(* Syntactic ring coefficients. *)

Inductive Rcst :=
  | C0
  | C1
  | CQ (r : Q)
  | CZ (r : Z)
  | CPlus (r1 r2 : Rcst)
  | CMinus (r1  r2 : Rcst)
  | CMult (r1 r2 : Rcst)
  | CPow  (r1 : Rcst) (z:Z+nat)
  | CInv  (r : Rcst)
  | COpp  (r : Rcst).(* Syntactic ring coefficients. *)

Inductive Rcst :=
  | C0
  | C1
  | CQ (r : Q)
  | CZ (r : Z)
  | CPlus (r1 r2 : Rcst)
  | CMinus (r1  r2 : Rcst)
  | CMult (r1 r2 : Rcst)
  | CPow  (r1 : Rcst) (z:Z+nat)
  | CInv  (r : Rcst)
  | COpp  (r : Rcst).(* Syntactic ring coefficients. *)

Inductive Rcst :=
  | C0
  | C1
  | CQ (r : Q)
  | CZ (r : Z)
  | CPlus (r1 r2 : Rcst)
  | CMinus (r1  r2 : Rcst)
  | CMult (r1 r2 : Rcst)
  | CPow  (r1 : Rcst) (z:Z+nat)
  | CInv  (r : Rcst)
  | COpp  (r : Rcst).(* Syntactic ring coefficients. *)

Inductive Rcst :=
  | C0
  | C1
  | CQ (r : Q)
  | CZ (r : Z)
  | CPlus (r1 r2 : Rcst)
  | CMinus (r1  r2 : Rcst)
  | CMult (r1 r2 : Rcst)
  | CPow  (r1 : Rcst) (z:Z+nat)
  | CInv  (r : Rcst)
  | COpp  (r : Rcst).(* Syntactic ring coefficients. *)

Inductive Rcst :=
  | C0
  | C1
  | CQ (r : Q)
  | CZ (r : Z)
  | CPlus (r1 r2 : Rcst)
  | CMinus (r1  r2 : Rcst)
  | CMult (r1 r2 : Rcst)
  | CPow  (r1 : Rcst) (z:Z+nat)
  | CInv  (r : Rcst)
  | COpp  (r : Rcst).(* Syntactic ring coefficients. *)

Inductive Rcst :=
  | C0
  | C1
  | CQ (r : Q)
  | CZ (r : Z)
  | CPlus (r1 r2 : Rcst)
  | CMinus (r1  r2 : Rcst)
  | CMult (r1 r2 : Rcst)
  | CPow  (r1 : Rcst) (z:Z+nat)
  | CInv  (r : Rcst)
  | COpp  (r : Rcst).(* Syntactic ring coefficients. *)

Inductive Rcst :=
  | C0
  | C1
  | CQ (r : Q)
  | CZ (r : Z)
  | CPlus (r1 r2 : Rcst)
  | CMinus (r1  r2 : Rcst)
  | CMult (r1 r2 : Rcst)
  | CPow  (r1 : Rcst) (z:Z+nat)
  | CInv  (r : Rcst)
  | COpp  (r : Rcst).(* Syntactic ring coefficients. *)

Inductive Rcst :=
  | C0
  | C1
  | CQ (r : Q)
  | CZ (r : Z)
  | CPlus (r1 r2 : Rcst)
  | CMinus (r1  r2 : Rcst)
  | CMult (r1 r2 : Rcst)
  | CPow  (r1 : Rcst) (z:Z+nat)
  | CInv  (r : Rcst)
  | COpp  (r : Rcst).(* Syntactic ring coefficients. *)

Inductive Rcst :=
  | C0
  | C1
  | CQ (r : Q)
  | CZ (r : Z)
  | CPlus (r1 r2 : Rcst)
  | CMinus (r1  r2 : Rcst)
  | CMult (r1 r2 : Rcst)
  | CPow  (r1 : Rcst) (z:Z+nat)
  | CInv  (r : Rcst)
  | COpp  (r : Rcst).(* Syntactic ring coefficients. *)

Inductive Rcst :=
  | C0
  | C1
  | CQ (r : Q)
  | CZ (r : Z)
  | CPlus (r1 r2 : Rcst)
  | CMinus (r1  r2 : Rcst)
  | CMult (r1 r2 : Rcst)
  | CPow  (r1 : Rcst) (z:Z+nat)
  | CInv  (r : Rcst)
  | COpp  (r : Rcst).(* Syntactic ring coefficients. *)

Inductive Rcst :=
  | C0
  | C1
  | CQ (r : Q)
  | CZ (r : Z)
  | CPlus (r1 r2 : Rcst)
  | CMinus (r1  r2 : Rcst)
  | CMult (r1 r2 : Rcst)
  | CPow  (r1 : Rcst) (z:Z+nat)
  | CInv  (r : Rcst)
  | COpp  (r : Rcst).Definition z_of_exp (z : Z + nat) :=
  match z with
  | inl z => z
  | inr n => Z.of_nat n
  end.Fixpoint Q_of_Rcst (r : Rcst) : Q :=
  match r with
    | C0 => 0 # 1
    | C1 => 1 # 1
    | CZ z => z # 1
    | CQ q => q
    | CPlus r1 r2 => Qplus (Q_of_Rcst r1) (Q_of_Rcst r2)
    | CMinus r1 r2 => Qminus (Q_of_Rcst r1) (Q_of_Rcst r2)
    | CMult r1 r2  => Qmult (Q_of_Rcst r1) (Q_of_Rcst r2)
    | CPow  r1 z   => Qpower (Q_of_Rcst r1) (z_of_exp z)
    | CInv r       => Qinv (Q_of_Rcst r)
    | COpp r       => Qopp (Q_of_Rcst r)
  end.Definition is_neg (z: Z+nat) :=
  match z with
  | inl (Zneg _) => true
  |  _           => false
  end.

Lemma is_neg_false : forall z, is_neg z = false -> (z_of_exp z >= 0)%Z.