Ltac inv_ok := match goal with
| H:sort X.lt (_ :: _) |- _ => inversion_clear H; inv_ok
| H:sort X.lt nil |- _ => clear H; inv_ok
| H:sort X.lt ?l |- _ => change (Ok l) in H; inv_ok
| H:Ok _ |- _ => rewrite <- isok_iff in H; inv_ok
| |- Ok _ => rewrite <- isok_iff
| _ => idtac
end.
Ltac inv := invlist InA; inv_ok; invlist lelistA.
Ltac constructors := repeat constructor.
Ltac sort_inf_in := match goal with
| H:Inf ?x ?l, H':In ?y ?l |- _ =>
cut (X.lt x y); [ intro | apply Sort_Inf_In with l; auto]
| _ => fail
end.
Global Instance isok_Ok s `(isok s = true) : Ok s | 10.
Definition Equal s s' := forall a : elt, In a s <-> In a s'.
Definition Subset s s' := forall a : elt, In a s -> In a s'.
Definition Empty s := forall a : elt, ~ In a s.
Definition For_all (P : elt -> Prop) s := forall x, In x s -> P x.
Definition Exists (P : elt -> Prop) (s : t) := exists x, In x s /\ P x.

Lemma add_inf :
forall (s : t) (x a : elt), Inf a s -> X.lt a x -> Inf a (add x s).