Proof using.
intros x s P al. change (P (Cons x s) \/ eventually P (tl (Cons x s))). case al; auto.
Qed.

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