wneuper/isa: src/Tools/isac/Knowledge/PolyEq.thy@bd4f7a35e892

     1 (* theory collecting all knowledge

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

     3    for PolynomialEquations.

     4    alternative dependencies see Isac.thy

     5    created by: rlang

     6          date: 02.07

     7    changed by: rlang

     8    last change by: rlang

     9              date: 03.06.03

    10    (c) by Richard Lang, 2003

    11 *)

    13 theory PolyEq imports LinEq RootRatEq begin

    15 consts

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

    18   Complete'_square

    19              :: "[bool,real,

    20 		   bool list] => bool list"

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

    22                   (_))" 9)

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

    24   Normalize'_poly

    25              :: "[bool,real,

    26 		   bool list] => bool list"

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

    28                   (_))" 9)

    29   Solve'_d0'_polyeq'_equation

    30              :: "[bool,real,

    31 		   bool list] => bool list"

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

    33                   (_))" 9)

    34   Solve'_d1'_polyeq'_equation

    35              :: "[bool,real,

    36 		   bool list] => bool list"

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

    38                   (_))" 9)

    39   Solve'_d2'_polyeq'_equation

    40              :: "[bool,real,

    41 		   bool list] => bool list"

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

    43                   (_))" 9)

    44   Solve'_d2'_polyeq'_sqonly'_equation

    45              :: "[bool,real,

    46 		   bool list] => bool list"

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

    48                   (_))" 9)

    49   Solve'_d2'_polyeq'_bdvonly'_equation

    50              :: "[bool,real,

    51 		   bool list] => bool list"

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

    53                   (_))" 9)

    54   Solve'_d2'_polyeq'_pq'_equation

    55              :: "[bool,real,

    56 		   bool list] => bool list"

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

    58                   (_))" 9)

    59   Solve'_d2'_polyeq'_abc'_equation

    60              :: "[bool,real,

    61 		   bool list] => bool list"

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

    63                   (_))" 9)

    64   Solve'_d3'_polyeq'_equation

    65              :: "[bool,real,

    66 		   bool list] => bool list"

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

    68                   (_))" 9)

    69   Solve'_d4'_polyeq'_equation

    70              :: "[bool,real,

    71 		   bool list] => bool list"

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

    73                   (_))" 9)

    74   Biquadrat'_poly

    75              :: "[bool,real,

    76 		   bool list] => bool list"

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

    78                   (_))" 9)

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

    81 axioms

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   103 			                   (      1/c*bdv + bdv^^^2 = 0)"

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

   105 			                   (      1/c*bdv - bdv^^^2 = 0)"

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

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

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

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

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

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

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

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

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

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

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

   119 		         (    (1/2 + bdv)^^^2 = (1/2)^^^2)"

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

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

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

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

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

   127   square_explicit2      "(a - b^^^2 = c) = (-(b^^^2) = c - a)"

   129   bdv_explicit1         "(a + bdv = b) = (bdv = - a + b)"

   130   bdv_explicit2         "(a - bdv = b) = ((-1)*bdv = - a + b)"

   131   bdv_explicit3         "((-1)*bdv = b) = (bdv = (-1)*b)"

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

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

   136 (*-- normalize --*)

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

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

   139   makex1_x              "a^^^1  = a"

   140   real_assoc_1          "a+(b+c) = a+b+c"

   141   real_assoc_2          "a*(b*c) = a*b*c"

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

   144   d0_true               "(0=0) = True"

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

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

   147   d1_isolate_add1

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

   149   d1_isolate_add2

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

   151   d1_isolate_div

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

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

   154   d2_isolate_add1

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

   156   d2_isolate_add2

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

   158   d2_isolate_div

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

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

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

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

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

   165   (* eliminate degree 2 *)

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

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

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

   169   d2_sqrt_equation1_neg

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

   171   d2_sqrt_equation2     "(bdv^^^2=0) = (bdv=0)"

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

   173   d2_reduce_equation1   "(bdv*(a +b*bdv)=0) = ((bdv=0)|(a+b*bdv=0))"

   174   d2_reduce_equation2   "(bdv*(a +  bdv)=0) = ((bdv=0)|(a+  bdv=0))"

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   210   d2_pqformula9_neg

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

   212   d2_pqformula10

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

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

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

   216   d2_pqformula10_neg

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

   218   d2_abcformula1

   219    "[|0<=b^^^2 - 4*a*c|] ==> (c + b*bdv+a*bdv^^^2=0) =

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

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

   222   d2_abcformula1_neg

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

   224   d2_abcformula2

   225    "[|0<=1 - 4*a*c|]     ==> (c+    bdv+a*bdv^^^2=0) =

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

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

   228   d2_abcformula2_neg

   229    "[|1 - 4*a*c<0|]     ==> (c+    bdv+a*bdv^^^2=0) = False"

   230   d2_abcformula3

   231    "[|0<=b^^^2 - 4*1*c|] ==> (c + b*bdv+  bdv^^^2=0) =

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

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

   234   d2_abcformula3_neg

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

   236   d2_abcformula4

   237    "[|0<=1 - 4*1*c|] ==> (c +   bdv+  bdv^^^2=0) =

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

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

   240   d2_abcformula4_neg

   241    "[|1 - 4*1*c<0|] ==> (c +   bdv+  bdv^^^2=0) = False"

   242   d2_abcformula5

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

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

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

   246   d2_abcformula5_neg

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

   248   d2_abcformula6

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

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

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

   252   d2_abcformula6_neg

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

   254   d2_abcformula7

   255    "[|0<=b^^^2 - 0|]     ==> (    b*bdv+a*bdv^^^2=0) =

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

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

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

   259   d2_abcformula8

   260    "[|0<=b^^^2 - 0|] ==> (    b*bdv+  bdv^^^2=0) =

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

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

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

   264   d2_abcformula9

   265    "[|0<=1 - 0|]     ==> (      bdv+a*bdv^^^2=0) =

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

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

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

   269   d2_abcformula10

   270    "[|0<=1 - 0|] ==> (      bdv+  bdv^^^2=0) =

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

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

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

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

   276   d3_reduce_equation1

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

   278   d3_reduce_equation2

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

   280   d3_reduce_equation3

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

   282   d3_reduce_equation4

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

   284   d3_reduce_equation5

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

   286   d3_reduce_equation6

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

   288   d3_reduce_equation7

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

   290   d3_reduce_equation8

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

   292   d3_reduce_equation9

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

   294   d3_reduce_equation10

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

   296   d3_reduce_equation11

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

   298   d3_reduce_equation12

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

   300   d3_reduce_equation13

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

   302   d3_reduce_equation14

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

   304   d3_reduce_equation15

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

   306   d3_reduce_equation16

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

   308   d3_isolate_add1

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

   310   d3_isolate_add2

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

   312   d3_isolate_div

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

   314   d3_root_equation2

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

   316   d3_root_equation1

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

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

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

   321  d4_sub_u1

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

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

   325 (* ---- 7.3.02 von Termorder ---- *)

   327   bdv_collect_1       "l * bdv + m * bdv = (l + m) * bdv"

   328   bdv_collect_2       "bdv + m * bdv = (1 + m) * bdv"

   329   bdv_collect_3       "l * bdv + bdv = (l + 1) * bdv"

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

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

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

   334 *)

   335   bdv_collect_assoc1_1 "l * bdv + (m * bdv + k) = (l + m) * bdv + k"

   336   bdv_collect_assoc1_2 "bdv + (m * bdv + k) = (1 + m) * bdv + k"

   337   bdv_collect_assoc1_3 "l * bdv + (bdv + k) = (l + 1) * bdv + k"

   339   bdv_collect_assoc2_1 "k + l * bdv + m * bdv = k + (l + m) * bdv"

   340   bdv_collect_assoc2_2 "k + bdv + m * bdv = k + (1 + m) * bdv"

   341   bdv_collect_assoc2_3 "k + l * bdv + bdv = k + (l + 1) * bdv"

