Introduction to Singletons (Part 3)

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Welcome back! This article is part 3 of our journey through the singleton design pattern, and the great singletons library!

This post will be a continuation of Part 1 and Part 2, so if you haven’t read those first, now would be a good time to pause and do so (and also try to complete the exercises). Today we will be expanding on the ideas in those posts by working with more complex ways to restrict functions based on types. Like the previous posts, we will start by writing things “by hand”, and then jumping into the singletons library and seeing how the framework gives you tools to work with these ideas in a smoother way.

The first half of today’s post will introduce a new application and design pattern that the usage of singletons greatly enhances. The second part of today’s post deals directly with the lifting of functions to the type level, which is made practical by the usage of singletons and the singletons library.

Code in this post is built on GHC 8.6.1 with the nightly-2018-09-29 snapshot (so, singletons-2.5). However, unless noted, all of the code should still work with GHC 8.4 and singletons-2.4. Again, you can download the source for this file here, and, if stack is installed, you can drop into a ghci session with all of the bindings in scope executing it:

Review

In the first post we looked at the Door type, indexed with a phantom type of kind DoorState.

This gives us (at least) three distinct types Door 'Opened, Door 'Closed, and Door 'Locked, which can be used to represent opened, closed, and locked doors, respectively. We talked in previous posts about how we can use this for for a lot of enat tings, including enforcing type-safety, talking about how inputs relate to outputs, and uniting functions polymorphic on all door states.

Then we talked about situations where we want to “not care” about the door status in the type system, or when we want to return a door with a state that is not known statically, and must be determined dynamically at runtime. After going through many “analogous” and equivalent type, we arrived at the existential wrapper SomeDoor:

(We must be careful to pack the Sing s with the Door s, so that we can pattern match at runtime to determine what the original s was.)

For the rest of this post, SomeDoor will essentially be used as a stand-in for a Door s that we do not know the state (the s) of until runtime, because to use a SomeDoor, we pattern-match at runtime. In general you’ll encounter types at runtime in a variety of different situations (discussed more deeply in Part 2), but SomeDoor is a nice nugget that we can examine to demonstrate more general points.

A Need for More Expressive Restrictions

Let’s write a function that “knocks” on a door in IO:

Hm. This doesn’t feel right. We can’t knock on an opened door..can we? Is there a way we can restrict this function to only work on non-opened doors? Or, more generally, is there a way to be more expressive in the manner in which we can restrict functions?

There are a couple of ways of doing this — we’re going to look at two possible ways that singletons and the singletons library help with. Both of these methods allow us to write dependently typed functions that are “type-safe” in more expressive ways than before.

Note that we’ll be exploring ways that are “generalizable” — to different types of restrictions that might be more complicated than just “cannot be 'Opened”.

Dependently Typed Proofs

To look at our first way of tackling this restriction problem, we’re going to explore a fun new application of singletons and DataKinds.

This new application is the usage of the dependently-typed “proof” to prove that an operation is legal. Proofs (in the dependently typed/constructivist/Curry-Howard sense) are witnesses to some type-level predicate or proposition.

A value-level predicate in Haskell is (generally) a function of type a -> Bool. Given a value of type a, if the function returns True, then the predicate is satisfied. If it returns False, it is not.

A type-level predicate is (generally) a type constructor of kind k -> Type. Given a type of kind k, if a value exists of that type (or, if a value can be constructed), then the predicate is satisfied. If no value exists, it is not. That value, if it exists, is called a witness or a proof.1

We can define a predicate Knockable :: DoorState -> Type as a GADT that only has values if given 'Closed and 'Locked, but not 'Opened:

Now, we have a value of type Knockable 'Closed and Knockable 'Locked (KnockClosed and KnockLocked, respectively), but no value of type Knockable 'Opened. How can we use this?

Well, we can make a version of knock that requires a proof that s is Knockable:

knock can now only be called with Closed and Locked doors — do you see why? There is no way to call knock with s ~ 'Opened…because there is no way to pass a value of Knockable 'Opened. No such value exists! There’s no compiler error because it’s “not even wrong”!

This works well if we want to do things at compile-time

Let the compiler prove it for you

We can even make it more seamless to use by auto-generating proofs at compile-time, with a general class like Auto:

Such a typeclass exists in libraries like type-combinators (called Known) and decidable (called Auto). In dependently typed languages like Idris, auto is actually a built-in language keyword that does this automatically!

