Lightweight Proof by Reflection Using a Posteriori Simulation of Effectful Computation

Proof-by-reflection is a well-established technique that employs decision procedures to reduce the size of proof-terms. Currently, decision procedures can be written either in Type Theory—in a purely functional way that also ensures termination— or in an effectful programming language, where they are used as oracles for the certified checker. The first option offers strong correctness guarantees, while the second one permits more efficient implementations. We propose a novel technique for proof-by-reflection that marries, in Type Theory, an effectful language with (partial) proofs of correctness. The key to our approach is to use simulable monads, where a monad is simulable if, for all terminating reduction sequences in its equivalent effectful computational model, there exists a witness from which the same reduction may be simulated a posteriori by the monad. We encode several examples using simulable monads and demonstrate the advantages of the technique over previous approaches.