Class Cring {R:Type}`{Rr:Ring R} :=
 cring_mul_comm: forall x y:R, x * y == y * x.Class Cring {R:Type}`{Rr:Ring R} :=
 cring_mul_comm: forall x y:R, x * y == y * x.Ltac reify_goal lvar lexpr lterm:=
  (*idtac lvar; idtac lexpr; idtac lterm;*)
  match lexpr with
     nil => idtac
   | ?e1::?e2::_ =>
        match goal with
          |- (?op ?u1 ?u2) =>
           change (op
             (@Ring_polynom.PEeval
               _ zero one _+_ _*_ _-_ -_ Z Ncring_initial.gen_phiZ N (fun n:N => n)
               (@Ring_theory.pow_N _ 1 multiplication) lvar e1)
             (@Ring_polynom.PEeval
               _ zero one _+_ _*_ _-_ -_ Z Ncring_initial.gen_phiZ N (fun n:N => n)
               (@Ring_theory.pow_N _ 1 multiplication) lvar e2))
        end
  end.

Lemma cring_div_theory: div_theory _==_ Z.add Z.mul Ncring_initial.gen_phiZ Z.quotrem.