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.
Proof using.
cofix c.
intros [x s].
apply Always.
- unfold True_tl; trivial.
- apply c.
Qed.
Proof using.
intros P Q.
cofix c.
intros s alP alQ.
destruct alP.
destruct alQ.
apply Always.
- split; assumption.
- apply c; assumption.
Qed.
Proof using.
intro P.
cofix c.
constructor.
- auto.
- do 2 destruct s.
constructor; eauto using always_invar.
Qed.
Proof using.
intros P.
cofix alwn.
intros s a; case a; clear a s. intros (x, s); simpl. constructor.
- assumption.
- apply alwn; assumption.
Qed.
Proof using.
intros P.
cofix alwn. destruct 1. constructor; simpl.
- assumption.
- apply alwn; assumption.
Qed.

Lemma always_weak_until :
forall (J P : infseq T -> Prop) (s : infseq T), always J s -> weak_until J P s.