Module Nat2Pos.
(** [Pos.of_nat] is a bijection between non-zero [nat] and [positive], with [Pos.to_nat] as reciprocal. See [Pos2Nat.id] above for the dual equation. *)
(** [Pos.of_nat] is hence injective for non-zero numbers *)
(** Usual operations are morphisms with respect to [Pos.of_nat] for non-zero numbers. *)

Lemma inj_compare (n m : nat) : n<>0 -> m<>0 ->
(n ?= m) = (Pos.of_nat n ?= Pos.of_nat m)%positive.