src/HOL/Filter.thy
 author eberlm Mon Nov 02 11:56:28 2015 +0100 (2015-11-02) changeset 61531 ab2e862263e7 parent 61378 3e04c9ca001a child 61806 d2e62ae01cd8 permissions -rw-r--r--
Rounding function, uniform limits, cotangent, binomial identities
```     1 (*  Title:      HOL/Filter.thy
```
```     2     Author:     Brian Huffman
```
```     3     Author:     Johannes Hölzl
```
```     4 *)
```
```     5
```
```     6 section \<open>Filters on predicates\<close>
```
```     7
```
```     8 theory Filter
```
```     9 imports Set_Interval Lifting_Set
```
```    10 begin
```
```    11
```
```    12 subsection \<open>Filters\<close>
```
```    13
```
```    14 text \<open>
```
```    15   This definition also allows non-proper filters.
```
```    16 \<close>
```
```    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 \<open>Eventually\<close>
```
```    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 eventuallyI: "(\<And>x. P x) \<Longrightarrow> eventually P F"
```
```    67   by (auto intro: always_eventually)
```
```    68
```
```    69 lemma eventually_mono:
```
```    70   "(\<forall>x. P x \<longrightarrow> Q x) \<Longrightarrow> eventually P F \<Longrightarrow> eventually Q F"
```
```    71   unfolding eventually_def
```
```    72   by (rule is_filter.mono [OF is_filter_Rep_filter])
```
```    73
```
```    74 lemma eventually_conj:
```
```    75   assumes P: "eventually (\<lambda>x. P x) F"
```
```    76   assumes Q: "eventually (\<lambda>x. Q x) F"
```
```    77   shows "eventually (\<lambda>x. P x \<and> Q x) F"
```
```    78   using assms unfolding eventually_def
```
```    79   by (rule is_filter.conj [OF is_filter_Rep_filter])
```
```    80
```
```    81 lemma eventually_mp:
```
```    82   assumes "eventually (\<lambda>x. P x \<longrightarrow> Q x) F"
```
```    83   assumes "eventually (\<lambda>x. P x) F"
```
```    84   shows "eventually (\<lambda>x. Q x) F"
```
```    85 proof (rule eventually_mono)
```
```    86   show "\<forall>x. (P x \<longrightarrow> Q x) \<and> P x \<longrightarrow> Q x" by simp
```
```    87   show "eventually (\<lambda>x. (P x \<longrightarrow> Q x) \<and> P x) F"
```
```    88     using assms by (rule eventually_conj)
```
```    89 qed
```
```    90
```
```    91 lemma eventually_rev_mp:
```
```    92   assumes "eventually (\<lambda>x. P x) F"
```
```    93   assumes "eventually (\<lambda>x. P x \<longrightarrow> Q x) F"
```
```    94   shows "eventually (\<lambda>x. Q x) F"
```
```    95 using assms(2) assms(1) by (rule eventually_mp)
```
```    96
```
```    97 lemma eventually_conj_iff:
```
```    98   "eventually (\<lambda>x. P x \<and> Q x) F \<longleftrightarrow> eventually P F \<and> eventually Q F"
```
```    99   by (auto intro: eventually_conj elim: eventually_rev_mp)
```
```   100
```
```   101 lemma eventually_elim1:
```
```   102   assumes "eventually (\<lambda>i. P i) F"
```
```   103   assumes "\<And>i. P i \<Longrightarrow> Q i"
```
```   104   shows "eventually (\<lambda>i. Q i) F"
```
```   105   using assms by (auto elim!: eventually_rev_mp)
```
```   106
```
```   107 lemma eventually_elim2:
```
```   108   assumes "eventually (\<lambda>i. P i) F"
```
```   109   assumes "eventually (\<lambda>i. Q i) F"
```
```   110   assumes "\<And>i. P i \<Longrightarrow> Q i \<Longrightarrow> R i"
```
```   111   shows "eventually (\<lambda>i. R i) F"
```
```   112   using assms by (auto elim!: eventually_rev_mp)
```
```   113
```
```   114 lemma eventually_ball_finite_distrib:
```
```   115   "finite A \<Longrightarrow> (eventually (\<lambda>x. \<forall>y\<in>A. P x y) net) \<longleftrightarrow> (\<forall>y\<in>A. eventually (\<lambda>x. P x y) net)"
```
```   116   by (induction A rule: finite_induct) (auto simp: eventually_conj_iff)
```
```   117
```
```   118 lemma eventually_ball_finite:
```
```   119   "finite A \<Longrightarrow> \<forall>y\<in>A. eventually (\<lambda>x. P x y) net \<Longrightarrow> eventually (\<lambda>x. \<forall>y\<in>A. P x y) net"
```
```   120   by (auto simp: eventually_ball_finite_distrib)
```
```   121
```
```   122 lemma eventually_all_finite:
```
```   123   fixes P :: "'a \<Rightarrow> 'b::finite \<Rightarrow> bool"
```
```   124   assumes "\<And>y. eventually (\<lambda>x. P x y) net"
```
```   125   shows "eventually (\<lambda>x. \<forall>y. P x y) net"
```
```   126 using eventually_ball_finite [of UNIV P] assms by simp
```
```   127
```
```   128 lemma eventually_ex: "(\<forall>\<^sub>Fx in F. \<exists>y. P x y) \<longleftrightarrow> (\<exists>Y. \<forall>\<^sub>Fx in F. P x (Y x))"
```
```   129 proof
```
```   130   assume "\<forall>\<^sub>Fx in F. \<exists>y. P x y"
```
```   131   then have "\<forall>\<^sub>Fx in F. P x (SOME y. P x y)"
```
```   132     by (auto intro: someI_ex eventually_elim1)
```
```   133   then show "\<exists>Y. \<forall>\<^sub>Fx in F. P x (Y x)"
```
```   134     by auto
```
```   135 qed (auto intro: eventually_elim1)
```
```   136
```
```   137 lemma not_eventually_impI: "eventually P F \<Longrightarrow> \<not> eventually Q F \<Longrightarrow> \<not> eventually (\<lambda>x. P x \<longrightarrow> Q x) F"
```
```   138   by (auto intro: eventually_mp)
```
```   139
```
```   140 lemma not_eventuallyD: "\<not> eventually P F \<Longrightarrow> \<exists>x. \<not> P x"
```
```   141   by (metis always_eventually)
```
```   142
```
```   143 lemma eventually_subst:
```
```   144   assumes "eventually (\<lambda>n. P n = Q n) F"
```
```   145   shows "eventually P F = eventually Q F" (is "?L = ?R")
```
```   146 proof -
```
```   147   from assms have "eventually (\<lambda>x. P x \<longrightarrow> Q x) F"
```
```   148       and "eventually (\<lambda>x. Q x \<longrightarrow> P x) F"
```
```   149     by (auto elim: eventually_elim1)
```
```   150   then show ?thesis by (auto elim: eventually_elim2)
```
```   151 qed
```
```   152
```
```   153 subsection \<open> Frequently as dual to eventually \<close>
```
```   154
```
```   155 definition frequently :: "('a \<Rightarrow> bool) \<Rightarrow> 'a filter \<Rightarrow> bool"
```
```   156   where "frequently P F \<longleftrightarrow> \<not> eventually (\<lambda>x. \<not> P x) F"
```
```   157
```
```   158 syntax (xsymbols)
```
```   159   "_frequently"  :: "pttrn \<Rightarrow> 'a filter \<Rightarrow> bool \<Rightarrow> bool"      ("(3\<exists>\<^sub>F _ in _./ _)" [0, 0, 10] 10)
```
```   160
```
```   161 translations
```
```   162   "\<exists>\<^sub>Fx in F. P" == "CONST frequently (\<lambda>x. P) F"
```
```   163
```
```   164 lemma not_frequently_False [simp]: "\<not> (\<exists>\<^sub>Fx in F. False)"
```
```   165   by (simp add: frequently_def)
```
```   166
```
```   167 lemma frequently_ex: "\<exists>\<^sub>Fx in F. P x \<Longrightarrow> \<exists>x. P x"
```
```   168   by (auto simp: frequently_def dest: not_eventuallyD)
```
```   169
```
```   170 lemma frequentlyE: assumes "frequently P F" obtains x where "P x"
```
```   171   using frequently_ex[OF assms] by auto
```
```   172
```
```   173 lemma frequently_mp:
```
```   174   assumes ev: "\<forall>\<^sub>Fx in F. P x \<longrightarrow> Q x" and P: "\<exists>\<^sub>Fx in F. P x" shows "\<exists>\<^sub>Fx in F. Q x"
