--- a/src/HOL/Probability/Projective_Family.thy Wed Nov 07 11:33:27 2012 +0100
+++ b/src/HOL/Probability/Projective_Family.thy Wed Nov 07 14:41:49 2012 +0100
@@ -25,14 +25,14 @@
fixes I::"'i set" and P::"'i set \<Rightarrow> ('i \<Rightarrow> 'a) measure" and M::"('i \<Rightarrow> 'a measure)"
assumes projective: "\<And>J H X. J \<noteq> {} \<Longrightarrow> J \<subseteq> H \<Longrightarrow> H \<subseteq> I \<Longrightarrow> finite H \<Longrightarrow> X \<in> sets (PiM J M) \<Longrightarrow>
(P H) (prod_emb H M J X) = (P J) X"
+ assumes prob_space: "\<And>J. finite J \<Longrightarrow> prob_space (P J)"
assumes proj_space: "\<And>J. finite J \<Longrightarrow> space (P J) = space (PiM J M)"
assumes proj_sets: "\<And>J. finite J \<Longrightarrow> sets (P J) = sets (PiM J M)"
assumes proj_finite_measure: "\<And>J. finite J \<Longrightarrow> emeasure (P J) (space (PiM J M)) \<noteq> \<infinity>"
- assumes prob_space: "\<And>i. prob_space (M i)"
+ assumes measure_space: "\<And>i. prob_space (M i)"
begin
lemma emeasure_PiP:
- assumes "J \<noteq> {}"
assumes "finite J"
assumes "J \<subseteq> I"
assumes A: "\<And>i. i\<in>J \<Longrightarrow> A i \<in> sets (M i)"
@@ -49,30 +49,27 @@
emeasure (PiP J M P) (prod_emb J M J (Pi\<^isub>E J A))"
using assms(1-3) sets_into_space by (auto simp add: prod_emb_id Pi_def)
also have "\<dots> = emeasure (P J) (Pi\<^isub>E J A)"
- proof (rule emeasure_extend_measure[OF PiP_def, where i="(J, A)", simplified,
- of J M "P J" P])
- show "positive (sets (PiM J M)) (P J)" unfolding positive_def by auto
- show "countably_additive (sets (PiM J M)) (P J)" unfolding countably_additive_def
+ proof (rule emeasure_extend_measure_Pair[OF PiP_def])
+ show "positive (sets (PiP J M P)) (P J)" unfolding positive_def by auto
+ show "countably_additive (sets (PiP J M P)) (P J)" unfolding countably_additive_def
by (auto simp: suminf_emeasure proj_sets[OF `finite J`])
- show "(\<lambda>(Ja, X). prod_emb J M Ja (Pi\<^isub>E Ja X)) ` {(Ja, X). (Ja = {} \<longrightarrow> J = {}) \<and>
- finite Ja \<and> Ja \<subseteq> J \<and> X \<in> (\<Pi> j\<in>Ja. sets (M j))} \<subseteq> Pow (\<Pi> i\<in>J. space (M i)) \<and>
- (\<lambda>(Ja, X). prod_emb J M Ja (Pi\<^isub>E Ja X)) `
- {(Ja, X). (Ja = {} \<longrightarrow> J = {}) \<and> finite Ja \<and> Ja \<subseteq> J \<and> X \<in> (\<Pi> j\<in>Ja. sets (M j))} \<subseteq>
- Pow (extensional J)" by (auto simp: prod_emb_def)
- show "(J = {} \<longrightarrow> J = {}) \<and> finite J \<and> J \<subseteq> J \<and> A \<in> (\<Pi> j\<in>J. sets (M j))"
+ show "(J \<noteq> {} \<or> J = {}) \<and> finite J \<and> J \<subseteq> J \<and> A \<in> (\<Pi> j\<in>J. sets (M j))"
using assms by auto
- fix i
- assume
- "case i of (Ja, X) \<Rightarrow> (Ja = {} \<longrightarrow> J = {}) \<and> finite Ja \<and> Ja \<subseteq> J \<and> X \<in> (\<Pi> j\<in>Ja. sets (M j))"
- thus "emeasure (P J) (case i of (Ja, X) \<Rightarrow> prod_emb J M Ja (Pi\<^isub>E Ja X)) =
- (case i of (J, X) \<Rightarrow> emeasure (P J) (Pi\<^isub>E J X))" using assms
- by (cases i) (auto simp add: intro!: projective sets_PiM_I_finite)
+ fix K and X::"'i \<Rightarrow> 'a set"
+ show "prod_emb J M K (Pi\<^isub>E K X) \<in> Pow (\<Pi>\<^isub>E i\<in>J. space (M i))"
+ by (auto simp: prod_emb_def)
+ assume JX: "(K \<noteq> {} \<or> J = {}) \<and> finite K \<and> K \<subseteq> J \<and> X \<in> (\<Pi> j\<in>K. sets (M j))"
+ thus "emeasure (P J) (prod_emb J M K (Pi\<^isub>E K X)) = emeasure (P K) (Pi\<^isub>E K X)"
+ using assms
+ apply (cases "J = {}")
+ apply (simp add: prod_emb_id)
+ apply (fastforce simp add: intro!: projective sets_PiM_I_finite)
+ done
qed
finally show ?thesis .
qed
lemma PiP_finite:
- assumes "J \<noteq> {}"
assumes "finite J"
assumes "J \<subseteq> I"
shows "PiP J M P = P J" (is "?P = _")
@@ -108,6 +105,6 @@
end
sublocale projective_family \<subseteq> M: prob_space "M i" for i
- by (rule prob_space)
+ by (rule measure_space)
end