I am trying to make use of some of GADT parameter from runtime, assuming that I have used DataKinds extension to allow promoting data to types. i.e. having
data Num = Zero | Succ Num
data Something (len :: Num) where
Some :: Something len
I would like to have function
toNum :: Something len -> Num
That for any Some :: Something n will return n:
toNum (s :: Something n) = n
Which is invalid in Haskell. Is it possible to do so?
In Haskell this is impossible, since types are erased at runtime. That is, when the program will run, there is no information in memory about the value of the index let in the type.
To overcome this issue, we need to force Haskell to keep in memory that value, at runtime. This is usually done using a singleton auxiliary type:
data Num = Zero | Succ Num
data SNum (n :: Num) where
SZero :: SNum 'Zero
SSucc :: SNum n -> SNum ('Succ n)
data Something (len :: Num) where
Some :: SNum len -> Something len
Using this you can easily write
sToNum :: SNum n -> Num
sToNum SZero = Zero
sToNum (SSucc n) = Succ (sToNum n)
and then
toNum :: Something len -> Num
toNum (Some n) = sToNum n
If you look up "haskell singletons" you should find several examples. There's even a singletons library to partially automatize this.
If / when "dependent Haskell" will be released, we will have less cumbersome tools at our disposal. Currently, singletons work, but they are a hassle sometimes. Still, for the moment, we have to use them.
Related
I'm defining a type GosperInteger, representing the Eisenstein integers in a complex base, and I'd like to enter these numbers in the REPL and do operations on them. So I put the type in the Read and Show typeclasses. Here's the code (there's also an Internals module, see https://github.com/phma/gosperbase to run it):
module Data.GosperBase where
import Data.Array.Unboxed
import Data.Word
import Data.GosperBase.Internals
import qualified Data.Sequence as Seq
import Data.Sequence ((><), (<|), (|>), Seq((:<|)), Seq((:|>)))
import Data.Char
import Data.List
import Data.Maybe
{- This computes complex numbers in base 2.5-√(-3/4), called the Gosper base
because it is the scale factor from one Gosper island to the next bigger one.
The digits are cyclotomic:
2 3
6 0 1
4 5
For layout of all numbers up to 3 digits, see doc/GosperBase.ps .
-}
newtype GosperInteger = GosperInteger (Seq.Seq Word)
chunkDigitsInt :: Seq.Seq Char -> Maybe (Seq.Seq (Seq.Seq Char))
-- ^If the string ends in 'G', reverses the rest of the characters
-- and groups them into chunks of digitsPerLimb.
chunkDigitsInt (as:|>'G') = Just (Seq.reverse (Seq.chunksOf (fromIntegral digitsPerLimb) (Seq.reverse as)))
chunkDigitsInt as = Nothing
parseChunkRjust :: Seq.Seq Char -> Maybe Word
parseChunkRjust Seq.Empty = Just 0
parseChunkRjust (n:<|ns) =
let ms = parseChunkRjust ns
in case ms of
Just num -> if (n >= '0' && n < '7')
then Just (7 * num + fromIntegral (ord n - ord '0'))
else Nothing
Nothing -> Nothing
showLimb :: Word -> Word -> String
showLimb _ 0 = ""
showLimb val ndig = chr (fromIntegral ((val `div` 7 ^ (ndig-1)) `mod` 7) + ord '0') : (showLimb val (ndig-1))
parseRjust :: Seq.Seq Char -> Maybe (Seq.Seq Word)
parseRjust as =
let ns = chunkDigitsInt as
in case ns of
Just chunks -> traverse parseChunkRjust chunks
Nothing -> Nothing
showRjust' :: Seq.Seq Word -> String
showRjust' Seq.Empty = ""
showRjust' (a:<|as) = (showLimb a digitsPerLimb) ++ (showRjust' as)
showRjust :: Seq.Seq Word -> String
showRjust Seq.Empty = "0"
showRjust (a:<|as) = (showLimb a (snd (msdPosLimb a))) ++ (showRjust' as)
parse1InitTail :: (String, String) -> Maybe (GosperInteger, String)
parse1InitTail (a,b) =
let aParse = parseRjust (Seq.fromList a)
in case aParse of
Just mant -> Just (GosperInteger mant,b)
Nothing -> Nothing
parseGosperInteger :: String -> [(GosperInteger, String)]
parseGosperInteger str =
let its = zip (inits str) (tails str) -- TODO stop on invalid char
in catMaybes (fmap parse1InitTail its)
instance Read GosperInteger where
readsPrec _ str = parseGosperInteger str
instance Show GosperInteger where
show (GosperInteger m) = showRjust m ++ "G"
iAdd :: GosperInteger -> GosperInteger -> GosperInteger
iAdd (GosperInteger a) (GosperInteger b) =
GosperInteger (stripLeading0 (addRjust a b))
iMult :: GosperInteger -> GosperInteger -> GosperInteger
iMult (GosperInteger a) (GosperInteger b) =
GosperInteger (stripLeading0 (mulMant a b))
I'd like to do
> 425G * 256301G
16061525G
which requires putting GosperInteger in the Num typeclass, which I haven't done yet.
