Inductive MetaState (state: Type): Type :=
  | Error: MetaState state
  | NonTerminating: MetaState state
  | Terminating: state -> MetaState state.Inductive MetaState (state: Type): Type :=
  | Error: MetaState state
  | NonTerminating: MetaState state
  | Terminating: state -> MetaState state.Inductive MetaState (state: Type): Type :=
  | Error: MetaState state
  | NonTerminating: MetaState state
  | Terminating: state -> MetaState state.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).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)).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)).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)).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)).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).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).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).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)) _.Definition meta_state_measurable_set {state: Type} {state_sig: SigmaAlgebra state} (P: measurable_set state) (error non_terminating: Prop): measurable_set (MetaState state).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 _
}.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 _
}.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.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).Definition non_branch_tstate {state: Type} {state_sigma: SigmaAlgebra state} (s: state): ProgState unit_space_domain state.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).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))
}.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))
}.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))
}.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))
}.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
}.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
}.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
}.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
}.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'.

(* Specifications: *)
Lemma RandomVarDomainStream_stable: forall (Omegas: RandomVarDomainStream) (dir: ConvergeDir Omegas) (n: nat) h, Omegas n h -> (forall m, m >= n -> ~ dir m h) -> (forall m, m >= n -> Omegas m h).
Lemma RandomVarDomainStream_same_covered: forall (Omegas: RandomVarDomainStream) (n1 n2: nat), same_covered_anti_chain (Omegas n1) (Omegas n2).
Lemma RandomVarDomainStream_mono: forall (Omegas: RandomVarDomainStream) (n1 n2: nat), n1 <= n2 -> future_anti_chain (Omegas n1) (Omegas n2).
Lemma RandomVarDomainStream_hered: forall (Omegas: RandomVarDomainStream) (n1 n2: nat) h1, n1 <= n2 -> Omegas n1 h1 -> exists h2, prefix_history h1 h2 /\ Omegas n2 h2.
Lemma limit_raw_domain_covered: forall (Omegas: RandomVarDomainStream) h n, limit_raw_domain Omegas h -> covered_by h (Omegas n).
Lemma limit_raw_domain_legal: forall (Omegas: RandomVarDomainStream), LegalHistoryAntiChain (limit_raw_domain Omegas).