# [Haskell-cafe] dependent types, singleton types....

Nicholls, Mark nicholls.mark at vimn.com
Tue Apr 28 14:45:54 UTC 2015

```Can someone check my answer (no I'm not doing an assessment...I'm actually learning stuff out of interest!)

working through

still there is a section about singleton types and the exercise is

"Exercise: Define the binary tree type and implement its singleton type."

Ok, I think I'm probably wrong....a binary tree is something like...

> data BTree a = Leaf | Branch a (BTree a) (BTree a)

With DataKind

My logic goes...
Leaf is an uninhabited type, so I need a value isomorphic to it....

Easy?

> data SBTree a where
>   SLeaf :: SBTree Leaf

Things like
Branch Integer Leaf  (Branch String Leaf Leaf)
Are uninhabited...so I need to add

>   SBranch :: (a :: *) -> (SBTree (b :: BTree *)) -> (SBTree (c :: BTree *)) -> SBTree (Branch a b c)

?

It compiles...but....is it actually correct?
Things like

> y = SBranch (SS (SS SZ)) SLeaf SLeaf
> z = SBranch (SS (SS SZ)) (SBranch SZ SLeaf SLeaf) SLeaf

Seem to make sense ish.

From: Nicholls, Mark
Sent: 28 April 2015 9:33 AM
To: Nicholls, Mark
Subject: sds

Hello,

working through

but a bit stuck...with an error...

> {-# LANGUAGE DataKinds, TypeFamilies, TypeOperators, UndecidableInstances, GADTs, StandaloneDeriving #-}

> data Nat = Z | S Nat

> data Vector a n where
>   Nil  :: Vector a Z
>   (:-) :: a -> Vector a n -> Vector a (S n)
> infixr 5 :-

I assume init...is a bit like tail but take n - 1 elements from the front....but...

> init' :: Vector a ('S n) -> Vector a n
> init' (x :- Nil) = Nil
> init' (x :- xs@(_ :- _)) = x :- (init' xs)

> zipWithSame :: (a -> b -> c) -> Vector a n -> Vector b n -> Vector c n
> zipWithSame f Nil Nil = Nil
> zipWithSame f (x :- xs) (y :- xs@(_ :- _)) = Nil

Mark Nicholls | Senior Technical Director, Programmes & Development - Viacom International Media Networks
A: 17-29 Hawley Crescent London NW1 8TT | e: Nicholls.Mark at vimn.com<mailto:mik at vimn.com> T: +44 (0)203 580 2223

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