> In either case I believe people who can put AI to the most value are the mathematicians themselves
The net output of math will increase, and mathematicians have more work now to unravel all this, and make it useful. AI plays the role of a monkey in the infinite monkey theorem [1]. We now need an LLM corollary - Something like: A finite number of LLM agents will almost surely find all theorems given an infinite token budget.
It's impossible for finite number of LLMs to solve all theorems. This would imply that the busy beaver sequence is computable which implies the halting problem is decidable.
"Given infinite thinking time a finite number of humans will solve all theorems"
I also love the angle that this was not intelligence just brute force. As if the mathematicians didn't reeaaally want to solve this they were just too lazy to give it a good try.
What does AI have to actually do before you realize these things are actually smart?
Machines have a much higher capacity for work than human beings. Saying that these proofs did not require equivelant intelligence, but benefitted from sheer volume, does not strike me as unreasonable.
It feels goalpost-movey to downplay exploring a large search space efficiently in regards to "intelligence". If we dug into a human genius's brain and found it was somehow trying out a million ways to solve a a problem at once, no one would seeiously suggest the person isn't actually intelligent.
And our brains must something like that at some physical level. You can't have a "turtles all the way down" of reasoning - the building blocks must be simpler. It must reduce to something like pathfinding and brute force at some point, weighted by factors in the system and maybe some randomness.
Finally, you might be wondering about the token cost. I wasn’t running this project in a particularly token-efficient way and have repeatedly maxed out my 20x Pro subscriptions for both Claude and ChatGPT every week. I also briefly had access to a prerelease model in the last few days, which did not have a usage cap. I was not tracking my actual token usage consistently. Some AI analysis from the recovered logs roughly estimates that we’re totaling around 40 billion tokens, of which around 210 million were output tokens. Over 95% were cache reads.
Who can pay for a 20x Pro subscriptions and also get prerelease models? something weird is going on here.
Several of my coworkers — it’s not that unusual that if you can max out a couple accounts, the companies will obviously notice you (as a high cost customer), and sometimes offer more
A wonderfully made introduction to the surreal numbers and their surrounding game theoretic concepts is this video on Hackenbush[0], a winner in 3Blue1Brown's Summer of Math competition.
>I’ve emailed some of the mathematicians with a few proposed typo fixes, and I got confirmation that at least a few of those fixes seemed real. However, some of the problems that weren’t backed by Lean also turned out to be misunderstandings.
I think this project is really neat, but is it appropriate to cold email specialists before you've put in enough hours of effort to describe yourself as more than an "amateur"? OP's emails may have been helpful, but billions of people use these LLMs to wade into new areas and email is already low signal-to-noise.
Yeah it's a pretty tough question! I've resisted doing that until I had a relatively high certainty that their published results contained minor mistakes, which I assumed they would want to know about. I've also been explicitly apologetic and tried to keep it super brief.
Here's my conjecture. Large Language Models are the great filter. They represent a local maximum in the technological advancement of a species from which we will not escape.
People will just limit publishing valid works to avoid becoming a hapless plagiarism victim class. Same thing happened to tech bloggers ripped off by low-effort you-tube content makers.
Isomorphic plagiarism makes people feel 23% smarter, but it also provably degrades core skills by 17%.
LLM are great at context search, but are also trivially proven degenerative under recursive self improvement scenarios. We look forwards to stripping their assets at a heavy discount.
Also, we shouldn't kink shame peoples cognitive dildo choices. =3
> On the second day, there are two gaps: “between nothing and zero” and “between zero and nothing”. Two numbers spawn in those two gaps. Call them –1 and 1.
Got lost here. I think I'm officially too dumb for math.
I don't think this is your fault; the description isn't very explicit. Let me try to do a bit better. (I'll also try to go somewhat further, and you should not be discouraged if at some point it stops making sense.)
You can think of the "surreal numbers" as being built up step by step. We start out with no numbers at all, and then we repeatedly do a construction that makes some new numbers.
A surreal number is made from two sets of (pre-existing) surreal numbers. We typically call them L and R, for "left" and "right", and sometimes write it as L|R or {L|R} or something like that. The "left" numbers have to be smaller than the "right" numbers. The resulting number will turn out to be, in a certain sense, the "simplest" number in between all the left numbers and all the right numbers.
