One of the most common gripes people have when learning Haskell is the fact that typeclass “laws” are only laws by convention, and aren’t enforced by the language and compiler. When asked why, the typical response is “Haskell can’t do that”, followed by a well-intentioned redirection to quickcheck or some other fuzzing library.
But, to any experienced Haskeller, “Haskell’s type system can’t express X” is always interpreted as a (personal) challenge.
GHC Haskell’s type system has been advanced enough to provide verified typeclasses for a long time, since the introduction of data kinds and associated types. And with the singletons library, it’s now as easy as ever.
(The code for this post is available here if you want to follow along! Some of the examples here involving
Demote and relying on its injectivity will only work with singletons HEAD, even though the necessary patches were made seven months ago1)
Let’s start simple – everyone’s favorite structural addition to magmas, semigroups. A semigroup is a type with an associative binary operation,
class Semigroup a where (<>) :: a -> a -> a
Its one law is associativity:
(x <> y) <> z = x <> (y <> z)
But, this class stinks, because it’s super easy to write bad instances:
data List a = Nil | Cons a (List a) deriving Show infixr 5 `Cons` instance Semigroup (List a) where Nil <> ys = ys Cons x xs <> ys = Cons x (ys <> xs)
This instance isn’t associative:
ghci> ((1 `Cons` 2 `Cons` Nil) <> (3 `Cons` 4 `Cons` Nil)) <> (5 `Cons` 6 `Cons` Nil) 1 `Cons` 5 `Cons` 3 `Cons` 6 `Cons` 2 `Cons` 4 `Cons` Nil ghci> (1 `Cons` 2 `Cons` Nil) <> ((3 `Cons` 4 `Cons` Nil) <> (5 `Cons` 6 `Cons` Nil)) 1 `Cons` 3 `Cons` 2 `Cons` 5 `Cons` 4 `Cons` 6 `Cons` Nil
But if you try to compile it, GHC doesn’t complain at all. Is this an error on the part of Haskell? Not quite; it’s an error on the part of the
Semigroup typeclass not requiring proofs that the instance is indeed associative.
Let’s try again.
Verify me, Captain
We will now define
Semigroup on the kind
-XDataKinds, instead of the type.
class Semigroup a where type (x :: a) <> (y :: a) :: a (%<>) :: Sing (x :: a) -> Sing (y :: a) -> Sing (x <> y) appendAssoc :: Sing (x :: a) -> Sing (y :: a) -> Sing (z :: a) -> ((x <> y) <> z) :~: (x <> (y <> z))
<> exists not as a function on values, but as a function on types.
%<> is a function that performs
<> at the value level, written to work with singletons representing the input types, so that GHC can verify that it is identical to the type family
<>. (it’s 100% boilerplate and should pretty much exactly match the
<> type family).2 Finally,
appendAssoc is a proof that the type family
<> is associative, using
:~: (type equality witness) from
This means that, if a type is an instance of
Semigroup, it not only has to provide
%<>, but also a proof that they are associative. You can’t write the full instance without it!
Semigroup is a “kind-class”, because it is a bunch of methods and types associated with a certain kind. Which
<> is dispatched when you do something like
x <> y depends on the kind of
y. GHC does “kind inference” and uses the
<> corresponding to the kinds of
SingKind typeclass from the singletons library, we can move back and forth from
Sing x and
x, and get our original (value-level)
(<>) :: (SingKind m, Semigroup m) => Demote m -> Demote m -> Demote m x <> y = withSomeSing x $ \sX -> withSomeSing y $ \sY -> fromSing (sX %<> sY)
(This works best with singletons HEAD at the moment, because
Demote is injective. On 2.2 or lower, using this would require an explicit type application or annotation at any place you use
Now, let’s write the instance for
List. First, we need to define the singletons:
data instance Sing (xs :: List a) where SNil :: Sing Nil SCons :: Sing x -> Sing xs -> Sing (Cons x xs)
Then, we can define the instance, using the traditional
(++) appending that lists famously have:
instance Semigroup (List a) where type Nil <> ys = ys type Cons x xs <> ys = Cons x (xs <> ys) SNil %<> ys = ys SCons x xs %<> ys = SCons x (xs %<> ys) appendAssoc = \case SNil -> \_ _ -> Refl SCons x xs -> \ys zs -> case appendAssoc xs ys zs of Refl -> Refl
Like I promised,
%<> is a boilerplate re-implementation of
<>, to manipulate value-level witnesses.
