HOL-Probability: use tree datastructure from Libary/Tree; hide left/right/var projections
(* Title: HOL/Probability/Tree_Space.thy
Author: Johannes Hölzl, CMU *)
theory Tree_Space
imports Analysis "~~/src/HOL/Library/Tree"
begin
lemma countable_lfp:
assumes step: "\<And>Y. countable Y \<Longrightarrow> countable (F Y)"
and cont: "Order_Continuity.sup_continuous F"
shows "countable (lfp F)"
by(subst sup_continuous_lfp[OF cont])(simp add: countable_funpow[OF step])
lemma countable_lfp_apply:
assumes step: "\<And>Y x. (\<And>x. countable (Y x)) \<Longrightarrow> countable (F Y x)"
and cont: "Order_Continuity.sup_continuous F"
shows "countable (lfp F x)"
proof -
{ fix n
have "\<And>x. countable ((F ^^ n) bot x)"
by(induct n)(auto intro: step) }
thus ?thesis using cont by(simp add: sup_continuous_lfp)
qed
primrec left :: "'a tree \<Rightarrow> 'a tree"
where
"left (Node l v r) = l"
| "left Leaf = Leaf"
primrec right :: "'a tree \<Rightarrow> 'a tree"
where
"right (Node l v r) = r"
| "right Leaf = Leaf"
primrec val :: "'a tree \<Rightarrow> 'a"
where
"val (Node l v r) = v"
| "val Leaf = undefined"
inductive_set trees :: "'a set \<Rightarrow> 'a tree set" for S :: "'a set" where
[intro!]: "Leaf \<in> trees S"
| "l \<in> trees S \<Longrightarrow> r \<in> trees S \<Longrightarrow> v \<in> S \<Longrightarrow> Node l v r \<in> trees S"
lemma Node_in_trees_iff[simp]: "Node l v r \<in> trees S \<longleftrightarrow> (l \<in> trees S \<and> v \<in> S \<and> r \<in> trees S)"
by (subst trees.simps) auto
lemma trees_sub_lfp: "trees S \<subseteq> lfp (\<lambda>T. T \<union> {Leaf} \<union> (\<Union>l\<in>T. (\<Union>v\<in>S. (\<Union>r\<in>T. {Node l v r}))))"
proof
have mono: "mono (\<lambda>T. T \<union> {Leaf} \<union> (\<Union>l\<in>T. (\<Union>v\<in>S. (\<Union>r\<in>T. {Node l v r}))))"
by (auto simp: mono_def)
fix t assume "t \<in> trees S" then show "t \<in> lfp (\<lambda>T. T \<union> {Leaf} \<union> (\<Union>l\<in>T. (\<Union>v\<in>S. (\<Union>r\<in>T. {Node l v r}))))"
proof induction
case 1 then show ?case
by (subst lfp_unfold[OF mono]) auto
next
case 2 then show ?case
by (subst lfp_unfold[OF mono]) auto
qed
qed
lemma countable_trees: "countable A \<Longrightarrow> countable (trees A)"
apply (rule countable_subset[OF trees_sub_lfp])
apply (rule countable_lfp)
subgoal by auto
apply (intro sup_continuous_sup sup_continuous_const)
subgoal by (simp add: sup_continuous_def)
subgoal apply (auto simp add: sup_continuous_def)
subgoal premises prems for M x c a y d
using prems(3,5) prems(2)[THEN incseqD, of x "max x y"] prems(2)[THEN incseqD, of y "max x y"]
by (intro exI[of _ "max x y"]) auto
done
done
lemma trees_UNIV[simp]: "trees UNIV = UNIV"
proof -
have "t \<in> trees UNIV" for t :: "'a tree"
by (induction t) (auto intro: trees.intros(2))
then show ?thesis by auto
qed
instance tree :: (countable) countable
proof
have "countable (UNIV :: 'a tree set)"
by (subst trees_UNIV[symmetric]) (intro countable_trees[OF countableI_type])
then show "\<exists>to_nat::'a tree \<Rightarrow> nat. inj to_nat"
by (auto simp: countable_def)
qed
