   moreover assume "B \<inter> U \<noteq> {}" then obtain y where y: "y \<in> U" "y \<in> B" by auto
```
```  2629   moreover note not_disjoint[of B A y x] not_disjoint[of A B x y]
```
```  2630   ultimately show False by (cases x y rule: linorder_cases) auto
```
```  2631 qed
```
```  2632
```
```  2633 lemma connected_iff_interval:
```
```  2634   fixes U :: "'a :: linear_continuum_topology set"
```
```  2635   shows "connected U \<longleftrightarrow> (\<forall>x\<in>U. \<forall>y\<in>U. \<forall>z. x \<le> z \<longrightarrow> z \<le> y \<longrightarrow> z \<in> U)"
```
```  2636   by (auto intro: connectedI_interval dest: connectedD_interval)
```
```  2637
```
```  2638 lemma connected_UNIV[simp]: "connected (UNIV::'a::linear_continuum_topology set)"
```
```  2639   unfolding connected_iff_interval by auto
```
```  2640
```
```  2641 lemma connected_Ioi[simp]: "connected {a::'a::linear_continuum_topology <..}"
```
```  2642   unfolding connected_iff_interval by auto
```
```  2643
```
```  2644 lemma connected_Ici[simp]: "connected {a::'a::linear_continuum_topology ..}"
```
```  2645   unfolding connected_iff_interval by auto
```
```  2646
```
```  2647 lemma connected_Iio[simp]: "connected {..< a::'a::linear_continuum_topology}"
```
```  2648   unfolding connected_iff_interval by auto
```
```  2649
```
```  2650 lemma connected_Iic[simp]: "connected {.. a::'a::linear_continuum_topology}"
```
```  2651   unfolding connected_iff_interval by auto
```
```  2652
```
```  2653 lemma connected_Ioo[simp]: "connected {a <..< b::'a::linear_continuum_topology}"
```
```  2654   unfolding connected_iff_interval by auto
```
```  2655
```
```  2656 lemma connected_Ioc[simp]: "connected {a <.. b::'a::linear_continuum_topology}"
```
```  2657   unfolding connected_iff_interval by auto
```
```  2658
```
```  2659 lemma connected_Ico[simp]: "connected {a ..< b::'a::linear_continuum_topology}"
```
```  2660   unfolding connected_iff_interval by auto
```
```  2661
```
```  2662 lemma connected_Icc[simp]: "connected {a .. b::'a::linear_continuum_topology}"
```
```  2663   unfolding connected_iff_interval by auto
```
```  2664
```
```  2665 lemma connected_contains_Ioo:
```
```  2666   fixes A :: "'a :: linorder_topology set"
```
```  2667   assumes A: "connected A" "a \<in> A" "b \<in> A" shows "{a <..< b} \<subseteq> A"
```
```  2668   using connectedD_interval[OF A] by (simp add: subset_eq Ball_def less_imp_le)
```
```  2669
```
```  2670 subsection {* Intermediate Value Theorem *}
```
```  2671
```
```  2672 lemma IVT':
```
```  2673   fixes f :: "'a :: linear_continuum_topology \<Rightarrow> 'b :: linorder_topology"
```
```  2674   assumes y: "f a \<le> y" "y \<le> f b" "a \<le> b"
```
```  2675   assumes *: "continuous_on {a .. b} f"
```
```  2676   shows "\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = y"
```
```  2677 proof -
```
```  2678   have "connected {a..b}"
```
```  2679     unfolding connected_iff_interval by auto
```
```  2680   from connected_continuous_image[OF * this, THEN connectedD_interval, of "f a" "f b" y] y
```
```  2681   show ?thesis
```
```  2682     by (auto simp add: atLeastAtMost_def atLeast_def atMost_def)
```
```  2683 qed
```
```  2684
```
```  2685 lemma IVT2':
```
```  2686   fixes f :: "'a :: linear_continuum_topology \<Rightarrow> 'b :: linorder_topology"
```
```  2687   assumes y: "f b \<le> y" "y \<le> f a" "a \<le> b"
```
```  2688   assumes *: "continuous_on {a .. b} f"
