-- {-# LANGUAGE UndecidableInstances #-}

-- | Semirings.  Original: Agda.Terminatio.Semiring

module Semiring
  ( HasZero(..), SemiRing(..)
  , Semiring(..)
--  , semiringInvariant
  , integerSemiring
  , boolSemiring
  ) where

import Data.Monoid


{- | SemiRing type class.  Additive monoid with multiplication operation.
Inherit addition and zero from Monoid. -}

class (Eq a, Monoid a) => SemiRing a where
--  isZero   :: a -> Bool
  multiply :: a -> a -> a


-- | @HasZero@ is needed for sparse matrices, to tell which is the element
--   that does not have to be stored.
--   It is a cut-down version of @SemiRing@ which is definable
--   without the implicit @?cutoff@.
class Eq a => HasZero a where
  zeroElement :: a

-- | Semirings.

data Semiring a
  = Semiring { add  :: a -> a -> a  -- ^ Addition.
             , mul  :: a -> a -> a  -- ^ Multiplication.
             , zero :: a            -- ^ Zero.
-- The one is never used in matrix multiplication
--             , one  :: a            -- ^ One.
             }

-- | Semiring invariant.

-- I think it's OK to use the same x, y, z triple for all the
-- properties below.

{-
semiringInvariant :: (Arbitrary a, Eq a, Show a)
                  => Semiring a
                  -> a -> a -> a -> Bool
semiringInvariant (Semiring { add = (+), mul = (*)
                            , zero = zero --, one = one
                            }) = \x y z ->
  associative (+)           x y z &&
  identity zero (+)         x     &&
  commutative (+)           x y   &&
  associative (*)           x y z &&
--  identity one (*)          x     &&
  leftDistributive (*) (+)  x y z &&
  rightDistributive (*) (+) x y z &&
  isZero zero (*)           x
-}

------------------------------------------------------------------------
-- Specific semirings

-- | The standard semiring on 'Integer's.

instance HasZero Integer where
  zeroElement = 0

-- The following (unused) instance has been removed for GHC-8.4 compatibility
-- (Semigroup instance missing).
--
-- instance Monoid Integer where
--   mempty = 0
--   mappend = (+)

-- instance SemiRing Integer where
--   multiply = (*)


integerSemiring :: Semiring Integer
integerSemiring = Semiring { add = (+), mul = (*), zero = 0 } -- , one = 1 }

-- prop_integerSemiring = semiringInvariant integerSemiring

-- | The standard semiring on 'Bool's.

boolSemiring :: Semiring Bool
boolSemiring =
  Semiring { add = (||), mul = (&&), zero = False } --, one = True }

-- prop_boolSemiring = semiringInvariant boolSemiring

------------------------------------------------------------------------
-- All tests

{-
tests :: IO Bool
tests = runTests "Agda.Termination.Semiring"
  [ quickCheck' prop_integerSemiring
  , quickCheck' prop_boolSemiring
  ]
-}