src/HOL/Multivariate_Analysis/Extended_Real_Limits.thy
 author wenzelm Sat, 07 Apr 2012 16:41:59 +0200 changeset 47389 e8552cba702d parent 45051 c478d1876371 child 47761 dfe747e72fa8 permissions -rw-r--r--
explicit checks stable_finished_theory/stable_command allow parallel asynchronous command transactions; tuned;
```
(*  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 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 \<in> A \<Longrightarrow> 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)"
unfolding open_ereal_def
proof (intro conjI impI)
obtain x y where S: "open (ereal -` S)"
"\<infinity> \<in> S \<Longrightarrow> {ereal x<..} \<subseteq> S" "-\<infinity> \<in> S \<Longrightarrow> {..< ereal y} \<subseteq> S"
using `open S` unfolding open_ereal_def by auto
have "ereal -` uminus ` S = uminus ` (ereal -` S)"
proof safe
fix x y assume "ereal x = - y" "y \<in> S"
then show "x \<in> uminus ` ereal -` S" by (cases y) auto
next
fix x assume "ereal x \<in> S"
then show "- x \<in> ereal -` uminus ` S"
by (auto intro: image_eqI[of _ _ "ereal x"])
qed
then show "open (ereal -` uminus ` S)"
using S by (auto intro: open_negations)
{ assume "\<infinity> \<in> uminus ` S"
then have "-\<infinity> \<in> S" by (metis image_iff ereal_uminus_uminus)
then have "uminus ` {..<ereal y} \<subseteq> uminus ` S" using S by (intro image_mono) auto
then show "\<exists>x. {ereal x<..} \<subseteq> uminus ` S" using ereal_uminus_lessThan by auto }
{ assume "-\<infinity> \<in> uminus ` S"
then have "\<infinity> : S" by (metis image_iff ereal_uminus_uminus)
then have "uminus ` {ereal x<..} <= uminus ` S" using S by (intro image_mono) auto
then show "\<exists>y. {..<ereal y} <= uminus ` S" using ereal_uminus_greaterThan by auto }
qed

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
using ereal_open_uminus[of "- S"] ereal_uminus_complement by auto

instance ereal :: perfect_space
proof (default, rule)
fix a :: ereal assume a: "open {a}"
show False
proof (cases a)
case MInf
then obtain y where "{..<ereal y} <= {a}" using a open_MInfty2[of "{a}"] by auto
hence "ereal(y - 1):{a}" apply (subst subsetD[of "{..<ereal y}"]) by auto
then show False using `a=(-\<infinity>)` by auto
next
case PInf
then obtain y where "{ereal y<..} <= {a}" using a open_PInfty2[of "{a}"] by auto
hence "ereal(y+1):{a}" apply (subst subsetD[of "{ereal y<..}"]) by auto
then show False using `a=\<infinity>` by auto
next
case (real r) then have fin: "\<bar>a\<bar> \<noteq> \<infinity>" by simp
from ereal_open_cont_interval[OF a singletonI this] guess e . note e = this
then obtain b where b_def: "a<b & b<a+e"
using fin ereal_between ereal_dense[of a "a+e"] by auto
then have "b: {a-e <..< a+e}" using fin ereal_between[of a e] e by auto
then show False using b_def e by auto
qed
qed

lemma ereal_closed_contains_Inf:
fixes S :: "ereal set"
assumes "closed S" "S ~= {}"
shows "Inf S : S"
proof(rule ccontr)
assume "Inf S \<notin> S" hence a: "open (-S)" "Inf S:(- S)" using assms by auto
show False
proof (cases "Inf S")
case MInf hence "(-\<infinity>) : - S" using a by auto
then obtain y where "{..<ereal y} <= (-S)" using a open_MInfty2[of "- S"] by auto
hence "ereal y <= Inf S" by (metis Compl_anti_mono Compl_lessThan atLeast_iff
complete_lattice_class.Inf_greatest double_complement set_rev_mp)
then show False using MInf by auto
next
case PInf then have "S={\<infinity>}" by (metis Inf_eq_PInfty assms(2))
then show False using `Inf S ~: S` by (simp add: top_ereal_def)
next
case (real r) then have fin: "\<bar>Inf S\<bar> \<noteq> \<infinity>" by simp
from ereal_open_cont_interval[OF a this] guess e . note e = this
{ fix x assume "x:S" hence "x>=Inf S" by (rule complete_lattice_class.Inf_lower)
hence *: "x>Inf S-e" using e by (metis fin ereal_between(1) order_less_le_trans)
{ assume "x<Inf S+e" hence "x:{Inf S-e <..< Inf S+e}" using * by auto
hence False using e `x:S` by auto
} hence "x>=Inf S+e" by (metis linorder_le_less_linear)
} hence "Inf S + e <= Inf S" by (metis le_Inf_iff)
then show False using real e by (cases e) auto
qed
qed

lemma ereal_closed_contains_Sup:
fixes S :: "ereal set"
assumes "closed S" "S ~= {}"
shows "Sup S : S"
proof-
have "closed (uminus ` S)" by (metis assms(1) ereal_closed_uminus)
hence "Inf (uminus ` S) : uminus ` S" using assms ereal_closed_contains_Inf[of "uminus ` S"] by auto
hence "- Sup S : uminus ` S" using ereal_Sup_uminus_image_eq[of "uminus ` S"] by (auto simp: image_image)
thus ?thesis by (metis imageI ereal_uminus_uminus ereal_minus_minus_image)
qed

lemma ereal_open_closed_aux:
fixes S :: "ereal set"
assumes "open S" "closed S"
assumes S: "(-\<infinity>) ~: S"
shows "S = {}"
proof(rule ccontr)
assume "S ~= {}"
hence *: "(Inf S):S" by (metis assms(2) ereal_closed_contains_Inf)
{ assume "Inf S=(-\<infinity>)" hence False using * assms(3) by auto }
moreover
{ assume "Inf S=\<infinity>" hence "S={\<infinity>}" by (metis Inf_eq_PInfty `S ~= {}`)
hence 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] guess e . note e = this
then obtain b where b_def: "Inf S-e<b & b<Inf S"
using fin ereal_between[of "Inf S" e] ereal_dense[of "Inf S-e"] by auto
hence "b: {Inf S-e <..< Inf S+e}" using e fin ereal_between[of "Inf S" e]
by auto
hence "b:S" using e by auto
hence False using b_def 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 "(-\<infinity>) ~: S" hence "S={}" using lhs ereal_open_closed_aux by auto }
moreover
{ assume "(-\<infinity>) : S" hence "(- S)={}" using lhs ereal_open_closed_aux[of "-S"] by auto }
ultimately have "S = {} | S = UNIV" by auto
} thus ?thesis by auto
qed

lemma ereal_open_affinity_pos:
fixes S :: "ereal set"
assumes "open S" and m: "m \<noteq> \<infinity>" "0 < m" and t: "\<bar>t\<bar> \<noteq> \<infinity>"
shows "open ((\<lambda>x. 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 \<noteq> 0" "0 < r" and m': "m \<noteq> \<infinity>" "m \<noteq> -\<infinity>" "m \<noteq> 0" using m by auto
from `open S`[THEN ereal_openE] guess l u . note T = this
let ?f = "(\<lambda>x. m * x + t)"
show ?thesis unfolding open_ereal_def
proof (intro conjI impI exI subsetI)
