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-rw-r--r--theories/Sets/Relations_2_facts.v18
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.