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Introduction to Singletons (Part 4)
Introduction to Singletons
functional programming, dependent types, haskell, singletons, types
2018/09/28 22:02:02
2018/10/22 04:06:55
singletons-4
introduction-to-singletons-4

Hi again! Welcome back; let's jump right into the fourth and final part of our journey through the singleton design pattern and the great singletons library.

Please check out the first three parts of the series and make sure you are comfortable with them before reading on. I definitely also recommend trying out some or all of the exercises, since we are going to be building on the concepts in those posts in a pretty heavy way.

Today we're going to jump straight into functional programming at the type level. 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.

## Review

Just as a quick review, this entire series we have been working with a Door type:

!!!singletons/Door4.hs "\$(singletons " "data Door " "mkDoor"

And we talked about using Sing s, or SDoorState s, to represent the state of the door (in its type) as a run-time value. We've been using a wrapper to existentially hide the door state type, but also stuffing in a singleton to let us recover the type information once we want it again:

data SomeDoor :: Type where
MkSomeDoor :: Sing s -> Door s -> SomeDoor

mkSomeDoor :: DoorState -> String -> SomeDoor
mkSomeDoor ds mat = withSomeSing ds \$ \dsSing ->
MkSomeDoor dsSing (mkDoor dsSing mat)

In Part 3 we talked about a Pass data type that we used to talk about whether or not we can walk through or knock on a door:

\$(singletons [d|
data Pass = Obstruct | Allow
deriving (Show, Eq, Ord)
|])

And we defined type-level functions on it using singletons Template Haskell:

\$(singletons [d|
statePass :: DoorState -> Pass
statePass Opened = Allow
statePass Closed = Obstruct
statePass Locked = Obstruct
|])

This essentially generates these three things:

statePass :: DoorState -> Pass
statePass Opened = Allow
statePass Closed = Obstruct
statePass Locked = Obstruct

type family StatePass (s :: DoorState) :: Pass where
StatePass 'Opened = 'Allow
StatePass 'Closed = 'Obstruct
StatePass 'Locked = 'Obstruct

sStatePass :: Sing s -> Sing (StatePass s)
sStatePass = \case
SOpened -> SAllow
SClosed -> SObstruct
SLocked -> SObstruct

And we can use StatePass as a type-level function while using sStatePass to manipulate the singletons representing s and StatePass s.

We used this as a constraint to restrict how we can call our functions:

!!!singletons/Door3.hs "knockP"

But then we wondered...is there a way to not only restrict our functions, but to describe how the inputs and outputs are related to each other?

## Inputs and Outputs

In the past we have settled with very simple relationships, like:

closeDoor :: Door 'Opened -> Door 'Closed

This means that the relationship between the input and output is that the input is opened...and is then closed.

However, armed with promotion of type-level functions, writing more complex relationships becomes fairly straightforward!

We can write a function mergeDoor that "merges" two doors together, in sequence:

mergeDoor :: Door s -> Door t -> Door ????
mergeDoor d e = UnsafeMkDoor \$ doorMaterial d ++ " and " ++ doorMaterial e

A merged door will have a material that is composite of the original materials. But, what will the new DoorState be? What goes in the ??? above?

Well, if we can write the function as a normal function in values...singletons lets us use it as a function on types. Let's write that relationship. Let's say merging takes on the higher "security" option --- merging opened with locked is locked, merging closed with opened is closed, merging locked with closed is locked.

\$(singletons [d|
mergeState :: DoorState -> DoorState -> DoorState
mergeState Opened d      = d
mergeState Closed Opened = Closed
mergeState Closed Closed = Closed
mergeState Closed Locked = Locked
mergeState Locked _      = Locked
|])

-- Alternatively, taking advantage of the derived Ord instance:
\$(singletons [d|
mergeState :: DoorState -> DoorState -> DoorState
mergeState = max
|])

This makes writing mergeDoor's type clean to read:

!!!singletons/Door4.hs "mergeDoor"

And, with the help of singletons, we can also write this for our doors where we don't know the types until runtime:

!!!singletons/Door4.hs "mergeSomeDoor"

To see why this typechecks properly, compare the types of sMergeState and mergeDoor:

sMergeState :: Sing s -> Sing t -> Sing (MergeState s t)
mergeDoor   :: Door s -> Door t -> Sing (MergeState s t)

MkSomeDoor  :: Sing (MergeState s t) -> Door (MergeState s t) -> SomeDoor

Because the results both create types MergeState s t, MkSomeDoor is happy to apply them to each other, and everything typechecks. However, if, say, we directly stuffed s or t into MkSomeDoor, things would fall apart and not typecheck.

