摘要翻译：
通过对(LIM+)定理的一个面向人的形式例子证明，即极限之和是极限之和，它本身具有参考价值，我们展示了β步和Delta+-步的不可置换性（根据Smullyan的分类），这种不可置换性在非自由化的Delta+-规则中是不可见的，在进一步自由化的Delta++规则中也是不严重的。除了对(LIM+)的证明的仔细介绍和几个教学意图之外，主要的主题是解释为什么β步的顺序在一些微积分中起着如此重要的作用。
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英文标题：
《lim+, delta+, and Non-Permutability of beta-Steps》
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作者：
Claus-Peter Wirth
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最新提交年份：
2009
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分类信息：

一级分类：Computer Science        计算机科学
二级分类：Artificial Intelligence        人工智能
分类描述：Covers all areas of AI except Vision, Robotics, Machine Learning, Multiagent Systems, and Computation and Language (Natural Language Processing), which have separate subject areas. In particular, includes Expert Systems, Theorem Proving (although this may overlap with Logic in Computer Science), Knowledge Representation, Planning, and Uncertainty in AI. Roughly includes material in ACM Subject Classes I.2.0, I.2.1, I.2.3, I.2.4, I.2.8, and I.2.11.
涵盖了人工智能的所有领域，除了视觉、机器人、机器学习、多智能体系统以及计算和语言（自然语言处理），这些领域有独立的学科领域。特别地，包括专家系统，定理证明（尽管这可能与计算机科学中的逻辑重叠），知识表示，规划，和人工智能中的不确定性。大致包括ACM学科类I.2.0、I.2.1、I.2.3、I.2.4、I.2.8和I.2.11中的材料。
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一级分类：Computer Science        计算机科学
二级分类：Logic in Computer Science        计算机科学中的逻辑
分类描述：Covers all aspects of logic in computer science, including finite model theory, logics of programs, modal logic, and program verification. Programming language semantics should have Programming Languages as the primary subject area. Roughly includes material in ACM Subject Classes D.2.4, F.3.1, F.4.0, F.4.1, and F.4.2; some material in F.4.3 (formal languages) may also be appropriate here, although Computational Complexity is typically the more appropriate subject area.
涵盖计算机科学中逻辑的所有方面，包括有限模型理论，程序逻辑，模态逻辑和程序验证。程序设计语言语义学应该把程序设计语言作为主要的学科领域。大致包括ACM学科类D.2.4、F.3.1、F.4.0、F.4.1和F.4.2中的材料；F.4.3（形式语言）中的一些材料在这里也可能是合适的，尽管计算复杂性通常是更合适的主题领域。
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英文摘要：
  Using a human-oriented formal example proof of the (lim+) theorem, i.e. that the sum of limits is the limit of the sum, which is of value for reference on its own, we exhibit a non-permutability of beta-steps and delta+-steps (according to Smullyan's classification), which is not visible with non-liberalized delta-rules and not serious with further liberalized delta-rules, such as the delta++-rule. Besides a careful presentation of the search for a proof of (lim+) with several pedagogical intentions, the main subject is to explain why the order of beta-steps plays such a practically important role in some calculi.
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PDF链接：
https://arxiv.org/pdf/0902.3635