(* 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
then have "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
then have "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"
then have a: "open (-S)" "Inf S:(- S)" using assms by auto
show False
proof (cases "Inf S")
case MInf
then have "(-\<infinity>) : - S" using a by auto
then obtain y where "{..<ereal y} <= (-S)" using a open_MInfty2[of "- S"] by auto
then have "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" then have "x>=Inf S" by (rule complete_lattice_class.Inf_lower)
then have *: "x>Inf S-e" using e by (metis fin ereal_between(1) order_less_le_trans)
{ assume "x<Inf S+e"
then have "x:{Inf S-e <..< Inf S+e}" using * by auto
then have False using e `x:S` by auto
} then have "x>=Inf S+e" by (metis linorder_le_less_linear)
}
then have "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)
then have "Inf (uminus ` S) : uminus ` S"
using assms ereal_closed_contains_Inf[of "uminus ` S"] by auto
then have "- Sup S : uminus ` S"
using ereal_Sup_uminus_image_eq[of "uminus ` S"] by (auto simp: image_image)
then show ?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"
and S: "(-\<infinity>) ~: S"
shows "S = {}"
proof (rule ccontr)
assume "S ~= {}"
then have *: "(Inf S):S" by (metis assms(2) ereal_closed_contains_Inf)
{ assume "Inf S=(-\<infinity>)"
then have False using * assms(3) by auto }
moreover
{ assume "Inf S=\<infinity>"
then have "S={\<infinity>}" by (metis Inf_eq_PInfty `S ~= {}`)
then have False by (metis assms(1) not_open_singleton) }
moreover
{ assume fin: "\<bar>Inf S\<bar> \<noteq> \<infinity>"
from ereal_open_cont_interval[OF assms(1) * fin] 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
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_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"
then have "S={}" using lhs ereal_open_closed_aux by auto }
moreover
{ assume "(-\<infinity>) : 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 \<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)"]
by (auto simp add: algebra_simps)
lemma Lim_bounded2_ereal:
assumes lim:"f ----> (l :: ereal)"
and ge: "ALL n>=N. f n >= C"
shows "l>=C"
using ge
by (intro tendsto_le[OF trivial_limit_sequentially lim tendsto_const])
(auto simp: eventually_sequentially)
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"
shows "(open S \<and> mono_set S) \<longleftrightarrow> (S = UNIV \<or> S = {Inf S <..})"
by (metis Inf_UNIV atLeast_eq_UNIV_iff ereal_open_atLeast
ereal_open_closed mono_set_iff open_ereal_greaterThan)
lemma ereal_closed_mono_set:
fixes S :: "ereal set"
shows "(closed S \<and> mono_set S) \<longleftrightarrow> (S = {} \<or> S = {Inf S ..})"
by (metis Inf_UNIV atLeast_eq_UNIV_iff closed_ereal_atLeast
ereal_open_closed mono_empty mono_set_iff open_ereal_greaterThan)
lemma ereal_Liminf_Sup_monoset:
fixes f :: "'a => 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" "INFI (Collect P) f \<in> S"
{ fix x assume "P x"
then have "INFI (Collect P) f \<le> f x"
by (intro complete_lattice_class.INF_lower) simp
with S have "f x \<in> S"
by (simp add: mono_set) }
with P show "eventually (\<lambda>x. f x \<in> S) net"
by (auto elim: eventually_elim1)
next
fix y l
assume S: "\<forall>S. open S \<longrightarrow> mono_set S \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) net"
assume P: "\<forall>P. eventually P net \<longrightarrow> INFI (Collect P) f \<le> y"
show "l \<le> y"
proof (rule ereal_le_ereal)
fix B assume "B < l"
then have "eventually (\<lambda>x. f x \<in> {B <..}) net"
by (intro S[rule_format]) auto
then have "INFI {x. B < f x} f \<le> y"
using P by auto
moreover have "B \<le> INFI {x. B < f x} f"
by (intro INF_greatest) auto
