src/HOL/Filter.thy
 author paulson Mon Dec 07 16:44:26 2015 +0000 (2015-12-07) changeset 61806 d2e62ae01cd8 parent 61531 ab2e862263e7 child 61810 3c5040d5694a permissions -rw-r--r--
Cauchy's integral formula for circles. Starting to fix eventually_mono.
```     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_mono':
```
```    75   "\<lbrakk>eventually P F; \<And>x. P x \<Longrightarrow> Q x\<rbrakk> \<Longrightarrow> eventually Q F"
```
```    76   unfolding eventually_def
```
```    77   by (blast intro: is_filter.mono [OF is_filter_Rep_filter])
```
```    78
```
```    79 lemma eventually_conj:
```
```    80   assumes P: "eventually (\<lambda>x. P x) F"
```
```    81   assumes Q: "eventually (\<lambda>x. Q x) F"
```
```    82   shows "eventually (\<lambda>x. P x \<and> Q x) F"
```
```    83   using assms unfolding eventually_def
```
```    84   by (rule is_filter.conj [OF is_filter_Rep_filter])
```
```    85
```
```    86 lemma eventually_mp:
```
```    87   assumes "eventually (\<lambda>x. P x \<longrightarrow> Q x) F"
```
```    88   assumes "eventually (\<lambda>x. P x) F"
```
```    89   shows "eventually (\<lambda>x. Q x) F"
```
```    90 proof -
```
```    91   have "eventually (\<lambda>x. (P x \<longrightarrow> Q x) \<and> P x) F"
```
```    92     using assms by (rule eventually_conj)
```
```    93   then show ?thesis
```
```    94     by (blast intro: eventually_mono')
```
```    95 qed
```
```    96
```
```    97 lemma eventually_rev_mp:
```
```    98   assumes "eventually (\<lambda>x. P x) F"
```
```    99   assumes "eventually (\<lambda>x. P x \<longrightarrow> Q x) F"
```
```   100   shows "eventually (\<lambda>x. Q x) F"
```
```   101 using assms(2) assms(1) by (rule eventually_mp)
```
```   102
```
```   103 lemma eventually_conj_iff:
```
```   104   "eventually (\<lambda>x. P x \<and> Q x) F \<longleftrightarrow> eventually P F \<and> eventually Q F"
```
```   105   by (auto intro: eventually_conj elim: eventually_rev_mp)
```
```   106
```
```   107 lemma eventually_elim1:
```
```   108   assumes "eventually (\<lambda>i. P i) F"
```
```   109   assumes "\<And>i. P i \<Longrightarrow> Q i"
```
```   110   shows "eventually (\<lambda>i. Q i) F"
```
```   111   using assms by (auto elim!: eventually_rev_mp)
```
```   112
```
```   113 lemma eventually_elim2:
```
```   114   assumes "eventually (\<lambda>i. P i) F"
```
```   115   assumes "eventually (\<lambda>i. Q i) F"
```
```   116   assumes "\<And>i. P i \<Longrightarrow> Q i \<Longrightarrow> R i"
```
```   117   shows "eventually (\<lambda>i. R i) F"
```
```   118   using assms by (auto elim!: eventually_rev_mp)
```
```   119
```
```   120 lemma eventually_ball_finite_distrib:
```
```   121   "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)"
```
```   122   by (induction A rule: finite_induct) (auto simp: eventually_conj_iff)
```
```   123
```
```   124 lemma eventually_ball_finite:
```
```   125   "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"
```
```   126   by (auto simp: eventually_ball_finite_distrib)
```
```   127
```
```   128 lemma eventually_all_finite:
```
```   129   fixes P :: "'a \<Rightarrow> 'b::finite \<Rightarrow> bool"
```
```   130   assumes "\<And>y. eventually (\<lambda>x. P x y) net"
```
```   131   shows "eventually (\<lambda>x. \<forall>y. P x y) net"
```
```   132 using eventually_ball_finite [of UNIV P] assms by simp
```
```   133
```
```   134 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))"
```
```   135 proof
```
```   136   assume "\<forall>\<^sub>Fx in F. \<exists>y. P x y"
```
```   137   then have "\<forall>\<^sub>Fx in F. P x (SOME y. P x y)"
```
```   138     by (auto intro: someI_ex eventually_elim1)
```
```   139   then show "\<exists>Y. \<forall>\<^sub>Fx in F. P x (Y x)"
```
```   140     by auto
```
```   141 qed (auto intro: eventually_elim1)
```
```   142
```
```   143 lemma not_eventually_impI: "eventually P F \<Longrightarrow> \<not> eventually Q F \<Longrightarrow> \<not> eventually (\<lambda>x. P x \<longrightarrow> Q x) F"
```
```   144   by (auto intro: eventually_mp)
```
```   145
```
```   146 lemma not_eventuallyD: "\<not> eventually P F \<Longrightarrow> \<exists>x. \<not> P x"
```
```   147   by (metis always_eventually)
```
```   148
```
```   149 lemma eventually_subst:
```
```   150   assumes "eventually (\<lambda>n. P n = Q n) F"
```
```   151   shows "eventually P F = eventually Q F" (is "?L = ?R")
```
```   152 proof -
```
```   153   from assms have "eventually (\<lambda>x. P x \<longrightarrow> Q x) F"
```
```   154       and "eventually (\<lambda>x. Q x \<longrightarrow> P x) F"
```
```   155     by (auto elim: eventually_elim1)
```
```   156   then show ?thesis by (auto elim: eventually_elim2)
```
```   157 qed
```
```   158
```
```   159 subsection \<open> Frequently as dual to eventually \<close>
```
```   160
```
```   161 definition frequently :: "('a \<Rightarrow> bool) \<Rightarrow> 'a filter \<Rightarrow> bool"
```
```   162   where "frequently P F \<longleftrightarrow> \<not> eventually (\<lambda>x. \<not> P x) F"
```
```   163
```
```   164 syntax (xsymbols)
```
```   165   "_frequently"  :: "pttrn \<Rightarrow> 'a filter \<Rightarrow> bool \<Rightarrow> bool"      ("(3\<exists>\<^sub>F _ in _./ _)" [0, 0, 10] 10)
```
```   166
```
```   167 translations
```
```   168   "\<exists>\<^sub>Fx in F. P" == "CONST frequently (\<lambda>x. P) F"
```
```   169
```
```   170 lemma not_frequently_False [simp]: "\<not> (\<exists>\<^sub>Fx in F. False)"
```
```   171   by (simp add: frequently_def)
```
```   172
```
```   173 lemma frequently_ex: "\<exists>\<^sub>Fx in F. P x \<Longrightarrow> \<exists>x. P x"
```
```   174   by (auto simp: frequently_def dest: not_eventuallyD)
```
```   175
```
```   176 lemma frequentlyE: assumes "frequently P F" obtains x where "P x"
```
```   177   using frequently_ex[OF assms] by auto
```
```   178
```
```   179 lemma frequently_mp:
```
```   180   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"
```
```   181 proof -
```
```   182   from ev have "eventually (\<lambda>x. \<not> Q x \<longrightarrow> \<not> P x) F"
```
```   183     by (rule eventually_rev_mp) (auto intro!: always_eventually)
```
```   184   from eventually_mp[OF this] P show ?thesis
```
```   185     by (auto simp: frequently_def)
```
```   186 qed
```
```   187
```
```   188 lemma frequently_rev_mp:
```
```   189   assumes "\<exists>\<^sub>Fx in F. P x"
```
```   190   assumes "\<forall>\<^sub>Fx in F. P x \<longrightarrow> Q x"
```
```   191   shows "\<exists>\<^sub>Fx in F. Q x"
```
```   192 using assms(2) assms(1) by (rule frequently_mp)
```
```   193
```
```   194 lemma frequently_mono: "(\<forall>x. P x \<longrightarrow> Q x) \<Longrightarrow> frequently P F \<Longrightarrow> frequently Q F"
```
```   195   using frequently_mp[of P Q] by (simp add: always_eventually)
```
```   196
```
```   197 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"
```
```   198   by (metis frequently_mono)
```
```   199
```
```   200 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)"
```
```   201   by (simp add: frequently_def eventually_conj_iff)
```
```   202
```
```   203 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"
```
```   204   by (simp add: frequently_disj_iff)
```
```   205
```
```   206 lemma frequently_bex_finite_distrib:
```
```   207   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)"
```
```   208   using assms by induction (auto simp: frequently_disj_iff)
```
```   209
```
```   210 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"
```
```   211   by (simp add: frequently_bex_finite_distrib)
```
```   212
```
```   213 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))"
```
```   214   using eventually_ex[of "\<lambda>x y. \<not> P x y" F] by (simp add: frequently_def)
```
```   215
```
```   216 lemma
```
```   217   shows not_eventually: "\<not> eventually P F \<longleftrightarrow> (\<exists>\<^sub>Fx in F. \<not> P x)"
```
```   218     and not_frequently: "\<not> frequently P F \<longleftrightarrow> (\<forall>\<^sub>Fx in F. \<not> P x)"
```
```   219   by (auto simp: frequently_def)
```
```   220
```
```   221 lemma frequently_imp_iff:
```
```   222   "(\<exists>\<^sub>Fx in F. P x \<longrightarrow> Q x) \<longleftrightarrow> (eventually P F \<longrightarrow> frequently Q F)"
```
```   223   unfolding imp_conv_disj frequently_disj_iff not_eventually[symmetric] ..