Decidable Predicates

However, all of this only works if you know what s is at compile-time. What if you don’t? What if you are retrieving s at runtime (like from a SomeDoor or withSomeSing), or you are forced to handle all possible ss?

To do this, we’re going to take advantage of a property of some predicates called “decidability”. We say that a predicate is decidable if, for any input type, we can say whether or not the predicate is satisfiable.

We say that a predicate P in Haskell is decidable if we can always prove, for any input, if the predicate holds or does not hold. Concretely, it means that we can write a total function:

Where:

Decision a is like a Maybe a, except instead of Nothing, we include a proof that the predicate is not true.

The a -> Void idiom (often called Not a, or Refuted a) is type we use in Haskell and other languages to represent the fact that it is impossible to construct a value of type a. That’s because if you could, then you could give it to an a -> Void to get a value of type Void, which is impossible to have. So, if a possible function a -> Void exists, it necessarily means that a value of type a cannot exist.

It’s a lot to handle all at once, so let’s look at an example. Is Knockable a decidable predicate? Yes!

We need to write a function:

I recommend taking a moment and trying to implement this yourself. Remember to enable -Werror=incomplete-patterns2 (or at least -Wall) to make sure you’re handling all potential pattern matching cases.

Are you ready? Here’s a solution:

This definition should seem pretty straightforward for the SClosed and SLocked branches. isKnockable SClosed :: Decision (Knockable 'Closed), we give Proved KnockClosed, which gives us just that!

However, isKnockable SOpened :: Decision (Knockable 'Opened). We can’t use Proved :: a -> Decision a, because no such value of type Knockable 'Opened exists. So, we have to say that we disprove it: we have to prove to GHC that no such type could possibly exist. We do this by providing a function of type Refuted (Knockable 'Opened), or type Knockable 'Opened -> Void.

We can write it like this:

And we’re good to go!

How does this work?

Well, remember, we have to pattern match on the possible inputs. However, we can’t use any of the “legal” patterns:

If you include either of those patterns, GHC will complain. So, there is no valid pattern to match on… so disproveOpened = \case {} is enough to write the function Knockable 'Opened -> Void, since there is no constructor for a value of type Knockable 'Opened to match on. This only works because disproveOpened is a complete pattern match, and therefore total.

We can use this decision function, finally, to handle an arbitrary Door whose status we not know until runtime:

While arguably less useful than one provable predicates, a typeclass for decidable predicates is also possible; the aptly-named decidable package offers such a typeclass, called Decidable!

Decision

The definition of the Decision data type might be surprising if you’re first seeing it. You want to prove something…so why would you care about the case where it’s “not true”? Why not just have something like Maybe, where you have data Decision a = Proved a | Disproved?

In other words, why do we care about proving both true or false, when it looks like we only ever use the true situation? After all, we ignore the Refuted (Knockable s) in our implementation of knockSomeDoor.

One answer is that we do use the contents of Disproved in practice. In knockSomeDoor, we matched on Disproved _ and threw away the counter-proof…however, as we see later in the exercises, there are situations where the contents of Disproved are used.

However, a deeper answer to me is that it keeps the author of the function accountable. You can’t just say “this predicate isn’t true”…you have to earn it. And often, the act of trying to earn your disproof (or not being able to) helps you iron out bad assumptions you’ve made.

For example, if we used Maybe instead of Decision, we could write:

We might falsely claim that SClosed is not knockable. So, if the user of our bad isKnockable gets Nothing, they don’t know if their input is not knockable or knockable…they know nothing about the knockability status of 'Opened or 'Closed.

However, we can’t write this bad implementation with Decision:

There is no valid thing you can put in the ????! That’s because you need to write a function of type Knocked 'Closed -> Void…but no such (total or non-partial) function exists. We can’t write \case {}, because that’s an incomplete pattern match — it’s missing a match on the KnockClosed pattern.

Note also that this is why it’s very important to always have -Werror=incomplete-patterns (or at least -Wall — warn all) on when writing dependently typed proofs, to ensure that GHC warns you when your pattern matches are incomplete and you know your proof is invalid.

In the examples, we see two more non-trivial examples of decision functions (and p q and or p q) that are impossible to implement incorrectly, due to the structure of the predicates.

Perspective on Proofs

We just briefly touched on a very simple version of a dependently typed proof, and how to “prove” properties.

