profunctorial container constraint characterizations

Profunctorial.hs

module Profunctorial where

import Control.Applicative
import Data.Function
import Data.Functor
import Data.Functor.Contravariant
import Data.Profunctor
import Witherable

class Foldable f where
  foldP ::
    ( Strong p, Choice p
    , forall x. Applicative (p x)
    , forall x. Contravariant (p x)
    ) => p a b -> p (f a) (f b)

class (Functor f, Foldable f) => Traversable f where
  -- prop> fmap = traverseP
  -- prop> foldP = traverseP
  traverseP ::
    ( Strong p, Choice p
    , forall x. Applicative (p x)
    ) => p a b -> p (f a) (f b)

class Applicative f => Cotraversable f where
  cotraverseP ::
    ( Closed p
    , forall x. Applicative (p x)
    ) => p a b -> p (f a) (f b)

replicateP ::
  ( Traversable f
  , Cotraversable f
  , Profunctor p
  , forall x. Applicative (p x)
  ) => p a b -> p (f a) (f b)
replicateP p =
  traverse (\f -> lmap f p) (cotraverse id id)
  where
    cotraverse = runCostar . cotraverseP . Costar
    traverse = runStar . traverseP . Star

witherP ::
  ( Traversable f
  , Filterable f
  , Strong p, Choice p
  , forall x. Alternative (p x)
  ) => p a b -> p (f a) (f b)
witherP = rmap catMaybes . traverseP . optional