(* 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 \ 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\<^marker>\<open>tag unimportant\<close> 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 (simp add: open_enat_def)
qed
instance\<^marker>\<open>tag unimportant\<close> ereal :: second_countable_topology
proof (standard, intro exI conjI)
let ?B = "(\r\\. {{..< 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 = {.. b = {e<..}"
by auto
moreover have "{..{{.. \ \ x < e}" "{e<..} = \{{x<..}|x. x \ \ \ 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 = "(\r\\. {{..< 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 = {.. b = {e<..}"
by auto
moreover have "{..{{.. \ \ x < e}" "{e<..} = \{{x<..}|x. x \ \ \ 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 \<open>S \<noteq> {}\<close>)
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<.. 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 = -\"
by (simp add: bot_ereal_def atLeast_eq_UNIV_iff)
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 \<open>S \<noteq> {}\<close> \<open>closed S\<close>
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"
by (simp add: not_le)
{
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: 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 \<open>0 < d\<close>) 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 \<inter> ball x d - {x})) \<le> Inf (f ` (Collect P))"
by (intro exI[of _ "\y. 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 \<open>0 < d\<close>) 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)) \<le> Sup (f ` (S \<inter> ball x d - {x}))"
by (intro exI[of _ "\y. 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)
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 "(\i\I. a) = a * card I"
apply (cases "finite I", induct set: finite, simp_all)
apply (cases a, auto, metis (no_types, hide_lams) add.commute mult.commute semiring_normalization_rules(3))
done
lemma real_lim_then_eventually_real:
assumes "(u \ ereal l) F"
shows "eventually (\n. 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 (\n. 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 "(\x::ereal. c * x) (Inf {x::ereal. P x}) = Inf ((\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 _ "\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 \ -\"
assumes f: "(f \ \) F" and g: "(g \ y) F"
shows "((\x. f x + g x) \ \) F"
proof -
have "\C. eventually (\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 (\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 (\x. g x > ereal(real_of_ereal(y - 1))) F" by simp
then show ?thesis by auto
next
case (PInf)
have "eventually (\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 (\x. g x > ereal C) F" by auto
{
fix M::real
have "eventually (\x. f x > ereal(M - C)) F" using f by (simp add: tendsto_PInfty)
then have "eventually (\x. (f x > ereal (M-C)) \ (g x > ereal C)) F"
by (auto simp add: ge eventually_conj_iff)
moreover have "\x. ((f x > ereal (M-C)) \ (g x > ereal C)) \ (f x + g x > ereal M)"
using ereal_add_strict_mono2 by fastforce
ultimately have "eventually (\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 \ \"
assumes f: "(f \ -\) F" and g: "(g \ y) F"
shows "((\x. f x + g x) \ -\) F"
proof -
have "\C. eventually (\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 (\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 (\x. g x < ereal(real_of_ereal(y + 1))) F" by simp
then show ?thesis by auto
next
case (MInf)
have "eventually (\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 (\x. g x < ereal C) F" by auto
{
fix M::real
have "eventually (\x. f x < ereal(M - C)) F" using f by (simp add: tendsto_MInfty)
then have "eventually (\x. (f x < ereal (M- C)) \ (g x < ereal C)) F"
by (auto simp add: ge eventually_conj_iff)
