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-rw-r--r--theories/Arith/Lt.v21
1 files changed, 10 insertions, 11 deletions
diff --git a/theories/Arith/Lt.v b/theories/Arith/Lt.v
index 004274fe..8559b782 100644
--- a/theories/Arith/Lt.v
+++ b/theories/Arith/Lt.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: Lt.v 14641 2011-11-06 11:59:10Z herbelin $ i*)
-
(** Theorems about [lt] in nat. [lt] is defined in library [Init/Peano.v] as:
<<
Definition lt (n m:nat) := S n <= m.
@@ -16,7 +14,7 @@ Infix "<" := lt : nat_scope.
*)
Require Import Le.
-Open Local Scope nat_scope.
+Local Open Scope nat_scope.
Implicit Types m n p : nat.
@@ -53,7 +51,7 @@ Qed.
Theorem lt_not_le : forall n m, n < m -> ~ m <= n.
Proof.
- red in |- *; intros n m Lt Le; exact (le_not_lt m n Le Lt).
+ red; intros n m Lt Le; exact (le_not_lt m n Le Lt).
Qed.
Hint Immediate le_not_lt lt_not_le: arith v62.
@@ -96,9 +94,9 @@ Proof.
Qed.
Hint Resolve lt_0_Sn: arith v62.
-Theorem lt_n_O : forall n, ~ n < 0.
-Proof le_Sn_O.
-Hint Resolve lt_n_O: arith v62.
+Theorem lt_n_0 : forall n, ~ n < 0.
+Proof le_Sn_0.
+Hint Resolve lt_n_0: arith v62.
(** * Predecessor *)
@@ -109,12 +107,12 @@ Qed.
Lemma lt_pred : forall n m, S n < m -> n < pred m.
Proof.
-induction 1; simpl in |- *; auto with arith.
+induction 1; simpl; auto with arith.
Qed.
Hint Immediate lt_pred: arith v62.
Lemma lt_pred_n_n : forall n, 0 < n -> pred n < n.
-destruct 1; simpl in |- *; auto with arith.
+destruct 1; simpl; auto with arith.
Qed.
Hint Resolve lt_pred_n_n: arith v62.
@@ -161,7 +159,7 @@ Hint Immediate lt_le_weak: arith v62.
Theorem le_or_lt : forall n m, n <= m \/ m < n.
Proof.
- intros n m; pattern n, m in |- *; apply nat_double_ind; auto with arith.
+ intros n m; pattern n, m; apply nat_double_ind; auto with arith.
induction 1; auto with arith.
Qed.
@@ -192,4 +190,5 @@ Hint Immediate lt_0_neq: arith v62.
Notation lt_O_Sn := lt_0_Sn (only parsing).
Notation neq_O_lt := neq_0_lt (only parsing).
Notation lt_O_neq := lt_0_neq (only parsing).
+Notation lt_n_O := lt_n_0 (only parsing).
(* end hide *)