Proof using.
intros f P Q eP eQ QP.
apply eventually_map_conv.
- apply extensional_always; assumption.
- apply extensional_always; assumption.
- apply always_map_conv; assumption.
Qed.
(* Some corollaries *)

Lemma eventually_now_map :
forall (f: A->B) (P: B->Prop) (s: infseq A),
eventually (now (fun x => P (f x))) s -> eventually (now P) (map f s).