And so now we have full expressiveness in determining input and output relationships! Once we unlock the power of type-level functions with singletons, writing type-level relationships become as simple as writing value-level ones. If you can write a value-level function, you can write a type-level function.

### Kicking it up a notch

How far we can really take this?

Let's make a data type that represents a series of hallways, each linked by a door. A hallway is either an empty stretch with no door, or two hallways linked by a door. We'll structure it like a linked list, and store the list of all door states as a type-level list as a type parameter:

!!!singletons/Door4.hs "data Hallway"

(If you need a refresher on type-level lists, check out the quick introduction in Part 1 and Exercise 4 in Part 2)

So we might have:

ghci> let door1 = mkDoor SClosed "Oak"
ghci> let door2 = mkDoor SOpened "Spruce"
ghci> let door3 = mkDoor SLocked "Acacia"
ghci> :t door1 :<# door2 :<# door3 :<# HEnd
Hallway '[ 'Closed, 'Opened, 'Locked ]

That is, a Hallway '[ s, t, u ] is a hallway consisting of a Door s, a Door t, and a Door u, constructed like a linked list in Haskell.

Now, let's write a function to collapse all doors in a hallway down to a single door:

collapseHallway :: Hallway ss -> Door ?????

Basically, we want to merge all of the doors one after the other, collapsing it until we have a single door state. Luckily, MergeState is both commutative and associative and has an identity, so this can be defined sensibly.

First, let's think about the type we want. What will the result of merging ss be?

We can pattern match and collapse an entire list down item-by-item:

\$(singletons [d|
mergeStateList :: [DoorState] -> DoorState
mergeStateList []     = Opened               -- ^ the identity of mergeState
mergeStateList (s:ss) = s `mergeState` mergeStateList ss
|])

Again, remember that this also defines the type family MergeStateList and the singleton function sMergeStateList :: Sing ss -> Sing (MergeStateList ss).

With this, we can write collapseHallway:

!!!singletons/Door4.hs "collapseHallway"

Now, because the structure of collapseHallway perfectly mirrors the structure of mergeStateList, this all typechecks, and we're done!

ghci> collapseHallway (door1 :<# door2 :<# door3 :<# HEnd)
UnsafeMkDoor "Oak and Spruce and Acacia and End of Hallway"
:: Door 'Locked

Note one nice benefit -- the door state of collapseHallway (door1 :<# door2 :<# door3 :<# HEnd) is known at compile-time to be Door 'Locked, if the types of all of the component doors are also known!

## Functional Programming

We went over that all a bit fast, but some of you might have noticed that the definition of mergeStateList bears a really strong resemblance to a very common Haskell list processing pattern:

mergeStateList :: [DoorState] -> DoorState
mergeStateList []     = Opened               -- ^ the identity of mergeState
mergeStateList (s:ss) = s `mergeState` mergeStateList ss

The algorithm is to basically [] with Opened, and all (:) with mergeState. If this sounds familiar, that's because this is exactly a right fold! (In fact, hlint actually made this suggestion to me while I was writing this)

mergeStateList :: [DoorState] -> DoorState
mergeStateList = foldr mergeState Opened

In Haskell, we are always encouraged to use higher-order functions whenever possible instead of explicit recursion, both because explicit recursion opens you up to a lot of potential bugs, and also because using established higher-order functions make your code more readable.

So, as Haskellers, let us hold ourselves to a higher standard and not be satisfied with a MergeState written using explicit recursion. Let us instead go full fold --- ONWARD HO!

### The Problem

Initial attempts to write a higher-order type-level function as a type family, however, serve to temper our enthusiasm.

type family Foldr (f :: j -> k -> k) (z :: k) (xs :: [j]) :: k where
Foldr f z '[]       = z
Foldr f z (x ': xs) = f x (Foldr f z xs)

So far so good right? So we should expect to be able to write MergeStateList using Foldr, MergeState, and 'Opened

type MergeStateList ss = Foldr MergeState 'Opened ss

Ah, but the compiler is here to tell you this isn't allowed in Haskell:

• The type family ‘MergeState’ should have 2 arguments, but has been given none
• In the equations for closed type family ‘MergeStateList’
In the type family declaration for ‘MergeStateList’

What happened? To figure out, we have to remember that pesky restriction on type synonyms and type families: they can not be used partially applied ("unsaturated"), and must always be fully applied ("saturated"). For the most part, only type constructors (like Maybe, Either, IO) and lifted DataKinds data constructors (like 'Just, '(:)) in Haskell can ever be partially applied at the type level. We therefore can't use MergeState as an argument to Foldr, because MergeState must always be fully applied.

