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(* modify it under the terms of the GNU Lesser General Public License *)
(* as published by the Free Software Foundation; either version 2.1   *)
(* of the License, or (at your option) any later version.             *)
(*                                                                    *)
(* This program is distributed in the hope that it will be useful,    *)
(* but WITHOUT ANY WARRANTY; without even the implied warranty of     *)
(* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the      *)
(* GNU General Public License for more details.                       *)
(*                                                                    *)
(* You should have received a copy of the GNU Lesser General Public   *)
(* License along with this program; if not, write to the Free         *)
(* Software Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA *)
(* 02110-1301 USA                                                     *)
Require Export Qhomographic_sign.
Require Export Qquadratic_sign.
Require Export Qhomographic_sign_properties.
Definition same_ratio (a b c d e f g h : Z) :=
(a * f)%Z = (b * e)%Z /\
(b * g)%Z = (c * f)%Z /\
(c * h)%Z = (d * g)%Z /\
(a * g)%Z = (c * e)%Z /\ (a * h)%Z = (d * e)%Z /\ (b * h)%Z = (d * f)%Z.
(* Proof by case analysis: 81 cases in 6 line :-) *)

Lemma Qquadratic_sign_neg_2 :
forall (a b c d e f g h : Z) (p1 p2 : Qpositive)
(H_Qquadratic_sg_denom_nonzero : Qquadratic_sg_denom_nonzero e f g h p1 p2)
(na nb nc nd ne nf ng nh : Z) (np1 np2 : Qpositive),
Qquadratic_sign a b c d e f g h p1 p2 H_Qquadratic_sg_denom_nonzero =
((-1)%Z, (na, (nb, (nc, nd)), (ne, (nf, (ng, nh))), (np1, np2))) ->
{(0 <= na)%Z /\
(0 <= nb)%Z /\
(0 <= nc)%Z /\
(0 <= nd)%Z /\ (ne <= 0)%Z /\ (nf <= 0)%Z /\ (ng <= 0)%Z /\ (nh <= 0)%Z} +
{(na <= 0)%Z /\
(nb <= 0)%Z /\
(nc <= 0)%Z /\
(nd <= 0)%Z /\ (0 <= ne)%Z /\ (0 <= nf)%Z /\ (0 <= ng)%Z /\ (0 <= nh)%Z} +
{np1 = One /\
(0 <= na + nc)%Z /\
(0 <= nb + nd)%Z /\ (ne + ng <= 0)%Z /\ (nf + nh <= 0)%Z} +
{np1 = One /\
(na + nc <= 0)%Z /\
(nb + nd <= 0)%Z /\ (0 <= ne + ng)%Z /\ (0 <= nf + nh)%Z} +
{np1 = One /\ np2 = One} +
{np1 <> One /\
np2 = One /\
(0 <= na + nb)%Z /\
(0 <= nc + nd)%Z /\ (ne + nf <= 0)%Z /\ (ng + nh <= 0)%Z} +
{np1 <> One /\
np2 = One /\
(na + nb <= 0)%Z /\
(nc + nd <= 0)%Z /\ (0 <= ne + nf)%Z /\ (0 <= ng + nh)%Z}.