ultimately show "B \<le> y"
by simp
qed
qed
lemma ereal_Limsup_Inf_monoset:
fixes f :: "'a => 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)" "SUPR (Collect P) f \<in> S"
{ fix x assume "P x"
then have "f x \<le> 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 \<in> S"
by (simp add: inj_image_mem_iff) }
with P show "eventually (\<lambda>x. f x \<in> S) net"
by (auto elim: eventually_elim1)
next
fix y l
assume S: "\<forall>S. open S \<longrightarrow> mono_set (uminus ` S) \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) net"
assume P: "\<forall>P. eventually P net \<longrightarrow> y \<le> SUPR (Collect P) f"
show "y \<le> l"
proof (rule ereal_ge_ereal, safe)
fix B assume "l < B"
then have "eventually (\<lambda>x. f x \<in> {..< B}) net"
by (intro S[rule_format]) auto
then have "y \<le> SUPR {x. f x < B} f"
using P by auto
moreover have "SUPR {x. f x < B} f \<le> B"
by (intro SUP_least) auto
ultimately show "y \<le> B"
by simp
qed
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
apply (erule_tac x="uminus`S" in allE)
apply (auto simp: image_image)
done }
note add_uminus_image = this
{ 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
by (subst add_uminus_image) (simp add: open_uminus_iff remove_uminus_image)
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"
and 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 convergent_ereal_limsup:
fixes X :: "nat \<Rightarrow> ereal"
shows "convergent X \<Longrightarrow> limsup X = lim X"
by (auto simp: convergent_def limI lim_imp_Limsup)
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>"
show "(f ---> \<infinity>) net"
unfolding tendsto_PInfty
proof
fix r :: real
have "ereal r < top" unfolding top_ereal_def by simp
with rhs obtain P where "eventually P net" "r < INFI (Collect P) f"
unfolding Liminf_def SUP_eq_top_iff top_ereal_def[symmetric] by auto
then show "eventually (\<lambda>x. ereal r < f x) net"
by (auto elim!: eventually_elim1 dest: less_INF_D)
qed
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"
and 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_def Limsup_def less_SUP_iff INF_less_iff
by (auto intro!: eventually_conj elim: eventually_elim1 dest: less_INF_D SUP_lessD)
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 convergent_ereal:
fixes X :: "nat \<Rightarrow> ereal"
shows "convergent X \<longleftrightarrow> limsup X = liminf X"
using ereal_Liminf_eq_Limsup_iff[of sequentially]
by (auto simp: convergent_def)
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"
and "X ----> x" "Y ----> y"
shows "x <= y"
using assms(1) by (intro LIMSEQ_le[OF assms(2,3)]) auto
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" then have "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
then have "B<x0" using om by auto
then have "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 lim_ereal_increasing:
assumes "\<And>n m. n \<ge> m \<Longrightarrow> f n \<ge> 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_ereal_decreasing:
assumes "\<And>n m. n \<ge> m \<Longrightarrow> f n \<le> 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_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)"
by (auto simp add: monoseq_def)
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> '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: "decseq X \<Longrightarrow> X ----> l \<Longrightarrow> (INF n. X n) = (l::ereal)"
using SUP_Lim_ereal[of "\<lambda>i. - X i" "- l"]
by (simp add: ereal_SUPR_uminus ereal_lim_uminus)
lemma LIMSEQ_ereal_INFI: "decseq X \<Longrightarrow> X ----> (INF n. X n :: ereal)"
using LIMSEQ_SUP[of "\<lambda>i. - X i"]
by (simp add: ereal_SUPR_uminus ereal_lim_uminus)
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_SUP[OF inc]
