src/HOL/Filter.thy
 author hoelzl Sun Apr 12 11:33:19 2015 +0200 (2015-04-12) changeset 60036 218fcc645d22 child 60037 071a99649dde permissions -rw-r--r--
move filters to their own theory
```     1 (*  Title:      HOL/Filter.thy
```
```     2     Author:     Brian Huffman
```
```     3     Author:     Johannes Hölzl
```
```     4 *)
```
```     5
```
```     6 section {* Filters on predicates *}
```
```     7
```
```     8 theory Filter
```
```     9 imports Set_Interval Lifting_Set
```
```    10 begin
```
```    11
```
```    12 subsection {* Filters *}
```
```    13
```
```    14 text {*
```
```    15   This definition also allows non-proper filters.
```
```    16 *}
```
```    17
```
```    18 locale is_filter =
```
```    19   fixes F :: "('a \<Rightarrow> bool) \<Rightarrow> bool"
```
```    20   assumes True: "F (\<lambda>x. True)"
```
```    21   assumes conj: "F (\<lambda>x. P x) \<Longrightarrow> F (\<lambda>x. Q x) \<Longrightarrow> F (\<lambda>x. P x \<and> Q x)"
```
```    22   assumes mono: "\<forall>x. P x \<longrightarrow> Q x \<Longrightarrow> F (\<lambda>x. P x) \<Longrightarrow> F (\<lambda>x. Q x)"
```
```    23
```
```    24 typedef 'a filter = "{F :: ('a \<Rightarrow> bool) \<Rightarrow> bool. is_filter F}"
```
```    25 proof
```
```    26   show "(\<lambda>x. True) \<in> ?filter" by (auto intro: is_filter.intro)
```
```    27 qed
```
```    28
```
```    29 lemma is_filter_Rep_filter: "is_filter (Rep_filter F)"
```
```    30   using Rep_filter [of F] by simp
```
```    31
```
```    32 lemma Abs_filter_inverse':
```
```    33   assumes "is_filter F" shows "Rep_filter (Abs_filter F) = F"
```
```    34   using assms by (simp add: Abs_filter_inverse)
```
```    35
```
```    36
```
```    37 subsubsection {* Eventually *}
```
```    38
```
```    39 definition eventually :: "('a \<Rightarrow> bool) \<Rightarrow> 'a filter \<Rightarrow> bool"
```
```    40   where "eventually P F \<longleftrightarrow> Rep_filter F P"
```
```    41
```
```    42 lemma eventually_Abs_filter:
```
```    43   assumes "is_filter F" shows "eventually P (Abs_filter F) = F P"
```
```    44   unfolding eventually_def using assms by (simp add: Abs_filter_inverse)
```
```    45
```
```    46 lemma filter_eq_iff:
```
```    47   shows "F = F' \<longleftrightarrow> (\<forall>P. eventually P F = eventually P F')"
```
```    48   unfolding Rep_filter_inject [symmetric] fun_eq_iff eventually_def ..
```
```    49
```
```    50 lemma eventually_True [simp]: "eventually (\<lambda>x. True) F"
```
```    51   unfolding eventually_def
```
```    52   by (rule is_filter.True [OF is_filter_Rep_filter])
```
```    53
```
```    54 lemma always_eventually: "\<forall>x. P x \<Longrightarrow> eventually P F"
```
```    55 proof -
```
```    56   assume "\<forall>x. P x" hence "P = (\<lambda>x. True)" by (simp add: ext)
```
```    57   thus "eventually P F" by simp
```
```    58 qed
```
```    59
```
```    60 lemma eventually_mono:
```
```    61   "(\<forall>x. P x \<longrightarrow> Q x) \<Longrightarrow> eventually P F \<Longrightarrow> eventually Q F"
```
```    62   unfolding eventually_def
```
```    63   by (rule is_filter.mono [OF is_filter_Rep_filter])
```
```    64
```
```    65 lemma eventually_conj:
```
```    66   assumes P: "eventually (\<lambda>x. P x) F"
```
```    67   assumes Q: "eventually (\<lambda>x. Q x) F"
```
```    68   shows "eventually (\<lambda>x. P x \<and> Q x) F"
```
```    69   using assms unfolding eventually_def
```
```    70   by (rule is_filter.conj [OF is_filter_Rep_filter])
```
```    71
```
```    72 lemma eventually_Ball_finite:
```
```    73   assumes "finite A" and "\<forall>y\<in>A. eventually (\<lambda>x. P x y) net"
```
```    74   shows "eventually (\<lambda>x. \<forall>y\<in>A. P x y) net"
```
```    75 using assms by (induct set: finite, simp, simp add: eventually_conj)
```
```    76
```
```    77 lemma eventually_all_finite:
```
```    78   fixes P :: "'a \<Rightarrow> 'b::finite \<Rightarrow> bool"
```
```    79   assumes "\<And>y. eventually (\<lambda>x. P x y) net"
```
```    80   shows "eventually (\<lambda>x. \<forall>y. P x y) net"
```
```    81 using eventually_Ball_finite [of UNIV P] assms by simp
```
```    82
```
```    83 lemma eventually_mp:
```
```    84   assumes "eventually (\<lambda>x. P x \<longrightarrow> Q x) F"
```
```    85   assumes "eventually (\<lambda>x. P x) F"
```
```    86   shows "eventually (\<lambda>x. Q x) F"
```
```    87 proof (rule eventually_mono)
```
```    88   show "\<forall>x. (P x \<longrightarrow> Q x) \<and> P x \<longrightarrow> Q x" by simp
```
```    89   show "eventually (\<lambda>x. (P x \<longrightarrow> Q x) \<and> P x) F"
```
```    90     using assms by (rule eventually_conj)
```
```    91 qed
```
```    92
```
```    93 lemma eventually_rev_mp:
```
```    94   assumes "eventually (\<lambda>x. P x) F"
```
```    95   assumes "eventually (\<lambda>x. P x \<longrightarrow> Q x) F"
```
```    96   shows "eventually (\<lambda>x. Q x) F"
```
```    97 using assms(2) assms(1) by (rule eventually_mp)
```
```    98
```
```    99 lemma eventually_conj_iff:
```
```   100   "eventually (\<lambda>x. P x \<and> Q x) F \<longleftrightarrow> eventually P F \<and> eventually Q F"
```
```   101   by (auto intro: eventually_conj elim: eventually_rev_mp)
```
```   102
```
```   103 lemma eventually_elim1:
```
```   104   assumes "eventually (\<lambda>i. P i) F"
```
```   105   assumes "\<And>i. P i \<Longrightarrow> Q i"
```
```   106   shows "eventually (\<lambda>i. Q i) F"
```
```   107   using assms by (auto elim!: eventually_rev_mp)
```
```   108
```
```   109 lemma eventually_elim2:
```
```   110   assumes "eventually (\<lambda>i. P i) F"
```
```   111   assumes "eventually (\<lambda>i. Q i) F"
```
```   112   assumes "\<And>i. P i \<Longrightarrow> Q i \<Longrightarrow> R i"
```
```   113   shows "eventually (\<lambda>i. R i) F"
```
```   114   using assms by (auto elim!: eventually_rev_mp)
```
```   115
```
```   116 lemma not_eventually_impI: "eventually P F \<Longrightarrow> \<not> eventually Q F \<Longrightarrow> \<not> eventually (\<lambda>x. P x \<longrightarrow> Q x) F"
```
```   117   by (auto intro: eventually_mp)
```
```   118
```
```   119 lemma not_eventuallyD: "\<not> eventually P F \<Longrightarrow> \<exists>x. \<not> P x"
```
```   120   by (metis always_eventually)
```
```   121
```
```   122 lemma eventually_subst:
```
```   123   assumes "eventually (\<lambda>n. P n = Q n) F"
```
```   124   shows "eventually P F = eventually Q F" (is "?L = ?R")
