# Unordered Function Application In Haskell

Here is a little puzzle: How might a function like `reorderArgs`

be defined, such that the code below compiles?

```
{-# LANGUAGE OverloadedStrings #-}
import qualified Data.Text as T
import Control.Apply.Unordered.Mono (reorderArgs)
-- Each example in this group is equal to "hello"
--
-- T.cons :: Char -> T.Text -> T.Text
ex1 = T.cons 'h' ello
ex2 = reorderArgs T.cons 'h' ello
ex3 = reorderArgs T.cons ello 'h'
-- Each example in this group is equal to "hhhello".
--
-- T.justifyRight :: Int -> Char -> T.Text -> T.Text
ex4 = T.justifyRight seven 'h' ello
ex5 = reorderArgs T.justifyRight ello 'h' seven
ex6 = reorderArgs T.justifyRight 'h' ello seven
ello :: T.Text
ello = "ello"
seven :: Int
seven = 7
```

I found it quite surprising that this is possible to define. One thing that makes it feasible is a restriction that all arguments are monomorphic. Solving this puzzle is a decent didactic exercise, as it involves a variety of type-level techniques:

Using closed type families to direct instance selection. This is described in this post.

Using custom type errors to improve the error messaging. This is described in the next post.

Using typeclasses to implement polyvariadic functions. This will be described in a future post.

The functionality described in these posts is available on Hackage, in the apply-unordered-mono package. It is also possible to generalize this to polymorphic functions, via a GHC compiler plugin. This is demonstrated by the related apply-unordered package.

## Type-directed application of a single argument

A step towards a solution to this is to write a function which implements type-directed application of a single argument. Lets name this function `(?)`

, used like this:

`(?)`

takes any function `f`

, and an argument `a`

that matches the type of one of its parameters. It will then partially apply `f`

to this argument. So, the example above is equivalent to:

`(?)`

should also be able to provide a value for the first parameter:

The plan is to automatically construct this code by using typeclass machinery!

## A failed attempt

One way to approach this might be a typeclass with an associated type family for the result:

The type-level algorithm implemented by these instances should check if the first parameter of `f`

matches `a`

. If it does, then apply the function:

If it doesn't, then it should recurse down the right-hand-side of the `(->)`

type:

```
instance ApplyByType a r => ApplyByType a (b -> r) where
type ApplyByTypeResult a (b -> r) = b -> ApplyByTypeResult a r
f ? y = \x -> f x ? y
```

However, this doesn't work! GHC complains:

```
Conflicting family instance declarations:
ApplyByTypeResult a (a -> r) = r
ApplyByTypeResult a (b -> r) = b -> ApplyByTypeResult a r
```

The problem is that open type families do not allow overlap in their instances.

The code that results in this error is didactic/V1.hs on github.

## Attempted fix: use a closed type family

Closed type families permit such overlap, and so provide a nice workaround for this problem. Here's what using a closed type family might look like:

```
type family ApplyByTypeResult a f where
ApplyByTypeResult a (a -> r) = r
ApplyByTypeResult a (b -> r) = b -> ApplyByTypeResult a r
```

This can be tried out in GHCi, by using `:k!`

to ask GHC to normalize the type – expanding synonyms and applying type functions:

```
> :k! ApplyByTypeResult Text (Char -> Text -> Text)
ApplyByTypeResult Text (Char -> Text -> Text) :: *
= Char -> Text
```

Awesome! It's figured out that `Text`

matches the second parameter, and so unordered partial application results in a `Char -> Text`

type.

Trying it along with the typeclass:

```
class ApplyByType a f where
(?) :: f -> a -> ApplyByTypeResult a f
instance ApplyByType a (a -> r) where
f ? x = f x
instance ApplyByType a r => ApplyByType a (b -> r) where
f ? y = \x -> f x ? y
```

Argh! Foiled again:

```
• Couldn't match expected type ‘ApplyByTypeResult a (b -> r)’
with actual type ‘b -> ApplyByTypeResult a r’
• The lambda expression ‘\ x -> f x ? y’
has one argument,
but its type ‘ApplyByTypeResult a (b -> r)’ has none
...
```

I think the problem here is that GHC has insufficient information to choose between the different cases of the closed type family. In order to reject the first equation, it would need to know that `a`

does not unify with `b`

.

The code that results in this error is didactic/V2.hs on github.

## Fix: use a closed type family to choose the instance!

A well-known trick among typeclass machinery enthusiasts is to use a closed type family to choose an instance. Here's how this works:

A type parameter is added to the typeclass, which is used to select the instance.

A

`Proxy`

parameter is added to the method(s), used to specify the type parameter when using the class.A closed type family is used to compute the type to pass in via the

`Proxy`

.

Here's what it looks like for this application:

```
data MatchArgResult
= Matches
| Doesn'tMatch
| NoArgToMatch
type family MatchFirstArg a f :: MatchArgResult where
MatchFirstArg a (a -> r) = 'Matches
MatchFirstArg a (b -> r) = 'Doesn'tMatch
MatchFirstArg _ _ = 'NoArgToMatch
```

The `MatchArgResult`

constructors are used at the type-level via the `DataKinds`

extension. This is not really necessary, but it's rather nice for having a proper type-level datatype to use for the result of the discriminating closed type family.

Trying it out in GHCi:

```
> :k! MatchFirstArg Char (Char -> Text -> Text)
MatchFirstArg Char (Char -> Text -> Text) :: MatchArgResult
= 'Matches
> :k! MatchFirstArg Text (Char -> Text -> Text)
MatchFirstArg Text (Char -> Text -> Text) :: MatchArgResult
= 'Doesn'tMatch
> :k! MatchFirstArg Char Text
MatchFirstArg Char Text :: MatchArgResult
= 'NoArgToMatch
```

It works! When the argument type matches the first parameter, the result is `Matches`

, and when it does not, the result is `Doesn'tMatch`

. When it isn't a function at all, the result is `NoArgToMatch`

.