If you have heard things about dependently typed programming before, you might have heard that a lot of it involves “proving properties about your programs” and “forcing you to provide proofs for all of your actions”. The idea of a “proof” might seem a bit scary and “mathematical” to those coming from a software development world.

However, as we just saw, working with proofs and decisions of proofs can be as simple as a couple lines of GADTs and dependent pattern matches.

So, when we see a function like:

We can read the type signature as: “Knocking requires both a Door s and a proof that the door’s status is Knockable”. It makes it impossible for us to run knock on a status that is not Knockable, like, say, 'Opened.

In this light, the role of a proof is like a “key” that a type (like 'Closed) must provide to “unlock” functions like knock.3 A decision function is a function to generate these proofs (or prove that they are impossible) for given types.

On one level, you can think of proofs as “compiler tricks”, or things that exist only to appease the compiler. In fact, compilers of languages that encourage heavy usage of proofs (like Agda, Coq, Idris) actually implement something called proof erasure. That is, in those languages, values like KnockClosed and KnockLocked might never exist at runtime, since they never actually do anything at runtime. They only exist as ways to limit or enable specific programs from compiling, and serve no purpose after compilation. GHC Haskell does not implement proof erasure at the time of this post (current GHC version 8.6), but if proofs like this become commonplace, you might be reading this during a time where GHC Haskell erases proofs like Knockable witnesses!4

The Role of Singletons

Proofs themselves might not play a role at run-time, but generating/deciding them with types requires being able to pattern match and work with types at run-time. Because of this, singletons play an important practical role in working with proofs in Haskell.

After all, remember the type of our decision function:

The Sing allows isKnockable to pattern match and inspect the type a to create your proof.

In this light, the singletons library provides many tools for working with proofs and decisions. In fact, the entire Data.Singletons.Decide module is dedicated to working with proofs and decisions. It provides the Decision data type and Refuted type synonym, both featured above.

It also re-exports a particularly useful predicate from base, propositional equality:

Like how Knockable is a predicate that a given status is “knockable”, ('Blah :~:) is a predicate that a given type is equal to 'Blah. A value of type Knockable s is a proof that s is knockable, and a value of type 'Blah :~: a is a proof that a is equal to 'Blah.

To see how, note the constructors that it allows. Remember that we limit Knockable s to only having “knockable” s by only allowing two constructors, so we can only construct valid values. The same thing happens here – ('Blah :~:) only has one single constructor: Refl :: 'Blah :~: 'Blah. The only valid constructor is one where the left hand side is equal to the right hand side. I like to use Refl with type application syntax, like Refl @'Blah, so it’s always clear exactly what we are saying is the same.

It also offers the “kindclass” SDecide, which provides decision functions for the (a :~:) predicate:

For example, Bool is an instance of SDecide, so we have a function:

which is a decision function to check if b is equal to 'True. You can sort of imagine SDecide as a type-level Eq typeclass, but for “type equality”.

Type Level Functions

We’re now going to look at a different method useful for restricting how we can call functions. Something we can do is define a type that expresses knockable-or-not-knockable, as a value:5

And we can write a type-level function (implemented as type family) from DoorState to a Pass:

We’ve briefly touched on type families before (in talking about SingKind), but, as a quick review: type families act a bit like type-level functions. They take types as input arguments and return types in return.

We can inspect how type families are applied by using the :kind! command in ghci:

(Note that :kind! is different from :kind: :kind is the ghci command to report the kind of a type expression, and :kind! is the ghci command to evaluate type families in type expressions.)

Like type synonyms, type families can’t be partially applied (“unsaturated”). They only ever make sense in “fully applied” (or “saturated”) form, with all arguments given syntactically.

Armed with this type family, we can write a new version of knock:

a ~ b is a constraint for type equality. This constraint means that calling knock requires that StatePass s is equal to (or unifies with) 'Allow. So, if we attempt to call knock with a 'Locked door, then because StatePass 'Locked is 'Allow, the constraint is satisfied and everyone is happy. If we attempt to call knock with an 'Opened door, StatePass 'Opened is 'Obstruct, so the constraint is not satisfied and everyone is sad.

Deciding at Runtime

One nice thing is that, if we know s at compile-time, we can call this function without having to pass any manual proofs. However, we have to deal with the same issue as before: what happens if we don’t know s until runtime? How do we prove to the compiler that Passable s is 'Allow?

