# Library Coq.Sets.Relations_3

Require Export Relations_1.
Require Export Relations_2.
Section Relations_3.
Variable U : Type.
Variable R : Relation U.
Definition coherent (x y:U) : Prop :=
exists z : _, Rstar U R x z /\ Rstar U R y z.
Definition locally_confluent (x:U) : Prop :=
forall y z:U, R x y -> R x z -> coherent y z.
Definition Locally_confluent : Prop := forall x:U, locally_confluent x.
Definition confluent (x:U) : Prop :=
forall y z:U, Rstar U R x y -> Rstar U R x z -> coherent y z.
Definition Confluent : Prop := forall x:U, confluent x.
Inductive noetherian (x: U) : Prop :=
definition_of_noetherian :
(forall y:U, R x y -> noetherian y) -> noetherian x.
Definition Noetherian : Prop := forall x:U, noetherian x.
End Relations_3.
Hint Unfold coherent: sets v62.
Hint Unfold locally_confluent: sets v62.
Hint Unfold confluent: sets v62.
Hint Unfold Confluent: sets v62.
Hint Resolve definition_of_noetherian: sets v62.
Hint Unfold Noetherian: sets v62.