wneuper/isa: test/Tools/isac/Knowledge/rational.sml@61cccc3f2f56

     1 (* Title: tests for rationals

     2    Author: Stefan Karnel

     3    Copyright (c) Stefan Karnel 2002

     4    Use is subject to license terms.

     6 12345678901234567890123456789012345678901234567890123456789012345678901234567890

     7         10        20        30        40        50        60        70        80

     8 LEGEND

     9 WN120317.TODO postponed test/../ratinal,ratinal2.sml to joint work with dmeind

    10 WN070906

    11    nonterm.SK   marks non-terminating examples

    12    ord.SK       PARTIALLY marks crucial ordering examples

    13    *SK*         of some (secondary) interest (on 070906)

    14 WN060104 transfer examples marked with (*SR..*) to the exp-collection

    15 *)

    17 "-----------------------------------------------------------------------------";

    18 "-----------------------------------------------------------------------------";

    19 "table of contents -----------------------------------------------------------";

    20 "-----------------------------------------------------------------------------";

    21 "-------- fun poly_of_term ---------------------------------------------------";

    22 "-------- fun is_poly --------------------------------------------------------";

    23 "-------- fun term_of_poly ---------------------------------------------------";

    24 "-------- fun factout_p_ -----------------------------------------------------";

    25 "-------- fun cancel_p_ ------------------------------------------------------";

    26 "-------- fun common_nominator_p_ --------------------------------------------";

    27 "-------- fun add_fraction_p_ ------------------------------------------------";

    28 "-------- TODO ---------------------------------------------------------------";

    29 "-------- and app_rev --------------------------------------------------------";

    30 "-------- external calculating functions test -----------";

    31 "-------- cancel from: Mathematik 1 Schalk Reniets Verlag";

    32 "-------- common_nominator_p ----------------------------";

    33 "-------- reverse rewrite -------------------------------";

    34 "-------- 'reverse-ruleset' cancel_p --------------------";

    35 "-------- norm_Rational ---------------------------------";

    36 "-------- numeral rationals -----------------------------";

    37 "-------- cancellation ----------------------------------";

    38 "-------- common denominator ----------------------------";

    39 "-------- multiply and cancel ---------------------------";

    40 "-------- common denominator and multiplication ---------";

    41 "-------- double fractions ------------------------------";

    42 "-------- crucial examples ------------------------------";

    43 "-------- examples for Stefan Karnels thesis ------------";

    44 "-------- me Schalk I No.186 ----------------------------";

    45 "-------- interSteps ..Simp_Rat_Double_No-1.xml ---------";

    46 "-------- interSteps ..Simp_Rat_Cancel_No-1.xml ---------";

    47 "-------- investigate rulesets for cancel_p -------------";

    48 "-------- investigate format of factout_ and factout_p_ -";

    49 "-------- how to stepwise construct Scripts -------------";

    50 "----------- get_denominator ----------------------------";

    51 "--------- several errpats in complicated term -------------------";

    52 "--------------------------------------------------------";

    53 "-------- nonterminating cancel_p, norm_Rational 2002 ---";

    54 "--------------------------------------------------------";

    55 "--------------------------------------------------------";

    58 "-------- fun poly_of_term ---------------------------------------------------";

    59 "-------- fun poly_of_term ---------------------------------------------------";

    60 "-------- fun poly_of_term ---------------------------------------------------";

    61 val vs = "12 * x^^^3 * y^^^4 * z^^^6" |> str2term |> vars |> map free2str |> sort string_ord;

    63 if poly_of_term vs (str2term "12::real") = SOME [(12, [0, 0, 0])]

    64 then () else error "poly_of_term 1 changed";

    65 if poly_of_term vs (str2term "x::real") = SOME [(1, [1, 0, 0])]

    66 then () else error "poly_of_term 2 changed";

    67 if poly_of_term vs (str2term "12 * x^^^3") = SOME [(12, [3, 0, 0])]

    68 then () else error "poly_of_term 3 changed";

    69 if poly_of_term vs (str2term "12 * x^^^3 * y^^^4 * z^^^6") = SOME [(12, [3, 4, 6])]

    70 then () else error "poly_of_term 4 changed";

