Module N2Nat.
(** [N.to_nat] is a bijection between [N] and [nat], with [Pos.of_nat] as reciprocal. See [Nat2N.id] below for the dual equation. *)
(** [N.to_nat] is hence injective *)
(** Interaction of this translation and usual operations. *)

Lemma inj_compare a a' :
(a ?= a')%N = (N.to_nat a ?= N.to_nat a').