have "ereal -` ?f ` S = (\<lambda>x. r * x + p) ` (ereal -` S)"
proof safe
fix x y assume "ereal y = m * x + t" "x \<in> S"
then show "y \<in> (\<lambda>x. r * x + p) ` ereal -` S"
using `r \<noteq> 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 \<noteq> 0`] by (auto simp: ac_simps)
next
assume "\<infinity> \<in> ?f`S" with `0 < r` have "\<infinity> \<in> S" by auto
fix x assume "x \<in> {ereal (r * l + p)<..}"
then have [simp]: "ereal (r * l + p) < x" by auto
show "x \<in> ?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 \<in> S" using T(2)[OF `\<infinity> \<in> S`] by auto
qed
next
assume "-\<infinity> \<in> ?f`S" with `0 < r` have "-\<infinity> \<in> S" by auto
fix x assume "x \<in> {..<ereal (r * u + p)}"
then have [simp]: "x < ereal (r * u + p)" by auto
show "x \<in> ?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 \<in> S" using T(3)[OF `-\<infinity> \<in> S`] by auto
qed
qed
qed

lemma ereal_open_affinity:
fixes S :: "ereal set"
assumes "open S" and m: "\<bar>m\<bar> \<noteq> \<infinity>" "m \<noteq> 0" and t: "\<bar>t\<bar> \<noteq> \<infinity>"
shows "open ((\<lambda>x. 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 "\<not> 0 < m" then
have "0 < -m" using `m \<noteq> 0` by (cases m) auto
then have m: "-m \<noteq> \<infinity>" "0 < -m" using `\<bar>m\<bar> \<noteq> \<infinity>`
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 \<Rightarrow> ereal"
assumes lim: "(X ---> L) net" and a: "\<bar>a\<bar> \<noteq> \<infinity>"
shows "((\<lambda>i. a * X i) ---> a * L) net"
proof cases
assume "a \<noteq> 0"
show ?thesis
proof (rule topological_tendstoI)
fix S assume "open S" "a * L \<in> S"
have "a * L / a = L"
using `a \<noteq> 0` a by (cases rule: ereal2_cases[of a L]) auto
then have L: "L \<in> ((\<lambda>x. x / a) ` S)"
using `a * L \<in> S` by (force simp: image_iff)
moreover have "open ((\<lambda>x. x / a) ` S)"
using ereal_open_affinity[OF `open S`, of "inverse a" 0] `a \<noteq> 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 \<noteq> 0` have "a * (x / a) = x"
by (cases rule: ereal2_cases[of a x]) auto }
note this[simp]
show "eventually (\<lambda>x. a * X x \<in> S) net"
by (rule eventually_mono[OF _ *]) auto
qed
qed auto

lemma ereal_lim_uminus:
fixes X :: "'a \<Rightarrow> ereal" shows "((\<lambda>i. - X i) ---> -L) net \<longleftrightarrow> (X ---> L) net"
using ereal_lim_mult[of X L net "ereal (-1)"]
ereal_lim_mult[of "(\<lambda>i. - X i)" "-L" net "ereal (-1)"]

lemma Lim_bounded2_ereal:
assumes lim:"f ----> (l :: ereal)"
and ge: "ALL n>=N. f n >= C"
shows "l>=C"
proof-
def g == "(%i. -(f i))"
{ fix n assume "n>=N" hence "g n <= -C" using assms ereal_minus_le_minus g_def by auto }
hence "ALL n>=N. g n <= -C" by auto
moreover have limg: "g ----> (-l)" using g_def ereal_lim_uminus lim by auto
ultimately have "-l <= -C" using Lim_bounded_ereal[of g "-l" _ "-C"] by auto
from this show ?thesis using ereal_minus_le_minus 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 ereal_open_mono_set:
fixes S :: "ereal set"
defines "a \<equiv> Inf S"
shows "(open S \<and> mono_set S) \<longleftrightarrow> (S = UNIV \<or> S = {a <..})"
by (metis Inf_UNIV a_def 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 => 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}"
unfolding Liminf_Sup
proof (intro arg_cong[where f="\<lambda>P. Sup (Collect P)"] ext iffI allI impI)
fix l S assume ev: "\<forall>y<l. eventually (\<lambda>x. y < f x) net" and "open S" "mono_set S" "l \<in> S"
then have "S = UNIV \<or> S = {Inf S <..}"
using ereal_open_mono_set[of S] by auto
then show "eventually (\<lambda>x. f x \<in> S) net"
proof
assume S: "S = {Inf S<..}"
then have "Inf S < l" using `l \<in> S` by auto
then have "eventually (\<lambda>x. Inf S < f x) net" using ev by auto
then show "eventually (\<lambda>x. f x \<in> S) net" by (subst S) auto
qed auto
next
fix l y assume S: "\<forall>S. open S \<longrightarrow> mono_set S \<longrightarrow> l \<in> S \<longrightarrow> eventually  (\<lambda>x. f x \<in> S) net" "y < l"
have "eventually  (\<lambda>x. f x \<in> {y <..}) net"
using `y < l` by (intro S[rule_format]) auto
then show "eventually (\<lambda>x. y < f x) net" by auto
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}"
unfolding Limsup_Inf
proof (intro arg_cong[where f="\<lambda>P. Inf (Collect P)"] ext iffI allI impI)
fix l S assume ev: "\<forall>y>l. eventually (\<lambda>x. f x < y) net" and "open S" "mono_set (uminus`S)" "l \<in> S"
then have "open (uminus`S) \<and> mono_set (uminus`S)" by (simp add: ereal_open_uminus)
then have "S = UNIV \<or> S = {..< Sup S}"
unfolding ereal_open_mono_set ereal_Inf_uminus_image_eq ereal_image_uminus_shift by simp
then show "eventually (\<lambda>x. f x \<in> S) net"
proof
assume S: "S = {..< Sup S}"
then have "l < Sup S" using `l \<in> S` by auto
then have "eventually (\<lambda>x. f x < Sup S) net" using ev by auto
then show "eventually (\<lambda>x. f x \<in> S) net"  by (subst S) auto
qed auto
next
fix l y assume S: "\<forall>S. open S \<longrightarrow> mono_set (uminus`S) \<longrightarrow> l \<in> S \<longrightarrow> eventually  (\<lambda>x. f x \<in> S) net" "l < y"
have "eventually  (\<lambda>x. f x \<in> {..< y}) net"
using `l < y` by (intro S[rule_format]) auto
then show "eventually (\<lambda>x. f x < y) net" by auto
qed

lemma open_uminus_iff: "open (uminus ` S) \<longleftrightarrow> open (S::ereal set)"
using ereal_open_uminus[of S] ereal_open_uminus[of "uminus`S"] by auto

lemma ereal_Limsup_uminus:
fixes f :: "'a => ereal"
shows "Limsup net (\<lambda>x. - (f x)) = -(Liminf net f)"
proof -
{ fix P l have "(\<exists>x. (l::ereal) = -x \<and> P x) \<longleftrightarrow> P (-l)" by (auto intro!: exI[of _ "-l"]) }
note Ex_cancel = this
{ fix P :: "ereal set \<Rightarrow> bool" have "(\<forall>S. P S) \<longleftrightarrow> (\<forall>S. P (uminus`S))"
apply auto by (erule_tac x="uminus`S" in allE) (auto simp: image_image) }
{ fix x S have "(x::ereal) \<in> uminus`S \<longleftrightarrow> -x\<in>S" by (auto intro!: image_eqI[of _ _ "-x"]) }
note remove_uminus_image = this
