(* Title: src/HOL/Library/Extended_Reals.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_Reals
imports Complex_Main
begin
text {*
For more lemmas about the extended real numbers go to
@{text "src/HOL/Multivaraite_Analysis/Extended_Real_Limits.thy"}
*}
lemma (in complete_lattice) atLeast_eq_UNIV_iff: "{x..} = UNIV \<longleftrightarrow> x = bot"
proof
assume "{x..} = UNIV"
show "x = bot"
proof (rule ccontr)
assume "x \<noteq> bot" then have "bot \<notin> {x..}" by (simp add: le_less)
then show False using `{x..} = UNIV` by simp
qed
qed auto
lemma SUPR_pair:
"(SUP i : A. SUP j : B. f i j) = (SUP p : A \<times> B. f (fst p) (snd p))"
by (rule antisym) (auto intro!: SUP_leI le_SUPI2)
lemma INFI_pair:
"(INF i : A. INF j : B. f i j) = (INF p : A \<times> B. f (fst p) (snd p))"
by (rule antisym) (auto intro!: le_INFI INF_leI2)
subsection {* Definition and basic properties *}
datatype extreal = extreal real | PInfty | MInfty
notation (xsymbols)
PInfty ("\<infinity>")
notation (HTML output)
PInfty ("\<infinity>")
declare [[coercion "extreal :: real \<Rightarrow> extreal"]]
instantiation extreal :: uminus
begin
fun uminus_extreal where
"- (extreal r) = extreal (- r)"
| "- \<infinity> = MInfty"
| "- MInfty = \<infinity>"
instance ..
end
lemma inj_extreal[simp]: "inj_on extreal A"
unfolding inj_on_def by auto
lemma MInfty_neq_PInfty[simp]:
"\<infinity> \<noteq> - \<infinity>" "- \<infinity> \<noteq> \<infinity>" by simp_all
lemma MInfty_neq_extreal[simp]:
"extreal r \<noteq> - \<infinity>" "- \<infinity> \<noteq> extreal r" by simp_all
lemma MInfinity_cases[simp]:
"(case - \<infinity> of extreal r \<Rightarrow> f r | \<infinity> \<Rightarrow> y | MInfinity \<Rightarrow> z) = z"
by simp
lemma extreal_uminus_uminus[simp]:
fixes a :: extreal shows "- (- a) = a"
by (cases a) simp_all
lemma MInfty_eq[simp]:
"MInfty = - \<infinity>" by simp
declare uminus_extreal.simps(2)[simp del]
lemma extreal_cases[case_names real PInf MInf, cases type: extreal]:
assumes "\<And>r. x = extreal r \<Longrightarrow> P"
assumes "x = \<infinity> \<Longrightarrow> P"
assumes "x = -\<infinity> \<Longrightarrow> P"
shows P
using assms by (cases x) auto
lemmas extreal2_cases = extreal_cases[case_product extreal_cases]
lemmas extreal3_cases = extreal2_cases[case_product extreal_cases]
lemma extreal_uminus_eq_iff[simp]:
fixes a b :: extreal shows "-a = -b \<longleftrightarrow> a = b"
by (cases rule: extreal2_cases[of a b]) simp_all
function of_extreal :: "extreal \<Rightarrow> real" where
"of_extreal (extreal r) = r" |
"of_extreal \<infinity> = 0" |
"of_extreal (-\<infinity>) = 0"
by (auto intro: extreal_cases)
termination proof qed (rule wf_empty)
defs (overloaded)
real_of_extreal_def [code_unfold]: "real \<equiv> of_extreal"
lemma real_of_extreal[simp]:
"real (- x :: extreal) = - (real x)"
"real (extreal r) = r"
"real \<infinity> = 0"
by (cases x) (simp_all add: real_of_extreal_def)
lemma range_extreal[simp]: "range extreal = UNIV - {\<infinity>, -\<infinity>}"
proof safe
fix x assume "x \<notin> range extreal" "x \<noteq> \<infinity>"
then show "x = -\<infinity>" by (cases x) auto
qed auto
lemma extreal_range_uminus[simp]: "range uminus = (UNIV::extreal set)"
proof safe
fix x :: extreal show "x \<in> range uminus" by (intro image_eqI[of _ _ "-x"]) auto
qed auto
instantiation extreal :: number
begin
definition [simp]: "number_of x = extreal (number_of x)"
instance proof qed
end
instantiation extreal :: abs
begin
function abs_extreal where
"\<bar>extreal r\<bar> = extreal \<bar>r\<bar>"
| "\<bar>-\<infinity>\<bar> = \<infinity>"
| "\<bar>\<infinity>\<bar> = \<infinity>"
by (auto intro: extreal_cases)
termination proof qed (rule wf_empty)
instance ..
end
lemma abs_eq_infinity_cases[elim!]: "\<lbrakk> \<bar>x\<bar> = \<infinity> ; x = \<infinity> \<Longrightarrow> P ; x = -\<infinity> \<Longrightarrow> P \<rbrakk> \<Longrightarrow> P"
by (cases x) auto
lemma abs_neq_infinity_cases[elim!]: "\<lbrakk> \<bar>x\<bar> \<noteq> \<infinity> ; \<And>r. x = extreal r \<Longrightarrow> P \<rbrakk> \<Longrightarrow> P"
by (cases x) auto
lemma abs_extreal_uminus[simp]: "\<bar>- x\<bar> = \<bar>x::extreal\<bar>"
by (cases x) auto
subsubsection "Addition"
instantiation extreal :: comm_monoid_add
begin
definition "0 = extreal 0"
function plus_extreal where
"extreal r + extreal p = extreal (r + p)" |
"\<infinity> + a = \<infinity>" |
"a + \<infinity> = \<infinity>" |
"extreal r + -\<infinity> = - \<infinity>" |
"-\<infinity> + extreal p = -\<infinity>" |
"-\<infinity> + -\<infinity> = -\<infinity>"
proof -
case (goal1 P x)
moreover then obtain a b where "x = (a, b)" by (cases x) auto
ultimately show P
by (cases rule: extreal2_cases[of a b]) auto
qed auto
termination proof qed (rule wf_empty)
lemma Infty_neq_0[simp]:
"\<infinity> \<noteq> 0" "0 \<noteq> \<infinity>"
"-\<infinity> \<noteq> 0" "0 \<noteq> -\<infinity>"
by (simp_all add: zero_extreal_def)
lemma extreal_eq_0[simp]:
"extreal r = 0 \<longleftrightarrow> r = 0"
"0 = extreal r \<longleftrightarrow> r = 0"
unfolding zero_extreal_def by simp_all
instance
proof
fix a :: extreal show "0 + a = a"
by (cases a) (simp_all add: zero_extreal_def)
fix b :: extreal show "a + b = b + a"
by (cases rule: extreal2_cases[of a b]) simp_all
fix c :: extreal show "a + b + c = a + (b + c)"
by (cases rule: extreal3_cases[of a b c]) simp_all
qed
end
lemma abs_extreal_zero[simp]: "\<bar>0\<bar> = (0::extreal)"
unfolding zero_extreal_def abs_extreal.simps by simp
lemma extreal_uminus_zero[simp]:
"- 0 = (0::extreal)"
by (simp add: zero_extreal_def)
lemma extreal_uminus_zero_iff[simp]:
fixes a :: extreal shows "-a = 0 \<longleftrightarrow> a = 0"
by (cases a) simp_all
lemma extreal_plus_eq_PInfty[simp]:
shows "a + b = \<infinity> \<longleftrightarrow> a = \<infinity> \<or> b = \<infinity>"
by (cases rule: extreal2_cases[of a b]) auto
lemma extreal_plus_eq_MInfty[simp]:
shows "a + b = -\<infinity> \<longleftrightarrow>
(a = -\<infinity> \<or> b = -\<infinity>) \<and> a \<noteq> \<infinity> \<and> b \<noteq> \<infinity>"
by (cases rule: extreal2_cases[of a b]) auto
lemma extreal_add_cancel_left:
assumes "a \<noteq> -\<infinity>"
shows "a + b = a + c \<longleftrightarrow> (a = \<infinity> \<or> b = c)"
using assms by (cases rule: extreal3_cases[of a b c]) auto
lemma extreal_add_cancel_right:
assumes "a \<noteq> -\<infinity>"
shows "b + a = c + a \<longleftrightarrow> (a = \<infinity> \<or> b = c)"
using assms by (cases rule: extreal3_cases[of a b c]) auto
lemma extreal_real:
"extreal (real x) = (if \<bar>x\<bar> = \<infinity> then 0 else x)"
by (cases x) simp_all
lemma real_of_extreal_add:
fixes a b :: extreal
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: extreal2_cases[of a b]) auto
subsubsection "Linear order on @{typ extreal}"
instantiation extreal :: linorder
begin
function less_extreal where
"extreal x < extreal y \<longleftrightarrow> x < y" |
" \<infinity> < a \<longleftrightarrow> False" |
" a < -\<infinity> \<longleftrightarrow> False" |
"extreal x < \<infinity> \<longleftrightarrow> True" |
" -\<infinity> < extreal r \<longleftrightarrow> True" |
" -\<infinity> < \<infinity> \<longleftrightarrow> True"
proof -
case (goal1 P x)
moreover then obtain a b where "x = (a,b)" by (cases x) auto
ultimately show P by (cases rule: extreal2_cases[of a b]) auto
qed simp_all
termination by (relation "{}") simp
definition "x \<le> (y::extreal) \<longleftrightarrow> x < y \<or> x = y"
lemma extreal_infty_less[simp]:
"x < \<infinity> \<longleftrightarrow> (x \<noteq> \<infinity>)"
"-\<infinity> < x \<longleftrightarrow> (x \<noteq> -\<infinity>)"
by (cases x, simp_all) (cases x, simp_all)
lemma extreal_infty_less_eq[simp]:
"\<infinity> \<le> x \<longleftrightarrow> x = \<infinity>"
"x \<le> -\<infinity> \<longleftrightarrow> x = -\<infinity>"
by (auto simp add: less_eq_extreal_def)
lemma extreal_less[simp]:
"extreal r < 0 \<longleftrightarrow> (r < 0)"
"0 < extreal r \<longleftrightarrow> (0 < r)"
"0 < \<infinity>"
"-\<infinity> < 0"
by (simp_all add: zero_extreal_def)
lemma extreal_less_eq[simp]:
"x \<le> \<infinity>"
"-\<infinity> \<le> x"
"extreal r \<le> extreal p \<longleftrightarrow> r \<le> p"
"extreal r \<le> 0 \<longleftrightarrow> r \<le> 0"
"0 \<le> extreal r \<longleftrightarrow> 0 \<le> r"
by (auto simp add: less_eq_extreal_def zero_extreal_def)
lemma extreal_infty_less_eq2:
"a \<le> b \<Longrightarrow> a = \<infinity> \<Longrightarrow> b = \<infinity>"
"a \<le> b \<Longrightarrow> b = -\<infinity> \<Longrightarrow> a = -\<infinity>"
by simp_all
instance
proof
fix x :: extreal show "x \<le> x"
by (cases x) simp_all
fix y :: extreal show "x < y \<longleftrightarrow> x \<le> y \<and> \<not> y \<le> x"
by (cases rule: extreal2_cases[of x y]) auto
show "x \<le> y \<or> y \<le> x "
by (cases rule: extreal2_cases[of x y]) auto
{ assume "x \<le> y" "y \<le> x" then show "x = y"
by (cases rule: extreal2_cases[of x y]) auto }
{ fix z assume "x \<le> y" "y \<le> z" then show "x \<le> z"
by (cases rule: extreal3_cases[of x y z]) auto }
qed
end
instance extreal :: ordered_ab_semigroup_add
proof
fix a b c :: extreal assume "a \<le> b" then show "c + a \<le> c + b"
by (cases rule: extreal3_cases[of a b c]) auto
qed
lemma extreal_MInfty_lessI[intro, simp]:
"a \<noteq> -\<infinity> \<Longrightarrow> -\<infinity> < a"
by (cases a) auto
lemma extreal_less_PInfty[intro, simp]:
"a \<noteq> \<infinity> \<Longrightarrow> a < \<infinity>"
by (cases a) auto
lemma extreal_less_extreal_Ex:
fixes a b :: extreal
shows "x < extreal r \<longleftrightarrow> x = -\<infinity> \<or> (\<exists>p. p < r \<and> x = extreal p)"
by (cases x) auto
lemma less_PInf_Ex_of_nat: "x \<noteq> \<infinity> \<longleftrightarrow> (\<exists>n::nat. x < extreal (real n))"
proof (cases x)
case (real r) then show ?thesis
using reals_Archimedean2[of r] by simp
qed simp_all
lemma extreal_add_mono:
fixes a b c d :: extreal assumes "a \<le> b" "c \<le> d" shows "a + c \<le> b + d"
using assms
apply (cases a)
apply (cases rule: extreal3_cases[of b c d], auto)
apply (cases rule: extreal3_cases[of b c d], auto)
done
lemma extreal_minus_le_minus[simp]:
fixes a b :: extreal shows "- a \<le> - b \<longleftrightarrow> b \<le> a"
by (cases rule: extreal2_cases[of a b]) auto
lemma extreal_minus_less_minus[simp]:
fixes a b :: extreal shows "- a < - b \<longleftrightarrow> b < a"
by (cases rule: extreal2_cases[of a b]) auto
lemma extreal_le_real_iff:
"x \<le> real y \<longleftrightarrow> ((\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> extreal x \<le> y) \<and> (\<bar>y\<bar> = \<infinity> \<longrightarrow> x \<le> 0))"
by (cases y) auto
lemma real_le_extreal_iff:
"real y \<le> x \<longleftrightarrow> ((\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> y \<le> extreal x) \<and> (\<bar>y\<bar> = \<infinity> \<longrightarrow> 0 \<le> x))"
by (cases y) auto
lemma extreal_less_real_iff:
"x < real y \<longleftrightarrow> ((\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> extreal x < y) \<and> (\<bar>y\<bar> = \<infinity> \<longrightarrow> x < 0))"
by (cases y) auto
lemma real_less_extreal_iff:
"real y < x \<longleftrightarrow> ((\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> y < extreal x) \<and> (\<bar>y\<bar> = \<infinity> \<longrightarrow> 0 < x))"
by (cases y) auto
lemma real_of_extreal_positive_mono:
assumes "x \<noteq> \<infinity>" "y \<noteq> \<infinity>" "0 \<le> x" "x \<le> y"
shows "real x \<le> real y"
using assms by (cases rule: extreal2_cases[of x y]) auto
lemma real_of_extreal_pos:
