Mixfix Operators

A type name, function name, or constructor name can comprise one or more name parts if we separate them with underscore characters _, and the resulting name can be used as an operator. From left to right, each argument goes in the place of each underscore _.

For instance, we can join with underscores the name parts if, then, and else into a single name if_then_else_. The application of the function name if_then_else_ to some arguments named x, y, and z can still be written as:

  • a standard application by using the full name if_then_else_ x y z
  • an operator application by placing the arguments between the name parts if x then y else z, leaving a space between arguments and part names
  • other sections of the full name, for instance leaving one or two underscores:
    • (if_then y else z) x
    • (if x then_else z) y
    • if x then y else_ z
    • if x then_else_ y z
    • if_then y else_ x z
    • (if_then_else z) x y

Examples of type names, function names, and constructor names as mixfix operators:

-- Example type name _⇒_
_⇒_   : Bool  Bool  Bool
true  ⇒ b = b
false ⇒ _ = true

-- Example function name _and_
_and_ : Bool  Bool  Bool
true and x = x
false and _ = false

-- Example function name if_then_else_
if_then_else_ : {A : Set}  Bool  A  A  A
if true then x else y = x
if false then x else y = y

-- Example constructor name _∷_
data List (A : Set) : Set where
  nil  : List A
  _∷_ : A  List A  List A

Precedence

Consider the expression true and false false. Depending on which of _and_ and _⇒_ has more precedence, it can either be read as (false and true) false = true, or as false and (true false) = true.

Each operator is associated to a precedence, which is a floating point number (can be negative and fractional!). The default precedence for an operator is 20.

Note

Please note that -> is directly handled in the parser. As a result, the precedence of -> is lower than any precedence you may declare with infixl and infixr.

If we give _and_ more precedence than _⇒_, then we will get the first result:

infix 30 _and_
-- infix 20 _⇒_ (default)

p-and : {x y z : Bool}   x and y ⇒ z  ≡  (x and y) ⇒ z
p-and = refl

e-and : false and true ⇒ false  ≡  true
e-and = refl

But, if we declare a new operator _and’_ and give it less precedence than _⇒_, then we will get the second result:

_and’_ : Bool  Bool  Bool
_and’_ = _and_
infix 15 _and’_
-- infix 20 _⇒_ (default)

p-⇒ : {x y z : Bool}   x and’ y ⇒ z  ≡  x and’ (y ⇒ z)
p-⇒ = refl

e-⇒ : false and’ true ⇒ false  ≡  false
e-⇒ = refl

Fixities can be changed when importing with a renaming directive:

open M using (_∙_)
open M renaming (_∙_ to infixl 10 _*_)

This code brings two instances of the operator _∙_ in scope:

  • the first named _∙_ and with its original fixity
  • the second named _*_ and with the fixity changed to act like a left associative operator of precedence 10.

Associativity

Consider the expression true false false. Depending on whether _⇒_ associates to the left or to the right, it can be read as (false true) false = false, or false (true false) = true, respectively.

If we declare an operator _⇒_ as infixr, it will associate to the right:

infixr 20 _⇒_

p-right : {x y z : Bool}   x ⇒ y ⇒ z  ≡  x ⇒ (y ⇒ z)
p-right = refl

e-right : false ⇒ true ⇒ false  ≡  true
e-right = refl

If we declare an operator _⇒’_ as infixl, it will associate to the left:

infixl 20 _⇒’_

_⇒’_ : Bool  Bool  Bool
_⇒’_ = _⇒_

p-left : {x y z : Bool}   x ⇒’ y ⇒’ z  ≡  (x ⇒’ y) ⇒’ z
p-left = refl

e-left : false ⇒’ true ⇒’ false  ≡  false
e-left = refl

Ambiguity and Scope

If you have not yet declared the fixity of an operator, Agda will complain if you try to use ambiguously:

e-ambiguous : Bool
e-ambiguous = true ⇒ true ⇒ true
Could not parse the application true ⇒ true ⇒ true
Operators used in the grammar:
  ⇒ (infix operator, level 20)

Fixity declarations may appear anywhere in a module that other declarations may appear. They then apply to the entire scope in which they appear (i.e. before and after, but not outside).