src/HOL/Filter.thy
 author hoelzl Sun Apr 12 11:33:50 2015 +0200 (2015-04-12) changeset 60039 d55937a8f97e parent 60038 ca431cbce2a3 child 60040 1fa1023b13b9 permissions -rw-r--r--
```     1 (*  Title:      HOL/Filter.thy
```
```     2     Author:     Brian Huffman
```
```     3     Author:     Johannes Hölzl
```
```     4 *)
```
```     5
```
```     6 section {* Filters on predicates *}
```
```     7
```
```     8 theory Filter
```
```     9 imports Set_Interval Lifting_Set
```
```    10 begin
```
```    11
```
```    12 subsection {* Filters *}
```
```    13
```
```    14 text {*
```
```    15   This definition also allows non-proper filters.
```
```    16 *}
```
```    17
```
```    18 locale is_filter =
```
```    19   fixes F :: "('a \<Rightarrow> bool) \<Rightarrow> bool"
```
```    20   assumes True: "F (\<lambda>x. True)"
```
```    21   assumes conj: "F (\<lambda>x. P x) \<Longrightarrow> F (\<lambda>x. Q x) \<Longrightarrow> F (\<lambda>x. P x \<and> Q x)"
```
```    22   assumes mono: "\<forall>x. P x \<longrightarrow> Q x \<Longrightarrow> F (\<lambda>x. P x) \<Longrightarrow> F (\<lambda>x. Q x)"
```
```    23
```
```    24 typedef 'a filter = "{F :: ('a \<Rightarrow> bool) \<Rightarrow> bool. is_filter F}"
```
```    25 proof
```
```    26   show "(\<lambda>x. True) \<in> ?filter" by (auto intro: is_filter.intro)
```
```    27 qed
```
```    28
```
```    29 lemma is_filter_Rep_filter: "is_filter (Rep_filter F)"
```
```    30   using Rep_filter [of F] by simp
```
```    31
```
```    32 lemma Abs_filter_inverse':
```
```    33   assumes "is_filter F" shows "Rep_filter (Abs_filter F) = F"
```
```    34   using assms by (simp add: Abs_filter_inverse)
```
```    35
```
```    36
```
```    37 subsubsection {* Eventually *}
```
```    38
```
```    39 definition eventually :: "('a \<Rightarrow> bool) \<Rightarrow> 'a filter \<Rightarrow> bool"
```
```    40   where "eventually P F \<longleftrightarrow> Rep_filter F P"
```
```    41
```
```    42 syntax (xsymbols)
```
```    43   "_eventually"  :: "pttrn => 'a filter => bool => bool"      ("(3\<forall>\<^sub>F _ in _./ _)" [0, 0, 10] 10)
```
```    44
```
```    45 translations
```
```    46   "\<forall>\<^sub>Fx in F. P" == "CONST eventually (\<lambda>x. P) F"
```
```    47
```
```    48 lemma eventually_Abs_filter:
```
```    49   assumes "is_filter F" shows "eventually P (Abs_filter F) = F P"
```
```    50   unfolding eventually_def using assms by (simp add: Abs_filter_inverse)
```
```    51
```
```    52 lemma filter_eq_iff:
```
```    53   shows "F = F' \<longleftrightarrow> (\<forall>P. eventually P F = eventually P F')"
```
```    54   unfolding Rep_filter_inject [symmetric] fun_eq_iff eventually_def ..
```
```    55
```
```    56 lemma eventually_True [simp]: "eventually (\<lambda>x. True) F"
```
```    57   unfolding eventually_def
```
```    58   by (rule is_filter.True [OF is_filter_Rep_filter])
```
```    59
```
```    60 lemma always_eventually: "\<forall>x. P x \<Longrightarrow> eventually P F"
```
```    61 proof -
```
```    62   assume "\<forall>x. P x" hence "P = (\<lambda>x. True)" by (simp add: ext)
```
```    63   thus "eventually P F" by simp
```
```    64 qed
```
```    65
```
```    66 lemma eventually_mono:
```
```    67   "(\<forall>x. P x \<longrightarrow> Q x) \<Longrightarrow> eventually P F \<Longrightarrow> eventually Q F"
```
```    68   unfolding eventually_def
```
```    69   by (rule is_filter.mono [OF is_filter_Rep_filter])
```
```    70
```
```    71 lemma eventually_conj:
```
```    72   assumes P: "eventually (\<lambda>x. P x) F"
```
```    73   assumes Q: "eventually (\<lambda>x. Q x) F"
```
```    74   shows "eventually (\<lambda>x. P x \<and> Q x) F"
```
```    75   using assms unfolding eventually_def
```
```    76   by (rule is_filter.conj [OF is_filter_Rep_filter])
```
```    77
```
```    78 lemma eventually_Ball_finite:
```
```    79   assumes "finite A" and "\<forall>y\<in>A. eventually (\<lambda>x. P x y) net"
```
```    80   shows "eventually (\<lambda>x. \<forall>y\<in>A. P x y) net"
```
```    81 using assms by (induct set: finite, simp, simp add: eventually_conj)
```
```    82
```
```    83 lemma eventually_all_finite:
```
```    84   fixes P :: "'a \<Rightarrow> 'b::finite \<Rightarrow> bool"
```
```    85   assumes "\<And>y. eventually (\<lambda>x. P x y) net"
```
```    86   shows "eventually (\<lambda>x. \<forall>y. P x y) net"
```
```    87 using eventually_Ball_finite [of UNIV P] assms by simp
```
```    88
```
```    89 lemma eventually_mp:
```
```    90   assumes "eventually (\<lambda>x. P x \<longrightarrow> Q x) F"
```
```    91   assumes "eventually (\<lambda>x. P x) F"
```
```    92   shows "eventually (\<lambda>x. Q x) F"
```
```    93 proof (rule eventually_mono)
```
```    94   show "\<forall>x. (P x \<longrightarrow> Q x) \<and> P x \<longrightarrow> Q x" by simp
```
```    95   show "eventually (\<lambda>x. (P x \<longrightarrow> Q x) \<and> P x) F"
```
```    96     using assms by (rule eventually_conj)
```
```    97 qed
```
```    98
```
```    99 lemma eventually_rev_mp:
```
```   100   assumes "eventually (\<lambda>x. P x) F"
```
```   101   assumes "eventually (\<lambda>x. P x \<longrightarrow> Q x) F"
```
```   102   shows "eventually (\<lambda>x. Q x) F"
```
```   103 using assms(2) assms(1) by (rule eventually_mp)
```
```   104
```
```   105 lemma eventually_conj_iff:
```
```   106   "eventually (\<lambda>x. P x \<and> Q x) F \<longleftrightarrow> eventually P F \<and> eventually Q F"
```
```   107   by (auto intro: eventually_conj elim: eventually_rev_mp)
```
```   108
```
```   109 lemma eventually_elim1:
```
```   110   assumes "eventually (\<lambda>i. P i) F"
```
```   111   assumes "\<And>i. P i \<Longrightarrow> Q i"
```
```   112   shows "eventually (\<lambda>i. Q i) F"
```
```   113   using assms by (auto elim!: eventually_rev_mp)
```
```   114
```
```   115 lemma eventually_elim2:
```
```   116   assumes "eventually (\<lambda>i. P i) F"
```
```   117   assumes "eventually (\<lambda>i. Q i) F"
```
```   118   assumes "\<And>i. P i \<Longrightarrow> Q i \<Longrightarrow> R i"
```
```   119   shows "eventually (\<lambda>i. R i) F"
```
```   120   using assms by (auto elim!: eventually_rev_mp)
```
```   121
```
```   122 lemma not_eventually_impI: "eventually P F \<Longrightarrow> \<not> eventually Q F \<Longrightarrow> \<not> eventually (\<lambda>x. P x \<longrightarrow> Q x) F"
```
```   123   by (auto intro: eventually_mp)
```
```   124
```
```   125 lemma not_eventuallyD: "\<not> eventually P F \<Longrightarrow> \<exists>x. \<not> P x"
```
```   126   by (metis always_eventually)
```
```   127
```
```   128 lemma eventually_subst:
```
```   129   assumes "eventually (\<lambda>n. P n = Q n) F"
```
```   130   shows "eventually P F = eventually Q F" (is "?L = ?R")
```
```   131 proof -
```
```   132   from assms have "eventually (\<lambda>x. P x \<longrightarrow> Q x) F"
```
```   133       and "eventually (\<lambda>x. Q x \<longrightarrow> P x) F"
```
```   134     by (auto elim: eventually_elim1)
```
```   135   then show ?thesis by (auto elim: eventually_elim2)
```
```   136 qed
```
```   137
```
```   138 ML {*
```
```   139   fun eventually_elim_tac ctxt facts = SUBGOAL_CASES (fn (goal, i) =>
```
```   140     let
```