```
```  2689   shows "\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = y"
```
```  2690 proof -
```
```  2691   have "connected {a..b}"
```
```  2692     unfolding connected_iff_interval by auto
```
```  2693   from connected_continuous_image[OF * this, THEN connectedD_interval, of "f b" "f a" y] y
```
```  2694   show ?thesis
```
```  2695     by (auto simp add: atLeastAtMost_def atLeast_def atMost_def)
```
```  2696 qed
```
```  2697
```
```  2698 lemma IVT:
```
```  2699   fixes f :: "'a :: linear_continuum_topology \<Rightarrow> 'b :: linorder_topology"
```
```  2700   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> \<exists>x. a \<le> x \<and> x \<le> b \<and> f x = y"
```
```  2701   by (rule IVT') (auto intro: continuous_at_imp_continuous_on)
```
```  2702
```
```  2703 lemma IVT2:
```
```  2704   fixes f :: "'a :: linear_continuum_topology \<Rightarrow> 'b :: linorder_topology"
```
```  2705   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> \<exists>x. a \<le> x \<and> x \<le> b \<and> f x = y"
```
```  2706   by (rule IVT2') (auto intro: continuous_at_imp_continuous_on)
```
```  2707
```
```  2708 lemma continuous_inj_imp_mono:
```
```  2709   fixes f :: "'a::linear_continuum_topology \<Rightarrow> 'b :: linorder_topology"
```
```  2710   assumes x: "a < x" "x < b"
```
```  2711   assumes cont: "continuous_on {a..b} f"
```
```  2712   assumes inj: "inj_on f {a..b}"
```
```  2713   shows "(f a < f x \<and> f x < f b) \<or> (f b < f x \<and> f x < f a)"
```
```  2714 proof -
```
```  2715   note I = inj_on_iff[OF inj]
```
```  2716   { assume "f x < f a" "f x < f b"
```
```  2717     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"
```
```  2718       using IVT'[of f x "min (f a) (f b)" b] IVT2'[of f x "min (f a) (f b)" a] x
```
```  2719       by (auto simp: continuous_on_subset[OF cont] less_imp_le)
```
```  2720     with x I have False by auto }
```
```  2721   moreover
```
```  2722   { assume "f a < f x" "f b < f x"
```
```  2723     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"
```
```  2724       using IVT'[of f a "max (f a) (f b)" x] IVT2'[of f b "max (f a) (f b)" x] x
```
```  2725       by (auto simp: continuous_on_subset[OF cont] less_imp_le)
```
```  2726     with x I have False by auto }
```
```  2727   ultimately show ?thesis
```
```  2728     using I[of a x] I[of x b] x less_trans[OF x] by (auto simp add: le_less less_imp_neq neq_iff)
```
```  2729 qed
```
```  2730
```
```  2731 subsection {* Setup @{typ "'a filter"} for lifting and transfer *}
```
```  2732
```
```  2733 context begin interpretation lifting_syntax .
```
```  2734
```
```  2735 definition rel_filter :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a filter \<Rightarrow> 'b filter \<Rightarrow> bool"
```
```  2736 where "rel_filter R F G = ((R ===> op =) ===> op =) (Rep_filter F) (Rep_filter G)"
```
```  2737
```
```  2738 lemma rel_filter_eventually:
```
```  2739   "rel_filter R F G \<longleftrightarrow>
```
```  2740   ((R ===> op =) ===> op =) (\<lambda>P. eventually P F) (\<lambda>P. eventually P G)"
```
```  2741 by(simp add: rel_filter_def eventually_def)
```
```  2742
```
```  2743 lemma filtermap_id [simp, id_simps]: "filtermap id = id"
```
```  2744 by(simp add: fun_eq_iff id_def filtermap_ident)
```
```  2745
```
```  2746 lemma filtermap_id' [simp]: "filtermap (\<lambda>x. x) = (\<lambda>F. F)"
```
```  2747 using filtermap_id unfolding id_def .
