We present prophecy variables for forward formulations of program analysis problems that require information about the future execution of the program. We specify prophecy and history variables via a domain specific language that augments the step rules of the base operational semantics with subset inclusion constraints over the prophecy and history variables. This tight coupling between the prophecy and history variable specification and the operational semantics promotes the construction of correctness and optimality proofs for program transformations, with the proofs structured as forward simulations between the original and transformed versions of the program. In comparison with traditional dataflow approaches, this approach eliminates mechanisms such as explicit control flow graphs, abstraction functions, concretization functions, Galois connections, and separate backward and forward analyses. We present a verified implementation of prophecy and history variables and use the implementation to prove correctness and optimality properties of two classic transformations, partial dead code elimination and lazy code motion, that use both prophecy and history variables. To the best of our knowledge, these proofs are the first machine checked correctness and optimality proofs for these transformations.