[frotaur]

I don't know much about Lean, but I attended an introductory talk at some point and I was particularly bothered by these partial function definitions. The example was sqrt, which would give 0 on the negatives.

Now, of course, if you're careful with the definitions you use, there is no problem. But in the (increasingly relevant) context of automatic theorem proving with LLMs, this seems to defeat the 'groundtruthness' of Lean!

How do you make sure that the LLM doesn't reward hack a proof using these workarounds?

[akoboldfrying]

I don't understand why they would make such footgun functions either, especially because (IIUC, and I probably don't) in a way the whole point of Lean's dependent type system is to be able to express arbitrary constraints on the inputs of these functions so that they can be total -- e.g., to be able to define a subtraction function on the nonnegative integers that takes one integer and one {integer that is less than or equal to the first integer}. And to even call this function, you (or perhaps Lean itself) would need to first prove that its second argument is less than or equal to its first.

[Smaug123]

You can express those constraints; it just turns out to be less ergonomic in practice if you do. (You can even do so in terms of the junk-valued total functions! Just define `actual_subtraction` to call straight through to `junky_subtraction`, but `actual_subtraction` has these constraints on its domain.)

The mathlib way to do things is to push those requirements out to the one who wishes to use the theorem. If you find that you're depending on a junk value in a way that's incompatible with what you wanted to prove, then you've simply discovered that you forgot to restrict your own domain to exclude the junk. (And if your desired usage lines up with the junk, then great, you get to omit an annoying busywork hypothesis.) A sqrt function that gives 0 on the negatives isn't breaking any of sqrt's properties on the positives!

The mathlib way means that instead of every function having to express these constraints and pass proofs down the line, only some functions have to.

[akoboldfrying]

Thanks.

> If you find that you're depending on a junk value in a way that's incompatible with what you wanted to prove

This is the part I'm struggling with. How would you actually know/realise that you were doing this? It seems like "the mathlib way" you describe is choosing to rely on programmer discipline for something that could be enforced automatically.

My fear is that relying on the junk values of functions (values where their "proper" partial counterparts are not defined) is somehow unsound (could lead to proving something untrue). But perhaps my intuition is off here? If so, I think the specific junk values chosen must not matter at all -- e.g., having sqrt return 42 for negative x values should work just as well, am I right?

[markusde]

You can't prove something untrue (in the sense that it implies false) without proving that the theorem prover is is unsound, which I think at the moment is not known to be possible in Lean.

But you're exactly right. There's nothing linking theorem prover definitions to pen and paper definitions in any formal system.

[yccs27]

It is enforced automatically for most purposes: If you're writing a proof involving e.g. the sqrt function, you want to use theorems about it, e.g. that (sqrt(x))^2 = x. Almost all of those theorems have x>=0 as a precondition, so you do need to prove it when it matters.

[markusde]

This is a topic of contention in formalized math with no universal right answer. Some libraries go heavy on the dependent types, and some like mathlib try to avoid them. I do math in both Rocq and Lean and I find I like the latter style a lot more for my work for a couple reasons:

- Fewer side conditions: Setting a / 0 = 0 means that some laws hold even when a denominator is 0, and so you don't need to prove the denominator is nonzero. This is super nice when the denominator is horrible. I heard once that if you set the junk value for a non-converging Riemann integral to the average of the lim sup and lim inf you can obliterate a huge number of integrability side conditions (though I didn't track down this paper to find out for sure).

- Some of the wacky junk arithmetic values, especially as it relates to extended reals, do show up in measure theory. Point being: "junk arithmetic" is a different mathematical theory than normal math, but it's no less legitimate, and is closely related.

