Proof using.
intros P s ioP.
apply not_always_eventually_not in ioP.
induction ioP.
- apply not_eventually_always_not in H.
apply E0.
assumption.
- apply E_next.
assumption.
Qed.

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