Proof using.
intros P Q s cnyP.
induction cnyP as [s alP|].
- intro cnyQ.
induction cnyQ.
apply E0.
apply always_and_tl; trivial.
apply E_next.
apply IHcnyQ.
apply always_invar in alP; assumption.
- intro cnyQ.
apply E_next.
apply IHcnyP.
apply continuously_invar in cnyQ; assumption.
Qed.

(* Specifications: *)
Lemma continuously_inf_often :
forall (P : infseq T -> Prop) (s : infseq T),
continuously P s -> inf_often P s.