We present the system $\mathtt{d}$, an extended type system with lambda-typed lambda-expressions. It is related to type systems originating from the Automath project. $\mathtt{d}$ extends existing lambda-typed systems by an existential abstraction operator as well as propositional operators. $\beta$-reduction is extended to also normalize negated expressions using a subset of the laws of classical negation, hence $\mathtt{d}$ is normalizing both proofs and formulas which are handled uniformly as functional expressions. $\mathtt{d}$ is using a reflexive type axiom for a constant $\tau$ to which no function can be typed. Some properties are shown including confluence, subject reduction, uniqueness of types, strong normalization, and consistency. We illustrate how, when using $\mathtt{d}$, due to its limited logical strength, additional axioms must be added both for negation and for the mathematical structures whose deductions are to be formalized.

中文翻译：

---

具有 lambda 类型的 lambda 表达式的扩展类型系统

我们展示了系统 $\mathtt{d}$，这是一个带有 lambda 类型的 lambda 表达式的扩展类型系统。它与源自 Automath 项目的类型系统有关。$\mathtt{d}$ 通过存在抽象运算符和命题运算符扩展现有的 lambda 类型系统。$\beta$-reduction 被扩展为还使用经典否定定律的子集对否定表达式进行规范化，因此 $\mathtt{d}$ 正在规范化作为函数表达式统一处理的证明和公式。$\mathtt{d}$ 使用自反类型公理来表示常量 $\tau$ 不能输入任何函数。显示了一些属性，包括汇合、主题减少、类型的唯一性、强规范化和一致性。我们举例说明，当使用 $\mathtt{d}$ 时，由于其有限的逻辑强度，