(* Title: HOL/Analysis/Extended_Real_Limits.thy
Author: Johannes Hölzl, TU München
Author: Robert Himmelmann, TU München
Author: Armin Heller, TU München
Author: Bogdan Grechuk, University of Edinburgh
*)
section \<open>Limits on the Extended Real Number Line\<close> (* TO FIX: perhaps put all Nonstandard Analysis related
topics together? *)
theory Extended_Real_Limits
imports
Topology_Euclidean_Space
"HOL-Library.Extended_Real"
"HOL-Library.Extended_Nonnegative_Real"
"HOL-Library.Indicator_Function"
begin
lemma compact_UNIV:
"compact (UNIV :: 'a::{complete_linorder,linorder_topology,second_countable_topology} set)"
using compact_complete_linorder
by (auto simp: seq_compact_eq_compact[symmetric] seq_compact_def)
lemma compact_eq_closed:
fixes S :: "'a::{complete_linorder,linorder_topology,second_countable_topology} set"
shows "compact S \<longleftrightarrow> closed S"
using closed_Int_compact[of S, OF _ compact_UNIV] compact_imp_closed
by auto
lemma closed_contains_Sup_cl:
fixes S :: "'a::{complete_linorder,linorder_topology,second_countable_topology} set"
assumes "closed S"
and "S \<noteq> {}"
shows "Sup S \<in> S"
proof -
from compact_eq_closed[of S] compact_attains_sup[of S] assms
obtain s where S: "s \<in> S" "\<forall>t\<in>S. t \<le> s"
by auto
then have "Sup S = s"
by (auto intro!: Sup_eqI)
with S show ?thesis
by simp
qed
lemma closed_contains_Inf_cl:
fixes S :: "'a::{complete_linorder,linorder_topology,second_countable_topology} set"
assumes "closed S"
and "S \<noteq> {}"
shows "Inf S \<in> S"
proof -
from compact_eq_closed[of S] compact_attains_inf[of S] assms
obtain s where S: "s \<in> S" "\<forall>t\<in>S. s \<le> t"
by auto
then have "Inf S = s"
by (auto intro!: Inf_eqI)
with S show ?thesis
by simp
qed
instance\<^marker>\<open>tag unimportant\<close> enat :: second_countable_topology
proof
show "\<exists>B::enat set set. countable B \<and> open = generate_topology B"
proof (intro exI conjI)
show "countable (range lessThan \<union> range greaterThan::enat set set)"
by auto
qed (simp add: open_enat_def)
qed
instance\<^marker>\<open>tag unimportant\<close> ereal :: second_countable_topology
proof (standard, intro exI conjI)
let ?B = "(\<Union>r\<in>\<rat>. {{..< r}, {r <..}} :: ereal set set)"
show "countable ?B"
by (auto intro: countable_rat)
show "open = generate_topology ?B"
proof (intro ext iffI)
fix S :: "ereal set"
assume "open S"
then show "generate_topology ?B S"
unfolding open_generated_order
proof induct
case (Basis b)
then obtain e where "b = {..<e} \<or> b = {e<..}"
by auto
moreover have "{..<e} = \<Union>{{..<x}|x. x \<in> \<rat> \<and> x < e}" "{e<..} = \<Union>{{x<..}|x. x \<in> \<rat> \<and> e < x}"
by (auto dest: ereal_dense3
simp del: ex_simps
simp add: ex_simps[symmetric] conj_commute Rats_def image_iff)
ultimately show ?case
by (auto intro: generate_topology.intros)
qed (auto intro: generate_topology.intros)
next
fix S
assume "generate_topology ?B S"
then show "open S"
by induct auto
qed
qed
text \<open>This is a copy from \<open>ereal :: second_countable_topology\<close>. Maybe find a common super class of
topological spaces where the rational numbers are densely embedded ?\<close>
instance ennreal :: second_countable_topology
proof (standard, intro exI conjI)
let ?B = "(\<Union>r\<in>\<rat>. {{..< r}, {r <..}} :: ennreal set set)"
show "countable ?B"
by (auto intro: countable_rat)
show "open = generate_topology ?B"
proof (intro ext iffI)
fix S :: "ennreal set"
assume "open S"
then show "generate_topology ?B S"
unfolding open_generated_order
proof induct
case (Basis b)
then obtain e where "b = {..<e} \<or> b = {e<..}"
by auto
moreover have "{..<e} = \<Union>{{..<x}|x. x \<in> \<rat> \<and> x < e}" "{e<..} = \<Union>{{x<..}|x. x \<in> \<rat> \<and> e < x}"
by (auto dest: ennreal_rat_dense
simp del: ex_simps
simp add: ex_simps[symmetric] conj_commute Rats_def image_iff)
ultimately show ?case
by (auto intro: generate_topology.intros)
qed (auto intro: generate_topology.intros)
next
fix S
assume "generate_topology ?B S"
then show "open S"
by induct auto
qed
qed
lemma ereal_open_closed_aux:
fixes S :: "ereal set"
assumes "open S"
and "closed S"
and S: "(-\<infinity>) \<notin> S"
shows "S = {}"
proof (rule ccontr)
assume "\<not> ?thesis"
then have *: "Inf S \<in> S"
by (metis assms(2) closed_contains_Inf_cl)
{
assume "Inf S = -\<infinity>"
then have False
using * assms(3) by auto
}
moreover
{
assume "Inf S = \<infinity>"
then have "S = {\<infinity>}"
by (metis Inf_eq_PInfty \<open>S \<noteq> {}\<close>)
then have False
by (metis assms(1) not_open_singleton)
}
moreover
{
assume fin: "\<bar>Inf S\<bar> \<noteq> \<infinity>"
from ereal_open_cont_interval[OF assms(1) * fin]
obtain e where e: "e > 0" "{Inf S - e<..<Inf S + e} \<subseteq> S" .
then obtain b where b: "Inf S - e < b" "b < Inf S"
using fin ereal_between[of "Inf S" e] dense[of "Inf S - e"]
by auto
then have "b \<in> {Inf S - e <..< Inf S + e}"
using e fin ereal_between[of "Inf S" e]
by auto
then have "b \<in> S"
using e by auto
then have False
using b by (metis complete_lattice_class.Inf_lower leD)
}
ultimately show False
by auto
qed
lemma ereal_open_closed:
fixes S :: "ereal set"
shows "open S \<and> closed S \<longleftrightarrow> S = {} \<or> S = UNIV"
proof -
{
assume lhs: "open S \<and> closed S"
{
assume "-\<infinity> \<notin> S"
then have "S = {}"
using lhs ereal_open_closed_aux by auto
}
moreover
{
assume "-\<infinity> \<in> S"
then have "- S = {}"
using lhs ereal_open_closed_aux[of "-S"] by auto
}
ultimately have "S = {} \<or> S = UNIV"
by auto
}
then show ?thesis
by auto
qed
lemma ereal_open_atLeast:
fixes x :: ereal
shows "open {x..} \<longleftrightarrow> x = -\<infinity>"
proof
assume "x = -\<infinity>"
then have "{x..} = UNIV"
by auto
then show "open {x..}"
by auto
next
assume "open {x..}"
then have "open {x..} \<and> closed {x..}"
by auto
then have "{x..} = UNIV"
unfolding ereal_open_closed by auto
then show "x = -\<infinity>"
by (simp add: bot_ereal_def atLeast_eq_UNIV_iff)
qed
lemma mono_closed_real:
fixes S :: "real set"
assumes mono: "\<forall>y z. y \<in> S \<and> y \<le> z \<longrightarrow> z \<in> S"
and "closed S"
shows "S = {} \<or> S = UNIV \<or> (\<exists>a. S = {a..})"
proof -
{
assume "S \<noteq> {}"
{ assume ex: "\<exists>B. \<forall>x\<in>S. B \<le> x"
then have *: "\<forall>x\<in>S. Inf S \<le> x"
using cInf_lower[of _ S] ex by (metis bdd_below_def)
then have "Inf S \<in> S"
apply (subst closed_contains_Inf)
using ex \<open>S \<noteq> {}\<close> \<open>closed S\<close>
apply auto
done
then have "\<forall>x. Inf S \<le> x \<longleftrightarrow> x \<in> S"
using mono[rule_format, of "Inf S"] *
by auto
then have "S = {Inf S ..}"
by auto
then have "\<exists>a. S = {a ..}"
by auto
}
moreover
{
assume "\<not> (\<exists>B. \<forall>x\<in>S. B \<le> x)"
then have nex: "\<forall>B. \<exists>x\<in>S. x < B"
by (simp add: not_le)
{
fix y
obtain x where "x\<in>S" and "x < y"
using nex by auto
then have "y \<in> S"
using mono[rule_format, of x y] by auto
}
then have "S = UNIV"
by auto
}
ultimately have "S = UNIV \<or> (\<exists>a. S = {a ..})"
by blast
}
then show ?thesis
by blast
qed
lemma mono_closed_ereal:
fixes S :: "real set"
assumes mono: "\<forall>y z. y \<in> S \<and> y \<le> z \<longrightarrow> z \<in> S"
and "closed S"
shows "\<exists>a. S = {x. a \<le> ereal x}"
proof -
{
assume "S = {}"
then have ?thesis
apply (rule_tac x=PInfty in exI)
apply auto
done
}
moreover
{
assume "S = UNIV"
then have ?thesis
apply (rule_tac x="-\<infinity>" in exI)
apply auto
done
}
moreover
{
assume "\<exists>a. S = {a ..}"
then obtain a where "S = {a ..}"
by auto
then have ?thesis
apply (rule_tac x="ereal a" in exI)
apply auto
done
}
ultimately show ?thesis
using mono_closed_real[of S] assms by auto
qed
lemma Liminf_within:
fixes f :: "'a::metric_space \<Rightarrow> 'b::complete_lattice"
shows "Liminf (at x within S) f = (SUP e\<in>{0<..}. INF y\<in>(S \<inter> ball x e - {x}). f y)"
unfolding Liminf_def eventually_at
proof (rule SUP_eq, simp_all add: Ball_def Bex_def, safe)
fix P d
assume "0 < d" and "\<forall>y. y \<in> S \<longrightarrow> y \<noteq> x \<and> dist y x < d \<longrightarrow> P y"
then have "S \<inter> ball x d - {x} \<subseteq> {x. P x}"
by (auto simp: dist_commute)
then show "\<exists>r>0. Inf (f ` (Collect P)) \<le> Inf (f ` (S \<inter> ball x r - {x}))"
by (intro exI[of _ d] INF_mono conjI \<open>0 < d\<close>) auto
next
fix d :: real
assume "0 < d"
then show "\<exists>P. (\<exists>d>0. \<forall>xa. xa \<in> S \<longrightarrow> xa \<noteq> x \<and> dist xa x < d \<longrightarrow> P xa) \<and>
Inf (f ` (S \<inter> ball x d - {x})) \<le> Inf (f ` (Collect P))"
by (intro exI[of _ "\<lambda>y. y \<in> S \<inter> ball x d - {x}"])
(auto intro!: INF_mono exI[of _ d] simp: dist_commute)
qed
lemma Limsup_within:
fixes f :: "'a::metric_space \<Rightarrow> 'b::complete_lattice"
shows "Limsup (at x within S) f = (INF e\<in>{0<..}. SUP y\<in>(S \<inter> ball x e - {x}). f y)"
unfolding Limsup_def eventually_at
proof (rule INF_eq, simp_all add: Ball_def Bex_def, safe)
fix P d
assume "0 < d" and "\<forall>y. y \<in> S \<longrightarrow> y \<noteq> x \<and> dist y x < d \<longrightarrow> P y"
then have "S \<inter> ball x d - {x} \<subseteq> {x. P x}"
by (auto simp: dist_commute)
then show "\<exists>r>0. Sup (f ` (S \<inter> ball x r - {x})) \<le> Sup (f ` (Collect P))"
by (intro exI[of _ d] SUP_mono conjI \<open>0 < d\<close>) auto
next
fix d :: real
assume "0 < d"
then show "\<exists>P. (\<exists>d>0. \<forall>xa. xa \<in> S \<longrightarrow> xa \<noteq> x \<and> dist xa x < d \<longrightarrow> P xa) \<and>
Sup (f ` (Collect P)) \<le> Sup (f ` (S \<inter> ball x d - {x}))"
by (intro exI[of _ "\<lambda>y. y \<in> S \<inter> ball x d - {x}"])
(auto intro!: SUP_mono exI[of _ d] simp: dist_commute)
qed
lemma Liminf_at:
fixes f :: "'a::metric_space \<Rightarrow> 'b::complete_lattice"
shows "Liminf (at x) f = (SUP e\<in>{0<..}. INF y\<in>(ball x e - {x}). f y)"
using Liminf_within[of x UNIV f] by simp
lemma Limsup_at:
fixes f :: "'a::metric_space \<Rightarrow> 'b::complete_lattice"
shows "Limsup (at x) f = (INF e\<in>{0<..}. SUP y\<in>(ball x e - {x}). f y)"
using Limsup_within[of x UNIV f] by simp
lemma min_Liminf_at:
fixes f :: "'a::metric_space \<Rightarrow> 'b::complete_linorder"
shows "min (f x) (Liminf (at x) f) = (SUP e\<in>{0<..}. INF y\<in>ball x e. f y)"
apply (simp add: inf_min [symmetric] Liminf_at)
apply (subst inf_commute)
apply (subst SUP_inf)
apply auto
apply (metis (no_types, lifting) INF_insert centre_in_ball greaterThan_iff image_cong inf_commute insert_Diff)
done
subsection \<open>Extended-Real.thy\<close> (*FIX ME change title *)
lemma sum_constant_ereal:
fixes a::ereal
shows "(\<Sum>i\<in>I. a) = a * card I"
apply (cases "finite I", induct set: finite, simp_all)
apply (cases a, auto, metis (no_types, opaque_lifting) add.commute mult.commute semiring_normalization_rules(3))
done
lemma real_lim_then_eventually_real:
assumes "(u \<longlongrightarrow> ereal l) F"
shows "eventually (\<lambda>n. u n = ereal(real_of_ereal(u n))) F"
proof -
have "ereal l \<in> {-\<infinity><..<(\<infinity>::ereal)}" by simp
moreover have "open {-\<infinity><..<(\<infinity>::ereal)}" by simp
ultimately have "eventually (\<lambda>n. u n \<in> {-\<infinity><..<(\<infinity>::ereal)}) F" using assms tendsto_def by blast
moreover have "\<And>x. x \<in> {-\<infinity><..<(\<infinity>::ereal)} \<Longrightarrow> x = ereal(real_of_ereal x)" using ereal_real by auto
ultimately show ?thesis by (metis (mono_tags, lifting) eventually_mono)
qed
lemma ereal_Inf_cmult:
assumes "c>(0::real)"
shows "Inf {ereal c * x |x. P x} = ereal c * Inf {x. P x}"
proof -
have "(\<lambda>x::ereal. c * x) (Inf {x::ereal. P x}) = Inf ((\<lambda>x::ereal. c * x)`{x::ereal. P x})"
apply (rule mono_bij_Inf)
apply (simp add: assms ereal_mult_left_mono less_imp_le mono_def)
apply (rule bij_betw_byWitness[of _ "\<lambda>x. (x::ereal) / c"], auto simp add: assms ereal_mult_divide)
using assms ereal_divide_eq apply auto
done
then show ?thesis by (simp only: setcompr_eq_image[symmetric])
qed
subsubsection\<^marker>\<open>tag important\<close> \<open>Continuity of addition\<close>
text \<open>The next few lemmas remove an unnecessary assumption in \<open>tendsto_add_ereal\<close>, culminating
in \<open>tendsto_add_ereal_general\<close> which essentially says that the addition
is continuous on ereal times ereal, except at \<open>(-\<infinity>, \<infinity>)\<close> and \<open>(\<infinity>, -\<infinity>)\<close>.