fixes x :: extreal shows "0 \<le> x \<Longrightarrow> 0 \<le> real x" by (cases x) auto
lemmas real_of_extreal_ord_simps =
extreal_le_real_iff real_le_extreal_iff extreal_less_real_iff real_less_extreal_iff
lemma extreal_dense:
fixes x y :: extreal assumes "x < y"
shows "EX z. x < z & z < y"
proof -
{ assume a: "x = (-\<infinity>)"
{ assume "y = \<infinity>" hence ?thesis using a by (auto intro!: exI[of _ "0"]) }
moreover
{ assume "y ~= \<infinity>"
with `x < y` obtain r where r: "y = extreal r" by (cases y) auto
hence ?thesis using `x < y` a by (auto intro!: exI[of _ "extreal (r - 1)"])
} ultimately have ?thesis by auto
}
moreover
{ assume "x ~= (-\<infinity>)"
with `x < y` obtain p where p: "x = extreal p" by (cases x) auto
{ assume "y = \<infinity>" hence ?thesis using `x < y` p
by (auto intro!: exI[of _ "extreal (p + 1)"]) }
moreover
{ assume "y ~= \<infinity>"
with `x < y` obtain r where r: "y = extreal r" by (cases y) auto
with p `x < y` have "p < r" by auto
with dense obtain z where "p < z" "z < r" by auto
hence ?thesis using r p by (auto intro!: exI[of _ "extreal z"])
} ultimately have ?thesis by auto
} ultimately show ?thesis by auto
qed
lemma extreal_dense2:
fixes x y :: extreal assumes "x < y"
shows "EX z. x < extreal z & extreal z < y"
by (metis extreal_dense[OF `x < y`] extreal_cases less_extreal.simps(2,3))
lemma extreal_add_strict_mono:
fixes a b c d :: extreal
assumes "a = b" "0 \<le> a" "a \<noteq> \<infinity>" "c < d"
shows "a + c < b + d"
using assms by (cases rule: extreal3_cases[case_product extreal_cases, of a b c d]) auto
lemma extreal_less_add: "\<bar>a\<bar> \<noteq> \<infinity> \<Longrightarrow> c < b \<Longrightarrow> a + c < a + b"
by (cases rule: extreal2_cases[of b c]) auto
lemma extreal_uminus_eq_reorder: "- a = b \<longleftrightarrow> a = (-b::extreal)" by auto
lemma extreal_uminus_less_reorder: "- a < b \<longleftrightarrow> -b < (a::extreal)"
by (subst (3) extreal_uminus_uminus[symmetric]) (simp only: extreal_minus_less_minus)
lemma extreal_uminus_le_reorder: "- a \<le> b \<longleftrightarrow> -b \<le> (a::extreal)"
by (subst (3) extreal_uminus_uminus[symmetric]) (simp only: extreal_minus_le_minus)
lemmas extreal_uminus_reorder =
extreal_uminus_eq_reorder extreal_uminus_less_reorder extreal_uminus_le_reorder
lemma extreal_bot:
fixes x :: extreal assumes "\<And>B. x \<le> extreal 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 extreal_top:
fixes x :: extreal assumes "\<And>B. x \<ge> extreal 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 extreal_max[simp]: "extreal (max x y) = max (extreal x) (extreal y)"
and extreal_min[simp]: "extreal (min x y) = min (extreal x) (extreal y)"
by (simp_all add: min_def max_def)
lemma extreal_max_0: "max 0 (extreal r) = extreal (max 0 r)"
by (auto simp: zero_extreal_def)
lemma
fixes f :: "nat \<Rightarrow> extreal"
shows incseq_uminus[simp]: "incseq (\<lambda>x. - f x) \<longleftrightarrow> decseq f"
and decseq_uminus[simp]: "decseq (\<lambda>x. - f x) \<longleftrightarrow> incseq f"
unfolding decseq_def incseq_def by auto
lemma extreal_add_nonneg_nonneg:
fixes a b :: extreal 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 extreal :: "{comm_monoid_mult, sgn}"
begin
definition "1 = extreal 1"
function sgn_extreal where
"sgn (extreal r) = extreal (sgn r)"
| "sgn \<infinity> = 1"
| "sgn (-\<infinity>) = -1"
by (auto intro: extreal_cases)
termination proof qed (rule wf_empty)
function times_extreal where
"extreal r * extreal p = extreal (r * p)" |
"extreal r * \<infinity> = (if r = 0 then 0 else if r > 0 then \<infinity> else -\<infinity>)" |
"\<infinity> * extreal r = (if r = 0 then 0 else if r > 0 then \<infinity> else -\<infinity>)" |
"extreal r * -\<infinity> = (if r = 0 then 0 else if r > 0 then -\<infinity> else \<infinity>)" |
"-\<infinity> * extreal r = (if r = 0 then 0 else if r > 0 then -\<infinity> else \<infinity>)" |
"\<infinity> * \<infinity> = \<infinity>" |
"-\<infinity> * \<infinity> = -\<infinity>" |
"\<infinity> * -\<infinity> = -\<infinity>" |
"-\<infinity> * -\<infinity> = \<infinity>"
proof -
case (goal1 P x)
moreover then obtain a b where "x = (a, b)" by (cases x) auto
ultimately show P by (cases rule: extreal2_cases[of a b]) auto
qed simp_all
termination by (relation "{}") simp
instance
proof
fix a :: extreal show "1 * a = a"
by (cases a) (simp_all add: one_extreal_def)
fix b :: extreal show "a * b = b * a"
by (cases rule: extreal2_cases[of a b]) simp_all
fix c :: extreal show "a * b * c = a * (b * c)"
by (cases rule: extreal3_cases[of a b c])
(simp_all add: zero_extreal_def zero_less_mult_iff)
qed
end
lemma abs_extreal_one[simp]: "\<bar>1\<bar> = (1::extreal)"
unfolding one_extreal_def by simp
lemma extreal_mult_zero[simp]:
fixes a :: extreal shows "a * 0 = 0"
by (cases a) (simp_all add: zero_extreal_def)
lemma extreal_zero_mult[simp]:
fixes a :: extreal shows "0 * a = 0"
by (cases a) (simp_all add: zero_extreal_def)
lemma extreal_m1_less_0[simp]:
"-(1::extreal) < 0"
by (simp add: zero_extreal_def one_extreal_def)
lemma extreal_zero_m1[simp]:
"1 \<noteq> (0::extreal)"
by (simp add: zero_extreal_def one_extreal_def)
lemma extreal_times_0[simp]:
fixes x :: extreal shows "0 * x = 0"
by (cases x) (auto simp: zero_extreal_def)
lemma extreal_times[simp]:
"1 \<noteq> \<infinity>" "\<infinity> \<noteq> 1"
"1 \<noteq> -\<infinity>" "-\<infinity> \<noteq> 1"
by (auto simp add: times_extreal_def one_extreal_def)
lemma extreal_plus_1[simp]:
"1 + extreal r = extreal (r + 1)" "extreal r + 1 = extreal (r + 1)"
"1 + -\<infinity> = -\<infinity>" "-\<infinity> + 1 = -\<infinity>"
unfolding one_extreal_def by auto
lemma extreal_zero_times[simp]:
fixes a b :: extreal shows "a * b = 0 \<longleftrightarrow> a = 0 \<or> b = 0"
by (cases rule: extreal2_cases[of a b]) auto
lemma extreal_mult_eq_PInfty[simp]:
shows "a * b = \<infinity> \<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: extreal2_cases[of a b]) auto
lemma extreal_mult_eq_MInfty[simp]:
shows "a * b = -\<infinity> \<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: extreal2_cases[of a b]) auto
lemma extreal_0_less_1[simp]: "0 < (1::extreal)"
by (simp_all add: zero_extreal_def one_extreal_def)
lemma extreal_zero_one[simp]: "0 \<noteq> (1::extreal)"
by (simp_all add: zero_extreal_def one_extreal_def)
lemma extreal_mult_minus_left[simp]:
fixes a b :: extreal shows "-a * b = - (a * b)"
by (cases rule: extreal2_cases[of a b]) auto
lemma extreal_mult_minus_right[simp]:
fixes a b :: extreal shows "a * -b = - (a * b)"
by (cases rule: extreal2_cases[of a b]) auto
lemma extreal_mult_infty[simp]:
"a * \<infinity> = (if a = 0 then 0 else if 0 < a then \<infinity> else - \<infinity>)"
by (cases a) auto
lemma extreal_infty_mult[simp]:
"\<infinity> * a = (if a = 0 then 0 else if 0 < a then \<infinity> else - \<infinity>)"
by (cases a) auto
lemma extreal_mult_strict_right_mono:
assumes "a < b" and "0 < c" "c < \<infinity>"
shows "a * c < b * c"
using assms
by (cases rule: extreal3_cases[of a b c])
(auto simp: zero_le_mult_iff extreal_less_PInfty)
lemma extreal_mult_strict_left_mono:
"\<lbrakk> a < b ; 0 < c ; c < \<infinity>\<rbrakk> \<Longrightarrow> c * a < c * b"
using extreal_mult_strict_right_mono by (simp add: mult_commute[of c])
lemma extreal_mult_right_mono:
fixes a b c :: extreal shows "\<lbrakk>a \<le> b; 0 \<le> c\<rbrakk> \<Longrightarrow> a*c \<le> b*c"
using assms
apply (cases "c = 0") apply simp
by (cases rule: extreal3_cases[of a b c])
(auto simp: zero_le_mult_iff extreal_less_PInfty)
lemma extreal_mult_left_mono:
fixes a b c :: extreal shows "\<lbrakk>a \<le> b; 0 \<le> c\<rbrakk> \<Longrightarrow> c * a \<le> c * b"
using extreal_mult_right_mono by (simp add: mult_commute[of c])
lemma zero_less_one_extreal[simp]: "0 \<le> (1::extreal)"
by (simp add: one_extreal_def zero_extreal_def)
lemma extreal_0_le_mult[simp]: "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> a * (b :: extreal)"
by (cases rule: extreal2_cases[of a b]) (auto simp: mult_nonneg_nonneg)
lemma extreal_right_distrib:
fixes r a b :: extreal shows "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> r * (a + b) = r * a + r * b"
by (cases rule: extreal3_cases[of r a b]) (simp_all add: field_simps)
lemma extreal_left_distrib:
fixes r a b :: extreal shows "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> (a + b) * r = a * r + b * r"
by (cases rule: extreal3_cases[of r a b]) (simp_all add: field_simps)
lemma extreal_mult_le_0_iff:
fixes a b :: extreal
shows "a * b \<le> 0 \<longleftrightarrow> (0 \<le> a \<and> b \<le> 0) \<or> (a \<le> 0 \<and> 0 \<le> b)"
by (cases rule: extreal2_cases[of a b]) (simp_all add: mult_le_0_iff)
lemma extreal_zero_le_0_iff:
fixes a b :: extreal
shows "0 \<le> a * b \<longleftrightarrow> (0 \<le> a \<and> 0 \<le> b) \<or> (a \<le> 0 \<and> b \<le> 0)"
by (cases rule: extreal2_cases[of a b]) (simp_all add: zero_le_mult_iff)
lemma extreal_mult_less_0_iff:
fixes a b :: extreal
shows "a * b < 0 \<longleftrightarrow> (0 < a \<and> b < 0) \<or> (a < 0 \<and> 0 < b)"
by (cases rule: extreal2_cases[of a b]) (simp_all add: mult_less_0_iff)
lemma extreal_zero_less_0_iff:
fixes a b :: extreal
shows "0 < a * b \<longleftrightarrow> (0 < a \<and> 0 < b) \<or> (a < 0 \<and> b < 0)"
by (cases rule: extreal2_cases[of a b]) (simp_all add: zero_less_mult_iff)
lemma extreal_distrib:
fixes a b c :: extreal
assumes "a \<noteq> \<infinity> \<or> b \<noteq> -\<infinity>" "a \<noteq> -\<infinity> \<or> b \<noteq> \<infinity>" "\<bar>c\<bar> \<noteq> \<infinity>"
shows "(a + b) * c = a * c + b * c"
using assms
by (cases rule: extreal3_cases[of a b c]) (simp_all add: field_simps)
lemma extreal_le_epsilon:
fixes x y :: extreal
assumes "ALL e. 0 < e --> x <= y + e"
shows "x <= y"
proof-
{ assume a: "EX r. y = extreal r"
from this obtain r where r_def: "y = extreal r" by auto
{ assume "x=(-\<infinity>)" hence ?thesis by auto }
moreover
{ assume "~(x=(-\<infinity>))"
from this obtain p where p_def: "x = extreal 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 "extreal e"] p_def r_def by auto }
hence "p <= r" apply (subst field_le_epsilon) by auto
hence ?thesis using r_def p_def by auto
} ultimately have ?thesis by blast
}
moreover
{ assume "y=(-\<infinity>) | y=\<infinity>" hence ?thesis
using assms[rule_format, of 1] by (cases x) auto
} ultimately show ?thesis by (cases y) auto
qed
lemma extreal_le_epsilon2:
fixes x y :: extreal
assumes "ALL e. 0 < e --> x <= y + extreal e"
shows "x <= y"
proof-
{ fix e :: extreal assume "e>0"
{ assume "e=\<infinity>" hence "x<=y+e" by auto }
moreover
{ assume "e~=\<infinity>"
from this obtain r where "e = extreal r" using `e>0` apply (cases e) by auto
hence "x<=y+e" using assms[rule_format, of r] `e>0` by auto
} ultimately have "x<=y+e" by blast
} from this show ?thesis using extreal_le_epsilon by auto
qed
lemma extreal_le_real:
fixes x y :: extreal
assumes "ALL z. x <= extreal z --> y <= extreal z"
shows "y <= x"
by (metis assms extreal.exhaust extreal_bot extreal_less_eq(1)
extreal_less_eq(2) order_refl uminus_extreal.simps(2))
lemma extreal_le_extreal:
fixes x y :: extreal
assumes "\<And>B. B < x \<Longrightarrow> B <= y"
shows "x <= y"
by (metis assms extreal_dense leD linorder_le_less_linear)
lemma extreal_ge_extreal:
fixes x y :: extreal
assumes "ALL B. B>x --> B >= y"
shows "x >= y"
by (metis assms extreal_dense leD linorder_le_less_linear)
subsubsection {* Power *}
instantiation extreal :: power
begin
primrec power_extreal where
"power_extreal x 0 = 1" |
"power_extreal x (Suc n) = x * x ^ n"
instance ..
