induction 1; intuition; subst.
econstructor.
rewrite <- H1.
trivial.
econstructor.
eapply IHsumList_rel.
intuition.
eapply eqRat_refl.
trivial.
eapply H2.
simpl.
intuition.
eauto.
rewrite <- H3.
trivial.
Qed.

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