```
```   175 proof -
```
```   176   from ev have "eventually (\<lambda>x. \<not> Q x \<longrightarrow> \<not> P x) F"
```
```   177     by (rule eventually_rev_mp) (auto intro!: always_eventually)
```
```   178   from eventually_mp[OF this] P show ?thesis
```
```   179     by (auto simp: frequently_def)
```
```   180 qed
```
```   181
```
```   182 lemma frequently_rev_mp:
```
```   183   assumes "\<exists>\<^sub>Fx in F. P x"
```
```   184   assumes "\<forall>\<^sub>Fx in F. P x \<longrightarrow> Q x"
```
```   185   shows "\<exists>\<^sub>Fx in F. Q x"
```
```   186 using assms(2) assms(1) by (rule frequently_mp)
```
```   187
```
```   188 lemma frequently_mono: "(\<forall>x. P x \<longrightarrow> Q x) \<Longrightarrow> frequently P F \<Longrightarrow> frequently Q F"
```
```   189   using frequently_mp[of P Q] by (simp add: always_eventually)
```
```   190
```
```   191 lemma frequently_elim1: "\<exists>\<^sub>Fx in F. P x \<Longrightarrow> (\<And>i. P i \<Longrightarrow> Q i) \<Longrightarrow> \<exists>\<^sub>Fx in F. Q x"
```
```   192   by (metis frequently_mono)
```
```   193
```
```   194 lemma frequently_disj_iff: "(\<exists>\<^sub>Fx in F. P x \<or> Q x) \<longleftrightarrow> (\<exists>\<^sub>Fx in F. P x) \<or> (\<exists>\<^sub>Fx in F. Q x)"
```
```   195   by (simp add: frequently_def eventually_conj_iff)
```
```   196
```
```   197 lemma frequently_disj: "\<exists>\<^sub>Fx in F. P x \<Longrightarrow> \<exists>\<^sub>Fx in F. Q x \<Longrightarrow> \<exists>\<^sub>Fx in F. P x \<or> Q x"
```
```   198   by (simp add: frequently_disj_iff)
```
```   199
```
```   200 lemma frequently_bex_finite_distrib:
```
```   201   assumes "finite A" shows "(\<exists>\<^sub>Fx in F. \<exists>y\<in>A. P x y) \<longleftrightarrow> (\<exists>y\<in>A. \<exists>\<^sub>Fx in F. P x y)"
```
```   202   using assms by induction (auto simp: frequently_disj_iff)
```
```   203
```
```   204 lemma frequently_bex_finite: "finite A \<Longrightarrow> \<exists>\<^sub>Fx in F. \<exists>y\<in>A. P x y \<Longrightarrow> \<exists>y\<in>A. \<exists>\<^sub>Fx in F. P x y"
```
```   205   by (simp add: frequently_bex_finite_distrib)
```
```   206
```
```   207 lemma frequently_all: "(\<exists>\<^sub>Fx in F. \<forall>y. P x y) \<longleftrightarrow> (\<forall>Y. \<exists>\<^sub>Fx in F. P x (Y x))"
```
```   208   using eventually_ex[of "\<lambda>x y. \<not> P x y" F] by (simp add: frequently_def)
```
```   209
```
```   210 lemma
```
```   211   shows not_eventually: "\<not> eventually P F \<longleftrightarrow> (\<exists>\<^sub>Fx in F. \<not> P x)"
```
```   212     and not_frequently: "\<not> frequently P F \<longleftrightarrow> (\<forall>\<^sub>Fx in F. \<not> P x)"
```
```   213   by (auto simp: frequently_def)
```
```   214
```
```   215 lemma frequently_imp_iff:
```
```   216   "(\<exists>\<^sub>Fx in F. P x \<longrightarrow> Q x) \<longleftrightarrow> (eventually P F \<longrightarrow> frequently Q F)"
```
```   217   unfolding imp_conv_disj frequently_disj_iff not_eventually[symmetric] ..
```
```   218
```
```   219 lemma eventually_frequently_const_simps:
```
```   220   "(\<exists>\<^sub>Fx in F. P x \<and> C) \<longleftrightarrow> (\<exists>\<^sub>Fx in F. P x) \<and> C"
```
```   221   "(\<exists>\<^sub>Fx in F. C \<and> P x) \<longleftrightarrow> C \<and> (\<exists>\<^sub>Fx in F. P x)"
```
```   222   "(\<forall>\<^sub>Fx in F. P x \<or> C) \<longleftrightarrow> (\<forall>\<^sub>Fx in F. P x) \<or> C"
```
```   223   "(\<forall>\<^sub>Fx in F. C \<or> P x) \<longleftrightarrow> C \<or> (\<forall>\<^sub>Fx in F. P x)"
```
```   224   "(\<forall>\<^sub>Fx in F. P x \<longrightarrow> C) \<longleftrightarrow> ((\<exists>\<^sub>Fx in F. P x) \<longrightarrow> C)"
```
```   225   "(\<forall>\<^sub>Fx in F. C \<longrightarrow> P x) \<longleftrightarrow> (C \<longrightarrow> (\<forall>\<^sub>Fx in F. P x))"
```
```   226   by (cases C; simp add: not_frequently)+
```
```   227
```
```   228 lemmas eventually_frequently_simps =
```
```   229   eventually_frequently_const_simps
```
```   230   not_eventually
```
```   231   eventually_conj_iff
```
```   232   eventually_ball_finite_distrib
```
```   233   eventually_ex
```
```   234   not_frequently
```
```   235   frequently_disj_iff
```
```   236   frequently_bex_finite_distrib
```
```   237   frequently_all
```
```   238   frequently_imp_iff
```
```   239
```
```   240 ML \<open>
```
```   241   fun eventually_elim_tac ctxt facts = SUBGOAL_CASES (fn (goal, i) =>
```
```   242     let
```
```   243       val mp_thms = facts RL @{thms eventually_rev_mp}
```
```   244       val raw_elim_thm =
```
```   245         (@{thm allI} RS @{thm always_eventually})
```
```   246         |> fold (fn thm1 => fn thm2 => thm2 RS thm1) mp_thms
```
```   247         |> fold (fn _ => fn thm => @{thm impI} RS thm) facts
```
```   248       val cases_prop =
```
```   249         Thm.prop_of
```
```   250           (Rule_Cases.internalize_params (raw_elim_thm RS Goal.init (Thm.cterm_of ctxt goal)))
```
```   251       val cases = Rule_Cases.make_common ctxt cases_prop [(("elim", []), [])]
```
```   252     in
```
```   253       CASES cases (resolve_tac ctxt [raw_elim_thm] i)
```
```   254     end)
```
```   255 \<close>
```
```   256
```
```   257 method_setup eventually_elim = \<open>
```
```   258   Scan.succeed (fn ctxt => METHOD_CASES (HEADGOAL o eventually_elim_tac ctxt))
```
```   259 \<close> "elimination of eventually quantifiers"
```
```   260
```
```   261 subsubsection \<open>Finer-than relation\<close>
```
```   262
```
```   263 text \<open>@{term "F \<le> F'"} means that filter @{term F} is finer than
```
```   264 filter @{term F'}.\<close>
```
```   265
```
```   266 instantiation filter :: (type) complete_lattice
```
```   267 begin
```
```   268
```
```   269 definition le_filter_def:
```
```   270   "F \<le> F' \<longleftrightarrow> (\<forall>P. eventually P F' \<longrightarrow> eventually P F)"
```
```   271
```
```   272 definition
```
```   273   "(F :: 'a filter) < F' \<longleftrightarrow> F \<le> F' \<and> \<not> F' \<le> F"
```
```   274
```
```   275 definition
```
```   276   "top = Abs_filter (\<lambda>P. \<forall>x. P x)"
```
```   277
```
```   278 definition
```
```   279   "bot = Abs_filter (\<lambda>P. True)"
```
```   280
```
```   281 definition
```
```   282   "sup F F' = Abs_filter (\<lambda>P. eventually P F \<and> eventually P F')"
```
```   283
```
```   284 definition
```
```   285   "inf F F' = Abs_filter
```
```   286       (\<lambda>P. \<exists>Q R. eventually Q F \<and> eventually R F' \<and> (\<forall>x. Q x \<and> R x \<longrightarrow> P x))"
```
```   287
```
```   288 definition
```
```   289   "Sup S = Abs_filter (\<lambda>P. \<forall>F\<in>S. eventually P F)"
```
```   290
```
```   291 definition
```
```   292   "Inf S = Sup {F::'a filter. \<forall>F'\<in>S. F \<le> F'}"
```
```   293
```
```   294 lemma eventually_top [simp]: "eventually P top \<longleftrightarrow> (\<forall>x. P x)"
```
```   295   unfolding top_filter_def
```
```   296   by (rule eventually_Abs_filter, rule is_filter.intro, auto)
```
```   297
```
```   298 lemma eventually_bot [simp]: "eventually P bot"
