We design hypersequent calculus proof systems for the theories of Riesz spaces and modal Riesz spaces and prove the key theorems: soundness, completeness and cut elimination. These are then used to obtain completely syntactic proofs of some interesting results concerning the two theories. Most notably, we prove a novel result: the theory of modal Riesz spaces is decidable. This work has applications in the field of logics of probabilistic programs since modal Riesz spaces provide the algebraic semantics of the Riesz modal logic underlying the probabilistic mu-calculus.

翻译：我们为Riesz空间和模型Riesz空间的理论设计了超序列计算校验系统,并证明了关键理论:正确性、完整性和切除性。然后,这些理论被用来获得关于这两个理论的一些有趣的结果的完全综合证据。最显著的是,我们证明了一个新的结果:模型Riesz空间的理论是可以变分的。这项工作在概率程序逻辑领域具有应用性,因为模型Riesz空间提供了Riesz模式逻辑的代数语义,作为概率肌肉计算基础。