# Theory Extended_Real_Limits

```(*  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 ‹Limits on the Extended Real Number Line› (* 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 ⟷ 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 ≠ {}"
shows "Sup S ∈ S"
proof -
from compact_eq_closed[of S] compact_attains_sup[of S] assms
obtain s where S: "s ∈ S" "∀t∈S. t ≤ 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 ≠ {}"
shows "Inf S ∈ S"
proof -
from compact_eq_closed[of S] compact_attains_inf[of S] assms
obtain s where S: "s ∈ S" "∀t∈S. s ≤ t"
by auto
then have "Inf S = s"
by (auto intro!: Inf_eqI)
with S show ?thesis
by simp
qed

instance✐‹tag unimportant› enat :: second_countable_topology
proof
show "∃B::enat set set. countable B ∧ open = generate_topology B"
proof (intro exI conjI)
show "countable (range lessThan ∪ range greaterThan::enat set set)"
by auto
qed

instance✐‹tag unimportant› ereal :: second_countable_topology
proof (standard, intro exI conjI)
let ?B = "(⋃r∈ℚ. {{..< 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} ∨ b = {e<..}"
by auto
moreover have "{..<e} = ⋃{{..<x}|x. x ∈ ℚ ∧ x < e}" "{e<..} = ⋃{{x<..}|x. x ∈ ℚ ∧ 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 ‹This is a copy from ‹ereal :: second_countable_topology›. Maybe find a common super class of
topological spaces where the rational numbers are densely embedded ?›
instance ennreal :: second_countable_topology
proof (standard, intro exI conjI)
let ?B = "(⋃r∈ℚ. {{..< 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} ∨ b = {e<..}"
by auto
moreover have "{..<e} = ⋃{{..<x}|x. x ∈ ℚ ∧ x < e}" "{e<..} = ⋃{{x<..}|x. x ∈ ℚ ∧ 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: "(-∞) ∉ S"
shows "S = {}"
proof (rule ccontr)
assume "¬ ?thesis"
then have *: "Inf S ∈ S"

by (metis assms(2) closed_contains_Inf_cl)
{
assume "Inf S = -∞"
then have False
using * assms(3) by auto
}
moreover
{
assume "Inf S = ∞"
then have "S = {∞}"
by (metis Inf_eq_PInfty ‹S ≠ {}›)
then have False
by (metis assms(1) not_open_singleton)
}
moreover
{
assume fin: "¦Inf S¦ ≠ ∞"
from ereal_open_cont_interval[OF assms(1) * fin]
obtain e where e: "e > 0" "{Inf S - e<..<Inf S + e} ⊆ 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 ∈ {Inf S - e <..< Inf S + e}"
using e fin ereal_between[of "Inf S" e]
by auto
then have "b ∈ 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 ∧ closed S ⟷ S = {} ∨ S = UNIV"
using ereal_open_closed_aux open_closed by auto

lemma ereal_open_atLeast:
fixes x :: ereal
shows "open {x..} ⟷ x = -∞"
by (metis atLeast_eq_UNIV_iff bot_ereal_def closed_atLeast ereal_open_closed not_Ici_eq_empty)

lemma mono_closed_real:
fixes S :: "real set"
assumes mono: "∀y z. y ∈ S ∧ y ≤ z ⟶ z ∈ S"
and "closed S"
shows "S = {} ∨ S = UNIV ∨ (∃a. S = {a..})"
proof -
{
assume "S ≠ {}"
{ assume ex: "∃B. ∀x∈S. B ≤ x"
then have *: "∀x∈S. Inf S ≤ x"
using cInf_lower[of _ S] ex by (metis bdd_below_def)
then have "Inf S ∈ S"
by (meson ‹S ≠ {}› assms(2) bdd_belowI closed_contains_Inf)
then have "∀x. Inf S ≤ x ⟷ x ∈ S"
using mono[rule_format, of "Inf S"] *
by auto
then have "S = {Inf S ..}"
by auto
then have "∃a. S = {a ..}"
by auto
}
moreover
{
assume "¬ (∃B. ∀x∈S. B ≤ x)"
then have nex: "∀B. ∃x∈S. x < B"
{
fix y
obtain x where "x∈S" and "x < y"
using nex by auto
then have "y ∈ S"
using mono[rule_format, of x y] by auto
}
then have "S = UNIV"
by auto
}
ultimately have "S = UNIV ∨ (∃a. S = {a ..})"
by blast
}
then show ?thesis
by blast
qed

lemma mono_closed_ereal:
fixes S :: "real set"
assumes mono: "∀y z. y ∈ S ∧ y ≤ z ⟶ z ∈ S"
and "closed S"
shows "∃a. S = {x. a ≤ ereal x}"
proof -
consider "S = {} ∨ S = UNIV" | "(∃a. S = {a..})"
using assms(2) mono mono_closed_real by blast
then show ?thesis
proof cases
case 1
then show ?thesis
by (meson PInfty_neq_ereal(1) UNIV_eq_I bot.extremum empty_Collect_eq ereal_infty_less_eq(1) mem_Collect_eq)
next
case 2
then show ?thesis
by (metis atLeast_iff ereal_less_eq(3) mem_Collect_eq subsetI subset_antisym)
qed
qed

lemma Liminf_within:
fixes f :: "'a::metric_space ⇒ 'b::complete_lattice"
shows "Liminf (at x within S) f = (SUP e∈{0<..}. INF y∈(S ∩ 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 "∀y. y ∈ S ⟶ y ≠ x ∧ dist y x < d ⟶ P y"
then have "S ∩ ball x d - {x} ⊆ {x. P x}"