```
```   299   unfolding bot_filter_def
```
```   300   by (subst eventually_Abs_filter, rule is_filter.intro, auto)
```
```   301
```
```   302 lemma eventually_sup:
```
```   303   "eventually P (sup F F') \<longleftrightarrow> eventually P F \<and> eventually P F'"
```
```   304   unfolding sup_filter_def
```
```   305   by (rule eventually_Abs_filter, rule is_filter.intro)
```
```   306      (auto elim!: eventually_rev_mp)
```
```   307
```
```   308 lemma eventually_inf:
```
```   309   "eventually P (inf F F') \<longleftrightarrow>
```
```   310    (\<exists>Q R. eventually Q F \<and> eventually R F' \<and> (\<forall>x. Q x \<and> R x \<longrightarrow> P x))"
```
```   311   unfolding inf_filter_def
```
```   312   apply (rule eventually_Abs_filter, rule is_filter.intro)
```
```   313   apply (fast intro: eventually_True)
```
```   314   apply clarify
```
```   315   apply (intro exI conjI)
```
```   316   apply (erule (1) eventually_conj)
```
```   317   apply (erule (1) eventually_conj)
```
```   318   apply simp
```
```   319   apply auto
```
```   320   done
```
```   321
```
```   322 lemma eventually_Sup:
```
```   323   "eventually P (Sup S) \<longleftrightarrow> (\<forall>F\<in>S. eventually P F)"
```
```   324   unfolding Sup_filter_def
```
```   325   apply (rule eventually_Abs_filter, rule is_filter.intro)
```
```   326   apply (auto intro: eventually_conj elim!: eventually_rev_mp)
```
```   327   done
```
```   328
```
```   329 instance proof
```
```   330   fix F F' F'' :: "'a filter" and S :: "'a filter set"
```
```   331   { show "F < F' \<longleftrightarrow> F \<le> F' \<and> \<not> F' \<le> F"
```
```   332     by (rule less_filter_def) }
```
```   333   { show "F \<le> F"
```
```   334     unfolding le_filter_def by simp }
```
```   335   { assume "F \<le> F'" and "F' \<le> F''" thus "F \<le> F''"
```
```   336     unfolding le_filter_def by simp }
```
```   337   { assume "F \<le> F'" and "F' \<le> F" thus "F = F'"
```
```   338     unfolding le_filter_def filter_eq_iff by fast }
```
```   339   { show "inf F F' \<le> F" and "inf F F' \<le> F'"
```
```   340     unfolding le_filter_def eventually_inf by (auto intro: eventually_True) }
```
```   341   { assume "F \<le> F'" and "F \<le> F''" thus "F \<le> inf F' F''"
```
```   342     unfolding le_filter_def eventually_inf
```
```   343     by (auto elim!: eventually_mono intro: eventually_conj) }
```
```   344   { show "F \<le> sup F F'" and "F' \<le> sup F F'"
```
```   345     unfolding le_filter_def eventually_sup by simp_all }
```
```   346   { assume "F \<le> F''" and "F' \<le> F''" thus "sup F F' \<le> F''"
```
```   347     unfolding le_filter_def eventually_sup by simp }
```
```   348   { assume "F'' \<in> S" thus "Inf S \<le> F''"
```
```   349     unfolding le_filter_def Inf_filter_def eventually_Sup Ball_def by simp }
```
```   350   { assume "\<And>F'. F' \<in> S \<Longrightarrow> F \<le> F'" thus "F \<le> Inf S"
```
```   351     unfolding le_filter_def Inf_filter_def eventually_Sup Ball_def by simp }
```
```   352   { assume "F \<in> S" thus "F \<le> Sup S"
```
```   353     unfolding le_filter_def eventually_Sup by simp }
```
```   354   { assume "\<And>F. F \<in> S \<Longrightarrow> F \<le> F'" thus "Sup S \<le> F'"
```
```   355     unfolding le_filter_def eventually_Sup by simp }
```
```   356   { show "Inf {} = (top::'a filter)"
```
```   357     by (auto simp: top_filter_def Inf_filter_def Sup_filter_def)
```
```   358       (metis (full_types) top_filter_def always_eventually eventually_top) }
```
```   359   { show "Sup {} = (bot::'a filter)"
```
```   360     by (auto simp: bot_filter_def Sup_filter_def) }
```
```   361 qed
```
```   362
```
```   363 end
```
```   364
```
```   365 lemma filter_leD:
```
```   366   "F \<le> F' \<Longrightarrow> eventually P F' \<Longrightarrow> eventually P F"
```
```   367   unfolding le_filter_def by simp
```
```   368
```
```   369 lemma filter_leI:
```
```   370   "(\<And>P. eventually P F' \<Longrightarrow> eventually P F) \<Longrightarrow> F \<le> F'"
```
```   371   unfolding le_filter_def by simp
```
```   372
```
```   373 lemma eventually_False:
```
```   374   "eventually (\<lambda>x. False) F \<longleftrightarrow> F = bot"
```
```   375   unfolding filter_eq_iff by (auto elim: eventually_rev_mp)
```
```   376
```
```   377 lemma eventually_frequently: "F \<noteq> bot \<Longrightarrow> eventually P F \<Longrightarrow> frequently P F"
```
```   378   using eventually_conj[of P F "\<lambda>x. \<not> P x"]
```
```   379   by (auto simp add: frequently_def eventually_False)
```
```   380
```
```   381 lemma eventually_const_iff: "eventually (\<lambda>x. P) F \<longleftrightarrow> P \<or> F = bot"
```
```   382   by (cases P) (auto simp: eventually_False)
```
```   383
```
```   384 lemma eventually_const[simp]: "F \<noteq> bot \<Longrightarrow> eventually (\<lambda>x. P) F \<longleftrightarrow> P"
```
```   385   by (simp add: eventually_const_iff)
```
```   386
```
```   387 lemma frequently_const_iff: "frequently (\<lambda>x. P) F \<longleftrightarrow> P \<and> F \<noteq> bot"
```
```   388   by (simp add: frequently_def eventually_const_iff)
```
```   389
```
```   390 lemma frequently_const[simp]: "F \<noteq> bot \<Longrightarrow> frequently (\<lambda>x. P) F \<longleftrightarrow> P"
```
```   391   by (simp add: frequently_const_iff)
```
```   392
```
```   393 lemma eventually_happens: "eventually P net \<Longrightarrow> net = bot \<or> (\<exists>x. P x)"
```
```   394   by (metis frequentlyE eventually_frequently)
```
```   395
```
```   396 lemma eventually_happens':
```
```   397   assumes "F \<noteq> bot" "eventually P F"
```
```   398   shows   "\<exists>x. P x"
```
```   399   using assms eventually_frequently frequentlyE by blast
```
```   400
```
```   401 abbreviation (input) trivial_limit :: "'a filter \<Rightarrow> bool"
```
```   402   where "trivial_limit F \<equiv> F = bot"
```
```   403
```
```   404 lemma trivial_limit_def: "trivial_limit F \<longleftrightarrow> eventually (\<lambda>x. False) F"
```
```   405   by (rule eventually_False [symmetric])
```
```   406
```
```   407 lemma eventually_Inf: "eventually P (Inf B) \<longleftrightarrow> (\<exists>X\<subseteq>B. finite X \<and> eventually P (Inf X))"
```
```   408 proof -
```
```   409   let ?F = "\<lambda>P. \<exists>X\<subseteq>B. finite X \<and> eventually P (Inf X)"
```
```   410
```
```   411   { fix P have "eventually P (Abs_filter ?F) \<longleftrightarrow> ?F P"
```
```   412     proof (rule eventually_Abs_filter is_filter.intro)+
```
```   413       show "?F (\<lambda>x. True)"
```
```   414         by (rule exI[of _ "{}"]) (simp add: le_fun_def)
```
```   415     next
```
```   416       fix P Q
```
```   417       assume "?F P" then guess X ..
```
```   418       moreover
```
```   419       assume "?F Q" then guess Y ..
```
```   420       ultimately show "?F (\<lambda>x. P x \<and> Q x)"
```
```   421         by (intro exI[of _ "X \<union> Y"])
```
```   422            (auto simp: Inf_union_distrib eventually_inf)
```
```   423     next
```
```   424       fix P Q
```
```   425       assume "?F P" then guess X ..
