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(************************************************************************)
(* * The Coq Proof Assistant / The Coq Development Team *)
(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *)
(* <O___,, * (see CREDITS file for the list of authors) *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(* * (see LICENSE file for the text of the license) *)
(************************************************************************)
(****************************************************************************)
(* *)
(* 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. *)
(****************************************************************************)
Require Export Relations_1.
Require Export Relations_2.
Section Relations_3.
Variable U : Type.
Variable R : Relation U.
Definition coherent (x y:U) : Prop :=
exists z : _, Rstar U R x z /\ Rstar U R y z.
Definition locally_confluent (x:U) : Prop :=
forall y z:U, R x y -> R x z -> coherent y z.
Definition Locally_confluent : Prop := forall x:U, locally_confluent x.
Definition confluent (x:U) : Prop :=
forall y z:U, Rstar U R x y -> Rstar U R x z -> coherent y z.
Definition Confluent : Prop := forall x:U, confluent x.
Inductive noetherian (x: U) : Prop :=
definition_of_noetherian :
(forall y:U, R x y -> noetherian y) -> noetherian x.
Definition Noetherian : Prop := forall x:U, noetherian x.
End Relations_3.
Hint Unfold coherent: sets.
Hint Unfold locally_confluent: sets.
Hint Unfold confluent: sets.
Hint Unfold Confluent: sets.
Hint Resolve definition_of_noetherian: sets.
Hint Unfold Noetherian: sets.
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