```   141       val mp_thms = facts RL @{thms eventually_rev_mp}
```
```   142       val raw_elim_thm =
```
```   143         (@{thm allI} RS @{thm always_eventually})
```
```   144         |> fold (fn thm1 => fn thm2 => thm2 RS thm1) mp_thms
```
```   145         |> fold (fn _ => fn thm => @{thm impI} RS thm) facts
```
```   146       val cases_prop = Thm.prop_of (raw_elim_thm RS Goal.init (Thm.cterm_of ctxt goal))
```
```   147       val cases = Rule_Cases.make_common ctxt cases_prop [(("elim", []), [])]
```
```   148     in
```
```   149       CASES cases (rtac raw_elim_thm i)
```
```   150     end)
```
```   151 *}
```
```   152
```
```   153 method_setup eventually_elim = {*
```
```   154   Scan.succeed (fn ctxt => METHOD_CASES (HEADGOAL o eventually_elim_tac ctxt))
```
```   155 *} "elimination of eventually quantifiers"
```
```   156
```
```   157 subsection \<open> Frequently as dual to eventually \<close>
```
```   158
```
```   159 definition frequently :: "('a \<Rightarrow> bool) \<Rightarrow> 'a filter \<Rightarrow> bool"
```
```   160   where "frequently P F \<longleftrightarrow> \<not> eventually (\<lambda>x. \<not> P x) F"
```
```   161
```
```   162 syntax (xsymbols)
```
```   163   "_frequently"  :: "pttrn \<Rightarrow> 'a filter \<Rightarrow> bool \<Rightarrow> bool"      ("(3\<exists>\<^sub>F _ in _./ _)" [0, 0, 10] 10)
```
```   164
```
```   165 translations
```
```   166   "\<exists>\<^sub>Fx in F. P" == "CONST frequently (\<lambda>x. P) F"
```
```   167
```
```   168 lemma not_frequently_False [simp]: "\<not> frequently (\<lambda>x. False) F"
```
```   169   by (simp add: frequently_def)
```
```   170
```
```   171 lemma frequently_ex: "frequently P F \<Longrightarrow> \<exists>x. P x"
```
```   172   by (auto simp: frequently_def dest: not_eventuallyD)
```
```   173
```
```   174 lemma frequently_mp:
```
```   175   assumes ev: "eventually (\<lambda>x. P x \<longrightarrow> Q x) F" and P: "frequently (\<lambda>x. P x) F"
```
```   176   shows "frequently (\<lambda>x. Q x) F"
```
```   177 proof -
```
```   178   from ev have "eventually (\<lambda>x. \<not> Q x \<longrightarrow> \<not> P x) F"
```
```   179     by (rule eventually_rev_mp) (auto intro!: always_eventually)
```
```   180   from eventually_mp[OF this] P show ?thesis
```
```   181     by (auto simp: frequently_def)
```
```   182 qed
```
```   183
```
```   184 lemma frequently_rev_mp:
```
```   185   assumes "frequently (\<lambda>x. P x) F"
```
```   186   assumes "eventually (\<lambda>x. P x \<longrightarrow> Q x) F"
```
```   187   shows "frequently (\<lambda>x. Q x) F"
```
```   188 using assms(2) assms(1) by (rule frequently_mp)
```
```   189
```
```   190 lemma frequently_mono: "(\<forall>x. P x \<longrightarrow> Q x) \<Longrightarrow> frequently P F \<Longrightarrow> frequently Q F"
```
```   191   using frequently_mp[of P Q] by (simp add: always_eventually)
```
```   192
```
```   193 lemma frequently_disj_iff:
```
```   194   "frequently (\<lambda>x. P x \<or> Q x) F \<longleftrightarrow> frequently (\<lambda>x. P x) F \<or> frequently (\<lambda>x. Q x) F"
```
```   195   by (simp add: frequently_def eventually_conj_iff)
```
```   196
```
```   197 lemma frequently_disj:
```
```   198   "frequently (\<lambda>x. P x) F \<Longrightarrow> frequently (\<lambda>x. Q x) F \<Longrightarrow> frequently (\<lambda>x. P x \<or> Q x) F"
```
```   199   by (simp add: frequently_disj_iff)
```
```   200
```
```   201 lemma frequently_Bex_finite:
```
```   202   assumes "finite A" shows "frequently (\<lambda>x. \<exists>y\<in>A. P x y) net \<longleftrightarrow> (\<exists>y\<in>A. frequently (\<lambda>x. P x y) net)"
```
```   203   using assms by induction (auto simp: frequently_disj_iff)
```
```   204
```
```   205 subsubsection {* Finer-than relation *}
```
```   206
```
```   207 text {* @{term "F \<le> F'"} means that filter @{term F} is finer than
```
```   208 filter @{term F'}. *}
```
```   209
```
```   210 instantiation filter :: (type) complete_lattice
```
```   211 begin
```
```   212
```
```   213 definition le_filter_def:
```
```   214   "F \<le> F' \<longleftrightarrow> (\<forall>P. eventually P F' \<longrightarrow> eventually P F)"
```
```   215
```
```   216 definition
```
```   217   "(F :: 'a filter) < F' \<longleftrightarrow> F \<le> F' \<and> \<not> F' \<le> F"
```
```   218
```
```   219 definition
```
```   220   "top = Abs_filter (\<lambda>P. \<forall>x. P x)"
```
```   221
```
```   222 definition
```
```   223   "bot = Abs_filter (\<lambda>P. True)"
```
```   224
```
```   225 definition
```
```   226   "sup F F' = Abs_filter (\<lambda>P. eventually P F \<and> eventually P F')"
```
```   227
```
```   228 definition
```
```   229   "inf F F' = Abs_filter
```
```   230       (\<lambda>P. \<exists>Q R. eventually Q F \<and> eventually R F' \<and> (\<forall>x. Q x \<and> R x \<longrightarrow> P x))"
```
```   231
```
```   232 definition
```
```   233   "Sup S = Abs_filter (\<lambda>P. \<forall>F\<in>S. eventually P F)"
```
```   234
```
```   235 definition
```
```   236   "Inf S = Sup {F::'a filter. \<forall>F'\<in>S. F \<le> F'}"
```
```   237
```
```   238 lemma eventually_top [simp]: "eventually P top \<longleftrightarrow> (\<forall>x. P x)"
```
```   239   unfolding top_filter_def
```
```   240   by (rule eventually_Abs_filter, rule is_filter.intro, auto)
```
```   241
```
```   242 lemma eventually_bot [simp]: "eventually P bot"
```
```   243   unfolding bot_filter_def
```
```   244   by (subst eventually_Abs_filter, rule is_filter.intro, auto)
```
```   245
```
```   246 lemma eventually_sup:
```
```   247   "eventually P (sup F F') \<longleftrightarrow> eventually P F \<and> eventually P F'"
```
```   248   unfolding sup_filter_def
```
```   249   by (rule eventually_Abs_filter, rule is_filter.intro)
```
```   250      (auto elim!: eventually_rev_mp)
```
```   251
```
```   252 lemma eventually_inf:
```
```   253   "eventually P (inf F F') \<longleftrightarrow>
```
```   254    (\<exists>Q R. eventually Q F \<and> eventually R F' \<and> (\<forall>x. Q x \<and> R x \<longrightarrow> P x))"
```
```   255   unfolding inf_filter_def
```
```   256   apply (rule eventually_Abs_filter, rule is_filter.intro)
```
```   257   apply (fast intro: eventually_True)
```
```   258   apply clarify
```
```   259   apply (intro exI conjI)
```
```   260   apply (erule (1) eventually_conj)
```
```   261   apply (erule (1) eventually_conj)
```
```   262   apply simp
```
```   263   apply auto
```
```   264   done
```
```   265
```
```   266 lemma eventually_Sup:
```
```   267   "eventually P (Sup S) \<longleftrightarrow> (\<forall>F\<in>S. eventually P F)"
```
```   268   unfolding Sup_filter_def
```
```   269   apply (rule eventually_Abs_filter, rule is_filter.intro)
```
```   270   apply (auto intro: eventually_conj elim!: eventually_rev_mp)
```
```   271   done
```
```   272
```
```   273 instance proof
```
```   274   fix F F' F'' :: "'a filter" and S :: "'a filter set"
```
```   275   { show "F < F' \<longleftrightarrow> F \<le> F' \<and> \<not> F' \<le> F"
```
```   276     by (rule less_filter_def) }
```
```   277   { show "F \<le> F"
```
```   278     unfolding le_filter_def by simp }
```
```   279   { assume "F \<le> F'" and "F' \<le> F''" thus "F \<le> F''"
```
