[Haskell-cafe] Infer Nat type from Integer argument

[Jake]

{-# LANGUAGE DataKinds, KindSignatures, ScopedTypeVariables, GADTs,
AllowAmbiguousTypes #-}

import GHC.TypeLits
import Data.Proxy
import Data.Type.Equality

data NatString (n :: Nat) = NatString String deriving Show

showNS :: KnownNat n => NatString n -> (String, Integer)
showNS b@(NatString s) = (s, natVal b)

In this example, we use NatString like this:

> showNS (NatString "hey" :: NatString 4)
-- ("hey", 4)

We can then dynamically make NatStrings from Integers. For example, we can
read an Integer from stdin and use it to create a NatString

main :: IO ()
main = do
  i <- readLn
  case someNatVal i of
    Just (SomeNat (p :: Proxy n)) ->
      let ns :: NatString n
          ns = NatString "hello!"
      in print $ showNS NatString

However, if I have trouble when I try to refactor out the middle part to
create a function with type forall n. KnownNat n => Integer -> Bar n.
I use sameNat to convince the compiler that the Nat we get from the Integer
is the same type as the the type n of the output.

fromN :: forall n. KnownNat n => Integer -> NatString n
fromN i =
 case someNatVal i of
   Just (SomeNat p) ->
     case sameNat p (Proxy :: Proxy n) of
       Just (Refl) ->
         let ns :: NatString n
             ns = NatString "hello!" in ns

This compiles, but if you try and use it the compiler complains.

> showNS (fromN 3)
-- No instance for (KnownNat n0) arising from a use of ‘showNS’

If we manually add the type annotation, everything is fine, but that kind
of defeats the entire purpose.

> showNS (fromN 3 :: NatString 3)
-- ("hello!",3)

Strangely enough, when the NatString is immediately consumed instead of
being returned, the compiler does its job and infers the correct type.

showFromN :: Integer -> (String, Integer)
showFromN i =
 case someNatVal i of
   Just (SomeNat (p :: Proxy n)) ->
     let ns :: NatString n
         ns = NatString "hello!" in showNS ns

> showFromN 3
-- ("hello!", 3)

Clearly the intermediate value of ns inside of showFromN has a type
NatString 3 because showNS is using that fact.
How can I make the compiler infer that fromN 3 :: NatString 3 separately?
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