(** This tactic solves the goal by contradiction, if in the hypothesis there are two terms `a<c` and `c<=a` *)
Ltac Order_mismatch :=
  Clear_eq_;
   match goal with
   | id1:(?X1 < ?X2)%Z,id2:(?X2 <= ?X1)%Z |- _ =>
       match goal with
       | id1:(?X1 < ?X2)%Z,id2:(?X2 <= ?X1)%Z |- _ =>
           first [ apply False_ind | apply False_rec ]; apply (Z.lt_irrefl X1);
            apply Z.lt_le_trans with X2; assumption
       end
   | id2:(?X1 <= ?X2)%Z,id1:(?X2 < ?X1)%Z |- _ =>
       match goal with
       | id1:(?X1 < ?X2)%Z,id2:(?X2 <= ?X1)%Z |- _ =>
           first [ apply False_ind | apply False_rec ]; apply (Z.lt_irrefl X1);
            apply Z.lt_le_trans with X2; assumption
       end
   end.

Lemma Qhomographic_Qpositive_to_Q_3 : forall (a b c d : Z) (p : Qpositive) (H_Qhomographic_sg_denom_nonzero : Qhomographic_sg_denom_nonzero c d p), (a * d)%Z <> (b * c)%Z -> h_sign a b c d p H_Qhomographic_sg_denom_nonzero = 1%Z -> forall hyp_homographicAcc : homographicAcc (new_a a b c d p H_Qhomographic_sg_denom_nonzero) (new_b a b c d p H_Qhomographic_sg_denom_nonzero) (new_c a b c d p H_Qhomographic_sg_denom_nonzero) (new_d a b c d p H_Qhomographic_sg_denom_nonzero) (new_p a b c d p H_Qhomographic_sg_denom_nonzero), (0 < Z.sgn (new_a a b c d p H_Qhomographic_sg_denom_nonzero + new_b a b c d p H_Qhomographic_sg_denom_nonzero))%Z -> Qhomographic_Qpositive_to_Q a b c d p H_Qhomographic_sg_denom_nonzero = Qpos (Qhomographic_Qpositive_to_Qpositive (new_a a b c d p H_Qhomographic_sg_denom_nonzero) (new_b a b c d p H_Qhomographic_sg_denom_nonzero) (new_c a b c d p H_Qhomographic_sg_denom_nonzero) (new_d a b c d p H_Qhomographic_sg_denom_nonzero) (new_p a b c d p H_Qhomographic_sg_denom_nonzero) hyp_homographicAcc).