(* This code is copyrighted by its authors; it is distributed under  *)
(* the terms of the LGPL license (see LICENSE and description files) *)
(****************************************************************************  Buchberger : reduction  Laurent Thery May 97 (revised April 01)  ****************************************************************************)
Require Export Pspminus.
Section Preduce.
Load "hCoefStructure".
Load "hOrderStructure".
Load "hSpminus".
Inductive inPolySet : list (Term A n) -> list (poly A0 eqA ltM) -> Prop :=
| incons :
forall (a : Term A n) (p : list (Term A n))
(H : canonical A0 eqA ltM (pX a p)) (P : list (poly A0 eqA ltM)),
inPolySet (pX a p)
(exist (fun l0 : list (Term A n) => canonical A0 eqA ltM l0)
(pX a p) H :: P)
| inskip :
forall (a : poly A0 eqA ltM) (p : list (Term A n))
(P : list (poly A0 eqA ltM)), inPolySet p P -> inPolySet p (a :: P).
Hint Resolve incons inskip.

Lemma inPolySet_imp_canonical :
forall (p : list (Term A n)) (L : list (poly A0 eqA ltM)),
inPolySet p L -> canonical A0 eqA ltM p.