Cats Law Checking with Discipline

As I was working through Underscore's book Advanced Scala With Cats, I got a bit confused with the Monad typeclass method tailRecM. This method is a topic for a different post, but as I tried to figure this out, I decided this was a good opportunity to dig in to how Cats defines and checks laws using Discipline. If Cats' laws for the Monad typeclass include any laws for tailRecM, this would help me learn to implement that method correctly.

Getting this to work took a little more effort than I expected, but the payoff is so very nice.

The pieces are spread around a little, and there is not really clear documentation in one place, so I had to look at a lot of example code and even a few StackOverflow answers.

I Know Some of Those Words

Cats is a Scala library for functional programming. Cats is heavily based around the typeclass pattern, which allows a developer to add behavior to an object in a sort of external "has-a" relationship.

In Java, we would typically make a class extend some interface, which would allow other algorithms to use those types generically:

public final class Integer extends Number implements Comparable<Integer> { ... }

static <T extends Comparable<? super T>> void sort(List<T> list) { ... }

With the typeclass pattern, we implement an external interface, which we can then use in combination with our class. The Java Comparable interface is a step in this direction:

static <T> void sort(List<T> list, Comparator<? super T> c) { ... }

In Scala, the language feature of implicit parameters makes this even more powerful. With this, we can define our typeclass instance as an implicit val or def, and let the compiler pass it implicitly to our other algorithms.

def sorted[B >: A](implicit ord: math.Ordering[B]): List[A] 
scala> List(2,3,1).sorted
res3: List[Int] = List(1, 2, 3)

scala> case class MyInt(x: Int)
defined class MyInt

scala> List(MyInt(2), MyInt(3), MyInt(1))
res4: List[MyInt] = List(MyInt(2), MyInt(3), MyInt(1))

scala> res4.sorted
<console>:11: error: No implicit Ordering defined for MyInt.

Cats provides several cool things for us: first, it defines a number of typeclasses, including many concepts from mathematics (specifically category theory) such as monoids, functors, and monads; some structural capabilities such as traversability (Traversable) or foldability (Foldable); and some other basic capabilities like string formatting (Show) and equivalence testing (Eq). Cats also provides instances of many of these typeclasses for common library types, such as a monoid instance for Int, a monad instance for Option, or a Show instance for a tuple. Cats provides syntactic support for using these so that their use is natural and invisible, for example, adding an extension method so that you can call map on any class for which you have an implicit Functor instance, as if it implemented map directly. And Cats provides laws and checking to make sure these instances are well-behaved.


If we want to make use of an interface in another algorithm, we would like to know what assumptions we can make about that interface. For example, we'd like to know whether order matters when we combine objects together. These assumptions can be expressed as mathematical laws which a type must follow.

For example, a Monoid is a structure which has two components: a zero element and a combination function. To be well-behaved, it must satisfy two laws:

If these hold, order doesn't matter when we combine elements, and the zero element preserves results. So we could imagine algorithms that might, for example, do work in parallel, or return zero for nominal results, such as counting "go" vs "no-go" answers where we don't care how many "go" answers we get but want to track how many "no-go"). If we know our monoid is well-behaved, we know that making these assumptions is safe and won't introduce bugs. So how can we tell if our monoid is well-behaved?

Cats implements tests against these laws using Discipline. Using this approach, the laws can be specified as a form of test code. Discipline uses ScalaCheck to run tests against these laws. This uses random data and multiple iterations to check whether the law holds for a wide range of inputs.

It turns out we can piggyback on this to make sure our own typeclass instances are lawful using Cats' own laws.