moreover have "\x. ((f x < ereal (M-C)) \ (g x < ereal C)) \ (f x + g x < ereal M)"
using ereal_add_strict_mono2 by fastforce
ultimately have "eventually (\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: "\y\ \ \"
assumes f: "(f \ x) F" and g: "(g \ y) F"
shows "((\x. 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
lemma tendsto_add_ereal_general2:
fixes x y :: ereal
assumes x: "\x\ \ \"
and f: "(f \ x) F" and g: "(g \ y) F"
shows "((\x. f x + g x) \ x + y) F"
proof -
have "((\x. g x + f x) \ x + y) F"
using tendsto_add_ereal_general1[OF x, OF g, OF f] add.commute[of "y", of "x"] by simp
moreover have "\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 "\((x=\ \ y=-\) \ (x=-\ \ y=\))"
and f: "(f \ x) F" and g: "(g \ y) F"
shows "((\x. f x + g x) \ 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 \ -\" 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 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 \ ereal l) F" "(v \ ereal m) F"
shows "((\n. u n * v n) \ ereal l * ereal m) F"
proof -
have ureal: "eventually (\n. u n = ereal(real_of_ereal(u n))) F" by (rule real_lim_then_eventually_real[OF assms(1)])
then have "((\n. ereal(real_of_ereal(u n))) \ ereal l) F" using assms by auto
then have limu: "((\n. real_of_ereal(u n)) \ l) F" by auto
have vreal: "eventually (\n. v n = ereal(real_of_ereal(v n))) F" by (rule real_lim_then_eventually_real[OF assms(2)])
then have "((\n. ereal(real_of_ereal(v n))) \ ereal m) F" using assms by auto
then have limv: "((\n. 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 (\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 "((\n. real_of_ereal(u n) * real_of_ereal(v n)) \ l * m) F" using tendsto_mult[OF limu limv] by auto
then have "((\n. 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 "((\x. 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 (\x. 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 \<open>ereal a > 0\<close> 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 \<open>ereal a > 0\<close> 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 (\x. g x > M/a) F" using assms(3) by (simp add: tendsto_PInfty)
then have "eventually (\x. (f x > a) \ (g x > M/a)) F"
using * by (auto simp add: eventually_conj_iff)
then have "eventually (\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::"_ \ ereal"
assumes "(f \ l) F" "(g \ m) F" "l>0" "m>0"
shows "((\x. 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 "((\x. 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 \<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 \ (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::"_ \ ereal"
assumes "(f \ l) F" "(g \ m) F" "\((l=0 \ abs(m) = \) \ (m=0 \ abs(l) = \))"
shows "((\x. 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 "\(l=0 \ m=0)"
then have "l \ 0" "m \ 0" by auto
then have "abs(l) > 0" "abs(m) > 0"
by (metis abs_ereal_ge0 abs_ereal_less0 abs_ereal_pos ereal_uminus_uminus ereal_uminus_zero less_le not_less)+
then have "sgn(l) * l > 0" "sgn(m) * m > 0" using ereal_sgn_abs by auto
moreover have "((\x. sgn(l) * f x) \ (sgn(l) * l)) F"
by (rule tendsto_cmult_ereal, auto simp add: sgn_finite assms(1))
moreover have "((\x. sgn(m) * g x) \ (sgn(m) * m)) F"
by (rule tendsto_cmult_ereal, auto simp add: sgn_finite assms(2))
ultimately have *: "((\x. (sgn(l) * f x) * (sgn(m) * g x)) \ (sgn(l) * l) * (sgn(m) * m)) F"
using tendsto_mult_ereal_pos by force
have "((\x. (sgn(l) * sgn(m)) * ((sgn(l) * f x) * (sgn(m) * g x))) \ (sgn(l) * sgn(m)) * ((sgn(l) * l) * (sgn(m) * m))) F"
by (rule tendsto_cmult_ereal, auto simp add: sgn_finite2 *)
moreover have "\x. (sgn(l) * sgn(m)) * ((sgn(l) * f x) * (sgn(m) * g x)) = f x * g x"