- Definition with Hilbert's epsilon operator. If I want to define a function that takes eg. a measurable set S as an argument, I could do the dependent types way

def MyDef (S) (H : measurable S) := /-- real definition -/

but then I need to write all of my theorems in terms of (MyDef S H) and this can cause annoying unification problems (moreso in Rocq than in Lean, assuming H is a Prop). Alternatively, I could use junk math

def MyDef' (S) := if (choose (H : measurable S)) then /-- real definition -/ else /-- junk -/

I can prove (MyDef' S = MyDef S H) when I have access to (H : measurable S). And the property H here can be be really complex, convergence properties, existence properties, etc. It's nice to avoid trucking them around everywhere.

[xigoi]

> if you set the junk value for a non-converging Riemann integral to the average of the lim sup and lim inf you can obliterate a huge number of integrability side conditions

Wouldn’t this still cause problems if the lim sup is ∞ and the lim inf is -∞?

[danabramov]

There’s a good blog post on this by Kevin Buzzard. I suggest to give it a read: https://xenaproject.wordpress.com/2020/07/05/division-by-zer...

I found the last section especially helpful.

[oersted]

This is a really good explanation, but it reinforces my understanding that these “junk maths” are literally undefined behavior as in C and such. They are not defined (in maths), you are not supposed to trigger them, so they can be anything. Great…

This is horrible for a language whose whole purpose I thought was that to be foolproof and that if it compiles its true. Having very subtly different definitions of common operations is such a footgun.

Of course, I understand that this doesn’t bother mathematicians because they are used to not having any guardrails anyways. Just like C programmers have the attitude that if you fall on such a trap, you deserve it and you are not a “real programmer”. But Lean is supposed to be the other extreme isn’t it? Take nothing for granted and verify it from the ground up.

I suppose I am falling for that “Twitter confusion” the post is referring to. I never had any issues with this when actually using Lean. I just don’t like the burden of having to be paranoid about it, I thought Lean had my back and I could use it fairly mechanically by transforming abstract structures without thinking about the underlying semantics too much.

Anyway, despite the annoyance, I do assume that the designers know better and that it is a pragmatic and necessary compromise if it’s such a common pattern. But there must be a better solution, if having the exception makes it uncomfortable to prove, then design the language so that it is comfortable to prove such a thing. Don’t just remove the exception because 99% of the time it doesn’t matter. If we are happy with 99% we wouldn’t be reaching for formal verification, there are much more practical means to check correctness.

[tux3]

There is still a guardrail. The blog post explains that it is just using different functions and notation which might allow things like 0/0. But at the end of the day, different notation still cannot be used to prove false things.

In other words, you can use all these junk theorems to build strange results on the side, but you can never build something that disagrees with normal math or that contradicts itself. There is no footgun, because the weird results you obtain are just notation. They look weird to a human, but they don't allow you to actually break any rules or to prove 1=0.

[oersted]

I understand that, but if "/", and other common operators, don't mean what they means on paper, you can prove things that would be untrue if copied onto paper (kinda). You can indeed prove "1/0 = 0", which is not that far off from redefining "=" and proving "1=0".

More importantly, the other way around, it seems too easy to copy a proposition from paper onto Lean and falsely prove it without realising they don't express the same thing. A human probably wouldn't but there's increased usage of AI and other automatic methods with Lean.

I do understand I'm being purist and that it doesn't matter that much in practice. I've used Lean seriously for a while and I've never encountered any of this.

[zarzavat]

I feel like this aged like milk because it assumes a human mathematician writing the proof but many people are now generating Lean proofs with LLMs.

[akoboldfrying]

Thank you! This hit the nail on the head for me, though I probably need to try out a few more examples to fully convince myself.

TL;DR: It's actually harmless (and often convenient) to "inflate" the domains of partial functions to make them total (by making them return arbitrary junk values where the original function is undefined), provided that every theorem you want to apply still comes with the original, full restrictions.

Kevin's example is good. My stupider example would be: We can define a set that contains the integers ..., -2, -1, 0, 1, 2, ..., plus the extra element "banana". If we define the result of any addition, subtraction or multiplication involving a banana to be 42, and to have their usual results otherwise, then, provided that we add the condition "None of the variables involved is banana" to the theorem "x+y = y+x", and to every other theorem about arithmetic, anything that we can prove about arithmetic on elements of this set is also true of arithmetic on integers.