show ?thesis
unfolding ereal_Limsup_Inf_monoset ereal_Liminf_Sup_monoset
unfolding ereal_Inf_uminus_image_eq[symmetric] image_Collect Ex_cancel
qed

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

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

lemma lim_imp_Limsup:
fixes f :: "'a => ereal"
assumes "\<not> trivial_limit net"
assumes lim: "(f ---> f0) net"
shows "Limsup net f = f0"
using ereal_Lim_uminus[of f f0] lim_imp_Liminf[of net "(%x. -(f x))" "-f0"]
ereal_Liminf_uminus[of net f] assms by simp

lemma Liminf_PInfty:
fixes f :: "'a \<Rightarrow> ereal"
assumes "\<not> trivial_limit net"
shows "(f ---> \<infinity>) net \<longleftrightarrow> Liminf net f = \<infinity>"
proof (intro lim_imp_Liminf iffI assms)
assume rhs: "Liminf net f = \<infinity>"
{ fix S :: "ereal set" assume "open S & \<infinity> : S"
then obtain m where "{ereal m<..} <= S" using open_PInfty2 by auto
moreover have "eventually (\<lambda>x. f x \<in> {ereal m<..}) net"
using rhs unfolding Liminf_Sup top_ereal_def[symmetric] Sup_eq_top_iff
by (auto elim!: allE[where x="ereal m"] simp: top_ereal_def)
ultimately have "eventually (%x. f x : S) net" apply (subst eventually_mono) by auto
} then show "(f ---> \<infinity>) net" unfolding tendsto_def by auto
qed

lemma Limsup_MInfty:
fixes f :: "'a \<Rightarrow> ereal"
assumes "\<not> trivial_limit net"
shows "(f ---> -\<infinity>) net \<longleftrightarrow> Limsup net f = -\<infinity>"
using assms ereal_Lim_uminus[of f "-\<infinity>"] Liminf_PInfty[of _ "\<lambda>x. - (f x)"]
ereal_Liminf_uminus[of _ f] by (auto simp: ereal_uminus_eq_reorder)

lemma ereal_Liminf_eq_Limsup:
fixes f :: "'a \<Rightarrow> ereal"
assumes ntriv: "\<not> trivial_limit net"
assumes lim: "Liminf net f = f0" "Limsup net f = f0"
shows "(f ---> f0) net"
proof (cases f0)
case PInf then show ?thesis using Liminf_PInfty[OF ntriv] lim by auto
next
case MInf then show ?thesis using Limsup_MInfty[OF ntriv] lim by auto
next
case (real r)
show "(f ---> f0) net"
proof (rule topological_tendstoI)
fix S assume "open S""f0 \<in> S"
then obtain a b where "a < Liminf net f" "Limsup net f < b" "{a<..<b} \<subseteq> S"
using ereal_open_cont_interval2[of S f0] real lim by auto
then have "eventually (\<lambda>x. f x \<in> {a<..<b}) net"
unfolding Liminf_Sup Limsup_Inf less_Sup_iff Inf_less_iff
by (auto intro!: eventually_conj)
with `{a<..<b} \<subseteq> S` show "eventually (%x. f x : S) net"
by (rule_tac eventually_mono) auto
qed
qed

lemma ereal_Liminf_eq_Limsup_iff:
fixes f :: "'a \<Rightarrow> ereal"
assumes "\<not> trivial_limit net"
shows "(f ---> f0) net \<longleftrightarrow> Liminf net f = f0 \<and> Limsup net f = f0"
by (metis assms ereal_Liminf_eq_Limsup lim_imp_Liminf lim_imp_Limsup)

lemma limsup_INFI_SUPR:
fixes f :: "nat \<Rightarrow> ereal"
shows "limsup f = (INF n. SUP m:{n..}. f m)"
using ereal_Limsup_uminus[of sequentially "\<lambda>x. - f x"]
by (simp add: liminf_SUPR_INFI ereal_INFI_uminus ereal_SUPR_uminus)

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

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

lemma ereal_lim_mono:
fixes X Y :: "nat => ereal"
assumes "\<And>n. N \<le> n \<Longrightarrow> X n <= Y n"
assumes "X ----> x" "Y ----> y"
shows "x <= y"
by (metis ereal_Liminf_eq_Limsup_iff[OF trivial_limit_sequentially] assms liminf_mono)

lemma incseq_le_ereal:
fixes X :: "nat \<Rightarrow> ereal"
assumes inc: "incseq X" and lim: "X ----> L"
shows "X N \<le> 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::ereal)" shows "X N >= L"
using dec
by (intro ereal_lim_mono[of N, OF _ lim tendsto_const]) (simp add: decseq_def)

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:S"
{ assume "S = UNIV" hence "EX N. (ALL n>=N. x n : S)" by auto }
moreover
{ assume "~(S=UNIV)"
then obtain B where B_def: "S = {B<..}" using om ereal_open_mono_set by auto
hence "B<x0" using om by auto
hence "EX N. ALL n>=N. x n : S" unfolding B_def using `x0 \<le> liminf x` liminf_bounded_iff by auto
} ultimately have "EX N. (ALL n>=N. x n : S)" by auto
} then show "?P x0" by auto
qed

lemma limsup_subseq_mono:
fixes X :: "nat \<Rightarrow> ereal"
assumes "subseq r"
shows "limsup (X \<circ> r) \<le> limsup X"
proof-
have "(\<lambda>n. - X n) \<circ> r = (\<lambda>n. - (X \<circ> r) n)" by (simp add: fun_eq_iff)
then have "- limsup X \<le> - limsup (X \<circ> r)"
using liminf_subseq_mono[of r "(%n. - X n)"]
ereal_Liminf_uminus[of sequentially X]
ereal_Liminf_uminus[of sequentially "X o r"] assms by auto
then show ?thesis by auto
qed

lemma bounded_abs:
assumes "(a::real)<=x" "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 bounded_increasing_convergent2: fixes f::"nat => real"
assumes "ALL n. f n <= B"  "ALL n m. n>=m --> f n >= f m"
shows "EX l. (f ---> l) sequentially"
proof-
def N == "max (abs (f 0)) (abs B)"
{ fix n have "abs (f n) <= N" unfolding N_def apply (subst bounded_abs) using assms by auto }
hence "bounded {f n| n::nat. True}" unfolding bounded_real by auto
from this show ?thesis apply(rule Topology_Euclidean_Space.bounded_increasing_convergent)
using assms by auto
qed
lemma lim_ereal_increasing: assumes "\<And>n m. n >= m \<Longrightarrow> f n >= f m"
obtains l where "f ----> (l::ereal)"
proof(cases "f = (\<lambda>x. - \<infinity>)")
case True then show thesis using tendsto_const[of "- \<infinity>" sequentially] by (intro that[of "-\<infinity>"]) auto
next
case False
from this obtain N where N_def: "f N > (-\<infinity>)" by (auto simp: fun_eq_iff)
have "ALL n>=N. f n >= f N" using assms by auto
hence minf: "ALL n>=N. f n > (-\<infinity>)" using N_def by auto
def Y == "(%n. (if n>=N then f n else f N))"
hence incy: "!!n m. n>=m ==> Y n >= Y m" using assms by auto
from minf have minfy: "ALL n. Y n ~= (-\<infinity>)" using Y_def by auto
show thesis
proof(cases "EX B. ALL n. f n < ereal B")
case False thus thesis apply- apply(rule that[of \<infinity>]) unfolding Lim_PInfty not_ex not_all
apply safe apply(erule_tac x=B in allE,safe) apply(rule_tac x=x in exI,safe)
apply(rule order_trans[OF _ assms[rule_format]]) by auto
next case True then guess B ..
