Definition Complement (U : Type) (R : Relation U) : 
  Relation U := fun x y : U => ~ R x y.

(* Specifications: *)
Theorem Rsym_imp_notRsym : forall (U : Type) (R : Relation U), Symmetric U R -> Symmetric U (Complement U R).
Theorem Equiv_from_preorder : forall (U : Type) (R : Relation U), Preorder U R -> Equivalence U (fun x y : U => R x y /\ R y x).
Theorem Equiv_from_order : forall (U : Type) (R : Relation U), Order U R -> Equivalence U (fun x y : U => R x y /\ R y x).
Theorem contains_is_preorder : forall U : Type, Preorder (Relation U) (contains U).
Theorem same_relation_is_equivalence : forall U : Type, Equivalence (Relation U) (same_relation U).