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
*)


header {* Limits on the Extended real number line *}

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

lemma convergent_limsup_cl:
fixes X :: "nat => 'a::{complete_linorder,linorder_topology}"
shows "convergent X ==> limsup X = lim X"
by (auto simp: convergent_def limI lim_imp_Limsup)

lemma lim_increasing_cl:
assumes "!!n m. n ≥ m ==> f n ≥ f m"
obtains l where "f ----> (l::'a::{complete_linorder,linorder_topology})"
proof
show "f ----> (SUP n. f n)"
using assms
by (intro increasing_tendsto)
(auto simp: SUP_upper eventually_sequentially less_SUP_iff intro: less_le_trans)
qed

lemma lim_decreasing_cl:
assumes "!!n m. n ≥ m ==> f n ≤ f m"
obtains l where "f ----> (l::'a::{complete_linorder,linorder_topology})"
proof
show "f ----> (INF n. f n)"
using assms
by (intro decreasing_tendsto)
(auto simp: INF_lower eventually_sequentially INF_less_iff intro: le_less_trans)
qed

lemma compact_complete_linorder:
fixes X :: "nat => 'a::{complete_linorder,linorder_topology}"
shows "∃l r. subseq r ∧ (X o r) ----> l"
proof -
obtain r where "subseq r" and mono: "monoseq (X o r)"
using seq_monosub[of X]
unfolding comp_def
by auto
then have "(∀n m. m ≤ n --> (X o r) m ≤ (X o r) n) ∨ (∀n m. m ≤ n --> (X o r) n ≤ (X o r) m)"
by (auto simp add: monoseq_def)
then obtain l where "(X o r) ----> l"
using lim_increasing_cl[of "X o r"] lim_decreasing_cl[of "X o r"]
by auto
then show ?thesis
using `subseq r` by auto
qed

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

lemma ereal_dense3:
fixes x y :: ereal
shows "x < y ==> ∃r::rat. x < real_of_rat r ∧ real_of_rat r < y"
proof (cases x y rule: ereal2_cases, simp_all)
fix r q :: real
assume "r < q"
from Rats_dense_in_real[OF this] show "∃x. r < real_of_rat x ∧ real_of_rat x < q"
by (fastforce simp: Rats_def)
next
fix r :: real
show "∃x. r < real_of_rat x" "∃x. real_of_rat x < r"
using gt_ex[of r] lt_ex[of r] Rats_dense_in_real
by (auto simp: Rats_def)
qed

instance ereal :: second_countable_topology
proof (default, intro exI conjI)
let ?B = "(\<Union>r∈\<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} ∨ b = {e<..}"
by auto
moreover have "{..<e} = \<Union>{{..<x}|x. x ∈ \<rat> ∧ x < e}" "{e<..} = \<Union>{{x<..}|x. x ∈ \<rat> ∧ 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 continuous_on_ereal[intro, simp]: "continuous_on A ereal"
unfolding continuous_on_topological open_ereal_def
by auto

lemma continuous_at_ereal[intro, simp]: "continuous (at x) ereal"
using continuous_on_eq_continuous_at[of UNIV]
by auto

lemma continuous_within_ereal[intro, simp]: "x ∈ A ==> continuous (at x within A) ereal"
using continuous_on_eq_continuous_within[of A]
by auto

lemma ereal_open_uminus:
fixes S :: "ereal set"
assumes "open S"
shows "open (uminus ` S)"
using `open S`[unfolded open_generated_order]
proof induct
have "range uminus = (UNIV :: ereal set)"
by (auto simp: image_iff ereal_uminus_eq_reorder)
then show "open (range uminus :: ereal set)"
by simp
qed (auto simp add: image_Union image_Int)

lemma ereal_uminus_complement:
fixes S :: "ereal set"
shows "uminus ` (- S) = - uminus ` S"
by (auto intro!: bij_image_Compl_eq surjI[of _ uminus] simp: bij_betw_def)

lemma ereal_closed_uminus:
fixes S :: "ereal set"
assumes "closed S"
shows "closed (uminus ` S)"
using assms
unfolding closed_def ereal_uminus_complement[symmetric]
by (rule ereal_open_uminus)

