(* -*- coding: utf-8 -*- *)
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(*         *      The Rocq Prover / The Rocq Development Team           *)
(*  v      *         Copyright INRIA, CNRS and contributors             *)
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(*         *     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 *)

Theorem inj_pred p : (1 < p)%positive ->
to_nat (pred p) = Nat.pred (to_nat p).