Theory Stream_Space

theory Stream_Space
imports Infinite_Product_Measure Linear_Temporal_Logic_on_Streams
```(*  Title:      HOL/Probability/Stream_Space.thy
Author:     Johannes Hölzl, TU München *)

theory Stream_Space
imports
Infinite_Product_Measure
"HOL-Library.Stream"
"HOL-Library.Linear_Temporal_Logic_on_Streams"
begin

lemma stream_eq_Stream_iff: "s = x ## t ⟷ (shd s = x ∧ stl s = t)"
by (cases s) simp

lemma Stream_snth: "(x ## s) !! n = (case n of 0 ⇒ x | Suc n ⇒ s !! n)"
by (cases n) simp_all

definition to_stream :: "(nat ⇒ 'a) ⇒ 'a stream" where
"to_stream X = smap X nats"

lemma to_stream_nat_case: "to_stream (case_nat x X) = x ## to_stream X"
unfolding to_stream_def
by (subst siterate.ctr) (simp add: smap_siterate[symmetric] stream.map_comp comp_def)

lemma to_stream_in_streams: "to_stream X ∈ streams S ⟷ (∀n. X n ∈ S)"

definition stream_space :: "'a measure ⇒ 'a stream measure" where
"stream_space M =
distr (Π⇩M i∈UNIV. M) (vimage_algebra (streams (space M)) snth (Π⇩M i∈UNIV. M)) to_stream"

lemma space_stream_space: "space (stream_space M) = streams (space M)"

lemma streams_stream_space[intro]: "streams (space M) ∈ sets (stream_space M)"
using sets.top[of "stream_space M"] by (simp add: space_stream_space)

lemma stream_space_Stream:
"x ## ω ∈ space (stream_space M) ⟷ x ∈ space M ∧ ω ∈ space (stream_space M)"

lemma stream_space_eq_distr: "stream_space M = distr (Π⇩M i∈UNIV. M) (stream_space M) to_stream"
unfolding stream_space_def by (rule distr_cong) auto

lemma sets_stream_space_cong[measurable_cong]:
"sets M = sets N ⟹ sets (stream_space M) = sets (stream_space N)"
using sets_eq_imp_space_eq[of M N] by (simp add: stream_space_def vimage_algebra_def cong: sets_PiM_cong)

lemma measurable_snth_PiM: "(λω n. ω !! n) ∈ measurable (stream_space M) (Π⇩M i∈UNIV. M)"
by (auto intro!: measurable_vimage_algebra1
simp: space_PiM streams_iff_sset sset_range image_subset_iff stream_space_def)

lemma measurable_snth[measurable]: "(λω. ω !! n) ∈ measurable (stream_space M) M"
using measurable_snth_PiM measurable_component_singleton by (rule measurable_compose) simp

lemma measurable_shd[measurable]: "shd ∈ measurable (stream_space M) M"
using measurable_snth[of 0] by simp

lemma measurable_stream_space2:
assumes f_snth: "⋀n. (λx. f x !! n) ∈ measurable N M"
shows "f ∈ measurable N (stream_space M)"
unfolding stream_space_def measurable_distr_eq2
proof (rule measurable_vimage_algebra2)
show "f ∈ space N → streams (space M)"
using f_snth[THEN measurable_space] by (auto simp add: streams_iff_sset sset_range)
show "(λx. op !! (f x)) ∈ measurable N (Pi⇩M UNIV (λi. M))"
proof (rule measurable_PiM_single')
show "(λx. op !! (f x)) ∈ space N → UNIV →⇩E space M"
using f_snth[THEN measurable_space] by auto
qed (rule f_snth)
qed

lemma measurable_stream_coinduct[consumes 1, case_names shd stl, coinduct set: measurable]:
assumes "F f"
assumes h: "⋀f. F f ⟹ (λx. shd (f x)) ∈ measurable N M"
assumes t: "⋀f. F f ⟹ F (λx. stl (f x))"
shows "f ∈ measurable N (stream_space M)"
proof (rule measurable_stream_space2)
fix n show "(λx. f x !! n) ∈ measurable N M"
using ‹F f› by (induction n arbitrary: f) (auto intro: h t)
qed

lemma measurable_sdrop[measurable]: "sdrop n ∈ measurable (stream_space M) (stream_space M)"
by (rule measurable_stream_space2) (simp add: sdrop_snth)

lemma measurable_stl[measurable]: "(λω. stl ω) ∈ measurable (stream_space M) (stream_space M)"
