--- a/NEWS Mon Dec 06 19:18:02 2010 +0100
+++ b/NEWS Fri Dec 03 15:25:14 2010 +0100
@@ -334,8 +334,8 @@
of euclidean spaces the real and complex numbers are instantiated to
be euclidean_spaces. INCOMPATIBILITY.
-* Probability: Introduced pinfreal as real numbers with infinity. Use
-pinfreal as value for measures. Introduce the Radon-Nikodym
+* Probability: Introduced pextreal as positive extended real numbers.
+Use pextreal as value for measures. Introduce the Radon-Nikodym
derivative, product spaces and Fubini's theorem for arbitrary sigma
finite measures. Introduces Lebesgue measure based on the integral in
Multivariate Analysis. INCOMPATIBILITY.
--- a/src/HOL/IsaMakefile Mon Dec 06 19:18:02 2010 +0100
+++ b/src/HOL/IsaMakefile Fri Dec 03 15:25:14 2010 +0100
@@ -1183,7 +1183,7 @@
Probability/ex/Koepf_Duermuth_Countermeasure.thy \
Probability/Information.thy Probability/Lebesgue_Integration.thy \
Probability/Lebesgue_Measure.thy Probability/Measure.thy \
- Probability/Positive_Infinite_Real.thy \
+ Probability/Positive_Extended_Real.thy \
Probability/Probability_Space.thy Probability/Probability.thy \
Probability/Product_Measure.thy Probability/Radon_Nikodym.thy \
Probability/ROOT.ML Probability/Sigma_Algebra.thy \
--- a/src/HOL/Probability/Borel_Space.thy Mon Dec 06 19:18:02 2010 +0100
+++ b/src/HOL/Probability/Borel_Space.thy Fri Dec 03 15:25:14 2010 +0100
@@ -3,7 +3,7 @@
header {*Borel spaces*}
theory Borel_Space
- imports Sigma_Algebra Positive_Infinite_Real Multivariate_Analysis
+ imports Sigma_Algebra Positive_Extended_Real Multivariate_Analysis
begin
lemma LIMSEQ_max:
@@ -1012,10 +1012,10 @@
lemma borel_Real_measurable:
"A \<in> sets borel \<Longrightarrow> Real -` A \<in> sets borel"
proof (rule borel_measurable_translate)
- fix B :: "pinfreal set" assume "open B"
+ fix B :: "pextreal set" assume "open B"
then obtain T x where T: "open T" "Real ` (T \<inter> {0..}) = B - {\<omega>}" and
x: "\<omega> \<in> B \<Longrightarrow> 0 \<le> x" "\<omega> \<in> B \<Longrightarrow> {Real x <..} \<subseteq> B"
- unfolding open_pinfreal_def by blast
+ unfolding open_pextreal_def by blast
have "Real -` B = Real -` (B - {\<omega>})" by auto
also have "\<dots> = Real -` (Real ` (T \<inter> {0..}))" using T by simp
also have "\<dots> = (if 0 \<in> T then T \<union> {.. 0} else T \<inter> {0..})"
@@ -1027,7 +1027,7 @@
qed simp
lemma borel_real_measurable:
- "A \<in> sets borel \<Longrightarrow> (real -` A :: pinfreal set) \<in> sets borel"
+ "A \<in> sets borel \<Longrightarrow> (real -` A :: pextreal set) \<in> sets borel"
proof (rule borel_measurable_translate)
fix B :: "real set" assume "open B"
{ fix x have "0 < real x \<longleftrightarrow> (\<exists>r>0. x = Real r)" by (cases x) auto }
@@ -1035,10 +1035,10 @@
have *: "real -` B = (if 0 \<in> B then real -` (B \<inter> {0 <..}) \<union> {0, \<omega>} else real -` (B \<inter> {0 <..}))"
by (force simp: Ex_less_real)
- have "open (real -` (B \<inter> {0 <..}) :: pinfreal set)"
- unfolding open_pinfreal_def using `open B`
+ have "open (real -` (B \<inter> {0 <..}) :: pextreal set)"
+ unfolding open_pextreal_def using `open B`
by (auto intro!: open_Int exI[of _ "B \<inter> {0 <..}"] simp: image_iff Ex_less_real)
- then show "(real -` B :: pinfreal set) \<in> sets borel" unfolding * by auto
+ then show "(real -` B :: pextreal set) \<in> sets borel" unfolding * by auto
qed simp
lemma (in sigma_algebra) borel_measurable_Real[intro, simp]:
@@ -1046,7 +1046,7 @@
shows "(\<lambda>x. Real (f x)) \<in> borel_measurable M"
unfolding in_borel_measurable_borel
proof safe
- fix S :: "pinfreal set" assume "S \<in> sets borel"
+ fix S :: "pextreal set" assume "S \<in> sets borel"
from borel_Real_measurable[OF this]
have "(Real \<circ> f) -` S \<inter> space M \<in> sets M"
using assms
@@ -1056,7 +1056,7 @@
qed
lemma (in sigma_algebra) borel_measurable_real[intro, simp]:
- fixes f :: "'a \<Rightarrow> pinfreal"
+ fixes f :: "'a \<Rightarrow> pextreal"
assumes "f \<in> borel_measurable M"
shows "(\<lambda>x. real (f x)) \<in> borel_measurable M"
unfolding in_borel_measurable_borel
@@ -1085,7 +1085,7 @@
by (simp cong: measurable_cong)
qed auto
-lemma (in sigma_algebra) borel_measurable_pinfreal_eq_real:
+lemma (in sigma_algebra) borel_measurable_pextreal_eq_real:
"f \<in> borel_measurable M \<longleftrightarrow>
((\<lambda>x. real (f x)) \<in> borel_measurable M \<and> f -` {\<omega>} \<inter> space M \<in> sets M)"
proof safe
@@ -1130,8 +1130,8 @@
ultimately show "{x\<in>space M. a \<le> f x} \<in> sets M" by auto
qed
-lemma (in sigma_algebra) less_eq_le_pinfreal_measurable:
- fixes f :: "'a \<Rightarrow> pinfreal"
+lemma (in sigma_algebra) less_eq_le_pextreal_measurable:
+ fixes f :: "'a \<Rightarrow> pextreal"
shows "(\<forall>a. {x\<in>space M. a < f x} \<in> sets M) \<longleftrightarrow> (\<forall>a. {x\<in>space M. a \<le> f x} \<in> sets M)"
proof
assume a: "\<forall>a. {x\<in>space M. a \<le> f x} \<in> sets M"
@@ -1143,9 +1143,9 @@
have "{x\<in>space M. a < f x} = (\<Union>i. {x\<in>space M. a + inverse (of_nat (Suc i)) \<le> f x})"
proof safe
fix x assume "a < f x" and [simp]: "x \<in> space M"
- with ex_pinfreal_inverse_of_nat_Suc_less[of "f x - a"]
+ with ex_pextreal_inverse_of_nat_Suc_less[of "f x - a"]
obtain n where "a + inverse (of_nat (Suc n)) < f x"
- by (cases "f x", auto simp: pinfreal_minus_order)
+ by (cases "f x", auto simp: pextreal_minus_order)
then have "a + inverse (of_nat (Suc n)) \<le> f x" by simp
then show "x \<in> (\<Union>i. {x \<in> space M. a + inverse (of_nat (Suc i)) \<le> f x})"
by auto
@@ -1174,7 +1174,7 @@
have "{x\<in>space M. f x < a} = (\<Union>i. {x\<in>space M. f x \<le> a - inverse (of_nat (Suc i))})"
proof safe
fix x assume "f x < a" and [simp]: "x \<in> space M"
- with ex_pinfreal_inverse_of_nat_Suc_less[of "a - f x"]
+ with ex_pextreal_inverse_of_nat_Suc_less[of "a - f x"]
obtain n where "inverse (of_nat (Suc n)) < a - f x"
using preal by (cases "f x") auto
then have "f x \<le> a - inverse (of_nat (Suc n)) "
@@ -1197,7 +1197,7 @@
show "f x = \<omega>" proof (rule ccontr)
assume "f x \<noteq> \<omega>"
with real_arch_lt[of "real (f x)"] obtain n where "f x < of_nat n"
- by (auto simp: pinfreal_noteq_omega_Ex)
+ by (auto simp: pextreal_noteq_omega_Ex)
with *[THEN spec, of n] show False by auto
qed
qed
@@ -1209,8 +1209,8 @@
qed
qed
-lemma (in sigma_algebra) borel_measurable_pinfreal_iff_greater:
- "(f::'a \<Rightarrow> pinfreal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. {x\<in>space M. a < f x} \<in> sets M)"
+lemma (in sigma_algebra) borel_measurable_pextreal_iff_greater:
+ "(f::'a \<Rightarrow> pextreal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. {x\<in>space M. a < f x} \<in> sets M)"
proof safe
fix a assume f: "f \<in> borel_measurable M"
have "{x\<in>space M. a < f x} = f -` {a <..} \<inter> space M" by auto
@@ -1219,9 +1219,9 @@
next
assume *: "\<forall>a. {x\<in>space M. a < f x} \<in> sets M"
hence **: "\<forall>a. {x\<in>space M. f x < a} \<in> sets M"
- unfolding less_eq_le_pinfreal_measurable
+ unfolding less_eq_le_pextreal_measurable
unfolding greater_eq_le_measurable .
- show "f \<in> borel_measurable M" unfolding borel_measurable_pinfreal_eq_real borel_measurable_iff_greater
+ show "f \<in> borel_measurable M" unfolding borel_measurable_pextreal_eq_real borel_measurable_iff_greater
proof safe
have "f -` {\<omega>} \<inter> space M = space M - {x\<in>space M. f x < \<omega>}" by auto
then show \<omega>: "f -` {\<omega>} \<inter> space M \<in> sets M" using ** by auto
@@ -1242,28 +1242,28 @@
qed
qed
-lemma (in sigma_algebra) borel_measurable_pinfreal_iff_less:
- "(f::'a \<Rightarrow> pinfreal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. {x\<in>space M. f x < a} \<in> sets M)"
- using borel_measurable_pinfreal_iff_greater unfolding less_eq_le_pinfreal_measurable greater_eq_le_measurable .
+lemma (in sigma_algebra) borel_measurable_pextreal_iff_less:
+ "(f::'a \<Rightarrow> pextreal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. {x\<in>space M. f x < a} \<in> sets M)"
+ using borel_measurable_pextreal_iff_greater unfolding less_eq_le_pextreal_measurable greater_eq_le_measurable .
-lemma (in sigma_algebra) borel_measurable_pinfreal_iff_le:
- "(f::'a \<Rightarrow> pinfreal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. {x\<in>space M. f x \<le> a} \<in> sets M)"
- using borel_measurable_pinfreal_iff_greater unfolding less_eq_ge_measurable .
+lemma (in sigma_algebra) borel_measurable_pextreal_iff_le:
+ "(f::'a \<Rightarrow> pextreal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. {x\<in>space M. f x \<le> a} \<in> sets M)"
+ using borel_measurable_pextreal_iff_greater unfolding less_eq_ge_measurable .
-lemma (in sigma_algebra) borel_measurable_pinfreal_iff_ge:
- "(f::'a \<Rightarrow> pinfreal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. {x\<in>space M. a \<le> f x} \<in> sets M)"
- using borel_measurable_pinfreal_iff_greater unfolding less_eq_le_pinfreal_measurable .
+lemma (in sigma_algebra) borel_measurable_pextreal_iff_ge:
+ "(f::'a \<Rightarrow> pextreal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. {x\<in>space M. a \<le> f x} \<in> sets M)"
+ using borel_measurable_pextreal_iff_greater unfolding less_eq_le_pextreal_measurable .
-lemma (in sigma_algebra) borel_measurable_pinfreal_eq_const:
- fixes f :: "'a \<Rightarrow> pinfreal" assumes "f \<in> borel_measurable M"
+lemma (in sigma_algebra) borel_measurable_pextreal_eq_const:
+ fixes f :: "'a \<Rightarrow> pextreal" assumes "f \<in> borel_measurable M"
shows "{x\<in>space M. f x = c} \<in> sets M"
proof -
have "{x\<in>space M. f x = c} = (f -` {c} \<inter> space M)" by auto
then show ?thesis using assms by (auto intro!: measurable_sets)
qed
-lemma (in sigma_algebra) borel_measurable_pinfreal_neq_const:
- fixes f :: "'a \<Rightarrow> pinfreal"
+lemma (in sigma_algebra) borel_measurable_pextreal_neq_const:
+ fixes f :: "'a \<Rightarrow> pextreal"
assumes "f \<in> borel_measurable M"
shows "{x\<in>space M. f x \<noteq> c} \<in> sets M"
proof -
@@ -1271,8 +1271,8 @@
then show ?thesis using assms by (auto intro!: measurable_sets)
qed
-lemma (in sigma_algebra) borel_measurable_pinfreal_less[intro,simp]:
- fixes f g :: "'a \<Rightarrow> pinfreal"
+lemma (in sigma_algebra) borel_measurable_pextreal_less[intro,simp]:
+ fixes f g :: "'a \<Rightarrow> pextreal"
assumes f: "f \<in> borel_measurable M"
assumes g: "g \<in> borel_measurable M"
shows "{x \<in> space M. f x < g x} \<in> sets M"
@@ -1282,17 +1282,17 @@
using assms by (auto intro!: borel_measurable_real)
from borel_measurable_less[OF this]
have "{x \<in> space M. real (f x) < real (g x)} \<in> sets M" .
- moreover have "{x \<in> space M. f x \<noteq> \<omega>} \<in> sets M" using f by (rule borel_measurable_pinfreal_neq_const)
- moreover have "{x \<in> space M. g x = \<omega>} \<in> sets M" using g by (rule borel_measurable_pinfreal_eq_const)
- moreover have "{x \<in> space M. g x \<noteq> \<omega>} \<in> sets M" using g by (rule borel_measurable_pinfreal_neq_const)
+ moreover have "{x \<in> space M. f x \<noteq> \<omega>} \<in> sets M" using f by (rule borel_measurable_pextreal_neq_const)
+ moreover have "{x \<in> space M. g x = \<omega>} \<in> sets M" using g by (rule borel_measurable_pextreal_eq_const)
+ moreover have "{x \<in> space M. g x \<noteq> \<omega>} \<in> sets M" using g by (rule borel_measurable_pextreal_neq_const)
moreover have "{x \<in> space M. f x < g x} = ({x \<in> space M. g x = \<omega>} \<inter> {x \<in> space M. f x \<noteq> \<omega>}) \<union>
({x \<in> space M. g x \<noteq> \<omega>} \<inter> {x \<in> space M. f x \<noteq> \<omega>} \<inter> {x \<in> space M. real (f x) < real (g x)})"
- by (auto simp: real_of_pinfreal_strict_mono_iff)
+ by (auto simp: real_of_pextreal_strict_mono_iff)
ultimately show ?thesis by auto
qed
-lemma (in sigma_algebra) borel_measurable_pinfreal_le[intro,simp]:
- fixes f :: "'a \<Rightarrow> pinfreal"
+lemma (in sigma_algebra) borel_measurable_pextreal_le[intro,simp]:
+ fixes f :: "'a \<Rightarrow> pextreal"
assumes f: "f \<in> borel_measurable M"
assumes g: "g \<in> borel_measurable M"
shows "{x \<in> space M. f x \<le> g x} \<in> sets M"
@@ -1301,8 +1301,8 @@
then show ?thesis using g f by auto
qed
-lemma (in sigma_algebra) borel_measurable_pinfreal_eq[intro,simp]:
- fixes f :: "'a \<Rightarrow> pinfreal"
+lemma (in sigma_algebra) borel_measurable_pextreal_eq[intro,simp]:
+ fixes f :: "'a \<Rightarrow> pextreal"
assumes f: "f \<in> borel_measurable M"
assumes g: "g \<in> borel_measurable M"
shows "{w \<in> space M. f w = g w} \<in> sets M"
@@ -1311,8 +1311,8 @@
then show ?thesis using g f by auto
qed
-lemma (in sigma_algebra) borel_measurable_pinfreal_neq[intro,simp]:
- fixes f :: "'a \<Rightarrow> pinfreal"
+lemma (in sigma_algebra) borel_measurable_pextreal_neq[intro,simp]:
+ fixes f :: "'a \<Rightarrow> pextreal"
assumes f: "f \<in> borel_measurable M"
assumes g: "g \<in> borel_measurable M"
shows "{w \<in> space M. f w \<noteq> g w} \<in> sets M"
@@ -1321,32 +1321,32 @@
thus ?thesis using f g by auto
qed
-lemma (in sigma_algebra) borel_measurable_pinfreal_add[intro, simp]:
- fixes f :: "'a \<Rightarrow> pinfreal"
+lemma (in sigma_algebra) borel_measurable_pextreal_add[intro, simp]:
+ fixes f :: "'a \<Rightarrow> pextreal"
assumes measure: "f \<in> borel_measurable M" "g \<in> borel_measurable M"
shows "(\<lambda>x. f x + g x) \<in> borel_measurable M"
proof -
have *: "(\<lambda>x. f x + g x) =
(\<lambda>x. if f x = \<omega> then \<omega> else if g x = \<omega> then \<omega> else Real (real (f x) + real (g x)))"
- by (auto simp: fun_eq_iff pinfreal_noteq_omega_Ex)
+ by (auto simp: fun_eq_iff pextreal_noteq_omega_Ex)
show ?thesis using assms unfolding *
by (auto intro!: measurable_If)
qed
-lemma (in sigma_algebra) borel_measurable_pinfreal_times[intro, simp]:
- fixes f :: "'a \<Rightarrow> pinfreal" assumes "f \<in> borel_measurable M" "g \<in> borel_measurable M"
+lemma (in sigma_algebra) borel_measurable_pextreal_times[intro, simp]:
+ fixes f :: "'a \<Rightarrow> pextreal" assumes "f \<in> borel_measurable M" "g \<in> borel_measurable M"
shows "(\<lambda>x. f x * g x) \<in> borel_measurable M"
proof -
have *: "(\<lambda>x. f x * g x) =
(\<lambda>x. if f x = 0 then 0 else if g x = 0 then 0 else if f x = \<omega> then \<omega> else if g x = \<omega> then \<omega> else
Real (real (f x) * real (g x)))"
- by (auto simp: fun_eq_iff pinfreal_noteq_omega_Ex)
+ by (auto simp: fun_eq_iff pextreal_noteq_omega_Ex)
show ?thesis using assms unfolding *
by (auto intro!: measurable_If)
qed
-lemma (in sigma_algebra) borel_measurable_pinfreal_setsum[simp, intro]:
- fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> pinfreal"
+lemma (in sigma_algebra) borel_measurable_pextreal_setsum[simp, intro]:
+ fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> pextreal"
assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M"
shows "(\<lambda>x. \<Sum>i\<in>S. f i x) \<in> borel_measurable M"
proof cases
@@ -1355,56 +1355,56 @@
by induct auto
qed (simp add: borel_measurable_const)
-lemma (in sigma_algebra) borel_measurable_pinfreal_min[simp, intro]:
- fixes f g :: "'a \<Rightarrow> pinfreal"
+lemma (in sigma_algebra) borel_measurable_pextreal_min[simp, intro]:
+ fixes f g :: "'a \<Rightarrow> pextreal"
assumes "f \<in> borel_measurable M"
assumes "g \<in> borel_measurable M"
shows "(\<lambda>x. min (g x) (f x)) \<in> borel_measurable M"
using assms unfolding min_def by (auto intro!: measurable_If)
-lemma (in sigma_algebra) borel_measurable_pinfreal_max[simp, intro]:
- fixes f g :: "'a \<Rightarrow> pinfreal"
+lemma (in sigma_algebra) borel_measurable_pextreal_max[simp, intro]:
+ fixes f g :: "'a \<Rightarrow> pextreal"
assumes "f \<in> borel_measurable M"
and "g \<in> borel_measurable M"
shows "(\<lambda>x. max (g x) (f x)) \<in> borel_measurable M"
using assms unfolding max_def by (auto intro!: measurable_If)
lemma (in sigma_algebra) borel_measurable_SUP[simp, intro]:
- fixes f :: "'d\<Colon>countable \<Rightarrow> 'a \<Rightarrow> pinfreal"
+ fixes f :: "'d\<Colon>countable \<Rightarrow> 'a \<Rightarrow> pextreal"
assumes "\<And>i. i \<in> A \<Longrightarrow> f i \<in> borel_measurable M"
shows "(SUP i : A. f i) \<in> borel_measurable M" (is "?sup \<in> borel_measurable M")
- unfolding borel_measurable_pinfreal_iff_greater
+ unfolding borel_measurable_pextreal_iff_greater
proof safe
fix a
have "{x\<in>space M. a < ?sup x} = (\<Union>i\<in>A. {x\<in>space M. a < f i x})"
- by (auto simp: less_Sup_iff SUPR_def[where 'a=pinfreal] SUPR_fun_expand[where 'c=pinfreal])
+ by (auto simp: less_Sup_iff SUPR_def[where 'a=pextreal] SUPR_fun_expand[where 'c=pextreal])
then show "{x\<in>space M. a < ?sup x} \<in> sets M"
using assms by auto
qed
lemma (in sigma_algebra) borel_measurable_INF[simp, intro]:
- fixes f :: "'d :: countable \<Rightarrow> 'a \<Rightarrow> pinfreal"
+ fixes f :: "'d :: countable \<Rightarrow> 'a \<Rightarrow> pextreal"
assumes "\<And>i. i \<in> A \<Longrightarrow> f i \<in> borel_measurable M"
shows "(INF i : A. f i) \<in> borel_measurable M" (is "?inf \<in> borel_measurable M")
- unfolding borel_measurable_pinfreal_iff_less
+ unfolding borel_measurable_pextreal_iff_less
proof safe
fix a
have "{x\<in>space M. ?inf x < a} = (\<Union>i\<in>A. {x\<in>space M. f i x < a})"
- by (auto simp: Inf_less_iff INFI_def[where 'a=pinfreal] INFI_fun_expand)
+ by (auto simp: Inf_less_iff INFI_def[where 'a=pextreal] INFI_fun_expand)
then show "{x\<in>space M. ?inf x < a} \<in> sets M"
using assms by auto
qed
-lemma (in sigma_algebra) borel_measurable_pinfreal_diff[simp, intro]:
- fixes f g :: "'a \<Rightarrow> pinfreal"
+lemma (in sigma_algebra) borel_measurable_pextreal_diff[simp, intro]:
+ fixes f g :: "'a \<Rightarrow> pextreal"
assumes "f \<in> borel_measurable M"
assumes "g \<in> borel_measurable M"
shows "(\<lambda>x. f x - g x) \<in> borel_measurable M"
- unfolding borel_measurable_pinfreal_iff_greater
+ unfolding borel_measurable_pextreal_iff_greater
proof safe
fix a
have "{x \<in> space M. a < f x - g x} = {x \<in> space M. g x + a < f x}"
- by (simp add: pinfreal_less_minus_iff)
+ by (simp add: pextreal_less_minus_iff)
then show "{x \<in> space M. a < f x - g x} \<in> sets M"
using assms by auto
qed
--- a/src/HOL/Probability/Caratheodory.thy Mon Dec 06 19:18:02 2010 +0100
+++ b/src/HOL/Probability/Caratheodory.thy Fri Dec 03 15:25:14 2010 +0100
@@ -1,14 +1,14 @@
header {*Caratheodory Extension Theorem*}
theory Caratheodory
- imports Sigma_Algebra Positive_Infinite_Real
+ imports Sigma_Algebra Positive_Extended_Real
begin
text{*From the Hurd/Coble measure theory development, translated by Lawrence Paulson.*}
subsection {* Measure Spaces *}
-definition "positive f \<longleftrightarrow> f {} = (0::pinfreal)" -- "Positive is enforced by the type"
+definition "positive f \<longleftrightarrow> f {} = (0::pextreal)" -- "Positive is enforced by the type"
definition
additive where
@@ -58,7 +58,7 @@
{r . \<exists>A. range A \<subseteq> sets M \<and> disjoint_family A \<and> X \<subseteq> (\<Union>i. A i) \<and> (\<Sum>\<^isub>\<infinity> i. f (A i)) = r}"
locale measure_space = sigma_algebra +
- fixes \<mu> :: "'a set \<Rightarrow> pinfreal"
+ fixes \<mu> :: "'a set \<Rightarrow> pextreal"
assumes empty_measure [simp]: "\<mu> {} = 0"
and ca: "countably_additive M \<mu>"
@@ -148,7 +148,7 @@
by (simp add: lambda_system_def)
lemma (in algebra) lambda_system_Compl:
- fixes f:: "'a set \<Rightarrow> pinfreal"
+ fixes f:: "'a set \<Rightarrow> pextreal"
assumes x: "x \<in> lambda_system M f"
shows "space M - x \<in> lambda_system M f"
proof -
@@ -161,7 +161,7 @@
qed
lemma (in algebra) lambda_system_Int:
- fixes f:: "'a set \<Rightarrow> pinfreal"
+ fixes f:: "'a set \<Rightarrow> pextreal"
assumes xl: "x \<in> lambda_system M f" and yl: "y \<in> lambda_system M f"
shows "x \<inter> y \<in> lambda_system M f"
proof -
@@ -196,7 +196,7 @@
lemma (in algebra) lambda_system_Un:
- fixes f:: "'a set \<Rightarrow> pinfreal"
+ fixes f:: "'a set \<Rightarrow> pextreal"
assumes xl: "x \<in> lambda_system M f" and yl: "y \<in> lambda_system M f"
shows "x \<union> y \<in> lambda_system M f"
proof -
@@ -295,7 +295,7 @@
by (auto simp add: countably_subadditive_def o_def)
lemma (in algebra) increasing_additive_bound:
- fixes A:: "nat \<Rightarrow> 'a set" and f :: "'a set \<Rightarrow> pinfreal"
+ fixes A:: "nat \<Rightarrow> 'a set" and f :: "'a set \<Rightarrow> pextreal"
assumes f: "positive f" and ad: "additive M f"
and inc: "increasing M f"
and A: "range A \<subseteq> sets M"
@@ -315,7 +315,7 @@
by (simp add: increasing_def lambda_system_def)
lemma (in algebra) lambda_system_strong_sum:
- fixes A:: "nat \<Rightarrow> 'a set" and f :: "'a set \<Rightarrow> pinfreal"
+ fixes A:: "nat \<Rightarrow> 'a set" and f :: "'a set \<Rightarrow> pextreal"
assumes f: "positive f" and a: "a \<in> sets M"
and A: "range A \<subseteq> lambda_system M f"
and disj: "disjoint_family A"
@@ -497,7 +497,7 @@
assumes posf: "positive f" and ca: "countably_additive M f"
and s: "s \<in> sets M"
shows "Inf (measure_set M f s) = f s"
- unfolding Inf_pinfreal_def
+ unfolding Inf_pextreal_def
proof (safe intro!: Greatest_equality)
fix z
assume z: "z \<in> measure_set M f s"
@@ -608,8 +608,8 @@
shows "countably_subadditive (| space = space M, sets = Pow (space M) |)
(\<lambda>x. Inf (measure_set M f x))"
unfolding countably_subadditive_def o_def
-proof (safe, simp, rule pinfreal_le_epsilon)
- fix A :: "nat \<Rightarrow> 'a set" and e :: pinfreal
+proof (safe, simp, rule pextreal_le_epsilon)
+ fix A :: "nat \<Rightarrow> 'a set" and e :: pextreal
let "?outer n" = "Inf (measure_set M f (A n))"
assume A: "range A \<subseteq> Pow (space M)"
@@ -688,8 +688,8 @@
by blast
have "Inf (measure_set M f (s\<inter>x)) + Inf (measure_set M f (s-x))
\<le> Inf (measure_set M f s)"
- proof (rule pinfreal_le_epsilon)
- fix e :: pinfreal
+ proof (rule pextreal_le_epsilon)
+ fix e :: pextreal
assume e: "0 < e"
from inf_measure_close [of f, OF posf e s]
obtain A where A: "range A \<subseteq> sets M" and disj: "disjoint_family A"
@@ -760,7 +760,7 @@
theorem (in algebra) caratheodory:
assumes posf: "positive f" and ca: "countably_additive M f"
- shows "\<exists>\<mu> :: 'a set \<Rightarrow> pinfreal. (\<forall>s \<in> sets M. \<mu> s = f s) \<and> measure_space (sigma M) \<mu>"
+ shows "\<exists>\<mu> :: 'a set \<Rightarrow> pextreal. (\<forall>s \<in> sets M. \<mu> s = f s) \<and> measure_space (sigma M) \<mu>"
proof -
have inc: "increasing M f"
by (metis additive_increasing ca countably_additive_additive posf)
--- a/src/HOL/Probability/Complete_Measure.thy Mon Dec 06 19:18:02 2010 +0100
+++ b/src/HOL/Probability/Complete_Measure.thy Fri Dec 03 15:25:14 2010 +0100
@@ -243,7 +243,7 @@
qed
lemma (in completeable_measure_space) completion_ex_borel_measurable:
- fixes g :: "'a \<Rightarrow> pinfreal"
+ fixes g :: "'a \<Rightarrow> pextreal"
assumes g: "g \<in> borel_measurable completion"
shows "\<exists>g'\<in>borel_measurable M. (AE x. g x = g' x)"
proof -
--- a/src/HOL/Probability/Information.thy Mon Dec 06 19:18:02 2010 +0100
+++ b/src/HOL/Probability/Information.thy Fri Dec 03 15:25:14 2010 +0100
@@ -210,7 +210,7 @@
have ms: "measure_space M \<nu>" by fact
show "(\<Sum>x \<in> space M. log b (real (RN_deriv \<nu> x)) * real (\<nu> {x})) = ?sum"
using RN_deriv_finite_measure[OF ms ac]
- by (auto intro!: setsum_cong simp: field_simps real_of_pinfreal_mult[symmetric])
+ by (auto intro!: setsum_cong simp: field_simps real_of_pextreal_mult[symmetric])
qed
lemma (in finite_prob_space) KL_divergence_positive_finite:
@@ -285,7 +285,7 @@
note jd_commute = this
{ fix A assume A: "A \<in> sets (sigma (pair_algebra T S))"
- have *: "\<And>x y. indicator ((\<lambda>(x, y). (y, x)) ` A) (x, y) = (indicator A (y, x) :: pinfreal)"
+ have *: "\<And>x y. indicator ((\<lambda>(x, y). (y, x)) ` A) (x, y) = (indicator A (y, x) :: pextreal)"
unfolding indicator_def by auto
have "ST.pair_measure ((\<lambda>(x, y). (y, x)) ` A) = TS.pair_measure A"
unfolding ST.pair_measure_def TS.pair_measure_def
@@ -361,7 +361,7 @@
show ?sum
unfolding Let_def mutual_information_def
by (subst XY.KL_divergence_eq_finite[OF P_ms finite_variables_absolutely_continuous[OF MX MY]])
- (auto simp add: pair_algebra_def setsum_cartesian_product' real_of_pinfreal_mult[symmetric])
+ (auto simp add: pair_algebra_def setsum_cartesian_product' real_of_pextreal_mult[symmetric])
show ?positive
using XY.KL_divergence_positive_finite[OF P_ps finite_variables_absolutely_continuous[OF MX MY] b_gt_1]
@@ -463,7 +463,7 @@
by (auto simp: simple_function_def)
also have "\<dots> = log b (\<Sum>x\<in>X`space M. if ?d x \<noteq> 0 then 1 else 0)"
using distribution_finite[OF `simple_function X`[THEN simple_function_imp_random_variable], simplified]
- by (intro arg_cong[where f="\<lambda>X. log b X"] setsum_cong) (auto simp: real_of_pinfreal_eq_0)
+ by (intro arg_cong[where f="\<lambda>X. log b X"] setsum_cong) (auto simp: real_of_pextreal_eq_0)
finally show ?thesis
using `simple_function X` by (auto simp: setsum_cases real_eq_of_nat simple_function_def)
qed
@@ -610,14 +610,14 @@
then have *: "?XYZ x y z / (?XZ x z * ?YZdZ y z) =
(?XYZ x y z / (?X x * ?YZ y z)) / (?XZ x z / (?X x * ?Z z))"
using order1(3)
- by (auto simp: real_of_pinfreal_mult[symmetric] real_of_pinfreal_eq_0)
+ by (auto simp: real_of_pextreal_mult[symmetric] real_of_pextreal_eq_0)
show "?L x y z = ?R x y z"
proof cases
assume "?XYZ x y z \<noteq> 0"
with space b_gt_1 order1 order2 show ?thesis unfolding *
by (subst log_divide)
- (auto simp: zero_less_divide_iff zero_less_real_of_pinfreal
- real_of_pinfreal_eq_0 zero_less_mult_iff)
+ (auto simp: zero_less_divide_iff zero_less_real_of_pextreal
+ real_of_pextreal_eq_0 zero_less_mult_iff)
qed simp
qed
also have "\<dots> = (\<Sum>(x, y, z)\<in>?S. ?XYZ x y z * log b (?XYZ x y z / (?X x * ?YZ y z))) -
@@ -721,7 +721,7 @@
have "- conditional_mutual_information b MX MY MZ X Y Z = - (\<Sum>(x, y, z) \<in> ?M. ?dXYZ {(x, y, z)} *
log b (?dXYZ {(x, y, z)} / (?dXZ {(x, z)} * ?dYZ {(y,z)} / ?dZ {z})))"
unfolding conditional_mutual_information_generic_eq[OF assms] neg_equal_iff_equal
- by (intro setsum_cong) (auto intro!: arg_cong[where f="log b"] simp: real_of_pinfreal_mult[symmetric])
+ by (intro setsum_cong) (auto intro!: arg_cong[where f="log b"] simp: real_of_pextreal_mult[symmetric])
also have "\<dots> \<le> log b (\<Sum>(x, y, z) \<in> ?M. ?dXZ {(x, z)} * ?dYZ {(y,z)} / ?dZ {z})"
unfolding split_beta
proof (rule log_setsum_divide)
@@ -743,15 +743,15 @@
fix x assume "x \<in> ?M"
let ?x = "(fst x, fst (snd x), snd (snd x))"
- show "0 \<le> ?dXYZ {?x}" using real_pinfreal_nonneg .
+ show "0 \<le> ?dXYZ {?x}" using real_pextreal_nonneg .
show "0 \<le> ?dXZ {(fst x, snd (snd x))} * ?dYZ {(fst (snd x), snd (snd x))} / ?dZ {snd (snd x)}"
- by (simp add: real_pinfreal_nonneg mult_nonneg_nonneg divide_nonneg_nonneg)
+ by (simp add: real_pextreal_nonneg mult_nonneg_nonneg divide_nonneg_nonneg)
assume *: "0 < ?dXYZ {?x}"
with `x \<in> ?M` show "0 < ?dXZ {(fst x, snd (snd x))} * ?dYZ {(fst (snd x), snd (snd x))} / ?dZ {snd (snd x)}"
using finite order
by (cases x)
- (auto simp add: zero_less_real_of_pinfreal zero_less_mult_iff zero_less_divide_iff)
+ (auto simp add: zero_less_real_of_pextreal zero_less_mult_iff zero_less_divide_iff)
qed
also have "(\<Sum>(x, y, z) \<in> ?M. ?dXZ {(x, z)} * ?dYZ {(y,z)} / ?dZ {z}) = (\<Sum>z\<in>space MZ. ?dZ {z})"
apply (simp add: setsum_cartesian_product')
@@ -817,11 +817,11 @@
also have "\<dots> = real (?XZ x z) * ?f x x z"
using `x \<in> space MX` by (simp add: setsum_cases[OF MX.finite_space])
also have "\<dots> = real (?XZ x z) * log b (real (?Z z) / real (?XZ x z))"
- by (auto simp: real_of_pinfreal_mult[symmetric])
+ by (auto simp: real_of_pextreal_mult[symmetric])
also have "\<dots> = - real (?XZ x z) * log b (real (?XZ x z) / real (?Z z))"
using assms[THEN finite_distribution_finite]
using finite_distribution_order(6)[OF MX MZ]
- by (auto simp: log_simps field_simps zero_less_mult_iff zero_less_real_of_pinfreal real_of_pinfreal_eq_0)
+ by (auto simp: log_simps field_simps zero_less_mult_iff zero_less_real_of_pextreal real_of_pextreal_eq_0)
finally have "(\<Sum>x'\<in>space MX. real (?XXZ x x' z) * ?f x x' z) =
- real (?XZ x z) * log b (real (?XZ x z) / real (?Z z))" . }
note * = this
@@ -830,7 +830,7 @@
unfolding conditional_entropy_def
unfolding conditional_mutual_information_generic_eq[OF MX MX MZ]
by (auto simp: setsum_cartesian_product' setsum_negf[symmetric]
- setsum_commute[of _ "space MZ"] * simp del: divide_pinfreal_def
+ setsum_commute[of _ "space MZ"] * simp del: divide_pextreal_def
intro!: setsum_cong)
qed
@@ -853,7 +853,7 @@
using finite_distribution_finite[OF finite_random_variable_pairI[OF assms[THEN simple_function_imp_finite_random_variable]]]
using finite_distribution_order(5,6)[OF assms[THEN simple_function_imp_finite_random_variable]]
using finite_distribution_finite[OF Y[THEN simple_function_imp_finite_random_variable]]
- by (auto simp: setsum_cartesian_product' setsum_commute[of _ "Y`space M"] setsum_right_distrib real_of_pinfreal_eq_0
+ by (auto simp: setsum_cartesian_product' setsum_commute[of _ "Y`space M"] setsum_right_distrib real_of_pextreal_eq_0
intro!: setsum_cong)
lemma (in information_space) conditional_entropy_eq_cartesian_product:
@@ -880,8 +880,8 @@
{ fix x z assume "x \<in> X`space M" "z \<in> Z`space M"
have "?XZ x z * log b (?XZ x z / (?X x * ?Z z)) =
?XZ x z * log b (?XZ x z / ?Z z) - ?XZ x z * log b (?X x)"
- by (auto simp: log_simps real_of_pinfreal_mult[symmetric] zero_less_mult_iff
- zero_less_real_of_pinfreal field_simps real_of_pinfreal_eq_0 abs_mult) }
+ by (auto simp: log_simps real_of_pextreal_mult[symmetric] zero_less_mult_iff
+ zero_less_real_of_pextreal field_simps real_of_pextreal_eq_0 abs_mult) }
note * = this
show ?thesis
unfolding entropy_eq[OF X] conditional_entropy_eq[OF X Z] mutual_information_eq[OF X Z]
@@ -913,8 +913,8 @@
{ fix x y assume "x \<in> X`space M" "y \<in> Y`space M"
have "?XY x y * log b (?XY x y / ?X x) =
?XY x y * log b (?XY x y) - ?XY x y * log b (?X x)"
- by (auto simp: log_simps real_of_pinfreal_mult[symmetric] zero_less_mult_iff
- zero_less_real_of_pinfreal field_simps real_of_pinfreal_eq_0 abs_mult) }
+ by (auto simp: log_simps real_of_pextreal_mult[symmetric] zero_less_mult_iff
+ zero_less_real_of_pextreal field_simps real_of_pextreal_eq_0 abs_mult) }
note * = this
show ?thesis
using setsum_real_joint_distribution_singleton[OF fY fX]
--- a/src/HOL/Probability/Lebesgue_Integration.thy Mon Dec 06 19:18:02 2010 +0100
+++ b/src/HOL/Probability/Lebesgue_Integration.thy Fri Dec 03 15:25:14 2010 +0100
@@ -54,7 +54,7 @@
qed
lemma (in sigma_algebra) simple_function_indicator_representation:
- fixes f ::"'a \<Rightarrow> pinfreal"
+ fixes f ::"'a \<Rightarrow> pextreal"
assumes f: "simple_function f" and x: "x \<in> space M"
shows "f x = (\<Sum>y \<in> f ` space M. y * indicator (f -` {y} \<inter> space M) x)"
(is "?l = ?r")
@@ -69,7 +69,7 @@
qed
lemma (in measure_space) simple_function_notspace:
- "simple_function (\<lambda>x. h x * indicator (- space M) x::pinfreal)" (is "simple_function ?h")
+ "simple_function (\<lambda>x. h x * indicator (- space M) x::pextreal)" (is "simple_function ?h")
proof -
have "?h ` space M \<subseteq> {0}" unfolding indicator_def by auto
hence [simp, intro]: "finite (?h ` space M)" by (auto intro: finite_subset)
@@ -212,7 +212,7 @@
qed
lemma (in sigma_algebra) borel_measurable_implies_simple_function_sequence:
- fixes u :: "'a \<Rightarrow> pinfreal"
+ fixes u :: "'a \<Rightarrow> pextreal"
assumes u: "u \<in> borel_measurable M"
shows "\<exists>f. (\<forall>i. simple_function (f i) \<and> (\<forall>x\<in>space M. f i x \<noteq> \<omega>)) \<and> f \<up> u"
proof -
@@ -265,7 +265,7 @@
qed simp }
note f_upper = this
- let "?g j x" = "of_nat (f x j) / 2^j :: pinfreal"
+ let "?g j x" = "of_nat (f x j) / 2^j :: pextreal"
show ?thesis unfolding simple_function_def isoton_fun_expand unfolding isoton_def
proof (safe intro!: exI[of _ ?g])
fix j
@@ -350,7 +350,7 @@
hence mono: "mono (\<lambda>i. ?g i t)" unfolding mono_iff_le_Suc by auto
show "(SUP j. of_nat (f t j) / 2 ^ j) = u t"
- proof (rule pinfreal_SUPI)
+ proof (rule pextreal_SUPI)
fix j show "of_nat (f t j) / 2 ^ j \<le> u t"
proof (rule fI)
assume "of_nat j \<le> u t" thus "of_nat (j * 2 ^ j) / 2 ^ j \<le> u t"
@@ -362,7 +362,7 @@
(auto simp: power_le_zero_eq divide_real_def[symmetric] field_simps zero_le_mult_iff)
qed
next
- fix y :: pinfreal assume *: "\<And>j. j \<in> UNIV \<Longrightarrow> of_nat (f t j) / 2 ^ j \<le> y"
+ fix y :: pextreal assume *: "\<And>j. j \<in> UNIV \<Longrightarrow> of_nat (f t j) / 2 ^ j \<le> y"
show "u t \<le> y"
proof (cases "u t")
case (preal r)
@@ -404,7 +404,7 @@
qed
lemma (in sigma_algebra) borel_measurable_implies_simple_function_sequence':
- fixes u :: "'a \<Rightarrow> pinfreal"
+ fixes u :: "'a \<Rightarrow> pextreal"
assumes "u \<in> borel_measurable M"
obtains (x) f where "f \<up> u" "\<And>i. simple_function (f i)" "\<And>i. \<omega>\<notin>f i`space M"
proof -
@@ -416,7 +416,7 @@
qed
lemma (in sigma_algebra) simple_function_eq_borel_measurable:
- fixes f :: "'a \<Rightarrow> pinfreal"
+ fixes f :: "'a \<Rightarrow> pextreal"
shows "simple_function f \<longleftrightarrow>
finite (f`space M) \<and> f \<in> borel_measurable M"
using simple_function_borel_measurable[of f]
@@ -424,7 +424,7 @@
by (fastsimp simp: simple_function_def)
lemma (in measure_space) simple_function_restricted:
- fixes f :: "'a \<Rightarrow> pinfreal" assumes "A \<in> sets M"
+ fixes f :: "'a \<Rightarrow> pextreal" assumes "A \<in> sets M"
shows "sigma_algebra.simple_function (restricted_space A) f \<longleftrightarrow> simple_function (\<lambda>x. f x * indicator A x)"
(is "sigma_algebra.simple_function ?R f \<longleftrightarrow> simple_function ?f")
proof -
@@ -448,7 +448,7 @@
using `A \<in> sets M` sets_into_space by (auto intro!: bexI[of _ x])
next
fix x
- assume "indicator A x \<noteq> (0::pinfreal)"
+ assume "indicator A x \<noteq> (0::pextreal)"
then have "x \<in> A" by (auto simp: indicator_def split: split_if_asm)
moreover assume "x \<in> space M" "\<forall>y\<in>A. ?f x \<noteq> f y"
ultimately show "f x = 0" by auto
@@ -472,7 +472,7 @@
by auto
lemma (in sigma_algebra) simple_function_vimage:
- fixes g :: "'a \<Rightarrow> pinfreal" and f :: "'d \<Rightarrow> 'a"
+ fixes g :: "'a \<Rightarrow> pextreal" and f :: "'d \<Rightarrow> 'a"
assumes g: "simple_function g" and f: "f \<in> S \<rightarrow> space M"
shows "sigma_algebra.simple_function (vimage_algebra S f) (\<lambda>x. g (f x))"
proof -
@@ -751,7 +751,7 @@
assume "space M = {}" hence "A = {}" using sets_into_space[OF assms] by auto
thus ?thesis unfolding simple_integral_def using `space M = {}` by auto
next
- assume "space M \<noteq> {}" hence "(\<lambda>x. 1) ` space M = {1::pinfreal}" by auto
+ assume "space M \<noteq> {}" hence "(\<lambda>x. 1) ` space M = {1::pextreal}" by auto
thus ?thesis
using simple_integral_indicator[OF assms simple_function_const[of 1]]
using sets_into_space[OF assms]
@@ -762,7 +762,7 @@
assumes "simple_function u" "N \<in> null_sets"
shows "simple_integral (\<lambda>x. u x * indicator N x) = 0"
proof -
- have "AE x. indicator N x = (0 :: pinfreal)"
+ have "AE x. indicator N x = (0 :: pextreal)"
using `N \<in> null_sets` by (auto simp: indicator_def intro!: AE_I[of _ N])
then have "simple_integral (\<lambda>x. u x * indicator N x) = simple_integral (\<lambda>x. 0)"
using assms by (intro simple_integral_cong_AE) (auto intro!: AE_disjI2)
@@ -806,7 +806,7 @@
by (auto simp: indicator_def split: split_if_asm)
then show "f x * \<mu> (f -` {f x} \<inter> A) =
f x * \<mu> (?f -` {f x} \<inter> space M)"
- unfolding pinfreal_mult_cancel_left by auto
+ unfolding pextreal_mult_cancel_left by auto
qed
lemma (in measure_space) simple_integral_subalgebra[simp]:
@@ -816,7 +816,7 @@
unfolding measure_space.simple_integral_def_raw[OF assms] by simp
lemma (in measure_space) simple_integral_vimage:
- fixes g :: "'a \<Rightarrow> pinfreal" and f :: "'d \<Rightarrow> 'a"
+ fixes g :: "'a \<Rightarrow> pextreal" and f :: "'d \<Rightarrow> 'a"
assumes f: "bij_betw f S (space M)"
shows "simple_integral g =
measure_space.simple_integral (vimage_algebra S f) (\<lambda>A. \<mu> (f ` A)) (\<lambda>x. g (f x))"
@@ -893,7 +893,7 @@
using `\<mu> ?G \<noteq> 0` by (auto simp: indicator_def split: split_if_asm)
have "x < (of_nat n / (if \<mu> ?G = \<omega> then 1 else \<mu> ?G)) * \<mu> ?G"
using n `\<mu> ?G \<noteq> 0` `0 < n`
- by (auto simp: pinfreal_noteq_omega_Ex field_simps)
+ by (auto simp: pextreal_noteq_omega_Ex field_simps)
also have "\<dots> = simple_integral ?g" using g `space M \<noteq> {}`
by (subst simple_integral_indicator)
(auto simp: image_constant ac_simps dest: simple_functionD)
@@ -950,7 +950,7 @@
assumes "simple_function f"
shows "positive_integral f = simple_integral f"
unfolding positive_integral_def
-proof (safe intro!: pinfreal_SUPI)
+proof (safe intro!: pextreal_SUPI)
fix g assume "simple_function g" "g \<le> f"
with assms show "simple_integral g \<le> simple_integral f"
by (auto intro!: simple_integral_mono simp: le_fun_def)
@@ -1017,7 +1017,7 @@
using assms by blast
lemma (in measure_space) positive_integral_vimage:
- fixes g :: "'a \<Rightarrow> pinfreal" and f :: "'d \<Rightarrow> 'a"
+ fixes g :: "'a \<Rightarrow> pextreal" and f :: "'d \<Rightarrow> 'a"
assumes f: "bij_betw f S (space M)"
shows "positive_integral g =
measure_space.positive_integral (vimage_algebra S f) (\<lambda>A. \<mu> (f ` A)) (\<lambda>x. g (f x))"
@@ -1039,14 +1039,14 @@
show ?thesis
unfolding positive_integral_alt1 T.positive_integral_alt1 SUPR_def * image_compose
proof (safe intro!: arg_cong[where f=Sup] image_set_cong, simp_all add: comp_def)
- fix g' :: "'a \<Rightarrow> pinfreal" assume "simple_function g'" "\<forall>x\<in>space M. g' x \<le> g x \<and> g' x \<noteq> \<omega>"
+ fix g' :: "'a \<Rightarrow> pextreal" assume "simple_function g'" "\<forall>x\<in>space M. g' x \<le> g x \<and> g' x \<noteq> \<omega>"
then show "\<exists>h. T.simple_function h \<and> (\<forall>x\<in>S. h x \<le> g (f x) \<and> h x \<noteq> \<omega>) \<and>
T.simple_integral (\<lambda>x. g' (f x)) = T.simple_integral h"
using f unfolding bij_betw_def
by (auto intro!: exI[of _ "\<lambda>x. g' (f x)"]
simp add: le_fun_def simple_function_vimage[OF _ f_fun])
next
- fix g' :: "'d \<Rightarrow> pinfreal" assume g': "T.simple_function g'" "\<forall>x\<in>S. g' x \<le> g (f x) \<and> g' x \<noteq> \<omega>"
+ fix g' :: "'d \<Rightarrow> pextreal" assume g': "T.simple_function g'" "\<forall>x\<in>S. g' x \<le> g (f x) \<and> g' x \<noteq> \<omega>"
let ?g = "\<lambda>x. g' (the_inv_into S f x)"
show "\<exists>h. simple_function h \<and> (\<forall>x\<in>space M. h x \<le> g x \<and> h x \<noteq> \<omega>) \<and>
T.simple_integral g' = T.simple_integral (\<lambda>x. h (f x))"
@@ -1068,7 +1068,7 @@
qed
lemma (in measure_space) positive_integral_vimage_inv:
- fixes g :: "'d \<Rightarrow> pinfreal" and f :: "'d \<Rightarrow> 'a"
+ fixes g :: "'d \<Rightarrow> pextreal" and f :: "'d \<Rightarrow> 'a"
assumes f: "bij_betw f S (space M)"
shows "measure_space.positive_integral (vimage_algebra S f) (\<lambda>A. \<mu> (f ` A)) g =
positive_integral (\<lambda>x. g (the_inv_into S f x))"
@@ -1087,8 +1087,8 @@
and "simple_function u"
and le: "u \<le> s" and real: "\<omega> \<notin> u`space M"
shows "simple_integral u \<le> (SUP i. positive_integral (f i))" (is "_ \<le> ?S")
-proof (rule pinfreal_le_mult_one_interval)
- fix a :: pinfreal assume "0 < a" "a < 1"
+proof (rule pextreal_le_mult_one_interval)
+ fix a :: pextreal assume "0 < a" "a < 1"
hence "a \<noteq> 0" by auto
let "?B i" = "{x \<in> space M. a * u x \<le> f i x}"
have B: "\<And>i. ?B i \<in> sets M"
@@ -1117,7 +1117,7 @@
next
assume "u x \<noteq> 0"
with `a < 1` real `x \<in> space M`
- have "a * u x < 1 * u x" by (rule_tac pinfreal_mult_strict_right_mono) (auto simp: image_iff)
+ have "a * u x < 1 * u x" by (rule_tac pextreal_mult_strict_right_mono) (auto simp: image_iff)
also have "\<dots> \<le> (SUP i. f i x)" using le `f \<up> s`
unfolding isoton_fun_expand by (auto simp: isoton_def le_fun_def)
finally obtain i where "a * u x < f i x" unfolding SUPR_def
@@ -1130,7 +1130,7 @@
have "simple_integral u = (SUP i. simple_integral (?uB i))"
unfolding simple_integral_indicator[OF B `simple_function u`]
- proof (subst SUPR_pinfreal_setsum, safe)
+ proof (subst SUPR_pextreal_setsum, safe)
fix x n assume "x \<in> space M"
have "\<mu> (u -` {u x} \<inter> space M \<inter> {x \<in> space M. a * u x \<le> f n x})
\<le> \<mu> (u -` {u x} \<inter> space M \<inter> {x \<in> space M. a * u x \<le> f (Suc n) x})"
@@ -1142,11 +1142,11 @@
show "simple_integral u =
(\<Sum>i\<in>u ` space M. SUP n. i * \<mu> (u -` {i} \<inter> space M \<inter> ?B n))"
using measure_conv unfolding simple_integral_def isoton_def
- by (auto intro!: setsum_cong simp: pinfreal_SUP_cmult)
+ by (auto intro!: setsum_cong simp: pextreal_SUP_cmult)
qed
moreover
have "a * (SUP i. simple_integral (?uB i)) \<le> ?S"
- unfolding pinfreal_SUP_cmult[symmetric]
+ unfolding pextreal_SUP_cmult[symmetric]
proof (safe intro!: SUP_mono bexI)
fix i
have "a * simple_integral (?uB i) = simple_integral (\<lambda>x. a * ?uB i x)"
@@ -1306,7 +1306,7 @@
case (insert i P)
have "f i \<in> borel_measurable M"
"(\<lambda>x. \<Sum>i\<in>P. f i x) \<in> borel_measurable M"
- using insert by (auto intro!: borel_measurable_pinfreal_setsum)
+ using insert by (auto intro!: borel_measurable_pextreal_setsum)
from positive_integral_add[OF this]
show ?case using insert by auto
qed simp
@@ -1319,7 +1319,7 @@
shows "positive_integral (\<lambda>x. f x - g x) = positive_integral f - positive_integral g"
proof -
have borel: "(\<lambda>x. f x - g x) \<in> borel_measurable M"
- using f g by (rule borel_measurable_pinfreal_diff)
+ using f g by (rule borel_measurable_pextreal_diff)
have "positive_integral (\<lambda>x. f x - g x) + positive_integral g =
positive_integral f"
unfolding positive_integral_add[OF borel g, symmetric]
@@ -1329,7 +1329,7 @@
by (cases "f x", cases "g x", simp, simp, cases "g x", auto)
qed
with mono show ?thesis
- by (subst minus_pinfreal_eq2[OF _ fin]) (auto intro!: positive_integral_mono)
+ by (subst minus_pextreal_eq2[OF _ fin]) (auto intro!: positive_integral_mono)
qed
lemma (in measure_space) positive_integral_psuminf:
@@ -1338,7 +1338,7 @@
proof -
have "(\<lambda>i. positive_integral (\<lambda>x. \<Sum>i<i. f i x)) \<up> positive_integral (\<lambda>x. \<Sum>\<^isub>\<infinity>i. f i x)"
by (rule positive_integral_isoton)
- (auto intro!: borel_measurable_pinfreal_setsum assms positive_integral_mono
+ (auto intro!: borel_measurable_pextreal_setsum assms positive_integral_mono
arg_cong[where f=Sup]
simp: isoton_def le_fun_def psuminf_def fun_eq_iff SUPR_def Sup_fun_def)
thus ?thesis
@@ -1347,7 +1347,7 @@
text {* Fatou's lemma: convergence theorem on limes inferior *}
lemma (in measure_space) positive_integral_lim_INF:
- fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> pinfreal"
+ fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> pextreal"
assumes "\<And>i. u i \<in> borel_measurable M"
shows "positive_integral (SUP n. INF m. u (m + n)) \<le>
(SUP n. INF m. positive_integral (u (m + n)))"
@@ -1421,7 +1421,7 @@
from G(2) have "(\<lambda>i x. f x * G i x) \<up> (\<lambda>x. f x * g x)"
unfolding isoton_fun_expand by (auto intro!: isoton_cmult_right)
then have "(\<lambda>i. positive_integral (\<lambda>x. f x * G i x)) \<up> positive_integral (\<lambda>x. f x * g x)"
- using assms(1) G_borel by (auto intro!: positive_integral_isoton borel_measurable_pinfreal_times)
+ using assms(1) G_borel by (auto intro!: positive_integral_isoton borel_measurable_pextreal_times)
with eq_Tg show "T.positive_integral g = positive_integral (\<lambda>x. f x * g x)"
unfolding isoton_def by simp
qed
@@ -1493,7 +1493,7 @@
next
fix n x assume "1 \<le> of_nat n * u x"
also have "\<dots> \<le> of_nat (Suc n) * u x"
- by (subst (1 2) mult_commute) (auto intro!: pinfreal_mult_cancel)
+ by (subst (1 2) mult_commute) (auto intro!: pextreal_mult_cancel)
finally show "1 \<le> of_nat (Suc n) * u x" .
