(* theory collecting all knowledge 

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

   for PolynomialEquations.

   alternative dependencies see @{theory "Isac_Knowledge"}

   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 

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

(* type real enforced by op " \<up> " *)

axiomatization where

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

			                   (a/c + b/c*bdv + bdv \<up> 2 = 0)" and

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

			                   (a/c - b/c*bdv + bdv \<up> 2 = 0)" and

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

			                   (a/c + b/c*bdv - bdv \<up> 2 = 0)" and

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

			                   (a/c + 1/c*bdv + bdv \<up> 2 = 0)" and

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

			                   (a/c - 1/c*bdv + bdv \<up> 2 = 0)" and

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

			                   (a/c + 1/c*bdv - bdv \<up> 2 = 0)" and

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

			                   (    b/c*bdv + bdv \<up> 2 = 0)" and

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

			                   (    b/c*bdv - bdv \<up> 2 = 0)" and

  cancel_leading_coeff9: "Not (c =!= 0) ==> (      bdv + c*bdv \<up> 2 = 0) = 

			                   (      1/c*bdv + bdv \<up> 2 = 0)" and

  cancel_leading_coeff10:"Not (c =!= 0) ==> (      bdv - c*bdv \<up> 2 = 0) = 

			                   (      1/c*bdv - bdv \<up> 2 = 0)" and

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

			                   (a/b +      bdv \<up> 2 = 0)" and

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

			                   (a/b -      bdv \<up> 2 = 0)" and

  cancel_leading_coeff13:"Not (c =!= 0) ==> (         b*bdv \<up> 2 = 0) = 

			                   (           bdv \<up> 2 = 0/b)" and

  complete_square1:      "(q + p*bdv + bdv \<up> 2 = 0) = 

		         (q + (p/2 + bdv) \<up> 2 = (p/2) \<up> 2)" and

  complete_square2:      "(    p*bdv + bdv \<up> 2 = 0) = 

		         (    (p/2 + bdv) \<up> 2 = (p/2) \<up> 2)" and

  complete_square3:      "(      bdv + bdv \<up> 2 = 0) = 

		         (    (1/2 + bdv) \<up> 2 = (1/2) \<up> 2)" and

  complete_square4:      "(q - p*bdv + bdv \<up> 2 = 0) = 

		         (q + (p/2 - bdv) \<up> 2 = (p/2) \<up> 2)" and

  complete_square5:      "(q + p*bdv - bdv \<up> 2 = 0) = 

		         (q + (p/2 - bdv) \<up> 2 = (p/2) \<up> 2)" and

  square_explicit1:      "(a + b \<up> 2 = c) = ( b \<up> 2 = c - a)" and

  square_explicit2:      "(a - b \<up> 2 = c) = (-(b \<up> 2) = c - a)" and

  (*bdv_explicit* required type constrain to real in --- (-8 - 2*x + x \<up> 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

(*-- normalise --*)

  (*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\<up>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 \<up> 2=0) = (b*bdv \<up> 2= (-1)*a)" and

  d2_isolate_add2:

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

  d2_isolate_div:

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

  d2_prescind1:          "(a*bdv + b*bdv \<up> 2 = 0) = (bdv*(a +b*bdv)=0)" and

  d2_prescind2:          "(a*bdv +   bdv \<up> 2 = 0) = (bdv*(a +  bdv)=0)" and

  d2_prescind3:          "(  bdv + b*bdv \<up> 2 = 0) = (bdv*(1+b*bdv)=0)" and

  d2_prescind4:          "(  bdv +   bdv \<up> 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 \<up> 2=c) = ((bdv=sqrt c) | (bdv=(-1)*sqrt c ))" and

 d2_sqrt_equation1_neg:

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

  d2_sqrt_equation2:     "(bdv \<up> 2=0) = (bdv=0)" and

  d2_sqrt_equation3:     "(b*bdv \<up> 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 " \<up> " 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 \<up> 2 - 4*q|] ==> (q+p*bdv+   bdv \<up> 2=0) =

                           ((bdv= (-1)*(p/2) + sqrt(p \<up> 2 - 4*q)/2) 

                          | (bdv= (-1)*(p/2) - sqrt(p \<up> 2 - 4*q)/2))" and

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

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

                           ((bdv= (-1)*(p/2) + sqrt(p \<up> 2 - 4*q)/2) 

                          | (bdv= (-1)*(p/2) - sqrt(p \<up> 2 - 4*q)/2))" and

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

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

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

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

                           ((bdv= (-1)*(p/2) + sqrt(p \<up> 2 - 0)/2) 

                          | (bdv= (-1)*(p/2) - sqrt(p \<up> 2 - 0)/2))" and

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

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

                           ((bdv= (-1)*(p/2) + sqrt(p \<up> 2 - 0)/2) 

                          | (bdv= (-1)*(p/2) - sqrt(p \<up> 2 - 0)/2))" and

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

   d2_pqformula7:        "[|0<=1 - 0|] ==> (    bdv+   bdv \<up> 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 \<up> 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 \<up> 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 \<up> 2=0) = False" and

  d2_pqformula10:

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

  d2_abcformula1:

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

           ((bdv=( -b + sqrt(b \<up> 2 - 4*a*c))/(2*a)) 

          | (bdv=( -b - sqrt(b \<up> 2 - 4*a*c))/(2*a)))" and

  d2_abcformula1_neg:

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

  d2_abcformula2:

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

  d2_abcformula3:

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

           ((bdv=( -b + sqrt(b \<up> 2 - 4*1*c))/(2*1)) 

          | (bdv=( -b - sqrt(b \<up> 2 - 4*1*c))/(2*1)))" and

  d2_abcformula3_neg:

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

  d2_abcformula4:

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

  d2_abcformula5:

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

  d2_abcformula6:

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

  d2_abcformula7:

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

           ((bdv=( -b + sqrt(b \<up> 2 - 0))/(2*a)) 

          | (bdv=( -b - sqrt(b \<up> 2 - 0))/(2*a)))" and

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

  d2_abcformula8:

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

           ((bdv=( -b + sqrt(b \<up> 2 - 0))/(2*1)) 

          | (bdv=( -b - sqrt(b \<up> 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 \<up> 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 \<up> 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 \<up> 2 + c*bdv \<up> 3=0) = (bdv=0 | (a + b*bdv + c*bdv \<up> 2=0))" and

  d3_reduce_equation2:

  "(  bdv + b*bdv \<up> 2 + c*bdv \<up> 3=0) = (bdv=0 | (1 + b*bdv + c*bdv \<up> 2=0))" and

  d3_reduce_equation3:

  "(a*bdv +   bdv \<up> 2 + c*bdv \<up> 3=0) = (bdv=0 | (a +   bdv + c*bdv \<up> 2=0))" and

  d3_reduce_equation4:

  "(  bdv +   bdv \<up> 2 + c*bdv \<up> 3=0) = (bdv=0 | (1 +   bdv + c*bdv \<up> 2=0))" and

  d3_reduce_equation5:

  "(a*bdv + b*bdv \<up> 2 +   bdv \<up> 3=0) = (bdv=0 | (a + b*bdv +   bdv \<up> 2=0))" and

  d3_reduce_equation6:

  "(  bdv + b*bdv \<up> 2 +   bdv \<up> 3=0) = (bdv=0 | (1 + b*bdv +   bdv \<up> 2=0))" and

  d3_reduce_equation7:

  "(a*bdv +   bdv \<up> 2 +   bdv \<up> 3=0) = (bdv=0 | (1 +   bdv +   bdv \<up> 2=0))" and

  d3_reduce_equation8:

  "(  bdv +   bdv \<up> 2 +   bdv \<up> 3=0) = (bdv=0 | (1 +   bdv +   bdv \<up> 2=0))" and

  d3_reduce_equation9:

  "(a*bdv             + c*bdv \<up> 3=0) = (bdv=0 | (a         + c*bdv \<up> 2=0))" and

  d3_reduce_equation10:

  "(  bdv             + c*bdv \<up> 3=0) = (bdv=0 | (1         + c*bdv \<up> 2=0))" and

  d3_reduce_equation11:

  "(a*bdv             +   bdv \<up> 3=0) = (bdv=0 | (a         +   bdv \<up> 2=0))" and

  d3_reduce_equation12:

  "(  bdv             +   bdv \<up> 3=0) = (bdv=0 | (1         +   bdv \<up> 2=0))" and

  d3_reduce_equation13:

  "(        b*bdv \<up> 2 + c*bdv \<up> 3=0) = (bdv=0 | (    b*bdv + c*bdv \<up> 2=0))" and

  d3_reduce_equation14:

  "(          bdv \<up> 2 + c*bdv \<up> 3=0) = (bdv=0 | (      bdv + c*bdv \<up> 2=0))" and

  d3_reduce_equation15:

  "(        b*bdv \<up> 2 +   bdv \<up> 3=0) = (bdv=0 | (    b*bdv +   bdv \<up> 2=0))" and

  d3_reduce_equation16:

  "(          bdv \<up> 2 +   bdv \<up> 3=0) = (bdv=0 | (      bdv +   bdv \<up> 2=0))" and

  d3_isolate_add1:

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

  d3_isolate_add2:

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

  d3_isolate_div:

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

  d3_root_equation2:

  "(bdv \<up> 3=0) = (bdv=0)" and

  d3_root_equation1:

  "(bdv \<up> 3=c) = (bdv = nroot 3 c)" and

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

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

 d4_sub_u1:

 "(c+b*bdv \<up> 2+a*bdv \<up> 4=0) =

   ((a*u \<up> 2+b*u+c=0) & (bdv \<up> 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 \<up> n + m * bdv \<up> n = (l + m) * bdv \<up> n" and

  bdv_n_collect_2:     " bdv \<up> n + m * bdv \<up> n = (1 + m) * bdv \<up> n" and

  bdv_n_collect_3:     "l * bdv \<up> n + bdv \<up> n = (l + 1) * bdv \<up> n" (*order!*) and

  bdv_n_collect_assoc1_1:

                      "l * bdv \<up> n + (m * bdv \<up> n + k) = (l + m) * bdv \<up> n + k" and

  bdv_n_collect_assoc1_2: "bdv \<up> n + (m * bdv \<up> n + k) = (1 + m) * bdv \<up> n + k" and

  bdv_n_collect_assoc1_3: "l * bdv \<up> n + (bdv \<up> n + k) = (l + 1) * bdv \<up> n + k" and

  bdv_n_collect_assoc2_1: "k + l * bdv \<up> n + m * bdv \<up> n = k +(l + m) * bdv \<up> n" and

  bdv_n_collect_assoc2_2: "k + bdv \<up> n + m * bdv \<up> n = k + (1 + m) * bdv \<up> n" and

  bdv_n_collect_assoc2_3: "k + l * bdv \<up> n + bdv \<up> n = k + (l + 1) * bdv \<up> 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 \<up> n) / b = (a / b) * bdv \<up> n" and

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

  separate_1_bdv_n:       "bdv \<up> n / b = (1 / b) * bdv \<up> n"

ML \<open>

val thy = @{theory};

