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 one_plus_reduceplus : forall (Q : list (poly A0 eqA ltM)) (p1 p2 : list (Term A n)), reduce A A0 A1 eqA invA minusA multA divA eqA_dec n ltM ltM_dec Q p1 p2 -> forall r : list (Term A n), canonical A0 eqA ltM p1 -> canonical A0 eqA ltM r -> exists s : list (Term A n), reduceplus Q (pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec p1 r) s /\ reduceplus Q (pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec p2 r) s.