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

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

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

   348   bdv_n_collect_assoc1_1 "l * bdv^^^n + (m * bdv^^^n + k) = (l + m) * bdv^^^n + k"

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

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

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

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

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

   356 (*WN.14.3.03*)

   357   real_minus_div         "- (a / b) = (-1 * a) / b"

   359   separate_bdv           "(a * bdv) / b = (a / b) * bdv"

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

   361   separate_1_bdv         "bdv / b = (1 / b) * bdv"

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

   364 ML {*

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

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

   367   append_rls "PolyEq_prls" e_rls

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

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

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

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

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

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

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

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

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

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

   378 	      Calc ("op =",eval_equal "#equal_"),

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

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

   381 	      Thm ("not_true",num_str @{not_true),

   382 	      Thm ("not_false",num_str @{not_false),

   383 	      Thm ("and_true",num_str @{and_true),

   384 	      Thm ("and_false",num_str @{and_false),

   385 	      Thm ("or_true",num_str @{or_true),

   386 	      Thm ("or_false",num_str @{or_false)

   387 	       ];

   389 val PolyEq_erls =

   390     merge_rls "PolyEq_erls" LinEq_erls

   391     (append_rls "ops_preds" calculate_Rational

   392 		[Calc ("op =",eval_equal "#equal_"),

   393 		 Thm ("plus_leq", num_str @{plus_leq),

   394 		 Thm ("minus_leq", num_str @{minus_leq),

   395 		 Thm ("rat_leq1", num_str @{rat_leq1),

   396 		 Thm ("rat_leq2", num_str @{rat_leq2),

   397 		 Thm ("rat_leq3", num_str @{rat_leq3)

   398 		 ]);

   400 val PolyEq_crls =

   401     merge_rls "PolyEq_crls" LinEq_crls

   402     (append_rls "ops_preds" calculate_Rational

   403 		[Calc ("op =",eval_equal "#equal_"),

   404 		 Thm ("plus_leq", num_str @{plus_leq),

   405 		 Thm ("minus_leq", num_str @{minus_leq),

   406 		 Thm ("rat_leq1", num_str @{rat_leq1),

   407 		 Thm ("rat_leq2", num_str @{rat_leq2),

   408 		 Thm ("rat_leq3", num_str @{rat_leq3)

   409 		 ]);

   411 val cancel_leading_coeff = prep_rls(

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

   413        rew_ord = ("e_rew_ord",e_rew_ord),

   414       erls = PolyEq_erls, srls = Erls, calc = [], (*asm_thm = [],*)

   415       rules = [Thm ("cancel_leading_coeff1",num_str @{cancel_leading_coeff1),

   416 	       Thm ("cancel_leading_coeff2",num_str @{cancel_leading_coeff2),

   417 	       Thm ("cancel_leading_coeff3",num_str @{cancel_leading_coeff3),

   418 	       Thm ("cancel_leading_coeff4",num_str @{cancel_leading_coeff4),

   419 	       Thm ("cancel_leading_coeff5",num_str @{cancel_leading_coeff5),

   420 	       Thm ("cancel_leading_coeff6",num_str @{cancel_leading_coeff6),

   421 	       Thm ("cancel_leading_coeff7",num_str @{cancel_leading_coeff7),

   422 	       Thm ("cancel_leading_coeff8",num_str @{cancel_leading_coeff8),

   423 	       Thm ("cancel_leading_coeff9",num_str @{cancel_leading_coeff9),

   424 	       Thm ("cancel_leading_coeff10",num_str @{cancel_leading_coeff10),

   425 	       Thm ("cancel_leading_coeff11",num_str @{cancel_leading_coeff11),

   426 	       Thm ("cancel_leading_coeff12",num_str @{cancel_leading_coeff12),

   427 	       Thm ("cancel_leading_coeff13",num_str @{cancel_leading_coeff13)

   428 	       ],

   429       scr = Script ((term_of o the o (parse thy))

   430       "empty_script")

   431       }:rls);

   433 val complete_square = prep_rls(

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

   435        rew_ord = ("e_rew_ord",e_rew_ord),

   436       erls = PolyEq_erls, srls = Erls, calc = [], (*asm_thm = [],*)

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

   438 	       Thm ("complete_square2",num_str @{complete_square2),

   439 	       Thm ("complete_square3",num_str @{complete_square3),

   440 	       Thm ("complete_square4",num_str @{complete_square4),

   441 	       Thm ("complete_square5",num_str @{complete_square5)

   442 	       ],

   443       scr = Script ((term_of o the o (parse thy))

   444       "empty_script")

   445       }:rls);

   447 val polyeq_simplify = prep_rls(

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

   449        rew_ord = ("termlessI",termlessI),

   450        erls = PolyEq_erls,

   451        srls = Erls,

   452        calc = [],

   453        (*asm_thm = [],*)

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

   455 		Thm  ("real_assoc_2",num_str @{real_assoc_2),

   456 		Thm  ("real_diff_minus",num_str @{real_diff_minus),

   457 		Thm  ("real_unari_minus",num_str @{real_unari_minus),

   458 		Thm  ("realpow_multI",num_str @{realpow_multI),

   459 		Calc ("op +",eval_binop "#add_"),

   460 		Calc ("op -",eval_binop "#sub_"),

   461 		Calc ("op *",eval_binop "#mult_"),

   462 		Calc ("HOL.divide", eval_cancel "#divide_"),

   463 		Calc ("Root.sqrt",eval_sqrt "#sqrt_"),

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

   465                 Rls_ reduce_012

   466                 ],

   467        scr = Script ((term_of o the o (parse thy)) "empty_script")

   468        }:rls);

   470 ruleset' := overwritelthy @{theory} (!ruleset',

   471 		[("cancel_leading_coeff",cancel_leading_coeff),

   472 		 ("complete_square",complete_square),

   473 		 ("PolyEq_erls",PolyEq_erls),(*FIXXXME:del with rls.rls'*)

   474 		 ("polyeq_simplify",polyeq_simplify)]);

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

   478 (* -- d0 -- *)

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

   480 val d0_polyeq_simplify = prep_rls(

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

   482        rew_ord = ("e_rew_ord",e_rew_ord),

   483        erls = PolyEq_erls,

   484        srls = Erls,

   485        calc = [],

   486        (*asm_thm = [],*)

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

   488 		Thm("d0_false",num_str @{d0_false)

   489 		],

   490        scr = Script ((term_of o the o (parse thy)) "empty_script")

   491        }:rls);

   493 (* -- d1 -- *)

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

   495 val d1_polyeq_simplify = prep_rls(

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

   497        rew_ord = ("e_rew_ord",e_rew_ord),

   498        erls = PolyEq_erls,

   499        srls = Erls,

   500        calc = [],

   501        (*asm_thm = [("d1_isolate_div","")],*)

   502        rules = [

   503 		Thm("d1_isolate_add1",num_str @{d1_isolate_add1),

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

   505 		Thm("d1_isolate_add2",num_str @{d1_isolate_add2),

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

   507 		Thm("d1_isolate_div",num_str @{d1_isolate_div)

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

   509 		],

   510        scr = Script ((term_of o the o (parse thy)) "empty_script")

   511        }:rls);

   513 (* -- d2 -- *)

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

   515    'bdv' is a meta-constant*)

   516 val d2_polyeq_bdv_only_simplify = prep_rls(

   517   Rls {id = "d2_polyeq_bdv_only_simplify", preconds = [],

   518        rew_ord = ("e_rew_ord",e_rew_ord),

   519        erls = PolyEq_erls,

   520        srls = Erls,

   521        calc = [],

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

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

   524        rules = [Thm("d2_prescind1",num_str @{d2_prescind1),

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

   526 		Thm("d2_prescind2",num_str @{d2_prescind2),

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

   528 		Thm("d2_prescind3",num_str @{d2_prescind3),

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

   530 		Thm("d2_prescind4",num_str @{d2_prescind4),

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

   532 		Thm("d2_sqrt_equation1",num_str @{d2_sqrt_equation1),

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

   534 		Thm("d2_sqrt_equation1_neg",num_str @{d2_sqrt_equation1_neg),

   535                 (* [0<c] x^2=c  -> [] *)