```
```   426       moreover assume "\<forall>x. P x \<longrightarrow> Q x"
```
```   427       ultimately show "?F Q"
```
```   428         by (intro exI[of _ X]) (auto elim: eventually_elim1)
```
```   429     qed }
```
```   430   note eventually_F = this
```
```   431
```
```   432   have "Inf B = Abs_filter ?F"
```
```   433   proof (intro antisym Inf_greatest)
```
```   434     show "Inf B \<le> Abs_filter ?F"
```
```   435       by (auto simp: le_filter_def eventually_F dest: Inf_superset_mono)
```
```   436   next
```
```   437     fix F assume "F \<in> B" then show "Abs_filter ?F \<le> F"
```
```   438       by (auto simp add: le_filter_def eventually_F intro!: exI[of _ "{F}"])
```
```   439   qed
```
```   440   then show ?thesis
```
```   441     by (simp add: eventually_F)
```
```   442 qed
```
```   443
```
```   444 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))"
```
```   445   unfolding INF_def[of B] eventually_Inf[of P "F`B"]
```
```   446   by (metis Inf_image_eq finite_imageI image_mono finite_subset_image)
```
```   447
```
```   448 lemma Inf_filter_not_bot:
```
```   449   fixes B :: "'a filter set"
```
```   450   shows "(\<And>X. X \<subseteq> B \<Longrightarrow> finite X \<Longrightarrow> Inf X \<noteq> bot) \<Longrightarrow> Inf B \<noteq> bot"
```
```   451   unfolding trivial_limit_def eventually_Inf[of _ B]
```
```   452     bot_bool_def [symmetric] bot_fun_def [symmetric] bot_unique by simp
```
```   453
```
```   454 lemma INF_filter_not_bot:
```
```   455   fixes F :: "'i \<Rightarrow> 'a filter"
```
```   456   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"
```
```   457   unfolding trivial_limit_def eventually_INF[of _ B]
```
```   458     bot_bool_def [symmetric] bot_fun_def [symmetric] bot_unique by simp
```
```   459
```
```   460 lemma eventually_Inf_base:
```
```   461   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"
```
```   462   shows "eventually P (Inf B) \<longleftrightarrow> (\<exists>b\<in>B. eventually P b)"
```
```   463 proof (subst eventually_Inf, safe)
```
```   464   fix X assume "finite X" "X \<subseteq> B"
```
```   465   then have "\<exists>b\<in>B. \<forall>x\<in>X. b \<le> x"
```
```   466   proof induct
```
```   467     case empty then show ?case
```
```   468       using \<open>B \<noteq> {}\<close> by auto
```
```   469   next
```
```   470     case (insert x X)
```
```   471     then obtain b where "b \<in> B" "\<And>x. x \<in> X \<Longrightarrow> b \<le> x"
```
```   472       by auto
```
```   473     with \<open>insert x X \<subseteq> B\<close> base[of b x] show ?case
```
```   474       by (auto intro: order_trans)
```
```   475   qed
```
```   476   then obtain b where "b \<in> B" "b \<le> Inf X"
```
```   477     by (auto simp: le_Inf_iff)
```
```   478   then show "eventually P (Inf X) \<Longrightarrow> Bex B (eventually P)"
```
```   479     by (intro bexI[of _ b]) (auto simp: le_filter_def)
```
```   480 qed (auto intro!: exI[of _ "{x}" for x])
```
```   481
```
```   482 lemma eventually_INF_base:
```
```   483   "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>
```
```   484     eventually P (INF b:B. F b) \<longleftrightarrow> (\<exists>b\<in>B. eventually P (F b))"
```
```   485   unfolding INF_def by (subst eventually_Inf_base) auto
```
```   486
```
```   487
```
```   488 subsubsection \<open>Map function for filters\<close>
```
```   489
```
```   490 definition filtermap :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a filter \<Rightarrow> 'b filter"
```
```   491   where "filtermap f F = Abs_filter (\<lambda>P. eventually (\<lambda>x. P (f x)) F)"
```
```   492
```
```   493 lemma eventually_filtermap:
```
```   494   "eventually P (filtermap f F) = eventually (\<lambda>x. P (f x)) F"
```
```   495   unfolding filtermap_def
```
```   496   apply (rule eventually_Abs_filter)
```
```   497   apply (rule is_filter.intro)
```
```   498   apply (auto elim!: eventually_rev_mp)
```
```   499   done
```
```   500
```
```   501 lemma filtermap_ident: "filtermap (\<lambda>x. x) F = F"
```
```   502   by (simp add: filter_eq_iff eventually_filtermap)
```
```   503
```
```   504 lemma filtermap_filtermap:
```
```   505   "filtermap f (filtermap g F) = filtermap (\<lambda>x. f (g x)) F"
```
```   506   by (simp add: filter_eq_iff eventually_filtermap)
```
```   507
```
```   508 lemma filtermap_mono: "F \<le> F' \<Longrightarrow> filtermap f F \<le> filtermap f F'"
```
```   509   unfolding le_filter_def eventually_filtermap by simp
```
```   510
```
```   511 lemma filtermap_bot [simp]: "filtermap f bot = bot"
```
```   512   by (simp add: filter_eq_iff eventually_filtermap)
```
```   513
```
```   514 lemma filtermap_sup: "filtermap f (sup F1 F2) = sup (filtermap f F1) (filtermap f F2)"
```
```   515   by (auto simp: filter_eq_iff eventually_filtermap eventually_sup)
```
```   516
```
```   517 lemma filtermap_inf: "filtermap f (inf F1 F2) \<le> inf (filtermap f F1) (filtermap f F2)"
```
```   518   by (auto simp: le_filter_def eventually_filtermap eventually_inf)
```
```   519
```
```   520 lemma filtermap_INF: "filtermap f (INF b:B. F b) \<le> (INF b:B. filtermap f (F b))"
```
```   521 proof -
```
```   522   { fix X :: "'c set" assume "finite X"
```
```   523     then have "filtermap f (INFIMUM X F) \<le> (INF b:X. filtermap f (F b))"
```
```   524     proof induct
```
```   525       case (insert x X)
```
```   526       have "filtermap f (INF a:insert x X. F a) \<le> inf (filtermap f (F x)) (filtermap f (INF a:X. F a))"
```
```   527         by (rule order_trans[OF _ filtermap_inf]) simp
```
```   528       also have "\<dots> \<le> inf (filtermap f (F x)) (INF a:X. filtermap f (F a))"
```
```   529         by (intro inf_mono insert order_refl)
```
```   530       finally show ?case
```
```   531         by simp
```
```   532     qed simp }
```
```   533   then show ?thesis
```
```   534     unfolding le_filter_def eventually_filtermap
```
```   535     by (subst (1 2) eventually_INF) auto
```
```   536 qed
```
```   537 subsubsection \<open>Standard filters\<close>
```
```   538
```
```   539 definition principal :: "'a set \<Rightarrow> 'a filter" where
```
```   540   "principal S = Abs_filter (\<lambda>P. \<forall>x\<in>S. P x)"
```
```   541
```
```   542 lemma eventually_principal: "eventually P (principal S) \<longleftrightarrow> (\<forall>x\<in>S. P x)"
```
```   543   unfolding principal_def
```
```   544   by (rule eventually_Abs_filter, rule is_filter.intro) auto
```
```   545
```
```   546 lemma eventually_inf_principal: "eventually P (inf F (principal s)) \<longleftrightarrow> eventually (\<lambda>x. x \<in> s \<longrightarrow> P x) F"
```
```   547   unfolding eventually_inf eventually_principal by (auto elim: eventually_elim1)
```
```   548
```
```   549 lemma principal_UNIV[simp]: "principal UNIV = top"
```
```   550   by (auto simp: filter_eq_iff eventually_principal)
```
```   551
```
```   552 lemma principal_empty[simp]: "principal {} = bot"
```
```   553   by (auto simp: filter_eq_iff eventually_principal)
```
```   554
```
```   555 lemma principal_eq_bot_iff: "principal X = bot \<longleftrightarrow> X = {}"
```
```   556   by (auto simp add: filter_eq_iff eventually_principal)
```
```   557
```
```   558 lemma principal_le_iff[iff]: "principal A \<le> principal B \<longleftrightarrow> A \<subseteq> B"
```
```   559   by (auto simp: le_filter_def eventually_principal)
```
```   560
```
```   561 lemma le_principal: "F \<le> principal A \<longleftrightarrow> eventually (\<lambda>x. x \<in> A) F"
```
```   562   unfolding le_filter_def eventually_principal
```
```   563   apply safe
```
```   564   apply (erule_tac x="\<lambda>x. x \<in> A" in allE)
```
```   565   apply (auto elim: eventually_elim1)
```
```   566   done
```
```   567
```
```   568 lemma principal_inject[iff]: "principal A = principal B \<longleftrightarrow> A = B"
```
```   569   unfolding eq_iff by simp
```
```   570
```
```   571 lemma sup_principal[simp]: "sup (principal A) (principal B) = principal (A \<union> B)"
```
```   572   unfolding filter_eq_iff eventually_sup eventually_principal by auto
```
```   573
```
```   574 lemma inf_principal[simp]: "inf (principal A) (principal B) = principal (A \<inter> B)"
```
```   575   unfolding filter_eq_iff eventually_inf eventually_principal
```
```   576   by (auto intro: exI[of _ "\<lambda>x. x \<in> A"] exI[of _ "\<lambda>x. x \<in> B"])
```
```   577
```
```   578 lemma SUP_principal[simp]: "(SUP i : I. principal (A i)) = principal (\<Union>i\<in>I. A i)"
```
```   579   unfolding filter_eq_iff eventually_Sup SUP_def by (auto simp: eventually_principal)
```
```   580
```
```   581 lemma INF_principal_finite: "finite X \<Longrightarrow> (INF x:X. principal (f x)) = principal (\<Inter>x\<in>X. f x)"
```
```   582   by (induct X rule: finite_induct) auto
```
```   583
```
```   584 lemma filtermap_principal[simp]: "filtermap f (principal A) = principal (f ` A)"
```
```   585   unfolding filter_eq_iff eventually_filtermap eventually_principal by simp
```
```   586
```
```   587 subsubsection \<open>Order filters\<close>
```
```   588
```
```   589 definition at_top :: "('a::order) filter"
```
```   590   where "at_top = (INF k. principal {k ..})"
```
```   591
```
```   592 lemma at_top_sub: "at_top = (INF k:{c::'a::linorder..}. principal {k ..})"
```
```   593   by (auto intro!: INF_eq max.cobounded1 max.cobounded2 simp: at_top_def)
```
```   594
```
```   595 lemma eventually_at_top_linorder: "eventually P at_top \<longleftrightarrow> (\<exists>N::'a::linorder. \<forall>n\<ge>N. P n)"
```
```   596   unfolding at_top_def
```
```   597   by (subst eventually_INF_base) (auto simp: eventually_principal intro: max.cobounded1 max.cobounded2)
```
```   598
```
```   599 lemma eventually_ge_at_top:
```
```   600   "eventually (\<lambda>x. (c::_::linorder) \<le> x) at_top"
```
```   601   unfolding eventually_at_top_linorder by auto
```
```   602
```
```   603 lemma eventually_at_top_dense: "eventually P at_top \<longleftrightarrow> (\<exists>N::'a::{no_top, linorder}. \<forall>n>N. P n)"
```
```   604 proof -
```
```   605   have "eventually P (INF k. principal {k <..}) \<longleftrightarrow> (\<exists>N::'a. \<forall>n>N. P n)"
```
```   606     by (subst eventually_INF_base) (auto simp: eventually_principal intro: max.cobounded1 max.cobounded2)
```
```   607   also have "(INF k. principal {k::'a <..}) = at_top"
```
```   608     unfolding at_top_def
```
```   609     by (intro INF_eq) (auto intro: less_imp_le simp: Ici_subset_Ioi_iff gt_ex)
```
```   610   finally show ?thesis .