show "f ----> x" by auto }
qed
lemma Liminf_within:
fixes f :: "'a::metric_space \<Rightarrow> 'b::complete_lattice"
shows "Liminf (at x within S) f = (SUP e:{0<..}. INF y:(S \<inter> ball x e - {x}). f y)"
unfolding Liminf_def eventually_within
proof (rule SUPR_eq, simp_all add: Ball_def Bex_def, safe)
fix P d assume "0 < d" "\<forall>y. y \<in> S \<longrightarrow> 0 < dist y x \<and> dist y x < d \<longrightarrow> P y"
then have "S \<inter> ball x d - {x} \<subseteq> {x. P x}"
by (auto simp: zero_less_dist_iff dist_commute)
then show "\<exists>r>0. INFI (Collect P) f \<le> INFI (S \<inter> 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 "\<exists>P. (\<exists>d>0. \<forall>xa. xa \<in> S \<longrightarrow> 0 < dist xa x \<and> dist xa x < d \<longrightarrow> P xa) \<and>
INFI (S \<inter> ball x d - {x}) f \<le> INFI (Collect P) f"
by (intro exI[of _ "\<lambda>y. y \<in> S \<inter> ball x d - {x}"])
(auto intro!: INF_mono exI[of _ d] simp: dist_commute)
qed
lemma Limsup_within:
fixes f :: "'a::metric_space => 'b::complete_lattice"
shows "Limsup (at x within S) f = (INF e:{0<..}. SUP y:(S \<inter> ball x e - {x}). f y)"
unfolding Limsup_def eventually_within
proof (rule INFI_eq, simp_all add: Ball_def Bex_def, safe)
fix P d assume "0 < d" "\<forall>y. y \<in> S \<longrightarrow> 0 < dist y x \<and> dist y x < d \<longrightarrow> P y"
then have "S \<inter> ball x d - {x} \<subseteq> {x. P x}"
by (auto simp: zero_less_dist_iff dist_commute)
then show "\<exists>r>0. SUPR (S \<inter> ball x r - {x}) f \<le> SUPR (Collect P) f"
by (intro exI[of _ d] SUP_mono conjI `0 < d`) auto
next
fix d :: real assume "0 < d"
then show "\<exists>P. (\<exists>d>0. \<forall>xa. xa \<in> S \<longrightarrow> 0 < dist xa x \<and> dist xa x < d \<longrightarrow> P xa) \<and>
SUPR (Collect P) f \<le> SUPR (S \<inter> ball x d - {x}) f"
by (intro exI[of _ "\<lambda>y. y \<in> S \<inter> ball x d - {x}"])
(auto intro!: SUP_mono exI[of _ d] simp: dist_commute)
qed
lemma Liminf_within_UNIV:
fixes f :: "'a::metric_space => _"
shows "Liminf (at x) f = Liminf (at x within UNIV) f"
by simp (* TODO: delete *)
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_within_UNIV:
fixes f :: "'a::metric_space => _"
shows "Limsup (at x) f = Limsup (at x within UNIV) f"
by simp (* TODO: delete *)
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 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"
and "~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"
and "~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
then have "y : (S Int ball x e - {x})"
unfolding ball_def by (simp add: dist_commute)
then have "S Int ball x e - {x} ~= {}" by blast
} then have "?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
then have "y:(S-{x}) & dist y x < e"
unfolding ball_def by (simp add: dist_commute)
then have "EX y:(S-{x}). dist y x < e" by auto
}
then have "?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"
and "x ----> x0"
shows "(f o x) ----> (f x0)"
proof -
{ fix S
assume "open S & (f x0):S"
then obtain T where T_def: "open T & x0 : T & (ALL x:T. f x : S)"
using assms continuous_at_open by metis
then have "(EX N. ALL n>=N. x n : T)"
using assms tendsto_explicit T_def by auto
then have "(EX N. ALL n>=N. f(x n) : S)" using T_def by auto
}
then 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)"
then 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}"
then have "x0 islimpt X"
unfolding islimpt_def using T_def by auto
then obtain x where x_def: "(ALL n. x n : X) & x ----> x0"
using islimpt_sequential[of x0 X] by auto
then have "~(f o x) ----> (f x0)"
unfolding tendsto_explicit using X_def T_def by auto
then have "EX x. x ----> x0 & (~(f o x) ----> (f x0))" using x_def by auto
