variables :: Expr a -> [String]
variables x = case x of
    Var y -> [y]
    _ -> composOpFold [] (++) variables x



zeroCount :: Expr a -> Int
zeroCount x = case x of
    Div y (Val 0) -> 1 + zeroCount y
    _ -> composOpFold 0 (+) zeroCount x

simplify :: Expr a -> Expr a
simplify x = case x of
    Sub  a b -> simplify $ Add (simplify a) (Neg (simplify b))
    Add  a b -> case  (simplify a, simplify b) of
                      (a',b')  | a' == b'   -> Mul (Val 2) a'
                               | otherwise  -> Add a' b'
    _ -> composOp simplify x

rename :: Tree c -> Tree c
rename t = case t of
    V x -> V ("_" ++ x)
    _   -> composOp rename t


symbols :: Tree c -> [(Tree Var, Tree Typ)]
symbols x = case x of
    SDecl t v -> [(v,t)]
    _ -> composOpMonoid symbols t


constFold :: Tree c -> Tree c
constFold e = case e of
    EAdd x y -> case  (f x, f y) of
                      (EInt n, EInt m) -> EInt (n+m)
                      (x',y') -> EAdd x' y'
    _ -> composOp constFold e

increase :: Integer -> Tree c -> Tree c
increase k c = case c of
    S s -> S (s + k)
    _ -> composOp (increase k) c


incrOne :: Name -> Float -> Tree c -> Tree c
incrOne d k x = case x of
    D n _ _ | n == d -> increase k x
    _ -> composOp (incrOne d k) x


salaryBill :: Tree c -> Integer
salaryBill x = case x of
    S s -> s
    _ -> composOpFold 0 (+) salaryBill x