```
```  2748
```
```  2749 lemma Quotient_filter [quot_map]:
```
```  2750   assumes Q: "Quotient R Abs Rep T"
```
```  2751   shows "Quotient (rel_filter R) (filtermap Abs) (filtermap Rep) (rel_filter T)"
```
```  2752 unfolding Quotient_alt_def
```
```  2753 proof(intro conjI strip)
```
```  2754   from Q have *: "\<And>x y. T x y \<Longrightarrow> Abs x = y"
```
```  2755     unfolding Quotient_alt_def by blast
```
```  2756
```
```  2757   fix F G
```
```  2758   assume "rel_filter T F G"
```
```  2759   thus "filtermap Abs F = G" unfolding filter_eq_iff
```
```  2760     by(auto simp add: eventually_filtermap rel_filter_eventually * rel_funI del: iffI elim!: rel_funD)
```
```  2761 next
```
```  2762   from Q have *: "\<And>x. T (Rep x) x" unfolding Quotient_alt_def by blast
```
```  2763
```
```  2764   fix F
```
```  2765   show "rel_filter T (filtermap Rep F) F"
```
```  2766     by(auto elim: rel_funD intro: * intro!: ext arg_cong[where f="\<lambda>P. eventually P F"] rel_funI
```
```  2767             del: iffI simp add: eventually_filtermap rel_filter_eventually)
```
```  2768 qed(auto simp add: map_fun_def o_def eventually_filtermap filter_eq_iff fun_eq_iff rel_filter_eventually
```
```  2769          fun_quotient[OF fun_quotient[OF Q identity_quotient] identity_quotient, unfolded Quotient_alt_def])
```
```  2770
```
```  2771 lemma eventually_parametric [transfer_rule]:
```
```  2772   "((A ===> op =) ===> rel_filter A ===> op =) eventually eventually"
```
```  2773 by(simp add: rel_fun_def rel_filter_eventually)
```
```  2774
```
```  2775 lemma rel_filter_eq [relator_eq]: "rel_filter op = = op ="
```
```  2776 by(auto simp add: rel_filter_eventually rel_fun_eq fun_eq_iff filter_eq_iff)
```
```  2777
```
```  2778 lemma rel_filter_mono [relator_mono]:
```
```  2779   "A \<le> B \<Longrightarrow> rel_filter A \<le> rel_filter B"
```
```  2780 unfolding rel_filter_eventually[abs_def]
```
```  2781 by(rule le_funI)+(intro fun_mono fun_mono[THEN le_funD, THEN le_funD] order.refl)
```
```  2782
```
```  2783 lemma rel_filter_conversep [simp]: "rel_filter A\<inverse>\<inverse> = (rel_filter A)\<inverse>\<inverse>"
```
```  2784 by(auto simp add: rel_filter_eventually fun_eq_iff rel_fun_def)
```
```  2785
```
```  2786 lemma is_filter_parametric_aux:
```
```  2787   assumes "is_filter F"
```
```  2788   assumes [transfer_rule]: "bi_total A" "bi_unique A"
```
```  2789   and [transfer_rule]: "((A ===> op =) ===> op =) F G"
```
```  2790   shows "is_filter G"
```
```  2791 proof -
```
```  2792   interpret is_filter F by fact
```
```  2793   show ?thesis
```
```  2794   proof
```
```  2795     have "F (\<lambda>_. True) = G (\<lambda>x. True)" by transfer_prover
```
```  2796     thus "G (\<lambda>x. True)" by(simp add: True)
```
```  2797   next
```
```  2798     fix P' Q'
```
```  2799     assume "G P'" "G Q'"
```
```  2800     moreover
```
```  2801     from bi_total_fun[OF `bi_unique A` bi_total_eq, unfolded bi_total_def]
```
```  2802     obtain P Q where [transfer_rule]: "(A ===> op =) P P'" "(A ===> op =) Q Q'" by blast
```
```  2803     have "F P = G P'" "F Q = G Q'" by transfer_prover+
```
```  2804     ultimately have "F (\<lambda>x. P x \<and> Q x)" by(simp add: conj)
```
```  2805     moreover have "F (\<lambda>x. P x \<and> Q x) = G (\<lambda>x. P' x \<and> Q' x)" by transfer_prover
```
```  2806     ultimately show "G (\<lambda>x. P' x \<and> Q' x)" by simp
```
```  2807   next
```
```  2808     fix P' Q'
```
