induction lsa1; intuition; simpl in *.
econstructor.
inversion H; clear H; subst.
econstructor.
eapply IHlsa1; eauto.
trivial.
inversion H; clear H; subst.
eapply IHlsa1; eauto.
Qed.

Lemma rel_map_map_inv : forall (A B C : Type)(rel : B -> C -> Prop)(f : A -> B)(lsa : list A)(lsc : list C),
rel_map rel (map f lsa) lsc ->
rel_map (fun a c => rel (f a) c) lsa lsc.