Positive integer type

Question

In many articles about Haskell they say it allows to make some checks during compile time instead of run time. So, I want to implement the simplest check possible - allow a

Answer 1

I know that this was answered a long time ago and I already provided an answer of my own, but I wanted to draw attention to a new solution that became available in the interim: Liquid Haskell, which you can read an introduction to here.

In this case, you can specify that a given value must be positive by writing:

{-@ myValue :: {v: Int | v > 0} #-}
myValue = 5

Similarly, you can specify that a function f requires only positive arguments like this:

{-@ f :: {v: Int | v > 0 } -> Int @-}

Liquid Haskell will verify at compile-time that the given constraints are satisfied.

Answer 2

Type-level natural numbers are planned for GHC 7.6.1: https://ghc.haskell.org/trac/ghc/ticket/4385

Using this feature it's trivial to write a "natural number" type, and gives a performance you could never achieve (e.g. with a manually written Peano number type).

Answer 3

I would be failing in my duty as his supervisor if I failed to plug Adam Gundry's Inch preprocessor, which manages integer constraints for Haskell.

Smart constructors and abstraction barriers are all very well, but they push too much testing to run time and don't allow for the possibility that you might actually know what you're doing in a way that checks out statically, with no need for Maybe padding. (A pedant writes. The author of another answer appears to suggest that 0 is positive, which some might consider contentious. Of course, the truth is that we have uses for a variety of lower bounds, 0 and 1 both occurring often. We also have some use for upper bounds.)

In the tradition of Xi's DML, Adam's preprocessor adds an extra layer of precision on top of what Haskell natively offers but the resulting code erases to Haskell as is. It would be great if what he's done could be better integrated with GHC, in coordination with the work on type level natural numbers that Iavor Diatchki has been doing. We're keen to figure out what's possible.

To return to the general point, Haskell is currently not sufficiently dependently typed to allow the construction of subtypes by comprehension (e.g., elements of Integer greater than 0), but you can often refactor the types to a more indexed version which admits static constraint. Currently, the singleton type construction is the cleanest of the available unpleasant ways to achieve this. You'd need a kind of "static" integers, then inhabitants of kind Integer -> * capture properties of particular integers such as "having a dynamic representation" (that's the singleton construction, giving each static thing a unique dynamic counterpart) but also more specific things like "being positive".

Inch represents an imagining of what it would be like if you didn't need to bother with the singleton construction in order to work with some reasonably well behaved subsets of the integers. Dependently typed programming is often possible in Haskell, but is currently more complicated than necessary. The appropriate sentiment toward this situation is embarrassment, and I for one feel it most keenly.

Answer 4

This—or actually, the similar desire for a type of natural numbers (including 0)—is actually a common complaints about Haskell's numeric class hierarchy, which makes it impossible to provide a really clean solution to this.

Why? Look at the definition of Num:

class (Eq a, Show a) => Num a where
    (+) :: a -> a -> a
    (*) :: a -> a -> a
    (-) :: a -> a -> a
    negate :: a -> a
    abs :: a -> a
    signum :: a -> a
    fromInteger :: Integer -> a

Unless you revert to using error (which is a bad practice), there is no way you can provide definitions for (-), negate and fromInteger.

Answer 5

module Positive (toPositive, Positive(unPositive)) where

newtype Positive = Positive { unPositive :: Int }

toPositive :: Int -> Maybe Positive
toPositive n = if (n < 0) then Nothing else Just (Positive n)

The above module doesn't export the constructor, so the only way to build a value of type Positive is to supply toPositive with a positive integer, which you can then unwrap using unPositive to access the actual value.

You can then write a function that only accepts positive integers using:

positiveInputsOnly :: Positive -> ...

Answer 6

Haskell can perform some checks at compile time that other languages perform at runtime. Your question seems to imply you are hoping for arbitrary checks to be lifted to compile time, which isn't possible without a large potential for proof obligations (which could mean you, the programmer, would need to prove the property is true for all uses).

In the below, I don't feel like I'm saying anything more than what pigworker touched on while mentioning the very cool sounding Inch tool. Hopefully the additional words on each topic will clarify some of the solution space for you.

What People Mean (when speaking of Haskell's static guarantees)

Typically when I hear people talk about the static guarantees provided by Haskell they are talking about the Hindley Milner style static type checking. This means one type can not be confused for another - any such misuse is caught at compile time (ex: let x = "5" in x + 1 is invalid). Obviously, this only scratches the surface and we can discuss some more aspects of static checking in Haskell.

Smart Constructors: Check once at runtime, ensure safety via types

Gabriel's solution is to have a type, Positive, that can only be positive. Building positive values still requires a check at runtime but once you have a positive there are no checks required by consuming functions - the static (compile time) type checking can be leveraged from here.

This is a good solution for many many problems. I recommended the same thing when discussing golden numbers. Never-the-less, I don't think this is what you are fishing for.

Exact Representations

dflemstr commented that you can use a type, Word, which is unable to represent negative numbers (a slightly different issue than representing positives). In this manner you really don't need to use a guarded constructor (as above) because there is no inhabitant of the type that violates your invariant.

A more common example of using proper representations is non-empty lists. If you want a type that can never be empty then you could just make a non-empty list type:

data NonEmptyList a = Single a | Cons a (NonEmptyList a)

This is in contrast to the traditional list definition using Nil instead of Single a.

Going back to the positive example, you could use a form of Peano numbers:

data NonNegative = One | S NonNegative

Or user GADTs to build unsigned binary numbers (and you can add Num, and other instances, allowing functions like +):

{-# LANGUAGE GADTs #-}

data Zero
data NonZero
data Binary a where
  I :: Binary a -> Binary NonZero
  O :: Binary a -> Binary a
  Z :: Binary Zero
  N :: Binary NonZero

instance Show (Binary a) where
        show (I x) = "1" ++ show x
        show (O x) = "0" ++ show x
        show (Z)   = "0" 
        show (N)   = "1"

External Proofs

While not part of the Haskell universe, it is possible to generate Haskell using alternate systems (such as Coq) that allow richer properties to be stated and proven. In this manner the Haskell code can simply omit checks like x > 0 but the fact that x will always be greater than 0 will be a static guarantee (again: the safety is not due to Haskell).

From what pigworker said, I would classify Inch in this category. Haskell has not grown sufficiently to perform your desired tasks, but tools to generate Haskell (in this case, very thin layers over Haskell) continue to make progress.

Research on More Descriptive Static Properties

The research community that works with Haskell is wonderful. While too immature for general use, people have developed tools to do things like statically check function partiality and contracts. If you look around you'll find it's a rich field.