diff options
Diffstat (limited to 'theories/Numbers/Integer/Abstract/ZMul.v')
| -rw-r--r-- | theories/Numbers/Integer/Abstract/ZMul.v | 17 |
1 files changed, 10 insertions, 7 deletions
diff --git a/theories/Numbers/Integer/Abstract/ZMul.v b/theories/Numbers/Integer/Abstract/ZMul.v index 83dc0e10..36f9c3d5 100644 --- a/theories/Numbers/Integer/Abstract/ZMul.v +++ b/theories/Numbers/Integer/Abstract/ZMul.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: ZMul.v 14641 2011-11-06 11:59:10Z herbelin $ i*) - Require Export ZAdd. -Module ZMulPropFunct (Import Z : ZAxiomsSig'). -Include ZAddPropFunct Z. +Module ZMulProp (Import Z : ZAxiomsMiniSig'). +Include ZAddProp Z. (** A note on naming: right (correspondingly, left) distributivity happens when the sum is multiplied by a number on the right @@ -41,7 +39,7 @@ Qed. Theorem mul_opp_l : forall n m, (- n) * m == - (n * m). Proof. -intros n m. apply -> add_move_0_r. +intros n m. apply add_move_0_r. now rewrite <- mul_add_distr_r, add_opp_diag_l, mul_0_l. Qed. @@ -55,6 +53,11 @@ Proof. intros n m; now rewrite mul_opp_l, mul_opp_r, opp_involutive. Qed. +Theorem mul_opp_comm : forall n m, (- n) * m == n * (- m). +Proof. +intros n m. now rewrite mul_opp_l, <- mul_opp_r. +Qed. + Theorem mul_sub_distr_l : forall n m p, n * (m - p) == n * m - n * p. Proof. intros n m p. do 2 rewrite <- add_opp_r. rewrite mul_add_distr_l. @@ -67,6 +70,6 @@ intros n m p; rewrite (mul_comm (n - m) p), (mul_comm n p), (mul_comm m p); now apply mul_sub_distr_l. Qed. -End ZMulPropFunct. +End ZMulProp. |