(* 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
*)
header {* Extended real number line *}
theory Extended_Real
imports Complex_Main Extended_Nat Liminf_Limsup
begin
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 "(\<infinity>::ereal) = PInfty"
instance ..
end
declare [[coercion "ereal :: real \<Rightarrow> ereal"]]
lemma ereal_uminus_uminus[simp]:
fixes a :: ereal
shows "- (- a) = a"
by (cases a) simp_all
lemma
shows PInfty_eq_infinity[simp]: "PInfty = \<infinity>"
and MInfty_eq_minfinity[simp]: "MInfty = - \<infinity>"
and MInfty_neq_PInfty[simp]: "\<infinity> \<noteq> - (\<infinity>::ereal)" "- \<infinity> \<noteq> (\<infinity>::ereal)"
and MInfty_neq_ereal[simp]: "ereal r \<noteq> - \<infinity>" "- \<infinity> \<noteq> ereal r"
and PInfty_neq_ereal[simp]: "ereal r \<noteq> \<infinity>" "\<infinity> \<noteq> ereal r"
and PInfty_cases[simp]: "(case \<infinity> of ereal r \<Rightarrow> f r | PInfty \<Rightarrow> y | MInfty \<Rightarrow> z) = y"
and MInfty_cases[simp]: "(case - \<infinity> of ereal r \<Rightarrow> f r | PInfty \<Rightarrow> y | MInfty \<Rightarrow> z) = z"
by (simp_all add: infinity_ereal_def)
declare
PInfty_eq_infinity[code_post]
MInfty_eq_minfinity[code_post]
lemma [code_unfold]:
"\<infinity> = 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 = \<infinity>"
| (MInf) "x = -\<infinity>"
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_uminus_eq_iff[simp]:
fixes a b :: ereal
shows "-a = -b \<longleftrightarrow> a = b"
by (cases rule: ereal2_cases[of a b]) simp_all
function of_ereal :: "ereal \<Rightarrow> real" where
"of_ereal (ereal r) = r"
| "of_ereal \<infinity> = 0"
| "of_ereal (-\<infinity>) = 0"
by (auto intro: ereal_cases)
termination by default (rule wf_empty)
defs (overloaded)
real_of_ereal_def [code_unfold]: "real \<equiv> of_ereal"
lemma real_of_ereal[simp]:
"real (- x :: ereal) = - (real x)"
"real (ereal r) = r"
"real (\<infinity>::ereal) = 0"
by (cases x) (simp_all add: real_of_ereal_def)
lemma range_ereal[simp]: "range ereal = UNIV - {\<infinity>, -\<infinity>}"
proof safe
fix x
assume "x \<notin> range ereal" "x \<noteq> \<infinity>"
then show "x = -\<infinity>"
by (cases x) auto
qed auto
lemma ereal_range_uminus[simp]: "range uminus = (UNIV::ereal set)"
proof safe
fix x :: ereal
show "x \<in> range uminus"
by (intro image_eqI[of _ _ "-x"]) auto
qed auto
instantiation ereal :: abs
begin
function abs_ereal where
"\<bar>ereal r\<bar> = ereal \<bar>r\<bar>"
| "\<bar>-\<infinity>\<bar> = (\<infinity>::ereal)"
| "\<bar>\<infinity>\<bar> = (\<infinity>::ereal)"
by (auto intro: ereal_cases)
termination proof qed (rule wf_empty)
instance ..
end
lemma abs_eq_infinity_cases[elim!]:
fixes x :: ereal
assumes "\<bar>x\<bar> = \<infinity>"
obtains "x = \<infinity>" | "x = -\<infinity>"
using assms by (cases x) auto
lemma abs_neq_infinity_cases[elim!]:
fixes x :: ereal
assumes "\<bar>x\<bar> \<noteq> \<infinity>"
obtains r where "x = ereal r"
using assms by (cases x) auto
lemma abs_ereal_uminus[simp]:
fixes x :: ereal
shows "\<bar>- x\<bar> = \<bar>x\<bar>"
by (cases x) auto
lemma ereal_infinity_cases:
fixes a :: ereal
shows "a \<noteq> \<infinity> \<Longrightarrow> a \<noteq> -\<infinity> \<Longrightarrow> \<bar>a\<bar> \<noteq> \<infinity>"
by auto
subsubsection "Addition"
instantiation ereal :: "{one,comm_monoid_add,zero_neq_one}"
begin
definition "0 = ereal 0"
definition "1 = ereal 1"
function plus_ereal where
"ereal r + ereal p = ereal (r + p)"
| "\<infinity> + a = (\<infinity>::ereal)"
| "a + \<infinity> = (\<infinity>::ereal)"
| "ereal r + -\<infinity> = - \<infinity>"
| "-\<infinity> + ereal p = -(\<infinity>::ereal)"
| "-\<infinity> + -\<infinity> = -(\<infinity>::ereal)"
proof -
case (goal1 P x)
then obtain a b where "x = (a, b)"
by (cases x) auto
with goal1 show P
by (cases rule: ereal2_cases[of a b]) auto
qed auto
termination by default (rule wf_empty)
lemma Infty_neq_0[simp]:
"(\<infinity>::ereal) \<noteq> 0" "0 \<noteq> (\<infinity>::ereal)"
"-(\<infinity>::ereal) \<noteq> 0" "0 \<noteq> -(\<infinity>::ereal)"
by (simp_all add: zero_ereal_def)
lemma ereal_eq_0[simp]:
"ereal r = 0 \<longleftrightarrow> r = 0"
"0 = ereal r \<longleftrightarrow> r = 0"
unfolding zero_ereal_def by simp_all
lemma ereal_eq_1[simp]:
"ereal r = 1 \<longleftrightarrow> r = 1"
"1 = ereal r \<longleftrightarrow> 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 \<noteq> (1::ereal)"
by (simp add: one_ereal_def zero_ereal_def)
qed
end
instance ereal :: numeral ..
lemma real_of_ereal_0[simp]: "real (0::ereal) = 0"
unfolding real_of_ereal_def zero_ereal_def by simp
lemma abs_ereal_zero[simp]: "\<bar>0\<bar> = (0::ereal)"
unfolding zero_ereal_def abs_ereal.simps by simp
lemma ereal_uminus_zero[simp]: "- 0 = (0::ereal)"
by (simp add: zero_ereal_def)
lemma ereal_uminus_zero_iff[simp]:
fixes a :: ereal
shows "-a = 0 \<longleftrightarrow> a = 0"
by (cases a) simp_all
lemma ereal_plus_eq_PInfty[simp]:
fixes a b :: ereal
shows "a + b = \<infinity> \<longleftrightarrow> a = \<infinity> \<or> b = \<infinity>"
by (cases rule: ereal2_cases[of a b]) auto
lemma ereal_plus_eq_MInfty[simp]:
fixes a b :: ereal
shows "a + b = -\<infinity> \<longleftrightarrow> (a = -\<infinity> \<or> b = -\<infinity>) \<and> a \<noteq> \<infinity> \<and> b \<noteq> \<infinity>"
by (cases rule: ereal2_cases[of a b]) auto
lemma ereal_add_cancel_left:
fixes a b :: ereal
assumes "a \<noteq> -\<infinity>"
shows "a + b = a + c \<longleftrightarrow> a = \<infinity> \<or> b = c"
using assms by (cases rule: ereal3_cases[of a b c]) auto
lemma ereal_add_cancel_right:
fixes a b :: ereal
assumes "a \<noteq> -\<infinity>"
shows "b + a = c + a \<longleftrightarrow> a = \<infinity> \<or> b = c"
using assms by (cases rule: ereal3_cases[of a b c]) auto
lemma ereal_real: "ereal (real x) = (if \<bar>x\<bar> = \<infinity> then 0 else x)"
by (cases x) simp_all
lemma real_of_ereal_add:
fixes a b :: ereal
shows "real (a + b) =
(if (\<bar>a\<bar> = \<infinity>) \<and> (\<bar>b\<bar> = \<infinity>) \<or> (\<bar>a\<bar> \<noteq> \<infinity>) \<and> (\<bar>b\<bar> \<noteq> \<infinity>) then real a + real 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 \<longleftrightarrow> x < y"
| "(\<infinity>::ereal) < a \<longleftrightarrow> False"
| " a < -(\<infinity>::ereal) \<longleftrightarrow> False"
| "ereal x < \<infinity> \<longleftrightarrow> True"
| " -\<infinity> < ereal r \<longleftrightarrow> True"
| " -\<infinity> < (\<infinity>::ereal) \<longleftrightarrow> True"
proof -
case (goal1 P x)
then obtain a b where "x = (a,b)" by (cases x) auto
with goal1 show P by (cases rule: ereal2_cases[of a b]) auto
qed simp_all
termination by (relation "{}") simp
definition "x \<le> (y::ereal) \<longleftrightarrow> x < y \<or> x = y"
lemma ereal_infty_less[simp]:
fixes x :: ereal
shows "x < \<infinity> \<longleftrightarrow> (x \<noteq> \<infinity>)"
"-\<infinity> < x \<longleftrightarrow> (x \<noteq> -\<infinity>)"
by (cases x, simp_all) (cases x, simp_all)
lemma ereal_infty_less_eq[simp]:
fixes x :: ereal
shows "\<infinity> \<le> x \<longleftrightarrow> x = \<infinity>"
and "x \<le> -\<infinity> \<longleftrightarrow> x = -\<infinity>"
by (auto simp add: less_eq_ereal_def)
lemma ereal_less[simp]:
"ereal r < 0 \<longleftrightarrow> (r < 0)"
"0 < ereal r \<longleftrightarrow> (0 < r)"
"ereal r < 1 \<longleftrightarrow> (r < 1)"
"1 < ereal r \<longleftrightarrow> (1 < r)"
"0 < (\<infinity>::ereal)"
"-(\<infinity>::ereal) < 0"
by (simp_all add: zero_ereal_def one_ereal_def)
lemma ereal_less_eq[simp]:
"x \<le> (\<infinity>::ereal)"
"-(\<infinity>::ereal) \<le> x"
"ereal r \<le> ereal p \<longleftrightarrow> r \<le> p"
"ereal r \<le> 0 \<longleftrightarrow> r \<le> 0"
"0 \<le> ereal r \<longleftrightarrow> 0 \<le> r"
"ereal r \<le> 1 \<longleftrightarrow> r \<le> 1"
"1 \<le> ereal r \<longleftrightarrow> 1 \<le> r"
by (auto simp add: less_eq_ereal_def zero_ereal_def one_ereal_def)
lemma ereal_infty_less_eq2:
"a \<le> b \<Longrightarrow> a = \<infinity> \<Longrightarrow> b = (\<infinity>::ereal)"
"a \<le> b \<Longrightarrow> b = -\<infinity> \<Longrightarrow> a = -(\<infinity>::ereal)"
by simp_all
instance
proof
fix x y z :: ereal
show "x \<le> x"
by (cases x) simp_all
show "x < y \<longleftrightarrow> x \<le> y \<and> \<not> y \<le> x"
by (cases rule: ereal2_cases[of x y]) auto
show "x \<le> y \<or> y \<le> x "
by (cases rule: ereal2_cases[of x y]) auto
{
assume "x \<le> y" "y \<le> x"
then show "x = y"
by (cases rule: ereal2_cases[of x y]) auto
}
{
assume "x \<le> y" "y \<le> z"
then show "x \<le> z"
by (cases rule: ereal3_cases[of x y z]) auto
}
qed
end
lemma ereal_dense2: "x < y \<Longrightarrow> \<exists>z. x < ereal z \<and> ereal z < y"
using lt_ex gt_ex dense by (cases x y rule: ereal2_cases) auto
instance ereal :: dense_linorder
by default (blast dest: ereal_dense2)
instance ereal :: ordered_ab_semigroup_add
proof
fix a b c :: ereal
assume "a \<le> b"
then show "c + a \<le> 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 \<le> x \<Longrightarrow> x \<le> y \<Longrightarrow> y \<noteq> \<infinity> \<Longrightarrow> real x \<le> real y"
by (cases rule: ereal2_cases[of x y]) auto
lemma ereal_MInfty_lessI[intro, simp]:
fixes a :: ereal
