## Add (->) representation and the Invariant class to GHC.Generics

We currently disallow any use of the parameter in the domain of (->).

```
newtype F a = F ((a -> Int) -> Int) deriving Generic1
<interactive>:4:38:
Can't make a derived instance of `Generic1 (F g)':
Constructor `F' must use the last type parameter only as the last argument of a data type, newtype, or (->)
In the data declaration for `F'
```

!DeriveFunctor succeeds for this F.

I'd like to add this representation type to GHC.Generics and !DeriveGeneric.

`newtype (f :->: g) a = FArrow1 (f a -> g a)`

We could then represent the first example above. We could also derive the more interesting Generic1 (F g).

```
newtype F g a = F (g a -> Int) deriving Generic1
type instance Rep1 (F g) = Rec1 g :->: Rec0 Int
instance Generic1 (F g) where
to x = F $ unRec0 . unArrow1 x . Rec1
from (F x) = FArrow1 $ Rec0 . x . unRec1
```

Admittedly, there's not many generic definitions impeded by not having (:->:). Contra- and in-variant types are uncommon.

I'm suggesting this feature without strong motivating examples because I think this would streamline the implementation of -XDeriveGenerics in some ways while also making it more general — assuming that we added the Invariant class to base or ghc-prim.

```
class Invariant t where
invmap :: (a -> b) -> (b -> a) -> t a -> t b
invmap_covariant :: Functor t => (a -> b) -> (b -> a) -> t a -> t b
invmap_covariant f _ = fmap f
instance (Invariant f,Invariant g) => Invariant (FArrow f g) where
invmap co contra (FArrow h) = FArrow $ invmap co contra . h . invmap contra co
```

(Of course, Invariant should be a super class of Functor. :/ )

Now we can handle quite involved examples:

```
newtype F g h a = F (g (h a)) deriving Generic1
instance Invariant g => Generic1 (F g h) where
to x = invmap unRec1 Rec1 $ unComp1 x
from (F x) = Comp1 $ invmap Rec1 unRec1
```

All of that said, I'm mostly opening this ticket so I can get feedback on difficulties I might not be anticipating and have a place to reference from the compiler source code comments.

## Trac metadata

Trac field | Value |
---|---|

Version | 7.7 |

Type | FeatureRequest |

TypeOfFailure | OtherFailure |

Priority | low |

Resolution | Unresolved |

Component | Compiler (Type checker) |

Test case | |

Differential revisions | |

BlockedBy | |

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Blocking | |

CC | |

Operating system | |

Architecture |