# Theory Extended_Real

theory Extended_Real
imports Extended_Nat Liminf_Limsup
(*  Title:      HOL/Library/Extended_Real.thy
Author:     Johannes Hölzl, TU München
Author:     Robert Himmelmann, TU München
Author:     Armin Heller, TU München
Author:     Bogdan Grechuk, University of Edinburgh
Author:     Manuel Eberl, TU München
*)

section ‹Extended real number line›

theory Extended_Real
imports Complex_Main Extended_Nat Liminf_Limsup
begin

text ‹

This should be part of @{theory Extended_Nat} or @{theory Order_Continuity}, but then the
AFP-entry ‹Jinja_Thread› fails, as it does overload certain named from @{theory Complex_Main}.

›

lemma continuous_at_left_imp_sup_continuous:
fixes f :: "'a ⇒ 'a::{complete_linorder, linorder_topology}"
assumes "mono f" "⋀x. continuous (at_left x) f"
shows "sup_continuous f"
unfolding sup_continuous_def
proof safe
fix M :: "nat ⇒ 'a" assume "incseq M" then show "f (SUP i. M i) = (SUP i. f (M i))"
using continuous_at_Sup_mono[OF assms, of "range M"] by simp
qed

lemma sup_continuous_at_left:
fixes f :: "'a ⇒ 'a::{complete_linorder, linorder_topology, first_countable_topology}"
assumes f: "sup_continuous f"
shows "continuous (at_left x) f"
proof cases
assume "x = bot" then show ?thesis
next
assume x: "x ≠ bot"
show ?thesis
unfolding continuous_within
proof (intro tendsto_at_left_sequentially[of bot])
fix S :: "nat ⇒ 'a" assume S: "incseq S" and S_x: "S ⇢ x"
from S_x have x_eq: "x = (SUP i. S i)"
by (rule LIMSEQ_unique) (intro LIMSEQ_SUP S)
show "(λn. f (S n)) ⇢ f x"
unfolding x_eq sup_continuousD[OF f S]
using S sup_continuous_mono[OF f] by (intro LIMSEQ_SUP) (auto simp: mono_def)
qed (insert x, auto simp: bot_less)
qed

lemma sup_continuous_iff_at_left:
fixes f :: "'a ⇒ 'a::{complete_linorder, linorder_topology, first_countable_topology}"
shows "sup_continuous f ⟷ (∀x. continuous (at_left x) f) ∧ mono f"
using sup_continuous_at_left[of f] continuous_at_left_imp_sup_continuous[of f]
sup_continuous_mono[of f] by auto

lemma continuous_at_right_imp_inf_continuous:
fixes f :: "'a ⇒ 'a::{complete_linorder, linorder_topology}"
assumes "mono f" "⋀x. continuous (at_right x) f"
shows "inf_continuous f"
unfolding inf_continuous_def
proof safe
fix M :: "nat ⇒ 'a" assume "decseq M" then show "f (INF i. M i) = (INF i. f (M i))"
using continuous_at_Inf_mono[OF assms, of "range M"] by simp
qed

lemma inf_continuous_at_right:
fixes f :: "'a ⇒ 'a::{complete_linorder, linorder_topology, first_countable_topology}"
assumes f: "inf_continuous f"
shows "continuous (at_right x) f"
proof cases
assume "x = top" then show ?thesis
next
assume x: "x ≠ top"
show ?thesis
unfolding continuous_within
proof (intro tendsto_at_right_sequentially[of _ top])
fix S :: "nat ⇒ 'a" assume S: "decseq S" and S_x: "S ⇢ x"
from S_x have x_eq: "x = (INF i. S i)"
by (rule LIMSEQ_unique) (intro LIMSEQ_INF S)
show "(λn. f (S n)) ⇢ f x"
unfolding x_eq inf_continuousD[OF f S]
using S inf_continuous_mono[OF f] by (intro LIMSEQ_INF) (auto simp: mono_def antimono_def)
qed (insert x, auto simp: less_top)
qed

lemma inf_continuous_iff_at_right:
fixes f :: "'a ⇒ 'a::{complete_linorder, linorder_topology, first_countable_topology}"
shows "inf_continuous f ⟷ (∀x. continuous (at_right x) f) ∧ mono f"
using inf_continuous_at_right[of f] continuous_at_right_imp_inf_continuous[of f]
inf_continuous_mono[of f] by auto

instantiation enat :: linorder_topology
begin

definition open_enat :: "enat set ⇒ bool" where
"open_enat = generate_topology (range lessThan ∪ range greaterThan)"

instance
proof qed (rule open_enat_def)

end

lemma open_enat: "open {enat n}"
proof (cases n)
case 0
then have "{enat n} = {..< eSuc 0}"
by (auto simp: enat_0)
then show ?thesis
by simp
next
case (Suc n')
then have "{enat n} = {enat n' <..< enat (Suc n)}"
apply auto
apply (case_tac x)
apply auto
done
then show ?thesis
by simp
qed

lemma open_enat_iff:
fixes A :: "enat set"
shows "open A ⟷ (∞ ∈ A ⟶ (∃n::nat. {n <..} ⊆ A))"
proof safe
assume "∞ ∉ A"
then have "A = (⋃n∈{n. enat n ∈ A}. {enat n})"
apply auto
apply (case_tac x)
apply auto
done
moreover have "open …"
by (auto intro: open_enat)
ultimately show "open A"
by simp
next
fix n assume "{enat n <..} ⊆ A"
then have "A = (⋃n∈{n. enat n ∈ A}. {enat n}) ∪ {enat n <..}"
apply auto
apply (case_tac x)
apply auto
done
moreover have "open …"
by (intro open_Un open_UN ballI open_enat open_greaterThan)
ultimately show "open A"
by simp
next
assume "open A" "∞ ∈ A"
then have "generate_topology (range lessThan ∪ range greaterThan) A" "∞ ∈ A"
unfolding open_enat_def by auto
then show "∃n::nat. {n <..} ⊆ A"
proof induction
case (Int A B)
then obtain n m where "{enat n<..} ⊆ A" "{enat m<..} ⊆ B"
by auto
then have "{enat (max n m) <..} ⊆ A ∩ B"
by (auto simp add: subset_eq Ball_def max_def enat_ord_code(1)[symmetric] simp del: enat_ord_code(1))
then show ?case
by auto
next
case (UN K)
then obtain k where "k ∈ K" "∞ ∈ k"
by auto
with UN.IH[OF this] show ?case
by auto
qed auto
qed

text ‹

For more lemmas about the extended real numbers go to
@{file "~~/src/HOL/Multivariate_Analysis/Extended_Real_Limits.thy"}

›

subsection ‹Definition and basic properties›

datatype ereal = ereal real | PInfty | MInfty

instantiation ereal :: uminus
begin

fun uminus_ereal where
"- (ereal r) = ereal (- r)"
| "- PInfty = MInfty"
| "- MInfty = PInfty"

instance ..

end

instantiation ereal :: infinity
begin

definition "(∞::ereal) = PInfty"
instance ..

end

declare [[coercion "ereal :: real ⇒ ereal"]]

lemma ereal_uminus_uminus[simp]:
fixes a :: ereal
shows "- (- a) = a"
by (cases a) simp_all

lemma
shows PInfty_eq_infinity[simp]: "PInfty = ∞"
and MInfty_eq_minfinity[simp]: "MInfty = - ∞"
and MInfty_neq_PInfty[simp]: "∞ ≠ - (∞::ereal)" "- ∞ ≠ (∞::ereal)"
and MInfty_neq_ereal[simp]: "ereal r ≠ - ∞" "- ∞ ≠ ereal r"
and PInfty_neq_ereal[simp]: "ereal r ≠ ∞" "∞ ≠ ereal r"
and PInfty_cases[simp]: "(case ∞ of ereal r ⇒ f r | PInfty ⇒ y | MInfty ⇒ z) = y"
and MInfty_cases[simp]: "(case - ∞ of ereal r ⇒ f r | PInfty ⇒ y | MInfty ⇒ z) = z"

declare
PInfty_eq_infinity[code_post]
MInfty_eq_minfinity[code_post]

lemma [code_unfold]:
"∞ = PInfty"
"- PInfty = MInfty"
by simp_all

lemma inj_ereal[simp]: "inj_on ereal A"
unfolding inj_on_def by auto

lemma ereal_cases[cases type: ereal]:
obtains (real) r where "x = ereal r"
| (PInf) "x = ∞"
| (MInf) "x = -∞"
using assms by (cases x) auto

lemmas ereal2_cases = ereal_cases[case_product ereal_cases]
lemmas ereal3_cases = ereal2_cases[case_product ereal_cases]

lemma ereal_all_split: "⋀P. (∀x::ereal. P x) ⟷ P ∞ ∧ (∀x. P (ereal x)) ∧ P (-∞)"
by (metis ereal_cases)

lemma ereal_ex_split: "⋀P. (∃x::ereal. P x) ⟷ P ∞ ∨ (∃x. P (ereal x)) ∨ P (-∞)"
by (metis ereal_cases)

lemma ereal_uminus_eq_iff[simp]:
fixes a b :: ereal
shows "-a = -b ⟷ a = b"
by (cases rule: ereal2_cases[of a b]) simp_all

function real_of_ereal :: "ereal ⇒ real" where
"real_of_ereal (ereal r) = r"
| "real_of_ereal ∞ = 0"
| "real_of_ereal (-∞) = 0"
by (auto intro: ereal_cases)
termination by standard (rule wf_empty)

lemma real_of_ereal[simp]:
"real_of_ereal (- x :: ereal) = - (real_of_ereal x)"
by (cases x) simp_all

lemma range_ereal[simp]: "range ereal = UNIV - {∞, -∞}"
proof safe
fix x
assume "x ∉ range ereal" "x ≠ ∞"
then show "x = -∞"
by (cases x) auto
qed auto

lemma ereal_range_uminus[simp]: "range uminus = (UNIV::ereal set)"
proof safe
fix x :: ereal
show "x ∈ range uminus"
by (intro image_eqI[of _ _ "-x"]) auto
qed auto

instantiation ereal :: abs
begin

function abs_ereal where
"¦ereal r¦ = ereal ¦r¦"
| "¦-∞¦ = (∞::ereal)"
| "¦∞¦ = (∞::ereal)"
by (auto intro: ereal_cases)
termination proof qed (rule wf_empty)

instance ..

end

lemma abs_eq_infinity_cases[elim!]:
fixes x :: ereal
assumes "¦x¦ = ∞"
obtains "x = ∞" | "x = -∞"
using assms by (cases x) auto

lemma abs_neq_infinity_cases[elim!]:
fixes x :: ereal
assumes "¦x¦ ≠ ∞"
obtains r where "x = ereal r"
using assms by (cases x) auto

lemma abs_ereal_uminus[simp]:
fixes x :: ereal
shows "¦- x¦ = ¦x¦"
by (cases x) auto

lemma ereal_infinity_cases:
fixes a :: ereal
shows "a ≠ ∞ ⟹ a ≠ -∞ ⟹ ¦a¦ ≠ ∞"
by auto

begin

definition "0 = ereal 0"
definition "1 = ereal 1"

function plus_ereal where
"ereal r + ereal p = ereal (r + p)"
| "∞ + a = (∞::ereal)"
| "a + ∞ = (∞::ereal)"
| "ereal r + -∞ = - ∞"
| "-∞ + ereal p = -(∞::ereal)"
| "-∞ + -∞ = -(∞::ereal)"
proof goal_cases
case prems: (1 P x)
then obtain a b where "x = (a, b)"
by (cases x) auto
with prems show P
by (cases rule: ereal2_cases[of a b]) auto
qed auto
termination by standard (rule wf_empty)

lemma Infty_neq_0[simp]:
"(∞::ereal) ≠ 0" "0 ≠ (∞::ereal)"
"-(∞::ereal) ≠ 0" "0 ≠ -(∞::ereal)"

lemma ereal_eq_0[simp]:
"ereal r = 0 ⟷ r = 0"
"0 = ereal r ⟷ r = 0"
unfolding zero_ereal_def by simp_all

lemma ereal_eq_1[simp]:
"ereal r = 1 ⟷ r = 1"
"1 = ereal r ⟷ r = 1"
unfolding one_ereal_def by simp_all

instance
proof
fix a b c :: ereal
show "0 + a = a"
by (cases a) (simp_all add: zero_ereal_def)
show "a + b = b + a"
by (cases rule: ereal2_cases[of a b]) simp_all
show "a + b + c = a + (b + c)"
by (cases rule: ereal3_cases[of a b c]) simp_all
show "0 ≠ (1::ereal)"
qed

end

lemma ereal_0_plus [simp]: "ereal 0 + x = x"
and plus_ereal_0 [simp]: "x + ereal 0 = x"

instance ereal :: numeral ..

lemma real_of_ereal_0[simp]: "real_of_ereal (0::ereal) = 0"
unfolding zero_ereal_def by simp

lemma abs_ereal_zero[simp]: "¦0¦ = (0::ereal)"
unfolding zero_ereal_def abs_ereal.simps by simp

lemma ereal_uminus_zero[simp]: "- 0 = (0::ereal)"

lemma ereal_uminus_zero_iff[simp]:
fixes a :: ereal
shows "-a = 0 ⟷ a = 0"
by (cases a) simp_all

lemma ereal_plus_eq_PInfty[simp]:
fixes a b :: ereal
shows "a + b = ∞ ⟷ a = ∞ ∨ b = ∞"
by (cases rule: ereal2_cases[of a b]) auto

lemma ereal_plus_eq_MInfty[simp]:
fixes a b :: ereal
shows "a + b = -∞ ⟷ (a = -∞ ∨ b = -∞) ∧ a ≠ ∞ ∧ b ≠ ∞"
by (cases rule: ereal2_cases[of a b]) auto

fixes a b :: ereal
assumes "a ≠ -∞"
shows "a + b = a + c ⟷ a = ∞ ∨ b = c"
using assms by (cases rule: ereal3_cases[of a b c]) auto

fixes a b :: ereal
assumes "a ≠ -∞"
shows "b + a = c + a ⟷ a = ∞ ∨ b = c"
using assms by (cases rule: ereal3_cases[of a b c]) auto

lemma ereal_real: "ereal (real_of_ereal x) = (if ¦x¦ = ∞ then 0 else x)"
by (cases x) simp_all

fixes a b :: ereal
shows "real_of_ereal (a + b) =
(if (¦a¦ = ∞) ∧ (¦b¦ = ∞) ∨ (¦a¦ ≠ ∞) ∧ (¦b¦ ≠ ∞) then real_of_ereal a + real_of_ereal b else 0)"
by (cases rule: ereal2_cases[of a b]) auto

subsubsection "Linear order on @{typ ereal}"

instantiation ereal :: linorder
begin

function less_ereal
where
"   ereal x < ereal y     ⟷ x < y"
| "(∞::ereal) < a           ⟷ False"
| "         a < -(∞::ereal) ⟷ False"
| "ereal x    < ∞           ⟷ True"
| "        -∞ < ereal r     ⟷ True"
| "        -∞ < (∞::ereal) ⟷ True"
proof goal_cases
case prems: (1 P x)
then obtain a b where "x = (a,b)" by (cases x) auto
with prems show P by (cases rule: ereal2_cases[of a b]) auto
qed simp_all
termination by (relation "{}") simp

definition "x ≤ (y::ereal) ⟷ x < y ∨ x = y"

lemma ereal_infty_less[simp]:
fixes x :: ereal
shows "x < ∞ ⟷ (x ≠ ∞)"
"-∞ < x ⟷ (x ≠ -∞)"
by (cases x, simp_all) (cases x, simp_all)

lemma ereal_infty_less_eq[simp]:
fixes x :: ereal
shows "∞ ≤ x ⟷ x = ∞"
and "x ≤ -∞ ⟷ x = -∞"

lemma ereal_less[simp]:
"ereal r < 0 ⟷ (r < 0)"
"0 < ereal r ⟷ (0 < r)"
"ereal r < 1 ⟷ (r < 1)"
"1 < ereal r ⟷ (1 < r)"
"0 < (∞::ereal)"
"-(∞::ereal) < 0"

lemma ereal_less_eq[simp]:
"x ≤ (∞::ereal)"
"-(∞::ereal) ≤ x"
"ereal r ≤ ereal p ⟷ r ≤ p"
"ereal r ≤ 0 ⟷ r ≤ 0"
"0 ≤ ereal r ⟷ 0 ≤ r"
"ereal r ≤ 1 ⟷ r ≤ 1"
"1 ≤ ereal r ⟷ 1 ≤ r"
by (auto simp add: less_eq_ereal_def zero_ereal_def one_ereal_def)

lemma ereal_infty_less_eq2:
"a ≤ b ⟹ a = ∞ ⟹ b = (∞::ereal)"
"a ≤ b ⟹ b = -∞ ⟹ a = -(∞::ereal)"
by simp_all

instance
proof
fix x y z :: ereal
show "x ≤ x"
by (cases x) simp_all
show "x < y ⟷ x ≤ y ∧ ¬ y ≤ x"
by (cases rule: ereal2_cases[of x y]) auto
show "x ≤ y ∨ y ≤ x "
by (cases rule: ereal2_cases[of x y]) auto
{
assume "x ≤ y" "y ≤ x"
then show "x = y"
by (cases rule: ereal2_cases[of x y]) auto
}
{
assume "x ≤ y" "y ≤ z"
then show "x ≤ z"
by (cases rule: ereal3_cases[of x y z]) auto
}
qed

end

lemma ereal_dense2: "x < y ⟹ ∃z. x < ereal z ∧ ereal z < y"
using lt_ex gt_ex dense by (cases x y rule: ereal2_cases) auto

instance ereal :: dense_linorder
by standard (blast dest: ereal_dense2)