hence "ALL n. Y n < ereal B" using Y_def by auto note B = this[rule_format]
{ fix n have "Y n < \<infinity>" using B[of n] apply (subst less_le_trans) by auto
hence "Y n ~= \<infinity> & Y n ~= (-\<infinity>)" using minfy by auto
} hence *: "ALL n. \<bar>Y n\<bar> \<noteq> \<infinity>" by auto
{ fix n have "real (Y n) < B" proof- case goal1 thus ?case
using B[of n] apply-apply(subst(asm) ereal_real'[THEN sym]) defer defer
unfolding ereal_less using * by auto
qed
}
hence B': "ALL n. (real (Y n) <= B)" using less_imp_le by auto
have "EX l. (%n. real (Y n)) ----> l"
apply(rule bounded_increasing_convergent2)
proof safe show "!!n. real (Y n) <= B" using B' by auto
fix n m::nat assume "n<=m"
hence "ereal (real (Y n)) <= ereal (real (Y m))"
using incy[rule_format,of n m] apply(subst ereal_real)+
using *[rule_format, of n] *[rule_format, of m] by auto
thus "real (Y n) <= real (Y m)" by auto
qed then guess l .. note l=this
have "Y ----> ereal l" using l apply-apply(subst(asm) lim_ereal[THEN sym])
unfolding ereal_real using * by auto
thus thesis apply-apply(rule that[of "ereal l"])
apply (subst tail_same_limit[of Y _ N]) using Y_def by auto
qed
qed

lemma lim_ereal_decreasing: assumes "\<And>n m. n >= m \<Longrightarrow> f n <= f m"
obtains l where "f ----> (l::ereal)"
proof -
from lim_ereal_increasing[of "\<lambda>x. - f x"] assms
obtain l where "(\<lambda>x. - f x) ----> l" by auto
from ereal_lim_mult[OF this, of "- 1"] show thesis
by (intro that[of "-l"]) (simp add: ereal_uminus_eq_reorder)
qed

lemma compact_ereal:
fixes X :: "nat \<Rightarrow> ereal"
shows "\<exists>l r. subseq r \<and> (X \<circ> r) ----> l"
proof -
obtain r where "subseq r" and mono: "monoseq (X \<circ> r)"
using seq_monosub[of X] unfolding comp_def by auto
then have "(\<forall>n m. m \<le> n \<longrightarrow> (X \<circ> r) m \<le> (X \<circ> r) n) \<or> (\<forall>n m. m \<le> n \<longrightarrow> (X \<circ> r) n \<le> (X \<circ> r) m)"
then obtain l where "(X\<circ>r) ----> l"
using lim_ereal_increasing[of "X \<circ> r"] lim_ereal_decreasing[of "X \<circ> r"] by auto
then show ?thesis using `subseq r` by auto
qed

lemma ereal_Sup_lim:
assumes "\<And>n. b n \<in> s" "b ----> (a::ereal)"
shows "a \<le> Sup s"
by (metis Lim_bounded_ereal assms complete_lattice_class.Sup_upper)

lemma ereal_Inf_lim:
assumes "\<And>n. b n \<in> s" "b ----> (a::ereal)"
shows "Inf s \<le> a"
by (metis Lim_bounded2_ereal assms complete_lattice_class.Inf_lower)

lemma SUP_Lim_ereal:
fixes X :: "nat \<Rightarrow> ereal" assumes "incseq X" "X ----> l" shows "(SUP n. X n) = l"
proof (rule ereal_SUPI)
fix n from assms show "X n \<le> l"
by (intro incseq_le_ereal) (simp add: incseq_def)
next
fix y assume "\<And>n. n \<in> UNIV \<Longrightarrow> X n \<le> y"
with ereal_Sup_lim[OF _ `X ----> l`, of "{..y}"]
show "l \<le> y" by auto
qed

lemma LIMSEQ_ereal_SUPR:
fixes X :: "nat \<Rightarrow> ereal" assumes "incseq X" shows "X ----> (SUP n. X n)"
proof (rule lim_ereal_increasing)
fix n m :: nat assume "m \<le> n" then show "X m \<le> X n"
using `incseq X` by (simp add: incseq_def)
next
fix l assume "X ----> l"
with SUP_Lim_ereal[of X, OF assms this] show ?thesis by simp
qed

lemma INF_Lim_ereal: "decseq X \<Longrightarrow> X ----> l \<Longrightarrow> (INF n. X n) = (l::ereal)"
using SUP_Lim_ereal[of "\<lambda>i. - X i" "- l"]

lemma LIMSEQ_ereal_INFI: "decseq X \<Longrightarrow> X ----> (INF n. X n :: ereal)"
using LIMSEQ_ereal_SUPR[of "\<lambda>i. - X i"]

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

lemma Liminf_within:
fixes f :: "'a::metric_space => ereal"
shows "Liminf (at x within S) f = (SUP e:{0<..}. INF y:(S Int ball x e - {x}). f y)"
proof-
let ?l="(SUP e:{0<..}. INF y:(S Int ball x e - {x}). f y)"
{ fix T assume T_def: "open T & mono_set T & ?l:T"
have "EX d>0. ALL y:S. 0 < dist y x & dist y x < d --> f y : T"
proof-
{ assume "T=UNIV" hence ?thesis by (simp add: gt_ex) }
moreover
{ assume "~(T=UNIV)"
then obtain B where "T={B<..}" using T_def ereal_open_mono_set[of T] by auto
hence "B<?l" using T_def by auto
then obtain d where d_def: "0<d & B<(INF y:(S Int ball x d - {x}). f y)"
unfolding less_SUP_iff by auto
{ fix y assume "y:S & 0 < dist y x & dist y x < d"
hence "y:(S Int ball x d - {x})" unfolding ball_def by (auto simp add: dist_commute)
hence "f y:T" using d_def INF_lower[of y "S Int ball x d - {x}" f] `T={B<..}` by auto
} hence ?thesis apply(rule_tac x="d" in exI) using d_def by auto
} ultimately show ?thesis by auto
qed
}
moreover
{ fix z
assume a: "ALL T. open T --> mono_set T --> z : T -->
(EX d>0. ALL y:S. 0 < dist y x & dist y x < d --> f y : T)"
{ fix B assume "B<z"
then obtain d where d_def: "d>0 & (ALL y:S. 0 < dist y x & dist y x < d --> B < f y)"
using a[rule_format, of "{B<..}"] mono_greaterThan by auto
{ fix y assume "y:(S Int ball x d - {x})"