```
```   611 qed
```
```   612
```
```   613 lemma eventually_at_top_not_equal: "eventually (\<lambda>x::'a::{no_top, linorder}. x \<noteq> c) at_top"
```
```   614   unfolding eventually_at_top_dense by auto
```
```   615
```
```   616 lemma eventually_gt_at_top: "eventually (\<lambda>x. (c::_::{no_top, linorder}) < x) at_top"
```
```   617   unfolding eventually_at_top_dense by auto
```
```   618
```
```   619 lemma eventually_all_ge_at_top:
```
```   620   assumes "eventually P (at_top :: ('a :: linorder) filter)"
```
```   621   shows   "eventually (\<lambda>x. \<forall>y\<ge>x. P y) at_top"
```
```   622 proof -
```
```   623   from assms obtain x where "\<And>y. y \<ge> x \<Longrightarrow> P y" by (auto simp: eventually_at_top_linorder)
```
```   624   hence "\<forall>z\<ge>y. P z" if "y \<ge> x" for y using that by simp
```
```   625   thus ?thesis by (auto simp: eventually_at_top_linorder)
```
```   626 qed
```
```   627
```
```   628 definition at_bot :: "('a::order) filter"
```
```   629   where "at_bot = (INF k. principal {.. k})"
```
```   630
```
```   631 lemma at_bot_sub: "at_bot = (INF k:{.. c::'a::linorder}. principal {.. k})"
```
```   632   by (auto intro!: INF_eq min.cobounded1 min.cobounded2 simp: at_bot_def)
```
```   633
```
```   634 lemma eventually_at_bot_linorder:
```
```   635   fixes P :: "'a::linorder \<Rightarrow> bool" shows "eventually P at_bot \<longleftrightarrow> (\<exists>N. \<forall>n\<le>N. P n)"
```
```   636   unfolding at_bot_def
```
```   637   by (subst eventually_INF_base) (auto simp: eventually_principal intro: min.cobounded1 min.cobounded2)
```
```   638
```
```   639 lemma eventually_le_at_bot:
```
```   640   "eventually (\<lambda>x. x \<le> (c::_::linorder)) at_bot"
```
```   641   unfolding eventually_at_bot_linorder by auto
```
```   642
```
```   643 lemma eventually_at_bot_dense: "eventually P at_bot \<longleftrightarrow> (\<exists>N::'a::{no_bot, linorder}. \<forall>n<N. P n)"
```
```   644 proof -
```
```   645   have "eventually P (INF k. principal {..< k}) \<longleftrightarrow> (\<exists>N::'a. \<forall>n<N. P n)"
```
```   646     by (subst eventually_INF_base) (auto simp: eventually_principal intro: min.cobounded1 min.cobounded2)
```
```   647   also have "(INF k. principal {..< k::'a}) = at_bot"
```
```   648     unfolding at_bot_def
```
```   649     by (intro INF_eq) (auto intro: less_imp_le simp: Iic_subset_Iio_iff lt_ex)
```
```   650   finally show ?thesis .
```
```   651 qed
```
```   652
```
```   653 lemma eventually_at_bot_not_equal: "eventually (\<lambda>x::'a::{no_bot, linorder}. x \<noteq> c) at_bot"
```
```   654   unfolding eventually_at_bot_dense by auto
```
```   655
```
```   656 lemma eventually_gt_at_bot:
```
```   657   "eventually (\<lambda>x. x < (c::_::unbounded_dense_linorder)) at_bot"
```
```   658   unfolding eventually_at_bot_dense by auto
```
```   659
```
```   660 lemma trivial_limit_at_bot_linorder: "\<not> trivial_limit (at_bot ::('a::linorder) filter)"
```
```   661   unfolding trivial_limit_def
```
```   662   by (metis eventually_at_bot_linorder order_refl)
```
```   663
```
```   664 lemma trivial_limit_at_top_linorder: "\<not> trivial_limit (at_top ::('a::linorder) filter)"
```
```   665   unfolding trivial_limit_def
```
```   666   by (metis eventually_at_top_linorder order_refl)
```
```   667
```
```   668 subsection \<open>Sequentially\<close>
```
```   669
```
```   670 abbreviation sequentially :: "nat filter"
```
```   671   where "sequentially \<equiv> at_top"
```
```   672
```
```   673 lemma eventually_sequentially:
```
```   674   "eventually P sequentially \<longleftrightarrow> (\<exists>N. \<forall>n\<ge>N. P n)"
```
```   675   by (rule eventually_at_top_linorder)
```
```   676
```
```   677 lemma sequentially_bot [simp, intro]: "sequentially \<noteq> bot"
```
```   678   unfolding filter_eq_iff eventually_sequentially by auto
```
```   679
```
```   680 lemmas trivial_limit_sequentially = sequentially_bot
```
```   681
```
```   682 lemma eventually_False_sequentially [simp]:
```
```   683   "\<not> eventually (\<lambda>n. False) sequentially"
```
```   684   by (simp add: eventually_False)
```
```   685
```
```   686 lemma le_sequentially:
```
```   687   "F \<le> sequentially \<longleftrightarrow> (\<forall>N. eventually (\<lambda>n. N \<le> n) F)"
```
```   688   by (simp add: at_top_def le_INF_iff le_principal)
```
```   689
```
```   690 lemma eventually_sequentiallyI [intro?]:
```
```   691   assumes "\<And>x. c \<le> x \<Longrightarrow> P x"
```
```   692   shows "eventually P sequentially"
```
```   693 using assms by (auto simp: eventually_sequentially)
```
```   694
```
```   695 lemma eventually_sequentially_Suc: "eventually (\<lambda>i. P (Suc i)) sequentially \<longleftrightarrow> eventually P sequentially"
```
```   696   unfolding eventually_sequentially by (metis Suc_le_D Suc_le_mono le_Suc_eq)
```
```   697
```
```   698 lemma eventually_sequentially_seg: "eventually (\<lambda>n. P (n + k)) sequentially \<longleftrightarrow> eventually P sequentially"
```
```   699   using eventually_sequentially_Suc[of "\<lambda>n. P (n + k)" for k] by (induction k) auto
```
```   700
```
```   701 subsection \<open> The cofinite filter \<close>
```
```   702
```
```   703 definition "cofinite = Abs_filter (\<lambda>P. finite {x. \<not> P x})"
```
```   704
```
```   705 abbreviation Inf_many :: "('a \<Rightarrow> bool) \<Rightarrow> bool"  (binder "INFM " 10) where
```
```   706   "Inf_many P \<equiv> frequently P cofinite"
```
```   707
```
```   708 abbreviation Alm_all :: "('a \<Rightarrow> bool) \<Rightarrow> bool"  (binder "MOST " 10) where
```
```   709   "Alm_all P \<equiv> eventually P cofinite"
```
```   710
```
```   711 notation (xsymbols)
```
```   712   Inf_many  (binder "\<exists>\<^sub>\<infinity>" 10) and
```
```   713   Alm_all  (binder "\<forall>\<^sub>\<infinity>" 10)
```
```   714
```
```   715 lemma eventually_cofinite: "eventually P cofinite \<longleftrightarrow> finite {x. \<not> P x}"
```
```   716   unfolding cofinite_def
```
```   717 proof (rule eventually_Abs_filter, rule is_filter.intro)
```
```   718   fix P Q :: "'a \<Rightarrow> bool" assume "finite {x. \<not> P x}" "finite {x. \<not> Q x}"
```
```   719   from finite_UnI[OF this] show "finite {x. \<not> (P x \<and> Q x)}"
```
```   720     by (rule rev_finite_subset) auto
```
```   721 next
```
```   722   fix P Q :: "'a \<Rightarrow> bool" assume P: "finite {x. \<not> P x}" and *: "\<forall>x. P x \<longrightarrow> Q x"
```
```   723   from * show "finite {x. \<not> Q x}"
```
```   724     by (intro finite_subset[OF _ P]) auto
```
```   725 qed simp
```
```   726
```
```   727 lemma frequently_cofinite: "frequently P cofinite \<longleftrightarrow> \<not> finite {x. P x}"
```
```   728   by (simp add: frequently_def eventually_cofinite)
```
```   729
```
```   730 lemma cofinite_bot[simp]: "cofinite = (bot::'a filter) \<longleftrightarrow> finite (UNIV :: 'a set)"
```
```   731   unfolding trivial_limit_def eventually_cofinite by simp
```
```   732
```
```   733 lemma cofinite_eq_sequentially: "cofinite = sequentially"
```
```   734   unfolding filter_eq_iff eventually_sequentially eventually_cofinite
```
```   735 proof safe
```
```   736   fix P :: "nat \<Rightarrow> bool" assume [simp]: "finite {x. \<not> P x}"
```
```   737   show "\<exists>N. \<forall>n\<ge>N. P n"
```
```   738   proof cases
```
```   739     assume "{x. \<not> P x} \<noteq> {}" then show ?thesis
```
```   740       by (intro exI[of _ "Suc (Max {x. \<not> P x})"]) (auto simp: Suc_le_eq)
```
```   741   qed auto
```
```   742 next
```
```   743   fix P :: "nat \<Rightarrow> bool" and N :: nat assume "\<forall>n\<ge>N. P n"
```
```   744   then have "{x. \<not> P x} \<subseteq> {..< N}"
```
```   745     by (auto simp: not_le)
```
```   746   then show "finite {x. \<not> P x}"
```
```   747     by (blast intro: finite_subset)
```
```   748 qed
```
```   749
```
```   750 subsection \<open>Limits\<close>
```
```   751
```
```   752 definition filterlim :: "('a \<Rightarrow> 'b) \<Rightarrow> 'b filter \<Rightarrow> 'a filter \<Rightarrow> bool" where
```
```   753   "filterlim f F2 F1 \<longleftrightarrow> filtermap f F1 \<le> F2"
```
```   754
```
```   755 syntax
```
```   756   "_LIM" :: "pttrns \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> bool" ("(3LIM (_)/ (_)./ (_) :> (_))" [1000, 10, 0, 10] 10)
```
```   757
```
```   758 translations
```
```   759   "LIM x F1. f :> F2"   == "CONST filterlim (%x. f) F2 F1"
```
```   760
```
```   761 lemma filterlim_iff:
```
```   762   "(LIM x F1. f x :> F2) \<longleftrightarrow> (\<forall>P. eventually P F2 \<longrightarrow> eventually (\<lambda>x. P (f x)) F1)"
```
```   763   unfolding filterlim_def le_filter_def eventually_filtermap ..