proof
fix a b c :: ereal
assume "a ≤ b"
then show "c + a ≤ c + b"
by (cases rule: ereal3_cases[of a b c]) auto
qed

lemma real_of_ereal_positive_mono:
fixes x y :: ereal
shows "0 ≤ x ⟹ x ≤ y ⟹ y ≠ ∞ ⟹ real_of_ereal x ≤ real_of_ereal y"
by (cases rule: ereal2_cases[of x y]) auto

lemma ereal_MInfty_lessI[intro, simp]:
fixes a :: ereal
shows "a ≠ -∞ ⟹ -∞ < a"
by (cases a) auto

lemma ereal_less_PInfty[intro, simp]:
fixes a :: ereal
shows "a ≠ ∞ ⟹ a < ∞"
by (cases a) auto

lemma ereal_less_ereal_Ex:
fixes a b :: ereal
shows "x < ereal r ⟷ x = -∞ ∨ (∃p. p < r ∧ x = ereal p)"
by (cases x) auto

lemma less_PInf_Ex_of_nat: "x ≠ ∞ ⟷ (∃n::nat. x < ereal (real n))"
proof (cases x)
case (real r)
then show ?thesis
using reals_Archimedean2[of r] by simp
qed simp_all

fixes a b c d :: ereal
assumes "a ≤ b"
and "c ≤ d"
shows "a + c ≤ b + d"
using assms
apply (cases a)
apply (cases rule: ereal3_cases[of b c d], auto)
apply (cases rule: ereal3_cases[of b c d], auto)
done

lemma ereal_minus_le_minus[simp]:
fixes a b :: ereal
shows "- a ≤ - b ⟷ b ≤ a"
by (cases rule: ereal2_cases[of a b]) auto

lemma ereal_minus_less_minus[simp]:
fixes a b :: ereal
shows "- a < - b ⟷ b < a"
by (cases rule: ereal2_cases[of a b]) auto

lemma ereal_le_real_iff:
"x ≤ real_of_ereal y ⟷ (¦y¦ ≠ ∞ ⟶ ereal x ≤ y) ∧ (¦y¦ = ∞ ⟶ x ≤ 0)"
by (cases y) auto

lemma real_le_ereal_iff:
"real_of_ereal y ≤ x ⟷ (¦y¦ ≠ ∞ ⟶ y ≤ ereal x) ∧ (¦y¦ = ∞ ⟶ 0 ≤ x)"
by (cases y) auto

lemma ereal_less_real_iff:
"x < real_of_ereal y ⟷ (¦y¦ ≠ ∞ ⟶ ereal x < y) ∧ (¦y¦ = ∞ ⟶ x < 0)"
by (cases y) auto

lemma real_less_ereal_iff:
"real_of_ereal y < x ⟷ (¦y¦ ≠ ∞ ⟶ y < ereal x) ∧ (¦y¦ = ∞ ⟶ 0 < x)"
by (cases y) auto

lemma real_of_ereal_pos:
fixes x :: ereal
shows "0 ≤ x ⟹ 0 ≤ real_of_ereal x" by (cases x) auto

lemmas real_of_ereal_ord_simps =
ereal_le_real_iff real_le_ereal_iff ereal_less_real_iff real_less_ereal_iff

lemma abs_ereal_ge0[simp]: "0 ≤ x ⟹ ¦x :: ereal¦ = x"
by (cases x) auto

lemma abs_ereal_less0[simp]: "x < 0 ⟹ ¦x :: ereal¦ = -x"
by (cases x) auto

lemma abs_ereal_pos[simp]: "0 ≤ ¦x :: ereal¦"
by (cases x) auto

lemma ereal_abs_leI:
fixes x y :: ereal
shows "⟦ x ≤ y; -x ≤ y ⟧ ⟹ ¦x¦ ≤ y"
by(cases x y rule: ereal2_cases)(simp_all)

lemma real_of_ereal_le_0[simp]: "real_of_ereal (x :: ereal) ≤ 0 ⟷ x ≤ 0 ∨ x = ∞"
by (cases x) auto

lemma abs_real_of_ereal[simp]: "¦real_of_ereal (x :: ereal)¦ = real_of_ereal ¦x¦"
by (cases x) auto

lemma zero_less_real_of_ereal:
fixes x :: ereal
shows "0 < real_of_ereal x ⟷ 0 < x ∧ x ≠ ∞"
by (cases x) auto

lemma ereal_0_le_uminus_iff[simp]:
fixes a :: ereal
shows "0 ≤ - a ⟷ a ≤ 0"
by (cases rule: ereal2_cases[of a]) auto

lemma ereal_uminus_le_0_iff[simp]:
fixes a :: ereal
shows "- a ≤ 0 ⟷ 0 ≤ a"
by (cases rule: ereal2_cases[of a]) auto

fixes a b c d :: ereal
assumes "a ≤ b"
and "0 ≤ a"
and "a ≠ ∞"
and "c < d"
shows "a + c < b + d"
using assms
by (cases rule: ereal3_cases[case_product ereal_cases, of a b c d]) auto

fixes a b c :: ereal
shows "¦a¦ ≠ ∞ ⟹ c < b ⟹ a + c < a + b"
by (cases rule: ereal2_cases[of b c]) auto

fixes a b :: ereal
shows "0 ≤ a ⟹ 0 ≤ b ⟹ a + b = 0 ⟷ a = 0 ∧ b = 0"
by (cases a b rule: ereal2_cases) auto

lemma ereal_uminus_eq_reorder: "- a = b ⟷ a = (-b::ereal)"
by auto

lemma ereal_uminus_less_reorder: "- a < b ⟷ -b < (a::ereal)"
by (subst (3) ereal_uminus_uminus[symmetric]) (simp only: ereal_minus_less_minus)

lemma ereal_less_uminus_reorder: "a < - b ⟷ b < - (a::ereal)"
by (subst (3) ereal_uminus_uminus[symmetric]) (simp only: ereal_minus_less_minus)

lemma ereal_uminus_le_reorder: "- a ≤ b ⟷ -b ≤ (a::ereal)"
by (subst (3) ereal_uminus_uminus[symmetric]) (simp only: ereal_minus_le_minus)

lemmas ereal_uminus_reorder =
ereal_uminus_eq_reorder ereal_uminus_less_reorder ereal_uminus_le_reorder

lemma ereal_bot:
fixes x :: ereal
assumes "⋀B. x ≤ ereal B"
shows "x = - ∞"
proof (cases x)
case (real r)
with assms[of "r - 1"] show ?thesis
by auto
next
case PInf
with assms[of 0] show ?thesis
by auto
next
case MInf
then show ?thesis
by simp
qed

lemma ereal_top:
fixes x :: ereal
assumes "⋀B. x ≥ ereal B"
shows "x = ∞"
proof (cases x)
case (real r)
with assms[of "r + 1"] show ?thesis
by auto
next
case MInf
with assms[of 0] show ?thesis
by auto
next
case PInf
then show ?thesis
by simp
qed

lemma
shows ereal_max[simp]: "ereal (max x y) = max (ereal x) (ereal y)"
and ereal_min[simp]: "ereal (min x y) = min (ereal x) (ereal y)"

lemma ereal_max_0: "max 0 (ereal r) = ereal (max 0 r)"
by (auto simp: zero_ereal_def)

lemma
fixes f :: "nat ⇒ ereal"
shows ereal_incseq_uminus[simp]: "incseq (λx. - f x) ⟷ decseq f"
and ereal_decseq_uminus[simp]: "decseq (λx. - f x) ⟷ incseq f"
unfolding decseq_def incseq_def by auto

lemma incseq_ereal: "incseq f ⟹ incseq (λx. ereal (f x))"
unfolding incseq_def by auto

fixes a b :: ereal
shows "0 ≤ a ⟹ 0 ≤ b ⟹ 0 ≤ a + b"
using add_mono[of 0 a 0 b] by simp

lemma image_eqD: "f ` A = B ⟹ ∀x∈A. f x ∈ B"
by auto

lemma incseq_setsumI:
assumes "⋀i. 0 ≤ f i"
shows "incseq (λi. setsum f {..< i})"
proof (intro incseq_SucI)
fix n
have "setsum f {..< n} + 0 ≤ setsum f {..<n} + f n"
then show "setsum f {..< n} ≤ setsum f {..< Suc n}"
by auto
qed

lemma incseq_setsumI2:
assumes "⋀n. n ∈ A ⟹ incseq (f n)"
shows "incseq (λi. ∑n∈A. f n i)"
using assms
unfolding incseq_def by (auto intro: setsum_mono)

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

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

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

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

lemma setsum_ereal_0:
fixes f :: "'a ⇒ ereal"
assumes "finite A"
and "⋀i. i ∈ A ⟹ 0 ≤ f i"
shows "(∑x∈A. f x) = 0 ⟷ (∀i∈A. f i = 0)"
proof
assume "setsum f A = 0" with assms show "∀i∈A. f i = 0"
proof (induction A)
case (insert a A)
then have "f a = 0 ∧ (∑a∈A. f a) = 0"
with insert show ?case
by simp
qed simp
qed auto

subsubsection "Multiplication"

instantiation ereal :: "{comm_monoid_mult,sgn}"
begin

function sgn_ereal :: "ereal ⇒ ereal" where
"sgn (ereal r) = ereal (sgn r)"
| "sgn (∞::ereal) = 1"
| "sgn (-∞::ereal) = -1"
by (auto intro: ereal_cases)
termination by standard (rule wf_empty)

function times_ereal where
"ereal r * ereal p = ereal (r * p)"
| "ereal r * ∞ = (if r = 0 then 0 else if r > 0 then ∞ else -∞)"
| "∞ * ereal r = (if r = 0 then 0 else if r > 0 then ∞ else -∞)"
| "ereal r * -∞ = (if r = 0 then 0 else if r > 0 then -∞ else ∞)"
| "-∞ * ereal r = (if r = 0 then 0 else if r > 0 then -∞ else ∞)"
| "(∞::ereal) * ∞ = ∞"
| "-(∞::ereal) * ∞ = -∞"
| "(∞::ereal) * -∞ = -∞"
| "-(∞::ereal) * -∞ = ∞"
proof goal_cases
case prems: (1 P x)
then obtain a b where "x = (a, b)"
by (cases x) auto
with prems show P
by (cases rule: ereal2_cases[of a b]) auto
qed simp_all
termination by (relation "{}") simp

instance
proof
fix a b c :: ereal
show "1 * a = a"
by (cases a) (simp_all add: one_ereal_def)
show "a * b = b * a"
by (cases rule: ereal2_cases[of a b]) simp_all
show "a * b * c = a * (b * c)"
by (cases rule: ereal3_cases[of a b c])
qed

end

lemma [simp]:
shows ereal_1_times: "ereal 1 * x = x"
and times_ereal_1: "x * ereal 1 = x"

lemma one_not_le_zero_ereal[simp]: "¬ (1 ≤ (0::ereal))"

lemma real_ereal_1[simp]: "real_of_ereal (1::ereal) = 1"
unfolding one_ereal_def by simp

lemma real_of_ereal_le_1:
fixes a :: ereal
shows "a ≤ 1 ⟹ real_of_ereal a ≤ 1"
by (cases a) (auto simp: one_ereal_def)

lemma abs_ereal_one[simp]: "¦1¦ = (1::ereal)"
unfolding one_ereal_def by simp

lemma ereal_mult_zero[simp]:
fixes a :: ereal
shows "a * 0 = 0"
by (cases a) (simp_all add: zero_ereal_def)

lemma ereal_zero_mult[simp]:
fixes a :: ereal
shows "0 * a = 0"
by (cases a) (simp_all add: zero_ereal_def)

lemma ereal_m1_less_0[simp]: "-(1::ereal) < 0"

lemma ereal_times[simp]:
"1 ≠ (∞::ereal)" "(∞::ereal) ≠ 1"
"1 ≠ -(∞::ereal)" "-(∞::ereal) ≠ 1"
by (auto simp: one_ereal_def)

lemma ereal_plus_1[simp]:
"1 + ereal r = ereal (r + 1)"
"ereal r + 1 = ereal (r + 1)"
"1 + -(∞::ereal) = -∞"
"-(∞::ereal) + 1 = -∞"
unfolding one_ereal_def by auto

lemma ereal_zero_times[simp]:
fixes a b :: ereal
shows "a * b = 0 ⟷ a = 0 ∨ b = 0"
by (cases rule: ereal2_cases[of a b]) auto

lemma ereal_mult_eq_PInfty[simp]:
"a * b = (∞::ereal) ⟷
(a = ∞ ∧ b > 0) ∨ (a > 0 ∧ b = ∞) ∨ (a = -∞ ∧ b < 0) ∨ (a < 0 ∧ b = -∞)"
by (cases rule: ereal2_cases[of a b]) auto

lemma ereal_mult_eq_MInfty[simp]:
"a * b = -(∞::ereal) ⟷
(a = ∞ ∧ b < 0) ∨ (a < 0 ∧ b = ∞) ∨ (a = -∞ ∧ b > 0) ∨ (a > 0 ∧ b = -∞)"
by (cases rule: ereal2_cases[of a b]) auto

lemma ereal_abs_mult: "¦x * y :: ereal¦ = ¦x¦ * ¦y¦"
by (cases x y rule: ereal2_cases) (auto simp: abs_mult)

lemma ereal_0_less_1[simp]: "0 < (1::ereal)"

lemma ereal_mult_minus_left[simp]:
fixes a b :: ereal
shows "-a * b = - (a * b)"
by (cases rule: ereal2_cases[of a b]) auto

lemma ereal_mult_minus_right[simp]:
fixes a b :: ereal
shows "a * -b = - (a * b)"
by (cases rule: ereal2_cases[of a b]) auto

lemma ereal_mult_infty[simp]:
"a * (∞::ereal) = (if a = 0 then 0 else if 0 < a then ∞ else - ∞)"
by (cases a) auto

lemma ereal_infty_mult[simp]:
"(∞::ereal) * a = (if a = 0 then 0 else if 0 < a then ∞ else - ∞)"
by (cases a) auto

lemma ereal_mult_strict_right_mono:
assumes "a < b"
and "0 < c"
and "c < (∞::ereal)"
shows "a * c < b * c"
using assms
by (cases rule: ereal3_cases[of a b c]) (auto simp: zero_le_mult_iff)

lemma ereal_mult_strict_left_mono:
"a < b ⟹ 0 < c ⟹ c < (∞::ereal) ⟹ c * a < c * b"
using ereal_mult_strict_right_mono

lemma ereal_mult_right_mono:
fixes a b c :: ereal
shows "a ≤ b ⟹ 0 ≤ c ⟹ a * c ≤ b * c"
using assms
apply (cases "c = 0")
apply simp
apply (cases rule: ereal3_cases[of a b c])
apply (auto simp: zero_le_mult_iff)
done

lemma ereal_mult_left_mono:
fixes a b c :: ereal
shows "a ≤ b ⟹ 0 ≤ c ⟹ c * a ≤ c * b"
using ereal_mult_right_mono

lemma zero_less_one_ereal[simp]: "0 ≤ (1::ereal)"

lemma ereal_0_le_mult[simp]: "0 ≤ a ⟹ 0 ≤ b ⟹ 0 ≤ a * (b :: ereal)"
by (cases rule: ereal2_cases[of a b]) auto

lemma ereal_right_distrib:
fixes r a b :: ereal
shows "0 ≤ a ⟹ 0 ≤ b ⟹ r * (a + b) = r * a + r * b"
by (cases rule: ereal3_cases[of r a b]) (simp_all add: field_simps)

lemma ereal_left_distrib:
fixes r a b :: ereal
shows "0 ≤ a ⟹ 0 ≤ b ⟹ (a + b) * r = a * r + b * r"
by (cases rule: ereal3_cases[of r a b]) (simp_all add: field_simps)

lemma ereal_mult_le_0_iff:
fixes a b :: ereal
shows "a * b ≤ 0 ⟷ (0 ≤ a ∧ b ≤ 0) ∨ (a ≤ 0 ∧ 0 ≤ b)"
by (cases rule: ereal2_cases[of a b]) (simp_all add: mult_le_0_iff)

lemma ereal_zero_le_0_iff:
fixes a b :: ereal
shows "0 ≤ a * b ⟷ (0 ≤ a ∧ 0 ≤ b) ∨ (a ≤ 0 ∧ b ≤ 0)"
by (cases rule: ereal2_cases[of a b]) (simp_all add: zero_le_mult_iff)

lemma ereal_mult_less_0_iff:
fixes a b :: ereal
shows "a * b < 0 ⟷ (0 < a ∧ b < 0) ∨ (a < 0 ∧ 0 < b)"
by (cases rule: ereal2_cases[of a b]) (simp_all add: mult_less_0_iff)

lemma ereal_zero_less_0_iff:
fixes a b :: ereal
shows "0 < a * b ⟷ (0 < a ∧ 0 < b) ∨ (a < 0 ∧ b < 0)"
by (cases rule: ereal2_cases[of a b]) (simp_all add: zero_less_mult_iff)

lemma ereal_left_mult_cong:
fixes a b c :: ereal
shows  "c = d ⟹ (d ≠ 0 ⟹ a = b) ⟹ a * c = b * d"
by (cases "c = 0") simp_all

lemma ereal_right_mult_cong:
fixes a b c :: ereal
shows "c = d ⟹ (d ≠ 0 ⟹ a = b) ⟹ c * a = d * b"
by (cases "c = 0") simp_all

lemma ereal_distrib:
fixes a b c :: ereal
assumes "a ≠ ∞ ∨ b ≠ -∞"
and "a ≠ -∞ ∨ b ≠ ∞"
and "¦c¦ ≠ ∞"
shows "(a + b) * c = a * c + b * c"
using assms
by (cases rule: ereal3_cases[of a b c]) (simp_all add: field_simps)

