induction ls; intuition;
unfold sumList in *; simpl in *.
intuition.
eapply leRat_trans.
2:{
eapply eqRat_impl_leRat.
symmetry.
rewrite fold_add_body_eq.
2:{
eapply ratAdd_comm.
}
eapply fold_add_init.
intuition.
eapply eqRat_refl.
}
destruct (P a); simpl.
rewrite fold_add_body_eq.
2:{
eapply ratAdd_comm.
}
2:{
intuition.
eapply eqRat_refl.
}
rewrite fold_add_init.
eapply ratAdd_leRat_compat; intuition.
rewrite ratAdd_0_l.
eapply ratAdd_leRat_compat; intuition.
eapply rat0_le_all.
Qed.

(* Specifications: *)
Lemma sumList_filter_partition : forall (A : Set)(P : A -> bool)(ls : list A)(f : A -> Rat),
sumList ls f == (sumList (filter P ls) f + (sumList (filter (fun a => negb (P a)) ls) f)).