使用羊羔型羊羔表情的扩大型系统 (An extended type system with lambda-typed lambda-expressions)

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.

翻译：我们展示了这个系统$matht{d}$。这个系统是一个使用 lambda 类型 lambda 表达式的扩展型系统。它与来自 Automath 项目的型号系统有关。 $\mathtt{d} 美元通过存在式抽象操作员和提议式操作员扩展了现有的 lambda 类型系统。 $\beta$ 削减范围扩大到使用经典否定法则中的一组标准表达式的普通化, 因此$\mathtt{d} 美元正在将统一处理为功能表达式的证明和公式标准化。 $\ mathtt{d} $\ d$ 正在使用一种反射式的轴承, 常数 $\ tau$ 无法输入。某些属性被显示, 包括共作用、减小、类型的独特性、强烈的正常化和一致性。我们说明, 当使用 $\ matht{d} 美元时, 由于其有限的逻辑强度, 如何必须添加额外的轴式, 既用于消化, 也用于数学结构, 其推算要正规化。