Module Coq.Sets.Relations_1

Section Relations_1.
   Variable U: Type.
   
   Definition Relation := U -> U -> Prop.
   Variable R: Relation.
   
   Definition Reflexive : Prop := (x: U) (R x x).
   
   Definition Transitive : Prop := (x,y,z: U) (R x y) -> (R y z) -> (R x z).
   
   Definition Symmetric : Prop := (x,y: U) (R x y) -> (R y x).
   
   Definition Antisymmetric : Prop :=
      (x: U) (y: U) (R x y) -> (R y x) -> x == y.
   
   Definition contains : Relation -> Relation -> Prop :=
      [R,R': Relation] (x: U) (y: U) (R' x y) -> (R x y).
   
   Definition same_relation : Relation -> Relation -> Prop :=
      [R,R': Relation] (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.
Hints Unfold Reflexive Transitive Antisymmetric Symmetric contains
        same_relation : sets v62.
Hints Resolve Definition_of_preorder Definition_of_order
        Definition_of_equivalence Definition_of_PER : sets v62.

Index