This paper explores the proof theory necessary for recommending an expressive but decidable first-order system, named MAV1, featuring a de Morgan dual pair of nominal quantifiers. These nominal quantifiers called `new' and `wen' are distinct from the self-dual Gabbay-Pitts and Miller-Tiu nominal quantifiers. The novelty of these nominal quantifiers is they are polarised in the sense that `new' distributes over positive operators while `wen' distributes over negative operators. This greater control of bookkeeping enables private names to be modelled in processes embedded as formulae in MAV1. The technical challenge is to establish a cut elimination result, from which essential properties including the transitivity of implication follow. Since the system is defined using the calculus of structures, a generalisation of the sequent calculus, novel techniques are employed. The proof relies on an intricately designed multiset-based measure of the size of a proof, which is used to guide a normalisation technique called splitting. The presence of equivariance, which swaps successive quantifiers, induces complex inter-dependencies between nominal quantifiers, additive conjunction and multiplicative operators in the proof of splitting. Every rule is justified by an example demonstrating why the rule is necessary for soundly embedding processes and ensuring that cut elimination holds.

中文翻译：

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De Morgan 对偶名义量词在非交换逻辑中对私有名称进行建模

本文探讨了推荐具有表现力但可判定的一阶系统 MAV1 所需的证明理论，该系统具有 de Morgan 对偶名义量词。这些称为“new”和“wen”的名义量词不同于自对偶的 Gabbay-Pitts 和 Miller-Tiu 名义量词。这些名义量词的新颖之处在于它们是两极化的，即“new”分布在正算子上，而“wen”分布在负算子上。这种对簿记的更大控制使私有名称能够在嵌入为 MAV1 中的公式的过程中建模。技术挑战是建立一个割消除结果，从该结果中可以得到包括蕴涵传递性在内的基本性质。由于系统是使用结构演算定义的，对连续演算的概括，采用了新技术。该证明依赖于复杂设计的基于多集的证明大小度量，该度量用于指导称为拆分的归一化技术。交换连续量词的等方差的存在在分裂证明中导致名义量词、加法合取和乘法运算符之间复杂的相互依赖关系。每个规则都通过一个例子来证明为什么该规则对于健全的嵌入过程和确保削减消除成立是必要的。在分裂证明中引入名义量词、加法连词和乘法运算符之间复杂的相互依赖关系。每个规则都通过一个例子来证明为什么该规则对于健全的嵌入过程和确保削减消除成立是必要的。在分裂证明中引入名义量词、加法连词和乘法运算符之间复杂的相互依赖关系。每个规则都通过一个例子来证明为什么该规则对于健全的嵌入过程和确保削减消除成立是必要的。