Inductive reduceplus (Q : list (poly A0 eqA ltM)) :
list (Term A n) -> list (Term A n) -> Prop :=
  | Rstar_0 :
      forall x y : list (Term A n), eqP A eqA n x y -> reduceplus Q x y
  | Rstar_n :
      forall x y z : list (Term A n),
      reduce A A0 A1 eqA invA minusA multA divA eqA_dec n ltM ltM_dec Q x y ->
      reduceplus Q y z -> reduceplus Q x z.Inductive reduceplus (Q : list (poly A0 eqA ltM)) :
list (Term A n) -> list (Term A n) -> Prop :=
  | Rstar_0 :
      forall x y : list (Term A n), eqP A eqA n x y -> reduceplus Q x y
  | Rstar_n :
      forall x y z : list (Term A n),
      reduce A A0 A1 eqA invA minusA multA divA eqA_dec n ltM ltM_dec Q x y ->
      reduceplus Q y z -> reduceplus Q x z.Inductive reduceplus (Q : list (poly A0 eqA ltM)) :
list (Term A n) -> list (Term A n) -> Prop :=
  | Rstar_0 :
      forall x y : list (Term A n), eqP A eqA n x y -> reduceplus Q x y
  | Rstar_n :
      forall x y z : list (Term A n),
      reduce A A0 A1 eqA invA minusA multA divA eqA_dec n ltM ltM_dec Q x y ->
      reduceplus Q y z -> reduceplus Q x z.

Theorem rep_plus_zero_reduce : forall (Q : list (poly A0 eqA ltM)) (s t : list (Term A n)), reduceplus Q s t -> canonical A0 eqA ltM s -> forall p q : list (Term A n), eqP A eqA n s (pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec p q) -> canonical A0 eqA ltM p -> canonical A0 eqA ltM q -> eqP A eqA n t (pO A n) -> exists r1 : list (Term A n), reduceplus Q p r1 /\ reduceplus Q q (mults (A:=A) multA (n:=n) (invTerm (A:=A) invA (n:=n) (T1 A1 n)) r1).