  We combine a pair of independent Weyl fermions to compose a Dirac fermion on
the four-dimensional Euclidean lattice. The obtained Dirac operator is
antihermitian and does not reproduce anomaly under the usual chiral
transformation. To simulate the correct chiral anomaly, we modify the chiral
transformation. We also show that chiral gauge theories can be constructed
nonperturbatively with exact gauge invariance. The formulation is based on a
doubler-free lattice derivative, which is a simple matrix defined as a discrete
Fourier transform of momentum with antiperiodic boundary conditions. Long-range
fermion hopping interactions are truncated using the Lanczos factor.
