We show that the normal form of the Taylor expansion of a $\lambda$-term is isomorphic to its B\"ohm tree, improving Ehrhard and Regnier's original proof along three independent directions. First, we simplify the final step of the proof by following the left reduction strategy directly in the resource calculus, avoiding to introduce an abstract machine ad hoc. We also introduce a groupoid of permutations of copies of arguments in a rigid variant of the resource calculus, and relate the coefficients of Taylor expansion with this structure, while Ehrhard and Regnier worked with groups of permutations of occurrences of variables. Finally, we extend all the results to a nondeterministic setting: by contrast with previous attempts, we show that the uniformity property that was crucial in Ehrhard and Regnier's approach can be preserved in this setting.
翻译:我们显示泰勒扩大美元(lambda)条件的正常形式与其B\'ohm树是形态化的,它沿三个独立方向改进了Ehrhard和Regnier的原始证据。首先,我们简化了证据的最后一步,直接在资源微积分中遵循左边削减战略,避免采用抽象机器临时使用。我们还在资源微积分的僵硬变体中引入一组参数,并将泰勒扩张的系数与这一结构联系起来,而Ehrhard和Regnier则与各种变数的变数组合合作。最后,我们将所有结果推广到非定式环境:与以往的尝试相比,我们表明在Ehrhard和Regnier方法中至关重要的统一性财产可以在这种环境中保存。