```
```   764
```
```   765 lemma filterlim_compose:
```
```   766   "filterlim g F3 F2 \<Longrightarrow> filterlim f F2 F1 \<Longrightarrow> filterlim (\<lambda>x. g (f x)) F3 F1"
```
```   767   unfolding filterlim_def filtermap_filtermap[symmetric] by (metis filtermap_mono order_trans)
```
```   768
```
```   769 lemma filterlim_mono:
```
```   770   "filterlim f F2 F1 \<Longrightarrow> F2 \<le> F2' \<Longrightarrow> F1' \<le> F1 \<Longrightarrow> filterlim f F2' F1'"
```
```   771   unfolding filterlim_def by (metis filtermap_mono order_trans)
```
```   772
```
```   773 lemma filterlim_ident: "LIM x F. x :> F"
```
```   774   by (simp add: filterlim_def filtermap_ident)
```
```   775
```
```   776 lemma filterlim_cong:
```
```   777   "F1 = F1' \<Longrightarrow> F2 = F2' \<Longrightarrow> eventually (\<lambda>x. f x = g x) F2 \<Longrightarrow> filterlim f F1 F2 = filterlim g F1' F2'"
```
```   778   by (auto simp: filterlim_def le_filter_def eventually_filtermap elim: eventually_elim2)
```
```   779
```
```   780 lemma filterlim_mono_eventually:
```
```   781   assumes "filterlim f F G" and ord: "F \<le> F'" "G' \<le> G"
```
```   782   assumes eq: "eventually (\<lambda>x. f x = f' x) G'"
```
```   783   shows "filterlim f' F' G'"
```
```   784   apply (rule filterlim_cong[OF refl refl eq, THEN iffD1])
```
```   785   apply (rule filterlim_mono[OF _ ord])
```
```   786   apply fact
```
```   787   done
```
```   788
```
```   789 lemma filtermap_mono_strong: "inj f \<Longrightarrow> filtermap f F \<le> filtermap f G \<longleftrightarrow> F \<le> G"
```
```   790   apply (auto intro!: filtermap_mono) []
```
```   791   apply (auto simp: le_filter_def eventually_filtermap)
```
```   792   apply (erule_tac x="\<lambda>x. P (inv f x)" in allE)
```
```   793   apply auto
```
```   794   done
```
```   795
```
```   796 lemma filtermap_eq_strong: "inj f \<Longrightarrow> filtermap f F = filtermap f G \<longleftrightarrow> F = G"
```
```   797   by (simp add: filtermap_mono_strong eq_iff)
```
```   798
```
```   799 lemma filtermap_fun_inverse:
```
```   800   assumes g: "filterlim g F G"
```
```   801   assumes f: "filterlim f G F"
```
```   802   assumes ev: "eventually (\<lambda>x. f (g x) = x) G"
```
```   803   shows "filtermap f F = G"
```
```   804 proof (rule antisym)
```
```   805   show "filtermap f F \<le> G"
```
```   806     using f unfolding filterlim_def .
```
```   807   have "G = filtermap f (filtermap g G)"
```
```   808     using ev by (auto elim: eventually_elim2 simp: filter_eq_iff eventually_filtermap)
```
```   809   also have "\<dots> \<le> filtermap f F"
```
```   810     using g by (intro filtermap_mono) (simp add: filterlim_def)
```
```   811   finally show "G \<le> filtermap f F" .
```
```   812 qed
```
```   813
```
```   814 lemma filterlim_principal:
```
```   815   "(LIM x F. f x :> principal S) \<longleftrightarrow> (eventually (\<lambda>x. f x \<in> S) F)"
```
```   816   unfolding filterlim_def eventually_filtermap le_principal ..
```
```   817
```
```   818 lemma filterlim_inf:
```
```   819   "(LIM x F1. f x :> inf F2 F3) \<longleftrightarrow> ((LIM x F1. f x :> F2) \<and> (LIM x F1. f x :> F3))"
```
```   820   unfolding filterlim_def by simp
```
```   821
```
```   822 lemma filterlim_INF:
```
```   823   "(LIM x F. f x :> (INF b:B. G b)) \<longleftrightarrow> (\<forall>b\<in>B. LIM x F. f x :> G b)"
```
```   824   unfolding filterlim_def le_INF_iff ..
```
```   825
```
```   826 lemma filterlim_INF_INF:
```
```   827   "(\<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)"
```
```   828   unfolding filterlim_def by (rule order_trans[OF filtermap_INF INF_mono])
```
```   829
```
```   830 lemma filterlim_base:
```
```   831   "(\<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>
```
```   832     LIM x (INF i:I. principal (F i)). f x :> (INF j:J. principal (G j))"
```
```   833   by (force intro!: filterlim_INF_INF simp: image_subset_iff)
```
```   834
```
```   835 lemma filterlim_base_iff:
```
```   836   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"
```
```   837   shows "(LIM x (INF i:I. principal (F i)). f x :> INF j:J. principal (G j)) \<longleftrightarrow>
```
```   838     (\<forall>j\<in>J. \<exists>i\<in>I. \<forall>x\<in>F i. f x \<in> G j)"
```
```   839   unfolding filterlim_INF filterlim_principal
```
```   840 proof (subst eventually_INF_base)
```
```   841   fix i j assume "i \<in> I" "j \<in> I"
```
```   842   with chain[OF this] show "\<exists>x\<in>I. principal (F x) \<le> inf (principal (F i)) (principal (F j))"
```
```   843     by auto
```
```   844 qed (auto simp: eventually_principal \<open>I \<noteq> {}\<close>)
```
```   845
```
```   846 lemma filterlim_filtermap: "filterlim f F1 (filtermap g F2) = filterlim (\<lambda>x. f (g x)) F1 F2"
```
```   847   unfolding filterlim_def filtermap_filtermap ..
