## Competitive Programming in Haskell: Scanner

In my previous post I explored solving a simple competitive programming problem in Haskell. The input of the problem just consisted of a bunch of lines containing specific data, so that we could parse it using lines and words. There is another common class of problems, however, which follow this pattern:

The first line of the input consists of an integer $T$. Each of the next $T$ lines consists of…

That is, the input contains integers which are not input data per se but just tell you how many things are to follow. This is really easy to process in an imperative language like Java or C++. For example, in Java we might write code like this:

Scanner in = new Scanner(System.in);
int T = in.nextInt();
for (int i = 0; i < T; i++) {
// process each line
}

Occasionally, we can get away with completely ignoring the extra information in Haskell. For example, if the input consists of a number $T$ followed by $T$ lines, each of which contains a number $n$ followed by a list of $n$ numbers, we can just write

main = interact $lines >>> drop 1 >>> map (words >>> drop 1 >>> map read) >>> ... That is, we can ignore the first line containing $T$ since the end-of-file will tell us how many lines there are; and we can ignore the $n$ at the beginning of each line, since the newline character tells us when the list on that line is done. Sometimes, however, this isn’t possible, especially when there are multiple test cases, or when a single test case has multiple parts, each of which can have a variable length. For example, consider Popular Vote, which describes its input as follows: The first line of input contains a single positive integer $T \leq 500$ indicating the number of test cases. The first line of each test case also contains a single positive integer $n$ indicating the number of candidates in the election. This is followed by $n$ lines, with the $i$th line containing a single nonnegative integer indicating the number of votes candidate $i$ received. How would we parse this? We could still ignore $T$—just keep reading until the end of the file—but there’s no way we can ignore the $n$ values. Since the values for each test case are all on separate lines instead of on one line, there’s otherwise no way to know when one test case ends and the next begins. Once upon a time, I would have done this using splitAt and explicit recursion, like so: type Election = [Int] readInput :: String -> [Election] readInput = lines >>> drop 1 {- ignore T -} >>> map read >>> go where go :: [Int] -> [Election] go [] = [] go (n:xs) = votes : go rest where (votes,rest) = splitAt n xs However, this is really annoying to write and easy to get wrong. There are way too many variable names to keep track of (n, xs, votes, rest, go) and for more complex inputs it becomes simply unmanageable. You might think we should switch to using a real parser combinator library—parsec is indeed installed in the environment Kattis uses to run Haskell solutions—and although sometimes a full-blown parser combinator library is needed, in this case it’s quite a bit more heavyweight than we would like. I can never remember which modules I have to import to get parsec set up; there’s a bunch of boilerplate needed to set up a lexer; and so on. Using parsec is only worth it if we’re parsing something really complex. ## Scanner The heart of the issue is that we want to be able to specify a high-level description of the sequence of things we expect to see in the input, without worrying about managing the stream of tokens explicitly. Another key insight is that 99% of the time, we don’t need the ability to deal with parse failure or the ability to parse multiple alternatives. With these insights in mind, we can create a very simple Scanner abstraction, which is just a Stateful computation over a list of tokens: type Scanner = State [String] runScanner :: Scanner a -> String -> a runScanner s = evalState s . words To run a scanner, we just feed it the entire input as a String, which gets chopped into tokens using words. (Of course in some scenarios we might want to use lines instead of words, or even do more complex tokenization.) Note since Scanner is just a type synonym for State [String], it is automatically an instance of Functor, Applicative, and Monad (but not Alternative). So let’s develop a little Scanner DSL. The most fundamental thing we can do is read the next token. str :: Scanner String str = get >>= \case { s:ss -> put ss >> return s } (This uses the LambdaCase extension, though we could easily rewrite it without.) str gets the current list of tokens, puts it back without the first token, and returns the first token. Note that I purposely didn’t include a case for the empty list. You might think we want to include a case for the empty token list and have it return the empty string or something like that. But since the input will always be properly formatted, if this scenario ever happens it means my program has a bug—e.g. perhaps I misunderstood the description of the input format. In this scenario I want it to crash loudly, as soon as possible, rather than continuing on with some bogus data. We can now add some scanners for reading specific token types other than String, simply by mapping the read function over the output of str: int :: Scanner Int int = read <$> str

integer :: Scanner Integer
integer = read <$> str double :: Scanner Double double = read <$> str

Again, these will crash if they see a token in an unexpected format, and that is a very deliberate choice.

