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grid :: Int -> Int -> [(Int,Int)]
grid m n = [(x,y) | x <- [0..m], y <- [0..n]]

square :: Int -> [(Int,Int)]
square n = [(x,y) | (x,y) <- grid n n, x /= y]

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replicate :: Int -> a -> [a]
replicate n x = [x | _ <- [1..n]]

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pyths :: Int -> [(Int,Int,Int)]
pyths n = [(a,b,c) | c <- [1..n],
                     b <- [1..c],
                     a <- [1..b],
                     a^2 + b^2 == c^2]

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factors :: Int -> [Int]
factors n = [x | x <- [1..n], n `mod` x == 0]

perfects :: Int -> [Int]
perfects n = [x | x <- [1..n],
                  x == sum (factors x) - ]

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list1 :: [(Int,Int)]
list1 =  concat [[(x,y) | y <- [3,4]] | x <- [1,2]]

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find :: Eq a => a -> [(a,b)] -> [b]
find k t = [v | (k',v) <- t, k == k']

positions :: Eq a => a -> [a] -> [Int]
positions x xs = [i | (x',i) <- zip xs [0..], x == x']

positions' :: Eq a => a -> [a] -> [Int]
positions' x xs = find x (zip xs [0..])

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chisqr :: [Float] -> [Float] -> Float
chisqr os es = sum [((o-e)^2)/e | (o,e) <- zip os es]

scalarproduct :: [Int] -> [Int] -> Int
scalarproduct xs ys = sum [x*y | (x,y) <- zip xs ys]

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-- We assume that the input is not only upper case letters.
import Data.Char

-- Returns a list of indices of the specifed value
positions :: Eq a => a -> [a] -> [Int]
positions x xs = [i | (x',i) <- zip xs [0..], x == x']

-- Returns the number of lowercase characters
lowers :: String -> Int
lowers xs = length [x | x <- xs, 'a' <= x && x <= 'z']

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facRecursive :: Int -> Int
facRecursive n
    | n < 0     = 0
    | n == 0    = 1
    | otherwise = n * facRecursive (n - 1)

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sumdown :: Int -> Int
sumdown 0 = 0
sumdown n = n + sumdown (n - 1)