How do I provide implicit arguments explicitly in Coq?

Question

Suppose I have a definition f : x -> y -> z where x can be easily inferred. I therefore choose to make x an implicit argument using Arguments.

Consider the following example:

Definition id : forall (S : Set), S -> S :=
fun S s => s.

Arguments id {_} s.

Check (id 1).

Clearly S = nat can be and is inferred by Coq, and Coq replies:

id 1
     : nat

However, at a later time, I want to make the implicit argument explicit, say, for readability.

In other words, I would like something like:

Definition foo :=
 id {nat} 1. (* We want to make it clear that the 2nd argument is nat*)

Is this possible at all? If so, what is the appropriate syntax?

Answer 1

You can prepend the name with @ to remove all implicits and provide them explicitly:

Check @id nat 1.

You can also use (a:=v) to pass an implicit argument by name. This can both clarify what argument is being passed and also allows you to pass some implicits without passing _ for the others:

Check id (S:=nat) 1.

Definition third {A B C:Type} (a:A) (b:B) (c:C) := c.
Check third (B:=nat) (A:=unit) tt 1 2.

Answer 2

One possible way is by prepending @ to the definition. For example:

Definition id : forall (S : Set), S -> S :=
fun S s => s.

Arguments id {_} s.

Check @id nat 1.

Which results in:

id 1
     : nat