(* Words on N : initial structure for almost-rings. *)
Definition Nword := list N.Definition NwO : Nword := nil.Definition NwI : Nword := 1%N :: nil.Definition Nwcons n (w : Nword) : Nword :=
  match w, n with
  | nil, 0%N => nil
  | _, _ => n :: w
  end.Fixpoint Nwadd (w1 w2 : Nword) {struct w1} : Nword :=
  match w1, w2 with
  | n1::w1', n2:: w2' => (n1+n2)%N :: Nwadd w1' w2'
  | nil, _ => w2
  | _, nil => w1
  end.Definition Nwopp (w:Nword) : Nword := Nwcons 0%N w.Definition Nwsub w1 w2 := Nwadd w1 (Nwopp w2).Fixpoint Nwscal (n : N) (w : Nword) {struct w} : Nword :=
  match w with
  | m :: w' => (n*m)%N :: Nwscal n w'
  | nil => nil
  end.Fixpoint Nwmul (w1 w2 : Nword) {struct w1} : Nword :=
  match w1 with
  | 0%N::w1' => Nwopp (Nwmul w1' w2)
  | n1::w1' => Nwsub (Nwscal n1 w2) (Nwmul w1' w2)
  | nil => nil
  end.Fixpoint Nw_is0 (w : Nword) : bool :=
  match w with
  | nil => true
  | 0%N :: w' => Nw_is0 w'
  | _ => false
  end.Fixpoint Nweq_bool (w1 w2 : Nword) {struct w1} : bool :=
  match w1, w2 with
  | n1::w1', n2::w2' =>
     if N.eqb n1 n2 then Nweq_bool w1' w2' else false
  | nil, _ => Nw_is0 w2
  | _, nil => Nw_is0 w1
  end.Add Parametric Relation : R req
       reflexivity  proved by (@Equivalence_Reflexive _ _ Rsth)
       symmetry     proved by (@Equivalence_Symmetric _ _ Rsth)
       transitivity proved by (@Equivalence_Transitive _ _ Rsth)
      as R_setoid5.Add Morphism radd with signature (req ==> req ==> req) as radd_ext5.Add Morphism rmul with signature (req ==> req ==> req) as rmul_ext5.Add Morphism ropp with signature (req ==> req) as ropp_ext5.Add Morphism rsub with signature (req ==> req ==> req) as rsub_ext7.Fixpoint gen_phiNword (w : Nword) : R :=
  match w with
  | nil => 0
  | n :: nil => gen_phiN rO rI radd rmul n
  | N0 :: w' => - gen_phiNword w'
  | n::w' => gen_phiN rO rI radd rmul n - gen_phiNword w'
  end.

Lemma Nwscal_ok : forall n x, gen_phiNword (Nwscal n x) == gen_phiN rO rI radd rmul n * gen_phiNword x.