Xia Li-yao

7 Jun 2018

•

12 min read

With the help of GHC’s many extensions, we can encode simple forms of dependent types, allowing us to enforce more expressive invariants in our programs at compile time.

In this post, I will walk you through an implementation of heterogeneous lists to demonstrate the features and techniques commonly associated with the current flavor of “dependent Haskell”.

My Literate Haskell blog posts render the whole source, including the extensions. In this post, I will mention most of them when they are used, and explain them a bit.

```
{-# LANGUAGE
AllowAmbiguousTypes,
ConstraintKinds,
DataKinds,
FlexibleContexts,
FlexibleInstances,
GADTs,
InstanceSigs,
MultiParamTypeClasses,
PolyKinds,
RankNTypes,
ScopedTypeVariables,
TypeApplications,
TypeFamilies,
TypeOperators,
UndecidableInstances #-}
module HLists where
```

As for those that won’t be mentioned later, let’s take care of them now briefly.

InstanceSigs allows type signatures to appear in instance declarations, so that we don’t need to go back to the type class to remember what is being defined. This is also consistent with the common practice of annotating all toplevel bindings.

ConstraintKinds, FlexibleContexts, FlexibleInstances, ScopedTypeVariables and UndecidableInstances lift restrictions that are mostly historical. They are quite benign extensions that might be unnoticeable when enabled if you don’t already know about them.

The GHC user guide also documents all the available extensions.

We will construct lists using the infix pair type below as a cons, and the unit type () as a nil. The infix type constructor (:*) is allowed by the TypeOperators extension (not to be confused with the identically-named data constructor (:*), which is allowed by the standard).

```
data a :* b = a :* b deriving Show
infixr 1 :*
```

Another common candidate for heterogeneous lists is a GADT indexed by a list of types, but I prefer (:*) for its simplicity.

Here is an example of a list of three elements:

```
hlistExample1 :: Int :* String :* String :* ()
hlistExample1 = 1 :* "One" :* "1" :* ()
```

You might wonder, why even have a nil, since there can be any type to the right of (:*)? This is clearly shorter:

```
hlistExample2 :: Int :* String :* String
hlistExample2 = 1 :* "One" :* "1"
```

The problem is that it introduces an odd corner case, where the last element of a list can never be a heterogeneous list, since (:*) will be interpreted as a tail of the list. Is the above list a list of three elements, or is it a list of two (one Int and one String :* String)?

Such a choice would be reflected in the rest of the implementation as an unnecessary complication compared to having lists be clearly terminated by ().

Hence, it is possible to accidentally construct these ill-formed lists, but something will break at compile time anyway, so that’s okay.

We will implement a function to extract an element of a given type in the list. It is declared as part of the following two-parameter type class. This requires the MultiParamTypeClasses extension.

```
class Get a bs where
get :: bs -> a
```

For example, we can get the string out of this triple as follows:

```
getExample1 :: String
getExample1 =
get @String ((1 :: Int) :* "One" :* "1" :* ())
-- "One"
```

In the simple examples of this post, we can always infer the output type of get from the context where it is applied. But in general, an explicit annotation will often be necessary, and that is nicely done with the TypeApplications extension as shown in the above example. get has two type arguments, in the order of the type class arguments.1 Here we explicitly apply it to its first type argument, which is the type of the element we want to extract.

We’ll go through three progressively more elaborate implementations for Get. To avoid conflicts between these attempts, we will declare a new copy of that class at the beginning of each section.

```
class Get0 a bs where
get0 :: bs -> a
```

The simplest solution is to write two overlapping instances. First, when the type of the head of the list doesn’t match, we keep looking in the tail, calling get0 recursively.

```
instance Get0 a bs => Get0 a (b :* bs) where
get0 :: (b :* bs) -> a
get0 (_ :* ys) = (get0 :: bs -> a) ys
```

Second, when the types match, we return the head of the list.

```
instance {-# OVERLAPPING #-} Get0 a (a :* bs) where
get0 :: (a :* bs) -> a
get0 (x :* _) = x
```

By default, overlapping instances are forbidden: often enough, they are caused by a mistake. We indicate that the overlap is intentional using the OVERLAPPING pragma on the more specific instance.2 The instance (Get0 a (a :* bs)) is more specific than (Get0 a (b :* bs)) because we can substitute variables in the latter to obtain the former: replace b with a.

