Section Elt.
Variable elt : Type.
Notation eqk := (eqk (elt:=elt)).
Notation eqke := (eqke (elt:=elt)).
Notation ltk := (ltk (elt:=elt)).
Notation MapsTo := (MapsTo (elt:=elt)).
Notation In := (In (elt:=elt)).
Notation Sort := (sort ltk).
Notation Inf := (lelistA (ltk)).
(** * [empty] *)
Definition empty : t elt := nil.
Definition Empty m := forall (a : key)(e:elt) , ~ MapsTo a e m.
Hint Resolve empty_1 : core.
(** * [is_empty] *)
Definition is_empty (l : t elt) : bool := if l then true else false.
(** * [mem] *)
Function mem (k : key) (s : t elt) {struct s} : bool :=
match s with
| nil => false
| (k',_) :: l =>
match X.compare k k' with
| LT _ => false
| EQ _ => true
| GT _ => mem k l
end
end.

Lemma mem_1 : forall m (Hm:Sort m) x, In x m -> mem x m = true.