Ltac QltCleanAbsurdCases :=
  match goal with
  | id1:(Qlt Zero (Qneg _)) |- _ => inversion id1
  | id1:(Qlt (Qpos _) Zero) |- _ =>
      inversion id1
  | id1:(Qlt (Qpos _) (Qneg _)) |- _ =>
      inversion id1
  | id1:(Qle Zero (Qneg _)) |- _ =>
      apply False_ind; apply id1; auto with *
  | id1:(Qle (Qpos _) Zero) |- _ =>
      apply False_ind; apply id1; auto with *
  | id1:(Qle (Qpos _) (Qneg _)) |- _ =>
      apply False_ind; apply id1; auto with *
  end.

Lemma Qpos_POS_1 : forall (m : Z) (qp : Qpositive), Z_to_Q m = Qpos qp -> (0 < m)%Z.