end
lemma extreal_power[simp]: "(extreal x) ^ n = extreal (x^n)"
by (induct n) (auto simp: one_extreal_def)
lemma extreal_power_PInf[simp]: "\<infinity> ^ n = (if n = 0 then 1 else \<infinity>)"
by (induct n) (auto simp: one_extreal_def)
lemma extreal_power_uminus[simp]:
fixes x :: extreal
shows "(- x) ^ n = (if even n then x ^ n else - (x^n))"
by (induct n) (auto simp: one_extreal_def)
lemma extreal_power_number_of[simp]:
"(number_of num :: extreal) ^ n = extreal (number_of num ^ n)"
by (induct n) (auto simp: one_extreal_def)
lemma zero_le_power_extreal[simp]:
fixes a :: extreal assumes "0 \<le> a"
shows "0 \<le> a ^ n"
using assms by (induct n) (auto simp: extreal_zero_le_0_iff)
subsubsection {* Subtraction *}
lemma extreal_minus_minus_image[simp]:
fixes S :: "extreal set"
shows "uminus ` uminus ` S = S"
by (auto simp: image_iff)
lemma extreal_uminus_lessThan[simp]:
fixes a :: extreal shows "uminus ` {..<a} = {-a<..}"
proof (safe intro!: image_eqI)
fix x assume "-a < x"
then have "- x < - (- a)" by (simp del: extreal_uminus_uminus)
then show "- x < a" by simp
qed auto
lemma extreal_uminus_greaterThan[simp]:
"uminus ` {(a::extreal)<..} = {..<-a}"
by (metis extreal_uminus_lessThan extreal_uminus_uminus
extreal_minus_minus_image)
instantiation extreal :: minus
begin
definition "x - y = x + -(y::extreal)"
instance ..
end
lemma extreal_minus[simp]:
"extreal r - extreal p = extreal (r - p)"
"-\<infinity> - extreal r = -\<infinity>"
"extreal r - \<infinity> = -\<infinity>"
"\<infinity> - x = \<infinity>"
"-\<infinity> - \<infinity> = -\<infinity>"
"x - -y = x + y"
"x - 0 = x"
"0 - x = -x"
by (simp_all add: minus_extreal_def)
lemma extreal_x_minus_x[simp]:
"x - x = (if \<bar>x\<bar> = \<infinity> then \<infinity> else 0)"
by (cases x) simp_all
lemma extreal_eq_minus_iff:
fixes x y z :: extreal
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: extreal3_cases[of x y z]) auto
lemma extreal_eq_minus:
fixes x y z :: extreal
shows "\<bar>y\<bar> \<noteq> \<infinity> \<Longrightarrow> x = z - y \<longleftrightarrow> x + y = z"
by (auto simp: extreal_eq_minus_iff)
lemma extreal_less_minus_iff:
fixes x y z :: extreal
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: extreal3_cases[of x y z]) auto
lemma extreal_less_minus:
fixes x y z :: extreal
shows "\<bar>y\<bar> \<noteq> \<infinity> \<Longrightarrow> x < z - y \<longleftrightarrow> x + y < z"
by (auto simp: extreal_less_minus_iff)
lemma extreal_le_minus_iff:
fixes x y z :: extreal
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: extreal3_cases[of x y z]) auto
lemma extreal_le_minus:
fixes x y z :: extreal
shows "\<bar>y\<bar> \<noteq> \<infinity> \<Longrightarrow> x \<le> z - y \<longleftrightarrow> x + y \<le> z"
by (auto simp: extreal_le_minus_iff)
lemma extreal_minus_less_iff:
fixes x y z :: extreal
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: extreal3_cases[of x y z]) auto
lemma extreal_minus_less:
fixes x y z :: extreal
shows "\<bar>y\<bar> \<noteq> \<infinity> \<Longrightarrow> x - y < z \<longleftrightarrow> x < z + y"
by (auto simp: extreal_minus_less_iff)
lemma extreal_minus_le_iff:
fixes x y z :: extreal
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: extreal3_cases[of x y z]) auto
lemma extreal_minus_le:
fixes x y z :: extreal
shows "\<bar>y\<bar> \<noteq> \<infinity> \<Longrightarrow> x - y \<le> z \<longleftrightarrow> x \<le> z + y"
by (auto simp: extreal_minus_le_iff)
lemma extreal_minus_eq_minus_iff:
fixes a b c :: extreal
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: extreal3_cases[of a b c]) auto
lemma extreal_add_le_add_iff:
"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: extreal3_cases[of a b c]) (simp_all add: field_simps)
lemma extreal_mult_le_mult_iff:
"\<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: extreal3_cases[of a b c]) (simp_all add: mult_le_cancel_left)
lemma extreal_minus_mono:
fixes A B C D :: extreal assumes "A \<le> B" "D \<le> C"
shows "A - C \<le> B - D"
using assms
by (cases rule: extreal3_cases[case_product extreal_cases, of A B C D]) simp_all
lemma real_of_extreal_minus:
"real (a - b) = (if \<bar>a\<bar> = \<infinity> \<or> \<bar>b\<bar> = \<infinity> then 0 else real a - real b)"
by (cases rule: extreal2_cases[of a b]) auto
lemma extreal_diff_positive:
fixes a b :: extreal shows "a \<le> b \<Longrightarrow> 0 \<le> b - a"
by (cases rule: extreal2_cases[of a b]) auto
lemma extreal_between:
fixes x e :: extreal
assumes "\<bar>x\<bar> \<noteq> \<infinity>" "0 < e"
shows "x - e < x" "x < x + e"
using assms apply (cases x, cases e) apply auto
using assms by (cases x, cases e) auto
subsubsection {* Division *}
instantiation extreal :: inverse
begin
function inverse_extreal where
"inverse (extreal r) = (if r = 0 then \<infinity> else extreal (inverse r))" |
"inverse \<infinity> = 0" |
"inverse (-\<infinity>) = 0"
by (auto intro: extreal_cases)
termination by (relation "{}") simp
definition "x / y = x * inverse (y :: extreal)"
instance proof qed
end
lemma extreal_inverse[simp]:
"inverse 0 = \<infinity>"
"inverse (1::extreal) = 1"
by (simp_all add: one_extreal_def zero_extreal_def)
lemma extreal_divide[simp]:
"extreal r / extreal p = (if p = 0 then extreal r * \<infinity> else extreal (r / p))"
unfolding divide_extreal_def by (auto simp: divide_real_def)
lemma extreal_divide_same[simp]:
"x / x = (if \<bar>x\<bar> = \<infinity> \<or> x = 0 then 0 else 1)"
by (cases x)
(simp_all add: divide_real_def divide_extreal_def one_extreal_def)
lemma extreal_inv_inv[simp]:
"inverse (inverse x) = (if x \<noteq> -\<infinity> then x else \<infinity>)"
by (cases x) auto
lemma extreal_inverse_minus[simp]:
"inverse (- x) = (if x = 0 then \<infinity> else -inverse x)"
by (cases x) simp_all
lemma extreal_uminus_divide[simp]:
fixes x y :: extreal shows "- x / y = - (x / y)"
unfolding divide_extreal_def by simp
lemma extreal_divide_Infty[simp]:
"x / \<infinity> = 0" "x / -\<infinity> = 0"
unfolding divide_extreal_def by simp_all
lemma extreal_divide_one[simp]:
"x / 1 = (x::extreal)"
unfolding divide_extreal_def by simp
lemma extreal_divide_extreal[simp]:
"\<infinity> / extreal r = (if 0 \<le> r then \<infinity> else -\<infinity>)"
unfolding divide_extreal_def by simp
lemma zero_le_divide_extreal[simp]:
fixes a :: extreal assumes "0 \<le> a" "0 \<le> b"
shows "0 \<le> a / b"
using assms by (cases rule: extreal2_cases[of a b]) (auto simp: zero_le_divide_iff)
lemma extreal_le_divide_pos:
"x > 0 \<Longrightarrow> x \<noteq> \<infinity> \<Longrightarrow> y \<le> z / x \<longleftrightarrow> x * y \<le> z"
by (cases rule: extreal3_cases[of x y z]) (auto simp: field_simps)
lemma extreal_divide_le_pos:
"x > 0 \<Longrightarrow> x \<noteq> \<infinity> \<Longrightarrow> z / x \<le> y \<longleftrightarrow> z \<le> x * y"
by (cases rule: extreal3_cases[of x y z]) (auto simp: field_simps)
lemma extreal_le_divide_neg:
"x < 0 \<Longrightarrow> x \<noteq> -\<infinity> \<Longrightarrow> y \<le> z / x \<longleftrightarrow> z \<le> x * y"
by (cases rule: extreal3_cases[of x y z]) (auto simp: field_simps)
lemma extreal_divide_le_neg:
"x < 0 \<Longrightarrow> x \<noteq> -\<infinity> \<Longrightarrow> z / x \<le> y \<longleftrightarrow> x * y \<le> z"
by (cases rule: extreal3_cases[of x y z]) (auto simp: field_simps)
lemma extreal_inverse_antimono_strict:
fixes x y :: extreal
shows "0 \<le> x \<Longrightarrow> x < y \<Longrightarrow> inverse y < inverse x"
by (cases rule: extreal2_cases[of x y]) auto
lemma extreal_inverse_antimono:
fixes x y :: extreal
shows "0 \<le> x \<Longrightarrow> x <= y \<Longrightarrow> inverse y <= inverse x"
by (cases rule: extreal2_cases[of x y]) auto
lemma inverse_inverse_Pinfty_iff[simp]:
"inverse x = \<infinity> \<longleftrightarrow> x = 0"
by (cases x) auto
lemma extreal_inverse_eq_0:
"inverse x = 0 \<longleftrightarrow> x = \<infinity> \<or> x = -\<infinity>"
by (cases x) auto
lemma extreal_0_gt_inverse:
fixes x :: extreal shows "0 < inverse x \<longleftrightarrow> x \<noteq> \<infinity> \<and> 0 \<le> x"
by (cases x) auto
lemma extreal_mult_less_right:
assumes "b * a < c * a" "0 < a" "a < \<infinity>"
shows "b < c"
using assms
by (cases rule: extreal3_cases[of a b c])
(auto split: split_if_asm simp: zero_less_mult_iff zero_le_mult_iff)
lemma extreal_power_divide:
"y \<noteq> 0 \<Longrightarrow> (x / y :: extreal) ^ n = x^n / y^n"
by (cases rule: extreal2_cases[of x y])
(auto simp: one_extreal_def zero_extreal_def power_divide not_le
power_less_zero_eq zero_le_power_iff)
lemma extreal_le_mult_one_interval:
fixes x y :: extreal
assumes y: "y \<noteq> -\<infinity>"
assumes z: "\<And>z. \<lbrakk> 0 < z ; z < 1 \<rbrakk> \<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_extreal_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 "extreal z"]
show "z * r \<le> p" using p r by (auto simp: zero_le_mult_iff one_extreal_def)
qed
then show "x \<le> y" using p r by simp
qed (insert y, simp_all)
qed simp
subsection "Complete lattice"
instantiation extreal :: lattice
begin
definition [simp]: "sup x y = (max x y :: extreal)"
definition [simp]: "inf x y = (min x y :: extreal)"
instance proof qed simp_all
end
instantiation extreal :: complete_lattice
begin
definition "bot = -\<infinity>"
definition "top = \<infinity>"
definition "Sup S = (LEAST z. ALL x:S. x <= z :: extreal)"
definition "Inf S = (GREATEST z. ALL x:S. z <= x :: extreal)"
lemma extreal_complete_Sup:
fixes S :: "extreal set" assumes "S \<noteq> {}"
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
assume "\<exists>x. \<forall>a\<in>S. a \<le> extreal x"
then obtain y where y: "\<And>a. a\<in>S \<Longrightarrow> a \<le> extreal y" by auto
then have "\<infinity> \<notin> S" by force
show ?thesis
proof cases
assume "S = {-\<infinity>}"
then show ?thesis by (auto intro!: exI[of _ "-\<infinity>"])
next
assume "S \<noteq> {-\<infinity>}"
with `S \<noteq> {}` `\<infinity> \<notin> S` obtain x where "x \<in> S - {-\<infinity>}" "x \<noteq> \<infinity>" by auto
