Ltac QSign_mismatch_ :=
  apply False_ind;
   match goal with
   | id1:(?X1 = 1%Z),id2:(?X1 = 0%Z) |- _ =>
       rewrite id2 in id1; discriminate id1
   | id1:(?X1 = (-1)%Z),id2:(?X1 = 0%Z) |- _ =>
       rewrite id2 in id1; discriminate id1
   | id1:(?X1 = 1%Z),id2:(?X1 = (-1)%Z) |- _ =>
       rewrite id2 in id1; discriminate id1

   | id2:(?X1 = 0%Z),id1:(?X1 = 1%Z) |- _ =>
       rewrite id2 in id1; discriminate id1
   | id1:(?X1 = 0%Z),id2:(?X1 = (-1)%Z) |- _ =>
       rewrite id2 in id1; discriminate id1
   | id1:(?X1 = (-1)%Z),id2:(?X1 = 1%Z) |- _ =>
       rewrite id2 in id1; discriminate id1

   end.(* The tactique Absurd_q_sign_ solves the goal by contradiction if in the context we have:

(C1)  id1:(q_sign a b c d e f g h p1 p2 H)=1
     id2:(Z.sgn new_a+new_b+new_c+new_d)=0 
     ----------------------------------
   
or 

(C2)  id1:(q_sign a b c d e f g h p1 p2 H)=1
     id2:(Z.sgn new_e+new_f+new_g+new_h)=0 
     ----------------------------------

or

(C3)  id1:(q_sign a b c d e f g h p1 p2 H)=-1
     id2:(Z.sgn new_a+new_b+new_c+new_d)=0 
     ----------------------------------
   
or 

(C4)  id1:(q_sign a b c d e f g h p1 p2 H)=-1
     id2:(Z.sgn new_e+new_f+new_g+new_h)=0 
     ----------------------------------
   
otherwise if in the context we have 

(C5)  id1:(q_sign a b c d e f g h p1 p2 H)=0
     id2:(Z.sgn ?)=0 
     ----------------------------------

it calls the QSign_mismatch_ tactic.

and otherwise it fails.

In the next files we used this tactic 4 times and all of them are instances of case (C1) or (C3) above.
*)

Ltac Absurd_q_sign_ :=
  match goal with
  | id1:(q_sign ?X1 ?X2 ?X3 ?X4 ?X5 ?X6 ?X7 ?X8 ?X9 ?X10 ?X11 = ?X0),id2:
  (Z.sgn _ = 0%Z) |- _ =>
      match constr:(X0) with
      | 0%Z => QSign_mismatch_
      end ||
        (apply False_ind;
          assert
           (HH :
            Qquadratic_sign X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11 =
            (X0,
            (qnew_a X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11,
            (qnew_b X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11,
            (qnew_c X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11,
            qnew_d X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11)),
            (qnew_e X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11,
            (qnew_f X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11,
            (qnew_g X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11,
            qnew_h X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11))),
            (qnew_p1 X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11,
            qnew_p2 X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11))));
          [ rewrite <- id1;
             unfold qnew_a, qnew_b, qnew_c, qnew_d, qnew_e, qnew_f, qnew_g,
              qnew_h, qnew_p1, qnew_p2 in |- *;
             replace (q_sign X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11) with
              (fst (Qquadratic_sign X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11));
             [ idtac | reflexivity ]; repeat rewrite <- pair_1; 
             reflexivity
          | idtac ];
          match constr:(X0) with
          | 1%Z =>
              generalize
               (Qquadratic_sign_pos_1 X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11
                  (qnew_a X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11)
                  (qnew_b X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11)
                  (qnew_c X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11)
                  (qnew_d X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11)
                  (qnew_e X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11)
                  (qnew_f X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11)
                  (qnew_g X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11)
                  (qnew_h X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11)
                  (qnew_p1 X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11)
                  (qnew_p2 X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11) HH)
          | (-1)%Z =>
              generalize
               (Qquadratic_sign_neg_1 X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11
                  (qnew_a X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11)
                  (qnew_b X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11)
                  (qnew_c X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11)
                  (qnew_d X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11)
                  (qnew_e X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11)
                  (qnew_f X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11)
                  (qnew_g X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11)
                  (qnew_h X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11)
                  (qnew_p1 X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11)
                  (qnew_p2 X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11) HH)
          end; intros [(na_nb_nc_nd_pos, _)| (na_nb_nc_nd_neg, _)];
          generalize (Zsgn_2 _ id2); [ apply sym_not_eq | idtac ];
          apply Zorder.Zlt_not_eq; assumption)
  end.

Lemma Qquadratic_Qpositive_to_Q_4 : 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), ~ same_ratio a b c d e f g h -> q_sign a b c d e f g h p1 p2 H_Qquadratic_sg_denom_nonzero = 0%Z -> Qquadratic_Qpositive_to_Q a b c d e f g h p1 p2 H_Qquadratic_sg_denom_nonzero = Zero.