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 Z_to_Qmult_POS_NEG : forall p1 p2 : positive, Z_to_Q (Zpos p1 * Zneg p2) = Qmult (Z_to_Q (Zpos p1)) (Z_to_Q (Zneg p2)).