```
```   125 proof -
```
```   126   from assms have "eventually (\<lambda>x. P x \<longrightarrow> Q x) F"
```
```   127       and "eventually (\<lambda>x. Q x \<longrightarrow> P x) F"
```
```   128     by (auto elim: eventually_elim1)
```
```   129   then show ?thesis by (auto elim: eventually_elim2)
```
```   130 qed
```
```   131
```
```   132 ML {*
```
```   133   fun eventually_elim_tac ctxt facts = SUBGOAL_CASES (fn (goal, i) =>
```
```   134     let
```
```   135       val mp_thms = facts RL @{thms eventually_rev_mp}
```
```   136       val raw_elim_thm =
```
```   137         (@{thm allI} RS @{thm always_eventually})
```
```   138         |> fold (fn thm1 => fn thm2 => thm2 RS thm1) mp_thms
```
```   139         |> fold (fn _ => fn thm => @{thm impI} RS thm) facts
```
```   140       val cases_prop = Thm.prop_of (raw_elim_thm RS Goal.init (Thm.cterm_of ctxt goal))
```
```   141       val cases = Rule_Cases.make_common ctxt cases_prop [(("elim", []), [])]
```
```   142     in
```
```   143       CASES cases (rtac raw_elim_thm i)
```
```   144     end)
```
```   145 *}
```
```   146
```
```   147 method_setup eventually_elim = {*
```
```   148   Scan.succeed (fn ctxt => METHOD_CASES (HEADGOAL o eventually_elim_tac ctxt))
```
```   149 *} "elimination of eventually quantifiers"
```
```   150
```
```   151
```
```   152 subsubsection {* Finer-than relation *}
```
```   153
```
```   154 text {* @{term "F \<le> F'"} means that filter @{term F} is finer than
```
```   155 filter @{term F'}. *}
```
```   156
```
```   157 instantiation filter :: (type) complete_lattice
```
```   158 begin
```
```   159
```
```   160 definition le_filter_def:
```
```   161   "F \<le> F' \<longleftrightarrow> (\<forall>P. eventually P F' \<longrightarrow> eventually P F)"
```
```   162
```
```   163 definition
```
```   164   "(F :: 'a filter) < F' \<longleftrightarrow> F \<le> F' \<and> \<not> F' \<le> F"
```
```   165
```
```   166 definition
```
```   167   "top = Abs_filter (\<lambda>P. \<forall>x. P x)"
```
```   168
```
```   169 definition
```
```   170   "bot = Abs_filter (\<lambda>P. True)"
```
```   171
```
```   172 definition
```
```   173   "sup F F' = Abs_filter (\<lambda>P. eventually P F \<and> eventually P F')"
```
```   174
```
```   175 definition
```
```   176   "inf F F' = Abs_filter
```
```   177       (\<lambda>P. \<exists>Q R. eventually Q F \<and> eventually R F' \<and> (\<forall>x. Q x \<and> R x \<longrightarrow> P x))"
```
```   178
```
```   179 definition
```
```   180   "Sup S = Abs_filter (\<lambda>P. \<forall>F\<in>S. eventually P F)"
```
```   181
```
```   182 definition
```
```   183   "Inf S = Sup {F::'a filter. \<forall>F'\<in>S. F \<le> F'}"
```
```   184
```
```   185 lemma eventually_top [simp]: "eventually P top \<longleftrightarrow> (\<forall>x. P x)"
```
```   186   unfolding top_filter_def
```
```   187   by (rule eventually_Abs_filter, rule is_filter.intro, auto)
```
```   188
```
```   189 lemma eventually_bot [simp]: "eventually P bot"
```
```   190   unfolding bot_filter_def
```
```   191   by (subst eventually_Abs_filter, rule is_filter.intro, auto)
```
```   192
```
```   193 lemma eventually_sup:
```
```   194   "eventually P (sup F F') \<longleftrightarrow> eventually P F \<and> eventually P F'"
```
```   195   unfolding sup_filter_def
```
```   196   by (rule eventually_Abs_filter, rule is_filter.intro)
```
```   197      (auto elim!: eventually_rev_mp)
```
```   198
```
```   199 lemma eventually_inf:
```
```   200   "eventually P (inf F F') \<longleftrightarrow>
```
```   201    (\<exists>Q R. eventually Q F \<and> eventually R F' \<and> (\<forall>x. Q x \<and> R x \<longrightarrow> P x))"
```
```   202   unfolding inf_filter_def
```
```   203   apply (rule eventually_Abs_filter, rule is_filter.intro)
```
```   204   apply (fast intro: eventually_True)
```
```   205   apply clarify
```
```   206   apply (intro exI conjI)
```
```   207   apply (erule (1) eventually_conj)
```
```   208   apply (erule (1) eventually_conj)
```
```   209   apply simp
```
```   210   apply auto
```
```   211   done
```
```   212
```
```   213 lemma eventually_Sup:
```
```   214   "eventually P (Sup S) \<longleftrightarrow> (\<forall>F\<in>S. eventually P F)"
```
```   215   unfolding Sup_filter_def
```
```   216   apply (rule eventually_Abs_filter, rule is_filter.intro)
```
```   217   apply (auto intro: eventually_conj elim!: eventually_rev_mp)
```
```   218   done
```
```   219
```
```   220 instance proof
```
```   221   fix F F' F'' :: "'a filter" and S :: "'a filter set"
```
```   222   { show "F < F' \<longleftrightarrow> F \<le> F' \<and> \<not> F' \<le> F"
```
```   223     by (rule less_filter_def) }
```
```   224   { show "F \<le> F"
```
```   225     unfolding le_filter_def by simp }
```
```   226   { assume "F \<le> F'" and "F' \<le> F''" thus "F \<le> F''"
```
```   227     unfolding le_filter_def by simp }
```
```   228   { assume "F \<le> F'" and "F' \<le> F" thus "F = F'"
```
```   229     unfolding le_filter_def filter_eq_iff by fast }
```
```   230   { show "inf F F' \<le> F" and "inf F F' \<le> F'"
```
```   231     unfolding le_filter_def eventually_inf by (auto intro: eventually_True) }
```
```   232   { assume "F \<le> F'" and "F \<le> F''" thus "F \<le> inf F' F''"
```
```   233     unfolding le_filter_def eventually_inf
```
```   234     by (auto elim!: eventually_mono intro: eventually_conj) }
```
```   235   { show "F \<le> sup F F'" and "F' \<le> sup F F'"
```
```   236     unfolding le_filter_def eventually_sup by simp_all }
```
```   237   { assume "F \<le> F''" and "F' \<le> F''" thus "sup F F' \<le> F''"
```
```   238     unfolding le_filter_def eventually_sup by simp }
```
```   239   { assume "F'' \<in> S" thus "Inf S \<le> F''"
```
```   240     unfolding le_filter_def Inf_filter_def eventually_Sup Ball_def by simp }
```
```   241   { assume "\<And>F'. F' \<in> S \<Longrightarrow> F \<le> F'" thus "F \<le> Inf S"
```
```   242     unfolding le_filter_def Inf_filter_def eventually_Sup Ball_def by simp }
```
```   243   { assume "F \<in> S" thus "F \<le> Sup S"
```
```   244     unfolding le_filter_def eventually_Sup by simp }
```
```   245   { assume "\<And>F. F \<in> S \<Longrightarrow> F \<le> F'" thus "Sup S \<le> F'"
```
```   246     unfolding le_filter_def eventually_Sup by simp }
```
```   247   { show "Inf {} = (top::'a filter)"
```
```   248     by (auto simp: top_filter_def Inf_filter_def Sup_filter_def)