by (metis mult.left_neutral sgn_squared_ereal[OF \<open>l \<noteq> 0\<close>] sgn_squared_ereal[OF \<open>m \<noteq> 0\<close>] mult.assoc mult.commute)
moreover have "(sgn(l) * sgn(m)) * ((sgn(l) * l) * (sgn(m) * m)) = l * m"
by (metis mult.left_neutral sgn_squared_ereal[OF \<open>l \<noteq> 0\<close>] sgn_squared_ereal[OF \<open>m \<noteq> 0\<close>] mult.assoc mult.commute)
ultimately show ?thesis by auto
qed
lemma tendsto_cmult_ereal_general [tendsto_intros]:
fixes f::"_ \ ereal" and c::ereal
assumes "(f \ l) F" "\ (l=0 \ abs(c) = \)"
shows "((\x. c * f x) \ 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::"_ \ ereal"
assumes "(u \ \) F"
shows "((\x. 1/ u x) \ 0) F"
proof -
{
fix e::real assume "e>0"
have "1/e < \" by auto
then have "eventually (\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 \ 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, hide_lams) 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 \ 0" "1/z < e" by auto
}
ultimately have "eventually (\n. 1/u nn. 1/u n\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 (\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 (\n. 1/u n < e) F" using *(1) by auto
then show "eventually (\n. 1/u n < a) F" using \a>e\ 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::"_ \ real"
shows "(u \ l) F \ l \ 0 \ ((\x. 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 "((\x. 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 = (\n. real_of_ereal(u n))"
have ureal: "eventually (\n. u n = ereal(v n)) F" unfolding v_def using real_lim_then_eventually_real assms(1) real by auto
then have "((\n. ereal(v n)) \ ereal r) F" using assms real v_def by auto
then have *: "((\n. v n) \ r) F" by auto
then have "((\n. 1/v n) \ 1/r) F" using \r \ 0\ tendsto_inverse_real by auto
then have lim: "((\n. 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 (\n. v n \ -{0}) F" using * using topological_tendstoD by blast
then have "eventually (\n. 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 (\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 (\n. u n < 0) F" by (metis (no_types) MInf assms(1) tendsto_MInfty zero_ereal_def)
ultimately have *: "eventually (\n. -1/-u n = 1/u n) F" by (simp add: eventually_mono)
define v where "v = (\n. - u n)"
have "(v \ \) F" unfolding v_def using MInf assms(1) tendsto_uminus_ereal by fastforce
then have "((\n. 1/v n) \ 0) F" using tendsto_inverse_ereal_PInf by auto
then have "((\n. -1/v n) \ 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::"_ \ ereal"
assumes "(f \ l) F" "(g \ m) F" "m \ 0" "\(abs(l) = \ \ abs(m) = \)"
shows "((\x. f x / g x) \ l / m) F"
proof -
define h where "h = (\x. 1/ g x)"
have *: "(h \ 1/m) F" unfolding h_def using assms(2) assms(3) tendsto_inverse_ereal by auto
have "((\x. f x * h x) \ 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 \ ereal"
assumes "(u \ l) F" "(v \ m) F" "\((l = \ \ m = \) \ (l = -\ \ m = -\))"
shows "((\n. u n - v n) \ l - m) F"
proof -
have "((\n. u n + (-v n)) \ 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]:
"(\ n::nat. real n) \ \"
by (simp add: filterlim_real_sequentially tendsto_PInfty_eq_at_top)
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 (\N. u N \ n) sequentially" using assms
by (simp add: filterlim_at_top)
then have *: "{N. u N \ n} \ {}" by force
have "N > C" if "u N \ n" for N
proof (rule ccontr)
assume "\(N > C)"
have "u N \ Max {u n| n. n \ 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 \ n} \ {C..}" by fastforce
have "Inf {N. u N \ n} \ C"
by (metis "*" "**" Inf_nat_def1 atLeast_iff subset_eq)
}
then have "eventually (\n. 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 (\N. u N \ n+1) sequentially" using assms
by (simp add: filterlim_at_top)
then obtain N1 where N1: "\N. N \ N1 \ u N \ n + 1"
using eventually_sequentially by auto
have "{N. u N \ n} \ {..
