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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(****************************************************************************)
(* *)
(* Naive Set Theory in Coq *)
(* *)
(* INRIA INRIA *)
(* Rocquencourt Sophia-Antipolis *)
(* *)
(* Coq V6.1 *)
(* *)
(* Gilles Kahn *)
(* Gerard Huet *)
(* *)
(* *)
(* *)
(* Acknowledgments: This work was started in July 1993 by F. Prost. Thanks *)
(* to the Newton Institute for providing an exceptional work environment *)
(* in Summer 1995. Several developments by E. Ledinot were an inspiration. *)
(****************************************************************************)
(*i $Id: Relations_2.v,v 1.1.2.1 2004/07/16 19:31:40 herbelin Exp $ i*)
Require Export Relations_1.
Section Relations_2.
Variable U: Type.
Variable R: (Relation U).
Inductive Rstar : (Relation U) :=
Rstar_0: (x: U) (Rstar x x)
| Rstar_n: (x, y, z: U) (R x y) -> (Rstar y z) -> (Rstar x z).
Inductive Rstar1 : (Relation U) :=
Rstar1_0: (x: U) (Rstar1 x x)
| Rstar1_1: (x: U) (y: U) (R x y) -> (Rstar1 x y)
| Rstar1_n: (x, y, z: U) (Rstar1 x y) -> (Rstar1 y z) -> (Rstar1 x z).
Inductive Rplus : (Relation U) :=
Rplus_0: (x, y: U) (R x y) -> (Rplus x y)
| Rplus_n: (x, y, z: U) (R x y) -> (Rplus y z) -> (Rplus x z).
Definition Strongly_confluent : Prop :=
(x, a, b: U) (R x a) -> (R x b) -> (exT U [z: U] (R a z) /\ (R b z)).
End Relations_2.
Hints Resolve Rstar_0 : sets v62.
Hints Resolve Rstar1_0 : sets v62.
Hints Resolve Rstar1_1 : sets v62.
Hints Resolve Rplus_0 : sets v62.
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