induction 1; intuition; simpl in *.
intuition; subst.
left; eauto.
right; eauto.
Qed.

Lemma rel_map_left_total : forall (A B : Type)(rel : A -> B -> Prop)(lsa : list A),
(forall a, exists b, rel a b) ->
exists lsb, rel_map rel lsa lsb.