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## Type-Level Literals

Currently, we support two forms of type-level literals: natural numbers, and symbol constants.
Natural number literals are a family of types, all of which belong to the kind `Nat`

. Similarly,
symbol literals are types that belong to the kind `Symbol`

:

0, 1, 2, ... :: Nat "hello", "world", "some string literal" :: Symbol

Both numeric and symbol literal types are empty---they have no inhabitants. However, they may be used as parameters to other type constructors, which makes them useful.

## Singleton Types

We use this idea to link the type-level literals to specific run-time values via *singleton types*.
The singleton types and some useful functions for working with them are defined in module `GHC.TypeLits`

:

module GHC.TypeLits where

A singleton type is simply a type that has only one interesting inhabitant. We define a whole family of singleton types, parameterized by type-level literals:

newtype Sing :: a -> *

For example, `Sing 0`

, `Sing 127`

, `Sing "hello"`

, `Sing "this also`

}, are all
singleton types. The intuition is that the only inhabitant of `Sing n`

is the value `n`

. Notice
that `Sing`

has a *polymorphic kind* because sometimes we apply it to numbers (which are of
kind `Nat`

) and sometimes we apply it to symbols (which are of kind `Symbol`

).

But, if we have a value of type `Sing a`

, how do we get the actual integer or string?
We can do this with the function `fromSing`

:

fromSing :: Sing a -> SingRep a type family SingRep a type instance SingRep (n :: Nat) = Integer type instance SingRep (n :: Symbol) = String

The function `fromSing`

has an interesting type: it maps singletons to ordinary values,
but the type of the result depends on the *kind* of the singleton parameter.
So, if we apply it to a value of type `Sing 3`

we get the *number* `3`

, but,
if we apply it to a value of type `Sing "hello"`

we get the *string* `"hello"`

.

So, how do we make values of type `Sing n`

in the first place? This is done with
the special overloaded constant `sing`

:

class SingI a where sing :: Sing a -- Built-in instances for all type-literals. instance SingI 0 where sing = ... the singleton value representing 0 ... instance SingI 1 where sing = ... the singleton value representing 1 ... instance SingI "hello" where sing = ... the singleton value representing "hello" ... // ... etc.

Here are some examples on the GHCi prompt to get a feel of how `sing`

works:

> :set -XDataKinds > sing :: Sing 1 > 1 > sing :: Sing "hello" > "hello"

The name *SingI* is a mnemonic for the different uses of the class:

- It is the
*introduction*construct for 'Sing' values, - It is an
*implicit*singleton parameter (this is discussed in more detail bellow)

Notice that GHCi could display values of type `Sing`

, so they have a `Show`

instance. As another example, here
is the definition of the `Show`

instance:

instance Show (SingRep a) => Show (Sing a) where showsPrec p = showsPrec p . fromSing

Easy! We just convert the singleton into an ordinary value (integer or string), and use *its* `Show`

instance to display it.

Next, we show two functions which make it easier to work with singleton types:

withSing :: SingI a => (Sing a -> b) -> b withSing f = f sing singThat :: SingI a => (SingRep a -> Bool) -> Maybe (Sing a) singThat p = withSing $ \x -> if p (fromSing x) then Just x else Nothing

The first function, `withSing`

, is useful when we want to work with the same singleton value multiple times.
The constant `sing`

is polymorphic, so every time we use it in a program, it may refer to a potentially
different singleton, so to ensure that two singleton values are the same we have to resort to
explicit type signatures, which just adds noise to a definition. By using, `withSing`

we avoid this problem
because we get an explicit (monomorphic) name for a given singleton, and so we can use the name many times
without any type signatures. This technique is shown in the definition of the second function, `singThat`

.

The function `singThat`

is similar to the constant `sing`

in that it defines new singleton values. However,
it allows us to specify a predicate on (the representation of) the value and it only succeeds if this predicate
holds. Here are some examples of how that works:

> singThat (== 1) :: Maybe (Sing 1) > Just 1 > singThat (== 1) :: Maybe (Sing 2) > Nothing > singThat ("he" `isPrefixOf`) :: Maybe (Sing "hello") > Just "hello"

Now, using `singThat`

we can show the definition of the `Read`

instance for singletons:

instance (SingI a, Read (SingRep a), Eq (SingRep a)) => Read (Sing a) where readsPrec p cs = do (x,ys) <- readsPrec p cs case singThat (== x) of Just y -> [(y,ys)] Nothing -> []

We use the `Read`

instance of the representation for the singletons to parse a value,
and then, we use `singThat`

to make sure that this was the value corresponding to
the singleton.

## Implicit vs. Explicit Parameters

There are two different styles of writing functions that use singleton types. To illustrate the two style consider a type for working with C-style arrays:

newtype ArrPtr (n :: Nat) a = ArrPtr (Ptr a)

Now consider the definition of the function `memset`

, which sets all elemnts
of the array to a given fixed value.

One approach is to use an explicit singleton parameter. For example:

memset_c :: Storable a => ArrPtr n a -> a -> Sing n -> IO () memset_c (ArrPtr p) a size = mapM_ (\i -> pokeElemOff p i a) [ 0 .. fromIntegral (fromSing size - 1) ]

This style is, basically, a more typed version of what is found in many standard C libraries.
Callers of this function have to pass the size of the array explicitly, and the type system checks that the
size matches that of the array. Note that in the implementation of `memset_c`

we used `fromSing`

to get the concrete integer associated with the singleton type, so that we know how many elements
to set with the given value/

While the explicit `Sing`

parameter is convenient when we define the function, it is a bit
tedious to have to provide it all the time---it is easier to let the compiler infer the value,
based on the type of the array:

memset :: (Storable a, SingI n) => ArrPtr n a -> a -> IO () memset ptr a = withSing (memset_c ptr a)