# HG changeset patch
# User hoelzl
# Date 1296825408 -3600
# Node ID 8c539202f85426b5993bbbf6bfe66d4d6257b03f
# Parent  f69bb9077b02f97238f987252f438e73d69ded56
add borel_fubini_integrable; remove unused bijectivity rules for measureable functions

diff -r f69bb9077b02 -r 8c539202f854 src/HOL/Probability/Lebesgue_Measure.thy
--- a/src/HOL/Probability/Lebesgue_Measure.thy	Wed Feb 02 22:48:24 2011 +0100
+++ b/src/HOL/Probability/Lebesgue_Measure.thy	Fri Feb 04 14:16:48 2011 +0100
@@ -756,13 +756,6 @@
   "p2e (e2p x) = (x::'a::ordered_euclidean_space)"
   by (auto simp: euclidean_eq[where 'a='a] p2e_def e2p_def)
 
-lemma bij_inv_p2e_e2p:
-  shows "bij_inv ({..<DIM('a)} \<rightarrow>\<^isub>E UNIV) (UNIV :: 'a::ordered_euclidean_space set)
-     p2e e2p" (is "bij_inv ?P ?U _ _")
-proof (rule bij_invI)
-  show "p2e \<in> ?P \<rightarrow> ?U" "e2p \<in> ?U \<rightarrow> ?P" by (auto simp: e2p_def)
-qed auto
-
 declare restrict_extensional[intro]
 
 lemma e2p_extensional[intro]:"e2p (y::'a::ordered_euclidean_space) \<in> extensional {..<DIM('a)}"
@@ -850,10 +843,15 @@
   then show "p2e -` A \<inter> space ?P \<in> sets ?P" by auto
 qed simp
 
+lemma inj_e2p[intro, simp]: "inj e2p"
+proof (intro inj_onI conjI allI impI euclidean_eq[where 'a='a, THEN iffD2])
+  fix x y ::'a and i assume "e2p x = e2p y" "i < DIM('a)"
+  then show "x $$ i= y $$ i"
+    by (auto simp: e2p_def restrict_def fun_eq_iff elim!: allE[where x=i])
+qed
+
 lemma e2p_Int:"e2p ` A \<inter> e2p ` B = e2p ` (A \<inter> B)" (is "?L = ?R")
-  apply(rule image_Int[THEN sym])
-  using bij_inv_p2e_e2p[THEN bij_inv_bij_betw(2)]
-  unfolding bij_betw_def by auto
+  by (auto simp: image_Int[THEN sym])
 
 lemma Int_stable_cuboids: fixes x::"'a::ordered_euclidean_space"
   shows "Int_stable \<lparr>space = UNIV, sets = range (\<lambda>(a, b::'a). e2p ` {a..b})\<rparr>"
@@ -947,6 +945,31 @@
     using lmeasure_measure_eq_borel_prod[OF A] by (simp add: range_e2p)
 qed
 
+lemma borel_fubini_integrable:
+  fixes f :: "'a::ordered_euclidean_space \<Rightarrow> real"
+  shows "integrable lborel f \<longleftrightarrow>
+    integrable (lborel_space.P TYPE('a)) (\<lambda>x. f (p2e x))"
+    (is "_ \<longleftrightarrow> integrable ?B ?f")
+proof
+  assume "integrable lborel f"
+  moreover then have f: "f \<in> borel_measurable borel"
+    by auto
+  moreover with measurable_p2e
+  have "f \<circ> p2e \<in> borel_measurable ?B"
+    by (rule measurable_comp)
+  ultimately show "integrable ?B ?f"
+    by (simp add: comp_def borel_fubini_positiv_integral integrable_def)
+next
+  assume "integrable ?B ?f"
+  moreover then
+  have "?f \<circ> e2p \<in> borel_measurable (borel::'a algebra)"
+    by (auto intro!: measurable_e2p measurable_comp)
+  then have "f \<in> borel_measurable borel"
+    by (simp cong: measurable_cong)
+  ultimately show "integrable lborel f"
+    by (simp add: comp_def borel_fubini_positiv_integral integrable_def)
+qed
+
 lemma borel_fubini:
   fixes f :: "'a::ordered_euclidean_space \<Rightarrow> real"
   assumes f: "f \<in> borel_measurable borel"
diff -r f69bb9077b02 -r 8c539202f854 src/HOL/Probability/Sigma_Algebra.thy
--- a/src/HOL/Probability/Sigma_Algebra.thy	Wed Feb 02 22:48:24 2011 +0100
+++ b/src/HOL/Probability/Sigma_Algebra.thy	Fri Feb 04 14:16:48 2011 +0100
@@ -769,48 +769,6 @@
   show ?thesis by (simp add: comp_def)
 qed
 
