Monadic W in Coq

Abstract

Though mechanized proofs for the type inference algorithm W have already been presented in the literature, none of them can be said to be similar to a usual functional implementation as they are not monadic and they carry extra parameters. Also, the proofs of those formalizations rely on axioms because, at the time, the formal certifications of the unification algorithm was still to be done. Recent developments have shown how to reason within proof assistants using monadic/effectful frameworks and how they may simplify parts of the certification. In this paper, we present a complete monadic formalization in Coq of algorithm W and Damas-Milner type system, which includes proofs of correctness and completeness of type inference, as well soundness, completeness, and termination of the unification algorithm. Our approach uses an extension of the Hoare State Monad to simply proofs.