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 v62.

Hint Resolve Definition_of_preorder Definition_of_order

  Definition_of_equivalence Definition_of_PER: sets v62.