A Lean companion to Analysis I | Hacker News Reader

A Lean companion to Analysis I

terrytao.wordpress.com

292 pointsjeremyscanvic42 comments

To me, the most exciting aspect of teaching mathematics using Lean is the immediate feedback. If a student's proof is wrong, it simply won't compile.

Previously, the only feedback students could receive would be from another human such as a TA, instructor, or other knowledgeable expert. Now they can receive rapid feedback from the Lean compiler.

In the future I hope there is an option for more instructive feedback from Lean's compiler in the spirit of how the Rust compiler offers suggestions to correct code. (This would probably require the use of dedicated LLMs though.)

I’m yes on this, almost completely.

But. I’m also thoughtful about proving things — my own math experience was decades ago, but I spent a lot of ‘slow thought’ time mulling over whatever my assignments were, trying things on paper, soaking up second hand smoke and coffee, your basic math paradise stuff.

I wonder if using Lean here could lead to some flailing / random checking / spewing. I haven’t worked with Lean much, although I did do a few rounds with coq five or ten years ago; my memory is that I mostly futzed with it and tried things.

Upshot - a solver might be great for a lot of things. But I wonder if it will cut out some of this slow thoughtful back-and-forth that leads to internalization, conceptualization, and sometimes new ideas. Any thoughts on this?

Jim Portegies (TU/e, Netherlands) and Jelle Wemmenhobe have done a lot of research on this, using their “waterproof” (controlled natural language compiled to coa) to test this directly in class. The results are very interesting, and indeed actively messing around is still a very important part of the learning experience, but you can see at least some benefits in also having a theorem prover to check if your proofs are correct.

What I was surprised is that the students learn some patterns of proof properly, but only if you make sure that they are explicitly exposed by the proof assistant (so the more automation the less learning also in this case).

You can find a lot of the work summarized in Jelle’s PhD thesis at https://research.tue.nl/nl/publications/waterproof-transform...

In the future I hope there is an option for more instructive feedback from Lean's compiler in the spirit of how the Rust compiler offers suggestions to correct code.

This is how Acorn works, so that when proving fails but you are "close", you get suggestions in VS Code like:

  Try this:
    reduce(r.num, r.denom) = reduce(a, b)
    cross_equals(a, b, r.num, r.denom)
    r.denom * a = r.num * b

It doesn't use LLMs, though, there's a small local model running inside the VS Code extension. One day hopefully that small local model can be superhumanly strong. For more info: https://acornprover.org/docs/tutorial/proving-a-theorem/

This looks nice. I wasn't aware of Acorn--how much adoption does it have in the mathematics community (or formal methods / robotics / other communities)? I feel like most are rallying around Lean.

It's a much newer project than Lean, and Lean has more adoption. But the vast majority of mathematicians aren't using formal methods at all, so perhaps the space is still oepn.

I'm super excited about this. Hope it gets moved to its own repo so it's easier to find and send to other people. I was always curious about math, and Tao's Analysis was the first textbook that really showed me how it can be constructed in a rigoruous way my programming brain was hoping for. Then I got a bit into Lean, and similarly it was very satisfying but Mathlib is a bit complicated to learn math concepts from. So it's nice to see a bridge from the book to the tool.

Same here. I learnt about convergence, Cauchy seq etc. it was published by a local non profit publisher -- Hindustan Book Agency so it was super affordable too.

It's nice to see theorem proving gain some momentum in a mainstream mathematics topic such as analysis.

In PLT, we already had a flagship book (The Formal Semantics of Programming Languages by Winskel) formally verified (kind of, it's not a 1-to-1 transcription) using Isabelle (http://concrete-semantics.org) back in the mid 2010s, when tools began to be very polished.

IMHO, if someone is interested in theorem proving, that's a much simpler starting point. Theorems in analysis are already pretty hard on their own.

I wouldn't be too surprised if PL proofs were simpler to start with. Part of what I hear people say is that they also are a lot more routine. Do structural induction, apply the IH to show an invariant holds, continue. I haven't done much theorem proving, nor have I done any "mathematical" (e.g. analysis) proofs with a theorem prover, but it makes me wonder how much skill transfer there is between them if "mathematical" proofs require a much different approach.

I will also mention Software Foundations in Rocq (perhaps there is a Lean port). I worked through some of the first parts of it and found it quite pleasant.

Kevin Buzzard said something like the PL proofs are about deep structures on simple objects (mostly integers), while modern math mainly concerns complex objects. If you already have the definitions, the properties usually don't involve a lot of recursion and case analyses.

