module FCT where import Data.List f x = x^2 + 1 list2fct :: Eq a => [(a,b)] -> a -> b list2fct [] _ = error "function not total" list2fct ((u,v):uvs) x | x == u = v | otherwise = list2fct uvs x fct2list :: (a -> b) -> [a] -> [(a,b)] fct2list f xs = [ (x, f x) | x <- xs ] ranPairs :: Eq b => [(a,b)] -> [b] ranPairs f = nub [ y | (_,y) <- f ] listValues :: Enum a => (a -> b) -> a -> [b] listValues f i = (f i) : listValues f (succ i) listRange :: (Bounded a, Enum a) => (a -> b) -> [b] listRange f = [ f i | i <- [minBound..maxBound] ] curry3 :: ((a,b,c) -> d) -> a -> b -> c -> d curry3 f x y z = f (x,y,z) uncurry3 :: (a -> b -> c -> d) -> (a,b,c) -> d uncurry3 f (x,y,z) = f x y z f1 x = x^2 + 2 * x + 1 g1 x = (x + 1)^2 f1' = \x -> x^2 + 2 * x + 1 g1' = \x -> (x + 1)^2 g 0 = 0 g n = g (n-1) + n g' n = ((n + 1) * n ) / 2 h 0 = 0 h n = h (n-1) + (2*n) k 0 = 0 k n = k (n-1) + (2*n-1) fac 0 = 1 fac n = fac (n-1) * n fac' n = product [1..n] restrict :: Eq a => (a -> b) -> [a] -> a -> b restrict f xs x | elem x xs = f x | otherwise = error "argument not in domain" restrictPairs :: Eq a => [(a,b)] -> [a] -> [(a,b)] restrictPairs xys xs = [ (x,y) | (x,y) <- xys, elem x xs ] image :: Eq b => (a -> b) -> [a] -> [b] image f xs = nub [ f x | x <- xs ] coImage :: Eq b => (a -> b) -> [a] -> [b] -> [a] coImage f xs ys = [ x | x <- xs, elem (f x) ys ] imagePairs :: (Eq a, Eq b) => [(a,b)] -> [a] -> [b] imagePairs f xs = nub [ y | (x,y) <- f, elem x xs] coImagePairs :: (Eq a, Eq b) => [(a,b)] -> [b] -> [a] coImagePairs f ys = [ x | (x,y) <- f, elem y ys] injective :: Eq b => (a -> b) -> [a] -> Bool injective f [] = True injective f (x:xs) = notElem (f x) (image f xs) && injective f xs surjective :: Eq b => (a -> b) -> [a] -> [b] -> Bool surjective f xs [] = True surjective f xs (y:ys) = elem y (image f xs) && surjective f xs ys c2f, f2c :: Int -> Int c2f x = div (9 * x) 5 + 32 f2c x = div (5 * (x - 32)) 9 succ1 :: Integer -> Integer succ1 = \ x -> if x < 0 then error "argument out of range" else x+1 succ2 :: Integer -> [Integer] succ2 = \ x -> if x < 0 then [] else [x+1] pcomp :: (b -> [c]) -> (a -> [b]) -> a -> [c] pcomp g f = \ x -> concat [ g y | y <- f x ] succ3 :: Integer -> Maybe Integer succ3 = \ x -> if x < 0 then Nothing else Just (x+1) mcomp :: (b -> Maybe c) -> (a -> Maybe b) -> a -> Maybe c mcomp g f = (maybe Nothing g) . f part2error :: (a -> Maybe b) -> a -> b part2error f = (maybe (error "value undefined") id) . f fct2equiv :: Eq a => (b -> a) -> b -> b -> Bool fct2equiv f x y = (f x) == (f y) block :: Eq b => (a -> b) -> a -> [a] -> [a] block f x list = [ y | y <- list, f x == f y ]