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.
Index
This page has been generated by coqdoc