session-qualified theory imports: isabelle imports -U -i -d '~~/src/Benchmarks' -a;
(* 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>
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 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 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: {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:{0<..}. INF y:(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: zero_less_dist_iff dist_commute)
then show "\<exists>r>0. INFIMUM (Collect P) f \<le> INFIMUM (S \<inter> ball x r - {x}) f"
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>
INFIMUM (S \<inter> ball x d - {x}) f \<le> INFIMUM (Collect P) f"
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:{0<..}. SUP y:(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: zero_less_dist_iff dist_commute)
then show "\<exists>r>0. SUPREMUM (S \<inter> ball x r - {x}) f \<le> SUPREMUM (Collect P) f"
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>
SUPREMUM (Collect P) f \<le> SUPREMUM (S \<inter> ball x d - {x}) f"
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:{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 \<Rightarrow> '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 \<Rightarrow> '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 \<open>monoset\<close>
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" "INFIMUM (Collect P) f \<in> S"
{
fix x
assume "P x"
then have "INFIMUM (Collect P) f \<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> INFIMUM (Collect P) f \<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 "INFIMUM {x. B < f x} f \<le> y"
using P by auto
moreover have "B \<le> INFIMUM {x. B < f x} f"
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)" "SUPREMUM (Collect P) f \<in> S"
{
fix x
assume "P x"
then have "f x \<le> 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 \<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> SUPREMUM (Collect P) f"
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> SUPREMUM {x. f x < B} f"
using P by auto
moreover have "SUPREMUM {x. f x < B} f \<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
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