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(****************************************************************************)
(*                 The Calculus of Inductive Constructions                  *)
(*                                                                          *)
(*                                Projet Coq                                *)
(*                                                                          *)
(*                     INRIA                        ENS-CNRS                *)
(*              Rocquencourt                        Lyon                    *)
(*                                                                          *)
(*                                 Coq V6.1                                 *)
(*                               Oct 1st 1996                               *)
(*                                                                          *)
(****************************************************************************)
(*                               Noetherian.v                               *)
(****************************************************************************)

Require Import Relations.

Set Implicit Arguments.
Unset Strict Implicit.

Section Noether.
   Variable U : Type.
   Variable R : Relation U.
   
   Inductive noetherian : U -> Prop :=
       Build_noetherian :
         forall x : U, (forall y : U, R x y -> noetherian y) -> noetherian x.
   
   Definition Noetherian : Prop := forall x : U, noetherian x.
   
End Noether.

Hint Resolve Build_noetherian.

Theorem Noetherian_contains_Noetherian :
 forall (U : Type) (R R' : Relation U),
 Noetherian R -> Contains R R' -> Noetherian R'.
Proof.
(* Goal: forall (U : Type) (R R' : Relation U) (_ : @Noetherian U R) (_ : @Contains U R R'), @Noetherian U R' *)
unfold Noetherian at 2 in |- *.
(* Goal: forall (U : Type) (R R' : Relation U) (_ : @Noetherian U R) (_ : @Contains U R R') (x : U), @noetherian U R' x *)
intros U R R' H' H'0 x.
(* Goal: @noetherian U R' x *)
elim H' with x; auto.
Qed.




(* $Id$ *)