# 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 since unary representation is inefficient, they are more often used to do type arithmetic due to their simplicity.

## Overview

### 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)

See an example implementation.

### 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)

See type arithmetic for other functions and encodings.

## Applications

### Lazy counting

The lazy nature of Peano numbers rehabilitates the use of list functions which count list elements. As described in Things to avoid one should not write

length xs == 0

because length constructs all list nodes, although after you have seen only one, it is obvious that the list is not empty. The above expression can be simply replaced by null, but with Peano numbers you achieve the same

genericLength xs == Zero
-- or
genericLength xs == (0::Peano)

The expression

length xs == length ys

is harder to make lazy without Peano numbers. With them (and an appropriate Num instance) this becomes rather simple, because

genericLength xs == (genericLength ys :: Peano)

constructs only as much list nodes, as the shorter list has.

### Equation solving

Lazy Peano numbers can also be used within "Tying the Knot" techniques. There is a package for determining the right order for solving equations, where an equation is only solved if only one of its variables is still indeterminate.