3

I am using:

$ coqtop -v
The Coq Proof Assistant, version 8.4pl5 (February 2015)
compiled on Feb 06 2015 17:44:41 with OCaml 4.02.1

I defined the following CoInductive type, stream:

$ coqtop
Welcome to Coq 8.4pl5 (February 2015)

Coq < CoInductive stream (A : Type) : Type := 
Coq < | Cons : A -> stream A -> stream A.
stream is defined

Coq < Check stream.
stream
     : Type -> Type

Coq < Check Cons.
Cons
     : forall A : Type, A -> stream A -> stream A

Then, I tried to define the following CoFixpoint definition, zeros:

Coq < CoFixpoint zeros : stream nat := Cons 0 zeros.

However, I got the following error instead:

Toplevel input, characters 38-39:
> CoFixpoint zeros : stream nat := Cons 0 zeros.
>                                       ^
Error: In environment
zeros : stream nat
The term "0" has type "nat" while it is expected to have type 
"Type".

I figured out that I have to explicitly instantiate the variable A:

Coq < CoFixpoint zeros : stream nat := Cons nat 0 zeros.
zeros is corecursively defined

Why doesn't the Coq infer the type of A by itself? How do I make it infer the type of A?

Anton Trunov
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Aadit M Shah
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1 Answers1

4

You need to declare A as implicit to ask Coq to infer it for you. There are a few ways of doing it:

  1. Add the following declaration to your file: Set Implicit Arguments.. This will cause Coq to turn on automatic inference for arguments such as A for Cons, allowing you to write Cons 0 zeros.

  2. Turn on implicit arguments just for Cons, without affecting the rest of the file:

    Arguments Cons {A} _ _.

    This declaration marks A as implicit and leaves the other two arguments as explicit.

  3. Mark A as implicit in the definition of stream:

    CoInductive stream {A : Type} : Type := 
| Cons : A -> stream A -> stream A.

    I personally do not recommend doing this, however, as it will mark A as implicit for stream as well.

You can find more information about implicit arguments in the reference manual

Arthur Azevedo De Amorim
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    Side note: items 2. and 3. declare `A` as a [maximally-inserted](https://coq.inria.fr/refman/Reference-Manual004.html#Implicit%20Arguments) argument. Implicit arguments inferred with the `Set Implicit Arguments` directive are not declared to be insertable maximally. So, when applicable, one might want to add `Set Maximal Implicit Insertion.`. – Anton Trunov Jun 02 '16 at 14:00