lemma numeral_eq_ereal [simp]: "numeral w = ereal (numeral w)"
apply (induct w rule: num_induct)
apply (simp only: numeral_One one_ereal_def)
apply (simp only: numeral_inc ereal_plus_1)
done

lemma distrib_left_ereal_nn:
"c ≥ 0 ⟹ (x + y) * ereal c = x * ereal c + y * ereal c"
by(cases x y rule: ereal2_cases)(simp_all add: ring_distribs)

lemma setsum_ereal_right_distrib:
fixes f :: "'a ⇒ ereal"
shows "(⋀i. i ∈ A ⟹ 0 ≤ f i) ⟹ r * setsum f A = (∑n∈A. r * f n)"
by (induct A rule: infinite_finite_induct)  (auto simp: ereal_right_distrib setsum_nonneg)

lemma setsum_ereal_left_distrib:
"(⋀i. i ∈ A ⟹ 0 ≤ f i) ⟹ setsum f A * r = (∑n∈A. f n * r :: ereal)"
using setsum_ereal_right_distrib[of A f r] by (simp add: mult_ac)

lemma setsum_left_distrib_ereal:
"c ≥ 0 ⟹ setsum f A * ereal c = (∑x∈A. f x * c :: ereal)"
by(subst setsum_comp_morphism[where h="λx. x * ereal c", symmetric])(simp_all add: distrib_left_ereal_nn)

lemma ereal_le_epsilon:
fixes x y :: ereal
assumes "∀e. 0 < e ⟶ x ≤ y + e"
shows "x ≤ y"
proof -
{
assume a: "∃r. y = ereal r"
then obtain r where r_def: "y = ereal r"
by auto
{
assume "x = -∞"
then have ?thesis by auto
}
moreover
{
assume "x ≠ -∞"
then obtain p where p_def: "x = ereal p"
using a assms[rule_format, of 1]
by (cases x) auto
{
fix e
have "0 < e ⟶ p ≤ r + e"
using assms[rule_format, of "ereal e"] p_def r_def by auto
}
then have "p ≤ r"
apply (subst field_le_epsilon)
apply auto
done
then have ?thesis
using r_def p_def by auto
}
ultimately have ?thesis
by blast
}
moreover
{
assume "y = -∞ | y = ∞"
then have ?thesis
using assms[rule_format, of 1] by (cases x) auto
}
ultimately show ?thesis
by (cases y) auto
qed

lemma ereal_le_epsilon2:
fixes x y :: ereal
assumes "∀e. 0 < e ⟶ x ≤ y + ereal e"
shows "x ≤ y"
proof -
{
fix e :: ereal
assume "e > 0"
{
assume "e = ∞"
then have "x ≤ y + e"
by auto
}
moreover
{
assume "e ≠ ∞"
then obtain r where "e = ereal r"
using ‹e > 0› by (cases e) auto
then have "x ≤ y + e"
using assms[rule_format, of r] ‹e>0› by auto
}
ultimately have "x ≤ y + e"
by blast
}
then show ?thesis
using ereal_le_epsilon by auto
qed

lemma ereal_le_real:
fixes x y :: ereal
assumes "∀z. x ≤ ereal z ⟶ y ≤ ereal z"
shows "y ≤ x"
by (metis assms ereal_bot ereal_cases ereal_infty_less_eq(2) ereal_less_eq(1) linorder_le_cases)

lemma setprod_ereal_0:
fixes f :: "'a ⇒ ereal"
shows "(∏i∈A. f i) = 0 ⟷ finite A ∧ (∃i∈A. f i = 0)"
proof (cases "finite A")
case True
then show ?thesis by (induct A) auto
next
case False
then show ?thesis by auto
qed

lemma setprod_ereal_pos:
fixes f :: "'a ⇒ ereal"
assumes pos: "⋀i. i ∈ I ⟹ 0 ≤ f i"
shows "0 ≤ (∏i∈I. f i)"
proof (cases "finite I")
case True
from this pos show ?thesis
by induct auto
next
case False
then show ?thesis by simp
qed

lemma setprod_PInf:
fixes f :: "'a ⇒ ereal"
assumes "⋀i. i ∈ I ⟹ 0 ≤ f i"
shows "(∏i∈I. f i) = ∞ ⟷ finite I ∧ (∃i∈I. f i = ∞) ∧ (∀i∈I. f i ≠ 0)"
proof (cases "finite I")
case True
from this assms show ?thesis
proof (induct I)
case (insert i I)
then have pos: "0 ≤ f i" "0 ≤ setprod f I"
by (auto intro!: setprod_ereal_pos)
from insert have "(∏j∈insert i I. f j) = ∞ ⟷ setprod f I * f i = ∞"
by auto
also have "… ⟷ (setprod f I = ∞ ∨ f i = ∞) ∧ f i ≠ 0 ∧ setprod f I ≠ 0"
using setprod_ereal_pos[of I f] pos
by (cases rule: ereal2_cases[of "f i" "setprod f I"]) auto
also have "… ⟷ finite (insert i I) ∧ (∃j∈insert i I. f j = ∞) ∧ (∀j∈insert i I. f j ≠ 0)"
using insert by (auto simp: setprod_ereal_0)
finally show ?case .
qed simp
next
case False
then show ?thesis by simp
qed

lemma setprod_ereal: "(∏i∈A. ereal (f i)) = ereal (setprod f A)"
proof (cases "finite A")
case True
then show ?thesis
by induct (auto simp: one_ereal_def)
next
case False
then show ?thesis
qed

subsubsection ‹Power›

lemma ereal_power[simp]: "(ereal x) ^ n = ereal (x^n)"
by (induct n) (auto simp: one_ereal_def)

lemma ereal_power_PInf[simp]: "(∞::ereal) ^ n = (if n = 0 then 1 else ∞)"
by (induct n) (auto simp: one_ereal_def)

lemma ereal_power_uminus[simp]:
fixes x :: ereal
shows "(- x) ^ n = (if even n then x ^ n else - (x^n))"
by (induct n) (auto simp: one_ereal_def)

lemma ereal_power_numeral[simp]:
"(numeral num :: ereal) ^ n = ereal (numeral num ^ n)"
by (induct n) (auto simp: one_ereal_def)

lemma zero_le_power_ereal[simp]:
fixes a :: ereal
assumes "0 ≤ a"
shows "0 ≤ a ^ n"
using assms by (induct n) (auto simp: ereal_zero_le_0_iff)

subsubsection ‹Subtraction›

lemma ereal_minus_minus_image[simp]:
fixes S :: "ereal set"
shows "uminus ` uminus ` S = S"
by (auto simp: image_iff)

lemma ereal_uminus_lessThan[simp]:
fixes a :: ereal
shows "uminus ` {..<a} = {-a<..}"
proof -
{
fix x
assume "-a < x"
then have "- x < - (- a)"
by (simp del: ereal_uminus_uminus)
then have "- x < a"
by simp
}
then show ?thesis
by force
qed

lemma ereal_uminus_greaterThan[simp]: "uminus ` {(a::ereal)<..} = {..<-a}"
by (metis ereal_uminus_lessThan ereal_uminus_uminus ereal_minus_minus_image)

instantiation ereal :: minus
begin

definition "x - y = x + -(y::ereal)"
instance ..

end

lemma ereal_minus[simp]:
"ereal r - ereal p = ereal (r - p)"
"-∞ - ereal r = -∞"
"ereal r - ∞ = -∞"
"(∞::ereal) - x = ∞"
"-(∞::ereal) - ∞ = -∞"
"x - -y = x + y"
"x - 0 = x"
"0 - x = -x"

lemma ereal_x_minus_x[simp]: "x - x = (if ¦x¦ = ∞ then ∞ else 0::ereal)"
by (cases x) simp_all

lemma ereal_eq_minus_iff:
fixes x y z :: ereal
shows "x = z - y ⟷
(¦y¦ ≠ ∞ ⟶ x + y = z) ∧
(y = -∞ ⟶ x = ∞) ∧
(y = ∞ ⟶ z = ∞ ⟶ x = ∞) ∧
(y = ∞ ⟶ z ≠ ∞ ⟶ x = -∞)"
by (cases rule: ereal3_cases[of x y z]) auto

lemma ereal_eq_minus:
fixes x y z :: ereal
shows "¦y¦ ≠ ∞ ⟹ x = z - y ⟷ x + y = z"
by (auto simp: ereal_eq_minus_iff)

lemma ereal_less_minus_iff:
fixes x y z :: ereal
shows "x < z - y ⟷
(y = ∞ ⟶ z = ∞ ∧ x ≠ ∞) ∧
(y = -∞ ⟶ x ≠ ∞) ∧
(¦y¦ ≠ ∞⟶ x + y < z)"
by (cases rule: ereal3_cases[of x y z]) auto

lemma ereal_less_minus:
fixes x y z :: ereal
shows "¦y¦ ≠ ∞ ⟹ x < z - y ⟷ x + y < z"
by (auto simp: ereal_less_minus_iff)

lemma ereal_le_minus_iff:
fixes x y z :: ereal
shows "x ≤ z - y ⟷ (y = ∞ ⟶ z ≠ ∞ ⟶ x = -∞) ∧ (¦y¦ ≠ ∞ ⟶ x + y ≤ z)"
by (cases rule: ereal3_cases[of x y z]) auto

lemma ereal_le_minus:
fixes x y z :: ereal
shows "¦y¦ ≠ ∞ ⟹ x ≤ z - y ⟷ x + y ≤ z"
by (auto simp: ereal_le_minus_iff)

lemma ereal_minus_less_iff:
fixes x y z :: ereal
shows "x - y < z ⟷ y ≠ -∞ ∧ (y = ∞ ⟶ x ≠ ∞ ∧ z ≠ -∞) ∧ (y ≠ ∞ ⟶ x < z + y)"
by (cases rule: ereal3_cases[of x y z]) auto

lemma ereal_minus_less:
fixes x y z :: ereal
shows "¦y¦ ≠ ∞ ⟹ x - y < z ⟷ x < z + y"
by (auto simp: ereal_minus_less_iff)

lemma ereal_minus_le_iff:
fixes x y z :: ereal
shows "x - y ≤ z ⟷
(y = -∞ ⟶ z = ∞) ∧
(y = ∞ ⟶ x = ∞ ⟶ z = ∞) ∧
(¦y¦ ≠ ∞ ⟶ x ≤ z + y)"
by (cases rule: ereal3_cases[of x y z]) auto

lemma ereal_minus_le:
fixes x y z :: ereal
shows "¦y¦ ≠ ∞ ⟹ x - y ≤ z ⟷ x ≤ z + y"
by (auto simp: ereal_minus_le_iff)

lemma ereal_minus_eq_minus_iff:
fixes a b c :: ereal
shows "a - b = a - c ⟷
b = c ∨ a = ∞ ∨ (a = -∞ ∧ b ≠ -∞ ∧ c ≠ -∞)"
by (cases rule: ereal3_cases[of a b c]) auto

fixes a b c :: ereal
shows "c + a ≤ c + b ⟷
a ≤ b ∨ c = ∞ ∨ (c = -∞ ∧ a ≠ ∞ ∧ b ≠ ∞)"
by (cases rule: ereal3_cases[of a b c]) (simp_all add: field_simps)

fixes a b c :: ereal
shows "a + c ≤ b + c ⟷ a ≤ b ∨ c = ∞ ∨ (c = -∞ ∧ a ≠ ∞ ∧ b ≠ ∞)"
by(cases rule: ereal3_cases[of a b c])(simp_all add: field_simps)

lemma ereal_mult_le_mult_iff:
fixes a b c :: ereal
shows "¦c¦ ≠ ∞ ⟹ c * a ≤ c * b ⟷ (0 < c ⟶ a ≤ b) ∧ (c < 0 ⟶ b ≤ a)"
by (cases rule: ereal3_cases[of a b c]) (simp_all add: mult_le_cancel_left)

lemma ereal_minus_mono:
fixes A B C D :: ereal assumes "A ≤ B" "D ≤ C"
shows "A - C ≤ B - D"
using assms
by (cases rule: ereal3_cases[case_product ereal_cases, of A B C D]) simp_all

lemma real_of_ereal_minus:
fixes a b :: ereal
shows "real_of_ereal (a - b) = (if ¦a¦ = ∞ ∨ ¦b¦ = ∞ then 0 else real_of_ereal a - real_of_ereal b)"
by (cases rule: ereal2_cases[of a b]) auto

lemma real_of_ereal_minus': "¦x¦ = ∞ ⟷ ¦y¦ = ∞ ⟹ real_of_ereal x - real_of_ereal y = real_of_ereal (x - y :: ereal)"
by(subst real_of_ereal_minus) auto

lemma ereal_diff_positive:
fixes a b :: ereal shows "a ≤ b ⟹ 0 ≤ b - a"
by (cases rule: ereal2_cases[of a b]) auto

lemma ereal_between:
fixes x e :: ereal
assumes "¦x¦ ≠ ∞"
and "0 < e"
shows "x - e < x"
and "x < x + e"
using assms
apply (cases x, cases e)
apply auto
using assms
apply (cases x, cases e)
apply auto
done

lemma ereal_minus_eq_PInfty_iff:
fixes x y :: ereal
shows "x - y = ∞ ⟷ y = -∞ ∨ x = ∞"
by (cases x y rule: ereal2_cases) simp_all

fixes x y z :: ereal
shows "¦y¦ ≠ ∞ ⟹ x - (y + z) = x - y - z"
by(cases x y z rule: ereal3_cases) simp_all

fixes x y z :: ereal
shows "x + y - z = x - z + y"
by(cases x y z rule: ereal3_cases) simp_all

lemma ereal_add_uminus_conv_diff: fixes x y z :: ereal shows "- x + y = y - x"
by(cases x y rule: ereal2_cases) simp_all

lemma ereal_minus_diff_eq:
fixes x y :: ereal
shows "⟦ x = ∞ ⟶ y ≠ ∞; x = -∞ ⟶ y ≠ - ∞ ⟧ ⟹ - (x - y) = y - x"
by(cases x y rule: ereal2_cases) simp_all

lemma ediff_le_self [simp]: "x - y ≤ (x :: enat)"
by(cases x y rule: enat.exhaust[case_product enat.exhaust]) simp_all

subsubsection ‹Division›

instantiation ereal :: inverse
begin

function inverse_ereal where
"inverse (ereal r) = (if r = 0 then ∞ else ereal (inverse r))"
| "inverse (∞::ereal) = 0"
| "inverse (-∞::ereal) = 0"
by (auto intro: ereal_cases)
termination by (relation "{}") simp

definition "x div y = x * inverse (y :: ereal)"

instance ..

end

lemma real_of_ereal_inverse[simp]:
fixes a :: ereal
shows "real_of_ereal (inverse a) = 1 / real_of_ereal a"
by (cases a) (auto simp: inverse_eq_divide)

lemma ereal_inverse[simp]:
"inverse (0::ereal) = ∞"
"inverse (1::ereal) = 1"

lemma ereal_divide[simp]:
"ereal r / ereal p = (if p = 0 then ereal r * ∞ else ereal (r / p))"
unfolding divide_ereal_def by (auto simp: divide_real_def)

lemma ereal_divide_same[simp]:
fixes x :: ereal
shows "x / x = (if ¦x¦ = ∞ ∨ x = 0 then 0 else 1)"
by (cases x) (simp_all add: divide_real_def divide_ereal_def one_ereal_def)

lemma ereal_inv_inv[simp]:
fixes x :: ereal
shows "inverse (inverse x) = (if x ≠ -∞ then x else ∞)"
by (cases x) auto

lemma ereal_inverse_minus[simp]:
fixes x :: ereal
shows "inverse (- x) = (if x = 0 then ∞ else -inverse x)"
by (cases x) simp_all

lemma ereal_uminus_divide[simp]:
fixes x y :: ereal
shows "- x / y = - (x / y)"
unfolding divide_ereal_def by simp

lemma ereal_divide_Infty[simp]:
fixes x :: ereal
shows "x / ∞ = 0" "x / -∞ = 0"
unfolding divide_ereal_def by simp_all

lemma ereal_divide_one[simp]: "x / 1 = (x::ereal)"
unfolding divide_ereal_def by simp

lemma ereal_divide_ereal[simp]: "∞ / ereal r = (if 0 ≤ r then ∞ else -∞)"
unfolding divide_ereal_def by simp

lemma ereal_inverse_nonneg_iff: "0 ≤ inverse (x :: ereal) ⟷ 0 ≤ x ∨ x = -∞"
by (cases x) auto

lemma inverse_ereal_ge0I: "0 ≤ (x :: ereal) ⟹ 0 ≤ inverse x"
by(cases x) simp_all

lemma zero_le_divide_ereal[simp]:
fixes a :: ereal
assumes "0 ≤ a"
and "0 ≤ b"
shows "0 ≤ a / b"
using assms by (cases rule: ereal2_cases[of a b]) (auto simp: zero_le_divide_iff)

lemma ereal_le_divide_pos:
fixes x y z :: ereal
shows "x > 0 ⟹ x ≠ ∞ ⟹ y ≤ z / x ⟷ x * y ≤ z"
by (cases rule: ereal3_cases[of x y z]) (auto simp: field_simps)

lemma ereal_divide_le_pos:
fixes x y z :: ereal
shows "x > 0 ⟹ x ≠ ∞ ⟹ z / x ≤ y ⟷ z ≤ x * y"
by (cases rule: ereal3_cases[of x y z]) (auto simp: field_simps)

lemma ereal_le_divide_neg:
fixes x y z :: ereal
shows "x < 0 ⟹ x ≠ -∞ ⟹ y ≤ z / x ⟷ z ≤ x * y"
by (cases rule: ereal3_cases[of x y z]) (auto simp: field_simps)

