infix properties of operators in Haskell

Question

I've done a Haskell library allowing to manipulate multivariate polynomials. It uses some instances defined by the numeric-prelude library, e.g. 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.

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?

Still looking at the code, I see:

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

What does that mean?

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

Finally, when the package author defines the instance e.g. 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?

Questions 2, 3, and 4 should be asked separately.

chepner
– chepner

2022-07-29 20:59:34 +00:00
Commented Jul 29, 2022 at 20:59 — chepner
– chepner, Commented Jul 29, 2022 at 20:59

Fyodor Soikin · Accepted Answer · 2022-07-29 21:25:31Z

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.
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 
```
Not sure.
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?

Thank you. For 3) I mean there is the substraction, as for numbers, x - y, and there is the opposite - x = 0 - x. I'm wondering how to define the 'opposite' operator?
The opposite operator would be negate wouldn't it? Normally an operator takes two argument so that wouldn't be a operator unless you use the PostfixOperators extension. This lets you define (-.) = negate written parenthesised: (x -.).
Re: (3) the negation operation -foo is special built-in syntax; it's not possible for users to define their own prefix operators. You can use RebindableSyntax to give it a different meaning, but you can only have one meaning per module that way.

chepner · Accepted Answer · 2022-07-29 21:05:58Z

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.

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