{-# LANGUAGE FlexibleContexts, UndecidableInstances, FlexibleInstances #-}

-- Lifting Numeric Prelude to Applicative
module ApplicativeAlgebra where

import Prelude hiding ((+), (*), (/), signum, fromInteger, abs)
  
import Control.Applicative

import Algebra.Additive
import Algebra.Field
import Algebra.Ring

-- The property that allows the lifting.  We can't lift all
-- Applicatives, so we need to bless the ones we can.

class Applicative a => NumericApp a  


instance (Algebra.Additive.C n, NumericApp a)
         => Algebra.Additive.C (a n) where
  (+) = liftA2 (+)
  zero = pure zero
  
instance (Algebra.Ring.C n, NumericApp a)
         => Algebra.Ring.C (a n) where
  (*) = liftA2 (*)

instance (Algebra.Field.C n, NumericApp a)
         => Algebra.Field.C (a n) where
  (/) = liftA2 (/)


{- test
  
data Pair n = Pair n n deriving (Eq, Show)

instance Functor Pair where
  fmap f (Pair a b) = Pair (f a) (f b)
  
instance Applicative Pair where
  pure f = Pair f f
  (<*>) (Pair fa fb) (Pair a b) = Pair (fa a) (fb b)

-- Allow lifting
instance NumericApp Pair 

-}