(** Z2R function (Z -> R) *)
Fixpoint P2R (p : positive) :=
  match p with
  | xH => 1%R
  | xO xH => 2%R
  | xO t => (2 * P2R t)%R
  | xI xH => 3%R
  | xI t => (1 + 2 * P2R t)%R
  end.Definition Z2R n :=
  match n with
  | Zpos p => P2R p
  | Zneg p => Ropp (P2R p)
  | Z0 => 0%R
  end.

Theorem lt_Z2R : forall m n, (Z2R m < Z2R n)%R -> (m < n)%Z.