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This package provides
Naperian functors, a more powerful form of
Distributive functor which is equal in power to a
Representable functor (for
Rep), but which can be implemented asymptotically more efficiently for
instances which don't support random access.
Distributive functors allow distribution of
distribute :: (Distributive f, Functor g) => g (f a) -> f (g a)
Distributive, you can, for example, zip two containers by distributing
data Pair a = Pair a a deriving Functor zipDistributive :: Distributive f => f a -> f a -> f (a, a) zipDistributive xs ys = fmap f $ distribute (Pair xs ys) where f (Pair x y) = (x, y)
Note that the two containers must have elements of the same type.
however, allows the containers to have elements of different types:
zipNaperian :: Naperian f => f a -> f b -> f (a, b)
It does so by allowing distribution of
Functor1s, where a
Functor1 is a
Hask -> Hask to
class Functor1 w where map1 :: (forall a. f a -> g a) -> w f -> w g distribute1 :: (Naperian f, Functor1 w) => w f -> f (w Identity)
The more polymorphic zip can then be implemented by distributing the
data Pair1 a b f = Pair1 (f a) (f b) instance Functor1 (Pair1 a b) where ... zipNaperian :: Naperian f => f a -> f b -> f (a, b) zipNaperian as bs = fmap f $ distribute1 (Pair1 as bs) where f (Pair1 (Identity a) (Identity b)) = (a, b)
Naperian functors can be shown to be equivalent to
Rep, by selecting
Rep f = ∀x. f x -> x. That is, a position in a
Naperian container can be represented as a function which gets the value at
tabulate can then be derived using the
newtype TabulateArg a f = TabulateArg ((forall x. f x -> x) -> a)
The rest is left as an exercise for the reader.