Relations.v

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Require Export Relation_Definitions.
Require Export Relation_Operators.
Require Export Operators_Properties.

Lemma inverse_image_of_equivalence :
  forall (A B:Type) (f:A -> B) (r:relation B),
    equivalence B r -> equivalence A (fun x y:A => r (f x) (f y)).forall (A B : Type) (f : A -> B) (r : relation B),
equivalence B r ->
equivalence A (fun x y : A => r (f x) (f y))
Proof.forall (A B : Type) (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; auto.A, B:Type
f:A -> B
r:relation B
H:equivalence B r
reflexive B r ->
transitive B r ->
symmetric B r ->
forall x y z : A,
r (f x) (f y) -> r (f y) (f z) -> r (f x) (f z)
  intros _ equiv_trans _ x y z H0 H1; apply equiv_trans with (f y); assumption.
Qed.

Lemma inverse_image_of_eq :
  forall (A B:Type) (f:A -> B), equivalence A (fun x y:A => f x = f y).forall (A B : Type) (f : A -> B),
equivalence A (fun x y : A => f x = f y)
Proof.forall (A B : Type) (f : A -> B),
equivalence A (fun x y : A => f x = f y)
  split; red;
    [  (* reflexivity *) reflexivity
      |  (* transitivity *) intros; transitivity (f y); assumption
      |  (* symmetry *) intros; symmetry ; assumption ].
Qed.