```
```   249       (metis (full_types) top_filter_def always_eventually eventually_top) }
```
```   250   { show "Sup {} = (bot::'a filter)"
```
```   251     by (auto simp: bot_filter_def Sup_filter_def) }
```
```   252 qed
```
```   253
```
```   254 end
```
```   255
```
```   256 lemma filter_leD:
```
```   257   "F \<le> F' \<Longrightarrow> eventually P F' \<Longrightarrow> eventually P F"
```
```   258   unfolding le_filter_def by simp
```
```   259
```
```   260 lemma filter_leI:
```
```   261   "(\<And>P. eventually P F' \<Longrightarrow> eventually P F) \<Longrightarrow> F \<le> F'"
```
```   262   unfolding le_filter_def by simp
```
```   263
```
```   264 lemma eventually_False:
```
```   265   "eventually (\<lambda>x. False) F \<longleftrightarrow> F = bot"
```
```   266   unfolding filter_eq_iff by (auto elim: eventually_rev_mp)
```
```   267
```
```   268 abbreviation (input) trivial_limit :: "'a filter \<Rightarrow> bool"
```
```   269   where "trivial_limit F \<equiv> F = bot"
```
```   270
```
```   271 lemma trivial_limit_def: "trivial_limit F \<longleftrightarrow> eventually (\<lambda>x. False) F"
```
```   272   by (rule eventually_False [symmetric])
```
```   273
```
```   274 lemma eventually_const: "\<not> trivial_limit net \<Longrightarrow> eventually (\<lambda>x. P) net \<longleftrightarrow> P"
```
```   275   by (cases P) (simp_all add: eventually_False)
```
```   276
```
```   277 lemma eventually_Inf: "eventually P (Inf B) \<longleftrightarrow> (\<exists>X\<subseteq>B. finite X \<and> eventually P (Inf X))"
```
```   278 proof -
```
```   279   let ?F = "\<lambda>P. \<exists>X\<subseteq>B. finite X \<and> eventually P (Inf X)"
```
```   280
```
```   281   { fix P have "eventually P (Abs_filter ?F) \<longleftrightarrow> ?F P"
```
```   282     proof (rule eventually_Abs_filter is_filter.intro)+
```
```   283       show "?F (\<lambda>x. True)"
```
```   284         by (rule exI[of _ "{}"]) (simp add: le_fun_def)
```
```   285     next
```
```   286       fix P Q
```
```   287       assume "?F P" then guess X ..
```
```   288       moreover
```
```   289       assume "?F Q" then guess Y ..
```
```   290       ultimately show "?F (\<lambda>x. P x \<and> Q x)"
```
```   291         by (intro exI[of _ "X \<union> Y"])
```
```   292            (auto simp: Inf_union_distrib eventually_inf)
```
```   293     next
```
```   294       fix P Q
```
```   295       assume "?F P" then guess X ..
```
```   296       moreover assume "\<forall>x. P x \<longrightarrow> Q x"
```
```   297       ultimately show "?F Q"
```
```   298         by (intro exI[of _ X]) (auto elim: eventually_elim1)
```
```   299     qed }
```
```   300   note eventually_F = this
```
```   301
```
```   302   have "Inf B = Abs_filter ?F"
```
```   303   proof (intro antisym Inf_greatest)
```
```   304     show "Inf B \<le> Abs_filter ?F"
```
```   305       by (auto simp: le_filter_def eventually_F dest: Inf_superset_mono)
```
```   306   next
```
```   307     fix F assume "F \<in> B" then show "Abs_filter ?F \<le> F"
```
```   308       by (auto simp add: le_filter_def eventually_F intro!: exI[of _ "{F}"])
```
```   309   qed
```
```   310   then show ?thesis
```
```   311     by (simp add: eventually_F)
```
```   312 qed
```
```   313
```
```   314 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))"
```
```   315   unfolding INF_def[of B] eventually_Inf[of P "F`B"]
```
```   316   by (metis Inf_image_eq finite_imageI image_mono finite_subset_image)
```
```   317
```
```   318 lemma Inf_filter_not_bot:
```
```   319   fixes B :: "'a filter set"
```
```   320   shows "(\<And>X. X \<subseteq> B \<Longrightarrow> finite X \<Longrightarrow> Inf X \<noteq> bot) \<Longrightarrow> Inf B \<noteq> bot"
```
```   321   unfolding trivial_limit_def eventually_Inf[of _ B]
```
```   322     bot_bool_def [symmetric] bot_fun_def [symmetric] bot_unique by simp
```
```   323
```
```   324 lemma INF_filter_not_bot:
```
```   325   fixes F :: "'i \<Rightarrow> 'a filter"
```
```   326   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"
```
```   327   unfolding trivial_limit_def eventually_INF[of _ B]
```
```   328     bot_bool_def [symmetric] bot_fun_def [symmetric] bot_unique by simp
```
```   329
```
```   330 lemma eventually_Inf_base:
```
```   331   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"
```
```   332   shows "eventually P (Inf B) \<longleftrightarrow> (\<exists>b\<in>B. eventually P b)"
```
```   333 proof (subst eventually_Inf, safe)
```
```   334   fix X assume "finite X" "X \<subseteq> B"
```
```   335   then have "\<exists>b\<in>B. \<forall>x\<in>X. b \<le> x"
```
```   336   proof induct
```
```   337     case empty then show ?case
```
```   338       using `B \<noteq> {}` by auto
```
```   339   next
```
```   340     case (insert x X)
```
```   341     then obtain b where "b \<in> B" "\<And>x. x \<in> X \<Longrightarrow> b \<le> x"
```
```   342       by auto
```
```   343     with `insert x X \<subseteq> B` base[of b x] show ?case
```
```   344       by (auto intro: order_trans)
```
```   345   qed
```
```   346   then obtain b where "b \<in> B" "b \<le> Inf X"
```
```   347     by (auto simp: le_Inf_iff)
```
```   348   then show "eventually P (Inf X) \<Longrightarrow> Bex B (eventually P)"
```
```   349     by (intro bexI[of _ b]) (auto simp: le_filter_def)
```
```   350 qed (auto intro!: exI[of _ "{x}" for x])
```
```   351
```
```   352 lemma eventually_INF_base:
```
```   353   "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>
```
```   354     eventually P (INF b:B. F b) \<longleftrightarrow> (\<exists>b\<in>B. eventually P (F b))"
```
```   355   unfolding INF_def by (subst eventually_Inf_base) auto
```
```   356
```
```   357
```
```   358 subsubsection {* Map function for filters *}
```
```   359
```
```   360 definition filtermap :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a filter \<Rightarrow> 'b filter"
```
```   361   where "filtermap f F = Abs_filter (\<lambda>P. eventually (\<lambda>x. P (f x)) F)"
```
```   362
```
```   363 lemma eventually_filtermap:
```
```   364   "eventually P (filtermap f F) = eventually (\<lambda>x. P (f x)) F"
```
```   365   unfolding filtermap_def
```
```   366   apply (rule eventually_Abs_filter)
```
```   367   apply (rule is_filter.intro)
```
```   368   apply (auto elim!: eventually_rev_mp)
```
```   369   done
```
```   370
```
```   371 lemma filtermap_ident: "filtermap (\<lambda>x. x) F = F"
```
