Some Junk Theorems in Lean

[andyjohnson0]

TIL that "junk theorems" are a thing in mathematics. Not being a mathematician myself, I found this [1] article a useful primer.

[1] https://www.cantorsparadise.com/what-are-junk-theorems-29868...

[ttctciyf]

Maybe I'm too familiar with the set theoretic construction of the natural numbers (0 is the empty set, 1 = {0}, ..., 5 = {0,1,2,3,4}, etc.) but their example of "3 ∩ 4 = 3" or "4 intersect 3 is 3" doesn't seem weird, problematic or even useless to me, it just looks like a handy set theoretic implementation of the min() function.

[oersted]

This is very interesting. What happens if you keep pulling the thread and construct large theories on such abstraction-layer-breaking theorems? Would we arrive at interesting things like pulling the thread on sqrt(-1) for imaginary numbers? Or is it somehow “undefined behavior”, quirks of the various implementation substrates of abstract mathematics that should be (informally) ignored? My gut says the former.

Are the various alternative axiomatic foundations also equivalent at this level or not? I suppose they are since they can implement/emulate each other, not sure.

[akoboldfrying]

This was helpful, thanks.

[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.

[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.

[tensegrist]

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

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

[414owen]

Wow, okay. I would imagine this makes mathematicians quite angry? I guess you're responsible for all the operations you use in your proof being well-behaved.

It sounds like subtraction over Nats needs to be split into `sub?`, and `sub!`, the former returning an option, and the latter crashing, on underflow, as is the Lean convention?

To use the default `sub`, you should need to provide a witness that the minuend is >= the subtrahend...

The version with silent underflow is still useful, it should just be called `saturatingSub`, or something, so that mathematicians using it know what they're getting themselves into...

[teiferer]

> Theorem 1. The third coordinate of the rational number 1/2 is a bijection.

What is a coordinate in the context of a rational number? How many coordinates does it have?

[pron]

I don't think anyone minds this. The purpose of a formal foundation is to prove useful theorems. Junk theorems are just a side effect. But I'm always tickled by logicians who think type theories are philosophically superior to set theories because in some of the latter you might be able to prove something like 1 ∈ 2, whereas type theories, even without their own junk theorems, have a pragmatic difficulty with division (hence they tend to define 1/0 = 0). Junk theorems just come with the territory, and foundations need to be considered based on their utility, not philosophical purity, which is never achieved anyway (at least not without a cost to utility).

[falcor84]

> But I'm always tickled by logicians who think type theories are philosophically superior to set theories because in some of the latter you might be able to prove something like 1 ∈ 2

Note that this is actually how the basic ZF construction works, where 0 = {} and successor(n) = n ∪ {n}, so you immediately get 2 = {0, 1} and thus 1 ∈ 2 , without any need for a proof.

[jhanschoo]

Note that the word "coordinate" used here feels a bit disingenuous to me, because that's how one might refer to the nth property defining a mathematical object or another.

For example: The third coordinate of the rational number 1/2 is a bijection.

Coordinate here actually means: third property in the definition of a rational number in Lean. Here, this property is the statement that the denominator 2 is not zero. This is not so absurd, if we define a rational number as a tuple consisting of a natural number for the numerator (property 1) and an integer for the denominator (property 2), with the added restrictions that the denominator is not the integer zero (property 3), and that the numerator and denominator are in least terms (property 4).

But the part where the proof that the denominator is nonzero can be viewed as a bijective function, is to me indeed type-theoretic weirdness. If I'm not wrong, it's just the proof viewed as a zero-argument function. (proofs for theorems that begin with e.g. forall are functions that take arguments).

[dernett]

Lean defines a != b as a = b => False, so it seems that we have a function from proofs of a = b to proofs of False. I guess this being bijective means that there are no proofs of a = b, since there are no proofs of False, which is an equivalent way of looking at a != b.

[emil-lp]

I don't understand. What does this mean?

    Theorem 6. The following are equivalent: The binary expansion of 7.

[tux3]

This is a junk theorem, it's trying to prove something that will sound strange or meaningless but is technically allowed by the details of the foundations.

Here it's building a list with one element and saying all elements of this list are equivalent. S̶o̶ ̶t̶h̶e̶ ̶f̶o̶l̶l̶o̶w̶i̶n̶g̶ ̶e̶l̶e̶m̶e̶n̶t̶s̶ ̶o̶f̶ ̶t̶h̶e̶ ̶l̶i̶s̶t̶ ̶a̶r̶e̶ ̶a̶l̶l̶ ̶e̶q̶u̶i̶v̶a̶l̶e̶n̶t̶ ̶t̶o̶ ̶e̶a̶c̶h̶ ̶o̶t̶h̶e̶r̶ ̶(̶t̶h̶e̶r̶e̶ ̶i̶s̶ ̶a̶ ̶s̶i̶n̶g̶l̶e̶ ̶e̶l̶e̶m̶e̶n̶t̶ ̶i̶n̶ ̶t̶h̶e̶ ̶l̶i̶s̶t̶)̶

[bzax]

It doesn't mean anything. The point is that the language of lean, and its proof derivation system, are able to express (and prove) statements that do not correspond to any meaningful mathematics.

[SkiFire13]

It's more like "the bits of 7 are all equivalent", which is kinda obvious when you notice that they are `111`

[homeless_engi]

As I think another commenter hinted, the binary expansion of 7 is 111. And indeed, 1 = 1 = 1

[silasdavis]

The following are equivalent:

[cmrx64]

List.TFAE is a helper definition and it’s invoked on a funny looking term when translated directly into english. I don’t know what I think, yeah it’s kinda junky but not in the way that 57 \mem 100 in a set encoding of the naturals.

    theorem TFAE_7_binary : List.TFAE (7).bits := by
  unfold Nat.bits Nat.binaryRec Nat.binaryRec; simp!

[bjt12345]

I'm surprised to learn that lean defines the natural number 1/0 as 0.

[zodiac]

Here’s a good document defending the merits of this design. https://xenaproject.wordpress.com/2020/07/05/division-by-zer...

[istjohn]

Doesn't this allow one to prove x=y for any x, y?

x/0 = x(1/0) = x*0 = 0, so x/0 = 0 for all x.

So x/0 = y/0.

Multiply both sides by 0: x = y.

[fsmv]

It seems to me that junk theorems are fundamentally a manifestation of leaky abstractions. It happens when you can see the implementation details from inside the abstraction.

[proof_by_vibes]

I've been writing [libsodium](https://doc.libsodium.org/) bindings in Lean4 and have ended up using `native_decide` quite liberally, mostly as a convenience. Can any Lean devs provide a more thorough interrogation of this? Should I go back and try to scrub its usage out of my library? Logically it seems consistent with what I'm trying to do with Lean4's FFI (i.e. you really do need to explicitly trust the Lean kernel since I'm adding nontrivial computation using a foreign cryptography library) but I'm curious if this isn't necessary and whether Lean devs would push back on its use.