    71 if poly_of_term vs (str2term "1 + 2 * x^^^3 * y^^^4 * z^^^6 + y") =

    72   SOME [(1, [0, 0, 0]), (1, [0, 1, 0]), (2, [3, 4, 6])]

    73 then () else error "poly_of_term 5 changed";

    75 (*poly_of_term is quite liberal:*)

    76 (*the coefficient may be somewhere, the order of variables and the parentheses

    77   within a monomial are arbitrary*)

    78 if poly_of_term vs (str2term "y^^^4 * (x^^^3 * 12 * z^^^6)") = SOME [(12, [3, 4, 6])]

    79 then () else error "poly_of_term 6 changed";

    81 (*there may even be more than 1 coefficient:*)

    82 if poly_of_term vs (str2term "2 * y^^^4 * (x^^^3 * 6 * z^^^6)") = SOME [(12, [3, 4, 6])]

    83 then () else error "poly_of_term 7 changed";

    85 (*the order and the parentheses within monomials are arbitrary:*)

    86 if poly_of_term vs (str2term "2 * x^^^3 * y^^^4 * z^^^6 + (7 * y^^^8 + 1)")

    87   = SOME [(1, [0, 0, 0]), (7, [0, 8, 0]), (2, [3, 4, 6])]

    88 then () else error "poly_of_term 8 changed";

    90 "-------- fun is_poly --------------------------------------------------------";

    91 "-------- fun is_poly --------------------------------------------------------";

    92 "-------- fun is_poly --------------------------------------------------------";

    93 if is_poly (str2term "2 * x^^^3 * y^^^4 * z^^^6 + 7 * y^^^8 + 1")

    94 then () else error "is_poly 1 changed";

    95 if not (is_poly (str2term "2 * (x^^^3 * y^^^4 * z^^^6 + 7) * y^^^8 + 1"))

    96 then () else error "is_poly 2 changed";

    98 "-------- fun term_of_poly ---------------------------------------------------";

    99 "-------- fun term_of_poly ---------------------------------------------------";

   100 "-------- fun term_of_poly ---------------------------------------------------";

   101 val expT = HOLogic.realT

   102 val Free (_, baseT) = (hd o vars o str2term) "12 * x^^^3 * y^^^4 * z^^^6";

   103 val p = [(1, [0, 0, 0]), (7, [0, 8, 0]), (2, [3, 4, 5])]

   104 val vs = ["x","y","z"]

   105 (*precondition for [(c, es),...]: legth es = length vs*)

   106 ;

   107 if term2str (term_of_poly baseT expT vs p) = "1 + 7 * y ^^^ 8 + 2 * x ^^^ 3 * y ^^^ 4 * z ^^^ 5"

   108 then () else error "term_of_poly 1 changed";

   110 "-------- fun factout_p_ -----------------------------------------------------";

   111 "-------- fun factout_p_ -----------------------------------------------------";

   112 "-------- fun factout_p_ -----------------------------------------------------";

   113 val t = str2term "(x^^^2 + -1*y^^^2) / (x^^^2 + -1*x*y)"

   114 val SOME (t', asm) = factout_p_ thy t;

   115 if term2str t' = "(x + y) * (x + -1 * y) / (x * (x + -1 * y))"

   116 then () else error ("factout_p_ term 1 changed: " ^ term2str t')

   117 ;

   118 if terms2str asm = "[\"x ~= 0\",\"x + -1 * y ~= 0\"]"

   119 then () else error "factout_p_ asm 1 changed"

   120 ;

   121 val t = str2term "nothing + to_cancel ::real";

   122 if NONE = factout_p_ thy t then () else error "factout_p_ doesn't report non-applicable";

   123 ;

   124 val t = str2term "((3 * x^^^2 + 6 *x + 3) / (2*x + 2))";

   125 val SOME (t', asm) = factout_p_ thy t;

   126 if term2str t' = "(3 + 3 * x) * (1 + x) / (2 * (1 + x))" andalso

   127   terms2str asm = "[\"1 + x ~= 0\"]"

   128 then () else error "factout_p_ 1 changed";

   130 "-------- fun cancel_p_ ------------------------------------------------------";

   131 "-------- fun cancel_p_ ------------------------------------------------------";

   132 "-------- fun cancel_p_ ------------------------------------------------------";

   133 val t = str2term "(x^^^2 + -1*y^^^2) / (x^^^2 + -1*x*y)"

