We have proof automation now

I've long had a soft spot for dependently-typed languages like Coq Rocq and Lean. They offer the possibility of a type system capable of encoding and enforcing arbitrarily subtle invariants. The sort of thing that, in regular languages, ends up (at best) as a comment, and which quickly gets lost as the size of the team grows. Then you get subtle misunderstandings and components that don't quite fit together. It's often the case that those components have grown to a sufficient size that, when the problem is noticed, aligning either of them is a wearying prospect. Perhaps, say dependent types seductively, you could write those invariants formally and have a machine check them.

(p.s. Coq changed its name! I remember many years ago at a Coq conference in Princeton, I tried suggesting that, in an English-speaking world, having a programming language called Coq was an impediment. I don't think the audience agreed at the time. I also joked that many of the talks there sounded like a speech by Tyrion Lannister, there being so many Coqs and Hoares. A joke that was hilarious and timely, even though it fell completely flat, coming as it did before the final season of that show and our collective memory-holing of it.)

The problem has always been that with great type-system power comes great proof effort. I can certainly attest to entire days spent proving really quite simple things. Doing proofs is actually quite fun: it's challenging, interactive, and there's a clear goal. But gosh, does it take a lot of time, especially if, like me, you don't know what you're doing. There's also the periodic, galling experience, at the end of many hours of effort, where you realise that the goal that you're trying to prove is, in fact, false . The classic result here is the retrospective from the seL4 effort that found that, even though the project was large enough for the engineers to develop considerable experience, they spent about 10 times as much time proving as they did designing and implementing. They ended up with more than 20 times as many lines of proof code as they did C code.

That overhead has made programming in dependently-typed languages extremely niche. It has also spurred people to try and automate it away. The attempt I'm passingly familiar with is F*, where the system tries to have an SMT solver automatically discharge the obligations. That certainly works for simple cases, but it's very easy to craft something that causes the SMT solver to go off into space and run for hours, leaving you wondering whether it's ever going to finish. I've seen that people who use these languages a lot have to develop a sixth sense for what is going to make the solver happy, and then craft everything around that. It can help, but to an extent it converts the problem into mysticism: you end up serving a complex and fickle god.

A critical fact is that, at least in theory, once the statement is correct, the contents of its proof are irrelevant: only its existence matters. This is not entirely true because of two complicating factors: first, what the seL4 group called “proof engineering”: the need to structure proofs so that the effort of realigning them after code changes is reduced. And, second, sufficiently complicated proofs can cause even type checkers to blow up and consume vast amounts of memory.

We now have LLMs which, combined with proof irrelevance, promise to be an extremely capable form of proof automation. With sufficient amounts of automation perhaps you don't need to worry about proof engineering nearly so much. You still need to avoid blowing up the type checker but, in my limited tests, LLMs can avoid that. Potentially, LLMs suddenly make dependent-type systems dramatically more practical. I wanted to play around with this so built a Zstandard decompressor in Lean, mostly because I was also curious about Zstandard.

Zstandard seems like it's winning the competition to replace gzip as the canonical compression utility. It's another LZ77-style compressor, but it offers better entropy coding and a careful design that allows it to achieve very impressive decompression speeds. It will never be as beautiful as bzip2, but the shining elegance of the Burrows–Wheeler transform doesn't count for too much in the face of significant practical advantages:

zstd bzip2 gzip lzma (XZ/LZMA2) 50 100 200 500 1000 2000 zstd bzip2 gzip lzma (XZ/LZMA2) 70 72 74 76 78 80 82 84 86 Compression tradeoff on 64 MiB of Lean/mathlib source Space saved (%) — farther right is more compression Decompression throughput (MiB/s, log scale) — higher is faster (Measurements taken on the standard reference computer, i.e. whatever the author was using at the time. And note the log scale on the y-axis: gzip and Zstandard are in their own speed class. This is an Apple machine and Apple's gzip is especially optimised; expect gzip to be slower elsewhere.)

Zstandard (by Yann Collet, building on the seminal ANS work by Jarek Duda) has an RFC , but it is quite terse. It contains all the information you need to implement a decompressor, but unless you're already quite familiar with compression, I think you'll need to re-read it a few times to understand what's going on. I, at least, had to read section 4.1 half a dozen times before I felt that I had a decent grasp of it. Too late into this process, I discovered that my colleague, Nigel Tao, has written a better write-up of Zstandard than I was going to manage anyway. So, if you want to understand Zstandard, you should read that. I'm just going to give an explanation of the most interesting bit, the entropy encoder, and mix that in with some evangelism about Lean.

