Proof using.
intros P Q J K PQ JK.
cofix cf. intros [x s] rl.
generalize (release_Cons x s J P rl); simpl.
intros [Pxs rlCJP].
case rlCJP; intros rlJP.
- apply R0.
* apply PQ; assumption.
* apply JK; assumption.
- apply R_tl.
* apply PQ; assumption.
* simpl.
apply cf. assumption.
Qed.

(* Specifications: *)
Lemma eventually_monotonic :
forall (P Q J: infseq T -> Prop),
(forall x s, J (Cons x s) -> J s) ->
(forall s, J s -> P s -> Q s) ->
forall s, J s -> eventually P s -> eventually Q s.