Open Scope Z.
(* Note: x must be non-zero, y is allowed to be zero (because x is a constant in all usages, its non-zero-check seems to be removed by the parser). *)
Definition mul_with_table(x y: Z)(pow: list int)(log: list Z): int :=
if Int.eq (if Int.eq (Int.repr y) Int.zero then Int.zero else Int.one) Int.zero
then Int.zero
else Znth (Int.unsigned
(Int.mods (Int.repr (Znth x log + Znth y log))
(Int.repr 255))) pow.

Lemma mul_equiv: forall (x y: Z) (pow : list int) (log : list Z),
(forall j : Z, 0 <= j < 256 -> Znth j pow = pow3 j) ->
Zlength pow = 256 ->
(forall j : Z, 1 <= j < 256 -> Znth j log = log3 (Int.repr j)) ->
Zlength log = 256 ->
1 <= x < 256 ->
0 <= y < 256 ->
mul_with_table x y pow log = mul (Int.repr x) (Int.repr y).