   536 		Thm("d2_sqrt_equation2",num_str @{d2_sqrt_equation2),

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

   538 		Thm("d2_reduce_equation1",num_str @{d2_reduce_equation1),

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

   540 		Thm("d2_reduce_equation2",num_str @{d2_reduce_equation2),

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

   542 		Thm("d2_isolate_div",num_str @{d2_isolate_div)

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

   544 		],

   545        scr = Script ((term_of o the o (parse thy)) "empty_script")

   546        }:rls);

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

   549    'bdv' is a meta-constant*)

   550 val d2_polyeq_sq_only_simplify = prep_rls(

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

   552        rew_ord = ("e_rew_ord",e_rew_ord),

   553        erls = PolyEq_erls,

   554        srls = Erls,

   555        calc = [],

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

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

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

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

   560 		Thm("d2_isolate_add2",num_str @{d2_isolate_add2),

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

   562 		Thm("d2_sqrt_equation2",num_str @{d2_sqrt_equation2),

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

   564 		Thm("d2_sqrt_equation1",num_str @{d2_sqrt_equation1),

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

   566 		Thm("d2_sqrt_equation1_neg",num_str @{d2_sqrt_equation1_neg),

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

   568 		Thm("d2_isolate_div",num_str @{d2_isolate_div)

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

   570 		],

   571        scr = Script ((term_of o the o (parse thy)) "empty_script")

   572        }:rls);

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

   575    'bdv' is a meta-constant*)

   576 val d2_polyeq_pqFormula_simplify = prep_rls(

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

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

   579        srls = Erls, calc = [],

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

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

   582 		Thm("d2_pqformula1_neg",num_str @{d2_pqformula1_neg),

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

   584 		Thm("d2_pqformula2",num_str @{d2_pqformula2),

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

   586 		Thm("d2_pqformula2_neg",num_str @{d2_pqformula2_neg),

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

   588 		Thm("d2_pqformula3",num_str @{d2_pqformula3),

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

   590 		Thm("d2_pqformula3_neg",num_str @{d2_pqformula3_neg),

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

   592 		Thm("d2_pqformula4",num_str @{d2_pqformula4),

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

   594 		Thm("d2_pqformula4_neg",num_str @{d2_pqformula4_neg),

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

   596 		Thm("d2_pqformula5",num_str @{d2_pqformula5),

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

   598 		Thm("d2_pqformula6",num_str @{d2_pqformula6),

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

   600 		Thm("d2_pqformula7",num_str @{d2_pqformula7),

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

   602 		Thm("d2_pqformula8",num_str @{d2_pqformula8),

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

   604 		Thm("d2_pqformula9",num_str @{d2_pqformula9),

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

   606 		Thm("d2_pqformula9_neg",num_str @{d2_pqformula9_neg),

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

   608 		Thm("d2_pqformula10",num_str @{d2_pqformula10),

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

   610 		Thm("d2_pqformula10_neg",num_str @{d2_pqformula10_neg),

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

   612 		Thm("d2_sqrt_equation2",num_str @{d2_sqrt_equation2),

   613                 (*       x^2=0 *)

   614 		Thm("d2_sqrt_equation3",num_str @{d2_sqrt_equation3)

   615                (*      1x^2=0 *)

   616 	       ],

   617        scr = Script ((term_of o the o (parse thy)) "empty_script")

   618        }:rls);

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

   621    'bdv' is a meta-constant*)

   622 val d2_polyeq_abcFormula_simplify = prep_rls(

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

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

   625        srls = Erls, calc = [],

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

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

   628 		Thm("d2_abcformula1_neg",num_str @{d2_abcformula1_neg),

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

   630 		Thm("d2_abcformula2",num_str @{d2_abcformula2),

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

   632 		Thm("d2_abcformula2_neg",num_str @{d2_abcformula2_neg),

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

   634 		Thm("d2_abcformula3",num_str @{d2_abcformula3),

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

   636 		Thm("d2_abcformula3_neg",num_str @{d2_abcformula3_neg),

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

   638 		Thm("d2_abcformula4",num_str @{d2_abcformula4),

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

   640 		Thm("d2_abcformula4_neg",num_str @{d2_abcformula4_neg),

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

   642 		Thm("d2_abcformula5",num_str @{d2_abcformula5),

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

   644 		Thm("d2_abcformula5_neg",num_str @{d2_abcformula5_neg),

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

   646 		Thm("d2_abcformula6",num_str @{d2_abcformula6),

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

   648 		Thm("d2_abcformula6_neg",num_str @{d2_abcformula6_neg),

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

   650 		Thm("d2_abcformula7",num_str @{d2_abcformula7),

   651                 (*  bx+ax^2=0 *)

   652 		Thm("d2_abcformula8",num_str @{d2_abcformula8),

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

   654 		Thm("d2_abcformula9",num_str @{d2_abcformula9),

   655                 (*   x+ax^2=0 *)

   656 		Thm("d2_abcformula10",num_str @{d2_abcformula10),

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

   658 		Thm("d2_sqrt_equation2",num_str @{d2_sqrt_equation2),

   659                 (*      x^2=0 *)

   660 		Thm("d2_sqrt_equation3",num_str @{d2_sqrt_equation3)

   661                (*     bx^2=0 *)

   662 	       ],

   663        scr = Script ((term_of o the o (parse thy)) "empty_script")

   664        }:rls);

   666 (* isolate the bound variable in an d2 equation;

   667    'bdv' is a meta-constant*)

   668 val d2_polyeq_simplify = prep_rls(

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

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

   671        srls = Erls, calc = [],

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

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

   674 		Thm("d2_pqformula1_neg",num_str @{d2_pqformula1_neg),

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

   676 		Thm("d2_pqformula2",num_str @{d2_pqformula2),

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

   678 		Thm("d2_pqformula2_neg",num_str @{d2_pqformula2_neg),

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

   680 		Thm("d2_pqformula3",num_str @{d2_pqformula3),

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

   682 		Thm("d2_pqformula3_neg",num_str @{d2_pqformula3_neg),

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

   684 		Thm("d2_pqformula4",num_str @{d2_pqformula4),

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

   686 		Thm("d2_pqformula4_neg",num_str @{d2_pqformula4_neg),

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

   688 		Thm("d2_abcformula1",num_str @{d2_abcformula1),

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

   690 		Thm("d2_abcformula1_neg",num_str @{d2_abcformula1_neg),

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

   692 		Thm("d2_abcformula2",num_str @{d2_abcformula2),

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

   694 		Thm("d2_abcformula2_neg",num_str @{d2_abcformula2_neg),

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

   696 		Thm("d2_prescind1",num_str @{d2_prescind1),

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

   698 		Thm("d2_prescind2",num_str @{d2_prescind2),

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

   700 		Thm("d2_prescind3",num_str @{d2_prescind3),

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

   702 		Thm("d2_prescind4",num_str @{d2_prescind4),

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

   704 		Thm("d2_isolate_add1",num_str @{d2_isolate_add1),

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

   706 		Thm("d2_isolate_add2",num_str @{d2_isolate_add2),

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

   708 		Thm("d2_sqrt_equation1",num_str @{d2_sqrt_equation1),

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

   710 		Thm("d2_sqrt_equation1_neg",num_str @{d2_sqrt_equation1_neg),

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

   712 		Thm("d2_sqrt_equation2",num_str @{d2_sqrt_equation2),

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

   714 		Thm("d2_reduce_equation1",num_str @{d2_reduce_equation1),

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

   716 		Thm("d2_reduce_equation2",num_str @{d2_reduce_equation2),

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

   718 		Thm("d2_isolate_div",num_str @{d2_isolate_div)

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

   720 	       ],

   721        scr = Script ((term_of o the o (parse thy)) "empty_script")

   722       }:rls);

   724 (* -- d3 -- *)