Unfortunately for us, this makes our Foldr effectively useless. That's because we're always going to want to pass in type families (like MergeState), so there's pretty much literally no way to ever actually call Foldr except with type constructors or lifted DataKinds data constructors.

So...back to the drawing board?

## Defunctionalization

I like to mentally think of the singletons library as having two parts: the first is linking lifted DataKinds types with run-time values to allow us to manipulate types at runtime as first-class values. The second is a system for effective functional programming at the type level.

To make a working Foldr, we're going to have to jump into that second half: defunctionalization.

Defunctionalization is a technique invented in the early 70's as a way of compiling higher-order functions into first-order functions in target languages. The main idea is:

• Instead of working with functions, work with symbols representing functions.
• Build your final functions and values by composing and combining these symbols.
• At the end of it all, have a single Apply function interpret all of your symbols and produce the value you want.

In singletons these symbols are implemented as "dummy" empty data constructors, and Apply is a type family.

To help us understand singleton's defunctionalization system better, let's build our own defunctionalization system from scratch.

First, a little trick to make things easier to read:

!!!singletons/Defunctionalization.hs "data TyFun" "infixr 0 ~>"

### Our First Symbols

Now we can define a dummy data type like Id, which represents the identity function id:

!!!singletons/Defunctionalization.hs "data Id"1

The "actual" kind of Id is Id :: TyFun a a -> Type; you can imagine TyFun a a as a phantom parameter that signifies that Id represents a function from a to a. It's essentially a nice trick to allow you to write Id :: a ~> a as a kind signature.

Now, Id is not a function...it's a dummy type constructor that represents a function a -> a. A type constructor of kind a ~> a represents a defunctionalization symbol -- a type constructor that represents a function from a to a.

To interpret it, we need to write our global interpreter function:

!!!singletons/Defunctionalization.hs "type family Apply"

That's the syntax for the definition of an open type family in Haskell: users are free to add their own instances, just like how type classes are normally open in Haskell.

Let's tell Apply how to interpret Id:

!!!singletons/Defunctionalization.hs "type instance Apply Id"

The above is the actual function definition, like writing id x = x. We can now call Id to get an actual type in return:

ghci> :kind! Apply Id 'True
'True

(Remember, :kind! is the ghci command to evaluate a type family)

Let's define another one! We'll implement Not:

!!!singletons/Defunctionalization.hs "data Not"

We can try it out:

ghci> :kind! Apply Not 'True
'False
ghci> :kind! Apply Not 'False
'True

It can be convenient to define an infix synonym for Apply:

!!!singletons/Defunctionalization.hs "type f @@ a" "infixl 9 @@"

Then we can write:

ghci> :kind! Not @@ 'False
'True
ghci> :kind! Id @@ 'True
'True

Remember, Id and Not are not actual functions --- they're just dummy data types ("defunctionalization symbols"), and we define the functions they represent through the global Apply type function.

### A Bit of Principle

So we've got the basics of defunctionalization --- instead of using functions directly, use dummy symbols that encode your functions that are interpreted using Apply. Let's add a bit of principle to make this all a bit more scalable.

The singletons library adopts a few conventions for linking all of these together. Using the Not function as an example, if we wanted to lift the function:

not :: Bool -> Bool
not False = True
not True  = Flse

We already know about the type family and singleton function this would produce:

type family Not (x :: Bool) :: Bool where
Not 'False = 'True
Not 'True  = 'False

sNot :: Sing x -> Sing (Not x)
sNot SFalse = STrue
sNot STrue  = SFalse

But the singletons library also produces the following defunctionalization symbols, according to a naming convention:

data NotSym0 :: Bool ~> Bool
type instance Apply NotSym0 x = Not x

-- also generated for consistency
type NotSym1 x = Not x

NotSym0 is the defunctionalization symbol associated with the Not type family, defined so that NotSym0 @@ x = Not x. Its purpose is to allow us to pass in Not as an un-applied function. The Sym0 suffix is a naming convention, and the 0 stands for "expects 0 arguments". Similarly for NotSym1 -- the 1 stands for "expects 1 argument".