It is much more convenient in many situations, see for instance the proof of
\<open>tendsto_sum_ereal\<close> below.\<close>
lemma tendsto_add_ereal_PInf:
fixes y :: ereal
assumes y: "y \<noteq> -\<infinity>"
assumes f: "(f \<longlongrightarrow> \<infinity>) F" and g: "(g \<longlongrightarrow> y) F"
shows "((\<lambda>x. f x + g x) \<longlongrightarrow> \<infinity>) F"
proof -
have "\<exists>C. eventually (\<lambda>x. g x > ereal C) F"
proof (cases y)
case (real r)
have "y > y-1" using y real by (simp add: ereal_between(1))
then have "eventually (\<lambda>x. g x > y - 1) F" using g y order_tendsto_iff by auto
moreover have "y-1 = ereal(real_of_ereal(y-1))"
by (metis real ereal_eq_1(1) ereal_minus(1) real_of_ereal.simps(1))
ultimately have "eventually (\<lambda>x. g x > ereal(real_of_ereal(y - 1))) F" by simp
then show ?thesis by auto
next
case (PInf)
have "eventually (\<lambda>x. g x > ereal 0) F" using g PInf by (simp add: tendsto_PInfty)
then show ?thesis by auto
qed (simp add: y)
then obtain C::real where ge: "eventually (\<lambda>x. g x > ereal C) F" by auto
{
fix M::real
have "eventually (\<lambda>x. f x > ereal(M - C)) F" using f by (simp add: tendsto_PInfty)
then have "eventually (\<lambda>x. (f x > ereal (M-C)) \<and> (g x > ereal C)) F"
by (auto simp add: ge eventually_conj_iff)
moreover have "\<And>x. ((f x > ereal (M-C)) \<and> (g x > ereal C)) \<Longrightarrow> (f x + g x > ereal M)"
using ereal_add_strict_mono2 by fastforce
ultimately have "eventually (\<lambda>x. f x + g x > ereal M) F" using eventually_mono by force
}
then show ?thesis by (simp add: tendsto_PInfty)
qed
text\<open>One would like to deduce the next lemma from the previous one, but the fact
that \<open>- (x + y)\<close> is in general different from \<open>(- x) + (- y)\<close> in ereal creates difficulties,
so it is more efficient to copy the previous proof.\<close>
lemma tendsto_add_ereal_MInf:
fixes y :: ereal
assumes y: "y \<noteq> \<infinity>"
assumes f: "(f \<longlongrightarrow> -\<infinity>) F" and g: "(g \<longlongrightarrow> y) F"
shows "((\<lambda>x. f x + g x) \<longlongrightarrow> -\<infinity>) F"
proof -
have "\<exists>C. eventually (\<lambda>x. g x < ereal C) F"
proof (cases y)
case (real r)
have "y < y+1" using y real by (simp add: ereal_between(1))
then have "eventually (\<lambda>x. g x < y + 1) F" using g y order_tendsto_iff by force
moreover have "y+1 = ereal(real_of_ereal (y+1))" by (simp add: real)
ultimately have "eventually (\<lambda>x. g x < ereal(real_of_ereal(y + 1))) F" by simp
then show ?thesis by auto
next
case (MInf)
have "eventually (\<lambda>x. g x < ereal 0) F" using g MInf by (simp add: tendsto_MInfty)
then show ?thesis by auto
qed (simp add: y)
then obtain C::real where ge: "eventually (\<lambda>x. g x < ereal C) F" by auto
{
fix M::real
have "eventually (\<lambda>x. f x < ereal(M - C)) F" using f by (simp add: tendsto_MInfty)
then have "eventually (\<lambda>x. (f x < ereal (M- C)) \<and> (g x < ereal C)) F"
by (auto simp add: ge eventually_conj_iff)
moreover have "\<And>x. ((f x < ereal (M-C)) \<and> (g x < ereal C)) \<Longrightarrow> (f x + g x < ereal M)"
using ereal_add_strict_mono2 by fastforce
ultimately have "eventually (\<lambda>x. f x + g x < ereal M) F" using eventually_mono by force
}
then show ?thesis by (simp add: tendsto_MInfty)
qed
lemma tendsto_add_ereal_general1:
fixes x y :: ereal
assumes y: "\<bar>y\<bar> \<noteq> \<infinity>"
assumes f: "(f \<longlongrightarrow> x) F" and g: "(g \<longlongrightarrow> y) F"
shows "((\<lambda>x. f x + g x) \<longlongrightarrow> x + y) F"
proof (cases x)
case (real r)
have a: "\<bar>x\<bar> \<noteq> \<infinity>" by (simp add: real)
show ?thesis by (rule tendsto_add_ereal[OF a, OF y, OF f, OF g])
next
case PInf
then show ?thesis using tendsto_add_ereal_PInf assms by force
next
case MInf
then show ?thesis using tendsto_add_ereal_MInf assms
by (metis abs_ereal.simps(3) ereal_MInfty_eq_plus)
qed
lemma tendsto_add_ereal_general2:
fixes x y :: ereal
assumes x: "\<bar>x\<bar> \<noteq> \<infinity>"
and f: "(f \<longlongrightarrow> x) F" and g: "(g \<longlongrightarrow> y) F"
shows "((\<lambda>x. f x + g x) \<longlongrightarrow> x + y) F"
proof -
have "((\<lambda>x. g x + f x) \<longlongrightarrow> x + y) F"
using tendsto_add_ereal_general1[OF x, OF g, OF f] add.commute[of "y", of "x"] by simp
moreover have "\<And>x. g x + f x = f x + g x" using add.commute by auto
ultimately show ?thesis by simp
qed
text \<open>The next lemma says that the addition is continuous on \<open>ereal\<close>, except at
the pairs \<open>(-\<infinity>, \<infinity>)\<close> and \<open>(\<infinity>, -\<infinity>)\<close>.\<close>
lemma tendsto_add_ereal_general [tendsto_intros]:
fixes x y :: ereal
assumes "\<not>((x=\<infinity> \<and> y=-\<infinity>) \<or> (x=-\<infinity> \<and> y=\<infinity>))"
and f: "(f \<longlongrightarrow> x) F" and g: "(g \<longlongrightarrow> y) F"
shows "((\<lambda>x. f x + g x) \<longlongrightarrow> x + y) F"
proof (cases x)
case (real r)
show ?thesis
apply (rule tendsto_add_ereal_general2) using real assms by auto
next
case (PInf)
then have "y \<noteq> -\<infinity>" using assms by simp
then show ?thesis using tendsto_add_ereal_PInf PInf assms by auto
next
case (MInf)
then have "y \<noteq> \<infinity>" using assms by simp
then show ?thesis using tendsto_add_ereal_MInf MInf f g by (metis ereal_MInfty_eq_plus)
qed
subsubsection\<^marker>\<open>tag important\<close> \<open>Continuity of multiplication\<close>
text \<open>In the same way as for addition, we prove that the multiplication is continuous on
ereal times ereal, except at \<open>(\<infinity>, 0)\<close> and \<open>(-\<infinity>, 0)\<close> and \<open>(0, \<infinity>)\<close> and \<open>(0, -\<infinity>)\<close>,
starting with specific situations.\<close>
lemma tendsto_mult_real_ereal:
assumes "(u \<longlongrightarrow> ereal l) F" "(v \<longlongrightarrow> ereal m) F"
shows "((\<lambda>n. u n * v n) \<longlongrightarrow> ereal l * ereal m) F"
proof -
have ureal: "eventually (\<lambda>n. u n = ereal(real_of_ereal(u n))) F" by (rule real_lim_then_eventually_real[OF assms(1)])
then have "((\<lambda>n. ereal(real_of_ereal(u n))) \<longlongrightarrow> ereal l) F" using assms by auto
then have limu: "((\<lambda>n. real_of_ereal(u n)) \<longlongrightarrow> l) F" by auto
have vreal: "eventually (\<lambda>n. v n = ereal(real_of_ereal(v n))) F" by (rule real_lim_then_eventually_real[OF assms(2)])
then have "((\<lambda>n. ereal(real_of_ereal(v n))) \<longlongrightarrow> ereal m) F" using assms by auto
then have limv: "((\<lambda>n. real_of_ereal(v n)) \<longlongrightarrow> m) F" by auto
{
fix n assume "u n = ereal(real_of_ereal(u n))" "v n = ereal(real_of_ereal(v n))"
then have "ereal(real_of_ereal(u n) * real_of_ereal(v n)) = u n * v n" by (metis times_ereal.simps(1))
}
then have *: "eventually (\<lambda>n. ereal(real_of_ereal(u n) * real_of_ereal(v n)) = u n * v n) F"
using eventually_elim2[OF ureal vreal] by auto
have "((\<lambda>n. real_of_ereal(u n) * real_of_ereal(v n)) \<longlongrightarrow> l * m) F" using tendsto_mult[OF limu limv] by auto
then have "((\<lambda>n. ereal(real_of_ereal(u n)) * real_of_ereal(v n)) \<longlongrightarrow> ereal(l * m)) F" by auto
then show ?thesis using * filterlim_cong by fastforce
qed
lemma tendsto_mult_ereal_PInf:
fixes f g::"_ \<Rightarrow> ereal"
assumes "(f \<longlongrightarrow> l) F" "l>0" "(g \<longlongrightarrow> \<infinity>) F"
shows "((\<lambda>x. f x * g x) \<longlongrightarrow> \<infinity>) F"
proof -
obtain a::real where "0 < ereal a" "a < l" using assms(2) using ereal_dense2 by blast
have *: "eventually (\<lambda>x. f x > a) F" using \<open>a < l\<close> assms(1) by (simp add: order_tendsto_iff)
{
fix K::real
define M where "M = max K 1"
then have "M > 0" by simp
then have "ereal(M/a) > 0" using \<open>ereal a > 0\<close> by simp
then have "\<And>x. ((f x > a) \<and> (g x > M/a)) \<Longrightarrow> (f x * g x > ereal a * ereal(M/a))"
using ereal_mult_mono_strict'[where ?c = "M/a", OF \<open>0 < ereal a\<close>] by auto
moreover have "ereal a * ereal(M/a) = M" using \<open>ereal a > 0\<close> by simp
ultimately have "\<And>x. ((f x > a) \<and> (g x > M/a)) \<Longrightarrow> (f x * g x > M)" by simp
moreover have "M \<ge> K" unfolding M_def by simp
ultimately have Imp: "\<And>x. ((f x > a) \<and> (g x > M/a)) \<Longrightarrow> (f x * g x > K)"
using ereal_less_eq(3) le_less_trans by blast
have "eventually (\<lambda>x. g x > M/a) F" using assms(3) by (simp add: tendsto_PInfty)
then have "eventually (\<lambda>x. (f x > a) \<and> (g x > M/a)) F"
using * by (auto simp add: eventually_conj_iff)
then have "eventually (\<lambda>x. f x * g x > K) F" using eventually_mono Imp by force
}
then show ?thesis by (auto simp add: tendsto_PInfty)
qed
lemma tendsto_mult_ereal_pos:
fixes f g::"_ \<Rightarrow> ereal"
assumes "(f \<longlongrightarrow> l) F" "(g \<longlongrightarrow> m) F" "l>0" "m>0"
shows "((\<lambda>x. f x * g x) \<longlongrightarrow> l * m) F"
proof (cases)
assume *: "l = \<infinity> \<or> m = \<infinity>"
then show ?thesis
proof (cases)
assume "m = \<infinity>"
then show ?thesis using tendsto_mult_ereal_PInf assms by auto
next
assume "\<not>(m = \<infinity>)"
then have "l = \<infinity>" using * by simp
then have "((\<lambda>x. g x * f x) \<longlongrightarrow> l * m) F" using tendsto_mult_ereal_PInf assms by auto
moreover have "\<And>x. g x * f x = f x * g x" using mult.commute by auto
ultimately show ?thesis by simp
qed
next
assume "\<not>(l = \<infinity> \<or> m = \<infinity>)"
then have "l < \<infinity>" "m < \<infinity>" by auto
then obtain lr mr where "l = ereal lr" "m = ereal mr"
using \<open>l>0\<close> \<open>m>0\<close> by (metis ereal_cases ereal_less(6) not_less_iff_gr_or_eq)
then show ?thesis using tendsto_mult_real_ereal assms by auto
qed
text \<open>We reduce the general situation to the positive case by multiplying by suitable signs.
