Yes, it's possible, but it's rather painful. Here is my working attempt, written to be compatible with GHC 7.6.3. Better ones may be possible, but I'm doubtful.

{-# LANGUAGE TemplateHaskell, RankNTypes, TypeFamilies, TypeOperators, DataKinds, ScopedTypeVariables, GADTs, PolyKinds #-}

module ListNat where

import Data.Singletons

$(singletons [d| data Nat = Zero | Succ Nat deriving Eq |])

-- in HEAD, these next two are defined in Data.Type.Equality data a :~: b where Refl :: a :~: a

gcastWith :: (a :~: b) -> ((a ~ b) => r) -> r gcastWith Refl x = x

-- functionality that subsumes this will be in the next release of singletons reifyNatEq :: forall (m :: Nat) (n :: Nat). ((m :==: n) ~ True, SingI m, SingI n) => m :~: n reifyNatEq = case (sing :: Sing m, sing :: Sing n) of (SZero, SZero) -> Refl (SSucc (_ :: Sing m'), SSucc (_ :: Sing n')) -> gcastWith (reifyNatEq :: (m' :~: n')) Refl _ -> bugInGHC -- this is necessary to prevent a spurious warning from GHC

data family X (n::Nat) :: *

data L (n::Nat) where Q :: L (Succ n) -> X n -> L n E :: L n

extract :: SingI n => L Zero -> X n extract = aux where aux :: forall m n. (SingI m, SingI n) => L m -> X n aux list = case ((sing :: Sing m) %==% (sing :: Sing n), list) of (_, E) -> error "Desired element does not exist" (STrue, Q _ datum) -> gcastWith (reifyNatEq :: (m :~: n)) datum (SFalse, Q rest _) -> aux rest

update :: forall n. SingI n => L Zero -> (X n -> X n) -> L Zero update list upd = aux list where aux :: forall m. SingI m => L m -> L m aux list = case ((sing :: Sing m) %==% (sing :: Sing n), list) of (_, E) -> error "Desired element does not exist" (STrue, Q rest datum) -> gcastWith (reifyNatEq :: (m :~: n)) (Q rest (upd datum)) (SFalse, Q rest datum) -> Q (aux rest) datum

Why is this so hard? There are two related sources of difficulty. The first is that `extract` and `update` require *runtime* information about the *type* parameter `n`. But, types are generally erased during compilation. So, the way to get the data you need is to use a typeclass (as your subject line suggests). The other source of difficulty is that you need to convince GHC that you've arrived at the right element when you get there; otherwise, your code won't type-check. The way to do this is, in my view, singletons. For better or worse, your example requires checking the equality of numbers at a value other than Zero. The singletons library doesn't do a great job of this, which is why we need the very clunky reifyNatEq. I'm hoping to add better support for equality-oriented operations in the next release of singletons. I'm happy to explain the details of the code above, but it might be better as Q&A instead of me just trying to walk through it -- there's a lot of gunk to stare at there! I hope this helps, Richard On Oct 12, 2013, at 4:41 AM, Paolino wrote:

Hello everyone,

I'm still trying to resolve my problem. I try to restate it in a simpler way. Is it possible to write extract and update functions for L ?

import Data.Nat

data family X (n::Nat) :: *

data L (n::Nat) where Q :: L (Succ n) -> X n -> L n E :: L n

extract :: L Zero -> X n extract = undefined

update :: L Zero -> (X n -> X n) -> L Zero update = undefined

Thanks for hints and help

paolino

2013/10/7 Paolino Hello, I'm trying to use a type class to select an element from a list. I would like to have a String "CC" as a value for l10'.

{-# LANGUAGE MultiParamTypeClasses, GADTs,FlexibleInstances, DataKinds ,TypeFamilies, KindSignatures, FlexibleContexts, OverlappingInstances, StandaloneDeriving, UndecidableInstances #-}

import Data.Nat import Data.Monoid

data family X (n::Nat) :: *

data L (n::Nat) where Q :: (Monoid (X n), Show (X n)) => L (Succ n) -> X n -> L n E :: Monoid (X n) => L n

deriving instance Show (L n) data instance X n = String String

instance Monoid (X n) where String x `mappend` String y = String $ x `mappend` y mempty = String "" deriving instance Show (X n)

class Compose n n' where compose :: L n -> L n -> X n'

instance Compose n n where compose (Q _ x) (Q _ y) = x `mappend` y compose _ _ = mempty

instance Compose n n' where compose (Q x _) (Q y _) = compose x y compose _ _ = mempty

l0 :: L Zero l0 = Q (Q E $ String "C") $ String "A"

l0' :: L Zero l0' = Q (Q E $ String "C") $ String "B"

l10' :: X (Succ Zero) l10' = compose l0 l0'

l00' :: X Zero l00' = compose l0 l0' {-

*Main> l00' String "AB" *Main> l10' String ""

-}

Thanks for help.

paolino

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