Before going in more details, let’s give this implementation a try. We can reproduce the above example successfully:

```
get0Example1 :: String
get0Example1 =
get0 @String ((1 :: Int) :* "One" :* "1" :* ())
-- "One"
```

What about this more abstract operation: get the third element of any sufficiently long list.

```
getThird0 :: (a :* b :* c :* ds) -> c
getThird0 = get0
-- Type error!
```

The constraint solver gets stuck on solving Get0 c (a :* b :* c :* ds) because it cannot determine whether a is equal to c, so it can’t choose one of the above instances.

If instances are allowed to overlap, which instance should it select? I won’t go into the rationale here, but the rule is to always prefer the most specific instance, and only pick a less specific one when there is no way to instantiate type variables in the environment to make the other more specific instance(s) match. In this case, since a can be made equal to c, the OVERLAPPING instance can still potentially match, hence the error.

The GHC user guide actually has a section on overlapping instances which is a pretty good reference about this algorithm if you can get past the jargon.

One way out is to add the unresolved constraint to the signature of getThird0, and let it be resolved at use sites with concrete types.

```
getThird0
:: Get0 c (a :* b :* c :* ds)
=> (a :* b :* c :* ds)
-> c
getThird0 = get0
```

However, if c is equal to a or b, this function won’t be doing what its name and type advertise!

```
-- a, b, c all equal to String, ds ~ (String :* ())
get0Example2 :: String
get0Example2 =
getThird0 ("One" :* "Two" :* "Three" :* "Four" :* ())
-- "One"
-- We would like "Three"
```

This problem is of the kind that has many solutions, and the best one would really depend on the purpose of this get function. But the one I want to share here is to write the constraint that c is not equal to a or b. This makes the type of getThird less general to allow the Get constraint above to be solved at the definition of getThird.

```
class Get1 a bs where
get1 :: bs -> a
```

Immediately, we face an obstacle: there is an equality constraint (~), but no primitive disequality constraint “(/~)”, or any general way to negate a constraint (such a thing would be incompatible with the open-world principle of type classes).

Fortunately, we can define equality differently: as a boolean-valued closed TypeFamily. (Are we playing language extensions bingo?)

The type-level function (==) has two clauses, it is equal to True if the operands are equal, False otherwise.

```
type family a == b :: Bool where
a == a = 'True
a == b = 'False
```

Note that type families allow nonlinear pattern-matching: the first clause matches twice on the same variable. This is not allowed at the term level, because there is no universal way of comparing values, especially infinite values, functions, and values of abstract types such as IO.3 But at the type level, it is possible.

Data constructors such as True and False are allowed at (or promoted to) the type level by the DataKinds extension. They are disambiguated from type constructors using the single-quote prefix, since they can use the same names even in a single module. DataKinds is the kind4 of extension that is so intuitive one doesn’t even notice when they’re using it.

Now we can match on the outcome of that comparison operation. The constraint 'False ~ (a == b) holds if and only if a and b are not equal. Conversely, one could also write 'True ~ (a == b) in the case of an equality, but the more common a ~ b works as well, if not better.5 Note that a ~ b implies 'True ~ (a == b), but the type checker is not aware of the converse! From the type checker’s point of view, (==) is a type-level function like any other, and it doesn’t make any particular effort to reason about its output.

We introduce a new type class GetIf, with one more parameter to carry the comparison result. The superclass constraint is a minor safety net, ensuring there’s only one way to use that type class: we must set the boolean aeqb to a == b, and instances must also satisfy that constraint.

```
class (aeqb ~ (a == b)) => GetIf aeqb a b bs where
getIf :: (b :* bs) -> a
```

We have two cases, depending on the value of the boolean aeqb. Either a is equal to b, then we get the head of the list.

```
instance (a ~ b) => GetIf 'True a b bs where
getIf :: (a :* bs) -> a
getIf (x :* _) = x
```

Or a is not equal to b, then we keep looking in the tail. Since instances must satisfy their superclass constraints, we must add 'False ~ (a == b).

```
instance ('False ~ (a == b), Get1 a bs)
=> GetIf 'False a b bs where
getIf :: (b :* bs) -> a
getIf (_ :* ys) = get1 ys
```

Finally, Get1 and GetIf are connected by this instance of Get1 which immediately defers to GetIf.

```
instance GetIf (a == b) a b bs => Get1 a (b :* bs) where
get1 = getIf
```

The main benefit of this technique over the previous one is that it avoids overlapping instances, which can be difficult to understand. The overlap still exists, as part of the definition of (==), but that is contained in a general and reusable construct, and the rule of matching clauses sequentially for closed type families may be more intuitive.6