hence "y:S & 0 < dist y x & dist y x < d" unfolding ball_def apply (simp add: dist_commute)
by (metis dist_eq_0_iff less_le zero_le_dist)
hence "B <= f y" using d_def by auto
} hence "B <= INFI (S Int ball x d - {x}) f" apply (subst INF_greatest) by auto
also have "...<=?l" apply (subst SUP_upper) using d_def by auto
finally have "B<=?l" by auto
} hence "z <= ?l" using ereal_le_ereal[of z "?l"] by auto
}
ultimately show ?thesis unfolding ereal_Liminf_Sup_monoset eventually_within
apply (subst ereal_SupI[of _ "(SUP e:{0<..}. INFI (S Int ball x e - {x}) f)"]) by auto
qed

lemma Limsup_within:
fixes f :: "'a::metric_space => ereal"
shows "Limsup (at x within S) f = (INF e:{0<..}. SUP y:(S Int ball x e - {x}). f y)"
proof-
let ?l="(INF e:{0<..}. SUP y:(S Int ball x e - {x}). f y)"
{ fix T assume T_def: "open T & mono_set (uminus ` T) & ?l:T"
have "EX d>0. ALL y:S. 0 < dist y x & dist y x < d --> f y : T"
proof-
{ assume "T=UNIV" hence ?thesis by (simp add: gt_ex) }
moreover
{ assume "~(T=UNIV)" hence "~(uminus ` T = UNIV)"
by (metis Int_UNIV_right Int_absorb1 image_mono ereal_minus_minus_image subset_UNIV)
hence "uminus ` T = {Inf (uminus ` T)<..}" using T_def ereal_open_mono_set[of "uminus ` T"]
ereal_open_uminus[of T] by auto
then obtain B where "T={..<B}"
unfolding ereal_Inf_uminus_image_eq ereal_uminus_lessThan[symmetric]
unfolding inj_image_eq_iff[OF ereal_inj_on_uminus] by simp
hence "?l<B" using T_def by auto
then obtain d where d_def: "0<d & (SUP y:(S Int ball x d - {x}). f y)<B"
unfolding INF_less_iff by auto
{ fix y assume "y:S & 0 < dist y x & dist y x < d"
hence "y:(S Int ball x d - {x})" unfolding ball_def by (auto simp add: dist_commute)
hence "f y:T" using d_def SUP_upper[of y "S Int ball x d - {x}" f] `T={..<B}` by auto
} hence ?thesis apply(rule_tac x="d" in exI) using d_def by auto
} ultimately show ?thesis by auto
qed
}
moreover
{ fix z
assume a: "ALL T. open T --> mono_set (uminus ` T) --> z : T -->
(EX d>0. ALL y:S. 0 < dist y x & dist y x < d --> f y : T)"
{ fix B assume "z<B"
then obtain d where d_def: "d>0 & (ALL y:S. 0 < dist y x & dist y x < d --> f y<B)"
using a[rule_format, of "{..<B}"] by auto
{ fix y assume "y:(S Int ball x d - {x})"
hence "y:S & 0 < dist y x & dist y x < d" unfolding ball_def apply (simp add: dist_commute)
by (metis dist_eq_0_iff less_le zero_le_dist)
hence "f y <= B" using d_def by auto
} hence "SUPR (S Int ball x d - {x}) f <= B" apply (subst SUP_least) by auto
moreover have "?l<=SUPR (S Int ball x d - {x}) f" apply (subst INF_lower) using d_def by auto
ultimately have "?l<=B" by auto
} hence "?l <= z" using ereal_ge_ereal[of z "?l"] by auto
}
ultimately show ?thesis unfolding ereal_Limsup_Inf_monoset eventually_within
apply (subst ereal_InfI) by auto
qed

lemma Liminf_within_UNIV:
fixes f :: "'a::metric_space => ereal"
shows "Liminf (at x) f = Liminf (at x within UNIV) f"
by simp (* TODO: delete *)

lemma Liminf_at:
fixes f :: "'a::metric_space => ereal"
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_within_UNIV:
fixes f :: "'a::metric_space => ereal"
shows "Limsup (at x) f = Limsup (at x within UNIV) f"
by simp (* TODO: delete *)

lemma Limsup_at:
fixes f :: "'a::metric_space => ereal"
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 Lim_within_constant:
assumes "ALL y:S. f y = C"
shows "(f ---> C) (at x within S)"
unfolding tendsto_def Limits.eventually_within eventually_at_topological
using assms by simp (metis open_UNIV UNIV_I)

lemma Liminf_within_constant:
fixes f :: "'a::topological_space \<Rightarrow> ereal"
assumes "ALL y:S. f y = C"
assumes "~trivial_limit (at x within S)"
shows "Liminf (at x within S) f = C"
by (metis Lim_within_constant assms lim_imp_Liminf)

lemma Limsup_within_constant:
fixes f :: "'a::topological_space \<Rightarrow> ereal"
assumes "ALL y:S. f y = C"
assumes "~trivial_limit (at x within S)"
shows "Limsup (at x within S) f = C"
by (metis Lim_within_constant assms lim_imp_Limsup)

lemma islimpt_punctured:
"x islimpt S = x islimpt (S-{x})"
unfolding islimpt_def by blast

lemma islimpt_in_closure:
"(x islimpt S) = (x:closure(S-{x}))"
unfolding closure_def using islimpt_punctured by blast

lemma not_trivial_limit_within:
"~trivial_limit (at x within S) = (x:closure(S-{x}))"
using islimpt_in_closure by (metis trivial_limit_within)

lemma not_trivial_limit_within_ball:
"(~trivial_limit (at x within S)) = (ALL e>0. S Int ball x e - {x} ~= {})"
(is "?lhs = ?rhs")
proof-
{ assume "?lhs"
{ fix e :: real assume "e>0"
then obtain y where "y:(S-{x}) & dist y x < e"
using `?lhs` not_trivial_limit_within[of x S] closure_approachable[of x "S - {x}"] by auto
hence "y : (S Int ball x e - {x})" unfolding ball_def by (simp add: dist_commute)
hence "S Int ball x e - {x} ~= {}" by blast
} hence "?rhs" by auto
}
moreover
{ assume "?rhs"
{ fix e :: real assume "e>0"