by (rule measurable_stream_space2) (simp del: snth.simps add: snth.simps[symmetric])

lemma measurable_to_stream[measurable]: "to_stream ∈ measurable (Π⇩M i∈UNIV. M) (stream_space M)"
by (rule measurable_stream_space2) (simp add: to_stream_def)

lemma measurable_Stream[measurable (raw)]:
assumes f[measurable]: "f ∈ measurable N M"
assumes g[measurable]: "g ∈ measurable N (stream_space M)"
shows "(λx. f x ## g x) ∈ measurable N (stream_space M)"
by (rule measurable_stream_space2) (simp add: Stream_snth)

lemma measurable_smap[measurable]:
assumes X[measurable]: "X ∈ measurable N M"
shows "smap X ∈ measurable (stream_space N) (stream_space M)"
by (rule measurable_stream_space2) simp

lemma measurable_stake[measurable]:
"stake i ∈ measurable (stream_space (count_space UNIV)) (count_space (UNIV :: 'a::countable list set))"
by (induct i) auto

lemma measurable_shift[measurable]:
assumes f: "f ∈ measurable N (stream_space M)"
assumes [measurable]: "g ∈ measurable N (stream_space M)"
shows "(λx. stake n (f x) @- g x) ∈ measurable N (stream_space M)"
using f by (induction n arbitrary: f) simp_all

lemma measurable_case_stream_replace[measurable (raw)]:
"(λx. f x (shd (g x)) (stl (g x))) ∈ measurable M N ⟹ (λx. case_stream (f x) (g x)) ∈ measurable M N"
unfolding stream.case_eq_if .

lemma measurable_ev_at[measurable]:
assumes [measurable]: "Measurable.pred (stream_space M) P"
shows "Measurable.pred (stream_space M) (ev_at P n)"
by (induction n) auto

lemma measurable_alw[measurable]:
"Measurable.pred (stream_space M) P ⟹ Measurable.pred (stream_space M) (alw P)"
unfolding alw_def
by (coinduction rule: measurable_gfp_coinduct) (auto simp: inf_continuous_def)

lemma measurable_ev[measurable]:
"Measurable.pred (stream_space M) P ⟹ Measurable.pred (stream_space M) (ev P)"
unfolding ev_def
by (coinduction rule: measurable_lfp_coinduct) (auto simp: sup_continuous_def)

lemma measurable_until:
assumes [measurable]: "Measurable.pred (stream_space M) φ" "Measurable.pred (stream_space M) ψ"
shows "Measurable.pred (stream_space M) (φ until ψ)"
unfolding UNTIL_def
by (coinduction rule: measurable_gfp_coinduct) (simp_all add: inf_continuous_def fun_eq_iff)

lemma measurable_holds [measurable]: "Measurable.pred M P ⟹ Measurable.pred (stream_space M) (holds P)"
unfolding holds.simps[abs_def]
by (rule measurable_compose[OF measurable_shd]) simp

lemma measurable_hld[measurable]: assumes [measurable]: "t ∈ sets M" shows "Measurable.pred (stream_space M) (HLD t)"
unfolding HLD_def by measurable

lemma measurable_nxt[measurable (raw)]:
"Measurable.pred (stream_space M) P ⟹ Measurable.pred (stream_space M) (nxt P)"
unfolding nxt.simps[abs_def] by simp

lemma measurable_suntil[measurable]:
assumes [measurable]: "Measurable.pred (stream_space M) Q" "Measurable.pred (stream_space M) P"
shows "Measurable.pred (stream_space M) (Q suntil P)"
unfolding suntil_def by (coinduction rule: measurable_lfp_coinduct) (auto simp: sup_continuous_def)

lemma measurable_szip:
"(λ(ω1, ω2). szip ω1 ω2) ∈ measurable (stream_space M ⨂⇩M stream_space N) (stream_space (M ⨂⇩M N))"
proof (rule measurable_stream_space2)
fix n
have "(λx. (case x of (ω1, ω2) ⇒ szip ω1 ω2) !! n) = (λ(ω1, ω2). (ω1 !! n, ω2 !! n))"
by auto
also have "… ∈ measurable (stream_space M ⨂⇩M stream_space N) (M ⨂⇩M N)"
by measurable
finally show "(λx. (case x of (ω1, ω2) ⇒ szip ω1 ω2) !! n) ∈ measurable (stream_space M ⨂⇩M stream_space N) (M ⨂⇩M N)"
.