```
```   224
```
```   225 lemma eventually_frequently_const_simps:
```
```   226   "(\<exists>\<^sub>Fx in F. P x \<and> C) \<longleftrightarrow> (\<exists>\<^sub>Fx in F. P x) \<and> C"
```
```   227   "(\<exists>\<^sub>Fx in F. C \<and> P x) \<longleftrightarrow> C \<and> (\<exists>\<^sub>Fx in F. P x)"
```
```   228   "(\<forall>\<^sub>Fx in F. P x \<or> C) \<longleftrightarrow> (\<forall>\<^sub>Fx in F. P x) \<or> C"
```
```   229   "(\<forall>\<^sub>Fx in F. C \<or> P x) \<longleftrightarrow> C \<or> (\<forall>\<^sub>Fx in F. P x)"
```
```   230   "(\<forall>\<^sub>Fx in F. P x \<longrightarrow> C) \<longleftrightarrow> ((\<exists>\<^sub>Fx in F. P x) \<longrightarrow> C)"
```
```   231   "(\<forall>\<^sub>Fx in F. C \<longrightarrow> P x) \<longleftrightarrow> (C \<longrightarrow> (\<forall>\<^sub>Fx in F. P x))"
```
```   232   by (cases C; simp add: not_frequently)+
```
```   233
```
```   234 lemmas eventually_frequently_simps =
```
```   235   eventually_frequently_const_simps
```
```   236   not_eventually
```
```   237   eventually_conj_iff
```
```   238   eventually_ball_finite_distrib
```
```   239   eventually_ex
```
```   240   not_frequently
```
```   241   frequently_disj_iff
```
```   242   frequently_bex_finite_distrib
```
```   243   frequently_all
```
```   244   frequently_imp_iff
```
```   245
```
```   246 ML \<open>
```
```   247   fun eventually_elim_tac ctxt facts = SUBGOAL_CASES (fn (goal, i) =>
```
```   248     let
```
```   249       val mp_thms = facts RL @{thms eventually_rev_mp}
```
```   250       val raw_elim_thm =
```
```   251         (@{thm allI} RS @{thm always_eventually})
```
```   252         |> fold (fn thm1 => fn thm2 => thm2 RS thm1) mp_thms
```
```   253         |> fold (fn _ => fn thm => @{thm impI} RS thm) facts
```
```   254       val cases_prop =
```
```   255         Thm.prop_of
```
```   256           (Rule_Cases.internalize_params (raw_elim_thm RS Goal.init (Thm.cterm_of ctxt goal)))
```
```   257       val cases = Rule_Cases.make_common ctxt cases_prop [(("elim", []), [])]
```
```   258     in
```
```   259       CASES cases (resolve_tac ctxt [raw_elim_thm] i)
```
```   260     end)
```
```   261 \<close>
```
```   262
```
```   263 method_setup eventually_elim = \<open>
```
```   264   Scan.succeed (fn ctxt => METHOD_CASES (HEADGOAL o eventually_elim_tac ctxt))
```
```   265 \<close> "elimination of eventually quantifiers"
```
```   266
```
```   267 subsubsection \<open>Finer-than relation\<close>
```
```   268
```
```   269 text \<open>@{term "F \<le> F'"} means that filter @{term F} is finer than
```
```   270 filter @{term F'}.\<close>
```
```   271
```
```   272 instantiation filter :: (type) complete_lattice
```
```   273 begin
```
```   274
```
```   275 definition le_filter_def:
```
```   276   "F \<le> F' \<longleftrightarrow> (\<forall>P. eventually P F' \<longrightarrow> eventually P F)"
```
```   277
```
```   278 definition
```
```   279   "(F :: 'a filter) < F' \<longleftrightarrow> F \<le> F' \<and> \<not> F' \<le> F"
```
```   280
```
```   281 definition
```
```   282   "top = Abs_filter (\<lambda>P. \<forall>x. P x)"
```
```   283
```
```   284 definition
```
```   285   "bot = Abs_filter (\<lambda>P. True)"
```
```   286
```
```   287 definition
```
```   288   "sup F F' = Abs_filter (\<lambda>P. eventually P F \<and> eventually P F')"
```
```   289
```
```   290 definition
```
```   291   "inf F F' = Abs_filter
```
```   292       (\<lambda>P. \<exists>Q R. eventually Q F \<and> eventually R F' \<and> (\<forall>x. Q x \<and> R x \<longrightarrow> P x))"
```
```   293
```
```   294 definition
```
```   295   "Sup S = Abs_filter (\<lambda>P. \<forall>F\<in>S. eventually P F)"
```
```   296
```
```   297 definition
```
```   298   "Inf S = Sup {F::'a filter. \<forall>F'\<in>S. F \<le> F'}"
```
```   299
```
```   300 lemma eventually_top [simp]: "eventually P top \<longleftrightarrow> (\<forall>x. P x)"
```
```   301   unfolding top_filter_def
```
```   302   by (rule eventually_Abs_filter, rule is_filter.intro, auto)
```
```   303
```
```   304 lemma eventually_bot [simp]: "eventually P bot"
```
```   305   unfolding bot_filter_def
```
```   306   by (subst eventually_Abs_filter, rule is_filter.intro, auto)
```
```   307
```
```   308 lemma eventually_sup:
```
```   309   "eventually P (sup F F') \<longleftrightarrow> eventually P F \<and> eventually P F'"
```
```   310   unfolding sup_filter_def
```
```   311   by (rule eventually_Abs_filter, rule is_filter.intro)
```
```   312      (auto elim!: eventually_rev_mp)
```
```   313
```
```   314 lemma eventually_inf:
```
```   315   "eventually P (inf F F') \<longleftrightarrow>
```
```   316    (\<exists>Q R. eventually Q F \<and> eventually R F' \<and> (\<forall>x. Q x \<and> R x \<longrightarrow> P x))"
```
```   317   unfolding inf_filter_def
```
```   318   apply (rule eventually_Abs_filter, rule is_filter.intro)
```
```   319   apply (fast intro: eventually_True)
```
```   320   apply clarify
```
```   321   apply (intro exI conjI)
```
```   322   apply (erule (1) eventually_conj)
```
```   323   apply (erule (1) eventually_conj)
```
```   324   apply simp
```
```   325   apply auto
```
```   326   done
```
```   327
```
```   328 lemma eventually_Sup:
```
```   329   "eventually P (Sup S) \<longleftrightarrow> (\<forall>F\<in>S. eventually P F)"
```
```   330   unfolding Sup_filter_def
```
```   331   apply (rule eventually_Abs_filter, rule is_filter.intro)
```
```   332   apply (auto intro: eventually_conj elim!: eventually_rev_mp)
```
```   333   done
```
```   334
```
```   335 instance proof
```
```   336   fix F F' F'' :: "'a filter" and S :: "'a filter set"
```
```   337   { show "F < F' \<longleftrightarrow> F \<le> F' \<and> \<not> F' \<le> F"
```
```   338     by (rule less_filter_def) }
```
```   339   { show "F \<le> F"
```
```   340     unfolding le_filter_def by simp }
```
```   341   { assume "F \<le> F'" and "F' \<le> F''" thus "F \<le> F''"
```
```   342     unfolding le_filter_def by simp }
```
```   343   { assume "F \<le> F'" and "F' \<le> F" thus "F = F'"
```
```   344     unfolding le_filter_def filter_eq_iff by fast }
```
```   345   { show "inf F F' \<le> F" and "inf F F' \<le> F'"
```
```   346     unfolding le_filter_def eventually_inf by (auto intro: eventually_True) }
```
```   347   { assume "F \<le> F'" and "F \<le> F''" thus "F \<le> inf F' F''"
```
```   348     unfolding le_filter_def eventually_inf
```
```   349     by (auto intro: eventually_mono' [OF eventually_conj]) }
```
```   350   { show "F \<le> sup F F'" and "F' \<le> sup F F'"
```
```   351     unfolding le_filter_def eventually_sup by simp_all }
```
```   352   { assume "F \<le> F''" and "F' \<le> F''" thus "sup F F' \<le> F''"
```
```   353     unfolding le_filter_def eventually_sup by simp }
```
```   354   { assume "F'' \<in> S" thus "Inf S \<le> F''"
```
```   355     unfolding le_filter_def Inf_filter_def eventually_Sup Ball_def by simp }
```
```   356   { assume "\<And>F'. F' \<in> S \<Longrightarrow> F \<le> F'" thus "F \<le> Inf S"
```
```   357     unfolding le_filter_def Inf_filter_def eventually_Sup Ball_def by simp }
```
```   358   { assume "F \<in> S" thus "F \<le> Sup S"
```
```   359     unfolding le_filter_def eventually_Sup by simp }
```
```   360   { assume "\<And>F. F \<in> S \<Longrightarrow> F \<le> F'" thus "Sup S \<le> F'"