For clarity, we can define a “class synonym” for type disequality. For constraints, the advantage of this method over type synonyms is that type synonyms can’t be partially applied, while “class synonyms” can.

```
class ('False ~ (a == b)) => a /~ b
instance ('False ~ (a == b)) => a /~ b
```

We can thus give getThird the following type, ensuring that the c we get is indeed the third element in the list.

```
getThird1 :: (c /~ a, c /~ b) => (a :* b :* c :* ds) -> c
getThird1 = get1
```

Now using getThird1 with types for the first or second element equal to the third results in a type error. Whether that is good or bad depends on what you actually want to do with those heterogeneous lists.

```
getThird1Example :: String
getThird1Example =
getThird1 @String ("One" :* "Two" :* "Three" :* "Four" :* ())
-- Type error!
```### Type-level conditionals
One last issue to address about that solution is that it is quite cumbersome to implement. We have to define a new type class, and the original class does nothing but call the new one immediately. Conceptually, all that work is only to match on a boolean, one of the simplest data types (or, data kinds?). We’d like a similarly simple abstraction to do that. How about if? We can certainly make it a type family7:
```

type family If (b :: Bool) (c :: k) (d :: k) :: k where
If 'True c * = c
If 'False * d = d

```
This family is polykinded, which is just another name for polymorphism of type-level constructs: If :: forall k. Bool -> k -> k -> k. (PolyKinds extension; kinds are to types as types are to terms.)
It will take a bit of setup to be able to use If, so let’s put it aside for the moment.
We also need a type class to pattern match on the type-level boolean and decide what to do at run time, as opposed to type families, which only live at compile time.
Here’s a first attempt.
```

class IsBool0 b where _If0 :: forall r. r -> r -> r

instance IsBool0 'True where
*If0 x * = x

instance IsBool0 'False where
*If0 * y = y

```
Note that b does not appear in the type of _If0, meaning that b cannot be inferred from the context where _If0 is applied. Historically, this would always result in ambiguity errors, thus, such a type was forbidden, forcing us to resolve the ambiguity by adding a trivial argument or tagging the function somehow.
The AllowAmbiguousTypes extension tells the compiler to accept definitions like _If0 above, keeping signatures tidy; the ambiguity can be resolved at use sites with the TypeApplications extension, as it makes type arguments explicit (corresponding to those variables under forall).
Some examples of _If0 in action:
```

_If0Examples :: [String] _If0Examples = [ _If0 @'True "This" "That" -- "This"

, _If0 @'False "This" "That" -- "That"

, _If0 @(Int == Bool) "Int is equal to Bool" "Int is not equal to Bool" -- "Int is not equal to Bool" ]

```
However we will have trouble implementing Get. To see why, below is what the final instance will roughly look like. There are two cases as always. If the type a we are looking for is equal to b, then we return the head y of the list, otherwise we keep looking in the tail ys.
```

instance (???) => Get a (b :* bs) where
get :: b :* bs -> a
get (y :* ys) =
_If @(a == b)
y -- When (a == b) ~ 'True, i.e., a ~ b
(get ys) -- When (a == b) ~ 'False, i.e., a /~ b

```
The two branches of _If make assumptions that the type checker (currently) rejects. In the True branch, we return y :: b when the expected type is a, this assumes that a is equal to b. In the False branch, we call get on the tail, which requires a Get a bs constraint. We could add it to the context ((???) above), but that would imply a traversal of the whole type list, whereas the behavior of the previous implementations was to stop as soon as we found the element we are looking for.
One way to view this problem is that the type of _If doesn’t encode the condition that the boolean is True in one branch, and False in the other. It turns out we can apply that intuition quite literally in order to fix _If:
```

class IsBool b where _If :: forall r . (('True ~ b) => r) -> (('False ~ b) => r) -> r

```
The RankNTypes extension allows quantifiers and constraints to occur in a type signature in more locations than only the beginning.8 Thus, the two arguments of _If are now parameterized by constraints. The function _If can only use its first argument if b is True, and its second argument if b is False, whereas those arguments can use the constraints to deduce other useful facts locally. Interestingly, there is thus exactly one way to write each of the two IsBool instances:
```

instance IsBool 'True where
*If :: forall r. r -> (('False ~ 'True) => r) -> r
_If x * = x

instance IsBool 'False where
*If :: forall r. (('True ~ 'False) => r) -> r -> r
_If * y = y

