(* Title: HOL/Library/Liminf_Limsup.thy
Author: Johannes Hölzl, TU München
*)
header {* Liminf and Limsup on complete lattices *}
theory Liminf_Limsup
imports Complex_Main
begin
lemma le_Sup_iff_less:
fixes x :: "'a :: {complete_linorder, dense_linorder}"
shows "x \<le> (SUP i:A. f i) \<longleftrightarrow> (\<forall>y<x. \<exists>i\<in>A. y \<le> f i)" (is "?lhs = ?rhs")
unfolding le_SUP_iff
by (blast intro: less_imp_le less_trans less_le_trans dest: dense)
lemma Inf_le_iff_less:
fixes x :: "'a :: {complete_linorder, dense_linorder}"
shows "(INF i:A. f i) \<le> x \<longleftrightarrow> (\<forall>y>x. \<exists>i\<in>A. f i \<le> y)"
unfolding INF_le_iff
by (blast intro: less_imp_le less_trans le_less_trans dest: dense)
lemma SUPR_pair:
fixes f :: "_ \<Rightarrow> _ \<Rightarrow> _ :: complete_lattice"
shows "(SUP i : A. SUP j : B. f i j) = (SUP p : A \<times> B. f (fst p) (snd p))"
by (rule antisym) (auto intro!: SUP_least SUP_upper2)
lemma INFI_pair:
fixes f :: "_ \<Rightarrow> _ \<Rightarrow> _ :: complete_lattice"
shows "(INF i : A. INF j : B. f i j) = (INF p : A \<times> B. f (fst p) (snd p))"
by (rule antisym) (auto intro!: INF_greatest INF_lower2)
subsubsection {* @{text Liminf} and @{text Limsup} *}
definition
"Liminf F f = (SUP P:{P. eventually P F}. INF x:{x. P x}. f x)"
definition
"Limsup F f = (INF P:{P. eventually P F}. SUP x:{x. P x}. f x)"
abbreviation "liminf \<equiv> Liminf sequentially"
abbreviation "limsup \<equiv> Limsup sequentially"
lemma Liminf_eqI:
fixes f :: "_ \<Rightarrow> _ :: complete_lattice"
shows "(\<And>P. eventually P F \<Longrightarrow> INFI (Collect P) f \<le> x) \<Longrightarrow>
(\<And>y. (\<And>P. eventually P F \<Longrightarrow> INFI (Collect P) f \<le> y) \<Longrightarrow> x \<le> y) \<Longrightarrow> Liminf F f = x"
unfolding Liminf_def by (auto intro!: SUP_eqI)
lemma Limsup_eqI:
fixes f :: "_ \<Rightarrow> _ :: complete_lattice"
shows "(\<And>P. eventually P F \<Longrightarrow> x \<le> SUPR (Collect P) f) \<Longrightarrow>
(\<And>y. (\<And>P. eventually P F \<Longrightarrow> y \<le> SUPR (Collect P) f) \<Longrightarrow> y \<le> x) \<Longrightarrow> Limsup F f = x"
unfolding Limsup_def by (auto intro!: INF_eqI)
lemma liminf_SUPR_INFI:
fixes f :: "nat \<Rightarrow> 'a :: complete_lattice"
shows "liminf f = (SUP n. INF m:{n..}. f m)"
unfolding Liminf_def eventually_sequentially
by (rule SUPR_eq) (auto simp: atLeast_def intro!: INF_mono)
lemma limsup_INFI_SUPR:
fixes f :: "nat \<Rightarrow> 'a :: complete_lattice"
shows "limsup f = (INF n. SUP m:{n..}. f m)"
unfolding Limsup_def eventually_sequentially
by (rule INFI_eq) (auto simp: atLeast_def intro!: SUP_mono)
lemma Limsup_const:
assumes ntriv: "\<not> trivial_limit F"
shows "Limsup F (\<lambda>x. c) = (c::'a::complete_lattice)"
proof -
have *: "\<And>P. Ex P \<longleftrightarrow> P \<noteq> (\<lambda>x. False)" by auto
have "\<And>P. eventually P F \<Longrightarrow> (SUP x : {x. P x}. c) = c"
using ntriv by (intro SUP_const) (auto simp: eventually_False *)
