# Type arithmetic

(Difference between revisions)

Type arithmetic is calculations on types using fundeps as functions.

A simple example is 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)
```

However, many other representations of numbers are possible, including binary and balanced base three. Type arithmetic may also include type representations of boolean values and so on.

## 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 2000 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:

```   ___         ___ _
/ _ \ /\  /\/ __(_)
/ /_\// /_/ / /  | |      GHC Interactive, version 5.04, for Haskell 98.
/ /_\\/ __  / /___| |      http://www.haskell.org/ghc/
\____/\/ /_/\____/|_|      Type :? for help.