Notation PExpr := (PExpr C).(** Definition of polynomial expressions *)
#[universes(template)]
Inductive PExpr {C} : Type :=
| PEc : C -> PExpr
| PEX : positive -> PExpr
| PEadd : PExpr -> PExpr -> PExpr
| PEsub : PExpr -> PExpr -> PExpr
| PEmul : PExpr -> PExpr -> PExpr
| PEopp : PExpr -> PExpr
| PEpow : PExpr -> N -> PExpr.(** Definition of polynomial expressions *)
#[universes(template)]
Inductive PExpr {C} : Type :=
| PEc : C -> PExpr
| PEX : positive -> PExpr
| PEadd : PExpr -> PExpr -> PExpr
| PEsub : PExpr -> PExpr -> PExpr
| PEmul : PExpr -> PExpr -> PExpr
| PEopp : PExpr -> PExpr
| PEpow : PExpr -> N -> PExpr.(** Definition of polynomial expressions *)
#[universes(template)]
Inductive PExpr {C} : Type :=
| PEc : C -> PExpr
| PEX : positive -> PExpr
| PEadd : PExpr -> PExpr -> PExpr
| PEsub : PExpr -> PExpr -> PExpr
| PEmul : PExpr -> PExpr -> PExpr
| PEopp : PExpr -> PExpr
| PEpow : PExpr -> N -> PExpr.(** Definition of polynomial expressions *)
#[universes(template)]
Inductive PExpr {C} : Type :=
| PEc : C -> PExpr
| PEX : positive -> PExpr
| PEadd : PExpr -> PExpr -> PExpr
| PEsub : PExpr -> PExpr -> PExpr
| PEmul : PExpr -> PExpr -> PExpr
| PEopp : PExpr -> PExpr
| PEpow : PExpr -> N -> PExpr.(** Definition of polynomial expressions *)
#[universes(template)]
Inductive PExpr {C} : Type :=
| PEc : C -> PExpr
| PEX : positive -> PExpr
| PEadd : PExpr -> PExpr -> PExpr
| PEsub : PExpr -> PExpr -> PExpr
| PEmul : PExpr -> PExpr -> PExpr
| PEopp : PExpr -> PExpr
| PEpow : PExpr -> N -> PExpr.(** Definition of polynomial expressions *)
#[universes(template)]
Inductive PExpr {C} : Type :=
| PEc : C -> PExpr
| PEX : positive -> PExpr
| PEadd : PExpr -> PExpr -> PExpr
| PEsub : PExpr -> PExpr -> PExpr
| PEmul : PExpr -> PExpr -> PExpr
| PEopp : PExpr -> PExpr
| PEpow : PExpr -> N -> PExpr.(** Definition of polynomial expressions *)
#[universes(template)]
Inductive PExpr {C} : Type :=
| PEc : C -> PExpr
| PEX : positive -> PExpr
| PEadd : PExpr -> PExpr -> PExpr
| PEsub : PExpr -> PExpr -> PExpr
| PEmul : PExpr -> PExpr -> PExpr
| PEopp : PExpr -> PExpr
| PEpow : PExpr -> N -> PExpr.Notation Pol := (Pol C).(* Definition of multivariable polynomials with coefficients in C :
    Type [Pol] represents [X1 ... Xn].
    The representation is Horner's where a [n] variable polynomial
    (C[X1..Xn]) is seen as a polynomial on [X1] which coefficients
    are polynomials with [n-1] variables (C[X2..Xn]).
    There are several optimisations to make the repr compacter:
    - [Pc c] is the constant polynomial of value c
       == c*X1^0*..*Xn^0
    - [Pinj j Q] is a polynomial constant w.r.t the [j] first variables.
        variable indices are shifted of j in Q.
       == X1^0 *..* Xj^0 * Q{X1 <- Xj+1;..; Xn-j <- Xn}
    - [PX P i Q] is an optimised Horner form of P*X^i + Q
        with P not the null polynomial
       == P * X1^i + Q{X1 <- X2; ..; Xn-1 <- Xn}

    In addition:
    - polynomials of the form (PX (PX P i (Pc 0)) j Q) are forbidden
      since they can be represented by the simpler form (PX P (i+j) Q)
    - (Pinj i (Pinj j P)) is (Pinj (i+j) P)
    - (Pinj i (Pc c)) is (Pc c)
 *)