#### Two-Argument Functions

Let's look at a slightly more complicated example -- a two-argument function. Let's define the boolean "and":

\$(singletons [d|
and :: Bool -> (Bool -> Bool)
and False _ = False
and True  x = x
])

this will generate:

type family And (x :: Bool) (y :: Bool) :: Bool where
And 'False x = 'False
And 'True  x = x

sAnd :: Sing x -> Sing y -> Sing (And x y)
sAnd SFalse x = SFalse
sAnd STrue  x = x

And the defunctionalization symbols:

data AndSym0 :: Bool ~> (Bool ~> Bool)
type instance Apply AndSym0 x = AndSym1 x

data AndSym1 (x :: Bool) :: (Bool ~> Bool)
-- or
data AndSym1 :: Bool -> (Bool ~> Bool)
type instance Apply (AndSym1 x) y = And x y

type AndSym2 x y = And x y

AndSym0 is a defunctionalization symbol representing a "fully unapplied" ("completely unsaturated") version of And. AndSym1 x is a defunctionalization symbol representing a "partially applied" version of And --- partially applied to x (its kind is AndSym1 :: Bool -> (Bool ~> Bool)).

The application of AndSym0 to x gives you AndSym1 x:

ghci> :kind! AndSym0 @@ 'False
AndSym1 'False

Remember its kind AndSym0 :: Bool ~> (Bool ~> Bool) (or just AndSym0 :: Bool ~> Bool ~> Bool): it takes a Bool, and returns a Bool ~> Bool defunctionalization symbol.

The application of AndSym1 x to y gives you And x y:

ghci> :kind! AndSym1 'False @@ 'True
'False              -- or FalseSym0, which is a synonym for 'False
ghci> :kind! AndSym1 'True  @@ 'True
'True

A note to remember: AndSym1 'True is the defunctionalization symbol, and not AndSym1 itself. AndSym1 has kind Bool -> (Bool ~> Bool), but AndSym1 'True has kind Bool ~> Bool --- the kind of a defunctionalization symbol. AndSym1 is a sort of "defunctionalization symbol constructor".

Also note here that we encounter the fact that singletons also provides "defunctionalization symbols" for "nullary" type functions like False and True, where:

type FalseSym0 = 'False
type TrueSym0  = 'True

Just like how it defines AndSym0 for consistency, as well.

#### Symbols for type constructors

One extra interesting defunctionalization symbol we can write: we turn lift any type constructor into a "free" defunctionalization symbol:

!!!singletons/Defunctionalization.hs "data TyCon1" "type instance Apply (TyCon1 t)"

Basically the Apply instance just applies the type constructor t to its input a.

ghci> :kind! TyCon1 Maybe @@ Int
Maybe Int
ghci> :kind! TyCon1 'Right @@ 'False
'Right 'False

We can use this to give a normal j -> k type constructor to a function that expects a j ~> k defunctionalization symbol.

## Bring Me a Higher Order

Okay, so now we have these tokens that represent "unapplied" versions of functions. So what?

Well, remember the problem with our implementation of Foldr? We couldn't pass in a type family, since type families must be passed fully applied. So, instead of having Foldr expect a type family...we can make it expect a defunctionalization symbol instead. Remember, defunctionalization symbols represent the "unapplied" versions of type families, so they are exactly the tools we need!

!!!singletons/Defunctionalization.hs "type family Foldr"

The difference is that instead of taking a type family or type constructor f :: j -> k -> k, we have it take the defunctionalization symbol f :: j ~> (k ~> k).

Instead of taking a type family or type constructor, we take that dummy type constructor.

Now we just need to have our defunctionalization symbols for MergeStateList:

!!!singletons/Defunctionalization.hs "data MergeStateSym0" "data MergeStateSym1" "type MergeStateSym2"

And now we can write MergeStateList:

!!!singletons/Defunctionalization.hs "type MergeStateList"

(If you "see" MergeStateSym0, you should read it was MergeState, but partially applied)

This compiles!

ghci> :kind! MergeStateList '[ 'Closed, 'Opened, 'Locked ]
'Locked
ghci> :kind! MergeStateList '[ 'Closed, 'Opened ]
'Closed
!!!singletons/Defunctionalization.hs "collapseHallway"

(Note: Unfortunately, we do have to use our our own Foldr here, that we just defined, instead of using the one that comes with singletons, because of some outstanding issues with how the singletons TH processes alternative implementations of foldr from Prelude. In general, the issue is that we should only expect type families to work with singletons if the definition of the type family perfectly matches the structure of how we implement our value-level functions like collapseHallway)

### Singletons to make things nicer

Admittedly this is all a huge mess of boilerplate. The code we had to write more than tripled, and we also have an unsightly number of defunctionalization symbols and Apply instance boilerplate for every function.