shows "a \<noteq> -\<infinity> \<Longrightarrow> -\<infinity> < a"
by (cases a) auto
lemma ereal_less_PInfty[intro, simp]:
fixes a :: ereal
shows "a \<noteq> \<infinity> \<Longrightarrow> a < \<infinity>"
by (cases a) auto
lemma ereal_less_ereal_Ex:
fixes a b :: ereal
shows "x < ereal r \<longleftrightarrow> x = -\<infinity> \<or> (\<exists>p. p < r \<and> x = ereal p)"
by (cases x) auto
lemma less_PInf_Ex_of_nat: "x \<noteq> \<infinity> \<longleftrightarrow> (\<exists>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
lemma ereal_add_mono:
fixes a b c d :: ereal
assumes "a \<le> b"
and "c \<le> d"
shows "a + c \<le> 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 \<le> - b \<longleftrightarrow> b \<le> a"
by (cases rule: ereal2_cases[of a b]) auto
lemma ereal_minus_less_minus[simp]:
fixes a b :: ereal
shows "- a < - b \<longleftrightarrow> b < a"
by (cases rule: ereal2_cases[of a b]) auto
lemma ereal_le_real_iff:
"x \<le> real y \<longleftrightarrow> (\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> ereal x \<le> y) \<and> (\<bar>y\<bar> = \<infinity> \<longrightarrow> x \<le> 0)"
by (cases y) auto
lemma real_le_ereal_iff:
"real y \<le> x \<longleftrightarrow> (\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> y \<le> ereal x) \<and> (\<bar>y\<bar> = \<infinity> \<longrightarrow> 0 \<le> x)"
by (cases y) auto
lemma ereal_less_real_iff:
"x < real y \<longleftrightarrow> (\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> ereal x < y) \<and> (\<bar>y\<bar> = \<infinity> \<longrightarrow> x < 0)"
by (cases y) auto
lemma real_less_ereal_iff:
"real y < x \<longleftrightarrow> (\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> y < ereal x) \<and> (\<bar>y\<bar> = \<infinity> \<longrightarrow> 0 < x)"
by (cases y) auto
lemma real_of_ereal_pos:
fixes x :: ereal
shows "0 \<le> x \<Longrightarrow> 0 \<le> real 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 \<le> x \<Longrightarrow> \<bar>x :: ereal\<bar> = x"
by (cases x) auto
lemma abs_ereal_less0[simp]: "x < 0 \<Longrightarrow> \<bar>x :: ereal\<bar> = -x"
by (cases x) auto
lemma abs_ereal_pos[simp]: "0 \<le> \<bar>x :: ereal\<bar>"
by (cases x) auto
lemma real_of_ereal_le_0[simp]: "real (x :: ereal) \<le> 0 \<longleftrightarrow> x \<le> 0 \<or> x = \<infinity>"
by (cases x) auto
lemma abs_real_of_ereal[simp]: "\<bar>real (x :: ereal)\<bar> = real \<bar>x\<bar>"
by (cases x) auto
lemma zero_less_real_of_ereal:
fixes x :: ereal
shows "0 < real x \<longleftrightarrow> 0 < x \<and> x \<noteq> \<infinity>"
by (cases x) auto
lemma ereal_0_le_uminus_iff[simp]:
fixes a :: ereal
shows "0 \<le> - a \<longleftrightarrow> a \<le> 0"
by (cases rule: ereal2_cases[of a]) auto
lemma ereal_uminus_le_0_iff[simp]:
fixes a :: ereal
shows "- a \<le> 0 \<longleftrightarrow> 0 \<le> a"
by (cases rule: ereal2_cases[of a]) auto
lemma ereal_add_strict_mono:
fixes a b c d :: ereal
assumes "a \<le> b"
and "0 \<le> a"
and "a \<noteq> \<infinity>"
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
lemma ereal_less_add:
fixes a b c :: ereal
shows "\<bar>a\<bar> \<noteq> \<infinity> \<Longrightarrow> c < b \<Longrightarrow> a + c < a + b"
by (cases rule: ereal2_cases[of b c]) auto
lemma ereal_add_nonneg_eq_0_iff:
fixes a b :: ereal
shows "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> a + b = 0 \<longleftrightarrow> a = 0 \<and> b = 0"
by (cases a b rule: ereal2_cases) auto
lemma ereal_uminus_eq_reorder: "- a = b \<longleftrightarrow> a = (-b::ereal)"
by auto
lemma ereal_uminus_less_reorder: "- a < b \<longleftrightarrow> -b < (a::ereal)"
by (subst (3) ereal_uminus_uminus[symmetric]) (simp only: ereal_minus_less_minus)
lemma ereal_uminus_le_reorder: "- a \<le> b \<longleftrightarrow> -b \<le> (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 "\<And>B. x \<le> ereal B"
shows "x = - \<infinity>"
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 "\<And>B. x \<ge> ereal B"
shows "x = \<infinity>"
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)"
by (simp_all add: min_def max_def)
lemma ereal_max_0: "max 0 (ereal r) = ereal (max 0 r)"
by (auto simp: zero_ereal_def)
lemma
fixes f :: "nat \<Rightarrow> ereal"
shows ereal_incseq_uminus[simp]: "incseq (\<lambda>x. - f x) \<longleftrightarrow> decseq f"
and ereal_decseq_uminus[simp]: "decseq (\<lambda>x. - f x) \<longleftrightarrow> incseq f"
unfolding decseq_def incseq_def by auto
lemma incseq_ereal: "incseq f \<Longrightarrow> incseq (\<lambda>x. ereal (f x))"
unfolding incseq_def by auto
lemma ereal_add_nonneg_nonneg[simp]:
fixes a b :: ereal
shows "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> a + b"
using add_mono[of 0 a 0 b] by simp
lemma image_eqD: "f ` A = B \<Longrightarrow> \<forall>x\<in>A. f x \<in> B"
by auto
lemma incseq_setsumI:
fixes f :: "nat \<Rightarrow> 'a::{comm_monoid_add,ordered_ab_semigroup_add}"
assumes "\<And>i. 0 \<le> f i"
shows "incseq (\<lambda>i. setsum f {..< i})"
proof (intro incseq_SucI)
fix n
have "setsum f {..< n} + 0 \<le> setsum f {..<n} + f n"
using assms by (rule add_left_mono)
then show "setsum f {..< n} \<le> setsum f {..< Suc n}"
by auto
qed
lemma incseq_setsumI2:
fixes f :: "'i \<Rightarrow> nat \<Rightarrow> 'a::{comm_monoid_add,ordered_ab_semigroup_add}"
assumes "\<And>n. n \<in> A \<Longrightarrow> incseq (f n)"
shows "incseq (\<lambda>i. \<Sum>n\<in>A. f n i)"
using assms
unfolding incseq_def by (auto intro: setsum_mono)
subsubsection "Multiplication"
instantiation ereal :: "{comm_monoid_mult,sgn}"
begin
function sgn_ereal :: "ereal \<Rightarrow> ereal" where
"sgn (ereal r) = ereal (sgn r)"
| "sgn (\<infinity>::ereal) = 1"
| "sgn (-\<infinity>::ereal) = -1"
by (auto intro: ereal_cases)
termination by default (rule wf_empty)
function times_ereal where
"ereal r * ereal p = ereal (r * p)"
| "ereal r * \<infinity> = (if r = 0 then 0 else if r > 0 then \<infinity> else -\<infinity>)"
| "\<infinity> * ereal r = (if r = 0 then 0 else if r > 0 then \<infinity> else -\<infinity>)"
| "ereal r * -\<infinity> = (if r = 0 then 0 else if r > 0 then -\<infinity> else \<infinity>)"
| "-\<infinity> * ereal r = (if r = 0 then 0 else if r > 0 then -\<infinity> else \<infinity>)"
| "(\<infinity>::ereal) * \<infinity> = \<infinity>"
| "-(\<infinity>::ereal) * \<infinity> = -\<infinity>"
| "(\<infinity>::ereal) * -\<infinity> = -\<infinity>"
| "-(\<infinity>::ereal) * -\<infinity> = \<infinity>"
proof -
case (goal1 P x)
then obtain a b where "x = (a, b)"
by (cases x) auto
with goal1 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])
(simp_all add: zero_ereal_def zero_less_mult_iff)
qed
end
lemma real_ereal_1[simp]: "real (1::ereal) = 1"
unfolding one_ereal_def by simp
lemma real_of_ereal_le_1:
fixes a :: ereal
shows "a \<le> 1 \<Longrightarrow> real a \<le> 1"
by (cases a) (auto simp: one_ereal_def)
lemma abs_ereal_one[simp]: "\<bar>1\<bar> = (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"
by (simp add: zero_ereal_def one_ereal_def)
lemma ereal_times[simp]:
"1 \<noteq> (\<infinity>::ereal)" "(\<infinity>::ereal) \<noteq> 1"
"1 \<noteq> -(\<infinity>::ereal)" "-(\<infinity>::ereal) \<noteq> 1"
by (auto simp add: times_ereal_def one_ereal_def)
lemma ereal_plus_1[simp]:
"1 + ereal r = ereal (r + 1)"
"ereal r + 1 = ereal (r + 1)"
"1 + -(\<infinity>::ereal) = -\<infinity>"
"-(\<infinity>::ereal) + 1 = -\<infinity>"
unfolding one_ereal_def by auto
lemma ereal_zero_times[simp]:
fixes a b :: ereal
shows "a * b = 0 \<longleftrightarrow> a = 0 \<or> b = 0"
by (cases rule: ereal2_cases[of a b]) auto
lemma ereal_mult_eq_PInfty[simp]:
"a * b = (\<infinity>::ereal) \<longleftrightarrow>
(a = \<infinity> \<and> b > 0) \<or> (a > 0 \<and> b = \<infinity>) \<or> (a = -\<infinity> \<and> b < 0) \<or> (a < 0 \<and> b = -\<infinity>)"
by (cases rule: ereal2_cases[of a b]) auto
lemma ereal_mult_eq_MInfty[simp]:
"a * b = -(\<infinity>::ereal) \<longleftrightarrow>
(a = \<infinity> \<and> b < 0) \<or> (a < 0 \<and> b = \<infinity>) \<or> (a = -\<infinity> \<and> b > 0) \<or> (a > 0 \<and> b = -\<infinity>)"
by (cases rule: ereal2_cases[of a b]) auto
lemma ereal_abs_mult: "\<bar>x * y :: ereal\<bar> = \<bar>x\<bar> * \<bar>y\<bar>"
by (cases x y rule: ereal2_cases) (auto simp: abs_mult)
lemma ereal_0_less_1[simp]: "0 < (1::ereal)"
by (simp_all add: zero_ereal_def one_ereal_def)
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 * (\<infinity>::ereal) = (if a = 0 then 0 else if 0 < a then \<infinity> else - \<infinity>)"
by (cases a) auto
lemma ereal_infty_mult[simp]:
"(\<infinity>::ereal) * a = (if a = 0 then 0 else if 0 < a then \<infinity> else - \<infinity>)"
by (cases a) auto
lemma ereal_mult_strict_right_mono:
assumes "a < b"
and "0 < c"
and "c < (\<infinity>::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 \<Longrightarrow> 0 < c \<Longrightarrow> c < (\<infinity>::ereal) \<Longrightarrow> c * a < c * b"
using ereal_mult_strict_right_mono
by (simp add: mult_commute[of c])
lemma ereal_mult_right_mono:
fixes a b c :: ereal
shows "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> 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 \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> c * a \<le> c * b"
using ereal_mult_right_mono
by (simp add: mult_commute[of c])
lemma zero_less_one_ereal[simp]: "0 \<le> (1::ereal)"
