Coq.Relations.Relations

# Module Coq.Relations.Relations

``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; Apply equiv_trans with (f y); Assumption.
`Save`.

`Lemma` inverse_image_of_eq : (A,B:Set)(f:A->B)
(equivalence A [x,y:A](f x)=(f y)).
Split; Red;
[ Reflexivity
| Intros; Transitivity (f y); Assumption
| Intros; Symmetry; Assumption
].
`Save`.

`

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