Now, as I explained earlier, a very common pattern is to have an integer $n$ followed by $n$ copies of something. So let’s make a combinator to encapsulate that pattern:

numberOf :: Scanner a -> Scanner [a]
numberOf s = int >>= flip replicateM s

numberOf s expects to first see an Int value $n$, and then it runs the provided scanner $n$ times, returning a list of the results.

It’s also sometimes useful to have a way to repeat a Scanner some unknown number of times until encountering EOF (for example, the input for some problems doesn’t specify the number of test cases up front the way that Popular Vote does). This is similar to the many combinator from Alternative.

many :: Scanner a -> Scanner [a]
many s = get >>= \case { [] -> return []; _ -> (:) <$> s <*> many s } many s repeats the scanner s as many times as it can, returning a list of the results. In particular it first peeks at the current token list to see if it is empty. If so, it returns the empty list of results; if there are more tokens, it runs s once and then recursively calls many s, consing the results together. Finally, it’s quite common to want to parse a specific small number of something, e.g. two double values representing a 2D coordinate pair. We could just write replicateM 2 double, but this is common enough that I find it helpful to define dedicated combinators with short names: two, three, four :: Scanner a -> Scanner [a] [two, three, four] = map replicateM [2..4] The complete file can be found on GitHub. As I continue this series I’ll be putting more code into that repository. Note I do not intend to make this into a Hackage package, since that wouldn’t be useful: you can’t tell Kattis to go download a package from Hackage before running your submission. However, it is possible to submit multiple files at once, so you can include Scanner.hs in your submission and just import Scanner at the top of your main module. ## Examples So what have we gained? Writing the parser for Popular Vote is now almost trivial: type Election = [Int] main = interact$ runScanner elections >>> ...

elections :: Scanner [Election]
elections = numberOf (numberOf int)

In practice I would probably just inline the definition of elections directly: interact $runScanner (numberOf (numberOf int)) >>> ... As a slightly more involved example, chosen almost at random, consider Board Wrapping: On the first line of input there is one integer, $N \leq 50$, giving the number of test cases (moulds) in the input. After this line, $N$ test cases follow. Each test case starts with a line containing one integer $n, 1 \leq n \leq 600$, which is the number of boards in the mould. Then $n$ lines follow, each with five floating point numbers $x,y,w,h,v$ where $0 \leq x,y,w,h \leq 10000$ and $-90^{\circ} < v \leq 90^{\circ}$. The $x$ and $y$ are the coordinates of the center of the board and $w$ and $h$ are the width and height of the board, respectively. $v$ is the angle between the height axis of the board to the $y$-axis in degrees, positive clockwise. Here’s how I would set up the input, using Scanner and a custom data type to represent boards. import Scanner type V = [Double] -- 2D vectors/points newtype A = A Double -- angle (radians) -- newtype helps avoid conversion errors fromDeg :: Double -> A fromDeg d = A (d * pi / 180) data Board = Board { boardLoc :: V, boardDims :: V, boardAngle :: A } board :: Scanner Board board = Board <$> two double
<*> two double
<*> ((fromDeg . negate) <$> double) main = interact$
runScanner (numberOf (numberOf board)) >>> ...

## Lightweight, efficiently sampleable enumerations in Haskell

For another project I’m working on, I needed a way to enumerate and randomly sample values from various potentially infinite collections. There are plenty of packages in this space, but none of them quite fit my needs:

• universe (and related packages) is very nice, but it’s focused on enumerating values of Haskell data types, not arbitrary sets: since it uses type classes, you have to make a new Haskell type for each thing you want to enumerate. It also uses actual Haskell lists of values, which doesn’t play nicely with sampling.
• enumerable has not been updated in a long time and seems to be superseded by universe.
• enumerate is likewise focused on generating values of Haskell data types, with accompanying generic deriving machinery.
• size-based is used as the basis for the venerable testing-feat library, but these are again focused on generating values of Haskell data types. I’m also not sure I need the added complexity of size-indexed enumerations.
• enumeration looks super interesting, and I might be able to use it for what I want, but (a) I’m not sure whether it’s maintained anymore, and (b) it seems rather more complex than I need.

