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.

#[global]
Hint Resolve Rstar_0: sets.
#[global]
Hint Resolve Rstar1_0: sets.
#[global]
Hint Resolve Rstar1_1: sets.
#[global]
Hint Resolve Rplus_0: sets.