Inductive MetaState (state: Type): Type :=
| Error: MetaState state
| NonTerminating: MetaState state
| Terminating: state -> MetaState state.
Instance MetaState_SigmaAlgebra (state: Type) {state_sig: SigmaAlgebra state}: SigmaAlgebra (MetaState state).
apply (Build_SigmaAlgebra _ (fun P => is_measurable_set (fun s => P (Terminating _ s)))).
+ hnf; intros; simpl.
apply is_measurable_set_proper.
destruct H; split; unfold Included, In in *; hnf; intros; auto.
+ eapply is_measurable_set_proper; [| apply full_measurable].
split; hnf; unfold In; intros; constructor.
+ intros.
change (fun s : state => Complement (MetaState state) P (Terminating state s))
with (Complement _ (fun s : state => P (Terminating state s))).
apply complement_measurable; auto.
+ intros.
change (fun s : state => Countable_Union (MetaState state) P (Terminating state s))
with (Countable_Union _ (fun i s => P i (Terminating state s))).
apply countable_union_measurable.
auto.
Defined.
Inductive raw_MetaState_pair_left (cmd state: Type) (c: cmd): MetaState state -> MetaState (cmd * state) -> Prop :=
| Error_pair_left: raw_MetaState_pair_left cmd state c (Error _) (Error _)
| NonTerminating_pair_left: raw_MetaState_pair_left cmd state c (NonTerminating _) (NonTerminating _)
| Terminating_pair_left: forall s, raw_MetaState_pair_left cmd state c (Terminating _ s) (Terminating _ (c, s)).
Definition MetaState_pair_left {cmd state: Type} {state_sig: SigmaAlgebra state} (c: cmd): @MeasurableFunction (MetaState state) (MetaState (cmd * state)) _ (@MetaState_SigmaAlgebra _ (left_discreste_prod_sigma_alg cmd state)).
apply (Build_MeasurableFunction _ _ _ _ (raw_MetaState_pair_left cmd state c)).
+ intros.
inversion H; inversion H0; try congruence.
+ intros.
destruct a.
- exists (Error _); constructor.
- exists (NonTerminating _); constructor.
- exists (Terminating _ (c, s)); constructor.
+ simpl.
intros.
destruct P as [P ?H].
simpl in *.
eapply is_measurable_set_proper; [| exact (H c)].
simpl.
split; hnf; unfold In; intros.
- apply H0; constructor.
- inversion H1; auto.
Defined.
Inductive raw_MetaState_snd (cmd state: Type): MetaState (cmd * state) -> MetaState state -> Prop :=
| Error_snd: raw_MetaState_snd cmd state (Error _) (Error _)
| NonTerminating_snd: raw_MetaState_snd cmd state (NonTerminating _) (NonTerminating _)
| Terminating_snd: forall c s, raw_MetaState_snd cmd state (Terminating _ (c, s)) (Terminating _ s).
Definition MetaState_snd {cmd state: Type} {state_sig: SigmaAlgebra state}: @MeasurableFunction (MetaState (cmd * state)) (MetaState state) (@MetaState_SigmaAlgebra _ (left_discreste_prod_sigma_alg cmd state)) _.
apply (Build_MeasurableFunction _ _ _ _ (raw_MetaState_snd cmd state)).
+ intros.
inversion H; inversion H0; try congruence.
+ intros.
destruct a as [ | | [? ?]].
- exists (Error _); constructor.
- exists (NonTerminating _); constructor.
- exists (Terminating _ s); constructor.
+ simpl.
intros.
destruct P as [P ?H].
eapply is_measurable_set_proper; [| exact H].
simpl.
split; hnf; unfold In; intros.
- apply H0; constructor.
- inversion H1; auto.
Defined.
Definition meta_state_measurable_set {state: Type} {state_sig: SigmaAlgebra state} (P: measurable_set state) (error non_terminating: Prop): measurable_set (MetaState state).
exists (fun x => match x with | Error => error | NonTerminating => non_terminating | Terminating s => P s end).
simpl.
apply (proj2_sig P).
Defined.
Section ProgState.
Context {ora: RandomOracle} {SFo: SigmaAlgebraFamily RandomHistory} {HBSFo: HistoryBasedSigF ora}.
Record ProgState (Omega: RandomVarDomain) (state: Type) {state_sigma: SigmaAlgebra state} : Type := {
raw_state: RandomVariable Omega (MetaState state);
inf_history_sound: forall (h: RandomHistory) (ms: MetaState state), is_inf_history h -> raw_state h ms -> ms = NonTerminating _
}.
