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Diffstat (limited to 'theories7/Relations/Relations.v')
| -rwxr-xr-x | theories7/Relations/Relations.v | 28 |
1 files changed, 28 insertions, 0 deletions
diff --git a/theories7/Relations/Relations.v b/theories7/Relations/Relations.v new file mode 100755 index 00000000..694d0eec --- /dev/null +++ b/theories7/Relations/Relations.v @@ -0,0 +1,28 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Relations.v,v 1.1.2.1 2004/07/16 19:31:38 herbelin Exp $ i*) + +Require Export Relation_Definitions. +Require Export Relation_Operators. +Require Export Operators_Properties. + +Lemma inverse_image_of_equivalence : (A,B:Set)(f:A->B) + (r:(relation B))(equivalence B r)->(equivalence A [x,y:A](r (f x) (f y))). +Intros; Split; Elim H; Red; Auto. +Intros _ equiv_trans _ x y z H0 H1; Apply equiv_trans with (f y); Assumption. +Qed. + +Lemma inverse_image_of_eq : (A,B:Set)(f:A->B) + (equivalence A [x,y:A](f x)=(f y)). +Split; Red; +[ (* reflexivity *) Reflexivity +| (* transitivity *) Intros; Transitivity (f y); Assumption +| (* symmetry *) Intros; Symmetry; Assumption +]. +Qed. |