by (auto simp: dist_commute)
then show "∃r>0. Inf (f ` (Collect P)) ≤ Inf (f ` (S ∩ ball x r - {x}))"
by (intro exI[of _ d] INF_mono conjI ‹0 < d›) auto
next
fix d :: real
assume "0 < d"
then show "∃P. (∃d>0. ∀xa. xa ∈ S ⟶ xa ≠ x ∧ dist xa x < d ⟶ P xa) ∧
Inf (f ` (S ∩ ball x d - {x})) ≤ Inf (f ` (Collect P))"
by (intro exI[of _ "λy. y ∈ S ∩ ball x d - {x}"])
(auto intro!: INF_mono exI[of _ d] simp: dist_commute)
qed

lemma Limsup_within:
fixes f :: "'a::metric_space ⇒ 'b::complete_lattice"
shows "Limsup (at x within S) f = (INF e∈{0<..}. SUP y∈(S ∩ 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 "∀y. y ∈ S ⟶ y ≠ x ∧ dist y x < d ⟶ P y"
then have "S ∩ ball x d - {x} ⊆ {x. P x}"
by (auto simp: dist_commute)
then show "∃r>0. Sup (f ` (S ∩ ball x r - {x})) ≤ Sup (f ` (Collect P))"
by (intro exI[of _ d] SUP_mono conjI ‹0 < d›) auto
next
fix d :: real
assume "0 < d"
then show "∃P. (∃d>0. ∀xa. xa ∈ S ⟶ xa ≠ x ∧ dist xa x < d ⟶ P xa) ∧
Sup (f ` (Collect P)) ≤ Sup (f ` (S ∩ ball x d - {x}))"
by (intro exI[of _ "λy. y ∈ S ∩ ball x d - {x}"])
(auto intro!: SUP_mono exI[of _ d] simp: dist_commute)
qed

lemma Liminf_at:
fixes f :: "'a::metric_space ⇒ 'b::complete_lattice"
shows "Liminf (at x) f = (SUP e∈{0<..}. INF y∈(ball x e - {x}). f y)"
using Liminf_within[of x UNIV f] by simp

lemma Limsup_at:
fixes f :: "'a::metric_space ⇒ 'b::complete_lattice"
shows "Limsup (at x) f = (INF e∈{0<..}. SUP y∈(ball x e - {x}). f y)"
using Limsup_within[of x UNIV f] by simp

lemma min_Liminf_at:
fixes f :: "'a::metric_space ⇒ 'b::complete_linorder"
shows "min (f x) (Liminf (at x) f) = (SUP e∈{0<..}. INF y∈ball x e. f y)"
apply (simp add: inf_min [symmetric] Liminf_at inf_commute [of "f x"] SUP_inf)
apply (metis (no_types, lifting) INF_insert centre_in_ball greaterThan_iff image_cong inf_commute insert_Diff)
done

subsection ‹Extended-Real.thy› (*FIX ME change title *)

lemma sum_constant_ereal:
fixes a::ereal
shows "(∑i∈I. a) = a * card I"
proof (induction I rule: infinite_finite_induct)
case (insert i I)
then show ?case
by (simp add: ereal_right_distrib flip: plus_ereal.simps)
qed auto

lemma real_lim_then_eventually_real:
assumes "(u ⤏ ereal l) F"
shows "eventually (λn. u n = ereal(real_of_ereal(u n))) F"
proof -
have "ereal l ∈ {-∞<..<(∞::ereal)}" by simp
moreover have "open {-∞<..<(∞::ereal)}" by simp
ultimately have "eventually (λn. u n ∈ {-∞<..<(∞::ereal)}) F" using assms tendsto_def by blast
moreover have "⋀x. x ∈ {-∞<..<(∞::ereal)} ⟹ 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 "bij ((*) (ereal c))"
apply (rule bij_betw_byWitness[of _ "λx. (x::ereal) / c"], auto simp: assms ereal_mult_divide)
using assms ereal_divide_eq by auto
then have "ereal c * Inf {x. P x} = Inf ((*) (ereal c) ` {x. P x})"
by (simp add: assms ereal_mult_left_mono less_imp_le mono_def mono_bij_Inf)
then show ?thesis
qed

text ‹The next few lemmas remove an unnecessary assumption in ‹tendsto_add_ereal›, culminating
is continuous on ereal times ereal, except at ‹(-∞, ∞)› and ‹(∞, -∞)›.
It is much more convenient in many situations, see for instance the proof of
‹tendsto_sum_ereal› below.›

fixes y :: ereal
assumes y: "y ≠ -∞"
assumes f: "(f ⤏ ∞) F" and g: "(g ⤏ y) F"
shows "((λx. f x + g x) ⤏ ∞) F"
proof -
have "∃C. eventually (λx. 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 (λx. 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 (λx. g x > ereal(real_of_ereal(y - 1))) F" by simp
then show ?thesis by auto
next
case (PInf)
have "eventually (λx. g x > ereal 0) F" using g PInf by (simp add: tendsto_PInfty)
then show ?thesis by auto
then obtain C::real where ge: "eventually (λx. g x > ereal C) F" by auto

{
fix M::real
have "eventually (λx. f x > ereal(M - C)) F" using f by (simp add: tendsto_PInfty)
then have "eventually (λx. (f x > ereal (M-C)) ∧ (g x > ereal C)) F"
by (auto simp: ge eventually_conj_iff)
moreover have "⋀x. ((f x > ereal (M-C)) ∧ (g x > ereal C)) ⟹ (f x + g x > ereal M)"
ultimately have "eventually (λx. f x + g x > ereal M) F" using eventually_mono by force
}
then show ?thesis by (simp add: tendsto_PInfty)
qed

text‹One would like to deduce the next lemma from the previous one, but the fact
that ‹- (x + y)› is in general different from ‹(- x) + (- y)› in ereal creates difficulties,
