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(** *    Injection from N to R                            *)
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Fixpoint INR (n:nat) : R :=
  match n with
  | O => 0
  | S O => 1
  | S n => INR n + 1
  end.

(* Specifications: *)
Lemma Rrepr_INR : forall n : nat, CRealEq (Rrepr (INR n)) (inject_Z (Z.of_nat n)).
Lemma Rrepr_IPR2 : forall n : positive, CRealEq (Rrepr (IPR_2 n)) (inject_Z (Z.pos (n~0))).
Lemma Rrepr_IPR : forall n : positive, CRealEq (Rrepr (IPR n)) (inject_Z (Z.pos n)).
Lemma Rrepr_IZR : forall n : Z, CRealEq (Rrepr (IZR n)) (inject_Z n).
Lemma archimed : forall r:R, IZR (up r) > r /\ IZR (up r) - r <= 1.