Extending Mart\'in Escard\'o's effectful forcing technique, we give a new proof of a well-known result: Brouwer's monotone bar theorem holds for any bar that can be realized by a functional of type $(\mathbb{N} \to \mathbb{N}) \to \mathbb{N}$ in G\"odel's System T. Effectful forcing is an elementary alternative to standard sheaf-theoretic forcing arguments, using ideas from programming languages, including computational effects, monads, the algebra interpretation of call-by-name ${\lambda}$-calculus, and logical relations. Our argument proceeds by interpreting System T programs as well-founded dialogue trees whose nodes branch on a query to an oracle of type $\mathbb{N}\to\mathbb{N}$, lifted to higher type along a call-by-name translation. To connect this interpretation to the bar theorem, we then show that Brouwer's famous "mental constructions" of barhood constitute an invariant form of these dialogue trees in which queries to the oracle are made maximally and in order.
翻译：在 Escard\'o 有效强制技术中, 我们提供了一个新的证据, 证明一个众所周知的结果: 布鲁韦尔的单色条形体理论为任何能用一种功能( mathbb{N}\ to\ mathbb{N}) 实现的任何条形($) (mathbb{N})\to\\\\mathbb{N}N}) 在 G\'odel 系统中的\\\\'mathb{N} $。 效果强迫是标准语系理论强制辩论的基本替代物, 使用来自编程语言的想法, 包括计算效果、 monads、 调用名( $_ lambda) $ 的代数解释和逻辑关系。 我们的论点通过解释系统 T 程序以及有根有根的对话树来进行, 其节点分支在查询 $\\\\'mathb{N\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
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