陶哲轩博客写数学问题时，通常先把对象定义清楚，再给直觉、反例和证明轮廓。把「A Lean companion to “Analysis I”」改写成可阅读的中文笔记，重点是：问题在问什么、已知到哪一步、下一步最容易走偏在哪。原站广告、分享条和导航已去掉。

问题在问什么

Almost 20 years ago, I wrote a textbook in real analysis called “ Analysis I “. It was intended to complement the many good available analysis textbooks out there by focusing more on foundational issues, such as the construction of the natural numbers, integers, rational numbers, and reals, as well as providing enough set theory and logic to allow students to develop proofs at high levels of rigor.

While some proof assistants such as Coq or Agda were well established when the book was written, formal verification was not on my radar at the time. However, now that I have had some experience with this subject, I realize that the content of this book is in fact very compatible with such proof assistants; in particular, the ‘naive type theory’ that I was implicitly using to do things like construct the standard number systems, dovetails well with the dependent type theory o

已知结果和反例

I have therefore decided to launch a Lean companion to “Analysis I” , which is a “translation” of many of the definitions, theorems, and exercises of the text into Lean. In particular, this gives an alternate way to perform the exercises in the book, by instead filling in the corresponding “sorries” in the Lean code. (I do not however plan on hosting “official” solutions to the exercises in this companion; instead, feel free to create forks of the repository in which these so

Currently, the following sections of the text have been translated into Lean: Section 2.1: The natural numbers Section 2.2: Addition Section 2.3: Multiplication Chapter 2 epilogue: Isomorphism with the Mathlib natural numbers Section 3.1: Basic set theory Section 4.1: The integers

证明或构造的主线

The formalization has been deliberately designed to be separate from the standard Lean math library Mathlib at some places, but reliant on it at others. For instance, Mathlib already has a standard notion of the natural numbers . In the Lean formalization, I first develop “by hand” an alternate construction Chapter2.Nat of the natural numbers (or just Nat , if one is working in the Chapter2 namespace), setting up many of the basic results about these alternate natural numbers

The code in this repository compiles in Lean, but I have not tested whether all of the (numerous) “sorries” in the code can actually be filled (i.e., if all the exercises can actually be solved in Lean). I would be interested in having volunteers “playtest” the companion to see if this can actually be done (and if the helper lemmas or “API” provided in the Lean files are sufficient to fill in the sorries in a conceptually straightforward manner without having to rely on more

阅读时建议盯住的点

[UPDATE, May 31: moved the companion to a standalone repository.]

值得单独记下的条目

• Section 2.1: The natural numbers
• Section 2.3: Multiplication
• Chapter 2 epilogue: Isomorphism with the Mathlib natural numbers
• Section 3.1: Basic set theory
• Section 4.1: The integers

阅读和落地时建议先做的 5 件事

1. 用自己的语言重写定义和结论，不看原文能不能说清对象是什么。
2. 找一个最小反例或边界情形，确认假设少一条会怎样。
3. 把证明拆成可独立检验的引理，每步只保留一个新想法。
4. 若涉及计算或形式化，先写可复现的小例子，再谈一般情形。
5. 记下尚未解决的缺口：缺估计、缺构造，还是缺正确的范畴。

和智能体、形式化工具怎么接

龙虾PRO做 OpenClaw 落地时，数学笔记最有用的部分往往是「可检验的步骤」：定义、反例、引理边界。智能体适合帮忙展开计算和检索，不适合代替你决定哪条假设能扔。

本文侧重全链路风控方法论。落地时请用自身业务单据做回放验证，不要把示例阈值直接当生产策略。 相关：风控体检 · 方案资源