Showing a number works, and calling read on a string works, but reading a number typed into the REPL does not. Why?
> read "45G" :: GosperInteger
45G
> 45G
<interactive>:2:3: error: Data constructor not in scope: G
It is not possible to do that in a proper way (you can probably bodge this by writing an odd Num instance).
I think a better approach would be to just write that num instance, then you can write:
ghci> 425 * 256301 :: GosperInteger
16061525
If you don't want to have to write that :: GosperInteger signature you can do a few things:
Use ghci> default (GosperInteger, Double) that will mean it will automatically pick your GosperInteger type if there is ambiguity. You can also use this in normal source files.
Define a function g :: GosperInteger -> GosperInteger; g = id which you can use to disambiguate manually with less syntactic overhead:
ghci> g (425 * 256301)
16061525
The GHCi repl doesn't simply call read on the text that you type in. Instead, it has a much more complicated parser that separates your text into various tokens. One type of token is numeric: any integral number you type in will get "read" as an Integer. Of course, if you type 32 and want it to be an Int, not an Integer, this would be a problem, so the Num type class has a super convenient fromInteger function. With this, an Integer token can be converted into any instance of the Num class.
But, you want something slightly different: you want the parser to group together the numeric token along with the G token and treat them as one unit. For full support, you'd need to make an extension to the GHC parser, much like how if you type 2e7 into the prompt, you correctly get a floating point number. This isn't a simple change you can address in your source file or GHCi settings.
With all that said, there are some hacks we can play with. As Noughtmare mentions, "you can probably bodge this by writing an odd Num instance", and indeed you can! Fair warning: you probably don't want to do this, but let's explore it anyway.
The problem is that the parser returned two tokens, one that's numeric and the other that's G. Since it's uppercase, that G token is being interpreted as a data constructor (your error message pointed that out too: " Data constructor not in scope: G"). The key is to use this to our advantage.
Consider the following:
data G = G
deriving Show
instance Num (G -> GosperInteger) where
fromInteger i G = integerToGosperInteger i
Now, assuming you wrote that function integerToGosperInteger, this instance would let you type, e.g., 45G and produce a GosperInteger 45G. Hurrah! You can even do 425G * 256301G and it will work as expected. Furthermore, if you cleverly omit a fromInteger definition from your Num GosperInteger class, then you'll get a runtime error if you try to simply use a number like 425 as as GosperInteger (that is, you'll get an error for implicit coercions that don't have the G).
There are some problems.
If you try this, you'll find that type inference is pretty terrible. It probably won't work right at the prompt unless you set default (GosperInteger, Double), and you'll probably want to use lots of type annotations in your source files.
If you leave out the G, you'll get terrible type error messages or, even worse, runtime errors.
You'll get a warning that your Num instance for G -> GosperInteger is incomplete. It is incomplete, but there's no sensible definitions for anything else. You could suppress the warning or set all of the missing methods to error "This isn't how this is supposed to be used" or something, but it's still a bit of a blemish in the code.
But, if you can deal with the problems and you squint hard enough, it sorta kinda gets you what you want.
I am studying Haskell currently and try to understand a project that uses Haskell to implement cryptographic algorithms. After reading Learn You a Haskell for Great Good online, I begin to understand the code in that project. Then I found I am stuck at the following code with the "#" symbol:
-- | Generate an #n#-dimensional secret key over #rq#.
genKey :: forall rq rnd n . (MonadRandom rnd, Random rq, Reflects n Int)
=> rnd (PRFKey n rq)
genKey = fmap Key $ randomMtx 1 $ value #n
Here the randomMtx is defined as follows:
-- | A random matrix having a given number of rows and columns.
randomMtx :: (MonadRandom rnd, Random a) => Int -> Int -> rnd (Matrix a)
randomMtx r c = M.fromList r c <$> replicateM (r*c) getRandom
And PRFKey is defined below:
-- | A PRF secret key of dimension #n# over ring #a#.
newtype PRFKey n a = Key { key :: Matrix a }
All information sources I can find say that # is the as-pattern, but this piece of code is apparently not that case. I have checked the online tutorial, blogs and even the Haskell 2010 language report at https://www.haskell.org/definition/haskell2010.pdf. There is simply no answer to this question.