I really want something like Racket’s nice data/enumerate package, but nothing like that seems to exist in Haskell. So, of course, I made my own! For now you can find it on GitHub.1 Here’s the package in a nutshell:

• Enumerations are represented by the parameterized type Enumeration, which is an instance of Functor, Applicative, and Alternative (but not Monad).
• Enumerations keep track of their cardinality, which could be either countably infinite or a specific natural number.
• Enumerations are represented as functions from index to value, so they can be efficiently indexed (which also enables efficient random sampling).
• The provided combinators will always do something sensible so that every value in the resulting enumeration occurs at a finite index. For example, if you take the disjoint union of two infinite enumerations, the resulting enumeration will alternate between values from the two inputs.

I wrote about something similar a few years ago. The main difference is that in that post I limited myself to only finite enumerations. There’s a lot more I could say but for now I think I will just show some examples:

>>> enumerate empty
[]
>>> enumerate unit
[()]
>>> enumerate $empty <|> unit <|> unit [(),()] >>> enumerate$ finite 4 >< finiteList [27,84,17]
[(0,27),(0,84),(0,17),(1,27),(1,84),(1,17),(2,27),(2,84),(2,17),(3,27),(3,84),(3,17)]

>>> select (finite 4000000000000 >< finite 123456789) 0
(0,0)
>>> select (finite 4000000000000 >< finite 123456789) 196598723084073
(1592449,82897812)
>>> card (finite 4000000000000 >< finite 123456789)
Finite 493827156000000000000

>>> :set -XTypeApplications
>>> enumerate $takeE 26 . dropE 65$ boundedEnum @Char
"ABCDEFGHIJKLMNOPQRSTUVWXYZ"

>>> take 10 . enumerate $nat >< nat [(0,0),(0,1),(1,0),(0,2),(1,1),(2,0),(0,3),(1,2),(2,1),(3,0)] >>> take 10 . enumerate$ cw
[1 % 1,1 % 2,2 % 1,1 % 3,3 % 2,2 % 3,3 % 1,1 % 4,4 % 3,3 % 5]

>>> take 15 . enumerate $listOf nat [[],[0],[0,0],[1],[0,0,0],[1,0],[2],[0,1],[1,0,0],[2,0],[3],[0,0,0,0],[1,1],[2,0,0],[3,0]] data Tree = L | B Tree Tree deriving (Eq, Show) trees :: Enumeration Tree trees = infinite$ singleton L <|> B <$> trees <*> trees >>> take 3 . enumerate$ trees
[L,B L L,B L (B L L)]
>>> select trees 87239862967296
B (B (B (B (B L L) (B (B (B L L) L) L)) (B L (B L (B L L)))) (B (B (B L (B L (B L L))) (B (B L L) (B L L))) (B (B L (B L (B L L))) L))) (B (B L (B (B (B L (B L L)) (B L L)) L)) (B (B (B L (B L L)) L) L))

