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 reduceplus_trans : forall (Q : list (poly A0 eqA ltM)) (x y z : list (Term A n)), reduceplus Q x y -> canonical A0 eqA ltM x -> reduceplus Q y z -> reduceplus Q x z.