with y `\<infinity> \<notin> S` have "\<forall>z\<in>real ` (S - {-\<infinity>}). z \<le> y"
by (auto simp: real_of_extreal_ord_simps)
with reals_complete2[of "real ` (S - {-\<infinity>})"] `x \<in> S - {-\<infinity>}`
obtain s where s:
"\<forall>y\<in>S - {-\<infinity>}. real y \<le> s" "\<And>z. (\<forall>y\<in>S - {-\<infinity>}. real y \<le> z) \<Longrightarrow> s \<le> z"
by auto
show ?thesis
proof (safe intro!: exI[of _ "extreal s"])
fix z assume "z \<in> S" with `\<infinity> \<notin> S` show "z \<le> extreal s"
proof (cases z)
case (real r)
then show ?thesis
using s(1)[rule_format, of z] `z \<in> S` `z = extreal r` by auto
qed auto
next
fix z assume *: "\<forall>y\<in>S. y \<le> z"
with `S \<noteq> {-\<infinity>}` `S \<noteq> {}` show "extreal s \<le> z"
proof (cases z)
case (real u)
with * have "s \<le> u"
by (intro s(2)[of u]) (auto simp: real_of_extreal_ord_simps)
then show ?thesis using real by simp
qed auto
qed
qed
next
assume *: "\<not> (\<exists>x. \<forall>a\<in>S. a \<le> extreal x)"
show ?thesis
proof (safe intro!: exI[of _ \<infinity>])
fix y assume **: "\<forall>z\<in>S. z \<le> y"
with * show "\<infinity> \<le> y"
proof (cases y)
case MInf with * ** show ?thesis by (force simp: not_le)
qed auto
qed simp
qed
lemma extreal_complete_Inf:
fixes S :: "extreal set" assumes "S ~= {}"
shows "EX x. (ALL y:S. x <= y) & (ALL z. (ALL y:S. z <= y) --> z <= x)"
proof-
def S1 == "uminus ` S"
hence "S1 ~= {}" using assms by auto
from this obtain x where x_def: "(ALL y:S1. y <= x) & (ALL z. (ALL y:S1. y <= z) --> x <= z)"
using extreal_complete_Sup[of S1] by auto
{ fix z assume "ALL y:S. z <= y"
hence "ALL y:S1. y <= -z" unfolding S1_def by auto
hence "x <= -z" using x_def by auto
hence "z <= -x"
apply (subst extreal_uminus_uminus[symmetric])
unfolding extreal_minus_le_minus . }
moreover have "(ALL y:S. -x <= y)"
using x_def unfolding S1_def
apply simp
apply (subst (3) extreal_uminus_uminus[symmetric])
unfolding extreal_minus_le_minus by simp
ultimately show ?thesis by auto
qed
lemma extreal_complete_uminus_eq:
fixes S :: "extreal 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 extreal_minus_le_minus extreal_uminus_uminus)
lemma extreal_Sup_uminus_image_eq:
fixes S :: "extreal set"
shows "Sup (uminus ` S) = - Inf S"
proof cases
assume "S = {}"
moreover have "(THE x. All (op \<le> x)) = (-\<infinity>::extreal)"
by (rule the_equality) (auto intro!: extreal_bot)
moreover have "(SOME x. \<forall>y. y \<le> x) = (\<infinity>::extreal)"
by (rule some_equality) (auto intro!: extreal_top)
ultimately show ?thesis unfolding Inf_extreal_def Sup_extreal_def
Least_def Greatest_def GreatestM_def by simp
next
assume "S \<noteq> {}"
with extreal_complete_Sup[of "uminus`S"]
obtain x where x: "(\<forall>y\<in>S. -x \<le> y) \<and> (\<forall>z. (\<forall>y\<in>S. z \<le> y) \<longrightarrow> z \<le> -x)"
unfolding extreal_complete_uminus_eq by auto
show "Sup (uminus ` S) = - Inf S"
unfolding Inf_extreal_def Greatest_def GreatestM_def
proof (intro someI2[of _ _ "\<lambda>x. Sup (uminus`S) = - x"])
show "(\<forall>y\<in>S. -x \<le> y) \<and> (\<forall>y. (\<forall>z\<in>S. y \<le> z) \<longrightarrow> y \<le> -x)"
using x .
fix x' assume "(\<forall>y\<in>S. x' \<le> y) \<and> (\<forall>y. (\<forall>z\<in>S. y \<le> z) \<longrightarrow> y \<le> x')"
then have "(\<forall>y\<in>uminus`S. y \<le> - x') \<and> (\<forall>y. (\<forall>z\<in>uminus`S. z \<le> y) \<longrightarrow> - x' \<le> y)"
unfolding extreal_complete_uminus_eq by simp
then show "Sup (uminus ` S) = -x'"
unfolding Sup_extreal_def extreal_uminus_eq_iff
by (intro Least_equality) auto
qed
qed
instance
proof
{ fix x :: extreal and A
show "bot <= x" by (cases x) (simp_all add: bot_extreal_def)
show "x <= top" by (simp add: top_extreal_def) }
{ fix x :: extreal and A assume "x : A"
with extreal_complete_Sup[of A]
obtain s where s: "\<forall>y\<in>A. y <= s" "\<forall>z. (\<forall>y\<in>A. y <= z) \<longrightarrow> s <= z" by auto
hence "x <= s" using `x : A` by auto
also have "... = Sup A" using s unfolding Sup_extreal_def
by (auto intro!: Least_equality[symmetric])
finally show "x <= Sup A" . }
note le_Sup = this
{ fix x :: extreal and A assume *: "!!z. (z : A ==> z <= x)"
show "Sup A <= x"
proof (cases "A = {}")
case True
hence "Sup A = -\<infinity>" unfolding Sup_extreal_def
by (auto intro!: Least_equality)
thus "Sup A <= x" by simp
next
case False
with extreal_complete_Sup[of A]
obtain s where s: "\<forall>y\<in>A. y <= s" "\<forall>z. (\<forall>y\<in>A. y <= z) \<longrightarrow> s <= z" by auto
hence "Sup A = s"
unfolding Sup_extreal_def by (auto intro!: Least_equality)
also have "s <= x" using * s by auto
finally show "Sup A <= x" .
qed }
note Sup_le = this
{ fix x :: extreal and A assume "x \<in> A"
with le_Sup[of "-x" "uminus`A"] show "Inf A \<le> x"
unfolding extreal_Sup_uminus_image_eq by simp }
{ fix x :: extreal and A assume *: "!!z. (z : A ==> x <= z)"
with Sup_le[of "uminus`A" "-x"] show "x \<le> Inf A"
unfolding extreal_Sup_uminus_image_eq by force }
qed
end
lemma extreal_SUPR_uminus:
fixes f :: "'a => extreal"
shows "(SUP i : R. -(f i)) = -(INF i : R. f i)"
unfolding SUPR_def INFI_def
using extreal_Sup_uminus_image_eq[of "f`R"]
by (simp add: image_image)
lemma extreal_INFI_uminus:
fixes f :: "'a => extreal"
shows "(INF i : R. -(f i)) = -(SUP i : R. f i)"
using extreal_SUPR_uminus[of _ "\<lambda>x. - f x"] by simp
lemma extreal_Inf_uminus_image_eq: "Inf (uminus ` S) = - Sup (S::extreal set)"
using extreal_Sup_uminus_image_eq[of "uminus ` S"] by (simp add: image_image)
lemma extreal_inj_on_uminus[intro, simp]: "inj_on uminus (A :: extreal set)"
by (auto intro!: inj_onI)
lemma extreal_image_uminus_shift:
fixes X Y :: "extreal 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_extreal_iff:
fixes z :: extreal
shows "(!!x. x:X ==> z <= x) ==> (EX x:X. x<y) <-> 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 :: "extreal set" shows "Sup S = -\<infinity> \<longleftrightarrow> S = {} \<or> S = {-\<infinity>}"
proof
assume a: "Sup S = -\<infinity>"
with complete_lattice_class.Sup_upper[of _ S]
show "S={} \<or> S={-\<infinity>}" by auto
next
assume "S={} \<or> S={-\<infinity>}" then show "Sup S = -\<infinity>"
unfolding Sup_extreal_def by (auto intro!: Least_equality)
qed
lemma Inf_eq_PInfty:
fixes S :: "extreal set" shows "Inf S = \<infinity> \<longleftrightarrow> S = {} \<or> S = {\<infinity>}"
using Sup_eq_MInfty[of "uminus`S"]
unfolding extreal_Sup_uminus_image_eq extreal_image_uminus_shift by simp
lemma Inf_eq_MInfty: "-\<infinity> : S ==> Inf S = -\<infinity>"
unfolding Inf_extreal_def
by (auto intro!: Greatest_equality)
lemma Sup_eq_PInfty: "\<infinity> : S ==> Sup S = \<infinity>"
unfolding Sup_extreal_def
by (auto intro!: Least_equality)
lemma extreal_SUPI:
fixes x :: extreal
assumes "!!i. i : A ==> f i <= x"
assumes "!!y. (!!i. i : A ==> f i <= y) ==> x <= y"
shows "(SUP i:A. f i) = x"
unfolding SUPR_def Sup_extreal_def
using assms by (auto intro!: Least_equality)
lemma extreal_INFI:
fixes x :: extreal
assumes "!!i. i : A ==> f i >= x"
assumes "!!y. (!!i. i : A ==> f i >= y) ==> x >= y"
shows "(INF i:A. f i) = x"
unfolding INFI_def Inf_extreal_def
using assms by (auto intro!: Greatest_equality)
lemma Sup_extreal_close:
fixes e :: extreal
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_extreal_close:
fixes e :: extreal assumes "\<bar>Inf X\<bar> \<noteq> \<infinity>" "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_eq_top_iff:
fixes A :: "'a::{complete_lattice, linorder} set"
shows "Sup A = top \<longleftrightarrow> (\<forall>x<top. \<exists>i\<in>A. x < i)"
proof
assume *: "Sup A = top"
show "(\<forall>x<top. \<exists>i\<in>A. x < i)" unfolding *[symmetric]
proof (intro allI impI)
fix x assume "x < Sup A" then show "\<exists>i\<in>A. x < i"
unfolding less_Sup_iff by auto
qed
next
assume *: "\<forall>x<top. \<exists>i\<in>A. x < i"
show "Sup A = top"
proof (rule ccontr)
assume "Sup A \<noteq> top"
with top_greatest[of "Sup A"]
have "Sup A < top" unfolding le_less by auto
then have "Sup A < Sup A"
using * unfolding less_Sup_iff by auto
then show False by auto
qed
qed
lemma SUP_eq_top_iff:
fixes f :: "'a \<Rightarrow> 'b::{complete_lattice, linorder}"
shows "(SUP i:A. f i) = top \<longleftrightarrow> (\<forall>x<top. \<exists>i\<in>A. x < f i)"
unfolding SUPR_def Sup_eq_top_iff by auto
lemma SUP_nat_Infty: "(SUP i::nat. extreal (real i)) = \<infinity>"
proof -
{ fix x assume "x \<noteq> \<infinity>"
then have "\<exists>k::nat. x < extreal (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. extreal (real n)"]
by (auto simp: top_extreal_def)
qed
lemma extreal_le_Sup:
fixes x :: extreal
shows "(x <= (SUP i:A. f i)) <-> (ALL y. y < x --> (EX i. i : A & y <= f i))"
(is "?lhs <-> ?rhs")
proof-
{ assume "?rhs"
{ assume "~(x <= (SUP i:A. f i))" hence "(SUP i:A. f i)<x" by (simp add: not_le)
from this obtain y where y_def: "(SUP i:A. f i)<y & y<x" using extreal_dense by auto
from this obtain i where "i : A & y <= f i" using `?rhs` by auto
hence "y <= (SUP i:A. f i)" using le_SUPI[of i A f] by auto
hence False using y_def by auto
} hence "?lhs" by auto
}
moreover
{ assume "?lhs" hence "?rhs"
by (metis Collect_def Collect_mem_eq SUP_leI assms atLeastatMost_empty atLeastatMost_empty_iff
inf_sup_ord(4) linorder_le_cases sup_absorb1 xt1(8))
} ultimately show ?thesis by auto
qed
lemma extreal_Inf_le:
fixes x :: extreal
shows "((INF i:A. f i) <= x) <-> (ALL y. x < y --> (EX i. i : A & f i <= y))"
(is "?lhs <-> ?rhs")
proof-
{ assume "?rhs"
{ assume "~((INF i:A. f i) <= x)" hence "x < (INF i:A. f i)" by (simp add: not_le)
from this obtain y where y_def: "x<y & y<(INF i:A. f i)" using extreal_dense by auto
from this obtain i where "i : A & f i <= y" using `?rhs` by auto
hence "(INF i:A. f i) <= y" using INF_leI[of i A f] by auto
hence False using y_def by auto
} hence "?lhs" by auto
}
moreover
