Proof using.
cofix cf.
intros J J' P s Hweak Halways.
destruct s.
inversion Hweak.
- now eauto using W0.
- inversion Halways.
eapply W_tl.
+ now unfold and_tl.
+ simpl. now eauto.
Qed.

(* Specifications: *)
Lemma until_weak_until :
forall (J P : infseq T -> Prop) (s : infseq T),
until J P s -> weak_until J P s.