I believe that at the value level, Peano Nats can be optimised using pattern synonyms. The primary interest of Peano Nats, however, is in their use at the kind level - so I think two things are possible: 1. With -XDependentHaskell, one can define a pattern synonym allowing the optimisation of Peano Nats, 2. With type families, it _might_ be possible to define a weaker form of (1), provided it is possible to put a type synonym on the LHS of a type family. Question: Can a TypeApplication be used on the LHS of an injective type family? On Wed, Mar 14, 2018 at 9:05 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.