{ assume "?lhs" hence "?rhs"
by (metis Collect_def Collect_mem_eq le_INFI assms atLeastatMost_empty atLeastatMost_empty_iff
inf_sup_ord(4) linorder_le_cases sup_absorb1 xt1(8))
} ultimately show ?thesis by auto
qed
lemma Inf_less:
fixes x :: extreal
assumes "(INF i:A. f i) < x"
shows "EX i. i : A & f i <= x"
proof(rule ccontr)
assume "~ (EX i. i : A & f i <= x)"
hence "ALL i:A. f i > x" by auto
hence "(INF i:A. f i) >= x" apply (subst le_INFI) by auto
thus False using assms by auto
qed
lemma same_INF:
assumes "ALL e:A. f e = g e"
shows "(INF e:A. f e) = (INF e:A. g e)"
proof-
have "f ` A = g ` A" unfolding image_def using assms by auto
thus ?thesis unfolding INFI_def by auto
qed
lemma same_SUP:
assumes "ALL e:A. f e = g e"
shows "(SUP e:A. f e) = (SUP e:A. g e)"
proof-
have "f ` A = g ` A" unfolding image_def using assms by auto
thus ?thesis unfolding SUPR_def by auto
qed
lemma SUPR_eq:
assumes "\<forall>i\<in>A. \<exists>j\<in>B. f i \<le> g j"
assumes "\<forall>j\<in>B. \<exists>i\<in>A. g j \<le> f i"
shows "(SUP i:A. f i) = (SUP j:B. g j)"
proof (intro antisym)
show "(SUP i:A. f i) \<le> (SUP j:B. g j)"
using assms by (metis SUP_leI le_SUPI2)
show "(SUP i:B. g i) \<le> (SUP j:A. f j)"
using assms by (metis SUP_leI le_SUPI2)
qed
lemma SUP_extreal_le_addI:
assumes "\<And>i. f i + y \<le> z" and "y \<noteq> -\<infinity>"
shows "SUPR 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: extreal_le_minus_iff)
then have "SUPR UNIV f \<le> z - y" by (rule SUP_leI)
then show ?thesis using real by (simp add: extreal_le_minus_iff)
qed (insert assms, auto)
lemma SUPR_extreal_add:
fixes f g :: "nat \<Rightarrow> extreal"
assumes "incseq f" "incseq g" and pos: "\<And>i. f i \<noteq> -\<infinity>" "\<And>i. g i \<noteq> -\<infinity>"
shows "(SUP i. f i + g i) = SUPR UNIV f + SUPR UNIV g"
proof (rule extreal_SUPI)
fix y assume *: "\<And>i. i \<in> UNIV \<Longrightarrow> f i + g i \<le> y"
have f: "SUPR UNIV f \<noteq> -\<infinity>" using pos
unfolding SUPR_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 "SUPR UNIV f + g j \<le> y"
using assms(4)[of j] by (intro SUP_extreal_le_addI) auto
then have "g j + SUPR UNIV f \<le> y" by (simp add: ac_simps) }
then have "SUPR UNIV g + SUPR UNIV f \<le> y"
using f by (rule SUP_extreal_le_addI)
then show "SUPR UNIV f + SUPR UNIV g \<le> y" by (simp add: ac_simps)
qed (auto intro!: add_mono le_SUPI)
lemma SUPR_extreal_add_pos:
fixes f g :: "nat \<Rightarrow> extreal"
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) = SUPR UNIV f + SUPR UNIV g"
proof (intro SUPR_extreal_add inc)
fix i show "f i \<noteq> -\<infinity>" "g i \<noteq> -\<infinity>" using pos[of i] by auto
qed
lemma SUPR_extreal_setsum:
fixes f g :: "'a \<Rightarrow> nat \<Rightarrow> extreal"
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. SUPR UNIV (f n))"
proof cases
assume "finite A" then show ?thesis using assms
by induct (auto simp: incseq_setsumI2 setsum_nonneg SUPR_extreal_add_pos)
qed simp
lemma SUPR_extreal_cmult:
fixes f :: "nat \<Rightarrow> extreal" assumes "\<And>i. 0 \<le> f i" "0 \<le> c"
shows "(SUP i. c * f i) = c * SUPR UNIV f"
proof (rule extreal_SUPI)
fix i have "f i \<le> SUPR UNIV f" by (rule le_SUPI) auto
then show "c * f i \<le> c * SUPR UNIV f"
using `0 \<le> c` by (rule extreal_mult_left_mono)
next
fix y assume *: "\<And>i. i \<in> UNIV \<Longrightarrow> c * f i \<le> y"
show "c * SUPR UNIV f \<le> y"
proof cases
assume c: "0 < c \<and> c \<noteq> \<infinity>"
with * have "SUPR UNIV f \<le> y / c"
by (intro SUP_leI) (auto simp: extreal_le_divide_pos)
with c show ?thesis
by (auto simp: extreal_le_divide_pos)
next
{ assume "c = \<infinity>" have ?thesis
proof cases
assume "\<forall>i. f i = 0"
moreover then have "range f = {0}" by auto
ultimately show "c * SUPR UNIV f \<le> y" using * by (auto simp: SUPR_def)
next
assume "\<not> (\<forall>i. f i = 0)"
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 assume "\<not> (0 < c \<and> c \<noteq> \<infinity>)"
ultimately show ?thesis using * `0 \<le> c` by auto
qed
qed
lemma SUP_PInfty:
fixes f :: "'a \<Rightarrow> extreal"
assumes "\<And>n::nat. \<exists>i\<in>A. extreal (real n) \<le> f i"
shows "(SUP i:A. f i) = \<infinity>"
unfolding SUPR_def Sup_eq_top_iff[where 'a=extreal, unfolded top_extreal_def]
apply simp
proof safe
fix x 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 < extreal (real n)" by auto
moreover from assms[of n] guess i ..
ultimately show ?thesis
by (auto intro!: bexI[of _ i])
qed
qed
lemma Sup_countable_SUPR:
assumes "A \<noteq> {}"
shows "\<exists>f::nat \<Rightarrow> extreal. range f \<subseteq> A \<and> Sup A = SUPR UNIV f"
proof (cases "Sup A")
case (real r)
have "\<forall>n::nat. \<exists>x. x \<in> A \<and> Sup A < x + 1 / extreal (real n)"
proof
fix n ::nat have "\<exists>x\<in>A. Sup A - 1 / extreal (real n) < x"
using assms real by (intro Sup_extreal_close) (auto simp: one_extreal_def)
then guess x ..
then show "\<exists>x. x \<in> A \<and> Sup A < x + 1 / extreal (real n)"
by (auto intro!: exI[of _ x] simp: extreal_minus_less_iff)
qed
from choice[OF this] guess f .. note f = this
have "SUPR UNIV f = Sup A"
proof (rule extreal_SUPI)
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 extreal_le_epsilon, intro allI impI)
fix e :: extreal assume "0 < e"
show "Sup A \<le> y + e"
proof (cases e)
case (real r)
hence "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 / extreal (real n)" using f[THEN spec, of n] by auto
also have "1 / extreal (real n) \<le> e" using real * by (auto simp: one_extreal_def )
with bound have "f n + 1 / extreal (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
assume "\<infinity> \<in> A"
moreover then have "\<infinity> \<le> Sup A" by (intro complete_lattice_class.Sup_upper)
ultimately show ?thesis by (auto intro!: exI[of _ "\<lambda>x. \<infinity>"])
next
assume "\<infinity> \<notin> A"
have "\<exists>x\<in>A. 0 \<le> x"
by (metis Infty_neq_0 PInf complete_lattice_class.Sup_least extreal_infty_less_eq2 linorder_linear)
then obtain x where "x \<in> A" "0 \<le> x" by auto
have "\<forall>n::nat. \<exists>f. f \<in> A \<and> x + extreal (real n) \<le> f"
proof (rule ccontr)
assume "\<not> ?thesis"
then have "\<exists>n::nat. Sup A \<le> x + extreal (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] guess f .. note f = this
have "SUPR UNIV f = \<infinity>"
proof (rule SUP_PInfty)
fix n :: nat show "\<exists>i\<in>UNIV. extreal (real n) \<le> f i"
using f[THEN spec, of n] `0 \<le> x`
by (cases rule: extreal2_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 SUPR_countable_SUPR:
"A \<noteq> {} \<Longrightarrow> \<exists>f::nat \<Rightarrow> extreal. range f \<subseteq> g`A \<and> SUPR A g = SUPR UNIV f"
using Sup_countable_SUPR[of "g`A"] by (auto simp: SUPR_def)
lemma Sup_extreal_cadd:
fixes A :: "extreal set" assumes "A \<noteq> {}" and "a \<noteq> -\<infinity>"
shows "Sup ((\<lambda>x. a + x) ` A) = a + Sup A"
proof (rule antisym)
have *: "\<And>a::extreal. \<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)
next
case (real r)
then have **: "op + (- a) ` op + a ` A = A"
by (auto simp: image_iff ac_simps zero_extreal_def[symmetric])
from *[of "-a" "(\<lambda>x. a + x) ` A"] real show ?thesis unfolding **
by (cases rule: extreal2_cases[of "Sup A" "Sup (op + a ` A)"]) auto
qed (insert `a \<noteq> -\<infinity>`, auto)
qed
lemma Sup_extreal_cminus:
fixes A :: "extreal set" assumes "A \<noteq> {}" and "a \<noteq> -\<infinity>"
shows "Sup ((\<lambda>x. a - x) ` A) = a - Inf A"
using Sup_extreal_cadd[of "uminus ` A" a] assms
by (simp add: comp_def image_image minus_extreal_def
extreal_Sup_uminus_image_eq)
lemma SUPR_extreal_cminus:
fixes A assumes "A \<noteq> {}" and "a \<noteq> -\<infinity>"
shows "(SUP x:A. a - f x) = a - (INF x:A. f x)"
using Sup_extreal_cminus[of "f`A" a] assms
unfolding SUPR_def INFI_def image_image by auto
lemma Inf_extreal_cminus:
fixes A :: "extreal 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 }
moreover then have "(\<lambda>x. -a - x)`uminus`A = uminus ` (\<lambda>x. a - x) ` A"
by (auto simp: image_image)
ultimately show ?thesis
using Sup_extreal_cminus[of "uminus ` A" "-a"] assms
by (auto simp add: extreal_Sup_uminus_image_eq extreal_Inf_uminus_image_eq)
qed
lemma INFI_extreal_cminus:
fixes A assumes "A \<noteq> {}" and "\<bar>a\<bar> \<noteq> \<infinity>"
shows "(INF x:A. a - f x) = a - (SUP x:A. f x)"
using Inf_extreal_cminus[of "f`A" a] assms
unfolding SUPR_def INFI_def image_image
by auto
subsection "Limits on @{typ extreal}"
subsubsection "Topological space"
instantiation extreal :: topological_space
begin
definition "open A \<longleftrightarrow> open (extreal -` A)
\<and> (\<infinity> \<in> A \<longrightarrow> (\<exists>x. {extreal x <..} \<subseteq> A))
\<and> (-\<infinity> \<in> A \<longrightarrow> (\<exists>x. {..<extreal x} \<subseteq> A))"
lemma open_PInfty: "open A \<Longrightarrow> \<infinity> \<in> A \<Longrightarrow> (\<exists>x. {extreal x<..} \<subseteq> A)"
unfolding open_extreal_def by auto
lemma open_MInfty: "open A \<Longrightarrow> -\<infinity> \<in> A \<Longrightarrow> (\<exists>x. {..<extreal x} \<subseteq> A)"
unfolding open_extreal_def by auto
lemma open_PInfty2: assumes "open A" "\<infinity> \<in> A" obtains x where "{extreal x<..} \<subseteq> A"
using open_PInfty[OF assms] by auto
lemma open_MInfty2: assumes "open A" "-\<infinity> \<in> A" obtains x where "{..<extreal x} \<subseteq> A"
using open_MInfty[OF assms] by auto
lemma extreal_openE: assumes "open A" obtains x y where
"open (extreal -` A)"
"\<infinity> \<in> A \<Longrightarrow> {extreal x<..} \<subseteq> A"
"-\<infinity> \<in> A \<Longrightarrow> {..<extreal y} \<subseteq> A"
using assms open_extreal_def by auto
instance
proof
let ?U = "UNIV::extreal set"
show "open ?U" unfolding open_extreal_def
by (auto intro!: exI[of _ 0])
next
fix S T::"extreal set" assume "open S" and "open T"
from `open S`[THEN extreal_openE] guess xS yS .