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_affinity_pos:
fixes S :: "ereal set"
assumes "open S"
and m: "m ≠ ∞" "0 < m"
and t: "¦t¦ ≠ ∞"
shows "open ((λx. m * x + t) ` S)"
proof -
obtain r where r[simp]: "m = ereal r"
using m by (cases m) auto
obtain p where p[simp]: "t = ereal p"
using t by auto
have "r ≠ 0" "0 < r" and m': "m ≠ ∞" "m ≠ -∞" "m ≠ 0"
using m by auto
from `open S` [THEN ereal_openE] guess l u . note T = this
let ?f = "(λx. m * x + t)"
show ?thesis
unfolding open_ereal_def
proof (intro conjI impI exI subsetI)
have "ereal -` ?f ` S = (λx. r * x + p) ` (ereal -` S)"
proof safe
fix x y
assume "ereal y = m * x + t" "x ∈ S"
then show "y ∈ (λx. r * x + p) ` ereal -` S"
using `r ≠ 0` by (cases x) (auto intro!: image_eqI[of _ _ "real x"] split: split_if_asm)
qed force
then show "open (ereal -` ?f ` S)"
using open_affinity[OF T(1) `r ≠ 0`]
by (auto simp: ac_simps)
next
assume "∞ ∈ ?f`S"
with `0 < r` have "∞ ∈ S"
by auto
fix x
assume "x ∈ {ereal (r * l + p)<..}"
then have [simp]: "ereal (r * l + p) < x"
by auto
show "x ∈ ?f`S"
proof (rule image_eqI)
show "x = m * ((x - t) / m) + t"
using m t
by (cases rule: ereal3_cases[of m x t]) auto
have "ereal l < (x - t) / m"
using m t
by (simp add: ereal_less_divide_pos ereal_less_minus)
then show "(x - t) / m ∈ S"
using T(2)[OF `∞ ∈ S`] by auto
qed
next
assume "-∞ ∈ ?f ` S"
with `0 < r` have "-∞ ∈ S"
by auto
fix x assume "x ∈ {..<ereal (r * u + p)}"
then have [simp]: "x < ereal (r * u + p)"
by auto
show "x ∈ ?f`S"
proof (rule image_eqI)
show "x = m * ((x - t) / m) + t"
using m t
by (cases rule: ereal3_cases[of m x t]) auto
have "(x - t)/m < ereal u"
using m t
by (simp add: ereal_divide_less_pos ereal_minus_less)
then show "(x - t)/m ∈ S"
using T(3)[OF `-∞ ∈ S`]
by auto
qed
qed
qed

lemma ereal_open_affinity:
fixes S :: "ereal set"
assumes "open S"
and m: "¦m¦ ≠ ∞" "m ≠ 0"
and t: "¦t¦ ≠ ∞"
shows "open ((λx. m * x + t) ` S)"
proof cases
assume "0 < m"
then show ?thesis
using ereal_open_affinity_pos[OF `open S` _ _ t, of m] m
by auto
next
assume "¬ 0 < m" then
have "0 < -m"
using `m ≠ 0`
by (cases m) auto
then have m: "-m ≠ ∞" "0 < -m"
using `¦m¦ ≠ ∞`
by (auto simp: ereal_uminus_eq_reorder)
from ereal_open_affinity_pos[OF ereal_open_uminus[OF `open S`] m t] show ?thesis
unfolding image_image by simp
qed

lemma ereal_lim_mult:
fixes X :: "'a => ereal"
assumes lim: "(X ---> L) net"
and a: "¦a¦ ≠ ∞"
shows "((λi. a * X i) ---> a * L) net"
proof cases
assume "a ≠ 0"
show ?thesis
proof (rule topological_tendstoI)
fix S
assume "open S" and "a * L ∈ S"
have "a * L / a = L"
using `a ≠ 0` a
by (cases rule: ereal2_cases[of a L]) auto
then have L: "L ∈ ((λx. x / a) ` S)"
using `a * L ∈ S`
by (force simp: image_iff)
moreover have "open ((λx. x / a) ` S)"
using ereal_open_affinity[OF `open S`, of "inverse a" 0] `a ≠ 0` a
by (auto simp: ereal_divide_eq ereal_inverse_eq_0 divide_ereal_def ac_simps)
note * = lim[THEN topological_tendstoD, OF this L]
{
fix x
from a `a ≠ 0` have "a * (x / a) = x"
by (cases rule: ereal2_cases[of a x]) auto
}
note this[simp]
show "eventually (λx. a * X x ∈ S) net"
by (rule eventually_mono[OF _ *]) auto
qed
qed auto

lemma ereal_lim_uminus:
fixes X :: "'a => ereal"
shows "((λi. - X i) ---> - L) net <-> (X ---> L) net"
using ereal_lim_mult[of X L net "ereal (-1)"]
ereal_lim_mult[of "(λi. - X i)" "-L" net "ereal (-1)"]
by (auto simp add: algebra_simps)

