+
+definition cotransitive: \forall A:Type.\forall R:relation A.Prop
+\def
+\lambda A.\lambda R.\forall x,y:A.R x y \to \forall z:A. R x z \lor R z y.
+
+definition tight_apart: \forall A:Type.\forall eq,ap:relation A.Prop
+\def
+\lambda A.\lambda eq,ap.\forall x,y:A. (\not (ap x y) \to eq x y) \land
+(eq x y \to \not (ap x y)).
+
+definition antisymmetric: \forall A:Type.\forall R:relation A.Prop
+\def
+\lambda A.\lambda R.\forall x,y:A. R x y \to \not (R y x).
+
+