diff options
Diffstat (limited to 'theories/Relations/Relations.v')
| -rw-r--r-- | theories/Relations/Relations.v | 25 |
1 files changed, 14 insertions, 11 deletions
diff --git a/theories/Relations/Relations.v b/theories/Relations/Relations.v index 2df0317b..9b2f4057 100644 --- a/theories/Relations/Relations.v +++ b/theories/Relations/Relations.v @@ -6,23 +6,26 @@ (* * GNU Lesser General Public License Version 2.1 *) (************************************************************************) -(*i $Id: Relations.v 8642 2006-03-17 10:09:02Z notin $ i*) +(*i $Id: Relations.v 9245 2006-10-17 12:53:34Z notin $ i*) Require Export Relation_Definitions. Require Export Relation_Operators. Require Export Operators_Properties. Lemma inverse_image_of_equivalence : - forall (A B:Set) (f:A -> B) (r:relation B), - equivalence B r -> equivalence A (fun x y:A => r (f x) (f y)). -intros; split; elim H; red in |- *; auto. -intros _ equiv_trans _ x y z H0 H1; apply equiv_trans with (f y); assumption. + forall (A B:Set) (f:A -> B) (r:relation B), + equivalence B r -> equivalence A (fun x y:A => r (f x) (f y)). +Proof. + intros; split; elim H; red in |- *; auto. + intros _ equiv_trans _ x y z H0 H1; apply equiv_trans with (f y); assumption. Qed. Lemma inverse_image_of_eq : - forall (A B:Set) (f:A -> B), equivalence A (fun x y:A => f x = f y). -split; red in |- *; - [ (* reflexivity *) reflexivity - | (* transitivity *) intros; transitivity (f y); assumption - | (* symmetry *) intros; symmetry in |- *; assumption ]. -Qed.
\ No newline at end of file + forall (A B:Set) (f:A -> B), equivalence A (fun x y:A => f x = f y). +Proof. + split; red in |- *; + [ (* reflexivity *) reflexivity + | (* transitivity *) intros; transitivity (f y); assumption + | (* symmetry *) intros; symmetry in |- *; assumption ]. +Qed. + |