Proof using.
intros f P Q QP s.
apply (always_map_conv_ext f P Q True_tl); auto.
Qed.

Lemma weak_until_map :
forall (f: A->B) (J P: infseq A->Prop) (K Q: infseq B->Prop),
(forall s, J s -> K (map f s)) ->
(forall s, P s -> Q (map f s)) ->
forall (s: infseq A),
weak_until J P s -> weak_until K Q (map f s).