lemma ereal_divide_le_neg:
fixes x y z :: ereal
shows "x < 0 ⟹ x ≠ -∞ ⟹ z / x ≤ y ⟷ x * y ≤ z"
by (cases rule: ereal3_cases[of x y z]) (auto simp: field_simps)

lemma ereal_inverse_antimono_strict:
fixes x y :: ereal
shows "0 ≤ x ⟹ x < y ⟹ inverse y < inverse x"
by (cases rule: ereal2_cases[of x y]) auto

lemma ereal_inverse_antimono:
fixes x y :: ereal
shows "0 ≤ x ⟹ x ≤ y ⟹ inverse y ≤ inverse x"
by (cases rule: ereal2_cases[of x y]) auto

lemma inverse_inverse_Pinfty_iff[simp]:
fixes x :: ereal
shows "inverse x = ∞ ⟷ x = 0"
by (cases x) auto

lemma ereal_inverse_eq_0:
fixes x :: ereal
shows "inverse x = 0 ⟷ x = ∞ ∨ x = -∞"
by (cases x) auto

lemma ereal_0_gt_inverse:
fixes x :: ereal
shows "0 < inverse x ⟷ x ≠ ∞ ∧ 0 ≤ x"
by (cases x) auto

lemma ereal_inverse_le_0_iff:
fixes x :: ereal
shows "inverse x ≤ 0 ⟷ x < 0 ∨ x = ∞"
by(cases x) auto

lemma ereal_divide_eq_0_iff: "x / y = 0 ⟷ x = 0 ∨ ¦y :: ereal¦ = ∞"
by(cases x y rule: ereal2_cases) simp_all

lemma ereal_mult_less_right:
fixes a b c :: ereal
assumes "b * a < c * a"
and "0 < a"
and "a < ∞"
shows "b < c"
using assms
by (cases rule: ereal3_cases[of a b c])
(auto split: split_if_asm simp: zero_less_mult_iff zero_le_mult_iff)

lemma ereal_mult_divide: fixes a b :: ereal shows "0 < b ⟹ b < ∞ ⟹ b * (a / b) = a"
by (cases a b rule: ereal2_cases) auto

lemma ereal_power_divide:
fixes x y :: ereal
shows "y ≠ 0 ⟹ (x / y) ^ n = x^n / y^n"
by (cases rule: ereal2_cases [of x y])
(auto simp: one_ereal_def zero_ereal_def power_divide zero_le_power_eq)

lemma ereal_le_mult_one_interval:
fixes x y :: ereal
assumes y: "y ≠ -∞"
assumes z: "⋀z. 0 < z ⟹ z < 1 ⟹ z * x ≤ y"
shows "x ≤ y"
proof (cases x)
case PInf
with z[of "1 / 2"] show "x ≤ y"
next
case (real r)
note r = this
show "x ≤ y"
proof (cases y)
case (real p)
note p = this
have "r ≤ p"
proof (rule field_le_mult_one_interval)
fix z :: real
assume "0 < z" and "z < 1"
with z[of "ereal z"] show "z * r ≤ p"
using p r by (auto simp: zero_le_mult_iff one_ereal_def)
qed
then show "x ≤ y"
using p r by simp
qed (insert y, simp_all)
qed simp

lemma ereal_divide_right_mono[simp]:
fixes x y z :: ereal
assumes "x ≤ y"
and "0 < z"
shows "x / z ≤ y / z"
using assms by (cases x y z rule: ereal3_cases) (auto intro: divide_right_mono)

lemma ereal_divide_left_mono[simp]:
fixes x y z :: ereal
assumes "y ≤ x"
and "0 < z"
and "0 < x * y"
shows "z / x ≤ z / y"
using assms
by (cases x y z rule: ereal3_cases)
(auto intro: divide_left_mono simp: field_simps zero_less_mult_iff mult_less_0_iff split: split_if_asm)

lemma ereal_divide_zero_left[simp]:
fixes a :: ereal
shows "0 / a = 0"
by (cases a) (auto simp: zero_ereal_def)

lemma ereal_times_divide_eq_left[simp]:
fixes a b c :: ereal
shows "b / c * a = b * a / c"
by (cases a b c rule: ereal3_cases) (auto simp: field_simps zero_less_mult_iff mult_less_0_iff)

lemma ereal_times_divide_eq: "a * (b / c :: ereal) = a * b / c"
by (cases a b c rule: ereal3_cases)
(auto simp: field_simps zero_less_mult_iff)

lemma ereal_inverse_real: "¦z¦ ≠ ∞ ⟹ z ≠ 0 ⟹ ereal (inverse (real_of_ereal z)) = inverse z"
by (cases z) simp_all

lemma ereal_inverse_mult:
"a ≠ 0 ⟹ b ≠ 0 ⟹ inverse (a * (b::ereal)) = inverse a * inverse b"
by (cases a; cases b) auto

subsection "Complete lattice"

instantiation ereal :: lattice
begin

definition [simp]: "sup x y = (max x y :: ereal)"
definition [simp]: "inf x y = (min x y :: ereal)"
instance by standard simp_all

end

instantiation ereal :: complete_lattice
begin

definition "bot = (-∞::ereal)"
definition "top = (∞::ereal)"

definition "Sup S = (SOME x :: ereal. (∀y∈S. y ≤ x) ∧ (∀z. (∀y∈S. y ≤ z) ⟶ x ≤ z))"
definition "Inf S = (SOME x :: ereal. (∀y∈S. x ≤ y) ∧ (∀z. (∀y∈S. z ≤ y) ⟶ z ≤ x))"

lemma ereal_complete_Sup:
fixes S :: "ereal set"
shows "∃x. (∀y∈S. y ≤ x) ∧ (∀z. (∀y∈S. y ≤ z) ⟶ x ≤ z)"
proof (cases "∃x. ∀a∈S. a ≤ ereal x")
case True
then obtain y where y: "⋀a. a∈S ⟹ a ≤ ereal y"
by auto
then have "∞ ∉ S"
by force
show ?thesis
proof (cases "S ≠ {-∞} ∧ S ≠ {}")
case True
with ‹∞ ∉ S› obtain x where x: "x ∈ S" "¦x¦ ≠ ∞"
by auto
obtain s where s: "∀x∈ereal -` S. x ≤ s" "⋀z. (∀x∈ereal -` S. x ≤ z) ⟹ s ≤ z"
proof (atomize_elim, rule complete_real)
show "∃x. x ∈ ereal -` S"
using x by auto
show "∃z. ∀x∈ereal -` S. x ≤ z"
by (auto dest: y intro!: exI[of _ y])
qed
show ?thesis
proof (safe intro!: exI[of _ "ereal s"])
fix y
assume "y ∈ S"
with s ‹∞ ∉ S› show "y ≤ ereal s"
by (cases y) auto
next
fix z
assume "∀y∈S. y ≤ z"
with ‹S ≠ {-∞} ∧ S ≠ {}› show "ereal s ≤ z"
by (cases z) (auto intro!: s)
qed
next
case False
then show ?thesis
by (auto intro!: exI[of _ "-∞"])
qed
next
case False
then show ?thesis
by (fastforce intro!: exI[of _ ] ereal_top intro: order_trans dest: less_imp_le simp: not_le)
qed

lemma ereal_complete_uminus_eq:
fixes S :: "ereal set"
shows "(∀y∈uminus`S. y ≤ x) ∧ (∀z. (∀y∈uminus`S. y ≤ z) ⟶ x ≤ z)
⟷ (∀y∈S. -x ≤ y) ∧ (∀z. (∀y∈S. z ≤ y) ⟶ z ≤ -x)"
by simp (metis ereal_minus_le_minus ereal_uminus_uminus)

lemma ereal_complete_Inf:
"∃x. (∀y∈S::ereal set. x ≤ y) ∧ (∀z. (∀y∈S. z ≤ y) ⟶ z ≤ x)"
using ereal_complete_Sup[of "uminus ` S"]
unfolding ereal_complete_uminus_eq
by auto

instance
proof
show "Sup {} = (bot::ereal)"
apply (auto simp: bot_ereal_def Sup_ereal_def)
apply (rule some1_equality)
apply (metis ereal_bot ereal_less_eq(2))
apply (metis ereal_less_eq(2))
done
show "Inf {} = (top::ereal)"
apply (auto simp: top_ereal_def Inf_ereal_def)
apply (rule some1_equality)
apply (metis ereal_top ereal_less_eq(1))
apply (metis ereal_less_eq(1))
done
qed (auto intro: someI2_ex ereal_complete_Sup ereal_complete_Inf
simp: Sup_ereal_def Inf_ereal_def bot_ereal_def top_ereal_def)

end

instance ereal :: complete_linorder ..

instance ereal :: linear_continuum
proof
show "∃a b::ereal. a ≠ b"
using zero_neq_one by blast
qed

subsubsection "Topological space"

instantiation ereal :: linear_continuum_topology
begin

definition "open_ereal" :: "ereal set ⇒ bool" where
open_ereal_generated: "open_ereal = generate_topology (range lessThan ∪ range greaterThan)"

instance

end

lemma continuous_on_ereal[continuous_intros]:
assumes f: "continuous_on s f" shows "continuous_on s (λx. ereal (f x))"
by (rule continuous_on_compose2 [OF continuous_onI_mono[of ereal UNIV] f]) auto

lemma tendsto_ereal[tendsto_intros, simp, intro]: "(f ⤏ x) F ⟹ ((λx. ereal (f x)) ⤏ ereal x) F"
using isCont_tendsto_compose[of x ereal f F] continuous_on_ereal[of UNIV "λx. x"]

lemma tendsto_uminus_ereal[tendsto_intros, simp, intro]: "(f ⤏ x) F ⟹ ((λx. - f x::ereal) ⤏ - x) F"
apply (rule tendsto_compose[where g=uminus])
apply (auto intro!: order_tendstoI simp: eventually_at_topological)
apply (rule_tac x="{..< -a}" in exI)
apply (auto split: ereal.split simp: ereal_less_uminus_reorder) []
apply (rule_tac x="{- a <..}" in exI)
apply (auto split: ereal.split simp: ereal_uminus_reorder) []
done

lemma at_infty_ereal_eq_at_top: "at ∞ = filtermap ereal at_top"
unfolding filter_eq_iff eventually_at_filter eventually_at_top_linorder eventually_filtermap
top_ereal_def[symmetric]
apply (subst eventually_nhds_top[of 0])
apply (auto simp: top_ereal_def less_le ereal_all_split ereal_ex_split)
apply (metis PInfty_neq_ereal(2) ereal_less_eq(3) ereal_top le_cases order_trans)
done

lemma ereal_Lim_uminus: "(f ⤏ f0) net ⟷ ((λx. - f x::ereal) ⤏ - f0) net"
using tendsto_uminus_ereal[of f f0 net] tendsto_uminus_ereal[of "λx. - f x" "- f0" net]
by auto

lemma ereal_divide_less_iff: "0 < (c::ereal) ⟹ c < ∞ ⟹ a / c < b ⟷ a < b * c"
by (cases a b c rule: ereal3_cases) (auto simp: field_simps)

lemma ereal_less_divide_iff: "0 < (c::ereal) ⟹ c < ∞ ⟹ a < b / c ⟷ a * c < b"
by (cases a b c rule: ereal3_cases) (auto simp: field_simps)

lemma tendsto_cmult_ereal[tendsto_intros, simp, intro]:
assumes c: "¦c¦ ≠ ∞" and f: "(f ⤏ x) F" shows "((λx. c * f x::ereal) ⤏ c * x) F"
proof -
{ fix c :: ereal assume "0 < c" "c < ∞"
then have "((λx. c * f x::ereal) ⤏ c * x) F"
apply (intro tendsto_compose[OF _ f])
apply (auto intro!: order_tendstoI simp: eventually_at_topological)
apply (rule_tac x="{a/c <..}" in exI)
apply (auto split: ereal.split simp: ereal_divide_less_iff mult.commute) []
apply (rule_tac x="{..< a/c}" in exI)
apply (auto split: ereal.split simp: ereal_less_divide_iff mult.commute) []
done }
note * = this

have "((0 < c ∧ c < ∞) ∨ (-∞ < c ∧ c < 0) ∨ c = 0)"
using c by (cases c) auto
then show ?thesis
proof (elim disjE conjE)
assume "- ∞ < c" "c < 0"
then have "0 < - c" "- c < ∞"
by (auto simp: ereal_uminus_reorder ereal_less_uminus_reorder[of 0])
then have "((λx. (- c) * f x) ⤏ (- c) * x) F"
by (rule *)
from tendsto_uminus_ereal[OF this] show ?thesis
by simp
qed (auto intro!: *)
qed

lemma tendsto_cmult_ereal_not_0[tendsto_intros, simp, intro]:
assumes "x ≠ 0" and f: "(f ⤏ x) F" shows "((λx. c * f x::ereal) ⤏ c * x) F"
proof cases
assume "¦c¦ = ∞"
show ?thesis
proof (rule filterlim_cong[THEN iffD1, OF refl refl _ tendsto_const])
have "0 < x ∨ x < 0"
using ‹x ≠ 0› by (auto simp add: neq_iff)
then show "eventually (λx'. c * x = c * f x') F"
proof
assume "0 < x" from order_tendstoD(1)[OF f this] show ?thesis
by eventually_elim (insert ‹0<x› ‹¦c¦ = ∞›, auto)
next
assume "x < 0" from order_tendstoD(2)[OF f this] show ?thesis
by eventually_elim (insert ‹x<0› ‹¦c¦ = ∞›, auto)
qed
qed
qed (rule tendsto_cmult_ereal[OF _ f])

assumes c: "y ≠ - ∞" "x ≠ - ∞" and f: "(f ⤏ x) F" shows "((λx. f x + y::ereal) ⤏ x + y) F"
apply (intro tendsto_compose[OF _ f])
apply (auto intro!: order_tendstoI simp: eventually_at_topological)
apply (rule_tac x="{a - y <..}" in exI)
apply (auto split: ereal.split simp: ereal_minus_less_iff c) []
apply (rule_tac x="{..< a - y}" in exI)
apply (auto split: ereal.split simp: ereal_less_minus_iff c) []
done

assumes c: "¦y¦ ≠ ∞" and f: "(f ⤏ x) F" shows "((λx. f x + y::ereal) ⤏ x + y) F"
apply (intro tendsto_compose[OF _ f])
apply (auto intro!: order_tendstoI simp: eventually_at_topological)
apply (rule_tac x="{a - y <..}" in exI)
apply (insert c, auto split: ereal.split simp: ereal_minus_less_iff) []
apply (rule_tac x="{..< a - y}" in exI)
apply (auto split: ereal.split simp: ereal_less_minus_iff c) []
done

lemma continuous_at_ereal[continuous_intros]: "continuous F f ⟹ continuous F (λx. ereal (f x))"
unfolding continuous_def by auto

lemma ereal_Sup:
assumes *: "¦SUP a:A. ereal a¦ ≠ ∞"
shows "ereal (Sup A) = (SUP a:A. ereal a)"
proof (rule continuous_at_Sup_mono)
obtain r where r: "ereal r = (SUP a:A. ereal a)" "A ≠ {}"
using * by (force simp: bot_ereal_def)
then show "bdd_above A" "A ≠ {}"
by (auto intro!: SUP_upper bdd_aboveI[of _ r] simp add: ereal_less_eq(3)[symmetric] simp del: ereal_less_eq)
qed (auto simp: mono_def continuous_at_imp_continuous_at_within continuous_at_ereal)

lemma ereal_SUP: "¦SUP a:A. ereal (f a)¦ ≠ ∞ ⟹ ereal (SUP a:A. f a) = (SUP a:A. ereal (f a))"
using ereal_Sup[of "f`A"] by auto

lemma ereal_Inf:
assumes *: "¦INF a:A. ereal a¦ ≠ ∞"
shows "ereal (Inf A) = (INF a:A. ereal a)"
proof (rule continuous_at_Inf_mono)
obtain r where r: "ereal r = (INF a:A. ereal a)" "A ≠ {}"
using * by (force simp: top_ereal_def)
then show "bdd_below A" "A ≠ {}"
by (auto intro!: INF_lower bdd_belowI[of _ r] simp add: ereal_less_eq(3)[symmetric] simp del: ereal_less_eq)
qed (auto simp: mono_def continuous_at_imp_continuous_at_within continuous_at_ereal)

lemma ereal_Inf':
assumes *: "bdd_below A" "A ≠ {}"
shows "ereal (Inf A) = (INF a:A. ereal a)"
proof (rule ereal_Inf)
from * obtain l u where "⋀x. x ∈ A ⟹ l ≤ x" "u ∈ A"
by (auto simp: bdd_below_def)
then have "l ≤ (INF x:A. ereal x)" "(INF x:A. ereal x) ≤ u"
by (auto intro!: INF_greatest INF_lower)
then show "¦INF a:A. ereal a¦ ≠ ∞"
by auto
qed

lemma ereal_INF: "¦INF a:A. ereal (f a)¦ ≠ ∞ ⟹ ereal (INF a:A. f a) = (INF a:A. ereal (f a))"
using ereal_Inf[of "f`A"] by auto

lemma ereal_Sup_uminus_image_eq: "Sup (uminus ` S::ereal set) = - Inf S"
by (auto intro!: SUP_eqI
simp: Ball_def[symmetric] ereal_uminus_le_reorder le_Inf_iff
intro!: complete_lattice_class.Inf_lower2)

lemma ereal_SUP_uminus_eq:
fixes f :: "'a ⇒ ereal"
shows "(SUP x:S. uminus (f x)) = - (INF x:S. f x)"
using ereal_Sup_uminus_image_eq [of "f ` S"] by (simp add: comp_def)

lemma ereal_inj_on_uminus[intro, simp]: "inj_on uminus (A :: ereal set)"
by (auto intro!: inj_onI)

