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