Definition ProgState_RandomVariable (Omega: RandomVarDomain) (state: Type) {state_sigma: SigmaAlgebra state} (s: ProgState Omega state): RandomVariable Omega (MetaState state) := @raw_state Omega state _ s.
Global Coercion ProgState_RandomVariable: ProgState >-> RandomVariable.
Definition ProgState_pair_left {cmd state: Type} {state_sigma: SigmaAlgebra state} (c: cmd) {Omega: RandomVarDomain} (s: ProgState Omega state): @ProgState Omega (cmd * state) (left_discreste_prod_sigma_alg cmd state).
refine (Build_ProgState Omega _ _ (RandomVarMap (MetaState_pair_left c) (raw_state _ _ s)) _).
intros.
rewrite RandomVarMap_sound in H0.
destruct H0 as [? [? ?]].
pose proof inf_history_sound _ _ s h x H H0.
inversion H1; subst; congruence.
Defined.
Definition non_branch_tstate {state: Type} {state_sigma: SigmaAlgebra state} (s: state): ProgState unit_space_domain state.
refine (Build_ProgState _ _ _ (unit_space_var (Terminating _ s)) _).
intros.
exfalso.
apply PrFamily.rf_sound in H0.
specialize (H0 0); simpl in H0.
specialize (H 0).
rewrite <- H0 in H; apply H; auto.
Defined.
Definition Terminating_raw_domain {Omega: RandomVarDomain} {state: Type} {state_sigma: SigmaAlgebra state} (s: ProgState Omega state): MeasurableSubset Omega :=
PrFamily.PreImage_MSet s (meta_state_measurable_set (Full_MSet _) False False).
End ProgState.
Section Limit.
Context {ora: RandomOracle} {SFo: SigmaAlgebraFamily RandomHistory} {HBSFo: HistoryBasedSigF ora}.
Record RandomVarDomainStream : Type := {
raw_domains: nat -> RandomVarDomain;
rdom_same_covered: forall n, same_covered_anti_chain (raw_domains n) (raw_domains (S n));
rdom_forward: forall n, future_anti_chain (raw_domains n) (raw_domains (S n))
}.
Global Coercion raw_domains: RandomVarDomainStream >-> Funclass.
Record ConvergeDir (Omegas: RandomVarDomainStream): Type := {
raw_direction: forall n, MeasurableSubset (Omegas n);
rdir_forward: forall n, future_anti_chain (raw_direction n) (raw_direction (S n));
rdir_slow: forall n h, Omegas n h -> ~ Omegas (S n) h -> raw_direction n h
}.
Global Coercion raw_direction: ConvergeDir >-> Funclass.
Definition ProgStateStream (Omegas: RandomVarDomainStream) (state: Type) {state_sigma: SigmaAlgebra state}: Type := forall n: nat, ProgState (Omegas n) state.
Definition is_limit_domain (Omegas: RandomVarDomainStream) (lim_Omega: RandomVarDomain) : Prop :=
forall h,
lim_Omega h <->
(forall n h_low, is_fin_history h_low -> prefix_history h_low h ->
exists n' h', n' > n /\ prefix_history h_low h' /\ prefix_history h' h /\ Omegas n' h').
Definition is_limit {Omegas: RandomVarDomainStream} {lim_Omega: RandomVarDomain} {state: Type} {state_sigma: SigmaAlgebra state} (l: ProgStateStream Omegas state) (dir: ConvergeDir Omegas) (lim: ProgState lim_Omega state) : Prop :=
forall h s,
lim h s <->
(exists n, (l n) h s /\ forall n', n' >= n -> ~ dir n' h) \/
(s = NonTerminating _ /\
forall n h_low, is_fin_history h_low -> prefix_history h_low h ->
exists n' h', n' > n /\ prefix_history h_low h' /\ prefix_history h' h /\ dir n' h').
Definition limit_raw_domain (Omegas: RandomVarDomainStream): RandomHistory -> Prop :=
fun h =>
forall n h_low, is_fin_history h_low -> prefix_history h_low h ->
exists n' h', n' > n /\ prefix_history h_low h' /\ prefix_history h' h /\ Omegas n' h'.

Lemma limit_domain_anti_chain_hered: forall (Omegas: RandomVarDomainStream) (n: nat) h,
Omegas n h ->
exists h_limit,
prefix_history h h_limit /\ limit_domain_anti_chain Omegas h_limit.