(*
File: Going_To_Filter.thy
Author: Manuel Eberl, TU München
A filter describing the points x such that f(x) tends to some other filter.
*)
section \<open>The \<open>going_to\<close> filter\<close>
theory Going_To_Filter
imports Complex_Main
begin
definition going_to_within :: "('a \<Rightarrow> 'b) \<Rightarrow> 'b filter \<Rightarrow> 'a set \<Rightarrow> 'a filter"
("(_)/ going'_to (_)/ within (_)" [1000,60,60] 60) where
"f going_to F within A = inf (filtercomap f F) (principal A)"
abbreviation going_to :: "('a \<Rightarrow> 'b) \<Rightarrow> 'b filter \<Rightarrow> 'a filter"
(infix "going'_to" 60)
where "f going_to F \<equiv> f going_to F within UNIV"
text \<open>
The \<open>going_to\<close> filter is, in a sense, the opposite of @{term filtermap}.
It corresponds to the intuition of, given a function $f: A \to B$ and a filter $F$ on the
range of $B$, looking at such values of $x$ that $f(x)$ approaches $F$. This can be
written as @{term "f going_to F"}.
A classic example is the @{term "at_infinity"} filter, which describes the neigbourhood
of infinity (i.\,e.\ all values sufficiently far away from the zero). This can also be written
as @{term "norm going_to at_top"}.
Additionally, the \<open>going_to\<close> filter can be restricted with an optional `within' parameter.
For instance, if one would would want to consider the filter of complex numbers near infinity
that do not lie on the negative real line, one could write
@{term "norm going_to at_top within - complex_of_real ` {..0}"}.
A third, less mathematical example lies in the complexity analysis of algorithms.
Suppose we wanted to say that an algorithm on lists takes $O(n^2)$ time where $n$ is
the length of the input list. We can write this using the Landau symbols from the AFP,
where the underlying filter is @{term "length going_to at_top"}. If, on the other hand,
we want to look the complexity of the algorithm on sorted lists, we could use the filter
@{term "length going_to at_top within {xs. sorted xs}"}.
\<close>
lemma going_to_def: "f going_to F = filtercomap f F"
by (simp add: going_to_within_def)
lemma eventually_going_toI [intro]:
assumes "eventually P F"
shows "eventually (\<lambda>x. P (f x)) (f going_to F)"
using assms by (auto simp: going_to_def)
lemma filterlim_going_toI_weak [intro]: "filterlim f F (f going_to F within A)"
unfolding going_to_within_def
by (meson filterlim_filtercomap filterlim_iff inf_le1 le_filter_def)
lemma going_to_mono: "F \<le> G \<Longrightarrow> A \<subseteq> B \<Longrightarrow> f going_to F within A \<le> f going_to G within B"
unfolding going_to_within_def by (intro inf_mono filtercomap_mono) simp_all
lemma going_to_inf:
"f going_to (inf F G) within A = inf (f going_to F within A) (f going_to G within A)"
by (simp add: going_to_within_def filtercomap_inf inf_assoc inf_commute inf_left_commute)
lemma going_to_sup:
"f going_to (sup F G) within A \<ge> sup (f going_to F within A) (f going_to G within A)"
by (auto simp: going_to_within_def intro!: inf.coboundedI1 filtercomap_sup filtercomap_mono)
lemma going_to_top [simp]: "f going_to top within A = principal A"
by (simp add: going_to_within_def)
lemma going_to_bot [simp]: "f going_to bot within A = bot"
by (simp add: going_to_within_def)
lemma going_to_principal:
"f going_to principal A within B = principal (f -` A \<inter> B)"
by (simp add: going_to_within_def)
lemma going_to_within_empty [simp]: "f going_to F within {} = bot"
by (simp add: going_to_within_def)
lemma going_to_within_union [simp]:
"f going_to F within (A \<union> B) = sup (f going_to F within A) (f going_to F within B)"
by (simp add: going_to_within_def flip: inf_sup_distrib1)
lemma eventually_going_to_at_top_linorder:
fixes f :: "'a \<Rightarrow> 'b :: linorder"
shows "eventually P (f going_to at_top within A) \<longleftrightarrow> (\<exists>C. \<forall>x\<in>A. f x \<ge> C \<longrightarrow> P x)"
unfolding going_to_within_def eventually_filtercomap
eventually_inf_principal eventually_at_top_linorder by fast
lemma eventually_going_to_at_bot_linorder:
fixes f :: "'a \<Rightarrow> 'b :: linorder"
shows "eventually P (f going_to at_bot within A) \<longleftrightarrow> (\<exists>C. \<forall>x\<in>A. f x \<le> C \<longrightarrow> P x)"
unfolding going_to_within_def eventually_filtercomap
eventually_inf_principal eventually_at_bot_linorder by fast
lemma eventually_going_to_at_top_dense:
fixes f :: "'a \<Rightarrow> 'b :: {linorder,no_top}"
shows "eventually P (f going_to at_top within A) \<longleftrightarrow> (\<exists>C. \<forall>x\<in>A. f x > C \<longrightarrow> P x)"
unfolding going_to_within_def eventually_filtercomap
eventually_inf_principal eventually_at_top_dense by fast
lemma eventually_going_to_at_bot_dense:
fixes f :: "'a \<Rightarrow> 'b :: {linorder,no_bot}"
shows "eventually P (f going_to at_bot within A) \<longleftrightarrow> (\<exists>C. \<forall>x\<in>A. f x < C \<longrightarrow> P x)"
unfolding going_to_within_def eventually_filtercomap
eventually_inf_principal eventually_at_bot_dense by fast
lemma eventually_going_to_nhds:
"eventually P (f going_to nhds a within A) \<longleftrightarrow>
(\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>A. f x \<in> S \<longrightarrow> P x))"
unfolding going_to_within_def eventually_filtercomap eventually_inf_principal
eventually_nhds by fast
lemma eventually_going_to_at:
"eventually P (f going_to (at a within B) within A) \<longleftrightarrow>
(\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>A. f x \<in> B \<inter> S - {a} \<longrightarrow> P x))"
unfolding at_within_def going_to_inf eventually_inf_principal
eventually_going_to_nhds going_to_principal by fast
lemma norm_going_to_at_top_eq: "norm going_to at_top = at_infinity"
by (simp add: eventually_at_infinity eventually_going_to_at_top_linorder filter_eq_iff)
lemmas at_infinity_altdef = norm_going_to_at_top_eq [symmetric]
end