Library Coq.micromega.EnvRing


Set Implicit Arguments.
Require Import Setoid Morphisms Env BinPos BinNat BinInt.
Require Export Ring_theory.

Local Open Scope positive_scope.
Import RingSyntax.

Section MakeRingPol.

 Variable R:Type.
 Variable (rO rI : R) (radd rmul rsub: R->R->R) (ropp : R->R).
 Variable req : R -> R -> Prop.

 Variable Rsth : Equivalence req.
 Variable Reqe : ring_eq_ext radd rmul ropp req.
 Variable ARth : almost_ring_theory rO rI radd rmul rsub ropp req.

 Variable C: Type.
 Variable (cO cI: C) (cadd cmul csub : C->C->C) (copp : C->C).
 Variable ceqb : C->C->bool.
 Variable phi : C -> R.
 Variable CRmorph : ring_morph rO rI radd rmul rsub ropp req
                                cO cI cadd cmul csub copp ceqb phi.

 Variable Cpow : Type.
 Variable Cp_phi : N -> Cpow.
 Variable rpow : R -> Cpow -> R.
 Variable pow_th : power_theory rI rmul req Cp_phi rpow.

 Notation "0" := rO. Notation "1" := rI.
 Infix "+" := radd. Infix "*" := rmul.
 Infix "-" := rsub. Notation "- x" := (ropp x).
 Infix "==" := req.
 Infix "^" := (pow_pos rmul).

 Infix "+!" := cadd. Infix "*!" := cmul.
 Infix "-! " := csub. Notation "-! x" := (copp x).
 Infix "?=!" := ceqb. Notation "[ x ]" := (phi x).

 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).


 Inductive Pol : Type :=
  | Pc : C -> Pol
  | Pinj : positive -> Pol -> Pol
  | PX : Pol -> positive -> Pol -> 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.

 Section PopI.

  Variable Pop : Pol -> Pol -> Pol.
  Variable Q : Pol.

  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.

 Variable P' : Pol.

 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.

 End PopI.

 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' (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.

 Section PmulI.
  Variable Pmul : Pol -> Pol -> Pol.
  Variable Q : Pol.
  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.

 End PmulI.

 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.

 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.

 Reserved Notation "P @ l " (at level 10, no associativity).
 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 Peq_ok P P' : (P ?== P') = true -> forall l, P@l == P'@ l.

 Lemma Peq_spec P P' :
   BoolSpec (forall l, P@l == P'@l) True (P ?== P').

 Lemma Pphi0 l : P0@l == 0.

 Lemma Pphi1 l : P1@l == 1.

Lemma env_morph p e1 e2 :
  (forall x, e1 x = e2 x) -> p @ e1 = p @ e2.

Lemma Pjump_add P i j l :
  P @ (jump (i + j) l) = P @ (jump j (jump i l)).

Lemma Pjump_xO_tail P p l :
  P @ (jump (xO p) (tail l)) = P @ (jump (xI p) l).

Lemma Pjump_pred_double P p l :
  P @ (jump (Pos.pred_double p) (tail l)) = P @ (jump (xO p) l).

 Lemma mkPinj_ok j l P : (mkPinj j P)@l == P@(jump j l).

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

 Lemma ceqb_spec c c' : BoolSpec ([c] == [c']) True (c ?=! c').

 Lemma mkPX_ok l P i Q :
  (mkPX P i Q)@l == P@l * (hd l)^i + Q@(tail l).

 Hint Rewrite
  Pphi0
  Pphi1
  mkPinj_ok
  mkPX_ok
  (morph0 CRmorph)
  (morph1 CRmorph)
  (morph0 CRmorph)
  (morph_add CRmorph)
  (morph_mul CRmorph)
  (morph_sub CRmorph)
  (morph_opp CRmorph)
  : Esimpl.

 Ltac Esimpl := try rewrite_db Esimpl; rsimpl; simpl.

 Lemma PaddC_ok c P l : (PaddC P c)@l == P@l + [c].

 Lemma PsubC_ok c P l : (PsubC P c)@l == P@l - [c].