Luckily, the singletons library is here to help. You can just write:

\$(singletons [d|
data DoorState = Opened | Closed | Locked
deriving (Show, Eq, Ord)

mergeState :: DoorState -> DoorState -> DoorState
mergeState = max

foldr :: (a -> b -> b) -> b -> [a] -> b
foldr _ z []     = z
foldr f z (x:xs) = f x (foldr f z xs)

mergeStateList :: [DoorState] -> DoorState
mergeStateList = foldr mergeState Opened
|])

And all of these defunctionalization symbols are generated for you; singletons is also able to recognize that foldr is a higher-order function and translate its lifted version to take a defunctionalization symbol a ~> b ~> b.

That the template haskell also generates SingI instances for all of your defunctionalization symbols, too (more on that in a bit).

It's okay to stay "in the world of singletons" for the most part, and let singletons handle the composition of functions for you. However, it's still important to know what the singletons library generates, because sometimes it's still useful to manually create defunctionalization symbols and work with them.

The naming convention for non-symbolic names (non-operators) like myFunction are just to call them MyFunctionSym0 for the completely unapplied defunctionalization symbol, MyFunctionSym1 for the type constructor that expects one argument before returning a defunctionalization symbol, MyFunctionSym2 for the type constructor that expects two arguments before returning a defunctionalization symbol, etc.

For operator names like ++, the naming convention is to have ++@#@\$ be the completely unapplied defunctionalization symbol, ++@#@\$\$ be the type constructor that expects one argument before returning a defunctionalization symbol, ++@#@\$\$\$ be the type constructor that takes two arguments before returning a defunctionalization symbol, etc.

Another helpful thing that singletons does is that it also generates defunctionalization symbols for type families and type synonyms you define in the Template Haskell, as well --- so if you write

\$(singletons [d|
type MyTypeFamily (b :: Bool) :: Type where
MyTypeFamily 'False = Int
MyTypeFamily 'True  = String
|])

and

\$(singletons [d|
type MyTypeSynonym a = (a, [a])
|])

singletons will generate:

data MyTypeFamilySym0 :: Bool ~> Type
type instance Apply MyTypeFamilySym0 b = MyTypeFamily b

type MyTypeFamilySym1 b = MyTypeFamily b

and

data MyTypeSynonymSym0 :: Type ~> Type
type instance Apply MyTypeSynonym b = MyTypeSynonym a

type MyTypeSynonymSym1 a = MyTypeSynonym a

#### Bringing it All Together

Just to show off the library, remember that singletons also promotes typeclasses?

Because DoorState is a monoid with respect to merging, we can actually write and promote a Monoid instance: (requires singletons-2.5 or higher)

\$(singletons [d|
instance Semigroup DoorState where
(<>) = mergeState
instance Monoid DoorState where
mempty  = Opened
mappend = (<>)
|])

We can promote fold:

\$(singletons [d|
fold :: Monoid b => [b] -> b
fold []     = mempty
fold (x:xs) = x <> fold xs
|])

And we can write collapseHallway in terms of those instead :)

!!!singletons/Door4Final.hs "collapseHallway'" "collapseSomeHallway'"

(Note again unfortunately that we have to define our own fold instead of using the one from singletons and the SFoldable typeclass, because of issue #339)

## Thoughts on Symbols

Defunctionalization symbols may feel like a bit of a mess, and the naming convention is arguably less than aesthetically satisfying. But, as you work with them more and more, you start to appreciate them on a deeper level.

At the end of the day, you can compare defunctionalization as turning "functions" into just constructors you can match on, just like any other data or type constructor. That's because they are just type constructors!

In a sense, defining defunctionalization symbols is a lot like working with pattern synonyms of your functions, instead of directly passing the functions themselves. At the type family and type class level, you can "pattern match" on these functions.

For a comparison at the value level -- you can't pattern match on (+), (-), (*), and (/):

-- Doesn't work like you think it does
invertOperation :: (Double -> Dobule -> Double) -> (Double -> Double -> Double)
invertOperation (+) = (-)
invertOperation (-) = (+)
invertOperation (*) = (/)
invertOperation (/) = (*)

You can't quite match on the equality of functions to some list of patterns. But, what you can do is create constructors representing your functions, and match on those.

This essentially fixes the "type lambda problem" of type inference and typeclass resolution. You can't match on arbitrary lambdas, but you can match on dummy constructors representing type functions.

And a bit of the magic here, also, is the fact that you don't always need to make our own defunctionalization symbols from scratch --- you can create them based on other ones in a compositional way. This is the basis of libraries like decidable.

For example, suppose we wanted to build defunctionalization symbols for MergeStateList. We can actually build them directly from defunctionalization symbols for Foldr.