Unfortunately, as ereal is not a ring, all the neat sign lemmas are not available there. We
give the bare minimum we need.\<close>
lemma ereal_sgn_abs:
fixes l::ereal
shows "sgn(l) * l = abs(l)"
apply (cases l) by (auto simp add: sgn_if ereal_less_uminus_reorder)
lemma sgn_squared_ereal:
assumes "l \<noteq> (0::ereal)"
shows "sgn(l) * sgn(l) = 1"
apply (cases l) using assms by (auto simp add: one_ereal_def sgn_if)
lemma tendsto_mult_ereal [tendsto_intros]:
fixes f g::"_ \<Rightarrow> ereal"
assumes "(f \<longlongrightarrow> l) F" "(g \<longlongrightarrow> m) F" "\<not>((l=0 \<and> abs(m) = \<infinity>) \<or> (m=0 \<and> abs(l) = \<infinity>))"
shows "((\<lambda>x. f x * g x) \<longlongrightarrow> l * m) F"
proof (cases)
assume "l=0 \<or> m=0"
then have "abs(l) \<noteq> \<infinity>" "abs(m) \<noteq> \<infinity>" using assms(3) by auto
then obtain lr mr where "l = ereal lr" "m = ereal mr" by auto
then show ?thesis using tendsto_mult_real_ereal assms by auto
next
have sgn_finite: "\<And>a::ereal. abs(sgn a) \<noteq> \<infinity>"
by (metis MInfty_neq_ereal(2) PInfty_neq_ereal(2) abs_eq_infinity_cases ereal_times(1) ereal_times(3) ereal_uminus_eq_reorder sgn_ereal.elims)
then have sgn_finite2: "\<And>a b::ereal. abs(sgn a * sgn b) \<noteq> \<infinity>"
by (metis abs_eq_infinity_cases abs_ereal.simps(2) abs_ereal.simps(3) ereal_mult_eq_MInfty ereal_mult_eq_PInfty)
assume "\<not>(l=0 \<or> m=0)"
then have "l \<noteq> 0" "m \<noteq> 0" by auto
then have "abs(l) > 0" "abs(m) > 0"
by (metis abs_ereal_ge0 abs_ereal_less0 abs_ereal_pos ereal_uminus_uminus ereal_uminus_zero less_le not_less)+
then have "sgn(l) * l > 0" "sgn(m) * m > 0" using ereal_sgn_abs by auto
moreover have "((\<lambda>x. sgn(l) * f x) \<longlongrightarrow> (sgn(l) * l)) F"
by (rule tendsto_cmult_ereal, auto simp add: sgn_finite assms(1))
moreover have "((\<lambda>x. sgn(m) * g x) \<longlongrightarrow> (sgn(m) * m)) F"
by (rule tendsto_cmult_ereal, auto simp add: sgn_finite assms(2))
ultimately have *: "((\<lambda>x. (sgn(l) * f x) * (sgn(m) * g x)) \<longlongrightarrow> (sgn(l) * l) * (sgn(m) * m)) F"
using tendsto_mult_ereal_pos by force
have "((\<lambda>x. (sgn(l) * sgn(m)) * ((sgn(l) * f x) * (sgn(m) * g x))) \<longlongrightarrow> (sgn(l) * sgn(m)) * ((sgn(l) * l) * (sgn(m) * m))) F"
by (rule tendsto_cmult_ereal, auto simp add: sgn_finite2 *)
moreover have "\<And>x. (sgn(l) * sgn(m)) * ((sgn(l) * f x) * (sgn(m) * g x)) = f x * g x"
by (metis mult.left_neutral sgn_squared_ereal[OF \<open>l \<noteq> 0\<close>] sgn_squared_ereal[OF \<open>m \<noteq> 0\<close>] mult.assoc mult.commute)
moreover have "(sgn(l) * sgn(m)) * ((sgn(l) * l) * (sgn(m) * m)) = l * m"
by (metis mult.left_neutral sgn_squared_ereal[OF \<open>l \<noteq> 0\<close>] sgn_squared_ereal[OF \<open>m \<noteq> 0\<close>] mult.assoc mult.commute)
ultimately show ?thesis by auto
qed
lemma tendsto_cmult_ereal_general [tendsto_intros]:
fixes f::"_ \<Rightarrow> ereal" and c::ereal
assumes "(f \<longlongrightarrow> l) F" "\<not> (l=0 \<and> abs(c) = \<infinity>)"
shows "((\<lambda>x. c * f x) \<longlongrightarrow> c * l) F"
by (cases "c = 0", auto simp add: assms tendsto_mult_ereal)
subsubsection\<^marker>\<open>tag important\<close> \<open>Continuity of division\<close>
lemma tendsto_inverse_ereal_PInf:
fixes u::"_ \<Rightarrow> ereal"
assumes "(u \<longlongrightarrow> \<infinity>) F"
shows "((\<lambda>x. 1/ u x) \<longlongrightarrow> 0) F"
proof -
{
fix e::real assume "e>0"
have "1/e < \<infinity>" by auto
then have "eventually (\<lambda>n. u n > 1/e) F" using assms(1) by (simp add: tendsto_PInfty)
moreover
{
fix z::ereal assume "z>1/e"
then have "z>0" using \<open>e>0\<close> using less_le_trans not_le by fastforce
then have "1/z \<ge> 0" by auto
moreover have "1/z < e" using \<open>e>0\<close> \<open>z>1/e\<close>
apply (cases z) apply auto
by (metis (mono_tags, opaque_lifting) less_ereal.simps(2) less_ereal.simps(4) divide_less_eq ereal_divide_less_pos ereal_less(4)
ereal_less_eq(4) less_le_trans mult_eq_0_iff not_le not_one_less_zero times_ereal.simps(1))
ultimately have "1/z \<ge> 0" "1/z < e" by auto
}
ultimately have "eventually (\<lambda>n. 1/u n<e) F" "eventually (\<lambda>n. 1/u n\<ge>0) F" by (auto simp add: eventually_mono)
} note * = this
show ?thesis
proof (subst order_tendsto_iff, auto)
fix a::ereal assume "a<0"
then show "eventually (\<lambda>n. 1/u n > a) F" using *(2) eventually_mono less_le_trans linordered_field_no_ub by fastforce
next
fix a::ereal assume "a>0"
then obtain e::real where "e>0" "a>e" using ereal_dense2 ereal_less(2) by blast
then have "eventually (\<lambda>n. 1/u n < e) F" using *(1) by auto
then show "eventually (\<lambda>n. 1/u n < a) F" using \<open>a>e\<close> by (metis (mono_tags, lifting) eventually_mono less_trans)
qed
qed
text \<open>The next lemma deserves to exist by itself, as it is so common and useful.\<close>
lemma tendsto_inverse_real [tendsto_intros]:
fixes u::"_ \<Rightarrow> real"
shows "(u \<longlongrightarrow> l) F \<Longrightarrow> l \<noteq> 0 \<Longrightarrow> ((\<lambda>x. 1/ u x) \<longlongrightarrow> 1/l) F"
using tendsto_inverse unfolding inverse_eq_divide .
