Section TemporalOperators_Ind.
(* Temporal operators with inductive types: definitions and properties *)
Variable S : Set.                (* States *)
Variable tr : S -> Label -> S -> Prop.  (* 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.
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(*                       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<>(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.
(**************************************************************************)
(*                            Some Properties                             *)
(**************************************************************************)

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.