Section sec_modal_op_lemmas.
Variable T : Type.
Proof using.
now destruct ex.
Qed.
(* always facts *)
Proof using.
intros P invP.
cofix c.
intros [x s] Pxs; apply Always; trivial.
apply c; apply invP in Pxs.
assumption.
Qed.
Proof using.
intros x s P al. change (P (Cons x s) /\ always P (tl (Cons x s))).
destruct al. split; assumption.
Qed.
Proof using.
intros x s P al. case (always_Cons x s P al); trivial.
Qed.
Proof using.
destruct ex.
apply always_now.
Qed.
Proof using.
intros x s P al. case (always_Cons x s P al); trivial.
Qed.
Proof using.
intros (x, s). simpl. apply always_invar.
Qed.
Proof using.
cofix c.
intros [x s].
apply Always.
- unfold not_tl, False_tl; auto.
- apply c.
Qed.

Lemma always_true :
forall s : infseq T, always True_tl s.