moreover from `open T`[THEN extreal_openE] guess xT yT .
ultimately have
"open (extreal -` (S \<inter> T))"
"\<infinity> \<in> S \<inter> T \<Longrightarrow> {extreal (max xS xT) <..} \<subseteq> S \<inter> T"
"-\<infinity> \<in> S \<inter> T \<Longrightarrow> {..< extreal (min yS yT)} \<subseteq> S \<inter> T"
by auto
then show "open (S Int T)" unfolding open_extreal_def by blast
next
fix K :: "extreal set set" assume "\<forall>S\<in>K. open S"
then have *: "\<forall>S. \<exists>x y. S \<in> K \<longrightarrow> open (extreal -` S) \<and>
(\<infinity> \<in> S \<longrightarrow> {extreal x <..} \<subseteq> S) \<and> (-\<infinity> \<in> S \<longrightarrow> {..< extreal y} \<subseteq> S)"
by (auto simp: open_extreal_def)
then show "open (Union K)" unfolding open_extreal_def
proof (intro conjI impI)
show "open (extreal -` \<Union>K)"
using *[THEN choice] by (auto simp: vimage_Union)
qed ((metis UnionE Union_upper subset_trans *)+)
qed
end
lemma open_extreal: "open S \<Longrightarrow> open (extreal ` S)"
by (auto simp: inj_vimage_image_eq open_extreal_def)
lemma open_extreal_vimage: "open S \<Longrightarrow> open (extreal -` S)"
unfolding open_extreal_def by auto
lemma open_extreal_lessThan[intro, simp]: "open {..< a :: extreal}"
proof -
have "\<And>x. extreal -` {..<extreal x} = {..< x}"
"extreal -` {..< \<infinity>} = UNIV" "extreal -` {..< -\<infinity>} = {}" by auto
then show ?thesis by (cases a) (auto simp: open_extreal_def)
qed
lemma open_extreal_greaterThan[intro, simp]:
"open {a :: extreal <..}"
proof -
have "\<And>x. extreal -` {extreal x<..} = {x<..}"
"extreal -` {\<infinity><..} = {}" "extreal -` {-\<infinity><..} = UNIV" by auto
then show ?thesis by (cases a) (auto simp: open_extreal_def)
qed
lemma extreal_open_greaterThanLessThan[intro, simp]: "open {a::extreal <..< b}"
unfolding greaterThanLessThan_def by auto
lemma closed_extreal_atLeast[simp, intro]: "closed {a :: extreal ..}"
proof -
have "- {a ..} = {..< a}" by auto
then show "closed {a ..}"
unfolding closed_def using open_extreal_lessThan by auto
qed
lemma closed_extreal_atMost[simp, intro]: "closed {.. b :: extreal}"
proof -
have "- {.. b} = {b <..}" by auto
then show "closed {.. b}"
unfolding closed_def using open_extreal_greaterThan by auto
qed
lemma closed_extreal_atLeastAtMost[simp, intro]:
shows "closed {a :: extreal .. b}"
unfolding atLeastAtMost_def by auto
lemma closed_extreal_singleton:
"closed {a :: extreal}"
by (metis atLeastAtMost_singleton closed_extreal_atLeastAtMost)
lemma extreal_open_cont_interval:
assumes "open S" "x \<in> S" "\<bar>x\<bar> \<noteq> \<infinity>"
obtains e where "e>0" "{x-e <..< x+e} \<subseteq> S"
proof-
from `open S` have "open (extreal -` S)" by (rule extreal_openE)
then obtain e where "0 < e" and e: "\<And>y. dist y (real x) < e \<Longrightarrow> extreal y \<in> S"
using assms unfolding open_dist by force
show thesis
proof (intro that subsetI)
show "0 < extreal e" using `0 < e` by auto
fix y assume "y \<in> {x - extreal e<..<x + extreal e}"
with assms obtain t where "y = extreal t" "dist t (real x) < e"
apply (cases y) by (auto simp: dist_real_def)
then show "y \<in> S" using e[of t] by auto
qed
qed
lemma extreal_open_cont_interval2:
assumes "open S" "x \<in> S" and x: "\<bar>x\<bar> \<noteq> \<infinity>"
obtains a b where "a < x" "x < b" "{a <..< b} \<subseteq> S"
proof-
guess e using extreal_open_cont_interval[OF assms] .
with that[of "x-e" "x+e"] extreal_between[OF x, of e]
show thesis by auto
qed
instance extreal :: t2_space
proof
fix x y :: extreal assume "x ~= y"
let "?P x (y::extreal)" = "EX U V. open U & open V & x : U & y : V & U Int V = {}"
{ fix x y :: extreal assume "x < y"
from extreal_dense[OF this] obtain z where z: "x < z" "z < y" by auto
have "?P x y"
apply (rule exI[of _ "{..<z}"])
apply (rule exI[of _ "{z<..}"])
using z by auto }
note * = this
from `x ~= y`
show "EX U V. open U & open V & x : U & y : V & U Int V = {}"
proof (cases rule: linorder_cases)
assume "x = y" with `x ~= y` show ?thesis by simp
next assume "x < y" from *[OF this] show ?thesis by auto
next assume "y < x" from *[OF this] show ?thesis by auto
qed
qed
subsubsection {* Convergent sequences *}
lemma lim_extreal[simp]:
"((\<lambda>n. extreal (f n)) ---> extreal x) net \<longleftrightarrow> (f ---> x) net" (is "?l = ?r")
proof (intro iffI topological_tendstoI)
fix S assume "?l" "open S" "x \<in> S"
then show "eventually (\<lambda>x. f x \<in> S) net"
using `?l`[THEN topological_tendstoD, OF open_extreal, OF `open S`]
by (simp add: inj_image_mem_iff)
next
fix S assume "?r" "open S" "extreal x \<in> S"
show "eventually (\<lambda>x. extreal (f x) \<in> S) net"
using `?r`[THEN topological_tendstoD, OF open_extreal_vimage, OF `open S`]
using `extreal x \<in> S` by auto
qed
lemma lim_real_of_extreal[simp]:
assumes lim: "(f ---> extreal x) net"
shows "((\<lambda>x. real (f x)) ---> x) net"
proof (intro topological_tendstoI)
fix S assume "open S" "x \<in> S"
then have S: "open S" "extreal x \<in> extreal ` S"
by (simp_all add: inj_image_mem_iff)
have "\<forall>x. f x \<in> extreal ` S \<longrightarrow> real (f x) \<in> S" by auto
from this lim[THEN topological_tendstoD, OF open_extreal, OF S]
show "eventually (\<lambda>x. real (f x) \<in> S) net"
by (rule eventually_mono)
qed
lemma Lim_PInfty: "f ----> \<infinity> <-> (ALL B. EX N. ALL n>=N. f n >= extreal B)" (is "?l = ?r")
proof assume ?r show ?l apply(rule topological_tendstoI)
unfolding eventually_sequentially
proof- fix S assume "open S" "\<infinity> : S"
from open_PInfty[OF this] guess B .. note B=this
from `?r`[rule_format,of "B+1"] guess N .. note N=this
show "EX N. ALL n>=N. f n : S" apply(rule_tac x=N in exI)
proof safe case goal1
have "extreal B < extreal (B + 1)" by auto
also have "... <= f n" using goal1 N by auto
finally show ?case using B by fastsimp
qed
qed
next assume ?l show ?r
proof fix B::real have "open {extreal B<..}" "\<infinity> : {extreal B<..}" by auto
from topological_tendstoD[OF `?l` this,unfolded eventually_sequentially]
guess N .. note N=this
show "EX N. ALL n>=N. extreal B <= f n" apply(rule_tac x=N in exI) using N by auto
qed
qed
lemma Lim_MInfty: "f ----> (-\<infinity>) <-> (ALL B. EX N. ALL n>=N. f n <= extreal B)" (is "?l = ?r")
proof assume ?r show ?l apply(rule topological_tendstoI)
unfolding eventually_sequentially
proof- fix S assume "open S" "(-\<infinity>) : S"
from open_MInfty[OF this] guess B .. note B=this
from `?r`[rule_format,of "B-(1::real)"] guess N .. note N=this
show "EX N. ALL n>=N. f n : S" apply(rule_tac x=N in exI)
proof safe case goal1
have "extreal (B - 1) >= f n" using goal1 N by auto
also have "... < extreal B" by auto
finally show ?case using B by fastsimp
qed
qed
next assume ?l show ?r
proof fix B::real have "open {..<extreal B}" "(-\<infinity>) : {..<extreal B}" by auto
from topological_tendstoD[OF `?l` this,unfolded eventually_sequentially]
guess N .. note N=this
show "EX N. ALL n>=N. extreal B >= f n" apply(rule_tac x=N in exI) using N by auto
qed
qed
lemma Lim_bounded_PInfty: assumes lim:"f ----> l" and "!!n. f n <= extreal B" shows "l ~= \<infinity>"
proof(rule ccontr,unfold not_not) let ?B = "B + 1" assume as:"l=\<infinity>"
from lim[unfolded this Lim_PInfty,rule_format,of "?B"]
guess N .. note N=this[rule_format,OF le_refl]
hence "extreal ?B <= extreal B" using assms(2)[of N] by(rule order_trans)
hence "extreal ?B < extreal ?B" apply (rule le_less_trans) by auto
thus False by auto
qed
lemma Lim_bounded_MInfty: assumes lim:"f ----> l" and "!!n. f n >= extreal B" shows "l ~= (-\<infinity>)"
proof(rule ccontr,unfold not_not) let ?B = "B - 1" assume as:"l=(-\<infinity>)"
from lim[unfolded this Lim_MInfty,rule_format,of "?B"]
guess N .. note N=this[rule_format,OF le_refl]
hence "extreal B <= extreal ?B" using assms(2)[of N] order_trans[of "extreal B" "f N" "extreal(B - 1)"] by blast
thus False by auto
qed
lemma tendsto_explicit:
"f ----> f0 <-> (ALL S. open S --> f0 : S --> (EX N. ALL n>=N. f n : S))"
unfolding tendsto_def eventually_sequentially by auto
lemma tendsto_obtains_N:
assumes "f ----> f0"
assumes "open S" "f0 : S"
obtains N where "ALL n>=N. f n : S"
using tendsto_explicit[of f f0] assms by auto
lemma tail_same_limit:
fixes X Y N
assumes "X ----> L" "ALL n>=N. X n = Y n"
shows "Y ----> L"
proof-
{ fix S assume "open S" and "L:S"
from this obtain N1 where "ALL n>=N1. X n : S"
using assms unfolding tendsto_def eventually_sequentially by auto
hence "ALL n>=max N N1. Y n : S" using assms by auto
hence "EX N. ALL n>=N. Y n : S" apply(rule_tac x="max N N1" in exI) by auto
}
thus ?thesis using tendsto_explicit by auto
qed
lemma Lim_bounded_PInfty2:
assumes lim:"f ----> l" and "ALL n>=N. f n <= extreal B"
shows "l ~= \<infinity>"
proof-
def g == "(%n. if n>=N then f n else extreal B)"
hence "g ----> l" using tail_same_limit[of f l N g] lim by auto
moreover have "!!n. g n <= extreal B" using g_def assms by auto
ultimately show ?thesis using Lim_bounded_PInfty by auto
qed
lemma Lim_bounded_extreal:
assumes lim:"f ----> (l :: extreal)"
and "ALL n>=M. f n <= C"
shows "l<=C"
proof-
{ assume "l=(-\<infinity>)" hence ?thesis by auto }
moreover
{ assume "~(l=(-\<infinity>))"
{ assume "C=\<infinity>" hence ?thesis by auto }
moreover
{ assume "C=(-\<infinity>)" hence "ALL n>=M. f n = (-\<infinity>)" using assms by auto
hence "l=(-\<infinity>)" using assms
