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zo$eFWq}+DO14;6$>_LK=>ODtAIM zbw`OFtscne+o4;!t*G#!RC3Xli*VVlD6wSWabyvyL^=G51|FpdxF)4w`f2LiantSc z_HV}bf;gMy=8nA)N*UHSAT5(^haTt8*lq=+lb a +fst (x, _) = x + +snd :: (a, b) -> b +snd (_, y) = y + +-- Funktionskomposition +(.) :: (b -> c) -> (a -> b) -> a -> c +(f . g) x = f (g x) + +-- Identitätsfunktion +id :: a -> a +id x = x + +data Bool = True | False deriving (Prelude.Show, Prelude.Eq) + +foldb :: a -> a -> Bool -> a +foldb a b True = a +foldb a b False = b + +data Nat = Zero | Succ Nat deriving (Prelude.Eq) + +foldn :: a -> (a -> a) -> Nat -> a +foldn c g Zero = c +foldn c g (Succ n) = g (foldn c g n) + +zero = Zero +one = Succ Zero +two = Succ one +three = Succ two +four = Succ three + +add :: Nat -> Nat -> Nat +add m = foldn m Succ + +mult :: Nat -> Nat -> Nat +mult m = foldn Zero (add m) + +exp :: Nat -> Nat -> Nat +exp m = foldn (Succ Zero) (mult m) + +data List a = Nil | Cons a (List a) deriving (Prelude.Show, Prelude.Eq) + +-- fold für Listen +foldL :: b -> (a -> b -> b) -> List a -> b +foldL c g Nil = c +foldL c g (Cons x xs) = g x (foldL c g xs) + +-- scan für Listen +scanL :: b -> (a -> b -> b) -> List a -> List b +scanL c g Nil = Cons c Nil +scanL c g (Cons x xs) = Cons (g x y) ys + where ys@(Cons y _) = scanL c g xs + +-- alternativ ohne @-Syntax: +scanL' :: b -> (a -> b -> b) -> List a -> List b +scanL' c g xs = snd (scanLHelper c g xs) + where + scanLHelper c g Nil = (c, Cons c Nil) + scanLHelper c g (Cons x xs) = (z, Cons z (snd ys)) + where + ys = scanLHelper c g xs + z = g x (fst ys) + +length :: List a -> Nat +length l = foldL Zero (\x -> Succ) l + +snoc :: a -> List a -> List a +snoc a l = foldL (Cons a Nil) Cons l + +reverse :: List a -> List a +reverse l = foldL Nil snoc l + + +-- cat ohne foldr +concat :: List a -> List a -> List a +concat Nil = id +concat (Cons x xs) = \ys -> Cons x (concat xs ys) + +-- cat mit foldr +cat' :: List a -> List a -> List a +cat' xs ys = foldL ys Cons xs + +data Tree a = Leaf a | Node (Tree a) (Tree a) deriving (Prelude.Show, Prelude.Eq) + +-- fold für Bäume +foldt :: (a -> b) -> (b -> b -> b) -> Tree a -> b +foldt c g (Leaf x) = c x +foldt c g (Node x y) = g (foldt c g x) (foldt c g y) + +-- front ohne foldt +front :: Tree a -> List a +front (Leaf x) = Cons x Nil +front (Node x y) = concat (front x) (front y) + +-- front mit foldt +front' :: Tree a -> List a +front' = foldt (\x -> Cons x Nil) concat + +-- Fakultätsfunktion aus der Vorlesung +fact :: Nat -> Nat +fact = f . foldn c h + where + f = snd + c = (Zero, one) + h = (\(x,y) -> (Succ x, mult (Succ x) y))