module Pandora.Pattern.Functor.Traversable where

import Pandora.Pattern.Functor.Covariant (Covariant)
import Pandora.Pattern.Functor.Monoidal (Monoidal)
import Pandora.Pattern.Morphism.Straight (Straight)
import Pandora.Paradigm.Algebraic.Product ((:*:))

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

> When providing a new instance, you should ensure it satisfies:
> * Numeratority of traversing: g . (f <<--) ≡ (g . f <<--)
> * Numeratority of sequencing: f . (identity <<--)= (identity <<--) . (f <-|-)
> * Preserving point: p (point x) ≡ point x
> * Preserving apply: f (x <.-*- y) ≡ f x <.-*- f y
-}

infixl 4 <<-<<-

infixl 1 <<-------
infixl 2 <<------
infixl 3 <<-----
infixl 4 <<----
infixl 5 <<---
infixl 6 <<--
infixl 7 <<-

class Covariant source target t => Traversable source target t where
	(<<-) :: (Covariant source target u, Monoidal (Straight source) (Straight target) (:*:) (:*:) u) => source a (u b) -> target (t a) (u (t b))

	(<<-------), (<<------), (<<-----), (<<----), (<<---), (<<--)
		:: (Covariant source target u, Monoidal (Straight source) (Straight target) (:*:) (:*:) u)
		=> source a (u b) -> target (t a) (u (t b))
	(<<-------) = (<<-)
	(<<------) = (<<-)
	(<<-----) = (<<-)
	(<<----) = (<<-)
	(<<---) = (<<-)
	(<<--) = (<<-)

(<<-<<-) :: forall t u v category a b .
	(Traversable category category t, Covariant category category u, Monoidal (Straight category) (Straight category) (:*:) (:*:) u, Traversable category category v)
	=> category a (u b) -> category (v (t a)) (u (v (t b)))
(<<-<<-) f = ((<<-) ((<<-) @category @category @t f))