then show ?thesis
unfolding Limsup_def using eventually_True
by (subst INF_cong[where D="\<lambda>x. c"])
(auto intro!: INF_const simp del: eventually_True)
qed
lemma Liminf_const:
assumes ntriv: "\<not> trivial_limit F"
shows "Liminf F (\<lambda>x. c) = (c::'a::complete_lattice)"
proof -
have *: "\<And>P. Ex P \<longleftrightarrow> P \<noteq> (\<lambda>x. False)" by auto
have "\<And>P. eventually P F \<Longrightarrow> (INF x : {x. P x}. c) = c"
using ntriv by (intro INF_const) (auto simp: eventually_False *)
then show ?thesis
unfolding Liminf_def using eventually_True
by (subst SUP_cong[where D="\<lambda>x. c"])
(auto intro!: SUP_const simp del: eventually_True)
qed
lemma Liminf_mono:
fixes f g :: "'a => 'b :: complete_lattice"
assumes ev: "eventually (\<lambda>x. f x \<le> g x) F"
shows "Liminf F f \<le> Liminf F g"
unfolding Liminf_def
proof (safe intro!: SUP_mono)
fix P assume "eventually P F"
with ev have "eventually (\<lambda>x. f x \<le> g x \<and> P x) F" (is "eventually ?Q F") by (rule eventually_conj)
then show "\<exists>Q\<in>{P. eventually P F}. INFI (Collect P) f \<le> INFI (Collect Q) g"
by (intro bexI[of _ ?Q]) (auto intro!: INF_mono)
qed
lemma Liminf_eq:
fixes f g :: "'a \<Rightarrow> 'b :: complete_lattice"
assumes "eventually (\<lambda>x. f x = g x) F"
shows "Liminf F f = Liminf F g"
by (intro antisym Liminf_mono eventually_mono[OF _ assms]) auto
lemma Limsup_mono:
fixes f g :: "'a \<Rightarrow> 'b :: complete_lattice"
assumes ev: "eventually (\<lambda>x. f x \<le> g x) F"
shows "Limsup F f \<le> Limsup F g"
unfolding Limsup_def
proof (safe intro!: INF_mono)
fix P assume "eventually P F"
with ev have "eventually (\<lambda>x. f x \<le> g x \<and> P x) F" (is "eventually ?Q F") by (rule eventually_conj)
then show "\<exists>Q\<in>{P. eventually P F}. SUPR (Collect Q) f \<le> SUPR (Collect P) g"
by (intro bexI[of _ ?Q]) (auto intro!: SUP_mono)
qed
lemma Limsup_eq:
fixes f g :: "'a \<Rightarrow> 'b :: complete_lattice"
assumes "eventually (\<lambda>x. f x = g x) net"
shows "Limsup net f = Limsup net g"
by (intro antisym Limsup_mono eventually_mono[OF _ assms]) auto
lemma Liminf_le_Limsup:
fixes f :: "'a \<Rightarrow> 'b::complete_lattice"
assumes ntriv: "\<not> trivial_limit F"
shows "Liminf F f \<le> Limsup F f"
unfolding Limsup_def Liminf_def
apply (rule complete_lattice_class.SUP_least)
apply (rule complete_lattice_class.INF_greatest)
proof safe
fix P Q assume "eventually P F" "eventually Q F"
then have "eventually (\<lambda>x. P x \<and> Q x) F" (is "eventually ?C F") by (rule eventually_conj)
then have not_False: "(\<lambda>x. P x \<and> Q x) \<noteq> (\<lambda>x. False)"
using ntriv by (auto simp add: eventually_False)
have "INFI (Collect P) f \<le> INFI (Collect ?C) f"
by (rule INF_mono) auto
also have "\<dots> \<le> SUPR (Collect ?C) f"
using not_False by (intro INF_le_SUP) auto
also have "\<dots> \<le> SUPR (Collect Q) f"
by (rule SUP_mono) auto
finally show "INFI (Collect P) f \<le> SUPR (Collect Q) f" .