```
```   361     unfolding le_filter_def eventually_Sup by simp }
```
```   362   { show "Inf {} = (top::'a filter)"
```
```   363     by (auto simp: top_filter_def Inf_filter_def Sup_filter_def)
```
```   364       (metis (full_types) top_filter_def always_eventually eventually_top) }
```
```   365   { show "Sup {} = (bot::'a filter)"
```
```   366     by (auto simp: bot_filter_def Sup_filter_def) }
```
```   367 qed
```
```   368
```
```   369 end
```
```   370
```
```   371 lemma filter_leD:
```
```   372   "F \<le> F' \<Longrightarrow> eventually P F' \<Longrightarrow> eventually P F"
```
```   373   unfolding le_filter_def by simp
```
```   374
```
```   375 lemma filter_leI:
```
```   376   "(\<And>P. eventually P F' \<Longrightarrow> eventually P F) \<Longrightarrow> F \<le> F'"
```
```   377   unfolding le_filter_def by simp
```
```   378
```
```   379 lemma eventually_False:
```
```   380   "eventually (\<lambda>x. False) F \<longleftrightarrow> F = bot"
```
```   381   unfolding filter_eq_iff by (auto elim: eventually_rev_mp)
```
```   382
```
```   383 lemma eventually_frequently: "F \<noteq> bot \<Longrightarrow> eventually P F \<Longrightarrow> frequently P F"
```
```   384   using eventually_conj[of P F "\<lambda>x. \<not> P x"]
```
```   385   by (auto simp add: frequently_def eventually_False)
```
```   386
```
```   387 lemma eventually_const_iff: "eventually (\<lambda>x. P) F \<longleftrightarrow> P \<or> F = bot"
```
```   388   by (cases P) (auto simp: eventually_False)
```
```   389
```
```   390 lemma eventually_const[simp]: "F \<noteq> bot \<Longrightarrow> eventually (\<lambda>x. P) F \<longleftrightarrow> P"
```
```   391   by (simp add: eventually_const_iff)
```
```   392
```
```   393 lemma frequently_const_iff: "frequently (\<lambda>x. P) F \<longleftrightarrow> P \<and> F \<noteq> bot"
```
```   394   by (simp add: frequently_def eventually_const_iff)
```
```   395
```
```   396 lemma frequently_const[simp]: "F \<noteq> bot \<Longrightarrow> frequently (\<lambda>x. P) F \<longleftrightarrow> P"
```
```   397   by (simp add: frequently_const_iff)
```
```   398
```
```   399 lemma eventually_happens: "eventually P net \<Longrightarrow> net = bot \<or> (\<exists>x. P x)"
```
```   400   by (metis frequentlyE eventually_frequently)
```
```   401
```
```   402 lemma eventually_happens':
```
```   403   assumes "F \<noteq> bot" "eventually P F"
```
```   404   shows   "\<exists>x. P x"
```
```   405   using assms eventually_frequently frequentlyE by blast
```
```   406
```
```   407 abbreviation (input) trivial_limit :: "'a filter \<Rightarrow> bool"
```
```   408   where "trivial_limit F \<equiv> F = bot"
```
```   409
```
```   410 lemma trivial_limit_def: "trivial_limit F \<longleftrightarrow> eventually (\<lambda>x. False) F"
```
```   411   by (rule eventually_False [symmetric])
```
```   412
```
```   413 lemma False_imp_not_eventually: "(\<forall>x. \<not> P x ) \<Longrightarrow> \<not> trivial_limit net \<Longrightarrow> \<not> eventually (\<lambda>x. P x) net"
```
```   414   by (simp add: eventually_False)
```
```   415
```
```   416 lemma eventually_Inf: "eventually P (Inf B) \<longleftrightarrow> (\<exists>X\<subseteq>B. finite X \<and> eventually P (Inf X))"
```
```   417 proof -
```
```   418   let ?F = "\<lambda>P. \<exists>X\<subseteq>B. finite X \<and> eventually P (Inf X)"
```
```   419
```
```   420   { fix P have "eventually P (Abs_filter ?F) \<longleftrightarrow> ?F P"
```
```   421     proof (rule eventually_Abs_filter is_filter.intro)+
```
```   422       show "?F (\<lambda>x. True)"
```
```   423         by (rule exI[of _ "{}"]) (simp add: le_fun_def)
```
```   424     next
```
```   425       fix P Q
```
```   426       assume "?F P" then guess X ..
```
```   427       moreover
```
```   428       assume "?F Q" then guess Y ..
```
```   429       ultimately show "?F (\<lambda>x. P x \<and> Q x)"
```
```   430         by (intro exI[of _ "X \<union> Y"])
```
```   431            (auto simp: Inf_union_distrib eventually_inf)
```
```   432     next
```
```   433       fix P Q
```
```   434       assume "?F P" then guess X ..
```
```   435       moreover assume "\<forall>x. P x \<longrightarrow> Q x"
```
```   436       ultimately show "?F Q"
```
```   437         by (intro exI[of _ X]) (auto elim: eventually_elim1)
```
```   438     qed }
```
```   439   note eventually_F = this
```
```   440
```
```   441   have "Inf B = Abs_filter ?F"
```
```   442   proof (intro antisym Inf_greatest)
```
```   443     show "Inf B \<le> Abs_filter ?F"
```
```   444       by (auto simp: le_filter_def eventually_F dest: Inf_superset_mono)
```
```   445   next
```
```   446     fix F assume "F \<in> B" then show "Abs_filter ?F \<le> F"
```
```   447       by (auto simp add: le_filter_def eventually_F intro!: exI[of _ "{F}"])
```
```   448   qed
```
```   449   then show ?thesis
```
```   450     by (simp add: eventually_F)
```
```   451 qed
```
```   452
```
```   453 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))"
```
```   454   unfolding INF_def[of B] eventually_Inf[of P "F`B"]
```
```   455   by (metis Inf_image_eq finite_imageI image_mono finite_subset_image)
```
```   456
```
```   457 lemma Inf_filter_not_bot:
```
```   458   fixes B :: "'a filter set"
```
```   459   shows "(\<And>X. X \<subseteq> B \<Longrightarrow> finite X \<Longrightarrow> Inf X \<noteq> bot) \<Longrightarrow> Inf B \<noteq> bot"
```
```   460   unfolding trivial_limit_def eventually_Inf[of _ B]
```
```   461     bot_bool_def [symmetric] bot_fun_def [symmetric] bot_unique by simp
```
```   462
```
```   463 lemma INF_filter_not_bot:
```
```   464   fixes F :: "'i \<Rightarrow> 'a filter"
```
```   465   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"
```
```   466   unfolding trivial_limit_def eventually_INF[of _ B]
```
```   467     bot_bool_def [symmetric] bot_fun_def [symmetric] bot_unique by simp
```
```   468
```
```   469 lemma eventually_Inf_base:
```
```   470   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"
```
```   471   shows "eventually P (Inf B) \<longleftrightarrow> (\<exists>b\<in>B. eventually P b)"
```
```   472 proof (subst eventually_Inf, safe)
```
```   473   fix X assume "finite X" "X \<subseteq> B"
```
```   474   then have "\<exists>b\<in>B. \<forall>x\<in>X. b \<le> x"
```
```   475   proof induct
```
```   476     case empty then show ?case
```
```   477       using \<open>B \<noteq> {}\<close> by auto
```
```   478   next
```
```   479     case (insert x X)
```
```   480     then obtain b where "b \<in> B" "\<And>x. x \<in> X \<Longrightarrow> b \<le> x"
```
```   481       by auto
```
```   482     with \<open>insert x X \<subseteq> B\<close> base[of b x] show ?case
```
```   483       by (auto intro: order_trans)
```
```   484   qed
```
```   485   then obtain b where "b \<in> B" "b \<le> Inf X"
```
```   486     by (auto simp: le_Inf_iff)
```
```   487   then show "eventually P (Inf X) \<Longrightarrow> Bex B (eventually P)"
```
```   488     by (intro bexI[of _ b]) (auto simp: le_filter_def)
```
```   489 qed (auto intro!: exI[of _ "{x}" for x])
```
```   490
```
```   491 lemma eventually_INF_base:
```
```   492   "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>
```
```   493     eventually P (INF b:B. F b) \<longleftrightarrow> (\<exists>b\<in>B. eventually P (F b))"
```
