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.
intros p L H'; elim H'; auto.
Qed.
Hint Resolve inPolySet_imp_canonical.
intros Q p H'; elim H'; auto.
intros a p0 H'0 H'1; red in |- *; intros H'2; inversion H'2.
Qed.
intros Q H'; red in |- *; intros H'0.
lapply (not_nil_in_polySet_elm Q (pO A n));
[ intros H'3; apply H'3 || elim H'3 | idtac ]; auto.
Qed.

Lemma inPolySet_is_pX :
forall (p : list (Term A n)) (L : list (poly A0 eqA ltM)),
inPolySet p L ->
exists a : Term A n, (exists q : list (Term A n), p = pX a q).