#[universes(template)]
Inductive Pol {C} : Type :=
| Pc : C -> Pol
| Pinj : positive -> Pol -> Pol
| PX : Pol -> positive -> Pol -> Pol.(* Definition of multivariable polynomials with coefficients in C :
    Type [Pol] represents [X1 ... Xn].
    The representation is Horner's where a [n] variable polynomial
    (C[X1..Xn]) is seen as a polynomial on [X1] which coefficients
    are polynomials with [n-1] variables (C[X2..Xn]).
    There are several optimisations to make the repr compacter:
    - [Pc c] is the constant polynomial of value c
       == c*X1^0*..*Xn^0
    - [Pinj j Q] is a polynomial constant w.r.t the [j] first variables.
        variable indices are shifted of j in Q.
       == X1^0 *..* Xj^0 * Q{X1 <- Xj+1;..; Xn-j <- Xn}
    - [PX P i Q] is an optimised Horner form of P*X^i + Q
        with P not the null polynomial
       == P * X1^i + Q{X1 <- X2; ..; Xn-1 <- Xn}

    In addition:
    - polynomials of the form (PX (PX P i (Pc 0)) j Q) are forbidden
      since they can be represented by the simpler form (PX P (i+j) Q)
    - (Pinj i (Pinj j P)) is (Pinj (i+j) P)
    - (Pinj i (Pc c)) is (Pc c)
 *)

#[universes(template)]
Inductive Pol {C} : Type :=
| Pc : C -> Pol
| Pinj : positive -> Pol -> Pol
| PX : Pol -> positive -> Pol -> Pol.(* Definition of multivariable polynomials with coefficients in C :
    Type [Pol] represents [X1 ... Xn].
    The representation is Horner's where a [n] variable polynomial
    (C[X1..Xn]) is seen as a polynomial on [X1] which coefficients
    are polynomials with [n-1] variables (C[X2..Xn]).
    There are several optimisations to make the repr compacter:
    - [Pc c] is the constant polynomial of value c
       == c*X1^0*..*Xn^0
    - [Pinj j Q] is a polynomial constant w.r.t the [j] first variables.
        variable indices are shifted of j in Q.
       == X1^0 *..* Xj^0 * Q{X1 <- Xj+1;..; Xn-j <- Xn}
    - [PX P i Q] is an optimised Horner form of P*X^i + Q
        with P not the null polynomial
       == P * X1^i + Q{X1 <- X2; ..; Xn-1 <- Xn}

    In addition:
    - polynomials of the form (PX (PX P i (Pc 0)) j Q) are forbidden
      since they can be represented by the simpler form (PX P (i+j) Q)
    - (Pinj i (Pinj j P)) is (Pinj (i+j) P)
    - (Pinj i (Pc c)) is (Pc c)
 *)

