Module MO:=OrderedTypeFacts O.
Local Notation t := (list O.t).
Local Notation In := (InA O.eq).
Definition eq s s' := forall x, In x s <-> In x s'.
Instance eq_equiv : Equivalence eq := _.
Inductive lt_list : t -> t -> Prop :=
| lt_nil : forall x s, lt_list nil (x :: s)
| lt_cons_lt : forall x y s s',
O.lt x y -> lt_list (x :: s) (y :: s')
| lt_cons_eq : forall x y s s',
O.eq x y -> lt_list s s' -> lt_list (x :: s) (y :: s').
Hint Constructors lt_list : core.
Definition lt := lt_list.
Hint Unfold lt : core.
Instance lt_strorder : StrictOrder lt.
Instance lt_compat' :
Proper (eqlistA O.eq==>eqlistA O.eq==>iff) lt.

Lemma eq_cons :
forall l1 l2 x y,
O.eq x y -> eq l1 l2 -> eq (x :: l1) (y :: l2).