```   494   unfolding INF_def by (subst eventually_Inf_base) auto
```
```   495
```
```   496
```
```   497 subsubsection \<open>Map function for filters\<close>
```
```   498
```
```   499 definition filtermap :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a filter \<Rightarrow> 'b filter"
```
```   500   where "filtermap f F = Abs_filter (\<lambda>P. eventually (\<lambda>x. P (f x)) F)"
```
```   501
```
```   502 lemma eventually_filtermap:
```
```   503   "eventually P (filtermap f F) = eventually (\<lambda>x. P (f x)) F"
```
```   504   unfolding filtermap_def
```
```   505   apply (rule eventually_Abs_filter)
```
```   506   apply (rule is_filter.intro)
```
```   507   apply (auto elim!: eventually_rev_mp)
```
```   508   done
```
```   509
```
```   510 lemma filtermap_ident: "filtermap (\<lambda>x. x) F = F"
```
```   511   by (simp add: filter_eq_iff eventually_filtermap)
```
```   512
```
```   513 lemma filtermap_filtermap:
```
```   514   "filtermap f (filtermap g F) = filtermap (\<lambda>x. f (g x)) F"
```
```   515   by (simp add: filter_eq_iff eventually_filtermap)
```
```   516
```
```   517 lemma filtermap_mono: "F \<le> F' \<Longrightarrow> filtermap f F \<le> filtermap f F'"
```
```   518   unfolding le_filter_def eventually_filtermap by simp
```
```   519
```
```   520 lemma filtermap_bot [simp]: "filtermap f bot = bot"
```
```   521   by (simp add: filter_eq_iff eventually_filtermap)
```
```   522
```
```   523 lemma filtermap_sup: "filtermap f (sup F1 F2) = sup (filtermap f F1) (filtermap f F2)"
```
```   524   by (auto simp: filter_eq_iff eventually_filtermap eventually_sup)
```
```   525
```
```   526 lemma filtermap_inf: "filtermap f (inf F1 F2) \<le> inf (filtermap f F1) (filtermap f F2)"
```
```   527   by (auto simp: le_filter_def eventually_filtermap eventually_inf)
```
```   528
```
```   529 lemma filtermap_INF: "filtermap f (INF b:B. F b) \<le> (INF b:B. filtermap f (F b))"
```
```   530 proof -
```
```   531   { fix X :: "'c set" assume "finite X"
```
```   532     then have "filtermap f (INFIMUM X F) \<le> (INF b:X. filtermap f (F b))"
```
```   533     proof induct
```
```   534       case (insert x X)
```
```   535       have "filtermap f (INF a:insert x X. F a) \<le> inf (filtermap f (F x)) (filtermap f (INF a:X. F a))"
```
```   536         by (rule order_trans[OF _ filtermap_inf]) simp
```
```   537       also have "\<dots> \<le> inf (filtermap f (F x)) (INF a:X. filtermap f (F a))"
```
```   538         by (intro inf_mono insert order_refl)
```
```   539       finally show ?case
```
```   540         by simp
```
```   541     qed simp }
```
```   542   then show ?thesis
```
```   543     unfolding le_filter_def eventually_filtermap
```
```   544     by (subst (1 2) eventually_INF) auto
```
```   545 qed
```
```   546 subsubsection \<open>Standard filters\<close>
```
```   547
```
```   548 definition principal :: "'a set \<Rightarrow> 'a filter" where
```
```   549   "principal S = Abs_filter (\<lambda>P. \<forall>x\<in>S. P x)"
```
```   550
```
```   551 lemma eventually_principal: "eventually P (principal S) \<longleftrightarrow> (\<forall>x\<in>S. P x)"
```
```   552   unfolding principal_def
```
```   553   by (rule eventually_Abs_filter, rule is_filter.intro) auto
```
```   554
```
```   555 lemma eventually_inf_principal: "eventually P (inf F (principal s)) \<longleftrightarrow> eventually (\<lambda>x. x \<in> s \<longrightarrow> P x) F"
```
```   556   unfolding eventually_inf eventually_principal by (auto elim: eventually_elim1)
```
```   557
```
```   558 lemma principal_UNIV[simp]: "principal UNIV = top"
```
```   559   by (auto simp: filter_eq_iff eventually_principal)
```
```   560
```
```   561 lemma principal_empty[simp]: "principal {} = bot"
```
```   562   by (auto simp: filter_eq_iff eventually_principal)
```
```   563
```
```   564 lemma principal_eq_bot_iff: "principal X = bot \<longleftrightarrow> X = {}"
```
```   565   by (auto simp add: filter_eq_iff eventually_principal)
```
```   566
```
```   567 lemma principal_le_iff[iff]: "principal A \<le> principal B \<longleftrightarrow> A \<subseteq> B"
```
```   568   by (auto simp: le_filter_def eventually_principal)
```
```   569
```
```   570 lemma le_principal: "F \<le> principal A \<longleftrightarrow> eventually (\<lambda>x. x \<in> A) F"
```
```   571   unfolding le_filter_def eventually_principal
```
```   572   apply safe
```
```   573   apply (erule_tac x="\<lambda>x. x \<in> A" in allE)
```
```   574   apply (auto elim: eventually_elim1)
```
```   575   done
```
```   576
```
```   577 lemma principal_inject[iff]: "principal A = principal B \<longleftrightarrow> A = B"
```
```   578   unfolding eq_iff by simp
```
```   579
```
```   580 lemma sup_principal[simp]: "sup (principal A) (principal B) = principal (A \<union> B)"
```
```   581   unfolding filter_eq_iff eventually_sup eventually_principal by auto
```
```   582
```
```   583 lemma inf_principal[simp]: "inf (principal A) (principal B) = principal (A \<inter> B)"
```
```   584   unfolding filter_eq_iff eventually_inf eventually_principal
```
```   585   by (auto intro: exI[of _ "\<lambda>x. x \<in> A"] exI[of _ "\<lambda>x. x \<in> B"])
```
```   586
```
```   587 lemma SUP_principal[simp]: "(SUP i : I. principal (A i)) = principal (\<Union>i\<in>I. A i)"
```
```   588   unfolding filter_eq_iff eventually_Sup SUP_def by (auto simp: eventually_principal)
```
```   589
```
```   590 lemma INF_principal_finite: "finite X \<Longrightarrow> (INF x:X. principal (f x)) = principal (\<Inter>x\<in>X. f x)"
```
```   591   by (induct X rule: finite_induct) auto
```
```   592
```
```   593 lemma filtermap_principal[simp]: "filtermap f (principal A) = principal (f ` A)"
```
```   594   unfolding filter_eq_iff eventually_filtermap eventually_principal by simp
```
```   595
```
```   596 subsubsection \<open>Order filters\<close>
```
```   597
```
```   598 definition at_top :: "('a::order) filter"
```
```   599   where "at_top = (INF k. principal {k ..})"
```
```   600
```
```   601 lemma at_top_sub: "at_top = (INF k:{c::'a::linorder..}. principal {k ..})"
```
```   602   by (auto intro!: INF_eq max.cobounded1 max.cobounded2 simp: at_top_def)
```
```   603
```
```   604 lemma eventually_at_top_linorder: "eventually P at_top \<longleftrightarrow> (\<exists>N::'a::linorder. \<forall>n\<ge>N. P n)"
```
```   605   unfolding at_top_def
```
```   606   by (subst eventually_INF_base) (auto simp: eventually_principal intro: max.cobounded1 max.cobounded2)
```
```   607
```
```   608 lemma eventually_ge_at_top:
```
```   609   "eventually (\<lambda>x. (c::_::linorder) \<le> x) at_top"
```
```   610   unfolding eventually_at_top_linorder by auto
```
```   611
```
```   612 lemma eventually_at_top_dense: "eventually P at_top \<longleftrightarrow> (\<exists>N::'a::{no_top, linorder}. \<forall>n>N. P n)"
```
```   613 proof -
```
```   614   have "eventually P (INF k. principal {k <..}) \<longleftrightarrow> (\<exists>N::'a. \<forall>n>N. P n)"
```
```   615     by (subst eventually_INF_base) (auto simp: eventually_principal intro: max.cobounded1 max.cobounded2)
```
```   616   also have "(INF k. principal {k::'a <..}) = at_top"
```
```   617     unfolding at_top_def
```
```   618     by (intro INF_eq) (auto intro: less_imp_le simp: Ici_subset_Ioi_iff gt_ex)
```
```   619   finally show ?thesis .
