A short Haskell program using QuickCheck to identify whether a defined relation R is an equivalence relation or partial ordering

properties.hs

import Test.QuickCheck

-- unfortunately, function names cannot start with uppercase letters
r :: Positive Integer -> Positive Integer -> Bool
r a b =
    let (x, y) = (getPositive a, getPositive b) in
    y `mod` (x * x) == 0

implies p q = not p || q

prop_reflexive :: Positive Integer -> Bool
prop_reflexive a = a `r` a

prop_symmetric :: Positive Integer -> Positive Integer -> Bool
prop_symmetric a b = a `r` b && b `r` a

prop_antisymmetric :: Positive Integer -> Positive Integer -> Bool
prop_antisymmetric a b = ((a `r` b) && (b `r` a)) `implies` (b == a)

prop_transitive :: Positive Integer -> Positive Integer -> Positive Integer -> Bool
prop_transitive x y z = ((x `r` y) && (y `r` z)) `implies` (x `r` z)

-- Output whether a property is satisfied, or otherwise which cases don't satisfy the property
main :: IO ()
main = do
    quickCheck prop_transitive
    quickCheck prop_antisymmetric
    quickCheck prop_symmetric
    quickCheck prop_reflexive