#[universes(template)]
Inductive Pol {C} : Type :=
| Pc : C -> Pol
| Pinj : positive -> Pol -> Pol
| PX : Pol -> positive -> Pol -> Pol.(* R notations *)
 Notation "0" := rO.Notation "1" := rI.Infix "+" := radd.Infix "*" := rmul.Infix "-" := rsub.Notation "- x" := (ropp x).Infix "==" := req.Infix "^" := (pow_pos rmul).(* C notations *)
 Infix "+!" := cadd.Infix "*!" := cmul.Infix "-! " := csub.Notation "-! x" := (copp x).Infix "?=!" := ceqb.Notation "[ x ]" := (phi x).(* Useful tactics *)
 Add Morphism radd with signature (req ==> req ==> req) as radd_ext.Add Morphism rmul with signature (req ==> req ==> req) as rmul_ext.Add Morphism ropp with signature (req ==> req) as ropp_ext.Add Morphism rsub with signature (req ==> req ==> req) as rsub_ext.Ltac rsimpl := gen_srewrite Rsth Reqe ARth.Ltac add_push := gen_add_push radd Rsth Reqe ARth.Ltac mul_push := gen_mul_push rmul Rsth Reqe ARth.Ltac add_permut_rec t :=
   match t with
   | ?x + ?y => add_permut_rec y || add_permut_rec x
   | _ => add_push t; apply (Radd_ext Reqe); [|reflexivity]
   end.Ltac add_permut :=
  repeat (reflexivity ||
    match goal with |- ?t == _ => add_permut_rec t end).Ltac mul_permut_rec t :=
   match t with
   | ?x * ?y => mul_permut_rec y || mul_permut_rec x
   | _ => mul_push t; apply (Rmul_ext Reqe); [|reflexivity]
   end.Ltac mul_permut :=
  repeat (reflexivity ||
    match goal with |- ?t == _ => mul_permut_rec t end).Implicit Types pe : PExpr.Implicit Types P : Pol.Definition P0 := Pc cO.Definition P1 := Pc cI.Fixpoint Peq (P P' : Pol) {struct P'} : bool :=
  match P, P' with
  | Pc c, Pc c' => c ?=! c'
  | Pinj j Q, Pinj j' Q' =>
    match j ?= j' with
    | Eq => Peq Q Q'
    | _ => false
    end
  | PX P i Q, PX P' i' Q' =>
    match i ?= i' with
    | Eq => if Peq P P' then Peq Q Q' else false
    | _ => false
    end
  | _, _ => false
  end.Infix "?==" := Peq.Definition mkPinj j P :=
  match P with
  | Pc _ => P
  | Pinj j' Q => Pinj (j + j') Q
  | _ => Pinj j P
  end.Definition mkPinj_pred j P :=
  match j with
  | xH => P
  | xO j => Pinj (Pos.pred_double j) P
  | xI j => Pinj (xO j) P
  end.Definition mkPX P i Q :=
  match P with
  | Pc c => if c ?=! cO then mkPinj xH Q else PX P i Q
  | Pinj _ _ => PX P i Q
  | PX P' i' Q' => if Q' ?== P0 then PX P' (i' + i) Q else PX P i Q
  end.Definition mkXi i := PX P1 i P0.Definition mkX := mkXi 1.(** Opposite of addition *)

 Fixpoint Popp (P:Pol) : Pol :=
  match P with
  | Pc c => Pc (-! c)
  | Pinj j Q => Pinj j (Popp Q)
  | PX P i Q => PX (Popp P) i (Popp Q)
  end.Notation "-- P" := (Popp P).(** Addition et subtraction *)

