```{-# LANGUAGE CPP, TypeOperators, TemplateHaskell, ScopedTypeVariables, FlexibleContexts #-}
{-# OPTIONS_HADDOCK hide #-}
-----------------------------------------------------------------------------
-- |
-- Module      :  Numeric.AD.Internal.Jet
-- Copyright   :  (c) Edward Kmett 2010
-- License     :  BSD3
-- Maintainer  :  ekmett@gmail.com
-- Stability   :  experimental
-- Portability :  GHC only
--
-----------------------------------------------------------------------------
( Jet(..)
, tailJet
, jet
) where

#ifndef MIN_VERSION_base
#define MIN_VERSION_base(x,y,z) 1
#endif

import Control.Applicative
import Data.Foldable
import Data.Traversable
import Data.Monoid
#if MIN_VERSION_base(4,4,0)
import Data.Typeable (Typeable1(..), TyCon, mkTyCon3, mkTyConApp)
#else
import Data.Typeable (Typeable1(..), TyCon, mkTyCon, mkTyConApp)
#endif

infixl 3 :-

-- | A 'Jet' is a tower of all (higher order) partial derivatives of a function
--
-- At each step, a @'Jet' f@ is wrapped in another layer worth of @f@.
--
-- > a :- f a :- f (f a) :- f (f (f a)) :- ...
data Jet f a = a :- Jet f (f a)

-- | Used to sidestep the need for UndecidableInstances.
newtype Showable = Showable (Int -> String -> String)

instance Show Showable where
showsPrec d (Showable f) = f d

showable :: Show a => a -> Showable
showable a = Showable (\d -> showsPrec d a)

-- Polymorphic recursion precludes 'Data' in its current form, as no Data1 class exists
-- Polymorphic recursion also breaks 'show' for 'Jet'!
-- factor Show1 out of Lifted?
instance (Functor f, Show (f Showable), Show a) => Show (Jet f a) where
showsPrec d (a :- as) = showParen (d > 3) \$
showsPrec 4 a . showString " :- " . showsPrec 3 (fmap showable <\$> as)

instance Functor f => Functor (Jet f) where
fmap f (a :- as) = f a :- fmap (fmap f) as

instance Foldable f => Foldable (Jet f) where
foldMap f (a :- as) = f a `mappend` foldMap (foldMap f) as

instance Traversable f => Traversable (Jet f) where
traverse f (a :- as) = (:-) <\$> f a <*> traverse (traverse f) as

-- | Take the tail of a 'Jet'.
tailJet :: Jet f a -> Jet f (f a)
tailJet (_ :- as) = as
{-# INLINE tailJet #-}

-- | Take the head of a 'Jet'.
headJet :: Jet f a -> a
headJet (a :- _) = a
{-# INLINE headJet #-}

-- | Construct a 'Jet' by unzipping the layers of a 'Cofree' 'Comonad'.
jet :: Functor f => Cofree f a -> Jet f a
jet (a :< as) = a :- dist (jet <\$> as)
where
dist :: Functor f => f (Jet f a) -> Jet f (f a)
dist x = (headJet <\$> x) :- dist (tailJet <\$> x)

instance Typeable1 f => Typeable1 (Jet f) where
typeOf1 tfa = mkTyConApp jetTyCon [typeOf1 (undefined `asArgsType` tfa)]
where asArgsType :: f a -> t f a -> f a
asArgsType = const

jetTyCon :: TyCon
#if MIN_VERSION_base(4,4,0)