# Type arithmetic

**Type arithmetic** (or type-level computation) are calculations on
the type-level, often implemented in Haskell using functional
dependencies to represent functions.

A simple example of type-level computation are operations on Peano numbers:

```
data Zero
data Succ a
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)
```

Many other representations of numbers are possible, including binary and balanced base tree. Type-level computation may also include type representations of boolean values, lists, trees and so on. It is closely connected to theorem proving, via the Curry-Howard isomorphism.

A decimal representation was put forward by Oleg Kiselyov in "Number-Paramterized Types" in the fifth issue of The Monad Reader. There is an implementation in the type-level package, but unfortunately the arithmetic is really slow, because in fact it simulates Peano arithmetic with decimal numbers.

## Contents

## Library support

Robert Dockins has gone as far as to write a library for type level arithmetic, supporting the following operations on type level naturals: addition, subtraction, multiplication, division, remainder, GCD, and also contains the following predicates: test for zero, test for equality and < > <= >=

This library uses a binary representation and can handle numbers at the order of 10^15 (at least). It also contains a test suite to help validate the somewhat unintuitive algorithms.

## More type hackery

Not to be outdone, Oleg Kiselyov has written on invertible, terminating, 3-place addition, multiplication, exponentiation relations on type-level Peano numerals, where any two operands determine the third. He also shows the invertible factorial relation. Thus providing all common arithmetic operations on Peano numerals, including n-base discrete logarithm, n-th root, and the inverse of factorial. The inverting method can work with any representation of (type-level) numerals, binary or decimal.

Oleg says, "The implementation of RSA on the type level is left for future work".

## Djinn

Somewhat related is Lennart Augustsson's tool Djinn, a theorem prover/coding wizard, that generates Haskell code from a given Haskell type declaration.

Djinn interprets a Haskell type as a logic formula using the Curry-Howard isomorphism and then a decision procedure for Intuitionistic Propositional Calculus.

## An Advanced Example : Type-Level Quicksort

An advanced example: quicksort on the type level.

Here is a complete example of advanced type level computation, kindly provided by Roman Leshchinskiy. For further information consult Thomas Hallgren's 2001 paper Fun with Functional Dependencies.

```
module Sort where
-- natural numbers
data Zero
data Succ a
-- booleans
data True
data False
-- lists
data Nil
data Cons a b
-- shortcuts
type One = Succ Zero
type Two = Succ One
type Three = Succ Two
type Four = Succ Three
-- example list
list1 :: Cons Three (Cons Two (Cons Four (Cons One Nil)))
list1 = undefined
-- utilities
numPred :: Succ a -> a
numPred = const undefined
class Number a where
numValue :: a -> Int
instance Number Zero where
numValue = const 0
instance Number x => Number (Succ x) where
numValue x = numValue (numPred x) + 1
numlHead :: Cons a b -> a
numlHead = const undefined
numlTail :: Cons a b -> b
numlTail = const undefined
class NumList l where
listValue :: l -> [Int]
instance NumList Nil where
listValue = const []
instance (Number x, NumList xs) => NumList (Cons x xs) where
listValue l = numValue (numlHead l) : listValue (numlTail l)
-- comparisons
data Less
data Equal
data Greater
class Cmp x y c | x y -> c
instance Cmp Zero Zero Equal
instance Cmp Zero (Succ x) Less
instance Cmp (Succ x) Zero Greater
instance Cmp x y c => Cmp (Succ x) (Succ y) c
-- put a value into one of three lists according to a pivot element
class Pick c x ls eqs gs ls' eqs' gs' | c x ls eqs gs -> ls' eqs' gs'
instance Pick Less x ls eqs gs (Cons x ls) eqs gs
instance Pick Equal x ls eqs gs ls (Cons x eqs) gs
instance Pick Greater x ls eqs gs ls eqs (Cons x gs)
-- split a list into three parts according to a pivot element
class Split n xs ls eqs gs | n xs -> ls eqs gs
instance Split n Nil Nil Nil Nil
instance (Split n xs ls' eqs' gs',
Cmp x n c,
Pick c x ls' eqs' gs' ls eqs gs) =>
Split n (Cons x xs) ls eqs gs
listSplit :: Split n xs ls eqs gs => (n, xs) -> (ls, eqs, gs)
listSplit = const (undefined, undefined, undefined)
-- zs = xs ++ ys
class App xs ys zs | xs ys -> zs
instance App Nil ys ys
instance App xs ys zs => App (Cons x xs) ys (Cons x zs)
-- zs = xs ++ [n] ++ ys
-- this is needed because
--
-- class CCons x xs xss | x xs -> xss
-- instance CCons x xs (Cons x xs)
--
-- doesn't work
class App' xs n ys zs | xs n ys -> zs
instance App' Nil n ys (Cons n ys)
instance (App' xs n ys zs) => App' (Cons x xs) n ys (Cons x zs)
-- quicksort
class QSort xs ys | xs -> ys
instance QSort Nil Nil
instance (Split x xs ls eqs gs,
QSort ls ls',
QSort gs gs',
App eqs gs' geqs,
App' ls' x geqs ys) =>
QSort (Cons x xs) ys
listQSort :: QSort xs ys => xs -> ys
listQSort = const undefined
```

And we need to be able to run this somehow, in the typechecker. So fire up ghci:

```
> :t listQSort list1
Cons
(Succ Zero)
(Cons (Succ One) (Cons (Succ Two) (Cons (Succ Three) Nil)))
```

## A Really Advanced Example : Type-Level Lambda Calculus

Again, thanks to Roman Leshchinskiy, we present a simple lambda calculus encoded in the type system (and with non-terminating typechecking fun!)

Below is an example which encodes a stripped-down version of the lambda calculus (with only one variable):

```
{-# OPTIONS -fglasgow-exts #-}
data X
data App t u
data Lam t
class Subst s t u | s t -> u
instance Subst X u u
instance (Subst s u s', Subst t u t') => Subst (App s t) u (App s' t')
instance Subst (Lam t) u (Lam t)
class Apply s t u | s t -> u
instance (Subst s t u, Eval u u') => Apply (Lam s) t u'
class Eval t u | t -> u
instance Eval X X
instance Eval (Lam t) (Lam t)
instance (Eval s s', Apply s' t u) => Eval (App s t) u
```

Now, lets evaluate some lambda expressions:

```
> :t undefined :: Eval (App (Lam X) X) u => u
undefined :: Eval (App (Lam X) X) u => u :: X
```

Ok good, and:

```
> :t undefined :: Eval (App (Lam (App X X)) (Lam (App X X)) ) u => u
^CInterrupted.
```

diverges ;)

## Turing-completeness

It's possible to embed the Turing-complete SK combinator calculus at the type level.

## Theory

See also dependent type theory.

## Practice

Extensible records (which are used e.g. in type safe, declarative relational algebra approaches to database management)