qed
also have "\<dots> = \<mu> ?A"
@@ -1774,7 +1774,7 @@
using mono by (rule AE_mp) (auto intro!: AE_cong)
ultimately show ?thesis using fg
by (auto simp: integral_def integrable_def diff_minus
- intro!: add_mono real_of_pinfreal_mono positive_integral_mono_AE)
+ intro!: add_mono real_of_pextreal_mono positive_integral_mono_AE)
qed
lemma (in measure_space) integral_mono:
@@ -1861,7 +1861,7 @@
also have "\<dots> \<le> positive_integral (\<lambda>x. Real (f x))"
using f by (auto intro!: positive_integral_mono)
also have "\<dots> < \<omega>"
- using `integrable f` unfolding integrable_def by (auto simp: pinfreal_less_\<omega>)
+ using `integrable f` unfolding integrable_def by (auto simp: pextreal_less_\<omega>)
finally have pos: "positive_integral (\<lambda>x. Real (g x)) < \<omega>" .
have "positive_integral (\<lambda>x. Real (- g x)) \<le> positive_integral (\<lambda>x. Real (\<bar>g x\<bar>))"
@@ -1869,7 +1869,7 @@
also have "\<dots> \<le> positive_integral (\<lambda>x. Real (f x))"
using f by (auto intro!: positive_integral_mono)
also have "\<dots> < \<omega>"
- using `integrable f` unfolding integrable_def by (auto simp: pinfreal_less_\<omega>)
+ using `integrable f` unfolding integrable_def by (auto simp: pextreal_less_\<omega>)
finally have neg: "positive_integral (\<lambda>x. Real (- g x)) < \<omega>" .
from neg pos borel show ?thesis
@@ -2018,7 +2018,7 @@
"positive_integral (\<lambda>x. Real \<bar>f x\<bar>) \<noteq> \<omega>" unfolding integrable_def by auto
from positive_integral_0_iff[OF this(1)] this(2)
show ?thesis unfolding integral_def *
- by (simp add: real_of_pinfreal_eq_0)
+ by (simp add: real_of_pextreal_eq_0)
qed
lemma (in measure_space) positive_integral_omega:
@@ -2125,8 +2125,8 @@
by (auto intro!: positive_integral_lim_INF)
also have "\<dots> = positive_integral (\<lambda>x. Real (2 * w x)) -
(INF n. SUP m. positive_integral (\<lambda>x. Real \<bar>u (m + n) x - u' x\<bar>))"
- unfolding PI_diff pinfreal_INF_minus[OF I2w_fin] pinfreal_SUP_minus ..
- finally show ?thesis using neq_0 I2w_fin by (rule pinfreal_le_minus_imp_0)
+ unfolding PI_diff pextreal_INF_minus[OF I2w_fin] pextreal_SUP_minus ..
+ finally show ?thesis using neq_0 I2w_fin by (rule pextreal_le_minus_imp_0)
qed
have [simp]: "\<And>n m x. Real (- \<bar>u (m + n) x - u' x\<bar>) = 0" by auto
@@ -2260,7 +2260,7 @@
also have "\<dots> = \<bar>enum r\<bar> * real (\<mu> (?A r))"
using f fin by (simp add: borel_measurable_vimage integral_cmul_indicator)
finally have "integral (\<lambda>x. \<bar>?F r x\<bar>) = \<bar>enum r * real (\<mu> (?A r))\<bar>"
- by (simp add: abs_mult_pos real_pinfreal_pos) }
+ by (simp add: abs_mult_pos real_pextreal_pos) }
note int_abs_F = this
have 1: "\<And>i. integrable (\<lambda>x. ?F i x)"
@@ -2329,8 +2329,8 @@
show "integrable f" using finite_space finite_measure
by (simp add: setsum_\<omega> integrable_def)
show ?I using finite_measure
- apply (simp add: integral_def real_of_pinfreal_setsum[symmetric]
- real_of_pinfreal_mult[symmetric] setsum_subtractf[symmetric])
+ apply (simp add: integral_def real_of_pextreal_setsum[symmetric]
+ real_of_pextreal_mult[symmetric] setsum_subtractf[symmetric])
by (rule setsum_cong) (simp_all split: split_if)
qed
--- a/src/HOL/Probability/Lebesgue_Measure.thy Mon Dec 06 19:18:02 2010 +0100
+++ b/src/HOL/Probability/Lebesgue_Measure.thy Fri Dec 03 15:25:14 2010 +0100
@@ -357,7 +357,7 @@
qed
lemma lebesgue_simple_function_indicator:
- fixes f::"'a::ordered_euclidean_space \<Rightarrow> pinfreal"
+ fixes f::"'a::ordered_euclidean_space \<Rightarrow> pextreal"
assumes f:"lebesgue.simple_function f"
shows "f = (\<lambda>x. (\<Sum>y \<in> f ` UNIV. y * indicator (f -` {y}) x))"
apply(rule,subst lebesgue.simple_function_indicator_representation[OF f]) by auto
@@ -421,7 +421,7 @@
lemma lmeasure_singleton[simp]:
fixes a :: "'a::ordered_euclidean_space" shows "lmeasure {a} = 0"
- using has_gmeasure_interval[of a a] unfolding zero_pinfreal_def
+ using has_gmeasure_interval[of a a] unfolding zero_pextreal_def
by (intro has_gmeasure_lmeasure)
(simp add: content_closed_interval DIM_positive)
@@ -475,7 +475,7 @@
qed
lemma simple_function_has_integral:
- fixes f::"'a::ordered_euclidean_space \<Rightarrow> pinfreal"
+ fixes f::"'a::ordered_euclidean_space \<Rightarrow> pextreal"
assumes f:"lebesgue.simple_function f"
and f':"\<forall>x. f x \<noteq> \<omega>"
and om:"\<forall>x\<in>range f. lmeasure (f -` {x} \<inter> UNIV) = \<omega> \<longrightarrow> x = 0"
@@ -486,9 +486,9 @@
have *:"\<And>x. \<forall>y\<in>range f. y * indicator (f -` {y}) x \<noteq> \<omega>"
"\<forall>x\<in>range f. x * lmeasure (f -` {x} \<inter> UNIV) \<noteq> \<omega>"
using f' om unfolding indicator_def by auto
- show ?case unfolding space_lebesgue real_of_pinfreal_setsum'[OF *(1),THEN sym]
- unfolding real_of_pinfreal_setsum'[OF *(2),THEN sym]
- unfolding real_of_pinfreal_setsum space_lebesgue
+ show ?case unfolding space_lebesgue real_of_pextreal_setsum'[OF *(1),THEN sym]
+ unfolding real_of_pextreal_setsum'[OF *(2),THEN sym]
+ unfolding real_of_pextreal_setsum space_lebesgue
apply(rule has_integral_setsum)
proof safe show "finite (range f)" using f by (auto dest: lebesgue.simple_functionD)
fix y::'a show "((\<lambda>x. real (f y * indicator (f -` {f y}) x)) has_integral
@@ -496,8 +496,8 @@
proof(cases "f y = 0") case False
have mea:"gmeasurable (f -` {f y})" apply(rule lmeasure_finite_gmeasurable)
using assms unfolding lebesgue.simple_function_def using False by auto
- have *:"\<And>x. real (indicator (f -` {f y}) x::pinfreal) = (if x \<in> f -` {f y} then 1 else 0)" by auto
- show ?thesis unfolding real_of_pinfreal_mult[THEN sym]
+ have *:"\<And>x. real (indicator (f -` {f y}) x::pextreal) = (if x \<in> f -` {f y} then 1 else 0)" by auto
+ show ?thesis unfolding real_of_pextreal_mult[THEN sym]
apply(rule has_integral_cmul[where 'b=real, unfolded real_scaleR_def])
unfolding Int_UNIV_right lmeasure_gmeasure[OF mea,THEN sym]
unfolding measure_integral_univ[OF mea] * apply(rule integrable_integral)
@@ -510,7 +510,7 @@
using assms by auto
lemma simple_function_has_integral':
- fixes f::"'a::ordered_euclidean_space \<Rightarrow> pinfreal"
+ fixes f::"'a::ordered_euclidean_space \<Rightarrow> pextreal"
assumes f:"lebesgue.simple_function f"
and i: "lebesgue.simple_integral f \<noteq> \<omega>"
shows "((\<lambda>x. real (f x)) has_integral (real (lebesgue.simple_integral f))) UNIV"
@@ -544,7 +544,7 @@
qed
lemma (in measure_space) positive_integral_monotone_convergence:
- fixes f::"nat \<Rightarrow> 'a \<Rightarrow> pinfreal"
+ fixes f::"nat \<Rightarrow> 'a \<Rightarrow> pextreal"
assumes i: "\<And>i. f i \<in> borel_measurable M" and mono: "\<And>x. mono (\<lambda>n. f n x)"
and lim: "\<And>x. (\<lambda>i. f i x) ----> u x"
shows "u \<in> borel_measurable M"
@@ -552,7 +552,7 @@
proof -
from positive_integral_isoton[unfolded isoton_fun_expand isoton_iff_Lim_mono, of f u]
show ?ilim using mono lim i by auto
- have "(SUP i. f i) = u" using mono lim SUP_Lim_pinfreal
+ have "(SUP i. f i) = u" using mono lim SUP_Lim_pextreal
unfolding fun_eq_iff SUPR_fun_expand mono_def by auto
moreover have "(SUP i. f i) \<in> borel_measurable M"
using i by (rule borel_measurable_SUP)
@@ -560,7 +560,7 @@
qed
lemma positive_integral_has_integral:
- fixes f::"'a::ordered_euclidean_space => pinfreal"
+ fixes f::"'a::ordered_euclidean_space => pextreal"
assumes f:"f \<in> borel_measurable lebesgue"
and int_om:"lebesgue.positive_integral f \<noteq> \<omega>"
and f_om:"\<forall>x. f x \<noteq> \<omega>" (* TODO: remove this *)
@@ -581,14 +581,14 @@
have "(\<lambda>x. real (f x)) integrable_on UNIV \<and>
(\<lambda>k. Integration.integral UNIV (\<lambda>x. real (u k x))) ----> Integration.integral UNIV (\<lambda>x. real (f x))"
apply(rule monotone_convergence_increasing) apply(rule,rule,rule u_int)
- proof safe case goal1 show ?case apply(rule real_of_pinfreal_mono) using u(2,3) by auto
+ proof safe case goal1 show ?case apply(rule real_of_pextreal_mono) using u(2,3) by auto
next case goal2 show ?case using u(3) apply(subst lim_Real[THEN sym])
prefer 3 apply(subst Real_real') defer apply(subst Real_real')
using isotone_Lim[OF u(3)[unfolded isoton_fun_expand, THEN spec]] using f_om u by auto
next case goal3
show ?case apply(rule bounded_realI[where B="real (lebesgue.positive_integral f)"])
apply safe apply(subst abs_of_nonneg) apply(rule integral_nonneg,rule) apply(rule u_int)
- unfolding integral_unique[OF u_int] defer apply(rule real_of_pinfreal_mono[OF _ int_u_le])
+ unfolding integral_unique[OF u_int] defer apply(rule real_of_pextreal_mono[OF _ int_u_le])
using u int_om by auto
qed note int = conjunctD2[OF this]
@@ -921,7 +921,7 @@
qed
lemma borel_fubini_positiv_integral:
- fixes f :: "'a::ordered_euclidean_space \<Rightarrow> pinfreal"
+ fixes f :: "'a::ordered_euclidean_space \<Rightarrow> pextreal"
assumes f: "f \<in> borel_measurable borel"
shows "borel.positive_integral f =
borel_product.product_positive_integral {..<DIM('a)} (f \<circ> p2e)"
--- a/src/HOL/Probability/Measure.thy Mon Dec 06 19:18:02 2010 +0100
+++ b/src/HOL/Probability/Measure.thy Fri Dec 03 15:25:14 2010 +0100
@@ -103,7 +103,7 @@
by (rule additiveD [OF additive]) (auto simp add: s)
finally have "\<mu> (space M) = \<mu> s + \<mu> (space M - s)" .
thus ?thesis
- unfolding minus_pinfreal_eq2[OF s_less_space fin]
+ unfolding minus_pextreal_eq2[OF s_less_space fin]
by (simp add: ac_simps)
qed
@@ -117,7 +117,7 @@
have "\<mu> ((A - B) \<union> B) = \<mu> (A - B) + \<mu> B"
using measurable by (rule_tac measure_additive[symmetric]) auto
thus ?thesis unfolding * using `\<mu> B \<noteq> \<omega>`
- by (simp add: pinfreal_cancel_plus_minus)
+ by (simp add: pextreal_cancel_plus_minus)
qed
lemma (in measure_space) measure_countable_increasing:
@@ -225,7 +225,7 @@
by (rule INF_leI) simp
have "\<mu> (A 0) - (INF n. \<mu> (A n)) = (SUP n. \<mu> (A 0 - A n))"
- unfolding pinfreal_SUP_minus[symmetric]
+ unfolding pextreal_SUP_minus[symmetric]
using mono A finite by (subst measure_Diff) auto
also have "\<dots> = \<mu> (\<Union>i. A 0 - A i)"
proof (rule continuity_from_below)
@@ -237,7 +237,7 @@
also have "\<dots> = \<mu> (A 0) - \<mu> (\<Inter>i. A i)"
using mono A finite * by (simp, subst measure_Diff) auto
finally show ?thesis
- by (rule pinfreal_diff_eq_diff_imp_eq[OF finite[of 0] le_IM le_MI])
+ by (rule pextreal_diff_eq_diff_imp_eq[OF finite[of 0] le_IM le_MI])
qed
lemma (in measure_space) measure_down:
@@ -516,7 +516,7 @@
also have "\<dots> \<le> \<mu> (T - S) + \<mu> (S \<inter> T)"
using assms by (auto intro!: measure_subadditive)
also have "\<dots> < \<mu> (T - S) + \<mu> S"
- by (rule pinfreal_less_add[OF not_\<omega>]) (insert contr, auto)
+ by (rule pextreal_less_add[OF not_\<omega>]) (insert contr, auto)
also have "\<dots> = \<mu> (T \<union> S)"
using assms by (subst measure_additive) auto
also have "\<dots> \<le> \<mu> (space M)"
@@ -962,8 +962,8 @@
fix i
have "\<mu> (A i \<inter> S) \<le> \<mu> (A i)"
using `range A \<subseteq> sets M` `S \<in> sets M` by (auto intro!: measure_mono)
- also have "\<dots> < \<omega>" using `\<mu> (A i) \<noteq> \<omega>` by (auto simp: pinfreal_less_\<omega>)
- finally show "\<mu> (A i \<inter> S) \<noteq> \<omega>" by (auto simp: pinfreal_less_\<omega>)
+ also have "\<dots> < \<omega>" using `\<mu> (A i) \<noteq> \<omega>` by (auto simp: pextreal_less_\<omega>)
+ finally show "\<mu> (A i \<inter> S) \<noteq> \<omega>" by (auto simp: pextreal_less_\<omega>)
qed
qed
@@ -1051,14 +1051,14 @@
and measurable: "A \<in> sets M" "B \<in> sets M" "A \<inter> B = {}"
shows "real (\<mu> (A \<union> B)) = real (\<mu> A) + real (\<mu> B)"
unfolding measure_additive[symmetric, OF measurable]
- using finite by (auto simp: real_of_pinfreal_add)
+ using finite by (auto simp: real_of_pextreal_add)
lemma (in measure_space) real_measure_finite_Union:
assumes measurable:
"finite S" "\<And>i. i \<in> S \<Longrightarrow> A i \<in> sets M" "disjoint_family_on A S"
assumes finite: "\<And>i. i \<in> S \<Longrightarrow> \<mu> (A i) \<noteq> \<omega>"
shows "real (\<mu> (\<Union>i\<in>S. A i)) = (\<Sum>i\<in>S. real (\<mu> (A i)))"
- using real_of_pinfreal_setsum[of S, OF finite]
+ using real_of_pextreal_setsum[of S, OF finite]
measure_finitely_additive''[symmetric, OF measurable]
by simp
@@ -1093,9 +1093,9 @@
shows "real (\<mu> (A \<union> B)) \<le> real (\<mu> A) + real (\<mu> B)"
proof -
have "real (\<mu> (A \<union> B)) \<le> real (\<mu> A + \<mu> B)"
- using measure_subadditive[OF measurable] fin by (auto intro!: real_of_pinfreal_mono)
+ using measure_subadditive[OF measurable] fin by (auto intro!: real_of_pextreal_mono)
also have "\<dots> = real (\<mu> A) + real (\<mu> B)"
- using fin by (simp add: real_of_pinfreal_add)
+ using fin by (simp add: real_of_pextreal_add)
finally show ?thesis .
qed
@@ -1104,7 +1104,7 @@
shows "real (\<mu> (\<Union>i. f i)) \<le> (\<Sum> i. real (\<mu> (f i)))"
proof -
have "real (\<mu> (\<Union>i. f i)) \<le> real (\<Sum>\<^isub>\<infinity> i. \<mu> (f i))"
- using assms by (auto intro!: real_of_pinfreal_mono measure_countably_subadditive)
+ using assms by (auto intro!: real_of_pextreal_mono measure_countably_subadditive)
also have "\<dots> = (\<Sum> i. real (\<mu> (f i)))"
using assms by (auto intro!: sums_unique psuminf_imp_suminf)
finally show ?thesis .
@@ -1114,7 +1114,7 @@
assumes S: "finite S" "\<And>x. x \<in> S \<Longrightarrow> {x} \<in> sets M"
and fin: "\<And>x. x \<in> S \<Longrightarrow> \<mu> {x} \<noteq> \<omega>"
shows "real (\<mu> S) = (\<Sum>x\<in>S. real (\<mu> {x}))"
- using measure_finite_singleton[OF S] fin by (simp add: real_of_pinfreal_setsum)
+ using measure_finite_singleton[OF S] fin by (simp add: real_of_pextreal_setsum)
lemma (in measure_space) real_continuity_from_below:
assumes A: "range A \<subseteq> sets M" "\<And>i. A i \<subseteq> A (Suc i)" and "\<mu> (\<Union>i. A i) \<noteq> \<omega>"
@@ -1126,7 +1126,7 @@
note this[simp]
show "mono (\<lambda>i. real (\<mu> (A i)))" unfolding mono_iff_le_Suc using A
- by (auto intro!: real_of_pinfreal_mono measure_mono)
+ by (auto intro!: real_of_pextreal_mono measure_mono)
show "(SUP n. Real (real (\<mu> (A n)))) = Real (real (\<mu> (\<Union>i. A i)))"
using continuity_from_below[OF A(1), OF A(2)]
@@ -1145,7 +1145,7 @@
note this[simp]
show "mono (\<lambda>i. - real (\<mu> (A i)))" unfolding mono_iff_le_Suc using assms
- by (auto intro!: real_of_pinfreal_mono measure_mono)
+ by (auto intro!: real_of_pextreal_mono measure_mono)
show "(INF n. Real (real (\<mu> (A n)))) =
Real (real (\<mu> (\<Inter>i. A i)))"
@@ -1171,8 +1171,8 @@
hence "\<mu> A \<le> \<mu> (space M)"
using assms top by (rule measure_mono)
also have "\<dots> < \<omega>"
- using finite_measure_of_space unfolding pinfreal_less_\<omega> .
- finally show ?thesis unfolding pinfreal_less_\<omega> .
+ using finite_measure_of_space unfolding pextreal_less_\<omega> .
+ finally show ?thesis unfolding pextreal_less_\<omega> .
qed
lemma (in finite_measure) restricted_finite_measure:
@@ -1226,7 +1226,7 @@
lemma (in finite_measure) real_measure_mono:
"A \<in> sets M \<Longrightarrow> B \<in> sets M \<Longrightarrow> A \<subseteq> B \<Longrightarrow> real (\<mu> A) \<le> real (\<mu> B)"
- by (auto intro!: measure_mono real_of_pinfreal_mono finite_measure)
+ by (auto intro!: measure_mono real_of_pextreal_mono finite_measure)
lemma (in finite_measure) real_finite_measure_subadditive:
assumes measurable: "A \<in> sets M" "B \<in> sets M"
@@ -1449,13 +1449,13 @@
assumes "disjoint_family A" "x \<in> A i"
shows "(\<Sum>\<^isub>\<infinity> n. f n * indicator (A n) x) = f i"
proof -
- have **: "\<And>n. f n * indicator (A n) x = (if n = i then f n else 0 :: pinfreal)"
+ have **: "\<And>n. f n * indicator (A n) x = (if n = i then f n else 0 :: pextreal)"
using `x \<in> A i` assms unfolding disjoint_family_on_def indicator_def by auto
- then have "\<And>n. (\<Sum>j<n. f j * indicator (A j) x) = (if i < n then f i else 0 :: pinfreal)"
+ then have "\<And>n. (\<Sum>j<n. f j * indicator (A j) x) = (if i < n then f i else 0 :: pextreal)"
by (auto simp: setsum_cases)
- moreover have "(SUP n. if i < n then f i else 0) = (f i :: pinfreal)"
- proof (rule pinfreal_SUPI)
- fix y :: pinfreal assume "\<And>n. n \<in> UNIV \<Longrightarrow> (if i < n then f i else 0) \<le> y"
+ moreover have "(SUP n. if i < n then f i else 0) = (f i :: pextreal)"
+ proof (rule pextreal_SUPI)
+ fix y :: pextreal assume "\<And>n. n \<in> UNIV \<Longrightarrow> (if i < n then f i else 0) \<le> y"
from this[of "Suc i"] show "f i \<le> y" by auto
qed simp
ultimately show ?thesis using `x \<in> A i` unfolding psuminf_def by auto
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Probability/Positive_Extended_Real.thy Fri Dec 03 15:25:14 2010 +0100
@@ -0,0 +1,2775 @@
+(* Author: Johannes Hoelzl, TU Muenchen *)
+
+header {* A type for positive real numbers with infinity *}
+
+theory Positive_Extended_Real
+ imports Complex_Main Nat_Bijection Multivariate_Analysis
+begin
+
+lemma (in complete_lattice) Sup_start:
+ assumes *: "\<And>x. f x \<le> f 0"
+ shows "(SUP n. f n) = f 0"
+proof (rule antisym)
+ show "f 0 \<le> (SUP n. f n)" by (rule le_SUPI) auto
+ show "(SUP n. f n) \<le> f 0" by (rule SUP_leI[OF *])
+qed
+
+lemma (in complete_lattice) Inf_start:
+ assumes *: "\<And>x. f 0 \<le> f x"
+ shows "(INF n. f n) = f 0"
+proof (rule antisym)
+ show "(INF n. f n) \<le> f 0" by (rule INF_leI) simp
+ show "f 0 \<le> (INF n. f n)" by (rule le_INFI[OF *])
+qed
+
+lemma (in complete_lattice) Sup_mono_offset:
+ fixes f :: "'b :: {ordered_ab_semigroup_add,monoid_add} \<Rightarrow> 'a"
+ assumes *: "\<And>x y. x \<le> y \<Longrightarrow> f x \<le> f y" and "0 \<le> k"
+ shows "(SUP n . f (k + n)) = (SUP n. f n)"
+proof (rule antisym)
+ show "(SUP n. f (k + n)) \<le> (SUP n. f n)"
+ by (auto intro!: Sup_mono simp: SUPR_def)
+ { fix n :: 'b
+ have "0 + n \<le> k + n" using `0 \<le> k` by (rule add_right_mono)
+ with * have "f n \<le> f (k + n)" by simp }
+ thus "(SUP n. f n) \<le> (SUP n. f (k + n))"
+ by (auto intro!: Sup_mono exI simp: SUPR_def)
+qed
+
+lemma (in complete_lattice) Sup_mono_offset_Suc:
+ assumes *: "\<And>x. f x \<le> f (Suc x)"
+ shows "(SUP n . f (Suc n)) = (SUP n. f n)"
+ unfolding Suc_eq_plus1
+ apply (subst add_commute)
+ apply (rule Sup_mono_offset)
+ by (auto intro!: order.lift_Suc_mono_le[of "op \<le>" "op <" f, OF _ *]) default
+
+lemma (in complete_lattice) Inf_mono_offset:
+ fixes f :: "'b :: {ordered_ab_semigroup_add,monoid_add} \<Rightarrow> 'a"
+ assumes *: "\<And>x y. x \<le> y \<Longrightarrow> f y \<le> f x" and "0 \<le> k"
+ shows "(INF n . f (k + n)) = (INF n. f n)"
+proof (rule antisym)
+ show "(INF n. f n) \<le> (INF n. f (k + n))"
+ by (auto intro!: Inf_mono simp: INFI_def)
+ { fix n :: 'b
+ have "0 + n \<le> k + n" using `0 \<le> k` by (rule add_right_mono)
+ with * have "f (k + n) \<le> f n" by simp }
+ thus "(INF n. f (k + n)) \<le> (INF n. f n)"
+ by (auto intro!: Inf_mono exI simp: INFI_def)
+qed
+
+lemma (in complete_lattice) isotone_converge:
+ fixes f :: "nat \<Rightarrow> 'a" assumes "\<And>x y. x \<le> y \<Longrightarrow> f x \<le> f y "
+ shows "(INF n. SUP m. f (n + m)) = (SUP n. INF m. f (n + m))"
+proof -
+ have "\<And>n. (SUP m. f (n + m)) = (SUP n. f n)"
+ apply (rule Sup_mono_offset)
+ apply (rule assms)
+ by simp_all
+ moreover
+ { fix n have "(INF m. f (n + m)) = f n"
+ using Inf_start[of "\<lambda>m. f (n + m)"] assms by simp }
+ ultimately show ?thesis by simp
+qed
+
+lemma (in complete_lattice) antitone_converges:
+ fixes f :: "nat \<Rightarrow> 'a" assumes "\<And>x y. x \<le> y \<Longrightarrow> f y \<le> f x"
+ shows "(INF n. SUP m. f (n + m)) = (SUP n. INF m. f (n + m))"
+proof -
+ have "\<And>n. (INF m. f (n + m)) = (INF n. f n)"
+ apply (rule Inf_mono_offset)
+ apply (rule assms)
+ by simp_all
+ moreover
+ { fix n have "(SUP m. f (n + m)) = f n"
+ using Sup_start[of "\<lambda>m. f (n + m)"] assms by simp }
+ ultimately show ?thesis by simp
+qed
+
+lemma (in complete_lattice) lim_INF_le_lim_SUP:
+ fixes f :: "nat \<Rightarrow> 'a"
+ shows "(SUP n. INF m. f (n + m)) \<le> (INF n. SUP m. f (n + m))"
+proof (rule SUP_leI, rule le_INFI)
+ fix i j show "(INF m. f (i + m)) \<le> (SUP m. f (j + m))"
+ proof (cases rule: le_cases)
+ assume "i \<le> j"
+ have "(INF m. f (i + m)) \<le> f (i + (j - i))" by (rule INF_leI) simp
+ also have "\<dots> = f (j + 0)" using `i \<le> j` by auto
+ also have "\<dots> \<le> (SUP m. f (j + m))" by (rule le_SUPI) simp
+ finally show ?thesis .
+ next
+ assume "j \<le> i"
+ have "(INF m. f (i + m)) \<le> f (i + 0)" by (rule INF_leI) simp
+ also have "\<dots> = f (j + (i - j))" using `j \<le> i` by auto
+ also have "\<dots> \<le> (SUP m. f (j + m))" by (rule le_SUPI) simp
+ finally show ?thesis .
+ qed
+qed
+
+text {*
+
+We introduce the the positive real numbers as needed for measure theory.
+
+*}
+
+typedef pextreal = "(Some ` {0::real..}) \<union> {None}"
+ by (rule exI[of _ None]) simp
+
+subsection "Introduce @{typ pextreal} similar to a datatype"
+
+definition "Real x = Abs_pextreal (Some (sup 0 x))"
+definition "\<omega> = Abs_pextreal None"
+
+definition "pextreal_case f i x = (if x = \<omega> then i else f (THE r. 0 \<le> r \<and> x = Real r))"
+
+definition "of_pextreal = pextreal_case (\<lambda>x. x) 0"
+
+defs (overloaded)
+ real_of_pextreal_def [code_unfold]: "real == of_pextreal"
+
+lemma pextreal_Some[simp]: "0 \<le> x \<Longrightarrow> Some x \<in> pextreal"
+ unfolding pextreal_def by simp
+
+lemma pextreal_Some_sup[simp]: "Some (sup 0 x) \<in> pextreal"
+ by (simp add: sup_ge1)
+
+lemma pextreal_None[simp]: "None \<in> pextreal"
+ unfolding pextreal_def by simp
+
+lemma Real_inj[simp]:
+ assumes "0 \<le> x" and "0 \<le> y"
+ shows "Real x = Real y \<longleftrightarrow> x = y"
+ unfolding Real_def assms[THEN sup_absorb2]
+ using assms by (simp add: Abs_pextreal_inject)
+
+lemma Real_neq_\<omega>[simp]:
+ "Real x = \<omega> \<longleftrightarrow> False"
+ "\<omega> = Real x \<longleftrightarrow> False"
+ by (simp_all add: Abs_pextreal_inject \<omega>_def Real_def)
+
+lemma Real_neg: "x < 0 \<Longrightarrow> Real x = Real 0"
+ unfolding Real_def by (auto simp add: Abs_pextreal_inject intro!: sup_absorb1)
+
+lemma pextreal_cases[case_names preal infinite, cases type: pextreal]:
+ assumes preal: "\<And>r. x = Real r \<Longrightarrow> 0 \<le> r \<Longrightarrow> P" and inf: "x = \<omega> \<Longrightarrow> P"
+ shows P
+proof (cases x rule: pextreal.Abs_pextreal_cases)
+ case (Abs_pextreal y)
+ hence "y = None \<or> (\<exists>x \<ge> 0. y = Some x)"
+ unfolding pextreal_def by auto
+ thus P
+ proof (rule disjE)
+ assume "\<exists>x\<ge>0. y = Some x" then guess x ..
+ thus P by (simp add: preal[of x] Real_def Abs_pextreal(1) sup_absorb2)
+ qed (simp add: \<omega>_def Abs_pextreal(1) inf)
+qed
+
+lemma pextreal_case_\<omega>[simp]: "pextreal_case f i \<omega> = i"
+ unfolding pextreal_case_def by simp
+
+lemma pextreal_case_Real[simp]: "pextreal_case f i (Real x) = (if 0 \<le> x then f x else f 0)"
+proof (cases "0 \<le> x")
+ case True thus ?thesis unfolding pextreal_case_def by (auto intro: theI2)
+next
+ case False
+ moreover have "(THE r. 0 \<le> r \<and> Real 0 = Real r) = 0"
+ by (auto intro!: the_equality)
+ ultimately show ?thesis unfolding pextreal_case_def by (simp add: Real_neg)
+qed
+
+lemma pextreal_case_cancel[simp]: "pextreal_case (\<lambda>c. i) i x = i"
+ by (cases x) simp_all
+
+lemma pextreal_case_split:
+ "P (pextreal_case f i x) = ((x = \<omega> \<longrightarrow> P i) \<and> (\<forall>r\<ge>0. x = Real r \<longrightarrow> P (f r)))"
+ by (cases x) simp_all
+
+lemma pextreal_case_split_asm:
+ "P (pextreal_case f i x) = (\<not> (x = \<omega> \<and> \<not> P i \<or> (\<exists>r. r \<ge> 0 \<and> x = Real r \<and> \<not> P (f r))))"
+ by (cases x) auto
+
+lemma pextreal_case_cong[cong]:
+ assumes eq: "x = x'" "i = i'" and cong: "\<And>r. 0 \<le> r \<Longrightarrow> f r = f' r"
+ shows "pextreal_case f i x = pextreal_case f' i' x'"
+ unfolding eq using cong by (cases x') simp_all
+
+lemma real_Real[simp]: "real (Real x) = (if 0 \<le> x then x else 0)"
+ unfolding real_of_pextreal_def of_pextreal_def by simp
+
+lemma Real_real_image:
+ assumes "\<omega> \<notin> A" shows "Real ` real ` A = A"
+proof safe
+ fix x assume "x \<in> A"
+ hence *: "x = Real (real x)"
+ using `\<omega> \<notin> A` by (cases x) auto
+ show "x \<in> Real ` real ` A"
+ using `x \<in> A` by (subst *) (auto intro!: imageI)
+next
+ fix x assume "x \<in> A"
+ thus "Real (real x) \<in> A"
+ using `\<omega> \<notin> A` by (cases x) auto
+qed
+
+lemma real_pextreal_nonneg[simp, intro]: "0 \<le> real (x :: pextreal)"
+ unfolding real_of_pextreal_def of_pextreal_def
+ by (cases x) auto
+
+lemma real_\<omega>[simp]: "real \<omega> = 0"
+ unfolding real_of_pextreal_def of_pextreal_def by simp
+
+lemma pextreal_noteq_omega_Ex: "X \<noteq> \<omega> \<longleftrightarrow> (\<exists>r\<ge>0. X = Real r)" by (cases X) auto
+
+subsection "@{typ pextreal} is a monoid for addition"
+
+instantiation pextreal :: comm_monoid_add
+begin
+
+definition "0 = Real 0"
+definition "x + y = pextreal_case (\<lambda>r. pextreal_case (\<lambda>p. Real (r + p)) \<omega> y) \<omega> x"
+
+lemma pextreal_plus[simp]:
+ "Real r + Real p = (if 0 \<le> r then if 0 \<le> p then Real (r + p) else Real r else Real p)"
+ "x + 0 = x"
+ "0 + x = x"
+ "x + \<omega> = \<omega>"
+ "\<omega> + x = \<omega>"
+ by (simp_all add: plus_pextreal_def Real_neg zero_pextreal_def split: pextreal_case_split)
+
+lemma \<omega>_neq_0[simp]:
+ "\<omega> = 0 \<longleftrightarrow> False"
+ "0 = \<omega> \<longleftrightarrow> False"
+ by (simp_all add: zero_pextreal_def)
+
+lemma Real_eq_0[simp]:
+ "Real r = 0 \<longleftrightarrow> r \<le> 0"
+ "0 = Real r \<longleftrightarrow> r \<le> 0"
+ by (auto simp add: Abs_pextreal_inject zero_pextreal_def Real_def sup_real_def)
+
+lemma Real_0[simp]: "Real 0 = 0" by (simp add: zero_pextreal_def)
+
+instance
+proof
+ fix a :: pextreal
+ show "0 + a = a" by (cases a) simp_all
+
+ fix b show "a + b = b + a"
+ by (cases a, cases b) simp_all
+
+ fix c show "a + b + c = a + (b + c)"
+ by (cases a, cases b, cases c) simp_all
+qed
+end
+
+lemma pextreal_plus_eq_\<omega>[simp]: "(a :: pextreal) + b = \<omega> \<longleftrightarrow> a = \<omega> \<or> b = \<omega>"
+ by (cases a, cases b) auto
+
+lemma pextreal_add_cancel_left:
+ "a + b = a + c \<longleftrightarrow> (a = \<omega> \<or> b = c)"
+ by (cases a, cases b, cases c, simp_all, cases c, simp_all)
+
+lemma pextreal_add_cancel_right:
+ "b + a = c + a \<longleftrightarrow> (a = \<omega> \<or> b = c)"
+ by (cases a, cases b, cases c, simp_all, cases c, simp_all)
+
+lemma Real_eq_Real:
+ "Real a = Real b \<longleftrightarrow> (a = b \<or> (a \<le> 0 \<and> b \<le> 0))"
+proof (cases "a \<le> 0 \<or> b \<le> 0")
+ case False with Real_inj[of a b] show ?thesis by auto
+next
+ case True
+ thus ?thesis
+ proof
+ assume "a \<le> 0"
+ hence *: "Real a = 0" by simp
+ show ?thesis using `a \<le> 0` unfolding * by auto
+ next
+ assume "b \<le> 0"
+ hence *: "Real b = 0" by simp
+ show ?thesis using `b \<le> 0` unfolding * by auto
+ qed
+qed
+
+lemma real_pextreal_0[simp]: "real (0 :: pextreal) = 0"
+ unfolding zero_pextreal_def real_Real by simp
+
+lemma real_of_pextreal_eq_0: "real X = 0 \<longleftrightarrow> (X = 0 \<or> X = \<omega>)"
+ by (cases X) auto
+
+lemma real_of_pextreal_eq: "real X = real Y \<longleftrightarrow>
+ (X = Y \<or> (X = 0 \<and> Y = \<omega>) \<or> (Y = 0 \<and> X = \<omega>))"
+ by (cases X, cases Y) (auto simp add: real_of_pextreal_eq_0)
+
+lemma real_of_pextreal_add: "real X + real Y =
+ (if X = \<omega> then real Y else if Y = \<omega> then real X else real (X + Y))"
+ by (auto simp: pextreal_noteq_omega_Ex)
+
+subsection "@{typ pextreal} is a monoid for multiplication"
+
+instantiation pextreal :: comm_monoid_mult
+begin
+
+definition "1 = Real 1"
+definition "x * y = (if x = 0 \<or> y = 0 then 0 else
+ pextreal_case (\<lambda>r. pextreal_case (\<lambda>p. Real (r * p)) \<omega> y) \<omega> x)"
+
+lemma pextreal_times[simp]:
+ "Real r * Real p = (if 0 \<le> r \<and> 0 \<le> p then Real (r * p) else 0)"
+ "\<omega> * x = (if x = 0 then 0 else \<omega>)"
+ "x * \<omega> = (if x = 0 then 0 else \<omega>)"
+ "0 * x = 0"
+ "x * 0 = 0"
+ "1 = \<omega> \<longleftrightarrow> False"
+ "\<omega> = 1 \<longleftrightarrow> False"
+ by (auto simp add: times_pextreal_def one_pextreal_def)
+
+lemma pextreal_one_mult[simp]:
+ "Real x + 1 = (if 0 \<le> x then Real (x + 1) else 1)"
+ "1 + Real x = (if 0 \<le> x then Real (1 + x) else 1)"
+ unfolding one_pextreal_def by simp_all
+
+instance
+proof
+ fix a :: pextreal show "1 * a = a"
+ by (cases a) (simp_all add: one_pextreal_def)
+
+ fix b show "a * b = b * a"
+ by (cases a, cases b) (simp_all add: mult_nonneg_nonneg)
+
+ fix c show "a * b * c = a * (b * c)"
+ apply (cases a, cases b, cases c)
+ apply (simp_all add: mult_nonneg_nonneg not_le mult_pos_pos)
+ apply (cases b, cases c)
+ apply (simp_all add: mult_nonneg_nonneg not_le mult_pos_pos)
+ done
+qed
+end
+
+lemma pextreal_mult_cancel_left:
+ "a * b = a * c \<longleftrightarrow> (a = 0 \<or> b = c \<or> (a = \<omega> \<and> b \<noteq> 0 \<and> c \<noteq> 0))"
+ by (cases a, cases b, cases c, auto simp: Real_eq_Real mult_le_0_iff, cases c, auto)
+
+lemma pextreal_mult_cancel_right:
+ "b * a = c * a \<longleftrightarrow> (a = 0 \<or> b = c \<or> (a = \<omega> \<and> b \<noteq> 0 \<and> c \<noteq> 0))"
+ by (cases a, cases b, cases c, auto simp: Real_eq_Real mult_le_0_iff, cases c, auto)
+
+lemma Real_1[simp]: "Real 1 = 1" by (simp add: one_pextreal_def)
+
+lemma real_pextreal_1[simp]: "real (1 :: pextreal) = 1"
+ unfolding one_pextreal_def real_Real by simp
+
+lemma real_of_pextreal_mult: "real X * real Y = real (X * Y :: pextreal)"
+ by (cases X, cases Y) (auto simp: zero_le_mult_iff)
+
+lemma Real_mult_nonneg: assumes "x \<ge> 0" "y \<ge> 0"
+ shows "Real (x * y) = Real x * Real y" using assms by auto
+
+lemma Real_setprod: assumes "\<forall>x\<in>A. f x \<ge> 0" shows "Real (setprod f A) = setprod (\<lambda>x. Real (f x)) A"
+proof(cases "finite A")
+ case True thus ?thesis using assms
+ proof(induct A) case (insert x A)
+ have "0 \<le> setprod f A" apply(rule setprod_nonneg) using insert by auto
+ thus ?case unfolding setprod_insert[OF insert(1-2)] apply-
+ apply(subst Real_mult_nonneg) prefer 3 apply(subst insert(3)[THEN sym])
+ using insert by auto
+ qed auto
+qed auto
+
+subsection "@{typ pextreal} is a linear order"
+
+instantiation pextreal :: linorder
+begin
+
+definition "x < y \<longleftrightarrow> pextreal_case (\<lambda>i. pextreal_case (\<lambda>j. i < j) True y) False x"
+definition "x \<le> y \<longleftrightarrow> pextreal_case (\<lambda>j. pextreal_case (\<lambda>i. i \<le> j) False x) True y"
+
+lemma pextreal_less[simp]:
+ "Real r < \<omega>"
+ "Real r < Real p \<longleftrightarrow> (if 0 \<le> r \<and> 0 \<le> p then r < p else 0 < p)"
+ "\<omega> < x \<longleftrightarrow> False"
+ "0 < \<omega>"
+ "0 < Real r \<longleftrightarrow> 0 < r"
+ "x < 0 \<longleftrightarrow> False"
+ "0 < (1::pextreal)"
+ by (simp_all add: less_pextreal_def zero_pextreal_def one_pextreal_def del: Real_0 Real_1)
+
+lemma pextreal_less_eq[simp]:
+ "x \<le> \<omega>"
+ "Real r \<le> Real p \<longleftrightarrow> (if 0 \<le> r \<and> 0 \<le> p then r \<le> p else r \<le> 0)"
+ "0 \<le> x"
+ by (simp_all add: less_eq_pextreal_def zero_pextreal_def del: Real_0)
+
+lemma pextreal_\<omega>_less_eq[simp]:
+ "\<omega> \<le> x \<longleftrightarrow> x = \<omega>"
+ by (cases x) (simp_all add: not_le less_eq_pextreal_def)
+
+lemma pextreal_less_eq_zero[simp]:
+ "(x::pextreal) \<le> 0 \<longleftrightarrow> x = 0"
+ by (cases x) (simp_all add: zero_pextreal_def del: Real_0)
+
+instance
+proof
+ fix x :: pextreal
+ show "x \<le> x" by (cases x) simp_all
+ fix y
+ show "(x < y) = (x \<le> y \<and> \<not> y \<le> x)"
+ by (cases x, cases y) auto
+ show "x \<le> y \<or> y \<le> x "
+ by (cases x, cases y) auto
+ { assume "x \<le> y" "y \<le> x" thus "x = y"
+ by (cases x, cases y) auto }
+ { fix z assume "x \<le> y" "y \<le> z"
+ thus "x \<le> z" by (cases x, cases y, cases z) auto }
+qed
+end
+
+lemma pextreal_zero_lessI[intro]:
+ "(a :: pextreal) \<noteq> 0 \<Longrightarrow> 0 < a"
+ by (cases a) auto
+
+lemma pextreal_less_omegaI[intro, simp]:
+ "a \<noteq> \<omega> \<Longrightarrow> a < \<omega>"
+ by (cases a) auto
+
+lemma pextreal_plus_eq_0[simp]: "(a :: pextreal) + b = 0 \<longleftrightarrow> a = 0 \<and> b = 0"
+ by (cases a, cases b) auto
+
+lemma pextreal_le_add1[simp, intro]: "n \<le> n + (m::pextreal)"
+ by (cases n, cases m) simp_all
+
+lemma pextreal_le_add2: "(n::pextreal) + m \<le> k \<Longrightarrow> m \<le> k"
+ by (cases n, cases m, cases k) simp_all
+
+lemma pextreal_le_add3: "(n::pextreal) + m \<le> k \<Longrightarrow> n \<le> k"
+ by (cases n, cases m, cases k) simp_all
+
+lemma pextreal_less_\<omega>: "x < \<omega> \<longleftrightarrow> x \<noteq> \<omega>"
+ by (cases x) auto
+
+lemma pextreal_0_less_mult_iff[simp]:
+ fixes x y :: pextreal shows "0 < x * y \<longleftrightarrow> 0 < x \<and> 0 < y"
+ by (cases x, cases y) (auto simp: zero_less_mult_iff)
+
+lemma pextreal_ord_one[simp]:
+ "Real p < 1 \<longleftrightarrow> p < 1"
+ "Real p \<le> 1 \<longleftrightarrow> p \<le> 1"
+ "1 < Real p \<longleftrightarrow> 1 < p"
+ "1 \<le> Real p \<longleftrightarrow> 1 \<le> p"
+ by (simp_all add: one_pextreal_def del: Real_1)
+
+subsection {* @{text "x - y"} on @{typ pextreal} *}
+
+instantiation pextreal :: minus
+begin
+definition "x - y = (if y < x then THE d. x = y + d else 0 :: pextreal)"
+
+lemma minus_pextreal_eq:
+ "(x - y = (z :: pextreal)) \<longleftrightarrow> (if y < x then x = y + z else z = 0)"
+ (is "?diff \<longleftrightarrow> ?if")
+proof
+ assume ?diff
+ thus ?if
+ proof (cases "y < x")
+ case True
+ then obtain p where p: "y = Real p" "0 \<le> p" by (cases y) auto
+
+ show ?thesis unfolding `?diff`[symmetric] if_P[OF True] minus_pextreal_def
+ proof (rule theI2[where Q="\<lambda>d. x = y + d"])
+ show "x = y + pextreal_case (\<lambda>r. Real (r - real y)) \<omega> x" (is "x = y + ?d")
+ using `y < x` p by (cases x) simp_all
+
+ fix d assume "x = y + d"
+ thus "d = ?d" using `y < x` p by (cases d, cases x) simp_all
+ qed simp
+ qed (simp add: minus_pextreal_def)
+next
+ assume ?if
+ thus ?diff
+ proof (cases "y < x")
+ case True
+ then obtain p where p: "y = Real p" "0 \<le> p" by (cases y) auto
+
+ from True `?if` have "x = y + z" by simp
+
+ show ?thesis unfolding minus_pextreal_def if_P[OF True] unfolding `x = y + z`
+ proof (rule the_equality)
+ fix d :: pextreal assume "y + z = y + d"
+ thus "d = z" using `y < x` p
+ by (cases d, cases z) simp_all
+ qed simp
+ qed (simp add: minus_pextreal_def)
+qed
+
+instance ..