so it is more efficient to copy the previous proof.›

fixes y :: ereal
assumes y: "y ≠ ∞"
assumes f: "(f ⤏ -∞) F" and g: "(g ⤏ y) F"
shows "((λx. f x + g x) ⤏ -∞) F"
proof -
have "∃C. eventually (λx. 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 (λx. 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 (λx. g x < ereal(real_of_ereal(y + 1))) F" by simp
then show ?thesis by auto
next
case (MInf)
have "eventually (λx. g x < ereal 0) F" using g MInf by (simp add: tendsto_MInfty)
then show ?thesis by auto
then obtain C::real where ge: "eventually (λx. g x < ereal C) F" by auto

{
fix M::real
have "eventually (λx. f x < ereal(M - C)) F" using f by (simp add: tendsto_MInfty)
then have "eventually (λx. (f x < ereal (M- C)) ∧ (g x < ereal C)) F"
by (auto simp: ge eventually_conj_iff)
moreover have "⋀x. ((f x < ereal (M-C)) ∧ (g x < ereal C)) ⟹ (f x + g x < ereal M)"
ultimately have "eventually (λx. f x + g x < ereal M) F" using eventually_mono by force
}
then show ?thesis by (simp add: tendsto_MInfty)
qed

fixes x y :: ereal
assumes y: "¦y¦ ≠ ∞"
assumes f: "(f ⤏ x) F" and g: "(g ⤏ y) F"
shows "((λx. f x + g x) ⤏ x + y) F"
proof (cases x)
case (real r)
have a: "¦x¦ ≠ ∞" 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

fixes x y :: ereal
assumes x: "¦x¦ ≠ ∞"
and f: "(f ⤏ x) F" and g: "(g ⤏ y) F"
shows "((λx. f x + g x) ⤏ x + y) F"
proof -
have "((λx. g x + f x) ⤏ x + y) F"
moreover have "⋀x. g x + f x = f x + g x" using add.commute by auto
ultimately show ?thesis by simp
qed

text ‹The next lemma says that the addition is continuous on ‹ereal›, except at
the pairs ‹(-∞, ∞)› and ‹(∞, -∞)›.›

fixes x y :: ereal
assumes "¬((x=∞ ∧ y=-∞) ∨ (x=-∞ ∧ y=∞))"
and f: "(f ⤏ x) F" and g: "(g ⤏ y) F"
shows "((λx. f x + g x) ⤏ x + y) F"
proof (cases x)
case (real r)
show ?thesis
using real assms
next
case (PInf)
then have "y ≠ -∞" using assms by simp
then show ?thesis using tendsto_add_ereal_PInf PInf assms by auto
next
case (MInf)
then have "y ≠ ∞" using assms by simp
then show ?thesis
by (metis ereal_MInfty_eq_plus tendsto_add_ereal_MInf MInf f g)
qed

subsubsection✐‹tag important› ‹Continuity of multiplication›

text ‹In the same way as for addition, we prove that the multiplication is continuous on
ereal times ereal, except at ‹(∞, 0)› and ‹(-∞, 0)› and ‹(0, ∞)› and ‹(0, -∞)›,
starting with specific situations.›

lemma tendsto_mult_real_ereal:
assumes "(u ⤏ ereal l) F" "(v ⤏ ereal m) F"
shows "((λn. u n * v n) ⤏ ereal l * ereal m) F"
proof -
have ureal: "eventually (λn. u n = ereal(real_of_ereal(u n))) F" by (rule real_lim_then_eventually_real[OF assms(1)])
then have "((λn. ereal(real_of_ereal(u n))) ⤏ ereal l) F" using assms by auto
then have limu: "((λn. real_of_ereal(u n)) ⤏ l) F" by auto
have vreal: "eventually (λn. v n = ereal(real_of_ereal(v n))) F" by (rule real_lim_then_eventually_real[OF assms(2)])
then have "((λn. ereal(real_of_ereal(v n))) ⤏ ereal m) F" using assms by auto
then have limv: "((λn. real_of_ereal(v n)) ⤏ 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 (λn. 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 "((λn. real_of_ereal(u n) * real_of_ereal(v n)) ⤏ l * m) F"
using tendsto_mult[OF limu limv] by auto
then have "((λn. ereal(real_of_ereal(u n)) * real_of_ereal(v n)) ⤏ ereal(l * m)) F"
by auto
then show ?thesis using * filterlim_cong by fastforce
qed

lemma tendsto_mult_ereal_PInf:
fixes f g::"_ ⇒ ereal"
assumes "(f ⤏ l) F" "l>0" "(g ⤏ ∞) F"
shows "((λx. f x * g x) ⤏ ∞) F"
proof -
obtain a::real where "0 < ereal a" "a < l"
using assms(2) using ereal_dense2 by blast
have *: "eventually (λx. f x > a) F"
using ‹a < l› 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 ‹ereal a > 0› by simp
then have "⋀x. ((f x > a) ∧ (g x > M/a)) ⟹ (f x * g x > ereal a * ereal(M/a))"
using ereal_mult_mono_strict'[where ?c = "M/a", OF ‹0 < ereal a›] by auto
moreover have "ereal a * ereal(M/a) = M" using ‹ereal a > 0› by simp
ultimately have "⋀x. ((f x > a) ∧ (g x > M/a)) ⟹ (f x * g x > M)" by simp
moreover have "M ≥ K" unfolding M_def by simp
ultimately have Imp: "⋀x. ((f x > a) ∧ (g x > M/a)) ⟹ (f x * g x > K)"
using ereal_less_eq(3) le_less_trans by blast

have "eventually (λx. g x > M/a) F" using assms(3) by (simp add: tendsto_PInfty)
