(* Temporal notions for discrete time *)

Infix "^" := Cons.(* increase clocks of states   *) 
 
  Notation SPath := (Stream S) (only parsing).CoInductive ForAllS (P : Stream S -> Prop) : Stream S -> Prop :=
      forallS :
        forall (x : Stream S) (s : S),
        P (s ^ x) -> ForAllS P x -> ForAllS P (s ^ x).CoInductive ForAllS (P : Stream S -> Prop) : Stream S -> Prop :=
      forallS :
        forall (x : Stream S) (s : S),
        P (s ^ x) -> ForAllS P x -> ForAllS P (s ^ x).Inductive ExistsS (P : Stream S -> Prop) : Stream S -> Prop :=
    | Here : forall x : Stream S, P x -> ExistsS P x
    | Further :
        forall (x : Stream S) (s : S), ExistsS P x -> ExistsS P (s ^ x).Inductive ExistsS (P : Stream S -> Prop) : Stream S -> Prop :=
    | Here : forall x : Stream S, P x -> ExistsS P x
    | Further :
        forall (x : Stream S) (s : S), ExistsS P x -> ExistsS P (s ^ x).Inductive ExistsS (P : Stream S -> Prop) : Stream S -> Prop :=
    | Here : forall x : Stream S, P x -> ExistsS P x
    | Further :
        forall (x : Stream S) (s : S), ExistsS P x -> ExistsS P (s ^ x).(* Ejecution traces *)
  
  CoInductive isTrace : Stream S -> Prop :=
    | isTraceTick :
        forall (x : Stream S) (s : S),
        inv (inc s tick) ->
        isTrace (inc s tick ^ x) -> isTrace (s ^ inc s tick ^ x)
    | isTraceDisc :
        forall (x : Stream S) (s1 s2 : S) (l : Label),
        tr s1 l s2 -> inv s2 -> isTrace (s2 ^ x) -> isTrace (s1 ^ s2 ^ x).(* Ejecution traces *)
  
  CoInductive isTrace : Stream S -> Prop :=
    | isTraceTick :
        forall (x : Stream S) (s : S),
        inv (inc s tick) ->
        isTrace (inc s tick ^ x) -> isTrace (s ^ inc s tick ^ x)
    | isTraceDisc :
        forall (x : Stream S) (s1 s2 : S) (l : Label),
        tr s1 l s2 -> inv s2 -> isTrace (s2 ^ x) -> isTrace (s1 ^ s2 ^ x).(* Ejecution traces *)
  
  CoInductive isTrace : Stream S -> Prop :=
    | isTraceTick :
        forall (x : Stream S) (s : S),
        inv (inc s tick) ->
        isTrace (inc s tick ^ x) -> isTrace (s ^ inc s tick ^ x)
    | isTraceDisc :
        forall (x : Stream S) (s1 s2 : S) (l : Label),
        tr s1 l s2 -> inv s2 -> isTrace (s2 ^ x) -> isTrace (s1 ^ s2 ^ x).Definition isTraceFrom (Sini : S) (x : Stream S) :=
    Sini = hd x /\ isTrace x.(* operator Until *)

  Inductive Until (P Q : Stream S -> Prop) : Stream S -> Prop :=
    | UntilFurther :
        forall (s : S) (x : Stream S),
        P (s ^ x) -> Until P Q x -> Until P Q (s ^ x)
    | UntilHere : forall x : Stream S, Q x -> Until P Q x.(* operator Until *)

  Inductive Until (P Q : Stream S -> Prop) : Stream S -> Prop :=
    | UntilFurther :
        forall (s : S) (x : Stream S),
        P (s ^ x) -> Until P Q x -> Until P Q (s ^ x)
    | UntilHere : forall x : Stream S, Q x -> Until P Q x.(* operator Until *)

  Inductive Until (P Q : Stream S -> Prop) : Stream S -> Prop :=
    | UntilFurther :
        forall (s : S) (x : Stream S),
        P (s ^ x) -> Until P Q x -> Until P Q (s ^ x)
    | UntilHere : forall x : Stream S, Q x -> Until P Q x.Inductive EX_Until (Sini : S) (P Q : Stream S -> Prop) : Prop :=
      ExUntil :
        forall x : Stream S,
        isTraceFrom Sini x -> Until P Q x -> EX_Until Sini P Q.Inductive EX_Until (Sini : S) (P Q : Stream S -> Prop) : Prop :=
      ExUntil :
        forall x : Stream S,
        isTraceFrom Sini x -> Until P Q x -> EX_Until Sini P Q.Definition FA_Until (Sini : S) (P Q : Stream S -> Prop) :=
    forall x : Stream S, isTraceFrom Sini x -> Until P Q x.(* Init => FA[] P *)

  Definition Always (Sini : S) (P : Stream S -> Prop) :=
    forall x : Stream S, isTraceFrom Sini x -> ForAllS P x.(* Init => FA<> P *)

  Definition Inevitable (Sini : S) (P : Stream S -> Prop) :=
    forall x : Stream S, isTraceFrom Sini x -> ExistsS P x.(* Init => EX<> P *)

  Inductive Posible (Sini : S) (P : Stream S -> Prop) : Prop :=
      posible :
        forall x : Stream S,
        isTraceFrom Sini x -> ExistsS P x -> Posible Sini P.(* Init => EX<> P *)

  Inductive Posible (Sini : S) (P : Stream S -> Prop) : Prop :=
      posible :
        forall x : Stream S,
        isTraceFrom Sini x -> ExistsS P x -> Posible Sini P.(* Init => EX[] P *)

  Inductive SafePath (Sini : S) (P : Stream S -> Prop) : Prop :=
      safePath :
        forall x : Stream S,
        isTraceFrom Sini x -> ForAllS P x -> SafePath Sini P.(* Init => EX[] P *)

  Inductive SafePath (Sini : S) (P : Stream S -> Prop) : Prop :=
      safePath :
        forall x : Stream S,
        isTraceFrom Sini x -> ForAllS P x -> SafePath Sini P.

Theorem Mon_I_S : forall (x : Stream S) (Pg Pp : Stream S -> Prop), ForAllS Pg x -> (forall s : Stream S, Pg s -> Pp s) -> ForAllS Pp x.