Random nat stream and subset types in Coq

问题

Yo!

I need a random stream of nats with guaranteed subset types, like this stream will only give 0 < nat < 10. Anyone up for helping me with this?

I found this function for generating random numbers:

CoFixpoint rand (seed n1 n2 : Z) : Stream Z :=
    let seed' := Zmod seed n2 in Cons seed' (rand (seed' * n1) n1 n2).

I want to replace Z with any subset type, e.g.

Definition Z_gt0 := { Z | Z > 0}.

So we have:

CoFixpoint rand (seed n1 n2 : Z_gt0) : Stream Z_gt0 :=
    let seed' := Zmod seed n2 in Cons seed' (rand (seed' * n1) n1 n2).

Now the problem is that Zmod does accept Z but not Z_gt0.

Do I have to redefine all functions? Or is there already a library function ready to use?

TO MOD: Please add a tag for subset-types or refinement-types.

回答1:

The issue with your type is that Zmod seed n2 is a positive integer that can be 0, so seed' can be 0, which means that seed' * n1 can be 0 too.

In the end your CoFixpoint is not typable, the seed should be in some Z_ge0 type, not in Z_gt0.

EDIT: to answer the part about the library, you might be interested by the positive type, which is the type of binary integer strictly greater than 0. In fact, Z is defined as:

Inductive Z : Set :=
    Z0 : Z (* 0 *)
  | Zpos : positive -> Z (* z > 0 *)
  | Zneg : positive -> Z (* z < 0 *)

However the problem is still the same: taking the modulo of positive integer can escape positive since you can end up with 0.

来源：https://stackoverflow.com/questions/26475258/random-nat-stream-and-subset-types-in-coq