The Taylor expansion, which stems from Linear Logic and its differential extensions, is an approximation framework for the $\lambda$-calculus (and many of its variants). The reduction of the approximants of a $\lambda$-term induces a reduction on the $\lambda$-term itself, which enjoys a simulation property: whenever a term reduces to another, the approximants reduce accordingly. In recent work, we extended this result to an infinitary $\lambda$-calculus (namely, $\Lambda_{\infty}^{001}$). This short paper solves the question whether the converse property also holds: if the approximants of some term reduce to the approximants of another term, is there a $\beta$-reduction between these terms? This happens to be true for the $\lambda$-calculus, as we show, but our proof fails in the infinitary case. We exhibit a counter-example, refuting the conservativity for $\Lambda_{\infty}^{001}$.

翻译：泰勒展开源于线性逻辑及其微分扩展，是λ演算（及其众多变体）的一种近似框架。λ项近似子的归约会诱导该λ项自身的归约，该归约具有模拟性质：每当一个项归约为另一个项时，其近似子也随之归约。在近期工作中，我们将此结果推广至无穷λ演算（即Λ_{\infty}^{001}）。本文短篇论文解决了其逆命题是否成立的问题：若某项的近似子归约为另一项的近似子，则这两项之间是否存在β-归约？我们证明，对于λ演算该结论成立，但我们的证明在无穷情形下失效。我们构造了一个反例，否定了Λ_{\infty}^{001}的守恒性。