  Let $(X,\Delta)$ be a normal pair with a projective morphism $X \to Z$ and
let $A$ be a relatively ample $\mathbb{R}$-divisor on $X$. We establish the
minimal model theory for $(X,\Delta+A)/Z$ under assumptions that the non-nef
locus of $K_{X}+\Delta+A$ over $Z$ does not intersect the non-lc locus of
$(X,\Delta)$ and that the restriction of $K_{X}+\Delta+A$ to the non-lc locus
of $(X,\Delta)$ is semi-ample over $Z$.
