diff options
Diffstat (limited to 'theories/Arith/Mult.v')
| -rw-r--r-- | theories/Arith/Mult.v | 19 |
1 files changed, 10 insertions, 9 deletions
diff --git a/theories/Arith/Mult.v b/theories/Arith/Mult.v index 5dd61d67..479138a9 100644 --- a/theories/Arith/Mult.v +++ b/theories/Arith/Mult.v @@ -1,13 +1,11 @@ (************************************************************************) (* 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 *) (************************************************************************) -(*i $Id: Mult.v 14641 2011-11-06 11:59:10Z herbelin $ i*) - Require Export Plus. Require Export Minus. Require Export Lt. @@ -177,19 +175,22 @@ Qed. Lemma mult_S_lt_compat_l : forall n m p, m < p -> S n * m < S n * p. Proof. induction n; intros; simpl in *. - rewrite <- 2! plus_n_O; assumption. + rewrite <- 2 plus_n_O; assumption. auto using plus_lt_compat. Qed. Hint Resolve mult_S_lt_compat_l: arith. +Lemma mult_lt_compat_l : forall n m p, n < m -> 0 < p -> p * n < p * m. +Proof. + intros m n p H Hp. destruct p. elim (lt_irrefl _ Hp). + now apply mult_S_lt_compat_l. +Qed. + Lemma mult_lt_compat_r : forall n m p, n < m -> 0 < p -> n * p < m * p. Proof. - intros m n p H H0. - induction p. - elim (lt_irrefl _ H0). - rewrite mult_comm. - replace (n * S p) with (S p * n); auto with arith. + intros m n p H Hp. destruct p. elim (lt_irrefl _ Hp). + rewrite (mult_comm m), (mult_comm n). now apply mult_S_lt_compat_l. Qed. Lemma mult_S_le_reg_l : forall n m p, S n * m <= S n * p -> m <= p. |