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

   726 val d3_polyeq_simplify = prep_rls(

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

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

   729        srls = Erls, calc = [],

   730        rules =

   731        [Thm("d3_reduce_equation1",num_str @{d3_reduce_equation1),

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

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

   734 	Thm("d3_reduce_equation2",num_str @{d3_reduce_equation2),

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

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

   737 	Thm("d3_reduce_equation3",num_str @{d3_reduce_equation3),

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

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

   740 	Thm("d3_reduce_equation4",num_str @{d3_reduce_equation4),

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

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

   743 	Thm("d3_reduce_equation5",num_str @{d3_reduce_equation5),

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

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

   746 	Thm("d3_reduce_equation6",num_str @{d3_reduce_equation6),

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

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

   749 	Thm("d3_reduce_equation7",num_str @{d3_reduce_equation7),

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

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

   752 	Thm("d3_reduce_equation8",num_str @{d3_reduce_equation8),

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

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

   755 	Thm("d3_reduce_equation9",num_str @{d3_reduce_equation9),

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

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

   758 	Thm("d3_reduce_equation10",num_str @{d3_reduce_equation10),

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

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

   761 	Thm("d3_reduce_equation11",num_str @{d3_reduce_equation11),

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

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

   764 	Thm("d3_reduce_equation12",num_str @{d3_reduce_equation12),

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

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

   767 	Thm("d3_reduce_equation13",num_str @{d3_reduce_equation13),

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

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

   770 	Thm("d3_reduce_equation14",num_str @{d3_reduce_equation14),

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

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

   773 	Thm("d3_reduce_equation15",num_str @{d3_reduce_equation15),

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

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

   776 	Thm("d3_reduce_equation16",num_str @{d3_reduce_equation16),

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

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

   779 	Thm("d3_isolate_add1",num_str @{d3_isolate_add1),

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

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

   782 	Thm("d3_isolate_add2",num_str @{d3_isolate_add2),

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

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

   785 	Thm("d3_isolate_div",num_str @{d3_isolate_div),

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

   787         Thm("d3_root_equation2",num_str @{d3_root_equation2),

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

   789 	Thm("d3_root_equation1",num_str @{d3_root_equation1)

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

   791        ],

   792        scr = Script ((term_of o the o (parse thy)) "empty_script")

   793       }:rls);

   795 (* -- d4 -- *)

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

   797 val d4_polyeq_simplify = prep_rls(

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

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

   800        srls = Erls, calc = [],

   801        rules =

   802        [Thm("d4_sub_u1",num_str @{d4_sub_u1)

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

   804        ],

   805        scr = Script ((term_of o the o (parse thy)) "empty_script")

   806       }:rls);

   808 ruleset' :=

   809 overwritelthy @{theory}

   810               (!ruleset',

   811                [("d0_polyeq_simplify", d0_polyeq_simplify),

   812                 ("d1_polyeq_simplify", d1_polyeq_simplify),

   813                 ("d2_polyeq_simplify", d2_polyeq_simplify),

   814                 ("d2_polyeq_bdv_only_simplify", d2_polyeq_bdv_only_simplify),

   815                 ("d2_polyeq_sq_only_simplify", d2_polyeq_sq_only_simplify),

   816                 ("d2_polyeq_pqFormula_simplify", d2_polyeq_pqFormula_simplify),

   817                 ("d2_polyeq_abcFormula_simplify",

   818                  d2_polyeq_abcFormula_simplify),

   819                 ("d3_polyeq_simplify", d3_polyeq_simplify),

   820 		("d4_polyeq_simplify", d4_polyeq_simplify)

   821 	      ]);

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

   824 (*

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

   826 show_ptyps();

   827 *)

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

   830 store_pbt

   831  (prep_pbt (theory "PolyEq") "pbl_equ_univ_poly" [] e_pblID

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

   833   [("#Given" ,["equality e_","solveFor v_"]),

   834    ("#Where" ,["~((e_::bool) is_ratequation_in (v_::real))",

   835 	       "~((lhs e_) is_rootTerm_in (v_::real))",

   836 	       "~((rhs e_) is_rootTerm_in (v_::real))"]),

   837    ("#Find"  ,["solutions v_i_"])

   838    ],

   839   PolyEq_prls, SOME "solve (e_::bool, v_)",

   840   []));

   841 (*--- d0 ---*)

   842 store_pbt

   843  (prep_pbt (theory "PolyEq") "pbl_equ_univ_poly_deg0" [] e_pblID

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

   845   [("#Given" ,["equality e_","solveFor v_"]),

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

   847 	       "(lhs e_) is_poly_in v_",

   848 	       "((lhs e_) has_degree_in v_ ) = 0"

   849 	      ]),

   850    ("#Find"  ,["solutions v_i_"])

   851   ],

   852   PolyEq_prls, SOME "solve (e_::bool, v_)",

   853   [["PolyEq","solve_d0_polyeq_equation"]]));

   855 (*--- d1 ---*)

   856 store_pbt

   857  (prep_pbt (theory "PolyEq") "pbl_equ_univ_poly_deg1" [] e_pblID

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

   859   [("#Given" ,["equality e_","solveFor v_"]),

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

   861 	       "(lhs e_) is_poly_in v_",

   862 	       "((lhs e_) has_degree_in v_ ) = 1"

   863 	      ]),

   864    ("#Find"  ,["solutions v_i_"])

   865   ],

   866   PolyEq_prls, SOME "solve (e_::bool, v_)",

   867   [["PolyEq","solve_d1_polyeq_equation"]]));

   869 (*--- d2 ---*)

   870 store_pbt

   871  (prep_pbt (theory "PolyEq") "pbl_equ_univ_poly_deg2" [] e_pblID

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

   873   [("#Given" ,["equality e_","solveFor v_"]),

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

   875 	       "(lhs e_) is_poly_in v_ ",

   876 	       "((lhs e_) has_degree_in v_ ) = 2"]),

   877    ("#Find"  ,["solutions v_i_"])

   878   ],

   879   PolyEq_prls, SOME "solve (e_::bool, v_)",

   880   [["PolyEq","solve_d2_polyeq_equation"]]));

   882  store_pbt

   883  (prep_pbt (theory "PolyEq") "pbl_equ_univ_poly_deg2_sqonly" [] e_pblID

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

   885   [("#Given" ,["equality e_","solveFor v_"]),

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

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

   888 	       "matches (         ?v_^^^2 = 0) e_ | " ^

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

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

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

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

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

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

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

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

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

   898    ("#Find"  ,["solutions v_i_"])

   899   ],

   900   PolyEq_prls, SOME "solve (e_::bool, v_)",

   901   [["PolyEq","solve_d2_polyeq_sqonly_equation"]]));

   903 store_pbt

   904  (prep_pbt (theory "PolyEq") "pbl_equ_univ_poly_deg2_bdvonly" [] e_pblID

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

   906   [("#Given" ,["equality e_","solveFor v_"]),

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

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

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

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

   911 	       "matches (            ?v_^^^2 = 0) e_ | " ^

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

   913    ("#Find"  ,["solutions v_i_"])

   914   ],

   915   PolyEq_prls, SOME "solve (e_::bool, v_)",

   916   [["PolyEq","solve_d2_polyeq_bdvonly_equation"]]));

   918 store_pbt

   919  (prep_pbt (theory "PolyEq") "pbl_equ_univ_poly_deg2_pq" [] e_pblID

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

   921   [("#Given" ,["equality e_","solveFor v_"]),

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

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

   924    ("#Find"  ,["solutions v_i_"])

   925   ],

   926   PolyEq_prls, SOME "solve (e_::bool, v_)",

   927   [["PolyEq","solve_d2_polyeq_pq_equation"]]));

   929 store_pbt

   930  (prep_pbt (theory "PolyEq") "pbl_equ_univ_poly_deg2_abc" [] e_pblID

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

   932   [("#Given" ,["equality e_","solveFor v_"]),

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

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

   935    ("#Find"  ,["solutions v_i_"])

