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A monoid leveraging addition for the associative operation

Illustration

Number types may be completed into monoids in different ways. The sum monoid favors addition for the associative operation of the monoid. We illustrate the sum monoid in Language:Haskell on the grounds of the type class Monoid with the first two members needed for a minimal complete definition: 

-- The type class Monoid
class Monoid a
  where
    mempty :: a -- neutral element
    mappend :: a -> a -> a -- associative operation
    mconcat :: [a] -> a -- fold
    mconcat = foldr mappend mempty

The sum monoid relies on a designated type which essentially wraps a number type: 

-- The type of the sum monoid
newtype Sum a = Sum { getSum :: a }

Here is the type-class instance for the sum monoid: 

-- The Monoid instance for numbers under addition
instance Num a => Monoid (Sum a)
  where
    mempty = Sum 0
    x `mappend` y = Sum (getSum x + getSum y)

For further illustration, we can reconstruct the standard sum function in a monoidal way. To this end, we first review the normal definition in terms of foldr: 

-- A foldr-based definition of sum
sum' :: Num a => [a] -> a
sum' = foldr (+) 0


-- A monoidal definition of sum
sum'' :: Num a => [a] -> a
sum'' = getSum . mconcat . map Sum

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