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Is this something that belongs to core libraries (it has other names in the wild, :~>, Natural..)

type f1 ~> f2 = forall a. f1 a -> f2 a

I use it all the time and end up redefining it (it is such a short type that maybe it's not worth it, ‘Fairbairn threshold’ something something)

unLift         :: Applicative f 
               => Lift f a -> f a
mapLift        :: (f a -> g a) 
               -> Lift f a -> Lift g a
mapFreeT       :: (Functor f, Functor m) 
               => (forall a. m a -> m' a) 
               -> FreeT f m a -> FreeT f m' a
vmap           :: (a -> a') 
               -> Vec a n -> Vec a' n
liftIO         :: MonadIO m
               => IO a -> m a
hoist          :: Monad m 
               => (forall a. m a -> n a) 
               -> t m b -> t n b
trans          :: (Monad m, Monad m') 
               => (forall a. m a -> m' a)
               -> Bundle m v a -> Bundle m' v a
process        :: Monad m 
               => (forall a. k a -> i -> a) 
               -> MachineT m k o -> ProcessT m i o
runAlt         :: Alternative g 
               => (forall x. f x -> g x) 
               -> Alt f a -> g a
hoistAlt       :: (forall a. f a -> g a) 
               -> Alt f b -> Alt g b
fromCurried    :: Functor f 
               => (forall a. k a -> Curried f h a) 
               -> Day f k b -> h b
hoistScope     :: Functor f
               => (forall x. f x -> g x) 
               -> Scope b f a -> Scope b g a
haddockWithGhc :: (forall a. [Flag] -> Ghc a -> IO a) 
               -> [String] -> IO ()

newtype Alt f a = Alt { _runAlt :: forall g. Alternative g => (forall x. f x -> g x) -> g a }

newtype Builder = Builder (forall r. BuildStep r -> BuildStep r)

data Zipper t a = Zipper (forall b. Seq b -> t b) {-# UNPACK #-} !Int !(Seq a)

class MonadCatch m => MonadMask m where
  mask :: ((forall a. m a -> m a) -> m b) -> m b

becomes

unLift         :: Applicative f 
               => Lift f ~> f
mapLift        :: f ~> g
               -> Lift f ~> Lift g
mapFreeT       :: (Functor f, Functor m) 
               => m ~> m'
               -> FreeT f m ~> FreeT f m'
vmap           :: (a -> a')
               -> Vec a ~> Vec a'
liftIO         :: MonadIO m
               => IO ~> m
hoist          :: Monad m 
               => m ~> n 
               -> t m ~> t n
trans          :: (Monad m, Monad m') 
               => m ~> m'
               -> Bundle m v ~> Bundle m' v
process        :: Monad m 
               => k ~> (i -> ) 
               -> MachineT m k ~> ProcessT m i
runAlt         :: Alternative g 
               => f ~> g 
               -> Alt f ~> g
hoistAlt       :: f ~> g 
               -> Alt f ~> Alt g
fromCurried    :: Functor f 
               => k ~> Curried f h 
               -> Day f k ~> h
hoistScope     :: Functor f
               => f ~> g 
               -> Scope b f ~> Scope b g
haddockWithGhc :: ([Flag] -> Ghc ~> IO) 
               -> [String] -> IO ()

newtype Alt f a = Alt { _runAlt :: forall g. Alternative g => (f ~> g) -> g a }

newtype Builder = Builder (BuildStep ~> BuildStep)

data Zipper t a = Zipper (Seq ~> t) {-# UNPACK #-} !Int !(Seq a)

class MonadCatch m => MonadMask m where
  mask :: ((m ~> m) -> m b) -> m b

these examples are pretty similar.

---

Same for

mapBlock       :: (forall e x. n e x -> n' e x) 
               -> Block n e x -> Block n' e x
mapGraph       :: (forall e x. n e x -> n' e x) 
               -> Graph n e x -> Graph n' e x
mapGraphNodes1 :: (forall e x. CmmNode e x -> CmmNode e x) 
               -> CmmGraph -> CmmGraph
foldCat        :: (Catenated t, Category s) 
               => (forall a b. r a b -> s a b) 
               -> t r a b -> s a b
mapCat         :: Catenated t 
               => (forall a b. r a b -> s a b) 
               -> t r a b -> t s a b

newtype Copastro p a b = Copastro { runCopastro :: forall r. Costrong r => (forall x y. p x y -> r x y) -> r a b }
newtype CopastroSum p a b = CopastroSum { runCopastroSum :: forall r. Cochoice r => (forall x y. p x y -> r x y) -> r a b }

vs

type f1 ~~> f2 = forall a b. f1 a b -> f2 a b

mapBlock       :: n ~~> n' 
               -> Block n ~~> Block n'
mapGraph       :: n ~~> n' 
               -> Graph n ~~> Graph n'
mapGraphNodes1 :: CmmNode ~~> CmmNode 
               -> CmmGraph -> CmmGraph
foldCat        :: (Catenated t, Category s) 
               => r   ~~> s 
               -> t r ~~> s
mapCat         :: Catenated t 
               => r   ~~> s 
               -> t r ~~> t s


newtype Copastro p a b = Copastro { runCopastro :: forall r. Costrong r => (p ~~> r) -> r a b }
newtype CopastroSum p a b = CopastroSum { runCopastroSum :: forall r. Cochoice r => (p ~~> r) -> r a b }