by (simp add: one_ereal_def zero_ereal_def)
lemma ereal_0_le_mult[simp]: "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> a * (b :: ereal)"
by (cases rule: ereal2_cases[of a b]) auto
lemma ereal_right_distrib:
fixes r a b :: ereal
shows "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 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 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> (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 \<le> 0 \<longleftrightarrow> (0 \<le> a \<and> b \<le> 0) \<or> (a \<le> 0 \<and> 0 \<le> 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 \<le> a * b \<longleftrightarrow> (0 \<le> a \<and> 0 \<le> b) \<or> (a \<le> 0 \<and> b \<le> 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 \<longleftrightarrow> (0 < a \<and> b < 0) \<or> (a < 0 \<and> 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 \<longleftrightarrow> (0 < a \<and> 0 < b) \<or> (a < 0 \<and> 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 \<noteq> 0 \<Longrightarrow> a = b) \<Longrightarrow> c * a = c * b"
by (cases "c = 0") simp_all
lemma ereal_right_mult_cong:
fixes a b c :: ereal
shows "(c \<noteq> 0 \<Longrightarrow> a = b) \<Longrightarrow> a * c = b * c"
by (cases "c = 0") simp_all
lemma ereal_distrib:
fixes a b c :: ereal
assumes "a \<noteq> \<infinity> \<or> b \<noteq> -\<infinity>"
and "a \<noteq> -\<infinity> \<or> b \<noteq> \<infinity>"
and "\<bar>c\<bar> \<noteq> \<infinity>"
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 ereal_le_epsilon:
fixes x y :: ereal
assumes "\<forall>e. 0 < e \<longrightarrow> x \<le> y + e"
shows "x \<le> y"
proof -
{
assume a: "\<exists>r. y = ereal r"
then obtain r where r_def: "y = ereal r"
by auto
{
assume "x = -\<infinity>"
then have ?thesis by auto
}
moreover
{
assume "x \<noteq> -\<infinity>"
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 \<longrightarrow> p \<le> r + e"
using assms[rule_format, of "ereal e"] p_def r_def by auto
}
then have "p \<le> 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 = -\<infinity> | y = \<infinity>"
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 "\<forall>e. 0 < e \<longrightarrow> x \<le> y + ereal e"
shows "x \<le> y"
proof -
{
fix e :: ereal
assume "e > 0"
{
assume "e = \<infinity>"
then have "x \<le> y + e"
by auto
}
moreover
{
assume "e \<noteq> \<infinity>"
then obtain r where "e = ereal r"
using `e > 0` by (cases e) auto
then have "x \<le> y + e"
using assms[rule_format, of r] `e>0` by auto
}
ultimately have "x \<le> y + e"
by blast
}
then show ?thesis
using ereal_le_epsilon by auto
qed
lemma ereal_le_real:
fixes x y :: ereal
assumes "\<forall>z. x \<le> ereal z \<longrightarrow> y \<le> ereal z"
shows "y \<le> 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 \<Rightarrow> ereal"
shows "(\<Prod>i\<in>A. f i) = 0 \<longleftrightarrow> finite A \<and> (\<exists>i\<in>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 \<Rightarrow> ereal"
assumes pos: "\<And>i. i \<in> I \<Longrightarrow> 0 \<le> f i"
shows "0 \<le> (\<Prod>i\<in>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 \<Rightarrow> ereal"
assumes "\<And>i. i \<in> I \<Longrightarrow> 0 \<le> f i"
shows "(\<Prod>i\<in>I. f i) = \<infinity> \<longleftrightarrow> finite I \<and> (\<exists>i\<in>I. f i = \<infinity>) \<and> (\<forall>i\<in>I. f i \<noteq> 0)"
proof (cases "finite I")
case True
from this assms show ?thesis
proof (induct I)
case (insert i I)
then have pos: "0 \<le> f i" "0 \<le> setprod f I"
by (auto intro!: setprod_ereal_pos)
from insert have "(\<Prod>j\<in>insert i I. f j) = \<infinity> \<longleftrightarrow> setprod f I * f i = \<infinity>"
by auto
also have "\<dots> \<longleftrightarrow> (setprod f I = \<infinity> \<or> f i = \<infinity>) \<and> f i \<noteq> 0 \<and> setprod f I \<noteq> 0"
using setprod_ereal_pos[of I f] pos
by (cases rule: ereal2_cases[of "f i" "setprod f I"]) auto
also have "\<dots> \<longleftrightarrow> finite (insert i I) \<and> (\<exists>j\<in>insert i I. f j = \<infinity>) \<and> (\<forall>j\<in>insert i I. f j \<noteq> 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: "(\<Prod>i\<in>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
by (simp add: one_ereal_def)
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]: "(\<infinity>::ereal) ^ n = (if n = 0 then 1 else \<infinity>)"
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 \<le> a"
shows "0 \<le> 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)"
"-\<infinity> - ereal r = -\<infinity>"
"ereal r - \<infinity> = -\<infinity>"
"(\<infinity>::ereal) - x = \<infinity>"
"-(\<infinity>::ereal) - \<infinity> = -\<infinity>"
"x - -y = x + y"
"x - 0 = x"
"0 - x = -x"
by (simp_all add: minus_ereal_def)
lemma ereal_x_minus_x[simp]: "x - x = (if \<bar>x\<bar> = \<infinity> then \<infinity> else 0::ereal)"
by (cases x) simp_all
lemma ereal_eq_minus_iff:
fixes x y z :: ereal
shows "x = z - y \<longleftrightarrow>
(\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> x + y = z) \<and>
(y = -\<infinity> \<longrightarrow> x = \<infinity>) \<and>
(y = \<infinity> \<longrightarrow> z = \<infinity> \<longrightarrow> x = \<infinity>) \<and>
(y = \<infinity> \<longrightarrow> z \<noteq> \<infinity> \<longrightarrow> x = -\<infinity>)"
by (cases rule: ereal3_cases[of x y z]) auto
lemma ereal_eq_minus:
fixes x y z :: ereal
shows "\<bar>y\<bar> \<noteq> \<infinity> \<Longrightarrow> x = z - y \<longleftrightarrow> x + y = z"
by (auto simp: ereal_eq_minus_iff)
lemma ereal_less_minus_iff:
fixes x y z :: ereal
shows "x < z - y \<longleftrightarrow>
(y = \<infinity> \<longrightarrow> z = \<infinity> \<and> x \<noteq> \<infinity>) \<and>
(y = -\<infinity> \<longrightarrow> x \<noteq> \<infinity>) \<and>
(\<bar>y\<bar> \<noteq> \<infinity>\<longrightarrow> x + y < z)"
by (cases rule: ereal3_cases[of x y z]) auto
lemma ereal_less_minus:
fixes x y z :: ereal
shows "\<bar>y\<bar> \<noteq> \<infinity> \<Longrightarrow> x < z - y \<longleftrightarrow> x + y < z"
by (auto simp: ereal_less_minus_iff)
lemma ereal_le_minus_iff:
fixes x y z :: ereal
shows "x \<le> z - y \<longleftrightarrow> (y = \<infinity> \<longrightarrow> z \<noteq> \<infinity> \<longrightarrow> x = -\<infinity>) \<and> (\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> x + y \<le> z)"
by (cases rule: ereal3_cases[of x y z]) auto
lemma ereal_le_minus:
fixes x y z :: ereal
shows "\<bar>y\<bar> \<noteq> \<infinity> \<Longrightarrow> x \<le> z - y \<longleftrightarrow> x + y \<le> z"
by (auto simp: ereal_le_minus_iff)
lemma ereal_minus_less_iff:
fixes x y z :: ereal
shows "x - y < z \<longleftrightarrow> y \<noteq> -\<infinity> \<and> (y = \<infinity> \<longrightarrow> x \<noteq> \<infinity> \<and> z \<noteq> -\<infinity>) \<and> (y \<noteq> \<infinity> \<longrightarrow> x < z + y)"
by (cases rule: ereal3_cases[of x y z]) auto
lemma ereal_minus_less:
fixes x y z :: ereal
shows "\<bar>y\<bar> \<noteq> \<infinity> \<Longrightarrow> x - y < z \<longleftrightarrow> x < z + y"
by (auto simp: ereal_minus_less_iff)
lemma ereal_minus_le_iff:
fixes x y z :: ereal
shows "x - y \<le> z \<longleftrightarrow>
(y = -\<infinity> \<longrightarrow> z = \<infinity>) \<and>
(y = \<infinity> \<longrightarrow> x = \<infinity> \<longrightarrow> z = \<infinity>) \<and>
(\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> x \<le> z + y)"
by (cases rule: ereal3_cases[of x y z]) auto
lemma ereal_minus_le:
fixes x y z :: ereal
shows "\<bar>y\<bar> \<noteq> \<infinity> \<Longrightarrow> x - y \<le> z \<longleftrightarrow> x \<le> 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 \<longleftrightarrow>
b = c \<or> a = \<infinity> \<or> (a = -\<infinity> \<and> b \<noteq> -\<infinity> \<and> c \<noteq> -\<infinity>)"
by (cases rule: ereal3_cases[of a b c]) auto
lemma ereal_add_le_add_iff:
fixes a b c :: ereal
shows "c + a \<le> c + b \<longleftrightarrow>
a \<le> b \<or> c = \<infinity> \<or> (c = -\<infinity> \<and> a \<noteq> \<infinity> \<and> b \<noteq> \<infinity>)"
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 "\<bar>c\<bar> \<noteq> \<infinity> \<Longrightarrow> c * a \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 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 \<le> B" "D \<le> C"
shows "A - C \<le> 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 (a - b) = (if \<bar>a\<bar> = \<infinity> \<or> \<bar>b\<bar> = \<infinity> then 0 else real a - real b)"
by (cases rule: ereal2_cases[of a b]) auto
lemma ereal_diff_positive:
fixes a b :: ereal shows "a \<le> b \<Longrightarrow> 0 \<le> b - a"
by (cases rule: ereal2_cases[of a b]) auto
lemma ereal_between:
fixes x e :: ereal
assumes "\<bar>x\<bar> \<noteq> \<infinity>"
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 = \<infinity> \<longleftrightarrow> y = -\<infinity> \<or> x = \<infinity>"
by (cases x y rule: ereal2_cases) simp_all
subsubsection {* Division *}
instantiation ereal :: inverse
begin
function inverse_ereal where
"inverse (ereal r) = (if r = 0 then \<infinity> else ereal (inverse r))"
| "inverse (\<infinity>::ereal) = 0"
| "inverse (-\<infinity>::ereal) = 0"
by (auto intro: ereal_cases)
termination by (relation "{}") simp
definition "x / y = x * inverse (y :: ereal)"
instance ..
