```module Pandora.Pattern.Functor.Traversable (Traversable (..)) where

import Pandora.Core.Functor (type (:.), type (>))
import Pandora.Core.Morphism (identity, (.))
import Pandora.Pattern.Functor.Covariant (Covariant)
import Pandora.Pattern.Functor.Applicative (Applicative)
import Pandora.Pattern.Functor.Pointable (Pointable)

{- |
> Let f :: (Applicative t, Applicative g) => t a -> u a
> Let p :: (Pointable t, Pointable g) => t a -> u a

> When providing a new instance, you should ensure it satisfies the four laws:
> * Naturality of traversing: g . traverse f ≡ traverse (g . f)
> * Naturality of sequencing: f . sequence = sequence . comap f
> * Preserving point: p (point x) ≡ point x
> * Preserving apply: f (x <*> y) ≡ f x <*> f y
-}

infixl 5 ->>, ->>>, ->>>>, ->>>>>

class Covariant t => Traversable t where
{-# MINIMAL (->>) #-}
-- | Infix version of 'traverse'
(->>) :: (Pointable u, Applicative u) => t a -> (a -> u b) -> u :. t > b

-- | Prefix version of '->>'
traverse :: (Pointable u, Applicative u) => (a -> u b) -> t a -> u :. t > b
traverse f t = t ->> f
-- | The dual of 'distribute'
sequence :: (Pointable u, Applicative u) => (t :. u) a -> u :. t > a
sequence t = t ->> identity

-- | Infix versions of `traverse` with various nesting levels
(->>>) :: (Pointable u, Applicative u, Traversable v)
=> v :. t > a -> (a -> u b) -> u :. v :. t > b
x ->>> f = (traverse . traverse) f x
(->>>>) :: (Pointable u, Applicative u, Traversable v, Traversable w)
=> w :. v :. t > a -> (a -> u b) -> u :. w :. v :. t > b
x ->>>> f = (traverse . traverse . traverse) f x
(->>>>>) :: (Pointable u, Applicative u, Traversable v, Traversable w, Traversable j)
=> j :. w :. v :. t > a -> (a -> u b) -> u :. j :. w :. v :. t > b
x ->>>>> f = (traverse . traverse . traverse . traverse) f x
```