Question

Here’s a simple function. It takes an input Int and returns a (possibly empty) list of (Int, Int) pairs, where the input Int is the sum of the cubed elements of any of the pairs.

cubeDecomposition :: Int -> [(Int, Int)]
cubeDecomposition n = [(x, y) | x <- [1..m], y <- [x..m], x^3 + y^3 == n] 
  where m = truncate $ fromIntegral n ** (1/3)

-- cubeDecomposition 1729
-- [(1,12),(9,10)]

I want to test the property that the above is true; if I cube each element and sum any of the return tuples, then I get my input back:

import Control.Arrow 

cubedElementsSumToN :: Int -> Bool
cubedElementsSumToN n = all (== n) d
    where d = map (uncurry (+) . ((^3) *** (^3))) (cubeDecomposition n)

For runtime considerations, I’d like to limit the input Ints to a certain size when testing this with QuickCheck. I can define an appropriate type and Arbitrary instance:

{-# LANGUAGE GeneralizedNewtypeDeriving #-}

import Test.QuickCheck

newtype SmallInt = SmallInt Int
    deriving (Show, Eq, Enum, Ord, Num, Real, Integral)

instance Arbitrary SmallInt where
    arbitrary = fmap SmallInt (choose (-10000000, 10000000))

And then I guess I have to define versions of the function and property that use SmallInt rather than Int:

cubeDecompositionQC :: SmallInt -> [(SmallInt, SmallInt)]
cubeDecompositionQC n = [(x, y) | x <- [1..m], y <- [x..m], x^3 + y^3 == n] 
  where m = truncate $ fromIntegral n ** (1/3)

cubedElementsSumToN' :: SmallInt -> Bool
cubedElementsSumToN' n = all (== n) d
    where d = map (uncurry (+) . ((^3) *** (^3))) (cubeDecompositionQC n)

-- cubeDecompositionQC 1729
-- [(SmallInt 1,SmallInt 12),(SmallInt 9,SmallInt 10)]

This works fine, and the standard 100 tests pass as expected. But it seems unnecessary to define a new type, instance, and function when all I really need is a custom generator. So I tried this:

smallInts :: Gen Int
smallInts = choose (-10000000, 10000000)

cubedElementsSumToN'' :: Int -> Property
cubedElementsSumToN'' n = forAll smallInts $ \m -> all (== n) (d m)
    where d =   map (uncurry (+) . ((^3) *** (^3)))
              . cubeDecomposition

Now, the first few times I ran this, everything worked fine, and all tests pass. But on subsequent runs I observed failures. Bumping up the test size reliably finds one:

*** Failed! Falsifiable (after 674 tests and 1 shrink):  
0
8205379

I’m a bit confused here due to the presence of two shrunken inputs – 0 and 8205379 – returned from QuickCheck, where I would intuitively expect one. Also, those inputs work as predicted (on my show-able property, at least):

*Main> cubedElementsSumToN 0
True
*Main> cubedElementsSumToN 8205379
True

So it seems like obviously there’s a problem in the property that uses the custom Gen I defined.

What have I done wrong?

Answer 1

I quickly realized that the property as I’ve written it is obviously incorrect. Here’s the proper way to do it, using the original cubedElementsSumToN property:

quickCheck (forAll smallInts cubedElementsSumToN)

which reads quite naturally.