Definition CInvR0 (r : Rcst) := Qeq_bool (Q_of_Rcst r) (0 # 1).Definition CPowR0 (z : Z) (r : Rcst) :=
  Z.ltb z Z0 && Qeq_bool (Q_of_Rcst r) (0 # 1).Fixpoint R_of_Rcst (r : Rcst) : R :=
  match r with
    | C0 => R0
    | C1 => R1
    | CZ z => IZR z
    | CQ q => Q2R q
    | CPlus r1 r2  => (R_of_Rcst r1) + (R_of_Rcst r2)
    | CMinus r1 r2 => (R_of_Rcst r1) - (R_of_Rcst r2)
    | CMult r1 r2  => (R_of_Rcst r1) * (R_of_Rcst r2)
    | CPow r1 z    =>
      match z with
      | inl z =>
        if CPowR0 z r1
        then R0
        else  powerRZ (R_of_Rcst r1) z
      | inr n => pow (R_of_Rcst r1) n
      end
    | CInv r       =>
      if CInvR0 r then R0
      else Rinv (R_of_Rcst r)
    | COpp r       => - (R_of_Rcst r)
  end.Add Morphism Q2R with signature Qeq ==> @eq R as Q2R_m.

Lemma Q_of_RcstR : forall c, Q2R (Q_of_Rcst c) = R_of_Rcst c.