More code snippets can be found in this project using # in this way too:
-- | Generate public parameters (\( \mathbf{A}_0 \) and \(
-- \mathbf{A}_1 \)) for #n#-dimensional secret keys over a ring #rq#
-- for gadget indicated by #gad#.
genParams :: forall gad rq rnd n .
(MonadRandom rnd, Random rq, Reflects n Int, Gadget gad rq)
=> rnd (PRFParams n gad rq)
genParams = let len = length $ gadget #gad #rq
n = value #n
in Params <$> (randomMtx n (n*len)) <*> (randomMtx n (n*len))
I deeply appreciate any help on this.
That #n is an advanced feature of modern Haskell, which is usually not covered by tutorials like LYAH, nor can be found the the Report.
It's called a type application and is a GHC language extension. To understand it, consider this simple polymorphic function
dup :: forall a . a -> (a, a)
dup x = (x, x)
Intuitively calling dup works as follows:
the caller chooses a type a
the caller chooses a value x of the previously chosen type a
dup then answers with a value of type (a,a)
In a sense, dup takes two arguments: the type a and the value x :: a. However, GHC is usually able to infer the type a (e.g. from x, or from the context where we are using dup), so we usually pass only one argument to dup, namely x. For instance, we have
dup True :: (Bool, Bool)
dup "hello" :: (String, String)
...
Now, what if we want to pass a explicitly? Well, in that case we can turn on the TypeApplications extension, and write
dup #Bool True :: (Bool, Bool)
dup #String "hello" :: (String, String)
...
Note the #... arguments carrying types (not values). Those are something that exists at compile time, only -- at runtime the argument does not exist.
Why do we want that? Well, sometimes there is no x around, and we want to prod the compiler to choose the right a. E.g.
dup #Bool :: Bool -> (Bool, Bool)
dup #String :: String -> (String, String)
...
Type applications are often useful in combination with some other extensions which make type inference unfeasible for GHC, like ambiguous types or type families. I won't discuss those, but you can simply understand that sometimes you really need to help the compiler, especially when using powerful type-level features.
Now, about your specific case. I don't have all the details, I don't know the library, but it's very likely that your n represents a kind of natural-number value at the type level. Here we are diving in rather advanced extensions, like the above-mentioned ones plus DataKinds, maybe GADTs, and some typeclass machinery. While I can't explain everything, hopefully I can provide some basic insight. Intuitively,
foo :: forall n . some type using n
takes as argument #n, a kind-of compile-time natural, which is not passed at runtime. Instead,
foo :: forall n . C n => some type using n
takes #n (compile-time), together with a proof that n satisfies constraint C n. The latter is a run-time argument, which might expose the actual value of n. Indeed, in your case, I guess you have something vaguely resembling
value :: forall n . Reflects n Int => Int
which essentially allows the code to bring the type-level natural to the term-level, essentially accessing the "type" as a "value". (The above type is considered an "ambiguous" one, by the way -- you really need #n to disambiguate.)
Finally: why should one want to pass n at the type level if we then later on convert that to the term level? Wouldn't be easier to simply write out functions like
foo :: Int -> ...
foo n ... = ... use n
instead of the more cumbersome
foo :: forall n . Reflects n Int => ...
foo ... = ... use (value #n)
The honest answer is: yes, it would be easier. However, having n at the type level allows the compiler to perform more static checks. For instance, you might want a type to represent "integers modulo n", and allow adding those. Having
data Mod = Mod Int -- Int modulo some n
foo :: Int -> Mod -> Mod -> Mod
foo n (Mod x) (Mod y) = Mod ((x+y) `mod` n)
works, but there is no check that x and y are of the same modulus. We might add apples and oranges, if we are not careful. We could instead write
data Mod n = Mod Int -- Int modulo n
foo :: Int -> Mod n -> Mod n -> Mod n
foo n (Mod x) (Mod y) = Mod ((x+y) `mod` n)
which is better, but still allows to call foo 5 x y even when n is not 5. Not good. Instead,
data Mod n = Mod Int -- Int modulo n
-- a lot of type machinery omitted here
foo :: forall n . SomeConstraint n => Mod n -> Mod n -> Mod n
foo (Mod x) (Mod y) = Mod ((x+y) `mod` (value #n))
prevents things to go wrong. The compiler statically checks everything. The code is harder to use, yes, but in a sense making it harder to use is the whole point: we want to make it impossible for the user to try adding something of the wrong modulus.