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 open_uminus_iff:
fixes S :: "ereal set"
shows "open (uminus ` S) <-> open S"
using ereal_open_uminus[of S] ereal_open_uminus[of "uminus ` S"]
by auto

lemma ereal_Liminf_uminus:
fixes f :: "'a => ereal"
shows "Liminf net (λx. - (f x)) = - Limsup net f"
using ereal_Limsup_uminus[of _ "(λx. - (f x))"] by auto

lemma ereal_Lim_uminus:
fixes f :: "'a => ereal"
shows "(f ---> f0) net <-> ((λx. - f x) ---> - f0) net"
using
ereal_lim_mult[of f f0 net "- 1"]
ereal_lim_mult[of "λx. - (f x)" "-f0" net "- 1"]
by (auto simp: ereal_uminus_reorder)

lemma Liminf_PInfty:
fixes f :: "'a => ereal"
assumes "¬ trivial_limit net"
shows "(f ---> ∞) net <-> Liminf net f = ∞"
unfolding tendsto_iff_Liminf_eq_Limsup[OF assms]
using Liminf_le_Limsup[OF assms, of f]
by auto

lemma Limsup_MInfty:
fixes f :: "'a => ereal"
assumes "¬ trivial_limit net"
shows "(f ---> -∞) net <-> Limsup net f = -∞"
unfolding tendsto_iff_Liminf_eq_Limsup[OF assms]
using Liminf_le_Limsup[OF assms, of f]
by auto

lemma convergent_ereal:
fixes X :: "nat => 'a :: {complete_linorder,linorder_topology}"
shows "convergent X <-> limsup X = liminf X"
using tendsto_iff_Liminf_eq_Limsup[of sequentially]
by (auto simp: convergent_def)

lemma liminf_PInfty:
fixes X :: "nat => ereal"
shows "X ----> ∞ <-> liminf X = ∞"
by (metis Liminf_PInfty trivial_limit_sequentially)

lemma limsup_MInfty:
fixes X :: "nat => ereal"
shows "X ----> -∞ <-> limsup X = -∞"
by (metis Limsup_MInfty trivial_limit_sequentially)

lemma ereal_lim_mono:
fixes X Y :: "nat => 'a::linorder_topology"
assumes "!!n. N ≤ n ==> X n ≤ Y n"
and "X ----> x"
and "Y ----> y"
shows "x ≤ y"
using assms(1) by (intro LIMSEQ_le[OF assms(2,3)]) auto

lemma incseq_le_ereal:
fixes X :: "nat => 'a::linorder_topology"
assumes inc: "incseq X"
and lim: "X ----> L"
shows "X N ≤ L"
using inc
by (intro ereal_lim_mono[of N, OF _ tendsto_const lim]) (simp add: incseq_def)

lemma decseq_ge_ereal:
assumes dec: "decseq X"
and lim: "X ----> (L::'a::linorder_topology)"
shows "X N ≥ L"
using dec by (intro ereal_lim_mono[of N, OF _ lim tendsto_const]) (simp add: decseq_def)

lemma bounded_abs:
fixes a :: real
assumes "a ≤ x"
and "x ≤ b"
shows "abs x ≤ max (abs a) (abs b)"
by (metis abs_less_iff assms leI le_max_iff_disj
less_eq_real_def less_le_not_le less_minus_iff minus_minus)

lemma ereal_Sup_lim:
fixes a :: "'a::{complete_linorder,linorder_topology}"
assumes "!!n. b n ∈ s"
and "b ----> a"
shows "a ≤ Sup s"
by (metis Lim_bounded_ereal assms complete_lattice_class.Sup_upper)

lemma ereal_Inf_lim:
fixes a :: "'a::{complete_linorder,linorder_topology}"
assumes "!!n. b n ∈ s"
and "b ----> a"
shows "Inf s ≤ a"
by (metis Lim_bounded2_ereal assms complete_lattice_class.Inf_lower)

