Existence of flips for kawamata log terminal pairs

Let $(X,\Delta)$ be a kawamata log terminal $\mathbb{Q}$-factorial pair of dimension $n$, where $\Delta$ is a $K$-divisor. Let $f\colon X\to Z$ be a flipping contraction, so that $-(K_X+\Delta)$ is relatively ample, and $f$ is a small contraction of relative Picard number one.

Existence of flips. The flip of $f$ exists.

This is one of the two main ingredients in the minimal model program. Existence was known for threefold kawamata log terminal pairs and, at the time of the source, in dimension four; the general statement remains open in higher dimensions.