then have "eventually (λx. (f x > a) ∧ (g x > M/a)) F"
using * by (auto simp: eventually_conj_iff)
then have "eventually (λx. f x * g x > K) F" using eventually_mono Imp by force
}
then show ?thesis by (auto simp: tendsto_PInfty)
qed

lemma tendsto_mult_ereal_pos:
fixes f g::"_ ⇒ ereal"
assumes "(f ⤏ l) F" "(g ⤏ m) F" "l>0" "m>0"
shows "((λx. f x * g x) ⤏ l * m) F"
proof (cases)
assume *: "l = ∞ ∨ m = ∞"
then show ?thesis
proof (cases)
assume "m = ∞"
then show ?thesis using tendsto_mult_ereal_PInf assms by auto
next
assume "¬(m = ∞)"
then have "l = ∞" using * by simp
then have "((λx. g x * f x) ⤏ l * m) F" using tendsto_mult_ereal_PInf assms by auto
moreover have "⋀x. g x * f x = f x * g x" using mult.commute by auto
ultimately show ?thesis by simp
qed
next
assume "¬(l = ∞ ∨ m = ∞)"
then have "l < ∞" "m < ∞" by auto
then obtain lr mr where "l = ereal lr" "m = ereal mr"
using ‹l>0› ‹m>0› 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 ‹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.›

lemma ereal_sgn_abs:
fixes l::ereal
shows "sgn(l) * l = abs(l)"
by (cases l, auto simp: sgn_if ereal_less_uminus_reorder)

lemma sgn_squared_ereal:
assumes "l ≠ (0::ereal)"
shows "sgn(l) * sgn(l) = 1"
using assms by (cases l, auto simp: one_ereal_def sgn_if)

lemma tendsto_mult_ereal [tendsto_intros]:
fixes f g::"_ ⇒ ereal"
assumes "(f ⤏ l) F" "(g ⤏ m) F" "¬((l=0 ∧ abs(m) = ∞) ∨ (m=0 ∧ abs(l) = ∞))"
shows "((λx. f x * g x) ⤏ l * m) F"
proof (cases)
assume "l=0 ∨ m=0"
then have "abs(l) ≠ ∞" "abs(m) ≠ ∞" 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: "⋀a::ereal. abs(sgn a) ≠ ∞"
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: "⋀a b::ereal. abs(sgn a * sgn b) ≠ ∞"
by (metis abs_eq_infinity_cases abs_ereal.simps(2) abs_ereal.simps(3) ereal_mult_eq_MInfty ereal_mult_eq_PInfty)
assume "¬(l=0 ∨ m=0)"
then have "l ≠ 0" "m ≠ 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 "((λx. sgn(l) * f x) ⤏ (sgn(l) * l)) F"
by (rule tendsto_cmult_ereal, auto simp: sgn_finite assms(1))
moreover have "((λx. sgn(m) * g x) ⤏ (sgn(m) * m)) F"
by (rule tendsto_cmult_ereal, auto simp: sgn_finite assms(2))
ultimately have *: "((λx. (sgn(l) * f x) * (sgn(m) * g x)) ⤏ (sgn(l) * l) * (sgn(m) * m)) F"
using tendsto_mult_ereal_pos by force
have "((λx. (sgn(l) * sgn(m)) * ((sgn(l) * f x) * (sgn(m) * g x))) ⤏ (sgn(l) * sgn(m)) * ((sgn(l) * l) * (sgn(m) * m))) F"
by (rule tendsto_cmult_ereal, auto simp: sgn_finite2 *)
moreover have "⋀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 ‹l ≠ 0›] sgn_squared_ereal[OF ‹m ≠ 0›] 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 ‹l ≠ 0›] sgn_squared_ereal[OF ‹m ≠ 0›] mult.assoc mult.commute)
ultimately show ?thesis by auto
qed

lemma tendsto_cmult_ereal_general [tendsto_intros]:
fixes f::"_ ⇒ ereal" and c::ereal
assumes "(f ⤏ l) F" "¬ (l=0 ∧ abs(c) = ∞)"
shows "((λx. c * f x) ⤏ c * l) F"
by (cases "c = 0", auto simp: assms tendsto_mult_ereal)

subsubsection✐‹tag important› ‹Continuity of division›

lemma tendsto_inverse_ereal_PInf:
fixes u::"_ ⇒ ereal"
assumes "(u ⤏ ∞) F"
shows "((λx. 1/ u x) ⤏ 0) F"
proof -
{
fix e::real assume "e>0"
have "1/e < ∞" by auto
then have "eventually (λn. 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 ‹e>0› using less_le_trans not_le by fastforce
then have "1/z ≥ 0" by auto
moreover have "1/z < e"
proof (cases z)
case (real r)
then show ?thesis
using ‹0 < e› ‹0 < z› ‹ereal (1 / e) < z› divide_less_eq ereal_divide_less_pos by fastforce
qed (use ‹0 < e› ‹0 < z› in auto)
ultimately have "1/z ≥ 0" "1/z < e" by auto
}
ultimately have "eventually (λn. 1/u n<e) F" "eventually (λn. 1/u n≥0) F" by (auto simp: eventually_mono)
} note * = this
show ?thesis
proof (subst order_tendsto_iff, auto)
fix a::ereal assume "a<0"
then show "eventually (λn. 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 (λn. 1/u n < e) F" using *(1) by auto
then show "eventually (λn. 1/u n < a) F" using ‹a>e› by (metis (mono_tags, lifting) eventually_mono less_trans)
qed
qed

text ‹The next lemma deserves to exist by itself, as it is so common and useful.›

lemma tendsto_inverse_real [tendsto_intros]:
fixes u::"_ ⇒ real"
shows "(u ⤏ l) F ⟹ l ≠ 0 ⟹ ((λx. 1/ u x) ⤏ 1/l) F"
using tendsto_inverse unfolding inverse_eq_divide .