Now, as I said, we start out with no numbers at all. It might seem like that gives us no way to proceed, but it does: even given no numbers at all, we can still make a set of numbers, namely the empty set! So we can use that for both L and R, getting ∅|∅. Empty sets on both sides. We call this 0, and it will turn out to behave in the way you'd expect the number 0 to behave.
Now we suddenly have another set available, namely {0}, the set containing only zero. Which means that instead of being able to make one number, maybe we can make four: ∅|∅, ∅|{0}, {0}|∅, {0}|{0}. The first of these we already knew about. The last isn't actually admissible -- remember that the "left" numbers have to be smaller than the "right" numbers, which is "vacuously" true when one of those sets is empty (it means "if you have a number x in the left set, and a number y in the right set, then x<y", and if there are no numbers in the left set or no numbers in the right set then that's trivially true) but isn't true when both sets contain 0 because 0<0 is false.
So actually we get two new numbers: ∅|{0} and {0}|∅. The first fits into what OP calls the gap "between nothing and zero". The second first into what OP calls "the gap between zero and nothing". In both cases, "zero" means a number and "nothing" means a space where we don't yet have any numbers.
The number ∅|{0} is called -1 (it has to lie to the left of 0, and there's no constraint on its left, and -1 is "the simplest number less than 0") and the number {0}|∅ is called +1 (it has to lie to the right of 0, and there's no constraint on its right, and +1 is "the simplest number greater than 0").
I should explicitly acknowledge that I haven't defined what "less than" and "greater than" actually mean for these numbers, nor anything else about how they relate to one another that could possibly justify giving these things the specific names 0, -1, and +1. But there are definitions for "less than" and "greater than" and "plus" and "minus" and so forth, and the whole thing does turn out to work very nicely.
Anyway, once we've got these numbers we have eight possible sets that can go on the left or on the right. The requirement for left-things to be smaller than right-things reduces the possibilities somewhat, and the actual new numbers we get next time around are: ∅|{-1}, which turns out to be -2; {-1}|{0} which turns out to be -1/2; {0}|{+1} which turns out to be +1/2; {+1}|∅ which turns out to be +2. We also get some already-existing numbers in new ways; for instance, {-1}|{+1} is actually equal to 0 ("0 is the simplest number between -1 and +1"). Again, I should explicitly acknowlege that I haven't said anything about how you determine when two of these things are actually equal; again, it does all turn out to work properly.
If you keep going with this construction, you produce all the integers, two at a time, and also all the "dyadic rationals", meaning fractions where the denominator is a power of 2. And then, once you've got all those, at the next stage of construction you abruptly get all the real numbers -- e.g., the square root of 2 is L|R where L = {dyadic rational numbers that are negative or have a square smaller than 2} and R = {dyadic rational numbers that are positive and have a square larger than 2} -- and you also get {0,1,2,3,4,...}|∅, conventionally written as a lower-case Greek letter omega, which is an infinite number, larger than all the integers. (And its negation.) And {0}|{1,1/2,1/3,1/4,...} which is an infinitesimal number, positive but smaller than any ratio of positive integers. And you can then proceed further and construct a vast infinitude of numbers, including all the real numbers (which we've already made) and all of the so-called infinite ordinals (which you can kinda think of as being a sort of "infinite positive integer", though there's more to them than that) and much more, all in a system that lets you do arithmetic and suchlike. It's very elegant, if your brain has been twisted into the mathematician-y shape that finds such things elegant.
What an interesting construction. Thank you from a curious layman for your write-up. I thought it was pretty easy to follow. I'd heard of the surreal numbers before and never knew about the construction mind-game behind them.
I think it's just we don't have a way to say/write these numbers. So you make up a way to write them (-1 and 1) and continue.
The numbers don't matter and you could replace -1 and 1 with anything. It's just easier to begin your new fake number at - 1 and 1. Because position does matter.
Basically what I take away is that we're inventing a new number system from scratch. So we're not "proving" that 1 is a number between 0 and the empty set. We're defining it as such, and it just so happens that a number system defined this way works out in convergent ways with other mathematics.
There has to be some procedure how to come up with "new" numbers though, if you want to have more in the end than just a fancy binary tree - in particular if you want to map your "fake numbers" to the reals, infinity, etc.
This procedure is enough. If you define addition and other operations in a certain way (as Conway did), it turns out that on the omega-th day (i.e. after initial infinite steps), all reals will be born.