```
```   620 qed
```
```   621
```
```   622 lemma eventually_at_top_not_equal: "eventually (\<lambda>x::'a::{no_top, linorder}. x \<noteq> c) at_top"
```
```   623   unfolding eventually_at_top_dense by auto
```
```   624
```
```   625 lemma eventually_gt_at_top: "eventually (\<lambda>x. (c::_::{no_top, linorder}) < x) at_top"
```
```   626   unfolding eventually_at_top_dense by auto
```
```   627
```
```   628 lemma eventually_all_ge_at_top:
```
```   629   assumes "eventually P (at_top :: ('a :: linorder) filter)"
```
```   630   shows   "eventually (\<lambda>x. \<forall>y\<ge>x. P y) at_top"
```
```   631 proof -
```
```   632   from assms obtain x where "\<And>y. y \<ge> x \<Longrightarrow> P y" by (auto simp: eventually_at_top_linorder)
```
```   633   hence "\<forall>z\<ge>y. P z" if "y \<ge> x" for y using that by simp
```
```   634   thus ?thesis by (auto simp: eventually_at_top_linorder)
```
```   635 qed
```
```   636
```
```   637 definition at_bot :: "('a::order) filter"
```
```   638   where "at_bot = (INF k. principal {.. k})"
```
```   639
```
```   640 lemma at_bot_sub: "at_bot = (INF k:{.. c::'a::linorder}. principal {.. k})"
```
```   641   by (auto intro!: INF_eq min.cobounded1 min.cobounded2 simp: at_bot_def)
```
```   642
```
```   643 lemma eventually_at_bot_linorder:
```
```   644   fixes P :: "'a::linorder \<Rightarrow> bool" shows "eventually P at_bot \<longleftrightarrow> (\<exists>N. \<forall>n\<le>N. P n)"
```
```   645   unfolding at_bot_def
```
```   646   by (subst eventually_INF_base) (auto simp: eventually_principal intro: min.cobounded1 min.cobounded2)
```
```   647
```
```   648 lemma eventually_le_at_bot:
```
```   649   "eventually (\<lambda>x. x \<le> (c::_::linorder)) at_bot"
```
```   650   unfolding eventually_at_bot_linorder by auto
```
```   651
```
```   652 lemma eventually_at_bot_dense: "eventually P at_bot \<longleftrightarrow> (\<exists>N::'a::{no_bot, linorder}. \<forall>n<N. P n)"
```
```   653 proof -
```
```   654   have "eventually P (INF k. principal {..< k}) \<longleftrightarrow> (\<exists>N::'a. \<forall>n<N. P n)"
```
```   655     by (subst eventually_INF_base) (auto simp: eventually_principal intro: min.cobounded1 min.cobounded2)
```
```   656   also have "(INF k. principal {..< k::'a}) = at_bot"
```
```   657     unfolding at_bot_def
```
```   658     by (intro INF_eq) (auto intro: less_imp_le simp: Iic_subset_Iio_iff lt_ex)
```
```   659   finally show ?thesis .
```
```   660 qed
```
```   661
```
```   662 lemma eventually_at_bot_not_equal: "eventually (\<lambda>x::'a::{no_bot, linorder}. x \<noteq> c) at_bot"
```
```   663   unfolding eventually_at_bot_dense by auto
```
```   664
```
```   665 lemma eventually_gt_at_bot:
```
```   666   "eventually (\<lambda>x. x < (c::_::unbounded_dense_linorder)) at_bot"
```
```   667   unfolding eventually_at_bot_dense by auto
```
```   668
```
```   669 lemma trivial_limit_at_bot_linorder: "\<not> trivial_limit (at_bot ::('a::linorder) filter)"
```
```   670   unfolding trivial_limit_def
```
```   671   by (metis eventually_at_bot_linorder order_refl)
```
```   672
```
```   673 lemma trivial_limit_at_top_linorder: "\<not> trivial_limit (at_top ::('a::linorder) filter)"
```
```   674   unfolding trivial_limit_def
```
```   675   by (metis eventually_at_top_linorder order_refl)
```
```   676
```
```   677 subsection \<open>Sequentially\<close>
```
```   678
```
```   679 abbreviation sequentially :: "nat filter"
```
```   680   where "sequentially \<equiv> at_top"
```
```   681
```
```   682 lemma eventually_sequentially:
```
```   683   "eventually P sequentially \<longleftrightarrow> (\<exists>N. \<forall>n\<ge>N. P n)"
```
```   684   by (rule eventually_at_top_linorder)
```
```   685
```
```   686 lemma sequentially_bot [simp, intro]: "sequentially \<noteq> bot"
```
```   687   unfolding filter_eq_iff eventually_sequentially by auto
```
```   688
```
```   689 lemmas trivial_limit_sequentially = sequentially_bot
```
```   690
```
```   691 lemma eventually_False_sequentially [simp]:
```
```   692   "\<not> eventually (\<lambda>n. False) sequentially"
```
```   693   by (simp add: eventually_False)
```
```   694
```
```   695 lemma le_sequentially:
```
```   696   "F \<le> sequentially \<longleftrightarrow> (\<forall>N. eventually (\<lambda>n. N \<le> n) F)"
```
```   697   by (simp add: at_top_def le_INF_iff le_principal)
```
```   698
```
```   699 lemma eventually_sequentiallyI [intro?]:
```
```   700   assumes "\<And>x. c \<le> x \<Longrightarrow> P x"
```
```   701   shows "eventually P sequentially"
```
```   702 using assms by (auto simp: eventually_sequentially)
```
```   703
```
```   704 lemma eventually_sequentially_Suc: "eventually (\<lambda>i. P (Suc i)) sequentially \<longleftrightarrow> eventually P sequentially"
```
```   705   unfolding eventually_sequentially by (metis Suc_le_D Suc_le_mono le_Suc_eq)
```
```   706
```
```   707 lemma eventually_sequentially_seg: "eventually (\<lambda>n. P (n + k)) sequentially \<longleftrightarrow> eventually P sequentially"
```
```   708   using eventually_sequentially_Suc[of "\<lambda>n. P (n + k)" for k] by (induction k) auto
```
```   709
```
```   710 subsection \<open> The cofinite filter \<close>
```
```   711
```
```   712 definition "cofinite = Abs_filter (\<lambda>P. finite {x. \<not> P x})"
```
```   713
```
```   714 abbreviation Inf_many :: "('a \<Rightarrow> bool) \<Rightarrow> bool"  (binder "INFM " 10) where
```
```   715   "Inf_many P \<equiv> frequently P cofinite"
```
```   716
```
```   717 abbreviation Alm_all :: "('a \<Rightarrow> bool) \<Rightarrow> bool"  (binder "MOST " 10) where
```
```   718   "Alm_all P \<equiv> eventually P cofinite"
```
```   719
```
```   720 notation (xsymbols)
```
```   721   Inf_many  (binder "\<exists>\<^sub>\<infinity>" 10) and
```
```   722   Alm_all  (binder "\<forall>\<^sub>\<infinity>" 10)
```
```   723
```
```   724 lemma eventually_cofinite: "eventually P cofinite \<longleftrightarrow> finite {x. \<not> P x}"
```
```   725   unfolding cofinite_def
```
```   726 proof (rule eventually_Abs_filter, rule is_filter.intro)
```
```   727   fix P Q :: "'a \<Rightarrow> bool" assume "finite {x. \<not> P x}" "finite {x. \<not> Q x}"
```
```   728   from finite_UnI[OF this] show "finite {x. \<not> (P x \<and> Q x)}"
```
```   729     by (rule rev_finite_subset) auto
```
```   730 next
```
```   731   fix P Q :: "'a \<Rightarrow> bool" assume P: "finite {x. \<not> P x}" and *: "\<forall>x. P x \<longrightarrow> Q x"
```
```   732   from * show "finite {x. \<not> Q x}"
```
```   733     by (intro finite_subset[OF _ P]) auto
```
```   734 qed simp
```
```   735
```
```   736 lemma frequently_cofinite: "frequently P cofinite \<longleftrightarrow> \<not> finite {x. P x}"
```
```   737   by (simp add: frequently_def eventually_cofinite)
```
```   738
```
```   739 lemma cofinite_bot[simp]: "cofinite = (bot::'a filter) \<longleftrightarrow> finite (UNIV :: 'a set)"
```
```   740   unfolding trivial_limit_def eventually_cofinite by simp
```
```   741
```
```   742 lemma cofinite_eq_sequentially: "cofinite = sequentially"
```
```   743   unfolding filter_eq_iff eventually_sequentially eventually_cofinite
```
```   744 proof safe
```
```   745   fix P :: "nat \<Rightarrow> bool" assume [simp]: "finite {x. \<not> P x}"
```
```   746   show "\<exists>N. \<forall>n\<ge>N. P n"
```
```   747   proof cases
```
```   748     assume "{x. \<not> P x} \<noteq> {}" then show ?thesis
```
```   749       by (intro exI[of _ "Suc (Max {x. \<not> P x})"]) (auto simp: Suc_le_eq)
```
```   750   qed auto
```
```   751 next
```
```   752   fix P :: "nat \<Rightarrow> bool" and N :: nat assume "\<forall>n\<ge>N. P n"
```
```   753   then have "{x. \<not> P x} \<subseteq> {..< N}"
```
```   754     by (auto simp: not_le)
```
```   755   then show "finite {x. \<not> P x}"
```
```   756     by (blast intro: finite_subset)
```
```   757 qed
```
```   758
```
```   759 subsection \<open>Limits\<close>
```
```   760
```
```   761 definition filterlim :: "('a \<Rightarrow> 'b) \<Rightarrow> 'b filter \<Rightarrow> 'a filter \<Rightarrow> bool" where
```
```   762   "filterlim f F2 F1 \<longleftrightarrow> filtermap f F1 \<le> F2"
```
```   763
```
```   764 syntax
```
```   765   "_LIM" :: "pttrns \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> bool" ("(3LIM (_)/ (_)./ (_) :> (_))" [1000, 10, 0, 10] 10)
```
```   766
```
```   767 translations
```
```   768   "LIM x F1. f :> F2"   == "CONST filterlim (%x. f) F2 F1"
```
```   769
```
```   770 lemma filterlim_iff:
```
```   771   "(LIM x F1. f x :> F2) \<longleftrightarrow> (\<forall>P. eventually P F2 \<longrightarrow> eventually (\<lambda>x. P (f x)) F1)"
```
```   772   unfolding filterlim_def le_filter_def eventually_filtermap ..