lemma tendsto_inverse_ereal [tendsto_intros]:
fixes u::"_ \<Rightarrow> ereal"
assumes "(u \<longlongrightarrow> l) F" "l \<noteq> 0"
shows "((\<lambda>x. 1/ u x) \<longlongrightarrow> 1/l) F"
proof (cases l)
case (real r)
then have "r \<noteq> 0" using assms(2) by auto
then have "1/l = ereal(1/r)" using real by (simp add: one_ereal_def)
define v where "v = (\<lambda>n. real_of_ereal(u n))"
have ureal: "eventually (\<lambda>n. u n = ereal(v n)) F" unfolding v_def using real_lim_then_eventually_real assms(1) real by auto
then have "((\<lambda>n. ereal(v n)) \<longlongrightarrow> ereal r) F" using assms real v_def by auto
then have *: "((\<lambda>n. v n) \<longlongrightarrow> r) F" by auto
then have "((\<lambda>n. 1/v n) \<longlongrightarrow> 1/r) F" using \<open>r \<noteq> 0\<close> tendsto_inverse_real by auto
then have lim: "((\<lambda>n. ereal(1/v n)) \<longlongrightarrow> 1/l) F" using \<open>1/l = ereal(1/r)\<close> by auto
have "r \<in> -{0}" "open (-{(0::real)})" using \<open>r \<noteq> 0\<close> by auto
then have "eventually (\<lambda>n. v n \<in> -{0}) F" using * using topological_tendstoD by blast
then have "eventually (\<lambda>n. v n \<noteq> 0) F" by auto
moreover
{
fix n assume H: "v n \<noteq> 0" "u n = ereal(v n)"
then have "ereal(1/v n) = 1/ereal(v n)" by (simp add: one_ereal_def)
then have "ereal(1/v n) = 1/u n" using H(2) by simp
}
ultimately have "eventually (\<lambda>n. ereal(1/v n) = 1/u n) F" using ureal eventually_elim2 by force
with Lim_transform_eventually[OF lim this] show ?thesis by simp
next
case (PInf)
then have "1/l = 0" by auto
then show ?thesis using tendsto_inverse_ereal_PInf assms PInf by auto
next
case (MInf)
then have "1/l = 0" by auto
have "1/z = -1/ -z" if "z < 0" for z::ereal
apply (cases z) using divide_ereal_def \<open> z < 0 \<close> by auto
moreover have "eventually (\<lambda>n. u n < 0) F" by (metis (no_types) MInf assms(1) tendsto_MInfty zero_ereal_def)
ultimately have *: "eventually (\<lambda>n. -1/-u n = 1/u n) F" by (simp add: eventually_mono)
define v where "v = (\<lambda>n. - u n)"
have "(v \<longlongrightarrow> \<infinity>) F" unfolding v_def using MInf assms(1) tendsto_uminus_ereal by fastforce
then have "((\<lambda>n. 1/v n) \<longlongrightarrow> 0) F" using tendsto_inverse_ereal_PInf by auto
then have "((\<lambda>n. -1/v n) \<longlongrightarrow> 0) F" using tendsto_uminus_ereal by fastforce
then show ?thesis unfolding v_def using Lim_transform_eventually[OF _ *] \<open> 1/l = 0 \<close> by auto
qed
lemma tendsto_divide_ereal [tendsto_intros]:
fixes f g::"_ \<Rightarrow> ereal"
assumes "(f \<longlongrightarrow> l) F" "(g \<longlongrightarrow> m) F" "m \<noteq> 0" "\<not>(abs(l) = \<infinity> \<and> abs(m) = \<infinity>)"
shows "((\<lambda>x. f x / g x) \<longlongrightarrow> l / m) F"
proof -
define h where "h = (\<lambda>x. 1/ g x)"
have *: "(h \<longlongrightarrow> 1/m) F" unfolding h_def using assms(2) assms(3) tendsto_inverse_ereal by auto
have "((\<lambda>x. f x * h x) \<longlongrightarrow> l * (1/m)) F"
apply (rule tendsto_mult_ereal[OF assms(1) *]) using assms(3) assms(4) by (auto simp add: divide_ereal_def)
moreover have "f x * h x = f x / g x" for x unfolding h_def by (simp add: divide_ereal_def)
moreover have "l * (1/m) = l/m" by (simp add: divide_ereal_def)
ultimately show ?thesis unfolding h_def using Lim_transform_eventually by auto
qed
subsubsection \<open>Further limits\<close>
text \<open>The assumptions of @{thm tendsto_diff_ereal} are too strong, we weaken them here.\<close>
lemma tendsto_diff_ereal_general [tendsto_intros]:
fixes u v::"'a \<Rightarrow> ereal"
assumes "(u \<longlongrightarrow> l) F" "(v \<longlongrightarrow> m) F" "\<not>((l = \<infinity> \<and> m = \<infinity>) \<or> (l = -\<infinity> \<and> m = -\<infinity>))"
shows "((\<lambda>n. u n - v n) \<longlongrightarrow> l - m) F"
proof -
have "((\<lambda>n. u n + (-v n)) \<longlongrightarrow> l + (-m)) F"
apply (intro tendsto_intros assms) using assms by (auto simp add: ereal_uminus_eq_reorder)
then show ?thesis by (simp add: minus_ereal_def)
qed
lemma id_nat_ereal_tendsto_PInf [tendsto_intros]:
"(\<lambda> n::nat. real n) \<longlonglongrightarrow> \<infinity>"
by (simp add: filterlim_real_sequentially tendsto_PInfty_eq_at_top)
lemma tendsto_at_top_pseudo_inverse [tendsto_intros]:
fixes u::"nat \<Rightarrow> nat"
assumes "LIM n sequentially. u n :> at_top"
shows "LIM n sequentially. Inf {N. u N \<ge> n} :> at_top"
proof -
{
fix C::nat
define M where "M = Max {u n| n. n \<le> C}+1"
{
fix n assume "n \<ge> M"
have "eventually (\<lambda>N. u N \<ge> n) sequentially" using assms
by (simp add: filterlim_at_top)
then have *: "{N. u N \<ge> n} \<noteq> {}" by force
have "N > C" if "u N \<ge> n" for N
proof (rule ccontr)
assume "\<not>(N > C)"
have "u N \<le> Max {u n| n. n \<le> C}"
apply (rule Max_ge) using \<open>\<not>(N > C)\<close> by auto
then show False using \<open>u N \<ge> n\<close> \<open>n \<ge> M\<close> unfolding M_def by auto
qed
then have **: "{N. u N \<ge> n} \<subseteq> {C..}" by fastforce
have "Inf {N. u N \<ge> n} \<ge> C"
by (metis "*" "**" Inf_nat_def1 atLeast_iff subset_eq)
}
then have "eventually (\<lambda>n. Inf {N. u N \<ge> n} \<ge> C) sequentially"
using eventually_sequentially by auto
}
then show ?thesis using filterlim_at_top by auto
qed
lemma pseudo_inverse_finite_set:
fixes u::"nat \<Rightarrow> nat"
assumes "LIM n sequentially. u n :> at_top"
shows "finite {N. u N \<le> n}"
proof -
fix n
have "eventually (\<lambda>N. u N \<ge> n+1) sequentially" using assms
by (simp add: filterlim_at_top)
then obtain N1 where N1: "\<And>N. N \<ge> N1 \<Longrightarrow> u N \<ge> n + 1"
using eventually_sequentially by auto
have "{N. u N \<le> n} \<subseteq> {..<N1}"
apply auto using N1 by (metis Suc_eq_plus1 not_less not_less_eq_eq)
then show "finite {N. u N \<le> n}" by (simp add: finite_subset)
qed
lemma tendsto_at_top_pseudo_inverse2 [tendsto_intros]:
fixes u::"nat \<Rightarrow> nat"
assumes "LIM n sequentially. u n :> at_top"
shows "LIM n sequentially. Max {N. u N \<le> n} :> at_top"
proof -
{
fix N0::nat
have "N0 \<le> Max {N. u N \<le> n}" if "n \<ge> u N0" for n
apply (rule Max.coboundedI) using pseudo_inverse_finite_set[OF assms] that by auto
then have "eventually (\<lambda>n. N0 \<le> Max {N. u N \<le> n}) sequentially"
using eventually_sequentially by blast
}
then show ?thesis using filterlim_at_top by auto
qed
lemma ereal_truncation_top [tendsto_intros]:
fixes x::ereal
shows "(\<lambda>n::nat. min x n) \<longlonglongrightarrow> x"
proof (cases x)
case (real r)
then obtain K::nat where "K>0" "K > abs(r)" using reals_Archimedean2 gr0I by auto
then have "min x n = x" if "n \<ge> K" for n apply (subst real, subst real, auto) using that eq_iff by fastforce
then have "eventually (\<lambda>n. min x n = x) sequentially" using eventually_at_top_linorder by blast
then show ?thesis by (simp add: tendsto_eventually)
next
case (PInf)
then have "min x n = n" for n::nat by (auto simp add: min_def)
then show ?thesis using id_nat_ereal_tendsto_PInf PInf by auto
next
case (MInf)
then have "min x n = x" for n::nat by (auto simp add: min_def)
then show ?thesis by auto
qed
lemma ereal_truncation_real_top [tendsto_intros]:
fixes x::ereal
assumes "x \<noteq> - \<infinity>"
shows "(\<lambda>n::nat. real_of_ereal(min x n)) \<longlonglongrightarrow> x"
proof (cases x)
case (real r)
then obtain K::nat where "K>0" "K > abs(r)" using reals_Archimedean2 gr0I by auto
then have "min x n = x" if "n \<ge> K" for n apply (subst real, subst real, auto) using that eq_iff by fastforce
then have "real_of_ereal(min x n) = r" if "n \<ge> K" for n using real that by auto
then have "eventually (\<lambda>n. real_of_ereal(min x n) = r) sequentially" using eventually_at_top_linorder by blast
then have "(\<lambda>n. real_of_ereal(min x n)) \<longlonglongrightarrow> r" by (simp add: tendsto_eventually)
then show ?thesis using real by auto
next
case (PInf)
then have "real_of_ereal(min x n) = n" for n::nat by (auto simp add: min_def)
then show ?thesis using id_nat_ereal_tendsto_PInf PInf by auto
qed (simp add: assms)
lemma ereal_truncation_bottom [tendsto_intros]:
fixes x::ereal
shows "(\<lambda>n::nat. max x (- real n)) \<longlonglongrightarrow> x"
proof (cases x)
case (real r)
then obtain K::nat where "K>0" "K > abs(r)" using reals_Archimedean2 gr0I by auto
then have "max x (-real n) = x" if "n \<ge> K" for n apply (subst real, subst real, auto) using that eq_iff by fastforce
then have "eventually (\<lambda>n. max x (-real n) = x) sequentially" using eventually_at_top_linorder by blast
then show ?thesis by (simp add: tendsto_eventually)
next
case (MInf)
then have "max x (-real n) = (-1)* ereal(real n)" for n::nat by (auto simp add: max_def)
moreover have "(\<lambda>n. (-1)* ereal(real n)) \<longlonglongrightarrow> -\<infinity>"
using tendsto_cmult_ereal[of "-1", OF _ id_nat_ereal_tendsto_PInf] by (simp add: one_ereal_def)
ultimately show ?thesis using MInf by auto
next
case (PInf)
then have "max x (-real n) = x" for n::nat by (auto simp add: max_def)
then show ?thesis by auto
qed
lemma ereal_truncation_real_bottom [tendsto_intros]:
fixes x::ereal
assumes "x \<noteq> \<infinity>"
shows "(\<lambda>n::nat. real_of_ereal(max x (- real n))) \<longlonglongrightarrow> x"
proof (cases x)
case (real r)
then obtain K::nat where "K>0" "K > abs(r)" using reals_Archimedean2 gr0I by auto
then have "max x (-real n) = x" if "n \<ge> K" for n apply (subst real, subst real, auto) using that eq_iff by fastforce
then have "real_of_ereal(max x (-real n)) = r" if "n \<ge> K" for n using real that by auto
then have "eventually (\<lambda>n. real_of_ereal(max x (-real n)) = r) sequentially" using eventually_at_top_linorder by blast
then have "(\<lambda>n. real_of_ereal(max x (-real n))) \<longlonglongrightarrow> r" by (simp add: tendsto_eventually)
then show ?thesis using real by auto
next
case (MInf)
then have "real_of_ereal(max x (-real n)) = (-1)* ereal(real n)" for n::nat by (auto simp add: max_def)
moreover have "(\<lambda>n. (-1)* ereal(real n)) \<longlonglongrightarrow> -\<infinity>"
using tendsto_cmult_ereal[of "-1", OF _ id_nat_ereal_tendsto_PInf] by (simp add: one_ereal_def)
ultimately show ?thesis using MInf by auto
qed (simp add: assms)
text \<open>the next one is copied from \<open>tendsto_sum\<close>.\<close>
lemma tendsto_sum_ereal [tendsto_intros]:
fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> ereal"
assumes "\<And>i. i \<in> S \<Longrightarrow> (f i \<longlongrightarrow> a i) F"
"\<And>i. abs(a i) \<noteq> \<infinity>"
shows "((\<lambda>x. \<Sum>i\<in>S. f i x) \<longlongrightarrow> (\<Sum>i\<in>S. a i)) F"
proof (cases "finite S")
assume "finite S" then show ?thesis using assms
by (induct, simp, simp add: tendsto_add_ereal_general2 assms)
qed(simp)
lemma continuous_ereal_abs:
"continuous_on (UNIV::ereal set) abs"
proof -
have "continuous_on ({..0} \<union> {(0::ereal)..}) abs"
apply (rule continuous_on_closed_Un, auto)
apply (rule iffD1[OF continuous_on_cong, of "{..0}" _ "\<lambda>x. -x"])
using less_eq_ereal_def apply (auto simp add: continuous_uminus_ereal)
apply (rule iffD1[OF continuous_on_cong, of "{0..}" _ "\<lambda>x. x"])
apply (auto)
done
moreover have "(UNIV::ereal set) = {..0} \<union> {(0::ereal)..}" by auto
ultimately show ?thesis by auto
qed
lemmas continuous_on_compose_ereal_abs[continuous_intros] =
continuous_on_compose2[OF continuous_ereal_abs _ subset_UNIV]
lemma tendsto_abs_ereal [tendsto_intros]:
assumes "(u \<longlongrightarrow> (l::ereal)) F"
shows "((\<lambda>n. abs(u n)) \<longlongrightarrow> abs l) F"
using continuous_ereal_abs assms by (metis UNIV_I continuous_on tendsto_compose)
lemma ereal_minus_real_tendsto_MInf [tendsto_intros]:
"(\<lambda>x. ereal (- real x)) \<longlonglongrightarrow> - \<infinity>"
by (subst uminus_ereal.simps(1)[symmetric], intro tendsto_intros)
subsection \<open>Extended-Nonnegative-Real.thy\<close> (*FIX title *)
lemma tendsto_diff_ennreal_general [tendsto_intros]:
fixes u v::"'a \<Rightarrow> ennreal"
assumes "(u \<longlongrightarrow> l) F" "(v \<longlongrightarrow> m) F" "\<not>(l = \<infinity> \<and> m = \<infinity>)"
shows "((\<lambda>n. u n - v n) \<longlongrightarrow> l - m) F"
proof -
have "((\<lambda>n. e2ennreal(enn2ereal(u n) - enn2ereal(v n))) \<longlongrightarrow> e2ennreal(enn2ereal l - enn2ereal m)) F"
apply (intro tendsto_intros) using assms by auto
then show ?thesis by auto
qed
lemma tendsto_mult_ennreal [tendsto_intros]:
fixes l m::ennreal
assumes "(u \<longlongrightarrow> l) F" "(v \<longlongrightarrow> m) F" "\<not>((l = 0 \<and> m = \<infinity>) \<or> (l = \<infinity> \<and> m = 0))"
shows "((\<lambda>n. u n * v n) \<longlongrightarrow> l * m) F"
proof -
have "((\<lambda>n. e2ennreal(enn2ereal (u n) * enn2ereal (v n))) \<longlongrightarrow> e2ennreal(enn2ereal l * enn2ereal m)) F"
apply (intro tendsto_intros) using assms apply auto
using enn2ereal_inject zero_ennreal.rep_eq by fastforce+
moreover have "e2ennreal(enn2ereal (u n) * enn2ereal (v n)) = u n * v n" for n
by (subst times_ennreal.abs_eq[symmetric], auto simp add: eq_onp_same_args)
moreover have "e2ennreal(enn2ereal l * enn2ereal m) = l * m"
by (subst times_ennreal.abs_eq[symmetric], auto simp add: eq_onp_same_args)
ultimately show ?thesis
by auto
qed
subsection \<open>monoset\<close> (*FIX ME title *)
definition (in order) mono_set:
"mono_set S \<longleftrightarrow> (\<forall>x y. x \<le> y \<longrightarrow> x \<in> S \<longrightarrow> y \<in> S)"
lemma (in order) mono_greaterThan [intro, simp]: "mono_set {B<..}" unfolding mono_set by auto
lemma (in order) mono_atLeast [intro, simp]: "mono_set {B..}" unfolding mono_set by auto
lemma (in order) mono_UNIV [intro, simp]: "mono_set UNIV" unfolding mono_set by auto
lemma (in order) mono_empty [intro, simp]: "mono_set {}" unfolding mono_set by auto
lemma (in complete_linorder) mono_set_iff:
fixes S :: "'a set"
defines "a \<equiv> Inf S"
shows "mono_set S \<longleftrightarrow> S = {a <..} \<or> S = {a..}" (is "_ = ?c")
proof
assume "mono_set S"
then have mono: "\<And>x y. x \<le> y \<Longrightarrow> x \<in> S \<Longrightarrow> y \<in> S"
by (auto simp: mono_set)
show ?c
proof cases
assume "a \<in> S"
show ?c
using mono[OF _ \<open>a \<in> S\<close>]
by (auto intro: Inf_lower simp: a_def)
next
assume "a \<notin> S"
have "S = {a <..}"
proof safe
fix x assume "x \<in> S"
then have "a \<le> x"
unfolding a_def by (rule Inf_lower)
then show "a < x"
using \<open>x \<in> S\<close> \<open>a \<notin> S\<close> by (cases "a = x") auto
next
fix x assume "a < x"
then obtain y where "y < x" "y \<in> S"
unfolding a_def Inf_less_iff ..