This would make a lot more sense to me if "nothing" and "nothing" were instead "-inf" and "+inf"
Some other comments clarify that "nothing" is more accurately "the empty set". This is helpful because at first I wrongly synonomized "nothing" with zero. But now I get tripped up on the "between" language. Maybe it's a lack of background in sets, but I don't know what "between" implies for an integer (zero) and a set (the empty set).
I didn't want to introduce the notion of infinity because there are actual "infinite numbers" on the surreal number line. I've kind of tried to have both the simplicity of set-theoretic definition and the intuition of the number line, and slightly bungled the exposition. I hope the newly added diagram helps.
I'm not a mathematician and "nothing" doesn't really make sense for me either. But I guess the problem with inf might be that it'd be strange to get +2 as the next thing between +1 and +inf, while getting +1.5 between +1 and +2.
Wow thanks. I'm a mathematician and also got lost at the step. This illustration makes the construction much more clear. The text isn't really describing this process well
I think he just wrote it in a confusing way. The quote before says:
> (crucially, “to the left of all” and “to the right of all” also count as “gaps”)
So there are two "nothings" here, left of "all" - i.e. the zero - and right of it.
Though I'm not quite sure how you'd get infinite or irrational numbers by this procedure. Wouldn't you simply get the rational numbers by this?
(unless the "put a number" step is doing more work here than it seems. He doesn't really say which number to put there. In the examples, he mostly did "new number = (left number + right number) / 2", with special cases if any number is "nothing" - but he never actually wrote what the rules are here.)
Yeah I'm curious about the difference between "nothing" and "0", but I just decided to roll with it. Until the Greek letters made my head start to spin as they usually do.
(Apparently I was extremely unclear with this text. For clarity: if you want to actually understand surreal numbers, go and read On Numbers and Games, by Conway, which is a delightful book; or get an LLM to talk you through Wikipedia. Original text follows.)
It’s a terrible explanation. A surreal number is defined as a pair of sets of surreal numbers (where you fiddle around the recursion in that definition by defining them in waves, so strictly speaking you’re defining “the surreal numbers born at time T” for each individual T given access to the surreal numbers born at all earlier times, and then you “take the union across all times”, scare quotes because there are too many times for this to result in a set). Zero is a surreal number but the LLM is using the word “zero” to mean “the set containing just the surreal number 0”; “nothing” here is the LLM’s obtuse word for the empty set. Wikipedia may actually be easier to follow.
I mean, I was intending to supply the words that would link the LLM’s explanation to a more normal one, not to explain it; apparently that was extremely unclear. An actual explanation is much longer, as indeed I attempted to indicate by pointing to Wikipedia and saying that it might be more clear.
“Doing better than a totally useless explanation in fewer characters” is in general impossible, of course, eg if the first explanation has only one character.
I'm hoping someone develops an interactive tutor that can teach any subject to any depth.
The tutor should optimize its pedagogy. It should use online RL to adapt to a learner's ideal learning style, model what the student understands and to what degree, and understand what the gaps and next steps are.
Plot twist: they were both sold on early investment in companies that survived the .com bust. Now they’re VCs that everybody worships as business geniuses even though they’re just lucky idiots, and the sycophantic chatbots finally let them feel as smart as everyone says they are, and a whole bunch of hype-drunk fans are feeding into it.
The Claude output in the first one-shot counterexample attempt is hilarious. I hate its writing most of the time but this stuff is next level deep-fried slop.
> And the control column confirms the resonance-necessity conjecture empirically: break the skeleton alignment and the joint kernel dies at the constrained window, exactly as the transversality heuristic predicted.
That makes perfect sense. Using language in unorthodox ways to convey very specific concepts that only make sense to you and sound like bullshit to others.
> Instead, I have a more complicated view, which I actually expressed in my essay The Two Cultures of Mathematics a quarter of a century ago, and which can be summarized by saying that there is a spectrum of attitudes in mathematics to the relationship between problem-solving and conceptual understanding. At one end of the spectrum you have mathematicians who are primarily motivated by the wish to solve problems, who see conceptual understanding as a very important means to that end. At the other you have mathematicians who are primarily motivated by the wish to attain conceptual understanding, who see problem-solving as a very important means to that end.
Before, understanding and problem-solving-ability were so interdependent that distinguishing between the two was practically very difficult and probably wouldn’t have changed anyone’s research agenda. Now, they’re not connected, and this guy just did the ultimate meta-experiment of seriously undertaking a project that is intentionally 100% problem-solving and 0% understanding to prove it (maybe 99% and 1% but pretty close. In his transcripts, he never asks ChatGPT about the math, only about its opinions of the math).