treesOfDepthUpTo :: Int -> Enumeration Tree
treesOfDepthUpTo 0 = singleton L
treesOfDepthUpTo n = singleton L <|> B <$> t' <*> t' where t' = treesOfDepthUpTo (n-1) >>> card (treesOfDepthUpTo 0) Finite 1 >>> card (treesOfDepthUpTo 1) Finite 2 >>> card (treesOfDepthUpTo 3) Finite 26 >>> card (treesOfDepthUpTo 10) Finite 14378219780015246281818710879551167697596193767663736497089725524386087657390556152293078723153293423353330879856663164406809615688082297859526620035327291442156498380795040822304677 >>> select (treesOfDepthUpTo 10) (2^50) B L (B L (B L (B (B L (B (B L (B (B L L) L)) (B (B (B (B L L) (B L L)) (B L (B L L))) (B (B (B L L) L) (B (B L L) L))))) (B (B (B (B (B (B L L) L) (B (B L L) L)) (B L L)) (B (B (B (B L L) L) (B L (B L L))) (B (B (B L L) (B L L)) L))) (B (B (B (B L L) (B L L)) (B (B (B L L) L) L)) (B (B L L) (B (B (B L L) L) (B (B L L) L)))))))) Comments, questions, suggestions for additional features, etc. are all very welcome! 1. I chose the name enumeration before I realized there was already a package of that name on Hackage! So now I have to come up with another name that’s not already taken. Suggestions welcome… Posted in combinatorics, haskell, projects | Tagged , , , , | 9 Comments ## Code style and moral absolutes In my previous post about my basic setup for solving competitive programming problems with Haskell, I (somewhat provocatively) used lists to represent pairs, and wrote a partial function to process them. Commenter Yom responded with a proposed alternative that was (less) partial. I was glad for the comment, because it gave me a good opportunity to think more about why I wrote the code in the way I did, and how it fits into larger issues of good coding practices and the reasons behind them. ## Good code style as moral behavior What is good code style? You probably have some opinions about this. In fact, I’m willing to bet you might even have some very strong opinions about this; I know I do. Whether consciously or not, we tend to frame good coding practices as a moral issue. Following good coding practices makes us feel virtuous; ignoring them makes us feel guilty. I can guess that this is why Yom said “I don’t think I could bring myself to be satisfied with partial functions” [emphasis added]. And this is why we say “good code style”, not “optimal” or “rational” or “best practice” code style. Why is this? Partly, it is just human: we like to have right and wrong ways to do everything (load the dishwasher, enforce grammar “rules”, use a text editor, etc.), and we naturally create and enforce community standards via subtle and not-so-subtle social cues. In the case of coding practices, I think we also sometimes do it consciously and explicitly, because the benefits can be unintuitive or only manifest in the long term. So the only way to get our students—or ourselves—to follow practices that are in our rational self-interest is by framing them in moral terms; rational arguments do not work in and of themselves. For example, I cannot get my students to write good comments by explaining to them how it will be beneficial to them in the future. It seems obvious to them that they will remember perfectly how their code works in the future, so any argument claiming the opposite falls on deaf ears. The only way to get them to write comments is to make it a moral issue: they should feel bad (i.e. lose points, lose respect, feel like they are “taking shortcuts”) if they don’t. Of course I do this “for their own good”: I trust that in the future they will come to appreciate this ingrained behavior on its own merits. The problem is that things framed in moral terms become absolutes, and it is then difficult for us to assess them rationally. My students will never be able to write a [function without comments, partial function, goto statement, …] without feeling bad about it, and they probably won’t stop to think about why. ## Good code style as rational behavior I ask again: what is good code style—and why? I have identified a few reasons for various “good” coding practices. Ultimately, we want our code to have properties such as: • Robustness: it should handle unexpected or invalid inputs gracefully. • Readability: it should be easy for others (or us in the future) to read and understand the program. • Maintainability: it should be easy to modify the program as requirements change. • Efficiency: in general, programs should not do anything obviously redundant, or use data structures with a lot of overhead when faster ones are available (e.g. String vs Text or ByteString). Even in scenarios where one might initially think these properties are not needed (e.g. writing a one-off script for some sysadmin or data processing task), they often end up being important anyway (e.g. that one-off script gets copied and mutated until it becomes a key piece of some production system). And this is exactly one of the reasons for framing good coding style in moral terms! I won’t write comments or use good function decomposition in my one-off script just because I know, rationally, that it might end up in a production system someday. (I “know” that this particular script really is just a one-off script!) But I just might follow good coding practices anyway if I feel bad about not doing it (e.g. I would feel ashamed if other people saw it). It seems to me that most things we would typically think of as good code style are geared towards producing code with some or all of the above properties (and perhaps some other properties as well), and most scenarios in which code is being written really do benefit from these properties. ## “Good” code style is