Haskell 中运算符的中缀属性

Question

I've done a Haskell library allowing to manipulate multivariate polynomials. It uses some instances defined by the numeric-prelude library, eg an additive group instance. The addition for this instance is denoted + in this library. I find this a bit annoying because there's already the usual + for numbers, so I defined:

import Algebra.Additive as AlgAdd

(^+^) :: Polynomial a -> Polynomial a -> Polynomial a
(^+^) p q = p AlgAdd.+ q

But it seems to me that by doing that, the "infix properties" of + are lost. I never dealt with the infix properties, and I have a couple of questions. I want to keep the same infix properties.

1. Looking at the code I see this line:

    infixl 6 +, -

This line is isolated, at the beginning of the code. So should I similarly include the line

infixl 6 ^+^, ^-^

at the beginning of my code? In this blog post the author defines the infix property of a function just before defining this function. Is it another, equivalent way to proceed? And is it not possible to do something like (^+^) = (AlgAdd.+) in such a way that the infix properties are automatically copied?

2. Still looking at the code, I see:

    {-# MINIMAL zero, (+), ((-) | negate) #-}

What does that mean?

3. I'm also wondering how to define the opposite of a polynomial. I have the substraction p ^-^ q but how to define ^-^ p ?

4. Finally, when the package author defines the instance eg for Double , he/she writes:

    instance C Double where {-# INLINE zero #-} {-# INLINE negate #-} {-# INLINE (+) #-} {-# INLINE (-) #-} zero = P.fromInteger 0 negate = P.negate (+) = (P.+) (-) = (P.-)

What is the purpose of INLINE ?

Answer 1

1. You can write the infix declaration above the definition, but it's not a requirement.

    infixl 6 ^+^, ^-^ (^+^):: Polynomial a -> Polynomial a -> Polynomial a (^+^) = (AlgAdd.+)

   The infix properties cannot be copied.

2. The Minimal pragma refers to the "minimal set of methods that must be defined" for a class definition to be total. In your case it means you can either write a definition for the methods " zero and (+) and (-) " or " zero and (+) and negate " since you can define negate in terms of the first group

    negate a = zero - a

   and (-) in terms of the second. So it doesn't matter which one you define.

    n - m = n + negate m

3. Not sure.

4. The Inline pragma forces ghc to inline (unfold) the definition. Here is the answer to Is there ever a reason to not mark a monadic bind operator (>>=) as inline?

Answer 2

Without a fixity declaration, your operators ^+^ and ^-^ will behave as if declared with

infixl 9 ^+^, ^-^

that is, left-associative with extremely high precedence. If you want them to behave more or less like "regular" addition and subtraction, then you should explicitly declare them to have precedence level 6 with

infixl 6 ^+^, ^-^

The associativity and precedence levels are not "transferable" from another operator; you'll just have to look up the behavior of the operator you want to be similar to and declare yours appropriately.