--- a/src/HOL/Library/Extended_Real.thy Wed Sep 25 12:29:06 2013 +0200
+++ b/src/HOL/Library/Extended_Real.thy Wed Sep 25 12:52:21 2013 +0200
@@ -24,23 +24,29 @@
instantiation ereal :: uminus
begin
- fun uminus_ereal where
- "- (ereal r) = ereal (- r)"
- | "- PInfty = MInfty"
- | "- MInfty = PInfty"
- instance ..
+
+fun uminus_ereal where
+ "- (ereal r) = ereal (- r)"
+| "- PInfty = MInfty"
+| "- MInfty = PInfty"
+
+instance ..
+
end
instantiation ereal :: infinity
begin
- definition "(\<infinity>::ereal) = PInfty"
- instance ..
+
+definition "(\<infinity>::ereal) = PInfty"
+instance ..
+
end
declare [[coercion "ereal :: real \<Rightarrow> ereal"]]
lemma ereal_uminus_uminus[simp]:
- fixes a :: ereal shows "- (- a) = a"
+ fixes a :: ereal
+ shows "- (- a) = a"
by (cases a) simp_all
lemma
@@ -59,7 +65,7 @@
lemma [code_unfold]:
"\<infinity> = PInfty"
- "-PInfty = MInfty"
+ "- PInfty = MInfty"
by simp_all
lemma inj_ereal[simp]: "inj_on ereal A"
@@ -76,77 +82,97 @@
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"
+ 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"
+ "of_ereal (ereal r) = r"
+| "of_ereal \<infinity> = 0"
+| "of_ereal (-\<infinity>) = 0"
by (auto intro: ereal_cases)
-termination proof qed (rule wf_empty)
+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"
+ "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
+ 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
+ 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 ..
+
+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!]: "\<lbrakk> \<bar>x :: ereal\<bar> = \<infinity> ; x = \<infinity> \<Longrightarrow> P ; x = -\<infinity> \<Longrightarrow> P \<rbrakk> \<Longrightarrow> P"
- by (cases x) auto
+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!]: "\<lbrakk> \<bar>x :: ereal\<bar> \<noteq> \<infinity> ; \<And>r. x = ereal r \<Longrightarrow> P \<rbrakk> \<Longrightarrow> P"
+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 abs_ereal_uminus[simp]: "\<bar>- x\<bar> = \<bar>x::ereal\<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
-lemma ereal_infinity_cases: "(a::ereal) \<noteq> \<infinity> \<Longrightarrow> a \<noteq> -\<infinity> \<Longrightarrow> \<bar>a\<bar> \<noteq> \<infinity>"
- by auto
subsubsection "Addition"
-instantiation ereal :: "{one, comm_monoid_add}"
+instantiation ereal :: "{one,comm_monoid_add}"
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)"
+ "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
+ 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
@@ -172,6 +198,7 @@
show "a + b + c = a + (b + c)"
by (cases rule: ereal3_cases[of a b c]) simp_all
qed
+
end
instance ereal :: numeral ..