lemma map_in_trees[intro]: "(\<And>x. x \<in> set_tree t \<Longrightarrow> f x \<in> S) \<Longrightarrow> map_tree f t \<in> trees S"
by (induction t) (auto intro: trees.intros(2))
primrec trees_cyl :: "'a set tree \<Rightarrow> 'a tree set" where
"trees_cyl Leaf = {Leaf} "
| "trees_cyl (Node l v r) = (\<Union>l'\<in>trees_cyl l. (\<Union>v'\<in>v. (\<Union>r'\<in>trees_cyl r. {Node l' v' r'})))"
definition tree_sigma :: "'a measure \<Rightarrow> 'a tree measure"
where
"tree_sigma M = sigma (trees (space M)) (trees_cyl ` trees (sets M))"
lemma Node_in_trees_cyl: "Node l' v' r' \<in> trees_cyl t \<longleftrightarrow>
(\<exists>l v r. t = Node l v r \<and> l' \<in> trees_cyl l \<and> r' \<in> trees_cyl r \<and> v' \<in> v)"
by (cases t) auto
lemma trees_cyl_sub_trees:
assumes "t \<in> trees A" "A \<subseteq> Pow B" shows "trees_cyl t \<subseteq> trees B"
using assms(1)
proof induction
case (2 l v r) with \<open>A \<subseteq> Pow B\<close> show ?case
by (auto intro!: trees.intros(2))
qed auto
lemma trees_cyl_sets_in_space: "trees_cyl ` trees (sets M) \<subseteq> Pow (trees (space M))"
using trees_cyl_sub_trees[OF _ sets.space_closed, of _ M] by auto
lemma space_tree_sigma: "space (tree_sigma M) = trees (space M)"
unfolding tree_sigma_def by (rule space_measure_of_conv)
lemma sets_tree_sigma_eq: "sets (tree_sigma M) = sigma_sets (trees (space M)) (trees_cyl ` trees (sets M))"
unfolding tree_sigma_def by (rule sets_measure_of) (rule trees_cyl_sets_in_space)
lemma Leaf_in_tree_sigma[measurable]: "{Leaf} \<in> sets (tree_sigma M)"
unfolding sets_tree_sigma_eq
by (rule sigma_sets.Basic) (auto intro: trees.intros(2) image_eqI[where x=Leaf])
lemma trees_cyl_map_treeI: "t \<in> trees_cyl (map_tree (\<lambda>x. A) t)" if *: "t \<in> trees A"
using * by induction auto
lemma trees_cyl_map_in_sets:
"(\<And>x. x \<in> set_tree t \<Longrightarrow> f x \<in> sets M) \<Longrightarrow> trees_cyl (map_tree f t) \<in> sets (tree_sigma M)"
by (subst sets_tree_sigma_eq) auto
lemma Node_in_tree_sigma:
assumes L: "X \<in> sets (M \<Otimes>\<^sub>M (tree_sigma M \<Otimes>\<^sub>M tree_sigma M))"
shows "{Node l v r | l v r. (v, l, r) \<in> X} \<in> sets (tree_sigma M)"
proof -
let ?E = "\<lambda>s::unit tree. trees_cyl (map_tree (\<lambda>_. space M) s)"
have 1: "countable (range ?E)"
by (intro countable_image countableI_type)
have 2: "trees_cyl ` trees (sets M) \<subseteq> Pow (space (tree_sigma M))"
using trees_cyl_sets_in_space[of M] by (simp add: space_tree_sigma)
have 3: "sets (tree_sigma M) = sigma_sets (space (tree_sigma M)) (trees_cyl ` trees (sets M))"
unfolding sets_tree_sigma_eq by (simp add: space_tree_sigma)
have 4: "(\<Union>s. ?E s) = space (tree_sigma M)"
proof (safe; clarsimp simp: space_tree_sigma)
fix t s assume "t \<in> trees_cyl (map_tree (\<lambda>_::unit. space M) s)"
then show "t \<in> trees (space M)"
by (induction s arbitrary: t) auto
next
fix t assume "t \<in> trees (space M)"
then show "\<exists>t'. t \<in> ?E t'"
by (intro exI[of _ "map_tree (\<lambda>_. ()) t"])
(auto simp: tree.map_comp comp_def intro: trees_cyl_map_treeI)
qed
have 5: "range ?E \<subseteq> trees_cyl ` trees (sets M)" by auto
