Fixpoint lowDistApproxFromTree(A : Set)(eqd : eq_dec A)(t : DistApproxTree A)(a : A) :=
  match t with
    | dat_leaf o =>
      match o with
        | None => 0
        | Some a' => if (eqd a a') then 1 else 0
      end
    | dat_internal t1 t2 =>
      (lowDistApproxFromTree eqd t1 a) * (1 / 2) + (lowDistApproxFromTree eqd t2 a) * (1 / 2)
  end.

Lemma lowDistApprox_ls_impl : forall n (A : Set)(c : Comp A) a r, lowDistApprox c a n r -> lowDistApprox_ls c a n nil r.