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-rw-r--r--theories/Arith/Le.v17
1 files changed, 6 insertions, 11 deletions
diff --git a/theories/Arith/Le.v b/theories/Arith/Le.v
index b73959e7..1febb76b 100644
--- a/theories/Arith/Le.v
+++ b/theories/Arith/Le.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 1999-2012 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
-(*i $Id: Le.v 14641 2011-11-06 11:59:10Z herbelin $ i*)
-
(** Order on natural numbers. [le] is defined in [Init/Peano.v] as:
<<
Inductive le (n:nat) : nat -> Prop :=
@@ -18,7 +16,7 @@ where "n <= m" := (le n m) : nat_scope.
>>
*)
-Open Local Scope nat_scope.
+Local Open Scope nat_scope.
Implicit Types m n p : nat.
@@ -48,8 +46,8 @@ Qed.
Theorem le_Sn_0 : forall n, ~ S n <= 0.
Proof.
- red in |- *; intros n H.
- change (IsSucc 0) in |- *; elim H; simpl in |- *; auto with arith.
+ red; intros n H.
+ change (IsSucc 0); elim H; simpl; auto with arith.
Qed.
Hint Resolve le_0_n le_Sn_0: arith v62.
@@ -84,8 +82,7 @@ Hint Immediate le_Sn_le: arith v62.
Theorem le_S_n : forall n m, S n <= S m -> n <= m.
Proof.
- intros n m H; change (pred (S n) <= pred (S m)) in |- *.
- destruct H; simpl; auto with arith.
+ exact Peano.le_S_n.
Qed.
Hint Immediate le_S_n: arith v62.
@@ -105,11 +102,9 @@ Hint Resolve le_pred_n: arith v62.
Theorem le_pred : forall n m, n <= m -> pred n <= pred m.
Proof.
- destruct n; simpl; auto with arith.
- destruct m; simpl; auto with arith.
+ exact Peano.le_pred.
Qed.
-
(** * [le] is a order on [nat] *)
(** Antisymmetry *)