end
lemma real_of_ereal_inverse[simp]:
fixes a :: ereal
shows "real (inverse a) = 1 / real a"
by (cases a) (auto simp: inverse_eq_divide)
lemma ereal_inverse[simp]:
"inverse (0::ereal) = \<infinity>"
"inverse (1::ereal) = 1"
by (simp_all add: one_ereal_def zero_ereal_def)
lemma ereal_divide[simp]:
"ereal r / ereal p = (if p = 0 then ereal r * \<infinity> 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 \<bar>x\<bar> = \<infinity> \<or> 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 \<noteq> -\<infinity> then x else \<infinity>)"
by (cases x) auto
lemma ereal_inverse_minus[simp]:
fixes x :: ereal
shows "inverse (- x) = (if x = 0 then \<infinity> 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 / \<infinity> = 0" "x / -\<infinity> = 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]: "\<infinity> / ereal r = (if 0 \<le> r then \<infinity> else -\<infinity>)"
unfolding divide_ereal_def by simp
lemma zero_le_divide_ereal[simp]:
fixes a :: ereal
assumes "0 \<le> a"
and "0 \<le> b"
shows "0 \<le> 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 \<Longrightarrow> x \<noteq> \<infinity> \<Longrightarrow> y \<le> z / x \<longleftrightarrow> x * y \<le> 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 \<Longrightarrow> x \<noteq> \<infinity> \<Longrightarrow> z / x \<le> y \<longleftrightarrow> z \<le> 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 \<Longrightarrow> x \<noteq> -\<infinity> \<Longrightarrow> y \<le> z / x \<longleftrightarrow> z \<le> 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 \<Longrightarrow> x \<noteq> -\<infinity> \<Longrightarrow> z / x \<le> y \<longleftrightarrow> x * y \<le> 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 \<le> x \<Longrightarrow> x < y \<Longrightarrow> inverse y < inverse x"
by (cases rule: ereal2_cases[of x y]) auto
lemma ereal_inverse_antimono:
fixes x y :: ereal
shows "0 \<le> x \<Longrightarrow> x \<le> y \<Longrightarrow> inverse y \<le> inverse x"
by (cases rule: ereal2_cases[of x y]) auto
lemma inverse_inverse_Pinfty_iff[simp]:
fixes x :: ereal
shows "inverse x = \<infinity> \<longleftrightarrow> x = 0"
by (cases x) auto
lemma ereal_inverse_eq_0:
fixes x :: ereal
shows "inverse x = 0 \<longleftrightarrow> x = \<infinity> \<or> x = -\<infinity>"
by (cases x) auto
lemma ereal_0_gt_inverse:
fixes x :: ereal
shows "0 < inverse x \<longleftrightarrow> x \<noteq> \<infinity> \<and> 0 \<le> x"
by (cases x) auto
lemma ereal_mult_less_right:
fixes a b c :: ereal
assumes "b * a < c * a"
and "0 < a"
and "a < \<infinity>"
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_power_divide:
fixes x y :: ereal
shows "y \<noteq> 0 \<Longrightarrow> (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 not_le
power_less_zero_eq zero_le_power_iff)
lemma ereal_le_mult_one_interval:
fixes x y :: ereal
assumes y: "y \<noteq> -\<infinity>"
assumes z: "\<And>z. 0 < z \<Longrightarrow> z < 1 \<Longrightarrow> z * x \<le> y"
shows "x \<le> y"
proof (cases x)
case PInf
with z[of "1 / 2"] show "x \<le> y"
by (simp add: one_ereal_def)
next
case (real r)
note r = this
show "x \<le> y"
proof (cases y)
case (real p)
note p = this
have "r \<le> 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 \<le> p"
using p r by (auto simp: zero_le_mult_iff one_ereal_def)
qed
then show "x \<le> 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 \<le> y"
and "0 < z"
shows "x / z \<le> 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 \<le> x"
and "0 < z"
and "0 < x * y"
shows "z / x \<le> 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)
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 default simp_all
end
instantiation ereal :: complete_lattice
begin
definition "bot = (-\<infinity>::ereal)"
definition "top = (\<infinity>::ereal)"
definition "Sup S = (SOME x :: ereal. (\<forall>y\<in>S. y \<le> x) \<and> (\<forall>z. (\<forall>y\<in>S. y \<le> z) \<longrightarrow> x \<le> z))"
definition "Inf S = (SOME x :: ereal. (\<forall>y\<in>S. x \<le> y) \<and> (\<forall>z. (\<forall>y\<in>S. z \<le> y) \<longrightarrow> z \<le> x))"
lemma ereal_complete_Sup:
fixes S :: "ereal set"
shows "\<exists>x. (\<forall>y\<in>S. y \<le> x) \<and> (\<forall>z. (\<forall>y\<in>S. y \<le> z) \<longrightarrow> x \<le> z)"
proof (cases "\<exists>x. \<forall>a\<in>S. a \<le> ereal x")
case True
then obtain y where y: "\<And>a. a\<in>S \<Longrightarrow> a \<le> ereal y"
by auto
then have "\<infinity> \<notin> S"
by force
show ?thesis
proof (cases "S \<noteq> {-\<infinity>} \<and> S \<noteq> {}")
case True
with `\<infinity> \<notin> S` obtain x where x: "x \<in> S" "\<bar>x\<bar> \<noteq> \<infinity>"
by auto
obtain s where s: "\<forall>x\<in>ereal -` S. x \<le> s" "\<And>z. (\<forall>x\<in>ereal -` S. x \<le> z) \<Longrightarrow> s \<le> z"
proof (atomize_elim, rule complete_real)
show "\<exists>x. x \<in> ereal -` S"
using x by auto
show "\<exists>z. \<forall>x\<in>ereal -` S. x \<le> z"
by (auto dest: y intro!: exI[of _ y])
qed
show ?thesis
proof (safe intro!: exI[of _ "ereal s"])
fix y
assume "y \<in> S"
with s `\<infinity> \<notin> S` show "y \<le> ereal s"
by (cases y) auto
next
fix z
assume "\<forall>y\<in>S. y \<le> z"
with `S \<noteq> {-\<infinity>} \<and> S \<noteq> {}` show "ereal s \<le> z"
by (cases z) (auto intro!: s)
qed
next
case False
then show ?thesis
by (auto intro!: exI[of _ "-\<infinity>"])
qed
next
case False
then show ?thesis
by (fastforce intro!: exI[of _ \<infinity>] ereal_top intro: order_trans dest: less_imp_le simp: not_le)
qed
lemma ereal_complete_uminus_eq:
fixes S :: "ereal set"
shows "(\<forall>y\<in>uminus`S. y \<le> x) \<and> (\<forall>z. (\<forall>y\<in>uminus`S. y \<le> z) \<longrightarrow> x \<le> z)
\<longleftrightarrow> (\<forall>y\<in>S. -x \<le> y) \<and> (\<forall>z. (\<forall>y\<in>S. z \<le> y) \<longrightarrow> z \<le> -x)"
by simp (metis ereal_minus_le_minus ereal_uminus_uminus)
lemma ereal_complete_Inf:
"\<exists>x. (\<forall>y\<in>S::ereal set. x \<le> y) \<and> (\<forall>z. (\<forall>y\<in>S. z \<le> y) \<longrightarrow> z \<le> 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 "\<exists>a b::ereal. a \<noteq> b"
using zero_neq_one by blast
qed
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 \<Rightarrow> 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 \<Rightarrow> ereal"
shows "(INF x:S. uminus (f x)) = - (SUP x:S. f x)"
using ereal_Inf_uminus_image_eq [of "f ` S"] by (simp add: comp_def)
lemma ereal_SUP_not_infty:
fixes f :: "_ \<Rightarrow> ereal"
shows "A \<noteq> {} \<Longrightarrow> l \<noteq> -\<infinity> \<Longrightarrow> u \<noteq> \<infinity> \<Longrightarrow> \<forall>a\<in>A. l \<le> f a \<and> f a \<le> u \<Longrightarrow> \<bar>SUPREMUM A f\<bar> \<noteq> \<infinity>"
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 :: "_ \<Rightarrow> ereal"
shows "A \<noteq> {} \<Longrightarrow> l \<noteq> -\<infinity> \<Longrightarrow> u \<noteq> \<infinity> \<Longrightarrow> \<forall>a\<in>A. l \<le> f a \<and> f a \<le> u \<Longrightarrow> \<bar>INFIMUM A f\<bar> \<noteq> \<infinity>"
using INF_lower2[of _ A f u] INF_greatest[of A l f]
by (cases "INFIMUM A f") auto
lemma ereal_SUP_uminus:
fixes f :: "'a \<Rightarrow> ereal"
