(* -*- coding: utf-8 -*- *)
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(*         *      The Rocq Prover / The Rocq Development Team           *)
(*  v      *         Copyright INRIA, CNRS and contributors             *)
(* <O___,, * (see version control and CREDITS file for authors & dates) *)
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(*    //   *    This file is distributed under the terms of the         *)
(*         *     GNU Lesser General Public License Version 2.1          *)
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(** Properties of the injection from binary positive numbers to Peano natural numbers *)
(** Original development by Pierre Crégut, CNET, Lannion, France *)
Local Open Scope positive_scope.
Local Open Scope nat_scope.
Module Pos2Nat.
(** [Pos.to_nat] is a morphism for successor, addition, multiplication *)

(** [Pos.to_nat] is a morphism for [lt], [le], etc *)
Lemma inj_lt p q : (p < q)%positive <-> to_nat p < to_nat q.