 Fixpoint PaddC (P:Pol) (c:C) : Pol :=
  match P with
  | Pc c1 => Pc (c1 +! c)
  | Pinj j Q => Pinj j (PaddC Q c)
  | PX P i Q => PX P i (PaddC Q c)
  end.Fixpoint PsubC (P:Pol) (c:C) : Pol :=
  match P with
  | Pc c1 => Pc (c1 -! c)
  | Pinj j Q => Pinj j (PsubC Q c)
  | PX P i Q => PX P i (PsubC Q c)
  end.Fixpoint PaddI (j:positive) (P:Pol) : Pol :=
   match P with
   | Pc c => mkPinj j (PaddC Q c)
   | Pinj j' Q' =>
     match Z.pos_sub j' j with
     | Zpos k =>  mkPinj j (Pop (Pinj k Q') Q)
     | Z0 => mkPinj j (Pop Q' Q)
     | Zneg k => mkPinj j' (PaddI k Q')
     end
   | PX P i Q' =>
     match j with
     | xH => PX P i (Pop Q' Q)
     | xO j => PX P i (PaddI (Pos.pred_double j) Q')
     | xI j => PX P i (PaddI (xO j) Q')
     end
   end.Fixpoint PsubI (j:positive) (P:Pol) : Pol :=
   match P with
   | Pc c => mkPinj j (PaddC (--Q) c)
   | Pinj j' Q' =>
     match Z.pos_sub j' j with
     | Zpos k =>  mkPinj j (Pop (Pinj k Q') Q)
     | Z0 => mkPinj j (Pop Q' Q)
     | Zneg k => mkPinj j' (PsubI k Q')
     end
   | PX P i Q' =>
     match j with
     | xH => PX P i (Pop Q' Q)
     | xO j => PX P i (PsubI (Pos.pred_double j) Q')
     | xI j => PX P i (PsubI (xO j) Q')
     end
   end.Fixpoint PaddX (i':positive) (P:Pol) : Pol :=
  match P with
  | Pc c => PX P' i' P
  | Pinj j Q' =>
    match j with
    | xH =>  PX P' i' Q'
    | xO j => PX P' i' (Pinj (Pos.pred_double j) Q')
    | xI j => PX P' i' (Pinj (xO j) Q')
    end
  | PX P i Q' =>
    match Z.pos_sub i i' with
    | Zpos k => mkPX (Pop (PX P k P0) P') i' Q'
    | Z0 => mkPX (Pop P P') i Q'
    | Zneg k => mkPX (PaddX k P) i Q'
    end
  end.Fixpoint PsubX (i':positive) (P:Pol) : Pol :=
  match P with
  | Pc c => PX (--P') i' P
  | Pinj j Q' =>
    match j with
    | xH =>  PX (--P') i' Q'
    | xO j => PX (--P') i' (Pinj (Pos.pred_double j) Q')
    | xI j => PX (--P') i' (Pinj (xO j) Q')
    end
  | PX P i Q' =>
    match Z.pos_sub i i' with
    | Zpos k => mkPX (Pop (PX P k P0) P') i' Q'
    | Z0 => mkPX (Pop P P') i Q'
    | Zneg k => mkPX (PsubX k P) i Q'
    end
  end.Fixpoint Padd (P P': Pol) {struct P'} : Pol :=
  match P' with
  | Pc c' => PaddC P c'
  | Pinj j' Q' => PaddI Padd Q' j' P
  | PX P' i' Q' =>
    match P with
    | Pc c => PX P' i' (PaddC Q' c)
    | Pinj j Q =>
      match j with
      | xH => PX P' i' (Padd Q Q')
      | xO j => PX P' i' (Padd (Pinj (Pos.pred_double j) Q) Q')
      | xI j => PX P' i' (Padd (Pinj (xO j) Q) Q')
      end
    | PX P i Q =>
      match Z.pos_sub i i' with
      | Zpos k => mkPX (Padd (PX P k P0) P') i' (Padd Q Q')
      | Z0 => mkPX (Padd P P') i (Padd Q Q')
      | Zneg k => mkPX (PaddX Padd P' k P) i (Padd Q Q')
      end
    end
  end.Infix "++" := Padd.Fixpoint Psub (P P': Pol) {struct P'} : Pol :=
  match P' with
  | Pc c' => PsubC P c'
  | Pinj j' Q' => PsubI Psub Q' j' P
  | PX P' i' Q' =>
    match P with
    | Pc c => PX (--P') i' (*(--(PsubC Q' c))*) (PaddC (--Q') c)
    | Pinj j Q =>
      match j with
      | xH => PX (--P') i' (Psub Q Q')
      | xO j => PX (--P') i' (Psub (Pinj (Pos.pred_double j) Q) Q')
      | xI j => PX (--P') i' (Psub (Pinj (xO j) Q) Q')
      end
    | PX P i Q =>
      match Z.pos_sub i i' with
      | Zpos k => mkPX (Psub (PX P k P0) P') i' (Psub Q Q')
      | Z0 => mkPX (Psub P P') i (Psub Q Q')
      | Zneg k => mkPX (PsubX Psub P' k P) i (Psub Q Q')
      end
    end
  end.Infix "--" := Psub.(** Multiplication *)

