Section SIGMA_TABLE.
Variable Site : Set.
Hypothesis eq_site_dec : eq_dec Site.
Variable LS : list Site.
(* This is generic code for any kind of tables hopefully *)
Variable Data : Set.
Let Table := Site -> Data.
Variable t : Table.
Variable s0 : Site.
Variable d0 : Data.
Variable f : Site -> Data -> Z.
(* Just an auxiliary one that is more convenient to use *)
Variable f1 : Site -> Data -> Z.
Variable f2 : Site -> Data -> Z.
Definition fun_sum_site (f1 f2 : Site -> Data -> Z)
(s : Site) (a : Data) := (f1 s a + f2 s a)%Z.
Definition fun_minus_site (f1 f2 : Site -> Data -> Z)
(s : Site) (a : Data) := (f1 s a - f2 s a)%Z.
Definition Z_id (s : Site) (x : Z) := x.

Lemma sigma_table_simpl_property :
forall (t1 t2 : Site -> Data) (P : (Site -> Data) -> Site -> Set),
((forall d : Site, P t2 d) -> t1 = t2) ->
(forall d : Site, P t2 d) ->
sigma_table Site LS Data f t1 = sigma_table Site LS Data f t2.