```  2809     assume "\<forall>x. P' x \<longrightarrow> Q' x" "G P'"
```
```  2810     moreover
```
```  2811     from bi_total_fun[OF `bi_unique A` bi_total_eq, unfolded bi_total_def]
```
```  2812     obtain P Q where [transfer_rule]: "(A ===> op =) P P'" "(A ===> op =) Q Q'" by blast
```
```  2813     have "F P = G P'" by transfer_prover
```
```  2814     moreover have "(\<forall>x. P x \<longrightarrow> Q x) \<longleftrightarrow> (\<forall>x. P' x \<longrightarrow> Q' x)" by transfer_prover
```
```  2815     ultimately have "F Q" by(simp add: mono)
```
```  2816     moreover have "F Q = G Q'" by transfer_prover
```
```  2817     ultimately show "G Q'" by simp
```
```  2818   qed
```
```  2819 qed
```
```  2820
```
```  2821 lemma is_filter_parametric [transfer_rule]:
```
```  2822   "\<lbrakk> bi_total A; bi_unique A \<rbrakk>
```
```  2823   \<Longrightarrow> (((A ===> op =) ===> op =) ===> op =) is_filter is_filter"
```
```  2824 apply(rule rel_funI)
```
```  2825 apply(rule iffI)
```
```  2826  apply(erule (3) is_filter_parametric_aux)
```
```  2827 apply(erule is_filter_parametric_aux[where A="conversep A"])
```
```  2828 apply(auto simp add: rel_fun_def)
```
```  2829 done
```
```  2830
```
```  2831 lemma left_total_rel_filter [transfer_rule]:
```
```  2832   assumes [transfer_rule]: "bi_total A" "bi_unique A"
```
```  2833   shows "left_total (rel_filter A)"
```
```  2834 proof(rule left_totalI)
```
```  2835   fix F :: "'a filter"
```
```  2836   from bi_total_fun[OF bi_unique_fun[OF `bi_total A` bi_unique_eq] bi_total_eq]
```
```  2837   obtain G where [transfer_rule]: "((A ===> op =) ===> op =) (\<lambda>P. eventually P F) G"
```
```  2838     unfolding  bi_total_def by blast
```
```  2839   moreover have "is_filter (\<lambda>P. eventually P F) \<longleftrightarrow> is_filter G" by transfer_prover
```
```  2840   hence "is_filter G" by(simp add: eventually_def is_filter_Rep_filter)
```
```  2841   ultimately have "rel_filter A F (Abs_filter G)"
```
```  2842     by(simp add: rel_filter_eventually eventually_Abs_filter)
```
```  2843   thus "\<exists>G. rel_filter A F G" ..
```
```  2844 qed
```
```  2845
```
```  2846 lemma right_total_rel_filter [transfer_rule]:
```
```  2847   "\<lbrakk> bi_total A; bi_unique A \<rbrakk> \<Longrightarrow> right_total (rel_filter A)"
```
```  2848 using left_total_rel_filter[of "A\<inverse>\<inverse>"] by simp
```
```  2849
```
```  2850 lemma bi_total_rel_filter [transfer_rule]:
```
```  2851   assumes "bi_total A" "bi_unique A"
```
```  2852   shows "bi_total (rel_filter A)"
```
```  2853 unfolding bi_total_alt_def using assms
```
```  2854 by(simp add: left_total_rel_filter right_total_rel_filter)
```
```  2855
```
```  2856 lemma left_unique_rel_filter [transfer_rule]:
```
```  2857   assumes "left_unique A"
```
```  2858   shows "left_unique (rel_filter A)"
```
```  2859 proof(rule left_uniqueI)
```
```  2860   fix F F' G
```
```  2861   assume [transfer_rule]: "rel_filter A F G" "rel_filter A F' G"
```
```  2862   show "F = F'"
```
```  2863     unfolding filter_eq_iff
```
```  2864   proof
```
```  2865     fix P :: "'a \<Rightarrow> bool"
```
```  2866     obtain P' where [transfer_rule]: "(A ===> op =) P P'"
```
```  2867       using left_total_fun[OF assms left_total_eq] unfolding left_total_def by blast
```
```  2868     have "eventually P F = eventually P' G"
```
```  2869       and "eventually P F' = eventually P' G" by transfer_prover+
```
```  2870     thus "eventually P F = eventually P F'" by simp
