Proof using.
intros P s alP evP.
induction evP.
- unfold not_tl in H.
contradict H.
destruct s as [x s].
apply always_now in alP.
assumption.
- contradict IHevP.
apply always_invar in alP.
assumption.
Qed.

Lemma continuously_not_inf_often :
forall (P : infseq T -> Prop) (s : infseq T),
continuously (~_ P) s -> ~ inf_often P s.