apply auto using N1 by (metis Suc_eq_plus1 not_less not_less_eq_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
apply (rule Max.coboundedI) using pseudo_inverse_finite_set[OF assms] that by auto
then have "eventually (\n. 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 "(\n::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 (\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 \ - \"
shows "(\n::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 (\n. real_of_ereal(min x n) = r) sequentially" using eventually_at_top_linorder by blast
then have "(\n. 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 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 "(\n::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 (\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 "(\n. (-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 add: max_def)
then show ?thesis by auto
qed
lemma ereal_truncation_real_bottom [tendsto_intros]:
fixes x::ereal
assumes "x \ \"
shows "(\n::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 (\n. real_of_ereal(max x (-real n)) = r) sequentially" using eventually_at_top_linorder by blast
then have "(\n. 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 add: max_def)
moreover have "(\n. (-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
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 \ 'b \ ereal"
assumes "\i. i \ S \ (f i \ a i) F"
"\i. abs(a i) \ \"
shows "((\x. \i\S. f i x) \ (\i\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} \ {(0::ereal)..}) abs"
apply (rule continuous_on_closed_Un, auto)
apply (rule iffD1[OF continuous_on_cong, of "{..0}" _ "\x. -x"])
using less_eq_ereal_def apply (auto simp add: continuous_uminus_ereal)
apply (rule iffD1[OF continuous_on_cong, of "{0..}" _ "\x. x"])
apply (auto)
done
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 "((\n. 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]:
"(\x. ereal (- real x)) \ - \"
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 \ ennreal"
assumes "(u \ l) F" "(v \ m) F" "\(l = \ \ m = \)"
shows "((\n. u n - v n) \ l - m) F"
proof -
have "((\n. e2ennreal(enn2ereal(u n) - enn2ereal(v n))) \ 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 \ l) F" "(v \ m) F" "\((l = 0 \ m = \) \ (l = \ \ m = 0))"
shows "((\n. u n * v n) \ l * m) F"
proof -
have "((\n. e2ennreal(enn2ereal (u n) * enn2ereal (v n))) \ 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 \ (\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 _ \<open>a \<in> S\<close>]
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 \<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 \ 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. \<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)) \ 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"
by (simp add: mono_set)
}
with P show "eventually (\x. f x \ S) net"
by (auto elim: eventually_mono)
next
fix y l
assume S: "\S. open S \ mono_set S \ l \ S \ eventually (\x. 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 (\x. 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. \<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)) \ 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"
by (simp add: inj_image_mem_iff) }
with P show "eventually (\x. 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 (\x. 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 (\x. 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"
{
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 \<open>x0 \<le> liminf x\<close> liminf_bounded_iff
by auto
}
ultimately have "\N. \n\N. x n \ S"
by auto
}
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 (\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: "\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 add: *)
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 \<open>C = ereal(real_of_ereal C)\<close> 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 (\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: "\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 add: *)
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 \<open>C = ereal(real_of_ereal C)\<close> 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 (\n. u (n+1)) = limsup u"
proof -
have "(SUP m\{n+1..}. u m) = (SUP m\{n..}. u (m + 1))" for n
apply (rule SUP_eq) using Suc_le_D by 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))"
apply (rule INF_eq) using Suc_le_D by auto
have "(INF n\{1..}. v n) = (INF n. v n)" if "decseq v" for v::"nat \ 'a"
apply (rule INF_eq) using \<open>decseq v\<close> decseq_Suc_iff by auto
moreover have "decseq (\n. (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 (\n. u (n+k)) = limsup u"
proof (induction k)
case (Suc k)