lemma ereal_Inf_uminus_image_eq: "Inf (uminus ` S::ereal set) = - Sup S"
using ereal_Sup_uminus_image_eq[of "uminus ` S"] by simp

lemma ereal_INF_uminus_eq:
fixes f :: "'a ⇒ ereal"
shows "(INF x:S. - f x) = - (SUP x:S. f x)"
using ereal_Inf_uminus_image_eq [of "f ` S"] by (simp add: comp_def)

lemma ereal_SUP_uminus:
fixes f :: "'a ⇒ ereal"
shows "(SUP i : R. - f i) = - (INF i : R. f i)"
using ereal_Sup_uminus_image_eq[of "f`R"]

lemma ereal_SUP_not_infty:
fixes f :: "_ ⇒ ereal"
shows "A ≠ {} ⟹ l ≠ -∞ ⟹ u ≠ ∞ ⟹ ∀a∈A. l ≤ f a ∧ f a ≤ u ⟹ ¦SUPREMUM A f¦ ≠ ∞"
using SUP_upper2[of _ A l f] SUP_least[of A f u]
by (cases "SUPREMUM A f") auto

lemma ereal_INF_not_infty:
fixes f :: "_ ⇒ ereal"
shows "A ≠ {} ⟹ l ≠ -∞ ⟹ u ≠ ∞ ⟹ ∀a∈A. l ≤ f a ∧ f a ≤ u ⟹ ¦INFIMUM A f¦ ≠ ∞"
using INF_lower2[of _ A f u] INF_greatest[of A l f]
by (cases "INFIMUM A f") auto

lemma ereal_image_uminus_shift:
fixes X Y :: "ereal set"
shows "uminus ` X = Y ⟷ X = uminus ` Y"
proof
assume "uminus ` X = Y"
then have "uminus ` uminus ` X = uminus ` Y"
then show "X = uminus ` Y"

lemma Sup_eq_MInfty:
fixes S :: "ereal set"
shows "Sup S = -∞ ⟷ S = {} ∨ S = {-∞}"
unfolding bot_ereal_def[symmetric] by auto

lemma Inf_eq_PInfty:
fixes S :: "ereal set"
shows "Inf S = ∞ ⟷ S = {} ∨ S = {∞}"
using Sup_eq_MInfty[of "uminus`S"]
unfolding ereal_Sup_uminus_image_eq ereal_image_uminus_shift by simp

lemma Inf_eq_MInfty:
fixes S :: "ereal set"
shows "-∞ ∈ S ⟹ Inf S = -∞"
unfolding bot_ereal_def[symmetric] by auto

lemma Sup_eq_PInfty:
fixes S :: "ereal set"
shows "∞ ∈ S ⟹ Sup S = ∞"
unfolding top_ereal_def[symmetric] by auto

lemma not_MInfty_nonneg[simp]: "0 ≤ (x::ereal) ⟹ x ≠ - ∞"
by auto

lemma Sup_ereal_close:
fixes e :: ereal
assumes "0 < e"
and S: "¦Sup S¦ ≠ ∞" "S ≠ {}"
shows "∃x∈S. Sup S - e < x"
using assms by (cases e) (auto intro!: less_Sup_iff[THEN iffD1])

lemma Inf_ereal_close:
fixes e :: ereal
assumes "¦Inf X¦ ≠ ∞"
and "0 < e"
shows "∃x∈X. x < Inf X + e"
proof (rule Inf_less_iff[THEN iffD1])
show "Inf X < Inf X + e"
using assms by (cases e) auto
qed

lemma SUP_PInfty:
"(⋀n::nat. ∃i∈A. ereal (real n) ≤ f i) ⟹ (SUP i:A. f i :: ereal) = ∞"
unfolding top_ereal_def[symmetric] SUP_eq_top_iff
by (metis MInfty_neq_PInfty(2) PInfty_neq_ereal(2) less_PInf_Ex_of_nat less_ereal.elims(2) less_le_trans)

lemma SUP_nat_Infty: "(SUP i::nat. ereal (real i)) = ∞"
by (rule SUP_PInfty) auto

assumes "I ≠ {}" "c ≠ -∞"
shows "(SUP i:I. f i + c :: ereal) = (SUP i:I. f i) + c"
proof cases
assume "(SUP i:I. f i) = - ∞"
moreover then have "⋀i. i ∈ I ⟹ f i = -∞"
unfolding Sup_eq_MInfty Sup_image_eq[symmetric] by auto
ultimately show ?thesis
by (cases c) (auto simp: ‹I ≠ {}›)
next
assume "(SUP i:I. f i) ≠ - ∞" then show ?thesis
unfolding Sup_image_eq[symmetric]
by (subst continuous_at_Sup_mono[where f="λx. x + c"])
(auto simp: continuous_at_imp_continuous_at_within continuous_at mono_def ereal_add_mono ‹I ≠ {}› ‹c ≠ -∞›)
qed

fixes c :: ereal
shows "I ≠ {} ⟹ c ≠ -∞ ⟹ (SUP i:I. c + f i) = c + (SUP i:I. f i)"

lemma SUP_ereal_minus_right:
assumes "I ≠ {}" "c ≠ -∞"
shows "(SUP i:I. c - f i :: ereal) = c - (INF i:I. f i)"
using SUP_ereal_add_right[OF assms, of "λi. - f i"]

lemma SUP_ereal_minus_left:
assumes "I ≠ {}" "c ≠ ∞"
shows "(SUP i:I. f i - c:: ereal) = (SUP i:I. f i) - c"
using SUP_ereal_add_left[OF ‹I ≠ {}›, of "-c" f] by (simp add: ‹c ≠ ∞› minus_ereal_def)

lemma INF_ereal_minus_right:
assumes "I ≠ {}" and "¦c¦ ≠ ∞"
shows "(INF i:I. c - f i) = c - (SUP i:I. f i::ereal)"
proof -
{ fix b have "(-c) + b = - (c - b)"
using ‹¦c¦ ≠ ∞› by (cases c b rule: ereal2_cases) auto }
note * = this
show ?thesis
using SUP_ereal_add_right[OF ‹I ≠ {}›, of "-c" f] ‹¦c¦ ≠ ∞›
by (auto simp add: * ereal_SUP_uminus_eq)
qed

fixes f :: "'i ⇒ ereal"
assumes "⋀i. f i + y ≤ z" and "y ≠ -∞"
shows "SUPREMUM UNIV f + y ≤ z"
unfolding SUP_ereal_add_left[OF UNIV_not_empty ‹y ≠ -∞›, symmetric]
by (rule SUP_least assms)+

lemma SUP_combine:
fixes f :: "'a::semilattice_sup ⇒ 'a::semilattice_sup ⇒ 'b::complete_lattice"
assumes mono: "⋀a b c d. a ≤ b ⟹ c ≤ d ⟹ f a c ≤ f b d"
shows "(SUP i:UNIV. SUP j:UNIV. f i j) = (SUP i. f i i)"
proof (rule antisym)
show "(SUP i j. f i j) ≤ (SUP i. f i i)"
by (rule SUP_least SUP_upper2[where i="sup i j" for i j] UNIV_I mono sup_ge1 sup_ge2)+
show "(SUP i. f i i) ≤ (SUP i j. f i j)"
by (rule SUP_least SUP_upper2 UNIV_I mono order_refl)+
qed

fixes f g :: "nat ⇒ ereal"
assumes inc: "incseq f" "incseq g"
and pos: "⋀i. f i ≠ -∞" "⋀i. g i ≠ -∞"
shows "(SUP i. f i + g i) = SUPREMUM UNIV f + SUPREMUM UNIV g"
apply (metis SUP_upper UNIV_I assms(4) ereal_infty_less_eq(2))
apply (subst SUP_ereal_add_left[symmetric, OF UNIV_not_empty assms(3)])
apply (rule SUP_combine[symmetric] ereal_add_mono inc[THEN monoD] | assumption)+
done

fixes f :: "nat ⇒ ereal"
assumes "decseq f" "decseq g"
and fin: "⋀i. f i ≠ ∞" "⋀i. g i ≠ ∞"
shows "(INF i. f i + g i) = INFIMUM UNIV f + INFIMUM UNIV g"
proof -
have INF_less: "(INF i. f i) < ∞" "(INF i. g i) < ∞"
using assms unfolding INF_less_iff by auto
{ fix a b :: ereal assume "a ≠ ∞" "b ≠ ∞"
then have "- ((- a) + (- b)) = a + b"
by (cases a b rule: ereal2_cases) auto }
note * = this
have "(INF i. f i + g i) = (INF i. - ((- f i) + (- g i)))"
also have "… = INFIMUM UNIV f + INFIMUM UNIV g"
unfolding ereal_INF_uminus_eq
using assms INF_less
by (subst SUP_ereal_add) (auto simp: ereal_SUP_uminus fin *)
finally show ?thesis .
qed

fixes f g :: "nat ⇒ ereal"
assumes inc: "incseq f" "incseq g"
and pos: "⋀i. 0 ≤ f i" "⋀i. 0 ≤ g i"
shows "(SUP i. f i + g i) = SUPREMUM UNIV f + SUPREMUM UNIV g"
fix i
show "f i ≠ -∞" "g i ≠ -∞"
using pos[of i] by auto
qed

lemma SUP_ereal_setsum:
fixes f g :: "'a ⇒ nat ⇒ ereal"
assumes "⋀n. n ∈ A ⟹ incseq (f n)"
and pos: "⋀n i. n ∈ A ⟹ 0 ≤ f n i"
shows "(SUP i. ∑n∈A. f n i) = (∑n∈A. SUPREMUM UNIV (f n))"
proof (cases "finite A")
case True
then show ?thesis using assms
by induct (auto simp: incseq_setsumI2 setsum_nonneg SUP_ereal_add_pos)
next
case False
then show ?thesis by simp
qed

lemma SUP_ereal_mult_left:
fixes f :: "'a ⇒ ereal"
assumes "I ≠ {}"
assumes f: "⋀i. i ∈ I ⟹ 0 ≤ f i" and c: "0 ≤ c"
shows "(SUP i:I. c * f i) = c * (SUP i:I. f i)"
proof cases
assume "(SUP i: I. f i) = 0"
moreover then have "⋀i. i ∈ I ⟹ f i = 0"
by (metis SUP_upper f antisym)
ultimately show ?thesis
by simp
next
assume "(SUP i:I. f i) ≠ 0" then show ?thesis
unfolding SUP_def
by (subst continuous_at_Sup_mono[where f="λx. c * x"])
(auto simp: mono_def continuous_at continuous_at_imp_continuous_at_within ‹I ≠ {}›
intro!: ereal_mult_left_mono c)
qed

lemma countable_approach:
fixes x :: ereal
assumes "x ≠ -∞"
shows "∃f. incseq f ∧ (∀i::nat. f i < x) ∧ (f ⇢ x)"
proof (cases x)
case (real r)
moreover have "(λn. r - inverse (real (Suc n))) ⇢ r - 0"
by (intro tendsto_intros LIMSEQ_inverse_real_of_nat)
ultimately show ?thesis
by (intro exI[of _ "λn. x - inverse (Suc n)"]) (auto simp: incseq_def)
next
case PInf with LIMSEQ_SUP[of "λn::nat. ereal (real n)"] show ?thesis
by (intro exI[of _ "λn. ereal (real n)"]) (auto simp: incseq_def SUP_nat_Infty)

lemma Sup_countable_SUP:
assumes "A ≠ {}"
shows "∃f::nat ⇒ ereal. incseq f ∧ range f ⊆ A ∧ Sup A = (SUP i. f i)"
proof cases
assume "Sup A = -∞"
with ‹A ≠ {}› have "A = {-∞}"
by (auto simp: Sup_eq_MInfty)
then show ?thesis
by (auto intro!: exI[of _ "λ_. -∞"] simp: bot_ereal_def)
next
assume "Sup A ≠ -∞"
then obtain l where "incseq l" and l: "⋀i::nat. l i < Sup A" and l_Sup: "l ⇢ Sup A"
by (auto dest: countable_approach)

have "∃f. ∀n. (f n ∈ A ∧ l n ≤ f n) ∧ (f n ≤ f (Suc n))"
proof (rule dependent_nat_choice)
show "∃x. x ∈ A ∧ l 0 ≤ x"
using l[of 0] by (auto simp: less_Sup_iff)
next
fix x n assume "x ∈ A ∧ l n ≤ x"
moreover from l[of "Suc n"] obtain y where "y ∈ A" "l (Suc n) < y"
by (auto simp: less_Sup_iff)
ultimately show "∃y. (y ∈ A ∧ l (Suc n) ≤ y) ∧ x ≤ y"
by (auto intro!: exI[of _ "max x y"] split: split_max)
qed
then guess f .. note f = this
then have "range f ⊆ A" "incseq f"
by (auto simp: incseq_Suc_iff)
moreover
have "(SUP i. f i) = Sup A"
proof (rule tendsto_unique)
show "f ⇢ (SUP i. f i)"
by (rule LIMSEQ_SUP ‹incseq f›)+
show "f ⇢ Sup A"
using l f
by (intro tendsto_sandwich[OF _ _ l_Sup tendsto_const])
(auto simp: Sup_upper)
qed simp
ultimately show ?thesis
by auto
qed

lemma SUP_countable_SUP:
"A ≠ {} ⟹ ∃f::nat ⇒ ereal. range f ⊆ g`A ∧ SUPREMUM A g = SUPREMUM UNIV f"
using Sup_countable_SUP [of "g`A"] by auto

subsection "Relation to @{typ enat}"

definition "ereal_of_enat n = (case n of enat n ⇒ ereal (real n) | ∞ ⇒ ∞)"

declare [[coercion "ereal_of_enat :: enat ⇒ ereal"]]
declare [[coercion "(λn. ereal (real n)) :: nat ⇒ ereal"]]

lemma ereal_of_enat_simps[simp]:
"ereal_of_enat (enat n) = ereal n"
"ereal_of_enat ∞ = ∞"

lemma ereal_of_enat_le_iff[simp]: "ereal_of_enat m ≤ ereal_of_enat n ⟷ m ≤ n"
by (cases m n rule: enat2_cases) auto

lemma ereal_of_enat_less_iff[simp]: "ereal_of_enat m < ereal_of_enat n ⟷ m < n"
by (cases m n rule: enat2_cases) auto

lemma numeral_le_ereal_of_enat_iff[simp]: "numeral m ≤ ereal_of_enat n ⟷ numeral m ≤ n"
by (cases n) (auto)

lemma numeral_less_ereal_of_enat_iff[simp]: "numeral m < ereal_of_enat n ⟷ numeral m < n"
by (cases n) auto

lemma ereal_of_enat_ge_zero_cancel_iff[simp]: "0 ≤ ereal_of_enat n ⟷ 0 ≤ n"
by (cases n) (auto simp: enat_0[symmetric])

lemma ereal_of_enat_gt_zero_cancel_iff[simp]: "0 < ereal_of_enat n ⟷ 0 < n"
by (cases n) (auto simp: enat_0[symmetric])

lemma ereal_of_enat_zero[simp]: "ereal_of_enat 0 = 0"
by (auto simp: enat_0[symmetric])

lemma ereal_of_enat_inf[simp]: "ereal_of_enat n = ∞ ⟷ n = ∞"
by (cases n) auto

lemma ereal_of_enat_add: "ereal_of_enat (m + n) = ereal_of_enat m + ereal_of_enat n"
by (cases m n rule: enat2_cases) auto

lemma ereal_of_enat_sub:
assumes "n ≤ m"
shows "ereal_of_enat (m - n) = ereal_of_enat m - ereal_of_enat n "
using assms by (cases m n rule: enat2_cases) auto

lemma ereal_of_enat_mult:
"ereal_of_enat (m * n) = ereal_of_enat m * ereal_of_enat n"
by (cases m n rule: enat2_cases) auto

lemmas ereal_of_enat_pushin = ereal_of_enat_add ereal_of_enat_sub ereal_of_enat_mult
lemmas ereal_of_enat_pushout = ereal_of_enat_pushin[symmetric]

lemma ereal_of_enat_nonneg: "ereal_of_enat n ≥ 0"
by(cases n) simp_all

lemma ereal_of_enat_Sup:
assumes "A ≠ {}" shows "ereal_of_enat (Sup A) = (SUP a : A. ereal_of_enat a)"
proof (intro antisym mono_Sup)
show "ereal_of_enat (Sup A) ≤ (SUP a : A. ereal_of_enat a)"
proof cases
assume "finite A"
with ‹A ≠ {}› obtain a where "a ∈ A" "ereal_of_enat (Sup A) = ereal_of_enat a"
using Max_in[of A] by (auto simp: Sup_enat_def simp del: Max_in)
then show ?thesis
by (auto intro: SUP_upper)
next
assume "¬ finite A"
have [simp]: "(SUP a : A. ereal_of_enat a) = top"
unfolding SUP_eq_top_iff
proof safe
fix x :: ereal assume "x < top"
then obtain n :: nat where "x < n"
using less_PInf_Ex_of_nat top_ereal_def by auto
obtain a where "a ∈ A - enat ` {.. n}"
by (metis ‹¬ finite A› all_not_in_conv finite_Diff2 finite_atMost finite_imageI finite.emptyI)
then have "a ∈ A" "ereal n ≤ ereal_of_enat a"
by (auto simp: image_iff Ball_def)
(metis enat_iless enat_ord_simps(1) ereal_of_enat_less_iff ereal_of_enat_simps(1) less_le not_less)
with ‹x < n› show "∃i∈A. x < ereal_of_enat i"
by (auto intro!: bexI[of _ a])
qed
show ?thesis
by simp
qed

lemma ereal_of_enat_SUP:
"A ≠ {} ⟹ ereal_of_enat (SUP a:A. f a) = (SUP a : A. ereal_of_enat (f a))"
using ereal_of_enat_Sup[of "f`A"] by auto