then obtain y where "y : (S Int ball x e - {x})" using `?rhs` by blast
hence "y:(S-{x}) & dist y x < e" unfolding ball_def by (simp add: dist_commute)
hence "EX y:(S-{x}). dist y x < e" by auto
} hence "?lhs" using not_trivial_limit_within[of x S] closure_approachable[of x "S - {x}"] by auto
} ultimately show ?thesis by auto
qed

lemma liminf_ereal_cminus:
fixes f :: "nat \<Rightarrow> ereal" assumes "c \<noteq> -\<infinity>"
shows "liminf (\<lambda>x. 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 \<noteq> -\<infinity>`, simp)

subsubsection {* Continuity *}

lemma continuous_imp_tendsto:
assumes "continuous (at x0) f"
assumes "x ----> x0"
shows "(f o x) ----> (f x0)"
proof-
{ fix S assume "open S & (f x0):S"
from this obtain T where T_def: "open T & x0 : T & (ALL x:T. f x : S)"
using assms continuous_at_open by metis
hence "(EX N. ALL n>=N. x n : T)" using assms tendsto_explicit T_def by auto
hence "(EX N. ALL n>=N. f(x n) : S)" using T_def by auto
} from this show ?thesis using tendsto_explicit[of "f o x" "f x0"] by auto
qed

lemma continuous_at_sequentially2:
fixes f :: "'a::metric_space => 'b:: topological_space"
shows "continuous (at x0) f <-> (ALL x. (x ----> x0) --> (f o x) ----> (f x0))"
proof-
{ assume "~(continuous (at x0) f)"
from this obtain T where T_def:
"open T & f x0 : T & (ALL S. (open S & x0 : S) --> (EX x':S. f x' ~: T))"
using continuous_at_open[of x0 f] by metis
def X == "{x'. f x' ~: T}" hence "x0 islimpt X" unfolding islimpt_def using T_def by auto
from this obtain x where x_def: "(ALL n. x n : X) & x ----> x0"
using islimpt_sequential[of x0 X] by auto
hence "~(f o x) ----> (f x0)" unfolding tendsto_explicit using X_def T_def by auto
hence "EX x. x ----> x0 & (~(f o x) ----> (f x0))" using x_def by auto
}
from this show ?thesis using continuous_imp_tendsto by auto
qed

lemma continuous_at_of_ereal:
fixes x0 :: ereal
assumes "\<bar>x0\<bar> \<noteq> \<infinity>"
shows "continuous (at x0) real"
proof-
{ fix T assume T_def: "open T & real x0 : T"
def S == "ereal ` T"
hence "ereal (real x0) : S" using T_def by auto
hence "x0 : S" using assms ereal_real by auto
moreover have "open S" using open_ereal S_def T_def by auto
moreover have "ALL y:S. real y : T" using S_def T_def by auto
ultimately have "EX S. x0 : S & open S & (ALL y:S. real y : T)" by auto
} from this 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" hence "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)"
hence "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"
fixes A 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)

lemma continuous_on_real: "continuous_on (UNIV-{\<infinity>,(-\<infinity>::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 "\<And>x. x \<in> A \<Longrightarrow> \<bar>f x\<bar> \<noteq> \<infinity>"
shows "continuous_on A f \<longleftrightarrow> continuous_on A (real \<circ> f)"
proof-
have "f ` A <= UNIV-{\<infinity>,(-\<infinity>)}" using assms by force
hence *: "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" hence "continuous_on A (real o f)"
apply (subst continuous_on_compose) using * by auto
}
moreover
{ assume "continuous_on A (real o f)"
hence "continuous_on A (ereal o (real o f))"
apply (subst continuous_on_compose) using ** by auto
hence "continuous_on A f"
apply (subst continuous_on_eq[of A "ereal o (real o f)" f])
using assms ereal_real by auto
}
ultimately show ?thesis by auto
qed

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

lemma closure_contains_Inf:
fixes S :: "real set"
assumes "S ~= {}" "EX B. ALL x:S. B<=x"
shows "Inf S : closure S"
proof-
have *: "ALL x:S. Inf S <= x" using Inf_lower_EX[of _ S] assms by metis
{ fix e assume "e>(0 :: real)"
from this obtain x where x_def: "x:S & x < Inf S + e" using Inf_close `S ~= {}` by auto
moreover hence "x > Inf S - e" using * by auto
ultimately have "abs (x - Inf S) < e" by (simp add: abs_diff_less_iff)
hence "EX x:S. abs (x - Inf S) < e" using x_def by auto
} from this show ?thesis apply (subst closure_approachable) unfolding dist_norm by auto
qed

lemma closed_contains_Inf:
fixes S :: "real set"
assumes "S ~= {}" "EX B. ALL x:S. B<=x"
assumes "closed S"
shows "Inf S : S"
by (metis closure_contains_Inf closure_closed assms)

lemma mono_closed_real:
fixes S :: "real set"
assumes mono: "ALL y z. y:S & y<=z --> z:S"
assumes "closed S"
shows "S = {} | S = UNIV | (EX a. S = {a ..})"
proof-
{ assume "S ~= {}"
{ assume ex: "EX B. ALL x:S. B<=x"
hence *: "ALL x:S. Inf S <= x" using Inf_lower_EX[of _ S] ex by metis
hence "Inf S : S" apply (subst closed_contains_Inf) using ex `S ~= {}` `closed S` by auto
hence "ALL x. (Inf S <= x <-> x:S)" using mono[rule_format, of "Inf S"] * by auto
hence "S = {Inf S ..}" by auto
hence "EX a. S = {a ..}" by auto
}
moreover
{ assume "~(EX B. ALL x:S. B<=x)"
hence nex: "ALL B. EX x:S. x<B" by (simp add: not_le)