shows "(SUP i : R. -(f i)) = -(INF i : R. f i)"
using ereal_Sup_uminus_image_eq[of "f`R"]
by (simp add: image_image)
lemma ereal_INF_uminus:
fixes f :: "'a \<Rightarrow> ereal"
shows "(INF i : R. - f i) = - (SUP i : R. f i)"
using ereal_SUP_uminus [of _ "\<lambda>x. - f x"] by simp
lemma ereal_image_uminus_shift:
fixes X Y :: "ereal set"
shows "uminus ` X = Y \<longleftrightarrow> X = uminus ` Y"
proof
assume "uminus ` X = Y"
then have "uminus ` uminus ` X = uminus ` Y"
by (simp add: inj_image_eq_iff)
then show "X = uminus ` Y"
by (simp add: image_image)
qed (simp add: image_image)
lemma Inf_ereal_iff:
fixes z :: ereal
shows "(\<And>x. x \<in> X \<Longrightarrow> z \<le> x) \<Longrightarrow> (\<exists>x\<in>X. x < y) \<longleftrightarrow> Inf X < y"
by (metis complete_lattice_class.Inf_greatest complete_lattice_class.Inf_lower
less_le_not_le linear order_less_le_trans)
lemma Sup_eq_MInfty:
fixes S :: "ereal set"
shows "Sup S = -\<infinity> \<longleftrightarrow> S = {} \<or> S = {-\<infinity>}"
unfolding bot_ereal_def[symmetric] by auto
lemma Inf_eq_PInfty:
fixes S :: "ereal set"
shows "Inf S = \<infinity> \<longleftrightarrow> S = {} \<or> S = {\<infinity>}"
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 "-\<infinity> \<in> S \<Longrightarrow> Inf S = -\<infinity>"
unfolding bot_ereal_def[symmetric] by auto
lemma Sup_eq_PInfty:
fixes S :: "ereal set"
shows "\<infinity> \<in> S \<Longrightarrow> Sup S = \<infinity>"
unfolding top_ereal_def[symmetric] by auto
lemma Sup_ereal_close:
fixes e :: ereal
assumes "0 < e"
and S: "\<bar>Sup S\<bar> \<noteq> \<infinity>" "S \<noteq> {}"
shows "\<exists>x\<in>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 "\<bar>Inf X\<bar> \<noteq> \<infinity>"
and "0 < e"
shows "\<exists>x\<in>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_nat_Infty: "(SUP i::nat. ereal (real i)) = \<infinity>"
proof -
{
fix x :: ereal
assume "x \<noteq> \<infinity>"
then have "\<exists>k::nat. x < ereal (real k)"
proof (cases x)
case MInf
then show ?thesis
by (intro exI[of _ 0]) auto
next
case (real r)
moreover obtain k :: nat where "r < real k"
using ex_less_of_nat by (auto simp: real_eq_of_nat)
ultimately show ?thesis
by auto
qed simp
}
then show ?thesis
using SUP_eq_top_iff[of UNIV "\<lambda>n::nat. ereal (real n)"]
by (auto simp: top_ereal_def)
qed
lemma Inf_less:
fixes x :: ereal
assumes "(INF i:A. f i) < x"
shows "\<exists>i. i \<in> A \<and> f i \<le> x"
proof (rule ccontr)
assume "\<not> ?thesis"
then have "\<forall>i\<in>A. f i > x"
by auto
then have "(INF i:A. f i) \<ge> x"
by (subst INF_greatest) auto
then show False
using assms by auto
qed
lemma SUP_ereal_le_addI:
fixes f :: "'i \<Rightarrow> ereal"
assumes "\<And>i. f i + y \<le> z"
and "y \<noteq> -\<infinity>"
shows "SUPREMUM UNIV f + y \<le> z"
proof (cases y)
case (real r)
then have "\<And>i. f i \<le> z - y"
using assms by (simp add: ereal_le_minus_iff)
then have "SUPREMUM UNIV f \<le> z - y"
by (rule SUP_least)
then show ?thesis
using real by (simp add: ereal_le_minus_iff)
qed (insert assms, auto)
lemma SUP_ereal_add:
fixes f g :: "nat \<Rightarrow> ereal"
assumes "incseq f"
and "incseq g"
and pos: "\<And>i. f i \<noteq> -\<infinity>" "\<And>i. g i \<noteq> -\<infinity>"
shows "(SUP i. f i + g i) = SUPREMUM UNIV f + SUPREMUM UNIV g"
proof (rule SUP_eqI)
fix y
assume *: "\<And>i. i \<in> UNIV \<Longrightarrow> f i + g i \<le> y"
have f: "SUPREMUM UNIV f \<noteq> -\<infinity>"
using pos
unfolding SUP_def Sup_eq_MInfty
by (auto dest: image_eqD)
{
fix j
{
fix i
have "f i + g j \<le> f i + g (max i j)"
using `incseq g`[THEN incseqD]
by (rule add_left_mono) auto
also have "\<dots> \<le> f (max i j) + g (max i j)"
using `incseq f`[THEN incseqD]
by (rule add_right_mono) auto
also have "\<dots> \<le> y" using * by auto
finally have "f i + g j \<le> y" .
}
then have "SUPREMUM UNIV f + g j \<le> y"
using assms(4)[of j] by (intro SUP_ereal_le_addI) auto
then have "g j + SUPREMUM UNIV f \<le> y" by (simp add: ac_simps)
}
then have "SUPREMUM UNIV g + SUPREMUM UNIV f \<le> y"
using f by (rule SUP_ereal_le_addI)
then show "SUPREMUM UNIV f + SUPREMUM UNIV g \<le> y"
by (simp add: ac_simps)
qed (auto intro!: add_mono SUP_upper)
lemma SUP_ereal_add_pos:
fixes f g :: "nat \<Rightarrow> ereal"
assumes inc: "incseq f" "incseq g"
and pos: "\<And>i. 0 \<le> f i" "\<And>i. 0 \<le> g i"
shows "(SUP i. f i + g i) = SUPREMUM UNIV f + SUPREMUM UNIV g"
proof (intro SUP_ereal_add inc)
fix i
show "f i \<noteq> -\<infinity>" "g i \<noteq> -\<infinity>"
using pos[of i] by auto
qed
lemma SUP_ereal_setsum:
fixes f g :: "'a \<Rightarrow> nat \<Rightarrow> ereal"
assumes "\<And>n. n \<in> A \<Longrightarrow> incseq (f n)"
and pos: "\<And>n i. n \<in> A \<Longrightarrow> 0 \<le> f n i"
shows "(SUP i. \<Sum>n\<in>A. f n i) = (\<Sum>n\<in>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_cmult:
fixes f :: "nat \<Rightarrow> ereal"
assumes "\<And>i. 0 \<le> f i"
and "0 \<le> c"
shows "(SUP i. c * f i) = c * SUPREMUM UNIV f"
proof (rule SUP_eqI)
fix i
have "f i \<le> SUPREMUM UNIV f"
by (rule SUP_upper) auto
then show "c * f i \<le> c * SUPREMUM UNIV f"
using `0 \<le> c` by (rule ereal_mult_left_mono)
next
fix y
assume "\<And>i. i \<in> UNIV \<Longrightarrow> c * f i \<le> y"
then have *: "\<And>i. c * f i \<le> y" by simp
show "c * SUPREMUM UNIV f \<le> y"
proof (cases "0 < c \<and> c \<noteq> \<infinity>")
case True
with * have "SUPREMUM UNIV f \<le> y / c"
by (intro SUP_least) (auto simp: ereal_le_divide_pos)
with True show ?thesis
by (auto simp: ereal_le_divide_pos)
next
case False
{
assume "c = \<infinity>"
have ?thesis
proof (cases "\<forall>i. f i = 0")
case True
then have "range f = {0}"
by auto
with True show "c * SUPREMUM UNIV f \<le> y"
using * by auto
next
case False
then obtain i where "f i \<noteq> 0"
by auto
with *[of i] `c = \<infinity>` `0 \<le> f i` show ?thesis
by (auto split: split_if_asm)
qed
}
moreover note False
ultimately show ?thesis
using * `0 \<le> c` by auto
qed
qed
lemma SUP_PInfty:
fixes f :: "'a \<Rightarrow> ereal"
assumes "\<And>n::nat. \<exists>i\<in>A. ereal (real n) \<le> f i"
shows "(SUP i:A. f i) = \<infinity>"
unfolding SUP_def Sup_eq_top_iff[where 'a=ereal, unfolded top_ereal_def]
apply simp
proof safe
fix x :: ereal
assume "x \<noteq> \<infinity>"
show "\<exists>i\<in>A. x < f i"
proof (cases x)
case PInf
with `x \<noteq> \<infinity>` show ?thesis
by simp
next
case MInf
with assms[of "0"] show ?thesis
by force
next
case (real r)
with less_PInf_Ex_of_nat[of x] obtain n :: nat where "x < ereal (real n)"
by auto
moreover obtain i where "i \<in> A" "ereal (real n) \<le> f i"
using assms ..
ultimately show ?thesis
by (auto intro!: bexI[of _ i])
qed
qed
lemma Sup_countable_SUP:
assumes "A \<noteq> {}"
shows "\<exists>f::nat \<Rightarrow> ereal. range f \<subseteq> A \<and> Sup A = SUPREMUM UNIV f"
proof (cases "Sup A")
case (real r)
have "\<forall>n::nat. \<exists>x. x \<in> A \<and> Sup A < x + 1 / ereal (real n)"
proof
fix n :: nat
have "\<exists>x\<in>A. Sup A - 1 / ereal (real n) < x"
using assms real by (intro Sup_ereal_close) (auto simp: one_ereal_def)
then obtain x where "x \<in> A" "Sup A - 1 / ereal (real n) < x" ..
