Proof using.
intros J P.
cofix c.
intros s.
case (classic (P s)).
- intros Ps un.
contradict un.
apply U0.
assumption.
- intros Ps.
case (classic (J s)); destruct s as [x s].
* intros Js un.
apply W_tl.
+ unfold and_tl, not_tl.
split; trivial.
+ simpl.
apply c.
intros unn.
contradict un.
apply U_next; trivial.
* intros Js un.
apply W0.
split; trivial.
Qed.

(* Specifications: *)
Lemma not_weak_until_until :
forall (J P : infseq T -> Prop) (s : infseq T),
~ weak_until J P s -> until (J /\_ ~_ P) (~_ J /\_ ~_ P) s.