[Haskell-beginners] Programming with Singletons

[Quentin Liu]

Thank you so much! Indeed changing the definition of `Add` helps solve the problem.

Following this idea, I changed the definition of `Minus` and `Min` also. Now they are defined as

> type family (Sub (a :: Nat) (b :: Nat)) :: Nat
> type instance Sub (Succ a) Zero       = Succ a
> type instance Sub Zero     b               = Zero
> type instance Sub (Succ a) (Succ b) = Sub a b
>
> type family (Min (a :: Nat) (b :: Nat)) :: Nat
> type instance Min Zero     Zero      = Zero
> type instance Min Zero     (Succ b)  = Zero
> type instance Min (Succ a) Zero      = Zero
> type instance Min (Succ a) (Succ b)  = Succ (Min a b)

The change, however, breaks the `tail` and `drop` function.

> drop :: SNat a -> Vec s b -> Vec s (Sub b a)
> drop SZero     vcons        = vcons
> drop (SSucc a) (VCons x xs) = drop a xs
>
> tail :: ((Zero :< a) ~ True) => Vec s a -> Vec s (Sub a (Succ Zero))
> tail (VCons x xs) = xs

So why does the code break and what would be the solution? The error message seems to confirm that even right now GHD still does not support type expansion.

Regards,
Qingbo Liu

On Nov 15, 2017, 19:51 -0500, mukesh tiwari <mukeshtiwari.iiitm at gmail.com>, wrote:
> Hi Quentin,
> I changed your pattern little bit in Add function and it is working
> fine. I think the problem was that type of (VCons x xs) ++++ b is Vec
> v (Add (Succ m1) + n) which was not present in your
> Add function pattern.
>
>
>
> {-# LANGUAGE TypeFamilies #-}
> {-# LANGUAGE DataKinds #-}
> {-# LANGUAGE KindSignatures #-}
> {-# LANGUAGE GADTs #-}
> {-# LANGUAGE InstanceSigs #-}
> module Tmp where
>
> data Nat = Zero | Succ Nat
>
> data SNat a where
>   SZero :: SNat Zero
>   SSucc :: SNat a -> SNat (Succ a)
>
> data Vec a n where
>   VNil :: Vec a Zero
>   VCons :: a -> Vec a n -> Vec a (Succ n)
>
> type family (Add (a :: Nat) (b :: Nat)) :: Nat
> type instance Add Zero b = b
> type instance Add (Succ a) b = Succ (Add a b)
>
> (++++) :: Vec v m -> Vec v n -> Vec v (Add m n)
> VNil ++++ b = b
> (VCons x xs) ++++ b = VCons x $ xs ++++ b
>
> On Thu, Nov 16, 2017 at 11:21 AM, Quentin Liu
> <quentin.liu.0415 at gmail.com> wrote:
> > Hi all,
> >
> > I was doing the “Singletons” problem at codewars[1]. The basic idea is to
> > use dependent types to encode the length of the vector in types.
> >
> > It uses
> >  data Nat = Zero | Succ Nat
> >
> >  data SNat a where
> >   SZero :: SNat Zero
> >   SSucc :: SNat a -> SNat (Succ a)
> > to do the encoding.
> >
> > The vector is defined based on the natural number encoding:
> >  data Vec a n where
> >   VNil :: Vec a Zero
> >   VCons :: a -> Vec a n -> Vec a (Succ n)
> >
> >
> > There are some type families declared for manipulating the natural numbers,
> > and one of them that is relevant to the question is
> >  type family (Add (a :: Nat) (b :: Nat)) :: Nat
> >   type instance Add Zero b = b
> >   type instance Add a Zero = a
> >   type instance Add (Succ a) (Succ b) = Succ (Succ (Add a b))
> > where the `Add` function adds natural numbers.
> >
> > The problem I am stuck with is the concatenation of two vectors:
> >  (++) :: Vec v m -> Vec v n -> Vec v (Add m n)
> >  VNil ++ b = b
> >  (VCons x xs) ++ b = VCons x $ xs ++ b
> >
> > The program would not compile because the compiler found that `VCons x $ xs
> > ++ b`gives type `Vec v (Succ (Add n1 n))`, which does not follow the
> > declared type `Vec v (Add m n)`. Is it because ghc does not expand `Add m n’
> > that the type does not match? I read Brent Yorgey’s blog on type-level
> > programming[2] and he mentioned that would not automatically expand types.
> > But the posted time of the blog is 2010 and I am wondering if there is any
> > improvement to the situation since then? Besides, what would be the solution
> > to this problem
> >
> >
> > Warm Regards,
> > Qingbo Liu
> >
> > [1] https://www.codewars.com/kata/singletons/train/haskell
> > [2]
> > https://byorgey.wordpress.com/2010/07/06/typed-type-level-programming-in-haskell-part-ii-type-families/
> >
> >
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> >
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