then show "\<exists>x. x \<in> A \<and> Sup A < x + 1 / ereal (real n)"
by (auto intro!: exI[of _ x] simp: ereal_minus_less_iff)
qed
from choice[OF this] obtain f :: "nat \<Rightarrow> ereal"
where f: "\<forall>x. f x \<in> A \<and> Sup A < f x + 1 / ereal (real x)" ..
have "SUPREMUM UNIV f = Sup A"
proof (rule SUP_eqI)
fix i
show "f i \<le> Sup A"
using f by (auto intro!: complete_lattice_class.Sup_upper)
next
fix y
assume bound: "\<And>i. i \<in> UNIV \<Longrightarrow> f i \<le> y"
show "Sup A \<le> y"
proof (rule ereal_le_epsilon, intro allI impI)
fix e :: ereal
assume "0 < e"
show "Sup A \<le> y + e"
proof (cases e)
case (real r)
then have "0 < r"
using `0 < e` by auto
then obtain n :: nat where *: "1 / real n < r" "0 < n"
using ex_inverse_of_nat_less
by (auto simp: real_eq_of_nat inverse_eq_divide)
have "Sup A \<le> f n + 1 / ereal (real n)"
using f[THEN spec, of n]
by auto
also have "1 / ereal (real n) \<le> e"
using real *
by (auto simp: one_ereal_def )
with bound have "f n + 1 / ereal (real n) \<le> y + e"
by (rule add_mono) simp
finally show "Sup A \<le> y + e" .
qed (insert `0 < e`, auto)
qed
qed
with f show ?thesis
by (auto intro!: exI[of _ f])
next
case PInf
from `A \<noteq> {}` obtain x where "x \<in> A"
by auto
show ?thesis
proof (cases "\<infinity> \<in> A")
case True
then have "\<infinity> \<le> Sup A"
by (intro complete_lattice_class.Sup_upper)
with True show ?thesis
by (auto intro!: exI[of _ "\<lambda>x. \<infinity>"])
next
case False
have "\<exists>x\<in>A. 0 \<le> x"
by (metis Infty_neq_0(2) PInf complete_lattice_class.Sup_least ereal_infty_less_eq2(1) linorder_linear)
then obtain x where "x \<in> A" and "0 \<le> x"
by auto
have "\<forall>n::nat. \<exists>f. f \<in> A \<and> x + ereal (real n) \<le> f"
proof (rule ccontr)
assume "\<not> ?thesis"
then have "\<exists>n::nat. Sup A \<le> x + ereal (real n)"
by (simp add: Sup_le_iff not_le less_imp_le Ball_def) (metis less_imp_le)
then show False using `x \<in> A` `\<infinity> \<notin> A` PInf
by (cases x) auto
qed
from choice[OF this] obtain f :: "nat \<Rightarrow> ereal"
where f: "\<forall>z. f z \<in> A \<and> x + ereal (real z) \<le> f z" ..
have "SUPREMUM UNIV f = \<infinity>"
proof (rule SUP_PInfty)
fix n :: nat
show "\<exists>i\<in>UNIV. ereal (real n) \<le> f i"
using f[THEN spec, of n] `0 \<le> x`
by (cases rule: ereal2_cases[of "f n" x]) (auto intro!: exI[of _ n])
qed
then show ?thesis
using f PInf by (auto intro!: exI[of _ f])
qed
next
case MInf
with `A \<noteq> {}` have "A = {-\<infinity>}"
by (auto simp: Sup_eq_MInfty)
then show ?thesis
using MInf by (auto intro!: exI[of _ "\<lambda>x. -\<infinity>"])
qed
lemma SUP_countable_SUP:
"A \<noteq> {} \<Longrightarrow> \<exists>f::nat \<Rightarrow> ereal. range f \<subseteq> g`A \<and> SUPREMUM A g = SUPREMUM UNIV f"
using Sup_countable_SUP [of "g`A"]
by auto
lemma Sup_ereal_cadd:
fixes A :: "ereal set"
assumes "A \<noteq> {}"
and "a \<noteq> -\<infinity>"
shows "Sup ((\<lambda>x. a + x) ` A) = a + Sup A"
proof (rule antisym)
have *: "\<And>a::ereal. \<And>A. Sup ((\<lambda>x. a + x) ` A) \<le> a + Sup A"
by (auto intro!: add_mono complete_lattice_class.SUP_least complete_lattice_class.Sup_upper)
then show "Sup ((\<lambda>x. a + x) ` A) \<le> a + Sup A" .
show "a + Sup A \<le> Sup ((\<lambda>x. a + x) ` A)"
proof (cases a)
case PInf with `A \<noteq> {}`
show ?thesis
by (auto simp: image_constant max.absorb1)
next
case (real r)
then have **: "op + (- a) ` op + a ` A = A"
by (auto simp: image_iff ac_simps zero_ereal_def[symmetric])
from *[of "-a" "(\<lambda>x. a + x) ` A"] real show ?thesis
unfolding **
by (cases rule: ereal2_cases[of "Sup A" "Sup (op + a ` A)"]) auto
qed (insert `a \<noteq> -\<infinity>`, auto)
qed
lemma Sup_ereal_cminus:
fixes A :: "ereal set"
assumes "A \<noteq> {}"
and "a \<noteq> -\<infinity>"
shows "Sup ((\<lambda>x. a - x) ` A) = a - Inf A"
using Sup_ereal_cadd [of "uminus ` A" a] assms
unfolding image_image minus_ereal_def by (simp add: ereal_SUP_uminus_eq)
lemma SUP_ereal_cminus:
fixes f :: "'i \<Rightarrow> ereal"
fixes A
assumes "A \<noteq> {}"
and "a \<noteq> -\<infinity>"
shows "(SUP x:A. a - f x) = a - (INF x:A. f x)"
using Sup_ereal_cminus[of "f`A" a] assms
unfolding SUP_def INF_def image_image by auto
lemma Inf_ereal_cminus:
fixes A :: "ereal set"
assumes "A \<noteq> {}"
and "\<bar>a\<bar> \<noteq> \<infinity>"
shows "Inf ((\<lambda>x. a - x) ` A) = a - Sup A"
proof -
{
fix x
have "-a - -x = -(a - x)"
using assms by (cases x) auto
} note * = this
then have "(\<lambda>x. -a - x)`uminus`A = uminus ` (\<lambda>x. a - x) ` A"
by (auto simp: image_image)
with * show ?thesis
using Sup_ereal_cminus [of "uminus ` A" "- a"] assms
by (auto simp add: ereal_INF_uminus_eq ereal_SUP_uminus_eq)
qed
lemma INF_ereal_cminus:
fixes a :: ereal
assumes "A \<noteq> {}"
and "\<bar>a\<bar> \<noteq> \<infinity>"
shows "(INF x:A. a - f x) = a - (SUP x:A. f x)"
using Inf_ereal_cminus[of "f`A" a] assms
unfolding SUP_def INF_def image_image
by auto
lemma uminus_ereal_add_uminus_uminus:
fixes a b :: ereal
shows "a \<noteq> \<infinity> \<Longrightarrow> b \<noteq> \<infinity> \<Longrightarrow> - (- a + - b) = a + b"
by (cases rule: ereal2_cases[of a b]) auto
lemma INF_ereal_add:
fixes f :: "nat \<Rightarrow> ereal"
assumes "decseq f" "decseq g"
and fin: "\<And>i. f i \<noteq> \<infinity>" "\<And>i. g i \<noteq> \<infinity>"
shows "(INF i. f i + g i) = INFIMUM UNIV f + INFIMUM UNIV g"
proof -
have INF_less: "(INF i. f i) < \<infinity>" "(INF i. g i) < \<infinity>"
using assms unfolding INF_less_iff by auto
{
fix i
from fin[of i] have "- ((- f i) + (- g i)) = f i + g i"
by (rule uminus_ereal_add_uminus_uminus)
}
then have "(INF i. f i + g i) = (INF i. - ((- f i) + (- g i)))"
by simp
also have "\<dots> = INFIMUM UNIV f + INFIMUM UNIV g"
unfolding ereal_INF_uminus
using assms INF_less
by (subst SUP_ereal_add)
(auto simp: ereal_SUP_uminus intro!: uminus_ereal_add_uminus_uminus)
finally show ?thesis .
