Proof using.
intros f P s. apply eventually_map_conv.
- apply extensional_now.
- apply extensional_now.
- clear s. intros (x, s). repeat rewrite map_Cons. simpl. trivial.
Qed.

Lemma eventually_map_now_eq :
forall (f: A -> B) a s, eventually (now (eq a)) s ->
eventually (now (eq (f a))) (map f s).