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{-# LANGUAGE AllowAmbiguousTypes #-}
{-# LANGUAGE ConstraintKinds #-}
{-# LANGUAGE GADTs #-}
{-# LANGUAGE RankNTypes #-}
{-# LANGUAGE ScopedTypeVariables #-}
{-# LANGUAGE TemplateHaskell #-}
{-# LANGUAGE Trustworthy #-}
{-# LANGUAGE TypeApplications #-}
{-# LANGUAGE TypeFamilyDependencies #-}
{-# LANGUAGE DataKinds #-}
{-# LANGUAGE PolyKinds #-}
{-# LANGUAGE TypeOperators #-}
module T13910 where
import Data.Kind
import Data.Type.Equality
data family Sing (a :: k)
class SingKind k where
type Demote k = (r :: Type) | r -> k
fromSing :: Sing (a :: k) -> Demote k
toSing :: Demote k -> SomeSing k
data SomeSing k where
SomeSing :: Sing (a :: k) -> SomeSing k
withSomeSing :: forall k r
. SingKind k
=> Demote k
-> (forall (a :: k). Sing a -> r)
-> r
withSomeSing x f =
case toSing x of
SomeSing x' -> f x'
data TyFun :: Type -> Type -> Type
type a ~> b = TyFun a b -> Type
infixr 0 ~>
type family Apply (f :: k1 ~> k2) (x :: k1) :: k2
type a @@ b = Apply a b
infixl 9 @@
data FunArrow = (:->) | (:~>)
class FunType (arr :: FunArrow) where
type Fun (k1 :: Type) arr (k2 :: Type) :: Type
class FunType arr => AppType (arr :: FunArrow) where
type App k1 arr k2 (f :: Fun k1 arr k2) (x :: k1) :: k2
type FunApp arr = (FunType arr, AppType arr)
instance FunType (:->) where
type Fun k1 (:->) k2 = k1 -> k2
$(return []) -- This is only necessary for GHC 8.0 -- GHC 8.2 is smarter
instance AppType (:->) where
type App k1 (:->) k2 (f :: k1 -> k2) x = f x
instance FunType (:~>) where
type Fun k1 (:~>) k2 = k1 ~> k2
$(return [])
instance AppType (:~>) where
type App k1 (:~>) k2 (f :: k1 ~> k2) x = f @@ x
infixr 0 -?>
type (-?>) (k1 :: Type) (k2 :: Type) (arr :: FunArrow) = Fun k1 arr k2
data instance Sing (z :: a :~: b) where
SRefl :: Sing Refl
instance SingKind (a :~: b) where
type Demote (a :~: b) = a :~: b
fromSing SRefl = Refl
toSing Refl = SomeSing SRefl
(~>:~:) :: forall (k :: Type) (a :: k) (b :: k) (r :: a :~: b) (p :: forall (y :: k). a :~: y ~> Type).
Sing r
-> p @@ Refl
-> p @@ r
(~>:~:) SRefl pRefl = pRefl
type WhyReplacePoly (arr :: FunArrow) (from :: t) (p :: (t -?> Type) arr)
(y :: t) (e :: from :~: y) = App t arr Type p y
data WhyReplacePolySym (arr :: FunArrow) (from :: t) (p :: (t -?> Type) arr)
:: forall (y :: t). from :~: y ~> Type
type instance Apply (WhyReplacePolySym arr from p :: from :~: y ~> Type) x
= WhyReplacePoly arr from p y x
replace :: forall (t :: Type) (from :: t) (to :: t) (p :: t -> Type).
p from
-> from :~: to
-> p to
replace = replacePoly @(:->)
replaceTyFun :: forall (t :: Type) (from :: t) (to :: t) (p :: t ~> Type).
p @@ from
-> from :~: to
-> p @@ to
replaceTyFun = replacePoly @(:~>) @_ @_ @_ @p
replacePoly :: forall (arr :: FunArrow) (t :: Type) (from :: t) (to :: t)
(p :: (t -?> Type) arr).
FunApp arr
=> App t arr Type p from
-> from :~: to
-> App t arr Type p to
replacePoly from eq =
withSomeSing eq $ \(singEq :: Sing r) ->
(~>:~:) @t @from @to @r @(WhyReplacePolySym arr from p) singEq from
type WhyLeibnizPoly (arr :: FunArrow) (f :: (t -?> Type) arr) (a :: t) (z :: t)
= App t arr Type f a -> App t arr Type f z
data WhyLeibnizPolySym (arr :: FunArrow) (f :: (t -?> Type) arr) (a :: t)
:: t ~> Type
type instance Apply (WhyLeibnizPolySym arr f a) z = WhyLeibnizPoly arr f a z
leibnizPoly :: forall (arr :: FunArrow) (t :: Type) (f :: (t -?> Type) arr)
(a :: t) (b :: t).
FunApp arr
=> a :~: b
-> App t arr Type f a
-> App t arr Type f b
leibnizPoly = replaceTyFun @t @a @b @(WhyLeibnizPolySym arr f a) id
leibniz :: forall (t :: Type) (f :: t -> Type) (a :: t) (b :: t).
a :~: b
-> f a
-> f b
leibniz = replaceTyFun @t @a @b @(WhyLeibnizPolySym (:->) f a) id
-- The line above is what you get if you inline the definition of leibnizPoly.
-- It causes a panic, however.
--
-- An equivalent implementation is commented out below, which does *not*
-- cause GHC to panic.
--
-- leibniz = leibnizPoly @(:->)
leibnizTyFun :: forall (t :: Type) (f :: t ~> Type) (a :: t) (b :: t).
a :~: b
-> f @@ a
-> f @@ b
leibnizTyFun = leibnizPoly @(:~>) @_ @f
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