Concluding: these are very advanced extensions. If you're a beginner, you will need to slowly progress towards these techniques. Don't be discouraged if you can't grasp them after only a short study, it does take some time. Make a small step at a time, solve some exercises for each feature to understand the point of it. And you'll always have StackOverflow when you are stuck :-)
I am trying to represent expressions with type families, but I cannot seem to figure out how to write the constraints that I want, and I'm starting to feel like it's just not possible. Here is my code:
class Evaluable c where
type Return c :: *
evaluate :: c -> Return c
data Negate n = Negate n
instance (Evaluable n, Return n ~ Int) => Evaluable (Negate n) where
type Return (Negate n) = Return n
evaluate (Negate n) = negate (evaluate n)
This all compiles fine, but it doesn't express exactly what I want. In the constraints of the Negate instance of Evaluable, I say that the return type of the expression inside Negate must be an Int (with Return n ~ Int) so that I can call negate on it, but that is too restrictive. The return type actually only needs to be an instance of the Num type class which has the negate function. That way Doubles, Integers, or any other instance of Num could also be negated and not just Ints. But I can't just write
Return n ~ Num
instead because Num is a type class and Return n is a type. I also cannot put
Num (Return n)
instead because Return n is a type not a type variable.
Is what I'm trying to do even possible with Haskell? If not, should it be, or am I misunderstanding some theory behind it? I feel like Java could add a constraint like this. Let me know if this question could be clearer.
Edit: Thanks guys, the responses are helping and are getting at what I suspected. It appears that the type checker isn't able to handle what I'd like to do without UndecidableInstances, so my question is, is what I'd like to express really undecidable? It is to the Haskell compiler, but is it in general? i.e. could a constraint even exist that means "check that Return n is an instance of Num" which is decidable to a more advanced type checker?
Actually, you can do exactly what you mentioned:
{-# LANGUAGE TypeFamilies, FlexibleContexts, UndecidableInstances #-}
class Evaluable c where
type Return c :: *
evaluate :: c -> Return c
data Negate n = Negate n
instance (Evaluable n, Num (Return n)) => Evaluable (Negate n) where
type Return (Negate n) = Return n
evaluate (Negate n) = negate (evaluate n)
Return n certainly is a type, which can be an instance of a class just like Int can. Your confusion might be about what can be the argument of a constraint. The answer is "anything with the correct kind". The kind of Int is *, as is the kind of Return n. Num has kind * -> Constraint, so anything of kind * can be its argument. It perfectly legal (though vacuous) to write Num Int as a constraint, in the same way that Num (a :: *) is legal.
To complement Eric's answer, let me suggest one possible alternative: using a functional dependency instead of a type family:
class EvaluableFD r c | c -> r where
evaluate :: c -> r
data Negate n = Negate n
instance (EvaluableFD r n, Num r) => EvaluableFD r (Negate n) where
evaluate (Negate n) = negate (evaluate n)
This makes it a bit easier to talk about the result type, I think. For instance, you can write
foo :: EvaluableFD Int a => Negate a -> Int
foo x = evaluate x + 12
You can also use ConstraintKinds to apply this partially (which is why I put the arguments in that funny-looking order):
type GivesInt = EvaluableFD Int
You could do this with your class as well, but it would be more annoying:
type GivesInt x = (Evaluable x, Result x ~ Int)
In some languages (#racket/typed, for example), the programmer can specify a union type without discriminating against it, for instance, the type (U Integer String) captures integers and strings, without tagging them (I Integer) (S String) in a data IntOrStringUnion = ... form or anything like that.
Is there a way to do the same in Haskell?
Either is what you're looking for... ish.
In Haskell terms, I'd describe what you're looking for as an anonymous sum type. By anonymous, I mean that it doesn't have a defined name (like something with a data declaration). By sum type, I mean a data type that can have one of several (distinguishable) types; a tagged union or such. (If you're not familiar with this terminology, try Wikipedia for starters.)
