Remark sigma_change_S :
 sigma E L (change_x0 E Z eq_E_dec f x0 (f x0 + 1)%Z) = (sigma E L f + 1)%Z.

(* Specifications: *)
Lemma sigma__S : forall g : E -> Z, g x0 = (f x0 + 1)%Z -> (forall x : E, x <> x0 -> g x = f x) -> sigma E L g = (sigma E L f + 1)%Z.