@@ -182,35 +209,37 @@
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)"
+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"
+ 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>"
+ 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>"
+ 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)"
+ 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)"
+ 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)"
+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:
@@ -219,6 +248,7 @@
(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
@@ -250,7 +280,7 @@
lemma ereal_infty_less_eq[simp]:
fixes x :: ereal
shows "\<infinity> \<le> x \<longleftrightarrow> x = \<infinity>"
- "x \<le> -\<infinity> \<longleftrightarrow> x = -\<infinity>"
+ and "x \<le> -\<infinity> \<longleftrightarrow> x = -\<infinity>"
by (auto simp add: less_eq_ereal_def)
lemma ereal_less[simp]:
@@ -282,10 +312,16 @@
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 }
+ {
+ 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
@@ -298,20 +334,25 @@
instance ereal :: ordered_ab_semigroup_add
proof
- fix a b c :: ereal assume "a \<le> b" then show "c + a \<le> c + b"
+ 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 "\<lbrakk>0 \<le> x; x \<le> y; y \<noteq> \<infinity>\<rbrakk> \<Longrightarrow> real x \<le> real y"
+ 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"
+ 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>"
+ fixes a :: ereal
+ shows "a \<noteq> \<infinity> \<Longrightarrow> a < \<infinity>"
by (cases a) auto
lemma ereal_less_ereal_Ex:
@@ -321,12 +362,16 @@
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
+ 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" "c \<le> d" shows "a + c \<le> b + d"
+ 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)
@@ -334,31 +379,34 @@
done
lemma ereal_minus_le_minus[simp]:
- fixes a b :: ereal shows "- a \<le> - b \<longleftrightarrow> b \<le> a"
+ 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"
+ 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))"
+ "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))"
+ "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))"
+ "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))"
+ "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
+ 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
@@ -372,35 +420,44 @@
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>)"
+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>)"
+ 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"
+ 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"
+ 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 = b" "0 \<le> a" "a \<noteq> \<infinity>" "c < d"
+ assumes "a = 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
+ 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"
+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_uminus_eq_reorder: "- a = b \<longleftrightarrow> a = (-b::ereal)" by 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)
@@ -412,23 +469,39 @@
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>"
+ 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
+ case (real r)
+ with assms[of "r - 1"] show ?thesis
+ by auto
next
- case PInf with assms[of 0] show ?thesis by auto
+ case PInf
+ with assms[of 0] show ?thesis
+ by auto
next
- case MInf then show ?thesis by simp
+ case MInf
+ then show ?thesis
+ by simp
qed
lemma ereal_top:
- fixes x :: ereal assumes "\<And>B. x \<ge> ereal B" shows "x = \<infinity>"
+ 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
+ case (real r)
+ with assms[of "r + 1"] show ?thesis
+ by auto
next
- case MInf with assms[of 0] show ?thesis by auto
+ case MInf
+ with assms[of 0] show ?thesis
+ by auto
next
- case PInf then show ?thesis by simp
+ case PInf
+ then show ?thesis
+ by simp
qed
lemma
@@ -449,32 +522,36 @@
unfolding incseq_def by auto
lemma ereal_add_nonneg_nonneg:
- fixes a b :: ereal shows "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> a + b"
+ 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)"
+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}"
+ 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"
+ 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}"
+ 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)
+ using assms
+ unfolding incseq_def by (auto intro: setsum_mono)
+
subsubsection "Multiplication"
-instantiation ereal :: "{comm_monoid_mult, sgn}"
+instantiation ereal :: "{comm_monoid_mult,sgn}"
begin
function sgn_ereal :: "ereal \<Rightarrow> ereal" where
@@ -482,28 +559,31 @@
| "sgn (\<infinity>::ereal) = 1"
| "sgn (-\<infinity>::ereal) = -1"
by (auto intro: ereal_cases)
-termination proof qed (rule wf_empty)
+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>"
