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How long until an AI can do the formalization for a proof like this one fully automatically?

For example, the "direct proof" in this paper is six paragraphs long.

> you're assuming a lot. Including a notion of mathematical existence which bears little relation to any concept that most lay people have of what mathematical existence might mean.

John Horton Conway:

> It's a funny thing that happens with mathematicians. What's the ontology of mathematical things? How do they exist? In what sense do they exist? There's no doubt that they do exist but you can't poke and prod them except by thinking about them. It's quite astonishing and I still don't understand it, having been a mathematician all my life. How can things be there without actually being there? There's no doubt that 2 is there or 3 or the square root of omega. They're very real things. I still don't know the sense in which mathematical objects exist, but they do. Of course, it's hard to say in what sense a cat is out there, too, but we know it is, very definitely. Cats have a stubborn reality but maybe numbers are stubborner still. You can't push a cat in a direction it doesn't want to go. You can't do it with a number either.

Here's the abstract of this earlier paper:

> Given the recent explosion of interest in streaming data and online algorithms, clustering of time series subsequences, extracted via a sliding window, has received much attention. In this work we make a surprising claim. Clustering of time series subsequences is meaningless. More concretely, clusters extracted from these time series are forced to obey a certain constraint that is pathologically unlikely to be satisfied by any dataset, and because of this, the clusters extracted by any clustering algorithm are essentially random. While this constraint can be intuitively demonstrated with a simple illustration and is simple to prove, it has never appeared in the literature. We can justify calling our claim surprising, since it invalidates the contribution of dozens of previously published papers. We will justify our claim with a theorem, illustrative examples, and a comprehensive set of experiments on reimplementations of previous work. Although the primary contribution of our work is to draw attention to the fact that an apparent solution to an important problem is incorrect and should no longer be used, we also introduce a novel method which, based on the concept of time series motifs, is able to meaningfully cluster subsequences on some time series datasets.

Several commenters here seem to ask "okay, so then what's the right way to cluster windows of timeseries??" Perhaps the final sentence of this abstract suggests a solution in that direction?

My understanding is that people would mostly try this approach when the samples are periodic/regular.

In case anyone was curious, the Internet archive on my parent commenter's link shows large dozen egg prices of: $7.90 March 2024, $7.50 November 2023, $6.50 February 2023.

I don't think "policy discretion to deny" is what the word "welfare" means.

But in any case, Medicare is literally denied to prisoners, which is an example of ... denying welfare benefits for antisocial behavior?

In case anyone is curious, the $1 is the increase in the maximum SNAP benefit per month for an individual, from $291/month to $292/month. (The increases for larger households are similarly small.)

This is not the actual increase of the benefit amount. In particular, it appears the cost of living adjustment this year is 2.5%. I have been unable to find statistics on how many people/households actually receive the maximum amount, but I don't have a particular reason to believe it is large. (The average benefit amounts are significantly below the maxima.)

Tldr: the average SNAP benefit amount received by people has increased and will increase by significantly more than $1/month.