subsection "Limits on @{typ ereal}"

lemma open_PInfty: "open A ⟹ ∞ ∈ A ⟹ (∃x. {ereal x<..} ⊆ A)"
unfolding open_ereal_generated
proof (induct rule: generate_topology.induct)
case (Int A B)
then obtain x z where "∞ ∈ A ⟹ {ereal x <..} ⊆ A" "∞ ∈ B ⟹ {ereal z <..} ⊆ B"
by auto
with Int show ?case
by (intro exI[of _ "max x z"]) fastforce
next
case (Basis S)
{
fix x
have "x ≠ ∞ ⟹ ∃t. x ≤ ereal t"
by (cases x) auto
}
moreover note Basis
ultimately show ?case
by (auto split: ereal.split)

lemma open_MInfty: "open A ⟹ -∞ ∈ A ⟹ (∃x. {..<ereal x} ⊆ A)"
unfolding open_ereal_generated
proof (induct rule: generate_topology.induct)
case (Int A B)
then obtain x z where "-∞ ∈ A ⟹ {..< ereal x} ⊆ A" "-∞ ∈ B ⟹ {..< ereal z} ⊆ B"
by auto
with Int show ?case
by (intro exI[of _ "min x z"]) fastforce
next
case (Basis S)
{
fix x
have "x ≠ - ∞ ⟹ ∃t. ereal t ≤ x"
by (cases x) auto
}
moreover note Basis
ultimately show ?case
by (auto split: ereal.split)

lemma open_ereal_vimage: "open S ⟹ open (ereal -` S)"
by (intro open_vimage continuous_intros)

lemma open_ereal: "open S ⟹ open (ereal ` S)"
unfolding open_generated_order[where 'a=real]
proof (induct rule: generate_topology.induct)
case (Basis S)
moreover {
fix x
have "ereal ` {..< x} = { -∞ <..< ereal x }"
apply auto
apply (case_tac xa)
apply auto
done
}
moreover {
fix x
have "ereal ` {x <..} = { ereal x <..< ∞ }"
apply auto
apply (case_tac xa)
apply auto
done
}
ultimately show ?case
by auto
qed (auto simp add: image_Union image_Int)

lemma eventually_finite:
fixes x :: ereal
assumes "¦x¦ ≠ ∞" "(f ⤏ x) F"
shows "eventually (λx. ¦f x¦ ≠ ∞) F"
proof -
have "(f ⤏ ereal (real_of_ereal x)) F"
using assms by (cases x) auto
then have "eventually (λx. f x ∈ ereal ` UNIV) F"
by (rule topological_tendstoD) (auto intro: open_ereal)
also have "(λx. f x ∈ ereal ` UNIV) = (λx. ¦f x¦ ≠ ∞)"
by auto
finally show ?thesis .
qed

lemma open_ereal_def:
"open A ⟷ open (ereal -` A) ∧ (∞ ∈ A ⟶ (∃x. {ereal x <..} ⊆ A)) ∧ (-∞ ∈ A ⟶ (∃x. {..<ereal x} ⊆ A))"
(is "open A ⟷ ?rhs")
proof
assume "open A"
then show ?rhs
using open_PInfty open_MInfty open_ereal_vimage by auto
next
assume "?rhs"
then obtain x y where A: "open (ereal -` A)" "∞ ∈ A ⟹ {ereal x<..} ⊆ A" "-∞ ∈ A ⟹ {..< ereal y} ⊆ A"
by auto
have *: "A = ereal ` (ereal -` A) ∪ (if ∞ ∈ A then {ereal x<..} else {}) ∪ (if -∞ ∈ A then {..< ereal y} else {})"
using A(2,3) by auto
from open_ereal[OF A(1)] show "open A"
by (subst *) (auto simp: open_Un)
qed

lemma open_PInfty2:
assumes "open A"
and "∞ ∈ A"
obtains x where "{ereal x<..} ⊆ A"
using open_PInfty[OF assms] by auto

lemma open_MInfty2:
assumes "open A"
and "-∞ ∈ A"
obtains x where "{..<ereal x} ⊆ A"
using open_MInfty[OF assms] by auto

lemma ereal_openE:
assumes "open A"
obtains x y where "open (ereal -` A)"
and "∞ ∈ A ⟹ {ereal x<..} ⊆ A"
and "-∞ ∈ A ⟹ {..<ereal y} ⊆ A"
using assms open_ereal_def by auto

lemmas open_ereal_lessThan = open_lessThan[where 'a=ereal]
lemmas open_ereal_greaterThan = open_greaterThan[where 'a=ereal]
lemmas ereal_open_greaterThanLessThan = open_greaterThanLessThan[where 'a=ereal]
lemmas closed_ereal_atLeast = closed_atLeast[where 'a=ereal]
lemmas closed_ereal_atMost = closed_atMost[where 'a=ereal]
lemmas closed_ereal_atLeastAtMost = closed_atLeastAtMost[where 'a=ereal]
lemmas closed_ereal_singleton = closed_singleton[where 'a=ereal]

lemma ereal_open_cont_interval:
fixes S :: "ereal set"
assumes "open S"
and "x ∈ S"
and "¦x¦ ≠ ∞"
obtains e where "e > 0" and "{x-e <..< x+e} ⊆ S"
proof -
from ‹open S›
have "open (ereal -` S)"
by (rule ereal_openE)
then obtain e where "e > 0" and e: "⋀y. dist y (real_of_ereal x) < e ⟹ ereal y ∈ S"
using assms unfolding open_dist by force
show thesis
proof (intro that subsetI)
show "0 < ereal e"
using ‹0 < e› by auto
fix y
assume "y ∈ {x - ereal e<..<x + ereal e}"
with assms obtain t where "y = ereal t" "dist t (real_of_ereal x) < e"
by (cases y) (auto simp: dist_real_def)
then show "y ∈ S"
using e[of t] by auto
qed
qed

lemma ereal_open_cont_interval2:
fixes S :: "ereal set"
assumes "open S"
and "x ∈ S"
and x: "¦x¦ ≠ ∞"
obtains a b where "a < x" and "x < b" and "{a <..< b} ⊆ S"
proof -
obtain e where "0 < e" "{x - e<..<x + e} ⊆ S"
using assms by (rule ereal_open_cont_interval)
with that[of "x - e" "x + e"] ereal_between[OF x, of e]
show thesis
by auto
qed

subsubsection ‹Convergent sequences›

lemma lim_real_of_ereal[simp]:
assumes lim: "(f ⤏ ereal x) net"
shows "((λx. real_of_ereal (f x)) ⤏ x) net"
proof (intro topological_tendstoI)
fix S
assume "open S" and "x ∈ S"
then have S: "open S" "ereal x ∈ ereal ` S"
show "eventually (λx. real_of_ereal (f x) ∈ S) net"
by (auto intro: eventually_mono [OF lim[THEN topological_tendstoD, OF open_ereal, OF S]])
qed

lemma lim_ereal[simp]: "((λn. ereal (f n)) ⤏ ereal x) net ⟷ (f ⤏ x) net"
by (auto dest!: lim_real_of_ereal)

lemma convergent_real_imp_convergent_ereal:
assumes "convergent a"
shows "convergent (λn. ereal (a n))" and "lim (λn. ereal (a n)) = ereal (lim a)"
proof -
from assms obtain L where L: "a ⇢ L" unfolding convergent_def ..
hence lim: "(λn. ereal (a n)) ⇢ ereal L" using lim_ereal by auto
thus "convergent (λn. ereal (a n))" unfolding convergent_def ..
thus "lim (λn. ereal (a n)) = ereal (lim a)" using lim L limI by metis
qed

lemma tendsto_PInfty: "(f ⤏ ∞) F ⟷ (∀r. eventually (λx. ereal r < f x) F)"
proof -
{
fix l :: ereal
assume "∀r. eventually (λx. ereal r < f x) F"
from this[THEN spec, of "real_of_ereal l"] have "l ≠ ∞ ⟹ eventually (λx. l < f x) F"
by (cases l) (auto elim: eventually_mono)
}
then show ?thesis
by (auto simp: order_tendsto_iff)
qed

lemma tendsto_PInfty': "(f ⤏ ∞) F = (∀r>c. eventually (λx. ereal r < f x) F)"
proof (subst tendsto_PInfty, intro iffI allI impI)
assume A: "∀r>c. eventually (λx. ereal r < f x) F"
fix r :: real
from A have A: "eventually (λx. ereal r < f x) F" if "r > c" for r using that by blast
show "eventually (λx. ereal r < f x) F"
proof (cases "r > c")
case False
hence B: "ereal r ≤ ereal (c + 1)" by simp
have "c < c + 1" by simp
from A[OF this] show "eventually (λx. ereal r < f x) F"
by eventually_elim (rule le_less_trans[OF B])
qed simp

lemma tendsto_PInfty_eq_at_top:
"((λz. ereal (f z)) ⤏ ∞) F ⟷ (LIM z F. f z :> at_top)"
unfolding tendsto_PInfty filterlim_at_top_dense by simp

lemma tendsto_MInfty: "(f ⤏ -∞) F ⟷ (∀r. eventually (λx. f x < ereal r) F)"
unfolding tendsto_def
proof safe
fix S :: "ereal set"
assume "open S" "-∞ ∈ S"
from open_MInfty[OF this] obtain B where "{..<ereal B} ⊆ S" ..
moreover
assume "∀r::real. eventually (λz. f z < r) F"
then have "eventually (λz. f z ∈ {..< B}) F"
by auto
ultimately show "eventually (λz. f z ∈ S) F"
by (auto elim!: eventually_mono)
next
fix x
assume "∀S. open S ⟶ -∞ ∈ S ⟶ eventually (λx. f x ∈ S) F"
from this[rule_format, of "{..< ereal x}"] show "eventually (λy. f y < ereal x) F"
by auto
qed

lemma tendsto_MInfty': "(f ⤏ -∞) F = (∀r<c. eventually (λx. ereal r > f x) F)"
proof (subst tendsto_MInfty, intro iffI allI impI)
assume A: "∀r<c. eventually (λx. ereal r > f x) F"
fix r :: real
from A have A: "eventually (λx. ereal r > f x) F" if "r < c" for r using that by blast
show "eventually (λx. ereal r > f x) F"
proof (cases "r < c")
case False
hence B: "ereal r ≥ ereal (c - 1)" by simp
have "c > c - 1" by simp
from A[OF this] show "eventually (λx. ereal r > f x) F"
by eventually_elim (erule less_le_trans[OF _ B])
qed simp

lemma Lim_PInfty: "f ⇢ ∞ ⟷ (∀B. ∃N. ∀n≥N. f n ≥ ereal B)"
unfolding tendsto_PInfty eventually_sequentially
proof safe
fix r
assume "∀r. ∃N. ∀n≥N. ereal r ≤ f n"
then obtain N where "∀n≥N. ereal (r + 1) ≤ f n"
by blast
moreover have "ereal r < ereal (r + 1)"
by auto
ultimately show "∃N. ∀n≥N. ereal r < f n"
by (blast intro: less_le_trans)
qed (blast intro: less_imp_le)

lemma Lim_MInfty: "f ⇢ -∞ ⟷ (∀B. ∃N. ∀n≥N. ereal B ≥ f n)"
unfolding tendsto_MInfty eventually_sequentially
proof safe
fix r
assume "∀r. ∃N. ∀n≥N. f n ≤ ereal r"
then obtain N where "∀n≥N. f n ≤ ereal (r - 1)"
by blast
moreover have "ereal (r - 1) < ereal r"
by auto
ultimately show "∃N. ∀n≥N. f n < ereal r"
by (blast intro: le_less_trans)
qed (blast intro: less_imp_le)

lemma Lim_bounded_PInfty: "f ⇢ l ⟹ (⋀n. f n ≤ ereal B) ⟹ l ≠ ∞"
using LIMSEQ_le_const2[of f l "ereal B"] by auto

lemma Lim_bounded_MInfty: "f ⇢ l ⟹ (⋀n. ereal B ≤ f n) ⟹ l ≠ -∞"
using LIMSEQ_le_const[of f l "ereal B"] by auto

lemma tendsto_zero_erealI:
assumes "⋀e. e > 0 ⟹ eventually (λx. ¦f x¦ < ereal e) F"
shows   "(f ⤏ 0) F"
proof (subst filterlim_cong[OF refl refl])
from assms[OF zero_less_one] show "eventually (λx. f x = ereal (real_of_ereal (f x))) F"
by eventually_elim (auto simp: ereal_real)
hence "eventually (λx. abs (real_of_ereal (f x)) < e) F" if "e > 0" for e using assms[OF that]
by eventually_elim (simp add: real_less_ereal_iff that)
hence "((λx. real_of_ereal (f x)) ⤏ 0) F" unfolding tendsto_iff
by (auto simp: tendsto_iff dist_real_def)
thus "((λx. ereal (real_of_ereal (f x))) ⤏ 0) F" by (simp add: zero_ereal_def)
qed

lemma tendsto_explicit:
"f ⇢ f0 ⟷ (∀S. open S ⟶ f0 ∈ S ⟶ (∃N. ∀n≥N. f n ∈ S))"
unfolding tendsto_def eventually_sequentially by auto

lemma Lim_bounded_PInfty2: "f ⇢ l ⟹ ∀n≥N. f n ≤ ereal B ⟹ l ≠ ∞"
using LIMSEQ_le_const2[of f l "ereal B"] by fastforce

lemma Lim_bounded_ereal: "f ⇢ (l :: 'a::linorder_topology) ⟹ ∀n≥M. f n ≤ C ⟹ l ≤ C"
by (intro LIMSEQ_le_const2) auto

lemma Lim_bounded2_ereal:
assumes lim:"f ⇢ (l :: 'a::linorder_topology)"
and ge: "∀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 real_of_ereal_mult[simp]:
fixes a b :: ereal
shows "real_of_ereal (a * b) = real_of_ereal a * real_of_ereal b"
by (cases rule: ereal2_cases[of a b]) auto

lemma real_of_ereal_eq_0:
fixes x :: ereal
shows "real_of_ereal x = 0 ⟷ x = ∞ ∨ x = -∞ ∨ x = 0"
by (cases x) auto

lemma tendsto_ereal_realD:
fixes f :: "'a ⇒ ereal"
assumes "x ≠ 0"
and tendsto: "((λx. ereal (real_of_ereal (f x))) ⤏ x) net"
shows "(f ⤏ x) net"
proof (intro topological_tendstoI)
fix S
assume S: "open S" "x ∈ S"
with ‹x ≠ 0› have "open (S - {0})" "x ∈ S - {0}"
by auto
from tendsto[THEN topological_tendstoD, OF this]
show "eventually (λx. f x ∈ S) net"
by (rule eventually_rev_mp) (auto simp: ereal_real)
qed

lemma tendsto_ereal_realI:
fixes f :: "'a ⇒ ereal"
assumes x: "¦x¦ ≠ ∞" and tendsto: "(f ⤏ x) net"
shows "((λx. ereal (real_of_ereal (f x))) ⤏ x) net"
proof (intro topological_tendstoI)
fix S
assume "open S" and "x ∈ S"
with x have "open (S - {∞, -∞})" "x ∈ S - {∞, -∞}"
by auto
from tendsto[THEN topological_tendstoD, OF this]
show "eventually (λx. ereal (real_of_ereal (f x)) ∈ S) net"
by (elim eventually_mono) (auto simp: ereal_real)
qed

lemma ereal_mult_cancel_left:
fixes a b c :: ereal
shows "a * b = a * c ⟷ (¦a¦ = ∞ ∧ 0 < b * c) ∨ a = 0 ∨ b = c"
by (cases rule: ereal3_cases[of a b c]) (simp_all add: zero_less_mult_iff)

fixes x y :: ereal
assumes x: "¦x¦ ≠ ∞" and y: "¦y¦ ≠ ∞"
assumes f: "(f ⤏ x) F" and g: "(g ⤏ y) F"
shows "((λx. f x + g x) ⤏ x + y) F"
proof -
from x obtain r where x': "x = ereal r" by (cases x) auto
with f have "((λi. real_of_ereal (f i)) ⤏ r) F" by simp
moreover
from y obtain p where y': "y = ereal p" by (cases y) auto
with g have "((λi. real_of_ereal (g i)) ⤏ p) F" by simp
ultimately have "((λi. real_of_ereal (f i) + real_of_ereal (g i)) ⤏ r + p) F"
moreover
from eventually_finite[OF x f] eventually_finite[OF y g]
have "eventually (λx. f x + g x = ereal (real_of_ereal (f x) + real_of_ereal (g x))) F"
by eventually_elim auto
ultimately show ?thesis
by (simp add: x' y' cong: filterlim_cong)
qed

lemma ereal_inj_affinity:
fixes m t :: ereal
assumes "¦m¦ ≠ ∞"
and "m ≠ 0"
and "¦t¦ ≠ ∞"
shows "inj_on (λx. m * x + t) A"
using assms
by (cases rule: ereal2_cases[of m t])
(auto intro!: inj_onI simp: ereal_add_cancel_right ereal_mult_cancel_left)

lemma ereal_PInfty_eq_plus[simp]:
fixes a b :: ereal
shows "∞ = a + b ⟷ a = ∞ ∨ b = ∞"
by (cases rule: ereal2_cases[of a b]) auto

lemma ereal_MInfty_eq_plus[simp]:
fixes a b :: ereal
shows "-∞ = a + b ⟷ (a = -∞ ∧ b ≠ ∞) ∨ (b = -∞ ∧ a ≠ ∞)"
by (cases rule: ereal2_cases[of a b]) auto

lemma ereal_less_divide_pos:
fixes x y :: ereal
shows "x > 0 ⟹ x ≠ ∞ ⟹ y < z / x ⟷ x * y < z"
by (cases rule: ereal3_cases[of x y z]) (auto simp: field_simps)

