  We discuss how to construct anomaly-free chiral gauge theories on the lattice
with exact gauge invariance in the framework of domain wall fermion. Chiral
gauge coupling is realized by introducing a five-dimensional gauge field which
interpolates between two different four-dimensional gauge fields at boundaries.
The five-dimensional dependence is compensated by a local and gauge-invariant
counter term. The cohomology problem to obtain the counter term is formulated
in 5+1 dimensional space, using the Chern-Simons current induced from the
five-dimensional Wilson fermion. We clarify the connection to the invariant
construction based on the Ginsparg-Wilson relation using overlap Dirac
operator. Formula for the measure and the effective action of Weyl fermions are
obtained in terms of five-dimensional lattice quantities.
