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.