Technology advances give us the hope of driving without human error, reducing
vehicle emissions and simplify an everyday task with the future of self-driving cars. Self-driving cars have come a long way since their creation and making sure these vehicles are safe is very important to the continuation of this field. In this paper, we formalize the Responsibility-Sensitive Safety model (RSS) for self-driving cars and prove the safety and optimality of this model in the longitudinal direction. We utilize the hybrid systems theorem prover KeYmaera~X as a tool to formalize RSS as a hybrid system with both its non-deterministic control constraints and motion model, and prove absence of collisions. We then illustrate the practicality of RSS through refinement proofs that turn the verified non-deterministic controllers into deterministic ones and further verified compilation to Python. The refinement and compilation are safety-preserving; as a result, safety proofs of the formal model transfer to the compiled code, while counterexamples discovered in testing the code transfer back to the formal model. The resulting Python code allows to test the behavior of cars following the motion model of RSS in simulation, to measure agreement between the model and simulation with monitors that are derived from the formal model, and to report counterexamples from simulation back to the formal model.