Check out the defunctionalization symbols for Foldr:

!!!singletons/Defunctionalization.hs "data FoldrSym0" "data FoldrSym1" "data FoldrSym2" "type FoldrSym3"

We can actually use these to define our MergeStateList defunctionalization symbols, since defunctionalization symbols are first-class:

!!!singletons/Defunctionalization.hs "type MergeStateListSym0"

And you can just write collapseHallway as:

collapseHallway :: Hallway ss -> Door (MergeStateListSym0 @@ ss)
-- or
collapseHallway :: Hallway ss -> Door (FoldrSym2 MergeStateSym0 'Opened @@ ss)

You never have to actually define MergeStateList as a function or type family!

The whole time, we're just building defunctionalization symbols in terms of other defunctionalization symbols. And, at the end, when we finally want to interpret the complex function we construct, we use Apply, or @@.

You can think of FoldrSym1 and FoldrSym2 as defunctionalization symbol constructors -- they're combinators that take in defunctionalization symbols (like MergeStateSym0) and return new ones.

### Sigma

Let's look at a nice tool that is made possible using defunctionalization symbols: dependent pairs. I talk a bit about dependent pairs (or dependent sums) in part 2 of this series, and also in my dependent types in Haskell series.

Essentially, a dependent pair is a tuple where the type of the second field depends on the value of the first one. This is basically what SomeDoor was:

data SomeDoor :: Type where
MkSomeDoor :: Sing x -> Door x -> SomeDoor

The type of the Door x depends on the value of the Sing x, which you can read as essentially storing the x.

We made SomeDoor pretty ad-hoc. But what if we wanted to make some other predicate? Well, we can make a generic dependent pair by parameterizing it on the dependence between the first and second field. Singletons provides the Sigma type, in the Data.Singletons.Sigma module:

data Sigma k :: (k ~> Type) -> Type where
(:&:) :: Sing x -> (f @@ x) -> Sigma k f

-- also available through fancy type synonym
type Σ k = Sigma k

If you squint carefully, you can see that Sigma k is just SomeDoor, but parameterized over Door. Instead of always holding Door x, we can have it parameterized on an arbitrary function f and have it hold an f @@ x.

We can actually define SomeDoor in terms of Sigma:

!!!singletons/Door4Final.hs "type SomeDoor" "mkSomeDoor"

(Remember TyCon1 is the defunctionalization symbol constructor that turns any normal type constructor j -> k into a defunctionalization symbol j ~> k)

That's because a Sigma DoorState (TyCon1 Door) contains a Sing (x :: DoorState) and a TyCon1 Door @@ x, or a Door x.

This is a simple relationship, but one can imagine a Sigma parameterized on an even more complex type-level function. We'll explore more of these in the exercises.

For some context, Sigma is an interesting data type (the "dependent sum") that is ubiquitous in dependently typed programming.

### Singletons of Defunctionalization Symbols

One last thing to tie it all together -- let's write collapseHallway in a way that we don't know the types of the doors.

!!!singletons/Door4Final.hs "type SomeHallway"

The easy way would be to just use sMergeStateList that we defined:

!!!singletons/Door4Final.hs "collapseSomeHallway"

But what if we didn't write sMergeStateList, and we constructed our defunctionalization symbols from scratch?

!!!singletons/Door4Final.hs "collapseHallway''"

collapseSomeHallway'' :: SomeHallway -> SomeDoor
collapseSomeHallway'' (ss :&: d) = ???    -- what goes here?
:&: collapseHallway'' d

This will be our final defunctionalization lesson. How do we turn a singleton of ss into a singleton of FoldrSym2 MergeStateSym0 'Opened @@ s ?

First -- we have Foldr at the value level, as sFoldr. We glossed over this earlier, but singletons generates the following function for us:

type family Foldr (f :: j ~> k ~> k) (z :: k) (xs :: [j]) :: k where
Foldr f z '[]       = z
Foldr f z (x ': xs) = (f @@ x) @@ Foldr f z xs

sFoldr
:: Sing (f :: j ~> k ~> k)
-> Sing (z :: k)
-> Sing (xs :: [j])
-> Sing (Foldr f z xs :: k)
sFoldr f z SNil           = z
sFoldr f z (x `SCons` xs) = (f @@ x) @@ sFoldr f z xs

Where (@@) :: Sing f -> Sing x -> Sing (f @@ x) (or applySing) is the singleton/value-level counterpart of Apply or (@@).[^slambda]

[^slambda]: (@@) (and as we see shortly, the singFun functions) are all implemented in terms of SLambda, the "singleton" for functions. Understanding the details of the implementation of SLambda aren't particularly important for the purposes of this introduction.

