  According to the current cosmological cold dark matter paradigm, the Galactic
halo could have been the result of the assemblage of smaller structures. Here
we explore the hypothesis that the classical and ultra-faint dwarf spheroidal
satellites of the Milky Way have been the building blocks of the Galactic halo
by comparing their [$\alpha$/Fe] and [Ba/Fe] versus [Fe/H] patterns with the
ones observed in Galactic halo stars. The $\alpha$ elements deviate
substantially from the observed abundances in the Galactic halo stars for
[Fe/H] values larger than -2 dex, while they overlap for lower metallicities.
On the other hand, for the [Ba/Fe] ratio the discrepancy is extended at all
[Fe/H] values, suggesting that the majority of stars in the halo are likely to
have been formed in situ. Therefore, we suggest that [Ba/Fe] ratios are a
better diagnostic than [$\alpha$/Fe] ratios. Moreover, for the first time we
consider the effects of an enriched infall of gas with the same chemical
abundances as the matter ejected and/or stripped from dwarf satellites of the
Milky Way on the chemical evolution of the Galactic halo. We find that the
resulting chemical abundances of the halo stars depend on the assumed infall
time scale, and the presence of a threshold in the gas for star formation. In
particular, in models with an infall timescale for the halo around 0.8 Gyr
coupled with a threshold in the surface gas density for the star formation (4
$\mathrm{M}_{\odot}\,\mathrm{pc}^{-2}$), and the enriched infall from dwarf
spheroidal satellites, the first halo stars formed show [Fe/H]$>$-2.4 dex. In
this case, to explain [$\alpha$/Fe] data for stars with [Fe/H]$<$-2.4 dex we
need stars formed in dSph systems.
