Gaster's Algebraic hierarchy

GasterAlgebraicHierarchy97.hs

-- http://benedictgaster.org/wp-content/uploads/2012/08/haskwork97.pdf
-- Polymorphic Extensible Records for Haskell (page 3)

type Monoid v r = Rec { plus :: v -> v -> v , id :: v | r }

type Group  v r = Monoid v { inv  :: v -> v | r }

type Ring   v r = Group  v { mult :: v -> v -> v , one :: v | r }

iMonoid :: Monoid Int {}
iMonoid = ( plus = (+) , id = 0 )

iGroup  :: Group  Int {}
iGroup  = (inv = negate | iMonoid)

iRing   :: Ring   Int {}
iRing   = ( mult = (*) , one = 1 | iGroup )

--

sum :: Monoid a r -> [a] -> a
sum monoid = foldr ( monoid.plus ) ( monoid.id )