  We aim to study the statistical properties of dusty star-forming galaxies,
such as their number counts, luminosity functions (LF) and dust-obscured
star-formation rate density (SFRD). We use state-of-the-art de-blended Herschel
catalogue in the COSMOS field, generated by combining the Bayesian source
extraction tool XID+ and informative prior on the spectral energy
distributions, to measure the number counts and LFs at far-infrared (FIR) and
sub-millimetre (sub-mm) wavelengths.
  Thanks to our de-confusion technique and deep multi-wavelength photometry, we
are able to achieve more accurate measurements while probing ten times below
the confusion limit. Our number counts at 250 microns agree well with previous
Herschel studies. However, our counts at 350 and 500 microns are considerably
below previous Herschel results. This is due to previous studies suffering from
source confusion which worsens towards longer wavelength. Our number counts at
450 and 870 microns show excellent agreement with previous determinations
derived from single dish and interferometric observations. Our measurements of
the monochromatic LF and the total IR LF agree well with previous results. The
increased dynamic range of our measurements allows us to better measure the
faint-end slope of the LF and measure the dust-obscured SFRD out to z~6. We
find that the fraction of dust obscured star-formation activity is at its
highest around z~1 which then decreases towards both low and high redshift. We
do not find a shift of balance between z~3 and z~4 in the cosmic star-formation
history from being dominated by unobscured star formation at higher redshift to
obscured star formation at lower redshift. However, we do find 3<z<4 to be an
interesting transition period as the fraction of the total SFRD that is
obscured by dust is significantly lower at higher redshifts.
