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People want to rely on optimization algorithms for complex decisions but verifying the optimality of the solutions can then become a valid concern, particularly for critical decisions taken by non-experts in optimization. One example is the shortest-path problem on a network, occurring in many contexts from transportation to logistics to telecommunications. While the standard shortest-path problem is both solvable in polynomial time and certifiable by duality, introducing side constraints makes solving and certifying the solutions much harder. We propose a proof system for constrained shortest-path problems, which gives a set of logical rules to derive new facts about feasible solutions. The key trait of the proposed proof system is that it specifically includes high-level graph concepts within its reasoning steps (such as connectivity or path structure), in contrast to, e.g., using linear combinations of model constraints. Thus, using our proof system, we can provide a step-by-step, human-auditable explanation showing that the path given by an external solver cannot be improved. Additionally, to maximize the advantages of this setup, we propose a proof search procedure that specifically aims to find small proofs of this form using a procedure similar to A* search. We evaluate our proof system on constrained shortest path instances generated from real-world road networks and experimentally show that we may indeed derive more interpretable proofs compared to an integer programming approach, in some cases leading to much smaller proofs.