As we (as a society) sit around asking ourselves what mathematicians (and software engineers, and anyone in deep technical fields) should be doing all day, we now have this case study to show us how wide our range of options has become.
> So, assuming my proof doesn’t rely on a Lean kernel bug, it’s likely to be legit too.
He lacks the understanding to verify his solution properly, and has to lean on those who do have the understanding to verify it, only being able to say himself that it's "likely" to be correct. (And what do those mathematicians get for laboriously checking the generated proof? 40 grand?)
Seems to me problem solving is as dependent on understanding as ever.
Moverover, the version I linked above is intentionally paranoid so it doesn't use any third-party code except Mathlib. If you allow usage of CombinatorialGames and trust its definitions, the part that needs to be checked narrows down to exactly 20 lines of code: https://github.com/gaearon/conway-refinement/blob/264445c93b...
What is your point, exactly? Increasing number of people working in and around mathematics are relying on Lean kernel's correctness. That's kind of the point of tools like Lean. Why is it a problem for me to publish a result that relies on it? How do you think other Lean proofs work?
I don't know why the author could claim this is "their" proof, and they kept saying "they" did this, "they" built that. but in reality everything is done by the LLM and the author is merely asking it to do things. i guess they did contribute money at least...
Author here! My impression is that it's customary in the mathematical community to take responsibility for the result with your name, regardless of whether it came from LLM etc (as long as you disclose LLM usage). I am perfectly fine calling it "LLM's proof" or somehow else, but it's "my" in the sense that "if there is a mistake in it, it is my mistake".
Someone had to choose the problem, steer the model, and check the output - it's clearly taken a lot of time. That's authorship with a powerful tool, same as it's always been.
When you use a drill to put a hole in the wall, do you take credit for it, or do you credit the drill? Without intent, a tool, whether it be a drill or an LLM, is just an inert object.
He seems genuinely interested in Surreal numbers. Seems like he found value in exploring them and the Lean process, devoting time and interest to it and understanding what it's like to be a mathematician. I think a lot of people who go into Computer Science may have been mathematicians in the 30s before they became separate majors at the university level.
Every human has to go through this in modern times.
I got quite frustrated and disappointed when taking pictures because everyone was doing it and my picture of x was similiar to others taking picture of x.
Either you learn from it and accept that and still do it, or you don't.
The net output of math will increase, and mathematicians have more work now to unravel all this, and make it useful. AI plays the role of a monkey in the infinite monkey theorem [1]. We now need an LLM corollary - Something like: A finite number of LLM agents will almost surely find all theorems given an infinite token budget.
[1] https://en.wikipedia.org/wiki/Infinite_monkey_theorem
"Given infinite thinking time a finite number of humans will solve all theorems"
I also love the angle that this was not intelligence just brute force. As if the mathematicians didn't reeaaally want to solve this they were just too lazy to give it a good try.
What does AI have to actually do before you realize these things are actually smart?
And our brains must something like that at some physical level. You can't have a "turtles all the way down" of reasoning - the building blocks must be simpler. It must reduce to something like pathfinding and brute force at some point, weighted by factors in the system and maybe some randomness.
Who can pay for a 20x Pro subscriptions and also get prerelease models? something weird is going on here.
[0]https://www.google.com/search?q=video+introduction+to+surrea...
I think this project is really neat, but is it appropriate to cold email specialists before you've put in enough hours of effort to describe yourself as more than an "amateur"? OP's emails may have been helpful, but billions of people use these LLMs to wade into new areas and email is already low signal-to-noise.
Isomorphic plagiarism makes people feel 23% smarter, but it also provably degrades core skills by 17%.
LLM are great at context search, but are also trivially proven degenerative under recursive self improvement scenarios. We look forwards to stripping their assets at a heavy discount.
Also, we shouldn't kink shame peoples cognitive dildo choices. =3
Got lost here. I think I'm officially too dumb for math.
You can think of the "surreal numbers" as being built up step by step. We start out with no numbers at all, and then we repeatedly do a construction that makes some new numbers.
A surreal number is made from two sets of (pre-existing) surreal numbers. We typically call them L and R, for "left" and "right", and sometimes write it as L|R or {L|R} or something like that. The "left" numbers have to be smaller than the "right" numbers. The resulting number will turn out to be, in a certain sense, the "simplest" number in between all the left numbers and all the right numbers.