```   372   by (simp add: filter_eq_iff eventually_filtermap)
```
```   373
```
```   374 lemma filtermap_filtermap:
```
```   375   "filtermap f (filtermap g F) = filtermap (\<lambda>x. f (g x)) F"
```
```   376   by (simp add: filter_eq_iff eventually_filtermap)
```
```   377
```
```   378 lemma filtermap_mono: "F \<le> F' \<Longrightarrow> filtermap f F \<le> filtermap f F'"
```
```   379   unfolding le_filter_def eventually_filtermap by simp
```
```   380
```
```   381 lemma filtermap_bot [simp]: "filtermap f bot = bot"
```
```   382   by (simp add: filter_eq_iff eventually_filtermap)
```
```   383
```
```   384 lemma filtermap_sup: "filtermap f (sup F1 F2) = sup (filtermap f F1) (filtermap f F2)"
```
```   385   by (auto simp: filter_eq_iff eventually_filtermap eventually_sup)
```
```   386
```
```   387 lemma filtermap_inf: "filtermap f (inf F1 F2) \<le> inf (filtermap f F1) (filtermap f F2)"
```
```   388   by (auto simp: le_filter_def eventually_filtermap eventually_inf)
```
```   389
```
```   390 lemma filtermap_INF: "filtermap f (INF b:B. F b) \<le> (INF b:B. filtermap f (F b))"
```
```   391 proof -
```
```   392   { fix X :: "'c set" assume "finite X"
```
```   393     then have "filtermap f (INFIMUM X F) \<le> (INF b:X. filtermap f (F b))"
```
```   394     proof induct
```
```   395       case (insert x X)
```
```   396       have "filtermap f (INF a:insert x X. F a) \<le> inf (filtermap f (F x)) (filtermap f (INF a:X. F a))"
```
```   397         by (rule order_trans[OF _ filtermap_inf]) simp
```
```   398       also have "\<dots> \<le> inf (filtermap f (F x)) (INF a:X. filtermap f (F a))"
```
```   399         by (intro inf_mono insert order_refl)
```
```   400       finally show ?case
```
```   401         by simp
```
```   402     qed simp }
```
```   403   then show ?thesis
```
```   404     unfolding le_filter_def eventually_filtermap
```
```   405     by (subst (1 2) eventually_INF) auto
```
```   406 qed
```
```   407 subsubsection {* Standard filters *}
```
```   408
```
```   409 definition principal :: "'a set \<Rightarrow> 'a filter" where
```
```   410   "principal S = Abs_filter (\<lambda>P. \<forall>x\<in>S. P x)"
```
```   411
```
```   412 lemma eventually_principal: "eventually P (principal S) \<longleftrightarrow> (\<forall>x\<in>S. P x)"
```
```   413   unfolding principal_def
```
```   414   by (rule eventually_Abs_filter, rule is_filter.intro) auto
```
```   415
```
```   416 lemma eventually_inf_principal: "eventually P (inf F (principal s)) \<longleftrightarrow> eventually (\<lambda>x. x \<in> s \<longrightarrow> P x) F"
```
```   417   unfolding eventually_inf eventually_principal by (auto elim: eventually_elim1)
```
```   418
```
```   419 lemma principal_UNIV[simp]: "principal UNIV = top"
```
```   420   by (auto simp: filter_eq_iff eventually_principal)
```
```   421
```
```   422 lemma principal_empty[simp]: "principal {} = bot"
```
```   423   by (auto simp: filter_eq_iff eventually_principal)
```
```   424
```
```   425 lemma principal_eq_bot_iff: "principal X = bot \<longleftrightarrow> X = {}"
```
```   426   by (auto simp add: filter_eq_iff eventually_principal)
```
```   427
```
```   428 lemma principal_le_iff[iff]: "principal A \<le> principal B \<longleftrightarrow> A \<subseteq> B"
```
```   429   by (auto simp: le_filter_def eventually_principal)
```
```   430
```
```   431 lemma le_principal: "F \<le> principal A \<longleftrightarrow> eventually (\<lambda>x. x \<in> A) F"
```
```   432   unfolding le_filter_def eventually_principal
```
```   433   apply safe
```
```   434   apply (erule_tac x="\<lambda>x. x \<in> A" in allE)
```
```   435   apply (auto elim: eventually_elim1)
```
```   436   done
```
```   437
```
```   438 lemma principal_inject[iff]: "principal A = principal B \<longleftrightarrow> A = B"
```
```   439   unfolding eq_iff by simp
```
```   440
```
```   441 lemma sup_principal[simp]: "sup (principal A) (principal B) = principal (A \<union> B)"
```
```   442   unfolding filter_eq_iff eventually_sup eventually_principal by auto
```
```   443
```
```   444 lemma inf_principal[simp]: "inf (principal A) (principal B) = principal (A \<inter> B)"
```
```   445   unfolding filter_eq_iff eventually_inf eventually_principal
```
```   446   by (auto intro: exI[of _ "\<lambda>x. x \<in> A"] exI[of _ "\<lambda>x. x \<in> B"])
```
```   447
```
```   448 lemma SUP_principal[simp]: "(SUP i : I. principal (A i)) = principal (\<Union>i\<in>I. A i)"
```
```   449   unfolding filter_eq_iff eventually_Sup SUP_def by (auto simp: eventually_principal)
```
```   450
```
```   451 lemma INF_principal_finite: "finite X \<Longrightarrow> (INF x:X. principal (f x)) = principal (\<Inter>x\<in>X. f x)"
```
```   452   by (induct X rule: finite_induct) auto
```
```   453
```
```   454 lemma filtermap_principal[simp]: "filtermap f (principal A) = principal (f ` A)"
```
```   455   unfolding filter_eq_iff eventually_filtermap eventually_principal by simp
```
```   456
```
```   457 subsubsection {* Order filters *}
```
```   458
```
```   459 definition at_top :: "('a::order) filter"
```
```   460   where "at_top = (INF k. principal {k ..})"
```
```   461
```
```   462 lemma at_top_sub: "at_top = (INF k:{c::'a::linorder..}. principal {k ..})"
```
```   463   by (auto intro!: INF_eq max.cobounded1 max.cobounded2 simp: at_top_def)
```
```   464
```
```   465 lemma eventually_at_top_linorder: "eventually P at_top \<longleftrightarrow> (\<exists>N::'a::linorder. \<forall>n\<ge>N. P n)"
```
```   466   unfolding at_top_def
```
```   467   by (subst eventually_INF_base) (auto simp: eventually_principal intro: max.cobounded1 max.cobounded2)
```
```   468
```
```   469 lemma eventually_ge_at_top:
```
```   470   "eventually (\<lambda>x. (c::_::linorder) \<le> x) at_top"
```
```   471   unfolding eventually_at_top_linorder by auto
```
```   472
```
```   473 lemma eventually_at_top_dense: "eventually P at_top \<longleftrightarrow> (\<exists>N::'a::{no_top, linorder}. \<forall>n>N. P n)"
```
```   474 proof -
```
```   475   have "eventually P (INF k. principal {k <..}) \<longleftrightarrow> (\<exists>N::'a. \<forall>n>N. P n)"
```
```   476     by (subst eventually_INF_base) (auto simp: eventually_principal intro: max.cobounded1 max.cobounded2)
```
```   477   also have "(INF k. principal {k::'a <..}) = at_top"
```
```   478     unfolding at_top_def
```
```   479     by (intro INF_eq) (auto intro: less_imp_le simp: Ici_subset_Ioi_iff gt_ex)
```
```   480   finally show ?thesis .