lemma tendsto_inverse_ereal [tendsto_intros]:
fixes u::"_ ⇒ ereal"
assumes "(u ⤏ l) F" "l ≠ 0"
shows "((λx. 1/ u x) ⤏ 1/l) F"
proof (cases l)
case (real r)
then have "r ≠ 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 = (λn. real_of_ereal(u n))"
have ureal: "eventually (λn. u n = ereal(v n)) F" unfolding v_def using real_lim_then_eventually_real assms(1) real by auto
then have "((λn. ereal(v n)) ⤏ ereal r) F" using assms real v_def by auto
then have *: "((λn. v n) ⤏ r) F" by auto
then have "((λn. 1/v n) ⤏ 1/r) F" using ‹r ≠ 0› tendsto_inverse_real by auto
then have lim: "((λn. ereal(1/v n)) ⤏ 1/l) F" using ‹1/l = ereal(1/r)› by auto

have "r ∈ -{0}" "open (-{(0::real)})" using ‹r ≠ 0› by auto
then have "eventually (λn. v n ∈ -{0}) F" using * using topological_tendstoD by blast
then have "eventually (λn. v n ≠ 0) F" by auto
moreover
{
fix n assume H: "v n ≠ 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 (λn. 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 ‹ z < 0 › by auto
moreover have "eventually (λn. u n < 0) F" by (metis (no_types) MInf assms(1) tendsto_MInfty zero_ereal_def)
ultimately have *: "eventually (λn. -1/-u n = 1/u n) F" by (simp add: eventually_mono)

define v where "v = (λn. - u n)"
have "(v ⤏ ∞) F" unfolding v_def using MInf assms(1) tendsto_uminus_ereal by fastforce
then have "((λn. 1/v n) ⤏ 0) F" using tendsto_inverse_ereal_PInf by auto
then have "((λn. -1/v n) ⤏ 0) F" using tendsto_uminus_ereal by fastforce
then show ?thesis unfolding v_def using Lim_transform_eventually[OF _ *] ‹ 1/l = 0 › by auto
qed

lemma tendsto_divide_ereal [tendsto_intros]:
fixes f g::"_ ⇒ ereal"
assumes "(f ⤏ l) F" "(g ⤏ m) F" "m ≠ 0" "¬(abs(l) = ∞ ∧ abs(m) = ∞)"
shows "((λx. f x / g x) ⤏ l / m) F"
proof -
define h where "h = (λx. 1/ g x)"
have *: "(h ⤏ 1/m) F" unfolding h_def using assms(2) assms(3) tendsto_inverse_ereal by auto
have "((λx. f x * h x) ⤏ l * (1/m)) F"
apply (rule tendsto_mult_ereal[OF assms(1) *]) using assms(3) assms(4) by (auto simp: 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 ‹Further limits›

text ‹The assumptions of @{thm tendsto_diff_ereal} are too strong, we weaken them here.›

lemma tendsto_diff_ereal_general [tendsto_intros]:
fixes u v::"'a ⇒ ereal"
assumes "(u ⤏ l) F" "(v ⤏ m) F" "¬((l = ∞ ∧ m = ∞) ∨ (l = -∞ ∧ m = -∞))"
shows "((λn. u n - v n) ⤏ l - m) F"
proof -
have "∞ = l ⟶ ((λa. u a + - v a) ⤏ l + - m) F"
by (metis (no_types) assms ereal_uminus_uminus tendsto_add_ereal_general tendsto_uminus_ereal)
then have "((λn. u n + (-v n)) ⤏ l + (-m)) F"
then show ?thesis
qed

lemma id_nat_ereal_tendsto_PInf [tendsto_intros]:
"(λ n::nat. real n) ⇢ ∞"

lemma tendsto_at_top_pseudo_inverse [tendsto_intros]:
fixes u::"nat ⇒ nat"
assumes "LIM n sequentially. u n :> at_top"
shows "LIM n sequentially. Inf {N. u N ≥ n} :> at_top"
proof -
{
fix C::nat
define M where "M = Max {u n| n. n ≤ C}+1"
{
fix n assume "n ≥ M"
have "eventually (λN. u N ≥ n) sequentially" using assms
then have *: "{N. u N ≥ n} ≠ {}" by force

have "N > C" if "u N ≥ n" for N
proof (rule ccontr)
assume "¬(N > C)"
then have "u N ≤ Max {u n| n. n ≤ C}"
using Max_ge by (simp add: setcompr_eq_image not_less)
then show False using ‹u N ≥ n› ‹n ≥ M› unfolding M_def by auto
qed
then have **: "{N. u N ≥ n} ⊆ {C..}" by fastforce
have "Inf {N. u N ≥ n} ≥ C"
by (metis "*" "**" Inf_nat_def1 atLeast_iff subset_eq)
}
then have "eventually (λn. Inf {N. u N ≥ n} ≥ 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 ⇒ nat"
assumes "LIM n sequentially. u n :> at_top"
shows "finite {N. u N ≤ n}"
proof -
fix n
have "eventually (λN. u N ≥ n+1) sequentially" using assms
then obtain N1 where N1: "⋀N. N ≥ N1 ⟹ u N ≥ n + 1"
using eventually_sequentially by auto
have "{N. u N ≤ n} ⊆ {..<N1}"
by (metis (no_types, lifting) N1 Suc_eq_plus1 lessThan_iff less_le_not_le mem_Collect_eq nle_le not_less_eq_eq subset_eq)
then show "finite {N. u N ≤ n}" by (simp add: finite_subset)
qed

lemma tendsto_at_top_pseudo_inverse2 [tendsto_intros]:
fixes u::"nat ⇒ nat"
assumes "LIM n sequentially. u n :> at_top"
shows "LIM n sequentially. Max {N. u N ≤ n} :> at_top"
proof -
{
fix N0::nat
have "N0 ≤ Max {N. u N ≤ n}" if "n ≥ u N0" for n
by (simp add: assms pseudo_inverse_finite_set that)
then have "eventually (λn. N0 ≤ Max {N. u N ≤ 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 "(λn::nat. min x n) ⇢ 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 ≥ K" for n apply (subst real, subst real, auto) using that eq_iff by fastforce
then have "eventually (λn. 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: 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: min_def)
