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.TypeNats where

Singleton Types

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

data Nat (n :: Nat)

The only value of type Nat n is the number n. (Technically, there is also an undefined element.)

class NatI n where
  nat :: Nat n

natToInteger :: Nat n -> Integer

Type-Level Operations

type family m ^ n :: Nat
type family m * n :: Nat
type family m + n :: Nat
class m <= n

Natural Numbers

data Natural = forall n . Natural !(Nat n)

data NaturalInteger
  = Negative Natural
  | NonNegative Natural

toNaturalInteger :: Integer -> NaturalInteger

subNatural :: Natural -> Natural -> NaturalInteger