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：泰勒展开源自线性逻辑及其微分扩展，是 $\lambda$演算（及其众多变体）的一种近似框架。$\lambda$项的近似项的约化会诱发该$\lambda$项本身的约化，且满足模拟性质：即当某一项约化为另一项时，其近似项也相应约化。在近期工作中，我们将此结果推广至无穷$\lambda$演算（即$\Lambda_{\infty}^{001}$）。本文解答了逆命题是否成立的问题：若某一项的近似项约化为另一项的近似项，这两个项之间是否存在$\beta$约化？我们证明，该结论对$\lambda$演算成立，但我们的证明无法推广至无穷情形。我们通过反例否定了$\Lambda_{\infty}^{001}$的保守性。