+ "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
+ 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"
+ 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
@@ -511,36 +591,39 @@
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"
+ 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"
+ 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"
+ fixes a :: ereal
+ shows "0 * a = 0"
by (cases a) (simp_all add: zero_ereal_def)
-lemma ereal_m1_less_0[simp]:
- "-(1::ereal) < 0"
+lemma ereal_m1_less_0[simp]: "-(1::ereal) < 0"
by (simp add: zero_ereal_def one_ereal_def)
-lemma ereal_zero_m1[simp]:
- "1 \<noteq> (0::ereal)"
+lemma ereal_zero_m1[simp]: "1 \<noteq> (0::ereal)"
by (simp add: zero_ereal_def one_ereal_def)
lemma ereal_times_0[simp]:
- fixes x :: ereal shows "0 * x = 0"
+ fixes x :: ereal
+ shows "0 * x = 0"
by (cases x) (auto simp: zero_ereal_def)
lemma ereal_times[simp]:
@@ -549,21 +632,24 @@
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>"
+ "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"
+ 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]:
- shows "a * b = (\<infinity>::ereal) \<longleftrightarrow>
+ "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]:
- shows "a * b = -(\<infinity>::ereal) \<longleftrightarrow>
+ "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
@@ -574,11 +660,13 @@
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)"
+ 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)"
+ fixes a b :: ereal
+ shows "a * -b = - (a * b)"
by (cases rule: ereal2_cases[of a b]) auto
lemma ereal_mult_infty[simp]:
@@ -590,26 +678,33 @@
by (cases a) auto
lemma ereal_mult_strict_right_mono:
- assumes "a < b" and "0 < c" "c < (\<infinity>::ereal)"
+ 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)
+ by (cases rule: ereal3_cases[of a b c]) (auto simp: zero_le_mult_iff)
lemma ereal_mult_strict_left_mono:
- "\<lbrakk> a < b ; 0 < c ; c < (\<infinity>::ereal)\<rbrakk> \<Longrightarrow> c * a < c * b"
- using ereal_mult_strict_right_mono by (simp add: mult_commute[of c])
+ "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 "\<lbrakk>a \<le> b; 0 \<le> c\<rbrakk> \<Longrightarrow> a*c \<le> b*c"
+ 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
- by (cases rule: ereal3_cases[of a b c])
- (auto simp: zero_le_mult_iff)
+ 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 "\<lbrakk>a \<le> b; 0 \<le> c\<rbrakk> \<Longrightarrow> c * a \<le> c * b"
- using ereal_mult_right_mono by (simp add: mult_commute[of c])
+ 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)
@@ -618,11 +713,13 @@
by (cases rule: ereal2_cases[of a b]) (auto simp: mult_nonneg_nonneg)
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"
+ 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"
+ 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:
@@ -657,7 +754,9 @@
lemma ereal_distrib:
fixes a b c :: ereal
- assumes "a \<noteq> \<infinity> \<or> b \<noteq> -\<infinity>" "a \<noteq> -\<infinity> \<or> b \<noteq> \<infinity>" "\<bar>c\<bar> \<noteq> \<infinity>"
+ 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)
@@ -670,74 +769,119 @@
lemma ereal_le_epsilon:
fixes x y :: ereal
- assumes "ALL e. 0 < e --> x <= y + e"
- shows "x <= y"
-proof-
-{ assume a: "EX r. y = ereal r"
- then obtain r where r_def: "y = ereal r" by auto
- { assume "x=(-\<infinity>)" hence ?thesis by auto }
+ 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 "~(x=(-\<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 --> p <= r + e"
- using assms[rule_format, of "ereal 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
+ {
+ 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 "ALL e. 0 < e --> x <= y + ereal e"
- shows "x <= y"
-proof-
-{ fix e :: ereal assume "e>0"
- { assume "e=\<infinity>" hence "x<=y+e" by auto }
- moreover
- { assume "e~=\<infinity>"
- then obtain r where "e = ereal 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
-} then show ?thesis using ereal_le_epsilon by auto
+ 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 "ALL z. x <= ereal z --> y <= ereal z"
- shows "y <= x"
-by (metis assms ereal_bot ereal_cases ereal_infty_less_eq(2) ereal_less_eq(1) linorder_le_cases)
+ 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
- assume "finite A"
+ 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
-qed 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
- assume "finite I" from this pos show ?thesis by induct auto
-qed simp
+ 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
- assume "finite I" from this assms show ?thesis
+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
+ 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
@@ -745,13 +889,22 @@
using insert by (auto simp: setprod_ereal_0)
finally show ?case .
