-----
-- More examples from lecture

data Nat = Zero | Succ Nat  deriving (Show, Eq)

plus Zero n = n
plus (Succ k) n = Succ (plus k n)

mult Zero n = Zero
mult (Succ k) n = plus n (mult n k)


num Zero = 0
num (Succ k) = 1 + (num k)

nat 0 = Zero
nat n | n>0 = Succ (nat (n-1))

-- Nat is a *type*, Zero and Succ are *constructors*

data List = Nil | Cons Int List  deriving (Show, Eq)

hlist Nil = []
hlist (Cons a ls) = a : (hlist ls)

flist [] = Nil
flist (x:xs) = Cons x (flist xs)


data Boolean = Troo | Phalse


knot Troo = Phalse
knot Phalse = Troo

-------
-- some functions on lists

hd (x:_) = x

tl (_:xs) = xs

null' [] = True
null' (_:_) = False

last' (x:xs) | xs == [] = x
             | otherwise = last' xs
            


sum' []  = 0
sum' (n:ns)  = n + sum'  ns

--         ^     ^ ^^^
--         v     v vvvv

prod' [] = 1
prod' (n:ns) = n * prod' ns


fold f u [] = u
fold f u (x:xs) = x `f` fold f u xs