--- a/src/HOL/Complete_Lattice.thy Mon Mar 14 15:17:10 2011 +0100
+++ b/src/HOL/Complete_Lattice.thy Mon Mar 14 15:29:10 2011 +0100
@@ -89,25 +89,45 @@
by (auto intro: Sup_least dest: Sup_upper)
lemma Inf_mono:
- assumes "\<And>b. b \<in> B \<Longrightarrow> \<exists>a\<in>A. a \<le> b"
- shows "Inf A \<le> Inf B"
+ assumes "\<And>b. b \<in> B \<Longrightarrow> \<exists>a\<in>A. a \<sqsubseteq> b"
+ shows "Inf A \<sqsubseteq> Inf B"
proof (rule Inf_greatest)
fix b assume "b \<in> B"
- with assms obtain a where "a \<in> A" and "a \<le> b" by blast
- from `a \<in> A` have "Inf A \<le> a" by (rule Inf_lower)
- with `a \<le> b` show "Inf A \<le> b" by auto
+ with assms obtain a where "a \<in> A" and "a \<sqsubseteq> b" by blast
+ from `a \<in> A` have "Inf A \<sqsubseteq> a" by (rule Inf_lower)
+ with `a \<sqsubseteq> b` show "Inf A \<sqsubseteq> b" by auto
qed
lemma Sup_mono:
- assumes "\<And>a. a \<in> A \<Longrightarrow> \<exists>b\<in>B. a \<le> b"
- shows "Sup A \<le> Sup B"
+ assumes "\<And>a. a \<in> A \<Longrightarrow> \<exists>b\<in>B. a \<sqsubseteq> b"
+ shows "Sup A \<sqsubseteq> Sup B"
proof (rule Sup_least)
fix a assume "a \<in> A"
- with assms obtain b where "b \<in> B" and "a \<le> b" by blast
- from `b \<in> B` have "b \<le> Sup B" by (rule Sup_upper)
- with `a \<le> b` show "a \<le> Sup B" by auto
+ with assms obtain b where "b \<in> B" and "a \<sqsubseteq> b" by blast
+ from `b \<in> B` have "b \<sqsubseteq> Sup B" by (rule Sup_upper)
+ with `a \<sqsubseteq> b` show "a \<sqsubseteq> Sup B" by auto
qed
+lemma top_le:
+ "top \<sqsubseteq> x \<Longrightarrow> x = top"
+ by (rule antisym) auto
+
+lemma le_bot:
+ "x \<sqsubseteq> bot \<Longrightarrow> x = bot"
+ by (rule antisym) auto
+
+lemma not_less_bot[simp]: "\<not> (x \<sqsubset> bot)"
+ using bot_least[of x] by (auto simp: le_less)
+
+lemma not_top_less[simp]: "\<not> (top \<sqsubset> x)"
+ using top_greatest[of x] by (auto simp: le_less)
+
+lemma Sup_upper2: "u \<in> A \<Longrightarrow> v \<sqsubseteq> u \<Longrightarrow> v \<sqsubseteq> Sup A"
+ using Sup_upper[of u A] by auto
+
+lemma Inf_lower2: "u \<in> A \<Longrightarrow> u \<sqsubseteq> v \<Longrightarrow> Inf A \<sqsubseteq> v"
+ using Inf_lower[of u A] by auto
+
definition INFI :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where
"INFI A f = \<Sqinter> (f ` A)"
@@ -146,15 +166,27 @@
context complete_lattice
begin
+lemma SUP_cong: "(\<And>x. x \<in> A \<Longrightarrow> f x = g x) \<Longrightarrow> SUPR A f = SUPR A g"
+ by (simp add: SUPR_def cong: image_cong)
+
+lemma INF_cong: "(\<And>x. x \<in> A \<Longrightarrow> f x = g x) \<Longrightarrow> INFI A f = INFI A g"
+ by (simp add: INFI_def cong: image_cong)
+
lemma le_SUPI: "i : A \<Longrightarrow> M i \<sqsubseteq> (SUP i:A. M i)"
by (auto simp add: SUPR_def intro: Sup_upper)
+lemma le_SUPI2: "i \<in> A \<Longrightarrow> u \<sqsubseteq> M i \<Longrightarrow> u \<sqsubseteq> (SUP i:A. M i)"
+ using le_SUPI[of i A M] by auto
+
lemma SUP_leI: "(\<And>i. i : A \<Longrightarrow> M i \<sqsubseteq> u) \<Longrightarrow> (SUP i:A. M i) \<sqsubseteq> u"
by (auto simp add: SUPR_def intro: Sup_least)
lemma INF_leI: "i : A \<Longrightarrow> (INF i:A. M i) \<sqsubseteq> M i"
by (auto simp add: INFI_def intro: Inf_lower)
+lemma INF_leI2: "i \<in> A \<Longrightarrow> M i \<sqsubseteq> u \<Longrightarrow> (INF i:A. M i) \<sqsubseteq> u"
+ using INF_leI[of i A M] by auto
+
lemma le_INFI: "(\<And>i. i : A \<Longrightarrow> u \<sqsubseteq> M i) \<Longrightarrow> u \<sqsubseteq> (INF i:A. M i)"
by (auto simp add: INFI_def intro: Inf_greatest)
--- a/src/HOL/IsaMakefile Mon Mar 14 15:17:10 2011 +0100
+++ b/src/HOL/IsaMakefile Mon Mar 14 15:29:10 2011 +0100
@@ -437,10 +437,10 @@
Library/ContNotDenum.thy Library/Continuity.thy Library/Convex.thy \
Library/Countable.thy Library/Diagonalize.thy Library/Dlist.thy \
Library/Efficient_Nat.thy Library/Eval_Witness.thy \
- Library/Executable_Set.thy Library/Float.thy \
- Library/Formal_Power_Series.thy Library/Fraction_Field.thy \
- Library/FrechetDeriv.thy Library/Cset.thy Library/FuncSet.thy \
- Library/Function_Algebras.thy \
+ Library/Executable_Set.thy Library/Extended_Reals.thy \
+ Library/Float.thy Library/Formal_Power_Series.thy \
+ Library/Fraction_Field.thy Library/FrechetDeriv.thy Library/Cset.thy \
+ Library/FuncSet.thy Library/Function_Algebras.thy \
Library/Fundamental_Theorem_Algebra.thy Library/Glbs.thy \
Library/Indicator_Function.thy Library/Infinite_Set.thy \
Library/Inner_Product.thy Library/Kleene_Algebra.thy \
@@ -619,7 +619,7 @@
Number_Theory/UniqueFactorization.thy \
Number_Theory/ROOT.ML
@$(ISABELLE_TOOL) usedir -g true $(OUT)/HOL Number_Theory
-
+
## HOL-Old_Number_Theory
@@ -1154,6 +1154,7 @@
Multivariate_Analysis/Derivative.thy \
Multivariate_Analysis/Determinants.thy \
Multivariate_Analysis/Euclidean_Space.thy \
+ Multivariate_Analysis/Extended_Real_Limits.thy \
Multivariate_Analysis/Fashoda.thy \
Multivariate_Analysis/Finite_Cartesian_Product.thy \
Multivariate_Analysis/Integration.certs \
@@ -1167,9 +1168,10 @@
Multivariate_Analysis/Topology_Euclidean_Space.thy \
Multivariate_Analysis/document/root.tex \
Multivariate_Analysis/normarith.ML Library/Glbs.thy \
- Library/Indicator_Function.thy Library/Inner_Product.thy \
- Library/Numeral_Type.thy Library/Convex.thy Library/FrechetDeriv.thy \
- Library/Product_Vector.thy Library/Product_plus.thy
+ Library/Extended_Reals.thy Library/Indicator_Function.thy \
+ Library/Inner_Product.thy Library/Numeral_Type.thy Library/Convex.thy \
+ Library/FrechetDeriv.thy Library/Product_Vector.thy \
+ Library/Product_plus.thy
@cd Multivariate_Analysis; $(ISABELLE_TOOL) usedir -b -g true $(OUT)/HOL HOL-Multivariate_Analysis
@@ -1184,7 +1186,6 @@
Probability/ex/Koepf_Duermuth_Countermeasure.thy \
Probability/Information.thy Probability/Lebesgue_Integration.thy \
Probability/Lebesgue_Measure.thy Probability/Measure.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 \
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Library/Extended_Reals.thy Mon Mar 14 15:29:10 2011 +0100
@@ -0,0 +1,2438 @@
+(* Title: src/HOL/Library/Extended_Reals.thy
+ Author: Johannes Hölzl; TU München
+ Author: Robert Himmelmann; TU München
+ Author: Armin Heller; TU München
+ Author: Bogdan Grechuk; University of Edinburgh *)
+
+header {* Extended real number line *}
+
+theory Extended_Reals
+ imports Complex_Main
+begin
+
+text {*
+
+For more lemmas about the extended real numbers go to
+ @{text "src/HOL/Multivaraite_Analysis/Extended_Real_Limits.thy"}
+
+*}
+
+lemma (in complete_lattice) atLeast_eq_UNIV_iff: "{x..} = UNIV \<longleftrightarrow> x = bot"
+proof
+ assume "{x..} = UNIV"
+ show "x = bot"
+ proof (rule ccontr)
+ assume "x \<noteq> bot" then have "bot \<notin> {x..}" by (simp add: le_less)
+ then show False using `{x..} = UNIV` by simp
+ qed
+qed auto
+
+lemma SUPR_pair:
+ "(SUP i : A. SUP j : B. f i j) = (SUP p : A \<times> B. f (fst p) (snd p))"
+ by (rule antisym) (auto intro!: SUP_leI le_SUPI2)
+
+lemma INFI_pair:
+ "(INF i : A. INF j : B. f i j) = (INF p : A \<times> B. f (fst p) (snd p))"
+ by (rule antisym) (auto intro!: le_INFI INF_leI2)
+
+subsection {* Definition and basic properties *}
+
+datatype extreal = extreal real | PInfty | MInfty
+
+notation (xsymbols)
+ PInfty ("\<infinity>")
+
+notation (HTML output)
+ PInfty ("\<infinity>")
+
+declare [[coercion "extreal :: real \<Rightarrow> extreal"]]
+
+instantiation extreal :: uminus
+begin
+ fun uminus_extreal where
+ "- (extreal r) = extreal (- r)"
+ | "- \<infinity> = MInfty"
+ | "- MInfty = \<infinity>"
+ instance ..
+end
+
+lemma inj_extreal[simp]: "inj_on extreal A"
+ unfolding inj_on_def by auto
+
+lemma MInfty_neq_PInfty[simp]:
+ "\<infinity> \<noteq> - \<infinity>" "- \<infinity> \<noteq> \<infinity>" by simp_all
+
+lemma MInfty_neq_extreal[simp]:
+ "extreal r \<noteq> - \<infinity>" "- \<infinity> \<noteq> extreal r" by simp_all
+
+lemma MInfinity_cases[simp]:
+ "(case - \<infinity> of extreal r \<Rightarrow> f r | \<infinity> \<Rightarrow> y | MInfinity \<Rightarrow> z) = z"
+ by simp
+
+lemma extreal_uminus_uminus[simp]:
+ fixes a :: extreal shows "- (- a) = a"
+ by (cases a) simp_all
+
+lemma MInfty_eq[simp]:
+ "MInfty = - \<infinity>" by simp
+
+declare uminus_extreal.simps(2)[simp del]
+
+lemma extreal_cases[case_names real PInf MInf, cases type: extreal]:
+ assumes "\<And>r. x = extreal r \<Longrightarrow> P"
+ assumes "x = \<infinity> \<Longrightarrow> P"
+ assumes "x = -\<infinity> \<Longrightarrow> P"
+ shows P
+ using assms by (cases x) auto
+
+lemmas extreal2_cases = extreal_cases[case_product extreal_cases]
+lemmas extreal3_cases = extreal2_cases[case_product extreal_cases]
+
+lemma extreal_uminus_eq_iff[simp]:
+ fixes a b :: extreal shows "-a = -b \<longleftrightarrow> a = b"
+ by (cases rule: extreal2_cases[of a b]) simp_all
+
+function of_extreal :: "extreal \<Rightarrow> real" where
+"of_extreal (extreal r) = r" |
+"of_extreal \<infinity> = 0" |
+"of_extreal (-\<infinity>) = 0"
+ by (auto intro: extreal_cases)
+termination proof qed (rule wf_empty)
+
+defs (overloaded)
+ real_of_extreal_def [code_unfold]: "real \<equiv> of_extreal"
+
+lemma real_of_extreal[simp]:
+ "real (- x :: extreal) = - (real x)"
+ "real (extreal r) = r"
+ "real \<infinity> = 0"
+ by (cases x) (simp_all add: real_of_extreal_def)
+
+lemma range_extreal[simp]: "range extreal = UNIV - {\<infinity>, -\<infinity>}"
+proof safe
+ fix x assume "x \<notin> range extreal" "x \<noteq> \<infinity>"
+ then show "x = -\<infinity>" by (cases x) auto
+qed auto
+
+lemma extreal_range_uminus[simp]: "range uminus = (UNIV::extreal set)"
+proof safe
+ fix x :: extreal show "x \<in> range uminus" by (intro image_eqI[of _ _ "-x"]) auto
+qed auto
+
+instantiation extreal :: number
+begin
+definition [simp]: "number_of x = extreal (number_of x)"
+instance proof qed
+end
+
+instantiation extreal :: abs
+begin
+ function abs_extreal where
+ "\<bar>extreal r\<bar> = extreal \<bar>r\<bar>"
+ | "\<bar>-\<infinity>\<bar> = \<infinity>"
+ | "\<bar>\<infinity>\<bar> = \<infinity>"
+ by (auto intro: extreal_cases)
+ termination proof qed (rule wf_empty)
+ instance ..
+end
+
+lemma abs_eq_infinity_cases[elim!]: "\<lbrakk> \<bar>x\<bar> = \<infinity> ; x = \<infinity> \<Longrightarrow> P ; x = -\<infinity> \<Longrightarrow> P \<rbrakk> \<Longrightarrow> P"
+ by (cases x) auto
+
+lemma abs_neq_infinity_cases[elim!]: "\<lbrakk> \<bar>x\<bar> \<noteq> \<infinity> ; \<And>r. x = extreal r \<Longrightarrow> P \<rbrakk> \<Longrightarrow> P"
+ by (cases x) auto
+
+lemma abs_extreal_uminus[simp]: "\<bar>- x\<bar> = \<bar>x::extreal\<bar>"
+ by (cases x) auto
+
+subsubsection "Addition"
+
+instantiation extreal :: comm_monoid_add
+begin
+
+definition "0 = extreal 0"
+
+function plus_extreal where
+"extreal r + extreal p = extreal (r + p)" |
+"\<infinity> + a = \<infinity>" |
+"a + \<infinity> = \<infinity>" |
+"extreal r + -\<infinity> = - \<infinity>" |
+"-\<infinity> + extreal p = -\<infinity>" |
+"-\<infinity> + -\<infinity> = -\<infinity>"
+proof -
+ case (goal1 P x)
+ moreover then obtain a b where "x = (a, b)" by (cases x) auto
+ ultimately show P
+ by (cases rule: extreal2_cases[of a b]) auto
+qed auto
+termination proof qed (rule wf_empty)
+
+lemma Infty_neq_0[simp]:
+ "\<infinity> \<noteq> 0" "0 \<noteq> \<infinity>"
+ "-\<infinity> \<noteq> 0" "0 \<noteq> -\<infinity>"
+ by (simp_all add: zero_extreal_def)
+
+lemma extreal_eq_0[simp]:
+ "extreal r = 0 \<longleftrightarrow> r = 0"
+ "0 = extreal r \<longleftrightarrow> r = 0"
+ unfolding zero_extreal_def by simp_all
+
+instance
+proof
+ fix a :: extreal show "0 + a = a"
+ by (cases a) (simp_all add: zero_extreal_def)
+ fix b :: extreal show "a + b = b + a"
+ by (cases rule: extreal2_cases[of a b]) simp_all
+ fix c :: extreal show "a + b + c = a + (b + c)"
+ by (cases rule: extreal3_cases[of a b c]) simp_all
+qed
+end
+
+lemma abs_extreal_zero[simp]: "\<bar>0\<bar> = (0::extreal)"
+ unfolding zero_extreal_def abs_extreal.simps by simp
+
+lemma extreal_uminus_zero[simp]:
+ "- 0 = (0::extreal)"
+ by (simp add: zero_extreal_def)
+
+lemma extreal_uminus_zero_iff[simp]:
+ fixes a :: extreal shows "-a = 0 \<longleftrightarrow> a = 0"
+ by (cases a) simp_all
+
+lemma extreal_plus_eq_PInfty[simp]:
+ shows "a + b = \<infinity> \<longleftrightarrow> a = \<infinity> \<or> b = \<infinity>"
+ by (cases rule: extreal2_cases[of a b]) auto
+
+lemma extreal_plus_eq_MInfty[simp]:
+ shows "a + b = -\<infinity> \<longleftrightarrow>
+ (a = -\<infinity> \<or> b = -\<infinity>) \<and> a \<noteq> \<infinity> \<and> b \<noteq> \<infinity>"
+ by (cases rule: extreal2_cases[of a b]) auto
+
+lemma extreal_add_cancel_left:
+ assumes "a \<noteq> -\<infinity>"
+ shows "a + b = a + c \<longleftrightarrow> (a = \<infinity> \<or> b = c)"
+ using assms by (cases rule: extreal3_cases[of a b c]) auto
+
+lemma extreal_add_cancel_right:
+ assumes "a \<noteq> -\<infinity>"
+ shows "b + a = c + a \<longleftrightarrow> (a = \<infinity> \<or> b = c)"
+ using assms by (cases rule: extreal3_cases[of a b c]) auto
+
+lemma extreal_real:
+ "extreal (real x) = (if \<bar>x\<bar> = \<infinity> then 0 else x)"
+ by (cases x) simp_all
+
+lemma real_of_extreal_add:
+ fixes a b :: extreal
+ shows "real (a + b) = (if (\<bar>a\<bar> = \<infinity>) \<and> (\<bar>b\<bar> = \<infinity>) \<or> (\<bar>a\<bar> \<noteq> \<infinity>) \<and> (\<bar>b\<bar> \<noteq> \<infinity>) then real a + real b else 0)"
+ by (cases rule: extreal2_cases[of a b]) auto
+
+subsubsection "Linear order on @{typ extreal}"
+
+instantiation extreal :: linorder
+begin
+
+function less_extreal where
+"extreal x < extreal y \<longleftrightarrow> x < y" |
+" \<infinity> < a \<longleftrightarrow> False" |
+" a < -\<infinity> \<longleftrightarrow> False" |
+"extreal x < \<infinity> \<longleftrightarrow> True" |
+" -\<infinity> < extreal r \<longleftrightarrow> True" |
+" -\<infinity> < \<infinity> \<longleftrightarrow> True"
+proof -
+ case (goal1 P x)
+ moreover then obtain a b where "x = (a,b)" by (cases x) auto
+ ultimately show P by (cases rule: extreal2_cases[of a b]) auto
+qed simp_all
+termination by (relation "{}") simp
+
+definition "x \<le> (y::extreal) \<longleftrightarrow> x < y \<or> x = y"
+
+lemma extreal_infty_less[simp]:
+ "x < \<infinity> \<longleftrightarrow> (x \<noteq> \<infinity>)"
+ "-\<infinity> < x \<longleftrightarrow> (x \<noteq> -\<infinity>)"
+ by (cases x, simp_all) (cases x, simp_all)
+
+lemma extreal_infty_less_eq[simp]:
+ "\<infinity> \<le> x \<longleftrightarrow> x = \<infinity>"
+ "x \<le> -\<infinity> \<longleftrightarrow> x = -\<infinity>"
+ by (auto simp add: less_eq_extreal_def)
+
+lemma extreal_less[simp]:
+ "extreal r < 0 \<longleftrightarrow> (r < 0)"
+ "0 < extreal r \<longleftrightarrow> (0 < r)"
+ "0 < \<infinity>"
+ "-\<infinity> < 0"
+ by (simp_all add: zero_extreal_def)
+
+lemma extreal_less_eq[simp]:
+ "x \<le> \<infinity>"
+ "-\<infinity> \<le> x"
+ "extreal r \<le> extreal p \<longleftrightarrow> r \<le> p"
+ "extreal r \<le> 0 \<longleftrightarrow> r \<le> 0"
+ "0 \<le> extreal r \<longleftrightarrow> 0 \<le> r"
+ by (auto simp add: less_eq_extreal_def zero_extreal_def)
+
+lemma extreal_infty_less_eq2:
+ "a \<le> b \<Longrightarrow> a = \<infinity> \<Longrightarrow> b = \<infinity>"
+ "a \<le> b \<Longrightarrow> b = -\<infinity> \<Longrightarrow> a = -\<infinity>"
+ by simp_all
+
+instance
+proof
+ fix x :: extreal show "x \<le> x"
+ by (cases x) simp_all
+ fix y :: extreal show "x < y \<longleftrightarrow> x \<le> y \<and> \<not> y \<le> x"
+ by (cases rule: extreal2_cases[of x y]) auto
+ show "x \<le> y \<or> y \<le> x "
+ by (cases rule: extreal2_cases[of x y]) auto
+ { assume "x \<le> y" "y \<le> x" then show "x = y"
+ by (cases rule: extreal2_cases[of x y]) auto }
+ { fix z assume "x \<le> y" "y \<le> z" then show "x \<le> z"
+ by (cases rule: extreal3_cases[of x y z]) auto }
+qed
+end
+
+instance extreal :: ordered_ab_semigroup_add
+proof
+ fix a b c :: extreal assume "a \<le> b" then show "c + a \<le> c + b"
+ by (cases rule: extreal3_cases[of a b c]) auto
+qed
+
+lemma extreal_MInfty_lessI[intro, simp]:
+ "a \<noteq> -\<infinity> \<Longrightarrow> -\<infinity> < a"
+ by (cases a) auto
+
+lemma extreal_less_PInfty[intro, simp]:
+ "a \<noteq> \<infinity> \<Longrightarrow> a < \<infinity>"
+ by (cases a) auto
+
+lemma extreal_less_extreal_Ex:
+ fixes a b :: extreal
+ shows "x < extreal r \<longleftrightarrow> x = -\<infinity> \<or> (\<exists>p. p < r \<and> x = extreal p)"
+ by (cases x) auto
+
+lemma less_PInf_Ex_of_nat: "x \<noteq> \<infinity> \<longleftrightarrow> (\<exists>n::nat. x < extreal (real n))"
+proof (cases x)
+ case (real r) then show ?thesis
+ using reals_Archimedean2[of r] by simp
+qed simp_all
+
+lemma extreal_add_mono:
+ fixes a b c d :: extreal assumes "a \<le> b" "c \<le> d" shows "a + c \<le> b + d"
+ using assms
+ apply (cases a)
+ apply (cases rule: extreal3_cases[of b c d], auto)
+ apply (cases rule: extreal3_cases[of b c d], auto)
+ done
+
+lemma extreal_minus_le_minus[simp]:
+ fixes a b :: extreal shows "- a \<le> - b \<longleftrightarrow> b \<le> a"
+ by (cases rule: extreal2_cases[of a b]) auto
+
+lemma extreal_minus_less_minus[simp]:
+ fixes a b :: extreal shows "- a < - b \<longleftrightarrow> b < a"
+ by (cases rule: extreal2_cases[of a b]) auto
+
+lemma extreal_le_real_iff:
+ "x \<le> real y \<longleftrightarrow> ((\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> extreal x \<le> y) \<and> (\<bar>y\<bar> = \<infinity> \<longrightarrow> x \<le> 0))"
+ by (cases y) auto
+
+lemma real_le_extreal_iff:
+ "real y \<le> x \<longleftrightarrow> ((\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> y \<le> extreal x) \<and> (\<bar>y\<bar> = \<infinity> \<longrightarrow> 0 \<le> x))"
+ by (cases y) auto
+
+lemma extreal_less_real_iff:
+ "x < real y \<longleftrightarrow> ((\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> extreal x < y) \<and> (\<bar>y\<bar> = \<infinity> \<longrightarrow> x < 0))"
+ by (cases y) auto
+
+lemma real_less_extreal_iff:
+ "real y < x \<longleftrightarrow> ((\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> y < extreal x) \<and> (\<bar>y\<bar> = \<infinity> \<longrightarrow> 0 < x))"
+ by (cases y) auto
+
+lemma real_of_extreal_positive_mono:
+ assumes "x \<noteq> \<infinity>" "y \<noteq> \<infinity>" "0 \<le> x" "x \<le> y"
+ shows "real x \<le> real y"
+ using assms by (cases rule: extreal2_cases[of x y]) auto
+
+lemma real_of_extreal_pos:
+ fixes x :: extreal shows "0 \<le> x \<Longrightarrow> 0 \<le> real x" by (cases x) auto
+
+lemmas real_of_extreal_ord_simps =
+ extreal_le_real_iff real_le_extreal_iff extreal_less_real_iff real_less_extreal_iff
+
+lemma extreal_dense:
+ fixes x y :: extreal assumes "x < y"
+ shows "EX z. x < z & z < y"
+proof -
+{ assume a: "x = (-\<infinity>)"
+ { assume "y = \<infinity>" hence ?thesis using a by (auto intro!: exI[of _ "0"]) }
+ moreover
+ { assume "y ~= \<infinity>"
+ with `x < y` obtain r where r: "y = extreal r" by (cases y) auto
+ hence ?thesis using `x < y` a by (auto intro!: exI[of _ "extreal (r - 1)"])
+ } ultimately have ?thesis by auto
+}
+moreover
+{ assume "x ~= (-\<infinity>)"
+ with `x < y` obtain p where p: "x = extreal p" by (cases x) auto
+ { assume "y = \<infinity>" hence ?thesis using `x < y` p
+ by (auto intro!: exI[of _ "extreal (p + 1)"]) }
+ moreover
+ { assume "y ~= \<infinity>"
+ with `x < y` obtain r where r: "y = extreal r" by (cases y) auto
+ with p `x < y` have "p < r" by auto
+ with dense obtain z where "p < z" "z < r" by auto
+ hence ?thesis using r p by (auto intro!: exI[of _ "extreal z"])
+ } ultimately have ?thesis by auto
+} ultimately show ?thesis by auto
+qed
+
+lemma extreal_dense2:
+ fixes x y :: extreal assumes "x < y"
+ shows "EX z. x < extreal z & extreal z < y"
+ by (metis extreal_dense[OF `x < y`] extreal_cases less_extreal.simps(2,3))
+
+lemma extreal_add_strict_mono:
+ fixes a b c d :: extreal
+ assumes "a = b" "0 \<le> a" "a \<noteq> \<infinity>" "c < d"
+ shows "a + c < b + d"
+ using assms by (cases rule: extreal3_cases[case_product extreal_cases, of a b c d]) auto
+
+lemma extreal_less_add: "\<bar>a\<bar> \<noteq> \<infinity> \<Longrightarrow> c < b \<Longrightarrow> a + c < a + b"
+ by (cases rule: extreal2_cases[of b c]) auto
+
+lemma extreal_uminus_eq_reorder: "- a = b \<longleftrightarrow> a = (-b::extreal)" by auto
+
+lemma extreal_uminus_less_reorder: "- a < b \<longleftrightarrow> -b < (a::extreal)"
+ by (subst (3) extreal_uminus_uminus[symmetric]) (simp only: extreal_minus_less_minus)
+
+lemma extreal_uminus_le_reorder: "- a \<le> b \<longleftrightarrow> -b \<le> (a::extreal)"
+ by (subst (3) extreal_uminus_uminus[symmetric]) (simp only: extreal_minus_le_minus)
+
+lemmas extreal_uminus_reorder =
+ extreal_uminus_eq_reorder extreal_uminus_less_reorder extreal_uminus_le_reorder
+
+lemma extreal_bot:
+ fixes x :: extreal assumes "\<And>B. x \<le> extreal B" shows "x = - \<infinity>"
+proof (cases x)
+ case (real r) with assms[of "r - 1"] show ?thesis by auto
+next case PInf with assms[of 0] show ?thesis by auto
+next case MInf then show ?thesis by simp
+qed
+
+lemma extreal_top:
+ fixes x :: extreal assumes "\<And>B. x \<ge> extreal B" shows "x = \<infinity>"
+proof (cases x)
+ case (real r) with assms[of "r + 1"] show ?thesis by auto
+next case MInf with assms[of 0] show ?thesis by auto
+next case PInf then show ?thesis by simp
+qed
+
+lemma
+ shows extreal_max[simp]: "extreal (max x y) = max (extreal x) (extreal y)"
+ and extreal_min[simp]: "extreal (min x y) = min (extreal x) (extreal y)"
+ by (simp_all add: min_def max_def)
+
+lemma extreal_max_0: "max 0 (extreal r) = extreal (max 0 r)"
+ by (auto simp: zero_extreal_def)
+
+lemma
+ fixes f :: "nat \<Rightarrow> extreal"
+ shows incseq_uminus[simp]: "incseq (\<lambda>x. - f x) \<longleftrightarrow> decseq f"
+ and decseq_uminus[simp]: "decseq (\<lambda>x. - f x) \<longleftrightarrow> incseq f"
+ unfolding decseq_def incseq_def by auto
+
+lemma extreal_add_nonneg_nonneg:
+ fixes a b :: extreal shows "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> a + b"
+ using add_mono[of 0 a 0 b] by simp
+
+lemma image_eqD: "f ` A = B \<Longrightarrow> (\<forall>x\<in>A. f x \<in> B)"
+ by auto
+
+lemma incseq_setsumI:
+ fixes f :: "nat \<Rightarrow> 'a::{comm_monoid_add, ordered_ab_semigroup_add}"
+ assumes "\<And>i. 0 \<le> f i"
+ shows "incseq (\<lambda>i. setsum f {..< i})"
+proof (intro incseq_SucI)
+ fix n have "setsum f {..< n} + 0 \<le> setsum f {..<n} + f n"
+ using assms by (rule add_left_mono)
+ then show "setsum f {..< n} \<le> setsum f {..< Suc n}"
+ by auto
+qed
+
+lemma incseq_setsumI2:
+ fixes f :: "'i \<Rightarrow> nat \<Rightarrow> 'a::{comm_monoid_add, ordered_ab_semigroup_add}"
+ assumes "\<And>n. n \<in> A \<Longrightarrow> incseq (f n)"
+ shows "incseq (\<lambda>i. \<Sum>n\<in>A. f n i)"
+ using assms unfolding incseq_def by (auto intro: setsum_mono)
+
+subsubsection "Multiplication"
+
+instantiation extreal :: "{comm_monoid_mult, sgn}"
+begin
+
+definition "1 = extreal 1"
+
+function sgn_extreal where
+ "sgn (extreal r) = extreal (sgn r)"
+| "sgn \<infinity> = 1"
+| "sgn (-\<infinity>) = -1"
+by (auto intro: extreal_cases)
+termination proof qed (rule wf_empty)
+
+function times_extreal where
+"extreal r * extreal p = extreal (r * p)" |
+"extreal r * \<infinity> = (if r = 0 then 0 else if r > 0 then \<infinity> else -\<infinity>)" |
+"\<infinity> * extreal r = (if r = 0 then 0 else if r > 0 then \<infinity> else -\<infinity>)" |
+"extreal r * -\<infinity> = (if r = 0 then 0 else if r > 0 then -\<infinity> else \<infinity>)" |
+"-\<infinity> * extreal r = (if r = 0 then 0 else if r > 0 then -\<infinity> else \<infinity>)" |
+"\<infinity> * \<infinity> = \<infinity>" |
+"-\<infinity> * \<infinity> = -\<infinity>" |
+"\<infinity> * -\<infinity> = -\<infinity>" |
+"-\<infinity> * -\<infinity> = \<infinity>"
+proof -
+ case (goal1 P x)
+ moreover then obtain a b where "x = (a, b)" by (cases x) auto
+ ultimately show P by (cases rule: extreal2_cases[of a b]) auto
+qed simp_all
+termination by (relation "{}") simp
+
+instance
+proof
+ fix a :: extreal show "1 * a = a"
+ by (cases a) (simp_all add: one_extreal_def)
+ fix b :: extreal show "a * b = b * a"
+ by (cases rule: extreal2_cases[of a b]) simp_all
+ fix c :: extreal show "a * b * c = a * (b * c)"
+ by (cases rule: extreal3_cases[of a b c])
+ (simp_all add: zero_extreal_def zero_less_mult_iff)
+qed
+end
+
+lemma abs_extreal_one[simp]: "\<bar>1\<bar> = (1::extreal)"
+ unfolding one_extreal_def by simp
+
+lemma extreal_mult_zero[simp]:
+ fixes a :: extreal shows "a * 0 = 0"
+ by (cases a) (simp_all add: zero_extreal_def)
+
+lemma extreal_zero_mult[simp]:
+ fixes a :: extreal shows "0 * a = 0"
+ by (cases a) (simp_all add: zero_extreal_def)
+
+lemma extreal_m1_less_0[simp]:
+ "-(1::extreal) < 0"
+ by (simp add: zero_extreal_def one_extreal_def)
+
+lemma extreal_zero_m1[simp]:
+ "1 \<noteq> (0::extreal)"
+ by (simp add: zero_extreal_def one_extreal_def)
+
+lemma extreal_times_0[simp]:
+ fixes x :: extreal shows "0 * x = 0"
+ by (cases x) (auto simp: zero_extreal_def)
+
+lemma extreal_times[simp]:
+ "1 \<noteq> \<infinity>" "\<infinity> \<noteq> 1"
+ "1 \<noteq> -\<infinity>" "-\<infinity> \<noteq> 1"
+ by (auto simp add: times_extreal_def one_extreal_def)
+
+lemma extreal_plus_1[simp]:
+ "1 + extreal r = extreal (r + 1)" "extreal r + 1 = extreal (r + 1)"
+ "1 + -\<infinity> = -\<infinity>" "-\<infinity> + 1 = -\<infinity>"
+ unfolding one_extreal_def by auto
+
+lemma extreal_zero_times[simp]:
+ fixes a b :: extreal shows "a * b = 0 \<longleftrightarrow> a = 0 \<or> b = 0"
+ by (cases rule: extreal2_cases[of a b]) auto
+
+lemma extreal_mult_eq_PInfty[simp]:
+ shows "a * b = \<infinity> \<longleftrightarrow>
+ (a = \<infinity> \<and> b > 0) \<or> (a > 0 \<and> b = \<infinity>) \<or> (a = -\<infinity> \<and> b < 0) \<or> (a < 0 \<and> b = -\<infinity>)"
+ by (cases rule: extreal2_cases[of a b]) auto
+
+lemma extreal_mult_eq_MInfty[simp]:
+ shows "a * b = -\<infinity> \<longleftrightarrow>
+ (a = \<infinity> \<and> b < 0) \<or> (a < 0 \<and> b = \<infinity>) \<or> (a = -\<infinity> \<and> b > 0) \<or> (a > 0 \<and> b = -\<infinity>)"
+ by (cases rule: extreal2_cases[of a b]) auto
+
+lemma extreal_0_less_1[simp]: "0 < (1::extreal)"
+ by (simp_all add: zero_extreal_def one_extreal_def)
+
+lemma extreal_zero_one[simp]: "0 \<noteq> (1::extreal)"
+ by (simp_all add: zero_extreal_def one_extreal_def)
+
+lemma extreal_mult_minus_left[simp]:
+ fixes a b :: extreal shows "-a * b = - (a * b)"
+ by (cases rule: extreal2_cases[of a b]) auto
+
+lemma extreal_mult_minus_right[simp]:
+ fixes a b :: extreal shows "a * -b = - (a * b)"
+ by (cases rule: extreal2_cases[of a b]) auto
+
+lemma extreal_mult_infty[simp]:
+ "a * \<infinity> = (if a = 0 then 0 else if 0 < a then \<infinity> else - \<infinity>)"
+ by (cases a) auto
+
+lemma extreal_infty_mult[simp]:
+ "\<infinity> * a = (if a = 0 then 0 else if 0 < a then \<infinity> else - \<infinity>)"
+ by (cases a) auto
+
+lemma extreal_mult_strict_right_mono:
+ assumes "a < b" and "0 < c" "c < \<infinity>"
+ shows "a * c < b * c"
+ using assms
+ by (cases rule: extreal3_cases[of a b c])
+ (auto simp: zero_le_mult_iff extreal_less_PInfty)
+
+lemma extreal_mult_strict_left_mono:
+ "\<lbrakk> a < b ; 0 < c ; c < \<infinity>\<rbrakk> \<Longrightarrow> c * a < c * b"
+ using extreal_mult_strict_right_mono by (simp add: mult_commute[of c])
+
+lemma extreal_mult_right_mono:
+ fixes a b c :: extreal shows "\<lbrakk>a \<le> b; 0 \<le> c\<rbrakk> \<Longrightarrow> a*c \<le> b*c"
+ using assms
+ apply (cases "c = 0") apply simp
+ by (cases rule: extreal3_cases[of a b c])
+ (auto simp: zero_le_mult_iff extreal_less_PInfty)
+
+lemma extreal_mult_left_mono:
+ fixes a b c :: extreal shows "\<lbrakk>a \<le> b; 0 \<le> c\<rbrakk> \<Longrightarrow> c * a \<le> c * b"
+ using extreal_mult_right_mono by (simp add: mult_commute[of c])
+
+lemma zero_less_one_extreal[simp]: "0 \<le> (1::extreal)"
+ by (simp add: one_extreal_def zero_extreal_def)
+
+lemma extreal_0_le_mult[simp]: "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> a * (b :: extreal)"
+ by (cases rule: extreal2_cases[of a b]) (auto simp: mult_nonneg_nonneg)
+
+lemma extreal_right_distrib:
+ fixes r a b :: extreal shows "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> r * (a + b) = r * a + r * b"
+ by (cases rule: extreal3_cases[of r a b]) (simp_all add: field_simps)
+
+lemma extreal_left_distrib:
+ fixes r a b :: extreal shows "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> (a + b) * r = a * r + b * r"
+ by (cases rule: extreal3_cases[of r a b]) (simp_all add: field_simps)
+
+lemma extreal_mult_le_0_iff:
+ fixes a b :: extreal
+ shows "a * b \<le> 0 \<longleftrightarrow> (0 \<le> a \<and> b \<le> 0) \<or> (a \<le> 0 \<and> 0 \<le> b)"
+ by (cases rule: extreal2_cases[of a b]) (simp_all add: mult_le_0_iff)
+
+lemma extreal_zero_le_0_iff:
+ fixes a b :: extreal
+ shows "0 \<le> a * b \<longleftrightarrow> (0 \<le> a \<and> 0 \<le> b) \<or> (a \<le> 0 \<and> b \<le> 0)"
+ by (cases rule: extreal2_cases[of a b]) (simp_all add: zero_le_mult_iff)
+
+lemma extreal_mult_less_0_iff:
+ fixes a b :: extreal
+ shows "a * b < 0 \<longleftrightarrow> (0 < a \<and> b < 0) \<or> (a < 0 \<and> 0 < b)"
+ by (cases rule: extreal2_cases[of a b]) (simp_all add: mult_less_0_iff)
+
+lemma extreal_zero_less_0_iff:
+ fixes a b :: extreal
+ shows "0 < a * b \<longleftrightarrow> (0 < a \<and> 0 < b) \<or> (a < 0 \<and> b < 0)"
+ by (cases rule: extreal2_cases[of a b]) (simp_all add: zero_less_mult_iff)
+
+lemma extreal_distrib:
+ fixes a b c :: extreal
+ assumes "a \<noteq> \<infinity> \<or> b \<noteq> -\<infinity>" "a \<noteq> -\<infinity> \<or> b \<noteq> \<infinity>" "\<bar>c\<bar> \<noteq> \<infinity>"
+ shows "(a + b) * c = a * c + b * c"
+ using assms
+ by (cases rule: extreal3_cases[of a b c]) (simp_all add: field_simps)
+
+lemma extreal_le_epsilon:
+ fixes x y :: extreal
+ assumes "ALL e. 0 < e --> x <= y + e"
+ shows "x <= y"
+proof-
+{ assume a: "EX r. y = extreal r"
+ from this obtain r where r_def: "y = extreal r" by auto
+ { assume "x=(-\<infinity>)" hence ?thesis by auto }
+ moreover
+ { assume "~(x=(-\<infinity>))"
+ from this obtain p where p_def: "x = extreal p"
+ using a assms[rule_format, of 1] by (cases x) auto
+ { fix e have "0 < e --> p <= r + e"
+ using assms[rule_format, of "extreal e"] p_def r_def by auto }
+ hence "p <= r" apply (subst field_le_epsilon) by auto
+ hence ?thesis using r_def p_def by auto
+ } ultimately have ?thesis by blast
+}
+moreover
+{ assume "y=(-\<infinity>) | y=\<infinity>" hence ?thesis
+ using assms[rule_format, of 1] by (cases x) auto
+} ultimately show ?thesis by (cases y) auto
+qed
+
+
+lemma extreal_le_epsilon2:
+ fixes x y :: extreal
+ assumes "ALL e. 0 < e --> x <= y + extreal e"
+ shows "x <= y"
+proof-
+{ fix e :: extreal assume "e>0"
+ { assume "e=\<infinity>" hence "x<=y+e" by auto }
+ moreover
+ { assume "e~=\<infinity>"
+ from this obtain r where "e = extreal r" using `e>0` apply (cases e) by auto
+ hence "x<=y+e" using assms[rule_format, of r] `e>0` by auto
+ } ultimately have "x<=y+e" by blast
+} from this show ?thesis using extreal_le_epsilon by auto
+qed
+
+lemma extreal_le_real:
+ fixes x y :: extreal
+ assumes "ALL z. x <= extreal z --> y <= extreal z"
+ shows "y <= x"
+by (metis assms extreal.exhaust extreal_bot extreal_less_eq(1)
+ extreal_less_eq(2) order_refl uminus_extreal.simps(2))
+
+lemma extreal_le_extreal:
+ fixes x y :: extreal
+ assumes "\<And>B. B < x \<Longrightarrow> B <= y"
+ shows "x <= y"
+by (metis assms extreal_dense leD linorder_le_less_linear)
+
+lemma extreal_ge_extreal:
+ fixes x y :: extreal
+ assumes "ALL B. B>x --> B >= y"
+ shows "x >= y"
+by (metis assms extreal_dense leD linorder_le_less_linear)
+
+subsubsection {* Power *}
+
+instantiation extreal :: power
+begin
+primrec power_extreal where
+ "power_extreal x 0 = 1" |
+ "power_extreal x (Suc n) = x * x ^ n"
+instance ..
+end
+
+lemma extreal_power[simp]: "(extreal x) ^ n = extreal (x^n)"
+ by (induct n) (auto simp: one_extreal_def)
+
+lemma extreal_power_PInf[simp]: "\<infinity> ^ n = (if n = 0 then 1 else \<infinity>)"
+ by (induct n) (auto simp: one_extreal_def)
+
+lemma extreal_power_uminus[simp]:
+ fixes x :: extreal
+ shows "(- x) ^ n = (if even n then x ^ n else - (x^n))"
+ by (induct n) (auto simp: one_extreal_def)
+
+lemma extreal_power_number_of[simp]:
+ "(number_of num :: extreal) ^ n = extreal (number_of num ^ n)"
+ by (induct n) (auto simp: one_extreal_def)
+
+lemma zero_le_power_extreal[simp]:
+ fixes a :: extreal assumes "0 \<le> a"
+ shows "0 \<le> a ^ n"
+ using assms by (induct n) (auto simp: extreal_zero_le_0_iff)
+
+subsubsection {* Subtraction *}
+
+lemma extreal_minus_minus_image[simp]:
+ fixes S :: "extreal set"
+ shows "uminus ` uminus ` S = S"
+ by (auto simp: image_iff)
+
+lemma extreal_uminus_lessThan[simp]:
+ fixes a :: extreal shows "uminus ` {..<a} = {-a<..}"
+proof (safe intro!: image_eqI)
+ fix x assume "-a < x"
+ then have "- x < - (- a)" by (simp del: extreal_uminus_uminus)
+ then show "- x < a" by simp
+qed auto
+
+lemma extreal_uminus_greaterThan[simp]:
+ "uminus ` {(a::extreal)<..} = {..<-a}"
+ by (metis extreal_uminus_lessThan extreal_uminus_uminus
+ extreal_minus_minus_image)
+
+instantiation extreal :: minus
+begin
+definition "x - y = x + -(y::extreal)"
+instance ..
+end
+
+lemma extreal_minus[simp]:
+ "extreal r - extreal p = extreal (r - p)"
+ "-\<infinity> - extreal r = -\<infinity>"
+ "extreal r - \<infinity> = -\<infinity>"
+ "\<infinity> - x = \<infinity>"
+ "-\<infinity> - \<infinity> = -\<infinity>"
+ "x - -y = x + y"
+ "x - 0 = x"
+ "0 - x = -x"
+ by (simp_all add: minus_extreal_def)
+
+lemma extreal_x_minus_x[simp]:
+ "x - x = (if \<bar>x\<bar> = \<infinity> then \<infinity> else 0)"
+ by (cases x) simp_all
+
+lemma extreal_eq_minus_iff:
+ fixes x y z :: extreal
+ shows "x = z - y \<longleftrightarrow>
+ (\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> x + y = z) \<and>
+ (y = -\<infinity> \<longrightarrow> x = \<infinity>) \<and>
+ (y = \<infinity> \<longrightarrow> z = \<infinity> \<longrightarrow> x = \<infinity>) \<and>
+ (y = \<infinity> \<longrightarrow> z \<noteq> \<infinity> \<longrightarrow> x = -\<infinity>)"
+ by (cases rule: extreal3_cases[of x y z]) auto
+
+lemma extreal_eq_minus:
+ fixes x y z :: extreal
+ shows "\<bar>y\<bar> \<noteq> \<infinity> \<Longrightarrow> x = z - y \<longleftrightarrow> x + y = z"
+ by (auto simp: extreal_eq_minus_iff)
+
+lemma extreal_less_minus_iff:
+ fixes x y z :: extreal
+ shows "x < z - y \<longleftrightarrow>
+ (y = \<infinity> \<longrightarrow> z = \<infinity> \<and> x \<noteq> \<infinity>) \<and>
+ (y = -\<infinity> \<longrightarrow> x \<noteq> \<infinity>) \<and>
+ (\<bar>y\<bar> \<noteq> \<infinity>\<longrightarrow> x + y < z)"
+ by (cases rule: extreal3_cases[of x y z]) auto
+
+lemma extreal_less_minus:
+ fixes x y z :: extreal
+ shows "\<bar>y\<bar> \<noteq> \<infinity> \<Longrightarrow> x < z - y \<longleftrightarrow> x + y < z"
+ by (auto simp: extreal_less_minus_iff)
+
+lemma extreal_le_minus_iff:
+ fixes x y z :: extreal
+ shows "x \<le> z - y \<longleftrightarrow>
+ (y = \<infinity> \<longrightarrow> z \<noteq> \<infinity> \<longrightarrow> x = -\<infinity>) \<and>
+ (\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> x + y \<le> z)"
+ by (cases rule: extreal3_cases[of x y z]) auto
+
+lemma extreal_le_minus:
+ fixes x y z :: extreal
+ shows "\<bar>y\<bar> \<noteq> \<infinity> \<Longrightarrow> x \<le> z - y \<longleftrightarrow> x + y \<le> z"
+ by (auto simp: extreal_le_minus_iff)
+
+lemma extreal_minus_less_iff:
+ fixes x y z :: extreal
+ shows "x - y < z \<longleftrightarrow>
+ y \<noteq> -\<infinity> \<and> (y = \<infinity> \<longrightarrow> x \<noteq> \<infinity> \<and> z \<noteq> -\<infinity>) \<and>
+ (y \<noteq> \<infinity> \<longrightarrow> x < z + y)"
+ by (cases rule: extreal3_cases[of x y z]) auto
+
+lemma extreal_minus_less:
+ fixes x y z :: extreal
+ shows "\<bar>y\<bar> \<noteq> \<infinity> \<Longrightarrow> x - y < z \<longleftrightarrow> x < z + y"
+ by (auto simp: extreal_minus_less_iff)
+
+lemma extreal_minus_le_iff:
+ fixes x y z :: extreal
+ shows "x - y \<le> z \<longleftrightarrow>
+ (y = -\<infinity> \<longrightarrow> z = \<infinity>) \<and>
+ (y = \<infinity> \<longrightarrow> x = \<infinity> \<longrightarrow> z = \<infinity>) \<and>
+ (\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> x \<le> z + y)"
+ by (cases rule: extreal3_cases[of x y z]) auto
+
+lemma extreal_minus_le:
+ fixes x y z :: extreal
+ shows "\<bar>y\<bar> \<noteq> \<infinity> \<Longrightarrow> x - y \<le> z \<longleftrightarrow> x \<le> z + y"
+ by (auto simp: extreal_minus_le_iff)
+
+lemma extreal_minus_eq_minus_iff:
+ fixes a b c :: extreal
+ shows "a - b = a - c \<longleftrightarrow>
+ b = c \<or> a = \<infinity> \<or> (a = -\<infinity> \<and> b \<noteq> -\<infinity> \<and> c \<noteq> -\<infinity>)"
+ by (cases rule: extreal3_cases[of a b c]) auto
+
+lemma extreal_add_le_add_iff:
+ "c + a \<le> c + b \<longleftrightarrow>
+ a \<le> b \<or> c = \<infinity> \<or> (c = -\<infinity> \<and> a \<noteq> \<infinity> \<and> b \<noteq> \<infinity>)"
+ by (cases rule: extreal3_cases[of a b c]) (simp_all add: field_simps)
+
+lemma extreal_mult_le_mult_iff:
+ "\<bar>c\<bar> \<noteq> \<infinity> \<Longrightarrow> c * a \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)"
+ by (cases rule: extreal3_cases[of a b c]) (simp_all add: mult_le_cancel_left)
+
+lemma extreal_minus_mono:
+ fixes A B C D :: extreal assumes "A \<le> B" "D \<le> C"
+ shows "A - C \<le> B - D"
+ using assms
+ by (cases rule: extreal3_cases[case_product extreal_cases, of A B C D]) simp_all
+
+lemma real_of_extreal_minus:
+ "real (a - b) = (if \<bar>a\<bar> = \<infinity> \<or> \<bar>b\<bar> = \<infinity> then 0 else real a - real b)"
+ by (cases rule: extreal2_cases[of a b]) auto
+
+lemma extreal_diff_positive:
+ fixes a b :: extreal shows "a \<le> b \<Longrightarrow> 0 \<le> b - a"
+ by (cases rule: extreal2_cases[of a b]) auto
+
+lemma extreal_between:
+ fixes x e :: extreal
+ assumes "\<bar>x\<bar> \<noteq> \<infinity>" "0 < e"
+ shows "x - e < x" "x < x + e"
+using assms apply (cases x, cases e) apply auto
+using assms by (cases x, cases e) auto
+
+subsubsection {* Division *}
+
+instantiation extreal :: inverse
+begin
+
+function inverse_extreal where
+"inverse (extreal r) = (if r = 0 then \<infinity> else extreal (inverse r))" |
+"inverse \<infinity> = 0" |
+"inverse (-\<infinity>) = 0"
+ by (auto intro: extreal_cases)
+termination by (relation "{}") simp
+
+definition "x / y = x * inverse (y :: extreal)"
+
+instance proof qed
+end
+
+lemma extreal_inverse[simp]:
+ "inverse 0 = \<infinity>"
+ "inverse (1::extreal) = 1"
+ by (simp_all add: one_extreal_def zero_extreal_def)
+
+lemma extreal_divide[simp]:
+ "extreal r / extreal p = (if p = 0 then extreal r * \<infinity> else extreal (r / p))"
+ unfolding divide_extreal_def by (auto simp: divide_real_def)
+
+lemma extreal_divide_same[simp]:
+ "x / x = (if \<bar>x\<bar> = \<infinity> \<or> x = 0 then 0 else 1)"
+ by (cases x)
+ (simp_all add: divide_real_def divide_extreal_def one_extreal_def)
+
+lemma extreal_inv_inv[simp]:
+ "inverse (inverse x) = (if x \<noteq> -\<infinity> then x else \<infinity>)"
+ by (cases x) auto
+
+lemma extreal_inverse_minus[simp]:
+ "inverse (- x) = (if x = 0 then \<infinity> else -inverse x)"
+ by (cases x) simp_all
+
+lemma extreal_uminus_divide[simp]:
+ fixes x y :: extreal shows "- x / y = - (x / y)"
+ unfolding divide_extreal_def by simp
+
+lemma extreal_divide_Infty[simp]:
+ "x / \<infinity> = 0" "x / -\<infinity> = 0"
+ unfolding divide_extreal_def by simp_all
+
+lemma extreal_divide_one[simp]:
+ "x / 1 = (x::extreal)"
+ unfolding divide_extreal_def by simp
+
+lemma extreal_divide_extreal[simp]:
+ "\<infinity> / extreal r = (if 0 \<le> r then \<infinity> else -\<infinity>)"
+ unfolding divide_extreal_def by simp
+
+lemma zero_le_divide_extreal[simp]:
+ fixes a :: extreal assumes "0 \<le> a" "0 \<le> b"
+ shows "0 \<le> a / b"
+ using assms by (cases rule: extreal2_cases[of a b]) (auto simp: zero_le_divide_iff)
+
+lemma extreal_le_divide_pos:
+ "x > 0 \<Longrightarrow> x \<noteq> \<infinity> \<Longrightarrow> y \<le> z / x \<longleftrightarrow> x * y \<le> z"
+ by (cases rule: extreal3_cases[of x y z]) (auto simp: field_simps)
+
+lemma extreal_divide_le_pos:
+ "x > 0 \<Longrightarrow> x \<noteq> \<infinity> \<Longrightarrow> z / x \<le> y \<longleftrightarrow> z \<le> x * y"
+ by (cases rule: extreal3_cases[of x y z]) (auto simp: field_simps)
+
+lemma extreal_le_divide_neg:
+ "x < 0 \<Longrightarrow> x \<noteq> -\<infinity> \<Longrightarrow> y \<le> z / x \<longleftrightarrow> z \<le> x * y"
+ by (cases rule: extreal3_cases[of x y z]) (auto simp: field_simps)
+
+lemma extreal_divide_le_neg:
+ "x < 0 \<Longrightarrow> x \<noteq> -\<infinity> \<Longrightarrow> z / x \<le> y \<longleftrightarrow> x * y \<le> z"
+ by (cases rule: extreal3_cases[of x y z]) (auto simp: field_simps)
+
+lemma extreal_inverse_antimono_strict:
+ fixes x y :: extreal
+ shows "0 \<le> x \<Longrightarrow> x < y \<Longrightarrow> inverse y < inverse x"
+ by (cases rule: extreal2_cases[of x y]) auto
+
+lemma extreal_inverse_antimono:
+ fixes x y :: extreal
+ shows "0 \<le> x \<Longrightarrow> x <= y \<Longrightarrow> inverse y <= inverse x"
+ by (cases rule: extreal2_cases[of x y]) auto
+
+lemma inverse_inverse_Pinfty_iff[simp]:
+ "inverse x = \<infinity> \<longleftrightarrow> x = 0"
+ by (cases x) auto
+
+lemma extreal_inverse_eq_0:
+ "inverse x = 0 \<longleftrightarrow> x = \<infinity> \<or> x = -\<infinity>"
+ by (cases x) auto
+
+lemma extreal_0_gt_inverse:
+ fixes x :: extreal shows "0 < inverse x \<longleftrightarrow> x \<noteq> \<infinity> \<and> 0 \<le> x"
+ by (cases x) auto
+
+lemma extreal_mult_less_right:
+ assumes "b * a < c * a" "0 < a" "a < \<infinity>"
+ shows "b < c"
+ using assms
+ by (cases rule: extreal3_cases[of a b c])
+ (auto split: split_if_asm simp: zero_less_mult_iff zero_le_mult_iff)
+
+lemma extreal_power_divide:
+ "y \<noteq> 0 \<Longrightarrow> (x / y :: extreal) ^ n = x^n / y^n"
+ by (cases rule: extreal2_cases[of x y])
+ (auto simp: one_extreal_def zero_extreal_def power_divide not_le
+ power_less_zero_eq zero_le_power_iff)
+
+lemma extreal_le_mult_one_interval:
+ fixes x y :: extreal
+ assumes y: "y \<noteq> -\<infinity>"
+ assumes z: "\<And>z. \<lbrakk> 0 < z ; z < 1 \<rbrakk> \<Longrightarrow> z * x \<le> y"
+ shows "x \<le> y"
+proof (cases x)
+ case PInf with z[of "1 / 2"] show "x \<le> y" by (simp add: one_extreal_def)
+next
+ case (real r) note r = this
+ show "x \<le> y"
+ proof (cases y)
+ case (real p) note p = this
+ have "r \<le> p"
+ proof (rule field_le_mult_one_interval)
+ fix z :: real assume "0 < z" and "z < 1"
+ with z[of "extreal z"]
+ show "z * r \<le> p" using p r by (auto simp: zero_le_mult_iff one_extreal_def)
+ qed
+ then show "x \<le> y" using p r by simp
+ qed (insert y, simp_all)
+qed simp
+
+subsection "Complete lattice"
+
+instantiation extreal :: lattice
+begin
+definition [simp]: "sup x y = (max x y :: extreal)"
+definition [simp]: "inf x y = (min x y :: extreal)"
+instance proof qed simp_all
+end
+
+instantiation extreal :: complete_lattice
+begin
+
+definition "bot = -\<infinity>"
+definition "top = \<infinity>"
+
+definition "Sup S = (LEAST z. ALL x:S. x <= z :: extreal)"
+definition "Inf S = (GREATEST z. ALL x:S. z <= x :: extreal)"
+
+lemma extreal_complete_Sup:
+ fixes S :: "extreal set" assumes "S \<noteq> {}"
+ shows "\<exists>x. (\<forall>y\<in>S. y \<le> x) \<and> (\<forall>z. (\<forall>y\<in>S. y \<le> z) \<longrightarrow> x \<le> z)"
+proof cases
+ assume "\<exists>x. \<forall>a\<in>S. a \<le> extreal x"
+ then obtain y where y: "\<And>a. a\<in>S \<Longrightarrow> a \<le> extreal y" by auto
+ then have "\<infinity> \<notin> S" by force
+ show ?thesis
+ proof cases
+ assume "S = {-\<infinity>}"
+ then show ?thesis by (auto intro!: exI[of _ "-\<infinity>"])
+ next
+ assume "S \<noteq> {-\<infinity>}"
+ with `S \<noteq> {}` `\<infinity> \<notin> S` obtain x where "x \<in> S - {-\<infinity>}" "x \<noteq> \<infinity>" by auto
+ with y `\<infinity> \<notin> S` have "\<forall>z\<in>real ` (S - {-\<infinity>}). z \<le> y"
+ by (auto simp: real_of_extreal_ord_simps)
+ with reals_complete2[of "real ` (S - {-\<infinity>})"] `x \<in> S - {-\<infinity>}`
+ obtain s where s:
+ "\<forall>y\<in>S - {-\<infinity>}. real y \<le> s" "\<And>z. (\<forall>y\<in>S - {-\<infinity>}. real y \<le> z) \<Longrightarrow> s \<le> z"
+ by auto
+ show ?thesis
+ proof (safe intro!: exI[of _ "extreal s"])
+ fix z assume "z \<in> S" with `\<infinity> \<notin> S` show "z \<le> extreal s"
+ proof (cases z)
+ case (real r)
+ then show ?thesis
+ using s(1)[rule_format, of z] `z \<in> S` `z = extreal r` by auto
+ qed auto
+ next
+ fix z assume *: "\<forall>y\<in>S. y \<le> z"
+ with `S \<noteq> {-\<infinity>}` `S \<noteq> {}` show "extreal s \<le> z"
+ proof (cases z)
+ case (real u)
+ with * have "s \<le> u"
+ by (intro s(2)[of u]) (auto simp: real_of_extreal_ord_simps)
+ then show ?thesis using real by simp
+ qed auto
+ qed
+ qed
+next
+ assume *: "\<not> (\<exists>x. \<forall>a\<in>S. a \<le> extreal x)"
+ show ?thesis
+ proof (safe intro!: exI[of _ \<infinity>])
+ fix y assume **: "\<forall>z\<in>S. z \<le> y"
+ with * show "\<infinity> \<le> y"
+ proof (cases y)
+ case MInf with * ** show ?thesis by (force simp: not_le)
+ qed auto
+ qed simp
+qed
+
+lemma extreal_complete_Inf:
+ fixes S :: "extreal set" assumes "S ~= {}"
+ shows "EX x. (ALL y:S. x <= y) & (ALL z. (ALL y:S. z <= y) --> z <= x)"
+proof-
+def S1 == "uminus ` S"
+hence "S1 ~= {}" using assms by auto
+from this obtain x where x_def: "(ALL y:S1. y <= x) & (ALL z. (ALL y:S1. y <= z) --> x <= z)"
+ using extreal_complete_Sup[of S1] by auto
+{ fix z assume "ALL y:S. z <= y"
+ hence "ALL y:S1. y <= -z" unfolding S1_def by auto
+ hence "x <= -z" using x_def by auto
+ hence "z <= -x"
+ apply (subst extreal_uminus_uminus[symmetric])
+ unfolding extreal_minus_le_minus . }
+moreover have "(ALL y:S. -x <= y)"
+ using x_def unfolding S1_def
+ apply simp
+ apply (subst (3) extreal_uminus_uminus[symmetric])
+ unfolding extreal_minus_le_minus by simp
+ultimately show ?thesis by auto
+qed
+
+lemma extreal_complete_uminus_eq:
+ fixes S :: "extreal set"
+ shows "(\<forall>y\<in>uminus`S. y \<le> x) \<and> (\<forall>z. (\<forall>y\<in>uminus`S. y \<le> z) \<longrightarrow> x \<le> z)
+ \<longleftrightarrow> (\<forall>y\<in>S. -x \<le> y) \<and> (\<forall>z. (\<forall>y\<in>S. z \<le> y) \<longrightarrow> z \<le> -x)"
+ by simp (metis extreal_minus_le_minus extreal_uminus_uminus)
+
+lemma extreal_Sup_uminus_image_eq:
+ fixes S :: "extreal set"
+ shows "Sup (uminus ` S) = - Inf S"
+proof cases
+ assume "S = {}"
+ moreover have "(THE x. All (op \<le> x)) = (-\<infinity>::extreal)"
+ by (rule the_equality) (auto intro!: extreal_bot)
+ moreover have "(SOME x. \<forall>y. y \<le> x) = (\<infinity>::extreal)"
+ by (rule some_equality) (auto intro!: extreal_top)
+ ultimately show ?thesis unfolding Inf_extreal_def Sup_extreal_def
+ Least_def Greatest_def GreatestM_def by simp
+next
+ assume "S \<noteq> {}"
+ with extreal_complete_Sup[of "uminus`S"]
+ obtain x where x: "(\<forall>y\<in>S. -x \<le> y) \<and> (\<forall>z. (\<forall>y\<in>S. z \<le> y) \<longrightarrow> z \<le> -x)"
+ unfolding extreal_complete_uminus_eq by auto
+ show "Sup (uminus ` S) = - Inf S"
+ unfolding Inf_extreal_def Greatest_def GreatestM_def
+ proof (intro someI2[of _ _ "\<lambda>x. Sup (uminus`S) = - x"])
+ show "(\<forall>y\<in>S. -x \<le> y) \<and> (\<forall>y. (\<forall>z\<in>S. y \<le> z) \<longrightarrow> y \<le> -x)"
+ using x .
+ fix x' assume "(\<forall>y\<in>S. x' \<le> y) \<and> (\<forall>y. (\<forall>z\<in>S. y \<le> z) \<longrightarrow> y \<le> x')"
+ then have "(\<forall>y\<in>uminus`S. y \<le> - x') \<and> (\<forall>y. (\<forall>z\<in>uminus`S. z \<le> y) \<longrightarrow> - x' \<le> y)"
+ unfolding extreal_complete_uminus_eq by simp
+ then show "Sup (uminus ` S) = -x'"
+ unfolding Sup_extreal_def extreal_uminus_eq_iff
+ by (intro Least_equality) auto
+ qed
+qed
+
+instance
+proof
+ { fix x :: extreal and A
+ show "bot <= x" by (cases x) (simp_all add: bot_extreal_def)
+ show "x <= top" by (simp add: top_extreal_def) }
+
+ { fix x :: extreal and A assume "x : A"
+ with extreal_complete_Sup[of A]
+ obtain s where s: "\<forall>y\<in>A. y <= s" "\<forall>z. (\<forall>y\<in>A. y <= z) \<longrightarrow> s <= z" by auto
+ hence "x <= s" using `x : A` by auto
+ also have "... = Sup A" using s unfolding Sup_extreal_def
+ by (auto intro!: Least_equality[symmetric])
+ finally show "x <= Sup A" . }
+ note le_Sup = this
+
+ { fix x :: extreal and A assume *: "!!z. (z : A ==> z <= x)"
+ show "Sup A <= x"
+ proof (cases "A = {}")
+ case True
+ hence "Sup A = -\<infinity>" unfolding Sup_extreal_def
+ by (auto intro!: Least_equality)
+ thus "Sup A <= x" by simp
+ next
+ case False
+ with extreal_complete_Sup[of A]
+ obtain s where s: "\<forall>y\<in>A. y <= s" "\<forall>z. (\<forall>y\<in>A. y <= z) \<longrightarrow> s <= z" by auto
+ hence "Sup A = s"
+ unfolding Sup_extreal_def by (auto intro!: Least_equality)
+ also have "s <= x" using * s by auto
+ finally show "Sup A <= x" .
+ qed }
+ note Sup_le = this
+
+ { fix x :: extreal and A assume "x \<in> A"
+ with le_Sup[of "-x" "uminus`A"] show "Inf A \<le> x"
+ unfolding extreal_Sup_uminus_image_eq by simp }
+
+ { fix x :: extreal and A assume *: "!!z. (z : A ==> x <= z)"
+ with Sup_le[of "uminus`A" "-x"] show "x \<le> Inf A"
+ unfolding extreal_Sup_uminus_image_eq by force }
+qed
+end
+
+lemma extreal_SUPR_uminus:
+ fixes f :: "'a => extreal"
+ shows "(SUP i : R. -(f i)) = -(INF i : R. f i)"
+ unfolding SUPR_def INFI_def
+ using extreal_Sup_uminus_image_eq[of "f`R"]
+ by (simp add: image_image)
+
+lemma extreal_INFI_uminus:
+ fixes f :: "'a => extreal"
+ shows "(INF i : R. -(f i)) = -(SUP i : R. f i)"
+ using extreal_SUPR_uminus[of _ "\<lambda>x. - f x"] by simp
+
+lemma extreal_Inf_uminus_image_eq: "Inf (uminus ` S) = - Sup (S::extreal set)"
+ using extreal_Sup_uminus_image_eq[of "uminus ` S"] by (simp add: image_image)
+
+lemma extreal_inj_on_uminus[intro, simp]: "inj_on uminus (A :: extreal set)"
+ by (auto intro!: inj_onI)
+
+lemma extreal_image_uminus_shift:
+ fixes X Y :: "extreal set" shows "uminus ` X = Y \<longleftrightarrow> X = uminus ` Y"
+proof
+ assume "uminus ` X = Y"
+ then have "uminus ` uminus ` X = uminus ` Y"
+ by (simp add: inj_image_eq_iff)
+ then show "X = uminus ` Y" by (simp add: image_image)
+qed (simp add: image_image)
+
+lemma Inf_extreal_iff:
+ fixes z :: extreal
+ shows "(!!x. x:X ==> z <= x) ==> (EX x:X. x<y) <-> Inf X < y"
+ by (metis complete_lattice_class.Inf_greatest complete_lattice_class.Inf_lower less_le_not_le linear
+ order_less_le_trans)
+
+lemma Sup_eq_MInfty:
+ fixes S :: "extreal set" shows "Sup S = -\<infinity> \<longleftrightarrow> S = {} \<or> S = {-\<infinity>}"
+proof
+ assume a: "Sup S = -\<infinity>"
+ with complete_lattice_class.Sup_upper[of _ S]
+ show "S={} \<or> S={-\<infinity>}" by auto
+next
+ assume "S={} \<or> S={-\<infinity>}" then show "Sup S = -\<infinity>"
+ unfolding Sup_extreal_def by (auto intro!: Least_equality)
+qed
+
+lemma Inf_eq_PInfty:
+ fixes S :: "extreal set" shows "Inf S = \<infinity> \<longleftrightarrow> S = {} \<or> S = {\<infinity>}"
+ using Sup_eq_MInfty[of "uminus`S"]
+ unfolding extreal_Sup_uminus_image_eq extreal_image_uminus_shift by simp
+
+lemma Inf_eq_MInfty: "-\<infinity> : S ==> Inf S = -\<infinity>"
+ unfolding Inf_extreal_def
+ by (auto intro!: Greatest_equality)
+
+lemma Sup_eq_PInfty: "\<infinity> : S ==> Sup S = \<infinity>"
+ unfolding Sup_extreal_def
+ by (auto intro!: Least_equality)
+
+lemma extreal_SUPI:
+ fixes x :: extreal
+ assumes "!!i. i : A ==> f i <= x"
+ assumes "!!y. (!!i. i : A ==> f i <= y) ==> x <= y"
+ shows "(SUP i:A. f i) = x"
+ unfolding SUPR_def Sup_extreal_def
+ using assms by (auto intro!: Least_equality)
+
+lemma extreal_INFI:
+ fixes x :: extreal
+ assumes "!!i. i : A ==> f i >= x"
+ assumes "!!y. (!!i. i : A ==> f i >= y) ==> x >= y"
+ shows "(INF i:A. f i) = x"
+ unfolding INFI_def Inf_extreal_def
+ using assms by (auto intro!: Greatest_equality)
+
+lemma Sup_extreal_close:
+ fixes e :: extreal
+ assumes "0 < e" and S: "\<bar>Sup S\<bar> \<noteq> \<infinity>" "S \<noteq> {}"
+ shows "\<exists>x\<in>S. Sup S - e < x"
+ using assms by (cases e) (auto intro!: less_Sup_iff[THEN iffD1])
+
+lemma Inf_extreal_close:
+ fixes e :: extreal assumes "\<bar>Inf X\<bar> \<noteq> \<infinity>" "0 < e"
+ shows "\<exists>x\<in>X. x < Inf X + e"
+proof (rule Inf_less_iff[THEN iffD1])
+ show "Inf X < Inf X + e" using assms
+ by (cases e) auto
+qed
+
+lemma Sup_eq_top_iff:
+ fixes A :: "'a::{complete_lattice, linorder} set"
+ shows "Sup A = top \<longleftrightarrow> (\<forall>x<top. \<exists>i\<in>A. x < i)"
+proof
+ assume *: "Sup A = top"
+ show "(\<forall>x<top. \<exists>i\<in>A. x < i)" unfolding *[symmetric]
+ proof (intro allI impI)
+ fix x assume "x < Sup A" then show "\<exists>i\<in>A. x < i"
+ unfolding less_Sup_iff by auto
+ qed
+next
+ assume *: "\<forall>x<top. \<exists>i\<in>A. x < i"
+ show "Sup A = top"
+ proof (rule ccontr)
+ assume "Sup A \<noteq> top"
+ with top_greatest[of "Sup A"]
+ have "Sup A < top" unfolding le_less by auto
+ then have "Sup A < Sup A"
+ using * unfolding less_Sup_iff by auto
+ then show False by auto
+ qed
+qed
+
+lemma SUP_eq_top_iff:
+ fixes f :: "'a \<Rightarrow> 'b::{complete_lattice, linorder}"
+ shows "(SUP i:A. f i) = top \<longleftrightarrow> (\<forall>x<top. \<exists>i\<in>A. x < f i)"
+ unfolding SUPR_def Sup_eq_top_iff by auto
+
+lemma SUP_nat_Infty: "(SUP i::nat. extreal (real i)) = \<infinity>"
+proof -
+ { fix x assume "x \<noteq> \<infinity>"
+ then have "\<exists>k::nat. x < extreal (real k)"
+ proof (cases x)
+ case MInf then show ?thesis by (intro exI[of _ 0]) auto
+ next
+ case (real r)
+ moreover obtain k :: nat where "r < real k"
+ using ex_less_of_nat by (auto simp: real_eq_of_nat)
+ ultimately show ?thesis by auto
+ qed simp }
+ then show ?thesis
+ using SUP_eq_top_iff[of UNIV "\<lambda>n::nat. extreal (real n)"]
+ by (auto simp: top_extreal_def)
+qed
+
+lemma extreal_le_Sup:
+ fixes x :: extreal
+ shows "(x <= (SUP i:A. f i)) <-> (ALL y. y < x --> (EX i. i : A & y <= f i))"
+(is "?lhs <-> ?rhs")
+proof-
+{ assume "?rhs"
+ { assume "~(x <= (SUP i:A. f i))" hence "(SUP i:A. f i)<x" by (simp add: not_le)
+ from this obtain y where y_def: "(SUP i:A. f i)<y & y<x" using extreal_dense by auto
+ from this obtain i where "i : A & y <= f i" using `?rhs` by auto
+ hence "y <= (SUP i:A. f i)" using le_SUPI[of i A f] by auto
+ hence False using y_def by auto
+ } hence "?lhs" by auto
+}
+moreover
+{ assume "?lhs" hence "?rhs"
+ by (metis Collect_def Collect_mem_eq SUP_leI assms atLeastatMost_empty atLeastatMost_empty_iff
+ inf_sup_ord(4) linorder_le_cases sup_absorb1 xt1(8))
+} ultimately show ?thesis by auto
+qed
+
+lemma extreal_Inf_le:
+ fixes x :: extreal
+ shows "((INF i:A. f i) <= x) <-> (ALL y. x < y --> (EX i. i : A & f i <= y))"
+(is "?lhs <-> ?rhs")
+proof-
+{ assume "?rhs"
+ { assume "~((INF i:A. f i) <= x)" hence "x < (INF i:A. f i)" by (simp add: not_le)
+ from this obtain y where y_def: "x<y & y<(INF i:A. f i)" using extreal_dense by auto
+ from this obtain i where "i : A & f i <= y" using `?rhs` by auto
+ hence "(INF i:A. f i) <= y" using INF_leI[of i A f] by auto
+ hence False using y_def by auto
+ } hence "?lhs" by auto
+}
+moreover
+{ assume "?lhs" hence "?rhs"
+ by (metis Collect_def Collect_mem_eq le_INFI assms atLeastatMost_empty atLeastatMost_empty_iff
+ inf_sup_ord(4) linorder_le_cases sup_absorb1 xt1(8))
+} ultimately show ?thesis by auto
+qed
+
+lemma Inf_less:
+ fixes x :: extreal
+ assumes "(INF i:A. f i) < x"
+ shows "EX i. i : A & f i <= x"
+proof(rule ccontr)
+ assume "~ (EX i. i : A & f i <= x)"
+ hence "ALL i:A. f i > x" by auto
+ hence "(INF i:A. f i) >= x" apply (subst le_INFI) by auto
+ thus False using assms by auto
+qed
+
+lemma same_INF:
+ assumes "ALL e:A. f e = g e"
+ shows "(INF e:A. f e) = (INF e:A. g e)"
+proof-
+have "f ` A = g ` A" unfolding image_def using assms by auto
+thus ?thesis unfolding INFI_def by auto
+qed
+
+lemma same_SUP:
+ assumes "ALL e:A. f e = g e"
+ shows "(SUP e:A. f e) = (SUP e:A. g e)"
+proof-
+have "f ` A = g ` A" unfolding image_def using assms by auto
+thus ?thesis unfolding SUPR_def by auto
+qed
+
+lemma SUPR_eq:
+ assumes "\<forall>i\<in>A. \<exists>j\<in>B. f i \<le> g j"
+ assumes "\<forall>j\<in>B. \<exists>i\<in>A. g j \<le> f i"
+ shows "(SUP i:A. f i) = (SUP j:B. g j)"
+proof (intro antisym)
+ show "(SUP i:A. f i) \<le> (SUP j:B. g j)"
+ using assms by (metis SUP_leI le_SUPI2)
+ show "(SUP i:B. g i) \<le> (SUP j:A. f j)"
+ using assms by (metis SUP_leI le_SUPI2)
+qed
+
+lemma SUP_extreal_le_addI:
+ assumes "\<And>i. f i + y \<le> z" and "y \<noteq> -\<infinity>"
+ shows "SUPR UNIV f + y \<le> z"
+proof (cases y)
+ case (real r)
+ then have "\<And>i. f i \<le> z - y" using assms by (simp add: extreal_le_minus_iff)
+ then have "SUPR UNIV f \<le> z - y" by (rule SUP_leI)
+ then show ?thesis using real by (simp add: extreal_le_minus_iff)
+qed (insert assms, auto)
+
+lemma SUPR_extreal_add:
+ fixes f g :: "nat \<Rightarrow> extreal"
+ assumes "incseq f" "incseq g" and pos: "\<And>i. f i \<noteq> -\<infinity>" "\<And>i. g i \<noteq> -\<infinity>"
+ shows "(SUP i. f i + g i) = SUPR UNIV f + SUPR UNIV g"
+proof (rule extreal_SUPI)
+ fix y assume *: "\<And>i. i \<in> UNIV \<Longrightarrow> f i + g i \<le> y"
+ have f: "SUPR UNIV f \<noteq> -\<infinity>" using pos
+ unfolding SUPR_def Sup_eq_MInfty by (auto dest: image_eqD)
+ { fix j
+ { fix i
+ have "f i + g j \<le> f i + g (max i j)"
+ using `incseq g`[THEN incseqD] by (rule add_left_mono) auto
+ also have "\<dots> \<le> f (max i j) + g (max i j)"
+ using `incseq f`[THEN incseqD] by (rule add_right_mono) auto
+ also have "\<dots> \<le> y" using * by auto
+ finally have "f i + g j \<le> y" . }
+ then have "SUPR UNIV f + g j \<le> y"
+ using assms(4)[of j] by (intro SUP_extreal_le_addI) auto
+ then have "g j + SUPR UNIV f \<le> y" by (simp add: ac_simps) }
+ then have "SUPR UNIV g + SUPR UNIV f \<le> y"
+ using f by (rule SUP_extreal_le_addI)
+ then show "SUPR UNIV f + SUPR UNIV g \<le> y" by (simp add: ac_simps)
+qed (auto intro!: add_mono le_SUPI)
+
+lemma SUPR_extreal_add_pos:
+ fixes f g :: "nat \<Rightarrow> extreal"
+ assumes inc: "incseq f" "incseq g" and pos: "\<And>i. 0 \<le> f i" "\<And>i. 0 \<le> g i"
+ shows "(SUP i. f i + g i) = SUPR UNIV f + SUPR UNIV g"
+proof (intro SUPR_extreal_add inc)
+ fix i show "f i \<noteq> -\<infinity>" "g i \<noteq> -\<infinity>" using pos[of i] by auto
+qed
+
+lemma SUPR_extreal_setsum:
+ fixes f g :: "'a \<Rightarrow> nat \<Rightarrow> extreal"
+ assumes "\<And>n. n \<in> A \<Longrightarrow> incseq (f n)" and pos: "\<And>n i. n \<in> A \<Longrightarrow> 0 \<le> f n i"
+ shows "(SUP i. \<Sum>n\<in>A. f n i) = (\<Sum>n\<in>A. SUPR UNIV (f n))"
+proof cases
+ assume "finite A" then show ?thesis using assms
+ by induct (auto simp: incseq_setsumI2 setsum_nonneg SUPR_extreal_add_pos)
+qed simp
+
+lemma SUPR_extreal_cmult:
+ fixes f :: "nat \<Rightarrow> extreal" assumes "\<And>i. 0 \<le> f i" "0 \<le> c"
+ shows "(SUP i. c * f i) = c * SUPR UNIV f"
+proof (rule extreal_SUPI)
+ fix i have "f i \<le> SUPR UNIV f" by (rule le_SUPI) auto
+ then show "c * f i \<le> c * SUPR UNIV f"
+ using `0 \<le> c` by (rule extreal_mult_left_mono)
+next
+ fix y assume *: "\<And>i. i \<in> UNIV \<Longrightarrow> c * f i \<le> y"
+ show "c * SUPR UNIV f \<le> y"
+ proof cases
+ assume c: "0 < c \<and> c \<noteq> \<infinity>"
+ with * have "SUPR UNIV f \<le> y / c"
+ by (intro SUP_leI) (auto simp: extreal_le_divide_pos)
+ with c show ?thesis
+ by (auto simp: extreal_le_divide_pos)
+ next
+ { assume "c = \<infinity>" have ?thesis
+ proof cases
+ assume "\<forall>i. f i = 0"
+ moreover then have "range f = {0}" by auto
+ ultimately show "c * SUPR UNIV f \<le> y" using * by (auto simp: SUPR_def)
+ next
+ assume "\<not> (\<forall>i. f i = 0)"
+ then obtain i where "f i \<noteq> 0" by auto
+ with *[of i] `c = \<infinity>` `0 \<le> f i` show ?thesis by (auto split: split_if_asm)
+ qed }
+ moreover assume "\<not> (0 < c \<and> c \<noteq> \<infinity>)"
+ ultimately show ?thesis using * `0 \<le> c` by auto
+ qed
+qed
+
+lemma SUP_PInfty:
+ fixes f :: "'a \<Rightarrow> extreal"
+ assumes "\<And>n::nat. \<exists>i\<in>A. extreal (real n) \<le> f i"
+ shows "(SUP i:A. f i) = \<infinity>"
+ unfolding SUPR_def Sup_eq_top_iff[where 'a=extreal, unfolded top_extreal_def]
+ apply simp
+proof safe
+ fix x assume "x \<noteq> \<infinity>"
+ show "\<exists>i\<in>A. x < f i"
+ proof (cases x)
+ case PInf with `x \<noteq> \<infinity>` show ?thesis by simp
+ next
+ case MInf with assms[of "0"] show ?thesis by force
+ next
+ case (real r)
+ with less_PInf_Ex_of_nat[of x] obtain n :: nat where "x < extreal (real n)" by auto
+ moreover from assms[of n] guess i ..
+ ultimately show ?thesis
+ by (auto intro!: bexI[of _ i])
+ qed
+qed
+
+lemma Sup_countable_SUPR:
+ assumes "A \<noteq> {}"
+ shows "\<exists>f::nat \<Rightarrow> extreal. range f \<subseteq> A \<and> Sup A = SUPR UNIV f"
+proof (cases "Sup A")
+ case (real r)
+ have "\<forall>n::nat. \<exists>x. x \<in> A \<and> Sup A < x + 1 / extreal (real n)"
+ proof
+ fix n ::nat have "\<exists>x\<in>A. Sup A - 1 / extreal (real n) < x"
+ using assms real by (intro Sup_extreal_close) (auto simp: one_extreal_def)
+ then guess x ..
+ then show "\<exists>x. x \<in> A \<and> Sup A < x + 1 / extreal (real n)"
+ by (auto intro!: exI[of _ x] simp: extreal_minus_less_iff)
+ qed
+ from choice[OF this] guess f .. note f = this
+ have "SUPR UNIV f = Sup A"
+ proof (rule extreal_SUPI)
+ fix i show "f i \<le> Sup A" using f
+ by (auto intro!: complete_lattice_class.Sup_upper)
+ next
+ fix y assume bound: "\<And>i. i \<in> UNIV \<Longrightarrow> f i \<le> y"
+ show "Sup A \<le> y"
+ proof (rule extreal_le_epsilon, intro allI impI)
+ fix e :: extreal assume "0 < e"
+ show "Sup A \<le> y + e"
+ proof (cases e)
+ case (real r)
+ hence "0 < r" using `0 < e` by auto
+ then obtain n ::nat where *: "1 / real n < r" "0 < n"
+ using ex_inverse_of_nat_less by (auto simp: real_eq_of_nat inverse_eq_divide)
+ have "Sup A \<le> f n + 1 / extreal (real n)" using f[THEN spec, of n] by auto
+ also have "1 / extreal (real n) \<le> e" using real * by (auto simp: one_extreal_def )
+ with bound have "f n + 1 / extreal (real n) \<le> y + e" by (rule add_mono) simp
+ finally show "Sup A \<le> y + e" .
+ qed (insert `0 < e`, auto)
+ qed
+ qed
+ with f show ?thesis by (auto intro!: exI[of _ f])
+next
+ case PInf
+ from `A \<noteq> {}` obtain x where "x \<in> A" by auto
+ show ?thesis
+ proof cases
+ assume "\<infinity> \<in> A"
+ moreover then have "\<infinity> \<le> Sup A" by (intro complete_lattice_class.Sup_upper)
+ ultimately show ?thesis by (auto intro!: exI[of _ "\<lambda>x. \<infinity>"])
+ next
+ assume "\<infinity> \<notin> A"
+ have "\<exists>x\<in>A. 0 \<le> x"
+ by (metis Infty_neq_0 PInf complete_lattice_class.Sup_least extreal_infty_less_eq2 linorder_linear)
+ then obtain x where "x \<in> A" "0 \<le> x" by auto
+ have "\<forall>n::nat. \<exists>f. f \<in> A \<and> x + extreal (real n) \<le> f"
+ proof (rule ccontr)
+ assume "\<not> ?thesis"
+ then have "\<exists>n::nat. Sup A \<le> x + extreal (real n)"
+ by (simp add: Sup_le_iff not_le less_imp_le Ball_def) (metis less_imp_le)
+ then show False using `x \<in> A` `\<infinity> \<notin> A` PInf
+ by(cases x) auto
+ qed
+ from choice[OF this] guess f .. note f = this
+ have "SUPR UNIV f = \<infinity>"
+ proof (rule SUP_PInfty)
+ fix n :: nat show "\<exists>i\<in>UNIV. extreal (real n) \<le> f i"
+ using f[THEN spec, of n] `0 \<le> x`
+ by (cases rule: extreal2_cases[of "f n" x]) (auto intro!: exI[of _ n])
+ qed
+ then show ?thesis using f PInf by (auto intro!: exI[of _ f])
+ qed
+next
+ case MInf
+ with `A \<noteq> {}` have "A = {-\<infinity>}" by (auto simp: Sup_eq_MInfty)
+ then show ?thesis using MInf by (auto intro!: exI[of _ "\<lambda>x. -\<infinity>"])
+qed
+
+lemma SUPR_countable_SUPR:
+ "A \<noteq> {} \<Longrightarrow> \<exists>f::nat \<Rightarrow> extreal. range f \<subseteq> g`A \<and> SUPR A g = SUPR UNIV f"
+ using Sup_countable_SUPR[of "g`A"] by (auto simp: SUPR_def)
+
+
+lemma Sup_extreal_cadd:
+ fixes A :: "extreal set" assumes "A \<noteq> {}" and "a \<noteq> -\<infinity>"
+ shows "Sup ((\<lambda>x. a + x) ` A) = a + Sup A"
+proof (rule antisym)
+ have *: "\<And>a::extreal. \<And>A. Sup ((\<lambda>x. a + x) ` A) \<le> a + Sup A"
+ by (auto intro!: add_mono complete_lattice_class.Sup_least complete_lattice_class.Sup_upper)
+ then show "Sup ((\<lambda>x. a + x) ` A) \<le> a + Sup A" .
+ show "a + Sup A \<le> Sup ((\<lambda>x. a + x) ` A)"
+ proof (cases a)
+ case PInf with `A \<noteq> {}` show ?thesis by (auto simp: image_constant)
+ next
+ case (real r)
+ then have **: "op + (- a) ` op + a ` A = A"
+ by (auto simp: image_iff ac_simps zero_extreal_def[symmetric])
+ from *[of "-a" "(\<lambda>x. a + x) ` A"] real show ?thesis unfolding **
+ by (cases rule: extreal2_cases[of "Sup A" "Sup (op + a ` A)"]) auto
+ qed (insert `a \<noteq> -\<infinity>`, auto)
+qed
+
+lemma Sup_extreal_cminus:
+ fixes A :: "extreal set" assumes "A \<noteq> {}" and "a \<noteq> -\<infinity>"
+ shows "Sup ((\<lambda>x. a - x) ` A) = a - Inf A"
+ using Sup_extreal_cadd[of "uminus ` A" a] assms
+ by (simp add: comp_def image_image minus_extreal_def
+ extreal_Sup_uminus_image_eq)
+
+lemma SUPR_extreal_cminus:
+ fixes A assumes "A \<noteq> {}" and "a \<noteq> -\<infinity>"
+ shows "(SUP x:A. a - f x) = a - (INF x:A. f x)"
+ using Sup_extreal_cminus[of "f`A" a] assms
+ unfolding SUPR_def INFI_def image_image by auto
+
+lemma Inf_extreal_cminus:
+ fixes A :: "extreal set" assumes "A \<noteq> {}" and "\<bar>a\<bar> \<noteq> \<infinity>"
+ shows "Inf ((\<lambda>x. a - x) ` A) = a - Sup A"
+proof -
+ { fix x have "-a - -x = -(a - x)" using assms by (cases x) auto }
+ moreover then have "(\<lambda>x. -a - x)`uminus`A = uminus ` (\<lambda>x. a - x) ` A"
+ by (auto simp: image_image)
+ ultimately show ?thesis
+ using Sup_extreal_cminus[of "uminus ` A" "-a"] assms
+ by (auto simp add: extreal_Sup_uminus_image_eq extreal_Inf_uminus_image_eq)
+qed
+
+lemma INFI_extreal_cminus:
+ fixes A assumes "A \<noteq> {}" and "\<bar>a\<bar> \<noteq> \<infinity>"
+ shows "(INF x:A. a - f x) = a - (SUP x:A. f x)"
+ using Inf_extreal_cminus[of "f`A" a] assms
+ unfolding SUPR_def INFI_def image_image
+ by auto
+
+subsection "Limits on @{typ extreal}"
+
+subsubsection "Topological space"
+
+instantiation extreal :: topological_space
+begin
+
+definition "open A \<longleftrightarrow> open (extreal -` A)
+ \<and> (\<infinity> \<in> A \<longrightarrow> (\<exists>x. {extreal x <..} \<subseteq> A))
+ \<and> (-\<infinity> \<in> A \<longrightarrow> (\<exists>x. {..<extreal x} \<subseteq> A))"
+
+lemma open_PInfty: "open A \<Longrightarrow> \<infinity> \<in> A \<Longrightarrow> (\<exists>x. {extreal x<..} \<subseteq> A)"
+ unfolding open_extreal_def by auto
+
+lemma open_MInfty: "open A \<Longrightarrow> -\<infinity> \<in> A \<Longrightarrow> (\<exists>x. {..<extreal x} \<subseteq> A)"
+ unfolding open_extreal_def by auto
+
+lemma open_PInfty2: assumes "open A" "\<infinity> \<in> A" obtains x where "{extreal x<..} \<subseteq> A"
+ using open_PInfty[OF assms] by auto
+
+lemma open_MInfty2: assumes "open A" "-\<infinity> \<in> A" obtains x where "{..<extreal x} \<subseteq> A"
+ using open_MInfty[OF assms] by auto
+
+lemma extreal_openE: assumes "open A" obtains x y where
+ "open (extreal -` A)"
+ "\<infinity> \<in> A \<Longrightarrow> {extreal x<..} \<subseteq> A"
+ "-\<infinity> \<in> A \<Longrightarrow> {..<extreal y} \<subseteq> A"
+ using assms open_extreal_def by auto
+
+instance
+proof
+ let ?U = "UNIV::extreal set"
+ show "open ?U" unfolding open_extreal_def
+ by (auto intro!: exI[of _ 0])
+next
+ fix S T::"extreal set" assume "open S" and "open T"
+ from `open S`[THEN extreal_openE] guess xS yS .
+ moreover from `open T`[THEN extreal_openE] guess xT yT .
+ ultimately have
+ "open (extreal -` (S \<inter> T))"
+ "\<infinity> \<in> S \<inter> T \<Longrightarrow> {extreal (max xS xT) <..} \<subseteq> S \<inter> T"
+ "-\<infinity> \<in> S \<inter> T \<Longrightarrow> {..< extreal (min yS yT)} \<subseteq> S \<inter> T"
+ by auto
+ then show "open (S Int T)" unfolding open_extreal_def by blast
+next
+ fix K :: "extreal set set" assume "\<forall>S\<in>K. open S"
+ then have *: "\<forall>S. \<exists>x y. S \<in> K \<longrightarrow> open (extreal -` S) \<and>
+ (\<infinity> \<in> S \<longrightarrow> {extreal x <..} \<subseteq> S) \<and> (-\<infinity> \<in> S \<longrightarrow> {..< extreal y} \<subseteq> S)"
+ by (auto simp: open_extreal_def)
+ then show "open (Union K)" unfolding open_extreal_def
+ proof (intro conjI impI)
+ show "open (extreal -` \<Union>K)"
+ using *[THEN choice] by (auto simp: vimage_Union)
+ qed ((metis UnionE Union_upper subset_trans *)+)
+qed
+end
+
+lemma open_extreal: "open S \<Longrightarrow> open (extreal ` S)"
+ by (auto simp: inj_vimage_image_eq open_extreal_def)
+
+lemma open_extreal_vimage: "open S \<Longrightarrow> open (extreal -` S)"
+ unfolding open_extreal_def by auto
+
+lemma open_extreal_lessThan[intro, simp]: "open {..< a :: extreal}"
+proof -
+ have "\<And>x. extreal -` {..<extreal x} = {..< x}"
+ "extreal -` {..< \<infinity>} = UNIV" "extreal -` {..< -\<infinity>} = {}" by auto
+ then show ?thesis by (cases a) (auto simp: open_extreal_def)
+qed
+
+lemma open_extreal_greaterThan[intro, simp]:
+ "open {a :: extreal <..}"
+proof -
+ have "\<And>x. extreal -` {extreal x<..} = {x<..}"
+ "extreal -` {\<infinity><..} = {}" "extreal -` {-\<infinity><..} = UNIV" by auto
+ then show ?thesis by (cases a) (auto simp: open_extreal_def)
+qed
+
+lemma extreal_open_greaterThanLessThan[intro, simp]: "open {a::extreal <..< b}"
+ unfolding greaterThanLessThan_def by auto
+
+lemma closed_extreal_atLeast[simp, intro]: "closed {a :: extreal ..}"
+proof -
+ have "- {a ..} = {..< a}" by auto
+ then show "closed {a ..}"
+ unfolding closed_def using open_extreal_lessThan by auto
+qed
+
+lemma closed_extreal_atMost[simp, intro]: "closed {.. b :: extreal}"
+proof -
+ have "- {.. b} = {b <..}" by auto
+ then show "closed {.. b}"
+ unfolding closed_def using open_extreal_greaterThan by auto
+qed
+
+lemma closed_extreal_atLeastAtMost[simp, intro]:
+ shows "closed {a :: extreal .. b}"
+ unfolding atLeastAtMost_def by auto
+
+lemma closed_extreal_singleton:
+ "closed {a :: extreal}"
+by (metis atLeastAtMost_singleton closed_extreal_atLeastAtMost)
+
+lemma extreal_open_cont_interval:
+ assumes "open S" "x \<in> S" "\<bar>x\<bar> \<noteq> \<infinity>"
+ obtains e where "e>0" "{x-e <..< x+e} \<subseteq> S"
+proof-
+ from `open S` have "open (extreal -` S)" by (rule extreal_openE)
+ then obtain e where "0 < e" and e: "\<And>y. dist y (real x) < e \<Longrightarrow> extreal y \<in> S"
+ using assms unfolding open_dist by force
+ show thesis
+ proof (intro that subsetI)
+ show "0 < extreal e" using `0 < e` by auto
+ fix y assume "y \<in> {x - extreal e<..<x + extreal e}"
+ with assms obtain t where "y = extreal t" "dist t (real x) < e"
+ apply (cases y) by (auto simp: dist_real_def)
+ then show "y \<in> S" using e[of t] by auto
+ qed
+qed
+
+lemma extreal_open_cont_interval2:
+ assumes "open S" "x \<in> S" and x: "\<bar>x\<bar> \<noteq> \<infinity>"
+ obtains a b where "a < x" "x < b" "{a <..< b} \<subseteq> S"
+proof-
+ guess e using extreal_open_cont_interval[OF assms] .
+ with that[of "x-e" "x+e"] extreal_between[OF x, of e]
+ show thesis by auto
+qed
+
+instance extreal :: t2_space
+proof
+ fix x y :: extreal assume "x ~= y"
+ let "?P x (y::extreal)" = "EX U V. open U & open V & x : U & y : V & U Int V = {}"
+
+ { fix x y :: extreal assume "x < y"
+ from extreal_dense[OF this] obtain z where z: "x < z" "z < y" by auto
+ have "?P x y"
+ apply (rule exI[of _ "{..<z}"])
+ apply (rule exI[of _ "{z<..}"])
+ using z by auto }
+ note * = this
+
+ from `x ~= y`
+ show "EX U V. open U & open V & x : U & y : V & U Int V = {}"
+ proof (cases rule: linorder_cases)
+ assume "x = y" with `x ~= y` show ?thesis by simp
+ next assume "x < y" from *[OF this] show ?thesis by auto
+ next assume "y < x" from *[OF this] show ?thesis by auto
+ qed
+qed
+
+subsubsection {* Convergent sequences *}
+
+lemma lim_extreal[simp]:
+ "((\<lambda>n. extreal (f n)) ---> extreal x) net \<longleftrightarrow> (f ---> x) net" (is "?l = ?r")
+proof (intro iffI topological_tendstoI)
+ fix S assume "?l" "open S" "x \<in> S"
+ then show "eventually (\<lambda>x. f x \<in> S) net"
+ using `?l`[THEN topological_tendstoD, OF open_extreal, OF `open S`]
+ by (simp add: inj_image_mem_iff)
+next
+ fix S assume "?r" "open S" "extreal x \<in> S"
+ show "eventually (\<lambda>x. extreal (f x) \<in> S) net"
+ using `?r`[THEN topological_tendstoD, OF open_extreal_vimage, OF `open S`]
+ using `extreal x \<in> S` by auto
+qed
+
+lemma lim_real_of_extreal[simp]:
+ assumes lim: "(f ---> extreal x) net"
+ shows "((\<lambda>x. real (f x)) ---> x) net"
+proof (intro topological_tendstoI)
+ fix S assume "open S" "x \<in> S"
+ then have S: "open S" "extreal x \<in> extreal ` S"
+ by (simp_all add: inj_image_mem_iff)
+ have "\<forall>x. f x \<in> extreal ` S \<longrightarrow> real (f x) \<in> S" by auto
+ from this lim[THEN topological_tendstoD, OF open_extreal, OF S]
+ show "eventually (\<lambda>x. real (f x) \<in> S) net"
+ by (rule eventually_mono)
+qed
+
+lemma Lim_PInfty: "f ----> \<infinity> <-> (ALL B. EX N. ALL n>=N. f n >= extreal B)" (is "?l = ?r")
+proof assume ?r show ?l apply(rule topological_tendstoI)
+ unfolding eventually_sequentially
+ proof- fix S assume "open S" "\<infinity> : S"
+ from open_PInfty[OF this] guess B .. note B=this
+ from `?r`[rule_format,of "B+1"] guess N .. note N=this
+ show "EX N. ALL n>=N. f n : S" apply(rule_tac x=N in exI)
+ proof safe case goal1
+ have "extreal B < extreal (B + 1)" by auto
+ also have "... <= f n" using goal1 N by auto
+ finally show ?case using B by fastsimp
+ qed
+ qed
+next assume ?l show ?r
+ proof fix B::real have "open {extreal B<..}" "\<infinity> : {extreal B<..}" by auto
+ from topological_tendstoD[OF `?l` this,unfolded eventually_sequentially]
+ guess N .. note N=this
+ show "EX N. ALL n>=N. extreal B <= f n" apply(rule_tac x=N in exI) using N by auto
+ qed
+qed
+
+
+lemma Lim_MInfty: "f ----> (-\<infinity>) <-> (ALL B. EX N. ALL n>=N. f n <= extreal B)" (is "?l = ?r")
+proof assume ?r show ?l apply(rule topological_tendstoI)
+ unfolding eventually_sequentially
+ proof- fix S assume "open S" "(-\<infinity>) : S"
+ from open_MInfty[OF this] guess B .. note B=this
+ from `?r`[rule_format,of "B-(1::real)"] guess N .. note N=this
+ show "EX N. ALL n>=N. f n : S" apply(rule_tac x=N in exI)
+ proof safe case goal1
+ have "extreal (B - 1) >= f n" using goal1 N by auto
+ also have "... < extreal B" by auto
+ finally show ?case using B by fastsimp
+ qed
+ qed
+next assume ?l show ?r
+ proof fix B::real have "open {..<extreal B}" "(-\<infinity>) : {..<extreal B}" by auto
+ from topological_tendstoD[OF `?l` this,unfolded eventually_sequentially]
+ guess N .. note N=this
+ show "EX N. ALL n>=N. extreal B >= f n" apply(rule_tac x=N in exI) using N by auto
+ qed
+qed
+
+
+lemma Lim_bounded_PInfty: assumes lim:"f ----> l" and "!!n. f n <= extreal B" shows "l ~= \<infinity>"
+proof(rule ccontr,unfold not_not) let ?B = "B + 1" assume as:"l=\<infinity>"
+ from lim[unfolded this Lim_PInfty,rule_format,of "?B"]
+ guess N .. note N=this[rule_format,OF le_refl]
+ hence "extreal ?B <= extreal B" using assms(2)[of N] by(rule order_trans)
+ hence "extreal ?B < extreal ?B" apply (rule le_less_trans) by auto
+ thus False by auto
+qed
+
+
+lemma Lim_bounded_MInfty: assumes lim:"f ----> l" and "!!n. f n >= extreal B" shows "l ~= (-\<infinity>)"
+proof(rule ccontr,unfold not_not) let ?B = "B - 1" assume as:"l=(-\<infinity>)"
+ from lim[unfolded this Lim_MInfty,rule_format,of "?B"]
+ guess N .. note N=this[rule_format,OF le_refl]
+ hence "extreal B <= extreal ?B" using assms(2)[of N] order_trans[of "extreal B" "f N" "extreal(B - 1)"] by blast
+ thus False by auto
+qed
+
+
+lemma tendsto_explicit:
+ "f ----> f0 <-> (ALL S. open S --> f0 : S --> (EX N. ALL n>=N. f n : S))"
+ unfolding tendsto_def eventually_sequentially by auto
+
+
+lemma tendsto_obtains_N:
+ assumes "f ----> f0"
+ assumes "open S" "f0 : S"
+ obtains N where "ALL n>=N. f n : S"
+ using tendsto_explicit[of f f0] assms by auto
+
+
+lemma tail_same_limit:
+ fixes X Y N
+ assumes "X ----> L" "ALL n>=N. X n = Y n"
+ shows "Y ----> L"
+proof-
+{ fix S assume "open S" and "L:S"
+ from this obtain N1 where "ALL n>=N1. X n : S"
+ using assms unfolding tendsto_def eventually_sequentially by auto
+ hence "ALL n>=max N N1. Y n : S" using assms by auto
+ hence "EX N. ALL n>=N. Y n : S" apply(rule_tac x="max N N1" in exI) by auto
+}
+thus ?thesis using tendsto_explicit by auto
+qed
+
+
+lemma Lim_bounded_PInfty2:
+assumes lim:"f ----> l" and "ALL n>=N. f n <= extreal B"
+shows "l ~= \<infinity>"
+proof-
+ def g == "(%n. if n>=N then f n else extreal B)"
+ hence "g ----> l" using tail_same_limit[of f l N g] lim by auto
+ moreover have "!!n. g n <= extreal B" using g_def assms by auto
+ ultimately show ?thesis using Lim_bounded_PInfty by auto
+qed
+
+lemma Lim_bounded_extreal:
+ assumes lim:"f ----> (l :: extreal)"
+ and "ALL n>=M. f n <= C"
+ shows "l<=C"
+proof-
+{ assume "l=(-\<infinity>)" hence ?thesis by auto }
+moreover
+{ assume "~(l=(-\<infinity>))"
+ { assume "C=\<infinity>" hence ?thesis by auto }
+ moreover
+ { assume "C=(-\<infinity>)" hence "ALL n>=M. f n = (-\<infinity>)" using assms by auto
+ hence "l=(-\<infinity>)" using assms
+ tendsto_unique[OF trivial_limit_sequentially] tail_same_limit[of "\<lambda>n. -\<infinity>" "-\<infinity>" M f, OF tendsto_const] by auto
+ hence ?thesis by auto }
+ moreover
+ { assume "EX B. C = extreal B"
+ from this obtain B where B_def: "C=extreal B" by auto
+ hence "~(l=\<infinity>)" using Lim_bounded_PInfty2 assms by auto
+ from this obtain m where m_def: "extreal m=l" using `~(l=(-\<infinity>))` by (cases l) auto
+ from this obtain N where N_def: "ALL n>=N. f n : {extreal(m - 1) <..< extreal(m+1)}"
+ apply (subst tendsto_obtains_N[of f l "{extreal(m - 1) <..< extreal(m+1)}"]) using assms by auto
+ { fix n assume "n>=N"
+ hence "EX r. extreal r = f n" using N_def by (cases "f n") auto
+ } from this obtain g where g_def: "ALL n>=N. extreal (g n) = f n" by metis
+ hence "(%n. extreal (g n)) ----> l" using tail_same_limit[of f l N] assms by auto
+ hence *: "(%n. g n) ----> m" using m_def by auto
+ { fix n assume "n>=max N M"
+ hence "extreal (g n) <= extreal B" using assms g_def B_def by auto
+ hence "g n <= B" by auto
+ } hence "EX N. ALL n>=N. g n <= B" by blast
+ hence "m<=B" using * LIMSEQ_le_const2[of g m B] by auto
+ hence ?thesis using m_def B_def by auto
+ } ultimately have ?thesis by (cases C) auto
+} ultimately show ?thesis by blast
+qed
+
+lemma real_of_extreal_0[simp]: "real (0::extreal) = 0"
+ unfolding real_of_extreal_def zero_extreal_def by simp
+
+lemma real_of_extreal_mult[simp]:
+ fixes a b :: extreal shows "real (a * b) = real a * real b"
+ by (cases rule: extreal2_cases[of a b]) auto
+
+lemma real_of_extreal_eq_0:
+ "real x = 0 \<longleftrightarrow> x = \<infinity> \<or> x = -\<infinity> \<or> x = 0"
+ by (cases x) auto
+
+lemma tendsto_extreal_realD:
+ fixes f :: "'a \<Rightarrow> extreal"
+ assumes "x \<noteq> 0" and tendsto: "((\<lambda>x. extreal (real (f x))) ---> x) net"
+ shows "(f ---> x) net"
+proof (intro topological_tendstoI)
+ fix S assume S: "open S" "x \<in> S"
+ with `x \<noteq> 0` have "open (S - {0})" "x \<in> S - {0}" by auto
+ from tendsto[THEN topological_tendstoD, OF this]
+ show "eventually (\<lambda>x. f x \<in> S) net"
+ by (rule eventually_rev_mp) (auto simp: extreal_real real_of_extreal_0)
+qed
+
+lemma tendsto_extreal_realI:
+ fixes f :: "'a \<Rightarrow> extreal"
+ assumes x: "\<bar>x\<bar> \<noteq> \<infinity>" and tendsto: "(f ---> x) net"
+ shows "((\<lambda>x. extreal (real (f x))) ---> x) net"
+proof (intro topological_tendstoI)
+ fix S assume "open S" "x \<in> S"
+ with x have "open (S - {\<infinity>, -\<infinity>})" "x \<in> S - {\<infinity>, -\<infinity>}" by auto
+ from tendsto[THEN topological_tendstoD, OF this]
+ show "eventually (\<lambda>x. extreal (real (f x)) \<in> S) net"
+ by (elim eventually_elim1) (auto simp: extreal_real)
+qed
+
+lemma extreal_mult_cancel_left:
+ fixes a b c :: extreal shows "a * b = a * c \<longleftrightarrow>
+ ((\<bar>a\<bar> = \<infinity> \<and> 0 < b * c) \<or> a = 0 \<or> b = c)"
+ by (cases rule: extreal3_cases[of a b c])
+ (simp_all add: zero_less_mult_iff)
+
+lemma extreal_inj_affinity:
+ assumes "\<bar>m\<bar> \<noteq> \<infinity>" "m \<noteq> 0" "\<bar>t\<bar> \<noteq> \<infinity>"
+ shows "inj_on (\<lambda>x. m * x + t) A"
+ using assms
+ by (cases rule: extreal2_cases[of m t])
+ (auto intro!: inj_onI simp: extreal_add_cancel_right extreal_mult_cancel_left)
+
+lemma extreal_PInfty_eq_plus[simp]:
+ shows "\<infinity> = a + b \<longleftrightarrow> a = \<infinity> \<or> b = \<infinity>"
+ by (cases rule: extreal2_cases[of a b]) auto
+
+lemma extreal_MInfty_eq_plus[simp]:
+ shows "-\<infinity> = a + b \<longleftrightarrow> (a = -\<infinity> \<and> b \<noteq> \<infinity>) \<or> (b = -\<infinity> \<and> a \<noteq> \<infinity>)"
+ by (cases rule: extreal2_cases[of a b]) auto
+
+lemma extreal_less_divide_pos:
+ "x > 0 \<Longrightarrow> x \<noteq> \<infinity> \<Longrightarrow> y < z / x \<longleftrightarrow> x * y < z"
+ by (cases rule: extreal3_cases[of x y z]) (auto simp: field_simps)
+
+lemma extreal_divide_less_pos:
+ "x > 0 \<Longrightarrow> x \<noteq> \<infinity> \<Longrightarrow> y / x < z \<longleftrightarrow> y < x * z"
+ by (cases rule: extreal3_cases[of x y z]) (auto simp: field_simps)
+
+lemma extreal_divide_eq:
+ "b \<noteq> 0 \<Longrightarrow> \<bar>b\<bar> \<noteq> \<infinity> \<Longrightarrow> a / b = c \<longleftrightarrow> a = b * c"
+ by (cases rule: extreal3_cases[of a b c])
+ (simp_all add: field_simps)
+
+lemma extreal_inverse_not_MInfty[simp]: "inverse a \<noteq> -\<infinity>"
+ by (cases a) auto
+
+lemma extreal_mult_m1[simp]: "x * extreal (-1) = -x"
+ by (cases x) auto
+
+lemma extreal_LimI_finite:
+ assumes "\<bar>x\<bar> \<noteq> \<infinity>"
+ assumes "!!r. 0 < r ==> EX N. ALL n>=N. u n < x + r & x < u n + r"
+ shows "u ----> x"
+proof (rule topological_tendstoI, unfold eventually_sequentially)
+ obtain rx where rx_def: "x=extreal rx" using assms by (cases x) auto
+ fix S assume "open S" "x : S"
+ then have "open (extreal -` S)" unfolding open_extreal_def by auto
+ with `x \<in> S` obtain r where "0 < r" and dist: "!!y. dist y rx < r ==> extreal y \<in> S"
+ unfolding open_real_def rx_def by auto
+ then obtain n where
+ upper: "!!N. n <= N ==> u N < x + extreal r" and
+ lower: "!!N. n <= N ==> x < u N + extreal r" using assms(2)[of "extreal r"] by auto
+ show "EX N. ALL n>=N. u n : S"
+ proof (safe intro!: exI[of _ n])
+ fix N assume "n <= N"
+ from upper[OF this] lower[OF this] assms `0 < r`
+ have "u N ~: {\<infinity>,(-\<infinity>)}" by auto
+ from this obtain ra where ra_def: "(u N) = extreal ra" by (cases "u N") auto
+ hence "rx < ra + r" and "ra < rx + r"
+ using rx_def assms `0 < r` lower[OF `n <= N`] upper[OF `n <= N`] by auto
+ hence "dist (real (u N)) rx < r"
+ using rx_def ra_def
+ by (auto simp: dist_real_def abs_diff_less_iff field_simps)
+ from dist[OF this] show "u N : S" using `u N ~: {\<infinity>, -\<infinity>}`
+ by (auto simp: extreal_real split: split_if_asm)
+ qed
+qed
+
+lemma extreal_LimI_finite_iff:
+ assumes "\<bar>x\<bar> \<noteq> \<infinity>"
+ shows "u ----> x <-> (ALL r. 0 < r --> (EX N. ALL n>=N. u n < x + r & x < u n + r))"
+ (is "?lhs <-> ?rhs")
+proof
+ assume lim: "u ----> x"
+ { fix r assume "(r::extreal)>0"
+ from this obtain N where N_def: "ALL n>=N. u n : {x - r <..< x + r}"
+ apply (subst tendsto_obtains_N[of u x "{x - r <..< x + r}"])
+ using lim extreal_between[of x r] assms `r>0` by auto
+ hence "EX N. ALL n>=N. u n < x + r & x < u n + r"
+ using extreal_minus_less[of r x] by (cases r) auto
+ } then show "?rhs" by auto
+next
+ assume ?rhs then show "u ----> x"
+ using extreal_LimI_finite[of x] assms by auto
+qed
+
+
+subsubsection {* @{text Liminf} and @{text Limsup} *}
+
+definition
+ "Liminf net f = (GREATEST l. \<forall>y<l. eventually (\<lambda>x. y < f x) net)"
+
+definition
+ "Limsup net f = (LEAST l. \<forall>y>l. eventually (\<lambda>x. f x < y) net)"
+
+lemma Liminf_Sup:
+ fixes f :: "'a => 'b::{complete_lattice, linorder}"
+ shows "Liminf net f = Sup {l. \<forall>y<l. eventually (\<lambda>x. y < f x) net}"
+ by (auto intro!: Greatest_equality complete_lattice_class.Sup_upper simp: less_Sup_iff Liminf_def)
+
+lemma Limsup_Inf:
+ fixes f :: "'a => 'b::{complete_lattice, linorder}"
+ shows "Limsup net f = Inf {l. \<forall>y>l. eventually (\<lambda>x. f x < y) net}"
+ by (auto intro!: Least_equality complete_lattice_class.Inf_lower simp: Inf_less_iff Limsup_def)
+
+lemma extreal_SupI:
+ fixes x :: extreal
+ assumes "\<And>y. y \<in> A \<Longrightarrow> y \<le> x"
+ assumes "\<And>y. (\<And>z. z \<in> A \<Longrightarrow> z \<le> y) \<Longrightarrow> x \<le> y"
+ shows "Sup A = x"
+ unfolding Sup_extreal_def
+ using assms by (auto intro!: Least_equality)
+
+lemma extreal_InfI:
+ fixes x :: extreal
+ assumes "\<And>i. i \<in> A \<Longrightarrow> x \<le> i"
+ assumes "\<And>y. (\<And>i. i \<in> A \<Longrightarrow> y \<le> i) \<Longrightarrow> y \<le> x"
+ shows "Inf A = x"
+ unfolding Inf_extreal_def
+ using assms by (auto intro!: Greatest_equality)
+
+lemma Limsup_const:
+ fixes c :: "'a::{complete_lattice, linorder}"
+ assumes ntriv: "\<not> trivial_limit net"
+ shows "Limsup net (\<lambda>x. c) = c"
+ unfolding Limsup_Inf
+proof (safe intro!: antisym complete_lattice_class.Inf_greatest complete_lattice_class.Inf_lower)
+ fix x assume *: "\<forall>y>x. eventually (\<lambda>_. c < y) net"
+ show "c \<le> x"
+ proof (rule ccontr)
+ assume "\<not> c \<le> x" then have "x < c" by auto
+ then show False using ntriv * by (auto simp: trivial_limit_def)
+ qed
+qed auto
+
+lemma Liminf_const:
+ fixes c :: "'a::{complete_lattice, linorder}"
+ assumes ntriv: "\<not> trivial_limit net"
+ shows "Liminf net (\<lambda>x. c) = c"
+ unfolding Liminf_Sup
+proof (safe intro!: antisym complete_lattice_class.Sup_least complete_lattice_class.Sup_upper)
+ fix x assume *: "\<forall>y<x. eventually (\<lambda>_. y < c) net"
+ show "x \<le> c"
+ proof (rule ccontr)
+ assume "\<not> x \<le> c" then have "c < x" by auto
+ then show False using ntriv * by (auto simp: trivial_limit_def)
+ qed
+qed auto
+
+lemma mono_set:
+ fixes S :: "('a::order) set"
+ shows "mono S \<longleftrightarrow> (\<forall>x y. x \<le> y \<longrightarrow> x \<in> S \<longrightarrow> y \<in> S)"
+ by (auto simp: mono_def mem_def)
+
+lemma mono_greaterThan[intro, simp]: "mono {B<..}" unfolding mono_set by auto
+lemma mono_atLeast[intro, simp]: "mono {B..}" unfolding mono_set by auto
+lemma mono_UNIV[intro, simp]: "mono UNIV" unfolding mono_set by auto
+lemma mono_empty[intro, simp]: "mono {}" unfolding mono_set by auto
+
+lemma mono_set_iff:
+ fixes S :: "'a::{linorder,complete_lattice} set"
+ defines "a \<equiv> Inf S"
+ shows "mono S \<longleftrightarrow> (S = {a <..} \<or> S = {a..})" (is "_ = ?c")
+proof
+ assume "mono S"
+ then have mono: "\<And>x y. x \<le> y \<Longrightarrow> x \<in> S \<Longrightarrow> y \<in> S" by (auto simp: mono_set)
+ show ?c
+ proof cases
+ assume "a \<in> S"
+ show ?c
+ using mono[OF _ `a \<in> S`]
+ by (auto intro: complete_lattice_class.Inf_lower simp: a_def)
+ next
+ assume "a \<notin> S"
+ have "S = {a <..}"
+ proof safe
+ fix x assume "x \<in> S"
+ then have "a \<le> x" unfolding a_def by (rule complete_lattice_class.Inf_lower)
+ then show "a < x" using `x \<in> S` `a \<notin> S` by (cases "a = x") auto
+ next
+ fix x assume "a < x"
+ then obtain y where "y < x" "y \<in> S" unfolding a_def Inf_less_iff ..
+ with mono[of y x] show "x \<in> S" by auto
+ qed
+ then show ?c ..
+ qed
+qed auto
+
+lemma lim_imp_Liminf:
+ fixes f :: "'a \<Rightarrow> extreal"
+ assumes ntriv: "\<not> trivial_limit net"
+ assumes lim: "(f ---> f0) net"
+ shows "Liminf net f = f0"
+ unfolding Liminf_Sup
+proof (safe intro!: extreal_SupI)
+ fix y assume *: "\<forall>y'<y. eventually (\<lambda>x. y' < f x) net"
+ show "y \<le> f0"
+ proof (rule extreal_le_extreal)
+ fix B assume "B < y"
+ { assume "f0 < B"
+ then have "eventually (\<lambda>x. f x < B \<and> B < f x) net"
+ using topological_tendstoD[OF lim, of "{..<B}"] *[rule_format, OF `B < y`]
+ by (auto intro: eventually_conj)
+ also have "(\<lambda>x. f x < B \<and> B < f x) = (\<lambda>x. False)" by (auto simp: fun_eq_iff)
+ finally have False using ntriv[unfolded trivial_limit_def] by auto
+ } then show "B \<le> f0" by (metis linorder_le_less_linear)
+ qed
+next
+ fix y assume *: "\<forall>z. z \<in> {l. \<forall>y<l. eventually (\<lambda>x. y < f x) net} \<longrightarrow> z \<le> y"
+ show "f0 \<le> y"
+ proof (safe intro!: *[rule_format])
+ fix y assume "y < f0" then show "eventually (\<lambda>x. y < f x) net"
+ using lim[THEN topological_tendstoD, of "{y <..}"] by auto
+ qed
+qed
+
+lemma extreal_Liminf_le_Limsup:
+ fixes f :: "'a \<Rightarrow> extreal"
+ assumes ntriv: "\<not> trivial_limit net"
+ shows "Liminf net f \<le> Limsup net f"
+ unfolding Limsup_Inf Liminf_Sup
+proof (safe intro!: complete_lattice_class.Inf_greatest complete_lattice_class.Sup_least)
+ fix u v assume *: "\<forall>y<u. eventually (\<lambda>x. y < f x) net" "\<forall>y>v. eventually (\<lambda>x. f x < y) net"
+ show "u \<le> v"
+ proof (rule ccontr)
+ assume "\<not> u \<le> v"
+ then obtain t where "t < u" "v < t"
+ using extreal_dense[of v u] by (auto simp: not_le)
+ then have "eventually (\<lambda>x. t < f x \<and> f x < t) net"
+ using * by (auto intro: eventually_conj)
+ also have "(\<lambda>x. t < f x \<and> f x < t) = (\<lambda>x. False)" by (auto simp: fun_eq_iff)
+ finally show False using ntriv by (auto simp: trivial_limit_def)
+ qed
+qed
+
+lemma Liminf_mono:
+ fixes f g :: "'a => extreal"
+ assumes ev: "eventually (\<lambda>x. f x \<le> g x) net"
+ shows "Liminf net f \<le> Liminf net g"
+ unfolding Liminf_Sup
+proof (safe intro!: Sup_mono bexI)
+ fix a y assume "\<forall>y<a. eventually (\<lambda>x. y < f x) net" and "y < a"
+ then have "eventually (\<lambda>x. y < f x) net" by auto
+ then show "eventually (\<lambda>x. y < g x) net"
+ by (rule eventually_rev_mp) (rule eventually_mono[OF _ ev], auto)
+qed simp
+
+lemma Liminf_eq:
+ fixes f g :: "'a \<Rightarrow> extreal"
+ assumes "eventually (\<lambda>x. f x = g x) net"
+ shows "Liminf net f = Liminf net g"
+ by (intro antisym Liminf_mono eventually_mono[OF _ assms]) auto
+
+lemma Liminf_mono_all:
+ fixes f g :: "'a \<Rightarrow> extreal"
+ assumes "\<And>x. f x \<le> g x"
+ shows "Liminf net f \<le> Liminf net g"
+ using assms by (intro Liminf_mono always_eventually) auto
+
+lemma Limsup_mono:
+ fixes f g :: "'a \<Rightarrow> extreal"
+ assumes ev: "eventually (\<lambda>x. f x \<le> g x) net"
+ shows "Limsup net f \<le> Limsup net g"
+ unfolding Limsup_Inf
+proof (safe intro!: Inf_mono bexI)
+ fix a y assume "\<forall>y>a. eventually (\<lambda>x. g x < y) net" and "a < y"
+ then have "eventually (\<lambda>x. g x < y) net" by auto
+ then show "eventually (\<lambda>x. f x < y) net"
+ by (rule eventually_rev_mp) (rule eventually_mono[OF _ ev], auto)
+qed simp
+
+lemma Limsup_mono_all:
+ fixes f g :: "'a \<Rightarrow> extreal"
+ assumes "\<And>x. f x \<le> g x"
+ shows "Limsup net f \<le> Limsup net g"
+ using assms by (intro Limsup_mono always_eventually) auto
+
+lemma Limsup_eq:
+ fixes f g :: "'a \<Rightarrow> extreal"
+ assumes "eventually (\<lambda>x. f x = g x) net"
+ shows "Limsup net f = Limsup net g"
+ by (intro antisym Limsup_mono eventually_mono[OF _ assms]) auto
+
+abbreviation "liminf \<equiv> Liminf sequentially"
+
+abbreviation "limsup \<equiv> Limsup sequentially"
+
+lemma (in complete_lattice) less_INFD:
+ assumes "y < INFI A f"" i \<in> A" shows "y < f i"
+proof -
+ note `y < INFI A f`
+ also have "INFI A f \<le> f i" using `i \<in> A` by (rule INF_leI)
+ finally show "y < f i" .
+qed
+
+lemma liminf_SUPR_INFI:
+ fixes f :: "nat \<Rightarrow> extreal"
+ shows "liminf f = (SUP n. INF m:{n..}. f m)"
+ unfolding Liminf_Sup eventually_sequentially
+proof (safe intro!: antisym complete_lattice_class.Sup_least)
+ fix x assume *: "\<forall>y<x. \<exists>N. \<forall>n\<ge>N. y < f n" show "x \<le> (SUP n. INF m:{n..}. f m)"
+ proof (rule extreal_le_extreal)
+ fix y assume "y < x"
+ with * obtain N where "\<And>n. N \<le> n \<Longrightarrow> y < f n" by auto
+ then have "y \<le> (INF m:{N..}. f m)" by (force simp: le_INF_iff)
+ also have "\<dots> \<le> (SUP n. INF m:{n..}. f m)" by (intro le_SUPI) auto
+ finally show "y \<le> (SUP n. INF m:{n..}. f m)" .