lemma SUP_Lim_ereal:
fixes X :: "nat => 'a::{complete_linorder,linorder_topology}"
assumes inc: "incseq X"
and l: "X ----> l"
shows "(SUP n. X n) = l"
using LIMSEQ_SUP[OF inc] tendsto_unique[OF trivial_limit_sequentially l]
by simp

lemma INF_Lim_ereal:
fixes X :: "nat => 'a::{complete_linorder,linorder_topology}"
assumes dec: "decseq X"
and l: "X ----> l"
shows "(INF n. X n) = l"
using LIMSEQ_INF[OF dec] tendsto_unique[OF trivial_limit_sequentially l]
by simp

lemma SUP_eq_LIMSEQ:
assumes "mono f"
shows "(SUP n. ereal (f n)) = ereal x <-> f ----> x"
proof
have inc: "incseq (λi. ereal (f i))"
using `mono f` unfolding mono_def incseq_def by auto
{
assume "f ----> x"
then have "(λi. ereal (f i)) ----> ereal x"
by auto
from SUP_Lim_ereal[OF inc this] show "(SUP n. ereal (f n)) = ereal x" .
next
assume "(SUP n. ereal (f n)) = ereal x"
with LIMSEQ_SUP[OF inc] show "f ----> x" by auto
}
qed

lemma liminf_ereal_cminus:
fixes f :: "nat => ereal"
assumes "c ≠ -∞"
shows "liminf (λx. c - f x) = c - limsup f"
proof (cases c)
case PInf
then show ?thesis
by (simp add: Liminf_const)
next
case (real r)
then show ?thesis
unfolding liminf_SUPR_INFI limsup_INFI_SUPR
apply (subst INFI_ereal_cminus)
apply auto
apply (subst SUPR_ereal_cminus)
apply auto
done
qed (insert `c ≠ -∞`, simp)


subsubsection {* Continuity *}

lemma continuous_at_of_ereal:
fixes x0 :: ereal
assumes "¦x0¦ ≠ ∞"
shows "continuous (at x0) real"
proof -
{
fix T
assume T: "open T" "real x0 ∈ T"
def S "ereal ` T"
then have "ereal (real x0) ∈ S"
using T by auto
then have "x0 ∈ S"
using assms ereal_real by auto
moreover have "open S"
using open_ereal S_def T by auto
moreover have "∀y∈S. real y ∈ T"
using S_def T by auto
ultimately have "∃S. x0 ∈ S ∧ open S ∧ (∀y∈S. real y ∈ T)"
by auto
}
then show ?thesis
unfolding continuous_at_open by blast
qed

lemma continuous_at_iff_ereal:
fixes f :: "'a::t2_space => real"
shows "continuous (at x0) f <-> continuous (at x0) (ereal o f)"
proof -
{
assume "continuous (at x0) f"
then have "continuous (at x0) (ereal o f)"
using continuous_at_ereal continuous_at_compose[of x0 f ereal]
by auto
}
moreover
{
assume "continuous (at x0) (ereal o f)"
then have "continuous (at x0) (real o (ereal o f))"
using continuous_at_of_ereal
by (intro continuous_at_compose[of x0 "ereal o f"]) auto
moreover have "real o (ereal o f) = f"
using real_ereal_id by (simp add: o_assoc)
ultimately have "continuous (at x0) f"
by auto
}
ultimately show ?thesis
by auto
qed


lemma continuous_on_iff_ereal:
fixes f :: "'a::t2_space => real"
assumes "open A"
shows "continuous_on A f <-> continuous_on A (ereal o f)"
using continuous_at_iff_ereal assms
by (auto simp add: continuous_on_eq_continuous_at cong del: continuous_on_cong)

lemma continuous_on_real: "continuous_on (UNIV - {∞, -∞::ereal}) real"
using continuous_at_of_ereal continuous_on_eq_continuous_at open_image_ereal
by auto