with mono[of y x] show "x \<in> S"
by auto
qed
then show ?c ..
qed
qed auto
lemma ereal_open_mono_set:
fixes S :: "ereal set"
shows "open S \<and> mono_set S \<longleftrightarrow> S = UNIV \<or> S = {Inf S <..}"
by (metis Inf_UNIV atLeast_eq_UNIV_iff ereal_open_atLeast
ereal_open_closed mono_set_iff open_ereal_greaterThan)
lemma ereal_closed_mono_set:
fixes S :: "ereal set"
shows "closed S \<and> mono_set S \<longleftrightarrow> S = {} \<or> S = {Inf S ..}"
by (metis Inf_UNIV atLeast_eq_UNIV_iff closed_ereal_atLeast
ereal_open_closed mono_empty mono_set_iff open_ereal_greaterThan)
lemma ereal_Liminf_Sup_monoset:
fixes f :: "'a \<Rightarrow> ereal"
shows "Liminf net f =
Sup {l. \<forall>S. open S \<longrightarrow> mono_set S \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) net}"
(is "_ = Sup ?A")
proof (safe intro!: Liminf_eqI complete_lattice_class.Sup_upper complete_lattice_class.Sup_least)
fix P
assume P: "eventually P net"
fix S
assume S: "mono_set S" "Inf (f ` (Collect P)) \<in> S"
{
fix x
assume "P x"
then have "Inf (f ` (Collect P)) \<le> f x"
by (intro complete_lattice_class.INF_lower) simp
with S have "f x \<in> S"
by (simp add: mono_set)
}
with P show "eventually (\<lambda>x. f x \<in> S) net"
by (auto elim: eventually_mono)
next
fix y l
assume S: "\<forall>S. open S \<longrightarrow> mono_set S \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) net"
assume P: "\<forall>P. eventually P net \<longrightarrow> Inf (f ` (Collect P)) \<le> y"
show "l \<le> y"
proof (rule dense_le)
fix B
assume "B < l"
then have "eventually (\<lambda>x. f x \<in> {B <..}) net"
by (intro S[rule_format]) auto
then have "Inf (f ` {x. B < f x}) \<le> y"
using P by auto
moreover have "B \<le> Inf (f ` {x. B < f x})"
by (intro INF_greatest) auto
ultimately show "B \<le> y"
by simp
qed
qed
lemma ereal_Limsup_Inf_monoset:
fixes f :: "'a \<Rightarrow> ereal"
shows "Limsup net f =
Inf {l. \<forall>S. open S \<longrightarrow> mono_set (uminus ` S) \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) net}"
(is "_ = Inf ?A")
proof (safe intro!: Limsup_eqI complete_lattice_class.Inf_lower complete_lattice_class.Inf_greatest)
fix P
assume P: "eventually P net"
fix S
assume S: "mono_set (uminus`S)" "Sup (f ` (Collect P)) \<in> S"
{
fix x
assume "P x"
then have "f x \<le> Sup (f ` (Collect P))"
by (intro complete_lattice_class.SUP_upper) simp
with S(1)[unfolded mono_set, rule_format, of "- Sup (f ` (Collect P))" "- f x"] S(2)
have "f x \<in> S"
by (simp add: inj_image_mem_iff) }
with P show "eventually (\<lambda>x. f x \<in> S) net"
by (auto elim: eventually_mono)
next
fix y l
assume S: "\<forall>S. open S \<longrightarrow> mono_set (uminus ` S) \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) net"
assume P: "\<forall>P. eventually P net \<longrightarrow> y \<le> Sup (f ` (Collect P))"
show "y \<le> l"
proof (rule dense_ge)
fix B
assume "l < B"
then have "eventually (\<lambda>x. f x \<in> {..< B}) net"
by (intro S[rule_format]) auto
then have "y \<le> Sup (f ` {x. f x < B})"
using P by auto
moreover have "Sup (f ` {x. f x < B}) \<le> B"
by (intro SUP_least) auto
ultimately show "y \<le> B"
by simp
qed
qed
lemma liminf_bounded_open:
fixes x :: "nat \<Rightarrow> ereal"
shows "x0 \<le> liminf x \<longleftrightarrow> (\<forall>S. open S \<longrightarrow> mono_set S \<longrightarrow> x0 \<in> S \<longrightarrow> (\<exists>N. \<forall>n\<ge>N. x n \<in> S))"
(is "_ \<longleftrightarrow> ?P x0")
proof
assume "?P x0"
then show "x0 \<le> liminf x"
unfolding ereal_Liminf_Sup_monoset eventually_sequentially
by (intro complete_lattice_class.Sup_upper) auto
next
assume "x0 \<le> liminf x"
{
fix S :: "ereal set"
assume om: "open S" "mono_set S" "x0 \<in> S"
{
assume "S = UNIV"
then have "\<exists>N. \<forall>n\<ge>N. x n \<in> S"
by auto
}
moreover
{
assume "S \<noteq> UNIV"
then obtain B where B: "S = {B<..}"
using om ereal_open_mono_set by auto
then have "B < x0"
using om by auto
then have "\<exists>N. \<forall>n\<ge>N. x n \<in> S"
unfolding B
using \<open>x0 \<le> liminf x\<close> liminf_bounded_iff
by auto
}
ultimately have "\<exists>N. \<forall>n\<ge>N. x n \<in> S"
by auto
}
then show "?P x0"
by auto
qed
lemma limsup_finite_then_bounded:
fixes u::"nat \<Rightarrow> real"
assumes "limsup u < \<infinity>"
shows "\<exists>C. \<forall>n. u n \<le> C"
proof -
obtain C where C: "limsup u < C" "C < \<infinity>" using assms ereal_dense2 by blast
then have "C = ereal(real_of_ereal C)" using ereal_real by force
have "eventually (\<lambda>n. u n < C) sequentially" using C(1) unfolding Limsup_def
apply (auto simp add: INF_less_iff)
using SUP_lessD eventually_mono by fastforce
then obtain N where N: "\<And>n. n \<ge> N \<Longrightarrow> u n < C" using eventually_sequentially by auto
define D where "D = max (real_of_ereal C) (Max {u n |n. n \<le> N})"
have "\<And>n. u n \<le> D"
proof -
fix n show "u n \<le> D"
proof (cases)
assume *: "n \<le> N"
have "u n \<le> Max {u n |n. n \<le> N}" by (rule Max_ge, auto simp add: *)
then show "u n \<le> D" unfolding D_def by linarith
next
assume "\<not>(n \<le> N)"
then have "n \<ge> N" by simp
then have "u n < C" using N by auto
then have "u n < real_of_ereal C" using \<open>C = ereal(real_of_ereal C)\<close> less_ereal.simps(1) by fastforce
then show "u n \<le> D" unfolding D_def by linarith
qed
qed
then show ?thesis by blast
qed
lemma liminf_finite_then_bounded_below:
fixes u::"nat \<Rightarrow> real"
assumes "liminf u > -\<infinity>"
shows "\<exists>C. \<forall>n. u n \<ge> C"
proof -
obtain C where C: "liminf u > C" "C > -\<infinity>" using assms using ereal_dense2 by blast
then have "C = ereal(real_of_ereal C)" using ereal_real by force
have "eventually (\<lambda>n. u n > C) sequentially" using C(1) unfolding Liminf_def
apply (auto simp add: less_SUP_iff)
using eventually_elim2 less_INF_D by fastforce
then obtain N where N: "\<And>n. n \<ge> N \<Longrightarrow> u n > C" using eventually_sequentially by auto
define D where "D = min (real_of_ereal C) (Min {u n |n. n \<le> N})"
have "\<And>n. u n \<ge> D"
proof -
fix n show "u n \<ge> D"
proof (cases)
assume *: "n \<le> N"
have "u n \<ge> Min {u n |n. n \<le> N}" by (rule Min_le, auto simp add: *)
then show "u n \<ge> D" unfolding D_def by linarith
next
assume "\<not>(n \<le> N)"
then have "n \<ge> N" by simp
then have "u n > C" using N by auto
then have "u n > real_of_ereal C" using \<open>C = ereal(real_of_ereal C)\<close> less_ereal.simps(1) by fastforce
then show "u n \<ge> D" unfolding D_def by linarith
qed
qed
then show ?thesis by blast
qed
lemma liminf_upper_bound:
fixes u:: "nat \<Rightarrow> ereal"
assumes "liminf u < l"
shows "\<exists>N>k. u N < l"
by (metis assms gt_ex less_le_trans liminf_bounded_iff not_less)
lemma limsup_shift:
"limsup (\<lambda>n. u (n+1)) = limsup u"
proof -
have "(SUP m\<in>{n+1..}. u m) = (SUP m\<in>{n..}. u (m + 1))" for n
apply (rule SUP_eq) using Suc_le_D by auto
then have a: "(INF n. SUP m\<in>{n..}. u (m + 1)) = (INF n. (SUP m\<in>{n+1..}. u m))" by auto
have b: "(INF n. (SUP m\<in>{n+1..}. u m)) = (INF n\<in>{1..}. (SUP m\<in>{n..}. u m))"
apply (rule INF_eq) using Suc_le_D by auto
have "(INF n\<in>{1..}. v n) = (INF n. v n)" if "decseq v" for v::"nat \<Rightarrow> 'a"
apply (rule INF_eq) using \<open>decseq v\<close> decseq_Suc_iff by auto
moreover have "decseq (\<lambda>n. (SUP m\<in>{n..}. u m))" by (simp add: SUP_subset_mono decseq_def)
ultimately have c: "(INF n\<in>{1..}. (SUP m\<in>{n..}. u m)) = (INF n. (SUP m\<in>{n..}. u m))" by simp
have "(INF n. Sup (u ` {n..})) = (INF n. SUP m\<in>{n..}. u (m + 1))" using a b c by simp
then show ?thesis by (auto cong: limsup_INF_SUP)
qed
lemma limsup_shift_k:
"limsup (\<lambda>n. u (n+k)) = limsup u"
proof (induction k)
case (Suc k)
have "limsup (\<lambda>n. u (n+k+1)) = limsup (\<lambda>n. u (n+k))" using limsup_shift[where ?u="\<lambda>n. u(n+k)"] by simp
then show ?case using Suc.IH by simp
qed (auto)
lemma liminf_shift:
"liminf (\<lambda>n. u (n+1)) = liminf u"
proof -
have "(INF m\<in>{n+1..}. u m) = (INF m\<in>{n..}. u (m + 1))" for n
apply (rule INF_eq) using Suc_le_D by (auto)
then have a: "(SUP n. INF m\<in>{n..}. u (m + 1)) = (SUP n. (INF m\<in>{n+1..}. u m))" by auto
have b: "(SUP n. (INF m\<in>{n+1..}. u m)) = (SUP n\<in>{1..}. (INF m\<in>{n..}. u m))"
apply (rule SUP_eq) using Suc_le_D by (auto)
have "(SUP n\<in>{1..}. v n) = (SUP n. v n)" if "incseq v" for v::"nat \<Rightarrow> 'a"
apply (rule SUP_eq) using \<open>incseq v\<close> incseq_Suc_iff by auto
moreover have "incseq (\<lambda>n. (INF m\<in>{n..}. u m))" by (simp add: INF_superset_mono mono_def)
ultimately have c: "(SUP n\<in>{1..}. (INF m\<in>{n..}. u m)) = (SUP n. (INF m\<in>{n..}. u m))" by simp
have "(SUP n. Inf (u ` {n..})) = (SUP n. INF m\<in>{n..}. u (m + 1))" using a b c by simp
then show ?thesis by (auto cong: liminf_SUP_INF)
qed
lemma liminf_shift_k:
"liminf (\<lambda>n. u (n+k)) = liminf u"
proof (induction k)
case (Suc k)
have "liminf (\<lambda>n. u (n+k+1)) = liminf (\<lambda>n. u (n+k))" using liminf_shift[where ?u="\<lambda>n. u(n+k)"] by simp
then show ?case using Suc.IH by simp
qed (auto)
lemma Limsup_obtain:
fixes u::"_ \<Rightarrow> 'a :: complete_linorder"
assumes "Limsup F u > c"
shows "\<exists>i. u i > c"
proof -
have "(INF P\<in>{P. eventually P F}. SUP x\<in>{x. P x}. u x) > c" using assms by (simp add: Limsup_def)
then show ?thesis by (metis eventually_True mem_Collect_eq less_INF_D less_SUP_iff)
qed
text \<open>The next lemma is extremely useful, as it often makes it possible to reduce statements
about limsups to statements about limits.\<close>
lemma limsup_subseq_lim:
