Proof using.
intros f J P K Q KJ QP. cofix cf.
intros (x, s) rl.
rewrite map_Cons in rl; case (release_Cons rl); clear rl; rewrite <- map_Cons; intros QC orR; case orR; intro cR.
- apply R0.
* apply QP. assumption.
* apply KJ. assumption.
- apply R_tl.
* apply QP. assumption.
* apply cf. assumption.
Qed.

Lemma eventually_map :
forall (f: A->B) (P: infseq A->Prop) (Q: infseq B->Prop),
(forall s, P s -> Q (map f s)) ->
forall s, eventually P s -> eventually Q (map f s).