   936   ],

   937   PolyEq_prls, SOME "solve (e_::bool, v_)",

   938   [["PolyEq","solve_d2_polyeq_abc_equation"]]));

   940 (*--- d3 ---*)

   941 store_pbt

   942  (prep_pbt (theory "PolyEq") "pbl_equ_univ_poly_deg3" [] e_pblID

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

   944   [("#Given" ,["equality e_","solveFor v_"]),

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

   946 	       "(lhs e_) is_poly_in v_ ",

   947 	       "((lhs e_) has_degree_in v_) = 3"]),

   948    ("#Find"  ,["solutions v_i_"])

   949   ],

   950   PolyEq_prls, SOME "solve (e_::bool, v_)",

   951   [["PolyEq","solve_d3_polyeq_equation"]]));

   953 (*--- d4 ---*)

   954 store_pbt

   955  (prep_pbt (theory "PolyEq") "pbl_equ_univ_poly_deg4" [] e_pblID

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

   957   [("#Given" ,["equality e_","solveFor v_"]),

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

   959 	       "(lhs e_) is_poly_in v_ ",

   960 	       "((lhs e_) has_degree_in v_) = 4"]),

   961    ("#Find"  ,["solutions v_i_"])

   962   ],

   963   PolyEq_prls, SOME "solve (e_::bool, v_)",

   964   [(*["PolyEq","solve_d4_polyeq_equation"]*)]));

   966 (*--- normalize ---*)

   967 store_pbt

   968  (prep_pbt (theory "PolyEq") "pbl_equ_univ_poly_norm" [] e_pblID

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

   970   [("#Given" ,["equality e_","solveFor v_"]),

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

   972 	       "(Not(((lhs e_) is_poly_in v_)))"]),

   973    ("#Find"  ,["solutions v_i_"])

   974   ],

   975   PolyEq_prls, SOME "solve (e_::bool, v_)",

   976   [["PolyEq","normalize_poly"]]));

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

   978 store_pbt

   979  (prep_pbt (theory "PolyEq") "pbl_equ_univ_expand" [] e_pblID

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

   981   [("#Given" ,["equality e_","solveFor v_"]),

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

   983 	       "(lhs e_) is_expanded_in v_ "]),

   984    ("#Find"  ,["solutions v_i_"])

   985    ],

   986   PolyEq_prls, SOME "solve (e_::bool, v_)",

   987   []));

   989 (*--- d2 ---*)

   990 store_pbt

   991  (prep_pbt (theory "PolyEq") "pbl_equ_univ_expand_deg2" [] e_pblID

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

   993   [("#Given" ,["equality e_","solveFor v_"]),

   994    ("#Where" ,["((lhs e_) has_degree_in v_) = 2"]),

   995    ("#Find"  ,["solutions v_i_"])

   996   ],

   997   PolyEq_prls, SOME "solve (e_::bool, v_)",

   998   [["PolyEq","complete_square"]]));

  1001 "-------------------------methods-----------------------";

  1002 store_met

  1003  (prep_met (theory "PolyEq") "met_polyeq" [] e_metID

  1004  (["PolyEq"],

  1005    [],

  1006    {rew_ord'="tless_true",rls'=Atools_erls,calc = [], srls = e_rls, prls=e_rls,

  1007     crls=PolyEq_crls, nrls=norm_Rational}, "empty_script"));

  1009 store_met

  1010  (prep_met (theory "PolyEq") "met_polyeq_norm" [] e_metID

  1011  (["PolyEq","normalize_poly"],

  1012    [("#Given" ,["equality e_","solveFor v_"]),

  1013    ("#Where" ,["(Not((matches (?a = 0 ) e_ ))) |" ^

  1014 	       "(Not(((lhs e_) is_poly_in v_)))"]),

  1015    ("#Find"  ,["solutions v_i_"])

  1016   ],

  1017    {rew_ord'="termlessI",

  1018     rls'=PolyEq_erls,

  1019     srls=e_rls,

  1020     prls=PolyEq_prls,

  1021     calc=[],

  1022     crls=PolyEq_crls, nrls=norm_Rational

  1023     "Script Normalize_poly (e_::bool) (v_::real) =                     " ^

  1024     "(let e_ =((Try         (Rewrite     all_left          False)) @@  " ^

  1025     "          (Try (Repeat (Rewrite     makex1_x         False))) @@  " ^

  1026     "          (Try (Repeat (Rewrite_Set expand_binoms    False))) @@  " ^

  1027     "          (Try (Repeat (Rewrite_Set_Inst [(bdv,v_::real)]         " ^

  1028     "                                 make_ratpoly_in     False))) @@  " ^

  1029     "          (Try (Repeat (Rewrite_Set polyeq_simplify  False)))) e_ " ^

  1030     " in (SubProblem (PolyEq_,[polynomial,univariate,equation],        " ^

  1031     "                [no_met]) [bool_ e_, real_ v_]))"

  1032    ));

  1034 store_met

  1035  (prep_met (theory "PolyEq") "met_polyeq_d0" [] e_metID

  1036  (["PolyEq","solve_d0_polyeq_equation"],

  1037    [("#Given" ,["equality e_","solveFor v_"]),

  1038    ("#Where" ,["(lhs e_) is_poly_in v_ ",

  1039 	       "((lhs e_) has_degree_in v_) = 0"]),

  1040    ("#Find"  ,["solutions v_i_"])

  1041   ],

  1042    {rew_ord'="termlessI",

  1043     rls'=PolyEq_erls,

  1044     srls=e_rls,

  1045     prls=PolyEq_prls,

  1046     calc=[("sqrt", ("Root.sqrt", eval_sqrt "#sqrt_"))],

  1047     crls=PolyEq_crls, nrls=norm_Rational},

  1048    "Script Solve_d0_polyeq_equation  (e_::bool) (v_::real)  = " ^

  1049     "(let e_ =  ((Try (Rewrite_Set_Inst [(bdv,v_::real)]      " ^

  1050     "                  d0_polyeq_simplify  False))) e_        " ^

  1051     " in ((Or_to_List e_)::bool list))"

  1052  ));

  1054 store_met

  1055  (prep_met (theory "PolyEq") "met_polyeq_d1" [] e_metID

  1056  (["PolyEq","solve_d1_polyeq_equation"],

  1057    [("#Given" ,["equality e_","solveFor v_"]),

  1058    ("#Where" ,["(lhs e_) is_poly_in v_ ",

  1059 	       "((lhs e_) has_degree_in v_) = 1"]),

  1060    ("#Find"  ,["solutions v_i_"])

  1061   ],

  1062    {rew_ord'="termlessI",

  1063     rls'=PolyEq_erls,

  1064     srls=e_rls,

  1065     prls=PolyEq_prls,

  1066     calc=[("sqrt", ("Root.sqrt", eval_sqrt "#sqrt_"))],

  1067     crls=PolyEq_crls, nrls=norm_Rational(*,

  1068     (*    asm_rls=["d1_polyeq_simplify"],*)

  1069     asm_rls=[],

  1070     asm_thm=[("d1_isolate_div","")]*)},

  1071    "Script Solve_d1_polyeq_equation  (e_::bool) (v_::real)  =   " ^

  1072     "(let e_ =  ((Try (Rewrite_Set_Inst [(bdv,v_::real)]        " ^

  1073     "                  d1_polyeq_simplify   True))          @@  " ^

  1074     "            (Try (Rewrite_Set polyeq_simplify  False)) @@  " ^

  1075     "            (Try (Rewrite_Set norm_Rational_parenthesized False))) e_;" ^

  1076     " (L_::bool list) = ((Or_to_List e_)::bool list)            " ^

  1077     " in Check_elementwise L_ {(v_::real). Assumptions} )"

  1078  ));

  1080 store_met

  1081  (prep_met (theory "PolyEq") "met_polyeq_d22" [] e_metID

  1082  (["PolyEq","solve_d2_polyeq_equation"],

  1083    [("#Given" ,["equality e_","solveFor v_"]),

  1084    ("#Where" ,["(lhs e_) is_poly_in v_ ",

  1085 	       "((lhs e_) has_degree_in v_) = 2"]),

  1086    ("#Find"  ,["solutions v_i_"])

