ThProg-Merkzettel/folds.hs

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2018-10-08 16:57:32 +02:00
import qualified Prelude
-- Paare
fst :: (a, b) -> 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))