-lemma (in sigma_algebra) vimage_vimage_inv:
-  assumes f: "bij_betw f S (space M)"
-  assumes [simp]: "\<And>x. x \<in> space M \<Longrightarrow> f (g x) = x" and g: "g \<in> space M \<rightarrow> S"
-  shows "sigma_algebra.vimage_algebra (vimage_algebra S f) (space M) g = M"
-proof -
-  interpret T: sigma_algebra "vimage_algebra S f"
-    using f by (safe intro!: sigma_algebra_vimage bij_betw_imp_funcset)
-
-  have inj: "inj_on f S" and [simp]: "f`S = space M"
-    using f unfolding bij_betw_def by auto
-
-  { fix A assume A: "A \<in> sets M"
-    have "g -` f -` A \<inter> g -` S \<inter> space M = (f \<circ> g) -` A \<inter> space M"
-      using g by auto
-    also have "\<dots> = A"
-      using `A \<in> sets M`[THEN sets_into_space] by auto
-    finally have "g -` f -` A \<inter> g -` S \<inter> space M = A" . }
-  note X = this
-  show ?thesis
-    unfolding T.vimage_algebra_def unfolding vimage_algebra_def
-    by (simp add: image_compose[symmetric] comp_def X cong: image_cong)
-qed
-
-lemma (in sigma_algebra) measurable_vimage_iff:
-  fixes f :: "'b \<Rightarrow> 'a" assumes f: "bij_betw f S (space M)"
-  shows "g \<in> measurable M M' \<longleftrightarrow> (g \<circ> f) \<in> measurable (vimage_algebra S f) M'"
-proof
-  assume "g \<in> measurable M M'"
-  from measurable_vimage[OF this f[THEN bij_betw_imp_funcset]]
-  show "(g \<circ> f) \<in> measurable (vimage_algebra S f) M'" unfolding comp_def .
-next
-  interpret v: sigma_algebra "vimage_algebra S f"
-    using f[THEN bij_betw_imp_funcset] by (rule sigma_algebra_vimage)
-  note f' = f[THEN bij_betw_the_inv_into]
-  assume "g \<circ> f \<in> measurable (vimage_algebra S f) M'"
-  from v.measurable_vimage[OF this, unfolded space_vimage_algebra, OF f'[THEN bij_betw_imp_funcset]]
-  show "g \<in> measurable M M'"
-    using f f'[THEN bij_betw_imp_funcset] f[unfolded bij_betw_def]
-    by (subst (asm) vimage_vimage_inv)
-       (simp_all add: f_the_inv_into_f cong: measurable_cong)
-qed
-
 lemma sigma_sets_vimage:
   assumes "f \<in> S' \<rightarrow> S" and "A \<subseteq> Pow S"
   shows "sigma_sets S' ((\<lambda>X. f -` X \<inter> S') ` A) = (\<lambda>X. f -` X \<inter> S') ` sigma_sets S A"
@@ -1417,93 +1375,10 @@
     by (auto simp: image_space_def)
 qed
 
