(* theory collecting all knowledge 

    (predicates 'is_rootEq_in', 'is_sqrt_in', 'is_ratEq_in')

    for PolynomialEquations.

    alternative dependencies see @{theory "Isac"}

    created by: rlang 

          date: 02.07

    changed by: rlang

    last change by: rlang

              date: 03.06.03

    (c) by Richard Lang, 2003

 *)

 theory PolyEq imports LinEq RootRatEq begin 

 consts

 (*---------scripts--------------------------*)

   Complete'_square

              :: "[bool,real, 

 		   bool list] => bool list"

                ("((Script Complete'_square (_ _ =))// 

                   (_))" 9)

  (*----- poly ----- *)	 

   Normalize'_poly

              :: "[bool,real, 

 		   bool list] => bool list"

                ("((Script Normalize'_poly (_ _=))// 

                   (_))" 9)

   Solve'_d0'_polyeq'_equation

              :: "[bool,real, 

 		   bool list] => bool list"

                ("((Script Solve'_d0'_polyeq'_equation (_ _ =))// 

                   (_))" 9)

   Solve'_d1'_polyeq'_equation

              :: "[bool,real, 

 		   bool list] => bool list"

                ("((Script Solve'_d1'_polyeq'_equation (_ _ =))// 

                   (_))" 9)

   Solve'_d2'_polyeq'_equation

              :: "[bool,real, 

 		   bool list] => bool list"

                ("((Script Solve'_d2'_polyeq'_equation (_ _ =))// 

                   (_))" 9)

   Solve'_d2'_polyeq'_sqonly'_equation

              :: "[bool,real, 

 		   bool list] => bool list"

                ("((Script Solve'_d2'_polyeq'_sqonly'_equation (_ _ =))// 

                   (_))" 9)

   Solve'_d2'_polyeq'_bdvonly'_equation

              :: "[bool,real, 

 		   bool list] => bool list"

                ("((Script Solve'_d2'_polyeq'_bdvonly'_equation (_ _ =))// 

                   (_))" 9)

   Solve'_d2'_polyeq'_pq'_equation

              :: "[bool,real, 

 		   bool list] => bool list"

                ("((Script Solve'_d2'_polyeq'_pq'_equation (_ _ =))// 

                   (_))" 9)

   Solve'_d2'_polyeq'_abc'_equation

              :: "[bool,real, 

 		   bool list] => bool list"

                ("((Script Solve'_d2'_polyeq'_abc'_equation (_ _ =))// 

                   (_))" 9)

   Solve'_d3'_polyeq'_equation

              :: "[bool,real, 

 		   bool list] => bool list"

                ("((Script Solve'_d3'_polyeq'_equation (_ _ =))// 

                   (_))" 9)

   Solve'_d4'_polyeq'_equation

              :: "[bool,real, 

 		   bool list] => bool list"

                ("((Script Solve'_d4'_polyeq'_equation (_ _ =))// 

                   (_))" 9)

   Biquadrat'_poly

              :: "[bool,real, 

 		   bool list] => bool list"

                ("((Script Biquadrat'_poly (_ _=))// 

                   (_))" 9)

 (*-------------------- rules -------------------------------------------------*)

 (* type real enforced by op "^^^" *)

 axiomatization where

   cancel_leading_coeff1: "Not (c =!= 0) ==> (a + b*bdv + c*bdv^^^2 = 0) = 

 			                   (a/c + b/c*bdv + bdv^^^2 = 0)" and

   cancel_leading_coeff2: "Not (c =!= 0) ==> (a - b*bdv + c*bdv^^^2 = 0) = 

 			                   (a/c - b/c*bdv + bdv^^^2 = 0)" and

   cancel_leading_coeff3: "Not (c =!= 0) ==> (a + b*bdv - c*bdv^^^2 = 0) = 

 			                   (a/c + b/c*bdv - bdv^^^2 = 0)" and

   cancel_leading_coeff4: "Not (c =!= 0) ==> (a +   bdv + c*bdv^^^2 = 0) = 

 			                   (a/c + 1/c*bdv + bdv^^^2 = 0)" and

   cancel_leading_coeff5: "Not (c =!= 0) ==> (a -   bdv + c*bdv^^^2 = 0) = 

 			                   (a/c - 1/c*bdv + bdv^^^2 = 0)" and

   cancel_leading_coeff6: "Not (c =!= 0) ==> (a +   bdv - c*bdv^^^2 = 0) = 

 			                   (a/c + 1/c*bdv - bdv^^^2 = 0)" and

   cancel_leading_coeff7: "Not (c =!= 0) ==> (    b*bdv + c*bdv^^^2 = 0) = 

 			                   (    b/c*bdv + bdv^^^2 = 0)" and

   cancel_leading_coeff8: "Not (c =!= 0) ==> (    b*bdv - c*bdv^^^2 = 0) = 

 			                   (    b/c*bdv - bdv^^^2 = 0)" and

   cancel_leading_coeff9: "Not (c =!= 0) ==> (      bdv + c*bdv^^^2 = 0) = 

 			                   (      1/c*bdv + bdv^^^2 = 0)" and

   cancel_leading_coeff10:"Not (c =!= 0) ==> (      bdv - c*bdv^^^2 = 0) = 

 			                   (      1/c*bdv - bdv^^^2 = 0)" and

   cancel_leading_coeff11:"Not (c =!= 0) ==> (a +      b*bdv^^^2 = 0) = 

 			                   (a/b +      bdv^^^2 = 0)" and

   cancel_leading_coeff12:"Not (c =!= 0) ==> (a -      b*bdv^^^2 = 0) = 

 			                   (a/b -      bdv^^^2 = 0)" and

   cancel_leading_coeff13:"Not (c =!= 0) ==> (         b*bdv^^^2 = 0) = 

 			                   (           bdv^^^2 = 0/b)" and

   complete_square1:      "(q + p*bdv + bdv^^^2 = 0) = 

 		         (q + (p/2 + bdv)^^^2 = (p/2)^^^2)" and

   complete_square2:      "(    p*bdv + bdv^^^2 = 0) = 

 		         (    (p/2 + bdv)^^^2 = (p/2)^^^2)" and

   complete_square3:      "(      bdv + bdv^^^2 = 0) = 

 		         (    (1/2 + bdv)^^^2 = (1/2)^^^2)" and

   complete_square4:      "(q - p*bdv + bdv^^^2 = 0) = 

 		         (q + (p/2 - bdv)^^^2 = (p/2)^^^2)" and

   complete_square5:      "(q + p*bdv - bdv^^^2 = 0) = 

 		         (q + (p/2 - bdv)^^^2 = (p/2)^^^2)" and

   square_explicit1:      "(a + b^^^2 = c) = ( b^^^2 = c - a)" and

   square_explicit2:      "(a - b^^^2 = c) = (-(b^^^2) = c - a)" and

   (*bdv_explicit* required type constrain to real in --- (-8 - 2*x + x^^^2 = 0),  by rewriting ---*)

   bdv_explicit1:         "(a + bdv = b) = (bdv = - a + (b::real))" and

   bdv_explicit2:         "(a - bdv = b) = ((-1)*bdv = - a + (b::real))" and

   bdv_explicit3:         "((-1)*bdv = b) = (bdv = (-1)*(b::real))" and

   plus_leq:              "(0 <= a + b) = ((-1)*b <= a)"(*Isa?*) and

   minus_leq:             "(0 <= a - b) = (     b <= a)"(*Isa?*) and

 (*-- normalize --*)

   (*WN0509 compare LinEq.all_left "[|Not(b=!=0)|] ==> (a=b) = (a+(-1)*b=0)"*)

   all_left:              "[|Not(b=!=0)|] ==> (a = b) = (a - b = 0)" and

   makex1_x:              "a^^^1  = a"   and

   real_assoc_1:          "a+(b+c) = a+b+c" and

   real_assoc_2:          "a*(b*c) = a*b*c" and

 (* ---- degree 0 ----*)

   d0_true:               "(0=0) = True" and

   d0_false:              "[|Not(bdv occurs_in a);Not(a=0)|] ==> (a=0) = False" and

 (* ---- degree 1 ----*)

   d1_isolate_add1:

    "[|Not(bdv occurs_in a)|] ==> (a + b*bdv = 0) = (b*bdv = (-1)*a)" and

   d1_isolate_add2:

    "[|Not(bdv occurs_in a)|] ==> (a +   bdv = 0) = (  bdv = (-1)*a)" and

   d1_isolate_div:

    "[|Not(b=0);Not(bdv occurs_in c)|] ==> (b*bdv = c) = (bdv = c/b)" and

 (* ---- degree 2 ----*)

   d2_isolate_add1:

    "[|Not(bdv occurs_in a)|] ==> (a + b*bdv^^^2=0) = (b*bdv^^^2= (-1)*a)" and

   d2_isolate_add2:

    "[|Not(bdv occurs_in a)|] ==> (a +   bdv^^^2=0) = (  bdv^^^2= (-1)*a)" and

   d2_isolate_div:

    "[|Not(b=0);Not(bdv occurs_in c)|] ==> (b*bdv^^^2=c) = (bdv^^^2=c/b)" and

   d2_prescind1:          "(a*bdv + b*bdv^^^2 = 0) = (bdv*(a +b*bdv)=0)" and

   d2_prescind2:          "(a*bdv +   bdv^^^2 = 0) = (bdv*(a +  bdv)=0)" and

   d2_prescind3:          "(  bdv + b*bdv^^^2 = 0) = (bdv*(1+b*bdv)=0)" and

   d2_prescind4:          "(  bdv +   bdv^^^2 = 0) = (bdv*(1+  bdv)=0)" and

   (* eliminate degree 2 *)

   (* thm for neg arguments in sqroot have postfix _neg *)

   d2_sqrt_equation1:     "[|(0<=c);Not(bdv occurs_in c)|] ==> 

                          (bdv^^^2=c) = ((bdv=sqrt c) | (bdv=(-1)*sqrt c ))" and

  d2_sqrt_equation1_neg:

   "[|(c<0);Not(bdv occurs_in c)|] ==> (bdv^^^2=c) = False" and

   d2_sqrt_equation2:     "(bdv^^^2=0) = (bdv=0)" and

   d2_sqrt_equation3:     "(b*bdv^^^2=0) = (bdv=0)"

 axiomatization where (*AK..if replaced by "and" we get errors:

   exception PTREE "nth _ []" raised 

   (line 783 of "/usr/local/isabisac/src/Tools/isac/Interpret/ctree.sml"):

     'fun nth _ []      = raise PTREE "nth _ []"'

 and

   exception Bind raised 

   (line 1097 of "/usr/local/isabisac/test/Tools/isac/Frontend/interface.sml"):

     'val (Form f, tac, asms) = pt_extract (pt, p);' *)

   (* WN120315 these 2 thms need "::real", because no "^^^" constrains type as

      required in test --- rls d2_polyeq_bdv_only_simplify --- *)

   d2_reduce_equation1:   "(bdv*(a +b*bdv)=0) = ((bdv=0)|(a+b*bdv=(0::real)))" and

   d2_reduce_equation2:   "(bdv*(a +  bdv)=0) = ((bdv=0)|(a+  bdv=(0::real)))"

 axiomatization where (*..if replaced by "and" we get errors:

   exception PTREE "nth _ []" raised 

   (line 783 of "/usr/local/isabisac/src/Tools/isac/Interpret/ctree.sml"):

     'fun nth _ []      = raise PTREE "nth _ []"'

 and

   exception Bind raised 

   (line 1097 of "/usr/local/isabisac/test/Tools/isac/Frontend/interface.sml"):

     'val (Form f, tac, asms) = pt_extract (pt, p);' *)

 d2_pqformula1:         "[|0<=p^^^2 - 4*q|] ==> (q+p*bdv+   bdv^^^2=0) =

                            ((bdv= (-1)*(p/2) + sqrt(p^^^2 - 4*q)/2) 

                           | (bdv= (-1)*(p/2) - sqrt(p^^^2 - 4*q)/2))" and

   d2_pqformula1_neg:     "[|p^^^2 - 4*q<0|] ==> (q+p*bdv+   bdv^^^2=0) = False" and

   d2_pqformula2:         "[|0<=p^^^2 - 4*q|] ==> (q+p*bdv+1*bdv^^^2=0) = 

                            ((bdv= (-1)*(p/2) + sqrt(p^^^2 - 4*q)/2) 

                           | (bdv= (-1)*(p/2) - sqrt(p^^^2 - 4*q)/2))" and

   d2_pqformula2_neg:     "[|p^^^2 - 4*q<0|] ==> (q+p*bdv+1*bdv^^^2=0) = False" and

   d2_pqformula3:         "[|0<=1 - 4*q|] ==> (q+  bdv+   bdv^^^2=0) = 

                            ((bdv= (-1)*(1/2) + sqrt(1 - 4*q)/2) 

                           | (bdv= (-1)*(1/2) - sqrt(1 - 4*q)/2))" and

   d2_pqformula3_neg:     "[|1 - 4*q<0|] ==> (q+  bdv+   bdv^^^2=0) = False" and

   d2_pqformula4:         "[|0<=1 - 4*q|] ==> (q+  bdv+1*bdv^^^2=0) = 

                            ((bdv= (-1)*(1/2) + sqrt(1 - 4*q)/2) 

                           | (bdv= (-1)*(1/2) - sqrt(1 - 4*q)/2))" and

   d2_pqformula4_neg:     "[|1 - 4*q<0|] ==> (q+  bdv+1*bdv^^^2=0) = False" and

   d2_pqformula5:         "[|0<=p^^^2 - 0|] ==> (  p*bdv+   bdv^^^2=0) =

                            ((bdv= (-1)*(p/2) + sqrt(p^^^2 - 0)/2) 

                           | (bdv= (-1)*(p/2) - sqrt(p^^^2 - 0)/2))" and

  (* d2_pqformula5_neg not need p^2 never less zero in R *)

   d2_pqformula6:         "[|0<=p^^^2 - 0|] ==> (  p*bdv+1*bdv^^^2=0) = 

                            ((bdv= (-1)*(p/2) + sqrt(p^^^2 - 0)/2) 

                           | (bdv= (-1)*(p/2) - sqrt(p^^^2 - 0)/2))" and

   (* d2_pqformula6_neg not need p^2 never less zero in R *)

    d2_pqformula7:        "[|0<=1 - 0|] ==> (    bdv+   bdv^^^2=0) = 

                            ((bdv= (-1)*(1/2) + sqrt(1 - 0)/2) 

                           | (bdv= (-1)*(1/2) - sqrt(1 - 0)/2))" and

   (* d2_pqformula7_neg not need, because 1<0 ==> False*)

   d2_pqformula8:        "[|0<=1 - 0|] ==> (    bdv+1*bdv^^^2=0) = 

                            ((bdv= (-1)*(1/2) + sqrt(1 - 0)/2) 

                           | (bdv= (-1)*(1/2) - sqrt(1 - 0)/2))" and

   (* d2_pqformula8_neg not need, because 1<0 ==> False*)

   d2_pqformula9:        "[|Not(bdv occurs_in q); 0<= (-1)*4*q|] ==> 

                            (q+    1*bdv^^^2=0) = ((bdv= 0 + sqrt(0 - 4*q)/2) 

                                                 | (bdv= 0 - sqrt(0 - 4*q)/2))" and

   d2_pqformula9_neg:

    "[|Not(bdv occurs_in q); (-1)*4*q<0|] ==> (q+    1*bdv^^^2=0) = False" and

   d2_pqformula10:

    "[|Not(bdv occurs_in q); 0<= (-1)*4*q|] ==> (q+     bdv^^^2=0) = 

            ((bdv= 0 + sqrt(0 - 4*q)/2) 

           | (bdv= 0 - sqrt(0 - 4*q)/2))" and

   d2_pqformula10_neg:

    "[|Not(bdv occurs_in q); (-1)*4*q<0|] ==> (q+     bdv^^^2=0) = False" and

   d2_abcformula1:

    "[|0<=b^^^2 - 4*a*c|] ==> (c + b*bdv+a*bdv^^^2=0) =

            ((bdv=( -b + sqrt(b^^^2 - 4*a*c))/(2*a)) 

           | (bdv=( -b - sqrt(b^^^2 - 4*a*c))/(2*a)))" and

   d2_abcformula1_neg:

    "[|b^^^2 - 4*a*c<0|] ==> (c + b*bdv+a*bdv^^^2=0) = False" and

   d2_abcformula2:

    "[|0<=1 - 4*a*c|]     ==> (c+    bdv+a*bdv^^^2=0) = 

            ((bdv=( -1 + sqrt(1 - 4*a*c))/(2*a)) 

           | (bdv=( -1 - sqrt(1 - 4*a*c))/(2*a)))" and

   d2_abcformula2_neg:

    "[|1 - 4*a*c<0|]     ==> (c+    bdv+a*bdv^^^2=0) = False" and

   d2_abcformula3:

    "[|0<=b^^^2 - 4*1*c|] ==> (c + b*bdv+  bdv^^^2=0) =

            ((bdv=( -b + sqrt(b^^^2 - 4*1*c))/(2*1)) 

           | (bdv=( -b - sqrt(b^^^2 - 4*1*c))/(2*1)))" and

   d2_abcformula3_neg:

    "[|b^^^2 - 4*1*c<0|] ==> (c + b*bdv+  bdv^^^2=0) = False" and

   d2_abcformula4:

    "[|0<=1 - 4*1*c|] ==> (c +   bdv+  bdv^^^2=0) =

            ((bdv=( -1 + sqrt(1 - 4*1*c))/(2*1)) 

           | (bdv=( -1 - sqrt(1 - 4*1*c))/(2*1)))" and

   d2_abcformula4_neg:

    "[|1 - 4*1*c<0|] ==> (c +   bdv+  bdv^^^2=0) = False" and

   d2_abcformula5:

    "[|Not(bdv occurs_in c); 0<=0 - 4*a*c|] ==> (c +  a*bdv^^^2=0) =

            ((bdv=( 0 + sqrt(0 - 4*a*c))/(2*a)) 

           | (bdv=( 0 - sqrt(0 - 4*a*c))/(2*a)))" and

   d2_abcformula5_neg:

    "[|Not(bdv occurs_in c); 0 - 4*a*c<0|] ==> (c +  a*bdv^^^2=0) = False" and

   d2_abcformula6:

    "[|Not(bdv occurs_in c); 0<=0 - 4*1*c|]     ==> (c+    bdv^^^2=0) = 

            ((bdv=( 0 + sqrt(0 - 4*1*c))/(2*1)) 

           | (bdv=( 0 - sqrt(0 - 4*1*c))/(2*1)))" and

   d2_abcformula6_neg:

    "[|Not(bdv occurs_in c); 0 - 4*1*c<0|]     ==> (c+    bdv^^^2=0) = False" and

   d2_abcformula7:

    "[|0<=b^^^2 - 0|]     ==> (    b*bdv+a*bdv^^^2=0) = 

            ((bdv=( -b + sqrt(b^^^2 - 0))/(2*a)) 

           | (bdv=( -b - sqrt(b^^^2 - 0))/(2*a)))" and

   (* d2_abcformula7_neg not need b^2 never less zero in R *)

   d2_abcformula8:

    "[|0<=b^^^2 - 0|] ==> (    b*bdv+  bdv^^^2=0) =

            ((bdv=( -b + sqrt(b^^^2 - 0))/(2*1)) 

           | (bdv=( -b - sqrt(b^^^2 - 0))/(2*1)))" and

   (* d2_abcformula8_neg not need b^2 never less zero in R *)

   d2_abcformula9:

    "[|0<=1 - 0|]     ==> (      bdv+a*bdv^^^2=0) = 

            ((bdv=( -1 + sqrt(1 - 0))/(2*a)) 

           | (bdv=( -1 - sqrt(1 - 0))/(2*a)))" and

   (* d2_abcformula9_neg not need, because 1<0 ==> False*)

   d2_abcformula10:

    "[|0<=1 - 0|] ==> (      bdv+  bdv^^^2=0) =

            ((bdv=( -1 + sqrt(1 - 0))/(2*1)) 

           | (bdv=( -1 - sqrt(1 - 0))/(2*1)))" and

   (* d2_abcformula10_neg not need, because 1<0 ==> False*)

 (* ---- degree 3 ----*)

   d3_reduce_equation1:

   "(a*bdv + b*bdv^^^2 + c*bdv^^^3=0) = (bdv=0 | (a + b*bdv + c*bdv^^^2=0))" and

   d3_reduce_equation2:

   "(  bdv + b*bdv^^^2 + c*bdv^^^3=0) = (bdv=0 | (1 + b*bdv + c*bdv^^^2=0))" and

   d3_reduce_equation3:

   "(a*bdv +   bdv^^^2 + c*bdv^^^3=0) = (bdv=0 | (a +   bdv + c*bdv^^^2=0))" and

   d3_reduce_equation4:

   "(  bdv +   bdv^^^2 + c*bdv^^^3=0) = (bdv=0 | (1 +   bdv + c*bdv^^^2=0))" and

   d3_reduce_equation5:

   "(a*bdv + b*bdv^^^2 +   bdv^^^3=0) = (bdv=0 | (a + b*bdv +   bdv^^^2=0))" and

   d3_reduce_equation6:

   "(  bdv + b*bdv^^^2 +   bdv^^^3=0) = (bdv=0 | (1 + b*bdv +   bdv^^^2=0))" and

   d3_reduce_equation7:

   "(a*bdv +   bdv^^^2 +   bdv^^^3=0) = (bdv=0 | (1 +   bdv +   bdv^^^2=0))" and

   d3_reduce_equation8:

   "(  bdv +   bdv^^^2 +   bdv^^^3=0) = (bdv=0 | (1 +   bdv +   bdv^^^2=0))" and

   d3_reduce_equation9:

   "(a*bdv             + c*bdv^^^3=0) = (bdv=0 | (a         + c*bdv^^^2=0))" and

   d3_reduce_equation10:

   "(  bdv             + c*bdv^^^3=0) = (bdv=0 | (1         + c*bdv^^^2=0))" and

   d3_reduce_equation11:

   "(a*bdv             +   bdv^^^3=0) = (bdv=0 | (a         +   bdv^^^2=0))" and

   d3_reduce_equation12:

   "(  bdv             +   bdv^^^3=0) = (bdv=0 | (1         +   bdv^^^2=0))" and

   d3_reduce_equation13:

   "(        b*bdv^^^2 + c*bdv^^^3=0) = (bdv=0 | (    b*bdv + c*bdv^^^2=0))" and

   d3_reduce_equation14:

   "(          bdv^^^2 + c*bdv^^^3=0) = (bdv=0 | (      bdv + c*bdv^^^2=0))" and

   d3_reduce_equation15:

   "(        b*bdv^^^2 +   bdv^^^3=0) = (bdv=0 | (    b*bdv +   bdv^^^2=0))" and

   d3_reduce_equation16:

   "(          bdv^^^2 +   bdv^^^3=0) = (bdv=0 | (      bdv +   bdv^^^2=0))" and

   d3_isolate_add1:

   "[|Not(bdv occurs_in a)|] ==> (a + b*bdv^^^3=0) = (b*bdv^^^3= (-1)*a)" and

   d3_isolate_add2:

   "[|Not(bdv occurs_in a)|] ==> (a +   bdv^^^3=0) = (  bdv^^^3= (-1)*a)" and

   d3_isolate_div:

    "[|Not(b=0);Not(bdv occurs_in a)|] ==> (b*bdv^^^3=c) = (bdv^^^3=c/b)" and

   d3_root_equation2:

   "(bdv^^^3=0) = (bdv=0)" and

   d3_root_equation1:

   "(bdv^^^3=c) = (bdv = nroot 3 c)" and

 (* ---- degree 4 ----*)

  (* RL03.FIXME es wir nicht getestet ob u>0 *)

  d4_sub_u1:

  "(c+b*bdv^^^2+a*bdv^^^4=0) =

    ((a*u^^^2+b*u+c=0) & (bdv^^^2=u))" and

 (* ---- 7.3.02 von Termorder ---- *)

   bdv_collect_1:      "l * bdv + m * bdv = (l + m) * bdv" and

   bdv_collect_2:      "bdv + m * bdv = (1 + m) * bdv" and

   bdv_collect_3:      "l * bdv + bdv = (l + 1) * bdv" and

 (*  bdv_collect_assoc0_1 "l * bdv + m * bdv + k = (l + m) * bdv + k"

     bdv_collect_assoc0_2 "bdv + m * bdv + k = (1 + m) * bdv + k"

     bdv_collect_assoc0_3 "l * bdv + bdv + k = (l + 1) * bdv + k"

 *)

   bdv_collect_assoc1_1: "l * bdv + (m * bdv + k) = (l + m) * bdv + k" and

   bdv_collect_assoc1_2: "bdv + (m * bdv + k) = (1 + m) * bdv + k" and

   bdv_collect_assoc1_3: "l * bdv + (bdv + k) = (l + 1) * bdv + k" and

   bdv_collect_assoc2_1: "k + l * bdv + m * bdv = k + (l + m) * bdv" and

   bdv_collect_assoc2_2: "k + bdv + m * bdv = k + (1 + m) * bdv" and

   bdv_collect_assoc2_3: "k + l * bdv + bdv = k + (l + 1) * bdv" and

   bdv_n_collect_1:     "l * bdv^^^n + m * bdv^^^n = (l + m) * bdv^^^n" and

   bdv_n_collect_2:     " bdv^^^n + m * bdv^^^n = (1 + m) * bdv^^^n" and

   bdv_n_collect_3:     "l * bdv^^^n + bdv^^^n = (l + 1) * bdv^^^n" (*order!*) and

   bdv_n_collect_assoc1_1:

                       "l * bdv^^^n + (m * bdv^^^n + k) = (l + m) * bdv^^^n + k" and

   bdv_n_collect_assoc1_2: "bdv^^^n + (m * bdv^^^n + k) = (1 + m) * bdv^^^n + k" and

   bdv_n_collect_assoc1_3: "l * bdv^^^n + (bdv^^^n + k) = (l + 1) * bdv^^^n + k" and

   bdv_n_collect_assoc2_1: "k + l * bdv^^^n + m * bdv^^^n = k +(l + m) * bdv^^^n" and

   bdv_n_collect_assoc2_2: "k + bdv^^^n + m * bdv^^^n = k + (1 + m) * bdv^^^n" and

   bdv_n_collect_assoc2_3: "k + l * bdv^^^n + bdv^^^n = k + (l + 1) * bdv^^^n" and

 (*WN.14.3.03*)

   real_minus_div:         "- (a / b) = (-1 * a) / b" and

   separate_bdv:           "(a * bdv) / b = (a / b) * (bdv::real)" and

   separate_bdv_n:         "(a * bdv ^^^ n) / b = (a / b) * bdv ^^^ n" and

   separate_1_bdv:         "bdv / b = (1 / b) * (bdv::real)" and

   separate_1_bdv_n:       "bdv ^^^ n / b = (1 / b) * bdv ^^^ n"

 ML {*

 val thy = @{theory};

 (*-------------------------rulse-------------------------*)

 val PolyEq_prls = (*3.10.02:just the following order due to subterm evaluation*)

   append_rls "PolyEq_prls" e_rls 

 	     [Calc ("Atools.ident",eval_ident "#ident_"),

 	      Calc ("Tools.matches",eval_matches ""),

 	      Calc ("Tools.lhs"    ,eval_lhs ""),

 	      Calc ("Tools.rhs"    ,eval_rhs ""),

 	      Calc ("Poly.is'_expanded'_in",eval_is_expanded_in ""),

 	      Calc ("Poly.is'_poly'_in",eval_is_poly_in ""),

 	      Calc ("Poly.has'_degree'_in",eval_has_degree_in ""),    

               Calc ("Poly.is'_polyrat'_in",eval_is_polyrat_in ""),

 	      (*Calc ("Atools.occurs'_in",eval_occurs_in ""),   *) 

 	      (*Calc ("Atools.is'_const",eval_const "#is_const_"),*)

 	      Calc ("HOL.eq",eval_equal "#equal_"),

               Calc ("RootEq.is'_rootTerm'_in",eval_is_rootTerm_in ""),

 	      Calc ("RatEq.is'_ratequation'_in",eval_is_ratequation_in ""),

 	      Thm ("not_true",num_str @{thm not_true}),

 	      Thm ("not_false",num_str @{thm not_false}),

 	      Thm ("and_true",num_str @{thm and_true}),

 	      Thm ("and_false",num_str @{thm and_false}),

 	      Thm ("or_true",num_str @{thm or_true}),

 	      Thm ("or_false",num_str @{thm or_false})

 	       ];

 val PolyEq_erls = 

     merge_rls "PolyEq_erls" LinEq_erls

     (append_rls "ops_preds" calculate_Rational

 		[Calc ("HOL.eq",eval_equal "#equal_"),

 		 Thm ("plus_leq", num_str @{thm plus_leq}),

 		 Thm ("minus_leq", num_str @{thm minus_leq}),

 		 Thm ("rat_leq1", num_str @{thm rat_leq1}),

 		 Thm ("rat_leq2", num_str @{thm rat_leq2}),

 		 Thm ("rat_leq3", num_str @{thm rat_leq3})

 		 ]);

 val PolyEq_crls = 

     merge_rls "PolyEq_crls" LinEq_crls

     (append_rls "ops_preds" calculate_Rational

 		[Calc ("HOL.eq",eval_equal "#equal_"),

 		 Thm ("plus_leq", num_str @{thm plus_leq}),

 		 Thm ("minus_leq", num_str @{thm minus_leq}),

 		 Thm ("rat_leq1", num_str @{thm rat_leq1}),

 		 Thm ("rat_leq2", num_str @{thm rat_leq2}),

 		 Thm ("rat_leq3", num_str @{thm rat_leq3})

 		 ]);

 val cancel_leading_coeff = prep_rls'(

   Rls {id = "cancel_leading_coeff", preconds = [], 

        rew_ord = ("e_rew_ord",e_rew_ord),

       erls = PolyEq_erls, srls = Erls, calc = [], errpatts = [],

       rules = 

       [Thm ("cancel_leading_coeff1",num_str @{thm cancel_leading_coeff1}),

        Thm ("cancel_leading_coeff2",num_str @{thm cancel_leading_coeff2}),

        Thm ("cancel_leading_coeff3",num_str @{thm cancel_leading_coeff3}),

        Thm ("cancel_leading_coeff4",num_str @{thm cancel_leading_coeff4}),

        Thm ("cancel_leading_coeff5",num_str @{thm cancel_leading_coeff5}),

        Thm ("cancel_leading_coeff6",num_str @{thm cancel_leading_coeff6}),

        Thm ("cancel_leading_coeff7",num_str @{thm cancel_leading_coeff7}),

        Thm ("cancel_leading_coeff8",num_str @{thm cancel_leading_coeff8}),

        Thm ("cancel_leading_coeff9",num_str @{thm cancel_leading_coeff9}),

        Thm ("cancel_leading_coeff10",num_str @{thm cancel_leading_coeff10}),

        Thm ("cancel_leading_coeff11",num_str @{thm cancel_leading_coeff11}),

        Thm ("cancel_leading_coeff12",num_str @{thm cancel_leading_coeff12}),

        Thm ("cancel_leading_coeff13",num_str @{thm cancel_leading_coeff13})

        ],scr = Prog ((Thm.term_of o the o (parse thy)) "empty_script")}:rls);

 val prep_rls' = prep_rls @{theory};

 *}

 ML{*

 val complete_square = prep_rls'(

   Rls {id = "complete_square", preconds = [], 

        rew_ord = ("e_rew_ord",e_rew_ord),

       erls = PolyEq_erls, srls = Erls, calc = [],  errpatts = [],

       rules = [Thm ("complete_square1",num_str @{thm complete_square1}),

 	       Thm ("complete_square2",num_str @{thm complete_square2}),

 	       Thm ("complete_square3",num_str @{thm complete_square3}),

 	       Thm ("complete_square4",num_str @{thm complete_square4}),

 	       Thm ("complete_square5",num_str @{thm complete_square5})

 	       ],

       scr = Prog ((Thm.term_of o the o (parse thy)) "empty_script")

       }:rls);

 val polyeq_simplify = prep_rls'(

   Rls {id = "polyeq_simplify", preconds = [], 

        rew_ord = ("termlessI",termlessI), 

        erls = PolyEq_erls, 

        srls = Erls, 

        calc = [], errpatts = [],

        rules = [Thm  ("real_assoc_1",num_str @{thm real_assoc_1}),

 		Thm  ("real_assoc_2",num_str @{thm real_assoc_2}),

 		Thm  ("real_diff_minus",num_str @{thm real_diff_minus}),

 		Thm  ("real_unari_minus",num_str @{thm real_unari_minus}),

 		Thm  ("realpow_multI",num_str @{thm realpow_multI}),

 		Calc ("Groups.plus_class.plus",eval_binop "#add_"),

 		Calc ("Groups.minus_class.minus",eval_binop "#sub_"),

 		Calc ("Groups.times_class.times",eval_binop "#mult_"),

 		Calc ("Fields.inverse_class.divide", eval_cancel "#divide_e"),

 		Calc ("NthRoot.sqrt",eval_sqrt "#sqrt_"),

 		Calc ("Atools.pow" ,eval_binop "#power_"),

                 Rls_ reduce_012

                 ],

        scr = Prog ((Thm.term_of o the o (parse thy)) "empty_script")

        }:rls);

 *}

 setup {* KEStore_Elems.add_rlss 

   [("cancel_leading_coeff", (Context.theory_name @{theory}, cancel_leading_coeff)), 

   ("complete_square", (Context.theory_name @{theory}, complete_square)), 

   ("PolyEq_erls", (Context.theory_name @{theory}, PolyEq_erls)), 

   ("polyeq_simplify", (Context.theory_name @{theory}, polyeq_simplify))] *}

 ML{*

 (* ------------- polySolve ------------------ *)

 (* -- d0 -- *)

 (*isolate the bound variable in an d0 equation; 'bdv' is a meta-constant*)

 val d0_polyeq_simplify = prep_rls'(

   Rls {id = "d0_polyeq_simplify", preconds = [],

        rew_ord = ("e_rew_ord",e_rew_ord),

        erls = PolyEq_erls,

        srls = Erls, 

        calc = [], errpatts = [],

        rules = [Thm("d0_true",num_str @{thm d0_true}),

 		Thm("d0_false",num_str @{thm  d0_false})

 		],

        scr = Prog ((Thm.term_of o the o (parse thy)) "empty_script")

        }:rls);

 (* -- d1 -- *)

 (*isolate the bound variable in an d1 equation; 'bdv' is a meta-constant*)

 val d1_polyeq_simplify = prep_rls'(

   Rls {id = "d1_polyeq_simplify", preconds = [],

        rew_ord = ("e_rew_ord",e_rew_ord),

        erls = PolyEq_erls,

        srls = Erls, 

        calc = [], errpatts = [],

        rules = [

 		Thm("d1_isolate_add1",num_str @{thm d1_isolate_add1}), 

 		(* a+bx=0 -> bx=-a *)

 		Thm("d1_isolate_add2",num_str @{thm d1_isolate_add2}), 

 		(* a+ x=0 ->  x=-a *)

 		Thm("d1_isolate_div",num_str @{thm d1_isolate_div})    

 		(*   bx=c -> x=c/b *)  

 		],

        scr = Prog ((Thm.term_of o the o (parse thy)) "empty_script")

        }:rls);

 *}

 subsection {* degree 2 *}

 ML{*

 (* isolate the bound variable in an d2 equation with bdv only;

   "bdv" is a meta-constant substituted for the "x" below by isac's rewriter. *)

 val d2_polyeq_bdv_only_simplify = prep_rls'(

   Rls {id = "d2_polyeq_bdv_only_simplify", preconds = [], rew_ord = ("e_rew_ord",e_rew_ord),

     erls = PolyEq_erls, srls = Erls, calc = [], errpatts = [],

     rules =

       [Thm ("d2_prescind1", num_str @{thm d2_prescind1}), (*   ax+bx^2=0 -> x(a+bx)=0 *)

        Thm ("d2_prescind2", num_str @{thm d2_prescind2}), (*   ax+ x^2=0 -> x(a+ x)=0 *)

        Thm ("d2_prescind3", num_str @{thm d2_prescind3}), (*    x+bx^2=0 -> x(1+bx)=0 *)

        Thm ("d2_prescind4", num_str @{thm d2_prescind4}), (*    x+ x^2=0 -> x(1+ x)=0 *)

        Thm ("d2_sqrt_equation1", num_str @{thm d2_sqrt_equation1}),    (* x^2=c   -> x=+-sqrt(c) *)

        Thm ("d2_sqrt_equation1_neg", num_str @{thm d2_sqrt_equation1_neg}), (* [0<c] x^2=c  -> []*)

        Thm ("d2_sqrt_equation2", num_str @{thm d2_sqrt_equation2}),    (*  x^2=0 ->    x=0       *)

        Thm ("d2_reduce_equation1", num_str @{thm d2_reduce_equation1}),(* x(a+bx)=0 -> x=0 |a+bx=0*)

        Thm ("d2_reduce_equation2", num_str @{thm d2_reduce_equation2}),(* x(a+ x)=0 -> x=0 |a+ x=0*)

        Thm ("d2_isolate_div", num_str @{thm d2_isolate_div})           (* bx^2=c -> x^2=c/b      *)

        ],

        scr = Prog ((Thm.term_of o the o (parse thy)) "empty_script")

        }:rls);

 *}

 ML{*

 (* isolate the bound variable in an d2 equation with sqrt only; 

    'bdv' is a meta-constant*)

 val d2_polyeq_sq_only_simplify = prep_rls'(

   Rls {id = "d2_polyeq_sq_only_simplify", preconds = [],

        rew_ord = ("e_rew_ord",e_rew_ord),

        erls = PolyEq_erls,

        srls = Erls, 

        calc = [], errpatts = [],

        (*asm_thm = [("d2_sqrt_equation1",""),("d2_sqrt_equation1_neg",""),

                   ("d2_isolate_div","")],*)

        rules = [Thm("d2_isolate_add1",num_str @{thm d2_isolate_add1}),

                 (* a+   bx^2=0 -> bx^2=(-1)a*)

 		Thm("d2_isolate_add2",num_str @{thm d2_isolate_add2}),

                 (* a+    x^2=0 ->  x^2=(-1)a*)

 		Thm("d2_sqrt_equation2",num_str @{thm d2_sqrt_equation2}),

                 (*  x^2=0 ->    x=0    *)

 		Thm("d2_sqrt_equation1",num_str @{thm d2_sqrt_equation1}),

                 (* x^2=c   -> x=+-sqrt(c)*)

 		Thm("d2_sqrt_equation1_neg",num_str @{thm d2_sqrt_equation1_neg}),

                 (* [c<0] x^2=c  -> x=[] *)

 		Thm("d2_isolate_div",num_str @{thm d2_isolate_div})

                  (* bx^2=c -> x^2=c/b*)

 		],

        scr = Prog ((Thm.term_of o the o (parse thy)) "empty_script")

        }:rls);

 *}

 ML{*

 (* isolate the bound variable in an d2 equation with pqFormula;

    'bdv' is a meta-constant*)

 val d2_polyeq_pqFormula_simplify = prep_rls'(

   Rls {id = "d2_polyeq_pqFormula_simplify", preconds = [],

        rew_ord = ("e_rew_ord",e_rew_ord), erls = PolyEq_erls,

        srls = Erls, calc = [], errpatts = [],

        rules = [Thm("d2_pqformula1",num_str @{thm d2_pqformula1}),

                 (* q+px+ x^2=0 *)

 		Thm("d2_pqformula1_neg",num_str @{thm d2_pqformula1_neg}),

                 (* q+px+ x^2=0 *)

 		Thm("d2_pqformula2",num_str @{thm d2_pqformula2}), 

                 (* q+px+1x^2=0 *)

 		Thm("d2_pqformula2_neg",num_str @{thm d2_pqformula2_neg}),

                 (* q+px+1x^2=0 *)

 		Thm("d2_pqformula3",num_str @{thm d2_pqformula3}),

                 (* q+ x+ x^2=0 *)

 		Thm("d2_pqformula3_neg",num_str @{thm d2_pqformula3_neg}), 

                 (* q+ x+ x^2=0 *)

 		Thm("d2_pqformula4",num_str @{thm d2_pqformula4}),

                 (* q+ x+1x^2=0 *)

 		Thm("d2_pqformula4_neg",num_str @{thm d2_pqformula4_neg}),

                 (* q+ x+1x^2=0 *)

 		Thm("d2_pqformula5",num_str @{thm d2_pqformula5}),

                 (*   qx+ x^2=0 *)

 		Thm("d2_pqformula6",num_str @{thm d2_pqformula6}),

                 (*   qx+1x^2=0 *)

 		Thm("d2_pqformula7",num_str @{thm d2_pqformula7}),

                 (*    x+ x^2=0 *)

 		Thm("d2_pqformula8",num_str @{thm d2_pqformula8}),

                 (*    x+1x^2=0 *)

 		Thm("d2_pqformula9",num_str @{thm d2_pqformula9}),

                 (* q   +1x^2=0 *)

 		Thm("d2_pqformula9_neg",num_str @{thm d2_pqformula9_neg}),

                 (* q   +1x^2=0 *)

 		Thm("d2_pqformula10",num_str @{thm d2_pqformula10}),

                 (* q   + x^2=0 *)

 		Thm("d2_pqformula10_neg",num_str @{thm d2_pqformula10_neg}),

                 (* q   + x^2=0 *)

 		Thm("d2_sqrt_equation2",num_str @{thm d2_sqrt_equation2}),

                 (*       x^2=0 *)

 		Thm("d2_sqrt_equation3",num_str @{thm d2_sqrt_equation3})

                (*      1x^2=0 *)

 	       ],scr = Prog ((Thm.term_of o the o (parse thy)) "empty_script")

        }:rls);

 *}

 ML{*

 (* isolate the bound variable in an d2 equation with abcFormula; 

    'bdv' is a meta-constant*)

 val d2_polyeq_abcFormula_simplify = prep_rls'(

   Rls {id = "d2_polyeq_abcFormula_simplify", preconds = [],

        rew_ord = ("e_rew_ord",e_rew_ord), erls = PolyEq_erls,

        srls = Erls, calc = [], errpatts = [],

        rules = [Thm("d2_abcformula1",num_str @{thm d2_abcformula1}),

                 (*c+bx+cx^2=0 *)

 		Thm("d2_abcformula1_neg",num_str @{thm d2_abcformula1_neg}),

                 (*c+bx+cx^2=0 *)

 		Thm("d2_abcformula2",num_str @{thm d2_abcformula2}),

                 (*c+ x+cx^2=0 *)

 		Thm("d2_abcformula2_neg",num_str @{thm d2_abcformula2_neg}),

                 (*c+ x+cx^2=0 *)

 		Thm("d2_abcformula3",num_str @{thm d2_abcformula3}), 

                 (*c+bx+ x^2=0 *)

 		Thm("d2_abcformula3_neg",num_str @{thm d2_abcformula3_neg}),

                 (*c+bx+ x^2=0 *)

 		Thm("d2_abcformula4",num_str @{thm d2_abcformula4}),

                 (*c+ x+ x^2=0 *)

 		Thm("d2_abcformula4_neg",num_str @{thm d2_abcformula4_neg}),

                 (*c+ x+ x^2=0 *)

 		Thm("d2_abcformula5",num_str @{thm d2_abcformula5}),

                 (*c+   cx^2=0 *)

 		Thm("d2_abcformula5_neg",num_str @{thm d2_abcformula5_neg}),

                 (*c+   cx^2=0 *)

 		Thm("d2_abcformula6",num_str @{thm d2_abcformula6}),

                 (*c+    x^2=0 *)

 		Thm("d2_abcformula6_neg",num_str @{thm d2_abcformula6_neg}),

                 (*c+    x^2=0 *)

 		Thm("d2_abcformula7",num_str @{thm d2_abcformula7}),

                 (*  bx+ax^2=0 *)

 		Thm("d2_abcformula8",num_str @{thm d2_abcformula8}),

                 (*  bx+ x^2=0 *)

 		Thm("d2_abcformula9",num_str @{thm d2_abcformula9}),

                 (*   x+ax^2=0 *)

 		Thm("d2_abcformula10",num_str @{thm d2_abcformula10}),

                 (*   x+ x^2=0 *)

 		Thm("d2_sqrt_equation2",num_str @{thm d2_sqrt_equation2}),

                 (*      x^2=0 *)  

 		Thm("d2_sqrt_equation3",num_str @{thm d2_sqrt_equation3})

                (*     bx^2=0 *)  

 	       ],

        scr = Prog ((Thm.term_of o the o (parse thy)) "empty_script")

        }:rls);

 *}

 ML{*

 (* isolate the bound variable in an d2 equation; 

    'bdv' is a meta-constant*)

 val d2_polyeq_simplify = prep_rls'(

   Rls {id = "d2_polyeq_simplify", preconds = [],

        rew_ord = ("e_rew_ord",e_rew_ord), erls = PolyEq_erls,

        srls = Erls, calc = [], errpatts = [],

        rules = [Thm("d2_pqformula1",num_str @{thm d2_pqformula1}),

                 (* p+qx+ x^2=0 *)

 		Thm("d2_pqformula1_neg",num_str @{thm d2_pqformula1_neg}),

                 (* p+qx+ x^2=0 *)

 		Thm("d2_pqformula2",num_str @{thm d2_pqformula2}),

                 (* p+qx+1x^2=0 *)

 		Thm("d2_pqformula2_neg",num_str @{thm d2_pqformula2_neg}),

                 (* p+qx+1x^2=0 *)

 		Thm("d2_pqformula3",num_str @{thm d2_pqformula3}),

                 (* p+ x+ x^2=0 *)

 		Thm("d2_pqformula3_neg",num_str @{thm d2_pqformula3_neg}),

                 (* p+ x+ x^2=0 *)

 		Thm("d2_pqformula4",num_str @{thm d2_pqformula4}), 

                 (* p+ x+1x^2=0 *)

 		Thm("d2_pqformula4_neg",num_str @{thm d2_pqformula4_neg}),

                 (* p+ x+1x^2=0 *)

 		Thm("d2_abcformula1",num_str @{thm d2_abcformula1}),

                 (* c+bx+cx^2=0 *)

 		Thm("d2_abcformula1_neg",num_str @{thm d2_abcformula1_neg}),

                 (* c+bx+cx^2=0 *)

 		Thm("d2_abcformula2",num_str @{thm d2_abcformula2}),

                 (* c+ x+cx^2=0 *)

 		Thm("d2_abcformula2_neg",num_str @{thm d2_abcformula2_neg}),

                 (* c+ x+cx^2=0 *)

 		Thm("d2_prescind1",num_str @{thm d2_prescind1}),

                 (*   ax+bx^2=0 -> x(a+bx)=0 *)

 		Thm("d2_prescind2",num_str @{thm d2_prescind2}),

                 (*   ax+ x^2=0 -> x(a+ x)=0 *)

 		Thm("d2_prescind3",num_str @{thm d2_prescind3}),

                 (*    x+bx^2=0 -> x(1+bx)=0 *)

 		Thm("d2_prescind4",num_str @{thm d2_prescind4}),

                 (*    x+ x^2=0 -> x(1+ x)=0 *)

 		Thm("d2_isolate_add1",num_str @{thm d2_isolate_add1}),

                 (* a+   bx^2=0 -> bx^2=(-1)a*)

 		Thm("d2_isolate_add2",num_str @{thm d2_isolate_add2}),

                 (* a+    x^2=0 ->  x^2=(-1)a*)

 		Thm("d2_sqrt_equation1",num_str @{thm d2_sqrt_equation1}),

                 (* x^2=c   -> x=+-sqrt(c)*)

 		Thm("d2_sqrt_equation1_neg",num_str @{thm d2_sqrt_equation1_neg}),

                 (* [c<0] x^2=c   -> x=[]*)

 		Thm("d2_sqrt_equation2",num_str @{thm d2_sqrt_equation2}),

                 (*  x^2=0 ->    x=0    *)

 		Thm("d2_reduce_equation1",num_str @{thm d2_reduce_equation1}),

                 (* x(a+bx)=0 -> x=0 | a+bx=0*)

 		Thm("d2_reduce_equation2",num_str @{thm d2_reduce_equation2}),

                 (* x(a+ x)=0 -> x=0 | a+ x=0*)

 		Thm("d2_isolate_div",num_str @{thm d2_isolate_div})

                (* bx^2=c -> x^2=c/b*)

 	       ],

        scr = Prog ((Thm.term_of o the o (parse thy)) "empty_script")

       }:rls);

 *}

 ML{*

 (* -- d3 -- *)

 (* isolate the bound variable in an d3 equation; 'bdv' is a meta-constant *)

 val d3_polyeq_simplify = prep_rls'(

   Rls {id = "d3_polyeq_simplify", preconds = [],

        rew_ord = ("e_rew_ord",e_rew_ord), erls = PolyEq_erls,

        srls = Erls, calc = [], errpatts = [],

        rules = 

        [Thm("d3_reduce_equation1",num_str @{thm d3_reduce_equation1}),

 	(*a*bdv + b*bdv^^^2 + c*bdv^^^3=0) = 

         (bdv=0 | (a + b*bdv + c*bdv^^^2=0)*)

 	Thm("d3_reduce_equation2",num_str @{thm d3_reduce_equation2}),

 	(*  bdv + b*bdv^^^2 + c*bdv^^^3=0) = 

         (bdv=0 | (1 + b*bdv + c*bdv^^^2=0)*)

 	Thm("d3_reduce_equation3",num_str @{thm d3_reduce_equation3}),

 	(*a*bdv +   bdv^^^2 + c*bdv^^^3=0) = 

         (bdv=0 | (a +   bdv + c*bdv^^^2=0)*)

 	Thm("d3_reduce_equation4",num_str @{thm d3_reduce_equation4}),

 	(*  bdv +   bdv^^^2 + c*bdv^^^3=0) = 

         (bdv=0 | (1 +   bdv + c*bdv^^^2=0)*)

 	Thm("d3_reduce_equation5",num_str @{thm d3_reduce_equation5}),

 	(*a*bdv + b*bdv^^^2 +   bdv^^^3=0) = 

         (bdv=0 | (a + b*bdv +   bdv^^^2=0)*)

 	Thm("d3_reduce_equation6",num_str @{thm d3_reduce_equation6}),

 	(*  bdv + b*bdv^^^2 +   bdv^^^3=0) = 

         (bdv=0 | (1 + b*bdv +   bdv^^^2=0)*)

 	Thm("d3_reduce_equation7",num_str @{thm d3_reduce_equation7}),

 	     (*a*bdv +   bdv^^^2 +   bdv^^^3=0) = 

              (bdv=0 | (1 +   bdv +   bdv^^^2=0)*)

 	Thm("d3_reduce_equation8",num_str @{thm d3_reduce_equation8}),

 	     (*  bdv +   bdv^^^2 +   bdv^^^3=0) = 

              (bdv=0 | (1 +   bdv +   bdv^^^2=0)*)

 	Thm("d3_reduce_equation9",num_str @{thm d3_reduce_equation9}),

 	     (*a*bdv             + c*bdv^^^3=0) = 

              (bdv=0 | (a         + c*bdv^^^2=0)*)

 	Thm("d3_reduce_equation10",num_str @{thm d3_reduce_equation10}),

 	     (*  bdv             + c*bdv^^^3=0) = 

              (bdv=0 | (1         + c*bdv^^^2=0)*)

 	Thm("d3_reduce_equation11",num_str @{thm d3_reduce_equation11}),

 	     (*a*bdv             +   bdv^^^3=0) = 

              (bdv=0 | (a         +   bdv^^^2=0)*)

 	Thm("d3_reduce_equation12",num_str @{thm d3_reduce_equation12}),

 	     (*  bdv             +   bdv^^^3=0) = 

              (bdv=0 | (1         +   bdv^^^2=0)*)

 	Thm("d3_reduce_equation13",num_str @{thm d3_reduce_equation13}),

 	     (*        b*bdv^^^2 + c*bdv^^^3=0) = 

              (bdv=0 | (    b*bdv + c*bdv^^^2=0)*)

 	Thm("d3_reduce_equation14",num_str @{thm d3_reduce_equation14}),

 	     (*          bdv^^^2 + c*bdv^^^3=0) = 

              (bdv=0 | (      bdv + c*bdv^^^2=0)*)

 	Thm("d3_reduce_equation15",num_str @{thm d3_reduce_equation15}),

 	     (*        b*bdv^^^2 +   bdv^^^3=0) = 

              (bdv=0 | (    b*bdv +   bdv^^^2=0)*)

 	Thm("d3_reduce_equation16",num_str @{thm d3_reduce_equation16}),

 	     (*          bdv^^^2 +   bdv^^^3=0) = 

              (bdv=0 | (      bdv +   bdv^^^2=0)*)

 	Thm("d3_isolate_add1",num_str @{thm d3_isolate_add1}),

 	     (*[|Not(bdv occurs_in a)|] ==> (a + b*bdv^^^3=0) = 

               (bdv=0 | (b*bdv^^^3=a)*)

 	Thm("d3_isolate_add2",num_str @{thm d3_isolate_add2}),

              (*[|Not(bdv occurs_in a)|] ==> (a +   bdv^^^3=0) = 

               (bdv=0 | (  bdv^^^3=a)*)

 	Thm("d3_isolate_div",num_str @{thm d3_isolate_div}),

         (*[|Not(b=0)|] ==> (b*bdv^^^3=c) = (bdv^^^3=c/b*)

         Thm("d3_root_equation2",num_str @{thm d3_root_equation2}),

         (*(bdv^^^3=0) = (bdv=0) *)

 	Thm("d3_root_equation1",num_str @{thm d3_root_equation1})

        (*bdv^^^3=c) = (bdv = nroot 3 c*)

        ],

        scr = Prog ((Thm.term_of o the o (parse thy)) "empty_script")

       }:rls);

 *}

 ML{*

 (* -- d4 -- *)

 (*isolate the bound variable in an d4 equation; 'bdv' is a meta-constant*)

 val d4_polyeq_simplify = prep_rls'(

   Rls {id = "d4_polyeq_simplify", preconds = [],

        rew_ord = ("e_rew_ord",e_rew_ord), erls = PolyEq_erls,

        srls = Erls, calc = [], errpatts = [],

        rules = 

        [Thm("d4_sub_u1",num_str @{thm d4_sub_u1})  

        (* ax^4+bx^2+c=0 -> x=+-sqrt(ax^2+bx^+c) *)

        ],

        scr = Prog ((Thm.term_of o the o (parse thy)) "empty_script")

       }:rls);

 *}

 setup {* KEStore_Elems.add_rlss 

   [("d0_polyeq_simplify", (Context.theory_name @{theory}, d0_polyeq_simplify)), 

   ("d1_polyeq_simplify", (Context.theory_name @{theory}, d1_polyeq_simplify)), 

   ("d2_polyeq_simplify", (Context.theory_name @{theory}, d2_polyeq_simplify)), 

   ("d2_polyeq_bdv_only_simplify", (Context.theory_name @{theory}, d2_polyeq_bdv_only_simplify)), 

   ("d2_polyeq_sq_only_simplify", (Context.theory_name @{theory}, d2_polyeq_sq_only_simplify)),

   ("d2_polyeq_pqFormula_simplify",

     (Context.theory_name @{theory}, d2_polyeq_pqFormula_simplify)), 

   ("d2_polyeq_abcFormula_simplify",

     (Context.theory_name @{theory}, d2_polyeq_abcFormula_simplify)), 

   ("d3_polyeq_simplify", (Context.theory_name @{theory}, d3_polyeq_simplify)), 

   ("d4_polyeq_simplify", (Context.theory_name @{theory}, d4_polyeq_simplify))] *}

 ML{*

 (*------------------------problems------------------------*)

 (*

 (get_pbt ["degree_2","polynomial","univariate","equation"]);

 show_ptyps(); 

 *)

 *}

 setup {* KEStore_Elems.add_pbts

   [(Specify.prep_pbt thy "pbl_equ_univ_poly" [] e_pblID

       (["polynomial","univariate","equation"],

         [("#Given" ,["equality e_e","solveFor v_v"]),

           ("#Where" ,["~((e_e::bool) is_ratequation_in (v_v::real))",

 	          "~((lhs e_e) is_rootTerm_in (v_v::real))",

 	          "~((rhs e_e) is_rootTerm_in (v_v::real))"]),

           ("#Find"  ,["solutions v_v'i'"])],

         PolyEq_prls, SOME "solve (e_e::bool, v_v)", [])),

     (*--- d0 ---*)

     (Specify.prep_pbt thy "pbl_equ_univ_poly_deg0" [] e_pblID

       (["degree_0","polynomial","univariate","equation"],

         [("#Given" ,["equality e_e","solveFor v_v"]),

           ("#Where" ,["matches (?a = 0) e_e",

 	          "(lhs e_e) is_poly_in v_v",

 	          "((lhs e_e) has_degree_in v_v ) = 0"]),

           ("#Find"  ,["solutions v_v'i'"])],

         PolyEq_prls, SOME "solve (e_e::bool, v_v)", [["PolyEq","solve_d0_polyeq_equation"]])),

     (*--- d1 ---*)

     (Specify.prep_pbt thy "pbl_equ_univ_poly_deg1" [] e_pblID

       (["degree_1","polynomial","univariate","equation"],

         [("#Given" ,["equality e_e","solveFor v_v"]),

           ("#Where" ,["matches (?a = 0) e_e",

 	          "(lhs e_e) is_poly_in v_v",

 	          "((lhs e_e) has_degree_in v_v ) = 1"]),

           ("#Find"  ,["solutions v_v'i'"])],

         PolyEq_prls, SOME "solve (e_e::bool, v_v)", [["PolyEq","solve_d1_polyeq_equation"]])),

     (*--- d2 ---*)

     (Specify.prep_pbt thy "pbl_equ_univ_poly_deg2" [] e_pblID

       (["degree_2","polynomial","univariate","equation"],

         [("#Given" ,["equality e_e","solveFor v_v"]),

           ("#Where" ,["matches (?a = 0) e_e",

 	          "(lhs e_e) is_poly_in v_v ",

 	          "((lhs e_e) has_degree_in v_v ) = 2"]),

           ("#Find"  ,["solutions v_v'i'"])],

         PolyEq_prls, SOME "solve (e_e::bool, v_v)", [["PolyEq","solve_d2_polyeq_equation"]])),

     (Specify.prep_pbt thy "pbl_equ_univ_poly_deg2_sqonly" [] e_pblID

       (["sq_only","degree_2","polynomial","univariate","equation"],

         [("#Given" ,["equality e_e","solveFor v_v"]),

           ("#Where" ,["matches ( ?a +    ?v_^^^2 = 0) e_e | " ^

 	          "matches ( ?a + ?b*?v_^^^2 = 0) e_e | " ^

             "matches (         ?v_^^^2 = 0) e_e | " ^

             "matches (      ?b*?v_^^^2 = 0) e_e" ,

             "Not (matches (?a +    ?v_ +    ?v_^^^2 = 0) e_e) &" ^

             "Not (matches (?a + ?b*?v_ +    ?v_^^^2 = 0) e_e) &" ^

             "Not (matches (?a +    ?v_ + ?c*?v_^^^2 = 0) e_e) &" ^

             "Not (matches (?a + ?b*?v_ + ?c*?v_^^^2 = 0) e_e) &" ^

             "Not (matches (        ?v_ +    ?v_^^^2 = 0) e_e) &" ^

             "Not (matches (     ?b*?v_ +    ?v_^^^2 = 0) e_e) &" ^

             "Not (matches (        ?v_ + ?c*?v_^^^2 = 0) e_e) &" ^

             "Not (matches (     ?b*?v_ + ?c*?v_^^^2 = 0) e_e)"]),

           ("#Find"  ,["solutions v_v'i'"])],

         PolyEq_prls, SOME "solve (e_e::bool, v_v)",

         [["PolyEq","solve_d2_polyeq_sqonly_equation"]])),

     (Specify.prep_pbt thy "pbl_equ_univ_poly_deg2_bdvonly" [] e_pblID

       (["bdv_only","degree_2","polynomial","univariate","equation"],

         [("#Given", ["equality e_e","solveFor v_v"]),

           ("#Where", ["matches (?a*?v_ +    ?v_^^^2 = 0) e_e | " ^

             "matches (   ?v_ +    ?v_^^^2 = 0) e_e | " ^

             "matches (   ?v_ + ?b*?v_^^^2 = 0) e_e | " ^

             "matches (?a*?v_ + ?b*?v_^^^2 = 0) e_e | " ^

             "matches (            ?v_^^^2 = 0) e_e | " ^

             "matches (         ?b*?v_^^^2 = 0) e_e "]),

           ("#Find", ["solutions v_v'i'"])],

         PolyEq_prls, SOME "solve (e_e::bool, v_v)",

         [["PolyEq","solve_d2_polyeq_bdvonly_equation"]])),

     (Specify.prep_pbt thy "pbl_equ_univ_poly_deg2_pq" [] e_pblID

       (["pqFormula","degree_2","polynomial","univariate","equation"],

         [("#Given", ["equality e_e","solveFor v_v"]),

           ("#Where", ["matches (?a + 1*?v_^^^2 = 0) e_e | " ^

 	          "matches (?a +   ?v_^^^2 = 0) e_e"]),

           ("#Find", ["solutions v_v'i'"])],

         PolyEq_prls, SOME "solve (e_e::bool, v_v)", [["PolyEq","solve_d2_polyeq_pq_equation"]])),

     (Specify.prep_pbt thy "pbl_equ_univ_poly_deg2_abc" [] e_pblID

       (["abcFormula","degree_2","polynomial","univariate","equation"],

         [("#Given", ["equality e_e","solveFor v_v"]),

           ("#Where", ["matches (?a +    ?v_^^^2 = 0) e_e | " ^

 	          "matches (?a + ?b*?v_^^^2 = 0) e_e"]),

           ("#Find", ["solutions v_v'i'"])],

         PolyEq_prls, SOME "solve (e_e::bool, v_v)", [["PolyEq","solve_d2_polyeq_abc_equation"]])),

     (*--- d3 ---*)

     (Specify.prep_pbt thy "pbl_equ_univ_poly_deg3" [] e_pblID

       (["degree_3","polynomial","univariate","equation"],

         [("#Given", ["equality e_e","solveFor v_v"]),

           ("#Where", ["matches (?a = 0) e_e",

 	          "(lhs e_e) is_poly_in v_v ",

 	          "((lhs e_e) has_degree_in v_v) = 3"]),

           ("#Find", ["solutions v_v'i'"])],

         PolyEq_prls, SOME "solve (e_e::bool, v_v)", [["PolyEq","solve_d3_polyeq_equation"]])),

     (*--- d4 ---*)

     (Specify.prep_pbt thy "pbl_equ_univ_poly_deg4" [] e_pblID

       (["degree_4","polynomial","univariate","equation"],

         [("#Given", ["equality e_e","solveFor v_v"]),

           ("#Where", ["matches (?a = 0) e_e",

 	          "(lhs e_e) is_poly_in v_v ",

 	          "((lhs e_e) has_degree_in v_v) = 4"]),

           ("#Find", ["solutions v_v'i'"])],

         PolyEq_prls, SOME "solve (e_e::bool, v_v)", [(*["PolyEq","solve_d4_polyeq_equation"]*)])),

     (*--- normalize ---*)

     (Specify.prep_pbt thy "pbl_equ_univ_poly_norm" [] e_pblID

       (["normalize","polynomial","univariate","equation"],

         [("#Given", ["equality e_e","solveFor v_v"]),

           ("#Where", ["(Not((matches (?a = 0 ) e_e ))) |" ^

 	          "(Not(((lhs e_e) is_poly_in v_v)))"]),

           ("#Find", ["solutions v_v'i'"])],

         PolyEq_prls, SOME "solve (e_e::bool, v_v)", [["PolyEq","normalize_poly"]])),

     (*-------------------------expanded-----------------------*)

     (Specify.prep_pbt thy "pbl_equ_univ_expand" [] e_pblID

       (["expanded","univariate","equation"],

         [("#Given", ["equality e_e","solveFor v_v"]),

           ("#Where", ["matches (?a = 0) e_e",

 	          "(lhs e_e) is_expanded_in v_v "]),

           ("#Find", ["solutions v_v'i'"])],

         PolyEq_prls, SOME "solve (e_e::bool, v_v)", [])),

     (*--- d2 ---*)

     (Specify.prep_pbt thy "pbl_equ_univ_expand_deg2" [] e_pblID

       (["degree_2","expanded","univariate","equation"],

         [("#Given", ["equality e_e","solveFor v_v"]),

           ("#Where", ["((lhs e_e) has_degree_in v_v) = 2"]),

           ("#Find", ["solutions v_v'i'"])],

          PolyEq_prls, SOME "solve (e_e::bool, v_v)", [["PolyEq","complete_square"]]))] *}

 ML{*

 val scr =     

     "Script Normalize_poly (e_e::bool) (v_v::real) =                     " ^

     "(let e_e =((Try         (Rewrite     all_left          False)) @@  " ^ 

     "          (Try (Repeat (Rewrite     makex1_x         False))) @@  " ^ 

     "          (Try (Repeat (Rewrite_Set expand_binoms    False))) @@  " ^ 

     "          (Try (Repeat (Rewrite_Set_Inst [(bdv,v_v::real)]         " ^

     "                                 make_ratpoly_in     False))) @@  " ^

     "          (Try (Repeat (Rewrite_Set polyeq_simplify  False)))) e_e " ^

     " in (SubProblem (PolyEq',[polynomial,univariate,equation], [no_met])   " ^

     "                 [BOOL e_e, REAL v_v]))";

 parse thy scr;

 *}

 text {* "-------------------------methods-----------------------" *}

 setup {* KEStore_Elems.add_mets

   [Specify.prep_met thy "met_polyeq" [] e_metID

       (["PolyEq"], [],

         {rew_ord'="tless_true",rls'=Atools_erls,calc = [], srls = e_rls, prls=e_rls,

           crls=PolyEq_crls, errpats = [], nrls = norm_Rational},

         "empty_script"),

     Specify.prep_met thy "met_polyeq_norm" [] e_metID

       (["PolyEq","normalize_poly"],

         [("#Given" ,["equality e_e","solveFor v_v"]),

           ("#Where" ,["(Not((matches (?a = 0 ) e_e ))) |(Not(((lhs e_e) is_poly_in v_v)))"]),

           ("#Find"  ,["solutions v_v'i'"])],

         {rew_ord'="termlessI", rls'=PolyEq_erls, srls=e_rls, prls=PolyEq_prls, calc=[],

           crls=PolyEq_crls, errpats = [], nrls = norm_Rational},

         "Script Normalize_poly (e_e::bool) (v_v::real) =                     " ^

           "(let e_e =((Try         (Rewrite     all_left          False)) @@  " ^ 

           "          (Try (Repeat (Rewrite     makex1_x         False))) @@  " ^ 

           "          (Try (Repeat (Rewrite_Set expand_binoms    False))) @@  " ^ 

           "          (Try (Repeat (Rewrite_Set_Inst [(bdv,v_v::real)]         " ^

           "                                 make_ratpoly_in     False))) @@  " ^

           "          (Try (Repeat (Rewrite_Set polyeq_simplify  False)))) e_e " ^

           " in (SubProblem (PolyEq',[polynomial,univariate,equation], [no_met])   " ^

           "                 [BOOL e_e, REAL v_v]))"),

     Specify.prep_met thy "met_polyeq_d0" [] e_metID

       (["PolyEq","solve_d0_polyeq_equation"],

         [("#Given" ,["equality e_e","solveFor v_v"]),

           ("#Where" ,["(lhs e_e) is_poly_in v_v ", "((lhs e_e) has_degree_in v_v) = 0"]),

           ("#Find"  ,["solutions v_v'i'"])],

         {rew_ord'="termlessI", rls'=PolyEq_erls, srls=e_rls, prls=PolyEq_prls,

           calc=[("sqrt", ("NthRoot.sqrt", eval_sqrt "#sqrt_"))], crls=PolyEq_crls, errpats = [],

           nrls = norm_Rational},

         "Script Solve_d0_polyeq_equation  (e_e::bool) (v_v::real)  = " ^

           "(let e_e =  ((Try (Rewrite_Set_Inst [(bdv,v_v::real)]      " ^

           "                  d0_polyeq_simplify  False))) e_e        " ^

           " in ((Or_to_List e_e)::bool list))"),

     Specify.prep_met thy "met_polyeq_d1" [] e_metID

       (["PolyEq","solve_d1_polyeq_equation"],

         [("#Given" ,["equality e_e","solveFor v_v"]),

           ("#Where" ,["(lhs e_e) is_poly_in v_v ", "((lhs e_e) has_degree_in v_v) = 1"]),

           ("#Find"  ,["solutions v_v'i'"])],

         {rew_ord'="termlessI", rls'=PolyEq_erls, srls=e_rls, prls=PolyEq_prls,

           calc=[("sqrt", ("NthRoot.sqrt", eval_sqrt "#sqrt_"))], crls=PolyEq_crls, errpats = [],

           nrls = norm_Rational},

         "Script Solve_d1_polyeq_equation  (e_e::bool) (v_v::real)  =   " ^

           "(let e_e =  ((Try (Rewrite_Set_Inst [(bdv,v_v::real)]        " ^

           "                  d1_polyeq_simplify   True))          @@  " ^

           "            (Try (Rewrite_Set polyeq_simplify  False)) @@  " ^

           "            (Try (Rewrite_Set norm_Rational_parenthesized False))) e_e;" ^

           " (L_L::bool list) = ((Or_to_List e_e)::bool list)            " ^

           " in Check_elementwise L_L {(v_v::real). Assumptions} )"),

     Specify.prep_met thy "met_polyeq_d22" [] e_metID

       (["PolyEq","solve_d2_polyeq_equation"],

         [("#Given" ,["equality e_e","solveFor v_v"]),

           ("#Where" ,["(lhs e_e) is_poly_in v_v ", "((lhs e_e) has_degree_in v_v) = 2"]),

           ("#Find"  ,["solutions v_v'i'"])],

         {rew_ord'="termlessI", rls'=PolyEq_erls, srls=e_rls, prls=PolyEq_prls,

           calc=[("sqrt", ("NthRoot.sqrt", eval_sqrt "#sqrt_"))], crls=PolyEq_crls, errpats = [],

           nrls = norm_Rational},

           "Script Solve_d2_polyeq_equation  (e_e::bool) (v_v::real) =      " ^

             "  (let e_e = ((Try (Rewrite_Set_Inst [(bdv,v_v::real)]         " ^

             "                    d2_polyeq_simplify           True)) @@   " ^

             "             (Try (Rewrite_Set polyeq_simplify   False)) @@  " ^

             "             (Try (Rewrite_Set_Inst [(bdv,v_v::real)]         " ^

             "                    d1_polyeq_simplify            True)) @@  " ^

             "            (Try (Rewrite_Set polyeq_simplify    False)) @@  " ^

             "            (Try (Rewrite_Set norm_Rational_parenthesized False))) e_e;" ^

             " (L_L::bool list) = ((Or_to_List e_e)::bool list)              " ^

             " in Check_elementwise L_L {(v_v::real). Assumptions} )"),

     Specify.prep_met thy "met_polyeq_d2_bdvonly" [] e_metID

       (["PolyEq","solve_d2_polyeq_bdvonly_equation"],

         [("#Given" ,["equality e_e","solveFor v_v"]),

           ("#Where" ,["(lhs e_e) is_poly_in v_v ", "((lhs e_e) has_degree_in v_v) = 2"]),

           ("#Find"  ,["solutions v_v'i'"])],

         {rew_ord'="termlessI", rls'=PolyEq_erls, srls=e_rls, prls=PolyEq_prls,

           calc=[("sqrt", ("NthRoot.sqrt", eval_sqrt "#sqrt_"))], crls=PolyEq_crls, errpats = [],

           nrls = norm_Rational},

           "Script Solve_d2_polyeq_bdvonly_equation  (e_e::bool) (v_v::real) =" ^

             "  (let e_e = ((Try (Rewrite_Set_Inst [(bdv,v_v::real)]         " ^

             "                   d2_polyeq_bdv_only_simplify    True)) @@  " ^

             "             (Try (Rewrite_Set polyeq_simplify   False)) @@  " ^

             "             (Try (Rewrite_Set_Inst [(bdv,v_v::real)]         " ^

             "                   d1_polyeq_simplify             True)) @@  " ^

             "            (Try (Rewrite_Set polyeq_simplify    False)) @@  " ^

             "            (Try (Rewrite_Set norm_Rational_parenthesized False))) e_e;" ^

             " (L_L::bool list) = ((Or_to_List e_e)::bool list)              " ^

             " in Check_elementwise L_L {(v_v::real). Assumptions} )"),

     Specify.prep_met thy "met_polyeq_d2_sqonly" [] e_metID

       (["PolyEq","solve_d2_polyeq_sqonly_equation"],

         [("#Given" ,["equality e_e","solveFor v_v"]),

           ("#Where" ,["(lhs e_e) is_poly_in v_v ", "((lhs e_e) has_degree_in v_v) = 2"]),

           ("#Find"  ,["solutions v_v'i'"])],

         {rew_ord'="termlessI", rls'=PolyEq_erls, srls=e_rls, prls=PolyEq_prls,

           calc=[("sqrt", ("NthRoot.sqrt", eval_sqrt "#sqrt_"))], crls=PolyEq_crls, errpats = [],

           nrls = norm_Rational},

           "Script Solve_d2_polyeq_sqonly_equation  (e_e::bool) (v_v::real) =" ^

             "  (let e_e = ((Try (Rewrite_Set_Inst [(bdv,v_v::real)]          " ^

             "                   d2_polyeq_sq_only_simplify     True)) @@   " ^

             "            (Try (Rewrite_Set polyeq_simplify    False)) @@   " ^

             "            (Try (Rewrite_Set norm_Rational_parenthesized False))) e_e; " ^

             " (L_L::bool list) = ((Or_to_List e_e)::bool list)               " ^

             " in Check_elementwise L_L {(v_v::real). Assumptions} )"),

     Specify.prep_met thy "met_polyeq_d2_pq" [] e_metID

       (["PolyEq","solve_d2_polyeq_pq_equation"],

         [("#Given" ,["equality e_e","solveFor v_v"]),

           ("#Where" ,["(lhs e_e) is_poly_in v_v ", "((lhs e_e) has_degree_in v_v) = 2"]),

           ("#Find"  ,["solutions v_v'i'"])],

         {rew_ord'="termlessI", rls'=PolyEq_erls, srls=e_rls, prls=PolyEq_prls,

           calc=[("sqrt", ("NthRoot.sqrt", eval_sqrt "#sqrt_"))], crls=PolyEq_crls, errpats = [],

           nrls = norm_Rational},

           "Script Solve_d2_polyeq_pq_equation  (e_e::bool) (v_v::real) =   " ^

             "  (let e_e = ((Try (Rewrite_Set_Inst [(bdv,v_v::real)]         " ^

             "                   d2_polyeq_pqFormula_simplify   True)) @@  " ^

             "            (Try (Rewrite_Set polyeq_simplify    False)) @@  " ^

             "            (Try (Rewrite_Set norm_Rational_parenthesized False))) e_e;" ^

             " (L_L::bool list) = ((Or_to_List e_e)::bool list)              " ^

             " in Check_elementwise L_L {(v_v::real). Assumptions} )"),

     Specify.prep_met thy "met_polyeq_d2_abc" [] e_metID

       (["PolyEq","solve_d2_polyeq_abc_equation"],

         [("#Given" ,["equality e_e","solveFor v_v"]),

           ("#Where" ,["(lhs e_e) is_poly_in v_v ", "((lhs e_e) has_degree_in v_v) = 2"]),

           ("#Find"  ,["solutions v_v'i'"])],

         {rew_ord'="termlessI", rls'=PolyEq_erls,srls=e_rls, prls=PolyEq_prls,

           calc=[("sqrt", ("NthRoot.sqrt", eval_sqrt "#sqrt_"))], crls=PolyEq_crls, errpats = [],

           nrls = norm_Rational},

           "Script Solve_d2_polyeq_abc_equation  (e_e::bool) (v_v::real) =   " ^

             "  (let e_e = ((Try (Rewrite_Set_Inst [(bdv,v_v::real)]          " ^

             "                   d2_polyeq_abcFormula_simplify   True)) @@  " ^

             "            (Try (Rewrite_Set polyeq_simplify     False)) @@  " ^

             "            (Try (Rewrite_Set norm_Rational_parenthesized False))) e_e;" ^

             " (L_L::bool list) = ((Or_to_List e_e)::bool list)               " ^

             " in Check_elementwise L_L {(v_v::real). Assumptions} )"),

     Specify.prep_met thy "met_polyeq_d3" [] e_metID

       (["PolyEq","solve_d3_polyeq_equation"],

         [("#Given" ,["equality e_e","solveFor v_v"]),

           ("#Where" ,["(lhs e_e) is_poly_in v_v ", "((lhs e_e) has_degree_in v_v) = 3"]),

           ("#Find"  ,["solutions v_v'i'"])],

         {rew_ord'="termlessI", rls'=PolyEq_erls, srls=e_rls, prls=PolyEq_prls,

           calc=[("sqrt", ("NthRoot.sqrt", eval_sqrt "#sqrt_"))], crls=PolyEq_crls, errpats = [],

           nrls = norm_Rational},

         "Script Solve_d3_polyeq_equation  (e_e::bool) (v_v::real) =     " ^

           "  (let e_e = ((Try (Rewrite_Set_Inst [(bdv,v_v::real)]        " ^

           "                    d3_polyeq_simplify           True)) @@  " ^

           "             (Try (Rewrite_Set polyeq_simplify  False)) @@  " ^

           "             (Try (Rewrite_Set_Inst [(bdv,v_v::real)]        " ^

           "                    d2_polyeq_simplify           True)) @@  " ^

           "             (Try (Rewrite_Set polyeq_simplify  False)) @@  " ^

           "             (Try (Rewrite_Set_Inst [(bdv,v_v::real)]        " ^   

           "                    d1_polyeq_simplify           True)) @@  " ^

           "             (Try (Rewrite_Set polyeq_simplify  False)) @@  " ^

           "             (Try (Rewrite_Set norm_Rational_parenthesized False))) e_e;" ^

           " (L_L::bool list) = ((Or_to_List e_e)::bool list)             " ^

           " in Check_elementwise L_L {(v_v::real). Assumptions} )"),

     (*.solves all expanded (ie. normalized) terms of degree 2.*) 

     (*Oct.02 restriction: 'eval_true 0 =< discriminant' ony for integer values

       by 'PolyEq_erls'; restricted until Float.thy is implemented*)

     Specify.prep_met thy "met_polyeq_complsq" [] e_metID

       (["PolyEq","complete_square"],

         [("#Given" ,["equality e_e","solveFor v_v"]),

           ("#Where" ,["matches (?a = 0) e_e", "((lhs e_e) has_degree_in v_v) = 2"]),

           ("#Find"  ,["solutions v_v'i'"])],

         {rew_ord'="termlessI",rls'=PolyEq_erls,srls=e_rls,prls=PolyEq_prls,

           calc=[("sqrt", ("NthRoot.sqrt", eval_sqrt "#sqrt_"))], crls=PolyEq_crls, errpats = [],

           nrls = norm_Rational},

         "Script Complete_square (e_e::bool) (v_v::real) =                         " ^

           "(let e_e = " ^ 

           "    ((Try (Rewrite_Set_Inst [(bdv,v_v)] cancel_leading_coeff True)) " ^

           "        @@ (Try (Rewrite_Set_Inst [(bdv,v_v)] complete_square True))     " ^

           "        @@ (Try (Rewrite square_explicit1 False))                       " ^

           "        @@ (Try (Rewrite square_explicit2 False))                       " ^

           "        @@ (Rewrite root_plus_minus True)                               " ^

           "        @@ (Try (Repeat (Rewrite_Inst [(bdv,v_v)] bdv_explicit1 False))) " ^

           "        @@ (Try (Repeat (Rewrite_Inst [(bdv,v_v)] bdv_explicit2 False))) " ^

           "        @@ (Try (Repeat                                                 " ^

           "                  (Rewrite_Inst [(bdv,v_v)] bdv_explicit3 False)))       " ^

           "        @@ (Try (Rewrite_Set calculate_RootRat False))                  " ^

           "        @@ (Try (Repeat (Calculate SQRT)))) e_e                         " ^

           " in ((Or_to_List e_e)::bool list))")]

 *}

 ML{*

 (* termorder hacked by MG *)

 local (*. for make_polynomial_in .*)

 open Term;  (* for type order = EQUAL | LESS | GREATER *)

 fun pr_ord EQUAL = "EQUAL"

   | pr_ord LESS  = "LESS"

   | pr_ord GREATER = "GREATER";

 fun dest_hd' x (Const (a, T)) = (((a, 0), T), 0)

   | dest_hd' x (t as Free (a, T)) =

     if x = t then ((("|||||||||||||", 0), T), 0)                        (*WN*)

     else (((a, 0), T), 1)

   | dest_hd' x (Var v) = (v, 2)

   | dest_hd' x (Bound i) = ((("", i), dummyT), 3)

   | dest_hd' x (Abs (_, T, _)) = ((("", 0), T), 4);

 fun size_of_term' x (Const ("Atools.pow",_) $ Free (var,_) $ Free (pot,_)) =

     (case x of                                                          (*WN*)

 	    (Free (xstr,_)) => 

 		(if xstr = var then 1000*(the (int_of_str pot)) else 3)

 	  | _ => error ("size_of_term' called with subst = "^

 			      (term2str x)))

   | size_of_term' x (Free (subst,_)) =

     (case x of

 	    (Free (xstr,_)) => (if xstr = subst then 1000 else 1)

 	  | _ => error ("size_of_term' called with subst = "^

 			  (term2str x)))

   | size_of_term' x (Abs (_,_,body)) = 1 + size_of_term' x body

   | size_of_term' x (f$t) = size_of_term' x f  +  size_of_term' x t

   | size_of_term' x _ = 1;

 fun term_ord' x pr thy (Abs (_, T, t), Abs(_, U, u)) =       (* ~ term.ML *)

     (case term_ord' x pr thy (t, u) of EQUAL => Term_Ord.typ_ord (T, U) | ord => ord)

   | term_ord' x pr thy (t, u) =

     (if pr

      then 

        let

          val (f, ts) = strip_comb t and (g, us) = strip_comb u;

          val _ = tracing ("t= f@ts= \"" ^ term_to_string''' thy f ^ "\" @ \"[" ^

            commas (map (term_to_string''' thy) ts) ^ "]\"");

          val _ = tracing ("u= g@us= \"" ^ term_to_string''' thy g ^ "\" @ \"[" ^

            commas(map (term_to_string''' thy) us) ^ "]\"");

          val _ = tracing ("size_of_term(t,u)= (" ^ string_of_int (size_of_term' x t) ^ ", " ^

            string_of_int (size_of_term' x u) ^ ")");

          val _ = tracing ("hd_ord(f,g)      = " ^ (pr_ord o (hd_ord x)) (f,g));

          val _ = tracing ("terms_ord(ts,us) = " ^ (pr_ord o (terms_ord x) str false) (ts, us));

          val _ = tracing ("-------");

        in () end

      else ();

 	  case int_ord (size_of_term' x t, size_of_term' x u) of

 	    EQUAL =>

 	      let val (f, ts) = strip_comb t and (g, us) = strip_comb u 

         in

 	        (case hd_ord x (f, g) of 

 	           EQUAL => (terms_ord x str pr) (ts, us) 

 	         | ord => ord)

 	      end

 	 | ord => ord)

 and hd_ord x (f, g) =                                        (* ~ term.ML *)

   prod_ord (prod_ord Term_Ord.indexname_ord Term_Ord.typ_ord) 

             int_ord (dest_hd' x f, dest_hd' x g)

 and terms_ord x str pr (ts, us) = 

     list_ord (term_ord' x pr (assoc_thy "Isac"))(ts, us);

 in

 fun ord_make_polynomial_in (pr:bool) thy subst tu = 

     let

 	(* val _=tracing("*** subs variable is: "^(subst2str subst)); *)

     in

 	case subst of

 	    (_,x)::_ => (term_ord' x pr thy tu = LESS)

 	  | _ => error ("ord_make_polynomial_in called with subst = "^

 			  (subst2str subst))

     end;

 end;(*local*)

 *}

 ML{*

 val order_add_mult_in = prep_rls'(

   Rls{id = "order_add_mult_in", preconds = [], 

       rew_ord = ("ord_make_polynomial_in",

 		 ord_make_polynomial_in false @{theory "Poly"}),

       erls = e_rls,srls = Erls,

       calc = [], errpatts = [],

       rules = [Thm ("mult_commute",num_str @{thm mult.commute}),

 	       (* z * w = w * z *)

 	       Thm ("real_mult_left_commute",num_str @{thm real_mult_left_commute}),

 	       (*z1.0 * (z2.0 * z3.0) = z2.0 * (z1.0 * z3.0)*)

 	       Thm ("mult_assoc",num_str @{thm mult.assoc}),		

 	       (*z1.0 * z2.0 * z3.0 = z1.0 * (z2.0 * z3.0)*)

 	       Thm ("add_commute",num_str @{thm add.commute}),	

 	       (*z + w = w + z*)

 	       Thm ("add_left_commute",num_str @{thm add.left_commute}),

 	       (*x + (y + z) = y + (x + z)*)

 	       Thm ("add_assoc",num_str @{thm add.assoc})	               

 	       (*z1.0 + z2.0 + z3.0 = z1.0 + (z2.0 + z3.0)*)

 	       ], scr = EmptyScr}:rls);

 *}

 ML{*

 val collect_bdv = prep_rls'(

   Rls{id = "collect_bdv", preconds = [], 

       rew_ord = ("dummy_ord", dummy_ord),

       erls = e_rls,srls = Erls,

       calc = [], errpatts = [],

       rules = [Thm ("bdv_collect_1",num_str @{thm bdv_collect_1}),

 	       Thm ("bdv_collect_2",num_str @{thm bdv_collect_2}),

 	       Thm ("bdv_collect_3",num_str @{thm bdv_collect_3}),

 	       Thm ("bdv_collect_assoc1_1",num_str @{thm bdv_collect_assoc1_1}),

 	       Thm ("bdv_collect_assoc1_2",num_str @{thm bdv_collect_assoc1_2}),

 	       Thm ("bdv_collect_assoc1_3",num_str @{thm bdv_collect_assoc1_3}),

 	       Thm ("bdv_collect_assoc2_1",num_str @{thm bdv_collect_assoc2_1}),

 	       Thm ("bdv_collect_assoc2_2",num_str @{thm bdv_collect_assoc2_2}),

 	       Thm ("bdv_collect_assoc2_3",num_str @{thm bdv_collect_assoc2_3}),

 	       Thm ("bdv_n_collect_1",num_str @{thm bdv_n_collect_1}),

 	       Thm ("bdv_n_collect_2",num_str @{thm bdv_n_collect_2}),

 	       Thm ("bdv_n_collect_3",num_str @{thm bdv_n_collect_3}),

 	       Thm ("bdv_n_collect_assoc1_1",num_str @{thm bdv_n_collect_assoc1_1}),

 	       Thm ("bdv_n_collect_assoc1_2",num_str @{thm bdv_n_collect_assoc1_2}),

 	       Thm ("bdv_n_collect_assoc1_3",num_str @{thm bdv_n_collect_assoc1_3}),

 	       Thm ("bdv_n_collect_assoc2_1",num_str @{thm bdv_n_collect_assoc2_1}),

 	       Thm ("bdv_n_collect_assoc2_2",num_str @{thm bdv_n_collect_assoc2_2}),

 	       Thm ("bdv_n_collect_assoc2_3",num_str @{thm bdv_n_collect_assoc2_3})

 	       ], scr = EmptyScr}:rls);

 *}

 ML{*

 (*.transforms an arbitrary term without roots to a polynomial [4] 

    according to knowledge/Poly.sml.*) 

 val make_polynomial_in = prep_rls'(

   Seq {id = "make_polynomial_in", preconds = []:term list, 

        rew_ord = ("dummy_ord", dummy_ord),

       erls = Atools_erls, srls = Erls,

       calc = [], errpatts = [],

       rules = [Rls_ expand_poly,

 	       Rls_ order_add_mult_in,

 	       Rls_ simplify_power,

 	       Rls_ collect_numerals,

 	       Rls_ reduce_012,

 	       Thm ("realpow_oneI",num_str @{thm realpow_oneI}),

 	       Rls_ discard_parentheses,

 	       Rls_ collect_bdv

 	       ],

       scr = EmptyScr

       }:rls);     

 *}

 ML{*

 val separate_bdvs = 

     append_rls "separate_bdvs"

 	       collect_bdv

 	       [Thm ("separate_bdv", num_str @{thm separate_bdv}),

 		(*"?a * ?bdv / ?b = ?a / ?b * ?bdv"*)

 		Thm ("separate_bdv_n", num_str @{thm separate_bdv_n}),

 		Thm ("separate_1_bdv", num_str @{thm separate_1_bdv}),

 		(*"?bdv / ?b = (1 / ?b) * ?bdv"*)

 		Thm ("separate_1_bdv_n", num_str @{thm separate_1_bdv_n}),

 		(*"?bdv ^^^ ?n / ?b = 1 / ?b * ?bdv ^^^ ?n"*)

 		Thm ("add_divide_distrib", 

 		     num_str @{thm add_divide_distrib})

 		(*"(?x + ?y) / ?z = ?x / ?z + ?y / ?z"

 		      WN051031 DOES NOT BELONG TO HERE*)

 		];

 *}

 ML{*

 val make_ratpoly_in = prep_rls'(

   Seq {id = "make_ratpoly_in", preconds = []:term list, 

        rew_ord = ("dummy_ord", dummy_ord),

       erls = Atools_erls, srls = Erls,

       calc = [], errpatts = [],

       rules = [Rls_ norm_Rational,

 	       Rls_ order_add_mult_in,

 	       Rls_ discard_parentheses,

 	       Rls_ separate_bdvs,

 	       (* Rls_ rearrange_assoc, WN060916 why does cancel_p not work?*)

 	       Rls_ cancel_p

 	       (*Calc ("Fields.inverse_class.divide"  ,eval_cancel "#divide_e") too weak!*)

 	       ],

       scr = EmptyScr}:rls);      

 *}

 setup {* KEStore_Elems.add_rlss 

   [("order_add_mult_in", (Context.theory_name @{theory}, order_add_mult_in)), 

   ("collect_bdv", (Context.theory_name @{theory}, collect_bdv)), 

   ("make_polynomial_in", (Context.theory_name @{theory}, make_polynomial_in)), 

   ("make_ratpoly_in", (Context.theory_name @{theory}, make_ratpoly_in)), 

   ("separate_bdvs", (Context.theory_name @{theory}, separate_bdvs))] *}

 end