```
```   848
```
```   849 lemma filterlim_sup:
```
```   850   "filterlim f F F1 \<Longrightarrow> filterlim f F F2 \<Longrightarrow> filterlim f F (sup F1 F2)"
```
```   851   unfolding filterlim_def filtermap_sup by auto
```
```   852
```
```   853 lemma filterlim_sequentially_Suc:
```
```   854   "(LIM x sequentially. f (Suc x) :> F) \<longleftrightarrow> (LIM x sequentially. f x :> F)"
```
```   855   unfolding filterlim_iff by (subst eventually_sequentially_Suc) simp
```
```   856
```
```   857 lemma filterlim_Suc: "filterlim Suc sequentially sequentially"
```
```   858   by (simp add: filterlim_iff eventually_sequentially) (metis le_Suc_eq)
```
```   859
```
```   860 lemma filterlim_If:
```
```   861   "LIM x inf F (principal {x. P x}). f x :> G \<Longrightarrow>
```
```   862     LIM x inf F (principal {x. \<not> P x}). g x :> G \<Longrightarrow>
```
```   863     LIM x F. if P x then f x else g x :> G"
```
```   864   unfolding filterlim_iff eventually_inf_principal by (auto simp: eventually_conj_iff)
```
```   865
```
```   866 subsection \<open>Limits to @{const at_top} and @{const at_bot}\<close>
```
```   867
```
```   868 lemma filterlim_at_top:
```
```   869   fixes f :: "'a \<Rightarrow> ('b::linorder)"
```
```   870   shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. Z \<le> f x) F)"
```
```   871   by (auto simp: filterlim_iff eventually_at_top_linorder elim!: eventually_elim1)
```
```   872
```
```   873 lemma filterlim_at_top_mono:
```
```   874   "LIM x F. f x :> at_top \<Longrightarrow> eventually (\<lambda>x. f x \<le> (g x::'a::linorder)) F \<Longrightarrow>
```
```   875     LIM x F. g x :> at_top"
```
```   876   by (auto simp: filterlim_at_top elim: eventually_elim2 intro: order_trans)
```
```   877
```
```   878 lemma filterlim_at_top_dense:
```
```   879   fixes f :: "'a \<Rightarrow> ('b::unbounded_dense_linorder)"
```
```   880   shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. Z < f x) F)"
```
```   881   by (metis eventually_elim1[of _ F] eventually_gt_at_top order_less_imp_le
```
```   882             filterlim_at_top[of f F] filterlim_iff[of f at_top F])
```
```   883
```
```   884 lemma filterlim_at_top_ge:
```
```   885   fixes f :: "'a \<Rightarrow> ('b::linorder)" and c :: "'b"
```
```   886   shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z\<ge>c. eventually (\<lambda>x. Z \<le> f x) F)"
```
```   887   unfolding at_top_sub[of c] filterlim_INF by (auto simp add: filterlim_principal)
```
```   888
```
```   889 lemma filterlim_at_top_at_top:
```
```   890   fixes f :: "'a::linorder \<Rightarrow> 'b::linorder"
```
```   891   assumes mono: "\<And>x y. Q x \<Longrightarrow> Q y \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y"
```
```   892   assumes bij: "\<And>x. P x \<Longrightarrow> f (g x) = x" "\<And>x. P x \<Longrightarrow> Q (g x)"
```
```   893   assumes Q: "eventually Q at_top"
```
```   894   assumes P: "eventually P at_top"
```
```   895   shows "filterlim f at_top at_top"
```
```   896 proof -
```
```   897   from P obtain x where x: "\<And>y. x \<le> y \<Longrightarrow> P y"
```
```   898     unfolding eventually_at_top_linorder by auto
```
```   899   show ?thesis
```
```   900   proof (intro filterlim_at_top_ge[THEN iffD2] allI impI)
```
```   901     fix z assume "x \<le> z"
```
```   902     with x have "P z" by auto
```
```   903     have "eventually (\<lambda>x. g z \<le> x) at_top"
```
```   904       by (rule eventually_ge_at_top)
```
```   905     with Q show "eventually (\<lambda>x. z \<le> f x) at_top"
```
```   906       by eventually_elim (metis mono bij \<open>P z\<close>)
```
```   907   qed
```
```   908 qed
```
```   909
```
```   910 lemma filterlim_at_top_gt:
```
```   911   fixes f :: "'a \<Rightarrow> ('b::unbounded_dense_linorder)" and c :: "'b"
```
```   912   shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z>c. eventually (\<lambda>x. Z \<le> f x) F)"
```
```   913   by (metis filterlim_at_top order_less_le_trans gt_ex filterlim_at_top_ge)
```
```   914
```
```   915 lemma filterlim_at_bot:
```
```   916   fixes f :: "'a \<Rightarrow> ('b::linorder)"
```
```   917   shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. f x \<le> Z) F)"
```
```   918   by (auto simp: filterlim_iff eventually_at_bot_linorder elim!: eventually_elim1)
```
```   919
```
```   920 lemma filterlim_at_bot_dense:
```
```   921   fixes f :: "'a \<Rightarrow> ('b::{dense_linorder, no_bot})"
```
```   922   shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. f x < Z) F)"
```
```   923 proof (auto simp add: filterlim_at_bot[of f F])
```
```   924   fix Z :: 'b
```
```   925   from lt_ex [of Z] obtain Z' where 1: "Z' < Z" ..
```
```   926   assume "\<forall>Z. eventually (\<lambda>x. f x \<le> Z) F"
```
```   927   hence "eventually (\<lambda>x. f x \<le> Z') F" by auto
```
```   928   thus "eventually (\<lambda>x. f x < Z) F"
```
```   929     apply (rule eventually_mono[rotated])
```
```   930     using 1 by auto
```
```   931   next
```
```   932     fix Z :: 'b
```
```   933     show "\<forall>Z. eventually (\<lambda>x. f x < Z) F \<Longrightarrow> eventually (\<lambda>x. f x \<le> Z) F"
```
```   934       by (drule spec [of _ Z], erule eventually_mono[rotated], auto simp add: less_imp_le)
```
```   935 qed
```
```   936
```
```   937 lemma filterlim_at_bot_le:
```
```   938   fixes f :: "'a \<Rightarrow> ('b::linorder)" and c :: "'b"
```
```   939   shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z\<le>c. eventually (\<lambda>x. Z \<ge> f x) F)"
```
```   940   unfolding filterlim_at_bot
```
```   941 proof safe
```
```   942   fix Z assume *: "\<forall>Z\<le>c. eventually (\<lambda>x. Z \<ge> f x) F"
```
```   943   with *[THEN spec, of "min Z c"] show "eventually (\<lambda>x. Z \<ge> f x) F"
```
```   944     by (auto elim!: eventually_elim1)
```
```   945 qed simp
```
```   946
```
```   947 lemma filterlim_at_bot_lt:
```
```   948   fixes f :: "'a \<Rightarrow> ('b::unbounded_dense_linorder)" and c :: "'b"
```
```   949   shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z<c. eventually (\<lambda>x. Z \<ge> f x) F)"
```
```   950   by (metis filterlim_at_bot filterlim_at_bot_le lt_ex order_le_less_trans)
```
```   951
```
```   952
```
```   953 subsection \<open>Setup @{typ "'a filter"} for lifting and transfer\<close>
```
```   954
```
```   955 context begin interpretation lifting_syntax .
```
```   956
```
```   957 definition rel_filter :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a filter \<Rightarrow> 'b filter \<Rightarrow> bool"
```
```   958 where "rel_filter R F G = ((R ===> op =) ===> op =) (Rep_filter F) (Rep_filter G)"
```
```   959
```
```   960 lemma rel_filter_eventually:
```
```   961   "rel_filter R F G \<longleftrightarrow>
```
```   962   ((R ===> op =) ===> op =) (\<lambda>P. eventually P F) (\<lambda>P. eventually P G)"
```
```   963 by(simp add: rel_filter_def eventually_def)
```
```   964
```
```   965 lemma filtermap_id [simp, id_simps]: "filtermap id = id"
```
```   966 by(simp add: fun_eq_iff id_def filtermap_ident)
```
```   967
```
```   968 lemma filtermap_id' [simp]: "filtermap (\<lambda>x. x) = (\<lambda>F. F)"
```
```   969 using filtermap_id unfolding id_def .
```
```   970
```
```   971 lemma Quotient_filter [quot_map]:
```
```   972   assumes Q: "Quotient R Abs Rep T"
```
```   973   shows "Quotient (rel_filter R) (filtermap Abs) (filtermap Rep) (rel_filter T)"
```
```   974 unfolding Quotient_alt_def
```
```   975 proof(intro conjI strip)
```
```   976   from Q have *: "\<And>x y. T x y \<Longrightarrow> Abs x = y"
```
```   977     unfolding Quotient_alt_def by blast
```
```   978
```
```   979   fix F G
```
```   980   assume "rel_filter T F G"
```
```   981   thus "filtermap Abs F = G" unfolding filter_eq_iff
```
```   982     by(auto simp add: eventually_filtermap rel_filter_eventually * rel_funI del: iffI elim!: rel_funD)
```
```   983 next
```
```   984   from Q have *: "\<And>x. T (Rep x) x" unfolding Quotient_alt_def by blast
```
```   985
```
```   986   fix F
```
```   987   show "rel_filter T (filtermap Rep F) F"
```
```   988     by(auto elim: rel_funD intro: * intro!: ext arg_cong[where f="\<lambda>P. eventually P F"] rel_funI
```
```   989             del: iffI simp add: eventually_filtermap rel_filter_eventually)
```
```   990 qed(auto simp add: map_fun_def o_def eventually_filtermap filter_eq_iff fun_eq_iff rel_filter_eventually
```
```   991          fun_quotient[OF fun_quotient[OF Q identity_quotient] identity_quotient, unfolded Quotient_alt_def])
```
```   992
```
```   993 lemma eventually_parametric [transfer_rule]:
```
```   994   "((A ===> op =) ===> rel_filter A ===> op =) eventually eventually"
```
```   995 by(simp add: rel_fun_def rel_filter_eventually)
```
```   996
```
```   997 lemma frequently_parametric [transfer_rule]:
```
```   998   "((A ===> op =) ===> rel_filter A ===> op =) frequently frequently"
```
```   999   unfolding frequently_def[abs_def] by transfer_prover
```
```  1000
```
