Fixed point algebras (To be Updated)

algebra.hs

-- Algebra gives the ability to perform calculaitons with number and symbols
-- algebra treat symbols as elements of a vector space, multiplied by scalar quantity and added to each other

type Algebra f a = f a -> a
type SimpleA = Algebra ExprF Int

data ExprF a = Cons Int
    | Add a a
    | Mul a a

newtype Fix f = In (f (Fix f))

type Expr = Fix ExprF
val :: Expr
val = In (Cons 12)

testExpr = In $ (In $ (In $ Cons 2) `Add` 
                (In $ Cons 3)) `Mul` (In $ Cons 4)


instance Functor ExprF where
    fmap f (Cons i) = Cons i
    fmap f (Add a b) = Add (f a) (f b)
    fmap f (Mul a b) = Mul (f a) (f b)


alg :: ExprF Int -> Int
alg (Cons i) = i
alg (Add a b) = a + b
alg (Mul a b) = a * b

algf :: SimpleA
algf (Cons i) = i
algf (Add a b) = a + b
algf (Mul a b) = a * b

-- Samples
-- algf $ fmap (+1) $ Add 7 5
-- alg $ Add 7 5
-- alg $ fmap (*2) $ Add 7 5
-- data Expr = Cons Int
--     | Add Expr Expr
--     | Mul Expr Expr


-- type Name = [Char]