+ qed
+next
+ show "(SUP n. INF m:{n..}. f m) \<le> Sup {l. \<forall>y<l. \<exists>N. \<forall>n\<ge>N. y < f n}"
+ proof (unfold SUPR_def, safe intro!: Sup_mono bexI)
+ fix y n assume "y < INFI {n..} f"
+ from less_INFD[OF this] show "\<exists>N. \<forall>n\<ge>N. y < f n" by (intro exI[of _ n]) auto
+ qed (rule order_refl)
+qed
+
+lemma tail_same_limsup:
+ fixes X Y :: "nat => extreal"
+ assumes "\<And>n. N \<le> n \<Longrightarrow> X n = Y n"
+ shows "limsup X = limsup Y"
+ using Limsup_eq[of X Y sequentially] eventually_sequentially assms by auto
+
+lemma tail_same_liminf:
+ fixes X Y :: "nat => extreal"
+ assumes "\<And>n. N \<le> n \<Longrightarrow> X n = Y n"
+ shows "liminf X = liminf Y"
+ using Liminf_eq[of X Y sequentially] eventually_sequentially assms by auto
+
+lemma liminf_mono:
+ fixes X Y :: "nat \<Rightarrow> extreal"
+ assumes "\<And>n. N \<le> n \<Longrightarrow> X n <= Y n"
+ shows "liminf X \<le> liminf Y"
+ using Liminf_mono[of X Y sequentially] eventually_sequentially assms by auto
+
+lemma limsup_mono:
+ fixes X Y :: "nat => extreal"
+ assumes "\<And>n. N \<le> n \<Longrightarrow> X n <= Y n"
+ shows "limsup X \<le> limsup Y"
+ using Limsup_mono[of X Y sequentially] eventually_sequentially assms by auto
+
+declare trivial_limit_sequentially[simp]
+
+lemma
+ fixes X :: "nat \<Rightarrow> extreal"
+ shows extreal_incseq_uminus[simp]: "incseq (\<lambda>i. - X i) = decseq X"
+ and extreal_decseq_uminus[simp]: "decseq (\<lambda>i. - X i) = incseq X"
+ unfolding incseq_def decseq_def by auto
+
+lemma liminf_bounded:
+ fixes X Y :: "nat \<Rightarrow> extreal"
+ assumes "\<And>n. N \<le> n \<Longrightarrow> C \<le> X n"
+ shows "C \<le> liminf X"
+ using liminf_mono[of N "\<lambda>n. C" X] assms Liminf_const[of sequentially C] by simp
+
+lemma limsup_bounded:
+ fixes X Y :: "nat => extreal"
+ assumes "\<And>n. N \<le> n \<Longrightarrow> X n <= C"
+ shows "limsup X \<le> C"
+ using limsup_mono[of N X "\<lambda>n. C"] assms Limsup_const[of sequentially C] by simp
+
+lemma liminf_bounded_iff:
+ fixes x :: "nat \<Rightarrow> extreal"
+ shows "C \<le> liminf x \<longleftrightarrow> (\<forall>B<C. \<exists>N. \<forall>n\<ge>N. B < x n)" (is "?lhs <-> ?rhs")
+proof safe
+ fix B assume "B < C" "C \<le> liminf x"
+ then have "B < liminf x" by auto
+ then obtain N where "B < (INF m:{N..}. x m)"
+ unfolding liminf_SUPR_INFI SUPR_def less_Sup_iff by auto
+ from less_INFD[OF this] show "\<exists>N. \<forall>n\<ge>N. B < x n" by auto
+next
+ assume *: "\<forall>B<C. \<exists>N. \<forall>n\<ge>N. B < x n"
+ { fix B assume "B<C"
+ then obtain N where "\<forall>n\<ge>N. B < x n" using `?rhs` by auto
+ hence "B \<le> (INF m:{N..}. x m)" by (intro le_INFI) auto
+ also have "... \<le> liminf x" unfolding liminf_SUPR_INFI by (intro le_SUPI) simp
+ finally have "B \<le> liminf x" .
+ } then show "?lhs" by (metis * leD liminf_bounded linorder_le_less_linear)
+qed
+
+lemma liminf_subseq_mono:
+ fixes X :: "nat \<Rightarrow> extreal"
+ assumes "subseq r"
+ shows "liminf X \<le> liminf (X \<circ> r) "
+proof-
+ have "\<And>n. (INF m:{n..}. X m) \<le> (INF m:{n..}. (X \<circ> r) m)"
+ proof (safe intro!: INF_mono)
+ fix n m :: nat assume "n \<le> m" then show "\<exists>ma\<in>{n..}. X ma \<le> (X \<circ> r) m"
+ using seq_suble[OF `subseq r`, of m] by (intro bexI[of _ "r m"]) auto
+ qed
+ then show ?thesis by (auto intro!: SUP_mono simp: liminf_SUPR_INFI comp_def)
+qed
+
+lemma extreal_real': assumes "\<bar>x\<bar> \<noteq> \<infinity>" shows "extreal (real x) = x"
+ using assms by auto
+
+lemma extreal_le_extreal_bounded:
+ fixes x y z :: extreal
+ assumes "z \<le> y"
+ assumes *: "\<And>B. z < B \<Longrightarrow> B < x \<Longrightarrow> B \<le> y"
+ shows "x \<le> y"
+proof (rule extreal_le_extreal)
+ fix B assume "B < x"
+ show "B \<le> y"
+ proof cases
+ assume "z < B" from *[OF this `B < x`] show "B \<le> y" .
+ next
+ assume "\<not> z < B" with `z \<le> y` show "B \<le> y" by auto
+ qed
+qed
+
+lemma fixes x y :: extreal
+ shows Sup_atMost[simp]: "Sup {.. y} = y"
+ and Sup_lessThan[simp]: "Sup {..< y} = y"
+ and Sup_atLeastAtMost[simp]: "x \<le> y \<Longrightarrow> Sup { x .. y} = y"
+ and Sup_greaterThanAtMost[simp]: "x < y \<Longrightarrow> Sup { x <.. y} = y"
+ and Sup_atLeastLessThan[simp]: "x < y \<Longrightarrow> Sup { x ..< y} = y"
+ by (auto simp: Sup_extreal_def intro!: Least_equality
+ intro: extreal_le_extreal extreal_le_extreal_bounded[of x])
+
+lemma Sup_greaterThanlessThan[simp]:
+ fixes x y :: extreal assumes "x < y" shows "Sup { x <..< y} = y"
+ unfolding Sup_extreal_def
+proof (intro Least_equality extreal_le_extreal_bounded[of _ _ y])
+ fix z assume z: "\<forall>u\<in>{x<..<y}. u \<le> z"
+ from extreal_dense[OF `x < y`] guess w .. note w = this
+ with z[THEN bspec, of w] show "x \<le> z" by auto
+qed auto
+
+lemma real_extreal_id: "real o extreal = id"
+proof-
+{ fix x have "(real o extreal) x = id x" by auto }
+from this show ?thesis using ext by blast
+qed
+
+
+lemma open_image_extreal: "open(UNIV-{\<infinity>,(-\<infinity>)})"
+by (metis range_extreal open_extreal open_UNIV)
+
+lemma extreal_le_distrib:
+ fixes a b c :: extreal shows "c * (a + b) \<le> c * a + c * b"
+ by (cases rule: extreal3_cases[of a b c])
+ (auto simp add: field_simps not_le mult_le_0_iff mult_less_0_iff)
+
+lemma extreal_pos_distrib:
+ fixes a b c :: extreal assumes "0 \<le> c" "c \<noteq> \<infinity>" shows "c * (a + b) = c * a + c * b"
+ using assms by (cases rule: extreal3_cases[of a b c])
+ (auto simp add: field_simps not_le mult_le_0_iff mult_less_0_iff)
+
+lemma extreal_pos_le_distrib:
+fixes a b c :: extreal
+assumes "c>=0"
+shows "c * (a + b) <= c * a + c * b"
+ using assms by (cases rule: extreal3_cases[of a b c])
+ (auto simp add: field_simps)
+
+lemma extreal_max_mono:
+ "[| (a::extreal) <= b; c <= d |] ==> max a c <= max b d"
+ by (metis sup_extreal_def sup_mono)
+
+
+lemma extreal_max_least:
+ "[| (a::extreal) <= x; c <= x |] ==> max a c <= x"
+ by (metis sup_extreal_def sup_least)
+
+end
--- a/src/HOL/Limits.thy Mon Mar 14 15:17:10 2011 +0100
+++ b/src/HOL/Limits.thy Mon Mar 14 15:29:10 2011 +0100
@@ -103,7 +103,6 @@
shows "eventually (\<lambda>i. R i) net"
using assms by (auto elim!: eventually_rev_mp)
-
subsection {* Finer-than relation *}
text {* @{term "net \<le> net'"} means that @{term net} is finer than
@@ -231,7 +230,6 @@
"eventually (\<lambda>x. False) net \<longleftrightarrow> net = bot"
unfolding expand_net_eq by (auto elim: eventually_rev_mp)
-
subsection {* Map function for nets *}
definition netmap :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a net \<Rightarrow> 'b net" where
@@ -287,6 +285,13 @@
by (safe, fast, drule_tac x=N in spec, auto elim: eventually_rev_mp)
+definition
+ trivial_limit :: "'a net \<Rightarrow> bool" where
+ "trivial_limit net \<longleftrightarrow> eventually (\<lambda>x. False) net"
+
+lemma trivial_limit_sequentially[intro]: "\<not> trivial_limit sequentially"
+ by (auto simp add: trivial_limit_def eventually_sequentially)
+
subsection {* Standard Nets *}
definition within :: "'a net \<Rightarrow> 'a set \<Rightarrow> 'a net" (infixr "within" 70) where
@@ -827,4 +832,29 @@
\<Longrightarrow> ((\<lambda>x. f x / g x) ---> a / b) net"
by (simp add: mult.tendsto tendsto_inverse divide_inverse)
+lemma tendsto_unique:
+ fixes f :: "'a \<Rightarrow> 'b::t2_space"
+ assumes "\<not> trivial_limit net" "(f ---> l) net" "(f ---> l') net"
+ shows "l = l'"
+proof (rule ccontr)
+ assume "l \<noteq> l'"
+ obtain U V where "open U" "open V" "l \<in> U" "l' \<in> V" "U \<inter> V = {}"
+ using hausdorff [OF `l \<noteq> l'`] by fast
+ have "eventually (\<lambda>x. f x \<in> U) net"
+ using `(f ---> l) net` `open U` `l \<in> U` by (rule topological_tendstoD)
+ moreover
+ have "eventually (\<lambda>x. f x \<in> V) net"
+ using `(f ---> l') net` `open V` `l' \<in> V` by (rule topological_tendstoD)
+ ultimately
+ have "eventually (\<lambda>x. False) net"
+ proof (rule eventually_elim2)
+ fix x
+ assume "f x \<in> U" "f x \<in> V"
+ hence "f x \<in> U \<inter> V" by simp
+ with `U \<inter> V = {}` show "False" by simp
+ qed
+ with `\<not> trivial_limit net` show "False"
+ by (simp add: trivial_limit_def)
+qed
+
end
--- a/src/HOL/Multivariate_Analysis/Derivative.thy Mon Mar 14 15:17:10 2011 +0100
+++ b/src/HOL/Multivariate_Analysis/Derivative.thy Mon Mar 14 15:29:10 2011 +0100
@@ -1129,11 +1129,11 @@
show "bounded_linear (g' x)" unfolding linear_linear linear_def apply(rule,rule,rule) defer proof(rule,rule)
fix x' y z::"'m" and c::real
note lin = assms(2)[rule_format,OF `x\<in>s`,THEN derivative_linear]
- show "g' x (c *\<^sub>R x') = c *\<^sub>R g' x x'" apply(rule Lim_unique[OF trivial_limit_sequentially])
+ show "g' x (c *\<^sub>R x') = c *\<^sub>R g' x x'" apply(rule tendsto_unique[OF trivial_limit_sequentially])
apply(rule lem3[rule_format])
unfolding lin[unfolded bounded_linear_def bounded_linear_axioms_def,THEN conjunct2,THEN conjunct1,rule_format]
apply(rule Lim_cmul) by(rule lem3[rule_format])
- show "g' x (y + z) = g' x y + g' x z" apply(rule Lim_unique[OF trivial_limit_sequentially])
+ show "g' x (y + z) = g' x y + g' x z" apply(rule tendsto_unique[OF trivial_limit_sequentially])
apply(rule lem3[rule_format]) unfolding lin[unfolded bounded_linear_def additive_def,THEN conjunct1,rule_format]
apply(rule Lim_add) by(rule lem3[rule_format])+ qed
show "\<forall>e>0. \<exists>d>0. \<forall>y\<in>s. norm (y - x) < d \<longrightarrow> norm (g y - g x - g' x (y - x)) \<le> e * norm (y - x)" proof(rule,rule) case goal1
--- a/src/HOL/Multivariate_Analysis/Euclidean_Space.thy Mon Mar 14 15:17:10 2011 +0100
+++ b/src/HOL/Multivariate_Analysis/Euclidean_Space.thy Mon Mar 14 15:29:10 2011 +0100
@@ -326,42 +326,6 @@
text{* Hence more metric properties. *}
-lemma dist_triangle_alt:
- fixes x y z :: "'a::metric_space"
- shows "dist y z <= dist x y + dist x z"
-by (rule dist_triangle3)
-
-lemma dist_pos_lt:
- fixes x y :: "'a::metric_space"
- shows "x \<noteq> y ==> 0 < dist x y"
-by (simp add: zero_less_dist_iff)
-
-lemma dist_nz:
- fixes x y :: "'a::metric_space"
- shows "x \<noteq> y \<longleftrightarrow> 0 < dist x y"
-by (simp add: zero_less_dist_iff)
-
-lemma dist_triangle_le:
- fixes x y z :: "'a::metric_space"
- shows "dist x z + dist y z <= e \<Longrightarrow> dist x y <= e"
-by (rule order_trans [OF dist_triangle2])
-
-lemma dist_triangle_lt:
- fixes x y z :: "'a::metric_space"
- shows "dist x z + dist y z < e ==> dist x y < e"
-by (rule le_less_trans [OF dist_triangle2])
-
-lemma dist_triangle_half_l:
- fixes x1 x2 y :: "'a::metric_space"
- shows "dist x1 y < e / 2 \<Longrightarrow> dist x2 y < e / 2 \<Longrightarrow> dist x1 x2 < e"
-by (rule dist_triangle_lt [where z=y], simp)
-
-lemma dist_triangle_half_r:
- fixes x1 x2 y :: "'a::metric_space"
- shows "dist y x1 < e / 2 \<Longrightarrow> dist y x2 < e / 2 \<Longrightarrow> dist x1 x2 < e"
-by (rule dist_triangle_half_l, simp_all add: dist_commute)
-
-
lemma norm_triangle_half_r:
shows "norm (y - x1) < e / 2 \<Longrightarrow> norm (y - x2) < e / 2 \<Longrightarrow> norm (x1 - x2) < e"
using dist_triangle_half_r unfolding dist_norm[THEN sym] by auto
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Multivariate_Analysis/Extended_Real_Limits.thy Mon Mar 14 15:29:10 2011 +0100
@@ -0,0 +1,1261 @@
+(* Title: src/HOL/Multivariate_Analysis/Extended_Reals.thy
+ Author: Johannes Hölzl; TU München
+ Author: Robert Himmelmann; TU München
+ Author: Armin Heller; TU München
+ Author: Bogdan Grechuk; University of Edinburgh *)
+
+header {* Limits on the Extended real number line *}
+
+theory Extended_Real_Limits
+ imports Topology_Euclidean_Space "~~/src/HOL/Library/Extended_Reals"
+begin
+
+lemma continuous_on_extreal[intro, simp]: "continuous_on A extreal"
+ unfolding continuous_on_topological open_extreal_def by auto
+
+lemma continuous_at_extreal[intro, simp]: "continuous (at x) extreal"
+ using continuous_on_eq_continuous_at[of UNIV] by auto
+
+lemma continuous_within_extreal[intro, simp]: "x \<in> A \<Longrightarrow> continuous (at x within A) extreal"
+ using continuous_on_eq_continuous_within[of A] by auto
+
+lemma extreal_open_uminus:
+ fixes S :: "extreal set"
+ assumes "open S"
+ shows "open (uminus ` S)"
+ unfolding open_extreal_def
+proof (intro conjI impI)
+ obtain x y where S: "open (extreal -` S)"
+ "\<infinity> \<in> S \<Longrightarrow> {extreal x<..} \<subseteq> S" "-\<infinity> \<in> S \<Longrightarrow> {..< extreal y} \<subseteq> S"
+ using `open S` unfolding open_extreal_def by auto
+ have "extreal -` uminus ` S = uminus ` (extreal -` S)"
+ proof safe
+ fix x y assume "extreal x = - y" "y \<in> S"
+ then show "x \<in> uminus ` extreal -` S" by (cases y) auto
+ next
+ fix x assume "extreal x \<in> S"
+ then show "- x \<in> extreal -` uminus ` S"
+ by (auto intro: image_eqI[of _ _ "extreal x"])
+ qed
+ then show "open (extreal -` uminus ` S)"
+ using S by (auto intro: open_negations)
+ { assume "\<infinity> \<in> uminus ` S"
+ then have "-\<infinity> \<in> S" by (metis image_iff extreal_uminus_uminus)
+ then have "uminus ` {..<extreal y} \<subseteq> uminus ` S" using S by (intro image_mono) auto
+ then show "\<exists>x. {extreal x<..} \<subseteq> uminus ` S" using extreal_uminus_lessThan by auto }
+ { assume "-\<infinity> \<in> uminus ` S"
+ then have "\<infinity> : S" by (metis image_iff extreal_uminus_uminus)
+ then have "uminus ` {extreal x<..} <= uminus ` S" using S by (intro image_mono) auto
+ then show "\<exists>y. {..<extreal y} <= uminus ` S" using extreal_uminus_greaterThan by auto }
+qed
+
+lemma extreal_uminus_complement:
+ fixes S :: "extreal set"
+ shows "uminus ` (- S) = - uminus ` S"
+ by (auto intro!: bij_image_Compl_eq surjI[of _ uminus] simp: bij_betw_def)
+
+lemma extreal_closed_uminus:
+ fixes S :: "extreal set"
+ assumes "closed S"
+ shows "closed (uminus ` S)"
+using assms unfolding closed_def
+using extreal_open_uminus[of "- S"] extreal_uminus_complement by auto
+
+lemma not_open_extreal_singleton:
+ "\<not> (open {a :: extreal})"
+proof(rule ccontr)
+ assume "\<not> \<not> open {a}" hence a: "open {a}" by auto
+ show False
+ proof (cases a)
+ case MInf
+ then obtain y where "{..<extreal y} <= {a}" using a open_MInfty2[of "{a}"] by auto
+ hence "extreal(y - 1):{a}" apply (subst subsetD[of "{..<extreal y}"]) by auto
+ then show False using `a=(-\<infinity>)` by auto
+ next
+ case PInf
+ then obtain y where "{extreal y<..} <= {a}" using a open_PInfty2[of "{a}"] by auto
+ hence "extreal(y+1):{a}" apply (subst subsetD[of "{extreal y<..}"]) by auto
+ then show False using `a=\<infinity>` by auto
+ next
+ case (real r) then have fin: "\<bar>a\<bar> \<noteq> \<infinity>" by simp
+ from extreal_open_cont_interval[OF a singletonI this] guess e . note e = this
+ then obtain b where b_def: "a<b & b<a+e"
+ using fin extreal_between extreal_dense[of a "a+e"] by auto
+ then have "b: {a-e <..< a+e}" using fin extreal_between[of a e] e by auto
+ then show False using b_def e by auto
+ qed
+qed
+
+lemma extreal_closed_contains_Inf:
+ fixes S :: "extreal set"
+ assumes "closed S" "S ~= {}"
+ shows "Inf S : S"
+proof(rule ccontr)
+ assume "Inf S \<notin> S" hence a: "open (-S)" "Inf S:(- S)" using assms by auto
+ show False
+ proof (cases "Inf S")
+ case MInf hence "(-\<infinity>) : - S" using a by auto
+ then obtain y where "{..<extreal y} <= (-S)" using a open_MInfty2[of "- S"] by auto
+ hence "extreal y <= Inf S" by (metis Compl_anti_mono Compl_lessThan atLeast_iff
+ complete_lattice_class.Inf_greatest double_complement set_rev_mp)
+ then show False using MInf by auto
+ next
+ case PInf then have "S={\<infinity>}" by (metis Inf_eq_PInfty assms(2))
+ then show False by (metis `Inf S ~: S` insert_code mem_def PInf)
+ next
+ case (real r) then have fin: "\<bar>Inf S\<bar> \<noteq> \<infinity>" by simp
+ from extreal_open_cont_interval[OF a this] guess e . note e = this
+ { fix x assume "x:S" hence "x>=Inf S" by (rule complete_lattice_class.Inf_lower)
+ hence *: "x>Inf S-e" using e by (metis fin extreal_between(1) order_less_le_trans)
+ { assume "x<Inf S+e" hence "x:{Inf S-e <..< Inf S+e}" using * by auto
+ hence False using e `x:S` by auto
+ } hence "x>=Inf S+e" by (metis linorder_le_less_linear)
+ } hence "Inf S + e <= Inf S" by (metis le_Inf_iff)
+ then show False using real e by (cases e) auto
+ qed
+qed
+
+lemma extreal_closed_contains_Sup:
+ fixes S :: "extreal set"
+ assumes "closed S" "S ~= {}"
+ shows "Sup S : S"
+proof-
+ have "closed (uminus ` S)" by (metis assms(1) extreal_closed_uminus)
+ hence "Inf (uminus ` S) : uminus ` S" using assms extreal_closed_contains_Inf[of "uminus ` S"] by auto
+ hence "- Sup S : uminus ` S" using extreal_Sup_uminus_image_eq[of "uminus ` S"] by (auto simp: image_image)
+ thus ?thesis by (metis imageI extreal_uminus_uminus extreal_minus_minus_image)
+qed
+
+lemma extreal_open_closed_aux:
+ fixes S :: "extreal set"
+ assumes "open S" "closed S"
+ assumes S: "(-\<infinity>) ~: S"
+ shows "S = {}"
+proof(rule ccontr)
+ assume "S ~= {}"
+ hence *: "(Inf S):S" by (metis assms(2) extreal_closed_contains_Inf)
+ { assume "Inf S=(-\<infinity>)" hence False using * assms(3) by auto }
+ moreover
+ { assume "Inf S=\<infinity>" hence "S={\<infinity>}" by (metis Inf_eq_PInfty `S ~= {}`)
+ hence False by (metis assms(1) not_open_extreal_singleton) }
+ moreover
+ { assume fin: "\<bar>Inf S\<bar> \<noteq> \<infinity>"
+ from extreal_open_cont_interval[OF assms(1) * fin] guess e . note e = this
+ then obtain b where b_def: "Inf S-e<b & b<Inf S"
+ using fin extreal_between[of "Inf S" e] extreal_dense[of "Inf S-e"] by auto
+ hence "b: {Inf S-e <..< Inf S+e}" using e fin extreal_between[of "Inf S" e] by auto
+ hence "b:S" using e by auto
+ hence False using b_def by (metis complete_lattice_class.Inf_lower leD)
+ } ultimately show False by auto
+qed
+
+lemma extreal_open_closed:
+ fixes S :: "extreal set"
+ shows "(open S & closed S) <-> (S = {} | S = UNIV)"
+proof-
+{ assume lhs: "open S & closed S"
+ { assume "(-\<infinity>) ~: S" hence "S={}" using lhs extreal_open_closed_aux by auto }
+ moreover
+ { assume "(-\<infinity>) : S" hence "(- S)={}" using lhs extreal_open_closed_aux[of "-S"] by auto }
+ ultimately have "S = {} | S = UNIV" by auto
+} thus ?thesis by auto
+qed
+
+lemma extreal_open_affinity_pos:
+ assumes "open S" and m: "m \<noteq> \<infinity>" "0 < m" and t: "\<bar>t\<bar> \<noteq> \<infinity>"
+ shows "open ((\<lambda>x. m * x + t) ` S)"
+proof -
+ obtain r where r[simp]: "m = extreal r" using m by (cases m) auto
+ obtain p where p[simp]: "t = extreal p" using t by auto
+ have "r \<noteq> 0" "0 < r" and m': "m \<noteq> \<infinity>" "m \<noteq> -\<infinity>" "m \<noteq> 0" using m by auto
+ from `open S`[THEN extreal_openE] guess l u . note T = this
+ let ?f = "(\<lambda>x. m * x + t)"
+ show ?thesis unfolding open_extreal_def
+ proof (intro conjI impI exI subsetI)
+ have "extreal -` ?f ` S = (\<lambda>x. r * x + p) ` (extreal -` S)"
+ proof safe
+ fix x y assume "extreal y = m * x + t" "x \<in> S"
+ then show "y \<in> (\<lambda>x. r * x + p) ` extreal -` S"
+ using `r \<noteq> 0` by (cases x) (auto intro!: image_eqI[of _ _ "real x"] split: split_if_asm)
+ qed force
+ then show "open (extreal -` ?f ` S)"
+ using open_affinity[OF T(1) `r \<noteq> 0`] by (auto simp: ac_simps)
+ next
+ assume "\<infinity> \<in> ?f`S" with `0 < r` have "\<infinity> \<in> S" by auto
+ fix x assume "x \<in> {extreal (r * l + p)<..}"
+ then have [simp]: "extreal (r * l + p) < x" by auto
+ show "x \<in> ?f`S"
+ proof (rule image_eqI)
+ show "x = m * ((x - t) / m) + t"
+ using m t by (cases rule: extreal3_cases[of m x t]) auto
+ have "extreal l < (x - t)/m"
+ using m t by (simp add: extreal_less_divide_pos extreal_less_minus)
+ then show "(x - t)/m \<in> S" using T(2)[OF `\<infinity> \<in> S`] by auto
+ qed
+ next
+ assume "-\<infinity> \<in> ?f`S" with `0 < r` have "-\<infinity> \<in> S" by auto
+ fix x assume "x \<in> {..<extreal (r * u + p)}"
+ then have [simp]: "x < extreal (r * u + p)" by auto
+ show "x \<in> ?f`S"
+ proof (rule image_eqI)
+ show "x = m * ((x - t) / m) + t"
+ using m t by (cases rule: extreal3_cases[of m x t]) auto
+ have "(x - t)/m < extreal u"
+ using m t by (simp add: extreal_divide_less_pos extreal_minus_less)
+ then show "(x - t)/m \<in> S" using T(3)[OF `-\<infinity> \<in> S`] by auto
+ qed
+ qed
+qed
+
+lemma extreal_open_affinity:
+ assumes "open S" and m: "\<bar>m\<bar> \<noteq> \<infinity>" "m \<noteq> 0" and t: "\<bar>t\<bar> \<noteq> \<infinity>"
+ shows "open ((\<lambda>x. m * x + t) ` S)"
+proof cases
+ assume "0 < m" then show ?thesis
+ using extreal_open_affinity_pos[OF `open S` _ _ t, of m] m by auto
+next
+ assume "\<not> 0 < m" then
+ have "0 < -m" using `m \<noteq> 0` by (cases m) auto
+ then have m: "-m \<noteq> \<infinity>" "0 < -m" using `\<bar>m\<bar> \<noteq> \<infinity>`
+ by (auto simp: extreal_uminus_eq_reorder)
+ from extreal_open_affinity_pos[OF extreal_open_uminus[OF `open S`] m t]
+ show ?thesis unfolding image_image by simp
+qed
+
+lemma extreal_lim_mult:
+ fixes X :: "'a \<Rightarrow> extreal"
+ assumes lim: "(X ---> L) net" and a: "\<bar>a\<bar> \<noteq> \<infinity>"
+ shows "((\<lambda>i. a * X i) ---> a * L) net"
+proof cases
+ assume "a \<noteq> 0"
+ show ?thesis
+ proof (rule topological_tendstoI)
+ fix S assume "open S" "a * L \<in> S"
+ have "a * L / a = L"
+ using `a \<noteq> 0` a by (cases rule: extreal2_cases[of a L]) auto
+ then have L: "L \<in> ((\<lambda>x. x / a) ` S)"
+ using `a * L \<in> S` by (force simp: image_iff)
+ moreover have "open ((\<lambda>x. x / a) ` S)"
+ using extreal_open_affinity[OF `open S`, of "inverse a" 0] `a \<noteq> 0` a
+ by (auto simp: extreal_divide_eq extreal_inverse_eq_0 divide_extreal_def ac_simps)
+ note * = lim[THEN topological_tendstoD, OF this L]
+ { fix x from a `a \<noteq> 0` have "a * (x / a) = x"
+ by (cases rule: extreal2_cases[of a x]) auto }
+ note this[simp]
+ show "eventually (\<lambda>x. a * X x \<in> S) net"
+ by (rule eventually_mono[OF _ *]) auto
+ qed
+qed auto
+
+lemma extreal_lim_uminus:
+ fixes X :: "'a \<Rightarrow> extreal" shows "((\<lambda>i. - X i) ---> -L) net \<longleftrightarrow> (X ---> L) net"
+ using extreal_lim_mult[of X L net "extreal (-1)"]
+ extreal_lim_mult[of "(\<lambda>i. - X i)" "-L" net "extreal (-1)"]
+ by (auto simp add: algebra_simps)
+
+lemma Lim_bounded2_extreal:
+ assumes lim:"f ----> (l :: extreal)"
+ and ge: "ALL n>=N. f n >= C"
+ shows "l>=C"
+proof-
+def g == "(%i. -(f i))"
+{ fix n assume "n>=N" hence "g n <= -C" using assms extreal_minus_le_minus g_def by auto }
+hence "ALL n>=N. g n <= -C" by auto
+moreover have limg: "g ----> (-l)" using g_def extreal_lim_uminus lim by auto
+ultimately have "-l <= -C" using Lim_bounded_extreal[of g "-l" _ "-C"] by auto
+from this show ?thesis using extreal_minus_le_minus by auto
+qed
+
+
+lemma extreal_open_atLeast: "open {x..} \<longleftrightarrow> x = -\<infinity>"
+proof
+ assume "x = -\<infinity>" then have "{x..} = UNIV" by auto
+ then show "open {x..}" by auto
+next
+ assume "open {x..}"
+ then have "open {x..} \<and> closed {x..}" by auto
+ then have "{x..} = UNIV" unfolding extreal_open_closed by auto
+ then show "x = -\<infinity>" by (simp add: bot_extreal_def atLeast_eq_UNIV_iff)
+qed
+
+lemma extreal_open_mono_set:
+ fixes S :: "extreal set"
+ defines "a \<equiv> Inf S"
+ shows "(open S \<and> mono S) \<longleftrightarrow> (S = UNIV \<or> S = {a <..})"
+ by (metis Inf_UNIV a_def atLeast_eq_UNIV_iff extreal_open_atLeast
+ extreal_open_closed mono_set_iff open_extreal_greaterThan)
+
+lemma extreal_closed_mono_set:
+ fixes S :: "extreal set"
+ shows "(closed S \<and> mono S) \<longleftrightarrow> (S = {} \<or> S = {Inf S ..})"
+ by (metis Inf_UNIV atLeast_eq_UNIV_iff closed_extreal_atLeast
+ extreal_open_closed mono_empty mono_set_iff open_extreal_greaterThan)
+
+lemma extreal_Liminf_Sup_monoset:
+ fixes f :: "'a => extreal"
+ shows "Liminf net f = Sup {l. \<forall>S. open S \<longrightarrow> mono S \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) net}"
+ unfolding Liminf_Sup
+proof (intro arg_cong[where f="\<lambda>P. Sup (Collect P)"] ext iffI allI impI)
+ fix l S assume ev: "\<forall>y<l. eventually (\<lambda>x. y < f x) net" and "open S" "mono S" "l \<in> S"
+ then have "S = UNIV \<or> S = {Inf S <..}"
+ using extreal_open_mono_set[of S] by auto
+ then show "eventually (\<lambda>x. f x \<in> S) net"
+ proof
+ assume S: "S = {Inf S<..}"
+ then have "Inf S < l" using `l \<in> S` by auto
+ then have "eventually (\<lambda>x. Inf S < f x) net" using ev by auto
+ then show "eventually (\<lambda>x. f x \<in> S) net" by (subst S) auto
+ qed auto
+next
+ fix l y assume S: "\<forall>S. open S \<longrightarrow> mono S \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) net" "y < l"
+ have "eventually (\<lambda>x. f x \<in> {y <..}) net"
+ using `y < l` by (intro S[rule_format]) auto
+ then show "eventually (\<lambda>x. y < f x) net" by auto
+qed
+
+lemma extreal_Limsup_Inf_monoset:
+ fixes f :: "'a => extreal"
+ shows "Limsup net f = Inf {l. \<forall>S. open S \<longrightarrow> mono (uminus ` S) \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) net}"
+ unfolding Limsup_Inf
+proof (intro arg_cong[where f="\<lambda>P. Inf (Collect P)"] ext iffI allI impI)
+ fix l S assume ev: "\<forall>y>l. eventually (\<lambda>x. f x < y) net" and "open S" "mono (uminus`S)" "l \<in> S"
+ then have "open (uminus`S) \<and> mono (uminus`S)" by (simp add: extreal_open_uminus)
+ then have "S = UNIV \<or> S = {..< Sup S}"
+ unfolding extreal_open_mono_set extreal_Inf_uminus_image_eq extreal_image_uminus_shift by simp
+ then show "eventually (\<lambda>x. f x \<in> S) net"
+ proof
+ assume S: "S = {..< Sup S}"
+ then have "l < Sup S" using `l \<in> S` by auto
+ then have "eventually (\<lambda>x. f x < Sup S) net" using ev by auto
+ then show "eventually (\<lambda>x. f x \<in> S) net" by (subst S) auto
+ qed auto
+next
+ fix l y assume S: "\<forall>S. open S \<longrightarrow> mono (uminus`S) \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) net" "l < y"
+ have "eventually (\<lambda>x. f x \<in> {..< y}) net"
+ using `l < y` by (intro S[rule_format]) auto
+ then show "eventually (\<lambda>x. f x < y) net" by auto
+qed
+
+
+lemma open_uminus_iff: "open (uminus ` S) \<longleftrightarrow> open (S::extreal set)"
+ using extreal_open_uminus[of S] extreal_open_uminus[of "uminus`S"] by auto
+
+lemma extreal_Limsup_uminus:
+ fixes f :: "'a => extreal"
+ shows "Limsup net (\<lambda>x. - (f x)) = -(Liminf net f)"
+proof -
+ { fix P l have "(\<exists>x. (l::extreal) = -x \<and> P x) \<longleftrightarrow> P (-l)" by (auto intro!: exI[of _ "-l"]) }
+ note Ex_cancel = this
+ { fix P :: "extreal set \<Rightarrow> bool" have "(\<forall>S. P S) \<longleftrightarrow> (\<forall>S. P (uminus`S))"
+ apply auto by (erule_tac x="uminus`S" in allE) (auto simp: image_image) }
+ note add_uminus_image = this
+ { fix x S have "(x::extreal) \<in> uminus`S \<longleftrightarrow> -x\<in>S" by (auto intro!: image_eqI[of _ _ "-x"]) }
+ note remove_uminus_image = this
+ show ?thesis
+ unfolding extreal_Limsup_Inf_monoset extreal_Liminf_Sup_monoset
+ unfolding extreal_Inf_uminus_image_eq[symmetric] image_Collect Ex_cancel
+ by (subst add_uminus_image) (simp add: open_uminus_iff remove_uminus_image)
+qed
+
+lemma extreal_Liminf_uminus:
+ fixes f :: "'a => extreal"
+ shows "Liminf net (\<lambda>x. - (f x)) = -(Limsup net f)"
+ using extreal_Limsup_uminus[of _ "(\<lambda>x. - (f x))"] by auto
+
+lemma extreal_Lim_uminus:
+ fixes f :: "'a \<Rightarrow> extreal" shows "(f ---> f0) net \<longleftrightarrow> ((\<lambda>x. - f x) ---> - f0) net"
+ using
+ extreal_lim_mult[of f f0 net "- 1"]
+ extreal_lim_mult[of "\<lambda>x. - (f x)" "-f0" net "- 1"]
+ by (auto simp: extreal_uminus_reorder)
+
+lemma lim_imp_Limsup:
+ fixes f :: "'a => extreal"
+ assumes "\<not> trivial_limit net"
+ assumes lim: "(f ---> f0) net"
+ shows "Limsup net f = f0"
+ using extreal_Lim_uminus[of f f0] lim_imp_Liminf[of net "(%x. -(f x))" "-f0"]
+ extreal_Liminf_uminus[of net f] assms by simp
+
+lemma Liminf_PInfty:
+ fixes f :: "'a \<Rightarrow> extreal"
+ assumes "\<not> trivial_limit net"
+ shows "(f ---> \<infinity>) net \<longleftrightarrow> Liminf net f = \<infinity>"
+proof (intro lim_imp_Liminf iffI assms)
+ assume rhs: "Liminf net f = \<infinity>"
+ { fix S assume "open S & \<infinity> : S"
+ then obtain m where "{extreal m<..} <= S" using open_PInfty2 by auto
+ moreover have "eventually (\<lambda>x. f x \<in> {extreal m<..}) net"
+ using rhs unfolding Liminf_Sup top_extreal_def[symmetric] Sup_eq_top_iff
+ by (auto elim!: allE[where x="extreal m"] simp: top_extreal_def)
+ ultimately have "eventually (%x. f x : S) net" apply (subst eventually_mono) by auto
+ } then show "(f ---> \<infinity>) net" unfolding tendsto_def by auto
+qed
+
+lemma Limsup_MInfty:
+ fixes f :: "'a \<Rightarrow> extreal"
+ assumes "\<not> trivial_limit net"
+ shows "(f ---> -\<infinity>) net \<longleftrightarrow> Limsup net f = -\<infinity>"
+ using assms extreal_Lim_uminus[of f "-\<infinity>"] Liminf_PInfty[of _ "\<lambda>x. - (f x)"]
+ extreal_Liminf_uminus[of _ f] by (auto simp: extreal_uminus_eq_reorder)
+
+lemma extreal_Liminf_eq_Limsup:
+ fixes f :: "'a \<Rightarrow> extreal"
+ assumes ntriv: "\<not> trivial_limit net"
+ assumes lim: "Liminf net f = f0" "Limsup net f = f0"
+ shows "(f ---> f0) net"
+proof (cases f0)
+ case PInf then show ?thesis using Liminf_PInfty[OF ntriv] lim by auto
+next
+ case MInf then show ?thesis using Limsup_MInfty[OF ntriv] lim by auto
+next
+ case (real r)
+ show "(f ---> f0) net"
+ proof (rule topological_tendstoI)
+ fix S assume "open S""f0 \<in> S"
+ then obtain a b where "a < Liminf net f" "Limsup net f < b" "{a<..<b} \<subseteq> S"
+ using extreal_open_cont_interval2[of S f0] real lim by auto
+ then have "eventually (\<lambda>x. f x \<in> {a<..<b}) net"
+ unfolding Liminf_Sup Limsup_Inf less_Sup_iff Inf_less_iff
+ by (auto intro!: eventually_conj simp add: greaterThanLessThan_iff)
+ with `{a<..<b} \<subseteq> S` show "eventually (%x. f x : S) net"
+ by (rule_tac eventually_mono) auto
+ qed
+qed
+
+lemma extreal_Liminf_eq_Limsup_iff:
+ fixes f :: "'a \<Rightarrow> extreal"
+ assumes "\<not> trivial_limit net"
+ shows "(f ---> f0) net \<longleftrightarrow> Liminf net f = f0 \<and> Limsup net f = f0"
+ by (metis assms extreal_Liminf_eq_Limsup lim_imp_Liminf lim_imp_Limsup)
+
+lemma limsup_INFI_SUPR:
+ fixes f :: "nat \<Rightarrow> extreal"
+ shows "limsup f = (INF n. SUP m:{n..}. f m)"
+ using extreal_Limsup_uminus[of sequentially "\<lambda>x. - f x"]
+ by (simp add: liminf_SUPR_INFI extreal_INFI_uminus extreal_SUPR_uminus)
+
+lemma liminf_PInfty:
+ fixes X :: "nat => extreal"
+ shows "X ----> \<infinity> <-> liminf X = \<infinity>"
+by (metis Liminf_PInfty trivial_limit_sequentially)
+
+lemma limsup_MInfty:
+ fixes X :: "nat => extreal"
+ shows "X ----> (-\<infinity>) <-> limsup X = (-\<infinity>)"
+by (metis Limsup_MInfty trivial_limit_sequentially)
+
+lemma extreal_lim_mono:
+ fixes X Y :: "nat => extreal"
+ assumes "\<And>n. N \<le> n \<Longrightarrow> X n <= Y n"
+ assumes "X ----> x" "Y ----> y"
+ shows "x <= y"
+ by (metis extreal_Liminf_eq_Limsup_iff[OF trivial_limit_sequentially] assms liminf_mono)
+
+lemma incseq_le_extreal:
+ fixes X :: "nat \<Rightarrow> extreal"
+ assumes inc: "incseq X" and lim: "X ----> L"
+ shows "X N \<le> L"
+ using inc
+ by (intro extreal_lim_mono[of N, OF _ Lim_const lim]) (simp add: incseq_def)
+
+lemma decseq_ge_extreal: assumes dec: "decseq X"
+ and lim: "X ----> (L::extreal)" shows "X N >= L"
+ using dec
+ by (intro extreal_lim_mono[of N, OF _ lim Lim_const]) (simp add: decseq_def)
+
+lemma liminf_bounded_open:
+ fixes x :: "nat \<Rightarrow> extreal"
+ shows "x0 \<le> liminf x \<longleftrightarrow> (\<forall>S. open S \<longrightarrow> mono S \<longrightarrow> x0 \<in> S \<longrightarrow> (\<exists>N. \<forall>n\<ge>N. x n \<in> S))"
+ (is "_ \<longleftrightarrow> ?P x0")
+proof
+ assume "?P x0" then show "x0 \<le> liminf x"
+ unfolding extreal_Liminf_Sup_monoset eventually_sequentially
+ by (intro complete_lattice_class.Sup_upper) auto
+next
+ assume "x0 \<le> liminf x"
+ { fix S :: "extreal set" assume om: "open S & mono S & x0:S"
+ { assume "S = UNIV" hence "EX N. (ALL n>=N. x n : S)" by auto }
+ moreover
+ { assume "~(S=UNIV)"
+ then obtain B where B_def: "S = {B<..}" using om extreal_open_mono_set by auto
+ hence "B<x0" using om by auto
+ hence "EX N. ALL n>=N. x n : S" unfolding B_def using `x0 \<le> liminf x` liminf_bounded_iff by auto
+ } ultimately have "EX N. (ALL n>=N. x n : S)" by auto
+ } then show "?P x0" by auto
+qed
+
+lemma limsup_subseq_mono:
+ fixes X :: "nat \<Rightarrow> extreal"
+ assumes "subseq r"
+ shows "limsup (X \<circ> r) \<le> limsup X"
+proof-
+ have "(\<lambda>n. - X n) \<circ> r = (\<lambda>n. - (X \<circ> r) n)" by (simp add: fun_eq_iff)
+ then have "- limsup X \<le> - limsup (X \<circ> r)"
+ using liminf_subseq_mono[of r "(%n. - X n)"]
+ extreal_Liminf_uminus[of sequentially X]
+ extreal_Liminf_uminus[of sequentially "X o r"] assms by auto
+ then show ?thesis by auto
+qed
+
+lemma bounded_abs:
+ assumes "(a::real)<=x" "x<=b"
+ shows "abs x <= max (abs a) (abs b)"
+by (metis abs_less_iff assms leI le_max_iff_disj less_eq_real_def less_le_not_le less_minus_iff minus_minus)
+
+lemma bounded_increasing_convergent2: fixes f::"nat => real"
+ assumes "ALL n. f n <= B" "ALL n m. n>=m --> f n >= f m"
+ shows "EX l. (f ---> l) sequentially"
+proof-
+def N == "max (abs (f 0)) (abs B)"
+{ fix n have "abs (f n) <= N" unfolding N_def apply (subst bounded_abs) using assms by auto }
+hence "bounded {f n| n::nat. True}" unfolding bounded_real by auto
+from this show ?thesis apply(rule Topology_Euclidean_Space.bounded_increasing_convergent)
+ using assms by auto
+qed
+lemma lim_extreal_increasing: assumes "\<And>n m. n >= m \<Longrightarrow> f n >= f m"
+ obtains l where "f ----> (l::extreal)"
+proof(cases "f = (\<lambda>x. - \<infinity>)")
+ case True then show thesis using Lim_const[of "- \<infinity>" sequentially] by (intro that[of "-\<infinity>"]) auto
+next
+ case False
+ from this obtain N where N_def: "f N > (-\<infinity>)" by (auto simp: fun_eq_iff)
+ have "ALL n>=N. f n >= f N" using assms by auto
+ hence minf: "ALL n>=N. f n > (-\<infinity>)" using N_def by auto
+ def Y == "(%n. (if n>=N then f n else f N))"
+ hence incy: "!!n m. n>=m ==> Y n >= Y m" using assms by auto
+ from minf have minfy: "ALL n. Y n ~= (-\<infinity>)" using Y_def by auto
+ show thesis
+ proof(cases "EX B. ALL n. f n < extreal B")
+ case False thus thesis apply- apply(rule that[of \<infinity>]) unfolding Lim_PInfty 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 ..
+ hence "ALL n. Y n < extreal B" using Y_def by auto note B = this[rule_format]
+ { fix n have "Y n < \<infinity>" using B[of n] apply (subst less_le_trans) by auto
+ hence "Y n ~= \<infinity> & Y n ~= (-\<infinity>)" using minfy by auto
+ } hence *: "ALL n. \<bar>Y n\<bar> \<noteq> \<infinity>" by auto
+ { fix n have "real (Y n) < B" proof- case goal1 thus ?case
+ using B[of n] apply-apply(subst(asm) extreal_real'[THEN sym]) defer defer
+ unfolding extreal_less using * by auto
+ qed
+ }
+ hence B': "ALL n. (real (Y n) <= B)" using less_imp_le by auto
+ have "EX l. (%n. real (Y n)) ----> l"
+ apply(rule bounded_increasing_convergent2)
+ proof safe show "!!n. real (Y n) <= B" using B' by auto
+ fix n m::nat assume "n<=m"
+ hence "extreal (real (Y n)) <= extreal (real (Y m))"
+ using incy[rule_format,of n m] apply(subst extreal_real)+
+ using *[rule_format, of n] *[rule_format, of m] by auto
+ thus "real (Y n) <= real (Y m)" by auto
+ qed then guess l .. note l=this
+ have "Y ----> extreal l" using l apply-apply(subst(asm) lim_extreal[THEN sym])
+ unfolding extreal_real using * by auto
+ thus thesis apply-apply(rule that[of "extreal l"])
+ apply (subst tail_same_limit[of Y _ N]) using Y_def by auto
+ qed
+qed
+
+lemma lim_extreal_decreasing: assumes "\<And>n m. n >= m \<Longrightarrow> f n <= f m"
+ obtains l where "f ----> (l::extreal)"
+proof -
+ from lim_extreal_increasing[of "\<lambda>x. - f x"] assms
+ obtain l where "(\<lambda>x. - f x) ----> l" by auto
+ from extreal_lim_mult[OF this, of "- 1"] show thesis
+ by (intro that[of "-l"]) (simp add: extreal_uminus_eq_reorder)
+qed
+
+lemma compact_extreal:
+ fixes X :: "nat \<Rightarrow> extreal"
+ shows "\<exists>l r. subseq r \<and> (X \<circ> r) ----> l"
+proof -
+ obtain r where "subseq r" and mono: "monoseq (X \<circ> r)"
+ using seq_monosub[of X] unfolding comp_def by auto
+ then have "(\<forall>n m. m \<le> n \<longrightarrow> (X \<circ> r) m \<le> (X \<circ> r) n) \<or> (\<forall>n m. m \<le> n \<longrightarrow> (X \<circ> r) n \<le> (X \<circ> r) m)"
+ by (auto simp add: monoseq_def)
+ then obtain l where "(X\<circ>r) ----> l"
+ using lim_extreal_increasing[of "X \<circ> r"] lim_extreal_decreasing[of "X \<circ> r"] by auto
+ then show ?thesis using `subseq r` by auto
+qed
+
+lemma extreal_Sup_lim:
+ assumes "\<And>n. b n \<in> s" "b ----> (a::extreal)"
+ shows "a \<le> Sup s"
+by (metis Lim_bounded_extreal assms complete_lattice_class.Sup_upper)
+
+lemma extreal_Inf_lim:
+ assumes "\<And>n. b n \<in> s" "b ----> (a::extreal)"
+ shows "Inf s \<le> a"
+by (metis Lim_bounded2_extreal assms complete_lattice_class.Inf_lower)
+
+lemma SUP_Lim_extreal:
+ fixes X :: "nat \<Rightarrow> extreal" assumes "incseq X" "X ----> l" shows "(SUP n. X n) = l"
+proof (rule extreal_SUPI)
+ fix n from assms show "X n \<le> l"
+ by (intro incseq_le_extreal) (simp add: incseq_def)
+next
+ fix y assume "\<And>n. n \<in> UNIV \<Longrightarrow> X n \<le> y"
+ with extreal_Sup_lim[OF _ `X ----> l`, of "{..y}"]
+ show "l \<le> y" by auto
+qed
+
+lemma LIMSEQ_extreal_SUPR:
+ fixes X :: "nat \<Rightarrow> extreal" assumes "incseq X" shows "X ----> (SUP n. X n)"
+proof (rule lim_extreal_increasing)
+ fix n m :: nat assume "m \<le> n" then show "X m \<le> X n"
+ using `incseq X` by (simp add: incseq_def)
+next
+ fix l assume "X ----> l"
+ with SUP_Lim_extreal[of X, OF assms this] show ?thesis by simp
+qed
+
+lemma INF_Lim_extreal: "decseq X \<Longrightarrow> X ----> l \<Longrightarrow> (INF n. X n) = (l::extreal)"
+ using SUP_Lim_extreal[of "\<lambda>i. - X i" "- l"]
+ by (simp add: extreal_SUPR_uminus extreal_lim_uminus)
+
+lemma LIMSEQ_extreal_INFI: "decseq X \<Longrightarrow> X ----> (INF n. X n :: extreal)"
+ using LIMSEQ_extreal_SUPR[of "\<lambda>i. - X i"]
+ by (simp add: extreal_SUPR_uminus extreal_lim_uminus)
+
+lemma SUP_eq_LIMSEQ:
+ assumes "mono f"
+ shows "(SUP n. extreal (f n)) = extreal x \<longleftrightarrow> f ----> x"
+proof
+ have inc: "incseq (\<lambda>i. extreal (f i))"
+ using `mono f` unfolding mono_def incseq_def by auto
+ { assume "f ----> x"
+ then have "(\<lambda>i. extreal (f i)) ----> extreal x" by auto
+ from SUP_Lim_extreal[OF inc this]
+ show "(SUP n. extreal (f n)) = extreal x" . }
+ { assume "(SUP n. extreal (f n)) = extreal x"
+ with LIMSEQ_extreal_SUPR[OF inc]
+ show "f ----> x" by auto }
+qed
+
+lemma Liminf_within:
+ fixes f :: "'a::metric_space => extreal"
+ shows "Liminf (at x within S) f = (SUP e:{0<..}. INF y:(S Int ball x e - {x}). f y)"
+proof-
+let ?l="(SUP e:{0<..}. INF y:(S Int ball x e - {x}). f y)"
+{ fix T assume T_def: "open T & mono T & ?l:T"
+ have "EX d>0. ALL y:S. 0 < dist y x & dist y x < d --> f y : T"
+ proof-
+ { assume "T=UNIV" hence ?thesis by (simp add: gt_ex) }
+ moreover
+ { assume "~(T=UNIV)"
+ then obtain B where "T={B<..}" using T_def extreal_open_mono_set[of T] by auto
+ hence "B<?l" using T_def by auto
+ then obtain d where d_def: "0<d & B<(INF y:(S Int ball x d - {x}). f y)"
+ unfolding less_SUP_iff by auto
+ { fix y assume "y:S & 0 < dist y x & dist y x < d"
+ hence "y:(S Int ball x d - {x})" unfolding ball_def by (auto simp add: dist_commute)
+ hence "f y:T" using d_def INF_leI[of y "S Int ball x d - {x}" f] `T={B<..}` by auto
+ } hence ?thesis apply(rule_tac x="d" in exI) using d_def by auto
+ } ultimately show ?thesis by auto
+ qed
+}
+moreover
+{ fix z
+ assume a: "ALL T. open T --> mono T --> z : T -->
+ (EX d>0. ALL y:S. 0 < dist y x & dist y x < d --> f y : T)"
+ { fix B assume "B<z"
+ then obtain d where d_def: "d>0 & (ALL y:S. 0 < dist y x & dist y x < d --> B < f y)"
+ using a[rule_format, of "{B<..}"] mono_greaterThan by auto
+ { fix y assume "y:(S Int ball x d - {x})"
+ hence "y:S & 0 < dist y x & dist y x < d" unfolding ball_def apply (simp add: dist_commute)
+ by (metis dist_eq_0_iff real_less_def zero_le_dist)
+ hence "B <= f y" using d_def by auto
+ } hence "B <= INFI (S Int ball x d - {x}) f" apply (subst le_INFI) by auto
+ also have "...<=?l" apply (subst le_SUPI) using d_def by auto
+ finally have "B<=?l" by auto
+ } hence "z <= ?l" using extreal_le_extreal[of z "?l"] by auto
+}
+ultimately show ?thesis unfolding extreal_Liminf_Sup_monoset eventually_within
+ apply (subst extreal_SupI[of _ "(SUP e:{0<..}. INFI (S Int ball x e - {x}) f)"]) by auto
+qed
+
+lemma Limsup_within:
+ fixes f :: "'a::metric_space => extreal"
+ shows "Limsup (at x within S) f = (INF e:{0<..}. SUP y:(S Int ball x e - {x}). f y)"
+proof-
+let ?l="(INF e:{0<..}. SUP y:(S Int ball x e - {x}). f y)"
+{ fix T assume T_def: "open T & mono (uminus ` T) & ?l:T"
+ have "EX d>0. ALL y:S. 0 < dist y x & dist y x < d --> f y : T"
+ proof-
+ { assume "T=UNIV" hence ?thesis by (simp add: gt_ex) }
+ moreover
+ { assume "~(T=UNIV)" hence "~(uminus ` T = UNIV)"
+ by (metis Int_UNIV_right Int_absorb1 image_mono extreal_minus_minus_image subset_UNIV)
+ hence "uminus ` T = {Inf (uminus ` T)<..}" using T_def extreal_open_mono_set[of "uminus ` T"]
+ extreal_open_uminus[of T] by auto
+ then obtain B where "T={..<B}"
+ unfolding extreal_Inf_uminus_image_eq extreal_uminus_lessThan[symmetric]
+ unfolding inj_image_eq_iff[OF extreal_inj_on_uminus] by simp
+ hence "?l<B" using T_def by auto
+ then obtain d where d_def: "0<d & (SUP y:(S Int ball x d - {x}). f y)<B"
+ unfolding INF_less_iff by auto
+ { fix y assume "y:S & 0 < dist y x & dist y x < d"
+ hence "y:(S Int ball x d - {x})" unfolding ball_def by (auto simp add: dist_commute)
+ hence "f y:T" using d_def le_SUPI[of y "S Int ball x d - {x}" f] `T={..<B}` by auto
+ } hence ?thesis apply(rule_tac x="d" in exI) using d_def by auto
+ } ultimately show ?thesis by auto
+ qed
+}
+moreover
+{ fix z
+ assume a: "ALL T. open T --> mono (uminus ` T) --> z : T -->
+ (EX d>0. ALL y:S. 0 < dist y x & dist y x < d --> f y : T)"
+ { fix B assume "z<B"
+ then obtain d where d_def: "d>0 & (ALL y:S. 0 < dist y x & dist y x < d --> f y<B)"
+ using a[rule_format, of "{..<B}"] by auto
+ { fix y assume "y:(S Int ball x d - {x})"
+ hence "y:S & 0 < dist y x & dist y x < d" unfolding ball_def apply (simp add: dist_commute)
+ by (metis dist_eq_0_iff real_less_def zero_le_dist)
+ hence "f y <= B" using d_def by auto
+ } hence "SUPR (S Int ball x d - {x}) f <= B" apply (subst SUP_leI) by auto
+ moreover have "?l<=SUPR (S Int ball x d - {x}) f" apply (subst INF_leI) using d_def by auto
+ ultimately have "?l<=B" by auto
+ } hence "?l <= z" using extreal_ge_extreal[of z "?l"] by auto
+}
+ultimately show ?thesis unfolding extreal_Limsup_Inf_monoset eventually_within
+ apply (subst extreal_InfI) by auto
+qed
+
+
+lemma Liminf_within_UNIV:
+ fixes f :: "'a::metric_space => extreal"
+ shows "Liminf (at x) f = Liminf (at x within UNIV) f"
+by (metis within_UNIV)
+
+
+lemma Liminf_at:
+ fixes f :: "'a::metric_space => extreal"
+ shows "Liminf (at x) f = (SUP e:{0<..}. INF y:(ball x e - {x}). f y)"
+using Liminf_within[of x UNIV f] Liminf_within_UNIV[of x f] by auto
+
+
+lemma Limsup_within_UNIV:
+ fixes f :: "'a::metric_space => extreal"
+ shows "Limsup (at x) f = Limsup (at x within UNIV) f"
+by (metis within_UNIV)
+
+
+lemma Limsup_at:
+ fixes f :: "'a::metric_space => extreal"
+ shows "Limsup (at x) f = (INF e:{0<..}. SUP y:(ball x e - {x}). f y)"
+using Limsup_within[of x UNIV f] Limsup_within_UNIV[of x f] by auto
+
+lemma Lim_within_constant:
+ fixes f :: "'a::metric_space => 'b::topological_space"
+ assumes "ALL y:S. f y = C"
+ shows "(f ---> C) (at x within S)"
+unfolding tendsto_def eventually_within
+by (metis assms(1) linorder_le_less_linear n_not_Suc_n real_of_nat_le_zero_cancel_iff)
+
+lemma Liminf_within_constant:
+ fixes f :: "'a::metric_space => extreal"
+ assumes "ALL y:S. f y = C"
+ assumes "~trivial_limit (at x within S)"
+ shows "Liminf (at x within S) f = C"
+by (metis Lim_within_constant assms lim_imp_Liminf)
+
+lemma Limsup_within_constant:
+ fixes f :: "'a::metric_space => extreal"
+ assumes "ALL y:S. f y = C"
+ assumes "~trivial_limit (at x within S)"
+ shows "Limsup (at x within S) f = C"
+by (metis Lim_within_constant assms lim_imp_Limsup)
+
+lemma islimpt_punctured:
+"x islimpt S = x islimpt (S-{x})"
+unfolding islimpt_def by blast
+
+
+lemma islimpt_in_closure:
+"(x islimpt S) = (x:closure(S-{x}))"
+unfolding closure_def using islimpt_punctured by blast
+
+
+lemma not_trivial_limit_within:
+ "~trivial_limit (at x within S) = (x:closure(S-{x}))"
+using islimpt_in_closure by (metis trivial_limit_within)
+
+
+lemma not_trivial_limit_within_ball:
+ "(~trivial_limit (at x within S)) = (ALL e>0. S Int ball x e - {x} ~= {})"
+ (is "?lhs = ?rhs")
+proof-
+{ assume "?lhs"
+ { fix e :: real assume "e>0"
+ then obtain y where "y:(S-{x}) & dist y x < e"
+ using `?lhs` not_trivial_limit_within[of x S] closure_approachable[of x "S - {x}"] by auto
+ hence "y : (S Int ball x e - {x})" unfolding ball_def by (simp add: dist_commute)
+ hence "S Int ball x e - {x} ~= {}" by blast
+ } hence "?rhs" by auto
+}
+moreover
+{ assume "?rhs"
+ { fix e :: real assume "e>0"
+ then obtain y where "y : (S Int ball x e - {x})" using `?rhs` by blast
+ hence "y:(S-{x}) & dist y x < e" unfolding ball_def by (simp add: dist_commute)
+ hence "EX y:(S-{x}). dist y x < e" by auto
+ } hence "?lhs" using not_trivial_limit_within[of x S] closure_approachable[of x "S - {x}"] by auto
+} ultimately show ?thesis by auto
+qed
+
+subsubsection {* Continuity *}
+
+lemma continuous_imp_tendsto:
+ assumes "continuous (at x0) f"
+ assumes "x ----> x0"
+ shows "(f o x) ----> (f x0)"
+proof-
+{ fix S assume "open S & (f x0):S"
+ from this obtain T where T_def: "open T & x0 : T & (ALL x:T. f x : S)"
+ using assms continuous_at_open by metis
+ hence "(EX N. ALL n>=N. x n : T)" using assms tendsto_explicit T_def by auto
+ hence "(EX N. ALL n>=N. f(x n) : S)" using T_def by auto
+} from this show ?thesis using tendsto_explicit[of "f o x" "f x0"] by auto
+qed
+
+
+lemma continuous_at_sequentially2:
+fixes f :: "'a::metric_space => 'b:: topological_space"
+shows "continuous (at x0) f <-> (ALL x. (x ----> x0) --> (f o x) ----> (f x0))"
+proof-
+{ assume "~(continuous (at x0) f)"
+ from this obtain T where T_def:
+ "open T & f x0 : T & (ALL S. (open S & x0 : S) --> (EX x':S. f x' ~: T))"
+ using continuous_at_open[of x0 f] by metis
+ def X == "{x'. f x' ~: T}" hence "x0 islimpt X" unfolding islimpt_def using T_def by auto
+ from this obtain x where x_def: "(ALL n. x n : X) & x ----> x0"
+ using islimpt_sequential[of x0 X] by auto
+ hence "~(f o x) ----> (f x0)" unfolding tendsto_explicit using X_def T_def by auto
+ hence "EX x. x ----> x0 & (~(f o x) ----> (f x0))" using x_def by auto
+}
+from this show ?thesis using continuous_imp_tendsto by auto
+qed
+
+lemma continuous_at_of_extreal:
+ fixes x0 :: extreal
+ assumes "\<bar>x0\<bar> \<noteq> \<infinity>"
+ shows "continuous (at x0) real"
+proof-
+{ fix T assume T_def: "open T & real x0 : T"
+ def S == "extreal ` T"
+ hence "extreal (real x0) : S" using T_def by auto
+ hence "x0 : S" using assms extreal_real by auto
+ moreover have "open S" using open_extreal S_def T_def by auto
+ moreover have "ALL y:S. real y : T" using S_def T_def by auto
+ ultimately have "EX S. x0 : S & open S & (ALL y:S. real y : T)" by auto
+} from this show ?thesis unfolding continuous_at_open by blast
+qed
+
+
+lemma continuous_at_iff_extreal:
+fixes f :: "'a::t2_space => real"
+shows "continuous (at x0) f <-> continuous (at x0) (extreal o f)"
+proof-
+{ assume "continuous (at x0) f" hence "continuous (at x0) (extreal o f)"
+ using continuous_at_extreal continuous_at_compose[of x0 f extreal] by auto
+}
+moreover
+{ assume "continuous (at x0) (extreal o f)"
+ hence "continuous (at x0) (real o (extreal o f))"
+ using continuous_at_of_extreal by (intro continuous_at_compose[of x0 "extreal o f"]) auto
+ moreover have "real o (extreal o f) = f" using real_extreal_id by (simp add: o_assoc)
+ ultimately have "continuous (at x0) f" by auto
+} ultimately show ?thesis by auto
+qed
+
+
+lemma continuous_on_iff_extreal:
+fixes f :: "'a::t2_space => real"
+fixes A assumes "open A"
+shows "continuous_on A f <-> continuous_on A (extreal o f)"
+ using continuous_at_iff_extreal assms by (auto simp add: continuous_on_eq_continuous_at)
+
+
+lemma continuous_on_real: "continuous_on (UNIV-{\<infinity>,(-\<infinity>)}) real"
+ using continuous_at_of_extreal continuous_on_eq_continuous_at open_image_extreal by auto
+
+
+lemma continuous_on_iff_real:
+ fixes f :: "'a::t2_space => extreal"
+ assumes "\<And>x. x \<in> A \<Longrightarrow> \<bar>f x\<bar> \<noteq> \<infinity>"
+ shows "continuous_on A f \<longleftrightarrow> continuous_on A (real \<circ> f)"
+proof-
+ have "f ` A <= UNIV-{\<infinity>,(-\<infinity>)}" using assms by force
+ hence *: "continuous_on (f ` A) real"
+ using continuous_on_real by (simp add: continuous_on_subset)
+have **: "continuous_on ((real o f) ` A) extreal"
+ using continuous_on_extreal continuous_on_subset[of "UNIV" "extreal" "(real o f) ` A"] by blast
+{ assume "continuous_on A f" hence "continuous_on A (real o f)"
+ apply (subst continuous_on_compose) using * by auto
+}
+moreover
+{ assume "continuous_on A (real o f)"
+ hence "continuous_on A (extreal o (real o f))"
+ apply (subst continuous_on_compose) using ** by auto
+ hence "continuous_on A f"
+ apply (subst continuous_on_eq[of A "extreal o (real o f)" f])
+ using assms extreal_real by auto
+}
+ultimately show ?thesis by auto
+qed
+
+
+lemma continuous_at_const:
+ fixes f :: "'a::t2_space => extreal"
+ assumes "ALL x. (f x = C)"
+ shows "ALL x. continuous (at x) f"
+unfolding continuous_at_open using assms t1_space by auto
+
+
+lemma closure_contains_Inf:
+ fixes S :: "real set"
+ assumes "S ~= {}" "EX B. ALL x:S. B<=x"
+ shows "Inf S : closure S"
+proof-
+have *: "ALL x:S. Inf S <= x" using Inf_lower_EX[of _ S] assms by metis
+{ fix e assume "e>(0 :: real)"
+ from this obtain x where x_def: "x:S & x < Inf S + e" using Inf_close `S ~= {}` by auto
+ moreover hence "x > Inf S - e" using * by auto
+ ultimately have "abs (x - Inf S) < e" by (simp add: abs_diff_less_iff)
+ hence "EX x:S. abs (x - Inf S) < e" using x_def by auto
+} from this show ?thesis apply (subst closure_approachable) unfolding dist_norm by auto
+qed
+
+
+lemma closed_contains_Inf:
+ fixes S :: "real set"
+ assumes "S ~= {}" "EX B. ALL x:S. B<=x"
+ assumes "closed S"
+ shows "Inf S : S"
+by (metis closure_contains_Inf closure_closed assms)
+
+
+lemma mono_closed_real:
+ fixes S :: "real set"
+ assumes mono: "ALL y z. y:S & y<=z --> z:S"
+ assumes "closed S"
+ shows "S = {} | S = UNIV | (EX a. S = {a ..})"
+proof-
+{ assume "S ~= {}"
+ { assume ex: "EX B. ALL x:S. B<=x"
+ hence *: "ALL x:S. Inf S <= x" using Inf_lower_EX[of _ S] ex by metis
+ hence "Inf S : S" apply (subst closed_contains_Inf) using ex `S ~= {}` `closed S` by auto
+ hence "ALL x. (Inf S <= x <-> x:S)" using mono[rule_format, of "Inf S"] * by auto
+ hence "S = {Inf S ..}" by auto
+ hence "EX a. S = {a ..}" by auto
+ }
+ moreover
+ { assume "~(EX B. ALL x:S. B<=x)"
+ hence nex: "ALL B. EX x:S. x<B" by (simp add: not_le)
+ { fix y obtain x where "x:S & x < y" using nex by auto
+ hence "y:S" using mono[rule_format, of x y] by auto
+ } hence "S = UNIV" by auto
+ } ultimately have "S = UNIV | (EX a. S = {a ..})" by blast
+} from this show ?thesis by blast
+qed
+
+
+lemma mono_closed_extreal:
+ fixes S :: "real set"
+ assumes mono: "ALL y z. y:S & y<=z --> z:S"
+ assumes "closed S"
+ shows "EX a. S = {x. a <= extreal x}"
+proof-
+{ assume "S = {}" hence ?thesis apply(rule_tac x=PInfty in exI) by auto }
+moreover
+{ assume "S = UNIV" hence ?thesis apply(rule_tac x="-\<infinity>" in exI) by auto }
+moreover
+{ assume "EX a. S = {a ..}"
+ from this obtain a where "S={a ..}" by auto
+ hence ?thesis apply(rule_tac x="extreal a" in exI) by auto
+} ultimately show ?thesis using mono_closed_real[of S] assms by auto
+qed
+
+subsection {* Sums *}
+
+lemma setsum_extreal[simp]:
+ "(\<Sum>x\<in>A. extreal (f x)) = extreal (\<Sum>x\<in>A. f x)"
+proof cases
+ assume "finite A" then show ?thesis by induct auto
+qed simp
+
+lemma setsum_Pinfty: "(\<Sum>x\<in>P. f x) = \<infinity> \<longleftrightarrow> (finite P \<and> (\<exists>i\<in>P. f i = \<infinity>))"
+proof safe
+ assume *: "setsum f P = \<infinity>"
+ show "finite P"
+ proof (rule ccontr) assume "infinite P" with * show False by auto qed
+ show "\<exists>i\<in>P. f i = \<infinity>"
+ proof (rule ccontr)
+ assume "\<not> ?thesis" then have "\<And>i. i \<in> P \<Longrightarrow> f i \<noteq> \<infinity>" by auto
+ from `finite P` this have "setsum f P \<noteq> \<infinity>"
+ by induct auto
+ with * show False by auto
+ qed
+next
+ fix i assume "finite P" "i \<in> P" "f i = \<infinity>"
+ thus "setsum f P = \<infinity>"
+ proof induct
+ case (insert x A)
+ show ?case using insert by (cases "x = i") auto
+ qed simp
+qed
+
+lemma setsum_Inf:
+ shows "\<bar>setsum f A\<bar> = \<infinity> \<longleftrightarrow> (finite A \<and> (\<exists>i\<in>A. \<bar>f i\<bar> = \<infinity>))"
+proof
+ assume *: "\<bar>setsum f A\<bar> = \<infinity>"
+ have "finite A" by (rule ccontr) (insert *, auto)
+ moreover have "\<exists>i\<in>A. \<bar>f i\<bar> = \<infinity>"
+ proof (rule ccontr)
+ assume "\<not> ?thesis" then have "\<forall>i\<in>A. \<exists>r. f i = extreal r" by auto
+ from bchoice[OF this] guess r ..
+ with * show False by (auto simp: setsum_extreal)
+ qed
+ ultimately show "finite A \<and> (\<exists>i\<in>A. \<bar>f i\<bar> = \<infinity>)" by auto
+next
+ assume "finite A \<and> (\<exists>i\<in>A. \<bar>f i\<bar> = \<infinity>)"
+ then obtain i where "finite A" "i \<in> A" "\<bar>f i\<bar> = \<infinity>" by auto
+ then show "\<bar>setsum f A\<bar> = \<infinity>"
+ proof induct
+ case (insert j A) then show ?case
+ by (cases rule: extreal3_cases[of "f i" "f j" "setsum f A"]) auto
+ qed simp
+qed
+
+lemma setsum_real_of_extreal:
+ assumes "\<And>x. x \<in> S \<Longrightarrow> \<bar>f x\<bar> \<noteq> \<infinity>"
+ shows "(\<Sum>x\<in>S. real (f x)) = real (setsum f S)"
+proof -
+ have "\<forall>x\<in>S. \<exists>r. f x = extreal r"
+ proof
+ fix x assume "x \<in> S"
+ from assms[OF this] show "\<exists>r. f x = extreal r" by (cases "f x") auto
+ qed
+ from bchoice[OF this] guess r ..
+ then show ?thesis by simp
+qed
+
+lemma setsum_extreal_0:
+ fixes f :: "'a \<Rightarrow> extreal" assumes "finite A" "\<And>i. i \<in> A \<Longrightarrow> 0 \<le> f i"
+ shows "(\<Sum>x\<in>A. f x) = 0 \<longleftrightarrow> (\<forall>i\<in>A. f i = 0)"
+proof
+ assume *: "(\<Sum>x\<in>A. f x) = 0"
+ then have "(\<Sum>x\<in>A. f x) \<noteq> \<infinity>" by auto
+ then have "\<forall>i\<in>A. \<bar>f i\<bar> \<noteq> \<infinity>" using assms by (force simp: setsum_Pinfty)
+ then have "\<forall>i\<in>A. \<exists>r. f i = extreal r" by auto
+ from bchoice[OF this] * assms show "\<forall>i\<in>A. f i = 0"
+ using setsum_nonneg_eq_0_iff[of A "\<lambda>i. real (f i)"] by auto
+qed (rule setsum_0')
+
+
+lemma setsum_extreal_right_distrib:
+ fixes f :: "'a \<Rightarrow> extreal" assumes "\<And>i. i \<in> A \<Longrightarrow> 0 \<le> f i"
+ shows "r * setsum f A = (\<Sum>n\<in>A. r * f n)"
+proof cases
+ assume "finite A" then show ?thesis using assms
+ by induct (auto simp: extreal_right_distrib setsum_nonneg)
+qed simp
+
+lemma sums_extreal_positive:
+ fixes f :: "nat \<Rightarrow> extreal" assumes "\<And>i. 0 \<le> f i" shows "f sums (SUP n. \<Sum>i<n. f i)"
+proof -
+ have "incseq (\<lambda>i. \<Sum>j=0..<i. f j)"
+ using extreal_add_mono[OF _ assms] by (auto intro!: incseq_SucI)
+ from LIMSEQ_extreal_SUPR[OF this]
+ show ?thesis unfolding sums_def by (simp add: atLeast0LessThan)
+qed
+
+lemma summable_extreal_pos:
+ fixes f :: "nat \<Rightarrow> extreal" assumes "\<And>i. 0 \<le> f i" shows "summable f"
+ using sums_extreal_positive[of f, OF assms] unfolding summable_def by auto
+
+lemma suminf_extreal_eq_SUPR:
+ fixes f :: "nat \<Rightarrow> extreal" assumes "\<And>i. 0 \<le> f i"
+ shows "(\<Sum>x. f x) = (SUP n. \<Sum>i<n. f i)"
+ using sums_extreal_positive[of f, OF assms, THEN sums_unique] by simp
+
+lemma sums_extreal:
+ "(\<lambda>x. extreal (f x)) sums extreal x \<longleftrightarrow> f sums x"
+ unfolding sums_def by simp
+
+lemma suminf_bound:
+ fixes f :: "nat \<Rightarrow> extreal"
+ assumes "\<forall>N. (\<Sum>n<N. f n) \<le> x" and pos: "\<And>n. 0 \<le> f n"
+ shows "suminf f \<le> x"
+proof (rule Lim_bounded_extreal)
+ have "summable f" using pos[THEN summable_extreal_pos] .
+ then show "(\<lambda>N. \<Sum>n<N. f n) ----> suminf f"
+ by (auto dest!: summable_sums simp: sums_def atLeast0LessThan)
+ show "\<forall>n\<ge>0. setsum f {..<n} \<le> x"
+ using assms by auto
+qed
+
+lemma suminf_bound_add:
+ fixes f :: "nat \<Rightarrow> extreal"
+ assumes "\<forall>N. (\<Sum>n<N. f n) + y \<le> x" and pos: "\<And>n. 0 \<le> f n" and "y \<noteq> -\<infinity>"
+ shows "suminf f + y \<le> x"
+proof (cases y)
+ case (real r) then have "\<forall>N. (\<Sum>n<N. f n) \<le> x - y"
+ using assms by (simp add: extreal_le_minus)
+ then have "(\<Sum> n. f n) \<le> x - y" using pos by (rule suminf_bound)
+ then show "(\<Sum> n. f n) + y \<le> x"
+ using assms real by (simp add: extreal_le_minus)
+qed (insert assms, auto)
+
+lemma sums_finite:
+ assumes "\<forall>N\<ge>n. f N = 0"
+ shows "f sums (\<Sum>N<n. f N)"
+proof -
+ { fix i have "(\<Sum>N<i + n. f N) = (\<Sum>N<n. f N)"
+ by (induct i) (insert assms, auto) }
+ note this[simp]
+ show ?thesis unfolding sums_def
+ by (rule LIMSEQ_offset[of _ n]) (auto simp add: atLeast0LessThan)
+qed
+
+lemma suminf_finite:
+ fixes f :: "nat \<Rightarrow> 'a::{comm_monoid_add,t2_space}" assumes "\<forall>N\<ge>n. f N = 0"
+ shows "suminf f = (\<Sum>N<n. f N)"
+ using sums_finite[OF assms, THEN sums_unique] by simp
+
+lemma suminf_extreal_0[simp]: "(\<Sum>i. 0) = (0::'a::{comm_monoid_add,t2_space})"
+ using suminf_finite[of 0 "\<lambda>x. 0"] by simp
+
+lemma suminf_upper:
+ fixes f :: "nat \<Rightarrow> extreal" assumes "\<And>n. 0 \<le> f n"
+ shows "(\<Sum>n<N. f n) \<le> (\<Sum>n. f n)"
+ unfolding suminf_extreal_eq_SUPR[OF assms] SUPR_def
+ by (auto intro: complete_lattice_class.Sup_upper image_eqI)
+
+lemma suminf_0_le:
+ fixes f :: "nat \<Rightarrow> extreal" assumes "\<And>n. 0 \<le> f n"
+ shows "0 \<le> (\<Sum>n. f n)"
+ using suminf_upper[of f 0, OF assms] by simp
+
+lemma suminf_le_pos:
+ fixes f g :: "nat \<Rightarrow> extreal"
+ assumes "\<And>N. f N \<le> g N" "\<And>N. 0 \<le> f N"
+ shows "suminf f \<le> suminf g"
+proof (safe intro!: suminf_bound)
+ fix n { fix N have "0 \<le> g N" using assms(2,1)[of N] by auto }
+ have "setsum f {..<n} \<le> setsum g {..<n}" using assms by (auto intro: setsum_mono)
+ also have "... \<le> suminf g" using `\<And>N. 0 \<le> g N` by (rule suminf_upper)
+ finally show "setsum f {..<n} \<le> suminf g" .
+qed (rule assms(2))
+
+lemma suminf_half_series_extreal: "(\<Sum>n. (1/2 :: extreal)^Suc n) = 1"
+ using sums_extreal[THEN iffD2, OF power_half_series, THEN sums_unique, symmetric]
+ by (simp add: one_extreal_def)
+
+lemma suminf_add_extreal:
+ fixes f g :: "nat \<Rightarrow> extreal"
+ assumes "\<And>i. 0 \<le> f i" "\<And>i. 0 \<le> g i"
+ shows "(\<Sum>i. f i + g i) = suminf f + suminf g"
+ apply (subst (1 2 3) suminf_extreal_eq_SUPR)
+ unfolding setsum_addf
+ by (intro assms extreal_add_nonneg_nonneg SUPR_extreal_add_pos incseq_setsumI setsum_nonneg ballI)+
+
+lemma suminf_cmult_extreal:
+ fixes f g :: "nat \<Rightarrow> extreal"
+ assumes "\<And>i. 0 \<le> f i" "0 \<le> a"
+ shows "(\<Sum>i. a * f i) = a * suminf f"
+ by (auto simp: setsum_extreal_right_distrib[symmetric] assms
+ extreal_zero_le_0_iff setsum_nonneg suminf_extreal_eq_SUPR
+ intro!: SUPR_extreal_cmult )
+
+lemma suminf_PInfty:
+ assumes "\<And>i. 0 \<le> f i" "suminf f \<noteq> \<infinity>"
+ shows "f i \<noteq> \<infinity>"
+proof -
+ from suminf_upper[of f "Suc i", OF assms(1)] assms(2)
+ have "(\<Sum>i<Suc i. f i) \<noteq> \<infinity>" by auto
+ then show ?thesis
+ unfolding setsum_Pinfty by simp
+qed
+
+lemma suminf_PInfty_fun:
+ assumes "\<And>i. 0 \<le> f i" "suminf f \<noteq> \<infinity>"
+ shows "\<exists>f'. f = (\<lambda>x. extreal (f' x))"
+proof -
+ have "\<forall>i. \<exists>r. f i = extreal r"
+ proof
+ fix i show "\<exists>r. f i = extreal r"
+ using suminf_PInfty[OF assms] assms(1)[of i] by (cases "f i") auto
+ qed
+ from choice[OF this] show ?thesis by auto
+qed
+
+lemma summable_extreal:
+ assumes "\<And>i. 0 \<le> f i" "(\<Sum>i. extreal (f i)) \<noteq> \<infinity>"
+ shows "summable f"
+proof -
+ have "0 \<le> (\<Sum>i. extreal (f i))"
+ using assms by (intro suminf_0_le) auto
+ with assms obtain r where r: "(\<Sum>i. extreal (f i)) = extreal r"
+ by (cases "\<Sum>i. extreal (f i)") auto
+ from summable_extreal_pos[of "\<lambda>x. extreal (f x)"]
+ have "summable (\<lambda>x. extreal (f x))" using assms by auto
+ from summable_sums[OF this]
+ have "(\<lambda>x. extreal (f x)) sums (\<Sum>x. extreal (f x))" by auto
+ then show "summable f"
+ unfolding r sums_extreal summable_def ..
+qed
+
+lemma suminf_extreal:
+ assumes "\<And>i. 0 \<le> f i" "(\<Sum>i. extreal (f i)) \<noteq> \<infinity>"
+ shows "(\<Sum>i. extreal (f i)) = extreal (suminf f)"
+proof (rule sums_unique[symmetric])
+ from summable_extreal[OF assms]
+ show "(\<lambda>x. extreal (f x)) sums (extreal (suminf f))"
+ unfolding sums_extreal using assms by (intro summable_sums summable_extreal)
+qed
+
+lemma suminf_extreal_minus:
+ fixes f g :: "nat \<Rightarrow> extreal"
+ assumes ord: "\<And>i. g i \<le> f i" "\<And>i. 0 \<le> g i" and fin: "suminf f \<noteq> \<infinity>" "suminf g \<noteq> \<infinity>"
+ shows "(\<Sum>i. f i - g i) = suminf f - suminf g"
+proof -
+ { fix i have "0 \<le> f i" using ord[of i] by auto }
+ moreover
+ from suminf_PInfty_fun[OF `\<And>i. 0 \<le> f i` fin(1)] guess f' .. note this[simp]
+ from suminf_PInfty_fun[OF `\<And>i. 0 \<le> g i` fin(2)] guess g' .. note this[simp]
+ { fix i have "0 \<le> f i - g i" using ord[of i] by (auto simp: extreal_le_minus_iff) }
+ moreover
+ have "suminf (\<lambda>i. f i - g i) \<le> suminf f"
+ using assms by (auto intro!: suminf_le_pos simp: field_simps)
+ then have "suminf (\<lambda>i. f i - g i) \<noteq> \<infinity>" using fin by auto
+ ultimately show ?thesis using assms `\<And>i. 0 \<le> f i`
+ apply simp
+ by (subst (1 2 3) suminf_extreal)
+ (auto intro!: suminf_diff[symmetric] summable_extreal)
+qed
+
+lemma suminf_extreal_PInf[simp]:
+ "(\<Sum>x. \<infinity>) = \<infinity>"
+proof -
+ have "(\<Sum>i<Suc 0. \<infinity>) \<le> (\<Sum>x. \<infinity>)" by (rule suminf_upper) auto
+ then show ?thesis by simp
+qed
+
+lemma summable_real_of_extreal:
+ assumes f: "\<And>i. 0 \<le> f i" and fin: "(\<Sum>i. f i) \<noteq> \<infinity>"
+ shows "summable (\<lambda>i. real (f i))"
+proof (rule summable_def[THEN iffD2])
+ have "0 \<le> (\<Sum>i. f i)" using assms by (auto intro: suminf_0_le)
+ with fin obtain r where r: "extreal r = (\<Sum>i. f i)" by (cases "(\<Sum>i. f i)") auto
+ { fix i have "f i \<noteq> \<infinity>" using f by (intro suminf_PInfty[OF _ fin]) auto
+ then have "\<bar>f i\<bar> \<noteq> \<infinity>" using f[of i] by auto }
+ note fin = this
+ have "(\<lambda>i. extreal (real (f i))) sums (\<Sum>i. extreal (real (f i)))"
+ using f by (auto intro!: summable_extreal_pos summable_sums simp: extreal_le_real_iff zero_extreal_def)
+ also have "\<dots> = extreal r" using fin r by (auto simp: extreal_real)
+ finally show "\<exists>r. (\<lambda>i. real (f i)) sums r" by (auto simp: sums_extreal)
+qed
+
+end
\ No newline at end of file
--- a/src/HOL/Multivariate_Analysis/Integration.thy Mon Mar 14 15:17:10 2011 +0100
+++ b/src/HOL/Multivariate_Analysis/Integration.thy Mon Mar 14 15:29:10 2011 +0100
@@ -1,4 +1,3 @@
-
header {* Kurzweil-Henstock Gauge Integration in many dimensions. *}
(* Author: John Harrison
Translation from HOL light: Robert Himmelmann, TU Muenchen *)
@@ -3780,7 +3779,7 @@
shows "f x = y"
proof- have "{x \<in> s. f x \<in> {y}} = {} \<or> {x \<in> s. f x \<in> {y}} = s"
apply(rule assms(1)[unfolded connected_clopen,rule_format]) apply rule defer
- apply(rule continuous_closed_in_preimage[OF assms(4) closed_sing])
+ apply(rule continuous_closed_in_preimage[OF assms(4) closed_singleton])
apply(rule open_openin_trans[OF assms(2)]) unfolding open_contains_ball
proof safe fix x assume "x\<in>s"
from assms(2)[unfolded open_contains_ball,rule_format,OF this] guess e .. note e=conjunctD2[OF this]
--- a/src/HOL/Multivariate_Analysis/Multivariate_Analysis.thy Mon Mar 14 15:17:10 2011 +0100
+++ b/src/HOL/Multivariate_Analysis/Multivariate_Analysis.thy Mon Mar 14 15:29:10 2011 +0100
@@ -1,5 +1,5 @@
theory Multivariate_Analysis
-imports Fashoda
+imports Fashoda Extended_Real_Limits
begin
end
--- a/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy Mon Mar 14 15:17:10 2011 +0100
+++ b/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy Mon Mar 14 15:29:10 2011 +0100
@@ -424,80 +424,6 @@
lemma connected_empty[simp, intro]: "connected {}"
by (simp add: connected_def)
-subsection{* Hausdorff and other separation properties *}
-
-class t0_space = topological_space +
- assumes t0_space: "x \<noteq> y \<Longrightarrow> \<exists>U. open U \<and> \<not> (x \<in> U \<longleftrightarrow> y \<in> U)"
-
-class t1_space = topological_space +
- assumes t1_space: "x \<noteq> y \<Longrightarrow> \<exists>U. open U \<and> x \<in> U \<and> y \<notin> U"
-
-instance t1_space \<subseteq> t0_space
-proof qed (fast dest: t1_space)
-
-lemma separation_t1:
- fixes x y :: "'a::t1_space"
- shows "x \<noteq> y \<longleftrightarrow> (\<exists>U. open U \<and> x \<in> U \<and> y \<notin> U)"
- using t1_space[of x y] by blast
-
-lemma closed_sing:
- fixes a :: "'a::t1_space"
- shows "closed {a}"
-proof -
- let ?T = "\<Union>{S. open S \<and> a \<notin> S}"
- have "open ?T" by (simp add: open_Union)
- also have "?T = - {a}"
- by (simp add: set_eq_iff separation_t1, auto)
- finally show "closed {a}" unfolding closed_def .
-qed
-
-lemma closed_insert [simp]:
- fixes a :: "'a::t1_space"
- assumes "closed S" shows "closed (insert a S)"
-proof -
- from closed_sing assms
- have "closed ({a} \<union> S)" by (rule closed_Un)
- thus "closed (insert a S)" by simp
-qed
-
-lemma finite_imp_closed:
- fixes S :: "'a::t1_space set"
- shows "finite S \<Longrightarrow> closed S"
-by (induct set: finite, simp_all)
-
-text {* T2 spaces are also known as Hausdorff spaces. *}
-
-class t2_space = topological_space +
- assumes hausdorff: "x \<noteq> y \<Longrightarrow> \<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {}"
-
-instance t2_space \<subseteq> t1_space
-proof qed (fast dest: hausdorff)
-
-instance metric_space \<subseteq> t2_space
-proof
- fix x y :: "'a::metric_space"
- assume xy: "x \<noteq> y"
- let ?U = "ball x (dist x y / 2)"
- let ?V = "ball y (dist x y / 2)"
- have th0: "\<And>d x y z. (d x z :: real) <= d x y + d y z \<Longrightarrow> d y z = d z y
- ==> ~(d x y * 2 < d x z \<and> d z y * 2 < d x z)" by arith
- have "open ?U \<and> open ?V \<and> x \<in> ?U \<and> y \<in> ?V \<and> ?U \<inter> ?V = {}"
- using dist_pos_lt[OF xy] th0[of dist,OF dist_triangle dist_commute]
- by (auto simp add: set_eq_iff)
- then show "\<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {}"
- by blast
-qed
-
-lemma separation_t2:
- fixes x y :: "'a::t2_space"
- shows "x \<noteq> y \<longleftrightarrow> (\<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {})"
- using hausdorff[of x y] by blast
-
-lemma separation_t0:
- fixes x y :: "'a::t0_space"
- shows "x \<noteq> y \<longleftrightarrow> (\<exists>U. open U \<and> ~(x\<in>U \<longleftrightarrow> y\<in>U))"
- using t0_space[of x y] by blast
-
subsection{* Limit points *}
definition
@@ -994,10 +920,6 @@
text {* Identify Trivial limits, where we can't approach arbitrarily closely. *}
-definition
- trivial_limit :: "'a net \<Rightarrow> bool" where
- "trivial_limit net \<longleftrightarrow> eventually (\<lambda>x. False) net"
-
lemma trivial_limit_within:
shows "trivial_limit (at a within S) \<longleftrightarrow> \<not> a islimpt S"
proof
@@ -1040,9 +962,6 @@
apply (simp add: norm_sgn)
done
-lemma trivial_limit_sequentially[intro]: "\<not> trivial_limit sequentially"
- by (auto simp add: trivial_limit_def eventually_sequentially)
-
text {* Some property holds "sufficiently close" to the limit point. *}
lemma eventually_at: (* FIXME: this replaces Limits.eventually_at *)
@@ -1074,6 +993,7 @@
lemma eventually_False: "eventually (\<lambda>x. False) net \<longleftrightarrow> trivial_limit net"
unfolding trivial_limit_def ..
+
lemma trivial_limit_eq: "trivial_limit net \<longleftrightarrow> (\<forall>P. eventually P net)"
apply (safe elim!: trivial_limit_eventually)
apply (simp add: eventually_False [symmetric])
@@ -1417,35 +1337,10 @@
text{* Uniqueness of the limit, when nontrivial. *}
-lemma Lim_unique:
- fixes f :: "'a \<Rightarrow> 'b::t2_space"
- assumes "\<not> trivial_limit net" "(f ---> l) net" "(f ---> l') net"
- shows "l = l'"
-proof (rule ccontr)
- assume "l \<noteq> l'"
- obtain U V where "open U" "open V" "l \<in> U" "l' \<in> V" "U \<inter> V = {}"
- using hausdorff [OF `l \<noteq> l'`] by fast
- have "eventually (\<lambda>x. f x \<in> U) net"
- using `(f ---> l) net` `open U` `l \<in> U` by (rule topological_tendstoD)
- moreover
- have "eventually (\<lambda>x. f x \<in> V) net"
- using `(f ---> l') net` `open V` `l' \<in> V` by (rule topological_tendstoD)
- ultimately
- have "eventually (\<lambda>x. False) net"
- proof (rule eventually_elim2)
- fix x
- assume "f x \<in> U" "f x \<in> V"
- hence "f x \<in> U \<inter> V" by simp
- with `U \<inter> V = {}` show "False" by simp
- qed
- with `\<not> trivial_limit net` show "False"
- by (simp add: eventually_False)
-qed
-
lemma tendsto_Lim:
fixes f :: "'a \<Rightarrow> 'b::t2_space"
shows "~(trivial_limit net) \<Longrightarrow> (f ---> l) net ==> Lim net f = l"
- unfolding Lim_def using Lim_unique[of net f] by auto
+ unfolding Lim_def using tendsto_unique[of net f] by auto
text{* Limit under bilinear function *}
@@ -1518,7 +1413,7 @@
apply (rule some_equality)
apply (rule Lim_at_within)
apply (rule Lim_ident_at)
-apply (erule Lim_unique [OF assms])
+apply (erule tendsto_unique [OF assms])
apply (rule Lim_at_within)
apply (rule Lim_ident_at)
done
@@ -2558,7 +2453,7 @@
unfolding islimpt_sequential by auto
then obtain l where l: "l\<in>s" "(f ---> l) sequentially"
using `complete s`[unfolded complete_def] using convergent_imp_cauchy[of f x] by auto
- hence "x \<in> s" using Lim_unique[of sequentially f l x] trivial_limit_sequentially f(2) by auto
+ hence "x \<in> s" using tendsto_unique[of sequentially f l x] trivial_limit_sequentially f(2) by auto
}
thus "closed s" unfolding closed_limpt by auto
qed
@@ -3131,7 +3026,7 @@
using assms(1) unfolding uniformly_convergent_eq_cauchy[THEN sym] by auto
moreover
{ fix x assume "P x"
- hence "l x = l' x" using Lim_unique[OF trivial_limit_sequentially, of "\<lambda>n. s n x" "l x" "l' x"]
+ hence "l x = l' x" using tendsto_unique[OF trivial_limit_sequentially, of "\<lambda>n. s n x" "l x" "l' x"]
using l and assms(2) unfolding Lim_sequentially by blast }
ultimately show ?thesis by auto
qed
@@ -5954,7 +5849,7 @@
hence "((snd \<circ> h \<circ> r) ---> g a) sequentially" unfolding continuous_on_sequentially
apply (rule allE[where x="\<lambda>n. (fst \<circ> h \<circ> r) n"]) apply (erule ballE[where x=a])
using lima unfolding h_def o_def using fs[OF `x\<in>s`] by (auto simp add: y_def)
- hence "g a = a" using Lim_unique[OF trivial_limit_sequentially limb, of "g a"]
+ hence "g a = a" using tendsto_unique[OF trivial_limit_sequentially limb, of "g a"]
unfolding `a=b` and o_assoc by auto
moreover
{ fix x assume "x\<in>s" "g x = x" "x\<noteq>a"
--- a/src/HOL/Probability/Borel_Space.thy Mon Mar 14 15:17:10 2011 +0100
+++ b/src/HOL/Probability/Borel_Space.thy Mon Mar 14 15:29:10 2011 +0100
@@ -3,13 +3,9 @@
header {*Borel spaces*}
theory Borel_Space
- imports Sigma_Algebra Positive_Extended_Real Multivariate_Analysis
+ imports Sigma_Algebra Multivariate_Analysis
begin
-lemma LIMSEQ_max:
- "u ----> (x::real) \<Longrightarrow> (\<lambda>i. max (u i) 0) ----> max x 0"
- by (fastsimp intro!: LIMSEQ_I dest!: LIMSEQ_D)
-
section "Generic Borel spaces"
definition "borel = sigma \<lparr> space = UNIV::'a::topological_space set, sets = open\<rparr>"
@@ -57,7 +53,7 @@
shows "f -` {x} \<inter> space M \<in> sets M"
proof (cases "x \<in> f ` space M")
case True then obtain y where "x = f y" by auto
- from closed_sing[of "f y"]
+ from closed_singleton[of "f y"]
have "{f y} \<in> sets borel" by (rule borel_closed)
with assms show ?thesis
unfolding in_borel_measurable_borel `x = f y` by auto
@@ -81,7 +77,7 @@
shows "A \<in> sets borel \<Longrightarrow> insert x A \<in> sets borel"
proof (rule borel.insert_in_sets)
show "{x} \<in> sets borel"
- using closed_sing[of x] by (rule borel_closed)
+ using closed_singleton[of x] by (rule borel_closed)
qed simp
lemma (in sigma_algebra) borel_measurable_const[simp, intro]:
@@ -112,26 +108,8 @@
ultimately show "?I \<in> borel_measurable M" by auto
qed
-lemma borel_measurable_translate:
- assumes "A \<in> sets borel" and trans: "\<And>B. open B \<Longrightarrow> f -` B \<in> sets borel"
- shows "f -` A \<in> sets borel"
-proof -
- have "A \<in> sigma_sets UNIV open" using assms
- by (simp add: borel_def sigma_def)
- thus ?thesis
- proof (induct rule: sigma_sets.induct)
- case (Basic a) thus ?case using trans[of a] by (simp add: mem_def)
- next
- case (Compl a)
- moreover have "UNIV \<in> sets borel"
- using borel.top by simp
- ultimately show ?case
- by (auto simp: vimage_Diff borel.Diff)
- qed (auto simp add: vimage_UN)
-qed
-
lemma (in sigma_algebra) borel_measurable_restricted:
- fixes f :: "'a \<Rightarrow> 'x\<Colon>{topological_space, semiring_1}" assumes "A \<in> sets M"
+ fixes f :: "'a \<Rightarrow> extreal" assumes "A \<in> sets M"
shows "f \<in> borel_measurable (restricted_space A) \<longleftrightarrow>
(\<lambda>x. f x * indicator A x) \<in> borel_measurable M"
(is "f \<in> borel_measurable ?R \<longleftrightarrow> ?f \<in> borel_measurable M")
@@ -142,7 +120,7 @@
show ?thesis unfolding *
unfolding in_borel_measurable_borel
proof (simp, safe)
- fix S :: "'x set" assume "S \<in> sets borel"
+ fix S :: "extreal set" assume "S \<in> sets borel"
"\<forall>S\<in>sets borel. ?f -` S \<inter> A \<in> op \<inter> A ` sets M"
then have "?f -` S \<inter> A \<in> op \<inter> A ` sets M" by auto
then have f: "?f -` S \<inter> A \<in> sets M"
@@ -161,7 +139,7 @@
then show ?thesis using f by auto
qed
next
- fix S :: "'x set" assume "S \<in> sets borel"
+ fix S :: "extreal set" assume "S \<in> sets borel"
"\<forall>S\<in>sets borel. ?f -` S \<inter> space M \<in> sets M"
then have f: "?f -` S \<inter> space M \<in> sets M" by auto
then show "?f -` S \<inter> A \<in> op \<inter> A ` sets M"
@@ -1024,103 +1002,6 @@
using borel_measurable_euclidean_component
unfolding nth_conv_component by auto
-section "Borel space over the real line with infinity"
-
-lemma borel_Real_measurable:
- "A \<in> sets borel \<Longrightarrow> Real -` A \<in> sets borel"
-proof (rule borel_measurable_translate)
- 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_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..})"
- apply (auto simp add: Real_eq_Real image_iff)
- apply (rule_tac x="max 0 x" in bexI)
- by (auto simp: max_def)
- finally show "Real -` B \<in> sets borel"
- using `open T` by auto
-qed simp
-
-lemma borel_real_measurable:
- "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 }
- note Ex_less_real = this
- 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 <..}) :: 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 :: pextreal set) \<in> sets borel" unfolding * by auto
-qed simp
-
-lemma (in sigma_algebra) borel_measurable_Real[intro, simp]:
- assumes "f \<in> borel_measurable M"
- shows "(\<lambda>x. Real (f x)) \<in> borel_measurable M"
- unfolding in_borel_measurable_borel
-proof safe
- 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
- unfolding vimage_compose in_borel_measurable_borel
- by auto
- thus "(\<lambda>x. Real (f x)) -` S \<inter> space M \<in> sets M" by (simp add: comp_def)
-qed
-
-lemma (in sigma_algebra) borel_measurable_real[intro, simp]:
- 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
-proof safe
- fix S :: "real 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
- unfolding vimage_compose in_borel_measurable_borel
- by auto
- thus "(\<lambda>x. real (f x)) -` S \<inter> space M \<in> sets M" by (simp add: comp_def)
-qed
-
-lemma (in sigma_algebra) borel_measurable_Real_eq:
- assumes "\<And>x. x \<in> space M \<Longrightarrow> 0 \<le> f x"
- shows "(\<lambda>x. Real (f x)) \<in> borel_measurable M \<longleftrightarrow> f \<in> borel_measurable M"
-proof
- have [simp]: "(\<lambda>x. Real (f x)) -` {\<omega>} \<inter> space M = {}"
- by auto
- assume "(\<lambda>x. Real (f x)) \<in> borel_measurable M"
- hence "(\<lambda>x. real (Real (f x))) \<in> borel_measurable M"
- by (rule borel_measurable_real)
- moreover have "\<And>x. x \<in> space M \<Longrightarrow> real (Real (f x)) = f x"
- using assms by auto
- ultimately show "f \<in> borel_measurable M"
- by (simp cong: measurable_cong)
-qed auto
-
-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
- assume "f \<in> borel_measurable M"
- then show "(\<lambda>x. real (f x)) \<in> borel_measurable M" "f -` {\<omega>} \<inter> space M \<in> sets M"
- by (auto intro: borel_measurable_vimage borel_measurable_real)
-next
- assume *: "(\<lambda>x. real (f x)) \<in> borel_measurable M" "f -` {\<omega>} \<inter> space M \<in> sets M"
- have "f -` {\<omega>} \<inter> space M = {x\<in>space M. f x = \<omega>}" by auto
- with * have **: "{x\<in>space M. f x = \<omega>} \<in> sets M" by simp
- have f: "f = (\<lambda>x. if f x = \<omega> then \<omega> else Real (real (f x)))"
- by (simp add: fun_eq_iff Real_real)
- show "f \<in> borel_measurable M"
- apply (subst f)
- apply (rule measurable_If)
- using * ** by auto
-qed
-
lemma borel_measurable_continuous_on1:
fixes f :: "'a::topological_space \<Rightarrow> 'b::t1_space"
assumes "continuous_on UNIV f"
@@ -1187,206 +1068,213 @@
using measurable_comp[OF f borel_measurable_borel_log[OF `1 < b`]]
by (simp add: comp_def)
+subsection "Borel space on the extended reals"
+
+lemma borel_measurable_extreal_borel:
+ "extreal \<in> borel_measurable borel"
+ unfolding borel_def[where 'a=extreal]
+proof (rule borel.measurable_sigma)
+ fix X :: "extreal set" assume "X \<in> sets \<lparr>space = UNIV, sets = open \<rparr>"
+ then have "open X" by (auto simp: mem_def)
+ then have "open (extreal -` X \<inter> space borel)"
+ by (simp add: open_extreal_vimage)
+ then show "extreal -` X \<inter> space borel \<in> sets borel" by auto
+qed auto
+
+lemma (in sigma_algebra) borel_measurable_extreal[simp, intro]:
+ assumes f: "f \<in> borel_measurable M" shows "(\<lambda>x. extreal (f x)) \<in> borel_measurable M"
+ using measurable_comp[OF f borel_measurable_extreal_borel] unfolding comp_def .
+
+lemma borel_measurable_real_of_extreal_borel:
+ "(real :: extreal \<Rightarrow> real) \<in> borel_measurable borel"
+ unfolding borel_def[where 'a=real]
+proof (rule borel.measurable_sigma)
+ fix B :: "real set" assume "B \<in> sets \<lparr>space = UNIV, sets = open \<rparr>"
+ then have "open B" by (auto simp: mem_def)
+ have *: "extreal -` real -` (B - {0}) = B - {0}" by auto
+ have open_real: "open (real -` (B - {0}) :: extreal set)"
+ unfolding open_extreal_def * using `open B` by auto
+ show "(real -` B \<inter> space borel :: extreal set) \<in> sets borel"
+ proof cases
+ assume "0 \<in> B"
+ then have *: "real -` B = real -` (B - {0}) \<union> {-\<infinity>, \<infinity>, 0}"
+ by (auto simp add: real_of_extreal_eq_0)
+ then show "(real -` B :: extreal set) \<inter> space borel \<in> sets borel"
+ using open_real by auto
+ next
+ assume "0 \<notin> B"
+ then have *: "(real -` B :: extreal set) = real -` (B - {0})"
+ by (auto simp add: real_of_extreal_eq_0)
+ then show "(real -` B :: extreal set) \<inter> space borel \<in> sets borel"
+ using open_real by auto
+ qed
+qed auto
+
+lemma (in sigma_algebra) borel_measurable_real_of_extreal[simp, intro]:
+ assumes f: "f \<in> borel_measurable M" shows "(\<lambda>x. real (f x :: extreal)) \<in> borel_measurable M"
+ using measurable_comp[OF f borel_measurable_real_of_extreal_borel] unfolding comp_def .
+
+lemma (in sigma_algebra) borel_measurable_extreal_iff:
+ shows "(\<lambda>x. extreal (f x)) \<in> borel_measurable M \<longleftrightarrow> f \<in> borel_measurable M"
+proof
+ assume "(\<lambda>x. extreal (f x)) \<in> borel_measurable M"
+ from borel_measurable_real_of_extreal[OF this]
+ show "f \<in> borel_measurable M" by auto
+qed auto
+
+lemma (in sigma_algebra) borel_measurable_extreal_iff_real:
+ "f \<in> borel_measurable M \<longleftrightarrow>
+ ((\<lambda>x. real (f x)) \<in> borel_measurable M \<and> f -` {\<infinity>} \<inter> space M \<in> sets M \<and> f -` {-\<infinity>} \<inter> space M \<in> sets M)"
+proof safe
+ assume *: "(\<lambda>x. real (f x)) \<in> borel_measurable M" "f -` {\<infinity>} \<inter> space M \<in> sets M" "f -` {-\<infinity>} \<inter> space M \<in> sets M"
+ have "f -` {\<infinity>} \<inter> space M = {x\<in>space M. f x = \<infinity>}" "f -` {-\<infinity>} \<inter> space M = {x\<in>space M. f x = -\<infinity>}" by auto
+ with * have **: "{x\<in>space M. f x = \<infinity>} \<in> sets M" "{x\<in>space M. f x = -\<infinity>} \<in> sets M" by simp_all
+ let "?f x" = "if f x = \<infinity> then \<infinity> else if f x = -\<infinity> then -\<infinity> else extreal (real (f x))"
+ have "?f \<in> borel_measurable M" using * ** by (intro measurable_If) auto
+ also have "?f = f" by (auto simp: fun_eq_iff extreal_real)
+ finally show "f \<in> borel_measurable M" .
+qed (auto intro: measurable_sets borel_measurable_real_of_extreal)
lemma (in sigma_algebra) less_eq_ge_measurable:
fixes f :: "'a \<Rightarrow> 'c::linorder"
- shows "{x\<in>space M. a < f x} \<in> sets M \<longleftrightarrow> {x\<in>space M. f x \<le> a} \<in> sets M"
+ shows "f -` {a <..} \<inter> space M \<in> sets M \<longleftrightarrow> f -` {..a} \<inter> space M \<in> sets M"
proof
- assume "{x\<in>space M. f x \<le> a} \<in> sets M"
- moreover have "{x\<in>space M. a < f x} = space M - {x\<in>space M. f x \<le> a}" by auto
- ultimately show "{x\<in>space M. a < f x} \<in> sets M" by auto
+ assume "f -` {a <..} \<inter> space M \<in> sets M"
+ moreover have "f -` {..a} \<inter> space M = space M - f -` {a <..} \<inter> space M" by auto
+ ultimately show "f -` {..a} \<inter> space M \<in> sets M" by auto
next
- assume "{x\<in>space M. a < f x} \<in> sets M"
- moreover have "{x\<in>space M. f x \<le> a} = space M - {x\<in>space M. a < f x}" by auto
- ultimately show "{x\<in>space M. f x \<le> a} \<in> sets M" by auto
+ assume "f -` {..a} \<inter> space M \<in> sets M"
+ moreover have "f -` {a <..} \<inter> space M = space M - f -` {..a} \<inter> space M" by auto
+ ultimately show "f -` {a <..} \<inter> space M \<in> sets M" by auto
qed
lemma (in sigma_algebra) greater_eq_le_measurable:
fixes f :: "'a \<Rightarrow> 'c::linorder"
- shows "{x\<in>space M. f x < a} \<in> sets M \<longleftrightarrow> {x\<in>space M. a \<le> f x} \<in> sets M"
+ shows "f -` {..< a} \<inter> space M \<in> sets M \<longleftrightarrow> f -` {a ..} \<inter> space M \<in> sets M"
proof
- assume "{x\<in>space M. a \<le> f x} \<in> sets M"
- moreover have "{x\<in>space M. f x < a} = space M - {x\<in>space M. a \<le> f x}" by auto
- ultimately show "{x\<in>space M. f x < a} \<in> sets M" by auto
+ assume "f -` {a ..} \<inter> space M \<in> sets M"
+ moreover have "f -` {..< a} \<inter> space M = space M - f -` {a ..} \<inter> space M" by auto
+ ultimately show "f -` {..< a} \<inter> space M \<in> sets M" by auto
next
- assume "{x\<in>space M. f x < a} \<in> sets M"
- moreover have "{x\<in>space M. a \<le> f x} = space M - {x\<in>space M. f x < a}" by auto
- ultimately show "{x\<in>space M. a \<le> f x} \<in> sets M" by auto
+ assume "f -` {..< a} \<inter> space M \<in> sets M"
+ moreover have "f -` {a ..} \<inter> space M = space M - f -` {..< a} \<inter> space M" by auto
+ ultimately show "f -` {a ..} \<inter> space M \<in> sets M" by auto
qed
-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)"
+lemma (in sigma_algebra) borel_measurable_uminus_borel_extreal:
+ "(uminus :: extreal \<Rightarrow> extreal) \<in> borel_measurable borel"
+proof (subst borel_def, rule borel.measurable_sigma)
+ fix X :: "extreal set" assume "X \<in> sets \<lparr>space = UNIV, sets = open\<rparr>"
+ then have "open X" by (simp add: mem_def)
+ have "uminus -` X = uminus ` X" by (force simp: image_iff)
+ then have "open (uminus -` X)" using `open X` extreal_open_uminus by auto
+ then show "uminus -` X \<inter> space borel \<in> sets borel" by auto
+qed auto
+
+lemma (in sigma_algebra) borel_measurable_uminus_extreal[intro]:
+ assumes "f \<in> borel_measurable M"
+ shows "(\<lambda>x. - f x :: extreal) \<in> borel_measurable M"
+ using measurable_comp[OF assms borel_measurable_uminus_borel_extreal] by (simp add: comp_def)
+
+lemma (in sigma_algebra) borel_measurable_uminus_eq_extreal[simp]:
+ "(\<lambda>x. - f x :: extreal) \<in> borel_measurable M \<longleftrightarrow> f \<in> borel_measurable M" (is "?l = ?r")
proof
- assume a: "\<forall>a. {x\<in>space M. a \<le> f x} \<in> sets M"
- show "\<forall>a. {x \<in> space M. a < f x} \<in> sets M"
- proof
- fix a show "{x \<in> space M. a < f x} \<in> sets M"
- proof (cases a)
- case (preal r)
- 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_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: 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
- next
- fix i x assume [simp]: "x \<in> space M"
- have "a < a + inverse (of_nat (Suc i))" using preal by auto
- also assume "a + inverse (of_nat (Suc i)) \<le> f x"
- finally show "a < f x" .
- qed
- with a show ?thesis by auto
- qed simp
+ assume ?l from borel_measurable_uminus_extreal[OF this] show ?r by simp
+qed auto
+
+lemma (in sigma_algebra) borel_measurable_eq_atMost_extreal:
+ "(f::'a \<Rightarrow> extreal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. f -` {..a} \<inter> space M \<in> sets M)"
+proof (intro iffI allI)
+ assume pos[rule_format]: "\<forall>a. f -` {..a} \<inter> space M \<in> sets M"
+ show "f \<in> borel_measurable M"
+ unfolding borel_measurable_extreal_iff_real borel_measurable_iff_le
+ proof (intro conjI allI)
+ fix a :: real
+ { fix x :: extreal assume *: "\<forall>i::nat. real i < x"
+ have "x = \<infinity>"
+ proof (rule extreal_top)
+ fix B from real_arch_lt[of B] guess n ..
+ then have "extreal B < real n" by auto
+ with * show "B \<le> x" by (metis less_trans less_imp_le)
+ qed }
+ then have "f -` {\<infinity>} \<inter> space M = space M - (\<Union>i::nat. f -` {.. real i} \<inter> space M)"
+ by (auto simp: not_le)
+ then show "f -` {\<infinity>} \<inter> space M \<in> sets M" using pos by (auto simp del: UN_simps intro!: Diff)
+ moreover
+ have "{-\<infinity>} = {..-\<infinity>}" by auto
+ then show "f -` {-\<infinity>} \<inter> space M \<in> sets M" using pos by auto
+ moreover have "{x\<in>space M. f x \<le> extreal a} \<in> sets M"
+ using pos[of "extreal a"] by (simp add: vimage_def Int_def conj_commute)
+ moreover have "{w \<in> space M. real (f w) \<le> a} =
+ (if a < 0 then {w \<in> space M. f w \<le> extreal a} - f -` {-\<infinity>} \<inter> space M
+ else {w \<in> space M. f w \<le> extreal a} \<union> (f -` {\<infinity>} \<inter> space M) \<union> (f -` {-\<infinity>} \<inter> space M))" (is "?l = ?r")
+ proof (intro set_eqI) fix x show "x \<in> ?l \<longleftrightarrow> x \<in> ?r" by (cases "f x") auto qed
+ ultimately show "{w \<in> space M. real (f w) \<le> a} \<in> sets M" by auto
qed
-next
- assume a': "\<forall>a. {x \<in> space M. a < f x} \<in> sets M"
- then have a: "\<forall>a. {x \<in> space M. f x \<le> a} \<in> sets M" unfolding less_eq_ge_measurable .
- show "\<forall>a. {x \<in> space M. a \<le> f x} \<in> sets M" unfolding greater_eq_le_measurable[symmetric]
- proof
- fix a show "{x \<in> space M. f x < a} \<in> sets M"
- proof (cases a)
- case (preal r)
- show ?thesis
- proof cases
- assume "a = 0" then show ?thesis by simp
- next
- assume "a \<noteq> 0"
- 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_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)) "
- using preal by (cases "f x") (auto split: split_if_asm)
- then show "x \<in> (\<Union>i. {x \<in> space M. f x \<le> a - inverse (of_nat (Suc i))})"
- by auto
- next
- fix i x assume [simp]: "x \<in> space M"
- assume "f x \<le> a - inverse (of_nat (Suc i))"
- also have "\<dots> < a" using `a \<noteq> 0` preal by auto
- finally show "f x < a" .
- qed
- with a show ?thesis by auto
- qed
- next
- case infinite
- have "f -` {\<omega>} \<inter> space M = (\<Inter>n. {x\<in>space M. of_nat n < f x})"
- proof (safe, simp_all, safe)
- fix x assume *: "\<forall>n::nat. Real (real n) < f x"
- 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: pextreal_noteq_omega_Ex)
- with *[THEN spec, of n] show False by auto
- qed
- qed
- with a' have \<omega>: "f -` {\<omega>} \<inter> space M \<in> sets M" by auto
- moreover have "{x \<in> space M. f x < a} = space M - f -` {\<omega>} \<inter> space M"
- using infinite by auto
- ultimately show ?thesis by auto
- qed
- qed
-qed
+qed (simp add: measurable_sets)
-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
- with f show "{x\<in>space M. a < f x} \<in> sets M"
- by (auto intro!: measurable_sets)
-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_pextreal_measurable
- unfolding greater_eq_le_measurable .
- 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
- fix a
- have "{w \<in> space M. a < real (f w)} =
- (if 0 \<le> a then {w\<in>space M. Real a < f w} - (f -` {\<omega>} \<inter> space M) else space M)"
- proof (split split_if, safe del: notI)
- fix x assume "0 \<le> a"
- { assume "a < real (f x)" then show "Real a < f x" "x \<notin> f -` {\<omega>} \<inter> space M"
- using `0 \<le> a` by (cases "f x", auto) }
- { assume "Real a < f x" "x \<notin> f -` {\<omega>}" then show "a < real (f x)"
- using `0 \<le> a` by (cases "f x", auto) }
- next
- fix x assume "\<not> 0 \<le> a" then show "a < real (f x)" by (cases "f x") auto
- qed
- then show "{w \<in> space M. a < real (f w)} \<in> sets M"
- using \<omega> * by (auto intro!: Diff)
- qed
-qed
+lemma (in sigma_algebra) borel_measurable_eq_atLeast_extreal:
+ "(f::'a \<Rightarrow> extreal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. f -` {a..} \<inter> space M \<in> sets M)"
+proof
+ assume pos: "\<forall>a. f -` {a..} \<inter> space M \<in> sets M"
+ moreover have "\<And>a. (\<lambda>x. - f x) -` {..a} = f -` {-a ..}"
+ by (auto simp: extreal_uminus_le_reorder)
+ ultimately have "(\<lambda>x. - f x) \<in> borel_measurable M"
+ unfolding borel_measurable_eq_atMost_extreal by auto
+ then show "f \<in> borel_measurable M" by simp
+qed (simp add: measurable_sets)
-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_extreal_iff_less:
+ "(f::'a \<Rightarrow> extreal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. f -` {..< a} \<inter> space M \<in> sets M)"
+ unfolding borel_measurable_eq_atLeast_extreal greater_eq_le_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_extreal_iff_ge:
+ "(f::'a \<Rightarrow> extreal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. f -` {a <..} \<inter> space M \<in> sets M)"
+ unfolding borel_measurable_eq_atMost_extreal less_eq_ge_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_pextreal_eq_const:
- fixes f :: "'a \<Rightarrow> pextreal" assumes "f \<in> borel_measurable M"
+lemma (in sigma_algebra) borel_measurable_extreal_eq_const:
+ fixes f :: "'a \<Rightarrow> extreal" 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_pextreal_neq_const:
- fixes f :: "'a \<Rightarrow> pextreal"
- assumes "f \<in> borel_measurable M"
+lemma (in sigma_algebra) borel_measurable_extreal_neq_const:
+ fixes f :: "'a \<Rightarrow> extreal" assumes "f \<in> borel_measurable M"
shows "{x\<in>space M. f x \<noteq> c} \<in> sets M"
proof -
have "{x\<in>space M. f x \<noteq> c} = space M - (f -` {c} \<inter> space M)" by auto
then show ?thesis using assms by (auto intro!: measurable_sets)
qed
-lemma (in sigma_algebra) borel_measurable_pextreal_less[intro,simp]:
- fixes f g :: "'a \<Rightarrow> pextreal"
+lemma (in sigma_algebra) borel_measurable_extreal_le[intro,simp]:
+ fixes f g :: "'a \<Rightarrow> extreal"
+ 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"
+proof -
+ have "{x \<in> space M. f x \<le> g x} =
+ {x \<in> space M. real (f x) \<le> real (g x)} - (f -` {\<infinity>, -\<infinity>} \<inter> space M \<union> g -` {\<infinity>, -\<infinity>} \<inter> space M) \<union>
+ f -` {-\<infinity>} \<inter> space M \<union> g -` {\<infinity>} \<inter> space M" (is "?l = ?r")
+ proof (intro set_eqI)
+ fix x show "x \<in> ?l \<longleftrightarrow> x \<in> ?r" by (cases rule: extreal2_cases[of "f x" "g x"]) auto
+ qed
+ with f g show ?thesis by (auto intro!: Un simp: measurable_sets)
+qed
+
+lemma (in sigma_algebra) borel_measurable_extreal_less[intro,simp]:
+ fixes f :: "'a \<Rightarrow> extreal"
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"
proof -
- have "(\<lambda>x. real (f x)) \<in> borel_measurable M"
- "(\<lambda>x. real (g x)) \<in> borel_measurable M"
- 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_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_pextreal_strict_mono_iff)
- ultimately show ?thesis by auto
-qed
-
-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"
-proof -
- have "{x \<in> space M. f x \<le> g x} = space M - {x \<in> space M. g x < f x}" by auto
+ have "{x \<in> space M. f x < g x} = space M - {x \<in> space M. g x \<le> f x}" by auto
then show ?thesis using g f by auto
qed
-lemma (in sigma_algebra) borel_measurable_pextreal_eq[intro,simp]:
- fixes f :: "'a \<Rightarrow> pextreal"
+lemma (in sigma_algebra) borel_measurable_extreal_eq[intro,simp]:
+ fixes f :: "'a \<Rightarrow> extreal"
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"
@@ -1395,8 +1283,8 @@
then show ?thesis using g f by auto
qed
-lemma (in sigma_algebra) borel_measurable_pextreal_neq[intro,simp]:
- fixes f :: "'a \<Rightarrow> pextreal"
+lemma (in sigma_algebra) borel_measurable_extreal_neq[intro,simp]:
+ fixes f :: "'a \<Rightarrow> extreal"
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"
@@ -1405,20 +1293,28 @@
thus ?thesis using f g by auto
qed
-lemma (in sigma_algebra) borel_measurable_pextreal_add[intro, simp]:
- fixes f :: "'a \<Rightarrow> pextreal"
+lemma (in sigma_algebra) split_sets:
+ "{x\<in>space M. P x \<or> Q x} = {x\<in>space M. P x} \<union> {x\<in>space M. Q x}"
+ "{x\<in>space M. P x \<and> Q x} = {x\<in>space M. P x} \<inter> {x\<in>space M. Q x}"
+ by auto
+
+lemma (in sigma_algebra) borel_measurable_extreal_add[intro, simp]:
+ fixes f :: "'a \<Rightarrow> extreal"
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 = \<omega> then \<omega> else if g x = \<omega> then \<omega> else Real (real (f x) + real (g x)))"
- by (auto simp: fun_eq_iff pextreal_noteq_omega_Ex)
- show ?thesis using assms unfolding *
- by (auto intro!: measurable_If)
+ { fix x assume "x \<in> space M" then have "f x + g x =
+ (if f x = \<infinity> \<or> g x = \<infinity> then \<infinity>
+ else if f x = -\<infinity> \<or> g x = -\<infinity> then -\<infinity>
+ else extreal (real (f x) + real (g x)))"
+ by (cases rule: extreal2_cases[of "f x" "g x"]) auto }
+ with assms show ?thesis
+ by (auto cong: measurable_cong simp: split_sets
+ intro!: Un measurable_If measurable_sets)
qed
-lemma (in sigma_algebra) borel_measurable_pextreal_setsum[simp, intro]:
- fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> pextreal"
+lemma (in sigma_algebra) borel_measurable_extreal_setsum[simp, intro]:
+ fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> extreal"
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
@@ -1427,20 +1323,49 @@
by induct auto
qed (simp add: borel_measurable_const)
-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"
+lemma abs_extreal_ge0[simp]: "0 \<le> x \<Longrightarrow> \<bar>x :: extreal\<bar> = x"
+ by (cases x) auto
+
+lemma abs_extreal_less0[simp]: "x < 0 \<Longrightarrow> \<bar>x :: extreal\<bar> = -x"
+ by (cases x) auto
+
+lemma abs_extreal_pos[simp]: "0 \<le> \<bar>x :: extreal\<bar>"
+ by (cases x) auto
+
+lemma (in sigma_algebra) borel_measurable_extreal_abs[intro, simp]:
+ fixes f :: "'a \<Rightarrow> extreal" assumes "f \<in> borel_measurable M"
+ shows "(\<lambda>x. \<bar>f x\<bar>) \<in> borel_measurable M"
+proof -
+ { fix x have "\<bar>f x\<bar> = (if 0 \<le> f x then f x else - f x)" by auto }
+ then show ?thesis using assms by (auto intro!: measurable_If)
+qed
+
+lemma (in sigma_algebra) borel_measurable_extreal_times[intro, simp]:
+ fixes f :: "'a \<Rightarrow> extreal" assumes "f \<in> borel_measurable M" "g \<in> borel_measurable M"
shows "(\<lambda>x. f x * g x) \<in> borel_measurable M"
proof -
+ { fix f g :: "'a \<Rightarrow> extreal"
+ assume b: "f \<in> borel_measurable M" "g \<in> borel_measurable M"
+ and pos: "\<And>x. 0 \<le> f x" "\<And>x. 0 \<le> g x"
+ { fix x have *: "f x * g x = (if f x = 0 \<or> g x = 0 then 0
+ else if f x = \<infinity> \<or> g x = \<infinity> then \<infinity>
+ else extreal (real (f x) * real (g x)))"
+ apply (cases rule: extreal2_cases[of "f x" "g x"])
+ using pos[of x] by auto }
+ with b have "(\<lambda>x. f x * g x) \<in> borel_measurable M"
+ by (auto cong: measurable_cong simp: split_sets
+ intro!: Un measurable_If measurable_sets) }
+ note pos_times = this
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 pextreal_noteq_omega_Ex)
+ (\<lambda>x. if 0 \<le> f x \<and> 0 \<le> g x \<or> f x < 0 \<and> g x < 0 then \<bar>f x\<bar> * \<bar>g x\<bar> else - (\<bar>f x\<bar> * \<bar>g x\<bar>))"
+ by (auto simp: fun_eq_iff)
show ?thesis using assms unfolding *
- by (auto intro!: measurable_If)
+ by (intro measurable_If pos_times borel_measurable_uminus_extreal)
+ (auto simp: split_sets intro!: Int)
qed
-lemma (in sigma_algebra) borel_measurable_pextreal_setprod[simp, intro]:
- fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> pextreal"
+lemma (in sigma_algebra) borel_measurable_extreal_setprod[simp, intro]:
+ fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> extreal"
assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M"
shows "(\<lambda>x. \<Prod>i\<in>S. f i x) \<in> borel_measurable M"
proof cases
@@ -1448,64 +1373,73 @@
thus ?thesis using assms by induct auto
qed simp
-lemma (in sigma_algebra) borel_measurable_pextreal_min[simp, intro]:
- fixes f g :: "'a \<Rightarrow> pextreal"
+lemma (in sigma_algebra) borel_measurable_extreal_min[simp, intro]:
+ fixes f g :: "'a \<Rightarrow> extreal"
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_pextreal_max[simp, intro]:
- fixes f g :: "'a \<Rightarrow> pextreal"
+lemma (in sigma_algebra) borel_measurable_extreal_max[simp, intro]:
+ fixes f g :: "'a \<Rightarrow> extreal"
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> pextreal"
+ fixes f :: "'d\<Colon>countable \<Rightarrow> 'a \<Rightarrow> extreal"
assumes "\<And>i. i \<in> A \<Longrightarrow> f i \<in> borel_measurable M"
shows "(\<lambda>x. SUP i : A. f i x) \<in> borel_measurable M" (is "?sup \<in> borel_measurable M")
- unfolding borel_measurable_pextreal_iff_greater
-proof safe
+ unfolding borel_measurable_extreal_iff_ge
+proof
fix a
- have "{x\<in>space M. a < ?sup x} = (\<Union>i\<in>A. {x\<in>space M. a < f i x})"
+ have "?sup -` {a<..} \<inter> space M = (\<Union>i\<in>A. {x\<in>space M. a < f i x})"
by (auto simp: less_SUP_iff SUPR_apply)
- then show "{x\<in>space M. a < ?sup x} \<in> sets M"
+ then show "?sup -` {a<..} \<inter> space M \<in> sets M"
using assms by auto
qed
lemma (in sigma_algebra) borel_measurable_INF[simp, intro]:
- fixes f :: "'d :: countable \<Rightarrow> 'a \<Rightarrow> pextreal"
+ fixes f :: "'d :: countable \<Rightarrow> 'a \<Rightarrow> extreal"
assumes "\<And>i. i \<in> A \<Longrightarrow> f i \<in> borel_measurable M"
shows "(\<lambda>x. INF i : A. f i x) \<in> borel_measurable M" (is "?inf \<in> borel_measurable M")
- unfolding borel_measurable_pextreal_iff_less
-proof safe
+ unfolding borel_measurable_extreal_iff_less
+proof
fix a
- have "{x\<in>space M. ?inf x < a} = (\<Union>i\<in>A. {x\<in>space M. f i x < a})"
+ have "?inf -` {..<a} \<inter> space M = (\<Union>i\<in>A. {x\<in>space M. f i x < a})"
by (auto simp: INF_less_iff INFI_apply)
- then show "{x\<in>space M. ?inf x < a} \<in> sets M"
+ then show "?inf -` {..<a} \<inter> space M \<in> sets M"
using assms by auto
qed
-lemma (in sigma_algebra) borel_measurable_pextreal_diff[simp, intro]:
- fixes f g :: "'a \<Rightarrow> pextreal"
+lemma (in sigma_algebra) borel_measurable_liminf[simp, intro]:
+ fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> extreal"
+ assumes "\<And>i. f i \<in> borel_measurable M"
+ shows "(\<lambda>x. liminf (\<lambda>i. f i x)) \<in> borel_measurable M"
+ unfolding liminf_SUPR_INFI using assms by auto
+
+lemma (in sigma_algebra) borel_measurable_limsup[simp, intro]:
+ fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> extreal"
+ assumes "\<And>i. f i \<in> borel_measurable M"
+ shows "(\<lambda>x. limsup (\<lambda>i. f i x)) \<in> borel_measurable M"
+ unfolding limsup_INFI_SUPR using assms by auto
+
+lemma (in sigma_algebra) borel_measurable_extreal_diff[simp, intro]:
+ fixes f g :: "'a \<Rightarrow> extreal"
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_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: pextreal_less_minus_iff)
- then show "{x \<in> space M. a < f x - g x} \<in> sets M"
- using assms by auto
-qed
+ unfolding minus_extreal_def using assms by auto
lemma (in sigma_algebra) borel_measurable_psuminf[simp, intro]:
- assumes "\<And>i. f i \<in> borel_measurable M"
- shows "(\<lambda>x. (\<Sum>\<^isub>\<infinity> i. f i x)) \<in> borel_measurable M"
- using assms unfolding psuminf_def by auto
+ fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> extreal"
+ assumes "\<And>i. f i \<in> borel_measurable M" and pos: "\<And>i x. x \<in> space M \<Longrightarrow> 0 \<le> f i x"
+ shows "(\<lambda>x. (\<Sum>i. f i x)) \<in> borel_measurable M"
+ apply (subst measurable_cong)
+ apply (subst suminf_extreal_eq_SUPR)
+ apply (rule pos)
+ using assms by auto
section "LIMSEQ is borel measurable"
@@ -1515,28 +1449,11 @@
and u: "\<And>i. u i \<in> borel_measurable M"
shows "u' \<in> borel_measurable M"
proof -
- let "?pu x i" = "max (u i x) 0"
- let "?nu x i" = "max (- u i x) 0"
- { fix x assume x: "x \<in> space M"
- have "(?pu x) ----> max (u' x) 0"
- "(?nu x) ----> max (- u' x) 0"
- using u'[OF x] by (auto intro!: LIMSEQ_max LIMSEQ_minus)
- from LIMSEQ_imp_lim_INF[OF _ this(1)] LIMSEQ_imp_lim_INF[OF _ this(2)]
- have "(SUP n. INF m. Real (u (n + m) x)) = Real (u' x)"
- "(SUP n. INF m. Real (- u (n + m) x)) = Real (- u' x)"
- by (simp_all add: Real_max'[symmetric]) }
- note eq = this
- have *: "\<And>x. real (Real (u' x)) - real (Real (- u' x)) = u' x"
+ have "\<And>x. x \<in> space M \<Longrightarrow> liminf (\<lambda>n. extreal (u n x)) = extreal (u' x)"
+ using u' by (simp add: lim_imp_Liminf trivial_limit_sequentially lim_extreal)
+ moreover from u have "(\<lambda>x. liminf (\<lambda>n. extreal (u n x))) \<in> borel_measurable M"
by auto
- have "(\<lambda>x. SUP n. INF m. Real (u (n + m) x)) \<in> borel_measurable M"
- "(\<lambda>x. SUP n. INF m. Real (- u (n + m) x)) \<in> borel_measurable M"
- using u by auto
- with eq[THEN measurable_cong, of M "\<lambda>x. x" borel]
- have "(\<lambda>x. Real (u' x)) \<in> borel_measurable M"
- "(\<lambda>x. Real (- u' x)) \<in> borel_measurable M" by auto
- note this[THEN borel_measurable_real]
- from borel_measurable_diff[OF this]
- show ?thesis unfolding * .