Remember that type families take types as inputs, so we can’t write:

because s, a value, can’t be given to StatePass.

What we really want to do is pass s (the singleton representing a type) to StatePass (the type family). And then, we want to match on the resulting type, so we can decide what to do based on the result.

If you think about this predicament long enough, you might begin to see a solution. Essentially, we want a function that takes a singleton of s, and return a singleton of StatePass s.

What we want, in the end, is a mirror of the type-level function at the value level. We need to write a function of type Sing s -> Sing (StatePass s): given a singleton of a type, return a singleton of the type family applied to the type.

We have to be very careful with how we define sStatePass, because GHC isn’t too smart. It’ll reject any definition that isn’t structurally identical to the type family it’s mirroring.

With our new tool, we can now write:

First we use sStatePass s to check the “pass” of the s. Then, we match on the Pass: if it’s 'Obstruct, like the type signature of knock requires, we can run knock. If not, then we cannot!

Singletons Library to the Rescue

At the high level, we defined a “function” on types (StatePass), using type families.

And, just like we have to define singletons (SOpened, SClosed, etc.) at the value level to mirror what is happening at the type level, we also have to define singleton functions (sStatePass) at the value level to mirror what is happening at the type level.

Defining singletons for our types is a tedious and mechanical process. Defining singletonized functions for our type families is also similarly tedious and mechanical. This is where the singletons library comes in: it provides us Template Haskell tools to automatically define type families and their associated singleton functions:

The above declaration would normally declare only the value-level function statePass with the type DoorSate -> Pass.

However, with singleton’s template haskell, this also generates:6

  • The type family StatePass (s :: DoorState) :: Pass, like we defined above
  • The singleton function sStatePass, with the type Sing s -> Sing (StatePass s), like we defined above.

The naming convention for functions with non-symbol names takes a function like myFunction and generates the type family MyFunction and the singleton function sMyFunction.

The naming convention for functions with symbolic names (operators) takes an operator like ++ and generates the type family ++ (keeping the identical name) and the singleton function %++.7

A Comparison

We went over two methods of using phantom types with the singleton library and dependent types to restrict how certain functions can be called, on a more non-trivial level.

Our first method was leveraging “dependently typed proofs”. These are useful because they are constructed to exploit the “structure” of the types you create. Essentially, we create a data type (predicate) in a way so that it is impossible to create an “invalid” proof. And, often, if we write our proofs in a clever enough way, we can actually use and combine proofs to generate new proofs. (More examples in the exercises)

Personally, I find this to be the source of a lot of the “fun” of dependently typed programming — our proofs become first class values, and if we define them in a nice enough way, we can use manipulate them to create new proofs. (A full exploration of this is well beyond the scope of this post)

However, in practice, carefully constructing predicates and proofs (ones more complicated than the one we just looked at) requires some up-front cost in thinking about how to best express your predicate, and is sometimes not straightforward.

I consider the second method, using type-level functions, to be the more “mechanical” way, with less upfront cost in thinking time. For the most part, if you can write a normal term-level function (something that most Haskell programmers are comfortable doing), you can write a type-level function. This is even made simpler with singletons — you can just write your term-level relationship as a normal function, and you can now just directly use your function at the type level.

In fact, consider if there were more than two Pass (maybe allow, obstruct, or partial?). In that case, we can easily restrict a function based on the Pass being equal to any of the three or more by using the ~ constraint. Using the dependently typed proof version, though, we would have to create a new GADT for each situation.

In a way, type-level functions deliver on the promise of blurring the line between type and value. Our term-level functions are now type-level functions! We just need to remember to switch our capitalizations!

But this strength is also its weakness. Remember that the problem of normal term-level functions was that they are potentially “incorrect”, and not directly verifiable. So, if you just lift your potentially incorrect term-level functions to the type level…what you get is potentially incorrect type-level functions! You get the same logic errors. Really, writing type-level functions (unsurprisingly) brings all of the error-proneness of writing at the term-level.

In contrast, if you use dependently typed proofs correctly, these proofs can compose, and GHC can check that these proofs compose correctly, or that the compositions of your proofs are also valid proofs. That’s because this is enforced at the structural level. (We’ll look at some examples in the exercises) GHC can’t do that directly with functions; it can’t check that the composition of functions gives correct answers.

