  Employing a Mathematica symbolic computer algebra package called xTensor, we
present $(1+3)$-covariant special case proofs of the shear-free perfect fluid
conjecture in General Relativity. We first present the case where the pressure
is constant, and where the acceleration is parallel to the vorticity vector.
These cases were first presented in their covariant form by Senovilla et. al.
We then provide a covariant proof for the case where the acceleration and
vorticity vectors are orthogonal, which leads to the existence of a Killing
vector along the vorticity. This Killing vector satisfies the new constraint
equations resulting from the vanishing of the shear. Furthermore, it is shown
that in order for the conjecture to be true, this Killing vector must have a
vanishing spatially projected directional covariant derivative along the
velocity vector field. This in turn implies the existence of another
\textit{basic} vector field along the direction of the vorticity for the
conjecture to hold. Finally, we show that in general, there exist a
\textit{basic} vector field parallel to the acceleration for which the
conjecture is true.
