Why doesn't TypeSynonymInstances allow partially applied type synonyms to be used in instance heads?

Question

I know that TypeSynomymInstances only allows fully applied type synonyms to be used in instance heads, but it seems like it would be handy if I could use paritally applied t

Answer 1

Partially applied type synonyms are not allowed in Haskell at all. A partially applied synonym is effectively a function whose inputs are the un-applied types and whose output is a type. For example, here is an encoding of boolean logic:

type True x y = x
type False x y = y
type Not b x y = b y x
type And b1 b2 x y = b1 (b2 x y) y
type Or b1 b2 x y = b1 x (b2 x y)

To decide whether two partially applied type synonyms are equal, the type checker would have to decide whether functions are equal. This is a hard problem, and in general it is undecidable.

Answer 2

Another issue with allowing partially applied type synonyms is that they would make type inference and instance selection essentially impossible. For example, suppose in the context of some program I wanted to use thingy at the type Int -> String -> Int -> (Int, String). thingy has type forall a b e. a -> b -> e a b, so we can unify a with Int and b with String, but if e is allowed to be a partially applied type synonym, we could have

e = FuncSynonym (,)

or

e = FuncSynonym' Int (,) where type FuncSynonym' x f a b = x -> f a b

or even

e = Const2 (Int -> (Int, String)) where Const2 a x y = a

The problem of type inference would become even worse than deciding equality of functions; it would require considering all functions with specified output on a particular input, or similar more complicated problems (imagine simply trying to unify a b with Int).

Answer 3

As we known Maybe's kind is *->*.

So it could be a instance of Functor

   instance Functor Maybe where
       fmap :: f -> Maybe a -> Maybe b
       fmap f Nothing = Nothing
       fmap f (Maybe x)  = Maybe (f x)

---

The first example:

   {-# LANGUAGE TypeSynonymInstances #-}

   type MaybeAlias a = Maybe

   instance {-# OVERLAPPING #-} Functor (MaybeAlias Int) where
       fmap f functor = undefined

Under the effect of TypeSynonymInstances extension (almost like a String replace), it equals

      instance {-# OVERLAPPING #-} Functor Maybe where
       fmap f functor = undefined

It is ok, because allow fully applied type synonyms to be used in instance heads

---

See the other example:

   {-# LANGUAGE TypeSynonymInstances #-}

   type MaybeAlias a b = Maybe

What's the kind of MaybeAlias Int now? It's kind is *->*->*.

Why?

As @heatsink comment above:

A partially applied synonym is effectively a function whose inputs are the un-applied types and whose output is a type

Explain it now:

Under the defintion of type MaybeAlias a b = Maybe :

MaybeAlias like a partially applied function:

(MaybeAlias) :: a -> b -> Maybe

MaybeAlias Int like a partially applied function:

(MaybeAlias Int) :: b -> Maybe

The Maybe 's kind is * -> *, b 's kind is *.

So MaybeAlias Int 's kind is * -> (* -> *) .

And * -> (* -> *) equals * -> * -> *.

The root cause why the below code not working, because Functor typeclass only accept type that has kind * -> *, not * -> * ->*!

   {-# LANGUAGE TypeSynonymInstances #-}

   type MaybeAlias a b = Maybe

   instance {-# OVERLAPPING #-} Functor (MaybeAlias Int) where
       fmap f functor = undefined

---

Why the below code not working?

class Example e where
  thingy :: a -> b -> e a b

-- legit, but awkward
newtype FuncWrapper e a b = FuncWrapper { ap :: a -> e a b }
instance (Example e) => Example (FuncWrapper e) where
  thingy _ = FuncWrapper . flip thingy
funcWrapperUse :: (Example e) => e Int String
funcWrapperUse = thingy 1 "two" `ap` 3 `ap` 4 `ap` 5

-- not legal, but a little easier to use
type FuncSynonym e a b = a -> e a b
instance (Example e) => Example (FuncSynonym e) where
  thingy _ = flip thingy
funcSynonymUse :: (Example e) => e Int String
funcSynonymUse = thingy 1 "two" 3 4 5

Example typeclass accept a type that has kind * -> * -> *

FuncSynonym like a partially applied function:

   FuncSynonym :: e -> a -> b -> (a -> e a b)

FuncSynonym e like a partially applied function:

   (FuncSynonym e):: a -> b -> ( a -> e a b)

a 's kind is *,

b 's kind is *,

a -> e a b 's kind *

(FuncSynonym e)'s kind is * -> * -> *

Example typeclass accept a type that has kind * -> * -> *, but why still not work?

It's the other reason in ghc issue 785 and comment