induction ls; intuition.
inversion H; clear H; subst.
rewrite H2.
eapply rat0_le_all.
inversion H; clear H; subst.
inversion H0; clear H0; subst.
rewrite H7.
eapply leRat_trans.
2:{
eapply eqRat_impl_leRat.
symmetry.
eapply H9.
}
eapply ratAdd_leRat_compat.
eapply H1.
simpl.
left.
eauto.
trivial.
trivial.
eapply IHls.
eauto.
eauto.
intuition.
eapply H1.
simpl.
right.
eauto.
trivial.
trivial.
Qed.

Lemma sumList_filter_le : forall (A : Set)(ls : list A)(f : A -> Rat)(P : A -> bool),
sumList (filter P ls) f <= sumList ls f.