Consider two widely used definitions of equality. That of Leibniz: one value equals another if any predicate that holds of the first holds of the second. And that of Martin-Löf: the type identifying one value with another is occupied if the two values are identical. The former dates back several centuries, while the latter is widely used in proof systems such as Agda and Coq. Here we show that the two definitions are isomorphic: we can convert any proof of Leibniz equality to one of Martin-Löf identity and vice versa, and each conversion followed by the other is the identity. One direction of the isomorphism depends crucially on values of the type corresponding to Leibniz equality satisfying functional extensionality and Reynolds’ notion of parametricity. The existence of the conversions is widely known (meaning that if one can prove one equality then one can prove the other), but that the two conversions form an isomorphism (internally) in the presence of parametricity and functional extensionality is, we believe, new. Our result is a special case of a more general relation that holds between inductive families and their Church encodings. Our proofs are given inside type theory, rather than meta-theoretically. Our paper is a literate Agda script.

中文翻译：

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Leibniz 等式同构于 Martin-Löf 恒等式，参数化

考虑两个广泛使用的平等定义。莱布尼茨：一个值等于另一个值，如果任何谓词满足第一个条件，第二个条件成立。和 Martin-Löf 的一样：如果两个值相同，则使用将一个值标识为另一个值的类型。前者可以追溯到几个世纪前，而后者则广泛用于 Agda 和 Coq 等证明系统。在这里，我们证明了这两个定义是同构的：我们可以将任何莱布尼茨等式的证明转换为 Martin-Löf 恒等式之一，并且反之亦然，并且每个转换后跟另一个是恒等式。同构的一个方向主要取决于满足函数外延性和雷诺参数概念的莱布尼茨等式对应的类型值。转换的存在是广为人知的（意味着如果一个人可以证明一个相等，那么一个人可以证明另一个），但是在参数性和功能外延性的存在下，两个转换形成了一个同构（内部），我们相信，新的. 我们的结果是归纳家庭和他们的 Church 编码之间存在更一般关系的一个特例。我们的证明是在类型理论中给出的，而不是在元理论中给出的。我们的论文是一个识字的 Agda 脚本。