```   280     unfolding le_filter_def by simp }
```
```   281   { assume "F \<le> F'" and "F' \<le> F" thus "F = F'"
```
```   282     unfolding le_filter_def filter_eq_iff by fast }
```
```   283   { show "inf F F' \<le> F" and "inf F F' \<le> F'"
```
```   284     unfolding le_filter_def eventually_inf by (auto intro: eventually_True) }
```
```   285   { assume "F \<le> F'" and "F \<le> F''" thus "F \<le> inf F' F''"
```
```   286     unfolding le_filter_def eventually_inf
```
```   287     by (auto elim!: eventually_mono intro: eventually_conj) }
```
```   288   { show "F \<le> sup F F'" and "F' \<le> sup F F'"
```
```   289     unfolding le_filter_def eventually_sup by simp_all }
```
```   290   { assume "F \<le> F''" and "F' \<le> F''" thus "sup F F' \<le> F''"
```
```   291     unfolding le_filter_def eventually_sup by simp }
```
```   292   { assume "F'' \<in> S" thus "Inf S \<le> F''"
```
```   293     unfolding le_filter_def Inf_filter_def eventually_Sup Ball_def by simp }
```
```   294   { assume "\<And>F'. F' \<in> S \<Longrightarrow> F \<le> F'" thus "F \<le> Inf S"
```
```   295     unfolding le_filter_def Inf_filter_def eventually_Sup Ball_def by simp }
```
```   296   { assume "F \<in> S" thus "F \<le> Sup S"
```
```   297     unfolding le_filter_def eventually_Sup by simp }
```
```   298   { assume "\<And>F. F \<in> S \<Longrightarrow> F \<le> F'" thus "Sup S \<le> F'"
```
```   299     unfolding le_filter_def eventually_Sup by simp }
```
```   300   { show "Inf {} = (top::'a filter)"
```
```   301     by (auto simp: top_filter_def Inf_filter_def Sup_filter_def)
```
```   302       (metis (full_types) top_filter_def always_eventually eventually_top) }
```
```   303   { show "Sup {} = (bot::'a filter)"
```
```   304     by (auto simp: bot_filter_def Sup_filter_def) }
```
```   305 qed
```
```   306
```
```   307 end
```
```   308
```
```   309 lemma filter_leD:
```
```   310   "F \<le> F' \<Longrightarrow> eventually P F' \<Longrightarrow> eventually P F"
```
```   311   unfolding le_filter_def by simp
```
```   312
```
```   313 lemma filter_leI:
```
```   314   "(\<And>P. eventually P F' \<Longrightarrow> eventually P F) \<Longrightarrow> F \<le> F'"
```
```   315   unfolding le_filter_def by simp
```
```   316
```
```   317 lemma eventually_False:
```
```   318   "eventually (\<lambda>x. False) F \<longleftrightarrow> F = bot"
```
```   319   unfolding filter_eq_iff by (auto elim: eventually_rev_mp)
```
```   320
```
```   321 abbreviation (input) trivial_limit :: "'a filter \<Rightarrow> bool"
```
```   322   where "trivial_limit F \<equiv> F = bot"
```
```   323
```
```   324 lemma trivial_limit_def: "trivial_limit F \<longleftrightarrow> eventually (\<lambda>x. False) F"
```
```   325   by (rule eventually_False [symmetric])
```
```   326
```
```   327 lemma eventually_const: "\<not> trivial_limit net \<Longrightarrow> eventually (\<lambda>x. P) net \<longleftrightarrow> P"
```
```   328   by (cases P) (simp_all add: eventually_False)
```
```   329
```
```   330 lemma eventually_Inf: "eventually P (Inf B) \<longleftrightarrow> (\<exists>X\<subseteq>B. finite X \<and> eventually P (Inf X))"
```
```   331 proof -
```
```   332   let ?F = "\<lambda>P. \<exists>X\<subseteq>B. finite X \<and> eventually P (Inf X)"
```
```   333
```
```   334   { fix P have "eventually P (Abs_filter ?F) \<longleftrightarrow> ?F P"
```
```   335     proof (rule eventually_Abs_filter is_filter.intro)+
```
```   336       show "?F (\<lambda>x. True)"
```
```   337         by (rule exI[of _ "{}"]) (simp add: le_fun_def)
```
```   338     next
```
```   339       fix P Q
```
```   340       assume "?F P" then guess X ..
```
```   341       moreover
```
```   342       assume "?F Q" then guess Y ..
```
```   343       ultimately show "?F (\<lambda>x. P x \<and> Q x)"
```
```   344         by (intro exI[of _ "X \<union> Y"])
```
```   345            (auto simp: Inf_union_distrib eventually_inf)
```
```   346     next
```
```   347       fix P Q
```
```   348       assume "?F P" then guess X ..
```
```   349       moreover assume "\<forall>x. P x \<longrightarrow> Q x"
```
```   350       ultimately show "?F Q"
```
```   351         by (intro exI[of _ X]) (auto elim: eventually_elim1)
```
```   352     qed }
```
```   353   note eventually_F = this
```
```   354
```
```   355   have "Inf B = Abs_filter ?F"
```
```   356   proof (intro antisym Inf_greatest)
```
```   357     show "Inf B \<le> Abs_filter ?F"
```
```   358       by (auto simp: le_filter_def eventually_F dest: Inf_superset_mono)
```
```   359   next
```
```   360     fix F assume "F \<in> B" then show "Abs_filter ?F \<le> F"
```
```   361       by (auto simp add: le_filter_def eventually_F intro!: exI[of _ "{F}"])
```
```   362   qed
```
```   363   then show ?thesis
```
```   364     by (simp add: eventually_F)
```
```   365 qed
```
```   366
```
```   367 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))"
```
```   368   unfolding INF_def[of B] eventually_Inf[of P "F`B"]
```
```   369   by (metis Inf_image_eq finite_imageI image_mono finite_subset_image)
```
```   370
```
```   371 lemma Inf_filter_not_bot:
```
```   372   fixes B :: "'a filter set"
```
```   373   shows "(\<And>X. X \<subseteq> B \<Longrightarrow> finite X \<Longrightarrow> Inf X \<noteq> bot) \<Longrightarrow> Inf B \<noteq> bot"
```
```   374   unfolding trivial_limit_def eventually_Inf[of _ B]
```
```   375     bot_bool_def [symmetric] bot_fun_def [symmetric] bot_unique by simp
```
```   376
```
```   377 lemma INF_filter_not_bot:
```
```   378   fixes F :: "'i \<Rightarrow> 'a filter"
```
```   379   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"
```
```   380   unfolding trivial_limit_def eventually_INF[of _ B]
```
```   381     bot_bool_def [symmetric] bot_fun_def [symmetric] bot_unique by simp
```
```   382
```
```   383 lemma eventually_Inf_base:
```
```   384   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"
```
```   385   shows "eventually P (Inf B) \<longleftrightarrow> (\<exists>b\<in>B. eventually P b)"
```
```   386 proof (subst eventually_Inf, safe)
```
```   387   fix X assume "finite X" "X \<subseteq> B"
```
```   388   then have "\<exists>b\<in>B. \<forall>x\<in>X. b \<le> x"
```
```   389   proof induct
```
```   390     case empty then show ?case
```
```   391       using `B \<noteq> {}` by auto
```
```   392   next
```
```   393     case (insert x X)
```
```   394     then obtain b where "b \<in> B" "\<And>x. x \<in> X \<Longrightarrow> b \<le> x"
```
```   395       by auto
```
```   396     with `insert x X \<subseteq> B` base[of b x] show ?case
```
```   397       by (auto intro: order_trans)
```
```   398   qed
```
```   399   then obtain b where "b \<in> B" "b \<le> Inf X"
```
```   400     by (auto simp: le_Inf_iff)
```
```   401   then show "eventually P (Inf X) \<Longrightarrow> Bex B (eventually P)"
```
```   402     by (intro bexI[of _ b]) (auto simp: le_filter_def)
```
```   403 qed (auto intro!: exI[of _ "{x}" for x])
```
```   404
```
```   405 lemma eventually_INF_base:
```
```   406   "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>
```
```   407     eventually P (INF b:B. F b) \<longleftrightarrow> (\<exists>b\<in>B. eventually P (F b))"
```
```   408   unfolding INF_def by (subst eventually_Inf_base) auto