appendAssoc is the interesting bit: It’s our proof. It reads like this:
If the first list is
-- left hand side (Nil <> ys) <> zs = ys <> zs -- definition of `(Nil <>)` -- right hand side Nil <> (ys <> zs) = ys <> zs -- definition of `(Nil <>)`
So, no work needed. QED! (Or, as we say in Haskell,
If the first list is
Cons x xs:
-- left hand side (Cons x xs <> ys) <> zs = (Cons x (xs <> ys)) <> zs -- definition of `(Cons x xs <>)` = Cons x ((xs <> ys) <> zs) -- definition of `(Cons x xs <>)` -- right hand side Cons x xs <> (ys <> zs) = Cons x (xs <> (ys <> zs)) -- definition of `(Cons x xs <>)`
So, the problem reduces to proving that
(xs <> ys) <> zsis equal to
xs <> (ys <> zs). If we can do that, then we can prove that the whole things are equal. We generate that proof using
appendAssoc xs ys zs, and, wit that proof in scope…QED!
And, we’re done!
Note that if you had tried any non-associative implementation of
%<>), GHC would reject it because you wouldn’t have been able to write the proof!
SingKind and both versions of
<> is kind of tedious, so it’s useful to use template haskell to do it all for us:
$(singletons [d| data List a = Nil | Cons a (List a) deriving (Show) infixr 5 `Cons` appendList :: List a -> List a -> List a appendList Nil ys = ys appendList (Cons x xs) ys = Cons x (appendList xs ys) |]) instance Semigroup (List a) where type xs <> ys = AppendList xs ys (%<>) = sAppendList appendAssoc = \case SNil -> \_ _ -> Refl SCons _ xs -> \ys zs -> case appendAssoc xs ys zs of Refl -> Refl
The boilerplate of re-defining
%<> goes away!
And now, we we can do:
ghci> print $ ((1::Integer) `Cons` 2 `Cons` Nil) <> (3 `Cons` 4 `Cons` Nil) 1 `Cons` 2 `Cons` 3 `Cons` 4 `Cons` Nil
Now that we have our basic infrastructure, let’s implement some other famous semigroups:
First, the inductive nats,
data N = Z | S N:
$(singletons [d| data N = Z | S N deriving (Show) plus :: N -> N -> N plus Z y = y plus (S x) y = S (plus x y) |]) instance Semigroup N where type xs <> ys = Plus xs ys (%<>) = sPlus appendAssoc = \case SZ -> \_ _ -> Refl SS x -> \y z -> case appendAssoc x y z of Refl -> Refl
And the standard instance for
Maybe, which lifts the underlying semigroup:
$(singletons [d| data Option a = None | Some a deriving (Show) |]) instance Semigroup a => Semigroup (Option a) where type None <> y = y type x <> None = x type Some x <> Some y = Some (x <> y) SNone %<> y = y x %<> SNone = x SSome x %<> SSome y = SSome (x %<> y) appendAssoc = \case SNone -> \_ _ -> Refl SSome x -> \case SNone -> \_ -> Refl SSome y -> \case SNone -> Refl SSome z -> case appendAssoc x y z of Refl -> Refl
ghci> print $ S (S Z) <> S Z S (S (S Z)) ghci> print $ Some (S Z) <> Some (S (S (S Z))) Some (S (S (S (S Z)))) ghci> print $ None <> Some (S (S (S Z))) Some (S (S (S Z)))
Of course, we can now introduce the
Monoid typeclass, which introduces a new element
empty, along with the laws that appending with empty leaves things unchanged:
class Semigroup a => Monoid a where type Empty a :: a sEmpty :: Sing (Empty a) emptyIdentLeft :: Sing x -> (Empty a <> x) :~: x emptyIdentRight :: Sing x -> (x <> Empty a) :~: x empty :: (SingKind m, Monoid m) => Demote m empty = fromSing sEmpty
Because working implicitly return-type polymorphism at the type level can be annoying sometimes, we have
Empty take the kind
a as a parameter, instead of having it be inferred through kind inference like we did for
<>. That is,
Empty (List a) is
Empty for the kind
As usual in Haskell, the instances write themselves!
instance Monoid (List a) where type Empty (List a) = Nil sEmpty = SNil emptyIdentLeft _ = Refl emptyIdentRight = \case SNil -> Refl SCons _ xs -> case emptyIdentRight xs of Refl -> Refl instance Monoid N where type Empty N = Z sEmpty = SZ emptyIdentLeft _ = Refl emptyIdentRight = \case SZ -> Refl SS x -> case emptyIdentRight x of Refl -> Refl instance Semigroup a => Monoid (Option a) where type Empty (Option a) = None sEmpty = SNone emptyIdentLeft _ = Refl emptyIdentRight _ = Refl
Play that Funcy Music
How about some higher-kinded typeclasses?
