induction ls; inversion 1; intuition; subst; simpl in *.
f_equal.
inversion H4; clear H4; subst.
inversion H0; clear H0; subst.
simpl in *.
inversion H6; clear H6; subst.
trivial.
inversion H0; clear H0; subst; simpl in *.
inversion H6.
inversion H; clear H; subst.
eapply IHls.
trivial.
Qed.

Theorem lowDistApprox_Ret_inv : forall (A : Set)(eqd : eq_dec A)(a a' : A) n v,
lowDistApprox (Ret eqd a) a' n v ->
v == (if (eqd a a') then 1 else 0).