tendsto_unique[OF trivial_limit_sequentially] tail_same_limit[of "\<lambda>n. -\<infinity>" "-\<infinity>" M f, OF tendsto_const] by auto
hence ?thesis by auto }
moreover
{ assume "EX B. C = extreal B"
from this obtain B where B_def: "C=extreal B" by auto
hence "~(l=\<infinity>)" using Lim_bounded_PInfty2 assms by auto
from this obtain m where m_def: "extreal m=l" using `~(l=(-\<infinity>))` by (cases l) auto
from this obtain N where N_def: "ALL n>=N. f n : {extreal(m - 1) <..< extreal(m+1)}"
apply (subst tendsto_obtains_N[of f l "{extreal(m - 1) <..< extreal(m+1)}"]) using assms by auto
{ fix n assume "n>=N"
hence "EX r. extreal r = f n" using N_def by (cases "f n") auto
} from this obtain g where g_def: "ALL n>=N. extreal (g n) = f n" by metis
hence "(%n. extreal (g n)) ----> l" using tail_same_limit[of f l N] assms by auto
hence *: "(%n. g n) ----> m" using m_def by auto
{ fix n assume "n>=max N M"
hence "extreal (g n) <= extreal B" using assms g_def B_def by auto
hence "g n <= B" by auto
} hence "EX N. ALL n>=N. g n <= B" by blast
hence "m<=B" using * LIMSEQ_le_const2[of g m B] by auto
hence ?thesis using m_def B_def by auto
} ultimately have ?thesis by (cases C) auto
} ultimately show ?thesis by blast
qed
lemma real_of_extreal_0[simp]: "real (0::extreal) = 0"
unfolding real_of_extreal_def zero_extreal_def by simp
lemma real_of_extreal_mult[simp]:
fixes a b :: extreal shows "real (a * b) = real a * real b"
by (cases rule: extreal2_cases[of a b]) auto
lemma real_of_extreal_eq_0:
"real x = 0 \<longleftrightarrow> x = \<infinity> \<or> x = -\<infinity> \<or> x = 0"
by (cases x) auto
lemma tendsto_extreal_realD:
fixes f :: "'a \<Rightarrow> extreal"
assumes "x \<noteq> 0" and tendsto: "((\<lambda>x. extreal (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: extreal_real real_of_extreal_0)
qed
lemma tendsto_extreal_realI:
fixes f :: "'a \<Rightarrow> extreal"
assumes x: "\<bar>x\<bar> \<noteq> \<infinity>" and tendsto: "(f ---> x) net"
shows "((\<lambda>x. extreal (real (f x))) ---> x) net"
proof (intro topological_tendstoI)
fix S assume "open S" "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. extreal (real (f x)) \<in> S) net"
by (elim eventually_elim1) (auto simp: extreal_real)
qed
lemma extreal_mult_cancel_left:
fixes a b c :: extreal shows "a * b = a * c \<longleftrightarrow>
((\<bar>a\<bar> = \<infinity> \<and> 0 < b * c) \<or> a = 0 \<or> b = c)"
by (cases rule: extreal3_cases[of a b c])
(simp_all add: zero_less_mult_iff)
lemma extreal_inj_affinity:
assumes "\<bar>m\<bar> \<noteq> \<infinity>" "m \<noteq> 0" "\<bar>t\<bar> \<noteq> \<infinity>"
shows "inj_on (\<lambda>x. m * x + t) A"
using assms
by (cases rule: extreal2_cases[of m t])
(auto intro!: inj_onI simp: extreal_add_cancel_right extreal_mult_cancel_left)
lemma extreal_PInfty_eq_plus[simp]:
shows "\<infinity> = a + b \<longleftrightarrow> a = \<infinity> \<or> b = \<infinity>"
by (cases rule: extreal2_cases[of a b]) auto
lemma extreal_MInfty_eq_plus[simp]:
shows "-\<infinity> = a + b \<longleftrightarrow> (a = -\<infinity> \<and> b \<noteq> \<infinity>) \<or> (b = -\<infinity> \<and> a \<noteq> \<infinity>)"
by (cases rule: extreal2_cases[of a b]) auto
lemma extreal_less_divide_pos:
"x > 0 \<Longrightarrow> x \<noteq> \<infinity> \<Longrightarrow> y < z / x \<longleftrightarrow> x * y < z"
by (cases rule: extreal3_cases[of x y z]) (auto simp: field_simps)
lemma extreal_divide_less_pos:
"x > 0 \<Longrightarrow> x \<noteq> \<infinity> \<Longrightarrow> y / x < z \<longleftrightarrow> y < x * z"
by (cases rule: extreal3_cases[of x y z]) (auto simp: field_simps)
lemma extreal_divide_eq:
"b \<noteq> 0 \<Longrightarrow> \<bar>b\<bar> \<noteq> \<infinity> \<Longrightarrow> a / b = c \<longleftrightarrow> a = b * c"
by (cases rule: extreal3_cases[of a b c])
(simp_all add: field_simps)
lemma extreal_inverse_not_MInfty[simp]: "inverse a \<noteq> -\<infinity>"
by (cases a) auto
lemma extreal_mult_m1[simp]: "x * extreal (-1) = -x"
by (cases x) auto
lemma extreal_LimI_finite:
assumes "\<bar>x\<bar> \<noteq> \<infinity>"
assumes "!!r. 0 < r ==> EX N. ALL n>=N. u n < x + r & x < u n + r"
shows "u ----> x"
proof (rule topological_tendstoI, unfold eventually_sequentially)
obtain rx where rx_def: "x=extreal rx" using assms by (cases x) auto
fix S assume "open S" "x : S"
then have "open (extreal -` S)" unfolding open_extreal_def by auto
with `x \<in> S` obtain r where "0 < r" and dist: "!!y. dist y rx < r ==> extreal y \<in> S"
unfolding open_real_def rx_def by auto
then obtain n where
upper: "!!N. n <= N ==> u N < x + extreal r" and
lower: "!!N. n <= N ==> x < u N + extreal r" using assms(2)[of "extreal r"] by auto
show "EX N. ALL 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 ~: {\<infinity>,(-\<infinity>)}" by auto
from this obtain ra where ra_def: "(u N) = extreal ra" by (cases "u N") auto
hence "rx < ra + r" and "ra < rx + r"
using rx_def assms `0 < r` lower[OF `n <= N`] upper[OF `n <= N`] by auto
hence "dist (real (u N)) rx < r"
using rx_def 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 ~: {\<infinity>, -\<infinity>}`
by (auto simp: extreal_real split: split_if_asm)
qed
qed
lemma extreal_LimI_finite_iff:
assumes "\<bar>x\<bar> \<noteq> \<infinity>"
shows "u ----> x <-> (ALL r. 0 < r --> (EX N. ALL n>=N. u n < x + r & x < u n + r))"
(is "?lhs <-> ?rhs")
proof
assume lim: "u ----> x"
{ fix r assume "(r::extreal)>0"
from this obtain N where N_def: "ALL n>=N. u n : {x - r <..< x + r}"
apply (subst tendsto_obtains_N[of u x "{x - r <..< x + r}"])
using lim extreal_between[of x r] assms `r>0` by auto
hence "EX N. ALL n>=N. u n < x + r & x < u n + r"
using extreal_minus_less[of r x] by (cases r) auto
} then show "?rhs" by auto
next
assume ?rhs then show "u ----> x"
using extreal_LimI_finite[of x] assms by auto
qed
subsubsection {* @{text Liminf} and @{text Limsup} *}
definition
"Liminf net f = (GREATEST l. \<forall>y<l. eventually (\<lambda>x. y < f x) net)"
definition
"Limsup net f = (LEAST l. \<forall>y>l. eventually (\<lambda>x. f x < y) net)"
lemma Liminf_Sup:
fixes f :: "'a => 'b::{complete_lattice, linorder}"
shows "Liminf net f = Sup {l. \<forall>y<l. eventually (\<lambda>x. y < f x) net}"
by (auto intro!: Greatest_equality complete_lattice_class.Sup_upper simp: less_Sup_iff Liminf_def)
lemma Limsup_Inf:
fixes f :: "'a => 'b::{complete_lattice, linorder}"
shows "Limsup net f = Inf {l. \<forall>y>l. eventually (\<lambda>x. f x < y) net}"
by (auto intro!: Least_equality complete_lattice_class.Inf_lower simp: Inf_less_iff Limsup_def)
lemma extreal_SupI:
fixes x :: extreal
assumes "\<And>y. y \<in> A \<Longrightarrow> y \<le> x"
assumes "\<And>y. (\<And>z. z \<in> A \<Longrightarrow> z \<le> y) \<Longrightarrow> x \<le> y"
shows "Sup A = x"
unfolding Sup_extreal_def
using assms by (auto intro!: Least_equality)
lemma extreal_InfI:
fixes x :: extreal
assumes "\<And>i. i \<in> A \<Longrightarrow> x \<le> i"
assumes "\<And>y. (\<And>i. i \<in> A \<Longrightarrow> y \<le> i) \<Longrightarrow> y \<le> x"
shows "Inf A = x"
unfolding Inf_extreal_def
using assms by (auto intro!: Greatest_equality)
lemma Limsup_const:
fixes c :: "'a::{complete_lattice, linorder}"
assumes ntriv: "\<not> trivial_limit net"
shows "Limsup net (\<lambda>x. c) = c"
unfolding Limsup_Inf
proof (safe intro!: antisym complete_lattice_class.Inf_greatest complete_lattice_class.Inf_lower)
fix x assume *: "\<forall>y>x. eventually (\<lambda>_. c < y) net"
show "c \<le> x"
proof (rule ccontr)
assume "\<not> c \<le> x" then have "x < c" by auto
then show False using ntriv * by (auto simp: trivial_limit_def)
qed
qed auto
lemma Liminf_const:
fixes c :: "'a::{complete_lattice, linorder}"
assumes ntriv: "\<not> trivial_limit net"
shows "Liminf net (\<lambda>x. c) = c"
unfolding Liminf_Sup
proof (safe intro!: antisym complete_lattice_class.Sup_least complete_lattice_class.Sup_upper)
fix x assume *: "\<forall>y<x. eventually (\<lambda>_. y < c) net"
show "x \<le> c"
proof (rule ccontr)
assume "\<not> x \<le> c" then have "c < x" by auto
then show False using ntriv * by (auto simp: trivial_limit_def)
qed
qed auto
lemma mono_set:
fixes S :: "('a::order) set"
shows "mono S \<longleftrightarrow> (\<forall>x y. x \<le> y \<longrightarrow> x \<in> S \<longrightarrow> y \<in> S)"
by (auto simp: mono_def mem_def)
lemma mono_greaterThan[intro, simp]: "mono {B<..}" unfolding mono_set by auto
lemma mono_atLeast[intro, simp]: "mono {B..}" unfolding mono_set by auto
lemma mono_UNIV[intro, simp]: "mono UNIV" unfolding mono_set by auto
lemma mono_empty[intro, simp]: "mono {}" unfolding mono_set by auto
lemma mono_set_iff:
fixes S :: "'a::{linorder,complete_lattice} set"
defines "a \<equiv> Inf S"
shows "mono S \<longleftrightarrow> (S = {a <..} \<or> S = {a..})" (is "_ = ?c")
proof
assume "mono S"
then have mono: "\<And>x y. x \<le> y \<Longrightarrow> x \<in> S \<Longrightarrow> y \<in> S" by (auto simp: mono_set)
show ?c
proof cases
assume "a \<in> S"
show ?c
using mono[OF _ `a \<in> S`]
by (auto intro: complete_lattice_class.Inf_lower simp: a_def)
next
assume "a \<notin> S"
have "S = {a <..}"
proof safe
fix x assume "x \<in> S"
then have "a \<le> x" unfolding a_def by (rule complete_lattice_class.Inf_lower)
then show "a < x" using `x \<in> S` `a \<notin> S` by (cases "a = x") auto
next
fix x assume "a < x"
then obtain y where "y < x" "y \<in> S" unfolding a_def Inf_less_iff ..
with mono[of y x] show "x \<in> S" by auto
qed
then show ?c ..