```
```   409
```
```   410
```
```   411 subsubsection {* Map function for filters *}
```
```   412
```
```   413 definition filtermap :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a filter \<Rightarrow> 'b filter"
```
```   414   where "filtermap f F = Abs_filter (\<lambda>P. eventually (\<lambda>x. P (f x)) F)"
```
```   415
```
```   416 lemma eventually_filtermap:
```
```   417   "eventually P (filtermap f F) = eventually (\<lambda>x. P (f x)) F"
```
```   418   unfolding filtermap_def
```
```   419   apply (rule eventually_Abs_filter)
```
```   420   apply (rule is_filter.intro)
```
```   421   apply (auto elim!: eventually_rev_mp)
```
```   422   done
```
```   423
```
```   424 lemma filtermap_ident: "filtermap (\<lambda>x. x) F = F"
```
```   425   by (simp add: filter_eq_iff eventually_filtermap)
```
```   426
```
```   427 lemma filtermap_filtermap:
```
```   428   "filtermap f (filtermap g F) = filtermap (\<lambda>x. f (g x)) F"
```
```   429   by (simp add: filter_eq_iff eventually_filtermap)
```
```   430
```
```   431 lemma filtermap_mono: "F \<le> F' \<Longrightarrow> filtermap f F \<le> filtermap f F'"
```
```   432   unfolding le_filter_def eventually_filtermap by simp
```
```   433
```
```   434 lemma filtermap_bot [simp]: "filtermap f bot = bot"
```
```   435   by (simp add: filter_eq_iff eventually_filtermap)
```
```   436
```
```   437 lemma filtermap_sup: "filtermap f (sup F1 F2) = sup (filtermap f F1) (filtermap f F2)"
```
```   438   by (auto simp: filter_eq_iff eventually_filtermap eventually_sup)
```
```   439
```
```   440 lemma filtermap_inf: "filtermap f (inf F1 F2) \<le> inf (filtermap f F1) (filtermap f F2)"
```
```   441   by (auto simp: le_filter_def eventually_filtermap eventually_inf)
```
```   442
```
```   443 lemma filtermap_INF: "filtermap f (INF b:B. F b) \<le> (INF b:B. filtermap f (F b))"
```
```   444 proof -
```
```   445   { fix X :: "'c set" assume "finite X"
```
```   446     then have "filtermap f (INFIMUM X F) \<le> (INF b:X. filtermap f (F b))"
```
```   447     proof induct
```
```   448       case (insert x X)
```
```   449       have "filtermap f (INF a:insert x X. F a) \<le> inf (filtermap f (F x)) (filtermap f (INF a:X. F a))"
```
```   450         by (rule order_trans[OF _ filtermap_inf]) simp
```
```   451       also have "\<dots> \<le> inf (filtermap f (F x)) (INF a:X. filtermap f (F a))"
```
```   452         by (intro inf_mono insert order_refl)
```
```   453       finally show ?case
```
```   454         by simp
```
```   455     qed simp }
```
```   456   then show ?thesis
```
```   457     unfolding le_filter_def eventually_filtermap
```
```   458     by (subst (1 2) eventually_INF) auto
```
```   459 qed
```
```   460 subsubsection {* Standard filters *}
```
```   461
```
```   462 definition principal :: "'a set \<Rightarrow> 'a filter" where
```
```   463   "principal S = Abs_filter (\<lambda>P. \<forall>x\<in>S. P x)"
```
```   464
```
```   465 lemma eventually_principal: "eventually P (principal S) \<longleftrightarrow> (\<forall>x\<in>S. P x)"
```
```   466   unfolding principal_def
```
```   467   by (rule eventually_Abs_filter, rule is_filter.intro) auto
```
```   468
```
```   469 lemma eventually_inf_principal: "eventually P (inf F (principal s)) \<longleftrightarrow> eventually (\<lambda>x. x \<in> s \<longrightarrow> P x) F"
```
```   470   unfolding eventually_inf eventually_principal by (auto elim: eventually_elim1)
```
```   471
```
```   472 lemma principal_UNIV[simp]: "principal UNIV = top"
```
```   473   by (auto simp: filter_eq_iff eventually_principal)
```
```   474
```
```   475 lemma principal_empty[simp]: "principal {} = bot"
```
```   476   by (auto simp: filter_eq_iff eventually_principal)
```
```   477
```
```   478 lemma principal_eq_bot_iff: "principal X = bot \<longleftrightarrow> X = {}"
```
```   479   by (auto simp add: filter_eq_iff eventually_principal)
```
```   480
```
```   481 lemma principal_le_iff[iff]: "principal A \<le> principal B \<longleftrightarrow> A \<subseteq> B"
```
```   482   by (auto simp: le_filter_def eventually_principal)
```
```   483
```
```   484 lemma le_principal: "F \<le> principal A \<longleftrightarrow> eventually (\<lambda>x. x \<in> A) F"
```
```   485   unfolding le_filter_def eventually_principal
```
```   486   apply safe
```
```   487   apply (erule_tac x="\<lambda>x. x \<in> A" in allE)
```
```   488   apply (auto elim: eventually_elim1)
```
```   489   done
```
```   490
```
```   491 lemma principal_inject[iff]: "principal A = principal B \<longleftrightarrow> A = B"
```
```   492   unfolding eq_iff by simp
```
```   493
```
```   494 lemma sup_principal[simp]: "sup (principal A) (principal B) = principal (A \<union> B)"
```
```   495   unfolding filter_eq_iff eventually_sup eventually_principal by auto
```
```   496
```
```   497 lemma inf_principal[simp]: "inf (principal A) (principal B) = principal (A \<inter> B)"
```
```   498   unfolding filter_eq_iff eventually_inf eventually_principal
```
```   499   by (auto intro: exI[of _ "\<lambda>x. x \<in> A"] exI[of _ "\<lambda>x. x \<in> B"])
```
```   500
```
```   501 lemma SUP_principal[simp]: "(SUP i : I. principal (A i)) = principal (\<Union>i\<in>I. A i)"
```
```   502   unfolding filter_eq_iff eventually_Sup SUP_def by (auto simp: eventually_principal)
```
```   503
```
```   504 lemma INF_principal_finite: "finite X \<Longrightarrow> (INF x:X. principal (f x)) = principal (\<Inter>x\<in>X. f x)"
```
```   505   by (induct X rule: finite_induct) auto
```
```   506
```
```   507 lemma filtermap_principal[simp]: "filtermap f (principal A) = principal (f ` A)"
```
```   508   unfolding filter_eq_iff eventually_filtermap eventually_principal by simp
```
```   509
```
```   510 subsubsection {* Order filters *}
```
```   511
```
```   512 definition at_top :: "('a::order) filter"
```
```   513   where "at_top = (INF k. principal {k ..})"
```
```   514
```
```   515 lemma at_top_sub: "at_top = (INF k:{c::'a::linorder..}. principal {k ..})"
```
```   516   by (auto intro!: INF_eq max.cobounded1 max.cobounded2 simp: at_top_def)
```
```   517
```
```   518 lemma eventually_at_top_linorder: "eventually P at_top \<longleftrightarrow> (\<exists>N::'a::linorder. \<forall>n\<ge>N. P n)"
```
```   519   unfolding at_top_def
```
```   520   by (subst eventually_INF_base) (auto simp: eventually_principal intro: max.cobounded1 max.cobounded2)
```
```   521
```
```   522 lemma eventually_ge_at_top:
```
```   523   "eventually (\<lambda>x. (c::_::linorder) \<le> x) at_top"
```
```   524   unfolding eventually_at_top_linorder by auto
```
```   525
```
```   526 lemma eventually_at_top_dense: "eventually P at_top \<longleftrightarrow> (\<exists>N::'a::{no_top, linorder}. \<forall>n>N. P n)"
```
```   527 proof -
```
```   528   have "eventually P (INF k. principal {k <..}) \<longleftrightarrow> (\<exists>N::'a. \<forall>n>N. P n)"
```
```   529     by (subst eventually_INF_base) (auto simp: eventually_principal intro: max.cobounded1 max.cobounded2)
```
```   530   also have "(INF k. principal {k::'a <..}) = at_top"
```
```   531     unfolding at_top_def
```
```   532     by (intro INF_eq) (auto intro: less_imp_le simp: Ici_subset_Ioi_iff gt_ex)
```
```   533   finally show ?thesis .