```
The last piece of the puzzle is to complete the (???) context in the Get instance sketched above.
* _If requires an IsBool instance in any case.
* We shall use the If type family we defined previously to construct a context which depends on the value of (a == b).
In each branch of _If, (a == b) is assumed to be either True or False, which allows the compiler to evaluate If (a == b) _ _, uncovering the constraint that each case exactly needs, (a ~ b) in one, (Get a bs) in the other.
We define a type synonym If' to collapse the IsBool and If constraints together.
```

type If' b c d = (IsBool b, If b c d)

instance If' (a == b) (a ~ b) (Get a bs)
=> Get a (b :* bs) where
get :: b :* bs -> a
get (y :* ys) =
*If @(a == b)
y -- (If 'True (a ~ b) *) becomes (a ~ b)
(get ys) -- (If 'False _ (Get a bs)) becomes (Get a bs)

```
We’re now reaching the end of this heterogeneous list example. From the outside, get should behave identically to get1, but the definition of get was refactored to a single instance. Do you find that easier to read than the previous versions?
## Appendix: Dependently typed programming with singletons
A central reference in Haskell’s dependently-typed landscape is the singletons library. One common point between this blog post and singletons is that, in fact, the class IsBool is the Church encoding of the Bool singleton.
A singleton is a type with a single inhabitant. A useful class of singletons can be defined as Generalized Algebraic Data Types (GADTs extension). GADTs are types whose parameters depend on the constructors, and vice versa.
The parameter b of SBool below is tied to the constructor: STrue is the unique inhabitant of SBool 'True and SFalse is the unique inhabitant of SBool 'False. The point of such a type is that by pattern matching on an SBool b, we can refine the value of b in each branch, similarly to the _If combinator.
```

data SBool (b :: Bool) where STrue :: SBool 'True SFalse :: SBool 'False

```
There is an isomorphism between SBool and IsBool instances, that we can witness by desugaring IsBool instances to the polymorphic functions that they contain.
```

-- Desugar IsBool by unwrapping *If
type IsBool* b
= forall r
. (('True ~ b) => r)
-> (('False ~ b) => r)
-> r

*SBool_to_IsBool :: forall b. SBool b -> IsBool* b
*SBool_to_IsBool sb = case sb of
STrue -> (\x * -> x)
SFalse -> (_ y -> y)

*IsBool_to_SBool :: forall b. IsBool* b -> SBool b
_IsBool_to_SBool _If = _If STrue SFalse

```
It is easy to check that _IsBool_to_SBool . _SBool_to_IsBool = id. The converse _SBool_to_IsBool . _IsBool_to_SBool = id is also true, but the proof relies on a more involved argument (parametricity, free theorems).
The “Church encoding” claim above can be made more explicit with the following equivalent9 reformulation of SBool. This also shows another characteristic of GADTs: constructors can be given arbitrary function types, especially polymorphic functions with constraints, as long as the result matches the type being defined. The function constraints and arguments become constructor fields.
```

data SBool' (b :: Bool) where STrue' :: ('True ~ b) => SBool' b SFalse' :: ('False ~ b) => SBool' b

```
It should be compared to the Either data type (which can be defined using GADT notation to make explicit the types of its constructors) and its Church encoding.
```

data Either a b where Left :: a -> Either a b Right :: b -> Either a b

type Either_ a b = forall r. (a -> r) -> (b -> r) -> r

```
Dependent types in Haskell, talk by Stephanie Weirich.
1. Practical dependent types in Haskell, blog post by Justin Le.
I don’t know whether they can be reordered without changing the order of the class arguments.
1. There are two more variants, OVERLAPPABLE and OVERLAPS. You can read more about them in the GHC user guide.↩
1. We could imagine using an Eq constraint to implement nonlinear pattern-matching; I haven’t really considered the implications.↩
1. The fact that “type”, “kind” and “sort” are commonly used in this area of programming languages makes it sometimes difficult to talk about things.↩
1. That type family is also defined in base: Data.Type.Equality.(==). However, its definition has recently changed (since base 4.11, i.e., GHC 8.4). I don’t like that it is not reflexive anymore, but I haven’t gathered any concrete data about the pros and cons yet. (Previous version, changelog (Ctrl-F “Equality”).)↩
1. It’s a bit more complex than that in general, for example Data.Type.Bool defines the type-level boolean (&&) and (||) in a way that simplifies ideally: True && a = a = a && True.↩
1. Data.Type.Bool.If in base.↩
1. Both RankNTypes and ScopedTypeVariables enable the forall quantifier/keyword, that allows us to explicitly bind type variables. Otherwise, the quantifier gets inserted implicitly at the beginning of a type signature.↩
1. Up to internal boxing shenanigans.↩
```

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