induction ls_c; intuition.
inversion H; subst.
simpl.
econstructor.
inversion H; clear H; subst.
simpl.
econstructor; eauto.
Qed.

Lemma rel_map_length : forall (A B : Type)(lsa : list A)(P : A -> B -> Prop)(lsb : list B),
rel_map P lsa lsb ->
length lsa = length lsb.