--- 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