Proof using.
intros P.
intros s alP evP.
induction evP.
* destruct s as [e s].
apply always_Cons in alP.
destruct alP as [nP alP].
unfold not_tl in nP.
contradict nP; assumption.
* apply always_invar in alP.
contradict IHevP; assumption.
Qed.

(* Specifications: *)
Lemma eventually_not_always :
forall (P : infseq T -> Prop) (s : infseq T),
eventually (~_ P) s -> ~ always P s.