These two approaches aren’t necessarily mutually exclusive, and you often might mix the two. It’s good to understand the trade-offs in up-front cost, expressiveness, and correctness! But, however way you play, the singletons library is here to make our life easier.

Singleton Library Functions

As we have seen, working with type-level functions with singletons involves at least two parts — the type family working on the type-level values, and the singleton functions mirroring the type family, working on the term-level singletons.

The singletons library offers template haskell to make working with these things pretty seamless. In fact, a good portion of Prelude and base is promoted and exported by singletons!

You can find most of these in the Data.Singletons.Prelude module namespace. So, with singletons, you get functions like:

and

and

But, how can we promote functions like (==) and max, which are typeclass-polymorphic?

With kindclasses (typeclasses for kinds), of course!

Let’s remember what we need for these promoted functions to work: the type families, and the singleton functions.

The singletons library handles this by providing each of these in a separate typeclass. Let’s look at the humble Eq typeclass as an example:

The singletons library promotes this as:

The naming convention is to just add P for the “promoted” type family functions, and S for the singleton functions.

In fact, you can even promote your own custom typeclasses:

This would create:

  1. The typeclass MyClass with method myFunc :: MyClass a => a -> a
  2. The promoted typeclass PMyClass with associated type/type family MyFunc (x :: a) :: a
  3. The singletonized typeclass SMyClass with method sMyFunc :: Sing x -> Sing (MyFunc x).

Automatically Promoting Instances

The singletons library is smart enough to automatically promote instances, as well, including derived ones!

This automatically also generates PEq and SEq instances for Pass:

But, you can also just write:

And this works as well!

Next Steps

In this article we tackled the problem of more expressive ways to restrict the ways users can manipulate our data types. We talked about “dependently typed proofs” (a staple tool of dependently typed programming) and about “type level functions” (a familiar friend in a new context), their trade-offs, and how the singletons library provides tools to make working with both easier.

When we first looked at the idea of phantom type parameters, using them to restrict how functions are called was definitely one of the promises I made. By now, this promise has hopefully been fully realized.

However, the other promise we made about the usefulness of phantom type parameters is that we can use them be more expressive in what our functions do. One huge benefit of using phantom types in this way is that we can express how our input values relate to our output values in ways that we couldn’t before. (as a simple example, we had previously written closeDoor :: Door 'Opened -> Door 'Closed, which we know closes a door just by looking at its type)

This goes beyond simple restrictions, and we will begin discussing this in the next post! We’ll explore using type-level functions to express more non-trivial and complex relationships, and also talk about code re-use using higher-order functions via singleton’s defunctionalization system.

That’s it for now — check out the exercises, and feel free to ask any questions in the comments, or in freenode #haskell, where I idle as jle`!

Exercises

Here are some exercises to help cement your understanding of the concepts here! Feel free to start from the sample source code; it contains all of the solutions, but you can delete everything after the comment -- Exercises if you wish to start on your own!

Remember to enable -Werror=incomplete-patterns or -Wall to ensure that all of your functions are total! None of these implementations should require any incomplete pattern matches!

  1. We talk about predicates as type constructors with type k -> Type. This fits a lot of things we’ve seen before (all instances of Functor, for example), but some predicates are more interesting than others.

    What is the interpretation of SDoorState as a predicate? (remember, SDoorState s is the type synonym for Sing (s :: DoorState)) What “traditional” (that is, a -> Bool) predicate does it correspond to?

    What is the type of its decision function? Can you implement it?

    Solution available here!

  2. Now let’s practice working with predicates, singletons, and negation via Refuted together.

    You may have heard of the principle of “double negation”, where not (not p) implies p. So, we should be able to say that Refuted (Refuted (Knockable s)) implies Knockable s.8 If something is not “not knockable”, then it must be knockable, right?

    Try writing refuteRefuteKnockable to verify this principle — at least for the Knockable predicate.

    While not required, I recommend using isKnockable and writing your implementation in terms of it! Use sing to give isKnockable the singleton it needs.

    Solution available here!

    Hint: You might find absurd (from Data.Void) helpful:

    If you have a Void, you can make a value of any type!9

  3. (This next one is fairly difficult compared to the others, and is only tangentially related to singletons, so feel free to skip it!)

    Type-level predicates are logical constructs, so we should be able to define concepts like “and” and “or” with them.