Testing my Monad

I started with a Monad instance for a clone of Option (I used this only as an exercise - there was no reason I really wanted my own Option). My option has the following structure:

sealed trait MyOption[+A]
case object MyNone extends MyOption[Nothing]
final case class MySome[A](value: A) extends MyOption[A]

To define a Monad instance for this, I needed to implement the three Cats Monad methods: pure, flatMap and tailRecM:

implicit val myOptionMonad = new Monad[MyOption] {
  def pure[A](value: A): MyOption[A] = MySome(value)

  def flatMap[A, B](opt: MyOption[A])(fn: A => MyOption[B]): MyOption[B] = opt match {
    case MyNone => MyNone
    case MySome(value) => fn(value)

  def tailRecM[A, B](a: A)(f: A => MyOption[Either[A, B]]): MyOption[B] = f(a) match {
    case MyNone => MyNone
    case MySome(Left(l)) => tailRecM(l)(f)
    case MySome(Right(r)) => MySome(r)

What I want to do, is to test this Monad instance using the Cats monad laws

First, I'm going to need to figure out how to even get Discipline into my test. This turns out to be pretty easy. Discipline provides a trait which can be mixed into your ScalaTest suite, with two minor caveats: first, you must use a FunSuite, and second, you need to have the right versions of ScalaTest and ScalaCheck or you'll run into a nasty exception. For newest versions of Cats, you'll need to use the newest (3.0.1) version of ScalaTest, and everything will work fine (I did not try back-revving ScalaCheck for earlier ScalaTest).

The restriction on FunSuite seems like it would be fairly easy to overcome, as the Discipline trait is pretty lightweight - Circe builds on FlatSpec in their base CirceSuite. Discipline also includes some bindings for Specs2 if that's more your thing.

My test therefore looks something like this:

class MyOptionSpec extends FunSuite with Matchers with Discipline {
  checkAll("MyOption[Int]", MonadTests[MyOption].monad[Int, Int, Int])

What's all that MonadTests stuff? I thought I wanted to run the tests in MonadLaws?

If you look at the Discipline trait, you see that it takes the laws as a Laws#RuleSet. As Cats implements it, the laws themselves are kept separate from test classes which build those laws into the RuleSet needed by Discipline. These test classes are in a nearby package, cats.laws.discipline to the laws, in cats.laws.

What about all the type parameters? In the case of the MonadLaws tests, we have three separate parameters to specify the type of the value of the instance (M[A]), the type of a second value to be created by map and flatMap (A => B or A => M[B], respectively), and a third type to be used in function composition (A => B => C).

Getting It To Work

In order to compile and run this, we need some other implicit instances. MonadTests.monad takes 14 implicit parameters! Some of these we can get some help with from Cats, but there are a couple we'll need to provide ourselves. I worked my way through these by iteratively trying to compile and letting the compiler point out the next missing thing.


In order to generate random instances of our data types, ScalaCheck uses Generators to create instances. These create random data, although biased towards edge cases like 0, maxInt, and so on. Generators can be composed, so we can both rely on multiple generators to build a larger structure, and rely on an implicit generator to fill a type hole we have.

Put another way, our job is to provide a generator for OurType[T], not for OurType[SpecificType], and we can rely on other generators to generate the values for T. By making our generator parametric, we can avoid having to hand-roll generators for the several types which we'll need to test the laws (for example, the MonadTest will need both an Arbitrary[M[A]] and an Arbitrary[M[A => B]], and even worse if we want to use different types for A and B).

Building an Arbitrary instance for MyOption was reasonably straightforward. Instances are either MyNone or MySome containing values from a different Arbitrary instance:

implicit def arbMyOption[T](implicit a: Arbitrary[T]): Arbitrary[MyOption[T]] =
      arbitrary[T].map(v => MySome(v)),

There isn't really a non-testing reason I'd need an Arbitrary instance for my type, so I'll leave this in my test scope.


The next piece I needed was Eq[MyOption[T]]. The Eq typeclass lets us check for equality between two items. Again we can use the trick of relying on an implicit Eq to be provided for checking the values of the type T. My implementation also makes use of Cats' syntax support, so instead of calling the eqv(x, y) method directly, I can just use == and the compiler will swap these in appropriately (there are some cases where this might not call exactly what you think, but as long as you're not defining equivalence strangely its probably not a concern).

