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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.                       *)
(*                                                                    *)
(* 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_Qpositive_to_Qpositive.
Definition quadratic_top_more (a b c d e f g h : Z) :=
(e <= a)%Z /\ (f <= b)%Z /\ (g <= c)%Z /\ (h < d)%Z \/
(e <= a)%Z /\ (f <= b)%Z /\ (g < c)%Z /\ (h <= d)%Z \/
(e <= a)%Z /\ (f < b)%Z /\ (g <= c)%Z /\ (h <= d)%Z \/
(e < a)%Z /\ (f <= b)%Z /\ (g <= c)%Z /\ (h <= d)%Z.
intros.
assert
((a + b + c + d + (e - a) + (f - b) + (g - c) + (h - d))%Z =
(0 + e + f + g + h)%Z).
abstract ring.
rewrite H0.
repeat rewrite Zplus_assoc_reverse with (n := (a + b + c + d)%Z).
repeat rewrite Zplus_assoc_reverse with (n := 0%Z).
apply Zplus_lt_le_compat.
assumption.
apply Z.le_refl.
Qed.
Inductive quadraticAcc :
Z -> Z -> Z -> Z -> Z -> Z -> Z -> Z -> Qpositive -> Qpositive -> Prop :=
| quadraticacc0 :
forall (a b c d e f g h : Z) (p1 p2 : Qpositive),
p1 = One ->
homographicAcc (a + c) (b + d) (e + g) (f + h) p2 ->
quadraticAcc a b c d e f g h p1 p2
| quadraticacc0' :
forall (a b c d e f g h : Z) (p1 p2 : Qpositive),
p1 <> One ->
p2 = One ->
homographicAcc (a + b) (c + d) (e + f) (g + h) p1 ->
quadraticAcc a b c d e f g h p1 p2
| quadraticacc1 :
forall (a b c d e f g h : Z) (p1 p2 : Qpositive),
p1 <> One ->
p2 <> One ->
quadratic_top_more a b c d e f g h ->
quadraticAcc (a - e)%Z (b - f)%Z (c - g)%Z (d - h)%Z e f g h p1 p2 ->
quadraticAcc a b c d e f g h p1 p2
| quadraticacc2 :
forall (a b c d e f g h : Z) (p1 p2 : Qpositive),
p1 <> One ->
p2 <> One ->
~ quadratic_top_more a b c d e f g h ->
quadratic_top_more e f g h a b c d ->
quadraticAcc a b c d (e - a)%Z (f - b)%Z (g - c)%Z (h - d)%Z p1 p2 ->
quadraticAcc a b c d e f g h p1 p2
| quadraticacc3_II :
forall (a b c d e f g h : Z) (xs ys : Qpositive),
~ quadratic_top_more a b c d e f g h ->
~ quadratic_top_more e f g h a b c d ->
quadraticAcc a (a + b)%Z (a + c)%Z (a + b + c + d)%Z e
(e + f)%Z (e + g)%Z (e + f + g + h)%Z xs ys ->
quadraticAcc a b c d e f g h (nR xs) (nR ys)
| quadraticacc3_IO :
forall (a b c d e f g h : Z) (xs ys : Qpositive),
~ quadratic_top_more a b c d e f g h ->
~ quadratic_top_more e f g h a b c d ->
quadraticAcc (a + b)%Z b (a + b + c + d)%Z (b + d)%Z
(e + f)%Z f (e + f + g + h)%Z (f + h)%Z xs ys ->
quadraticAcc a b c d e f g h (nR xs) (dL ys)
| quadraticacc3_OI :
forall (a b c d e f g h : Z) (xs ys : Qpositive),
~ quadratic_top_more a b c d e f g h ->
~ quadratic_top_more e f g h a b c d ->
quadraticAcc (a + c)%Z (a + b + c + d)%Z c (c + d)%Z
(e + g)%Z (e + f + g + h)%Z g (g + h)%Z xs ys ->
quadraticAcc a b c d e f g h (dL xs) (nR ys)
| quadraticacc3_OO :
forall (a b c d e f g h : Z) (xs ys : Qpositive),
~ quadratic_top_more a b c d e f g h ->
~ quadratic_top_more e f g h a b c d ->
quadraticAcc (a + b + c + d)%Z (b + d)%Z (c + d)%Z d
(e + f + g + h)%Z (f + h)%Z (g + h)%Z h xs ys ->
quadraticAcc a b c d e f g h (dL xs) (dL ys).
*)
Definition octet_lt (a b c d e f g h : Z) (p1 p2 : Qpositive)
(a' b' c' d' e' f' g' h' : Z) (p1' p2' : Qpositive) : Prop :=
bin_lt p1 p1' \/
p1 = p1' /\
(a + b + c + d + e + f + g + h < a' + b' + c' + d' + e' + f' + g' + h')%Z.
Definition octointegral_lt (a b c d e f g h a' b' c' d' e' f' g' h' : Z) :=
(a + b + c + d + e + f + g + h < a' + b' + c' + d' + e' + f' + g' + h')%Z.
Definition octointegral_eq (a b c d e f g h a' b' c' d' e' f' g' h' : Z) :=
a = a' /\
b = b' /\ c = c' /\ d = d' /\ e = e' /\ f = f' /\ g = g' /\ h = h'.
Record Z8 : Set :=
{z8crr :> Z * Z * (Z * Z) * (Z * Z * (Z * Z));
z8prf :
(0 <= fst (fst (fst z8crr)))%Z /\
(0 <= snd (fst (fst z8crr)))%Z /\
(0 <= fst (snd (fst z8crr)))%Z /\
(0 <= snd (snd (fst z8crr)))%Z /\
(0 <= fst (fst (snd z8crr)))%Z /\
(0 <= snd (fst (snd z8crr)))%Z /\
(0 <= fst (snd (snd z8crr)))%Z /\ (0 <= snd (snd (snd z8crr)))%Z}.
Definition Z8_lt : Z8 -> Z8 -> Prop.
intros x y.
case (z8crr x).
intros W1 W2.
case (z8crr y).
intros W3 W4.
case W1.
intros V1 V2.
case W2.
intros V3 V4.
case W3.
intros U1 U2.
case W4.
intros U3 U4.
case V1.
intros a b.
case V2.
intros c d.
case V3.
intros e f.
case V4.
intros g h.
case U1.
intros a' b'.
case U2.
intros c' d'.
case U3.
intros e' f'.
case U4.
intros g' h'.
exact (octointegral_lt a b c d e f g h a' b' c' d' e' f' g' h').
Defined.
Definition Z8_eq : Z8 -> Z8 -> Prop.
intros x y.
case (z8crr x).
intros W1 W2.
case (z8crr y).
intros W3 W4.
case W1.
intros V1 V2.
case W2.
intros V3 V4.
case W3.
intros U1 U2.
case W4.
intros U3 U4.
case V1.
intros a b.
case V2.
intros c d.
case V3.
intros e f.
case V4.
intros g h.
case U1.
intros a' b'.
case U2.
intros c' d'.
case U3.
intros e' f'.
case U4.
intros g' h'.
exact (octointegral_eq a b c d e f g h a' b' c' d' e' f' g' h').
Defined.

Lemma Z8_lt_is_transitive :
forall x y z : Z8, Z8_lt x y -> Z8_lt y z -> Z8_lt x z.