Definition PolZ := Pol Z.Definition PEZ := PExpr Z.Definition P0Z : PolZ := P0 (C:=Z) 0%Z.Definition PolZadd : PolZ -> PolZ -> PolZ :=
  @Padd  Z 0%Z Z.add Zeq_bool.Definition PolZmul : PolZ -> PolZ -> PolZ :=
  @Pmul  Z 0%Z 1%Z Z.add Z.mul Zeq_bool.Definition PolZeq := @Peq Z Zeq_bool.Definition norm :=
  @norm_aux Z 0%Z 1%Z Z.add Z.mul Z.sub Z.opp Zeq_bool.Fixpoint mult_l (la : list PEZ) (lp: list PolZ) : PolZ :=
 match la, lp with
 | a::la, p::lp => PolZadd (PolZmul (norm a) p) (mult_l la lp)
 | _, _ => P0Z
 end.Fixpoint compute_list (lla: list (list PEZ)) (lp:list PolZ) :=
 match lla with
 | List.nil => lp
 | la::lla => compute_list lla ((mult_l la lp)::lp)
 end.Definition check (lpe:list PEZ) (qe:PEZ) (certif: list (list PEZ) * list PEZ) :=
 let (lla, lq) := certif in
 let lp := List.map norm lpe in
 PolZeq (norm qe) (mult_l lq (compute_list lla lp)).(* Correction *)
Definition PhiR : list R -> PolZ -> R :=
  (Pphi ring0 add mul
    (InitialRing.gen_phiZ ring0 ring1 add mul opp)).Definition PEevalR : list R -> PEZ -> R :=
   PEeval ring0 ring1 add mul sub opp
    (gen_phiZ ring0 ring1 add mul opp)
         N.to_nat pow.

(* Specifications: *)
Lemma P0Z_correct : forall l, PhiR l P0Z = 0.
Lemma Rext: ring_eq_ext add mul opp _==_.
Lemma Rset : Setoid_Theory R _==_.