  We show that a nearest-neighbor singlet phase results (from an effective
Hamiltonian) for the one-dimensional Hubbard-Holstein model in the regime of
strong electron-electron and electron-phonon interactions and under
non-adiabatic conditions ($t/\omega_0 \leq 1$). By mapping the system of
nearest-neighbor singlets at a filling $N_p/N$ onto a hard-core-boson (HCB)
$t$-$V$ model at a filling $N_p/(N-N_p)$, we demonstrate explicitly that
superfluidity and charge-density-wave (CDW) occur mutually exclusively with the
diagonal long range order manifesting itself only at one-third filling.
Furthermore, we also show that the Bose-Einstein condensate (BEC) occupation
number $n_0$ for the singlet phase, similar to the $n_0$ for a HCB tight
binding model, scales as $\sqrt N$; however, the coefficient of $\sqrt N$ in
the $n_0$ for the interacting singlet phase is numerically demonstrated to be
smaller.
