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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   *)
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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.

(* Specifications: *)
Lemma octet_leq_inf :
forall 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)}.
Lemma quadratic_top_more_2 :
forall a b c d e f g h : Z,
quadratic_top_more a b c d e f g h -> (e + f + g + h < a + b + c + d)%Z.
Lemma quadratic_top_more_3 :
forall a b c d e f g h : Z,
(0 < e + f + g + h)%Z ->
(a - e + (b - f) + (c - g) + (d - h) < a + b + c + d)%Z.
Lemma quadratic_top_more_4_1 :
forall a b c d e f g h : Z, quadratic_top_more a b c d e f g h -> (e <= a)%Z.
Lemma quadratic_top_more_4_2 :
forall a b c d e f g h : Z, quadratic_top_more a b c d e f g h -> (f <= b)%Z.
Lemma quadratic_top_more_4_3 :
forall a b c d e f g h : Z, quadratic_top_more a b c d e f g h -> (g <= c)%Z.
Lemma quadratic_top_more_4_4 :
forall a b c d e f g h : Z, quadratic_top_more a b c d e f g h -> (h <= d)%Z.
Lemma quadratic_top_more_5 :
forall a b c d e f g h : Z,
(0 < e + f + g + h)%Z ->
(a - e + (b - f) + (c - g) + (d - h) + e + f + g + h <
a + b + c + d + e + f + g + h)%Z.
Lemma quadratic_top_more_5' :
forall a b c d e f g h : Z,
(0 < a + b + c + d)%Z ->
(a + b + c + d + (e - a) + (f - b) + (g - c) + (h - d) <
a + b + c + d + e + f + g + h)%Z.