fixes u::"nat \<Rightarrow> 'a :: {complete_linorder, linorder_topology}"
shows "\<exists>r::nat\<Rightarrow>nat. strict_mono r \<and> (u o r) \<longlonglongrightarrow> limsup u"
proof (cases)
assume "\<forall>n. \<exists>p>n. \<forall>m\<ge>p. u m \<le> u p"
then have "\<exists>r. \<forall>n. (\<forall>m\<ge>r n. u m \<le> u (r n)) \<and> r n < r (Suc n)"
by (intro dependent_nat_choice) (auto simp: conj_commute)
then obtain r :: "nat \<Rightarrow> nat" where "strict_mono r" and mono: "\<And>n m. r n \<le> m \<Longrightarrow> u m \<le> u (r n)"
by (auto simp: strict_mono_Suc_iff)
define umax where "umax = (\<lambda>n. (SUP m\<in>{n..}. u m))"
have "decseq umax" unfolding umax_def by (simp add: SUP_subset_mono antimono_def)
then have "umax \<longlonglongrightarrow> limsup u" unfolding umax_def by (metis LIMSEQ_INF limsup_INF_SUP)
then have *: "(umax o r) \<longlonglongrightarrow> limsup u" by (simp add: LIMSEQ_subseq_LIMSEQ \<open>strict_mono r\<close>)
have "\<And>n. umax(r n) = u(r n)" unfolding umax_def using mono
by (metis SUP_le_iff antisym atLeast_def mem_Collect_eq order_refl)
then have "umax o r = u o r" unfolding o_def by simp
then have "(u o r) \<longlonglongrightarrow> limsup u" using * by simp
then show ?thesis using \<open>strict_mono r\<close> by blast
next
assume "\<not> (\<forall>n. \<exists>p>n. (\<forall>m\<ge>p. u m \<le> u p))"
then obtain N where N: "\<And>p. p > N \<Longrightarrow> \<exists>m>p. u p < u m" by (force simp: not_le le_less)
have "\<exists>r. \<forall>n. N < r n \<and> r n < r (Suc n) \<and> (\<forall>i\<in> {N<..r (Suc n)}. u i \<le> u (r (Suc n)))"
proof (rule dependent_nat_choice)
fix x assume "N < x"
then have a: "finite {N<..x}" "{N<..x} \<noteq> {}" by simp_all
have "Max {u i |i. i \<in> {N<..x}} \<in> {u i |i. i \<in> {N<..x}}" apply (rule Max_in) using a by (auto)
then obtain p where "p \<in> {N<..x}" and upmax: "u p = Max{u i |i. i \<in> {N<..x}}" by auto
define U where "U = {m. m > p \<and> u p < u m}"
have "U \<noteq> {}" unfolding U_def using N[of p] \<open>p \<in> {N<..x}\<close> by auto
define y where "y = Inf U"
then have "y \<in> U" using \<open>U \<noteq> {}\<close> by (simp add: Inf_nat_def1)
have a: "\<And>i. i \<in> {N<..x} \<Longrightarrow> u i \<le> u p"
proof -
fix i assume "i \<in> {N<..x}"
then have "u i \<in> {u i |i. i \<in> {N<..x}}" by blast
then show "u i \<le> u p" using upmax by simp
qed
moreover have "u p < u y" using \<open>y \<in> U\<close> U_def by auto
ultimately have "y \<notin> {N<..x}" using not_le by blast
moreover have "y > N" using \<open>y \<in> U\<close> U_def \<open>p \<in> {N<..x}\<close> by auto
ultimately have "y > x" by auto
have "\<And>i. i \<in> {N<..y} \<Longrightarrow> u i \<le> u y"
proof -
fix i assume "i \<in> {N<..y}" show "u i \<le> u y"
proof (cases)
assume "i = y"
then show ?thesis by simp
next
assume "\<not>(i=y)"
then have i:"i \<in> {N<..<y}" using \<open>i \<in> {N<..y}\<close> by simp
have "u i \<le> u p"
proof (cases)
assume "i \<le> x"
then have "i \<in> {N<..x}" using i by simp
then show ?thesis using a by simp
next
assume "\<not>(i \<le> x)"
then have "i > x" by simp
then have *: "i > p" using \<open>p \<in> {N<..x}\<close> by simp
have "i < Inf U" using i y_def by simp
then have "i \<notin> U" using Inf_nat_def not_less_Least by auto
then show ?thesis using U_def * by auto
qed
then show "u i \<le> u y" using \<open>u p < u y\<close> by auto
qed
qed
then have "N < y \<and> x < y \<and> (\<forall>i\<in>{N<..y}. u i \<le> u y)" using \<open>y > x\<close> \<open>y > N\<close> by auto
then show "\<exists>y>N. x < y \<and> (\<forall>i\<in>{N<..y}. u i \<le> u y)" by auto
qed (auto)
then obtain r where r: "\<forall>n. N < r n \<and> r n < r (Suc n) \<and> (\<forall>i\<in> {N<..r (Suc n)}. u i \<le> u (r (Suc n)))" by auto
have "strict_mono r" using r by (auto simp: strict_mono_Suc_iff)
have "incseq (u o r)" unfolding o_def using r by (simp add: incseq_SucI order.strict_implies_order)
then have "(u o r) \<longlonglongrightarrow> (SUP n. (u o r) n)" using LIMSEQ_SUP by blast
then have "limsup (u o r) = (SUP n. (u o r) n)" by (simp add: lim_imp_Limsup)
moreover have "limsup (u o r) \<le> limsup u" using \<open>strict_mono r\<close> by (simp add: limsup_subseq_mono)
ultimately have "(SUP n. (u o r) n) \<le> limsup u" by simp
{
fix i assume i: "i \<in> {N<..}"
obtain n where "i < r (Suc n)" using \<open>strict_mono r\<close> using Suc_le_eq seq_suble by blast
then have "i \<in> {N<..r(Suc n)}" using i by simp
then have "u i \<le> u (r(Suc n))" using r by simp
then have "u i \<le> (SUP n. (u o r) n)" unfolding o_def by (meson SUP_upper2 UNIV_I)
}
then have "(SUP i\<in>{N<..}. u i) \<le> (SUP n. (u o r) n)" using SUP_least by blast
then have "limsup u \<le> (SUP n. (u o r) n)" unfolding Limsup_def
by (metis (mono_tags, lifting) INF_lower2 atLeast_Suc_greaterThan atLeast_def eventually_ge_at_top mem_Collect_eq)
then have "limsup u = (SUP n. (u o r) n)" using \<open>(SUP n. (u o r) n) \<le> limsup u\<close> by simp
then have "(u o r) \<longlonglongrightarrow> limsup u" using \<open>(u o r) \<longlonglongrightarrow> (SUP n. (u o r) n)\<close> by simp
then show ?thesis using \<open>strict_mono r\<close> by auto
qed
lemma liminf_subseq_lim:
fixes u::"nat \<Rightarrow> 'a :: {complete_linorder, linorder_topology}"
shows "\<exists>r::nat\<Rightarrow>nat. strict_mono r \<and> (u o r) \<longlonglongrightarrow> liminf u"
proof (cases)
assume "\<forall>n. \<exists>p>n. \<forall>m\<ge>p. u m \<ge> u p"
then have "\<exists>r. \<forall>n. (\<forall>m\<ge>r n. u m \<ge> u (r n)) \<and> r n < r (Suc n)"
by (intro dependent_nat_choice) (auto simp: conj_commute)
then obtain r :: "nat \<Rightarrow> nat" where "strict_mono r" and mono: "\<And>n m. r n \<le> m \<Longrightarrow> u m \<ge> u (r n)"
by (auto simp: strict_mono_Suc_iff)
define umin where "umin = (\<lambda>n. (INF m\<in>{n..}. u m))"
have "incseq umin" unfolding umin_def by (simp add: INF_superset_mono incseq_def)
then have "umin \<longlonglongrightarrow> liminf u" unfolding umin_def by (metis LIMSEQ_SUP liminf_SUP_INF)
then have *: "(umin o r) \<longlonglongrightarrow> liminf u" by (simp add: LIMSEQ_subseq_LIMSEQ \<open>strict_mono r\<close>)
have "\<And>n. umin(r n) = u(r n)" unfolding umin_def using mono
by (metis le_INF_iff antisym atLeast_def mem_Collect_eq order_refl)
then have "umin o r = u o r" unfolding o_def by simp
then have "(u o r) \<longlonglongrightarrow> liminf u" using * by simp
then show ?thesis using \<open>strict_mono r\<close> by blast
next
assume "\<not> (\<forall>n. \<exists>p>n. (\<forall>m\<ge>p. u m \<ge> u p))"
then obtain N where N: "\<And>p. p > N \<Longrightarrow> \<exists>m>p. u p > u m" by (force simp: not_le le_less)
have "\<exists>r. \<forall>n. N < r n \<and> r n < r (Suc n) \<and> (\<forall>i\<in> {N<..r (Suc n)}. u i \<ge> u (r (Suc n)))"
proof (rule dependent_nat_choice)
fix x assume "N < x"
then have a: "finite {N<..x}" "{N<..x} \<noteq> {}" by simp_all
have "Min {u i |i. i \<in> {N<..x}} \<in> {u i |i. i \<in> {N<..x}}" apply (rule Min_in) using a by (auto)
then obtain p where "p \<in> {N<..x}" and upmin: "u p = Min{u i |i. i \<in> {N<..x}}" by auto
define U where "U = {m. m > p \<and> u p > u m}"
have "U \<noteq> {}" unfolding U_def using N[of p] \<open>p \<in> {N<..x}\<close> by auto
define y where "y = Inf U"
then have "y \<in> U" using \<open>U \<noteq> {}\<close> by (simp add: Inf_nat_def1)
have a: "\<And>i. i \<in> {N<..x} \<Longrightarrow> u i \<ge> u p"
proof -
fix i assume "i \<in> {N<..x}"
then have "u i \<in> {u i |i. i \<in> {N<..x}}" by blast
then show "u i \<ge> u p" using upmin by simp
qed
moreover have "u p > u y" using \<open>y \<in> U\<close> U_def by auto
ultimately have "y \<notin> {N<..x}" using not_le by blast
moreover have "y > N" using \<open>y \<in> U\<close> U_def \<open>p \<in> {N<..x}\<close> by auto
ultimately have "y > x" by auto
have "\<And>i. i \<in> {N<..y} \<Longrightarrow> u i \<ge> u y"
proof -
fix i assume "i \<in> {N<..y}" show "u i \<ge> u y"
proof (cases)
assume "i = y"
then show ?thesis by simp
next
assume "\<not>(i=y)"
then have i:"i \<in> {N<..<y}" using \<open>i \<in> {N<..y}\<close> by simp
have "u i \<ge> u p"
proof (cases)
assume "i \<le> x"
then have "i \<in> {N<..x}" using i by simp
then show ?thesis using a by simp
next
assume "\<not>(i \<le> x)"
then have "i > x" by simp
then have *: "i > p" using \<open>p \<in> {N<..x}\<close> by simp
have "i < Inf U" using i y_def by simp
then have "i \<notin> U" using Inf_nat_def not_less_Least by auto
then show ?thesis using U_def * by auto
qed
then show "u i \<ge> u y" using \<open>u p > u y\<close> by auto
qed
qed
then have "N < y \<and> x < y \<and> (\<forall>i\<in>{N<..y}. u i \<ge> u y)" using \<open>y > x\<close> \<open>y > N\<close> by auto
then show "\<exists>y>N. x < y \<and> (\<forall>i\<in>{N<..y}. u i \<ge> u y)" by auto
qed (auto)
then obtain r :: "nat \<Rightarrow> nat"
where r: "\<forall>n. N < r n \<and> r n < r (Suc n) \<and> (\<forall>i\<in> {N<..r (Suc n)}. u i \<ge> u (r (Suc n)))" by auto
have "strict_mono r" using r by (auto simp: strict_mono_Suc_iff)
have "decseq (u o r)" unfolding o_def using r by (simp add: decseq_SucI order.strict_implies_order)
then have "(u o r) \<longlonglongrightarrow> (INF n. (u o r) n)" using LIMSEQ_INF by blast
then have "liminf (u o r) = (INF n. (u o r) n)" by (simp add: lim_imp_Liminf)
moreover have "liminf (u o r) \<ge> liminf u" using \<open>strict_mono r\<close> by (simp add: liminf_subseq_mono)
ultimately have "(INF n. (u o r) n) \<ge> liminf u" by simp
{
fix i assume i: "i \<in> {N<..}"