 Lemma PmulC_aux_ok c P l : (PmulC_aux P c)@l == P@l * [c].

 Lemma PmulC_ok c P l : (PmulC P c)@l == P@l * [c].

 Lemma Popp_ok P l : (--P)@l == - P@l.

 Hint Rewrite PaddC_ok PsubC_ok PmulC_ok Popp_ok : Esimpl.

 Lemma PaddX_ok P' P k l :
  (forall P l, (P++P')@l == P@l + P'@l) ->
  (PaddX Padd P' k P) @ l == P@l + P'@l * (hd l)^k.

 Lemma Padd_ok P' P l : (P ++ P')@l == P@l + P'@l.

 Lemma PsubX_ok P' P k l :
  (forall P l, (P--P')@l == P@l - P'@l) ->
  (PsubX Psub P' k P) @ l == P@l - P'@l * (hd l)^k.

 Lemma Psub_ok P' P l : (P -- P')@l == P@l - P'@l.

 Lemma PmulI_ok P' :
   (forall P l, (Pmul P P') @ l == P @ l * P' @ l) ->
   forall P p l, (PmulI Pmul P' p P) @ l == P @ l * P' @ (jump p l).

 Lemma Pmul_ok P P' l : (P**P')@l == P@l * P'@l.

 Lemma Psquare_ok P l : (Psquare P)@l == P@l * P@l.

 Lemma Mphi_morph M e1 e2 :
  (forall x, e1 x = e2 x) -> M @@ e1 = M @@ e2.

Lemma Mjump_xO_tail M p l :
  M @@ (jump (xO p) (tail l)) = M @@ (jump (xI p) l).

Lemma Mjump_pred_double M p l :
  M @@ (jump (Pos.pred_double p) (tail l)) = M @@ (jump (xO p) l).

Lemma Mjump_add M i j l :
  M @@ (jump (i + j) l) = M @@ (jump j (jump i l)).

 Lemma mkZmon_ok M j l :
   (mkZmon j M) @@ l == (zmon j M) @@ l.

 Lemma zmon_pred_ok M j l :
   (zmon_pred j M) @@ (tail l) == (zmon j M) @@ l.

 Lemma mkVmon_ok M i l :
   (mkVmon i M)@@l == M@@l * (hd l)^i.

 Ltac destr_mfactor R S := match goal with
  | H : context [MFactor ?P _] |- context [MFactor ?P ?M] =>
    specialize (H M); destruct MFactor as (R,S)
 end.

 Lemma Mphi_ok P M l :
   let (Q,R) := MFactor P M in
     P@l == Q@l + M@@l * R@l.

 Lemma POneSubst_ok P1 M1 P2 P3 l :
   POneSubst P1 M1 P2 = Some P3 -> M1@@l == P2@l ->
   P1@l == P3@l.

 Lemma PNSubst1_ok n P1 M1 P2 l :
    M1@@l == P2@l -> P1@l == (PNSubst1 P1 M1 P2 n)@l.

 Lemma PNSubst_ok n P1 M1 P2 l P3 :
    PNSubst P1 M1 P2 n = Some P3 -> M1@@l == P2@l -> P1@l == P3@l.

 Fixpoint MPcond (LM1: list (Mon * Pol)) (l: Env R) : Prop :=
   match LM1 with
   | cons (M1,P2) LM2 => (M1@@l == P2@l) /\ MPcond LM2 l
   | _ => True
   end.

 Lemma PSubstL1_ok n LM1 P1 l :
   MPcond LM1 l -> P1@l == (PSubstL1 P1 LM1 n)@l.

 Lemma PSubstL_ok n LM1 P1 P2 l :
   PSubstL P1 LM1 n = Some P2 -> MPcond LM1 l -> P1@l == P2@l.

 Lemma PNSubstL_ok m n LM1 P1 l :
    MPcond LM1 l -> P1@l == (PNSubstL P1 LM1 m n)@l.

