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).
Theorem cong_reflexive_same_relation : forall (U:Type) (R R':Relation U), same_relation U R R' -> Reflexive U R -> Reflexive U R'.
Theorem cong_symmetric_same_relation : forall (U:Type) (R R':Relation U), same_relation U R R' -> Symmetric U R -> Symmetric U R'.
Theorem cong_antisymmetric_same_relation : forall (U:Type) (R R':Relation U), same_relation U R R' -> Antisymmetric U R -> Antisymmetric U R'.
Theorem cong_transitive_same_relation : forall (U:Type) (R R':Relation U), same_relation U R R' -> Transitive U R -> Transitive U R'.