qed

lemma (in prob_space) prob_space_stream_space: "prob_space (stream_space M)"
proof -
interpret product_prob_space "λ_. M" UNIV ..
show ?thesis
by (subst stream_space_eq_distr) (auto intro!: P.prob_space_distr)
qed

lemma (in prob_space) nn_integral_stream_space:
assumes [measurable]: "f ∈ borel_measurable (stream_space M)"
shows "(∫⇧+X. f X ∂stream_space M) = (∫⇧+x. (∫⇧+X. f (x ## X) ∂stream_space M) ∂M)"
proof -
interpret S: sequence_space M ..
interpret P: pair_sigma_finite M "Π⇩M i::nat∈UNIV. M" ..

have "(∫⇧+X. f X ∂stream_space M) = (∫⇧+X. f (to_stream X) ∂S.S)"
by (subst stream_space_eq_distr) (simp add: nn_integral_distr)
also have "… = (∫⇧+X. f (to_stream ((λ(s, ω). case_nat s ω) X)) ∂(M ⨂⇩M S.S))"
by (subst S.PiM_iter[symmetric]) (simp add: nn_integral_distr)
also have "… = (∫⇧+x. ∫⇧+X. f (to_stream ((λ(s, ω). case_nat s ω) (x, X))) ∂S.S ∂M)"
by (subst S.nn_integral_fst) simp_all
also have "… = (∫⇧+x. ∫⇧+X. f (x ## to_stream X) ∂S.S ∂M)"
by (auto intro!: nn_integral_cong simp: to_stream_nat_case)
also have "… = (∫⇧+x. ∫⇧+X. f (x ## X) ∂stream_space M ∂M)"
by (subst stream_space_eq_distr)
finally show ?thesis .
qed

lemma (in prob_space) emeasure_stream_space:
assumes X[measurable]: "X ∈ sets (stream_space M)"
shows "emeasure (stream_space M) X = (∫⇧+t. emeasure (stream_space M) {x∈space (stream_space M). t ## x ∈ X } ∂M)"
proof -
have eq: "⋀x xs. xs ∈ space (stream_space M) ⟹ x ∈ space M ⟹
indicator X (x ## xs) = indicator {xs∈space (stream_space M). x ## xs ∈ X } xs"
by (auto split: split_indicator)
show ?thesis
using nn_integral_stream_space[of "indicator X"]
apply (auto intro!: nn_integral_cong)
apply (subst nn_integral_cong)
apply (rule eq)
apply simp_all
done
qed

lemma (in prob_space) prob_stream_space:
assumes P[measurable]: "{x∈space (stream_space M). P x} ∈ sets (stream_space M)"
shows "𝒫(x in stream_space M. P x) = (∫⇧+t. 𝒫(x in stream_space M. P (t ## x)) ∂M)"
proof -
interpret S: prob_space "stream_space M"
by (rule prob_space_stream_space)
show ?thesis
unfolding S.emeasure_eq_measure[symmetric]
by (subst emeasure_stream_space) (auto simp: stream_space_Stream intro!: nn_integral_cong)
qed

lemma (in prob_space) AE_stream_space:
assumes [measurable]: "Measurable.pred (stream_space M) P"
shows "(AE X in stream_space M. P X) = (AE x in M. AE X in stream_space M. P (x ## X))"
proof -
interpret stream: prob_space "stream_space M"
by (rule prob_space_stream_space)

have eq: "⋀x X. indicator {x. ¬ P x} (x ## X) = indicator {X. ¬ P (x ## X)} X"
by (auto split: split_indicator)
show ?thesis
apply (subst AE_iff_nn_integral, simp)
apply (subst nn_integral_stream_space, simp)
apply (subst eq)
apply (subst nn_integral_0_iff_AE, simp)
done
qed

lemma (in prob_space) AE_stream_all:
assumes [measurable]: "Measurable.pred M P" and P: "AE x in M. P x"
shows "AE x in stream_space M. stream_all P x"
proof -
{ fix n have "AE x in stream_space M. P (x !! n)"
proof (induct n)
case 0 with P show ?case
by (subst AE_stream_space) (auto elim!: eventually_mono)
next
case (Suc n) then show ?case
by (subst AE_stream_space) auto
qed }
then show ?thesis
unfolding stream_all_def by (simp add: AE_all_countable)
qed

lemma streams_sets:
assumes X[measurable]: "X ∈ sets M" shows "streams X ∈ sets (stream_space M)"
proof -
have "streams X = {x∈space (stream_space M). x ∈ streams X}"
using streams_mono[OF _ sets.sets_into_space[OF X]] by (auto simp: space_stream_space)
also have "… = {x∈space (stream_space M). gfp (λp x. shd x ∈ X ∧ p (stl x)) x}"
apply (simp add: set_eq_iff streams_def streamsp_def)
apply (intro allI conj_cong refl arg_cong2[where f=gfp] ext)
apply (case_tac xa)
apply auto
done
also have "… ∈ sets (stream_space M)"
apply (intro predE)
apply (coinduction rule: measurable_gfp_coinduct)
apply (auto simp: inf_continuous_def)
done
finally show ?thesis .