 Fixpoint PmulC_aux (P:Pol) (c:C) : Pol :=
  match P with
  | Pc c' => Pc (c' *! c)
  | Pinj j Q => mkPinj j (PmulC_aux Q c)
  | PX P i Q => mkPX (PmulC_aux P c) i (PmulC_aux Q c)
  end.Definition PmulC P c :=
  if c ?=! cO then P0 else
  if c ?=! cI then P else PmulC_aux P c.Fixpoint PmulI (j:positive) (P:Pol) : Pol :=
   match P with
   | Pc c => mkPinj j (PmulC Q c)
   | Pinj j' Q' =>
     match Z.pos_sub j' j with
     | Zpos k => mkPinj j (Pmul (Pinj k Q') Q)
     | Z0 => mkPinj j (Pmul Q' Q)
     | Zneg k => mkPinj j' (PmulI k Q')
     end
   | PX P' i' Q' =>
     match j with
     | xH => mkPX (PmulI xH P') i' (Pmul Q' Q)
     | xO j' => mkPX (PmulI j P') i' (PmulI (Pos.pred_double j') Q')
     | xI j' => mkPX (PmulI j P') i' (PmulI (xO j') Q')
     end
   end.Fixpoint Pmul (P P'' : Pol) {struct P''} : Pol :=
   match P'' with
   | Pc c => PmulC P c
   | Pinj j' Q' => PmulI Pmul Q' j' P
   | PX P' i' Q' =>
     match P with
     | Pc c => PmulC P'' c
     | Pinj j Q =>
       let QQ' :=
         match j with
         | xH => Pmul Q Q'
         | xO j => Pmul (Pinj (Pos.pred_double j) Q) Q'
         | xI j => Pmul (Pinj (xO j) Q) Q'
         end in
       mkPX (Pmul P P') i' QQ'
     | PX P i Q=>
       let QQ' := Pmul Q Q' in
       let PQ' := PmulI Pmul Q' xH P in
       let QP' := Pmul (mkPinj xH Q) P' in
       let PP' := Pmul P P' in
       (mkPX (mkPX PP' i P0 ++ QP') i' P0) ++ mkPX PQ' i QQ'
     end
  end.Infix "**" := Pmul.Fixpoint Psquare (P:Pol) : Pol :=
   match P with
   | Pc c => Pc (c *! c)
   | Pinj j Q => Pinj j (Psquare Q)
   | PX P i Q =>
     let twoPQ := Pmul P (mkPinj xH (PmulC Q (cI +! cI))) in
     let Q2 := Psquare Q in
     let P2 := Psquare P in
     mkPX (mkPX P2 i P0 ++ twoPQ) i Q2
   end.(** Monomial **)

 (** A monomial is X1^k1...Xi^ki. Its representation
     is a simplified version of the polynomial representation:

     - [mon0] correspond to the polynom [P1].
     - [(zmon j M)] corresponds to [(Pinj j ...)],
       i.e. skip j variable indices.
     - [(vmon i M)] is X^i*M with X the current variable,
       its corresponds to (PX P1 i ...)]
 *)

  Inductive Mon: Set :=
  | mon0: Mon
  | zmon: positive -> Mon -> Mon
  | vmon: positive -> Mon -> Mon.(** Monomial **)

 (** A monomial is X1^k1...Xi^ki. Its representation
     is a simplified version of the polynomial representation:

     - [mon0] correspond to the polynom [P1].
     - [(zmon j M)] corresponds to [(Pinj j ...)],
       i.e. skip j variable indices.
     - [(vmon i M)] is X^i*M with X the current variable,
       its corresponds to (PX P1 i ...)]
 *)

  Inductive Mon: Set :=
  | mon0: Mon
  | zmon: positive -> Mon -> Mon
  | vmon: positive -> Mon -> Mon.(** Monomial **)

 (** A monomial is X1^k1...Xi^ki. Its representation
     is a simplified version of the polynomial representation:

     - [mon0] correspond to the polynom [P1].
     - [(zmon j M)] corresponds to [(Pinj j ...)],
       i.e. skip j variable indices.
     - [(vmon i M)] is X^i*M with X the current variable,
       its corresponds to (PX P1 i ...)]
 *)

  Inductive Mon: Set :=
  | mon0: Mon
  | zmon: positive -> Mon -> Mon
  | vmon: positive -> Mon -> Mon.(** Monomial **)

 (** A monomial is X1^k1...Xi^ki. Its representation
     is a simplified version of the polynomial representation:

     - [mon0] correspond to the polynom [P1].
     - [(zmon j M)] corresponds to [(Pinj j ...)],
       i.e. skip j variable indices.
     - [(vmon i M)] is X^i*M with X the current variable,
       its corresponds to (PX P1 i ...)]
 *)