have "limsup (\n. u (n+k+1)) = limsup (\n. u (n+k))" using limsup_shift[where ?u="\n. u(n+k)"] by simp
then show ?case using Suc.IH by simp
qed (auto)
lemma liminf_shift:
"liminf (\n. u (n+1)) = liminf u"
proof -
have "(INF m\{n+1..}. u m) = (INF m\{n..}. u (m + 1))" for n
apply (rule INF_eq) using Suc_le_D by (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))"
apply (rule SUP_eq) using Suc_le_D by (auto)
have "(SUP n\{1..}. v n) = (SUP n. v n)" if "incseq v" for v::"nat \ 'a"
apply (rule SUP_eq) using \<open>incseq v\<close> incseq_Suc_iff by auto
moreover have "incseq (\n. (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 (\n. u (n+k)) = liminf u"
proof (induction k)
case (Suc k)
have "liminf (\n. u (n+k+1)) = liminf (\n. u (n+k))" using liminf_shift[where ?u="\n. 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 \<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 \ '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 = (\n. (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 \<open>strict_mono r\<close> 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 \<open>y \<in> U\<close> U_def by auto
ultimately have "y \ {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 "\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<..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 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 \ U" using Inf_nat_def not_less_Least by auto
then show ?thesis using U_def * by auto
qed
then show "u i \ u y" using \u p < u y\ by auto
qed
qed
then have "N < y \ x < y \ (\i\{N<..y}. u i \ u y)" using \y > x\ \y > N\ by auto
then show "\y>N. x < y \ (\i\{N<..y}. u i \ u y)" by auto
qed (auto)
then obtain r where r: "\n. N < r n \ r n < r (Suc n) \ (\i\ {N<..r (Suc n)}. u i \ 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) \ (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) \ limsup u" using \strict_mono r\ by (simp add: limsup_subseq_mono)
ultimately have "(SUP n. (u o r) n) \ limsup u" by simp
{
fix i assume i: "i \ {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 \ {N<..r(Suc n)}" using i by simp
then have "u i \ u (r(Suc n))" using r by simp
then have "u i \ (SUP n. (u o r) n)" unfolding o_def by (meson SUP_upper2 UNIV_I)
}
then have "(SUP i\{N<..}. u i) \ (SUP n. (u o r) n)" using SUP_least by blast
then have "limsup u \ (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) \ limsup u" using \(u o r) \ (SUP n. (u o r) n)\ by simp
then show ?thesis using \<open>strict_mono r\<close> by auto
qed
lemma liminf_subseq_lim:
fixes u::"nat \ 'a :: {complete_linorder, linorder_topology}"
shows "\r::nat\nat. strict_mono r \ (u o r) \ liminf u"
proof (cases)
assume "\n. \p>n. \m\p. u m \ u p"
then have "\r. \n. (\m\r n. u m \ u (r n)) \ r n < r (Suc n)"
by (intro dependent_nat_choice) (auto simp: conj_commute)
then obtain r :: "nat \ nat" where "strict_mono r" and mono: "\n m. r n \ m \ u m \ u (r n)"
by (auto simp: strict_mono_Suc_iff)
define umin where "umin = (\n. (INF m\{n..}. u m))"
have "incseq umin" unfolding umin_def by (simp add: INF_superset_mono incseq_def)
then have "umin \ liminf u" unfolding umin_def by (metis LIMSEQ_SUP liminf_SUP_INF)
then have *: "(umin o r) \ liminf u" by (simp add: LIMSEQ_subseq_LIMSEQ \strict_mono r\)
have "\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) \ liminf u" using * by simp
then show ?thesis using \<open>strict_mono r\<close> 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 "Min {u i |i. i \ {N<..x}} \ {u i |i. i \ {N<..x}}" apply (rule Min_in) using a by (auto)
then obtain p where "p \ {N<..x}" and upmin: "u p = Min{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 upmin by simp
qed
moreover have "u p > u y" using \<open>y \<in> U\<close> U_def by auto
ultimately have "y \ {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 "\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<..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 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 \ U" using Inf_nat_def not_less_Least by auto
then show ?thesis using U_def * by auto
qed
then show "u i \ u y" using \u p > u y\ by auto
qed
qed
then have "N < y \ x < y \ (\i\{N<..y}. u i \ u y)" using \y > x\ \y > N\ by auto
then show "\y>N. x < y \ (\i\{N<..y}. u i \ u y)" by auto
qed (auto)
then obtain r :: "nat \ nat"
where r: "\n. N < r n \ r n < r (Suc n) \ (\i\ {N<..r (Suc n)}. u i \ 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) \ (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) \ liminf u" using \strict_mono r\ by (simp add: liminf_subseq_mono)
ultimately have "(INF n. (u o r) n) \ liminf u" by simp
{
fix i assume i: "i \ {N<..}"
--> --------------------
--> maximum size reached
--> --------------------
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