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## Type-Level Naturals

There is a new kind, `Nat`. It is completely separate from GHC's hierarchy of sub-kinds, so `Nat` is only a sub-kind of itself.

The inhabitants of `Nat` are an infinite family of (empty) types, corresponding to the natural numbers:

0, 1, 2, ... :: Nat

These types are linked to the value world by a small library with the following API:

module GHC.TypeLits where

## Singleton Types

We relate type-level natural numbers to run-time values via a family of singleton types:

data TNat :: Nat -> * tNat :: NatI n => TNat n tNatInteger :: TNat n -> Integer -- Convenient derived functions tNatThat :: NatI n => (Integer -> Bool) -> Maybe (TNat n) withTNat :: NatI n => (TNat n -> a) -> a

The only "interesting" value of type `TNat n` is the number `n`. Technically, there is also an undefined element.
The value of a singleton type may be named using `tNat`, which is a bit like a "smart" constructor for `TNat n`.
Note that because `tNat` is polymorphic, we may have to use a type signature to specify which singleton we mean. For example:

> :set -XDataKinds > tNatInteger (tNat :: TNat 3) 3

One may think of the smart constructor `tNat` as being a method of a special built-in class, `NatI`:

class NatI n where tNat :: TNat n instance NatI 0 where tNat = ...singleton 0 value... instance NatI 1 where tNat = ...singleton 1 value... instance NatI 2 where tNat = ...singleton 2 value... etc.

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

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

## Examples

Here is how we can use the basic primitives to define a `Show` instance for singleton types:

instance Show (TNat n) where showsPrec p n = showsPrec p (tNatInteger n)

Here is how to define a `Read` instance:

instance NatI n => Read (Nat n) where readsPrec p x = do (x,xs) <- readsPrec p x case tNatThat (== x) of Just n -> [(n,xs)] Nothing -> []

The derived function `tNatThat` is similar to `tNat` except that it succeeds only if the integer representation
of the singleton type matches the given predicate. So, in the `Read` instance we parse an integer and then we check
if it is the expected integer for the singleton that we are trying to parse.

## Implicit vs. Explicit Parameters

There are two different styles of writing functions which need the integer corresponding to a type level natural. To illustrate the two style consider a type for working with C-style arrays:

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

One approach is to use an explicit parameter of type `TNat n`. For example:

memset_c :: Storable a => ArrPtr n a -> a -> TNat n -> IO () memset_c (ArrPtr p) a n = mapM_ (\i -> pokeElemOff p i a) [ 0 .. fromIntegral (tNatInteger n - 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 `tNatInteger`
to get the concrete integer associated with the singleton type.

While the explicit `TNat` 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 system infer the value,
based on the type of the array:

memset :: (Storable a, NatI n) => ArrPtr n a -> a -> IO () memset ptr a = withTNat (memset_c ptr a)

The function `withTNat` is useful when converting from the "explicit" to the "implicit" style
because it avoids ambiguity errors, scoped type-variables and other complications.