Ltac Inside_Square_Qsgn_ p1 p2 :=
  repeat
   match goal with
   | id1:(?X1 < ?X2)%Z |- _ =>
       unfold X2 in id1 || unfold X1 in id1;
        repeat
         match goal with
         | id1:(2 < outside_square ?X1 ?X2 ?X3 ?X4)%Z,id2:
         (~ (?X2 = 0%Z /\ ?X3 = 0%Z /\ ?X4 = 0%Z)) |- _ =>
             generalize
              (Qsgn_7 _ (outside_square_correct_1 X1 X2 X3 X4 p1 p2 id1 id2));
              clear id1
         | id1:(outside_square ?X1 ?X2 ?X3 ?X4 < -2)%Z,id2:
         (~ (?X2 = 0%Z /\ ?X3 = 0%Z /\ ?X4 = 0%Z)) |- _ =>
             generalize
              (Qsgn_8 _ (outside_square_correct_2 X1 X2 X3 X4 p1 p2 id1 id2));
              clear id1
         end
   end.

Lemma quadratic_positive_input : forall (a b c d e f g h : Z) (p1 p2 : Qpositive) (H_Qquadratic_sg_denom_nonzero : Qquadratic_sg_denom_nonzero e f g h p1 p2), Qquadratic_Qpositive_to_Q a b c d e f g h p1 p2 H_Qquadratic_sg_denom_nonzero = Qmult (Qplus (Qplus (Qplus (Qmult (Qmult a (Qpos p1)) (Qpos p2)) (Qmult b (Qpos p1))) (Qmult c (Qpos p2))) d) (Qinv (Qplus (Qplus (Qplus (Qmult (Qmult e (Qpos p1)) (Qpos p2)) (Qmult f (Qpos p1))) (Qmult g (Qpos p2))) h)).