then show ?thesis by auto
qed

lemma ereal_truncation_real_top [tendsto_intros]:
fixes x::ereal
assumes "x ≠ - ∞"
shows "(λn::nat. real_of_ereal(min x n)) ⇢ 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 ≥ 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 ≥ K" for n using real that by auto
then have "eventually (λn. real_of_ereal(min x n) = r) sequentially" using eventually_at_top_linorder by blast
then have "(λn. real_of_ereal(min x n)) ⇢ 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: min_def)
then show ?thesis using id_nat_ereal_tendsto_PInf PInf by auto

lemma ereal_truncation_bottom [tendsto_intros]:
fixes x::ereal
shows "(λn::nat. max x (- real n)) ⇢ 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 ≥ K" for n apply (subst real, subst real, auto) using that eq_iff by fastforce
then have "eventually (λn. 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: max_def)
moreover have "(λn. (-1)* ereal(real n)) ⇢ -∞"
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: max_def)
then show ?thesis by auto
qed

lemma ereal_truncation_real_bottom [tendsto_intros]:
fixes x::ereal
assumes "x ≠ ∞"
shows "(λn::nat. real_of_ereal(max x (- real n))) ⇢ 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 ≥ 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 ≥ K" for n using real that by auto
then have "eventually (λn. real_of_ereal(max x (-real n)) = r) sequentially" using eventually_at_top_linorder by blast
then have "(λn. real_of_ereal(max x (-real n))) ⇢ 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: max_def)
moreover have "(λn. (-1)* ereal(real n)) ⇢ -∞"
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

text ‹the next one is copied from ‹tendsto_sum›.›
lemma tendsto_sum_ereal [tendsto_intros]:
fixes f :: "'a ⇒ 'b ⇒ ereal"
assumes "⋀i. i ∈ S ⟹ (f i ⤏ a i) F"
"⋀i. abs(a i) ≠ ∞"
shows "((λx. ∑i∈S. f i x) ⤏ (∑i∈S. a i)) F"
proof (cases "finite S")
assume "finite S" then show ?thesis using assms
qed(simp)

lemma continuous_ereal_abs:
"continuous_on (UNIV::ereal set) abs"
proof -
have "continuous_on ({..0} ∪ {(0::ereal)..}) abs"
proof (intro continuous_on_closed_Un continuous_intros)
show "continuous_on {..0::ereal} abs"
by (metis abs_ereal_ge0 abs_ereal_less0 continuous_on_eq antisym_conv1 atMost_iff continuous_uminus_ereal ereal_uminus_zero)
show "continuous_on {0::ereal..} abs"
by (metis abs_ereal_ge0 atLeast_iff continuous_on_eq continuous_on_id)
qed
moreover have "(UNIV::ereal set) = {..0} ∪ {(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 ⤏ (l::ereal)) F"
shows "((λn. abs(u n)) ⤏ abs l) F"
using continuous_ereal_abs assms by (metis UNIV_I continuous_on tendsto_compose)

lemma ereal_minus_real_tendsto_MInf [tendsto_intros]:
"(λx. ereal (- real x)) ⇢ - ∞"
by (subst uminus_ereal.simps(1)[symmetric], intro tendsto_intros)

subsection ‹Extended-Nonnegative-Real.thy› (*FIX title *)

lemma tendsto_diff_ennreal_general [tendsto_intros]:
fixes u v::"'a ⇒ ennreal"
assumes "(u ⤏ l) F" "(v ⤏ m) F" "¬(l = ∞ ∧ m = ∞)"
shows "((λn. u n - v n) ⤏ l - m) F"
proof -
have "((λn. e2ennreal(enn2ereal(u n) - enn2ereal(v n))) ⤏ e2ennreal(enn2ereal l - enn2ereal m)) F"
by (intro tendsto_intros) (use assms in auto)
then show ?thesis by auto
qed

lemma tendsto_mult_ennreal [tendsto_intros]:
fixes l m::ennreal
assumes "(u ⤏ l) F" "(v ⤏ m) F" "¬((l = 0 ∧ m = ∞) ∨ (l = ∞ ∧ m = 0))"
shows "((λn. u n * v n) ⤏ l * m) F"
proof -
have "((λn. e2ennreal(enn2ereal (u n) * enn2ereal (v n))) ⤏ e2ennreal(enn2ereal l * enn2ereal m)) F"
by (intro tendsto_intros) (use assms enn2ereal_inject zero_ennreal.rep_eq in 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: eq_onp_same_args)
moreover have "e2ennreal(enn2ereal l * enn2ereal m)  = l * m"
by (subst times_ennreal.abs_eq[symmetric], auto simp: eq_onp_same_args)
ultimately show ?thesis
by auto
qed

subsection ‹monoset› (*FIX ME title *)

definition (in order) mono_set:
"mono_set S ⟷ (∀x y. x ≤ y ⟶ x ∈ S ⟶ y ∈ 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 ≡ Inf S"
shows "mono_set S ⟷ S = {a <..} ∨ S = {a..}" (is "_ = ?c")
proof
assume "mono_set S"
then have mono: "⋀x y. x ≤ y ⟹ x ∈ S ⟹ y ∈ S"
by (auto simp: mono_set)
show ?c
proof cases
assume "a ∈ S"
show ?c
using mono[OF _ ‹a ∈ S›]
by (auto intro: Inf_lower simp: a_def)
next
assume "a ∉ S"
have "S = {a <..}"
proof safe
fix x assume "x ∈ S"
then have "a ≤ x"
unfolding a_def by (rule Inf_lower)
then show "a < x"
using ‹x ∈ S› ‹a ∉ S› by (cases "a = x") auto
next
fix x assume "a < x"
then obtain y where "y < x" "y ∈ S"
unfolding a_def Inf_less_iff ..
with mono[of y x] show "x ∈ S"
by auto
qed
then show ?c ..
