## Concept:

# Foldable type

## Headline

A type for which a fold function can be defined

## Illustration

Obviously, a fold function can be defined for lists. See also the concept of Maybe type for another simple example of a foldable type.

In Language:Haskell, there is a type class of foldable types:

```
class Foldable t
where
fold :: Monoid m => t m -> m
foldMap :: Monoid m => (a -> m) -> t a -> m
foldr :: (a -> b -> b) -> b -> t a -> b
foldl :: (a -> b -> a) -> a -> t b -> a
foldr1 :: (a -> a -> a) -> t a -> a
foldl1 :: (a -> a -> a) -> t a -> a
```

The members *foldr* and *foldl* generalize the function signatures of the folklore fold functions for lists. It should be noted that a minimal complete definition requires either the definition of *foldr* or *foldMap*, as all other class members are then defined by appropriate defaults. Here is a particular attempt at such defaults:

```
class Foldable t
where
fold :: Monoid m => t m -> m
foldMap :: Monoid m => (a -> m) -> t a -> m
foldr :: (a -> b -> b) -> b -> t a -> b
foldl :: (a -> b -> a) -> a -> t b -> a
foldr1 :: (a -> a -> a) -> t a -> a
foldl1 :: (a -> a -> a) -> t a -> a
fold = foldr mappend mempty
foldMap f = foldr (mappend . f) mempty
foldr f z = foldr f z . toList
foldl f z q = foldr (\x g a -> g (f a x)) id q z
foldr1 f = foldr1 f . toList
foldl1 f = foldl1 f . toList
```

In a number of places, we leverage a conversion function *toList* for going from a foldable type over an element type to the list type over the same element type. In this manner, we can reduce some operations on foldables to operations on lists. This conversion function is easily defined by a *foldMap* application:

```
toList :: Foldable t => t a -> [a]
toList = foldMap (\x->[x])
```

Looking at the defaults again and their use of *toList*, there is obviously an "unsound" circularity within the definitions, which however would be soundly broken, when either *foldr* or *foldMap* was defined for any given foldable type.

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