class Functor f where type Fmap a b (g :: a ~> b) (x :: f a) :: f b sFmap :: Sing (g :: a ~> b) -> Sing (x :: f a ) -> Sing (Fmap a b g x :: f b ) -- | fmap id x == x fmapId :: Sing (x :: f a) -> Fmap a a IdSym0 x :~: x -- | fmap f (fmap g x) = fmap (f . g) x fmapCompose :: Sing (g :: b ~> c) -> Sing (h :: a ~> b) -> Sing (x :: f a ) -> Fmap b c g (Fmap a b h x) :~: Fmap a c (((:.$) @@ g) @@ h) x
Fmap a b g x maps the type-level function
g :: a ~> b over
x :: f a, and returns a type of kind
f b. Like with
Empty, to help with kind inference, we have
Fmap explicitly requre the kinds of the input and results of
b) so GHC doesn’t have to struggle to infer it implicitly.
And, of course, along with
sFmap (the singleton mirror of
Fmap), we have our laws:
fmap id x = x, and
fmap g (fmap h) x = fmap (g . h) x.
But, what are
a ~> b,
@@? They’re a part of the defunctionalization system that the singletons library uses. A
g :: a ~> b means that
g represents a type-level function taking a type of kind
a to a type of kind
b, but, importantly, encodes it in a way that makes Haskell happy. This hack is required because you can’t partially apply type families in Haskell. If
g was a regular old
a -> b type family, you wouldn’t be able to pass just
Fmap a b g (because it’d be partially applied, and type families always have to appear fully saturated).
You can convert a
g :: a ~> b back into a regular old
g :: a -> b using
Apply, or its convenient infix synonym
g @@ (x :: a) :: b
The singletons library provides
type family Id a where Id a = a, but we can’t pass in
Id directly into
Fmap. We have to pass in its “defunctionalized” encoding,
IdSym0 :: a ~> a.
For the composition law, we use
(:.$) (which is a defunctionalized type-level
.) and apply it to
h to get, essentially,
g :. h, where
:. is type-level function composition.
Now we Haskell.
$(singletons [d| mapOption :: (a -> b) -> Option a -> Option b mapOption _ None = None mapOption f (Some x) = Some (f x) mapList :: (a -> b) -> List a -> List b mapList _ Nil = Nil mapList f (Cons x xs) = Cons (f x) (mapList f xs) |]) instance Functor Option where type Fmap a b g x = MapOption g x sFmap = sMapOption fmapId = \case SNone -> Refl SSome _ -> Refl fmapCompose _ _ = \case SNone -> Refl SSome _ -> Refl instance Functor List where type Fmap a b g x = MapList g x sFmap = sMapList fmapId = \case SNil -> Refl SCons _ xs -> case fmapId xs of Refl -> Refl fmapCompose g h = \case SNil -> Refl SCons _ xs -> case fmapCompose g h xs of Refl -> Refl
And there you have it. A verified
Functor typeclass, ensuring that all instances are lawful. Never tell me that Haskell’s type system can’t do anything ever again!
Note that any mistakes in implementation (like, for example, having
mapOption _ _ = None) will cause a compile-time error now, because the proofs are impossible to provide.
As a side note, I’m not quite sure how to implement the value-level
fmap from this, since I can’t figure out how to promote functions nicely. Using
sFmap is the only way to work with this at the value level that I can see, but it’s probably because of my own lack of understanding. If anyone knows how to do this, please let me know!
Anyway, what an exciting journey and a wonderful conclusion. I hope you enjoyed this and will begin using this in your normal day-to-day Haskell. Goodbye, until next time!
Just one more
Hah! Of course we aren’t done. I wouldn’t let you down like that. I know that you probably saw that the entire last section’s only purpose was to build up to the pièce de résistance: the crown jewel of every Haskell article, the Monad.
class Functor f => Monad f where type Return a (x :: a) :: f a type Bind a b (m :: f a) (g :: a ~> f b) :: f b sReturn :: Sing (x :: a) -> Sing (Return a x :: f a) sBind :: Sing (m :: f a) -> Sing (g :: a ~> f b) -> Sing (Bind a b m g) -- | (return x >>= f) == f x returnIdentLeft :: Sing (x :: a) -> Sing (g :: a ~> f b) -> Bind a b (Return a x) g :~: (g @@ x) -- | (m >>= return) == m returnIdentRight :: Sing (m :: f a) -> Bind a a m ReturnSym0 :~: m -- | m >>= (\x -> f x >>= h) == (m >>= f) >>= h bindCompose :: Sing (m :: f a) -> Sing (g :: a ~> f b) -> Sing (h :: b ~> f c) -> Bind a c m (KCompSym2 a b c g h) :~: Bind b c (Bind a b m g) h data ReturnSym0 :: a ~> f a type instance Apply (ReturnSym0 :: a ~> f a) (x :: a) = Return a x type KComp a b c (g :: a ~> f b) (h :: b ~> f c) (x :: a) = Bind b c (g @@ x) h data KCompSym2 a b c g h :: (a ~> f c) type instance Apply (KCompSym2 a b c g h :: a ~> f c) (x :: a) = KComp a b c g h x return :: (SingKind a, SingKind (f a), Monad f) => Demote a -> Demote (f a) return x = withSomeSing x $ \sX -> fromSing (sReturn sX)
To help with kind inference, again, we provide explicit kind arguments for
Return (the kind of the thing that is being lifted) and
Bind (the original
a and the resulting
Some boilerplate exists there at the bottom — it’s the plumbing for the defunctionalization system.
