Inductive CombLinear (Q : list (poly A0 eqA ltM)) :
list (Term A n) -> Prop :=
  | CombLinear_0 : CombLinear Q (pO A n)
  | CombLinear_1 :
      forall (a : Term A n) (p q s : list (Term A n)),
      ~ zeroP (A:=A) A0 eqA (n:=n) a ->
      inPolySet A A0 eqA n ltM q Q ->
      CombLinear Q p ->
      eqP A eqA n s
        (pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
           (mults (A:=A) multA (n:=n) a q) p) -> CombLinear Q s.Inductive CombLinear (Q : list (poly A0 eqA ltM)) :
list (Term A n) -> Prop :=
  | CombLinear_0 : CombLinear Q (pO A n)
  | CombLinear_1 :
      forall (a : Term A n) (p q s : list (Term A n)),
      ~ zeroP (A:=A) A0 eqA (n:=n) a ->
      inPolySet A A0 eqA n ltM q Q ->
      CombLinear Q p ->
      eqP A eqA n s
        (pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
           (mults (A:=A) multA (n:=n) a q) p) -> CombLinear Q s.Inductive CombLinear (Q : list (poly A0 eqA ltM)) :
list (Term A n) -> Prop :=
  | CombLinear_0 : CombLinear Q (pO A n)
  | CombLinear_1 :
      forall (a : Term A n) (p q s : list (Term A n)),
      ~ zeroP (A:=A) A0 eqA (n:=n) a ->
      inPolySet A A0 eqA n ltM q Q ->
      CombLinear Q p ->
      eqP A eqA n s
        (pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
           (mults (A:=A) multA (n:=n) a q) p) -> CombLinear Q s.

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