+end
+
+lemma pextreal_minus[simp]:
+ "Real r - Real p = (if 0 \<le> r \<and> p < r then if 0 \<le> p then Real (r - p) else Real r else 0)"
+ "(A::pextreal) - A = 0"
+ "\<omega> - Real r = \<omega>"
+ "Real r - \<omega> = 0"
+ "A - 0 = A"
+ "0 - A = 0"
+ by (auto simp: minus_pextreal_eq not_less)
+
+lemma pextreal_le_epsilon:
+ fixes x y :: pextreal
+ assumes "\<And>e. 0 < e \<Longrightarrow> x \<le> y + e"
+ shows "x \<le> y"
+proof (cases y)
+ case (preal r)
+ then obtain p where x: "x = Real p" "0 \<le> p"
+ using assms[of 1] by (cases x) auto
+ { fix e have "0 < e \<Longrightarrow> p \<le> r + e"
+ using assms[of "Real e"] preal x by auto }
+ hence "p \<le> r" by (rule field_le_epsilon)
+ thus ?thesis using preal x by auto
+qed simp
+
+instance pextreal :: "{ordered_comm_semiring, comm_semiring_1}"
+proof
+ show "0 \<noteq> (1::pextreal)" unfolding zero_pextreal_def one_pextreal_def
+ by (simp del: Real_1 Real_0)
+
+ fix a :: pextreal
+ show "0 * a = 0" "a * 0 = 0" by simp_all
+
+ fix b c :: pextreal
+ show "(a + b) * c = a * c + b * c"
+ by (cases c, cases a, cases b)
+ (auto intro!: arg_cong[where f=Real] simp: field_simps not_le mult_le_0_iff mult_less_0_iff)
+
+ { assume "a \<le> b" thus "c + a \<le> c + b"
+ by (cases c, cases a, cases b) auto }
+
+ assume "a \<le> b" "0 \<le> c"
+ thus "c * a \<le> c * b"
+ apply (cases c, cases a, cases b)
+ by (auto simp: mult_left_mono mult_le_0_iff mult_less_0_iff not_le)
+qed
+
+lemma mult_\<omega>[simp]: "x * y = \<omega> \<longleftrightarrow> (x = \<omega> \<or> y = \<omega>) \<and> x \<noteq> 0 \<and> y \<noteq> 0"
+ by (cases x, cases y) auto
+
+lemma \<omega>_mult[simp]: "(\<omega> = x * y) = ((x = \<omega> \<or> y = \<omega>) \<and> x \<noteq> 0 \<and> y \<noteq> 0)"
+ by (cases x, cases y) auto
+
+lemma pextreal_mult_0[simp]: "x * y = 0 \<longleftrightarrow> x = 0 \<or> (y::pextreal) = 0"
+ by (cases x, cases y) (auto simp: mult_le_0_iff)
+
+lemma pextreal_mult_cancel:
+ fixes x y z :: pextreal
+ assumes "y \<le> z"
+ shows "x * y \<le> x * z"
+ using assms
+ by (cases x, cases y, cases z)
+ (auto simp: mult_le_cancel_left mult_le_0_iff mult_less_0_iff not_le)
+
+lemma Real_power[simp]:
+ "Real x ^ n = (if x \<le> 0 then (if n = 0 then 1 else 0) else Real (x ^ n))"
+ by (induct n) auto
+
+lemma Real_power_\<omega>[simp]:
+ "\<omega> ^ n = (if n = 0 then 1 else \<omega>)"
+ by (induct n) auto
+
+lemma pextreal_of_nat[simp]: "of_nat m = Real (real m)"
+ by (induct m) (auto simp: real_of_nat_Suc one_pextreal_def simp del: Real_1)
+
+lemma less_\<omega>_Ex_of_nat: "x < \<omega> \<longleftrightarrow> (\<exists>n. x < of_nat n)"
+proof safe
+ assume "x < \<omega>"
+ then obtain r where "0 \<le> r" "x = Real r" by (cases x) auto
+ moreover obtain n where "r < of_nat n" using ex_less_of_nat by auto
+ ultimately show "\<exists>n. x < of_nat n" by (auto simp: real_eq_of_nat)
+qed auto
+
+lemma real_of_pextreal_mono:
+ fixes a b :: pextreal
+ assumes "b \<noteq> \<omega>" "a \<le> b"
+ shows "real a \<le> real b"
+using assms by (cases b, cases a) auto
+
+lemma setprod_pextreal_0:
+ "(\<Prod>i\<in>I. f i) = (0::pextreal) \<longleftrightarrow> finite I \<and> (\<exists>i\<in>I. f i = 0)"
+proof cases
+ assume "finite I" then show ?thesis
+ proof (induct I)
+ case (insert i I)
+ then show ?case by simp
+ qed simp
+qed simp
+
+lemma setprod_\<omega>:
+ "(\<Prod>i\<in>I. f i) = \<omega> \<longleftrightarrow> finite I \<and> (\<exists>i\<in>I. f i = \<omega>) \<and> (\<forall>i\<in>I. f i \<noteq> 0)"
+proof cases
+ assume "finite I" then show ?thesis
+ proof (induct I)
+ case (insert i I) then show ?case
+ by (auto simp: setprod_pextreal_0)
+ qed simp
+qed simp
+
+instance pextreal :: "semiring_char_0"
+proof
+ fix m n
+ show "inj (of_nat::nat\<Rightarrow>pextreal)" by (auto intro!: inj_onI)
+qed
+
+subsection "@{typ pextreal} is a complete lattice"
+
+instantiation pextreal :: lattice
+begin
+definition [simp]: "sup x y = (max x y :: pextreal)"
+definition [simp]: "inf x y = (min x y :: pextreal)"
+instance proof qed simp_all
+end
+
+instantiation pextreal :: complete_lattice
+begin
+
+definition "bot = Real 0"
+definition "top = \<omega>"
+
+definition "Sup S = (LEAST z. \<forall>x\<in>S. x \<le> z :: pextreal)"
+definition "Inf S = (GREATEST z. \<forall>x\<in>S. z \<le> x :: pextreal)"
+
+lemma pextreal_complete_Sup:
+ fixes S :: "pextreal 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 "\<exists>x\<ge>0. \<forall>a\<in>S. a \<le> Real x")
+ case False
+ hence *: "\<And>x. x\<ge>0 \<Longrightarrow> \<exists>a\<in>S. \<not>a \<le> Real x" by simp
+ show ?thesis
+ proof (safe intro!: exI[of _ \<omega>])
+ fix y assume **: "\<forall>z\<in>S. z \<le> y"
+ show "\<omega> \<le> y" unfolding pextreal_\<omega>_less_eq
+ proof (rule ccontr)
+ assume "y \<noteq> \<omega>"
+ then obtain x where [simp]: "y = Real x" and "0 \<le> x" by (cases y) auto
+ from *[OF `0 \<le> x`] show False using ** by auto
+ qed
+ qed simp
+next
+ case True then obtain y where y: "\<And>a. a\<in>S \<Longrightarrow> a \<le> Real y" and "0 \<le> y" by auto
+ from y[of \<omega>] have "\<omega> \<notin> S" by auto
+
+ with `S \<noteq> {}` obtain x where "x \<in> S" and "x \<noteq> \<omega>" by auto
+
+ have bound: "\<forall>x\<in>real ` S. x \<le> y"
+ proof
+ fix z assume "z \<in> real ` S" then guess a ..
+ with y[of a] `\<omega> \<notin> S` `0 \<le> y` show "z \<le> y" by (cases a) auto
+ qed
+ with reals_complete2[of "real ` S"] `x \<in> S`
+ obtain s where s: "\<forall>y\<in>S. real y \<le> s" "\<forall>z. ((\<forall>y\<in>S. real y \<le> z) \<longrightarrow> s \<le> z)"
+ by auto
+
+ show ?thesis
+ proof (safe intro!: exI[of _ "Real s"])
+ fix z assume "z \<in> S" thus "z \<le> Real s"
+ using s `\<omega> \<notin> S` by (cases z) auto
+ next
+ fix z assume *: "\<forall>y\<in>S. y \<le> z"
+ show "Real s \<le> z"
+ proof (cases z)
+ case (preal u)
+ { fix v assume "v \<in> S"
+ hence "v \<le> Real u" using * preal by auto
+ hence "real v \<le> u" using `\<omega> \<notin> S` `0 \<le> u` by (cases v) auto }
+ hence "s \<le> u" using s(2) by auto
+ thus "Real s \<le> z" using preal by simp
+ qed simp
+ qed
+qed
+
+lemma pextreal_complete_Inf:
+ fixes S :: "pextreal set" assumes "S \<noteq> {}"
+ shows "\<exists>x. (\<forall>y\<in>S. x \<le> y) \<and> (\<forall>z. (\<forall>y\<in>S. z \<le> y) \<longrightarrow> z \<le> x)"
+proof (cases "S = {\<omega>}")
+ case True thus ?thesis by (auto intro!: exI[of _ \<omega>])
+next
+ case False with `S \<noteq> {}` have "S - {\<omega>} \<noteq> {}" by auto
+ hence not_empty: "\<exists>x. x \<in> uminus ` real ` (S - {\<omega>})" by auto
+ have bounds: "\<exists>x. \<forall>y\<in>uminus ` real ` (S - {\<omega>}). y \<le> x" by (auto intro!: exI[of _ 0])
+ from reals_complete2[OF not_empty bounds]
+ obtain s where s: "\<And>y. y\<in>S - {\<omega>} \<Longrightarrow> - real y \<le> s" "\<forall>z. ((\<forall>y\<in>S - {\<omega>}. - real y \<le> z) \<longrightarrow> s \<le> z)"
+ by auto
+
+ show ?thesis
+ proof (safe intro!: exI[of _ "Real (-s)"])
+ fix z assume "z \<in> S"
+ show "Real (-s) \<le> z"
+ proof (cases z)
+ case (preal r)
+ with s `z \<in> S` have "z \<in> S - {\<omega>}" by simp
+ hence "- r \<le> s" using preal s(1)[of z] by auto
+ hence "- s \<le> r" by (subst neg_le_iff_le[symmetric]) simp
+ thus ?thesis using preal by simp
+ qed simp
+ next
+ fix z assume *: "\<forall>y\<in>S. z \<le> y"
+ show "z \<le> Real (-s)"
+ proof (cases z)
+ case (preal u)
+ { fix v assume "v \<in> S-{\<omega>}"
+ hence "Real u \<le> v" using * preal by auto
+ hence "- real v \<le> - u" using `0 \<le> u` `v \<in> S - {\<omega>}` by (cases v) auto }
+ hence "u \<le> - s" using s(2) by (subst neg_le_iff_le[symmetric]) auto
+ thus "z \<le> Real (-s)" using preal by simp
+ next
+ case infinite
+ with * have "S = {\<omega>}" using `S \<noteq> {}` by auto
+ with `S - {\<omega>} \<noteq> {}` show ?thesis by simp
+ qed
+ qed
+qed
+
+instance
+proof
+ fix x :: pextreal and A
+
+ show "bot \<le> x" by (cases x) (simp_all add: bot_pextreal_def)
+ show "x \<le> top" by (simp add: top_pextreal_def)
+
+ { assume "x \<in> A"
+ with pextreal_complete_Sup[of A]
+ obtain s where s: "\<forall>y\<in>A. y \<le> s" "\<forall>z. (\<forall>y\<in>A. y \<le> z) \<longrightarrow> s \<le> z" by auto
+ hence "x \<le> s" using `x \<in> A` by auto
+ also have "... = Sup A" using s unfolding Sup_pextreal_def
+ by (auto intro!: Least_equality[symmetric])
+ finally show "x \<le> Sup A" . }
+
+ { assume "x \<in> A"
+ with pextreal_complete_Inf[of A]
+ obtain i where i: "\<forall>y\<in>A. i \<le> y" "\<forall>z. (\<forall>y\<in>A. z \<le> y) \<longrightarrow> z \<le> i" by auto
+ hence "Inf A = i" unfolding Inf_pextreal_def
+ by (auto intro!: Greatest_equality)
+ also have "i \<le> x" using i `x \<in> A` by auto
+ finally show "Inf A \<le> x" . }
+
+ { assume *: "\<And>z. z \<in> A \<Longrightarrow> z \<le> x"
+ show "Sup A \<le> x"
+ proof (cases "A = {}")
+ case True
+ hence "Sup A = 0" unfolding Sup_pextreal_def
+ by (auto intro!: Least_equality)
+ thus "Sup A \<le> x" by simp
+ next
+ case False
+ with pextreal_complete_Sup[of A]
+ obtain s where s: "\<forall>y\<in>A. y \<le> s" "\<forall>z. (\<forall>y\<in>A. y \<le> z) \<longrightarrow> s \<le> z" by auto
+ hence "Sup A = s"
+ unfolding Sup_pextreal_def by (auto intro!: Least_equality)
+ also have "s \<le> x" using * s by auto
+ finally show "Sup A \<le> x" .
+ qed }
+
+ { assume *: "\<And>z. z \<in> A \<Longrightarrow> x \<le> z"
+ show "x \<le> Inf A"
+ proof (cases "A = {}")
+ case True
+ hence "Inf A = \<omega>" unfolding Inf_pextreal_def
+ by (auto intro!: Greatest_equality)
+ thus "x \<le> Inf A" by simp
+ next
+ case False
+ with pextreal_complete_Inf[of A]
+ obtain i where i: "\<forall>y\<in>A. i \<le> y" "\<forall>z. (\<forall>y\<in>A. z \<le> y) \<longrightarrow> z \<le> i" by auto
+ have "x \<le> i" using * i by auto
+ also have "i = Inf A" using i
+ unfolding Inf_pextreal_def by (auto intro!: Greatest_equality[symmetric])
+ finally show "x \<le> Inf A" .
+ qed }
+qed
+end
+
+lemma Inf_pextreal_iff:
+ fixes z :: pextreal
+ 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 Inf_greater:
+ fixes z :: pextreal assumes "Inf X < z"
+ shows "\<exists>x \<in> X. x < z"
+proof -
+ have "X \<noteq> {}" using assms by (auto simp: Inf_empty top_pextreal_def)
+ with assms show ?thesis
+ by (metis Inf_pextreal_iff mem_def not_leE)
+qed
+
+lemma Inf_close:
+ fixes e :: pextreal assumes "Inf X \<noteq> \<omega>" "0 < e"
+ shows "\<exists>x \<in> X. x < Inf X + e"
+proof (rule Inf_greater)
+ show "Inf X < Inf X + e" using assms
+ by (cases "Inf X", cases e) auto
+qed
+
+lemma pextreal_SUPI:
+ fixes x :: pextreal
+ assumes "\<And>i. i \<in> A \<Longrightarrow> f i \<le> x"
+ assumes "\<And>y. (\<And>i. i \<in> A \<Longrightarrow> f i \<le> y) \<Longrightarrow> x \<le> y"
+ shows "(SUP i:A. f i) = x"
+ unfolding SUPR_def Sup_pextreal_def
+ using assms by (auto intro!: Least_equality)
+
+lemma Sup_pextreal_iff:
+ fixes z :: pextreal
+ shows "(\<And>x. x \<in> X \<Longrightarrow> x \<le> z) \<Longrightarrow> (\<exists>x\<in>X. y<x) \<longleftrightarrow> y < Sup X"
+ by (metis complete_lattice_class.Sup_least complete_lattice_class.Sup_upper less_le_not_le linear
+ order_less_le_trans)
+
+lemma Sup_lesser:
+ fixes z :: pextreal assumes "z < Sup X"
+ shows "\<exists>x \<in> X. z < x"
+proof -
+ have "X \<noteq> {}" using assms by (auto simp: Sup_empty bot_pextreal_def)
+ with assms show ?thesis
+ by (metis Sup_pextreal_iff mem_def not_leE)
+qed
+
+lemma Sup_eq_\<omega>: "\<omega> \<in> S \<Longrightarrow> Sup S = \<omega>"
+ unfolding Sup_pextreal_def
+ by (auto intro!: Least_equality)
+
+lemma Sup_close:
+ assumes "0 < e" and S: "Sup S \<noteq> \<omega>" "S \<noteq> {}"
+ shows "\<exists>X\<in>S. Sup S < X + e"
+proof cases
+ assume "Sup S = 0"
+ moreover obtain X where "X \<in> S" using `S \<noteq> {}` by auto
+ ultimately show ?thesis using `0 < e` by (auto intro!: bexI[OF _ `X\<in>S`])
+next
+ assume "Sup S \<noteq> 0"
+ have "\<exists>X\<in>S. Sup S - e < X"
+ proof (rule Sup_lesser)
+ show "Sup S - e < Sup S" using `0 < e` `Sup S \<noteq> 0` `Sup S \<noteq> \<omega>`
+ by (cases e) (auto simp: pextreal_noteq_omega_Ex)
+ qed
+ then guess X .. note X = this
+ with `Sup S \<noteq> \<omega>` Sup_eq_\<omega> have "X \<noteq> \<omega>" by auto
+ thus ?thesis using `Sup S \<noteq> \<omega>` X unfolding pextreal_noteq_omega_Ex
+ by (cases e) (auto intro!: bexI[OF _ `X\<in>S`] simp: split: split_if_asm)
+qed
+
+lemma Sup_\<omega>: "(SUP i::nat. Real (real i)) = \<omega>"
+proof (rule pextreal_SUPI)
+ fix y assume *: "\<And>i::nat. i \<in> UNIV \<Longrightarrow> Real (real i) \<le> y"
+ thus "\<omega> \<le> y"
+ proof (cases y)
+ case (preal r)
+ then obtain k :: nat where "r < real k"
+ using ex_less_of_nat by (auto simp: real_eq_of_nat)
+ with *[of k] preal show ?thesis by auto
+ qed simp
+qed simp
+
+lemma SUP_\<omega>: "(SUP i:A. f i) = \<omega> \<longleftrightarrow> (\<forall>x<\<omega>. \<exists>i\<in>A. x < f i)"
+proof
+ assume *: "(SUP i:A. f i) = \<omega>"
+ show "(\<forall>x<\<omega>. \<exists>i\<in>A. x < f i)" unfolding *[symmetric]
+ proof (intro allI impI)
+ fix x assume "x < SUPR A f" then show "\<exists>i\<in>A. x < f i"
+ unfolding less_SUP_iff by auto
+ qed
+next
+ assume *: "\<forall>x<\<omega>. \<exists>i\<in>A. x < f i"
+ show "(SUP i:A. f i) = \<omega>"
+ proof (rule pextreal_SUPI)
+ fix y assume **: "\<And>i. i \<in> A \<Longrightarrow> f i \<le> y"
+ show "\<omega> \<le> y"
+ proof cases
+ assume "y < \<omega>"
+ from *[THEN spec, THEN mp, OF this]
+ obtain i where "i \<in> A" "\<not> (f i \<le> y)" by auto
+ with ** show ?thesis by auto
+ qed auto
+ qed auto
+qed
+
+subsubsection {* Equivalence between @{text "f ----> x"} and @{text SUP} on @{typ pextreal} *}
+
+lemma monoseq_monoI: "mono f \<Longrightarrow> monoseq f"
+ unfolding mono_def monoseq_def by auto
+
+lemma incseq_mono: "mono f \<longleftrightarrow> incseq f"
+ unfolding mono_def incseq_def by auto
+
+lemma SUP_eq_LIMSEQ:
+ assumes "mono f" and "\<And>n. 0 \<le> f n" and "0 \<le> x"
+ shows "(SUP n. Real (f n)) = Real x \<longleftrightarrow> f ----> x"
+proof
+ assume x: "(SUP n. Real (f n)) = Real x"
+ { fix n
+ have "Real (f n) \<le> Real x" using x[symmetric] by (auto intro: le_SUPI)
+ hence "f n \<le> x" using assms by simp }
+ show "f ----> x"
+ proof (rule LIMSEQ_I)
+ fix r :: real assume "0 < r"
+ show "\<exists>no. \<forall>n\<ge>no. norm (f n - x) < r"
+ proof (rule ccontr)
+ assume *: "\<not> ?thesis"
+ { fix N
+ from * obtain n where "N \<le> n" "r \<le> x - f n"
+ using `\<And>n. f n \<le> x` by (auto simp: not_less)
+ hence "f N \<le> f n" using `mono f` by (auto dest: monoD)
+ hence "f N \<le> x - r" using `r \<le> x - f n` by auto
+ hence "Real (f N) \<le> Real (x - r)" and "r \<le> x" using `0 \<le> f N` by auto }
+ hence "(SUP n. Real (f n)) \<le> Real (x - r)"
+ and "Real (x - r) < Real x" using `0 < r` by (auto intro: SUP_leI)
+ hence "(SUP n. Real (f n)) < Real x" by (rule le_less_trans)
+ thus False using x by auto
+ qed
+ qed
+next
+ assume "f ----> x"
+ show "(SUP n. Real (f n)) = Real x"
+ proof (rule pextreal_SUPI)
+ fix n
+ from incseq_le[of f x] `mono f` `f ----> x`
+ show "Real (f n) \<le> Real x" using assms incseq_mono by auto
+ next
+ fix y assume *: "\<And>n. n\<in>UNIV \<Longrightarrow> Real (f n) \<le> y"
+ show "Real x \<le> y"
+ proof (cases y)
+ case (preal r)
+ with * have "\<exists>N. \<forall>n\<ge>N. f n \<le> r" using assms by fastsimp
+ from LIMSEQ_le_const2[OF `f ----> x` this]
+ show "Real x \<le> y" using `0 \<le> x` preal by auto
+ qed simp
+ qed
+qed
+
+lemma SUPR_bound:
+ assumes "\<forall>N. f N \<le> x"
+ shows "(SUP n. f n) \<le> x"
+ using assms by (simp add: SUPR_def Sup_le_iff)
+
+lemma pextreal_less_eq_diff_eq_sum:
+ fixes x y z :: pextreal
+ assumes "y \<le> x" and "x \<noteq> \<omega>"
+ shows "z \<le> x - y \<longleftrightarrow> z + y \<le> x"
+ using assms
+ apply (cases z, cases y, cases x)
+ by (simp_all add: field_simps minus_pextreal_eq)
+
+lemma Real_diff_less_omega: "Real r - x < \<omega>" by (cases x) auto
+
+subsubsection {* Numbers on @{typ pextreal} *}
+
+instantiation pextreal :: number
+begin
+definition [simp]: "number_of x = Real (number_of x)"
+instance proof qed
+end
+
+subsubsection {* Division on @{typ pextreal} *}
+
+instantiation pextreal :: inverse
+begin
+
+definition "inverse x = pextreal_case (\<lambda>x. if x = 0 then \<omega> else Real (inverse x)) 0 x"
+definition [simp]: "x / y = x * inverse (y :: pextreal)"
+
+instance proof qed
+end
+
+lemma pextreal_inverse[simp]:
+ "inverse 0 = \<omega>"
+ "inverse (Real x) = (if x \<le> 0 then \<omega> else Real (inverse x))"
+ "inverse \<omega> = 0"
+ "inverse (1::pextreal) = 1"
+ "inverse (inverse x) = x"
+ by (simp_all add: inverse_pextreal_def one_pextreal_def split: pextreal_case_split del: Real_1)
+
+lemma pextreal_inverse_le_eq:
+ assumes "x \<noteq> 0" "x \<noteq> \<omega>"
+ shows "y \<le> z / x \<longleftrightarrow> x * y \<le> (z :: pextreal)"
+proof -
+ from assms obtain r where r: "x = Real r" "0 < r" by (cases x) auto
+ { fix p q :: real assume "0 \<le> p" "0 \<le> q"
+ have "p \<le> q * inverse r \<longleftrightarrow> p \<le> q / r" by (simp add: divide_inverse)
+ also have "... \<longleftrightarrow> p * r \<le> q" using `0 < r` by (auto simp: field_simps)
+ finally have "p \<le> q * inverse r \<longleftrightarrow> p * r \<le> q" . }
+ with r show ?thesis
+ by (cases y, cases z, auto simp: zero_le_mult_iff field_simps)
+qed
+
+lemma inverse_antimono_strict:
+ fixes x y :: pextreal
+ assumes "x < y" shows "inverse y < inverse x"
+ using assms by (cases x, cases y) auto
+
+lemma inverse_antimono:
+ fixes x y :: pextreal
+ assumes "x \<le> y" shows "inverse y \<le> inverse x"
+ using assms by (cases x, cases y) auto
+
+lemma pextreal_inverse_\<omega>_iff[simp]: "inverse x = \<omega> \<longleftrightarrow> x = 0"
+ by (cases x) auto
+
+subsection "Infinite sum over @{typ pextreal}"
+
+text {*
+
+The infinite sum over @{typ pextreal} has the nice property that it is always
+defined.
+
+*}
+
+definition psuminf :: "(nat \<Rightarrow> pextreal) \<Rightarrow> pextreal" (binder "\<Sum>\<^isub>\<infinity>" 10) where
+ "(\<Sum>\<^isub>\<infinity> x. f x) = (SUP n. \<Sum>i<n. f i)"
+
+subsubsection {* Equivalence between @{text "\<Sum> n. f n"} and @{text "\<Sum>\<^isub>\<infinity> n. f n"} *}
+
+lemma setsum_Real:
+ assumes "\<forall>x\<in>A. 0 \<le> f x"
+ shows "(\<Sum>x\<in>A. Real (f x)) = Real (\<Sum>x\<in>A. f x)"
+proof (cases "finite A")
+ case True
+ thus ?thesis using assms
+ proof induct case (insert x s)
+ hence "0 \<le> setsum f s" apply-apply(rule setsum_nonneg) by auto
+ thus ?case using insert by auto
+ qed auto
+qed simp
+
+lemma setsum_Real':
+ assumes "\<forall>x. 0 \<le> f x"
+ shows "(\<Sum>x\<in>A. Real (f x)) = Real (\<Sum>x\<in>A. f x)"
+ apply(rule setsum_Real) using assms by auto
+
+lemma setsum_\<omega>:
+ "(\<Sum>x\<in>P. f x) = \<omega> \<longleftrightarrow> (finite P \<and> (\<exists>i\<in>P. f i = \<omega>))"
+proof safe
+ assume *: "setsum f P = \<omega>"
+ show "finite P"
+ proof (rule ccontr) assume "infinite P" with * show False by auto qed
+ show "\<exists>i\<in>P. f i = \<omega>"
+ proof (rule ccontr)
+ assume "\<not> ?thesis" hence "\<And>i. i\<in>P \<Longrightarrow> f i \<noteq> \<omega>" by auto
+ from `finite P` this have "setsum f P \<noteq> \<omega>"
+ by induct auto
+ with * show False by auto
+ qed
+next
+ fix i assume "finite P" "i \<in> P" "f i = \<omega>"
+ thus "setsum f P = \<omega>"
+ proof induct
+ case (insert x A)
+ show ?case using insert by (cases "x = i") auto
+ qed simp
+qed
+
+lemma real_of_pextreal_setsum:
+ assumes "\<And>x. x \<in> S \<Longrightarrow> f x \<noteq> \<omega>"
+ shows "(\<Sum>x\<in>S. real (f x)) = real (setsum f S)"
+proof cases
+ assume "finite S"
+ from this assms show ?thesis
+ by induct (simp_all add: real_of_pextreal_add setsum_\<omega>)
+qed simp
+
+lemma setsum_0:
+ fixes f :: "'a \<Rightarrow> pextreal" assumes "finite A"
+ shows "(\<Sum>x\<in>A. f x) = 0 \<longleftrightarrow> (\<forall>i\<in>A. f i = 0)"
+ using assms by induct auto
+
+lemma suminf_imp_psuminf:
+ assumes "f sums x" and "\<forall>n. 0 \<le> f n"
+ shows "(\<Sum>\<^isub>\<infinity> x. Real (f x)) = Real x"
+ unfolding psuminf_def setsum_Real'[OF assms(2)]
+proof (rule SUP_eq_LIMSEQ[THEN iffD2])
+ show "mono (\<lambda>n. setsum f {..<n})" (is "mono ?S")
+ unfolding mono_iff_le_Suc using assms by simp
+
+ { fix n show "0 \<le> ?S n"
+ using setsum_nonneg[of "{..<n}" f] assms by auto }
+
+ thus "0 \<le> x" "?S ----> x"
+ using `f sums x` LIMSEQ_le_const
+ by (auto simp: atLeast0LessThan sums_def)
+qed
+
+lemma psuminf_equality:
+ assumes "\<And>n. setsum f {..<n} \<le> x"
+ and "\<And>y. y \<noteq> \<omega> \<Longrightarrow> (\<And>n. setsum f {..<n} \<le> y) \<Longrightarrow> x \<le> y"
+ shows "psuminf f = x"
+ unfolding psuminf_def
+proof (safe intro!: pextreal_SUPI)
+ fix n show "setsum f {..<n} \<le> x" using assms(1) .
+next
+ fix y assume *: "\<forall>n. n \<in> UNIV \<longrightarrow> setsum f {..<n} \<le> y"
+ show "x \<le> y"
+ proof (cases "y = \<omega>")
+ assume "y \<noteq> \<omega>" from assms(2)[OF this] *
+ show "x \<le> y" by auto
+ qed simp
+qed
+
+lemma psuminf_\<omega>:
+ assumes "f i = \<omega>"
+ shows "(\<Sum>\<^isub>\<infinity> x. f x) = \<omega>"
+proof (rule psuminf_equality)
+ fix y assume *: "\<And>n. setsum f {..<n} \<le> y"
+ have "setsum f {..<Suc i} = \<omega>"
+ using assms by (simp add: setsum_\<omega>)
+ thus "\<omega> \<le> y" using *[of "Suc i"] by auto
+qed simp
+
+lemma psuminf_imp_suminf:
+ assumes "(\<Sum>\<^isub>\<infinity> x. f x) \<noteq> \<omega>"
+ shows "(\<lambda>x. real (f x)) sums real (\<Sum>\<^isub>\<infinity> x. f x)"
+proof -
+ have "\<forall>i. \<exists>r. f i = Real r \<and> 0 \<le> r"
+ proof
+ fix i show "\<exists>r. f i = Real r \<and> 0 \<le> r" using psuminf_\<omega> assms by (cases "f i") auto
+ qed
+ from choice[OF this] obtain r where f: "f = (\<lambda>i. Real (r i))"
+ and pos: "\<forall>i. 0 \<le> r i"
+ by (auto simp: fun_eq_iff)
+ hence [simp]: "\<And>i. real (f i) = r i" by auto
+
+ have "mono (\<lambda>n. setsum r {..<n})" (is "mono ?S")
+ unfolding mono_iff_le_Suc using pos by simp
+
+ { fix n have "0 \<le> ?S n"
+ using setsum_nonneg[of "{..<n}" r] pos by auto }
+
+ from assms obtain p where *: "(\<Sum>\<^isub>\<infinity> x. f x) = Real p" and "0 \<le> p"
+ by (cases "(\<Sum>\<^isub>\<infinity> x. f x)") auto
+ show ?thesis unfolding * using * pos `0 \<le> p` SUP_eq_LIMSEQ[OF `mono ?S` `\<And>n. 0 \<le> ?S n` `0 \<le> p`]
+ by (simp add: f atLeast0LessThan sums_def psuminf_def setsum_Real'[OF pos] f)
+qed
+
+lemma psuminf_bound:
+ assumes "\<forall>N. (\<Sum>n<N. f n) \<le> x"
+ shows "(\<Sum>\<^isub>\<infinity> n. f n) \<le> x"
+ using assms by (simp add: psuminf_def SUPR_def Sup_le_iff)
+
+lemma psuminf_bound_add:
+ assumes "\<forall>N. (\<Sum>n<N. f n) + y \<le> x"
+ shows "(\<Sum>\<^isub>\<infinity> n. f n) + y \<le> x"
+proof (cases "x = \<omega>")
+ have "y \<le> x" using assms by (auto intro: pextreal_le_add2)
+ assume "x \<noteq> \<omega>"
+ note move_y = pextreal_less_eq_diff_eq_sum[OF `y \<le> x` this]
+
+ have "\<forall>N. (\<Sum>n<N. f n) \<le> x - y" using assms by (simp add: move_y)
+ hence "(\<Sum>\<^isub>\<infinity> n. f n) \<le> x - y" by (rule psuminf_bound)
+ thus ?thesis by (simp add: move_y)
+qed simp
+
+lemma psuminf_finite:
+ assumes "\<forall>N\<ge>n. f N = 0"
+ shows "(\<Sum>\<^isub>\<infinity> n. f n) = (\<Sum>N<n. f N)"
+proof (rule psuminf_equality)
+ fix N
+ show "setsum f {..<N} \<le> setsum f {..<n}"
+ proof (cases rule: linorder_cases)
+ assume "N < n" thus ?thesis by (auto intro!: setsum_mono3)
+ next
+ assume "n < N"
+ hence *: "{..<N} = {..<n} \<union> {n..<N}" by auto
+ moreover have "setsum f {n..<N} = 0"
+ using assms by (auto intro!: setsum_0')
+ ultimately show ?thesis unfolding *
+ by (subst setsum_Un_disjoint) auto
+ qed simp
+qed simp
+
+lemma psuminf_upper:
+ shows "(\<Sum>n<N. f n) \<le> (\<Sum>\<^isub>\<infinity> n. f n)"
+ unfolding psuminf_def SUPR_def
+ by (auto intro: complete_lattice_class.Sup_upper image_eqI)
+
+lemma psuminf_le:
+ assumes "\<And>N. f N \<le> g N"
+ shows "psuminf f \<le> psuminf g"
+proof (safe intro!: psuminf_bound)
+ fix n
+ have "setsum f {..<n} \<le> setsum g {..<n}" using assms by (auto intro: setsum_mono)
+ also have "... \<le> psuminf g" by (rule psuminf_upper)
+ finally show "setsum f {..<n} \<le> psuminf g" .
+qed
+
+lemma psuminf_const[simp]: "psuminf (\<lambda>n. c) = (if c = 0 then 0 else \<omega>)" (is "_ = ?if")
+proof (rule psuminf_equality)
+ fix y assume *: "\<And>n :: nat. (\<Sum>n<n. c) \<le> y" and "y \<noteq> \<omega>"
+ then obtain r p where
+ y: "y = Real r" "0 \<le> r" and
+ c: "c = Real p" "0 \<le> p"
+ using *[of 1] by (cases c, cases y) auto
+ show "(if c = 0 then 0 else \<omega>) \<le> y"
+ proof (cases "p = 0")
+ assume "p = 0" with c show ?thesis by simp
+ next
+ assume "p \<noteq> 0"
+ with * c y have **: "\<And>n :: nat. real n \<le> r / p"
+ by (auto simp: zero_le_mult_iff field_simps)
+ from ex_less_of_nat[of "r / p"] guess n ..
+ with **[of n] show ?thesis by (simp add: real_eq_of_nat)
+ qed
+qed (cases "c = 0", simp_all)
+
+lemma psuminf_add[simp]: "psuminf (\<lambda>n. f n + g n) = psuminf f + psuminf g"
+proof (rule psuminf_equality)
+ fix n
+ from psuminf_upper[of f n] psuminf_upper[of g n]
+ show "(\<Sum>n<n. f n + g n) \<le> psuminf f + psuminf g"
+ by (auto simp add: setsum_addf intro!: add_mono)
+next
+ fix y assume *: "\<And>n. (\<Sum>n<n. f n + g n) \<le> y" and "y \<noteq> \<omega>"
+ { fix n m
+ have **: "(\<Sum>n<n. f n) + (\<Sum>n<m. g n) \<le> y"
+ proof (cases rule: linorder_le_cases)
+ assume "n \<le> m"
+ hence "(\<Sum>n<n. f n) + (\<Sum>n<m. g n) \<le> (\<Sum>n<m. f n) + (\<Sum>n<m. g n)"
+ by (auto intro!: add_right_mono setsum_mono3)
+ also have "... \<le> y"
+ using * by (simp add: setsum_addf)
+ finally show ?thesis .
+ next
+ assume "m \<le> n"
+ hence "(\<Sum>n<n. f n) + (\<Sum>n<m. g n) \<le> (\<Sum>n<n. f n) + (\<Sum>n<n. g n)"
+ by (auto intro!: add_left_mono setsum_mono3)
+ also have "... \<le> y"
+ using * by (simp add: setsum_addf)
+ finally show ?thesis .
+ qed }
+ hence "\<And>m. \<forall>n. (\<Sum>n<n. f n) + (\<Sum>n<m. g n) \<le> y" by simp
+ from psuminf_bound_add[OF this]
+ have "\<forall>m. (\<Sum>n<m. g n) + psuminf f \<le> y" by (simp add: ac_simps)
+ from psuminf_bound_add[OF this]
+ show "psuminf f + psuminf g \<le> y" by (simp add: ac_simps)
+qed
+
+lemma psuminf_0: "psuminf f = 0 \<longleftrightarrow> (\<forall>i. f i = 0)"
+proof safe
+ assume "\<forall>i. f i = 0"
+ hence "f = (\<lambda>i. 0)" by (simp add: fun_eq_iff)
+ thus "psuminf f = 0" using psuminf_const by simp
+next
+ fix i assume "psuminf f = 0"
+ hence "(\<Sum>n<Suc i. f n) = 0" using psuminf_upper[of f "Suc i"] by simp
+ thus "f i = 0" by simp
+qed
+
+lemma psuminf_cmult_right[simp]: "psuminf (\<lambda>n. c * f n) = c * psuminf f"
+proof (rule psuminf_equality)
+ fix n show "(\<Sum>n<n. c * f n) \<le> c * psuminf f"
+ by (auto simp: setsum_right_distrib[symmetric] intro: mult_left_mono psuminf_upper)
+next
+ fix y
+ assume "\<And>n. (\<Sum>n<n. c * f n) \<le> y"
+ hence *: "\<And>n. c * (\<Sum>n<n. f n) \<le> y" by (auto simp add: setsum_right_distrib)
+ thus "c * psuminf f \<le> y"
+ proof (cases "c = \<omega> \<or> c = 0")
+ assume "c = \<omega> \<or> c = 0"
+ thus ?thesis
+ using * by (fastsimp simp add: psuminf_0 setsum_0 split: split_if_asm)
+ next
+ assume "\<not> (c = \<omega> \<or> c = 0)"
+ hence "c \<noteq> 0" "c \<noteq> \<omega>" by auto
+ note rewrite_div = pextreal_inverse_le_eq[OF this, of _ y]
+ hence "\<forall>n. (\<Sum>n<n. f n) \<le> y / c" using * by simp
+ hence "psuminf f \<le> y / c" by (rule psuminf_bound)
+ thus ?thesis using rewrite_div by simp
+ qed
+qed
+
+lemma psuminf_cmult_left[simp]: "psuminf (\<lambda>n. f n * c) = psuminf f * c"
+ using psuminf_cmult_right[of c f] by (simp add: ac_simps)
+
+lemma psuminf_half_series: "psuminf (\<lambda>n. (1/2)^Suc n) = 1"
+ using suminf_imp_psuminf[OF power_half_series] by auto
+
+lemma setsum_pinfsum: "(\<Sum>\<^isub>\<infinity> n. \<Sum>m\<in>A. f n m) = (\<Sum>m\<in>A. (\<Sum>\<^isub>\<infinity> n. f n m))"
+proof (cases "finite A")
+ assume "finite A"
+ thus ?thesis by induct simp_all
+qed simp
+
+lemma psuminf_reindex:
+ fixes f:: "nat \<Rightarrow> nat" assumes "bij f"
+ shows "psuminf (g \<circ> f) = psuminf g"
+proof -
+ have [intro, simp]: "\<And>A. inj_on f A" using `bij f` unfolding bij_def by (auto intro: subset_inj_on)
+ have f[intro, simp]: "\<And>x. f (inv f x) = x"
+ using `bij f` unfolding bij_def by (auto intro: surj_f_inv_f)
+ show ?thesis
+ proof (rule psuminf_equality)
+ fix n
+ have "setsum (g \<circ> f) {..<n} = setsum g (f ` {..<n})"
+ by (simp add: setsum_reindex)
+ also have "\<dots> \<le> setsum g {..Max (f ` {..<n})}"
+ by (rule setsum_mono3) auto
+ also have "\<dots> \<le> psuminf g" unfolding lessThan_Suc_atMost[symmetric] by (rule psuminf_upper)
+ finally show "setsum (g \<circ> f) {..<n} \<le> psuminf g" .
+ next
+ fix y assume *: "\<And>n. setsum (g \<circ> f) {..<n} \<le> y"
+ show "psuminf g \<le> y"
+ proof (safe intro!: psuminf_bound)
+ fix N
+ have "setsum g {..<N} \<le> setsum g (f ` {..Max (inv f ` {..<N})})"
+ by (rule setsum_mono3) (auto intro!: image_eqI[where f="f", OF f[symmetric]])
+ also have "\<dots> = setsum (g \<circ> f) {..Max (inv f ` {..<N})}"
+ by (simp add: setsum_reindex)
+ also have "\<dots> \<le> y" unfolding lessThan_Suc_atMost[symmetric] by (rule *)
+ finally show "setsum g {..<N} \<le> y" .
+ qed
+ qed
+qed
+
+lemma pextreal_mult_less_right:
+ assumes "b * a < c * a" "0 < a" "a < \<omega>"
+ shows "b < c"
+ using assms
+ by (cases a, cases b, cases c) (auto split: split_if_asm simp: zero_less_mult_iff zero_le_mult_iff)
+
+lemma pextreal_\<omega>_eq_plus[simp]: "\<omega> = a + b \<longleftrightarrow> (a = \<omega> \<or> b = \<omega>)"
+ by (cases a, cases b) auto
+
+lemma pextreal_of_nat_le_iff:
+ "(of_nat k :: pextreal) \<le> of_nat m \<longleftrightarrow> k \<le> m" by auto
+
+lemma pextreal_of_nat_less_iff:
+ "(of_nat k :: pextreal) < of_nat m \<longleftrightarrow> k < m" by auto
+
+lemma pextreal_bound_add:
+ assumes "\<forall>N. f N + y \<le> (x::pextreal)"
+ shows "(SUP n. f n) + y \<le> x"
+proof (cases "x = \<omega>")
+ have "y \<le> x" using assms by (auto intro: pextreal_le_add2)
+ assume "x \<noteq> \<omega>"
+ note move_y = pextreal_less_eq_diff_eq_sum[OF `y \<le> x` this]
+
+ have "\<forall>N. f N \<le> x - y" using assms by (simp add: move_y)
+ hence "(SUP n. f n) \<le> x - y" by (rule SUPR_bound)
+ thus ?thesis by (simp add: move_y)
+qed simp
+
+lemma SUPR_pextreal_add:
+ fixes f g :: "nat \<Rightarrow> pextreal"
+ assumes f: "\<forall>n. f n \<le> f (Suc n)" and g: "\<forall>n. g n \<le> g (Suc n)"
+ shows "(SUP n. f n + g n) = (SUP n. f n) + (SUP n. g n)"
+proof (rule pextreal_SUPI)
+ fix n :: nat from le_SUPI[of n UNIV f] le_SUPI[of n UNIV g]
+ show "f n + g n \<le> (SUP n. f n) + (SUP n. g n)"
+ by (auto intro!: add_mono)
+next
+ fix y assume *: "\<And>n. n \<in> UNIV \<Longrightarrow> f n + g n \<le> y"
+ { fix n m
+ have "f n + g m \<le> y"
+ proof (cases rule: linorder_le_cases)
+ assume "n \<le> m"
+ hence "f n + g m \<le> f m + g m"
+ using f lift_Suc_mono_le by (auto intro!: add_right_mono)
+ also have "\<dots> \<le> y" using * by simp
+ finally show ?thesis .
+ next
+ assume "m \<le> n"
+ hence "f n + g m \<le> f n + g n"
+ using g lift_Suc_mono_le by (auto intro!: add_left_mono)
+ also have "\<dots> \<le> y" using * by simp
+ finally show ?thesis .
+ qed }
+ hence "\<And>m. \<forall>n. f n + g m \<le> y" by simp
+ from pextreal_bound_add[OF this]
+ have "\<forall>m. (g m) + (SUP n. f n) \<le> y" by (simp add: ac_simps)
+ from pextreal_bound_add[OF this]
+ show "SUPR UNIV f + SUPR UNIV g \<le> y" by (simp add: ac_simps)
+qed
+
+lemma SUPR_pextreal_setsum:
+ fixes f :: "'x \<Rightarrow> nat \<Rightarrow> pextreal"
+ assumes "\<And>i. i \<in> P \<Longrightarrow> \<forall>n. f i n \<le> f i (Suc n)"
+ shows "(SUP n. \<Sum>i\<in>P. f i n) = (\<Sum>i\<in>P. SUP n. f i n)"
+proof cases
+ assume "finite P" from this assms show ?thesis
+ proof induct
+ case (insert i P)
+ thus ?case
+ apply simp
+ apply (subst SUPR_pextreal_add)
+ by (auto intro!: setsum_mono)
+ qed simp
+qed simp
+
+lemma psuminf_SUP_eq:
+ assumes "\<And>n i. f n i \<le> f (Suc n) i"
+ shows "(\<Sum>\<^isub>\<infinity> i. SUP n::nat. f n i) = (SUP n::nat. \<Sum>\<^isub>\<infinity> i. f n i)"
+proof -
+ { fix n :: nat
+ have "(\<Sum>i<n. SUP k. f k i) = (SUP k. \<Sum>i<n. f k i)"
+ using assms by (auto intro!: SUPR_pextreal_setsum[symmetric]) }
+ note * = this
+ show ?thesis
+ unfolding psuminf_def
+ unfolding *
+ apply (subst SUP_commute) ..