```
```   481 qed
```
```   482
```
```   483 lemma eventually_gt_at_top:
```
```   484   "eventually (\<lambda>x. (c::_::unbounded_dense_linorder) < x) at_top"
```
```   485   unfolding eventually_at_top_dense by auto
```
```   486
```
```   487 definition at_bot :: "('a::order) filter"
```
```   488   where "at_bot = (INF k. principal {.. k})"
```
```   489
```
```   490 lemma at_bot_sub: "at_bot = (INF k:{.. c::'a::linorder}. principal {.. k})"
```
```   491   by (auto intro!: INF_eq min.cobounded1 min.cobounded2 simp: at_bot_def)
```
```   492
```
```   493 lemma eventually_at_bot_linorder:
```
```   494   fixes P :: "'a::linorder \<Rightarrow> bool" shows "eventually P at_bot \<longleftrightarrow> (\<exists>N. \<forall>n\<le>N. P n)"
```
```   495   unfolding at_bot_def
```
```   496   by (subst eventually_INF_base) (auto simp: eventually_principal intro: min.cobounded1 min.cobounded2)
```
```   497
```
```   498 lemma eventually_le_at_bot:
```
```   499   "eventually (\<lambda>x. x \<le> (c::_::linorder)) at_bot"
```
```   500   unfolding eventually_at_bot_linorder by auto
```
```   501
```
```   502 lemma eventually_at_bot_dense: "eventually P at_bot \<longleftrightarrow> (\<exists>N::'a::{no_bot, linorder}. \<forall>n<N. P n)"
```
```   503 proof -
```
```   504   have "eventually P (INF k. principal {..< k}) \<longleftrightarrow> (\<exists>N::'a. \<forall>n<N. P n)"
```
```   505     by (subst eventually_INF_base) (auto simp: eventually_principal intro: min.cobounded1 min.cobounded2)
```
```   506   also have "(INF k. principal {..< k::'a}) = at_bot"
```
```   507     unfolding at_bot_def
```
```   508     by (intro INF_eq) (auto intro: less_imp_le simp: Iic_subset_Iio_iff lt_ex)
```
```   509   finally show ?thesis .
```
```   510 qed
```
```   511
```
```   512 lemma eventually_gt_at_bot:
```
```   513   "eventually (\<lambda>x. x < (c::_::unbounded_dense_linorder)) at_bot"
```
```   514   unfolding eventually_at_bot_dense by auto
```
```   515
```
```   516 lemma trivial_limit_at_bot_linorder: "\<not> trivial_limit (at_bot ::('a::linorder) filter)"
```
```   517   unfolding trivial_limit_def
```
```   518   by (metis eventually_at_bot_linorder order_refl)
```
```   519
```
```   520 lemma trivial_limit_at_top_linorder: "\<not> trivial_limit (at_top ::('a::linorder) filter)"
```
```   521   unfolding trivial_limit_def
```
```   522   by (metis eventually_at_top_linorder order_refl)
```
```   523
```
```   524 subsection {* Sequentially *}
```
```   525
```
```   526 abbreviation sequentially :: "nat filter"
```
```   527   where "sequentially \<equiv> at_top"
```
```   528
```
```   529 lemma eventually_sequentially:
```
```   530   "eventually P sequentially \<longleftrightarrow> (\<exists>N. \<forall>n\<ge>N. P n)"
```
```   531   by (rule eventually_at_top_linorder)
```
```   532
```
```   533 lemma sequentially_bot [simp, intro]: "sequentially \<noteq> bot"
```
```   534   unfolding filter_eq_iff eventually_sequentially by auto
```
```   535
```
```   536 lemmas trivial_limit_sequentially = sequentially_bot
```
```   537
```
```   538 lemma eventually_False_sequentially [simp]:
```
```   539   "\<not> eventually (\<lambda>n. False) sequentially"
```
```   540   by (simp add: eventually_False)
```
```   541
```
```   542 lemma le_sequentially:
```
```   543   "F \<le> sequentially \<longleftrightarrow> (\<forall>N. eventually (\<lambda>n. N \<le> n) F)"
```
```   544   by (simp add: at_top_def le_INF_iff le_principal)
```
```   545
```
```   546 lemma eventually_sequentiallyI:
```
```   547   assumes "\<And>x. c \<le> x \<Longrightarrow> P x"
```
```   548   shows "eventually P sequentially"
```
```   549 using assms by (auto simp: eventually_sequentially)
```
```   550
```
```   551 lemma eventually_sequentially_seg:
```
```   552   "eventually (\<lambda>n. P (n + k)) sequentially \<longleftrightarrow> eventually P sequentially"
```
```   553   unfolding eventually_sequentially
```
```   554   apply safe
```
```   555    apply (rule_tac x="N + k" in exI)
```
```   556    apply rule
```
```   557    apply (erule_tac x="n - k" in allE)
```
```   558    apply auto []
```
```   559   apply (rule_tac x=N in exI)
```
```   560   apply auto []
```
```   561   done
```
```   562
```
```   563
```
```   564 subsection {* Limits *}
```
```   565
```
```   566 definition filterlim :: "('a \<Rightarrow> 'b) \<Rightarrow> 'b filter \<Rightarrow> 'a filter \<Rightarrow> bool" where
```
```   567   "filterlim f F2 F1 \<longleftrightarrow> filtermap f F1 \<le> F2"
```
```   568
```
```   569 syntax
```
```   570   "_LIM" :: "pttrns \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> bool" ("(3LIM (_)/ (_)./ (_) :> (_))" [1000, 10, 0, 10] 10)
```
```   571
```
```   572 translations
```
```   573   "LIM x F1. f :> F2"   == "CONST filterlim (%x. f) F2 F1"
```
```   574
```
```   575 lemma filterlim_iff:
```
```   576   "(LIM x F1. f x :> F2) \<longleftrightarrow> (\<forall>P. eventually P F2 \<longrightarrow> eventually (\<lambda>x. P (f x)) F1)"
```
```   577   unfolding filterlim_def le_filter_def eventually_filtermap ..
```
```   578
```
```   579 lemma filterlim_compose:
```
```   580   "filterlim g F3 F2 \<Longrightarrow> filterlim f F2 F1 \<Longrightarrow> filterlim (\<lambda>x. g (f x)) F3 F1"
```
```   581   unfolding filterlim_def filtermap_filtermap[symmetric] by (metis filtermap_mono order_trans)
```
```   582
```
```   583 lemma filterlim_mono:
```
```   584   "filterlim f F2 F1 \<Longrightarrow> F2 \<le> F2' \<Longrightarrow> F1' \<le> F1 \<Longrightarrow> filterlim f F2' F1'"
```
```   585   unfolding filterlim_def by (metis filtermap_mono order_trans)
```
```   586
```
```   587 lemma filterlim_ident: "LIM x F. x :> F"
```
```   588   by (simp add: filterlim_def filtermap_ident)
```
```   589
```
```   590 lemma filterlim_cong:
```
```   591   "F1 = F1' \<Longrightarrow> F2 = F2' \<Longrightarrow> eventually (\<lambda>x. f x = g x) F2 \<Longrightarrow> filterlim f F1 F2 = filterlim g F1' F2'"
```
```   592   by (auto simp: filterlim_def le_filter_def eventually_filtermap elim: eventually_elim2)
```
```   593
```
```   594 lemma filterlim_mono_eventually:
```
```   595   assumes "filterlim f F G" and ord: "F \<le> F'" "G' \<le> G"
```
```   596   assumes eq: "eventually (\<lambda>x. f x = f' x) G'"
```
```   597   shows "filterlim f' F' G'"
```
```   598   apply (rule filterlim_cong[OF refl refl eq, THEN iffD1])
```
```   599   apply (rule filterlim_mono[OF _ ord])
```
```   600   apply fact
```
```   601   done
```
```   602
```
```   603 lemma filtermap_mono_strong: "inj f \<Longrightarrow> filtermap f F \<le> filtermap f G \<longleftrightarrow> F \<le> G"
```
```   604   apply (auto intro!: filtermap_mono) []
```
```   605   apply (auto simp: le_filter_def eventually_filtermap)
```
```   606   apply (erule_tac x="\<lambda>x. P (inv f x)" in allE)
```
```   607   apply auto
```
```   608   done
```
```   609
```
```   610 lemma filtermap_eq_strong: "inj f \<Longrightarrow> filtermap f F = filtermap f G \<longleftrightarrow> F = G"
```
```   611   by (simp add: filtermap_mono_strong eq_iff)
```
```   612
```
```   613 lemma filterlim_principal:
```
```   614   "(LIM x F. f x :> principal S) \<longleftrightarrow> (eventually (\<lambda>x. f x \<in> S) F)"
```
```   615   unfolding filterlim_def eventually_filtermap le_principal ..
