(* This program is free software; you can redistribute it and/or      *)
(* 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 Qhomographic_Qpositive_to_Qpositive.
Require Export Qhomographic_sign_properties.
Definition new_a (a b c d : Z) (p : Qpositive)
(H_Qhomographic_sg_denom_nonzero : Qhomographic_sg_denom_nonzero c d p) :=
fst
(fst (snd (Qhomographic_sign a b c d p H_Qhomographic_sg_denom_nonzero))).
Definition new_b (a b c d : Z) (p : Qpositive)
(H_Qhomographic_sg_denom_nonzero : Qhomographic_sg_denom_nonzero c d p) :=
fst
(snd
(fst
(snd (Qhomographic_sign a b c d p H_Qhomographic_sg_denom_nonzero)))).
Definition new_c (a b c d : Z) (p : Qpositive)
(H_Qhomographic_sg_denom_nonzero : Qhomographic_sg_denom_nonzero c d p) :=
fst
(snd
(snd
(fst
(snd
(Qhomographic_sign a b c d p H_Qhomographic_sg_denom_nonzero))))).
Definition new_d (a b c d : Z) (p : Qpositive)
(H_Qhomographic_sg_denom_nonzero : Qhomographic_sg_denom_nonzero c d p) :=
snd
(snd
(snd
(fst
(snd
(Qhomographic_sign a b c d p H_Qhomographic_sg_denom_nonzero))))).
Definition new_p (a b c d : Z) (p : Qpositive)
(H_Qhomographic_sg_denom_nonzero : Qhomographic_sg_denom_nonzero c d p) :=
snd (snd (Qhomographic_sign a b c d p H_Qhomographic_sg_denom_nonzero)).

Lemma Qhomographic_Qpositive_to_Q_homographicAcc_pos_1 :
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 ->
(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 ->
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).