Module MD := OrderedTypeFacts(D).
Definition t := MapS.t D.t.
Definition cmp e e' := match D.compare e e' with EQ _ => true | _ => false end.
Fixpoint eq_list (m m' : list (X.t * D.t)) : Prop :=
match m, m' with
| nil, nil => True
| (x,e)::l, (x',e')::l' =>
match X.compare x x' with
| EQ _ => D.eq e e' /\ eq_list l l'
| _       => False
end
| _, _ => False
end.
Definition eq m m' := eq_list (this m) (this m').
Fixpoint lt_list (m m' : list (X.t * D.t)) : Prop :=
match m, m' with
| nil, nil => False
| nil, _  => True
| _, nil  => False
| (x,e)::l, (x',e')::l' =>
match X.compare x x' with
| LT _ => True
| GT _ => False
| EQ _ => D.lt e e' \/ (D.eq e e' /\ lt_list l l')
end
end.
Definition lt m m' := lt_list (this m) (this m').

Lemma lt_not_eq : forall m1 m2 : t, lt m1 m2 -> ~ eq m1 m2.