The derivation of Quickselect from Quicksort, see my blog post https://xcycl.wordpress.com/2019/01/02/fromquicksorttoquickselect/

Quickselect.hs

{-# LANGUAGE TypeApplications #-}
import Test.QuickCheck
import Data.List (sort)

selectOrigin k = (!! k) . sort 

select :: (Ord a) => Int -> [a] -> a
select k (x:xs) = case compare k n of
  LT -> select k ys 
  EQ -> x
  GT -> select (k-n-1) zs
  where
    (n, ys, zs) = partition' (< x) xs 

partition' :: (a -> Bool) -> [a] -> (Int, [a], [a])
partition' p = foldr op (0, [], [])
  where
    op x (n, ys, zs)
      | p x       = (1+n, x:ys, zs)
      | otherwise = (n,   ys, x:zs)

prop1 :: Ord a => NonEmptyList a -> Int -> Bool
prop1 (NonEmpty xs) k =
  let k' = k `mod` length xs
  in selectOrigin k' xs == select k' xs 

prop2 :: [Int] -> Int -> [Int] -> Int -> Bool
prop2 xs y ys n = 
  let m = length xs
      n' = n `mod` (m + length (y:ys))
  in (xs ++ y:ys) !! n' == 
     case compare n' m of
       LT -> xs !! n'
       EQ -> y
       GT -> ys !! (n' - m - 1)

main :: IO ()
main = quickCheck (prop1 @Int) 
      >> quickCheck prop2