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(*                                                                      *)
(* Micromega: A reflexive tactic using the Positivstellensatz           *)
(*                                                                      *)
(*  Frédéric Besson (Irisa/Inria) 2006-2008                             *)
(*                                                                      *)
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Require Setoid.
(*Declare ML Module "micromega_plugin".*)
Definition Rsrt : ring_theory R0 R1 Rplus Rmult Rminus Ropp (@eq R).
Open Scope R_scope.
Notation to_nat := N.to_nat.
(* 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.
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.
exact Qeq_eqR.
Qed.

Lemma Qpower0 : forall z, (z <> 0)%Z -> (0 ^ z == 0)%Q.