So we can write:

collapseSomeHallway'' :: SomeHallway -> SomeDoor
collapseSomeHallway'' (ss :&: d) = sFoldr ???? SOpened ss
:&: collapseHallwa''y d

But how do we get a Sing MergeStateSym0?

We can use the singFun family of functions:

singFun2 @MergeStateSym0 sMergeState
:: Sing MergeStateSym0

But, also, conveniently, the singletons library generates a SingI instance for MergeStateSym0, if you defined mergeState using the singletons template haskell:

sing :: Sing MergeStateSym0
-- or
sing @_ @MergeStateSym0         -- singletons 2.4
sing @MergeStateSym0            -- singletons 2.5

And finally, we get our answer:

!!!singletons/Door4Final.hs "collapseSomeHallway''"

## Closing Up

Woo! Congratulations, you've made it to the end of the this Introduction to Singletons tetralogy! This last and final part understandably ramps things up pretty quickly, so don't be afraid to re-read it a few times until it all sinks in before jumping into the exercises.

I hope you enjoyed this journey deep into the motivation, philosophy, mechanics, and usage of this great library. Hopefully these toy examples have been able to show you a lot of ways that type-level programming can help your programs today, both in type safety and in writing more expressive programs. And also, I hope that you can also see now how to leverage the full power of the singletons library to make those gains a reality.

There are a few corners of the library we haven't gone over (like the TypeLits- and TypeRep-based singletons -- if you're interested, check out this post where I talk a lot about them), but I'd like to hope as well that this series has equipped you to be able to dive into the library documentation and decipher what it holds, armed with the knowledge you now have. (We also look at TypeLits briefly in the exercises)

You can download the source code here --- [Door4Final.hs][source-final] contains the final versions of all our definitions, and [Defunctionalization.hs][] contains all of our defunctionalization-from-scratch work. These are designed as stack scripts that you can load into ghci. Just execute the scripts:

!!![Defunctionalization.hs]:singletons/Door3.hs

\$ ./Door4Final.hs
ghci>

And you'll be dropped into a ghci session with all of the definitions in scope.

As always, please try out the exercises, which are designed to help solidify the concepts we went over here! And if you ever have any future questions, feel free to leave a comment or find me on twitter or in freenode #haskell, where I idle as jle`.

### Looking Forward

Some final things to note before truly embracing singletons: remember that, as a library, singletons was always meant to become obsolete. It's a library that only exists because Haskell doesn't have real dependent types yet.

Dependent Haskell is coming some day! It's mostly driven by one solo man, Richard Eisenberg, but every year buzz does get bigger. In a recent progress report, we do know that we realistically won't have dependent types before 2020. That means that this tutorial will still remain relevant for at least another two years :)

How will things be different in a world of Haskell with real dependent types? Well, for a good guess, take a look at Richard Eisenberg's Dissertation!

One day, hopefully, we won't need singletons to work with types at the value-level; we would just be able to directly pattern match and manipulate the types within the language and use them as first-class values, with a nice story for dependent sums. And some day, I hope we won't need any more dances with defunctionalization symbols to write higher-order functions at the type level --- maybe we'll have a nicer way to work with partially applied type-level functions (maybe they'll just be normal functions?), and we don't need to think any different about higher-order or first-order functions.

So, as a final word --- Happy Haskelling, everyone! May you leverage the great singletons library to its full potential, and may we also all dream of a day where singletons becomes obsolete. But may we all enjoy the wonderful journey along the way.

Until next time!

## Exercises

Here are your final exercises for this series! Start from [this sample source code][source-final], which has all of the definitions that the exercises and their solutions require. Just make sure to delete all of the parts after the -- Exercises comment if you don't want to be spoiled. Remember again to enable -Werror=incomplete-patterns or -Wall to ensure that all of your functions are total.

!!![source-final]:singletons/Door4Final.hs

!!![solution1]:singletons/Door4Final.hs "-- | 1."1 !!![solution2]:singletons/Door4Final.hs "-- | 2."1 !!![solution3]:singletons/Door4Final.hs "-- | 3."1 !!![solution4]:singletons/Door4Final.hs "-- | 4."1 !!![solution5]:singletons/Door4Final.hs "-- | 5."1 !!![solution6]:singletons/Door4Final.hs "-- | 6."1

1. Let's try combining type families with proofs! In doing so, hopefully we can also see the value of using dependent proofs to show how we can manipulate proofs as first-class values that the compiler can verify.

Remember Knockable from Part 3?

!!!singletons/Door4Final.hs "data Knockable"

Closed and Locked doors are knockable. But, if you merge two knockable doors...is the result also always knockable?

I say yes, but don't take my word for it. Prove it using Knockable!

