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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.             *)
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(* 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.                       *)
(*                                                                    *)
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(* Software Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA *)
(* 02110-1301 USA                                                     *)
Definition outside_square (a b c d : Z) :=
(Z.sgn a + Z.sgn b + Z.sgn c + Z.sgn d)%Z.
Inductive Qquadratic_sg_denom_nonzero :
Z -> Z -> Z -> Z -> Qpositive -> Qpositive -> Prop :=
| Qquadratic_signok0 :
forall (e f g h : Z) (p1 p2 : Qpositive),
p2 = One ->
Qhomographic_sg_denom_nonzero (e + f) (g + h) p1 ->
Qquadratic_sg_denom_nonzero e f g h p1 p2
| Qquadratic_signok0' :
forall (e f g h : Z) (p1 p2 : Qpositive),
p1 = One ->
Qhomographic_sg_denom_nonzero (e + g) (f + h) p2 ->
Qquadratic_sg_denom_nonzero e f g h p1 p2
| Qquadratic_signok1 :
forall (e f g h : Z) (p1 p2 : Qpositive),
Qquadratic_sg_denom_nonzero e (e + f)%Z (e + g)%Z
(e + f + g + h)%Z p1 p2 ->
Qquadratic_sg_denom_nonzero e f g h (nR p1) (nR p2)
| Qquadratic_signok2 :
forall (e f g h : Z) (p1 p2 : Qpositive),
Qquadratic_sg_denom_nonzero (e + f)%Z f (e + f + g + h)%Z
(f + h)%Z p1 p2 ->
Qquadratic_sg_denom_nonzero e f g h (nR p1) (dL p2)
| Qquadratic_signok3 :
forall (e f g h : Z) (p1 p2 : Qpositive),
Qquadratic_sg_denom_nonzero (e + g)%Z (e + f + g + h)%Z g
(g + h)%Z p1 p2 ->
Qquadratic_sg_denom_nonzero e f g h (dL p1) (nR p2)
| Qquadratic_signok4 :
forall (e f g h : Z) (p1 p2 : Qpositive),
Qquadratic_sg_denom_nonzero (e + f + g + h)%Z
(f + h)%Z (g + h)%Z h p1 p2 ->
Qquadratic_sg_denom_nonzero e f g h (dL p1) (dL p2).

Lemma Qquadratic_sign_nRdL_nRdL_3 :
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),
p1 <> One ->
p2 <> One ->
b = 0%Z /\ c = 0%Z /\ d = 0%Z ->
~ (f = 0%Z /\ g = 0%Z /\ h = 0%Z) ->
~ (2 < outside_square e f g h)%Z ->
(outside_square e f g h < -2)%Z ->
Qquadratic_sign a b c d e f g h p1 p2 H_Qquadratic_sg_denom_nonzero =
((- Z.sgn a)%Z, (a, (b, (c, d)), (e, (f, (g, h))), (p1, p2))).