This was written about two years ago. Mostly inspired by Philip Wadler‘s note Recursive types for free.

### Inductive types

Inductive types are least fixed-points. An indcutive type μF is simulated by

` forall x . (F x -> x) -> x`

In Haskell, we need a constructor to wrap a “forall” quantifier.

```
> newtype LFix f = LF (forall x. (f x -> x) -> x)
> instance Show (LFix f) where
> show _ = "
```"
> foldL :: Functor f => (f x -> x) -> LFix f -> x
> foldL f (LF x) = x f
> inL :: Functor f => f (LFix f) -> LFix f
> inL fx = LF (\\f -> f . fmap (foldL f) $ fx)
> outL :: Functor f => LFix f -> f (LFix f)
> outL = foldL (fmap inL)

Example: lists.

```
> data F a x = L | R a x deriving Show
> instance Functor (F a) where
> fmap f L = L
> fmap f (R a x) = R a (f x)
> type ListL a = LFix (F a)
```

A `ListL`

can only be constructed out of nil and cons.

```
Main> consL 1 (consL 2 (consL 3 nilL))
```

```
> nilL = inL L
> consL a x = inL (R a x)
```

We can convert a ListL to a ‘real’ Haskell list.

```
Main> lList $ consL 1 (consL 2 (consL 3 nilL))
[1,2,3]
```

```
> lList :: LFix (F a) -> [a]
> lList = foldL ll
> where ll L = []
> ll (R a x) = a : x
```

However, a `ListL`

has to be explicitly built, so there is no infinite `ListL`

.

### Coinductive types

A coinductive type νF is simulatd by

```
exists x . (x -> F x, x)
```

We represent it in Haskell using the property `exists x . F x == forall y . (forall x . F x -> y) -> y`

:

```
> data GFix f = forall x . GF (x -> f x, x)
> instance Show (GFix f) where
> show _ = "
```"
> unfoldG :: Functor f => (x -> f x) -> x -> GFix f
> unfoldG g x = GF (g,x)
> outG :: Functor f => GFix f -> f (GFix f)
> outG (GF (g,x)) = fmap (unfoldG g) . g $ x
> inG :: Functor f => f (GFix f) -> GFix f
> inG = unfoldG (fmap outG)

Example:

```
> type ListG a = GFix (F a)
```

`ListG`

can be constructed out of `nil`

and `cons`

, as well as an unfold.

```
> nilG = inG L
> consG a x = inG (R a x)
> fromG :: Int -> ListG Int
> fromG m = unfoldG ft m
> where ft m = R m (m+1)
```

However, one can perform only a finite number of outG.

```
Main> fmap outG . outG . fromG $ 1
R 1 (R 2
```)

### Isomorphism

The function `force`

witnesses the isomorphism between `LFix`

and `GFix`

.

```
> force :: Functor f => GFix f -> LFix f
> force = inL . fmap force . outG -- recursion!
```

If `LFix`

and `GFix`

are isomorphic, we are allowed to build hylomorphisms:

```
> fromTo :: Int -> Int -> LFix (F Int)
> fromTo m n = takeLessThan n . force . fromG $ m
> takeLessThan :: Int -> LFix (F Int) -> LFix (F Int)
> takeLessThan n = foldL (tlt n)
> where tlt n L = nilL
> tlt n (R m x) | n <= m = nilL
> | otherwise = consL m x
Main> lList (fromTo 1 10)
[1,2,3,4,5,6,7,8,9]
```

However, `force`

has to be defined using general recursion. The price is that it is now possible to write programs that do not terminate.