blob: 144bce72b8a1062beaa196e284c09190fb28740f (
plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
|
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2015 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(* Evgeny Makarov, INRIA, 2007 *)
(************************************************************************)
Require Export NOrder.
Module NAddOrderProp (Import N : NAxiomsMiniSig').
Include NOrderProp 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 NAddOrderProp.
|