```
```   616
```
```   617 lemma filterlim_inf:
```
```   618   "(LIM x F1. f x :> inf F2 F3) \<longleftrightarrow> ((LIM x F1. f x :> F2) \<and> (LIM x F1. f x :> F3))"
```
```   619   unfolding filterlim_def by simp
```
```   620
```
```   621 lemma filterlim_INF:
```
```   622   "(LIM x F. f x :> (INF b:B. G b)) \<longleftrightarrow> (\<forall>b\<in>B. LIM x F. f x :> G b)"
```
```   623   unfolding filterlim_def le_INF_iff ..
```
```   624
```
```   625 lemma filterlim_INF_INF:
```
```   626   "(\<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)"
```
```   627   unfolding filterlim_def by (rule order_trans[OF filtermap_INF INF_mono])
```
```   628
```
```   629 lemma filterlim_base:
```
```   630   "(\<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>
```
```   631     LIM x (INF i:I. principal (F i)). f x :> (INF j:J. principal (G j))"
```
```   632   by (force intro!: filterlim_INF_INF simp: image_subset_iff)
```
```   633
```
```   634 lemma filterlim_base_iff:
```
```   635   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"
```
```   636   shows "(LIM x (INF i:I. principal (F i)). f x :> INF j:J. principal (G j)) \<longleftrightarrow>
```
```   637     (\<forall>j\<in>J. \<exists>i\<in>I. \<forall>x\<in>F i. f x \<in> G j)"
```
```   638   unfolding filterlim_INF filterlim_principal
```
```   639 proof (subst eventually_INF_base)
```
```   640   fix i j assume "i \<in> I" "j \<in> I"
```
```   641   with chain[OF this] show "\<exists>x\<in>I. principal (F x) \<le> inf (principal (F i)) (principal (F j))"
```
```   642     by auto
```
```   643 qed (auto simp: eventually_principal `I \<noteq> {}`)
```
```   644
```
```   645 lemma filterlim_filtermap: "filterlim f F1 (filtermap g F2) = filterlim (\<lambda>x. f (g x)) F1 F2"
```
```   646   unfolding filterlim_def filtermap_filtermap ..
```
```   647
```
```   648 lemma filterlim_sup:
```
```   649   "filterlim f F F1 \<Longrightarrow> filterlim f F F2 \<Longrightarrow> filterlim f F (sup F1 F2)"
```
```   650   unfolding filterlim_def filtermap_sup by auto
```
```   651
```
```   652 lemma eventually_sequentially_Suc: "eventually (\<lambda>i. P (Suc i)) sequentially \<longleftrightarrow> eventually P sequentially"
```
```   653   unfolding eventually_sequentially by (metis Suc_le_D Suc_le_mono le_Suc_eq)
```
```   654
```
```   655 lemma filterlim_sequentially_Suc:
```
```   656   "(LIM x sequentially. f (Suc x) :> F) \<longleftrightarrow> (LIM x sequentially. f x :> F)"
```
```   657   unfolding filterlim_iff by (subst eventually_sequentially_Suc) simp
```
```   658
```
```   659 lemma filterlim_Suc: "filterlim Suc sequentially sequentially"
```
```   660   by (simp add: filterlim_iff eventually_sequentially) (metis le_Suc_eq)
```
```   661
```
```   662
```
```   663 subsection {* Limits to @{const at_top} and @{const at_bot} *}
```
```   664
```
```   665 lemma filterlim_at_top:
```
```   666   fixes f :: "'a \<Rightarrow> ('b::linorder)"
```
```   667   shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. Z \<le> f x) F)"
```
```   668   by (auto simp: filterlim_iff eventually_at_top_linorder elim!: eventually_elim1)
```
```   669
```
```   670 lemma filterlim_at_top_mono:
```
```   671   "LIM x F. f x :> at_top \<Longrightarrow> eventually (\<lambda>x. f x \<le> (g x::'a::linorder)) F \<Longrightarrow>
```
```   672     LIM x F. g x :> at_top"
```
```   673   by (auto simp: filterlim_at_top elim: eventually_elim2 intro: order_trans)
```
```   674
```
```   675 lemma filterlim_at_top_dense:
```
```   676   fixes f :: "'a \<Rightarrow> ('b::unbounded_dense_linorder)"
```
```   677   shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. Z < f x) F)"
```
```   678   by (metis eventually_elim1[of _ F] eventually_gt_at_top order_less_imp_le
```
```   679             filterlim_at_top[of f F] filterlim_iff[of f at_top F])
```
```   680
```
```   681 lemma filterlim_at_top_ge:
```
```   682   fixes f :: "'a \<Rightarrow> ('b::linorder)" and c :: "'b"
```
```   683   shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z\<ge>c. eventually (\<lambda>x. Z \<le> f x) F)"
```
```   684   unfolding at_top_sub[of c] filterlim_INF by (auto simp add: filterlim_principal)
```
```   685
```
```   686 lemma filterlim_at_top_at_top:
```
```   687   fixes f :: "'a::linorder \<Rightarrow> 'b::linorder"
```
```   688   assumes mono: "\<And>x y. Q x \<Longrightarrow> Q y \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y"
```
```   689   assumes bij: "\<And>x. P x \<Longrightarrow> f (g x) = x" "\<And>x. P x \<Longrightarrow> Q (g x)"
```
```   690   assumes Q: "eventually Q at_top"
```
```   691   assumes P: "eventually P at_top"
```
```   692   shows "filterlim f at_top at_top"
```
```   693 proof -
```
```   694   from P obtain x where x: "\<And>y. x \<le> y \<Longrightarrow> P y"
```
```   695     unfolding eventually_at_top_linorder by auto
```
```   696   show ?thesis
```
```   697   proof (intro filterlim_at_top_ge[THEN iffD2] allI impI)
```
```   698     fix z assume "x \<le> z"
```
```   699     with x have "P z" by auto
```
```   700     have "eventually (\<lambda>x. g z \<le> x) at_top"
```
```   701       by (rule eventually_ge_at_top)
```
```   702     with Q show "eventually (\<lambda>x. z \<le> f x) at_top"
```
```   703       by eventually_elim (metis mono bij `P z`)
```
```   704   qed
```
```   705 qed
```
```   706
```
```   707 lemma filterlim_at_top_gt:
```
```   708   fixes f :: "'a \<Rightarrow> ('b::unbounded_dense_linorder)" and c :: "'b"
```
```   709   shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z>c. eventually (\<lambda>x. Z \<le> f x) F)"
```
```   710   by (metis filterlim_at_top order_less_le_trans gt_ex filterlim_at_top_ge)
```
```   711
```
```   712 lemma filterlim_at_bot:
```
```   713   fixes f :: "'a \<Rightarrow> ('b::linorder)"
```
```   714   shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. f x \<le> Z) F)"
```
```   715   by (auto simp: filterlim_iff eventually_at_bot_linorder elim!: eventually_elim1)
```
```   716
```
```   717 lemma filterlim_at_bot_dense:
```
```   718   fixes f :: "'a \<Rightarrow> ('b::{dense_linorder, no_bot})"
```
```   719   shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. f x < Z) F)"
```
```   720 proof (auto simp add: filterlim_at_bot[of f F])
```
```   721   fix Z :: 'b
```
```   722   from lt_ex [of Z] obtain Z' where 1: "Z' < Z" ..