let ?P = "{A \<times> B | A B. A \<in> trees_cyl ` trees (sets M) \<and> B \<in> trees_cyl ` trees (sets M)}"
have P: "sets (tree_sigma M \<Otimes>\<^sub>M tree_sigma M) = sets (sigma (space (tree_sigma M) \<times> space (tree_sigma M)) ?P)"
by (rule sets_pair_eq[OF 2 3 1 5 4 2 3 1 5 4])
have "sets (M \<Otimes>\<^sub>M (tree_sigma M \<Otimes>\<^sub>M tree_sigma M)) =
sets (sigma (space M \<times> space (tree_sigma M \<Otimes>\<^sub>M tree_sigma M)) {A \<times> BC | A BC. A \<in> sets M \<and> BC \<in> ?P})"
proof (rule sets_pair_eq)
show "sets M \<subseteq> Pow (space M)" "sets M = sigma_sets (space M) (sets M)"
by (auto simp: sets.sigma_sets_eq sets.space_closed)
show "countable {space M}" "{space M} \<subseteq> sets M" "\<Union>{space M} = space M"
by auto
show "?P \<subseteq> Pow (space (tree_sigma M \<Otimes>\<^sub>M tree_sigma M))"
using trees_cyl_sets_in_space[of M]
by (auto simp: space_pair_measure space_tree_sigma subset_eq)
then show "sets (tree_sigma M \<Otimes>\<^sub>M tree_sigma M) =
sigma_sets (space (tree_sigma M \<Otimes>\<^sub>M tree_sigma M)) ?P"
by (subst P, subst sets_measure_of) (auto simp: space_tree_sigma space_pair_measure)
show "countable ((\<lambda>(a, b). a \<times> b) ` (range ?E \<times> range ?E))"
by (intro countable_image countable_SIGMA countableI_type)
show "(\<lambda>(a, b). a \<times> b) ` (range ?E \<times> range ?E) \<subseteq> ?P"
by auto
qed (insert 4, auto simp: space_pair_measure space_tree_sigma set_eq_iff)
also have "\<dots> = sigma_sets (space M \<times> trees (space M) \<times> trees (space M))
{A \<times> trees_cyl B \<times> trees_cyl C | A B C. A \<in> sets M \<and> B \<in> trees (sets M) \<and> C \<in> trees (sets M) }"
apply (subst sets_measure_of)
subgoal
using sets.space_closed[of M] trees_cyl_sets_in_space[of M]
by (clarsimp simp: space_pair_measure space_tree_sigma) blast
apply (rule arg_cong2[where f=sigma_sets])
apply (auto simp: space_pair_measure space_tree_sigma)
subgoal premises prems for A B C
apply (rule exI conjI refl prems)+
using trees_cyl_sets_in_space[of M] prems
by auto
done
finally have "X \<in> sigma_sets (space M \<times> trees (space M) \<times> trees (space M))
{A \<times> trees_cyl B \<times> trees_cyl C | A B C. A \<in> sets M \<and> B \<in> trees (sets M) \<and> C \<in> trees (sets M) }"
using assms by auto
then show ?thesis
proof induction
case (Basic A')
then obtain A B C where "A' = A \<times> trees_cyl B \<times> trees_cyl C"
and *: "A \<in> sets M" "B \<in> trees (sets M)" "C \<in> trees (sets M)"
by auto
then have "{Node l v r |l v r. (v, l, r) \<in> A'} = trees_cyl (Node B A C)"
by auto
then show ?case
by (auto simp del: trees_cyl.simps simp: sets_tree_sigma_eq intro!: sigma_sets.Basic *)
next
case Empty show ?case by auto
next
case (Compl A)
have "{Node l v r |l v r. (v, l, r) \<in> space M \<times> trees (space M) \<times> trees (space M) - A} =
(space (tree_sigma M) - {Node l v r |l v r. (v, l, r) \<in> A}) - {Leaf}"
apply (auto simp: space_tree_sigma)
subgoal for t
by (cases t) auto
done
also have "\<dots> \<in> sets (tree_sigma M)"
by (intro sets.Diff Compl) auto
finally show ?case .