!!!singletons/Door4Final.hs "mergedIsKnockable"4

mergedIsKnockable is only implementable if the merging of two DoorStates that are knockable is also knockable. See if you can write the implementation!

[Solution here!][solution1]

2. Write a function to append two hallways together.

appendHallways
:: Hallway ss
-> Hallway ts
-> Hallway ????

from singletons --- implement any type families you might need from scratch!

Remember the important principle that your type family must mirror the implementation of the functions that use it.

Next, for fun, use appendHallways to implement appendSomeHallways:

!!!singletons/Door4Final.hs "type SomeHallway" "appendSomeHallways"4

[Solution here!][solution2]

3. Can you use Sigma to define a door that must be knockable?

To do this, try directly defining the defunctionalization symbol KnockableDoor :: DoorState ~> Type (or use singletons to generate it for you --- remember that singletons can also promote type families) so that:

type SomeKnockableDoor = Sigma DoorState KnockableDoor

will contain a Door that must be knockable.

Try doing it for both (a) the "dependent proof" version (with the Knockable data type) and for (b) the type family version (with the StatePass type family).

[Solutions here!][solution3] I gave four different ways of doing it, for a full range of manual vs. auto-promoted defunctionalization symbols and Knockable vs. Pass-based methods.

Hint: Look at the definition of SomeDoor in terms of Sigma:

type SomeDoor = Sigma DoorState (TyCon1 Door)

Hint: Try having KnockableDoor return a tuple.

4. Take a look at the API of the Data.Singletons.TypeLits module, based on the API exposed in GHC.TypeNats module from base.

Using this, you can use Sigma to create a predicate that a given Nat number is even:

data IsHalfOf :: Nat -> Nat ~> Type
type instance Apply (IsHalfOf n) m = n :~: (m * 2)

type IsEven n = Sigma Nat (IsHalfOf n)

(*) is multiplication from the Data.Singletons.Prelude.Num module. (You must have the -XNoStarIsType extension on for this to work in GHC 8.6+), and :~: is the predicate of equality from Part 3:

data (:~:) :: k -> k -> Type where
Refl :: a :~: a

(It's only possible to make a value of type a :~: b using Refl :: a :~: a, so it's only possible to make a value of that type when a and b are equal. I like to use Refl with type application syntax, like Refl @a, so it's clear what we are saying is the same on both sides; Refl @a :: a :~: a)

The only way to construct an IsEven n is to provide a number m where m * 2 is n. We can do this by using SNat @m, which is the singleton constructor for the Nat kind (just like how STrue and SFalse are the singleton constructors for the Bool kind):

tenIsEven :: IsEven 10
tenIsEven = SNat @5 :&: Refl @10
-- Refl is the constructor of type n :~: (m * 2)
-- here, we use it as Refl @10 :: 10 :~: 10

-- won't compile
sevenIsEven :: IsEven 10
sevenIsEven = SNat @4 :&: Refl
-- won't compile, because we need something of type `(4 * 2) :~: 7`,
-- but Refl must have type `a :~: a`; `8 :~: 7` is not constructable
-- using `Refl`.  Neither `Refl @8` nor `Refl @7` will work.

Write a similar type IsOdd n that can only be constructed if n is odd.

type IsOdd n = Sigma Nat (???? n)

And construct a proof that 7 is odd:

!!!singletons/Door4Final.hs "sevenIsOdd"1

[Solution here!][solution4]

On a sad note, one exercise I'd like to be able to add is to ask you to write decision functions and proofs for IsEven and IsOdd. Unfortunately, Nat is not rich enough to support this out of the box without a lot of extra tooling!

5. A common beginner Haskeller exercise is to implement map in terms of foldr:

map :: (a -> b) -> [a] _> [b]
map f = foldr ((:) . f) []

Let's do the same thing at the type level, manually.

Directly implement a type-level Map, with kind (j ~> k) -> [j] -> [k], in terms of Foldr:

type Map f xs = Foldr ???? ???? xs

Try to mirror the value-level definition, passing in (:) . f, and use the promoted version of (.) from the singletons library, in Data.Singletons.Prelude. You might find TyCon2 helpful!

[Solution here!][solution5]

6. Make a SomeHallway from a list of SomeDoor:

!!!singletons/Door4Final.hs "type SomeDoor" "type SomeHallway" "mkSomeHallway"1

Remember that the singleton constructors for list are SNil (for []) and SCons (for (:))!

[Solution here!][solution5]

## Special Thanks

None of this entire series would be possible without the hard work and effort of the amazing singletons library authors and maintainers --- especially Richard Eisenberg and Ryan Scott.

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!