Now, as I said, we start out with no numbers at all. It might seem like that gives us no way to proceed, but it does: even given no numbers at all, we can still make a set of numbers, namely the empty set! So we can use that for both L and R, getting ∅|∅. Empty sets on both sides. We call this 0, and it will turn out to behave in the way you'd expect the number 0 to behave.
Now we suddenly have another set available, namely {0}, the set containing only zero. Which means that instead of being able to make one number, maybe we can make four: ∅|∅, ∅|{0}, {0}|∅, {0}|{0}. The first of these we already knew about. The last isn't actually admissible -- remember that the "left" numbers have to be smaller than the "right" numbers, which is "vacuously" true when one of those sets is empty (it means "if you have a number x in the left set, and a number y in the right set, then x<y", and if there are no numbers in the left set or no numbers in the right set then that's trivially true) but isn't true when both sets contain 0 because 0<0 is false.
So actually we get two new numbers: ∅|{0} and {0}|∅. The first fits into what OP calls the gap "between nothing and zero". The second first into what OP calls "the gap between zero and nothing". In both cases, "zero" means a number and "nothing" means a space where we don't yet have any numbers.
The number ∅|{0} is called -1 (it has to lie to the left of 0, and there's no constraint on its left, and -1 is "the simplest number less than 0") and the number {0}|∅ is called +1 (it has to lie to the right of 0, and there's no constraint on its right, and +1 is "the simplest number greater than 0").
I should explicitly acknowledge that I haven't defined what "less than" and "greater than" actually mean for these numbers, nor anything else about how they relate to one another that could possibly justify giving these things the specific names 0, -1, and +1. But there are definitions for "less than" and "greater than" and "plus" and "minus" and so forth, and the whole thing does turn out to work very nicely.
Anyway, once we've got these numbers we have eight possible sets that can go on the left or on the right. The requirement for left-things to be smaller than right-things reduces the possibilities somewhat, and the actual new numbers we get next time around are: ∅|{-1}, which turns out to be -2; {-1}|{0} which turns out to be -1/2; {0}|{+1} which turns out to be +1/2; {+1}|∅ which turns out to be +2. We also get some already-existing numbers in new ways; for instance, {-1}|{+1} is actually equal to 0 ("0 is the simplest number between -1 and +1"). Again, I should explicitly acknowlege that I haven't said anything about how you determine when two of these things are actually equal; again, it does all turn out to work properly.
If you keep going with this construction, you produce all the integers, two at a time, and also all the "dyadic rationals", meaning fractions where the denominator is a power of 2. And then, once you've got all those, at the next stage of construction you abruptly get all the real numbers -- e.g., the square root of 2 is L|R where L = {dyadic rational numbers that are negative or have a square smaller than 2} and R = {dyadic rational numbers that are positive and have a square larger than 2} -- and you also get {0,1,2,3,4,...}|∅, conventionally written as a lower-case Greek letter omega, which is an infinite number, larger than all the integers. (And its negation.) And {0}|{1,1/2,1/3,1/4,...} which is an infinitesimal number, positive but smaller than any ratio of positive integers. And you can then proceed further and construct a vast infinitude of numbers, including all the real numbers (which we've already made) and all of the so-called infinite ordinals (which you can kinda think of as being a sort of "infinite positive integer", though there's more to them than that) and much more, all in a system that lets you do arithmetic and suchlike. It's very elegant, if your brain has been twisted into the mathematician-y shape that finds such things elegant.
The numbers don't matter and you could replace -1 and 1 with anything. It's just easier to begin your new fake number at - 1 and 1. Because position does matter.
Basically what I take away is that we're inventing a new number system from scratch. So we're not "proving" that 1 is a number between 0 and the empty set. We're defining it as such, and it just so happens that a number system defined this way works out in convergent ways with other mathematics.
Is that roughly right?
Only as a mental abstraction that's based on our experience/concept of space+time.
Sorry it was confusing.
Edit: the picture is now edited into the article.
Some other comments clarify that "nothing" is more accurately "the empty set". This is helpful because at first I wrongly synonomized "nothing" with zero. But now I get tripped up on the "between" language. Maybe it's a lack of background in sets, but I don't know what "between" implies for an integer (zero) and a set (the empty set).
My favorite intro to surreal numbers is https://www.infinitelymore.xyz/p/surreal-numbers, but it is behind a registration wall.
> (crucially, “to the left of all” and “to the right of all” also count as “gaps”)
So there are two "nothings" here, left of "all" - i.e. the zero - and right of it.
