  We show that if $\mathcal{F}$ is an algebraically integrable foliation on a
$\mathbb{Q}$-factorial normal projective variety $X$, $ A, B \geq 0$ are
$\mathbb{Q}$-divisors on $X$ with $A$ ample such that $(\mathcal{F}, B)$ is
foliated dlt and $K_{\mathcal{F}}+ A+B$ is nef, then $K_{\mathcal{F}}+A+B$ is
semiample. We also provide some applications of this and related results such
as contraction theorem for F-dlt pairs and a special case of the
b-semiampleness conjecture.
