intros Q H'; elim H'.
intros H'1.
apply SpolyQ0; auto.
intros p q H'0 H'2 H'3 H'4.
elim
reduce0_reducestar
with
(ltM_dec := ltM_dec)
(eqA_dec := eqA_dec)
(1 := cs)
(Q := Q)
(p := spolyf A A0 A1 eqA invA minusA multA divA eqA_dec n ltM ltM_dec p q
H'2 H'4); auto.
intros t E; apply Spoly_10 with (Cp := H'2) (Cq := H'4); auto.
apply
reducestar_eqp_com
with
(1 := cs)
(p := spolyf A A0 A1 eqA invA minusA multA divA eqA_dec n ltM ltM_dec p q
H'2 H'4)
(q := t); auto.
apply spolyf_canonical with (1 := cs); auto.
apply
H'1
with
(p := spolyf A A0 A1 eqA invA minusA multA divA eqA_dec n ltM ltM_dec p q
H'2 H'4); auto.
apply CombLinear_spoly; auto.
apply spolyf_canonical with (1 := cs); auto.
Qed.
Inductive ConfluentReduce (Q : list (poly A0 eqA ltM)) : Prop :=
ConfluentReduce0 :
(forall p : list (Term A n),
canonical A0 eqA ltM p ->
ReduStarConfluent A A0 A1 eqA invA minusA multA divA eqA_dec n ltM
ltM_dec Q p) -> ConfluentReduce Q.

(* Specifications: *)
Theorem SpolyQ_imp_ConfluentReduce :
forall Q : list (poly A0 eqA ltM),
SpolyQ A A0 A1 eqA invA minusA multA divA eqA_dec n ltM ltM_dec Q ->
ConfluentReduce Q.