# Theory Extended_Real_Limits

theory Extended_Real_Limits
imports Topology_Euclidean_Space Extended_Real
```(*  Title:      HOL/Multivariate_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›

theory Extended_Real_Limits
imports
Topology_Euclidean_Space
"~~/src/HOL/Library/Extended_Real"
"~~/src/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_inter_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 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

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"
proof -
{
assume lhs: "open S ∧ closed S"
{
assume "-∞ ∉ S"
then have "S = {}"
using lhs ereal_open_closed_aux by auto
}
moreover
{
assume "-∞ ∈ S"
then have "- S = {}"
using lhs ereal_open_closed_aux[of "-S"] by auto
}
ultimately have "S = {} ∨ S = UNIV"
by auto
}
then show ?thesis
by auto
qed

lemma ereal_open_atLeast:
fixes x :: ereal
shows "open {x..} ⟷ x = -∞"
proof
assume "x = -∞"
then have "{x..} = UNIV"
by auto
then show "open {x..}"
by auto
next
assume "open {x..}"
then have "open {x..} ∧ closed {x..}"
by auto
then have "{x..} = UNIV"
unfolding ereal_open_closed by auto
then show "x = -∞"
qed

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"
apply (subst closed_contains_Inf)
using ex ‹S ≠ {}› ‹closed S›
apply auto
done
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 -
{
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="-∞" in exI)
apply auto
done
}
moreover
{
assume "∃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 ⇒ '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: zero_less_dist_iff dist_commute)
then show "∃r>0. INFIMUM (Collect P) f ≤ INFIMUM (S ∩ ball x r - {x}) f"
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) ∧
INFIMUM (S ∩ ball x d - {x}) f ≤ INFIMUM (Collect P) f"
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: zero_less_dist_iff dist_commute)
then show "∃r>0. SUPREMUM (S ∩ ball x r - {x}) f ≤ SUPREMUM (Collect P) f"
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) ∧
SUPREMUM (Collect P) f ≤ SUPREMUM (S ∩ ball x d - {x}) f"
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)"
unfolding inf_min[symmetric] Liminf_at
apply (subst inf_commute)
apply (subst SUP_inf)
apply (intro SUP_cong[OF refl])
apply (cut_tac A="ball x xa - {x}" and B="{x}" and M=f in INF_union)
apply (drule sym)
apply auto
apply (metis INF_absorb centre_in_ball)
done

subsection ‹monoset›

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" "INFIMUM (Collect P) f ∈ S"
{
fix x
assume "P x"
then have "INFIMUM (Collect P) f ≤ 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 ⟶ INFIMUM (Collect P) f ≤ 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 "INFIMUM {x. B < f x} f ≤ y"
using P by auto
moreover have "B ≤ INFIMUM {x. B < f x} f"
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)" "SUPREMUM (Collect P) f ∈ S"
{
fix x
assume "P x"
then have "f x ≤ SUPREMUM (Collect P) f"
by (intro complete_lattice_class.SUP_upper) simp
with S(1)[unfolded mono_set, rule_format, of "- SUPREMUM (Collect P) f" "- 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 ≤ SUPREMUM (Collect P) f"
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 ≤ SUPREMUM {x. f x < B} f"
using P by auto
moreover have "SUPREMUM {x. f x < B} f ≤ 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"
{
assume "S = UNIV"
then have "∃N. ∀n≥N. x n ∈ S"
by auto
}
moreover
{
assume "S ≠ 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 "∃N. ∀n≥N. x n ∈ S"
unfolding B
using ‹x0 ≤ liminf x› liminf_bounded_iff
by auto
}
ultimately have "∃N. ∀n≥N. x n ∈ S"
by auto
}
then show "?P x0"
by auto
qed

subsection "Relate extended reals and the indicator function"

lemma ereal_indicator_le_0: "(indicator S x::ereal) ≤ 0 ⟷ x ∉ 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 ≤ (indicator A x ::ereal)"
unfolding indicator_def by auto

lemma indicator_inter_arith_ereal: "indicator A x * indicator B x = (indicator (A ∩ B) x :: ereal)"
by (simp split: split_indicator)

end
```