qed
lemma Liminf_bounded:
fixes X Y :: "'a \<Rightarrow> 'b::complete_lattice"
assumes ntriv: "\<not> trivial_limit F"
assumes le: "eventually (\<lambda>n. C \<le> X n) F"
shows "C \<le> Liminf F X"
using Liminf_mono[OF le] Liminf_const[OF ntriv, of C] by simp
lemma Limsup_bounded:
fixes X Y :: "'a \<Rightarrow> 'b::complete_lattice"
assumes ntriv: "\<not> trivial_limit F"
assumes le: "eventually (\<lambda>n. X n \<le> C) F"
shows "Limsup F X \<le> C"
using Limsup_mono[OF le] Limsup_const[OF ntriv, of C] by simp
lemma le_Liminf_iff:
fixes X :: "_ \<Rightarrow> _ :: complete_linorder"
shows "C \<le> Liminf F X \<longleftrightarrow> (\<forall>y<C. eventually (\<lambda>x. y < X x) F)"
proof -
{ fix y P assume "eventually P F" "y < INFI (Collect P) X"
then have "eventually (\<lambda>x. y < X x) F"
by (auto elim!: eventually_elim1 dest: less_INF_D) }
moreover
{ fix y P assume "y < C" and y: "\<forall>y<C. eventually (\<lambda>x. y < X x) F"
have "\<exists>P. eventually P F \<and> y < INFI (Collect P) X"
proof (cases "\<exists>z. y < z \<and> z < C")
case True
then obtain z where z: "y < z \<and> z < C" ..
moreover from z have "z \<le> INFI {x. z < X x} X"
by (auto intro!: INF_greatest)
ultimately show ?thesis
using y by (intro exI[of _ "\<lambda>x. z < X x"]) auto
next
case False
then have "C \<le> INFI {x. y < X x} X"
by (intro INF_greatest) auto
with `y < C` show ?thesis
using y by (intro exI[of _ "\<lambda>x. y < X x"]) auto
qed }
ultimately show ?thesis
unfolding Liminf_def le_SUP_iff by auto
qed
lemma lim_imp_Liminf:
fixes f :: "'a \<Rightarrow> _ :: {complete_linorder, linorder_topology}"
assumes ntriv: "\<not> trivial_limit F"
assumes lim: "(f ---> f0) F"
shows "Liminf F f = f0"
proof (intro Liminf_eqI)
fix P assume P: "eventually P F"
then have "eventually (\<lambda>x. INFI (Collect P) f \<le> f x) F"
by eventually_elim (auto intro!: INF_lower)
then show "INFI (Collect P) f \<le> f0"
by (rule tendsto_le[OF ntriv lim tendsto_const])
next
fix y assume upper: "\<And>P. eventually P F \<Longrightarrow> INFI (Collect P) f \<le> y"
show "f0 \<le> y"
proof cases
assume "\<exists>z. y < z \<and> z < f0"
then obtain z where "y < z \<and> z < f0" ..
moreover have "z \<le> INFI {x. z < f x} f"
by (rule INF_greatest) simp
ultimately show ?thesis
using lim[THEN topological_tendstoD, THEN upper, of "{z <..}"] by auto
next
assume discrete: "\<not> (\<exists>z. y < z \<and> z < f0)"
show ?thesis
proof (rule classical)
assume "\<not> f0 \<le> y"
then have "eventually (\<lambda>x. y < f x) F"
using lim[THEN topological_tendstoD, of "{y <..}"] by auto
then have "eventually (\<lambda>x. f0 \<le> f x) F"
using discrete by (auto elim!: eventually_elim1)
then have "INFI {x. f0 \<le> f x} f \<le> y"
by (rule upper)
moreover have "f0 \<le> INFI {x. f0 \<le> f x} f"
by (intro INF_greatest) simp
ultimately show "f0 \<le> y" by simp
qed
qed
qed
lemma lim_imp_Limsup:
fixes f :: "'a \<Rightarrow> _ :: {complete_linorder, linorder_topology}"
assumes ntriv: "\<not> trivial_limit F"
assumes lim: "(f ---> f0) F"
shows "Limsup F f = f0"
proof (intro Limsup_eqI)
fix P assume P: "eventually P F"
then have "eventually (\<lambda>x. f x \<le> SUPR (Collect P) f) F"
by eventually_elim (auto intro!: SUP_upper)
then show "f0 \<le> SUPR (Collect P) f"
by (rule tendsto_le[OF ntriv tendsto_const lim])
next
fix y assume lower: "\<And>P. eventually P F \<Longrightarrow> y \<le> SUPR (Collect P) f"
show "y \<le> f0"
proof (cases "\<exists>z. f0 < z \<and> z < y")
case True
then obtain z where "f0 < z \<and> z < y" ..
