The usual homogeneous form of equality type in Martin-L\"of 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 paper introduces a simple set of axioms for such types. The axioms are 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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典型的异构平等类型

Martin-L\" of Type Theory 中常见的等式类型的同质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元素之间的标识。本文介绍了一组简单的公理对于此类类型。公理等效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，以及 Streicher 的公理 K，或等效的唯一性原则证明。