Fixpoint Qquadratic_sign (a b c d e f g h : Z) (p1 p2 : Qpositive) {struct p1} :
  forall (H_Qquadratic_sg_denom_nonzero : Qquadratic_sg_denom_nonzero e f g h p1 p2),
  Z * (Z * (Z * (Z * Z)) * (Z * (Z * (Z * Z))) * (Qpositive * Qpositive)).Scheme Qquadratic_sg_denom_nonzero_inv_dep :=
  Induction for Qquadratic_sg_denom_nonzero Sort Prop.

Lemma Qquadratic_sign_dL_dL_4 : forall (a b c d e f g h : Z) (p1 xs p2 ys : Qpositive) (H_Qquadratic_sg_denom_nonzero : Qquadratic_sg_denom_nonzero e f g h p1 p2) (H_Qquadratic_sg_denom_nonzero_OO : Qquadratic_sg_denom_nonzero (e + f + g + h) (f + h) (g + h) h xs ys), p1 = dL xs -> p2 = dL ys -> b = 0%Z /\ c = 0%Z /\ d = 0%Z -> ~ (f = 0%Z /\ g = 0%Z /\ h = 0%Z) -> ~ (2 < outside_square e f g h)%Z -> ~ (outside_square e f g h < -2)%Z -> Qquadratic_sign a b c d e f g h p1 p2 H_Qquadratic_sg_denom_nonzero = Qquadratic_sign (a + b + c + d) (b + d) (c + d) d (e + f + g + h) (f + h) (g + h) h xs ys H_Qquadratic_sg_denom_nonzero_OO.