lemma continuous_on_iff_real:
fixes f :: "'a::t2_space => ereal"
assumes "!!x. x ∈ A ==> ¦f x¦ ≠ ∞"
shows "continuous_on A f <-> continuous_on A (real o f)"
proof -
have "f ` A ⊆ UNIV - {∞, -∞}"
using assms by force
then have *: "continuous_on (f ` A) real"
using continuous_on_real by (simp add: continuous_on_subset)
have **: "continuous_on ((real o f) ` A) ereal"
using continuous_on_ereal continuous_on_subset[of "UNIV" "ereal" "(real o f) ` A"]
by blast
{
assume "continuous_on A f"
then have "continuous_on A (real o f)"
apply (subst continuous_on_compose)
using *
apply auto
done
}
moreover
{
assume "continuous_on A (real o f)"
then have "continuous_on A (ereal o (real o f))"
apply (subst continuous_on_compose)
using **
apply auto
done
then have "continuous_on A f"
apply (subst continuous_on_eq[of A "ereal o (real o f)" f])
using assms ereal_real
apply auto
done
}
ultimately show ?thesis
by auto
qed

lemma continuous_at_const:
fixes f :: "'a::t2_space => ereal"
assumes "∀x. f x = C"
shows "∀x. continuous (at x) f"
unfolding continuous_at_open
using assms t1_space
by auto

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_EX[of _ S] ex by metis
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"
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


subsection {* Sums *}

lemma setsum_ereal[simp]: "(∑x∈A. ereal (f x)) = ereal (∑x∈A. f x)"
proof (cases "finite A")
case True
then show ?thesis by induct auto
next
case False
then show ?thesis by simp
qed

lemma setsum_Pinfty:
fixes f :: "'a => ereal"
shows "(∑x∈P. f x) = ∞ <-> finite P ∧ (∃i∈P. f i = ∞)"
proof safe
assume *: "setsum f P = ∞"
show "finite P"
proof (rule ccontr)
assume "infinite P"
with * show False
by auto
qed
show "∃i∈P. f i = ∞"
proof (rule ccontr)
assume "¬ ?thesis"
then have "!!i. i ∈ P ==> f i ≠ ∞"
by auto
with `finite P` have "setsum f P ≠ ∞"
by induct auto
with * show False
by auto
qed
next
fix i
assume "finite P" and "i ∈ P" and "f i = ∞"
then show "setsum f P = ∞"
proof induct
case (insert x A)
show ?case using insert by (cases "x = i") auto
qed simp
qed

lemma setsum_Inf:
fixes f :: "'a => ereal"
shows "¦setsum f A¦ = ∞ <-> finite A ∧ (∃i∈A. ¦f i¦ = ∞)"
proof
assume *: "¦setsum f A¦ = ∞"
have "finite A"
by (rule ccontr) (insert *, auto)
moreover have "∃i∈A. ¦f i¦ = ∞"
proof (rule ccontr)
assume "¬ ?thesis"
then have "∀i∈A. ∃r. f i = ereal r"
by auto
from bchoice[OF this] obtain r where "∀x∈A. f x = ereal (r x)" ..
with * show False
by auto
qed
ultimately show "finite A ∧ (∃i∈A. ¦f i¦ = ∞)"
by auto
next
assume "finite A ∧ (∃i∈A. ¦f i¦ = ∞)"
then obtain i where "finite A" "i ∈ A" and "¦f i¦ = ∞"
by auto
then show "¦setsum f A¦ = ∞"
proof induct
case (insert j A)
then show ?case
by (cases rule: ereal3_cases[of "f i" "f j" "setsum f A"]) auto
qed simp
qed

lemma setsum_real_of_ereal:
fixes f :: "'i => ereal"
assumes "!!x. x ∈ S ==> ¦f x¦ ≠ ∞"
shows "(∑x∈S. real (f x)) = real (setsum f S)"
proof -
have "∀x∈S. ∃r. f x = ereal r"
proof
fix x
assume "x ∈ S"
from assms[OF this] show "∃r. f x = ereal r"
by (cases "f x") auto
qed
from bchoice[OF this] obtain r where "∀x∈S. f x = ereal (r x)" ..
then show ?thesis
by simp
qed

lemma setsum_ereal_0:
fixes f :: "'a => ereal"
assumes "finite A"
and "!!i. i ∈ A ==> 0 ≤ f i"
shows "(∑x∈A. f x) = 0 <-> (∀i∈A. f i = 0)"
proof
assume *: "(∑x∈A. f x) = 0"
then have "(∑x∈A. f x) ≠ ∞"
by auto
then have "∀i∈A. ¦f i¦ ≠ ∞"
using assms by (force simp: setsum_Pinfty)
then have "∀i∈A. ∃r. f i = ereal r"
by auto
from bchoice[OF this] * assms show "∀i∈A. f i = 0"
using setsum_nonneg_eq_0_iff[of A "λi. real (f i)"] by auto
qed (rule setsum_0')