We have a well-known idiomatic anonymous product type, which is just a tuple. If you want to have both an Int and a String, you just smush them together with a comma: (Int, String). And tuples (seemingly) can go on forever--(Int, String, Double, Word), and you can pattern-match the same way. (There's a limit, but never mind.)
The well-known idiomatic anonymous sum type is Either, from Data.Either (and the Prelude):
data Either a b = Left a | Right b
deriving (Eq, Ord, Read, Show, Typeable)
It has some shortcomings, most prominently a Functor instance that favors Right in a way that's odd in this context. The problem is that chaining it introduces a lot of awkwardness: the type ends up like Either (Int (Either String (Either Double Word))). Pattern matching is even more awkward, as others have noted.
I just want to note that we can get closer to (what I understand to be) the Racket use case. From my extremely brief Googling, it looks like in Racket you can use functions like isNumber? to determine what type is actually in a given value of a union type. In Haskell, we usually do that with case analysis (pattern matching), but that's awkward with Either, and function using simple pattern-matching will likely end up hard-wired to a particular union type. We can do better.
IsNumber?
I'm going to write a function I think is an idiomatic Haskell stand-in for isNumber?. First, we don't like doing Boolean tests and then running functions that assume their result; instead, we tend to just convert to Maybe and go from there. So the function's type will end with -> Maybe Int. (Using Int as a stand-in for now.)
But what's on the left hand of the arrow? "Something that might be an Int -- or a String, or whatever other types we put in the union." Uh, okay. So it's going to be one of a number of types. That sounds like typeclass, so we'll put a constraint and a type variable on the left hand of the arrow: MightBeInt a => a -> Maybe Int. Okay, let's write out the class:
class MightBeInt a where
isInt :: a -> Maybe Int
fromInt :: Int -> a
Okay, now how do we write the instances? Well, we know if the first parameter to Either is Int, we're golden, so let's write that out. (Incidentally, if you want a nice exercise, only look at the instance ... where parts of these next three code blocks, and try to implement that class members yourself.)
instance MightBeInt (Either Int b) where
isInt (Left i) = Just i
isInt _ = Nothing
fromInt = Left
Fine. And ditto if Int is the second parameter:
instance MightBeInt (Either a Int) where
isInt (Right i) = Just i
isInt _ = Nothing
fromInt = Right
But what about Either String (Either Bool Int)? The trick is to recurse on the right hand type: if it's not Int, is it an instance of MightBeInt itself?
instance MightBeInt b => MightBeInt (Either a b) where
isInt (Right xs) = isInt xs
isInt _ = Nothing
fromInt = Right . fromInt
(Note that these all require FlexibleInstances and OverlappingInstances.) It took me a long time to get a feel for writing and reading these class instances; don't worry if this instance is surprising. The punch line is that we can now do this:
anInt1 :: Either Int String
anInt1 = fromInt 1
anInt2 :: Either String (Either Int Double)
anInt2 = fromInt 2
anInt3 :: Either String Int
anInt3 = fromInt 3
notAnInt :: Either String Int
notAnInt = Left "notint"
ghci> isInt anInt3
Just 3
ghci> isInt notAnInt
Nothing
Great!
Generalizing
Okay, but now do we need to write another type class for each type we want to look up? Nope! We can parameterize the class by the type we want to look up! It's a pretty mechanical translation; the only question is how to tell the compiler what type we're looking for, and that's where Proxy comes to the rescue. (If you don't want to install tagged or run base 4.7, just define data Proxy a = Proxy. It's nothing special, but you'll need PolyKinds.)
class MightBeA t a where
isA :: proxy t -> a -> Maybe t
fromA :: t -> a
instance MightBeA t t where
isA _ = Just
fromA = id
instance MightBeA t (Either t b) where
isA _ (Left i) = Just i
isA _ _ = Nothing
fromA = Left
instance MightBeA t b => MightBeA t (Either a b) where
isA p (Right xs) = isA p xs
isA _ _ = Nothing
fromA = Right . fromA
ghci> isA (Proxy :: Proxy Int) anInt3
Just 3
ghci> isA (Proxy :: Proxy String) notAnInt
Just "notint"
Now the usability situation is... better. The main thing we've lost, by the way, is the exhaustiveness checker.
