diff options
Diffstat (limited to 'theories/Arith/EqNat.v')
| -rw-r--r-- | theories/Arith/EqNat.v | 18 |
1 files changed, 9 insertions, 9 deletions
diff --git a/theories/Arith/EqNat.v b/theories/Arith/EqNat.v index 94986278..ce8eb478 100644 --- a/theories/Arith/EqNat.v +++ b/theories/Arith/EqNat.v @@ -1,6 +1,6 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010 *) +(* <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 *) @@ -8,7 +8,7 @@ (** Equality on natural numbers *) -Open Local Scope nat_scope. +Local Open Scope nat_scope. Implicit Types m n x y : nat. @@ -23,7 +23,7 @@ Fixpoint eq_nat n m : Prop := end. Theorem eq_nat_refl : forall n, eq_nat n n. - induction n; simpl in |- *; auto. + induction n; simpl; auto. Qed. Hint Resolve eq_nat_refl: arith v62. @@ -35,7 +35,7 @@ Qed. Hint Immediate eq_eq_nat: arith v62. Lemma eq_nat_eq : forall n m, eq_nat n m -> n = m. - induction n; induction m; simpl in |- *; contradiction || auto with arith. + induction n; induction m; simpl; contradiction || auto with arith. Qed. Hint Immediate eq_nat_eq: arith v62. @@ -55,11 +55,11 @@ Proof. induction n. destruct m as [| n]. auto with arith. - intros; right; red in |- *; trivial with arith. + intros; right; red; trivial with arith. destruct m as [| n0]. - right; red in |- *; auto with arith. + right; red; auto with arith. intros. - simpl in |- *. + simpl. apply IHn. Defined. @@ -76,12 +76,12 @@ Fixpoint beq_nat n m : bool := Lemma beq_nat_refl : forall n, true = beq_nat n n. Proof. - intro x; induction x; simpl in |- *; auto. + intro x; induction x; simpl; auto. Qed. Definition beq_nat_eq : forall x y, true = beq_nat x y -> x = y. Proof. - double induction x y; simpl in |- *. + double induction x y; simpl. reflexivity. intros n H1 H2. discriminate H2. intros n H1 H2. discriminate H2. |