Arend supports the following kinds of definitions: functions, data, records, classes, instances, and coercions. Every definition has a name which must be a valid identifier as described in Lexical structure.
Definitions can be referred by defcall expressions. If def is a name of a definition, defcall is an expression of the form: def e_1 … e_n, where e_1, …, e_n are expressions. Expressions e_1, …, e_n are called arguments of the defcall.
There are alternative notations in case a defcall has precisely one or two arguments. In case of two arguments, it is possible to use the infix notation: def e_1 e_2 is equivalent to e_1 `def` e_2. In case of one argument, it is possible to use the postfix notation: def e_1 is equivalent to e_1 `def.
The parsing of complex defcall expressions, containing several infix and postfix notations, is regulated by precedence of involved definitions: their priority and the type of associativity. Precedence of a definition can be specified by writing FIX N just before the name of the definition, where FIX is one of keywords \fixl, \fixr or \fix, which mark the definition as left, right associative or non-associative respectively, and N is the priority, which is an integer between 1 and 9. For example, \func \fixl 3 op (a b : Nat) => 0 defines a binary function named op which is left associative with priority 3. The default precedence is \fixr 10.
If op1 and op2 are two definitions and e1, e2, e3 are expressions, the expression e1 `op1` e2 `op2` e3 is parsed according to the following rules:
- If priorities of op1 and op2 are different and, say, the priority of op1 is higher, then the expression is parsed as (e1 `op1` e2) `op2` e3.
- Assume priorities of op1 and op2 are the same. If they are both left or both right associative, the expression is parsed as (e1 `op1` e2) `op2` e3 or e1 `op1` (e2 `op2` e3) respectively. If op1 and op2 have different types of associativity or are non-associative, then the parsing error is generated.
A definition can be labeled as an infix operator.
This means that its defcalls are parsed as infix notations even without
An infix operator is defined by specifying one of keywords \infixl, \infixr, \infix before the name of operator.
These keywords have the same syntax and semantics as keywords \fixl, \fixr, and \fix, which are described above.
An infix operator can be used in the prefix form as an ordinary definition. For example, if the function + is defined as \infixl 6 +, then it is allowed to write either + 1 2 or 1 + 2; these expressions are equivalent.
Finally, if f is an infix operator or an operator surrounded with
` `, then it is allowed to write e f and
this is equivalent to f e.
For example, the function that adds 1 to its argument can be written either as 1 + or as + 1.
The result of application of the first function to 2 is 1 + 2, the result of application of the second one to 2
is + 1 2, and as noted before these expressions are equivalent.
As noted above, it is possible to use a left section of an operator, that is x + is equivalent to \lam y => x + y. It is also possible to use right sections: if a postfix notation is applied to an argument from the right as in `+ y, then such an expression is equivalent to \lam x => x + y.