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.Pattern.Operation.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 1 <-/------
infixl 2 <-/-----
infixl 3 <-/----
infixl 4 <-/---, <-/-/-
infixl 5 <-/--
infixl 6 <-/-

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))