Section ForNotations.
Definition inf x l :=
match l with
| nil => true
| y::_ => match X.compare x y with Lt => true | _ => false end
end.
Fixpoint isok l :=
match l with
| nil => true
| x::l => inf x l && isok l
end.
Notation Sort l := (isok l = true).
Notation Inf := (lelistA X.lt).
Notation In := (InA X.eq).
Existing Instance X.eq_equiv.
Hint Extern 20 => solve [order] : core.
Definition IsOk s := Sort s.
Class Ok (s:t) : Prop := ok : Sort s.
Hint Resolve ok : core.
Hint Unfold Ok : core.
Instance Sort_Ok s `(Hs : Sort s) : Ok s := { ok := Hs }.

Lemma inf_iff : forall x l, Inf x l <-> inf x l = true.