```
```   534 qed
```
```   535
```
```   536 lemma eventually_gt_at_top:
```
```   537   "eventually (\<lambda>x. (c::_::unbounded_dense_linorder) < x) at_top"
```
```   538   unfolding eventually_at_top_dense by auto
```
```   539
```
```   540 definition at_bot :: "('a::order) filter"
```
```   541   where "at_bot = (INF k. principal {.. k})"
```
```   542
```
```   543 lemma at_bot_sub: "at_bot = (INF k:{.. c::'a::linorder}. principal {.. k})"
```
```   544   by (auto intro!: INF_eq min.cobounded1 min.cobounded2 simp: at_bot_def)
```
```   545
```
```   546 lemma eventually_at_bot_linorder:
```
```   547   fixes P :: "'a::linorder \<Rightarrow> bool" shows "eventually P at_bot \<longleftrightarrow> (\<exists>N. \<forall>n\<le>N. P n)"
```
```   548   unfolding at_bot_def
```
```   549   by (subst eventually_INF_base) (auto simp: eventually_principal intro: min.cobounded1 min.cobounded2)
```
```   550
```
```   551 lemma eventually_le_at_bot:
```
```   552   "eventually (\<lambda>x. x \<le> (c::_::linorder)) at_bot"
```
```   553   unfolding eventually_at_bot_linorder by auto
```
```   554
```
```   555 lemma eventually_at_bot_dense: "eventually P at_bot \<longleftrightarrow> (\<exists>N::'a::{no_bot, linorder}. \<forall>n<N. P n)"
```
```   556 proof -
```
```   557   have "eventually P (INF k. principal {..< k}) \<longleftrightarrow> (\<exists>N::'a. \<forall>n<N. P n)"
```
```   558     by (subst eventually_INF_base) (auto simp: eventually_principal intro: min.cobounded1 min.cobounded2)
```
```   559   also have "(INF k. principal {..< k::'a}) = at_bot"
```
```   560     unfolding at_bot_def
```
```   561     by (intro INF_eq) (auto intro: less_imp_le simp: Iic_subset_Iio_iff lt_ex)
```
```   562   finally show ?thesis .
```
```   563 qed
```
```   564
```
```   565 lemma eventually_gt_at_bot:
```
```   566   "eventually (\<lambda>x. x < (c::_::unbounded_dense_linorder)) at_bot"
```
```   567   unfolding eventually_at_bot_dense by auto
```
```   568
```
```   569 lemma trivial_limit_at_bot_linorder: "\<not> trivial_limit (at_bot ::('a::linorder) filter)"
```
```   570   unfolding trivial_limit_def
```
```   571   by (metis eventually_at_bot_linorder order_refl)
```
```   572
```
```   573 lemma trivial_limit_at_top_linorder: "\<not> trivial_limit (at_top ::('a::linorder) filter)"
```
```   574   unfolding trivial_limit_def
```
```   575   by (metis eventually_at_top_linorder order_refl)
```
```   576
```
```   577 subsection {* Sequentially *}
```
```   578
```
```   579 abbreviation sequentially :: "nat filter"
```
```   580   where "sequentially \<equiv> at_top"
```
```   581
```
```   582 lemma eventually_sequentially:
```
```   583   "eventually P sequentially \<longleftrightarrow> (\<exists>N. \<forall>n\<ge>N. P n)"
```
```   584   by (rule eventually_at_top_linorder)
```
```   585
```
```   586 lemma sequentially_bot [simp, intro]: "sequentially \<noteq> bot"
```
```   587   unfolding filter_eq_iff eventually_sequentially by auto
```
```   588
```
```   589 lemmas trivial_limit_sequentially = sequentially_bot
```
```   590
```
```   591 lemma eventually_False_sequentially [simp]:
```
```   592   "\<not> eventually (\<lambda>n. False) sequentially"
```
```   593   by (simp add: eventually_False)
```
```   594
```
```   595 lemma le_sequentially:
```
```   596   "F \<le> sequentially \<longleftrightarrow> (\<forall>N. eventually (\<lambda>n. N \<le> n) F)"
```
```   597   by (simp add: at_top_def le_INF_iff le_principal)
```
```   598
```
```   599 lemma eventually_sequentiallyI:
```
```   600   assumes "\<And>x. c \<le> x \<Longrightarrow> P x"
```
```   601   shows "eventually P sequentially"
```
```   602 using assms by (auto simp: eventually_sequentially)
```
```   603
```
```   604 lemma eventually_sequentially_seg:
```
```   605   "eventually (\<lambda>n. P (n + k)) sequentially \<longleftrightarrow> eventually P sequentially"
```
```   606   unfolding eventually_sequentially
```
```   607   apply safe
```
```   608    apply (rule_tac x="N + k" in exI)
```
```   609    apply rule
```
```   610    apply (erule_tac x="n - k" in allE)
```
```   611    apply auto []
```
```   612   apply (rule_tac x=N in exI)
```
```   613   apply auto []
```
```   614   done
```
```   615
```
```   616 subsection \<open> The cofinite filter \<close>
```
```   617
```
```   618 definition "cofinite = Abs_filter (\<lambda>P. finite {x. \<not> P x})"
```
```   619
```
```   620 lemma eventually_cofinite: "eventually P cofinite \<longleftrightarrow> finite {x. \<not> P x}"
```
```   621   unfolding cofinite_def
```
```   622 proof (rule eventually_Abs_filter, rule is_filter.intro)
```
```   623   fix P Q :: "'a \<Rightarrow> bool" assume "finite {x. \<not> P x}" "finite {x. \<not> Q x}"
```
```   624   from finite_UnI[OF this] show "finite {x. \<not> (P x \<and> Q x)}"
```
```   625     by (rule rev_finite_subset) auto
```
```   626 next
```
```   627   fix P Q :: "'a \<Rightarrow> bool" assume P: "finite {x. \<not> P x}" and *: "\<forall>x. P x \<longrightarrow> Q x"
```
```   628   from * show "finite {x. \<not> Q x}"
```
```   629     by (intro finite_subset[OF _ P]) auto
```
```   630 qed simp
```
```   631
```
```   632 lemma cofinite_bot[simp]: "cofinite = (bot::'a filter) \<longleftrightarrow> finite (UNIV :: 'a set)"
```
```   633   unfolding trivial_limit_def eventually_cofinite by simp
```
```   634
```
```   635 lemma cofinite_eq_sequentially: "cofinite = sequentially"
```
```   636   unfolding filter_eq_iff eventually_sequentially eventually_cofinite
```
```   637 proof safe
```
```   638   fix P :: "nat \<Rightarrow> bool" assume [simp]: "finite {x. \<not> P x}"
```
```   639   show "\<exists>N. \<forall>n\<ge>N. P n"
```
```   640   proof cases
```
```   641     assume "{x. \<not> P x} \<noteq> {}" then show ?thesis
```
```   642       by (intro exI[of _ "Suc (Max {x. \<not> P x})"]) (auto simp: Suc_le_eq)
```
```   643   qed auto
```
```   644 next
```
```   645   fix P :: "nat \<Rightarrow> bool" and N :: nat assume "\<forall>n\<ge>N. P n"
```
```   646   then have "{x. \<not> P x} \<subseteq> {..< N}"
```
```   647     by (auto simp: not_le)
```
```   648   then show "finite {x. \<not> P x}"
```
```   649     by (blast intro: finite_subset)
```
```   650 qed
```
```   651
```
```   652 subsection {* Limits *}
```
```   653
```
```   654 definition filterlim :: "('a \<Rightarrow> 'b) \<Rightarrow> 'b filter \<Rightarrow> 'a filter \<Rightarrow> bool" where
```
```   655   "filterlim f F2 F1 \<longleftrightarrow> filtermap f F1 \<le> F2"
```
```   656
```
```   657 syntax
```
```   658   "_LIM" :: "pttrns \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> bool" ("(3LIM (_)/ (_)./ (_) :> (_))" [1000, 10, 0, 10] 10)
```
```   659
```
```   660 translations
```
```   661   "LIM x F1. f :> F2"   == "CONST filterlim (%x. f) F2 F1"
```
```   662
```
```   663 lemma filterlim_iff:
```
```   664   "(LIM x F1. f x :> F2) \<longleftrightarrow> (\<forall>P. eventually P F2 \<longrightarrow> eventually (\<lambda>x. P (f x)) F1)"
```
```   665   unfolding filterlim_def le_filter_def eventually_filtermap ..
