diff options
Diffstat (limited to 'theories/Arith/Lt.v')
| -rw-r--r-- | theories/Arith/Lt.v | 21 |
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 *) |