lemma ereal_divide_less_pos:
fixes x y z :: ereal
shows "x > 0 ⟹ x ≠ ∞ ⟹ y / x < z ⟷ y < x * z"
by (cases rule: ereal3_cases[of x y z]) (auto simp: field_simps)

lemma ereal_divide_eq:
fixes a b c :: ereal
shows "b ≠ 0 ⟹ ¦b¦ ≠ ∞ ⟹ a / b = c ⟷ a = b * c"
by (cases rule: ereal3_cases[of a b c])

lemma ereal_inverse_not_MInfty[simp]: "inverse (a::ereal) ≠ -∞"
by (cases a) auto

lemma ereal_mult_m1[simp]: "x * ereal (-1) = -x"
by (cases x) auto

lemma ereal_real':
assumes "¦x¦ ≠ ∞"
shows "ereal (real_of_ereal x) = x"
using assms by auto

lemma real_ereal_id: "real_of_ereal ∘ ereal = id"
proof -
{
fix x
have "(real_of_ereal o ereal) x = id x"
by auto
}
then show ?thesis
using ext by blast
qed

lemma open_image_ereal: "open(UNIV-{ ∞ , (-∞ :: ereal)})"
by (metis range_ereal open_ereal open_UNIV)

lemma ereal_le_distrib:
fixes a b c :: ereal
shows "c * (a + b) ≤ c * a + c * b"
by (cases rule: ereal3_cases[of a b c])
(auto simp add: field_simps not_le mult_le_0_iff mult_less_0_iff)

lemma ereal_pos_distrib:
fixes a b c :: ereal
assumes "0 ≤ c"
and "c ≠ ∞"
shows "c * (a + b) = c * a + c * b"
using assms
by (cases rule: ereal3_cases[of a b c])
(auto simp add: field_simps not_le mult_le_0_iff mult_less_0_iff)

lemma ereal_max_mono: "(a::ereal) ≤ b ⟹ c ≤ d ⟹ max a c ≤ max b d"
by (metis sup_ereal_def sup_mono)

lemma ereal_max_least: "(a::ereal) ≤ x ⟹ c ≤ x ⟹ max a c ≤ x"
by (metis sup_ereal_def sup_least)

lemma ereal_LimI_finite:
fixes x :: ereal
assumes "¦x¦ ≠ ∞"
and "⋀r. 0 < r ⟹ ∃N. ∀n≥N. u n < x + r ∧ x < u n + r"
shows "u ⇢ x"
proof (rule topological_tendstoI, unfold eventually_sequentially)
obtain rx where rx: "x = ereal rx"
using assms by (cases x) auto
fix S
assume "open S" and "x ∈ S"
then have "open (ereal -` S)"
unfolding open_ereal_def by auto
with ‹x ∈ S› obtain r where "0 < r" and dist: "⋀y. dist y rx < r ⟹ ereal y ∈ S"
unfolding open_dist rx by auto
then obtain n where
upper: "⋀N. n ≤ N ⟹ u N < x + ereal r" and
lower: "⋀N. n ≤ N ⟹ x < u N + ereal r"
using assms(2)[of "ereal r"] by auto
show "∃N. ∀n≥N. u n ∈ S"
proof (safe intro!: exI[of _ n])
fix N
assume "n ≤ N"
from upper[OF this] lower[OF this] assms ‹0 < r›
have "u N ∉ {∞,(-∞)}"
by auto
then obtain ra where ra_def: "(u N) = ereal ra"
by (cases "u N") auto
then have "rx < ra + r" and "ra < rx + r"
using rx assms ‹0 < r› lower[OF ‹n ≤ N›] upper[OF ‹n ≤ N›]
by auto
then have "dist (real_of_ereal (u N)) rx < r"
using rx ra_def
by (auto simp: dist_real_def abs_diff_less_iff field_simps)
from dist[OF this] show "u N ∈ S"
using ‹u N  ∉ {∞, -∞}›
by (auto simp: ereal_real split: split_if_asm)
qed
qed

lemma tendsto_obtains_N:
assumes "f ⇢ f0"
assumes "open S"
and "f0 ∈ S"
obtains N where "∀n≥N. f n ∈ S"
using assms using tendsto_def
using tendsto_explicit[of f f0] assms by auto

lemma ereal_LimI_finite_iff:
fixes x :: ereal
assumes "¦x¦ ≠ ∞"
shows "u ⇢ x ⟷ (∀r. 0 < r ⟶ (∃N. ∀n≥N. u n < x + r ∧ x < u n + r))"
(is "?lhs ⟷ ?rhs")
proof
assume lim: "u ⇢ x"
{
fix r :: ereal
assume "r > 0"
then obtain N where "∀n≥N. u n ∈ {x - r <..< x + r}"
apply (subst tendsto_obtains_N[of u x "{x - r <..< x + r}"])
using lim ereal_between[of x r] assms ‹r > 0›
apply auto
done
then have "∃N. ∀n≥N. u n < x + r ∧ x < u n + r"
using ereal_minus_less[of r x]
by (cases r) auto
}
then show ?rhs
by auto
next
assume ?rhs
then show "u ⇢ x"
using ereal_LimI_finite[of x] assms by auto
qed

lemma ereal_Limsup_uminus:
fixes f :: "'a ⇒ ereal"
shows "Limsup net (λx. - (f x)) = - Liminf net f"
unfolding Limsup_def Liminf_def ereal_SUP_uminus ereal_INF_uminus_eq ..

lemma liminf_bounded_iff:
fixes x :: "nat ⇒ ereal"
shows "C ≤ liminf x ⟷ (∀B<C. ∃N. ∀n≥N. B < x n)"
(is "?lhs ⟷ ?rhs")
unfolding le_Liminf_iff eventually_sequentially ..

fixes f g :: "_ ⇒ ereal"
assumes F: "F ≠ bot"
assumes ev: "eventually (λx. 0 ≤ f x) F" "eventually (λx. 0 ≤ g x) F"
shows "Liminf F f + Liminf F g ≤ Liminf F (λx. f x + g x)"
unfolding Liminf_def
let ?F = "{P. eventually P F}"
let ?INF = "λP g. INFIMUM (Collect P) g"
show "?F ≠ {}"
by (auto intro: eventually_True)
show "(SUP P:?F. ?INF P g) ≠ - ∞"
unfolding bot_ereal_def[symmetric] SUP_bot_conv INF_eq_bot_iff
by (auto intro!: exI[of _ 0] ev simp: bot_ereal_def)
have "(SUP P:?F. ?INF P f + (SUP P:?F. ?INF P g)) ≤ (SUP P:?F. (SUP P':?F. ?INF P f + ?INF P' g))"
proof (safe intro!: SUP_mono bexI[of _ "λx. P x ∧ 0 ≤ f x" for P])
fix P let ?P' = "λx. P x ∧ 0 ≤ f x"
assume "eventually P F"
with ev show "eventually ?P' F"
by eventually_elim auto
have "?INF P f + (SUP P:?F. ?INF P g) ≤ ?INF ?P' f + (SUP P:?F. ?INF P g)"
also have "… = (SUP P':?F. ?INF ?P' f + ?INF P' g)"
show "INFIMUM {x. P x ∧ 0 ≤ f x} f ≠ - ∞"
unfolding bot_ereal_def[symmetric] INF_eq_bot_iff
by (auto intro!: exI[of _ 0] ev simp: bot_ereal_def)
qed fact
finally show "?INF P f + (SUP P:?F. ?INF P g) ≤ (SUP P':?F. ?INF ?P' f + ?INF P' g)" .
qed
also have "… ≤ (SUP P:?F. INF x:Collect P. f x + g x)"
proof (safe intro!: SUP_least)
fix P Q assume *: "eventually P F" "eventually Q F"
show "?INF P f + ?INF Q g ≤ (SUP P:?F. INF x:Collect P. f x + g x)"
proof (rule SUP_upper2)
show "(λx. P x ∧ Q x) ∈ ?F"
using * by (auto simp: eventually_conj)
show "?INF P f + ?INF Q g ≤ (INF x:{x. P x ∧ Q x}. f x + g x)"
by (intro INF_greatest ereal_add_mono) (auto intro: INF_lower)
qed
qed
finally show "(SUP P:?F. ?INF P f + (SUP P:?F. ?INF P g)) ≤ (SUP P:?F. INF x:Collect P. f x + g x)" .
qed

lemma Sup_ereal_mult_right':
assumes nonempty: "Y ≠ {}"
and x: "x ≥ 0"
shows "(SUP i:Y. f i) * ereal x = (SUP i:Y. f i * ereal x)" (is "?lhs = ?rhs")
proof(cases "x = 0")
case True thus ?thesis by(auto simp add: nonempty zero_ereal_def[symmetric])
next
case False
show ?thesis
proof(rule antisym)
show "?rhs ≤ ?lhs"
by(rule SUP_least)(simp add: ereal_mult_right_mono SUP_upper x)
next
have "?lhs / ereal x = (SUP i:Y. f i) * (ereal x / ereal x)" by(simp only: ereal_times_divide_eq)
also have "… = (SUP i:Y. f i)" using False by simp
also have "… ≤ ?rhs / x"
proof(rule SUP_least)
fix i
assume "i ∈ Y"
have "f i = f i * (ereal x / ereal x)" using False by simp
also have "… = f i * x / x" by(simp only: ereal_times_divide_eq)
also from ‹i ∈ Y› have "f i * x ≤ ?rhs" by(rule SUP_upper)
hence "f i * x / x ≤ ?rhs / x" using x False by simp
finally show "f i ≤ ?rhs / x" .
qed
finally have "(?lhs / x) * x ≤ (?rhs / x) * x"
also have "… = ?rhs" using False ereal_divide_eq mult.commute by force
also have "(?lhs / x) * x = ?lhs" using False ereal_divide_eq mult.commute by force
finally show "?lhs ≤ ?rhs" .
qed
qed

lemma Sup_ereal_mult_left':
"⟦ Y ≠ {}; x ≥ 0 ⟧ ⟹ ereal x * (SUP i:Y. f i) = (SUP i:Y. ereal x * f i)"
by(subst (1 2) mult.commute)(rule Sup_ereal_mult_right')

fixes f g :: "'a::complete_lattice ⇒ ereal"
assumes nn: "⋀x. 0 ≤ f x" "⋀x. 0 ≤ g x" and cont: "sup_continuous f" "sup_continuous g"
shows "sup_continuous (λx. f x + g x)"
unfolding sup_continuous_def
proof safe
fix M :: "nat ⇒ 'a" assume "incseq M"
then show "f (SUP i. M i) + g (SUP i. M i) = (SUP i. f (M i) + g (M i))"
using SUP_ereal_add_pos[of "λi. f (M i)" "λi. g (M i)"] nn
cont[THEN sup_continuous_mono] cont[THEN sup_continuousD]
by (auto simp: mono_def)
qed

lemma sup_continuous_mult_right[order_continuous_intros]:
"0 ≤ c ⟹ c < ∞ ⟹ sup_continuous f ⟹ sup_continuous (λx. f x * c :: ereal)"
by (cases c) (auto simp: sup_continuous_def fun_eq_iff Sup_ereal_mult_right')

lemma sup_continuous_mult_left[order_continuous_intros]:
"0 ≤ c ⟹ c < ∞ ⟹ sup_continuous f ⟹ sup_continuous (λx. c * f x :: ereal)"
using sup_continuous_mult_right[of c f] by (simp add: mult_ac)

lemma sup_continuous_ereal_of_enat[order_continuous_intros]:
assumes f: "sup_continuous f" shows "sup_continuous (λx. ereal_of_enat (f x))"
by (rule sup_continuous_compose[OF _ f])
(auto simp: sup_continuous_def ereal_of_enat_SUP)

subsubsection ‹Sums›

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

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

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

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

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

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

lemma suminf_upper:
fixes f :: "nat ⇒ ereal"
assumes "⋀n. 0 ≤ f n"
shows "(∑n<N. f n) ≤ (∑n. f n)"
unfolding suminf_ereal_eq_SUP [OF assms]
by (auto intro: complete_lattice_class.SUP_upper)

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

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

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

fixes f g :: "nat ⇒ ereal"
assumes "⋀i. 0 ≤ f i"
and "⋀i. 0 ≤ g i"
shows "(∑i. f i + g i) = suminf f + suminf g"
apply (subst (1 2 3) suminf_ereal_eq_SUP)
unfolding setsum.distrib
done

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

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

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

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

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

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

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

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

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

lemma suminf_setsum_ereal:
fixes f :: "_ ⇒ _ ⇒ ereal"
assumes nonneg: "⋀i a. a ∈ A ⟹ 0 ≤ f i a"
shows "(∑i. ∑a∈A. f i a) = (∑a∈A. ∑i. f i a)"
proof (cases "finite A")
case True
then show ?thesis
using nonneg
next
case False
then show ?thesis by simp
qed

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

lemma suminf_ereal_offset_le:
fixes f :: "nat ⇒ ereal"
assumes f: "⋀i. 0 ≤ f i"
shows "(∑i. f (i + k)) ≤ suminf f"
proof -
have "(λn. ∑i<n. f (i + k)) ⇢ (∑i. f (i + k))"
using summable_sums[OF summable_ereal_pos] by (simp add: sums_def atLeast0LessThan f)
moreover have "(λn. ∑i<n. f i) ⇢ (∑i. f i)"
using summable_sums[OF summable_ereal_pos] by (simp add: sums_def atLeast0LessThan f)
then have "(λn. ∑i<n + k. f i) ⇢ (∑i. f i)"
by (rule LIMSEQ_ignore_initial_segment)
ultimately show ?thesis
proof (rule LIMSEQ_le, safe intro!: exI[of _ k])
fix n assume "k ≤ n"
have "(∑i<n. f (i + k)) = (∑i<n. (f ∘ (λi. i + k)) i)"
by simp
also have "… = (∑i∈(λi. i + k) ` {..<n}. f i)"
by (subst setsum.reindex) auto
also have "… ≤ setsum f {..<n + k}"
by (intro setsum_mono3) (auto simp: f)
finally show "(∑i<n. f (i + k)) ≤ setsum f {..<n + k}" .
qed
qed

lemma sums_suminf_ereal: "f sums x ⟹ (∑i. ereal (f i)) = ereal x"
by (metis sums_ereal sums_unique)

lemma suminf_ereal': "summable f ⟹ (∑i. ereal (f i)) = ereal (∑i. f i)"
by (metis sums_ereal sums_unique summable_def)

lemma suminf_ereal_finite: "summable f ⟹ (∑i. ereal (f i)) ≠ ∞"
by (auto simp: sums_ereal[symmetric] summable_def sums_unique[symmetric])

lemma suminf_ereal_finite_neg:
assumes "summable f"
shows "(∑x. ereal (f x)) ≠ -∞"
proof-
from assms obtain x where "f sums x" by blast
hence "(λx. ereal (f x)) sums ereal x" by (simp add: sums_ereal)
from sums_unique[OF this] have "(∑x. ereal (f x)) = ereal x" ..
thus "(∑x. ereal (f x)) ≠ -∞" by simp_all
qed

fixes f g :: "'a ⇒ ereal"
assumes nonneg: "⋀i. i ∈ I ⟹ 0 ≤ f i" "⋀i. i ∈ I ⟹ 0 ≤ g i"
assumes directed: "⋀i j. i ∈ I ⟹ j ∈ I ⟹ ∃k∈I. f i + g j ≤ f k + g k"
shows "(SUP i:I. f i + g i) = (SUP i:I. f i) + (SUP i:I. g i)"
proof cases
assume "I = {}" then show ?thesis
next
assume "I ≠ {}"
show ?thesis
proof (rule antisym)
show "(SUP i:I. f i + g i) ≤ (SUP i:I. f i) + (SUP i:I. g i)"
by (rule SUP_least; intro ereal_add_mono SUP_upper)
next
have "bot < (SUP i:I. g i)"
using ‹I ≠ {}› nonneg(2) by (auto simp: bot_ereal_def less_SUP_iff)
then have "(SUP i:I. f i) + (SUP i:I. g i) = (SUP i:I. f i + (SUP i:I. g i))"
by (intro SUP_ereal_add_left[symmetric] ‹I ≠ {}›) auto
also have "… = (SUP i:I. (SUP j:I. f i + g j))"
using nonneg(1) by (intro SUP_cong refl SUP_ereal_add_right[symmetric] ‹I ≠ {}›) auto
also have "… ≤ (SUP i:I. f i + g i)"
using directed by (intro SUP_least) (blast intro: SUP_upper2)
finally show "(SUP i:I. f i) + (SUP i:I. g i) ≤ (SUP i:I. f i + g i)" .
qed
qed

lemma SUP_ereal_setsum_directed:
fixes f g :: "'a ⇒ 'b ⇒ ereal"
assumes "I ≠ {}"
assumes directed: "⋀N i j. N ⊆ A ⟹ i ∈ I ⟹ j ∈ I ⟹ ∃k∈I. ∀n∈N. f n i ≤ f n k ∧ f n j ≤ f n k"
assumes nonneg: "⋀n i. i ∈ I ⟹ n ∈ A ⟹ 0 ≤ f n i"
shows "(SUP i:I. ∑n∈A. f n i) = (∑n∈A. SUP i:I. f n i)"
proof -
have "N ⊆ A ⟹ (SUP i:I. ∑n∈N. f n i) = (∑n∈N. SUP i:I. f n i)" for N
proof (induction N rule: infinite_finite_induct)
case (insert n N)
moreover have "(SUP i:I. f n i + (∑l∈N. f l i)) = (SUP i:I. f n i) + (SUP i:I. ∑l∈N. f l i)"
fix i assume "i ∈ I" then show "0 ≤ f n i" "0 ≤ (∑l∈N. f l i)"
using insert by (auto intro!: setsum_nonneg nonneg)
next
fix i j assume "i ∈ I" "j ∈ I"
from directed[OF ‹insert n N ⊆ A› this] guess k ..
then show "∃k∈I. f n i + (∑l∈N. f l j) ≤ f n k + (∑l∈N. f l k)"
by (intro bexI[of _ k]) (auto intro!: ereal_add_mono setsum_mono)
qed
ultimately show ?case
by simp
qed (simp_all add: SUP_constant ‹I ≠ {}›)
from this[of A] show ?thesis by simp
qed

lemma suminf_SUP_eq_directed:
fixes f :: "_ ⇒ nat ⇒ ereal"
assumes "I ≠ {}"
assumes directed: "⋀N i j. i ∈ I ⟹ j ∈ I ⟹ finite N ⟹ ∃k∈I. ∀n∈N. f i n ≤ f k n ∧ f j n ≤ f k n"
assumes nonneg: "⋀n i. 0 ≤ f n i"
shows "(∑i. SUP n:I. f n i) = (SUP n:I. ∑i. f n i)"
proof (subst (1 2) suminf_ereal_eq_SUP)
show "⋀n i. 0 ≤ f n i" "⋀i. 0 ≤ (SUP n:I. f n i)"
using ‹I ≠ {}› nonneg by (auto intro: SUP_upper2)
show "(SUP n. ∑i<n. SUP n:I. f n i) = (SUP n:I. SUP j. ∑i<j. f n i)"
apply (subst SUP_commute)
apply (subst SUP_ereal_setsum_directed)
apply (auto intro!: assms simp: finite_subset)
done
qed