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

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

  Rule_Set.append_rules "PolyEq_prls" Rule_Set.empty 

	     [Rule.Eval ("Prog_Expr.ident", Prog_Expr.eval_ident "#ident_"),

	      Rule.Eval ("Prog_Expr.matches", Prog_Expr.eval_matches ""),

	      Rule.Eval ("Prog_Expr.lhs", Prog_Expr.eval_lhs ""),

	      Rule.Eval ("Prog_Expr.rhs", Prog_Expr.eval_rhs ""),

	      Rule.Eval ("Poly.is_expanded_in", eval_is_expanded_in ""),

	      Rule.Eval ("Poly.is_poly_in", eval_is_poly_in ""),

	      Rule.Eval ("Poly.has_degree_in", eval_has_degree_in ""),    

              Rule.Eval ("Poly.is_polyrat_in", eval_is_polyrat_in ""),

	      (*Rule.Eval ("Prog_Expr.occurs_in", Prog_Expr.eval_occurs_in ""),   *) 

	      (*Rule.Eval ("Prog_Expr.is_const", Prog_Expr.eval_const "#is_const_"),*)

	      Rule.Eval ("HOL.eq", Prog_Expr.eval_equal "#equal_"),

              Rule.Eval ("RootEq.is_rootTerm_in", eval_is_rootTerm_in ""),

	      Rule.Eval ("RatEq.is_ratequation_in", eval_is_ratequation_in ""),

	      Rule.Thm ("not_true",ThmC.numerals_to_Free @{thm not_true}),

	      Rule.Thm ("not_false",ThmC.numerals_to_Free @{thm not_false}),

	      Rule.Thm ("and_true",ThmC.numerals_to_Free @{thm and_true}),

	      Rule.Thm ("and_false",ThmC.numerals_to_Free @{thm and_false}),

	      Rule.Thm ("or_true",ThmC.numerals_to_Free @{thm or_true}),

	      Rule.Thm ("or_false",ThmC.numerals_to_Free @{thm or_false})

	       ];

val PolyEq_erls = 

    Rule_Set.merge "PolyEq_erls" LinEq_erls

    (Rule_Set.append_rules "ops_preds" calculate_Rational

		[Rule.Eval ("HOL.eq", Prog_Expr.eval_equal "#equal_"),

		 Rule.Thm ("plus_leq", ThmC.numerals_to_Free @{thm plus_leq}),

		 Rule.Thm ("minus_leq", ThmC.numerals_to_Free @{thm minus_leq}),

		 Rule.Thm ("rat_leq1", ThmC.numerals_to_Free @{thm rat_leq1}),

		 Rule.Thm ("rat_leq2", ThmC.numerals_to_Free @{thm rat_leq2}),

		 Rule.Thm ("rat_leq3", ThmC.numerals_to_Free @{thm rat_leq3})

		 ]);

val PolyEq_crls = 

    Rule_Set.merge "PolyEq_crls" LinEq_crls

    (Rule_Set.append_rules "ops_preds" calculate_Rational

		[Rule.Eval ("HOL.eq", Prog_Expr.eval_equal "#equal_"),

		 Rule.Thm ("plus_leq", ThmC.numerals_to_Free @{thm plus_leq}),

		 Rule.Thm ("minus_leq", ThmC.numerals_to_Free @{thm minus_leq}),

		 Rule.Thm ("rat_leq1", ThmC.numerals_to_Free @{thm rat_leq1}),

		 Rule.Thm ("rat_leq2", ThmC.numerals_to_Free @{thm rat_leq2}),

		 Rule.Thm ("rat_leq3", ThmC.numerals_to_Free @{thm rat_leq3})

		 ]);

val cancel_leading_coeff = prep_rls'(

  Rule_Def.Repeat {id = "cancel_leading_coeff", preconds = [], 

       rew_ord = ("e_rew_ord",Rewrite_Ord.e_rew_ord),

      erls = PolyEq_erls, srls = Rule_Set.Empty, calc = [], errpatts = [],

      rules = 

      [Rule.Thm ("cancel_leading_coeff1",ThmC.numerals_to_Free @{thm cancel_leading_coeff1}),

       Rule.Thm ("cancel_leading_coeff2",ThmC.numerals_to_Free @{thm cancel_leading_coeff2}),

       Rule.Thm ("cancel_leading_coeff3",ThmC.numerals_to_Free @{thm cancel_leading_coeff3}),

       Rule.Thm ("cancel_leading_coeff4",ThmC.numerals_to_Free @{thm cancel_leading_coeff4}),

       Rule.Thm ("cancel_leading_coeff5",ThmC.numerals_to_Free @{thm cancel_leading_coeff5}),

       Rule.Thm ("cancel_leading_coeff6",ThmC.numerals_to_Free @{thm cancel_leading_coeff6}),

       Rule.Thm ("cancel_leading_coeff7",ThmC.numerals_to_Free @{thm cancel_leading_coeff7}),

       Rule.Thm ("cancel_leading_coeff8",ThmC.numerals_to_Free @{thm cancel_leading_coeff8}),

       Rule.Thm ("cancel_leading_coeff9",ThmC.numerals_to_Free @{thm cancel_leading_coeff9}),

       Rule.Thm ("cancel_leading_coeff10",ThmC.numerals_to_Free @{thm cancel_leading_coeff10}),

       Rule.Thm ("cancel_leading_coeff11",ThmC.numerals_to_Free @{thm cancel_leading_coeff11}),

       Rule.Thm ("cancel_leading_coeff12",ThmC.numerals_to_Free @{thm cancel_leading_coeff12}),

       Rule.Thm ("cancel_leading_coeff13",ThmC.numerals_to_Free @{thm cancel_leading_coeff13})

       ],scr = Rule.Empty_Prog});

val prep_rls' = Auto_Prog.prep_rls @{theory};

\<close>

ML\<open>

val complete_square = prep_rls'(

  Rule_Def.Repeat {id = "complete_square", preconds = [], 

       rew_ord = ("e_rew_ord",Rewrite_Ord.e_rew_ord),

      erls = PolyEq_erls, srls = Rule_Set.Empty, calc = [],  errpatts = [],

      rules = [Rule.Thm ("complete_square1",ThmC.numerals_to_Free @{thm complete_square1}),

	       Rule.Thm ("complete_square2",ThmC.numerals_to_Free @{thm complete_square2}),

	       Rule.Thm ("complete_square3",ThmC.numerals_to_Free @{thm complete_square3}),

	       Rule.Thm ("complete_square4",ThmC.numerals_to_Free @{thm complete_square4}),

	       Rule.Thm ("complete_square5",ThmC.numerals_to_Free @{thm complete_square5})

	       ],

      scr = Rule.Empty_Prog

      });

val polyeq_simplify = prep_rls'(

  Rule_Def.Repeat {id = "polyeq_simplify", preconds = [], 

       rew_ord = ("termlessI",termlessI), 

       erls = PolyEq_erls, 

       srls = Rule_Set.Empty, 

       calc = [], errpatts = [],

       rules = [Rule.Thm  ("real_assoc_1",ThmC.numerals_to_Free @{thm real_assoc_1}),

		Rule.Thm  ("real_assoc_2",ThmC.numerals_to_Free @{thm real_assoc_2}),

		Rule.Thm  ("real_diff_minus",ThmC.numerals_to_Free @{thm real_diff_minus}),

		Rule.Thm  ("real_unari_minus",ThmC.numerals_to_Free @{thm real_unari_minus}),

		Rule.Thm  ("realpow_multI",ThmC.numerals_to_Free @{thm realpow_multI}),

		Rule.Eval ("Groups.plus_class.plus", (**)eval_binop "#add_"),

		Rule.Eval ("Groups.minus_class.minus", (**)eval_binop "#sub_"),

		Rule.Eval ("Groups.times_class.times", (**)eval_binop "#mult_"),

		Rule.Eval ("Rings.divide_class.divide", Prog_Expr.eval_cancel "#divide_e"),

		Rule.Eval ("NthRoot.sqrt", eval_sqrt "#sqrt_"),

		Rule.Eval ("Transcendental.powr" , (**)eval_binop "#power_"),

                Rule.Rls_ reduce_012

                ],

       scr = Rule.Empty_Prog

       });

\<close>

setup_rule

  cancel_leading_coeff = cancel_leading_coeff and

  complete_square = complete_square and

  PolyEq_erls = PolyEq_erls and

  polyeq_simplify = polyeq_simplify

ML\<open>

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

(* -- d0 -- *)

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

val d0_polyeq_simplify = prep_rls'(

  Rule_Def.Repeat {id = "d0_polyeq_simplify", preconds = [],

       rew_ord = ("e_rew_ord",Rewrite_Ord.e_rew_ord),

       erls = PolyEq_erls,

       srls = Rule_Set.Empty, 

       calc = [], errpatts = [],

       rules = [Rule.Thm("d0_true",ThmC.numerals_to_Free @{thm d0_true}),

		Rule.Thm("d0_false",ThmC.numerals_to_Free @{thm  d0_false})

		],

       scr = Rule.Empty_Prog

       });

(* -- d1 -- *)

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

val d1_polyeq_simplify = prep_rls'(

  Rule_Def.Repeat {id = "d1_polyeq_simplify", preconds = [],

       rew_ord = ("e_rew_ord",Rewrite_Ord.e_rew_ord),

       erls = PolyEq_erls,

       srls = Rule_Set.Empty, 

       calc = [], errpatts = [],

       rules = [

		Rule.Thm("d1_isolate_add1",ThmC.numerals_to_Free @{thm d1_isolate_add1}), 

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

		Rule.Thm("d1_isolate_add2",ThmC.numerals_to_Free @{thm d1_isolate_add2}), 

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

		Rule.Thm("d1_isolate_div",ThmC.numerals_to_Free @{thm d1_isolate_div})    

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

		],

       scr = Rule.Empty_Prog

       });

\<close>

subsection \<open>degree 2\<close>

ML\<open>

(* 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'(

  Rule_Def.Repeat {id = "d2_polyeq_bdv_only_simplify", preconds = [], rew_ord = ("e_rew_ord",Rewrite_Ord.e_rew_ord),

    erls = PolyEq_erls, srls = Rule_Set.Empty, calc = [], errpatts = [],

    rules =

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

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

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

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

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

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

       Rule.Thm ("d2_sqrt_equation2", ThmC.numerals_to_Free @{thm d2_sqrt_equation2}),    (*  x^2=0 ->    x=0       *)

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

       Rule.Thm ("d2_reduce_equation2", ThmC.numerals_to_Free @{thm d2_reduce_equation2}),(* x(a+ x)=0 -> x=0 |a+ x=0*)

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

       ],

       scr = Rule.Empty_Prog

       });

\<close>

ML\<open>

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

   'bdv' is a meta-constant*)

val d2_polyeq_sq_only_simplify = prep_rls'(

  Rule_Def.Repeat {id = "d2_polyeq_sq_only_simplify", preconds = [],

       rew_ord = ("e_rew_ord",Rewrite_Ord.e_rew_ord),

       erls = PolyEq_erls,

       srls = Rule_Set.Empty, 

       calc = [], errpatts = [],

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

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

       rules = [Rule.Thm("d2_isolate_add1",ThmC.numerals_to_Free @{thm d2_isolate_add1}),

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

		Rule.Thm("d2_isolate_add2",ThmC.numerals_to_Free @{thm d2_isolate_add2}),

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

		Rule.Thm("d2_sqrt_equation2",ThmC.numerals_to_Free @{thm d2_sqrt_equation2}),

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

		Rule.Thm("d2_sqrt_equation1",ThmC.numerals_to_Free @{thm d2_sqrt_equation1}),

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

		Rule.Thm("d2_sqrt_equation1_neg",ThmC.numerals_to_Free @{thm d2_sqrt_equation1_neg}),

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

		Rule.Thm("d2_isolate_div",ThmC.numerals_to_Free @{thm d2_isolate_div})

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

		],

       scr = Rule.Empty_Prog

       });

\<close>

ML\<open>

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

   'bdv' is a meta-constant*)

val d2_polyeq_pqFormula_simplify = prep_rls'(

  Rule_Def.Repeat {id = "d2_polyeq_pqFormula_simplify", preconds = [],

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

       srls = Rule_Set.Empty, calc = [], errpatts = [],

       rules = [Rule.Thm("d2_pqformula1",ThmC.numerals_to_Free @{thm d2_pqformula1}),

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

		Rule.Thm("d2_pqformula1_neg",ThmC.numerals_to_Free @{thm d2_pqformula1_neg}),

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

		Rule.Thm("d2_pqformula2",ThmC.numerals_to_Free @{thm d2_pqformula2}), 

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

		Rule.Thm("d2_pqformula2_neg",ThmC.numerals_to_Free @{thm d2_pqformula2_neg}),

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

		Rule.Thm("d2_pqformula3",ThmC.numerals_to_Free @{thm d2_pqformula3}),

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

		Rule.Thm("d2_pqformula3_neg",ThmC.numerals_to_Free @{thm d2_pqformula3_neg}), 

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

		Rule.Thm("d2_pqformula4",ThmC.numerals_to_Free @{thm d2_pqformula4}),

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

		Rule.Thm("d2_pqformula4_neg",ThmC.numerals_to_Free @{thm d2_pqformula4_neg}),

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

		Rule.Thm("d2_pqformula5",ThmC.numerals_to_Free @{thm d2_pqformula5}),

                (*   qx+ x^2=0 *)

		Rule.Thm("d2_pqformula6",ThmC.numerals_to_Free @{thm d2_pqformula6}),

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

		Rule.Thm("d2_pqformula7",ThmC.numerals_to_Free @{thm d2_pqformula7}),

                (*    x+ x^2=0 *)

		Rule.Thm("d2_pqformula8",ThmC.numerals_to_Free @{thm d2_pqformula8}),

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

		Rule.Thm("d2_pqformula9",ThmC.numerals_to_Free @{thm d2_pqformula9}),

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

		Rule.Thm("d2_pqformula9_neg",ThmC.numerals_to_Free @{thm d2_pqformula9_neg}),

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

		Rule.Thm("d2_pqformula10",ThmC.numerals_to_Free @{thm d2_pqformula10}),

                (* q   + x^2=0 *)

		Rule.Thm("d2_pqformula10_neg",ThmC.numerals_to_Free @{thm d2_pqformula10_neg}),

                (* q   + x^2=0 *)

		Rule.Thm("d2_sqrt_equation2",ThmC.numerals_to_Free @{thm d2_sqrt_equation2}),

                (*       x^2=0 *)

		Rule.Thm("d2_sqrt_equation3",ThmC.numerals_to_Free @{thm d2_sqrt_equation3})

               (*      1x^2=0 *)

	       ],scr = Rule.Empty_Prog

       });

\<close>

ML\<open>

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

   'bdv' is a meta-constant*)

val d2_polyeq_abcFormula_simplify = prep_rls'(

  Rule_Def.Repeat {id = "d2_polyeq_abcFormula_simplify", preconds = [],

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

       srls = Rule_Set.Empty, calc = [], errpatts = [],

       rules = [Rule.Thm("d2_abcformula1",ThmC.numerals_to_Free @{thm d2_abcformula1}),

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

		Rule.Thm("d2_abcformula1_neg",ThmC.numerals_to_Free @{thm d2_abcformula1_neg}),

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

		Rule.Thm("d2_abcformula2",ThmC.numerals_to_Free @{thm d2_abcformula2}),

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

		Rule.Thm("d2_abcformula2_neg",ThmC.numerals_to_Free @{thm d2_abcformula2_neg}),

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

		Rule.Thm("d2_abcformula3",ThmC.numerals_to_Free @{thm d2_abcformula3}), 

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

		Rule.Thm("d2_abcformula3_neg",ThmC.numerals_to_Free @{thm d2_abcformula3_neg}),

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

		Rule.Thm("d2_abcformula4",ThmC.numerals_to_Free @{thm d2_abcformula4}),

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

		Rule.Thm("d2_abcformula4_neg",ThmC.numerals_to_Free @{thm d2_abcformula4_neg}),

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

		Rule.Thm("d2_abcformula5",ThmC.numerals_to_Free @{thm d2_abcformula5}),

                (*c+   cx^2=0 *)

		Rule.Thm("d2_abcformula5_neg",ThmC.numerals_to_Free @{thm d2_abcformula5_neg}),

                (*c+   cx^2=0 *)

		Rule.Thm("d2_abcformula6",ThmC.numerals_to_Free @{thm d2_abcformula6}),

                (*c+    x^2=0 *)

		Rule.Thm("d2_abcformula6_neg",ThmC.numerals_to_Free @{thm d2_abcformula6_neg}),

                (*c+    x^2=0 *)

		Rule.Thm("d2_abcformula7",ThmC.numerals_to_Free @{thm d2_abcformula7}),

                (*  bx+ax^2=0 *)

		Rule.Thm("d2_abcformula8",ThmC.numerals_to_Free @{thm d2_abcformula8}),

                (*  bx+ x^2=0 *)

		Rule.Thm("d2_abcformula9",ThmC.numerals_to_Free @{thm d2_abcformula9}),

                (*   x+ax^2=0 *)

		Rule.Thm("d2_abcformula10",ThmC.numerals_to_Free @{thm d2_abcformula10}),

                (*   x+ x^2=0 *)

		Rule.Thm("d2_sqrt_equation2",ThmC.numerals_to_Free @{thm d2_sqrt_equation2}),

                (*      x^2=0 *)  

		Rule.Thm("d2_sqrt_equation3",ThmC.numerals_to_Free @{thm d2_sqrt_equation3})

               (*     bx^2=0 *)  

	       ],

       scr = Rule.Empty_Prog

       });

\<close>

ML\<open>

(* isolate the bound variable in an d2 equation; 

   'bdv' is a meta-constant*)

val d2_polyeq_simplify = prep_rls'(

  Rule_Def.Repeat {id = "d2_polyeq_simplify", preconds = [],

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

       srls = Rule_Set.Empty, calc = [], errpatts = [],

       rules = [Rule.Thm("d2_pqformula1",ThmC.numerals_to_Free @{thm d2_pqformula1}),

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

		Rule.Thm("d2_pqformula1_neg",ThmC.numerals_to_Free @{thm d2_pqformula1_neg}),

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

		Rule.Thm("d2_pqformula2",ThmC.numerals_to_Free @{thm d2_pqformula2}),

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

		Rule.Thm("d2_pqformula2_neg",ThmC.numerals_to_Free @{thm d2_pqformula2_neg}),

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

		Rule.Thm("d2_pqformula3",ThmC.numerals_to_Free @{thm d2_pqformula3}),

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

		Rule.Thm("d2_pqformula3_neg",ThmC.numerals_to_Free @{thm d2_pqformula3_neg}),

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

		Rule.Thm("d2_pqformula4",ThmC.numerals_to_Free @{thm d2_pqformula4}), 

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

		Rule.Thm("d2_pqformula4_neg",ThmC.numerals_to_Free @{thm d2_pqformula4_neg}),

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

		Rule.Thm("d2_abcformula1",ThmC.numerals_to_Free @{thm d2_abcformula1}),

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

		Rule.Thm("d2_abcformula1_neg",ThmC.numerals_to_Free @{thm d2_abcformula1_neg}),

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

		Rule.Thm("d2_abcformula2",ThmC.numerals_to_Free @{thm d2_abcformula2}),

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

		Rule.Thm("d2_abcformula2_neg",ThmC.numerals_to_Free @{thm d2_abcformula2_neg}),

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

		Rule.Thm("d2_prescind1",ThmC.numerals_to_Free @{thm d2_prescind1}),

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

		Rule.Thm("d2_prescind2",ThmC.numerals_to_Free @{thm d2_prescind2}),

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

		Rule.Thm("d2_prescind3",ThmC.numerals_to_Free @{thm d2_prescind3}),

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

		Rule.Thm("d2_prescind4",ThmC.numerals_to_Free @{thm d2_prescind4}),

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

		Rule.Thm("d2_isolate_add1",ThmC.numerals_to_Free @{thm d2_isolate_add1}),

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

		Rule.Thm("d2_isolate_add2",ThmC.numerals_to_Free @{thm d2_isolate_add2}),

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

		Rule.Thm("d2_sqrt_equation1",ThmC.numerals_to_Free @{thm d2_sqrt_equation1}),

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

		Rule.Thm("d2_sqrt_equation1_neg",ThmC.numerals_to_Free @{thm d2_sqrt_equation1_neg}),

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

		Rule.Thm("d2_sqrt_equation2",ThmC.numerals_to_Free @{thm d2_sqrt_equation2}),

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

		Rule.Thm("d2_reduce_equation1",ThmC.numerals_to_Free @{thm d2_reduce_equation1}),

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

		Rule.Thm("d2_reduce_equation2",ThmC.numerals_to_Free @{thm d2_reduce_equation2}),

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

		Rule.Thm("d2_isolate_div",ThmC.numerals_to_Free @{thm d2_isolate_div})

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

	       ],

       scr = Rule.Empty_Prog

      });

\<close>

ML\<open>

(* -- d3 -- *)

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

val d3_polyeq_simplify = prep_rls'(

  Rule_Def.Repeat {id = "d3_polyeq_simplify", preconds = [],

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

       srls = Rule_Set.Empty, calc = [], errpatts = [],

       rules = 

       [Rule.Thm("d3_reduce_equation1",ThmC.numerals_to_Free @{thm d3_reduce_equation1}),

	(*a*bdv + b*bdv \<up> 2 + c*bdv \<up> 3=0) = 

        (bdv=0 | (a + b*bdv + c*bdv \<up> 2=0)*)

	Rule.Thm("d3_reduce_equation2",ThmC.numerals_to_Free @{thm d3_reduce_equation2}),

	(*  bdv + b*bdv \<up> 2 + c*bdv \<up> 3=0) = 

        (bdv=0 | (1 + b*bdv + c*bdv \<up> 2=0)*)

	Rule.Thm("d3_reduce_equation3",ThmC.numerals_to_Free @{thm d3_reduce_equation3}),

	(*a*bdv +   bdv \<up> 2 + c*bdv \<up> 3=0) = 

        (bdv=0 | (a +   bdv + c*bdv \<up> 2=0)*)

	Rule.Thm("d3_reduce_equation4",ThmC.numerals_to_Free @{thm d3_reduce_equation4}),

	(*  bdv +   bdv \<up> 2 + c*bdv \<up> 3=0) = 

        (bdv=0 | (1 +   bdv + c*bdv \<up> 2=0)*)

	Rule.Thm("d3_reduce_equation5",ThmC.numerals_to_Free @{thm d3_reduce_equation5}),

	(*a*bdv + b*bdv \<up> 2 +   bdv \<up> 3=0) = 

        (bdv=0 | (a + b*bdv +   bdv \<up> 2=0)*)

	Rule.Thm("d3_reduce_equation6",ThmC.numerals_to_Free @{thm d3_reduce_equation6}),

	(*  bdv + b*bdv \<up> 2 +   bdv \<up> 3=0) = 

        (bdv=0 | (1 + b*bdv +   bdv \<up> 2=0)*)

	Rule.Thm("d3_reduce_equation7",ThmC.numerals_to_Free @{thm d3_reduce_equation7}),

	     (*a*bdv +   bdv \<up> 2 +   bdv \<up> 3=0) = 

             (bdv=0 | (1 +   bdv +   bdv \<up> 2=0)*)

	Rule.Thm("d3_reduce_equation8",ThmC.numerals_to_Free @{thm d3_reduce_equation8}),

	     (*  bdv +   bdv \<up> 2 +   bdv \<up> 3=0) = 

             (bdv=0 | (1 +   bdv +   bdv \<up> 2=0)*)

	Rule.Thm("d3_reduce_equation9",ThmC.numerals_to_Free @{thm d3_reduce_equation9}),

	     (*a*bdv             + c*bdv \<up> 3=0) = 

             (bdv=0 | (a         + c*bdv \<up> 2=0)*)

	Rule.Thm("d3_reduce_equation10",ThmC.numerals_to_Free @{thm d3_reduce_equation10}),

	     (*  bdv             + c*bdv \<up> 3=0) = 

             (bdv=0 | (1         + c*bdv \<up> 2=0)*)

	Rule.Thm("d3_reduce_equation11",ThmC.numerals_to_Free @{thm d3_reduce_equation11}),

	     (*a*bdv             +   bdv \<up> 3=0) = 

             (bdv=0 | (a         +   bdv \<up> 2=0)*)

	Rule.Thm("d3_reduce_equation12",ThmC.numerals_to_Free @{thm d3_reduce_equation12}),

	     (*  bdv             +   bdv \<up> 3=0) = 

             (bdv=0 | (1         +   bdv \<up> 2=0)*)

	Rule.Thm("d3_reduce_equation13",ThmC.numerals_to_Free @{thm d3_reduce_equation13}),

	     (*        b*bdv \<up> 2 + c*bdv \<up> 3=0) = 

             (bdv=0 | (    b*bdv + c*bdv \<up> 2=0)*)

	Rule.Thm("d3_reduce_equation14",ThmC.numerals_to_Free @{thm d3_reduce_equation14}),

	     (*          bdv \<up> 2 + c*bdv \<up> 3=0) = 

             (bdv=0 | (      bdv + c*bdv \<up> 2=0)*)

	Rule.Thm("d3_reduce_equation15",ThmC.numerals_to_Free @{thm d3_reduce_equation15}),

	     (*        b*bdv \<up> 2 +   bdv \<up> 3=0) = 

             (bdv=0 | (    b*bdv +   bdv \<up> 2=0)*)

	Rule.Thm("d3_reduce_equation16",ThmC.numerals_to_Free @{thm d3_reduce_equation16}),

	     (*          bdv \<up> 2 +   bdv \<up> 3=0) = 

             (bdv=0 | (      bdv +   bdv \<up> 2=0)*)

	Rule.Thm("d3_isolate_add1",ThmC.numerals_to_Free @{thm d3_isolate_add1}),

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

              (bdv=0 | (b*bdv \<up> 3=a)*)

	Rule.Thm("d3_isolate_add2",ThmC.numerals_to_Free @{thm d3_isolate_add2}),

             (*[|Not(bdv occurs_in a)|] ==> (a +   bdv \<up> 3=0) = 

              (bdv=0 | (  bdv \<up> 3=a)*)

	Rule.Thm("d3_isolate_div",ThmC.numerals_to_Free @{thm d3_isolate_div}),

        (*[|Not(b=0)|] ==> (b*bdv \<up> 3=c) = (bdv \<up> 3=c/b*)

        Rule.Thm("d3_root_equation2",ThmC.numerals_to_Free @{thm d3_root_equation2}),

        (*(bdv \<up> 3=0) = (bdv=0) *)

	Rule.Thm("d3_root_equation1",ThmC.numerals_to_Free @{thm d3_root_equation1})

       (*bdv \<up> 3=c) = (bdv = nroot 3 c*)

       ],

       scr = Rule.Empty_Prog

      });

\<close>

ML\<open>

(* -- d4 -- *)

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

val d4_polyeq_simplify = prep_rls'(

  Rule_Def.Repeat {id = "d4_polyeq_simplify", preconds = [],

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

       srls = Rule_Set.Empty, calc = [], errpatts = [],

       rules = 

       [Rule.Thm("d4_sub_u1",ThmC.numerals_to_Free @{thm d4_sub_u1})  

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

       ],

       scr = Rule.Empty_Prog

      });

\<close>

setup_rule

  d0_polyeq_simplify = d0_polyeq_simplify and

  d1_polyeq_simplify = d1_polyeq_simplify and

  d2_polyeq_simplify = d2_polyeq_simplify and

  d2_polyeq_bdv_only_simplify = d2_polyeq_bdv_only_simplify and

  d2_polyeq_sq_only_simplify = d2_polyeq_sq_only_simplify and

  d2_polyeq_pqFormula_simplify = d2_polyeq_pqFormula_simplify and

  d2_polyeq_abcFormula_simplify = d2_polyeq_abcFormula_simplify and

  d3_polyeq_simplify = d3_polyeq_simplify and

  d4_polyeq_simplify = d4_polyeq_simplify

setup \<open>KEStore_Elems.add_pbts

  [(Problem.prep_input thy "pbl_equ_univ_poly" [] Problem.id_empty

      (["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 ---*)

    (Problem.prep_input thy "pbl_equ_univ_poly_deg0" [] Problem.id_empty

      (["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 ---*)

    (Problem.prep_input thy "pbl_equ_univ_poly_deg1" [] Problem.id_empty

      (["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 ---*)

    (Problem.prep_input thy "pbl_equ_univ_poly_deg2" [] Problem.id_empty

      (["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"]])),

    (Problem.prep_input thy "pbl_equ_univ_poly_deg2_sqonly" [] Problem.id_empty

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

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

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

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

            "matches (         ?v_ \<up> 2 = 0) e_e | " ^

            "matches (      ?b*?v_ \<up> 2 = 0) e_e" ,

            "Not (matches (?a +    ?v_ +    ?v_ \<up> 2 = 0) e_e) &" ^

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

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

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

            "Not (matches (        ?v_ +    ?v_ \<up> 2 = 0) e_e) &" ^

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

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

            "Not (matches (     ?b*?v_ + ?c*?v_ \<up> 2 = 0) e_e)"]),

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

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

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

    (Problem.prep_input thy "pbl_equ_univ_poly_deg2_bdvonly" [] Problem.id_empty

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

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

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

            "matches (   ?v_ +    ?v_ \<up> 2 = 0) e_e | " ^

            "matches (   ?v_ + ?b*?v_ \<up> 2 = 0) e_e | " ^

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

            "matches (            ?v_ \<up> 2 = 0) e_e | " ^

            "matches (         ?b*?v_ \<up> 2 = 0) e_e "]),

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

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

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

    (Problem.prep_input thy "pbl_equ_univ_poly_deg2_pq" [] Problem.id_empty

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

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

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

	          "matches (?a +   ?v_ \<up> 2 = 0) e_e"]),

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

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

    (Problem.prep_input thy "pbl_equ_univ_poly_deg2_abc" [] Problem.id_empty

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

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

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

	          "matches (?a + ?b*?v_ \<up> 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 ---*)

    (Problem.prep_input thy "pbl_equ_univ_poly_deg3" [] Problem.id_empty

      (["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 ---*)

    (Problem.prep_input thy "pbl_equ_univ_poly_deg4" [] Problem.id_empty

      (["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"]*)])),

    (*--- normalise ---*)

    (Problem.prep_input thy "pbl_equ_univ_poly_norm" [] Problem.id_empty

      (["normalise", "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", "normalise_poly"]])),

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

    (Problem.prep_input thy "pbl_equ_univ_expand" [] Problem.id_empty

      (["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 ---*)

    (Problem.prep_input thy "pbl_equ_univ_expand_deg2" [] Problem.id_empty

      (["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"]]))]\<close>

text \<open>"-------------------------methods-----------------------"\<close>

setup \<open>KEStore_Elems.add_mets

    [MethodC.prep_input thy "met_polyeq" [] MethodC.id_empty

      (["PolyEq"], [],

        {rew_ord'="tless_true",rls'=Atools_erls,calc = [], srls = Rule_Set.empty, prls=Rule_Set.empty,

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

        @{thm refl})]

\<close>

partial_function (tailrec) normalize_poly_eq :: "bool \<Rightarrow> real \<Rightarrow> bool"

  where

"normalize_poly_eq e_e v_v = (

  let

    e_e = (

      (Try (Rewrite ''all_left'')) #>

      (Try (Repeat (Rewrite ''makex1_x''))) #>

      (Try (Repeat (Rewrite_Set ''expand_binoms''))) #>

      (Try (Repeat (Rewrite_Set_Inst [(''bdv'', v_v)] ''make_ratpoly_in''))) #>

      (Try (Repeat (Rewrite_Set ''polyeq_simplify''))) ) e_e

  in

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

      [BOOL e_e, REAL v_v])"

setup \<open>KEStore_Elems.add_mets

    [MethodC.prep_input thy "met_polyeq_norm" [] MethodC.id_empty

      (["PolyEq", "normalise_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=Rule_Set.empty, prls=PolyEq_prls, calc=[],

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

        @{thm normalize_poly_eq.simps})]

\<close>

partial_function (tailrec) solve_poly_equ :: "bool \<Rightarrow> real \<Rightarrow> bool list"

  where

"solve_poly_equ e_e v_v = (

  let

    e_e = (Try (Rewrite_Set_Inst [(''bdv'', v_v)] ''d0_polyeq_simplify'')) e_e   

  in

    Or_to_List e_e)"

setup \<open>KEStore_Elems.add_mets

    [MethodC.prep_input thy "met_polyeq_d0" [] MethodC.id_empty

      (["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=Rule_Set.empty, prls=PolyEq_prls,

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

          nrls = norm_Rational},

        @{thm solve_poly_equ.simps})]

\<close>

partial_function (tailrec) solve_poly_eq1 :: "bool \<Rightarrow> real \<Rightarrow> bool list"

  where

"solve_poly_eq1 e_e v_v = (

  let

    e_e = (

      (Try (Rewrite_Set_Inst [(''bdv'', v_v)] ''d1_polyeq_simplify'')) #>

      (Try (Rewrite_Set ''polyeq_simplify'')) #> 

      (Try (Rewrite_Set ''norm_Rational_parenthesized'')) ) e_e;

    L_L = Or_to_List e_e

  in

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

setup \<open>KEStore_Elems.add_mets

    [MethodC.prep_input thy "met_polyeq_d1" [] MethodC.id_empty

      (["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=Rule_Set.empty, prls=PolyEq_prls,

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

          nrls = norm_Rational},

        @{thm solve_poly_eq1.simps})]

\<close>

partial_function (tailrec) solve_poly_equ2 :: "bool \<Rightarrow> real \<Rightarrow> bool list"

  where

"solve_poly_equ2 e_e v_v = (

  let

    e_e = (

      (Try (Rewrite_Set_Inst [(''bdv'', v_v)] ''d2_polyeq_simplify'')) #>

      (Try (Rewrite_Set ''polyeq_simplify'')) #>

      (Try (Rewrite_Set_Inst [(''bdv'', v_v)] ''d1_polyeq_simplify'')) #>

      (Try (Rewrite_Set ''polyeq_simplify'')) #>

      (Try (Rewrite_Set ''norm_Rational_parenthesized'')) ) e_e;

    L_L =  Or_to_List e_e

  in

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

setup \<open>KEStore_Elems.add_mets

    [MethodC.prep_input thy "met_polyeq_d22" [] MethodC.id_empty

      (["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=Rule_Set.empty, prls=PolyEq_prls,

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

          nrls = norm_Rational},

        @{thm solve_poly_equ2.simps})]

\<close>