obtain n where "i < r (Suc n)" using \<open>strict_mono r\<close> using Suc_le_eq seq_suble by blast
then have "i \<in> {N<..r(Suc n)}" using i by simp
then have "u i \<ge> u (r(Suc n))" using r by simp
then have "u i \<ge> (INF n. (u o r) n)" unfolding o_def by (meson INF_lower2 UNIV_I)
}
then have "(INF i\<in>{N<..}. u i) \<ge> (INF n. (u o r) n)" using INF_greatest by blast
then have "liminf u \<ge> (INF n. (u o r) n)" unfolding Liminf_def
by (metis (mono_tags, lifting) SUP_upper2 atLeast_Suc_greaterThan atLeast_def eventually_ge_at_top mem_Collect_eq)
then have "liminf u = (INF n. (u o r) n)" using \<open>(INF n. (u o r) n) \<ge> liminf u\<close> by simp
then have "(u o r) \<longlonglongrightarrow> liminf u" using \<open>(u o r) \<longlonglongrightarrow> (INF n. (u o r) n)\<close> by simp
then show ?thesis using \<open>strict_mono r\<close> by auto
qed
text \<open>The following statement about limsups is reduced to a statement about limits using
subsequences thanks to \<open>limsup_subseq_lim\<close>. The statement for limits follows for instance from
\<open>tendsto_add_ereal_general\<close>.\<close>
lemma ereal_limsup_add_mono:
fixes u v::"nat \<Rightarrow> ereal"
shows "limsup (\<lambda>n. u n + v n) \<le> limsup u + limsup v"
proof (cases)
assume "(limsup u = \<infinity>) \<or> (limsup v = \<infinity>)"
then have "limsup u + limsup v = \<infinity>" by simp
then show ?thesis by auto
next
assume "\<not>((limsup u = \<infinity>) \<or> (limsup v = \<infinity>))"
then have "limsup u < \<infinity>" "limsup v < \<infinity>" by auto
define w where "w = (\<lambda>n. u n + v n)"
obtain r where r: "strict_mono r" "(w o r) \<longlonglongrightarrow> limsup w" using limsup_subseq_lim by auto
obtain s where s: "strict_mono s" "(u o r o s) \<longlonglongrightarrow> limsup (u o r)" using limsup_subseq_lim by auto
obtain t where t: "strict_mono t" "(v o r o s o t) \<longlonglongrightarrow> limsup (v o r o s)" using limsup_subseq_lim by auto
define a where "a = r o s o t"
have "strict_mono a" using r s t by (simp add: a_def strict_mono_o)
have l:"(w o a) \<longlonglongrightarrow> limsup w"
"(u o a) \<longlonglongrightarrow> limsup (u o r)"
"(v o a) \<longlonglongrightarrow> limsup (v o r o s)"
apply (metis (no_types, lifting) r(2) s(1) t(1) LIMSEQ_subseq_LIMSEQ a_def comp_assoc)
apply (metis (no_types, lifting) s(2) t(1) LIMSEQ_subseq_LIMSEQ a_def comp_assoc)
apply (metis (no_types, lifting) t(2) a_def comp_assoc)
done
have "limsup (u o r) \<le> limsup u" by (simp add: limsup_subseq_mono r(1))
then have a: "limsup (u o r) \<noteq> \<infinity>" using \<open>limsup u < \<infinity>\<close> by auto
have "limsup (v o r o s) \<le> limsup v"
by (simp add: comp_assoc limsup_subseq_mono r(1) s(1) strict_mono_o)
then have b: "limsup (v o r o s) \<noteq> \<infinity>" using \<open>limsup v < \<infinity>\<close> by auto
have "(\<lambda>n. (u o a) n + (v o a) n) \<longlonglongrightarrow> limsup (u o r) + limsup (v o r o s)"
using l tendsto_add_ereal_general a b by fastforce
moreover have "(\<lambda>n. (u o a) n + (v o a) n) = (w o a)" unfolding w_def by auto
ultimately have "(w o a) \<longlonglongrightarrow> limsup (u o r) + limsup (v o r o s)" by simp
then have "limsup w = limsup (u o r) + limsup (v o r o s)" using l(1) LIMSEQ_unique by blast
then have "limsup w \<le> limsup u + limsup v"
using \<open>limsup (u o r) \<le> limsup u\<close> \<open>limsup (v o r o s) \<le> limsup v\<close> add_mono by simp
then show ?thesis unfolding w_def by simp
qed
text \<open>There is an asymmetry between liminfs and limsups in \<open>ereal\<close>, as \<open>\<infinity> + (-\<infinity>) = \<infinity>\<close>.
This explains why there are more assumptions in the next lemma dealing with liminfs that in the
previous one about limsups.\<close>
lemma ereal_liminf_add_mono:
fixes u v::"nat \<Rightarrow> ereal"
assumes "\<not>((liminf u = \<infinity> \<and> liminf v = -\<infinity>) \<or> (liminf u = -\<infinity> \<and> liminf v = \<infinity>))"
shows "liminf (\<lambda>n. u n + v n) \<ge> liminf u + liminf v"
proof (cases)
assume "(liminf u = -\<infinity>) \<or> (liminf v = -\<infinity>)"
then have *: "liminf u + liminf v = -\<infinity>" using assms by auto
show ?thesis by (simp add: *)
next
assume "\<not>((liminf u = -\<infinity>) \<or> (liminf v = -\<infinity>))"
then have "liminf u > -\<infinity>" "liminf v > -\<infinity>" by auto
define w where "w = (\<lambda>n. u n + v n)"
obtain r where r: "strict_mono r" "(w o r) \<longlonglongrightarrow> liminf w" using liminf_subseq_lim by auto
obtain s where s: "strict_mono s" "(u o r o s) \<longlonglongrightarrow> liminf (u o r)" using liminf_subseq_lim by auto
obtain t where t: "strict_mono t" "(v o r o s o t) \<longlonglongrightarrow> liminf (v o r o s)" using liminf_subseq_lim by auto
define a where "a = r o s o t"
have "strict_mono a" using r s t by (simp add: a_def strict_mono_o)
have l:"(w o a) \<longlonglongrightarrow> liminf w"
"(u o a) \<longlonglongrightarrow> liminf (u o r)"
"(v o a) \<longlonglongrightarrow> liminf (v o r o s)"
apply (metis (no_types, lifting) r(2) s(1) t(1) LIMSEQ_subseq_LIMSEQ a_def comp_assoc)
apply (metis (no_types, lifting) s(2) t(1) LIMSEQ_subseq_LIMSEQ a_def comp_assoc)
apply (metis (no_types, lifting) t(2) a_def comp_assoc)
done
have "liminf (u o r) \<ge> liminf u" by (simp add: liminf_subseq_mono r(1))
then have a: "liminf (u o r) \<noteq> -\<infinity>" using \<open>liminf u > -\<infinity>\<close> by auto
have "liminf (v o r o s) \<ge> liminf v"
by (simp add: comp_assoc liminf_subseq_mono r(1) s(1) strict_mono_o)
then have b: "liminf (v o r o s) \<noteq> -\<infinity>" using \<open>liminf v > -\<infinity>\<close> by auto
have "(\<lambda>n. (u o a) n + (v o a) n) \<longlonglongrightarrow> liminf (u o r) + liminf (v o r o s)"
using l tendsto_add_ereal_general a b by fastforce
moreover have "(\<lambda>n. (u o a) n + (v o a) n) = (w o a)" unfolding w_def by auto
ultimately have "(w o a) \<longlonglongrightarrow> liminf (u o r) + liminf (v o r o s)" by simp
then have "liminf w = liminf (u o r) + liminf (v o r o s)" using l(1) LIMSEQ_unique by blast
then have "liminf w \<ge> liminf u + liminf v"
using \<open>liminf (u o r) \<ge> liminf u\<close> \<open>liminf (v o r o s) \<ge> liminf v\<close> add_mono by simp
then show ?thesis unfolding w_def by simp
qed
lemma ereal_limsup_lim_add:
fixes u v::"nat \<Rightarrow> ereal"
assumes "u \<longlonglongrightarrow> a" "abs(a) \<noteq> \<infinity>"
shows "limsup (\<lambda>n. u n + v n) = a + limsup v"
proof -
have "limsup u = a" using assms(1) using tendsto_iff_Liminf_eq_Limsup trivial_limit_at_top_linorder by blast
have "(\<lambda>n. -u n) \<longlonglongrightarrow> -a" using assms(1) by auto
then have "limsup (\<lambda>n. -u n) = -a" using tendsto_iff_Liminf_eq_Limsup trivial_limit_at_top_linorder by blast
have "limsup (\<lambda>n. u n + v n) \<le> limsup u + limsup v"
by (rule ereal_limsup_add_mono)
then have up: "limsup (\<lambda>n. u n + v n) \<le> a + limsup v" using \<open>limsup u = a\<close> by simp
have a: "limsup (\<lambda>n. (u n + v n) + (-u n)) \<le> limsup (\<lambda>n. u n + v n) + limsup (\<lambda>n. -u n)"
by (rule ereal_limsup_add_mono)
have "eventually (\<lambda>n. u n = ereal(real_of_ereal(u n))) sequentially" using assms
real_lim_then_eventually_real by auto
moreover have "\<And>x. x = ereal(real_of_ereal(x)) \<Longrightarrow> x + (-x) = 0"
by (metis plus_ereal.simps(1) right_minus uminus_ereal.simps(1) zero_ereal_def)
ultimately have "eventually (\<lambda>n. u n + (-u n) = 0) sequentially"
by (metis (mono_tags, lifting) eventually_mono)
moreover have "\<And>n. u n + (-u n) = 0 \<Longrightarrow> u n + v n + (-u n) = v n"
by (metis add.commute add.left_commute add.left_neutral)
ultimately have "eventually (\<lambda>n. u n + v n + (-u n) = v n) sequentially"
using eventually_mono by force
then have "limsup v = limsup (\<lambda>n. u n + v n + (-u n))" using Limsup_eq by force
then have "limsup v \<le> limsup (\<lambda>n. u n + v n) -a" using a \<open>limsup (\<lambda>n. -u n) = -a\<close> by (simp add: minus_ereal_def)
then have "limsup (\<lambda>n. u n + v n) \<ge> a + limsup v" using assms(2) by (metis add.commute ereal_le_minus)
then show ?thesis using up by simp
qed
lemma ereal_limsup_lim_mult:
fixes u v::"nat \<Rightarrow> ereal"
assumes "u \<longlonglongrightarrow> a" "a>0" "a \<noteq> \<infinity>"
shows "limsup (\<lambda>n. u n * v n) = a * limsup v"
proof -
define w where "w = (\<lambda>n. u n * v n)"
obtain r where r: "strict_mono r" "(v o r) \<longlonglongrightarrow> limsup v" using limsup_subseq_lim by auto
have "(u o r) \<longlonglongrightarrow> a" using assms(1) LIMSEQ_subseq_LIMSEQ r by auto
with tendsto_mult_ereal[OF this r(2)] have "(\<lambda>n. (u o r) n * (v o r) n) \<longlonglongrightarrow> a * limsup v" using assms(2) assms(3) by auto
moreover have "\<And>n. (w o r) n = (u o r) n * (v o r) n" unfolding w_def by auto
ultimately have "(w o r) \<longlonglongrightarrow> a * limsup v" unfolding w_def by presburger
then have "limsup (w o r) = a * limsup v" by (simp add: tendsto_iff_Liminf_eq_Limsup)
then have I: "limsup w \<ge> a * limsup v" by (metis limsup_subseq_mono r(1))
obtain s where s: "strict_mono s" "(w o s) \<longlonglongrightarrow> limsup w" using limsup_subseq_lim by auto
have *: "(u o s) \<longlonglongrightarrow> a" using assms(1) LIMSEQ_subseq_LIMSEQ s by auto
have "eventually (\<lambda>n. (u o s) n > 0) sequentially" using assms(2) * order_tendsto_iff by blast
moreover have "eventually (\<lambda>n. (u o s) n < \<infinity>) sequentially" using assms(3) * order_tendsto_iff by blast