+ ultimately show ?thesis by (simp cong: measurable_cong add: borel_measurable_extreal_iff)
qed
end
--- a/src/HOL/Probability/Caratheodory.thy Mon Mar 14 15:17:10 2011 +0100
+++ b/src/HOL/Probability/Caratheodory.thy Mon Mar 14 15:29:10 2011 +0100
@@ -1,36 +1,66 @@
header {*Caratheodory Extension Theorem*}
theory Caratheodory
- imports Sigma_Algebra Positive_Extended_Real
+ imports Sigma_Algebra Extended_Real_Limits
begin
+lemma suminf_extreal_2dimen:
+ fixes f:: "nat \<times> nat \<Rightarrow> extreal"
+ assumes pos: "\<And>p. 0 \<le> f p"
+ assumes "\<And>m. g m = (\<Sum>n. f (m,n))"
+ shows "(\<Sum>i. f (prod_decode i)) = suminf g"
+proof -
+ have g_def: "g = (\<lambda>m. (\<Sum>n. f (m,n)))"
+ using assms by (simp add: fun_eq_iff)
+ have reindex: "\<And>B. (\<Sum>x\<in>B. f (prod_decode x)) = setsum f (prod_decode ` B)"
+ by (simp add: setsum_reindex[OF inj_prod_decode] comp_def)
+ { fix n
+ let ?M = "\<lambda>f. Suc (Max (f ` prod_decode ` {..<n}))"
+ { fix a b x assume "x < n" and [symmetric]: "(a, b) = prod_decode x"
+ then have "a < ?M fst" "b < ?M snd"
+ by (auto intro!: Max_ge le_imp_less_Suc image_eqI) }
+ then have "setsum f (prod_decode ` {..<n}) \<le> setsum f ({..<?M fst} \<times> {..<?M snd})"
+ by (auto intro!: setsum_mono3 simp: pos)
+ then have "\<exists>a b. setsum f (prod_decode ` {..<n}) \<le> setsum f ({..<a} \<times> {..<b})" by auto }
+ moreover
+ { fix a b
+ let ?M = "prod_decode ` {..<Suc (Max (prod_encode ` ({..<a} \<times> {..<b})))}"
+ { fix a' b' assume "a' < a" "b' < b" then have "(a', b') \<in> ?M"
+ by (auto intro!: Max_ge le_imp_less_Suc image_eqI[where x="prod_encode (a', b')"]) }
+ then have "setsum f ({..<a} \<times> {..<b}) \<le> setsum f ?M"
+ by (auto intro!: setsum_mono3 simp: pos) }
+ ultimately
+ show ?thesis unfolding g_def using pos
+ by (auto intro!: SUPR_eq simp: setsum_cartesian_product reindex le_SUPI2
+ setsum_nonneg suminf_extreal_eq_SUPR SUPR_pair
+ SUPR_extreal_setsum[symmetric] incseq_setsumI setsum_nonneg)
+qed
+
text{*From the Hurd/Coble measure theory development, translated by Lawrence Paulson.*}
subsection {* Measure Spaces *}
record 'a measure_space = "'a algebra" +
- measure :: "'a set \<Rightarrow> pextreal"
+ measure :: "'a set \<Rightarrow> extreal"
-definition positive where "positive M f \<longleftrightarrow> f {} = (0::pextreal)"
- -- "Positive is enforced by the type"
+definition positive where "positive M f \<longleftrightarrow> f {} = (0::extreal) \<and> (\<forall>A\<in>sets M. 0 \<le> f A)"
definition additive where "additive M f \<longleftrightarrow>
(\<forall>x \<in> sets M. \<forall>y \<in> sets M. x \<inter> y = {} \<longrightarrow> f (x \<union> y) = f x + f y)"
-definition countably_additive where "countably_additive M f \<longleftrightarrow>
- (\<forall>A. range A \<subseteq> sets M \<longrightarrow> disjoint_family A \<longrightarrow> (\<Union>i. A i) \<in> sets M \<longrightarrow>
- (\<Sum>\<^isub>\<infinity> n. f (A n)) = f (\<Union>i. A i))"
+definition countably_additive :: "('a, 'b) algebra_scheme \<Rightarrow> ('a set \<Rightarrow> extreal) \<Rightarrow> bool" where
+ "countably_additive M f \<longleftrightarrow> (\<forall>A. range A \<subseteq> sets M \<longrightarrow> disjoint_family A \<longrightarrow> (\<Union>i. A i) \<in> sets M \<longrightarrow>
+ (\<Sum>i. f (A i)) = f (\<Union>i. A i))"
definition increasing where "increasing M f \<longleftrightarrow>
(\<forall>x \<in> sets M. \<forall>y \<in> sets M. x \<subseteq> y \<longrightarrow> f x \<le> f y)"
definition subadditive where "subadditive M f \<longleftrightarrow>
- (\<forall>x \<in> sets M. \<forall>y \<in> sets M. x \<inter> y = {} \<longrightarrow>
- f (x \<union> y) \<le> f x + f y)"
+ (\<forall>x \<in> sets M. \<forall>y \<in> sets M. x \<inter> y = {} \<longrightarrow> f (x \<union> y) \<le> f x + f y)"
definition countably_subadditive where "countably_subadditive M f \<longleftrightarrow>
(\<forall>A. range A \<subseteq> sets M \<longrightarrow> disjoint_family A \<longrightarrow> (\<Union>i. A i) \<in> sets M \<longrightarrow>
- f (\<Union>i. A i) \<le> (\<Sum>\<^isub>\<infinity> n. f (A n)))"
+ (f (\<Union>i. A i) \<le> (\<Sum>i. f (A i))))"
definition lambda_system where "lambda_system M f = {l \<in> sets M.
\<forall>x \<in> sets M. f (l \<inter> x) + f ((space M - l) \<inter> x) = f x}"
@@ -39,14 +69,19 @@
positive M f \<and> increasing M f \<and> countably_subadditive M f"
definition measure_set where "measure_set M f X = {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}"
+ \<exists>A. range A \<subseteq> sets M \<and> disjoint_family A \<and> X \<subseteq> (\<Union>i. A i) \<and> (\<Sum>i. f (A i)) = r}"
locale measure_space = sigma_algebra M for M :: "('a, 'b) measure_space_scheme" +
- assumes empty_measure [simp]: "measure M {} = 0"
+ assumes measure_positive: "positive M (measure M)"
and ca: "countably_additive M (measure M)"
abbreviation (in measure_space) "\<mu> \<equiv> measure M"
+lemma (in measure_space)
+ shows empty_measure[simp, intro]: "\<mu> {} = 0"
+ and positive_measure[simp, intro!]: "\<And>A. A \<in> sets M \<Longrightarrow> 0 \<le> \<mu> A"
+ using measure_positive unfolding positive_def by auto
+
lemma increasingD:
"increasing M f \<Longrightarrow> x \<subseteq> y \<Longrightarrow> x\<in>sets M \<Longrightarrow> y\<in>sets M \<Longrightarrow> f x \<le> f y"
by (auto simp add: increasing_def)
@@ -61,39 +96,30 @@
\<Longrightarrow> f (x \<union> y) = f x + f y"
by (auto simp add: additive_def)
-lemma countably_additiveD:
- "countably_additive M f \<Longrightarrow> range A \<subseteq> sets M \<Longrightarrow> disjoint_family A
- \<Longrightarrow> (\<Union>i. A i) \<in> sets M \<Longrightarrow> (\<Sum>\<^isub>\<infinity> n. f (A n)) = f (\<Union>i. A i)"
- by (simp add: countably_additive_def)
-
-lemma countably_subadditiveD:
- "countably_subadditive M f \<Longrightarrow> range A \<subseteq> sets M \<Longrightarrow> disjoint_family A \<Longrightarrow>
- (\<Union>i. A i) \<in> sets M \<Longrightarrow> f (\<Union>i. A i) \<le> psuminf (f o A)"
- by (auto simp add: countably_subadditive_def o_def)
-
lemma countably_additiveI:
- "(\<And>A. range A \<subseteq> sets M \<Longrightarrow> disjoint_family A \<Longrightarrow> (\<Union>i. A i) \<in> sets M
- \<Longrightarrow> (\<Sum>\<^isub>\<infinity> n. f (A n)) = f (\<Union>i. A i)) \<Longrightarrow> countably_additive M f"
- by (simp add: countably_additive_def)
+ assumes "\<And>A x. range A \<subseteq> sets M \<Longrightarrow> disjoint_family A \<Longrightarrow> (\<Union>i. A i) \<in> sets M
+ \<Longrightarrow> (\<Sum>i. f (A i)) = f (\<Union>i. A i)"
+ shows "countably_additive M f"
+ using assms by (simp add: countably_additive_def)
section "Extend binary sets"
lemma LIMSEQ_binaryset:
assumes f: "f {} = 0"
- shows "(\<lambda>n. \<Sum>i = 0..<n. f (binaryset A B i)) ----> f A + f B"
+ shows "(\<lambda>n. \<Sum>i<n. f (binaryset A B i)) ----> f A + f B"
proof -
- have "(\<lambda>n. \<Sum>i = 0..< Suc (Suc n). f (binaryset A B i)) = (\<lambda>n. f A + f B)"
+ have "(\<lambda>n. \<Sum>i < Suc (Suc n). f (binaryset A B i)) = (\<lambda>n. f A + f B)"
proof
fix n
- show "(\<Sum>i\<Colon>nat = 0\<Colon>nat..<Suc (Suc n). f (binaryset A B i)) = f A + f B"
+ show "(\<Sum>i < Suc (Suc n). f (binaryset A B i)) = f A + f B"
by (induct n) (auto simp add: binaryset_def f)
qed
moreover
have "... ----> f A + f B" by (rule LIMSEQ_const)
ultimately
- have "(\<lambda>n. \<Sum>i = 0..< Suc (Suc n). f (binaryset A B i)) ----> f A + f B"
+ have "(\<lambda>n. \<Sum>i< Suc (Suc n). f (binaryset A B i)) ----> f A + f B"
by metis
- hence "(\<lambda>n. \<Sum>i = 0..< n+2. f (binaryset A B i)) ----> f A + f B"
+ hence "(\<lambda>n. \<Sum>i< n+2. f (binaryset A B i)) ----> f A + f B"
by simp
thus ?thesis by (rule LIMSEQ_offset [where k=2])
qed
@@ -101,28 +127,13 @@
lemma binaryset_sums:
assumes f: "f {} = 0"
shows "(\<lambda>n. f (binaryset A B n)) sums (f A + f B)"
- by (simp add: sums_def LIMSEQ_binaryset [where f=f, OF f])
+ by (simp add: sums_def LIMSEQ_binaryset [where f=f, OF f] atLeast0LessThan)
lemma suminf_binaryset_eq:
- fixes f :: "'a set \<Rightarrow> real"
+ fixes f :: "'a set \<Rightarrow> 'b::{comm_monoid_add, t2_space}"
shows "f {} = 0 \<Longrightarrow> (\<Sum>n. f (binaryset A B n)) = f A + f B"
by (metis binaryset_sums sums_unique)
-lemma binaryset_psuminf:
- assumes "f {} = 0"
- shows "(\<Sum>\<^isub>\<infinity> n. f (binaryset A B n)) = f A + f B" (is "?suminf = ?sum")
-proof -
- have *: "{..<2} = {0, 1::nat}" by auto
- have "\<forall>n\<ge>2. f (binaryset A B n) = 0"
- unfolding binaryset_def
- using assms by auto
- hence "?suminf = (\<Sum>N<2. f (binaryset A B N))"
- by (rule psuminf_finite)
- also have "... = ?sum" unfolding * binaryset_def
- by simp
- finally show ?thesis .
-qed
-
subsection {* Lambda Systems *}
lemma (in algebra) lambda_system_eq:
@@ -144,7 +155,7 @@
by (simp add: lambda_system_def)
lemma (in algebra) lambda_system_Compl:
- fixes f:: "'a set \<Rightarrow> pextreal"
+ fixes f:: "'a set \<Rightarrow> extreal"
assumes x: "x \<in> lambda_system M f"
shows "space M - x \<in> lambda_system M f"
proof -
@@ -157,7 +168,7 @@
qed
lemma (in algebra) lambda_system_Int:
- fixes f:: "'a set \<Rightarrow> pextreal"
+ fixes f:: "'a set \<Rightarrow> extreal"
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 -
@@ -191,7 +202,7 @@
qed
lemma (in algebra) lambda_system_Un:
- fixes f:: "'a set \<Rightarrow> pextreal"
+ fixes f:: "'a set \<Rightarrow> extreal"
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 -
@@ -250,53 +261,54 @@
by (auto simp add: disjoint_family_on_def binaryset_def)
hence "range (binaryset x y) \<subseteq> sets M \<longrightarrow>
(\<Union>i. binaryset x y i) \<in> sets M \<longrightarrow>
- f (\<Union>i. binaryset x y i) \<le> (\<Sum>\<^isub>\<infinity> n. f (binaryset x y n))"
- using cs by (simp add: countably_subadditive_def)
+ f (\<Union>i. binaryset x y i) \<le> (\<Sum> n. f (binaryset x y n))"
+ using cs by (auto simp add: countably_subadditive_def)
hence "{x,y,{}} \<subseteq> sets M \<longrightarrow> x \<union> y \<in> sets M \<longrightarrow>
- f (x \<union> y) \<le> (\<Sum>\<^isub>\<infinity> n. f (binaryset x y n))"
+ f (x \<union> y) \<le> (\<Sum> n. f (binaryset x y n))"
by (simp add: range_binaryset_eq UN_binaryset_eq)
thus "f (x \<union> y) \<le> f x + f y" using f x y
- by (auto simp add: Un o_def binaryset_psuminf positive_def)
+ by (auto simp add: Un o_def suminf_binaryset_eq positive_def)
qed
lemma (in algebra) additive_sum:
fixes A:: "nat \<Rightarrow> 'a set"
- assumes f: "positive M f" and ad: "additive M f"
+ assumes f: "positive M f" and ad: "additive M f" and "finite S"
and A: "range A \<subseteq> sets M"
- and disj: "disjoint_family A"
- shows "setsum (f \<circ> A) {0..<n} = f (\<Union>i\<in>{0..<n}. A i)"
-proof (induct n)
- case 0 show ?case using f by (simp add: positive_def)
+ and disj: "disjoint_family_on A S"
+ shows "(\<Sum>i\<in>S. f (A i)) = f (\<Union>i\<in>S. A i)"
+using `finite S` disj proof induct
+ case empty show ?case using f by (simp add: positive_def)
next
- case (Suc n)
- have "A n \<inter> (\<Union>i\<in>{0..<n}. A i) = {}" using disj
- by (auto simp add: disjoint_family_on_def neq_iff) blast
+ case (insert s S)
+ then have "A s \<inter> (\<Union>i\<in>S. A i) = {}"
+ by (auto simp add: disjoint_family_on_def neq_iff)
moreover
- have "A n \<in> sets M" using A by blast
- moreover have "(\<Union>i\<in>{0..<n}. A i) \<in> sets M"
- by (metis A UNION_in_sets atLeast0LessThan)
+ have "A s \<in> sets M" using A by blast
+ moreover have "(\<Union>i\<in>S. A i) \<in> sets M"
+ using A `finite S` by auto
moreover
- ultimately have "f (A n \<union> (\<Union>i\<in>{0..<n}. A i)) = f (A n) + f(\<Union>i\<in>{0..<n}. A i)"
+ ultimately have "f (A s \<union> (\<Union>i\<in>S. A i)) = f (A s) + f(\<Union>i\<in>S. A i)"
using ad UNION_in_sets A by (auto simp add: additive_def)
- with Suc.hyps show ?case using ad
- by (auto simp add: atLeastLessThanSuc additive_def)
+ with insert show ?case using ad disjoint_family_on_mono[of S "insert s S" A]
+ by (auto simp add: additive_def subset_insertI)
qed
lemma (in algebra) increasing_additive_bound:
- fixes A:: "nat \<Rightarrow> 'a set" and f :: "'a set \<Rightarrow> pextreal"
+ fixes A:: "nat \<Rightarrow> 'a set" and f :: "'a set \<Rightarrow> extreal"
assumes f: "positive M f" and ad: "additive M f"
and inc: "increasing M f"
and A: "range A \<subseteq> sets M"
and disj: "disjoint_family A"
- shows "psuminf (f \<circ> A) \<le> f (space M)"
-proof (safe intro!: psuminf_bound)
+ shows "(\<Sum>i. f (A i)) \<le> f (space M)"
+proof (safe intro!: suminf_bound)
fix N
- have "setsum (f \<circ> A) {0..<N} = f (\<Union>i\<in>{0..<N}. A i)"
- by (rule additive_sum [OF f ad A disj])
+ note disj_N = disjoint_family_on_mono[OF _ disj, of "{..<N}"]
+ have "(\<Sum>i<N. f (A i)) = f (\<Union>i\<in>{..<N}. A i)"
+ by (rule additive_sum [OF f ad _ A]) (auto simp: disj_N)
also have "... \<le> f (space M)" using space_closed A
- by (blast intro: increasingD [OF inc] UNION_in_sets top)
- finally show "setsum (f \<circ> A) {..<N} \<le> f (space M)" by (simp add: atLeast0LessThan)
-qed
+ by (intro increasingD[OF inc] finite_UN) auto
+ finally show "(\<Sum>i<N. f (A i)) \<le> f (space M)" by simp
+qed (insert f A, auto simp: positive_def)
lemma lambda_system_increasing:
"increasing M f \<Longrightarrow> increasing (M (|sets := lambda_system M f|)) f"
@@ -307,7 +319,7 @@
by (simp add: positive_def lambda_system_def)
lemma (in algebra) lambda_system_strong_sum:
- fixes A:: "nat \<Rightarrow> 'a set" and f :: "'a set \<Rightarrow> pextreal"
+ fixes A:: "nat \<Rightarrow> 'a set" and f :: "'a set \<Rightarrow> extreal"
assumes f: "positive M f" and a: "a \<in> sets M"
and A: "range A \<subseteq> lambda_system M f"
and disj: "disjoint_family A"
@@ -331,7 +343,7 @@
assumes oms: "outer_measure_space M f"
and A: "range A \<subseteq> lambda_system M f"
and disj: "disjoint_family A"
- shows "(\<Union>i. A i) \<in> lambda_system M f \<and> psuminf (f \<circ> A) = f (\<Union>i. A i)"
+ shows "(\<Union>i. A i) \<in> lambda_system M f \<and> (\<Sum>i. f (A i)) = f (\<Union>i. A i)"
proof -
have pos: "positive M f" and inc: "increasing M f"
and csa: "countably_subadditive M f"
@@ -347,15 +359,16 @@
have U_in: "(\<Union>i. A i) \<in> sets M"
by (metis A'' countable_UN)
- have U_eq: "f (\<Union>i. A i) = psuminf (f o A)"
+ have U_eq: "f (\<Union>i. A i) = (\<Sum>i. f (A i))"
proof (rule antisym)
- show "f (\<Union>i. A i) \<le> psuminf (f \<circ> A)"
- by (rule countably_subadditiveD [OF csa A'' disj U_in])
- show "psuminf (f \<circ> A) \<le> f (\<Union>i. A i)"
- by (rule psuminf_bound, unfold atLeast0LessThan[symmetric])
- (metis algebra.additive_sum [OF alg_ls] pos disj UN_Un Un_UNIV_right
- lambda_system_positive lambda_system_additive
- subset_Un_eq increasingD [OF inc] A' A'' UNION_in_sets U_in)
+ show "f (\<Union>i. A i) \<le> (\<Sum>i. f (A i))"
+ using csa[unfolded countably_subadditive_def] A'' disj U_in by auto
+ have *: "\<And>i. 0 \<le> f (A i)" using pos A'' unfolding positive_def by auto
+ have dis: "\<And>N. disjoint_family_on A {..<N}" by (intro disjoint_family_on_mono[OF _ disj]) auto
+ show "(\<Sum>i. f (A i)) \<le> f (\<Union>i. A i)"
+ using algebra.additive_sum [OF alg_ls lambda_system_positive[OF pos] lambda_system_additive _ A' dis]
+ using A''
+ by (intro suminf_bound[OF _ *]) (auto intro!: increasingD[OF inc] allI countable_UN)
qed
{
fix a
@@ -373,15 +386,15 @@
have "a \<inter> (\<Union>i. A i) \<in> sets M"
by (metis Int U_in a)
ultimately
- have "f (a \<inter> (\<Union>i. A i)) \<le> psuminf (f \<circ> (\<lambda>i. a \<inter> i) \<circ> A)"
- using countably_subadditiveD [OF csa, of "(\<lambda>i. a \<inter> A i)"]
+ have "f (a \<inter> (\<Union>i. A i)) \<le> (\<Sum>i. f (a \<inter> A i))"
+ using csa[unfolded countably_subadditive_def, rule_format, of "(\<lambda>i. a \<inter> A i)"]
by (simp add: o_def)
hence "f (a \<inter> (\<Union>i. A i)) + f (a - (\<Union>i. A i)) \<le>
- psuminf (f \<circ> (\<lambda>i. a \<inter> i) \<circ> A) + f (a - (\<Union>i. A i))"
+ (\<Sum>i. f (a \<inter> A i)) + f (a - (\<Union>i. A i))"
by (rule add_right_mono)
moreover
- have "psuminf (f \<circ> (\<lambda>i. a \<inter> i) \<circ> A) + f (a - (\<Union>i. A i)) \<le> f a"
- proof (safe intro!: psuminf_bound_add)
+ have "(\<Sum>i. f (a \<inter> A i)) + f (a - (\<Union>i. A i)) \<le> f a"
+ proof (intro suminf_bound_add allI)
fix n
have UNION_in: "(\<Union>i\<in>{0..<n}. A i) \<in> sets M"
by (metis A'' UNION_in_sets)
@@ -395,8 +408,14 @@
have "f (a - (\<Union>i. A i)) \<le> f (a - (\<Union>i\<in>{0..<n}. A i))"
by (blast intro: increasingD [OF inc] UNION_eq_Union_image
UNION_in U_in)
- thus "setsum (f \<circ> (\<lambda>i. a \<inter> i) \<circ> A) {..<n} + f (a - (\<Union>i. A i)) \<le> f a"
+ thus "(\<Sum>i<n. f (a \<inter> A i)) + f (a - (\<Union>i. A i)) \<le> f a"
by (simp add: lambda_system_strong_sum pos A disj eq_fa add_left_mono atLeast0LessThan[symmetric])
+ next
+ have "\<And>i. a \<inter> A i \<in> sets M" using A'' by auto
+ then show "\<And>i. 0 \<le> f (a \<inter> A i)" using pos[unfolded positive_def] by auto
+ have "\<And>i. a - (\<Union>i. A i) \<in> sets M" using A'' by auto
+ then have "\<And>i. 0 \<le> f (a - (\<Union>i. A i))" using pos[unfolded positive_def] by auto
+ then show "f (a - (\<Union>i. A i)) \<noteq> -\<infinity>" by auto
qed
ultimately show "f (a \<inter> (\<Union>i. A i)) + f (a - (\<Union>i. A i)) \<le> f a"
by (rule order_trans)
@@ -443,12 +462,14 @@
proof (auto simp add: increasing_def)
fix x y
assume xy: "x \<in> sets M" "y \<in> sets M" "x \<subseteq> y"
- have "f x \<le> f x + f (y-x)" ..
+ then have "y - x \<in> sets M" by auto
+ then have "0 \<le> f (y-x)" using posf[unfolded positive_def] by auto
+ then have "f x + 0 \<le> f x + f (y-x)" by (intro add_left_mono) auto
also have "... = f (x \<union> (y-x))" using addf
by (auto simp add: additive_def) (metis Diff_disjoint Un_Diff_cancel Diff xy(1,2))
also have "... = f y"
by (metis Un_Diff_cancel Un_absorb1 xy(3))
- finally show "f x \<le> f y" .
+ finally show "f x \<le> f y" by simp
qed
lemma (in algebra) countably_additive_additive:
@@ -461,27 +482,27 @@
by (auto simp add: disjoint_family_on_def binaryset_def)
hence "range (binaryset x y) \<subseteq> sets M \<longrightarrow>
(\<Union>i. binaryset x y i) \<in> sets M \<longrightarrow>
- f (\<Union>i. binaryset x y i) = (\<Sum>\<^isub>\<infinity> n. f (binaryset x y n))"
+ f (\<Union>i. binaryset x y i) = (\<Sum> n. f (binaryset x y n))"
using ca
by (simp add: countably_additive_def)
hence "{x,y,{}} \<subseteq> sets M \<longrightarrow> x \<union> y \<in> sets M \<longrightarrow>
- f (x \<union> y) = (\<Sum>\<^isub>\<infinity> n. f (binaryset x y n))"
+ f (x \<union> y) = (\<Sum>n. f (binaryset x y n))"
by (simp add: range_binaryset_eq UN_binaryset_eq)
thus "f (x \<union> y) = f x + f y" using posf x y
- by (auto simp add: Un binaryset_psuminf positive_def)
+ by (auto simp add: Un suminf_binaryset_eq positive_def)
qed
lemma inf_measure_nonempty:
assumes f: "positive M f" and b: "b \<in> sets M" and a: "a \<subseteq> b" "{} \<in> sets M"
shows "f b \<in> measure_set M f a"
proof -
- have "psuminf (f \<circ> (\<lambda>i. {})(0 := b)) = setsum (f \<circ> (\<lambda>i. {})(0 := b)) {..<1::nat}"
- by (rule psuminf_finite) (simp add: f[unfolded positive_def])
+ let ?A = "\<lambda>i::nat. (if i = 0 then b else {})"
+ have "(\<Sum>i. f (?A i)) = (\<Sum>i<1::nat. f (?A i))"
+ by (rule suminf_finite) (simp add: f[unfolded positive_def])
also have "... = f b"
by simp
- finally have "psuminf (f \<circ> (\<lambda>i. {})(0 := b)) = f b" .
- thus ?thesis using assms
- by (auto intro!: exI [of _ "(\<lambda>i. {})(0 := b)"]
+ finally show ?thesis using assms
+ by (auto intro!: exI [of _ ?A]
simp: measure_set_def disjoint_family_on_def split_if_mem2 comp_def)
qed
@@ -489,19 +510,19 @@
assumes posf: "positive M f" and ca: "countably_additive M f"
and s: "s \<in> sets M"
shows "Inf (measure_set M f s) = f s"
- unfolding Inf_pextreal_def
+ unfolding Inf_extreal_def
proof (safe intro!: Greatest_equality)
fix z
assume z: "z \<in> measure_set M f s"
from this obtain A where
A: "range A \<subseteq> sets M" and disj: "disjoint_family A"
- and "s \<subseteq> (\<Union>x. A x)" and si: "psuminf (f \<circ> A) = z"
+ and "s \<subseteq> (\<Union>x. A x)" and si: "(\<Sum>i. f (A i)) = z"
by (auto simp add: measure_set_def comp_def)
hence seq: "s = (\<Union>i. A i \<inter> s)" by blast
have inc: "increasing M f"
by (metis additive_increasing ca countably_additive_additive posf)
- have sums: "psuminf (\<lambda>i. f (A i \<inter> s)) = f (\<Union>i. A i \<inter> s)"
- proof (rule countably_additiveD [OF ca])
+ have sums: "(\<Sum>i. f (A i \<inter> s)) = f (\<Union>i. A i \<inter> s)"
+ proof (rule ca[unfolded countably_additive_def, rule_format])
show "range (\<lambda>n. A n \<inter> s) \<subseteq> sets M" using A s
by blast
show "disjoint_family (\<lambda>n. A n \<inter> s)" using disj
@@ -509,12 +530,14 @@
show "(\<Union>i. A i \<inter> s) \<in> sets M" using A s
by (metis UN_extend_simps(4) s seq)
qed
- hence "f s = psuminf (\<lambda>i. f (A i \<inter> s))"
+ hence "f s = (\<Sum>i. f (A i \<inter> s))"
using seq [symmetric] by (simp add: sums_iff)
- also have "... \<le> psuminf (f \<circ> A)"
- proof (rule psuminf_le)
- fix n show "f (A n \<inter> s) \<le> (f \<circ> A) n" using A s
+ also have "... \<le> (\<Sum>i. f (A i))"
+ proof (rule suminf_le_pos)
+ fix n show "f (A n \<inter> s) \<le> f (A n)" using A s
by (force intro: increasingD [OF inc])
+ fix N have "A N \<inter> s \<in> sets M" using A s by auto
+ then show "0 \<le> f (A N \<inter> s)" using posf unfolding positive_def by auto
qed
also have "... = z" by (rule si)
finally show "f s \<le> z" .
@@ -525,18 +548,40 @@
by (blast intro: inf_measure_nonempty [of _ f, OF posf s subset_refl])
qed
+lemma measure_set_pos:
+ assumes posf: "positive M f" "r \<in> measure_set M f X"
+ shows "0 \<le> r"
+proof -
+ obtain A where "range A \<subseteq> sets M" and r: "r = (\<Sum>i. f (A i))"
+ using `r \<in> measure_set M f X` unfolding measure_set_def by auto
+ then show "0 \<le> r" using posf unfolding r positive_def
+ by (intro suminf_0_le) auto
+qed
+
+lemma inf_measure_pos:
+ assumes posf: "positive M f"
+ shows "0 \<le> Inf (measure_set M f X)"
+proof (rule complete_lattice_class.Inf_greatest)
+ fix r assume "r \<in> measure_set M f X" with posf show "0 \<le> r"
+ by (rule measure_set_pos)
+qed
+
lemma inf_measure_empty:
- assumes posf: "positive M f" "{} \<in> sets M"
+ assumes posf: "positive M f" and "{} \<in> sets M"
shows "Inf (measure_set M f {}) = 0"
proof (rule antisym)
show "Inf (measure_set M f {}) \<le> 0"
by (metis complete_lattice_class.Inf_lower `{} \<in> sets M`
inf_measure_nonempty[OF posf] subset_refl posf[unfolded positive_def])
-qed simp
+qed (rule inf_measure_pos[OF posf])
lemma (in algebra) inf_measure_positive:
- "positive M f \<Longrightarrow> positive M (\<lambda>x. Inf (measure_set M f x))"
- by (simp add: positive_def inf_measure_empty)
+ assumes p: "positive M f" and "{} \<in> sets M"
+ shows "positive M (\<lambda>x. Inf (measure_set M f x))"
+proof (unfold positive_def, intro conjI ballI)
+ show "Inf (measure_set M f {}) = 0" using inf_measure_empty[OF assms] by auto
+ fix A assume "A \<in> sets M"
+qed (rule inf_measure_pos[OF p])
lemma (in algebra) inf_measure_increasing:
assumes posf: "positive M f"
@@ -548,25 +593,25 @@
apply (clarsimp simp add: measure_set_def, rule_tac x=A in exI, blast)
done
-
lemma (in algebra) inf_measure_le:
assumes posf: "positive M f" and inc: "increasing M f"
- and x: "x \<in> {r . \<exists>A. range A \<subseteq> sets M \<and> s \<subseteq> (\<Union>i. A i) \<and> psuminf (f \<circ> A) = r}"
+ and x: "x \<in> {r . \<exists>A. range A \<subseteq> sets M \<and> s \<subseteq> (\<Union>i. A i) \<and> (\<Sum>i. f (A i)) = r}"
shows "Inf (measure_set M f s) \<le> x"
proof -
- from x
obtain A where A: "range A \<subseteq> sets M" and ss: "s \<subseteq> (\<Union>i. A i)"
- and xeq: "psuminf (f \<circ> A) = x"
- by auto
+ and xeq: "(\<Sum>i. f (A i)) = x"
+ using x by auto
have dA: "range (disjointed A) \<subseteq> sets M"
by (metis A range_disjointed_sets)
- have "\<forall>n.(f o disjointed A) n \<le> (f \<circ> A) n" unfolding comp_def
+ have "\<forall>n. f (disjointed A n) \<le> f (A n)"
by (metis increasingD [OF inc] UNIV_I dA image_subset_iff disjointed_subset A comp_def)
- hence sda: "psuminf (f o disjointed A) \<le> psuminf (f \<circ> A)"
- by (blast intro: psuminf_le)
- hence ley: "psuminf (f o disjointed A) \<le> x"
+ moreover have "\<forall>i. 0 \<le> f (disjointed A i)"
+ using posf dA unfolding positive_def by auto
+ ultimately have sda: "(\<Sum>i. f (disjointed A i)) \<le> (\<Sum>i. f (A i))"
+ by (blast intro!: suminf_le_pos)
+ hence ley: "(\<Sum>i. f (disjointed A i)) \<le> x"
by (metis xeq)
- hence y: "psuminf (f o disjointed A) \<in> measure_set M f s"
+ hence y: "(\<Sum>i. f (disjointed A i)) \<in> measure_set M f s"
apply (auto simp add: measure_set_def)
apply (rule_tac x="disjointed A" in exI)
apply (simp add: disjoint_family_disjointed UN_disjointed_eq ss dA comp_def)
@@ -576,13 +621,16 @@
qed
lemma (in algebra) inf_measure_close:
+ fixes e :: extreal
assumes posf: "positive M f" and e: "0 < e" and ss: "s \<subseteq> (space M)"
shows "\<exists>A. range A \<subseteq> sets M \<and> disjoint_family A \<and> s \<subseteq> (\<Union>i. A i) \<and>
- psuminf (f \<circ> A) \<le> Inf (measure_set M f s) + e"
-proof (cases "Inf (measure_set M f s) = \<omega>")
+ (\<Sum>i. f (A i)) \<le> Inf (measure_set M f s) + e"
+proof (cases "Inf (measure_set M f s) = \<infinity>")
case False
+ then have fin: "\<bar>Inf (measure_set M f s)\<bar> \<noteq> \<infinity>"
+ using inf_measure_pos[OF posf, of s] by auto
obtain l where "l \<in> measure_set M f s" "l \<le> Inf (measure_set M f s) + e"
- using Inf_close[OF False e] by auto
+ using Inf_extreal_close[OF fin e] by auto
thus ?thesis
by (auto intro!: exI[of _ l] simp: measure_set_def comp_def)
next
@@ -600,9 +648,8 @@
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 pextreal_le_epsilon)
- fix A :: "nat \<Rightarrow> 'a set" and e :: pextreal
-
+proof (safe, simp, rule extreal_le_epsilon, safe)
+ fix A :: "nat \<Rightarrow> 'a set" and e :: extreal
let "?outer n" = "Inf (measure_set M f (A n))"
assume A: "range A \<subseteq> Pow (space M)"
and disj: "disjoint_family A"
@@ -610,21 +657,27 @@
and e: "0 < e"
hence "\<exists>BB. \<forall>n. range (BB n) \<subseteq> sets M \<and> disjoint_family (BB n) \<and>
A n \<subseteq> (\<Union>i. BB n i) \<and>
- psuminf (f o BB n) \<le> ?outer n + e * (1/2)^(Suc n)"
- apply (safe intro!: choice inf_measure_close [of f, OF posf _])
- using e sb by (cases e, auto simp add: not_le mult_pos_pos)
+ (\<Sum>i. f (BB n i)) \<le> ?outer n + e * (1/2)^(Suc n)"
+ apply (safe intro!: choice inf_measure_close [of f, OF posf])
+ using e sb by (cases e) (auto simp add: not_le mult_pos_pos one_extreal_def)
then obtain BB
where BB: "\<And>n. (range (BB n) \<subseteq> sets M)"
and disjBB: "\<And>n. disjoint_family (BB n)"
and sbBB: "\<And>n. A n \<subseteq> (\<Union>i. BB n i)"
- and BBle: "\<And>n. psuminf (f o BB n) \<le> ?outer n + e * (1/2)^(Suc n)"
+ and BBle: "\<And>n. (\<Sum>i. f (BB n i)) \<le> ?outer n + e * (1/2)^(Suc n)"
by auto blast
- have sll: "(\<Sum>\<^isub>\<infinity> n. psuminf (f o BB n)) \<le> psuminf ?outer + e"
+ have sll: "(\<Sum>n. \<Sum>i. (f (BB n i))) \<le> suminf ?outer + e"
proof -
- have "(\<Sum>\<^isub>\<infinity> n. psuminf (f o BB n)) \<le> (\<Sum>\<^isub>\<infinity> n. ?outer n + e*(1/2) ^ Suc n)"
- by (rule psuminf_le[OF BBle])
- also have "... = psuminf ?outer + e"
- using psuminf_half_series by simp
+ have sum_eq_1: "(\<Sum>n. e*(1/2) ^ Suc n) = e"
+ using suminf_half_series_extreal e
+ by (simp add: extreal_zero_le_0_iff zero_le_divide_extreal suminf_cmult_extreal)
+ have "\<And>n i. 0 \<le> f (BB n i)" using posf[unfolded positive_def] BB by auto
+ then have "\<And>n. 0 \<le> (\<Sum>i. f (BB n i))" by (rule suminf_0_le)
+ then have "(\<Sum>n. \<Sum>i. (f (BB n i))) \<le> (\<Sum>n. ?outer n + e*(1/2) ^ Suc n)"
+ by (rule suminf_le_pos[OF BBle])
+ also have "... = suminf ?outer + e"
+ using sum_eq_1 inf_measure_pos[OF posf] e
+ by (subst suminf_add_extreal) (auto simp add: extreal_zero_le_0_iff)
finally show ?thesis .
qed
def C \<equiv> "(split BB) o prod_decode"
@@ -644,23 +697,25 @@
qed
have "(f \<circ> C) = (f \<circ> (\<lambda>(x, y). BB x y)) \<circ> prod_decode"
by (rule ext) (auto simp add: C_def)
- moreover have "psuminf ... = (\<Sum>\<^isub>\<infinity> n. psuminf (f o BB n))" using BBle
- by (force intro!: psuminf_2dimen simp: o_def)
- ultimately have Csums: "psuminf (f \<circ> C) = (\<Sum>\<^isub>\<infinity> n. psuminf (f o BB n))" by simp
- have "Inf (measure_set M f (\<Union>i. A i)) \<le> (\<Sum>\<^isub>\<infinity> n. psuminf (f o BB n))"
+ moreover have "suminf ... = (\<Sum>n. \<Sum>i. f (BB n i))" using BBle
+ using BB posf[unfolded positive_def]
+ by (force intro!: suminf_extreal_2dimen simp: o_def)
+ ultimately have Csums: "(\<Sum>i. f (C i)) = (\<Sum>n. \<Sum>i. f (BB n i))" by (simp add: o_def)
+ have "Inf (measure_set M f (\<Union>i. A i)) \<le> (\<Sum>n. \<Sum>i. f (BB n i))"
apply (rule inf_measure_le [OF posf(1) inc], auto)
apply (rule_tac x="C" in exI)
apply (auto simp add: C sbC Csums)
done
- also have "... \<le> (\<Sum>\<^isub>\<infinity>n. Inf (measure_set M f (A n))) + e" using sll
+ also have "... \<le> (\<Sum>n. Inf (measure_set M f (A n))) + e" using sll
by blast
- finally show "Inf (measure_set M f (\<Union>i. A i)) \<le> psuminf ?outer + e" .
+ finally show "Inf (measure_set M f (\<Union>i. A i)) \<le> suminf ?outer + e" .
qed
lemma (in algebra) inf_measure_outer:
"\<lbrakk> positive M f ; increasing M f \<rbrakk>
\<Longrightarrow> outer_measure_space \<lparr> space = space M, sets = Pow (space M) \<rparr>
(\<lambda>x. Inf (measure_set M f x))"
+ using inf_measure_pos[of M f]
by (simp add: outer_measure_space_def inf_measure_empty
inf_measure_increasing inf_measure_countably_subadditive positive_def)
@@ -680,13 +735,13 @@
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 pextreal_le_epsilon)
- fix e :: pextreal
+ proof (rule extreal_le_epsilon, intro allI impI)
+ fix e :: extreal
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"
and sUN: "s \<subseteq> (\<Union>i. A i)"
- and l: "psuminf (f \<circ> A) \<le> Inf (measure_set M f s) + e"
+ and l: "(\<Sum>i. f (A i)) \<le> Inf (measure_set M f s) + e"
by auto
have [simp]: "!!x. x \<in> sets M \<Longrightarrow>
(f o (\<lambda>z. z \<inter> (space M - x)) o A) = (f o (\<lambda>z. z - x) o A)"
@@ -698,9 +753,9 @@
assume u: "u \<in> sets M"
have [simp]: "\<And>n. f (A n \<inter> u) \<le> f (A n)"
by (simp add: increasingD [OF inc] u Int range_subsetD [OF A])
- have 2: "Inf (measure_set M f (s \<inter> u)) \<le> psuminf (f \<circ> (\<lambda>z. z \<inter> u) \<circ> A)"
+ have 2: "Inf (measure_set M f (s \<inter> u)) \<le> (\<Sum>i. f (A i \<inter> u))"
proof (rule complete_lattice_class.Inf_lower)
- show "psuminf (f \<circ> (\<lambda>z. z \<inter> u) \<circ> A) \<in> measure_set M f (s \<inter> u)"
+ show "(\<Sum>i. f (A i \<inter> u)) \<in> measure_set M f (s \<inter> u)"
apply (simp add: measure_set_def)
apply (rule_tac x="(\<lambda>z. z \<inter> u) o A" in exI)
apply (auto simp add: disjoint_family_subset [OF disj] o_def)
@@ -709,15 +764,16 @@
done
qed
} note lesum = this
- have inf1: "Inf (measure_set M f (s\<inter>x)) \<le> psuminf (f o (\<lambda>z. z\<inter>x) o A)"
+ have [simp]: "\<And>i. A i \<inter> (space M - x) = A i - x" using A sets_into_space by auto
+ have inf1: "Inf (measure_set M f (s\<inter>x)) \<le> (\<Sum>i. f (A i \<inter> x))"
and inf2: "Inf (measure_set M f (s \<inter> (space M - x)))
- \<le> psuminf (f o (\<lambda>z. z \<inter> (space M - x)) o A)"
+ \<le> (\<Sum>i. f (A i \<inter> (space M - x)))"
by (metis Diff lesum top x)+
hence "Inf (measure_set M f (s\<inter>x)) + Inf (measure_set M f (s-x))
- \<le> psuminf (f o (\<lambda>s. s\<inter>x) o A) + psuminf (f o (\<lambda>s. s-x) o A)"
- by (simp add: x add_mono)
- also have "... \<le> psuminf (f o A)"
- by (simp add: x psuminf_add[symmetric] o_def)
+ \<le> (\<Sum>i. f (A i \<inter> x)) + (\<Sum>i. f (A i - x))"
+ by (simp add: add_mono x)
+ also have "... \<le> (\<Sum>i. f (A i))" using posf[unfolded positive_def] A x
+ by (subst suminf_add_extreal[symmetric]) auto
also have "... \<le> Inf (measure_set M f s) + e"
by (rule l)
finally show "Inf (measure_set M f (s\<inter>x)) + Inf (measure_set M f (s-x))
@@ -732,7 +788,7 @@
also have "... \<le> Inf (measure_set M f (s\<inter>x)) + Inf (measure_set M f (s-x))"
apply (rule subadditiveD)
apply (rule algebra.countably_subadditive_subadditive[OF algebra_Pow])
- apply (simp add: positive_def inf_measure_empty[OF posf])
+ apply (simp add: positive_def inf_measure_empty[OF posf] inf_measure_pos[OF posf])
apply (rule inf_measure_countably_subadditive)
using s by (auto intro!: posf inc)
finally show ?thesis .
@@ -751,7 +807,7 @@
theorem (in algebra) caratheodory:
assumes posf: "positive M f" and ca: "countably_additive M f"
- shows "\<exists>\<mu> :: 'a set \<Rightarrow> pextreal. (\<forall>s \<in> sets M. \<mu> s = f s) \<and>
+ shows "\<exists>\<mu> :: 'a set \<Rightarrow> extreal. (\<forall>s \<in> sets M. \<mu> s = f s) \<and>
measure_space \<lparr> space = space M, sets = sets (sigma M), measure = \<mu> \<rparr>"
proof -
have inc: "increasing M f"
--- a/src/HOL/Probability/Complete_Measure.thy Mon Mar 14 15:17:10 2011 +0100
+++ b/src/HOL/Probability/Complete_Measure.thy Mon Mar 14 15:29:10 2011 +0100
@@ -1,7 +1,6 @@
-(* Title: HOL/Probability/Complete_Measure.thy
+(* Title: Complete_Measure.thy
Author: Robert Himmelmann, Johannes Hoelzl, TU Muenchen
*)
-
theory Complete_Measure
imports Product_Measure
begin
@@ -177,7 +176,8 @@
proof -
show "measure_space completion"
proof
- show "measure completion {} = 0" by (auto simp: completion_def)
+ show "positive completion (measure completion)"
+ by (auto simp: completion_def positive_def)
next
show "countably_additive completion (measure completion)"
proof (intro countably_additiveI)
@@ -189,9 +189,9 @@
using A by (subst (1 2) main_part_null_part_Un) auto
then show "main_part (A n) \<inter> main_part (A m) = {}" by auto
qed
- then have "(\<Sum>\<^isub>\<infinity>n. measure completion (A n)) = \<mu> (\<Union>i. main_part (A i))"
+ then have "(\<Sum>n. measure completion (A n)) = \<mu> (\<Union>i. main_part (A i))"
unfolding completion_def using A by (auto intro!: measure_countably_additive)
- then show "(\<Sum>\<^isub>\<infinity>n. measure completion (A n)) = measure completion (UNION UNIV A)"
+ then show "(\<Sum>n. measure completion (A n)) = measure completion (UNION UNIV A)"
by (simp add: completion_def \<mu>_main_part_UN[OF A(1)])
qed
qed
@@ -251,30 +251,52 @@
qed
qed
-lemma (in completeable_measure_space) completion_ex_borel_measurable:
- fixes g :: "'a \<Rightarrow> pextreal"
- assumes g: "g \<in> borel_measurable completion"
+lemma (in completeable_measure_space) completion_ex_borel_measurable_pos:
+ fixes g :: "'a \<Rightarrow> extreal"
+ assumes g: "g \<in> borel_measurable completion" and "\<And>x. 0 \<le> g x"
shows "\<exists>g'\<in>borel_measurable M. (AE x. g x = g' x)"
proof -
- from g[THEN completion.borel_measurable_implies_simple_function_sequence]
- obtain f where "\<And>i. simple_function completion (f i)" "f \<up> g" by auto
- then have "\<forall>i. \<exists>f'. simple_function M f' \<and> (AE x. f i x = f' x)"
- using completion_ex_simple_function by auto
+ from g[THEN completion.borel_measurable_implies_simple_function_sequence'] guess f . note f = this
+ from this(1)[THEN completion_ex_simple_function]
+ have "\<forall>i. \<exists>f'. simple_function M f' \<and> (AE x. f i x = f' x)" ..
from this[THEN choice] obtain f' where
sf: "\<And>i. simple_function M (f' i)" and
AE: "\<forall>i. AE x. f i x = f' i x" by auto
show ?thesis
proof (intro bexI)
- from AE[unfolded all_AE_countable]
+ from AE[unfolded AE_all_countable[symmetric]]
show "AE x. g x = (SUP i. f' i x)" (is "AE x. g x = ?f x")
proof (elim AE_mp, safe intro!: AE_I2)
fix x assume eq: "\<forall>i. f i x = f' i x"
- moreover have "g = SUPR UNIV f" using `f \<up> g` unfolding isoton_def by simp
- ultimately show "g x = ?f x" by (simp add: SUPR_apply)
+ moreover have "g x = (SUP i. f i x)"
+ unfolding f using `0 \<le> g x` by (auto split: split_max)
+ ultimately show "g x = ?f x" by auto
qed
show "?f \<in> borel_measurable M"
using sf by (auto intro: borel_measurable_simple_function)
qed
qed
+lemma (in completeable_measure_space) completion_ex_borel_measurable:
+ fixes g :: "'a \<Rightarrow> extreal"
+ assumes g: "g \<in> borel_measurable completion"
+ shows "\<exists>g'\<in>borel_measurable M. (AE x. g x = g' x)"
+proof -
+ have "(\<lambda>x. max 0 (g x)) \<in> borel_measurable completion" "\<And>x. 0 \<le> max 0 (g x)" using g by auto
+ from completion_ex_borel_measurable_pos[OF this] guess g_pos ..
+ moreover
+ have "(\<lambda>x. max 0 (- g x)) \<in> borel_measurable completion" "\<And>x. 0 \<le> max 0 (- g x)" using g by auto
+ from completion_ex_borel_measurable_pos[OF this] guess g_neg ..
+ ultimately
+ show ?thesis
+ proof (safe intro!: bexI[of _ "\<lambda>x. g_pos x - g_neg x"])
+ show "AE x. max 0 (- g x) = g_neg x \<longrightarrow> max 0 (g x) = g_pos x \<longrightarrow> g x = g_pos x - g_neg x"
+ proof (intro AE_I2 impI)
+ fix x assume g: "max 0 (- g x) = g_neg x" "max 0 (g x) = g_pos x"
+ show "g x = g_pos x - g_neg x" unfolding g[symmetric]
+ by (cases "g x") (auto split: split_max)
+ qed
+ qed auto
+qed
+
end
--- a/src/HOL/Probability/Information.thy Mon Mar 14 15:17:10 2011 +0100
+++ b/src/HOL/Probability/Information.thy Mon Mar 14 15:29:10 2011 +0100
@@ -2,9 +2,12 @@
imports
Probability_Space
"~~/src/HOL/Library/Convex"
- Lebesgue_Measure
begin
+lemma (in prob_space) not_zero_less_distribution[simp]:
+ "(\<not> 0 < distribution X A) \<longleftrightarrow> distribution X A = 0"
+ using distribution_positive[of X A] by arith
+
lemma log_le: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> x \<le> y \<Longrightarrow> log a x \<le> log a y"
by (subst log_le_cancel_iff) auto
@@ -238,7 +241,7 @@
have ms: "measure_space (M\<lparr>measure := \<nu>\<rparr>)" by default
show "(\<Sum>x \<in> space M. log b (real (RN_deriv M \<nu> x)) * real (\<nu> {x})) = ?sum"
using RN_deriv_finite_measure[OF ms ac]
- by (auto intro!: setsum_cong simp: field_simps real_of_pextreal_mult[symmetric])
+ by (auto intro!: setsum_cong simp: field_simps)
qed
lemma (in finite_prob_space) KL_divergence_positive_finite:
@@ -254,7 +257,8 @@
proof (subst KL_divergence_eq_finite[OF ms ac], safe intro!: log_setsum_divide not_empty)
show "finite (space M)" using finite_space by simp
show "1 < b" by fact
- show "(\<Sum>x\<in>space M. real (\<nu> {x})) = 1" using v.finite_sum_over_space_eq_1 by simp
+ show "(\<Sum>x\<in>space M. real (\<nu> {x})) = 1"
+ using v.finite_sum_over_space_eq_1 by (simp add: v.\<mu>'_def)
fix x assume "x \<in> space M"
then have x: "{x} \<in> sets M" unfolding sets_eq_Pow by auto
@@ -262,17 +266,19 @@
then have "\<nu> {x} \<noteq> 0" by auto
then have "\<mu> {x} \<noteq> 0"
using ac[unfolded absolutely_continuous_def, THEN bspec, of "{x}"] x by auto
- thus "0 < prob {x}" using finite_measure[of "{x}"] x by auto }
- qed auto
- thus "0 \<le> KL_divergence b M \<nu>" using finite_sum_over_space_eq_1 by simp
+ thus "0 < real (\<mu> {x})" using real_measure[OF x] by auto }
+ show "0 \<le> real (\<mu> {x})" "0 \<le> real (\<nu> {x})"
+ using real_measure[OF x] v.real_measure[of "{x}"] x by auto
+ qed
+ thus "0 \<le> KL_divergence b M \<nu>" using finite_sum_over_space_eq_1 by (simp add: \<mu>'_def)
qed
subsection {* Mutual Information *}
definition (in prob_space)
"mutual_information b S T X Y =
- KL_divergence b (S\<lparr>measure := distribution X\<rparr> \<Otimes>\<^isub>M T\<lparr>measure := distribution Y\<rparr>)
- (joint_distribution X Y)"
+ KL_divergence b (S\<lparr>measure := extreal\<circ>distribution X\<rparr> \<Otimes>\<^isub>M T\<lparr>measure := extreal\<circ>distribution Y\<rparr>)
+ (extreal\<circ>joint_distribution X Y)"
definition (in prob_space)
"entropy b s X = mutual_information b s s X X"
@@ -280,38 +286,33 @@
abbreviation (in information_space)
mutual_information_Pow ("\<I>'(_ ; _')") where
"\<I>(X ; Y) \<equiv> mutual_information b
- \<lparr> space = X`space M, sets = Pow (X`space M), measure = distribution X \<rparr>
- \<lparr> space = Y`space M, sets = Pow (Y`space M), measure = distribution Y \<rparr> X Y"
+ \<lparr> space = X`space M, sets = Pow (X`space M), measure = extreal\<circ>distribution X \<rparr>
+ \<lparr> space = Y`space M, sets = Pow (Y`space M), measure = extreal\<circ>distribution Y \<rparr> X Y"
lemma (in prob_space) finite_variables_absolutely_continuous:
assumes X: "finite_random_variable S X" and Y: "finite_random_variable T Y"
shows "measure_space.absolutely_continuous
- (S\<lparr>measure := distribution X\<rparr> \<Otimes>\<^isub>M T\<lparr>measure := distribution Y\<rparr>)
- (joint_distribution X Y)"
+ (S\<lparr>measure := extreal\<circ>distribution X\<rparr> \<Otimes>\<^isub>M T\<lparr>measure := extreal\<circ>distribution Y\<rparr>)
+ (extreal\<circ>joint_distribution X Y)"
proof -
- interpret X: finite_prob_space "S\<lparr>measure := distribution X\<rparr>"
+ interpret X: finite_prob_space "S\<lparr>measure := extreal\<circ>distribution X\<rparr>"
using X by (rule distribution_finite_prob_space)
- interpret Y: finite_prob_space "T\<lparr>measure := distribution Y\<rparr>"
+ interpret Y: finite_prob_space "T\<lparr>measure := extreal\<circ>distribution Y\<rparr>"
using Y by (rule distribution_finite_prob_space)
interpret XY: pair_finite_prob_space
- "S\<lparr>measure := distribution X\<rparr>" "T\<lparr> measure := distribution Y\<rparr>" by default
- interpret P: finite_prob_space "XY.P\<lparr> measure := joint_distribution X Y\<rparr>"
+ "S\<lparr>measure := extreal\<circ>distribution X\<rparr>" "T\<lparr> measure := extreal\<circ>distribution Y\<rparr>" by default
+ interpret P: finite_prob_space "XY.P\<lparr> measure := extreal\<circ>joint_distribution X Y\<rparr>"
using assms by (auto intro!: joint_distribution_finite_prob_space)
note rv = assms[THEN finite_random_variableD]
- show "XY.absolutely_continuous (joint_distribution X Y)"
+ show "XY.absolutely_continuous (extreal\<circ>joint_distribution X Y)"
proof (rule XY.absolutely_continuousI)
- show "finite_measure_space (XY.P\<lparr> measure := joint_distribution X Y\<rparr>)" by default
+ show "finite_measure_space (XY.P\<lparr> measure := extreal\<circ>joint_distribution X Y\<rparr>)" by default
fix x assume "x \<in> space XY.P" and "XY.\<mu> {x} = 0"
- then obtain a b where "(a, b) = x" and "a \<in> space S" "b \<in> space T"
- and distr: "distribution X {a} * distribution Y {b} = 0"
+ then obtain a b where "x = (a, b)"
+ and "distribution X {a} = 0 \<or> distribution Y {b} = 0"
by (cases x) (auto simp: space_pair_measure)
- with X.sets_eq_Pow Y.sets_eq_Pow
- joint_distribution_Times_le_fst[OF rv, of "{a}" "{b}"]
- joint_distribution_Times_le_snd[OF rv, of "{a}" "{b}"]
- have "joint_distribution X Y {x} \<le> distribution Y {b}"
- "joint_distribution X Y {x} \<le> distribution X {a}"
- by (auto simp del: X.sets_eq_Pow Y.sets_eq_Pow)
- with distr show "joint_distribution X Y {x} = 0" by auto
+ with finite_distribution_order(5,6)[OF X Y]
+ show "(extreal \<circ> joint_distribution X Y) {x} = 0" by auto
qed
qed
@@ -320,28 +321,28 @@
assumes MY: "finite_random_variable MY Y"
shows mutual_information_generic_eq:
"mutual_information b MX MY X Y = (\<Sum> (x,y) \<in> space MX \<times> space MY.
- real (joint_distribution X Y {(x,y)}) *
- log b (real (joint_distribution X Y {(x,y)}) /
- (real (distribution X {x}) * real (distribution Y {y}))))"
+ joint_distribution X Y {(x,y)} *
+ log b (joint_distribution X Y {(x,y)} /
+ (distribution X {x} * distribution Y {y})))"
(is ?sum)
and mutual_information_positive_generic:
"0 \<le> mutual_information b MX MY X Y" (is ?positive)
proof -
- interpret X: finite_prob_space "MX\<lparr>measure := distribution X\<rparr>"
+ interpret X: finite_prob_space "MX\<lparr>measure := extreal\<circ>distribution X\<rparr>"
using MX by (rule distribution_finite_prob_space)
- interpret Y: finite_prob_space "MY\<lparr>measure := distribution Y\<rparr>"
+ interpret Y: finite_prob_space "MY\<lparr>measure := extreal\<circ>distribution Y\<rparr>"
using MY by (rule distribution_finite_prob_space)
- interpret XY: pair_finite_prob_space "MX\<lparr>measure := distribution X\<rparr>" "MY\<lparr>measure := distribution Y\<rparr>" by default
- interpret P: finite_prob_space "XY.P\<lparr>measure := joint_distribution X Y\<rparr>"
+ interpret XY: pair_finite_prob_space "MX\<lparr>measure := extreal\<circ>distribution X\<rparr>" "MY\<lparr>measure := extreal\<circ>distribution Y\<rparr>" by default
+ interpret P: finite_prob_space "XY.P\<lparr>measure := extreal\<circ>joint_distribution X Y\<rparr>"
using assms by (auto intro!: joint_distribution_finite_prob_space)
- have P_ms: "finite_measure_space (XY.P\<lparr>measure :=joint_distribution X Y\<rparr>)" by default
- have P_ps: "finite_prob_space (XY.P\<lparr>measure := joint_distribution X Y\<rparr>)" by default
+ have P_ms: "finite_measure_space (XY.P\<lparr>measure := extreal\<circ>joint_distribution X Y\<rparr>)" by default
+ have P_ps: "finite_prob_space (XY.P\<lparr>measure := extreal\<circ>joint_distribution X Y\<rparr>)" by default
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: space_pair_measure setsum_cartesian_product' real_of_pextreal_mult[symmetric])
+ (auto simp add: space_pair_measure setsum_cartesian_product')
show ?positive
using XY.KL_divergence_positive_finite[OF P_ps finite_variables_absolutely_continuous[OF MX MY] b_gt_1]
@@ -351,10 +352,10 @@
lemma (in information_space) mutual_information_commute_generic:
assumes X: "random_variable S X" and Y: "random_variable T Y"
assumes ac: "measure_space.absolutely_continuous
- (S\<lparr>measure := distribution X\<rparr> \<Otimes>\<^isub>M T\<lparr>measure := distribution Y\<rparr>) (joint_distribution X Y)"
+ (S\<lparr>measure := extreal\<circ>distribution X\<rparr> \<Otimes>\<^isub>M T\<lparr>measure := extreal\<circ>distribution Y\<rparr>) (extreal\<circ>joint_distribution X Y)"
shows "mutual_information b S T X Y = mutual_information b T S Y X"
proof -
- let ?S = "S\<lparr>measure := distribution X\<rparr>" and ?T = "T\<lparr>measure := distribution Y\<rparr>"
+ let ?S = "S\<lparr>measure := extreal\<circ>distribution X\<rparr>" and ?T = "T\<lparr>measure := extreal\<circ>distribution Y\<rparr>"
interpret S: prob_space ?S using X by (rule distribution_prob_space)
interpret T: prob_space ?T using Y by (rule distribution_prob_space)
interpret P: pair_prob_space ?S ?T ..
@@ -363,13 +364,13 @@
unfolding mutual_information_def
proof (intro Q.KL_divergence_vimage[OF Q.measure_preserving_swap _ _ _ _ ac b_gt_1])
show "(\<lambda>(x,y). (y,x)) \<in> measure_preserving
- (P.P \<lparr> measure := joint_distribution X Y\<rparr>) (Q.P \<lparr> measure := joint_distribution Y X\<rparr>)"
+ (P.P \<lparr> measure := extreal\<circ>joint_distribution X Y\<rparr>) (Q.P \<lparr> measure := extreal\<circ>joint_distribution Y X\<rparr>)"
using X Y unfolding measurable_def
unfolding measure_preserving_def using P.pair_sigma_algebra_swap_measurable
- by (auto simp add: space_pair_measure distribution_def intro!: arg_cong[where f=\<mu>])
- have "prob_space (P.P\<lparr> measure := joint_distribution X Y\<rparr>)"
+ by (auto simp add: space_pair_measure distribution_def intro!: arg_cong[where f=\<mu>'])
+ have "prob_space (P.P\<lparr> measure := extreal\<circ>joint_distribution X Y\<rparr>)"
using X Y by (auto intro!: distribution_prob_space random_variable_pairI)
- then show "measure_space (P.P\<lparr> measure := joint_distribution X Y\<rparr>)"
+ then show "measure_space (P.P\<lparr> measure := extreal\<circ>joint_distribution X Y\<rparr>)"
unfolding prob_space_def by simp
qed auto
qed
@@ -389,8 +390,8 @@
lemma (in information_space) mutual_information_eq:
assumes "simple_function M X" "simple_function M Y"
shows "\<I>(X;Y) = (\<Sum> (x,y) \<in> X ` space M \<times> Y ` space M.
- real (distribution (\<lambda>x. (X x, Y x)) {(x,y)}) * log b (real (distribution (\<lambda>x. (X x, Y x)) {(x,y)}) /
- (real (distribution X {x}) * real (distribution Y {y}))))"
+ distribution (\<lambda>x. (X x, Y x)) {(x,y)} * log b (distribution (\<lambda>x. (X x, Y x)) {(x,y)} /
+ (distribution X {x} * distribution Y {y})))"
using assms by (simp add: mutual_information_generic_eq)
lemma (in information_space) mutual_information_generic_cong:
@@ -416,22 +417,27 @@
abbreviation (in information_space)
entropy_Pow ("\<H>'(_')") where
- "\<H>(X) \<equiv> entropy b \<lparr> space = X`space M, sets = Pow (X`space M), measure = distribution X \<rparr> X"
+ "\<H>(X) \<equiv> entropy b \<lparr> space = X`space M, sets = Pow (X`space M), measure = extreal\<circ>distribution X \<rparr> X"
lemma (in information_space) entropy_generic_eq:
+ fixes X :: "'a \<Rightarrow> 'c"
assumes MX: "finite_random_variable MX X"
- shows "entropy b MX X = -(\<Sum> x \<in> space MX. real (distribution X {x}) * log b (real (distribution X {x})))"
+ shows "entropy b MX X = -(\<Sum> x \<in> space MX. distribution X {x} * log b (distribution X {x}))"
proof -
- interpret MX: finite_prob_space "MX\<lparr>measure := distribution X\<rparr>"
+ interpret MX: finite_prob_space "MX\<lparr>measure := extreal\<circ>distribution X\<rparr>"
using MX by (rule distribution_finite_prob_space)
- let "?X x" = "real (distribution X {x})"
- let "?XX x y" = "real (joint_distribution X X {(x, y)})"
- { fix x y
- have "(\<lambda>x. (X x, X x)) -` {(x, y)} = (if x = y then X -` {x} else {})" by auto
+ let "?X x" = "distribution X {x}"
+ let "?XX x y" = "joint_distribution X X {(x, y)}"
+
+ { fix x y :: 'c
+ { assume "x \<noteq> y"
+ then have "(\<lambda>x. (X x, X x)) -` {(x,y)} \<inter> space M = {}" by auto
+ then have "joint_distribution X X {(x, y)} = 0" by (simp add: distribution_def) }
then have "?XX x y * log b (?XX x y / (?X x * ?X y)) =
(if x = y then - ?X y * log b (?X y) else 0)"
- unfolding distribution_def by (auto simp: log_simps zero_less_mult_iff) }
+ by (auto simp: log_simps zero_less_mult_iff) }
note remove_XX = this
+
show ?thesis
unfolding entropy_def mutual_information_generic_eq[OF MX MX]
unfolding setsum_cartesian_product[symmetric] setsum_negf[symmetric] remove_XX
@@ -440,7 +446,7 @@
lemma (in information_space) entropy_eq:
assumes "simple_function M X"
- shows "\<H>(X) = -(\<Sum> x \<in> X ` space M. real (distribution X {x}) * log b (real (distribution X {x})))"
+ shows "\<H>(X) = -(\<Sum> x \<in> X ` space M. distribution X {x} * log b (distribution X {x}))"
using assms by (simp add: entropy_generic_eq)
lemma (in information_space) entropy_positive:
@@ -448,63 +454,77 @@
unfolding entropy_def by (simp add: mutual_information_positive)
lemma (in information_space) entropy_certainty_eq_0:
- assumes "simple_function M X" and "x \<in> X ` space M" and "distribution X {x} = 1"
+ assumes X: "simple_function M X" and "x \<in> X ` space M" and "distribution X {x} = 1"
shows "\<H>(X) = 0"
proof -
- let ?X = "\<lparr> space = X ` space M, sets = Pow (X ` space M), measure = distribution X\<rparr>"
+ let ?X = "\<lparr> space = X ` space M, sets = Pow (X ` space M), measure = extreal\<circ>distribution X\<rparr>"
note simple_function_imp_finite_random_variable[OF `simple_function M X`]
- from distribution_finite_prob_space[OF this, of "\<lparr> measure = distribution X \<rparr>"]
+ from distribution_finite_prob_space[OF this, of "\<lparr> measure = extreal\<circ>distribution X \<rparr>"]
interpret X: finite_prob_space ?X by simp
have "distribution X (X ` space M - {x}) = distribution X (X ` space M) - distribution X {x}"
using X.measure_compl[of "{x}"] assms by auto
also have "\<dots> = 0" using X.prob_space assms by auto
finally have X0: "distribution X (X ` space M - {x}) = 0" by auto
- { fix y assume asm: "y \<noteq> x" "y \<in> X ` space M"
- hence "{y} \<subseteq> X ` space M - {x}" by auto
- from X.measure_mono[OF this] X0 asm
- have "distribution X {y} = 0" by auto }
- hence fi: "\<And> y. y \<in> X ` space M \<Longrightarrow> real (distribution X {y}) = (if x = y then 1 else 0)"
- using assms by auto
+ { fix y assume *: "y \<in> X ` space M"
+ { assume asm: "y \<noteq> x"
+ with * have "{y} \<subseteq> X ` space M - {x}" by auto
+ from X.measure_mono[OF this] X0 asm *
+ have "distribution X {y} = 0" by (auto intro: antisym) }
+ then have "distribution X {y} = (if x = y then 1 else 0)"
+ using assms by auto }
+ note fi = this
have y: "\<And>y. (if x = y then 1 else 0) * log b (if x = y then 1 else 0) = 0" by simp
show ?thesis unfolding entropy_eq[OF `simple_function M X`] by (auto simp: y fi)
qed
lemma (in information_space) entropy_le_card_not_0:
- assumes "simple_function M X"
- shows "\<H>(X) \<le> log b (real (card (X ` space M \<inter> {x . distribution X {x} \<noteq> 0})))"
+ assumes X: "simple_function M X"
+ shows "\<H>(X) \<le> log b (card (X ` space M \<inter> {x. distribution X {x} \<noteq> 0}))"
proof -
- let "?d x" = "distribution X {x}"
- let "?p x" = "real (?d x)"
+ let "?p x" = "distribution X {x}"
have "\<H>(X) = (\<Sum>x\<in>X`space M. ?p x * log b (1 / ?p x))"
- by (auto intro!: setsum_cong simp: entropy_eq[OF `simple_function M X`] setsum_negf[symmetric] log_simps not_less)
+ unfolding entropy_eq[OF X] setsum_negf[symmetric]
+ by (auto intro!: setsum_cong simp: log_simps)
also have "\<dots> \<le> log b (\<Sum>x\<in>X`space M. ?p x * (1 / ?p x))"
- apply (rule log_setsum')
- using not_empty b_gt_1 `simple_function M X` sum_over_space_real_distribution
- 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 M 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_pextreal_eq_0)
+ using not_empty b_gt_1 `simple_function M X` sum_over_space_real_distribution[OF X]
+ by (intro log_setsum') (auto simp: simple_function_def)
+ also have "\<dots> = log b (\<Sum>x\<in>X`space M. if ?p x \<noteq> 0 then 1 else 0)"
+ by (intro arg_cong[where f="\<lambda>X. log b X"] setsum_cong) auto
finally show ?thesis
using `simple_function M X` by (auto simp: setsum_cases real_eq_of_nat simple_function_def)
qed
+lemma (in prob_space) measure'_translate:
+ assumes X: "random_variable S X" and A: "A \<in> sets S"
+ shows "finite_measure.\<mu>' (S\<lparr> measure := extreal\<circ>distribution X \<rparr>) A = distribution X A"
+proof -
+ interpret S: prob_space "S\<lparr> measure := extreal\<circ>distribution X \<rparr>"
+ using distribution_prob_space[OF X] .
+ from A show "S.\<mu>' A = distribution X A"
+ unfolding S.\<mu>'_def by (simp add: distribution_def_raw \<mu>'_def)
+qed
+
lemma (in information_space) entropy_uniform_max:
- assumes "simple_function M X"
+ assumes X: "simple_function M X"
assumes "\<And>x y. \<lbrakk> x \<in> X ` space M ; y \<in> X ` space M \<rbrakk> \<Longrightarrow> distribution X {x} = distribution X {y}"
shows "\<H>(X) = log b (real (card (X ` space M)))"
proof -
- let ?X = "\<lparr> space = X ` space M, sets = Pow (X ` space M), measure = distribution X\<rparr>"
- note simple_function_imp_finite_random_variable[OF `simple_function M X`]
- from distribution_finite_prob_space[OF this, of "\<lparr> measure = distribution X \<rparr>"]
+ let ?X = "\<lparr> space = X ` space M, sets = Pow (X ` space M), measure = undefined\<rparr>\<lparr> measure := extreal\<circ>distribution X\<rparr>"
+ note frv = simple_function_imp_finite_random_variable[OF X]
+ from distribution_finite_prob_space[OF this, of "\<lparr> measure = extreal\<circ>distribution X \<rparr>"]
interpret X: finite_prob_space ?X by simp
+ note rv = finite_random_variableD[OF frv]
have card_gt0: "0 < card (X ` space M)" unfolding card_gt_0_iff
using `simple_function M X` not_empty by (auto simp: simple_function_def)
- { fix x assume "x \<in> X ` space M"
- hence "real (distribution X {x}) = 1 / real (card (X ` space M))"
- proof (rule X.uniform_prob[simplified])
- fix x y assume "x \<in> X`space M" "y \<in> X`space M"
- from assms(2)[OF this] show "real (distribution X {x}) = real (distribution X {y})" by simp
- qed }
+ { fix x assume "x \<in> space ?X"
+ moreover then have "X.\<mu>' {x} = 1 / card (space ?X)"
+ proof (rule X.uniform_prob)
+ fix x y assume "x \<in> space ?X" "y \<in> space ?X"
+ with assms(2)[of x y] show "X.\<mu>' {x} = X.\<mu>' {y}"
+ by (subst (1 2) measure'_translate[OF rv]) auto
+ qed
+ ultimately have "distribution X {x} = 1 / card (space ?X)"
+ by (subst (asm) measure'_translate[OF rv]) auto }
thus ?thesis
using not_empty X.finite_space b_gt_1 card_gt0
by (simp add: entropy_eq[OF `simple_function M X`] real_eq_of_nat[symmetric] log_simps)
@@ -552,8 +572,7 @@
lemma (in information_space) entropy_eq_cartesian_product:
assumes "simple_function M X" "simple_function M Y"
shows "\<H>(\<lambda>x. (X x, Y x)) = -(\<Sum>x\<in>X`space M. \<Sum>y\<in>Y`space M.
- real (joint_distribution X Y {(x,y)}) *
- log b (real (joint_distribution X Y {(x,y)})))"
+ joint_distribution X Y {(x,y)} * log b (joint_distribution X Y {(x,y)}))"
proof -
have sf: "simple_function M (\<lambda>x. (X x, Y x))"
using assms by (auto intro: simple_function_Pair)
@@ -576,9 +595,9 @@
abbreviation (in information_space)
conditional_mutual_information_Pow ("\<I>'( _ ; _ | _ ')") where
"\<I>(X ; Y | Z) \<equiv> conditional_mutual_information b
- \<lparr> space = X`space M, sets = Pow (X`space M), measure = distribution X \<rparr>
- \<lparr> space = Y`space M, sets = Pow (Y`space M), measure = distribution Y \<rparr>
- \<lparr> space = Z`space M, sets = Pow (Z`space M), measure = distribution Z \<rparr>
+ \<lparr> space = X`space M, sets = Pow (X`space M), measure = extreal\<circ>distribution X \<rparr>
+ \<lparr> space = Y`space M, sets = Pow (Y`space M), measure = extreal\<circ>distribution Y \<rparr>
+ \<lparr> space = Z`space M, sets = Pow (Z`space M), measure = extreal\<circ>distribution Z \<rparr>
X Y Z"
lemma (in information_space) conditional_mutual_information_generic_eq:
@@ -586,58 +605,44 @@
and MY: "finite_random_variable MY Y"
and MZ: "finite_random_variable MZ Z"
shows "conditional_mutual_information b MX MY MZ X Y Z = (\<Sum>(x, y, z) \<in> space MX \<times> space MY \<times> space MZ.
- real (distribution (\<lambda>x. (X x, Y x, Z x)) {(x, y, z)}) *
- log b (real (distribution (\<lambda>x. (X x, Y x, Z x)) {(x, y, z)}) /
- (real (joint_distribution X Z {(x, z)}) * real (joint_distribution Y Z {(y,z)} / distribution Z {z}))))"
- (is "_ = (\<Sum>(x, y, z)\<in>?S. ?XYZ x y z * log b (?XYZ x y z / (?XZ x z * ?YZdZ y z)))")
+ distribution (\<lambda>x. (X x, Y x, Z x)) {(x, y, z)} *
+ log b (distribution (\<lambda>x. (X x, Y x, Z x)) {(x, y, z)} /
+ (joint_distribution X Z {(x, z)} * (joint_distribution Y Z {(y,z)} / distribution Z {z}))))"
+ (is "_ = (\<Sum>(x, y, z)\<in>?S. ?XYZ x y z * log b (?XYZ x y z / (?XZ x z * (?YZ y z / ?Z z))))")
proof -
- let ?YZ = "\<lambda>y z. real (joint_distribution Y Z {(y, z)})"
- let ?X = "\<lambda>x. real (distribution X {x})"
- let ?Z = "\<lambda>z. real (distribution Z {z})"
-
- txt {* This proof is actually quiet easy, however we need to show that the
- distributions are finite and the joint distributions are zero when one of
- the variables distribution is also zero. *}
-
+ let ?X = "\<lambda>x. distribution X {x}"
note finite_var = MX MY MZ
- note random_var = finite_var[THEN finite_random_variableD]
-
- note space_simps = space_pair_measure space_sigma algebra.simps
-
note YZ = finite_random_variable_pairI[OF finite_var(2,3)]
+ note XYZ = finite_random_variable_pairI[OF MX YZ]
note XZ = finite_random_variable_pairI[OF finite_var(1,3)]
note ZX = finite_random_variable_pairI[OF finite_var(3,1)]
note YZX = finite_random_variable_pairI[OF finite_var(2) ZX]
note order1 =
- finite_distribution_order(5,6)[OF finite_var(1) YZ, simplified space_simps]
- finite_distribution_order(5,6)[OF finite_var(1,3), simplified space_simps]
+ finite_distribution_order(5,6)[OF finite_var(1) YZ]
+ finite_distribution_order(5,6)[OF finite_var(1,3)]
+ note random_var = finite_var[THEN finite_random_variableD]
note finite = finite_var(1) YZ finite_var(3) XZ YZX
- note finite[THEN finite_distribution_finite, simplified space_simps, simp]
have order2: "\<And>x y z. \<lbrakk>x \<in> space MX; y \<in> space MY; z \<in> space MZ; joint_distribution X Z {(x, z)} = 0\<rbrakk>
\<Longrightarrow> joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)} = 0"
unfolding joint_distribution_commute_singleton[of X]
unfolding joint_distribution_assoc_singleton[symmetric]
using finite_distribution_order(6)[OF finite_var(2) ZX]
- by (auto simp: space_simps)
+ by auto
- have "(\<Sum>(x, y, z)\<in>?S. ?XYZ x y z * log b (?XYZ x y z / (?XZ x z * ?YZdZ y z))) =
+ have "(\<Sum>(x, y, z)\<in>?S. ?XYZ x y z * log b (?XYZ x y z / (?XZ x z * (?YZ y z / ?Z z)))) =
(\<Sum>(x, y, z)\<in>?S. ?XYZ x y z * (log b (?XYZ x y z / (?X x * ?YZ y z)) - log b (?XZ x z / (?X x * ?Z z))))"
(is "(\<Sum>(x, y, z)\<in>?S. ?L x y z) = (\<Sum>(x, y, z)\<in>?S. ?R x y z)")
proof (safe intro!: setsum_cong)
fix x y z assume space: "x \<in> space MX" "y \<in> space MY" "z \<in> space MZ"
- 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_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_pextreal
- real_of_pextreal_eq_0 zero_less_mult_iff)
+ with space have "0 < ?X x" "0 < ?Z z" "0 < ?XZ x z" "0 < ?YZ y z" "0 < ?XYZ x y z"
+ using order1 order2 by (auto simp: less_le)
+ with b_gt_1 show ?thesis
+ by (simp add: log_mult log_divide zero_less_mult_iff zero_less_divide_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))) -
@@ -649,8 +654,8 @@
setsum_left_distrib[symmetric]
unfolding joint_distribution_commute_singleton[of X]
unfolding joint_distribution_assoc_singleton[symmetric]
- using setsum_real_joint_distribution_singleton[OF finite_var(2) ZX, unfolded space_simps]
- by (intro setsum_cong refl) simp
+ using setsum_joint_distribution_singleton[OF finite_var(2) ZX]
+ by (intro setsum_cong refl) (simp add: space_pair_measure)
also have "(\<Sum>(x, y, z)\<in>?S. ?XYZ x y z * log b (?XYZ x y z / (?X x * ?YZ y z))) -
(\<Sum>(x, z)\<in>space MX \<times> space MZ. ?XZ x z * log b (?XZ x z / (?X x * ?Z z))) =
conditional_mutual_information b MX MY MZ X Y Z"
@@ -664,11 +669,11 @@
lemma (in information_space) conditional_mutual_information_eq:
assumes "simple_function M X" "simple_function M Y" "simple_function M Z"
shows "\<I>(X;Y|Z) = (\<Sum>(x, y, z) \<in> X`space M \<times> Y`space M \<times> Z`space M.
- real (distribution (\<lambda>x. (X x, Y x, Z x)) {(x, y, z)}) *
- log b (real (distribution (\<lambda>x. (X x, Y x, Z x)) {(x, y, z)}) /
- (real (joint_distribution X Z {(x, z)}) * real (joint_distribution Y Z {(y,z)} / distribution Z {z}))))"
- using conditional_mutual_information_generic_eq[OF assms[THEN simple_function_imp_finite_random_variable]]
- by simp
+ distribution (\<lambda>x. (X x, Y x, Z x)) {(x, y, z)} *
+ log b (distribution (\<lambda>x. (X x, Y x, Z x)) {(x, y, z)} /
+ (joint_distribution X Z {(x, z)} * joint_distribution Y Z {(y,z)} / distribution Z {z})))"
+ by (subst conditional_mutual_information_generic_eq[OF assms[THEN simple_function_imp_finite_random_variable]])
+ simp
lemma (in information_space) conditional_mutual_information_eq_mutual_information:
assumes X: "simple_function M X" and Y: "simple_function M Y"
@@ -683,10 +688,10 @@
qed
lemma (in prob_space) distribution_unit[simp]: "distribution (\<lambda>x. ()) {()} = 1"
- unfolding distribution_def using measure_space_1 by auto
+ unfolding distribution_def using prob_space by auto
lemma (in prob_space) joint_distribution_unit[simp]: "distribution (\<lambda>x. (X x, ())) {(a, ())} = distribution X {a}"
- unfolding distribution_def by (auto intro!: arg_cong[where f=\<mu>])
+ unfolding distribution_def by (auto intro!: arg_cong[where f=\<mu>'])
lemma (in prob_space) setsum_distribution:
assumes X: "finite_random_variable MX X" shows "(\<Sum>a\<in>space MX. distribution X {a}) = 1"
@@ -695,12 +700,13 @@
lemma (in prob_space) setsum_real_distribution:
fixes MX :: "('c, 'd) measure_space_scheme"
- assumes X: "finite_random_variable MX X" shows "(\<Sum>a\<in>space MX. real (distribution X {a})) = 1"
- using setsum_real_joint_distribution[OF assms, of "\<lparr> space = UNIV, sets = Pow UNIV, measure = undefined \<rparr>" "\<lambda>x. ()" "{()}"]
- using sigma_algebra_Pow[of "UNIV::unit set" "\<lparr> measure = undefined \<rparr>"] by simp
+ assumes X: "finite_random_variable MX X" shows "(\<Sum>a\<in>space MX. distribution X {a}) = 1"
+ using setsum_joint_distribution[OF assms, of "\<lparr> space = UNIV, sets = Pow UNIV, measure = undefined \<rparr>" "\<lambda>x. ()" "{()}"]
+ using sigma_algebra_Pow[of "UNIV::unit set" "\<lparr> measure = undefined \<rparr>"]
+ by auto
lemma (in information_space) conditional_mutual_information_generic_positive:
- assumes "finite_random_variable MX X" and "finite_random_variable MY Y" and "finite_random_variable MZ Z"
+ assumes X: "finite_random_variable MX X" and Y: "finite_random_variable MY Y" and Z: "finite_random_variable MZ Z"
shows "0 \<le> conditional_mutual_information b MX MY MZ X Y Z"
proof (cases "space MX \<times> space MY \<times> space MZ = {}")
case True show ?thesis
@@ -708,43 +714,35 @@
by simp
next
case False
- let "?dXYZ A" = "real (distribution (\<lambda>x. (X x, Y x, Z x)) A)"
- let "?dXZ A" = "real (joint_distribution X Z A)"
- let "?dYZ A" = "real (joint_distribution Y Z A)"
- let "?dX A" = "real (distribution X A)"
- let "?dZ A" = "real (distribution Z A)"
+ let ?dXYZ = "distribution (\<lambda>x. (X x, Y x, Z x))"
+ let ?dXZ = "joint_distribution X Z"
+ let ?dYZ = "joint_distribution Y Z"
+ let ?dX = "distribution X"
+ let ?dZ = "distribution Z"
let ?M = "space MX \<times> space MY \<times> space MZ"
- have split_beta: "\<And>f. split f = (\<lambda>x. f (fst x) (snd x))" by (simp add: fun_eq_iff)
-
- note space_simps = space_pair_measure space_sigma algebra.simps
-
- note finite_var = assms
- note YZ = finite_random_variable_pairI[OF finite_var(2,3)]
- note XZ = finite_random_variable_pairI[OF finite_var(1,3)]
- note ZX = finite_random_variable_pairI[OF finite_var(3,1)]
- note YZ = finite_random_variable_pairI[OF finite_var(2,3)]
- note XYZ = finite_random_variable_pairI[OF finite_var(1) YZ]
- note finite = finite_var(3) YZ XZ XYZ
- note finite = finite[THEN finite_distribution_finite, simplified space_simps]
-
+ note YZ = finite_random_variable_pairI[OF Y Z]
+ note XZ = finite_random_variable_pairI[OF X Z]
+ note ZX = finite_random_variable_pairI[OF Z X]
+ note YZ = finite_random_variable_pairI[OF Y Z]
+ note XYZ = finite_random_variable_pairI[OF X YZ]
+ note finite = Z YZ XZ XYZ
have order: "\<And>x y z. \<lbrakk>x \<in> space MX; y \<in> space MY; z \<in> space MZ; joint_distribution X Z {(x, z)} = 0\<rbrakk>
\<Longrightarrow> joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)} = 0"
unfolding joint_distribution_commute_singleton[of X]
unfolding joint_distribution_assoc_singleton[symmetric]
- using finite_distribution_order(6)[OF finite_var(2) ZX]
- by (auto simp: space_simps)
+ using finite_distribution_order(6)[OF Y ZX]
+ by auto
note order = order
- finite_distribution_order(5,6)[OF finite_var(1) YZ, simplified space_simps]
- finite_distribution_order(5,6)[OF finite_var(2,3), simplified space_simps]
+ finite_distribution_order(5,6)[OF X YZ]
+ finite_distribution_order(5,6)[OF Y Z]
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_pextreal_mult[symmetric])
+ unfolding conditional_mutual_information_generic_eq[OF assms] neg_equal_iff_equal by auto
also have "\<dots> \<le> log b (\<Sum>(x, y, z) \<in> ?M. ?dXZ {(x, z)} * ?dYZ {(y,z)} / ?dZ {z})"
- unfolding split_beta
+ unfolding split_beta'
proof (rule log_setsum_divide)
show "?M \<noteq> {}" using False by simp
show "1 < b" using b_gt_1 .
@@ -757,33 +755,31 @@
unfolding setsum_commute[of _ "space MY"]
unfolding setsum_commute[of _ "space MZ"]
by (simp_all add: space_pair_measure
- setsum_real_joint_distribution_singleton[OF `finite_random_variable MX X` YZ]
- setsum_real_joint_distribution_singleton[OF `finite_random_variable MY Y` finite_var(3)]
- setsum_real_distribution[OF `finite_random_variable MZ Z`])
+ setsum_joint_distribution_singleton[OF X YZ]
+ setsum_joint_distribution_singleton[OF Y Z]
+ setsum_distribution[OF Z])
fix x assume "x \<in> ?M"
let ?x = "(fst x, fst (snd x), snd (snd x))"
- 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_pextreal_nonneg mult_nonneg_nonneg divide_nonneg_nonneg)
+ show "0 \<le> ?dXYZ {?x}"
+ "0 \<le> ?dXZ {(fst x, snd (snd x))} * ?dYZ {(fst (snd x), snd (snd x))} / ?dZ {snd (snd x)}"
+ by (simp_all add: 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_pextreal zero_less_mult_iff zero_less_divide_iff)
+ with `x \<in> ?M` finite order show "0 < ?dXZ {(fst x, snd (snd x))} * ?dYZ {(fst (snd x), snd (snd x))} / ?dZ {snd (snd x)}"
+ by (cases x) (auto simp add: zero_le_mult_iff zero_le_divide_iff less_le)
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')
apply (subst setsum_commute)
apply (subst (2) setsum_commute)
by (auto simp: setsum_divide_distrib[symmetric] setsum_product[symmetric]
- setsum_real_joint_distribution_singleton[OF finite_var(1,3)]
- setsum_real_joint_distribution_singleton[OF finite_var(2,3)]
+ setsum_joint_distribution_singleton[OF X Z]
+ setsum_joint_distribution_singleton[OF Y Z]
intro!: setsum_cong)
also have "log b (\<Sum>z\<in>space MZ. ?dZ {z}) = 0"
- unfolding setsum_real_distribution[OF finite_var(3)] by simp
+ unfolding setsum_real_distribution[OF Z] by simp
finally show ?thesis by simp
qed
@@ -800,57 +796,52 @@
abbreviation (in information_space)
conditional_entropy_Pow ("\<H>'(_ | _')") where
"\<H>(X | Y) \<equiv> conditional_entropy b
- \<lparr> space = X`space M, sets = Pow (X`space M), measure = distribution X \<rparr>
- \<lparr> space = Y`space M, sets = Pow (Y`space M), measure = distribution Y \<rparr> X Y"
+ \<lparr> space = X`space M, sets = Pow (X`space M), measure = extreal\<circ>distribution X \<rparr>
+ \<lparr> space = Y`space M, sets = Pow (Y`space M), measure = extreal\<circ>distribution Y \<rparr> X Y"
lemma (in information_space) conditional_entropy_positive:
"simple_function M X \<Longrightarrow> simple_function M Y \<Longrightarrow> 0 \<le> \<H>(X | Y)"
unfolding conditional_entropy_def by (auto intro!: conditional_mutual_information_positive)
-lemma (in measure_space) empty_measureI: "A = {} \<Longrightarrow> \<mu> A = 0" by simp
-
lemma (in information_space) conditional_entropy_generic_eq:
fixes MX :: "('c, 'd) measure_space_scheme" and MY :: "('e, 'f) measure_space_scheme"
assumes MX: "finite_random_variable MX X"
assumes MZ: "finite_random_variable MZ Z"
shows "conditional_entropy b MX MZ X Z =
- (\<Sum>(x, z)\<in>space MX \<times> space MZ.
- real (joint_distribution X Z {(x, z)}) *
- log b (real (joint_distribution X Z {(x, z)}) / real (distribution Z {z})))"
+ joint_distribution X Z {(x, z)} * log b (joint_distribution X Z {(x, z)} / distribution Z {z}))"
proof -
interpret MX: finite_sigma_algebra MX using MX by simp
interpret MZ: finite_sigma_algebra MZ using MZ by simp
let "?XXZ x y z" = "joint_distribution X (\<lambda>x. (X x, Z x)) {(x, y, z)}"
let "?XZ x z" = "joint_distribution X Z {(x, z)}"
let "?Z z" = "distribution Z {z}"
- let "?f x y z" = "log b (real (?XXZ x y z) / (real (?XZ x z) * real (?XZ y z / ?Z z)))"
+ let "?f x y z" = "log b (?XXZ x y z * ?Z z / (?XZ x z * ?XZ y z))"
{ fix x z have "?XXZ x x z = ?XZ x z"
- unfolding distribution_def by (auto intro!: arg_cong[where f=\<mu>]) }
+ unfolding distribution_def by (auto intro!: arg_cong[where f=\<mu>']) }
note this[simp]
{ fix x x' :: 'c and z assume "x' \<noteq> x"
then have "?XXZ x x' z = 0"
- by (auto simp: distribution_def intro!: arg_cong[where f=\<mu>] empty_measureI) }
+ by (auto simp: distribution_def empty_measure'[symmetric]
+ simp del: empty_measure' intro!: arg_cong[where f=\<mu>']) }
note this[simp]
{ fix x x' z assume *: "x \<in> space MX" "z \<in> space MZ"
- then have "(\<Sum>x'\<in>space MX. real (?XXZ x x' z) * ?f x x' z)
- = (\<Sum>x'\<in>space MX. if x = x' then real (?XZ x z) * ?f x x z else 0)"
+ then have "(\<Sum>x'\<in>space MX. ?XXZ x x' z * ?f x x' z)
+ = (\<Sum>x'\<in>space MX. if x = x' then ?XZ x z * ?f x x z else 0)"
by (auto intro!: setsum_cong)
- also have "\<dots> = real (?XZ x z) * ?f x x z"
+ also have "\<dots> = ?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_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]
+ also have "\<dots> = ?XZ x z * log b (?Z z / ?XZ x z)" by auto
+ also have "\<dots> = - ?XZ x z * log b (?XZ x z / ?Z z)"
using finite_distribution_order(6)[OF MX MZ]
- 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))" . }
+ by (auto simp: log_simps field_simps zero_less_mult_iff)
+ finally have "(\<Sum>x'\<in>space MX. ?XXZ x x' z * ?f x x' z) = - ?XZ x z * log b (?XZ x z / ?Z z)" . }
note * = this
show ?thesis
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_pextreal_def
+ setsum_commute[of _ "space MZ"] *
intro!: setsum_cong)
qed
@@ -858,29 +849,27 @@
assumes "simple_function M X" "simple_function M Z"
shows "\<H>(X | Z) =
- (\<Sum>(x, z)\<in>X ` space M \<times> Z ` space M.
- real (joint_distribution X Z {(x, z)}) *
- log b (real (joint_distribution X Z {(x, z)}) / real (distribution Z {z})))"
- using conditional_entropy_generic_eq[OF assms[THEN simple_function_imp_finite_random_variable]]
- by simp
+ joint_distribution X Z {(x, z)} *
+ log b (joint_distribution X Z {(x, z)} / distribution Z {z}))"
+ by (subst conditional_entropy_generic_eq[OF assms[THEN simple_function_imp_finite_random_variable]])
+ simp
lemma (in information_space) conditional_entropy_eq_ce_with_hypothesis:
assumes X: "simple_function M X" and Y: "simple_function M Y"
shows "\<H>(X | Y) =
- -(\<Sum>y\<in>Y`space M. real (distribution Y {y}) *
- (\<Sum>x\<in>X`space M. real (joint_distribution X Y {(x,y)}) / real (distribution Y {(y)}) *
- log b (real (joint_distribution X Y {(x,y)}) / real (distribution Y {(y)}))))"
+ -(\<Sum>y\<in>Y`space M. distribution Y {y} *
+ (\<Sum>x\<in>X`space M. joint_distribution X Y {(x,y)} / distribution Y {(y)} *
+ log b (joint_distribution X Y {(x,y)} / distribution Y {(y)})))"
unfolding conditional_entropy_eq[OF assms]
- 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_pextreal_eq_0
+ by (auto simp: setsum_cartesian_product' setsum_commute[of _ "Y`space M"] setsum_right_distrib
intro!: setsum_cong)
lemma (in information_space) conditional_entropy_eq_cartesian_product:
assumes "simple_function M X" "simple_function M Y"
shows "\<H>(X | Y) = -(\<Sum>x\<in>X`space M. \<Sum>y\<in>Y`space M.
- real (joint_distribution X Y {(x,y)}) *
- log b (real (joint_distribution X Y {(x,y)}) / real (distribution Y {y})))"
+ joint_distribution X Y {(x,y)} *
+ log b (joint_distribution X Y {(x,y)} / distribution Y {y}))"
unfolding conditional_entropy_eq[OF assms]
by (auto intro!: setsum_cong simp: setsum_cartesian_product')
@@ -890,24 +879,22 @@
assumes X: "simple_function M X" and Z: "simple_function M Z"
shows "\<I>(X ; Z) = \<H>(X) - \<H>(X | Z)"
proof -
- let "?XZ x z" = "real (joint_distribution X Z {(x, z)})"
- let "?Z z" = "real (distribution Z {z})"
- let "?X x" = "real (distribution X {x})"
+ let "?XZ x z" = "joint_distribution X Z {(x, z)}"
+ let "?Z z" = "distribution Z {z}"
+ let "?X x" = "distribution X {x}"
note fX = X[THEN simple_function_imp_finite_random_variable]
note fZ = Z[THEN simple_function_imp_finite_random_variable]
- note fX[THEN finite_distribution_finite, simp] and fZ[THEN finite_distribution_finite, simp]
note finite_distribution_order[OF fX fZ, simp]
{ 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_pextreal_mult[symmetric] zero_less_mult_iff
- zero_less_real_of_pextreal field_simps real_of_pextreal_eq_0 abs_mult) }
+ by (auto simp: log_simps zero_le_mult_iff field_simps less_le) }
note * = this
show ?thesis
unfolding entropy_eq[OF X] conditional_entropy_eq[OF X Z] mutual_information_eq[OF X Z]
- using setsum_real_joint_distribution_singleton[OF fZ fX, unfolded joint_distribution_commute_singleton[of Z X]]
+ using setsum_joint_distribution_singleton[OF fZ fX, unfolded joint_distribution_commute_singleton[of Z X]]
by (simp add: * setsum_cartesian_product' setsum_subtractf setsum_left_distrib[symmetric]
- setsum_real_distribution)
+ setsum_distribution)
qed
lemma (in information_space) conditional_entropy_less_eq_entropy:
@@ -923,21 +910,19 @@
assumes X: "simple_function M X" and Y: "simple_function M Y"
shows "\<H>(\<lambda>x. (X x, Y x)) = \<H>(X) + \<H>(Y|X)"
proof -
- let "?XY x y" = "real (joint_distribution X Y {(x, y)})"
- let "?Y y" = "real (distribution Y {y})"
- let "?X x" = "real (distribution X {x})"
+ let "?XY x y" = "joint_distribution X Y {(x, y)}"
+ let "?Y y" = "distribution Y {y}"
+ let "?X x" = "distribution X {x}"
note fX = X[THEN simple_function_imp_finite_random_variable]
note fY = Y[THEN simple_function_imp_finite_random_variable]
- note fX[THEN finite_distribution_finite, simp] and fY[THEN finite_distribution_finite, simp]
note finite_distribution_order[OF fX fY, simp]
{ 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_pextreal_mult[symmetric] zero_less_mult_iff
- zero_less_real_of_pextreal field_simps real_of_pextreal_eq_0 abs_mult) }
+ by (auto simp: log_simps zero_le_mult_iff field_simps less_le) }
note * = this
show ?thesis
- using setsum_real_joint_distribution_singleton[OF fY fX]
+ using setsum_joint_distribution_singleton[OF fY fX]
unfolding entropy_eq[OF X] conditional_entropy_eq_cartesian_product[OF Y X] entropy_eq_cartesian_product[OF X Y]
unfolding joint_distribution_commute_singleton[of Y X] setsum_commute[of _ "X`space M"]
by (simp add: * setsum_subtractf setsum_left_distrib[symmetric])
@@ -1063,23 +1048,21 @@
assumes svi: "subvimage (space M) X P"
shows "\<H>(X) = \<H>(P) + \<H>(X|P)"
proof -
- let "?XP x p" = "real (joint_distribution X P {(x, p)})"
- let "?X x" = "real (distribution X {x})"
- let "?P p" = "real (distribution P {p})"
+ let "?XP x p" = "joint_distribution X P {(x, p)}"
+ let "?X x" = "distribution X {x}"
+ let "?P p" = "distribution P {p}"
note fX = sf(1)[THEN simple_function_imp_finite_random_variable]
note fP = sf(2)[THEN simple_function_imp_finite_random_variable]
- note fX[THEN finite_distribution_finite, simp] and fP[THEN finite_distribution_finite, simp]
note finite_distribution_order[OF fX fP, simp]
- have "(\<Sum>x\<in>X ` space M. real (distribution X {x}) * log b (real (distribution X {x}))) =
- (\<Sum>y\<in>P `space M. \<Sum>x\<in>X ` space M.
- real (joint_distribution X P {(x, y)}) * log b (real (joint_distribution X P {(x, y)})))"
+ have "(\<Sum>x\<in>X ` space M. ?X x * log b (?X x)) =
+ (\<Sum>y\<in>P `space M. \<Sum>x\<in>X ` space M. ?XP x y * log b (?XP x y))"
proof (subst setsum_image_split[OF svi],
safe intro!: setsum_mono_zero_cong_left imageI)
show "finite (X ` space M)" "finite (X ` space M)" "finite (P ` space M)"
using sf unfolding simple_function_def by auto
next
fix p x assume in_space: "p \<in> space M" "x \<in> space M"
- assume "real (joint_distribution X P {(X x, P p)}) * log b (real (joint_distribution X P {(X x, P p)})) \<noteq> 0"
+ assume "?XP (X x) (P p) * log b (?XP (X x) (P p)) \<noteq> 0"
hence "(\<lambda>x. (X x, P x)) -` {(X x, P p)} \<inter> space M \<noteq> {}" by (auto simp: distribution_def)
with svi[unfolded subvimage_def, rule_format, OF `x \<in> space M`]
show "x \<in> P -` {P p}" by auto
@@ -1091,20 +1074,16 @@
by auto
hence "(\<lambda>x. (X x, P x)) -` {(X x, P p)} \<inter> space M = X -` {X x} \<inter> space M"
by auto
- thus "real (distribution X {X x}) * log b (real (distribution X {X x})) =
- real (joint_distribution X P {(X x, P p)}) *
- log b (real (joint_distribution X P {(X x, P p)}))"
+ thus "?X (X x) * log b (?X (X x)) = ?XP (X x) (P p) * log b (?XP (X x) (P p))"
by (auto simp: distribution_def)
qed
- moreover have "\<And>x y. real (joint_distribution X P {(x, y)}) *
- log b (real (joint_distribution X P {(x, y)}) / real (distribution P {y})) =
- real (joint_distribution X P {(x, y)}) * log b (real (joint_distribution X P {(x, y)})) -
- real (joint_distribution X P {(x, y)}) * log b (real (distribution P {y}))"
+ moreover have "\<And>x y. ?XP x y * log b (?XP x y / ?P y) =
+ ?XP x y * log b (?XP x y) - ?XP x y * log b (?P y)"
by (auto simp add: log_simps zero_less_mult_iff field_simps)
ultimately show ?thesis
unfolding sf[THEN entropy_eq] conditional_entropy_eq[OF sf]
- using setsum_real_joint_distribution_singleton[OF fX fP]
- by (simp add: setsum_cartesian_product' setsum_subtractf setsum_real_distribution
+ using setsum_joint_distribution_singleton[OF fX fP]
+ by (simp add: setsum_cartesian_product' setsum_subtractf setsum_distribution
setsum_left_distrib[symmetric] setsum_commute[where B="P`space M"])
qed
--- a/src/HOL/Probability/Lebesgue_Integration.thy Mon Mar 14 15:17:10 2011 +0100
+++ b/src/HOL/Probability/Lebesgue_Integration.thy Mon Mar 14 15:29:10 2011 +0100
@@ -6,6 +6,88 @@
imports Measure Borel_Space
begin
+lemma extreal_indicator_pos[simp,intro]: "0 \<le> (indicator A x ::extreal)"
+ unfolding indicator_def by auto
+
+lemma tendsto_real_max:
+ fixes x y :: real
+ assumes "(X ---> x) net"
+ assumes "(Y ---> y) net"
+ shows "((\<lambda>x. max (X x) (Y x)) ---> max x y) net"
+proof -
+ have *: "\<And>x y :: real. max x y = y + ((x - y) + norm (x - y)) / 2"
+ by (auto split: split_max simp: field_simps)
+ show ?thesis
+ unfolding *
+ by (intro tendsto_add assms tendsto_divide tendsto_norm tendsto_diff) auto
+qed
+
+lemma (in measure_space) measure_Union:
+ assumes "finite S" "S \<subseteq> sets M" "\<And>A B. A \<in> S \<Longrightarrow> B \<in> S \<Longrightarrow> A \<noteq> B \<Longrightarrow> A \<inter> B = {}"
+ shows "setsum \<mu> S = \<mu> (\<Union>S)"
+proof -
+ have "setsum \<mu> S = \<mu> (\<Union>i\<in>S. i)"
+ using assms by (intro measure_setsum[OF `finite S`]) (auto simp: disjoint_family_on_def)
+ also have "\<dots> = \<mu> (\<Union>S)" by (auto intro!: arg_cong[where f=\<mu>])
+ finally show ?thesis .
+qed
+
+lemma (in sigma_algebra) measurable_sets2[intro]:
+ assumes "f \<in> measurable M M'" "g \<in> measurable M M''"
+ and "A \<in> sets M'" "B \<in> sets M''"
+ shows "f -` A \<inter> g -` B \<inter> space M \<in> sets M"
+proof -
+ have "f -` A \<inter> g -` B \<inter> space M = (f -` A \<inter> space M) \<inter> (g -` B \<inter> space M)"
+ by auto
+ then show ?thesis using assms by (auto intro: measurable_sets)
+qed
+
+lemma incseq_extreal: "incseq f \<Longrightarrow> incseq (\<lambda>x. extreal (f x))"
+ unfolding incseq_def by auto
+
+lemma incseq_Suc_iff: "incseq f \<longleftrightarrow> (\<forall>n. f n \<le> f (Suc n))"
+proof
+ assume "\<forall>n. f n \<le> f (Suc n)" then show "incseq f" by (auto intro!: incseq_SucI)
+qed (auto simp: incseq_def)
+
+lemma borel_measurable_real_floor:
+ "(\<lambda>x::real. real \<lfloor>x\<rfloor>) \<in> borel_measurable borel"
+ unfolding borel.borel_measurable_iff_ge
+proof (intro allI)
+ fix a :: real
+ { fix x have "a \<le> real \<lfloor>x\<rfloor> \<longleftrightarrow> real \<lceil>a\<rceil> \<le> x"
+ using le_floor_eq[of "\<lceil>a\<rceil>" x] ceiling_le_iff[of a "\<lfloor>x\<rfloor>"]
+ unfolding real_eq_of_int by simp }
+ then have "{w::real \<in> space borel. a \<le> real \<lfloor>w\<rfloor>} = {real \<lceil>a\<rceil>..}" by auto
+ then show "{w::real \<in> space borel. a \<le> real \<lfloor>w\<rfloor>} \<in> sets borel" by auto
+qed
+
+lemma measure_preservingD2:
+ "f \<in> measure_preserving A B \<Longrightarrow> f \<in> measurable A B"
+ unfolding measure_preserving_def by auto
+
+lemma measure_preservingD3:
+ "f \<in> measure_preserving A B \<Longrightarrow> f \<in> space A \<rightarrow> space B"
+ unfolding measure_preserving_def measurable_def by auto
+
+lemma measure_preservingD:
+ "T \<in> measure_preserving A B \<Longrightarrow> X \<in> sets B \<Longrightarrow> measure A (T -` X \<inter> space A) = measure B X"
+ unfolding measure_preserving_def by auto
+
+lemma (in sigma_algebra) borel_measurable_real_natfloor[intro, simp]:
+ assumes "f \<in> borel_measurable M"
+ shows "(\<lambda>x. real (natfloor (f x))) \<in> borel_measurable M"
+proof -
+ have "\<And>x. real (natfloor (f x)) = max 0 (real \<lfloor>f x\<rfloor>)"
+ by (auto simp: max_def natfloor_def)
+ with borel_measurable_max[OF measurable_comp[OF assms borel_measurable_real_floor] borel_measurable_const]
+ show ?thesis by (simp add: comp_def)
+qed
+
+lemma (in measure_space) AE_not_in:
+ assumes N: "N \<in> null_sets" shows "AE x. x \<notin> N"
+ using N by (rule AE_I') auto
+
lemma sums_If_finite:
fixes f :: "nat \<Rightarrow> 'a::real_normed_vector"
assumes finite: "finite {r. P r}"
@@ -55,8 +137,17 @@
by (auto intro!: finite_UN simp del: UN_simps simp: simple_function_def)
qed
+lemma (in sigma_algebra) simple_function_measurable2[intro]:
+ assumes "simple_function M f" "simple_function M g"
+ shows "f -` A \<inter> g -` B \<inter> space M \<in> sets M"
+proof -
+ have "f -` A \<inter> g -` B \<inter> space M = (f -` A \<inter> space M) \<inter> (g -` B \<inter> space M)"
+ by auto
+ then show ?thesis using assms[THEN simple_functionD(2)] by auto
+qed
+
lemma (in sigma_algebra) simple_function_indicator_representation:
- fixes f ::"'a \<Rightarrow> pextreal"
+ fixes f ::"'a \<Rightarrow> extreal"
assumes f: "simple_function M 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")
@@ -71,7 +162,7 @@
qed
lemma (in measure_space) simple_function_notspace:
- "simple_function M (\<lambda>x. h x * indicator (- space M) x::pextreal)" (is "simple_function M ?h")
+ "simple_function M (\<lambda>x. h x * indicator (- space M) x::extreal)" (is "simple_function M ?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)
@@ -111,16 +202,22 @@
qed
lemma (in sigma_algebra) simple_function_borel_measurable:
- fixes f :: "'a \<Rightarrow> 'x::t2_space"
+ fixes f :: "'a \<Rightarrow> 'x::{t2_space}"
assumes "f \<in> borel_measurable M" and "finite (f ` space M)"
shows "simple_function M f"
using assms unfolding simple_function_def
by (auto intro: borel_measurable_vimage)
+lemma (in sigma_algebra) simple_function_eq_borel_measurable:
+ fixes f :: "'a \<Rightarrow> extreal"
+ shows "simple_function M f \<longleftrightarrow> finite (f`space M) \<and> f \<in> borel_measurable M"
+ using simple_function_borel_measurable[of f]
+ borel_measurable_simple_function[of f]
+ by (fastsimp simp: simple_function_def)
+
lemma (in sigma_algebra) simple_function_const[intro, simp]:
"simple_function M (\<lambda>x. c)"
by (auto intro: finite_subset simp: simple_function_def)
-
lemma (in sigma_algebra) simple_function_compose[intro, simp]:
assumes "simple_function M f"
shows "simple_function M (g \<circ> f)"
@@ -189,6 +286,7 @@
and simple_function_mult[intro, simp] = simple_function_compose2[where h="op *"]
and simple_function_div[intro, simp] = simple_function_compose2[where h="op /"]
and simple_function_inverse[intro, simp] = simple_function_compose[where g="inverse"]
+ and simple_function_max[intro, simp] = simple_function_compose2[where h=max]
lemma (in sigma_algebra) simple_function_setsum[intro, simp]:
assumes "\<And>i. i \<in> P \<Longrightarrow> simple_function M (f i)"
@@ -197,247 +295,168 @@
assume "finite P" from this assms show ?thesis by induct auto
qed auto
-lemma (in sigma_algebra) simple_function_le_measurable:
- assumes "simple_function M f" "simple_function M g"
- shows "{x \<in> space M. f x \<le> g x} \<in> sets M"
-proof -
- have *: "{x \<in> space M. f x \<le> g x} =
- (\<Union>(F, G)\<in>f`space M \<times> g`space M.
- if F \<le> G then (f -` {F} \<inter> space M) \<inter> (g -` {G} \<inter> space M) else {})"
- apply (auto split: split_if_asm)
- apply (rule_tac x=x in bexI)
- apply (rule_tac x=x in bexI)
- by simp_all
- have **: "\<And>x y. x \<in> space M \<Longrightarrow> y \<in> space M \<Longrightarrow>
- (f -` {f x} \<inter> space M) \<inter> (g -` {g y} \<inter> space M) \<in> sets M"
- using assms unfolding simple_function_def by auto
- have "finite (f`space M \<times> g`space M)"
- using assms unfolding simple_function_def by auto
- thus ?thesis unfolding *
- apply (rule finite_UN)
- using assms unfolding simple_function_def
- by (auto intro!: **)
-qed
+lemma (in sigma_algebra)
+ fixes f g :: "'a \<Rightarrow> real" assumes sf: "simple_function M f"
+ shows simple_function_extreal[intro, simp]: "simple_function M (\<lambda>x. extreal (f x))"
+ by (auto intro!: simple_function_compose1[OF sf])
+
+lemma (in sigma_algebra)
+ fixes f g :: "'a \<Rightarrow> nat" assumes sf: "simple_function M f"
+ shows simple_function_real_of_nat[intro, simp]: "simple_function M (\<lambda>x. real (f x))"
+ by (auto intro!: simple_function_compose1[OF sf])
lemma (in sigma_algebra) borel_measurable_implies_simple_function_sequence:
- fixes u :: "'a \<Rightarrow> pextreal"
+ fixes u :: "'a \<Rightarrow> extreal"
assumes u: "u \<in> borel_measurable M"
- shows "\<exists>f. (\<forall>i. simple_function M (f i) \<and> (\<forall>x\<in>space M. f i x \<noteq> \<omega>)) \<and> f \<up> u"
+ shows "\<exists>f. incseq f \<and> (\<forall>i. \<infinity> \<notin> range (f i) \<and> simple_function M (f i)) \<and>
+ (\<forall>x. (SUP i. f i x) = max 0 (u x)) \<and> (\<forall>i x. 0 \<le> f i x)"
proof -
- have "\<exists>f. \<forall>x j. (of_nat j \<le> u x \<longrightarrow> f x j = j*2^j) \<and>
- (u x < of_nat j \<longrightarrow> of_nat (f x j) \<le> u x * 2^j \<and> u x * 2^j < of_nat (Suc (f x j)))"
- (is "\<exists>f. \<forall>x j. ?P x j (f x j)")
- proof(rule choice, rule, rule choice, rule)
- fix x j show "\<exists>n. ?P x j n"
- proof cases
- assume *: "u x < of_nat j"
- then obtain r where r: "u x = Real r" "0 \<le> r" by (cases "u x") auto
- from reals_Archimedean6a[of "r * 2^j"]
- obtain n :: nat where "real n \<le> r * 2 ^ j" "r * 2 ^ j < real (Suc n)"
- using `0 \<le> r` by (auto simp: zero_le_power zero_le_mult_iff)
- thus ?thesis using r * by (auto intro!: exI[of _ n])
- qed auto
- qed
- then obtain f where top: "\<And>j x. of_nat j \<le> u x \<Longrightarrow> f x j = j*2^j" and
- upper: "\<And>j x. u x < of_nat j \<Longrightarrow> of_nat (f x j) \<le> u x * 2^j" and
- lower: "\<And>j x. u x < of_nat j \<Longrightarrow> u x * 2^j < of_nat (Suc (f x j))" by blast
-
- { fix j x P
- assume 1: "of_nat j \<le> u x \<Longrightarrow> P (j * 2^j)"
- assume 2: "\<And>k. \<lbrakk> u x < of_nat j ; of_nat k \<le> u x * 2^j ; u x * 2^j < of_nat (Suc k) \<rbrakk> \<Longrightarrow> P k"
- have "P (f x j)"
- proof cases
- assume "of_nat j \<le> u x" thus "P (f x j)"
- using top[of j x] 1 by auto
- next
- assume "\<not> of_nat j \<le> u x"
- hence "u x < of_nat j" "of_nat (f x j) \<le> u x * 2^j" "u x * 2^j < of_nat (Suc (f x j))"
- using upper lower by auto
- from 2[OF this] show "P (f x j)" .
- qed }
- note fI = this
-
- { fix j x
- have "f x j = j * 2 ^ j \<longleftrightarrow> of_nat j \<le> u x"
- by (rule fI, simp, cases "u x") (auto split: split_if_asm) }
- note f_eq = this
-
- { fix j x
- have "f x j \<le> j * 2 ^ j"
- proof (rule fI)
- fix k assume *: "u x < of_nat j"
- assume "of_nat k \<le> u x * 2 ^ j"
- also have "\<dots> \<le> of_nat (j * 2^j)"
- using * by (cases "u x") (auto simp: zero_le_mult_iff)
- finally show "k \<le> j*2^j" by (auto simp del: real_of_nat_mult)
- qed simp }
+ def f \<equiv> "\<lambda>x i. if real i \<le> u x then i * 2 ^ i else natfloor (real (u x) * 2 ^ i)"
+ { fix x j have "f x j \<le> j * 2 ^ j" unfolding f_def
+ proof (split split_if, intro conjI impI)
+ assume "\<not> real j \<le> u x"
+ then have "natfloor (real (u x) * 2 ^ j) \<le> natfloor (j * 2 ^ j)"
+ by (cases "u x") (auto intro!: natfloor_mono simp: mult_nonneg_nonneg)
+ moreover have "real (natfloor (j * 2 ^ j)) \<le> j * 2^j"
+ by (intro real_natfloor_le) (auto simp: mult_nonneg_nonneg)
+ ultimately show "natfloor (real (u x) * 2 ^ j) \<le> j * 2 ^ j"
+ unfolding real_of_nat_le_iff by auto
+ qed auto }
note f_upper = this
- 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
- have *: "?g j ` space M \<subseteq> (\<lambda>x. of_nat x / 2^j) ` {..j * 2^j}"
- using f_upper by auto
- thus "finite (?g j ` space M)" by (rule finite_subset) auto
- next
- fix j t assume "t \<in> space M"
- have **: "?g j -` {?g j t} \<inter> space M = {x \<in> space M. f t j = f x j}"
- by (auto simp: power_le_zero_eq Real_eq_Real mult_le_0_iff)
+ have real_f:
+ "\<And>i x. real (f x i) = (if real i \<le> u x then i * 2 ^ i else real (natfloor (real (u x) * 2 ^ i)))"
+ unfolding f_def by auto
- show "?g j -` {?g j t} \<inter> space M \<in> sets M"
- proof cases
- assume "of_nat j \<le> u t"
- hence "?g j -` {?g j t} \<inter> space M = {x\<in>space M. of_nat j \<le> u x}"
- unfolding ** f_eq[symmetric] by auto
- thus "?g j -` {?g j t} \<inter> space M \<in> sets M"
- using u by auto
- next
- assume not_t: "\<not> of_nat j \<le> u t"
- hence less: "f t j < j*2^j" using f_eq[symmetric] `f t j \<le> j*2^j` by auto
- have split_vimage: "?g j -` {?g j t} \<inter> space M =
- {x\<in>space M. of_nat (f t j)/2^j \<le> u x} \<inter> {x\<in>space M. u x < of_nat (Suc (f t j))/2^j}"
- unfolding **
- proof safe
- fix x assume [simp]: "f t j = f x j"
- have *: "\<not> of_nat j \<le> u x" using not_t f_eq[symmetric] by simp
- hence "of_nat (f t j) \<le> u x * 2^j \<and> u x * 2^j < of_nat (Suc (f t j))"
- using upper lower by auto
- hence "of_nat (f t j) / 2^j \<le> u x \<and> u x < of_nat (Suc (f t j))/2^j" using *
- by (cases "u x") (auto simp: zero_le_mult_iff power_le_zero_eq divide_real_def[symmetric] field_simps)
- thus "of_nat (f t j) / 2^j \<le> u x" "u x < of_nat (Suc (f t j))/2^j" by auto
+ let "?g j x" = "real (f x j) / 2^j :: extreal"
+ show ?thesis
+ proof (intro exI[of _ ?g] conjI allI ballI)
+ fix i
+ have "simple_function M (\<lambda>x. real (f x i))"
+ proof (intro simple_function_borel_measurable)
+ show "(\<lambda>x. real (f x i)) \<in> borel_measurable M"
+ using u by (auto intro!: measurable_If simp: real_f)
+ have "(\<lambda>x. real (f x i))`space M \<subseteq> real`{..i*2^i}"
+ using f_upper[of _ i] by auto
+ then show "finite ((\<lambda>x. real (f x i))`space M)"
+ by (rule finite_subset) auto
+ qed
+ then show "simple_function M (?g i)"
+ by (auto intro: simple_function_extreal simple_function_div)
+ next
+ show "incseq ?g"
+ proof (intro incseq_extreal incseq_SucI le_funI)
+ fix x and i :: nat
+ have "f x i * 2 \<le> f x (Suc i)" unfolding f_def
+ proof ((split split_if)+, intro conjI impI)
+ assume "extreal (real i) \<le> u x" "\<not> extreal (real (Suc i)) \<le> u x"
+ then show "i * 2 ^ i * 2 \<le> natfloor (real (u x) * 2 ^ Suc i)"
+ by (cases "u x") (auto intro!: le_natfloor)
next
- fix x
- assume "of_nat (f t j) / 2^j \<le> u x" "u x < of_nat (Suc (f t j))/2^j"
- hence "of_nat (f t j) \<le> u x * 2 ^ j \<and> u x * 2 ^ j < of_nat (Suc (f t j))"
- by (cases "u x") (auto simp: zero_le_mult_iff power_le_zero_eq divide_real_def[symmetric] field_simps)
- hence 1: "of_nat (f t j) \<le> u x * 2 ^ j" and 2: "u x * 2 ^ j < of_nat (Suc (f t j))" by auto
- note 2
- also have "\<dots> \<le> of_nat (j*2^j)"
- using less by (auto simp: zero_le_mult_iff simp del: real_of_nat_mult)
- finally have bound_ux: "u x < of_nat j"
- by (cases "u x") (auto simp: zero_le_mult_iff power_le_zero_eq)
- show "f t j = f x j"
- proof (rule antisym)
- from 1 lower[OF bound_ux]
- show "f t j \<le> f x j" by (cases "u x") (auto split: split_if_asm)
- from upper[OF bound_ux] 2
- show "f x j \<le> f t j" by (cases "u x") (auto split: split_if_asm)
+ assume "\<not> extreal (real i) \<le> u x" "extreal (real (Suc i)) \<le> u x"
+ then show "natfloor (real (u x) * 2 ^ i) * 2 \<le> Suc i * 2 ^ Suc i"
+ by (cases "u x") auto
+ next
+ assume "\<not> extreal (real i) \<le> u x" "\<not> extreal (real (Suc i)) \<le> u x"
+ have "natfloor (real (u x) * 2 ^ i) * 2 = natfloor (real (u x) * 2 ^ i) * natfloor 2"
+ by simp
+ also have "\<dots> \<le> natfloor (real (u x) * 2 ^ i * 2)"
+ proof cases
+ assume "0 \<le> u x" then show ?thesis
+ by (intro le_mult_natfloor) (cases "u x", auto intro!: mult_nonneg_nonneg)
+ next
+ assume "\<not> 0 \<le> u x" then show ?thesis
+ by (cases "u x") (auto simp: natfloor_neg mult_nonpos_nonneg)
qed
- qed
- show ?thesis unfolding split_vimage using u by auto
+ also have "\<dots> = natfloor (real (u x) * 2 ^ Suc i)"
+ by (simp add: ac_simps)
+ finally show "natfloor (real (u x) * 2 ^ i) * 2 \<le> natfloor (real (u x) * 2 ^ Suc i)" .
