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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2011 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(* Evgeny Makarov, INRIA, 2007 *)
(************************************************************************)
(*i $Id: NAddOrder.v 14641 2011-11-06 11:59:10Z herbelin $ i*)
Require Export NOrder.
Module NAddOrderPropFunct (Import N : NAxiomsSig').
Include NOrderPropFunct N.
(** Theorems true for natural numbers, not for integers *)
Theorem le_add_r : forall n m, n <= n + m.
Proof.
intro n; induct m.
rewrite add_0_r; now apply eq_le_incl.
intros m IH. rewrite add_succ_r; now apply le_le_succ_r.
Qed.
Theorem lt_lt_add_r : forall n m p, n < m -> n < m + p.
Proof.
intros n m p H; rewrite <- (add_0_r n).
apply add_lt_le_mono; [assumption | apply le_0_l].
Qed.
Theorem lt_lt_add_l : forall n m p, n < m -> n < p + m.
Proof.
intros n m p; rewrite add_comm; apply lt_lt_add_r.
Qed.
Theorem add_pos_l : forall n m, 0 < n -> 0 < n + m.
Proof.
intros; apply add_pos_nonneg. assumption. apply le_0_l.
Qed.
Theorem add_pos_r : forall n m, 0 < m -> 0 < n + m.
Proof.
intros; apply add_nonneg_pos. apply le_0_l. assumption.
Qed.
End NAddOrderPropFunct.
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