returnIdentRight requires a defunctionalized version of
Return, so we can provide that by defining
ReturnSym0, and writing an
Apply instance for it (which “applies” it the parameter
KComp (kleisli composition) and its defunctionalized version in order to express the third law, because we don’t yet have type-level lambdas in Haskell. The actual function it is expressing is
\x -> f x >>= g, and that definition is given on the
type KComp a b c ... = Bind ... line.
KCompSym2 is the defunctioanlized version, which is not a
a -> f c but rather an
a ~> f c, which allows it to be partially applied (like we do for
composeBind). And, finally, to hook all of this up into the defunctionalization system, we write an
Apply instance yet again.
And, again, if anyone knows how I can write a value-level
Bind, I’d definitely appreciate hearing!
Let’s see some sample implementations.
$(singletons [d| bindOption :: Option a -> (a -> Option b) -> Option b bindOption None _ = None bindOption (Some x) f = f x concatMapList :: (a -> List b) -> List a -> List b concatMapList _ Nil = Nil concatMapList f (Cons x xs) = f x `appendList` concatMapList f xs |]) instance Monad Option where type Return a x = Some x type Bind a b m g = BindOption m g sReturn = SSome sBind = sBindOption returnIdentLeft _ _ = Refl returnIdentRight = \case SNone -> Refl SSome x -> case sReturn x of SSome _ -> Refl bindCompose = \case SNone -> \_ _ -> Refl SSome _ -> \_ _ -> Refl instance Monad List where type Return a x = PureList x type Bind a b m g = ConcatMapList g m sReturn = sPureList sBind x f = sConcatMapList f x returnIdentLeft x g = case sReturn x of SCons y SNil -> case emptyIdentRight (unSingFun1 Proxy g y) of Refl -> Refl returnIdentRight = \case SNil -> Refl SCons _ xs -> case returnIdentRight xs of Refl -> Refl bindCompose = \case SNil -> \_ _ -> Refl SCons x xs -> \g h -> case bindCompose xs g h of Refl -> case unSingFun1 Proxy g x of SNil -> Refl SCons y ys -> let gxs = sConcatMapList g xs hgxs = sConcatMapList h gxs hy = unSingFun1 Proxy h y hys = sConcatMapList h ys in case distribConcatMap h ys gxs of Refl -> case appendAssoc hy hys hgxs of Refl -> Refl -- | Proving that concatMap distributes over <> distribConcatMap :: Sing (g :: a ~> List b) -> Sing (xs :: List a) -> Sing (ys :: List a) -> ConcatMapList g (xs <> ys) :~: (ConcatMapList g xs <> ConcatMapList g ys) distribConcatMap g = \case SNil -> \_ -> Refl SCons x xs -> \ys -> case distribConcatMap g xs ys of Refl -> let gx = unSingFun1 Proxy g x cmgxs = sConcatMapList g xs cmgys = sConcatMapList g ys in case appendAssoc gx cmgxs cmgys of Refl -> Refl
Here we use
unSingFun1, which converts a singleton of a type-level function into a value-level function on singletons:
unSingFun1 :: Proxy (f :: a ~> b) -> Sing (f :: a ~> b) -> Sing (x :: a) -> Sing (f @@ x :: b)
Proxy argument only exists for historical reasons, I believe. But, the crux is that, given a
Sing (f :: a ~> b) and a
Sing (x :: a), we can “apply” them to get
Sing (f @@ x :: b)
The proofs for the list instance is admittedly ugly to write, due to the fact that
List is a recursive type. It’s also tricky because Haskell has poor to little support for theorem proving and no real tools to help you write them efficiently. But, the proofs for
Option are really something, aren’t they? It’s kind of amazing how much GHC can do on its own without requiring any manual proving on the part of the user.
Don’t do this in actual code, please (why?). This post started off as an April Fools joke that accidentally compiled correctly for reasons which I cannot explain.
While I don’t recommend that you do this in actual code, but definitely do recommend that you do it for fun! The code in this post is available here if you want to play around!