  We introduce a new boundary condition which renders the flux-insertion
argument for the Lieb-Schultz-Mattis type theorems in two or higher dimensions
free from the specific choice of system sizes. It also enables a formulation of
the Lieb-Schultz-Mattis type theorems in arbitrary dimensions in terms of the
anomaly in field theories of $1+1$ dimensions with a bulk correspondence as a
BF-theory in 2+1 dimensions. Furthermore, we apply the anomaly-based
formulation to the constraints on a half-filled spinless fermion on a square
lattice with $\pi$ flux, utilizing time-reversal, the magnetic translation and
on-site internal $U(N)$ symmetries. This demonstrates the role of time-reversal
anomaly on the ingappabilities of a lattice model.