qed simp
-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
- assume "finite A" then show ?thesis
+proof (cases "finite A")
+ case True
+ then show ?thesis
by induct (auto simp: one_ereal_def)
-qed (simp add: one_ereal_def)
+next
+ case False
+ then show ?thesis
+ by (simp add: one_ereal_def)
+qed
+
subsubsection {* Power *}
@@ -771,10 +924,12 @@
by (induct n) (auto simp: one_ereal_def)
lemma zero_le_power_ereal[simp]:
- fixes a :: ereal assumes "0 \<le> a"
+ 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]:
@@ -783,25 +938,30 @@
by (auto simp: image_iff)
lemma ereal_uminus_lessThan[simp]:
- fixes a :: ereal shows "uminus ` {..<a} = {-a<..}"
+ 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
+ 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 (auto intro!: image_eqI)
+ then show ?thesis
+ by (auto intro!: image_eqI)
qed
-lemma ereal_uminus_greaterThan[simp]:
- "uminus ` {(a::ereal)<..} = {..<-a}"
- by (metis ereal_uminus_lessThan ereal_uminus_uminus
- ereal_minus_minus_image)
+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]:
@@ -815,8 +975,7 @@
"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)"
+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:
@@ -848,9 +1007,7 @@
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)"
+ 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:
@@ -860,9 +1017,7 @@
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)"
+ 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:
@@ -917,31 +1072,40 @@
lemma ereal_between:
fixes x e :: ereal
- 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 apply (cases x, cases e) apply auto
-done
+ 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>"
+ 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"
+ "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]:
@@ -959,53 +1123,61 @@
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)
+ 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>)"
+ 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)"
+ 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)"
+ 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"
+ 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)"
+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>)"
+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" "0 \<le> b"
+ 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"
+ 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"
+ 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"
+ 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"
+ 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:
@@ -1015,31 +1187,37 @@
lemma ereal_inverse_antimono:
fixes x y :: ereal
- shows "0 \<le> x \<Longrightarrow> x <= y \<Longrightarrow> inverse y <= inverse x"
+ 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"
+ 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>"
+ 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"
+ 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" "0 < a" "a < \<infinity>"
+ 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"
+ 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)
@@ -1047,36 +1225,47 @@
lemma ereal_le_mult_one_interval:
fixes x y :: ereal
assumes y: "y \<noteq> -\<infinity>"
- assumes z: "\<And>z. \<lbrakk> 0 < z ; z < 1 \<rbrakk> \<Longrightarrow> z * x \<le> y"
+ 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)
+ case PInf
+ with z[of "1 / 2"] show "x \<le> y"
+ by (simp add: one_ereal_def)
next
- case (real r) note r = this
+ case (real r)
+ note r = this
show "x \<le> y"
proof (cases y)
- case (real p) note p = this
+ 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)
+ 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
+ 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" "0 < z" shows "x / z \<le> y / z"
-using assms by (cases x y z rule: ereal3_cases) (auto intro: divide_right_mono)
+ 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" "0 < z" "0 < x * y"
+ 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 sign_simps split: split_if_asm)
+ using assms
+ by (cases x y z rule: ereal3_cases)
+ (auto intro: divide_left_mono simp: field_simps sign_simps split: split_if_asm)
lemma ereal_divide_zero_left[simp]:
fixes a :: ereal
@@ -1088,13 +1277,16 @@
shows "b / c * a = b * a / c"
by (cases a b c rule: ereal3_cases) (auto simp: field_simps sign_simps)
+
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
@@ -1109,29 +1301,46 @@
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
- assume "\<exists>x. \<forall>a\<in>S. a \<le> ereal x"
- then obtain y where y: "\<And>a. a\<in>S \<Longrightarrow> a \<le> ereal y" by auto
- then have "\<infinity> \<notin> S" by force
+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
- assume "S \<noteq> {-\<infinity>} \<and> S \<noteq> {}"
- with `\<infinity> \<notin> S` obtain x where x: "x \<in> S" "\<bar>x\<bar> \<noteq> \<infinity>" by auto
+ 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])
+ 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"
+ 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"
+ 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
- qed (auto intro!: exI[of _ "-\<infinity>"])
-qed (fastforce intro!: exI[of _ \<infinity>] ereal_top intro: order_trans dest: less_imp_le simp: not_le)
+ 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"
@@ -1141,23 +1350,24 @@
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
+ 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
-next
+ 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
