--------------------
-- Exercise 1
--------------------
prim c g 0 x = c x
prim c g n x = g (f (n - 1) x) (n - 1) x
    where
        f = prim c g

m2 :: Integer -> () -> Integer
m2 n x = prim (\_ -> 0) (\a -> \n -> \x -> a + 2) n x

-- >>> print (m2 8 ())
-- 16
--

e2 :: Integer -> () -> Integer
e2 n x = prim (\_ -> 1) (\a -> \n -> \x -> a * 2) n x
-- >>> print (e2 4 ())
-- 16
--

exp :: Integer -> Integer -> Integer
exp x n = prim (\_ -> 1) (\a -> \n -> \x -> a * x) n x
-- >>> print (Main.exp 2 3)
-- 8
--

fact :: Integer -> () -> Integer
fact n x = prim (\_ -> 1) (\a -> \n -> \x -> a * (n + 1)) n x
-- >>> print (fact 3 ())
-- 6
--

--------------------
-- Exercise 2
--------------------
f g x
    | x == 0 = g x
    | otherwise = g $ f g (x - 1)
-- Nicht endrekursiv

length xs = case xs of
    [] -> 0
    x : xs -> (+1) $ Main.length xs
-- ist endrekursiv

length' ls = aux
    $ map ( const 1)
    $ ls
    where
        aux ys = case ys of
            [] -> 0
            [x] -> x
            x : xs -> aux $ map (\y -> (+1) x ) xs
-- length' und aux sind nicht endrekursiv

--------------------
-- Exercise 3
--------------------
-- sieve :: ( a -> a -> Bool ) -> [ a ] -> [ a ]
-- sieve pred xs = case xs of
--     [] -> []
--     x : xs -> x :( sieve pred $ filter ( pred x ) xs )
sieve :: (a -> a -> Bool) -> [a] -> [a]
sieve pred xs = realSieve [] xs
    where
        realSieve acc [] = acc
        realSieve acc (x : xs) = realSieve (acc ++ [x]) (filter (pred x) xs)

-- >>> sieve (\ x -> \ y -> y > x) [1, 2, 3, 1, 2, 9, 7]
-- [1,2,3,9]
--
--------------------
-- Exercise 3 b) 1.
--------------------
-- >>> (\ n -> sieve (\ x -> \ y -> (y `mod` x) > 0) [2..n]) 100
-- [2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97]
--

sieveC :: (a -> a -> Bool) -> [a] -> [a]
sieveC pred xs = realSieve id xs
    where
        realSieve f [] = f []
        realSieve f (x : xs) = realSieve (\ items -> filter (pred x) (f items)) xs

-- >>> sieve (\ x -> \ y -> y > x) [6, 2, 3, 1, 2, 9, 7]
-- [6,9]
--
--------------------
-- Exercise 3 b) 2.
--------------------
-- >>> (\ n -> sieve (\ x -> \ y -> (y `mod` x) > 0) [2..n]) 100
-- [2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97]
--

--------------------
-- Exercise 4
--------------------
data Tree a = Tree a [Tree a]
depth :: Tree a -> Integer

-- depth (Tree _ subtrees) =
--     1 + (maximum $ 0 : (map depth subtrees))
depth tree = realDepth [tree] 0
    where
        realDepth [] x = x
        realDepth trees x = realDepth (concatMap (\ (Tree _ subtrees) -> subtrees) trees) (x + 1)

-- >>> depth (Tree 1 [Tree 1 []])
-- 2
--

-- Notes
-- realDepth [(Tree 1 [Tree 1 []])] 0
-- realDepth (concatMap (\ (Tree _ subtrees) -> subtrees) [Tree 1 [Tree 1 []]]) (0 + 1)
-- realDepth [Tree 1 []] 1
-- realDepth (concatMap (\ (Tree _ subtrees) -> subtrees) [Tree 1 []]) ((0 + 1) + 1)
-- realDepth [] 2
-- 2

fibonacci :: Integer -> Integer
fibonacci = fib 0 1
    where
        fib acc b 0 = acc
        fib acc b x = fib b (acc + b) (x - 1)

-- >>> fibonacci 6
-- 8
--

tribonacci :: Integer -> Integer
tribonacci = trib 0 0 1
    where
        trib acc b c 0 = acc
        trib acc b c x = trib b c (acc + b + c) (x - 1)

-- >>> tribonacci 7
-- 13
--