Dominant energy condition for a kinetic scalar field

For a scalar field with $X=-q^2/2$, Lagrangian $F(X)$ and stress-energy tensor $T_{ab}=F\prime q_a{q}_b+F{g}_{ab}$, the inequalities $A=F\prime \ge 0$ and $Y=XF\prime -F\ge 0$ imply the dominant energy condition for any gradient type. In a local Lorentz frame, $\rho =A(q_0^2+{\sum }_i{q}_i^2)/2+Y\ge 0$ and ${\rho }^2-{\sum }_i{T}_{0i}^2=A^2{X}^2+A(q_0^2+{\sum }_i{q}_i^2)Y+Y^2\ge 0$.