Lens, State is your father... and I can prove It!
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Lens, State is your father... and I can prove it!

About a year ago, we published "Lens, State Is Your Father" in this blog. It led us to our first contribution to Monocle and sowed the seeds of Stateless, the Scala library that we are currently working on. The post left some questions opened, which we'll try to address today. Particularly, we have set two goals:

  • Proving the major claim from the original post: MonadState[A, State[S, ?]] is yet another Lens[S, A] definition (besides concrete, van laarhoven, profunctor, etc.).
  • Contextualizing the relevance of a proof assistant like Coq in functional programming.

It doesn't matter if you haven't read the previous post, we'll help you to catch up with a brief recap. Let's go for it!

1. Recap

In the aforementioned post, we showed some connections between lens and state monad. So, first of all, we'll introduce these elements separately. Then, we'll bring those connections back.

A lens is just a pair of methods packed together, one of them to get a part from a bigger whole and another one to put a new version of the part in an old whole, returning a new whole. We, as programmers, are very familiar with those methods, since they correspond with the notions of getter and setter, respectively. However, to avoid name conflicts with other definitions, we'll refer to them as view and update. We can encode a lens in Scala as follows:

case class Lens[S, A](
  view: S => A,
  update: S => A => S)

(*) Notice that the "whole" data structure is represented with an S, while the "part" is represented with an A.

Lenses are extremely useful to update nested data structures, but we won't go further down that road today. Instead, we'll focus on the properties that a lens must hold to be considered very well-behaved:

law viewUpdate:   update(s)(view(s)) = s
law updateView:   view(update(s)(a)) = a
law updateUpdate: update(update(s)(a1))(a2) = update(s)(a2)

So, a very well-behaved lens is any instance of Lens whose view and update methods are compliant with the previous laws.

On the other hand, there is MonadState. It's a widespread typeclass that classifies all those effects which are able to access and manipulate an inner state which in turn, might be contextualized within a bigger one. MonadState supplies a pair of methods, the first of them to get the current inner state and the second one to replace it with another one, taken as argument. We'll be using the terms get and put to refer to these methods. Having said so, this is how we encode this typeclass in Scala:

trait MonadState[A, M[_]] extends Monad[M] {
  def get: M[A]
  def put: A => M[Unit]

As we can see, A refers to the inner state and the effect M provides the way to access and manipulate it. Whether A roles the whole state or it's just a part of it, it's something that is hidden by M, beyond our vision at this point.

As usual, this typeclass comes along with some laws that any instance should satisfy. We show them here:

law getGet: get >>= (a1 => get >>= (a2 => point((a1, a2)))) =
            get >>= (a => point((a, a)))
law getPut: get >>= put = point(())
law putGet: put a >>= get = put a >>= point(a)
law putPut: put a1 >> put a2 = put a2

(*) Notice that >>= and point are the Monad methods, which are also known as bind and return, respectively.

The most popular instance of this typeclass is MonadState[S, State[S, ?]], which is just a state transformation that also produces an additional output Out. In this particular case, the state S hidden by State[S, ?] is exactly the same as the one we are providing access to via MonadState, I mean, the inner state corresponds to the whole state:

case class State[S, Out](runState: S => (Out, S))

def stateMonadState[S] = new MonadState[S, State[S, ?]] {
  def get = State(s => (s, s))
  def put = s => State(_ => ((), s))

These are convenience methods for State that we'll be using in the upcoming definitions:

def evalState[S, A](st: State[S, A])(s: S): A = st.runState(s)._1
def execState[S, A](st: State[S, A])(s: S): S = st.runState(s)._2

Now, we have collected all the ingredients to state our claim. Perhaps, you might have noticed that Lens and MonadState show a high degree of similarity, which embraces not only their methods (view-get, update-put) but also their associated laws (viewUpdate-getPut, updateUpdate-putPut, etc.). However, they're quite different beasts: lenses deal with immutable data structures while MonadState handles monadic effects. In the original post, we related them by suggesting that MonadState generalizes Lens. Particularly, we claimed that any State instance of MonadState is just an alternative way of representing a lens:

type MSLens[S, A] = MonadState[A, State[S, ?]]