+qed
+
+lemma psuminf_commute:
+ shows "(\<Sum>\<^isub>\<infinity> i j. f i j) = (\<Sum>\<^isub>\<infinity> j i. f i j)"
+proof -
+ have "(SUP n. \<Sum> i < n. SUP m. \<Sum> j < m. f i j) = (SUP n. SUP m. \<Sum> i < n. \<Sum> j < m. f i j)"
+ apply (subst SUPR_pextreal_setsum)
+ by auto
+ also have "\<dots> = (SUP m n. \<Sum> j < m. \<Sum> i < n. f i j)"
+ apply (subst SUP_commute)
+ apply (subst setsum_commute)
+ by auto
+ also have "\<dots> = (SUP m. \<Sum> j < m. SUP n. \<Sum> i < n. f i j)"
+ apply (subst SUPR_pextreal_setsum)
+ by auto
+ finally show ?thesis
+ unfolding psuminf_def by auto
+qed
+
+lemma psuminf_2dimen:
+ fixes f:: "nat * nat \<Rightarrow> pextreal"
+ assumes fsums: "\<And>m. g m = (\<Sum>\<^isub>\<infinity> n. f (m,n))"
+ shows "psuminf (f \<circ> prod_decode) = psuminf g"
+proof (rule psuminf_equality)
+ fix n :: nat
+ let ?P = "prod_decode ` {..<n}"
+ have "setsum (f \<circ> prod_decode) {..<n} = setsum f ?P"
+ by (auto simp: setsum_reindex inj_prod_decode)
+ also have "\<dots> \<le> setsum f ({..Max (fst ` ?P)} \<times> {..Max (snd ` ?P)})"
+ proof (safe intro!: setsum_mono3 Max_ge image_eqI)
+ fix a b x assume "(a, b) = prod_decode x"
+ from this[symmetric] show "a = fst (prod_decode x)" "b = snd (prod_decode x)"
+ by simp_all
+ qed simp_all
+ also have "\<dots> = (\<Sum>m\<le>Max (fst ` ?P). (\<Sum>n\<le>Max (snd ` ?P). f (m,n)))"
+ unfolding setsum_cartesian_product by simp
+ also have "\<dots> \<le> (\<Sum>m\<le>Max (fst ` ?P). g m)"
+ by (auto intro!: setsum_mono psuminf_upper simp del: setsum_lessThan_Suc
+ simp: fsums lessThan_Suc_atMost[symmetric])
+ also have "\<dots> \<le> psuminf g"
+ by (auto intro!: psuminf_upper simp del: setsum_lessThan_Suc
+ simp: lessThan_Suc_atMost[symmetric])
+ finally show "setsum (f \<circ> prod_decode) {..<n} \<le> psuminf g" .
+next
+ fix y assume *: "\<And>n. setsum (f \<circ> prod_decode) {..<n} \<le> y"
+ have g: "g = (\<lambda>m. \<Sum>\<^isub>\<infinity> n. f (m,n))" unfolding fsums[symmetric] ..
+ show "psuminf g \<le> y" unfolding g
+ proof (rule psuminf_bound, unfold setsum_pinfsum[symmetric], safe intro!: psuminf_bound)
+ fix N M :: nat
+ let ?P = "{..<N} \<times> {..<M}"
+ let ?M = "Max (prod_encode ` ?P)"
+ have "(\<Sum>n<M. \<Sum>m<N. f (m, n)) \<le> (\<Sum>(m, n)\<in>?P. f (m, n))"
+ unfolding setsum_commute[of _ _ "{..<M}"] unfolding setsum_cartesian_product ..
+ also have "\<dots> \<le> (\<Sum>(m,n)\<in>(prod_decode ` {..?M}). f (m, n))"
+ by (auto intro!: setsum_mono3 image_eqI[where f=prod_decode, OF prod_encode_inverse[symmetric]])
+ also have "\<dots> \<le> y" using *[of "Suc ?M"]
+ by (simp add: lessThan_Suc_atMost[symmetric] setsum_reindex
+ inj_prod_decode del: setsum_lessThan_Suc)
+ finally show "(\<Sum>n<M. \<Sum>m<N. f (m, n)) \<le> y" .
+ qed
+qed
+
+lemma Real_max:
+ assumes "x \<ge> 0" "y \<ge> 0"
+ shows "Real (max x y) = max (Real x) (Real y)"
+ using assms unfolding max_def by (auto simp add:not_le)
+
+lemma Real_real: "Real (real x) = (if x = \<omega> then 0 else x)"
+ using assms by (cases x) auto
+
+lemma inj_on_real: "inj_on real (UNIV - {\<omega>})"
+proof (rule inj_onI)
+ fix x y assume mem: "x \<in> UNIV - {\<omega>}" "y \<in> UNIV - {\<omega>}" and "real x = real y"
+ thus "x = y" by (cases x, cases y) auto
+qed
+
+lemma inj_on_Real: "inj_on Real {0..}"
+ by (auto intro!: inj_onI)
+
+lemma range_Real[simp]: "range Real = UNIV - {\<omega>}"
+proof safe
+ fix x assume "x \<notin> range Real"
+ thus "x = \<omega>" by (cases x) auto
+qed auto
+
+lemma image_Real[simp]: "Real ` {0..} = UNIV - {\<omega>}"
+proof safe
+ fix x assume "x \<notin> Real ` {0..}"
+ thus "x = \<omega>" by (cases x) auto
+qed auto
+
+lemma pextreal_SUP_cmult:
+ fixes f :: "'a \<Rightarrow> pextreal"
+ shows "(SUP i : R. z * f i) = z * (SUP i : R. f i)"
+proof (rule pextreal_SUPI)
+ fix i assume "i \<in> R"
+ from le_SUPI[OF this]
+ show "z * f i \<le> z * (SUP i:R. f i)" by (rule pextreal_mult_cancel)
+next
+ fix y assume "\<And>i. i\<in>R \<Longrightarrow> z * f i \<le> y"
+ hence *: "\<And>i. i\<in>R \<Longrightarrow> z * f i \<le> y" by auto
+ show "z * (SUP i:R. f i) \<le> y"
+ proof (cases "\<forall>i\<in>R. f i = 0")
+ case True
+ show ?thesis
+ proof cases
+ assume "R \<noteq> {}" hence "f ` R = {0}" using True by auto
+ thus ?thesis by (simp add: SUPR_def)
+ qed (simp add: SUPR_def Sup_empty bot_pextreal_def)
+ next
+ case False then obtain i where i: "i \<in> R" and f0: "f i \<noteq> 0" by auto
+ show ?thesis
+ proof (cases "z = 0 \<or> z = \<omega>")
+ case True with f0 *[OF i] show ?thesis by auto
+ next
+ case False hence z: "z \<noteq> 0" "z \<noteq> \<omega>" by auto
+ note div = pextreal_inverse_le_eq[OF this, symmetric]
+ hence "\<And>i. i\<in>R \<Longrightarrow> f i \<le> y / z" using * by auto
+ thus ?thesis unfolding div SUP_le_iff by simp
+ qed
+ qed
+qed
+
+instantiation pextreal :: topological_space
+begin
+
+definition "open A \<longleftrightarrow>
+ (\<exists>T. open T \<and> (Real ` (T\<inter>{0..}) = A - {\<omega>})) \<and> (\<omega> \<in> A \<longrightarrow> (\<exists>x\<ge>0. {Real x <..} \<subseteq> A))"
+
+lemma open_omega: "open A \<Longrightarrow> \<omega> \<in> A \<Longrightarrow> (\<exists>x\<ge>0. {Real x<..} \<subseteq> A)"
+ unfolding open_pextreal_def by auto
+
+lemma open_omegaD: assumes "open A" "\<omega> \<in> A" obtains x where "x\<ge>0" "{Real x<..} \<subseteq> A"
+ using open_omega[OF assms] by auto
+
+lemma pextreal_openE: assumes "open A" obtains A' x where
+ "open A'" "Real ` (A' \<inter> {0..}) = A - {\<omega>}"
+ "x \<ge> 0" "\<omega> \<in> A \<Longrightarrow> {Real x<..} \<subseteq> A"
+ using assms open_pextreal_def by auto
+
+instance
+proof
+ let ?U = "UNIV::pextreal set"
+ show "open ?U" unfolding open_pextreal_def
+ by (auto intro!: exI[of _ "UNIV"] exI[of _ 0])
+next
+ fix S T::"pextreal set" assume "open S" and "open T"
+ from `open S`[THEN pextreal_openE] guess S' xS . note S' = this
+ from `open T`[THEN pextreal_openE] guess T' xT . note T' = this
+
+ from S'(1-3) T'(1-3)
+ show "open (S \<inter> T)" unfolding open_pextreal_def
+ proof (safe intro!: exI[of _ "S' \<inter> T'"] exI[of _ "max xS xT"])
+ fix x assume *: "Real (max xS xT) < x" and "\<omega> \<in> S" "\<omega> \<in> T"
+ from `\<omega> \<in> S`[THEN S'(4)] * show "x \<in> S"
+ by (cases x, auto simp: max_def split: split_if_asm)
+ from `\<omega> \<in> T`[THEN T'(4)] * show "x \<in> T"
+ by (cases x, auto simp: max_def split: split_if_asm)
+ next
+ fix x assume x: "x \<notin> Real ` (S' \<inter> T' \<inter> {0..})"
+ have *: "S' \<inter> T' \<inter> {0..} = (S' \<inter> {0..}) \<inter> (T' \<inter> {0..})" by auto
+ assume "x \<in> T" "x \<in> S"
+ with S'(2) T'(2) show "x = \<omega>"
+ using x[unfolded *] inj_on_image_Int[OF inj_on_Real] by auto
+ qed auto
+next
+ fix K assume openK: "\<forall>S \<in> K. open (S:: pextreal set)"
+ hence "\<forall>S\<in>K. \<exists>T. open T \<and> Real ` (T \<inter> {0..}) = S - {\<omega>}" by (auto simp: open_pextreal_def)
+ from bchoice[OF this] guess T .. note T = this[rule_format]
+
+ show "open (\<Union>K)" unfolding open_pextreal_def
+ proof (safe intro!: exI[of _ "\<Union>(T ` K)"])
+ fix x S assume "0 \<le> x" "x \<in> T S" "S \<in> K"
+ with T[OF `S \<in> K`] show "Real x \<in> \<Union>K" by auto
+ next
+ fix x S assume x: "x \<notin> Real ` (\<Union>T ` K \<inter> {0..})" "S \<in> K" "x \<in> S"
+ hence "x \<notin> Real ` (T S \<inter> {0..})"
+ by (auto simp: image_UN UN_simps[symmetric] simp del: UN_simps)
+ thus "x = \<omega>" using T[OF `S \<in> K`] `x \<in> S` by auto
+ next
+ fix S assume "\<omega> \<in> S" "S \<in> K"
+ from openK[rule_format, OF `S \<in> K`, THEN pextreal_openE] guess S' x .
+ from this(3, 4) `\<omega> \<in> S`
+ show "\<exists>x\<ge>0. {Real x<..} \<subseteq> \<Union>K"
+ by (auto intro!: exI[of _ x] bexI[OF _ `S \<in> K`])
+ next
+ from T[THEN conjunct1] show "open (\<Union>T`K)" by auto
+ qed auto
+qed
+end
+
+lemma open_pextreal_lessThan[simp]:
+ "open {..< a :: pextreal}"
+proof (cases a)
+ case (preal x) thus ?thesis unfolding open_pextreal_def
+ proof (safe intro!: exI[of _ "{..< x}"])
+ fix y assume "y < Real x"
+ moreover assume "y \<notin> Real ` ({..<x} \<inter> {0..})"
+ ultimately have "y \<noteq> Real (real y)" using preal by (cases y) auto
+ thus "y = \<omega>" by (auto simp: Real_real split: split_if_asm)
+ qed auto
+next
+ case infinite thus ?thesis
+ unfolding open_pextreal_def by (auto intro!: exI[of _ UNIV])
+qed
+
+lemma open_pextreal_greaterThan[simp]:
+ "open {a :: pextreal <..}"
+proof (cases a)
+ case (preal x) thus ?thesis unfolding open_pextreal_def
+ proof (safe intro!: exI[of _ "{x <..}"])
+ fix y assume "Real x < y"
+ moreover assume "y \<notin> Real ` ({x<..} \<inter> {0..})"
+ ultimately have "y \<noteq> Real (real y)" using preal by (cases y) auto
+ thus "y = \<omega>" by (auto simp: Real_real split: split_if_asm)
+ qed auto
+next
+ case infinite thus ?thesis
+ unfolding open_pextreal_def by (auto intro!: exI[of _ "{}"])
+qed
+
+lemma pextreal_open_greaterThanLessThan[simp]: "open {a::pextreal <..< b}"
+ unfolding greaterThanLessThan_def by auto
+
+lemma closed_pextreal_atLeast[simp, intro]: "closed {a :: pextreal ..}"
+proof -
+ have "- {a ..} = {..< a}" by auto
+ then show "closed {a ..}"
+ unfolding closed_def using open_pextreal_lessThan by auto
+qed
+
+lemma closed_pextreal_atMost[simp, intro]: "closed {.. b :: pextreal}"
+proof -
+ have "- {.. b} = {b <..}" by auto
+ then show "closed {.. b}"
+ unfolding closed_def using open_pextreal_greaterThan by auto
+qed
+
+lemma closed_pextreal_atLeastAtMost[simp, intro]:
+ shows "closed {a :: pextreal .. b}"
+ unfolding atLeastAtMost_def by auto
+
+lemma pextreal_dense:
+ fixes x y :: pextreal assumes "x < y"
+ shows "\<exists>z. x < z \<and> z < y"
+proof -
+ from `x < y` obtain p where p: "x = Real p" "0 \<le> p" by (cases x) auto
+ show ?thesis
+ proof (cases y)
+ case (preal r) with p `x < y` have "p < r" by auto
+ with dense obtain z where "p < z" "z < r" by auto
+ thus ?thesis using preal p by (auto intro!: exI[of _ "Real z"])
+ next
+ case infinite thus ?thesis using `x < y` p
+ by (auto intro!: exI[of _ "Real p + 1"])
+ qed
+qed
+
+instance pextreal :: t2_space
+proof
+ fix x y :: pextreal assume "x \<noteq> y"
+ let "?P x (y::pextreal)" = "\<exists> U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {}"
+
+ { fix x y :: pextreal assume "x < y"
+ from pextreal_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 \<noteq> y`
+ show "\<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {}"
+ proof (cases rule: linorder_cases)
+ assume "x = y" with `x \<noteq> 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
+
+definition (in complete_lattice) isoton :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> bool" (infix "\<up>" 50) where
+ "A \<up> X \<longleftrightarrow> (\<forall>i. A i \<le> A (Suc i)) \<and> (SUP i. A i) = X"
+
+definition (in complete_lattice) antiton (infix "\<down>" 50) where
+ "A \<down> X \<longleftrightarrow> (\<forall>i. A i \<ge> A (Suc i)) \<and> (INF i. A i) = X"
+
+lemma isotoneI[intro?]: "\<lbrakk> \<And>i. f i \<le> f (Suc i) ; (SUP i. f i) = F \<rbrakk> \<Longrightarrow> f \<up> F"
+ unfolding isoton_def by auto
+
+lemma (in complete_lattice) isotonD[dest]:
+ assumes "A \<up> X" shows "A i \<le> A (Suc i)" "(SUP i. A i) = X"
+ using assms unfolding isoton_def by auto
+
+lemma isotonD'[dest]:
+ assumes "(A::_=>_) \<up> X" shows "A i x \<le> A (Suc i) x" "(SUP i. A i) = X"
+ using assms unfolding isoton_def le_fun_def by auto
+
+lemma isoton_mono_le:
+ assumes "f \<up> x" "i \<le> j"
+ shows "f i \<le> f j"
+ using `f \<up> x`[THEN isotonD(1)] lift_Suc_mono_le[of f, OF _ `i \<le> j`] by auto
+
+lemma isoton_const:
+ shows "(\<lambda> i. c) \<up> c"
+unfolding isoton_def by auto
+
+lemma isoton_cmult_right:
+ assumes "f \<up> (x::pextreal)"
+ shows "(\<lambda>i. c * f i) \<up> (c * x)"
+ using assms unfolding isoton_def pextreal_SUP_cmult
+ by (auto intro: pextreal_mult_cancel)
+
+lemma isoton_cmult_left:
+ "f \<up> (x::pextreal) \<Longrightarrow> (\<lambda>i. f i * c) \<up> (x * c)"
+ by (subst (1 2) mult_commute) (rule isoton_cmult_right)
+
+lemma isoton_add:
+ assumes "f \<up> (x::pextreal)" and "g \<up> y"
+ shows "(\<lambda>i. f i + g i) \<up> (x + y)"
+ using assms unfolding isoton_def
+ by (auto intro: pextreal_mult_cancel add_mono simp: SUPR_pextreal_add)
+
+lemma isoton_fun_expand:
+ "f \<up> x \<longleftrightarrow> (\<forall>i. (\<lambda>j. f j i) \<up> (x i))"
+proof -
+ have "\<And>i. {y. \<exists>f'\<in>range f. y = f' i} = range (\<lambda>j. f j i)"
+ by auto
+ with assms show ?thesis
+ by (auto simp add: isoton_def le_fun_def Sup_fun_def SUPR_def)
+qed
+
+lemma isoton_indicator:
+ assumes "f \<up> g"
+ shows "(\<lambda>i x. f i x * indicator A x) \<up> (\<lambda>x. g x * indicator A x :: pextreal)"
+ using assms unfolding isoton_fun_expand by (auto intro!: isoton_cmult_left)
+
+lemma isoton_setsum:
+ fixes f :: "'a \<Rightarrow> nat \<Rightarrow> pextreal"
+ assumes "finite A" "A \<noteq> {}"
+ assumes "\<And> x. x \<in> A \<Longrightarrow> f x \<up> y x"
+ shows "(\<lambda> i. (\<Sum> x \<in> A. f x i)) \<up> (\<Sum> x \<in> A. y x)"
+using assms
+proof (induct A rule:finite_ne_induct)
+ case singleton thus ?case by auto
+next
+ case (insert a A) note asms = this
+ hence *: "(\<lambda> i. \<Sum> x \<in> A. f x i) \<up> (\<Sum> x \<in> A. y x)" by auto
+ have **: "(\<lambda> i. f a i) \<up> y a" using asms by simp
+ have "(\<lambda> i. f a i + (\<Sum> x \<in> A. f x i)) \<up> (y a + (\<Sum> x \<in> A. y x))"
+ using * ** isoton_add by auto
+ thus "(\<lambda> i. \<Sum> x \<in> insert a A. f x i) \<up> (\<Sum> x \<in> insert a A. y x)"
+ using asms by fastsimp
+qed
+
+lemma isoton_Sup:
+ assumes "f \<up> u"
+ shows "f i \<le> u"
+ using le_SUPI[of i UNIV f] assms
+ unfolding isoton_def by auto
+
+lemma isoton_mono:
+ assumes iso: "x \<up> a" "y \<up> b" and *: "\<And>n. x n \<le> y (N n)"
+ shows "a \<le> b"
+proof -
+ from iso have "a = (SUP n. x n)" "b = (SUP n. y n)"
+ unfolding isoton_def by auto
+ with * show ?thesis by (auto intro!: SUP_mono)
+qed
+
+lemma pextreal_le_mult_one_interval:
+ fixes x y :: pextreal
+ assumes "\<And>z. \<lbrakk> 0 < z ; z < 1 \<rbrakk> \<Longrightarrow> z * x \<le> y"
+ shows "x \<le> y"
+proof (cases x, cases y)
+ assume "x = \<omega>"
+ with assms[of "1 / 2"]
+ show "x \<le> y" by simp
+next
+ fix r p assume *: "y = Real p" "x = Real r" and **: "0 \<le> r" "0 \<le> p"
+ have "r \<le> p"
+ proof (rule field_le_mult_one_interval)
+ fix z :: real assume "0 < z" and "z < 1"
+ with assms[of "Real z"]
+ show "z * r \<le> p" using ** * by (auto simp: zero_le_mult_iff)
+ qed
+ thus "x \<le> y" using ** * by simp
+qed simp
+
+lemma pextreal_greater_0[intro]:
+ fixes a :: pextreal
+ assumes "a \<noteq> 0"
+ shows "a > 0"
+using assms apply (cases a) by auto
+
+lemma pextreal_mult_strict_right_mono:
+ assumes "a < b" and "0 < c" "c < \<omega>"
+ shows "a * c < b * c"
+ using assms
+ by (cases a, cases b, cases c)
+ (auto simp: zero_le_mult_iff pextreal_less_\<omega>)
+
+lemma minus_pextreal_eq2:
+ fixes x y z :: pextreal
+ assumes "y \<le> x" and "y \<noteq> \<omega>" shows "z = x - y \<longleftrightarrow> z + y = x"
+ using assms
+ apply (subst eq_commute)
+ apply (subst minus_pextreal_eq)
+ by (cases x, cases z, auto simp add: ac_simps not_less)
+
+lemma pextreal_diff_eq_diff_imp_eq:
+ assumes "a \<noteq> \<omega>" "b \<le> a" "c \<le> a"
+ assumes "a - b = a - c"
+ shows "b = c"
+ using assms
+ by (cases a, cases b, cases c) (auto split: split_if_asm)
+
+lemma pextreal_inverse_eq_0: "inverse x = 0 \<longleftrightarrow> x = \<omega>"
+ by (cases x) auto
+
+lemma pextreal_mult_inverse:
+ "\<lbrakk> x \<noteq> \<omega> ; x \<noteq> 0 \<rbrakk> \<Longrightarrow> x * inverse x = 1"
+ by (cases x) auto
+
+lemma pextreal_zero_less_diff_iff:
+ fixes a b :: pextreal shows "0 < a - b \<longleftrightarrow> b < a"
+ apply (cases a, cases b)
+ apply (auto simp: pextreal_noteq_omega_Ex pextreal_less_\<omega>)
+ apply (cases b)
+ by auto
+
+lemma pextreal_less_Real_Ex:
+ fixes a b :: pextreal shows "x < Real r \<longleftrightarrow> (\<exists>p\<ge>0. p < r \<and> x = Real p)"
+ by (cases x) auto
+
+lemma open_Real: assumes "open S" shows "open (Real ` ({0..} \<inter> S))"
+ unfolding open_pextreal_def apply(rule,rule,rule,rule assms) by auto
+
+lemma pextreal_zero_le_diff:
+ fixes a b :: pextreal shows "a - b = 0 \<longleftrightarrow> a \<le> b"
+ by (cases a, cases b, simp_all, cases b, auto)
+
+lemma lim_Real[simp]: assumes "\<forall>n. f n \<ge> 0" "m\<ge>0"
+ shows "(\<lambda>n. Real (f n)) ----> Real m \<longleftrightarrow> (\<lambda>n. f n) ----> m" (is "?l = ?r")
+proof assume ?l show ?r unfolding Lim_sequentially
+ proof safe fix e::real assume e:"e>0"
+ note open_ball[of m e] note open_Real[OF this]
+ note * = `?l`[unfolded tendsto_def,rule_format,OF this]
+ have "eventually (\<lambda>x. Real (f x) \<in> Real ` ({0..} \<inter> ball m e)) sequentially"
+ apply(rule *) unfolding image_iff using assms(2) e by auto
+ thus "\<exists>N. \<forall>n\<ge>N. dist (f n) m < e" unfolding eventually_sequentially
+ apply safe apply(rule_tac x=N in exI,safe) apply(erule_tac x=n in allE,safe)
+ proof- fix n x assume "Real (f n) = Real x" "0 \<le> x"
+ hence *:"f n = x" using assms(1) by auto
+ assume "x \<in> ball m e" thus "dist (f n) m < e" unfolding *
+ by (auto simp add:dist_commute)
+ qed qed
+next assume ?r show ?l unfolding tendsto_def eventually_sequentially
+ proof safe fix S assume S:"open S" "Real m \<in> S"
+ guess T y using S(1) apply-apply(erule pextreal_openE) . note T=this
+ have "m\<in>real ` (S - {\<omega>})" unfolding image_iff
+ apply(rule_tac x="Real m" in bexI) using assms(2) S(2) by auto
+ hence "m \<in> T" unfolding T(2)[THEN sym] by auto
+ from `?r`[unfolded tendsto_def eventually_sequentially,rule_format,OF T(1) this]
+ guess N .. note N=this[rule_format]
+ show "\<exists>N. \<forall>n\<ge>N. Real (f n) \<in> S" apply(rule_tac x=N in exI)
+ proof safe fix n assume n:"N\<le>n"
+ have "f n \<in> real ` (S - {\<omega>})" using N[OF n] assms unfolding T(2)[THEN sym]
+ unfolding image_iff apply-apply(rule_tac x="Real (f n)" in bexI)
+ unfolding real_Real by auto
+ then guess x unfolding image_iff .. note x=this
+ show "Real (f n) \<in> S" unfolding x apply(subst Real_real) using x by auto
+ qed
+ qed
+qed
+
+lemma pextreal_INFI:
+ fixes x :: pextreal
+ assumes "\<And>i. i \<in> A \<Longrightarrow> x \<le> f i"
+ assumes "\<And>y. (\<And>i. i \<in> A \<Longrightarrow> y \<le> f i) \<Longrightarrow> y \<le> x"
+ shows "(INF i:A. f i) = x"
+ unfolding INFI_def Inf_pextreal_def
+ using assms by (auto intro!: Greatest_equality)
+
+lemma real_of_pextreal_less:"x < y \<Longrightarrow> y\<noteq>\<omega> \<Longrightarrow> real x < real y"
+proof- case goal1
+ have *:"y = Real (real y)" "x = Real (real x)" using goal1 Real_real by auto
+ show ?case using goal1 apply- apply(subst(asm) *(1))apply(subst(asm) *(2))
+ unfolding pextreal_less by auto
+qed
+
+lemma not_less_omega[simp]:"\<not> x < \<omega> \<longleftrightarrow> x = \<omega>"
+ by (metis antisym_conv3 pextreal_less(3))
+
+lemma Real_real': assumes "x\<noteq>\<omega>" shows "Real (real x) = x"
+proof- have *:"(THE r. 0 \<le> r \<and> x = Real r) = real x"
+ apply(rule the_equality) using assms unfolding Real_real by auto
+ have "Real (THE r. 0 \<le> r \<and> x = Real r) = x" unfolding *
+ using assms unfolding Real_real by auto
+ thus ?thesis unfolding real_of_pextreal_def of_pextreal_def
+ unfolding pextreal_case_def using assms by auto
+qed
+
+lemma Real_less_plus_one:"Real x < Real (max (x + 1) 1)"
+ unfolding pextreal_less by auto
+
+lemma Lim_omega: "f ----> \<omega> \<longleftrightarrow> (\<forall>B. \<exists>N. \<forall>n\<ge>N. f n \<ge> Real B)" (is "?l = ?r")
+proof assume ?r show ?l apply(rule topological_tendstoI)
+ unfolding eventually_sequentially
+ proof- fix S assume "open S" "\<omega> \<in> S"
+ from open_omega[OF this] guess B .. note B=this
+ from `?r`[rule_format,of "(max B 0)+1"] guess N .. note N=this
+ show "\<exists>N. \<forall>n\<ge>N. f n \<in> S" apply(rule_tac x=N in exI)
+ proof safe case goal1
+ have "Real B < Real ((max B 0) + 1)" by auto
+ also have "... \<le> 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 {Real B<..}" "\<omega> \<in> {Real B<..}" by auto
+ from topological_tendstoD[OF `?l` this,unfolded eventually_sequentially]
+ guess N .. note N=this
+ show "\<exists>N. \<forall>n\<ge>N. Real B \<le> f n" apply(rule_tac x=N in exI) using N by auto
+ qed
+qed
+
+lemma Lim_bounded_omgea: assumes lim:"f ----> l" and "\<And>n. f n \<le> Real B" shows "l \<noteq> \<omega>"
+proof(rule ccontr,unfold not_not) let ?B = "max (B + 1) 1" assume as:"l=\<omega>"
+ from lim[unfolded this Lim_omega,rule_format,of "?B"]
+ guess N .. note N=this[rule_format,OF le_refl]
+ hence "Real ?B \<le> Real B" using assms(2)[of N] by(rule order_trans)
+ hence "Real ?B < Real ?B" using Real_less_plus_one[of B] by(rule le_less_trans)
+ thus False by auto
+qed
+
+lemma incseq_le_pextreal: assumes inc: "\<And>n m. n\<ge>m \<Longrightarrow> X n \<ge> X m"
+ and lim: "X ----> (L::pextreal)" shows "X n \<le> L"
+proof(cases "L = \<omega>")
+ case False have "\<forall>n. X n \<noteq> \<omega>"
+ proof(rule ccontr,unfold not_all not_not,safe)
+ case goal1 hence "\<forall>n\<ge>x. X n = \<omega>" using inc[of x] by auto
+ hence "X ----> \<omega>" unfolding tendsto_def eventually_sequentially
+ apply safe apply(rule_tac x=x in exI) by auto
+ note Lim_unique[OF trivial_limit_sequentially this lim]
+ with False show False by auto
+ qed note * =this[rule_format]
+
+ have **:"\<forall>m n. m \<le> n \<longrightarrow> Real (real (X m)) \<le> Real (real (X n))"
+ unfolding Real_real using * inc by auto
+ have "real (X n) \<le> real L" apply-apply(rule incseq_le) defer
+ apply(subst lim_Real[THEN sym]) apply(rule,rule,rule)
+ unfolding Real_real'[OF *] Real_real'[OF False]
+ unfolding incseq_def using ** lim by auto
+ hence "Real (real (X n)) \<le> Real (real L)" by auto
+ thus ?thesis unfolding Real_real using * False by auto
+qed auto
+
+lemma SUP_Lim_pextreal: assumes "\<And>n m. n\<ge>m \<Longrightarrow> f n \<ge> f m" "f ----> l"
+ shows "(SUP n. f n) = (l::pextreal)" unfolding SUPR_def Sup_pextreal_def
+proof (safe intro!: Least_equality)
+ fix n::nat show "f n \<le> l" apply(rule incseq_le_pextreal)
+ using assms by auto
+next fix y assume y:"\<forall>x\<in>range f. x \<le> y" show "l \<le> y"
+ proof(rule ccontr,cases "y=\<omega>",unfold not_le)
+ case False assume as:"y < l"
+ have l:"l \<noteq> \<omega>" apply(rule Lim_bounded_omgea[OF assms(2), of "real y"])
+ using False y unfolding Real_real by auto
+
+ have yl:"real y < real l" using as apply-
+ apply(subst(asm) Real_real'[THEN sym,OF `y\<noteq>\<omega>`])
+ apply(subst(asm) Real_real'[THEN sym,OF `l\<noteq>\<omega>`])
+ unfolding pextreal_less apply(subst(asm) if_P) by auto
+ hence "y + (y - l) * Real (1 / 2) < l" apply-
+ apply(subst Real_real'[THEN sym,OF `y\<noteq>\<omega>`]) apply(subst(2) Real_real'[THEN sym,OF `y\<noteq>\<omega>`])
+ apply(subst Real_real'[THEN sym,OF `l\<noteq>\<omega>`]) apply(subst(2) Real_real'[THEN sym,OF `l\<noteq>\<omega>`]) by auto
+ hence *:"l \<in> {y + (y - l) / 2<..}" by auto
+ have "open {y + (y-l)/2 <..}" by auto
+ note topological_tendstoD[OF assms(2) this *]
+ from this[unfolded eventually_sequentially] guess N .. note this[rule_format, of N]
+ hence "y + (y - l) * Real (1 / 2) < y" using y[rule_format,of "f N"] by auto
+ hence "Real (real y) + (Real (real y) - Real (real l)) * Real (1 / 2) < Real (real y)"
+ unfolding Real_real using `y\<noteq>\<omega>` `l\<noteq>\<omega>` by auto
+ thus False using yl by auto
+ qed auto
+qed
+
+lemma Real_max':"Real x = Real (max x 0)"
+proof(cases "x < 0") case True
+ hence *:"max x 0 = 0" by auto
+ show ?thesis unfolding * using True by auto
+qed auto
+
+lemma lim_pextreal_increasing: assumes "\<forall>n m. n\<ge>m \<longrightarrow> f n \<ge> f m"
+ obtains l where "f ----> (l::pextreal)"
+proof(cases "\<exists>B. \<forall>n. f n < Real B")
+ case False thus thesis apply- apply(rule that[of \<omega>]) unfolding Lim_omega not_ex not_all
+ apply safe apply(erule_tac x=B in allE,safe) apply(rule_tac x=x in exI,safe)
+ apply(rule order_trans[OF _ assms[rule_format]]) by auto
+next case True then guess B .. note B = this[rule_format]
+ hence *:"\<And>n. f n < \<omega>" apply-apply(rule less_le_trans,assumption) by auto
+ have *:"\<And>n. f n \<noteq> \<omega>" proof- case goal1 show ?case using *[of n] by auto qed
+ have B':"\<And>n. real (f n) \<le> max 0 B" proof- case goal1 thus ?case
+ using B[of n] apply-apply(subst(asm) Real_real'[THEN sym]) defer
+ apply(subst(asm)(2) Real_max') unfolding pextreal_less apply(subst(asm) if_P) using *[of n] by auto
+ qed
+ have "\<exists>l. (\<lambda>n. real (f n)) ----> l" apply(rule Topology_Euclidean_Space.bounded_increasing_convergent)
+ proof safe show "bounded {real (f n) |n. True}"
+ unfolding bounded_def apply(rule_tac x=0 in exI,rule_tac x="max 0 B" in exI)
+ using B' unfolding dist_norm by auto
+ fix n::nat have "Real (real (f n)) \<le> Real (real (f (Suc n)))"
+ using assms[rule_format,of n "Suc n"] apply(subst Real_real)+
+ using *[of n] *[of "Suc n"] by fastsimp
+ thus "real (f n) \<le> real (f (Suc n))" by auto
+ qed then guess l .. note l=this
+ have "0 \<le> l" apply(rule LIMSEQ_le_const[OF l])
+ by(rule_tac x=0 in exI,auto)
+
+ thus ?thesis apply-apply(rule that[of "Real l"])
+ using l apply-apply(subst(asm) lim_Real[THEN sym]) prefer 3
+ unfolding Real_real using * by auto
+qed
+
+lemma setsum_neq_omega: assumes "finite s" "\<And>x. x \<in> s \<Longrightarrow> f x \<noteq> \<omega>"
+ shows "setsum f s \<noteq> \<omega>" using assms
+proof induct case (insert x s)
+ show ?case unfolding setsum.insert[OF insert(1-2)]
+ using insert by auto
+qed auto
+
+
+lemma real_Real': "0 \<le> x \<Longrightarrow> real (Real x) = x"
+ unfolding real_Real by auto
+
+lemma real_pextreal_pos[intro]:
+ assumes "x \<noteq> 0" "x \<noteq> \<omega>"
+ shows "real x > 0"
+ apply(subst real_Real'[THEN sym,of 0]) defer
+ apply(rule real_of_pextreal_less) using assms by auto
+
+lemma Lim_omega_gt: "f ----> \<omega> \<longleftrightarrow> (\<forall>B. \<exists>N. \<forall>n\<ge>N. f n > Real B)" (is "?l = ?r")
+proof assume ?l thus ?r unfolding Lim_omega apply safe
+ apply(erule_tac x="max B 0 +1" in allE,safe)
+ apply(rule_tac x=N in exI,safe) apply(erule_tac x=n in allE,safe)
+ apply(rule_tac y="Real (max B 0 + 1)" in less_le_trans) by auto
+next assume ?r thus ?l unfolding Lim_omega apply safe
+ apply(erule_tac x=B in allE,safe) apply(rule_tac x=N in exI,safe) by auto
+qed
+
+lemma pextreal_minus_le_cancel:
+ fixes a b c :: pextreal
+ assumes "b \<le> a"
+ shows "c - a \<le> c - b"
+ using assms by (cases a, cases b, cases c, simp, simp, simp, cases b, cases c, simp_all)
+
+lemma pextreal_minus_\<omega>[simp]: "x - \<omega> = 0" by (cases x) simp_all
+
+lemma pextreal_minus_mono[intro]: "a - x \<le> (a::pextreal)"
+proof- have "a - x \<le> a - 0"
+ apply(rule pextreal_minus_le_cancel) by auto
+ thus ?thesis by auto
+qed
+
+lemma pextreal_minus_eq_\<omega>[simp]: "x - y = \<omega> \<longleftrightarrow> (x = \<omega> \<and> y \<noteq> \<omega>)"
+ by (cases x, cases y) (auto, cases y, auto)
+
+lemma pextreal_less_minus_iff:
+ fixes a b c :: pextreal
+ shows "a < b - c \<longleftrightarrow> c + a < b"
+ by (cases c, cases a, cases b, auto)
+
+lemma pextreal_minus_less_iff:
+ fixes a b c :: pextreal shows "a - c < b \<longleftrightarrow> (0 < b \<and> (c \<noteq> \<omega> \<longrightarrow> a < b + c))"
+ by (cases c, cases a, cases b, auto)
+
+lemma pextreal_le_minus_iff:
+ fixes a b c :: pextreal
+ shows "a \<le> c - b \<longleftrightarrow> ((c \<le> b \<longrightarrow> a = 0) \<and> (b < c \<longrightarrow> a + b \<le> c))"
+ by (cases a, cases c, cases b, auto simp: pextreal_noteq_omega_Ex)
+
+lemma pextreal_minus_le_iff:
+ fixes a b c :: pextreal
+ shows "a - c \<le> b \<longleftrightarrow> (c \<le> a \<longrightarrow> a \<le> b + c)"
+ by (cases a, cases c, cases b, auto simp: pextreal_noteq_omega_Ex)
+
+lemmas pextreal_minus_order = pextreal_minus_le_iff pextreal_minus_less_iff pextreal_le_minus_iff pextreal_less_minus_iff
+
+lemma pextreal_minus_strict_mono:
+ assumes "a > 0" "x > 0" "a\<noteq>\<omega>"
+ shows "a - x < (a::pextreal)"
+ using assms by(cases x, cases a, auto)
+
+lemma pextreal_minus':
+ "Real r - Real p = (if 0 \<le> r \<and> p \<le> r then if 0 \<le> p then Real (r - p) else Real r else 0)"
+ by (auto simp: minus_pextreal_eq not_less)
+
+lemma pextreal_minus_plus:
+ "x \<le> (a::pextreal) \<Longrightarrow> a - x + x = a"
+ by (cases a, cases x) auto
+
+lemma pextreal_cancel_plus_minus: "b \<noteq> \<omega> \<Longrightarrow> a + b - b = a"
+ by (cases a, cases b) auto
+
+lemma pextreal_minus_le_cancel_right:
+ fixes a b c :: pextreal
+ assumes "a \<le> b" "c \<le> a"
+ shows "a - c \<le> b - c"
+ using assms by (cases a, cases b, cases c, auto, cases c, auto)
+
+lemma real_of_pextreal_setsum':
+ assumes "\<forall>x \<in> S. f x \<noteq> \<omega>"
+ shows "(\<Sum>x\<in>S. real (f x)) = real (setsum f S)"
+proof cases
+ assume "finite S"
+ from this assms show ?thesis
+ by induct (simp_all add: real_of_pextreal_add setsum_\<omega>)
+qed simp
+
+lemma Lim_omega_pos: "f ----> \<omega> \<longleftrightarrow> (\<forall>B>0. \<exists>N. \<forall>n\<ge>N. f n \<ge> Real B)" (is "?l = ?r")
+ unfolding Lim_omega apply safe defer
+ apply(erule_tac x="max 1 B" in allE) apply safe defer
+ apply(rule_tac x=N in exI,safe) apply(erule_tac x=n in allE,safe)
+ apply(rule_tac y="Real (max 1 B)" in order_trans) by auto
+
+lemma pextreal_LimI_finite:
+ assumes "x \<noteq> \<omega>" "\<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)
+ fix S assume "open S" "x \<in> S"
+ then obtain A where "open A" and A_eq: "Real ` (A \<inter> {0..}) = S - {\<omega>}" by (auto elim!: pextreal_openE)
+ then have "x \<in> Real ` (A \<inter> {0..})" using `x \<in> S` `x \<noteq> \<omega>` by auto
+ then have "real x \<in> A" by auto
+ then obtain r where "0 < r" and dist: "\<And>y. dist y (real x) < r \<Longrightarrow> y \<in> A"
+ using `open A` unfolding open_real_def by auto
+ then obtain n where
+ upper: "\<And>N. n \<le> N \<Longrightarrow> u N < x + Real r" and
+ lower: "\<And>N. n \<le> N \<Longrightarrow> x < u N + Real r" using assms(2)[of "Real r"] by auto
+ show "\<exists>N. \<forall>n\<ge>N. u n \<in> S"
+ proof (safe intro!: exI[of _ n])
+ fix N assume "n \<le> N"
+ from upper[OF this] `x \<noteq> \<omega>` `0 < r`
+ have "u N \<noteq> \<omega>" by (force simp: pextreal_noteq_omega_Ex)
+ with `x \<noteq> \<omega>` `0 < r` lower[OF `n \<le> N`] upper[OF `n \<le> N`]
+ have "dist (real (u N)) (real x) < r" "u N \<noteq> \<omega>"
+ by (auto simp: pextreal_noteq_omega_Ex dist_real_def abs_diff_less_iff field_simps)
+ from dist[OF this(1)]
+ have "u N \<in> Real ` (A \<inter> {0..})" using `u N \<noteq> \<omega>`
+ by (auto intro!: image_eqI[of _ _ "real (u N)"] simp: pextreal_noteq_omega_Ex Real_real)
+ thus "u N \<in> S" using A_eq by simp
+ qed
+qed
+
+lemma real_Real_max:"real (Real x) = max x 0"
+ unfolding real_Real by auto
+
+lemma Sup_lim:
+ assumes "\<forall>n. b n \<in> s" "b ----> (a::pextreal)"
+ shows "a \<le> Sup s"
+proof(rule ccontr,unfold not_le)
+ assume as:"Sup s < a" hence om:"Sup s \<noteq> \<omega>" by auto
+ have s:"s \<noteq> {}" using assms by auto
+ { presume *:"\<forall>n. b n < a \<Longrightarrow> False"
+ show False apply(cases,rule *,assumption,unfold not_all not_less)
+ proof- case goal1 then guess n .. note n=this
+ thus False using complete_lattice_class.Sup_upper[OF assms(1)[rule_format,of n]]
+ using as by auto
+ qed
+ } assume b:"\<forall>n. b n < a"
+ show False
+ proof(cases "a = \<omega>")
+ case False have *:"a - Sup s > 0"
+ using False as by(auto simp: pextreal_zero_le_diff)
+ have "(a - Sup s) / 2 \<le> a / 2" unfolding divide_pextreal_def
+ apply(rule mult_right_mono) by auto
+ also have "... = Real (real (a / 2))" apply(rule Real_real'[THEN sym])
+ using False by auto
+ also have "... < Real (real a)" unfolding pextreal_less using as False
+ by(auto simp add: real_of_pextreal_mult[THEN sym])
+ also have "... = a" apply(rule Real_real') using False by auto
+ finally have asup:"a > (a - Sup s) / 2" .
+ have "\<exists>n. a - b n < (a - Sup s) / 2"
+ proof(rule ccontr,unfold not_ex not_less)
+ case goal1
+ have "(a - Sup s) * Real (1 / 2) > 0"
+ using * by auto
+ hence "a - (a - Sup s) * Real (1 / 2) < a"
+ apply-apply(rule pextreal_minus_strict_mono)
+ using False * by auto
+ hence *:"a \<in> {a - (a - Sup s) / 2<..}"using asup by auto
+ note topological_tendstoD[OF assms(2) open_pextreal_greaterThan,OF *]
+ from this[unfolded eventually_sequentially] guess n ..
+ note n = this[rule_format,of n]
+ have "b n + (a - Sup s) / 2 \<le> a"
+ using add_right_mono[OF goal1[rule_format,of n],of "b n"]
+ unfolding pextreal_minus_plus[OF less_imp_le[OF b[rule_format]]]
+ by(auto simp: add_commute)
+ hence "b n \<le> a - (a - Sup s) / 2" unfolding pextreal_le_minus_iff
+ using asup by auto
+ hence "b n \<notin> {a - (a - Sup s) / 2<..}" by auto
+ thus False using n by auto
+ qed
+ then guess n .. note n = this
+ have "Sup s < a - (a - Sup s) / 2"
+ using False as om by (cases a) (auto simp: pextreal_noteq_omega_Ex field_simps)
+ also have "... \<le> b n"
+ proof- note add_right_mono[OF less_imp_le[OF n],of "b n"]
+ note this[unfolded pextreal_minus_plus[OF less_imp_le[OF b[rule_format]]]]
+ hence "a - (a - Sup s) / 2 \<le> (a - Sup s) / 2 + b n - (a - Sup s) / 2"
+ apply(rule pextreal_minus_le_cancel_right) using asup by auto
+ also have "... = b n + (a - Sup s) / 2 - (a - Sup s) / 2"
+ by(auto simp add: add_commute)
+ also have "... = b n" apply(subst pextreal_cancel_plus_minus)
+ proof(rule ccontr,unfold not_not) case goal1
+ show ?case using asup unfolding goal1 by auto
+ qed auto
+ finally show ?thesis .
+ qed
+ finally show False
+ using complete_lattice_class.Sup_upper[OF assms(1)[rule_format,of n]] by auto
+ next case True
+ from assms(2)[unfolded True Lim_omega_gt,rule_format,of "real (Sup s)"]
+ guess N .. note N = this[rule_format,of N]
+ thus False using complete_lattice_class.Sup_upper[OF assms(1)[rule_format,of N]]
+ unfolding Real_real using om by auto
+ qed qed
+
+lemma Sup_mono_lim:
+ assumes "\<forall>a\<in>A. \<exists>b. \<forall>n. b n \<in> B \<and> b ----> (a::pextreal)"
+ shows "Sup A \<le> Sup B"
+ unfolding Sup_le_iff apply(rule) apply(drule assms[rule_format]) apply safe
+ apply(rule_tac b=b in Sup_lim) by auto
+
+lemma pextreal_less_add:
+ assumes "x \<noteq> \<omega>" "a < b"
+ shows "x + a < x + b"
+ using assms by (cases a, cases b, cases x) auto
+
+lemma SUPR_lim:
+ assumes "\<forall>n. b n \<in> B" "(\<lambda>n. f (b n)) ----> (f a::pextreal)"
+ shows "f a \<le> SUPR B f"
+ unfolding SUPR_def apply(rule Sup_lim[of "\<lambda>n. f (b n)"])
+ using assms by auto
+
+lemma SUP_\<omega>_imp:
+ assumes "(SUP i. f i) = \<omega>"
+ shows "\<exists>i. Real x < f i"
+proof (rule ccontr)
+ assume "\<not> ?thesis" hence "\<And>i. f i \<le> Real x" by (simp add: not_less)
+ hence "(SUP i. f i) \<le> Real x" unfolding SUP_le_iff by auto
+ with assms show False by auto
+qed
+
+lemma SUPR_mono_lim:
+ assumes "\<forall>a\<in>A. \<exists>b. \<forall>n. b n \<in> B \<and> (\<lambda>n. f (b n)) ----> (f a::pextreal)"
+ shows "SUPR A f \<le> SUPR B f"
+ unfolding SUPR_def apply(rule Sup_mono_lim)
+ apply safe apply(drule assms[rule_format],safe)
+ apply(rule_tac x="\<lambda>n. f (b n)" in exI) by auto
+
+lemma real_0_imp_eq_0:
+ assumes "x \<noteq> \<omega>" "real x = 0"
+ shows "x = 0"
+using assms by (cases x) auto
+
+lemma SUPR_mono:
+ assumes "\<forall>a\<in>A. \<exists>b\<in>B. f b \<ge> f a"
+ shows "SUPR A f \<le> SUPR B f"
+ unfolding SUPR_def apply(rule Sup_mono)
+ using assms by auto
+
+lemma less_add_Real:
+ fixes x :: real
+ fixes a b :: pextreal
+ assumes "x \<ge> 0" "a < b"
+ shows "a + Real x < b + Real x"
+using assms by (cases a, cases b) auto
+
+lemma le_add_Real:
+ fixes x :: real
+ fixes a b :: pextreal
+ assumes "x \<ge> 0" "a \<le> b"
+ shows "a + Real x \<le> b + Real x"
+using assms by (cases a, cases b) auto
+
+lemma le_imp_less_pextreal:
+ fixes x :: pextreal
+ assumes "x > 0" "a + x \<le> b" "a \<noteq> \<omega>"
+ shows "a < b"
+using assms by (cases x, cases a, cases b) auto
+
+lemma pextreal_INF_minus:
+ fixes f :: "nat \<Rightarrow> pextreal"
+ assumes "c \<noteq> \<omega>"
+ shows "(INF i. c - f i) = c - (SUP i. f i)"
+proof (cases "SUP i. f i")
+ case infinite
+ from `c \<noteq> \<omega>` obtain x where [simp]: "c = Real x" by (cases c) auto
+ from SUP_\<omega>_imp[OF infinite] obtain i where "Real x < f i" by auto
+ have "(INF i. c - f i) \<le> c - f i"
+ by (auto intro!: complete_lattice_class.INF_leI)
+ also have "\<dots> = 0" using `Real x < f i` by (auto simp: minus_pextreal_eq)
+ finally show ?thesis using infinite by auto
+next
+ case (preal r)
+ from `c \<noteq> \<omega>` obtain x where c: "c = Real x" by (cases c) auto
+
+ show ?thesis unfolding c
+ proof (rule pextreal_INFI)
+ fix i have "f i \<le> (SUP i. f i)" by (rule le_SUPI) simp
+ thus "Real x - (SUP i. f i) \<le> Real x - f i" by (rule pextreal_minus_le_cancel)
+ next
+ fix y assume *: "\<And>i. i \<in> UNIV \<Longrightarrow> y \<le> Real x - f i"
+ from this[of 0] obtain p where p: "y = Real p" "0 \<le> p"
+ by (cases "f 0", cases y, auto split: split_if_asm)
+ hence "\<And>i. Real p \<le> Real x - f i" using * by auto
+ hence *: "\<And>i. Real x \<le> f i \<Longrightarrow> Real p = 0"
+ "\<And>i. f i < Real x \<Longrightarrow> Real p + f i \<le> Real x"
+ unfolding pextreal_le_minus_iff by auto
+ show "y \<le> Real x - (SUP i. f i)" unfolding p pextreal_le_minus_iff
+ proof safe
+ assume x_less: "Real x \<le> (SUP i. f i)"
+ show "Real p = 0"
+ proof (rule ccontr)
+ assume "Real p \<noteq> 0"
+ hence "0 < Real p" by auto
+ from Sup_close[OF this, of "range f"]
+ obtain i where e: "(SUP i. f i) < f i + Real p"
+ using preal unfolding SUPR_def by auto
+ hence "Real x \<le> f i + Real p" using x_less by auto
+ show False
+ proof cases
+ assume "\<forall>i. f i < Real x"
+ hence "Real p + f i \<le> Real x" using * by auto
+ hence "f i + Real p \<le> (SUP i. f i)" using x_less by (auto simp: field_simps)
+ thus False using e by auto
+ next
+ assume "\<not> (\<forall>i. f i < Real x)"
+ then obtain i where "Real x \<le> f i" by (auto simp: not_less)
+ from *(1)[OF this] show False using `Real p \<noteq> 0` by auto
+ qed
+ qed
+ next
+ have "\<And>i. f i \<le> (SUP i. f i)" by (rule complete_lattice_class.le_SUPI) auto
+ also assume "(SUP i. f i) < Real x"
+ finally have "\<And>i. f i < Real x" by auto
+ hence *: "\<And>i. Real p + f i \<le> Real x" using * by auto
+ have "Real p \<le> Real x" using *[of 0] by (cases "f 0") (auto split: split_if_asm)
+
+ have SUP_eq: "(SUP i. f i) \<le> Real x - Real p"
+ proof (rule SUP_leI)
+ fix i show "f i \<le> Real x - Real p" unfolding pextreal_le_minus_iff
+ proof safe
+ assume "Real x \<le> Real p"
+ with *[of i] show "f i = 0"
+ by (cases "f i") (auto split: split_if_asm)
+ next
+ assume "Real p < Real x"
+ show "f i + Real p \<le> Real x" using * by (auto simp: field_simps)
+ qed
+ qed
+
+ show "Real p + (SUP i. f i) \<le> Real x"
+ proof cases
+ assume "Real x \<le> Real p"
+ with `Real p \<le> Real x` have [simp]: "Real p = Real x" by (rule antisym)
+ { fix i have "f i = 0" using *[of i] by (cases "f i") (auto split: split_if_asm) }
+ hence "(SUP i. f i) \<le> 0" by (auto intro!: SUP_leI)
+ thus ?thesis by simp
+ next
+ assume "\<not> Real x \<le> Real p" hence "Real p < Real x" unfolding not_le .