Definition of polynomial expressions

 Inductive PExpr : 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.

evaluation of polynomial expressions towards R
 Definition mk_X j := mkPinj_pred j mkX.

evaluation of polynomial expressions towards R

 Fixpoint PEeval (l:Env R) (pe:PExpr) : R :=
   match pe with
   | PEc c => phi c
   | PEX j => nth j l
   | PEadd pe1 pe2 => (PEeval l pe1) + (PEeval l pe2)
   | PEsub pe1 pe2 => (PEeval l pe1) - (PEeval l pe2)
   | PEmul pe1 pe2 => (PEeval l pe1) * (PEeval l pe2)
   | PEopp pe1 => - (PEeval l pe1)
   | PEpow pe1 n => rpow (PEeval l pe1) (Cp_phi n)
   end.

Correctness proofs

 Lemma mkX_ok p l : nth p l == (mk_X p) @ l.

 Hint Rewrite Padd_ok Psub_ok : Esimpl.

Section POWER.
  Variable subst_l : Pol -> Pol.
  Fixpoint Ppow_pos (res P:Pol) (p:positive) : Pol :=
   match p with
   | xH => subst_l (res ** P)
   | xO p => Ppow_pos (Ppow_pos res P p) P p
   | xI p => subst_l ((Ppow_pos (Ppow_pos res P p) P p) ** P)
   end.

  Definition Ppow_N P n :=
   match n with
   | N0 => P1
   | Npos p => Ppow_pos P1 P p
   end.

  Lemma Ppow_pos_ok l :
    (forall P, subst_l P@l == P@l) ->
    forall res P p, (Ppow_pos res P p)@l == res@l * (pow_pos Pmul P p)@l.

  Lemma Ppow_N_ok l :
    (forall P, subst_l P@l == P@l) ->
    forall P n, (Ppow_N P n)@l == (pow_N P1 Pmul P n)@l.

 End POWER.

Normalization and rewriting

 Section NORM_SUBST_REC.
  Variable n : nat.
  Variable lmp:list (Mon*Pol).
  Let subst_l P := PNSubstL P lmp n n.
  Let Pmul_subst P1 P2 := subst_l (Pmul P1 P2).
  Let Ppow_subst := Ppow_N subst_l.

  Fixpoint norm_aux (pe:PExpr) : Pol :=
   match pe with
   | PEc c => Pc c
   | PEX j => mk_X j
   | PEadd (PEopp pe1) pe2 => Psub (norm_aux pe2) (norm_aux pe1)
   | PEadd pe1 (PEopp pe2) =>
     Psub (norm_aux pe1) (norm_aux pe2)
   | PEadd pe1 pe2 => Padd (norm_aux pe1) (norm_aux pe2)
   | PEsub pe1 pe2 => Psub (norm_aux pe1) (norm_aux pe2)
   | PEmul pe1 pe2 => Pmul (norm_aux pe1) (norm_aux pe2)
   | PEopp pe1 => Popp (norm_aux pe1)
   | PEpow pe1 n => Ppow_N (fun p => p) (norm_aux pe1) n
   end.

  Definition norm_subst pe := subst_l (norm_aux pe).

Internally, norm_aux is expanded in a large number of cases. To speed-up proofs, we use an alternative definition.

  Definition get_PEopp pe :=
   match pe with
   | PEopp pe' => Some pe'
   | _ => None
   end.

  Lemma norm_aux_PEadd pe1 pe2 :
    norm_aux (PEadd pe1 pe2) =
    match get_PEopp pe1, get_PEopp pe2 with
    | Some pe1', _ => (norm_aux pe2) -- (norm_aux pe1')
    | None, Some pe2' => (norm_aux pe1) -- (norm_aux pe2')
    | None, None => (norm_aux pe1) ++ (norm_aux pe2)
    end.

  Lemma norm_aux_PEopp pe :
    match get_PEopp pe with
    | Some pe' => norm_aux pe = -- (norm_aux pe')
    | None => True
    end.

  Lemma norm_aux_spec l pe :
    PEeval l pe == (norm_aux pe)@l.

 End NORM_SUBST_REC.

End MakeRingPol.