lemma setsum_ereal_right_distrib:
fixes f :: "'a => ereal"
assumes "!!i. i ∈ A ==> 0 ≤ f i"
shows "r * setsum f A = (∑n∈A. r * f n)"
proof cases
assume "finite A"
then show ?thesis using assms
by induct (auto simp: ereal_right_distrib setsum_nonneg)
qed simp

lemma sums_ereal_positive:
fixes f :: "nat => ereal"
assumes "!!i. 0 ≤ f i"
shows "f sums (SUP n. ∑i<n. f i)"
proof -
have "incseq (λi. ∑j=0..<i. f j)"
using ereal_add_mono[OF _ assms]
by (auto intro!: incseq_SucI)
from LIMSEQ_SUP[OF this]
show ?thesis unfolding sums_def
by (simp add: atLeast0LessThan)
qed

lemma summable_ereal_pos:
fixes f :: "nat => ereal"
assumes "!!i. 0 ≤ f i"
shows "summable f"
using sums_ereal_positive[of f, OF assms]
unfolding summable_def
by auto

lemma suminf_ereal_eq_SUPR:
fixes f :: "nat => ereal"
assumes "!!i. 0 ≤ f i"
shows "(∑x. f x) = (SUP n. ∑i<n. f i)"
using sums_ereal_positive[of f, OF assms, THEN sums_unique]
by simp

lemma sums_ereal: "(λx. ereal (f x)) sums ereal x <-> f sums x"
unfolding sums_def by simp

lemma suminf_bound:
fixes f :: "nat => ereal"
assumes "∀N. (∑n<N. f n) ≤ x"
and pos: "!!n. 0 ≤ f n"
shows "suminf f ≤ x"
proof (rule Lim_bounded_ereal)
have "summable f" using pos[THEN summable_ereal_pos] .
then show "(λN. ∑n<N. f n) ----> suminf f"
by (auto dest!: summable_sums simp: sums_def atLeast0LessThan)
show "∀n≥0. setsum f {..<n} ≤ x"
using assms by auto
qed

lemma suminf_bound_add:
fixes f :: "nat => ereal"
assumes "∀N. (∑n<N. f n) + y ≤ x"
and pos: "!!n. 0 ≤ f n"
and "y ≠ -∞"
shows "suminf f + y ≤ x"
proof (cases y)
case (real r)
then have "∀N. (∑n<N. f n) ≤ x - y"
using assms by (simp add: ereal_le_minus)
then have "(∑ n. f n) ≤ x - y"
using pos by (rule suminf_bound)
then show "(∑ n. f n) + y ≤ x"
using assms real by (simp add: ereal_le_minus)
qed (insert assms, auto)

lemma suminf_upper:
fixes f :: "nat => ereal"
assumes "!!n. 0 ≤ f n"
shows "(∑n<N. f n) ≤ (∑n. f n)"
unfolding suminf_ereal_eq_SUPR[OF assms] SUP_def
by (auto intro: complete_lattice_class.Sup_upper)

lemma suminf_0_le:
fixes f :: "nat => ereal"
assumes "!!n. 0 ≤ f n"
shows "0 ≤ (∑n. f n)"
using suminf_upper[of f 0, OF assms]
by simp

lemma suminf_le_pos:
fixes f g :: "nat => ereal"
assumes "!!N. f N ≤ g N"
and "!!N. 0 ≤ f N"
shows "suminf f ≤ suminf g"
proof (safe intro!: suminf_bound)
fix n
{
fix N
have "0 ≤ g N"
using assms(2,1)[of N] by auto
}
have "setsum f {..<n} ≤ setsum g {..<n}"
using assms by (auto intro: setsum_mono)
also have "… ≤ suminf g"
using `!!N. 0 ≤ g N`
by (rule suminf_upper)
finally show "setsum f {..<n} ≤ suminf g" .
qed (rule assms(2))

lemma suminf_half_series_ereal: "(∑n. (1/2 :: ereal) ^ Suc n) = 1"
using sums_ereal[THEN iffD2, OF power_half_series, THEN sums_unique, symmetric]
by (simp add: one_ereal_def)

