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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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(** * Properties of orders and addition for modules implementing [NZOrdAxiomsSig']  This file defines the [NZAddOrderProp] functor type, meant to be [Include]d in a module implementing [NZOrdAxiomsSig'] (see [Stdlib.Numbers.NatInt.NZAxioms]).  This gives important basic compatibility lemmas between [add] and [lt], [le]. *)
From Stdlib.Numbers.NatInt Require Import NZAxioms NZBase NZMul NZOrder.
Module Type NZAddOrderProp (Import NZ : NZOrdAxiomsSig').
Include NZBaseProp NZ <+ NZMulProp NZ <+ NZOrderProp NZ.

(** Subtraction *)
(** We can prove the existence of a subtraction of any number by a smaller one *)
Lemma le_exists_sub : forall n m, n<=m -> exists p, m == p+n /\ 0<=p.
End NZAddOrderProp.