--- a/src/HOL/Probability/Probability_Mass_Function.thy Thu Mar 12 14:08:47 2015 +0100
+++ b/src/HOL/Probability/Probability_Mass_Function.thy Thu Mar 12 19:09:18 2015 +0100
@@ -884,6 +884,136 @@
map_pmf fst pq = p; map_pmf snd pq = q \<rbrakk>
\<Longrightarrow> rel_pmf R p q"
+lemma rel_pmfI:
+ assumes R: "rel_set R (set_pmf p) (set_pmf q)"
+ assumes eq: "\<And>x y. x \<in> set_pmf p \<Longrightarrow> y \<in> set_pmf q \<Longrightarrow> R x y \<Longrightarrow>
+ measure p {x. R x y} = measure q {y. R x y}"
+ shows "rel_pmf R p q"
+proof
+ let ?pq = "bind_pmf p (\<lambda>x. bind_pmf (cond_pmf q {y. R x y}) (\<lambda>y. return_pmf (x, y)))"
+ have "\<And>x. x \<in> set_pmf p \<Longrightarrow> set_pmf q \<inter> {y. R x y} \<noteq> {}"
+ using R by (auto simp: rel_set_def)
+ then show "\<And>x y. (x, y) \<in> set_pmf ?pq \<Longrightarrow> R x y"
+ by auto
+ show "map_pmf fst ?pq = p"
+ by (simp add: map_bind_pmf map_return_pmf bind_return_pmf')
+
+ show "map_pmf snd ?pq = q"
+ using R eq
+ apply (simp add: bind_cond_pmf_cancel map_bind_pmf map_return_pmf bind_return_pmf')
+ apply (rule bind_cond_pmf_cancel)
+ apply (auto simp: rel_set_def)
+ done
+qed
+
+lemma rel_pmf_imp_rel_set: "rel_pmf R p q \<Longrightarrow> rel_set R (set_pmf p) (set_pmf q)"
+ by (force simp add: rel_pmf.simps rel_set_def)
+
+lemma rel_pmfD_measure:
+ assumes rel_R: "rel_pmf R p q" and R: "\<And>a b. R a b \<Longrightarrow> R a y \<longleftrightarrow> R x b"
+ assumes "x \<in> set_pmf p" "y \<in> set_pmf q"
+ shows "measure p {x. R x y} = measure q {y. R x y}"
+proof -
+ from rel_R obtain pq where pq: "\<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> R x y"
+ and eq: "p = map_pmf fst pq" "q = map_pmf snd pq"
+ by (auto elim: rel_pmf.cases)
+ have "measure p {x. R x y} = measure pq {x. R (fst x) y}"
+ by (simp add: eq map_pmf_rep_eq measure_distr)
+ also have "\<dots> = measure pq {y. R x (snd y)}"
+ by (intro measure_pmf.finite_measure_eq_AE)
+ (auto simp: AE_measure_pmf_iff R dest!: pq)
+ also have "\<dots> = measure q {y. R x y}"
+ by (simp add: eq map_pmf_rep_eq measure_distr)
+ finally show "measure p {x. R x y} = measure q {y. R x y}" .
+qed
+
+lemma rel_pmf_iff_measure:
+ assumes "symp R" "transp R"
+ shows "rel_pmf R p q \<longleftrightarrow>
+ rel_set R (set_pmf p) (set_pmf q) \<and>
+ (\<forall>x\<in>set_pmf p. \<forall>y\<in>set_pmf q. R x y \<longrightarrow> measure p {x. R x y} = measure q {y. R x y})"
+ by (safe intro!: rel_pmf_imp_rel_set rel_pmfI)
+ (auto intro!: rel_pmfD_measure dest: sympD[OF \<open>symp R\<close>] transpD[OF \<open>transp R\<close>])
+
+lemma quotient_rel_set_disjoint:
+ "equivp R \<Longrightarrow> C \<in> UNIV // {(x, y). R x y} \<Longrightarrow> rel_set R A B \<Longrightarrow> A \<inter> C = {} \<longleftrightarrow> B \<inter> C = {}"
+ using in_quotient_imp_closed[of UNIV "{(x, y). R x y}" C]
+ by (auto 0 0 simp: equivp_equiv rel_set_def set_eq_iff elim: equivpE)
+ (blast dest: equivp_symp)+
+
+lemma quotientD: "equiv X R \<Longrightarrow> A \<in> X // R \<Longrightarrow> x \<in> A \<Longrightarrow> A = R `` {x}"
+ by (metis Image_singleton_iff equiv_class_eq_iff quotientE)
+
+lemma rel_pmf_iff_equivp:
+ assumes "equivp R"
+ shows "rel_pmf R p q \<longleftrightarrow> (\<forall>C\<in>UNIV // {(x, y). R x y}. measure p C = measure q C)"
