Section TemporalOperators_Ind.
(*** Temporal operators with inductive types: definition and properties ***)
Variable Label : Set.               (* labels: dicrete transitions *)
Variable S : Set.               (* states                      *)
Variable tr : S -> Label -> S -> Prop. (* transitions                 *)
Variable inv : S -> Prop.         (* location invariants         *)
Variable inc : S -> Instant -> S. (* increase clocks of states   *)
(*  Reachable states from "Sini" with transitions "tr"  *)
Inductive RState (Sini : S) : S -> Prop :=
| rsIni : RState Sini Sini
| rsNextTick :
forall s : S,
RState Sini s -> inv (inc s tick) -> RState Sini (inc s tick)
| rsNextDisc :
forall (s1 s2 : S) (l : Label),
RState Sini s1 -> tr s1 l s2 -> inv s2 -> RState Sini s2.
(*  Reachable states from "Sini" with transitions "tr" in time units  *)
Inductive RState_T (Sini : S) : S -> Instant -> Prop :=
| rsIni_T : RState_T Sini Sini time0
| rsNextTick_T :
forall (s : S) (t : Instant),
RState_T Sini s t ->
inv (inc s tick) -> RState_T Sini (inc s tick) (Inc t)
| rsNextDisc_T :
forall (s1 s2 : S) (l : Label) (t : Instant),
RState_T Sini s1 t -> tr s1 l s2 -> inv 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<>(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.