We use a family of singleton types to relate type-level naturals to runtime values.

In our design, we chose to provide as primitive an overloaded "smart" constructor and a polymorphic elimination construct:

newtypeT Nat (n :: Nat) = Nat Integer

class NatI n where
  tNat :: TNat n

instance NatI 0 where tNat = TNat 0
instance NatI 1 where tNat = TNat 1
...

tNatInteger :: TNat n -> Integer
tNatInteger (TNat n) = n

It is also possible to make the dual choice, where we provide a polymorphic constructor and an overloaded elimination construct:

data TNat (n :: Nat) = TNat

class NatE n where
  tNatInteger :: TNat n -> Integer

instance NatE 0 where tNatInteger TNat = 0
instance NatE 1 where tNatInteger TNat = 1
...

We made this choice, at least in part, because it made the implementation simpler: the evidence for class NatI is just an integer. Note that our choice does not loose any generality because we can define the alternative design in terms of it:

data TNat1 (n :: Nat) = TNat1

tNat1Integer :: NatI n => TNat1 n -> Integer
tNat1Integer = tNatInteger . cast
  where cast :: NatI n => TNat1 n -> TNat n
        cast TNat1 = tNat