There is a recent interest for the verification of monadic programs using proof assistants. This line of research raises the question of the integration of monad transformers, a standard technique to combine monads. In this paper, we extend Monae, a Coq library for monadic equational reasoning, with monad transformers and we explain the benefits of this extension. Our starting point is the existing theory of modular monad transformers, which provides a uniform treatment of operations. Using this theory, we simplify the formalization of models in Monae and we propose an approach to support monadic equational reasoning in the presence of monad transformers. We also use Monae to revisit the lifting theorems of modular monad transformers by providing equational proofs and explaining how to patch a known bug with a non-standard use of Coq that combines impredicative polymorphism and parametricity.

中文翻译：

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使用 Monad Transformers 扩展等式一元推理

最近有兴趣使用证明助手来验证 monadic 程序。这一系列研究提出了 monad 转换器的集成问题，这是一种组合 monad 的标准技术。在本文中，我们使用 monad 转换器扩展了 Monae，这是一个用于 monadic 方程推理的 Coq 库，我们解释了这种扩展的好处。我们的出发点是现有的模块化单子转换器理论，它提供了对操作的统一处理。使用这一理论，我们简化了 Monae 中模型的形式化，并提出了一种在存在 monad 转换器的情况下支持 monadic 方程推理的方法。