## More beautiful fold zipping

My previous post, *Another lovely example of type class morphisms*, placed some standard structure around Max Rabkin’s *Beautiful folding*, using type class morphisms to confirm that the `Functor`

and `Applicative`

instances agreed with their inevitable meanings.

In the process of working out the `Applicative`

case, I realized the essence of fold zipping was getting a bit obscured.
This post simplifies Max’s `Fold`

data type a bit and shows how to zip strict left folds in the simpler setting.
You can get the code described here.

**Edits**:

- 2008-11-15: I renamed the
`Pair`

and`Pair'`

classes to`Zip`

and`Zip'`

.

### A simpler fold

The strict left-folding functional is

```
foldl' :: (a -> b -> a) -> a -> [b] -> a
```

Max’s fold type packages up the first two arguments of `foldl'`

and adds on an post-fold step, as needed for `fmap`

:

```
data Fold b c = forall a. F (a -> b -> a) a (a -> c) -- clever version
```

A simpler, less clever version just has the two `foldl'`

arguments, and no `forall`

:

```
data Fold b a = F (a -> b -> a) a -- simple version
```

I’ll use this simpler version for zipping and later enhance it for `fmap`

.

The strict interpretation of `Fold`

simply unpacks and folds:

```
cfoldl' :: Fold b a -> [b] -> a
cfoldl' (F op e) = foldl' op e
```

Zipping is then just a simplification of the `(<*>)`

definition in the previous post.
As before, `P`

is a type and constructor of strict pairs.

```
data P c c' = P !c !c'
zipF' :: Fold b a -> Fold b a' -> Fold b (P a a')
F op e `zipF'` F op' e' = F op'' (P e e')
where
P a a' `op''` b = P (a `op` b) (a' `op'` b)
```

We can also define non-strict counterparts:

```
cfoldl :: Fold b a -> [b] -> a
cfoldl (F op e) = foldl op e
zipF :: Fold b a -> Fold b a' -> Fold b (a,a')
F op e `zipF` F op' e' = F op'' (e,e')
where
(a,a') `op''` b = (a `op` b, a' `op'` b)
```

### More type classes …

A previous post suggested a criterion for inevitability of interpretation functions like `cfoldl`

and `cfoldl'`

.
Whatever class instances exists for the source type (`Fold`

here), interpretation functions should be
“type class morphisms” for those classes, which loosely means that the meaning of a method is the same method on the meaning.

Where’s the type class for our simplified `Fold`

?
It lacks the magic ingredient Max added to make it a `Functor`

and an `Applicative`

, namely the post-fold step.

However, there’s another class that’s just the thing:

```
class Zip f where zip :: f a -> f b -> f (a,b)
```

(I’ve used this class with type representations and user interfaces.)

The `Zip`

class from `TypeCompose`

is almost identical to the `Zip`

class in `category-extras`

.
The latter class requires `Functor`

, however, which is too restrictive for the simplified `Fold`

type.

Our *non-strict* left folds have a `Zip`

instance:

```
instance Zip (Fold b) where zip = zipF
```

For strict folds, define a strict variant on `Zip`

:

```
class Zip' f where zip' :: f a -> f b -> f (P a b)
```

and a `Zip'`

instance for `Fold`

:

```
instance Zip' (Fold b) where zip' = zipF'
```

### … and more morphisms

These zipping classes have associated morphism properties.
A function `q`

mapping from one `Zip`

instance to another is a “`Zip`

morphism” when

```
q (f `zip` g) == q f `zip` q g
```

for all folds `f`

and `g`

.

Similarly for `Zip'`

morphisms:

```
q (f `zip'` g) == q f `zip'` q g
```

In other words, `q`

distributes over `zip`

or `zip'`

.

As I’ll show in another post, the two interpretation functions `cfoldl`

and `cfoldl'`

are `Zip`

and `Zip'`

morphisms, respectively, i.e.,

```
cfoldl (f `zip` g) == cfoldl f `zip` cfoldl g
cfoldl' (f `zip'` g) == cfoldl' f `zip'` cfoldl' g
```

which means that zip folding as defined above works correctly.

### What’s next?

I like to keep blog posts fairly short, so I’ll share two more pieces of this story in upcoming blog posts:

- Regaining the lost
`Functor`

and`Applicative`

instances. - The
`Zip`

and`Zip'`

morphism proofs for zip folding.

## Conal Elliott » Blog Archive » Enhancing a Zip:

[…] a pattern that’s been steering me lately toward natural (inevitable) design of libraries. * Part Two simplified that representation to help get to the essence of zipping, and in doing so lost the […]

15 November 2008, 5:10 pm## Conal Elliott » Blog Archive » Proofs for left fold zipping:

[…] « More beautiful fold zipping Enhancing a Zip […]

14 February 2009, 6:22 pm## Dan Piponi:

This zip for folds is a product in the category of certain F-algebras. More generally we can write:

which generalises this to all catamorphisms.

23 February 2009, 10:17 am## conal:

Thanks for the pointer, Dan. I hadn’t explored that corner of category-extras. Will look further.

Meanwhile, here’s a tweaked definition of

`prod`

:Or replace

23 February 2009, 10:37 am``zip``

with`&&&`

.## conal:

It recently hit me that

23 February 2009, 10:38 am`Zip`

and`zip`

aren’t such a great choice of names after all. I’d like the this notion to coincide with the`Monoidal`

operation in section 7 ofApplicative Programming with Effects, which must be consistent with`Applicative`

instance. So the operation on lists would be cross product, not`zip`

. On`ZipList`

and`Stream`

,`zip`

is fine.## Edward Kmett:

FYI- In the newly broken out category-extras, the Apply class in semigroupoids provides the ‘Applicative-consistent’ semantics you noticed that you want for Zip here, while the Zip class in ‘keys’ provides the zip-like semantics.

31 May 2011, 2:54 pm