{ fix y obtain x where "x:S & x < y" using nex by auto
hence "y:S" using mono[rule_format, of x y] by auto
} hence "S = UNIV" by auto
} ultimately have "S = UNIV | (EX a. S = {a ..})" by blast
} from this show ?thesis by blast
qed

lemma mono_closed_ereal:
fixes S :: "real set"
assumes mono: "ALL y z. y:S & y<=z --> z:S"
assumes "closed S"
shows "EX a. S = {x. a <= ereal x}"
proof-
{ assume "S = {}" hence ?thesis apply(rule_tac x=PInfty in exI) by auto }
moreover
{ assume "S = UNIV" hence ?thesis apply(rule_tac x="-\<infinity>" in exI) by auto }
moreover
{ assume "EX a. S = {a ..}"
from this obtain a where "S={a ..}" by auto
hence ?thesis apply(rule_tac x="ereal a" in exI) by auto
} ultimately show ?thesis using mono_closed_real[of S] assms by auto
qed

subsection {* Sums *}

lemma setsum_ereal[simp]:
"(\<Sum>x\<in>A. ereal (f x)) = ereal (\<Sum>x\<in>A. f x)"
proof cases
assume "finite A" then show ?thesis by induct auto
qed simp

lemma setsum_Pinfty:
fixes f :: "'a \<Rightarrow> ereal"
shows "(\<Sum>x\<in>P. f x) = \<infinity> \<longleftrightarrow> (finite P \<and> (\<exists>i\<in>P. f i = \<infinity>))"
proof safe
assume *: "setsum f P = \<infinity>"
show "finite P"
proof (rule ccontr) assume "infinite P" with * show False by auto qed
show "\<exists>i\<in>P. f i = \<infinity>"
proof (rule ccontr)
assume "\<not> ?thesis" then have "\<And>i. i \<in> P \<Longrightarrow> f i \<noteq> \<infinity>" by auto
from `finite P` this have "setsum f P \<noteq> \<infinity>"
by induct auto
with * show False by auto
qed
next
fix i assume "finite P" "i \<in> P" "f i = \<infinity>"
thus "setsum f P = \<infinity>"
proof induct
case (insert x A)
show ?case using insert by (cases "x = i") auto
qed simp
qed

lemma setsum_Inf:
fixes f :: "'a \<Rightarrow> ereal"
shows "\<bar>setsum f A\<bar> = \<infinity> \<longleftrightarrow> (finite A \<and> (\<exists>i\<in>A. \<bar>f i\<bar> = \<infinity>))"
proof
assume *: "\<bar>setsum f A\<bar> = \<infinity>"
have "finite A" by (rule ccontr) (insert *, auto)
moreover have "\<exists>i\<in>A. \<bar>f i\<bar> = \<infinity>"
proof (rule ccontr)
assume "\<not> ?thesis" then have "\<forall>i\<in>A. \<exists>r. f i = ereal r" by auto
from bchoice[OF this] guess r ..
with * show False by auto
qed
ultimately show "finite A \<and> (\<exists>i\<in>A. \<bar>f i\<bar> = \<infinity>)" by auto
next
assume "finite A \<and> (\<exists>i\<in>A. \<bar>f i\<bar> = \<infinity>)"
then obtain i where "finite A" "i \<in> A" "\<bar>f i\<bar> = \<infinity>" by auto
then show "\<bar>setsum f A\<bar> = \<infinity>"
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 \<Rightarrow> ereal"
assumes "\<And>x. x \<in> S \<Longrightarrow> \<bar>f x\<bar> \<noteq> \<infinity>"
shows "(\<Sum>x\<in>S. real (f x)) = real (setsum f S)"
proof -
have "\<forall>x\<in>S. \<exists>r. f x = ereal r"
proof
fix x assume "x \<in> S"
from assms[OF this] show "\<exists>r. f x = ereal r" by (cases "f x") auto
qed
from bchoice[OF this] guess r ..
then show ?thesis by simp
qed

lemma setsum_ereal_0:
fixes f :: "'a \<Rightarrow> ereal" assumes "finite A" "\<And>i. i \<in> A \<Longrightarrow> 0 \<le> f i"
shows "(\<Sum>x\<in>A. f x) = 0 \<longleftrightarrow> (\<forall>i\<in>A. f i = 0)"
proof
assume *: "(\<Sum>x\<in>A. f x) = 0"
then have "(\<Sum>x\<in>A. f x) \<noteq> \<infinity>" by auto
then have "\<forall>i\<in>A. \<bar>f i\<bar> \<noteq> \<infinity>" using assms by (force simp: setsum_Pinfty)
then have "\<forall>i\<in>A. \<exists>r. f i = ereal r" by auto
from bchoice[OF this] * assms show "\<forall>i\<in>A. f i = 0"
using setsum_nonneg_eq_0_iff[of A "\<lambda>i. real (f i)"] by auto
qed (rule setsum_0')

lemma setsum_ereal_right_distrib:
fixes f :: "'a \<Rightarrow> ereal" assumes "\<And>i. i \<in> A \<Longrightarrow> 0 \<le> f i"
shows "r * setsum f A = (\<Sum>n\<in>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 \<Rightarrow> ereal" assumes "\<And>i. 0 \<le> f i" shows "f sums (SUP n. \<Sum>i<n. f i)"
proof -
have "incseq (\<lambda>i. \<Sum>j=0..<i. f j)"
using ereal_add_mono[OF _ assms] by (auto intro!: incseq_SucI)
from LIMSEQ_ereal_SUPR[OF this]
show ?thesis unfolding sums_def by (simp add: atLeast0LessThan)
qed

lemma summable_ereal_pos:
fixes f :: "nat \<Rightarrow> ereal" assumes "\<And>i. 0 \<le> 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 \<Rightarrow> ereal" assumes "\<And>i. 0 \<le> f i"
shows "(\<Sum>x. f x) = (SUP n. \<Sum>i<n. f i)"
using sums_ereal_positive[of f, OF assms, THEN sums_unique] by simp

lemma sums_ereal:
"(\<lambda>x. ereal (f x)) sums ereal x \<longleftrightarrow> f sums x"
unfolding sums_def by simp

lemma suminf_bound:
fixes f :: "nat \<Rightarrow> ereal"
assumes "\<forall>N. (\<Sum>n<N. f n) \<le> x" and pos: "\<And>n. 0 \<le> f n"
shows "suminf f \<le> x"
proof (rule Lim_bounded_ereal)
have "summable f" using pos[THEN summable_ereal_pos] .
