Library Coq.Sets.Relations_1

Section Relations_1.
   Variable U : Type.

   Definition Relation := U -> U -> Prop.
   Variable R : Relation.

   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 Symmetric : Prop := forall x y:U, R x y -> R y x.

   Definition Antisymmetric : Prop := forall x y:U, R x y -> R y x -> x = y.

   Definition contains (R R':Relation) : Prop :=
     forall x y:U, R' x y -> R x y.

   Definition same_relation (R R':Relation) : Prop :=
     contains R R' /\ contains R' R.

   Inductive Preorder : Prop :=
       Definition_of_preorder : Reflexive -> Transitive -> Preorder.

   Inductive Order : Prop :=
       Definition_of_order :
         Reflexive -> Transitive -> Antisymmetric -> Order.

   Inductive Equivalence : Prop :=
       Definition_of_equivalence :
         Reflexive -> Transitive -> Symmetric -> Equivalence.

   Inductive PER : Prop :=
       Definition_of_PER : Symmetric -> Transitive -> PER.

End Relations_1.
Hint Unfold Reflexive Transitive Antisymmetric Symmetric contains
  same_relation: sets.
Hint Resolve Definition_of_preorder Definition_of_order
  Definition_of_equivalence Definition_of_PER: sets.