+ with SUP_eq show ?thesis unfolding pextreal_le_minus_iff by (auto simp: field_simps)
+ qed
+ qed
+ qed
+qed
+
+lemma pextreal_SUP_minus:
+ fixes f :: "nat \<Rightarrow> pextreal"
+ shows "(SUP i. c - f i) = c - (INF i. f i)"
+proof (rule pextreal_SUPI)
+ fix i have "(INF i. f i) \<le> f i" by (rule INF_leI) simp
+ thus "c - f i \<le> c - (INF i. f i)" by (rule pextreal_minus_le_cancel)
+next
+ fix y assume *: "\<And>i. i \<in> UNIV \<Longrightarrow> c - f i \<le> y"
+ show "c - (INF i. f i) \<le> y"
+ proof (cases y)
+ case (preal p)
+
+ show ?thesis unfolding pextreal_minus_le_iff preal
+ proof safe
+ assume INF_le_x: "(INF i. f i) \<le> c"
+ from * have *: "\<And>i. f i \<le> c \<Longrightarrow> c \<le> Real p + f i"
+ unfolding pextreal_minus_le_iff preal by auto
+
+ have INF_eq: "c - Real p \<le> (INF i. f i)"
+ proof (rule le_INFI)
+ fix i show "c - Real p \<le> f i" unfolding pextreal_minus_le_iff
+ proof safe
+ assume "Real p \<le> c"
+ show "c \<le> f i + Real p"
+ proof cases
+ assume "f i \<le> c" from *[OF this]
+ show ?thesis by (simp add: field_simps)
+ next
+ assume "\<not> f i \<le> c"
+ hence "c \<le> f i" by auto
+ also have "\<dots> \<le> f i + Real p" by auto
+ finally show ?thesis .
+ qed
+ qed
+ qed
+
+ show "c \<le> Real p + (INF i. f i)"
+ proof cases
+ assume "Real p \<le> c"
+ with INF_eq show ?thesis unfolding pextreal_minus_le_iff by (auto simp: field_simps)
+ next
+ assume "\<not> Real p \<le> c"
+ hence "c \<le> Real p" by auto
+ also have "Real p \<le> Real p + (INF i. f i)" by auto
+ finally show ?thesis .
+ qed
+ qed
+ qed simp
+qed
+
+lemma pextreal_le_minus_imp_0:
+ fixes a b :: pextreal
+ shows "a \<le> a - b \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> a \<noteq> \<omega> \<Longrightarrow> b = 0"
+ by (cases a, cases b, auto split: split_if_asm)
+
+lemma lim_INF_eq_lim_SUP:
+ fixes X :: "nat \<Rightarrow> real"
+ assumes "\<And>i. 0 \<le> X i" and "0 \<le> x"
+ and lim_INF: "(SUP n. INF m. Real (X (n + m))) = Real x" (is "(SUP n. ?INF n) = _")
+ and lim_SUP: "(INF n. SUP m. Real (X (n + m))) = Real x" (is "(INF n. ?SUP n) = _")
+ shows "X ----> x"
+proof (rule LIMSEQ_I)
+ fix r :: real assume "0 < r"
+ hence "0 \<le> r" by auto
+ from Sup_close[of "Real r" "range ?INF"]
+ obtain n where inf: "Real x < ?INF n + Real r"
+ unfolding SUPR_def lim_INF[unfolded SUPR_def] using `0 < r` by auto
+
+ from Inf_close[of "range ?SUP" "Real r"]
+ obtain n' where sup: "?SUP n' < Real x + Real r"
+ unfolding INFI_def lim_SUP[unfolded INFI_def] using `0 < r` by auto
+
+ show "\<exists>N. \<forall>n\<ge>N. norm (X n - x) < r"
+ proof (safe intro!: exI[of _ "max n n'"])
+ fix m assume "max n n' \<le> m" hence "n \<le> m" "n' \<le> m" by auto
+
+ note inf
+ also have "?INF n + Real r \<le> Real (X (n + (m - n))) + Real r"
+ by (rule le_add_Real, auto simp: `0 \<le> r` intro: INF_leI)
+ finally have up: "x < X m + r"
+ using `0 \<le> X m` `0 \<le> x` `0 \<le> r` `n \<le> m` by auto
+
+ have "Real (X (n' + (m - n'))) \<le> ?SUP n'"
+ by (auto simp: `0 \<le> r` intro: le_SUPI)
+ also note sup
+ finally have down: "X m < x + r"
+ using `0 \<le> X m` `0 \<le> x` `0 \<le> r` `n' \<le> m` by auto
+
+ show "norm (X m - x) < r" using up down by auto
+ qed
+qed
+
+lemma Sup_countable_SUPR:
+ assumes "Sup A \<noteq> \<omega>" "A \<noteq> {}"
+ shows "\<exists> f::nat \<Rightarrow> pextreal. range f \<subseteq> A \<and> Sup A = SUPR UNIV f"
+proof -
+ have "\<And>n. 0 < 1 / (of_nat n :: pextreal)" by auto
+ from Sup_close[OF this assms]
+ have "\<forall>n. \<exists>x. x \<in> A \<and> Sup A < x + 1 / of_nat n" by blast
+ from choice[OF this] obtain f where "range f \<subseteq> A" and
+ epsilon: "\<And>n. Sup A < f n + 1 / of_nat n" by blast
+ have "SUPR UNIV f = Sup A"
+ proof (rule pextreal_SUPI)
+ fix i show "f i \<le> Sup A" using `range f \<subseteq> A`
+ 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 pextreal_le_epsilon)
+ fix e :: pextreal assume "0 < e"
+ show "Sup A \<le> y + e"
+ proof (cases e)
+ case (preal r)
+ hence "0 < r" using `0 < e` by auto
+ then obtain n where *: "inverse (of_nat n) < r" "0 < n"
+ using ex_inverse_of_nat_less by auto
+ have "Sup A \<le> f n + 1 / of_nat n" using epsilon[of n] by auto
+ also have "1 / of_nat n \<le> e" using preal * by (auto simp: real_eq_of_nat)
+ with bound have "f n + 1 / of_nat n \<le> y + e" by (rule add_mono) simp
+ finally show "Sup A \<le> y + e" .
+ qed simp
+ qed
+ qed
+ with `range f \<subseteq> A` show ?thesis by (auto intro!: exI[of _ f])
+qed
+
+lemma SUPR_countable_SUPR:
+ assumes "SUPR A g \<noteq> \<omega>" "A \<noteq> {}"
+ shows "\<exists> f::nat \<Rightarrow> pextreal. range f \<subseteq> g`A \<and> SUPR A g = SUPR UNIV f"
+proof -
+ have "Sup (g`A) \<noteq> \<omega>" "g`A \<noteq> {}" using assms unfolding SUPR_def by auto
+ from Sup_countable_SUPR[OF this]
+ show ?thesis unfolding SUPR_def .
+qed
+
+lemma pextreal_setsum_subtractf:
+ assumes "\<And>i. i\<in>A \<Longrightarrow> g i \<le> f i" and "\<And>i. i\<in>A \<Longrightarrow> f i \<noteq> \<omega>"
+ shows "(\<Sum>i\<in>A. f i - g i) = (\<Sum>i\<in>A. f i) - (\<Sum>i\<in>A. g i)"
+proof cases
+ assume "finite A" from this assms show ?thesis
+ proof induct
+ case (insert x A)
+ hence hyp: "(\<Sum>i\<in>A. f i - g i) = (\<Sum>i\<in>A. f i) - (\<Sum>i\<in>A. g i)"
+ by auto
+ { fix i assume *: "i \<in> insert x A"
+ hence "g i \<le> f i" using insert by simp
+ also have "f i < \<omega>" using * insert by (simp add: pextreal_less_\<omega>)
+ finally have "g i \<noteq> \<omega>" by (simp add: pextreal_less_\<omega>) }
+ hence "setsum g A \<noteq> \<omega>" "g x \<noteq> \<omega>" by (auto simp: setsum_\<omega>)
+ moreover have "setsum f A \<noteq> \<omega>" "f x \<noteq> \<omega>" using insert by (auto simp: setsum_\<omega>)
+ moreover have "g x \<le> f x" using insert by auto
+ moreover have "(\<Sum>i\<in>A. g i) \<le> (\<Sum>i\<in>A. f i)" using insert by (auto intro!: setsum_mono)
+ ultimately show ?case using `finite A` `x \<notin> A` hyp
+ by (auto simp: pextreal_noteq_omega_Ex)
+ qed simp
+qed simp
+
+lemma real_of_pextreal_diff:
+ "y \<le> x \<Longrightarrow> x \<noteq> \<omega> \<Longrightarrow> real x - real y = real (x - y)"
+ by (cases x, cases y) auto
+
+lemma psuminf_minus:
+ assumes ord: "\<And>i. g i \<le> f i" and fin: "psuminf g \<noteq> \<omega>" "psuminf f \<noteq> \<omega>"
+ shows "(\<Sum>\<^isub>\<infinity> i. f i - g i) = psuminf f - psuminf g"
+proof -
+ have [simp]: "\<And>i. f i \<noteq> \<omega>" using fin by (auto intro: psuminf_\<omega>)
+ from fin have "(\<lambda>x. real (f x)) sums real (\<Sum>\<^isub>\<infinity>x. f x)"
+ and "(\<lambda>x. real (g x)) sums real (\<Sum>\<^isub>\<infinity>x. g x)"
+ by (auto intro: psuminf_imp_suminf)
+ from sums_diff[OF this]
+ have "(\<lambda>n. real (f n - g n)) sums (real ((\<Sum>\<^isub>\<infinity>x. f x) - (\<Sum>\<^isub>\<infinity>x. g x)))" using fin ord
+ by (subst (asm) (1 2) real_of_pextreal_diff) (auto simp: psuminf_\<omega> psuminf_le)
+ hence "(\<Sum>\<^isub>\<infinity> i. Real (real (f i - g i))) = Real (real ((\<Sum>\<^isub>\<infinity>x. f x) - (\<Sum>\<^isub>\<infinity>x. g x)))"
+ by (rule suminf_imp_psuminf) simp
+ thus ?thesis using fin by (simp add: Real_real psuminf_\<omega>)
+qed
+
+lemma INF_eq_LIMSEQ:
+ assumes "mono (\<lambda>i. - f i)" and "\<And>n. 0 \<le> f n" and "0 \<le> x"
+ shows "(INF n. Real (f n)) = Real x \<longleftrightarrow> f ----> x"
+proof
+ assume x: "(INF n. Real (f n)) = Real x"
+ { fix n
+ have "Real x \<le> Real (f n)" using x[symmetric] by (auto intro: INF_leI)
+ hence "x \<le> f n" using assms by simp
+ hence "\<bar>f n - x\<bar> = f n - x" by auto }
+ note abs_eq = this
+ show "f ----> x"
+ proof (rule LIMSEQ_I)
+ fix r :: real assume "0 < r"
+ show "\<exists>no. \<forall>n\<ge>no. norm (f n - x) < r"
+ proof (rule ccontr)
+ assume *: "\<not> ?thesis"
+ { fix N
+ from * obtain n where *: "N \<le> n" "r \<le> f n - x"
+ using abs_eq by (auto simp: not_less)
+ hence "x + r \<le> f n" by auto
+ also have "f n \<le> f N" using `mono (\<lambda>i. - f i)` * by (auto dest: monoD)
+ finally have "Real (x + r) \<le> Real (f N)" using `0 \<le> f N` by auto }
+ hence "Real x < Real (x + r)"
+ and "Real (x + r) \<le> (INF n. Real (f n))" using `0 < r` `0 \<le> x` by (auto intro: le_INFI)
+ hence "Real x < (INF n. Real (f n))" by (rule less_le_trans)
+ thus False using x by auto
+ qed
+ qed
+next
+ assume "f ----> x"
+ show "(INF n. Real (f n)) = Real x"
+ proof (rule pextreal_INFI)
+ fix n
+ from decseq_le[OF _ `f ----> x`] assms
+ show "Real x \<le> Real (f n)" unfolding decseq_eq_incseq incseq_mono by auto
+ next
+ fix y assume *: "\<And>n. n\<in>UNIV \<Longrightarrow> y \<le> Real (f n)"
+ thus "y \<le> Real x"
+ proof (cases y)
+ case (preal r)
+ with * have "\<exists>N. \<forall>n\<ge>N. r \<le> f n" using assms by fastsimp
+ from LIMSEQ_le_const[OF `f ----> x` this]
+ show "y \<le> Real x" using `0 \<le> x` preal by auto
+ qed simp
+ qed
+qed
+
+lemma INFI_bound:
+ assumes "\<forall>N. x \<le> f N"
+ shows "x \<le> (INF n. f n)"
+ using assms by (simp add: INFI_def le_Inf_iff)
+
+lemma LIMSEQ_imp_lim_INF:
+ assumes pos: "\<And>i. 0 \<le> X i" and lim: "X ----> x"
+ shows "(SUP n. INF m. Real (X (n + m))) = Real x"
+proof -
+ have "0 \<le> x" using assms by (auto intro!: LIMSEQ_le_const)
+
+ have "\<And>n. (INF m. Real (X (n + m))) \<le> Real (X (n + 0))" by (rule INF_leI) simp
+ also have "\<And>n. Real (X (n + 0)) < \<omega>" by simp
+ finally have "\<forall>n. \<exists>r\<ge>0. (INF m. Real (X (n + m))) = Real r"
+ by (auto simp: pextreal_less_\<omega> pextreal_noteq_omega_Ex)
+ from choice[OF this] obtain r where r: "\<And>n. (INF m. Real (X (n + m))) = Real (r n)" "\<And>n. 0 \<le> r n"
+ by auto
+
+ show ?thesis unfolding r
+ proof (subst SUP_eq_LIMSEQ)
+ show "mono r" unfolding mono_def
+ proof safe
+ fix x y :: nat assume "x \<le> y"
+ have "Real (r x) \<le> Real (r y)" unfolding r(1)[symmetric] using pos
+ proof (safe intro!: INF_mono bexI)
+ fix m have "x + (m + y - x) = y + m"
+ using `x \<le> y` by auto
+ thus "Real (X (x + (m + y - x))) \<le> Real (X (y + m))" by simp
+ qed simp
+ thus "r x \<le> r y" using r by auto
+ qed
+ show "\<And>n. 0 \<le> r n" by fact
+ show "0 \<le> x" by fact
+ show "r ----> x"
+ proof (rule LIMSEQ_I)
+ fix e :: real assume "0 < e"
+ hence "0 < e/2" by auto
+ from LIMSEQ_D[OF lim this] obtain N where *: "\<And>n. N \<le> n \<Longrightarrow> \<bar>X n - x\<bar> < e/2"
+ by auto
+ show "\<exists>N. \<forall>n\<ge>N. norm (r n - x) < e"
+ proof (safe intro!: exI[of _ N])
+ fix n assume "N \<le> n"
+ show "norm (r n - x) < e"
+ proof cases
+ assume "r n < x"
+ have "x - r n \<le> e/2"
+ proof cases
+ assume e: "e/2 \<le> x"
+ have "Real (x - e/2) \<le> Real (r n)" unfolding r(1)[symmetric]
+ proof (rule le_INFI)
+ fix m show "Real (x - e / 2) \<le> Real (X (n + m))"
+ using *[of "n + m"] `N \<le> n`
+ using pos by (auto simp: field_simps abs_real_def split: split_if_asm)
+ qed
+ with e show ?thesis using pos `0 \<le> x` r(2) by auto
+ next
+ assume "\<not> e/2 \<le> x" hence "x - e/2 < 0" by auto
+ with `0 \<le> r n` show ?thesis by auto
+ qed
+ with `r n < x` show ?thesis by simp
+ next
+ assume e: "\<not> r n < x"
+ have "Real (r n) \<le> Real (X (n + 0))" unfolding r(1)[symmetric]
+ by (rule INF_leI) simp
+ hence "r n - x \<le> X n - x" using r pos by auto
+ also have "\<dots> < e/2" using *[OF `N \<le> n`] by (auto simp: field_simps abs_real_def split: split_if_asm)
+ finally have "r n - x < e" using `0 < e` by auto
+ with e show ?thesis by auto
+ qed
+ qed
+ qed
+ qed
+qed
+
+lemma real_of_pextreal_strict_mono_iff:
+ "real a < real b \<longleftrightarrow> (b \<noteq> \<omega> \<and> ((a = \<omega> \<and> 0 < b) \<or> (a < b)))"
+proof (cases a)
+ case infinite thus ?thesis by (cases b) auto
+next
+ case preal thus ?thesis by (cases b) auto
+qed
+
+lemma real_of_pextreal_mono_iff:
+ "real a \<le> real b \<longleftrightarrow> (a = \<omega> \<or> (b \<noteq> \<omega> \<and> a \<le> b) \<or> (b = \<omega> \<and> a = 0))"
+proof (cases a)
+ case infinite thus ?thesis by (cases b) auto
+next
+ case preal thus ?thesis by (cases b) auto
+qed
+
+lemma ex_pextreal_inverse_of_nat_Suc_less:
+ fixes e :: pextreal assumes "0 < e" shows "\<exists>n. inverse (of_nat (Suc n)) < e"
+proof (cases e)
+ case (preal r)
+ with `0 < e` ex_inverse_of_nat_Suc_less[of r]
+ obtain n where "inverse (of_nat (Suc n)) < r" by auto
+ with preal show ?thesis
+ by (auto simp: real_eq_of_nat[symmetric])
+qed auto
+
+lemma Lim_eq_Sup_mono:
+ fixes u :: "nat \<Rightarrow> pextreal" assumes "mono u"
+ shows "u ----> (SUP i. u i)"
+proof -
+ from lim_pextreal_increasing[of u] `mono u`
+ obtain l where l: "u ----> l" unfolding mono_def by auto
+ from SUP_Lim_pextreal[OF _ this] `mono u`
+ have "(SUP i. u i) = l" unfolding mono_def by auto
+ with l show ?thesis by simp
+qed
+
+lemma isotone_Lim:
+ fixes x :: pextreal assumes "u \<up> x"
+ shows "u ----> x" (is ?lim) and "mono u" (is ?mono)
+proof -
+ show ?mono using assms unfolding mono_iff_le_Suc isoton_def by auto
+ from Lim_eq_Sup_mono[OF this] `u \<up> x`
+ show ?lim unfolding isoton_def by simp
+qed
+
+lemma isoton_iff_Lim_mono:
+ fixes u :: "nat \<Rightarrow> pextreal"
+ shows "u \<up> x \<longleftrightarrow> (mono u \<and> u ----> x)"
+proof safe
+ assume "mono u" and x: "u ----> x"
+ with SUP_Lim_pextreal[OF _ x]
+ show "u \<up> x" unfolding isoton_def
+ using `mono u`[unfolded mono_def]
+ using `mono u`[unfolded mono_iff_le_Suc]
+ by auto
+qed (auto dest: isotone_Lim)
+
+lemma pextreal_inverse_inverse[simp]:
+ fixes x :: pextreal
+ shows "inverse (inverse x) = x"
+ by (cases x) auto
+
+lemma atLeastAtMost_omega_eq_atLeast:
+ shows "{a .. \<omega>} = {a ..}"
+by auto
+
+lemma atLeast0AtMost_eq_atMost: "{0 :: pextreal .. a} = {.. a}" by auto
+
+lemma greaterThan_omega_Empty: "{\<omega> <..} = {}" by auto
+
+lemma lessThan_0_Empty: "{..< 0 :: pextreal} = {}" by auto
+
+lemma real_of_pextreal_inverse[simp]:
+ fixes X :: pextreal
+ shows "real (inverse X) = 1 / real X"
+ by (cases X) (auto simp: inverse_eq_divide)
+
+lemma real_of_pextreal_le_0[simp]: "real (X :: pextreal) \<le> 0 \<longleftrightarrow> (X = 0 \<or> X = \<omega>)"
+ by (cases X) auto
+
+lemma real_of_pextreal_less_0[simp]: "\<not> (real (X :: pextreal) < 0)"
+ by (cases X) auto
+
+lemma abs_real_of_pextreal[simp]: "\<bar>real (X :: pextreal)\<bar> = real X"
+ by simp
+
+lemma zero_less_real_of_pextreal: "0 < real (X :: pextreal) \<longleftrightarrow> X \<noteq> 0 \<and> X \<noteq> \<omega>"
+ by (cases X) auto
+
+end
--- a/src/HOL/Probability/Positive_Infinite_Real.thy Mon Dec 06 19:18:02 2010 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,2775 +0,0 @@
-(* Author: Johannes Hoelzl, TU Muenchen *)
-
-header {* A type for positive real numbers with infinity *}
-
-theory Positive_Infinite_Real
- imports Complex_Main Nat_Bijection Multivariate_Analysis
-begin
-
-lemma (in complete_lattice) Sup_start:
- assumes *: "\<And>x. f x \<le> f 0"
- shows "(SUP n. f n) = f 0"
-proof (rule antisym)
- show "f 0 \<le> (SUP n. f n)" by (rule le_SUPI) auto
- show "(SUP n. f n) \<le> f 0" by (rule SUP_leI[OF *])
-qed
-
-lemma (in complete_lattice) Inf_start:
- assumes *: "\<And>x. f 0 \<le> f x"
- shows "(INF n. f n) = f 0"
-proof (rule antisym)
- show "(INF n. f n) \<le> f 0" by (rule INF_leI) simp
- show "f 0 \<le> (INF n. f n)" by (rule le_INFI[OF *])
-qed
-
-lemma (in complete_lattice) Sup_mono_offset:
- fixes f :: "'b :: {ordered_ab_semigroup_add,monoid_add} \<Rightarrow> 'a"
- assumes *: "\<And>x y. x \<le> y \<Longrightarrow> f x \<le> f y" and "0 \<le> k"
- shows "(SUP n . f (k + n)) = (SUP n. f n)"
-proof (rule antisym)
- show "(SUP n. f (k + n)) \<le> (SUP n. f n)"
- by (auto intro!: Sup_mono simp: SUPR_def)
- { fix n :: 'b
- have "0 + n \<le> k + n" using `0 \<le> k` by (rule add_right_mono)
- with * have "f n \<le> f (k + n)" by simp }
- thus "(SUP n. f n) \<le> (SUP n. f (k + n))"
- by (auto intro!: Sup_mono exI simp: SUPR_def)
-qed
-
-lemma (in complete_lattice) Sup_mono_offset_Suc:
- assumes *: "\<And>x. f x \<le> f (Suc x)"
- shows "(SUP n . f (Suc n)) = (SUP n. f n)"
- unfolding Suc_eq_plus1
- apply (subst add_commute)
- apply (rule Sup_mono_offset)
- by (auto intro!: order.lift_Suc_mono_le[of "op \<le>" "op <" f, OF _ *]) default
-
-lemma (in complete_lattice) Inf_mono_offset:
- fixes f :: "'b :: {ordered_ab_semigroup_add,monoid_add} \<Rightarrow> 'a"
- assumes *: "\<And>x y. x \<le> y \<Longrightarrow> f y \<le> f x" and "0 \<le> k"
- shows "(INF n . f (k + n)) = (INF n. f n)"
-proof (rule antisym)
- show "(INF n. f n) \<le> (INF n. f (k + n))"
- by (auto intro!: Inf_mono simp: INFI_def)
- { fix n :: 'b
- have "0 + n \<le> k + n" using `0 \<le> k` by (rule add_right_mono)
- with * have "f (k + n) \<le> f n" by simp }
- thus "(INF n. f (k + n)) \<le> (INF n. f n)"
- by (auto intro!: Inf_mono exI simp: INFI_def)
-qed
-
-lemma (in complete_lattice) isotone_converge:
- fixes f :: "nat \<Rightarrow> 'a" assumes "\<And>x y. x \<le> y \<Longrightarrow> f x \<le> f y "
- shows "(INF n. SUP m. f (n + m)) = (SUP n. INF m. f (n + m))"
-proof -
- have "\<And>n. (SUP m. f (n + m)) = (SUP n. f n)"
- apply (rule Sup_mono_offset)
- apply (rule assms)
- by simp_all
- moreover
- { fix n have "(INF m. f (n + m)) = f n"
- using Inf_start[of "\<lambda>m. f (n + m)"] assms by simp }
- ultimately show ?thesis by simp
-qed
-
-lemma (in complete_lattice) antitone_converges:
- fixes f :: "nat \<Rightarrow> 'a" assumes "\<And>x y. x \<le> y \<Longrightarrow> f y \<le> f x"
- shows "(INF n. SUP m. f (n + m)) = (SUP n. INF m. f (n + m))"
-proof -
- have "\<And>n. (INF m. f (n + m)) = (INF n. f n)"
- apply (rule Inf_mono_offset)
- apply (rule assms)
- by simp_all
- moreover
- { fix n have "(SUP m. f (n + m)) = f n"
- using Sup_start[of "\<lambda>m. f (n + m)"] assms by simp }
- ultimately show ?thesis by simp
-qed
-
-lemma (in complete_lattice) lim_INF_le_lim_SUP:
- fixes f :: "nat \<Rightarrow> 'a"
- shows "(SUP n. INF m. f (n + m)) \<le> (INF n. SUP m. f (n + m))"
-proof (rule SUP_leI, rule le_INFI)
- fix i j show "(INF m. f (i + m)) \<le> (SUP m. f (j + m))"
- proof (cases rule: le_cases)
- assume "i \<le> j"
- have "(INF m. f (i + m)) \<le> f (i + (j - i))" by (rule INF_leI) simp
- also have "\<dots> = f (j + 0)" using `i \<le> j` by auto
- also have "\<dots> \<le> (SUP m. f (j + m))" by (rule le_SUPI) simp
- finally show ?thesis .
- next
- assume "j \<le> i"
- have "(INF m. f (i + m)) \<le> f (i + 0)" by (rule INF_leI) simp
- also have "\<dots> = f (j + (i - j))" using `j \<le> i` by auto
- also have "\<dots> \<le> (SUP m. f (j + m))" by (rule le_SUPI) simp
- finally show ?thesis .
- qed
-qed
-
-text {*
-
-We introduce the the positive real numbers as needed for measure theory.
-
-*}
-
-typedef pinfreal = "(Some ` {0::real..}) \<union> {None}"
- by (rule exI[of _ None]) simp
-
-subsection "Introduce @{typ pinfreal} similar to a datatype"
-
-definition "Real x = Abs_pinfreal (Some (sup 0 x))"
-definition "\<omega> = Abs_pinfreal None"
-
-definition "pinfreal_case f i x = (if x = \<omega> then i else f (THE r. 0 \<le> r \<and> x = Real r))"
-
-definition "of_pinfreal = pinfreal_case (\<lambda>x. x) 0"
-
-defs (overloaded)
- real_of_pinfreal_def [code_unfold]: "real == of_pinfreal"
-
-lemma pinfreal_Some[simp]: "0 \<le> x \<Longrightarrow> Some x \<in> pinfreal"
- unfolding pinfreal_def by simp
-
-lemma pinfreal_Some_sup[simp]: "Some (sup 0 x) \<in> pinfreal"
- by (simp add: sup_ge1)
-
-lemma pinfreal_None[simp]: "None \<in> pinfreal"
- unfolding pinfreal_def by simp
-
-lemma Real_inj[simp]:
- assumes "0 \<le> x" and "0 \<le> y"
- shows "Real x = Real y \<longleftrightarrow> x = y"
- unfolding Real_def assms[THEN sup_absorb2]
- using assms by (simp add: Abs_pinfreal_inject)
-
-lemma Real_neq_\<omega>[simp]:
- "Real x = \<omega> \<longleftrightarrow> False"
- "\<omega> = Real x \<longleftrightarrow> False"
- by (simp_all add: Abs_pinfreal_inject \<omega>_def Real_def)
-
-lemma Real_neg: "x < 0 \<Longrightarrow> Real x = Real 0"
- unfolding Real_def by (auto simp add: Abs_pinfreal_inject intro!: sup_absorb1)
-
-lemma pinfreal_cases[case_names preal infinite, cases type: pinfreal]:
- assumes preal: "\<And>r. x = Real r \<Longrightarrow> 0 \<le> r \<Longrightarrow> P" and inf: "x = \<omega> \<Longrightarrow> P"
- shows P
-proof (cases x rule: pinfreal.Abs_pinfreal_cases)
- case (Abs_pinfreal y)
- hence "y = None \<or> (\<exists>x \<ge> 0. y = Some x)"
- unfolding pinfreal_def by auto
- thus P
- proof (rule disjE)
- assume "\<exists>x\<ge>0. y = Some x" then guess x ..
- thus P by (simp add: preal[of x] Real_def Abs_pinfreal(1) sup_absorb2)
- qed (simp add: \<omega>_def Abs_pinfreal(1) inf)
-qed
-
-lemma pinfreal_case_\<omega>[simp]: "pinfreal_case f i \<omega> = i"
- unfolding pinfreal_case_def by simp
-
-lemma pinfreal_case_Real[simp]: "pinfreal_case f i (Real x) = (if 0 \<le> x then f x else f 0)"
-proof (cases "0 \<le> x")
- case True thus ?thesis unfolding pinfreal_case_def by (auto intro: theI2)
-next
- case False
- moreover have "(THE r. 0 \<le> r \<and> Real 0 = Real r) = 0"
- by (auto intro!: the_equality)
- ultimately show ?thesis unfolding pinfreal_case_def by (simp add: Real_neg)
-qed
-
-lemma pinfreal_case_cancel[simp]: "pinfreal_case (\<lambda>c. i) i x = i"
- by (cases x) simp_all
-
-lemma pinfreal_case_split:
- "P (pinfreal_case f i x) = ((x = \<omega> \<longrightarrow> P i) \<and> (\<forall>r\<ge>0. x = Real r \<longrightarrow> P (f r)))"
- by (cases x) simp_all
-
-lemma pinfreal_case_split_asm:
- "P (pinfreal_case f i x) = (\<not> (x = \<omega> \<and> \<not> P i \<or> (\<exists>r. r \<ge> 0 \<and> x = Real r \<and> \<not> P (f r))))"
- by (cases x) auto
-
-lemma pinfreal_case_cong[cong]:
- assumes eq: "x = x'" "i = i'" and cong: "\<And>r. 0 \<le> r \<Longrightarrow> f r = f' r"
- shows "pinfreal_case f i x = pinfreal_case f' i' x'"
- unfolding eq using cong by (cases x') simp_all
-
-lemma real_Real[simp]: "real (Real x) = (if 0 \<le> x then x else 0)"
- unfolding real_of_pinfreal_def of_pinfreal_def by simp
-
-lemma Real_real_image:
- assumes "\<omega> \<notin> A" shows "Real ` real ` A = A"
-proof safe
- fix x assume "x \<in> A"
- hence *: "x = Real (real x)"
- using `\<omega> \<notin> A` by (cases x) auto
- show "x \<in> Real ` real ` A"
- using `x \<in> A` by (subst *) (auto intro!: imageI)
-next
- fix x assume "x \<in> A"
- thus "Real (real x) \<in> A"
- using `\<omega> \<notin> A` by (cases x) auto
-qed
-
-lemma real_pinfreal_nonneg[simp, intro]: "0 \<le> real (x :: pinfreal)"
- unfolding real_of_pinfreal_def of_pinfreal_def
- by (cases x) auto
-
-lemma real_\<omega>[simp]: "real \<omega> = 0"
- unfolding real_of_pinfreal_def of_pinfreal_def by simp
-
-lemma pinfreal_noteq_omega_Ex: "X \<noteq> \<omega> \<longleftrightarrow> (\<exists>r\<ge>0. X = Real r)" by (cases X) auto
-
-subsection "@{typ pinfreal} is a monoid for addition"
-
-instantiation pinfreal :: comm_monoid_add
-begin
-
-definition "0 = Real 0"
-definition "x + y = pinfreal_case (\<lambda>r. pinfreal_case (\<lambda>p. Real (r + p)) \<omega> y) \<omega> x"
-
-lemma pinfreal_plus[simp]:
- "Real r + Real p = (if 0 \<le> r then if 0 \<le> p then Real (r + p) else Real r else Real p)"
- "x + 0 = x"
- "0 + x = x"
- "x + \<omega> = \<omega>"
- "\<omega> + x = \<omega>"
- by (simp_all add: plus_pinfreal_def Real_neg zero_pinfreal_def split: pinfreal_case_split)
-
-lemma \<omega>_neq_0[simp]:
- "\<omega> = 0 \<longleftrightarrow> False"
- "0 = \<omega> \<longleftrightarrow> False"
- by (simp_all add: zero_pinfreal_def)
-
-lemma Real_eq_0[simp]:
- "Real r = 0 \<longleftrightarrow> r \<le> 0"
- "0 = Real r \<longleftrightarrow> r \<le> 0"
- by (auto simp add: Abs_pinfreal_inject zero_pinfreal_def Real_def sup_real_def)
-
-lemma Real_0[simp]: "Real 0 = 0" by (simp add: zero_pinfreal_def)
-
-instance
-proof
- fix a :: pinfreal
- show "0 + a = a" by (cases a) simp_all
-
- fix b show "a + b = b + a"
- by (cases a, cases b) simp_all
-
- fix c show "a + b + c = a + (b + c)"
- by (cases a, cases b, cases c) simp_all
-qed
-end
-
-lemma pinfreal_plus_eq_\<omega>[simp]: "(a :: pinfreal) + b = \<omega> \<longleftrightarrow> a = \<omega> \<or> b = \<omega>"
- by (cases a, cases b) auto
-
-lemma pinfreal_add_cancel_left:
- "a + b = a + c \<longleftrightarrow> (a = \<omega> \<or> b = c)"
- by (cases a, cases b, cases c, simp_all, cases c, simp_all)
-
-lemma pinfreal_add_cancel_right:
- "b + a = c + a \<longleftrightarrow> (a = \<omega> \<or> b = c)"
- by (cases a, cases b, cases c, simp_all, cases c, simp_all)
-
-lemma Real_eq_Real:
- "Real a = Real b \<longleftrightarrow> (a = b \<or> (a \<le> 0 \<and> b \<le> 0))"
-proof (cases "a \<le> 0 \<or> b \<le> 0")
- case False with Real_inj[of a b] show ?thesis by auto
-next
- case True
- thus ?thesis
- proof
- assume "a \<le> 0"
- hence *: "Real a = 0" by simp
- show ?thesis using `a \<le> 0` unfolding * by auto
- next
- assume "b \<le> 0"
- hence *: "Real b = 0" by simp
- show ?thesis using `b \<le> 0` unfolding * by auto
- qed
-qed
-
-lemma real_pinfreal_0[simp]: "real (0 :: pinfreal) = 0"
- unfolding zero_pinfreal_def real_Real by simp
-
-lemma real_of_pinfreal_eq_0: "real X = 0 \<longleftrightarrow> (X = 0 \<or> X = \<omega>)"
- by (cases X) auto
-
-lemma real_of_pinfreal_eq: "real X = real Y \<longleftrightarrow>
- (X = Y \<or> (X = 0 \<and> Y = \<omega>) \<or> (Y = 0 \<and> X = \<omega>))"
- by (cases X, cases Y) (auto simp add: real_of_pinfreal_eq_0)
-
-lemma real_of_pinfreal_add: "real X + real Y =
- (if X = \<omega> then real Y else if Y = \<omega> then real X else real (X + Y))"
- by (auto simp: pinfreal_noteq_omega_Ex)
-
-subsection "@{typ pinfreal} is a monoid for multiplication"
-
-instantiation pinfreal :: comm_monoid_mult
-begin
-
-definition "1 = Real 1"
-definition "x * y = (if x = 0 \<or> y = 0 then 0 else
- pinfreal_case (\<lambda>r. pinfreal_case (\<lambda>p. Real (r * p)) \<omega> y) \<omega> x)"
-
-lemma pinfreal_times[simp]:
- "Real r * Real p = (if 0 \<le> r \<and> 0 \<le> p then Real (r * p) else 0)"
- "\<omega> * x = (if x = 0 then 0 else \<omega>)"
- "x * \<omega> = (if x = 0 then 0 else \<omega>)"
- "0 * x = 0"
- "x * 0 = 0"
- "1 = \<omega> \<longleftrightarrow> False"
- "\<omega> = 1 \<longleftrightarrow> False"
- by (auto simp add: times_pinfreal_def one_pinfreal_def)
-
-lemma pinfreal_one_mult[simp]:
- "Real x + 1 = (if 0 \<le> x then Real (x + 1) else 1)"
- "1 + Real x = (if 0 \<le> x then Real (1 + x) else 1)"
- unfolding one_pinfreal_def by simp_all
-
-instance
-proof
- fix a :: pinfreal show "1 * a = a"
- by (cases a) (simp_all add: one_pinfreal_def)
-
- fix b show "a * b = b * a"
- by (cases a, cases b) (simp_all add: mult_nonneg_nonneg)
-
- fix c show "a * b * c = a * (b * c)"
- apply (cases a, cases b, cases c)
- apply (simp_all add: mult_nonneg_nonneg not_le mult_pos_pos)
- apply (cases b, cases c)
- apply (simp_all add: mult_nonneg_nonneg not_le mult_pos_pos)
- done
-qed
-end
-
-lemma pinfreal_mult_cancel_left:
- "a * b = a * c \<longleftrightarrow> (a = 0 \<or> b = c \<or> (a = \<omega> \<and> b \<noteq> 0 \<and> c \<noteq> 0))"
- by (cases a, cases b, cases c, auto simp: Real_eq_Real mult_le_0_iff, cases c, auto)
-
-lemma pinfreal_mult_cancel_right:
- "b * a = c * a \<longleftrightarrow> (a = 0 \<or> b = c \<or> (a = \<omega> \<and> b \<noteq> 0 \<and> c \<noteq> 0))"
- by (cases a, cases b, cases c, auto simp: Real_eq_Real mult_le_0_iff, cases c, auto)
-
-lemma Real_1[simp]: "Real 1 = 1" by (simp add: one_pinfreal_def)
-
-lemma real_pinfreal_1[simp]: "real (1 :: pinfreal) = 1"
- unfolding one_pinfreal_def real_Real by simp
-
-lemma real_of_pinfreal_mult: "real X * real Y = real (X * Y :: pinfreal)"
- by (cases X, cases Y) (auto simp: zero_le_mult_iff)
-
-lemma Real_mult_nonneg: assumes "x \<ge> 0" "y \<ge> 0"
- shows "Real (x * y) = Real x * Real y" using assms by auto
-
-lemma Real_setprod: assumes "\<forall>x\<in>A. f x \<ge> 0" shows "Real (setprod f A) = setprod (\<lambda>x. Real (f x)) A"
-proof(cases "finite A")
- case True thus ?thesis using assms
- proof(induct A) case (insert x A)
- have "0 \<le> setprod f A" apply(rule setprod_nonneg) using insert by auto
- thus ?case unfolding setprod_insert[OF insert(1-2)] apply-
- apply(subst Real_mult_nonneg) prefer 3 apply(subst insert(3)[THEN sym])
- using insert by auto
- qed auto
-qed auto
-
-subsection "@{typ pinfreal} is a linear order"
-
-instantiation pinfreal :: linorder
-begin
-
-definition "x < y \<longleftrightarrow> pinfreal_case (\<lambda>i. pinfreal_case (\<lambda>j. i < j) True y) False x"
-definition "x \<le> y \<longleftrightarrow> pinfreal_case (\<lambda>j. pinfreal_case (\<lambda>i. i \<le> j) False x) True y"
-
-lemma pinfreal_less[simp]:
- "Real r < \<omega>"
- "Real r < Real p \<longleftrightarrow> (if 0 \<le> r \<and> 0 \<le> p then r < p else 0 < p)"
- "\<omega> < x \<longleftrightarrow> False"
- "0 < \<omega>"
- "0 < Real r \<longleftrightarrow> 0 < r"
- "x < 0 \<longleftrightarrow> False"
- "0 < (1::pinfreal)"
- by (simp_all add: less_pinfreal_def zero_pinfreal_def one_pinfreal_def del: Real_0 Real_1)
-
-lemma pinfreal_less_eq[simp]:
- "x \<le> \<omega>"
- "Real r \<le> Real p \<longleftrightarrow> (if 0 \<le> r \<and> 0 \<le> p then r \<le> p else r \<le> 0)"
- "0 \<le> x"
- by (simp_all add: less_eq_pinfreal_def zero_pinfreal_def del: Real_0)
-
-lemma pinfreal_\<omega>_less_eq[simp]:
- "\<omega> \<le> x \<longleftrightarrow> x = \<omega>"
- by (cases x) (simp_all add: not_le less_eq_pinfreal_def)
-
-lemma pinfreal_less_eq_zero[simp]:
- "(x::pinfreal) \<le> 0 \<longleftrightarrow> x = 0"
- by (cases x) (simp_all add: zero_pinfreal_def del: Real_0)
-
-instance
-proof
- fix x :: pinfreal
- show "x \<le> x" by (cases x) simp_all
- fix y
- show "(x < y) = (x \<le> y \<and> \<not> y \<le> x)"
- by (cases x, cases y) auto
- show "x \<le> y \<or> y \<le> x "
- by (cases x, cases y) auto
- { assume "x \<le> y" "y \<le> x" thus "x = y"
- by (cases x, cases y) auto }
- { fix z assume "x \<le> y" "y \<le> z"
- thus "x \<le> z" by (cases x, cases y, cases z) auto }
-qed
-end
-
-lemma pinfreal_zero_lessI[intro]:
- "(a :: pinfreal) \<noteq> 0 \<Longrightarrow> 0 < a"
- by (cases a) auto
-
-lemma pinfreal_less_omegaI[intro, simp]:
- "a \<noteq> \<omega> \<Longrightarrow> a < \<omega>"
- by (cases a) auto
-
-lemma pinfreal_plus_eq_0[simp]: "(a :: pinfreal) + b = 0 \<longleftrightarrow> a = 0 \<and> b = 0"
- by (cases a, cases b) auto
-
-lemma pinfreal_le_add1[simp, intro]: "n \<le> n + (m::pinfreal)"
- by (cases n, cases m) simp_all
-
-lemma pinfreal_le_add2: "(n::pinfreal) + m \<le> k \<Longrightarrow> m \<le> k"
- by (cases n, cases m, cases k) simp_all
-
-lemma pinfreal_le_add3: "(n::pinfreal) + m \<le> k \<Longrightarrow> n \<le> k"
- by (cases n, cases m, cases k) simp_all
-
-lemma pinfreal_less_\<omega>: "x < \<omega> \<longleftrightarrow> x \<noteq> \<omega>"
- by (cases x) auto
-
-lemma pinfreal_0_less_mult_iff[simp]:
- fixes x y :: pinfreal shows "0 < x * y \<longleftrightarrow> 0 < x \<and> 0 < y"
- by (cases x, cases y) (auto simp: zero_less_mult_iff)
-
-lemma pinfreal_ord_one[simp]:
- "Real p < 1 \<longleftrightarrow> p < 1"
- "Real p \<le> 1 \<longleftrightarrow> p \<le> 1"
- "1 < Real p \<longleftrightarrow> 1 < p"
- "1 \<le> Real p \<longleftrightarrow> 1 \<le> p"
- by (simp_all add: one_pinfreal_def del: Real_1)
-
-subsection {* @{text "x - y"} on @{typ pinfreal} *}
-
-instantiation pinfreal :: minus
-begin
-definition "x - y = (if y < x then THE d. x = y + d else 0 :: pinfreal)"
-
-lemma minus_pinfreal_eq:
- "(x - y = (z :: pinfreal)) \<longleftrightarrow> (if y < x then x = y + z else z = 0)"
- (is "?diff \<longleftrightarrow> ?if")
-proof
- assume ?diff
- thus ?if
- proof (cases "y < x")
- case True
- then obtain p where p: "y = Real p" "0 \<le> p" by (cases y) auto
-
- show ?thesis unfolding `?diff`[symmetric] if_P[OF True] minus_pinfreal_def
- proof (rule theI2[where Q="\<lambda>d. x = y + d"])
- show "x = y + pinfreal_case (\<lambda>r. Real (r - real y)) \<omega> x" (is "x = y + ?d")
- using `y < x` p by (cases x) simp_all
-
- fix d assume "x = y + d"
- thus "d = ?d" using `y < x` p by (cases d, cases x) simp_all
- qed simp
- qed (simp add: minus_pinfreal_def)
-next
- assume ?if
- thus ?diff
- proof (cases "y < x")
- case True
- then obtain p where p: "y = Real p" "0 \<le> p" by (cases y) auto
-
- from True `?if` have "x = y + z" by simp
-
- show ?thesis unfolding minus_pinfreal_def if_P[OF True] unfolding `x = y + z`
- proof (rule the_equality)
- fix d :: pinfreal assume "y + z = y + d"
- thus "d = z" using `y < x` p
- by (cases d, cases z) simp_all
- qed simp
- qed (simp add: minus_pinfreal_def)
-qed
-
-instance ..
