Definition total := forall x y, R x y || R y x.Definition transitive := forall y x z, R x y -> R y z -> R x z.Definition symmetric := forall x y, R x y = R y x.Definition antisymmetric := forall x y, R x y && R y x -> x = y.Definition pre_symmetric := forall x y, R x y -> R y x.

Lemma symmetric_from_pre : pre_symmetric -> symmetric.