  1087   ],

  1088    {rew_ord'="termlessI",

  1089     rls'=PolyEq_erls,

  1090     srls=e_rls,

  1091     prls=PolyEq_prls,

  1092     calc=[("sqrt", ("Root.sqrt", eval_sqrt "#sqrt_"))],

  1093     crls=PolyEq_crls, nrls=norm_Rational},

  1094    "Script Solve_d2_polyeq_equation  (e_::bool) (v_::real) =      " ^

  1095     "  (let e_ = ((Try (Rewrite_Set_Inst [(bdv,v_::real)]         " ^

  1096     "                    d2_polyeq_simplify           True)) @@   " ^

  1097     "             (Try (Rewrite_Set polyeq_simplify   False)) @@  " ^

  1098     "             (Try (Rewrite_Set_Inst [(bdv,v_::real)]         " ^

  1099     "                    d1_polyeq_simplify            True)) @@  " ^

  1100     "            (Try (Rewrite_Set polyeq_simplify    False)) @@  " ^

  1101     "            (Try (Rewrite_Set norm_Rational_parenthesized False))) e_;" ^

  1102     " (L_::bool list) = ((Or_to_List e_)::bool list)              " ^

  1103     " in Check_elementwise L_ {(v_::real). Assumptions} )"

  1104  ));

  1106 store_met

  1107  (prep_met (theory "PolyEq") "met_polyeq_d2_bdvonly" [] e_metID

  1108  (["PolyEq","solve_d2_polyeq_bdvonly_equation"],

  1109    [("#Given" ,["equality e_","solveFor v_"]),

  1110    ("#Where" ,["(lhs e_) is_poly_in v_ ",

  1111 	       "((lhs e_) has_degree_in v_) = 2"]),

  1112    ("#Find"  ,["solutions v_i_"])

  1113   ],

  1114    {rew_ord'="termlessI",

  1115     rls'=PolyEq_erls,

  1116     srls=e_rls,

  1117     prls=PolyEq_prls,

  1118     calc=[("sqrt", ("Root.sqrt", eval_sqrt "#sqrt_"))],

  1119     crls=PolyEq_crls, nrls=norm_Rational},

  1120    "Script Solve_d2_polyeq_bdvonly_equation  (e_::bool) (v_::real) =" ^

  1121     "  (let e_ = ((Try (Rewrite_Set_Inst [(bdv,v_::real)]         " ^

  1122     "                   d2_polyeq_bdv_only_simplify    True)) @@  " ^

  1123     "             (Try (Rewrite_Set polyeq_simplify   False)) @@  " ^

  1124     "             (Try (Rewrite_Set_Inst [(bdv,v_::real)]         " ^

  1125     "                   d1_polyeq_simplify             True)) @@  " ^

  1126     "            (Try (Rewrite_Set polyeq_simplify    False)) @@  " ^

  1127     "            (Try (Rewrite_Set norm_Rational_parenthesized False))) e_;" ^

  1128     " (L_::bool list) = ((Or_to_List e_)::bool list)              " ^

  1129     " in Check_elementwise L_ {(v_::real). Assumptions} )"

  1130  ));

  1132 store_met

  1133  (prep_met (theory "PolyEq") "met_polyeq_d2_sqonly" [] e_metID

  1134  (["PolyEq","solve_d2_polyeq_sqonly_equation"],

  1135    [("#Given" ,["equality e_","solveFor v_"]),

  1136    ("#Where" ,["(lhs e_) is_poly_in v_ ",

  1137 	       "((lhs e_) has_degree_in v_) = 2"]),

  1138    ("#Find"  ,["solutions v_i_"])

  1139   ],

  1140    {rew_ord'="termlessI",

  1141     rls'=PolyEq_erls,

  1142     srls=e_rls,

  1143     prls=PolyEq_prls,

  1144     calc=[("sqrt", ("Root.sqrt", eval_sqrt "#sqrt_"))],

  1145     crls=PolyEq_crls, nrls=norm_Rational},

  1146    "Script Solve_d2_polyeq_sqonly_equation  (e_::bool) (v_::real) =" ^

  1147     "  (let e_ = ((Try (Rewrite_Set_Inst [(bdv,v_::real)]          " ^

  1148     "                   d2_polyeq_sq_only_simplify     True)) @@   " ^

  1149     "            (Try (Rewrite_Set polyeq_simplify    False)) @@   " ^

  1150     "            (Try (Rewrite_Set norm_Rational_parenthesized False))) e_; " ^

  1151     " (L_::bool list) = ((Or_to_List e_)::bool list)               " ^

  1152     " in Check_elementwise L_ {(v_::real). Assumptions} )"

  1153  ));

  1155 store_met

  1156  (prep_met (theory "PolyEq") "met_polyeq_d2_pq" [] e_metID

  1157  (["PolyEq","solve_d2_polyeq_pq_equation"],

  1158    [("#Given" ,["equality e_","solveFor v_"]),

  1159    ("#Where" ,["(lhs e_) is_poly_in v_ ",

  1160 	       "((lhs e_) has_degree_in v_) = 2"]),

  1161    ("#Find"  ,["solutions v_i_"])

  1162   ],

  1163    {rew_ord'="termlessI",

  1164     rls'=PolyEq_erls,

  1165     srls=e_rls,

  1166     prls=PolyEq_prls,

  1167     calc=[("sqrt", ("Root.sqrt", eval_sqrt "#sqrt_"))],

  1168     crls=PolyEq_crls, nrls=norm_Rational},

  1169    "Script Solve_d2_polyeq_pq_equation  (e_::bool) (v_::real) =   " ^

  1170     "  (let e_ = ((Try (Rewrite_Set_Inst [(bdv,v_::real)]         " ^

  1171     "                   d2_polyeq_pqFormula_simplify   True)) @@  " ^

  1172     "            (Try (Rewrite_Set polyeq_simplify    False)) @@  " ^

  1173     "            (Try (Rewrite_Set norm_Rational_parenthesized False))) e_;" ^

  1174     " (L_::bool list) = ((Or_to_List e_)::bool list)              " ^

  1175     " in Check_elementwise L_ {(v_::real). Assumptions} )"

  1176  ));

  1178 store_met

  1179  (prep_met (theory "PolyEq") "met_polyeq_d2_abc" [] e_metID

  1180  (["PolyEq","solve_d2_polyeq_abc_equation"],

  1181    [("#Given" ,["equality e_","solveFor v_"]),

  1182    ("#Where" ,["(lhs e_) is_poly_in v_ ",

  1183 	       "((lhs e_) has_degree_in v_) = 2"]),

  1184    ("#Find"  ,["solutions v_i_"])

  1185   ],

  1186    {rew_ord'="termlessI",

  1187     rls'=PolyEq_erls,

  1188     srls=e_rls,

  1189     prls=PolyEq_prls,

  1190     calc=[("sqrt", ("Root.sqrt", eval_sqrt "#sqrt_"))],

  1191     crls=PolyEq_crls, nrls=norm_Rational},

  1192    "Script Solve_d2_polyeq_abc_equation  (e_::bool) (v_::real) =   " ^

  1193     "  (let e_ = ((Try (Rewrite_Set_Inst [(bdv,v_::real)]          " ^

  1194     "                   d2_polyeq_abcFormula_simplify   True)) @@  " ^

  1195     "            (Try (Rewrite_Set polyeq_simplify     False)) @@  " ^

  1196     "            (Try (Rewrite_Set norm_Rational_parenthesized False))) e_;" ^

  1197     " (L_::bool list) = ((Or_to_List e_)::bool list)               " ^

  1198     " in Check_elementwise L_ {(v_::real). Assumptions} )"

  1199  ));