```
```   666
```
```   667 lemma filterlim_compose:
```
```   668   "filterlim g F3 F2 \<Longrightarrow> filterlim f F2 F1 \<Longrightarrow> filterlim (\<lambda>x. g (f x)) F3 F1"
```
```   669   unfolding filterlim_def filtermap_filtermap[symmetric] by (metis filtermap_mono order_trans)
```
```   670
```
```   671 lemma filterlim_mono:
```
```   672   "filterlim f F2 F1 \<Longrightarrow> F2 \<le> F2' \<Longrightarrow> F1' \<le> F1 \<Longrightarrow> filterlim f F2' F1'"
```
```   673   unfolding filterlim_def by (metis filtermap_mono order_trans)
```
```   674
```
```   675 lemma filterlim_ident: "LIM x F. x :> F"
```
```   676   by (simp add: filterlim_def filtermap_ident)
```
```   677
```
```   678 lemma filterlim_cong:
```
```   679   "F1 = F1' \<Longrightarrow> F2 = F2' \<Longrightarrow> eventually (\<lambda>x. f x = g x) F2 \<Longrightarrow> filterlim f F1 F2 = filterlim g F1' F2'"
```
```   680   by (auto simp: filterlim_def le_filter_def eventually_filtermap elim: eventually_elim2)
```
```   681
```
```   682 lemma filterlim_mono_eventually:
```
```   683   assumes "filterlim f F G" and ord: "F \<le> F'" "G' \<le> G"
```
```   684   assumes eq: "eventually (\<lambda>x. f x = f' x) G'"
```
```   685   shows "filterlim f' F' G'"
```
```   686   apply (rule filterlim_cong[OF refl refl eq, THEN iffD1])
```
```   687   apply (rule filterlim_mono[OF _ ord])
```
```   688   apply fact
```
```   689   done
```
```   690
```
```   691 lemma filtermap_mono_strong: "inj f \<Longrightarrow> filtermap f F \<le> filtermap f G \<longleftrightarrow> F \<le> G"
```
```   692   apply (auto intro!: filtermap_mono) []
```
```   693   apply (auto simp: le_filter_def eventually_filtermap)
```
```   694   apply (erule_tac x="\<lambda>x. P (inv f x)" in allE)
```
```   695   apply auto
```
```   696   done
```
```   697
```
```   698 lemma filtermap_eq_strong: "inj f \<Longrightarrow> filtermap f F = filtermap f G \<longleftrightarrow> F = G"
```
```   699   by (simp add: filtermap_mono_strong eq_iff)
```
```   700
```
```   701 lemma filterlim_principal:
```
```   702   "(LIM x F. f x :> principal S) \<longleftrightarrow> (eventually (\<lambda>x. f x \<in> S) F)"
```
```   703   unfolding filterlim_def eventually_filtermap le_principal ..
```
```   704
```
```   705 lemma filterlim_inf:
```
```   706   "(LIM x F1. f x :> inf F2 F3) \<longleftrightarrow> ((LIM x F1. f x :> F2) \<and> (LIM x F1. f x :> F3))"
```
```   707   unfolding filterlim_def by simp
```
```   708
```
```   709 lemma filterlim_INF:
```
```   710   "(LIM x F. f x :> (INF b:B. G b)) \<longleftrightarrow> (\<forall>b\<in>B. LIM x F. f x :> G b)"
```
```   711   unfolding filterlim_def le_INF_iff ..
```
```   712
```
```   713 lemma filterlim_INF_INF:
```
```   714   "(\<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)"
```
```   715   unfolding filterlim_def by (rule order_trans[OF filtermap_INF INF_mono])
```
```   716
```
```   717 lemma filterlim_base:
```
```   718   "(\<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>
```
```   719     LIM x (INF i:I. principal (F i)). f x :> (INF j:J. principal (G j))"
```
```   720   by (force intro!: filterlim_INF_INF simp: image_subset_iff)
```
```   721
```
```   722 lemma filterlim_base_iff:
```
```   723   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"
```
```   724   shows "(LIM x (INF i:I. principal (F i)). f x :> INF j:J. principal (G j)) \<longleftrightarrow>
```
```   725     (\<forall>j\<in>J. \<exists>i\<in>I. \<forall>x\<in>F i. f x \<in> G j)"
```
```   726   unfolding filterlim_INF filterlim_principal
```
```   727 proof (subst eventually_INF_base)
```
```   728   fix i j assume "i \<in> I" "j \<in> I"
```
```   729   with chain[OF this] show "\<exists>x\<in>I. principal (F x) \<le> inf (principal (F i)) (principal (F j))"
```
```   730     by auto
```
```   731 qed (auto simp: eventually_principal `I \<noteq> {}`)
```
```   732
```
```   733 lemma filterlim_filtermap: "filterlim f F1 (filtermap g F2) = filterlim (\<lambda>x. f (g x)) F1 F2"
```
```   734   unfolding filterlim_def filtermap_filtermap ..
```
```   735
```
```   736 lemma filterlim_sup:
```
```   737   "filterlim f F F1 \<Longrightarrow> filterlim f F F2 \<Longrightarrow> filterlim f F (sup F1 F2)"
```
```   738   unfolding filterlim_def filtermap_sup by auto
```
```   739
```
```   740 lemma eventually_sequentially_Suc: "eventually (\<lambda>i. P (Suc i)) sequentially \<longleftrightarrow> eventually P sequentially"
```
```   741   unfolding eventually_sequentially by (metis Suc_le_D Suc_le_mono le_Suc_eq)
```
```   742
```
```   743 lemma filterlim_sequentially_Suc:
```
```   744   "(LIM x sequentially. f (Suc x) :> F) \<longleftrightarrow> (LIM x sequentially. f x :> F)"
```
```   745   unfolding filterlim_iff by (subst eventually_sequentially_Suc) simp
```
```   746
```
```   747 lemma filterlim_Suc: "filterlim Suc sequentially sequentially"
```
```   748   by (simp add: filterlim_iff eventually_sequentially) (metis le_Suc_eq)
```
```   749
```
```   750
```
```   751 subsection {* Limits to @{const at_top} and @{const at_bot} *}
```
```   752
```
```   753 lemma filterlim_at_top:
```
```   754   fixes f :: "'a \<Rightarrow> ('b::linorder)"
```
```   755   shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. Z \<le> f x) F)"
```
```   756   by (auto simp: filterlim_iff eventually_at_top_linorder elim!: eventually_elim1)
```
```   757
```
```   758 lemma filterlim_at_top_mono:
```
```   759   "LIM x F. f x :> at_top \<Longrightarrow> eventually (\<lambda>x. f x \<le> (g x::'a::linorder)) F \<Longrightarrow>
```
```   760     LIM x F. g x :> at_top"
```
```   761   by (auto simp: filterlim_at_top elim: eventually_elim2 intro: order_trans)
```
```   762
```
```   763 lemma filterlim_at_top_dense:
```
```   764   fixes f :: "'a \<Rightarrow> ('b::unbounded_dense_linorder)"
```
```   765   shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. Z < f x) F)"
```
```   766   by (metis eventually_elim1[of _ F] eventually_gt_at_top order_less_imp_le
```
```   767             filterlim_at_top[of f F] filterlim_iff[of f at_top F])
```
```   768
```
```   769 lemma filterlim_at_top_ge:
```
```   770   fixes f :: "'a \<Rightarrow> ('b::linorder)" and c :: "'b"
```
```   771   shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z\<ge>c. eventually (\<lambda>x. Z \<le> f x) F)"
```
```   772   unfolding at_top_sub[of c] filterlim_INF by (auto simp add: filterlim_principal)
```
```   773
```
```   774 lemma filterlim_at_top_at_top:
```
```   775   fixes f :: "'a::linorder \<Rightarrow> 'b::linorder"
```
```   776   assumes mono: "\<And>x y. Q x \<Longrightarrow> Q y \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y"
```
```   777   assumes bij: "\<And>x. P x \<Longrightarrow> f (g x) = x" "\<And>x. P x \<Longrightarrow> Q (g x)"
```
```   778   assumes Q: "eventually Q at_top"
```
```   779   assumes P: "eventually P at_top"
```
```   780   shows "filterlim f at_top at_top"
```
```   781 proof -
```
```   782   from P obtain x where x: "\<And>y. x \<le> y \<Longrightarrow> P y"
```
```   783     unfolding eventually_at_top_linorder by auto
```
```   784   show ?thesis
```
```   785   proof (intro filterlim_at_top_ge[THEN iffD2] allI impI)
```
```   786     fix z assume "x \<le> z"
```
```   787     with x have "P z" by auto
```
```   788     have "eventually (\<lambda>x. g z \<le> x) at_top"
```
```   789       by (rule eventually_ge_at_top)
```
```   790     with Q show "eventually (\<lambda>x. z \<le> f x) at_top"
```
```   791       by eventually_elim (metis mono bij `P z`)
```
```   792   qed
```
```   793 qed
```
```   794
```
```   795 lemma filterlim_at_top_gt:
```
```   796   fixes f :: "'a \<Rightarrow> ('b::unbounded_dense_linorder)" and c :: "'b"
```
```   797   shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z>c. eventually (\<lambda>x. Z \<le> f x) F)"
```
```   798   by (metis filterlim_at_top order_less_le_trans gt_ex filterlim_at_top_ge)
```
```   799
```
```   800 lemma filterlim_at_bot:
```
```   801   fixes f :: "'a \<Rightarrow> ('b::linorder)"
```
```   802   shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. f x \<le> Z) F)"
```
```   803   by (auto simp: filterlim_iff eventually_at_bot_linorder elim!: eventually_elim1)
```
```   804
```
```   805 lemma filterlim_at_bot_dense:
```
```   806   fixes f :: "'a \<Rightarrow> ('b::{dense_linorder, no_bot})"
```
```   807   shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. f x < Z) F)"
```
```   808 proof (auto simp add: filterlim_at_bot[of f F])
```
```   809   fix Z :: 'b
```
```   810   from lt_ex [of Z] obtain Z' where 1: "Z' < Z" ..
