A Hierarchy of Monadic Effects for Program Verification Using Equational Reasoning

One can perform equational reasoning about computational effects with a purely functional programming language thanks to monads. Even though equational reasoning for effectful programs is desirable, it is not yet mainstream. This is partly because it is difficult to maintain pencil-and-paper proofs of large examples. We propose a formalization of a hierarchy of effects using monads in the Coq proof assistant that makes equational reasoning practical. Our main idea is to formalize the hierarchy of effects and algebraic laws like it is done when formalizing hierarchy of traditional algebras. We can then take advantage of the sophisticated rewriting capabilities of Coq to achieve concise proofs of programs. We also show how to ensure the consistency of our hierarchy by providing rigorous models. We explain the various techniques we use to formalize a rich hierarchy of effects (with nondeterminism, state, probability, and more), to mechanize numerous examples from the literature, and we furthermore discuss extensions and new applications.