```
```   773
```
```   774 lemma filterlim_compose:
```
```   775   "filterlim g F3 F2 \<Longrightarrow> filterlim f F2 F1 \<Longrightarrow> filterlim (\<lambda>x. g (f x)) F3 F1"
```
```   776   unfolding filterlim_def filtermap_filtermap[symmetric] by (metis filtermap_mono order_trans)
```
```   777
```
```   778 lemma filterlim_mono:
```
```   779   "filterlim f F2 F1 \<Longrightarrow> F2 \<le> F2' \<Longrightarrow> F1' \<le> F1 \<Longrightarrow> filterlim f F2' F1'"
```
```   780   unfolding filterlim_def by (metis filtermap_mono order_trans)
```
```   781
```
```   782 lemma filterlim_ident: "LIM x F. x :> F"
```
```   783   by (simp add: filterlim_def filtermap_ident)
```
```   784
```
```   785 lemma filterlim_cong:
```
```   786   "F1 = F1' \<Longrightarrow> F2 = F2' \<Longrightarrow> eventually (\<lambda>x. f x = g x) F2 \<Longrightarrow> filterlim f F1 F2 = filterlim g F1' F2'"
```
```   787   by (auto simp: filterlim_def le_filter_def eventually_filtermap elim: eventually_elim2)
```
```   788
```
```   789 lemma filterlim_mono_eventually:
```
```   790   assumes "filterlim f F G" and ord: "F \<le> F'" "G' \<le> G"
```
```   791   assumes eq: "eventually (\<lambda>x. f x = f' x) G'"
```
```   792   shows "filterlim f' F' G'"
```
```   793   apply (rule filterlim_cong[OF refl refl eq, THEN iffD1])
```
```   794   apply (rule filterlim_mono[OF _ ord])
```
```   795   apply fact
```
```   796   done
```
```   797
```
```   798 lemma filtermap_mono_strong: "inj f \<Longrightarrow> filtermap f F \<le> filtermap f G \<longleftrightarrow> F \<le> G"
```
```   799   apply (auto intro!: filtermap_mono) []
```
```   800   apply (auto simp: le_filter_def eventually_filtermap)
```
```   801   apply (erule_tac x="\<lambda>x. P (inv f x)" in allE)
```
```   802   apply auto
```
```   803   done
```
```   804
```
```   805 lemma filtermap_eq_strong: "inj f \<Longrightarrow> filtermap f F = filtermap f G \<longleftrightarrow> F = G"
```
```   806   by (simp add: filtermap_mono_strong eq_iff)
```
```   807
```
```   808 lemma filtermap_fun_inverse:
```
```   809   assumes g: "filterlim g F G"
```
```   810   assumes f: "filterlim f G F"
```
```   811   assumes ev: "eventually (\<lambda>x. f (g x) = x) G"
```
```   812   shows "filtermap f F = G"
```
```   813 proof (rule antisym)
```
```   814   show "filtermap f F \<le> G"
```
```   815     using f unfolding filterlim_def .
```
```   816   have "G = filtermap f (filtermap g G)"
```
```   817     using ev by (auto elim: eventually_elim2 simp: filter_eq_iff eventually_filtermap)
```
```   818   also have "\<dots> \<le> filtermap f F"
```
```   819     using g by (intro filtermap_mono) (simp add: filterlim_def)
```
```   820   finally show "G \<le> filtermap f F" .
```
```   821 qed
```
```   822
```
```   823 lemma filterlim_principal:
```
```   824   "(LIM x F. f x :> principal S) \<longleftrightarrow> (eventually (\<lambda>x. f x \<in> S) F)"
```
```   825   unfolding filterlim_def eventually_filtermap le_principal ..
```
```   826
```
```   827 lemma filterlim_inf:
```
```   828   "(LIM x F1. f x :> inf F2 F3) \<longleftrightarrow> ((LIM x F1. f x :> F2) \<and> (LIM x F1. f x :> F3))"
```
```   829   unfolding filterlim_def by simp
```
```   830
```
```   831 lemma filterlim_INF:
```
```   832   "(LIM x F. f x :> (INF b:B. G b)) \<longleftrightarrow> (\<forall>b\<in>B. LIM x F. f x :> G b)"
```
```   833   unfolding filterlim_def le_INF_iff ..
```
```   834
```
```   835 lemma filterlim_INF_INF:
```
```   836   "(\<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)"
```
```   837   unfolding filterlim_def by (rule order_trans[OF filtermap_INF INF_mono])
```
```   838
```
```   839 lemma filterlim_base:
```
```   840   "(\<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>
```
```   841     LIM x (INF i:I. principal (F i)). f x :> (INF j:J. principal (G j))"
```
```   842   by (force intro!: filterlim_INF_INF simp: image_subset_iff)
```
```   843
```
```   844 lemma filterlim_base_iff:
```
```   845   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"
```
```   846   shows "(LIM x (INF i:I. principal (F i)). f x :> INF j:J. principal (G j)) \<longleftrightarrow>
```
```   847     (\<forall>j\<in>J. \<exists>i\<in>I. \<forall>x\<in>F i. f x \<in> G j)"
```
```   848   unfolding filterlim_INF filterlim_principal
```
```   849 proof (subst eventually_INF_base)
```
```   850   fix i j assume "i \<in> I" "j \<in> I"
```
```   851   with chain[OF this] show "\<exists>x\<in>I. principal (F x) \<le> inf (principal (F i)) (principal (F j))"
```
```   852     by auto
```
```   853 qed (auto simp: eventually_principal \<open>I \<noteq> {}\<close>)
```
```   854
```
```   855 lemma filterlim_filtermap: "filterlim f F1 (filtermap g F2) = filterlim (\<lambda>x. f (g x)) F1 F2"
```
```   856   unfolding filterlim_def filtermap_filtermap ..
```
```   857
```
```   858 lemma filterlim_sup:
```
```   859   "filterlim f F F1 \<Longrightarrow> filterlim f F F2 \<Longrightarrow> filterlim f F (sup F1 F2)"
```
```   860   unfolding filterlim_def filtermap_sup by auto
```
```   861
```
```   862 lemma filterlim_sequentially_Suc:
```
```   863   "(LIM x sequentially. f (Suc x) :> F) \<longleftrightarrow> (LIM x sequentially. f x :> F)"
```
```   864   unfolding filterlim_iff by (subst eventually_sequentially_Suc) simp
```
```   865
```
```   866 lemma filterlim_Suc: "filterlim Suc sequentially sequentially"
```
```   867   by (simp add: filterlim_iff eventually_sequentially) (metis le_Suc_eq)
```
```   868
```
```   869 lemma filterlim_If:
```
```   870   "LIM x inf F (principal {x. P x}). f x :> G \<Longrightarrow>
```
```   871     LIM x inf F (principal {x. \<not> P x}). g x :> G \<Longrightarrow>
```
```   872     LIM x F. if P x then f x else g x :> G"
```
```   873   unfolding filterlim_iff eventually_inf_principal by (auto simp: eventually_conj_iff)
```
```   874
```
```   875 subsection \<open>Limits to @{const at_top} and @{const at_bot}\<close>
```
```   876
```
```   877 lemma filterlim_at_top:
```
```   878   fixes f :: "'a \<Rightarrow> ('b::linorder)"
```
```   879   shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. Z \<le> f x) F)"
```
```   880   by (auto simp: filterlim_iff eventually_at_top_linorder elim!: eventually_elim1)
```
```   881
```
```   882 lemma filterlim_at_top_mono:
```
```   883   "LIM x F. f x :> at_top \<Longrightarrow> eventually (\<lambda>x. f x \<le> (g x::'a::linorder)) F \<Longrightarrow>
```
```   884     LIM x F. g x :> at_top"
```
```   885   by (auto simp: filterlim_at_top elim: eventually_elim2 intro: order_trans)
```
```   886
```
```   887 lemma filterlim_at_top_dense:
```
```   888   fixes f :: "'a \<Rightarrow> ('b::unbounded_dense_linorder)"
```
```   889   shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. Z < f x) F)"
```
```   890   by (metis eventually_elim1[of _ F] eventually_gt_at_top order_less_imp_le
```
```   891             filterlim_at_top[of f F] filterlim_iff[of f at_top F])
```
```   892
```
```   893 lemma filterlim_at_top_ge:
```
```   894   fixes f :: "'a \<Rightarrow> ('b::linorder)" and c :: "'b"
```
```   895   shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z\<ge>c. eventually (\<lambda>x. Z \<le> f x) F)"
```
```   896   unfolding at_top_sub[of c] filterlim_INF by (auto simp add: filterlim_principal)
```
```   897
```
```   898 lemma filterlim_at_top_at_top:
```
```   899   fixes f :: "'a::linorder \<Rightarrow> 'b::linorder"
```
```   900   assumes mono: "\<And>x y. Q x \<Longrightarrow> Q y \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y"
```
```   901   assumes bij: "\<And>x. P x \<Longrightarrow> f (g x) = x" "\<And>x. P x \<Longrightarrow> Q (g x)"
```
```   902   assumes Q: "eventually Q at_top"
```
```   903   assumes P: "eventually P at_top"
```
```   904   shows "filterlim f at_top at_top"
```
```   905 proof -
```
```   906   from P obtain x where x: "\<And>y. x \<le> y \<Longrightarrow> P y"
```
```   907     unfolding eventually_at_top_linorder by auto
```
```   908   show ?thesis
```
```   909   proof (intro filterlim_at_top_ge[THEN iffD2] allI impI)
```
```   910     fix z assume "x \<le> z"
```
```   911     with x have "P z" by auto
```
```   912     have "eventually (\<lambda>x. g z \<le> x) at_top"
```
```   913       by (rule eventually_ge_at_top)
```
```   914     with Q show "eventually (\<lambda>x. z \<le> f x) at_top"
```
```   915       by eventually_elim (metis mono bij \<open>P z\<close>)
```
```   916   qed
```
```   917 qed
```
```   918
```
```   919 lemma filterlim_at_top_gt:
```
```   920   fixes f :: "'a \<Rightarrow> ('b::unbounded_dense_linorder)" and c :: "'b"
```
```   921   shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z>c. eventually (\<lambda>x. Z \<le> f x) F)"
```
```   922   by (metis filterlim_at_top order_less_le_trans gt_ex filterlim_at_top_ge)
```
```   923
```
```   924 lemma filterlim_at_bot:
```
```   925   fixes f :: "'a \<Rightarrow> ('b::linorder)"
```
```   926   shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. f x \<le> Z) F)"
```
```   927   by (auto simp: filterlim_iff eventually_at_bot_linorder elim!: eventually_elim1)
```
```   928
```
```   929 lemma filterlim_at_bot_dense:
```
```   930   fixes f :: "'a \<Rightarrow> ('b::{dense_linorder, no_bot})"
```
```   931   shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. f x < Z) F)"
```
```   932 proof (auto simp add: filterlim_at_bot[of f F])
```
```   933   fix Z :: 'b
```
```   934   from lt_ex [of Z] obtain Z' where 1: "Z' < Z" ..