qed

lemma sets_stream_space_in_sets:
assumes space: "space N = streams (space M)"
assumes sets: "⋀i. (λx. x !! i) ∈ measurable N M"
shows "sets (stream_space M) ⊆ sets N"
unfolding stream_space_def sets_distr
by (auto intro!: sets_image_in_sets measurable_Sup2 measurable_vimage_algebra2 del: subsetI equalityI
simp add: sets_PiM_eq_proj snth_in space sets cong: measurable_cong_sets)

lemma sets_stream_space_eq: "sets (stream_space M) =
sets (SUP i:UNIV. vimage_algebra (streams (space M)) (λs. s !! i) M)"
by (auto intro!: sets_stream_space_in_sets sets_Sup_in_sets sets_image_in_sets
measurable_Sup1 snth_in measurable_vimage_algebra1 del: subsetI
simp: space_Sup_eq_UN space_stream_space)

lemma sets_restrict_stream_space:
assumes S[measurable]: "S ∈ sets M"
shows "sets (restrict_space (stream_space M) (streams S)) = sets (stream_space (restrict_space M S))"
using  S[THEN sets.sets_into_space]
apply (subst restrict_space_eq_vimage_algebra)
apply (subst vimage_algebra_cong[OF refl refl sets_stream_space_eq])
apply (subst sets_stream_space_eq)
apply (subst sets_vimage_Sup_eq[where Y="streams (space M)"])
apply simp
apply auto []
apply (auto intro: streams_mono) []
apply auto []
apply (simp add: vimage_algebra_cong[OF refl refl restrict_space_eq_vimage_algebra])
apply (subst (1 2) vimage_algebra_vimage_algebra_eq)
apply (auto simp: streams_mono snth_in )
done

primrec sstart :: "'a set ⇒ 'a list ⇒ 'a stream set" where
"sstart S [] = streams S"
| [simp del]: "sstart S (x # xs) = op ## x ` sstart S xs"

lemma in_sstart[simp]: "s ∈ sstart S (x # xs) ⟷ shd s = x ∧ stl s ∈ sstart S xs"
by (cases s) (auto simp: sstart.simps(2))

lemma sstart_in_streams: "xs ∈ lists S ⟹ sstart S xs ⊆ streams S"
by (induction xs) (auto simp: sstart.simps(2))

lemma sstart_eq: "x ∈ streams S ⟹ x ∈ sstart S xs = (∀i<length xs. x !! i = xs ! i)"
by (induction xs arbitrary: x) (auto simp: nth_Cons streams_stl split: nat.splits)

lemma sstart_sets: "sstart S xs ∈ sets (stream_space (count_space UNIV))"
proof (induction xs)
case (Cons x xs)
note Cons[measurable]
have "sstart S (x # xs) =
{s∈space (stream_space (count_space UNIV)). shd s = x ∧ stl s ∈ sstart S xs}"
also have "… ∈ sets (stream_space (count_space UNIV))"
by measurable
finally show ?case .

