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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) *)
(*   \VV/  **************************************************************)
(*    //   *    This file is distributed under the terms of the         *)
(*         *     GNU Lesser General Public License Version 2.1          *)
(*         *     (see LICENSE file for the text of the license)         *)
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(*                      Evgeny Makarov, INRIA, 2007                     *)
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Require Export Decidable.
Require Export NAxioms.
Module NBaseProp (Import N : NAxiomsMiniSig').
(** First, we import all known facts about both natural numbers and integers. *)
Include NZMulOrderProp N.
(** From [pred_0] and order facts, we can prove that 0 isn't a successor. *)

(* Specifications: *)
Theorem neq_succ_0 : forall n, S n ~= 0.
Theorem neq_0_succ : forall n, 0 ~= S n.
(** Next, we show that all numbers are nonnegative and recover regular induction from the bidirectional induction on NZ *)
Theorem le_0_l : forall n, 0 <= n.
Theorem induction :
forall A : N.t -> Prop, Proper (N.eq==>iff) A ->
A 0 -> (forall n, A n -> A (S n)) -> forall n, A n.