```
```   723   assume "\<forall>Z. eventually (\<lambda>x. f x \<le> Z) F"
```
```   724   hence "eventually (\<lambda>x. f x \<le> Z') F" by auto
```
```   725   thus "eventually (\<lambda>x. f x < Z) F"
```
```   726     apply (rule eventually_mono[rotated])
```
```   727     using 1 by auto
```
```   728   next
```
```   729     fix Z :: 'b
```
```   730     show "\<forall>Z. eventually (\<lambda>x. f x < Z) F \<Longrightarrow> eventually (\<lambda>x. f x \<le> Z) F"
```
```   731       by (drule spec [of _ Z], erule eventually_mono[rotated], auto simp add: less_imp_le)
```
```   732 qed
```
```   733
```
```   734 lemma filterlim_at_bot_le:
```
```   735   fixes f :: "'a \<Rightarrow> ('b::linorder)" and c :: "'b"
```
```   736   shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z\<le>c. eventually (\<lambda>x. Z \<ge> f x) F)"
```
```   737   unfolding filterlim_at_bot
```
```   738 proof safe
```
```   739   fix Z assume *: "\<forall>Z\<le>c. eventually (\<lambda>x. Z \<ge> f x) F"
```
```   740   with *[THEN spec, of "min Z c"] show "eventually (\<lambda>x. Z \<ge> f x) F"
```
```   741     by (auto elim!: eventually_elim1)
```
```   742 qed simp
```
```   743
```
```   744 lemma filterlim_at_bot_lt:
```
```   745   fixes f :: "'a \<Rightarrow> ('b::unbounded_dense_linorder)" and c :: "'b"
```
```   746   shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z<c. eventually (\<lambda>x. Z \<ge> f x) F)"
```
```   747   by (metis filterlim_at_bot filterlim_at_bot_le lt_ex order_le_less_trans)
```
```   748
```
```   749
```
```   750 subsection {* Setup @{typ "'a filter"} for lifting and transfer *}
```
```   751
```
```   752 context begin interpretation lifting_syntax .
```
```   753
```
```   754 definition rel_filter :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a filter \<Rightarrow> 'b filter \<Rightarrow> bool"
```
```   755 where "rel_filter R F G = ((R ===> op =) ===> op =) (Rep_filter F) (Rep_filter G)"
```
```   756
```
```   757 lemma rel_filter_eventually:
```
```   758   "rel_filter R F G \<longleftrightarrow>
```
```   759   ((R ===> op =) ===> op =) (\<lambda>P. eventually P F) (\<lambda>P. eventually P G)"
```
```   760 by(simp add: rel_filter_def eventually_def)
```
```   761
```
```   762 lemma filtermap_id [simp, id_simps]: "filtermap id = id"
```
```   763 by(simp add: fun_eq_iff id_def filtermap_ident)
```
```   764
```
```   765 lemma filtermap_id' [simp]: "filtermap (\<lambda>x. x) = (\<lambda>F. F)"
```
```   766 using filtermap_id unfolding id_def .
```
```   767
```
```   768 lemma Quotient_filter [quot_map]:
```
```   769   assumes Q: "Quotient R Abs Rep T"
```
```   770   shows "Quotient (rel_filter R) (filtermap Abs) (filtermap Rep) (rel_filter T)"
```
```   771 unfolding Quotient_alt_def
```
```   772 proof(intro conjI strip)
```
```   773   from Q have *: "\<And>x y. T x y \<Longrightarrow> Abs x = y"
```
```   774     unfolding Quotient_alt_def by blast
```
```   775
```
```   776   fix F G
```
```   777   assume "rel_filter T F G"
```
```   778   thus "filtermap Abs F = G" unfolding filter_eq_iff
```
```   779     by(auto simp add: eventually_filtermap rel_filter_eventually * rel_funI del: iffI elim!: rel_funD)
```
```   780 next
```
```   781   from Q have *: "\<And>x. T (Rep x) x" unfolding Quotient_alt_def by blast
```
```   782
```
```   783   fix F
```
```   784   show "rel_filter T (filtermap Rep F) F"
```
```   785     by(auto elim: rel_funD intro: * intro!: ext arg_cong[where f="\<lambda>P. eventually P F"] rel_funI
```
```   786             del: iffI simp add: eventually_filtermap rel_filter_eventually)
```
```   787 qed(auto simp add: map_fun_def o_def eventually_filtermap filter_eq_iff fun_eq_iff rel_filter_eventually
```
```   788          fun_quotient[OF fun_quotient[OF Q identity_quotient] identity_quotient, unfolded Quotient_alt_def])
```
```   789
```
```   790 lemma eventually_parametric [transfer_rule]:
```
```   791   "((A ===> op =) ===> rel_filter A ===> op =) eventually eventually"
```
```   792 by(simp add: rel_fun_def rel_filter_eventually)
```
```   793
```
```   794 lemma rel_filter_eq [relator_eq]: "rel_filter op = = op ="
```
```   795 by(auto simp add: rel_filter_eventually rel_fun_eq fun_eq_iff filter_eq_iff)
```
```   796
```
```   797 lemma rel_filter_mono [relator_mono]:
```
```   798   "A \<le> B \<Longrightarrow> rel_filter A \<le> rel_filter B"
```
```   799 unfolding rel_filter_eventually[abs_def]
```
```   800 by(rule le_funI)+(intro fun_mono fun_mono[THEN le_funD, THEN le_funD] order.refl)
```
```   801
```
```   802 lemma rel_filter_conversep [simp]: "rel_filter A\<inverse>\<inverse> = (rel_filter A)\<inverse>\<inverse>"
```
```   803 by(auto simp add: rel_filter_eventually fun_eq_iff rel_fun_def)
```
```   804
```
```   805 lemma is_filter_parametric_aux:
```
```   806   assumes "is_filter F"
```
```   807   assumes [transfer_rule]: "bi_total A" "bi_unique A"
```
```   808   and [transfer_rule]: "((A ===> op =) ===> op =) F G"
```
```   809   shows "is_filter G"
```
```   810 proof -
```
```   811   interpret is_filter F by fact
```
```   812   show ?thesis
```
```   813   proof
```
```   814     have "F (\<lambda>_. True) = G (\<lambda>x. True)" by transfer_prover
```
```   815     thus "G (\<lambda>x. True)" by(simp add: True)
```
```   816   next
```
```   817     fix P' Q'
```
```   818     assume "G P'" "G Q'"
```
```   819     moreover
```
```   820     from bi_total_fun[OF `bi_unique A` bi_total_eq, unfolded bi_total_def]
```
```   821     obtain P Q where [transfer_rule]: "(A ===> op =) P P'" "(A ===> op =) Q Q'" by blast
```
```   822     have "F P = G P'" "F Q = G Q'" by transfer_prover+
```
```   823     ultimately have "F (\<lambda>x. P x \<and> Q x)" by(simp add: conj)
```
```   824     moreover have "F (\<lambda>x. P x \<and> Q x) = G (\<lambda>x. P' x \<and> Q' x)" by transfer_prover
```
```   825     ultimately show "G (\<lambda>x. P' x \<and> Q' x)" by simp
```
```   826   next
```
```   827     fix P' Q'
```
```   828     assume "\<forall>x. P' x \<longrightarrow> Q' x" "G P'"
```
```   829     moreover
```
```   830     from bi_total_fun[OF `bi_unique A` bi_total_eq, unfolded bi_total_def]
```
```   831     obtain P Q where [transfer_rule]: "(A ===> op =) P P'" "(A ===> op =) Q Q'" by blast
```
```   832     have "F P = G P'" by transfer_prover
```
```   833     moreover have "(\<forall>x. P x \<longrightarrow> Q x) \<longleftrightarrow> (\<forall>x. P' x \<longrightarrow> Q' x)" by transfer_prover
```
```   834     ultimately have "F Q" by(simp add: mono)
```
```   835     moreover have "F Q = G Q'" by transfer_prover
```
```   836     ultimately show "G Q'" by simp
```
```   837   qed
```
```   838 qed
```
```   839
```
```   840 lemma is_filter_parametric [transfer_rule]:
```
```   841   "\<lbrakk> bi_total A; bi_unique A \<rbrakk>
```
```   842   \<Longrightarrow> (((A ===> op =) ===> op =) ===> op =) is_filter is_filter"
```
```   843 apply(rule rel_funI)
```
```   844 apply(rule iffI)
```
```   845  apply(erule (3) is_filter_parametric_aux)
```
```   846 apply(erule is_filter_parametric_aux[where A="conversep A"])
```
```   847 apply(auto simp add: rel_fun_def)
```
```   848 done
```
```   849
```
```   850 lemma left_total_rel_filter [transfer_rule]:
```
```   851   assumes [transfer_rule]: "bi_total A" "bi_unique A"
```
```   852   shows "left_total (rel_filter A)"
```
```   853 proof(rule left_totalI)
```
```   854   fix F :: "'a filter"
```
```   855   from bi_total_fun[OF bi_unique_fun[OF `bi_total A` bi_unique_eq] bi_total_eq]
```
```   856   obtain G where [transfer_rule]: "((A ===> op =) ===> op =) (\<lambda>P. eventually P F) G"
```
```   857     unfolding  bi_total_def by blast
```
```   858   moreover have "is_filter (\<lambda>P. eventually P F) \<longleftrightarrow> is_filter G" by transfer_prover
```
```   859   hence "is_filter G" by(simp add: eventually_def is_filter_Rep_filter)
```
```   860   ultimately have "rel_filter A F (Abs_filter G)"
```
```   861     by(simp add: rel_filter_eventually eventually_Abs_filter)
```
```   862   thus "\<exists>G. rel_filter A F G" ..
