# Library Coq.Sets.Relations_2

Require Export Relations_1.
Section Relations_2.
Variable U : Type.
Variable R : Relation U.
Inductive Rstar (x:U) : U -> Prop :=
| Rstar_0 : Rstar x x
| Rstar_n : forall y z:U, R x y -> Rstar y z -> Rstar x z.
Inductive Rstar1 (x:U) : U -> Prop :=
| Rstar1_0 : Rstar1 x x
| Rstar1_1 : forall y:U, R x y -> Rstar1 x y
| Rstar1_n : forall y z:U, Rstar1 x y -> Rstar1 y z -> Rstar1 x z.
Inductive Rplus (x:U) : U -> Prop :=
| Rplus_0 : forall y:U, R x y -> Rplus x y
| Rplus_n : forall y z:U, R x y -> Rplus y z -> Rplus x z.
Definition Strongly_confluent : Prop :=
forall x a b:U, R x a -> R x b -> ex (fun z:U => R a z /\ R b z).
End Relations_2.
Hint Resolve Rstar_0: sets v62.
Hint Resolve Rstar1_0: sets v62.
Hint Resolve Rstar1_1: sets v62.
Hint Resolve Rplus_0: sets v62.