+ (is "_ \<longleftrightarrow> (\<forall>C\<in>_//?R. _)")
+proof (subst rel_pmf_iff_measure, safe)
+ show "symp R" "transp R"
+ using assms by (auto simp: equivp_reflp_symp_transp)
+next
+ fix C assume C: "C \<in> UNIV // ?R" and R: "rel_set R (set_pmf p) (set_pmf q)"
+ assume eq: "\<forall>x\<in>set_pmf p. \<forall>y\<in>set_pmf q. R x y \<longrightarrow> measure p {x. R x y} = measure q {y. R x y}"
+
+ show "measure p C = measure q C"
+ proof cases
+ assume "p \<inter> C = {}"
+ moreover then have "q \<inter> C = {}"
+ using quotient_rel_set_disjoint[OF assms C R] by simp
+ ultimately show ?thesis
+ unfolding measure_pmf_zero_iff[symmetric] by simp
+ next
+ assume "p \<inter> C \<noteq> {}"
+ moreover then have "q \<inter> C \<noteq> {}"
+ using quotient_rel_set_disjoint[OF assms C R] by simp
+ ultimately obtain x y where in_set: "x \<in> set_pmf p" "y \<in> set_pmf q" and in_C: "x \<in> C" "y \<in> C"
+ by auto
+ then have "R x y"
+ using in_quotient_imp_in_rel[of UNIV ?R C x y] C assms
+ by (simp add: equivp_equiv)
+ with in_set eq have "measure p {x. R x y} = measure q {y. R x y}"
+ by auto
+ moreover have "{y. R x y} = C"
+ using assms `x \<in> C` C quotientD[of UNIV ?R C x] by (simp add: equivp_equiv)
+ moreover have "{x. R x y} = C"
+ using assms `y \<in> C` C quotientD[of UNIV "?R" C y] sympD[of R]
+ by (auto simp add: equivp_equiv elim: equivpE)
+ ultimately show ?thesis
+ by auto
+ qed
+next
+ assume eq: "\<forall>C\<in>UNIV // ?R. measure p C = measure q C"
+ show "rel_set R (set_pmf p) (set_pmf q)"
+ unfolding rel_set_def
+ proof safe
+ fix x assume x: "x \<in> set_pmf p"
+ have "{y. R x y} \<in> UNIV // ?R"
+ by (auto simp: quotient_def)
+ with eq have *: "measure q {y. R x y} = measure p {y. R x y}"
+ by auto
+ have "measure q {y. R x y} \<noteq> 0"
+ using x assms unfolding * by (auto simp: measure_pmf_zero_iff set_eq_iff dest: equivp_reflp)
+ then show "\<exists>y\<in>set_pmf q. R x y"
+ unfolding measure_pmf_zero_iff by auto
+ next
+ fix y assume y: "y \<in> set_pmf q"
+ have "{x. R x y} \<in> UNIV // ?R"
+ using assms by (auto simp: quotient_def dest: equivp_symp)
+ with eq have *: "measure p {x. R x y} = measure q {x. R x y}"
+ by auto
+ have "measure p {x. R x y} \<noteq> 0"
+ using y assms unfolding * by (auto simp: measure_pmf_zero_iff set_eq_iff dest: equivp_reflp)
+ then show "\<exists>x\<in>set_pmf p. R x y"
+ unfolding measure_pmf_zero_iff by auto
+ qed
+
+ fix x y assume "x \<in> set_pmf p" "y \<in> set_pmf q" "R x y"
+ have "{y. R x y} \<in> UNIV // ?R" "{x. R x y} = {y. R x y}"
+ using assms `R x y` by (auto simp: quotient_def dest: equivp_symp equivp_transp)
+ with eq show "measure p {x. R x y} = measure q {y. R x y}"
+ by auto
+qed
+
bnf pmf: "'a pmf" map: map_pmf sets: set_pmf bd : "natLeq" rel: rel_pmf
proof -
show "map_pmf id = id" by (rule map_pmf_id)
@@ -912,7 +1042,7 @@
and x: "x \<in> set_pmf p"
thus "f x = g x" by simp }
- fix R :: "'a => 'b \<Rightarrow> bool" and S :: "'b \<Rightarrow> 'c \<Rightarrow> bool"
+ fix R :: "'a \<Rightarrow> 'b \<Rightarrow> bool" and S :: "'b \<Rightarrow> 'c \<Rightarrow> bool"
{ fix p q r
assume pq: "rel_pmf R p q"
and qr:"rel_pmf S q r"
@@ -1130,46 +1260,31 @@
assumes 2: "rel_pmf R q p"
and refl: "reflp R" and trans: "transp R"
shows "rel_pmf (inf R R\<inverse>\<inverse>) p q"
-proof
+proof (subst rel_pmf_iff_equivp, safe)
