Higherorder QuickCheck
A QuickCheck extension for properties of higherorder values.
Examples
Higherorder properties are properties which may:
 quantify over functions;
 state equalities between functions.
Some examples:
fmap_dot :: forall a b c. (b > c) > (a > b) > Equation (Maybe a > Maybe c)
fmap_dot g f x = (fmap g . fmap f) :=: fmap (f . g)
callCC_bind :: forall r a. Cont r a > Equation (Cont r a)
callCC_bind m = callCC ((>>=) m) :=: m
quickcheckhigherorder makes it easy to define and test such properties.
main :: IO ()
main = do
quickCheck' (fmap_dot @Int @Int @Int)
quickCheck' (callCC_bind @Int @Int)
(Additional setup is required for the callCC
example.)
Summary
QuickCheck has a cute trick to implicitly convert functions
Thing > Bool
or Thing > Gen Bool
to testable properties,
provided Thing
is an instance of Arbitrary
and Show
,
i.e., there is a random generator, a shrinker, and a printer for Thing
.
Sadly, those constraints limit the range of types that Thing
can be.
In particular, they rule out functions and other values of "infinite size".
This library, quickcheckhigherorder, lifts that limitation, generalizing
that technique to arbitrary types, and provides other related qualityoflife
improvements for propertybased test suites.
The key idea is to separate the Thing
manipulated by the application under
test, of arbitrary structure, from its representation, which is manipulated
by QuickCheck and needs to be "concrete" enough to be possible to generate,
shrink, and show.
Constructible types
The Constructible
type class relates types a
to representations
Repr a
from which their values can be constructed.
Constraints for Arbitrary
and Show
are thus attached to those
representations instead of the raw type that will be used in properties.
class (Arbitrary (Repr a), Show (Repr a)) => Constructible a where
type Repr a :: Type
fromRepr :: Repr a > a
To illustrate what it enables, here's an example of higherorder property:
prop_bool :: (Bool > Bool) > Bool > Property
prop_bool f x =
f (f (f x)) === f x
In vanilla QuickCheck, it needs a little wrapping to actually run it:
main :: IO ()
main = quickCheck (\(Fn f) x > prop_bool f x)
The simpler expression quickCheck prop_bool
would not typecheck
because Bool > Bool
is not an instance of Show
.
With "higherorder" QuickCheck, that wrapping performed by Fn
is instead
taken care of by the Constructible
class, so we can write simply:
main :: IO ()
main = quickCheck' prop_bool
This is especially convenient when the function type is not
directly exposed in the type of the property (as in prop_bool
),
but may be hidden inside various data types or newtypes.
Testable equality
In a similar vein, the Eq
class is limited to types with
decidable equality,
which typically requires them to have values of "finite size".
Most notably, the type of functions a > b
cannot be an instance of Eq
in
general.
But testing can still be effective with a weaker constraint, dubbed
testable equality. To compare two functions, we can generate some random
arguments and compare their images.
That is useful even if we can't cover the whole domain:

if we find two inputs with distinct outputs, then the two functions
are definitely not equal, and we now have a very concrete counterexample to
contemplate;

if we don't find any difference, then we can't conclude for sure, but:
 we can always try harder (more inputs, or rerun the whole property from
scratch);
 in some situations, such as implementations of algebraic structures,
bugs cause extremely obvious inequalities. If only we would look at them.
The point of this new feature is to lower the bar for testing equations
between higherorder values in the first place.
This package introduces a new type class TestEq
, for testable equality.
class TestEq a where
(=?) :: a > a > Property
The codomain being Property
offers some notable capabilities:

as explained earlier, we can use randomness to choose finite subsets
of infinite values (such as functions) to compare;

we can also provide detailed context in the case of failure,
by reporting the observations which lead to unequal outcomes.
For example, we can rewrite prop_bool
as an algebraic property of functions
using TestEq
:
prop_bool :: (Bool > Bool) > Property
prop_bool f = (f . f . f) =? f
More types of properties
Many common properties are quite simple, like prop_bool
.
However, QuickCheck's way of declaring properties as functions with result type
Property
introduces some unexpected complexity in the types.
For example, try generalizing the property prop_bool
above to
arbitrary types instead of Bool
.
Since we use testable equality of functions a > a
, we incur constraints that
the domain must be Constructible
, and the codomain itself must have
testable equality.
prop_fun :: (Constructible a, TestEq a) => (a > a) > Property
prop_fun f = (f . f . f) =? f
This type tells us both too much and too little.
Too much, because the constraints leak details about the very specific
way in which the comparison is performed. Too little, because a Property
can do a lot of things besides testing the equality of two values;
in fact that is one cause for the previous concern.
A more precise formulation is the following:
prop_fun :: (a > a) > Equation (a > a)
prop_fun f = (f . f . f) :=: f
This does not actually do the comparison, but exposes just the necessary
amount of information to do it in whatever way one deems appropriate.
Indeed, Equation
is simply a type of pairs:
data Equation a = a :=: a
It is equipped with a Testable
instance that will require a TestEq
constraint indirectly at call sites only.
Full example
import Test.QuickCheck (quickCheck)
import Test.QuickCheck.HigherOrder (property', Equation((:=:)), CoArbitrary)
import Control.Monad.Cont (Cont, ContT(..), callCC)
 Example property
callCC_bind :: forall r a. Cont r a > Equation (Cont r a)
callCC_bind m = callCC ((>>=) m) :=: m
main :: IO ()
main = quickCheck' (callCC_bind @Int @Int)
 Newtype boilerplate
import Test.QuickCheck (Gen)
import Test.QuickCheck.HigherOrder (CoArbitrary, TestEq(..), Constructible(..))
 Constructible instances
instance (CoArbitrary Gen (m r), Constructible a, Constructible (m r)) => Constructible (ContT r m a) where
type Repr (ContT r m a) = Repr ((a > m r) > m r)
fromRepr = ContT . fromRepr
instance (TestEq ((a > m r) > m r)) => TestEq (ContT r m a) where
ContT f =? ContT g = f =? g