```
```   863 qed
```
```   864
```
```   865 lemma right_total_rel_filter [transfer_rule]:
```
```   866   "\<lbrakk> bi_total A; bi_unique A \<rbrakk> \<Longrightarrow> right_total (rel_filter A)"
```
```   867 using left_total_rel_filter[of "A\<inverse>\<inverse>"] by simp
```
```   868
```
```   869 lemma bi_total_rel_filter [transfer_rule]:
```
```   870   assumes "bi_total A" "bi_unique A"
```
```   871   shows "bi_total (rel_filter A)"
```
```   872 unfolding bi_total_alt_def using assms
```
```   873 by(simp add: left_total_rel_filter right_total_rel_filter)
```
```   874
```
```   875 lemma left_unique_rel_filter [transfer_rule]:
```
```   876   assumes "left_unique A"
```
```   877   shows "left_unique (rel_filter A)"
```
```   878 proof(rule left_uniqueI)
```
```   879   fix F F' G
```
```   880   assume [transfer_rule]: "rel_filter A F G" "rel_filter A F' G"
```
```   881   show "F = F'"
```
```   882     unfolding filter_eq_iff
```
```   883   proof
```
```   884     fix P :: "'a \<Rightarrow> bool"
```
```   885     obtain P' where [transfer_rule]: "(A ===> op =) P P'"
```
```   886       using left_total_fun[OF assms left_total_eq] unfolding left_total_def by blast
```
```   887     have "eventually P F = eventually P' G"
```
```   888       and "eventually P F' = eventually P' G" by transfer_prover+
```
```   889     thus "eventually P F = eventually P F'" by simp
```
```   890   qed
```
```   891 qed
```
```   892
```
```   893 lemma right_unique_rel_filter [transfer_rule]:
```
```   894   "right_unique A \<Longrightarrow> right_unique (rel_filter A)"
```
```   895 using left_unique_rel_filter[of "A\<inverse>\<inverse>"] by simp
```
```   896
```
```   897 lemma bi_unique_rel_filter [transfer_rule]:
```
```   898   "bi_unique A \<Longrightarrow> bi_unique (rel_filter A)"
```
```   899 by(simp add: bi_unique_alt_def left_unique_rel_filter right_unique_rel_filter)
```
```   900
```
```   901 lemma top_filter_parametric [transfer_rule]:
```
```   902   "bi_total A \<Longrightarrow> (rel_filter A) top top"
```
```   903 by(simp add: rel_filter_eventually All_transfer)
```
```   904
```
```   905 lemma bot_filter_parametric [transfer_rule]: "(rel_filter A) bot bot"
```
```   906 by(simp add: rel_filter_eventually rel_fun_def)
```
```   907
```
```   908 lemma sup_filter_parametric [transfer_rule]:
```
```   909   "(rel_filter A ===> rel_filter A ===> rel_filter A) sup sup"
```
```   910 by(fastforce simp add: rel_filter_eventually[abs_def] eventually_sup dest: rel_funD)
```
```   911
```
```   912 lemma Sup_filter_parametric [transfer_rule]:
```
```   913   "(rel_set (rel_filter A) ===> rel_filter A) Sup Sup"
```
```   914 proof(rule rel_funI)
```
```   915   fix S T
```
```   916   assume [transfer_rule]: "rel_set (rel_filter A) S T"
```
```   917   show "rel_filter A (Sup S) (Sup T)"
```
```   918     by(simp add: rel_filter_eventually eventually_Sup) transfer_prover
```
```   919 qed
```
```   920
```
```   921 lemma principal_parametric [transfer_rule]:
```
```   922   "(rel_set A ===> rel_filter A) principal principal"
```
```   923 proof(rule rel_funI)
```
```   924   fix S S'
```
```   925   assume [transfer_rule]: "rel_set A S S'"
```
```   926   show "rel_filter A (principal S) (principal S')"
```
```   927     by(simp add: rel_filter_eventually eventually_principal) transfer_prover
```
```   928 qed
```
```   929
```
```   930 context
```
```   931   fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"
```
```   932   assumes [transfer_rule]: "bi_unique A"
```
```   933 begin
```
```   934
```
```   935 lemma le_filter_parametric [transfer_rule]:
```
```   936   "(rel_filter A ===> rel_filter A ===> op =) op \<le> op \<le>"
```
```   937 unfolding le_filter_def[abs_def] by transfer_prover
```
```   938
```
```   939 lemma less_filter_parametric [transfer_rule]:
```
```   940   "(rel_filter A ===> rel_filter A ===> op =) op < op <"
```
```   941 unfolding less_filter_def[abs_def] by transfer_prover
```
```   942
```
```   943 context
```
```   944   assumes [transfer_rule]: "bi_total A"
```
```   945 begin
```
```   946
```
```   947 lemma Inf_filter_parametric [transfer_rule]:
```
```   948   "(rel_set (rel_filter A) ===> rel_filter A) Inf Inf"
```
```   949 unfolding Inf_filter_def[abs_def] by transfer_prover
```
```   950
```
```   951 lemma inf_filter_parametric [transfer_rule]:
```
```   952   "(rel_filter A ===> rel_filter A ===> rel_filter A) inf inf"
```
```   953 proof(intro rel_funI)+
```
```   954   fix F F' G G'
```
```   955   assume [transfer_rule]: "rel_filter A F F'" "rel_filter A G G'"
```
```   956   have "rel_filter A (Inf {F, G}) (Inf {F', G'})" by transfer_prover
```
```   957   thus "rel_filter A (inf F G) (inf F' G')" by simp
```
```   958 qed
```
```   959
```
```   960 end
```
```   961
```
```   962 end
```
```   963
```
```   964 end
```
```   965
```
`   966 end`