qed
qed auto

lemma ereal_open_mono_set:
fixes S :: "ereal set"
shows "open S ∧ mono_set S ⟷ S = UNIV ∨ 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 ∧ mono_set S ⟷ S = {} ∨ 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 ⇒ ereal"
shows "Liminf net f =
Sup {l. ∀S. open S ⟶ mono_set S ⟶ l ∈ S ⟶ eventually (λx. f x ∈ 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)) ∈ S"
{
fix x
assume "P x"
then have "Inf (f ` (Collect P)) ≤ f x"
by (intro complete_lattice_class.INF_lower) simp
with S have "f x ∈ S"
}
with P show "eventually (λx. f x ∈ S) net"
by (auto elim: eventually_mono)
next
fix y l
assume S: "∀S. open S ⟶ mono_set S ⟶ l ∈ S ⟶ eventually  (λx. f x ∈ S) net"
assume P: "∀P. eventually P net ⟶ Inf (f ` (Collect P)) ≤ y"
show "l ≤ y"
proof (rule dense_le)
fix B
assume "B < l"
then have "eventually (λx. f x ∈ {B <..}) net"
by (intro S[rule_format]) auto
then have "Inf (f ` {x. B < f x}) ≤ y"
using P by auto
moreover have "B ≤ Inf (f ` {x. B < f x})"
by (intro INF_greatest) auto
ultimately show "B ≤ y"
by simp
qed
qed

lemma ereal_Limsup_Inf_monoset:
fixes f :: "'a ⇒ ereal"
shows "Limsup net f =
Inf {l. ∀S. open S ⟶ mono_set (uminus ` S) ⟶ l ∈ S ⟶ eventually (λx. f x ∈ 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)) ∈ S"
{
fix x
assume "P x"
then have "f x ≤ 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 ∈ S"
with P show "eventually (λx. f x ∈ S) net"
by (auto elim: eventually_mono)
next
fix y l
assume S: "∀S. open S ⟶ mono_set (uminus ` S) ⟶ l ∈ S ⟶ eventually  (λx. f x ∈ S) net"
assume P: "∀P. eventually P net ⟶ y ≤ Sup (f ` (Collect P))"
show "y ≤ l"
proof (rule dense_ge)
fix B
assume "l < B"
then have "eventually (λx. f x ∈ {..< B}) net"
by (intro S[rule_format]) auto
then have "y ≤ Sup (f ` {x. f x < B})"
using P by auto
moreover have "Sup (f ` {x. f x < B}) ≤ B"
by (intro SUP_least) auto
ultimately show "y ≤ B"
by simp
qed
qed

lemma liminf_bounded_open:
fixes x :: "nat ⇒ ereal"
shows "x0 ≤ liminf x ⟷ (∀S. open S ⟶ mono_set S ⟶ x0 ∈ S ⟶ (∃N. ∀n≥N. x n ∈ S))"
(is "_ ⟷ ?P x0")
proof
assume "?P x0"
then show "x0 ≤ liminf x"
unfolding ereal_Liminf_Sup_monoset eventually_sequentially
by (intro complete_lattice_class.Sup_upper) auto
next
assume "x0 ≤ liminf x"
{
fix S :: "ereal set"
assume om: "open S" "mono_set S" "x0 ∈ S"
then have "∃N. ∀n≥N. x n ∈ S"
by (metis ‹x0 ≤ liminf x› ereal_open_mono_set greaterThan_iff liminf_bounded_iff om UNIV_I)
}
then show "?P x0"
by auto
qed

lemma limsup_finite_then_bounded:
fixes u::"nat ⇒ real"
assumes "limsup u < ∞"
shows "∃C. ∀n. u n ≤ C"
proof -
obtain C where C: "limsup u < C" "C < ∞" using assms ereal_dense2 by blast
then have "C = ereal(real_of_ereal C)" using ereal_real by force
have "eventually (λn. u n < C) sequentially"
using SUP_lessD eventually_mono C(1)
by (fastforce simp: INF_less_iff Limsup_def)
then obtain N where N: "⋀n. n ≥ N ⟹ u n < C" using eventually_sequentially by auto
define D where "D = max (real_of_ereal C) (Max {u n |n. n ≤ N})"
have "⋀n. u n ≤ D"
proof -
fix n show "u n ≤ D"
proof (cases)
assume *: "n ≤ N"
have "u n ≤ Max {u n |n. n ≤ N}" by (rule Max_ge, auto simp: *)
then show "u n ≤ D" unfolding D_def by linarith
next
assume "¬(n ≤ N)"
then have "n ≥ N" by simp
then have "u n < C" using N by auto
then have "u n < real_of_ereal C" using ‹C = ereal(real_of_ereal C)› less_ereal.simps(1) by fastforce
then show "u n ≤ D" unfolding D_def by linarith
qed
qed
then show ?thesis by blast
qed

lemma liminf_finite_then_bounded_below:
fixes u::"nat ⇒ real"
assumes "liminf u > -∞"
shows "∃C. ∀n. u n ≥ C"
proof -
obtain C where C: "liminf u > C" "C > -∞" using assms using ereal_dense2 by blast
then have "C = ereal(real_of_ereal C)" using ereal_real by force
have "eventually (λn. u n > C) sequentially"
using eventually_elim2 less_INF_D C(1)
by (fastforce simp: less_SUP_iff Liminf_def)
then obtain N where N: "⋀n. n ≥ N ⟹ u n > C" using eventually_sequentially by auto
define D where "D = min (real_of_ereal C) (Min {u n |n. n ≤ N})"
have "⋀n. u n ≥ D"
proof -
fix n show "u n ≥ D"
proof (cases)
assume *: "n ≤ N"
have "u n ≥ Min {u n |n. n ≤ N}" by (rule Min_le, auto simp: *)
then show "u n ≥ D" unfolding D_def by linarith
next
assume "¬(n ≤ N)"
then have "n ≥ N" by simp
then have "u n > C" using N by auto
then have "u n > real_of_ereal C"
using ‹C = ereal(real_of_ereal C)› less_ereal.simps(1) by fastforce
then show "u n ≥ D" unfolding D_def by linarith
qed
qed
then show ?thesis by blast
qed

lemma liminf_upper_bound:
fixes u:: "nat ⇒ ereal"
assumes "liminf u < l"
shows "∃N>k. u N < l"
by (metis assms gt_ex less_le_trans liminf_bounded_iff not_less)

