Section Relations.
Variable E : Type.
Definition relation (E : Type) := E -> E -> Prop.
Definition app_rel (R : relation E) (x y : E) := R x y.
Definition reflexive (R : relation E) : Prop := forall x : E, app_rel R x x.
Definition symmetric (R : relation E) : Prop :=
forall x y : E, app_rel R x y -> app_rel R y x.
Definition transitive (R : relation E) : Prop :=
forall x y z : E, app_rel R x y -> app_rel R y z -> app_rel R x z.
Comments "A partial equivalence on" E
" is a relation which is transitive and symmetric:".
Definition partial_equivalence (R : relation E) : Prop :=
transitive R /\ symmetric R.
Comments "An equivalence relation is reflexive, symmetric and transitive:".
Definition equivalence (R : relation E) : Prop :=
reflexive R /\ partial_equivalence R.
Comments "Some immediate properties:".

Lemma equiv_refl : forall R : relation E, equivalence R -> reflexive R.