lemma suminf_add_ereal:
fixes f g :: "nat => ereal"
assumes "!!i. 0 ≤ f i"
and "!!i. 0 ≤ g i"
shows "(∑i. f i + g i) = suminf f + suminf g"
apply (subst (1 2 3) suminf_ereal_eq_SUPR)
unfolding setsum_addf
apply (intro assms ereal_add_nonneg_nonneg SUPR_ereal_add_pos incseq_setsumI setsum_nonneg ballI)+
done

lemma suminf_cmult_ereal:
fixes f g :: "nat => ereal"
assumes "!!i. 0 ≤ f i"
and "0 ≤ a"
shows "(∑i. a * f i) = a * suminf f"
by (auto simp: setsum_ereal_right_distrib[symmetric] assms
ereal_zero_le_0_iff setsum_nonneg suminf_ereal_eq_SUPR
intro!: SUPR_ereal_cmult )

lemma suminf_PInfty:
fixes f :: "nat => ereal"
assumes "!!i. 0 ≤ f i"
and "suminf f ≠ ∞"
shows "f i ≠ ∞"
proof -
from suminf_upper[of f "Suc i", OF assms(1)] assms(2)
have "(∑i<Suc i. f i) ≠ ∞"
by auto
then show ?thesis
unfolding setsum_Pinfty by simp
qed

lemma suminf_PInfty_fun:
assumes "!!i. 0 ≤ f i"
and "suminf f ≠ ∞"
shows "∃f'. f = (λx. ereal (f' x))"
proof -
have "∀i. ∃r. f i = ereal r"
proof
fix i
show "∃r. f i = ereal r"
using suminf_PInfty[OF assms] assms(1)[of i]
by (cases "f i") auto
qed
from choice[OF this] show ?thesis
by auto
qed

lemma summable_ereal:
assumes "!!i. 0 ≤ f i"
and "(∑i. ereal (f i)) ≠ ∞"
shows "summable f"
proof -
have "0 ≤ (∑i. ereal (f i))"
using assms by (intro suminf_0_le) auto
with assms obtain r where r: "(∑i. ereal (f i)) = ereal r"
by (cases "∑i. ereal (f i)") auto
from summable_ereal_pos[of "λx. ereal (f x)"]
have "summable (λx. ereal (f x))"
using assms by auto
from summable_sums[OF this]
have "(λx. ereal (f x)) sums (∑x. ereal (f x))"
by auto
then show "summable f"
unfolding r sums_ereal summable_def ..
qed

lemma suminf_ereal:
assumes "!!i. 0 ≤ f i"
and "(∑i. ereal (f i)) ≠ ∞"
shows "(∑i. ereal (f i)) = ereal (suminf f)"
proof (rule sums_unique[symmetric])
from summable_ereal[OF assms]
show "(λx. ereal (f x)) sums (ereal (suminf f))"
unfolding sums_ereal
using assms
by (intro summable_sums summable_ereal)
qed

lemma suminf_ereal_minus:
fixes f g :: "nat => ereal"
assumes ord: "!!i. g i ≤ f i" "!!i. 0 ≤ g i"
and fin: "suminf f ≠ ∞" "suminf g ≠ ∞"
shows "(∑i. f i - g i) = suminf f - suminf g"
proof -
{
fix i
have "0 ≤ f i"
using ord[of i] by auto
}
moreover
from suminf_PInfty_fun[OF `!!i. 0 ≤ f i` fin(1)] obtain f' where [simp]: "f = (λx. ereal (f' x))" ..
from suminf_PInfty_fun[OF `!!i. 0 ≤ g i` fin(2)] obtain g' where [simp]: "g = (λx. ereal (g' x))" ..
{
fix i
have "0 ≤ f i - g i"
using ord[of i] by (auto simp: ereal_le_minus_iff)
}
moreover
have "suminf (λi. f i - g i) ≤ suminf f"
using assms by (auto intro!: suminf_le_pos simp: field_simps)
then have "suminf (λi. f i - g i) ≠ ∞"
using fin by auto
ultimately show ?thesis
using assms `!!i. 0 ≤ f i`
apply simp
apply (subst (1 2 3) suminf_ereal)
apply (auto intro!: suminf_diff[symmetric] summable_ereal)
done
qed

lemma suminf_ereal_PInf [simp]: "(∑x. ∞::ereal) = ∞"
proof -
have "(∑i<Suc 0. ∞) ≤ (∑x. ∞::ereal)"
by (rule suminf_upper) auto
then show ?thesis
by simp
qed

