Proof using.
destruct la; intros; simpl in *.
- subst. reflexivity.
- destruct lb; try congruence.
simpl in *. find_inversion.
find_higher_order_rewrite.
f_equal.
rewrite map_map.
erewrite map_ext; [symmetry; apply map_id|].
simpl in *. auto.
Qed.

Lemma In_notIn_implies_neq :
forall x y l,
In(A:=A) x l ->
~ In(A:=A) y l ->
x <> y.