diff options
Diffstat (limited to 'theories/Arith/Plus.v')
| -rw-r--r-- | theories/Arith/Plus.v | 31 |
1 files changed, 8 insertions, 23 deletions
diff --git a/theories/Arith/Plus.v b/theories/Arith/Plus.v index 12f12300..02975d8f 100644 --- a/theories/Arith/Plus.v +++ b/theories/Arith/Plus.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: Plus.v 14641 2011-11-06 11:59:10Z herbelin $ i*) - (** Properties of addition. [add] is defined in [Init/Peano.v] as: << Fixpoint plus (n m:nat) : nat := @@ -26,17 +24,10 @@ Open Local Scope nat_scope. Implicit Types m n p q : nat. -(** * Zero is neutral *) - -Lemma plus_0_l : forall n, 0 + n = n. -Proof. - reflexivity. -Qed. - -Lemma plus_0_r : forall n, n + 0 = n. -Proof. - intro; symmetry in |- *; apply plus_n_O. -Qed. +(** * Zero is neutral +Deprecated : Already in Init/Peano.v *) +Notation plus_0_l := plus_O_n (only parsing). +Definition plus_0_r n := eq_sym (plus_n_O n). (** * Commutativity *) @@ -49,14 +40,8 @@ Hint Immediate plus_comm: arith v62. (** * Associativity *) -Lemma plus_Snm_nSm : forall n m, S n + m = n + S m. -Proof. - intros. - simpl in |- *. - rewrite (plus_comm n m). - rewrite (plus_comm n (S m)). - trivial with arith. -Qed. +Definition plus_Snm_nSm : forall n m, S n + m = n + S m:= + plus_n_Sm. Lemma plus_assoc : forall n m p, n + (m + p) = n + m + p. Proof. |