```
```  2871   qed
```
```  2872 qed
```
```  2873
```
```  2874 lemma right_unique_rel_filter [transfer_rule]:
```
```  2875   "right_unique A \<Longrightarrow> right_unique (rel_filter A)"
```
```  2876 using left_unique_rel_filter[of "A\<inverse>\<inverse>"] by simp
```
```  2877
```
```  2878 lemma bi_unique_rel_filter [transfer_rule]:
```
```  2879   "bi_unique A \<Longrightarrow> bi_unique (rel_filter A)"
```
```  2880 by(simp add: bi_unique_alt_def left_unique_rel_filter right_unique_rel_filter)
```
```  2881
```
```  2882 lemma top_filter_parametric [transfer_rule]:
```
```  2883   "bi_total A \<Longrightarrow> (rel_filter A) top top"
```
```  2884 by(simp add: rel_filter_eventually All_transfer)
```
```  2885
```
```  2886 lemma bot_filter_parametric [transfer_rule]: "(rel_filter A) bot bot"
```
```  2887 by(simp add: rel_filter_eventually rel_fun_def)
```
```  2888
```
```  2889 lemma sup_filter_parametric [transfer_rule]:
```
```  2890   "(rel_filter A ===> rel_filter A ===> rel_filter A) sup sup"
```
```  2891 by(fastforce simp add: rel_filter_eventually[abs_def] eventually_sup dest: rel_funD)
```
```  2892
```
```  2893 lemma Sup_filter_parametric [transfer_rule]:
```
```  2894   "(rel_set (rel_filter A) ===> rel_filter A) Sup Sup"
```
```  2895 proof(rule rel_funI)
```
```  2896   fix S T
```
```  2897   assume [transfer_rule]: "rel_set (rel_filter A) S T"
```
```  2898   show "rel_filter A (Sup S) (Sup T)"
```
```  2899     by(simp add: rel_filter_eventually eventually_Sup) transfer_prover
```
```  2900 qed
```
```  2901
```
```  2902 lemma principal_parametric [transfer_rule]:
```
```  2903   "(rel_set A ===> rel_filter A) principal principal"
```
```  2904 proof(rule rel_funI)
```
```  2905   fix S S'
```
```  2906   assume [transfer_rule]: "rel_set A S S'"
```
```  2907   show "rel_filter A (principal S) (principal S')"
```
```  2908     by(simp add: rel_filter_eventually eventually_principal) transfer_prover
```
```  2909 qed
```
```  2910
```
```  2911 context
```
```  2912   fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"
```
```  2913   assumes [transfer_rule]: "bi_unique A"
```
```  2914 begin
```
```  2915
```
```  2916 lemma le_filter_parametric [transfer_rule]:
```
```  2917   "(rel_filter A ===> rel_filter A ===> op =) op \<le> op \<le>"
```
```  2918 unfolding le_filter_def[abs_def] by transfer_prover
```
```  2919
```
```  2920 lemma less_filter_parametric [transfer_rule]:
```
```  2921   "(rel_filter A ===> rel_filter A ===> op =) op < op <"
```
```  2922 unfolding less_filter_def[abs_def] by transfer_prover
```
```  2923
```
```  2924 context
```
```  2925   assumes [transfer_rule]: "bi_total A"
```
```  2926 begin
```
```  2927
```
```  2928 lemma Inf_filter_parametric [transfer_rule]:
```
```  2929   "(rel_set (rel_filter A) ===> rel_filter A) Inf Inf"
```
```  2930 unfolding Inf_filter_def[abs_def] by transfer_prover
```
```  2931
```
```  2932 lemma inf_filter_parametric [transfer_rule]:
```
```  2933   "(rel_filter A ===> rel_filter A ===> rel_filter A) inf inf"
```
```  2934 proof(intro rel_funI)+
```
```  2935   fix F F' G G'
```
```  2936   assume [transfer_rule]: "rel_filter A F F'" "rel_filter A G G'"
```
```  2937   have "rel_filter A (Inf {F, G}) (Inf {F', G'})" by transfer_prover
```
```  2938   thus "rel_filter A (inf F G) (inf F' G')" by simp
```
```  2939 qed
```
```  2940
```
```  2941 end
```
```  2942
```
```  2943 end
```
```  2944
```
```  2945 end
```
```  2946
```
```  2947 end
```