-subsection "Bijective functions with inverse"
-
-definition "bij_inv A B f g \<longleftrightarrow>
-  f \<in> A \<rightarrow> B \<and> g \<in> B \<rightarrow> A \<and> (\<forall>x\<in>A. g (f x) = x) \<and> (\<forall>x\<in>B. f (g x) = x)"
-
-lemma bij_inv_symmetric[sym]: "bij_inv A B f g \<Longrightarrow> bij_inv B A g f"
-  unfolding bij_inv_def by auto
-
-lemma bij_invI:
-  assumes "f \<in> A \<rightarrow> B" "g \<in> B \<rightarrow> A"
-  and "\<And>x. x \<in> A \<Longrightarrow> g (f x) = x"
-  and "\<And>x. x \<in> B \<Longrightarrow> f (g x) = x"
-  shows "bij_inv A B f g"
-  using assms unfolding bij_inv_def by auto
-
-lemma bij_invE:
-  assumes "bij_inv A B f g"
-    "\<lbrakk> f \<in> A \<rightarrow> B ; g \<in> B \<rightarrow> A ;
-        (\<And>x. x \<in> A \<Longrightarrow> g (f x) = x) ;
-        (\<And>x. x \<in> B \<Longrightarrow> f (g x) = x) \<rbrakk> \<Longrightarrow> P"
-  shows P
-  using assms unfolding bij_inv_def by auto
-
-lemma bij_inv_bij_betw:
-  assumes "bij_inv A B f g"
-  shows "bij_betw f A B" "bij_betw g B A"
-  using assms by (auto intro: bij_betwI elim!: bij_invE)
-
-lemma bij_inv_vimage_vimage:
-  assumes "bij_inv A B f e"
-  shows "f -` (e -` X \<inter> B) \<inter> A = X \<inter> A"
-  using assms by (auto elim!: bij_invE)
-
-lemma (in sigma_algebra) measurable_vimage_iff_inv:
-  fixes f :: "'b \<Rightarrow> 'a" assumes "bij_inv S (space M) f g"
-  shows "h \<in> measurable (vimage_algebra S f) M' \<longleftrightarrow> (\<lambda>x. h (g x)) \<in> measurable M M'"
-  unfolding measurable_vimage_iff[OF bij_inv_bij_betw(1), OF assms]
-proof (rule measurable_cong)
-  fix w assume "w \<in> space (vimage_algebra S f)"
-  then have "w \<in> S" by auto
-  then show "h w = ((\<lambda>x. h (g x)) \<circ> f) w"
-    using assms by (auto elim: bij_invE)
-qed
-
-lemma vimage_algebra_sigma:
-  assumes bi: "bij_inv (space (sigma F)) (space (sigma E)) f e"
-    and "sets E \<subseteq> Pow (space E)" and F: "sets F \<subseteq> Pow (space F)"
-    and "more E = more F"
-    and "f \<in> measurable F E" "e \<in> measurable E F"
-  shows "sigma_algebra.vimage_algebra (sigma E) (space (sigma F)) f = sigma F"
-proof -
-  interpret sigma_algebra "sigma E"
-    using assms by (intro sigma_algebra_sigma) auto
-  have eq: "sets F = (\<lambda>X. f -` X \<inter> space F) ` sets E"
-  proof safe
-    fix X assume "X \<in> sets F"
-    then have "e -` X \<inter> space E \<in> sets E"
-      using `e \<in> measurable E F` unfolding measurable_def by auto
-    then show "X \<in>(\<lambda>Y. f -` Y \<inter> space F) ` sets E"
-      apply (rule rev_image_eqI)
-      unfolding bij_inv_vimage_vimage[OF bi[simplified]]
-      using F `X \<in> sets F` by auto
-  next
-    fix X assume "X \<in> sets E" then show "f -` X \<inter> space F \<in> sets F"
-      using `f \<in> measurable F E` unfolding measurable_def by auto
-  qed
-  show "vimage_algebra (space (sigma F)) f = sigma F"
-    unfolding vimage_algebra_def using assms
-    by (auto simp: bij_inv_def eq sigma_sets_vimage[symmetric] sigma_def)
-qed
-
 lemma measurable_sigma_sigma:
   assumes M: "sets M \<subseteq> Pow (space M)" and N: "sets N \<subseteq> Pow (space N)"
   shows "f \<in> measurable M N \<Longrightarrow> f \<in> measurable (sigma M) (sigma N)"
   using sigma_algebra.measurable_subset[OF sigma_algebra_sigma[OF M], of N]
   using measurable_up_sigma[of M N] N by auto
 
-lemma bij_inv_the_inv_into:
-  assumes "bij_betw f A B" shows "bij_inv A B f (the_inv_into A f)"
-proof (rule bij_invI)
-  show "the_inv_into A f \<in> B \<rightarrow> A"
-    using bij_betw_the_inv_into[OF assms] by (rule bij_betw_imp_funcset)
-  show "f \<in> A \<rightarrow> B" using assms by (rule bij_betw_imp_funcset)
-  show "\<And>x. x \<in> A \<Longrightarrow> the_inv_into A f (f x) = x"
-    "\<And>x. x \<in> B \<Longrightarrow> f (the_inv_into A f x) = x"
-    using the_inv_into_f_f[of f A] f_the_inv_into_f[of f A]
-    using assms by (auto simp: bij_betw_def)
-qed
-
 end