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

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

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

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

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

lemma ereal_open_affinity_pos:
fixes S :: "ereal set"
assumes "open S"
and m: "m ≠ ∞" "0 < m"
and t: "¦t¦ ≠ ∞"
shows "open ((λx. m * x + t) ` S)"
proof -
have "open ((λx. inverse m * (x + -t)) -` S)"
using m t
apply (intro open_vimage ‹open S›)
apply (intro continuous_at_imp_continuous_on ballI tendsto_cmult_ereal continuous_at[THEN iffD2]
apply auto
done
also have "(λx. inverse m * (x + -t)) -` S = (λx. (x - t) / m) -` S"
using m t by (auto simp: divide_ereal_def mult.commute uminus_ereal.simps[symmetric] minus_ereal_def
simp del: uminus_ereal.simps)
also have "(λx. (x - t) / m) -` S = (λx. m * x + t) ` S"
using m t
(metis abs_ereal_less0 abs_ereal_uminus ereal_divide_eq ereal_eq_minus ereal_minus(7,8)
ereal_minus_less_minus ereal_mult_eq_PInfty ereal_uminus_uminus ereal_zero_mult)
finally show ?thesis .
qed

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

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

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

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

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

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

lemma limsup_le_liminf_real:
fixes X :: "nat ⇒ real" and L :: real
assumes 1: "limsup X ≤ L" and 2: "L ≤ liminf X"
shows "X ⇢ L"
proof -
from 1 2 have "limsup X ≤ liminf X" by auto
hence 3: "limsup X = liminf X"
apply (subst eq_iff, rule conjI)
by (rule Liminf_le_Limsup, auto)
hence 4: "convergent (λn. ereal (X n))"
by (subst convergent_ereal)
hence "limsup X = lim (λn. ereal(X n))"
by (rule convergent_limsup_cl)
also from 1 2 3 have "limsup X = L" by auto
finally have "lim (λn. ereal(X n)) = L" ..
hence "(λn. ereal (X n)) ⇢ L"
apply (elim subst)
by (subst convergent_LIMSEQ_iff [symmetric], rule 4)
thus ?thesis by simp
qed

lemma liminf_PInfty:
fixes X :: "nat ⇒ ereal"
shows "X ⇢ ∞ ⟷ liminf X = ∞"
by (metis Liminf_PInfty trivial_limit_sequentially)

lemma limsup_MInfty:
fixes X :: "nat ⇒ ereal"
shows "X ⇢ -∞ ⟷ limsup X = -∞"
by (metis Limsup_MInfty trivial_limit_sequentially)

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

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

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

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

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

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

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

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

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

lemma liminf_ereal_cminus:
fixes f :: "nat ⇒ ereal"
assumes "c ≠ -∞"
shows "liminf (λx. c - f x) = c - limsup f"
proof (cases c)
case PInf
then show ?thesis
next
case (real r)
then show ?thesis
unfolding liminf_SUP_INF limsup_INF_SUP
apply (subst INF_ereal_minus_right)
apply auto
apply (subst SUP_ereal_minus_right)
apply auto
done
qed (insert ‹c ≠ -∞›, simp)

subsubsection ‹Continuity›

lemma continuous_at_of_ereal:
"¦x0 :: ereal¦ ≠ ∞ ⟹ continuous (at x0) real_of_ereal"
unfolding continuous_at
by (rule lim_real_of_ereal) (simp add: ereal_real)

lemma nhds_ereal: "nhds (ereal r) = filtermap ereal (nhds r)"
by (simp add: filtermap_nhds_open_map open_ereal continuous_at_of_ereal)

lemma at_ereal: "at (ereal r) = filtermap ereal (at r)"
by (simp add: filter_eq_iff eventually_at_filter nhds_ereal eventually_filtermap)

lemma at_left_ereal: "at_left (ereal r) = filtermap ereal (at_left r)"
by (simp add: filter_eq_iff eventually_at_filter nhds_ereal eventually_filtermap)

lemma at_right_ereal: "at_right (ereal r) = filtermap ereal (at_right r)"
by (simp add: filter_eq_iff eventually_at_filter nhds_ereal eventually_filtermap)

lemma
shows at_left_PInf: "at_left ∞ = filtermap ereal at_top"
and at_right_MInf: "at_right (-∞) = filtermap ereal at_bot"
unfolding filter_eq_iff eventually_filtermap eventually_at_top_dense eventually_at_bot_dense
eventually_at_left[OF ereal_less(5)] eventually_at_right[OF ereal_less(6)]
by (auto simp add: ereal_all_split ereal_ex_split)

lemma ereal_tendsto_simps1:
"((f ∘ real_of_ereal) ⤏ y) (at_left (ereal x)) ⟷ (f ⤏ y) (at_left x)"
"((f ∘ real_of_ereal) ⤏ y) (at_right (ereal x)) ⟷ (f ⤏ y) (at_right x)"
"((f ∘ real_of_ereal) ⤏ y) (at_left (∞::ereal)) ⟷ (f ⤏ y) at_top"
"((f ∘ real_of_ereal) ⤏ y) (at_right (-∞::ereal)) ⟷ (f ⤏ y) at_bot"
unfolding tendsto_compose_filtermap at_left_ereal at_right_ereal at_left_PInf at_right_MInf
by (auto simp: filtermap_filtermap filtermap_ident)

lemma ereal_tendsto_simps2:
"((ereal ∘ f) ⤏ ereal a) F ⟷ (f ⤏ a) F"
"((ereal ∘ f) ⤏ ∞) F ⟷ (LIM x F. f x :> at_top)"
"((ereal ∘ f) ⤏ -∞) F ⟷ (LIM x F. f x :> at_bot)"
unfolding tendsto_PInfty filterlim_at_top_dense tendsto_MInfty filterlim_at_bot_dense
using lim_ereal by (simp_all add: comp_def)

lemma inverse_infty_ereal_tendsto_0: "inverse ─∞→ (0::ereal)"
proof -
have **: "((λx. ereal (inverse x)) ⤏ ereal 0) at_infinity"
by (intro tendsto_intros tendsto_inverse_0)

show ?thesis
by (simp add: at_infty_ereal_eq_at_top tendsto_compose_filtermap[symmetric] comp_def)
(auto simp: eventually_at_top_linorder exI[of _ 1] zero_ereal_def at_top_le_at_infinity
intro!: filterlim_mono_eventually[OF **])
qed

lemma inverse_ereal_tendsto_pos:
fixes x :: ereal assumes "0 < x"
shows "inverse ─x→ inverse x"
proof (cases x)
case (real r)
with ‹0 < x› have **: "(λx. ereal (inverse x)) ─r→ ereal (inverse r)"
by (auto intro!: tendsto_inverse)
from real ‹0 < x› show ?thesis
by (auto simp: at_ereal tendsto_compose_filtermap[symmetric] eventually_at_filter
intro!: Lim_transform_eventually[OF _ **] t1_space_nhds)
qed (insert ‹0 < x›, auto intro!: inverse_infty_ereal_tendsto_0)

lemma inverse_ereal_tendsto_at_right_0: "(inverse ⤏ ∞) (at_right (0::ereal))"
unfolding tendsto_compose_filtermap[symmetric] at_right_ereal zero_ereal_def
by (subst filterlim_cong[OF refl refl, where g="λx. ereal (inverse x)"])
(auto simp: eventually_at_filter tendsto_PInfty_eq_at_top filterlim_inverse_at_top_right)

lemmas ereal_tendsto_simps = ereal_tendsto_simps1 ereal_tendsto_simps2

lemma continuous_at_iff_ereal:
fixes f :: "'a::t2_space ⇒ real"
shows "continuous (at x0 within s) f ⟷ continuous (at x0 within s) (ereal ∘ f)"
unfolding continuous_within comp_def lim_ereal ..

lemma continuous_on_iff_ereal:
fixes f :: "'a::t2_space => real"
assumes "open A"
shows "continuous_on A f ⟷ continuous_on A (ereal ∘ f)"
unfolding continuous_on_def comp_def lim_ereal ..

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

lemma continuous_on_iff_real:
fixes f :: "'a::t2_space ⇒ ereal"
assumes *: "⋀x. x ∈ A ⟹ ¦f x¦ ≠ ∞"
shows "continuous_on A f ⟷ continuous_on A (real_of_ereal ∘ f)"
proof -
have "f ` A ⊆ UNIV - {∞, -∞}"
using assms by force
then have *: "continuous_on (f ` A) real_of_ereal"
using continuous_on_real by (simp add: continuous_on_subset)
have **: "continuous_on ((real_of_ereal ∘ f) ` A) ereal"
by (intro continuous_on_ereal continuous_on_id)
{
assume "continuous_on A f"
then have "continuous_on A (real_of_ereal ∘ f)"
apply (subst continuous_on_compose)
using *
apply auto
done
}
moreover
{
assume "continuous_on A (real_of_ereal ∘ f)"
then have "continuous_on A (ereal ∘ (real_of_ereal ∘ f))"
apply (subst continuous_on_compose)
using **
apply auto
done
then have "continuous_on A f"
apply (subst continuous_on_cong[of _ A _ "ereal ∘ (real_of_ereal ∘ f)"])
using assms ereal_real
apply auto
done
}
ultimately show ?thesis
by auto
qed

lemma continuous_uminus_ereal [continuous_intros]: "continuous_on (A :: ereal set) uminus"
unfolding continuous_on_def
by (intro ballI tendsto_uminus_ereal[of "λx. x::ereal"]) simp

lemma ereal_uminus_atMost [simp]: "uminus ` {..(a::ereal)} = {-a..}"
proof (intro equalityI subsetI)
fix x :: ereal assume "x ∈ {-a..}"
hence "-(-x) ∈ uminus ` {..a}" by (intro imageI) (simp add: ereal_uminus_le_reorder)
thus "x ∈ uminus ` {..a}" by simp
qed auto

lemma continuous_on_inverse_ereal [continuous_intros]:
"continuous_on {0::ereal ..} inverse"
unfolding continuous_on_def
proof clarsimp
fix x :: ereal assume "0 ≤ x"
moreover have "at 0 within {0 ..} = at_right (0::ereal)"
by (auto simp: filter_eq_iff eventually_at_filter le_less)
moreover have "at x within {0 ..} = at x" if "0 < x"
using that by (intro at_within_nhd[of _ "{0<..}"]) auto
ultimately show "(inverse ⤏ inverse x) (at x within {0..})"
by (auto simp: le_less inverse_ereal_tendsto_at_right_0 inverse_ereal_tendsto_pos)
qed

lemma continuous_inverse_ereal_nonpos: "continuous_on ({..<0} :: ereal set) inverse"
proof (subst continuous_on_cong[OF refl])
have "continuous_on {(0::ereal)<..} inverse"
by (rule continuous_on_subset[OF continuous_on_inverse_ereal]) auto
thus "continuous_on {..<(0::ereal)} (uminus ∘ inverse ∘ uminus)"
by (intro continuous_intros) simp_all
qed simp

lemma tendsto_inverse_ereal:
assumes "(f ⤏ (c :: ereal)) F"
assumes "eventually (λx. f x ≥ 0) F"
shows   "((λx. inverse (f x)) ⤏ inverse c) F"
by (cases "F = bot")
(auto intro!: tendsto_le_const[of F] assms
continuous_on_tendsto_compose[OF continuous_on_inverse_ereal])

subsubsection ‹liminf and limsup›

lemma Limsup_ereal_mult_right:
assumes "F ≠ bot" "(c::real) ≥ 0"
shows   "Limsup F (λn. f n * ereal c) = Limsup F f * ereal c"
proof (rule Limsup_compose_continuous_mono)
from assms show "continuous_on UNIV (λa. a * ereal c)"
using tendsto_cmult_ereal[of "ereal c" "λx. x" ]
by (force simp: continuous_on_def mult_ac)
qed (insert assms, auto simp: mono_def ereal_mult_right_mono)

lemma Liminf_ereal_mult_right:
assumes "F ≠ bot" "(c::real) ≥ 0"
shows   "Liminf F (λn. f n * ereal c) = Liminf F f * ereal c"
proof (rule Liminf_compose_continuous_mono)
from assms show "continuous_on UNIV (λa. a * ereal c)"
using tendsto_cmult_ereal[of "ereal c" "λx. x" ]
by (force simp: continuous_on_def mult_ac)
qed (insert assms, auto simp: mono_def ereal_mult_right_mono)

lemma Limsup_ereal_mult_left:
assumes "F ≠ bot" "(c::real) ≥ 0"
shows   "Limsup F (λn. ereal c * f n) = ereal c * Limsup F f"
using Limsup_ereal_mult_right[OF assms] by (subst (1 2) mult.commute)

lemma limsup_ereal_mult_right:
"(c::real) ≥ 0 ⟹ limsup (λn. f n * ereal c) = limsup f * ereal c"
by (rule Limsup_ereal_mult_right) simp_all

lemma limsup_ereal_mult_left:
"(c::real) ≥ 0 ⟹ limsup (λn. ereal c * f n) = ereal c * limsup f"
by (subst (1 2) mult.commute, rule limsup_ereal_mult_right) simp_all

"F ≠ bot ⟹ abs c ≠ ∞ ⟹ Limsup F (λn. g n + (c :: ereal)) = Limsup F g + c"
by (rule Limsup_compose_continuous_mono) (auto simp: mono_def ereal_add_mono continuous_on_def)

"F ≠ bot ⟹ abs c ≠ ∞ ⟹ Limsup F (λn. (c :: ereal) + g n) = c + Limsup F g"

"F ≠ bot ⟹ abs c ≠ ∞ ⟹ Liminf F (λn. g n + (c :: ereal)) = Liminf F g + c"
by (rule Liminf_compose_continuous_mono) (auto simp: mono_def ereal_add_mono continuous_on_def)

"F ≠ bot ⟹ abs c ≠ ∞ ⟹ Liminf F (λn. (c :: ereal) + g n) = c + Liminf F g"

lemma
assumes "F ≠ bot"
assumes nonneg: "eventually (λx. f x ≥ (0::ereal)) F"
shows   Liminf_inverse_ereal: "Liminf F (λx. inverse (f x)) = inverse (Limsup F f)"
and     Limsup_inverse_ereal: "Limsup F (λx. inverse (f x)) = inverse (Liminf F f)"
proof -
def inv  "λx. if x ≤ 0 then ∞ else inverse x :: ereal"
have "continuous_on ({..0} ∪ {0..}) inv" unfolding inv_def
by (intro continuous_on_If) (auto intro!: continuous_intros)
also have "{..0} ∪ {0..} = (UNIV :: ereal set)" by auto
finally have cont: "continuous_on UNIV inv" .
have antimono: "antimono inv" unfolding inv_def antimono_def
by (auto intro!: ereal_inverse_antimono)

have "Liminf F (λx. inverse (f x)) = Liminf F (λx. inv (f x))" using nonneg
by (auto intro!: Liminf_eq elim!: eventually_mono simp: inv_def)
also have "... = inv (Limsup F f)"
by (simp add: assms(1) Liminf_compose_continuous_antimono[OF cont antimono])
also from assms have "Limsup F f ≥ 0" by (intro le_Limsup) simp_all
hence "inv (Limsup F f) = inverse (Limsup F f)" by (simp add: inv_def)
finally show "Liminf F (λx. inverse (f x)) = inverse (Limsup F f)" .

have "Limsup F (λx. inverse (f x)) = Limsup F (λx. inv (f x))" using nonneg
by (auto intro!: Limsup_eq elim!: eventually_mono simp: inv_def)
also have "... = inv (Liminf F f)"
by (simp add: assms(1) Limsup_compose_continuous_antimono[OF cont antimono])
also from assms have "Liminf F f ≥ 0" by (intro Liminf_bounded) simp_all
hence "inv (Liminf F f) = inverse (Liminf F f)" by (simp add: inv_def)
finally show "Limsup F (λx. inverse (f x)) = inverse (Liminf F f)" .
qed

subsubsection ‹Tests for code generator›

(* A small list of simple arithmetic expressions *)

value "- ∞ :: ereal"
value "¦-∞¦ :: ereal"
value "4 + 5 / 4 - ereal 2 :: ereal"
value "ereal 3 < ∞"
value "real_of_ereal (∞::ereal) = 0"

end