+ qed simp
+ then show "?g i x \<le> ?g (Suc i) x"
+ by (auto simp: field_simps)
qed
next
- fix j t assume "?g j t = \<omega>" thus False by (auto simp: power_le_zero_eq)
- next
- fix t
- { fix i
- have "f t i * 2 \<le> f t (Suc i)"
- proof (rule fI)
- assume "of_nat (Suc i) \<le> u t"
- hence "of_nat i \<le> u t" by (cases "u t") auto
- thus "f t i * 2 \<le> Suc i * 2 ^ Suc i" unfolding f_eq[symmetric] by simp
- next
- fix k
- assume *: "u t * 2 ^ Suc i < of_nat (Suc k)"
- show "f t i * 2 \<le> k"
- proof (rule fI)
- assume "of_nat i \<le> u t"
- hence "of_nat (i * 2^Suc i) \<le> u t * 2^Suc i"
- by (cases "u t") (auto simp: zero_le_mult_iff power_le_zero_eq)
- also have "\<dots> < of_nat (Suc k)" using * by auto
- finally show "i * 2 ^ i * 2 \<le> k"
- by (auto simp del: real_of_nat_mult)
- next
- fix j assume "of_nat j \<le> u t * 2 ^ i"
- with * show "j * 2 \<le> k" by (cases "u t") (auto simp: zero_le_mult_iff power_le_zero_eq)
+ fix x show "(SUP i. ?g i x) = max 0 (u x)"
+ proof (rule extreal_SUPI)
+ fix i show "?g i x \<le> max 0 (u x)" unfolding max_def f_def
+ by (cases "u x") (auto simp: field_simps real_natfloor_le natfloor_neg
+ mult_nonpos_nonneg mult_nonneg_nonneg)
+ next
+ fix y assume *: "\<And>i. i \<in> UNIV \<Longrightarrow> ?g i x \<le> y"
+ have "\<And>i. 0 \<le> ?g i x" by (auto simp: divide_nonneg_pos)
+ from order_trans[OF this *] have "0 \<le> y" by simp
+ show "max 0 (u x) \<le> y"
+ proof (cases y)
+ case (real r)
+ with * have *: "\<And>i. f x i \<le> r * 2^i" by (auto simp: divide_le_eq)
+ from real_arch_lt[of r] * have "u x \<noteq> \<infinity>" by (auto simp: f_def) (metis less_le_not_le)
+ then have "\<exists>p. max 0 (u x) = extreal p \<and> 0 \<le> p" by (cases "u x") (auto simp: max_def)
+ then guess p .. note ux = this
+ obtain m :: nat where m: "p < real m" using real_arch_lt ..
+ have "p \<le> r"
+ proof (rule ccontr)
+ assume "\<not> p \<le> r"
+ with LIMSEQ_inverse_realpow_zero[unfolded LIMSEQ_iff, rule_format, of 2 "p - r"]
+ obtain N where "\<forall>n\<ge>N. r * 2^n < p * 2^n - 1" by (auto simp: inverse_eq_divide field_simps)
+ then have "r * 2^max N m < p * 2^max N m - 1" by simp
+ moreover
+ have "real (natfloor (p * 2 ^ max N m)) \<le> r * 2 ^ max N m"
+ using *[of "max N m"] m unfolding real_f using ux
+ by (cases "0 \<le> u x") (simp_all add: max_def mult_nonneg_nonneg split: split_if_asm)
+ then have "p * 2 ^ max N m - 1 < r * 2 ^ max N m"
+ by (metis real_natfloor_gt_diff_one less_le_trans)
+ ultimately show False by auto
qed
- qed
- thus "?g i t \<le> ?g (Suc i) t"
- by (auto simp: zero_le_mult_iff power_le_zero_eq divide_real_def[symmetric] field_simps) }
- hence mono: "mono (\<lambda>i. ?g i t)" unfolding mono_iff_le_Suc by auto
+ then show "max 0 (u x) \<le> y" using real ux by simp
+ qed (insert `0 \<le> y`, auto)
+ qed
+ qed (auto simp: divide_nonneg_pos)
+qed
- show "(SUP j. of_nat (f t j) / 2 ^ j) = u t"
- 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"
- by (cases "u t") (auto simp: power_le_zero_eq divide_real_def[symmetric] field_simps)
- next
- fix k assume "of_nat k \<le> u t * 2 ^ j"
- thus "of_nat k / 2 ^ j \<le> u t"
- by (cases "u t")
- (auto simp: power_le_zero_eq divide_real_def[symmetric] field_simps zero_le_mult_iff)
- qed
- next
- 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)
- show ?thesis
- proof (rule ccontr)
- assume "\<not> u t \<le> y"
- then obtain p where p: "y = Real p" "0 \<le> p" "p < r" using preal by (cases y) auto
- with LIMSEQ_inverse_realpow_zero[of 2, unfolded LIMSEQ_iff, rule_format, of "r - p"]
- obtain n where n: "\<And>N. n \<le> N \<Longrightarrow> inverse (2^N) < r - p" by auto
- let ?N = "max n (natfloor r + 1)"
- have "u t < of_nat ?N" "n \<le> ?N"
- using ge_natfloor_plus_one_imp_gt[of r n] preal
- using real_natfloor_add_one_gt
- by (auto simp: max_def real_of_nat_Suc)
- from lower[OF this(1)]
- have "r < (real (f t ?N) + 1) / 2 ^ ?N" unfolding less_divide_eq
- using preal by (auto simp add: power_le_zero_eq zero_le_mult_iff real_of_nat_Suc split: split_if_asm)
- hence "u t < of_nat (f t ?N) / 2 ^ ?N + 1 / 2 ^ ?N"
- using preal by (auto simp: field_simps divide_real_def[symmetric])
- with n[OF `n \<le> ?N`] p preal *[of ?N]
- show False
- by (cases "f t ?N") (auto simp: power_le_zero_eq split: split_if_asm)
- qed
- next
- case infinite
- { fix j have "f t j = j*2^j" using top[of j t] infinite by simp
- hence "of_nat j \<le> y" using *[of j]
- by (cases y) (auto simp: power_le_zero_eq zero_le_mult_iff field_simps) }
- note all_less_y = this
- show ?thesis unfolding infinite
- proof (rule ccontr)
- assume "\<not> \<omega> \<le> y" then obtain r where r: "y = Real r" "0 \<le> r" by (cases y) auto
- moreover obtain n ::nat where "r < real n" using ex_less_of_nat by (auto simp: real_eq_of_nat)
- with all_less_y[of n] r show False by auto
- qed
- qed
- qed
+lemma (in sigma_algebra) borel_measurable_implies_simple_function_sequence':
+ fixes u :: "'a \<Rightarrow> extreal"
+ assumes u: "u \<in> borel_measurable M"
+ obtains f where "\<And>i. simple_function M (f i)" "incseq f" "\<And>i. \<infinity> \<notin> range (f i)"
+ "\<And>x. (SUP i. f i x) = max 0 (u x)" "\<And>i x. 0 \<le> f i x"
+ using borel_measurable_implies_simple_function_sequence[OF u] by auto
+
+lemma (in sigma_algebra) simple_function_If_set:
+ assumes sf: "simple_function M f" "simple_function M g" and A: "A \<inter> space M \<in> sets M"
+ shows "simple_function M (\<lambda>x. if x \<in> A then f x else g x)" (is "simple_function M ?IF")
+proof -
+ def F \<equiv> "\<lambda>x. f -` {x} \<inter> space M" and G \<equiv> "\<lambda>x. g -` {x} \<inter> space M"
+ show ?thesis unfolding simple_function_def
+ proof safe
+ have "?IF ` space M \<subseteq> f ` space M \<union> g ` space M" by auto
+ from finite_subset[OF this] assms
+ show "finite (?IF ` space M)" unfolding simple_function_def by auto
+ next
+ fix x assume "x \<in> space M"
+ then have *: "?IF -` {?IF x} \<inter> space M = (if x \<in> A
+ then ((F (f x) \<inter> (A \<inter> space M)) \<union> (G (f x) - (G (f x) \<inter> (A \<inter> space M))))
+ else ((F (g x) \<inter> (A \<inter> space M)) \<union> (G (g x) - (G (g x) \<inter> (A \<inter> space M)))))"
+ using sets_into_space[OF A] by (auto split: split_if_asm simp: G_def F_def)
+ have [intro]: "\<And>x. F x \<in> sets M" "\<And>x. G x \<in> sets M"
+ unfolding F_def G_def using sf[THEN simple_functionD(2)] by auto
+ show "?IF -` {?IF x} \<inter> space M \<in> sets M" unfolding * using A by auto
qed
qed
-lemma (in sigma_algebra) borel_measurable_implies_simple_function_sequence':
- fixes u :: "'a \<Rightarrow> pextreal"
- assumes "u \<in> borel_measurable M"
- obtains (x) f where "f \<up> u" "\<And>i. simple_function M (f i)" "\<And>i. \<omega>\<notin>f i`space M"
+lemma (in sigma_algebra) simple_function_If:
+ assumes sf: "simple_function M f" "simple_function M g" and P: "{x\<in>space M. P x} \<in> sets M"
+ shows "simple_function M (\<lambda>x. if P x then f x else g x)"
proof -
- from borel_measurable_implies_simple_function_sequence[OF assms]
- obtain f where x: "\<And>i. simple_function M (f i)" "f \<up> u"
- and fin: "\<And>i. \<And>x. x\<in>space M \<Longrightarrow> f i x \<noteq> \<omega>" by auto
- { fix i from fin[of _ i] have "\<omega> \<notin> f i`space M" by fastsimp }
- with x show thesis by (auto intro!: that[of f])
+ have "{x\<in>space M. P x} = {x. P x} \<inter> space M" by auto
+ with simple_function_If_set[OF sf, of "{x. P x}"] P show ?thesis by simp
qed
-lemma (in sigma_algebra) simple_function_eq_borel_measurable:
- fixes f :: "'a \<Rightarrow> pextreal"
- shows "simple_function M f \<longleftrightarrow>
- finite (f`space M) \<and> f \<in> borel_measurable M"
- using simple_function_borel_measurable[of f]
- borel_measurable_simple_function[of f]
- by (fastsimp simp: simple_function_def)
-
lemma (in measure_space) simple_function_restricted:
- fixes f :: "'a \<Rightarrow> pextreal" assumes "A \<in> sets M"
+ fixes f :: "'a \<Rightarrow> extreal" assumes "A \<in> sets M"
shows "simple_function (restricted_space A) f \<longleftrightarrow> simple_function M (\<lambda>x. f x * indicator A x)"
(is "simple_function ?R f \<longleftrightarrow> simple_function M ?f")
proof -
interpret R: sigma_algebra ?R by (rule restricted_sigma_algebra[OF `A \<in> sets M`])
- have "finite (f`A) \<longleftrightarrow> finite (?f`space M)"
+ have f: "finite (f`A) \<longleftrightarrow> finite (?f`space M)"
proof cases
assume "A = space M"
then have "f`A = ?f`space M" by (fastsimp simp: image_iff)
@@ -456,7 +475,7 @@
using `A \<in> sets M` sets_into_space by (auto intro!: bexI[of _ x])
next
fix x
- assume "indicator A x \<noteq> (0::pextreal)"
+ assume "indicator A x \<noteq> (0::extreal)"
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
@@ -467,7 +486,8 @@
unfolding simple_function_eq_borel_measurable
R.simple_function_eq_borel_measurable
unfolding borel_measurable_restricted[OF `A \<in> sets M`]
- by auto
+ using assms(1)[THEN sets_into_space]
+ by (auto simp: indicator_def)
qed
lemma (in sigma_algebra) simple_function_subalgebra:
@@ -504,7 +524,7 @@
"integral\<^isup>S M f = (\<Sum>x \<in> f ` space M. x * measure M (f -` {x} \<inter> space M))"
syntax
- "_simple_integral" :: "'a \<Rightarrow> ('a \<Rightarrow> pextreal) \<Rightarrow> ('a, 'b) measure_space_scheme \<Rightarrow> pextreal" ("\<integral>\<^isup>S _. _ \<partial>_" [60,61] 110)
+ "_simple_integral" :: "'a \<Rightarrow> ('a \<Rightarrow> extreal) \<Rightarrow> ('a, 'b) measure_space_scheme \<Rightarrow> extreal" ("\<integral>\<^isup>S _. _ \<partial>_" [60,61] 110)
translations
"\<integral>\<^isup>S x. f \<partial>M" == "CONST integral\<^isup>S M (%x. f)"
@@ -540,7 +560,7 @@
qed
lemma (in measure_space) simple_function_partition:
- assumes "simple_function M f" and "simple_function M g"
+ assumes f: "simple_function M f" and g: "simple_function M g"
shows "integral\<^isup>S M f = (\<Sum>A\<in>(\<lambda>x. f -` {f x} \<inter> g -` {g x} \<inter> space M) ` space M. the_elem (f`A) * \<mu> A)"
(is "_ = setsum _ (?p ` space M)")
proof-
@@ -559,23 +579,16 @@
hence "finite (?p ` (A \<inter> space M))"
by (rule finite_subset) auto }
note this[intro, simp]
+ note sets = simple_function_measurable2[OF f g]
{ fix x assume "x \<in> space M"
have "\<Union>(?sub (f x)) = (f -` {f x} \<inter> space M)" by auto
- moreover {
- fix x y
- have *: "f -` {f x} \<inter> g -` {g x} \<inter> space M
- = (f -` {f x} \<inter> space M) \<inter> (g -` {g x} \<inter> space M)" by auto
- assume "x \<in> space M" "y \<in> space M"
- hence "f -` {f x} \<inter> g -` {g x} \<inter> space M \<in> sets M"
- using assms unfolding simple_function_def * by auto }
- ultimately
- have "\<mu> (f -` {f x} \<inter> space M) = setsum (\<mu>) (?sub (f x))"
- by (subst measure_finitely_additive) auto }
+ with sets have "\<mu> (f -` {f x} \<inter> space M) = setsum \<mu> (?sub (f x))"
+ by (subst measure_Union) auto }
hence "integral\<^isup>S M f = (\<Sum>(x,A)\<in>?SIGMA. x * \<mu> A)"
- unfolding simple_integral_def
- by (subst setsum_Sigma[symmetric],
- auto intro!: setsum_cong simp: setsum_right_distrib[symmetric])
+ unfolding simple_integral_def using f sets
+ by (subst setsum_Sigma[symmetric])
+ (auto intro!: setsum_cong setsum_extreal_right_distrib)
also have "\<dots> = (\<Sum>A\<in>?p ` space M. the_elem (f`A) * \<mu> A)"
proof -
have [simp]: "\<And>x. x \<in> space M \<Longrightarrow> f ` ?p x = {f x}" by (auto intro!: imageI)
@@ -595,7 +608,7 @@
qed
lemma (in measure_space) simple_integral_add[simp]:
- assumes "simple_function M f" and "simple_function M g"
+ assumes f: "simple_function M f" and "\<And>x. 0 \<le> f x" and g: "simple_function M g" and "\<And>x. 0 \<le> g x"
shows "(\<integral>\<^isup>Sx. f x + g x \<partial>M) = integral\<^isup>S M f + integral\<^isup>S M g"
proof -
{ fix x let ?S = "g -` {g x} \<inter> f -` {f x} \<inter> space M"
@@ -603,63 +616,43 @@
hence "(\<lambda>a. f a + g a) ` ?S = {f x + g x}" "f ` ?S = {f x}" "g ` ?S = {g x}"
"(\<lambda>x. (f x, g x)) -` {(f x, g x)} \<inter> space M = ?S"
by auto }
- thus ?thesis
+ with assms show ?thesis
unfolding
- simple_function_partition[OF simple_function_add[OF assms] simple_function_Pair[OF assms]]
- simple_function_partition[OF `simple_function M f` `simple_function M g`]
- simple_function_partition[OF `simple_function M g` `simple_function M f`]
- apply (subst (3) Int_commute)
- by (auto simp add: field_simps setsum_addf[symmetric] intro!: setsum_cong)
+ simple_function_partition[OF simple_function_add[OF f g] simple_function_Pair[OF f g]]
+ simple_function_partition[OF f g]
+ simple_function_partition[OF g f]
+ by (subst (3) Int_commute)
+ (auto simp add: extreal_left_distrib setsum_addf[symmetric] intro!: setsum_cong)
qed
lemma (in measure_space) simple_integral_setsum[simp]:
+ assumes "\<And>i x. i \<in> P \<Longrightarrow> 0 \<le> f i x"
assumes "\<And>i. i \<in> P \<Longrightarrow> simple_function M (f i)"
shows "(\<integral>\<^isup>Sx. (\<Sum>i\<in>P. f i x) \<partial>M) = (\<Sum>i\<in>P. integral\<^isup>S M (f i))"
proof cases
assume "finite P"
from this assms show ?thesis
- by induct (auto simp: simple_function_setsum simple_integral_add)
+ by induct (auto simp: simple_function_setsum simple_integral_add setsum_nonneg)
qed auto
lemma (in measure_space) simple_integral_mult[simp]:
- assumes "simple_function M f"
+ assumes f: "simple_function M f" "\<And>x. 0 \<le> f x"
shows "(\<integral>\<^isup>Sx. c * f x \<partial>M) = c * integral\<^isup>S M f"
proof -
- note mult = simple_function_mult[OF simple_function_const[of c] assms]
+ note mult = simple_function_mult[OF simple_function_const[of c] f(1)]
{ fix x let ?S = "f -` {f x} \<inter> (\<lambda>x. c * f x) -` {c * f x} \<inter> space M"
assume "x \<in> space M"
hence "(\<lambda>x. c * f x) ` ?S = {c * f x}" "f ` ?S = {f x}"
by auto }
- thus ?thesis
- unfolding simple_function_partition[OF mult assms]
- simple_function_partition[OF assms mult]
- by (auto simp: setsum_right_distrib field_simps intro!: setsum_cong)
-qed
-
-lemma (in sigma_algebra) simple_function_If:
- assumes sf: "simple_function M f" "simple_function M g" and A: "A \<in> sets M"
- shows "simple_function M (\<lambda>x. if x \<in> A then f x else g x)" (is "simple_function M ?IF")
-proof -
- def F \<equiv> "\<lambda>x. f -` {x} \<inter> space M" and G \<equiv> "\<lambda>x. g -` {x} \<inter> space M"
- show ?thesis unfolding simple_function_def
- proof safe
- have "?IF ` space M \<subseteq> f ` space M \<union> g ` space M" by auto
- from finite_subset[OF this] assms
- show "finite (?IF ` space M)" unfolding simple_function_def by auto
- next
- fix x assume "x \<in> space M"
- then have *: "?IF -` {?IF x} \<inter> space M = (if x \<in> A
- then ((F (f x) \<inter> A) \<union> (G (f x) - (G (f x) \<inter> A)))
- else ((F (g x) \<inter> A) \<union> (G (g x) - (G (g x) \<inter> A))))"
- using sets_into_space[OF A] by (auto split: split_if_asm simp: G_def F_def)
- have [intro]: "\<And>x. F x \<in> sets M" "\<And>x. G x \<in> sets M"
- unfolding F_def G_def using sf[THEN simple_functionD(2)] by auto
- show "?IF -` {?IF x} \<inter> space M \<in> sets M" unfolding * using A by auto
- qed
+ with assms show ?thesis
+ unfolding simple_function_partition[OF mult f(1)]
+ simple_function_partition[OF f(1) mult]
+ by (subst setsum_extreal_right_distrib)
+ (auto intro!: extreal_0_le_mult setsum_cong simp: mult_assoc)
qed
lemma (in measure_space) simple_integral_mono_AE:
- assumes "simple_function M f" and "simple_function M g"
+ assumes f: "simple_function M f" and g: "simple_function M g"
and mono: "AE x. f x \<le> g x"
shows "integral\<^isup>S M f \<le> integral\<^isup>S M g"
proof -
@@ -668,14 +661,16 @@
"\<And>x. f -` {f x} \<inter> g -` {g x} \<inter> space M = ?S x" by auto
show ?thesis
unfolding *
- simple_function_partition[OF `simple_function M f` `simple_function M g`]
- simple_function_partition[OF `simple_function M g` `simple_function M f`]
+ simple_function_partition[OF f g]
+ simple_function_partition[OF g f]
proof (safe intro!: setsum_mono)
fix x assume "x \<in> space M"
then have *: "f ` ?S x = {f x}" "g ` ?S x = {g x}" by auto
show "the_elem (f`?S x) * \<mu> (?S x) \<le> the_elem (g`?S x) * \<mu> (?S x)"
proof (cases "f x \<le> g x")
- case True then show ?thesis using * by (auto intro!: mult_right_mono)
+ case True then show ?thesis
+ using * assms(1,2)[THEN simple_functionD(2)]
+ by (auto intro!: extreal_mult_right_mono)
next
case False
obtain N where N: "{x\<in>space M. \<not> f x \<le> g x} \<subseteq> N" "N \<in> sets M" "\<mu> N = 0"
@@ -685,7 +680,10 @@
by (rule_tac Int) (auto intro!: simple_functionD)
ultimately have "\<mu> (?S x) \<le> \<mu> N"
using `N \<in> sets M` by (auto intro!: measure_mono)
- then show ?thesis using `\<mu> N = 0` by auto
+ moreover have "0 \<le> \<mu> (?S x)"
+ using assms(1,2)[THEN simple_functionD(2)] by auto
+ ultimately have "\<mu> (?S x) = 0" using `\<mu> N = 0` by auto
+ then show ?thesis by simp
qed
qed
qed
@@ -697,7 +695,8 @@
using assms by (intro simple_integral_mono_AE) auto
lemma (in measure_space) simple_integral_cong_AE:
- assumes "simple_function M f" "simple_function M g" and "AE x. f x = g x"
+ assumes "simple_function M f" and "simple_function M g"
+ and "AE x. f x = g x"
shows "integral\<^isup>S M f = integral\<^isup>S M g"
using assms by (auto simp: eq_iff intro!: simple_integral_mono_AE)
@@ -765,7 +764,7 @@
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::pextreal}" by auto
+ assume "space M \<noteq> {}" hence "(\<lambda>x. 1) ` space M = {1::extreal}" by auto
thus ?thesis
using simple_integral_indicator[OF assms simple_function_const[of 1]]
using sets_into_space[OF assms]
@@ -773,13 +772,13 @@
qed
lemma (in measure_space) simple_integral_null_set:
- assumes "simple_function M u" "N \<in> null_sets"
+ assumes "simple_function M u" "\<And>x. 0 \<le> u x" and "N \<in> null_sets"
shows "(\<integral>\<^isup>Sx. u x * indicator N x \<partial>M) = 0"
proof -
- have "AE x. indicator N x = (0 :: pextreal)"
+ have "AE x. indicator N x = (0 :: extreal)"
using `N \<in> null_sets` by (auto simp: indicator_def intro!: AE_I[of _ N])
then have "(\<integral>\<^isup>Sx. u x * indicator N x \<partial>M) = (\<integral>\<^isup>Sx. 0 \<partial>M)"
- using assms by (intro simple_integral_cong_AE) auto
+ using assms apply (intro simple_integral_cong_AE) by auto
then show ?thesis by simp
qed
@@ -813,7 +812,7 @@
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 pextreal_mult_cancel_left by auto
+ unfolding extreal_mult_cancel_left by auto
qed
lemma (in measure_space) simple_integral_subalgebra:
@@ -821,10 +820,6 @@
shows "integral\<^isup>S N = integral\<^isup>S M"
unfolding simple_integral_def_raw by simp
-lemma measure_preservingD:
- "T \<in> measure_preserving A B \<Longrightarrow> X \<in> sets B \<Longrightarrow> measure A (T -` X \<inter> space A) = measure B X"
- unfolding measure_preserving_def by auto
-
lemma (in measure_space) simple_integral_vimage:
assumes T: "sigma_algebra M'" "T \<in> measure_preserving M M'"
and f: "simple_function M' f"
@@ -853,196 +848,164 @@
qed
qed
+lemma (in measure_space) simple_integral_cmult_indicator:
+ assumes A: "A \<in> sets M"
+ shows "(\<integral>\<^isup>Sx. c * indicator A x \<partial>M) = c * \<mu> A"
+ using simple_integral_mult[OF simple_function_indicator[OF A]]
+ unfolding simple_integral_indicator_only[OF A] by simp
+
+lemma (in measure_space) simple_integral_positive:
+ assumes f: "simple_function M f" and ae: "AE x. 0 \<le> f x"
+ shows "0 \<le> integral\<^isup>S M f"
+proof -
+ have "integral\<^isup>S M (\<lambda>x. 0) \<le> integral\<^isup>S M f"
+ using simple_integral_mono_AE[OF _ f ae] by auto
+ then show ?thesis by simp
+qed
+
section "Continuous positive integration"
definition positive_integral_def:
- "integral\<^isup>P M f = (SUP g : {g. simple_function M g \<and> g \<le> f}. integral\<^isup>S M g)"
+ "integral\<^isup>P M f = (SUP g : {g. simple_function M g \<and> g \<le> max 0 \<circ> f}. integral\<^isup>S M g)"
syntax
- "_positive_integral" :: "'a \<Rightarrow> ('a \<Rightarrow> pextreal) \<Rightarrow> ('a, 'b) measure_space_scheme \<Rightarrow> pextreal" ("\<integral>\<^isup>+ _. _ \<partial>_" [60,61] 110)
+ "_positive_integral" :: "'a \<Rightarrow> ('a \<Rightarrow> extreal) \<Rightarrow> ('a, 'b) measure_space_scheme \<Rightarrow> extreal" ("\<integral>\<^isup>+ _. _ \<partial>_" [60,61] 110)
translations
"\<integral>\<^isup>+ x. f \<partial>M" == "CONST integral\<^isup>P M (%x. f)"
-lemma (in measure_space) positive_integral_alt: "integral\<^isup>P M f =
- (SUP g : {g. simple_function M g \<and> g \<le> f \<and> \<omega> \<notin> g`space M}. integral\<^isup>S M g)"
- (is "_ = ?alt")
-proof (rule antisym SUP_leI)
- show "integral\<^isup>P M f \<le> ?alt" unfolding positive_integral_def
- proof (safe intro!: SUP_leI)
- fix g assume g: "simple_function M g" "g \<le> f"
- let ?G = "g -` {\<omega>} \<inter> space M"
- show "integral\<^isup>S M g \<le>
- (SUP h : {i. simple_function M i \<and> i \<le> f \<and> \<omega> \<notin> i ` space M}. integral\<^isup>S M h)"
- (is "integral\<^isup>S M g \<le> SUPR ?A _")
- proof cases
- let ?g = "\<lambda>x. indicator (space M - ?G) x * g x"
- have g': "simple_function M ?g"
- using g by (auto intro: simple_functionD)
- moreover
- assume "\<mu> ?G = 0"
- then have "AE x. g x = ?g x" using g
- by (intro AE_I[where N="?G"])
- (auto intro: simple_functionD simp: indicator_def)
- with g(1) g' have "integral\<^isup>S M g = integral\<^isup>S M ?g"
- by (rule simple_integral_cong_AE)
- moreover have "?g \<le> g" by (auto simp: le_fun_def indicator_def)
- from this `g \<le> f` have "?g \<le> f" by (rule order_trans)
- moreover have "\<omega> \<notin> ?g ` space M"
- by (auto simp: indicator_def split: split_if_asm)
- ultimately show ?thesis by (auto intro!: le_SUPI)
- next
- assume "\<mu> ?G \<noteq> 0"
- then have "?G \<noteq> {}" by auto
- then have "\<omega> \<in> g`space M" by force
- then have "space M \<noteq> {}" by auto
- have "SUPR ?A (integral\<^isup>S M) = \<omega>"
- proof (intro SUP_\<omega>[THEN iffD2] allI impI)
- fix x assume "x < \<omega>"
- then guess n unfolding less_\<omega>_Ex_of_nat .. note n = this
- then have "0 < n" by (intro neq0_conv[THEN iffD1] notI) simp
- let ?g = "\<lambda>x. (of_nat n / (if \<mu> ?G = \<omega> then 1 else \<mu> ?G)) * indicator ?G x"
- show "\<exists>i\<in>?A. x < integral\<^isup>S M i"
- proof (intro bexI impI CollectI conjI)
- show "simple_function M ?g" using g
- by (auto intro!: simple_functionD simple_function_add)
- have "?g \<le> g" by (auto simp: le_fun_def indicator_def)
- from this g(2) show "?g \<le> f" by (rule order_trans)
- show "\<omega> \<notin> ?g ` space M"
- 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: pextreal_noteq_omega_Ex field_simps)
- also have "\<dots> = integral\<^isup>S M ?g" using g `space M \<noteq> {}`
- by (subst simple_integral_indicator)
- (auto simp: image_constant ac_simps dest: simple_functionD)
- finally show "x < integral\<^isup>S M ?g" .
- qed
- qed
- then show ?thesis by simp
- qed
- qed
-qed (auto intro!: SUP_subset simp: positive_integral_def)
-
lemma (in measure_space) positive_integral_cong_measure:
assumes "\<And>A. A \<in> sets M \<Longrightarrow> measure N A = \<mu> A" "sets N = sets M" "space N = space M"
shows "integral\<^isup>P N f = integral\<^isup>P M f"
-proof -
- interpret v: measure_space N
- by (rule measure_space_cong) fact+
- with assms show ?thesis
- unfolding positive_integral_def SUPR_def
- by (auto intro!: arg_cong[where f=Sup] image_cong
- simp: simple_integral_cong_measure[OF assms]
- simple_function_cong_algebra[OF assms(2,3)])
-qed
+ unfolding positive_integral_def
+ unfolding simple_function_cong_algebra[OF assms(2,3), symmetric]
+ using AE_cong_measure[OF assms]
+ using simple_integral_cong_measure[OF assms]
+ by (auto intro!: SUP_cong)
+
+lemma (in measure_space) positive_integral_positive:
+ "0 \<le> integral\<^isup>P M f"
+ by (auto intro!: le_SUPI2[of "\<lambda>x. 0"] simp: positive_integral_def le_fun_def)
-lemma (in measure_space) positive_integral_alt1:
- "integral\<^isup>P M f =
- (SUP g : {g. simple_function M g \<and> (\<forall>x\<in>space M. g x \<le> f x \<and> g x \<noteq> \<omega>)}. integral\<^isup>S M g)"
- unfolding positive_integral_alt SUPR_def
-proof (safe intro!: arg_cong[where f=Sup])
- fix g let ?g = "\<lambda>x. if x \<in> space M then g x else f x"
- assume "simple_function M g" "\<forall>x\<in>space M. g x \<le> f x \<and> g x \<noteq> \<omega>"
- hence "?g \<le> f" "simple_function M ?g" "integral\<^isup>S M g = integral\<^isup>S M ?g"
- "\<omega> \<notin> g`space M"
- unfolding le_fun_def by (auto cong: simple_function_cong simple_integral_cong)
- thus "integral\<^isup>S M g \<in> integral\<^isup>S M ` {g. simple_function M g \<and> g \<le> f \<and> \<omega> \<notin> g`space M}"
- by auto
-next
- fix g assume "simple_function M g" "g \<le> f" "\<omega> \<notin> g`space M"
- hence "simple_function M g" "\<forall>x\<in>space M. g x \<le> f x \<and> g x \<noteq> \<omega>"
- by (auto simp add: le_fun_def image_iff)
- thus "integral\<^isup>S M g \<in> integral\<^isup>S M ` {g. simple_function M g \<and> (\<forall>x\<in>space M. g x \<le> f x \<and> g x \<noteq> \<omega>)}"
- by auto
-qed
+lemma (in measure_space) positive_integral_def_finite:
+ "integral\<^isup>P M f = (SUP g : {g. simple_function M g \<and> g \<le> max 0 \<circ> f \<and> range g \<subseteq> {0 ..< \<infinity>}}. integral\<^isup>S M g)"
+ (is "_ = SUPR ?A ?f")
+ unfolding positive_integral_def
+proof (safe intro!: antisym SUP_leI)
+ fix g assume g: "simple_function M g" "g \<le> max 0 \<circ> f"
+ let ?G = "{x \<in> space M. \<not> g x \<noteq> \<infinity>}"
+ note gM = g(1)[THEN borel_measurable_simple_function]
+ have \<mu>G_pos: "0 \<le> \<mu> ?G" using gM by auto
+ let "?g y x" = "if g x = \<infinity> then y else max 0 (g x)"
+ from g gM have g_in_A: "\<And>y. 0 \<le> y \<Longrightarrow> y \<noteq> \<infinity> \<Longrightarrow> ?g y \<in> ?A"
+ apply (safe intro!: simple_function_max simple_function_If)
+ apply (force simp: max_def le_fun_def split: split_if_asm)+
+ done
+ show "integral\<^isup>S M g \<le> SUPR ?A ?f"
+ proof cases
+ have g0: "?g 0 \<in> ?A" by (intro g_in_A) auto
+ assume "\<mu> ?G = 0"
+ with gM have "AE x. x \<notin> ?G" by (simp add: AE_iff_null_set)
+ with gM g show ?thesis
+ by (intro le_SUPI2[OF g0] simple_integral_mono_AE)
+ (auto simp: max_def intro!: simple_function_If)
+ next
+ assume \<mu>G: "\<mu> ?G \<noteq> 0"
+ have "SUPR ?A (integral\<^isup>S M) = \<infinity>"
+ proof (intro SUP_PInfty)
+ fix n :: nat
+ let ?y = "extreal (real n) / (if \<mu> ?G = \<infinity> then 1 else \<mu> ?G)"
+ have "0 \<le> ?y" "?y \<noteq> \<infinity>" using \<mu>G \<mu>G_pos by (auto simp: extreal_divide_eq)
+ then have "?g ?y \<in> ?A" by (rule g_in_A)
+ have "real n \<le> ?y * \<mu> ?G"
+ using \<mu>G \<mu>G_pos by (cases "\<mu> ?G") (auto simp: field_simps)
+ also have "\<dots> = (\<integral>\<^isup>Sx. ?y * indicator ?G x \<partial>M)"
+ using `0 \<le> ?y` `?g ?y \<in> ?A` gM
+ by (subst simple_integral_cmult_indicator) auto
+ also have "\<dots> \<le> integral\<^isup>S M (?g ?y)" using `?g ?y \<in> ?A` gM
+ by (intro simple_integral_mono) auto
+ finally show "\<exists>i\<in>?A. real n \<le> integral\<^isup>S M i"
+ using `?g ?y \<in> ?A` by blast
+ qed
+ then show ?thesis by simp
+ qed
+qed (auto intro: le_SUPI)
-lemma (in measure_space) positive_integral_cong:
- assumes "\<And>x. x \<in> space M \<Longrightarrow> f x = g x"
- shows "integral\<^isup>P M f = integral\<^isup>P M g"
-proof -
- have "\<And>h. (\<forall>x\<in>space M. h x \<le> f x \<and> h x \<noteq> \<omega>) = (\<forall>x\<in>space M. h x \<le> g x \<and> h x \<noteq> \<omega>)"
- using assms by auto
- thus ?thesis unfolding positive_integral_alt1 by auto
+lemma (in measure_space) positive_integral_mono_AE:
+ assumes ae: "AE x. u x \<le> v x" shows "integral\<^isup>P M u \<le> integral\<^isup>P M v"
+ unfolding positive_integral_def
+proof (safe intro!: SUP_mono)
+ fix n assume n: "simple_function M n" "n \<le> max 0 \<circ> u"
+ from ae[THEN AE_E] guess N . note N = this
+ then have ae_N: "AE x. x \<notin> N" by (auto intro: AE_not_in)
+ let "?n x" = "n x * indicator (space M - N) x"
+ have "AE x. n x \<le> ?n x" "simple_function M ?n"
+ using n N ae_N by auto
+ moreover
+ { fix x have "?n x \<le> max 0 (v x)"
+ proof cases
+ assume x: "x \<in> space M - N"
+ with N have "u x \<le> v x" by auto
+ with n(2)[THEN le_funD, of x] x show ?thesis
+ by (auto simp: max_def split: split_if_asm)
+ qed simp }
+ then have "?n \<le> max 0 \<circ> v" by (auto simp: le_funI)
+ moreover have "integral\<^isup>S M n \<le> integral\<^isup>S M ?n"
+ using ae_N N n by (auto intro!: simple_integral_mono_AE)
+ ultimately show "\<exists>m\<in>{g. simple_function M g \<and> g \<le> max 0 \<circ> v}. integral\<^isup>S M n \<le> integral\<^isup>S M m"
+ by force
qed
-lemma (in measure_space) positive_integral_eq_simple_integral:
- assumes "simple_function M f"
- shows "integral\<^isup>P M f = integral\<^isup>S M f"
- unfolding positive_integral_def
-proof (safe intro!: pextreal_SUPI)
- fix g assume "simple_function M g" "g \<le> f"
- with assms show "integral\<^isup>S M g \<le> integral\<^isup>S M f"
- by (auto intro!: simple_integral_mono simp: le_fun_def)
-next
- fix y assume "\<forall>x. x\<in>{g. simple_function M g \<and> g \<le> f} \<longrightarrow> integral\<^isup>S M x \<le> y"
- with assms show "integral\<^isup>S M f \<le> y" by auto
-qed
-
-lemma (in measure_space) positive_integral_mono_AE:
- assumes ae: "AE x. u x \<le> v x"
- shows "integral\<^isup>P M u \<le> integral\<^isup>P M v"
- unfolding positive_integral_alt1
-proof (safe intro!: SUPR_mono)
- fix a assume a: "simple_function M a" and mono: "\<forall>x\<in>space M. a x \<le> u x \<and> a x \<noteq> \<omega>"
- from ae obtain N where N: "{x\<in>space M. \<not> u x \<le> v x} \<subseteq> N" "N \<in> sets M" "\<mu> N = 0"
- by (auto elim!: AE_E)
- have "simple_function M (\<lambda>x. a x * indicator (space M - N) x)"
- using `N \<in> sets M` a by auto
- with a show "\<exists>b\<in>{g. simple_function M g \<and> (\<forall>x\<in>space M. g x \<le> v x \<and> g x \<noteq> \<omega>)}.
- integral\<^isup>S M a \<le> integral\<^isup>S M b"
- proof (safe intro!: bexI[of _ "\<lambda>x. a x * indicator (space M - N) x"]
- simple_integral_mono_AE)
- show "AE x. a x \<le> a x * indicator (space M - N) x"
- proof (rule AE_I, rule subset_refl)
- have *: "{x \<in> space M. \<not> a x \<le> a x * indicator (space M - N) x} =
- N \<inter> {x \<in> space M. a x \<noteq> 0}" (is "?N = _")
- using `N \<in> sets M`[THEN sets_into_space] by (auto simp: indicator_def)
- then show "?N \<in> sets M"
- using `N \<in> sets M` `simple_function M a`[THEN borel_measurable_simple_function]
- by (auto intro!: measure_mono Int)
- then have "\<mu> ?N \<le> \<mu> N"
- unfolding * using `N \<in> sets M` by (auto intro!: measure_mono)
- then show "\<mu> ?N = 0" using `\<mu> N = 0` by auto
- qed
- next
- fix x assume "x \<in> space M"
- show "a x * indicator (space M - N) x \<le> v x"
- proof (cases "x \<in> N")
- case True then show ?thesis by simp
- next
- case False
- with N mono have "a x \<le> u x" "u x \<le> v x" using `x \<in> space M` by auto
- with False `x \<in> space M` show "a x * indicator (space M - N) x \<le> v x" by auto
- qed
- assume "a x * indicator (space M - N) x = \<omega>"
- with mono `x \<in> space M` show False
- by (simp split: split_if_asm add: indicator_def)
- qed
-qed
+lemma (in measure_space) positive_integral_mono:
+ "(\<And>x. x \<in> space M \<Longrightarrow> u x \<le> v x) \<Longrightarrow> integral\<^isup>P M u \<le> integral\<^isup>P M v"
+ by (auto intro: positive_integral_mono_AE)
lemma (in measure_space) positive_integral_cong_AE:
"AE x. u x = v x \<Longrightarrow> integral\<^isup>P M u = integral\<^isup>P M v"
by (auto simp: eq_iff intro!: positive_integral_mono_AE)
-lemma (in measure_space) positive_integral_mono:
- "(\<And>x. x \<in> space M \<Longrightarrow> u x \<le> v x) \<Longrightarrow> integral\<^isup>P M u \<le> integral\<^isup>P M v"
- by (auto intro: positive_integral_mono_AE)
+lemma (in measure_space) positive_integral_cong:
+ "(\<And>x. x \<in> space M \<Longrightarrow> u x = v x) \<Longrightarrow> integral\<^isup>P M u = integral\<^isup>P M v"
+ by (auto intro: positive_integral_cong_AE)
-lemma image_set_cong:
- assumes A: "\<And>x. x \<in> A \<Longrightarrow> \<exists>y\<in>B. f x = g y"
- assumes B: "\<And>y. y \<in> B \<Longrightarrow> \<exists>x\<in>A. g y = f x"
- shows "f ` A = g ` B"
- using assms by blast
+lemma (in measure_space) positive_integral_eq_simple_integral:
+ assumes f: "simple_function M f" "\<And>x. 0 \<le> f x" shows "integral\<^isup>P M f = integral\<^isup>S M f"
+proof -
+ let "?f x" = "f x * indicator (space M) x"
+ have f': "simple_function M ?f" using f by auto
+ with f(2) have [simp]: "max 0 \<circ> ?f = ?f"
+ by (auto simp: fun_eq_iff max_def split: split_indicator)
+ have "integral\<^isup>P M ?f \<le> integral\<^isup>S M ?f" using f'
+ by (force intro!: SUP_leI simple_integral_mono simp: le_fun_def positive_integral_def)
+ moreover have "integral\<^isup>S M ?f \<le> integral\<^isup>P M ?f"
+ unfolding positive_integral_def
+ using f' by (auto intro!: le_SUPI)
+ ultimately show ?thesis
+ by (simp cong: positive_integral_cong simple_integral_cong)
+qed
+
+lemma (in measure_space) positive_integral_eq_simple_integral_AE:
+ assumes f: "simple_function M f" "AE x. 0 \<le> f x" shows "integral\<^isup>P M f = integral\<^isup>S M f"
+proof -
+ have "AE x. f x = max 0 (f x)" using f by (auto split: split_max)
+ with f have "integral\<^isup>P M f = integral\<^isup>S M (\<lambda>x. max 0 (f x))"
+ by (simp cong: positive_integral_cong_AE simple_integral_cong_AE
+ add: positive_integral_eq_simple_integral)
+ with assms show ?thesis
+ by (auto intro!: simple_integral_cong_AE split: split_max)
+qed
lemma (in measure_space) positive_integral_SUP_approx:
- assumes "f \<up> s"
- and f: "\<And>i. f i \<in> borel_measurable M"
- and "simple_function M u"
- and le: "u \<le> s" and real: "\<omega> \<notin> u`space M"
+ assumes f: "incseq f" "\<And>i. f i \<in> borel_measurable M" "\<And>i x. 0 \<le> f i x"
+ and u: "simple_function M u" "u \<le> (SUP i. f i)" "u`space M \<subseteq> {0..<\<infinity>}"
shows "integral\<^isup>S M u \<le> (SUP i. integral\<^isup>P M (f i))" (is "_ \<le> ?S")
-proof (rule pextreal_le_mult_one_interval)
- fix a :: pextreal assume "0 < a" "a < 1"
+proof (rule extreal_le_mult_one_interval)
+ have "0 \<le> (SUP i. integral\<^isup>P M (f i))"
+ using f(3) by (auto intro!: le_SUPI2 positive_integral_positive)
+ then show "(SUP i. integral\<^isup>P M (f i)) \<noteq> -\<infinity>" by auto
+ have u_range: "\<And>x. x \<in> space M \<Longrightarrow> 0 \<le> u x \<and> u x \<noteq> \<infinity>"
+ using u(3) by auto
+ fix a :: extreal 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"
@@ -1054,203 +1017,269 @@
proof safe
fix i x assume "a * u x \<le> f i x"
also have "\<dots> \<le> f (Suc i) x"
- using `f \<up> s` unfolding isoton_def le_fun_def by auto
+ using `incseq f`[THEN incseq_SucD] unfolding le_fun_def by auto
finally show "a * u x \<le> f (Suc i) x" .
qed }
note B_mono = this
- have u: "\<And>x. x \<in> space M \<Longrightarrow> u -` {u x} \<inter> space M \<in> sets M"
- using `simple_function M u` by (auto simp add: simple_function_def)
+ note B_u = Int[OF u(1)[THEN simple_functionD(2)] B]
- have "\<And>i. (\<lambda>n. (u -` {i} \<inter> space M) \<inter> ?B n) \<up> (u -` {i} \<inter> space M)" using B_mono unfolding isoton_def
- proof safe
- fix x i assume "x \<in> space M"
- show "x \<in> (\<Union>i. (u -` {u x} \<inter> space M) \<inter> ?B i)"
- proof cases
- assume "u x = 0" thus ?thesis using `x \<in> space M` by simp
- next
- assume "u x \<noteq> 0"
- with `a < 1` real `x \<in> space M`
- 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
- by (auto simp add: less_Sup_iff)
- hence "a * u x \<le> f i x" by auto
- thus ?thesis using `x \<in> space M` by auto
+ let "?B' i n" = "(u -` {i} \<inter> space M) \<inter> ?B n"
+ have measure_conv: "\<And>i. \<mu> (u -` {i} \<inter> space M) = (SUP n. \<mu> (?B' i n))"
+ proof -
+ fix i
+ have 1: "range (?B' i) \<subseteq> sets M" using B_u by auto
+ have 2: "incseq (?B' i)" using B_mono by (auto intro!: incseq_SucI)
+ have "(\<Union>n. ?B' i n) = u -` {i} \<inter> space M"
+ proof safe
+ fix x i assume x: "x \<in> space M"
+ show "x \<in> (\<Union>i. ?B' (u x) i)"
+ proof cases
+ assume "u x = 0" thus ?thesis using `x \<in> space M` f(3) by simp
+ next
+ assume "u x \<noteq> 0"
+ with `a < 1` u_range[OF `x \<in> space M`]
+ have "a * u x < 1 * u x"
+ by (intro extreal_mult_strict_right_mono) (auto simp: image_iff)
+ also have "\<dots> \<le> (SUP i. f i x)" using u(2) by (auto simp: le_fun_def SUPR_apply)
+ finally obtain i where "a * u x < f i x" unfolding SUPR_def
+ by (auto simp add: less_Sup_iff)
+ hence "a * u x \<le> f i x" by auto
+ thus ?thesis using `x \<in> space M` by auto
+ qed
qed
- qed auto
- note measure_conv = measure_up[OF Int[OF u B] this]
+ then show "?thesis i" using continuity_from_below[OF 1 2] by simp
+ qed
have "integral\<^isup>S M u = (SUP i. integral\<^isup>S M (?uB i))"
unfolding simple_integral_indicator[OF B `simple_function M u`]
- proof (subst SUPR_pextreal_setsum, safe)
+ proof (subst SUPR_extreal_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})"
- using B_mono Int[OF u[OF `x \<in> space M`] B] by (auto intro!: measure_mono)
- thus "u x * \<mu> (u -` {u x} \<inter> space M \<inter> ?B n)
- \<le> u x * \<mu> (u -` {u x} \<inter> space M \<inter> ?B (Suc n))"
- by (auto intro: mult_left_mono)
+ with u_range show "incseq (\<lambda>i. u x * \<mu> (?B' (u x) i))" "\<And>i. 0 \<le> u x * \<mu> (?B' (u x) i)"
+ using B_mono B_u by (auto intro!: measure_mono extreal_mult_left_mono incseq_SucI simp: extreal_zero_le_0_iff)
next
- show "integral\<^isup>S M 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: pextreal_SUP_cmult)
+ show "integral\<^isup>S M u = (\<Sum>i\<in>u ` space M. SUP n. i * \<mu> (?B' i n))"
+ using measure_conv u_range B_u unfolding simple_integral_def
+ by (auto intro!: setsum_cong SUPR_extreal_cmult[symmetric])
qed
moreover
have "a * (SUP i. integral\<^isup>S M (?uB i)) \<le> ?S"
- unfolding pextreal_SUP_cmult[symmetric]
+ apply (subst SUPR_extreal_cmult[symmetric])
proof (safe intro!: SUP_mono bexI)
fix i
have "a * integral\<^isup>S M (?uB i) = (\<integral>\<^isup>Sx. a * ?uB i x \<partial>M)"
- using B `simple_function M u`
- by (subst simple_integral_mult[symmetric]) (auto simp: field_simps)
+ using B `simple_function M u` u_range
+ by (subst simple_integral_mult) (auto split: split_indicator)
also have "\<dots> \<le> integral\<^isup>P M (f i)"
proof -
- have "\<And>x. a * ?uB i x \<le> f i x" unfolding indicator_def by auto
- hence *: "simple_function M (\<lambda>x. a * ?uB i x)" using B assms(3)
- by (auto intro!: simple_integral_mono)
- show ?thesis unfolding positive_integral_eq_simple_integral[OF *, symmetric]
- by (auto intro!: positive_integral_mono simp: indicator_def)
+ have *: "simple_function M (\<lambda>x. a * ?uB i x)" using B `0 < a` u(1) by auto
+ show ?thesis using f(3) * u_range `0 < a`
+ by (subst positive_integral_eq_simple_integral[symmetric])
+ (auto intro!: positive_integral_mono split: split_indicator)
qed
finally show "a * integral\<^isup>S M (?uB i) \<le> integral\<^isup>P M (f i)"
by auto
- qed simp
+ next
+ fix i show "0 \<le> \<integral>\<^isup>S x. ?uB i x \<partial>M" using B `0 < a` u(1) u_range
+ by (intro simple_integral_positive) (auto split: split_indicator)
+ qed (insert `0 < a`, auto)
ultimately show "a * integral\<^isup>S M u \<le> ?S" by simp
qed
+lemma (in measure_space) incseq_positive_integral:
+ assumes "incseq f" shows "incseq (\<lambda>i. integral\<^isup>P M (f i))"
+proof -
+ have "\<And>i x. f i x \<le> f (Suc i) x"
+ using assms by (auto dest!: incseq_SucD simp: le_fun_def)
+ then show ?thesis
+ by (auto intro!: incseq_SucI positive_integral_mono)
+qed
+
text {* Beppo-Levi monotone convergence theorem *}
-lemma (in measure_space) positive_integral_isoton:
- assumes "f \<up> u" "\<And>i. f i \<in> borel_measurable M"
- shows "(\<lambda>i. integral\<^isup>P M (f i)) \<up> integral\<^isup>P M u"
- unfolding isoton_def
-proof safe
- fix i show "integral\<^isup>P M (f i) \<le> integral\<^isup>P M (f (Suc i))"
- apply (rule positive_integral_mono)
- using `f \<up> u` unfolding isoton_def le_fun_def by auto
+lemma (in measure_space) positive_integral_monotone_convergence_SUP:
+ assumes f: "incseq f" "\<And>i. f i \<in> borel_measurable M" "\<And>i x. 0 \<le> f i x"
+ shows "(\<integral>\<^isup>+ x. (SUP i. f i x) \<partial>M) = (SUP i. integral\<^isup>P M (f i))"
+proof (rule antisym)
+ show "(SUP j. integral\<^isup>P M (f j)) \<le> (\<integral>\<^isup>+ x. (SUP i. f i x) \<partial>M)"
+ by (auto intro!: SUP_leI le_SUPI positive_integral_mono)
next
- have u: "u = (SUP i. f i)" using `f \<up> u` unfolding isoton_def by auto
- show "(SUP i. integral\<^isup>P M (f i)) = integral\<^isup>P M u"
- proof (rule antisym)
- from `f \<up> u`[THEN isoton_Sup, unfolded le_fun_def]
- show "(SUP j. integral\<^isup>P M (f j)) \<le> integral\<^isup>P M u"
- by (auto intro!: SUP_leI positive_integral_mono)
- next
- show "integral\<^isup>P M u \<le> (SUP i. integral\<^isup>P M (f i))"
- unfolding positive_integral_alt[of u]
- by (auto intro!: SUP_leI positive_integral_SUP_approx[OF assms])
+ show "(\<integral>\<^isup>+ x. (SUP i. f i x) \<partial>M) \<le> (SUP j. integral\<^isup>P M (f j))"
+ unfolding positive_integral_def_finite[of "\<lambda>x. SUP i. f i x"]
+ proof (safe intro!: SUP_leI)
+ fix g assume g: "simple_function M g"
+ and "g \<le> max 0 \<circ> (\<lambda>x. SUP i. f i x)" "range g \<subseteq> {0..<\<infinity>}"
+ moreover then have "\<And>x. 0 \<le> (SUP i. f i x)" and g': "g`space M \<subseteq> {0..<\<infinity>}"
+ using f by (auto intro!: le_SUPI2)
+ ultimately show "integral\<^isup>S M g \<le> (SUP j. integral\<^isup>P M (f j))"
+ by (intro positive_integral_SUP_approx[OF f g _ g'])
+ (auto simp: le_fun_def max_def SUPR_apply)
qed
qed
-lemma (in measure_space) positive_integral_monotone_convergence_SUP:
- assumes "\<And>i x. x \<in> space M \<Longrightarrow> f i x \<le> f (Suc i) x"
- assumes "\<And>i. f i \<in> borel_measurable M"
- shows "(SUP i. integral\<^isup>P M (f i)) = (\<integral>\<^isup>+ x. (SUP i. f i x) \<partial>M)"
- (is "_ = integral\<^isup>P M ?u")
+lemma (in measure_space) positive_integral_monotone_convergence_SUP_AE:
+ assumes f: "\<And>i. AE x. f i x \<le> f (Suc i) x \<and> 0 \<le> f i x" "\<And>i. f i \<in> borel_measurable M"
+ shows "(\<integral>\<^isup>+ x. (SUP i. f i x) \<partial>M) = (SUP i. integral\<^isup>P M (f i))"
proof -
- show ?thesis
- proof (rule antisym)
- show "(SUP j. integral\<^isup>P M (f j)) \<le> integral\<^isup>P M ?u"
- by (auto intro!: SUP_leI positive_integral_mono le_SUPI)
- next
- def rf \<equiv> "\<lambda>i. \<lambda>x\<in>space M. f i x" and ru \<equiv> "\<lambda>x\<in>space M. ?u x"
- have "\<And>i. rf i \<in> borel_measurable M" unfolding rf_def
- using assms by (simp cong: measurable_cong)
- moreover have iso: "rf \<up> ru" using assms unfolding rf_def ru_def
- unfolding isoton_def le_fun_def fun_eq_iff SUPR_apply
- using SUP_const[OF UNIV_not_empty]
- by (auto simp: restrict_def le_fun_def fun_eq_iff)
- ultimately have "integral\<^isup>P M ru \<le> (SUP i. integral\<^isup>P M (rf i))"
- unfolding positive_integral_alt[of ru]
- by (auto simp: le_fun_def intro!: SUP_leI positive_integral_SUP_approx)
- then show "integral\<^isup>P M ?u \<le> (SUP i. integral\<^isup>P M (f i))"
- unfolding ru_def rf_def by (simp cong: positive_integral_cong)
+ from f have "AE x. \<forall>i. f i x \<le> f (Suc i) x \<and> 0 \<le> f i x"
+ by (simp add: AE_all_countable)
+ from this[THEN AE_E] guess N . note N = this
+ let "?f i x" = "if x \<in> space M - N then f i x else 0"
+ have f_eq: "AE x. \<forall>i. ?f i x = f i x" using N by (auto intro!: AE_I[of _ N])
+ then have "(\<integral>\<^isup>+ x. (SUP i. f i x) \<partial>M) = (\<integral>\<^isup>+ x. (SUP i. ?f i x) \<partial>M)"
+ by (auto intro!: positive_integral_cong_AE)
+ also have "\<dots> = (SUP i. (\<integral>\<^isup>+ x. ?f i x \<partial>M))"
+ proof (rule positive_integral_monotone_convergence_SUP)
+ show "incseq ?f" using N(1) by (force intro!: incseq_SucI le_funI)
+ { fix i show "(\<lambda>x. if x \<in> space M - N then f i x else 0) \<in> borel_measurable M"
+ using f N(3) by (intro measurable_If_set) auto
+ fix x show "0 \<le> ?f i x"
+ using N(1) by auto }
qed
+ also have "\<dots> = (SUP i. (\<integral>\<^isup>+ x. f i x \<partial>M))"
+ using f_eq by (force intro!: arg_cong[where f="SUPR UNIV"] positive_integral_cong_AE ext)
+ finally show ?thesis .
+qed
+
+lemma (in measure_space) positive_integral_monotone_convergence_SUP_AE_incseq:
+ assumes f: "incseq f" "\<And>i. AE x. 0 \<le> f i x" and borel: "\<And>i. f i \<in> borel_measurable M"
+ shows "(\<integral>\<^isup>+ x. (SUP i. f i x) \<partial>M) = (SUP i. integral\<^isup>P M (f i))"
+ using f[unfolded incseq_Suc_iff le_fun_def]
+ by (intro positive_integral_monotone_convergence_SUP_AE[OF _ borel])
+ auto
+
+lemma (in measure_space) positive_integral_monotone_convergence_simple:
+ assumes f: "incseq f" "\<And>i x. 0 \<le> f i x" "\<And>i. simple_function M (f i)"
+ shows "(SUP i. integral\<^isup>S M (f i)) = (\<integral>\<^isup>+x. (SUP i. f i x) \<partial>M)"
+ using assms unfolding positive_integral_monotone_convergence_SUP[OF f(1)
+ f(3)[THEN borel_measurable_simple_function] f(2)]
+ by (auto intro!: positive_integral_eq_simple_integral[symmetric] arg_cong[where f="SUPR UNIV"] ext)
+
+lemma positive_integral_max_0:
+ "(\<integral>\<^isup>+x. max 0 (f x) \<partial>M) = integral\<^isup>P M f"
+ by (simp add: le_fun_def positive_integral_def)
+
+lemma (in measure_space) positive_integral_cong_pos:
+ assumes "\<And>x. x \<in> space M \<Longrightarrow> f x \<le> 0 \<and> g x \<le> 0 \<or> f x = g x"
+ shows "integral\<^isup>P M f = integral\<^isup>P M g"
+proof -
+ have "integral\<^isup>P M (\<lambda>x. max 0 (f x)) = integral\<^isup>P M (\<lambda>x. max 0 (g x))"
+ proof (intro positive_integral_cong)
+ fix x assume "x \<in> space M"
+ from assms[OF this] show "max 0 (f x) = max 0 (g x)"
+ by (auto split: split_max)
+ qed
+ then show ?thesis by (simp add: positive_integral_max_0)
qed
lemma (in measure_space) SUP_simple_integral_sequences:
- assumes f: "f \<up> u" "\<And>i. simple_function M (f i)"
- and g: "g \<up> u" "\<And>i. simple_function M (g i)"
+ assumes f: "incseq f" "\<And>i x. 0 \<le> f i x" "\<And>i. simple_function M (f i)"
+ and g: "incseq g" "\<And>i x. 0 \<le> g i x" "\<And>i. simple_function M (g i)"
+ and eq: "AE x. (SUP i. f i x) = (SUP i. g i x)"
shows "(SUP i. integral\<^isup>S M (f i)) = (SUP i. integral\<^isup>S M (g i))"
(is "SUPR _ ?F = SUPR _ ?G")
proof -
- have "(SUP i. ?F i) = (SUP i. integral\<^isup>P M (f i))"
- using assms by (simp add: positive_integral_eq_simple_integral)
- also have "\<dots> = integral\<^isup>P M u"
- using positive_integral_isoton[OF f(1) borel_measurable_simple_function[OF f(2)]]
- unfolding isoton_def by simp
- also have "\<dots> = (SUP i. integral\<^isup>P M (g i))"
- using positive_integral_isoton[OF g(1) borel_measurable_simple_function[OF g(2)]]
- unfolding isoton_def by simp
+ have "(SUP i. integral\<^isup>S M (f i)) = (\<integral>\<^isup>+x. (SUP i. f i x) \<partial>M)"
+ using f by (rule positive_integral_monotone_convergence_simple)
+ also have "\<dots> = (\<integral>\<^isup>+x. (SUP i. g i x) \<partial>M)"
+ unfolding eq[THEN positive_integral_cong_AE] ..
also have "\<dots> = (SUP i. ?G i)"
- using assms by (simp add: positive_integral_eq_simple_integral)
- finally show ?thesis .
+ using g by (rule positive_integral_monotone_convergence_simple[symmetric])
+ finally show ?thesis by simp
qed
lemma (in measure_space) positive_integral_const[simp]:
- "(\<integral>\<^isup>+ x. c \<partial>M) = c * \<mu> (space M)"
+ "0 \<le> c \<Longrightarrow> (\<integral>\<^isup>+ x. c \<partial>M) = c * \<mu> (space M)"
by (subst positive_integral_eq_simple_integral) auto
-lemma (in measure_space) positive_integral_isoton_simple:
- assumes "f \<up> u" and e: "\<And>i. simple_function M (f i)"
- shows "(\<lambda>i. integral\<^isup>S M (f i)) \<up> integral\<^isup>P M u"
- using positive_integral_isoton[OF `f \<up> u` e(1)[THEN borel_measurable_simple_function]]
- unfolding positive_integral_eq_simple_integral[OF e] .
-
-lemma measure_preservingD2:
- "f \<in> measure_preserving A B \<Longrightarrow> f \<in> measurable A B"
- unfolding measure_preserving_def by auto
-
lemma (in measure_space) positive_integral_vimage:
- assumes T: "sigma_algebra M'" "T \<in> measure_preserving M M'" and f: "f \<in> borel_measurable M'"
+ assumes T: "sigma_algebra M'" "T \<in> measure_preserving M M'"
+ and f: "f \<in> borel_measurable M'"
shows "integral\<^isup>P M' f = (\<integral>\<^isup>+ x. f (T x) \<partial>M)"
proof -
interpret T: measure_space M' by (rule measure_space_vimage[OF T])
- obtain f' where f': "f' \<up> f" "\<And>i. simple_function M' (f' i)"
- using T.borel_measurable_implies_simple_function_sequence[OF f] by blast
- then have f: "(\<lambda>i x. f' i (T x)) \<up> (\<lambda>x. f (T x))" "\<And>i. simple_function M (\<lambda>x. f' i (T x))"
- using simple_function_vimage[OF T(1) measure_preservingD2[OF T(2)]] unfolding isoton_fun_expand by auto
+ from T.borel_measurable_implies_simple_function_sequence'[OF f]
+ guess f' . note f' = this
+ let "?f i x" = "f' i (T x)"
+ have inc: "incseq ?f" using f' by (force simp: le_fun_def incseq_def)
+ have sup: "\<And>x. (SUP i. ?f i x) = max 0 (f (T x))"
+ using f'(4) .
+ have sf: "\<And>i. simple_function M (\<lambda>x. f' i (T x))"
+ using simple_function_vimage[OF T(1) measure_preservingD2[OF T(2)] f'(1)] .
show "integral\<^isup>P M' f = (\<integral>\<^isup>+ x. f (T x) \<partial>M)"
- using positive_integral_isoton_simple[OF f]
- using T.positive_integral_isoton_simple[OF f']
- by (simp add: simple_integral_vimage[OF T f'(2)] isoton_def)
+ using
+ T.positive_integral_monotone_convergence_simple[OF f'(2,5,1)]
+ positive_integral_monotone_convergence_simple[OF inc f'(5) sf]
+ by (simp add: positive_integral_max_0 simple_integral_vimage[OF T f'(1)] f')
qed
lemma (in measure_space) positive_integral_linear:
- assumes f: "f \<in> borel_measurable M"
- and g: "g \<in> borel_measurable M"
+ assumes f: "f \<in> borel_measurable M" "\<And>x. 0 \<le> f x" and "0 \<le> a"
+ and g: "g \<in> borel_measurable M" "\<And>x. 0 \<le> g x"
shows "(\<integral>\<^isup>+ x. a * f x + g x \<partial>M) = a * integral\<^isup>P M f + integral\<^isup>P M g"
(is "integral\<^isup>P M ?L = _")
proof -
- from borel_measurable_implies_simple_function_sequence'[OF f] guess u .
- note u = this positive_integral_isoton_simple[OF this(1-2)]
- from borel_measurable_implies_simple_function_sequence'[OF g] guess v .
- note v = this positive_integral_isoton_simple[OF this(1-2)]
+ from borel_measurable_implies_simple_function_sequence'[OF f(1)] guess u .
+ note u = positive_integral_monotone_convergence_simple[OF this(2,5,1)] this
+ from borel_measurable_implies_simple_function_sequence'[OF g(1)] guess v .
+ note v = positive_integral_monotone_convergence_simple[OF this(2,5,1)] this
let "?L' i x" = "a * u i x + v i x"
- have "?L \<in> borel_measurable M"
- using assms by simp
+ have "?L \<in> borel_measurable M" using assms by auto
from borel_measurable_implies_simple_function_sequence'[OF this] guess l .
- note positive_integral_isoton_simple[OF this(1-2)] and l = this
- moreover have "(SUP i. integral\<^isup>S M (l i)) = (SUP i. integral\<^isup>S M (?L' i))"
- proof (rule SUP_simple_integral_sequences[OF l(1-2)])
- show "?L' \<up> ?L" "\<And>i. simple_function M (?L' i)"
- using u v by (auto simp: isoton_fun_expand isoton_add isoton_cmult_right)
+ note l = positive_integral_monotone_convergence_simple[OF this(2,5,1)] this
+
+ have inc: "incseq (\<lambda>i. a * integral\<^isup>S M (u i))" "incseq (\<lambda>i. integral\<^isup>S M (v i))"
+ using u v `0 \<le> a`
+ by (auto simp: incseq_Suc_iff le_fun_def
+ intro!: add_mono extreal_mult_left_mono simple_integral_mono)
+ have pos: "\<And>i. 0 \<le> integral\<^isup>S M (u i)" "\<And>i. 0 \<le> integral\<^isup>S M (v i)" "\<And>i. 0 \<le> a * integral\<^isup>S M (u i)"
+ using u v `0 \<le> a` by (auto simp: simple_integral_positive)
+ { fix i from pos[of i] have "a * integral\<^isup>S M (u i) \<noteq> -\<infinity>" "integral\<^isup>S M (v i) \<noteq> -\<infinity>"
+ by (auto split: split_if_asm) }
+ note not_MInf = this
+
+ have l': "(SUP i. integral\<^isup>S M (l i)) = (SUP i. integral\<^isup>S M (?L' i))"
+ proof (rule SUP_simple_integral_sequences[OF l(3,6,2)])
+ show "incseq ?L'" "\<And>i x. 0 \<le> ?L' i x" "\<And>i. simple_function M (?L' i)"
+ using u v `0 \<le> a` unfolding incseq_Suc_iff le_fun_def
+ by (auto intro!: add_mono extreal_mult_left_mono extreal_add_nonneg_nonneg)
+ { fix x
+ { fix i have "a * u i x \<noteq> -\<infinity>" "v i x \<noteq> -\<infinity>" "u i x \<noteq> -\<infinity>" using `0 \<le> a` u(6)[of i x] v(6)[of i x]
+ by auto }
+ then have "(SUP i. a * u i x + v i x) = a * (SUP i. u i x) + (SUP i. v i x)"
+ using `0 \<le> a` u(3) v(3) u(6)[of _ x] v(6)[of _ x]
+ by (subst SUPR_extreal_cmult[symmetric, OF u(6) `0 \<le> a`])
+ (auto intro!: SUPR_extreal_add
+ simp: incseq_Suc_iff le_fun_def add_mono extreal_mult_left_mono extreal_add_nonneg_nonneg) }
+ then show "AE x. (SUP i. l i x) = (SUP i. ?L' i x)"
+ unfolding l(5) using `0 \<le> a` u(5) v(5) l(5) f(2) g(2)
+ by (intro AE_I2) (auto split: split_max simp add: extreal_add_nonneg_nonneg)
qed
- moreover from u v have L'_isoton:
- "(\<lambda>i. integral\<^isup>S M (?L' i)) \<up> a * integral\<^isup>P M f + integral\<^isup>P M g"
- by (simp add: isoton_add isoton_cmult_right)
- ultimately show ?thesis by (simp add: isoton_def)
+ also have "\<dots> = (SUP i. a * integral\<^isup>S M (u i) + integral\<^isup>S M (v i))"
+ using u(2, 6) v(2, 6) `0 \<le> a` by (auto intro!: arg_cong[where f="SUPR UNIV"] ext)
+ finally have "(\<integral>\<^isup>+ x. max 0 (a * f x + g x) \<partial>M) = a * (\<integral>\<^isup>+x. max 0 (f x) \<partial>M) + (\<integral>\<^isup>+x. max 0 (g x) \<partial>M)"
+ unfolding l(5)[symmetric] u(5)[symmetric] v(5)[symmetric]
+ unfolding l(1)[symmetric] u(1)[symmetric] v(1)[symmetric]
+ apply (subst SUPR_extreal_cmult[symmetric, OF pos(1) `0 \<le> a`])
+ apply (subst SUPR_extreal_add[symmetric, OF inc not_MInf]) .
+ then show ?thesis by (simp add: positive_integral_max_0)
qed
lemma (in measure_space) positive_integral_cmult:
- assumes "f \<in> borel_measurable M"
+ assumes f: "f \<in> borel_measurable M" "AE x. 0 \<le> f x" "0 \<le> c"
shows "(\<integral>\<^isup>+ x. c * f x \<partial>M) = c * integral\<^isup>P M f"
- using positive_integral_linear[OF assms, of "\<lambda>x. 0" c] by auto
+proof -
+ have [simp]: "\<And>x. c * max 0 (f x) = max 0 (c * f x)" using `0 \<le> c`
+ by (auto split: split_max simp: extreal_zero_le_0_iff)
+ have "(\<integral>\<^isup>+ x. c * f x \<partial>M) = (\<integral>\<^isup>+ x. c * max 0 (f x) \<partial>M)"
+ by (simp add: positive_integral_max_0)
+ then show ?thesis
+ using positive_integral_linear[OF _ _ `0 \<le> c`, of "\<lambda>x. max 0 (f x)" "\<lambda>x. 0"] f
+ by (auto simp: positive_integral_max_0)
+qed
lemma (in measure_space) positive_integral_multc:
- assumes "f \<in> borel_measurable M"
+ assumes "f \<in> borel_measurable M" "AE x. 0 \<le> f x" "0 \<le> c"
shows "(\<integral>\<^isup>+ x. f x * c \<partial>M) = integral\<^isup>P M f * c"
unfolding mult_commute[of _ c] positive_integral_cmult[OF assms] by simp
@@ -1260,143 +1289,172 @@
(auto simp: simple_function_indicator simple_integral_indicator)
lemma (in measure_space) positive_integral_cmult_indicator:
- "A \<in> sets M \<Longrightarrow> (\<integral>\<^isup>+ x. c * indicator A x \<partial>M) = c * \<mu> A"
+ "0 \<le> c \<Longrightarrow> A \<in> sets M \<Longrightarrow> (\<integral>\<^isup>+ x. c * indicator A x \<partial>M) = c * \<mu> A"
by (subst positive_integral_eq_simple_integral)
(auto simp: simple_function_indicator simple_integral_indicator)
lemma (in measure_space) positive_integral_add:
- assumes "f \<in> borel_measurable M" "g \<in> borel_measurable M"
+ assumes f: "f \<in> borel_measurable M" "AE x. 0 \<le> f x"
+ and g: "g \<in> borel_measurable M" "AE x. 0 \<le> g x"
shows "(\<integral>\<^isup>+ x. f x + g x \<partial>M) = integral\<^isup>P M f + integral\<^isup>P M g"
- using positive_integral_linear[OF assms, of 1] by simp
+proof -
+ have ae: "AE x. max 0 (f x) + max 0 (g x) = max 0 (f x + g x)"
+ using assms by (auto split: split_max simp: extreal_add_nonneg_nonneg)
+ have "(\<integral>\<^isup>+ x. f x + g x \<partial>M) = (\<integral>\<^isup>+ x. max 0 (f x + g x) \<partial>M)"
+ by (simp add: positive_integral_max_0)
+ also have "\<dots> = (\<integral>\<^isup>+ x. max 0 (f x) + max 0 (g x) \<partial>M)"
+ unfolding ae[THEN positive_integral_cong_AE] ..
+ also have "\<dots> = (\<integral>\<^isup>+ x. max 0 (f x) \<partial>M) + (\<integral>\<^isup>+ x. max 0 (g x) \<partial>M)"
+ using positive_integral_linear[of "\<lambda>x. max 0 (f x)" 1 "\<lambda>x. max 0 (g x)"] f g
+ by auto
+ finally show ?thesis
+ by (simp add: positive_integral_max_0)
+qed
lemma (in measure_space) positive_integral_setsum:
- assumes "\<And>i. i\<in>P \<Longrightarrow> f i \<in> borel_measurable M"
+ assumes "\<And>i. i\<in>P \<Longrightarrow> f i \<in> borel_measurable M" "\<And>i. i\<in>P \<Longrightarrow> AE x. 0 \<le> f i x"
shows "(\<integral>\<^isup>+ x. (\<Sum>i\<in>P. f i x) \<partial>M) = (\<Sum>i\<in>P. integral\<^isup>P M (f i))"
proof cases
- assume "finite P"
- from this assms show ?thesis
+ assume f: "finite P"
+ from assms have "AE x. \<forall>i\<in>P. 0 \<le> f i x" unfolding AE_finite_all[OF f] by auto
+ from f this assms(1) show ?thesis
proof induct
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_pextreal_setsum)
+ then have "f i \<in> borel_measurable M" "AE x. 0 \<le> f i x"
+ "(\<lambda>x. \<Sum>i\<in>P. f i x) \<in> borel_measurable M" "AE x. 0 \<le> (\<Sum>i\<in>P. f i x)"
+ by (auto intro!: borel_measurable_extreal_setsum setsum_nonneg)
from positive_integral_add[OF this]
show ?case using insert by auto
qed simp
qed simp
+lemma (in measure_space) positive_integral_Markov_inequality:
+ assumes u: "u \<in> borel_measurable M" "AE x. 0 \<le> u x" and "A \<in> sets M" and c: "0 \<le> c" "c \<noteq> \<infinity>"
+ shows "\<mu> ({x\<in>space M. 1 \<le> c * u x} \<inter> A) \<le> c * (\<integral>\<^isup>+ x. u x * indicator A x \<partial>M)"
+ (is "\<mu> ?A \<le> _ * ?PI")
+proof -
+ have "?A \<in> sets M"
+ using `A \<in> sets M` u by auto
+ hence "\<mu> ?A = (\<integral>\<^isup>+ x. indicator ?A x \<partial>M)"
+ using positive_integral_indicator by simp
+ also have "\<dots> \<le> (\<integral>\<^isup>+ x. c * (u x * indicator A x) \<partial>M)" using u c
+ by (auto intro!: positive_integral_mono_AE
+ simp: indicator_def extreal_zero_le_0_iff)
+ also have "\<dots> = c * (\<integral>\<^isup>+ x. u x * indicator A x \<partial>M)"
+ using assms
+ by (auto intro!: positive_integral_cmult borel_measurable_indicator simp: extreal_zero_le_0_iff)
+ finally show ?thesis .
+qed
+
+lemma (in measure_space) positive_integral_noteq_infinite:
+ assumes g: "g \<in> borel_measurable M" "AE x. 0 \<le> g x"
+ and "integral\<^isup>P M g \<noteq> \<infinity>"
+ shows "AE x. g x \<noteq> \<infinity>"
+proof (rule ccontr)
+ assume c: "\<not> (AE x. g x \<noteq> \<infinity>)"
+ have "\<mu> {x\<in>space M. g x = \<infinity>} \<noteq> 0"
+ using c g by (simp add: AE_iff_null_set)
+ moreover have "0 \<le> \<mu> {x\<in>space M. g x = \<infinity>}" using g by (auto intro: measurable_sets)
+ ultimately have "0 < \<mu> {x\<in>space M. g x = \<infinity>}" by auto
+ then have "\<infinity> = \<infinity> * \<mu> {x\<in>space M. g x = \<infinity>}" by auto
+ also have "\<dots> \<le> (\<integral>\<^isup>+x. \<infinity> * indicator {x\<in>space M. g x = \<infinity>} x \<partial>M)"
+ using g by (subst positive_integral_cmult_indicator) auto
+ also have "\<dots> \<le> integral\<^isup>P M g"
+ using assms by (auto intro!: positive_integral_mono_AE simp: indicator_def)
+ finally show False using `integral\<^isup>P M g \<noteq> \<infinity>` by auto
+qed
+
lemma (in measure_space) positive_integral_diff:
- assumes f: "f \<in> borel_measurable M" and g: "g \<in> borel_measurable M"
- and fin: "integral\<^isup>P M g \<noteq> \<omega>"
- and mono: "\<And>x. x \<in> space M \<Longrightarrow> g x \<le> f x"
+ assumes f: "f \<in> borel_measurable M"
+ and g: "g \<in> borel_measurable M" "AE x. 0 \<le> g x"
+ and fin: "integral\<^isup>P M g \<noteq> \<infinity>"
+ and mono: "AE x. g x \<le> f x"
shows "(\<integral>\<^isup>+ x. f x - g x \<partial>M) = integral\<^isup>P M f - integral\<^isup>P M g"
proof -
- have borel: "(\<lambda>x. f x - g x) \<in> borel_measurable M"
- using f g by (rule borel_measurable_pextreal_diff)
- have "(\<integral>\<^isup>+x. f x - g x \<partial>M) + integral\<^isup>P M g = integral\<^isup>P M f"
- unfolding positive_integral_add[OF borel g, symmetric]
- proof (rule positive_integral_cong)
- fix x assume "x \<in> space M"
- from mono[OF this] show "f x - g x + g x = f x"
- by (cases "f x", cases "g x", simp, simp, cases "g x", auto)
- qed
- with mono show ?thesis
- by (subst minus_pextreal_eq2[OF _ fin]) (auto intro!: positive_integral_mono)
+ have diff: "(\<lambda>x. f x - g x) \<in> borel_measurable M" "AE x. 0 \<le> f x - g x"
+ using assms by (auto intro: extreal_diff_positive)
+ have pos_f: "AE x. 0 \<le> f x" using mono g by auto
+ { fix a b :: extreal assume "0 \<le> a" "a \<noteq> \<infinity>" "0 \<le> b" "a \<le> b" then have "b - a + a = b"
+ by (cases rule: extreal2_cases[of a b]) auto }
+ note * = this
+ then have "AE x. f x = f x - g x + g x"
+ using mono positive_integral_noteq_infinite[OF g fin] assms by auto
+ then have **: "integral\<^isup>P M f = (\<integral>\<^isup>+x. f x - g x \<partial>M) + integral\<^isup>P M g"
+ unfolding positive_integral_add[OF diff g, symmetric]
+ by (rule positive_integral_cong_AE)
+ show ?thesis unfolding **
+ using fin positive_integral_positive[of g]
+ by (cases rule: extreal2_cases[of "\<integral>\<^isup>+ x. f x - g x \<partial>M" "integral\<^isup>P M g"]) auto
qed
-lemma (in measure_space) positive_integral_psuminf:
- assumes "\<And>i. f i \<in> borel_measurable M"
- shows "(\<integral>\<^isup>+ x. (\<Sum>\<^isub>\<infinity> i. f i x) \<partial>M) = (\<Sum>\<^isub>\<infinity> i. integral\<^isup>P M (f i))"
+lemma (in measure_space) positive_integral_suminf:
+ assumes f: "\<And>i. f i \<in> borel_measurable M" "\<And>i. AE x. 0 \<le> f i x"
+ shows "(\<integral>\<^isup>+ x. (\<Sum>i. f i x) \<partial>M) = (\<Sum>i. integral\<^isup>P M (f i))"
proof -
- have "(\<lambda>i. (\<integral>\<^isup>+x. (\<Sum>i<i. f i x) \<partial>M)) \<up> (\<integral>\<^isup>+x. (\<Sum>\<^isub>\<infinity>i. f i x) \<partial>M)"
- by (rule positive_integral_isoton)
- (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
- by (auto simp: isoton_def psuminf_def positive_integral_setsum[OF assms])
+ have all_pos: "AE x. \<forall>i. 0 \<le> f i x"
+ using assms by (auto simp: AE_all_countable)
+ have "(\<Sum>i. integral\<^isup>P M (f i)) = (SUP n. \<Sum>i<n. integral\<^isup>P M (f i))"
+ using positive_integral_positive by (rule suminf_extreal_eq_SUPR)
+ also have "\<dots> = (SUP n. \<integral>\<^isup>+x. (\<Sum>i<n. f i x) \<partial>M)"
+ unfolding positive_integral_setsum[OF f] ..
+ also have "\<dots> = \<integral>\<^isup>+x. (SUP n. \<Sum>i<n. f i x) \<partial>M" using f all_pos
+ by (intro positive_integral_monotone_convergence_SUP_AE[symmetric])
+ (elim AE_mp, auto simp: setsum_nonneg simp del: setsum_lessThan_Suc intro!: AE_I2 setsum_mono3)
+ also have "\<dots> = \<integral>\<^isup>+x. (\<Sum>i. f i x) \<partial>M" using all_pos
+ by (intro positive_integral_cong_AE) (auto simp: suminf_extreal_eq_SUPR)
+ finally show ?thesis by simp
qed
text {* Fatou's lemma: convergence theorem on limes inferior *}
lemma (in measure_space) positive_integral_lim_INF:
- fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> pextreal"
- assumes "\<And>i. u i \<in> borel_measurable M"
- shows "(\<integral>\<^isup>+ x. (SUP n. INF m. u (m + n) x) \<partial>M) \<le>
- (SUP n. INF m. integral\<^isup>P M (u (m + n)))"
+ fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> extreal"
+ assumes u: "\<And>i. u i \<in> borel_measurable M" "\<And>i. AE x. 0 \<le> u i x"
+ shows "(\<integral>\<^isup>+ x. liminf (\<lambda>n. u n x) \<partial>M) \<le> liminf (\<lambda>n. integral\<^isup>P M (u n))"
proof -
- have "(\<integral>\<^isup>+x. (SUP n. INF m. u (m + n) x) \<partial>M)
- = (SUP n. (\<integral>\<^isup>+x. (INF m. u (m + n) x) \<partial>M))"
- using assms
- by (intro positive_integral_monotone_convergence_SUP[symmetric] INF_mono)
- (auto simp del: add_Suc simp add: add_Suc[symmetric])
- also have "\<dots> \<le> (SUP n. INF m. integral\<^isup>P M (u (m + n)))"
- by (auto intro!: SUP_mono bexI le_INFI positive_integral_mono INF_leI)
+ have pos: "AE x. \<forall>i. 0 \<le> u i x" using u by (auto simp: AE_all_countable)
+ have "(\<integral>\<^isup>+ x. liminf (\<lambda>n. u n x) \<partial>M) =
+ (SUP n. \<integral>\<^isup>+ x. (INF i:{n..}. u i x) \<partial>M)"
+ unfolding liminf_SUPR_INFI using pos u
+ by (intro positive_integral_monotone_convergence_SUP_AE)
+ (elim AE_mp, auto intro!: AE_I2 intro: le_INFI INF_subset)
+ also have "\<dots> \<le> liminf (\<lambda>n. integral\<^isup>P M (u n))"
+ unfolding liminf_SUPR_INFI
+ by (auto intro!: SUP_mono exI le_INFI positive_integral_mono INF_leI)
finally show ?thesis .
qed
lemma (in measure_space) measure_space_density:
- assumes borel: "u \<in> borel_measurable M"
+ assumes u: "u \<in> borel_measurable M" "AE x. 0 \<le> u x"
and M'[simp]: "M' = (M\<lparr>measure := \<lambda>A. (\<integral>\<^isup>+ x. u x * indicator A x \<partial>M)\<rparr>)"
shows "measure_space M'"
proof -
interpret M': sigma_algebra M' by (intro sigma_algebra_cong) auto
show ?thesis
proof
- show "measure M' {} = 0" unfolding M' by simp
+ have pos: "\<And>A. AE x. 0 \<le> u x * indicator A x"
+ using u by (auto simp: extreal_zero_le_0_iff)
+ then show "positive M' (measure M')" unfolding M'
+ using u(1) by (auto simp: positive_def intro!: positive_integral_positive)
show "countably_additive M' (measure M')"
proof (intro countably_additiveI)
fix A :: "nat \<Rightarrow> 'a set" assume "range A \<subseteq> sets M'"
- then have "\<And>i. (\<lambda>x. u x * indicator (A i) x) \<in> borel_measurable M"
- using borel by (auto intro: borel_measurable_indicator)
- moreover assume "disjoint_family A"
- note psuminf_indicator[OF this]
- ultimately show "(\<Sum>\<^isub>\<infinity>n. measure M' (A n)) = measure M' (\<Union>x. A x)"
- by (simp add: positive_integral_psuminf[symmetric])
+ then have *: "\<And>i. (\<lambda>x. u x * indicator (A i) x) \<in> borel_measurable M"
+ using u by (auto intro: borel_measurable_indicator)
+ assume disj: "disjoint_family A"
+ have "(\<Sum>n. measure M' (A n)) = (\<integral>\<^isup>+ x. (\<Sum>n. u x * indicator (A n) x) \<partial>M)"
+ unfolding M' using u(1) *
+ by (simp add: positive_integral_suminf[OF _ pos, symmetric])
+ also have "\<dots> = (\<integral>\<^isup>+ x. u x * (\<Sum>n. indicator (A n) x) \<partial>M)" using u
+ by (intro positive_integral_cong_AE)
+ (elim AE_mp, auto intro!: AE_I2 suminf_cmult_extreal)
+ also have "\<dots> = (\<integral>\<^isup>+ x. u x * indicator (\<Union>n. A n) x \<partial>M)"
+ unfolding suminf_indicator[OF disj] ..
+ finally show "(\<Sum>n. measure M' (A n)) = measure M' (\<Union>x. A x)"
+ unfolding M' by simp
qed
qed
qed
-lemma (in measure_space) positive_integral_translated_density:
- assumes "f \<in> borel_measurable M" "g \<in> borel_measurable M"
- and M': "M' = (M\<lparr> measure := \<lambda>A. (\<integral>\<^isup>+ x. f x * indicator A x \<partial>M)\<rparr>)"
- shows "integral\<^isup>P M' g = (\<integral>\<^isup>+ x. f x * g x \<partial>M)"
-proof -
- from measure_space_density[OF assms(1) M']
- interpret T: measure_space M' .
- have borel[simp]:
- "borel_measurable M' = borel_measurable M"
- "simple_function M' = simple_function M"
- unfolding measurable_def simple_function_def_raw by (auto simp: M')
- from borel_measurable_implies_simple_function_sequence[OF assms(2)]
- obtain G where G: "\<And>i. simple_function M (G i)" "G \<up> g" by blast
- note G_borel = borel_measurable_simple_function[OF this(1)]
- from T.positive_integral_isoton[unfolded borel, OF `G \<up> g` G_borel]
- have *: "(\<lambda>i. integral\<^isup>P M' (G i)) \<up> integral\<^isup>P M' g" .
- { fix i
- have [simp]: "finite (G i ` space M)"
- using G(1) unfolding simple_function_def by auto
- have "integral\<^isup>P M' (G i) = integral\<^isup>S M' (G i)"
- using G T.positive_integral_eq_simple_integral by simp
- also have "\<dots> = (\<integral>\<^isup>+x. f x * (\<Sum>y\<in>G i`space M. y * indicator (G i -` {y} \<inter> space M) x) \<partial>M)"
- apply (simp add: simple_integral_def M')
- apply (subst positive_integral_cmult[symmetric])
- using G_borel assms(1) apply (fastsimp intro: borel_measurable_vimage)
- apply (subst positive_integral_setsum[symmetric])
- using G_borel assms(1) apply (fastsimp intro: borel_measurable_vimage)
- by (simp add: setsum_right_distrib field_simps)
- also have "\<dots> = (\<integral>\<^isup>+x. f x * G i x \<partial>M)"
- by (auto intro!: positive_integral_cong
- simp: indicator_def if_distrib setsum_cases)
- finally have "integral\<^isup>P M' (G i) = (\<integral>\<^isup>+x. f x * G i x \<partial>M)" . }
- with * have eq_Tg: "(\<lambda>i. (\<integral>\<^isup>+x. f x * G i x \<partial>M)) \<up> integral\<^isup>P M' g" by simp
- 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. (\<integral>\<^isup>+x. f x * G i x \<partial>M)) \<up> (\<integral>\<^isup>+x. f x * g x \<partial>M)"
- using assms(1) G_borel by (auto intro!: positive_integral_isoton borel_measurable_pextreal_times)
- with eq_Tg show "integral\<^isup>P M' g = (\<integral>\<^isup>+x. f x * g x \<partial>M)"
- unfolding isoton_def by simp
-qed
-
lemma (in measure_space) positive_integral_null_set:
assumes "N \<in> null_sets" shows "(\<integral>\<^isup>+ x. u x * indicator N x \<partial>M) = 0"
proof -
@@ -1410,144 +1468,199 @@
then show ?thesis by simp
qed
-lemma (in measure_space) positive_integral_Markov_inequality:
- assumes borel: "u \<in> borel_measurable M" and "A \<in> sets M" and c: "c \<noteq> \<omega>"
- shows "\<mu> ({x\<in>space M. 1 \<le> c * u x} \<inter> A) \<le> c * (\<integral>\<^isup>+ x. u x * indicator A x \<partial>M)"
- (is "\<mu> ?A \<le> _ * ?PI")
+lemma (in measure_space) positive_integral_translated_density:
+ assumes f: "f \<in> borel_measurable M" "AE x. 0 \<le> f x"
+ assumes g: "g \<in> borel_measurable M" "AE x. 0 \<le> g x"
+ and M': "M' = (M\<lparr> measure := \<lambda>A. (\<integral>\<^isup>+ x. f x * indicator A x \<partial>M)\<rparr>)"
+ shows "integral\<^isup>P M' g = (\<integral>\<^isup>+ x. f x * g x \<partial>M)"
proof -
- have "?A \<in> sets M"
- using `A \<in> sets M` borel by auto
- hence "\<mu> ?A = (\<integral>\<^isup>+ x. indicator ?A x \<partial>M)"
- using positive_integral_indicator by simp
- also have "\<dots> \<le> (\<integral>\<^isup>+ x. c * (u x * indicator A x) \<partial>M)"
- proof (rule positive_integral_mono)
- fix x assume "x \<in> space M"
- show "indicator ?A x \<le> c * (u x * indicator A x)"
- by (cases "x \<in> ?A") auto
- qed
- also have "\<dots> = c * (\<integral>\<^isup>+ x. u x * indicator A x \<partial>M)"
- using assms
- by (auto intro!: positive_integral_cmult borel_measurable_indicator)
- finally show ?thesis .
+ from measure_space_density[OF f M']
+ interpret T: measure_space M' .
+ have borel[simp]:
+ "borel_measurable M' = borel_measurable M"
+ "simple_function M' = simple_function M"
+ unfolding measurable_def simple_function_def_raw by (auto simp: M')
+ from borel_measurable_implies_simple_function_sequence'[OF g(1)] guess G . note G = this
+ note G' = borel_measurable_simple_function[OF this(1)] simple_functionD[OF G(1)]
+ note G'(2)[simp]
+ { fix P have "AE x. P x \<Longrightarrow> AE x in M'. P x"
+ using positive_integral_null_set[of _ f]
+ unfolding T.almost_everywhere_def almost_everywhere_def
+ by (auto simp: M') }
+ note ac = this
+ from G(4) g(2) have G_M': "AE x in M'. (SUP i. G i x) = g x"
+ by (auto intro!: ac split: split_max)
+ { fix i
+ let "?I y x" = "indicator (G i -` {y} \<inter> space M) x"
+ { fix x assume *: "x \<in> space M" "0 \<le> f x" "0 \<le> g x"
+ then have [simp]: "G i ` space M \<inter> {y. G i x = y \<and> x \<in> space M} = {G i x}" by auto
+ from * G' G have "(\<Sum>y\<in>G i`space M. y * (f x * ?I y x)) = f x * (\<Sum>y\<in>G i`space M. (y * ?I y x))"
+ by (subst setsum_extreal_right_distrib) (auto simp: ac_simps)
+ also have "\<dots> = f x * G i x"
+ by (simp add: indicator_def if_distrib setsum_cases)
+ finally have "(\<Sum>y\<in>G i`space M. y * (f x * ?I y x)) = f x * G i x" . }
+ note to_singleton = this
+ have "integral\<^isup>P M' (G i) = integral\<^isup>S M' (G i)"
+ using G T.positive_integral_eq_simple_integral by simp
+ also have "\<dots> = (\<Sum>y\<in>G i`space M. y * (\<integral>\<^isup>+x. f x * ?I y x \<partial>M))"
+ unfolding simple_integral_def M' by simp
+ also have "\<dots> = (\<Sum>y\<in>G i`space M. (\<integral>\<^isup>+x. y * (f x * ?I y x) \<partial>M))"
+ using f G' G by (auto intro!: setsum_cong positive_integral_cmult[symmetric])
+ also have "\<dots> = (\<integral>\<^isup>+x. (\<Sum>y\<in>G i`space M. y * (f x * ?I y x)) \<partial>M)"
+ using f G' G by (auto intro!: positive_integral_setsum[symmetric])
+ finally have "integral\<^isup>P M' (G i) = (\<integral>\<^isup>+x. f x * G i x \<partial>M)"
+ using f g G' to_singleton by (auto intro!: positive_integral_cong_AE) }
+ note [simp] = this
+ have "integral\<^isup>P M' g = (SUP i. integral\<^isup>P M' (G i))" using G'(1) G_M'(1) G
+ using T.positive_integral_monotone_convergence_SUP[symmetric, OF `incseq G`]
+ by (simp cong: T.positive_integral_cong_AE)
+ also have "\<dots> = (SUP i. (\<integral>\<^isup>+x. f x * G i x \<partial>M))" by simp
+ also have "\<dots> = (\<integral>\<^isup>+x. (SUP i. f x * G i x) \<partial>M)"
+ using f G' G(2)[THEN incseq_SucD] G
+ by (intro positive_integral_monotone_convergence_SUP_AE[symmetric])
+ (auto simp: extreal_mult_left_mono le_fun_def extreal_zero_le_0_iff)
+ also have "\<dots> = (\<integral>\<^isup>+x. f x * g x \<partial>M)" using f G' G g
+ by (intro positive_integral_cong_AE)
+ (auto simp add: SUPR_extreal_cmult split: split_max)
+ finally show "integral\<^isup>P M' g = (\<integral>\<^isup>+x. f x * g x \<partial>M)" .
qed
lemma (in measure_space) positive_integral_0_iff:
- assumes borel: "u \<in> borel_measurable M"
+ assumes u: "u \<in> borel_measurable M" and pos: "AE x. 0 \<le> u x"
shows "integral\<^isup>P M u = 0 \<longleftrightarrow> \<mu> {x\<in>space M. u x \<noteq> 0} = 0"
(is "_ \<longleftrightarrow> \<mu> ?A = 0")
proof -
- have A: "?A \<in> sets M" using borel by auto
- have u: "(\<integral>\<^isup>+ x. u x * indicator ?A x \<partial>M) = integral\<^isup>P M u"
+ have u_eq: "(\<integral>\<^isup>+ x. u x * indicator ?A x \<partial>M) = integral\<^isup>P M u"
by (auto intro!: positive_integral_cong simp: indicator_def)
-
show ?thesis
proof
assume "\<mu> ?A = 0"
- hence "?A \<in> null_sets" using `?A \<in> sets M` by auto
- from positive_integral_null_set[OF this]
- have "0 = (\<integral>\<^isup>+ x. u x * indicator ?A x \<partial>M)" by simp
- thus "integral\<^isup>P M u = 0" unfolding u by simp
+ with positive_integral_null_set[of ?A u] u
+ show "integral\<^isup>P M u = 0" by (simp add: u_eq)
next
+ { fix r :: extreal and n :: nat assume gt_1: "1 \<le> real n * r"
+ then have "0 < real n * r" by (cases r) (auto split: split_if_asm simp: one_extreal_def)
+ then have "0 \<le> r" by (auto simp add: extreal_zero_less_0_iff) }
+ note gt_1 = this
assume *: "integral\<^isup>P M u = 0"
- let "?M n" = "{x \<in> space M. 1 \<le> of_nat n * u x}"
+ let "?M n" = "{x \<in> space M. 1 \<le> real (n::nat) * u x}"
have "0 = (SUP n. \<mu> (?M n \<inter> ?A))"
proof -
- { fix n
- from positive_integral_Markov_inequality[OF borel `?A \<in> sets M`, of "of_nat n"]
- have "\<mu> (?M n \<inter> ?A) = 0" unfolding * u by simp }
+ { fix n :: nat
+ from positive_integral_Markov_inequality[OF u pos, of ?A "extreal (real n)"]
+ have "\<mu> (?M n \<inter> ?A) \<le> 0" unfolding u_eq * using u by simp
+ moreover have "0 \<le> \<mu> (?M n \<inter> ?A)" using u by auto
+ ultimately have "\<mu> (?M n \<inter> ?A) = 0" by auto }
thus ?thesis by simp
qed
also have "\<dots> = \<mu> (\<Union>n. ?M n \<inter> ?A)"
proof (safe intro!: continuity_from_below)
fix n show "?M n \<inter> ?A \<in> sets M"
- using borel by (auto intro!: Int)
+ using u by (auto intro!: Int)
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!: pextreal_mult_cancel)
- finally show "1 \<le> of_nat (Suc n) * u x" .
+ show "incseq (\<lambda>n. {x \<in> space M. 1 \<le> real n * u x} \<inter> {x \<in> space M. u x \<noteq> 0})"
+ proof (safe intro!: incseq_SucI)
+ fix n :: nat and x
+ assume *: "1 \<le> real n * u x"
+ also from gt_1[OF this] have "real n * u x \<le> real (Suc n) * u x"
+ using `0 \<le> u x` by (auto intro!: extreal_mult_right_mono)
+ finally show "1 \<le> real (Suc n) * u x" by auto
+ qed
qed
- also have "\<dots> = \<mu> ?A"
- proof (safe intro!: arg_cong[where f="\<mu>"])
- fix x assume "u x \<noteq> 0" and [simp, intro]: "x \<in> space M"
+ also have "\<dots> = \<mu> {x\<in>space M. 0 < u x}"
+ proof (safe intro!: arg_cong[where f="\<mu>"] dest!: gt_1)
+ fix x assume "0 < u x" and [simp, intro]: "x \<in> space M"
show "x \<in> (\<Union>n. ?M n \<inter> ?A)"
proof (cases "u x")
- case (preal r)
- obtain j where "1 / r \<le> of_nat j" using ex_le_of_nat ..
- hence "1 / r * r \<le> of_nat j * r" using preal unfolding mult_le_cancel_right by auto
- hence "1 \<le> of_nat j * r" using preal `u x \<noteq> 0` by auto
- thus ?thesis using `u x \<noteq> 0` preal by (auto simp: real_of_nat_def[symmetric])
- qed auto
- qed
- finally show "\<mu> ?A = 0" by simp
+ case (real r) with `0 < u x` have "0 < r" by auto
+ obtain j :: nat where "1 / r \<le> real j" using real_arch_simple ..
+ hence "1 / r * r \<le> real j * r" unfolding mult_le_cancel_right using `0 < r` by auto
+ hence "1 \<le> real j * r" using real `0 < r` by auto
+ thus ?thesis using `0 < r` real by (auto simp: one_extreal_def)
+ qed (insert `0 < u x`, auto)
+ qed auto
+ finally have "\<mu> {x\<in>space M. 0 < u x} = 0" by simp
+ moreover
+ from pos have "AE x. \<not> (u x < 0)" by auto
+ then have "\<mu> {x\<in>space M. u x < 0} = 0"
+ using AE_iff_null_set u by auto
+ moreover have "\<mu> {x\<in>space M. u x \<noteq> 0} = \<mu> {x\<in>space M. u x < 0} + \<mu> {x\<in>space M. 0 < u x}"
+ using u by (subst measure_additive) (auto intro!: arg_cong[where f=\<mu>])
+ ultimately show "\<mu> ?A = 0" by simp
qed
qed
lemma (in measure_space) positive_integral_0_iff_AE:
assumes u: "u \<in> borel_measurable M"
- shows "integral\<^isup>P M u = 0 \<longleftrightarrow> (AE x. u x = 0)"
+ shows "integral\<^isup>P M u = 0 \<longleftrightarrow> (AE x. u x \<le> 0)"
proof -
- have sets: "{x\<in>space M. u x \<noteq> 0} \<in> sets M"
+ have sets: "{x\<in>space M. max 0 (u x) \<noteq> 0} \<in> sets M"
using u by auto
- then show ?thesis
- using positive_integral_0_iff[OF u] AE_iff_null_set[OF sets] by auto
+ from positive_integral_0_iff[of "\<lambda>x. max 0 (u x)"]
+ have "integral\<^isup>P M u = 0 \<longleftrightarrow> (AE x. max 0 (u x) = 0)"
+ unfolding positive_integral_max_0
+ using AE_iff_null_set[OF sets] u by auto
+ also have "\<dots> \<longleftrightarrow> (AE x. u x \<le> 0)" by (auto split: split_max)
+ finally show ?thesis .
qed
lemma (in measure_space) positive_integral_restricted:
- assumes "A \<in> sets M"
+ assumes A: "A \<in> sets M"
shows "integral\<^isup>P (restricted_space A) f = (\<integral>\<^isup>+ x. f x * indicator A x \<partial>M)"
(is "integral\<^isup>P ?R f = integral\<^isup>P M ?f")
proof -
- have msR: "measure_space ?R" by (rule restricted_measure_space[OF `A \<in> sets M`])
- then interpret R: measure_space ?R .
- have saR: "sigma_algebra ?R" by fact
- have *: "integral\<^isup>P ?R f = integral\<^isup>P ?R ?f"
- by (intro R.positive_integral_cong) auto
+ interpret R: measure_space ?R
+ by (rule restricted_measure_space) fact
+ let "?I g x" = "g x * indicator A x :: extreal"
show ?thesis
- unfolding * positive_integral_def
- unfolding simple_function_restricted[OF `A \<in> sets M`]
- apply (simp add: SUPR_def)
- apply (rule arg_cong[where f=Sup])
- proof (auto simp add: image_iff simple_integral_restricted[OF `A \<in> sets M`])
- fix g assume "simple_function M (\<lambda>x. g x * indicator A x)"
- "g \<le> f"
- then show "\<exists>x. simple_function M x \<and> x \<le> (\<lambda>x. f x * indicator A x) \<and>
- (\<integral>\<^isup>Sx. g x * indicator A x \<partial>M) = integral\<^isup>S M x"
- apply (rule_tac exI[of _ "\<lambda>x. g x * indicator A x"])
- by (auto simp: indicator_def le_fun_def)
+ unfolding positive_integral_def
+ unfolding simple_function_restricted[OF A]
+ unfolding AE_restricted[OF A]
+ proof (safe intro!: SUPR_eq)
+ fix g assume g: "simple_function M (?I g)" and le: "g \<le> max 0 \<circ> f"
+ show "\<exists>j\<in>{g. simple_function M g \<and> g \<le> max 0 \<circ> ?I f}.
+ integral\<^isup>S (restricted_space A) g \<le> integral\<^isup>S M j"
+ proof (safe intro!: bexI[of _ "?I g"])
+ show "integral\<^isup>S (restricted_space A) g \<le> integral\<^isup>S M (?I g)"
+ using g A by (simp add: simple_integral_restricted)
+ show "?I g \<le> max 0 \<circ> ?I f"
+ using le by (auto simp: le_fun_def max_def indicator_def split: split_if_asm)
+ qed fact
next
- fix g assume g: "simple_function M g" "g \<le> (\<lambda>x. f x * indicator A x)"
- then have *: "(\<lambda>x. g x * indicator A x) = g"
- "\<And>x. g x * indicator A x = g x"
- "\<And>x. g x \<le> f x"
- by (auto simp: le_fun_def fun_eq_iff indicator_def split: split_if_asm)
- from g show "\<exists>x. simple_function M (\<lambda>xa. x xa * indicator A xa) \<and> x \<le> f \<and>
- integral\<^isup>S M g = integral\<^isup>S M (\<lambda>xa. x xa * indicator A xa)"
- using `A \<in> sets M`[THEN sets_into_space]
- apply (rule_tac exI[of _ "\<lambda>x. g x * indicator A x"])
- by (fastsimp simp: le_fun_def *)
+ fix g assume g: "simple_function M g" and le: "g \<le> max 0 \<circ> ?I f"
+ show "\<exists>i\<in>{g. simple_function M (?I g) \<and> g \<le> max 0 \<circ> f}.
+ integral\<^isup>S M g \<le> integral\<^isup>S (restricted_space A) i"
+ proof (safe intro!: bexI[of _ "?I g"])
+ show "?I g \<le> max 0 \<circ> f"
+ using le by (auto simp: le_fun_def max_def indicator_def split: split_if_asm)
+ from le have "\<And>x. g x \<le> ?I (?I g) x"
+ by (auto simp: le_fun_def max_def indicator_def split: split_if_asm)
+ then show "integral\<^isup>S M g \<le> integral\<^isup>S (restricted_space A) (?I g)"
+ using A g by (auto intro!: simple_integral_mono simp: simple_integral_restricted)
+ show "simple_function M (?I (?I g))" using g A by auto
+ qed
qed
qed
lemma (in measure_space) positive_integral_subalgebra:
- assumes borel: "f \<in> borel_measurable N"
+ assumes f: "f \<in> borel_measurable N" "AE x in N. 0 \<le> f x"
and N: "sets N \<subseteq> sets M" "space N = space M" "\<And>A. A \<in> sets N \<Longrightarrow> measure N A = \<mu> A"
and sa: "sigma_algebra N"
shows "integral\<^isup>P N f = integral\<^isup>P M f"
proof -
interpret N: measure_space N using measure_space_subalgebra[OF sa N] .
- from N.borel_measurable_implies_simple_function_sequence[OF borel]
- obtain fs where Nsf: "\<And>i. simple_function N (fs i)" and "fs \<up> f" by blast
- then have sf: "\<And>i. simple_function M (fs i)"
- using simple_function_subalgebra[OF _ N(1,2)] by blast
- from N.positive_integral_isoton_simple[OF `fs \<up> f` Nsf]
+ from N.borel_measurable_implies_simple_function_sequence'[OF f(1)] guess fs . note fs = this
+ note sf = simple_function_subalgebra[OF fs(1) N(1,2)]
+ from N.positive_integral_monotone_convergence_simple[OF fs(2,5,1), symmetric]
have "integral\<^isup>P N f = (SUP i. \<Sum>x\<in>fs i ` space M. x * N.\<mu> (fs i -` {x} \<inter> space M))"
- unfolding isoton_def simple_integral_def `space N = space M` by simp
+ unfolding fs(4) positive_integral_max_0
+ unfolding simple_integral_def `space N = space M` by simp
also have "\<dots> = (SUP i. \<Sum>x\<in>fs i ` space M. x * \<mu> (fs i -` {x} \<inter> space M))"
- using N N.simple_functionD(2)[OF Nsf] unfolding `space N = space M` by auto
+ using N N.simple_functionD(2)[OF fs(1)] unfolding `space N = space M` by auto
also have "\<dots> = integral\<^isup>P M f"
- using positive_integral_isoton_simple[OF `fs \<up> f` sf]
- unfolding isoton_def simple_integral_def `space N = space M` by simp
+ using positive_integral_monotone_convergence_simple[OF fs(2,5) sf, symmetric]
+ unfolding fs(4) positive_integral_max_0
+ unfolding simple_integral_def `space N = space M` by simp
finally show ?thesis .
qed
@@ -1555,16 +1668,15 @@
definition integrable where
"integrable M f \<longleftrightarrow> f \<in> borel_measurable M \<and>
- (\<integral>\<^isup>+ x. Real (f x) \<partial>M) \<noteq> \<omega> \<and>
- (\<integral>\<^isup>+ x. Real (- f x) \<partial>M) \<noteq> \<omega>"
+ (\<integral>\<^isup>+ x. extreal (f x) \<partial>M) \<noteq> \<infinity> \<and> (\<integral>\<^isup>+ x. extreal (- f x) \<partial>M) \<noteq> \<infinity>"
lemma integrableD[dest]:
assumes "integrable M f"
- shows "f \<in> borel_measurable M" "(\<integral>\<^isup>+ x. Real (f x) \<partial>M) \<noteq> \<omega>" "(\<integral>\<^isup>+ x. Real (- f x) \<partial>M) \<noteq> \<omega>"
+ shows "f \<in> borel_measurable M" "(\<integral>\<^isup>+ x. extreal (f x) \<partial>M) \<noteq> \<infinity>" "(\<integral>\<^isup>+ x. extreal (- f x) \<partial>M) \<noteq> \<infinity>"
using assms unfolding integrable_def by auto
definition lebesgue_integral_def:
- "integral\<^isup>L M f = real ((\<integral>\<^isup>+ x. Real (f x) \<partial>M)) - real ((\<integral>\<^isup>+ x. Real (- f x) \<partial>M))"
+ "integral\<^isup>L M f = real ((\<integral>\<^isup>+ x. extreal (f x) \<partial>M)) - real ((\<integral>\<^isup>+ x. extreal (- f x) \<partial>M))"
syntax
"_lebesgue_integral" :: "'a \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> ('a, 'b) measure_space_scheme \<Rightarrow> real" ("\<integral> _. _ \<partial>_" [60,61] 110)
@@ -1572,6 +1684,17 @@
translations
"\<integral> x. f \<partial>M" == "CONST integral\<^isup>L M (%x. f)"
+lemma (in measure_space) integrableE:
+ assumes "integrable M f"
+ obtains r q where
+ "(\<integral>\<^isup>+x. extreal (f x)\<partial>M) = extreal r"
+ "(\<integral>\<^isup>+x. extreal (-f x)\<partial>M) = extreal q"
+ "f \<in> borel_measurable M" "integral\<^isup>L M f = r - q"
+ using assms unfolding integrable_def lebesgue_integral_def
+ using positive_integral_positive[of "\<lambda>x. extreal (f x)"]
+ using positive_integral_positive[of "\<lambda>x. extreal (-f x)"]
+ by (cases rule: extreal2_cases[of "(\<integral>\<^isup>+x. extreal (-f x)\<partial>M)" "(\<integral>\<^isup>+x. extreal (f x)\<partial>M)"]) auto
+
lemma (in measure_space) integral_cong:
assumes "\<And>x. x \<in> space M \<Longrightarrow> f x = g x"
shows "integral\<^isup>L M f = integral\<^isup>L M g"
@@ -1580,21 +1703,16 @@
lemma (in measure_space) integral_cong_measure:
assumes "\<And>A. A \<in> sets M \<Longrightarrow> measure N A = \<mu> A" "sets N = sets M" "space N = space M"
shows "integral\<^isup>L N f = integral\<^isup>L M f"
-proof -
- interpret v: measure_space N
- by (rule measure_space_cong) fact+
- show ?thesis
- by (simp add: positive_integral_cong_measure[OF assms] lebesgue_integral_def)
-qed
+ by (simp add: positive_integral_cong_measure[OF assms] lebesgue_integral_def)
lemma (in measure_space) integral_cong_AE:
assumes cong: "AE x. f x = g x"
shows "integral\<^isup>L M f = integral\<^isup>L M g"
proof -
- have "AE x. Real (f x) = Real (g x)" using cong by auto
- moreover have "AE x. Real (- f x) = Real (- g x)" using cong by auto
- ultimately show ?thesis
- by (simp cong: positive_integral_cong_AE add: lebesgue_integral_def)
+ have *: "AE x. extreal (f x) = extreal (g x)"
+ "AE x. extreal (- f x) = extreal (- g x)" using cong by auto
+ show ?thesis
+ unfolding *[THEN positive_integral_cong_AE] lebesgue_integral_def ..
qed
lemma (in measure_space) integrable_cong:
@@ -1602,11 +1720,14 @@
by (simp cong: positive_integral_cong measurable_cong add: integrable_def)
lemma (in measure_space) integral_eq_positive_integral:
- assumes "\<And>x. 0 \<le> f x"
- shows "integral\<^isup>L M f = real (\<integral>\<^isup>+ x. Real (f x) \<partial>M)"
+ assumes f: "\<And>x. 0 \<le> f x"
+ shows "integral\<^isup>L M f = real (\<integral>\<^isup>+ x. extreal (f x) \<partial>M)"
proof -
- have "\<And>x. Real (- f x) = 0" using assms by simp
- thus ?thesis by (simp del: Real_eq_0 add: lebesgue_integral_def)
+ { fix x have "max 0 (extreal (- f x)) = 0" using f[of x] by (simp split: split_max) }
+ then have "0 = (\<integral>\<^isup>+ x. max 0 (extreal (- f x)) \<partial>M)" by simp
+ also have "\<dots> = (\<integral>\<^isup>+ x. extreal (- f x) \<partial>M)" unfolding positive_integral_max_0 ..
+ finally show ?thesis
+ unfolding lebesgue_integral_def by simp
qed
lemma (in measure_space) integral_vimage:
@@ -1616,7 +1737,7 @@
proof -
interpret T: measure_space M' by (rule measure_space_vimage[OF T])
from measurable_comp[OF measure_preservingD2[OF T(2)], of f borel]
- have borel: "(\<lambda>x. Real (f x)) \<in> borel_measurable M'" "(\<lambda>x. Real (- f x)) \<in> borel_measurable M'"
+ have borel: "(\<lambda>x. extreal (f x)) \<in> borel_measurable M'" "(\<lambda>x. extreal (- f x)) \<in> borel_measurable M'"
and "(\<lambda>x. f (T x)) \<in> borel_measurable M"
using f by (auto simp: comp_def)
then show ?thesis
@@ -1631,7 +1752,7 @@
proof -
interpret T: measure_space M' by (rule measure_space_vimage[OF T])
from measurable_comp[OF measure_preservingD2[OF T(2)], of f borel]
- have borel: "(\<lambda>x. Real (f x)) \<in> borel_measurable M'" "(\<lambda>x. Real (- f x)) \<in> borel_measurable M'"
+ have borel: "(\<lambda>x. extreal (f x)) \<in> borel_measurable M'" "(\<lambda>x. extreal (- f x)) \<in> borel_measurable M'"
and "(\<lambda>x. f (T x)) \<in> borel_measurable M"
using f by (auto simp: comp_def)
then show ?thesis
@@ -1649,10 +1770,10 @@
and f_def: "\<And>x. f x = u x - v x" and pos: "\<And>x. 0 \<le> u x" "\<And>x. 0 \<le> v x"
shows "integrable M f" and "integral\<^isup>L M f = integral\<^isup>L M u - integral\<^isup>L M v"
proof -
- let "?f x" = "Real (f x)"
- let "?mf x" = "Real (- f x)"
- let "?u x" = "Real (u x)"
- let "?v x" = "Real (v x)"
+ let "?f x" = "max 0 (extreal (f x))"
+ let "?mf x" = "max 0 (extreal (- f x))"
+ let "?u x" = "max 0 (extreal (u x))"
+ let "?v x" = "max 0 (extreal (v x))"
from borel_measurable_diff[of u v] integrable
have f_borel: "?f \<in> borel_measurable M" and
@@ -1662,73 +1783,62 @@
"f \<in> borel_measurable M"
by (auto simp: f_def[symmetric] integrable_def)
- have "(\<integral>\<^isup>+ x. Real (u x - v x) \<partial>M) \<le> integral\<^isup>P M ?u"
- using pos by (auto intro!: positive_integral_mono)
- moreover have "(\<integral>\<^isup>+ x. Real (v x - u x) \<partial>M) \<le> integral\<^isup>P M ?v"
- using pos by (auto intro!: positive_integral_mono)
+ have "(\<integral>\<^isup>+ x. extreal (u x - v x) \<partial>M) \<le> integral\<^isup>P M ?u"
+ using pos by (auto intro!: positive_integral_mono simp: positive_integral_max_0)
+ moreover have "(\<integral>\<^isup>+ x. extreal (v x - u x) \<partial>M) \<le> integral\<^isup>P M ?v"
+ using pos by (auto intro!: positive_integral_mono simp: positive_integral_max_0)
ultimately show f: "integrable M f"
using `integrable M u` `integrable M v` `f \<in> borel_measurable M`
- by (auto simp: integrable_def f_def)
- hence mf: "integrable M (\<lambda>x. - f x)" ..
-
- have *: "\<And>x. Real (- v x) = 0" "\<And>x. Real (- u x) = 0"
- using pos by auto
+ by (auto simp: integrable_def f_def positive_integral_max_0)
have "\<And>x. ?u x + ?mf x = ?v x + ?f x"
- unfolding f_def using pos by simp
- hence "(\<integral>\<^isup>+ x. ?u x + ?mf x \<partial>M) = (\<integral>\<^isup>+ x. ?v x + ?f x \<partial>M)" by simp
- hence "real (integral\<^isup>P M ?u + integral\<^isup>P M ?mf) =
+ unfolding f_def using pos by (simp split: split_max)
+ then have "(\<integral>\<^isup>+ x. ?u x + ?mf x \<partial>M) = (\<integral>\<^isup>+ x. ?v x + ?f x \<partial>M)" by simp
+ then have "real (integral\<^isup>P M ?u + integral\<^isup>P M ?mf) =
real (integral\<^isup>P M ?v + integral\<^isup>P M ?f)"
- using positive_integral_add[OF u_borel mf_borel]
- using positive_integral_add[OF v_borel f_borel]
+ using positive_integral_add[OF u_borel _ mf_borel]
+ using positive_integral_add[OF v_borel _ f_borel]
by auto
then show "integral\<^isup>L M f = integral\<^isup>L M u - integral\<^isup>L M v"
- using f mf `integrable M u` `integrable M v`
- unfolding lebesgue_integral_def integrable_def *
- by (cases "integral\<^isup>P M ?f", cases "integral\<^isup>P M ?mf", cases "integral\<^isup>P M ?v", cases "integral\<^isup>P M ?u")
- (auto simp add: field_simps)
+ unfolding positive_integral_max_0
+ unfolding pos[THEN integral_eq_positive_integral]
+ using integrable f by (auto elim!: integrableE)
qed
lemma (in measure_space) integral_linear:
assumes "integrable M f" "integrable M g" and "0 \<le> a"
shows "integrable M (\<lambda>t. a * f t + g t)"
- and "(\<integral> t. a * f t + g t \<partial>M) = a * integral\<^isup>L M f + integral\<^isup>L M g"
+ and "(\<integral> t. a * f t + g t \<partial>M) = a * integral\<^isup>L M f + integral\<^isup>L M g" (is ?EQ)
proof -
- let ?PI = "integral\<^isup>P M"
- let "?f x" = "Real (f x)"
- let "?g x" = "Real (g x)"
- let "?mf x" = "Real (- f x)"
- let "?mg x" = "Real (- g x)"
+ let "?f x" = "max 0 (extreal (f x))"
+ let "?g x" = "max 0 (extreal (g x))"
+ let "?mf x" = "max 0 (extreal (- f x))"
+ let "?mg x" = "max 0 (extreal (- g x))"
let "?p t" = "max 0 (a * f t) + max 0 (g t)"
let "?n t" = "max 0 (- (a * f t)) + max 0 (- g t)"
- have pos: "?f \<in> borel_measurable M" "?g \<in> borel_measurable M"
- and neg: "?mf \<in> borel_measurable M" "?mg \<in> borel_measurable M"
- and p: "?p \<in> borel_measurable M"
- and n: "?n \<in> borel_measurable M"
- using assms by (simp_all add: integrable_def)
+ from assms have linear:
+ "(\<integral>\<^isup>+ x. extreal a * ?f x + ?g x \<partial>M) = extreal a * integral\<^isup>P M ?f + integral\<^isup>P M ?g"
+ "(\<integral>\<^isup>+ x. extreal a * ?mf x + ?mg x \<partial>M) = extreal a * integral\<^isup>P M ?mf + integral\<^isup>P M ?mg"
+ by (auto intro!: positive_integral_linear simp: integrable_def)
- have *: "\<And>x. Real (?p x) = Real a * ?f x + ?g x"
- "\<And>x. Real (?n x) = Real a * ?mf x + ?mg x"
- "\<And>x. Real (- ?p x) = 0"
- "\<And>x. Real (- ?n x) = 0"
- using `0 \<le> a` by (auto simp: max_def min_def zero_le_mult_iff mult_le_0_iff add_nonpos_nonpos)
-
- note linear =
- positive_integral_linear[OF pos]
- positive_integral_linear[OF neg]
+ have *: "(\<integral>\<^isup>+x. extreal (- ?p x) \<partial>M) = 0" "(\<integral>\<^isup>+x. extreal (- ?n x) \<partial>M) = 0"
+ using `0 \<le> a` assms by (auto simp: positive_integral_0_iff_AE integrable_def)
+ have **: "\<And>x. extreal a * ?f x + ?g x = max 0 (extreal (?p x))"
+ "\<And>x. extreal a * ?mf x + ?mg x = max 0 (extreal (?n x))"
+ using `0 \<le> a` by (auto split: split_max simp: zero_le_mult_iff mult_le_0_iff)
have "integrable M ?p" "integrable M ?n"
"\<And>t. a * f t + g t = ?p t - ?n t" "\<And>t. 0 \<le> ?p t" "\<And>t. 0 \<le> ?n t"
- using assms p n unfolding integrable_def * linear by auto
+ using linear assms unfolding integrable_def ** *
+ by (auto simp: positive_integral_max_0)
note diff = integral_of_positive_diff[OF this]
show "integrable M (\<lambda>t. a * f t + g t)" by (rule diff)
-
- from assms show "(\<integral> t. a * f t + g t \<partial>M) = a * integral\<^isup>L M f + integral\<^isup>L M g"
- unfolding diff(2) unfolding lebesgue_integral_def * linear integrable_def
- by (cases "?PI ?f", cases "?PI ?mf", cases "?PI ?g", cases "?PI ?mg")
- (auto simp add: field_simps zero_le_mult_iff)
+ from assms linear show ?EQ
+ unfolding diff(2) ** positive_integral_max_0
+ unfolding lebesgue_integral_def *
+ by (auto elim!: integrableE simp: field_simps)
qed
lemma (in measure_space) integral_add[simp, intro]:
@@ -1772,13 +1882,13 @@
and mono: "AE t. f t \<le> g t"
shows "integral\<^isup>L M f \<le> integral\<^isup>L M g"
proof -
- have "AE x. Real (f x) \<le> Real (g x)"
+ have "AE x. extreal (f x) \<le> extreal (g x)"
using mono by auto
- moreover have "AE x. Real (- g x) \<le> Real (- f x)"
+ moreover have "AE x. extreal (- g x) \<le> extreal (- f x)"
using mono by auto
ultimately show ?thesis using fg
- by (auto simp: lebesgue_integral_def integrable_def diff_minus
- intro!: add_mono real_of_pextreal_mono positive_integral_mono_AE)
+ by (auto intro!: add_mono positive_integral_mono_AE real_of_extreal_positive_mono
+ simp: positive_integral_positive lebesgue_integral_def diff_minus)
qed
lemma (in measure_space) integral_mono:
@@ -1795,20 +1905,21 @@
by auto
lemma (in measure_space) integral_indicator[simp, intro]:
- assumes "a \<in> sets M" and "\<mu> a \<noteq> \<omega>"
- shows "integral\<^isup>L M (indicator a) = real (\<mu> a)" (is ?int)
- and "integrable M (indicator a)" (is ?able)
+ assumes "A \<in> sets M" and "\<mu> A \<noteq> \<infinity>"
+ shows "integral\<^isup>L M (indicator A) = real (\<mu> A)" (is ?int)
+ and "integrable M (indicator A)" (is ?able)
proof -
- have *:
- "\<And>A x. Real (indicator A x) = indicator A x"
- "\<And>A x. Real (- indicator A x) = 0" unfolding indicator_def by auto
+ from `A \<in> sets M` have *:
+ "\<And>x. extreal (indicator A x) = indicator A x"
+ "(\<integral>\<^isup>+x. extreal (- indicator A x) \<partial>M) = 0"
+ by (auto split: split_indicator simp: positive_integral_0_iff_AE one_extreal_def)
show ?int ?able
using assms unfolding lebesgue_integral_def integrable_def
by (auto simp: * positive_integral_indicator borel_measurable_indicator)
qed
lemma (in measure_space) integral_cmul_indicator:
- assumes "A \<in> sets M" and "c \<noteq> 0 \<Longrightarrow> \<mu> A \<noteq> \<omega>"
+ assumes "A \<in> sets M" and "c \<noteq> 0 \<Longrightarrow> \<mu> A \<noteq> \<infinity>"
shows "integrable M (\<lambda>x. c * indicator A x)" (is ?P)
and "(\<integral>x. c * indicator A x \<partial>M) = c * real (\<mu> A)" (is ?I)
proof -
@@ -1840,15 +1951,11 @@
assumes "integrable M f"
shows "integrable M (\<lambda> x. \<bar>f x\<bar>)"
proof -
- have *:
- "\<And>x. Real \<bar>f x\<bar> = Real (f x) + Real (- f x)"
- "\<And>x. Real (- \<bar>f x\<bar>) = 0" by auto
- have abs: "(\<lambda>x. \<bar>f x\<bar>) \<in> borel_measurable M" and
- f: "(\<lambda>x. Real (f x)) \<in> borel_measurable M"
- "(\<lambda>x. Real (- f x)) \<in> borel_measurable M"
- using assms unfolding integrable_def by auto
- from abs assms show ?thesis unfolding integrable_def *
- using positive_integral_linear[OF f, of 1] by simp
+ from assms have *: "(\<integral>\<^isup>+x. extreal (- \<bar>f x\<bar>)\<partial>M) = 0"
+ "\<And>x. extreal \<bar>f x\<bar> = max 0 (extreal (f x)) + max 0 (extreal (- f x))"
+ by (auto simp: integrable_def positive_integral_0_iff_AE split: split_max)
+ with assms show ?thesis
+ by (simp add: positive_integral_add positive_integral_max_0 integrable_def)
qed
lemma (in measure_space) integral_subalgebra:
@@ -1858,12 +1965,13 @@
and "integral\<^isup>L N f = integral\<^isup>L M f" (is ?I)
proof -
interpret N: measure_space N using measure_space_subalgebra[OF sa N] .