-end
-
-lemma pinfreal_minus[simp]:
- "Real r - Real p = (if 0 \<le> r \<and> p < r then if 0 \<le> p then Real (r - p) else Real r else 0)"
- "(A::pinfreal) - A = 0"
- "\<omega> - Real r = \<omega>"
- "Real r - \<omega> = 0"
- "A - 0 = A"
- "0 - A = 0"
- by (auto simp: minus_pinfreal_eq not_less)
-
-lemma pinfreal_le_epsilon:
- fixes x y :: pinfreal
- assumes "\<And>e. 0 < e \<Longrightarrow> x \<le> y + e"
- shows "x \<le> y"
-proof (cases y)
- case (preal r)
- then obtain p where x: "x = Real p" "0 \<le> p"
- using assms[of 1] by (cases x) auto
- { fix e have "0 < e \<Longrightarrow> p \<le> r + e"
- using assms[of "Real e"] preal x by auto }
- hence "p \<le> r" by (rule field_le_epsilon)
- thus ?thesis using preal x by auto
-qed simp
-
-instance pinfreal :: "{ordered_comm_semiring, comm_semiring_1}"
-proof
- show "0 \<noteq> (1::pinfreal)" unfolding zero_pinfreal_def one_pinfreal_def
- by (simp del: Real_1 Real_0)
-
- fix a :: pinfreal
- show "0 * a = 0" "a * 0 = 0" by simp_all
-
- fix b c :: pinfreal
- show "(a + b) * c = a * c + b * c"
- by (cases c, cases a, cases b)
- (auto intro!: arg_cong[where f=Real] simp: field_simps not_le mult_le_0_iff mult_less_0_iff)
-
- { assume "a \<le> b" thus "c + a \<le> c + b"
- by (cases c, cases a, cases b) auto }
-
- assume "a \<le> b" "0 \<le> c"
- thus "c * a \<le> c * b"
- apply (cases c, cases a, cases b)
- by (auto simp: mult_left_mono mult_le_0_iff mult_less_0_iff not_le)
-qed
-
-lemma mult_\<omega>[simp]: "x * y = \<omega> \<longleftrightarrow> (x = \<omega> \<or> y = \<omega>) \<and> x \<noteq> 0 \<and> y \<noteq> 0"
- by (cases x, cases y) auto
-
-lemma \<omega>_mult[simp]: "(\<omega> = x * y) = ((x = \<omega> \<or> y = \<omega>) \<and> x \<noteq> 0 \<and> y \<noteq> 0)"
- by (cases x, cases y) auto
-
-lemma pinfreal_mult_0[simp]: "x * y = 0 \<longleftrightarrow> x = 0 \<or> (y::pinfreal) = 0"
- by (cases x, cases y) (auto simp: mult_le_0_iff)
-
-lemma pinfreal_mult_cancel:
- fixes x y z :: pinfreal
- assumes "y \<le> z"
- shows "x * y \<le> x * z"
- using assms
- by (cases x, cases y, cases z)
- (auto simp: mult_le_cancel_left mult_le_0_iff mult_less_0_iff not_le)
-
-lemma Real_power[simp]:
- "Real x ^ n = (if x \<le> 0 then (if n = 0 then 1 else 0) else Real (x ^ n))"
- by (induct n) auto
-
-lemma Real_power_\<omega>[simp]:
- "\<omega> ^ n = (if n = 0 then 1 else \<omega>)"
- by (induct n) auto
-
-lemma pinfreal_of_nat[simp]: "of_nat m = Real (real m)"
- by (induct m) (auto simp: real_of_nat_Suc one_pinfreal_def simp del: Real_1)
-
-lemma less_\<omega>_Ex_of_nat: "x < \<omega> \<longleftrightarrow> (\<exists>n. x < of_nat n)"
-proof safe
- assume "x < \<omega>"
- then obtain r where "0 \<le> r" "x = Real r" by (cases x) auto
- moreover obtain n where "r < of_nat n" using ex_less_of_nat by auto
- ultimately show "\<exists>n. x < of_nat n" by (auto simp: real_eq_of_nat)
-qed auto
-
-lemma real_of_pinfreal_mono:
- fixes a b :: pinfreal
- assumes "b \<noteq> \<omega>" "a \<le> b"
- shows "real a \<le> real b"
-using assms by (cases b, cases a) auto
-
-lemma setprod_pinfreal_0:
- "(\<Prod>i\<in>I. f i) = (0::pinfreal) \<longleftrightarrow> finite I \<and> (\<exists>i\<in>I. f i = 0)"
-proof cases
- assume "finite I" then show ?thesis
- proof (induct I)
- case (insert i I)
- then show ?case by simp
- qed simp
-qed simp
-
-lemma setprod_\<omega>:
- "(\<Prod>i\<in>I. f i) = \<omega> \<longleftrightarrow> finite I \<and> (\<exists>i\<in>I. f i = \<omega>) \<and> (\<forall>i\<in>I. f i \<noteq> 0)"
-proof cases
- assume "finite I" then show ?thesis
- proof (induct I)
- case (insert i I) then show ?case
- by (auto simp: setprod_pinfreal_0)
- qed simp
-qed simp
-
-instance pinfreal :: "semiring_char_0"
-proof
- fix m n
- show "inj (of_nat::nat\<Rightarrow>pinfreal)" by (auto intro!: inj_onI)
-qed
-
-subsection "@{typ pinfreal} is a complete lattice"
-
-instantiation pinfreal :: lattice
-begin
-definition [simp]: "sup x y = (max x y :: pinfreal)"
-definition [simp]: "inf x y = (min x y :: pinfreal)"
-instance proof qed simp_all
-end
-
-instantiation pinfreal :: complete_lattice
-begin
-
-definition "bot = Real 0"
-definition "top = \<omega>"
-
-definition "Sup S = (LEAST z. \<forall>x\<in>S. x \<le> z :: pinfreal)"
-definition "Inf S = (GREATEST z. \<forall>x\<in>S. z \<le> x :: pinfreal)"
-
-lemma pinfreal_complete_Sup:
- fixes S :: "pinfreal 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 "\<exists>x\<ge>0. \<forall>a\<in>S. a \<le> Real x")
- case False
- hence *: "\<And>x. x\<ge>0 \<Longrightarrow> \<exists>a\<in>S. \<not>a \<le> Real x" by simp
- show ?thesis
- proof (safe intro!: exI[of _ \<omega>])
- fix y assume **: "\<forall>z\<in>S. z \<le> y"
- show "\<omega> \<le> y" unfolding pinfreal_\<omega>_less_eq
- proof (rule ccontr)
- assume "y \<noteq> \<omega>"
- then obtain x where [simp]: "y = Real x" and "0 \<le> x" by (cases y) auto
- from *[OF `0 \<le> x`] show False using ** by auto
- qed
- qed simp
-next
- case True then obtain y where y: "\<And>a. a\<in>S \<Longrightarrow> a \<le> Real y" and "0 \<le> y" by auto
- from y[of \<omega>] have "\<omega> \<notin> S" by auto
-
- with `S \<noteq> {}` obtain x where "x \<in> S" and "x \<noteq> \<omega>" by auto
-
- have bound: "\<forall>x\<in>real ` S. x \<le> y"
- proof
- fix z assume "z \<in> real ` S" then guess a ..
- with y[of a] `\<omega> \<notin> S` `0 \<le> y` show "z \<le> y" by (cases a) auto
- qed
- with reals_complete2[of "real ` S"] `x \<in> S`
- obtain s where s: "\<forall>y\<in>S. real y \<le> s" "\<forall>z. ((\<forall>y\<in>S. real y \<le> z) \<longrightarrow> s \<le> z)"
- by auto
-
- show ?thesis
- proof (safe intro!: exI[of _ "Real s"])
- fix z assume "z \<in> S" thus "z \<le> Real s"
- using s `\<omega> \<notin> S` by (cases z) auto
- next
- fix z assume *: "\<forall>y\<in>S. y \<le> z"
- show "Real s \<le> z"
- proof (cases z)
- case (preal u)
- { fix v assume "v \<in> S"
- hence "v \<le> Real u" using * preal by auto
- hence "real v \<le> u" using `\<omega> \<notin> S` `0 \<le> u` by (cases v) auto }
- hence "s \<le> u" using s(2) by auto
- thus "Real s \<le> z" using preal by simp
- qed simp
- qed
-qed
-
-lemma pinfreal_complete_Inf:
- fixes S :: "pinfreal set" assumes "S \<noteq> {}"
- shows "\<exists>x. (\<forall>y\<in>S. x \<le> y) \<and> (\<forall>z. (\<forall>y\<in>S. z \<le> y) \<longrightarrow> z \<le> x)"
-proof (cases "S = {\<omega>}")
- case True thus ?thesis by (auto intro!: exI[of _ \<omega>])
-next
- case False with `S \<noteq> {}` have "S - {\<omega>} \<noteq> {}" by auto
- hence not_empty: "\<exists>x. x \<in> uminus ` real ` (S - {\<omega>})" by auto
- have bounds: "\<exists>x. \<forall>y\<in>uminus ` real ` (S - {\<omega>}). y \<le> x" by (auto intro!: exI[of _ 0])
- from reals_complete2[OF not_empty bounds]
- obtain s where s: "\<And>y. y\<in>S - {\<omega>} \<Longrightarrow> - real y \<le> s" "\<forall>z. ((\<forall>y\<in>S - {\<omega>}. - real y \<le> z) \<longrightarrow> s \<le> z)"
- by auto
-
- show ?thesis
- proof (safe intro!: exI[of _ "Real (-s)"])
- fix z assume "z \<in> S"
- show "Real (-s) \<le> z"
- proof (cases z)
- case (preal r)
- with s `z \<in> S` have "z \<in> S - {\<omega>}" by simp
- hence "- r \<le> s" using preal s(1)[of z] by auto
- hence "- s \<le> r" by (subst neg_le_iff_le[symmetric]) simp
- thus ?thesis using preal by simp
- qed simp
- next
- fix z assume *: "\<forall>y\<in>S. z \<le> y"
- show "z \<le> Real (-s)"
- proof (cases z)
- case (preal u)
- { fix v assume "v \<in> S-{\<omega>}"
- hence "Real u \<le> v" using * preal by auto
- hence "- real v \<le> - u" using `0 \<le> u` `v \<in> S - {\<omega>}` by (cases v) auto }
- hence "u \<le> - s" using s(2) by (subst neg_le_iff_le[symmetric]) auto
- thus "z \<le> Real (-s)" using preal by simp
- next
- case infinite
- with * have "S = {\<omega>}" using `S \<noteq> {}` by auto
- with `S - {\<omega>} \<noteq> {}` show ?thesis by simp
- qed
- qed
-qed
-
-instance
-proof
- fix x :: pinfreal and A
-
- show "bot \<le> x" by (cases x) (simp_all add: bot_pinfreal_def)
- show "x \<le> top" by (simp add: top_pinfreal_def)
-
- { assume "x \<in> A"
- with pinfreal_complete_Sup[of A]
- obtain s where s: "\<forall>y\<in>A. y \<le> s" "\<forall>z. (\<forall>y\<in>A. y \<le> z) \<longrightarrow> s \<le> z" by auto
- hence "x \<le> s" using `x \<in> A` by auto
- also have "... = Sup A" using s unfolding Sup_pinfreal_def
- by (auto intro!: Least_equality[symmetric])
- finally show "x \<le> Sup A" . }
-
- { assume "x \<in> A"
- with pinfreal_complete_Inf[of A]
- obtain i where i: "\<forall>y\<in>A. i \<le> y" "\<forall>z. (\<forall>y\<in>A. z \<le> y) \<longrightarrow> z \<le> i" by auto
- hence "Inf A = i" unfolding Inf_pinfreal_def
- by (auto intro!: Greatest_equality)
- also have "i \<le> x" using i `x \<in> A` by auto
- finally show "Inf A \<le> x" . }
-
- { assume *: "\<And>z. z \<in> A \<Longrightarrow> z \<le> x"
- show "Sup A \<le> x"
- proof (cases "A = {}")
- case True
- hence "Sup A = 0" unfolding Sup_pinfreal_def
- by (auto intro!: Least_equality)
- thus "Sup A \<le> x" by simp
- next
- case False
- with pinfreal_complete_Sup[of A]
- obtain s where s: "\<forall>y\<in>A. y \<le> s" "\<forall>z. (\<forall>y\<in>A. y \<le> z) \<longrightarrow> s \<le> z" by auto
- hence "Sup A = s"
- unfolding Sup_pinfreal_def by (auto intro!: Least_equality)
- also have "s \<le> x" using * s by auto
- finally show "Sup A \<le> x" .
- qed }
-
- { assume *: "\<And>z. z \<in> A \<Longrightarrow> x \<le> z"
- show "x \<le> Inf A"
- proof (cases "A = {}")
- case True
- hence "Inf A = \<omega>" unfolding Inf_pinfreal_def
- by (auto intro!: Greatest_equality)
- thus "x \<le> Inf A" by simp
- next
- case False
- with pinfreal_complete_Inf[of A]
- obtain i where i: "\<forall>y\<in>A. i \<le> y" "\<forall>z. (\<forall>y\<in>A. z \<le> y) \<longrightarrow> z \<le> i" by auto
- have "x \<le> i" using * i by auto
- also have "i = Inf A" using i
- unfolding Inf_pinfreal_def by (auto intro!: Greatest_equality[symmetric])
- finally show "x \<le> Inf A" .
- qed }
-qed
-end
-
-lemma Inf_pinfreal_iff:
- fixes z :: pinfreal
- 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 Inf_greater:
- fixes z :: pinfreal assumes "Inf X < z"
- shows "\<exists>x \<in> X. x < z"
-proof -
- have "X \<noteq> {}" using assms by (auto simp: Inf_empty top_pinfreal_def)
- with assms show ?thesis
- by (metis Inf_pinfreal_iff mem_def not_leE)
-qed
-
-lemma Inf_close:
- fixes e :: pinfreal assumes "Inf X \<noteq> \<omega>" "0 < e"
- shows "\<exists>x \<in> X. x < Inf X + e"
-proof (rule Inf_greater)
- show "Inf X < Inf X + e" using assms
- by (cases "Inf X", cases e) auto
-qed
-
-lemma pinfreal_SUPI:
- fixes x :: pinfreal
- assumes "\<And>i. i \<in> A \<Longrightarrow> f i \<le> x"
- assumes "\<And>y. (\<And>i. i \<in> A \<Longrightarrow> f i \<le> y) \<Longrightarrow> x \<le> y"
- shows "(SUP i:A. f i) = x"
- unfolding SUPR_def Sup_pinfreal_def
- using assms by (auto intro!: Least_equality)
-
-lemma Sup_pinfreal_iff:
- fixes z :: pinfreal
- shows "(\<And>x. x \<in> X \<Longrightarrow> x \<le> z) \<Longrightarrow> (\<exists>x\<in>X. y<x) \<longleftrightarrow> y < Sup X"
- by (metis complete_lattice_class.Sup_least complete_lattice_class.Sup_upper less_le_not_le linear
- order_less_le_trans)
-
-lemma Sup_lesser:
- fixes z :: pinfreal assumes "z < Sup X"
- shows "\<exists>x \<in> X. z < x"
-proof -
- have "X \<noteq> {}" using assms by (auto simp: Sup_empty bot_pinfreal_def)
- with assms show ?thesis
- by (metis Sup_pinfreal_iff mem_def not_leE)
-qed
-
-lemma Sup_eq_\<omega>: "\<omega> \<in> S \<Longrightarrow> Sup S = \<omega>"
- unfolding Sup_pinfreal_def
- by (auto intro!: Least_equality)
-
-lemma Sup_close:
- assumes "0 < e" and S: "Sup S \<noteq> \<omega>" "S \<noteq> {}"
- shows "\<exists>X\<in>S. Sup S < X + e"
-proof cases
- assume "Sup S = 0"
- moreover obtain X where "X \<in> S" using `S \<noteq> {}` by auto
- ultimately show ?thesis using `0 < e` by (auto intro!: bexI[OF _ `X\<in>S`])
-next
- assume "Sup S \<noteq> 0"
- have "\<exists>X\<in>S. Sup S - e < X"
- proof (rule Sup_lesser)
- show "Sup S - e < Sup S" using `0 < e` `Sup S \<noteq> 0` `Sup S \<noteq> \<omega>`
- by (cases e) (auto simp: pinfreal_noteq_omega_Ex)
- qed
- then guess X .. note X = this
- with `Sup S \<noteq> \<omega>` Sup_eq_\<omega> have "X \<noteq> \<omega>" by auto
- thus ?thesis using `Sup S \<noteq> \<omega>` X unfolding pinfreal_noteq_omega_Ex
- by (cases e) (auto intro!: bexI[OF _ `X\<in>S`] simp: split: split_if_asm)
-qed
-
-lemma Sup_\<omega>: "(SUP i::nat. Real (real i)) = \<omega>"
-proof (rule pinfreal_SUPI)
- fix y assume *: "\<And>i::nat. i \<in> UNIV \<Longrightarrow> Real (real i) \<le> y"
- thus "\<omega> \<le> y"
- proof (cases y)
- case (preal r)
- then obtain k :: nat where "r < real k"
- using ex_less_of_nat by (auto simp: real_eq_of_nat)
- with *[of k] preal show ?thesis by auto
- qed simp
-qed simp
-
-lemma SUP_\<omega>: "(SUP i:A. f i) = \<omega> \<longleftrightarrow> (\<forall>x<\<omega>. \<exists>i\<in>A. x < f i)"
-proof
- assume *: "(SUP i:A. f i) = \<omega>"
- show "(\<forall>x<\<omega>. \<exists>i\<in>A. x < f i)" unfolding *[symmetric]
- proof (intro allI impI)
- fix x assume "x < SUPR A f" then show "\<exists>i\<in>A. x < f i"
- unfolding less_SUP_iff by auto
- qed
-next
- assume *: "\<forall>x<\<omega>. \<exists>i\<in>A. x < f i"
- show "(SUP i:A. f i) = \<omega>"
- proof (rule pinfreal_SUPI)
- fix y assume **: "\<And>i. i \<in> A \<Longrightarrow> f i \<le> y"
- show "\<omega> \<le> y"
- proof cases
- assume "y < \<omega>"
- from *[THEN spec, THEN mp, OF this]
- obtain i where "i \<in> A" "\<not> (f i \<le> y)" by auto
- with ** show ?thesis by auto
- qed auto
- qed auto
-qed
-
-subsubsection {* Equivalence between @{text "f ----> x"} and @{text SUP} on @{typ pinfreal} *}
-
-lemma monoseq_monoI: "mono f \<Longrightarrow> monoseq f"
- unfolding mono_def monoseq_def by auto
-
-lemma incseq_mono: "mono f \<longleftrightarrow> incseq f"
- unfolding mono_def incseq_def by auto
-
-lemma SUP_eq_LIMSEQ:
- assumes "mono f" and "\<And>n. 0 \<le> f n" and "0 \<le> x"
- shows "(SUP n. Real (f n)) = Real x \<longleftrightarrow> f ----> x"
-proof
- assume x: "(SUP n. Real (f n)) = Real x"
- { fix n
- have "Real (f n) \<le> Real x" using x[symmetric] by (auto intro: le_SUPI)
- hence "f n \<le> x" using assms by simp }
- show "f ----> x"
- proof (rule LIMSEQ_I)
- fix r :: real assume "0 < r"
- show "\<exists>no. \<forall>n\<ge>no. norm (f n - x) < r"
- proof (rule ccontr)
- assume *: "\<not> ?thesis"
- { fix N
- from * obtain n where "N \<le> n" "r \<le> x - f n"
- using `\<And>n. f n \<le> x` by (auto simp: not_less)
- hence "f N \<le> f n" using `mono f` by (auto dest: monoD)
- hence "f N \<le> x - r" using `r \<le> x - f n` by auto
- hence "Real (f N) \<le> Real (x - r)" and "r \<le> x" using `0 \<le> f N` by auto }
- hence "(SUP n. Real (f n)) \<le> Real (x - r)"
- and "Real (x - r) < Real x" using `0 < r` by (auto intro: SUP_leI)
- hence "(SUP n. Real (f n)) < Real x" by (rule le_less_trans)
- thus False using x by auto
- qed
- qed
-next
- assume "f ----> x"
- show "(SUP n. Real (f n)) = Real x"
- proof (rule pinfreal_SUPI)
- fix n
- from incseq_le[of f x] `mono f` `f ----> x`
- show "Real (f n) \<le> Real x" using assms incseq_mono by auto
- next
- fix y assume *: "\<And>n. n\<in>UNIV \<Longrightarrow> Real (f n) \<le> y"
- show "Real x \<le> y"
- proof (cases y)
- case (preal r)
- with * have "\<exists>N. \<forall>n\<ge>N. f n \<le> r" using assms by fastsimp
- from LIMSEQ_le_const2[OF `f ----> x` this]
- show "Real x \<le> y" using `0 \<le> x` preal by auto
- qed simp
- qed
-qed
-
-lemma SUPR_bound:
- assumes "\<forall>N. f N \<le> x"
- shows "(SUP n. f n) \<le> x"
- using assms by (simp add: SUPR_def Sup_le_iff)
-
-lemma pinfreal_less_eq_diff_eq_sum:
- fixes x y z :: pinfreal
- assumes "y \<le> x" and "x \<noteq> \<omega>"
- shows "z \<le> x - y \<longleftrightarrow> z + y \<le> x"
- using assms
- apply (cases z, cases y, cases x)
- by (simp_all add: field_simps minus_pinfreal_eq)
-
-lemma Real_diff_less_omega: "Real r - x < \<omega>" by (cases x) auto
-
-subsubsection {* Numbers on @{typ pinfreal} *}
-
-instantiation pinfreal :: number
-begin
-definition [simp]: "number_of x = Real (number_of x)"
-instance proof qed
-end
-
-subsubsection {* Division on @{typ pinfreal} *}
-
-instantiation pinfreal :: inverse
-begin
-
-definition "inverse x = pinfreal_case (\<lambda>x. if x = 0 then \<omega> else Real (inverse x)) 0 x"
-definition [simp]: "x / y = x * inverse (y :: pinfreal)"
-
-instance proof qed
-end
-
-lemma pinfreal_inverse[simp]:
- "inverse 0 = \<omega>"
- "inverse (Real x) = (if x \<le> 0 then \<omega> else Real (inverse x))"
- "inverse \<omega> = 0"
- "inverse (1::pinfreal) = 1"
- "inverse (inverse x) = x"
- by (simp_all add: inverse_pinfreal_def one_pinfreal_def split: pinfreal_case_split del: Real_1)
-
-lemma pinfreal_inverse_le_eq:
- assumes "x \<noteq> 0" "x \<noteq> \<omega>"
- shows "y \<le> z / x \<longleftrightarrow> x * y \<le> (z :: pinfreal)"
-proof -
- from assms obtain r where r: "x = Real r" "0 < r" by (cases x) auto
- { fix p q :: real assume "0 \<le> p" "0 \<le> q"
- have "p \<le> q * inverse r \<longleftrightarrow> p \<le> q / r" by (simp add: divide_inverse)
- also have "... \<longleftrightarrow> p * r \<le> q" using `0 < r` by (auto simp: field_simps)
- finally have "p \<le> q * inverse r \<longleftrightarrow> p * r \<le> q" . }
- with r show ?thesis
- by (cases y, cases z, auto simp: zero_le_mult_iff field_simps)
-qed
-
-lemma inverse_antimono_strict:
- fixes x y :: pinfreal
- assumes "x < y" shows "inverse y < inverse x"
- using assms by (cases x, cases y) auto
-
-lemma inverse_antimono:
- fixes x y :: pinfreal
- assumes "x \<le> y" shows "inverse y \<le> inverse x"
- using assms by (cases x, cases y) auto
-
-lemma pinfreal_inverse_\<omega>_iff[simp]: "inverse x = \<omega> \<longleftrightarrow> x = 0"
- by (cases x) auto
-
-subsection "Infinite sum over @{typ pinfreal}"
-
-text {*
-
-The infinite sum over @{typ pinfreal} has the nice property that it is always
-defined.
-
-*}
-
-definition psuminf :: "(nat \<Rightarrow> pinfreal) \<Rightarrow> pinfreal" (binder "\<Sum>\<^isub>\<infinity>" 10) where
- "(\<Sum>\<^isub>\<infinity> x. f x) = (SUP n. \<Sum>i<n. f i)"
-
-subsubsection {* Equivalence between @{text "\<Sum> n. f n"} and @{text "\<Sum>\<^isub>\<infinity> n. f n"} *}
-
-lemma setsum_Real:
- assumes "\<forall>x\<in>A. 0 \<le> f x"
- shows "(\<Sum>x\<in>A. Real (f x)) = Real (\<Sum>x\<in>A. f x)"
-proof (cases "finite A")
- case True
- thus ?thesis using assms
- proof induct case (insert x s)
- hence "0 \<le> setsum f s" apply-apply(rule setsum_nonneg) by auto
- thus ?case using insert by auto
- qed auto
-qed simp
-
-lemma setsum_Real':
- assumes "\<forall>x. 0 \<le> f x"
- shows "(\<Sum>x\<in>A. Real (f x)) = Real (\<Sum>x\<in>A. f x)"
- apply(rule setsum_Real) using assms by auto
-
-lemma setsum_\<omega>:
- "(\<Sum>x\<in>P. f x) = \<omega> \<longleftrightarrow> (finite P \<and> (\<exists>i\<in>P. f i = \<omega>))"
-proof safe
- assume *: "setsum f P = \<omega>"
- show "finite P"
- proof (rule ccontr) assume "infinite P" with * show False by auto qed
- show "\<exists>i\<in>P. f i = \<omega>"
- proof (rule ccontr)
- assume "\<not> ?thesis" hence "\<And>i. i\<in>P \<Longrightarrow> f i \<noteq> \<omega>" by auto
- from `finite P` this have "setsum f P \<noteq> \<omega>"
- by induct auto
- with * show False by auto
- qed
-next
- fix i assume "finite P" "i \<in> P" "f i = \<omega>"
- thus "setsum f P = \<omega>"
- proof induct
- case (insert x A)
- show ?case using insert by (cases "x = i") auto
- qed simp
-qed
-
-lemma real_of_pinfreal_setsum:
- assumes "\<And>x. x \<in> S \<Longrightarrow> f x \<noteq> \<omega>"
- shows "(\<Sum>x\<in>S. real (f x)) = real (setsum f S)"
-proof cases
- assume "finite S"
- from this assms show ?thesis
- by induct (simp_all add: real_of_pinfreal_add setsum_\<omega>)
-qed simp
-
-lemma setsum_0:
- fixes f :: "'a \<Rightarrow> pinfreal" assumes "finite A"
- shows "(\<Sum>x\<in>A. f x) = 0 \<longleftrightarrow> (\<forall>i\<in>A. f i = 0)"
- using assms by induct auto
-
-lemma suminf_imp_psuminf:
- assumes "f sums x" and "\<forall>n. 0 \<le> f n"
- shows "(\<Sum>\<^isub>\<infinity> x. Real (f x)) = Real x"
- unfolding psuminf_def setsum_Real'[OF assms(2)]
-proof (rule SUP_eq_LIMSEQ[THEN iffD2])
- show "mono (\<lambda>n. setsum f {..<n})" (is "mono ?S")
- unfolding mono_iff_le_Suc using assms by simp
-
- { fix n show "0 \<le> ?S n"
- using setsum_nonneg[of "{..<n}" f] assms by auto }
-
- thus "0 \<le> x" "?S ----> x"
- using `f sums x` LIMSEQ_le_const
- by (auto simp: atLeast0LessThan sums_def)
-qed
-
-lemma psuminf_equality:
- assumes "\<And>n. setsum f {..<n} \<le> x"
- and "\<And>y. y \<noteq> \<omega> \<Longrightarrow> (\<And>n. setsum f {..<n} \<le> y) \<Longrightarrow> x \<le> y"
- shows "psuminf f = x"
- unfolding psuminf_def
-proof (safe intro!: pinfreal_SUPI)
- fix n show "setsum f {..<n} \<le> x" using assms(1) .
-next
- fix y assume *: "\<forall>n. n \<in> UNIV \<longrightarrow> setsum f {..<n} \<le> y"
- show "x \<le> y"
- proof (cases "y = \<omega>")
- assume "y \<noteq> \<omega>" from assms(2)[OF this] *
- show "x \<le> y" by auto
- qed simp
-qed
-
-lemma psuminf_\<omega>:
- assumes "f i = \<omega>"
- shows "(\<Sum>\<^isub>\<infinity> x. f x) = \<omega>"
-proof (rule psuminf_equality)
- fix y assume *: "\<And>n. setsum f {..<n} \<le> y"
- have "setsum f {..<Suc i} = \<omega>"
- using assms by (simp add: setsum_\<omega>)
- thus "\<omega> \<le> y" using *[of "Suc i"] by auto
-qed simp
-
-lemma psuminf_imp_suminf:
- assumes "(\<Sum>\<^isub>\<infinity> x. f x) \<noteq> \<omega>"
- shows "(\<lambda>x. real (f x)) sums real (\<Sum>\<^isub>\<infinity> x. f x)"
-proof -
- have "\<forall>i. \<exists>r. f i = Real r \<and> 0 \<le> r"
- proof
- fix i show "\<exists>r. f i = Real r \<and> 0 \<le> r" using psuminf_\<omega> assms by (cases "f i") auto
- qed
- from choice[OF this] obtain r where f: "f = (\<lambda>i. Real (r i))"
- and pos: "\<forall>i. 0 \<le> r i"
- by (auto simp: fun_eq_iff)
- hence [simp]: "\<And>i. real (f i) = r i" by auto
-
- have "mono (\<lambda>n. setsum r {..<n})" (is "mono ?S")
- unfolding mono_iff_le_Suc using pos by simp
-
- { fix n have "0 \<le> ?S n"
- using setsum_nonneg[of "{..<n}" r] pos by auto }
-
- from assms obtain p where *: "(\<Sum>\<^isub>\<infinity> x. f x) = Real p" and "0 \<le> p"
- by (cases "(\<Sum>\<^isub>\<infinity> x. f x)") auto
- show ?thesis unfolding * using * pos `0 \<le> p` SUP_eq_LIMSEQ[OF `mono ?S` `\<And>n. 0 \<le> ?S n` `0 \<le> p`]
- by (simp add: f atLeast0LessThan sums_def psuminf_def setsum_Real'[OF pos] f)
-qed
-
-lemma psuminf_bound:
- assumes "\<forall>N. (\<Sum>n<N. f n) \<le> x"
- shows "(\<Sum>\<^isub>\<infinity> n. f n) \<le> x"
- using assms by (simp add: psuminf_def SUPR_def Sup_le_iff)
-
-lemma psuminf_bound_add:
- assumes "\<forall>N. (\<Sum>n<N. f n) + y \<le> x"
- shows "(\<Sum>\<^isub>\<infinity> n. f n) + y \<le> x"
-proof (cases "x = \<omega>")
- have "y \<le> x" using assms by (auto intro: pinfreal_le_add2)
- assume "x \<noteq> \<omega>"
- note move_y = pinfreal_less_eq_diff_eq_sum[OF `y \<le> x` this]
-
- have "\<forall>N. (\<Sum>n<N. f n) \<le> x - y" using assms by (simp add: move_y)
- hence "(\<Sum>\<^isub>\<infinity> n. f n) \<le> x - y" by (rule psuminf_bound)
- thus ?thesis by (simp add: move_y)
-qed simp
-
-lemma psuminf_finite:
- assumes "\<forall>N\<ge>n. f N = 0"
- shows "(\<Sum>\<^isub>\<infinity> n. f n) = (\<Sum>N<n. f N)"
-proof (rule psuminf_equality)
- fix N
- show "setsum f {..<N} \<le> setsum f {..<n}"
- proof (cases rule: linorder_cases)
- assume "N < n" thus ?thesis by (auto intro!: setsum_mono3)
- next
- assume "n < N"
- hence *: "{..<N} = {..<n} \<union> {n..<N}" by auto
- moreover have "setsum f {n..<N} = 0"
- using assms by (auto intro!: setsum_0')
- ultimately show ?thesis unfolding *
- by (subst setsum_Un_disjoint) auto
- qed simp
-qed simp
-
-lemma psuminf_upper:
- shows "(\<Sum>n<N. f n) \<le> (\<Sum>\<^isub>\<infinity> n. f n)"
- unfolding psuminf_def SUPR_def
- by (auto intro: complete_lattice_class.Sup_upper image_eqI)
-
-lemma psuminf_le:
- assumes "\<And>N. f N \<le> g N"
- shows "psuminf f \<le> psuminf g"
-proof (safe intro!: psuminf_bound)
- fix n
- have "setsum f {..<n} \<le> setsum g {..<n}" using assms by (auto intro: setsum_mono)
- also have "... \<le> psuminf g" by (rule psuminf_upper)
- finally show "setsum f {..<n} \<le> psuminf g" .
-qed
-
-lemma psuminf_const[simp]: "psuminf (\<lambda>n. c) = (if c = 0 then 0 else \<omega>)" (is "_ = ?if")
-proof (rule psuminf_equality)
- fix y assume *: "\<And>n :: nat. (\<Sum>n<n. c) \<le> y" and "y \<noteq> \<omega>"
- then obtain r p where
- y: "y = Real r" "0 \<le> r" and
- c: "c = Real p" "0 \<le> p"
- using *[of 1] by (cases c, cases y) auto
- show "(if c = 0 then 0 else \<omega>) \<le> y"
- proof (cases "p = 0")
- assume "p = 0" with c show ?thesis by simp
- next
- assume "p \<noteq> 0"
- with * c y have **: "\<And>n :: nat. real n \<le> r / p"
- by (auto simp: zero_le_mult_iff field_simps)
- from ex_less_of_nat[of "r / p"] guess n ..
- with **[of n] show ?thesis by (simp add: real_eq_of_nat)
- qed
-qed (cases "c = 0", simp_all)
-
-lemma psuminf_add[simp]: "psuminf (\<lambda>n. f n + g n) = psuminf f + psuminf g"
-proof (rule psuminf_equality)
- fix n
- from psuminf_upper[of f n] psuminf_upper[of g n]
- show "(\<Sum>n<n. f n + g n) \<le> psuminf f + psuminf g"
- by (auto simp add: setsum_addf intro!: add_mono)
-next
- fix y assume *: "\<And>n. (\<Sum>n<n. f n + g n) \<le> y" and "y \<noteq> \<omega>"
- { fix n m
- have **: "(\<Sum>n<n. f n) + (\<Sum>n<m. g n) \<le> y"
- proof (cases rule: linorder_le_cases)
- assume "n \<le> m"
- hence "(\<Sum>n<n. f n) + (\<Sum>n<m. g n) \<le> (\<Sum>n<m. f n) + (\<Sum>n<m. g n)"
- by (auto intro!: add_right_mono setsum_mono3)
- also have "... \<le> y"
- using * by (simp add: setsum_addf)
- finally show ?thesis .
- next
- assume "m \<le> n"
- hence "(\<Sum>n<n. f n) + (\<Sum>n<m. g n) \<le> (\<Sum>n<n. f n) + (\<Sum>n<n. g n)"
- by (auto intro!: add_left_mono setsum_mono3)
- also have "... \<le> y"
- using * by (simp add: setsum_addf)
- finally show ?thesis .
- qed }
- hence "\<And>m. \<forall>n. (\<Sum>n<n. f n) + (\<Sum>n<m. g n) \<le> y" by simp
- from psuminf_bound_add[OF this]
- have "\<forall>m. (\<Sum>n<m. g n) + psuminf f \<le> y" by (simp add: ac_simps)
- from psuminf_bound_add[OF this]
- show "psuminf f + psuminf g \<le> y" by (simp add: ac_simps)
-qed
-
-lemma psuminf_0: "psuminf f = 0 \<longleftrightarrow> (\<forall>i. f i = 0)"
-proof safe
- assume "\<forall>i. f i = 0"
- hence "f = (\<lambda>i. 0)" by (simp add: fun_eq_iff)
- thus "psuminf f = 0" using psuminf_const by simp
-next
- fix i assume "psuminf f = 0"
- hence "(\<Sum>n<Suc i. f n) = 0" using psuminf_upper[of f "Suc i"] by simp
- thus "f i = 0" by simp
-qed
-
-lemma psuminf_cmult_right[simp]: "psuminf (\<lambda>n. c * f n) = c * psuminf f"
-proof (rule psuminf_equality)
- fix n show "(\<Sum>n<n. c * f n) \<le> c * psuminf f"
- by (auto simp: setsum_right_distrib[symmetric] intro: mult_left_mono psuminf_upper)
-next
- fix y
- assume "\<And>n. (\<Sum>n<n. c * f n) \<le> y"
- hence *: "\<And>n. c * (\<Sum>n<n. f n) \<le> y" by (auto simp add: setsum_right_distrib)
- thus "c * psuminf f \<le> y"
- proof (cases "c = \<omega> \<or> c = 0")
- assume "c = \<omega> \<or> c = 0"
- thus ?thesis
- using * by (fastsimp simp add: psuminf_0 setsum_0 split: split_if_asm)
- next
- assume "\<not> (c = \<omega> \<or> c = 0)"
- hence "c \<noteq> 0" "c \<noteq> \<omega>" by auto
- note rewrite_div = pinfreal_inverse_le_eq[OF this, of _ y]
- hence "\<forall>n. (\<Sum>n<n. f n) \<le> y / c" using * by simp
- hence "psuminf f \<le> y / c" by (rule psuminf_bound)
- thus ?thesis using rewrite_div by simp
- qed
-qed
-
-lemma psuminf_cmult_left[simp]: "psuminf (\<lambda>n. f n * c) = psuminf f * c"
- using psuminf_cmult_right[of c f] by (simp add: ac_simps)
-
-lemma psuminf_half_series: "psuminf (\<lambda>n. (1/2)^Suc n) = 1"
- using suminf_imp_psuminf[OF power_half_series] by auto
-
-lemma setsum_pinfsum: "(\<Sum>\<^isub>\<infinity> n. \<Sum>m\<in>A. f n m) = (\<Sum>m\<in>A. (\<Sum>\<^isub>\<infinity> n. f n m))"
-proof (cases "finite A")
- assume "finite A"
- thus ?thesis by induct simp_all
-qed simp
-
-lemma psuminf_reindex:
- fixes f:: "nat \<Rightarrow> nat" assumes "bij f"
- shows "psuminf (g \<circ> f) = psuminf g"
-proof -
- have [intro, simp]: "\<And>A. inj_on f A" using `bij f` unfolding bij_def by (auto intro: subset_inj_on)
- have f[intro, simp]: "\<And>x. f (inv f x) = x"
- using `bij f` unfolding bij_def by (auto intro: surj_f_inv_f)
- show ?thesis
- proof (rule psuminf_equality)
- fix n
- have "setsum (g \<circ> f) {..<n} = setsum g (f ` {..<n})"
- by (simp add: setsum_reindex)
- also have "\<dots> \<le> setsum g {..Max (f ` {..<n})}"
- by (rule setsum_mono3) auto
- also have "\<dots> \<le> psuminf g" unfolding lessThan_Suc_atMost[symmetric] by (rule psuminf_upper)
- finally show "setsum (g \<circ> f) {..<n} \<le> psuminf g" .
- next
- fix y assume *: "\<And>n. setsum (g \<circ> f) {..<n} \<le> y"
- show "psuminf g \<le> y"
- proof (safe intro!: psuminf_bound)
- fix N
- have "setsum g {..<N} \<le> setsum g (f ` {..Max (inv f ` {..<N})})"
- by (rule setsum_mono3) (auto intro!: image_eqI[where f="f", OF f[symmetric]])
- also have "\<dots> = setsum (g \<circ> f) {..Max (inv f ` {..<N})}"
- by (simp add: setsum_reindex)
- also have "\<dots> \<le> y" unfolding lessThan_Suc_atMost[symmetric] by (rule *)
- finally show "setsum g {..<N} \<le> y" .
- qed
- qed
-qed
-
-lemma pinfreal_mult_less_right:
- assumes "b * a < c * a" "0 < a" "a < \<omega>"
- shows "b < c"
- using assms
- by (cases a, cases b, cases c) (auto split: split_if_asm simp: zero_less_mult_iff zero_le_mult_iff)
-
-lemma pinfreal_\<omega>_eq_plus[simp]: "\<omega> = a + b \<longleftrightarrow> (a = \<omega> \<or> b = \<omega>)"
- by (cases a, cases b) auto
-
-lemma pinfreal_of_nat_le_iff:
- "(of_nat k :: pinfreal) \<le> of_nat m \<longleftrightarrow> k \<le> m" by auto
-
-lemma pinfreal_of_nat_less_iff:
- "(of_nat k :: pinfreal) < of_nat m \<longleftrightarrow> k < m" by auto
-
-lemma pinfreal_bound_add:
- assumes "\<forall>N. f N + y \<le> (x::pinfreal)"
- shows "(SUP n. f n) + y \<le> x"
-proof (cases "x = \<omega>")
- have "y \<le> x" using assms by (auto intro: pinfreal_le_add2)
- assume "x \<noteq> \<omega>"
- note move_y = pinfreal_less_eq_diff_eq_sum[OF `y \<le> x` this]
-
- have "\<forall>N. f N \<le> x - y" using assms by (simp add: move_y)
- hence "(SUP n. f n) \<le> x - y" by (rule SUPR_bound)
- thus ?thesis by (simp add: move_y)
-qed simp
-
-lemma SUPR_pinfreal_add:
- fixes f g :: "nat \<Rightarrow> pinfreal"
- assumes f: "\<forall>n. f n \<le> f (Suc n)" and g: "\<forall>n. g n \<le> g (Suc n)"
- shows "(SUP n. f n + g n) = (SUP n. f n) + (SUP n. g n)"
-proof (rule pinfreal_SUPI)
- fix n :: nat from le_SUPI[of n UNIV f] le_SUPI[of n UNIV g]
- show "f n + g n \<le> (SUP n. f n) + (SUP n. g n)"
- by (auto intro!: add_mono)
-next
- fix y assume *: "\<And>n. n \<in> UNIV \<Longrightarrow> f n + g n \<le> y"
- { fix n m
- have "f n + g m \<le> y"
- proof (cases rule: linorder_le_cases)
- assume "n \<le> m"
- hence "f n + g m \<le> f m + g m"
- using f lift_Suc_mono_le by (auto intro!: add_right_mono)
- also have "\<dots> \<le> y" using * by simp
- finally show ?thesis .
- next
- assume "m \<le> n"
- hence "f n + g m \<le> f n + g n"
- using g lift_Suc_mono_le by (auto intro!: add_left_mono)
- also have "\<dots> \<le> y" using * by simp
- finally show ?thesis .
- qed }
- hence "\<And>m. \<forall>n. f n + g m \<le> y" by simp
- from pinfreal_bound_add[OF this]
- have "\<forall>m. (g m) + (SUP n. f n) \<le> y" by (simp add: ac_simps)
- from pinfreal_bound_add[OF this]
- show "SUPR UNIV f + SUPR UNIV g \<le> y" by (simp add: ac_simps)
-qed
-
-lemma SUPR_pinfreal_setsum:
- fixes f :: "'x \<Rightarrow> nat \<Rightarrow> pinfreal"
- assumes "\<And>i. i \<in> P \<Longrightarrow> \<forall>n. f i n \<le> f i (Suc n)"
- shows "(SUP n. \<Sum>i\<in>P. f i n) = (\<Sum>i\<in>P. SUP n. f i n)"
-proof cases
- assume "finite P" from this assms show ?thesis
- proof induct
- case (insert i P)
- thus ?case
- apply simp
- apply (subst SUPR_pinfreal_add)
- by (auto intro!: setsum_mono)
- qed simp
-qed simp
-
-lemma psuminf_SUP_eq:
- assumes "\<And>n i. f n i \<le> f (Suc n) i"
- shows "(\<Sum>\<^isub>\<infinity> i. SUP n::nat. f n i) = (SUP n::nat. \<Sum>\<^isub>\<infinity> i. f n i)"
-proof -
- { fix n :: nat
- have "(\<Sum>i<n. SUP k. f k i) = (SUP k. \<Sum>i<n. f k i)"
- using assms by (auto intro!: SUPR_pinfreal_setsum[symmetric]) }
- note * = this
- show ?thesis
- unfolding psuminf_def
- unfolding *
- apply (subst SUP_commute) ..
-qed
-
-lemma psuminf_commute:
- shows "(\<Sum>\<^isub>\<infinity> i j. f i j) = (\<Sum>\<^isub>\<infinity> j i. f i j)"
-proof -
- have "(SUP n. \<Sum> i < n. SUP m. \<Sum> j < m. f i j) = (SUP n. SUP m. \<Sum> i < n. \<Sum> j < m. f i j)"
- apply (subst SUPR_pinfreal_setsum)
- by auto
- also have "\<dots> = (SUP m n. \<Sum> j < m. \<Sum> i < n. f i j)"
- apply (subst SUP_commute)
- apply (subst setsum_commute)
- by auto
- also have "\<dots> = (SUP m. \<Sum> j < m. SUP n. \<Sum> i < n. f i j)"
- apply (subst SUPR_pinfreal_setsum)
- by auto
- finally show ?thesis
- unfolding psuminf_def by auto
-qed
-
-lemma psuminf_2dimen:
- fixes f:: "nat * nat \<Rightarrow> pinfreal"
- assumes fsums: "\<And>m. g m = (\<Sum>\<^isub>\<infinity> n. f (m,n))"
- shows "psuminf (f \<circ> prod_decode) = psuminf g"
-proof (rule psuminf_equality)
- fix n :: nat
- let ?P = "prod_decode ` {..<n}"
- have "setsum (f \<circ> prod_decode) {..<n} = setsum f ?P"
- by (auto simp: setsum_reindex inj_prod_decode)
- also have "\<dots> \<le> setsum f ({..Max (fst ` ?P)} \<times> {..Max (snd ` ?P)})"
- proof (safe intro!: setsum_mono3 Max_ge image_eqI)
- fix a b x assume "(a, b) = prod_decode x"
- from this[symmetric] show "a = fst (prod_decode x)" "b = snd (prod_decode x)"
- by simp_all
- qed simp_all
- also have "\<dots> = (\<Sum>m\<le>Max (fst ` ?P). (\<Sum>n\<le>Max (snd ` ?P). f (m,n)))"
- unfolding setsum_cartesian_product by simp
- also have "\<dots> \<le> (\<Sum>m\<le>Max (fst ` ?P). g m)"
- by (auto intro!: setsum_mono psuminf_upper simp del: setsum_lessThan_Suc
- simp: fsums lessThan_Suc_atMost[symmetric])
- also have "\<dots> \<le> psuminf g"
- by (auto intro!: psuminf_upper simp del: setsum_lessThan_Suc
- simp: lessThan_Suc_atMost[symmetric])
- finally show "setsum (f \<circ> prod_decode) {..<n} \<le> psuminf g" .
-next
- fix y assume *: "\<And>n. setsum (f \<circ> prod_decode) {..<n} \<le> y"
- have g: "g = (\<lambda>m. \<Sum>\<^isub>\<infinity> n. f (m,n))" unfolding fsums[symmetric] ..
- show "psuminf g \<le> y" unfolding g
- proof (rule psuminf_bound, unfold setsum_pinfsum[symmetric], safe intro!: psuminf_bound)
- fix N M :: nat
- let ?P = "{..<N} \<times> {..<M}"
- let ?M = "Max (prod_encode ` ?P)"
- have "(\<Sum>n<M. \<Sum>m<N. f (m, n)) \<le> (\<Sum>(m, n)\<in>?P. f (m, n))"
- unfolding setsum_commute[of _ _ "{..<M}"] unfolding setsum_cartesian_product ..
- also have "\<dots> \<le> (\<Sum>(m,n)\<in>(prod_decode ` {..?M}). f (m, n))"
- by (auto intro!: setsum_mono3 image_eqI[where f=prod_decode, OF prod_encode_inverse[symmetric]])
- also have "\<dots> \<le> y" using *[of "Suc ?M"]
- by (simp add: lessThan_Suc_atMost[symmetric] setsum_reindex
- inj_prod_decode del: setsum_lessThan_Suc)
- finally show "(\<Sum>n<M. \<Sum>m<N. f (m, n)) \<le> y" .
- qed
-qed
-
-lemma Real_max:
- assumes "x \<ge> 0" "y \<ge> 0"
- shows "Real (max x y) = max (Real x) (Real y)"
- using assms unfolding max_def by (auto simp add:not_le)
-
-lemma Real_real: "Real (real x) = (if x = \<omega> then 0 else x)"
- using assms by (cases x) auto
-
-lemma inj_on_real: "inj_on real (UNIV - {\<omega>})"
-proof (rule inj_onI)
- fix x y assume mem: "x \<in> UNIV - {\<omega>}" "y \<in> UNIV - {\<omega>}" and "real x = real y"
- thus "x = y" by (cases x, cases y) auto
-qed
-
-lemma inj_on_Real: "inj_on Real {0..}"
- by (auto intro!: inj_onI)
-
-lemma range_Real[simp]: "range Real = UNIV - {\<omega>}"
-proof safe
- fix x assume "x \<notin> range Real"
- thus "x = \<omega>" by (cases x) auto
-qed auto
-
-lemma image_Real[simp]: "Real ` {0..} = UNIV - {\<omega>}"
-proof safe
- fix x assume "x \<notin> Real ` {0..}"
- thus "x = \<omega>" by (cases x) auto
-qed auto
-
-lemma pinfreal_SUP_cmult:
- fixes f :: "'a \<Rightarrow> pinfreal"
- shows "(SUP i : R. z * f i) = z * (SUP i : R. f i)"
-proof (rule pinfreal_SUPI)
- fix i assume "i \<in> R"
- from le_SUPI[OF this]
- show "z * f i \<le> z * (SUP i:R. f i)" by (rule pinfreal_mult_cancel)
-next
- fix y assume "\<And>i. i\<in>R \<Longrightarrow> z * f i \<le> y"
- hence *: "\<And>i. i\<in>R \<Longrightarrow> z * f i \<le> y" by auto
- show "z * (SUP i:R. f i) \<le> y"
- proof (cases "\<forall>i\<in>R. f i = 0")
- case True
- show ?thesis
- proof cases
- assume "R \<noteq> {}" hence "f ` R = {0}" using True by auto
- thus ?thesis by (simp add: SUPR_def)
- qed (simp add: SUPR_def Sup_empty bot_pinfreal_def)
- next
- case False then obtain i where i: "i \<in> R" and f0: "f i \<noteq> 0" by auto
- show ?thesis
- proof (cases "z = 0 \<or> z = \<omega>")
- case True with f0 *[OF i] show ?thesis by auto
- next
- case False hence z: "z \<noteq> 0" "z \<noteq> \<omega>" by auto
- note div = pinfreal_inverse_le_eq[OF this, symmetric]
- hence "\<And>i. i\<in>R \<Longrightarrow> f i \<le> y / z" using * by auto
- thus ?thesis unfolding div SUP_le_iff by simp
- qed
- qed
-qed
-
-instantiation pinfreal :: topological_space
-begin
-
-definition "open A \<longleftrightarrow>
- (\<exists>T. open T \<and> (Real ` (T\<inter>{0..}) = A - {\<omega>})) \<and> (\<omega> \<in> A \<longrightarrow> (\<exists>x\<ge>0. {Real x <..} \<subseteq> A))"
-
-lemma open_omega: "open A \<Longrightarrow> \<omega> \<in> A \<Longrightarrow> (\<exists>x\<ge>0. {Real x<..} \<subseteq> A)"
- unfolding open_pinfreal_def by auto
-
-lemma open_omegaD: assumes "open A" "\<omega> \<in> A" obtains x where "x\<ge>0" "{Real x<..} \<subseteq> A"
- using open_omega[OF assms] by auto
-
-lemma pinfreal_openE: assumes "open A" obtains A' x where
- "open A'" "Real ` (A' \<inter> {0..}) = A - {\<omega>}"
- "x \<ge> 0" "\<omega> \<in> A \<Longrightarrow> {Real x<..} \<subseteq> A"
- using assms open_pinfreal_def by auto
-
-instance
-proof
- let ?U = "UNIV::pinfreal set"
- show "open ?U" unfolding open_pinfreal_def
- by (auto intro!: exI[of _ "UNIV"] exI[of _ 0])
-next
- fix S T::"pinfreal set" assume "open S" and "open T"
- from `open S`[THEN pinfreal_openE] guess S' xS . note S' = this
- from `open T`[THEN pinfreal_openE] guess T' xT . note T' = this
-
- from S'(1-3) T'(1-3)
- show "open (S \<inter> T)" unfolding open_pinfreal_def
- proof (safe intro!: exI[of _ "S' \<inter> T'"] exI[of _ "max xS xT"])
- fix x assume *: "Real (max xS xT) < x" and "\<omega> \<in> S" "\<omega> \<in> T"
- from `\<omega> \<in> S`[THEN S'(4)] * show "x \<in> S"
- by (cases x, auto simp: max_def split: split_if_asm)
- from `\<omega> \<in> T`[THEN T'(4)] * show "x \<in> T"
- by (cases x, auto simp: max_def split: split_if_asm)
- next
- fix x assume x: "x \<notin> Real ` (S' \<inter> T' \<inter> {0..})"
- have *: "S' \<inter> T' \<inter> {0..} = (S' \<inter> {0..}) \<inter> (T' \<inter> {0..})" by auto
- assume "x \<in> T" "x \<in> S"
- with S'(2) T'(2) show "x = \<omega>"
- using x[unfolded *] inj_on_image_Int[OF inj_on_Real] by auto
- qed auto
-next
- fix K assume openK: "\<forall>S \<in> K. open (S:: pinfreal set)"
- hence "\<forall>S\<in>K. \<exists>T. open T \<and> Real ` (T \<inter> {0..}) = S - {\<omega>}" by (auto simp: open_pinfreal_def)
- from bchoice[OF this] guess T .. note T = this[rule_format]
-
- show "open (\<Union>K)" unfolding open_pinfreal_def
- proof (safe intro!: exI[of _ "\<Union>(T ` K)"])
- fix x S assume "0 \<le> x" "x \<in> T S" "S \<in> K"
- with T[OF `S \<in> K`] show "Real x \<in> \<Union>K" by auto
- next
- fix x S assume x: "x \<notin> Real ` (\<Union>T ` K \<inter> {0..})" "S \<in> K" "x \<in> S"
- hence "x \<notin> Real ` (T S \<inter> {0..})"
- by (auto simp: image_UN UN_simps[symmetric] simp del: UN_simps)
- thus "x = \<omega>" using T[OF `S \<in> K`] `x \<in> S` by auto
- next
- fix S assume "\<omega> \<in> S" "S \<in> K"
- from openK[rule_format, OF `S \<in> K`, THEN pinfreal_openE] guess S' x .