qed
qed auto
lemma lim_imp_Liminf:
fixes f :: "'a \<Rightarrow> extreal"
assumes ntriv: "\<not> trivial_limit net"
assumes lim: "(f ---> f0) net"
shows "Liminf net f = f0"
unfolding Liminf_Sup
proof (safe intro!: extreal_SupI)
fix y assume *: "\<forall>y'<y. eventually (\<lambda>x. y' < f x) net"
show "y \<le> f0"
proof (rule extreal_le_extreal)
fix B assume "B < y"
{ assume "f0 < B"
then have "eventually (\<lambda>x. f x < B \<and> B < f x) net"
using topological_tendstoD[OF lim, of "{..<B}"] *[rule_format, OF `B < y`]
by (auto intro: eventually_conj)
also have "(\<lambda>x. f x < B \<and> B < f x) = (\<lambda>x. False)" by (auto simp: fun_eq_iff)
finally have False using ntriv[unfolded trivial_limit_def] by auto
} then show "B \<le> f0" by (metis linorder_le_less_linear)
qed
next
fix y assume *: "\<forall>z. z \<in> {l. \<forall>y<l. eventually (\<lambda>x. y < f x) net} \<longrightarrow> z \<le> y"
show "f0 \<le> y"
proof (safe intro!: *[rule_format])
fix y assume "y < f0" then show "eventually (\<lambda>x. y < f x) net"
using lim[THEN topological_tendstoD, of "{y <..}"] by auto
qed
qed
lemma extreal_Liminf_le_Limsup:
fixes f :: "'a \<Rightarrow> extreal"
assumes ntriv: "\<not> trivial_limit net"
shows "Liminf net f \<le> Limsup net f"
unfolding Limsup_Inf Liminf_Sup
proof (safe intro!: complete_lattice_class.Inf_greatest complete_lattice_class.Sup_least)
fix u v assume *: "\<forall>y<u. eventually (\<lambda>x. y < f x) net" "\<forall>y>v. eventually (\<lambda>x. f x < y) net"
show "u \<le> v"
proof (rule ccontr)
assume "\<not> u \<le> v"
then obtain t where "t < u" "v < t"
using extreal_dense[of v u] by (auto simp: not_le)
then have "eventually (\<lambda>x. t < f x \<and> f x < t) net"
using * by (auto intro: eventually_conj)
also have "(\<lambda>x. t < f x \<and> f x < t) = (\<lambda>x. False)" by (auto simp: fun_eq_iff)
finally show False using ntriv by (auto simp: trivial_limit_def)
qed
qed
lemma Liminf_mono:
fixes f g :: "'a => extreal"
assumes ev: "eventually (\<lambda>x. f x \<le> g x) net"
shows "Liminf net f \<le> Liminf net g"
unfolding Liminf_Sup
proof (safe intro!: Sup_mono bexI)
fix a y assume "\<forall>y<a. eventually (\<lambda>x. y < f x) net" and "y < a"
then have "eventually (\<lambda>x. y < f x) net" by auto
then show "eventually (\<lambda>x. y < g x) net"
by (rule eventually_rev_mp) (rule eventually_mono[OF _ ev], auto)
qed simp
lemma Liminf_eq:
fixes f g :: "'a \<Rightarrow> extreal"
assumes "eventually (\<lambda>x. f x = g x) net"
shows "Liminf net f = Liminf net g"
by (intro antisym Liminf_mono eventually_mono[OF _ assms]) auto
lemma Liminf_mono_all:
fixes f g :: "'a \<Rightarrow> extreal"
assumes "\<And>x. f x \<le> g x"
shows "Liminf net f \<le> Liminf net g"
using assms by (intro Liminf_mono always_eventually) auto
lemma Limsup_mono:
fixes f g :: "'a \<Rightarrow> extreal"
assumes ev: "eventually (\<lambda>x. f x \<le> g x) net"
shows "Limsup net f \<le> Limsup net g"
unfolding Limsup_Inf
proof (safe intro!: Inf_mono bexI)
fix a y assume "\<forall>y>a. eventually (\<lambda>x. g x < y) net" and "a < y"
then have "eventually (\<lambda>x. g x < y) net" by auto
then show "eventually (\<lambda>x. f x < y) net"
by (rule eventually_rev_mp) (rule eventually_mono[OF _ ev], auto)
qed simp
lemma Limsup_mono_all:
fixes f g :: "'a \<Rightarrow> extreal"
assumes "\<And>x. f x \<le> g x"
shows "Limsup net f \<le> Limsup net g"
using assms by (intro Limsup_mono always_eventually) auto
lemma Limsup_eq:
fixes f g :: "'a \<Rightarrow> extreal"
assumes "eventually (\<lambda>x. f x = g x) net"
shows "Limsup net f = Limsup net g"
by (intro antisym Limsup_mono eventually_mono[OF _ assms]) auto
abbreviation "liminf \<equiv> Liminf sequentially"
abbreviation "limsup \<equiv> Limsup sequentially"
lemma (in complete_lattice) less_INFD:
assumes "y < INFI A f"" i \<in> A" shows "y < f i"
proof -
note `y < INFI A f`
also have "INFI A f \<le> f i" using `i \<in> A` by (rule INF_leI)
finally show "y < f i" .
qed
lemma liminf_SUPR_INFI:
fixes f :: "nat \<Rightarrow> extreal"
shows "liminf f = (SUP n. INF m:{n..}. f m)"
unfolding Liminf_Sup eventually_sequentially
proof (safe intro!: antisym complete_lattice_class.Sup_least)
fix x assume *: "\<forall>y<x. \<exists>N. \<forall>n\<ge>N. y < f n" show "x \<le> (SUP n. INF m:{n..}. f m)"
proof (rule extreal_le_extreal)
fix y assume "y < x"
with * obtain N where "\<And>n. N \<le> n \<Longrightarrow> y < f n" by auto
then have "y \<le> (INF m:{N..}. f m)" by (force simp: le_INF_iff)
also have "\<dots> \<le> (SUP n. INF m:{n..}. f m)" by (intro le_SUPI) auto
finally show "y \<le> (SUP n. INF m:{n..}. f m)" .
qed
next
show "(SUP n. INF m:{n..}. f m) \<le> Sup {l. \<forall>y<l. \<exists>N. \<forall>n\<ge>N. y < f n}"
proof (unfold SUPR_def, safe intro!: Sup_mono bexI)
fix y n assume "y < INFI {n..} f"
from less_INFD[OF this] show "\<exists>N. \<forall>n\<ge>N. y < f n" by (intro exI[of _ n]) auto
qed (rule order_refl)
qed
lemma tail_same_limsup:
fixes X Y :: "nat => extreal"
assumes "\<And>n. N \<le> n \<Longrightarrow> X n = Y n"
shows "limsup X = limsup Y"
using Limsup_eq[of X Y sequentially] eventually_sequentially assms by auto
lemma tail_same_liminf:
fixes X Y :: "nat => extreal"
assumes "\<And>n. N \<le> n \<Longrightarrow> X n = Y n"
shows "liminf X = liminf Y"
using Liminf_eq[of X Y sequentially] eventually_sequentially assms by auto
lemma liminf_mono:
fixes X Y :: "nat \<Rightarrow> extreal"
assumes "\<And>n. N \<le> n \<Longrightarrow> X n <= Y n"
shows "liminf X \<le> liminf Y"
using Liminf_mono[of X Y sequentially] eventually_sequentially assms by auto
lemma limsup_mono:
fixes X Y :: "nat => extreal"
assumes "\<And>n. N \<le> n \<Longrightarrow> X n <= Y n"
shows "limsup X \<le> limsup Y"
using Limsup_mono[of X Y sequentially] eventually_sequentially assms by auto
declare trivial_limit_sequentially[simp]
lemma
fixes X :: "nat \<Rightarrow> extreal"
shows extreal_incseq_uminus[simp]: "incseq (\<lambda>i. - X i) = decseq X"
and extreal_decseq_uminus[simp]: "decseq (\<lambda>i. - X i) = incseq X"
unfolding incseq_def decseq_def by auto
lemma liminf_bounded:
fixes X Y :: "nat \<Rightarrow> extreal"
assumes "\<And>n. N \<le> n \<Longrightarrow> C \<le> X n"
shows "C \<le> liminf X"
using liminf_mono[of N "\<lambda>n. C" X] assms Liminf_const[of sequentially C] by simp
lemma limsup_bounded:
fixes X Y :: "nat => extreal"
assumes "\<And>n. N \<le> n \<Longrightarrow> X n <= C"
shows "limsup X \<le> C"
using limsup_mono[of N X "\<lambda>n. C"] assms Limsup_const[of sequentially C] by simp
lemma liminf_bounded_iff:
fixes x :: "nat \<Rightarrow> extreal"
shows "C \<le> liminf x \<longleftrightarrow> (\<forall>B<C. \<exists>N. \<forall>n\<ge>N. B < x n)" (is "?lhs <-> ?rhs")
proof safe
fix B assume "B < C" "C \<le> liminf x"
then have "B < liminf x" by auto
then obtain N where "B < (INF m:{N..}. x m)"
unfolding liminf_SUPR_INFI SUPR_def less_Sup_iff by auto
from less_INFD[OF this] show "\<exists>N. \<forall>n\<ge>N. B < x n" by auto
next
assume *: "\<forall>B<C. \<exists>N. \<forall>n\<ge>N. B < x n"
{ fix B assume "B<C"
then obtain N where "\<forall>n\<ge>N. B < x n" using `?rhs` by auto
hence "B \<le> (INF m:{N..}. x m)" by (intro le_INFI) auto
also have "... \<le> liminf x" unfolding liminf_SUPR_INFI by (intro le_SUPI) simp
finally have "B \<le> liminf x" .
} then show "?lhs" by (metis * leD liminf_bounded linorder_le_less_linear)
qed
lemma liminf_subseq_mono:
fixes X :: "nat \<Rightarrow> extreal"
assumes "subseq r"
shows "liminf X \<le> liminf (X \<circ> r) "
proof-
have "\<And>n. (INF m:{n..}. X m) \<le> (INF m:{n..}. (X \<circ> r) m)"
proof (safe intro!: INF_mono)
fix n m :: nat assume "n \<le> m" then show "\<exists>ma\<in>{n..}. X ma \<le> (X \<circ> r) m"
using seq_suble[OF `subseq r`, of m] by (intro bexI[of _ "r m"]) auto
qed
then show ?thesis by (auto intro!: SUP_mono simp: liminf_SUPR_INFI comp_def)
qed
lemma extreal_real': assumes "\<bar>x\<bar> \<noteq> \<infinity>" shows "extreal (real x) = x"
using assms by auto
lemma extreal_le_extreal_bounded:
fixes x y z :: extreal
assumes "z \<le> y"
assumes *: "\<And>B. z < B \<Longrightarrow> B < x \<Longrightarrow> B \<le> y"
shows "x \<le> y"
proof (rule extreal_le_extreal)
fix B assume "B < x"
show "B \<le> y"
proof cases
assume "z < B" from *[OF this `B < x`] show "B \<le> y" .
next
assume "\<not> z < B" with `z \<le> y` show "B \<le> y" by auto
qed
qed
lemma fixes x y :: extreal
shows Sup_atMost[simp]: "Sup {.. y} = y"
and Sup_lessThan[simp]: "Sup {..< y} = y"
and Sup_atLeastAtMost[simp]: "x \<le> y \<Longrightarrow> Sup { x .. y} = y"
and Sup_greaterThanAtMost[simp]: "x < y \<Longrightarrow> Sup { x <.. y} = y"
and Sup_atLeastLessThan[simp]: "x < y \<Longrightarrow> Sup { x ..< y} = y"
by (auto simp: Sup_extreal_def intro!: Least_equality
intro: extreal_le_extreal extreal_le_extreal_bounded[of x])
lemma Sup_greaterThanlessThan[simp]:
fixes x y :: extreal assumes "x < y" shows "Sup { x <..< y} = y"
unfolding Sup_extreal_def
proof (intro Least_equality extreal_le_extreal_bounded[of _ _ y])
fix z assume z: "\<forall>u\<in>{x<..<y}. u \<le> z"
from extreal_dense[OF `x < y`] guess w .. note w = this
with z[THEN bspec, of w] show "x \<le> z" by auto
qed auto
lemma real_extreal_id: "real o extreal = id"
proof-
{ fix x have "(real o extreal) x = id x" by auto }
from this show ?thesis using ext by blast
qed
lemma open_image_extreal: "open(UNIV-{\<infinity>,(-\<infinity>)})"
by (metis range_extreal open_extreal open_UNIV)
lemma extreal_le_distrib:
fixes a b c :: extreal shows "c * (a + b) \<le> c * a + c * b"
by (cases rule: extreal3_cases[of a b c])
(auto simp add: field_simps not_le mult_le_0_iff mult_less_0_iff)
lemma extreal_pos_distrib:
fixes a b c :: extreal assumes "0 \<le> c" "c \<noteq> \<infinity>" shows "c * (a + b) = c * a + c * b"
using assms by (cases rule: extreal3_cases[of a b c])
(auto simp add: field_simps not_le mult_le_0_iff mult_less_0_iff)
lemma extreal_pos_le_distrib:
fixes a b c :: extreal
assumes "c>=0"
shows "c * (a + b) <= c * a + c * b"
using assms by (cases rule: extreal3_cases[of a b c])
(auto simp add: field_simps)
lemma extreal_max_mono:
"[| (a::extreal) <= b; c <= d |] ==> max a c <= max b d"
by (metis sup_extreal_def sup_mono)
lemma extreal_max_least:
"[| (a::extreal) <= x; c <= x |] ==> max a c <= x"
by (metis sup_extreal_def sup_least)
end