[Haskell-cafe] Type-level naturals & multiplication

Manuel M T Chakravarty chak at cse.unsw.edu.au
Sun Oct 11 05:24:34 EDT 2009

Brad Larsen:
> Suppose we implement type-level naturals as so:
> data Zero
> data Succ a
> Then, we can reflect the type-level naturals into a GADT as so (not
> sure if ``reflect'' is the right terminology here):
> data Nat :: * -> * where
>  Zero :: Nat Zero
>  Succ :: Nat a -> Nat (Succ a)
> Using type families, we can then proceed to define type-level  
> addition:
> type family Add a b :: *
> type instance Add Zero b = b
> type instance Add (Succ a) b = Succ (Add a b)
> Is there any way to define type-level multiplication without requiring
> undecidable instances?

No, not at the moment.  The reasons are explained in the paper "Type  
Checking with Open Type Functions" (ICFP'08):


We want to eventually add closed *type families* to the system (ie,  
families where you can't add new instances in other modules).  For  
such closed families, we should be able to admit more complex  
instances without requiring undecidable instances.


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