Testing higher-order properties with QuickCheck

Posted on February 24, 2020

I have just released two libraries to enhance QuickCheck for testing higher-order properties: quickcheck-higherorder and test-fun.

This is a summary of their purpose and main features. For more details, refer to the README and the implementations of the respective packages.


This project started from experiments to design laws for the mtl library. What makes a good law? I still don’t know the answer, but there is at least one sure sign of a bad law: find a counterexample! That’s precisely what property-based testing is useful for. As a byproduct, if you can’t find a counterexample after looking for it, that is some empirical evidence that the property is valid, especially if you expect counterexamples to be easy to find.

Ideally we would write down a property, and get some feedback from running it. Of course, complex applications will require extra effort for worthwhile results. But I believe that, once we have our property, the cost of entry to just start running test cases can be reduced to zero, and that many applications may benefit from it.

QuickCheck already offers a smooth user experience for testing simple “first-order properties”. quickcheck-higherorder extends that experience to higher-order properties.


A higher-order property is a property quantifying over functions. For example:

prop_bool :: (Bool -> Bool) -> Bool -> Property
prop_bool f x = f (f (f x)) === f x

Vanilla QuickCheck is sufficient to test such properties, provided you know where to find the necessary utilities. Indeed, simply passing the above property to the quickCheck runner results in a type error:

main :: IO ()
main = quickCheck prop_bool  -- Type error!

quickCheck tries to convert prop_bool to a Property, but that requires Bool -> Bool to be an instance of Show, which is of course absurd.1

Instead, functions must be wrapped in the Fun type:

prop_bool' :: Fun Bool Bool -> Bool -> Property
prop_bool' (Fn f) x = f (f (f x)) === f x

main :: IO ()
main = quickCheck prop_bool'  -- OK!

Compounded over many properties, this Fun/Fn boilerplate is repetitive. It becomes especially cumbersome when the functions are contained inside other data types.

quickcheck-higherorder moves that cruft out of sight. The quickCheck' runner replaces the original quickCheck, and infers that (->) should be replaced with Fun.

-- The first version
prop_bool :: (Bool -> Bool) -> Bool -> Property
prop_bool f x = f (f (f x)) === f x

main :: IO ()
main = quickCheck' prop_bool  -- OK!

Data and its representation

The general idea behind this is to distinguish the data that your application manipulates, from its representation that QuickCheck manipulates. The data can take any form, whatever is most convenient for the application, but its representation must be concrete enough so QuickCheck can randomly generate it, shrink it, and print it in the case of failure.

Vanilla QuickCheck handles the simplest case, where the data is identical to its representation, and gives up as soon as the representation has a different type, requiring us to manually modify the property to make the representation of its input data explicit. This is certainly not a problem that can generally be automated away, but the UX here still has room for improvement. quickcheck-higherorder provides a new way to associate data to its representation, via a type class Constructible, which quickCheck' uses implicitly.

class (Arbitrary (Repr a), Show (Repr a)) => Constructible a where
  type Repr a :: Type
  fromRepr :: Repr a -> a

Notably, we no longer require a itself to be an instance of Arbitrary and Show. Instead, we put those constraints on an associated type Repr a, which is thus inferred implicitly whenever values of type a are quantified over.

Testable equality

Aiming to make properties higher-level, more declarative, the prop_bool property above can also be written like this:

prop_bool :: (Bool -> Bool) -> Equation (Bool -> Bool)
prop_bool f = (f . f . f) :=: f

Where (:=:) is a simple constructor. That defers the choice of how to interpret the equation to the caller of prop_bool, leaving the above specification free of such operational details.

Behind the scenes, this exercises a new type class for testable equality,2 TestEq, turning equality into a first-class concept even for higher-order data (the main examples being functions and infinite lists).

class TestEq a where
  (=?) :: a -> a -> Property

For more details, see the README of quickcheck-higherorder.

Testable higher-order functions

QuickCheck offers a Fun type to express properties of arbitrary functions.3 However, Fun is limited to first-order functions. An example of type that cannot be represented is Cont.

The library test-fun implements a generalization of Fun which can represent higher-order functions. Any order!

It’s a very simple idea at its core, but it took quite a few iterations to get the design right. The end result is a lot of fun. The implementation exhibits the following characteristics, which are not obvious a priori:

As a consequence, test-fun can be reused entirely to work with Hedgehog. However, unlike with QuickCheck, some significant plumbing is required, which is work in progress. test-fun cannot just be specialized to Hedgehog’s Gen monad; it will only work with QuickCheck’s Gen,4 so we currently have to break into Hedgehog’s internals to build a compatible version of the “right” Gen.

test-fun implements core functionality for the internals of libraries like quickcheck-higherorder. Users are thus expected to only depend directly on quickcheck-higherorder (or the WIP hedgehog-higherorder linked above).

Generators as traversals

test-fun only requires an Applicative constraint in most cases, because intuitively a testable function has a fixed “shape”: we represent a function by a big table mapping every input to an output. To generate a random function, we can generate one output independently for each input, collect them together using (<*>), and build a table purely using (<$>).

However this view of “functions as tables” does not extend to higher-order functions, which may only make finite observations of their infinite inputs. A more general approach is to represent functions as decision trees over their inputs. “Function as tables” is the special case where those trees are maximal, such that there is a one-to-one correspondence between leaves and inputs. However, maximal trees don’t always exist. Then a random generator must preemptively terminate trees, and that requires stronger constraints such as Monad (intermediate ones like Alternative or Selective might be worth considering too).

For more details, see the README of test-fun.


These libraries are already used extensively in my project checkers-mtl, which is where most of the code originated from.

One future direction on my mind is to port this to Coq, as part of the QuickChick project. I’m curious about the challenges involved in making the implementation provably total, and in formalizing the correctness of testing higher-order properties.

I’m always looking for opportunities to make testing as easy as possible. I’d love to hear use cases for these libraries you can come up with!

  1. You could hack something in this case because Bool is a small type, but that does not scale to arbitrary types.↩︎

  2. This also exists in checkers as EqProp.↩︎

  3. Shrinking and showing functions (functional pearl), by Koen Claessen, in Haskell Symposium 2012.↩︎

  4. It must be lazy, in the right way. A random monad built on top of lazy State is no good either. As of now, QuickCheck’s Gen is the only monad I know which is useful for test-fun.↩︎