Fixpoint sigma_but (l1 : list E) : (E -> Z) -> Z :=
  fun f : E -> Z =>
  match l1 with
  | nil => 0%Z
  | e :: l2 =>
      if eq_E_dec e x0
      then sigma_but l2 f
      else (f e + sigma_but l2 f)%Z
  end.

Lemma sigma_but_simpl : forall (l : list E) (f g : E -> Z), (forall x : E, x <> x0 -> In x l -> f x = g x) -> sigma_but l f = sigma_but l g.