Module Data := D.Module MapS := Make(X).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.