- have "(\<lambda>x. Real (f x)) \<in> borel_measurable N" "(\<lambda>x. Real (- f x)) \<in> borel_measurable N"
- using borel by auto
- note * = this[THEN positive_integral_subalgebra[OF _ N sa]]
- have "f \<in> borel_measurable M \<longleftrightarrow> f \<in> borel_measurable N"
+ have "(\<integral>\<^isup>+ x. max 0 (extreal (f x)) \<partial>N) = (\<integral>\<^isup>+ x. max 0 (extreal (f x)) \<partial>M)"
+ "(\<integral>\<^isup>+ x. max 0 (extreal (- f x)) \<partial>N) = (\<integral>\<^isup>+ x. max 0 (extreal (- f x)) \<partial>M)"
+ using borel by (auto intro!: positive_integral_subalgebra N sa)
+ moreover have "f \<in> borel_measurable M \<longleftrightarrow> f \<in> borel_measurable N"
using assms unfolding measurable_def by auto
- then show ?P ?I by (auto simp: * integrable_def lebesgue_integral_def)
+ ultimately show ?P ?I
+ by (auto simp: integrable_def lebesgue_integral_def positive_integral_max_0)
qed
lemma (in measure_space) integrable_bound:
@@ -1873,21 +1981,21 @@
assumes borel: "g \<in> borel_measurable M"
shows "integrable M g"
proof -
- have "(\<integral>\<^isup>+ x. Real (g x) \<partial>M) \<le> (\<integral>\<^isup>+ x. Real \<bar>g x\<bar> \<partial>M)"
+ have "(\<integral>\<^isup>+ x. extreal (g x) \<partial>M) \<le> (\<integral>\<^isup>+ x. extreal \<bar>g x\<bar> \<partial>M)"
by (auto intro!: positive_integral_mono)
- also have "\<dots> \<le> (\<integral>\<^isup>+ x. Real (f x) \<partial>M)"
+ also have "\<dots> \<le> (\<integral>\<^isup>+ x. extreal (f x) \<partial>M)"
using f by (auto intro!: positive_integral_mono)
- also have "\<dots> < \<omega>"
- using `integrable M f` unfolding integrable_def by (auto simp: pextreal_less_\<omega>)
- finally have pos: "(\<integral>\<^isup>+ x. Real (g x) \<partial>M) < \<omega>" .
+ also have "\<dots> < \<infinity>"
+ using `integrable M f` unfolding integrable_def by auto
+ finally have pos: "(\<integral>\<^isup>+ x. extreal (g x) \<partial>M) < \<infinity>" .
- have "(\<integral>\<^isup>+ x. Real (- g x) \<partial>M) \<le> (\<integral>\<^isup>+ x. Real (\<bar>g x\<bar>) \<partial>M)"
+ have "(\<integral>\<^isup>+ x. extreal (- g x) \<partial>M) \<le> (\<integral>\<^isup>+ x. extreal (\<bar>g x\<bar>) \<partial>M)"
by (auto intro!: positive_integral_mono)
- also have "\<dots> \<le> (\<integral>\<^isup>+ x. Real (f x) \<partial>M)"
+ also have "\<dots> \<le> (\<integral>\<^isup>+ x. extreal (f x) \<partial>M)"
using f by (auto intro!: positive_integral_mono)
- also have "\<dots> < \<omega>"
- using `integrable M f` unfolding integrable_def by (auto simp: pextreal_less_\<omega>)
- finally have neg: "(\<integral>\<^isup>+ x. Real (- g x) \<partial>M) < \<omega>" .
+ also have "\<dots> < \<infinity>"
+ using `integrable M f` unfolding integrable_def by auto
+ finally have neg: "(\<integral>\<^isup>+ x. extreal (- g x) \<partial>M) < \<infinity>" .
from neg pos borel show ?thesis
unfolding integrable_def by auto
@@ -1959,41 +2067,34 @@
by (simp add: mono_def incseq_def) }
note pos_u = this
- hence [simp]: "\<And>i x. Real (- f i x) = 0" "\<And>x. Real (- u x) = 0"
- using pos by auto
+ have SUP_F: "\<And>x. (SUP n. extreal (f n x)) = extreal (u x)"
+ unfolding SUP_eq_LIMSEQ[OF mono] by (rule lim)
- have SUP_F: "\<And>x. (SUP n. Real (f n x)) = Real (u x)"
- using mono pos pos_u lim by (rule SUP_eq_LIMSEQ[THEN iffD2])
-
- have borel_f: "\<And>i. (\<lambda>x. Real (f i x)) \<in> borel_measurable M"
+ have borel_f: "\<And>i. (\<lambda>x. extreal (f i x)) \<in> borel_measurable M"
using i unfolding integrable_def by auto
- hence "(\<lambda>x. SUP i. Real (f i x)) \<in> borel_measurable M"
+ hence "(\<lambda>x. SUP i. extreal (f i x)) \<in> borel_measurable M"
by auto
hence borel_u: "u \<in> borel_measurable M"
- using pos_u by (auto simp: borel_measurable_Real_eq SUP_F)
+ by (auto simp: borel_measurable_extreal_iff SUP_F)
- have integral_eq: "\<And>n. (\<integral>\<^isup>+ x. Real (f n x) \<partial>M) = Real (integral\<^isup>L M (f n))"
- using i unfolding lebesgue_integral_def integrable_def by (auto simp: Real_real)
+ hence [simp]: "\<And>i. (\<integral>\<^isup>+x. extreal (- f i x) \<partial>M) = 0" "(\<integral>\<^isup>+x. extreal (- u x) \<partial>M) = 0"
+ using i borel_u pos pos_u by (auto simp: positive_integral_0_iff_AE integrable_def)
+
+ have integral_eq: "\<And>n. (\<integral>\<^isup>+ x. extreal (f n x) \<partial>M) = extreal (integral\<^isup>L M (f n))"
+ using i positive_integral_positive by (auto simp: extreal_real lebesgue_integral_def integrable_def)
have pos_integral: "\<And>n. 0 \<le> integral\<^isup>L M (f n)"
using pos i by (auto simp: integral_positive)
hence "0 \<le> x"
using LIMSEQ_le_const[OF ilim, of 0] by auto
- have "(\<lambda>i. (\<integral>\<^isup>+ x. Real (f i x) \<partial>M)) \<up> (\<integral>\<^isup>+ x. Real (u x) \<partial>M)"
- proof (rule positive_integral_isoton)
- from SUP_F mono pos
- show "(\<lambda>i x. Real (f i x)) \<up> (\<lambda>x. Real (u x))"
- unfolding isoton_fun_expand by (auto simp: isoton_def mono_def)
- qed (rule borel_f)
- hence pI: "(\<integral>\<^isup>+ x. Real (u x) \<partial>M) = (SUP n. (\<integral>\<^isup>+ x. Real (f n x) \<partial>M))"
- unfolding isoton_def by simp
- also have "\<dots> = Real x" unfolding integral_eq
+ from mono pos i have pI: "(\<integral>\<^isup>+ x. extreal (u x) \<partial>M) = (SUP n. (\<integral>\<^isup>+ x. extreal (f n x) \<partial>M))"
+ by (auto intro!: positive_integral_monotone_convergence_SUP
+ simp: integrable_def incseq_mono incseq_Suc_iff le_fun_def SUP_F[symmetric])
+ also have "\<dots> = extreal x" unfolding integral_eq
proof (rule SUP_eq_LIMSEQ[THEN iffD2])
show "mono (\<lambda>n. integral\<^isup>L M (f n))"
using mono i by (auto simp: mono_def intro!: integral_mono)
- show "\<And>n. 0 \<le> integral\<^isup>L M (f n)" using pos_integral .
- show "0 \<le> x" using `0 \<le> x` .
show "(\<lambda>n. integral\<^isup>L M (f n)) ----> x" using ilim .
qed
finally show "integrable M u" "integral\<^isup>L M u = x" using borel_u `0 \<le> x`
@@ -2028,61 +2129,72 @@
assumes "integrable M f"
shows "(\<integral>x. \<bar>f x\<bar> \<partial>M) = 0 \<longleftrightarrow> \<mu> {x\<in>space M. f x \<noteq> 0} = 0"
proof -
- have *: "\<And>x. Real (- \<bar>f x\<bar>) = 0" by auto
+ have *: "(\<integral>\<^isup>+x. extreal (- \<bar>f x\<bar>) \<partial>M) = 0"
+ using assms by (auto simp: positive_integral_0_iff_AE integrable_def)
have "integrable M (\<lambda>x. \<bar>f x\<bar>)" using assms by (rule integrable_abs)
- hence "(\<lambda>x. Real (\<bar>f x\<bar>)) \<in> borel_measurable M"
- "(\<integral>\<^isup>+ x. Real \<bar>f x\<bar> \<partial>M) \<noteq> \<omega>" unfolding integrable_def by auto
+ hence "(\<lambda>x. extreal (\<bar>f x\<bar>)) \<in> borel_measurable M"
+ "(\<integral>\<^isup>+ x. extreal \<bar>f x\<bar> \<partial>M) \<noteq> \<infinity>" unfolding integrable_def by auto
from positive_integral_0_iff[OF this(1)] this(2)
show ?thesis unfolding lebesgue_integral_def *
- by (simp add: real_of_pextreal_eq_0)
+ using positive_integral_positive[of "\<lambda>x. extreal \<bar>f x\<bar>"]
+ by (auto simp add: real_of_extreal_eq_0)
qed
-lemma (in measure_space) positive_integral_omega:
- assumes "f \<in> borel_measurable M"
- and "integral\<^isup>P M f \<noteq> \<omega>"
- shows "\<mu> (f -` {\<omega>} \<inter> space M) = 0"
+lemma (in measure_space) positive_integral_PInf:
+ assumes f: "f \<in> borel_measurable M"
+ and not_Inf: "integral\<^isup>P M f \<noteq> \<infinity>"
+ shows "\<mu> (f -` {\<infinity>} \<inter> space M) = 0"
proof -
- have "\<omega> * \<mu> (f -` {\<omega>} \<inter> space M) = (\<integral>\<^isup>+ x. \<omega> * indicator (f -` {\<omega>} \<inter> space M) x \<partial>M)"
- using positive_integral_cmult_indicator[OF borel_measurable_vimage, OF assms(1), of \<omega> \<omega>] by simp
- also have "\<dots> \<le> integral\<^isup>P M f"
- by (auto intro!: positive_integral_mono simp: indicator_def)
- finally show ?thesis
- using assms(2) by (cases ?thesis) auto
+ have "\<infinity> * \<mu> (f -` {\<infinity>} \<inter> space M) = (\<integral>\<^isup>+ x. \<infinity> * indicator (f -` {\<infinity>} \<inter> space M) x \<partial>M)"
+ using f by (subst positive_integral_cmult_indicator) (auto simp: measurable_sets)
+ also have "\<dots> \<le> integral\<^isup>P M (\<lambda>x. max 0 (f x))"
+ by (auto intro!: positive_integral_mono simp: indicator_def max_def)
+ finally have "\<infinity> * \<mu> (f -` {\<infinity>} \<inter> space M) \<le> integral\<^isup>P M f"
+ by (simp add: positive_integral_max_0)
+ moreover have "0 \<le> \<mu> (f -` {\<infinity>} \<inter> space M)"
+ using f by (simp add: measurable_sets)
+ ultimately show ?thesis
+ using assms by (auto split: split_if_asm)
qed
-lemma (in measure_space) positive_integral_omega_AE:
- assumes "f \<in> borel_measurable M" "integral\<^isup>P M f \<noteq> \<omega>" shows "AE x. f x \<noteq> \<omega>"
+lemma (in measure_space) positive_integral_PInf_AE:
+ assumes "f \<in> borel_measurable M" "integral\<^isup>P M f \<noteq> \<infinity>" shows "AE x. f x \<noteq> \<infinity>"
proof (rule AE_I)
- show "\<mu> (f -` {\<omega>} \<inter> space M) = 0"
- by (rule positive_integral_omega[OF assms])
- show "f -` {\<omega>} \<inter> space M \<in> sets M"
+ show "\<mu> (f -` {\<infinity>} \<inter> space M) = 0"
+ by (rule positive_integral_PInf[OF assms])
+ show "f -` {\<infinity>} \<inter> space M \<in> sets M"
using assms by (auto intro: borel_measurable_vimage)
qed auto
-lemma (in measure_space) simple_integral_omega:
- assumes "simple_function M f"
- and "integral\<^isup>S M f \<noteq> \<omega>"
- shows "\<mu> (f -` {\<omega>} \<inter> space M) = 0"
-proof (rule positive_integral_omega)
+lemma (in measure_space) simple_integral_PInf:
+ assumes "simple_function M f" "\<And>x. 0 \<le> f x"
+ and "integral\<^isup>S M f \<noteq> \<infinity>"
+ shows "\<mu> (f -` {\<infinity>} \<inter> space M) = 0"
+proof (rule positive_integral_PInf)
show "f \<in> borel_measurable M" using assms by (auto intro: borel_measurable_simple_function)
- show "integral\<^isup>P M f \<noteq> \<omega>"
+ show "integral\<^isup>P M f \<noteq> \<infinity>"
using assms by (simp add: positive_integral_eq_simple_integral)
qed
lemma (in measure_space) integral_real:
- fixes f :: "'a \<Rightarrow> pextreal"
- assumes [simp]: "AE x. f x \<noteq> \<omega>"
- shows "(\<integral>x. real (f x) \<partial>M) = real (integral\<^isup>P M f)" (is ?plus)
- and "(\<integral>x. - real (f x) \<partial>M) = - real (integral\<^isup>P M f)" (is ?minus)
-proof -
- have "(\<integral>\<^isup>+ x. Real (real (f x)) \<partial>M) = integral\<^isup>P M f"
- by (auto intro!: positive_integral_cong_AE simp: Real_real)
- moreover
- have "(\<integral>\<^isup>+ x. Real (- real (f x)) \<partial>M) = (\<integral>\<^isup>+ x. 0 \<partial>M)"
- by (intro positive_integral_cong) auto
- ultimately show ?plus ?minus
- by (auto simp: lebesgue_integral_def integrable_def)
-qed
+ "AE x. \<bar>f x\<bar> \<noteq> \<infinity> \<Longrightarrow> (\<integral>x. real (f x) \<partial>M) = real (integral\<^isup>P M f) - real (integral\<^isup>P M (\<lambda>x. - f x))"
+ using assms unfolding lebesgue_integral_def
+ by (subst (1 2) positive_integral_cong_AE) (auto simp add: extreal_real)
+
+lemma liminf_extreal_cminus:
+ fixes f :: "nat \<Rightarrow> extreal" assumes "c \<noteq> -\<infinity>"
+ shows "liminf (\<lambda>x. c - f x) = c - limsup f"
+proof (cases c)
+ case PInf then show ?thesis by (simp add: Liminf_const)
+next
+ case (real r) then show ?thesis
+ unfolding liminf_SUPR_INFI limsup_INFI_SUPR
+ apply (subst INFI_extreal_cminus)
+ apply auto
+ apply (subst SUPR_extreal_cminus)
+ apply auto
+ done
+qed (insert `c \<noteq> -\<infinity>`, simp)
lemma (in measure_space) integral_dominated_convergence:
assumes u: "\<And>i. integrable M (u i)" and bound: "\<And>x j. x\<in>space M \<Longrightarrow> \<bar>u j x\<bar> \<le> w x"
@@ -2129,61 +2241,76 @@
finally have "\<bar>u j x - u' x\<bar> \<le> 2 * w x" by simp }
note diff_less_2w = this
- have PI_diff: "\<And>m n. (\<integral>\<^isup>+ x. Real (?diff (m + n) x) \<partial>M) =
- (\<integral>\<^isup>+ x. Real (2 * w x) \<partial>M) - (\<integral>\<^isup>+ x. Real \<bar>u (m + n) x - u' x\<bar> \<partial>M)"
+ have PI_diff: "\<And>n. (\<integral>\<^isup>+ x. extreal (?diff n x) \<partial>M) =
+ (\<integral>\<^isup>+ x. extreal (2 * w x) \<partial>M) - (\<integral>\<^isup>+ x. extreal \<bar>u n x - u' x\<bar> \<partial>M)"
using diff w diff_less_2w w_pos
by (subst positive_integral_diff[symmetric])
(auto simp: integrable_def intro!: positive_integral_cong)
have "integrable M (\<lambda>x. 2 * w x)"
using w by (auto intro: integral_cmult)
- hence I2w_fin: "(\<integral>\<^isup>+ x. Real (2 * w x) \<partial>M) \<noteq> \<omega>" and
- borel_2w: "(\<lambda>x. Real (2 * w x)) \<in> borel_measurable M"
+ hence I2w_fin: "(\<integral>\<^isup>+ x. extreal (2 * w x) \<partial>M) \<noteq> \<infinity>" and
+ borel_2w: "(\<lambda>x. extreal (2 * w x)) \<in> borel_measurable M"
unfolding integrable_def by auto
- have "(INF n. SUP m. (\<integral>\<^isup>+ x. Real \<bar>u (m + n) x - u' x\<bar> \<partial>M)) = 0" (is "?lim_SUP = 0")
+ have "limsup (\<lambda>n. \<integral>\<^isup>+ x. extreal \<bar>u n x - u' x\<bar> \<partial>M) = 0" (is "limsup ?f = 0")
proof cases
- assume eq_0: "(\<integral>\<^isup>+ x. Real (2 * w x) \<partial>M) = 0"
- have "\<And>i. (\<integral>\<^isup>+ x. Real \<bar>u i x - u' x\<bar> \<partial>M) \<le> (\<integral>\<^isup>+ x. Real (2 * w x) \<partial>M)"
- proof (rule positive_integral_mono)
- fix i x assume "x \<in> space M" from diff_less_2w[OF this, of i]
- show "Real \<bar>u i x - u' x\<bar> \<le> Real (2 * w x)" by auto
- qed
- hence "\<And>i. (\<integral>\<^isup>+ x. Real \<bar>u i x - u' x\<bar> \<partial>M) = 0" using eq_0 by auto
- thus ?thesis by simp
+ assume eq_0: "(\<integral>\<^isup>+ x. max 0 (extreal (2 * w x)) \<partial>M) = 0" (is "?wx = 0")
+ { fix n
+ have "?f n \<le> ?wx" (is "integral\<^isup>P M ?f' \<le> _")
+ using diff_less_2w[of _ n] unfolding positive_integral_max_0
+ by (intro positive_integral_mono) auto
+ then have "?f n = 0"
+ using positive_integral_positive[of ?f'] eq_0 by auto }
+ then show ?thesis by (simp add: Limsup_const)
next
- assume neq_0: "(\<integral>\<^isup>+ x. Real (2 * w x) \<partial>M) \<noteq> 0"
- have "(\<integral>\<^isup>+ x. Real (2 * w x) \<partial>M) = (\<integral>\<^isup>+ x. (SUP n. INF m. Real (?diff (m + n) x)) \<partial>M)"
- proof (rule positive_integral_cong, subst add_commute)
+ assume neq_0: "(\<integral>\<^isup>+ x. max 0 (extreal (2 * w x)) \<partial>M) \<noteq> 0" (is "?wx \<noteq> 0")
+ have "0 = limsup (\<lambda>n. 0 :: extreal)" by (simp add: Limsup_const)
+ also have "\<dots> \<le> limsup (\<lambda>n. \<integral>\<^isup>+ x. extreal \<bar>u n x - u' x\<bar> \<partial>M)"
+ by (intro limsup_mono positive_integral_positive)
+ finally have pos: "0 \<le> limsup (\<lambda>n. \<integral>\<^isup>+ x. extreal \<bar>u n x - u' x\<bar> \<partial>M)" .
+ have "?wx = (\<integral>\<^isup>+ x. liminf (\<lambda>n. max 0 (extreal (?diff n x))) \<partial>M)"
+ proof (rule positive_integral_cong)
fix x assume x: "x \<in> space M"
- show "Real (2 * w x) = (SUP n. INF m. Real (?diff (n + m) x))"
- proof (rule LIMSEQ_imp_lim_INF[symmetric])
- fix j show "0 \<le> ?diff j x" using diff_less_2w[OF x, of j] by simp
- next
+ show "max 0 (extreal (2 * w x)) = liminf (\<lambda>n. max 0 (extreal (?diff n x)))"
+ unfolding extreal_max_0
+ proof (rule lim_imp_Liminf[symmetric], unfold lim_extreal)
have "(\<lambda>i. ?diff i x) ----> 2 * w x - \<bar>u' x - u' x\<bar>"
using u'[OF x] by (safe intro!: LIMSEQ_diff LIMSEQ_const LIMSEQ_imp_rabs)
- thus "(\<lambda>i. ?diff i x) ----> 2 * w x" by simp
- qed
+ then show "(\<lambda>i. max 0 (?diff i x)) ----> max 0 (2 * w x)"
+ by (auto intro!: tendsto_real_max simp add: lim_extreal)
+ qed (rule trivial_limit_sequentially)
qed
- also have "\<dots> \<le> (SUP n. INF m. (\<integral>\<^isup>+ x. Real (?diff (m + n) x) \<partial>M))"
+ also have "\<dots> \<le> liminf (\<lambda>n. \<integral>\<^isup>+ x. max 0 (extreal (?diff n x)) \<partial>M)"
using u'_borel w u unfolding integrable_def
- by (auto intro!: positive_integral_lim_INF)
- also have "\<dots> = (\<integral>\<^isup>+ x. Real (2 * w x) \<partial>M) -
- (INF n. SUP m. \<integral>\<^isup>+ x. Real \<bar>u (m + n) x - u' x\<bar> \<partial>M)"
- 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)
+ by (intro positive_integral_lim_INF) (auto intro!: positive_integral_lim_INF)
+ also have "\<dots> = (\<integral>\<^isup>+ x. extreal (2 * w x) \<partial>M) -
+ limsup (\<lambda>n. \<integral>\<^isup>+ x. extreal \<bar>u n x - u' x\<bar> \<partial>M)"
+ unfolding PI_diff positive_integral_max_0
+ using positive_integral_positive[of "\<lambda>x. extreal (2 * w x)"]
+ by (subst liminf_extreal_cminus) auto
+ finally show ?thesis
+ using neq_0 I2w_fin positive_integral_positive[of "\<lambda>x. extreal (2 * w x)"] pos
+ unfolding positive_integral_max_0
+ by (cases rule: extreal2_cases[of "\<integral>\<^isup>+ x. extreal (2 * w x) \<partial>M" "limsup (\<lambda>n. \<integral>\<^isup>+ x. extreal \<bar>u n x - u' x\<bar> \<partial>M)"])
+ auto
qed
- have [simp]: "\<And>n m x. Real (- \<bar>u (m + n) x - u' x\<bar>) = 0" by auto
-
- have [simp]: "\<And>n m. (\<integral>\<^isup>+ x. Real \<bar>u (m + n) x - u' x\<bar> \<partial>M) =
- Real ((\<integral>x. \<bar>u (n + m) x - u' x\<bar> \<partial>M))"
- using diff by (subst add_commute) (simp add: lebesgue_integral_def integrable_def Real_real)
-
- have "(SUP n. INF m. (\<integral>\<^isup>+ x. Real \<bar>u (m + n) x - u' x\<bar> \<partial>M)) \<le> ?lim_SUP"
- (is "?lim_INF \<le> _") by (subst (1 2) add_commute) (rule lim_INF_le_lim_SUP)
- hence "?lim_INF = Real 0" "?lim_SUP = Real 0" using `?lim_SUP = 0` by auto
- thus ?lim_diff using diff by (auto intro!: integral_positive lim_INF_eq_lim_SUP)
+ have "liminf ?f \<le> limsup ?f"
+ by (intro extreal_Liminf_le_Limsup trivial_limit_sequentially)
+ moreover
+ { have "0 = liminf (\<lambda>n. 0 :: extreal)" by (simp add: Liminf_const)
+ also have "\<dots> \<le> liminf ?f"
+ by (intro liminf_mono positive_integral_positive)
+ finally have "0 \<le> liminf ?f" . }
+ ultimately have liminf_limsup_eq: "liminf ?f = extreal 0" "limsup ?f = extreal 0"
+ using `limsup ?f = 0` by auto
+ have "\<And>n. (\<integral>\<^isup>+ x. extreal \<bar>u n x - u' x\<bar> \<partial>M) = extreal (\<integral>x. \<bar>u n x - u' x\<bar> \<partial>M)"
+ using diff positive_integral_positive
+ by (subst integral_eq_positive_integral) (auto simp: extreal_real integrable_def)
+ then show ?lim_diff
+ using extreal_Liminf_eq_Limsup[OF trivial_limit_sequentially liminf_limsup_eq]
+ by (simp add: lim_extreal)
show ?lim
proof (rule LIMSEQ_I)
@@ -2266,7 +2393,7 @@
assumes f: "f \<in> borel_measurable M"
and bij: "bij_betw enum S (f ` space M)"
and enum_zero: "enum ` (-S) \<subseteq> {0}"
- and fin: "\<And>x. x \<noteq> 0 \<Longrightarrow> \<mu> (f -` {x} \<inter> space M) \<noteq> \<omega>"
+ and fin: "\<And>x. x \<noteq> 0 \<Longrightarrow> \<mu> (f -` {x} \<inter> space M) \<noteq> \<infinity>"
and abs_summable: "summable (\<lambda>r. \<bar>enum r * real (\<mu> (f -` {enum r} \<inter> space M))\<bar>)"
shows "integrable M f"
and "(\<lambda>r. enum r * real (\<mu> (f -` {enum r} \<inter> space M))) sums integral\<^isup>L M f" (is ?sums)
@@ -2303,7 +2430,7 @@
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>x. \<bar>?F r x\<bar> \<partial>M) = \<bar>enum r * real (\<mu> (?A r))\<bar>"
- by (simp add: abs_mult_pos real_pextreal_pos) }
+ using f by (subst (2) abs_mult_pos[symmetric]) (auto intro!: real_of_extreal_pos measurable_sets) }
note int_abs_F = this
have 1: "\<And>i. integrable M (\<lambda>x. ?F i x)"
@@ -2323,7 +2450,7 @@
lemma (in measure_space) integral_on_finite:
assumes f: "f \<in> borel_measurable M" and finite: "finite (f`space M)"
- and fin: "\<And>x. x \<noteq> 0 \<Longrightarrow> \<mu> (f -` {x} \<inter> space M) \<noteq> \<omega>"
+ and fin: "\<And>x. x \<noteq> 0 \<Longrightarrow> \<mu> (f -` {x} \<inter> space M) \<noteq> \<infinity>"
shows "integrable M f"
and "(\<integral>x. f x \<partial>M) =
(\<Sum> r \<in> f`space M. r * real (\<mu> (f -` {r} \<inter> space M)))" (is "?integral")
@@ -2353,11 +2480,12 @@
by (auto intro: borel_measurable_simple_function)
lemma (in finite_measure_space) positive_integral_finite_eq_setsum:
- "integral\<^isup>P M f = (\<Sum>x \<in> space M. f x * \<mu> {x})"
+ assumes pos: "\<And>x. x \<in> space M \<Longrightarrow> 0 \<le> f x"
+ shows "integral\<^isup>P M f = (\<Sum>x \<in> space M. f x * \<mu> {x})"
proof -
have *: "integral\<^isup>P M f = (\<integral>\<^isup>+ x. (\<Sum>y\<in>space M. f y * indicator {y} x) \<partial>M)"
by (auto intro!: positive_integral_cong simp add: indicator_def if_distrib setsum_cases[OF finite_space])
- show ?thesis unfolding * using borel_measurable_finite[of f]
+ show ?thesis unfolding * using borel_measurable_finite[of f] pos
by (simp add: positive_integral_setsum positive_integral_cmult_indicator)
qed
@@ -2365,16 +2493,20 @@
shows "integrable M f"
and "integral\<^isup>L M f = (\<Sum>x \<in> space M. f x * real (\<mu> {x}))" (is ?I)
proof -
- have [simp]:
- "(\<integral>\<^isup>+ x. Real (f x) \<partial>M) = (\<Sum>x \<in> space M. Real (f x) * \<mu> {x})"
- "(\<integral>\<^isup>+ x. Real (- f x) \<partial>M) = (\<Sum>x \<in> space M. Real (- f x) * \<mu> {x})"
- unfolding positive_integral_finite_eq_setsum by auto
- show "integrable M f" using finite_space finite_measure
- by (simp add: setsum_\<omega> integrable_def)
- show ?I using finite_measure
- apply (simp add: lebesgue_integral_def real_of_pextreal_setsum[symmetric]
- real_of_pextreal_mult[symmetric] setsum_subtractf[symmetric])
- by (rule setsum_cong) (simp_all split: split_if)
+ have *:
+ "(\<integral>\<^isup>+ x. max 0 (extreal (f x)) \<partial>M) = (\<Sum>x \<in> space M. max 0 (extreal (f x)) * \<mu> {x})"
+ "(\<integral>\<^isup>+ x. max 0 (extreal (- f x)) \<partial>M) = (\<Sum>x \<in> space M. max 0 (extreal (- f x)) * \<mu> {x})"
+ by (simp_all add: positive_integral_finite_eq_setsum)
+ then show "integrable M f" using finite_space finite_measure
+ by (simp add: setsum_Pinfty integrable_def positive_integral_max_0
+ split: split_max)
+ show ?I using finite_measure *
+ apply (simp add: positive_integral_max_0 lebesgue_integral_def)
+ apply (subst (1 2) setsum_real_of_extreal[symmetric])
+ apply (simp_all split: split_max add: setsum_subtractf[symmetric])
+ apply (intro setsum_cong[OF refl])
+ apply (simp split: split_max)
+ done
qed
end
--- a/src/HOL/Probability/Lebesgue_Measure.thy Mon Mar 14 15:17:10 2011 +0100
+++ b/src/HOL/Probability/Lebesgue_Measure.thy Mon Mar 14 15:29:10 2011 +0100
@@ -48,12 +48,12 @@
lemma Pi_iff: "f \<in> Pi I X \<longleftrightarrow> (\<forall>i\<in>I. f i \<in> X i)"
unfolding Pi_def by auto
-subsection {* Lebesgue measure *}
+subsection {* Lebesgue measure *}
definition lebesgue :: "'a::ordered_euclidean_space measure_space" where
"lebesgue = \<lparr> space = UNIV,
sets = {A. \<forall>n. (indicator A :: 'a \<Rightarrow> real) integrable_on cube n},
- measure = \<lambda>A. SUP n. Real (integral (cube n) (indicator A)) \<rparr>"
+ measure = \<lambda>A. SUP n. extreal (integral (cube n) (indicator A)) \<rparr>"
lemma space_lebesgue[simp]: "space lebesgue = UNIV"
unfolding lebesgue_def by simp
@@ -114,10 +114,33 @@
qed (auto intro: LIMSEQ_indicator_UN simp: cube_def)
qed simp
+lemma suminf_SUP_eq:
+ fixes f :: "nat \<Rightarrow> nat \<Rightarrow> extreal"
+ assumes "\<And>i. incseq (\<lambda>n. f n i)" "\<And>n i. 0 \<le> f n i"
+ shows "(\<Sum>i. SUP n. f n i) = (SUP n. \<Sum>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_extreal_setsum[symmetric]) }
+ note * = this
+ show ?thesis using assms
+ apply (subst (1 2) suminf_extreal_eq_SUPR)
+ unfolding *
+ apply (auto intro!: le_SUPI2)
+ apply (subst SUP_commute) ..
+qed
+
interpretation lebesgue: measure_space lebesgue
proof
have *: "indicator {} = (\<lambda>x. 0 :: real)" by (simp add: fun_eq_iff)
- show "measure lebesgue {} = 0" by (simp add: integral_0 * lebesgue_def)
+ show "positive lebesgue (measure lebesgue)"
+ proof (unfold positive_def, safe)
+ show "measure lebesgue {} = 0" by (simp add: integral_0 * lebesgue_def)
+ fix A assume "A \<in> sets lebesgue"
+ then show "0 \<le> measure lebesgue A"
+ unfolding lebesgue_def
+ by (auto intro!: le_SUPI2 integral_nonneg)
+ qed
next
show "countably_additive lebesgue (measure lebesgue)"
proof (intro countably_additive_def[THEN iffD2] allI impI)
@@ -130,23 +153,17 @@
assume "(\<Union>i. A i) \<in> sets lebesgue"
then have UN_A[simp, intro]: "\<And>i n. (indicator (\<Union>i. A i) :: _ \<Rightarrow> real) integrable_on cube n"
by (auto dest: lebesgueD)
- show "(\<Sum>\<^isub>\<infinity>n. measure lebesgue (A n)) = measure lebesgue (\<Union>i. A i)"
- proof (simp add: lebesgue_def, subst psuminf_SUP_eq)
- fix n i show "Real (?m n i) \<le> Real (?m (Suc n) i)"
- using cube_subset[of n "Suc n"] by (auto intro!: integral_subset_le)
+ show "(\<Sum>n. measure lebesgue (A n)) = measure lebesgue (\<Union>i. A i)"
+ proof (simp add: lebesgue_def, subst suminf_SUP_eq, safe intro!: incseq_SucI)
+ fix i n show "extreal (?m n i) \<le> extreal (?m (Suc n) i)"
+ using cube_subset[of n "Suc n"] by (auto intro!: integral_subset_le incseq_SucI)
next
- show "(SUP n. \<Sum>\<^isub>\<infinity>i. Real (?m n i)) = (SUP n. Real (?M n UNIV))"
- unfolding psuminf_def
- proof (subst setsum_Real, (intro arg_cong[where f="SUPR UNIV"] ext ballI nn SUP_eq_LIMSEQ[THEN iffD2])+)
- fix n :: nat show "mono (\<lambda>m. \<Sum>x<m. ?m n x)"
- proof (intro mono_iff_le_Suc[THEN iffD2] allI)
- fix m show "(\<Sum>x<m. ?m n x) \<le> (\<Sum>x<Suc m. ?m n x)"
- using nn[of n m] by auto
- qed
- show "0 \<le> ?M n UNIV"
- using UN_A by (auto intro!: integral_nonneg)
- fix m show "0 \<le> (\<Sum>x<m. ?m n x)" by (auto intro!: setsum_nonneg)
- next
+ fix i n show "0 \<le> extreal (?m n i)"
+ using rA unfolding lebesgue_def
+ by (auto intro!: le_SUPI2 integral_nonneg)
+ next
+ show "(SUP n. \<Sum>i. extreal (?m n i)) = (SUP n. extreal (?M n UNIV))"
+ proof (intro arg_cong[where f="SUPR UNIV"] ext sums_unique[symmetric] sums_extreal[THEN iffD2] sums_def[THEN iffD2])
fix n
have "\<And>m. (UNION {..<m} A) \<in> sets lebesgue" using rA by auto
from lebesgueD[OF this]
@@ -171,8 +188,8 @@
ultimately show ?case
using Suc A by (simp add: integral_add[symmetric])
qed auto }
- ultimately show "(\<lambda>m. \<Sum>x<m. ?m n x) ----> ?M n UNIV"
- by simp
+ ultimately show "(\<lambda>m. \<Sum>x = 0..<m. ?m n x) ----> ?M n UNIV"
+ by (simp add: atLeast0LessThan)
qed
qed
qed
@@ -232,13 +249,11 @@
lemma lmeasure_iff_LIMSEQ:
assumes "A \<in> sets lebesgue" "0 \<le> m"
- shows "lebesgue.\<mu> A = Real m \<longleftrightarrow> (\<lambda>n. integral (cube n) (indicator A :: _ \<Rightarrow> real)) ----> m"
+ shows "lebesgue.\<mu> A = extreal m \<longleftrightarrow> (\<lambda>n. integral (cube n) (indicator A :: _ \<Rightarrow> real)) ----> m"
proof (simp add: lebesgue_def, intro SUP_eq_LIMSEQ)
show "mono (\<lambda>n. integral (cube n) (indicator A::_=>real))"
using cube_subset assms by (intro monoI integral_subset_le) (auto dest!: lebesgueD)
- fix n show "0 \<le> integral (cube n) (indicator A::_=>real)"
- using assms by (auto dest!: lebesgueD intro!: integral_nonneg)
-qed fact
+qed
lemma has_integral_indicator_UNIV:
fixes s A :: "'a::ordered_euclidean_space set" and x :: real
@@ -260,7 +275,7 @@
lemma lmeasure_finite_has_integral:
fixes s :: "'a::ordered_euclidean_space set"
- assumes "s \<in> sets lebesgue" "lebesgue.\<mu> s = Real m" "0 \<le> m"
+ assumes "s \<in> sets lebesgue" "lebesgue.\<mu> s = extreal m" "0 \<le> m"
shows "(indicator s has_integral m) UNIV"
proof -
let ?I = "indicator :: 'a set \<Rightarrow> 'a \<Rightarrow> real"
@@ -302,12 +317,14 @@
unfolding m by (intro integrable_integral **)
qed
-lemma lmeasure_finite_integrable: assumes "s \<in> sets lebesgue" "lebesgue.\<mu> s \<noteq> \<omega>"
+lemma lmeasure_finite_integrable: assumes s: "s \<in> sets lebesgue" and "lebesgue.\<mu> s \<noteq> \<infinity>"
shows "(indicator s :: _ \<Rightarrow> real) integrable_on UNIV"
proof (cases "lebesgue.\<mu> s")
- case (preal m) from lmeasure_finite_has_integral[OF `s \<in> sets lebesgue` this]
+ case (real m)
+ with lmeasure_finite_has_integral[OF `s \<in> sets lebesgue` this]
+ lebesgue.positive_measure[OF s]
show ?thesis unfolding integrable_on_def by auto
-qed (insert assms, auto)
+qed (insert assms lebesgue.positive_measure[OF s], auto)
lemma has_integral_lebesgue: assumes "((indicator s :: _\<Rightarrow>real) has_integral m) UNIV"
shows "s \<in> sets lebesgue"
@@ -321,7 +338,7 @@
qed
lemma has_integral_lmeasure: assumes "((indicator s :: _\<Rightarrow>real) has_integral m) UNIV"
- shows "lebesgue.\<mu> s = Real m"
+ shows "lebesgue.\<mu> s = extreal m"
proof (intro lmeasure_iff_LIMSEQ[THEN iffD2])
let ?I = "indicator :: 'a set \<Rightarrow> 'a \<Rightarrow> real"
show "s \<in> sets lebesgue" using has_integral_lebesgue[OF assms] .
@@ -346,28 +363,28 @@
qed
lemma has_integral_iff_lmeasure:
- "(indicator A has_integral m) UNIV \<longleftrightarrow> (A \<in> sets lebesgue \<and> 0 \<le> m \<and> lebesgue.\<mu> A = Real m)"
+ "(indicator A has_integral m) UNIV \<longleftrightarrow> (A \<in> sets lebesgue \<and> 0 \<le> m \<and> lebesgue.\<mu> A = extreal m)"
proof
assume "(indicator A has_integral m) UNIV"
with has_integral_lmeasure[OF this] has_integral_lebesgue[OF this]
- show "A \<in> sets lebesgue \<and> 0 \<le> m \<and> lebesgue.\<mu> A = Real m"
+ show "A \<in> sets lebesgue \<and> 0 \<le> m \<and> lebesgue.\<mu> A = extreal m"
by (auto intro: has_integral_nonneg)
next
- assume "A \<in> sets lebesgue \<and> 0 \<le> m \<and> lebesgue.\<mu> A = Real m"
+ assume "A \<in> sets lebesgue \<and> 0 \<le> m \<and> lebesgue.\<mu> A = extreal m"
then show "(indicator A has_integral m) UNIV" by (intro lmeasure_finite_has_integral) auto
qed
lemma lmeasure_eq_integral: assumes "(indicator s::_\<Rightarrow>real) integrable_on UNIV"
- shows "lebesgue.\<mu> s = Real (integral UNIV (indicator s))"
+ shows "lebesgue.\<mu> s = extreal (integral UNIV (indicator s))"
using assms unfolding integrable_on_def
proof safe
fix y :: real assume "(indicator s has_integral y) UNIV"
from this[unfolded has_integral_iff_lmeasure] integral_unique[OF this]
- show "lebesgue.\<mu> s = Real (integral UNIV (indicator s))" by simp
+ show "lebesgue.\<mu> s = extreal (integral UNIV (indicator s))" by simp
qed
lemma lebesgue_simple_function_indicator:
- fixes f::"'a::ordered_euclidean_space \<Rightarrow> pextreal"
+ fixes f::"'a::ordered_euclidean_space \<Rightarrow> extreal"
assumes f:"simple_function lebesgue f"
shows "f = (\<lambda>x. (\<Sum>y \<in> f ` UNIV. y * indicator (f -` {y}) x))"
by (rule, subst lebesgue.simple_function_indicator_representation[OF f]) auto
@@ -376,7 +393,7 @@
"(indicator s::_\<Rightarrow>real) integrable_on UNIV \<Longrightarrow> integral UNIV (indicator s) = real (lebesgue.\<mu> s)"
by (subst lmeasure_eq_integral) (auto intro!: integral_nonneg)
-lemma lmeasure_finite: assumes "(indicator s::_\<Rightarrow>real) integrable_on UNIV" shows "lebesgue.\<mu> s \<noteq> \<omega>"
+lemma lmeasure_finite: assumes "(indicator s::_\<Rightarrow>real) integrable_on UNIV" shows "lebesgue.\<mu> s \<noteq> \<infinity>"
using lmeasure_eq_integral[OF assms] by auto
lemma negligible_iff_lebesgue_null_sets:
@@ -409,37 +426,29 @@
shows "integral {a .. b} (\<lambda>x. c) = content {a .. b} *\<^sub>R c"
by (rule integral_unique) (rule has_integral_const)
-lemma lmeasure_UNIV[intro]: "lebesgue.\<mu> (UNIV::'a::ordered_euclidean_space set) = \<omega>"
-proof (simp add: lebesgue_def SUP_\<omega>, intro allI impI)
- fix x assume "x < \<omega>"
- then obtain r where r: "x = Real r" "0 \<le> r" by (cases x) auto
- then obtain n where n: "r < of_nat n" using ex_less_of_nat by auto
- show "\<exists>i. x < Real (integral (cube i) (indicator UNIV::'a\<Rightarrow>real))"
- proof (intro exI[of _ n])
- have [simp]: "indicator UNIV = (\<lambda>x. 1)" by (auto simp: fun_eq_iff)
- { fix m :: nat assume "0 < m" then have "real n \<le> (\<Prod>x<m. 2 * real n)"
- proof (induct m)
- case (Suc m)
- show ?case
- proof cases
- assume "m = 0" then show ?thesis by (simp add: lessThan_Suc)
- next
- assume "m \<noteq> 0" then have "real n \<le> (\<Prod>x<m. 2 * real n)" using Suc by auto
- then show ?thesis
- by (auto simp: lessThan_Suc field_simps mult_le_cancel_left1)
- qed
- qed auto } note this[OF DIM_positive[where 'a='a], simp]
- then have [simp]: "0 \<le> (\<Prod>x<DIM('a). 2 * real n)" using real_of_nat_ge_zero by arith
- have "x < Real (of_nat n)" using n r by auto
- also have "Real (of_nat n) \<le> Real (integral (cube n) (indicator UNIV::'a\<Rightarrow>real))"
- by (auto simp: real_eq_of_nat[symmetric] cube_def content_closed_interval_cases)
- finally show "x < Real (integral (cube n) (indicator UNIV::'a\<Rightarrow>real))" .
- qed
-qed
+lemma lmeasure_UNIV[intro]: "lebesgue.\<mu> (UNIV::'a::ordered_euclidean_space set) = \<infinity>"
+proof (simp add: lebesgue_def, intro SUP_PInfty bexI)
+ fix n :: nat
+ have "indicator UNIV = (\<lambda>x::'a. 1 :: real)" by auto
+ moreover
+ { have "real n \<le> (2 * real n) ^ DIM('a)"
+ proof (cases n)
+ case 0 then show ?thesis by auto
+ next
+ case (Suc n')
+ have "real n \<le> (2 * real n)^1" by auto
+ also have "(2 * real n)^1 \<le> (2 * real n) ^ DIM('a)"
+ using Suc DIM_positive[where 'a='a] by (intro power_increasing) (auto simp: real_of_nat_Suc)
+ finally show ?thesis .
+ qed }
+ ultimately show "extreal (real n) \<le> extreal (integral (cube n) (indicator UNIV::'a\<Rightarrow>real))"
+ using integral_const DIM_positive[where 'a='a]
+ by (auto simp: cube_def content_closed_interval_cases setprod_constant)
+qed simp
lemma
fixes a b ::"'a::ordered_euclidean_space"
- shows lmeasure_atLeastAtMost[simp]: "lebesgue.\<mu> {a..b} = Real (content {a..b})"
+ shows lmeasure_atLeastAtMost[simp]: "lebesgue.\<mu> {a..b} = extreal (content {a..b})"
proof -
have "(indicator (UNIV \<inter> {a..b})::_\<Rightarrow>real) integrable_on UNIV"
unfolding integrable_indicator_UNIV by (simp add: integrable_const indicator_def_raw)
@@ -467,7 +476,7 @@
lemma
fixes a b :: real
shows lmeasure_real_greaterThanAtMost[simp]:
- "lebesgue.\<mu> {a <.. b} = Real (if a \<le> b then b - a else 0)"
+ "lebesgue.\<mu> {a <.. b} = extreal (if a \<le> b then b - a else 0)"
proof cases
assume "a < b"
then have "lebesgue.\<mu> {a <.. b} = lebesgue.\<mu> {a .. b} - lebesgue.\<mu> {a}"
@@ -479,7 +488,7 @@
lemma
fixes a b :: real
shows lmeasure_real_atLeastLessThan[simp]:
- "lebesgue.\<mu> {a ..< b} = Real (if a \<le> b then b - a else 0)"
+ "lebesgue.\<mu> {a ..< b} = extreal (if a \<le> b then b - a else 0)"
proof cases
assume "a < b"
then have "lebesgue.\<mu> {a ..< b} = lebesgue.\<mu> {a .. b} - lebesgue.\<mu> {b}"
@@ -491,7 +500,7 @@
lemma
fixes a b :: real
shows lmeasure_real_greaterThanLessThan[simp]:
- "lebesgue.\<mu> {a <..< b} = Real (if a \<le> b then b - a else 0)"
+ "lebesgue.\<mu> {a <..< b} = extreal (if a \<le> b then b - a else 0)"
proof cases
assume "a < b"
then have "lebesgue.\<mu> {a <..< b} = lebesgue.\<mu> {a <.. b} - lebesgue.\<mu> {b}"
@@ -511,19 +520,16 @@
and measurable_lborel[simp]: "measurable lborel = measurable borel"
by (simp_all add: measurable_def_raw lborel_def)
-interpretation lborel: measure_space lborel
+interpretation lborel: measure_space "lborel :: ('a::ordered_euclidean_space) measure_space"
where "space lborel = UNIV"
and "sets lborel = sets borel"
and "measure lborel = lebesgue.\<mu>"
and "measurable lborel = measurable borel"
-proof -
- show "measure_space lborel"
- proof
- show "countably_additive lborel (measure lborel)"
- using lebesgue.ca unfolding countably_additive_def lborel_def
- apply safe apply (erule_tac x=A in allE) by auto
- qed (auto simp: lborel_def)
-qed simp_all
+proof (rule lebesgue.measure_space_subalgebra)
+ have "sigma_algebra (lborel::'a measure_space) \<longleftrightarrow> sigma_algebra (borel::'a algebra)"
+ unfolding sigma_algebra_iff2 lborel_def by simp
+ then show "sigma_algebra (lborel::'a measure_space)" by simp default
+qed auto
interpretation lborel: sigma_finite_measure lborel
where "space lborel = UNIV"
@@ -536,7 +542,7 @@
show "range cube \<subseteq> sets lborel" by (auto intro: borel_closed)
{ fix x have "\<exists>n. x\<in>cube n" using mem_big_cube by auto }
thus "(\<Union>i. cube i) = space lborel" by auto
- show "\<forall>i. measure lborel (cube i) \<noteq> \<omega>" by (simp add: cube_def)
+ show "\<forall>i. measure lborel (cube i) \<noteq> \<infinity>" by (simp add: cube_def)
qed
qed simp_all
@@ -544,171 +550,221 @@
proof
from lborel.sigma_finite guess A ..
moreover then have "range A \<subseteq> sets lebesgue" using lebesgueI_borel by blast
- ultimately show "\<exists>A::nat \<Rightarrow> 'b set. range A \<subseteq> sets lebesgue \<and> (\<Union>i. A i) = space lebesgue \<and> (\<forall>i. lebesgue.\<mu> (A i) \<noteq> \<omega>)"
+ ultimately show "\<exists>A::nat \<Rightarrow> 'b set. range A \<subseteq> sets lebesgue \<and> (\<Union>i. A i) = space lebesgue \<and> (\<forall>i. lebesgue.\<mu> (A i) \<noteq> \<infinity>)"
by auto
qed
subsection {* Lebesgue integrable implies Gauge integrable *}
+lemma positive_not_Inf:
+ "0 \<le> x \<Longrightarrow> x \<noteq> \<infinity> \<Longrightarrow> \<bar>x\<bar> \<noteq> \<infinity>"
+ by (cases x) auto
+
+lemma has_integral_cmult_real:
+ fixes c :: real
+ assumes "c \<noteq> 0 \<Longrightarrow> (f has_integral x) A"
+ shows "((\<lambda>x. c * f x) has_integral c * x) A"
+proof cases
+ assume "c \<noteq> 0"
+ from has_integral_cmul[OF assms[OF this], of c] show ?thesis
+ unfolding real_scaleR_def .
+qed simp
+
lemma simple_function_has_integral:
- fixes f::"'a::ordered_euclidean_space \<Rightarrow> pextreal"
+ fixes f::"'a::ordered_euclidean_space \<Rightarrow> extreal"
assumes f:"simple_function lebesgue f"
- and f':"\<forall>x. f x \<noteq> \<omega>"
- and om:"\<forall>x\<in>range f. lebesgue.\<mu> (f -` {x} \<inter> UNIV) = \<omega> \<longrightarrow> x = 0"
+ and f':"range f \<subseteq> {0..<\<infinity>}"
+ and om:"\<And>x. x \<in> range f \<Longrightarrow> lebesgue.\<mu> (f -` {x} \<inter> UNIV) = \<infinity> \<Longrightarrow> x = 0"
shows "((\<lambda>x. real (f x)) has_integral (real (integral\<^isup>S lebesgue f))) UNIV"
- unfolding simple_integral_def
- apply(subst lebesgue_simple_function_indicator[OF f])
-proof -
- case goal1
- have *:"\<And>x. \<forall>y\<in>range f. y * indicator (f -` {y}) x \<noteq> \<omega>"
- "\<forall>x\<in>range f. x * lebesgue.\<mu> (f -` {x} \<inter> UNIV) \<noteq> \<omega>"
- using f' om unfolding indicator_def by auto
- 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
- real (f y * lebesgue.\<mu> (f -` {f y} \<inter> UNIV))) UNIV"
- proof(cases "f y = 0") case False
- have mea:"(indicator (f -` {f y}) ::_\<Rightarrow>real) integrable_on UNIV"
- apply(rule lmeasure_finite_integrable)
- using assms unfolding simple_function_def using False by auto
- have *:"\<And>x. real (indicator (f -` {f y}) x::pextreal) = (indicator (f -` {f y}) x)"
- by (auto simp: indicator_def)
- 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_eq_integral[OF mea,THEN sym]
- unfolding integral_eq_lmeasure[OF mea, symmetric] *
- apply(rule integrable_integral) using mea .
- qed auto
+ unfolding simple_integral_def space_lebesgue
+proof (subst lebesgue_simple_function_indicator)
+ let "?M x" = "lebesgue.\<mu> (f -` {x} \<inter> UNIV)"
+ let "?F x" = "indicator (f -` {x})"
+ { fix x y assume "y \<in> range f"
+ from subsetD[OF f' this] have "y * ?F y x = extreal (real y * ?F y x)"
+ by (cases rule: extreal2_cases[of y "?F y x"])
+ (auto simp: indicator_def one_extreal_def split: split_if_asm) }
+ moreover
+ { fix x assume x: "x\<in>range f"
+ have "x * ?M x = real x * real (?M x)"
+ proof cases
+ assume "x \<noteq> 0" with om[OF x] have "?M x \<noteq> \<infinity>" by auto
+ with subsetD[OF f' x] f[THEN lebesgue.simple_functionD(2)] show ?thesis
+ by (cases rule: extreal2_cases[of x "?M x"]) auto
+ qed simp }
+ ultimately
+ have "((\<lambda>x. real (\<Sum>y\<in>range f. y * ?F y x)) has_integral real (\<Sum>x\<in>range f. x * ?M x)) UNIV \<longleftrightarrow>
+ ((\<lambda>x. \<Sum>y\<in>range f. real y * ?F y x) has_integral (\<Sum>x\<in>range f. real x * real (?M x))) UNIV"
+ by simp
+ also have \<dots>
+ proof (intro has_integral_setsum has_integral_cmult_real lmeasure_finite_has_integral
+ real_of_extreal_pos lebesgue.positive_measure ballI)
+ show *: "finite (range f)" "\<And>y. f -` {y} \<in> sets lebesgue" "\<And>y. f -` {y} \<inter> UNIV \<in> sets lebesgue"
+ using lebesgue.simple_functionD[OF f] by auto
+ fix y assume "real y \<noteq> 0" "y \<in> range f"
+ with * om[OF this(2)] show "lebesgue.\<mu> (f -` {y}) = extreal (real (?M y))"
+ by (auto simp: extreal_real)
qed
-qed
+ finally show "((\<lambda>x. real (\<Sum>y\<in>range f. y * ?F y x)) has_integral real (\<Sum>x\<in>range f. x * ?M x)) UNIV" .
+qed fact
lemma bounded_realI: assumes "\<forall>x\<in>s. abs (x::real) \<le> B" shows "bounded s"
unfolding bounded_def dist_real_def apply(rule_tac x=0 in exI)
using assms by auto
lemma simple_function_has_integral':
- fixes f::"'a::ordered_euclidean_space \<Rightarrow> pextreal"
- assumes f:"simple_function lebesgue f"
- and i: "integral\<^isup>S lebesgue f \<noteq> \<omega>"
+ fixes f::"'a::ordered_euclidean_space \<Rightarrow> extreal"
+ assumes f: "simple_function lebesgue f" "\<And>x. 0 \<le> f x"
+ and i: "integral\<^isup>S lebesgue f \<noteq> \<infinity>"
shows "((\<lambda>x. real (f x)) has_integral (real (integral\<^isup>S lebesgue f))) UNIV"
-proof- let ?f = "\<lambda>x. if f x = \<omega> then 0 else f x"
- { fix x have "real (f x) = real (?f x)" by (cases "f x") auto } note * = this
- have **:"{x. f x \<noteq> ?f x} = f -` {\<omega>}" by auto
- have **:"lebesgue.\<mu> {x\<in>space lebesgue. f x \<noteq> ?f x} = 0"
- using lebesgue.simple_integral_omega[OF assms] by(auto simp add:**)
- show ?thesis apply(subst lebesgue.simple_integral_cong'[OF f _ **])
- apply(rule lebesgue.simple_function_compose1[OF f])
- unfolding * defer apply(rule simple_function_has_integral)
- proof-
- show "simple_function lebesgue ?f"
- using lebesgue.simple_function_compose1[OF f] .
- show "\<forall>x. ?f x \<noteq> \<omega>" by auto
- show "\<forall>x\<in>range ?f. lebesgue.\<mu> (?f -` {x} \<inter> UNIV) = \<omega> \<longrightarrow> x = 0"
- proof (safe, simp, safe, rule ccontr)
- fix y assume "f y \<noteq> \<omega>" "f y \<noteq> 0"
- hence "(\<lambda>x. if f x = \<omega> then 0 else f x) -` {if f y = \<omega> then 0 else f y} = f -` {f y}"
- by (auto split: split_if_asm)
- moreover assume "lebesgue.\<mu> ((\<lambda>x. if f x = \<omega> then 0 else f x) -` {if f y = \<omega> then 0 else f y}) = \<omega>"
- ultimately have "lebesgue.\<mu> (f -` {f y}) = \<omega>" by simp
- moreover
- have "f y * lebesgue.\<mu> (f -` {f y}) \<noteq> \<omega>" using i f
- unfolding simple_integral_def setsum_\<omega> simple_function_def
- by auto
- ultimately have "f y = 0" by (auto split: split_if_asm)
- then show False using `f y \<noteq> 0` by simp
- qed
+proof -
+ let ?f = "\<lambda>x. if x \<in> f -` {\<infinity>} then 0 else f x"
+ note f(1)[THEN lebesgue.simple_functionD(2)]
+ then have [simp, intro]: "\<And>X. f -` X \<in> sets lebesgue" by auto
+ have f': "simple_function lebesgue ?f"
+ using f by (intro lebesgue.simple_function_If_set) auto
+ have rng: "range ?f \<subseteq> {0..<\<infinity>}" using f by auto
+ have "AE x in lebesgue. f x = ?f x"
+ using lebesgue.simple_integral_PInf[OF f i]
+ by (intro lebesgue.AE_I[where N="f -` {\<infinity>} \<inter> space lebesgue"]) auto
+ from f(1) f' this have eq: "integral\<^isup>S lebesgue f = integral\<^isup>S lebesgue ?f"
+ by (rule lebesgue.simple_integral_cong_AE)
+ have real_eq: "\<And>x. real (f x) = real (?f x)" by auto
+
+ show ?thesis
+ unfolding eq real_eq
+ proof (rule simple_function_has_integral[OF f' rng])
+ fix x assume x: "x \<in> range ?f" and inf: "lebesgue.\<mu> (?f -` {x} \<inter> UNIV) = \<infinity>"
+ have "x * lebesgue.\<mu> (?f -` {x} \<inter> UNIV) = (\<integral>\<^isup>S y. x * indicator (?f -` {x}) y \<partial>lebesgue)"
+ using f'[THEN lebesgue.simple_functionD(2)]
+ by (simp add: lebesgue.simple_integral_cmult_indicator)
+ also have "\<dots> \<le> integral\<^isup>S lebesgue f"
+ using f'[THEN lebesgue.simple_functionD(2)] f
+ by (intro lebesgue.simple_integral_mono lebesgue.simple_function_mult lebesgue.simple_function_indicator)
+ (auto split: split_indicator)
+ finally show "x = 0" unfolding inf using i subsetD[OF rng x] by (auto split: split_if_asm)
qed
qed
-lemma (in measure_space) positive_integral_monotone_convergence:
- 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"
- and "(\<lambda>i. integral\<^isup>P M (f i)) ----> integral\<^isup>P M u" (is ?ilim)
-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 "\<And>x. (SUP i. f i x) = u x" using mono lim SUP_Lim_pextreal
- unfolding fun_eq_iff mono_def by auto
- moreover have "(\<lambda>x. SUP i. f i x) \<in> borel_measurable M"
- using i by auto
- ultimately show "u \<in> borel_measurable M" by simp
-qed
+lemma real_of_extreal_positive_mono:
+ "\<lbrakk>0 \<le> x; x \<le> y; y \<noteq> \<infinity>\<rbrakk> \<Longrightarrow> real x \<le> real y"
+ by (cases rule: extreal2_cases[of x y]) auto
lemma positive_integral_has_integral:
- fixes f::"'a::ordered_euclidean_space => pextreal"
- assumes f:"f \<in> borel_measurable lebesgue"
- and int_om:"integral\<^isup>P lebesgue f \<noteq> \<omega>"
- and f_om:"\<forall>x. f x \<noteq> \<omega>" (* TODO: remove this *)
+ fixes f :: "'a::ordered_euclidean_space \<Rightarrow> extreal"
+ assumes f: "f \<in> borel_measurable lebesgue" "range f \<subseteq> {0..<\<infinity>}" "integral\<^isup>P lebesgue f \<noteq> \<infinity>"
shows "((\<lambda>x. real (f x)) has_integral (real (integral\<^isup>P lebesgue f))) UNIV"
-proof- let ?i = "integral\<^isup>P lebesgue f"
- from lebesgue.borel_measurable_implies_simple_function_sequence[OF f]
- guess u .. note conjunctD2[OF this,rule_format] note u = conjunctD2[OF this(1)] this(2)
- let ?u = "\<lambda>i x. real (u i x)" and ?f = "\<lambda>x. real (f x)"
- have u_simple:"\<And>k. integral\<^isup>S lebesgue (u k) = integral\<^isup>P lebesgue (u k)"
- apply(subst lebesgue.positive_integral_eq_simple_integral[THEN sym,OF u(1)]) ..
- have int_u_le:"\<And>k. integral\<^isup>S lebesgue (u k) \<le> integral\<^isup>P lebesgue f"
- unfolding u_simple apply(rule lebesgue.positive_integral_mono)
- using isoton_Sup[OF u(3)] unfolding le_fun_def by auto
- have u_int_om:"\<And>i. integral\<^isup>S lebesgue (u i) \<noteq> \<omega>"
- proof- case goal1 thus ?case using int_u_le[of i] int_om by auto qed
+proof -
+ from lebesgue.borel_measurable_implies_simple_function_sequence'[OF f(1)]
+ guess u . note u = this
+ have SUP_eq: "\<And>x. (SUP i. u i x) = f x"
+ using u(4) f(2)[THEN subsetD] by (auto split: split_max)
+ let "?u i x" = "real (u i x)"
+ note u_eq = lebesgue.positive_integral_eq_simple_integral[OF u(1,5), symmetric]
+ { fix i
+ note u_eq
+ also have "integral\<^isup>P lebesgue (u i) \<le> (\<integral>\<^isup>+x. max 0 (f x) \<partial>lebesgue)"
+ by (intro lebesgue.positive_integral_mono) (auto intro: le_SUPI simp: u(4)[symmetric])
+ finally have "integral\<^isup>S lebesgue (u i) \<noteq> \<infinity>"
+ unfolding positive_integral_max_0 using f by auto }
+ note u_fin = this
+ then have u_int: "\<And>i. (?u i has_integral real (integral\<^isup>S lebesgue (u i))) UNIV"
+ by (rule simple_function_has_integral'[OF u(1,5)])
+ have "\<forall>x. \<exists>r\<ge>0. f x = extreal r"
+ proof
+ fix x from f(2) have "0 \<le> f x" "f x \<noteq> \<infinity>" by (auto simp: subset_eq)
+ then show "\<exists>r\<ge>0. f x = extreal r" by (cases "f x") auto
+ qed
+ from choice[OF this] obtain f' where f': "f = (\<lambda>x. extreal (f' x))" "\<And>x. 0 \<le> f' x" by auto
+
+ have "\<forall>i. \<exists>r. \<forall>x. 0 \<le> r x \<and> u i x = extreal (r x)"
+ proof
+ fix i show "\<exists>r. \<forall>x. 0 \<le> r x \<and> u i x = extreal (r x)"
+ proof (intro choice allI)
+ fix i x have "u i x \<noteq> \<infinity>" using u(3)[of i] by (auto simp: image_iff) metis
+ then show "\<exists>r\<ge>0. u i x = extreal r" using u(5)[of i x] by (cases "u i x") auto
+ qed
+ qed
+ from choice[OF this] obtain u' where
+ u': "u = (\<lambda>i x. extreal (u' i x))" "\<And>i x. 0 \<le> u' i x" by (auto simp: fun_eq_iff)
- note u_int = simple_function_has_integral'[OF u(1) this]
- 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_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 (integral\<^isup>P lebesgue 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_pextreal_mono[OF _ int_u_le])
- using u int_om by auto
- qed note int = conjunctD2[OF this]
+ have convergent: "f' integrable_on UNIV \<and>
+ (\<lambda>k. integral UNIV (u' k)) ----> integral UNIV f'"
+ proof (intro monotone_convergence_increasing allI ballI)
+ show int: "\<And>k. (u' k) integrable_on UNIV"
+ using u_int unfolding integrable_on_def u' by auto
+ show "\<And>k x. u' k x \<le> u' (Suc k) x" using u(2,3,5)
+ by (auto simp: incseq_Suc_iff le_fun_def image_iff u' intro!: real_of_extreal_positive_mono)
+ show "\<And>x. (\<lambda>k. u' k x) ----> f' x"
+ using SUP_eq u(2)
+ by (intro SUP_eq_LIMSEQ[THEN iffD1]) (auto simp: u' f' incseq_mono incseq_Suc_iff le_fun_def)
+ show "bounded {integral UNIV (u' k)|k. True}"
+ proof (safe intro!: bounded_realI)
+ fix k
+ have "\<bar>integral UNIV (u' k)\<bar> = integral UNIV (u' k)"
+ by (intro abs_of_nonneg integral_nonneg int ballI u')
+ also have "\<dots> = real (integral\<^isup>S lebesgue (u k))"
+ using u_int[THEN integral_unique] by (simp add: u')
+ also have "\<dots> = real (integral\<^isup>P lebesgue (u k))"
+ using lebesgue.positive_integral_eq_simple_integral[OF u(1,5)] by simp
+ also have "\<dots> \<le> real (integral\<^isup>P lebesgue f)" using f
+ by (auto intro!: real_of_extreal_positive_mono lebesgue.positive_integral_positive
+ lebesgue.positive_integral_mono le_SUPI simp: SUP_eq[symmetric])
+ finally show "\<bar>integral UNIV (u' k)\<bar> \<le> real (integral\<^isup>P lebesgue f)" .
+ qed
+ qed
- have "(\<lambda>i. integral\<^isup>S lebesgue (u i)) ----> ?i" unfolding u_simple
- apply(rule lebesgue.positive_integral_monotone_convergence(2))
- apply(rule lebesgue.borel_measurable_simple_function[OF u(1)])
- using isotone_Lim[OF u(3)[unfolded isoton_fun_expand, THEN spec]] by auto
- hence "(\<lambda>i. real (integral\<^isup>S lebesgue (u i))) ----> real ?i" apply-
- apply(subst lim_Real[THEN sym]) prefer 3
- apply(subst Real_real') defer apply(subst Real_real')
- using u f_om int_om u_int_om by auto
- note * = LIMSEQ_unique[OF this int(2)[unfolded integral_unique[OF u_int]]]
- show ?thesis unfolding * by(rule integrable_integral[OF int(1)])
+ have "integral\<^isup>P lebesgue f = extreal (integral UNIV f')"
+ proof (rule tendsto_unique[OF trivial_limit_sequentially])
+ have "(\<lambda>i. integral\<^isup>S lebesgue (u i)) ----> (SUP i. integral\<^isup>P lebesgue (u i))"
+ unfolding u_eq by (intro LIMSEQ_extreal_SUPR lebesgue.incseq_positive_integral u)
+ also note lebesgue.positive_integral_monotone_convergence_SUP
+ [OF u(2) lebesgue.borel_measurable_simple_function[OF u(1)] u(5), symmetric]
+ finally show "(\<lambda>k. integral\<^isup>S lebesgue (u k)) ----> integral\<^isup>P lebesgue f"
+ unfolding SUP_eq .
+
+ { fix k
+ have "0 \<le> integral\<^isup>S lebesgue (u k)"
+ using u by (auto intro!: lebesgue.simple_integral_positive)
+ then have "integral\<^isup>S lebesgue (u k) = extreal (real (integral\<^isup>S lebesgue (u k)))"
+ using u_fin by (auto simp: extreal_real) }
+ note * = this
+ show "(\<lambda>k. integral\<^isup>S lebesgue (u k)) ----> extreal (integral UNIV f')"
+ using convergent using u_int[THEN integral_unique, symmetric]
+ by (subst *) (simp add: lim_extreal u')
+ qed
+ then show ?thesis using convergent by (simp add: f' integrable_integral)
qed
lemma lebesgue_integral_has_integral:
- fixes f::"'a::ordered_euclidean_space => real"
- assumes f:"integrable lebesgue f"
+ fixes f :: "'a::ordered_euclidean_space \<Rightarrow> real"
+ assumes f: "integrable lebesgue f"
shows "(f has_integral (integral\<^isup>L lebesgue f)) UNIV"
-proof- let ?n = "\<lambda>x. - min (f x) 0" and ?p = "\<lambda>x. max (f x) 0"
- have *:"f = (\<lambda>x. ?p x - ?n x)" apply rule by auto
- note f = integrableD[OF f]
- show ?thesis unfolding lebesgue_integral_def apply(subst *)
- proof(rule has_integral_sub) case goal1
- have *:"\<forall>x. Real (f x) \<noteq> \<omega>" by auto
- note lebesgue.borel_measurable_Real[OF f(1)]
- from positive_integral_has_integral[OF this f(2) *]
- show ?case unfolding real_Real_max .
- next case goal2
- have *:"\<forall>x. Real (- f x) \<noteq> \<omega>" by auto
- note lebesgue.borel_measurable_uminus[OF f(1)]
- note lebesgue.borel_measurable_Real[OF this]
- from positive_integral_has_integral[OF this f(3) *]
- show ?case unfolding real_Real_max minus_min_eq_max by auto
- qed
+proof -
+ let ?n = "\<lambda>x. real (extreal (max 0 (- f x)))" and ?p = "\<lambda>x. real (extreal (max 0 (f x)))"
+ have *: "f = (\<lambda>x. ?p x - ?n x)" by (auto simp del: extreal_max)
+ { fix f have "(\<integral>\<^isup>+ x. extreal (f x) \<partial>lebesgue) = (\<integral>\<^isup>+ x. extreal (max 0 (f x)) \<partial>lebesgue)"
+ by (intro lebesgue.positive_integral_cong_pos) (auto split: split_max) }
+ note eq = this
+ show ?thesis
+ unfolding lebesgue_integral_def
+ apply (subst *)
+ apply (rule has_integral_sub)
+ unfolding eq[of f] eq[of "\<lambda>x. - f x"]
+ apply (safe intro!: positive_integral_has_integral)
+ using integrableD[OF f]
+ by (auto simp: zero_extreal_def[symmetric] positive_integral_max_0 split: split_max
+ intro!: lebesgue.measurable_If lebesgue.borel_measurable_extreal)
qed
lemma lebesgue_positive_integral_eq_borel:
- "f \<in> borel_measurable borel \<Longrightarrow> integral\<^isup>P lebesgue f = integral\<^isup>P lborel f"
- by (auto intro!: lebesgue.positive_integral_subalgebra[symmetric]) default
+ assumes f: "f \<in> borel_measurable borel"
+ shows "integral\<^isup>P lebesgue f = integral\<^isup>P lborel f"
+proof -
+ from f have "integral\<^isup>P lebesgue (\<lambda>x. max 0 (f x)) = integral\<^isup>P lborel (\<lambda>x. max 0 (f x))"
+ by (auto intro!: lebesgue.positive_integral_subalgebra[symmetric]) default
+ then show ?thesis unfolding positive_integral_max_0 .
+qed
lemma lebesgue_integral_eq_borel:
assumes "f \<in> borel_measurable borel"
@@ -771,7 +827,7 @@
have "sets ?G = sets (\<Pi>\<^isub>M i\<in>I.
sigma \<lparr> space = UNIV::real set, sets = range lessThan, measure = lebesgue.\<mu> \<rparr>)"
by (subst sigma_product_algebra_sigma_eq[of I "\<lambda>_ i. {..<real i}" ])
- (auto intro!: measurable_sigma_sigma isotoneI real_arch_lt
+ (auto intro!: measurable_sigma_sigma incseq_SucI real_arch_lt
simp: product_algebra_def)
then show ?thesis
unfolding lborel_def borel_eq_lessThan lebesgue_def sigma_def by simp
@@ -838,9 +894,10 @@
let ?E = "\<lparr> space = UNIV :: 'a set, sets = range (\<lambda>(a,b). {a..b}) \<rparr>"
show "Int_stable ?E" using Int_stable_cuboids .
show "range cube \<subseteq> sets ?E" unfolding cube_def_raw by auto
+ show "incseq cube" using cube_subset_Suc by (auto intro!: incseq_SucI)
{ fix x have "\<exists>n. x \<in> cube n" using mem_big_cube[of x] by fastsimp }
- then show "cube \<up> space ?E" by (intro isotoneI cube_subset_Suc) auto
- { fix i show "lborel.\<mu> (cube i) \<noteq> \<omega>" unfolding cube_def by auto }
+ then show "(\<Union>i. cube i) = space ?E" by auto
+ { fix i show "lborel.\<mu> (cube i) \<noteq> \<infinity>" unfolding cube_def by auto }
show "A \<in> sets (sigma ?E)" "sets (sigma ?E) = sets lborel" "space ?E = space lborel"
using assms by (simp_all add: borel_eq_atLeastAtMost)
@@ -857,7 +914,7 @@
by (simp add: interval_ne_empty eucl_le[where 'a='a])
then have "lborel.\<mu> X = (\<Prod>x<DIM('a). lborel.\<mu> {a $$ x .. b $$ x})"
by (auto simp: content_closed_interval eucl_le[where 'a='a]
- intro!: Real_setprod )
+ intro!: setprod_extreal[symmetric])
also have "\<dots> = measure ?P (?T X)"
unfolding * by (subst lborel_space.measure_times) auto
finally show ?thesis .
@@ -882,7 +939,7 @@
using lborel_eq_lborel_space[OF A] by simp
lemma borel_fubini_positiv_integral:
- fixes f :: "'a::ordered_euclidean_space \<Rightarrow> pextreal"
+ fixes f :: "'a::ordered_euclidean_space \<Rightarrow> extreal"
assumes f: "f \<in> borel_measurable borel"
shows "integral\<^isup>P lborel f = \<integral>\<^isup>+x. f (p2e x) \<partial>(lborel_space.P DIM('a))"
proof (rule lborel_space.positive_integral_vimage[OF _ measure_preserving_p2e _])
--- a/src/HOL/Probability/Measure.thy Mon Mar 14 15:17:10 2011 +0100
+++ b/src/HOL/Probability/Measure.thy Mon Mar 14 15:29:10 2011 +0100
@@ -76,7 +76,7 @@
lemma (in measure_space) measure_countably_additive:
assumes "range A \<subseteq> sets M" "disjoint_family A"
- shows "psuminf (\<lambda>i. \<mu> (A i)) = \<mu> (\<Union>i. A i)"
+ shows "(\<Sum>i. \<mu> (A i)) = \<mu> (\<Union>i. A i)"
proof -
have "(\<Union> i. A i) \<in> sets M" using assms(1) by (rule countable_UN)
with ca assms show ?thesis by (simp add: countably_additive_def)
@@ -94,13 +94,13 @@
interpret N: sigma_algebra N by (intro sigma_algebra_cong assms)
show ?thesis
proof
- show "measure N {} = 0" using assms by auto
+ show "positive N (measure N)" using assms by (auto simp: positive_def)
show "countably_additive N (measure N)" unfolding countably_additive_def
proof safe
fix A :: "nat \<Rightarrow> 'a set" assume A: "range A \<subseteq> sets N" "disjoint_family A"
then have "\<And>i. A i \<in> sets M" "(UNION UNIV A) \<in> sets M" unfolding assms by auto
from measure_countably_additive[of A] A this[THEN assms(1)]
- show "(\<Sum>\<^isub>\<infinity>n. measure N (A n)) = measure N (UNION UNIV A)"
+ show "(\<Sum>n. measure N (A n)) = measure N (UNION UNIV A)"
unfolding assms by simp
qed
qed
@@ -124,51 +124,51 @@
have "b = a \<union> (b - a)" using assms by auto
moreover have "{} = a \<inter> (b - a)" by auto
ultimately have "\<mu> b = \<mu> a + \<mu> (b - a)"
- using measure_additive[of a "b - a"] local.Diff[of b a] assms by auto
- moreover have "\<mu> (b - a) \<ge> 0" using assms by auto
+ using measure_additive[of a "b - a"] Diff[of b a] assms by auto
+ moreover have "\<mu> a + 0 \<le> \<mu> a + \<mu> (b - a)" using assms by (intro add_mono) auto
ultimately show "\<mu> a \<le> \<mu> b" by auto
qed
lemma (in measure_space) measure_compl:
- assumes s: "s \<in> sets M" and fin: "\<mu> s \<noteq> \<omega>"
+ assumes s: "s \<in> sets M" and fin: "\<mu> s \<noteq> \<infinity>"
shows "\<mu> (space M - s) = \<mu> (space M) - \<mu> s"
proof -
have s_less_space: "\<mu> s \<le> \<mu> (space M)"
using s by (auto intro!: measure_mono sets_into_space)
-
+ from s have "0 \<le> \<mu> s" by auto
have "\<mu> (space M) = \<mu> (s \<union> (space M - s))" using s
by (metis Un_Diff_cancel Un_absorb1 s sets_into_space)
also have "... = \<mu> s + \<mu> (space M - s)"
by (rule additiveD [OF additive]) (auto simp add: s)
finally have "\<mu> (space M) = \<mu> s + \<mu> (space M - s)" .
- thus ?thesis
- unfolding minus_pextreal_eq2[OF s_less_space fin]
- by (simp add: ac_simps)
+ then show ?thesis
+ using fin `0 \<le> \<mu> s`
+ unfolding extreal_eq_minus_iff by (auto simp: ac_simps)
qed
lemma (in measure_space) measure_Diff:
- assumes finite: "\<mu> B \<noteq> \<omega>"
+ assumes finite: "\<mu> B \<noteq> \<infinity>"
and measurable: "A \<in> sets M" "B \<in> sets M" "B \<subseteq> A"
shows "\<mu> (A - B) = \<mu> A - \<mu> B"
proof -
- have *: "(A - B) \<union> B = A" using `B \<subseteq> A` by auto
-
- 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: pextreal_cancel_plus_minus)
+ have "0 \<le> \<mu> B" using assms by auto
+ have "(A - B) \<union> B = A" using `B \<subseteq> A` by auto
+ then have "\<mu> A = \<mu> ((A - B) \<union> B)" by simp
+ also have "\<dots> = \<mu> (A - B) + \<mu> B"
+ using measurable by (subst measure_additive[symmetric]) auto
+ finally show "\<mu> (A - B) = \<mu> A - \<mu> B"
+ unfolding extreal_eq_minus_iff
+ using finite `0 \<le> \<mu> B` by auto
qed
lemma (in measure_space) measure_countable_increasing:
assumes A: "range A \<subseteq> sets M"
and A0: "A 0 = {}"
- and ASuc: "\<And>n. A n \<subseteq> A (Suc n)"
+ and ASuc: "\<And>n. A n \<subseteq> A (Suc n)"
shows "(SUP n. \<mu> (A n)) = \<mu> (\<Union>i. A i)"
proof -
- {
- fix n
- have "\<mu> (A n) =
- setsum (\<mu> \<circ> (\<lambda>i. A (Suc i) - A i)) {..<n}"
+ { fix n
+ have "\<mu> (A n) = (\<Sum>i<n. \<mu> (A (Suc i) - A i))"
proof (induct n)
case 0 thus ?case by (auto simp add: A0)
next
@@ -199,92 +199,83 @@
by (metis A Diff range_subsetD)
have A2: "(\<Union>i. A (Suc i) - A i) \<in> sets M"
by (blast intro: range_subsetD [OF A])
- have "psuminf ( (\<lambda>i. \<mu> (A (Suc i) - A i))) = \<mu> (\<Union>i. A (Suc i) - A i)"
+ have "(SUP n. \<Sum>i<n. \<mu> (A (Suc i) - A i)) = (\<Sum>i. \<mu> (A (Suc i) - A i))"
+ using A by (auto intro!: suminf_extreal_eq_SUPR[symmetric])
+ also have "\<dots> = \<mu> (\<Union>i. A (Suc i) - A i)"
by (rule measure_countably_additive)
(auto simp add: disjoint_family_Suc ASuc A1 A2)
also have "... = \<mu> (\<Union>i. A i)"
by (simp add: Aeq)
- finally have "psuminf (\<lambda>i. \<mu> (A (Suc i) - A i)) = \<mu> (\<Union>i. A i)" .
- thus ?thesis
- by (auto simp add: Meq psuminf_def)
+ finally have "(SUP n. \<Sum>i<n. \<mu> (A (Suc i) - A i)) = \<mu> (\<Union>i. A i)" .
+ then show ?thesis by (auto simp add: Meq)
qed
lemma (in measure_space) continuity_from_below:
- assumes A: "range A \<subseteq> sets M"
- and ASuc: "!!n. A n \<subseteq> A (Suc n)"
+ assumes A: "range A \<subseteq> sets M" and "incseq A"
shows "(SUP n. \<mu> (A n)) = \<mu> (\<Union>i. A i)"
proof -
have *: "(SUP n. \<mu> (nat_case {} A (Suc n))) = (SUP n. \<mu> (nat_case {} A n))"
- apply (rule Sup_mono_offset_Suc)
- apply (rule measure_mono)
- using assms by (auto split: nat.split)
-
+ using A by (auto intro!: SUPR_eq exI split: nat.split)
have ueq: "(\<Union>i. nat_case {} A i) = (\<Union>i. A i)"
by (auto simp add: split: nat.splits)
have meq: "\<And>n. \<mu> (A n) = (\<mu> \<circ> nat_case {} A) (Suc n)"
by simp
have "(SUP n. \<mu> (nat_case {} A n)) = \<mu> (\<Union>i. nat_case {} A i)"
- by (rule measure_countable_increasing)
- (auto simp add: range_subsetD [OF A] subsetD [OF ASuc] split: nat.splits)
+ using range_subsetD[OF A] incseq_SucD[OF `incseq A`]
+ by (force split: nat.splits intro!: measure_countable_increasing)
also have "\<mu> (\<Union>i. nat_case {} A i) = \<mu> (\<Union>i. A i)"
by (simp add: ueq)
finally have "(SUP n. \<mu> (nat_case {} A n)) = \<mu> (\<Union>i. A i)" .
thus ?thesis unfolding meq * comp_def .
qed
-lemma (in measure_space) measure_up:
- assumes "\<And>i. B i \<in> sets M" "B \<up> P"
- shows "(\<lambda>i. \<mu> (B i)) \<up> \<mu> P"
- using assms unfolding isoton_def
- by (auto intro!: measure_mono continuity_from_below)
+lemma (in measure_space) measure_incseq:
+ assumes "range B \<subseteq> sets M" "incseq B"
+ shows "incseq (\<lambda>i. \<mu> (B i))"
+ using assms by (auto simp: incseq_def intro!: measure_mono)
-lemma (in measure_space) continuity_from_below':
- assumes A: "range A \<subseteq> sets M" "\<And>i. A i \<subseteq> A (Suc i)"
- shows "(\<lambda>i. (\<mu> (A i))) ----> (\<mu> (\<Union>i. A i))"
-proof- let ?A = "\<Union>i. A i"
- have " (\<lambda>i. \<mu> (A i)) \<up> \<mu> ?A" apply(rule measure_up)
- using assms unfolding complete_lattice_class.isoton_def by auto
- thus ?thesis by(rule isotone_Lim(1))
-qed
+lemma (in measure_space) continuity_from_below_Lim:
+ assumes A: "range A \<subseteq> sets M" "incseq A"
+ shows "(\<lambda>i. (\<mu> (A i))) ----> \<mu> (\<Union>i. A i)"
+ using LIMSEQ_extreal_SUPR[OF measure_incseq, OF A]
+ continuity_from_below[OF A] by simp
+
+lemma (in measure_space) measure_decseq:
+ assumes "range B \<subseteq> sets M" "decseq B"
+ shows "decseq (\<lambda>i. \<mu> (B i))"
+ using assms by (auto simp: decseq_def intro!: measure_mono)
lemma (in measure_space) continuity_from_above:
- assumes A: "range A \<subseteq> sets M"
- and mono_Suc: "\<And>n. A (Suc n) \<subseteq> A n"
- and finite: "\<And>i. \<mu> (A i) \<noteq> \<omega>"
+ assumes A: "range A \<subseteq> sets M" and "decseq A"
+ and finite: "\<And>i. \<mu> (A i) \<noteq> \<infinity>"
shows "(INF n. \<mu> (A n)) = \<mu> (\<Inter>i. A i)"
proof -
- { fix n have "A n \<subseteq> A 0" using mono_Suc by (induct n) auto }
- note mono = this
-
have le_MI: "\<mu> (\<Inter>i. A i) \<le> \<mu> (A 0)"
using A by (auto intro!: measure_mono)
- hence *: "\<mu> (\<Inter>i. A i) \<noteq> \<omega>" using finite[of 0] by auto
+ hence *: "\<mu> (\<Inter>i. A i) \<noteq> \<infinity>" using finite[of 0] by auto
+
+ have A0: "0 \<le> \<mu> (A 0)" using A by auto
- have le_IM: "(INF n. \<mu> (A n)) \<le> \<mu> (A 0)"
- by (rule INF_leI) simp
-
- have "\<mu> (A 0) - (INF n. \<mu> (A n)) = (SUP n. \<mu> (A 0 - A n))"
- unfolding pextreal_SUP_minus[symmetric]
- using mono A finite by (subst measure_Diff) auto
+ have "\<mu> (A 0) - (INF n. \<mu> (A n)) = \<mu> (A 0) + (SUP n. - \<mu> (A n))"
+ by (simp add: extreal_SUPR_uminus minus_extreal_def)
+ also have "\<dots> = (SUP n. \<mu> (A 0) - \<mu> (A n))"
+ unfolding minus_extreal_def using A0 assms
+ by (subst SUPR_extreal_add) (auto simp add: measure_decseq)
+ also have "\<dots> = (SUP n. \<mu> (A 0 - A n))"
+ using A finite `decseq A`[unfolded decseq_def] by (subst measure_Diff) auto
also have "\<dots> = \<mu> (\<Union>i. A 0 - A i)"
proof (rule continuity_from_below)
show "range (\<lambda>n. A 0 - A n) \<subseteq> sets M"
using A by auto
- show "\<And>n. A 0 - A n \<subseteq> A 0 - A (Suc n)"
- using mono_Suc by auto
+ show "incseq (\<lambda>n. A 0 - A n)"
+ using `decseq A` by (auto simp add: incseq_def decseq_def)
qed
also have "\<dots> = \<mu> (A 0) - \<mu> (\<Inter>i. A i)"
- using mono A finite * by (simp, subst measure_Diff) auto
+ using A finite * by (simp, subst measure_Diff) auto
finally show ?thesis
- by (rule pextreal_diff_eq_diff_imp_eq[OF finite[of 0] le_IM le_MI])
+ unfolding extreal_minus_eq_minus_iff using finite A0 by auto
qed
-lemma (in measure_space) measure_down:
- assumes "\<And>i. B i \<in> sets M" "B \<down> P"
- and finite: "\<And>i. \<mu> (B i) \<noteq> \<omega>"
- shows "(\<lambda>i. \<mu> (B i)) \<down> \<mu> P"
- using assms unfolding antiton_def
- by (auto intro!: measure_mono continuity_from_above)
lemma (in measure_space) measure_insert:
assumes sets: "{x} \<in> sets M" "A \<in> sets M" and "x \<notin> A"
shows "\<mu> (insert x A) = \<mu> {x} + \<mu> A"
@@ -293,109 +284,26 @@
from measure_additive[OF sets this] show ?thesis by simp
qed
-lemma (in measure_space) measure_finite_singleton:
- assumes fin: "finite S"
- and ssets: "\<And>x. x \<in> S \<Longrightarrow> {x} \<in> sets M"
- shows "\<mu> S = (\<Sum>x\<in>S. \<mu> {x})"
+lemma (in measure_space) measure_setsum:
+ assumes "finite S" and "\<And>i. i \<in> S \<Longrightarrow> A i \<in> sets M"
+ assumes disj: "disjoint_family_on A S"
+ shows "(\<Sum>i\<in>S. \<mu> (A i)) = \<mu> (\<Union>i\<in>S. A i)"
using assms proof induct
- case (insert x S)
- have *: "\<mu> S = (\<Sum>x\<in>S. \<mu> {x})" "{x} \<in> sets M"
- using insert.prems by (blast intro!: insert.hyps(3))+
-
- have "(\<Union>x\<in>S. {x}) \<in> sets M"
- using insert.prems `finite S` by (blast intro!: finite_UN)
- hence "S \<in> sets M" by auto
- from measure_insert[OF `{x} \<in> sets M` this `x \<notin> S`]
- show ?case using `x \<notin> S` `finite S` * by simp
+ case (insert i S)
+ then have "(\<Sum>i\<in>S. \<mu> (A i)) = \<mu> (\<Union>a\<in>S. A a)"
+ by (auto intro: disjoint_family_on_mono)
+ moreover have "A i \<inter> (\<Union>a\<in>S. A a) = {}"
+ using `disjoint_family_on A (insert i S)` `i \<notin> S`
+ by (auto simp: disjoint_family_on_def)
+ ultimately show ?case using insert
+ by (auto simp: measure_additive finite_UN)
qed simp
-lemma (in measure_space) measure_finitely_additive':
- assumes "f \<in> ({..< n :: nat} \<rightarrow> sets M)"
- assumes "\<And> a b. \<lbrakk>a < n ; b < n ; a \<noteq> b\<rbrakk> \<Longrightarrow> f a \<inter> f b = {}"
- assumes "s = \<Union> (f ` {..< n})"
- shows "(\<Sum>i<n. (\<mu> \<circ> f) i) = \<mu> s"
-proof -
- def f' == "\<lambda> i. (if i < n then f i else {})"
- have rf: "range f' \<subseteq> sets M" unfolding f'_def
- using assms empty_sets by auto
- have df: "disjoint_family f'" unfolding f'_def disjoint_family_on_def
- using assms by simp
- have "\<Union> range f' = (\<Union> i \<in> {..< n}. f i)"
- unfolding f'_def by auto
- then have "\<mu> s = \<mu> (\<Union> range f')"
- using assms by simp
- then have part1: "(\<Sum>\<^isub>\<infinity> i. \<mu> (f' i)) = \<mu> s"
- using df rf ca[unfolded countably_additive_def, rule_format, of f']
- by auto
-
- have "(\<Sum>\<^isub>\<infinity> i. \<mu> (f' i)) = (\<Sum> i< n. \<mu> (f' i))"
- by (rule psuminf_finite) (simp add: f'_def)
- also have "\<dots> = (\<Sum>i<n. \<mu> (f i))"
- unfolding f'_def by auto
- finally have part2: "(\<Sum>\<^isub>\<infinity> i. \<mu> (f' i)) = (\<Sum>i<n. \<mu> (f i))" by simp
- show ?thesis using part1 part2 by auto
-qed
-
-
-lemma (in measure_space) measure_finitely_additive:
- assumes "finite r"
- assumes "r \<subseteq> sets M"
- assumes d: "\<And> a b. \<lbrakk>a \<in> r ; b \<in> r ; a \<noteq> b\<rbrakk> \<Longrightarrow> a \<inter> b = {}"
- shows "(\<Sum> i \<in> r. \<mu> i) = \<mu> (\<Union> r)"
-using assms
-proof -
- (* counting the elements in r is enough *)
- let ?R = "{..<card r}"
- obtain f where f: "f ` ?R = r" "inj_on f ?R"
- using ex_bij_betw_nat_finite[unfolded bij_betw_def, OF `finite r`]
- unfolding atLeast0LessThan by auto
- hence f_into_sets: "f \<in> ?R \<rightarrow> sets M" using assms by auto
- have disj: "\<And> a b. \<lbrakk>a \<in> ?R ; b \<in> ?R ; a \<noteq> b\<rbrakk> \<Longrightarrow> f a \<inter> f b = {}"
- proof -
- fix a b assume asm: "a \<in> ?R" "b \<in> ?R" "a \<noteq> b"
- hence neq: "f a \<noteq> f b" using f[unfolded inj_on_def, rule_format] by blast
- from asm have "f a \<in> r" "f b \<in> r" using f by auto
- thus "f a \<inter> f b = {}" using d[of "f a" "f b"] f using neq by auto
- qed
- have "(\<Union> r) = (\<Union> i \<in> ?R. f i)"
- using f by auto
- hence "\<mu> (\<Union> r)= \<mu> (\<Union> i \<in> ?R. f i)" by simp
- also have "\<dots> = (\<Sum> i \<in> ?R. \<mu> (f i))"
- using measure_finitely_additive'[OF f_into_sets disj] by simp
- also have "\<dots> = (\<Sum> a \<in> r. \<mu> a)"
- using f[rule_format] setsum_reindex[of f ?R "\<lambda> a. \<mu> a"] by auto
- finally show ?thesis by simp
-qed
-
-lemma (in measure_space) measure_finitely_additive'':
- assumes "finite s"
- assumes "\<And> i. i \<in> s \<Longrightarrow> a i \<in> sets M"
- assumes d: "disjoint_family_on a s"
- shows "(\<Sum> i \<in> s. \<mu> (a i)) = \<mu> (\<Union> i \<in> s. a i)"
-using assms
-proof -
- (* counting the elements in s is enough *)
- let ?R = "{..< card s}"
- obtain f where f: "f ` ?R = s" "inj_on f ?R"
- using ex_bij_betw_nat_finite[unfolded bij_betw_def, OF `finite s`]
- unfolding atLeast0LessThan by auto
- hence f_into_sets: "a \<circ> f \<in> ?R \<rightarrow> sets M" using assms unfolding o_def by auto
- have disj: "\<And> i j. \<lbrakk>i \<in> ?R ; j \<in> ?R ; i \<noteq> j\<rbrakk> \<Longrightarrow> (a \<circ> f) i \<inter> (a \<circ> f) j = {}"
- proof -
- fix i j assume asm: "i \<in> ?R" "j \<in> ?R" "i \<noteq> j"
- hence neq: "f i \<noteq> f j" using f[unfolded inj_on_def, rule_format] by blast
- from asm have "f i \<in> s" "f j \<in> s" using f by auto
- thus "(a \<circ> f) i \<inter> (a \<circ> f) j = {}"
- using d f neq unfolding disjoint_family_on_def by auto
- qed
- have "(\<Union> i \<in> s. a i) = (\<Union> i \<in> f ` ?R. a i)" using f by auto
- hence "(\<Union> i \<in> s. a i) = (\<Union> i \<in> ?R. a (f i))" by auto
- hence "\<mu> (\<Union> i \<in> s. a i) = (\<Sum> i \<in> ?R. \<mu> (a (f i)))"
- using measure_finitely_additive'[OF f_into_sets disj] by simp
- also have "\<dots> = (\<Sum> i \<in> s. \<mu> (a i))"
- using f[rule_format] setsum_reindex[of f ?R "\<lambda> i. \<mu> (a i)"] by auto
- finally show ?thesis by simp
-qed
+lemma (in measure_space) measure_finite_singleton:
+ assumes "finite S" "\<And>x. x \<in> S \<Longrightarrow> {x} \<in> sets M"
+ shows "\<mu> S = (\<Sum>x\<in>S. \<mu> {x})"
+ using measure_setsum[of S "\<lambda>x. {x}", OF assms]
+ by (auto simp: disjoint_family_on_def)
lemma finite_additivity_sufficient:
assumes "sigma_algebra M"
@@ -405,7 +313,7 @@
interpret sigma_algebra M by fact
show ?thesis
proof
- show [simp]: "measure M {} = 0" using pos by (simp add: positive_def)
+ show [simp]: "positive M (measure M)" using pos by (simp add: positive_def)
show "countably_additive M (measure M)"
proof (auto simp add: countably_additive_def)
fix A :: "nat \<Rightarrow> 'a set"
@@ -434,12 +342,12 @@
by blast
qed
then obtain N where N: "\<forall>m\<ge>N. A m = {}" by blast
- then have "\<forall>m\<ge>N. measure M (A m) = 0" by simp
- then have "(\<Sum>\<^isub>\<infinity> n. measure M (A n)) = setsum (\<lambda>m. measure M (A m)) {..<N}"
- by (simp add: psuminf_finite)
+ then have "\<forall>m\<ge>N. measure M (A m) = 0" using pos[unfolded positive_def] by simp
+ then have "(\<Sum>n. measure M (A n)) = (\<Sum>m<N. measure M (A m))"
+ by (simp add: suminf_finite)
also have "... = measure M (\<Union>i<N. A i)"
proof (induct N)
- case 0 thus ?case by simp
+ case 0 thus ?case using pos[unfolded positive_def] by simp
next
case (Suc n)
have "measure M (A n \<union> (\<Union> x<n. A x)) = measure M (A n) + measure M (\<Union> i<n. A i)"
@@ -465,30 +373,25 @@
by auto (metis Int_absorb N disjoint_iff_not_equal lessThan_iff not_leE)
thus ?thesis by simp
qed
- finally show "(\<Sum>\<^isub>\<infinity> n. measure M (A n)) = measure M (\<Union>i. A i)" .
+ finally show "(\<Sum>n. measure M (A n)) = measure M (\<Union>i. A i)" .
qed
qed
qed
lemma (in measure_space) measure_setsum_split:
- assumes "finite r" and "a \<in> sets M" and br_in_M: "b ` r \<subseteq> sets M"
- assumes "(\<Union>i \<in> r. b i) = space M"
- assumes "disjoint_family_on b r"
- shows "\<mu> a = (\<Sum> i \<in> r. \<mu> (a \<inter> (b i)))"
+ assumes "finite S" and "A \<in> sets M" and br_in_M: "B ` S \<subseteq> sets M"
+ assumes "(\<Union>i\<in>S. B i) = space M"
+ assumes "disjoint_family_on B S"
+ shows "\<mu> A = (\<Sum>i\<in>S. \<mu> (A \<inter> (B i)))"
proof -
- have *: "\<mu> a = \<mu> (\<Union>i \<in> r. a \<inter> b i)"
+ have *: "\<mu> A = \<mu> (\<Union>i\<in>S. A \<inter> B i)"
using assms by auto
show ?thesis unfolding *
- proof (rule measure_finitely_additive''[symmetric])
- show "finite r" using `finite r` by auto
- { fix i assume "i \<in> r"
- hence "b i \<in> sets M" using br_in_M by auto
- thus "a \<inter> b i \<in> sets M" using `a \<in> sets M` by auto
- }
- show "disjoint_family_on (\<lambda>i. a \<inter> b i) r"
- using `disjoint_family_on b r`
+ proof (rule measure_setsum[symmetric])
+ show "disjoint_family_on (\<lambda>i. A \<inter> B i) S"
+ using `disjoint_family_on B S`
unfolding disjoint_family_on_def by auto
- qed
+ qed (insert assms, auto)
qed
lemma (in measure_space) measure_subadditive:
@@ -506,7 +409,7 @@
lemma (in measure_space) measure_eq_0:
assumes "N \<in> sets M" and "\<mu> N = 0" and "K \<subseteq> N" and "K \<in> sets M"
shows "\<mu> K = 0"
-using measure_mono[OF assms(3,4,1)] assms(2) by auto
+ using measure_mono[OF assms(3,4,1)] assms(2) positive_measure[OF assms(4)] by auto
lemma (in measure_space) measure_finitely_subadditive:
assumes "finite I" "A ` I \<subseteq> sets M"
@@ -523,35 +426,38 @@
lemma (in measure_space) measure_countably_subadditive:
assumes "range f \<subseteq> sets M"
- shows "\<mu> (\<Union>i. f i) \<le> (\<Sum>\<^isub>\<infinity> i. \<mu> (f i))"
+ shows "\<mu> (\<Union>i. f i) \<le> (\<Sum>i. \<mu> (f i))"
proof -
have "\<mu> (\<Union>i. f i) = \<mu> (\<Union>i. disjointed f i)"
unfolding UN_disjointed_eq ..
- also have "\<dots> = (\<Sum>\<^isub>\<infinity> i. \<mu> (disjointed f i))"
+ also have "\<dots> = (\<Sum>i. \<mu> (disjointed f i))"
using range_disjointed_sets[OF assms] measure_countably_additive
by (simp add: disjoint_family_disjointed comp_def)
- also have "\<dots> \<le> (\<Sum>\<^isub>\<infinity> i. \<mu> (f i))"
- proof (rule psuminf_le, rule measure_mono)
- fix i show "disjointed f i \<subseteq> f i" by (rule disjointed_subset)
- show "f i \<in> sets M" "disjointed f i \<in> sets M"
- using assms range_disjointed_sets[OF assms] by auto
- qed
+ also have "\<dots> \<le> (\<Sum>i. \<mu> (f i))"
+ using range_disjointed_sets[OF assms] assms
+ by (auto intro!: suminf_le_pos measure_mono positive_measure disjointed_subset)
finally show ?thesis .
qed
lemma (in measure_space) measure_UN_eq_0:
- assumes "\<And> i :: nat. \<mu> (N i) = 0" and "range N \<subseteq> sets M"
+ assumes "\<And>i::nat. \<mu> (N i) = 0" and "range N \<subseteq> sets M"
shows "\<mu> (\<Union> i. N i) = 0"
-using measure_countably_subadditive[OF assms(2)] assms(1) by auto
+proof -
+ have "0 \<le> \<mu> (\<Union> i. N i)" using assms by auto
+ moreover have "\<mu> (\<Union> i. N i) \<le> 0"
+ using measure_countably_subadditive[OF assms(2)] assms(1) by simp
+ ultimately show ?thesis by simp
+qed
lemma (in measure_space) measure_inter_full_set:
- assumes "S \<in> sets M" "T \<in> sets M" and not_\<omega>: "\<mu> (T - S) \<noteq> \<omega>"
+ assumes "S \<in> sets M" "T \<in> sets M" and fin: "\<mu> (T - S) \<noteq> \<infinity>"
assumes T: "\<mu> T = \<mu> (space M)"
shows "\<mu> (S \<inter> T) = \<mu> S"
proof (rule antisym)
show " \<mu> (S \<inter> T) \<le> \<mu> S"
using assms by (auto intro!: measure_mono)
+ have pos: "0 \<le> \<mu> (T - S)" using assms by auto
show "\<mu> S \<le> \<mu> (S \<inter> T)"
proof (rule ccontr)
assume contr: "\<not> ?thesis"
@@ -560,7 +466,7 @@
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 pextreal_less_add[OF not_\<omega>]) (insert contr, auto)
+ using fin contr pos by (intro extreal_less_add) auto
also have "\<dots> = \<mu> (T \<union> S)"
using assms by (subst measure_additive) auto
also have "\<dots> \<le> \<mu> (space M)"
@@ -572,11 +478,11 @@
lemma measure_unique_Int_stable:
fixes E :: "('a, 'b) algebra_scheme" and A :: "nat \<Rightarrow> 'a set"
assumes "Int_stable E"
- and A: "range A \<subseteq> sets E" "A \<up> space E"
+ and A: "range A \<subseteq> sets E" "incseq A" "(\<Union>i. A i) = space E"
and M: "measure_space \<lparr>space = space E, sets = sets (sigma E), measure = \<mu>\<rparr>" (is "measure_space ?M")
and N: "measure_space \<lparr>space = space E, sets = sets (sigma E), measure = \<nu>\<rparr>" (is "measure_space ?N")
and eq: "\<And>X. X \<in> sets E \<Longrightarrow> \<mu> X = \<nu> X"
- and finite: "\<And>i. \<mu> (A i) \<noteq> \<omega>"
+ and finite: "\<And>i. \<mu> (A i) \<noteq> \<infinity>"
assumes "X \<in> sets (sigma E)"
shows "\<mu> X = \<nu> X"
proof -
@@ -585,9 +491,9 @@
where "space ?M = space E" and "sets ?M = sets (sigma E)" and "measure ?M = \<mu>" by (simp_all add: M)
interpret N: measure_space ?N
where "space ?N = space E" and "sets ?N = sets (sigma E)" and "measure ?N = \<nu>" by (simp_all add: N)
- { fix F assume "F \<in> sets E" and "\<mu> F \<noteq> \<omega>"
+ { fix F assume "F \<in> sets E" and "\<mu> F \<noteq> \<infinity>"
then have [intro]: "F \<in> sets (sigma E)" by auto
- have "\<nu> F \<noteq> \<omega>" using `\<mu> F \<noteq> \<omega>` `F \<in> sets E` eq by simp
+ have "\<nu> F \<noteq> \<infinity>" using `\<mu> F \<noteq> \<infinity>` `F \<in> sets E` eq by simp
interpret D: dynkin_system "\<lparr>space=space E, sets=?D F\<rparr>"
proof (rule dynkin_systemI, simp_all)
fix A assume "A \<in> sets (sigma E) \<and> \<mu> (F \<inter> A) = \<nu> (F \<inter> A)"
@@ -602,14 +508,14 @@
and [intro]: "F \<inter> A \<in> sets (sigma E)"
using `F \<in> sets E` M.sets_into_space by auto
have "\<nu> (F \<inter> A) \<le> \<nu> F" by (auto intro!: N.measure_mono)
- then have "\<nu> (F \<inter> A) \<noteq> \<omega>" using `\<nu> F \<noteq> \<omega>` by auto
+ then have "\<nu> (F \<inter> A) \<noteq> \<infinity>" using `\<nu> F \<noteq> \<infinity>` by auto
have "\<mu> (F \<inter> A) \<le> \<mu> F" by (auto intro!: M.measure_mono)
- then have "\<mu> (F \<inter> A) \<noteq> \<omega>" using `\<mu> F \<noteq> \<omega>` by auto
+ then have "\<mu> (F \<inter> A) \<noteq> \<infinity>" using `\<mu> F \<noteq> \<infinity>` by auto
then have "\<mu> (F \<inter> (space (sigma E) - A)) = \<mu> F - \<mu> (F \<inter> A)" unfolding **
using `F \<inter> A \<in> sets (sigma E)` by (auto intro!: M.measure_Diff)
also have "\<dots> = \<nu> F - \<nu> (F \<inter> A)" using eq `F \<in> sets E` * by simp
also have "\<dots> = \<nu> (F \<inter> (space (sigma E) - A))" unfolding **
- using `F \<inter> A \<in> sets (sigma E)` `\<nu> (F \<inter> A) \<noteq> \<omega>` by (auto intro!: N.measure_Diff[symmetric])
+ using `F \<inter> A \<in> sets (sigma E)` `\<nu> (F \<inter> A) \<noteq> \<infinity>` by (auto intro!: N.measure_Diff[symmetric])
finally show "space E - A \<in> sets (sigma E) \<and> \<mu> (F \<inter> (space E - A)) = \<nu> (F \<inter> (space E - A))"
using * by auto
next
@@ -630,15 +536,13 @@
have "\<And>D. D \<in> sets (sigma E) \<Longrightarrow> \<mu> (F \<inter> D) = \<nu> (F \<inter> D)"
by (subst (asm) *) auto }
note * = this
- { fix i have "\<mu> (A i \<inter> X) = \<nu> (A i \<inter> X)"
+ let "?A i" = "A i \<inter> X"
+ have A': "range ?A \<subseteq> sets (sigma E)" "incseq ?A"
+ using A(1,2) `X \<in> sets (sigma E)` by (auto simp: incseq_def)
+ { fix i have "\<mu> (?A i) = \<nu> (?A i)"
using *[of "A i" X] `X \<in> sets (sigma E)` A finite by auto }
- moreover
- have "(\<lambda>i. A i \<inter> X) \<up> X"
- using `X \<in> sets (sigma E)` M.sets_into_space A
- by (auto simp: isoton_def)
- then have "(\<lambda>i. \<mu> (A i \<inter> X)) \<up> \<mu> X" "(\<lambda>i. \<nu> (A i \<inter> X)) \<up> \<nu> X"
- using `X \<in> sets (sigma E)` A by (auto intro!: M.measure_up N.measure_up M.Int simp: subset_eq)
- ultimately show ?thesis by (simp add: isoton_def)
+ with M.continuity_from_below[OF A'] N.continuity_from_below[OF A']
+ show ?thesis using A(3) `X \<in> sets (sigma E)` by auto
qed
section "@{text \<mu>}-null sets"
@@ -650,10 +554,10 @@
shows "N \<union> N' \<in> null_sets"
proof (intro conjI CollectI)
show "N \<union> N' \<in> sets M" using assms by auto
- have "\<mu> (N \<union> N') \<le> \<mu> N + \<mu> N'"
+ then have "0 \<le> \<mu> (N \<union> N')" by simp
+ moreover have "\<mu> (N \<union> N') \<le> \<mu> N + \<mu> N'"
using assms by (intro measure_subadditive) auto
- then show "\<mu> (N \<union> N') = 0"
- using assms by auto
+ ultimately show "\<mu> (N \<union> N') = 0" using assms by auto
qed
lemma UN_from_nat: "(\<Union>i. N i) = (\<Union>i. N (Countable.from_nat i))"
@@ -669,11 +573,11 @@
shows "(\<Union>i. N i) \<in> null_sets"
proof (intro conjI CollectI)
show "(\<Union>i. N i) \<in> sets M" using assms by auto
- have "\<mu> (\<Union>i. N i) \<le> (\<Sum>\<^isub>\<infinity> n. \<mu> (N (Countable.from_nat n)))"
+ then have "0 \<le> \<mu> (\<Union>i. N i)" by simp
+ moreover have "\<mu> (\<Union>i. N i) \<le> (\<Sum>n. \<mu> (N (Countable.from_nat n)))"
unfolding UN_from_nat[of N]
using assms by (intro measure_countably_subadditive) auto
- then show "\<mu> (\<Union>i. N i) = 0"
- using assms by auto
+ ultimately show "\<mu> (\<Union>i. N i) = 0" using assms by auto
qed
lemma (in measure_space) null_sets_finite_UN:
@@ -681,10 +585,10 @@
shows "(\<Union>i\<in>S. A i) \<in> null_sets"
proof (intro CollectI conjI)
show "(\<Union>i\<in>S. A i) \<in> sets M" using assms by (intro finite_UN) auto
- have "\<mu> (\<Union>i\<in>S. A i) \<le> (\<Sum>i\<in>S. \<mu> (A i))"
+ then have "0 \<le> \<mu> (\<Union>i\<in>S. A i)" by simp
+ moreover have "\<mu> (\<Union>i\<in>S. A i) \<le> (\<Sum>i\<in>S. \<mu> (A i))"
using assms by (intro measure_finitely_subadditive) auto
- then show "\<mu> (\<Union>i\<in>S. A i) = 0"
- using assms by auto
+ ultimately show "\<mu> (\<Union>i\<in>S. A i) = 0" using assms by auto
qed
lemma (in measure_space) null_set_Int1:
@@ -731,6 +635,23 @@
almost_everywhere :: "('a \<Rightarrow> bool) \<Rightarrow> bool" (binder "AE " 10) where
"almost_everywhere P \<longleftrightarrow> (\<exists>N\<in>null_sets. { x \<in> space M. \<not> P x } \<subseteq> N)"
+syntax
+ "_almost_everywhere" :: "'a \<Rightarrow> ('a, 'b) measure_space_scheme \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool" ("AE _ in _. _" [0,0,10] 10)
+
+translations
+ "AE x in M. P" == "CONST measure_space.almost_everywhere M (%x. P)"
+
+lemma (in measure_space) AE_cong_measure:
+ assumes "\<And>A. A \<in> sets M \<Longrightarrow> measure N A = \<mu> A" "sets N = sets M" "space N = space M"
+ shows "(AE x in N. P x) \<longleftrightarrow> (AE x. P x)"
+proof -
+ interpret N: measure_space N
+ by (rule measure_space_cong) fact+
+ show ?thesis
+ unfolding N.almost_everywhere_def almost_everywhere_def
+ by (auto simp: assms)
+qed
+
lemma (in measure_space) AE_I':
"N \<in> null_sets \<Longrightarrow> {x\<in>space M. \<not> P x} \<subseteq> N \<Longrightarrow> (AE x. P x)"
unfolding almost_everywhere_def by auto
@@ -741,13 +662,19 @@
proof
assume "AE x. P x" then obtain N where N: "N \<in> sets M" "?P \<subseteq> N" "\<mu> N = 0"
unfolding almost_everywhere_def by auto
+ have "0 \<le> \<mu> ?P" using assms by simp
moreover have "\<mu> ?P \<le> \<mu> N"
using assms N(1,2) by (auto intro: measure_mono)
- ultimately show "?P \<in> null_sets" using assms by auto
+ ultimately have "\<mu> ?P = 0" unfolding `\<mu> N = 0` by auto
+ then show "?P \<in> null_sets" using assms by simp
next
assume "?P \<in> null_sets" with assms show "AE x. P x" by (auto intro: AE_I')
qed
+lemma (in measure_space) AE_iff_measurable:
+ "N \<in> sets M \<Longrightarrow> {x\<in>space M. \<not> P x} = N \<Longrightarrow> (AE x. P x) \<longleftrightarrow> \<mu> N = 0"
+ using AE_iff_null_set[of P] by simp
+
lemma (in measure_space) AE_True[intro, simp]: "AE x. True"
unfolding almost_everywhere_def by auto
@@ -760,13 +687,9 @@
assumes "AE x. P x" "{x\<in>space M. P x} \<in> sets M"
shows "\<mu> {x\<in>space M. \<not> P x} = 0" (is "\<mu> ?P = 0")
proof -
- obtain A where A: "?P \<subseteq> A" "A \<in> sets M" "\<mu> A = 0"
- using assms by (auto elim!: AE_E)
- have "?P = space M - {x\<in>space M. P x}" by auto
- then have "?P \<in> sets M" using assms by auto
- with assms `?P \<subseteq> A` `A \<in> sets M` have "\<mu> ?P \<le> \<mu> A"
- by (auto intro!: measure_mono)
- then show "\<mu> ?P = 0" using A by simp
+ have "{x\<in>space M. \<not> P x} = space M - {x\<in>space M. P x}"
+ by auto
+ with AE_iff_null_set[of P] assms show ?thesis by auto
qed
lemma (in measure_space) AE_I:
@@ -788,8 +711,10 @@
show ?thesis
proof (intro AE_I)
- show "A \<union> B \<in> sets M" "\<mu> (A \<union> B) = 0" using A B
- using measure_subadditive[of A B] by auto
+ have "0 \<le> \<mu> (A \<union> B)" using A B by auto
+ moreover have "\<mu> (A \<union> B) \<le> 0"
+ using measure_subadditive[of A B] A B by auto
+ ultimately show "A \<union> B \<in> sets M" "\<mu> (A \<union> B) = 0" using A B by auto
show "{x\<in>space M. \<not> Q x} \<subseteq> A \<union> B"
using P imp by auto
qed
@@ -818,8 +743,8 @@
"(\<And>x. x \<in> space M \<Longrightarrow> P x \<longleftrightarrow> Q x) \<Longrightarrow> (AE x. P x) \<longleftrightarrow> (AE x. Q x)"
by auto
-lemma (in measure_space) all_AE_countable:
- "(\<forall>i::'i::countable. AE x. P i x) \<longleftrightarrow> (AE x. \<forall>i. P i x)"
+lemma (in measure_space) AE_all_countable:
+ "(AE x. \<forall>i. P i x) \<longleftrightarrow> (\<forall>i::'i::countable. AE x. P i x)"
proof
assume "\<forall>i. AE x. P i x"
from this[unfolded almost_everywhere_def Bex_def, THEN choice]
@@ -833,6 +758,10 @@
unfolding almost_everywhere_def by auto
qed auto
+lemma (in measure_space) AE_finite_all:
+ assumes f: "finite S" shows "(AE x. \<forall>i\<in>S. P i x) \<longleftrightarrow> (\<forall>i\<in>S. AE x. P i x)"
+ using f by induct auto
+
lemma (in measure_space) restricted_measure_space:
assumes "S \<in> sets M"
shows "measure_space (restricted_space S)"
@@ -840,7 +769,7 @@
unfolding measure_space_def measure_space_axioms_def
proof safe
show "sigma_algebra ?r" using restricted_sigma_algebra[OF assms] .
- show "measure ?r {} = 0" by simp
+ show "positive ?r (measure ?r)" using `S \<in> sets M` by (auto simp: positive_def)
show "countably_additive ?r (measure ?r)"
unfolding countably_additive_def
@@ -848,13 +777,38 @@
fix A :: "nat \<Rightarrow> 'a set"
assume *: "range A \<subseteq> sets ?r" and **: "disjoint_family A"
from restriction_in_sets[OF assms *[simplified]] **
- show "(\<Sum>\<^isub>\<infinity> n. measure ?r (A n)) = measure ?r (\<Union>i. A i)"
+ show "(\<Sum>n. measure ?r (A n)) = measure ?r (\<Union>i. A i)"
using measure_countably_additive by simp
qed
qed
+lemma (in measure_space) AE_restricted:
+ assumes "A \<in> sets M"
+ shows "(AE x in restricted_space A. P x) \<longleftrightarrow> (AE x. x \<in> A \<longrightarrow> P x)"
+proof -
+ interpret R: measure_space "restricted_space A"
+ by (rule restricted_measure_space[OF `A \<in> sets M`])
+ show ?thesis
+ proof
+ assume "AE x in restricted_space A. P x"
+ from this[THEN R.AE_E] guess N' .
+ then obtain N where "{x \<in> A. \<not> P x} \<subseteq> A \<inter> N" "\<mu> (A \<inter> N) = 0" "N \<in> sets M"
+ by auto
+ moreover then have "{x \<in> space M. \<not> (x \<in> A \<longrightarrow> P x)} \<subseteq> A \<inter> N"
+ using `A \<in> sets M` sets_into_space by auto
+ ultimately show "AE x. x \<in> A \<longrightarrow> P x"
+ using `A \<in> sets M` by (auto intro!: AE_I[where N="A \<inter> N"])
+ next
+ assume "AE x. x \<in> A \<longrightarrow> P x"
+ from this[THEN AE_E] guess N .
+ then show "AE x in restricted_space A. P x"
+ using null_set_Int1[OF _ `A \<in> sets M`] `A \<in> sets M`[THEN sets_into_space]
+ by (auto intro!: R.AE_I[where N="A \<inter> N"] simp: subset_eq)
+ qed
+qed
+
lemma (in measure_space) measure_space_subalgebra:
- assumes "sigma_algebra N" and [simp]: "sets N \<subseteq> sets M" "space N = space M"
+ assumes "sigma_algebra N" and "sets N \<subseteq> sets M" "space N = space M"
and measure[simp]: "\<And>X. X \<in> sets N \<Longrightarrow> measure N X = measure M X"
shows "measure_space N"
proof -
@@ -864,13 +818,26 @@
from `sets N \<subseteq> sets M` have "\<And>A. range A \<subseteq> sets N \<Longrightarrow> range A \<subseteq> sets M" by blast
then show "countably_additive N (measure N)"
by (auto intro!: measure_countably_additive simp: countably_additive_def subset_eq)
- qed simp
+ show "positive N (measure_space.measure N)"
+ using assms(2) by (auto simp add: positive_def)
+ qed
+qed
+
+lemma (in measure_space) AE_subalgebra:
+ assumes ae: "AE x in N. P x"
+ and N: "sets N \<subseteq> sets M" "space N = space M" "\<And>A. A \<in> sets N \<Longrightarrow> measure N A = \<mu> A"
+ and sa: "sigma_algebra N"
+ shows "AE x. P x"
+proof -
+ interpret N: measure_space N using measure_space_subalgebra[OF sa N] .
+ from ae[THEN N.AE_E] guess N .