    1. Define a predicate constructor And that takes two predicates and returns a new predicate. This new predicate is true (aka, has an inhabitant) if and only if the two original predicates are true (aka, have inhabitants)

    2. Define a predicate constructor Or that takes two predicates and returns a new predicate. This new predicate is true (aka, has an inhabitant) if and only if at least one of the two original predicates are true (aka, have inhabitants)

      There are potentially multiple non-trivial variations of this type.

      Do And and Or look similar to any types you might have encountered in the past? Maybe, perhaps, similiar to types that are a part of basic beginner Haskell concepts?

    3. Maybe surprisingly, And p q and Or p q are decidable if p and q are. Can we write the decision functions?

      These functions actually demonstrate, I feel, why Decision having both a Proved a and Disproved (Refuted a) branch is very useful. This is because, if you wrote the structure of And and Or correctly, it’s impossible to incorrectly define decideAnd and decideOr. You can’t accidentally say false when it’s true, or true when it’s false — your implementation is guarunteed correct.

    4. Now let’s use And and Or to prove some useful facts about Knockable and ('Opened :~:). We know that it’s impossible for something to be both Knockable and ('Opened :~:) (that is, both knockable and equal to 'Opened). Write such a witness:

      We also know that a given DoorState is either Knockable or ('Opened :~:) — at least one of these is always true. Write such a witness:

    Solutions available here!

  4. Instead of creating an entire Knocked type, we could have just said “as long as the door is not 'Opened, you can knock”. This means we could write knock as:

    Which we must pass a proof that s is not equal to 'Opened in order to open our door.

    Is this really the same thing? Is Refuted (s :~: 'Opened) the same thing as Knockable s?

    Let’s try to say that the two things are the same! Write the following functions to show that Refuted (s :~: 'Opened) is the same logical predicate as Knockable s!

    Solution available here!

    Note: knockedRefute is fairly straightforward, but refuteKnocked is definitely trickier, so don’t be discouraged!

    Hint: See the note about absurd from Exercise 2!

  5. On our type level function version of knock, we wrote, with a constraint:

    We can muddy the waters a bit, for fun, by having this take a proof of the constraint instead:

    Rewrite a version of knockSomeDoor in terms of knockRefl, called knockSomeDoorRefl:

    Remember not to use knock!

    Solution available here.

    Assume that DoorState has an instance of SDecide, so you can use (%~). This should be derived automatically as long as you derive Eq:

  6. With the function that inverts Pass:

    Implement knock in a way that lets you knock if invertPass is Allow:

    And write knockSomeDoor in terms of it:

    Remember again to implement it in terms of knockInv, not knock.

    Solution available here!

  7. Let’s work with a toy typeclass called Cycle, based on Enum

    next is like succ, but loops over to the first item after the last constructor. prev is like pred, but loops over to the last item if pred-ing the first item

    Can you manually promote this instance for DoorState to the type level?

    Solution available here!

Special Thanks

I am very humbled to be supported by an amazing community, who make it possible for me to devote time to researching and writing these posts. Very special thanks to my two supporters at the “Amazing” level on patreon, Sam Stites and Josh Vera! :)

Thanks also to Koz Ross for helping proofread this post!


  1. All of this is ignoring the “bottom” value that is an occupant of every type in Haskell. We can use bottom to subvert pretty much all proofs in Haskell, unfortunately, so the discussion from this point forward assumes we are talking about a subset of Haskell where all values are non-bottom and all functions are total.

  2. Thanks to Darwin226 on reddit for this tip!

  3. Sorry to mix up similar metaphors like this! Definitely not intentional :)

  4. Note, however, that we are a little lucky in our case. In the case of our implementation of knock, we match on a wildcard pattern, so the input proof is never evaluated.

  5. Really, we could just use Bool instead of defining a Pass type. We’re just going through a new type for the sake of example, and it can be useful because a type like Pass might potentially have even more constructors!

  6. In the spirit of full disclosure, the Template Haskell also generates some other things (known as defunctionalization symbols), which we will be talking about in the next part of this series.

  7. Note that this is a change since singletons-2.4. In previous versions, ++ would generate the type family :++ and the singleton function %:++.

  8. Double negation is not true in general, but it is true in the case that our predicate is decidable. That’s because Decision a is essentially a witness to the excluded middle for that specific predicate, from which double negation can be derived.

  9. It’s the good ol’ Principle of Explosion

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