lemma sigma_sets_singletons:
assumes "countable S"
shows "sigma_sets S ((λs. {s})`S) = Pow S"
proof safe
interpret sigma_algebra S "sigma_sets S ((λs. {s})`S)"
by (rule sigma_algebra_sigma_sets) auto
fix A assume "A ⊆ S"
with assms have "(⋃a∈A. {a}) ∈ sigma_sets S ((λs. {s})`S)"
by (intro countable_UN') (auto dest: countable_subset)
then show "A ∈ sigma_sets S ((λs. {s})`S)"
by simp
qed (auto dest: sigma_sets_into_sp[rotated])

lemma sets_count_space_eq_sigma:
"countable S ⟹ sets (count_space S) = sets (sigma S ((λs. {s})`S))"
by (subst sets_measure_of) (auto simp: sigma_sets_singletons)

lemma sets_stream_space_sstart:
assumes S[simp]: "countable S"
shows "sets (stream_space (count_space S)) = sets (sigma (streams S) (sstart S`lists S ∪ {{}}))"
proof
have [simp]: "sstart S ` lists S ⊆ Pow (streams S)"

let ?S = "sigma (streams S) (sstart S ` lists S ∪ {{}})"
{ fix i a assume "a ∈ S"
{ fix x have "(x !! i = a ∧ x ∈ streams S) ⟷ (∃xs∈lists S. length xs = i ∧ x ∈ sstart S (xs @ [a]))"
proof (induction i arbitrary: x)
case (Suc i) from this[of "stl x"] show ?case
by (simp add: length_Suc_conv Bex_def ex_simps[symmetric] del: ex_simps)
(metis stream.collapse streams_Stream)
qed (insert ‹a ∈ S›, auto intro: streams_stl in_streams) }
then have "(λx. x !! i) -` {a} ∩ streams S = (⋃xs∈{xs∈lists S. length xs = i}. sstart S (xs @ [a]))"
also have "… ∈ sets ?S"
using ‹a∈S› by (intro sets.countable_UN') (auto intro!: sigma_sets.Basic image_eqI)
finally have " (λx. x !! i) -` {a} ∩ streams S ∈ sets ?S" . }
then show "sets (stream_space (count_space S)) ⊆ sets (sigma (streams S) (sstart S`lists S ∪ {{}}))"
by (intro sets_stream_space_in_sets) (auto simp: measurable_count_space_eq_countable snth_in)

have "sigma_sets (space (stream_space (count_space S))) (sstart S`lists S ∪ {{}}) ⊆ sets (stream_space (count_space S))"
proof (safe intro!: sets.sigma_sets_subset)
fix xs assume "∀x∈set xs. x ∈ S"
then have "sstart S xs = {x∈space (stream_space (count_space S)). ∀i<length xs. x !! i = xs ! i}"
by (induction xs)
(auto simp: space_stream_space nth_Cons split: nat.split intro: in_streams streams_stl)
also have "… ∈ sets (stream_space (count_space S))"
by measurable
finally show "sstart S xs ∈ sets (stream_space (count_space S))" .
qed
then show "sets (sigma (streams S) (sstart S`lists S ∪ {{}})) ⊆ sets (stream_space (count_space S))"
qed

lemma Int_stable_sstart: "Int_stable (sstart S`lists S ∪ {{}})"
proof -
{ fix xs ys assume "xs ∈ lists S" "ys ∈ lists S"
then have "sstart S xs ∩ sstart S ys ∈ sstart S ` lists S ∪ {{}}"
proof (induction xs ys rule: list_induct2')
case (4 x xs y ys)
show ?case
proof cases
assume "x = y"
then have "sstart S (x # xs) ∩ sstart S (y # ys) = op ## x ` (sstart S xs ∩ sstart S ys)"
by (auto simp: image_iff intro!: stream.collapse[symmetric])
also have "… ∈ sstart S ` lists S ∪ {{}}"
using 4 by (auto simp: sstart.simps(2)[symmetric] del: in_listsD)
finally show ?case .
