(* Transitions *)

  (* Reachable states form "Sini" with transitions "tr" *)

  Inductive RState (Sini : S) : S -> Prop :=
    | rsIni : RState Sini Sini
    | rsNext :
        forall (s1 s2 : S) (l : Label),
        RState Sini s1 -> tr s1 l s2 -> RState Sini s2.(* Transitions *)

  (* Reachable states form "Sini" with transitions "tr" *)

  Inductive RState (Sini : S) : S -> Prop :=
    | rsIni : RState Sini Sini
    | rsNext :
        forall (s1 s2 : S) (l : Label),
        RState Sini s1 -> tr s1 l s2 -> RState Sini s2.(* Transitions *)

  (* Reachable states form "Sini" with transitions "tr" *)

  Inductive RState (Sini : S) : S -> Prop :=
    | rsIni : RState Sini Sini
    | rsNext :
        forall (s1 s2 : S) (l : Label),
        RState Sini s1 -> tr s1 l s2 -> RState Sini s2.(* Reachable states form "Sini" with transitions "tr" in time units *)

  Inductive RState_T (Sini : S) : S -> Instant -> Prop :=
    | rsIni_T : RState_T Sini Sini time0
    | rsTime_T :
        forall (s1 s2 : S) (t : Instant),
        RState_T Sini s1 t -> tr s1 Tick s2 -> RState_T Sini s2 (Inc t)
    | rsNoTime_T :
        forall (s1 s2 : S) (l : Label) (t : Instant),
        RState_T Sini s1 t -> l <> Tick -> tr s1 l s2 -> RState_T Sini s2 t.(* Reachable states form "Sini" with transitions "tr" in time units *)

  Inductive RState_T (Sini : S) : S -> Instant -> Prop :=
    | rsIni_T : RState_T Sini Sini time0
    | rsTime_T :
        forall (s1 s2 : S) (t : Instant),
        RState_T Sini s1 t -> tr s1 Tick s2 -> RState_T Sini s2 (Inc t)
    | rsNoTime_T :
        forall (s1 s2 : S) (l : Label) (t : Instant),
        RState_T Sini s1 t -> l <> Tick -> tr s1 l s2 -> RState_T Sini s2 t.(* Reachable states form "Sini" with transitions "tr" in time units *)

  Inductive RState_T (Sini : S) : S -> Instant -> Prop :=
    | rsIni_T : RState_T Sini Sini time0
    | rsTime_T :
        forall (s1 s2 : S) (t : Instant),
        RState_T Sini s1 t -> tr s1 Tick s2 -> RState_T Sini s2 (Inc t)
    | rsNoTime_T :
        forall (s1 s2 : S) (l : Label) (t : Instant),
        RState_T Sini s1 t -> l <> Tick -> tr s1 l s2 -> RState_T Sini s2 t.(* Reachable states form "Sini" with transitions "tr" in time units *)

  Inductive RState_T (Sini : S) : S -> Instant -> Prop :=
    | rsIni_T : RState_T Sini Sini time0
    | rsTime_T :
        forall (s1 s2 : S) (t : Instant),
        RState_T Sini s1 t -> tr s1 Tick s2 -> RState_T Sini s2 (Inc t)
    | rsNoTime_T :
        forall (s1 s2 : S) (l : Label) (t : Instant),
        RState_T Sini s1 t -> l <> Tick -> tr s1 l s2 -> RState_T Sini s2 t.(**************************************************************************)
  (*                       Invariants and Reachability                      *)
  (**************************************************************************)  

  (* Init => FA[] P *)

  Definition ForAll (Sini : S) (P : S -> Prop) :=
    forall s : S, RState Sini s -> P s.(* Init => FA[](bound t) P *)

  Definition ForAll_T (Sini : S) (P : S -> Prop) (bound : Instant -> Prop) :=
    forall (s : S) (t : Instant), bound t -> RState_T Sini s t -> P s.(* Init => EX<> P *)

  Inductive Exists (Sini : S) (P : S -> Prop) : Prop :=
      exists_ : forall s : S, RState Sini s -> P s -> Exists Sini P.(* Init => EX<> P *)

  Inductive Exists (Sini : S) (P : S -> Prop) : Prop :=
      exists_ : forall s : S, RState Sini s -> P s -> Exists Sini P.(* Init => EX<>(bound t) P *)

  Inductive Exists_T (Sini : S) (P : S -> Prop) (bound : Instant -> Prop) :
  Prop :=
      exists_T :
        forall (s : S) (t : Instant),
        bound t -> RState_T Sini s t -> P s -> Exists_T Sini P bound.(* Init => EX<>(bound t) P *)

  Inductive Exists_T (Sini : S) (P : S -> Prop) (bound : Instant -> Prop) :
  Prop :=
      exists_T :
        forall (s : S) (t : Instant),
        bound t -> RState_T Sini s t -> P s -> Exists_T Sini P bound.

(* Specifications: *)
Theorem Mon_I : forall (Sini : S) (Pg Pp : S -> Prop), ForAll Sini Pg -> (forall s : S, Pg s -> Pp s) -> ForAll Sini Pp.
Theorem Mon_I_T : forall (Sini : S) (Pg Pp : S -> Prop) (bound : Instant -> Prop), ForAll_T Sini Pg bound -> (forall s : S, Pg s -> Pp s) -> ForAll_T Sini Pp bound.
Theorem Conj : forall (Sini : S) (P1 P2 : S -> Prop), ForAll Sini P1 -> ForAll Sini P2 -> ForAll Sini (fun s : S => P1 s /\ P2 s).
Theorem Conj_T : forall (Sini : S) (P1 P2 : S -> Prop) (bound : Instant -> Prop), ForAll_T Sini P1 bound -> ForAll_T Sini P2 bound -> ForAll_T Sini (fun s : S => P1 s /\ P2 s) bound.
Theorem Mon_I_EX : forall (Sini : S) (Pg Pp : S -> Prop), Exists Sini Pg -> (forall s : S, Pg s -> Pp s) -> Exists Sini Pp.
Theorem Mon_I_EX_T : forall (Sini : S) (Pg Pp : S -> Prop) (bound : Instant -> Prop), Exists_T Sini Pg bound -> (forall s : S, Pg s -> Pp s) -> Exists_T Sini Pp bound.
Lemma RState_Trans : forall s1 s2 s3 : S, RState s1 s2 -> RState s2 s3 -> RState s1 s3.
Lemma RState_Trans_T : forall (s1 s2 s3 : S) (t1 t2 : Instant), RState_T s1 s2 t1 -> RState_T s2 s3 t2 -> RState_T s1 s3 (plus_Ck t1 t2).
Theorem StepsEX : forall (s1 s2 : S) (P : S -> Prop), RState s1 s2 -> Exists s2 P -> Exists s1 P.
Theorem ForAll_EX : forall (Sini : S) (P : S -> Prop), ForAll Sini P <-> ~ Exists Sini (fun s : S => ~ P s).
Theorem ForAll_EX_T : forall (Sini : S) (P : S -> Prop) (bound : Instant -> Prop), ForAll_T Sini P bound <-> ~ Exists_T Sini (fun s : S => ~ P s) bound.