+ 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)
@@ -1183,74 +1393,89 @@
using ereal_Sup_uminus_image_eq[of "uminus ` S"] by simp
lemma ereal_SUPR_uminus:
- fixes f :: "'a => ereal"
+ 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: SUP_def INF_def image_image)
lemma ereal_INFI_uminus:
- fixes f :: "'a => ereal"
- shows "(INF i : R. -(f i)) = -(SUP i : R. f i)"
+ fixes f :: "'a \<Rightarrow> ereal"
+ shows "(INF i : R. - f i) = - (SUP i : R. f i)"
using ereal_SUPR_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"
+ 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)
+ then show "X = uminus ` Y"
+ by (simp add: image_image)
qed (simp add: image_image)
lemma Inf_ereal_iff:
fixes z :: ereal
- 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)
+ 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>}"
+ 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>}"
+ 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>"
+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>"
+ 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> {}"
+ 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>" "0 < e"
+ 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
+ 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>"
+ {
+ 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
+ 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 }
+ 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)
@@ -1259,96 +1484,136 @@
lemma Inf_less:
fixes x :: ereal
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 INF_greatest) by auto
- thus False using assms by auto
+ 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>"
+ 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: ereal_le_minus_iff)
- then have "SUPR UNIV f \<le> z - y" by (rule SUP_least)
- then show ?thesis using real by (simp add: ereal_le_minus_iff)
+ then have "\<And>i. f i \<le> z - y"
+ using assms by (simp add: ereal_le_minus_iff)
+ then have "SUPR 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 SUPR_ereal_add:
fixes f g :: "nat \<Rightarrow> ereal"
- assumes "incseq f" "incseq g" and pos: "\<And>i. f i \<noteq> -\<infinity>" "\<And>i. g i \<noteq> -\<infinity>"
+ 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) = SUPR UNIV f + SUPR UNIV g"
proof (rule SUP_eqI)
- 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 SUP_def Sup_eq_MInfty by (auto dest: image_eqD)
- { fix j
- { fix i
+ 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 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
+ 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
+ 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" . }
+ 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_ereal_le_addI) auto
- then have "g j + SUPR UNIV f \<le> y" by (simp add: ac_simps) }
+ 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_ereal_le_addI)
- then show "SUPR UNIV f + SUPR UNIV g \<le> y" by (simp add: ac_simps)
+ then show "SUPR UNIV f + SUPR UNIV g \<le> y"
+ by (simp add: ac_simps)
qed (auto intro!: add_mono SUP_upper)
lemma SUPR_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"
+ 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_ereal_add inc)
- fix i show "f i \<noteq> -\<infinity>" "g i \<noteq> -\<infinity>" using pos[of i] by auto
+ fix i
+ show "f i \<noteq> -\<infinity>" "g i \<noteq> -\<infinity>"
+ using pos[of i] by auto
qed
lemma SUPR_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"
+ 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
+proof (cases "finite A")
+ case True
+ then show ?thesis using assms
by induct (auto simp: incseq_setsumI2 setsum_nonneg SUPR_ereal_add_pos)
-qed simp
+next
+ case False
+ then show ?thesis by simp
+qed
lemma SUPR_ereal_cmult:
- fixes f :: "nat \<Rightarrow> ereal" assumes "\<And>i. 0 \<le> f i" "0 \<le> c"
+ fixes f :: "nat \<Rightarrow> ereal"
+ assumes "\<And>i. 0 \<le> f i"
+ and "0 \<le> c"
shows "(SUP i. c * f i) = c * SUPR UNIV f"
proof (rule SUP_eqI)
- fix i have "f i \<le> SUPR UNIV f" by (rule SUP_upper) auto
+ fix i
+ have "f i \<le> SUPR UNIV f"
+ by (rule SUP_upper) auto
then show "c * f i \<le> c * SUPR 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"
+ 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>"
+ proof (cases "0 < c \<and> c \<noteq> \<infinity>")
+ case True
with * have "SUPR UNIV f \<le> y / c"
by (intro SUP_least) (auto simp: ereal_le_divide_pos)
- with c show ?thesis
+ with True show ?thesis
by (auto simp: ereal_le_divide_pos)
next
- { assume "c = \<infinity>" have ?thesis
- proof cases
- assume **: "\<forall>i. f i = 0"
- then have "range f = {0}" by auto
- with ** show "c * SUPR UNIV f \<le> y" using *
- by (auto simp: SUP_def min_max.sup_absorb1)
+ 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 * SUPR UNIV f \<le> y"
+ using * by (auto simp: SUP_def min_max.sup_absorb1)
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
+ 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
@@ -1359,15 +1624,21 @@
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>"
+ 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
+ case PInf
+ with `x \<noteq> \<infinity>` show ?thesis
+ by simp
next
- case MInf with assms[of "0"] show ?thesis by force
+ 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
+ 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
@@ -1382,7 +1653,8 @@
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"
+ 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)"
@@ -1392,48 +1664,63 @@
where f: "\<forall>x. f x \<in> A \<and> Sup A < f x + 1 / ereal (real x)" ..