}
then 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"
then have "ereal (real x0) : S" using T_def by auto
then have "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
}
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"
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
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 "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)"
then obtain x where x_def: "x:S & x < Inf S + e" using Inf_close `S ~= {}` by auto
moreover then have "x > Inf S - e" using * by auto
ultimately have "abs (x - Inf S) < e" by (simp add: abs_diff_less_iff)
then have "EX x:S. abs (x - Inf S) < e" using x_def by auto
}
then show ?thesis
apply (subst closure_approachable)
unfolding dist_norm apply auto
done
qed
lemma closed_contains_Inf:
fixes S :: "real set"
assumes "S ~= {}" "EX B. ALL x:S. B<=x"
and "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"
and "closed S"
shows "S = {} | S = UNIV | (EX a. S = {a ..})"
proof -
{ assume "S ~= {}"
{ assume ex: "EX B. ALL x:S. B<=x"
then have *: "ALL x:S. Inf S <= x" using Inf_lower_EX[of _ S] ex by metis
then have "Inf S : S" apply (subst closed_contains_Inf) using ex `S ~= {}` `closed S` by auto
then have "ALL x. (Inf S <= x <-> x:S)" using mono[rule_format, of "Inf S"] * by auto
then have "S = {Inf S ..}" by auto
then have "EX a. S = {a ..}" by auto
}
moreover
{ assume "~(EX B. ALL x:S. B<=x)"
then have 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
then have "y:S" using mono[rule_format, of x y] by auto
} then have "S = UNIV" by auto
}
ultimately have "S = UNIV | (EX a. S = {a ..})" by blast
} then show ?thesis by blast
qed
lemma mono_closed_ereal:
fixes S :: "real set"
assumes mono: "ALL y z. y:S & y<=z --> z:S"
and "closed S"
shows "EX a. S = {x. a <= ereal x}"
proof -
{ assume "S = {}"
then have ?thesis apply(rule_tac x=PInfty in exI) by auto }
moreover
{ assume "S = UNIV"
then have ?thesis apply(rule_tac x="-\<infinity>" in exI) by auto }
moreover
{ assume "EX a. S = {a ..}"
then obtain a where "S={a ..}" by auto
then have ?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>"
then show "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_SUP[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
lemma suminf_bound_add:
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 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]
by (simp add: one_ereal_def)
lemma suminf_add_ereal:
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)
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 \<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
apply (subst (1 2 3) suminf_ereal)
apply (auto intro!: suminf_diff[symmetric] summable_ereal)
done
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)
apply rule
done
qed
lemma suminf_setsum_ereal:
fixes f :: "_ \<Rightarrow> _ \<Rightarrow> ereal"
assumes nonneg: "\<And>i a. a \<in> A \<Longrightarrow> 0 \<le> f i a"
shows "(\<Sum>i. \<Sum>a\<in>A. f i a) = (\<Sum>a\<in>A. \<Sum>i. f i a)"
proof cases
assume "finite A"
then show ?thesis using nonneg
by induct (simp_all add: suminf_add_ereal setsum_nonneg)
qed simp
lemma suminf_ereal_eq_0:
fixes f :: "nat \<Rightarrow> ereal"
assumes nneg: "\<And>i. 0 \<le> f i"
shows "(\<Sum>i. f i) = 0 \<longleftrightarrow> (\<forall>i. f i = 0)"
proof
assume "(\<Sum>i. f i) = 0"
{ fix i assume "f i \<noteq> 0"
with nneg have "0 < f i" by (auto simp: less_le)
also have "f i = (\<Sum>j. if j = i then f i else 0)"
by (subst suminf_finite[where N="{i}"]) auto
also have "\<dots> \<le> (\<Sum>i. f i)"
using nneg by (auto intro!: suminf_le_pos)
finally have False using `(\<Sum>i. f i) = 0` by auto }
then show "\<forall>i. f i = 0" by auto
qed simp
end