qed auto
qed (simp_all add: sstart_in_streams inf.absorb1 inf.absorb2 image_eqI[where x="[]"]) }
then show ?thesis
by (auto simp: Int_stable_def)
qed

lemma stream_space_eq_sstart:
assumes S[simp]: "countable S"
assumes P: "prob_space M" "prob_space N"
assumes ae: "AE x in M. x ∈ streams S" "AE x in N. x ∈ streams S"
assumes sets_M: "sets M = sets (stream_space (count_space UNIV))"
assumes sets_N: "sets N = sets (stream_space (count_space UNIV))"
assumes *: "⋀xs. xs ≠ [] ⟹ xs ∈ lists S ⟹ emeasure M (sstart S xs) = emeasure N (sstart S xs)"
shows "M = N"
proof (rule measure_eqI_restrict_generator[OF Int_stable_sstart])
have [simp]: "sstart S ` lists S ⊆ Pow (streams S)"

interpret M: prob_space M by fact

show "sstart S ` lists S ∪ {{}} ⊆ Pow (streams S)"
by (auto dest: sstart_in_streams del: in_listsD)

{ fix M :: "'a stream measure" assume M: "sets M = sets (stream_space (count_space UNIV))"
have "sets (restrict_space M (streams S)) = sigma_sets (streams S) (sstart S ` lists S ∪ {{}})"
by (subst sets_restrict_space_cong[OF M])
(simp add: sets_restrict_stream_space restrict_count_space sets_stream_space_sstart) }
from this[OF sets_M] this[OF sets_N]
show "sets (restrict_space M (streams S)) = sigma_sets (streams S) (sstart S ` lists S ∪ {{}})"
"sets (restrict_space N (streams S)) = sigma_sets (streams S) (sstart S ` lists S ∪ {{}})"
by auto
show "{streams S} ⊆ sstart S ` lists S ∪ {{}}"
"⋃{streams S} = streams S" "⋀s. s ∈ {streams S} ⟹ emeasure M s ≠ ∞"
using M.emeasure_space_1 space_stream_space[of "count_space S"] sets_eq_imp_space_eq[OF sets_M]
by (auto simp add: image_eqI[where x="[]"])
show "sets M = sets N"
next
fix X assume "X ∈ sstart S ` lists S ∪ {{}}"
then obtain xs where "X = {} ∨ (xs ∈ lists S ∧ X = sstart S xs)"
by auto
moreover have "emeasure M (streams S) = 1"
using ae by (intro prob_space.emeasure_eq_1_AE[OF P(1)]) (auto simp: sets_M streams_sets)
moreover have "emeasure N (streams S) = 1"
using ae by (intro prob_space.emeasure_eq_1_AE[OF P(2)]) (auto simp: sets_N streams_sets)
ultimately show "emeasure M X = emeasure N X"
using P[THEN prob_space.emeasure_space_1]
by (cases "xs = []") (auto simp: * space_stream_space del: in_listsD)
qed (auto simp: * ae sets_M del: in_listsD intro!: streams_sets)

lemma sets_sstart[measurable]: "sstart Ω xs ∈ sets (stream_space (count_space UNIV))"
proof (induction xs)
case (Cons x xs)
note this[measurable]
have "sstart Ω (x # xs) = {ω∈space (stream_space (count_space UNIV)). ω ∈ sstart Ω (x # xs)}"
by (auto simp: space_stream_space)
also have "… ∈ sets (stream_space (count_space UNIV))"
unfolding in_sstart by measurable
finally show ?case .
qed (auto intro!: streams_sets)

primrec scylinder :: "'a set ⇒ 'a set list ⇒ 'a stream set"
where
"scylinder S [] = streams S"
| "scylinder S (A # As) = {ω∈streams S. shd ω ∈ A ∧ stl ω ∈ scylinder S As}"

lemma scylinder_streams: "scylinder S xs ⊆ streams S"
by (induction xs) auto

lemma sets_scylinder: "(∀x∈set xs. x ∈ sets S) ⟹ scylinder (space S) xs ∈ sets (stream_space S)"
by (induction xs) (auto simp: space_stream_space[symmetric])

lemma stream_space_eq_scylinder:
assumes P: "prob_space M" "prob_space N"
assumes "Int_stable G" and S: "sets S = sets (sigma (space S) G)"
and C: "countable C" "C ⊆ G" "⋃C = space S" and G: "G ⊆ Pow (space S)"
assumes sets_M: "sets M = sets (stream_space S)"
assumes sets_N: "sets N = sets (stream_space S)"
assumes *: "⋀xs. xs ≠ [] ⟹ xs ∈ lists G ⟹ emeasure M (scylinder (space S) xs) = emeasure N (scylinder (space S) xs)"
shows "M = N"
proof (rule measure_eqI_generator_eq)
interpret M: prob_space M by fact
interpret N: prob_space N by fact

let ?G = "scylinder (space S) ` lists G"
show sc_Pow: "?G ⊆ Pow (streams (space S))"
using scylinder_streams by auto

have "sets (stream_space S) = sets (sigma (streams (space S)) ?G)"
(is "?S = sets ?R")
proof (rule antisym)
let ?V = "λi. vimage_algebra (streams (space S)) (λs. s !! i) S"
show "?S ⊆ sets ?R"
unfolding sets_stream_space_eq
proof (safe intro!: sets_Sup_in_sets del: subsetI equalityI)
fix i :: nat
show "space (?V i) = space ?R"
using scylinder_streams by (subst space_measure_of) (auto simp: )
{ fix A assume "A ∈ G"
then have "scylinder (space S) (replicate i (space S) @ [A]) = (λs. s !! i) -` A ∩ streams (space S)"
by (induction i) (auto simp add: streams_shd streams_stl cong: conj_cong)
also have "scylinder (space S) (replicate i (space S) @ [A]) = (⋃xs∈{xs∈lists C. length xs = i}. scylinder (space S) (xs @ [A]))"
apply (induction i)
apply auto []
apply (simp add: length_Suc_conv set_eq_iff ex_simps(1,2)[symmetric] cong: conj_cong del: ex_simps(1,2))
apply rule
subgoal for i x
apply (cases x)
apply (subst (2) C(3)[symmetric])
apply (simp del: ex_simps(1,2) add: ex_simps(1,2)[symmetric] ac_simps Bex_def)
apply auto
done
done
finally have "(λs. s !! i) -` A ∩ streams (space S) = (⋃xs∈{xs∈lists C. length xs = i}. scylinder (space S) (xs @ [A]))"
..
