Definition now (P: T->Prop) (s: infseq T) : Prop :=
  match s with Cons x s => P x end.Definition next (P: infseq T -> Prop) (s: infseq T) : Prop :=
  match s with Cons x s => P s end.Definition consecutive  (R: T -> T -> Prop) (s: infseq T) : Prop :=
  match s with Cons x1 (Cons x2 s) => R x1 x2 end.CoInductive always1 (P: T->Prop) : infseq T -> Prop :=
  | Always1 : forall x s, P x -> always1 P s -> always1 P (Cons x s).CoInductive always1 (P: T->Prop) : infseq T -> Prop :=
  | Always1 : forall x s, P x -> always1 P s -> always1 P (Cons x s).CoInductive always (P: infseq T->Prop) : infseq T -> Prop :=
  | Always : forall s, P s -> always P (tl s) -> always P s.CoInductive always (P: infseq T->Prop) : infseq T -> Prop :=
  | Always : forall s, P s -> always P (tl s) -> always P s.CoInductive weak_until (J P: infseq T->Prop) : infseq T -> Prop :=
  | W0 : forall s, P s -> weak_until J P s
  | W_tl : forall s, J s -> weak_until J P (tl s) -> weak_until J P s.CoInductive weak_until (J P: infseq T->Prop) : infseq T -> Prop :=
  | W0 : forall s, P s -> weak_until J P s
  | W_tl : forall s, J s -> weak_until J P (tl s) -> weak_until J P s.CoInductive weak_until (J P: infseq T->Prop) : infseq T -> Prop :=
  | W0 : forall s, P s -> weak_until J P s
  | W_tl : forall s, J s -> weak_until J P (tl s) -> weak_until J P s.Inductive until (J P: infseq T->Prop) : infseq T -> Prop :=
  | U0 : forall s, P s -> until J P s
  | U_next : forall x s, J (Cons x s) -> until J P s -> until J P (Cons x s).Inductive until (J P: infseq T->Prop) : infseq T -> Prop :=
  | U0 : forall s, P s -> until J P s
  | U_next : forall x s, J (Cons x s) -> until J P s -> until J P (Cons x s).Inductive until (J P: infseq T->Prop) : infseq T -> Prop :=
  | U0 : forall s, P s -> until J P s
  | U_next : forall x s, J (Cons x s) -> until J P s -> until J P (Cons x s).CoInductive release (J P: infseq T->Prop) : infseq T -> Prop :=
  | R0 : forall s, P s -> J s -> release J P s
  | R_tl : forall s, P s -> release J P (tl s) -> release J P s.CoInductive release (J P: infseq T->Prop) : infseq T -> Prop :=
  | R0 : forall s, P s -> J s -> release J P s
  | R_tl : forall s, P s -> release J P (tl s) -> release J P s.CoInductive release (J P: infseq T->Prop) : infseq T -> Prop :=
  | R0 : forall s, P s -> J s -> release J P s
  | R_tl : forall s, P s -> release J P (tl s) -> release J P s.Inductive eventually (P: infseq T->Prop) : infseq T -> Prop :=
  | E0 : forall s, P s -> eventually P s
  | E_next : forall x s, eventually P s -> eventually P (Cons x s).Inductive eventually (P: infseq T->Prop) : infseq T -> Prop :=
  | E0 : forall s, P s -> eventually P s
  | E_next : forall x s, eventually P s -> eventually P (Cons x s).Inductive eventually (P: infseq T->Prop) : infseq T -> Prop :=
  | E0 : forall s, P s -> eventually P s
  | E_next : forall x s, eventually P s -> eventually P (Cons x s).Definition inf_often (P: infseq T->Prop) (s: infseq T) : Prop :=
  always (eventually P) s.Definition continuously (P: infseq T->Prop) (s: infseq T) : Prop :=  
  eventually (always P) s.(* temporal logic connectors *)
Definition impl_tl (P Q: infseq T -> Prop) : infseq T -> Prop :=
  fun s => P s -> Q s.Definition and_tl (P Q: infseq T -> Prop) : infseq T -> Prop :=
  fun s => P s /\ Q s.Definition or_tl (P Q: infseq T -> Prop) : infseq T -> Prop :=
  fun s => P s \/ Q s.Definition not_tl (P : infseq T -> Prop) : infseq T -> Prop := 
  fun s => ~ P s.(* constants *)
Definition True_tl : infseq T -> Prop := fun _ => True.Definition False_tl : infseq T -> Prop := fun _ => False.Notation "A ->_ B" := (impl_tl A B) (right associativity, at level 90).Notation "A /\_ B" := (and_tl A B) (right associativity, at level 80).Notation "A \/_ B" := (or_tl A B) (right associativity, at level 85).Notation "~_ A" := (not_tl A) (right associativity, at level 75).

Lemma weak_until_eventually : forall (P Q J: infseq T -> Prop), (forall s, J s -> P s -> Q s) -> forall s, J s -> weak_until J Q s -> eventually P s -> eventually Q s.