have "SUPR 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)
+ 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"
+ 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"
+ fix e :: ereal
+ 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 / ereal (real n)" using f[THEN spec, of n]
+ 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
+ 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])
+ with f show ?thesis
+ by (auto intro!: exI[of _ f])
next
case PInf
- from `A \<noteq> {}` obtain x where "x \<in> A" by auto
+ from `A \<noteq> {}` obtain x where "x \<in> A"
+ by auto
show ?thesis
- proof cases
- assume *: "\<infinity> \<in> A"
- then have "\<infinity> \<le> Sup A" by (intro complete_lattice_class.Sup_upper)
- with * show ?thesis by (auto intro!: exI[of _ "\<lambda>x. \<infinity>"])
+ 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
- assume "\<infinity> \<notin> A"
+ case False
have "\<exists>x\<in>A. 0 \<le> x"
- by (metis Infty_neq_0 PInf complete_lattice_class.Sup_least ereal_infty_less_eq2 linorder_linear)
- then obtain x where "x \<in> A" "0 \<le> x" by auto
+ by (metis Infty_neq_0 PInf complete_lattice_class.Sup_least
+ ereal_infty_less_eq2 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
+ 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" ..
@@ -1444,20 +1731,26 @@
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])
+ 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>"])
+ 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> ereal. range f \<subseteq> g`A \<and> SUPR A g = SUPR UNIV f"
- using Sup_countable_SUPR[of "g`A"] by (auto simp: SUP_def)
+ using Sup_countable_SUPR[of "g`A"]
+ by (auto simp: SUP_def)
lemma Sup_ereal_cadd:
- fixes A :: "ereal set" assumes "A \<noteq> {}" and "a \<noteq> -\<infinity>"
+ 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"
@@ -1465,37 +1758,46 @@
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 min_max.sup_absorb1)
+ case PInf with `A \<noteq> {}`
+ show ?thesis
+ by (auto simp: image_constant min_max.sup_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 **
+ 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>"
+ 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
- by (simp add: comp_def image_image minus_ereal_def
- ereal_Sup_uminus_image_eq)
+ by (simp add: comp_def image_image minus_ereal_def ereal_Sup_uminus_image_eq)
lemma SUPR_ereal_cminus:
fixes f :: "'i \<Rightarrow> ereal"
- fixes A assumes "A \<noteq> {}" and "a \<noteq> -\<infinity>"
+ 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>"
+ 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
+ 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)
@@ -1505,25 +1807,32 @@
qed
lemma INFI_ereal_cminus:
- fixes a :: ereal assumes "A \<noteq> {}" and "\<bar>a\<bar> \<noteq> \<infinity>"
+ 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"
+ 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 INFI_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>"
+ 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) = INFI UNIV f + INFI 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) }
+ {
+ 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> = INFI UNIV f + INFI UNIV g"
@@ -1534,6 +1843,7 @@
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>)"
@@ -1546,50 +1856,41 @@
"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_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 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]:
- shows "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_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]:
- shows "numeral m < ereal_of_enat n \<longleftrightarrow> numeral m < n"
-by (cases n) (auto simp: real_of_nat_less_iff[symmetric])
+lemma numeral_less_ereal_of_enat_iff[simp]: "numeral m < ereal_of_enat n \<longleftrightarrow> numeral m < n"
+ by (cases n) (auto simp: real_of_nat_less_iff[symmetric])
-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_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_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_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>"
+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_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
