  We study a product rule and a difference operator equipped with Leibniz rule
in a general framework of lattice field theory. It is shown that the difference
operator can be determined by the product rule and some initial data through
the Leibniz rule. This observation leads to a no-go theorem that it is
impossible to construct any difference operator and product rule on a lattice
with the properties of (i) translation invariance, (ii) locality and (iii)
Leibniz rule. We present a formalism to overcome the difficulty by an infinite
flavor extension or a matrix expression of a lattice field theory.