also have "… ∈ ?R"
using C(2) ‹A∈G›
by (intro sets.countable_UN' countable_Collect countable_lists C)
(auto intro!: in_measure_of[OF sc_Pow] imageI)
finally have "(λs. s !! i) -` A ∩ streams (space S) ∈ ?R" . }
then show "sets (?V i) ⊆ ?R"
apply (subst vimage_algebra_cong[OF refl refl S])
apply (subst vimage_algebra_sigma[OF G])
apply (subst sigma_le_sets)
apply auto
done
qed
have "G ⊆ sets S"
unfolding S using G by auto
with C(2) show "sets ?R ⊆ ?S"
unfolding sigma_le_sets[OF sc_Pow] by (auto intro!: sets_scylinder)
qed
then show "sets M = sigma_sets (streams (space S)) (scylinder (space S) ` lists G)"
"sets N = sigma_sets (streams (space S)) (scylinder (space S) ` lists G)"
unfolding sets_M sets_N by (simp_all add: sc_Pow)

show "Int_stable ?G"
proof (rule Int_stableI_image)
fix xs ys assume "xs ∈ lists G" "ys ∈ lists G"
then show "∃zs∈lists G. scylinder (space S) xs ∩ scylinder (space S) ys = scylinder (space S) zs"
proof (induction xs arbitrary: ys)
case Nil then show ?case
by (auto simp add: Int_absorb1 scylinder_streams)
next
case xs: (Cons x xs)
show ?case
proof (cases ys)
case Nil with xs.hyps show ?thesis
by (auto simp add: Int_absorb2 scylinder_streams intro!: bexI[of _ "x#xs"])
next
case ys: (Cons y ys')
with xs.IH[of ys'] xs.prems obtain zs where
"zs ∈ lists G" and eq: "scylinder (space S) xs ∩ scylinder (space S) ys' = scylinder (space S) zs"
by auto
show ?thesis
proof (intro bexI[of _ "(x ∩ y)#zs"])
show "x ∩ y # zs ∈ lists G"
using ‹zs∈lists G› ‹x∈G› ‹ys∈lists G› ys ‹Int_stable G›[THEN Int_stableD, of x y] by auto
show "scylinder (space S) (x # xs) ∩ scylinder (space S) ys = scylinder (space S) (x ∩ y # zs)"
by (auto simp add: eq[symmetric] ys)
qed
qed
qed
qed

show "range (λ_::nat. streams (space S)) ⊆ scylinder (space S) ` lists G"
"(⋃i. streams (space S)) = streams (space S)"
"emeasure M (streams (space S)) ≠ ∞"
by (auto intro!: image_eqI[of _ _ "[]"])

fix X assume "X ∈ scylinder (space S) ` lists G"
then obtain xs where xs: "xs ∈ lists G" and eq: "X = scylinder (space S) xs"
by auto
then show "emeasure M X = emeasure N X"
proof (cases "xs = []")
assume "xs = []" then show ?thesis
unfolding eq
using sets_M[THEN sets_eq_imp_space_eq] sets_N[THEN sets_eq_imp_space_eq]
M.emeasure_space_1 N.emeasure_space_1
next
assume "xs ≠ []" with xs show ?thesis
unfolding eq by (intro *)
qed
qed

lemma stream_space_coinduct:
fixes R :: "'a stream measure ⇒ 'a stream measure ⇒ bool"
assumes "R A B"
assumes R: "⋀A B. R A B ⟹ ∃K∈space (prob_algebra M).
∃A'∈M →⇩M prob_algebra (stream_space M). ∃B'∈M →⇩M prob_algebra (stream_space M).
