intros Q p q H'0 H'1 H'2.
case reduce_inv2 with (1 := cs) (3 := H'0); auto.
intros x H; elim H; intros a H0; elim H0; intros H1 H2; elim H2; intros H3 H4;
elim H4; intros H5 H6; clear H4 H2 H0 H.
apply CombLinear_1 with (a := a) (p := q) (q := x); auto.
apply (eqp_sym _ _ _ _ _ _ _ _ _ cs n).
cut (canonical A0 eqA ltM q);
[ intros Op1 | apply canonical_reduce with (1 := cs) (3 := H'0) ];
auto.
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
(mults (A:=A) multA (n:=n) a x)
(minuspf A A0 A1 eqA invA minusA multA eqA_dec n ltM ltM_dec p
(mults (A:=A) multA (n:=n) a x))); auto.
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
(minuspf A A0 A1 eqA invA minusA multA eqA_dec n ltM ltM_dec p
(mults (A:=A) multA (n:=n) a x))
(mults (A:=A) multA (n:=n) a x)); auto.
apply pluspf_minuspf_id with (1 := cs); auto.
Qed.

Theorem CombLinear_reduceplus :
forall (Q : list (poly A0 eqA ltM)) (p q : list (Term A n)),
reduceplus A A0 A1 eqA invA minusA multA divA eqA_dec n ltM ltM_dec Q p q ->
canonical A0 eqA ltM p -> CombLinear Q q -> CombLinear Q p.