  1201 store_met

  1202  (prep_met (theory "PolyEq") "met_polyeq_d3" [] e_metID

  1203  (["PolyEq","solve_d3_polyeq_equation"],

  1204    [("#Given" ,["equality e_","solveFor v_"]),

  1205    ("#Where" ,["(lhs e_) is_poly_in v_ ",

  1206 	       "((lhs e_) has_degree_in v_) = 3"]),

  1207    ("#Find"  ,["solutions v_i_"])

  1208   ],

  1209    {rew_ord'="termlessI",

  1210     rls'=PolyEq_erls,

  1211     srls=e_rls,

  1212     prls=PolyEq_prls,

  1213     calc=[("sqrt", ("Root.sqrt", eval_sqrt "#sqrt_"))],

  1214     crls=PolyEq_crls, nrls=norm_Rational},

  1215    "Script Solve_d3_polyeq_equation  (e_::bool) (v_::real) =     " ^

  1216     "  (let e_ = ((Try (Rewrite_Set_Inst [(bdv,v_::real)]        " ^

  1217     "                    d3_polyeq_simplify           True)) @@  " ^

  1218     "             (Try (Rewrite_Set polyeq_simplify  False)) @@  " ^

  1219     "             (Try (Rewrite_Set_Inst [(bdv,v_::real)]        " ^

  1220     "                    d2_polyeq_simplify           True)) @@  " ^

  1221     "             (Try (Rewrite_Set polyeq_simplify  False)) @@  " ^

  1222     "             (Try (Rewrite_Set_Inst [(bdv,v_::real)]        " ^

  1223     "                    d1_polyeq_simplify           True)) @@  " ^

  1224     "             (Try (Rewrite_Set polyeq_simplify  False)) @@  " ^

  1225     "             (Try (Rewrite_Set norm_Rational_parenthesized False))) e_;" ^

  1226     " (L_::bool list) = ((Or_to_List e_)::bool list)             " ^

  1227     " in Check_elementwise L_ {(v_::real). Assumptions} )"

  1228    ));

  1230  (*.solves all expanded (ie. normalized) terms of degree 2.*)

  1231  (*Oct.02 restriction: 'eval_true 0 =< discriminant' ony for integer values

  1232    by 'PolyEq_erls'; restricted until Float.thy is implemented*)

  1233 store_met

  1234  (prep_met (theory "PolyEq") "met_polyeq_complsq" [] e_metID

  1235  (["PolyEq","complete_square"],

  1236    [("#Given" ,["equality e_","solveFor v_"]),

  1237    ("#Where" ,["matches (?a = 0) e_",

  1238 	       "((lhs e_) has_degree_in v_) = 2"]),

  1239    ("#Find"  ,["solutions v_i_"])

  1240   ],

  1241    {rew_ord'="termlessI",rls'=PolyEq_erls,srls=e_rls,prls=PolyEq_prls,

  1242     calc=[("sqrt", ("Root.sqrt", eval_sqrt "#sqrt_"))],

  1243     crls=PolyEq_crls, nrls=norm_Rational},

  1244    "Script Complete_square (e_::bool) (v_::real) =                          " ^

  1245    "(let e_ = ((Try (Rewrite_Set_Inst [(bdv,v_)] cancel_leading_coeff True))" ^

  1246    "        @@ (Try (Rewrite_Set_Inst [(bdv,v_)] complete_square True))     " ^

  1247    "        @@ (Try (Rewrite square_explicit1 False))                       " ^

  1248    "        @@ (Try (Rewrite square_explicit2 False))                       " ^

  1249    "        @@ (Rewrite root_plus_minus True)                               " ^

  1250    "        @@ (Try (Repeat (Rewrite_Inst [(bdv,v_)] bdv_explicit1 False))) " ^

  1251    "        @@ (Try (Repeat (Rewrite_Inst [(bdv,v_)] bdv_explicit2 False))) " ^

  1252    "        @@ (Try (Repeat                                                 " ^

  1253    "                  (Rewrite_Inst [(bdv,v_)] bdv_explicit3 False)))       " ^

  1254    "        @@ (Try (Rewrite_Set calculate_RootRat False))                  " ^

  1255    "        @@ (Try (Repeat (Calculate sqrt_)))) e_                         " ^

  1256    " in ((Or_to_List e_)::bool list))"

  1257    ));

  1260 (* termorder hacked by MG *)

  1261 local (*. for make_polynomial_in .*)

  1263 open Term;  (* for type order = EQUAL | LESS | GREATER *)

  1265 fun pr_ord EQUAL = "EQUAL"

  1266   | pr_ord LESS  = "LESS"

  1267   | pr_ord GREATER = "GREATER";

  1269 fun dest_hd' x (Const (a, T)) = (((a, 0), T), 0)

  1270   | dest_hd' x (t as Free (a, T)) =

  1271     if x = t then ((("|||||||||||||", 0), T), 0)                        (*WN*)

  1272     else (((a, 0), T), 1)

  1273   | dest_hd' x (Var v) = (v, 2)

  1274   | dest_hd' x (Bound i) = ((("", i), dummyT), 3)

  1275   | dest_hd' x (Abs (_, T, _)) = ((("", 0), T), 4);

  1277 fun size_of_term' x (Const ("Atools.pow",_) $ Free (var,_) $ Free (pot,_)) =

  1278     (case x of                                                          (*WN*)

  1279 	    (Free (xstr,_)) =>

  1280 		(if xstr = var then 1000*(the (int_of_str pot)) else 3)

  1281 	  | _ => raise error ("size_of_term' called with subst = "^

  1282 			      (term2str x)))

  1283   | size_of_term' x (Free (subst,_)) =

  1284     (case x of

  1285 	    (Free (xstr,_)) => (if xstr = subst then 1000 else 1)

  1286 	  | _ => raise error ("size_of_term' called with subst = "^

  1287 			  (term2str x)))

  1288   | size_of_term' x (Abs (_,_,body)) = 1 + size_of_term' x body

  1289   | size_of_term' x (f$t) = size_of_term' x f  +  size_of_term' x t

  1290   | size_of_term' x _ = 1;

  1293 fun term_ord' x pr thy (Abs (_, T, t), Abs(_, U, u)) =       (* ~ term.ML *)

  1294       (case term_ord' x pr thy (t, u) of EQUAL => typ_ord (T, U) | ord => ord)

  1295   | term_ord' x pr thy (t, u) =

  1296       (if pr then

  1297 	 let

  1298 	   val (f, ts) = strip_comb t and (g, us) = strip_comb u;

  1299 	   val _=writeln("t= f@ts= \""^

  1300 	      ((Syntax.string_of_term (thy2ctxt thy)) f)^"\" @ \"["^

  1301 	      (commas(map(string_of_cterm o cterm_of(sign_of thy)) ts))^"]\"");

  1302 	   val _=writeln("u= g@us= \""^

  1303 	      ((Syntax.string_of_term (thy2ctxt thy)) g)^"\" @ \"["^

  1304 	      (commas(map(string_of_cterm o cterm_of(sign_of thy)) us))^"]\"");

  1305 	   val _=writeln("size_of_term(t,u)= ("^

  1306 	      (string_of_int(size_of_term' x t))^", "^

  1307 	      (string_of_int(size_of_term' x u))^")");

  1308 	   val _=writeln("hd_ord(f,g)      = "^((pr_ord o (hd_ord x))(f,g)));

  1309 	   val _=writeln("terms_ord(ts,us) = "^

  1310 			   ((pr_ord o (terms_ord x) str false)(ts,us)));

  1311 	   val _=writeln("-------");

  1312 	 in () end

  1313        else ();

  1314 	 case int_ord (size_of_term' x t, size_of_term' x u) of

  1315 	   EQUAL =>

  1316 	     let val (f, ts) = strip_comb t and (g, us) = strip_comb u in

  1317 	       (case hd_ord x (f, g) of EQUAL => (terms_ord x str pr) (ts, us)

  1318 	     | ord => ord)

  1319 	     end

  1320 	 | ord => ord)

  1321 and hd_ord x (f, g) =                                        (* ~ term.ML *)