This closed type family will allow us to choose between `ApplyByTypeResult`

instances, avoiding any overlap issues. The method also gets renamed to `applyByTypeImpl`

, as the `(?)`

function will get defined in terms of it:

```
class ApplyByType (matches :: MatchArgResult) a f where
type ApplyByTypeResult matches a f
applyByTypeImpl
:: Proxy matches
-> f
-> a
-> ApplyByTypeResult matches a f
instance ApplyByType 'Matches a (a -> r) where
type ApplyByTypeResult 'Matches a (a -> r) = r
applyByTypeImpl _ f x = f x
```

The recursive case gets a little trickier:

```
instance ApplyByType (MatchFirstArg a r) a r
=> ApplyByType 'Doesn'tMatch a (b -> r) where
type ApplyByTypeResult 'Doesn'tMatch a (b -> r) =
b -> ApplyByTypeResult (MatchFirstArg a r) a r
applyByTypeImpl _ f y =
\x -> applyByTypeImpl (Proxy :: Proxy (MatchFirstArg a r)) (f x) y
```

In 3 different places we need to apply the `MatchFirstArg`

type function in order to discriminate which instance to recurse into. It might be helpful to reference the old simpler (but non-functioning) definition:

```
instance ApplyByType a r
=> ApplyByType a (b -> r) where
type ApplyByTypeResult a (b -> r) =
b -> ApplyByTypeResult a r
f ? y =
\x -> f x ? y
```

Giving this new definition a try:

Woohoo! It figured out that `"ello" :: T.Text`

needs to be passed in as the second argument to `T.cons`

.

However, it's a bit inconvenient to need to pass in that `Proxy`

. The convenience of `(?)`

can be regained by writing a helper function which applies `MatchFirstArg`

:

```
(?)
:: forall matches a f.
( matches ~ MatchFirstArg a f
, ApplyByType matches a f
)
=> f -> a -> ApplyByTypeResult matches a f
(?) = applyByTypeImpl (Proxy :: Proxy matches)
```

Giving that a try:

Boom! Unordered, type-directed function application!

This version of the code is didactic/V3.hs on github. The next post in this series will improve the type errors that occur when the argument type mismatches with all of the parameters.

## Alternative approach: functional dependencies

In the discussion on reddit, u/lightandlight described a more elegant implementation using functional dependencies with overlapping instances:

```
class Apply f x y | f x -> y where
apply :: f -> x -> y
instance {-# overlappable #-} (x ~ (a -> b)) => Apply x a b where
apply = ($)
instance {-# overlapping #-} Apply b a b' => Apply (a' -> b) a (a' -> b') where
apply f a = \a' -> apply (f a')
```

Very cool! However, this approach leads to more obscure error messages. Lets say we supply the wrong argument type:

```
> cons ? True
<interactive>:1:1: error:
• Couldn't match type ‘Text’ with ‘Bool -> b'’
arising from a use of ‘?’
...
```

A bit of a head-scratcher! If something like "Inspecting Haskell Instance Resolution" was implemented in GHC, then this error might instead look like:

```
> cons ? True
<interactive>:11:1: error:
• Couldn't match type ‘Text’ with ‘Bool -> b'’
arising from a use of ‘?
• Due to the constraint (Text ~ (Bool -> b))
arising from superclass constraints:
(Text ~ (Bool -> b)) =>
Apply Text Bool b =>
Apply (Text -> Text) Bool (Text -> b) =>
Apply (Char -> Text -> Text) Bool (Char -> Text -> b)
...
```

This makes it a bit clearer that it recursed on the function type and errored out on attempting to satisfy the `(x ~ (a -> b))`

constraint.

With the implementation here, the error is instead:

```
λ cons ? True
<interactive>:1:1: error:
• No instance for (ApplyByType 'NoArgToMatch Bool Text)
arising from a use of ‘?’
...
```

Still not great, but I think a bit clearer. The next post describes using GHC custom type errors to further improve this circumstance.

## Appendix: Why?

You might be wondering why you would ever want to use this. Honestly, I can't think of many practical applications, and so the apply-unordered-mono package is in the `ACME`

category on Hackage. Even so, I think it's pretty cool that this is possible.

One potential use-case might be to handle shifting APIs in dependencies. If a library changes the order of its arguments, you might be able to use the `(?)`

operator to be compatible with versions before and after this change, without using the C preprocessor.

The inspiration for this package was speculation about a programming language where arguments are discriminated by type instead of position. To experiment with this, it turned out to be more straightforward to directly implement the idea in Haskell rather than writing a whole new language. Here are some reasons why I think this is an interesting idea for language design:

Parameter order is often arbitrary, and so requires effort to memorize.

It is safer to have distinct types for all arguments, to avoid bugs due to inadvertent transposition.

Named parameters can help with this, but then you need to memorize the names! Conventions where the name is the same as the type adds verbosity / redundancy.

In practice, many newtypes would be involved in usage of type-directed application. Newtypes might even be associated with the function definition, to provide something akin to named parameters.

Ordered arguments do have some nice advantages, though:

Polymorphism. Matching of arguments with parameters gets ambiguous when type variables are involved.

Ordered arguments provides more direct visual comparison of calls of the same function.

To me, this suggests an alternative implementation approach. Rather than incorporating type-directed application directly into the language, or as a library, it can be part of edit-time tooling. For example, the reformatting tool might handle reordering the arguments.