```
```   935   assume "\<forall>Z. eventually (\<lambda>x. f x \<le> Z) F"
```
```   936   hence "eventually (\<lambda>x. f x \<le> Z') F" by auto
```
```   937   thus "eventually (\<lambda>x. f x < Z) F"
```
```   938     apply (rule eventually_mono')
```
```   939     using 1 by auto
```
```   940   next
```
```   941     fix Z :: 'b
```
```   942     show "\<forall>Z. eventually (\<lambda>x. f x < Z) F \<Longrightarrow> eventually (\<lambda>x. f x \<le> Z) F"
```
```   943       by (drule spec [of _ Z], erule eventually_mono', auto simp add: less_imp_le)
```
```   944 qed
```
```   945
```
```   946 lemma filterlim_at_bot_le:
```
```   947   fixes f :: "'a \<Rightarrow> ('b::linorder)" and c :: "'b"
```
```   948   shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z\<le>c. eventually (\<lambda>x. Z \<ge> f x) F)"
```
```   949   unfolding filterlim_at_bot
```
```   950 proof safe
```
```   951   fix Z assume *: "\<forall>Z\<le>c. eventually (\<lambda>x. Z \<ge> f x) F"
```
```   952   with *[THEN spec, of "min Z c"] show "eventually (\<lambda>x. Z \<ge> f x) F"
```
```   953     by (auto elim!: eventually_elim1)
```
```   954 qed simp
```
```   955
```
```   956 lemma filterlim_at_bot_lt:
```
```   957   fixes f :: "'a \<Rightarrow> ('b::unbounded_dense_linorder)" and c :: "'b"
```
```   958   shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z<c. eventually (\<lambda>x. Z \<ge> f x) F)"
```
```   959   by (metis filterlim_at_bot filterlim_at_bot_le lt_ex order_le_less_trans)
```
```   960
```
```   961
```
```   962 subsection \<open>Setup @{typ "'a filter"} for lifting and transfer\<close>
```
```   963
```
```   964 context begin interpretation lifting_syntax .
```
```   965
```
```   966 definition rel_filter :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a filter \<Rightarrow> 'b filter \<Rightarrow> bool"
```
```   967 where "rel_filter R F G = ((R ===> op =) ===> op =) (Rep_filter F) (Rep_filter G)"
```
```   968
```
```   969 lemma rel_filter_eventually:
```
```   970   "rel_filter R F G \<longleftrightarrow>
```
```   971   ((R ===> op =) ===> op =) (\<lambda>P. eventually P F) (\<lambda>P. eventually P G)"
```
```   972 by(simp add: rel_filter_def eventually_def)
```
```   973
```
```   974 lemma filtermap_id [simp, id_simps]: "filtermap id = id"
```
```   975 by(simp add: fun_eq_iff id_def filtermap_ident)
```
```   976
```
```   977 lemma filtermap_id' [simp]: "filtermap (\<lambda>x. x) = (\<lambda>F. F)"
```
```   978 using filtermap_id unfolding id_def .
```
```   979
```
```   980 lemma Quotient_filter [quot_map]:
```
```   981   assumes Q: "Quotient R Abs Rep T"
```
```   982   shows "Quotient (rel_filter R) (filtermap Abs) (filtermap Rep) (rel_filter T)"
```
```   983 unfolding Quotient_alt_def
```
```   984 proof(intro conjI strip)
```
```   985   from Q have *: "\<And>x y. T x y \<Longrightarrow> Abs x = y"
```
```   986     unfolding Quotient_alt_def by blast
```
```   987
```
```   988   fix F G
```
```   989   assume "rel_filter T F G"
```
```   990   thus "filtermap Abs F = G" unfolding filter_eq_iff
```
```   991     by(auto simp add: eventually_filtermap rel_filter_eventually * rel_funI del: iffI elim!: rel_funD)
```
```   992 next
```
```   993   from Q have *: "\<And>x. T (Rep x) x" unfolding Quotient_alt_def by blast
```
```   994
```
```   995   fix F
```
```   996   show "rel_filter T (filtermap Rep F) F"
```
```   997     by(auto elim: rel_funD intro: * intro!: ext arg_cong[where f="\<lambda>P. eventually P F"] rel_funI
```
```   998             del: iffI simp add: eventually_filtermap rel_filter_eventually)
```
```   999 qed(auto simp add: map_fun_def o_def eventually_filtermap filter_eq_iff fun_eq_iff rel_filter_eventually
```
```  1000          fun_quotient[OF fun_quotient[OF Q identity_quotient] identity_quotient, unfolded Quotient_alt_def])
```
```  1001
```
```  1002 lemma eventually_parametric [transfer_rule]:
```
```  1003   "((A ===> op =) ===> rel_filter A ===> op =) eventually eventually"
```
```  1004 by(simp add: rel_fun_def rel_filter_eventually)
```
```  1005
```
```  1006 lemma frequently_parametric [transfer_rule]:
```
```  1007   "((A ===> op =) ===> rel_filter A ===> op =) frequently frequently"
```
```  1008   unfolding frequently_def[abs_def] by transfer_prover
```
```  1009
```
```  1010 lemma rel_filter_eq [relator_eq]: "rel_filter op = = op ="
```
```  1011 by(auto simp add: rel_filter_eventually rel_fun_eq fun_eq_iff filter_eq_iff)
```
```  1012
```
```  1013 lemma rel_filter_mono [relator_mono]:
```
```  1014   "A \<le> B \<Longrightarrow> rel_filter A \<le> rel_filter B"
```
```  1015 unfolding rel_filter_eventually[abs_def]
```
```  1016 by(rule le_funI)+(intro fun_mono fun_mono[THEN le_funD, THEN le_funD] order.refl)
```
```  1017
```
```  1018 lemma rel_filter_conversep [simp]: "rel_filter A\<inverse>\<inverse> = (rel_filter A)\<inverse>\<inverse>"
```
```  1019 apply (simp add: rel_filter_eventually fun_eq_iff rel_fun_def)
```
```  1020 apply (safe; metis)
```
```  1021 done
```
```  1022
```
```  1023 lemma is_filter_parametric_aux:
```
```  1024   assumes "is_filter F"
```
```  1025   assumes [transfer_rule]: "bi_total A" "bi_unique A"
```
```  1026   and [transfer_rule]: "((A ===> op =) ===> op =) F G"
```
```  1027   shows "is_filter G"
```
```  1028 proof -
```
```  1029   interpret is_filter F by fact
```
```  1030   show ?thesis
```
```  1031   proof
```
```  1032     have "F (\<lambda>_. True) = G (\<lambda>x. True)" by transfer_prover
```
```  1033     thus "G (\<lambda>x. True)" by(simp add: True)
```
```  1034   next
```
```  1035     fix P' Q'
```
```  1036     assume "G P'" "G Q'"
```
```  1037     moreover
```
```  1038     from bi_total_fun[OF \<open>bi_unique A\<close> bi_total_eq, unfolded bi_total_def]
```
```  1039     obtain P Q where [transfer_rule]: "(A ===> op =) P P'" "(A ===> op =) Q Q'" by blast
```
```  1040     have "F P = G P'" "F Q = G Q'" by transfer_prover+
```
```  1041     ultimately have "F (\<lambda>x. P x \<and> Q x)" by(simp add: conj)
```
```  1042     moreover have "F (\<lambda>x. P x \<and> Q x) = G (\<lambda>x. P' x \<and> Q' x)" by transfer_prover
```
```  1043     ultimately show "G (\<lambda>x. P' x \<and> Q' x)" by simp
```
```  1044   next
```
```  1045     fix P' Q'
```
```  1046     assume "\<forall>x. P' x \<longrightarrow> Q' x" "G P'"
```
```  1047     moreover
```
```  1048     from bi_total_fun[OF \<open>bi_unique A\<close> bi_total_eq, unfolded bi_total_def]
```
```  1049     obtain P Q where [transfer_rule]: "(A ===> op =) P P'" "(A ===> op =) Q Q'" by blast
```
```  1050     have "F P = G P'" by transfer_prover
```