[nextaccountic]

If your language isn't ergonomic then people will not use it (or use it less). Maybe another theorem prover will arise that makes "correct" definitions more ergonomic

Note that the same thing happens in Rust. Rather than putting trait bounds in structs (like struct Aa<T: Something> { .. }, people are encouraged to make the structs more generic (struct Aa<T> { .. }) and put restrictions on impls instead (impl<T: Something> Aa<T> { .. }). The rationale being that this is more ergonomic because it doesn't require you to repeat bounds in places you don't need them, and if every impl requires a Something bound, you will need the bound to make anything with this type (doubly so if the fields of Aa are private and so you need to build one using a method with T: Something)

[sebzim4500]

>How do you make sure that the LLM doesn't reward hack a proof using these workarounds?

I'm not sure what you mean exactly? There is no soundness issue here, the fact that `sqrt -1` is defined to be 0 does not have any impact on what statements about `sqrt x` can be proved when `x` is positive.

It just means that if you are working on an intermediate step of a proof and you need the result that `sqrt y >= 0` you don't need to provide a proof that `y >= 0`. If you wanted an intermediate result that `(sqrt y) * 2 = y` then you would still need to provide a proof that `y >= 0`, though.

[munchler]

If sqrt -1 = 0, then (by squaring both sides) -1 = 0, which is clearly unsound.

[sebzim4500]

Right but there isn't a theorem saying `(sqrt x)^2 = x`, there's a theorem saying `x >= 0 -> (sqrt x)^2 = x`

[munchler]

Ah, that makes sense. Thank you. As long as every use of sqrt has such a condition.

[tensegrist]

https://www.hillelwayne.com/post/divide-by-zero/

(discussion: https://news.ycombinator.com/item?id=17736046)

[teiferer]

I love that article of Hillel's!

[dnautics]

it's terrible advice for actual programmers though because often 0 is a sentinel value with special meaning for systems that you don't have control over (sometimes because of pre-digital conventions that shouldn't be lightly fucked with).

This is usually done by PL's that want to avoid crashes at all costs, but "turning crashes into subtle logic errors" seems like a really bad idea.

[teiferer]

Two quotes from the article you are criticizing:

"As a programmer, I don’t like it."

"As mentioned before, this is not a post about what’s practically a good idea. All I’m arguing is that mathematically, we can extend division in this way without leading to a contradiction. Programming languages are different from mathematical formalisms, and should be different. I prefer that 1/0 is an error, because I’m not using my program to prove theories."

Please do yourself a favor and actually read it.

Besides, 0 as a sentinental value on disk or on the wire is fine, but once you have values in a programming language, use option types. This is not 1980s anymore, you don't need to use 0 or -1 or 0xffff to express something special which sooner or later just falls on your feet.

[dnautics]

I read the article. As someone who was a math major I get why it's "fine". But nowhere in the article does hillel explain WHY it's bad for a real pl. (And defenders of e.g. pony gleam point to this article too to say it's okay. It's not.) I am adding context.

> This is not 1980s anymore, you don't need to use 0 or -1 or 0xffff to express something special which sooner or later just falls on your feet.

No. You missed the whole "real world systems". E.g. like stock trading, where zero stock trades are tombstones.

[teiferer]

> I am adding context.

Ah, that's apprecitated. Indeed, he didn't provide that "why" and tbf that wasn't the point of the article. But thanks for adding that context.

> You missed the whole "real world systems". E.g. like stock trading, where zero stock trades are tombstones.

Hm I don't think I missed that. This counts as "on the wire". Externally, there are surely good reasons for that representation, though I'd argue that internally it's better to represent this in the type system instead of special casing 0 everywhere which can be forgotten and then you get your (potential) division-by-0 issues. Avoiding them by construction is even better than failing explicitly (which I agree is in turn still better than silently returning 0).