+ with N show ?thesis unfolding almost_everywhere_def by auto
qed
section "@{text \<sigma>}-finite Measures"
locale sigma_finite_measure = measure_space +
- assumes sigma_finite: "\<exists>A::nat \<Rightarrow> 'a set. range A \<subseteq> sets M \<and> (\<Union>i. A i) = space M \<and> (\<forall>i. \<mu> (A i) \<noteq> \<omega>)"
+ assumes sigma_finite: "\<exists>A::nat \<Rightarrow> 'a set. range A \<subseteq> sets M \<and> (\<Union>i. A i) = space M \<and> (\<forall>i. \<mu> (A i) \<noteq> \<infinity>)"
lemma (in sigma_finite_measure) restricted_sigma_finite_measure:
assumes "S \<in> sets M"
@@ -881,9 +848,9 @@
show "measure_space ?r" using restricted_measure_space[OF assms] .
next
obtain A :: "nat \<Rightarrow> 'a set" where
- "range A \<subseteq> sets M" "(\<Union>i. A i) = space M" "\<And>i. \<mu> (A i) \<noteq> \<omega>"
+ "range A \<subseteq> sets M" "(\<Union>i. A i) = space M" "\<And>i. \<mu> (A i) \<noteq> \<infinity>"
using sigma_finite by auto
- show "\<exists>A::nat \<Rightarrow> 'a set. range A \<subseteq> sets ?r \<and> (\<Union>i. A i) = space ?r \<and> (\<forall>i. measure ?r (A i) \<noteq> \<omega>)"
+ show "\<exists>A::nat \<Rightarrow> 'a set. range A \<subseteq> sets ?r \<and> (\<Union>i. A i) = space ?r \<and> (\<forall>i. measure ?r (A i) \<noteq> \<infinity>)"
proof (safe intro!: exI[of _ "\<lambda>i. A i \<inter> S"] del: notI)
fix i
show "A i \<inter> S \<in> sets ?r"
@@ -897,8 +864,7 @@
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: pextreal_less_\<omega>)
- finally show "measure ?r (A i \<inter> S) \<noteq> \<omega>" by (auto simp: pextreal_less_\<omega>)
+ then show "measure ?r (A i \<inter> S) \<noteq> \<infinity>" using `\<mu> (A i) \<noteq> \<infinity>` by auto
qed
qed
@@ -909,7 +875,7 @@
interpret M': measure_space M' by (intro measure_space_cong cong)
from sigma_finite guess A .. note A = this
then have "\<And>i. A i \<in> sets M" by auto
- with A have fin: "(\<forall>i. measure M' (A i) \<noteq> \<omega>)" using cong by auto
+ with A have fin: "\<forall>i. measure M' (A i) \<noteq> \<infinity>" using cong by auto
show ?thesis
apply default
using A fin cong by auto
@@ -917,30 +883,30 @@
lemma (in sigma_finite_measure) disjoint_sigma_finite:
"\<exists>A::nat\<Rightarrow>'a set. range A \<subseteq> sets M \<and> (\<Union>i. A i) = space M \<and>
- (\<forall>i. \<mu> (A i) \<noteq> \<omega>) \<and> disjoint_family A"
+ (\<forall>i. \<mu> (A i) \<noteq> \<infinity>) \<and> disjoint_family A"
proof -
obtain A :: "nat \<Rightarrow> 'a set" where
range: "range A \<subseteq> sets M" and
space: "(\<Union>i. A i) = space M" and
- measure: "\<And>i. \<mu> (A i) \<noteq> \<omega>"
+ measure: "\<And>i. \<mu> (A i) \<noteq> \<infinity>"
using sigma_finite by auto
note range' = range_disjointed_sets[OF range] range
{ fix i
have "\<mu> (disjointed A i) \<le> \<mu> (A i)"
using range' disjointed_subset[of A i] by (auto intro!: measure_mono)
- then have "\<mu> (disjointed A i) \<noteq> \<omega>"
+ then have "\<mu> (disjointed A i) \<noteq> \<infinity>"
using measure[of i] by auto }
with disjoint_family_disjointed UN_disjointed_eq[of A] space range'
show ?thesis by (auto intro!: exI[of _ "disjointed A"])
qed
lemma (in sigma_finite_measure) sigma_finite_up:
- "\<exists>F. range F \<subseteq> sets M \<and> F \<up> space M \<and> (\<forall>i. \<mu> (F i) \<noteq> \<omega>)"
+ "\<exists>F. range F \<subseteq> sets M \<and> incseq F \<and> (\<Union>i. F i) = space M \<and> (\<forall>i. \<mu> (F i) \<noteq> \<infinity>)"
proof -
obtain F :: "nat \<Rightarrow> 'a set" where
- F: "range F \<subseteq> sets M" "(\<Union>i. F i) = space M" "\<And>i. \<mu> (F i) \<noteq> \<omega>"
+ F: "range F \<subseteq> sets M" "(\<Union>i. F i) = space M" "\<And>i. \<mu> (F i) \<noteq> \<infinity>"
using sigma_finite by auto
- then show ?thesis unfolding isoton_def
+ then show ?thesis
proof (intro exI[of _ "\<lambda>n. \<Union>i\<le>n. F i"] conjI allI)
from F have "\<And>x. x \<in> space M \<Longrightarrow> \<exists>i. x \<in> F i" by auto
then show "(\<Union>n. \<Union> i\<le>n. F i) = space M"
@@ -949,16 +915,16 @@
fix n
have "\<mu> (\<Union> i\<le>n. F i) \<le> (\<Sum>i\<le>n. \<mu> (F i))" using F
by (auto intro!: measure_finitely_subadditive)
- also have "\<dots> < \<omega>"
- using F by (auto simp: setsum_\<omega>)
- finally show "\<mu> (\<Union> i\<le>n. F i) \<noteq> \<omega>" by simp
- qed force+
+ also have "\<dots> < \<infinity>"
+ using F by (auto simp: setsum_Pinfty)
+ finally show "\<mu> (\<Union> i\<le>n. F i) \<noteq> \<infinity>" by simp
+ qed (force simp: incseq_def)+
qed
section {* Measure preserving *}
definition "measure_preserving A B =
- {f \<in> measurable A B. (\<forall>y \<in> sets B. measure A (f -` y \<inter> space A) = measure B y)}"
+ {f \<in> measurable A B. (\<forall>y \<in> sets B. measure B y = measure A (f -` y \<inter> space A))}"
lemma measure_preservingI[intro?]:
assumes "f \<in> measurable A B"
@@ -974,7 +940,8 @@
interpret M': sigma_algebra M' by fact
show ?thesis
proof
- show "measure M' {} = 0" using T by (force simp: measure_preserving_def)
+ show "positive M' (measure M')" using T
+ by (auto simp: measure_preserving_def positive_def measurable_sets)
show "countably_additive M' (measure M')"
proof (intro countably_additiveI)
@@ -986,7 +953,7 @@
using * by blast
moreover have **: "disjoint_family (\<lambda>i. T -` A i \<inter> space M)"
using `disjoint_family A` by (auto simp: disjoint_family_on_def)
- ultimately show "(\<Sum>\<^isub>\<infinity> i. measure M' (A i)) = measure M' (\<Union>i. A i)"
+ ultimately show "(\<Sum>i. measure M' (A i)) = measure M' (\<Union>i. A i)"
using measure_countably_additive[OF _ **] A T
by (auto simp: comp_def vimage_UN measure_preserving_def)
qed
@@ -1009,13 +976,13 @@
lemma measure_unique_Int_stable_vimage:
fixes A :: "nat \<Rightarrow> 'a set"
assumes E: "Int_stable E"
- and A: "range A \<subseteq> sets E" "A \<up> space E" "\<And>i. measure M (A i) \<noteq> \<omega>"
+ and A: "range A \<subseteq> sets E" "incseq A" "(\<Union>i. A i) = space E" "\<And>i. measure M (A i) \<noteq> \<infinity>"
and N: "measure_space N" "T \<in> measurable N M"
and M: "measure_space M" "sets (sigma E) = sets M" "space E = space M"
and eq: "\<And>X. X \<in> sets E \<Longrightarrow> measure M X = measure N (T -` X \<inter> space N)"
assumes X: "X \<in> sets (sigma E)"
shows "measure M X = measure N (T -` X \<inter> space N)"
-proof (rule measure_unique_Int_stable[OF E A(1,2) _ _ eq _ X])
+proof (rule measure_unique_Int_stable[OF E A(1,2,3) _ _ eq _ X])
interpret M: measure_space M by fact
interpret N: measure_space N by fact
let "?T X" = "T -` X \<inter> space N"
@@ -1028,12 +995,12 @@
show "T \<in> measure_preserving N ?E"
using `T \<in> measurable N M` by (auto simp: M measurable_def measure_preserving_def)
qed
- show "\<And>i. M.\<mu> (A i) \<noteq> \<omega>" by fact
+ show "\<And>i. M.\<mu> (A i) \<noteq> \<infinity>" by fact
qed
lemma (in measure_space) measure_preserving_Int_stable:
fixes A :: "nat \<Rightarrow> 'a set"
- assumes E: "Int_stable E" "range A \<subseteq> sets E" "A \<up> space E" "\<And>i. measure E (A i) \<noteq> \<omega>"
+ assumes E: "Int_stable E" "range A \<subseteq> sets E" "incseq A" "(\<Union>i. A i) = space E" "\<And>i. measure E (A i) \<noteq> \<infinity>"
and N: "measure_space (sigma E)"
and T: "T \<in> measure_preserving M E"
shows "T \<in> measure_preserving M (sigma E)"
@@ -1046,7 +1013,7 @@
show "\<mu> (T -` X \<inter> space M) = E.\<mu> X"
proof (rule measure_unique_Int_stable_vimage[symmetric])
show "sets (sigma E) = sets (sigma E)" "space E = space (sigma E)"
- "\<And>i. E.\<mu> (A i) \<noteq> \<omega>" using E by auto
+ "\<And>i. E.\<mu> (A i) \<noteq> \<infinity>" using E by auto
show "measure_space M" by default
next
fix X assume "X \<in> sets E" then show "E.\<mu> X = \<mu> (T -` X \<inter> space M)"
@@ -1057,28 +1024,28 @@
section "Real measure values"
lemma (in measure_space) real_measure_Union:
- assumes finite: "\<mu> A \<noteq> \<omega>" "\<mu> B \<noteq> \<omega>"
+ assumes finite: "\<mu> A \<noteq> \<infinity>" "\<mu> B \<noteq> \<infinity>"
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_pextreal_add)
+ using measurable(1,2)[THEN positive_measure]
+ using finite by (cases rule: extreal2_cases[of "\<mu> A" "\<mu> B"]) auto
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>"
+ assumes finite: "\<And>i. i \<in> S \<Longrightarrow> \<mu> (A i) \<noteq> \<infinity>"
shows "real (\<mu> (\<Union>i\<in>S. A i)) = (\<Sum>i\<in>S. real (\<mu> (A i)))"
- using real_of_pextreal_setsum[of S, OF finite]
- measure_finitely_additive''[symmetric, OF measurable]
- by simp
+ using finite measurable(2)[THEN positive_measure]
+ by (force intro!: setsum_real_of_extreal[symmetric]
+ simp: measure_setsum[OF measurable, symmetric])
lemma (in measure_space) real_measure_Diff:
- assumes finite: "\<mu> A \<noteq> \<omega>"
+ assumes finite: "\<mu> A \<noteq> \<infinity>"
and measurable: "A \<in> sets M" "B \<in> sets M" "B \<subseteq> A"
shows "real (\<mu> (A - B)) = real (\<mu> A) - real (\<mu> B)"
proof -
- have "\<mu> (A - B) \<le> \<mu> A"
- "\<mu> B \<le> \<mu> A"
+ have "\<mu> (A - B) \<le> \<mu> A" "\<mu> B \<le> \<mu> A"
using measurable by (auto intro!: measure_mono)
hence "real (\<mu> ((A - B) \<union> B)) = real (\<mu> (A - B)) + real (\<mu> B)"
using measurable finite by (rule_tac real_measure_Union) auto
@@ -1087,117 +1054,155 @@
lemma (in measure_space) real_measure_UNION:
assumes measurable: "range A \<subseteq> sets M" "disjoint_family A"
- assumes finite: "\<mu> (\<Union>i. A i) \<noteq> \<omega>"
+ assumes finite: "\<mu> (\<Union>i. A i) \<noteq> \<infinity>"
shows "(\<lambda>i. real (\<mu> (A i))) sums (real (\<mu> (\<Union>i. A i)))"
proof -
- have *: "(\<Sum>\<^isub>\<infinity> i. \<mu> (A i)) = \<mu> (\<Union>i. A i)"
- using measure_countably_additive[OF measurable] by (simp add: comp_def)
- hence "(\<Sum>\<^isub>\<infinity> i. \<mu> (A i)) \<noteq> \<omega>" using finite by simp
- from psuminf_imp_suminf[OF this]
- show ?thesis using * by simp
+ have "\<And>i. 0 \<le> \<mu> (A i)" using measurable by auto
+ with summable_sums[OF summable_extreal_pos, of "\<lambda>i. \<mu> (A i)"]
+ measure_countably_additive[OF measurable]
+ have "(\<lambda>i. \<mu> (A i)) sums (\<mu> (\<Union>i. A i))" by simp
+ moreover
+ { fix i
+ have "\<mu> (A i) \<le> \<mu> (\<Union>i. A i)"
+ using measurable by (auto intro!: measure_mono)
+ moreover have "0 \<le> \<mu> (A i)" using measurable by auto
+ ultimately have "\<mu> (A i) = extreal (real (\<mu> (A i)))"
+ using finite by (cases "\<mu> (A i)") auto }
+ moreover
+ have "0 \<le> \<mu> (\<Union>i. A i)" using measurable by auto
+ then have "\<mu> (\<Union>i. A i) = extreal (real (\<mu> (\<Union>i. A i)))"
+ using finite by (cases "\<mu> (\<Union>i. A i)") auto
+ ultimately show ?thesis
+ unfolding sums_extreal[symmetric] by simp
qed
lemma (in measure_space) real_measure_subadditive:
assumes measurable: "A \<in> sets M" "B \<in> sets M"
- and fin: "\<mu> A \<noteq> \<omega>" "\<mu> B \<noteq> \<omega>"
+ and fin: "\<mu> A \<noteq> \<infinity>" "\<mu> B \<noteq> \<infinity>"
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_pextreal_mono)
- also have "\<dots> = real (\<mu> A) + real (\<mu> B)"
- using fin by (simp add: real_of_pextreal_add)
- finally show ?thesis .
-qed
-
-lemma (in measure_space) real_measure_countably_subadditive:
- assumes "range f \<subseteq> sets M" and "(\<Sum>\<^isub>\<infinity> i. \<mu> (f i)) \<noteq> \<omega>"
- 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_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 .
+ have "0 \<le> \<mu> (A \<union> B)" using measurable by auto
+ then show "real (\<mu> (A \<union> B)) \<le> real (\<mu> A) + real (\<mu> B)"
+ using measure_subadditive[OF measurable] fin
+ by (cases rule: extreal3_cases[of "\<mu> (A \<union> B)" "\<mu> A" "\<mu> B"]) auto
qed
lemma (in measure_space) real_measure_setsum_singleton:
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>"
+ and fin: "\<And>x. x \<in> S \<Longrightarrow> \<mu> {x} \<noteq> \<infinity>"
shows "real (\<mu> S) = (\<Sum>x\<in>S. real (\<mu> {x}))"
- using measure_finite_singleton[OF S] fin by (simp add: real_of_pextreal_setsum)
+ using measure_finite_singleton[OF S] fin
+ using positive_measure[OF S(2)]
+ by (force intro!: setsum_real_of_extreal[symmetric])
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>"
+ assumes A: "range A \<subseteq> sets M" "incseq A" and fin: "\<mu> (\<Union>i. A i) \<noteq> \<infinity>"
shows "(\<lambda>i. real (\<mu> (A i))) ----> real (\<mu> (\<Union>i. A i))"
-proof (rule SUP_eq_LIMSEQ[THEN iffD1])
- { fix n have "\<mu> (A n) \<le> \<mu> (\<Union>i. A i)"
- using A by (auto intro!: measure_mono)
- hence "\<mu> (A n) \<noteq> \<omega>" using assms by auto }
- note this[simp]
+proof -
+ have "0 \<le> \<mu> (\<Union>i. A i)" using A by auto
+ then have "extreal (real (\<mu> (\<Union>i. A i))) = \<mu> (\<Union>i. A i)"
+ using fin by (auto intro: extreal_real')
+ then show ?thesis
+ using continuity_from_below_Lim[OF A]
+ by (intro lim_real_of_extreal) simp
+qed
- show "mono (\<lambda>i. real (\<mu> (A i)))" unfolding mono_iff_le_Suc using A
- 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)]
- using assms by (simp add: Real_real)
-qed simp_all
+lemma (in measure_space) continuity_from_above_Lim:
+ assumes A: "range A \<subseteq> sets M" "decseq A" and fin: "\<And>i. \<mu> (A i) \<noteq> \<infinity>"
+ shows "(\<lambda>i. (\<mu> (A i))) ----> \<mu> (\<Inter>i. A i)"
+ using LIMSEQ_extreal_INFI[OF measure_decseq, OF A]
+ using continuity_from_above[OF A fin] by simp
lemma (in measure_space) real_continuity_from_above:
- assumes A: "range A \<subseteq> sets M"
- and mono_Suc: "\<And>n. A (Suc n) \<subseteq> A n"
- and finite: "\<And>i. \<mu> (A i) \<noteq> \<omega>"
+ assumes A: "range A \<subseteq> sets M" "decseq A" and fin: "\<And>i. \<mu> (A i) \<noteq> \<infinity>"
shows "(\<lambda>n. real (\<mu> (A n))) ----> real (\<mu> (\<Inter>i. A i))"
-proof (rule INF_eq_LIMSEQ[THEN iffD1])
- { fix n have "\<mu> (\<Inter>i. A i) \<le> \<mu> (A n)"
- using A by (auto intro!: measure_mono)
- hence "\<mu> (\<Inter>i. A i) \<noteq> \<omega>" using assms by auto }
- note this[simp]
+proof -
+ have "0 \<le> \<mu> (\<Inter>i. A i)" using A by auto
+ moreover
+ have "\<mu> (\<Inter>i. A i) \<le> \<mu> (A 0)"
+ using A by (auto intro!: measure_mono)
+ ultimately have "extreal (real (\<mu> (\<Inter>i. A i))) = \<mu> (\<Inter>i. A i)"
+ using fin by (auto intro: extreal_real')
+ then show ?thesis
+ using continuity_from_above_Lim[OF A fin]
+ by (intro lim_real_of_extreal) simp
+qed
- show "mono (\<lambda>i. - real (\<mu> (A i)))" unfolding mono_iff_le_Suc using assms
- by (auto intro!: real_of_pextreal_mono measure_mono)
-
- show "(INF n. Real (real (\<mu> (A n)))) =
- Real (real (\<mu> (\<Inter>i. A i)))"
- using continuity_from_above[OF A, OF mono_Suc finite]
- using assms by (simp add: Real_real)
-qed simp_all
+lemma (in measure_space) real_measure_countably_subadditive:
+ assumes A: "range A \<subseteq> sets M" and fin: "(\<Sum>i. \<mu> (A i)) \<noteq> \<infinity>"
+ shows "real (\<mu> (\<Union>i. A i)) \<le> (\<Sum>i. real (\<mu> (A i)))"
+proof -
+ { fix i
+ have "0 \<le> \<mu> (A i)" using A by auto
+ moreover have "\<mu> (A i) \<noteq> \<infinity>" using A by (intro suminf_PInfty[OF _ fin]) auto
+ ultimately have "\<bar>\<mu> (A i)\<bar> \<noteq> \<infinity>" by auto }
+ moreover have "0 \<le> \<mu> (\<Union>i. A i)" using A by auto
+ ultimately have "extreal (real (\<mu> (\<Union>i. A i))) \<le> (\<Sum>i. extreal (real (\<mu> (A i))))"
+ using measure_countably_subadditive[OF A] by (auto simp: extreal_real)
+ also have "\<dots> = extreal (\<Sum>i. real (\<mu> (A i)))"
+ using A
+ by (auto intro!: sums_unique[symmetric] sums_extreal[THEN iffD2] summable_sums summable_real_of_extreal fin)
+ finally show ?thesis by simp
+qed
locale finite_measure = measure_space +
- assumes finite_measure_of_space: "\<mu> (space M) \<noteq> \<omega>"
+ assumes finite_measure_of_space: "\<mu> (space M) \<noteq> \<infinity>"
sublocale finite_measure < sigma_finite_measure
proof
- show "\<exists>A. range A \<subseteq> sets M \<and> (\<Union>i. A i) = space M \<and> (\<forall>i. \<mu> (A i) \<noteq> \<omega>)"
+ show "\<exists>A. range A \<subseteq> sets M \<and> (\<Union>i. A i) = space M \<and> (\<forall>i. \<mu> (A i) \<noteq> \<infinity>)"
using finite_measure_of_space by (auto intro!: exI[of _ "\<lambda>x. space M"])
qed
lemma (in finite_measure) finite_measure[simp, intro]:
assumes "A \<in> sets M"
- shows "\<mu> A \<noteq> \<omega>"
+ shows "\<mu> A \<noteq> \<infinity>"
proof -
from `A \<in> sets M` have "A \<subseteq> space M"
using sets_into_space by blast
- hence "\<mu> A \<le> \<mu> (space M)"
+ then have "\<mu> A \<le> \<mu> (space M)"
using assms top by (rule measure_mono)
- also have "\<dots> < \<omega>"
- using finite_measure_of_space unfolding pextreal_less_\<omega> .
- finally show ?thesis unfolding pextreal_less_\<omega> .
+ then show ?thesis
+ using finite_measure_of_space by auto
qed
+lemma (in finite_measure) measure_not_inf:
+ assumes A: "A \<in> sets M"
+ shows "\<bar>\<mu> A\<bar> \<noteq> \<infinity>"
+ using finite_measure[OF A] positive_measure[OF A] by auto
+
+definition (in finite_measure)
+ "\<mu>' A = (if A \<in> sets M then real (\<mu> A) else 0)"
+
+lemma (in finite_measure) finite_measure_eq: "A \<in> sets M \<Longrightarrow> \<mu> A = extreal (\<mu>' A)"
+ using measure_not_inf[of A] by (auto simp: \<mu>'_def)
+
+lemma (in finite_measure) positive_measure': "0 \<le> \<mu>' A"
+ unfolding \<mu>'_def by (auto simp: real_of_extreal_pos)
+
+lemma (in finite_measure) bounded_measure: "\<mu>' A \<le> \<mu>' (space M)"
+proof cases
+ assume "A \<in> sets M"
+ moreover then have "\<mu> A \<le> \<mu> (space M)"
+ using sets_into_space by (auto intro!: measure_mono)
+ ultimately show ?thesis
+ using measure_not_inf[of A] measure_not_inf[of "space M"]
+ by (auto simp: \<mu>'_def)
+qed (simp add: \<mu>'_def real_of_extreal_pos)
+
lemma (in finite_measure) restricted_finite_measure:
assumes "S \<in> sets M"
shows "finite_measure (restricted_space S)"
(is "finite_measure ?r")
unfolding finite_measure_def finite_measure_axioms_def
-proof (safe del: notI)
+proof (intro conjI)
show "measure_space ?r" using restricted_measure_space[OF assms] .
next
- show "measure ?r (space ?r) \<noteq> \<omega>" using finite_measure[OF `S \<in> sets M`] by auto
+ show "measure ?r (space ?r) \<noteq> \<infinity>" using finite_measure[OF `S \<in> sets M`] by auto
qed
lemma (in measure_space) restricted_to_finite_measure:
- assumes "S \<in> sets M" "\<mu> S \<noteq> \<omega>"
+ assumes "S \<in> sets M" "\<mu> S \<noteq> \<infinity>"
shows "finite_measure (restricted_space S)"
proof -
have "measure_space (restricted_space S)"
@@ -1207,202 +1212,128 @@
using assms by auto
qed
-lemma (in finite_measure) real_finite_measure_Diff:
- assumes "A \<in> sets M" "B \<in> sets M" "B \<subseteq> A"
- shows "real (\<mu> (A - B)) = real (\<mu> A) - real (\<mu> B)"
- using finite_measure[OF `A \<in> sets M`] by (rule real_measure_Diff[OF _ assms])
-
-lemma (in finite_measure) real_finite_measure_Union:
- assumes sets: "A \<in> sets M" "B \<in> sets M" and "A \<inter> B = {}"
- shows "real (\<mu> (A \<union> B)) = real (\<mu> A) + real (\<mu> B)"
- using sets by (auto intro!: real_measure_Union[OF _ _ assms] finite_measure)
+lemma (in finite_measure) finite_measure_Diff:
+ assumes sets: "A \<in> sets M" "B \<in> sets M" and "B \<subseteq> A"
+ shows "\<mu>' (A - B) = \<mu>' A - \<mu>' B"
+ using sets[THEN finite_measure_eq]
+ using Diff[OF sets, THEN finite_measure_eq]
+ using measure_Diff[OF _ assms] by simp
-lemma (in finite_measure) real_finite_measure_finite_Union:
- assumes "finite S" and dis: "disjoint_family_on A S"
- and *: "\<And>i. i \<in> S \<Longrightarrow> A i \<in> sets M"
- shows "real (\<mu> (\<Union>i\<in>S. A i)) = (\<Sum>i\<in>S. real (\<mu> (A i)))"
-proof (rule real_measure_finite_Union[OF `finite S` _ dis])
- fix i assume "i \<in> S" from *[OF this] show "A i \<in> sets M" .
- from finite_measure[OF this] show "\<mu> (A i) \<noteq> \<omega>" .
-qed
-
-lemma (in finite_measure) real_finite_measure_UNION:
- assumes measurable: "range A \<subseteq> sets M" "disjoint_family A"
- shows "(\<lambda>i. real (\<mu> (A i))) sums (real (\<mu> (\<Union>i. A i)))"
-proof (rule real_measure_UNION[OF assms])
- have "(\<Union>i. A i) \<in> sets M" using measurable(1) by (rule countable_UN)
- thus "\<mu> (\<Union>i. A i) \<noteq> \<omega>" by (rule finite_measure)
-qed
-
-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_pextreal_mono finite_measure)
-
-lemma (in finite_measure) real_finite_measure_subadditive:
- assumes measurable: "A \<in> sets M" "B \<in> sets M"
- shows "real (\<mu> (A \<union> B)) \<le> real (\<mu> A) + real (\<mu> B)"
- using measurable measurable[THEN finite_measure] by (rule real_measure_subadditive)
+lemma (in finite_measure) finite_measure_Union:
+ assumes sets: "A \<in> sets M" "B \<in> sets M" and "A \<inter> B = {}"
+ shows "\<mu>' (A \<union> B) = \<mu>' A + \<mu>' B"
+ using measure_additive[OF assms]
+ using sets[THEN finite_measure_eq]
+ using Un[OF sets, THEN finite_measure_eq]
+ by simp
-lemma (in finite_measure) real_finite_measure_countably_subadditive:
- assumes "range f \<subseteq> sets M" and "summable (\<lambda>i. real (\<mu> (f i)))"
- shows "real (\<mu> (\<Union>i. f i)) \<le> (\<Sum> i. real (\<mu> (f i)))"
-proof (rule real_measure_countably_subadditive[OF assms(1)])
- have *: "\<And>i. f i \<in> sets M" using assms by auto
- have "(\<lambda>i. real (\<mu> (f i))) sums (\<Sum>i. real (\<mu> (f i)))"
- using assms(2) by (rule summable_sums)
- from suminf_imp_psuminf[OF this]
- have "(\<Sum>\<^isub>\<infinity>i. \<mu> (f i)) = Real (\<Sum>i. real (\<mu> (f i)))"
- using * by (simp add: Real_real finite_measure)
- thus "(\<Sum>\<^isub>\<infinity>i. \<mu> (f i)) \<noteq> \<omega>" by simp
-qed
-
-lemma (in finite_measure) real_finite_measure_finite_singelton:
- assumes "finite S" and *: "\<And>x. x \<in> S \<Longrightarrow> {x} \<in> sets M"
- shows "real (\<mu> S) = (\<Sum>x\<in>S. real (\<mu> {x}))"
-proof (rule real_measure_setsum_singleton[OF `finite S`])
- fix x assume "x \<in> S" thus "{x} \<in> sets M" by (rule *)
- with finite_measure show "\<mu> {x} \<noteq> \<omega>" .
-qed
+lemma (in finite_measure) finite_measure_finite_Union:
+ assumes S: "finite S" "\<And>i. i \<in> S \<Longrightarrow> A i \<in> sets M"
+ and dis: "disjoint_family_on A S"
+ shows "\<mu>' (\<Union>i\<in>S. A i) = (\<Sum>i\<in>S. \<mu>' (A i))"
+ using measure_setsum[OF assms]
+ using finite_UN[of S A, OF S, THEN finite_measure_eq]
+ using S(2)[THEN finite_measure_eq]
+ by simp
-lemma (in finite_measure) real_finite_continuity_from_below:
- assumes "range A \<subseteq> sets M" "\<And>i. A i \<subseteq> A (Suc i)"
- shows "(\<lambda>i. real (\<mu> (A i))) ----> real (\<mu> (\<Union>i. A i))"
- using real_continuity_from_below[OF assms(1), OF assms(2) finite_measure] assms by auto
-
-lemma (in finite_measure) real_finite_continuity_from_above:
- assumes A: "range A \<subseteq> sets M"
- and mono_Suc: "\<And>n. A (Suc n) \<subseteq> A n"
- shows "(\<lambda>n. real (\<mu> (A n))) ----> real (\<mu> (\<Inter>i. A i))"
- using real_continuity_from_above[OF A, OF mono_Suc finite_measure] A
- by auto
-
-lemma (in finite_measure) real_finite_measure_finite_Union':
- assumes "finite S" "A`S \<subseteq> sets M" "disjoint_family_on A S"
- shows "real (\<mu> (\<Union>i\<in>S. A i)) = (\<Sum>i\<in>S. real (\<mu> (A i)))"
- using assms real_finite_measure_finite_Union[of S A] by auto
-
-lemma (in finite_measure) finite_measure_compl:
- assumes s: "s \<in> sets M"
- shows "\<mu> (space M - s) = \<mu> (space M) - \<mu> s"
- using measure_compl[OF s, OF finite_measure, OF s] .
-
-lemma (in finite_measure) finite_measure_inter_full_set:
- assumes "S \<in> sets M" "T \<in> sets M"
- assumes T: "\<mu> T = \<mu> (space M)"
- shows "\<mu> (S \<inter> T) = \<mu> S"
- using measure_inter_full_set[OF assms(1,2) finite_measure assms(3)] assms
+lemma (in finite_measure) finite_measure_UNION:
+ assumes A: "range A \<subseteq> sets M" "disjoint_family A"
+ shows "(\<lambda>i. \<mu>' (A i)) sums (\<mu>' (\<Union>i. A i))"
+ using real_measure_UNION[OF A]
+ using countable_UN[OF A(1), THEN finite_measure_eq]
+ using A(1)[THEN subsetD, THEN finite_measure_eq]
by auto
-lemma (in finite_measure) measure_preserving_lift:
- fixes f :: "'a \<Rightarrow> 'c" and A :: "('c, 'd) measure_space_scheme"
- assumes "algebra A" "finite_measure (sigma A)" (is "finite_measure ?sA")
- assumes fmp: "f \<in> measure_preserving M A"
- shows "f \<in> measure_preserving M (sigma A)"
+lemma (in finite_measure) finite_measure_mono:
+ assumes B: "B \<in> sets M" and "A \<subseteq> B" shows "\<mu>' A \<le> \<mu>' B"
+proof cases
+ assume "A \<in> sets M"
+ from this[THEN finite_measure_eq] B[THEN finite_measure_eq]
+ show ?thesis using measure_mono[OF `A \<subseteq> B` `A \<in> sets M` `B \<in> sets M`] by simp
+next
+ assume "A \<notin> sets M" then show ?thesis
+ using positive_measure'[of B] unfolding \<mu>'_def by auto
+qed
+
+lemma (in finite_measure) finite_measure_subadditive:
+ assumes m: "A \<in> sets M" "B \<in> sets M"
+ shows "\<mu>' (A \<union> B) \<le> \<mu>' A + \<mu>' B"
+ using measure_subadditive[OF m]
+ using m[THEN finite_measure_eq] Un[OF m, THEN finite_measure_eq] by simp
+
+lemma (in finite_measure) finite_measure_countably_subadditive:
+ assumes A: "range A \<subseteq> sets M" and sum: "summable (\<lambda>i. \<mu>' (A i))"
+ shows "\<mu>' (\<Union>i. A i) \<le> (\<Sum>i. \<mu>' (A i))"
proof -
- interpret sA: finite_measure ?sA by fact
- interpret A: algebra A by fact
- show ?thesis using fmp
- proof (clarsimp simp add: measure_preserving_def)
- assume fm: "f \<in> measurable M A"
- and "\<forall>y\<in>sets A. \<mu> (f -` y \<inter> space M) = measure A y"
- then have meq: "\<forall>y\<in>sets A. \<mu> (f -` y \<inter> space M) = sA.\<mu> y"
- by simp
- have f12: "f \<in> measurable M ?sA"
- using measurable_subset[OF A.sets_into_space] fm by auto
- hence ffn: "f \<in> space M \<rightarrow> space A"
- by (simp add: measurable_def)
- have "space M \<subseteq> f -` (space A)"
- by auto (metis PiE ffn)
- hence fveq [simp]: "(f -` (space A)) \<inter> space M = space M"
- by blast
- {
- fix y
- assume y: "y \<in> sets ?sA"
- have "sets ?sA = sigma_sets (space A) (sets A)" (is "_ = ?A") by (auto simp: sigma_def)
- also have "\<dots> \<subseteq> {s . \<mu> ((f -` s) \<inter> space M) = sA.\<mu> s}"
- proof (rule A.sigma_property_disjoint, safe)
- fix x assume "x \<in> sets A" then show "\<mu> (f -` x \<inter> space M) = sA.\<mu> x" by (simp add: meq)
- next
- fix s
- assume eq: "\<mu> (f -` s \<inter> space M) = sA.\<mu> s" and s: "s \<in> ?A"
- then have s': "s \<in> sets ?sA" by (simp add: sigma_def)
- show "\<mu> (f -` (space A - s) \<inter> space M) = measure ?sA (space A - s)"
- using sA.finite_measure_compl[OF s']
- using measurable_sets[OF f12 s'] meq[THEN bspec, OF A.top]
- by (simp add: vimage_Diff Diff_Int_distrib2 finite_measure_compl eq)
- next
- fix S
- assume S0: "S 0 = {}"
- and SSuc: "\<And>n. S n \<subseteq> S (Suc n)"
- and rS1: "range S \<subseteq> {s. \<mu> (f -` s \<inter> space M) = sA.\<mu> s} \<inter> ?A"
- hence rS2: "range S \<subseteq> sets ?sA" by (simp add: sigma_def)
- have eq1: "\<And>i. \<mu> (f -` S i \<inter> space M) = sA.\<mu> (S i)"
- using rS1 by blast
- have *: "(\<lambda>n. sA.\<mu> (S n)) = (\<lambda>n. \<mu> (f -` S n \<inter> space M))"
- by (simp add: eq1)
- have "(SUP n. ... n) = \<mu> (\<Union>i. f -` S i \<inter> space M)"
- proof (rule measure_countable_increasing)
- show "range (\<lambda>i. f -` S i \<inter> space M) \<subseteq> sets M"
- using f12 rS2 by (auto simp add: measurable_def)
- show "f -` S 0 \<inter> space M = {}" using S0
- by blast
- show "\<And>n. f -` S n \<inter> space M \<subseteq> f -` S (Suc n) \<inter> space M"
- using SSuc by auto
- qed
- also have "\<mu> (\<Union>i. f -` S i \<inter> space M) = \<mu> (f -` (\<Union>i. S i) \<inter> space M)"
- by (simp add: vimage_UN)
- finally have "(SUP n. sA.\<mu> (S n)) = \<mu> (f -` (\<Union>i. S i) \<inter> space M)" unfolding * .
- moreover
- have "(SUP n. sA.\<mu> (S n)) = sA.\<mu> (\<Union>i. S i)"
- by (rule sA.measure_countable_increasing[OF rS2, OF S0 SSuc])
- ultimately
- show "\<mu> (f -` (\<Union>i. S i) \<inter> space M) = sA.\<mu> (\<Union>i. S i)" by simp
- next
- fix S :: "nat \<Rightarrow> 'c set"
- assume dS: "disjoint_family S"
- and rS1: "range S \<subseteq> {s. \<mu> (f -` s \<inter> space M) = sA.\<mu> s} \<inter> ?A"
- hence rS2: "range S \<subseteq> sets ?sA" by (simp add: sigma_def)
- have "\<And>i. \<mu> (f -` S i \<inter> space M) = sA.\<mu> (S i)"
- using rS1 by blast
- hence *: "(\<lambda>i. sA.\<mu> (S i)) = (\<lambda>n. \<mu> (f -` S n \<inter> space M))"
- by simp
- have "psuminf ... = \<mu> (\<Union>i. f -` S i \<inter> space M)"
- proof (rule measure_countably_additive)
- show "range (\<lambda>i. f -` S i \<inter> space M) \<subseteq> sets M"
- using f12 rS2 by (auto simp add: measurable_def)
- show "disjoint_family (\<lambda>i. f -` S i \<inter> space M)" using dS
- by (auto simp add: disjoint_family_on_def)
- qed
- hence "(\<Sum>\<^isub>\<infinity> i. sA.\<mu> (S i)) = \<mu> (\<Union>i. f -` S i \<inter> space M)" unfolding * .
- with sA.measure_countably_additive [OF rS2 dS]
- have "\<mu> (\<Union>i. f -` S i \<inter> space M) = sA.\<mu> (\<Union>i. S i)"
- by simp
- moreover have "\<mu> (f -` (\<Union>i. S i) \<inter> space M) = \<mu> (\<Union>i. f -` S i \<inter> space M)"
- by (simp add: vimage_UN)
- ultimately show "\<mu> (f -` (\<Union>i. S i) \<inter> space M) = sA.\<mu> (\<Union>i. S i)"
- by metis
- qed
- finally have "sets ?sA \<subseteq> {s . \<mu> ((f -` s) \<inter> space M) = sA.\<mu> s}" .
- hence "\<mu> (f -` y \<inter> space M) = sA.\<mu> y" using y by force
- }
- thus "f \<in> measurable M ?sA \<and> (\<forall>y\<in>sets ?sA. \<mu> (f -` y \<inter> space M) = measure A y)"
- by simp_all (blast intro: f12)
- qed
+ note A[THEN subsetD, THEN finite_measure_eq, simp]
+ note countable_UN[OF A, THEN finite_measure_eq, simp]
+ from `summable (\<lambda>i. \<mu>' (A i))`
+ have "(\<lambda>i. extreal (\<mu>' (A i))) sums extreal (\<Sum>i. \<mu>' (A i))"
+ by (simp add: sums_extreal) (rule summable_sums)
+ from sums_unique[OF this, symmetric]
+ measure_countably_subadditive[OF A]
+ show ?thesis by simp
qed
+lemma (in finite_measure) finite_measure_finite_singleton:
+ assumes "finite S" and *: "\<And>x. x \<in> S \<Longrightarrow> {x} \<in> sets M"
+ shows "\<mu>' S = (\<Sum>x\<in>S. \<mu>' {x})"
+ using real_measure_setsum_singleton[OF assms]
+ using *[THEN finite_measure_eq]
+ using finite_UN[of S "\<lambda>x. {x}", OF assms, THEN finite_measure_eq]
+ by simp
+
+lemma (in finite_measure) finite_continuity_from_below:
+ assumes A: "range A \<subseteq> sets M" and "incseq A"
+ shows "(\<lambda>i. \<mu>' (A i)) ----> \<mu>' (\<Union>i. A i)"
+ using real_continuity_from_below[OF A, OF `incseq A` finite_measure] assms
+ using A[THEN subsetD, THEN finite_measure_eq]
+ using countable_UN[OF A, THEN finite_measure_eq]
+ by auto
+
+lemma (in finite_measure) finite_continuity_from_above:
+ assumes A: "range A \<subseteq> sets M" and "decseq A"
+ shows "(\<lambda>n. \<mu>' (A n)) ----> \<mu>' (\<Inter>i. A i)"
+ using real_continuity_from_above[OF A, OF `decseq A` finite_measure] assms
+ using A[THEN subsetD, THEN finite_measure_eq]
+ using countable_INT[OF A, THEN finite_measure_eq]
+ by auto
+
+lemma (in finite_measure) finite_measure_compl:
+ assumes S: "S \<in> sets M"
+ shows "\<mu>' (space M - S) = \<mu>' (space M) - \<mu>' S"
+ using measure_compl[OF S, OF finite_measure, OF S]
+ using S[THEN finite_measure_eq]
+ using compl_sets[OF S, THEN finite_measure_eq]
+ using top[THEN finite_measure_eq]
+ by simp
+
+lemma (in finite_measure) finite_measure_inter_full_set:
+ assumes S: "S \<in> sets M" "T \<in> sets M"
+ assumes T: "\<mu>' T = \<mu>' (space M)"
+ shows "\<mu>' (S \<inter> T) = \<mu>' S"
+ using measure_inter_full_set[OF S finite_measure]
+ using T Diff[OF S(2,1)] Diff[OF S, THEN finite_measure_eq]
+ using Int[OF S, THEN finite_measure_eq]
+ using S[THEN finite_measure_eq] top[THEN finite_measure_eq]
+ by simp
+
+lemma (in finite_measure) empty_measure'[simp]: "\<mu>' {} = 0"
+ unfolding \<mu>'_def by simp
+
section "Finite spaces"
locale finite_measure_space = measure_space + finite_sigma_algebra +
- assumes finite_single_measure[simp]: "\<And>x. x \<in> space M \<Longrightarrow> \<mu> {x} \<noteq> \<omega>"
+ assumes finite_single_measure[simp]: "\<And>x. x \<in> space M \<Longrightarrow> \<mu> {x} \<noteq> \<infinity>"
lemma (in finite_measure_space) sum_over_space: "(\<Sum>x\<in>space M. \<mu> {x}) = \<mu> (space M)"
- using measure_finitely_additive''[of "space M" "\<lambda>i. {i}"]
+ using measure_setsum[of "space M" "\<lambda>i. {i}"]
by (simp add: sets_eq_Pow disjoint_family_on_def finite_space)
lemma finite_measure_spaceI:
- assumes "finite (space M)" "sets M = Pow(space M)" and space: "measure M (space M) \<noteq> \<omega>"
+ assumes "finite (space M)" "sets M = Pow(space M)" and space: "measure M (space M) \<noteq> \<infinity>"
and add: "\<And>A B. A\<subseteq>space M \<Longrightarrow> B\<subseteq>space M \<Longrightarrow> A \<inter> B = {} \<Longrightarrow> measure M (A \<union> B) = measure M A + measure M B"
- and "measure M {} = 0"
+ and "measure M {} = 0" "\<And>A. A \<subseteq> space M \<Longrightarrow> 0 \<le> measure M A"
shows "finite_measure_space M"
unfolding finite_measure_space_def finite_measure_space_axioms_def
proof (intro allI impI conjI)
@@ -1414,7 +1345,7 @@
using sigma_algebra_Pow[of "space M" "algebra.more M"]
unfolding * .
show "finite (space M)" by fact
- show "positive M (measure M)" unfolding positive_def by fact
+ show "positive M (measure M)" unfolding positive_def using assms by auto
show "additive M (measure M)" unfolding additive_def using assms by simp
qed
then interpret measure_space M .
@@ -1425,19 +1356,20 @@
qed
{ fix x assume *: "x \<in> space M"
with add[of "{x}" "space M - {x}"] space
- show "\<mu> {x} \<noteq> \<omega>" by (auto simp: insert_absorb[OF *] Diff_subset) }
+ show "\<mu> {x} \<noteq> \<infinity>" by (auto simp: insert_absorb[OF *] Diff_subset) }
qed
sublocale finite_measure_space \<subseteq> finite_measure
proof
- show "\<mu> (space M) \<noteq> \<omega>"
- unfolding sum_over_space[symmetric] setsum_\<omega>
+ show "\<mu> (space M) \<noteq> \<infinity>"
+ unfolding sum_over_space[symmetric] setsum_Pinfty
using finite_space finite_single_measure by auto
qed
lemma finite_measure_space_iff:
"finite_measure_space M \<longleftrightarrow>
- finite (space M) \<and> sets M = Pow(space M) \<and> measure M (space M) \<noteq> \<omega> \<and> measure M {} = 0 \<and>
+ finite (space M) \<and> sets M = Pow(space M) \<and> measure M (space M) \<noteq> \<infinity> \<and>
+ measure M {} = 0 \<and> (\<forall>A\<subseteq>space M. 0 \<le> measure M A) \<and>
(\<forall>A\<subseteq>space M. \<forall>B\<subseteq>space M. A \<inter> B = {} \<longrightarrow> measure M (A \<union> B) = measure M A + measure M B)"
(is "_ = ?rhs")
proof (intro iffI)
@@ -1451,29 +1383,31 @@
by (auto intro!: finite_measure_spaceI)
qed
-lemma psuminf_cmult_indicator:
- assumes "disjoint_family A" "x \<in> A i"
- shows "(\<Sum>\<^isub>\<infinity> n. f n * indicator (A n) x) = f i"
+lemma suminf_cmult_indicator:
+ fixes f :: "nat \<Rightarrow> extreal"
+ assumes "disjoint_family A" "x \<in> A i" "\<And>i. 0 \<le> f i"
+ shows "(\<Sum>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 :: pextreal)"
+ have **: "\<And>n. f n * indicator (A n) x = (if n = i then f n else 0 :: extreal)"
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 :: pextreal)"
+ then have "\<And>n. (\<Sum>j<n. f j * indicator (A j) x) = (if i < n then f i else 0 :: extreal)"
by (auto simp: setsum_cases)
- 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"
+ moreover have "(SUP n. if i < n then f i else 0) = (f i :: extreal)"
+ proof (rule extreal_SUPI)
+ fix y :: extreal 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
+ qed (insert assms, simp)
+ ultimately show ?thesis using assms
+ by (subst suminf_extreal_eq_SUPR) (auto simp: indicator_def)
qed
-lemma psuminf_indicator:
+lemma suminf_indicator:
assumes "disjoint_family A"
- shows "(\<Sum>\<^isub>\<infinity> n. indicator (A n) x) = indicator (\<Union>i. A i) x"
+ shows "(\<Sum>n. indicator (A n) x :: extreal) = indicator (\<Union>i. A i) x"
proof cases
assume *: "x \<in> (\<Union>i. A i)"
then obtain i where "x \<in> A i" by auto
- from psuminf_cmult_indicator[OF assms, OF this, of "\<lambda>i. 1"]
+ from suminf_cmult_indicator[OF assms(1), OF `x \<in> A i`, of "\<lambda>k. 1"]
show ?thesis using * by simp
qed simp
--- a/src/HOL/Probability/Positive_Extended_Real.thy Mon Mar 14 15:17:10 2011 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,2908 +0,0 @@
-(* Author: Johannes Hoelzl, TU Muenchen *)
-
-header {* A type for positive real numbers with infinity *}
-
-theory Positive_Extended_Real
- imports Complex_Main "~~/src/HOL/Library/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 Real_minus_abs[simp]: "Real (- \<bar>x\<bar>) = 0"
- by simp
-
-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 minus_omega[simp]: "x - \<omega> = 0" by (cases x) auto
-
-lemma open_pextreal_alt: "open A \<longleftrightarrow>
- (\<forall>x\<in>A. \<exists>e>0. {x-e <..< x+e} \<subseteq> A) \<and> (\<omega> \<in> A \<longrightarrow> (\<exists>x\<ge>0. {Real x <..} \<subseteq> A))"
-proof -
- have *: "(\<exists>T. open T \<and> (Real ` (T\<inter>{0..}) = A - {\<omega>})) \<longleftrightarrow>
- open (real ` (A - {\<omega>}) \<union> {..<0})"
- proof safe
- fix T assume "open T" and A: "Real ` (T \<inter> {0..}) = A - {\<omega>}"
- have *: "(\<lambda>x. real (Real x)) ` (T \<inter> {0..}) = T \<inter> {0..}"
- by auto
- have **: "T \<inter> {0..} \<union> {..<0} = T \<union> {..<0}" by auto
- show "open (real ` (A - {\<omega>}) \<union> {..<0})"
- unfolding A[symmetric] image_image * ** using `open T` by auto
- next
- assume "open (real ` (A - {\<omega>}) \<union> {..<0})"
- moreover have "Real ` ((real ` (A - {\<omega>}) \<union> {..<0}) \<inter> {0..}) = A - {\<omega>}"
- apply auto
- apply (case_tac xb)
- apply auto
- apply (case_tac x)
- apply (auto simp: image_iff)
- apply (erule_tac x="Real r" in ballE)
- apply auto
- done
- ultimately show "\<exists>T. open T \<and> Real ` (T \<inter> {0..}) = A - {\<omega>}" by auto
- qed
- also have "\<dots> \<longleftrightarrow> (\<forall>x\<in>A. \<exists>e>0. {x-e <..< x+e} \<subseteq> A)"
- proof (intro iffI ballI open_subopen[THEN iffD2])
- fix x assume *: "\<forall>x\<in>A. \<exists>e>0. {x - e<..<x + e} \<subseteq> A" and x: "x \<in> real ` (A - {\<omega>}) \<union> {..<0}"
- show "\<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> real ` (A - {\<omega>}) \<union> {..<0}"
- proof (cases rule: linorder_cases)
- assume "x < 0" then show ?thesis by (intro exI[of _ "{..<0}"]) auto
- next
- assume "x = 0" with x
- have "0 \<in> A"
- apply auto by (case_tac x) auto
- with * obtain e where "e > 0" "{0 - e<..<0 + e} \<subseteq> A" by auto
- then have "{..<e} \<subseteq> A" using `0 \<in> A`
- apply auto
- apply (case_tac "x = 0")
- by (auto dest!: pextreal_zero_lessI)
- then have *: "{..<e} \<subseteq> A - {\<omega>}" by auto
- def e' \<equiv> "if e = \<omega> then 1 else real e"
- then have "0 < e'" using `e > 0` by (cases e) auto
- have "{0..<e'} \<subseteq> real ` (A - {\<omega>})"
- proof (cases e)
- case infinite
- then have "{..<e} = UNIV - {\<omega>}" by auto
- then have A: "A - {\<omega>} = UNIV - {\<omega>}" using * by auto
- show ?thesis unfolding e'_def infinite A
- apply safe
- apply (rule_tac x="Real x" in image_eqI)
- apply auto
- done
- next
- case (preal r)
- then show ?thesis unfolding e'_def using *
- apply safe
- apply (rule_tac x="Real x" in image_eqI)
- by (auto simp: subset_eq)
- qed
- then have "{0..<e'} \<union> {..<0} \<subseteq> real ` (A - {\<omega>}) \<union> {..<0}" by auto
- moreover have "{0..<e'} \<union> {..<0} = {..<e'}" using `0 < e'` by auto
- ultimately have "{..<e'} \<subseteq> real ` (A - {\<omega>}) \<union> {..<0}" by simp
- then show ?thesis using `0 < e'` `x = 0` by auto
- next
- assume "0 < x"
- with x have "Real x \<in> A" apply auto by (case_tac x) auto
- with * obtain e where "0 < e" and e: "{Real x - e<..<Real x + e} \<subseteq> A" by auto
- show ?thesis
- proof cases
- assume "e < Real x"
- with `0 < e` obtain r where r: "e = Real r" "0 < r" by (cases e) auto
- then have "r < x" using `e < Real x` `0 < e` by (auto split: split_if_asm)
- then have "{x - r <..< x + r} \<subseteq> real ` (A - {\<omega>}) \<union> {..<0}"
- using e unfolding r
- apply (auto simp: subset_eq)
- apply (rule_tac x="Real xa" in image_eqI)
- by auto
- then show ?thesis using `0 < r` by (intro exI[of _ "{x - r <..< x + r}"]) auto
- next
- assume "\<not> e < Real x"
- moreover then have "Real x - e = 0" by (cases e) auto
- moreover have "\<And>z. 0 < z \<Longrightarrow> z * 2 < 3 * x \<Longrightarrow> Real z < Real x + e"
- using `\<not> e < Real x` by (cases e) auto
- ultimately have "{0 <..< x + x / 2} \<subseteq> real ` (A - {\<omega>}) \<union> {..<0}"
- using e
- apply (auto simp: subset_eq)
- apply (erule_tac x="Real xa" in ballE)
- apply (auto simp: not_less)
- apply (rule_tac x="Real xa" in image_eqI)
- apply auto
- done
- moreover have "x \<in> {0 <..< x + x / 2}" using `0 < x` by auto
- ultimately show ?thesis by (intro exI[of _ "{0 <..< x + x / 2}"]) auto
- qed
- qed
- next
- fix x assume x: "x \<in> A" "open (real ` (A - {\<omega>}) \<union> {..<0})"
- then show "\<exists>e>0. {x - e<..<x + e} \<subseteq> A"
- proof (cases x)
- case infinite then show ?thesis by (intro exI[of _ 2]) auto
- next
- case (preal r)
- with `x \<in> A` have r: "r \<in> real ` (A - {\<omega>}) \<union> {..<0}" by force
- from x(2)[unfolded open_real, THEN bspec, OF r]
- obtain e where e: "0 < e" "\<And>x'. \<bar>x' - r\<bar> < e \<Longrightarrow> x' \<in> real ` (A - {\<omega>}) \<union> {..<0}"
- by auto
- show ?thesis using `0 < e` preal
- proof (auto intro!: exI[of _ "Real e"] simp: greaterThanLessThan_iff not_less
- split: split_if_asm)
- fix z assume *: "Real (r - e) < z" "z < Real (r + e)"
- then obtain q where [simp]: "z = Real q" "0 \<le> q" by (cases z) auto
- with * have "r - e < q" "q < r + e" by (auto split: split_if_asm)
- with e(2)[of q] have "q \<in> real ` (A - {\<omega>}) \<union> {..<0}" by auto
- then show "z \<in> A" using `0 \<le> q` apply auto apply (case_tac x) apply auto done
- next
- fix z assume *: "0 < z" "z < Real (r + e)" "r \<le> e"
- then obtain q where [simp]: "z = Real q" and "0 < q" by (cases z) auto
- with * have "q < r + e" by (auto split: split_if_asm)
- moreover have "r - e < q" using `r \<le> e` `0 < q` by auto
- ultimately have "q \<in> real ` (A - {\<omega>}) \<union> {..<0}" using e(2)[of q] by auto
- then show "z \<in> A" using `0 < q` apply auto apply (case_tac x) apply auto done
- qed
- qed
- qed
- finally show ?thesis unfolding open_pextreal_def by simp
-qed
-
-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/Probability.thy Mon Mar 14 15:17:10 2011 +0100
+++ b/src/HOL/Probability/Probability.thy Mon Mar 14 15:29:10 2011 +0100
@@ -1,6 +1,7 @@
theory Probability
imports
Complete_Measure
+ Lebesgue_Measure
Information
"ex/Dining_Cryptographers"
"ex/Koepf_Duermuth_Countermeasure"
--- a/src/HOL/Probability/Probability_Space.thy Mon Mar 14 15:17:10 2011 +0100
+++ b/src/HOL/Probability/Probability_Space.thy Mon Mar 14 15:29:10 2011 +0100
@@ -2,33 +2,33 @@
imports Lebesgue_Integration Radon_Nikodym Product_Measure
begin
-lemma real_of_pextreal_inverse[simp]:
- fixes X :: pextreal
+lemma real_of_extreal_inverse[simp]:
+ fixes X :: extreal
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>)"
+lemma real_of_extreal_le_0[simp]: "real (X :: extreal) \<le> 0 \<longleftrightarrow> (X \<le> 0 \<or> X = \<infinity>)"
+ by (cases X) auto
+
+lemma abs_real_of_extreal[simp]: "\<bar>real (X :: extreal)\<bar> = real \<bar>X\<bar>"
by (cases X) auto
-lemma real_of_pextreal_less_0[simp]: "\<not> (real (X :: pextreal) < 0)"
+lemma zero_less_real_of_extreal: "0 < real X \<longleftrightarrow> (0 < X \<and> X \<noteq> \<infinity>)"
by (cases X) auto
+lemma real_of_extreal_le_1: fixes X :: extreal shows "X \<le> 1 \<Longrightarrow> real X \<le> 1"
+ by (cases X) (auto simp: one_extreal_def)
+
locale prob_space = measure_space +
assumes measure_space_1: "measure M (space M) = 1"
-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
-
sublocale prob_space < finite_measure
proof
- from measure_space_1 show "\<mu> (space M) \<noteq> \<omega>" by simp
+ from measure_space_1 show "\<mu> (space M) \<noteq> \<infinity>" by simp
qed
abbreviation (in prob_space) "events \<equiv> sets M"
-abbreviation (in prob_space) "prob \<equiv> \<lambda>A. real (\<mu> A)"
+abbreviation (in prob_space) "prob \<equiv> \<mu>'"
abbreviation (in prob_space) "prob_preserving \<equiv> measure_preserving"
abbreviation (in prob_space) "random_variable M' X \<equiv> sigma_algebra M' \<and> X \<in> measurable M M'"
abbreviation (in prob_space) "expectation \<equiv> integral\<^isup>L M"
@@ -40,53 +40,57 @@
"indep_families F G \<longleftrightarrow> (\<forall> A \<in> F. \<forall> B \<in> G. indep A B)"
definition (in prob_space)
- "distribution X = (\<lambda>s. \<mu> ((X -` s) \<inter> (space M)))"
+ "distribution X A = \<mu>' (X -` A \<inter> space M)"
abbreviation (in prob_space)
"joint_distribution X Y \<equiv> distribution (\<lambda>x. (X x, Y x))"
+declare (in finite_measure) positive_measure'[intro, simp]
+
lemma (in prob_space) distribution_cong:
assumes "\<And>x. x \<in> space M \<Longrightarrow> X x = Y x"
shows "distribution X = distribution Y"
unfolding distribution_def fun_eq_iff
- using assms by (auto intro!: arg_cong[where f="\<mu>"])
+ using assms by (auto intro!: arg_cong[where f="\<mu>'"])
lemma (in prob_space) joint_distribution_cong:
assumes "\<And>x. x \<in> space M \<Longrightarrow> X x = X' x"
assumes "\<And>x. x \<in> space M \<Longrightarrow> Y x = Y' x"
shows "joint_distribution X Y = joint_distribution X' Y'"
unfolding distribution_def fun_eq_iff
- using assms by (auto intro!: arg_cong[where f="\<mu>"])
+ using assms by (auto intro!: arg_cong[where f="\<mu>'"])
lemma (in prob_space) distribution_id[simp]:
- assumes "N \<in> sets M" shows "distribution (\<lambda>x. x) N = \<mu> N"
- using assms by (auto simp: distribution_def intro!: arg_cong[where f=\<mu>])
+ "N \<in> events \<Longrightarrow> distribution (\<lambda>x. x) N = prob N"
+ by (auto simp: distribution_def intro!: arg_cong[where f=prob])
lemma (in prob_space) prob_space: "prob (space M) = 1"
- unfolding measure_space_1 by simp
+ using measure_space_1 unfolding \<mu>'_def by (simp add: one_extreal_def)
+
+lemma (in prob_space) prob_le_1[simp, intro]: "prob A \<le> 1"
+ using bounded_measure[of A] by (simp add: prob_space)
+
+lemma (in prob_space) distribution_positive[simp, intro]:
+ "0 \<le> distribution X A" unfolding distribution_def by auto
-lemma (in prob_space) measure_le_1[simp, intro]:
- assumes "A \<in> events" shows "\<mu> A \<le> 1"
-proof -
- have "\<mu> A \<le> \<mu> (space M)"
- using assms sets_into_space by(auto intro!: measure_mono)
- also note measure_space_1
- finally show ?thesis .
-qed
+lemma (in prob_space) joint_distribution_remove[simp]:
+ "joint_distribution X X {(x, x)} = distribution X {x}"
+ unfolding distribution_def by (auto intro!: arg_cong[where f=\<mu>'])
+
+lemma (in prob_space) distribution_1:
+ "distribution X A \<le> 1"
+ unfolding distribution_def by simp
lemma (in prob_space) prob_compl:
- assumes "A \<in> events"
+ assumes A: "A \<in> events"
shows "prob (space M - A) = 1 - prob A"
- using `A \<in> events`[THEN sets_into_space] `A \<in> events` measure_space_1
- by (subst real_finite_measure_Diff) auto
+ using finite_measure_compl[OF A] by (simp add: prob_space)
-lemma (in prob_space) indep_space:
- assumes "s \<in> events"
- shows "indep (space M) s"
- using assms prob_space by (simp add: indep_def)
+lemma (in prob_space) indep_space: "s \<in> events \<Longrightarrow> indep (space M) s"
+ by (simp add: indep_def prob_space)
lemma (in prob_space) prob_space_increasing: "increasing M prob"
- by (auto intro!: real_measure_mono simp: increasing_def)
+ by (auto intro!: finite_measure_mono simp: increasing_def)
lemma (in prob_space) prob_zero_union:
assumes "s \<in> events" "t \<in> events" "prob t = 0"
@@ -94,9 +98,9 @@
using assms
proof -
have "prob (s \<union> t) \<le> prob s"
- using real_finite_measure_subadditive[of s t] assms by auto
+ using finite_measure_subadditive[of s t] assms by auto
moreover have "prob (s \<union> t) \<ge> prob s"
- using assms by (blast intro: real_measure_mono)
+ using assms by (blast intro: finite_measure_mono)
ultimately show ?thesis by simp
qed
@@ -127,9 +131,9 @@
using assms
proof -
have a: "(\<lambda> i. prob (f i)) sums (prob (\<Union> i. f i))"
- by (rule real_finite_measure_UNION[OF assms(1,3)])
+ by (rule finite_measure_UNION[OF assms(1,3)])
have b: "(\<lambda> i. prob (g i)) sums (prob (\<Union> i. g i))"
- by (rule real_finite_measure_UNION[OF assms(2,4)])
+ by (rule finite_measure_UNION[OF assms(2,4)])
show ?thesis using sums_unique[OF b] sums_unique[OF a] assms by simp
qed
@@ -139,10 +143,9 @@
shows "prob (\<Union> i :: nat. c i) = 0"
proof (rule antisym)
show "prob (\<Union> i :: nat. c i) \<le> 0"
- using real_finite_measure_countably_subadditive[OF assms(1)]
+ using 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_pextreal_nonneg)
-qed
+qed simp
lemma (in prob_space) indep_sym:
"indep a b \<Longrightarrow> indep b a"
@@ -163,7 +166,7 @@
from someI_ex[OF this] assms
have prob_some: "\<And> x. x \<in> s \<Longrightarrow> prob {x} = prob {SOME y. y \<in> s}" by blast
have "prob s = (\<Sum> x \<in> s. prob {x})"
- using real_finite_measure_finite_singelton[OF s_finite] by simp
+ using finite_measure_finite_singleton[OF s_finite] by simp
also have "\<dots> = (\<Sum> x \<in> s. prob {SOME y. y \<in> s})" using prob_some by auto
also have "\<dots> = real (card s) * prob {(SOME x. x \<in> s)}"
using setsum_constant assms by (simp add: real_eq_of_nat)
@@ -182,7 +185,7 @@
using `e \<in> events` sets_into_space upper by blast
hence "prob e = prob (\<Union> i \<in> s. e \<inter> f i)" by simp
also have "\<dots> = (\<Sum> x \<in> s. prob (e \<inter> f x))"
- proof (rule real_finite_measure_finite_Union)
+ proof (rule finite_measure_finite_Union)
show "finite s" by fact
show "\<And>i. i \<in> s \<Longrightarrow> e \<inter> f i \<in> events" by fact
show "disjoint_family_on (\<lambda>i. e \<inter> f i) s"
@@ -193,64 +196,62 @@
lemma (in prob_space) distribution_prob_space:
assumes "random_variable S X"
- shows "prob_space (S\<lparr>measure := distribution X\<rparr>)"
+ shows "prob_space (S\<lparr>measure := extreal \<circ> distribution X\<rparr>)"
proof -
- interpret S: measure_space "S\<lparr>measure := distribution X\<rparr>"
- unfolding distribution_def using assms
- by (intro measure_space_vimage)
- (auto intro!: sigma_algebra.sigma_algebra_cong[of S] simp: measure_preserving_def)
+ interpret S: measure_space "S\<lparr>measure := extreal \<circ> distribution X\<rparr>"
+ proof (rule measure_space.measure_space_cong)
+ show "measure_space (S\<lparr> measure := \<lambda>A. \<mu> (X -` A \<inter> space M) \<rparr>)"
+ using assms by (auto intro!: measure_space_vimage simp: measure_preserving_def)
+ qed (insert assms, auto simp add: finite_measure_eq distribution_def measurable_sets)
show ?thesis
proof (default, simp)
have "X -` space S \<inter> space M = space M"
using `random_variable S X` by (auto simp: measurable_def)
- then show "distribution X (space S) = 1"
- using measure_space_1 by (simp add: distribution_def)
+ then show "extreal (distribution X (space S)) = 1"
+ by (simp add: distribution_def one_extreal_def prob_space)
qed
qed
lemma (in prob_space) AE_distribution:
- assumes X: "random_variable MX X" and "measure_space.almost_everywhere (MX\<lparr>measure := distribution X\<rparr>) (\<lambda>x. Q x)"
+ assumes X: "random_variable MX X" and "AE x in MX\<lparr>measure := extreal \<circ> distribution X\<rparr>. Q x"
shows "AE x. Q (X x)"
proof -
- interpret X: prob_space "MX\<lparr>measure := distribution X\<rparr>" using X by (rule distribution_prob_space)
+ interpret X: prob_space "MX\<lparr>measure := extreal \<circ> distribution X\<rparr>" using X by (rule distribution_prob_space)
obtain N where N: "N \<in> sets MX" "distribution X N = 0" "{x\<in>space MX. \<not> Q x} \<subseteq> N"
using assms unfolding X.almost_everywhere_def by auto
- show "AE x. Q (X x)"
- using X[unfolded measurable_def] N unfolding distribution_def
- by (intro AE_I'[where N="X -` N \<inter> space M"]) auto
+ from X[unfolded measurable_def] N show "AE x. Q (X x)"
+ by (intro AE_I'[where N="X -` N \<inter> space M"])
+ (auto simp: finite_measure_eq distribution_def measurable_sets)
qed
-lemma (in prob_space) distribution_lebesgue_thm1:
- assumes "random_variable s X"
- assumes "A \<in> sets s"
- shows "real (distribution X A) = expectation (indicator (X -` A \<inter> space M))"
-unfolding distribution_def
-using assms unfolding measurable_def
-using integral_indicator by auto
+lemma (in prob_space) distribution_eq_integral:
+ "random_variable S X \<Longrightarrow> A \<in> sets S \<Longrightarrow> distribution X A = expectation (indicator (X -` A \<inter> space M))"
+ using finite_measure_eq[of "X -` A \<inter> space M"]
+ by (auto simp: measurable_sets distribution_def)
-lemma (in prob_space) distribution_lebesgue_thm2:
- assumes "random_variable S X" and "A \<in> sets S"
- shows "distribution X A = integral\<^isup>P (S\<lparr>measure := distribution X\<rparr>) (indicator A)"
+lemma (in prob_space) distribution_eq_translated_integral:
+ assumes "random_variable S X" "A \<in> sets S"
+ shows "distribution X A = integral\<^isup>P (S\<lparr>measure := extreal \<circ> distribution X\<rparr>) (indicator A)"
proof -
- interpret S: prob_space "S\<lparr>measure := distribution X\<rparr>"
+ interpret S: prob_space "S\<lparr>measure := extreal \<circ> distribution X\<rparr>"
using assms(1) by (rule distribution_prob_space)
show ?thesis
- using S.positive_integral_indicator(1)
- using assms unfolding distribution_def by auto
+ using S.positive_integral_indicator(1)[of A] assms by simp
qed
lemma (in prob_space) finite_expectation1:
assumes f: "finite (X`space M)" and rv: "random_variable borel X"
- shows "expectation X = (\<Sum> r \<in> X ` space M. r * prob (X -` {r} \<inter> space M))"
-proof (rule integral_on_finite(2)[OF rv[THEN conjunct2] f])
- fix x have "X -` {x} \<inter> space M \<in> sets M"
- using rv unfolding measurable_def by auto
- thus "\<mu> (X -` {x} \<inter> space M) \<noteq> \<omega>" using finite_measure by simp
+ shows "expectation X = (\<Sum>r \<in> X ` space M. r * prob (X -` {r} \<inter> space M))" (is "_ = ?r")
+proof (subst integral_on_finite)
+ show "X \<in> borel_measurable M" "finite (X`space M)" using assms by auto
+ show "(\<Sum> r \<in> X ` space M. r * real (\<mu> (X -` {r} \<inter> space M))) = ?r"
+ "\<And>x. \<mu> (X -` {x} \<inter> space M) \<noteq> \<infinity>"
+ using finite_measure_eq[OF borel_measurable_vimage, of X] rv by auto
qed
lemma (in prob_space) finite_expectation:
assumes "finite (X`space M)" "random_variable borel X"
- shows "expectation X = (\<Sum> r \<in> X ` (space M). r * real (distribution X {r}))"
+ shows "expectation X = (\<Sum> r \<in> X ` (space M). r * distribution X {r})"
using assms unfolding distribution_def using finite_expectation1 by auto
lemma (in prob_space) prob_x_eq_1_imp_prob_y_eq_0:
@@ -267,23 +268,18 @@
lemma (in prob_space) distribution_space[simp]: "distribution X (X ` space M) = 1"
proof -
have "X -` X ` space M \<inter> space M = space M" by auto
- thus ?thesis unfolding distribution_def by (simp add: measure_space_1)
+ thus ?thesis unfolding distribution_def by (simp add: prob_space)
qed
lemma (in prob_space) distribution_one:
assumes "random_variable M' X" and "A \<in> sets M'"
shows "distribution X A \<le> 1"
proof -
- have "distribution X A \<le> \<mu> (space M)" unfolding distribution_def
- using assms[unfolded measurable_def] by (auto intro!: measure_mono)
- thus ?thesis by (simp add: measure_space_1)
+ have "distribution X A \<le> \<mu>' (space M)" unfolding distribution_def
+ using assms[unfolded measurable_def] by (auto intro!: finite_measure_mono)
+ thus ?thesis by (simp add: prob_space)
qed
-lemma (in prob_space) distribution_finite:
- assumes "random_variable M' X" and "A \<in> sets M'"
- shows "distribution X A \<noteq> \<omega>"
- using distribution_one[OF assms] by auto
-
lemma (in prob_space) distribution_x_eq_1_imp_distribution_y_eq_0:
assumes X: "random_variable \<lparr>space = X ` (space M), sets = Pow (X ` (space M))\<rparr> X"
(is "random_variable ?S X")
@@ -302,8 +298,8 @@
have "X -` {x} \<inter> space M - X -` {y} \<inter> space M = X -` {x} \<inter> space M"
"X -` {y} \<inter> space M \<inter> (X -` {x} \<inter> space M) = {}"
using `y \<noteq> x` by auto
- with measure_inter_full_set[OF single single, of x y] assms(2)
- show ?thesis unfolding distribution_def measure_space_1 by auto
+ with finite_measure_inter_full_set[OF single single, of x y] assms(2)
+ show ?thesis by (auto simp: distribution_def prob_space)
next
assume "{y} \<notin> sets ?S"
then have "X -` {y} \<inter> space M = {}" by auto
@@ -315,20 +311,17 @@
and A: "A \<in> sets MX" and B: "B \<in> sets MY"
shows "joint_distribution X Y (A \<times> B) \<le> distribution X A"
unfolding distribution_def
-proof (intro measure_mono)
+proof (intro finite_measure_mono)
show "(\<lambda>x. (X x, Y x)) -` (A \<times> B) \<inter> space M \<subseteq> X -` A \<inter> space M" by force
show "X -` A \<inter> space M \<in> events"
using X A unfolding measurable_def by simp
have *: "(\<lambda>x. (X x, Y x)) -` (A \<times> B) \<inter> space M =
(X -` A \<inter> space M) \<inter> (Y -` B \<inter> space M)" by auto
- show "(\<lambda>x. (X x, Y x)) -` (A \<times> B) \<inter> space M \<in> events"
- unfolding * apply (rule Int)
- using assms unfolding measurable_def by auto
qed
lemma (in prob_space) joint_distribution_commute:
"joint_distribution X Y x = joint_distribution Y X ((\<lambda>(x,y). (y,x))`x)"
- unfolding distribution_def by (auto intro!: arg_cong[where f=\<mu>])
+ unfolding distribution_def by (auto intro!: arg_cong[where f=\<mu>'])
lemma (in prob_space) joint_distribution_Times_le_snd:
assumes X: "random_variable MX X" and Y: "random_variable MY Y"
@@ -352,27 +345,14 @@
unfolding P.measurable_pair_iff[OF sa] using assms by (simp add: comp_def)
qed
-lemma (in prob_space) distribution_order:
- assumes "random_variable MX X" "random_variable MY Y"
- assumes "{x} \<in> sets MX" "{y} \<in> sets MY"
- shows "r \<le> joint_distribution X Y {(x, y)} \<Longrightarrow> r \<le> distribution X {x}"
- and "r \<le> joint_distribution X Y {(x, y)} \<Longrightarrow> r \<le> distribution Y {y}"
- and "r < joint_distribution X Y {(x, y)} \<Longrightarrow> r < distribution X {x}"
- and "r < joint_distribution X Y {(x, y)} \<Longrightarrow> r < distribution Y {y}"
- and "distribution X {x} = 0 \<Longrightarrow> joint_distribution X Y {(x, y)} = 0"
- and "distribution Y {y} = 0 \<Longrightarrow> joint_distribution X Y {(x, y)} = 0"
- using joint_distribution_Times_le_snd[OF assms]
- using joint_distribution_Times_le_fst[OF assms]
- by auto
-
lemma (in prob_space) joint_distribution_commute_singleton:
"joint_distribution X Y {(x, y)} = joint_distribution Y X {(y, x)}"
- unfolding distribution_def by (auto intro!: arg_cong[where f=\<mu>])
+ unfolding distribution_def by (auto intro!: arg_cong[where f=\<mu>'])
lemma (in prob_space) joint_distribution_assoc_singleton:
"joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)} =
joint_distribution (\<lambda>x. (X x, Y x)) Z {((x, y), z)}"
- unfolding distribution_def by (auto intro!: arg_cong[where f=\<mu>])
+ unfolding distribution_def by (auto intro!: arg_cong[where f=\<mu>'])
locale pair_prob_space = M1: prob_space M1 + M2: prob_space M2 for M1 M2
@@ -383,13 +363,13 @@
lemma countably_additiveI[case_names countably]:
assumes "\<And>A. \<lbrakk> range A \<subseteq> sets M ; disjoint_family A ; (\<Union>i. A i) \<in> sets M\<rbrakk> \<Longrightarrow>
- (\<Sum>\<^isub>\<infinity>n. \<mu> (A n)) = \<mu> (\<Union>i. A i)"
+ (\<Sum>n. \<mu> (A n)) = \<mu> (\<Union>i. A i)"
shows "countably_additive M \<mu>"
using assms unfolding countably_additive_def by auto
lemma (in prob_space) joint_distribution_prob_space:
assumes "random_variable MX X" "random_variable MY Y"
- shows "prob_space ((MX \<Otimes>\<^isub>M MY) \<lparr> measure := joint_distribution X Y\<rparr>)"
+ shows "prob_space ((MX \<Otimes>\<^isub>M MY) \<lparr> measure := extreal \<circ> joint_distribution X Y\<rparr>)"
using random_variable_pairI[OF assms] by (rule distribution_prob_space)
section "Probability spaces on finite sets"
@@ -407,20 +387,13 @@
lemma (in prob_space) distribution_finite_prob_space:
assumes "finite_random_variable MX X"
- shows "finite_prob_space (MX\<lparr>measure := distribution X\<rparr>)"
+ shows "finite_prob_space (MX\<lparr>measure := extreal \<circ> distribution X\<rparr>)"
proof -
- interpret X: prob_space "MX\<lparr>measure := distribution X\<rparr>"
+ interpret X: prob_space "MX\<lparr>measure := extreal \<circ> distribution X\<rparr>"
using assms[THEN finite_random_variableD] by (rule distribution_prob_space)
interpret MX: finite_sigma_algebra MX
using assms by auto
- show ?thesis
- proof (default, simp_all)
- fix x assume "x \<in> space MX"
- then have "X -` {x} \<inter> space M \<in> sets M"
- using assms unfolding measurable_def by simp
- then show "distribution X {x} \<noteq> \<omega>"
- unfolding distribution_def by simp
- qed (rule MX.finite_space)
+ show ?thesis by default (simp_all add: MX.finite_space)
qed
lemma (in prob_space) simple_function_imp_finite_random_variable[simp, intro]:
@@ -451,9 +424,9 @@
by (auto dest!: finite_random_variableD)
lemma (in prob_space) sum_over_space_real_distribution:
- "simple_function M X \<Longrightarrow> (\<Sum>x\<in>X`space M. real (distribution X {x})) = 1"
+ "simple_function M X \<Longrightarrow> (\<Sum>x\<in>X`space M. distribution X {x}) = 1"
unfolding distribution_def prob_space[symmetric]
- by (subst real_finite_measure_finite_Union[symmetric])
+ by (subst finite_measure_finite_Union[symmetric])
(auto simp add: disjoint_family_on_def simple_function_def
intro!: arg_cong[where f=prob])
@@ -475,7 +448,7 @@
"finite_random_variable MX X \<Longrightarrow> x \<in> space MX \<Longrightarrow> {x} \<in> sets MX"
unfolding finite_sigma_algebra_def finite_sigma_algebra_axioms_def by simp
-lemma (in prob_space) finite_random_variable_vimage:
+lemma (in prob_space) finite_random_variable_measurable:
assumes X: "finite_random_variable MX X" shows "X -` A \<inter> space M \<in> events"
proof -
interpret X: finite_sigma_algebra MX using X by simp
@@ -492,15 +465,12 @@
assumes X: "finite_random_variable MX X" and Y: "finite_random_variable MY Y"
shows "joint_distribution X Y (A \<times> B) \<le> distribution X A"
unfolding distribution_def
-proof (intro measure_mono)
+proof (intro finite_measure_mono)
show "(\<lambda>x. (X x, Y x)) -` (A \<times> B) \<inter> space M \<subseteq> X -` A \<inter> space M" by force
show "X -` A \<inter> space M \<in> events"
- using finite_random_variable_vimage[OF X] .
+ using finite_random_variable_measurable[OF X] .
have *: "(\<lambda>x. (X x, Y x)) -` (A \<times> B) \<inter> space M =
(X -` A \<inter> space M) \<inter> (Y -` B \<inter> space M)" by auto
- show "(\<lambda>x. (X x, Y x)) -` (A \<times> B) \<inter> space M \<in> events"
- unfolding * apply (rule Int)
- using assms[THEN finite_random_variable_vimage] by auto
qed
lemma (in prob_space) joint_distribution_finite_Times_le_snd:
@@ -511,6 +481,7 @@
(simp add: swap_product joint_distribution_finite_Times_le_fst)
lemma (in prob_space) finite_distribution_order:
+ fixes MX :: "('c, 'd) measure_space_scheme" and MY :: "('e, 'f) measure_space_scheme"
assumes "finite_random_variable MX X" "finite_random_variable MY Y"
shows "r \<le> joint_distribution X Y {(x, y)} \<Longrightarrow> r \<le> distribution X {x}"
and "r \<le> joint_distribution X Y {(x, y)} \<Longrightarrow> r \<le> distribution Y {y}"
@@ -520,26 +491,14 @@
and "distribution Y {y} = 0 \<Longrightarrow> joint_distribution X Y {(x, y)} = 0"
using joint_distribution_finite_Times_le_snd[OF assms, of "{x}" "{y}"]
using joint_distribution_finite_Times_le_fst[OF assms, of "{x}" "{y}"]
- by auto
-
-lemma (in prob_space) finite_distribution_finite:
- assumes "finite_random_variable M' X"
- shows "distribution X {x} \<noteq> \<omega>"
-proof -
- have "distribution X {x} \<le> \<mu> (space M)"
- unfolding distribution_def
- using finite_random_variable_vimage[OF assms]
- by (intro measure_mono) auto
- then show ?thesis
- by auto
-qed
+ by (auto intro: antisym)
lemma (in prob_space) setsum_joint_distribution:
assumes X: "finite_random_variable MX X"
assumes Y: "random_variable MY Y" "B \<in> sets MY"
shows "(\<Sum>a\<in>space MX. joint_distribution X Y ({a} \<times> B)) = distribution Y B"
unfolding distribution_def
-proof (subst measure_finitely_additive'')
+proof (subst finite_measure_finite_Union[symmetric])
interpret MX: finite_sigma_algebra MX using X by auto
show "finite (space MX)" using MX.finite_space .
let "?d i" = "(\<lambda>x. (X x, Y x)) -` ({i} \<times> B) \<inter> space M"
@@ -549,9 +508,8 @@
using measurable_sets[of X M MX] measurable_sets[of Y M MY B] X Y
using MX.sets_eq_Pow by auto }
show "disjoint_family_on ?d (space MX)" by (auto simp: disjoint_family_on_def)
- show "\<mu> (\<Union>i\<in>space MX. ?d i) = \<mu> (Y -` B \<inter> space M)"
- using X[unfolded measurable_def]
- by (auto intro!: arg_cong[where f=\<mu>])
+ show "\<mu>' (\<Union>i\<in>space MX. ?d i) = \<mu>' (Y -` B \<inter> space M)"
+ using X[unfolded measurable_def] by (auto intro!: arg_cong[where f=\<mu>'])
qed
lemma (in prob_space) setsum_joint_distribution_singleton:
@@ -562,28 +520,6 @@
finite_random_variableD[OF Y(1)]
finite_random_variable_imp_sets[OF Y]] by simp
-lemma (in prob_space) setsum_real_joint_distribution:
- fixes MX :: "('c, 'x) measure_space_scheme" and MY :: "('d, 'y) measure_space_scheme"
- assumes X: "finite_random_variable MX X"
- assumes Y: "random_variable MY Y" "B \<in> sets MY"
- shows "(\<Sum>a\<in>space MX. real (joint_distribution X Y ({a} \<times> B))) = real (distribution Y B)"
-proof -
- interpret MX: finite_sigma_algebra MX using X by auto
- 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_measure real_of_pextreal_setsum)
-qed
-
-lemma (in prob_space) setsum_real_joint_distribution_singleton:
- fixes MX :: "('c, 'x) measure_space_scheme" and MY :: "('d, 'y) measure_space_scheme"
- assumes X: "finite_random_variable MX X"
- assumes Y: "finite_random_variable MY Y" "b \<in> space MY"
- shows "(\<Sum>a\<in>space MX. real (joint_distribution X Y {(a,b)})) = real (distribution Y {b})"
- using setsum_real_joint_distribution[OF X
- finite_random_variableD[OF Y(1)]
- finite_random_variable_imp_sets[OF Y]] by simp
-
locale pair_finite_prob_space = M1: finite_prob_space M1 + M2: finite_prob_space M2 for M1 M2
sublocale pair_finite_prob_space \<subseteq> pair_prob_space M1 M2 by default
@@ -593,7 +529,7 @@
lemma (in prob_space) joint_distribution_finite_prob_space:
assumes X: "finite_random_variable MX X"
assumes Y: "finite_random_variable MY Y"
- shows "finite_prob_space ((MX \<Otimes>\<^isub>M MY)\<lparr> measure := joint_distribution X Y\<rparr>)"
+ shows "finite_prob_space ((MX \<Otimes>\<^isub>M MY)\<lparr> measure := extreal \<circ> joint_distribution X Y\<rparr>)"
by (intro distribution_finite_prob_space finite_random_variable_pairI X Y)
lemma finite_prob_space_eq:
@@ -602,18 +538,15 @@
by auto
lemma (in prob_space) not_empty: "space M \<noteq> {}"
- using prob_space empty_measure by auto
+ using prob_space empty_measure' by auto
lemma (in finite_prob_space) sum_over_space_eq_1: "(\<Sum>x\<in>space M. \<mu> {x}) = 1"
using measure_space_1 sum_over_space by simp
-lemma (in finite_prob_space) positive_distribution: "0 \<le> distribution X x"
- unfolding distribution_def by simp
-
lemma (in finite_prob_space) joint_distribution_restriction_fst:
"joint_distribution X Y A \<le> distribution X (fst ` A)"
unfolding distribution_def
-proof (safe intro!: measure_mono)
+proof (safe intro!: finite_measure_mono)
fix x assume "x \<in> space M" and *: "(X x, Y x) \<in> A"
show "x \<in> X -` fst ` A"
by (auto intro!: image_eqI[OF _ *])
@@ -622,7 +555,7 @@
lemma (in finite_prob_space) joint_distribution_restriction_snd:
"joint_distribution X Y A \<le> distribution Y (snd ` A)"
unfolding distribution_def
-proof (safe intro!: measure_mono)
+proof (safe intro!: finite_measure_mono)
fix x assume "x \<in> space M" and *: "(X x, Y x) \<in> A"
show "x \<in> Y -` snd ` A"
by (auto intro!: image_eqI[OF _ *])
@@ -637,17 +570,16 @@
and "r < joint_distribution X Y {(x, y)} \<Longrightarrow> r < distribution Y {y}"
and "distribution X {x} = 0 \<Longrightarrow> joint_distribution X Y {(x, y)} = 0"
and "distribution Y {y} = 0 \<Longrightarrow> joint_distribution X Y {(x, y)} = 0"
- using positive_distribution[of X x']
- positive_distribution[of "\<lambda>x. (X x, Y x)" "{(x, y)}"]
+ using
joint_distribution_restriction_fst[of X Y "{(x, y)}"]
joint_distribution_restriction_snd[of X Y "{(x, y)}"]
- by auto
+ by (auto intro: antisym)
lemma (in finite_prob_space) distribution_mono:
assumes "\<And>t. \<lbrakk> t \<in> space M ; X t \<in> x \<rbrakk> \<Longrightarrow> Y t \<in> y"
shows "distribution X x \<le> distribution Y y"
unfolding distribution_def
- using assms by (auto simp: sets_eq_Pow intro!: measure_mono)
+ using assms by (auto simp: sets_eq_Pow intro!: finite_measure_mono)
lemma (in finite_prob_space) distribution_mono_gt_0:
assumes gt_0: "0 < distribution X x"
@@ -657,49 +589,21 @@
lemma (in finite_prob_space) sum_over_space_distrib:
"(\<Sum>x\<in>X`space M. distribution X {x}) = 1"
- unfolding distribution_def measure_space_1[symmetric] using finite_space
- by (subst measure_finitely_additive'')
- (auto simp add: disjoint_family_on_def sets_eq_Pow intro!: arg_cong[where f=\<mu>])
+ unfolding distribution_def prob_space[symmetric] using finite_space
+ by (subst finite_measure_finite_Union[symmetric])
+ (auto simp add: disjoint_family_on_def sets_eq_Pow
+ intro!: arg_cong[where f=\<mu>'])
lemma (in finite_prob_space) sum_over_space_real_distribution:
- "(\<Sum>x\<in>X`space M. real (distribution X {x})) = 1"
+ "(\<Sum>x\<in>X`space M. distribution X {x}) = 1"
unfolding distribution_def prob_space[symmetric] using finite_space
- by (subst real_finite_measure_finite_Union[symmetric])
+ by (subst finite_measure_finite_Union[symmetric])
(auto simp add: disjoint_family_on_def sets_eq_Pow intro!: arg_cong[where f=prob])
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_pextreal_setsum sets_eq_Pow)
-
-lemma (in finite_prob_space) distribution_finite:
- "distribution X A \<noteq> \<omega>"
- using finite_measure[of "X -` A \<inter> space M"]
- unfolding distribution_def sets_eq_Pow by auto
-
-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_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_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_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_pextreal_def real_of_pextreal_mult[symmetric]
- using assms distribution_finite[of X x] by (cases "distribution X x") (auto intro!: mult_pos_pos)
+ "(\<Sum>x\<in>space M. prob {x}) = 1"
+ using prob_space finite_space
+ by (subst (asm) finite_measure_finite_singleton) auto
lemma (in prob_space) distribution_remove_const:
shows "joint_distribution X (\<lambda>x. ()) {(x, ())} = distribution X {x}"
@@ -707,8 +611,7 @@
and "joint_distribution X (\<lambda>x. (Y x, ())) {(x, y, ())} = joint_distribution X Y {(x, y)}"
and "joint_distribution X (\<lambda>x. ((), Y x)) {(x, (), y)} = joint_distribution X Y {(x, y)}"
and "distribution (\<lambda>x. ()) {()} = 1"
- unfolding measure_space_1[symmetric]
- by (auto intro!: arg_cong[where f="\<mu>"] simp: distribution_def)
+ by (auto intro!: arg_cong[where f=\<mu>'] simp: distribution_def prob_space[symmetric])
lemma (in finite_prob_space) setsum_distribution_gen:
assumes "Z -` {c} \<inter> space M = (\<Union>x \<in> X`space M. Y -` {f x}) \<inter> space M"
@@ -716,7 +619,7 @@
shows "(\<Sum>x \<in> X`space M. distribution Y {f x}) = distribution Z {c}"
unfolding distribution_def assms
using finite_space assms
- by (subst measure_finitely_additive'')
+ by (subst finite_measure_finite_Union[symmetric])
(auto simp add: disjoint_family_on_def sets_eq_Pow inj_on_def
intro!: arg_cong[where f=prob])
@@ -728,61 +631,17 @@
"(\<Sum>z \<in> Z`space M. joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)}) = joint_distribution X Y {(x, y)}"
by (auto intro!: inj_onI setsum_distribution_gen)
-lemma (in finite_prob_space) setsum_real_distribution_gen:
- assumes "Z -` {c} \<inter> space M = (\<Union>x \<in> X`space M. Y -` {f x}) \<inter> space M"
- and "inj_on f (X`space M)"
- shows "(\<Sum>x \<in> X`space M. real (distribution Y {f x})) = real (distribution Z {c})"
- unfolding distribution_def assms
- using finite_space assms
- by (subst real_finite_measure_finite_Union[symmetric])
- (auto simp add: disjoint_family_on_def sets_eq_Pow inj_on_def
- intro!: arg_cong[where f=prob])
-
-lemma (in finite_prob_space) setsum_real_distribution:
- "(\<Sum>x \<in> X`space M. real (joint_distribution X Y {(x, y)})) = real (distribution Y {y})"
- "(\<Sum>y \<in> Y`space M. real (joint_distribution X Y {(x, y)})) = real (distribution X {x})"
- "(\<Sum>x \<in> X`space M. real (joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)})) = real (joint_distribution Y Z {(y, z)})"
- "(\<Sum>y \<in> Y`space M. real (joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)})) = real (joint_distribution X Z {(x, z)})"
- "(\<Sum>z \<in> Z`space M. real (joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)})) = real (joint_distribution X Y {(x, y)})"
- by (auto intro!: inj_onI setsum_real_distribution_gen)
-
-lemma (in finite_prob_space) real_distribution_order:
- shows "r \<le> real (joint_distribution X Y {(x, y)}) \<Longrightarrow> r \<le> real (distribution X {x})"
- and "r \<le> real (joint_distribution X Y {(x, y)}) \<Longrightarrow> r \<le> real (distribution Y {y})"
- and "r < real (joint_distribution X Y {(x, y)}) \<Longrightarrow> r < real (distribution X {x})"
- 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_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]:
- "joint_distribution X X {(x, x)} = distribution X {x}"
- unfolding distribution_def by (auto intro!: arg_cong[where f="\<mu>"])
-
-lemma (in finite_prob_space) distribution_1:
- "distribution X A \<le> 1"
- unfolding distribution_def measure_space_1[symmetric]
- by (auto intro!: measure_mono simp: sets_eq_Pow)
-
-lemma (in finite_prob_space) real_distribution_1:
- "real (distribution X A) \<le> 1"
- 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"
assumes "\<And> x y. \<lbrakk>x \<in> space M ; y \<in> space M\<rbrakk> \<Longrightarrow> prob {x} = prob {y}"
- shows "prob {x} = 1 / real (card (space M))"
+ shows "prob {x} = 1 / card (space M)"
proof -
have prob_x: "\<And> y. y \<in> space M \<Longrightarrow> prob {y} = prob {x}"
using assms(2)[OF _ `x \<in> space M`] by blast
have "1 = prob (space M)"
using prob_space by auto
also have "\<dots> = (\<Sum> x \<in> space M. prob {x})"
- using real_finite_measure_finite_Union[of "space M" "\<lambda> x. {x}", simplified]
+ using finite_measure_finite_Union[of "space M" "\<lambda> x. {x}", simplified]
sets_eq_Pow inj_singleton[unfolded inj_on_def, rule_format]
finite_space unfolding disjoint_family_on_def prob_space[symmetric]
by (auto simp add:setsum_restrict_set)
@@ -809,7 +668,7 @@
qed
lemma (in prob_space) prob_space_of_restricted_space:
- assumes "\<mu> A \<noteq> 0" "\<mu> A \<noteq> \<omega>" "A \<in> sets M"
+ assumes "\<mu> A \<noteq> 0" "A \<in> sets M"
shows "prob_space (restricted_space A \<lparr>measure := \<lambda>S. \<mu> S / \<mu> A\<rparr>)"
(is "prob_space ?P")
proof -
@@ -820,29 +679,48 @@
show "prob_space ?P"
proof
show "measure ?P (space ?P) = 1"
- using `\<mu> A \<noteq> 0` `\<mu> A \<noteq> \<omega>` by (auto simp: pextreal_noteq_omega_Ex)
- show "measure ?P {} = 0" by auto
- have "\<And>S. \<mu> S / \<mu> A = inverse (\<mu> A) * \<mu> S" by (simp add: mult_commute)
- then show "countably_additive ?P (measure ?P)"
- unfolding countably_additive_def psuminf_cmult_right
- using A.measure_countably_additive by auto
+ using real_measure[OF `A \<in> events`] `\<mu> A \<noteq> 0` by auto
+ show "positive ?P (measure ?P)"
+ proof (simp add: positive_def, safe)
+ show "0 / \<mu> A = 0" using `\<mu> A \<noteq> 0` by (cases "\<mu> A") (auto simp: zero_extreal_def)
+ fix B assume "B \<in> events"
+ with real_measure[of "A \<inter> B"] real_measure[OF `A \<in> events`] `A \<in> sets M`
+ show "0 \<le> \<mu> (A \<inter> B) / \<mu> A" by (auto simp: Int)
+ qed
+ show "countably_additive ?P (measure ?P)"
+ proof (simp add: countably_additive_def, safe)
+ fix B and F :: "nat \<Rightarrow> 'a set"
+ assume F: "range F \<subseteq> op \<inter> A ` events" "disjoint_family F"
+ { fix i
+ from F have "F i \<in> op \<inter> A ` events" by auto
+ with `A \<in> events` have "F i \<in> events" by auto }
+ moreover then have "range F \<subseteq> events" by auto
+ moreover have "\<And>S. \<mu> S / \<mu> A = inverse (\<mu> A) * \<mu> S"
+ by (simp add: mult_commute divide_extreal_def)
+ moreover have "0 \<le> inverse (\<mu> A)"
+ using real_measure[OF `A \<in> events`] by auto
+ ultimately show "(\<Sum>i. \<mu> (F i) / \<mu> A) = \<mu> (\<Union>i. F i) / \<mu> A"
+ using measure_countably_additive[of F] F
+ by (auto simp: suminf_cmult_extreal)
+ qed
qed
qed
lemma finite_prob_spaceI:
- assumes "finite (space M)" "sets M = Pow(space M)" "measure M (space M) = 1" "measure M {} = 0"
+ assumes "finite (space M)" "sets M = Pow(space M)"
+ and "measure M (space M) = 1" "measure M {} = 0" "\<And>A. A \<subseteq> space M \<Longrightarrow> 0 \<le> measure M A"
and "\<And>A B. A\<subseteq>space M \<Longrightarrow> B\<subseteq>space M \<Longrightarrow> A \<inter> B = {} \<Longrightarrow> measure M (A \<union> B) = measure M A + measure M B"
shows "finite_prob_space M"
unfolding finite_prob_space_eq
proof
show "finite_measure_space M" using assms
- by (auto intro!: finite_measure_spaceI)
+ by (auto intro!: finite_measure_spaceI)
show "measure M (space M) = 1" by fact
qed
lemma (in finite_prob_space) finite_measure_space:
fixes X :: "'a \<Rightarrow> 'x"
- shows "finite_measure_space \<lparr>space = X ` space M, sets = Pow (X ` space M), measure = distribution X\<rparr>"
+ shows "finite_measure_space \<lparr>space = X ` space M, sets = Pow (X ` space M), measure = extreal \<circ> distribution X\<rparr>"
(is "finite_measure_space ?S")
proof (rule finite_measure_spaceI, simp_all)
show "finite (X ` space M)" using finite_space by simp
@@ -850,51 +728,41 @@
fix A B :: "'x set" assume "A \<inter> B = {}"
then show "distribution X (A \<union> B) = distribution X A + distribution X B"
unfolding distribution_def
- by (subst measure_additive)
- (auto intro!: arg_cong[where f=\<mu>] simp: sets_eq_Pow)
+ by (subst finite_measure_Union[symmetric])
+ (auto intro!: arg_cong[where f=\<mu>'] simp: sets_eq_Pow)
qed
lemma (in finite_prob_space) finite_prob_space_of_images:
- "finite_prob_space \<lparr> space = X ` space M, sets = Pow (X ` space M), measure = distribution X \<rparr>"
- by (simp add: finite_prob_space_eq finite_measure_space)
-
-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_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
+ "finite_prob_space \<lparr> space = X ` space M, sets = Pow (X ` space M), measure = extreal \<circ> distribution X \<rparr>"
+ by (simp add: finite_prob_space_eq finite_measure_space measure_space_1 one_extreal_def)
lemma (in finite_prob_space) finite_product_measure_space:
fixes X :: "'a \<Rightarrow> 'x" and Y :: "'a \<Rightarrow> 'y"
assumes "finite s1" "finite s2"
- shows "finite_measure_space \<lparr> space = s1 \<times> s2, sets = Pow (s1 \<times> s2), measure = joint_distribution X Y\<rparr>"
+ shows "finite_measure_space \<lparr> space = s1 \<times> s2, sets = Pow (s1 \<times> s2), measure = extreal \<circ> joint_distribution X Y\<rparr>"
(is "finite_measure_space ?M")
proof (rule finite_measure_spaceI, simp_all)
show "finite (s1 \<times> s2)"
using assms by auto
- show "joint_distribution X Y (s1\<times>s2) \<noteq> \<omega>"
- using distribution_finite .
next
fix A B :: "('x*'y) set" assume "A \<inter> B = {}"
then show "joint_distribution X Y (A \<union> B) = joint_distribution X Y A + joint_distribution X Y B"
unfolding distribution_def
- by (subst measure_additive)
- (auto intro!: arg_cong[where f=\<mu>] simp: sets_eq_Pow)
+ by (subst finite_measure_Union[symmetric])
+ (auto intro!: arg_cong[where f=\<mu>'] simp: sets_eq_Pow)
qed
lemma (in finite_prob_space) finite_product_measure_space_of_images:
shows "finite_measure_space \<lparr> space = X ` space M \<times> Y ` space M,
sets = Pow (X ` space M \<times> Y ` space M),
- measure = joint_distribution X Y \<rparr>"
+ measure = extreal \<circ> joint_distribution X Y \<rparr>"
using finite_space by (auto intro!: finite_product_measure_space)
lemma (in finite_prob_space) finite_product_prob_space_of_images:
"finite_prob_space \<lparr> space = X ` space M \<times> Y ` space M, sets = Pow (X ` space M \<times> Y ` space M),
- measure = joint_distribution X Y \<rparr>"
+ measure = extreal \<circ> joint_distribution X Y \<rparr>"
(is "finite_prob_space ?S")
-proof (simp add: finite_prob_space_eq finite_product_measure_space_of_images)
+proof (simp add: finite_prob_space_eq finite_product_measure_space_of_images one_extreal_def)
have "X -` X ` space M \<inter> Y -` Y ` space M \<inter> space M = space M" by auto
thus "joint_distribution X Y (X ` space M \<times> Y ` space M) = 1"
by (simp add: distribution_def prob_space vimage_Times comp_def measure_space_1)
@@ -903,11 +771,11 @@
section "Conditional Expectation and Probability"
lemma (in prob_space) conditional_expectation_exists:
- fixes X :: "'a \<Rightarrow> pextreal" and N :: "('a, 'b) measure_space_scheme"
- assumes borel: "X \<in> borel_measurable M"
+ fixes X :: "'a \<Rightarrow> extreal" and N :: "('a, 'b) measure_space_scheme"
+ assumes borel: "X \<in> borel_measurable M" "AE x. 0 \<le> X x"
and N: "sigma_algebra N" "sets N \<subseteq> sets M" "space N = space M" "\<And>A. measure N A = \<mu> A"
- shows "\<exists>Y\<in>borel_measurable N. \<forall>C\<in>sets N.
- (\<integral>\<^isup>+x. Y x * indicator C x \<partial>M) = (\<integral>\<^isup>+x. X x * indicator C x \<partial>M)"
+ shows "\<exists>Y\<in>borel_measurable N. (\<forall>x. 0 \<le> Y x) \<and> (\<forall>C\<in>sets N.
+ (\<integral>\<^isup>+x. Y x * indicator C x \<partial>M) = (\<integral>\<^isup>+x. X x * indicator C x \<partial>M))"
proof -
note N(4)[simp]
interpret P: prob_space N
@@ -926,34 +794,29 @@
unfolding P.absolutely_continuous_def
proof safe
fix A assume "A \<in> sets N" "P.\<mu> A = 0"
- moreover then have f_borel: "?f A \<in> borel_measurable M"
- using borel N by (auto intro: borel_measurable_indicator)
- moreover have "{x\<in>space M. ?f A x \<noteq> 0} = (?f A -` {0<..} \<inter> space M) \<inter> A"
- by (auto simp: indicator_def)
- moreover have "\<mu> \<dots> \<le> \<mu> A"
- using `A \<in> sets N` N f_borel
- by (auto intro!: measure_mono Int[of _ A] measurable_sets)
- ultimately show "?Q A = 0"
- by (simp add: positive_integral_0_iff)
+ then have f_borel: "?f A \<in> borel_measurable M" "AE x. x \<notin> A"
+ using borel N by (auto intro!: borel_measurable_indicator AE_not_in)
+ then show "?Q A = 0"
+ by (auto simp add: positive_integral_0_iff_AE)
qed
from P.Radon_Nikodym[OF Q this]
- obtain Y where Y: "Y \<in> borel_measurable N"
+ obtain Y where Y: "Y \<in> borel_measurable N" "\<And>x. 0 \<le> Y x"
"\<And>A. A \<in> sets N \<Longrightarrow> ?Q A =(\<integral>\<^isup>+x. Y x * indicator A x \<partial>N)"
by blast
with N(2) show ?thesis
- by (auto intro!: bexI[OF _ Y(1)] simp: positive_integral_subalgebra[OF _ N(2,3,4,1)])
+ by (auto intro!: bexI[OF _ Y(1)] simp: positive_integral_subalgebra[OF _ _ N(2,3,4,1)])
qed
definition (in prob_space)
- "conditional_expectation N X = (SOME Y. Y\<in>borel_measurable N
+ "conditional_expectation N X = (SOME Y. Y\<in>borel_measurable N \<and> (\<forall>x. 0 \<le> Y x)
\<and> (\<forall>C\<in>sets N. (\<integral>\<^isup>+x. Y x * indicator C x\<partial>M) = (\<integral>\<^isup>+x. X x * indicator C x\<partial>M)))"
abbreviation (in prob_space)
"conditional_prob N A \<equiv> conditional_expectation N (indicator A)"
lemma (in prob_space)
- fixes X :: "'a \<Rightarrow> pextreal" and N :: "('a, 'b) measure_space_scheme"
- assumes borel: "X \<in> borel_measurable M"
+ fixes X :: "'a \<Rightarrow> extreal" and N :: "('a, 'b) measure_space_scheme"
+ assumes borel: "X \<in> borel_measurable M" "AE x. 0 \<le> X x"
and N: "sigma_algebra N" "sets N \<subseteq> sets M" "space N = space M" "\<And>A. measure N A = \<mu> A"
shows borel_measurable_conditional_expectation:
"conditional_expectation N X \<in> borel_measurable N"
@@ -970,36 +833,23 @@
unfolding conditional_expectation_def by (rule someI2_ex) blast
qed
-lemma (in sigma_algebra) factorize_measurable_function:
- fixes Z :: "'a \<Rightarrow> pextreal" and Y :: "'a \<Rightarrow> 'c"
+lemma (in sigma_algebra) factorize_measurable_function_pos:
+ fixes Z :: "'a \<Rightarrow> extreal" 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))"
-proof safe
+ assumes Z: "Z \<in> borel_measurable (sigma_algebra.vimage_algebra M' (space M) Y)"
+ shows "\<exists>g\<in>borel_measurable M'. \<forall>x\<in>space M. max 0 (Z x) = g (Y x)"
+proof -
interpret M': sigma_algebra M' by fact
have Y: "Y \<in> space M \<rightarrow> space M'" using assms unfolding measurable_def by auto
from M'.sigma_algebra_vimage[OF this]
interpret va: sigma_algebra "M'.vimage_algebra (space M) Y" .
- { 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)
- moreover assume "\<forall>x\<in>space M. Z x = g (Y x)"
- then have "Z \<in> borel_measurable (M'.vimage_algebra (space M) Y) \<longleftrightarrow>
- g \<circ> Y \<in> borel_measurable (M'.vimage_algebra (space M) Y)"
- by (auto intro!: measurable_cong)
- ultimately show "Z \<in> borel_measurable (M'.vimage_algebra (space M) Y)"
- by simp }
-
- assume "Z \<in> borel_measurable (M'.vimage_algebra (space M) Y)"
- from va.borel_measurable_implies_simple_function_sequence[OF this]
- obtain f where f: "\<And>i. simple_function (M'.vimage_algebra (space M) Y) (f i)" and "f \<up> Z" by blast
+ from va.borel_measurable_implies_simple_function_sequence'[OF Z] guess f . note f = this
have "\<forall>i. \<exists>g. simple_function M' g \<and> (\<forall>x\<in>space M. f i x = g (Y x))"
proof
fix i
- from f[of i] have "finite (f i`space M)" and B_ex:
+ from f(1)[of i] have "finite (f i`space M)" and B_ex:
"\<forall>z\<in>(f i)`space M. \<exists>B. B \<in> sets M' \<and> (f i) -` {z} \<inter> space M = Y -` B \<inter> space M"
unfolding simple_function_def by auto
from B_ex[THEN bchoice] guess B .. note B = this
@@ -1011,23 +861,67 @@
show "simple_function M' ?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::pextreal)"
+ then have "\<And>z. z \<in> f i`space M \<Longrightarrow> indicator (B z) (Y x) = (indicator (f i -` {z} \<inter> space M) x::extreal)"
unfolding indicator_def using B by auto
- then show "f i x = ?g (Y x)" using `x \<in> space M` f[of i]
+ then show "f i x = ?g (Y x)" using `x \<in> space M` f(1)[of i]
by (subst va.simple_function_indicator_representation) auto
qed
qed
from choice[OF this] guess g .. note g = this
- show "\<exists>g\<in>borel_measurable M'. \<forall>x\<in>space M. Z x = g (Y x)"
+ show ?thesis
proof (intro ballI bexI)
show "(\<lambda>x. SUP i. g i x) \<in> borel_measurable M'"
using g by (auto intro: M'.borel_measurable_simple_function)
fix x assume "x \<in> space M"
- have "Z x = (SUP i. f i) x" using `f \<up> Z` unfolding isoton_def by simp
- also have "\<dots> = (SUP i. g i (Y x))" unfolding SUPR_apply
+ have "max 0 (Z x) = (SUP i. f i x)" using f by simp
+ also have "\<dots> = (SUP i. g i (Y x))"
using g `x \<in> space M` by simp
- finally show "Z x = (SUP i. g i (Y x))" .
+ finally show "max 0 (Z x) = (SUP i. g i (Y x))" .
+ qed
+qed
+
+lemma extreal_0_le_iff_le_0[simp]:
+ fixes a :: extreal shows "0 \<le> -a \<longleftrightarrow> a \<le> 0"
+ by (cases rule: extreal2_cases[of a]) auto
+
+lemma (in sigma_algebra) factorize_measurable_function:
+ fixes Z :: "'a \<Rightarrow> extreal" 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))"
+proof safe
+ interpret M': sigma_algebra M' by fact
+ have Y: "Y \<in> space M \<rightarrow> space M'" using assms unfolding measurable_def by auto
+ from M'.sigma_algebra_vimage[OF this]
+ interpret va: sigma_algebra "M'.vimage_algebra (space M) Y" .
+
+ { fix g :: "'c \<Rightarrow> extreal" 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)
+ moreover assume "\<forall>x\<in>space M. Z x = g (Y x)"
+ then have "Z \<in> borel_measurable (M'.vimage_algebra (space M) Y) \<longleftrightarrow>
+ g \<circ> Y \<in> borel_measurable (M'.vimage_algebra (space M) Y)"
+ by (auto intro!: measurable_cong)
+ ultimately show "Z \<in> borel_measurable (M'.vimage_algebra (space M) Y)"
+ by simp }
+
+ assume Z: "Z \<in> borel_measurable (M'.vimage_algebra (space M) Y)"
+ with assms have "(\<lambda>x. - Z x) \<in> borel_measurable M"
+ "(\<lambda>x. - Z x) \<in> borel_measurable (M'.vimage_algebra (space M) Y)"
+ by auto
+ from factorize_measurable_function_pos[OF assms(1,2) this] guess n .. note n = this
+ from factorize_measurable_function_pos[OF assms Z] guess p .. note p = this
+ let "?g x" = "p x - n x"
+ show "\<exists>g\<in>borel_measurable M'. \<forall>x\<in>space M. Z x = g (Y x)"
+ proof (intro bexI ballI)
+ show "?g \<in> borel_measurable M'" using p n by auto
+ fix x assume "x \<in> space M"
+ then have "p (Y x) = max 0 (Z x)" "n (Y x) = max 0 (- Z x)"
+ using p n by auto
+ then show "Z x = ?g (Y x)"
+ by (auto split: split_max)
qed
qed
--- a/src/HOL/Probability/Product_Measure.thy Mon Mar 14 15:17:10 2011 +0100
+++ b/src/HOL/Probability/Product_Measure.thy Mon Mar 14 15:29:10 2011 +0100
@@ -416,7 +416,7 @@
show "?f -` A \<inter> space ?P \<in> sets ?P" by simp
qed
-lemma (in pair_sigma_algebra) measurable_cut_fst:
+lemma (in pair_sigma_algebra) measurable_cut_fst[simp,intro]:
assumes "Q \<in> sets P" shows "Pair x -` Q \<in> sets M2"
proof -
let ?Q' = "{Q. Q \<subseteq> space P \<and> Pair x -` Q \<in> sets M2}"
@@ -504,15 +504,14 @@
fix A assume "A \<in> sets ?D"
with sets_into_space have "\<And>x. measure M2 (Pair x -` (space M1 \<times> space M2 - A)) =
(if x \<in> space M1 then measure M2 (space M2) - ?s A x else 0)"
- by (auto intro!: M2.finite_measure_compl measurable_cut_fst
- simp: vimage_Diff)
+ by (auto intro!: M2.measure_compl 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_pextreal_diff)
+ by (auto intro!: Diff M1.measurable_If M1.borel_measurable_extreal_diff)
next
fix F :: "nat \<Rightarrow> ('a\<times>'b) set" assume "disjoint_family F" "range F \<subseteq> sets ?D"
- moreover then have "\<And>x. measure M2 (\<Union>i. Pair x -` F i) = (\<Sum>\<^isub>\<infinity> i. ?s (F i) x)"
+ moreover then have "\<And>x. measure M2 (\<Union>i. Pair x -` F i) = (\<Sum>i. ?s (F i) x)"
by (intro M2.measure_countably_additive[symmetric])
- (auto intro!: measurable_cut_fst simp: disjoint_family_on_def)
+ (auto simp: disjoint_family_on_def)
ultimately show "(\<Union>i. F i) \<in> sets ?D"
by (auto simp: vimage_UN intro!: M1.borel_measurable_psuminf)
qed
@@ -546,7 +545,7 @@
have [intro]: "sigma_algebra M1" and [intro]: "sigma_algebra M2" by default+
have M1: "sigma_finite_measure M1" by default
from M2.disjoint_sigma_finite guess F .. note F = this
- then have "\<And>i. F i \<in> sets M2" by auto
+ then have F_sets: "\<And>i. F i \<in> sets M2" by auto
let "?C x i" = "F i \<inter> Pair x -` Q"
{ fix i
let ?R = "M2.restricted_space (F i)"
@@ -578,10 +577,10 @@
by simp }
moreover
{ fix x
- have "(\<Sum>\<^isub>\<infinity>i. measure M2 (?C x i)) = measure M2 (\<Union>i. ?C x i)"
+ have "(\<Sum>i. measure M2 (?C x i)) = measure M2 (\<Union>i. ?C x i)"
proof (intro M2.measure_countably_additive)
show "range (?C x) \<subseteq> sets M2"
- using F `Q \<in> sets P` by (auto intro!: M2.Int measurable_cut_fst)
+ using F `Q \<in> sets P` by (auto intro!: M2.Int)
have "disjoint_family F" using F by auto
show "disjoint_family (?C x)"
by (rule disjoint_family_on_bisimulation[OF `disjoint_family F`]) auto
@@ -589,10 +588,10 @@
also have "(\<Union>i. ?C x i) = Pair x -` Q"
using F sets_into_space `Q \<in> sets P`
by (auto simp: space_pair_measure)
- finally have "measure M2 (Pair x -` Q) = (\<Sum>\<^isub>\<infinity>i. measure M2 (?C x i))"
+ finally have "measure M2 (Pair x -` Q) = (\<Sum>i. measure M2 (?C x i))"
by simp }
- ultimately show ?thesis
- by (auto intro!: M1.borel_measurable_psuminf)
+ ultimately show ?thesis using `Q \<in> sets P` F_sets
+ by (auto intro!: M1.borel_measurable_psuminf M2.Int)
qed
lemma (in pair_sigma_finite) measure_cut_measurable_snd:
@@ -620,7 +619,7 @@
apply (simp add: pair_measure_def pair_measure_generator_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 :: pextreal)"
+ have *: "\<And>y. indicator A (x, y) = (indicator (Pair x -` A) y :: extreal)"
unfolding indicator_def by auto
show "(\<integral>\<^isup>+ y. indicator A (x, y) \<partial>M2) = measure M2 (Pair x -` A)"
unfolding *
@@ -639,18 +638,21 @@
by (simp add: M1.positive_integral_cmult_indicator ac_simps)
qed
+lemma (in measure_space) measure_not_negative[simp,intro]:
+ assumes A: "A \<in> sets M" shows "\<mu> A \<noteq> - \<infinity>"
+ using positive_measure[OF A] by auto
+
lemma (in pair_sigma_finite) sigma_finite_up_in_pair_measure_generator:
- "\<exists>F::nat \<Rightarrow> ('a \<times> 'b) set. range F \<subseteq> sets E \<and> F \<up> space E \<and>
- (\<forall>i. measure (M1 \<Otimes>\<^isub>M M2) (F i) \<noteq> \<omega>)"
+ "\<exists>F::nat \<Rightarrow> ('a \<times> 'b) set. range F \<subseteq> sets E \<and> incseq F \<and> (\<Union>i. F i) = space E \<and>
+ (\<forall>i. measure (M1 \<Otimes>\<^isub>M M2) (F i) \<noteq> \<infinity>)"
proof -
obtain F1 :: "nat \<Rightarrow> 'a set" and F2 :: "nat \<Rightarrow> 'b set" where
- F1: "range F1 \<subseteq> sets M1" "F1 \<up> space M1" "\<And>i. M1.\<mu> (F1 i) \<noteq> \<omega>" and
- F2: "range F2 \<subseteq> sets M2" "F2 \<up> space M2" "\<And>i. M2.\<mu> (F2 i) \<noteq> \<omega>"
+ F1: "range F1 \<subseteq> sets M1" "incseq F1" "(\<Union>i. F1 i) = space M1" "\<And>i. M1.\<mu> (F1 i) \<noteq> \<infinity>" and
+ F2: "range F2 \<subseteq> sets M2" "incseq F2" "(\<Union>i. F2 i) = space M2" "\<And>i. M2.\<mu> (F2 i) \<noteq> \<infinity>"
using M1.sigma_finite_up M2.sigma_finite_up by auto
- then have space: "space M1 = (\<Union>i. F1 i)" "space M2 = (\<Union>i. F2 i)"
- unfolding isoton_def by auto
+ then have space: "space M1 = (\<Union>i. F1 i)" "space M2 = (\<Union>i. F2 i)" by auto
let ?F = "\<lambda>i. F1 i \<times> F2 i"
- show ?thesis unfolding isoton_def space_pair_measure
+ show ?thesis unfolding space_pair_measure
proof (intro exI[of _ ?F] conjI allI)
show "range ?F \<subseteq> sets E" using F1 F2
by (fastsimp intro!: pair_measure_generatorI)
@@ -661,8 +663,8 @@
then obtain i j where "fst x \<in> F1 i" "snd x \<in> F2 j"
by (auto simp: space)
then have "fst x \<in> F1 (max i j)" "snd x \<in> F2 (max j i)"
- using `F1 \<up> space M1` `F2 \<up> space M2`
- by (auto simp: max_def dest: isoton_mono_le)
+ using `incseq F1` `incseq F2` unfolding incseq_def
+ by (force split: split_max)+
then have "(fst x, snd x) \<in> F1 (max i j) \<times> F2 (max i j)"
by (intro SigmaI) (auto simp add: min_max.sup_commute)
then show "x \<in> (\<Union>i. ?F i)" by auto
@@ -670,21 +672,22 @@
then show "(\<Union>i. ?F i) = space E"
using space by (auto simp: space pair_measure_generator_def)
next
- fix i show "F1 i \<times> F2 i \<subseteq> F1 (Suc i) \<times> F2 (Suc i)"
- using `F1 \<up> space M1` `F2 \<up> space M2` unfolding isoton_def
- by auto
+ fix i show "incseq (\<lambda>i. F1 i \<times> F2 i)"
+ using `incseq F1` `incseq F2` unfolding incseq_Suc_iff by auto
next
fix i
from F1 F2 have "F1 i \<in> sets M1" "F2 i \<in> sets M2" by auto
- with F1 F2 show "measure P (F1 i \<times> F2 i) \<noteq> \<omega>"
+ with F1 F2 M1.positive_measure[OF this(1)] M2.positive_measure[OF this(2)]
+ show "measure P (F1 i \<times> F2 i) \<noteq> \<infinity>"
by (simp add: pair_measure_times)
qed
qed
sublocale pair_sigma_finite \<subseteq> sigma_finite_measure P
proof
- show "measure P {} = 0"
- unfolding pair_measure_def pair_measure_generator_def sigma_def by auto
+ show "positive P (measure P)"
+ unfolding pair_measure_def pair_measure_generator_def sigma_def positive_def
+ by (auto intro: M1.positive_integral_positive M2.positive_integral_positive)
show "countably_additive P (measure P)"
unfolding countably_additive_def
@@ -697,20 +700,20 @@
moreover have "\<And>x. disjoint_family (\<lambda>i. Pair x -` F i)"
by (intro disjoint_family_on_bisimulation[OF F(2)]) auto
moreover have "\<And>x. x \<in> space M1 \<Longrightarrow> range (\<lambda>i. Pair x -` F i) \<subseteq> sets M2"
- using F by (auto intro!: measurable_cut_fst)
- ultimately show "(\<Sum>\<^isub>\<infinity>n. measure P (F n)) = measure P (\<Union>i. F i)"
- by (simp add: pair_measure_alt vimage_UN M1.positive_integral_psuminf[symmetric]
+ using F by auto
+ ultimately show "(\<Sum>n. measure P (F n)) = measure P (\<Union>i. F i)"
+ by (simp add: pair_measure_alt vimage_UN M1.positive_integral_suminf[symmetric]
M2.measure_countably_additive
cong: M1.positive_integral_cong)
qed
from sigma_finite_up_in_pair_measure_generator guess F :: "nat \<Rightarrow> ('a \<times> 'b) set" .. note F = this
- show "\<exists>F::nat \<Rightarrow> ('a \<times> 'b) set. range F \<subseteq> sets P \<and> (\<Union>i. F i) = space P \<and> (\<forall>i. measure P (F i) \<noteq> \<omega>)"
+ show "\<exists>F::nat \<Rightarrow> ('a \<times> 'b) set. range F \<subseteq> sets P \<and> (\<Union>i. F i) = space P \<and> (\<forall>i. measure P (F i) \<noteq> \<infinity>)"
proof (rule exI[of _ F], intro conjI)
show "range F \<subseteq> sets P" using F by (auto simp: pair_measure_def)
show "(\<Union>i. F i) = space P"
- using F by (auto simp: pair_measure_def pair_measure_generator_def isoton_def)
- show "\<forall>i. measure P (F i) \<noteq> \<omega>" using F by auto
+ using F by (auto simp: pair_measure_def pair_measure_generator_def)
+ show "\<forall>i. measure P (F i) \<noteq> \<infinity>" using F by auto
qed
qed
@@ -741,7 +744,7 @@
show "(\<lambda>(y, x). (x, y)) \<in> measurable Q.P P" by (rule Q.pair_sigma_algebra_swap_measurable)
show "sets (sigma E) = sets P" "space E = space P" "A \<in> sets (sigma E)"
using assms unfolding pair_measure_def by auto
- show "range F \<subseteq> sets E" "F \<up> space E" "\<And>i. \<mu> (F i) \<noteq> \<omega>"
+ show "range F \<subseteq> sets E" "incseq F" "(\<Union>i. F i) = space E" "\<And>i. \<mu> (F i) \<noteq> \<infinity>"
using F `A \<in> sets P` by (auto simp: pair_measure_def)
fix X assume "X \<in> sets E"
then obtain A B where X[simp]: "X = A \<times> B" and AB: "A \<in> sets M1" "B \<in> sets M2"
@@ -758,8 +761,8 @@
qed
lemma pair_sigma_algebra_sigma:
- assumes 1: "S1 \<up> (space E1)" "range S1 \<subseteq> sets E1" and E1: "sets E1 \<subseteq> Pow (space E1)"
- assumes 2: "S2 \<up> (space E2)" "range S2 \<subseteq> sets E2" and E2: "sets E2 \<subseteq> Pow (space E2)"
+ assumes 1: "incseq S1" "(\<Union>i. S1 i) = space E1" "range S1 \<subseteq> sets E1" and E1: "sets E1 \<subseteq> Pow (space E1)"
+ assumes 2: "decseq S2" "(\<Union>i. S2 i) = space E2" "range S2 \<subseteq> sets E2" and E2: "sets E2 \<subseteq> Pow (space E2)"
shows "sets (sigma (pair_measure_generator (sigma E1) (sigma E2))) = sets (sigma (pair_measure_generator E1 E2))"
(is "sets ?S = sets ?E")
proof -
@@ -771,22 +774,22 @@
using E1 E2 by (intro sigma_algebra_sigma) auto
{ fix A assume "A \<in> sets E1"
then have "fst -` A \<inter> space ?E = A \<times> (\<Union>i. S2 i)"
- using E1 2 unfolding isoton_def pair_measure_generator_def by auto
+ using E1 2 unfolding pair_measure_generator_def by auto
also have "\<dots> = (\<Union>i. A \<times> S2 i)" by auto
also have "\<dots> \<in> sets ?E" unfolding pair_measure_generator_def sets_sigma
using 2 `A \<in> sets E1`
by (intro sigma_sets.Union)
- (auto simp: image_subset_iff intro!: sigma_sets.Basic)
+ (force simp: image_subset_iff intro!: sigma_sets.Basic)
finally have "fst -` A \<inter> space ?E \<in> sets ?E" . }
moreover
{ fix B assume "B \<in> sets E2"
then have "snd -` B \<inter> space ?E = (\<Union>i. S1 i) \<times> B"
- using E2 1 unfolding isoton_def pair_measure_generator_def by auto
+ using E2 1 unfolding pair_measure_generator_def by auto
also have "\<dots> = (\<Union>i. S1 i \<times> B)" by auto
also have "\<dots> \<in> sets ?E"
using 1 `B \<in> sets E2` unfolding pair_measure_generator_def sets_sigma
by (intro sigma_sets.Union)
- (auto simp: image_subset_iff intro!: sigma_sets.Basic)
+ (force simp: image_subset_iff intro!: sigma_sets.Basic)
finally have "snd -` B \<inter> space ?E \<in> sets ?E" . }
ultimately have proj:
"fst \<in> measurable ?E (sigma E1) \<and> snd \<in> measurable ?E (sigma E2)"
@@ -819,12 +822,12 @@
section "Fubinis theorem"
lemma (in pair_sigma_finite) simple_function_cut:
- assumes f: "simple_function P f"
+ assumes f: "simple_function P f" "\<And>x. 0 \<le> f x"
shows "(\<lambda>x. \<integral>\<^isup>+y. f (x, y) \<partial>M2) \<in> borel_measurable M1"
and "(\<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. f (x, y) \<partial>M2) \<partial>M1) = integral\<^isup>P P f"
proof -
have f_borel: "f \<in> borel_measurable P"
- using f by (rule borel_measurable_simple_function)
+ using f(1) by (rule borel_measurable_simple_function)
let "?F z" = "f -` {z} \<inter> space P"
let "?F' x z" = "Pair x -` ?F z"
{ fix x assume "x \<in> space M1"
@@ -836,15 +839,15 @@
by (intro borel_measurable_vimage measurable_cut_fst)
ultimately have "simple_function M2 (\<lambda> y. f (x, y))"
apply (rule_tac M2.simple_function_cong[THEN iffD2, OF _])
- apply (rule simple_function_indicator_representation[OF f])
+ apply (rule simple_function_indicator_representation[OF f(1)])
using `x \<in> space M1` by (auto simp del: space_sigma) }
note M2_sf = this
{ fix x assume x: "x \<in> space M1"
then have "(\<integral>\<^isup>+y. f (x, y) \<partial>M2) = (\<Sum>z\<in>f ` space P. z * M2.\<mu> (?F' x z))"
- unfolding M2.positive_integral_eq_simple_integral[OF M2_sf[OF x]]
+ unfolding M2.positive_integral_eq_simple_integral[OF M2_sf[OF x] f(2)]
unfolding simple_integral_def
proof (safe intro!: setsum_mono_zero_cong_left)
- from f show "finite (f ` space P)" by (rule simple_functionD)
+ from f(1) show "finite (f ` space P)" by (rule simple_functionD)
next
fix y assume "y \<in> space M2" then show "f (x, y) \<in> f ` space P"
using `x \<in> space M1` by (auto simp: space_pair_measure)
@@ -862,11 +865,16 @@
by (auto intro!: f_borel borel_measurable_vimage simp del: space_sigma)
moreover then have "\<And>z. (\<lambda>x. M2.\<mu> (?F' x z)) \<in> borel_measurable M1"
by (auto intro!: measure_cut_measurable_fst simp del: vimage_Int)
+ moreover have *: "\<And>i x. 0 \<le> M2.\<mu> (Pair x -` (f -` {i} \<inter> space P))"
+ using f(1)[THEN simple_functionD(2)] f(2) by (intro M2.positive_measure measurable_cut_fst)
+ moreover { fix i assume "i \<in> f`space P"
+ with * have "\<And>x. 0 \<le> i * M2.\<mu> (Pair x -` (f -` {i} \<inter> space P))"
+ using f(2) by auto }
ultimately
show "(\<lambda>x. \<integral>\<^isup>+y. f (x, y) \<partial>M2) \<in> borel_measurable M1"
- and "(\<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. f (x, y) \<partial>M2) \<partial>M1) = integral\<^isup>P P f"
+ and "(\<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. f (x, y) \<partial>M2) \<partial>M1) = integral\<^isup>P P f" using f(2)
by (auto simp del: vimage_Int cong: measurable_cong
- intro!: M1.borel_measurable_pextreal_setsum
+ intro!: M1.borel_measurable_extreal_setsum setsum_cong
simp add: M1.positive_integral_setsum simple_integral_def
M1.positive_integral_cmult
M1.positive_integral_cong[OF eq]
@@ -880,42 +888,38 @@
(is "?C f \<in> borel_measurable M1")
and "(\<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. f (x, y) \<partial>M2) \<partial>M1) = integral\<^isup>P P f"
proof -
- from borel_measurable_implies_simple_function_sequence[OF f]
- obtain F where F: "\<And>i. simple_function P (F i)" "F \<up> f" by auto
+ from borel_measurable_implies_simple_function_sequence'[OF f] guess F . note F = this
then have F_borel: "\<And>i. F i \<in> borel_measurable P"
- and F_mono: "\<And>i x. F i x \<le> F (Suc i) x"
- and F_SUPR: "\<And>x. (SUP i. F i x) = f x"
- unfolding isoton_fun_expand unfolding isoton_def le_fun_def
by (auto intro: borel_measurable_simple_function)
- note sf = simple_function_cut[OF F(1)]
+ note sf = simple_function_cut[OF F(1,5)]
then have "(\<lambda>x. SUP i. ?C (F i) x) \<in> borel_measurable M1"
using F(1) by auto
moreover
{ fix x assume "x \<in> space M1"
- have isotone: "(\<lambda> i y. F i (x, y)) \<up> (\<lambda>y. f (x, y))"
- using `F \<up> f` unfolding isoton_fun_expand
- by (auto simp: isoton_def)
- note measurable_pair_image_snd[OF F_borel`x \<in> space M1`]
- from M2.positive_integral_isoton[OF isotone this]
- have "(SUP i. ?C (F i) x) = ?C f x"
- by (simp add: isoton_def) }
+ from F measurable_pair_image_snd[OF F_borel`x \<in> space M1`]
+ have "(\<integral>\<^isup>+y. (SUP i. F i (x, y)) \<partial>M2) = (SUP i. ?C (F i) x)"
+ by (intro M2.positive_integral_monotone_convergence_SUP)
+ (auto simp: incseq_Suc_iff le_fun_def)
+ then have "(SUP i. ?C (F i) x) = ?C f x"
+ unfolding F(4) positive_integral_max_0 by simp }
note SUPR_C = this
ultimately show "?C f \<in> borel_measurable M1"
by (simp cong: measurable_cong)
have "(\<integral>\<^isup>+x. (SUP i. F i x) \<partial>P) = (SUP i. integral\<^isup>P P (F i))"
- using F_borel F_mono
- by (auto intro!: positive_integral_monotone_convergence_SUP[symmetric])
+ using F_borel F
+ by (intro positive_integral_monotone_convergence_SUP) auto
also have "(SUP i. integral\<^isup>P P (F i)) = (SUP i. \<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. F i (x, y) \<partial>M2) \<partial>M1)"
unfolding sf(2) by simp
- also have "\<dots> = \<integral>\<^isup>+ x. (SUP i. \<integral>\<^isup>+ y. F i (x, y) \<partial>M2) \<partial>M1"
- by (auto intro!: M1.positive_integral_monotone_convergence_SUP[OF _ sf(1)]
- M2.positive_integral_mono F_mono)
+ also have "\<dots> = \<integral>\<^isup>+ x. (SUP i. \<integral>\<^isup>+ y. F i (x, y) \<partial>M2) \<partial>M1" using F sf(1)
+ by (intro M1.positive_integral_monotone_convergence_SUP[symmetric])
+ (auto intro!: M2.positive_integral_mono M2.positive_integral_positive
+ simp: incseq_Suc_iff le_fun_def)
also have "\<dots> = \<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. (SUP i. F i (x, y)) \<partial>M2) \<partial>M1"
- using F_borel F_mono
- by (auto intro!: M2.positive_integral_monotone_convergence_SUP
- M1.positive_integral_cong measurable_pair_image_snd)
+ using F_borel F(2,5)
+ by (auto intro!: M1.positive_integral_cong M2.positive_integral_monotone_convergence_SUP[symmetric]
+ simp: incseq_Suc_iff le_fun_def measurable_pair_image_snd)
finally show "(\<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. f (x, y) \<partial>M2) \<partial>M1) = integral\<^isup>P P f"
- unfolding F_SUPR by simp
+ using F by (simp add: positive_integral_max_0)
qed
lemma (in pair_sigma_finite) measure_preserving_swap:
@@ -963,8 +967,8 @@
unfolding positive_integral_fst_measurable[OF assms] ..
lemma (in pair_sigma_finite) AE_pair:
- assumes "almost_everywhere (\<lambda>x. Q x)"
- shows "M1.almost_everywhere (\<lambda>x. M2.almost_everywhere (\<lambda>y. Q (x, y)))"
+ assumes "AE x in P. Q x"
+ shows "AE x in M1. (AE y in M2. Q (x, y))"
proof -
obtain N where N: "N \<in> sets P" "\<mu> N = 0" "{x\<in>space P. \<not> Q x} \<subseteq> N"
using assms unfolding almost_everywhere_def by auto
@@ -972,9 +976,9 @@
proof (rule M1.AE_I)
from N measure_cut_measurable_fst[OF `N \<in> sets P`]
show "M1.\<mu> {x\<in>space M1. M2.\<mu> (Pair x -` N) \<noteq> 0} = 0"
- by (simp add: M1.positive_integral_0_iff pair_measure_alt vimage_def)
+ by (auto simp: pair_measure_alt M1.positive_integral_0_iff)
show "{x \<in> space M1. M2.\<mu> (Pair x -` N) \<noteq> 0} \<in> sets M1"
- by (intro M1.borel_measurable_pextreal_neq_const measure_cut_measurable_fst N)
+ by (intro M1.borel_measurable_extreal_neq_const measure_cut_measurable_fst N)
{ fix x assume "x \<in> space M1" "M2.\<mu> (Pair x -` N) = 0"
have "M2.almost_everywhere (\<lambda>y. Q (x, y))"
proof (rule M2.AE_I)
@@ -1036,41 +1040,52 @@
shows "M1.almost_everywhere (\<lambda>x. integrable M2 (\<lambda> y. f (x, y)))" (is "?AE")
and "(\<integral>x. (\<integral>y. f (x, y) \<partial>M2) \<partial>M1) = integral\<^isup>L P f" (is "?INT")
proof -
- let "?pf x" = "Real (f x)" and "?nf x" = "Real (- f x)"
+ let "?pf x" = "extreal (f x)" and "?nf x" = "extreal (- f x)"
have
borel: "?nf \<in> borel_measurable P""?pf \<in> borel_measurable P" and
- int: "integral\<^isup>P P ?nf \<noteq> \<omega>" "integral\<^isup>P P ?pf \<noteq> \<omega>"
+ int: "integral\<^isup>P P ?nf \<noteq> \<infinity>" "integral\<^isup>P P ?pf \<noteq> \<infinity>"
using assms by auto
- have "(\<integral>\<^isup>+x. (\<integral>\<^isup>+y. Real (f (x, y)) \<partial>M2) \<partial>M1) \<noteq> \<omega>"
- "(\<integral>\<^isup>+x. (\<integral>\<^isup>+y. Real (- f (x, y)) \<partial>M2) \<partial>M1) \<noteq> \<omega>"
+ have "(\<integral>\<^isup>+x. (\<integral>\<^isup>+y. extreal (f (x, y)) \<partial>M2) \<partial>M1) \<noteq> \<infinity>"
+ "(\<integral>\<^isup>+x. (\<integral>\<^isup>+y. extreal (- f (x, y)) \<partial>M2) \<partial>M1) \<noteq> \<infinity>"
using borel[THEN positive_integral_fst_measurable(1)] int
unfolding borel[THEN positive_integral_fst_measurable(2)] by simp_all
with borel[THEN positive_integral_fst_measurable(1)]
- have AE: "M1.almost_everywhere (\<lambda>x. (\<integral>\<^isup>+y. Real (f (x, y)) \<partial>M2) \<noteq> \<omega>)"
- "M1.almost_everywhere (\<lambda>x. (\<integral>\<^isup>+y. Real (- f (x, y)) \<partial>M2) \<noteq> \<omega>)"
- by (auto intro!: M1.positive_integral_omega_AE)
- then show ?AE using assms
+ have AE_pos: "AE x in M1. (\<integral>\<^isup>+y. extreal (f (x, y)) \<partial>M2) \<noteq> \<infinity>"
+ "AE x in M1. (\<integral>\<^isup>+y. extreal (- f (x, y)) \<partial>M2) \<noteq> \<infinity>"
+ by (auto intro!: M1.positive_integral_PInf_AE )
+ then have AE: "AE x in M1. \<bar>\<integral>\<^isup>+y. extreal (f (x, y)) \<partial>M2\<bar> \<noteq> \<infinity>"
+ "AE x in M1. \<bar>\<integral>\<^isup>+y. extreal (- f (x, y)) \<partial>M2\<bar> \<noteq> \<infinity>"
+ by (auto simp: M2.positive_integral_positive)
+ from AE_pos show ?AE using assms
by (simp add: measurable_pair_image_snd integrable_def)
- { fix f assume borel: "(\<lambda>x. Real (f x)) \<in> borel_measurable P"
- and int: "integral\<^isup>P P (\<lambda>x. Real (f x)) \<noteq> \<omega>"
- and AE: "M1.almost_everywhere (\<lambda>x. (\<integral>\<^isup>+y. Real (f (x, y)) \<partial>M2) \<noteq> \<omega>)"
- have "integrable M1 (\<lambda>x. real (\<integral>\<^isup>+y. Real (f (x, y)) \<partial>M2))" (is "integrable M1 ?f")
+ { fix f have "(\<integral>\<^isup>+ x. - \<integral>\<^isup>+ y. extreal (f x y) \<partial>M2 \<partial>M1) = (\<integral>\<^isup>+x. 0 \<partial>M1)"
+ using M2.positive_integral_positive
+ by (intro M1.positive_integral_cong_pos) (auto simp: extreal_uminus_le_reorder)
+ then have "(\<integral>\<^isup>+ x. - \<integral>\<^isup>+ y. extreal (f x y) \<partial>M2 \<partial>M1) = 0" by simp }
+ note this[simp]
+ { fix f assume borel: "(\<lambda>x. extreal (f x)) \<in> borel_measurable P"
+ and int: "integral\<^isup>P P (\<lambda>x. extreal (f x)) \<noteq> \<infinity>"
+ and AE: "M1.almost_everywhere (\<lambda>x. (\<integral>\<^isup>+y. extreal (f (x, y)) \<partial>M2) \<noteq> \<infinity>)"
+ have "integrable M1 (\<lambda>x. real (\<integral>\<^isup>+y. extreal (f (x, y)) \<partial>M2))" (is "integrable M1 ?f")
proof (intro integrable_def[THEN iffD2] conjI)
show "?f \<in> borel_measurable M1"
- using borel by (auto intro!: M1.borel_measurable_real positive_integral_fst_measurable)
- have "(\<integral>\<^isup>+x. Real (?f x) \<partial>M1) = (\<integral>\<^isup>+x. (\<integral>\<^isup>+y. Real (f (x, y)) \<partial>M2) \<partial>M1)"
- using AE by (auto intro!: M1.positive_integral_cong_AE simp: Real_real)
- then show "(\<integral>\<^isup>+x. Real (?f x) \<partial>M1) \<noteq> \<omega>"
+ using borel by (auto intro!: M1.borel_measurable_real_of_extreal positive_integral_fst_measurable)
+ have "(\<integral>\<^isup>+x. extreal (?f x) \<partial>M1) = (\<integral>\<^isup>+x. (\<integral>\<^isup>+y. extreal (f (x, y)) \<partial>M2) \<partial>M1)"
+ using AE M2.positive_integral_positive
+ by (auto intro!: M1.positive_integral_cong_AE simp: extreal_real)
+ then show "(\<integral>\<^isup>+x. extreal (?f x) \<partial>M1) \<noteq> \<infinity>"
using positive_integral_fst_measurable[OF borel] int by simp
- have "(\<integral>\<^isup>+x. Real (- ?f x) \<partial>M1) = (\<integral>\<^isup>+x. 0 \<partial>M1)"
- by (intro M1.positive_integral_cong) simp
- then show "(\<integral>\<^isup>+x. Real (- ?f x) \<partial>M1) \<noteq> \<omega>" by simp
+ have "(\<integral>\<^isup>+x. extreal (- ?f x) \<partial>M1) = (\<integral>\<^isup>+x. 0 \<partial>M1)"
+ by (intro M1.positive_integral_cong_pos)
+ (simp add: M2.positive_integral_positive real_of_extreal_pos)
+ then show "(\<integral>\<^isup>+x. extreal (- ?f x) \<partial>M1) \<noteq> \<infinity>" by simp
qed }
- from this[OF borel(1) int(1) AE(2)] this[OF borel(2) int(2) AE(1)]
+ with this[OF borel(1) int(1) AE_pos(2)] this[OF borel(2) int(2) AE_pos(1)]
show ?INT
unfolding lebesgue_integral_def[of P] lebesgue_integral_def[of M2]
borel[THEN positive_integral_fst_measurable(2), symmetric]
- using AE by (simp add: M1.integral_real)
+ using AE[THEN M1.integral_real]
+ by simp
qed
lemma (in pair_sigma_finite) integrable_snd_measurable:
@@ -1195,7 +1210,8 @@
lemma sigma_product_algebra_sigma_eq:
assumes "finite I"
- assumes isotone: "\<And>i. i \<in> I \<Longrightarrow> (S i) \<up> (space (E i))"
+ assumes mono: "\<And>i. i \<in> I \<Longrightarrow> incseq (S i)"
+ assumes union: "\<And>i. i \<in> I \<Longrightarrow> (\<Union>j. S i j) = space (E i)"
assumes sets_into: "\<And>i. i \<in> I \<Longrightarrow> range (S i) \<subseteq> sets (E i)"
and E: "\<And>i. sets (E i) \<subseteq> Pow (space (E i))"
shows "sets (\<Pi>\<^isub>M i\<in>I. sigma (E i)) = sets (\<Pi>\<^isub>M i\<in>I. E i)"
@@ -1214,13 +1230,13 @@
by (intro sigma_algebra.sigma_algebra_cong[OF sigma_algebra_sigma]) (auto dest: Pi_mem)
{ fix A i assume "i \<in> I" and A: "A \<in> sets (E i)"
then have "(\<lambda>x. x i) -` A \<inter> space ?E = (\<Pi>\<^isub>E j\<in>I. if j = i then A else \<Union>n. S j n) \<inter> space ?E"
- using isotone unfolding isoton_def space_product_algebra
+ using mono union unfolding incseq_Suc_iff space_product_algebra
by (auto dest: Pi_mem)
also have "\<dots> = (\<Union>n. (\<Pi>\<^isub>E j\<in>I. if j = i then A else S j n))"
unfolding space_product_algebra
apply simp
apply (subst Pi_UN[OF `finite I`])
- using isotone[THEN isoton_mono_le] apply simp
+ using mono[THEN incseqD] apply simp
apply (simp add: PiE_Int)
apply (intro PiE_cong)
using A sets_into by (auto intro!: into_space)
@@ -1324,7 +1340,7 @@
obtain E where E: "A = Pi\<^isub>E (I \<union> J) E" "E \<in> (\<Pi> i\<in>I \<union> J. sets (M i))" .
then have *: "?f -` A \<inter> space (?I \<Otimes>\<^isub>M ?J) = Pi\<^isub>E I E \<times> Pi\<^isub>E J E"
using sets_into_space `I \<inter> J = {}`
- by (auto simp add: space_pair_measure) blast+
+ by (auto simp add: space_pair_measure) fast+
then show "?f -` A \<inter> space (?I \<Otimes>\<^isub>M ?J) \<in> sets (?I \<Otimes>\<^isub>M ?J)"
using E unfolding * by (auto intro!: pair_measureI in_sigma product_algebra_generatorI
simp: product_algebra_def)
@@ -1360,6 +1376,44 @@
sublocale finite_product_sigma_finite \<subseteq> finite_product_sigma_algebra
by default (fact finite_index')
+lemma setprod_extreal_0:
+ fixes f :: "'a \<Rightarrow> extreal"
+ shows "(\<Prod>i\<in>A. f i) = 0 \<longleftrightarrow> (finite A \<and> (\<exists>i\<in>A. f i = 0))"
+proof cases
+ assume "finite A"
+ then show ?thesis by (induct A) auto
+qed auto
+
+lemma setprod_extreal_pos:
+ fixes f :: "'a \<Rightarrow> extreal" assumes pos: "\<And>i. i \<in> I \<Longrightarrow> 0 \<le> f i" shows "0 \<le> (\<Prod>i\<in>I. f i)"
+proof cases
+ assume "finite I" from this pos show ?thesis by induct auto
+qed simp
+
+lemma setprod_PInf:
+ assumes "\<And>i. i \<in> I \<Longrightarrow> 0 \<le> f i"
+ shows "(\<Prod>i\<in>I. f i) = \<infinity> \<longleftrightarrow> finite I \<and> (\<exists>i\<in>I. f i = \<infinity>) \<and> (\<forall>i\<in>I. f i \<noteq> 0)"
+proof cases
+ assume "finite I" from this assms show ?thesis
+ proof (induct I)
+ case (insert i I)
+ then have pos: "0 \<le> f i" "0 \<le> setprod f I" by (auto intro!: setprod_extreal_pos)
+ from insert have "(\<Prod>j\<in>insert i I. f j) = \<infinity> \<longleftrightarrow> setprod f I * f i = \<infinity>" by auto
+ also have "\<dots> \<longleftrightarrow> (setprod f I = \<infinity> \<or> f i = \<infinity>) \<and> f i \<noteq> 0 \<and> setprod f I \<noteq> 0"
+ using setprod_extreal_pos[of I f] pos
+ by (cases rule: extreal2_cases[of "f i" "setprod f I"]) auto
+ also have "\<dots> \<longleftrightarrow> finite (insert i I) \<and> (\<exists>j\<in>insert i I. f j = \<infinity>) \<and> (\<forall>j\<in>insert i I. f j \<noteq> 0)"
+ using insert by (auto simp: setprod_extreal_0)
+ finally show ?case .
