The usual homogeneous form of equality type in Martin-Löf Type Theory contains identifications between elements of the same type. By contrast, the heterogeneous form of equality contains identifications between elements of possibly different types. This short note introduces a simple set of axioms for such types. The axioms are shown to be equivalent to the combination of systematic elimination rules for both forms of equality, albeit with typal (also known as “propositional”) computation properties, together with Streicher’s Axiom K, or equivalently, the principle of uniqueness of identity proofs.

中文翻译：

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典型的异构等式类型

Martin-Löf 类型理论中的等式类型通常的齐次形式包含相同类型元素之间的标识。相比之下，等式的异质形式包含可能不同类型的元素之间的识别。这篇简短的说明介绍了这些类型的一组简单公理。这些公理与 Streicher 的公理 K 或等效的身份证明唯一性原理等价于两种形式的等式的系统消除规则的组合，尽管具有典型（也称为“命题”）计算属性.