Reduced distance based at singular time in the Ricci flow

In this paper, we define a reduced distance function based at a point at the singular time $T<\mathrm{\infty }$ of a Ricci flow. We also show the monotonicity of the corresponding reduced volume based at time T, with equality iff the Ricci flow is a gradient shrinking soliton. Our curvature bound assumption is more general than the type I condition.