quickcheck-higherorder alternatives and similar packages
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A QuickCheck extension for properties of higher-order values.
Higher-order properties are properties which may:
- quantify over functions;
- state equalities between functions.
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
quickcheck-higherorder 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
QuickCheck has a cute trick to implicitly convert functions
Thing -> Bool or
Thing -> Gen Bool to testable properties,
Thing is an instance of
i.e., there is a random generator, a shrinker, and a printer for
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, quickcheck-higherorder, lifts that limitation, generalizing that technique to arbitrary types, and provides other related quality-of-life improvements for property-based 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 type class relates types
a to representations
Repr a from which their values can be constructed.
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 higher-order 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
Bool -> Bool is not an instance of
With "higher-order" 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
but may be hidden inside various data types or newtypes.
In a similar vein, the
Eq class is limited to types with
which typically requires them to have values of "finite size".
Most notably, the type of functions
a -> b cannot be an instance of
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:
1. we can always try harder (more inputs, or rerun the whole property from scratch); 2. 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 higher-order 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
prop_bool :: (Bool -> Bool) -> Property prop_bool f = (f . f . f) =? f
More types of properties
Many common properties are quite simple, like
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
Since we use testable equality of functions
a -> a, we incur constraints that
the domain must be
Constructible, and the codomain itself must have
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
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.
Equation is simply a type of pairs:
data Equation a = a :=: a
It is equipped with a
Testable instance that will require a
constraint indirectly at call sites only.
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