It will be incredibly interesting to assess how the mainstream "textbook" approach to the subject compares to the one taken in the Mathlib. In general, formalized math libraries make it comparatively easier to state results with a maximum degree of generality and to refactor proof developments for straightforwardness and elegance.

(Refactoring is of course made easier because the system will always keep track of what follows logically from what. You don't have that when working with pen and paper, so opportunities to rework things are very often neglected.)

It is a natural question also to ask whether it makes sense to teach the Mathlib, "maximum generality" version of real analysis, or at least something approaching that, in an academic course. Same for any other field of proof-based math, of course.

As a mathematician and also someone who has programmed for quite awhile I think any programmatic formalism will fail at inculcating the underlying understating. My bias of course is that I learned mathematical concepts via academic papers.

I just feel that the overhead code presents is massive, since it often does not adhere to any semblance of style. I say say this as someone who has had to parse through other’s mathematical papers that were deemed incomprehensible. Code is 10x worse since there are virtually no standards with regards to comprehensibility.

Is there not an idiomatic way to write proofs in Lean/Coq/Agda though? Idiomatic in the sense that once you learn the common idioms/tactics proofs become a degree more readable.

Certainly not for introductory courses. There's already too much on the table - how to prove, how to program, and the base material itself.

I believe that's the experience of faculty who've tried as well - it's fine for advanced students, but a waste of instructional time for the average pupil.

He also has is own YouTube channel with a few videos where he uses Lean [1]. I don't know much about any of it, but seeing him at work, with and without LLMs, was cool.

[1] https://www.youtube.com/@TerenceTao27

I think this is a very nice project and a very nice approach for a foundational topic such as analysis.

A couple immediate worries:

1. Core analysis results in Mathlib work with limits in a general, unified way using the notion of filters. There are nevertheless some ad-hoc specializations of these results to their epsilon-delta form. I presume that Tao's Analysis uses a more traditional epsilon-delta approach.

2. Mathlib moves fast and breaks things. Things get renamed and refactored all the time. Downstream repos needs continual maintenance.

You can look for yourself. Much of the chapter on limits of real sequences is available in the sample pages. See here:

https://link.springer.com/chapter/10.1007/978-981-19-7261-4_...

I have a pretty radical opinion that math education should be focused on building Computer Algebra Systems like Mathematica and Theorem provers like Lean with a heavy focus on visualization and practical applications. Taken to the extreme this could mean never doing paper math while being able to prove everything you learned in Lean.

I feel our current system's focus on endless manual calculations that seem so useless is boring and tedious and makes people hate math.

A Lean textbook!

Why no HoTT, though?

"Should Type Theory (HoTT) Replace (ZFC) Set Theory as the Foundation of Math?" https://news.ycombinator.com/item?id=43196452

Additional Lean resources from HN this week:

"100 theorems in Lean" https://news.ycombinator.com/item?id=44075061

"Google-DeepMind/formal-conjectures: collection of formalized conjectures in lean" https://news.ycombinator.com/item?id=44119725

Why should there be HoTT, though?

A lot less work has gone into making HoTT theorem provers ergonomic to use. Also, the documentation is a lot more sparse. The benefits of HoTT are also unclear - it seems to only save work when dealing with very esoteric constructs in category theory.

It’s an existing textbook, which answers the “Why not HoTT?” question. Although another might be that people doubt its pedagogical value.

HoTT is a highly technical, highly niche topic and it doesn't make sense to tackle two ambitious projects at the same time like this. HoTT isn't even remotely close to being accepted as a reasonable standard, and it's kind of a non-starter for most people.

This is a bit like asking the developers of a Javascript framework why they they didn't write a framework for Elm or Haskell.

> HoTT is a highly technical, highly niche topic and it doesn't make sense to tackle two ambitious projects at the same time like this.

It's also a bit of a controversial topic in formalized mathematics. See the following relevant comments by Kevin Buzzard (an expert in Lean and also in the sort of abstract nonsense that might arguably benefit directly from HoTT) - Links courtesy of https://ncatlab.org/nlab/show/Kevin+Buzzard

* Is HoTT the way to do mathematics? (2020) https://www.cl.cam.ac.uk/events/owls/slides/buzzard.pdf ("Nobody knows because nobody tried")

* Grothendieck's use of equality (2024) https://arxiv.org/abs/2405.10387 - Discussed here https://news.ycombinator.com/item?id=40414404

> Why no HoTT, though?