+ 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
+ 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]
@@ -1607,6 +1908,7 @@
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)"
@@ -1618,8 +1920,13 @@
with Int show ?case
by (intro exI[of _ "max x z"]) fastforce
next
- { fix x have "x \<noteq> \<infinity> \<Longrightarrow> \<exists>t. x \<le> ereal t" by (cases x) auto }
- moreover case (Basis S)
+ 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)+
@@ -1633,8 +1940,13 @@
with Int show ?case
by (intro exI[of _ "min x z"]) fastforce
next
- { fix x have "x \<noteq> - \<infinity> \<Longrightarrow> \<exists>t. ereal t \<le> x" by (cases x) auto }
- moreover case (Basis S)
+ 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)+
@@ -1642,13 +1954,18 @@
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
+ case (Int A B)
+ then show ?case
+ by auto
next
- { fix x have
+ 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 case (Basis S)
+ by (induct x) auto
+ }
+ moreover note Basis
ultimately show ?case
by (auto split: ereal.split)
qed (fastforce simp add: vimage_Union)+
@@ -1657,16 +1974,32 @@
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 }" by auto (case_tac xa, auto) }
- moreover { fix x have "ereal ` {x <..} = { ereal x <..< \<infinity> }" by auto (case_tac xa, auto) }
+ 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 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))"
+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
+ assume "open A"
+ then show ?rhs
using open_PInfty open_MInfty open_ereal_vimage by auto
next
assume "?rhs"
@@ -1678,14 +2011,23 @@
by (subst *) (auto simp: open_Un)
qed
-lemma open_PInfty2: assumes "open A" "\<infinity> \<in> A" obtains x where "{ereal x<..} \<subseteq> A"
+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" "-\<infinity> \<in> A" obtains x where "{..<ereal x} \<subseteq> A"
+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)" "\<infinity> \<in> A \<Longrightarrow> {ereal x<..} \<subseteq> A" "-\<infinity> \<in> A \<Longrightarrow> {..<ereal y} \<subseteq> A"
+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]
@@ -1695,60 +2037,76 @@
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" "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 (ereal -` S)" by (rule ereal_openE)
- then obtain e where "0 < e" and e: "\<And>y. dist y (real x) < e \<Longrightarrow> ereal y \<in> S"
+ 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}"
+ 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"
- apply (cases y) by (auto simp: dist_real_def)
- then show "y \<in> S" using e[of t] by auto
+ 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" "x \<in> S" and x: "\<bar>x\<bar> \<noteq> \<infinity>"
- obtains a b where "a < x" "x < b" "{a <..< b} \<subseteq> S"
+ 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
+ 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")
+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" "open S" "x \<in> S"
+ 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" "open S" "ereal x \<in> S"
+ 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
+ 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" "x \<in> S"
+ 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
+ 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)
@@ -1756,10 +2114,12 @@
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) }
+ {
+ 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
@@ -1772,20 +2132,26 @@
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)
+ 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
+ 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
+ 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)
@@ -1793,9 +2159,12 @@
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
+ 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)
@@ -1807,38 +2176,43 @@
using LIMSEQ_le_const[of f l "ereal B"] by auto
lemma tendsto_explicit:
- "f ----> f0 <-> (ALL S. open S --> f0 : S --> (EX N. ALL n>=N. f n : S))"
+ "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> ALL n>=N. f n <= ereal B \<Longrightarrow> l ~= \<infinity>"
+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> ALL n>=M. f n <= C \<Longrightarrow> l<=C"