context-dependent But what if there was a scenario where these properties are actually, concretely of no benefit? As you can probably guess, I would argue that competitive programming is one such scenario: • Robustness: we do not care what our program does when given unexpected or invalid inputs, since we are absolutely, 100% guaranteed that our program will only ever be run on inputs that exactly follow the given specification. • Maintainability: the requirements for our program will never change. • Efficiency: if you haven’t done much competitive programming you might be surprised to learn that we often don’t care about efficiency either. That is, although we certainly do care about asymptotic efficiency, i.e. choosing a good algorithm, problem time limits are typically set in such a way that constant factors don’t matter very much. A program that runs within 5x-10x of the optimal speed will often fit comfortably within the time limit. So what do we care about? • Readability: the one thing from my previous list that we do care about is readability. Debugging becomes quite difficult if you can’t read and understand the code you wrote (this becomes even more important if you’re working on a team). And insofar as a solution represents particular insights or techniques, you may want be able to read it much later in order to remember or share what you learned. • Programmer time: programmer time is always valuable, of course, but with competitive programming this is taken to an extreme: it is almost always done under time pressure, so the whole point is to write a program to solve a given problem as fast as possible. The combination of optimizing for speed and not caring about things like robustness, maintainability, and efficiency leads to a number of “best practices” for competitive programming that fly in the face of typical standards. For example: • Adding code to deal gracefully with inputs that don’t follow the specification would just be a waste of time (a cardinal sin in this context!). My Haskell solutions are full of calls to partial functions like read, head, tail, fromJust, and so on, even though I would almost never use these functions in other contexts. This is also why I used a partial function that was only defined on lists of length two in my previous post (though as I argue in a comment, perhaps it’s not so much that the function is partial as that its type is too big). • I often just use String for text processing, even though something like Text or ByteString (depending on the scenario) would be faster or more robust. (The exception is problems with a large amount of I/O, when the overhead of String really does become a problem; more on this in a future post.) • Other than the simplest uses of foldr, foldl', and scanl, I don’t bother with generic recursion schemes; I tend to just write lots of explicit recursion, which I find quicker to write and easier to debug. There are similar things I do in Java as well. It has taken me quite a while to become comfortable with these things and stop feeling bad about them, and I think I finally understand why. I’m not sure I really have a main point, other than to encourage you to consider your coding practices, and why you consider certain practices to be good or bad (and whether it depends on the context!). Next time, back to your regularly scheduled competitive programming tips! Posted in haskell | Tagged , , | 4 Comments ## Competitive Programming in Haskell: Basic Setup I am the coach of my school’s competitive programming team and enjoy solving problems on Open Kattis. Since Kattis accepts submissions in a wide variety of languages (including Haskell, OCaml, Rust, Common Lisp, and even Prolog), I often enjoy submitting solutions in Haskell. Of the 946 problems I have solved on Kattis1, I used Haskell for 607 of them (I used Java for the rest, except for one in C++). After solving so many problems in Haskell, by now I’ve figured out some patterns that work well, identified some common pitfalls, developed some nice little libraries, and so forth. I thought it would be fun to write a series of blog posts sharing my experience for the benefit of others—and because I expect I will also learn things from the ensuing discussion! ## The Basics: I/O As a basic running example I’ll use the same example problem that Kattis uses in its help section, namely, A Different Problem. In this problem, we are told that the input will consist of a number of pairs of integers between $0$ and $10^{15}$, one pair per line, and we should output the absolute value of the difference between each pair. The given example is that if the input looks like this: 10 12 71293781758123 72784 1 12345677654321 then our program should produce output that looks like this: 2 71293781685339 12345677654320 Kattis problems are always set up this way, with input of a specified format provided on standard input, and output to be written to standard output. To do this in Haskell, one might think we will need to use things like getLine and putStrLn to read and write the input. But wait! There is a much better way. Haskell’s standard Prelude has a function interact :: (String -> String) -> IO () It takes a pure String -> String function, and creates an IO action which reads from standard input, feeds the input to the function, and then writes the function’s output to standard output. It uses lazy IO, so the reading and writing can be interleaved with computation of the function—a bit controversial and dangerous in general, but absolutely perfect for our use case! Every single Kattis problem I have ever solved begins with main = interact$ ...