Though I'm not quite sure how you'd get infinite or irrational numbers by this procedure. Wouldn't you simply get the rational numbers by this?
(unless the "put a number" step is doing more work here than it seems. He doesn't really say which number to put there. In the examples, he mostly did "new number = (left number + right number) / 2", with special cases if any number is "nothing" - but he never actually wrote what the rules are here.)
It’s a terrible explanation. A surreal number is defined as a pair of sets of surreal numbers (where you fiddle around the recursion in that definition by defining them in waves, so strictly speaking you’re defining “the surreal numbers born at time T” for each individual T given access to the surreal numbers born at all earlier times, and then you “take the union across all times”, scare quotes because there are too many times for this to result in a set). Zero is a surreal number but the LLM is using the word “zero” to mean “the set containing just the surreal number 0”; “nothing” here is the LLM’s obtuse word for the empty set. Wikipedia may actually be easier to follow.
“Doing better than a totally useless explanation in fewer characters” is in general impossible, of course, eg if the first explanation has only one character.
I'm hoping someone develops an interactive tutor that can teach any subject to any depth.
The tutor should optimize its pedagogy. It should use online RL to adapt to a learner's ideal learning style, model what the student understands and to what degree, and understand what the gaps and next steps are.
I'd subscribe in a heartbeat.
- https://github.com/mattpocock/skills/blob/main/skills/produc...
- https://www.youtube.com/watch?v=s5T5oQJcJ6U
ie. i am an expert at zig, explain this c++ in terms of zig
In each episode they make a major science-fiction style breakthrough and grapple with the consequences without revealing themselves.
and "proof map": https://gaearon.github.io/conway-refinement/#/map/conway-ref...
> And the control column confirms the resonance-necessity conjecture empirically: break the skeleton alignment and the joint kernel dies at the constrained window, exactly as the transversality heuristic predicted.
> The den has air in it.
> Drift fuel exists.
You can so easily imagine this shit being read in a 90s slam poetry coffeehouse. Trust me I was there
I'd imagine that in three months when we all have access to communicating agent swarms this should be easier
> Instead, I have a more complicated view, which I actually expressed in my essay The Two Cultures of Mathematics a quarter of a century ago, and which can be summarized by saying that there is a spectrum of attitudes in mathematics to the relationship between problem-solving and conceptual understanding. At one end of the spectrum you have mathematicians who are primarily motivated by the wish to solve problems, who see conceptual understanding as a very important means to that end. At the other you have mathematicians who are primarily motivated by the wish to attain conceptual understanding, who see problem-solving as a very important means to that end.
Before, understanding and problem-solving-ability were so interdependent that distinguishing between the two was practically very difficult and probably wouldn’t have changed anyone’s research agenda. Now, they’re not connected, and this guy just did the ultimate meta-experiment of seriously undertaking a project that is intentionally 100% problem-solving and 0% understanding to prove it (maybe 99% and 1% but pretty close. In his transcripts, he never asks ChatGPT about the math, only about its opinions of the math).
As we (as a society) sit around asking ourselves what mathematicians (and software engineers, and anyone in deep technical fields) should be doing all day, we now have this case study to show us how wide our range of options has become.
> So, assuming my proof doesn’t rely on a Lean kernel bug, it’s likely to be legit too.
He lacks the understanding to verify his solution properly, and has to lean on those who do have the understanding to verify it, only being able to say himself that it's "likely" to be correct. (And what do those mathematicians get for laboriously checking the generated proof? 40 grand?)
Seems to me problem solving is as dependent on understanding as ever.
The only thing that needs a check is this 500-line file: https://github.com/gaearon/conway-refinement/blob/264445c93b.... If this file is correct and Lean kernel is correct, the proof is correct.
Moverover, the version I linked above is intentionally paranoid so it doesn't use any third-party code except Mathlib. If you allow usage of CombinatorialGames and trust its definitions, the part that needs to be checked narrows down to exactly 20 lines of code: https://github.com/gaearon/conway-refinement/blob/264445c93b...
There are two ifs in this sentence.
> Me: btw how’s your mood overall?
LOL. mood??
I got quite frustrated and disappointed when taking pictures because everyone was doing it and my picture of x was similiar to others taking picture of x.
Either you learn from it and accept that and still do it, or you don't.
But its not new
Let’s not infuse good faith into what is clearly meant as a derogatory comment.