Proving inequalities related to Dijkstra's algorithm
Consider a shortest path from $s$ to $v$. Let $u \in \mathrm{IN}(v)$ be the vertex preceding $u$ in this shortest path. The path from $s$ to $u$ is also a shortest path, and so $\mathit{spdist}(s,v) = \mathit{spdist}(s,u) + w(u,v)$.