```  1051     moreover have "(\<forall>x. P x \<longrightarrow> Q x) \<longleftrightarrow> (\<forall>x. P' x \<longrightarrow> Q' x)" by transfer_prover
```
```  1052     ultimately have "F Q" by(simp add: mono)
```
```  1053     moreover have "F Q = G Q'" by transfer_prover
```
```  1054     ultimately show "G Q'" by simp
```
```  1055   qed
```
```  1056 qed
```
```  1057
```
```  1058 lemma is_filter_parametric [transfer_rule]:
```
```  1059   "\<lbrakk> bi_total A; bi_unique A \<rbrakk>
```
```  1060   \<Longrightarrow> (((A ===> op =) ===> op =) ===> op =) is_filter is_filter"
```
```  1061 apply(rule rel_funI)
```
```  1062 apply(rule iffI)
```
```  1063  apply(erule (3) is_filter_parametric_aux)
```
```  1064 apply(erule is_filter_parametric_aux[where A="conversep A"])
```
```  1065 apply (simp_all add: rel_fun_def)
```
```  1066 apply metis
```
```  1067 done
```
```  1068
```
```  1069 lemma left_total_rel_filter [transfer_rule]:
```
```  1070   assumes [transfer_rule]: "bi_total A" "bi_unique A"
```
```  1071   shows "left_total (rel_filter A)"
```
```  1072 proof(rule left_totalI)
```
```  1073   fix F :: "'a filter"
```
```  1074   from bi_total_fun[OF bi_unique_fun[OF \<open>bi_total A\<close> bi_unique_eq] bi_total_eq]
```
```  1075   obtain G where [transfer_rule]: "((A ===> op =) ===> op =) (\<lambda>P. eventually P F) G"
```
```  1076     unfolding  bi_total_def by blast
```
```  1077   moreover have "is_filter (\<lambda>P. eventually P F) \<longleftrightarrow> is_filter G" by transfer_prover
```
```  1078   hence "is_filter G" by(simp add: eventually_def is_filter_Rep_filter)
```
```  1079   ultimately have "rel_filter A F (Abs_filter G)"
```
```  1080     by(simp add: rel_filter_eventually eventually_Abs_filter)
```
```  1081   thus "\<exists>G. rel_filter A F G" ..
```
```  1082 qed
```
```  1083
```
```  1084 lemma right_total_rel_filter [transfer_rule]:
```
```  1085   "\<lbrakk> bi_total A; bi_unique A \<rbrakk> \<Longrightarrow> right_total (rel_filter A)"
```
```  1086 using left_total_rel_filter[of "A\<inverse>\<inverse>"] by simp
```
```  1087
```
```  1088 lemma bi_total_rel_filter [transfer_rule]:
```
```  1089   assumes "bi_total A" "bi_unique A"
```
```  1090   shows "bi_total (rel_filter A)"
```
```  1091 unfolding bi_total_alt_def using assms
```
```  1092 by(simp add: left_total_rel_filter right_total_rel_filter)
```
```  1093
```
```  1094 lemma left_unique_rel_filter [transfer_rule]:
```
```  1095   assumes "left_unique A"
```
```  1096   shows "left_unique (rel_filter A)"
```
```  1097 proof(rule left_uniqueI)
```
```  1098   fix F F' G
```
```  1099   assume [transfer_rule]: "rel_filter A F G" "rel_filter A F' G"
```
```  1100   show "F = F'"
```
```  1101     unfolding filter_eq_iff
```
```  1102   proof
```
```  1103     fix P :: "'a \<Rightarrow> bool"
```
```  1104     obtain P' where [transfer_rule]: "(A ===> op =) P P'"
```
```  1105       using left_total_fun[OF assms left_total_eq] unfolding left_total_def by blast
```
```  1106     have "eventually P F = eventually P' G"
```
```  1107       and "eventually P F' = eventually P' G" by transfer_prover+
```
```  1108     thus "eventually P F = eventually P F'" by simp
```
```  1109   qed
```
```  1110 qed
```
```  1111
```
```  1112 lemma right_unique_rel_filter [transfer_rule]:
```
```  1113   "right_unique A \<Longrightarrow> right_unique (rel_filter A)"
```
```  1114 using left_unique_rel_filter[of "A\<inverse>\<inverse>"] by simp
```
```  1115
```
```  1116 lemma bi_unique_rel_filter [transfer_rule]:
```
```  1117   "bi_unique A \<Longrightarrow> bi_unique (rel_filter A)"
```
```  1118 by(simp add: bi_unique_alt_def left_unique_rel_filter right_unique_rel_filter)
```
```  1119
```
```  1120 lemma top_filter_parametric [transfer_rule]:
```
```  1121   "bi_total A \<Longrightarrow> (rel_filter A) top top"
```
```  1122 by(simp add: rel_filter_eventually All_transfer)
```
```  1123
```
```  1124 lemma bot_filter_parametric [transfer_rule]: "(rel_filter A) bot bot"
```
```  1125 by(simp add: rel_filter_eventually rel_fun_def)
```
```  1126
```
```  1127 lemma sup_filter_parametric [transfer_rule]:
```
```  1128   "(rel_filter A ===> rel_filter A ===> rel_filter A) sup sup"
```
```  1129 by(fastforce simp add: rel_filter_eventually[abs_def] eventually_sup dest: rel_funD)
```
```  1130
```
```  1131 lemma Sup_filter_parametric [transfer_rule]:
```
```  1132   "(rel_set (rel_filter A) ===> rel_filter A) Sup Sup"
```
```  1133 proof(rule rel_funI)
```
```  1134   fix S T
```
```  1135   assume [transfer_rule]: "rel_set (rel_filter A) S T"
```
```  1136   show "rel_filter A (Sup S) (Sup T)"
```
```  1137     by(simp add: rel_filter_eventually eventually_Sup) transfer_prover
```
```  1138 qed
```
```  1139
```
```  1140 lemma principal_parametric [transfer_rule]:
```
```  1141   "(rel_set A ===> rel_filter A) principal principal"
```
```  1142 proof(rule rel_funI)
```
```  1143   fix S S'
```
```  1144   assume [transfer_rule]: "rel_set A S S'"
```
```  1145   show "rel_filter A (principal S) (principal S')"
```
```  1146     by(simp add: rel_filter_eventually eventually_principal) transfer_prover
```
```  1147 qed
```
```  1148
```
```  1149 context
```
```  1150   fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"
```
```  1151   assumes [transfer_rule]: "bi_unique A"
```
```  1152 begin
```
```  1153
```
```  1154 lemma le_filter_parametric [transfer_rule]:
```
```  1155   "(rel_filter A ===> rel_filter A ===> op =) op \<le> op \<le>"
```
```  1156 unfolding le_filter_def[abs_def] by transfer_prover
```
```  1157
```
```  1158 lemma less_filter_parametric [transfer_rule]:
```
```  1159   "(rel_filter A ===> rel_filter A ===> op =) op < op <"
```
```  1160 unfolding less_filter_def[abs_def] by transfer_prover
```
```  1161
```
```  1162 context
```
```  1163   assumes [transfer_rule]: "bi_total A"
```
```  1164 begin
```
```  1165
```
```  1166 lemma Inf_filter_parametric [transfer_rule]:
```
```  1167   "(rel_set (rel_filter A) ===> rel_filter A) Inf Inf"
```
```  1168 unfolding Inf_filter_def[abs_def] by transfer_prover
```
```  1169
```
```  1170 lemma inf_filter_parametric [transfer_rule]:
```
```  1171   "(rel_filter A ===> rel_filter A ===> rel_filter A) inf inf"
```
```  1172 proof(intro rel_funI)+
```
```  1173   fix F F' G G'
```
```  1174   assume [transfer_rule]: "rel_filter A F F'" "rel_filter A G G'"
```
```  1175   have "rel_filter A (Inf {F, G}) (Inf {F', G'})" by transfer_prover
```
```  1176   thus "rel_filter A (inf F G) (inf F' G')" by simp
```
```  1177 qed
```
```  1178
```
```  1179 end
```
```  1180
```
```  1181 end
```
```  1182
```
```  1183 end
```
```  1184
```
```  1185 end
```