(AE y in K. R (A' y) (B' y) ∨ A' y = B' y) ∧
A = do { y ← K; ω ← A' y; return (stream_space M) (y ## ω) } ∧
B = do { y ← K; ω ← B' y; return (stream_space M) (y ## ω) }"
shows "A = B"
proof (rule stream_space_eq_scylinder)
let ?step = "λK L. do { y ← K; ω ← L y; return (stream_space M) (y ## ω) }"
{ fix K A A' assume K: "K ∈ space (prob_algebra M)"
and A'[measurable]: "A' ∈ M →⇩M prob_algebra (stream_space M)" and A_eq: "A = ?step K A'"
have ps: "prob_space A"
unfolding A_eq by (rule prob_space_bind'[OF K]) measurable
have "sets A = sets (stream_space M)"
unfolding A_eq by (rule sets_bind'[OF K]) measurable
note ps this }
note ** = this

{ fix A B assume "R A B"
obtain K A' B' where K: "K ∈ space (prob_algebra M)"
and A': "A' ∈ M →⇩M prob_algebra (stream_space M)" "A = ?step K A'"
and B': "B' ∈ M →⇩M prob_algebra (stream_space M)" "B = ?step K B'"
using R[OF ‹R A B›] by blast
have "prob_space A" "prob_space B" "sets A = sets (stream_space M)" "sets B = sets (stream_space M)"
using **[OF K A'] **[OF K B'] by auto }
note R_D = this

show "prob_space A" "prob_space B" "sets A = sets (stream_space M)" "sets B = sets (stream_space M)"
using R_D[OF ‹R A B›] by auto

show "Int_stable (sets M)" "sets M = sets (sigma (space M) (sets M))" "countable {space M}"
"{space M} ⊆ sets M" "⋃{space M} = space M" "sets M ⊆ Pow (space M)"
using sets.space_closed[of M] by (auto simp: Int_stable_def)

{ fix A As L K assume K[measurable]: "K ∈ space (prob_algebra M)" and A: "A ∈ sets M" "As ∈ lists (sets M)"
and L[measurable]: "L ∈ M →⇩M prob_algebra (stream_space M)"
from A have [measurable]: "∀x∈set (A # As). x ∈ sets M" "∀x∈set As. x ∈ sets M"
by auto
have [simp]: "space K = space M" "sets K = sets M"
using K by (auto simp: space_prob_algebra intro!: sets_eq_imp_space_eq)
have [simp]: "x ∈ space M ⟹ sets (L x) = sets (stream_space M)" for x
using measurable_space[OF L] by (auto simp: space_prob_algebra)
note sets_scylinder[measurable]
have *: "indicator (scylinder (space M) (A # As)) (x ## ω) =
(indicator A x * indicator (scylinder (space M) As) ω :: ennreal)" for ω x
using scylinder_streams[of "space M" As] ‹A ∈ sets M›[THEN sets.sets_into_space]
by (auto split: split_indicator)
have "emeasure (?step K L) (scylinder (space M) (A # As)) = (∫⇧+y. L y (scylinder (space M) As) * indicator A y ∂K)"
apply (subst emeasure_bind_prob_algebra[OF K])
apply measurable
apply (rule nn_integral_cong)
apply (subst emeasure_bind_prob_algebra[OF L[THEN measurable_space]])
apply (simp_all add: ac_simps * nn_integral_cmult_indicator del: scylinder.simps)
apply measurable
done }
note emeasure_step = this

fix Xs assume "Xs ∈ lists (sets M)"
from this ‹R A B› show "emeasure A (scylinder (space M) Xs) = emeasure B (scylinder (space M) Xs)"
proof (induction Xs arbitrary: A B)
case (Cons X Xs)
obtain K A' B' where K: "K ∈ space (prob_algebra M)"
and A'[measurable]: "A' ∈ M →⇩M prob_algebra (stream_space M)" and A: "A = ?step K A'"
and B'[measurable]: "B' ∈ M →⇩M prob_algebra (stream_space M)" and B: "B = ?step K B'"
and AE_R: "AE x in K. R (A' x) (B' x) ∨ A' x = B' x"
using R[OF ‹R A B›] by blast

show ?case
unfolding A B emeasure_step[OF K Cons.hyps A'] emeasure_step[OF K Cons.hyps B']
apply (rule nn_integral_cong_AE)
using AE_R by eventually_elim (auto simp add: Cons.IH)
next
case Nil
note R_D[OF this]
from this(1,2)[THEN prob_space.emeasure_space_1] this(3,4)[THEN sets_eq_imp_space_eq]
show ?case