[Haskell-cafe] Question on inductive type-families

[Brandon Allbery]

There are some plugins like natnormalise that help with this kind of thing.
ghc itself is not so good at these kinds of deductions.

On Thu, Aug 22, 2019 at 4:33 AM Sylvain Henry <sylvain at haskus.fr> wrote:

> Hi Li-yao,
>
> Thanks for your answer. Sadly I haven't managed to make it work this way
> neither.
>
> In addition I need a "KnownNat (n-1)" while I only have "KnownNat n" (and
> the knowledge that n > 0). For the record, it works with an unsafeCoerce:
>
> data V2 (vs :: [k]) where
>     V2 :: forall vs (n :: Nat). KnownNat n => Proxy n -> Index n vs -> V2
> vs
>
> splitV2 :: forall v vs. V2 (v:vs) -> Either (V2 vs) v
> {-# INLINE splitV2 #-}
> splitV2 = \case
>     V2 (n1 :: Proxy n) v -> case sameNat n1 (Proxy @0) of
>        Just Refl -> Right v
>        Nothing   -> case someNatVal (natVal n1 - 1) of
>          SomeNat (n2 :: Proxy n') -> case unsafeCoerce Refl :: Index n
> (v:vs) :~: Index n' vs of
>             Refl -> Left (V2 n2 v)
>
> instance Show (V2 '[]) where
>    {-# INLINABLE show #-}
>    show = undefined
>
> instance (Show x, Show (V2 xs)) => Show (V2 (x ': xs)) where
>    {-# INLINABLE show #-}
>    show v = case splitV2 v of
>       Right x -> show x
>       Left xs -> show xs
>
> test :: V2 '[Int,Word,String] -> String
> test v = show v
>
> However the generated Core isn't as pretty as with the previous
> representation (which uses unsafeCoerce for the value, not for the index):
>
> data Variant (types :: [*]) = Variant {-# UNPACK #-} !Word Any
>
>
> I'll stick to this one for now.
>
> Sylvain
>
>
> On 21/08/2019 16:46, Li-yao Xia wrote:
>
> Hi Sylvain,
>
> The problem is that in the case where `n1` is not equal to `0`, this
> unequality is not made available as a constraint solver (whereas in the
> case where they are equal, the `Refl` constructor makes the corresponding
> equality constraint available).
>
> In fact, there is no built-in "unequality constraint" that acts as a
> counterpart to `~` (it does not seem quite straightforward how that would
> even work), but there is one way to encode it as a boolean type family `((n
> == 0) ~ 'False)`. That requires changing type families to pattern-match on
> `n == 0` instead of `n` directly, and possibly additional gymnastics to
> reflect the comparison at term-level:
>
>
> type Index n l = IndexIf (n == 0) n l
>
> type family IndexIf (b :: Bool) (n :: Nat) (l :: [k]) :: k where
>   IndexIf 'False n (x ': _ ) = x
>   IndexIf 'True  n (_ ': xs) = Index (n - 1) xs
>
>
> Cheers,
> Li-yao
>
> On 8/21/19 9:09 AM, Sylvain Henry wrote:
>
> Hi,
>
> Suppose I have the following type family:
>
> type family Index (n :: Nat) (l :: [k]) = (r :: k) where
>     Index 0 (x ': _ ) = x
>     Index n (_ ': xs) = Index (n-1) xs
>
> I'm using it to define a Variant type:
>
> data V2 (vs :: [k]) where
>     V2 :: forall vs (n :: Nat). KnownNat n => Proxy n -> Index n vs -> V2
> vs
>
> Now I want to match on the Variant retrieving either the head or the tail
> (another Variant):
>
> splitV2 :: forall v vs. V2 (v:vs) -> Either v (V2 vs)
> splitV2 = \case
>     V2 (n1 :: Proxy n) v -> case sameNat n1 (Proxy @0) of
>           Just Refl -> Left v
>           Nothing   -> Right (V2 (Proxy @(n-1)) v)
>
> It fails to typecheck with:
>
> Couldn't match type ‘Index n (v : vs)’ with ‘Index (n - 1) vs’
> Expected type: Index (n - 1) vs
> Actual type: Index n vs1
> NB: ‘Index’ is a non-injective type family
>
>
> Is there a way to make this work?
>
>
> This is highly speculative as I'm not very familiar with type-checking:
> would it be sound to make the inductive Index family "weakly" injective?
> I.e. somehow produce a typing rule for the type family:
>
>    forall n n' x xs xs'.
>      Solving: Index n (x ': xs) ~ Index (n'-1) xs'
>      Is equivalent to solving: if n /= 0 then (n ~ n', xs ~ xs') else (x ~
> Index (n'-1) xs')
>
> Here we could know that "n /= 0" thanks to the match on "sameNat n1 (Proxy
> @0)".
>
> ("weakly" because we have some form of injectivity on the shape of the
> list but not on its contents)
>
> Any pointer on this kind of stuff in research papers, Coq, Agda, etc. ?
>
> Thanks,
> Sylvain
>
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-- 
brandon s allbery kf8nh
allbery.b at gmail.com
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