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_informative : forall a b c d e f g h : Z, {quadratic_top_more a b c d e f g h} + {~ quadratic_top_more a b c d e f g h}.
Lemma quadratic_top_more_1 : forall a b c d e f g h : Z, quadratic_top_more a b c d e f g h -> (0 < a - e + (b - f) + (c - g) + (d - h))%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.