Definition top_more (a b c d : Z) :=
  (c <= a)%Z /\ (d < b)%Z \/ (c < a)%Z /\ (d <= b)%Z.

(* Specifications: *)
Lemma top_more_informative : forall a b c d : Z, {top_more a b c d} + {~ top_more a b c d}.
Lemma top_more_1 : forall a b c d : Z, top_more a b c d -> (0 < a - c + (b - d))%Z.
Lemma top_more_2 : forall a b c d : Z, top_more a b c d -> (c + d < a + b)%Z.
Lemma top_more_3 : forall a b c d : Z, (0 < c + d)%Z -> (a - c + (b - d) < a + b)%Z.
Lemma top_more_4 : forall a b c d : Z, top_more a b c d -> (c <= a)%Z.
Lemma top_more_4' : forall a b c d : Z, top_more a b c d -> (d <= b)%Z.
Lemma top_more_5 : forall a b c d : Z, (0 < c + d)%Z -> (a - c + (b - d) + c + d < a + b + c + d)%Z.
Lemma top_more_5' : forall a b c d : Z, (0 < a + b)%Z -> (a + b + (c - a) + (d - b) < a + b + c + d)%Z.