The job of an entropy encoder is, given a set of symbols with non-uniform probabilities, to encode a sequence of those symbols using the fewest number of bits. The classic entropy coder is a Huffman encoder. Huffman encoders build a binary tree with symbols at the leaf nodes, and Huffman showed that a very simple algorithm produces an optimal prefix-tree: you take the list of symbols, you find the two with the least probability, and you form a tree node with them as children. That tree node then has a probability that is the sum of its two children, and then you repeat the algorithm with two fewer symbols, but now with a tree node in the mix. Obviously each step of this algorithm reduces the size of the set of elements by one, so it terminates, and it also produces an optimal tree. Huffman trees are very fast because you can build a table indexed by the next n bits (where n is the length of the longest code). The table entry tells you what symbol you've decoded and how many bits to unread. The drawback of Huffman trees is that they can only use a whole number of bits for each symbol: if you have a symbol where -log 2 (p) = 2.3 then ideally you want to use 2.3 bits to encode it. But Huffman forces you either to round up to 3 bits or to round down, which will force some other symbols to consume more bits.

Zstandard uses Huffman trees, but it also has a higher-compression entropy encoder called FSE. FSE is a state machine. There are more states than symbols, and each symbol gets a fraction of the states that mirrors its probability of occurrence in the stream. So if there's some symbol that is expected to appear 50% of the time, it gets ~50% of the states. Each state has three values: the symbol for that state, a number of bits to read from the bitstream when in that state, and a baseline state number that is added to those bits to get the next state. Now, if you recall, the problem with Huffman trees was that they could only use a whole number of bits, and these states also read a whole number of bits. But the trick is that if you are aiming to read one and a half bits for a given symbol, then half of its states will read one bit and half of them will read two bits. Then you hit your target on average . The table of states is never transmitted. The RFC prescribes an algorithm for building the table from a list of symbol probabilities, and so only the probabilities need to be transmitted.

Let's do an example. Let's say we have four symbols and we're going to use 16 states. So we have to approximate the symbol probabilities in terms of 16ths. (If you want a more accurate approximation of the probabilities, you can use a larger number of states; zstd actually never uses fewer than 32 states.)

Any symbol may follow any other symbol, and a symbol might only have a single state. So every symbol must be able to reach every state. Take a look at state three, which is the only state for symbol D. Because it's the only one, it has to read four bits, which is sufficient to encode any other state. But if you look at a symbol like B, its states only demand that you read one or two bits. However, the set of 16 possible next states is exactly partitioned between those states for symbol B. So, for any particular state, there is exactly one state for symbol B that can reach it.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 A A B D A B C A B C A B C A A B 0–1 1 bit 2–3 1 bit 4–7 2 bits 8–11 2 bits 12–15 2 bits Again, consider symbol B, which we said had a probability of 5/16. The ideal number of bits to encode that symbol is -log 2 (5/16) = 1.68. There are three symbol B states that read two bits and two that read one bit. The states aren't used equally often and, weighted by how often they're used, the average comes out to almost exactly the right value for the quantised probabilities. If you want to capture the true symbol probabilities with more accuracy, use a bigger table.

The central trick is that, by giving multiple states to more common symbols, the encoder doesn't just pick a symbol: it also picks which of that symbol's states to land in, and that choice carries information forward to the next symbol. That's where the fractional bits of information go. But this entropy encoder is still just table based, and so it runs very quickly.

The wrinkle is that you can't work forwards. Assume that you want to encode C, D. Which C state do you start in? Well, D only has one state so it has to be the C state which can reach that one. If D had multiple states then you would need to worry about what came after D to know which of those you needed. FSE forces you to start at the end of the sequence and work backwards. (That's not too bad because you usually need to know the whole sequence in order to calculate the symbol probabilities anyway.) Furthermore, a Zstandard compressor thus encodes symbols back-to-front, but writes output incrementally, so the decompressor has to seek to the end of a block and read the bits backwards in order to straighten it out! That's getting into broader details of the format that I'm not going to cover; see Nigel's piece .

Basic entropy encoders do not care about inter-symbol probabilities. I.e. they can't use the fact that the letter Q is disproportionately followed by the letter U (in English). There has to be some other encoding that is exploiting those redundancies. In Zstandard, that's a traditional Lempel–Ziv structure where it encodes either literal bytes or back references to previously decoded data. So FSE is primarily used for efficiently encoding these back reference offsets and lengths.

Let's talk about Lean! Above I said that it's a dependently-typed language, and that is a concept better articulated in examples than in a complicated definition. So here's the type of a function that reads n bytes from a stream and, if it doesn't throw, returns a byte array that the type system knows is n bytes long.

def IO.FS.Stream.readExact (st : Stream) (n : Nat) : IO {ba : ByteArray // ba.size = n} := … Here is a function that returns two numbers and a byte array such that the first number is prime, the sum of the two numbers is divisible by six, and the byte array is at least as long as the smaller of those two numbers.

def getResult : IO (Σ a b : Nat, { bytes : ByteArray // Nat.Prime a ∧ 6 ∣ a + b ∧