# Lazy functors

## Question

I have a data type like

```
data Pair a = Pair a a
```

Shall I define `Functor`

and related instances
with lazy pattern matching or with strict pattern matching?

That is, shall I define

```
instance Functor Pair where
fmap f ~(Pair a b) = Pair (f a) (f b)
instance Applicative Pair where
pure a = Pair a a
~(Pair fa fb) <*> ~(Pair a b) = Pair (fa a) (fb b)
instance Fold.Foldable Pair where
foldMap = Trav.foldMapDefault
instance Trav.Traversable Pair where
sequenceA ~(Pair a b) = liftA2 Pair a b
```

or shall I define

```
instance Functor Pair where
fmap f (Pair a b) = Pair (f a) (f b)
instance Applicative Pair where
pure a = Pair a a
(Pair fa fb) <*> (Pair a b) = Pair (fa a) (fb b)
instance Fold.Foldable Pair where
foldMap = Trav.foldMapDefault
instance Trav.Traversable Pair where
sequenceA (Pair a b) = liftA2 Pair a b
```

?

## Answer

We can deduce the answers from the following laws applied to undefined values.

```
import Control.Monad.Identity (Identity(Identity))
fmap id x == x
pure id <*> x == x
f <*> pure x == pure ($x) <*> f
sequenceA (fmap Identity x) = Identity x
```

With the first definitions with lazy matching the laws are violated:

```
fmap id undefined == Pair undefined undefined
-- because of laziness in the second operand of <*> we get:
pure id <*> undefined == Pair undefined undefined
-- if the second operand is matched strictly, and the first one lazily,
-- then we get:
undefined <*> pure undefined == Pair undefined undefined
pure ($ undefined) <*> undefined == undefined
-- given that fmap matches strict now, since lazy matching is incorrect
sequenceA (fmap Identity undefined) == Identity (Pair undefined undefined)
```

In contrast to that the strict pattern matching is correct in this respect:

```
fmap id undefined == undefined
pure id <*> undefined == undefined
undefined <*> pure undefined == undefined
pure ($ undefined) <*> undefined == undefined
sequenceA (fmap Identity undefined) = Identity undefined
```

It is a good idea to comply with these laws since they minimize the surprise of the users of your data type, including yourself.

If you use strict record fields (denoted with `!`

)
then there is no `(Pair undefined undefined)`

,
only `undefined`

.
That is, in this case the laws would hold
independent of the mode of pattern matching.
However, this shall not suggest,
that using strict record fields is generally prefered

## See also

- For laws of Traversable see The Essence of the Iterator Pattern by Jeremy Gibbons and Bruno C. d. S. Oliveira, Section 5.2 "Sequential composition of traversals"