(*) Notice, that MonadState[A, State[S, ?]] differs from the instance we showed above, since the state S hidden by State[S, ?] is not the same as the focus A that we are providing access to via MonadState.

The equivalence is manifested by the following isomorphism:

// forward arrow
def ms_2_ln[S, A](ms: MSLens[S, A]): Lens[S, A] =
  Lens[S, A](
    view = s => evalState(get)(s),
    update = s => a => execState(put(a))(s))

// backward arrow
def ln_2_ms[S, A](ln: Lens[S, A]): MSLens[S, A] =
  new MonadState[A, State[S, ?]] {
    def get = State(s => (ln.view(s), s))
    def put = a => State(s => ((), ln.update(s)(a)))

Right now, we can't guarantee that the result of an invocation to ms_2_ln yields a very well-behaved lens. Indeed, the only fact that we can say about this definition is that it compiles (and that's saying a lot). Thereby, our mission consists on proving that the resulting lens is lawful. We'll do so in the next section. Later, in the concluding section, we'll argue why this proof is so important to us.

2. Proof of Paternity

If we want to show that State MonadState is the parent of Lens, we're gonna need proofs. Firstly, we'll do it informally, old school style (though using a pseudo-scala notation). Then, we'll formalize the same idea using Coq. Since the proof would take too much writing, we'll just show you a subpart of it, and we'll link the rest of them, just in case you are interested.

Informal Proof

Fortunately, we don't have to start from scratch to prove that our isomorphism is correct, we know that MonadState has some laws that we could exploit for this purpose. Prior to that, we're going to narrow the scope of the proof, to make things simpler. Instead of showing the whole correspondence, we'll simply show the following subproof of the forward function:

∀ ms ∈ MSLens[S, A], putPut ms -> updateUpdate (ms_2_ln ms)

that we can read as: "for any ms of type MSLens[S, A] (or MonadState[A, State[S, ?]]) where putPut holds, then passing ms through ms_2_ln must return a lens where updateUpdate holds". Here it's the informal proof:

= [1. def update]
= [2. Lemma execexec is >>]
  execState(put(a1) >> put(a2))(s)
= [3. putPut (MonadState)]
= [4. def update]

We start in with the hypothesis update(update(s)(a1))(a2), being our goal update(s)(a2). Given all the information in the context, that is MonadState laws and other information related to State, we should be able to reach that goal. Step 1 simply unfolds the definition of update that we provided in ms_2_lens. Now, in 2, we assume a lemma execexec is >> (which we invite you to prove) which manifests that sequencing executions of different stateful programs is the same as the single execution of the sequence of those programs. This leads to step 3 where we can apply putPut smoothly. Finally, we apply the definition of update again (step 4), but this time in a reversed way, to reach our desired goal. Seems that we're done here!

You can find the whole informal forward proof here. Please, notice that this linked version uses pseudo-haskell notation, which is more standard in computer science publications.

Coq Proof

I'm sure you've heard that Coq is a proof assistant but, did you know that it's also a purely functional language? A very nice one indeed, which is equipped with dependent types features. As in Scala, it's possible to define a lens using Coq:

Record lens (S A : Type) := mkLens
{ view : S -> A
; update : S -> A -> S

Now, we're going to encode putPut in Coq. Of course, you don't have to understand everything, I just want you to see that this definitions takes a lens as parameter and returns a law (or proposition) over it:

Definition update_update {S A : Type} (ln : lens S A) : Prop :=
  forall s a1 a2, update ln (update ln s a1) a2 = update ln s a2.

Now, we can read update_update ln as a proposition where we postulate that update_updateholds forln`.

