diff options
Diffstat (limited to 'theories/Sets/Partial_Order.v')
-rw-r--r-- | theories/Sets/Partial_Order.v | 24 |
1 files changed, 11 insertions, 13 deletions
diff --git a/theories/Sets/Partial_Order.v b/theories/Sets/Partial_Order.v index e819cafa..054164da 100644 --- a/theories/Sets/Partial_Order.v +++ b/theories/Sets/Partial_Order.v @@ -1,6 +1,6 @@ (************************************************************************) (* 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 *) @@ -24,8 +24,6 @@ (* in Summer 1995. Several developments by E. Ledinot were an inspiration. *) (****************************************************************************) -(*i $Id: Partial_Order.v 14641 2011-11-06 11:59:10Z herbelin $ i*) - Require Export Ensembles. Require Export Relations_1. @@ -65,13 +63,13 @@ Section Partial_order_facts. forall x y z:U, Strict_Rel_of U D x y -> Rel_of U D y z -> Strict_Rel_of U D x z. Proof. - unfold Strict_Rel_of at 1 in |- *. - red in |- *. - elim D; simpl in |- *. + unfold Strict_Rel_of at 1. + red. + elim D; simpl. intros C R H' H'0; elim H'0. intros H'1 H'2 H'3 x y z H'4 H'5; split. apply H'2 with (y := y); tauto. - red in |- *; intro H'6. + red; intro H'6. elim H'4; intros H'7 H'8; apply H'8; clear H'4. apply H'3; auto. rewrite H'6; tauto. @@ -81,22 +79,22 @@ Section Partial_order_facts. forall x y z:U, Rel_of U D x y -> Strict_Rel_of U D y z -> Strict_Rel_of U D x z. Proof. - unfold Strict_Rel_of at 1 in |- *. - red in |- *. - elim D; simpl in |- *. + unfold Strict_Rel_of at 1. + red. + elim D; simpl. intros C R H' H'0; elim H'0. intros H'1 H'2 H'3 x y z H'4 H'5; split. apply H'2 with (y := y); tauto. - red in |- *; intro H'6. + red; intro H'6. elim H'5; intros H'7 H'8; apply H'8; clear H'5. apply H'3; auto. rewrite <- H'6; auto. Qed. Lemma Strict_Rel_Transitive : Transitive U (Strict_Rel_of U D). - red in |- *. + red. intros x y z H' H'0. apply Strict_Rel_Transitive_with_Rel with (y := y); [ intuition | unfold Strict_Rel_of in H', H'0; intuition ]. Qed. -End Partial_order_facts.
\ No newline at end of file +End Partial_order_facts. |