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：源自Girard线性逻辑的Ehrhard与Regnier的λ-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。许多结果源于一项交换定理，该定理将项泰勒展开的范式与其Böhm树相关联。这促使我们考虑将该形式化方法扩展至无穷λ-演算，因为该演算的Λ∞^001版本以Böhm树作为范式，且似乎是重新表述交换定理的理想框架。我们给出了Λ∞^001的（共）归纳表示，定义了该演算上的泰勒展开，并指出无穷β-规约可通过此泰勒展开进行模拟。目标语言为通常的资源演算，其中资源规约保持有限性、合流性与终止性。最后，我们阐述了广义交换定理，并利用所得结果为无穷λ-演算中的若干规范性与合流性质提供了简洁证明。