Sort of a weird question to ask imo.

Terrence Tao has a couple of analysis textbooks and this is his companion to the first of those books in Lean. He doesn’t have a type theory textbook, so that’s why no higher-order type theory - it’s not what he’s trying to do at all.

If HoTT is already proved and sets, categories, and types are already proven; I agree that it's not necessary to prove same in an applied analysis book; though it is another opportunity to verify HoTT in actual application domains.

"Is this consistent with HoTT?" a tool could ask.

But none of this is what he’s trying to do.

He wrote a book to go with his course in undergraduate real analysis. <= this does not contain HoTT because HoTT is not part of undergraduate real analysis

He’s making a companion to that book for lean <= so this also doesn’t contain HoTT.

Just like it doesn’t contain anything about raising partridges or doing skydiving, it doesn’t have anything about HoTT because that’s not what he’s trying to write a book about. He’s writing a lean companion to his analysis textbook. I get that you are interested in HoTT. If so you should probably get a book on it. This isn’t that book.

He is a working mathematician showing how a proof assistant can be used as part of accomplishing a very specific mainstream mathematical task (ie proving the foundational theorems of analysis). He's not trying to write something about the theoretical basis for proof assistants.

Are there types presented therein?

Presumably, left-to-right MRO solves for diamond-inheritance because of a type theory.

I suppose it doesn't matter if HoTT is the most sufficient type / category / information-theoretic set theory for inductively-presented real analysis in classical spaces at all.

But then why present in Lean, if the precedent formalisms are irrelevant? (Fairly also, is HoTT proven as a basis for Lean itself, though?)

   Are there types presented therein?

No. Analysis typically is presented using naive set theory to build the natural numbers, integers, rationals and (via Dedekind cuts) real numbers. For avoidance of doubt in the standard construction of maths these are sets, not types. Then from there a standard first course in analysis deals with the topology of the reals, sequences and series, limits and continuity of real functions, derivatives and the integral. Then if people study analysis further it would typically involve complex numbers and functions, how you do calculus with them, Fourier analysis etc, but types and type theory wouldn’t form part of any standard treatment of analysis that I’m aware of.

Types are not a mainstream topic in pure maths. Type theory was part of Bertrand Russell’s attempt to solve the problems of set theory, which ended up as a bit of a sideshow because ZF/ZFC and naive set theory turned out to require far less of a rewrite of all the rest of maths and so became the standard approach. Types come up I think if people go deeply into formal logic or category theory (neither of which is a type of analysis), but category theory is touched on in abstract algebra after dealing with the more conventional topics (Groups, rings, fields, modules) sometimes. Most people I know who know about type theory came at it from a computer science angle rather than pure maths. You might learn some type theory if you do the type of computer science course where you learn lambda calculus.

   why present in Lean, if the precedent formalisms are irrelevant?

If someone gave a powerpoint presentation, do they have to use the presentation to talk about how powerpoint is made? If someone writes a paper in latex, does the paper have to be about latex and mathematical typesetting? Or are those tools that you’re allowed to use to accomplish some goal?

He’s presenting in Lean because that’s the proof assistant that he actually uses and he’s interested in proof assistants and other tools and how they help mathematicians. He’s not presenting about lean and how it’s made. He’s showing how you can use it to do proofs in analysis.

Well, FWIU, what you refer to as "topology" is founded in HoTT "type theory".

> for avoidance of doubt in the standard construction of maths these are sets, not types. Then from there a standard first course in analysis deals with the topology of the reals, sequences and series, limits and continuity of real functions, derivatives and the integral.

HoTT is precedent to these as well.

So, Lean isn't proven with HoTT either.

Is type theory useful in describing a Hilbert hotel infinite continuum of Reals or for describing quantum values like qubits and qudits?

I don't think I'm overselling HoTT as a useful axiomatic basis for things proven in absence of type and information-theoretic foundations.

From https://news.ycombinator.com/item?id=43191103#43201742 , which I may have already linked to in this thread :

> "Should Type Theory Replace Set Theory as the Foundation of Mathematics?" (2023) https://link.springer.com/article/10.1007/s10516-023-09676-0

That would include Real Analysis (which indeed one isn't at all obligated to prove with a prover and a language for proofs down to {Null, 0, 1} and <0.5|-0.8>).

Are types checked at runtime, with other Hoare logic preconditions and postconditions?

Formalization of many of the ideas from HoTT are currently happening in the Agda community. [1] It's out of my wheelhouse, so I don't know the exact motivations, but Agda is apparently a better way to formalize those ideas than in Lean.