Being a functional language as it is, Coq also enables typeclass definitions. This is how we represent MonadState. I invite you to come back to the Scala definition to spot the differences between them:

Class MonadState (A : Type) (m : Type -> Type) `{Monad m} : Type :=
{ get : m A
; put : A -> m unit

What comes next is quite interesting. We encode the laws of a typeclass in another class that depends on the original one. In this sense, when we create a new typeclass instance, we should provide not only an instance of the main class but also an instance of its associated laws, where we prove that the laws of the main instance hold. Having said so, this is how we encode MonadStateLaws in Coq:

Class MonadStateLaws (A : Type) (m : Type -> Type) `{MonadState A m} : Type :=
{ get_get : get >>= (fun s1 => get >>= (fun s2 => ret (s1, s2))) =
            get >>= (fun s => ret (s, s))
; get_put : get >>= put = ret tt
; put_get : forall s, put s >> get = put s >> ret s
; put_put : forall s1 s2, put s1 >> put s2 = put s2

The previous definition is rather strange. Usually, a method signature contains types exclusively, but MonadStateLaws methods include expressions in it! This is a nice consequence of having dependent types, which are just types that depend on plain values. In fact, the dependent type that we find here is =, which depends on two values: the left hand side expression and the right hand side expression of the particular equation (or law) that we want to declare.

Now, defining state turns out to be straightforward:

Record state (S A : Type) := mkState
{ runState : S -> A * S }.

Once we have defined the main structures, we can translate the function that maps any MonadState A (state S) into a lens S A, which we'll use in our proof.

Definition ms_2_lens {S A : Type} (ms : MonadState A (state S)) : lens S A :=
{| view s := evalState get s
;  update s a := execState (put a) s

And finally here it's the formal proof:

Theorem putput_leads_to_updateupdate :
    forall {S A : Type} (ms : MonadState A (state S)),
    (forall a1 a2, put a1 >> put a2 = put a2) -> update_update (ms_2_lens ms).
  rename H into putPut.
  unfold update_update.
  unfold ms_2_lens.
  rewrite -> execexec_is_gtgt.
  now rewrite -> putPut.

Reading the header of the proof (from Theorem to Proof) should be straightforward at this point. After Proof we use the tactics language to prove such a proposition. Again, you don't have to understand everything, but you could find some analogous elements from the informal proof. Particularly, the unfold is quite similar to the expansion of definitions (step 1 in the informal proof), rewrite -> execexec_is_gtgt corresponds directly to the lemma in step 2 and rewrite -> putPut is exactly the step 3. The final Qed (quod erat demonstrandum) evidences the positive Dna test result that we were searching for. As a consequence, we can conclude that MonadState is actually the father of Lens. ;)

You can find the sources associated to the formal proof here.

3. Further Insights and Conclusions

Ok, I buy it, MonadState A (state S) is a lens S A. So what? As I said previously, "Lens, State Is Your Father" was the starting point of Stateless, a library that we are currently working on. In this library, we try to take optics (lens, optional, traversal, etc.) beyond immutable data structures. Particularly, we want to take the algebra and design patterns from optics to more realistic settings, such as databases or microservices, since indeed it's not practical to keep the whole state of a modern application in memory. MonadState, distilling the algebraic essence of a lens, opens up these new possibilities, including, but not restricted to immutable data structures. In fact, we could say that MonadState is the father of many children, siblings of lens, which are about to be discovered in future posts. As a final note, we must say that we haven't found this connection between MonadState and Lens elsewhere, but it seems to be evident on previous work.

We've found it essential to prove that the foundations of Stateless are solid. However, it was becoming quite annoying to do this task, as proofs were becoming more complex. This situation inevitably led us to leap of faiths, you know, everything compiles with pen and paper. That's why we decided to jump into a proof assistant. Honestly, learning Coq will blow your mind, because it's such a different beast (although exactly the same at the end), but you start to be more confident after one fully immersed week (I recommend to follow the dark arrow itinerary from Logical Foundations). In my case, it was totally worth it, since there's no room for leap of faiths any more.

Proof assistants are the natural step towards more reliable systems. Given the nowadays society needs, unit testing is becoming completely obsolete. In this regard, property based testing is a nice alternative, but we can do better. If we can prove our software, there's no need for additional tests. It's correct. Period. Proof assistants are still very academic and they're not necessarily focused on functional programming, but dependent type languages like Idris are starting to emerge and they enable such capabilities. And here's a new claim that I hope to "prove" in the future: they're here to stay!