Ltac hypInv :=
      try (match goal with
          | [H: Some _ = Some _ |-_ ] => inversion H; clear H; subst
      end); try pairInv.Fixpoint lookupIndex (A : Set)(eqd : eq_dec A)(ls : list A)(a : A) def :=
  match ls with
    | nil => def
    | a' :: ls' =>
      if (eqd a a') then O else S (lookupIndex eqd ls' a def)
  end.

Lemma fold_add_nat_filter_partition : forall (A : Set)(P : A -> bool)(f : A -> nat)(ls : list A), fold_left (fun acc a => acc + (f a))%nat ls O = (plus (fold_left (fun acc a => acc + (f a))%nat (filter P ls) O) (fold_left (fun acc a => acc + (f a))%nat (filter (fun a => negb (P a)) ls) O) ).