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:从Girard的线性逻辑、Ehrhard和Regnier的Taylor的Linear逻辑推算,Ehrhard和Regnier的Taylor扩大美元-lumbda$-terms-termas, 被广泛用作一种工具,以大致使用美元- lambda$- calculs的几种变种。许多结果来自将泰勒的正常扩展期限与B\\"ohm'树的Brumbda$- calalal- loguls。这导致我们考虑将这种形式扩展扩大到Finite $\ lambda$- calcululs, 并且说,由于美元- 美元- beta- lex- laxalus 版本的缩放, 将B'ohm 树作为正常的形式, 似乎是重塑调制调制调制调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调时的理想框架。我们给出调调调调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调调调和调和调和调和调和调和调和调调调调的调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调,我们调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调,我们调调调调调调调调调调调调调调调调调调调调调