Originating in Girard's Linear logic, Ehrhard and Regnier's Taylor expansion of $\lambda$-terms has been broadly used as a tool to approximate the terms of several variants of the $\lambda$-calculus. Many results arise from a Commutation theorem relating the normal form of the Taylor expansion of a term to its B\"ohm tree. This led us to consider extending this formalism to the infinitary $\lambda$-calculus, since the $\Lambda_{\infty}^{001}$ version of this calculus has B\"ohm trees as normal forms and seems to be the ideal framework to reformulate the Commutation theorem. We give a (co-)inductive presentation of $\Lambda_{\infty}^{001}$. We define a Taylor expansion on this calculus, and state that the infinitary $\beta$-reduction can be simulated through this Taylor expansion. The target language is the usual resource calculus, and in particular the resource reduction remains finite, confluent and terminating. Finally, we state the generalised Commutation theorem and use our results to provide simple proofs of some normalisation and confluence properties in the infinitary $\lambda$-calculus.

翻译：起源于Girard线性逻辑的Ehrhard与Regnierλ-项泰勒展开，已被广泛用作逼近λ-演算若干变体中项的工具。许多结果源于连接项泰勒展开范式与其Böhm树的交换定理。这促使我们考虑将该形式主义扩展至无穷λ-演算——该演算的Λ_{\infty}^{001}版本以Böhm树为范式，且似乎是重构交换定理的理想框架。我们给出Λ_{\infty}^{001}的(余)归纳表示，定义该演算上的泰勒展开，并证明无穷β-约化可通过该泰勒展开模拟。目标语言为通常的资源演算，其资源约化保持有穷性、合流性与终止性。最终，我们陈述广义交换定理，并利用所得结果给出无穷λ-演算中若干规范化与合流性质的简洁证明。