next
case (Union I)
have *: "{Node l v r |l v r. (v, l, r) \<in> UNION UNIV I} =
(\<Union>i. {Node l v r |l v r. (v, l, r) \<in> I i})" by auto
show ?case unfolding * using Union(2) by (intro sets.countable_UN) auto
qed
qed
lemma measurable_left[measurable]: "left \<in> tree_sigma M \<rightarrow>\<^sub>M tree_sigma M"
proof (rule measurableI)
show "t \<in> space (tree_sigma M) \<Longrightarrow> left t \<in> space (tree_sigma M)" for t
by (cases t) (auto simp: space_tree_sigma)
fix A assume A: "A \<in> sets (tree_sigma M)"
from sets.sets_into_space[OF this]
have *: "left -` A \<inter> space (tree_sigma M) =
(if Leaf \<in> A then {Leaf} else {}) \<union>
{Node a v r | a v r. (v, a, r) \<in> space M \<times> A \<times> space (tree_sigma M)}"
by (auto simp: space_tree_sigma elim: trees.cases)
show "left -` A \<inter> space (tree_sigma M) \<in> sets (tree_sigma M)"
unfolding * using A by (intro sets.Un Node_in_tree_sigma pair_measureI) auto
qed
lemma measurable_right[measurable]: "right \<in> tree_sigma M \<rightarrow>\<^sub>M tree_sigma M"
proof (rule measurableI)
show "t \<in> space (tree_sigma M) \<Longrightarrow> right t \<in> space (tree_sigma M)" for t
by (cases t) (auto simp: space_tree_sigma)
fix A assume A: "A \<in> sets (tree_sigma M)"
from sets.sets_into_space[OF this]
have *: "right -` A \<inter> space (tree_sigma M) =
(if Leaf \<in> A then {Leaf} else {}) \<union>
{Node l v a | l v a. (v, l, a) \<in> space M \<times> space (tree_sigma M) \<times> A}"
by (auto simp: space_tree_sigma elim: trees.cases)
show "right -` A \<inter> space (tree_sigma M) \<in> sets (tree_sigma M)"
unfolding * using A by (intro sets.Un Node_in_tree_sigma pair_measureI) auto
qed
lemma measurable_val': "val \<in> restrict_space (tree_sigma M) (-{Leaf}) \<rightarrow>\<^sub>M M"
proof (rule measurableI)
show "t \<in> space (restrict_space (tree_sigma M) (- {Leaf})) \<Longrightarrow> val t \<in> space M" for t
by (cases t) (auto simp: space_restrict_space space_tree_sigma)
fix A assume A: "A \<in> sets M"
from sets.sets_into_space[OF this]
have "val -` A \<inter> space (restrict_space (tree_sigma M) (- {Leaf})) =
{Node l a r | l a r. (a, l, r) \<in> A \<times> space (tree_sigma M) \<times> space (tree_sigma M)}"
by (auto simp: space_tree_sigma space_restrict_space elim: trees.cases)
also have "\<dots> \<in> sets (tree_sigma M)"
using A by (intro sets.Un Node_in_tree_sigma pair_measureI) auto
finally show "val -` A \<inter> space (restrict_space (tree_sigma M) (- {Leaf})) \<in>
sets (restrict_space (tree_sigma M) (- {Leaf}))"
by (auto simp: sets_restrict_space_iff space_restrict_space)
qed
lemma measurable_restrict_mono:
assumes f: "f \<in> restrict_space M A \<rightarrow>\<^sub>M N" and "B \<subseteq> A"
shows "f \<in> restrict_space M B \<rightarrow>\<^sub>M N"
by (rule measurable_compose[OF measurable_restrict_space3 f])
(insert \<open>B \<subseteq> A\<close>, auto)
(*
lemma measurable_val[measurable (raw)]:
assumes "f \<in> X \<rightarrow>\<^sub>M tree_sigma M"
and "\<And>x. x \<in> space X \<Longrightarrow> f x \<noteq> Leaf"
shows "(\<lambda>\<omega>. val (f \<omega>)) \<in> X \<rightarrow>\<^sub>M M"
sorry
*)
lemma measurable_rec_tree[measurable (raw)]:
assumes t: "t \<in> B \<rightarrow>\<^sub>M tree_sigma M"
assumes l: "l \<in> B \<rightarrow>\<^sub>M A"
assumes n: "(\<lambda>(x, l, v, r, al, ar). n x l v r al ar) \<in>
(B \<Otimes>\<^sub>M tree_sigma M \<Otimes>\<^sub>M M \<Otimes>\<^sub>M tree_sigma M \<Otimes>\<^sub>M A \<Otimes>\<^sub>M A) \<rightarrow>\<^sub>M A" (is "?N \<in> ?M \<rightarrow>\<^sub>M A")
shows "(\<lambda>x. rec_tree (l x) (n x) (t x)) \<in> B \<rightarrow>\<^sub>M A"
proof (rule measurable_piecewise_restrict)
let ?C = "\<lambda>t. \<lambda>s::unit tree. t -` trees_cyl (map_tree (\<lambda>_. space M) s)"
show "countable (range (?C t))" by (intro countable_image countableI_type)
show "space B \<subseteq> (\<Union>s. ?C t s)"
proof (safe; clarsimp)
fix x assume x: "x \<in> space B" have "t x \<in> trees (space M)"
using t[THEN measurable_space, OF x] by (simp add: space_tree_sigma)
then show "\<exists>xa::unit tree. t x \<in> trees_cyl (map_tree (\<lambda>_. space M) xa)"
by (intro exI[of _ "map_tree (\<lambda>_. ()) (t x)"])
(simp add: tree.map_comp comp_def trees_cyl_map_treeI)
qed
fix \<Omega> assume "\<Omega> \<in> range (?C t)"
then obtain s :: "unit tree" where \<Omega>: "\<Omega> = ?C t s" by auto
then show "\<Omega> \<inter> space B \<in> sets B"
by (safe intro!: measurable_sets[OF t] trees_cyl_map_in_sets)
show "(\<lambda>x. rec_tree (l x) (n x) (t x)) \<in> restrict_space B \<Omega> \<rightarrow>\<^sub>M A"
unfolding \<Omega> using t
proof (induction s arbitrary: t)
case Leaf
show ?case
proof (rule measurable_cong[THEN iffD2])
fix \<omega> assume "\<omega> \<in> space (restrict_space B (?C t Leaf))"
then show "rec_tree (l \<omega>) (n \<omega>) (t \<omega>) = l \<omega>"
by (auto simp: space_restrict_space)
next
show "l \<in> restrict_space B (?C t Leaf) \<rightarrow>\<^sub>M A"
using l by (rule measurable_restrict_space1)
qed
next
case (Node ls u rs)
let ?F = "\<lambda>\<omega>. ?N (\<omega>, left (t \<omega>), val (t \<omega>), right (t \<omega>),
rec_tree (l \<omega>) (n \<omega>) (left (t \<omega>)), rec_tree (l \<omega>) (n \<omega>) (right (t \<omega>)))"
show ?case
proof (rule measurable_cong[THEN iffD2])
fix \<omega> assume "\<omega> \<in> space (restrict_space B (?C t (Node ls u rs)))"
then show "rec_tree (l \<omega>) (n \<omega>) (t \<omega>) = ?F \<omega>"
by (auto simp: space_restrict_space)
next
show "?F \<in> (restrict_space B (?C t (Node ls u rs))) \<rightarrow>\<^sub>M A"
apply (intro measurable_compose[OF _ n] measurable_Pair[rotated])
subgoal
apply (rule measurable_restrict_mono[OF Node(2)])
apply (rule measurable_compose[OF Node(3) measurable_right])
by auto
subgoal
apply (rule measurable_restrict_mono[OF Node(1)])
apply (rule measurable_compose[OF Node(3) measurable_left])
by auto
subgoal
by (rule measurable_restrict_space1)
(rule measurable_compose[OF Node(3) measurable_right])
subgoal
apply (rule measurable_compose[OF _ measurable_val'])
apply (rule measurable_restrict_space3[OF Node(3)])
by auto
subgoal
by (rule measurable_restrict_space1)
(rule measurable_compose[OF Node(3) measurable_left])
by (rule measurable_restrict_space1) auto
qed
qed
qed
hide_const (open) val
hide_const (open) left
hide_const (open) right
end