qed
subsection "Relation to @{typ enat}"
definition "ereal_of_enat n = (case n of enat n \<Rightarrow> ereal (real n) | \<infinity> \<Rightarrow> \<infinity>)"
declare [[coercion "ereal_of_enat :: enat \<Rightarrow> ereal"]]
declare [[coercion "(\<lambda>n. ereal (real n)) :: nat \<Rightarrow> ereal"]]
lemma ereal_of_enat_simps[simp]:
"ereal_of_enat (enat n) = ereal n"
"ereal_of_enat \<infinity> = \<infinity>"
by (simp_all add: ereal_of_enat_def)
lemma ereal_of_enat_le_iff[simp]: "ereal_of_enat m \<le> ereal_of_enat n \<longleftrightarrow> m \<le> n"
by (cases m n rule: enat2_cases) auto
lemma ereal_of_enat_less_iff[simp]: "ereal_of_enat m < ereal_of_enat n \<longleftrightarrow> m < n"
by (cases m n rule: enat2_cases) auto
lemma numeral_le_ereal_of_enat_iff[simp]: "numeral m \<le> ereal_of_enat n \<longleftrightarrow> numeral m \<le> n"
by (cases n) (auto dest: natceiling_le intro: natceiling_le_eq[THEN iffD1])
lemma numeral_less_ereal_of_enat_iff[simp]: "numeral m < ereal_of_enat n \<longleftrightarrow> numeral m < n"
by (cases n) auto
lemma ereal_of_enat_ge_zero_cancel_iff[simp]: "0 \<le> ereal_of_enat n \<longleftrightarrow> 0 \<le> n"
by (cases n) (auto simp: enat_0[symmetric])
lemma ereal_of_enat_gt_zero_cancel_iff[simp]: "0 < ereal_of_enat n \<longleftrightarrow> 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 = \<infinity> \<longleftrightarrow> n = \<infinity>"
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 \<le> 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]
subsection "Limits on @{typ ereal}"
subsubsection "Topological space"
instantiation ereal :: linear_continuum_topology
begin
definition "open_ereal" :: "ereal set \<Rightarrow> bool" where
open_ereal_generated: "open_ereal = generate_topology (range lessThan \<union> range greaterThan)"
instance
by default (simp add: open_ereal_generated)
end
lemma open_PInfty: "open A \<Longrightarrow> \<infinity> \<in> A \<Longrightarrow> (\<exists>x. {ereal x<..} \<subseteq> A)"
unfolding open_ereal_generated
proof (induct rule: generate_topology.induct)
case (Int A B)
then obtain x z where "\<infinity> \<in> A \<Longrightarrow> {ereal x <..} \<subseteq> A" "\<infinity> \<in> B \<Longrightarrow> {ereal z <..} \<subseteq> B"
by auto
with Int show ?case
by (intro exI[of _ "max x z"]) fastforce
next
case (Basis S)
{
fix x
have "x \<noteq> \<infinity> \<Longrightarrow> \<exists>t. x \<le> ereal t"
by (cases x) auto
}
moreover note Basis
ultimately show ?case
by (auto split: ereal.split)
qed (fastforce simp add: vimage_Union)+
lemma open_MInfty: "open A \<Longrightarrow> -\<infinity> \<in> A \<Longrightarrow> (\<exists>x. {..<ereal x} \<subseteq> A)"
unfolding open_ereal_generated
proof (induct rule: generate_topology.induct)
case (Int A B)
then obtain x z where "-\<infinity> \<in> A \<Longrightarrow> {..< ereal x} \<subseteq> A" "-\<infinity> \<in> B \<Longrightarrow> {..< ereal z} \<subseteq> B"
by auto
with Int show ?case
by (intro exI[of _ "min x z"]) fastforce
next
case (Basis S)
{
fix x
have "x \<noteq> - \<infinity> \<Longrightarrow> \<exists>t. ereal t \<le> x"
by (cases x) auto
}
moreover note Basis
ultimately show ?case
by (auto split: ereal.split)
qed (fastforce simp add: vimage_Union)+
lemma open_ereal_vimage: "open S \<Longrightarrow> open (ereal -` S)"
unfolding open_ereal_generated
proof (induct rule: generate_topology.induct)
case (Int A B)
then show ?case
by auto
next
case (Basis S)
{
fix x have
"ereal -` {..<x} = (case x of PInfty \<Rightarrow> UNIV | MInfty \<Rightarrow> {} | ereal r \<Rightarrow> {..<r})"
"ereal -` {x<..} = (case x of PInfty \<Rightarrow> {} | MInfty \<Rightarrow> UNIV | ereal r \<Rightarrow> {r<..})"
by (induct x) auto
}
moreover note Basis
ultimately show ?case
by (auto split: ereal.split)
qed (fastforce simp add: vimage_Union)+
lemma open_ereal: "open S \<Longrightarrow> 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} = { -\<infinity> <..< ereal x }"
apply auto
apply (case_tac xa)
apply auto
done
}
moreover {
fix x
have "ereal ` {x <..} = { ereal x <..< \<infinity> }"
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 "\<bar>x\<bar> \<noteq> \<infinity>" "(f ---> x) F"
shows "eventually (\<lambda>x. \<bar>f x\<bar> \<noteq> \<infinity>) F"
proof -
have "(f ---> ereal (real x)) F"
using assms by (cases x) auto
then have "eventually (\<lambda>x. f x \<in> ereal ` UNIV) F"
by (rule topological_tendstoD) (auto intro: open_ereal)
also have "(\<lambda>x. f x \<in> ereal ` UNIV) = (\<lambda>x. \<bar>f x\<bar> \<noteq> \<infinity>)"
by auto
finally show ?thesis .
qed
lemma open_ereal_def:
"open A \<longleftrightarrow> open (ereal -` A) \<and> (\<infinity> \<in> A \<longrightarrow> (\<exists>x. {ereal x <..} \<subseteq> A)) \<and> (-\<infinity> \<in> A \<longrightarrow> (\<exists>x. {..<ereal x} \<subseteq> A))"
(is "open A \<longleftrightarrow> ?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)" "\<infinity> \<in> A \<Longrightarrow> {ereal x<..} \<subseteq> A" "-\<infinity> \<in> A \<Longrightarrow> {..< ereal y} \<subseteq> A"
by auto
have *: "A = ereal ` (ereal -` A) \<union> (if \<infinity> \<in> A then {ereal x<..} else {}) \<union> (if -\<infinity> \<in> 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 "\<infinity> \<in> A"
obtains x where "{ereal x<..} \<subseteq> A"
using open_PInfty[OF assms] by auto
lemma open_MInfty2:
assumes "open A"
and "-\<infinity> \<in> A"
obtains x where "{..<ereal x} \<subseteq> A"
using open_MInfty[OF assms] by auto
lemma ereal_openE:
assumes "open A"
obtains x y where "open (ereal -` A)"
and "\<infinity> \<in> A \<Longrightarrow> {ereal x<..} \<subseteq> A"
and "-\<infinity> \<in> A \<Longrightarrow> {..<ereal y} \<subseteq> 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 \<in> S"
and "\<bar>x\<bar> \<noteq> \<infinity>"
obtains e where "e > 0" and "{x-e <..< x+e} \<subseteq> S"
proof -
from `open S`
have "open (ereal -` S)"
by (rule ereal_openE)
then obtain e where "e > 0" and e: "\<And>y. dist y (real x) < e \<Longrightarrow> ereal y \<in> 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 \<in> {x - ereal e<..<x + ereal e}"
with assms obtain t where "y = ereal t" "dist t (real x) < e"
by (cases y) (auto simp: dist_real_def)
then show "y \<in> S"
using e[of t] by auto
qed
qed
lemma ereal_open_cont_interval2:
fixes S :: "ereal set"
assumes "open S"
and "x \<in> S"
and x: "\<bar>x\<bar> \<noteq> \<infinity>"
obtains a b where "a < x" and "x < b" and "{a <..< b} \<subseteq> S"
proof -
obtain e where "0 < e" "{x - e<..<x + e} \<subseteq> 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_ereal[simp]: "((\<lambda>n. ereal (f n)) ---> ereal x) net \<longleftrightarrow> (f ---> x) net"
(is "?l = ?r")
proof (intro iffI topological_tendstoI)
fix S
assume "?l" and "open S" and "x \<in> S"
then show "eventually (\<lambda>x. f x \<in> S) net"
using `?l`[THEN topological_tendstoD, OF open_ereal, OF `open S`]
by (simp add: inj_image_mem_iff)
next
fix S
assume "?r" and "open S" and "ereal x \<in> S"
show "eventually (\<lambda>x. ereal (f x) \<in> S) net"
using `?r`[THEN topological_tendstoD, OF open_ereal_vimage, OF `open S`]
using `ereal x \<in> S`
by auto
qed
lemma lim_real_of_ereal[simp]:
assumes lim: "(f ---> ereal x) net"
shows "((\<lambda>x. real (f x)) ---> x) net"
proof (intro topological_tendstoI)
fix S
assume "open S" and "x \<in> S"
then have S: "open S" "ereal x \<in> ereal ` S"
by (simp_all add: inj_image_mem_iff)
have "\<forall>x. f x \<in> ereal ` S \<longrightarrow> real (f x) \<in> S"
by auto
from this lim[THEN topological_tendstoD, OF open_ereal, OF S]
show "eventually (\<lambda>x. real (f x) \<in> S) net"
by (rule eventually_mono)
qed
lemma tendsto_PInfty: "(f ---> \<infinity>) F \<longleftrightarrow> (\<forall>r. eventually (\<lambda>x. ereal r < f x) F)"
proof -
{
fix l :: ereal
assume "\<forall>r. eventually (\<lambda>x. ereal r < f x) F"
from this[THEN spec, of "real l"] have "l \<noteq> \<infinity> \<Longrightarrow> eventually (\<lambda>x. l < f x) F"
by (cases l) (auto elim: eventually_elim1)
}
then show ?thesis
by (auto simp: order_tendsto_iff)
qed
lemma tendsto_PInfty_eq_at_top:
"((\<lambda>z. ereal (f z)) ---> \<infinity>) F \<longleftrightarrow> (LIM z F. f z :> at_top)"
unfolding tendsto_PInfty filterlim_at_top_dense by simp
lemma tendsto_MInfty: "(f ---> -\<infinity>) F \<longleftrightarrow> (\<forall>r. eventually (\<lambda>x. f x < ereal r) F)"
unfolding tendsto_def
proof safe
fix S :: "ereal set"
assume "open S" "-\<infinity> \<in> S"
from open_MInfty[OF this] obtain B where "{..<ereal B} \<subseteq> S" ..
