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 <paolo.verone...@gmail.com>
> 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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