Section sec_classical.
Variable T : Type.
Proof using.
intros J P s.
case (classic (eventually P s)).
- intros evP wu.
left.
induction evP.
* apply U0. assumption.
* apply weak_until_Cons in wu.
case wu.
+ intros PC.
apply U0. assumption.
+ intros [Js wu'].
apply U_next; trivial.
apply IHevP.
assumption.
- intros evP wu.
right.
apply not_eventually_always_not in evP.
apply weak_until_always_not_always in wu; trivial.
Qed.
Proof using.
intros J P.
cofix c.
intros s.
case (classic (P s)).
- intros Ps un.
contradict un.
apply U0.
assumption.
- intros Ps.
case (classic (J s)); destruct s as [x s].
* intros Js un.
apply W_tl.
+ unfold and_tl, not_tl.
split; trivial.
+ simpl.
apply c.
intros unn.
contradict un.
apply U_next; trivial.
* intros Js un.
apply W0.
split; trivial.
Qed.
Proof using.
intros J P s wun.
case (classic (until (J /\_ ~_ P) (~_ J /\_ ~_ P) s)); trivial.
intros un.
contradict wun.
revert s un.
cofix c.
intros s.
case (classic (P s)).
- intros Ps un.
apply W0.
assumption.
- intros Ps.
case (classic (J s)); destruct s as [x s].
* intros Js un.
apply W_tl; trivial.
simpl.
apply c.
intros unn.
contradict un.
apply U_next; trivial.
split; trivial.
* intros Js un.
contradict un.
apply U0.
split; trivial.
Qed.
Proof using.
intros P.
cofix c.
intro s.
destruct s as [e s].
intros evnP.
apply Always.
- case (classic (P (Cons e s))); trivial.
intros orP.
apply (E0 _ (~_ P)) in orP.
contradict evnP.
assumption.
- apply c.
simpl.
intros evP.
contradict evnP.
apply E_next.
assumption.
Qed.
Proof using.
intros P s alP.
case (classic ((eventually (~_ P)) s)); trivial.
intros evP.
apply not_eventually_not_always in evP.
contradict alP.
assumption.
Qed.
Proof using.
intros J P.
cofix c.
intros s.
case (classic (J s)).
- intros Js un.
destruct s as [x s].
apply R0; trivial.
case (classic (P (Cons x s))); trivial.
intros Ps.
contradict un.
apply U0.
apply Ps.
- intros Js un.
destruct s as [x s].
apply R_tl.
* case (classic (P (Cons x s))); trivial.
intros Ps.
contradict un.
apply U0.
apply Ps.
* simpl.
apply c.
intros unn.
contradict un.
apply U_next; trivial.
Qed.
Proof using.
intros J P s rl.
case (classic (until J P s)); trivial.
intros un.
contradict rl.
revert s un.
cofix c.
intros s un.
case (classic (P s)).
- intros Ps.
contradict un.
apply U0.
assumption.
- intros Ps.
case (classic (J s)).
* intros Js.
destruct s as [x s].
apply R_tl; trivial.
simpl.
apply c.
intros unn.
contradict un.
apply U_next; trivial.
* intros Js.
destruct s as [x s].
apply R0; unfold not_tl; assumption.
Qed.
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.
Proof using.
intros P.
cofix c.
intros [x s] cnyP.
apply Always.
- unfold continuously in cnyP.
apply not_eventually_always_not in cnyP.
apply always_now in cnyP.
unfold not_tl in cnyP.
apply not_always_eventually_not in cnyP.
assumption.
- apply c.
intros cnynP.
contradict cnyP.
apply E_next.
assumption.
Qed.

Lemma not_tl_and_tl_or_tl :
forall (P Q : infseq T -> Prop) (s : infseq T),
(~_ (P /\_ Q)) s -> ((~_ P) \/_ (~_ Q)) s.