The Coq Standard Library

# 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.