- from this(3, 4) `\<omega> \<in> S`
- show "\<exists>x\<ge>0. {Real x<..} \<subseteq> \<Union>K"
- by (auto intro!: exI[of _ x] bexI[OF _ `S \<in> K`])
- next
- from T[THEN conjunct1] show "open (\<Union>T`K)" by auto
- qed auto
-qed
-end
-
-lemma open_pinfreal_lessThan[simp]:
- "open {..< a :: pinfreal}"
-proof (cases a)
- case (preal x) thus ?thesis unfolding open_pinfreal_def
- proof (safe intro!: exI[of _ "{..< x}"])
- fix y assume "y < Real x"
- moreover assume "y \<notin> Real ` ({..<x} \<inter> {0..})"
- ultimately have "y \<noteq> Real (real y)" using preal by (cases y) auto
- thus "y = \<omega>" by (auto simp: Real_real split: split_if_asm)
- qed auto
-next
- case infinite thus ?thesis
- unfolding open_pinfreal_def by (auto intro!: exI[of _ UNIV])
-qed
-
-lemma open_pinfreal_greaterThan[simp]:
- "open {a :: pinfreal <..}"
-proof (cases a)
- case (preal x) thus ?thesis unfolding open_pinfreal_def
- proof (safe intro!: exI[of _ "{x <..}"])
- fix y assume "Real x < y"
- moreover assume "y \<notin> Real ` ({x<..} \<inter> {0..})"
- ultimately have "y \<noteq> Real (real y)" using preal by (cases y) auto
- thus "y = \<omega>" by (auto simp: Real_real split: split_if_asm)
- qed auto
-next
- case infinite thus ?thesis
- unfolding open_pinfreal_def by (auto intro!: exI[of _ "{}"])
-qed
-
-lemma pinfreal_open_greaterThanLessThan[simp]: "open {a::pinfreal <..< b}"
- unfolding greaterThanLessThan_def by auto
-
-lemma closed_pinfreal_atLeast[simp, intro]: "closed {a :: pinfreal ..}"
-proof -
- have "- {a ..} = {..< a}" by auto
- then show "closed {a ..}"
- unfolding closed_def using open_pinfreal_lessThan by auto
-qed
-
-lemma closed_pinfreal_atMost[simp, intro]: "closed {.. b :: pinfreal}"
-proof -
- have "- {.. b} = {b <..}" by auto
- then show "closed {.. b}"
- unfolding closed_def using open_pinfreal_greaterThan by auto
-qed
-
-lemma closed_pinfreal_atLeastAtMost[simp, intro]:
- shows "closed {a :: pinfreal .. b}"
- unfolding atLeastAtMost_def by auto
-
-lemma pinfreal_dense:
- fixes x y :: pinfreal assumes "x < y"
- shows "\<exists>z. x < z \<and> z < y"
-proof -
- from `x < y` obtain p where p: "x = Real p" "0 \<le> p" by (cases x) auto
- show ?thesis
- proof (cases y)
- case (preal r) with p `x < y` have "p < r" by auto
- with dense obtain z where "p < z" "z < r" by auto
- thus ?thesis using preal p by (auto intro!: exI[of _ "Real z"])
- next
- case infinite thus ?thesis using `x < y` p
- by (auto intro!: exI[of _ "Real p + 1"])
- qed
-qed
-
-instance pinfreal :: t2_space
-proof
- fix x y :: pinfreal assume "x \<noteq> y"
- let "?P x (y::pinfreal)" = "\<exists> U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {}"
-
- { fix x y :: pinfreal assume "x < y"
- from pinfreal_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 \<noteq> y`
- show "\<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {}"
- proof (cases rule: linorder_cases)
- assume "x = y" with `x \<noteq> 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
-
-definition (in complete_lattice) isoton :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> bool" (infix "\<up>" 50) where
- "A \<up> X \<longleftrightarrow> (\<forall>i. A i \<le> A (Suc i)) \<and> (SUP i. A i) = X"
-
-definition (in complete_lattice) antiton (infix "\<down>" 50) where
- "A \<down> X \<longleftrightarrow> (\<forall>i. A i \<ge> A (Suc i)) \<and> (INF i. A i) = X"
-
-lemma isotoneI[intro?]: "\<lbrakk> \<And>i. f i \<le> f (Suc i) ; (SUP i. f i) = F \<rbrakk> \<Longrightarrow> f \<up> F"
- unfolding isoton_def by auto
-
-lemma (in complete_lattice) isotonD[dest]:
- assumes "A \<up> X" shows "A i \<le> A (Suc i)" "(SUP i. A i) = X"
- using assms unfolding isoton_def by auto
-
-lemma isotonD'[dest]:
- assumes "(A::_=>_) \<up> X" shows "A i x \<le> A (Suc i) x" "(SUP i. A i) = X"
- using assms unfolding isoton_def le_fun_def by auto
-
-lemma isoton_mono_le:
- assumes "f \<up> x" "i \<le> j"
- shows "f i \<le> f j"
- using `f \<up> x`[THEN isotonD(1)] lift_Suc_mono_le[of f, OF _ `i \<le> j`] by auto
-
-lemma isoton_const:
- shows "(\<lambda> i. c) \<up> c"
-unfolding isoton_def by auto
-
-lemma isoton_cmult_right:
- assumes "f \<up> (x::pinfreal)"
- shows "(\<lambda>i. c * f i) \<up> (c * x)"
- using assms unfolding isoton_def pinfreal_SUP_cmult
- by (auto intro: pinfreal_mult_cancel)
-
-lemma isoton_cmult_left:
- "f \<up> (x::pinfreal) \<Longrightarrow> (\<lambda>i. f i * c) \<up> (x * c)"
- by (subst (1 2) mult_commute) (rule isoton_cmult_right)
-
-lemma isoton_add:
- assumes "f \<up> (x::pinfreal)" and "g \<up> y"
- shows "(\<lambda>i. f i + g i) \<up> (x + y)"
- using assms unfolding isoton_def
- by (auto intro: pinfreal_mult_cancel add_mono simp: SUPR_pinfreal_add)
-
-lemma isoton_fun_expand:
- "f \<up> x \<longleftrightarrow> (\<forall>i. (\<lambda>j. f j i) \<up> (x i))"
-proof -
- have "\<And>i. {y. \<exists>f'\<in>range f. y = f' i} = range (\<lambda>j. f j i)"
- by auto
- with assms show ?thesis
- by (auto simp add: isoton_def le_fun_def Sup_fun_def SUPR_def)
-qed
-
-lemma isoton_indicator:
- assumes "f \<up> g"
- shows "(\<lambda>i x. f i x * indicator A x) \<up> (\<lambda>x. g x * indicator A x :: pinfreal)"
- using assms unfolding isoton_fun_expand by (auto intro!: isoton_cmult_left)
-
-lemma isoton_setsum:
- fixes f :: "'a \<Rightarrow> nat \<Rightarrow> pinfreal"
- assumes "finite A" "A \<noteq> {}"
- assumes "\<And> x. x \<in> A \<Longrightarrow> f x \<up> y x"
- shows "(\<lambda> i. (\<Sum> x \<in> A. f x i)) \<up> (\<Sum> x \<in> A. y x)"
-using assms
-proof (induct A rule:finite_ne_induct)
- case singleton thus ?case by auto
-next
- case (insert a A) note asms = this
- hence *: "(\<lambda> i. \<Sum> x \<in> A. f x i) \<up> (\<Sum> x \<in> A. y x)" by auto
- have **: "(\<lambda> i. f a i) \<up> y a" using asms by simp
- have "(\<lambda> i. f a i + (\<Sum> x \<in> A. f x i)) \<up> (y a + (\<Sum> x \<in> A. y x))"
- using * ** isoton_add by auto
- thus "(\<lambda> i. \<Sum> x \<in> insert a A. f x i) \<up> (\<Sum> x \<in> insert a A. y x)"
- using asms by fastsimp
-qed
-
-lemma isoton_Sup:
- assumes "f \<up> u"
- shows "f i \<le> u"
- using le_SUPI[of i UNIV f] assms
- unfolding isoton_def by auto
-
-lemma isoton_mono:
- assumes iso: "x \<up> a" "y \<up> b" and *: "\<And>n. x n \<le> y (N n)"
- shows "a \<le> b"
-proof -
- from iso have "a = (SUP n. x n)" "b = (SUP n. y n)"
- unfolding isoton_def by auto
- with * show ?thesis by (auto intro!: SUP_mono)
-qed
-
-lemma pinfreal_le_mult_one_interval:
- fixes x y :: pinfreal
- assumes "\<And>z. \<lbrakk> 0 < z ; z < 1 \<rbrakk> \<Longrightarrow> z * x \<le> y"
- shows "x \<le> y"
-proof (cases x, cases y)
- assume "x = \<omega>"
- with assms[of "1 / 2"]
- show "x \<le> y" by simp
-next
- fix r p assume *: "y = Real p" "x = Real r" and **: "0 \<le> r" "0 \<le> p"
- have "r \<le> p"
- proof (rule field_le_mult_one_interval)
- fix z :: real assume "0 < z" and "z < 1"
- with assms[of "Real z"]
- show "z * r \<le> p" using ** * by (auto simp: zero_le_mult_iff)
- qed
- thus "x \<le> y" using ** * by simp
-qed simp
-
-lemma pinfreal_greater_0[intro]:
- fixes a :: pinfreal
- assumes "a \<noteq> 0"
- shows "a > 0"
-using assms apply (cases a) by auto
-
-lemma pinfreal_mult_strict_right_mono:
- assumes "a < b" and "0 < c" "c < \<omega>"
- shows "a * c < b * c"
- using assms
- by (cases a, cases b, cases c)
- (auto simp: zero_le_mult_iff pinfreal_less_\<omega>)
-
-lemma minus_pinfreal_eq2:
- fixes x y z :: pinfreal
- assumes "y \<le> x" and "y \<noteq> \<omega>" shows "z = x - y \<longleftrightarrow> z + y = x"
- using assms
- apply (subst eq_commute)
- apply (subst minus_pinfreal_eq)
- by (cases x, cases z, auto simp add: ac_simps not_less)
-
-lemma pinfreal_diff_eq_diff_imp_eq:
- assumes "a \<noteq> \<omega>" "b \<le> a" "c \<le> a"
- assumes "a - b = a - c"
- shows "b = c"
- using assms
- by (cases a, cases b, cases c) (auto split: split_if_asm)
-
-lemma pinfreal_inverse_eq_0: "inverse x = 0 \<longleftrightarrow> x = \<omega>"
- by (cases x) auto
-
-lemma pinfreal_mult_inverse:
- "\<lbrakk> x \<noteq> \<omega> ; x \<noteq> 0 \<rbrakk> \<Longrightarrow> x * inverse x = 1"
- by (cases x) auto
-
-lemma pinfreal_zero_less_diff_iff:
- fixes a b :: pinfreal shows "0 < a - b \<longleftrightarrow> b < a"
- apply (cases a, cases b)
- apply (auto simp: pinfreal_noteq_omega_Ex pinfreal_less_\<omega>)
- apply (cases b)
- by auto
-
-lemma pinfreal_less_Real_Ex:
- fixes a b :: pinfreal shows "x < Real r \<longleftrightarrow> (\<exists>p\<ge>0. p < r \<and> x = Real p)"
- by (cases x) auto
-
-lemma open_Real: assumes "open S" shows "open (Real ` ({0..} \<inter> S))"
- unfolding open_pinfreal_def apply(rule,rule,rule,rule assms) by auto
-
-lemma pinfreal_zero_le_diff:
- fixes a b :: pinfreal shows "a - b = 0 \<longleftrightarrow> a \<le> b"
- by (cases a, cases b, simp_all, cases b, auto)
-
-lemma lim_Real[simp]: assumes "\<forall>n. f n \<ge> 0" "m\<ge>0"
- shows "(\<lambda>n. Real (f n)) ----> Real m \<longleftrightarrow> (\<lambda>n. f n) ----> m" (is "?l = ?r")
-proof assume ?l show ?r unfolding Lim_sequentially
- proof safe fix e::real assume e:"e>0"
- note open_ball[of m e] note open_Real[OF this]
- note * = `?l`[unfolded tendsto_def,rule_format,OF this]
- have "eventually (\<lambda>x. Real (f x) \<in> Real ` ({0..} \<inter> ball m e)) sequentially"
- apply(rule *) unfolding image_iff using assms(2) e by auto
- thus "\<exists>N. \<forall>n\<ge>N. dist (f n) m < e" unfolding eventually_sequentially
- apply safe apply(rule_tac x=N in exI,safe) apply(erule_tac x=n in allE,safe)
- proof- fix n x assume "Real (f n) = Real x" "0 \<le> x"
- hence *:"f n = x" using assms(1) by auto
- assume "x \<in> ball m e" thus "dist (f n) m < e" unfolding *
- by (auto simp add:dist_commute)
- qed qed
-next assume ?r show ?l unfolding tendsto_def eventually_sequentially
- proof safe fix S assume S:"open S" "Real m \<in> S"
- guess T y using S(1) apply-apply(erule pinfreal_openE) . note T=this
- have "m\<in>real ` (S - {\<omega>})" unfolding image_iff
- apply(rule_tac x="Real m" in bexI) using assms(2) S(2) by auto
- hence "m \<in> T" unfolding T(2)[THEN sym] by auto
- from `?r`[unfolded tendsto_def eventually_sequentially,rule_format,OF T(1) this]
- guess N .. note N=this[rule_format]
- show "\<exists>N. \<forall>n\<ge>N. Real (f n) \<in> S" apply(rule_tac x=N in exI)
- proof safe fix n assume n:"N\<le>n"
- have "f n \<in> real ` (S - {\<omega>})" using N[OF n] assms unfolding T(2)[THEN sym]
- unfolding image_iff apply-apply(rule_tac x="Real (f n)" in bexI)
- unfolding real_Real by auto
- then guess x unfolding image_iff .. note x=this
- show "Real (f n) \<in> S" unfolding x apply(subst Real_real) using x by auto
- qed
- qed
-qed
-
-lemma pinfreal_INFI:
- fixes x :: pinfreal
- assumes "\<And>i. i \<in> A \<Longrightarrow> x \<le> f i"
- assumes "\<And>y. (\<And>i. i \<in> A \<Longrightarrow> y \<le> f i) \<Longrightarrow> y \<le> x"
- shows "(INF i:A. f i) = x"
- unfolding INFI_def Inf_pinfreal_def
- using assms by (auto intro!: Greatest_equality)
-
-lemma real_of_pinfreal_less:"x < y \<Longrightarrow> y\<noteq>\<omega> \<Longrightarrow> real x < real y"
-proof- case goal1
- have *:"y = Real (real y)" "x = Real (real x)" using goal1 Real_real by auto
- show ?case using goal1 apply- apply(subst(asm) *(1))apply(subst(asm) *(2))
- unfolding pinfreal_less by auto
-qed
-
-lemma not_less_omega[simp]:"\<not> x < \<omega> \<longleftrightarrow> x = \<omega>"
- by (metis antisym_conv3 pinfreal_less(3))
-
-lemma Real_real': assumes "x\<noteq>\<omega>" shows "Real (real x) = x"
-proof- have *:"(THE r. 0 \<le> r \<and> x = Real r) = real x"
- apply(rule the_equality) using assms unfolding Real_real by auto
- have "Real (THE r. 0 \<le> r \<and> x = Real r) = x" unfolding *
- using assms unfolding Real_real by auto
- thus ?thesis unfolding real_of_pinfreal_def of_pinfreal_def
- unfolding pinfreal_case_def using assms by auto
-qed
-
-lemma Real_less_plus_one:"Real x < Real (max (x + 1) 1)"
- unfolding pinfreal_less by auto
-
-lemma Lim_omega: "f ----> \<omega> \<longleftrightarrow> (\<forall>B. \<exists>N. \<forall>n\<ge>N. f n \<ge> Real B)" (is "?l = ?r")
-proof assume ?r show ?l apply(rule topological_tendstoI)
- unfolding eventually_sequentially
- proof- fix S assume "open S" "\<omega> \<in> S"
- from open_omega[OF this] guess B .. note B=this
- from `?r`[rule_format,of "(max B 0)+1"] guess N .. note N=this
- show "\<exists>N. \<forall>n\<ge>N. f n \<in> S" apply(rule_tac x=N in exI)
- proof safe case goal1
- have "Real B < Real ((max B 0) + 1)" by auto
- also have "... \<le> 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 {Real B<..}" "\<omega> \<in> {Real B<..}" by auto
- from topological_tendstoD[OF `?l` this,unfolded eventually_sequentially]
- guess N .. note N=this
- show "\<exists>N. \<forall>n\<ge>N. Real B \<le> f n" apply(rule_tac x=N in exI) using N by auto
- qed
-qed
-
-lemma Lim_bounded_omgea: assumes lim:"f ----> l" and "\<And>n. f n \<le> Real B" shows "l \<noteq> \<omega>"
-proof(rule ccontr,unfold not_not) let ?B = "max (B + 1) 1" assume as:"l=\<omega>"
- from lim[unfolded this Lim_omega,rule_format,of "?B"]
- guess N .. note N=this[rule_format,OF le_refl]
- hence "Real ?B \<le> Real B" using assms(2)[of N] by(rule order_trans)
- hence "Real ?B < Real ?B" using Real_less_plus_one[of B] by(rule le_less_trans)
- thus False by auto
-qed
-
-lemma incseq_le_pinfreal: assumes inc: "\<And>n m. n\<ge>m \<Longrightarrow> X n \<ge> X m"
- and lim: "X ----> (L::pinfreal)" shows "X n \<le> L"
-proof(cases "L = \<omega>")
- case False have "\<forall>n. X n \<noteq> \<omega>"
- proof(rule ccontr,unfold not_all not_not,safe)
- case goal1 hence "\<forall>n\<ge>x. X n = \<omega>" using inc[of x] by auto
- hence "X ----> \<omega>" unfolding tendsto_def eventually_sequentially
- apply safe apply(rule_tac x=x in exI) by auto
- note Lim_unique[OF trivial_limit_sequentially this lim]
- with False show False by auto
- qed note * =this[rule_format]
-
- have **:"\<forall>m n. m \<le> n \<longrightarrow> Real (real (X m)) \<le> Real (real (X n))"
- unfolding Real_real using * inc by auto
- have "real (X n) \<le> real L" apply-apply(rule incseq_le) defer
- apply(subst lim_Real[THEN sym]) apply(rule,rule,rule)
- unfolding Real_real'[OF *] Real_real'[OF False]
- unfolding incseq_def using ** lim by auto
- hence "Real (real (X n)) \<le> Real (real L)" by auto
- thus ?thesis unfolding Real_real using * False by auto
-qed auto
-
-lemma SUP_Lim_pinfreal: assumes "\<And>n m. n\<ge>m \<Longrightarrow> f n \<ge> f m" "f ----> l"
- shows "(SUP n. f n) = (l::pinfreal)" unfolding SUPR_def Sup_pinfreal_def
-proof (safe intro!: Least_equality)
- fix n::nat show "f n \<le> l" apply(rule incseq_le_pinfreal)
- using assms by auto
-next fix y assume y:"\<forall>x\<in>range f. x \<le> y" show "l \<le> y"
- proof(rule ccontr,cases "y=\<omega>",unfold not_le)
- case False assume as:"y < l"
- have l:"l \<noteq> \<omega>" apply(rule Lim_bounded_omgea[OF assms(2), of "real y"])
- using False y unfolding Real_real by auto
-
- have yl:"real y < real l" using as apply-
- apply(subst(asm) Real_real'[THEN sym,OF `y\<noteq>\<omega>`])
- apply(subst(asm) Real_real'[THEN sym,OF `l\<noteq>\<omega>`])
- unfolding pinfreal_less apply(subst(asm) if_P) by auto
- hence "y + (y - l) * Real (1 / 2) < l" apply-
- apply(subst Real_real'[THEN sym,OF `y\<noteq>\<omega>`]) apply(subst(2) Real_real'[THEN sym,OF `y\<noteq>\<omega>`])
- apply(subst Real_real'[THEN sym,OF `l\<noteq>\<omega>`]) apply(subst(2) Real_real'[THEN sym,OF `l\<noteq>\<omega>`]) by auto
- hence *:"l \<in> {y + (y - l) / 2<..}" by auto
- have "open {y + (y-l)/2 <..}" by auto
- note topological_tendstoD[OF assms(2) this *]
- from this[unfolded eventually_sequentially] guess N .. note this[rule_format, of N]
- hence "y + (y - l) * Real (1 / 2) < y" using y[rule_format,of "f N"] by auto
- hence "Real (real y) + (Real (real y) - Real (real l)) * Real (1 / 2) < Real (real y)"
- unfolding Real_real using `y\<noteq>\<omega>` `l\<noteq>\<omega>` by auto
- thus False using yl by auto
- qed auto
-qed
-
-lemma Real_max':"Real x = Real (max x 0)"
-proof(cases "x < 0") case True
- hence *:"max x 0 = 0" by auto
- show ?thesis unfolding * using True by auto
-qed auto
-
-lemma lim_pinfreal_increasing: assumes "\<forall>n m. n\<ge>m \<longrightarrow> f n \<ge> f m"
- obtains l where "f ----> (l::pinfreal)"
-proof(cases "\<exists>B. \<forall>n. f n < Real B")
- case False thus thesis apply- apply(rule that[of \<omega>]) unfolding Lim_omega not_ex not_all
- apply safe apply(erule_tac x=B in allE,safe) apply(rule_tac x=x in exI,safe)
- apply(rule order_trans[OF _ assms[rule_format]]) by auto
-next case True then guess B .. note B = this[rule_format]
- hence *:"\<And>n. f n < \<omega>" apply-apply(rule less_le_trans,assumption) by auto
- have *:"\<And>n. f n \<noteq> \<omega>" proof- case goal1 show ?case using *[of n] by auto qed
- have B':"\<And>n. real (f n) \<le> max 0 B" proof- case goal1 thus ?case
- using B[of n] apply-apply(subst(asm) Real_real'[THEN sym]) defer
- apply(subst(asm)(2) Real_max') unfolding pinfreal_less apply(subst(asm) if_P) using *[of n] by auto
- qed
- have "\<exists>l. (\<lambda>n. real (f n)) ----> l" apply(rule Topology_Euclidean_Space.bounded_increasing_convergent)
- proof safe show "bounded {real (f n) |n. True}"
- unfolding bounded_def apply(rule_tac x=0 in exI,rule_tac x="max 0 B" in exI)
- using B' unfolding dist_norm by auto
- fix n::nat have "Real (real (f n)) \<le> Real (real (f (Suc n)))"
- using assms[rule_format,of n "Suc n"] apply(subst Real_real)+
- using *[of n] *[of "Suc n"] by fastsimp
- thus "real (f n) \<le> real (f (Suc n))" by auto
- qed then guess l .. note l=this
- have "0 \<le> l" apply(rule LIMSEQ_le_const[OF l])
- by(rule_tac x=0 in exI,auto)
-
- thus ?thesis apply-apply(rule that[of "Real l"])
- using l apply-apply(subst(asm) lim_Real[THEN sym]) prefer 3
- unfolding Real_real using * by auto
-qed
-
-lemma setsum_neq_omega: assumes "finite s" "\<And>x. x \<in> s \<Longrightarrow> f x \<noteq> \<omega>"
- shows "setsum f s \<noteq> \<omega>" using assms
-proof induct case (insert x s)
- show ?case unfolding setsum.insert[OF insert(1-2)]
- using insert by auto
-qed auto
-
-
-lemma real_Real': "0 \<le> x \<Longrightarrow> real (Real x) = x"
- unfolding real_Real by auto
-
-lemma real_pinfreal_pos[intro]:
- assumes "x \<noteq> 0" "x \<noteq> \<omega>"
- shows "real x > 0"
- apply(subst real_Real'[THEN sym,of 0]) defer
- apply(rule real_of_pinfreal_less) using assms by auto
-
-lemma Lim_omega_gt: "f ----> \<omega> \<longleftrightarrow> (\<forall>B. \<exists>N. \<forall>n\<ge>N. f n > Real B)" (is "?l = ?r")
-proof assume ?l thus ?r unfolding Lim_omega apply safe
- apply(erule_tac x="max B 0 +1" in allE,safe)
- apply(rule_tac x=N in exI,safe) apply(erule_tac x=n in allE,safe)
- apply(rule_tac y="Real (max B 0 + 1)" in less_le_trans) by auto
-next assume ?r thus ?l unfolding Lim_omega apply safe
- apply(erule_tac x=B in allE,safe) apply(rule_tac x=N in exI,safe) by auto
-qed
-
-lemma pinfreal_minus_le_cancel:
- fixes a b c :: pinfreal
- assumes "b \<le> a"
- shows "c - a \<le> c - b"
- using assms by (cases a, cases b, cases c, simp, simp, simp, cases b, cases c, simp_all)
-
-lemma pinfreal_minus_\<omega>[simp]: "x - \<omega> = 0" by (cases x) simp_all
-
-lemma pinfreal_minus_mono[intro]: "a - x \<le> (a::pinfreal)"
-proof- have "a - x \<le> a - 0"
- apply(rule pinfreal_minus_le_cancel) by auto
- thus ?thesis by auto
-qed
-
-lemma pinfreal_minus_eq_\<omega>[simp]: "x - y = \<omega> \<longleftrightarrow> (x = \<omega> \<and> y \<noteq> \<omega>)"
- by (cases x, cases y) (auto, cases y, auto)
-
-lemma pinfreal_less_minus_iff:
- fixes a b c :: pinfreal
- shows "a < b - c \<longleftrightarrow> c + a < b"
- by (cases c, cases a, cases b, auto)
-
-lemma pinfreal_minus_less_iff:
- fixes a b c :: pinfreal shows "a - c < b \<longleftrightarrow> (0 < b \<and> (c \<noteq> \<omega> \<longrightarrow> a < b + c))"
- by (cases c, cases a, cases b, auto)
-
-lemma pinfreal_le_minus_iff:
- fixes a b c :: pinfreal
- shows "a \<le> c - b \<longleftrightarrow> ((c \<le> b \<longrightarrow> a = 0) \<and> (b < c \<longrightarrow> a + b \<le> c))"
- by (cases a, cases c, cases b, auto simp: pinfreal_noteq_omega_Ex)
-
-lemma pinfreal_minus_le_iff:
- fixes a b c :: pinfreal
- shows "a - c \<le> b \<longleftrightarrow> (c \<le> a \<longrightarrow> a \<le> b + c)"
- by (cases a, cases c, cases b, auto simp: pinfreal_noteq_omega_Ex)
-
-lemmas pinfreal_minus_order = pinfreal_minus_le_iff pinfreal_minus_less_iff pinfreal_le_minus_iff pinfreal_less_minus_iff
-
-lemma pinfreal_minus_strict_mono:
- assumes "a > 0" "x > 0" "a\<noteq>\<omega>"
- shows "a - x < (a::pinfreal)"
- using assms by(cases x, cases a, auto)
-
-lemma pinfreal_minus':
- "Real r - Real p = (if 0 \<le> r \<and> p \<le> r then if 0 \<le> p then Real (r - p) else Real r else 0)"
- by (auto simp: minus_pinfreal_eq not_less)
-
-lemma pinfreal_minus_plus:
- "x \<le> (a::pinfreal) \<Longrightarrow> a - x + x = a"
- by (cases a, cases x) auto
-
-lemma pinfreal_cancel_plus_minus: "b \<noteq> \<omega> \<Longrightarrow> a + b - b = a"
- by (cases a, cases b) auto
-
-lemma pinfreal_minus_le_cancel_right:
- fixes a b c :: pinfreal
- assumes "a \<le> b" "c \<le> a"
- shows "a - c \<le> b - c"
- using assms by (cases a, cases b, cases c, auto, cases c, auto)
-
-lemma real_of_pinfreal_setsum':
- assumes "\<forall>x \<in> S. f x \<noteq> \<omega>"
- shows "(\<Sum>x\<in>S. real (f x)) = real (setsum f S)"
-proof cases
- assume "finite S"
- from this assms show ?thesis
- by induct (simp_all add: real_of_pinfreal_add setsum_\<omega>)
-qed simp
-
-lemma Lim_omega_pos: "f ----> \<omega> \<longleftrightarrow> (\<forall>B>0. \<exists>N. \<forall>n\<ge>N. f n \<ge> Real B)" (is "?l = ?r")
- unfolding Lim_omega apply safe defer
- apply(erule_tac x="max 1 B" in allE) apply safe defer
- apply(rule_tac x=N in exI,safe) apply(erule_tac x=n in allE,safe)
- apply(rule_tac y="Real (max 1 B)" in order_trans) by auto
-
-lemma pinfreal_LimI_finite:
- assumes "x \<noteq> \<omega>" "\<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)
- fix S assume "open S" "x \<in> S"
- then obtain A where "open A" and A_eq: "Real ` (A \<inter> {0..}) = S - {\<omega>}" by (auto elim!: pinfreal_openE)
- then have "x \<in> Real ` (A \<inter> {0..})" using `x \<in> S` `x \<noteq> \<omega>` by auto
- then have "real x \<in> A" by auto
- then obtain r where "0 < r" and dist: "\<And>y. dist y (real x) < r \<Longrightarrow> y \<in> A"
- using `open A` unfolding open_real_def by auto
- then obtain n where
- upper: "\<And>N. n \<le> N \<Longrightarrow> u N < x + Real r" and
- lower: "\<And>N. n \<le> N \<Longrightarrow> x < u N + Real r" using assms(2)[of "Real r"] by auto
- show "\<exists>N. \<forall>n\<ge>N. u n \<in> S"
- proof (safe intro!: exI[of _ n])
- fix N assume "n \<le> N"
- from upper[OF this] `x \<noteq> \<omega>` `0 < r`
- have "u N \<noteq> \<omega>" by (force simp: pinfreal_noteq_omega_Ex)
- with `x \<noteq> \<omega>` `0 < r` lower[OF `n \<le> N`] upper[OF `n \<le> N`]
- have "dist (real (u N)) (real x) < r" "u N \<noteq> \<omega>"
- by (auto simp: pinfreal_noteq_omega_Ex dist_real_def abs_diff_less_iff field_simps)
- from dist[OF this(1)]
- have "u N \<in> Real ` (A \<inter> {0..})" using `u N \<noteq> \<omega>`
- by (auto intro!: image_eqI[of _ _ "real (u N)"] simp: pinfreal_noteq_omega_Ex Real_real)
- thus "u N \<in> S" using A_eq by simp
- qed
-qed
-
-lemma real_Real_max:"real (Real x) = max x 0"
- unfolding real_Real by auto
-
-lemma Sup_lim:
- assumes "\<forall>n. b n \<in> s" "b ----> (a::pinfreal)"
- shows "a \<le> Sup s"
-proof(rule ccontr,unfold not_le)
- assume as:"Sup s < a" hence om:"Sup s \<noteq> \<omega>" by auto
- have s:"s \<noteq> {}" using assms by auto
- { presume *:"\<forall>n. b n < a \<Longrightarrow> False"
- show False apply(cases,rule *,assumption,unfold not_all not_less)
- proof- case goal1 then guess n .. note n=this
- thus False using complete_lattice_class.Sup_upper[OF assms(1)[rule_format,of n]]
- using as by auto
- qed
- } assume b:"\<forall>n. b n < a"
- show False
- proof(cases "a = \<omega>")
- case False have *:"a - Sup s > 0"
- using False as by(auto simp: pinfreal_zero_le_diff)
- have "(a - Sup s) / 2 \<le> a / 2" unfolding divide_pinfreal_def
- apply(rule mult_right_mono) by auto
- also have "... = Real (real (a / 2))" apply(rule Real_real'[THEN sym])
- using False by auto
- also have "... < Real (real a)" unfolding pinfreal_less using as False
- by(auto simp add: real_of_pinfreal_mult[THEN sym])
- also have "... = a" apply(rule Real_real') using False by auto
- finally have asup:"a > (a - Sup s) / 2" .
- have "\<exists>n. a - b n < (a - Sup s) / 2"
- proof(rule ccontr,unfold not_ex not_less)
- case goal1
- have "(a - Sup s) * Real (1 / 2) > 0"
- using * by auto
- hence "a - (a - Sup s) * Real (1 / 2) < a"
- apply-apply(rule pinfreal_minus_strict_mono)
- using False * by auto
- hence *:"a \<in> {a - (a - Sup s) / 2<..}"using asup by auto
- note topological_tendstoD[OF assms(2) open_pinfreal_greaterThan,OF *]
- from this[unfolded eventually_sequentially] guess n ..
- note n = this[rule_format,of n]
- have "b n + (a - Sup s) / 2 \<le> a"
- using add_right_mono[OF goal1[rule_format,of n],of "b n"]
- unfolding pinfreal_minus_plus[OF less_imp_le[OF b[rule_format]]]
- by(auto simp: add_commute)
- hence "b n \<le> a - (a - Sup s) / 2" unfolding pinfreal_le_minus_iff
- using asup by auto
- hence "b n \<notin> {a - (a - Sup s) / 2<..}" by auto
- thus False using n by auto
- qed
- then guess n .. note n = this
- have "Sup s < a - (a - Sup s) / 2"
- using False as om by (cases a) (auto simp: pinfreal_noteq_omega_Ex field_simps)
- also have "... \<le> b n"
- proof- note add_right_mono[OF less_imp_le[OF n],of "b n"]
- note this[unfolded pinfreal_minus_plus[OF less_imp_le[OF b[rule_format]]]]
- hence "a - (a - Sup s) / 2 \<le> (a - Sup s) / 2 + b n - (a - Sup s) / 2"
- apply(rule pinfreal_minus_le_cancel_right) using asup by auto
- also have "... = b n + (a - Sup s) / 2 - (a - Sup s) / 2"
- by(auto simp add: add_commute)
- also have "... = b n" apply(subst pinfreal_cancel_plus_minus)
- proof(rule ccontr,unfold not_not) case goal1
- show ?case using asup unfolding goal1 by auto
- qed auto
- finally show ?thesis .
- qed
- finally show False
- using complete_lattice_class.Sup_upper[OF assms(1)[rule_format,of n]] by auto
- next case True
- from assms(2)[unfolded True Lim_omega_gt,rule_format,of "real (Sup s)"]
- guess N .. note N = this[rule_format,of N]
- thus False using complete_lattice_class.Sup_upper[OF assms(1)[rule_format,of N]]
- unfolding Real_real using om by auto
- qed qed
-
-lemma Sup_mono_lim:
- assumes "\<forall>a\<in>A. \<exists>b. \<forall>n. b n \<in> B \<and> b ----> (a::pinfreal)"
- shows "Sup A \<le> Sup B"
- unfolding Sup_le_iff apply(rule) apply(drule assms[rule_format]) apply safe
- apply(rule_tac b=b in Sup_lim) by auto
-
-lemma pinfreal_less_add:
- assumes "x \<noteq> \<omega>" "a < b"
- shows "x + a < x + b"
- using assms by (cases a, cases b, cases x) auto
-
-lemma SUPR_lim:
- assumes "\<forall>n. b n \<in> B" "(\<lambda>n. f (b n)) ----> (f a::pinfreal)"
- shows "f a \<le> SUPR B f"
- unfolding SUPR_def apply(rule Sup_lim[of "\<lambda>n. f (b n)"])
- using assms by auto
-
-lemma SUP_\<omega>_imp:
- assumes "(SUP i. f i) = \<omega>"
- shows "\<exists>i. Real x < f i"
-proof (rule ccontr)
- assume "\<not> ?thesis" hence "\<And>i. f i \<le> Real x" by (simp add: not_less)
- hence "(SUP i. f i) \<le> Real x" unfolding SUP_le_iff by auto
- with assms show False by auto
-qed
-
-lemma SUPR_mono_lim:
- assumes "\<forall>a\<in>A. \<exists>b. \<forall>n. b n \<in> B \<and> (\<lambda>n. f (b n)) ----> (f a::pinfreal)"
- shows "SUPR A f \<le> SUPR B f"
- unfolding SUPR_def apply(rule Sup_mono_lim)
- apply safe apply(drule assms[rule_format],safe)
- apply(rule_tac x="\<lambda>n. f (b n)" in exI) by auto
-
-lemma real_0_imp_eq_0:
- assumes "x \<noteq> \<omega>" "real x = 0"
- shows "x = 0"
-using assms by (cases x) auto
-
-lemma SUPR_mono:
- assumes "\<forall>a\<in>A. \<exists>b\<in>B. f b \<ge> f a"
- shows "SUPR A f \<le> SUPR B f"
- unfolding SUPR_def apply(rule Sup_mono)
- using assms by auto
-
-lemma less_add_Real:
- fixes x :: real
- fixes a b :: pinfreal
- assumes "x \<ge> 0" "a < b"
- shows "a + Real x < b + Real x"
-using assms by (cases a, cases b) auto
-
-lemma le_add_Real:
- fixes x :: real
- fixes a b :: pinfreal
- assumes "x \<ge> 0" "a \<le> b"
- shows "a + Real x \<le> b + Real x"
-using assms by (cases a, cases b) auto
-
-lemma le_imp_less_pinfreal:
- fixes x :: pinfreal
- assumes "x > 0" "a + x \<le> b" "a \<noteq> \<omega>"
- shows "a < b"
-using assms by (cases x, cases a, cases b) auto
-
-lemma pinfreal_INF_minus:
- fixes f :: "nat \<Rightarrow> pinfreal"
- assumes "c \<noteq> \<omega>"
- shows "(INF i. c - f i) = c - (SUP i. f i)"
-proof (cases "SUP i. f i")
- case infinite
- from `c \<noteq> \<omega>` obtain x where [simp]: "c = Real x" by (cases c) auto
- from SUP_\<omega>_imp[OF infinite] obtain i where "Real x < f i" by auto
- have "(INF i. c - f i) \<le> c - f i"
- by (auto intro!: complete_lattice_class.INF_leI)
- also have "\<dots> = 0" using `Real x < f i` by (auto simp: minus_pinfreal_eq)
- finally show ?thesis using infinite by auto
-next
- case (preal r)
- from `c \<noteq> \<omega>` obtain x where c: "c = Real x" by (cases c) auto
-
- show ?thesis unfolding c
- proof (rule pinfreal_INFI)
- fix i have "f i \<le> (SUP i. f i)" by (rule le_SUPI) simp
- thus "Real x - (SUP i. f i) \<le> Real x - f i" by (rule pinfreal_minus_le_cancel)
- next
- fix y assume *: "\<And>i. i \<in> UNIV \<Longrightarrow> y \<le> Real x - f i"
- from this[of 0] obtain p where p: "y = Real p" "0 \<le> p"
- by (cases "f 0", cases y, auto split: split_if_asm)
- hence "\<And>i. Real p \<le> Real x - f i" using * by auto
- hence *: "\<And>i. Real x \<le> f i \<Longrightarrow> Real p = 0"
- "\<And>i. f i < Real x \<Longrightarrow> Real p + f i \<le> Real x"
- unfolding pinfreal_le_minus_iff by auto
- show "y \<le> Real x - (SUP i. f i)" unfolding p pinfreal_le_minus_iff
- proof safe
- assume x_less: "Real x \<le> (SUP i. f i)"
- show "Real p = 0"
- proof (rule ccontr)
- assume "Real p \<noteq> 0"
- hence "0 < Real p" by auto
- from Sup_close[OF this, of "range f"]
- obtain i where e: "(SUP i. f i) < f i + Real p"
- using preal unfolding SUPR_def by auto
- hence "Real x \<le> f i + Real p" using x_less by auto
- show False
- proof cases
- assume "\<forall>i. f i < Real x"
- hence "Real p + f i \<le> Real x" using * by auto
- hence "f i + Real p \<le> (SUP i. f i)" using x_less by (auto simp: field_simps)
- thus False using e by auto
- next
- assume "\<not> (\<forall>i. f i < Real x)"
- then obtain i where "Real x \<le> f i" by (auto simp: not_less)
- from *(1)[OF this] show False using `Real p \<noteq> 0` by auto
- qed
- qed
- next
- have "\<And>i. f i \<le> (SUP i. f i)" by (rule complete_lattice_class.le_SUPI) auto
- also assume "(SUP i. f i) < Real x"
- finally have "\<And>i. f i < Real x" by auto
- hence *: "\<And>i. Real p + f i \<le> Real x" using * by auto
- have "Real p \<le> Real x" using *[of 0] by (cases "f 0") (auto split: split_if_asm)
-
- have SUP_eq: "(SUP i. f i) \<le> Real x - Real p"
- proof (rule SUP_leI)
- fix i show "f i \<le> Real x - Real p" unfolding pinfreal_le_minus_iff
- proof safe
- assume "Real x \<le> Real p"
- with *[of i] show "f i = 0"
- by (cases "f i") (auto split: split_if_asm)
- next
- assume "Real p < Real x"
- show "f i + Real p \<le> Real x" using * by (auto simp: field_simps)
- qed
- qed
-
- show "Real p + (SUP i. f i) \<le> Real x"
- proof cases
- assume "Real x \<le> Real p"
- with `Real p \<le> Real x` have [simp]: "Real p = Real x" by (rule antisym)
- { fix i have "f i = 0" using *[of i] by (cases "f i") (auto split: split_if_asm) }
- hence "(SUP i. f i) \<le> 0" by (auto intro!: SUP_leI)
- thus ?thesis by simp
- next
- assume "\<not> Real x \<le> Real p" hence "Real p < Real x" unfolding not_le .
- with SUP_eq show ?thesis unfolding pinfreal_le_minus_iff by (auto simp: field_simps)
- qed
- qed
- qed
-qed
-
-lemma pinfreal_SUP_minus:
- fixes f :: "nat \<Rightarrow> pinfreal"
- shows "(SUP i. c - f i) = c - (INF i. f i)"
-proof (rule pinfreal_SUPI)
- fix i have "(INF i. f i) \<le> f i" by (rule INF_leI) simp
- thus "c - f i \<le> c - (INF i. f i)" by (rule pinfreal_minus_le_cancel)
-next
- fix y assume *: "\<And>i. i \<in> UNIV \<Longrightarrow> c - f i \<le> y"
- show "c - (INF i. f i) \<le> y"
- proof (cases y)
- case (preal p)
-
- show ?thesis unfolding pinfreal_minus_le_iff preal
- proof safe
- assume INF_le_x: "(INF i. f i) \<le> c"
- from * have *: "\<And>i. f i \<le> c \<Longrightarrow> c \<le> Real p + f i"
- unfolding pinfreal_minus_le_iff preal by auto
-
- have INF_eq: "c - Real p \<le> (INF i. f i)"
- proof (rule le_INFI)
- fix i show "c - Real p \<le> f i" unfolding pinfreal_minus_le_iff
- proof safe
- assume "Real p \<le> c"
- show "c \<le> f i + Real p"
- proof cases
- assume "f i \<le> c" from *[OF this]
- show ?thesis by (simp add: field_simps)
- next
- assume "\<not> f i \<le> c"
- hence "c \<le> f i" by auto
- also have "\<dots> \<le> f i + Real p" by auto
- finally show ?thesis .
- qed
- qed
- qed
-
- show "c \<le> Real p + (INF i. f i)"
- proof cases
- assume "Real p \<le> c"
- with INF_eq show ?thesis unfolding pinfreal_minus_le_iff by (auto simp: field_simps)
- next
- assume "\<not> Real p \<le> c"
- hence "c \<le> Real p" by auto
- also have "Real p \<le> Real p + (INF i. f i)" by auto
- finally show ?thesis .
- qed
- qed
- qed simp
-qed
-
-lemma pinfreal_le_minus_imp_0:
- fixes a b :: pinfreal
- shows "a \<le> a - b \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> a \<noteq> \<omega> \<Longrightarrow> b = 0"
- by (cases a, cases b, auto split: split_if_asm)
-
-lemma lim_INF_eq_lim_SUP:
- fixes X :: "nat \<Rightarrow> real"
- assumes "\<And>i. 0 \<le> X i" and "0 \<le> x"
- and lim_INF: "(SUP n. INF m. Real (X (n + m))) = Real x" (is "(SUP n. ?INF n) = _")
- and lim_SUP: "(INF n. SUP m. Real (X (n + m))) = Real x" (is "(INF n. ?SUP n) = _")
- shows "X ----> x"
-proof (rule LIMSEQ_I)
- fix r :: real assume "0 < r"
- hence "0 \<le> r" by auto
- from Sup_close[of "Real r" "range ?INF"]
- obtain n where inf: "Real x < ?INF n + Real r"
- unfolding SUPR_def lim_INF[unfolded SUPR_def] using `0 < r` by auto
-
- from Inf_close[of "range ?SUP" "Real r"]
- obtain n' where sup: "?SUP n' < Real x + Real r"
- unfolding INFI_def lim_SUP[unfolded INFI_def] using `0 < r` by auto
-
- show "\<exists>N. \<forall>n\<ge>N. norm (X n - x) < r"
- proof (safe intro!: exI[of _ "max n n'"])
- fix m assume "max n n' \<le> m" hence "n \<le> m" "n' \<le> m" by auto
-
- note inf
- also have "?INF n + Real r \<le> Real (X (n + (m - n))) + Real r"
- by (rule le_add_Real, auto simp: `0 \<le> r` intro: INF_leI)
- finally have up: "x < X m + r"
- using `0 \<le> X m` `0 \<le> x` `0 \<le> r` `n \<le> m` by auto
-
- have "Real (X (n' + (m - n'))) \<le> ?SUP n'"
- by (auto simp: `0 \<le> r` intro: le_SUPI)
- also note sup
- finally have down: "X m < x + r"
- using `0 \<le> X m` `0 \<le> x` `0 \<le> r` `n' \<le> m` by auto
-
- show "norm (X m - x) < r" using up down by auto
- qed
-qed
-
-lemma Sup_countable_SUPR:
- assumes "Sup A \<noteq> \<omega>" "A \<noteq> {}"
- shows "\<exists> f::nat \<Rightarrow> pinfreal. range f \<subseteq> A \<and> Sup A = SUPR UNIV f"
-proof -
- have "\<And>n. 0 < 1 / (of_nat n :: pinfreal)" by auto
- from Sup_close[OF this assms]
- have "\<forall>n. \<exists>x. x \<in> A \<and> Sup A < x + 1 / of_nat n" by blast
- from choice[OF this] obtain f where "range f \<subseteq> A" and
- epsilon: "\<And>n. Sup A < f n + 1 / of_nat n" by blast
- have "SUPR UNIV f = Sup A"
- proof (rule pinfreal_SUPI)
- fix i show "f i \<le> Sup A" using `range f \<subseteq> A`
- 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 pinfreal_le_epsilon)
- fix e :: pinfreal assume "0 < e"
- show "Sup A \<le> y + e"
- proof (cases e)
- case (preal r)
- hence "0 < r" using `0 < e` by auto
- then obtain n where *: "inverse (of_nat n) < r" "0 < n"
- using ex_inverse_of_nat_less by auto
- have "Sup A \<le> f n + 1 / of_nat n" using epsilon[of n] by auto
- also have "1 / of_nat n \<le> e" using preal * by (auto simp: real_eq_of_nat)
- with bound have "f n + 1 / of_nat n \<le> y + e" by (rule add_mono) simp
- finally show "Sup A \<le> y + e" .