  1322   prod_ord (prod_ord indexname_ord typ_ord) int_ord (dest_hd' x f,

  1323 						     dest_hd' x g)

  1324 and terms_ord x str pr (ts, us) =

  1325     list_ord (term_ord' x pr (assoc_thy "Isac.thy"))(ts, us);

  1326 in

  1328 fun ord_make_polynomial_in (pr:bool) thy subst tu =

  1329     let

  1330 	(* val _=writeln("*** subs variable is: "^(subst2str subst)); *)

  1331     in

  1332 	case subst of

  1333 	    (_,x)::_ => (term_ord' x pr thy tu = LESS)

  1334 	  | _ => raise error ("ord_make_polynomial_in called with subst = "^

  1335 			  (subst2str subst))

  1336     end;

  1337 end;

  1339 val order_add_mult_in = prep_rls(

  1340   Rls{id = "order_add_mult_in", preconds = [],

  1341       rew_ord = ("ord_make_polynomial_in",

  1342 		 ord_make_polynomial_in false Poly.thy),

  1343       erls = e_rls,srls = Erls,

  1344       calc = [],

  1345       (*asm_thm = [],*)

  1346       rules = [Thm ("real_mult_commute",num_str @{real_mult_commute),

  1347 	       (* z * w = w * z *)

  1348 	       Thm ("real_mult_left_commute",num_str @{real_mult_left_commute),

  1349 	       (*z1.0 * (z2.0 * z3.0) = z2.0 * (z1.0 * z3.0)*)

  1350 	       Thm ("real_mult_assoc",num_str @{real_mult_assoc),

  1351 	       (*z1.0 * z2.0 * z3.0 = z1.0 * (z2.0 * z3.0)*)

  1352 	       Thm ("add_commute",num_str @{thm add_commute}),

  1353 	       (*z + w = w + z*)

  1354 	       Thm ("add_left_commute",num_str @{thm add_left_commute}),

  1355 	       (*x + (y + z) = y + (x + z)*)

  1356 	       Thm ("add_assoc",num_str @{thm add_assoc})

  1357 	       (*z1.0 + z2.0 + z3.0 = z1.0 + (z2.0 + z3.0)*)

  1358 	       ], scr = EmptyScr}:rls);

  1360 val collect_bdv = prep_rls(

  1361   Rls{id = "collect_bdv", preconds = [],

  1362       rew_ord = ("dummy_ord", dummy_ord),

  1363       erls = e_rls,srls = Erls,

  1364       calc = [],

  1365       (*asm_thm = [],*)

  1366       rules = [Thm ("bdv_collect_1",num_str @{bdv_collect_1),

  1367 	       Thm ("bdv_collect_2",num_str @{bdv_collect_2),

  1368 	       Thm ("bdv_collect_3",num_str @{bdv_collect_3),

  1370 	       Thm ("bdv_collect_assoc1_1",num_str @{bdv_collect_assoc1_1),

  1371 	       Thm ("bdv_collect_assoc1_2",num_str @{bdv_collect_assoc1_2),

  1372 	       Thm ("bdv_collect_assoc1_3",num_str @{bdv_collect_assoc1_3),

  1374 	       Thm ("bdv_collect_assoc2_1",num_str @{bdv_collect_assoc2_1),

  1375 	       Thm ("bdv_collect_assoc2_2",num_str @{bdv_collect_assoc2_2),

  1376 	       Thm ("bdv_collect_assoc2_3",num_str @{bdv_collect_assoc2_3),

  1379 	       Thm ("bdv_n_collect_1",num_str @{bdv_n_collect_1),

  1380 	       Thm ("bdv_n_collect_2",num_str @{bdv_n_collect_2),

  1381 	       Thm ("bdv_n_collect_3",num_str @{bdv_n_collect_3),

  1383 	       Thm ("bdv_n_collect_assoc1_1",num_str @{bdv_n_collect_assoc1_1),

  1384 	       Thm ("bdv_n_collect_assoc1_2",num_str @{bdv_n_collect_assoc1_2),

  1385 	       Thm ("bdv_n_collect_assoc1_3",num_str @{bdv_n_collect_assoc1_3),

  1387 	       Thm ("bdv_n_collect_assoc2_1",num_str @{bdv_n_collect_assoc2_1),

  1388 	       Thm ("bdv_n_collect_assoc2_2",num_str @{bdv_n_collect_assoc2_2),

  1389 	       Thm ("bdv_n_collect_assoc2_3",num_str @{bdv_n_collect_assoc2_3)

  1390 	       ], scr = EmptyScr}:rls);

  1392 (*.transforms an arbitrary term without roots to a polynomial [4]

  1393    according to knowledge/Poly.sml.*)

  1394 val make_polynomial_in = prep_rls(

  1395   Seq {id = "make_polynomial_in", preconds = []:term list,

  1396        rew_ord = ("dummy_ord", dummy_ord),

  1397       erls = Atools_erls, srls = Erls,

  1398       calc = [], (*asm_thm = [],*)

  1399       rules = [Rls_ expand_poly,

  1400 	       Rls_ order_add_mult_in,

  1401 	       Rls_ simplify_power,

  1402 	       Rls_ collect_numerals,

  1403 	       Rls_ reduce_012,

  1404 	       Thm ("realpow_oneI",num_str @{realpow_oneI),

  1405 	       Rls_ discard_parentheses,

  1406 	       Rls_ collect_bdv

  1407 	       ],

  1408       scr = EmptyScr

  1409       }:rls);

  1411 val separate_bdvs =

  1412     append_rls "separate_bdvs"

  1413 	       collect_bdv

  1414 	       [Thm ("separate_bdv", num_str @{separate_bdv),

  1415 		(*"?a * ?bdv / ?b = ?a / ?b * ?bdv"*)

  1416 		Thm ("separate_bdv_n", num_str @{separate_bdv_n),

  1417 		Thm ("separate_1_bdv", num_str @{separate_1_bdv),

  1418 		(*"?bdv / ?b = (1 / ?b) * ?bdv"*)

  1419 		Thm ("separate_1_bdv_n", num_str @{separate_1_bdv_n),

  1420 		(*"?bdv ^^^ ?n / ?b = 1 / ?b * ?bdv ^^^ ?n"*)

  1421 		Thm ("nadd_divide_distrib",

  1422 		     num_str @{thm nadd_divide_distrib})

  1423 		(*"(?x + ?y) / ?z = ?x / ?z + ?y / ?z"

  1424 		      WN051031 DOES NOT BELONG TO HERE*)

  1425 		];

  1426 val make_ratpoly_in = prep_rls(

  1427   Seq {id = "make_ratpoly_in", preconds = []:term list,

  1428        rew_ord = ("dummy_ord", dummy_ord),

  1429       erls = Atools_erls, srls = Erls,

  1430       calc = [], (*asm_thm = [],*)

  1431       rules = [Rls_ norm_Rational,

  1432 	       Rls_ order_add_mult_in,

  1433 	       Rls_ discard_parentheses,

  1434 	       Rls_ separate_bdvs,

  1435 	       (* Rls_ rearrange_assoc, WN060916 why does cancel_p not work?*)

  1436 	       Rls_ cancel_p

  1437 	       (*Calc ("HOL.divide"  ,eval_cancel "#divide_") too weak!*)

  1438 	       ],

  1439       scr = EmptyScr}:rls);

  1442 ruleset' := overwritelthy @{theory} (!ruleset',

  1443   [("order_add_mult_in", order_add_mult_in),

  1444    ("collect_bdv", collect_bdv),

  1445    ("make_polynomial_in", make_polynomial_in),

  1446    ("make_ratpoly_in", make_ratpoly_in),

  1447    ("separate_bdvs", separate_bdvs)

  1448    ]);

  1449 *}

  1451 end

author	Walther Neuper <neuper@ist.tugraz.at>
	Tue, 31 Aug 2010 16:38:22 +0200
branch	isac-update-Isa09-2
changeset 37967	bd4f7a35e892
parent 37966	78938fc8e022
child 37969	81922154e742
permissions	-rw-r--r--