moreover
assume "\<forall>r::real. eventually (\<lambda>z. f z < r) F"
then have "eventually (\<lambda>z. f z \<in> {..< B}) F"
by auto
ultimately show "eventually (\<lambda>z. f z \<in> S) F"
by (auto elim!: eventually_elim1)
next
fix x
assume "\<forall>S. open S \<longrightarrow> -\<infinity> \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) F"
from this[rule_format, of "{..< ereal x}"] show "eventually (\<lambda>y. f y < ereal x) F"
by auto
qed
lemma Lim_PInfty: "f ----> \<infinity> \<longleftrightarrow> (\<forall>B. \<exists>N. \<forall>n\<ge>N. f n \<ge> ereal B)"
unfolding tendsto_PInfty eventually_sequentially
proof safe
fix r
assume "\<forall>r. \<exists>N. \<forall>n\<ge>N. ereal r \<le> f n"
then obtain N where "\<forall>n\<ge>N. ereal (r + 1) \<le> f n"
by blast
moreover have "ereal r < ereal (r + 1)"
by auto
ultimately show "\<exists>N. \<forall>n\<ge>N. ereal r < f n"
by (blast intro: less_le_trans)
qed (blast intro: less_imp_le)
lemma Lim_MInfty: "f ----> -\<infinity> \<longleftrightarrow> (\<forall>B. \<exists>N. \<forall>n\<ge>N. ereal B \<ge> f n)"
unfolding tendsto_MInfty eventually_sequentially
proof safe
fix r
assume "\<forall>r. \<exists>N. \<forall>n\<ge>N. f n \<le> ereal r"
then obtain N where "\<forall>n\<ge>N. f n \<le> ereal (r - 1)"
by blast
moreover have "ereal (r - 1) < ereal r"
by auto
ultimately show "\<exists>N. \<forall>n\<ge>N. f n < ereal r"
by (blast intro: le_less_trans)
qed (blast intro: less_imp_le)
lemma Lim_bounded_PInfty: "f ----> l \<Longrightarrow> (\<And>n. f n \<le> ereal B) \<Longrightarrow> l \<noteq> \<infinity>"
using LIMSEQ_le_const2[of f l "ereal B"] by auto
lemma Lim_bounded_MInfty: "f ----> l \<Longrightarrow> (\<And>n. ereal B \<le> f n) \<Longrightarrow> l \<noteq> -\<infinity>"
using LIMSEQ_le_const[of f l "ereal B"] by auto
lemma tendsto_explicit:
"f ----> f0 \<longleftrightarrow> (\<forall>S. open S \<longrightarrow> f0 \<in> S \<longrightarrow> (\<exists>N. \<forall>n\<ge>N. f n \<in> S))"
unfolding tendsto_def eventually_sequentially by auto
lemma Lim_bounded_PInfty2: "f ----> l \<Longrightarrow> \<forall>n\<ge>N. f n \<le> ereal B \<Longrightarrow> l \<noteq> \<infinity>"
using LIMSEQ_le_const2[of f l "ereal B"] by fastforce
lemma Lim_bounded_ereal: "f ----> (l :: 'a::linorder_topology) \<Longrightarrow> \<forall>n\<ge>M. f n \<le> C \<Longrightarrow> l \<le> C"
by (intro LIMSEQ_le_const2) auto
lemma Lim_bounded2_ereal:
assumes lim:"f ----> (l :: 'a::linorder_topology)"
and ge: "\<forall>n\<ge>N. f n \<ge> C"
shows "l \<ge> 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 (a * b) = real a * real b"
by (cases rule: ereal2_cases[of a b]) auto
lemma real_of_ereal_eq_0:
fixes x :: ereal
shows "real x = 0 \<longleftrightarrow> x = \<infinity> \<or> x = -\<infinity> \<or> x = 0"
by (cases x) auto
lemma tendsto_ereal_realD:
fixes f :: "'a \<Rightarrow> ereal"
assumes "x \<noteq> 0"
and tendsto: "((\<lambda>x. ereal (real (f x))) ---> x) net"
shows "(f ---> x) net"
proof (intro topological_tendstoI)
fix S
assume S: "open S" "x \<in> S"
with `x \<noteq> 0` have "open (S - {0})" "x \<in> S - {0}"
by auto
from tendsto[THEN topological_tendstoD, OF this]
show "eventually (\<lambda>x. f x \<in> S) net"
by (rule eventually_rev_mp) (auto simp: ereal_real)
qed
lemma tendsto_ereal_realI:
fixes f :: "'a \<Rightarrow> ereal"
assumes x: "\<bar>x\<bar> \<noteq> \<infinity>" and tendsto: "(f ---> x) net"
shows "((\<lambda>x. ereal (real (f x))) ---> x) net"
proof (intro topological_tendstoI)
fix S
assume "open S" and "x \<in> S"
with x have "open (S - {\<infinity>, -\<infinity>})" "x \<in> S - {\<infinity>, -\<infinity>}"
by auto
from tendsto[THEN topological_tendstoD, OF this]
show "eventually (\<lambda>x. ereal (real (f x)) \<in> S) net"
by (elim eventually_elim1) (auto simp: ereal_real)
qed
lemma ereal_mult_cancel_left:
fixes a b c :: ereal
shows "a * b = a * c \<longleftrightarrow> (\<bar>a\<bar> = \<infinity> \<and> 0 < b * c) \<or> a = 0 \<or> b = c"
by (cases rule: ereal3_cases[of a b c]) (simp_all add: zero_less_mult_iff)
lemma tendsto_add_ereal:
fixes x y :: ereal
assumes x: "\<bar>x\<bar> \<noteq> \<infinity>" and y: "\<bar>y\<bar> \<noteq> \<infinity>"
assumes f: "(f ---> x) F" and g: "(g ---> y) F"
shows "((\<lambda>x. f x + g x) ---> x + y) F"
proof -
from x obtain r where x': "x = ereal r" by (cases x) auto
with f have "((\<lambda>i. real (f i)) ---> r) F" by simp
moreover
from y obtain p where y': "y = ereal p" by (cases y) auto
with g have "((\<lambda>i. real (g i)) ---> p) F" by simp
ultimately have "((\<lambda>i. real (f i) + real (g i)) ---> r + p) F"
by (rule tendsto_add)
moreover
from eventually_finite[OF x f] eventually_finite[OF y g]
have "eventually (\<lambda>x. f x + g x = ereal (real (f x) + real (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 "\<bar>m\<bar> \<noteq> \<infinity>"
and "m \<noteq> 0"
and "\<bar>t\<bar> \<noteq> \<infinity>"
shows "inj_on (\<lambda>x. 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 "\<infinity> = a + b \<longleftrightarrow> a = \<infinity> \<or> b = \<infinity>"
by (cases rule: ereal2_cases[of a b]) auto
lemma ereal_MInfty_eq_plus[simp]:
fixes a b :: ereal
shows "-\<infinity> = a + b \<longleftrightarrow> (a = -\<infinity> \<and> b \<noteq> \<infinity>) \<or> (b = -\<infinity> \<and> a \<noteq> \<infinity>)"
by (cases rule: ereal2_cases[of a b]) auto
lemma ereal_less_divide_pos:
fixes x y :: ereal
shows "x > 0 \<Longrightarrow> x \<noteq> \<infinity> \<Longrightarrow> y < z / x \<longleftrightarrow> 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 \<Longrightarrow> x \<noteq> \<infinity> \<Longrightarrow> y / x < z \<longleftrightarrow> 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 \<noteq> 0 \<Longrightarrow> \<bar>b\<bar> \<noteq> \<infinity> \<Longrightarrow> a / b = c \<longleftrightarrow> a = b * c"
by (cases rule: ereal3_cases[of a b c])
(simp_all add: field_simps)
lemma ereal_inverse_not_MInfty[simp]: "inverse (a::ereal) \<noteq> -\<infinity>"
by (cases a) auto
lemma ereal_mult_m1[simp]: "x * ereal (-1) = -x"
by (cases x) auto
lemma ereal_real':
assumes "\<bar>x\<bar> \<noteq> \<infinity>"
shows "ereal (real x) = x"
using assms by auto
lemma real_ereal_id: "real \<circ> ereal = id"
proof -
{
fix x
have "(real o ereal) x = id x"
by auto
}
then show ?thesis
using ext by blast
qed
lemma open_image_ereal: "open(UNIV-{ \<infinity> , (-\<infinity> :: ereal)})"
by (metis range_ereal open_ereal open_UNIV)
lemma ereal_le_distrib:
fixes a b c :: ereal
shows "c * (a + b) \<le> 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 \<le> c"
and "c \<noteq> \<infinity>"
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_pos_le_distrib:
fixes a b c :: ereal
assumes "c \<ge> 0"
shows "c * (a + b) \<le> c * a + c * b"
using assms
by (cases rule: ereal3_cases[of a b c]) (auto simp add: field_simps)
lemma ereal_max_mono: "(a::ereal) \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> max a c \<le> max b d"
by (metis sup_ereal_def sup_mono)
lemma ereal_max_least: "(a::ereal) \<le> x \<Longrightarrow> c \<le> x \<Longrightarrow> max a c \<le> x"
by (metis sup_ereal_def sup_least)
lemma ereal_LimI_finite:
fixes x :: ereal
assumes "\<bar>x\<bar> \<noteq> \<infinity>"
and "\<And>r. 0 < r \<Longrightarrow> \<exists>N. \<forall>n\<ge>N. u n < x + r \<and> 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 \<in> S"
then have "open (ereal -` S)"
unfolding open_ereal_def by auto
with `x \<in> S` obtain r where "0 < r" and dist: "\<And>y. dist y rx < r \<Longrightarrow> ereal y \<in> S"
unfolding open_real_def rx by auto
then obtain n where
upper: "\<And>N. n \<le> N \<Longrightarrow> u N < x + ereal r" and
lower: "\<And>N. n \<le> N \<Longrightarrow> x < u N + ereal r"
using assms(2)[of "ereal r"] by auto
show "\<exists>N. \<forall>n\<ge>N. u n \<in> S"
proof (safe intro!: exI[of _ n])
fix N
assume "n \<le> N"
from upper[OF this] lower[OF this] assms `0 < r`
have "u N \<notin> {\<infinity>,(-\<infinity>)}"
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 \<le> N`] upper[OF `n \<le> N`]
by auto
then have "dist (real (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 \<in> S"
using `u N \<notin> {\<infinity>, -\<infinity>}`
by (auto simp: ereal_real split: split_if_asm)
qed
qed
lemma tendsto_obtains_N:
assumes "f ----> f0"
assumes "open S"
and "f0 \<in> S"
obtains N where "\<forall>n\<ge>N. f n \<in> S"
using assms using tendsto_def
using tendsto_explicit[of f f0] assms by auto
lemma ereal_LimI_finite_iff:
fixes x :: ereal
assumes "\<bar>x\<bar> \<noteq> \<infinity>"
shows "u ----> x \<longleftrightarrow> (\<forall>r. 0 < r \<longrightarrow> (\<exists>N. \<forall>n\<ge>N. u n < x + r \<and> x < u n + r))"
(is "?lhs \<longleftrightarrow> ?rhs")
proof
assume lim: "u ----> x"
{
fix r :: ereal
assume "r > 0"
then obtain N where "\<forall>n\<ge>N. u n \<in> {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 "\<exists>N. \<forall>n\<ge>N. u n < x + r \<and> 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 \<Rightarrow> ereal"
shows "Limsup net (\<lambda>x. - (f x)) = - Liminf net f"
unfolding Limsup_def Liminf_def ereal_SUP_uminus ereal_INF_uminus ..
lemma liminf_bounded_iff:
fixes x :: "nat \<Rightarrow> ereal"
shows "C \<le> liminf x \<longleftrightarrow> (\<forall>B<C. \<exists>N. \<forall>n\<ge>N. B < x n)"
(is "?lhs \<longleftrightarrow> ?rhs")
unfolding le_Liminf_iff eventually_sequentially ..
subsubsection {* Tests for code generator *}
(* A small list of simple arithmetic expressions *)
value "- \<infinity> :: ereal"
value "\<bar>-\<infinity>\<bar> :: ereal"
value "4 + 5 / 4 - ereal 2 :: ereal"
value "ereal 3 < \<infinity>"
value "real (\<infinity>::ereal) = 0"
end