moreover have "SUPR {x. f x < z} f \<le> z"
by (rule SUP_least) simp
ultimately show ?thesis
using lim[THEN topological_tendstoD, THEN lower, of "{..< z}"] by auto
next
case False
show ?thesis
proof (rule classical)
assume "\<not> y \<le> f0"
then have "eventually (\<lambda>x. f x < y) F"
using lim[THEN topological_tendstoD, of "{..< y}"] by auto
then have "eventually (\<lambda>x. f x \<le> f0) F"
using False by (auto elim!: eventually_elim1 simp: not_less)
then have "y \<le> SUPR {x. f x \<le> f0} f"
by (rule lower)
moreover have "SUPR {x. f x \<le> f0} f \<le> f0"
by (intro SUP_least) simp
ultimately show "y \<le> f0" by simp
qed
qed
qed
lemma Liminf_eq_Limsup:
fixes f0 :: "'a :: {complete_linorder, linorder_topology}"
assumes ntriv: "\<not> trivial_limit F"
and lim: "Liminf F f = f0" "Limsup F f = f0"
shows "(f ---> f0) F"
proof (rule order_tendstoI)
fix a assume "f0 < a"
with assms have "Limsup F f < a" by simp
then obtain P where "eventually P F" "SUPR (Collect P) f < a"
unfolding Limsup_def INF_less_iff by auto
then show "eventually (\<lambda>x. f x < a) F"
by (auto elim!: eventually_elim1 dest: SUP_lessD)
next
fix a assume "a < f0"
with assms have "a < Liminf F f" by simp
then obtain P where "eventually P F" "a < INFI (Collect P) f"
unfolding Liminf_def less_SUP_iff by auto
then show "eventually (\<lambda>x. a < f x) F"
by (auto elim!: eventually_elim1 dest: less_INF_D)
qed
lemma tendsto_iff_Liminf_eq_Limsup:
fixes f0 :: "'a :: {complete_linorder, linorder_topology}"
shows "\<not> trivial_limit F \<Longrightarrow> (f ---> f0) F \<longleftrightarrow> (Liminf F f = f0 \<and> Limsup F f = f0)"
by (metis Liminf_eq_Limsup lim_imp_Limsup lim_imp_Liminf)
lemma liminf_subseq_mono:
fixes X :: "nat \<Rightarrow> 'a :: complete_linorder"
assumes "subseq r"
shows "liminf X \<le> liminf (X \<circ> r) "
proof-
have "\<And>n. (INF m:{n..}. X m) \<le> (INF m:{n..}. (X \<circ> r) m)"
proof (safe intro!: INF_mono)
fix n m :: nat assume "n \<le> m" then show "\<exists>ma\<in>{n..}. X ma \<le> (X \<circ> r) m"
using seq_suble[OF `subseq r`, of m] by (intro bexI[of _ "r m"]) auto
qed
then show ?thesis by (auto intro!: SUP_mono simp: liminf_SUPR_INFI comp_def)
qed
lemma limsup_subseq_mono:
fixes X :: "nat \<Rightarrow> 'a :: complete_linorder"
assumes "subseq r"
shows "limsup (X \<circ> r) \<le> limsup X"
proof-
have "\<And>n. (SUP m:{n..}. (X \<circ> r) m) \<le> (SUP m:{n..}. X m)"
proof (safe intro!: SUP_mono)
fix n m :: nat assume "n \<le> m" then show "\<exists>ma\<in>{n..}. (X \<circ> r) m \<le> X ma"
using seq_suble[OF `subseq r`, of m] by (intro bexI[of _ "r m"]) auto
qed
then show ?thesis by (auto intro!: INF_mono simp: limsup_INFI_SUPR comp_def)
qed
end