Coq: Stuck using the subtype

Question

I have following definitions: (definition of positive integers as a subtype of nat)

Definition Z_pos_filter (p: nat) : bool :=
  if (beq_nat p 0) then false else         


        

Answer 1

Here is how I would do it in vanilla Coq. I'm assuming we can still tweak the definitions.

From Coq Require Import Arith.
Local Coercion is_true : bool >-> Sortclass.

Definition Z_pos: Set := {n : nat | 0  Z_pos -> Z_pos.
  intros [x xpos%Nat.ltb_lt] [y ypos%Nat.ltb_lt].
  refine (exist  _ (x * y) _).
  now apply Nat.ltb_lt, Nat.mul_pos_pos.
Defined.

Lemma compat: forall p q: Z_pos, Z_pos__N (Z_pos_mult p q) = Z_pos__N p * Z_pos__N q.
Proof. now intros [x xpos] [y ypos]. Qed.

Let me add that dealing with this kind of thing is much more pleasant in SSReflect/Mathcomp.