Definition R := DReal.Definition Rabst := DRealAbstr.Definition Rrepr := DRealRepr.Definition Rquot1 := DRealQuot1.Definition Rquot2 := DRealQuot2.Definition R0 : R := Rabst (inject_Q 0).Definition R1 : R := Rabst (inject_Q 1).Definition Rplus : R -> R -> R
    := fun x y : R => Rabst (CReal_plus (Rrepr x) (Rrepr y)).Definition Rmult : R -> R -> R
    := fun x y : R => Rabst (CReal_mult (Rrepr x) (Rrepr y)).Definition Ropp : R -> R
    := fun x : R => Rabst (CReal_opp (Rrepr x)).Definition Rlt : R -> R -> Prop
    := fun x y : R => CRealLtProp (Rrepr x) (Rrepr y).Definition R0_def := eq_refl R0.Definition R1_def := eq_refl R1.Definition Rplus_def := fun x y => eq_refl (Rplus x y).Definition Rmult_def := fun x y => eq_refl (Rmult x y).Definition Ropp_def := fun x => eq_refl (Ropp x).Definition Rlt_def := fun x y => eq_refl (Rlt x y).(* Keep the same names as before *)
Notation R := RbaseSymbolsImpl.R (only parsing).Notation R0 := RbaseSymbolsImpl.R0 (only parsing).Notation R1 := RbaseSymbolsImpl.R1 (only parsing).Notation Rplus := RbaseSymbolsImpl.Rplus (only parsing).Notation Rmult := RbaseSymbolsImpl.Rmult (only parsing).Notation Ropp := RbaseSymbolsImpl.Ropp (only parsing).Notation Rlt := RbaseSymbolsImpl.Rlt (only parsing).Infix "+" := Rplus : R_scope.Infix "*" := Rmult : R_scope.Notation "- x" := (Ropp x) : R_scope.Infix "<" := Rlt : R_scope.(***********************************************************)

(**********)
Definition Rgt (r1 r2:R) : Prop := r2 < r1.(**********)
Definition Rle (r1 r2:R) : Prop := r1 < r2 \/ r1 = r2.(**********)
Definition Rge (r1 r2:R) : Prop := Rgt r1 r2 \/ r1 = r2.(**********)
Definition Rminus (r1 r2:R) : R := r1 + - r2.(**********)

Infix "-" := Rminus : R_scope.Infix "<=" := Rle : R_scope.Infix ">=" := Rge : R_scope.Infix ">"  := Rgt : R_scope.Notation "x <= y <= z" := (x <= y /\ y <= z) : R_scope.Notation "x <= y < z"  := (x <= y /\ y <  z) : R_scope.Notation "x < y < z"   := (x <  y /\ y <  z) : R_scope.Notation "x < y <= z"  := (x <  y /\ y <= z) : R_scope.(**********************************************************)
(** *    Injection from [Z] to [R]                        *)
(**********************************************************)

(* compact representation for 2*p *)
Fixpoint IPR_2 (p:positive) : R :=
  match p with
  | xH => R1 + R1
  | xO p => (R1 + R1) * IPR_2 p
  | xI p => (R1 + R1) * (R1 + IPR_2 p)
  end.Definition IPR (p:positive) : R :=
  match p with
  | xH => R1
  | xO p => IPR_2 p
  | xI p => R1 + IPR_2 p
  end.(**********)
Definition IZR (z:Z) : R :=
  match z with
  | Z0 => R0
  | Zpos n => IPR n
  | Zneg n => - IPR n
  end.

Lemma Rrepr_appart_0 : forall x:R, (x < R0 \/ R0 < x) -> CReal_appart (Rrepr x) (inject_Q 0).