```
```   811   assume "\<forall>Z. eventually (\<lambda>x. f x \<le> Z) F"
```
```   812   hence "eventually (\<lambda>x. f x \<le> Z') F" by auto
```
```   813   thus "eventually (\<lambda>x. f x < Z) F"
```
```   814     apply (rule eventually_mono[rotated])
```
```   815     using 1 by auto
```
```   816   next
```
```   817     fix Z :: 'b
```
```   818     show "\<forall>Z. eventually (\<lambda>x. f x < Z) F \<Longrightarrow> eventually (\<lambda>x. f x \<le> Z) F"
```
```   819       by (drule spec [of _ Z], erule eventually_mono[rotated], auto simp add: less_imp_le)
```
```   820 qed
```
```   821
```
```   822 lemma filterlim_at_bot_le:
```
```   823   fixes f :: "'a \<Rightarrow> ('b::linorder)" and c :: "'b"
```
```   824   shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z\<le>c. eventually (\<lambda>x. Z \<ge> f x) F)"
```
```   825   unfolding filterlim_at_bot
```
```   826 proof safe
```
```   827   fix Z assume *: "\<forall>Z\<le>c. eventually (\<lambda>x. Z \<ge> f x) F"
```
```   828   with *[THEN spec, of "min Z c"] show "eventually (\<lambda>x. Z \<ge> f x) F"
```
```   829     by (auto elim!: eventually_elim1)
```
```   830 qed simp
```
```   831
```
```   832 lemma filterlim_at_bot_lt:
```
```   833   fixes f :: "'a \<Rightarrow> ('b::unbounded_dense_linorder)" and c :: "'b"
```
```   834   shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z<c. eventually (\<lambda>x. Z \<ge> f x) F)"
```
```   835   by (metis filterlim_at_bot filterlim_at_bot_le lt_ex order_le_less_trans)
```
```   836
```
```   837
```
```   838 subsection {* Setup @{typ "'a filter"} for lifting and transfer *}
```
```   839
```
```   840 context begin interpretation lifting_syntax .
```
```   841
```
```   842 definition rel_filter :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a filter \<Rightarrow> 'b filter \<Rightarrow> bool"
```
```   843 where "rel_filter R F G = ((R ===> op =) ===> op =) (Rep_filter F) (Rep_filter G)"
```
```   844
```
```   845 lemma rel_filter_eventually:
```
```   846   "rel_filter R F G \<longleftrightarrow>
```
```   847   ((R ===> op =) ===> op =) (\<lambda>P. eventually P F) (\<lambda>P. eventually P G)"
```
```   848 by(simp add: rel_filter_def eventually_def)
```
```   849
```
```   850 lemma filtermap_id [simp, id_simps]: "filtermap id = id"
```
```   851 by(simp add: fun_eq_iff id_def filtermap_ident)
```
```   852
```
```   853 lemma filtermap_id' [simp]: "filtermap (\<lambda>x. x) = (\<lambda>F. F)"
```
```   854 using filtermap_id unfolding id_def .
```
```   855
```
```   856 lemma Quotient_filter [quot_map]:
```
```   857   assumes Q: "Quotient R Abs Rep T"
```
```   858   shows "Quotient (rel_filter R) (filtermap Abs) (filtermap Rep) (rel_filter T)"
```
```   859 unfolding Quotient_alt_def
```
```   860 proof(intro conjI strip)
```
```   861   from Q have *: "\<And>x y. T x y \<Longrightarrow> Abs x = y"
```
```   862     unfolding Quotient_alt_def by blast
```
```   863
```
```   864   fix F G
```
```   865   assume "rel_filter T F G"
```
```   866   thus "filtermap Abs F = G" unfolding filter_eq_iff
```
```   867     by(auto simp add: eventually_filtermap rel_filter_eventually * rel_funI del: iffI elim!: rel_funD)
```
```   868 next
```
```   869   from Q have *: "\<And>x. T (Rep x) x" unfolding Quotient_alt_def by blast
```
```   870
```
```   871   fix F
```
```   872   show "rel_filter T (filtermap Rep F) F"
```
```   873     by(auto elim: rel_funD intro: * intro!: ext arg_cong[where f="\<lambda>P. eventually P F"] rel_funI
```
```   874             del: iffI simp add: eventually_filtermap rel_filter_eventually)
```
```   875 qed(auto simp add: map_fun_def o_def eventually_filtermap filter_eq_iff fun_eq_iff rel_filter_eventually
```
```   876          fun_quotient[OF fun_quotient[OF Q identity_quotient] identity_quotient, unfolded Quotient_alt_def])
```
```   877
```
```   878 lemma eventually_parametric [transfer_rule]:
```
```   879   "((A ===> op =) ===> rel_filter A ===> op =) eventually eventually"
```
```   880 by(simp add: rel_fun_def rel_filter_eventually)
```
```   881
```
```   882 lemma frequently_parametric [transfer_rule]:
```
```   883   "((A ===> op =) ===> rel_filter A ===> op =) frequently frequently"
```
```   884   unfolding frequently_def[abs_def] by transfer_prover
```
```   885
```
```   886 lemma rel_filter_eq [relator_eq]: "rel_filter op = = op ="
```
```   887 by(auto simp add: rel_filter_eventually rel_fun_eq fun_eq_iff filter_eq_iff)
```
```   888
```
```   889 lemma rel_filter_mono [relator_mono]:
```
```   890   "A \<le> B \<Longrightarrow> rel_filter A \<le> rel_filter B"
```
```   891 unfolding rel_filter_eventually[abs_def]
```
```   892 by(rule le_funI)+(intro fun_mono fun_mono[THEN le_funD, THEN le_funD] order.refl)
```
```   893
```
```   894 lemma rel_filter_conversep [simp]: "rel_filter A\<inverse>\<inverse> = (rel_filter A)\<inverse>\<inverse>"
```
```   895 by(auto simp add: rel_filter_eventually fun_eq_iff rel_fun_def)
```
```   896
```
```   897 lemma is_filter_parametric_aux:
```
```   898   assumes "is_filter F"
```
```   899   assumes [transfer_rule]: "bi_total A" "bi_unique A"
```
```   900   and [transfer_rule]: "((A ===> op =) ===> op =) F G"
```
```   901   shows "is_filter G"
```
```   902 proof -
```
```   903   interpret is_filter F by fact
```
```   904   show ?thesis
```
```   905   proof
```
```   906     have "F (\<lambda>_. True) = G (\<lambda>x. True)" by transfer_prover
```
```   907     thus "G (\<lambda>x. True)" by(simp add: True)
```
```   908   next
```
```   909     fix P' Q'
```
```   910     assume "G P'" "G Q'"
```
```   911     moreover
```
```   912     from bi_total_fun[OF `bi_unique A` bi_total_eq, unfolded bi_total_def]
```
```   913     obtain P Q where [transfer_rule]: "(A ===> op =) P P'" "(A ===> op =) Q Q'" by blast
```
```   914     have "F P = G P'" "F Q = G Q'" by transfer_prover+
```
```   915     ultimately have "F (\<lambda>x. P x \<and> Q x)" by(simp add: conj)
```
```   916     moreover have "F (\<lambda>x. P x \<and> Q x) = G (\<lambda>x. P' x \<and> Q' x)" by transfer_prover
```
```   917     ultimately show "G (\<lambda>x. P' x \<and> Q' x)" by simp
```
```   918   next
```
```   919     fix P' Q'
```
```   920     assume "\<forall>x. P' x \<longrightarrow> Q' x" "G P'"
```
```   921     moreover
```
```   922     from bi_total_fun[OF `bi_unique A` bi_total_eq, unfolded bi_total_def]
```
```   923     obtain P Q where [transfer_rule]: "(A ===> op =) P P'" "(A ===> op =) Q Q'" by blast
```
```   924     have "F P = G P'" by transfer_prover
```
