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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_3.v,v 1.1.2.1 2004/07/16 19:31:40 herbelin Exp $ i*)
Require Export Relations_1.
Require Export Relations_2.
Section Relations_3.
Variable U: Type.
Variable R: (Relation U).
Definition coherent : U -> U -> Prop :=
[x,y: U] (EXT z | (Rstar U R x z) /\ (Rstar U R y z)).
Definition locally_confluent : U -> Prop :=
[x: U] (y,z: U) (R x y) -> (R x z) -> (coherent y z).
Definition Locally_confluent : Prop := (x: U) (locally_confluent x).
Definition confluent : U -> Prop :=
[x: U] (y,z: U) (Rstar U R x y) -> (Rstar U R x z) -> (coherent y z).
Definition Confluent : Prop := (x: U) (confluent x).
Inductive noetherian : U -> Prop :=
definition_of_noetherian:
(x: U) ((y: U) (R x y) -> (noetherian y)) -> (noetherian x).
Definition Noetherian : Prop := (x: U) (noetherian x).
End Relations_3.
Hints Unfold coherent : sets v62.
Hints Unfold locally_confluent : sets v62.
Hints Unfold confluent : sets v62.
Hints Unfold Confluent : sets v62.
Hints Resolve definition_of_noetherian : sets v62.
Hints Unfold Noetherian : sets v62.
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