3.13. Precedence and Associativity

In order to parse and pretty print expressions in a non ambiguous manner we attach to some language symbols and all operators a precedence and an associativity. Associativity can be either left, right or nonassoc.

For expressions like a o₁ b o₂ c where the operators o₁ and o₂ have the same precedence, but different associativity, the operator with left associativity gets higher precedence.

For two subsequent operators in an operator expression one of the operators has always higher precedence. If the numerical precendece is the same, the associativity decides. Left associativity of the first gives the first a higher precedence. Right associativity of the first gives the first a lower precedence.

There are the following precedence levels from lowest to highest with some example operators/symbols:

level	left	right	noassoc
where
comma		,
assign		λ :=
colon		all :
arrow		→ =
or		\/ ∨ or
and		/\ ∧ and
not			¬ Not not
apply	|>	<|
composition	>>	<<
relation			= ≤ < …
add	+ -
mult	* / mod
exp		^
application	f a b c ..

Some examples of expressions and the corresponding parsing:

expression                          parsed as

a + b + c                           (a + b) + c

- a + b                             (- a) + b

- a * b                             - (a * b)

x |> f |> g                         (x |> f) |> g

f <| g <| x                         f <| (g <| x)

x |> f <| y                         (x |> f) <| y

x |> f >> g                         x |> (f >> g)

a :: b :: c                         a :: (b :: c)

A : B : C                           A : ( B : C)

λ x := a, b                         (λ x := a), b
--                                  comma < assign

λ x := a : T                        λ x := (a: T)
--                                  assign < colon

all x: T, e                         (all x: T), e
--                                  comma < colon

all y: T → U                        all y: (T → U)
--                                  colon < arrow