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

Lemma sigma__pred :
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.