  Starting from the {\em ab initio} many-body theory of electrons and phonons,
we go through a series of well defined simplifications to derive a set of
coupled equations of motion for the electronic occupations and polarizations,
nuclear displacements as well as phononic occupations and coherences. These are
the semiconductor electron-phonon equations (SEPE), sharing the same scaling
with system size and propagation time as the Boltzmann equations. At the core
of the SEPE is the {\em mirrored} Generalized Kadanoff-Baym ansatz (GKBA) for
the Green's functions, an alternative to the standard GKBA which we show to
lead to unstable equilibrium states. The SEPE treat coherent and incoherent
degrees of freedom on equal footing, widen the scope of the semiconductor Bloch
equations and Boltzmann equations, and reduce to them under additional
simplifications. The new features of the SEPE pave the way for first-principles
studies of phonon squeezed states and coherence effects in time-resolved
absorption and diffraction experiments.
