Peano numbers

Peano numbers are a simple way of representing the natural numbers using only a zero value and a successor function. In Haskell it is easy to create a type of Peano number values, but they are more often used to create a set of phantom types.

Peano number values

data Peano = Zero | Succ Peano

Here Zero and Succ are values (constructors). Zero has type Peano, and Succ has type Peano -> Peano. The natural number zero is represented by Zero, one by Succ Zero, two by Succ (Succ Zero) and so forth.

Here's a simple addition function:

add Zero b = b
add (Succ a) b = Succ (add a b)

Peano number types

data Zero

data Succ a

Here Zero and Succ are types. Zero has kind *, and Succ has kind * -> *. The natural numbers are represented by types (of kind *) Zero, Succ Zero, Succ (Succ Zero) etc.

Arithmetic can be done using fundeps:

class Add a b ab | a b -> ab
instance Add Zero b b
instance (Add a b ab) => Add (Succ a) b (Succ ab)

Note that classes express relationships between types, rather than functions from type to type. Accordingly, with the instance declarations above one can add another fundep to the class declaration to get subtraction for free:

class Add a b ab | a b -> ab, a ab -> b
instance Add Zero b b
instance (Add a b ab) => Add (Succ a) b (Succ ab)