+ qed simp
+qed simp
+
+lemma setprod_extreal: "(\<Prod>i\<in>A. extreal (f i)) = extreal (setprod f A)"
+proof cases
+ assume "finite A" then show ?thesis
+ by induct (auto simp: one_extreal_def)
+qed (simp add: one_extreal_def)
+
lemma (in finite_product_sigma_finite) product_algebra_generator_measure:
assumes "Pi\<^isub>E I F \<in> sets G"
shows "measure G (Pi\<^isub>E I F) = (\<Prod>i\<in>I. M.\<mu> i (F i))"
@@ -1373,13 +1427,13 @@
next
assume empty: "\<not> (\<forall>j\<in>I. F j \<noteq> {})"
then have "(\<Prod>j\<in>I. M.\<mu> j (F j)) = 0"
- by (auto simp: setprod_pextreal_0 intro!: bexI)
+ by (auto simp: setprod_extreal_0 intro!: bexI)
moreover
have "\<exists>j\<in>I. (SOME F'. Pi\<^isub>E I F = Pi\<^isub>E I F') j = {}"
by (rule someI2[where P="\<lambda>F'. Pi\<^isub>E I F = Pi\<^isub>E I F'"])
(insert empty, auto simp: Pi_eq_empty_iff[symmetric])
then have "(\<Prod>j\<in>I. M.\<mu> j ((SOME F'. Pi\<^isub>E I F = Pi\<^isub>E I F') j)) = 0"
- by (auto simp: setprod_pextreal_0 intro!: bexI)
+ by (auto simp: setprod_extreal_0 intro!: bexI)
ultimately show ?thesis
unfolding product_algebra_generator_def by simp
qed
@@ -1387,34 +1441,32 @@
lemma (in finite_product_sigma_finite) sigma_finite_pairs:
"\<exists>F::'i \<Rightarrow> nat \<Rightarrow> 'a set.
(\<forall>i\<in>I. range (F i) \<subseteq> sets (M i)) \<and>
- (\<forall>k. \<forall>i\<in>I. \<mu> i (F i k) \<noteq> \<omega>) \<and>
- (\<lambda>k. \<Pi>\<^isub>E i\<in>I. F i k) \<up> space G"
+ (\<forall>k. \<forall>i\<in>I. \<mu> i (F i k) \<noteq> \<infinity>) \<and> incseq (\<lambda>k. \<Pi>\<^isub>E i\<in>I. F i k) \<and>
+ (\<Union>k. \<Pi>\<^isub>E i\<in>I. F i k) = space G"
proof -
- have "\<forall>i::'i. \<exists>F::nat \<Rightarrow> 'a set. range F \<subseteq> sets (M i) \<and> F \<up> space (M i) \<and> (\<forall>k. \<mu> i (F k) \<noteq> \<omega>)"
+ have "\<forall>i::'i. \<exists>F::nat \<Rightarrow> 'a set. range F \<subseteq> sets (M i) \<and> incseq F \<and> (\<Union>i. F i) = space (M i) \<and> (\<forall>k. \<mu> i (F k) \<noteq> \<infinity>)"
using M.sigma_finite_up by simp
from choice[OF this] guess F :: "'i \<Rightarrow> nat \<Rightarrow> 'a set" ..
- then have "\<And>i. range (F i) \<subseteq> sets (M i)" "\<And>i. F i \<up> space (M i)" "\<And>i k. \<mu> i (F i k) \<noteq> \<omega>"
+ then have F: "\<And>i. range (F i) \<subseteq> sets (M i)" "\<And>i. incseq (F i)" "\<And>i. (\<Union>j. F i j) = space (M i)" "\<And>i k. \<mu> i (F i k) \<noteq> \<infinity>"
by auto
let ?F = "\<lambda>k. \<Pi>\<^isub>E i\<in>I. F i k"
note space_product_algebra[simp]
show ?thesis
- proof (intro exI[of _ F] conjI allI isotoneI set_eqI iffI ballI)
+ proof (intro exI[of _ F] conjI allI incseq_SucI set_eqI iffI ballI)
fix i show "range (F i) \<subseteq> sets (M i)" by fact
next
- fix i k show "\<mu> i (F i k) \<noteq> \<omega>" by fact
+ fix i k show "\<mu> i (F i k) \<noteq> \<infinity>" by fact
next
fix A assume "A \<in> (\<Union>i. ?F i)" then show "A \<in> space G"
using `\<And>i. range (F i) \<subseteq> sets (M i)` M.sets_into_space
by (force simp: image_subset_iff)
next
fix f assume "f \<in> space G"
- with Pi_UN[OF finite_index, of "\<lambda>k i. F i k"]
- `\<And>i. F i \<up> space (M i)`[THEN isotonD(2)]
- `\<And>i. F i \<up> space (M i)`[THEN isoton_mono_le]
- show "f \<in> (\<Union>i. ?F i)" by auto
+ with Pi_UN[OF finite_index, of "\<lambda>k i. F i k"] F
+ show "f \<in> (\<Union>i. ?F i)" by (auto simp: incseq_def)
next
fix i show "?F i \<subseteq> ?F (Suc i)"
- using `\<And>i. F i \<up> space (M i)`[THEN isotonD(1)] by auto
+ using `\<And>i. incseq (F i)`[THEN incseq_SucD] by auto
qed
qed
@@ -1438,7 +1490,7 @@
using `finite I` proof induct
case empty
interpret finite_product_sigma_finite M "{}" by default simp
- let ?\<nu> = "(\<lambda>A. if A = {} then 0 else 1) :: 'd set \<Rightarrow> pextreal"
+ let ?\<nu> = "(\<lambda>A. if A = {} then 0 else 1) :: 'd set \<Rightarrow> extreal"
show ?case
proof (intro exI conjI ballI)
have "sigma_algebra (sigma G \<lparr>measure := ?\<nu>\<rparr>)"
@@ -1488,22 +1540,23 @@
proof (unfold *, default, simp)
from I'.sigma_finite_pairs guess F :: "'i \<Rightarrow> nat \<Rightarrow> 'a set" ..
then have F: "\<And>j. j \<in> insert i I \<Longrightarrow> range (F j) \<subseteq> sets (M j)"
- "(\<lambda>k. \<Pi>\<^isub>E j \<in> insert i I. F j k) \<up> space I'.G"
- "\<And>k. \<And>j. j \<in> insert i I \<Longrightarrow> \<mu> j (F j k) \<noteq> \<omega>"
+ "incseq (\<lambda>k. \<Pi>\<^isub>E j \<in> insert i I. F j k)"
+ "(\<Union>k. \<Pi>\<^isub>E j \<in> insert i I. F j k) = space I'.G"
+ "\<And>k. \<And>j. j \<in> insert i I \<Longrightarrow> \<mu> j (F j k) \<noteq> \<infinity>"
by blast+
let "?F k" = "\<Pi>\<^isub>E j \<in> insert i I. F j k"
show "\<exists>F::nat \<Rightarrow> ('i \<Rightarrow> 'a) set. range F \<subseteq> sets I'.P \<and>
- (\<Union>i. F i) = (\<Pi>\<^isub>E i\<in>insert i I. space (M i)) \<and> (\<forall>i. ?m (F i) \<noteq> \<omega>)"
+ (\<Union>i. F i) = (\<Pi>\<^isub>E i\<in>insert i I. space (M i)) \<and> (\<forall>i. ?m (F i) \<noteq> \<infinity>)"
proof (intro exI[of _ ?F] conjI allI)
show "range ?F \<subseteq> sets I'.P" using F(1) by auto
next
- from F(2)[THEN isotonD(2)]
- show "(\<Union>i. ?F i) = (\<Pi>\<^isub>E i\<in>insert i I. space (M i))" by simp
+ from F(3) show "(\<Union>i. ?F i) = (\<Pi>\<^isub>E i\<in>insert i I. space (M i))" by simp
next
fix j
- show "?\<nu> (?F j) \<noteq> \<omega>"
- using F `finite I`
- by (subst product) (auto simp: setprod_\<omega>)
+ have "\<And>k. k \<in> insert i I \<Longrightarrow> 0 \<le> measure (M k) (F k j)"
+ using F(1) by auto
+ with F `finite I` setprod_PInf[of "insert i I", OF this] show "?\<nu> (?F j) \<noteq> \<infinity>"
+ by (subst product) auto
qed
qed
qed
@@ -1542,7 +1595,8 @@
by (simp_all add: sigma_def image_constant)
lemma (in product_sigma_finite) positive_integral_empty:
- "integral\<^isup>P (Pi\<^isub>M {} M) f = f (\<lambda>k. undefined)"
+ assumes pos: "0 \<le> f (\<lambda>k. undefined)"
+ shows "integral\<^isup>P (Pi\<^isub>M {} M) f = f (\<lambda>k. undefined)"
proof -
interpret finite_product_sigma_finite M "{}" by default (fact finite.emptyI)
have "\<And>A. measure (Pi\<^isub>M {} M) (Pi\<^isub>E {} A) = 1"
@@ -1550,10 +1604,10 @@
then show ?thesis
unfolding positive_integral_def simple_function_def simple_integral_def_raw
proof (simp add: P_empty, intro antisym)
- show "f (\<lambda>k. undefined) \<le> (SUP f:{g. g \<le> f}. f (\<lambda>k. undefined))"
- by (intro le_SUPI) auto
- show "(SUP f:{g. g \<le> f}. f (\<lambda>k. undefined)) \<le> f (\<lambda>k. undefined)"
- by (intro SUP_leI) (auto simp: le_fun_def)
+ show "f (\<lambda>k. undefined) \<le> (SUP f:{g. g \<le> max 0 \<circ> f}. f (\<lambda>k. undefined))"
+ by (intro le_SUPI) (auto simp: le_fun_def split: split_max)
+ show "(SUP f:{g. g \<le> max 0 \<circ> f}. f (\<lambda>k. undefined)) \<le> f (\<lambda>k. undefined)" using pos
+ by (intro SUP_leI) (auto simp: le_fun_def simp: max_def split: split_if_asm)
qed
qed
@@ -1572,8 +1626,9 @@
let "?X S" = "?g -` S \<inter> space P.P"
from IJ.sigma_finite_pairs obtain F where
F: "\<And>i. i\<in> I \<union> J \<Longrightarrow> range (F i) \<subseteq> sets (M i)"
- "(\<lambda>k. \<Pi>\<^isub>E i\<in>I \<union> J. F i k) \<up> space IJ.G"
- "\<And>k. \<forall>i\<in>I\<union>J. \<mu> i (F i k) \<noteq> \<omega>"
+ "incseq (\<lambda>k. \<Pi>\<^isub>E i\<in>I \<union> J. F i k)"
+ "(\<Union>k. \<Pi>\<^isub>E i\<in>I \<union> J. F i k) = space IJ.G"
+ "\<And>k. \<forall>i\<in>I\<union>J. \<mu> i (F i k) \<noteq> \<infinity>"
by auto
let ?F = "\<lambda>k. \<Pi>\<^isub>E i\<in>I \<union> J. F i k"
show "IJ.\<mu> A = P.\<mu> (?X A)"
@@ -1589,12 +1644,13 @@
show "range ?F \<subseteq> sets IJ.G" using F
by (simp add: image_subset_iff product_algebra_def
product_algebra_generator_def)
- show "?F \<up> space IJ.G " using F(2) by simp
- show "\<And>k. IJ.\<mu> (?F k) \<noteq> \<omega>"
- using `finite I` F
- by (subst IJ.measure_times) (auto simp add: setprod_\<omega>)
- show "?g \<in> measurable P.P IJ.P"
- using IJ by (rule measurable_merge)
+ show "incseq ?F" "(\<Union>i. ?F i) = space IJ.G " by fact+
+ next
+ fix k
+ have "\<And>j. j \<in> I \<union> J \<Longrightarrow> 0 \<le> measure (M j) (F j k)"
+ using F(1) by auto
+ with F `finite I` setprod_PInf[of "I \<union> J", OF this]
+ show "IJ.\<mu> (?F k) \<noteq> \<infinity>" by (subst IJ.measure_times) auto
next
fix A assume "A \<in> sets IJ.G"
then obtain F where A: "A = Pi\<^isub>E (I \<union> J) F"
@@ -1611,7 +1667,7 @@
using `finite J` `finite I` F unfolding A
by (intro IJ.measure_times[symmetric]) auto
finally show "IJ.\<mu> A = P.\<mu> (?X A)" by (rule sym)
- qed
+ qed (rule measurable_merge[OF IJ])
qed
lemma (in product_sigma_finite) measure_preserving_merge:
@@ -1692,8 +1748,9 @@
qed
lemma (in product_sigma_finite) product_positive_integral_setprod:
- fixes f :: "'i \<Rightarrow> 'a \<Rightarrow> pextreal"
+ fixes f :: "'i \<Rightarrow> 'a \<Rightarrow> extreal"
assumes "finite I" and borel: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> borel_measurable (M i)"
+ and pos: "\<And>i x. i \<in> I \<Longrightarrow> 0 \<le> f i x"
shows "(\<integral>\<^isup>+ x. (\<Prod>i\<in>I. f i (x i)) \<partial>Pi\<^isub>M I M) = (\<Prod>i\<in>I. integral\<^isup>P (M i) (f i))"
using assms proof induct
case empty
@@ -1707,12 +1764,15 @@
using insert by (auto intro!: setprod_cong)
have prod: "\<And>J. J \<subseteq> insert i I \<Longrightarrow> (\<lambda>x. (\<Prod>i\<in>J. f i (x i))) \<in> borel_measurable (Pi\<^isub>M J M)"
using sets_into_space insert
- by (intro sigma_algebra.borel_measurable_pextreal_setprod sigma_algebra_product_algebra
+ by (intro sigma_algebra.borel_measurable_extreal_setprod sigma_algebra_product_algebra
measurable_comp[OF measurable_component_singleton, unfolded comp_def])
auto
- show ?case
- by (simp add: product_positive_integral_insert[OF insert(1,2) prod])
- (simp add: insert I.positive_integral_cmult M.positive_integral_multc * prod subset_insertI)
+ then show ?case
+ apply (simp add: product_positive_integral_insert[OF insert(1,2) prod])
+ apply (simp add: insert * pos borel setprod_extreal_pos M.positive_integral_multc)
+ apply (subst I.positive_integral_cmult)
+ apply (auto simp add: pos borel insert setprod_extreal_pos positive_integral_positive)
+ done
qed
lemma (in product_sigma_finite) product_integral_singleton:
@@ -1720,8 +1780,8 @@
shows "(\<integral>x. f (x i) \<partial>Pi\<^isub>M {i} M) = integral\<^isup>L (M i) f"
proof -
interpret I: finite_product_sigma_finite M "{i}" by default simp
- have *: "(\<lambda>x. Real (f x)) \<in> borel_measurable (M i)"
- "(\<lambda>x. Real (- f x)) \<in> borel_measurable (M i)"
+ have *: "(\<lambda>x. extreal (f x)) \<in> borel_measurable (M i)"
+ "(\<lambda>x. extreal (- f x)) \<in> borel_measurable (M i)"
using assms by auto
show ?thesis
unfolding lebesgue_integral_def *[THEN product_positive_integral_singleton] ..
@@ -1795,15 +1855,17 @@
proof (unfold integrable_def, intro conjI)
show "(\<lambda>x. abs (?f x)) \<in> borel_measurable P"
using borel by auto
- have "(\<integral>\<^isup>+x. Real (abs (?f x)) \<partial>P) = (\<integral>\<^isup>+x. (\<Prod>i\<in>I. Real (abs (f i (x i)))) \<partial>P)"
- by (simp add: Real_setprod abs_setprod)
- also have "\<dots> = (\<Prod>i\<in>I. (\<integral>\<^isup>+x. Real (abs (f i x)) \<partial>M i))"
+ have "(\<integral>\<^isup>+x. extreal (abs (?f x)) \<partial>P) = (\<integral>\<^isup>+x. (\<Prod>i\<in>I. extreal (abs (f i (x i)))) \<partial>P)"
+ by (simp add: setprod_extreal abs_setprod)
+ also have "\<dots> = (\<Prod>i\<in>I. (\<integral>\<^isup>+x. extreal (abs (f i x)) \<partial>M i))"
using f by (subst product_positive_integral_setprod) auto
- also have "\<dots> < \<omega>"
+ also have "\<dots> < \<infinity>"
using integrable[THEN M.integrable_abs]
- unfolding pextreal_less_\<omega> setprod_\<omega> integrable_def by simp
- finally show "(\<integral>\<^isup>+x. Real (abs (?f x)) \<partial>P) \<noteq> \<omega>" by auto
- show "(\<integral>\<^isup>+x. Real (- abs (?f x)) \<partial>P) \<noteq> \<omega>" by simp
+ by (simp add: setprod_PInf integrable_def M.positive_integral_positive)
+ finally show "(\<integral>\<^isup>+x. extreal (abs (?f x)) \<partial>P) \<noteq> \<infinity>" by auto
+ have "(\<integral>\<^isup>+x. extreal (- abs (?f x)) \<partial>P) = (\<integral>\<^isup>+x. 0 \<partial>P)"
+ by (intro positive_integral_cong_pos) auto
+ then show "(\<integral>\<^isup>+x. extreal (- abs (?f x)) \<partial>P) \<noteq> \<infinity>" by simp
qed
ultimately show ?thesis
by (rule integrable_abs_iff[THEN iffD1])
--- a/src/HOL/Probability/Radon_Nikodym.thy Mon Mar 14 15:17:10 2011 +0100
+++ b/src/HOL/Probability/Radon_Nikodym.thy Mon Mar 14 15:29:10 2011 +0100
@@ -2,87 +2,81 @@
imports Lebesgue_Integration
begin
-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 (in sigma_finite_measure) Ex_finite_integrable_function:
- shows "\<exists>h\<in>borel_measurable M. integral\<^isup>P M h \<noteq> \<omega> \<and> (\<forall>x\<in>space M. 0 < h x \<and> h x < \<omega>)"
+ shows "\<exists>h\<in>borel_measurable M. integral\<^isup>P M h \<noteq> \<infinity> \<and> (\<forall>x\<in>space M. 0 < h x \<and> h x < \<infinity>) \<and> (\<forall>x. 0 \<le> h x)"
proof -
obtain A :: "nat \<Rightarrow> 'a set" where
range: "range A \<subseteq> sets M" and
space: "(\<Union>i. A i) = space M" and
- measure: "\<And>i. \<mu> (A i) \<noteq> \<omega>" and
+ measure: "\<And>i. \<mu> (A i) \<noteq> \<infinity>" and
disjoint: "disjoint_family A"
using disjoint_sigma_finite by auto
let "?B i" = "2^Suc i * \<mu> (A i)"
have "\<forall>i. \<exists>x. 0 < x \<and> x < inverse (?B i)"
proof
- fix i show "\<exists>x. 0 < x \<and> x < inverse (?B i)"
- proof cases
- assume "\<mu> (A i) = 0"
- then show ?thesis by (auto intro!: exI[of _ 1])
- 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 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: pextreal_noteq_omega_Ex zero_le_mult_iff zero_less_mult_iff mult_le_0_iff power_le_zero_eq)
- qed
+ fix i have Ai: "A i \<in> sets M" using range by auto
+ from measure positive_measure[OF this]
+ show "\<exists>x. 0 < x \<and> x < inverse (?B i)"
+ by (auto intro!: extreal_dense simp: extreal_0_gt_inverse)
qed
from choice[OF this] obtain n where n: "\<And>i. 0 < n i"
"\<And>i. n i < inverse (2^Suc i * \<mu> (A i))" by auto
- let "?h x" = "\<Sum>\<^isub>\<infinity> i. n i * indicator (A i) x"
+ { fix i have "0 \<le> n i" using n(1)[of i] by auto } note pos = this
+ let "?h x" = "\<Sum>i. n i * indicator (A i) x"
show ?thesis
proof (safe intro!: bexI[of _ ?h] del: notI)
have "\<And>i. A i \<in> sets M"
using range by fastsimp+
- then have "integral\<^isup>P M ?h = (\<Sum>\<^isub>\<infinity> i. n i * \<mu> (A i))"
- by (simp add: positive_integral_psuminf positive_integral_cmult_indicator)
- also have "\<dots> \<le> (\<Sum>\<^isub>\<infinity> i. Real ((1 / 2)^Suc i))"
- proof (rule psuminf_le)
- fix N show "n N * \<mu> (A N) \<le> Real ((1 / 2) ^ Suc N)"
+ then have "integral\<^isup>P M ?h = (\<Sum>i. n i * \<mu> (A i))" using pos
+ by (simp add: positive_integral_suminf positive_integral_cmult_indicator)
+ also have "\<dots> \<le> (\<Sum>i. (1 / 2)^Suc i)"
+ proof (rule suminf_le_pos)
+ fix N
+ have "n N * \<mu> (A N) \<le> inverse (2^Suc N * \<mu> (A N)) * \<mu> (A N)"
+ using positive_measure[OF `A N \<in> sets M`] n[of N]
+ by (intro extreal_mult_right_mono) auto
+ also have "\<dots> \<le> (1 / 2) ^ Suc N"
using measure[of N] n[of N]
- by (cases "n N")
- (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)
+ by (cases rule: extreal2_cases[of "n N" "\<mu> (A N)"])
+ (simp_all add: inverse_eq_divide power_divide one_extreal_def extreal_power_divide)
+ finally show "n N * \<mu> (A N) \<le> (1 / 2) ^ Suc N" .
+ show "0 \<le> n N * \<mu> (A N)" using n[of N] `A N \<in> sets M` by simp
qed
- also have "\<dots> = Real 1"
- by (rule suminf_imp_psuminf, rule power_half_series, auto)
- finally show "integral\<^isup>P M ?h \<noteq> \<omega>" by auto
+ finally show "integral\<^isup>P M ?h \<noteq> \<infinity>" unfolding suminf_half_series_extreal by auto
next
- fix x assume "x \<in> space M"
- then obtain i where "x \<in> A i" using space[symmetric] by auto
- from psuminf_cmult_indicator[OF disjoint, OF this]
- have "?h x = n i" by simp
- then show "0 < ?h x" and "?h x < \<omega>" using n[of i] by auto
+ { fix x assume "x \<in> space M"
+ then obtain i where "x \<in> A i" using space[symmetric] by auto
+ with disjoint n have "?h x = n i"
+ by (auto intro!: suminf_cmult_indicator intro: less_imp_le)
+ then show "0 < ?h x" and "?h x < \<infinity>" using n[of i] by auto }
+ note pos = this
+ fix x show "0 \<le> ?h x"
+ proof cases
+ assume "x \<in> space M" then show "0 \<le> ?h x" using pos by (auto intro: less_imp_le)
+ next
+ assume "x \<notin> space M" then have "\<And>i. x \<notin> A i" using space by auto
+ then show "0 \<le> ?h x" by auto
+ qed
next
- show "?h \<in> borel_measurable M" using range
- by (auto intro!: borel_measurable_psuminf borel_measurable_pextreal_times)
+ show "?h \<in> borel_measurable M" using range n
+ by (auto intro!: borel_measurable_psuminf borel_measurable_extreal_times extreal_0_le_mult intro: less_imp_le)
qed
qed
subsection "Absolutely continuous"
definition (in measure_space)
- "absolutely_continuous \<nu> = (\<forall>N\<in>null_sets. \<nu> N = (0 :: pextreal))"
+ "absolutely_continuous \<nu> = (\<forall>N\<in>null_sets. \<nu> N = (0 :: extreal))"
lemma (in measure_space) absolutely_continuous_AE:
assumes "measure_space M'" and [simp]: "sets M' = sets M" "space M' = space M"
and "absolutely_continuous (measure M')" "AE x. P x"
- shows "measure_space.almost_everywhere M' P"
+ shows "AE x in M'. P x"
proof -
interpret \<nu>: measure_space M' by fact
from `AE x. P x` obtain N where N: "N \<in> null_sets" and "{x\<in>space M. \<not> P x} \<subseteq> N"
unfolding almost_everywhere_def by auto
- show "\<nu>.almost_everywhere P"
+ show "AE x in M'. P x"
proof (rule \<nu>.AE_I')
show "{x\<in>space M'. \<not> P x} \<subseteq> N" by simp fact
from `absolutely_continuous (measure M')` show "N \<in> \<nu>.null_sets"
@@ -99,7 +93,7 @@
interpret v: finite_measure_space ?\<nu> by fact
have "\<nu> N = measure ?\<nu> (\<Union>x\<in>N. {x})" by simp
also have "\<dots> = (\<Sum>x\<in>N. measure ?\<nu> {x})"
- proof (rule v.measure_finitely_additive''[symmetric])
+ proof (rule v.measure_setsum[symmetric])
show "finite N" using `N \<subseteq> space M` finite_space by (auto intro: finite_subset)
show "disjoint_family_on (\<lambda>i. {i}) N" unfolding disjoint_family_on_def by auto
fix x assume "x \<in> N" thus "{x} \<in> sets ?\<nu>" using `N \<subseteq> space M` sets_eq_Pow by auto
@@ -107,8 +101,10 @@
also have "\<dots> = 0"
proof (safe intro!: setsum_0')
fix x assume "x \<in> N"
- hence "\<mu> {x} \<le> \<mu> N" using sets_eq_Pow `N \<subseteq> space M` by (auto intro!: measure_mono)
- hence "\<mu> {x} = 0" using `\<mu> N = 0` by simp
+ hence "\<mu> {x} \<le> \<mu> N" "0 \<le> \<mu> {x}"
+ using sets_eq_Pow `N \<subseteq> space M` positive_measure[of "{x}"]
+ by (auto intro!: measure_mono)
+ then have "\<mu> {x} = 0" using `\<mu> N = 0` by simp
thus "measure ?\<nu> {x} = 0" using v[of x] `x \<in> N` `N \<subseteq> space M` by auto
qed
finally show "\<nu> N = 0" by simp
@@ -125,12 +121,12 @@
lemma (in finite_measure) Radon_Nikodym_aux_epsilon:
fixes e :: real assumes "0 < e"
assumes "finite_measure (M\<lparr>measure := \<nu>\<rparr>)"
- shows "\<exists>A\<in>sets M. real (\<mu> (space M)) - real (\<nu> (space M)) \<le>
- real (\<mu> A) - real (\<nu> A) \<and>
- (\<forall>B\<in>sets M. B \<subseteq> A \<longrightarrow> - e < real (\<mu> B) - real (\<nu> B))"
+ shows "\<exists>A\<in>sets M. \<mu>' (space M) - finite_measure.\<mu>' (M\<lparr>measure := \<nu>\<rparr>) (space M) \<le>
+ \<mu>' A - finite_measure.\<mu>' (M\<lparr>measure := \<nu>\<rparr>) A \<and>
+ (\<forall>B\<in>sets M. B \<subseteq> A \<longrightarrow> - e < \<mu>' B - finite_measure.\<mu>' (M\<lparr>measure := \<nu>\<rparr>) B)"
proof -
- let "?d A" = "real (\<mu> A) - real (\<nu> A)"
interpret M': finite_measure "M\<lparr>measure := \<nu>\<rparr>" by fact
+ let "?d A" = "\<mu>' A - M'.\<mu>' A"
let "?A A" = "if (\<forall>B\<in>sets M. B \<subseteq> space M - A \<longrightarrow> -e < ?d B)
then {}
else (SOME B. B \<in> sets M \<and> B \<subseteq> space M - A \<and> ?d B \<le> -e)"
@@ -157,7 +153,7 @@
fix B assume "B \<in> sets M \<and> B \<subseteq> space M - A n \<and> ?d B \<le> - e"
hence "A n \<inter> B = {}" "B \<in> sets M" and dB: "?d B \<le> -e" by auto
hence "?d (A n \<union> B) = ?d (A n) + ?d B"
- using `A n \<in> sets M` real_finite_measure_Union M'.real_finite_measure_Union by simp
+ using `A n \<in> sets M` finite_measure_Union M'.finite_measure_Union by simp
also have "\<dots> \<le> ?d (A n) - e" using dB by simp
finally show "?d (A n \<union> B) \<le> ?d (A n) - e" .
qed }
@@ -186,11 +182,7 @@
fix n assume "?d (space M) \<le> ?d (space M - A n)"
also have "\<dots> \<le> ?d (space M - A (Suc n))"
using A_in_sets sets_into_space dA_mono[of n]
- real_finite_measure_Diff[of "space M"]
- real_finite_measure_Diff[of "space M"]
- M'.real_finite_measure_Diff[of "space M"]
- M'.real_finite_measure_Diff[of "space M"]
- by (simp del: A_simps)
+ by (simp del: A_simps add: finite_measure_Diff M'.finite_measure_Diff)
finally show "?d (space M) \<le> ?d (space M - A (Suc n))" .
qed simp
qed
@@ -200,13 +192,16 @@
{ fix n have "?d (A n) \<le> - real n * e"
proof (induct n)
case (Suc n) with dA_epsilon[of n, OF B] show ?case by (simp del: A_simps add: real_of_nat_Suc field_simps)
- qed simp } note dA_less = this
+ next
+ case 0 with M'.empty_measure show ?case by (simp add: zero_extreal_def)
+ qed } note dA_less = this
have decseq: "decseq (\<lambda>n. ?d (A n))" unfolding decseq_eq_incseq
proof (rule incseq_SucI)
fix n show "- ?d (A n) \<le> - ?d (A (Suc n))" using dA_mono[of n] by auto
qed
- from real_finite_continuity_from_below[of A] `range A \<subseteq> sets M`
- M'.real_finite_continuity_from_below[of A]
+ have A: "incseq A" by (auto intro!: incseq_SucI)
+ from finite_continuity_from_below[OF _ A] `range A \<subseteq> sets M`
+ M'.finite_continuity_from_below[OF _ A]
have convergent: "(\<lambda>i. ?d (A i)) ----> ?d (\<Union>i. A i)"
by (auto intro!: LIMSEQ_diff)
obtain n :: nat where "- ?d (\<Union>i. A i) / e < real n" using reals_Archimedean2 by auto
@@ -216,33 +211,55 @@
qed
qed
+lemma (in finite_measure) restricted_measure_subset:
+ assumes S: "S \<in> sets M" and X: "X \<subseteq> S"
+ shows "finite_measure.\<mu>' (restricted_space S) X = \<mu>' X"
+proof cases
+ note r = restricted_finite_measure[OF S]
+ { assume "X \<in> sets M" with S X show ?thesis
+ unfolding finite_measure.\<mu>'_def[OF r] \<mu>'_def by auto }
+ { assume "X \<notin> sets M"
+ moreover then have "S \<inter> X \<notin> sets M"
+ using X by (simp add: Int_absorb1)
+ ultimately show ?thesis
+ unfolding finite_measure.\<mu>'_def[OF r] \<mu>'_def using S by auto }
+qed
+
+lemma (in finite_measure) restricted_measure:
+ assumes X: "S \<in> sets M" "X \<in> sets (restricted_space S)"
+ shows "finite_measure.\<mu>' (restricted_space S) X = \<mu>' X"
+proof -
+ from X have "S \<in> sets M" "X \<subseteq> S" by auto
+ from restricted_measure_subset[OF this] show ?thesis .
+qed
+
lemma (in finite_measure) Radon_Nikodym_aux:
assumes "finite_measure (M\<lparr>measure := \<nu>\<rparr>)" (is "finite_measure ?M'")
- shows "\<exists>A\<in>sets M. real (\<mu> (space M)) - real (\<nu> (space M)) \<le>
- real (\<mu> A) - real (\<nu> A) \<and>
- (\<forall>B\<in>sets M. B \<subseteq> A \<longrightarrow> 0 \<le> real (\<mu> B) - real (\<nu> B))"
+ shows "\<exists>A\<in>sets M. \<mu>' (space M) - finite_measure.\<mu>' (M\<lparr>measure := \<nu>\<rparr>) (space M) \<le>
+ \<mu>' A - finite_measure.\<mu>' (M\<lparr>measure := \<nu>\<rparr>) A \<and>
+ (\<forall>B\<in>sets M. B \<subseteq> A \<longrightarrow> 0 \<le> \<mu>' B - finite_measure.\<mu>' (M\<lparr>measure := \<nu>\<rparr>) B)"
proof -
- let "?d A" = "real (\<mu> A) - real (\<nu> A)"
- let "?P A B n" = "A \<in> sets M \<and> A \<subseteq> B \<and> ?d B \<le> ?d A \<and> (\<forall>C\<in>sets M. C \<subseteq> A \<longrightarrow> - 1 / real (Suc n) < ?d C)"
interpret M': finite_measure ?M' where
"space ?M' = space M" and "sets ?M' = sets M" and "measure ?M' = \<nu>" by fact auto
+ let "?d A" = "\<mu>' A - M'.\<mu>' A"
+ let "?P A B n" = "A \<in> sets M \<and> A \<subseteq> B \<and> ?d B \<le> ?d A \<and> (\<forall>C\<in>sets M. C \<subseteq> A \<longrightarrow> - 1 / real (Suc n) < ?d C)"
let "?r S" = "restricted_space S"
- { fix S n
- assume S: "S \<in> sets M"
- hence **: "\<And>X. X \<in> op \<inter> S ` sets M \<longleftrightarrow> X \<in> sets M \<and> X \<subseteq> S" by auto
- have [simp]: "(restricted_space S\<lparr>measure := \<nu>\<rparr>) = M'.restricted_space S"
- by (cases M) simp
- from M'.restricted_finite_measure[of S] restricted_finite_measure[OF S] S
- have "finite_measure (?r S)" "0 < 1 / real (Suc n)"
+ { fix S n assume S: "S \<in> sets M"
+ note r = M'.restricted_finite_measure[of S] restricted_finite_measure[OF S] S
+ then have "finite_measure (?r S)" "0 < 1 / real (Suc n)"
"finite_measure (?r S\<lparr>measure := \<nu>\<rparr>)" by auto
- from finite_measure.Radon_Nikodym_aux_epsilon[OF this] guess X ..
- hence "?P X S n"
- proof (simp add: **, safe)
- fix C assume C: "C \<in> sets M" "C \<subseteq> X" "X \<subseteq> S" and
- *: "\<forall>B\<in>sets M. S \<inter> B \<subseteq> X \<longrightarrow> - (1 / real (Suc n)) < ?d (S \<inter> B)"
- hence "C \<subseteq> S" "C \<subseteq> X" "S \<inter> C = C" by auto
- with *[THEN bspec, OF `C \<in> sets M`]
- show "- (1 / real (Suc n)) < ?d C" by auto
+ from finite_measure.Radon_Nikodym_aux_epsilon[OF this] guess X .. note X = this
+ have "?P X S n"
+ proof (intro conjI ballI impI)
+ show "X \<in> sets M" "X \<subseteq> S" using X(1) `S \<in> sets M` by auto
+ have "S \<in> op \<inter> S ` sets M" using `S \<in> sets M` by auto
+ then show "?d S \<le> ?d X"
+ using S X restricted_measure[OF S] M'.restricted_measure[OF S] by simp
+ fix C assume "C \<in> sets M" "C \<subseteq> X"
+ then have "C \<in> sets (restricted_space S)" "C \<subseteq> X"
+ using `S \<in> sets M` `X \<subseteq> S` by auto
+ with X(2) show "- 1 / real (Suc n) < ?d C"
+ using S X restricted_measure[OF S] M'.restricted_measure[OF S] by auto
qed
hence "\<exists>A. ?P A S n" by auto }
note Ex_P = this
@@ -268,10 +285,11 @@
show ?thesis
proof (safe intro!: bexI[of _ "\<Inter>i. A i"])
show "(\<Inter>i. A i) \<in> sets M" using A_in_sets by auto
- from `range A \<subseteq> sets M` A_mono
- real_finite_continuity_from_above[of A]
- M'.real_finite_continuity_from_above[of A]
- have "(\<lambda>i. ?d (A i)) ----> ?d (\<Inter>i. A i)" by (auto intro!: LIMSEQ_diff)
+ have A: "decseq A" using A_mono by (auto intro!: decseq_SucI)
+ from
+ finite_continuity_from_above[OF `range A \<subseteq> sets M` A]
+ M'.finite_continuity_from_above[OF `range A \<subseteq> sets M` A]
+ have "(\<lambda>i. ?d (A i)) ----> ?d (\<Inter>i. A i)" by (intro LIMSEQ_diff)
thus "?d (space M) \<le> ?d (\<Inter>i. A i)" using mono_dA[THEN monoD, of 0 _]
by (rule_tac LIMSEQ_le_const) (auto intro!: exI)
next
@@ -290,6 +308,10 @@
qed
qed
+lemma (in finite_measure) real_measure:
+ assumes A: "A \<in> sets M" shows "\<exists>r. 0 \<le> r \<and> \<mu> A = extreal r"
+ using finite_measure[OF A] positive_measure[OF A] by (cases "\<mu> A") auto
+
lemma (in finite_measure) Radon_Nikodym_finite_measure:
assumes "finite_measure (M\<lparr> measure := \<nu>\<rparr>)" (is "finite_measure ?M'")
assumes "absolutely_continuous \<nu>"
@@ -298,7 +320,7 @@
interpret M': finite_measure ?M'
where "space ?M' = space M" and "sets ?M' = sets M" and "measure ?M' = \<nu>"
using assms(1) by auto
- def G \<equiv> "{g \<in> borel_measurable M. \<forall>A\<in>sets M. (\<integral>\<^isup>+x. g x * indicator A x \<partial>M) \<le> \<nu> A}"
+ def G \<equiv> "{g \<in> borel_measurable M. (\<forall>x. 0 \<le> g x) \<and> (\<forall>A\<in>sets M. (\<integral>\<^isup>+x. g x * indicator A x \<partial>M) \<le> \<nu> A)}"
have "(\<lambda>x. 0) \<in> G" unfolding G_def by auto
hence "G \<noteq> {}" by auto
{ fix f g assume f: "f \<in> G" and g: "g \<in> G"
@@ -324,24 +346,28 @@
also have "\<dots> = \<nu> A"
using M'.measure_additive[OF sets] union by auto
finally show "(\<integral>\<^isup>+x. max (g x) (f x) * indicator A x \<partial>M) \<le> \<nu> A" .
+ next
+ fix x show "0 \<le> max (g x) (f x)" using f g by (auto simp: G_def split: split_max)
qed }
note max_in_G = this
- { fix f g assume "f \<up> g" and f: "\<And>i. f i \<in> G"
- have "g \<in> G" unfolding G_def
+ { fix f assume "incseq f" and f: "\<And>i. f i \<in> G"
+ have "(\<lambda>x. SUP i. f i x) \<in> G" unfolding G_def
proof safe
- from `f \<up> g` have [simp]: "g = (\<lambda>x. SUP i. f i x)"
- unfolding isoton_def fun_eq_iff SUPR_apply by simp
- have f_borel: "\<And>i. f i \<in> borel_measurable M" using f unfolding G_def by simp
- thus "g \<in> borel_measurable M" by auto
+ show "(\<lambda>x. SUP i. f i x) \<in> borel_measurable M"
+ using f by (auto simp: G_def)
+ { fix x show "0 \<le> (SUP i. f i x)"
+ using f by (auto simp: G_def intro: le_SUPI2) }
+ next
fix A assume "A \<in> sets M"
- hence "\<And>i. (\<lambda>x. f i x * indicator A x) \<in> borel_measurable M"
- using f_borel by (auto intro!: borel_measurable_indicator)
- from positive_integral_isoton[OF isoton_indicator[OF `f \<up> g`] this]
- have SUP: "(\<integral>\<^isup>+x. g x * indicator A x \<partial>M) =
- (SUP i. (\<integral>\<^isup>+x. f i x * indicator A x \<partial>M))"
- unfolding isoton_def by simp
- show "(\<integral>\<^isup>+x. g x * indicator A x \<partial>M) \<le> \<nu> A" unfolding SUP
- using f `A \<in> sets M` unfolding G_def by (auto intro!: SUP_leI)
+ have "(\<integral>\<^isup>+x. (SUP i. f i x) * indicator A x \<partial>M) =
+ (\<integral>\<^isup>+x. (SUP i. f i x * indicator A x) \<partial>M)"
+ by (intro positive_integral_cong) (simp split: split_indicator)
+ also have "\<dots> = (SUP i. (\<integral>\<^isup>+x. f i x * indicator A x \<partial>M))"
+ using `incseq f` f `A \<in> sets M`
+ by (intro positive_integral_monotone_convergence_SUP)
+ (auto simp: G_def incseq_Suc_iff le_fun_def split: split_indicator)
+ finally show "(\<integral>\<^isup>+x. (SUP i. f i x) * indicator A x \<partial>M) \<le> \<nu> A"
+ using f `A \<in> sets M` by (auto intro!: SUP_leI simp: G_def)
qed }
note SUP_in_G = this
let ?y = "SUP g : G. integral\<^isup>P M g"
@@ -351,9 +377,8 @@
from this[THEN bspec, OF top] show "integral\<^isup>P M g \<le> \<nu> (space M)"
by (simp cong: positive_integral_cong)
qed
- hence "?y \<noteq> \<omega>" using M'.finite_measure_of_space by auto
- from SUPR_countable_SUPR[OF this `G \<noteq> {}`] guess ys .. note ys = this
- hence "\<forall>n. \<exists>g. g\<in>G \<and> integral\<^isup>P M g = ys n"
+ from SUPR_countable_SUPR[OF `G \<noteq> {}`, of "integral\<^isup>P M"] guess ys .. note ys = this
+ then have "\<forall>n. \<exists>g. g\<in>G \<and> integral\<^isup>P M g = ys n"
proof safe
fix n assume "range ys \<subseteq> integral\<^isup>P M ` G"
hence "ys n \<in> integral\<^isup>P M ` G" by auto
@@ -362,8 +387,9 @@
from choice[OF this] obtain gs where "\<And>i. gs i \<in> G" "\<And>n. integral\<^isup>P M (gs n) = ys n" by auto
hence y_eq: "?y = (SUP i. integral\<^isup>P M (gs i))" using ys by auto
let "?g i x" = "Max ((\<lambda>n. gs n x) ` {..i})"
- def f \<equiv> "SUP i. ?g i"
- have gs_not_empty: "\<And>i. (\<lambda>n. gs n x) ` {..i} \<noteq> {}" by auto
+ def f \<equiv> "\<lambda>x. SUP i. ?g i x"
+ let "?F A x" = "f x * indicator A x"
+ have gs_not_empty: "\<And>i x. (\<lambda>n. gs n x) ` {..i} \<noteq> {}" by auto
{ fix i have "?g i \<in> G"
proof (induct i)
case 0 thus ?case by simp fact
@@ -373,15 +399,13 @@
by (auto simp add: atMost_Suc intro!: max_in_G)
qed }
note g_in_G = this
- have "\<And>x. \<forall>i. ?g i x \<le> ?g (Suc i) x"
- using gs_not_empty by (simp add: atMost_Suc)
- hence isoton_g: "?g \<up> f" by (simp add: isoton_def le_fun_def f_def)
- from SUP_in_G[OF this g_in_G] have "f \<in> G" .
- hence [simp, intro]: "f \<in> borel_measurable M" unfolding G_def by auto
- have "(\<lambda>i. integral\<^isup>P M (?g i)) \<up> integral\<^isup>P M f"
- using isoton_g g_in_G by (auto intro!: positive_integral_isoton simp: G_def f_def)
- hence "integral\<^isup>P M f = (SUP i. integral\<^isup>P M (?g i))"
- unfolding isoton_def by simp
+ have "incseq ?g" using gs_not_empty
+ by (auto intro!: incseq_SucI le_funI simp add: atMost_Suc)
+ from SUP_in_G[OF this g_in_G] have "f \<in> G" unfolding f_def .
+ then have [simp, intro]: "f \<in> borel_measurable M" unfolding G_def by auto
+ have "integral\<^isup>P M f = (SUP i. integral\<^isup>P M (?g i))" unfolding f_def
+ using g_in_G `incseq ?g`
+ by (auto intro!: positive_integral_monotone_convergence_SUP simp: G_def)
also have "\<dots> = ?y"
proof (rule antisym)
show "(SUP i. integral\<^isup>P M (?g i)) \<le> ?y"
@@ -390,42 +414,57 @@
by (auto intro!: SUP_mono positive_integral_mono Max_ge)
qed
finally have int_f_eq_y: "integral\<^isup>P M f = ?y" .
- let "?t A" = "\<nu> A - (\<integral>\<^isup>+x. f x * indicator A x \<partial>M)"
+ have "\<And>x. 0 \<le> f x"
+ unfolding f_def using `\<And>i. gs i \<in> G`
+ by (auto intro!: le_SUPI2 Max_ge_iff[THEN iffD2] simp: G_def)
+ let "?t A" = "\<nu> A - (\<integral>\<^isup>+x. ?F A x \<partial>M)"
let ?M = "M\<lparr> measure := ?t\<rparr>"
interpret M: sigma_algebra ?M
by (intro sigma_algebra_cong) auto
+ have f_le_\<nu>: "\<And>A. A \<in> sets M \<Longrightarrow> (\<integral>\<^isup>+x. ?F A x \<partial>M) \<le> \<nu> A"
+ using `f \<in> G` unfolding G_def by auto
have fmM: "finite_measure ?M"
- proof (default, simp_all add: countably_additive_def, safe)
+ proof (default, simp_all add: countably_additive_def positive_def, safe del: notI)
fix A :: "nat \<Rightarrow> 'a set" assume A: "range A \<subseteq> sets M" "disjoint_family A"
- have "(\<Sum>\<^isub>\<infinity> n. (\<integral>\<^isup>+x. f x * indicator (A n) x \<partial>M))
- = (\<integral>\<^isup>+x. (\<Sum>\<^isub>\<infinity>n. f x * indicator (A n) x) \<partial>M)"
- using `range A \<subseteq> sets M`
- by (rule_tac positive_integral_psuminf[symmetric]) (auto intro!: borel_measurable_indicator)
- also have "\<dots> = (\<integral>\<^isup>+x. f x * indicator (\<Union>n. A n) x \<partial>M)"
- apply (rule positive_integral_cong)
- apply (subst psuminf_cmult_right)
- unfolding psuminf_indicator[OF `disjoint_family A`] ..
- finally have "(\<Sum>\<^isub>\<infinity> n. (\<integral>\<^isup>+x. f x * indicator (A n) x \<partial>M))
- = (\<integral>\<^isup>+x. f x * indicator (\<Union>n. A n) x \<partial>M)" .
- moreover have "(\<Sum>\<^isub>\<infinity>n. \<nu> (A n)) = \<nu> (\<Union>n. A n)"
+ have "(\<Sum>n. (\<integral>\<^isup>+x. ?F (A n) x \<partial>M)) = (\<integral>\<^isup>+x. (\<Sum>n. ?F (A n) x) \<partial>M)"
+ using `range A \<subseteq> sets M` `\<And>x. 0 \<le> f x`
+ by (intro positive_integral_suminf[symmetric]) auto
+ also have "\<dots> = (\<integral>\<^isup>+x. ?F (\<Union>n. A n) x \<partial>M)"
+ using `\<And>x. 0 \<le> f x`
+ by (intro positive_integral_cong) (simp add: suminf_cmult_extreal suminf_indicator[OF `disjoint_family A`])
+ finally have "(\<Sum>n. (\<integral>\<^isup>+x. ?F (A n) x \<partial>M)) = (\<integral>\<^isup>+x. ?F (\<Union>n. A n) x \<partial>M)" .
+ moreover have "(\<Sum>n. \<nu> (A n)) = \<nu> (\<Union>n. A n)"
using M'.measure_countably_additive A by (simp add: comp_def)
- moreover have "\<And>i. (\<integral>\<^isup>+x. f x * indicator (A i) x \<partial>M) \<le> \<nu> (A i)"
- using A `f \<in> G` unfolding G_def by auto
- moreover have v_fin: "\<nu> (\<Union>i. A i) \<noteq> \<omega>" using M'.finite_measure A by (simp add: countable_UN)
+ moreover have v_fin: "\<nu> (\<Union>i. A i) \<noteq> \<infinity>" using M'.finite_measure A by (simp add: countable_UN)
moreover {
- have "(\<integral>\<^isup>+x. f x * indicator (\<Union>i. A i) x \<partial>M) \<le> \<nu> (\<Union>i. A i)"
+ have "(\<integral>\<^isup>+x. ?F (\<Union>i. A i) x \<partial>M) \<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: pextreal_less_\<omega>)
- finally have "(\<integral>\<^isup>+x. f x * indicator (\<Union>i. A i) x \<partial>M) \<noteq> \<omega>"
- by (simp add: pextreal_less_\<omega>) }
+ also have "\<nu> (\<Union>i. A i) < \<infinity>" using v_fin by simp
+ finally have "(\<integral>\<^isup>+x. ?F (\<Union>i. A i) x \<partial>M) \<noteq> \<infinity>" by simp }
+ moreover have "\<And>i. (\<integral>\<^isup>+x. ?F (A i) x \<partial>M) \<le> \<nu> (A i)"
+ using A by (intro f_le_\<nu>) auto
ultimately
- show "(\<Sum>\<^isub>\<infinity> n. ?t (A n)) = ?t (\<Union>i. A i)"
- apply (subst psuminf_minus) by simp_all
+ show "(\<Sum>n. ?t (A n)) = ?t (\<Union>i. A i)"
+ by (subst suminf_extreal_minus) (simp_all add: positive_integral_positive)
+ next
+ fix A assume A: "A \<in> sets M" show "0 \<le> \<nu> A - \<integral>\<^isup>+ x. ?F A x \<partial>M"
+ using f_le_\<nu>[OF A] `f \<in> G` M'.finite_measure[OF A] by (auto simp: G_def extreal_le_minus_iff)
+ next
+ show "\<nu> (space M) - (\<integral>\<^isup>+ x. ?F (space M) x \<partial>M) \<noteq> \<infinity>" (is "?a - ?b \<noteq> _")
+ using positive_integral_positive[of "?F (space M)"]
+ by (cases rule: extreal2_cases[of ?a ?b]) auto
qed
then interpret M: finite_measure ?M
where "space ?M = space M" and "sets ?M = sets M" and "measure ?M = ?t"
by (simp_all add: fmM)
- have ac: "absolutely_continuous ?t" using `absolutely_continuous \<nu>` unfolding absolutely_continuous_def by auto
+ have ac: "absolutely_continuous ?t" unfolding absolutely_continuous_def
+ proof
+ fix N assume N: "N \<in> null_sets"
+ with `absolutely_continuous \<nu>` have "\<nu> N = 0" unfolding absolutely_continuous_def by auto
+ moreover with N have "(\<integral>\<^isup>+ x. ?F N x \<partial>M) \<le> \<nu> N" using `f \<in> G` by (auto simp: G_def)
+ ultimately show "\<nu> N - (\<integral>\<^isup>+ x. ?F N x \<partial>M) = 0"
+ using positive_integral_positive by (auto intro!: antisym)
+ qed
have upper_bound: "\<forall>A\<in>sets M. ?t A \<le> 0"
proof (rule ccontr)
assume "\<not> ?thesis"
@@ -436,43 +475,54 @@
using M.measure_mono[of A "space M"] A sets_into_space by simp
finally have pos_t: "0 < ?t (space M)" by simp
moreover
- hence pos_M: "0 < \<mu> (space M)"
- using ac top unfolding absolutely_continuous_def by auto
+ then have "\<mu> (space M) \<noteq> 0"
+ using ac unfolding absolutely_continuous_def by auto
+ then have pos_M: "0 < \<mu> (space M)"
+ using positive_measure[OF top] by (simp add: le_less)
moreover
have "(\<integral>\<^isup>+x. f x * indicator (space M) x \<partial>M) \<le> \<nu> (space M)"
using `f \<in> G` unfolding G_def by auto
- hence "(\<integral>\<^isup>+x. f x * indicator (space M) x \<partial>M) \<noteq> \<omega>"
+ hence "(\<integral>\<^isup>+x. f x * indicator (space M) x \<partial>M) \<noteq> \<infinity>"
using M'.finite_measure_of_space by auto
moreover
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: pextreal_inverse_eq_0 finite_measure_of_space)
+ ultimately have b: "b \<noteq> 0 \<and> 0 \<le> b \<and> b \<noteq> \<infinity>"
+ using M'.finite_measure_of_space positive_integral_positive[of "?F (space M)"]
+ by (cases rule: extreal3_cases[of "integral\<^isup>P M (?F (space M))" "\<nu> (space M)" "\<mu> (space M)"])
+ (simp_all add: field_simps)
+ then have b: "b \<noteq> 0" "0 \<le> b" "0 < b" "b \<noteq> \<infinity>" by auto
let ?Mb = "?M\<lparr>measure := \<lambda>A. b * \<mu> A\<rparr>"
interpret b: sigma_algebra ?Mb by (intro sigma_algebra_cong) auto
- have "finite_measure ?Mb"
- by default
- (insert finite_measure_of_space b measure_countably_additive,
- auto simp: psuminf_cmult_right countably_additive_def)
+ have Mb: "finite_measure ?Mb"
+ proof
+ show "positive ?Mb (measure ?Mb)"
+ using `0 \<le> b` by (auto simp: positive_def)
+ show "countably_additive ?Mb (measure ?Mb)"
+ using `0 \<le> b` measure_countably_additive
+ by (auto simp: countably_additive_def suminf_cmult_extreal subset_eq)
+ show "measure ?Mb (space ?Mb) \<noteq> \<infinity>"
+ using b by auto
+ qed
from M.Radon_Nikodym_aux[OF this]
obtain A0 where "A0 \<in> sets M" and
space_less_A0: "real (?t (space M)) - real (b * \<mu> (space M)) \<le> real (?t A0) - real (b * \<mu> A0)" and
- *: "\<And>B. \<lbrakk> B \<in> sets M ; B \<subseteq> A0 \<rbrakk> \<Longrightarrow> 0 \<le> real (?t B) - real (b * \<mu> B)" by auto
- { fix B assume "B \<in> sets M" "B \<subseteq> A0"
+ *: "\<And>B. \<lbrakk> B \<in> sets M ; B \<subseteq> A0 \<rbrakk> \<Longrightarrow> 0 \<le> real (?t B) - real (b * \<mu> B)"
+ unfolding M.\<mu>'_def finite_measure.\<mu>'_def[OF Mb] by auto
+ { fix B assume B: "B \<in> sets M" "B \<subseteq> A0"
with *[OF this] have "b * \<mu> B \<le> ?t B"
- using M'.finite_measure b finite_measure
- by (cases "b * \<mu> B", cases "?t B") (auto simp: field_simps) }
+ using M'.finite_measure b finite_measure M.positive_measure[OF B(1)]
+ by (cases rule: extreal2_cases[of "?t B" "b * \<mu> B"]) auto }
note bM_le_t = this
let "?f0 x" = "f x + b * indicator A0 x"
{ fix A assume A: "A \<in> sets M"
hence "A \<inter> A0 \<in> sets M" using `A0 \<in> sets M` by auto
have "(\<integral>\<^isup>+x. ?f0 x * indicator A x \<partial>M) =
(\<integral>\<^isup>+x. f x * indicator A x + b * indicator (A \<inter> A0) x \<partial>M)"
- by (auto intro!: positive_integral_cong simp: field_simps indicator_inter_arith)
+ by (auto intro!: positive_integral_cong split: split_indicator)
hence "(\<integral>\<^isup>+x. ?f0 x * indicator A x \<partial>M) =
(\<integral>\<^isup>+x. f x * indicator A x \<partial>M) + b * \<mu> (A \<inter> A0)"
- using `A0 \<in> sets M` `A \<inter> A0 \<in> sets M` A
- by (simp add: borel_measurable_indicator positive_integral_add positive_integral_cmult_indicator) }
+ using `A0 \<in> sets M` `A \<inter> A0 \<in> sets M` A b `f \<in> G`
+ by (simp add: G_def positive_integral_add positive_integral_cmult_indicator) }
note f0_eq = this
{ fix A assume A: "A \<in> sets M"
hence "A \<inter> A0 \<in> sets M" using `A0 \<in> sets M` by auto
@@ -487,39 +537,57 @@
using M.measure_mono[simplified, OF _ `A \<inter> A0 \<in> sets M` `A \<in> sets M`]
by (auto intro!: add_left_mono)
also have "\<dots> \<le> \<nu> A"
- using f_le_v M'.finite_measure[simplified, OF `A \<in> sets M`]
- by (cases "(\<integral>\<^isup>+x. f x * indicator A x \<partial>M)", cases "\<nu> A", auto)
+ using f_le_v M'.finite_measure[simplified, OF `A \<in> sets M`] positive_integral_positive[of "?F A"]
+ by (cases "\<integral>\<^isup>+x. ?F A x \<partial>M", cases "\<nu> A") auto
finally have "(\<integral>\<^isup>+x. ?f0 x * indicator A x \<partial>M) \<le> \<nu> A" . }
- hence "?f0 \<in> G" using `A0 \<in> sets M` unfolding G_def
- 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>"
+ hence "?f0 \<in> G" using `A0 \<in> sets M` b `f \<in> G` unfolding G_def
+ by (auto intro!: borel_measurable_indicator borel_measurable_extreal_add
+ borel_measurable_extreal_times extreal_add_nonneg_nonneg)
+ have real: "?t (space M) \<noteq> \<infinity>" "?t A0 \<noteq> \<infinity>"
+ "b * \<mu> (space M) \<noteq> \<infinity>" "b * \<mu> A0 \<noteq> \<infinity>"
using `A0 \<in> sets M` b
finite_measure[of A0] M.finite_measure[of A0]
finite_measure_of_space M.finite_measure_of_space
by auto
- have int_f_finite: "integral\<^isup>P M f \<noteq> \<omega>"
- using M'.finite_measure_of_space pos_t unfolding pextreal_zero_less_diff_iff
+ have int_f_finite: "integral\<^isup>P M f \<noteq> \<infinity>"
+ using M'.finite_measure_of_space pos_t unfolding extreal_less_minus_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 pextreal_mult_inverse)
+ have "0 < ?t (space M) - b * \<mu> (space M)" unfolding b_def
using finite_measure_of_space M'.finite_measure_of_space pos_t pos_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: pextreal_zero_less_diff_iff)
+ using positive_integral_positive[of "?F (space M)"]
+ by (cases rule: extreal3_cases[of "\<mu> (space M)" "\<nu> (space M)" "integral\<^isup>P M (?F (space M))"])
+ (auto simp: field_simps mult_less_cancel_left)
also have "\<dots> \<le> ?t A0 - b * \<mu> A0"
- 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 space_less_A0 b
+ using
+ `A0 \<in> sets M`[THEN M.real_measure]
+ top[THEN M.real_measure]
+ apply safe
+ apply simp
+ using
+ `A0 \<in> sets M`[THEN real_measure]
+ `A0 \<in> sets M`[THEN M'.real_measure]
+ top[THEN real_measure]
+ top[THEN M'.real_measure]
+ by (cases b) auto
+ finally have 1: "b * \<mu> A0 < ?t A0"
+ using
+ `A0 \<in> sets M`[THEN M.real_measure]
+ apply safe
+ apply simp
+ using
+ `A0 \<in> sets M`[THEN real_measure]
+ `A0 \<in> sets M`[THEN M'.real_measure]
+ by (cases b) auto
+ have "0 < ?t A0"
+ using b `A0 \<in> sets M` by (auto intro!: le_less_trans[OF _ 1])
+ then have "\<mu> A0 \<noteq> 0" 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 < integral\<^isup>P M f + b * \<mu> A0" unfolding int_f_eq_y
- by (rule pextreal_less_add)
+ then have "0 < \<mu> A0" using positive_measure[OF `A0 \<in> sets M`] by auto
+ hence "0 < b * \<mu> A0" using b by (auto simp: extreal_zero_less_0_iff)
+ with int_f_finite have "?y + 0 < integral\<^isup>P M f + b * \<mu> A0" unfolding int_f_eq_y
+ using `f \<in> G`
+ by (intro extreal_add_strict_mono) (auto intro!: le_SUPI2 positive_integral_positive)
also have "\<dots> = integral\<^isup>P M ?f0" using f0_eq[OF top] `A0 \<in> sets M` sets_into_space
by (simp cong: positive_integral_cong)
finally have "?y < integral\<^isup>P M ?f0" by simp
@@ -528,14 +596,15 @@
qed
show ?thesis
proof (safe intro!: bexI[of _ f])
- fix A assume "A\<in>sets M"
+ fix A assume A: "A\<in>sets M"
show "\<nu> A = (\<integral>\<^isup>+x. f x * indicator A x \<partial>M)"
proof (rule antisym)
show "(\<integral>\<^isup>+x. f x * indicator A x \<partial>M) \<le> \<nu> A"
using `f \<in> G` `A \<in> sets M` unfolding G_def by auto
show "\<nu> A \<le> (\<integral>\<^isup>+x. f x * indicator A x \<partial>M)"
using upper_bound[THEN bspec, OF `A \<in> sets M`]
- by (simp add: pextreal_zero_le_diff)
+ using M'.real_measure[OF A]
+ by (cases "integral\<^isup>P M (?F A)") auto
qed
qed simp
qed
@@ -543,22 +612,22 @@
lemma (in finite_measure) split_space_into_finite_sets_and_rest:
assumes "measure_space (M\<lparr>measure := \<nu>\<rparr>)" (is "measure_space ?N")
assumes ac: "absolutely_continuous \<nu>"
- shows "\<exists>\<Omega>0\<in>sets M. \<exists>\<Omega>::nat\<Rightarrow>'a set. disjoint_family \<Omega> \<and> range \<Omega> \<subseteq> sets M \<and> \<Omega>0 = space M - (\<Union>i. \<Omega> i) \<and>
- (\<forall>A\<in>sets M. A \<subseteq> \<Omega>0 \<longrightarrow> (\<mu> A = 0 \<and> \<nu> A = 0) \<or> (\<mu> A > 0 \<and> \<nu> A = \<omega>)) \<and>
- (\<forall>i. \<nu> (\<Omega> i) \<noteq> \<omega>)"
+ shows "\<exists>A0\<in>sets M. \<exists>B::nat\<Rightarrow>'a set. disjoint_family B \<and> range B \<subseteq> sets M \<and> A0 = space M - (\<Union>i. B i) \<and>
+ (\<forall>A\<in>sets M. A \<subseteq> A0 \<longrightarrow> (\<mu> A = 0 \<and> \<nu> A = 0) \<or> (\<mu> A > 0 \<and> \<nu> A = \<infinity>)) \<and>
+ (\<forall>i. \<nu> (B i) \<noteq> \<infinity>)"
proof -
interpret v: measure_space ?N
where "space ?N = space M" and "sets ?N = sets M" and "measure ?N = \<nu>"
by fact auto
- let ?Q = "{Q\<in>sets M. \<nu> Q \<noteq> \<omega>}"
+ let ?Q = "{Q\<in>sets M. \<nu> Q \<noteq> \<infinity>}"
let ?a = "SUP Q:?Q. \<mu> Q"
have "{} \<in> ?Q" using v.empty_measure by auto
then have Q_not_empty: "?Q \<noteq> {}" by blast
have "?a \<le> \<mu> (space M)" using sets_into_space
by (auto intro!: SUP_leI measure_mono top)
- then have "?a \<noteq> \<omega>" using finite_measure_of_space
+ then have "?a \<noteq> \<infinity>" using finite_measure_of_space
by auto
- from SUPR_countable_SUPR[OF this Q_not_empty]
+ from SUPR_countable_SUPR[OF Q_not_empty, of \<mu>]
obtain Q'' where "range Q'' \<subseteq> \<mu> ` ?Q" and a: "?a = (SUP i::nat. Q'' i)"
by auto
then have "\<forall>i. \<exists>Q'. Q'' i = \<mu> Q' \<and> Q' \<in> ?Q" by auto
@@ -569,7 +638,7 @@
have Union: "(SUP i. \<mu> (?O i)) = \<mu> (\<Union>i. ?O i)"
proof (rule continuity_from_below[of ?O])
show "range ?O \<subseteq> sets M" using Q' by (auto intro!: finite_UN)
- show "\<And>i. ?O i \<subseteq> ?O (Suc i)" by fastsimp
+ show "incseq ?O" by (fastsimp intro!: incseq_SucI)
qed
have Q'_sets: "\<And>i. Q' i \<in> sets M" using Q' by auto
have O_sets: "\<And>i. ?O i \<in> sets M"
@@ -580,8 +649,8 @@
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> pextreal_less_\<omega> using Q' by auto
- finally show "\<nu> (?O i) \<noteq> \<omega>" unfolding pextreal_less_\<omega> by auto
+ also have "\<dots> < \<infinity>" using Q' by (simp add: setsum_Pinfty)
+ finally show "\<nu> (?O i) \<noteq> \<infinity>" by simp
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]
@@ -592,7 +661,7 @@
proof (safe intro!: Sup_mono, unfold bex_simps)
fix i
have *: "(\<Union>Q' ` {..i}) = ?O i" by auto
- then show "\<exists>x. (x \<in> sets M \<and> \<nu> x \<noteq> \<omega>) \<and>
+ then show "\<exists>x. (x \<in> sets M \<and> \<nu> x \<noteq> \<infinity>) \<and>
\<mu> (\<Union>Q' ` {..i}) \<le> \<mu> x"
using O_in_G[of i] by (auto intro!: exI[of _ "?O i"])
qed
@@ -610,50 +679,52 @@
show "range Q \<subseteq> sets M"
using Q_sets by auto
{ fix A assume A: "A \<in> sets M" "A \<subseteq> space M - ?O_0"
- show "\<mu> A = 0 \<and> \<nu> A = 0 \<or> 0 < \<mu> A \<and> \<nu> A = \<omega>"
+ show "\<mu> A = 0 \<and> \<nu> A = 0 \<or> 0 < \<mu> A \<and> \<nu> A = \<infinity>"
proof (rule disjCI, simp)
- assume *: "0 < \<mu> A \<longrightarrow> \<nu> A \<noteq> \<omega>"
+ assume *: "0 < \<mu> A \<longrightarrow> \<nu> A \<noteq> \<infinity>"
show "\<mu> A = 0 \<and> \<nu> A = 0"
proof cases
assume "\<mu> A = 0" moreover with ac A have "\<nu> A = 0"
unfolding absolutely_continuous_def by auto
ultimately show ?thesis by simp
next
- assume "\<mu> A \<noteq> 0" with * have "\<nu> A \<noteq> \<omega>" by auto
+ assume "\<mu> A \<noteq> 0" with * have "\<nu> A \<noteq> \<infinity>" using positive_measure[OF A(1)] by auto
with A have "\<mu> ?O_0 + \<mu> A = \<mu> (?O_0 \<union> A)"
using Q' by (auto intro!: measure_additive countable_UN)
also have "\<dots> = (SUP i. \<mu> (?O i \<union> A))"
proof (rule continuity_from_below[of "\<lambda>i. ?O i \<union> A", symmetric, simplified])
show "range (\<lambda>i. ?O i \<union> A) \<subseteq> sets M"
- using `\<nu> A \<noteq> \<omega>` O_sets A by auto
- qed fastsimp
+ using `\<nu> A \<noteq> \<infinity>` O_sets A by auto
+ qed (fastsimp intro!: incseq_SucI)
also have "\<dots> \<le> ?a"
- proof (safe intro!: SUPR_bound)
+ proof (safe intro!: SUP_leI)
fix i have "?O i \<union> A \<in> ?Q"
proof (safe del: notI)
show "?O i \<union> A \<in> sets M" using O_sets A by auto
from O_in_G[of i]
moreover have "\<nu> (?O i \<union> A) \<le> \<nu> (?O i) + \<nu> A"
using v.measure_subadditive[of "?O i" A] A O_sets by auto
- ultimately show "\<nu> (?O i \<union> A) \<noteq> \<omega>"
- using `\<nu> A \<noteq> \<omega>` by auto
+ ultimately show "\<nu> (?O i \<union> A) \<noteq> \<infinity>"
+ using `\<nu> A \<noteq> \<infinity>` by auto
qed
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: pextreal_noteq_omega_Ex)
+ finally have "\<mu> A = 0"
+ unfolding a_eq using real_measure[OF `?O_0 \<in> sets M`] real_measure[OF A(1)] by auto
with `\<mu> A \<noteq> 0` show ?thesis by auto
qed
qed }
- { fix i show "\<nu> (Q i) \<noteq> \<omega>"
+ { fix i show "\<nu> (Q i) \<noteq> \<infinity>"
proof (cases i)
case 0 then show ?thesis
unfolding Q_def using Q'[of 0] by simp
next
case (Suc n)
then show ?thesis unfolding Q_def
- using `?O n \<in> ?Q` `?O (Suc n) \<in> ?Q` O_mono
- using v.measure_Diff[of "?O n" "?O (Suc n)"] by auto
+ using `?O n \<in> ?Q` `?O (Suc n) \<in> ?Q`
+ using v.measure_mono[OF O_mono, of n] v.positive_measure[of "?O n"] v.positive_measure[of "?O (Suc n)"]
+ using v.measure_Diff[of "?O n" "?O (Suc n)", OF _ _ _ O_mono]
+ by (cases rule: extreal2_cases[of "\<nu> (\<Union> x\<le>Suc n. Q' x)" "\<nu> (\<Union> i\<le>n. Q' i)"]) auto
qed }
show "space M - ?O_0 \<in> sets M" using Q'_sets by auto
{ fix j have "(\<Union>i\<le>j. ?O i) = (\<Union>i\<le>j. Q i)"
@@ -675,7 +746,7 @@
lemma (in finite_measure) Radon_Nikodym_finite_measure_infinite:
assumes "measure_space (M\<lparr>measure := \<nu>\<rparr>)" (is "measure_space ?N")
assumes "absolutely_continuous \<nu>"
- shows "\<exists>f \<in> borel_measurable M. \<forall>A\<in>sets M. \<nu> A = (\<integral>\<^isup>+x. f x * indicator A x \<partial>M)"
+ shows "\<exists>f \<in> borel_measurable M. (\<forall>x. 0 \<le> f x) \<and> (\<forall>A\<in>sets M. \<nu> A = (\<integral>\<^isup>+x. f x * indicator A x \<partial>M))"
proof -
interpret v: measure_space ?N
where "space ?N = space M" and "sets ?N = sets M" and "measure ?N = \<nu>"
@@ -684,14 +755,14 @@
obtain Q0 and Q :: "nat \<Rightarrow> 'a set"
where Q: "disjoint_family Q" "range Q \<subseteq> sets M"
and Q0: "Q0 \<in> sets M" "Q0 = space M - (\<Union>i. Q i)"
- and in_Q0: "\<And>A. A \<in> sets M \<Longrightarrow> A \<subseteq> Q0 \<Longrightarrow> \<mu> A = 0 \<and> \<nu> A = 0 \<or> 0 < \<mu> A \<and> \<nu> A = \<omega>"
- and Q_fin: "\<And>i. \<nu> (Q i) \<noteq> \<omega>" by force
+ and in_Q0: "\<And>A. A \<in> sets M \<Longrightarrow> A \<subseteq> Q0 \<Longrightarrow> \<mu> A = 0 \<and> \<nu> A = 0 \<or> 0 < \<mu> A \<and> \<nu> A = \<infinity>"
+ and Q_fin: "\<And>i. \<nu> (Q i) \<noteq> \<infinity>" by force
from Q have Q_sets: "\<And>i. Q i \<in> sets M" by auto
- have "\<forall>i. \<exists>f. f\<in>borel_measurable M \<and> (\<forall>A\<in>sets M.
+ have "\<forall>i. \<exists>f. f\<in>borel_measurable M \<and> (\<forall>x. 0 \<le> f x) \<and> (\<forall>A\<in>sets M.
\<nu> (Q i \<inter> A) = (\<integral>\<^isup>+x. f x * indicator (Q i \<inter> A) x \<partial>M))"
proof
fix i
- have indicator_eq: "\<And>f x A. (f x :: pextreal) * indicator (Q i \<inter> A) x * indicator (Q i) x
+ have indicator_eq: "\<And>f x A. (f x :: extreal) * 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))"
@@ -702,7 +773,7 @@
proof
show "measure_space ?Q"
using v.restricted_measure_space Q_sets[of i] by auto
- show "measure ?Q (space ?Q) \<noteq> \<omega>" using Q_fin by simp
+ show "measure ?Q (space ?Q) \<noteq> \<infinity>" using Q_fin by simp
qed
have "R.absolutely_continuous \<nu>"
using `absolutely_continuous \<nu>` `Q i \<in> sets M`
@@ -712,48 +783,40 @@
and f_int: "\<And>A. A\<in>sets M \<Longrightarrow> \<nu> (Q i \<inter> A) = (\<integral>\<^isup>+x. (f x * indicator (Q i) x) * indicator A x \<partial>M)"
unfolding Bex_def borel_measurable_restricted[OF `Q i \<in> sets M`]
positive_integral_restricted[OF `Q i \<in> sets M`] by (auto simp: indicator_eq)
- then show "\<exists>f. f\<in>borel_measurable M \<and> (\<forall>A\<in>sets M.
+ then show "\<exists>f. f\<in>borel_measurable M \<and> (\<forall>x. 0 \<le> f x) \<and> (\<forall>A\<in>sets M.
\<nu> (Q i \<inter> A) = (\<integral>\<^isup>+x. f x * indicator (Q i \<inter> A) x \<partial>M))"
- by (fastsimp intro!: exI[of _ "\<lambda>x. f x * indicator (Q i) x"] positive_integral_cong
- simp: indicator_def)
+ by (auto intro!: exI[of _ "\<lambda>x. max 0 (f x * indicator (Q i) x)"] positive_integral_cong_pos
+ split: split_indicator split_if_asm simp: max_def)
qed
- from choice[OF this] obtain f where borel: "\<And>i. f i \<in> borel_measurable M"
+ from choice[OF this] obtain f where borel: "\<And>i. f i \<in> borel_measurable M" "\<And>i x. 0 \<le> f i x"
and f: "\<And>A i. A \<in> sets M \<Longrightarrow>
\<nu> (Q i \<inter> A) = (\<integral>\<^isup>+x. f i x * indicator (Q i \<inter> A) x \<partial>M)"
by auto
- let "?f x" =
- "(\<Sum>\<^isub>\<infinity> i. f i x * indicator (Q i) x) + \<omega> * indicator Q0 x"
+ let "?f x" = "(\<Sum>i. f i x * indicator (Q i) x) + \<infinity> * indicator Q0 x"
show ?thesis
proof (safe intro!: bexI[of _ ?f])
- show "?f \<in> borel_measurable M"
- 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)
+ show "?f \<in> borel_measurable M" using Q0 borel Q_sets
+ by (auto intro!: measurable_If)
+ show "\<And>x. 0 \<le> ?f x" using borel by (auto intro!: suminf_0_le simp: indicator_def)
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 :: pextreal) =
- indicator (Q0 \<inter> A) x" by (auto simp: indicator_def)
- have "(\<integral>\<^isup>+x. ?f x * indicator A x \<partial>M) =
- (\<Sum>\<^isub>\<infinity> i. \<nu> (Q i \<inter> A)) + \<omega> * \<mu> (Q0 \<inter> A)"
- unfolding f[OF `A \<in> sets M`]
- 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)
- apply (subst positive_integral_cmult_indicator)
- apply (fastsimp intro: Q0 `A \<in> sets M`)
- unfolding psuminf_cmult_right[symmetric]
- apply (subst positive_integral_psuminf)
- apply (fastsimp intro: `A \<in> sets M` Q_sets borel)
- apply (simp add: *)
- done
- moreover have "(\<Sum>\<^isub>\<infinity>i. \<nu> (Q i \<inter> A)) = \<nu> ((\<Union>i. Q i) \<inter> A)"
+ have Qi: "\<And>i. Q i \<in> sets M" using Q by auto
+ have [intro,simp]: "\<And>i. (\<lambda>x. f i x * indicator (Q i \<inter> A) x) \<in> borel_measurable M"
+ "\<And>i. AE x. 0 \<le> f i x * indicator (Q i \<inter> A) x"
+ using borel Qi Q0(1) `A \<in> sets M` by (auto intro!: borel_measurable_extreal_times)
+ have "(\<integral>\<^isup>+x. ?f x * indicator A x \<partial>M) = (\<integral>\<^isup>+x. (\<Sum>i. f i x * indicator (Q i \<inter> A) x) + \<infinity> * indicator (Q0 \<inter> A) x \<partial>M)"
+ using borel by (intro positive_integral_cong) (auto simp: indicator_def)
+ also have "\<dots> = (\<integral>\<^isup>+x. (\<Sum>i. f i x * indicator (Q i \<inter> A) x) \<partial>M) + \<infinity> * \<mu> (Q0 \<inter> A)"
+ using borel Qi Q0(1) `A \<in> sets M`
+ by (subst positive_integral_add) (auto simp del: extreal_infty_mult
+ simp add: positive_integral_cmult_indicator Int intro!: suminf_0_le)
+ also have "\<dots> = (\<Sum>i. \<nu> (Q i \<inter> A)) + \<infinity> * \<mu> (Q0 \<inter> A)"
+ by (subst f[OF `A \<in> sets M`], subst positive_integral_suminf) auto
+ finally have "(\<integral>\<^isup>+x. ?f x * indicator A x \<partial>M) = (\<Sum>i. \<nu> (Q i \<inter> A)) + \<infinity> * \<mu> (Q0 \<inter> A)" .
+ moreover have "(\<Sum>i. \<nu> (Q i \<inter> A)) = \<nu> ((\<Union>i. Q i) \<inter> A)"
using Q Q_sets `A \<in> sets M`
by (intro v.measure_countably_additive[of "\<lambda>i. Q i \<inter> A", unfolded comp_def, simplified])
(auto simp: disjoint_family_on_def)
- moreover have "\<omega> * \<mu> (Q0 \<inter> A) = \<nu> (Q0 \<inter> A)"
+ moreover have "\<infinity> * \<mu> (Q0 \<inter> A) = \<nu> (Q0 \<inter> A)"
proof -
have "Q0 \<inter> A \<in> sets M" using Q0(1) `A \<in> sets M` by blast
from in_Q0[OF this] show ?thesis by auto
@@ -770,40 +833,43 @@
lemma (in sigma_finite_measure) Radon_Nikodym:
assumes "measure_space (M\<lparr>measure := \<nu>\<rparr>)" (is "measure_space ?N")
- assumes "absolutely_continuous \<nu>"
- shows "\<exists>f \<in> borel_measurable M. \<forall>A\<in>sets M. \<nu> A = (\<integral>\<^isup>+x. f x * indicator A x \<partial>M)"
+ assumes ac: "absolutely_continuous \<nu>"
+ shows "\<exists>f \<in> borel_measurable M. (\<forall>x. 0 \<le> f x) \<and> (\<forall>A\<in>sets M. \<nu> A = (\<integral>\<^isup>+x. f x * indicator A x \<partial>M))"
proof -
from Ex_finite_integrable_function
- obtain h where finite: "integral\<^isup>P M h \<noteq> \<omega>" and
+ obtain h where finite: "integral\<^isup>P M h \<noteq> \<infinity>" and
borel: "h \<in> borel_measurable M" and
+ nn: "\<And>x. 0 \<le> h x" and
pos: "\<And>x. x \<in> space M \<Longrightarrow> 0 < h x" and
- "\<And>x. x \<in> space M \<Longrightarrow> h x < \<omega>" by auto
+ "\<And>x. x \<in> space M \<Longrightarrow> h x < \<infinity>" by auto
let "?T A" = "(\<integral>\<^isup>+x. h x * indicator A x \<partial>M)"
let ?MT = "M\<lparr> measure := ?T \<rparr>"
- from measure_space_density[OF borel] finite
interpret T: finite_measure ?MT
where "space ?MT = space M" and "sets ?MT = sets M" and "measure ?MT = ?T"
- unfolding finite_measure_def finite_measure_axioms_def
- by (simp_all 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::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
- by (fastsimp simp: borel_measurable_indicator positive_integral_0_iff)
+ unfolding finite_measure_def finite_measure_axioms_def using borel finite nn
+ by (auto intro!: measure_space_density cong: positive_integral_cong)
+ have "T.absolutely_continuous \<nu>"
+ proof (unfold T.absolutely_continuous_def, safe)
+ fix N assume "N \<in> sets M" "(\<integral>\<^isup>+x. h x * indicator N x \<partial>M) = 0"
+ with borel ac pos have "AE x. x \<notin> N"
+ by (subst (asm) positive_integral_0_iff_AE) (auto split: split_indicator simp: not_le)
+ then have "N \<in> null_sets" using `N \<in> sets M` sets_into_space
+ by (subst (asm) AE_iff_measurable[OF `N \<in> sets M`]) auto
+ then show "\<nu> N = 0" using ac by (auto simp: absolutely_continuous_def)
+ qed
from T.Radon_Nikodym_finite_measure_infinite[simplified, OF assms(1) this]
- obtain f where f_borel: "f \<in> borel_measurable M" and
+ obtain f where f_borel: "f \<in> borel_measurable M" "\<And>x. 0 \<le> f x" and
fT: "\<And>A. A \<in> sets M \<Longrightarrow> \<nu> A = (\<integral>\<^isup>+ x. f x * indicator A x \<partial>?MT)"
by (auto simp: measurable_def)
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_pextreal_times)
+ using borel f_borel by (auto intro: borel_measurable_extreal_times)
+ show "\<And>x. 0 \<le> h x * f x" using nn f_borel by auto
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_pextreal_times borel_measurable_indicator)
- from positive_integral_translated_density[OF borel this]
- show "\<nu> A = (\<integral>\<^isup>+x. h x * f x * indicator A x \<partial>M)"
- unfolding fT[OF `A \<in> sets M`] by (simp add: field_simps)
+ then show "\<nu> A = (\<integral>\<^isup>+x. h x * f x * indicator A x \<partial>M)"
+ unfolding fT[OF `A \<in> sets M`] mult_assoc using nn borel f_borel
+ by (intro positive_integral_translated_density) auto
qed
qed
@@ -811,7 +877,8 @@
lemma (in measure_space) finite_density_unique:
assumes borel: "f \<in> borel_measurable M" "g \<in> borel_measurable M"
- and fin: "integral\<^isup>P M f < \<omega>"
+ assumes pos: "AE x. 0 \<le> f x" "AE x. 0 \<le> g x"
+ and fin: "integral\<^isup>P M f \<noteq> \<infinity>"
shows "(\<forall>A\<in>sets M. (\<integral>\<^isup>+x. f x * indicator A x \<partial>M) = (\<integral>\<^isup>+x. g x * indicator A x \<partial>M))
\<longleftrightarrow> (AE x. f x = g x)"
(is "(\<forall>A\<in>sets M. ?P f A = ?P g A) \<longleftrightarrow> _")
@@ -822,42 +889,38 @@
next
assume eq: "\<forall>A\<in>sets M. ?P f A = ?P g A"
from this[THEN bspec, OF top] fin
- have g_fin: "integral\<^isup>P M g < \<omega>" by (simp cong: positive_integral_cong)
+ have g_fin: "integral\<^isup>P M g \<noteq> \<infinity>" by (simp cong: positive_integral_cong)
{ fix f g assume borel: "f \<in> borel_measurable M" "g \<in> borel_measurable M"
- and g_fin: "integral\<^isup>P M g < \<omega>" and eq: "\<forall>A\<in>sets M. ?P f A = ?P g A"
+ and pos: "AE x. 0 \<le> f x" "AE x. 0 \<le> g x"
+ and g_fin: "integral\<^isup>P M g \<noteq> \<infinity>" and eq: "\<forall>A\<in>sets M. ?P f A = ?P g A"
let ?N = "{x\<in>space M. g x < f x}"
have N: "?N \<in> sets M" using borel by simp
+ have "?P g ?N \<le> integral\<^isup>P M g" using pos
+ by (intro positive_integral_mono_AE) (auto split: split_indicator)
+ then have Pg_fin: "?P g ?N \<noteq> \<infinity>" using g_fin by auto
have "?P (\<lambda>x. (f x - g x)) ?N = (\<integral>\<^isup>+x. f x * indicator ?N x - g x * indicator ?N x \<partial>M)"
by (auto intro!: positive_integral_cong simp: indicator_def)
also have "\<dots> = ?P f ?N - ?P g ?N"
proof (rule positive_integral_diff)
show "(\<lambda>x. f x * indicator ?N x) \<in> borel_measurable M" "(\<lambda>x. g x * indicator ?N x) \<in> borel_measurable M"
using borel N by auto
- have "?P g ?N \<le> integral\<^isup>P M g"
- by (auto intro!: positive_integral_mono simp: indicator_def)
- then show "?P g ?N \<noteq> \<omega>" using g_fin by auto
- fix x assume "x \<in> space M"
- show "g x * indicator ?N x \<le> f x * indicator ?N x"
- by (auto simp: indicator_def)
- qed
+ show "AE x. g x * indicator ?N x \<le> f x * indicator ?N x"
+ "AE x. 0 \<le> g x * indicator ?N x"
+ using pos by (auto split: split_indicator)
+ qed fact
also have "\<dots> = 0"
- using eq[THEN bspec, OF N] by simp
- 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: 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"
- using eq by (intro null_sets_Un) auto
- also have "{x\<in>space M. g x < f x} \<union> {x\<in>space M. f x < g x} = {x\<in>space M. f x \<noteq> g x}"
- by auto
- finally show "AE x. f x = g x"
- unfolding almost_everywhere_def by auto
+ unfolding eq[THEN bspec, OF N] using positive_integral_positive Pg_fin by auto
+ finally have "AE x. f x \<le> g x"
+ using pos borel positive_integral_PInf_AE[OF borel(2) g_fin]
+ by (subst (asm) positive_integral_0_iff_AE)
+ (auto split: split_indicator simp: not_less extreal_minus_le_iff) }
+ from this[OF borel pos g_fin eq] this[OF borel(2,1) pos(2,1) fin] eq
+ show "AE x. f x = g x" by auto
qed
lemma (in finite_measure) density_unique_finite_measure:
assumes borel: "f \<in> borel_measurable M" "f' \<in> borel_measurable M"
+ assumes pos: "AE x. 0 \<le> f x" "AE x. 0 \<le> f' x"
assumes f: "\<And>A. A \<in> sets M \<Longrightarrow> (\<integral>\<^isup>+x. f x * indicator A x \<partial>M) = (\<integral>\<^isup>+x. f' x * indicator A x \<partial>M)"
(is "\<And>A. A \<in> sets M \<Longrightarrow> ?P f A = ?P f' A")
shows "AE x. f x = f' x"
@@ -865,26 +928,26 @@
let "?\<nu> A" = "?P f A" and "?\<nu>' A" = "?P f' A"
let "?f A x" = "f x * indicator A x" and "?f' A x" = "f' x * indicator A x"
interpret M: measure_space "M\<lparr> measure := ?\<nu>\<rparr>"
- using borel(1) by (rule measure_space_density) simp
+ using borel(1) pos(1) by (rule measure_space_density) simp
have ac: "absolutely_continuous ?\<nu>"
using f by (rule density_is_absolutely_continuous)
from split_space_into_finite_sets_and_rest[OF `measure_space (M\<lparr> measure := ?\<nu>\<rparr>)` ac]
obtain Q0 and Q :: "nat \<Rightarrow> 'a set"
where Q: "disjoint_family Q" "range Q \<subseteq> sets M"
and Q0: "Q0 \<in> sets M" "Q0 = space M - (\<Union>i. Q i)"
- and in_Q0: "\<And>A. A \<in> sets M \<Longrightarrow> A \<subseteq> Q0 \<Longrightarrow> \<mu> A = 0 \<and> ?\<nu> A = 0 \<or> 0 < \<mu> A \<and> ?\<nu> A = \<omega>"
- and Q_fin: "\<And>i. ?\<nu> (Q i) \<noteq> \<omega>" by force
+ and in_Q0: "\<And>A. A \<in> sets M \<Longrightarrow> A \<subseteq> Q0 \<Longrightarrow> \<mu> A = 0 \<and> ?\<nu> A = 0 \<or> 0 < \<mu> A \<and> ?\<nu> A = \<infinity>"
+ and Q_fin: "\<And>i. ?\<nu> (Q i) \<noteq> \<infinity>" by force
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::pextreal. y * indicator (Q i) x * indicator A x = y * indicator (Q i \<inter> A) x"
+ have *: "\<And>i x A. \<And>y::extreal. y * indicator (Q i) x * indicator A x = y * indicator (Q i \<inter> A) x"
unfolding indicator_def by auto
- have "\<forall>i. AE x. ?f (Q i) x = ?f' (Q i) x" using borel Q_fin Q
+ have "\<forall>i. AE x. ?f (Q i) x = ?f' (Q i) x" using borel Q_fin Q pos
by (intro finite_density_unique[THEN iffD1] allI)
- (auto intro!: borel_measurable_pextreal_times f Int simp: *)
+ (auto intro!: borel_measurable_extreal_times f Int simp: *)
moreover have "AE x. ?f Q0 x = ?f' Q0 x"
proof (rule AE_I')
- { fix f :: "'a \<Rightarrow> pextreal" assume borel: "f \<in> borel_measurable M"
+ { fix f :: "'a \<Rightarrow> extreal" assume borel: "f \<in> borel_measurable M"
and eq: "\<And>A. A \<in> sets M \<Longrightarrow> ?\<nu> A = (\<integral>\<^isup>+x. f x * indicator A x \<partial>M)"
let "?A i" = "Q0 \<inter> {x \<in> space M. f x < of_nat i}"
have "(\<Union>i. ?A i) \<in> null_sets"
@@ -896,69 +959,74 @@
by (auto intro!: positive_integral_mono simp: indicator_def)
also have "\<dots> = of_nat i * \<mu> (?A i)"
using `?A i \<in> sets M` by (auto intro!: positive_integral_cmult_indicator)
- also have "\<dots> < \<omega>"
+ also have "\<dots> < \<infinity>"
using `?A i \<in> sets M`[THEN finite_measure] by auto
- finally have "?\<nu> (?A i) \<noteq> \<omega>" by simp
+ finally have "?\<nu> (?A i) \<noteq> \<infinity>" by simp
then show "?A i \<in> null_sets" using in_Q0[OF `?A i \<in> sets M`] `?A i \<in> sets M` by auto
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: pextreal_less_\<omega>) }
+ also have "(\<Union>i. ?A i) = Q0 \<inter> {x\<in>space M. f x \<noteq> \<infinity>}"
+ by (auto simp: less_PInf_Ex_of_nat real_eq_of_nat)
+ finally have "Q0 \<inter> {x\<in>space M. f x \<noteq> \<infinity>} \<in> null_sets" by simp }
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)
+ have "Q0 \<inter> {x\<in>space M. f x \<noteq> \<infinity>} \<in> null_sets" "Q0 \<inter> {x\<in>space M. f' x \<noteq> \<infinity>} \<in> null_sets" by simp_all
+ then show "(Q0 \<inter> {x\<in>space M. f x \<noteq> \<infinity>}) \<union> (Q0 \<inter> {x\<in>space M. f' x \<noteq> \<infinity>}) \<in> null_sets" by (rule null_sets_Un)
show "{x \<in> space M. ?f Q0 x \<noteq> ?f' Q0 x} \<subseteq>
- (Q0 \<inter> {x\<in>space M. f x \<noteq> \<omega>}) \<union> (Q0 \<inter> {x\<in>space M. f' x \<noteq> \<omega>})" by (auto simp: indicator_def)
+ (Q0 \<inter> {x\<in>space M. f x \<noteq> \<infinity>}) \<union> (Q0 \<inter> {x\<in>space M. f' x \<noteq> \<infinity>})" by (auto simp: indicator_def)
qed
moreover have "\<And>x. (?f Q0 x = ?f' Q0 x) \<longrightarrow> (\<forall>i. ?f (Q i) x = ?f' (Q i) x) \<longrightarrow>
?f (space M) x = ?f' (space M) x"
by (auto simp: indicator_def Q0)
ultimately have "AE x. ?f (space M) x = ?f' (space M) x"
- by (auto simp: all_AE_countable)
+ by (auto simp: AE_all_countable[symmetric])
then show "AE x. f x = f' x" by auto
qed
lemma (in sigma_finite_measure) density_unique:
- assumes borel: "f \<in> borel_measurable M" "f' \<in> borel_measurable M"
- assumes f: "\<And>A. A \<in> sets M \<Longrightarrow> (\<integral>\<^isup>+x. f x * indicator A x \<partial>M) = (\<integral>\<^isup>+x. f' x * indicator A x \<partial>M)"
+ assumes f: "f \<in> borel_measurable M" "AE x. 0 \<le> f x"
+ assumes f': "f' \<in> borel_measurable M" "AE x. 0 \<le> f' x"
+ assumes eq: "\<And>A. A \<in> sets M \<Longrightarrow> (\<integral>\<^isup>+x. f x * indicator A x \<partial>M) = (\<integral>\<^isup>+x. f' x * indicator A x \<partial>M)"
(is "\<And>A. A \<in> sets M \<Longrightarrow> ?P f A = ?P f' A")
shows "AE x. f x = f' x"
proof -
obtain h where h_borel: "h \<in> borel_measurable M"
- and fin: "integral\<^isup>P M h \<noteq> \<omega>" and pos: "\<And>x. x \<in> space M \<Longrightarrow> 0 < h x \<and> h x < \<omega>"
+ and fin: "integral\<^isup>P M h \<noteq> \<infinity>" and pos: "\<And>x. x \<in> space M \<Longrightarrow> 0 < h x \<and> h x < \<infinity>" "\<And>x. 0 \<le> h x"
using Ex_finite_integrable_function by auto
- interpret h: measure_space "M\<lparr> measure := \<lambda>A. (\<integral>\<^isup>+x. h x * indicator A x \<partial>M) \<rparr>"
- using h_borel by (rule measure_space_density) simp
+ then have h_nn: "AE x. 0 \<le> h x" by auto
+ let ?H = "M\<lparr> measure := \<lambda>A. (\<integral>\<^isup>+x. h x * indicator A x \<partial>M) \<rparr>"
+ have H: "measure_space ?H"
+ using h_borel h_nn by (rule measure_space_density) simp
+ then interpret h: measure_space ?H .
interpret h: finite_measure "M\<lparr> measure := \<lambda>A. (\<integral>\<^isup>+x. h x * indicator A x \<partial>M) \<rparr>"
by default (simp cong: positive_integral_cong add: fin)
let ?fM = "M\<lparr>measure := \<lambda>A. (\<integral>\<^isup>+x. f x * indicator A x \<partial>M)\<rparr>"
interpret f: measure_space ?fM
- using borel(1) by (rule measure_space_density) simp
+ using f by (rule measure_space_density) simp
let ?f'M = "M\<lparr>measure := \<lambda>A. (\<integral>\<^isup>+x. f' x * indicator A x \<partial>M)\<rparr>"
interpret f': measure_space ?f'M
- using borel(2) by (rule measure_space_density) simp
+ using f' by (rule measure_space_density) simp
{ fix A assume "A \<in> sets M"
- 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 "{x \<in> space M. h x * indicator A x \<noteq> 0} = A"
+ using pos(1) sets_into_space by (force simp: indicator_def)
then have "(\<integral>\<^isup>+x. h x * indicator A x \<partial>M) = 0 \<longleftrightarrow> A \<in> null_sets"
- using h_borel `A \<in> sets M` by (simp add: positive_integral_0_iff) }
+ using h_borel `A \<in> sets M` h_nn by (subst positive_integral_0_iff) auto }
note h_null_sets = this
{ fix A assume "A \<in> sets M"
- have "(\<integral>\<^isup>+x. h x * (f x * indicator A x) \<partial>M) = (\<integral>\<^isup>+x. h x * indicator A x \<partial>?fM)"
- using `A \<in> sets M` h_borel borel
- by (simp add: positive_integral_translated_density ac_simps cong: positive_integral_cong)
+ have "(\<integral>\<^isup>+x. f x * (h x * indicator A x) \<partial>M) = (\<integral>\<^isup>+x. h x * indicator A x \<partial>?fM)"
+ using `A \<in> sets M` h_borel h_nn f f'
+ by (intro positive_integral_translated_density[symmetric]) auto
also have "\<dots> = (\<integral>\<^isup>+x. h x * indicator A x \<partial>?f'M)"
- by (rule f'.positive_integral_cong_measure) (simp_all add: f)
- also have "\<dots> = (\<integral>\<^isup>+x. h x * (f' x * indicator A x) \<partial>M)"
- using `A \<in> sets M` h_borel borel
- by (simp add: positive_integral_translated_density ac_simps cong: positive_integral_cong)
- finally have "(\<integral>\<^isup>+x. h x * (f x * indicator A x) \<partial>M) = (\<integral>\<^isup>+x. h x * (f' x * indicator A x) \<partial>M)" . }
- then have "h.almost_everywhere (\<lambda>x. f x = f' x)"
- using h_borel borel
- apply (intro h.density_unique_finite_measure)
- apply (simp add: measurable_def)
- apply (simp add: measurable_def)
- by (simp add: positive_integral_translated_density)
+ by (rule f'.positive_integral_cong_measure) (simp_all add: eq)
+ also have "\<dots> = (\<integral>\<^isup>+x. f' x * (h x * indicator A x) \<partial>M)"
+ using `A \<in> sets M` h_borel h_nn f f'
+ by (intro positive_integral_translated_density) auto
+ finally have "(\<integral>\<^isup>+x. h x * (f x * indicator A x) \<partial>M) = (\<integral>\<^isup>+x. h x * (f' x * indicator A x) \<partial>M)"
+ by (simp add: ac_simps)
+ then have "(\<integral>\<^isup>+x. (f x * indicator A x) \<partial>?H) = (\<integral>\<^isup>+x. (f' x * indicator A x) \<partial>?H)"
+ using `A \<in> sets M` h_borel h_nn f f'
+ by (subst (asm) (1 2) positive_integral_translated_density[symmetric]) auto }
+ then have "AE x in ?H. f x = f' x" using h_borel h_nn f f'
+ by (intro h.density_unique_finite_measure absolutely_continuous_AE[OF H] density_is_absolutely_continuous)
+ simp_all
then show "AE x. f x = f' x"
unfolding h.almost_everywhere_def almost_everywhere_def
by (auto simp add: h_null_sets)
@@ -966,41 +1034,42 @@
lemma (in sigma_finite_measure) sigma_finite_iff_density_finite:
assumes \<nu>: "measure_space (M\<lparr>measure := \<nu>\<rparr>)" (is "measure_space ?N")
- and f: "f \<in> borel_measurable M"
+ and f: "f \<in> borel_measurable M" "AE x. 0 \<le> f x"
and eq: "\<And>A. A \<in> sets M \<Longrightarrow> \<nu> A = (\<integral>\<^isup>+x. f x * indicator A x \<partial>M)"
- shows "sigma_finite_measure (M\<lparr>measure := \<nu>\<rparr>) \<longleftrightarrow> (AE x. f x \<noteq> \<omega>)"
+ shows "sigma_finite_measure (M\<lparr>measure := \<nu>\<rparr>) \<longleftrightarrow> (AE x. f x \<noteq> \<infinity>)"
proof
assume "sigma_finite_measure ?N"
then interpret \<nu>: sigma_finite_measure ?N
where "borel_measurable ?N = borel_measurable M" and "measure ?N = \<nu>"
and "sets ?N = sets M" and "space ?N = space M" by (auto simp: measurable_def)
from \<nu>.Ex_finite_integrable_function obtain h where
- h: "h \<in> borel_measurable M" "integral\<^isup>P ?N h \<noteq> \<omega>"
- and fin: "\<forall>x\<in>space M. 0 < h x \<and> h x < \<omega>" by auto
- have "AE x. f x * h x \<noteq> \<omega>"
+ h: "h \<in> borel_measurable M" "integral\<^isup>P ?N h \<noteq> \<infinity>" and
+ h_nn: "\<And>x. 0 \<le> h x" and
+ fin: "\<forall>x\<in>space M. 0 < h x \<and> h x < \<infinity>" by auto
+ have "AE x. f x * h x \<noteq> \<infinity>"
proof (rule AE_I')
- have "integral\<^isup>P ?N h = (\<integral>\<^isup>+x. f x * h x \<partial>M)" using f h
+ have "integral\<^isup>P ?N h = (\<integral>\<^isup>+x. f x * h x \<partial>M)" using f h h_nn
by (subst \<nu>.positive_integral_cong_measure[symmetric,
of "M\<lparr> measure := \<lambda> A. \<integral>\<^isup>+x. f x * indicator A x \<partial>M \<rparr>"])
(auto intro!: positive_integral_translated_density simp: eq)
- then have "(\<integral>\<^isup>+x. f x * h x \<partial>M) \<noteq> \<omega>"
+ then have "(\<integral>\<^isup>+x. f x * h x \<partial>M) \<noteq> \<infinity>"
using h(2) by simp
- then show "(\<lambda>x. f x * h x) -` {\<omega>} \<inter> space M \<in> null_sets"
- using f h(1) by (auto intro!: positive_integral_omega borel_measurable_vimage)
+ then show "(\<lambda>x. f x * h x) -` {\<infinity>} \<inter> space M \<in> null_sets"
+ using f h(1) by (auto intro!: positive_integral_PInf borel_measurable_vimage)
qed auto
- then show "AE x. f x \<noteq> \<omega>"
+ then show "AE x. f x \<noteq> \<infinity>"
using fin by (auto elim!: AE_Ball_mp)
next
- assume AE: "AE x. f x \<noteq> \<omega>"
+ assume AE: "AE x. f x \<noteq> \<infinity>"
from sigma_finite guess Q .. note Q = this
interpret \<nu>: measure_space ?N
where "borel_measurable ?N = borel_measurable M" and "measure ?N = \<nu>"
and "sets ?N = sets M" and "space ?N = space M" using \<nu> by (auto simp: measurable_def)
- def A \<equiv> "\<lambda>i. f -` (case i of 0 \<Rightarrow> {\<omega>} | Suc n \<Rightarrow> {.. of_nat (Suc n)}) \<inter> space M"
+ def A \<equiv> "\<lambda>i. f -` (case i of 0 \<Rightarrow> {\<infinity>} | Suc n \<Rightarrow> {.. of_nat (Suc n)}) \<inter> space M"
{ fix i j have "A i \<inter> Q j \<in> sets M"
unfolding A_def using f Q
apply (rule_tac Int)
- by (cases i) (auto intro: measurable_sets[OF f]) }
+ by (cases i) (auto intro: measurable_sets[OF f(1)]) }
note A_in_sets = this
let "?A n" = "case prod_decode n of (i,j) \<Rightarrow> A i \<inter> Q j"
show "sigma_finite_measure ?N"
@@ -1021,18 +1090,21 @@
fix x assume x: "x \<in> space M"
show "x \<in> (\<Union>i. A i)"
proof (cases "f x")
- case infinite then show ?thesis using x unfolding A_def by (auto intro: exI[of _ 0])
+ case PInf with x show ?thesis unfolding A_def by (auto intro: exI[of _ 0])
next
- case (preal r)
- with less_\<omega>_Ex_of_nat[of "f x"] obtain n where "f x < of_nat n" by auto
- then show ?thesis using x preal unfolding A_def by (auto intro!: exI[of _ "Suc n"])
+ case (real r)
+ with less_PInf_Ex_of_nat[of "f x"] obtain n where "f x < of_nat n" by (auto simp: real_eq_of_nat)
+ then show ?thesis using x real unfolding A_def by (auto intro!: exI[of _ "Suc n"])
+ next
+ case MInf with x show ?thesis
+ unfolding A_def by (auto intro!: exI[of _ "Suc 0"])
qed
qed (auto simp: A_def)
finally show "(\<Union>i. ?A i) = space ?N" by simp
next
fix n obtain i j where
[simp]: "prod_decode n = (i, j)" by (cases "prod_decode n") auto
- have "(\<integral>\<^isup>+x. f x * indicator (A i \<inter> Q j) x \<partial>M) \<noteq> \<omega>"
+ have "(\<integral>\<^isup>+x. f x * indicator (A i \<inter> Q j) x \<partial>M) \<noteq> \<infinity>"
proof (cases i)
case 0
have "AE x. f x * indicator (A i \<inter> Q j) x = 0"
@@ -1045,11 +1117,11 @@
by (auto intro!: positive_integral_mono simp: indicator_def A_def)
also have "\<dots> = of_nat (Suc n) * \<mu> (Q j)"
using Q by (auto intro!: positive_integral_cmult_indicator)
- also have "\<dots> < \<omega>"
- using Q by auto
+ also have "\<dots> < \<infinity>"
+ using Q by (auto simp: real_eq_of_nat[symmetric])
finally show ?thesis by simp
qed
- then show "measure ?N (?A n) \<noteq> \<omega>"
+ then show "measure ?N (?A n) \<noteq> \<infinity>"
using A_in_sets Q eq by auto
qed
qed
@@ -1057,7 +1129,7 @@
section "Radon-Nikodym derivative"
definition
- "RN_deriv M \<nu> \<equiv> SOME f. f \<in> borel_measurable M \<and>
+ "RN_deriv M \<nu> \<equiv> SOME f. f \<in> borel_measurable M \<and> (\<forall>x. 0 \<le> f x) \<and>
(\<forall>A \<in> sets M. \<nu> A = (\<integral>\<^isup>+x. f x * indicator A x \<partial>M))"
lemma (in sigma_finite_measure) RN_deriv_cong:
@@ -1078,9 +1150,12 @@
shows "RN_deriv M \<nu> \<in> borel_measurable M" (is ?borel)
and "\<And>A. A \<in> sets M \<Longrightarrow> \<nu> A = (\<integral>\<^isup>+x. RN_deriv M \<nu> x * indicator A x \<partial>M)"
(is "\<And>A. _ \<Longrightarrow> ?int A")
+ and "0 \<le> RN_deriv M \<nu> x"
proof -
note Ex = Radon_Nikodym[OF assms, unfolded Bex_def]
- thus ?borel unfolding RN_deriv_def by (rule someI2_ex) auto
+ then show ?borel unfolding RN_deriv_def by (rule someI2_ex) auto
+ from Ex show "0 \<le> RN_deriv M \<nu> x" unfolding RN_deriv_def
+ by (rule someI2_ex) simp
fix A assume "A \<in> sets M"
from Ex show "?int A" unfolding RN_deriv_def
by (rule someI2_ex) (simp add: `A \<in> sets M`)
@@ -1092,22 +1167,28 @@
shows "integral\<^isup>P (M\<lparr>measure := \<nu>\<rparr>) f = (\<integral>\<^isup>+x. RN_deriv M \<nu> x * f x \<partial>M)"
proof -
interpret \<nu>: measure_space "M\<lparr>measure := \<nu>\<rparr>" by fact
- have "integral\<^isup>P (M\<lparr>measure := \<nu>\<rparr>) f =
- integral\<^isup>P (M\<lparr>measure := \<lambda>A. (\<integral>\<^isup>+x. RN_deriv M \<nu> x * indicator A x \<partial>M)\<rparr>) f"
- by (intro \<nu>.positive_integral_cong_measure[symmetric])
- (simp_all add: RN_deriv(2)[OF \<nu>, symmetric])
+ note RN = RN_deriv[OF \<nu>]
+ have "integral\<^isup>P (M\<lparr>measure := \<nu>\<rparr>) f = (\<integral>\<^isup>+x. max 0 (f x) \<partial>M\<lparr>measure := \<nu>\<rparr>)"
+ unfolding positive_integral_max_0 ..
+ also have "(\<integral>\<^isup>+x. max 0 (f x) \<partial>M\<lparr>measure := \<nu>\<rparr>) =
+ (\<integral>\<^isup>+x. max 0 (f x) \<partial>M\<lparr>measure := \<lambda>A. (\<integral>\<^isup>+x. RN_deriv M \<nu> x * indicator A x \<partial>M)\<rparr>)"
+ by (intro \<nu>.positive_integral_cong_measure[symmetric]) (simp_all add: RN(2))
+ also have "\<dots> = (\<integral>\<^isup>+x. RN_deriv M \<nu> x * max 0 (f x) \<partial>M)"
+ by (intro positive_integral_translated_density) (auto simp add: RN f)
also have "\<dots> = (\<integral>\<^isup>+x. RN_deriv M \<nu> x * f x \<partial>M)"
- by (intro positive_integral_translated_density)
- (simp_all add: RN_deriv[OF \<nu>] f)
+ using RN_deriv(3)[OF \<nu>]
+ by (auto intro!: positive_integral_cong_pos split: split_if_asm
+ simp: max_def extreal_mult_le_0_iff)
finally show ?thesis .
qed
lemma (in sigma_finite_measure) RN_deriv_unique:
assumes \<nu>: "measure_space (M\<lparr>measure := \<nu>\<rparr>)" "absolutely_continuous \<nu>"
- and f: "f \<in> borel_measurable M"
+ and f: "f \<in> borel_measurable M" "AE x. 0 \<le> f x"
and eq: "\<And>A. A \<in> sets M \<Longrightarrow> \<nu> A = (\<integral>\<^isup>+x. f x * indicator A x \<partial>M)"
shows "AE x. f x = RN_deriv M \<nu> x"
proof (rule density_unique[OF f RN_deriv(1)[OF \<nu>]])
+ show "AE x. 0 \<le> RN_deriv M \<nu> x" using RN_deriv[OF \<nu>] by auto
fix A assume A: "A \<in> sets M"
show "(\<integral>\<^isup>+x. f x * indicator A x \<partial>M) = (\<integral>\<^isup>+x. RN_deriv M \<nu> x * indicator A x \<partial>M)"
unfolding eq[OF A, symmetric] RN_deriv(2)[OF \<nu> A, symmetric] ..
