induction lsa; intuition; simpl in *.
inversion H.
econstructor.
inversion H; clear H; subst.
econstructor; eauto.
Qed.

Lemma rel_map_inverse : forall (A B : Type)(lsa : list A)(lsb : list B) rel,
rel_map rel lsa lsb ->
forall b,
In b lsb ->
exists a, In a lsa /\ rel a b.