Notational Parity With (U String Int Double)
For fun, in GHC 7.8 we can use DataKinds and TypeFamilies to eliminate the infix type constructors in favor of type-level lists. (In Haskell, you can't have one type constructor--like IO or Either--take a variable number of parameters, but a type-level list is just one parameter.) It's just a few lines, which I'm not really going to explain:
type family OneOf (as :: [*]) :: * where
OneOf '[] = Void
OneOf '[a] = a
OneOf (a ': as) = Either a (OneOf as)
Note that you'll need to import Data.Void. Now we can do this:
anInt4 :: OneOf '[Int, Double, Float, String]
anInt4 = fromInt 4
ghci> :kind! OneOf '[Int, Double, Float, String]
OneOf '[Int, Double, Float, String] :: *
= Either Int (Either Double (Either Float [Char]))
In other words, OneOf '[Int, Double, Float, String] is the same as Either Int (Either Double (Either Float [Char])).
You need some kind of tagging because you need to be able to check if a value is actually an Integer or a String to use it for anything. One way to alleviate having to create a custom ADT for every combination is to use a type such as
{-# LANGUAGE TypeOperators #-}
data a :+: b = L a | R b
infixr 6 :+:
returnsIntOrString :: Integer -> Integer :+: String
returnsIntOrString i
| i `rem` 2 == 0 = R "Even"
| otherwise = L (i * 2)
returnsOneOfThree :: Integer -> Integer :+: String :+: Bool
returnsOneOfThree i
| i `rem` 2 == 0 = (R . L) "Even"
| i `rem` 3 == 0 = (R . R) False
| otherwise = L (i * 2)
I've defined the following GADT:
data Vector v where
Zero :: Num a => Vector a
Scalar :: Num a => a -> Vector a
Vector :: Num a => [a] -> Vector [a]
TVector :: Num a => [a] -> Vector [a]
If it's not obvious, I'm trying to implement a simple vector space. All vector spaces need vector addition, so I want to implement this by making Vector and instance of Num. In a vector space, it doesn't make sense to add vectors of different lengths, and this is something I would like to enforce. One way I thought to do it would be using guards:
instance Num (Vector v) where
(Vector a) + (Vector b) | length a == length b =
Vector $ zipWith (+) a b
| otherwise =
error "Only add vectors with the same length."
There is nothing really wrong with this approach, but I feel like there has to be a way to do this with pattern matching. Perhaps one way to do it would be to define a new data type VectorLength, which would look something like this:
data Length l where
AnyLength :: Nat a => Length a
FixedLength :: Nat a -> Length a
Then, a length component could be added to the Vector data type, something like this:
data Vector (Length l) v where
Zero :: Num a => Vector AnyLength a
-- ...
Vector :: Num a => [a] -> Vector (length [a]) [a]
I know this isn't correct syntax, but this is the general idea I'm playing with. Finally, you could define addition to be
instance Num (Vector v) where
(Vector l a) + (Vector l b) = Vector $ zipWith (+) a b
Is such a thing possible, or is there any other way to use pattern matching for this purpose?
What you're looking for is something (in this instance confusingly) named a Vector as well. Generally, these are used in dependently typed languages where you'd write something like
data Vec (n :: Natural) a where
Nil :: Vec 0 a
Cons :: a -> Vec n a -> Vec (n + 1) a
But that's far from valid Haskell (or really any language). Some very recent extensions to GHC are beginning to enable this kind of expression but they're not there yet.
You might be interested in fixed-vector which does a best approximation of a fixed Vector available in relatively stable GHC. It uses a number of tricks between type families and continuations to create classes of fixed-size vectors.
Just to add to the example in the other answer - this nearly works already in GHC 7.6:
{-# LANGUAGE DataKinds, GADTs, KindSignatures, TypeOperators #-}
import GHC.TypeLits
data Vector (n :: Nat) a where
Nil :: Vector 0 a
Cons :: a -> Vector n a -> Vector (n + 1) a
That code compiles fine, it just doesn't work quite the way you'd hope. Let's check it out in ghci:
*Main> :t Nil
Nil :: Vector 0 a
Good so far...
*Main> :t Cons "foo" Nil
Cons "foo" Nil :: Vector (0 + 1) [Char]
Well, that's a little odd... Why does it say (0 + 1) instead of 1?
*Main> :t Cons "foo" Nil :: Vector 1 String
<interactive>:1:1:
Couldn't match type `0 + 1' with `1'
Expected type: Vector 1 String
Actual type: Vector (0 + 1) String
In the return type of a call of `Cons'
In the expression: Cons "foo" Nil :: Vector 1 String
Uh. Oops. That'd be why it says (0 + 1) instead of 1. It doesn't know that those are the same. This will be fixed (at least this case will) in GHC 7.8, which is due out... In a couple months, I think?