+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: "ALL n>=N. f n >= C"
- shows "l>=C"
+ 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"
+ 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"
+ 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"
+ 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
+ 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)
@@ -1849,22 +2223,25 @@
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" "x \<in> S"
- with x have "open (S - {\<infinity>, -\<infinity>})" "x \<in> S - {\<infinity>, -\<infinity>}" by auto
+ 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)
+ 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 ereal_inj_affinity:
fixes m t :: ereal
- assumes "\<bar>m\<bar> \<noteq> \<infinity>" "m \<noteq> 0" "\<bar>t\<bar> \<noteq> \<infinity>"
+ 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])
@@ -1902,108 +2279,136 @@
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"
+lemma ereal_real':
+ assumes "\<bar>x\<bar> \<noteq> \<infinity>"
+ shows "ereal (real x) = x"
using assms by auto
-lemma real_ereal_id: "real o ereal = id"
-proof-
- { fix x have "(real o ereal) x = id x" by auto }
- then show ?thesis using ext by blast
+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)
+ 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"
+ 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" "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)
+ 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>=0"
-shows "c * (a + b) <= c * a + c * b"
- using assms by (cases rule: ereal3_cases[of a b c])
- (auto simp add: field_simps)
+ 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) <= b; c <= d |] ==> max a c <= max b d"
+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) <= x; c <= x |] ==> max a c <= x"
+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>"
- assumes "!!r. 0 < r ==> EX N. ALL n>=N. u n < x + r & x < u n + r"
+ 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_def: "x=ereal rx" using assms by (cases x) auto
- fix S assume "open S" "x : S"
- then have "open (ereal -` S)" unfolding open_ereal_def by auto
- with `x \<in> S` obtain r where "0 < r" and dist: "!!y. dist y rx < r ==> ereal y \<in> S"
- unfolding open_real_def rx_def by auto
+ 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: "!!N. n <= N ==> u N < x + ereal r" and
- lower: "!!N. n <= N ==> x < u N + ereal r" using assms(2)[of "ereal r"] by auto
- show "EX N. ALL n>=N. u n : S"
+ 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 <= N"
+ fix N
+ assume "n \<le> N"
from upper[OF this] lower[OF this] assms `0 < r`
- have "u N ~: {\<infinity>,(-\<infinity>)}" by auto
- then obtain ra where ra_def: "(u N) = ereal 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
+ 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 : S" using `u N ~: {\<infinity>, -\<infinity>}`
+ 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" "f0 : S"
- obtains N where "ALL n>=N. f n : S"
+ 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 <-> (ALL r. 0 < r --> (EX N. ALL n>=N. u n < x + r & x < u n + r))"
- (is "?lhs <-> ?rhs")
+ 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 assume "(r::ereal)>0"
- then obtain N where N_def: "ALL n>=N. u n : {x - r <..< x + r}"
+ {
+ 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` by auto
- hence "EX N. ALL n>=N. u n < x + r & x < u n + r"
- using ereal_minus_less[of r x] by (cases r) auto
- } then show "?rhs" by auto
+ 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"
+ assume ?rhs
+ then show "u ----> x"
using ereal_LimI_finite[of x] assms by auto
qed
lemma ereal_Limsup_uminus:
- fixes f :: "'a => ereal"
- shows "Limsup net (\<lambda>x. - (f x)) = -(Liminf net f)"
+ fixes f :: "'a \<Rightarrow> ereal"
+ shows "Limsup net (\<lambda>x. - (f x)) = - Liminf net f"
unfolding Limsup_def Liminf_def ereal_SUPR_uminus ereal_INFI_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 <-> ?rhs")
+ 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 ..
lemma
@@ -2012,6 +2417,7 @@
and ereal_decseq_uminus[simp]: "decseq (\<lambda>i. - X i) = incseq X"
unfolding incseq_def decseq_def by auto
+
subsubsection {* Tests for code generator *}
(* A small list of simple arithmetic expressions *)