Let \(G(V,E)\) be an undirected, edge-weighted graph with integer weights. The weight of a path is the sum of the weights of the edges in that path. The length of a path is the number of edges in that path.
Let \(s \in V\) be a vertex in \(G\). For every \(u \in V\) and for every \(k \geq 0\), let \(d_k(u)\) denote the weight of a shortest path (in terms of weight) from \(s\) to \(u\) of length at most \(k\). If there is no path from \(s\) to \(u\) of length at most \(k\), then \(d_k(u) = \infty\).
Consider the statements:
S1: For every \(k \geq 0\) and \(u \in V\), \(d_{k+1}(u) \leq d_k(u)\).
S2: For every \((u,v) \in E\), if \((u,v)\) is part of a shortest path (in terms of weight) from \(s\) to \(v\), then for every \(k \geq 0\), \(d_k(u) \leq d_k(v)\).
Which one of the following options is correct?