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 preferred. In particular, strict record fields also violate the functor laws! The normal functor instance is not too lazy, and not too strict, and as such satisfies the functor laws.

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"