intros a b; case a; case b; simpl in |- *.
intros x c x0 c1 Q H'0.
case reduce_inv2 with (1 := cs) (3 := H'0); auto; intros c0 E; elim E;
intros d E0; elim E0; intros H'7 H'8; elim H'8; intros H'9 H'10;
elim H'10; intros H'11 H'12; clear H'10 H'8 E0 E.
apply
CombLinear_comp
with
(p := pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
x (mults (A:=A) multA (n:=n) d c0)); auto.
apply CombLinear_pluspf; auto.
generalize c H'0; case x; auto.
intros a0 l c2 H'.
apply CombLinear_id; auto.
change
(inPolySet A A0 eqA n ltM (pX a0 l)
(exist (canonical A0 eqA ltM) (pX a0 l) c2 :: Q))
in |- *.
apply incons with (a := a0) (p := l) (P := Q); auto.
apply CombLinear_mults1; auto.
apply CombLinear_id; auto.
apply inskip; 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 x0
(mults (A:=A) multA (n:=n) d c0))
(mults (A:=A) multA (n:=n) d c0)); auto.
apply pluspf_minuspf_id with (1 := cs); auto.
Qed.

Theorem reduceplus_cb2_lem :
forall (a b : list (Term A n)) (Q : list (poly A0 eqA ltM)),
reduceplus A A0 A1 eqA invA minusA multA divA eqA_dec n ltM ltM_dec Q a b ->
canonical A0 eqA ltM a ->
forall x : poly A0 eqA ltM,
s2p A A0 eqA n ltM x = b -> CombLinear (x :: Q) a.