+ show "equivp (inf R R\<inverse>\<inverse>)"
+ using trans refl by (auto simp: equivp_reflp_symp_transp intro: sympI transpI reflpI dest: transpD reflpD)
+
+ fix C assume "C \<in> UNIV // {(x, y). inf R R\<inverse>\<inverse> x y}"
+ then obtain x where C: "C = {y. R x y \<and> R y x}"
+ by (auto elim: quotientE)
+
let ?R = "\<lambda>x y. R x y \<and> R y x"
let ?\<mu>R = "\<lambda>y. measure q {x. ?R x y}"
- { fix x
- have "measure p {y. ?R x y} = measure p ({y. R x y} - {y. R x y \<and> \<not> R y x})"
- by(auto intro!: arg_cong[where f="measure p"])
- also have "\<dots> = measure p {y. R x y} - measure p {y. R x y \<and> \<not> R y x}"
- by (rule measure_pmf.finite_measure_Diff) auto
- also have "measure p {y. R x y \<and> \<not> R y x} = measure q {y. R x y \<and> \<not> R y x}"
- using 1 2 refl trans by(auto intro!: Orderings.antisym measure_Ioi)
- also have "measure p {y. R x y} = measure q {y. R x y}"
- using 1 2 refl trans by(auto intro!: Orderings.antisym measure_Ici)
- also have "measure q {y. R x y} - measure q {y. R x y \<and> ~ R y x} =
- measure q ({y. R x y} - {y. R x y \<and> \<not> R y x})"
- by(rule measure_pmf.finite_measure_Diff[symmetric]) auto
- also have "\<dots> = ?\<mu>R x"
- by(auto intro!: arg_cong[where f="measure q"])
- also note calculation }
- note eq = this
-
- def pq \<equiv> "bind_pmf p (\<lambda>x. bind_pmf (cond_pmf q {y. ?R y x}) (\<lambda>y. return_pmf (x, y)))"
-
- show "map_pmf fst pq = p"
- by(simp add: pq_def map_bind_pmf map_return_pmf bind_return_pmf')
-
- { fix y assume "y \<in> set_pmf p" then have "set_pmf q \<inter> {x. ?R x y} \<noteq> {}"
- unfolding measure_pmf_zero_iff[symmetric] eq[symmetric] by (auto simp: measure_pmf_zero_iff intro: reflpD[OF refl]) }
- note set_p = this
- moreover
- { fix x assume "x \<in> set_pmf q" then have "set_pmf p \<inter> {y. R x y \<and> R y x} \<noteq> {}"
- unfolding measure_pmf_zero_iff[symmetric] eq by (auto simp: measure_pmf_zero_iff intro: reflpD[OF refl]) }
- ultimately
- show "map_pmf snd pq = q"
- unfolding pq_def map_bind_pmf map_return_pmf bind_return_pmf' snd_conv
- by (subst bind_cond_pmf_cancel)
- (auto simp add: eq AE_measure_pmf_iff dest: transpD[OF trans]
- intro!: measure_pmf.finite_measure_eq_AE)
-
- fix x y assume "(x, y) \<in> set_pmf pq" with set_p show "inf R R\<inverse>\<inverse> x y"
- by (auto simp add: pq_def)
+ have "measure p {y. ?R x y} = measure p ({y. R x y} - {y. R x y \<and> \<not> R y x})"
+ by(auto intro!: arg_cong[where f="measure p"])
+ also have "\<dots> = measure p {y. R x y} - measure p {y. R x y \<and> \<not> R y x}"
+ by (rule measure_pmf.finite_measure_Diff) auto
+ also have "measure p {y. R x y \<and> \<not> R y x} = measure q {y. R x y \<and> \<not> R y x}"
+ using 1 2 refl trans by(auto intro!: Orderings.antisym measure_Ioi)
+ also have "measure p {y. R x y} = measure q {y. R x y}"
+ using 1 2 refl trans by(auto intro!: Orderings.antisym measure_Ici)
+ also have "measure q {y. R x y} - measure q {y. R x y \<and> \<not> R y x} =
+ measure q ({y. R x y} - {y. R x y \<and> \<not> R y x})"
+ by(rule measure_pmf.finite_measure_Diff[symmetric]) auto
+ also have "\<dots> = ?\<mu>R x"
+ by(auto intro!: arg_cong[where f="measure q"])
+ finally show "measure p C = measure q C"
+ by (simp add: C conj_commute)
qed
lemma rel_pmf_antisym: