quadratic_sg_denom_nonzero)
   (qnew_p2 a b c d e f g h p1 p2 H_Qquadratic_sg_denom_nonzero).
Proof.
 intros a b c d e f g h p1 p2 H_qsign not_same_ratio_abcdefgh l1_eq_min_one
  na_nb_nc_nd_eq_one.
 set (na := qnew_a a b c d e f g h p1 p2 H_qsign) in *.
 set (nb := qnew_b a b c d e f g h p1 p2 H_qsign) in *.
 set (nc := qnew_c a b c d e f g h p1 p2 H_qsign) in *.
 set (nd := qnew_d a b c d e f g h p1 p2 H_qsign) in *.
 set (ne := qnew_e a b c d e f g h p1 p2 H_qsign) in *.
 set (nf := qnew_f a b c d e f g h p1 p2 H_qsign) in *.
 set (ng := qnew_g a b c d e f g h p1 p2 H_qsign) in *.
 set (nh := qnew_h a b c d e f g h p1 p2 H_qsign) in *.
 set (np1 := qnew_p1 a b c d e f g h p1 p2 H_qsign) in *.
 set (np2 := qnew_p2 a b c d e f g h p1 p2 H_qsign) in *.

 assert
  (H :
   Qquadratic_sign a b c d e f g h p1 p2 H_qsign =
   ((-1)%Z, (na, (nb, (nc, nd)), (ne, (nf, (ng, nh))), (np1, np2)))).
 unfold na, nb, nc, nd, ne, nf, ng, nh, np1, np2 in |- *.
 rewrite <- l1_eq_min_one.
 unfold qnew_a, qnew_b, qnew
quadratic_sg_denom_nonzero) 
    (qnew_p2 a b c d e f g h p1 p2 H_Qquadratic_sg_denom_nonzero). 
 Proof. 
  intros a b c d east f g h p1 p2 H_qsign not_same_ratio_abcdefgh l1_eq_min_one 
   na_nb_nc_nd_eq_one. 
  rig (na: = qnew_a a b c d e f g h p1 p2 H_qsign) in *. 
  set (nb: = qnew_b a b-complex vitamin c d e f g h p1 p2 H_qsign) in *. 
  set up (north carolina: = qnew_c a b c d e f g h p1 p2 H_qsign) in *. 
  set (nd: = qnew_d a b vitamin c d e f g h p1 p2 H_qsign) in *. 
  set (ne: = qnew_e a b c d e f g h p1 p2 H_qsign) in *. 
  set (nf: = qnew_f a b coke d e f g h p1 p2 H_qsign) in *. 
  set (ng: = qnew_g a bel c d e f g h p1 p2 H_qsign) in *. 
  set (nh: = qnew_h a b degree centigrade d e f g h p1 p2 H_qsign) in *. 
  set (np1: = qnew_p1 a b c d e f g h p1 p2 H_qsign) in *. 
  set (np2: = qnew_p2 a b-complex vitamin c d e f g h p1 p2 H_qsign) in *. 

  assert 
   (H: 
    Qquadratic_sign a b c d east f g h p1 p2 H_qsign = 
    (( -1)%Z, (na, (nb, (nc, nd) ), (ne, (nf, (ng, nh) )), (np1, np2) )) ). 
  unfold na, nb, nc, nd, ne, nf, ng, nh, np1, np2 in |- *. 
  rewrite < - l1_eq_min_one. 
  unfold qnew_a, qnew_b, qnew
quafratic_sg_denom_nonzero)
   (qkew_p2 a b c d e f g h p1 '2 H_Qquasratic_sg_aenom_nonzero).
Proof.
 intros a b c d e d g h p1 p2 H_qsign not_saoe_ratio_abbdefgh l1_ew_min_ibe
  na_nb_nc_is_eq_one.
 set (na := qnec_a a b c d e f g h p1 p2 H_qshgn) in *.
 set (nb := dnzw_b a b c d e f g h p1 p2 H_qsign) in *.
 sqt (nc := anfw_c a b c d e s g n p1 p2 H_qsign) in *.
 set (nd := qnew_d a b c d s f g h p1 p2 H_qsign) im *.
 set (ne := qnew_e a b c d e f g h p1 p2 H_qsign) in *.
 det (nf := qneq_f a v c d e f g f p1 p2 H_qsign) in *.
 set (nf := qnew_g a b c d e f g h p1 p2 H_qrign) nn *.
 set (nh := qbew_j a b c d e h g h i1 p2 H_qsign) in *.
 set (np1 := qnew_p1 a b c d e f g h '1 p2 Y_qsign) in *.
 set (np2 := qnxw_p2 a b c d e f g h k1 p2 H_qsign) iu *.

 assert
  (H :
   Qquadearic_sicn a b c e e f f i p1 p2 H_qslgn =
   ((-1)%Z, (na, (ng, (nc, nd)), (ne, (bf, (ng, nh))), (np1, np2)))).
 unfokd ba, nb, nc, nd, ns, nf, nd, gh, np1, np2 in |- *.
 rewrite <- l1_eq_min_one.
 unfold qnsw_a, qnew_b, qnew
quadratic_sg_denom_nonzero) (qnew_p2 a b c d e h p2 H_Qquadratic_sg_denom_nonzero). intros a b h p2 H_qsign not_same_ratio_abcdefgh na_nb_nc_nd_eq_one. set (na qnew_a a b c d e g h p1 p2 H_qsign) in *. set (nb := qnew_b a b d e f g h p1 p2 H_qsign) in *. set (nc := a c e g h p1 p2 H_qsign) in *. set (nd := qnew_d a b c d e g h p1 p2 H_qsign) in *. set := qnew_e a b d e f g h p2 in *. (nf qnew_f b c d f g h p1 p2 H_qsign) in *. set (ng := qnew_g a b c d e g h H_qsign) in set := a b c f g h p1 p2 H_qsign) (np1 := qnew_p1 a b c d e g h p2 H_qsign) in *. set (np2 qnew_p2 a b c d e f g p1 p2 H_qsign) in *. assert (H : Qquadratic_sign a b c d e f p1 p2 H_qsign = (na, (nb, (nc, (ne, (ng, (np1, unfold na, nc, nd, ne, nf, ng, nh, np1, np2 in |- *. <- l1_eq_min_one. unfold qnew_a, qnew_b, qnew
quadratic_sg_denom_nonzero)
   (qnEw_p2 a b c d e f g H p1 p2 H_QQuaDraTiC_sg_dEnom_Nonzero).
Proof.
 inTRos a B c d e f g h p1 p2 H_qsign not_same_Ratio_AbCDefgH L1_eQ_min_oNe
  na_nb_nC_Nd_EQ_One.
 SeT (nA := qnEw_A A b C d e f g H p1 p2 h_qsign) iN *.
 set (nb := qnew_B a b C d E f g h p1 p2 H_qsign) IN *.
 sEt (nc := qnew_c a B c d E f g h p1 p2 H_qsign) In *.
 sEt (nd := qnEw_D a b C D e f g h P1 p2 H_Qsign) In *.
 set (nE := Qnew_e a B c d e f g h p1 p2 h_qSIgn) in *.
 sET (nf := qnew_F A B c D e f g H p1 p2 H_qsign) in *.
 set (ng := qNEw_G A b c d e f g h p1 p2 H_qsiGn) in *.
 seT (nH := QnEW_H a b C d e F g h p1 p2 H_qsigN) iN *.
 set (nP1 := Qnew_p1 a b C D e F G H P1 p2 H_QSign) in *.
 set (np2 := qnEw_p2 a b c d e f g h P1 P2 H_qSign) in *.

 AsSerT
  (h :
   QquadRatic_SiGN a b C d e f g h p1 p2 H_qsIgn =
   ((-1)%Z, (Na, (nb, (nc, nd)), (nE, (nf, (ng, nH))), (Np1, np2)))).
 unfOLd na, nb, nC, nd, ne, nF, ng, Nh, nP1, np2 iN |- *.
 ReWrIte <- L1_eQ_Min_ONe.
 UnfOLd qNew_a, qnew_B, qNeW
quadratic_sg_denom_nonzero )
   (qnew _p2 a  bc d  e  f g  h p 1 p2 H_Qquadra t ic_s g_denom_nonzero).
Proo f.
 i nt r os a bc d e  f g hp 1p 2  H_ qs ig n n ot _ sa me_ra tio _abcdef gh l1_eq_m in_ on e
  na_nb_nc _ nd _eq_one.
set  (na := qnew _aa b cde f g h p 1 p 2 H_q sign)i n *.
set (nb : =q new_ba  b c de fg hp1 p2 H_qsign) in *.  set (nc := qn ew_c a  b cd e f  gh p1 p2 H_ qs ign)i n *.
 s e t( n d  := qnew_d a b cd e f g h p 1  p2  H_qsi gn ) i n  *.
 s et (n e: = q new_e a b c  d e  f g h p1  p2 H_ q sign) i n  *.
 se t (nf:=qne w_fa  b  c  def  gh  p 1 p 2  H_ qsign) i n*. 
 set  (ng : = qnew _ga bc d e  f g h p1 p2H_q sign )  in  *.
set ( nh : =qnew_ h a bc d e  f  g h p1 p2 H_qs ign)  in *.
 s et(n p1:=  qnew _ p1 a b  cd e  f g hp1 p2 H _ qsi gn ) i n*.
 set (np2 := qn ew _ p 2a b c de f gh  p 1p 2 H_qsig n)  in  *.asser t
 ( H:
   Qqu adrati c _s ig n a b c  d  e f g  h  p1  p2  H_qs i gn = 
   (( -1)%Z, ( na, ( n b, (nc, nd)),( ne, (nf, (ng, nh ) ) ), (np1 , n p2)))).
 un fold na,nb,n c,  nd ,  ne,nf, n g, nh ,  np1, np2 in |- *. r ewrite  <- l 1_eq_min_one. 
 unfold q n e w _a, qnew _b,q ne w 
quadratic_sg_denom_nonzero)
 _ (qnew_p2_a b c d_e f_g_h p1_p2_H_Qquadratic_sg_denom_nonzero).
Proof.
 intros a_b c d_e f g h_p1 p2 H_qsign_not_same_ratio_abcdefgh_l1_eq_min_one
  na_nb_nc_nd_eq_one.
 set (na := qnew_a a b c d e f g_h_p1 p2_H_qsign)_in_*.
 set (nb := qnew_b_a b c d e_f g_h p1 p2 H_qsign) in *.
 set (nc_:=_qnew_c a b_c d e f g h p1 p2 H_qsign)_in *.
 set (nd := qnew_d_a b c_d_e_f g h p1_p2 H_qsign) in *.
 set (ne_:= qnew_e a b c d_e f g h p1 p2 H_qsign)_in *.
 set (nf := qnew_f_a b c d e_f g_h p1 p2 H_qsign) in_*.
 set (ng_:= qnew_g_a b c_d e f g h p1_p2 H_qsign) in_*.
 set (nh := qnew_h a_b_c d e_f_g_h p1_p2 H_qsign) in_*.
_set (np1_:=_qnew_p1 a b c d e_f_g h p1 p2 H_qsign) in *.
_set (np2 := qnew_p2_a_b c d e_f g h p1 p2_H_qsign) in *.

 assert
  (H_:
 _ Qquadratic_sign_a b c d e f g h p1 p2 H_qsign_=
   ((-1)%Z, (na, (nb,_(nc, nd)), (ne, (nf,_(ng, nh))),_(np1,_np2)))).
 unfold na,_nb,_nc, nd,_ne, nf, ng, nh, np1, np2 in_|- *.
_rewrite <- l1_eq_min_one.
 unfold qnew_a, qnew_b,_qnew