lemma summable_real_of_ereal:
fixes f :: "nat => ereal"
assumes f: "!!i. 0 ≤ f i"
and fin: "(∑i. f i) ≠ ∞"
shows "summable (λi. real (f i))"
proof (rule summable_def[THEN iffD2])
have "0 ≤ (∑i. f i)"
using assms by (auto intro: suminf_0_le)
with fin obtain r where r: "ereal r = (∑i. f i)"
by (cases "(∑i. f i)") auto
{
fix i
have "f i ≠ ∞"
using f by (intro suminf_PInfty[OF _ fin]) auto
then have "¦f i¦ ≠ ∞"
using f[of i] by auto
}
note fin = this
have "(λi. ereal (real (f i))) sums (∑i. ereal (real (f i)))"
using f
by (auto intro!: summable_ereal_pos summable_sums simp: ereal_le_real_iff zero_ereal_def)
also have "… = ereal r"
using fin r by (auto simp: ereal_real)
finally show "∃r. (λi. real (f i)) sums r"
by (auto simp: sums_ereal)
qed

lemma suminf_SUP_eq:
fixes f :: "nat => nat => ereal"
assumes "!!i. incseq (λn. f n i)"
and "!!n i. 0 ≤ f n i"
shows "(∑i. SUP n. f n i) = (SUP n. ∑i. f n i)"
proof -
{
fix n :: nat
have "(∑i<n. SUP k. f k i) = (SUP k. ∑i<n. f k i)"
using assms
by (auto intro!: SUPR_ereal_setsum[symmetric])
}
note * = this
show ?thesis
using assms
apply (subst (1 2) suminf_ereal_eq_SUPR)
unfolding *
apply (auto intro!: SUP_upper2)
apply (subst SUP_commute)
apply rule
done
qed

lemma suminf_setsum_ereal:
fixes f :: "_ => _ => ereal"
assumes nonneg: "!!i a. a ∈ A ==> 0 ≤ f i a"
shows "(∑i. ∑a∈A. f i a) = (∑a∈A. ∑i. f i a)"
proof (cases "finite A")
case True
then show ?thesis
using nonneg
by induct (simp_all add: suminf_add_ereal setsum_nonneg)
next
case False
then show ?thesis by simp
qed

lemma suminf_ereal_eq_0:
fixes f :: "nat => ereal"
assumes nneg: "!!i. 0 ≤ f i"
shows "(∑i. f i) = 0 <-> (∀i. f i = 0)"
proof
assume "(∑i. f i) = 0"
{
fix i
assume "f i ≠ 0"
with nneg have "0 < f i"
by (auto simp: less_le)
also have "f i = (∑j. if j = i then f i else 0)"
by (subst suminf_finite[where N="{i}"]) auto
also have "… ≤ (∑i. f i)"
using nneg
by (auto intro!: suminf_le_pos)
finally have False
using `(∑i. f i) = 0` by auto
}
then show "∀i. f i = 0"
by auto
qed simp

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 SUPR_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. INFI (Collect P) f ≤ INFI (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) ∧
INFI (S ∩ ball x d - {x}) f ≤ INFI (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 INFI_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. SUPR (S ∩ ball x r - {x}) f ≤ SUPR (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) ∧
SUPR (Collect P) f ≤ SUPR (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 => _"
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 => _"
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 b - {x}" and B="{x}" and M=f in INF_union)
apply (simp add: INF_def del: inf_ereal_def)
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" "INFI (Collect P) f ∈ S"
{
fix x
assume "P x"
then have "INFI (Collect P) f ≤ 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 (λx. f x ∈ S) net"
by (auto elim: eventually_elim1)
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 --> INFI (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 "INFI {x. B < f x} f ≤ y"
using P by auto
moreover have "B ≤ INFI {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)" "SUPR (Collect P) f ∈ S"
{
fix x
assume "P x"
then have "f x ≤ SUPR (Collect P) f"
by (intro complete_lattice_class.SUP_upper) simp
with S(1)[unfolded mono_set, rule_format, of "- SUPR (Collect P) f" "- f x"] S(2)
have "f x ∈ S"
by (simp add: inj_image_mem_iff) }
with P show "eventually (λx. f x ∈ S) net"
by (auto elim: eventually_elim1)
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 ≤ SUPR (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 ≤ SUPR {x. f x < B} f"
using P by auto
moreover have "SUPR {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

end