@@ -1143,7 +1224,7 @@
interpret M': sigma_finite_measure M'
proof
from sigma_finite guess F .. note F = this
- show "\<exists>A::nat \<Rightarrow> 'c set. range A \<subseteq> sets M' \<and> (\<Union>i. A i) = space M' \<and> (\<forall>i. M'.\<mu> (A i) \<noteq> \<omega>)"
+ show "\<exists>A::nat \<Rightarrow> 'c set. range A \<subseteq> sets M' \<and> (\<Union>i. A i) = space M' \<and> (\<forall>i. M'.\<mu> (A i) \<noteq> \<infinity>)"
proof (intro exI conjI allI)
show *: "range (\<lambda>i. T' -` F i \<inter> space M') \<subseteq> sets M'"
using F T' by (auto simp: measurable_def measure_preserving_def)
@@ -1157,7 +1238,7 @@
then have "T -` (T' -` F i \<inter> space M') \<inter> space M = F i"
using T inv sets_into_space[OF Fi]
by (auto simp: measurable_def measure_preserving_def)
- ultimately show "measure M' (T' -` F i \<inter> space M') \<noteq> \<omega>"
+ ultimately show "measure M' (T' -` F i \<inter> space M') \<noteq> \<infinity>"
using F by simp
qed
qed
@@ -1165,6 +1246,7 @@
by (intro measurable_comp[where b=M'] M'.RN_deriv(1) measure_preservingD2[OF T]) fact+
then show "(\<lambda>x. RN_deriv M' \<nu>' (T x)) \<in> borel_measurable M"
by (simp add: comp_def)
+ show "AE x. 0 \<le> RN_deriv M' \<nu>' (T x)" using M'.RN_deriv(3)[OF \<nu>'] by auto
fix A let ?A = "T' -` A \<inter> space M'"
assume A: "A \<in> sets M"
then have A': "?A \<in> sets M'" using T' unfolding measurable_def measure_preserving_def
@@ -1185,12 +1267,12 @@
lemma (in sigma_finite_measure) RN_deriv_finite:
assumes sfm: "sigma_finite_measure (M\<lparr>measure := \<nu>\<rparr>)" and ac: "absolutely_continuous \<nu>"
- shows "AE x. RN_deriv M \<nu> x \<noteq> \<omega>"
+ shows "AE x. RN_deriv M \<nu> x \<noteq> \<infinity>"
proof -
interpret \<nu>: sigma_finite_measure "M\<lparr>measure := \<nu>\<rparr>" by fact
have \<nu>: "measure_space (M\<lparr>measure := \<nu>\<rparr>)" by default
from sfm show ?thesis
- using sigma_finite_iff_density_finite[OF \<nu> RN_deriv[OF \<nu> ac]] by simp
+ using sigma_finite_iff_density_finite[OF \<nu> RN_deriv(1)[OF \<nu> ac]] RN_deriv(2,3)[OF \<nu> ac] by simp
qed
lemma (in sigma_finite_measure)
@@ -1203,22 +1285,24 @@
proof -
interpret \<nu>: sigma_finite_measure "M\<lparr>measure := \<nu>\<rparr>" by fact
have ms: "measure_space (M\<lparr>measure := \<nu>\<rparr>)" by default
- 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 minus_cong: "\<And>A B A' B'::extreal. 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
- have Nf: "f \<in> borel_measurable (M\<lparr>measure := \<nu>\<rparr>)" using f unfolding measurable_def by auto
+ have Nf: "f \<in> borel_measurable (M\<lparr>measure := \<nu>\<rparr>)" using f by simp
{ fix f :: "'a \<Rightarrow> real"
- { fix x assume *: "RN_deriv M \<nu> x \<noteq> \<omega>"
- have "Real (real (RN_deriv M \<nu> x)) * Real (f x) = Real (real (RN_deriv M \<nu> x) * f x)"
+ { fix x assume *: "RN_deriv M \<nu> x \<noteq> \<infinity>"
+ have "extreal (real (RN_deriv M \<nu> x)) * extreal (f x) = extreal (real (RN_deriv M \<nu> x) * f x)"
by (simp add: mult_le_0_iff)
- then have "RN_deriv M \<nu> x * Real (f x) = Real (real (RN_deriv M \<nu> x) * f x)"
- using * by (simp add: Real_real) }
- then have "(\<integral>\<^isup>+x. RN_deriv M \<nu> x * Real (f x) \<partial>M) = (\<integral>\<^isup>+x. Real (real (RN_deriv M \<nu> x) * f x) \<partial>M)"
- using RN_deriv_finite[OF \<nu>] by (auto intro: positive_integral_cong_AE) }
- with this this f f' Nf
+ then have "RN_deriv M \<nu> x * extreal (f x) = extreal (real (RN_deriv M \<nu> x) * f x)"
+ using RN_deriv(3)[OF ms \<nu>(2)] * by (auto simp add: extreal_real split: split_if_asm) }
+ then have "(\<integral>\<^isup>+x. extreal (real (RN_deriv M \<nu> x) * f x) \<partial>M) = (\<integral>\<^isup>+x. RN_deriv M \<nu> x * extreal (f x) \<partial>M)"
+ "(\<integral>\<^isup>+x. extreal (- (real (RN_deriv M \<nu> x) * f x)) \<partial>M) = (\<integral>\<^isup>+x. RN_deriv M \<nu> x * extreal (- f x) \<partial>M)"
+ using RN_deriv_finite[OF \<nu>] unfolding extreal_mult_minus_right uminus_extreal.simps(1)[symmetric]
+ by (auto intro!: positive_integral_cong_AE) }
+ note * = this
show ?integral ?integrable
- unfolding lebesgue_integral_def integrable_def
- by (auto intro!: RN_deriv(1)[OF ms \<nu>(2)] minus_cong
- simp: RN_deriv_positive_integral[OF ms \<nu>(2)])
+ unfolding lebesgue_integral_def integrable_def *
+ using f RN_deriv(1)[OF ms \<nu>(2)]
+ by (auto simp: RN_deriv_positive_integral[OF ms \<nu>(2)])
qed
lemma (in sigma_finite_measure) RN_deriv_singleton:
@@ -1231,7 +1315,7 @@
from deriv(2)[OF `{x} \<in> sets M`]
have "\<nu> {x} = (\<integral>\<^isup>+w. RN_deriv M \<nu> x * indicator {x} w \<partial>M)"
by (auto simp: indicator_def intro!: positive_integral_cong)
- thus ?thesis using positive_integral_cmult_indicator[OF `{x} \<in> sets M`]
+ thus ?thesis using positive_integral_cmult_indicator[OF _ `{x} \<in> sets M`] deriv(3)
by auto
qed
--- a/src/HOL/Probability/Sigma_Algebra.thy Mon Mar 14 15:17:10 2011 +0100
+++ b/src/HOL/Probability/Sigma_Algebra.thy Mon Mar 14 15:29:10 2011 +0100
@@ -1,7 +1,7 @@
-(* Title: HOL/Probability/Sigma_Algebra.thy
- Author: Stefan Richter
- Author: Markus Wenzel
- Author: Lawrence Paulson
+(* Title: Sigma_Algebra.thy
+ Author: Stefan Richter, Markus Wenzel, TU Muenchen
+ Plus material from the Hurd/Coble measure theory development,
+ translated by Lawrence Paulson.
*)
header {* Sigma Algebras *}
@@ -70,6 +70,16 @@
"finite X \<Longrightarrow> X \<subseteq> sets M \<Longrightarrow> Union X \<in> sets M"
by (induct set: finite) (auto simp add: Un)
+lemma (in algebra) finite_UN[intro]:
+ assumes "finite I" and "\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets M"
+ shows "(\<Union>i\<in>I. A i) \<in> sets M"
+ using assms by induct auto
+
+lemma (in algebra) finite_INT[intro]:
+ assumes "finite I" "I \<noteq> {}" "\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets M"
+ shows "(\<Inter>i\<in>I. A i) \<in> sets M"
+ using assms by (induct rule: finite_ne_induct) auto
+
lemma algebra_iff_Int:
"algebra M \<longleftrightarrow>
sets M \<subseteq> Pow (space M) & {} \<in> sets M &
@@ -149,11 +159,6 @@
ultimately show ?thesis by simp
qed
-lemma (in sigma_algebra) finite_UN:
- assumes "finite I" and "\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets M"
- shows "(\<Union>i\<in>I. A i) \<in> sets M"
- using assms by induct auto
-
lemma (in sigma_algebra) countable_INT [intro]:
fixes A :: "'i::countable \<Rightarrow> 'a set"
assumes A: "A`X \<subseteq> sets M" "X \<noteq> {}"
@@ -167,11 +172,6 @@
ultimately show ?thesis by metis
qed
-lemma (in sigma_algebra) finite_INT:
- assumes "finite I" "I \<noteq> {}" "\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets M"
- shows "(\<Inter>i\<in>I. A i) \<in> sets M"
- using assms by (induct rule: finite_ne_induct) auto
-
lemma algebra_Pow:
"algebra \<lparr> space = sp, sets = Pow sp, \<dots> = X \<rparr>"
by (auto simp add: algebra_def)
--- a/src/HOL/Probability/ex/Dining_Cryptographers.thy Mon Mar 14 15:17:10 2011 +0100
+++ b/src/HOL/Probability/ex/Dining_Cryptographers.thy Mon Mar 14 15:29:10 2011 +0100
@@ -1,5 +1,5 @@
theory Dining_Cryptographers
-imports Information
+imports "~~/src/HOL/Probability/Information"
begin
locale finite_space =
@@ -8,7 +8,7 @@
and not_empty[simp]: "S \<noteq> {}"
definition (in finite_space) "M = \<lparr> space = S, sets = Pow S,
- measure = (\<lambda>A. of_nat (card A) / of_nat (card S) :: pextreal) \<rparr>"
+ measure = \<lambda>A. extreal (card A / card S) \<rparr>"
lemma (in finite_space)
shows space_M[simp]: "space M = S"
@@ -19,13 +19,14 @@
proof (rule finite_measure_spaceI)
fix A B :: "'a set" assume "A \<inter> B = {}" "A \<subseteq> space M" "B \<subseteq> space M"
then show "measure M (A \<union> B) = measure M A + measure M B"
- by (simp add: inverse_eq_divide field_simps Real_real M_def
- divide_le_0_iff zero_le_divide_iff
- card_Un_disjoint finite_subset[OF _ finite])
-qed (auto simp: M_def)
+ by (simp add: M_def card_Un_disjoint finite_subset[OF _ finite] field_simps)
+qed (auto simp: M_def divide_nonneg_nonneg)
sublocale finite_space \<subseteq> information_space M 2
- by default (simp_all add: M_def)
+ by default (simp_all add: M_def one_extreal_def)
+
+lemma (in finite_space) \<mu>'_eq[simp]: "\<mu>' A = (if A \<subseteq> S then card A / card S else 0)"
+ unfolding \<mu>'_def by (auto simp: M_def)
lemma set_of_list_extend:
"{xs. length xs = Suc n \<and> (\<forall>x\<in>set xs. x \<in> A)} =
@@ -491,7 +492,7 @@
let ?dI = "distribution inversion"
{ have "\<H>(inversion|payer) =
- - (\<Sum>x\<in>inversion`dc_crypto. (\<Sum>z\<in>Some ` {0..<n}. real (?dIP {(x, z)}) * log 2 (real (?dIP {(x, z)}) / real (?dP {z}))))"
+ - (\<Sum>x\<in>inversion`dc_crypto. (\<Sum>z\<in>Some ` {0..<n}. ?dIP {(x, z)} * log 2 (?dIP {(x, z)} / ?dP {z})))"
unfolding conditional_entropy_eq[OF simple_function_finite simple_function_finite]
by (simp add: image_payer_dc_crypto setsum_Sigma)
also have "... =
@@ -505,30 +506,27 @@
moreover from x z obtain i where "z = Some i" and "i < n" by auto
moreover from x have "length x = n" by (auto simp: inversion_def_raw dc_crypto)
ultimately
- have "real (?dIP {(x, z)}) = 2 / (real n * 2^n)" using x
- unfolding distribution_def
- by (simp add: M_def card_dc_crypto card_payer_and_inversion
- inverse_eq_divide mult_le_0_iff zero_le_mult_iff power_le_zero_eq)
+ have "?dIP {(x, z)} = 2 / (real n * 2^n)" using x
+ by (auto simp add: card_dc_crypto card_payer_and_inversion distribution_def)
moreover
from z have "payer -` {z} \<inter> dc_crypto = {z} \<times> {xs. length xs = n}"
by (auto simp: dc_crypto payer_def)
hence "card (payer -` {z} \<inter> dc_crypto) = 2^n"
using card_list_length[where A="UNIV::bool set"]
by (simp add: card_cartesian_product_singleton)
- hence "real (?dP {z}) = 1 / real n" unfolding distribution_def
- by (simp add: card_dc_crypto field_simps inverse_eq_divide M_def
- mult_le_0_iff zero_le_mult_iff power_le_zero_eq)
+ hence "?dP {z} = 1 / real n"
+ by (simp add: distribution_def card_dc_crypto)
ultimately
- show "real (?dIP {(x,z)}) * log 2 (real (?dIP {(x,z)}) / real (?dP {z})) =
+ show "?dIP {(x,z)} * log 2 (?dIP {(x,z)} / ?dP {z}) =
2 / (real n * 2^n) * (1 - real n)"
- by (simp add: field_simps log_divide log_nat_power[of 2])
+ by (simp add: log_divide log_nat_power[of 2])
qed
also have "... = real n - 1"
using n finite_space
by (simp add: card_image_inversion card_image[OF inj_Some] field_simps real_eq_of_nat[symmetric])
finally have "\<H>(inversion|payer) = real n - 1" . }
moreover
- { have "\<H>(inversion) = - (\<Sum>x \<in> inversion`dc_crypto. real (?dI {x}) * log 2 (real (?dI {x})))"
+ { have "\<H>(inversion) = - (\<Sum>x \<in> inversion`dc_crypto. ?dI {x} * log 2 (?dI {x}))"
unfolding entropy_eq[OF simple_function_finite] by simp
also have "... = - (\<Sum>x \<in> inversion`dc_crypto. 2 * (1 - real n) / 2^n)"
unfolding neg_equal_iff_equal
@@ -536,10 +534,9 @@
fix x assume x_inv: "x \<in> inversion ` dc_crypto"
hence "length x = n" by (auto simp: inversion_def_raw dc_crypto)
moreover have "inversion -` {x} \<inter> dc_crypto = {dc \<in> dc_crypto. inversion dc = x}" by auto
- ultimately have "?dI {x} = 2 / 2^n" using `0 < n` unfolding distribution_def
- by (simp add: card_inversion[OF x_inv] card_dc_crypto M_def
- mult_le_0_iff zero_le_mult_iff power_le_zero_eq)
- thus "real (?dI {x}) * log 2 (real (?dI {x})) = 2 * (1 - real n) / 2^n"
+ ultimately have "?dI {x} = 2 / 2^n" using `0 < n`
+ by (auto simp: card_inversion[OF x_inv] card_dc_crypto distribution_def)
+ thus "?dI {x} * log 2 (?dI {x}) = 2 * (1 - real n) / 2^n"
by (simp add: log_divide log_nat_power power_le_zero_eq inverse_eq_divide)
qed
also have "... = real n - 1"
--- a/src/HOL/Probability/ex/Koepf_Duermuth_Countermeasure.thy Mon Mar 14 15:17:10 2011 +0100
+++ b/src/HOL/Probability/ex/Koepf_Duermuth_Countermeasure.thy Mon Mar 14 15:29:10 2011 +0100
@@ -3,7 +3,7 @@
header {* Formalization of a Countermeasure by Koepf \& Duermuth 2009 *}
theory Koepf_Duermuth_Countermeasure
- imports Information "~~/src/HOL/Library/Permutation"
+ imports "~~/src/HOL/Probability/Information" "~~/src/HOL/Library/Permutation"
begin
lemma
@@ -201,14 +201,17 @@
lemma (in finite_information) positive_p_sum[simp]: "0 \<le> setsum p X"
by (auto intro!: setsum_nonneg)
-sublocale finite_information \<subseteq> finite_measure_space "\<lparr> space = \<Omega>, sets = Pow \<Omega>, measure = \<lambda>S. Real (setsum p S)\<rparr>"
+sublocale finite_information \<subseteq> finite_measure_space "\<lparr> space = \<Omega>, sets = Pow \<Omega>, measure = \<lambda>S. extreal (setsum p S)\<rparr>"
by (rule finite_measure_spaceI) (simp_all add: setsum_Un_disjoint finite_subset)
-sublocale finite_information \<subseteq> finite_prob_space "\<lparr> space = \<Omega>, sets = Pow \<Omega>, measure = \<lambda>S. Real (setsum p S)\<rparr>"
+sublocale finite_information \<subseteq> finite_prob_space "\<lparr> space = \<Omega>, sets = Pow \<Omega>, measure = \<lambda>S. extreal (setsum p S)\<rparr>"
+ by default (simp add: one_extreal_def)
+
+sublocale finite_information \<subseteq> information_space "\<lparr> space = \<Omega>, sets = Pow \<Omega>, measure = \<lambda>S. extreal (setsum p S)\<rparr>" b
by default simp
-sublocale finite_information \<subseteq> information_space "\<lparr> space = \<Omega>, sets = Pow \<Omega>, measure = \<lambda>S. Real (setsum p S)\<rparr>" b
- by default simp
+lemma (in finite_information) \<mu>'_eq: "A \<subseteq> \<Omega> \<Longrightarrow> \<mu>' A = setsum p A"
+ unfolding \<mu>'_def by auto
locale koepf_duermuth = K: finite_information keys K b + M: finite_information messages M b
for b :: real
@@ -259,16 +262,6 @@
"snd ` (SIGMA x:f`X. f -` {x} \<inter> X) = X"
by (auto simp: image_iff)
-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"
- shows "(\<Prod>i\<in>X. Real (f i)) = Real (\<Prod>i\<in>X. f i)"
-proof cases
- assume "finite X"
- from this assms show ?thesis by induct (auto simp: mult_le_0_iff)
-qed simp
-
lemma inj_Cons[simp]: "\<And>X. inj_on (\<lambda>(xs, x). x#xs) X" by (auto intro!: inj_onI)
lemma setprod_setsum_distrib_lists:
@@ -323,10 +316,10 @@
"p A \<equiv> setsum P A"
abbreviation probability ("\<P>'(_') _") where
- "\<P>(X) x \<equiv> real (distribution X x)"
+ "\<P>(X) x \<equiv> distribution X x"
abbreviation joint_probability ("\<P>'(_, _') _") where
- "\<P>(X, Y) x \<equiv> real (joint_distribution X Y x)"
+ "\<P>(X, Y) x \<equiv> joint_distribution X Y x"
abbreviation conditional_probability ("\<P>'(_|_') _") where
"\<P>(X|Y) x \<equiv> \<P>(X, Y) x / \<P>(Y) (snd`x)"
@@ -355,11 +348,12 @@
from assms have *:
"fst -` {k} \<inter> msgs = {k}\<times>{ms. length ms = n \<and> (\<forall>M\<in>set ms. M \<in> messages)}"
unfolding msgs_def by auto
- show "\<P>(fst) {k} = K k" unfolding distribution_def
- apply (simp add: *) unfolding P_def
+ show "\<P>(fst) {k} = K k"
+ apply (simp add: \<mu>'_eq distribution_def)
+ apply (simp add: * P_def)
apply (simp add: setsum_cartesian_product')
- using setprod_setsum_distrib_lists[OF M.finite_space, of M n "\<lambda>x x. True"]
- by (simp add: setsum_right_distrib[symmetric] subset_eq setprod_1)
+ using setprod_setsum_distrib_lists[OF M.finite_space, of M n "\<lambda>x x. True"] `k \<in> keys`
+ by (auto simp add: setsum_right_distrib[symmetric] subset_eq setprod_1)
qed
lemma fst_image_msgs[simp]: "fst`msgs = keys"
@@ -390,7 +384,7 @@
have "\<P>(OB, fst) {(obs, k)} / K k =
p ({k}\<times>{ms. (k,ms) \<in> msgs \<and> OB (k,ms) = obs}) / K k"
- unfolding distribution_def by (auto intro!: arg_cong[where f=p])
+ apply (simp add: distribution_def \<mu>'_eq) by (auto intro!: arg_cong[where f=p])
also have "\<dots> =
(\<Prod>i<n. \<Sum>m\<in>{m\<in>messages. observe k m = obs ! i}. M m)"
unfolding P_def using `K k \<noteq> 0` `k \<in> keys`
@@ -415,7 +409,7 @@
unfolding disjoint_family_on_def by auto
have "\<P>(t\<circ>OB, fst) {(t obs, k)} = (\<Sum>obs'\<in>?S obs. \<P>(OB, fst) {(obs', k)})"
unfolding distribution_def comp_def
- using real_finite_measure_finite_Union[OF _ df]
+ using finite_measure_finite_Union[OF _ _ df]
by (force simp add: * intro!: setsum_nonneg)
also have "(\<Sum>obs'\<in>?S obs. \<P>(OB, fst) {(obs', k)}) = real (card (?S obs)) * \<P>(OB, fst) {(obs, k)}"
by (simp add: t_eq_imp[OF `k \<in> keys` `K k \<noteq> 0` obs] real_eq_of_nat)
@@ -433,11 +427,11 @@
then have "real (card ?S) \<noteq> 0" by auto
have "\<P>(fst | t\<circ>OB) {(k, t obs)} = \<P>(t\<circ>OB | fst) {(t obs, k)} * \<P>(fst) {k} / \<P>(t\<circ>OB) {t obs}"
- using real_distribution_order'[of fst k "t\<circ>OB" "t obs"]
+ using distribution_order(7,8)[where X=fst and x=k and Y="t\<circ>OB" and y="t obs"]
by (subst joint_distribution_commute) auto
also have "\<P>(t\<circ>OB) {t obs} = (\<Sum>k'\<in>keys. \<P>(t\<circ>OB|fst) {(t obs, k')} * \<P>(fst) {k'})"
- using setsum_real_distribution(2)[of "t\<circ>OB" fst "t obs", symmetric]
- using real_distribution_order'[of fst _ "t\<circ>OB" "t obs"] by (auto intro!: setsum_cong)
+ using setsum_distribution(2)[of "t\<circ>OB" fst "t obs", symmetric]
+ by (auto intro!: setsum_cong distribution_order(8))
also have "\<P>(t\<circ>OB | fst) {(t obs, k)} * \<P>(fst) {k} / (\<Sum>k'\<in>keys. \<P>(t\<circ>OB|fst) {(t obs, k')} * \<P>(fst) {k'}) =
\<P>(OB | fst) {(obs, k)} * \<P>(fst) {k} / (\<Sum>k'\<in>keys. \<P>(OB|fst) {(obs, k')} * \<P>(fst) {k'})"
using CP_t_K[OF `k\<in>keys` obs] CP_t_K[OF _ obs] `real (card ?S) \<noteq> 0`
@@ -445,11 +439,10 @@
mult_divide_mult_cancel_left[OF `real (card ?S) \<noteq> 0`]
cong: setsum_cong)
also have "(\<Sum>k'\<in>keys. \<P>(OB|fst) {(obs, k')} * \<P>(fst) {k'}) = \<P>(OB) {obs}"
- using setsum_real_distribution(2)[of OB fst obs, symmetric]
- using real_distribution_order'[of fst _ OB obs] by (auto intro!: setsum_cong)
+ using setsum_distribution(2)[of OB fst obs, symmetric]
+ by (auto intro!: setsum_cong distribution_order(8))
also have "\<P>(OB | fst) {(obs, k)} * \<P>(fst) {k} / \<P>(OB) {obs} = \<P>(fst | OB) {(k, obs)}"
- using real_distribution_order'[of fst k OB obs]
- by (subst joint_distribution_commute) auto
+ by (subst joint_distribution_commute) (auto intro!: distribution_order(8))
finally have "\<P>(fst | t\<circ>OB) {(k, t obs)} = \<P>(fst | OB) {(k, obs)}" . }
note CP_T_eq_CP_O = this
@@ -472,7 +465,7 @@
unfolding disjoint_family_on_def by auto
have "\<P>(t\<circ>OB) {t (OB x)} = (\<Sum>obs\<in>?S (OB x). \<P>(OB) {obs})"
unfolding distribution_def comp_def
- using real_finite_measure_finite_Union[OF _ df]
+ using finite_measure_finite_Union[OF _ _ df]
by (force simp add: * intro!: setsum_nonneg) }
note P_t_sum_P_O = this
--- a/src/HOL/RealVector.thy Mon Mar 14 15:17:10 2011 +0100
+++ b/src/HOL/RealVector.thy Mon Mar 14 15:29:10 2011 +0100
@@ -534,6 +534,34 @@
lemma dist_triangle3: "dist x y \<le> dist a x + dist a y"
using dist_triangle2 [of x y a] by (simp add: dist_commute)
+lemma dist_triangle_alt:
+ shows "dist y z <= dist x y + dist x z"
+by (rule dist_triangle3)
+
+lemma dist_pos_lt:
+ shows "x \<noteq> y ==> 0 < dist x y"
+by (simp add: zero_less_dist_iff)
+
+lemma dist_nz:
+ shows "x \<noteq> y \<longleftrightarrow> 0 < dist x y"
+by (simp add: zero_less_dist_iff)
+
+lemma dist_triangle_le:
+ shows "dist x z + dist y z <= e \<Longrightarrow> dist x y <= e"
+by (rule order_trans [OF dist_triangle2])
+
+lemma dist_triangle_lt:
+ shows "dist x z + dist y z < e ==> dist x y < e"
+by (rule le_less_trans [OF dist_triangle2])
+
+lemma dist_triangle_half_l:
+ shows "dist x1 y < e / 2 \<Longrightarrow> dist x2 y < e / 2 \<Longrightarrow> dist x1 x2 < e"
+by (rule dist_triangle_lt [where z=y], simp)
+
+lemma dist_triangle_half_r:
+ shows "dist y x1 < e / 2 \<Longrightarrow> dist y x2 < e / 2 \<Longrightarrow> dist x1 x2 < e"
+by (rule dist_triangle_half_l, simp_all add: dist_commute)
+
subclass topological_space
proof
have "\<exists>e::real. 0 < e"
@@ -554,6 +582,13 @@
unfolding open_dist by fast
qed
+lemma (in metric_space) open_ball: "open {y. dist x y < d}"
+proof (unfold open_dist, intro ballI)
+ fix y assume *: "y \<in> {y. dist x y < d}"
+ then show "\<exists>e>0. \<forall>z. dist z y < e \<longrightarrow> z \<in> {y. dist x y < d}"
+ by (auto intro!: exI[of _ "d - dist x y"] simp: field_simps dist_triangle_lt)
+qed
+
end
@@ -1060,4 +1095,78 @@
interpretation of_real: bounded_linear "\<lambda>r. of_real r"
unfolding of_real_def by (rule scaleR.bounded_linear_left)
+subsection{* Hausdorff and other separation properties *}
+
+class t0_space = topological_space +
+ assumes t0_space: "x \<noteq> y \<Longrightarrow> \<exists>U. open U \<and> \<not> (x \<in> U \<longleftrightarrow> y \<in> U)"
+
+class t1_space = topological_space +
+ assumes t1_space: "x \<noteq> y \<Longrightarrow> \<exists>U. open U \<and> x \<in> U \<and> y \<notin> U"
+
+instance t1_space \<subseteq> t0_space
+proof qed (fast dest: t1_space)
+
+lemma separation_t1:
+ fixes x y :: "'a::t1_space"
+ shows "x \<noteq> y \<longleftrightarrow> (\<exists>U. open U \<and> x \<in> U \<and> y \<notin> U)"
+ using t1_space[of x y] by blast
+
+lemma closed_singleton:
+ fixes a :: "'a::t1_space"
+ shows "closed {a}"
+proof -
+ let ?T = "\<Union>{S. open S \<and> a \<notin> S}"
+ have "open ?T" by (simp add: open_Union)
+ also have "?T = - {a}"
+ by (simp add: set_eq_iff separation_t1, auto)
+ finally show "closed {a}" unfolding closed_def .
+qed
+
+lemma closed_insert [simp]:
+ fixes a :: "'a::t1_space"
+ assumes "closed S" shows "closed (insert a S)"
+proof -
+ from closed_singleton assms
+ have "closed ({a} \<union> S)" by (rule closed_Un)
+ thus "closed (insert a S)" by simp
+qed
+
+lemma finite_imp_closed:
+ fixes S :: "'a::t1_space set"
+ shows "finite S \<Longrightarrow> closed S"
+by (induct set: finite, simp_all)
+
+text {* T2 spaces are also known as Hausdorff spaces. *}
+
+class t2_space = topological_space +
+ assumes hausdorff: "x \<noteq> y \<Longrightarrow> \<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {}"
+
+instance t2_space \<subseteq> t1_space
+proof qed (fast dest: hausdorff)
+
+instance metric_space \<subseteq> t2_space
+proof
+ fix x y :: "'a::metric_space"
+ assume xy: "x \<noteq> y"
+ let ?U = "{y'. dist x y' < dist x y / 2}"
+ let ?V = "{x'. dist y x' < dist x y / 2}"
+ have th0: "\<And>d x y z. (d x z :: real) \<le> d x y + d y z \<Longrightarrow> d y z = d z y
+ \<Longrightarrow> \<not>(d x y * 2 < d x z \<and> d z y * 2 < d x z)" by arith
+ have "open ?U \<and> open ?V \<and> x \<in> ?U \<and> y \<in> ?V \<and> ?U \<inter> ?V = {}"
+ using dist_pos_lt[OF xy] th0[of dist, OF dist_triangle dist_commute]
+ using open_ball[of _ "dist x y / 2"] by auto
+ then show "\<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {}"
+ by blast
+qed
+
+lemma separation_t2:
+ fixes x y :: "'a::t2_space"
+ shows "x \<noteq> y \<longleftrightarrow> (\<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {})"
+ using hausdorff[of x y] by blast
+
+lemma separation_t0:
+ fixes x y :: "'a::t0_space"
+ shows "x \<noteq> y \<longleftrightarrow> (\<exists>U. open U \<and> ~(x\<in>U \<longleftrightarrow> y\<in>U))"
+ using t0_space[of x y] by blast
+
end
--- a/src/HOL/SEQ.thy Mon Mar 14 15:17:10 2011 +0100
+++ b/src/HOL/SEQ.thy Mon Mar 14 15:29:10 2011 +0100
@@ -13,25 +13,7 @@
imports Limits RComplete
begin
-abbreviation
- LIMSEQ :: "[nat \<Rightarrow> 'a::topological_space, 'a] \<Rightarrow> bool"
- ("((_)/ ----> (_))" [60, 60] 60) where
- "X ----> L \<equiv> (X ---> L) sequentially"
-
-definition
- lim :: "(nat \<Rightarrow> 'a::metric_space) \<Rightarrow> 'a" where
- --{*Standard definition of limit using choice operator*}
- "lim X = (THE L. X ----> L)"
-
-definition
- convergent :: "(nat \<Rightarrow> 'a::metric_space) \<Rightarrow> bool" where
- --{*Standard definition of convergence*}
- "convergent X = (\<exists>L. X ----> L)"
-
-definition
- Bseq :: "(nat => 'a::real_normed_vector) => bool" where
- --{*Standard definition for bounded sequence*}
- "Bseq X = (\<exists>K>0.\<forall>n. norm (X n) \<le> K)"
+subsection {* Monotone sequences and subsequences *}
definition
monoseq :: "(nat \<Rightarrow> 'a::order) \<Rightarrow> bool" where
@@ -56,6 +38,171 @@
--{*Definition of subsequence*}
"subseq f \<longleftrightarrow> (\<forall>m. \<forall>n>m. f m < f n)"
+lemma incseq_mono: "mono f \<longleftrightarrow> incseq f"
+ unfolding mono_def incseq_def by auto
+
+lemma incseq_SucI:
+ "(\<And>n. X n \<le> X (Suc n)) \<Longrightarrow> incseq X"
+ using lift_Suc_mono_le[of X]
+ by (auto simp: incseq_def)
+
+lemma incseqD: "\<And>i j. incseq f \<Longrightarrow> i \<le> j \<Longrightarrow> f i \<le> f j"
+ by (auto simp: incseq_def)
+
+lemma incseq_SucD: "incseq A \<Longrightarrow> A i \<le> A (Suc i)"
+ using incseqD[of A i "Suc i"] by auto
+
+lemma incseq_Suc_iff: "incseq f \<longleftrightarrow> (\<forall>n. f n \<le> f (Suc n))"
+ by (auto intro: incseq_SucI dest: incseq_SucD)
+
+lemma incseq_const[simp, intro]: "incseq (\<lambda>x. k)"
+ unfolding incseq_def by auto
+
+lemma decseq_SucI:
+ "(\<And>n. X (Suc n) \<le> X n) \<Longrightarrow> decseq X"
+ using order.lift_Suc_mono_le[OF dual_order, of X]
+ by (auto simp: decseq_def)
+
+lemma decseqD: "\<And>i j. decseq f \<Longrightarrow> i \<le> j \<Longrightarrow> f j \<le> f i"
+ by (auto simp: decseq_def)
+
+lemma decseq_SucD: "decseq A \<Longrightarrow> A (Suc i) \<le> A i"
+ using decseqD[of A i "Suc i"] by auto
+
+lemma decseq_Suc_iff: "decseq f \<longleftrightarrow> (\<forall>n. f (Suc n) \<le> f n)"
+ by (auto intro: decseq_SucI dest: decseq_SucD)
+
+lemma decseq_const[simp, intro]: "decseq (\<lambda>x. k)"
+ unfolding decseq_def by auto
+
+lemma monoseq_iff: "monoseq X \<longleftrightarrow> incseq X \<or> decseq X"
+ unfolding monoseq_def incseq_def decseq_def ..
+
+lemma monoseq_Suc:
+ "monoseq X \<longleftrightarrow> (\<forall>n. X n \<le> X (Suc n)) \<or> (\<forall>n. X (Suc n) \<le> X n)"
+ unfolding monoseq_iff incseq_Suc_iff decseq_Suc_iff ..
+
+lemma monoI1: "\<forall>m. \<forall> n \<ge> m. X m \<le> X n ==> monoseq X"
+by (simp add: monoseq_def)
+
+lemma monoI2: "\<forall>m. \<forall> n \<ge> m. X n \<le> X m ==> monoseq X"
+by (simp add: monoseq_def)
+
+lemma mono_SucI1: "\<forall>n. X n \<le> X (Suc n) ==> monoseq X"
+by (simp add: monoseq_Suc)
+
+lemma mono_SucI2: "\<forall>n. X (Suc n) \<le> X n ==> monoseq X"
+by (simp add: monoseq_Suc)
+
+lemma monoseq_minus:
+ fixes a :: "nat \<Rightarrow> 'a::ordered_ab_group_add"
+ assumes "monoseq a"
+ shows "monoseq (\<lambda> n. - a n)"
+proof (cases "\<forall> m. \<forall> n \<ge> m. a m \<le> a n")
+ case True
+ hence "\<forall> m. \<forall> n \<ge> m. - a n \<le> - a m" by auto
+ thus ?thesis by (rule monoI2)
+next
+ case False
+ hence "\<forall> m. \<forall> n \<ge> m. - a m \<le> - a n" using `monoseq a`[unfolded monoseq_def] by auto
+ thus ?thesis by (rule monoI1)
+qed
+
+text{*Subsequence (alternative definition, (e.g. Hoskins)*}
+
+lemma subseq_Suc_iff: "subseq f = (\<forall>n. (f n) < (f (Suc n)))"
+apply (simp add: subseq_def)
+apply (auto dest!: less_imp_Suc_add)
+apply (induct_tac k)
+apply (auto intro: less_trans)
+done
+
+text{* for any sequence, there is a monotonic subsequence *}
+lemma seq_monosub:
+ fixes s :: "nat => 'a::linorder"
+ shows "\<exists>f. subseq f \<and> monoseq (\<lambda> n. (s (f n)))"
+proof cases
+ let "?P p n" = "p > n \<and> (\<forall>m\<ge>p. s m \<le> s p)"
+ assume *: "\<forall>n. \<exists>p. ?P p n"
+ def f \<equiv> "nat_rec (SOME p. ?P p 0) (\<lambda>_ n. SOME p. ?P p n)"
+ have f_0: "f 0 = (SOME p. ?P p 0)" unfolding f_def by simp
+ have f_Suc: "\<And>i. f (Suc i) = (SOME p. ?P p (f i))" unfolding f_def nat_rec_Suc ..
+ have P_0: "?P (f 0) 0" unfolding f_0 using *[rule_format] by (rule someI2_ex) auto
+ have P_Suc: "\<And>i. ?P (f (Suc i)) (f i)" unfolding f_Suc using *[rule_format] by (rule someI2_ex) auto
+ then have "subseq f" unfolding subseq_Suc_iff by auto
+ moreover have "monoseq (\<lambda>n. s (f n))" unfolding monoseq_Suc
+ proof (intro disjI2 allI)
+ fix n show "s (f (Suc n)) \<le> s (f n)"
+ proof (cases n)
+ case 0 with P_Suc[of 0] P_0 show ?thesis by auto
+ next
+ case (Suc m)
+ from P_Suc[of n] Suc have "f (Suc m) \<le> f (Suc (Suc m))" by simp
+ with P_Suc Suc show ?thesis by simp
+ qed
+ qed
+ ultimately show ?thesis by auto
+next
+ let "?P p m" = "m < p \<and> s m < s p"
+ assume "\<not> (\<forall>n. \<exists>p>n. (\<forall>m\<ge>p. s m \<le> s p))"
+ then obtain N where N: "\<And>p. p > N \<Longrightarrow> \<exists>m>p. s p < s m" by (force simp: not_le le_less)
+ def f \<equiv> "nat_rec (SOME p. ?P p (Suc N)) (\<lambda>_ n. SOME p. ?P p n)"
+ have f_0: "f 0 = (SOME p. ?P p (Suc N))" unfolding f_def by simp
+ have f_Suc: "\<And>i. f (Suc i) = (SOME p. ?P p (f i))" unfolding f_def nat_rec_Suc ..
+ have P_0: "?P (f 0) (Suc N)"
+ unfolding f_0 some_eq_ex[of "\<lambda>p. ?P p (Suc N)"] using N[of "Suc N"] by auto
+ { fix i have "N < f i \<Longrightarrow> ?P (f (Suc i)) (f i)"
+ unfolding f_Suc some_eq_ex[of "\<lambda>p. ?P p (f i)"] using N[of "f i"] . }
+ note P' = this
+ { fix i have "N < f i \<and> ?P (f (Suc i)) (f i)"
+ by (induct i) (insert P_0 P', auto) }
+ then have "subseq f" "monoseq (\<lambda>x. s (f x))"
+ unfolding subseq_Suc_iff monoseq_Suc by (auto simp: not_le intro: less_imp_le)
+ then show ?thesis by auto
+qed
+
+lemma seq_suble: assumes sf: "subseq f" shows "n \<le> f n"
+proof(induct n)
+ case 0 thus ?case by simp
+next
+ case (Suc n)
+ from sf[unfolded subseq_Suc_iff, rule_format, of n] Suc.hyps
+ have "n < f (Suc n)" by arith
+ thus ?case by arith
+qed
+
+lemma incseq_imp_monoseq: "incseq X \<Longrightarrow> monoseq X"
+ by (simp add: incseq_def monoseq_def)
+
+lemma decseq_imp_monoseq: "decseq X \<Longrightarrow> monoseq X"
+ by (simp add: decseq_def monoseq_def)
+
+lemma decseq_eq_incseq:
+ fixes X :: "nat \<Rightarrow> 'a::ordered_ab_group_add" shows "decseq X = incseq (\<lambda>n. - X n)"
+ by (simp add: decseq_def incseq_def)
+
+subsection {* Defintions of limits *}
+
+abbreviation
+ LIMSEQ :: "[nat \<Rightarrow> 'a::topological_space, 'a] \<Rightarrow> bool"
+ ("((_)/ ----> (_))" [60, 60] 60) where
+ "X ----> L \<equiv> (X ---> L) sequentially"
+
+definition
+ lim :: "(nat \<Rightarrow> 'a::metric_space) \<Rightarrow> 'a" where
+ --{*Standard definition of limit using choice operator*}
+ "lim X = (THE L. X ----> L)"
+
+definition
+ convergent :: "(nat \<Rightarrow> 'a::metric_space) \<Rightarrow> bool" where
+ --{*Standard definition of convergence*}
+ "convergent X = (\<exists>L. X ----> L)"
+
+definition
+ Bseq :: "(nat => 'a::real_normed_vector) => bool" where
+ --{*Standard definition for bounded sequence*}
+ "Bseq X = (\<exists>K>0.\<forall>n. norm (X n) \<le> K)"
+
definition
Cauchy :: "(nat \<Rightarrow> 'a::metric_space) \<Rightarrow> bool" where
--{*Standard definition of the Cauchy condition*}
@@ -502,53 +649,6 @@
qed
qed
-text{*Subsequence (alternative definition, (e.g. Hoskins)*}
-
-lemma subseq_Suc_iff: "subseq f = (\<forall>n. (f n) < (f (Suc n)))"
-apply (simp add: subseq_def)
-apply (auto dest!: less_imp_Suc_add)
-apply (induct_tac k)
-apply (auto intro: less_trans)
-done
-
-lemma monoseq_Suc:
- "monoseq X \<longleftrightarrow> (\<forall>n. X n \<le> X (Suc n)) \<or> (\<forall>n. X (Suc n) \<le> X n)"
-apply (simp add: monoseq_def)
-apply (auto dest!: le_imp_less_or_eq)
-apply (auto intro!: lessI [THEN less_imp_le] dest!: less_imp_Suc_add)
-apply (induct_tac "ka")
-apply (auto intro: order_trans)
-apply (erule contrapos_np)
-apply (induct_tac "k")
-apply (auto intro: order_trans)
-done
-
-lemma monoI1: "\<forall>m. \<forall> n \<ge> m. X m \<le> X n ==> monoseq X"
-by (simp add: monoseq_def)
-
-lemma monoI2: "\<forall>m. \<forall> n \<ge> m. X n \<le> X m ==> monoseq X"
-by (simp add: monoseq_def)
-
-lemma mono_SucI1: "\<forall>n. X n \<le> X (Suc n) ==> monoseq X"
-by (simp add: monoseq_Suc)
-
-lemma mono_SucI2: "\<forall>n. X (Suc n) \<le> X n ==> monoseq X"
-by (simp add: monoseq_Suc)
-
-lemma monoseq_minus:
- fixes a :: "nat \<Rightarrow> 'a::ordered_ab_group_add"
- assumes "monoseq a"
- shows "monoseq (\<lambda> n. - a n)"
-proof (cases "\<forall> m. \<forall> n \<ge> m. a m \<le> a n")
- case True
- hence "\<forall> m. \<forall> n \<ge> m. - a n \<le> - a m" by auto
- thus ?thesis by (rule monoI2)
-next
- case False
- hence "\<forall> m. \<forall> n \<ge> m. - a m \<le> - a n" using `monoseq a`[unfolded monoseq_def] by auto
- thus ?thesis by (rule monoI1)
-qed
-
lemma monoseq_le:
fixes a :: "nat \<Rightarrow> real"
assumes "monoseq a" and "a ----> x"
@@ -602,60 +702,6 @@
qed auto
qed
-text{* for any sequence, there is a monotonic subsequence *}
-lemma seq_monosub:
- fixes s :: "nat => 'a::linorder"
- shows "\<exists>f. subseq f \<and> monoseq (\<lambda> n. (s (f n)))"
-proof cases
- let "?P p n" = "p > n \<and> (\<forall>m\<ge>p. s m \<le> s p)"
- assume *: "\<forall>n. \<exists>p. ?P p n"
- def f \<equiv> "nat_rec (SOME p. ?P p 0) (\<lambda>_ n. SOME p. ?P p n)"
- have f_0: "f 0 = (SOME p. ?P p 0)" unfolding f_def by simp
- have f_Suc: "\<And>i. f (Suc i) = (SOME p. ?P p (f i))" unfolding f_def nat_rec_Suc ..
- have P_0: "?P (f 0) 0" unfolding f_0 using *[rule_format] by (rule someI2_ex) auto
- have P_Suc: "\<And>i. ?P (f (Suc i)) (f i)" unfolding f_Suc using *[rule_format] by (rule someI2_ex) auto
- then have "subseq f" unfolding subseq_Suc_iff by auto
- moreover have "monoseq (\<lambda>n. s (f n))" unfolding monoseq_Suc
- proof (intro disjI2 allI)
- fix n show "s (f (Suc n)) \<le> s (f n)"
- proof (cases n)
- case 0 with P_Suc[of 0] P_0 show ?thesis by auto
- next
- case (Suc m)
- from P_Suc[of n] Suc have "f (Suc m) \<le> f (Suc (Suc m))" by simp
- with P_Suc Suc show ?thesis by simp
- qed
- qed
- ultimately show ?thesis by auto
-next
- let "?P p m" = "m < p \<and> s m < s p"
- assume "\<not> (\<forall>n. \<exists>p>n. (\<forall>m\<ge>p. s m \<le> s p))"
- then obtain N where N: "\<And>p. p > N \<Longrightarrow> \<exists>m>p. s p < s m" by (force simp: not_le le_less)
- def f \<equiv> "nat_rec (SOME p. ?P p (Suc N)) (\<lambda>_ n. SOME p. ?P p n)"
- have f_0: "f 0 = (SOME p. ?P p (Suc N))" unfolding f_def by simp
- have f_Suc: "\<And>i. f (Suc i) = (SOME p. ?P p (f i))" unfolding f_def nat_rec_Suc ..
- have P_0: "?P (f 0) (Suc N)"
- unfolding f_0 some_eq_ex[of "\<lambda>p. ?P p (Suc N)"] using N[of "Suc N"] by auto
- { fix i have "N < f i \<Longrightarrow> ?P (f (Suc i)) (f i)"
- unfolding f_Suc some_eq_ex[of "\<lambda>p. ?P p (f i)"] using N[of "f i"] . }
- note P' = this
- { fix i have "N < f i \<and> ?P (f (Suc i)) (f i)"
- by (induct i) (insert P_0 P', auto) }
- then have "subseq f" "monoseq (\<lambda>x. s (f x))"
- unfolding subseq_Suc_iff monoseq_Suc by (auto simp: not_le intro: less_imp_le)
- then show ?thesis by auto
-qed
-
-lemma seq_suble: assumes sf: "subseq f" shows "n \<le> f n"
-proof(induct n)
- case 0 thus ?case by simp
-next
- case (Suc n)
- from sf[unfolded subseq_Suc_iff, rule_format, of n] Suc.hyps
- have "n < f (Suc n)" by arith
- thus ?case by arith
-qed
-
lemma LIMSEQ_subseq_LIMSEQ:
"\<lbrakk> X ----> L; subseq f \<rbrakk> \<Longrightarrow> (X o f) ----> L"
apply (rule topological_tendstoI)
@@ -810,9 +856,6 @@
subsubsection{*Increasing and Decreasing Series*}
-lemma incseq_imp_monoseq: "incseq X \<Longrightarrow> monoseq X"
- by (simp add: incseq_def monoseq_def)
-
lemma incseq_le:
fixes X :: "nat \<Rightarrow> real"
assumes inc: "incseq X" and lim: "X ----> L" shows "X n \<le> L"
@@ -834,32 +877,6 @@
by (blast intro: eq_refl X)
qed
-lemma incseq_SucI:
- assumes "\<And>n. X n \<le> X (Suc n)"
- shows "incseq X" unfolding incseq_def
-proof safe
- fix m n :: nat
- { fix d m :: nat
- have "X m \<le> X (m + d)"
- proof (induct d)
- case (Suc d)
- also have "X (m + d) \<le> X (m + Suc d)"
- using assms by simp
- finally show ?case .
- qed simp }
- note this[of m "n - m"]
- moreover assume "m \<le> n"
- ultimately show "X m \<le> X n" by simp
-qed
-
-lemma decseq_imp_monoseq: "decseq X \<Longrightarrow> monoseq X"
- by (simp add: decseq_def monoseq_def)
-
-lemma decseq_eq_incseq:
- fixes X :: "nat \<Rightarrow> 'a::ordered_ab_group_add" shows "decseq X = incseq (\<lambda>n. - X n)"
- by (simp add: decseq_def incseq_def)
-
-
lemma decseq_le:
fixes X :: "nat \<Rightarrow> real" assumes dec: "decseq X" and lim: "X ----> L" shows "L \<le> X n"
proof -
--- a/src/HOL/Series.thy Mon Mar 14 15:17:10 2011 +0100
+++ b/src/HOL/Series.thy Mon Mar 14 15:29:10 2011 +0100
@@ -5,7 +5,7 @@
Converted to Isar and polished by lcp
Converted to setsum and polished yet more by TNN
Additional contributions by Jeremy Avigad
-*)
+*)
header{*Finite Summation and Infinite Series*}
@@ -14,16 +14,16 @@
begin
definition
- sums :: "(nat \<Rightarrow> 'a::real_normed_vector) \<Rightarrow> 'a \<Rightarrow> bool"
+ sums :: "(nat \<Rightarrow> 'a::{topological_space, comm_monoid_add}) \<Rightarrow> 'a \<Rightarrow> bool"
(infixr "sums" 80) where
"f sums s = (%n. setsum f {0..<n}) ----> s"
definition
- summable :: "(nat \<Rightarrow> 'a::real_normed_vector) \<Rightarrow> bool" where
+ summable :: "(nat \<Rightarrow> 'a::{topological_space, comm_monoid_add}) \<Rightarrow> bool" where
"summable f = (\<exists>s. f sums s)"
definition
- suminf :: "(nat \<Rightarrow> 'a::real_normed_vector) \<Rightarrow> 'a" where
+ suminf :: "(nat \<Rightarrow> 'a::{topological_space, comm_monoid_add}) \<Rightarrow> 'a" where
"suminf f = (THE s. f sums s)"
syntax
@@ -81,62 +81,65 @@
"\<forall>n f. setsum f {0::nat..<n+k} = (\<Sum>m=0..<n. f (m+k)::real) + setsum f {0..<k}"
by (clarify, rule sumr_offset3)
-(*
-lemma sumr_from_1_from_0: "0 < n ==>
- (\<Sum>n=Suc 0 ..< Suc n. if even(n) then 0 else
- ((- 1) ^ ((n - (Suc 0)) div 2))/(real (fact n))) * a ^ n =
- (\<Sum>n=0..<Suc n. if even(n) then 0 else
- ((- 1) ^ ((n - (Suc 0)) div 2))/(real (fact n))) * a ^ n"
-by (rule_tac n1 = 1 in sumr_split_add [THEN subst], auto)
-*)
-
subsection{* Infinite Sums, by the Properties of Limits*}
(*----------------------
- suminf is the sum
+ suminf is the sum
---------------------*)
lemma sums_summable: "f sums l ==> summable f"
-by (simp add: sums_def summable_def, blast)
+ by (simp add: sums_def summable_def, blast)
-lemma summable_sums: "summable f ==> f sums (suminf f)"
-apply (simp add: summable_def suminf_def sums_def)
-apply (fast intro: theI LIMSEQ_unique)
-done
+lemma summable_sums:
+ fixes f :: "nat \<Rightarrow> 'a::{t2_space, comm_monoid_add}" assumes "summable f" shows "f sums (suminf f)"
+proof -
+ from assms guess s unfolding summable_def sums_def_raw .. note s = this
+ then show ?thesis unfolding sums_def_raw suminf_def
+ by (rule theI, auto intro!: tendsto_unique[OF trivial_limit_sequentially])
+qed
-lemma summable_sumr_LIMSEQ_suminf:
- "summable f ==> (%n. setsum f {0..<n}) ----> (suminf f)"
+lemma summable_sumr_LIMSEQ_suminf:
+ fixes f :: "nat \<Rightarrow> 'a::{t2_space, comm_monoid_add}"
+ shows "summable f \<Longrightarrow> (\<lambda>n. setsum f {0..<n}) ----> suminf f"
by (rule summable_sums [unfolded sums_def])
lemma suminf_eq_lim: "suminf f = lim (%n. setsum f {0..<n})"
- by (simp add: suminf_def sums_def lim_def)
+ by (simp add: suminf_def sums_def lim_def)
(*-------------------
- sum is unique
+ sum is unique
------------------*)
-lemma sums_unique: "f sums s ==> (s = suminf f)"
-apply (frule sums_summable [THEN summable_sums])
-apply (auto intro!: LIMSEQ_unique simp add: sums_def)
+lemma sums_unique:
+ fixes f :: "nat \<Rightarrow> 'a::{t2_space, comm_monoid_add}"
+ shows "f sums s \<Longrightarrow> (s = suminf f)"
+apply (frule sums_summable[THEN summable_sums])
+apply (auto intro!: tendsto_unique[OF trivial_limit_sequentially] simp add: sums_def)
done
-lemma sums_iff: "f sums x \<longleftrightarrow> summable f \<and> (suminf f = x)"
+lemma sums_iff:
+ fixes f :: "nat \<Rightarrow> 'a::{t2_space, comm_monoid_add}"
+ shows "f sums x \<longleftrightarrow> summable f \<and> (suminf f = x)"
by (metis summable_sums sums_summable sums_unique)
-lemma sums_split_initial_segment: "f sums s ==>
- (%n. f(n + k)) sums (s - (SUM i = 0..< k. f i))"
- apply (unfold sums_def);
- apply (simp add: sumr_offset);
+lemma sums_split_initial_segment:
+ fixes f :: "nat \<Rightarrow> 'a::real_normed_vector"
+ shows "f sums s ==> (\<lambda>n. f(n + k)) sums (s - (SUM i = 0..< k. f i))"
+ apply (unfold sums_def)
+ apply (simp add: sumr_offset)
apply (rule LIMSEQ_diff_const)
apply (rule LIMSEQ_ignore_initial_segment)
apply assumption
done
-lemma summable_ignore_initial_segment: "summable f ==>
- summable (%n. f(n + k))"
+lemma summable_ignore_initial_segment:
+ fixes f :: "nat \<Rightarrow> 'a::real_normed_vector"
+ shows "summable f ==> summable (%n. f(n + k))"
apply (unfold summable_def)
apply (auto intro: sums_split_initial_segment)
done
-lemma suminf_minus_initial_segment: "summable f ==>
+lemma suminf_minus_initial_segment:
+ fixes f :: "nat \<Rightarrow> 'a::real_normed_vector"
+ shows "summable f ==>
suminf f = s ==> suminf (%n. f(n + k)) = s - (SUM i = 0..< k. f i)"
apply (frule summable_ignore_initial_segment)
apply (rule sums_unique [THEN sym])
@@ -145,8 +148,10 @@
apply auto
done
-lemma suminf_split_initial_segment: "summable f ==>
- suminf f = (SUM i = 0..< k. f i) + suminf (%n. f(n + k))"
+lemma suminf_split_initial_segment:
+ fixes f :: "nat \<Rightarrow> 'a::real_normed_vector"
+ shows "summable f ==>
+ suminf f = (SUM i = 0..< k. f i) + (\<Sum>n. f(n + k))"
by (auto simp add: suminf_minus_initial_segment)
lemma suminf_exist_split: fixes r :: real assumes "0 < r" and "summable a"
@@ -158,31 +163,42 @@
by auto
qed
-lemma sums_Suc: assumes sumSuc: "(\<lambda> n. f (Suc n)) sums l" shows "f sums (l + f 0)"
+lemma sums_Suc:
+ fixes f :: "nat \<Rightarrow> 'a::real_normed_vector"
+ assumes sumSuc: "(\<lambda> n. f (Suc n)) sums l" shows "f sums (l + f 0)"
proof -
from sumSuc[unfolded sums_def]
have "(\<lambda>i. \<Sum>n = Suc 0..<Suc i. f n) ----> l" unfolding setsum_reindex[OF inj_Suc] image_Suc_atLeastLessThan[symmetric] comp_def .
- from LIMSEQ_add_const[OF this, where b="f 0"]
+ from LIMSEQ_add_const[OF this, where b="f 0"]
have "(\<lambda>i. \<Sum>n = 0..<Suc i. f n) ----> l + f 0" unfolding add_commute setsum_head_upt_Suc[OF zero_less_Suc] .
thus ?thesis unfolding sums_def by (rule LIMSEQ_imp_Suc)
qed
-lemma series_zero:
- "(\<forall>m. n \<le> m --> f(m) = 0) ==> f sums (setsum f {0..<n})"
-apply (simp add: sums_def LIMSEQ_iff diff_minus[symmetric], safe)
-apply (rule_tac x = n in exI)
-apply (clarsimp simp add:setsum_diff[symmetric] cong:setsum_ivl_cong)
-done
+lemma series_zero:
+ fixes f :: "nat \<Rightarrow> 'a::{t2_space, comm_monoid_add}"
+ assumes "\<forall>m. n \<le> m \<longrightarrow> f m = 0"
+ shows "f sums (setsum f {0..<n})"
+proof -
+ { fix k :: nat have "setsum f {0..<k + n} = setsum f {0..<n}"
+ using assms by (induct k) auto }
+ note setsum_const = this
+ show ?thesis
+ unfolding sums_def
+ apply (rule LIMSEQ_offset[of _ n])
+ unfolding setsum_const
+ apply (rule LIMSEQ_const)
+ done
+qed
-lemma sums_zero: "(\<lambda>n. 0) sums 0"
+lemma sums_zero[simp, intro]: "(\<lambda>n. 0) sums 0"
unfolding sums_def by (simp add: LIMSEQ_const)
-lemma summable_zero: "summable (\<lambda>n. 0)"
+lemma summable_zero[simp, intro]: "summable (\<lambda>n. 0)"
by (rule sums_zero [THEN sums_summable])
-lemma suminf_zero: "suminf (\<lambda>n. 0) = 0"
+lemma suminf_zero[simp]: "suminf (\<lambda>n. 0::'a::{t2_space, comm_monoid_add}) = 0"
by (rule sums_zero [THEN sums_unique, symmetric])
-
+
lemma (in bounded_linear) sums:
"(\<lambda>n. X n) sums a \<Longrightarrow> (\<lambda>n. f (X n)) sums (f a)"
unfolding sums_def by (drule LIMSEQ, simp only: setsum)
@@ -207,7 +223,7 @@
lemma suminf_mult:
fixes c :: "'a::real_normed_algebra"
- shows "summable f \<Longrightarrow> suminf (\<lambda>n. c * f n) = c * suminf f";
+ shows "summable f \<Longrightarrow> suminf (\<lambda>n. c * f n) = c * suminf f"
by (rule mult_right.suminf [symmetric])
lemma sums_mult2:
@@ -240,37 +256,54 @@
shows "summable f \<Longrightarrow> suminf (\<lambda>n. f n / c) = suminf f / c"
by (rule divide.suminf [symmetric])
-lemma sums_add: "\<lbrakk>X sums a; Y sums b\<rbrakk> \<Longrightarrow> (\<lambda>n. X n + Y n) sums (a + b)"
+lemma sums_add:
+ fixes a b :: "'a::real_normed_field"
+ shows "\<lbrakk>X sums a; Y sums b\<rbrakk> \<Longrightarrow> (\<lambda>n. X n + Y n) sums (a + b)"
unfolding sums_def by (simp add: setsum_addf LIMSEQ_add)
-lemma summable_add: "\<lbrakk>summable X; summable Y\<rbrakk> \<Longrightarrow> summable (\<lambda>n. X n + Y n)"
+lemma summable_add:
+ fixes X Y :: "nat \<Rightarrow> 'a::real_normed_field"
+ shows "\<lbrakk>summable X; summable Y\<rbrakk> \<Longrightarrow> summable (\<lambda>n. X n + Y n)"
unfolding summable_def by (auto intro: sums_add)
lemma suminf_add:
- "\<lbrakk>summable X; summable Y\<rbrakk> \<Longrightarrow> suminf X + suminf Y = (\<Sum>n. X n + Y n)"
+ fixes X Y :: "nat \<Rightarrow> 'a::real_normed_field"
+ shows "\<lbrakk>summable X; summable Y\<rbrakk> \<Longrightarrow> suminf X + suminf Y = (\<Sum>n. X n + Y n)"
by (intro sums_unique sums_add summable_sums)
-lemma sums_diff: "\<lbrakk>X sums a; Y sums b\<rbrakk> \<Longrightarrow> (\<lambda>n. X n - Y n) sums (a - b)"
+lemma sums_diff:
+ fixes X Y :: "nat \<Rightarrow> 'a::real_normed_field"
+ shows "\<lbrakk>X sums a; Y sums b\<rbrakk> \<Longrightarrow> (\<lambda>n. X n - Y n) sums (a - b)"
unfolding sums_def by (simp add: setsum_subtractf LIMSEQ_diff)
-lemma summable_diff: "\<lbrakk>summable X; summable Y\<rbrakk> \<Longrightarrow> summable (\<lambda>n. X n - Y n)"
+lemma summable_diff:
+ fixes X Y :: "nat \<Rightarrow> 'a::real_normed_field"
+ shows "\<lbrakk>summable X; summable Y\<rbrakk> \<Longrightarrow> summable (\<lambda>n. X n - Y n)"
unfolding summable_def by (auto intro: sums_diff)
lemma suminf_diff:
- "\<lbrakk>summable X; summable Y\<rbrakk> \<Longrightarrow> suminf X - suminf Y = (\<Sum>n. X n - Y n)"
+ fixes X Y :: "nat \<Rightarrow> 'a::real_normed_field"
+ shows "\<lbrakk>summable X; summable Y\<rbrakk> \<Longrightarrow> suminf X - suminf Y = (\<Sum>n. X n - Y n)"
by (intro sums_unique sums_diff summable_sums)
-lemma sums_minus: "X sums a ==> (\<lambda>n. - X n) sums (- a)"
+lemma sums_minus:
+ fixes X :: "nat \<Rightarrow> 'a::real_normed_field"
+ shows "X sums a ==> (\<lambda>n. - X n) sums (- a)"
unfolding sums_def by (simp add: setsum_negf LIMSEQ_minus)
-lemma summable_minus: "summable X \<Longrightarrow> summable (\<lambda>n. - X n)"
+lemma summable_minus:
+ fixes X :: "nat \<Rightarrow> 'a::real_normed_field"
+ shows "summable X \<Longrightarrow> summable (\<lambda>n. - X n)"
unfolding summable_def by (auto intro: sums_minus)
-lemma suminf_minus: "summable X \<Longrightarrow> (\<Sum>n. - X n) = - (\<Sum>n. X n)"
+lemma suminf_minus:
+ fixes X :: "nat \<Rightarrow> 'a::real_normed_field"
+ shows "summable X \<Longrightarrow> (\<Sum>n. - X n) = - (\<Sum>n. X n)"
by (intro sums_unique [symmetric] sums_minus summable_sums)
lemma sums_group:
- "[|summable f; 0 < k |] ==> (%n. setsum f {n*k..<n*k+k}) sums (suminf f)"
+ fixes f :: "nat \<Rightarrow> 'a::real_normed_field"
+ shows "[|summable f; 0 < k |] ==> (%n. setsum f {n*k..<n*k+k}) sums (suminf f)"
apply (drule summable_sums)
apply (simp only: sums_def sumr_group)
apply (unfold LIMSEQ_iff, safe)
@@ -290,19 +323,19 @@
assumes pos: "!!n. 0 \<le> f n" and le: "!!n. setsum f {0..<n} \<le> x"
shows "summable f"
proof -
- have "convergent (\<lambda>n. setsum f {0..<n})"
+ have "convergent (\<lambda>n. setsum f {0..<n})"
proof (rule Bseq_mono_convergent)
show "Bseq (\<lambda>n. setsum f {0..<n})"
by (rule f_inc_g_dec_Beq_f [of "(\<lambda>n. setsum f {0..<n})" "\<lambda>n. x"])
- (auto simp add: le pos)
- next
+ (auto simp add: le pos)
+ next
show "\<forall>m n. m \<le> n \<longrightarrow> setsum f {0..<m} \<le> setsum f {0..<n}"
- by (auto intro: setsum_mono2 pos)
+ by (auto intro: setsum_mono2 pos)
qed
then obtain L where "(%n. setsum f {0..<n}) ----> L"
by (blast dest: convergentD)
thus ?thesis
- by (force simp add: summable_def sums_def)
+ by (force simp add: summable_def sums_def)
qed
lemma series_pos_le:
@@ -382,7 +415,7 @@
by (rule geometric_sums [THEN sums_summable])
lemma half: "0 < 1 / (2::'a::{number_ring,linordered_field_inverse_zero})"
- by arith
+ by arith
lemma power_half_series: "(\<lambda>n. (1/2::real)^Suc n) sums 1"
proof -
@@ -400,7 +433,9 @@
"summable f = convergent (%n. setsum f {0..<n})"
by (simp add: summable_def sums_def convergent_def)
-lemma summable_LIMSEQ_zero: "summable f \<Longrightarrow> f ----> 0"
+lemma summable_LIMSEQ_zero:
+ fixes f :: "nat \<Rightarrow> 'a::real_normed_field"
+ shows "summable f \<Longrightarrow> f ----> 0"
apply (drule summable_convergent_sumr_iff [THEN iffD1])
apply (drule convergent_Cauchy)
apply (simp only: Cauchy_iff LIMSEQ_iff, safe)
@@ -413,10 +448,10 @@
lemma suminf_le:
fixes x :: real
shows "summable f \<Longrightarrow> (!!n. setsum f {0..<n} \<le> x) \<Longrightarrow> suminf f \<le> x"
- by (simp add: summable_convergent_sumr_iff suminf_eq_lim lim_le)
+ by (simp add: summable_convergent_sumr_iff suminf_eq_lim lim_le)
lemma summable_Cauchy:
- "summable (f::nat \<Rightarrow> 'a::banach) =
+ "summable (f::nat \<Rightarrow> 'a::banach) =
(\<forall>e > 0. \<exists>N. \<forall>m \<ge> N. \<forall>n. norm (setsum f {m..<n}) < e)"
apply (simp only: summable_convergent_sumr_iff Cauchy_convergent_iff [symmetric] Cauchy_iff, safe)
apply (drule spec, drule (1) mp)
@@ -522,7 +557,7 @@
moreover from sm have "summable f" .
ultimately have "suminf ?g \<le> suminf f" by (rule summable_le)
then show "0 \<le> suminf f" by (simp add: suminf_zero)
-qed
+qed
text{*Absolute convergence imples normal convergence*}
@@ -596,7 +631,7 @@
fixes f :: "nat \<Rightarrow> 'a::banach"
shows "\<lbrakk>c < 1; \<forall>n\<ge>N. norm (f (Suc n)) \<le> c * norm (f n)\<rbrakk> \<Longrightarrow> summable f"
apply (frule ratio_test_lemma2, auto)
-apply (rule_tac g = "%n. (norm (f N) / (c ^ N))*c ^ n"
+apply (rule_tac g = "%n. (norm (f N) / (c ^ N))*c ^ n"
in summable_comparison_test)
apply (rule_tac x = N in exI, safe)
apply (drule le_Suc_ex_iff [THEN iffD1])
@@ -605,7 +640,7 @@
apply (rule_tac y = "c * norm (f (N + n))" in order_trans)
apply (auto intro: mult_right_mono simp add: summable_def)
apply (rule_tac x = "norm (f N) * (1/ (1 - c)) / (c ^ N)" in exI)
-apply (rule sums_divide)
+apply (rule sums_divide)
apply (rule sums_mult)
apply (auto intro!: geometric_sums)
done
--- a/src/HOL/Transcendental.thy Mon Mar 14 15:17:10 2011 +0100
+++ b/src/HOL/Transcendental.thy Mon Mar 14 15:29:10 2011 +0100
@@ -22,14 +22,14 @@
lemma lemma_realpow_diff_sumr:
fixes y :: "'a::{comm_semiring_0,monoid_mult}" shows
- "(\<Sum>p=0..<Suc n. (x ^ p) * y ^ (Suc n - p)) =
+ "(\<Sum>p=0..<Suc n. (x ^ p) * y ^ (Suc n - p)) =
y * (\<Sum>p=0..<Suc n. (x ^ p) * y ^ (n - p))"
by (simp add: setsum_right_distrib lemma_realpow_diff mult_ac
del: setsum_op_ivl_Suc)
lemma lemma_realpow_diff_sumr2:
fixes y :: "'a::{comm_ring,monoid_mult}" shows
- "x ^ (Suc n) - y ^ (Suc n) =
+ "x ^ (Suc n) - y ^ (Suc n) =
(x - y) * (\<Sum>p=0..<Suc n. (x ^ p) * y ^ (n - p))"
apply (induct n, simp)
apply (simp del: setsum_op_ivl_Suc)
@@ -42,7 +42,7 @@
done
lemma lemma_realpow_rev_sumr:
- "(\<Sum>p=0..<Suc n. (x ^ p) * (y ^ (n - p))) =
+ "(\<Sum>p=0..<Suc n. (x ^ p) * (y ^ (n - p))) =
(\<Sum>p=0..<Suc n. (x ^ (n - p)) * (y ^ p))"
apply (rule setsum_reindex_cong [where f="\<lambda>i. n - i"])
apply (rule inj_onI, simp)
@@ -107,16 +107,16 @@
lemma powser_inside:
fixes f :: "nat \<Rightarrow> 'a::{real_normed_field,banach}" shows
- "[| summable (%n. f(n) * (x ^ n)); norm z < norm x |]
+ "[| summable (%n. f(n) * (x ^ n)); norm z < norm x |]
==> summable (%n. f(n) * (z ^ n))"
by (rule powser_insidea [THEN summable_norm_cancel])
lemma sum_split_even_odd: fixes f :: "nat \<Rightarrow> real" shows
- "(\<Sum> i = 0 ..< 2 * n. if even i then f i else g i) =
+ "(\<Sum> i = 0 ..< 2 * n. if even i then f i else g i) =
(\<Sum> i = 0 ..< n. f (2 * i)) + (\<Sum> i = 0 ..< n. g (2 * i + 1))"
proof (induct n)
case (Suc n)
- have "(\<Sum> i = 0 ..< 2 * Suc n. if even i then f i else g i) =
+ have "(\<Sum> i = 0 ..< 2 * Suc n. if even i then f i else g i) =
(\<Sum> i = 0 ..< n. f (2 * i)) + (\<Sum> i = 0 ..< n. g (2 * i + 1)) + (f (2 * n) + g (2 * n + 1))"
using Suc.hyps unfolding One_nat_def by auto
also have "\<dots> = (\<Sum> i = 0 ..< Suc n. f (2 * i)) + (\<Sum> i = 0 ..< Suc n. g (2 * i + 1))" by auto
@@ -133,7 +133,7 @@
let ?SUM = "\<lambda> m. \<Sum> i = 0 ..< m. if even i then 0 else g ((i - 1) div 2)"
{ fix m assume "m \<ge> 2 * no" hence "m div 2 \<ge> no" by auto
- have sum_eq: "?SUM (2 * (m div 2)) = setsum g { 0 ..< m div 2 }"
+ have sum_eq: "?SUM (2 * (m div 2)) = setsum g { 0 ..< m div 2 }"
using sum_split_even_odd by auto
hence "(norm (?SUM (2 * (m div 2)) - x) < r)" using no_eq unfolding sum_eq using `m div 2 \<ge> no` by auto
moreover
@@ -161,13 +161,13 @@
{ fix B T E have "(if B then (0 :: real) else E) + (if B then T else 0) = (if B then T else E)"
by (cases B) auto } note if_sum = this
have g_sums: "(\<lambda> n. if even n then 0 else g ((n - 1) div 2)) sums x" using sums_if'[OF `g sums x`] .
- {
+ {
have "?s 0 = 0" by auto
have Suc_m1: "\<And> n. Suc n - 1 = n" by auto
have if_eq: "\<And>B T E. (if \<not> B then T else E) = (if B then E else T)" by auto
have "?s sums y" using sums_if'[OF `f sums y`] .
- from this[unfolded sums_def, THEN LIMSEQ_Suc]
+ from this[unfolded sums_def, THEN LIMSEQ_Suc]
have "(\<lambda> n. if even n then f (n div 2) else 0) sums y"
unfolding sums_def setsum_shift_lb_Suc0_0_upt[where f="?s", OF `?s 0 = 0`, symmetric]
image_Suc_atLeastLessThan[symmetric] setsum_reindex[OF inj_Suc, unfolded comp_def]
@@ -181,7 +181,7 @@
lemma sums_alternating_upper_lower:
fixes a :: "nat \<Rightarrow> real"
assumes mono: "\<And>n. a (Suc n) \<le> a n" and a_pos: "\<And>n. 0 \<le> a n" and "a ----> 0"
- shows "\<exists>l. ((\<forall>n. (\<Sum>i=0..<2*n. -1^i*a i) \<le> l) \<and> (\<lambda> n. \<Sum>i=0..<2*n. -1^i*a i) ----> l) \<and>
+ shows "\<exists>l. ((\<forall>n. (\<Sum>i=0..<2*n. -1^i*a i) \<le> l) \<and> (\<lambda> n. \<Sum>i=0..<2*n. -1^i*a i) ----> l) \<and>
((\<forall>n. l \<le> (\<Sum>i=0..<2*n + 1. -1^i*a i)) \<and> (\<lambda> n. \<Sum>i=0..<2*n + 1. -1^i*a i) ----> l)"
(is "\<exists>l. ((\<forall>n. ?f n \<le> l) \<and> _) \<and> ((\<forall>n. l \<le> ?g n) \<and> _)")
proof -
@@ -194,21 +194,21 @@
proof fix n show "?g (Suc n) \<le> ?g n" using mono[of "Suc (2*n)"]
unfolding One_nat_def by auto qed
moreover
- have "\<forall> n. ?f n \<le> ?g n"
+ have "\<forall> n. ?f n \<le> ?g n"
proof fix n show "?f n \<le> ?g n" using fg_diff a_pos
unfolding One_nat_def by auto qed
moreover
have "(\<lambda> n. ?f n - ?g n) ----> 0" unfolding fg_diff
proof (rule LIMSEQ_I)
fix r :: real assume "0 < r"
- with `a ----> 0`[THEN LIMSEQ_D]
+ with `a ----> 0`[THEN LIMSEQ_D]
obtain N where "\<And> n. n \<ge> N \<Longrightarrow> norm (a n - 0) < r" by auto
hence "\<forall> n \<ge> N. norm (- a (2 * n) - 0) < r" by auto
thus "\<exists> N. \<forall> n \<ge> N. norm (- a (2 * n) - 0) < r" by auto
qed
ultimately
show ?thesis by (rule lemma_nest_unique)
-qed
+qed
lemma summable_Leibniz': fixes a :: "nat \<Rightarrow> real"
assumes a_zero: "a ----> 0" and a_pos: "\<And> n. 0 \<le> a n"
@@ -225,16 +225,16 @@
let "?g n" = "?P (2 * n + 1)"
obtain l :: real where below_l: "\<forall> n. ?f n \<le> l" and "?f ----> l" and above_l: "\<forall> n. l \<le> ?g n" and "?g ----> l"
using sums_alternating_upper_lower[OF a_monotone a_pos a_zero] by blast
-
+
let ?Sa = "\<lambda> m. \<Sum> n = 0..<m. ?S n"
have "?Sa ----> l"
proof (rule LIMSEQ_I)
fix r :: real assume "0 < r"
- with `?f ----> l`[THEN LIMSEQ_D]
+ with `?f ----> l`[THEN LIMSEQ_D]
obtain f_no where f: "\<And> n. n \<ge> f_no \<Longrightarrow> norm (?f n - l) < r" by auto
- from `0 < r` `?g ----> l`[THEN LIMSEQ_D]
+ from `0 < r` `?g ----> l`[THEN LIMSEQ_D]
obtain g_no where g: "\<And> n. n \<ge> g_no \<Longrightarrow> norm (?g n - l) < r" by auto
{ fix n :: nat
@@ -302,7 +302,7 @@
hence ?summable unfolding summable_def by auto
moreover
have "\<And> a b :: real. \<bar> - a - - b \<bar> = \<bar>a - b\<bar>" unfolding minus_diff_minus by auto
-
+
from suminf_minus[OF leibniz(1), unfolded mult_minus_right minus_minus]
have move_minus: "(\<Sum>n. - (-1 ^ n * a n)) = - (\<Sum>n. -1 ^ n * a n)" by auto
@@ -336,8 +336,9 @@
done
lemma diffs_equiv:
- "summable (%n. (diffs c)(n) * (x ^ n)) ==>
- (%n. of_nat n * c(n) * (x ^ (n - Suc 0))) sums
+ fixes x :: "'a::{real_normed_vector, ring_1}"
+ shows "summable (%n. (diffs c)(n) * (x ^ n)) ==>
+ (%n. of_nat n * c(n) * (x ^ (n - Suc 0))) sums
(\<Sum>n. (diffs c)(n) * (x ^ n))"
unfolding diffs_def
apply (drule summable_sums)
@@ -346,7 +347,7 @@
lemma lemma_termdiff1:
fixes z :: "'a :: {monoid_mult,comm_ring}" shows
- "(\<Sum>p=0..<m. (((z + h) ^ (m - p)) * (z ^ p)) - (z ^ m)) =
+ "(\<Sum>p=0..<m. (((z + h) ^ (m - p)) * (z ^ p)) - (z ^ m)) =
(\<Sum>p=0..<m. (z ^ p) * (((z + h) ^ (m - p)) - (z ^ (m - p))))"
by(auto simp add: algebra_simps power_add [symmetric])
@@ -535,7 +536,7 @@
apply (simp add: diffs_def)
apply (case_tac n, simp_all add: r_neq_0)
done
- finally have "summable
+ finally have "summable
(\<lambda>n. of_nat n * (of_nat (n - Suc 0) * norm (c n) * inverse r) * r ^ (n - Suc 0))"
by (rule diffs_equiv [THEN sums_summable])
also have
@@ -596,7 +597,7 @@
have C: "summable (\<lambda>n. diffs c n * x ^ n)"
by (rule powser_inside [OF 2 4])
show "((\<Sum>n. c n * (x + h) ^ n) - (\<Sum>n. c n * x ^ n)) / h
- - (\<Sum>n. diffs c n * x ^ n) =
+ - (\<Sum>n. diffs c n * x ^ n) =
(\<Sum>n. c n * (((x + h) ^ n - x ^ n) / h - of_nat n * x ^ (n - Suc 0)))"
apply (subst sums_unique [OF diffs_equiv [OF C]])
apply (subst suminf_diff [OF B A])
@@ -646,10 +647,10 @@
proof (rule LIM_I)
fix r :: real assume "0 < r" hence "0 < r/3" by auto
- obtain N_L where N_L: "\<And> n. N_L \<le> n \<Longrightarrow> \<bar> \<Sum> i. L (i + n) \<bar> < r/3"
+ obtain N_L where N_L: "\<And> n. N_L \<le> n \<Longrightarrow> \<bar> \<Sum> i. L (i + n) \<bar> < r/3"
using suminf_exist_split[OF `0 < r/3` `summable L`] by auto
- obtain N_f' where N_f': "\<And> n. N_f' \<le> n \<Longrightarrow> \<bar> \<Sum> i. f' x0 (i + n) \<bar> < r/3"
+ obtain N_f' where N_f': "\<And> n. N_f' \<le> n \<Longrightarrow> \<bar> \<Sum> i. f' x0 (i + n) \<bar> < r/3"
using suminf_exist_split[OF `0 < r/3` `summable (f' x0)`] by auto
let ?N = "Suc (max N_L N_f')"
@@ -672,7 +673,7 @@
proof (rule ballI)
fix x assume "x \<in> ?s ` {0..<?N}"
then obtain n where "x = ?s n" and "n \<in> {0..<?N}" using image_iff[THEN iffD1] by blast
- from DERIV_D[OF DERIV_f[where n=n], THEN LIM_D, OF `0 < ?r`, unfolded real_norm_def]
+ from DERIV_D[OF DERIV_f[where n=n], THEN LIM_D, OF `0 < ?r`, unfolded real_norm_def]
obtain s where s_bound: "0 < s \<and> (\<forall>x. x \<noteq> 0 \<and> \<bar>x\<bar> < s \<longrightarrow> \<bar>?diff n x - f' x0 n\<bar> < ?r)" by auto
have "0 < ?s n" by (rule someI2[where a=s], auto simp add: s_bound)
thus "0 < x" unfolding `x = ?s n` .
@@ -685,7 +686,7 @@
{ fix x assume "x \<noteq> 0" and "\<bar> x \<bar> < S"
hence x_in_I: "x0 + x \<in> { a <..< b }" using S_a S_b by auto
-
+
note diff_smbl = summable_diff[OF allf_summable[OF x_in_I] allf_summable[OF x0_in_I]]
note div_smbl = summable_divide[OF diff_smbl]
note all_smbl = summable_diff[OF div_smbl `summable (f' x0)`]
@@ -695,7 +696,7 @@
note all_shft_smbl = summable_diff[OF div_smbl ign[OF `summable (f' x0)`]]
{ fix n
- have "\<bar> ?diff (n + ?N) x \<bar> \<le> L (n + ?N) * \<bar> (x0 + x) - x0 \<bar> / \<bar> x \<bar>"
+ have "\<bar> ?diff (n + ?N) x \<bar> \<le> L (n + ?N) * \<bar> (x0 + x) - x0 \<bar> / \<bar> x \<bar>"
using divide_right_mono[OF L_def[OF x_in_I x0_in_I] abs_ge_zero] unfolding abs_divide .
hence "\<bar> ( \<bar> ?diff (n + ?N) x \<bar>) \<bar> \<le> L (n + ?N)" using `x \<noteq> 0` by auto
} note L_ge = summable_le2[OF allI[OF this] ign[OF `summable L`]]
@@ -709,7 +710,7 @@
fix n assume "n \<in> { 0 ..< ?N}"
have "\<bar> x \<bar> < S" using `\<bar> x \<bar> < S` .
also have "S \<le> S'" using `S \<le> S'` .
- also have "S' \<le> ?s n" unfolding S'_def
+ also have "S' \<le> ?s n" unfolding S'_def
proof (rule Min_le_iff[THEN iffD2])
have "?s n \<in> (?s ` {0..<?N}) \<and> ?s n \<le> ?s n" using `n \<in> { 0 ..< ?N}` by auto
thus "\<exists> a \<in> (?s ` {0..<?N}). a \<le> ?s n" by blast
@@ -727,16 +728,16 @@
finally have "\<bar>\<Sum>n \<in> { 0 ..< ?N}. ?diff n x - f' x0 n \<bar> < r / 3" (is "?diff_part < r / 3") .
from suminf_diff[OF allf_summable[OF x_in_I] allf_summable[OF x0_in_I]]
- have "\<bar> (suminf (f (x0 + x)) - (suminf (f x0))) / x - suminf (f' x0) \<bar> =
+ have "\<bar> (suminf (f (x0 + x)) - (suminf (f x0))) / x - suminf (f' x0) \<bar> =
\<bar> \<Sum>n. ?diff n x - f' x0 n \<bar>" unfolding suminf_diff[OF div_smbl `summable (f' x0)`, symmetric] using suminf_divide[OF diff_smbl, symmetric] by auto
also have "\<dots> \<le> ?diff_part + \<bar> (\<Sum>n. ?diff (n + ?N) x) - (\<Sum> n. f' x0 (n + ?N)) \<bar>" unfolding suminf_split_initial_segment[OF all_smbl, where k="?N"] unfolding suminf_diff[OF div_shft_smbl ign[OF `summable (f' x0)`]] by (rule abs_triangle_ineq)
also have "\<dots> \<le> ?diff_part + ?L_part + ?f'_part" using abs_triangle_ineq4 by auto
- also have "\<dots> < r /3 + r/3 + r/3"
+ also have "\<dots> < r /3 + r/3 + r/3"
using `?diff_part < r/3` `?L_part \<le> r/3` and `?f'_part < r/3`
by (rule add_strict_mono [OF add_less_le_mono])
finally have "\<bar> (suminf (f (x0 + x)) - (suminf (f x0))) / x - suminf (f' x0) \<bar> < r"
by auto
- } thus "\<exists> s > 0. \<forall> x. x \<noteq> 0 \<and> norm (x - 0) < s \<longrightarrow>
+ } thus "\<exists> s > 0. \<forall> x. x \<noteq> 0 \<and> norm (x - 0) < s \<longrightarrow>
norm (((\<Sum>n. f (x0 + x) n) - (\<Sum>n. f x0 n)) / x - (\<Sum>n. f' x0 n)) < r" using `0 < S`
unfolding real_norm_def diff_0_right by blast
qed
@@ -761,9 +762,9 @@
{ fix n x y assume "x \<in> {-R' <..< R'}" and "y \<in> {-R' <..< R'}"
show "\<bar>?f x n - ?f y n\<bar> \<le> \<bar>f n * real (Suc n) * R'^n\<bar> * \<bar>x-y\<bar>"
proof -
- have "\<bar>f n * x ^ (Suc n) - f n * y ^ (Suc n)\<bar> = (\<bar>f n\<bar> * \<bar>x-y\<bar>) * \<bar>\<Sum>p = 0..<Suc n. x ^ p * y ^ (n - p)\<bar>"
+ have "\<bar>f n * x ^ (Suc n) - f n * y ^ (Suc n)\<bar> = (\<bar>f n\<bar> * \<bar>x-y\<bar>) * \<bar>\<Sum>p = 0..<Suc n. x ^ p * y ^ (n - p)\<bar>"
unfolding right_diff_distrib[symmetric] lemma_realpow_diff_sumr2 abs_mult by auto
- also have "\<dots> \<le> (\<bar>f n\<bar> * \<bar>x-y\<bar>) * (\<bar>real (Suc n)\<bar> * \<bar>R' ^ n\<bar>)"
+ also have "\<dots> \<le> (\<bar>f n\<bar> * \<bar>x-y\<bar>) * (\<bar>real (Suc n)\<bar> * \<bar>R' ^ n\<bar>)"
proof (rule mult_left_mono)
have "\<bar>\<Sum>p = 0..<Suc n. x ^ p * y ^ (n - p)\<bar> \<le> (\<Sum>p = 0..<Suc n. \<bar>x ^ p * y ^ (n - p)\<bar>)" by (rule setsum_abs)
also have "\<dots> \<le> (\<Sum>p = 0..<Suc n. R' ^ n)"
@@ -809,7 +810,7 @@
next
case False
have "- ?R < 0" using assms by auto
- also have "\<dots> \<le> x0" using False by auto
+ also have "\<dots> \<le> x0" using False by auto
finally show ?thesis .
qed
hence "0 < ?R" "?R < R" "- ?R < x0" and "x0 < ?R" using assms by auto
@@ -871,7 +872,7 @@
unfolding exp_def by (rule summable_exp_generic [THEN summable_sums])
-lemma exp_fdiffs:
+lemma exp_fdiffs:
"diffs (%n. inverse(real (fact n))) = (%n. inverse(real (fact n)))"
by (simp add: diffs_def mult_assoc [symmetric] real_of_nat_def of_nat_mult
del: mult_Suc of_nat_Suc)
@@ -1081,7 +1082,7 @@
lemma lemma_exp_total: "1 \<le> y ==> \<exists>x. 0 \<le> x & x \<le> y - 1 & exp(x::real) = y"
apply (rule IVT)
apply (auto intro: isCont_exp simp add: le_diff_eq)
-apply (subgoal_tac "1 + (y - 1) \<le> exp (y - 1)")
+apply (subgoal_tac "1 + (y - 1) \<le> exp (y - 1)")
apply simp
apply (rule exp_ge_add_one_self_aux, simp)
done
@@ -1160,12 +1161,12 @@
qed
lemma ln_ge_zero_imp_ge_one:
- assumes ln: "0 \<le> ln x"
+ assumes ln: "0 \<le> ln x"
and x: "0 < x"
shows "1 \<le> x"
proof -
from ln have "ln 1 \<le> ln x" by simp
- thus ?thesis by (simp add: x del: ln_one)
+ thus ?thesis by (simp add: x del: ln_one)
qed
lemma ln_ge_zero_iff [simp]: "0 < x ==> (0 \<le> ln x) = (1 \<le> x)"
@@ -1183,12 +1184,12 @@
qed
lemma ln_gt_zero_imp_gt_one:
- assumes ln: "0 < ln x"
+ assumes ln: "0 < ln x"
and x: "0 < x"
shows "1 < x"
proof -
from ln have "ln 1 < ln x" by simp
- thus ?thesis by (simp add: x del: ln_one)
+ thus ?thesis by (simp add: x del: ln_one)
qed
lemma ln_gt_zero_iff [simp]: "0 < x ==> (0 < ln x) = (1 < x)"
@@ -1285,22 +1286,22 @@
done
lemma lemma_STAR_sin:
- "(if even n then 0
+ "(if even n then 0
else -1 ^ ((n - Suc 0) div 2)/(real (fact n))) * 0 ^ n = 0"
by (induct "n", auto)
lemma lemma_STAR_cos:
- "0 < n -->
+ "0 < n -->
-1 ^ (n div 2)/(real (fact n)) * 0 ^ n = 0"
by (induct "n", auto)
lemma lemma_STAR_cos1:
- "0 < n -->
+ "0 < n -->
(-1) ^ (n div 2)/(real (fact n)) * 0 ^ n = 0"
by (induct "n", auto)
lemma lemma_STAR_cos2:
- "(\<Sum>n=1..<n. if even n then -1 ^ (n div 2)/(real (fact n)) * 0 ^ n
+ "(\<Sum>n=1..<n. if even n then -1 ^ (n div 2)/(real (fact n)) * 0 ^ n
else 0) = 0"
apply (induct "n")
apply (case_tac [2] "n", auto)
@@ -1314,7 +1315,7 @@
lemma sin_fdiffs: "diffs sin_coeff = cos_coeff"
unfolding sin_coeff_def cos_coeff_def
-by (auto intro!: ext
+by (auto intro!: ext
simp add: diffs_def divide_inverse real_of_nat_def of_nat_mult
simp del: mult_Suc of_nat_Suc)
@@ -1323,7 +1324,7 @@
lemma cos_fdiffs: "diffs cos_coeff = (\<lambda>n. - sin_coeff n)"
unfolding sin_coeff_def cos_coeff_def
-by (auto intro!: ext
+by (auto intro!: ext
simp add: diffs_def divide_inverse odd_Suc_mult_two_ex real_of_nat_def of_nat_mult
simp del: mult_Suc of_nat_Suc)
@@ -1424,8 +1425,8 @@
"DERIV (%x. cos(x)*cos(x)) x :> -(2 * cos(x) * sin(x))"
by (auto intro!: DERIV_intros)
-lemma DERIV_sin_circle_all:
- "\<forall>x. DERIV (%x. (sin x)\<twosuperior> + (cos x)\<twosuperior>) x :>
+lemma DERIV_sin_circle_all:
+ "\<forall>x. DERIV (%x. (sin x)\<twosuperior> + (cos x)\<twosuperior>) x :>
(2*cos(x)*sin(x) - 2*cos(x)*sin(x))"
by (auto intro!: DERIV_intros)
@@ -1462,29 +1463,29 @@
by (rule power2_le_imp_le, simp_all add: sin_squared_eq)
lemma sin_ge_minus_one [simp]: "-1 \<le> sin x"
-apply (insert abs_sin_le_one [of x])
-apply (simp add: abs_le_iff del: abs_sin_le_one)
+apply (insert abs_sin_le_one [of x])
+apply (simp add: abs_le_iff del: abs_sin_le_one)
done
lemma sin_le_one [simp]: "sin x \<le> 1"
-apply (insert abs_sin_le_one [of x])
-apply (simp add: abs_le_iff del: abs_sin_le_one)
+apply (insert abs_sin_le_one [of x])
+apply (simp add: abs_le_iff del: abs_sin_le_one)
done
lemma abs_cos_le_one [simp]: "\<bar>cos x\<bar> \<le> 1"
by (rule power2_le_imp_le, simp_all add: cos_squared_eq)
lemma cos_ge_minus_one [simp]: "-1 \<le> cos x"
-apply (insert abs_cos_le_one [of x])
-apply (simp add: abs_le_iff del: abs_cos_le_one)
+apply (insert abs_cos_le_one [of x])
+apply (simp add: abs_le_iff del: abs_cos_le_one)
done
lemma cos_le_one [simp]: "cos x \<le> 1"
-apply (insert abs_cos_le_one [of x])
+apply (insert abs_cos_le_one [of x])
apply (simp add: abs_le_iff del: abs_cos_le_one)
done
-lemma DERIV_fun_pow: "DERIV g x :> m ==>
+lemma DERIV_fun_pow: "DERIV g x :> m ==>
DERIV (%x. (g x) ^ n) x :> real n * (g x) ^ (n - 1) * m"
unfolding One_nat_def
apply (rule lemma_DERIV_subst)
@@ -1515,15 +1516,15 @@
(* lemma *)
lemma lemma_DERIV_sin_cos_add:
- "\<forall>x.
- DERIV (%x. (sin (x + y) - (sin x * cos y + cos x * sin y)) ^ 2 +
+ "\<forall>x.
+ DERIV (%x. (sin (x + y) - (sin x * cos y + cos x * sin y)) ^ 2 +
(cos (x + y) - (cos x * cos y - sin x * sin y)) ^ 2) x :> 0"
by (auto intro!: DERIV_intros simp add: algebra_simps)
lemma sin_cos_add [simp]:
- "(sin (x + y) - (sin x * cos y + cos x * sin y)) ^ 2 +
+ "(sin (x + y) - (sin x * cos y + cos x * sin y)) ^ 2 +
(cos (x + y) - (cos x * cos y - sin x * sin y)) ^ 2 = 0"
-apply (cut_tac y = 0 and x = x and y7 = y
+apply (cut_tac y = 0 and x = x and y7 = y
in lemma_DERIV_sin_cos_add [THEN DERIV_isconst_all])
apply (auto simp add: numeral_2_eq_2)
done
@@ -1543,9 +1544,9 @@
by (auto intro!: DERIV_intros simp add: algebra_simps)
-lemma sin_cos_minus:
+lemma sin_cos_minus:
"(sin(-x) + (sin x)) ^ 2 + (cos(-x) - (cos x)) ^ 2 = 0"
-apply (cut_tac y = 0 and x = x
+apply (cut_tac y = 0 and x = x
in lemma_DERIV_sin_cos_minus [THEN DERIV_isconst_all])
apply simp
done
@@ -1582,27 +1583,27 @@
pi :: "real" where
"pi = 2 * (THE x. 0 \<le> (x::real) & x \<le> 2 & cos x = 0)"
-text{*Show that there's a least positive @{term x} with @{term "cos(x) = 0"};
+text{*Show that there's a least positive @{term x} with @{term "cos(x) = 0"};
hence define pi.*}
lemma sin_paired:
- "(%n. -1 ^ n /(real (fact (2 * n + 1))) * x ^ (2 * n + 1))
+ "(%n. -1 ^ n /(real (fact (2 * n + 1))) * x ^ (2 * n + 1))
sums sin x"
proof -
have "(\<lambda>n. \<Sum>k = n * 2..<n * 2 + 2. sin_coeff k * x ^ k) sums sin x"
unfolding sin_def
- by (rule sin_converges [THEN sums_summable, THEN sums_group], simp)
+ by (rule sin_converges [THEN sums_summable, THEN sums_group], simp)
thus ?thesis unfolding One_nat_def sin_coeff_def by (simp add: mult_ac)
qed
text {* FIXME: This is a long, ugly proof! *}
lemma sin_gt_zero: "[|0 < x; x < 2 |] ==> 0 < sin x"
-apply (subgoal_tac
+apply (subgoal_tac
"(\<lambda>n. \<Sum>k = n * 2..<n * 2 + 2.
- -1 ^ k / real (fact (2 * k + 1)) * x ^ (2 * k + 1))
+ -1 ^ k / real (fact (2 * k + 1)) * x ^ (2 * k + 1))
sums (\<Sum>n. -1 ^ n / real (fact (2 * n + 1)) * x ^ (2 * n + 1))")
prefer 2
- apply (rule sin_paired [THEN sums_summable, THEN sums_group], simp)
+ apply (rule sin_paired [THEN sums_summable, THEN sums_group], simp)
apply (rotate_tac 2)
apply (drule sin_paired [THEN sums_unique, THEN ssubst])
unfolding One_nat_def
@@ -1614,10 +1615,10 @@
apply (erule sums_summable)
apply (case_tac "m=0")
apply (simp (no_asm_simp))
-apply (subgoal_tac "6 * (x * (x * x) / real (Suc (Suc (Suc (Suc (Suc (Suc 0))))))) < 6 * x")
-apply (simp only: mult_less_cancel_left, simp)
+apply (subgoal_tac "6 * (x * (x * x) / real (Suc (Suc (Suc (Suc (Suc (Suc 0))))))) < 6 * x")
+apply (simp only: mult_less_cancel_left, simp)
apply (simp (no_asm_simp) add: numeral_2_eq_2 [symmetric] mult_assoc [symmetric])
-apply (subgoal_tac "x*x < 2*3", simp)
+apply (subgoal_tac "x*x < 2*3", simp)
apply (rule mult_strict_mono)
apply (auto simp add: real_0_less_add_iff real_of_nat_Suc simp del: fact_Suc)
apply (subst fact_Suc)
@@ -1630,15 +1631,15 @@
apply (subst real_of_nat_mult)
apply (simp (no_asm) add: divide_inverse del: fact_Suc)
apply (auto simp add: mult_assoc [symmetric] simp del: fact_Suc)
-apply (rule_tac c="real (Suc (Suc (4*m)))" in mult_less_imp_less_right)
+apply (rule_tac c="real (Suc (Suc (4*m)))" in mult_less_imp_less_right)
apply (auto simp add: mult_assoc simp del: fact_Suc)
-apply (rule_tac c="real (Suc (Suc (Suc (4*m))))" in mult_less_imp_less_right)
+apply (rule_tac c="real (Suc (Suc (Suc (4*m))))" in mult_less_imp_less_right)
apply (auto simp add: mult_assoc mult_less_cancel_left simp del: fact_Suc)
-apply (subgoal_tac "x * (x * x ^ (4*m)) = (x ^ (4*m)) * (x * x)")
+apply (subgoal_tac "x * (x * x ^ (4*m)) = (x ^ (4*m)) * (x * x)")
apply (erule ssubst)+
apply (auto simp del: fact_Suc)
apply (subgoal_tac "0 < x ^ (4 * m) ")
- prefer 2 apply (simp only: zero_less_power)
+ prefer 2 apply (simp only: zero_less_power)
apply (simp (no_asm_simp) add: mult_less_cancel_left)
apply (rule mult_strict_mono)
apply (simp_all (no_asm_simp))
@@ -1657,7 +1658,7 @@
proof -
have "(\<lambda>n. \<Sum>k = n * 2..<n * 2 + 2. cos_coeff k * x ^ k) sums cos x"
unfolding cos_def
- by (rule cos_converges [THEN sums_summable, THEN sums_group], simp)
+ by (rule cos_converges [THEN sums_summable, THEN sums_group], simp)
thus ?thesis unfolding cos_coeff_def by (simp add: mult_ac)
qed
@@ -1665,12 +1666,12 @@
by simp
lemma real_mult_inverse_cancel:
- "[|(0::real) < x; 0 < x1; x1 * y < x * u |]
+ "[|(0::real) < x; 0 < x1; x1 * y < x * u |]
==> inverse x * y < inverse x1 * u"
-apply (rule_tac c=x in mult_less_imp_less_left)
+apply (rule_tac c=x in mult_less_imp_less_left)
apply (auto simp add: mult_assoc [symmetric])
apply (simp (no_asm) add: mult_ac)
-apply (rule_tac c=x1 in mult_less_imp_less_right)
+apply (rule_tac c=x1 in mult_less_imp_less_right)
apply (auto simp add: mult_ac)
done
@@ -1687,7 +1688,7 @@
lemma cos_two_less_zero [simp]: "cos (2) < 0"
apply (cut_tac x = 2 in cos_paired)
apply (drule sums_minus)
-apply (rule neg_less_iff_less [THEN iffD1])
+apply (rule neg_less_iff_less [THEN iffD1])
apply (frule sums_unique, auto)
apply (rule_tac y =
"\<Sum>n=0..< Suc(Suc(Suc 0)). - (-1 ^ n / (real(fact (2*n))) * 2 ^ (2*n))"
@@ -1697,12 +1698,12 @@
apply (rule sumr_pos_lt_pair)
apply (erule sums_summable, safe)
unfolding One_nat_def
-apply (simp (no_asm) add: divide_inverse real_0_less_add_iff mult_assoc [symmetric]
+apply (simp (no_asm) add: divide_inverse real_0_less_add_iff mult_assoc [symmetric]
del: fact_Suc)
apply (rule real_mult_inverse_cancel2)
apply (rule real_of_nat_fact_gt_zero)+
apply (simp (no_asm) add: mult_assoc [symmetric] del: fact_Suc)
-apply (subst fact_lemma)
+apply (subst fact_lemma)
apply (subst fact_Suc [of "Suc (Suc (Suc (Suc (Suc (Suc (Suc (4 * d)))))))"])
apply (simp only: real_of_nat_mult)
apply (rule mult_strict_mono, force)
@@ -1822,7 +1823,7 @@
lemma cos_npi2 [simp]: "cos (pi * real n) = -1 ^ n"
proof -
have "cos (pi * real n) = cos (real n * pi)" by (simp only: mult_commute)
- also have "... = -1 ^ n" by (rule cos_npi)
+ also have "... = -1 ^ n" by (rule cos_npi)
finally show ?thesis .
qed
@@ -1832,7 +1833,7 @@
done
lemma sin_npi2 [simp]: "sin (pi * real (n::nat)) = 0"
-by (simp add: mult_commute [of pi])
+by (simp add: mult_commute [of pi])
lemma cos_two_pi [simp]: "cos (2 * pi) = 1"
by (simp add: cos_double)
@@ -1846,10 +1847,10 @@
apply (rule pi_half_less_two)
done
-lemma sin_less_zero:
+lemma sin_less_zero:
assumes lb: "- pi/2 < x" and "x < 0" shows "sin x < 0"
proof -
- have "0 < sin (- x)" using assms by (simp only: sin_gt_zero2)
+ have "0 < sin (- x)" using assms by (simp only: sin_gt_zero2)
thus ?thesis by simp
qed
@@ -1862,7 +1863,7 @@
apply (cut_tac cos_is_zero, safe)
apply (rename_tac y z)
apply (drule_tac x = y in spec)
-apply (drule_tac x = "pi/2" in spec, simp)
+apply (drule_tac x = "pi/2" in spec, simp)
done
lemma cos_gt_zero_pi: "[| -(pi/2) < x; x < pi/2 |] ==> 0 < cos x"
@@ -1871,10 +1872,10 @@
apply (rule cos_gt_zero)
apply (auto intro: cos_gt_zero)
done
-
+
lemma cos_ge_zero: "[| -(pi/2) \<le> x; x \<le> pi/2 |] ==> 0 \<le> cos x"
apply (auto simp add: order_le_less cos_gt_zero_pi)
-apply (subgoal_tac "x = pi/2", auto)
+apply (subgoal_tac "x = pi/2", auto)
done
lemma sin_gt_zero_pi: "[| 0 < x; x < pi |] ==> 0 < sin x"
@@ -1897,7 +1898,7 @@
qed
then obtain y where "pi < y" and "y < 2" and "y < 2 * pi" by blast
hence "0 < sin y" using sin_gt_zero by auto
- moreover
+ moreover
have "sin y < 0" using sin_gt_zero_pi[of "y - pi"] `pi < y` and `y < 2 * pi` sin_periodic_pi[of "y - pi"] by auto
ultimately show False by auto
qed
@@ -1914,7 +1915,7 @@
apply (drule_tac f = cos in Rolle)
apply (drule_tac [5] f = cos in Rolle)
apply (auto intro: order_less_imp_le DERIV_isCont DERIV_cos
- dest!: DERIV_cos [THEN DERIV_unique]
+ dest!: DERIV_cos [THEN DERIV_unique]
simp add: differentiable_def)
apply (auto dest: sin_gt_zero_pi [OF order_le_less_trans order_less_le_trans])
done
@@ -1925,32 +1926,32 @@
apply (subgoal_tac "\<forall>x. (- (pi/2) \<le> x & x \<le> pi/2 & (sin x = y)) = (0 \<le> (x + pi/2) & (x + pi/2) \<le> pi & (cos (x + pi/2) = -y))")
apply (erule contrapos_np)
apply (simp del: minus_sin_cos_eq [symmetric])
-apply (cut_tac y="-y" in cos_total, simp) apply simp
+apply (cut_tac y="-y" in cos_total, simp) apply simp
apply (erule ex1E)
apply (rule_tac a = "x - (pi/2)" in ex1I)
apply (simp (no_asm) add: add_assoc)
apply (rotate_tac 3)
-apply (drule_tac x = "xa + pi/2" in spec, safe, simp_all)
+apply (drule_tac x = "xa + pi/2" in spec, safe, simp_all)
done
lemma reals_Archimedean4:
"[| 0 < y; 0 \<le> x |] ==> \<exists>n. real n * y \<le> x & x < real (Suc n) * y"
apply (auto dest!: reals_Archimedean3)
-apply (drule_tac x = x in spec, clarify)
+apply (drule_tac x = x in spec, clarify)
apply (subgoal_tac "x < real(LEAST m::nat. x < real m * y) * y")
- prefer 2 apply (erule LeastI)
-apply (case_tac "LEAST m::nat. x < real m * y", simp)
+ prefer 2 apply (erule LeastI)
+apply (case_tac "LEAST m::nat. x < real m * y", simp)
apply (subgoal_tac "~ x < real nat * y")
- prefer 2 apply (rule not_less_Least, simp, force)
+ prefer 2 apply (rule not_less_Least, simp, force)
done
-(* Pre Isabelle99-2 proof was simpler- numerals arithmetic
+(* Pre Isabelle99-2 proof was simpler- numerals arithmetic
now causes some unwanted re-arrangements of literals! *)
lemma cos_zero_lemma:
- "[| 0 \<le> x; cos x = 0 |] ==>
+ "[| 0 \<le> x; cos x = 0 |] ==>
\<exists>n::nat. ~even n & x = real n * (pi/2)"
apply (drule pi_gt_zero [THEN reals_Archimedean4], safe)
-apply (subgoal_tac "0 \<le> x - real n * pi &
+apply (subgoal_tac "0 \<le> x - real n * pi &
(x - real n * pi) \<le> pi & (cos (x - real n * pi) = 0) ")
apply (auto simp add: algebra_simps real_of_nat_Suc)
prefer 2 apply (simp add: cos_diff)
@@ -1965,39 +1966,39 @@
done
lemma sin_zero_lemma:
- "[| 0 \<le> x; sin x = 0 |] ==>
+ "[| 0 \<le> x; sin x = 0 |] ==>
\<exists>n::nat. even n & x = real n * (pi/2)"
apply (subgoal_tac "\<exists>n::nat. ~ even n & x + pi/2 = real n * (pi/2) ")
apply (clarify, rule_tac x = "n - 1" in exI)
apply (force simp add: odd_Suc_mult_two_ex real_of_nat_Suc left_distrib)
apply (rule cos_zero_lemma)
-apply (simp_all add: add_increasing)
+apply (simp_all add: add_increasing)
done
lemma cos_zero_iff:
- "(cos x = 0) =
- ((\<exists>n::nat. ~even n & (x = real n * (pi/2))) |
+ "(cos x = 0) =
+ ((\<exists>n::nat. ~even n & (x = real n * (pi/2))) |
(\<exists>n::nat. ~even n & (x = -(real n * (pi/2)))))"
apply (rule iffI)
apply (cut_tac linorder_linear [of 0 x], safe)
apply (drule cos_zero_lemma, assumption+)
-apply (cut_tac x="-x" in cos_zero_lemma, simp, simp)
-apply (force simp add: minus_equation_iff [of x])
-apply (auto simp only: odd_Suc_mult_two_ex real_of_nat_Suc left_distrib)
+apply (cut_tac x="-x" in cos_zero_lemma, simp, simp)
+apply (force simp add: minus_equation_iff [of x])
+apply (auto simp only: odd_Suc_mult_two_ex real_of_nat_Suc left_distrib)
apply (auto simp add: cos_add)
done
(* ditto: but to a lesser extent *)
lemma sin_zero_iff:
- "(sin x = 0) =
- ((\<exists>n::nat. even n & (x = real n * (pi/2))) |
+ "(sin x = 0) =
+ ((\<exists>n::nat. even n & (x = real n * (pi/2))) |
(\<exists>n::nat. even n & (x = -(real n * (pi/2)))))"
apply (rule iffI)
apply (cut_tac linorder_linear [of 0 x], safe)
apply (drule sin_zero_lemma, assumption+)
apply (cut_tac x="-x" in sin_zero_lemma, simp, simp, safe)
-apply (force simp add: minus_equation_iff [of x])
+apply (force simp add: minus_equation_iff [of x])
apply (auto simp add: even_mult_two_ex)
done
@@ -2066,19 +2067,19 @@
lemma tan_periodic [simp]: "tan (x + 2*pi) = tan x"
by (simp add: tan_def)
-lemma lemma_tan_add1:
- "[| cos x \<noteq> 0; cos y \<noteq> 0 |]
+lemma lemma_tan_add1:
+ "[| cos x \<noteq> 0; cos y \<noteq> 0 |]
==> 1 - tan(x)*tan(y) = cos (x + y)/(cos x * cos y)"
apply (simp add: tan_def divide_inverse)
-apply (auto simp del: inverse_mult_distrib
+apply (auto simp del: inverse_mult_distrib
simp add: inverse_mult_distrib [symmetric] mult_ac)
apply (rule_tac c1 = "cos x * cos y" in real_mult_right_cancel [THEN subst])
-apply (auto simp del: inverse_mult_distrib
+apply (auto simp del: inverse_mult_distrib
simp add: mult_assoc left_diff_distrib cos_add)
done
-lemma add_tan_eq:
- "[| cos x \<noteq> 0; cos y \<noteq> 0 |]
+lemma add_tan_eq:
+ "[| cos x \<noteq> 0; cos y \<noteq> 0 |]
==> tan x + tan y = sin(x + y)/(cos x * cos y)"
apply (simp add: tan_def)
apply (rule_tac c1 = "cos x * cos y" in real_mult_right_cancel [THEN subst])
@@ -2087,27 +2088,27 @@
done
lemma tan_add:
- "[| cos x \<noteq> 0; cos y \<noteq> 0; cos (x + y) \<noteq> 0 |]
+ "[| cos x \<noteq> 0; cos y \<noteq> 0; cos (x + y) \<noteq> 0 |]
==> tan(x + y) = (tan(x) + tan(y))/(1 - tan(x) * tan(y))"
apply (simp (no_asm_simp) add: add_tan_eq lemma_tan_add1)
apply (simp add: tan_def)
done
lemma tan_double:
- "[| cos x \<noteq> 0; cos (2 * x) \<noteq> 0 |]
+ "[| cos x \<noteq> 0; cos (2 * x) \<noteq> 0 |]
==> tan (2 * x) = (2 * tan x)/(1 - (tan(x) ^ 2))"
-apply (insert tan_add [of x x])
-apply (simp add: mult_2 [symmetric])
+apply (insert tan_add [of x x])
+apply (simp add: mult_2 [symmetric])
apply (auto simp add: numeral_2_eq_2)
done
lemma tan_gt_zero: "[| 0 < x; x < pi/2 |] ==> 0 < tan x"
-by (simp add: tan_def zero_less_divide_iff sin_gt_zero2 cos_gt_zero_pi)
-
-lemma tan_less_zero:
+by (simp add: tan_def zero_less_divide_iff sin_gt_zero2 cos_gt_zero_pi)
+
+lemma tan_less_zero:
assumes lb: "- pi/2 < x" and "x < 0" shows "tan x < 0"
proof -
- have "0 < tan (- x)" using assms by (simp only: tan_gt_zero)
+ have "0 < tan (- x)" using assms by (simp only: tan_gt_zero)
thus ?thesis by simp
qed
@@ -2143,8 +2144,8 @@
apply (rule LIM_mult)
apply (rule_tac [2] inverse_1 [THEN subst])
apply (rule_tac [2] LIM_inverse)
-apply (simp_all add: divide_inverse [symmetric])
-apply (simp_all only: isCont_def [symmetric] cos_pi_half [symmetric] sin_pi_half [symmetric])
+apply (simp_all add: divide_inverse [symmetric])
+apply (simp_all only: isCont_def [symmetric] cos_pi_half [symmetric] sin_pi_half [symmetric])
apply (blast intro!: DERIV_isCont DERIV_sin DERIV_cos)+
done
@@ -2189,12 +2190,12 @@
apply (subgoal_tac "\<exists>z. xa < z & z < y & DERIV tan z :> 0")
apply (rule_tac [4] Rolle)
apply (rule_tac [2] Rolle)
-apply (auto intro!: DERIV_tan DERIV_isCont exI
+apply (auto intro!: DERIV_tan DERIV_isCont exI
simp add: differentiable_def)
txt{*Now, simulate TRYALL*}
apply (rule_tac [!] DERIV_tan asm_rl)
apply (auto dest!: DERIV_unique [OF _ DERIV_tan]
- simp add: cos_gt_zero_pi [THEN less_imp_neq, THEN not_sym])
+ simp add: cos_gt_zero_pi [THEN less_imp_neq, THEN not_sym])
done
lemma tan_monotone: assumes "- (pi / 2) < y" and "y < x" and "x < pi / 2"
@@ -2208,7 +2209,7 @@
have "cos x' \<noteq> 0" by auto
thus "DERIV tan x' :> inverse (cos x'^2)" by (rule DERIV_tan)
qed
- from MVT2[OF `y < x` this]
+ from MVT2[OF `y < x` this]
obtain z where "y < z" and "z < x" and tan_diff: "tan x - tan y = (x - y) * inverse ((cos z)\<twosuperior>)" by auto
hence "- (pi / 2) < z" and "z < pi / 2" using assms by auto
hence "0 < cos z" using cos_gt_zero_pi by auto
@@ -2228,7 +2229,7 @@
show "y < x"
proof (rule ccontr)
assume "\<not> y < x" hence "x \<le> y" by auto
- hence "tan x \<le> tan y"
+ hence "tan x \<le> tan y"
proof (cases "x = y")
case True thus ?thesis by auto
next
@@ -2241,10 +2242,10 @@
lemma tan_inverse: "1 / (tan y) = tan (pi / 2 - y)" unfolding tan_def sin_cos_eq[of y] cos_sin_eq[of y] by auto
-lemma tan_periodic_pi[simp]: "tan (x + pi) = tan x"
+lemma tan_periodic_pi[simp]: "tan (x + pi) = tan x"
by (simp add: tan_def)
-lemma tan_periodic_nat[simp]: fixes n :: nat shows "tan (x + real n * pi) = tan x"
+lemma tan_periodic_nat[simp]: fixes n :: nat shows "tan (x + real n * pi) = tan x"
proof (induct n arbitrary: x)
case (Suc n)
have split_pi_off: "x + real (Suc n) * pi = (x + real n * pi) + pi" unfolding Suc_eq_plus1 real_of_nat_add real_of_one left_distrib by auto
@@ -2275,18 +2276,18 @@
arccos :: "real => real" where
"arccos y = (THE x. 0 \<le> x & x \<le> pi & cos x = y)"
-definition
+definition
arctan :: "real => real" where
"arctan y = (THE x. -(pi/2) < x & x < pi/2 & tan x = y)"
lemma arcsin:
- "[| -1 \<le> y; y \<le> 1 |]
- ==> -(pi/2) \<le> arcsin y &
+ "[| -1 \<le> y; y \<le> 1 |]
+ ==> -(pi/2) \<le> arcsin y &
arcsin y \<le> pi/2 & sin(arcsin y) = y"
unfolding arcsin_def by (rule theI' [OF sin_total])
lemma arcsin_pi:
- "[| -1 \<le> y; y \<le> 1 |]
+ "[| -1 \<le> y; y \<le> 1 |]
==> -(pi/2) \<le> arcsin y & arcsin y \<le> pi & sin(arcsin y) = y"
apply (drule (1) arcsin)
apply (force intro: order_trans)
@@ -2294,7 +2295,7 @@
lemma sin_arcsin [simp]: "[| -1 \<le> y; y \<le> 1 |] ==> sin(arcsin y) = y"
by (blast dest: arcsin)
-
+
lemma arcsin_bounded:
"[| -1 \<le> y; y \<le> 1 |] ==> -(pi/2) \<le> arcsin y & arcsin y \<le> pi/2"
by (blast dest: arcsin)
@@ -2323,13 +2324,13 @@
done
lemma arccos:
- "[| -1 \<le> y; y \<le> 1 |]
+ "[| -1 \<le> y; y \<le> 1 |]
==> 0 \<le> arccos y & arccos y \<le> pi & cos(arccos y) = y"
unfolding arccos_def by (rule theI' [OF cos_total])
lemma cos_arccos [simp]: "[| -1 \<le> y; y \<le> 1 |] ==> cos(arccos y) = y"
by (blast dest: arccos)
-
+
lemma arccos_bounded: "[| -1 \<le> y; y \<le> 1 |] ==> 0 \<le> arccos y & arccos y \<le> pi"
by (blast dest: arccos)
@@ -2340,7 +2341,7 @@
by (blast dest: arccos)
lemma arccos_lt_bounded:
- "[| -1 < y; y < 1 |]
+ "[| -1 < y; y < 1 |]
==> 0 < arccos y & arccos y < pi"
apply (frule order_less_imp_le)
apply (frule_tac y = y in order_less_imp_le)
@@ -2400,7 +2401,7 @@
lemma arctan_ubound: "arctan y < pi/2"
by (auto simp only: arctan)
-lemma arctan_tan:
+lemma arctan_tan:
"[|-(pi/2) < x; x < pi/2 |] ==> arctan(tan x) = x"
apply (unfold arctan_def)
apply (rule the1_equality)
@@ -2415,7 +2416,7 @@
apply (case_tac "n")
apply (case_tac [3] "n")
apply (cut_tac [2] y = x in arctan_ubound)
-apply (cut_tac [4] y = x in arctan_lbound)
+apply (cut_tac [4] y = x in arctan_lbound)
apply (auto simp add: real_of_nat_Suc left_distrib mult_less_0_iff)
done
@@ -2423,7 +2424,7 @@
apply (rule power_inverse [THEN subst])
apply (rule_tac c1 = "(cos x)\<twosuperior>" in real_mult_right_cancel [THEN iffD1])
apply (auto dest: field_power_not_zero
- simp add: power_mult_distrib left_distrib power_divide tan_def
+ simp add: power_mult_distrib left_distrib power_divide tan_def
mult_assoc power_inverse [symmetric])
done
@@ -2588,7 +2589,7 @@
have "sin ((real n + 1/2) * pi) = cos (real n * pi)"
by (auto simp add: algebra_simps sin_add)
thus ?thesis
- by (simp add: real_of_nat_Suc left_distrib add_divide_distrib
+ by (simp add: real_of_nat_Suc left_distrib add_divide_distrib
mult_commute [of pi])
qed
@@ -2627,7 +2628,7 @@
proof -
obtain z where "- (pi / 2) < z" and "z < pi / 2" and "tan z = x" using tan_total by blast
have "tan (pi / 4) = 1" and "tan (- (pi / 4)) = - 1" using tan_45 tan_minus by auto
- have "z \<noteq> pi / 4"
+ have "z \<noteq> pi / 4"
proof (rule ccontr)
assume "\<not> (z \<noteq> pi / 4)" hence "z = pi / 4" by auto
have "tan z = 1" unfolding `z = pi / 4` `tan (pi / 4) = 1` ..
@@ -2644,11 +2645,11 @@
proof (rule ccontr)
assume "\<not> (z < pi / 4)" hence "pi / 4 < z" using `z \<noteq> pi / 4` by auto
have "- (pi / 2) < pi / 4" using m2pi_less_pi by auto
- from tan_monotone[OF this `pi / 4 < z` `z < pi / 2`]
+ from tan_monotone[OF this `pi / 4 < z` `z < pi / 2`]
have "1 < x" unfolding `tan z = x` `tan (pi / 4) = 1` .
thus False using `\<bar>x\<bar> < 1` by auto
qed
- moreover
+ moreover
have "-(pi / 4) < z"
proof (rule ccontr)
assume "\<not> (-(pi / 4) < z)" hence "z < - (pi / 4)" using `z \<noteq> - (pi / 4)` by auto
@@ -2677,14 +2678,14 @@
show ?thesis
proof (cases "x = 1")
case True hence "tan (pi/4) = x" using tan_45 by auto
- moreover
+ moreover
have "- pi \<le> pi" unfolding minus_le_self_iff by auto
hence "-(pi/4) \<le> pi/4" and "pi/4 \<le> pi/4" by auto
ultimately show ?thesis by blast
next
case False hence "x = -1" using `\<not> \<bar>x\<bar> < 1` and `\<bar>x\<bar> \<le> 1` by auto
hence "tan (-(pi/4)) = x" using tan_45 tan_minus by auto
- moreover
+ moreover
have "- pi \<le> pi" unfolding minus_le_self_iff by auto
hence "-(pi/4) \<le> pi/4" and "-(pi/4) \<le> -(pi/4)" by auto
ultimately show ?thesis by blast
@@ -2723,7 +2724,7 @@
have "\<bar>5 / 12\<bar> < (1 :: real)" by auto
from arctan_add[OF less_imp_le[OF this] this]
have "2 * arctan (5 / 12) = arctan (120 / 119)" by auto
- moreover
+ moreover
have "\<bar>1\<bar> \<le> (1::real)" and "\<bar>1 / 239\<bar> < (1::real)" by auto
from arctan_add[OF this]
have "arctan 1 + arctan (1 / 239) = arctan (120 / 119)" by auto
@@ -2749,7 +2750,7 @@
show "0 \<le> x ^ Suc (Suc n * 2)" by (rule zero_le_power) (simp add: `0 \<le> x`)
qed
} note mono = this
-
+
show ?thesis
proof (cases "0 \<le> x")
case True from mono[OF this `x \<le> 1`, THEN allI]
@@ -2793,7 +2794,7 @@
from mult_left_mono[OF less_imp_le[OF `\<bar>x\<bar> < 1`] abs_ge_zero[of x]]
have "\<bar> x^2 \<bar> < 1" using `\<bar> x \<bar> < 1` unfolding numeral_2_eq_2 power_Suc2 by auto
thus ?thesis using zero_le_power2 by auto
-qed
+qed
lemma DERIV_arctan_series: assumes "\<bar> x \<bar> < 1"
shows "DERIV (\<lambda> x'. \<Sum> k. (-1)^k * (1 / real (k*2+1) * x' ^ (k*2+1))) x :> (\<Sum> k. (-1)^k * x^(k*2))" (is "DERIV ?arctan _ :> ?Int")
@@ -2812,7 +2813,7 @@
{ fix f :: "nat \<Rightarrow> real"
have "\<And> x. f sums x = (\<lambda> n. if even n then f (n div 2) else 0) sums x"
proof
- fix x :: real assume "f sums x"
+ fix x :: real assume "f sums x"
from sums_if[OF sums_zero this]
show "(\<lambda> n. if even n then f (n div 2) else 0) sums x" by auto
next
@@ -2827,10 +2828,10 @@
by auto
{ fix x :: real
- have if_eq': "\<And> n. (if even n then -1 ^ (n div 2) * 1 / real (Suc n) else 0) * x ^ Suc n =
+ have if_eq': "\<And> n. (if even n then -1 ^ (n div 2) * 1 / real (Suc n) else 0) * x ^ Suc n =
(if even n then -1 ^ (n div 2) * (1 / real (Suc (2 * (n div 2))) * x ^ Suc (2 * (n div 2))) else 0)"
using n_even by auto
- have idx_eq: "\<And> n. n * 2 + 1 = Suc (2 * n)" by auto
+ have idx_eq: "\<And> n. n * 2 + 1 = Suc (2 * n)" by auto
have "(\<Sum> n. ?f n * x^(Suc n)) = ?arctan x" unfolding if_eq' idx_eq suminf_def sums_even[of "\<lambda> n. -1 ^ n * (1 / real (Suc (2 * n)) * x ^ Suc (2 * n))", symmetric]
by auto
} note arctan_eq = this
@@ -2893,10 +2894,10 @@
show "\<forall> y. a \<le> y \<and> y \<le> b \<longrightarrow> isCont (\<lambda> x. suminf (?c x) - arctan x) y" using DERIV_in_rball DERIV_isCont by auto
qed
qed
-
+
have suminf_arctan_zero: "suminf (?c 0) - arctan 0 = 0"
unfolding Suc_eq_plus1[symmetric] power_Suc2 mult_zero_right arctan_zero_zero suminf_zero by auto
-
+
have "suminf (?c x) - arctan x = 0"
proof (cases "x = 0")
case True thus ?thesis using suminf_arctan_zero by auto
@@ -2909,7 +2910,7 @@
have "suminf (?c x) - arctan x = suminf (?c (-\<bar>x\<bar>)) - arctan (-\<bar>x\<bar>)"
by (rule suminf_eq_arctan_bounded[where x="x" and a="-\<bar>x\<bar>" and b="\<bar>x\<bar>"])
(simp_all only: `\<bar>x\<bar> < r` `-\<bar>x\<bar> < \<bar>x\<bar>` neg_less_iff_less)
- ultimately
+ ultimately
show ?thesis using suminf_arctan_zero by auto
qed
thus ?thesis by auto
@@ -2971,16 +2972,16 @@
from this[unfolded LIMSEQ_rabs_zero diff_minus add_commute[of "arctan 1"], THEN LIMSEQ_add_const, of "- arctan 1", THEN LIMSEQ_minus]
have "(?c 1) sums (arctan 1)" unfolding sums_def by auto
hence "arctan 1 = (\<Sum> i. ?c 1 i)" by (rule sums_unique)
-
+
show ?thesis
proof (cases "x = 1", simp add: `arctan 1 = (\<Sum> i. ?c 1 i)`)
assume "x \<noteq> 1" hence "x = -1" using `\<bar>x\<bar> = 1` by auto
-
+
have "- (pi / 2) < 0" using pi_gt_zero by auto
have "- (2 * pi) < 0" using pi_gt_zero by auto
-
+
have c_minus_minus: "\<And> i. ?c (- 1) i = - ?c 1 i" unfolding One_nat_def by auto
-
+
have "arctan (- 1) = arctan (tan (-(pi / 4)))" unfolding tan_45 tan_minus ..
also have "\<dots> = - (pi / 4)" by (rule arctan_tan, auto simp add: order_less_trans[OF `- (pi / 2) < 0` pi_gt_zero])
also have "\<dots> = - (arctan (tan (pi / 4)))" unfolding neg_equal_iff_equal by (rule arctan_tan[symmetric], auto simp add: order_less_trans[OF `- (2 * pi) < 0` pi_gt_zero])
@@ -2999,7 +3000,7 @@
hence low2: "- (pi / 2) < y / 2" and high2: "y / 2 < pi / 2" by auto
have divide_nonzero_divide: "\<And> A B C :: real. C \<noteq> 0 \<Longrightarrow> A / B = (A / C) / (B / C)" by auto
-
+
have "0 < cos y" using cos_gt_zero_pi[OF low high] .
hence "cos y \<noteq> 0" and cos_sqrt: "sqrt ((cos y) ^ 2) = cos y" by auto
@@ -3032,14 +3033,14 @@
qed
lemma arctan_monotone': assumes "x \<le> y" shows "arctan x \<le> arctan y"
-proof (cases "x = y")
+proof (cases "x = y")
case False hence "x < y" using `x \<le> y` by auto from arctan_monotone[OF this] show ?thesis by auto
qed auto
-lemma arctan_minus: "arctan (- x) = - arctan x"
+lemma arctan_minus: "arctan (- x) = - arctan x"
proof -
obtain y where "- (pi / 2) < y" and "y < pi / 2" and "tan y = x" using tan_total by blast
- thus ?thesis unfolding `tan y = x`[symmetric] tan_minus[symmetric] using arctan_tan[of y] arctan_tan[of "-y"] by auto
+ thus ?thesis unfolding `tan y = x`[symmetric] tan_minus[symmetric] using arctan_tan[of y] arctan_tan[of "-y"] by auto
qed
lemma arctan_inverse: assumes "x \<noteq> 0" shows "arctan (1 / x) = sgn x * pi / 2 - arctan x"
@@ -3062,7 +3063,7 @@
case True from pos_y[OF this `y < pi / 2` `y = arctan x` `tan y = x`] show ?thesis .
next
case False hence "y \<le> 0" by auto
- moreover have "y \<noteq> 0"
+ moreover have "y \<noteq> 0"
proof (rule ccontr)
assume "\<not> y \<noteq> 0" hence "y = 0" by auto
have "x = 0" unfolding `tan y = x`[symmetric] `y = 0` tan_zero ..