Use False_rec to build an aliased list type in Coq

Question

I'd like to create a dependent function, but I am running into a type mismatch error on the False_rec term. I'm trying to do something similar to the following:

Definition env A := list A.

Fixpoint zip (xs ys : env nat) (H : length xs = length ys) : env (nat * nat).
Proof.
  refine
  (match xs, ys with
  | [], [] => []
  | x :: xs, y :: ys =>
      (x, y) :: zip xs ys _
  | _, _ => False_rec _ _
  end).
...

However, I receive the following error:

The term "False_rec ?P ?f" has type "?P" while it is expected to have type 
"env (nat * nat)" (unable to find a well-typed instantiation for 
"?P": cannot ensure that "Type" is a subtype of "Set").

The standard list, when it is given a Set, becomes a set itself, but it seems that it does not do so when aliased in this way. For example:

Check (list nat). (* list nat : Set *)
Check (env nat). (* env nat : Type *)

Is there a reason for this mismatch? And is there a way to work around it? (For example, can I convince Coq that (env nat) is a Set ? Or can I use a more general False_rec function that operates on Type instead of Set ?)

Answer 1

Is there a reason for this mismatch?

Yes. Type is an entity that can contain (big) things that won't fit into Set . Eg nat fits into Set , but Set does not fit into itself. I think there should be more than one excellent answers on SO explaining this issue.

(For example, can I convince Coq that (env nat) is a Set ?

You can use explicit type annotations:

Definition env (A : Set) : Set := list A.

Or can I use a more general False_rec function that operates on Type instead of Set ?

Yes, you can. It's called False_rect .

Observe that simple match -expression won't work in your case (regardless of Set / Type business), you will need to use dependent pattern matching. Another possibility is to use proof mode to define your function.