diff options
Diffstat (limited to 'theories/Sets/Relations_2_facts.v')
| -rw-r--r-- | theories/Sets/Relations_2_facts.v | 18 |
1 files changed, 8 insertions, 10 deletions
diff --git a/theories/Sets/Relations_2_facts.v b/theories/Sets/Relations_2_facts.v index 5ccdcb11..676fd719 100644 --- a/theories/Sets/Relations_2_facts.v +++ b/theories/Sets/Relations_2_facts.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: Relations_2_facts.v 14641 2011-11-06 11:59:10Z herbelin $ i*) - Require Export Relations_1. Require Export Relations_1_facts. Require Export Relations_2. @@ -45,13 +43,13 @@ Qed. Theorem Rstar_contains_R : forall (U:Type) (R:Relation U), contains U (Rstar U R) R. Proof. -intros U R; red in |- *; intros x y H'; apply Rstar_n with y; auto with sets. +intros U R; red; intros x y H'; apply Rstar_n with y; auto with sets. Qed. Theorem Rstar_contains_Rplus : forall (U:Type) (R:Relation U), contains U (Rstar U R) (Rplus U R). Proof. -intros U R; red in |- *. +intros U R; red. intros x y H'; elim H'. generalize Rstar_contains_R; intro T; red in T; auto with sets. intros x0 y0 z H'0 H'1 H'2; apply Rstar_n with y0; auto with sets. @@ -60,7 +58,7 @@ Qed. Theorem Rstar_transitive : forall (U:Type) (R:Relation U), Transitive U (Rstar U R). Proof. -intros U R; red in |- *. +intros U R; red. intros x y z H'; elim H'; auto with sets. intros x0 y0 z0 H'0 H'1 H'2 H'3; apply Rstar_n with y0; auto with sets. Qed. @@ -77,7 +75,7 @@ Theorem Rstar_equiv_Rstar1 : forall (U:Type) (R:Relation U), same_relation U (Rstar U R) (Rstar1 U R). Proof. generalize Rstar_contains_R; intro T; red in T. -intros U R; unfold same_relation, contains in |- *. +intros U R; unfold same_relation, contains. split; intros x y H'; elim H'; auto with sets. generalize Rstar_transitive; intro T1; red in T1. intros x0 y0 z H'0 H'1 H'2 H'3; apply T1 with y0; auto with sets. @@ -87,7 +85,7 @@ Qed. Theorem Rsym_imp_Rstarsym : forall (U:Type) (R:Relation U), Symmetric U R -> Symmetric U (Rstar U R). Proof. -intros U R H'; red in |- *. +intros U R H'; red. intros x y H'0; elim H'0; auto with sets. intros x0 y0 z H'1 H'2 H'3. generalize Rstar_transitive; intro T1; red in T1. @@ -99,7 +97,7 @@ Theorem Sstar_contains_Rstar : forall (U:Type) (R S:Relation U), contains U (Rstar U S) R -> contains U (Rstar U S) (Rstar U R). Proof. -unfold contains in |- *. +unfold contains. intros U R S H' x y H'0; elim H'0; auto with sets. generalize Rstar_transitive; intro T1; red in T1. intros x0 y0 z H'1 H'2 H'3; apply T1 with y0; auto with sets. @@ -150,4 +148,4 @@ elim (H'3 t); auto with sets. intros z1 H'5; elim H'5; intros H'8 H'10; try exact H'8; clear H'5. exists z1; split; [ idtac | assumption ]. apply Rstar_n with t; auto with sets. -Qed.
\ No newline at end of file +Qed. |