In the first part of this paper, we define two resource aware typing systems for the {\lambda}{\mu}-calculus based on non-idempotent intersection and union types. The non-idempotent approach provides very simple combinatorial arguments-based on decreasing measures of type derivations-to characterize head and strongly normalizing terms. Moreover, typability provides upper bounds for the lengths of the head reduction and the maximal reduction sequences to normal-form. In the second part of this paper, the {\lambda}{\mu}-calculus is refined to a small-step calculus called {\lambda}{\mu}s, which is inspired by the substitution at a distance paradigm. The {\lambda}{\mu}s-calculus turns out to be compatible with a natural extensionof the non-idempotent interpretations of {\lambda}{\mu}, i.e., {\lambda}{\mu}s-reduction preserves and decreases typing derivations in an extended appropriate typing system. We thus derive a simple arithmetical characterization of strongly {\lambda}{\mu}s-normalizing terms by means of typing.

中文翻译：

---

自然演绎式经典微积分的非幂等类型

在本文的第一部分，我们为基于非幂等交集和联合类型的 {\lambda}{\mu} 演算定义了两个资源感知类型系统。非幂等方法提供了非常简单的组合论证——基于类型推导的递减度量——来表征头部和强归一化项。此外，可打字性为头部缩减的长度和到正常形式的最大缩减序列提供了上限。在本文的第二部分，{\lambda}{\mu}-演算被细化为称为 {\lambda}{\mu}s 的小步演算，其灵感来自于距离范式的替换。{\lambda}{\mu}s 演算结果证明与 {\lambda}{\mu} 的非幂等解释的自然扩展兼容，即，{\lambda}{\mu}s-reduction 在扩展的适当类型系统中保留和减少类型派生。因此，我们通过打字推导出强 {\lambda}{\mu}s 归一化项的简单算术特征。