The Eq for MyOption does a tedious but obvious structural comparison with a pattern match, delegating to the other Eq instance to check the contents only when the structure already matches.

implicit def eqMyOption[T](implicit t: Eq[T]): Eq[MyOption[T]] = new Eq[MyOption[T]] {
  def eqv(x: MyOption[T], y: MyOption[T]): Boolean = x match {
    case MyNone => y match {
      case MyNone => true
      case _ => false
    case MySome(a) => y match {
      case MySome(b) if a == b => true
      case _ => false

Now this is a typeclass instance I might want to use in other production code, so its a good candidate to go into the implementation in compile scope, but I've left it in my test class for this exercise for illustrative and exercise purposes.

As I put this together, I needed an Eq instance for the tuple type (Int, Int, Int). Cats to the rescue! Cats has appropriate typeclass instances for this in the cats.instances.tuple._ package.

Run It!

I have my MyOptionSpec, and I have the Arbitrary, the Eq, and of course the Monad instance under test, all in scope, as well as some Cats standard instances for Int and tuple types. Now we can run the test as usual in our IDE or command line:

> test-only MyOptionSpec
[info] MyOptionSpec:
[info] - MyOption[Int].monad.ap consistent with product + map
[info] - MyOption[Int].monad.applicative homomorphism
[info] - MyOption[Int].monad.applicative identity
[info] - MyOption[Int].monad.applicative interchange
[info] - MyOption[Int].monad.applicative map
[info] - MyOption[Int].monad.apply composition
[info] - MyOption[Int].monad.cartesian associativity
[info] - MyOption[Int].monad.covariant composition
[info] - MyOption[Int].monad.covariant identity
[info] - MyOption[Int].monad.flatMap associativity
[info] - MyOption[Int].monad.flatMap consistent apply
[info] - MyOption[Int].monad.followedBy consistent flatMap
[info] - MyOption[Int].monad.invariant composition
[info] - MyOption[Int].monad.invariant identity
[info] - MyOption[Int] flatMap coherence
[info] - MyOption[Int].monad.monad left identity
[info] - MyOption[Int].monad.monad right identity
[info] - MyOption[Int].monad.monoidal left identity
[info] - MyOption[Int].monad.monoidal right identity
[info] - MyOption[Int].monad.mproduct consistent flatMap
[info] - MyOption[Int].monad.tailRecM consistent flatMap
[info] - MyOption[Int].monad.tailRecM stack safety
[info] ScalaTest
[info] Run completed in 774 milliseconds.
[info] Total number of tests run: 22
[info] Suites: completed 1, aborted 0
[info] Tests: succeeded 22, failed 0, canceled 0, ignored 0, pending 0
[info] All tests passed.
[info] Passed: Total 22, Failed 0, Errors 0, Passed 22
[success] Total time: 1 s, completed Dec 1, 2016 12:00:22 AM

Great! My monad instance is lawful, and we notice that the original questions: - are there any laws for tailRecM? - does my instance conform to those laws?

are both "yes".


Law testing is a powerful technique to make sure that your typeclass instances are lawful, and therefore likely to work as advertised in other algorithms, without introducing bugs. If I build on Cats' typeclasses I definitely want to make sure I follow Cats' laws as well as doing other appropriate testing for my data structures and functions. Despite a bit of learning curve, Discipline provides a powerful way to do this kind of testing.

If I wanted to reconstruct this approach in some other library without leveraging Cats, I think the biggest challenge would be building the structure which constructs the Laws#RuleSet from the individual Laws definitions. Cats test classes provide a model to follow here, but these seem like they would take a bit of work to replicate and get right. Of course, this would be something that you set up much less often than you build instances to be tested, and really, why not just leverage Cats?

If you want to see the complete example test, it can be found on github: - Type definition and Monad instance - Law test

Definitely check out Cats, Discipline, ScalaCheck, and Circe. All are great work and very useful. The Underscore book that prompted my investigation of Cats is also recommended.