- qed simp
- qed
- qed
- with `range f \<subseteq> A` show ?thesis by (auto intro!: exI[of _ f])
-qed
-
-lemma SUPR_countable_SUPR:
- assumes "SUPR A g \<noteq> \<omega>" "A \<noteq> {}"
- shows "\<exists> f::nat \<Rightarrow> pinfreal. range f \<subseteq> g`A \<and> SUPR A g = SUPR UNIV f"
-proof -
- have "Sup (g`A) \<noteq> \<omega>" "g`A \<noteq> {}" using assms unfolding SUPR_def by auto
- from Sup_countable_SUPR[OF this]
- show ?thesis unfolding SUPR_def .
-qed
-
-lemma pinfreal_setsum_subtractf:
- assumes "\<And>i. i\<in>A \<Longrightarrow> g i \<le> f i" and "\<And>i. i\<in>A \<Longrightarrow> f i \<noteq> \<omega>"
- shows "(\<Sum>i\<in>A. f i - g i) = (\<Sum>i\<in>A. f i) - (\<Sum>i\<in>A. g i)"
-proof cases
- assume "finite A" from this assms show ?thesis
- proof induct
- case (insert x A)
- hence hyp: "(\<Sum>i\<in>A. f i - g i) = (\<Sum>i\<in>A. f i) - (\<Sum>i\<in>A. g i)"
- by auto
- { fix i assume *: "i \<in> insert x A"
- hence "g i \<le> f i" using insert by simp
- also have "f i < \<omega>" using * insert by (simp add: pinfreal_less_\<omega>)
- finally have "g i \<noteq> \<omega>" by (simp add: pinfreal_less_\<omega>) }
- hence "setsum g A \<noteq> \<omega>" "g x \<noteq> \<omega>" by (auto simp: setsum_\<omega>)
- moreover have "setsum f A \<noteq> \<omega>" "f x \<noteq> \<omega>" using insert by (auto simp: setsum_\<omega>)
- moreover have "g x \<le> f x" using insert by auto
- moreover have "(\<Sum>i\<in>A. g i) \<le> (\<Sum>i\<in>A. f i)" using insert by (auto intro!: setsum_mono)
- ultimately show ?case using `finite A` `x \<notin> A` hyp
- by (auto simp: pinfreal_noteq_omega_Ex)
- qed simp
-qed simp
-
-lemma real_of_pinfreal_diff:
- "y \<le> x \<Longrightarrow> x \<noteq> \<omega> \<Longrightarrow> real x - real y = real (x - y)"
- by (cases x, cases y) auto
-
-lemma psuminf_minus:
- assumes ord: "\<And>i. g i \<le> f i" and fin: "psuminf g \<noteq> \<omega>" "psuminf f \<noteq> \<omega>"
- shows "(\<Sum>\<^isub>\<infinity> i. f i - g i) = psuminf f - psuminf g"
-proof -
- have [simp]: "\<And>i. f i \<noteq> \<omega>" using fin by (auto intro: psuminf_\<omega>)
- from fin have "(\<lambda>x. real (f x)) sums real (\<Sum>\<^isub>\<infinity>x. f x)"
- and "(\<lambda>x. real (g x)) sums real (\<Sum>\<^isub>\<infinity>x. g x)"
- by (auto intro: psuminf_imp_suminf)
- from sums_diff[OF this]
- have "(\<lambda>n. real (f n - g n)) sums (real ((\<Sum>\<^isub>\<infinity>x. f x) - (\<Sum>\<^isub>\<infinity>x. g x)))" using fin ord
- by (subst (asm) (1 2) real_of_pinfreal_diff) (auto simp: psuminf_\<omega> psuminf_le)
- hence "(\<Sum>\<^isub>\<infinity> i. Real (real (f i - g i))) = Real (real ((\<Sum>\<^isub>\<infinity>x. f x) - (\<Sum>\<^isub>\<infinity>x. g x)))"
- by (rule suminf_imp_psuminf) simp
- thus ?thesis using fin by (simp add: Real_real psuminf_\<omega>)
-qed
-
-lemma INF_eq_LIMSEQ:
- assumes "mono (\<lambda>i. - f i)" and "\<And>n. 0 \<le> f n" and "0 \<le> x"
- shows "(INF n. Real (f n)) = Real x \<longleftrightarrow> f ----> x"
-proof
- assume x: "(INF n. Real (f n)) = Real x"
- { fix n
- have "Real x \<le> Real (f n)" using x[symmetric] by (auto intro: INF_leI)
- hence "x \<le> f n" using assms by simp
- hence "\<bar>f n - x\<bar> = f n - x" by auto }
- note abs_eq = this
- show "f ----> x"
- proof (rule LIMSEQ_I)
- fix r :: real assume "0 < r"
- show "\<exists>no. \<forall>n\<ge>no. norm (f n - x) < r"
- proof (rule ccontr)
- assume *: "\<not> ?thesis"
- { fix N
- from * obtain n where *: "N \<le> n" "r \<le> f n - x"
- using abs_eq by (auto simp: not_less)
- hence "x + r \<le> f n" by auto
- also have "f n \<le> f N" using `mono (\<lambda>i. - f i)` * by (auto dest: monoD)
- finally have "Real (x + r) \<le> Real (f N)" using `0 \<le> f N` by auto }
- hence "Real x < Real (x + r)"
- and "Real (x + r) \<le> (INF n. Real (f n))" using `0 < r` `0 \<le> x` by (auto intro: le_INFI)
- hence "Real x < (INF n. Real (f n))" by (rule less_le_trans)
- thus False using x by auto
- qed
- qed
-next
- assume "f ----> x"
- show "(INF n. Real (f n)) = Real x"
- proof (rule pinfreal_INFI)
- fix n
- from decseq_le[OF _ `f ----> x`] assms
- show "Real x \<le> Real (f n)" unfolding decseq_eq_incseq incseq_mono by auto
- next
- fix y assume *: "\<And>n. n\<in>UNIV \<Longrightarrow> y \<le> Real (f n)"
- thus "y \<le> Real x"
- proof (cases y)
- case (preal r)
- with * have "\<exists>N. \<forall>n\<ge>N. r \<le> f n" using assms by fastsimp
- from LIMSEQ_le_const[OF `f ----> x` this]
- show "y \<le> Real x" using `0 \<le> x` preal by auto
- qed simp
- qed
-qed
-
-lemma INFI_bound:
- assumes "\<forall>N. x \<le> f N"
- shows "x \<le> (INF n. f n)"
- using assms by (simp add: INFI_def le_Inf_iff)
-
-lemma LIMSEQ_imp_lim_INF:
- assumes pos: "\<And>i. 0 \<le> X i" and lim: "X ----> x"
- shows "(SUP n. INF m. Real (X (n + m))) = Real x"
-proof -
- have "0 \<le> x" using assms by (auto intro!: LIMSEQ_le_const)
-
- have "\<And>n. (INF m. Real (X (n + m))) \<le> Real (X (n + 0))" by (rule INF_leI) simp
- also have "\<And>n. Real (X (n + 0)) < \<omega>" by simp
- finally have "\<forall>n. \<exists>r\<ge>0. (INF m. Real (X (n + m))) = Real r"
- by (auto simp: pinfreal_less_\<omega> pinfreal_noteq_omega_Ex)
- from choice[OF this] obtain r where r: "\<And>n. (INF m. Real (X (n + m))) = Real (r n)" "\<And>n. 0 \<le> r n"
- by auto
-
- show ?thesis unfolding r
- proof (subst SUP_eq_LIMSEQ)
- show "mono r" unfolding mono_def
- proof safe
- fix x y :: nat assume "x \<le> y"
- have "Real (r x) \<le> Real (r y)" unfolding r(1)[symmetric] using pos
- proof (safe intro!: INF_mono bexI)
- fix m have "x + (m + y - x) = y + m"
- using `x \<le> y` by auto
- thus "Real (X (x + (m + y - x))) \<le> Real (X (y + m))" by simp
- qed simp
- thus "r x \<le> r y" using r by auto
- qed
- show "\<And>n. 0 \<le> r n" by fact
- show "0 \<le> x" by fact
- show "r ----> x"
- proof (rule LIMSEQ_I)
- fix e :: real assume "0 < e"
- hence "0 < e/2" by auto
- from LIMSEQ_D[OF lim this] obtain N where *: "\<And>n. N \<le> n \<Longrightarrow> \<bar>X n - x\<bar> < e/2"
- by auto
- show "\<exists>N. \<forall>n\<ge>N. norm (r n - x) < e"
- proof (safe intro!: exI[of _ N])
- fix n assume "N \<le> n"
- show "norm (r n - x) < e"
- proof cases
- assume "r n < x"
- have "x - r n \<le> e/2"
- proof cases
- assume e: "e/2 \<le> x"
- have "Real (x - e/2) \<le> Real (r n)" unfolding r(1)[symmetric]
- proof (rule le_INFI)
- fix m show "Real (x - e / 2) \<le> Real (X (n + m))"
- using *[of "n + m"] `N \<le> n`
- using pos by (auto simp: field_simps abs_real_def split: split_if_asm)
- qed
- with e show ?thesis using pos `0 \<le> x` r(2) by auto
- next
- assume "\<not> e/2 \<le> x" hence "x - e/2 < 0" by auto
- with `0 \<le> r n` show ?thesis by auto
- qed
- with `r n < x` show ?thesis by simp
- next
- assume e: "\<not> r n < x"
- have "Real (r n) \<le> Real (X (n + 0))" unfolding r(1)[symmetric]
- by (rule INF_leI) simp
- hence "r n - x \<le> X n - x" using r pos by auto
- also have "\<dots> < e/2" using *[OF `N \<le> n`] by (auto simp: field_simps abs_real_def split: split_if_asm)
- finally have "r n - x < e" using `0 < e` by auto
- with e show ?thesis by auto
- qed
- qed
- qed
- qed
-qed
-
-lemma real_of_pinfreal_strict_mono_iff:
- "real a < real b \<longleftrightarrow> (b \<noteq> \<omega> \<and> ((a = \<omega> \<and> 0 < b) \<or> (a < b)))"
-proof (cases a)
- case infinite thus ?thesis by (cases b) auto
-next
- case preal thus ?thesis by (cases b) auto
-qed
-
-lemma real_of_pinfreal_mono_iff:
- "real a \<le> real b \<longleftrightarrow> (a = \<omega> \<or> (b \<noteq> \<omega> \<and> a \<le> b) \<or> (b = \<omega> \<and> a = 0))"
-proof (cases a)
- case infinite thus ?thesis by (cases b) auto
-next
- case preal thus ?thesis by (cases b) auto
-qed
-
-lemma ex_pinfreal_inverse_of_nat_Suc_less:
- fixes e :: pinfreal assumes "0 < e" shows "\<exists>n. inverse (of_nat (Suc n)) < e"
-proof (cases e)
- case (preal r)
- with `0 < e` ex_inverse_of_nat_Suc_less[of r]
- obtain n where "inverse (of_nat (Suc n)) < r" by auto
- with preal show ?thesis
- by (auto simp: real_eq_of_nat[symmetric])
-qed auto
-
-lemma Lim_eq_Sup_mono:
- fixes u :: "nat \<Rightarrow> pinfreal" assumes "mono u"
- shows "u ----> (SUP i. u i)"
-proof -
- from lim_pinfreal_increasing[of u] `mono u`
- obtain l where l: "u ----> l" unfolding mono_def by auto
- from SUP_Lim_pinfreal[OF _ this] `mono u`
- have "(SUP i. u i) = l" unfolding mono_def by auto
- with l show ?thesis by simp
-qed
-
-lemma isotone_Lim:
- fixes x :: pinfreal assumes "u \<up> x"
- shows "u ----> x" (is ?lim) and "mono u" (is ?mono)
-proof -
- show ?mono using assms unfolding mono_iff_le_Suc isoton_def by auto
- from Lim_eq_Sup_mono[OF this] `u \<up> x`
- show ?lim unfolding isoton_def by simp
-qed
-
-lemma isoton_iff_Lim_mono:
- fixes u :: "nat \<Rightarrow> pinfreal"
- shows "u \<up> x \<longleftrightarrow> (mono u \<and> u ----> x)"
-proof safe
- assume "mono u" and x: "u ----> x"
- with SUP_Lim_pinfreal[OF _ x]
- show "u \<up> x" unfolding isoton_def
- using `mono u`[unfolded mono_def]
- using `mono u`[unfolded mono_iff_le_Suc]
- by auto
-qed (auto dest: isotone_Lim)
-
-lemma pinfreal_inverse_inverse[simp]:
- fixes x :: pinfreal
- shows "inverse (inverse x) = x"
- by (cases x) auto
-
-lemma atLeastAtMost_omega_eq_atLeast:
- shows "{a .. \<omega>} = {a ..}"
-by auto
-
-lemma atLeast0AtMost_eq_atMost: "{0 :: pinfreal .. a} = {.. a}" by auto
-
-lemma greaterThan_omega_Empty: "{\<omega> <..} = {}" by auto
-
-lemma lessThan_0_Empty: "{..< 0 :: pinfreal} = {}" by auto
-
-lemma real_of_pinfreal_inverse[simp]:
- fixes X :: pinfreal
- shows "real (inverse X) = 1 / real X"
- by (cases X) (auto simp: inverse_eq_divide)
-
-lemma real_of_pinfreal_le_0[simp]: "real (X :: pinfreal) \<le> 0 \<longleftrightarrow> (X = 0 \<or> X = \<omega>)"
- by (cases X) auto
-
-lemma real_of_pinfreal_less_0[simp]: "\<not> (real (X :: pinfreal) < 0)"
- by (cases X) auto
-
-lemma abs_real_of_pinfreal[simp]: "\<bar>real (X :: pinfreal)\<bar> = real X"
- by simp
-
-lemma zero_less_real_of_pinfreal: "0 < real (X :: pinfreal) \<longleftrightarrow> X \<noteq> 0 \<and> X \<noteq> \<omega>"
- by (cases X) auto
-
-end
--- a/src/HOL/Probability/Probability_Space.thy Mon Dec 06 19:18:02 2010 +0100
+++ b/src/HOL/Probability/Probability_Space.thy Fri Dec 03 15:25:14 2010 +0100
@@ -2,24 +2,24 @@
imports Lebesgue_Integration Radon_Nikodym Product_Measure
begin
-lemma real_of_pinfreal_inverse[simp]:
- fixes X :: pinfreal
+lemma real_of_pextreal_inverse[simp]:
+ fixes X :: pextreal
shows "real (inverse X) = 1 / real X"
by (cases X) (auto simp: inverse_eq_divide)
-lemma real_of_pinfreal_le_0[simp]: "real (X :: pinfreal) \<le> 0 \<longleftrightarrow> (X = 0 \<or> X = \<omega>)"
+lemma real_of_pextreal_le_0[simp]: "real (X :: pextreal) \<le> 0 \<longleftrightarrow> (X = 0 \<or> X = \<omega>)"
by (cases X) auto
-lemma real_of_pinfreal_less_0[simp]: "\<not> (real (X :: pinfreal) < 0)"
+lemma real_of_pextreal_less_0[simp]: "\<not> (real (X :: pextreal) < 0)"
by (cases X) auto
locale prob_space = measure_space +
assumes measure_space_1: "\<mu> (space M) = 1"
-lemma abs_real_of_pinfreal[simp]: "\<bar>real (X :: pinfreal)\<bar> = real X"
+lemma abs_real_of_pextreal[simp]: "\<bar>real (X :: pextreal)\<bar> = real X"
by simp
-lemma zero_less_real_of_pinfreal: "0 < real (X :: pinfreal) \<longleftrightarrow> X \<noteq> 0 \<and> X \<noteq> \<omega>"
+lemma zero_less_real_of_pextreal: "0 < real (X :: pextreal) \<longleftrightarrow> X \<noteq> 0 \<and> X \<noteq> \<omega>"
by (cases X) auto
sublocale prob_space < finite_measure
@@ -141,7 +141,7 @@
show "prob (\<Union> i :: nat. c i) \<le> 0"
using real_finite_measure_countably_subadditive[OF assms(1)]
by (simp add: assms(2) suminf_zero summable_zero)
- show "0 \<le> prob (\<Union> i :: nat. c i)" by (rule real_pinfreal_nonneg)
+ show "0 \<le> prob (\<Union> i :: nat. c i)" by (rule real_pextreal_nonneg)
qed
lemma (in prob_space) indep_sym:
@@ -606,7 +606,7 @@
show ?thesis
unfolding setsum_joint_distribution[OF assms, symmetric]
using distribution_finite[OF random_variable_pairI[OF finite_random_variableD[OF X] Y(1)]] Y(2)
- by (simp add: space_pair_algebra in_sigma pair_algebraI MX.sets_eq_Pow real_of_pinfreal_setsum)
+ by (simp add: space_pair_algebra in_sigma pair_algebraI MX.sets_eq_Pow real_of_pextreal_setsum)
qed
lemma (in prob_space) setsum_real_joint_distribution_singleton:
@@ -721,7 +721,7 @@
lemma (in finite_prob_space) finite_sum_over_space_eq_1:
"(\<Sum>x\<in>space M. real (\<mu> {x})) = 1"
- using sum_over_space_eq_1 finite_measure by (simp add: real_of_pinfreal_setsum sets_eq_Pow)
+ using sum_over_space_eq_1 finite_measure by (simp add: real_of_pextreal_setsum sets_eq_Pow)
lemma (in finite_prob_space) distribution_finite:
"distribution X A \<noteq> \<omega>"
@@ -730,27 +730,27 @@
lemma (in finite_prob_space) real_distribution_gt_0[simp]:
"0 < real (distribution Y y) \<longleftrightarrow> 0 < distribution Y y"
- using assms by (auto intro!: real_pinfreal_pos distribution_finite)
+ using assms by (auto intro!: real_pextreal_pos distribution_finite)
lemma (in finite_prob_space) real_distribution_mult_pos_pos:
assumes "0 < distribution Y y"
and "0 < distribution X x"
shows "0 < real (distribution Y y * distribution X x)"
- unfolding real_of_pinfreal_mult[symmetric]
+ unfolding real_of_pextreal_mult[symmetric]
using assms by (auto intro!: mult_pos_pos)
lemma (in finite_prob_space) real_distribution_divide_pos_pos:
assumes "0 < distribution Y y"
and "0 < distribution X x"
shows "0 < real (distribution Y y / distribution X x)"
- unfolding divide_pinfreal_def real_of_pinfreal_mult[symmetric]
+ unfolding divide_pextreal_def real_of_pextreal_mult[symmetric]
using assms distribution_finite[of X x] by (cases "distribution X x") (auto intro!: mult_pos_pos)
lemma (in finite_prob_space) real_distribution_mult_inverse_pos_pos:
assumes "0 < distribution Y y"
and "0 < distribution X x"
shows "0 < real (distribution Y y * inverse (distribution X x))"
- unfolding divide_pinfreal_def real_of_pinfreal_mult[symmetric]
+ unfolding divide_pextreal_def real_of_pextreal_mult[symmetric]
using assms distribution_finite[of X x] by (cases "distribution X x") (auto intro!: mult_pos_pos)
lemma (in prob_space) distribution_remove_const:
@@ -805,9 +805,9 @@
and "r < real (joint_distribution X Y {(x, y)}) \<Longrightarrow> r < real (distribution Y {y})"
and "distribution X {x} = 0 \<Longrightarrow> real (joint_distribution X Y {(x, y)}) = 0"
and "distribution Y {y} = 0 \<Longrightarrow> real (joint_distribution X Y {(x, y)}) = 0"
- using real_of_pinfreal_mono[OF distribution_finite joint_distribution_restriction_fst, of X Y "{(x, y)}"]
- using real_of_pinfreal_mono[OF distribution_finite joint_distribution_restriction_snd, of X Y "{(x, y)}"]
- using real_pinfreal_nonneg[of "joint_distribution X Y {(x, y)}"]
+ using real_of_pextreal_mono[OF distribution_finite joint_distribution_restriction_fst, of X Y "{(x, y)}"]
+ using real_of_pextreal_mono[OF distribution_finite joint_distribution_restriction_snd, of X Y "{(x, y)}"]
+ using real_pextreal_nonneg[of "joint_distribution X Y {(x, y)}"]
by auto
lemma (in prob_space) joint_distribution_remove[simp]:
@@ -821,8 +821,8 @@
lemma (in finite_prob_space) real_distribution_1:
"real (distribution X A) \<le> 1"
- unfolding real_pinfreal_1[symmetric]
- by (rule real_of_pinfreal_mono[OF _ distribution_1]) simp
+ unfolding real_pextreal_1[symmetric]
+ by (rule real_of_pextreal_mono[OF _ distribution_1]) simp
lemma (in finite_prob_space) uniform_prob:
assumes "x \<in> space M"
@@ -865,7 +865,7 @@
unfolding prob_space_def prob_space_axioms_def
proof
show "\<mu> (space (restricted_space A)) / \<mu> A = 1"
- using `\<mu> A \<noteq> 0` `\<mu> A \<noteq> \<omega>` by (auto simp: pinfreal_noteq_omega_Ex)
+ using `\<mu> A \<noteq> 0` `\<mu> A \<noteq> \<omega>` by (auto simp: pextreal_noteq_omega_Ex)
have *: "\<And>S. \<mu> S / \<mu> A = inverse (\<mu> A) * \<mu> S" by (simp add: mult_commute)
interpret A: measure_space "restricted_space A" \<mu>
using `A \<in> sets M` by (rule restricted_measure_space)
@@ -910,9 +910,9 @@
lemma (in finite_prob_space) real_distribution_order':
shows "real (distribution X {x}) = 0 \<Longrightarrow> real (joint_distribution X Y {(x, y)}) = 0"
and "real (distribution Y {y}) = 0 \<Longrightarrow> real (joint_distribution X Y {(x, y)}) = 0"
- using real_of_pinfreal_mono[OF distribution_finite joint_distribution_restriction_fst, of X Y "{(x, y)}"]
- using real_of_pinfreal_mono[OF distribution_finite joint_distribution_restriction_snd, of X Y "{(x, y)}"]
- using real_pinfreal_nonneg[of "joint_distribution X Y {(x, y)}"]
+ using real_of_pextreal_mono[OF distribution_finite joint_distribution_restriction_fst, of X Y "{(x, y)}"]
+ using real_of_pextreal_mono[OF distribution_finite joint_distribution_restriction_snd, of X Y "{(x, y)}"]
+ using real_pextreal_nonneg[of "joint_distribution X Y {(x, y)}"]
by auto
lemma (in finite_prob_space) finite_product_measure_space:
@@ -952,7 +952,7 @@
section "Conditional Expectation and Probability"
lemma (in prob_space) conditional_expectation_exists:
- fixes X :: "'a \<Rightarrow> pinfreal"
+ fixes X :: "'a \<Rightarrow> pextreal"
assumes borel: "X \<in> borel_measurable M"
and N_subalgebra: "N \<subseteq> sets M" "sigma_algebra (M\<lparr> sets := N \<rparr>)"
shows "\<exists>Y\<in>borel_measurable (M\<lparr> sets := N \<rparr>). \<forall>C\<in>N.
@@ -999,7 +999,7 @@
"conditional_prob N A \<equiv> conditional_expectation N (indicator A)"
lemma (in prob_space)
- fixes X :: "'a \<Rightarrow> pinfreal"
+ fixes X :: "'a \<Rightarrow> pextreal"
assumes borel: "X \<in> borel_measurable M"
and N_subalgebra: "N \<subseteq> sets M" "sigma_algebra (M\<lparr> sets := N \<rparr>)"
shows borel_measurable_conditional_expectation:
@@ -1018,7 +1018,7 @@
qed
lemma (in sigma_algebra) factorize_measurable_function:
- fixes Z :: "'a \<Rightarrow> pinfreal" and Y :: "'a \<Rightarrow> 'c"
+ fixes Z :: "'a \<Rightarrow> pextreal" and Y :: "'a \<Rightarrow> 'c"
assumes "sigma_algebra M'" and "Y \<in> measurable M M'" "Z \<in> borel_measurable M"
shows "Z \<in> borel_measurable (sigma_algebra.vimage_algebra M' (space M) Y)
\<longleftrightarrow> (\<exists>g\<in>borel_measurable M'. \<forall>x\<in>space M. Z x = g (Y x))"
@@ -1028,7 +1028,7 @@
from M'.sigma_algebra_vimage[OF this]
interpret va: sigma_algebra "M'.vimage_algebra (space M) Y" .
- { fix g :: "'c \<Rightarrow> pinfreal" assume "g \<in> borel_measurable M'"
+ { fix g :: "'c \<Rightarrow> pextreal" assume "g \<in> borel_measurable M'"
with M'.measurable_vimage_algebra[OF Y]
have "g \<circ> Y \<in> borel_measurable (M'.vimage_algebra (space M) Y)"
by (rule measurable_comp)
@@ -1058,7 +1058,7 @@
show "M'.simple_function ?g" using B by auto
fix x assume "x \<in> space M"
- then have "\<And>z. z \<in> f i`space M \<Longrightarrow> indicator (B z) (Y x) = (indicator (f i -` {z} \<inter> space M) x::pinfreal)"
+ then have "\<And>z. z \<in> f i`space M \<Longrightarrow> indicator (B z) (Y x) = (indicator (f i -` {z} \<inter> space M) x::pextreal)"
unfolding indicator_def using B by auto
then show "f i x = ?g (Y x)" using `x \<in> space M` f[of i]
by (subst va.simple_function_indicator_representation) auto
--- a/src/HOL/Probability/Product_Measure.thy Mon Dec 06 19:18:02 2010 +0100
+++ b/src/HOL/Probability/Product_Measure.thy Fri Dec 03 15:25:14 2010 +0100
@@ -379,7 +379,7 @@
by (auto intro!: M2.finite_measure_compl measurable_cut_fst
simp: vimage_Diff)
with `A \<in> sets ?D` top show "space ?D - A \<in> sets ?D"
- by (auto intro!: Diff M1.measurable_If M1.borel_measurable_pinfreal_diff)
+ by (auto intro!: Diff M1.measurable_If M1.borel_measurable_pextreal_diff)
next
fix F :: "nat \<Rightarrow> ('a\<times>'b) set" assume "disjoint_family F" "range F \<subseteq> sets ?D"
moreover then have "\<And>x. \<mu>2 (\<Union>i. Pair x -` F i) = (\<Sum>\<^isub>\<infinity> i. ?s (F i) x)"
@@ -505,7 +505,7 @@
unfolding pair_measure_def
proof (rule M1.positive_integral_cong)
fix x assume "x \<in> space M1"
- have *: "\<And>y. indicator A (x, y) = (indicator (Pair x -` A) y :: pinfreal)"
+ have *: "\<And>y. indicator A (x, y) = (indicator (Pair x -` A) y :: pextreal)"
unfolding indicator_def by auto
show "M2.positive_integral (\<lambda>y. indicator A (x, y)) = \<mu>2 (Pair x -` A)"
unfolding *
@@ -703,7 +703,7 @@
and "M1.positive_integral (\<lambda>x. M2.positive_integral (\<lambda>y. f (x, y)))
= positive_integral f"
by (auto simp del: vimage_Int cong: measurable_cong
- intro!: M1.borel_measurable_pinfreal_setsum
+ intro!: M1.borel_measurable_pextreal_setsum
simp add: M1.positive_integral_setsum simple_integral_def
M1.positive_integral_cmult
M1.positive_integral_cong[OF eq]
@@ -805,7 +805,7 @@
show "\<mu>1 {x\<in>space M1. \<mu>2 (Pair x -` N) \<noteq> 0} = 0"
by (simp add: M1.positive_integral_0_iff pair_measure_alt vimage_def)
show "{x \<in> space M1. \<mu>2 (Pair x -` N) \<noteq> 0} \<in> sets M1"
- by (intro M1.borel_measurable_pinfreal_neq_const measure_cut_measurable_fst N)
+ by (intro M1.borel_measurable_pextreal_neq_const measure_cut_measurable_fst N)
{ fix x assume "x \<in> space M1" "\<mu>2 (Pair x -` N) = 0"
have "M2.almost_everywhere (\<lambda>y. Q (x, y))"
proof (rule M2.AE_I)
@@ -1201,7 +1201,7 @@
qed
locale product_sigma_finite =
- fixes M :: "'i \<Rightarrow> 'a algebra" and \<mu> :: "'i \<Rightarrow> 'a set \<Rightarrow> pinfreal"
+ fixes M :: "'i \<Rightarrow> 'a algebra" and \<mu> :: "'i \<Rightarrow> 'a set \<Rightarrow> pextreal"
assumes sigma_finite_measures: "\<And>i. sigma_finite_measure (M i) (\<mu> i)"
locale finite_product_sigma_finite = product_sigma_finite M \<mu> for M :: "'i \<Rightarrow> 'a algebra" and \<mu> +
@@ -1319,7 +1319,7 @@
qed
definition (in finite_product_sigma_finite)
- measure :: "('i \<Rightarrow> 'a) set \<Rightarrow> pinfreal" where
+ measure :: "('i \<Rightarrow> 'a) set \<Rightarrow> pextreal" where
"measure = (SOME \<nu>.
(\<forall>A\<in>\<Pi> i\<in>I. sets (M i). \<nu> (Pi\<^isub>E I A) = (\<Prod>i\<in>I. \<mu> i (A i))) \<and>
sigma_finite_measure P \<nu>)"
--- a/src/HOL/Probability/Radon_Nikodym.thy Mon Dec 06 19:18:02 2010 +0100
+++ b/src/HOL/Probability/Radon_Nikodym.thy Fri Dec 03 15:25:14 2010 +0100
@@ -29,10 +29,10 @@
next
assume not_0: "\<mu> (A i) \<noteq> 0"
then have "?B i \<noteq> \<omega>" using measure[of i] by auto
- then have "inverse (?B i) \<noteq> 0" unfolding pinfreal_inverse_eq_0 by simp
+ then have "inverse (?B i) \<noteq> 0" unfolding pextreal_inverse_eq_0 by simp
then show ?thesis using measure[of i] not_0
by (auto intro!: exI[of _ "inverse (?B i) / 2"]
- simp: pinfreal_noteq_omega_Ex zero_le_mult_iff zero_less_mult_iff mult_le_0_iff power_le_zero_eq)
+ simp: pextreal_noteq_omega_Ex zero_le_mult_iff zero_less_mult_iff mult_le_0_iff power_le_zero_eq)
qed
qed
from choice[OF this] obtain n where n: "\<And>i. 0 < n i"
@@ -49,7 +49,7 @@
fix N show "n N * \<mu> (A N) \<le> Real ((1 / 2) ^ Suc N)"
using measure[of N] n[of N]
by (cases "n N")
- (auto simp: pinfreal_noteq_omega_Ex field_simps zero_le_mult_iff
+ (auto simp: pextreal_noteq_omega_Ex field_simps zero_le_mult_iff
mult_le_0_iff mult_less_0_iff power_less_zero_eq
power_le_zero_eq inverse_eq_divide less_divide_eq
power_divide split: split_if_asm)
@@ -65,14 +65,14 @@
then show "0 < ?h x" and "?h x < \<omega>" using n[of i] by auto
next
show "?h \<in> borel_measurable M" using range
- by (auto intro!: borel_measurable_psuminf borel_measurable_pinfreal_times)
+ by (auto intro!: borel_measurable_psuminf borel_measurable_pextreal_times)
qed
qed
subsection "Absolutely continuous"
definition (in measure_space)
- "absolutely_continuous \<nu> = (\<forall>N\<in>null_sets. \<nu> N = (0 :: pinfreal))"
+ "absolutely_continuous \<nu> = (\<forall>N\<in>null_sets. \<nu> N = (0 :: pextreal))"
lemma (in sigma_finite_measure) absolutely_continuous_AE:
assumes "measure_space M \<nu>" "absolutely_continuous \<nu>" "AE x. P x"
@@ -409,9 +409,9 @@
moreover {
have "positive_integral (\<lambda>x. f x * indicator (\<Union>i. A i) x) \<le> \<nu> (\<Union>i. A i)"
using A `f \<in> G` unfolding G_def by (auto simp: countable_UN)
- also have "\<nu> (\<Union>i. A i) < \<omega>" using v_fin by (simp add: pinfreal_less_\<omega>)
+ also have "\<nu> (\<Union>i. A i) < \<omega>" using v_fin by (simp add: pextreal_less_\<omega>)
finally have "positive_integral (\<lambda>x. f x * indicator (\<Union>i. A i) x) \<noteq> \<omega>"
- by (simp add: pinfreal_less_\<omega>) }
+ by (simp add: pextreal_less_\<omega>) }
ultimately
show "(\<Sum>\<^isub>\<infinity> n. ?t (A n)) = ?t (\<Union>i. A i)"
apply (subst psuminf_minus) by simp_all
@@ -440,7 +440,7 @@
def b \<equiv> "?t (space M) / \<mu> (space M) / 2"
ultimately have b: "b \<noteq> 0" "b \<noteq> \<omega>"
using M'.finite_measure_of_space
- by (auto simp: pinfreal_inverse_eq_0 finite_measure_of_space)
+ by (auto simp: pextreal_inverse_eq_0 finite_measure_of_space)
have "finite_measure M (\<lambda>A. b * \<mu> A)" (is "finite_measure M ?b")
proof
show "?b {} = 0" by simp
@@ -486,7 +486,7 @@
by (cases "positive_integral (\<lambda>x. f x * indicator A x)", cases "\<nu> A", auto)
finally have "positive_integral (\<lambda>x. ?f0 x * indicator A x) \<le> \<nu> A" . }
hence "?f0 \<in> G" using `A0 \<in> sets M` unfolding G_def
- by (auto intro!: borel_measurable_indicator borel_measurable_pinfreal_add borel_measurable_pinfreal_times)
+ by (auto intro!: borel_measurable_indicator borel_measurable_pextreal_add borel_measurable_pextreal_times)
have real: "?t (space M) \<noteq> \<omega>" "?t A0 \<noteq> \<omega>"
"b * \<mu> (space M) \<noteq> \<omega>" "b * \<mu> A0 \<noteq> \<omega>"
using `A0 \<in> sets M` b
@@ -494,27 +494,27 @@
finite_measure_of_space M.finite_measure_of_space
by auto
have int_f_finite: "positive_integral f \<noteq> \<omega>"
- using M'.finite_measure_of_space pos_t unfolding pinfreal_zero_less_diff_iff
+ using M'.finite_measure_of_space pos_t unfolding pextreal_zero_less_diff_iff
by (auto cong: positive_integral_cong)
have "?t (space M) > b * \<mu> (space M)" unfolding b_def
apply (simp add: field_simps)
apply (subst mult_assoc[symmetric])
- apply (subst pinfreal_mult_inverse)
+ apply (subst pextreal_mult_inverse)
using finite_measure_of_space M'.finite_measure_of_space pos_t pos_M
- using pinfreal_mult_strict_right_mono[of "Real (1/2)" 1 "?t (space M)"]
+ using pextreal_mult_strict_right_mono[of "Real (1/2)" 1 "?t (space M)"]
by simp_all
hence "0 < ?t (space M) - b * \<mu> (space M)"
- by (simp add: pinfreal_zero_less_diff_iff)
+ by (simp add: pextreal_zero_less_diff_iff)
also have "\<dots> \<le> ?t A0 - b * \<mu> A0"
- using space_less_A0 pos_M pos_t b real[unfolded pinfreal_noteq_omega_Ex] by auto
- finally have "b * \<mu> A0 < ?t A0" unfolding pinfreal_zero_less_diff_iff .
+ using space_less_A0 pos_M pos_t b real[unfolded pextreal_noteq_omega_Ex] by auto
+ finally have "b * \<mu> A0 < ?t A0" unfolding pextreal_zero_less_diff_iff .
hence "0 < ?t A0" by auto
hence "0 < \<mu> A0" using ac unfolding absolutely_continuous_def
using `A0 \<in> sets M` by auto
hence "0 < b * \<mu> A0" using b by auto
from int_f_finite this
have "?y + 0 < positive_integral f + b * \<mu> A0" unfolding int_f_eq_y
- by (rule pinfreal_less_add)
+ by (rule pextreal_less_add)
also have "\<dots> = positive_integral ?f0" using f0_eq[OF top] `A0 \<in> sets M` sets_into_space
by (simp cong: positive_integral_cong)
finally have "?y < positive_integral ?f0" by simp
@@ -530,7 +530,7 @@
using `f \<in> G` `A \<in> sets M` unfolding G_def by auto
show "\<nu> A \<le> positive_integral (\<lambda>x. f x * indicator A x)"
using upper_bound[THEN bspec, OF `A \<in> sets M`]
- by (simp add: pinfreal_zero_le_diff)
+ by (simp add: pextreal_zero_le_diff)
qed
qed simp
qed
@@ -573,8 +573,8 @@
using Q' by (auto intro: finite_UN)
with v.measure_finitely_subadditive[of "{.. i}" Q']
have "\<nu> (?O i) \<le> (\<Sum>i\<le>i. \<nu> (Q' i))" by auto
- also have "\<dots> < \<omega>" unfolding setsum_\<omega> pinfreal_less_\<omega> using Q' by auto
- finally show "\<nu> (?O i) \<noteq> \<omega>" unfolding pinfreal_less_\<omega> by auto
+ also have "\<dots> < \<omega>" unfolding setsum_\<omega> pextreal_less_\<omega> using Q' by auto
+ finally show "\<nu> (?O i) \<noteq> \<omega>" unfolding pextreal_less_\<omega> by auto
qed auto
have O_mono: "\<And>n. ?O n \<subseteq> ?O (Suc n)" by fastsimp
have a_eq: "?a = \<mu> (\<Union>i. ?O i)" unfolding Union[symmetric]
@@ -634,7 +634,7 @@
then show "\<mu> (?O i \<union> A) \<le> ?a" by (rule le_SUPI)
qed
finally have "\<mu> A = 0" unfolding a_eq using finite_measure[OF `?O_0 \<in> sets M`]
- by (cases "\<mu> A") (auto simp: pinfreal_noteq_omega_Ex)
+ by (cases "\<mu> A") (auto simp: pextreal_noteq_omega_Ex)
with `\<mu> A \<noteq> 0` show ?thesis by auto
qed
qed }
@@ -682,7 +682,7 @@
\<nu> (Q i \<inter> A) = positive_integral (\<lambda>x. f x * indicator (Q i \<inter> A) x))"
proof
fix i
- have indicator_eq: "\<And>f x A. (f x :: pinfreal) * indicator (Q i \<inter> A) x * indicator (Q i) x
+ have indicator_eq: "\<And>f x A. (f x :: pextreal) * indicator (Q i \<inter> A) x * indicator (Q i) x
= (f x * indicator (Q i) x) * indicator A x"
unfolding indicator_def by auto
have fm: "finite_measure (restricted_space (Q i)) \<mu>"
@@ -718,19 +718,19 @@
show ?thesis
proof (safe intro!: bexI[of _ ?f])
show "?f \<in> borel_measurable M"
- by (safe intro!: borel_measurable_psuminf borel_measurable_pinfreal_times
- borel_measurable_pinfreal_add borel_measurable_indicator
+ by (safe intro!: borel_measurable_psuminf borel_measurable_pextreal_times
+ borel_measurable_pextreal_add borel_measurable_indicator
borel_measurable_const borel Q_sets Q0 Diff countable_UN)
fix A assume "A \<in> sets M"
have *:
"\<And>x i. indicator A x * (f i x * indicator (Q i) x) =
f i x * indicator (Q i \<inter> A) x"
- "\<And>x i. (indicator A x * indicator Q0 x :: pinfreal) =
+ "\<And>x i. (indicator A x * indicator Q0 x :: pextreal) =
indicator (Q0 \<inter> A) x" by (auto simp: indicator_def)
have "positive_integral (\<lambda>x. ?f x * indicator A x) =
(\<Sum>\<^isub>\<infinity> i. \<nu> (Q i \<inter> A)) + \<omega> * \<mu> (Q0 \<inter> A)"
unfolding f[OF `A \<in> sets M`]
- apply (simp del: pinfreal_times(2) add: field_simps *)
+ apply (simp del: pextreal_times(2) add: field_simps *)
apply (subst positive_integral_add)
apply (fastsimp intro: Q0 `A \<in> sets M`)
apply (fastsimp intro: Q_sets `A \<in> sets M` borel_measurable_psuminf borel)
@@ -775,7 +775,7 @@
interpret T: finite_measure M ?T
unfolding finite_measure_def finite_measure_axioms_def
by (simp cong: positive_integral_cong)
- have "\<And>N. N \<in> sets M \<Longrightarrow> {x \<in> space M. h x \<noteq> 0 \<and> indicator N x \<noteq> (0::pinfreal)} = N"
+ have "\<And>N. N \<in> sets M \<Longrightarrow> {x \<in> space M. h x \<noteq> 0 \<and> indicator N x \<noteq> (0::pextreal)} = N"
using sets_into_space pos by (force simp: indicator_def)
then have "T.absolutely_continuous \<nu>" using assms(2) borel
unfolding T.absolutely_continuous_def absolutely_continuous_def
@@ -786,10 +786,10 @@
show ?thesis
proof (safe intro!: bexI[of _ "\<lambda>x. h x * f x"])
show "(\<lambda>x. h x * f x) \<in> borel_measurable M"
- using borel f_borel by (auto intro: borel_measurable_pinfreal_times)
+ using borel f_borel by (auto intro: borel_measurable_pextreal_times)
fix A assume "A \<in> sets M"
then have "(\<lambda>x. f x * indicator A x) \<in> borel_measurable M"
- using f_borel by (auto intro: borel_measurable_pinfreal_times borel_measurable_indicator)
+ using f_borel by (auto intro: borel_measurable_pextreal_times borel_measurable_indicator)
from positive_integral_translated_density[OF borel this]
show "\<nu> A = positive_integral (\<lambda>x. h x * f x * indicator A x)"
unfolding fT[OF `A \<in> sets M`] by (simp add: field_simps)
@@ -834,7 +834,7 @@
finally have "\<mu> {x\<in>space M. (f x - g x) * indicator ?N x \<noteq> 0} = 0"
using borel N by (subst (asm) positive_integral_0_iff) auto
moreover have "{x\<in>space M. (f x - g x) * indicator ?N x \<noteq> 0} = ?N"
- by (auto simp: pinfreal_zero_le_diff)
+ by (auto simp: pextreal_zero_le_diff)
ultimately have "?N \<in> null_sets" using N by simp }
from this[OF borel g_fin eq] this[OF borel(2,1) fin]
have "{x\<in>space M. g x < f x} \<union> {x\<in>space M. f x < g x} \<in> null_sets"
@@ -866,15 +866,15 @@
from Q have Q_sets: "\<And>i. Q i \<in> sets M" by auto
let ?N = "{x\<in>space M. f x \<noteq> f' x}"
have "?N \<in> sets M" using borel by auto
- have *: "\<And>i x A. \<And>y::pinfreal. y * indicator (Q i) x * indicator A x = y * indicator (Q i \<inter> A) x"
+ have *: "\<And>i x A. \<And>y::pextreal. y * indicator (Q i) x * indicator A x = y * indicator (Q i \<inter> A) x"
unfolding indicator_def by auto
have 1: "\<forall>i. AE x. ?f (Q i) x = ?f' (Q i) x"
using borel Q_fin Q
by (intro finite_density_unique[THEN iffD1] allI)
- (auto intro!: borel_measurable_pinfreal_times f Int simp: *)
+ (auto intro!: borel_measurable_pextreal_times f Int simp: *)
have 2: "AE x. ?f Q0 x = ?f' Q0 x"
proof (rule AE_I')
- { fix f :: "'a \<Rightarrow> pinfreal" assume borel: "f \<in> borel_measurable M"
+ { fix f :: "'a \<Rightarrow> pextreal" assume borel: "f \<in> borel_measurable M"
and eq: "\<And>A. A \<in> sets M \<Longrightarrow> ?\<nu> A = positive_integral (\<lambda>x. f x * indicator A x)"
let "?A i" = "Q0 \<inter> {x \<in> space M. f x < of_nat i}"
have "(\<Union>i. ?A i) \<in> null_sets"
@@ -893,7 +893,7 @@
qed
also have "(\<Union>i. ?A i) = Q0 \<inter> {x\<in>space M. f x < \<omega>}"
by (auto simp: less_\<omega>_Ex_of_nat)
- finally have "Q0 \<inter> {x\<in>space M. f x \<noteq> \<omega>} \<in> null_sets" by (simp add: pinfreal_less_\<omega>) }
+ finally have "Q0 \<inter> {x\<in>space M. f x \<noteq> \<omega>} \<in> null_sets" by (simp add: pextreal_less_\<omega>) }
from this[OF borel(1) refl] this[OF borel(2) f]
have "Q0 \<inter> {x\<in>space M. f x \<noteq> \<omega>} \<in> null_sets" "Q0 \<inter> {x\<in>space M. f' x \<noteq> \<omega>} \<in> null_sets" by simp_all
then show "(Q0 \<inter> {x\<in>space M. f x \<noteq> \<omega>}) \<union> (Q0 \<inter> {x\<in>space M. f' x \<noteq> \<omega>}) \<in> null_sets" by (rule null_sets_Un)
@@ -927,7 +927,7 @@
interpret f': measure_space M "\<lambda>A. positive_integral (\<lambda>x. f' x * indicator A x)"
using borel(2) by (rule measure_space_density)
{ fix A assume "A \<in> sets M"
- then have " {x \<in> space M. h x \<noteq> 0 \<and> indicator A x \<noteq> (0::pinfreal)} = A"
+ then have " {x \<in> space M. h x \<noteq> 0 \<and> indicator A x \<noteq> (0::pextreal)} = A"
using pos sets_into_space by (force simp: indicator_def)
then have "positive_integral (\<lambda>xa. h xa * indicator A xa) = 0 \<longleftrightarrow> A \<in> null_sets"
using h_borel `A \<in> sets M` by (simp add: positive_integral_0_iff) }
@@ -1027,7 +1027,7 @@
apply (subst positive_integral_0_iff)
apply fast
apply (subst (asm) AE_iff_null_set)
- apply (intro borel_measurable_pinfreal_neq_const)
+ apply (intro borel_measurable_pextreal_neq_const)
apply fast
by simp
then show ?thesis by simp
@@ -1130,7 +1130,7 @@
using sf.RN_deriv(1)[OF \<nu>' ac]
unfolding measurable_vimage_iff_inv[OF f] comp_def .
fix A assume "A \<in> sets M"
- then have *: "\<And>x. x \<in> space M \<Longrightarrow> indicator (f -` A \<inter> S) (the_inv_into S f x) = (indicator A x :: pinfreal)"
+ then have *: "\<And>x. x \<in> space M \<Longrightarrow> indicator (f -` A \<inter> S) (the_inv_into S f x) = (indicator A x :: pextreal)"
using f[unfolded bij_betw_def]
unfolding indicator_def by (auto simp: f_the_inv_into_f the_inv_into_in)
have "\<nu> (f ` (f -` A \<inter> S)) = sf.positive_integral (\<lambda>x. sf.RN_deriv (\<lambda>A. \<nu> (f ` A)) x * indicator (f -` A \<inter> S) x)"
@@ -1160,7 +1160,7 @@
proof -
interpret \<nu>: sigma_finite_measure M \<nu> by fact
have ms: "measure_space M \<nu>" by default
- have minus_cong: "\<And>A B A' B'::pinfreal. A = A' \<Longrightarrow> B = B' \<Longrightarrow> real A - real B = real A' - real B'" by simp
+ have minus_cong: "\<And>A B A' B'::pextreal. A = A' \<Longrightarrow> B = B' \<Longrightarrow> real A - real B = real A' - real B'" by simp
have f': "(\<lambda>x. - f x) \<in> borel_measurable M" using f by auto
{ fix f :: "'a \<Rightarrow> real" assume "f \<in> borel_measurable M"
{ fix x assume *: "RN_deriv \<nu> x \<noteq> \<omega>"
--- a/src/HOL/Probability/ex/Dining_Cryptographers.thy Mon Dec 06 19:18:02 2010 +0100
+++ b/src/HOL/Probability/ex/Dining_Cryptographers.thy Fri Dec 03 15:25:14 2010 +0100
@@ -8,7 +8,7 @@
and not_empty[simp]: "S \<noteq> {}"
definition (in finite_space) "M = \<lparr> space = S, sets = Pow S \<rparr>"
-definition (in finite_space) \<mu>_def[simp]: "\<mu> A = (of_nat (card A) / of_nat (card S) :: pinfreal)"
+definition (in finite_space) \<mu>_def[simp]: "\<mu> A = (of_nat (card A) / of_nat (card S) :: pextreal)"
lemma (in finite_space)
shows space_M[simp]: "space M = S"
--- a/src/HOL/Probability/ex/Koepf_Duermuth_Countermeasure.thy Mon Dec 06 19:18:02 2010 +0100
+++ b/src/HOL/Probability/ex/Koepf_Duermuth_Countermeasure.thy Fri Dec 03 15:25:14 2010 +0100
@@ -274,7 +274,7 @@
"snd ` (SIGMA x:f`X. f -` {x} \<inter> X) = X"
by (auto simp: image_iff)
-lemma zero_le_eq_True: "0 \<le> (x::pinfreal) \<longleftrightarrow> True" by simp
+lemma zero_le_eq_True: "0 \<le> (x::pextreal) \<longleftrightarrow> True" by simp
lemma Real_setprod:
assumes"\<And>i. i\<in>X \<Longrightarrow> 0 \<le> f i"