Definition Relation := U -> U -> Prop.Definition Reflexive : Prop := forall x : U, R x x.Definition Transitive : Prop := forall x y z : U, R x y -> R y z -> R x z.Definition Antisymmetric : Prop := forall x y : U, R x y /\ R y x -> x = y.Definition Order : Prop := (Reflexive /\ Transitive) /\ Antisymmetric.Definition Symmetric : Prop := forall x y : U, R x y -> R y x.Definition Equivalence : Prop := (Reflexive /\ Symmetric) /\ Transitive.Definition PER : Prop := Symmetric /\ Transitive.

(* Specifications: *)
Theorem sym_not_P : forall (U : Type) (P : Relation U) (x y : U), Symmetric U P -> ~ P x y -> ~ P 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).