The algebraic properties of the combination of probabilistic choice and nondeterministic choice have long been a research topic in program semantics. This paper explains a formalization in the Coq proof assistant of a monad equipped with both choices: the geometrically convex monad. This formalization has an immediate application: it provides a model for a monad that implements a nontrivial interface, which allows for proofs by equational reasoning using probabilistic and nondeterministic effects. We explain the technical choices we made to go from the literature to a complete Coq formalization, from which we identify reusable theories about mathematical structures such as convex spaces and concrete categories, and that we integrate in a framework for monadic equational reasoning.

中文翻译：

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具有概率和非确定性的公理推理的可信单子

概率选择和非确定性选择组合的代数性质一直是程序语义学的研究课题。这篇论文解释了 Coq 证明助手中的一个具有两种选择的单子的形式化：几何凸单子。这种形式化有一个直接的应用：它为实现非平凡接口的 monad 提供了一个模型，它允许使用概率和非确定性效应通过等式推理进行证明。我们解释了我们从文献到完整的 Coq 形式化所做的技术选择，从中我们确定了关于数学结构（如凸空间和具体类别）的可重用理论，并将我们整合到一元方程推理的框架中。