Fixity, Associativity and Precedence

With normal, prefix operators (e.g. functions), the semantics of f g h is clear: f is a function that takes g and h as parameters. If we want f to take the result of applying g to h we write f (g h).

In the case of infix operators (e.g. symbol operators such as + and *, or functions surrounded by backticks, for example `elem`), it is less clear. What does x - y - z mean? Subtracting x from y first (i.e. (x - y) - z) generally yields different results than subtracting z from y first (i.e. x - (y - z)). In Daml, the subtraction operator - is defined as a left-assocative operator. That is, when we write x - y - z - ... the parser associates to the left, meaning the parser interprets this as ((x - y) - z) - ....

Some operators are right-associative. We have already encountered one: function application! A function signature of a -> b -> c -> ... is parsed as (a -> (b -> (c -> ...))).

Finally, some operators are non-associative. A good example are comparison operators such as == and >. This means any ambiguous usage of these operators (e.g. a == b == c or a > b > c) results in a parse error.

Note

Non-associative operators are not to be confused with operators that are both left- and right-associative, such as + (since (x + y) + z = x + (y + z))). To obtain a deterministic parser, such operators must be declared as one of either left-associative or right-associative. In Daml the + operator has been declared as left-associative

The precedence of operators defines, when combining different operators, which operator is processed first. For example, in general (and in Daml), multiplication takes precedence over addition. That is, x + y * z is parsed as x + (y * z). Operator precedence is expressed as a number, where a higher number indicates a higher precedence. Operators of same precedence are associated to the left (e.g. x + y - z is parsed as (x + y) - z.

The fixity and precedence of an operator are declared using the infixl, infix, and infixr keywords (denoting left-, non-, and right-associativity, respectfully) that take an integer between 0 and 9 inclusive and an operator the fixity applies to. For example, infixl 6 + declares that + is a left-associative operator with precedence 6. These keywords can be used for user-defined operators as well. The following table shows the fixity and precedence for operators that are built-in to the Daml language, such as + and -:

Precedence	Left-associative	Non-associative	Right-associative
9	`!!		.
8			<#&&>, ^, **
7	*, /, %
6	+, -		<>
5			`++, ::
4	<$, <$>, $>, <*, <*>, *>	==, /=, <, <=, >==, >, ===, =/=
3			&&, &&&
2			||, |||
1	>>, >>=, <&>		=<<, <=<, >=>
0			$