diff options
Diffstat (limited to 'theories/Numbers/Natural/Abstract/NMulOrder.v')
| -rw-r--r-- | theories/Numbers/Natural/Abstract/NMulOrder.v | 22 |
1 files changed, 12 insertions, 10 deletions
diff --git a/theories/Numbers/Natural/Abstract/NMulOrder.v b/theories/Numbers/Natural/Abstract/NMulOrder.v index bdd4b674..1d6e8ba0 100644 --- a/theories/Numbers/Natural/Abstract/NMulOrder.v +++ b/theories/Numbers/Natural/Abstract/NMulOrder.v @@ -1,6 +1,6 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2011 *) +(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010 *) (* \VV/ **************************************************************) (* // * This file is distributed under the terms of the *) (* * GNU Lesser General Public License Version 2.1 *) @@ -8,12 +8,10 @@ (* Evgeny Makarov, INRIA, 2007 *) (************************************************************************) -(*i $Id: NMulOrder.v 14641 2011-11-06 11:59:10Z herbelin $ i*) - Require Export NAddOrder. -Module NMulOrderPropFunct (Import N : NAxiomsSig'). -Include NAddOrderPropFunct N. +Module NMulOrderProp (Import N : NAxiomsMiniSig'). +Include NAddOrderProp N. (** Theorems that are either not valid on Z or have different proofs on N and Z *) @@ -55,7 +53,7 @@ Qed. Theorem lt_0_mul' : forall n m, n * m > 0 <-> n > 0 /\ m > 0. Proof. intros n m; split; [intro H | intros [H1 H2]]. -apply -> lt_0_mul in H. destruct H as [[H1 H2] | [H1 H2]]. now split. +apply lt_0_mul in H. destruct H as [[H1 H2] | [H1 H2]]. now split. false_hyp H1 nlt_0_r. now apply mul_pos_pos. Qed. @@ -67,14 +65,18 @@ Proof. intros n m. split; [| intros [H1 H2]; now rewrite H1, H2, mul_1_l]. intro H; destruct (lt_trichotomy n 1) as [H1 | [H1 | H1]]. -apply -> lt_1_r in H1. rewrite H1, mul_0_l in H. false_hyp H neq_0_succ. +apply lt_1_r in H1. rewrite H1, mul_0_l in H. order'. rewrite H1, mul_1_l in H; now split. destruct (eq_0_gt_0_cases m) as [H2 | H2]. -rewrite H2, mul_0_r in H; false_hyp H neq_0_succ. -apply -> (mul_lt_mono_pos_r m) in H1; [| assumption]. rewrite mul_1_l in H1. +rewrite H2, mul_0_r in H. order'. +apply (mul_lt_mono_pos_r m) in H1; [| assumption]. rewrite mul_1_l in H1. assert (H3 : 1 < n * m) by now apply (lt_1_l m). rewrite H in H3; false_hyp H3 lt_irrefl. Qed. -End NMulOrderPropFunct. +(** Alternative name : *) + +Definition mul_eq_1 := eq_mul_1. + +End NMulOrderProp. |