```   925     moreover have "(\<forall>x. P x \<longrightarrow> Q x) \<longleftrightarrow> (\<forall>x. P' x \<longrightarrow> Q' x)" by transfer_prover
```
```   926     ultimately have "F Q" by(simp add: mono)
```
```   927     moreover have "F Q = G Q'" by transfer_prover
```
```   928     ultimately show "G Q'" by simp
```
```   929   qed
```
```   930 qed
```
```   931
```
```   932 lemma is_filter_parametric [transfer_rule]:
```
```   933   "\<lbrakk> bi_total A; bi_unique A \<rbrakk>
```
```   934   \<Longrightarrow> (((A ===> op =) ===> op =) ===> op =) is_filter is_filter"
```
```   935 apply(rule rel_funI)
```
```   936 apply(rule iffI)
```
```   937  apply(erule (3) is_filter_parametric_aux)
```
```   938 apply(erule is_filter_parametric_aux[where A="conversep A"])
```
```   939 apply(auto simp add: rel_fun_def)
```
```   940 done
```
```   941
```
```   942 lemma left_total_rel_filter [transfer_rule]:
```
```   943   assumes [transfer_rule]: "bi_total A" "bi_unique A"
```
```   944   shows "left_total (rel_filter A)"
```
```   945 proof(rule left_totalI)
```
```   946   fix F :: "'a filter"
```
```   947   from bi_total_fun[OF bi_unique_fun[OF `bi_total A` bi_unique_eq] bi_total_eq]
```
```   948   obtain G where [transfer_rule]: "((A ===> op =) ===> op =) (\<lambda>P. eventually P F) G"
```
```   949     unfolding  bi_total_def by blast
```
```   950   moreover have "is_filter (\<lambda>P. eventually P F) \<longleftrightarrow> is_filter G" by transfer_prover
```
```   951   hence "is_filter G" by(simp add: eventually_def is_filter_Rep_filter)
```
```   952   ultimately have "rel_filter A F (Abs_filter G)"
```
```   953     by(simp add: rel_filter_eventually eventually_Abs_filter)
```
```   954   thus "\<exists>G. rel_filter A F G" ..
```
```   955 qed
```
```   956
```
```   957 lemma right_total_rel_filter [transfer_rule]:
```
```   958   "\<lbrakk> bi_total A; bi_unique A \<rbrakk> \<Longrightarrow> right_total (rel_filter A)"
```
```   959 using left_total_rel_filter[of "A\<inverse>\<inverse>"] by simp
```
```   960
```
```   961 lemma bi_total_rel_filter [transfer_rule]:
```
```   962   assumes "bi_total A" "bi_unique A"
```
```   963   shows "bi_total (rel_filter A)"
```
```   964 unfolding bi_total_alt_def using assms
```
```   965 by(simp add: left_total_rel_filter right_total_rel_filter)
```
```   966
```
```   967 lemma left_unique_rel_filter [transfer_rule]:
```
```   968   assumes "left_unique A"
```
```   969   shows "left_unique (rel_filter A)"
```
```   970 proof(rule left_uniqueI)
```
```   971   fix F F' G
```
```   972   assume [transfer_rule]: "rel_filter A F G" "rel_filter A F' G"
```
```   973   show "F = F'"
```
```   974     unfolding filter_eq_iff
```
```   975   proof
```
```   976     fix P :: "'a \<Rightarrow> bool"
```
```   977     obtain P' where [transfer_rule]: "(A ===> op =) P P'"
```
```   978       using left_total_fun[OF assms left_total_eq] unfolding left_total_def by blast
```
```   979     have "eventually P F = eventually P' G"
```
```   980       and "eventually P F' = eventually P' G" by transfer_prover+
```
```   981     thus "eventually P F = eventually P F'" by simp
```
```   982   qed
```
```   983 qed
```
```   984
```
```   985 lemma right_unique_rel_filter [transfer_rule]:
```
```   986   "right_unique A \<Longrightarrow> right_unique (rel_filter A)"
```
```   987 using left_unique_rel_filter[of "A\<inverse>\<inverse>"] by simp
```
```   988
```
```   989 lemma bi_unique_rel_filter [transfer_rule]:
```
```   990   "bi_unique A \<Longrightarrow> bi_unique (rel_filter A)"
```
```   991 by(simp add: bi_unique_alt_def left_unique_rel_filter right_unique_rel_filter)
```
```   992
```
```   993 lemma top_filter_parametric [transfer_rule]:
```
```   994   "bi_total A \<Longrightarrow> (rel_filter A) top top"
```
```   995 by(simp add: rel_filter_eventually All_transfer)
```
```   996
```
```   997 lemma bot_filter_parametric [transfer_rule]: "(rel_filter A) bot bot"
```
```   998 by(simp add: rel_filter_eventually rel_fun_def)
```
```   999
```
```  1000 lemma sup_filter_parametric [transfer_rule]:
```
```  1001   "(rel_filter A ===> rel_filter A ===> rel_filter A) sup sup"
```
```  1002 by(fastforce simp add: rel_filter_eventually[abs_def] eventually_sup dest: rel_funD)
```
```  1003
```
```  1004 lemma Sup_filter_parametric [transfer_rule]:
```
```  1005   "(rel_set (rel_filter A) ===> rel_filter A) Sup Sup"
```
```  1006 proof(rule rel_funI)
```
```  1007   fix S T
```
```  1008   assume [transfer_rule]: "rel_set (rel_filter A) S T"
```
```  1009   show "rel_filter A (Sup S) (Sup T)"
```
```  1010     by(simp add: rel_filter_eventually eventually_Sup) transfer_prover
```
```  1011 qed
```
```  1012
```
```  1013 lemma principal_parametric [transfer_rule]:
```
```  1014   "(rel_set A ===> rel_filter A) principal principal"
```
```  1015 proof(rule rel_funI)
```
```  1016   fix S S'
```
```  1017   assume [transfer_rule]: "rel_set A S S'"
```
```  1018   show "rel_filter A (principal S) (principal S')"
```
```  1019     by(simp add: rel_filter_eventually eventually_principal) transfer_prover
```
```  1020 qed
```
```  1021
```
```  1022 context
```
```  1023   fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"
```
```  1024   assumes [transfer_rule]: "bi_unique A"
```
```  1025 begin
```
```  1026
```
```  1027 lemma le_filter_parametric [transfer_rule]:
```
```  1028   "(rel_filter A ===> rel_filter A ===> op =) op \<le> op \<le>"
```
```  1029 unfolding le_filter_def[abs_def] by transfer_prover
```
```  1030
```
```  1031 lemma less_filter_parametric [transfer_rule]:
```
```  1032   "(rel_filter A ===> rel_filter A ===> op =) op < op <"
```
```  1033 unfolding less_filter_def[abs_def] by transfer_prover
```
```  1034
```
```  1035 context
```
```  1036   assumes [transfer_rule]: "bi_total A"
```
```  1037 begin
```
```  1038
```
```  1039 lemma Inf_filter_parametric [transfer_rule]:
```
```  1040   "(rel_set (rel_filter A) ===> rel_filter A) Inf Inf"
```
```  1041 unfolding Inf_filter_def[abs_def] by transfer_prover
```
```  1042
```
```  1043 lemma inf_filter_parametric [transfer_rule]:
```
```  1044   "(rel_filter A ===> rel_filter A ===> rel_filter A) inf inf"
```
```  1045 proof(intro rel_funI)+
```
```  1046   fix F F' G G'
```
```  1047   assume [transfer_rule]: "rel_filter A F F'" "rel_filter A G G'"
```
```  1048   have "rel_filter A (Inf {F, G}) (Inf {F', G'})" by transfer_prover
```
```  1049   thus "rel_filter A (inf F G) (inf F' G')" by simp
```
```  1050 qed
```
```  1051
```
```  1052 end
```
```  1053
```
```  1054 end
```
```  1055
```
```  1056 end
```
```  1057
```
`  1058 end`