moreover have "(w o s) n / (u o s) n = (v o s) n" if "(u o s) n > 0" "(u o s) n < \<infinity>" for n
unfolding w_def using that by (auto simp add: ereal_divide_eq)
ultimately have "eventually (\<lambda>n. (w o s) n / (u o s) n = (v o s) n) sequentially" using eventually_elim2 by force
moreover have "(\<lambda>n. (w o s) n / (u o s) n) \<longlonglongrightarrow> (limsup w) / a"
apply (rule tendsto_divide_ereal[OF s(2) *]) using assms(2) assms(3) by auto
ultimately have "(v o s) \<longlonglongrightarrow> (limsup w) / a" using Lim_transform_eventually by fastforce
then have "limsup (v o s) = (limsup w) / a" by (simp add: tendsto_iff_Liminf_eq_Limsup)
then have "limsup v \<ge> (limsup w) / a" by (metis limsup_subseq_mono s(1))
then have "a * limsup v \<ge> limsup w" using assms(2) assms(3) by (simp add: ereal_divide_le_pos)
then show ?thesis using I unfolding w_def by auto
qed
lemma ereal_liminf_lim_mult:
fixes u v::"nat \<Rightarrow> ereal"
assumes "u \<longlonglongrightarrow> a" "a>0" "a \<noteq> \<infinity>"
shows "liminf (\<lambda>n. u n * v n) = a * liminf v"
proof -
define w where "w = (\<lambda>n. u n * v n)"
obtain r where r: "strict_mono r" "(v o r) \<longlonglongrightarrow> liminf v" using liminf_subseq_lim by auto
have "(u o r) \<longlonglongrightarrow> a" using assms(1) LIMSEQ_subseq_LIMSEQ r by auto
with tendsto_mult_ereal[OF this r(2)] have "(\<lambda>n. (u o r) n * (v o r) n) \<longlonglongrightarrow> a * liminf v" using assms(2) assms(3) by auto
moreover have "\<And>n. (w o r) n = (u o r) n * (v o r) n" unfolding w_def by auto
ultimately have "(w o r) \<longlonglongrightarrow> a * liminf v" unfolding w_def by presburger
then have "liminf (w o r) = a * liminf v" by (simp add: tendsto_iff_Liminf_eq_Limsup)
then have I: "liminf w \<le> a * liminf v" by (metis liminf_subseq_mono r(1))
obtain s where s: "strict_mono s" "(w o s) \<longlonglongrightarrow> liminf w" using liminf_subseq_lim by auto
have *: "(u o s) \<longlonglongrightarrow> a" using assms(1) LIMSEQ_subseq_LIMSEQ s by auto
have "eventually (\<lambda>n. (u o s) n > 0) sequentially" using assms(2) * order_tendsto_iff by blast
moreover have "eventually (\<lambda>n. (u o s) n < \<infinity>) sequentially" using assms(3) * order_tendsto_iff by blast
moreover have "(w o s) n / (u o s) n = (v o s) n" if "(u o s) n > 0" "(u o s) n < \<infinity>" for n
unfolding w_def using that by (auto simp add: ereal_divide_eq)
ultimately have "eventually (\<lambda>n. (w o s) n / (u o s) n = (v o s) n) sequentially" using eventually_elim2 by force
moreover have "(\<lambda>n. (w o s) n / (u o s) n) \<longlonglongrightarrow> (liminf w) / a"
apply (rule tendsto_divide_ereal[OF s(2) *]) using assms(2) assms(3) by auto
ultimately have "(v o s) \<longlonglongrightarrow> (liminf w) / a" using Lim_transform_eventually by fastforce
then have "liminf (v o s) = (liminf w) / a" by (simp add: tendsto_iff_Liminf_eq_Limsup)
then have "liminf v \<le> (liminf w) / a" by (metis liminf_subseq_mono s(1))
then have "a * liminf v \<le> liminf w" using assms(2) assms(3) by (simp add: ereal_le_divide_pos)
then show ?thesis using I unfolding w_def by auto
qed
lemma ereal_liminf_lim_add:
fixes u v::"nat \<Rightarrow> ereal"
assumes "u \<longlonglongrightarrow> a" "abs(a) \<noteq> \<infinity>"
shows "liminf (\<lambda>n. u n + v n) = a + liminf v"
proof -
have "liminf u = a" using assms(1) tendsto_iff_Liminf_eq_Limsup trivial_limit_at_top_linorder by blast
then have *: "abs(liminf u) \<noteq> \<infinity>" using assms(2) by auto
have "(\<lambda>n. -u n) \<longlonglongrightarrow> -a" using assms(1) by auto
then have "liminf (\<lambda>n. -u n) = -a" using tendsto_iff_Liminf_eq_Limsup trivial_limit_at_top_linorder by blast
then have **: "abs(liminf (\<lambda>n. -u n)) \<noteq> \<infinity>" using assms(2) by auto
have "liminf (\<lambda>n. u n + v n) \<ge> liminf u + liminf v"
apply (rule ereal_liminf_add_mono) using * by auto
then have up: "liminf (\<lambda>n. u n + v n) \<ge> a + liminf v" using \<open>liminf u = a\<close> by simp
have a: "liminf (\<lambda>n. (u n + v n) + (-u n)) \<ge> liminf (\<lambda>n. u n + v n) + liminf (\<lambda>n. -u n)"
apply (rule ereal_liminf_add_mono) using ** by auto
have "eventually (\<lambda>n. u n = ereal(real_of_ereal(u n))) sequentially" using assms
real_lim_then_eventually_real by auto
moreover have "\<And>x. x = ereal(real_of_ereal(x)) \<Longrightarrow> x + (-x) = 0"
by (metis plus_ereal.simps(1) right_minus uminus_ereal.simps(1) zero_ereal_def)
ultimately have "eventually (\<lambda>n. u n + (-u n) = 0) sequentially"
by (metis (mono_tags, lifting) eventually_mono)
moreover have "\<And>n. u n + (-u n) = 0 \<Longrightarrow> u n + v n + (-u n) = v n"
by (metis add.commute add.left_commute add.left_neutral)
ultimately have "eventually (\<lambda>n. u n + v n + (-u n) = v n) sequentially"
using eventually_mono by force
then have "liminf v = liminf (\<lambda>n. u n + v n + (-u n))" using Liminf_eq by force
then have "liminf v \<ge> liminf (\<lambda>n. u n + v n) -a" using a \<open>liminf (\<lambda>n. -u n) = -a\<close> by (simp add: minus_ereal_def)
then have "liminf (\<lambda>n. u n + v n) \<le> a + liminf v" using assms(2) by (metis add.commute ereal_minus_le)
then show ?thesis using up by simp
qed
lemma ereal_liminf_limsup_add:
fixes u v::"nat \<Rightarrow> ereal"
shows "liminf (\<lambda>n. u n + v n) \<le> liminf u + limsup v"
proof (cases)
assume "limsup v = \<infinity> \<or> liminf u = \<infinity>"
then show ?thesis by auto
next
assume "\<not>(limsup v = \<infinity> \<or> liminf u = \<infinity>)"
then have "limsup v < \<infinity>" "liminf u < \<infinity>" by auto
define w where "w = (\<lambda>n. u n + v n)"
obtain r where r: "strict_mono r" "(u o r) \<longlonglongrightarrow> liminf u" using liminf_subseq_lim by auto
obtain s where s: "strict_mono s" "(w o r o s) \<longlonglongrightarrow> liminf (w o r)" using liminf_subseq_lim by auto
obtain t where t: "strict_mono t" "(v o r o s o t) \<longlonglongrightarrow> limsup (v o r o s)" using limsup_subseq_lim by auto
define a where "a = r o s o t"
have "strict_mono a" using r s t by (simp add: a_def strict_mono_o)
have l:"(u o a) \<longlonglongrightarrow> liminf u"
"(w o a) \<longlonglongrightarrow> liminf (w o r)"
"(v o a) \<longlonglongrightarrow> limsup (v o r o s)"
apply (metis (no_types, lifting) r(2) s(1) t(1) LIMSEQ_subseq_LIMSEQ a_def comp_assoc)
apply (metis (no_types, lifting) s(2) t(1) LIMSEQ_subseq_LIMSEQ a_def comp_assoc)
apply (metis (no_types, lifting) t(2) a_def comp_assoc)
done
have "liminf (w o r) \<ge> liminf w" by (simp add: liminf_subseq_mono r(1))
have "limsup (v o r o s) \<le> limsup v"
by (simp add: comp_assoc limsup_subseq_mono r(1) s(1) strict_mono_o)
then have b: "limsup (v o r o s) < \<infinity>" using \<open>limsup v < \<infinity>\<close> by auto
have "(\<lambda>n. (u o a) n + (v o a) n) \<longlonglongrightarrow> liminf u + limsup (v o r o s)"
apply (rule tendsto_add_ereal_general) using b \<open>liminf u < \<infinity>\<close> l(1) l(3) by force+
moreover have "(\<lambda>n. (u o a) n + (v o a) n) = (w o a)" unfolding w_def by auto
ultimately have "(w o a) \<longlonglongrightarrow> liminf u + limsup (v o r o s)" by simp
then have "liminf (w o r) = liminf u + limsup (v o r o s)" using l(2) using LIMSEQ_unique by blast
then have "liminf w \<le> liminf u + limsup v"
using \<open>liminf (w o r) \<ge> liminf w\<close> \<open>limsup (v o r o s) \<le> limsup v\<close>
by (metis add_mono_thms_linordered_semiring(2) le_less_trans not_less)
then show ?thesis unfolding w_def by simp
qed
lemma ereal_liminf_limsup_minus:
fixes u v::"nat \<Rightarrow> ereal"
shows "liminf (\<lambda>n. u n - v n) \<le> limsup u - limsup v"
unfolding minus_ereal_def
apply (subst add.commute)
apply (rule order_trans[OF ereal_liminf_limsup_add])
using ereal_Limsup_uminus[of sequentially "\<lambda>n. - v n"]
apply (simp add: add.commute)
done
lemma liminf_minus_ennreal:
fixes u v::"nat \<Rightarrow> ennreal"
shows "(\<And>n. v n \<le> u n) \<Longrightarrow> liminf (\<lambda>n. u n - v n) \<le> limsup u - limsup v"
unfolding liminf_SUP_INF limsup_INF_SUP
including ennreal.lifting
proof (transfer, clarsimp)
fix v u :: "nat \<Rightarrow> ereal" assume *: "\<forall>x. 0 \<le> v x" "\<forall>x. 0 \<le> u x" "\<And>n. v n \<le> u n"
moreover have "0 \<le> limsup u - limsup v"
using * by (intro ereal_diff_positive Limsup_mono always_eventually) simp
moreover have "0 \<le> Sup (u ` {x..})" for x
using * by (intro SUP_upper2[of x]) auto
moreover have "0 \<le> Sup (v ` {x..})" for x
using * by (intro SUP_upper2[of x]) auto
ultimately show "(SUP n. INF n\<in>{n..}. max 0 (u n - v n))
\<le> max 0 ((INF x. max 0 (Sup (u ` {x..}))) - (INF x. max 0 (Sup (v ` {x..}))))"
by (auto simp: * ereal_diff_positive max.absorb2 liminf_SUP_INF[symmetric] limsup_INF_SUP[symmetric] ereal_liminf_limsup_minus)
qed
subsection "Relate extended reals and the indicator function"
lemma ereal_indicator_le_0: "(indicator S x::ereal) \<le> 0 \<longleftrightarrow> x \<notin> S"
by (auto split: split_indicator simp: one_ereal_def)
lemma ereal_indicator: "ereal (indicator A x) = indicator A x"
by (auto simp: indicator_def one_ereal_def)
lemma ereal_mult_indicator: "ereal (x * indicator A y) = ereal x * indicator A y"
by (simp split: split_indicator)
lemma ereal_indicator_mult: "ereal (indicator A y * x) = indicator A y * ereal x"
by (simp split: split_indicator)
lemma ereal_indicator_nonneg[simp, intro]: "0 \<le> (indicator A x ::ereal)"
unfolding indicator_def by auto
lemma indicator_inter_arith_ereal: "indicator A x * indicator B x = (indicator (A \<inter> B) x :: ereal)"
by (simp split: split_indicator)
end