Also, there's a new textbook coming out later this year that's a more modern update to the original HoTT book [2] which also has an Agda formalization. [3]

[1] https://martinescardo.github.io/HoTT-UF-in-Agda-Lecture-Note...

[2] https://www.cambridge.org/core/books/introduction-to-homotop...

[3] https://github.com/HoTT-Intro/Agda

Very cool. Analysis I was the first "real" math textbook that I (an engineer, not a mathematician) felt like I could completely follow and work through, after a few attempts to get through others like Rudin. Hopefully the Lean companion will help make it even more accessible to people who are familiar with math and programming and looking to learn things rigorously.

Over the years there has been a steady stream of people attempting to formalize Tao's Analysis I book in Lean, i.e. doing exactly what Tao is doing now (unfortunately none of them go beyond the first few chapters -- I hope Tao can go further!). I was considering doing this myself, so that my Analysis I solutions blog [1] would be accompanied by formalized proofs of each exercise (which I thought people working through the book might find useful).

I already posted the following in the private Discord server for the book, but this seems like possibly a good public space to share the following, in case anyone here may find it useful:

- https://github.com/cruhland/lean4-analysis which pulls from https://github.com/cruhland/lean4-axiomatic

- https://github.com/Shaunticlair/tao-analysis-lean-practice

- https://github.com/vltanh/lean4-analysis-tao

- https://github.com/gabriel128/analysis_in_lean

- https://github.com/mk12/analysis-i

- https://github.com/melembroucarlitos/Tao_Analysis-LEAN

- https://github.com/leanprover-community/NNG4/ (this one does not follow Tao's book, but it's the Lean4 version of the natural numbers game, so has very similar content as Chapter 2)

- https://github.com/djvelleman/STG4/ (similar to the previous one, this is the Lean4 set theory game, so it's possibly similar content as Chapter 3; however, in https://github.com/djvelleman/STG4/blob/main/Game/Metadata.l... I see "import Mathlib.Data.Set.Basic" so this seems to just import the sets from Lean rather than defining it anew and setting down axioms, so this approach might mean that Lean will "know" too much about set theory, which is not good for our purposes)

- https://gist.github.com/kbuzzard/35bf66993e99cbcd8c9edc4914c... -- for constructing the integers

- https://github.com/ImperialCollegeLondon/IUM/blob/main/IUM/2... -- possibly the same file as above

- https://github.com/ImperialCollegeLondon/IUM/blob/main/IUM/2... -- for constructing the rationals

- https://lean-lang.org/theorem_proving_in_lean4/axioms_and_co... -- shows one way of defining a custom Set type

[1] https://taoanalysis.wordpress.com/

I am wondering how much this "prove isomorphism to Mathlib equivalent" is relevant. By this I mean, would anything change if the isomorphism part would be left out, i.e. is it actually used anywhere, for example for automatically translating theorems?

If nothing else it has pedagogical value, convincing yourself that the set and operations you established are the “same” as the ones you’re using for the rest of the book.

I loved that from the pedagogical perspective personally. When I dreamed about this textbook getting formalized, I was worried about how unwieldy the formalization could become if it strayed too far from Mathlib — but also was worried about losing the self-contained-ness if it just used Mathlib. This is a nice compromise imo.

I think that isomorphisms like this serve to establish

1. The development that you just did is equivalent to the corresponding object in Mathlib. Frequently, what is developed comes from concrete building blocks, whereas the corresponding definition in Mathlib might be a specialization of a complicated general class.

2. The basic notation and nomenclature for the object in Mathlib, which may differ.

He didn't really fully explain why he chose Lean over Coq or Agda though.

For this book specifically, I think the reason is just that Lean is the theorem prover he already knows and uses in his own work [1].

Now as to why he picked Lean a few years ago as his go-to theorem prover, I do think that would be an interesting backstory. As far as I know he hasn't written anything explicit about it.

[1] E.g., https://terrytao.wordpress.com/2023/11/18/formalizing-the-pr...

I would assume he chose Lean because it is what Kevin Buzzard has popularized and has been used to prove some very complex theories.

Does anyone know a good starting point to try Lean for Linear Algebra?

The canonical introduction is Mathematics in Lean

https://leanprover-community.github.io/mathematics_in_lean/i...

It doesn't get to linear algebra till chapter 9, and it tends to be cumulative, but you could try the first 4 chapters to get the basics and jump ahead to see how you get on

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