induction lsa; intuition.
inversion H; clear H; subst.
econstructor.
inversion H; clear H; subst.
econstructor.
eapply IHlsa; eauto.
intuition.
eapply H0.
simpl.
intuition.
trivial.
Qed.

(* Specifications: *)
Lemma rel_map_in_inv : forall (A B : Type)(rel : A -> B -> Prop) lsa lsb,
rel_map rel lsa lsb ->
forall a, In a lsa -> exists b, In b lsb /\ rel a b.