  Context: The Crab pulsar is a bright $\gamma$-ray source detected at photon
energies up to $\sim$1 TeV. Its phase-averaged and phase-resolved $\gamma$-ray
spectra below 10 GeV exhibit exponential cutoffs while those above 10 GeV
apparently follow simple power-laws. Aims: We re-visit the $\gamma$-ray
properties of the Crab pulsar with 10-year \emph{Fermi} Large Area Telescope
(LAT) data in the range of 60 MeV--500 GeV. With the phase-resolved spectra, we
investigate the origins and mechanisms responsible for the emissions. Methods:
The phaseograms are reconstructed for different energy bands and further
analysed using a wavelet decomposition. The phase-resolved energy spectra are
combined with the observations of ground-based instruments like MAGIC and
VERITAS to achieve a larger energy converage. We fit power-law models to the
overlapping energy spectra from 10 GeV to $\sim$1 TeV. We include in the fit a
relative cross-calibration of energy scales between air-shower based gamma-ray
telescopes with the orbital pair-production telescope of the Fermi mission.
Results: We confirm the energy-dependence of the $\gamma$-ray pulse shape, and
equivalently, the phase-dependence of the spectral shape for the Crab pulsar. A
relatively sharp cutoff at a relatively high energy of $\sim$8 GeV is observed
for the bridge-phase emission. The $E>$10 GeV spectrum observed for the second
pulse peak is harder than those for other phases. Conclusions: In view of the
diversity of phase-resolved spectral shapes of the Crab pulsar, we tentatively
propose a multi-origin scenario where the polar-cap, outer-gap and
relativistic-wind regions are involved.
