Relaxing all edges n minus one times finds shortest paths unless another relaxation reveals a negative cycle. The point is to make the formal expression readable enough to audit line by line.
Objects and notation
Weighted paths minimize \(\sum_{e\in P}w(e)\), flows use values \(f:E\to\mathbf R_{\ge0}\) with conservation, and matchings enforce degree at most one. Each problem has a dual obstruction.
The typography mirrors the proof: first declare \(\mathsf D\), then state \(\mathsf C\). The symbol \(\Longrightarrow\) below is a logical dependency, not extra mathematical structure.
Push the symbols
The computation below is not a second theorem. It is a checksum for the definitions and a place to inspect the difficult LaTeX at full size.
Structural reading
The formula is reusable precisely because it says which pieces are structural and which belong only to the worked example.
A hypothesis worth keeping
Dijkstra requires nonnegative edge weights, while flow capacities must be interpreted on a directed residual network even when the original graph is undirected.
A symbolic summary is trustworthy only because the example and limitation remain visible beside it. The box compresses the conclusion without hiding its origin.