  Inductive Mon: Set :=
  | mon0: Mon
  | zmon: positive -> Mon -> Mon
  | vmon: positive -> Mon -> Mon.Definition mkZmon j M :=
   match M with mon0 => mon0 | _ => zmon j M end.Definition zmon_pred j M :=
   match j with xH => M | _ => mkZmon (Pos.pred j) M end.Definition mkVmon i M :=
   match M with
   | mon0 => vmon i mon0
   | zmon j m => vmon i (zmon_pred j m)
   | vmon i' m => vmon (i+i') m
   end.Fixpoint MFactor (P: Pol) (M: Mon) : Pol * Pol :=
   match P, M with
        _, mon0 => (Pc cO, P)
   | Pc _, _    => (P, Pc cO)
   | Pinj j1 P1, zmon j2 M1 =>
      match (j1 ?= j2) with
        Eq => let (R,S) := MFactor P1 M1 in
                 (mkPinj j1 R, mkPinj j1 S)
      | Lt => let (R,S) := MFactor P1 (zmon (j2 - j1) M1) in
                 (mkPinj j1 R, mkPinj j1 S)
      | Gt => (P, Pc cO)
      end
  | Pinj _ _, vmon _ _ => (P, Pc cO)
  | PX P1 i Q1, zmon j M1 =>
             let M2 := zmon_pred j M1 in
             let (R1, S1) := MFactor P1 M in
             let (R2, S2) := MFactor Q1 M2 in
               (mkPX R1 i R2, mkPX S1 i S2)
  | PX P1 i Q1, vmon j M1 =>
      match (i ?= j) with
        Eq => let (R1,S1) := MFactor P1 (mkZmon xH M1) in
                 (mkPX R1 i Q1, S1)
      | Lt => let (R1,S1) := MFactor P1 (vmon (j - i) M1) in
                 (mkPX R1 i Q1, S1)
      | Gt => let (R1,S1) := MFactor P1 (mkZmon xH M1) in
                 (mkPX R1 i Q1, mkPX S1 (i-j) (Pc cO))
      end
   end.Definition POneSubst (P1: Pol) (M1: Mon) (P2: Pol): option Pol :=
    let (Q1,R1) := MFactor P1 M1 in
    match R1 with
     (Pc c) => if c ?=! cO then None
               else Some (Padd Q1 (Pmul P2 R1))
    | _ => Some (Padd Q1 (Pmul P2 R1))
    end.Fixpoint PNSubst1 (P1: Pol) (M1: Mon) (P2: Pol) (n: nat) : Pol :=
    match POneSubst P1 M1 P2 with
     Some P3 => match n with S n1 => PNSubst1 P3 M1 P2 n1 | _ => P3 end
    | _ => P1
    end.Definition PNSubst (P1: Pol) (M1: Mon) (P2: Pol) (n: nat): option Pol :=
    match POneSubst P1 M1 P2 with
     Some P3 => match n with S n1 => Some (PNSubst1 P3 M1 P2 n1) | _ => None end
    | _ => None
    end.Fixpoint PSubstL1 (P1: Pol) (LM1: list (Mon * Pol)) (n: nat) : Pol :=
    match LM1 with
     cons (M1,P2) LM2 => PSubstL1 (PNSubst1 P1 M1 P2 n) LM2 n
    | _ => P1
    end.Fixpoint PSubstL (P1: Pol) (LM1: list (Mon * Pol)) (n: nat) : option Pol :=
    match LM1 with
     cons (M1,P2) LM2 =>
      match PNSubst P1 M1 P2 n with
        Some P3 => Some (PSubstL1 P3 LM2 n)
     |  None => PSubstL P1 LM2 n
     end
    | _ => None
    end.Fixpoint PNSubstL (P1: Pol) (LM1: list (Mon * Pol)) (m n: nat) : Pol :=
    match PSubstL P1 LM1 n with
     Some P3 => match m with S m1 => PNSubstL P3 LM1 m1 n | _ => P3 end
    | _ => P1
    end.(** Evaluation of a polynomial towards R *)

 Fixpoint Pphi(l:Env R) (P:Pol) : R :=
  match P with
  | Pc c => [c]
  | Pinj j Q => Pphi (jump j l) Q
  | PX P i Q => Pphi l P * (hd l) ^ i + Pphi (tail l) Q
  end.Notation "P @ l " := (Pphi l P).(** Evaluation of a monomial towards R *)

 Fixpoint Mphi(l:Env R) (M: Mon) : R :=
  match M with
  | mon0 => rI
  | zmon j M1 => Mphi (jump j l) M1
  | vmon i M1 => Mphi (tail l) M1 * (hd l) ^ i
  end.Notation "M @@ l" := (Mphi l M) (at level 10, no associativity).(** Proofs *)

 Ltac destr_pos_sub :=
  match goal with |- context [Z.pos_sub ?x ?y] =>
   generalize (Z.pos_sub_discr x y); destruct (Z.pos_sub x y)
  end.

Lemma pow_pos_add x i j : x^(j + i) == x^i * x^j.