Troll Monoid

TrollMonoid.hs

{-# LANGUAGE GeneralizedNewtypeDeriving #-}

import Test.QuickCheck
import Data.Monoid
import Data.Word

newtype Troll = Troll Word64
  deriving (Eq, Num, Show, Arbitrary)

t :: Troll
t = 10 -- set this to anything, and it represents a new Monoid

instance Monoid Troll where
  mempty = 0
  mappend 0 x = x -- obey the identity law by definition
  mappend x 0 = x
  mappend x y = x + y + t
  -- these definitions are trivially commutative

  -- this obeys the associativity laws:
  -- first, with nothing equal to 0
  -- (x <> y) <> z
  -- (x + y + t) <> z
  -- (x + y + t) + z + t
  -- x + y + z + t + t
  --
  -- x <> (y <> z)
  -- x <> (y + z + t)
  -- x + (y + z + t) + t
  -- x + y + z + t + t
  --
  -- now, with one zero
  -- (0 <> x) <> y
  -- x <> y
  -- x + y + t
  --
  -- trivially equal to (x <> 0) <> y by commutativity
  --
  -- 0 <> (x <> y)
  -- x <> y    directly from our definition
  -- x + y + t
  --
  -- trivially equal to (x <> y) <> 0 by commutativity
  --
  -- now with two zeros
  --
  -- (0 <> x) <> 0
  -- x <> 0
  -- x
  --
  -- trivially equal to (x <> 0) <> 0, 0 <> (0 <> x), 0 <> (x <> 0), etc.
  --
  -- three zeros is trivial, any order of operations gives 0

identityLaws :: Troll -> Bool
identityLaws x = x <> mempty == x && mempty <> x == x

associativityLaws :: Troll -> Troll -> Troll -> Bool
associativityLaws x y z = (x <> y) <> z == x <> (y <> z)

main :: IO ()
main = do
  quickCheck identityLaws
  quickCheck associativityLaws