The subtyping rules for intersection types traditionally employ a transitivity rule (Barendregt et al. 1983), which means that subtyping does not satisfy the subformula property, making it more difficult to use in filter models for compiler verification. Laurent develops a sequent-style subtyping system, without transitivity, and proves transitivity via a sequence of six lemmas that culminate in cut-elimination (2018). This article develops a subtyping system in regular style that omits transitivity and provides a direct proof of transitivity, significantly reducing the length of the proof, exchanging the six lemmas for just one. Inspired by Laurent's system, the rule for function types is essentially the $\beta$-soundness property. The new system satisfies the "subformula conjunction property": every type occurring in the derivation of $A <: B$ is a subformula of $A$ or $B$, or an intersection of such subformulas. The article proves that the new subtyping system is equivalent to that of Barendregt, Coppo, and Dezani-Ciancaglini.

中文翻译：

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交叉类型子类型的传递性

交集类型的子类型规则传统上采用传递性规则 (Barendregt et al. 1983)，这意味着子类型不满足子公式属性，使其更难在过滤器模型中用于编译器验证。Laurent 开发了一个没有传递性的序列式子类型系统，并通过以切割消除（2018）告终的六个引理序列证明了传递性。本文开发了一个常规风格的子类型系统，它省略了传递性并提供了传递性的直接证明，显着减少了证明的长度，将六个引理交换为一个。受 Laurent 系统的启发，函数类型的规则本质上是 $\beta$-soundness 属性。新系统满足“子公式合取性”：$A <: B$ 的推导中出现的每种类型都是 $A$ 或 $B$ 的子公式，或这些子公式的交集。文章证明了新的子类型系统等同于 Barendregt、Coppo 和 Dezani-Ciancaglini 的子类型系统。