then show "(\<lambda>N. \<Sum>n<N. f n) ----> suminf f"
by (auto dest!: summable_sums simp: sums_def atLeast0LessThan)
show "\<forall>n\<ge>0. setsum f {..<n} \<le> x"
using assms by auto
qed

fixes f :: "nat \<Rightarrow> ereal"
assumes "\<forall>N. (\<Sum>n<N. f n) + y \<le> x" and pos: "\<And>n. 0 \<le> f n" and "y \<noteq> -\<infinity>"
shows "suminf f + y \<le> x"
proof (cases y)
case (real r) then have "\<forall>N. (\<Sum>n<N. f n) \<le> x - y"
using assms by (simp add: ereal_le_minus)
then have "(\<Sum> n. f n) \<le> x - y" using pos by (rule suminf_bound)
then show "(\<Sum> n. f n) + y \<le> x"
using assms real by (simp add: ereal_le_minus)
qed (insert assms, auto)

lemma sums_finite:
assumes "\<forall>N\<ge>n. f N = 0"
shows "f sums (\<Sum>N<n. f N)"
proof -
{ fix i have "(\<Sum>N<i + n. f N) = (\<Sum>N<n. f N)"
by (induct i) (insert assms, auto) }
note this[simp]
show ?thesis unfolding sums_def
by (rule LIMSEQ_offset[of _ n]) (auto simp add: atLeast0LessThan intro: tendsto_const)
qed

lemma suminf_finite:
fixes f :: "nat \<Rightarrow> 'a::{comm_monoid_add,t2_space}" assumes "\<forall>N\<ge>n. f N = 0"
shows "suminf f = (\<Sum>N<n. f N)"
using sums_finite[OF assms, THEN sums_unique] by simp

lemma suminf_upper:
fixes f :: "nat \<Rightarrow> ereal" assumes "\<And>n. 0 \<le> f n"
shows "(\<Sum>n<N. f n) \<le> (\<Sum>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 \<Rightarrow> ereal" assumes "\<And>n. 0 \<le> f n"
shows "0 \<le> (\<Sum>n. f n)"
using suminf_upper[of f 0, OF assms] by simp

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

lemma suminf_half_series_ereal: "(\<Sum>n. (1/2 :: ereal)^Suc n) = 1"
using sums_ereal[THEN iffD2, OF power_half_series, THEN sums_unique, symmetric]

fixes f g :: "nat \<Rightarrow> ereal"
assumes "\<And>i. 0 \<le> f i" "\<And>i. 0 \<le> g i"
shows "(\<Sum>i. f i + g i) = suminf f + suminf g"
apply (subst (1 2 3) suminf_ereal_eq_SUPR)

lemma suminf_cmult_ereal:
fixes f g :: "nat \<Rightarrow> ereal"
assumes "\<And>i. 0 \<le> f i" "0 \<le> a"
shows "(\<Sum>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 \<Rightarrow> ereal"
assumes "\<And>i. 0 \<le> f i" "suminf f \<noteq> \<infinity>"
shows "f i \<noteq> \<infinity>"
proof -
from suminf_upper[of f "Suc i", OF assms(1)] assms(2)
have "(\<Sum>i<Suc i. f i) \<noteq> \<infinity>" by auto
then show ?thesis
unfolding setsum_Pinfty by simp
qed

lemma suminf_PInfty_fun:
assumes "\<And>i. 0 \<le> f i" "suminf f \<noteq> \<infinity>"
shows "\<exists>f'. f = (\<lambda>x. ereal (f' x))"
proof -
have "\<forall>i. \<exists>r. f i = ereal r"
proof
fix i show "\<exists>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 "\<And>i. 0 \<le> f i" "(\<Sum>i. ereal (f i)) \<noteq> \<infinity>"
shows "summable f"
proof -
have "0 \<le> (\<Sum>i. ereal (f i))"
using assms by (intro suminf_0_le) auto
with assms obtain r where r: "(\<Sum>i. ereal (f i)) = ereal r"
by (cases "\<Sum>i. ereal (f i)") auto
from summable_ereal_pos[of "\<lambda>x. ereal (f x)"]
have "summable (\<lambda>x. ereal (f x))" using assms by auto
from summable_sums[OF this]
have "(\<lambda>x. ereal (f x)) sums (\<Sum>x. ereal (f x))" by auto
then show "summable f"
unfolding r sums_ereal summable_def ..
qed

lemma suminf_ereal:
assumes "\<And>i. 0 \<le> f i" "(\<Sum>i. ereal (f i)) \<noteq> \<infinity>"
shows "(\<Sum>i. ereal (f i)) = ereal (suminf f)"
proof (rule sums_unique[symmetric])
from summable_ereal[OF assms]
show "(\<lambda>x. 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 \<Rightarrow> ereal"
assumes ord: "\<And>i. g i \<le> f i" "\<And>i. 0 \<le> g i" and fin: "suminf f \<noteq> \<infinity>" "suminf g \<noteq> \<infinity>"
shows "(\<Sum>i. f i - g i) = suminf f - suminf g"
proof -
{ fix i have "0 \<le> f i" using ord[of i] by auto }
moreover
from suminf_PInfty_fun[OF `\<And>i. 0 \<le> f i` fin(1)] guess f' .. note this[simp]
from suminf_PInfty_fun[OF `\<And>i. 0 \<le> g i` fin(2)] guess g' .. note this[simp]
{ fix i have "0 \<le> f i - g i" using ord[of i] by (auto simp: ereal_le_minus_iff) }
moreover
have "suminf (\<lambda>i. f i - g i) \<le> suminf f"
using assms by (auto intro!: suminf_le_pos simp: field_simps)
then have "suminf (\<lambda>i. f i - g i) \<noteq> \<infinity>" using fin by auto
ultimately show ?thesis using assms `\<And>i. 0 \<le> f i`
apply simp
by (subst (1 2 3) suminf_ereal)
(auto intro!: suminf_diff[symmetric] summable_ereal)
qed

lemma suminf_ereal_PInf[simp]:
"(\<Sum>x. \<infinity>::ereal) = \<infinity>"
proof -
have "(\<Sum>i<Suc 0. \<infinity>) \<le> (\<Sum>x. \<infinity>::ereal)" by (rule suminf_upper) auto
then show ?thesis by simp
qed

lemma summable_real_of_ereal:
fixes f :: "nat \<Rightarrow> ereal"
assumes f: "\<And>i. 0 \<le> f i" and fin: "(\<Sum>i. f i) \<noteq> \<infinity>"
shows "summable (\<lambda>i. real (f i))"
proof (rule summable_def[THEN iffD2])
have "0 \<le> (\<Sum>i. f i)" using assms by (auto intro: suminf_0_le)
with fin obtain r where r: "ereal r = (\<Sum>i. f i)" by (cases "(\<Sum>i. f i)") auto
{ fix i have "f i \<noteq> \<infinity>" using f by (intro suminf_PInfty[OF _ fin]) auto
then have "\<bar>f i\<bar> \<noteq> \<infinity>" using f[of i] by auto }
note fin = this
have "(\<lambda>i. ereal (real (f i))) sums (\<Sum>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 "\<dots> = ereal r" using fin r by (auto simp: ereal_real)
finally show "\<exists>r. (\<lambda>i. real (f i)) sums r" by (auto simp: sums_ereal)
qed

lemma suminf_SUP_eq:
fixes f :: "nat \<Rightarrow> nat \<Rightarrow> ereal"
assumes "\<And>i. incseq (\<lambda>n. f n i)" "\<And>n i. 0 \<le> f n i"
shows "(\<Sum>i. SUP n. f n i) = (SUP n. \<Sum>i. f n i)"
proof -
{ fix n :: nat
have "(\<Sum>i<n. SUP k. f k i) = (SUP k. \<Sum>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) ..
qed

end
```