Re: Proposal: Add inductively-defined Nat to base

AFAIK, the existing naturals in base don't have any useful internal structure permitting this sort of thing. -Edward On Thu, Mar 15, 2018 at 3:48 PM, Carter Schonwald < carter.schonwald@gmail.com> wrote:

Could we provide a pattern synonym for treating naturals as in base already as being peano encoded ?

On Wed, Mar 14, 2018 at 11:51 PM Daniel Cartwright wrote:

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I just realised I made a typo. For full clarity, the function 'f' should be:

f : D -> N -> R where D = Data Structure isomorphic to Nat (or any numeric type) N = Number System Encoding R = Representation of the numeric type in N.

For Nats, the simplest example would be:

f : List () -> Binary -> BinaryEncodingOfNat

On Wed, Mar 14, 2018 at 9:47 PM, Daniel Cartwright

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wrote:

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I prefer Z and S, but wrote Zero and Succ for clarity (though the likelihood of being at all misunderstood was small).

Most recent definitions I see use Z and S.

On Wed, Mar 14, 2018 at 9:41 PM, David Feuer wrote:

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Another problem: different people like to call the constructors by different names. I personally prefer Z and S, because they're short. Some people like Zero and Succ or Suc.

On Mar 14, 2018 9:06 PM, "Daniel Cartwright" wrote:

The proposed addition is simple, add the following to base:

data Nat = Zero | Succ Nat,

i.e. Peano Nats - commonly used along with -XDataKinds.

I will list the pros/cons I see below:

Pros: - This datatype is commonly defined throughout many packages throughout Hackage. It would be good for it to have a central location - The inductive definition of 'Nat' is useful for correctness (e.g. safeHead :: Vec a (Succ n) -> a; safeHead (Cons a as) = a;) - -XDependentHaskell is likely to bring this into base anyway - I believe that it might be possible to eliminate a Peano Nat at some stage during/after typechecking. I'm not well-versed in GHC implementation, but something along the lines of possibly converting an inductive Nat to a GMP Integer using some sort of specialisation (Succ -> +1)? Another interesting, related approach (and this is a very top-level view, and perhaps not totally sensical) would be something like a function 'f', that given a data structure and a number system, outputs the representation of that data structure in that number system (Nat is isomorphic to List (), so f : List () -> Binary -> BinaryListRep)

Cons: - -XDependentHaskell will most likely obviate any benefit brought about by type families defined in base that directly involve Nat - Looking at base, I'm not sure where this would go. Having it in its own module seems a tad strange.

I am open to criticism concerning the usefulness of the idea, or if anyone sees a Pro(s)/Con(s) that I am missing.

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