diff options
Diffstat (limited to 'theories/Relations/Operators_Properties.v')
-rw-r--r-- | theories/Relations/Operators_Properties.v | 32 |
1 files changed, 15 insertions, 17 deletions
diff --git a/theories/Relations/Operators_Properties.v b/theories/Relations/Operators_Properties.v index 26c8ef59..779c3d9a 100644 --- a/theories/Relations/Operators_Properties.v +++ b/theories/Relations/Operators_Properties.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: Operators_Properties.v 14641 2011-11-06 11:59:10Z herbelin $ i*) - (************************************************************************) (** * Some properties of the operators on relations *) (************************************************************************) @@ -19,17 +17,17 @@ Require Import Relation_Operators. Section Properties. - Implicit Arguments clos_refl_trans [A]. - Implicit Arguments clos_refl_trans_1n [A]. - Implicit Arguments clos_refl_trans_n1 [A]. - Implicit Arguments clos_refl_sym_trans [A]. - Implicit Arguments clos_refl_sym_trans_1n [A]. - Implicit Arguments clos_refl_sym_trans_n1 [A]. - Implicit Arguments clos_trans [A]. - Implicit Arguments clos_trans_1n [A]. - Implicit Arguments clos_trans_n1 [A]. - Implicit Arguments inclusion [A]. - Implicit Arguments preorder [A]. + Arguments clos_refl_trans [A] R x _. + Arguments clos_refl_trans_1n [A] R x _. + Arguments clos_refl_trans_n1 [A] R x _. + Arguments clos_refl_sym_trans [A] R _ _. + Arguments clos_refl_sym_trans_1n [A] R x _. + Arguments clos_refl_sym_trans_n1 [A] R x _. + Arguments clos_trans [A] R x _. + Arguments clos_trans_1n [A] R x _. + Arguments clos_trans_n1 [A] R x _. + Arguments inclusion [A] R1 R2. + Arguments preorder [A] R. Variable A : Type. Variable R : relation A. @@ -52,7 +50,7 @@ Section Properties. Lemma clos_rt_idempotent : inclusion (R*)* R*. Proof. - red in |- *. + red. induction 1; auto with sets. intros. apply rt_trans with y; auto with sets. @@ -68,7 +66,7 @@ Section Properties. Lemma clos_rt_clos_rst : inclusion (clos_refl_trans R) (clos_refl_sym_trans R). Proof. - red in |- *. + red. induction 1; auto with sets. apply rst_trans with y; auto with sets. Qed. @@ -89,7 +87,7 @@ Section Properties. inclusion (clos_refl_sym_trans (clos_refl_sym_trans R)) (clos_refl_sym_trans R). Proof. - red in |- *. + red. induction 1; auto with sets. apply rst_trans with y; auto with sets. Qed. |