(or the equivalent for ByteString, more on that in a future post) and that is the only bit of IO in the entire program. Yay!

## From Input to Output

So now we need to write a pure function which transforms the input into the output. Of course, in true Haskell fashion, we will do this by constructing a chained pipeline of functions to do the job incrementally. The general plan of attack (for any Kattis problem) is as follows:

1. First, parse the input, that is, transform the raw String input into some more semantically meaningful representation—typically using a combination of functions like lines, words, read, map, and so on (or more sophisticated tools—see a later post).
2. Next, solve the problem, turning a semantically meaningful representation of the input into a semantically meaningful representation of the output.
3. Finally, format the output using things like show, unwords, unlines, and so on.

Idiomatic Haskell uses the composition operator (.) to combine functions. However, when solving competitive programming problems, I much prefer to use the reverse composition operator, (>>>) from Control.Arrow (that is, (>>>) = flip (.)). The reason is that since I often end up constructing long function pipelines, I want to be able to think about the process of transforming input to output and type from left to right at the same time; having to add functions from right to left would be tedious.

## A Full Solution

So here’s my solution to A Different Problem:

eigVals = toList . fst . eig $companion toReal (a :+ b) | abs b < 1e-10 = Just a -- This arbitrary value is annoying too! | otherwise = Nothing  Posted in haskell, math | | 12 Comments ## Worstsort Thanks for the responses to my previous post about finding roots of polynomials; I now have some new avenues to explore. But today I want to write about something completely different. I recently stumbled across this fun paper by Miguel Lerna. I realized a Haskell implementation would be very elegant, and I couldn’t pass up the opportunity to share. ## Badsort This is badsort. > import Data.List (permutations, insert) > > badsort :: Ord a => Integer -> [a] -> [a] > badsort 0 = foldr insert [] > badsort k = head . badsort (k-1) . permutations  Claim: badsort k is a correct sorting algorithm for any natural number k. Before reading further I recommend staring at that until you understand what it’s doing and why it works. ## How badsort works Badsort is very bad. Here’s how it works: • badsort 0 is just plain old insertion sort. • badsort k xs creates the list of all permutations of xs, sorts them into lexicographic order, and selects the first. This works because the lexicographically smallest permutation of xs is, in fact, the one which is sorted. Oh, and of course, sorting the permutations lexicographically is done by a recursive call to badsort (k-1). (As an aside, I like how seamless this is in Haskell with polymorphic recursion—each recursive call is at a different type.) Here are a few examples to show that it works: ghci> badsort 0 [3,1,2] [1,2,3] ghci> badsort 1 [3,1,2] -- generates 6 permutations [1,2,3] ghci> badsort 2 [3,1,2] -- generates 720 permutations of 6 permutations [1,2,3]  badsort 3 [3,1,2], if we tried it (not recommended!!), would generate all possible permutations of the list of 720 permutations of the list of 6 permutations of [3,1,2]. The number of such permutations is, of course, $720!$, which has $1747$ decimal digits; there is literally not enough space in the universe to store all those permutations. In general, badsort k is a correct sorting algorithm1 which takes time $\Omega((n!^{(k)})^2)$, where $n!^{(k)}$ denotes the $k$-fold iterated factorial of $n$, that is, $n!^{(0)} = n$ and $n!^{(k+1)} = (n!^{(k)})!$. (This doesn’t even take into account the time for accessing memory; at this scale we certainly can’t assume memory access takes constant time. Fetching memory from a data center in another galaxy takes a while, you know? =) ## It gets worse > worstsort :: Ord a => (Integer -> Integer) -> [a] -> [a] > worstsort f xs = badsort (f n) xs > where > n = fromIntegral$ length xs


Worstsort is parameterized by a function on natural numbers, and calls badsort with a recursion depth given by the function $f$ applied to the length of the list. Oh my.

Just for fun, let’s try, oh, say, the Ackermann function.

> ack :: Integer -> Integer -> Integer
> ack 0 n = n+1
> ack m 0 = ack (m-1) 1
> ack m n = ack (m-1) (ack m (n-1))
>
> diabolicalsort :: Ord a => [a] -> [a]
> diabolicalsort = worstsort (\n -> ack n n)


Here are some fun properties of diabolicalsort (and any other instantiation of worstsort):

• It will provably terminate in a finite amount of time for any input! Although probably the words “terminate” and “finite” should be in scare quotes.

• In some sense I can’t quite define formally but still believe in my heart, it “doesn’t cheat” in the sense that it is always “making real progress” towards sorting the input list. If you are trying to design a slow sorting algorithm, it would be cheating, for example, to make an algorithm that spins in a useless loop for a thousand years and then does insertion sort.

• It works in practice on lists of length 1 or 2, but length 3 is completely hopeless. ack 3 3 = 61, so we are looking at the 61-fold iterated factorial of 3, which is a… rather large number.

• ack 4 4 is $2^{2^{2^{65536}}} - 3$; there are not enough atoms in the universe to even write down this number in base 10. And then of course we take that number and iterate factorial that many times on $4$. Sheesh.

• Let us not even speak of lists of length 5.

The upshot of this, in the end, is that it is possible to make a “non-cheating” sorting algorithm whose running time grows faster than any computable function you care to choose (proof: take your chosen computable function and substitute it for f).

1. It might be a fun exercise to prove this formally using a proof assistant.

Posted in haskell, humor, math | Tagged , , , , , | 8 Comments