lemma limsup_shift:
"limsup (λn. u (n+1)) = limsup u"
proof -
have "(SUP m∈{n+1..}. u m) = (SUP m∈{n..}. u (m + 1))" for n
by (rule SUP_eq) (use Suc_le_D in auto)
then have a: "(INF n. SUP m∈{n..}. u (m + 1)) = (INF n. (SUP m∈{n+1..}. u m))" by auto
have b: "(INF n. (SUP m∈{n+1..}. u m)) = (INF n∈{1..}. (SUP m∈{n..}. u m))"
by (rule INF_eq) (use Suc_le_D in auto)
have "(INF n∈{1..}. v n) = (INF n. v n)" if "decseq v" for v::"nat ⇒ 'a"
by (rule INF_eq) (use ‹decseq v› decseq_Suc_iff in auto)
moreover have "decseq (λn. (SUP m∈{n..}. u m))" by (simp add: SUP_subset_mono decseq_def)
ultimately have c: "(INF n∈{1..}. (SUP m∈{n..}. u m)) = (INF n. (SUP m∈{n..}. u m))" by simp
have "(INF n. Sup (u ` {n..})) = (INF n. SUP m∈{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 (λn. u (n+k)) = limsup u"
proof (induction k)
case (Suc k)
have "limsup (λn. u (n+k+1)) = limsup (λn. u (n+k))" using limsup_shift[where ?u="λn. u(n+k)"] by simp
then show ?case using Suc.IH by simp
qed (auto)

lemma liminf_shift:
"liminf (λn. u (n+1)) = liminf u"
proof -
have "(INF m∈{n+1..}. u m) = (INF m∈{n..}. u (m + 1))" for n
by (rule INF_eq) (use Suc_le_D in auto)
then have a: "(SUP n. INF m∈{n..}. u (m + 1)) = (SUP n. (INF m∈{n+1..}. u m))" by auto
have b: "(SUP n. (INF m∈{n+1..}. u m)) = (SUP n∈{1..}. (INF m∈{n..}. u m))"
by (rule SUP_eq) (use Suc_le_D in auto)
have "(SUP n∈{1..}. v n) = (SUP n. v n)" if "incseq v" for v::"nat ⇒ 'a"
by (rule SUP_eq) (use ‹incseq v› incseq_Suc_iff in auto)
moreover have "incseq (λn. (INF m∈{n..}. u m))" by (simp add: INF_superset_mono mono_def)
ultimately have c: "(SUP n∈{1..}. (INF m∈{n..}. u m)) = (SUP n. (INF m∈{n..}. u m))" by simp
have "(SUP n. Inf (u ` {n..})) = (SUP n. INF m∈{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 (λn. u (n+k)) = liminf u"
proof (induction k)
case (Suc k)
have "liminf (λn. u (n+k+1)) = liminf (λn. u (n+k))"
using liminf_shift[where ?u="λn. u(n+k)"] by simp
then show ?case using Suc.IH by simp
qed (auto)

lemma Limsup_obtain:
fixes u::"_ ⇒ 'a :: complete_linorder"
assumes "Limsup F u > c"
shows "∃i. u i > c"
proof -
have "(INF P∈{P. eventually P F}. SUP x∈{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 ‹The next lemma is extremely useful, as it often makes it possible to reduce statements

lemma limsup_subseq_lim:
fixes u::"nat ⇒ 'a :: {complete_linorder, linorder_topology}"
shows "∃r::nat⇒nat. strict_mono r ∧ (u o r) ⇢ limsup u"
proof (cases)
assume "∀n. ∃p>n. ∀m≥p. u m ≤ u p"
then have "∃r. ∀n. (∀m≥r n. u m ≤ u (r n)) ∧ r n < r (Suc n)"
by (intro dependent_nat_choice) (auto simp: conj_commute)
then obtain r :: "nat ⇒ nat" where "strict_mono r" and mono: "⋀n m. r n ≤ m ⟹ u m ≤ u (r n)"
by (auto simp: strict_mono_Suc_iff)
define umax where "umax = (λn. (SUP m∈{n..}. u m))"
have "decseq umax" unfolding umax_def by (simp add: SUP_subset_mono antimono_def)
then have "umax ⇢ limsup u" unfolding umax_def by (metis LIMSEQ_INF limsup_INF_SUP)
then have *: "(umax o r) ⇢ limsup u" by (simp add: LIMSEQ_subseq_LIMSEQ ‹strict_mono r›)
have "⋀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) ⇢ limsup u" using * by simp
then show ?thesis using ‹strict_mono r› by blast
next
assume "¬ (∀n. ∃p>n. (∀m≥p. u m ≤ u p))"
then obtain N where N: "⋀p. p > N ⟹ ∃m>p. u p < u m" by (force simp: not_le le_less)
have "∃r. ∀n. N < r n ∧ r n < r (Suc n) ∧ (∀i∈ {N<..r (Suc n)}. u i ≤ u (r (Suc n)))"
proof (rule dependent_nat_choice)
fix x assume "N < x"
then have a: "finite {N<..x}" "{N<..x} ≠ {}" by simp_all
have "Max {u i |i. i ∈ {N<..x}} ∈ {u i |i. i ∈ {N<..x}}" apply (rule Max_in) using a by (auto)
then obtain p where "p ∈ {N<..x}" and upmax: "u p = Max{u i |i. i ∈ {N<..x}}" by auto
define U where "U = {m. m > p ∧ u p < u m}"
have "U ≠ {}" unfolding U_def using N[of p] ‹p ∈ {N<..x}› by auto
define y where "y = Inf U"
then have "y ∈ U" using ‹U ≠ {}› by (simp add: Inf_nat_def1)
have a: "⋀i. i ∈ {N<..x} ⟹ u i ≤ u p"
proof -
fix i assume "i ∈ {N<..x}"
then have "u i ∈ {u i |i. i ∈ {N<..x}}" by blast
then show "u i ≤ u p" using upmax by simp
qed
moreover have "u p < u y" using ‹y ∈ U› U_def by auto
ultimately have "y ∉ {N<..x}" using not_le by blast
moreover have "y > N" using ‹y ∈ U› U_def ‹p ∈ {N<..x}› by auto
ultimately have "y > x" by auto

have "⋀i. i ∈ {N<..y} ⟹ u i ≤ u y"
proof -
fix i assume "i ∈ {N<..y}" show "u i ≤ u y"
proof (cases)
assume "i = y"
then show ?thesis by simp
next
assume "¬(i=y)"
then have i:"i ∈ {N<..<y}" using ‹i ∈ {N<..y}› by simp
have "u i ≤ u p"
proof (cases)
assume "i ≤ x"
then have "i ∈ {N<..x}" using i by simp
then show ?thesis using a by simp
next
assume "¬(i ≤ x)"
then```