.. _type-level-literals:

Type-Level Literals
===================

GHC supports numeric and string literals at the type level, giving
convenient access to a large number of predefined type-level constants.
Numeric literals are of kind ``Nat``, while string literals are of kind
``Symbol``. This feature is enabled by the :extension:`DataKinds` language
extension.

The kinds of the literals and all other low-level operations for this
feature are defined in modules ``GHC.TypeLits`` and ``GHC.TypeNats``.
Note that these modules define some type-level operators that clash with their
value-level counterparts (e.g. ``(+)``). Import and export declarations
referring to these operators require an explicit namespace annotation (see
:ref:`explicit-namespaces`).

Here is an example of using type-level numeric literals to provide a
safe interface to a low-level function: ::

    import GHC.TypeLits
    import Data.Word
    import Foreign

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

    clearPage :: ArrPtr 4096 Word8 -> IO ()
    clearPage (ArrPtr p) = ...

Here is an example of using type-level string literals to simulate
simple record operations: ::

    data Label (l :: Symbol) = Get

    class Has a l b | a l -> b where
      from :: a -> Label l -> b

    data Point = Point Int Int deriving Show

    instance Has Point "x" Int where from (Point x _) _ = x
    instance Has Point "y" Int where from (Point _ y) _ = y

    example = from (Point 1 2) (Get :: Label "x")


.. _typelit-runtime:

Runtime Values for Type-Level Literals
--------------------------------------

Sometimes it is useful to access the value-level literal associated with
a type-level literal. This is done with the functions ``natVal`` and
``symbolVal``. For example: ::

    GHC.TypeLits> natVal (Proxy :: Proxy 2)
    2

These functions are overloaded because they need to return a different
result, depending on the type at which they are instantiated. ::

    natVal :: KnownNat n => proxy n -> Natural  -- from GHC.TypeNats
    natVal :: KnownNat n => proxy n -> Integer  -- from GHC.TypeLits

    -- instance KnownNat 0
    -- instance KnownNat 1
    -- instance KnownNat 2
    -- ...

GHC discharges the constraint as soon as it knows what concrete
type-level literal is being used in the program. Note that this works
only for *literals* and not arbitrary type expressions. For example, a
constraint of the form ``KnownNat (a + b)`` will *not* be simplified to
``(KnownNat a, KnownNat b)``; instead, GHC will keep the constraint as
is, until it can simplify ``a + b`` to a constant value.

It is also possible to convert a run-time integer or string value to the
corresponding type-level literal. Of course, the resulting type literal
will be unknown at compile-time, so it is hidden in an existential type.
The conversion may be performed using ``someNatVal`` for integers and
``someSymbolVal`` for strings: ::

    someNatVal :: Natural -> Maybe SomeNat  -- from GHC.TypeNats
    someNatVal :: Integer -> Maybe SomeNat  -- from GHC.TypeLits

    SomeNat    :: KnownNat n => Proxy n -> SomeNat

The operations on strings are similar.

.. _typelit-tyfuns:

Computing With Type-Level Naturals
----------------------------------

GHC 7.8 can evaluate arithmetic expressions involving type-level natural
numbers. Such expressions may be constructed using the type-families
``(+), (*), (^)`` for addition, multiplication, and exponentiation.
Numbers may be compared using ``(<=?)``, which returns a promoted
boolean value, or ``(<=)``, which compares numbers as a constraint. For
example:

.. code-block:: none

    GHC.TypeLits> natVal (Proxy :: Proxy (2 + 3))
    5

At present, GHC is quite limited in its reasoning about arithmetic: it
will only evaluate the arithmetic type functions and compare the
results--- in the same way that it does for any other type function. In
particular, it does not know more general facts about arithmetic, such
as the commutativity and associativity of ``(+)``, for example.

However, it is possible to perform a bit of "backwards" evaluation. For
example, here is how we could get GHC to compute arbitrary logarithms at
the type level:

.. code-block:: none

    lg :: Proxy base -> Proxy (base ^ pow) -> Proxy pow
    lg _ _ = Proxy

    GHC.TypeLits> natVal (lg (Proxy :: Proxy 2) (Proxy :: Proxy 8))
    3