```  1001 lemma rel_filter_eq [relator_eq]: "rel_filter op = = op ="
```
```  1002 by(auto simp add: rel_filter_eventually rel_fun_eq fun_eq_iff filter_eq_iff)
```
```  1003
```
```  1004 lemma rel_filter_mono [relator_mono]:
```
```  1005   "A \<le> B \<Longrightarrow> rel_filter A \<le> rel_filter B"
```
```  1006 unfolding rel_filter_eventually[abs_def]
```
```  1007 by(rule le_funI)+(intro fun_mono fun_mono[THEN le_funD, THEN le_funD] order.refl)
```
```  1008
```
```  1009 lemma rel_filter_conversep [simp]: "rel_filter A\<inverse>\<inverse> = (rel_filter A)\<inverse>\<inverse>"
```
```  1010 apply (simp add: rel_filter_eventually fun_eq_iff rel_fun_def)
```
```  1011 apply (safe; metis)
```
```  1012 done
```
```  1013
```
```  1014 lemma is_filter_parametric_aux:
```
```  1015   assumes "is_filter F"
```
```  1016   assumes [transfer_rule]: "bi_total A" "bi_unique A"
```
```  1017   and [transfer_rule]: "((A ===> op =) ===> op =) F G"
```
```  1018   shows "is_filter G"
```
```  1019 proof -
```
```  1020   interpret is_filter F by fact
```
```  1021   show ?thesis
```
```  1022   proof
```
```  1023     have "F (\<lambda>_. True) = G (\<lambda>x. True)" by transfer_prover
```
```  1024     thus "G (\<lambda>x. True)" by(simp add: True)
```
```  1025   next
```
```  1026     fix P' Q'
```
```  1027     assume "G P'" "G Q'"
```
```  1028     moreover
```
```  1029     from bi_total_fun[OF \<open>bi_unique A\<close> bi_total_eq, unfolded bi_total_def]
```
```  1030     obtain P Q where [transfer_rule]: "(A ===> op =) P P'" "(A ===> op =) Q Q'" by blast
```
```  1031     have "F P = G P'" "F Q = G Q'" by transfer_prover+
```
```  1032     ultimately have "F (\<lambda>x. P x \<and> Q x)" by(simp add: conj)
```
```  1033     moreover have "F (\<lambda>x. P x \<and> Q x) = G (\<lambda>x. P' x \<and> Q' x)" by transfer_prover
```
```  1034     ultimately show "G (\<lambda>x. P' x \<and> Q' x)" by simp
```
```  1035   next
```
```  1036     fix P' Q'
```
```  1037     assume "\<forall>x. P' x \<longrightarrow> Q' x" "G P'"
```
```  1038     moreover
```
```  1039     from bi_total_fun[OF \<open>bi_unique A\<close> bi_total_eq, unfolded bi_total_def]
```
```  1040     obtain P Q where [transfer_rule]: "(A ===> op =) P P'" "(A ===> op =) Q Q'" by blast
```
```  1041     have "F P = G P'" by transfer_prover
```
```  1042     moreover have "(\<forall>x. P x \<longrightarrow> Q x) \<longleftrightarrow> (\<forall>x. P' x \<longrightarrow> Q' x)" by transfer_prover
```
```  1043     ultimately have "F Q" by(simp add: mono)
```
```  1044     moreover have "F Q = G Q'" by transfer_prover
```
```  1045     ultimately show "G Q'" by simp
```
```  1046   qed
```
```  1047 qed
```
```  1048
```
```  1049 lemma is_filter_parametric [transfer_rule]:
```
```  1050   "\<lbrakk> bi_total A; bi_unique A \<rbrakk>
```
```  1051   \<Longrightarrow> (((A ===> op =) ===> op =) ===> op =) is_filter is_filter"
```
```  1052 apply(rule rel_funI)
```
```  1053 apply(rule iffI)
```
```  1054  apply(erule (3) is_filter_parametric_aux)
```
```  1055 apply(erule is_filter_parametric_aux[where A="conversep A"])
```
```  1056 apply (simp_all add: rel_fun_def)
```
```  1057 apply metis
```
```  1058 done
```
```  1059
```
```  1060 lemma left_total_rel_filter [transfer_rule]:
```
```  1061   assumes [transfer_rule]: "bi_total A" "bi_unique A"
```
```  1062   shows "left_total (rel_filter A)"
```
```  1063 proof(rule left_totalI)
```
```  1064   fix F :: "'a filter"
```
```  1065   from bi_total_fun[OF bi_unique_fun[OF \<open>bi_total A\<close> bi_unique_eq] bi_total_eq]
```
```  1066   obtain G where [transfer_rule]: "((A ===> op =) ===> op =) (\<lambda>P. eventually P F) G"
```
```  1067     unfolding  bi_total_def by blast
```
```  1068   moreover have "is_filter (\<lambda>P. eventually P F) \<longleftrightarrow> is_filter G" by transfer_prover
```
```  1069   hence "is_filter G" by(simp add: eventually_def is_filter_Rep_filter)
```
```  1070   ultimately have "rel_filter A F (Abs_filter G)"
```
```  1071     by(simp add: rel_filter_eventually eventually_Abs_filter)
```
```  1072   thus "\<exists>G. rel_filter A F G" ..
```
```  1073 qed
```
```  1074
```
```  1075 lemma right_total_rel_filter [transfer_rule]:
```
```  1076   "\<lbrakk> bi_total A; bi_unique A \<rbrakk> \<Longrightarrow> right_total (rel_filter A)"
```
```  1077 using left_total_rel_filter[of "A\<inverse>\<inverse>"] by simp
```
```  1078
```
```  1079 lemma bi_total_rel_filter [transfer_rule]:
```
```  1080   assumes "bi_total A" "bi_unique A"
```
```  1081   shows "bi_total (rel_filter A)"
```
```  1082 unfolding bi_total_alt_def using assms
```
```  1083 by(simp add: left_total_rel_filter right_total_rel_filter)
```
```  1084
```
```  1085 lemma left_unique_rel_filter [transfer_rule]:
```
```  1086   assumes "left_unique A"
```
```  1087   shows "left_unique (rel_filter A)"
```
```  1088 proof(rule left_uniqueI)
```
```  1089   fix F F' G
```
```  1090   assume [transfer_rule]: "rel_filter A F G" "rel_filter A F' G"
```
```  1091   show "F = F'"
```
```  1092     unfolding filter_eq_iff
```
```  1093   proof
```
```  1094     fix P :: "'a \<Rightarrow> bool"
```
```  1095     obtain P' where [transfer_rule]: "(A ===> op =) P P'"
```
```  1096       using left_total_fun[OF assms left_total_eq] unfolding left_total_def by blast
```
```  1097     have "eventually P F = eventually P' G"
```
```  1098       and "eventually P F' = eventually P' G" by transfer_prover+
```
```  1099     thus "eventually P F = eventually P F'" by simp
```
```  1100   qed
```
```  1101 qed
```
```  1102
```
```  1103 lemma right_unique_rel_filter [transfer_rule]:
```
```  1104   "right_unique A \<Longrightarrow> right_unique (rel_filter A)"
```
```  1105 using left_unique_rel_filter[of "A\<inverse>\<inverse>"] by simp
```
```  1106
```
```  1107 lemma bi_unique_rel_filter [transfer_rule]:
```
```  1108   "bi_unique A \<Longrightarrow> bi_unique (rel_filter A)"
```
```  1109 by(simp add: bi_unique_alt_def left_unique_rel_filter right_unique_rel_filter)
```
```  1110
```
```  1111 lemma top_filter_parametric [transfer_rule]:
```
```  1112   "bi_total A \<Longrightarrow> (rel_filter A) top top"
```
```  1113 by(simp add: rel_filter_eventually All_transfer)
```
```  1114
```
```  1115 lemma bot_filter_parametric [transfer_rule]: "(rel_filter A) bot bot"
```
```  1116 by(simp add: rel_filter_eventually rel_fun_def)
```
```  1117
```
```  1118 lemma sup_filter_parametric [transfer_rule]:
```
```  1119   "(rel_filter A ===> rel_filter A ===> rel_filter A) sup sup"
```
```  1120 by(fastforce simp add: rel_filter_eventually[abs_def] eventually_sup dest: rel_funD)
```
```  1121
```
```  1122 lemma Sup_filter_parametric [transfer_rule]:
```
```  1123   "(rel_set (rel_filter A) ===> rel_filter A) Sup Sup"
```
```  1124 proof(rule rel_funI)
```
```  1125   fix S T
```
```  1126   assume [transfer_rule]: "rel_set (rel_filter A) S T"
```
```  1127   show "rel_filter A (Sup S) (Sup T)"
```
```  1128     by(simp add: rel_filter_eventually eventually_Sup) transfer_prover
```
```  1129 qed
```
```  1130
```
```  1131 lemma principal_parametric [transfer_rule]:
```
```  1132   "(rel_set A ===> rel_filter A) principal principal"
```
```  1133 proof(rule rel_funI)
```
```  1134   fix S S'
```
```  1135   assume [transfer_rule]: "rel_set A S S'"
```
```  1136   show "rel_filter A (principal S) (principal S')"
```
```  1137     by(simp add: rel_filter_eventually eventually_principal) transfer_prover
```
```  1138 qed
```
```  1139
```
```  1140 context
```
```  1141   fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"
```
```  1142   assumes [transfer_rule]: "bi_unique A"
```
```  1143 begin
```
```  1144
```
```  1145 lemma le_filter_parametric [transfer_rule]:
```
```  1146   "(rel_filter A ===> rel_filter A ===> op =) op \<le> op \<le>"
```
```  1147 unfolding le_filter_def[abs_def] by transfer_prover
```
```  1148
```
```  1149 lemma less_filter_parametric [transfer_rule]:
```
```  1150   "(rel_filter A ===> rel_filter A ===> op =) op < op <"
```
```  1151 unfolding less_filter_def[abs_def] by transfer_prover
```
```  1152
```
```  1153 context
```
```  1154   assumes [transfer_rule]: "bi_total A"
```
```  1155 begin
```
```  1156
```
```  1157 lemma Inf_filter_parametric [transfer_rule]:
```
```  1158   "(rel_set (rel_filter A) ===> rel_filter A) Inf Inf"
```
```  1159 unfolding Inf_filter_def[abs_def] by transfer_prover
```
```  1160
```
```  1161 lemma inf_filter_parametric [transfer_rule]:
```
```  1162   "(rel_filter A ===> rel_filter A ===> rel_filter A) inf inf"
```
```  1163 proof(intro rel_funI)+
```
```  1164   fix F F' G G'
```
```  1165   assume [transfer_rule]: "rel_filter A F F'" "rel_filter A G G'"
```
```  1166   have "rel_filter A (Inf {F, G}) (Inf {F', G'})" by transfer_prover
```
```  1167   thus "rel_filter A (inf F G) (inf F' G')" by simp
```
```  1168 qed
```
```  1169
```
```  1170 end
```
```  1171
```
```  1172 end
```
```  1173
```
```  1174 end
```
```  1175
```
`  1176 end`