   134 val SOME (t', asm) = cancel_p_ thy t;

   135 if (term2str t', terms2str asm) = ("(x + y) / x", "[\"x ~= 0\"]")

   136 then () else error ("cancel_p_ (t', asm) 1 changed: " ^ term2str t')

   137 ;

   138 val t = str2term "nothing + to_cancel ::real";

   139 if NONE = cancel_p_ thy t then () else error "cancel_p_ doesn't report non-applicable";

   140 ;

   141 val t = str2term "((3 * x^^^2 + 6 *x + 3) / (2*x + 2))";

   142 val SOME (t', asm) = cancel_p_ thy t;

   143 if term2str t' = "(3 + 3 * x) / 2" andalso terms2str asm = "[]"

   144 then () else error "cancel_p_ 1 changed";

   146 "-------- fun common_nominator_p_ --------------------------------------------";

   147 "-------- fun common_nominator_p_ --------------------------------------------";

   148 "-------- fun common_nominator_p_ --------------------------------------------";

   149 val t = str2term ("y / (a*x + b*x + c*x) " ^

   150               (* n1    d1                   *)

   151                 "+ a / (x*y)");

   152               (* n2    d2                   *)

   153 val SOME (t', asm) = common_nominator_p_ thy t;

   154 if term2str t' =

   155       ("y * y / (x * ((a + b + c) * y)) " ^

   156   (*  n1  *d2'/ (c'* ( d1'        *d2')) *)

   157      "+ a * (a + b + c) / (x * ((a + b + c) * y))")

   158    (*  n2 * d1'         / (c'* ( d1'        *d2')) *)

   159 then () else error "common_nominator_p_ term 1 changed";

   160 if terms2str asm = "[\"a + b + c ~= 0\",\"y ~= 0\",\"x ~= 0\"]"

   161 then () else error "common_nominator_p_ asm 1 changed"

   163 "-------- example in mail Nipkow";

   164 val t = str2term "x/(x^^^2 + -1*y^^^2) + y/(x^^^2 + -1*x*y)";

   165 val SOME (t', asm) = common_nominator_p_ thy t;

   166 if term2str t' = "x * x / " ^

   167                  "((x + -1 * y) * ((x + y) * x))" ^

   168             " +\n" ^

   169                  "y * (x + y) / " ^

   170                  "((x + -1 * y) * ((x + y) * x))"

   171 then () else error "common_nominator_p_ term 2 changed"

   172 ;

   173 if terms2str asm = "[\"x + y ~= 0\",\"x ~= 0\",\"x + -1 * y ~= 0\"]"

   174 then () else error "common_nominator_p_ asm 2 changed"

   176 "-------- example: applicable tested by SML code";

   177 val t = str2term "nothing / to_add";

   178 if NONE = common_nominator_p_ thy t then () else error "common_nominator_p_ term 3 changed";

   179 ;

   180 val t = str2term "((x + (-1)) / (x + 1)) + ((x + 1) / (x + (-1)))";

   181 val SOME (t', asm) = common_nominator_p_ thy t;

   182 if term2str t' =

   183   "(x + -1) * (-1 + x) / (1 * ((1 + x) * (-1 + x))) +\n(x + 1) * (1 + x) / (1 * ((1 + x) * (-1 + x)))"

   184   andalso terms2str asm = "[\"1 + x ~= 0\",\"-1 + x ~= 0\"]"

   185 then () else error "common_nominator_p_ 3 changed";

   187 "-------- fun add_fraction_p_ ------------------------------------------------";

   188 "-------- fun add_fraction_p_ ------------------------------------------------";

   189 "-------- fun add_fraction_p_ ------------------------------------------------";

   190 val t = str2term "((x + (-1)) / (x + 1)) + ((x + 1) / (x + (-1)))";

   191 val SOME (t', asm) = add_fraction_p_ thy t;

   192 if term2str t' = "(2 + 2 * x ^^^ 2) / (-1 + x ^^^ 2)"

   193 then () else error "add_fraction_p_ 3 changed";

   194 ;

   195 if terms2str asm = "[\"-1 + x ^^^ 2 ~= 0\"]"

   196 then () else error "add_fraction_p_ 3 changed";

   197 ;

   198 val t = str2term "nothing / to_add";

   199 if NONE = add_fraction_p_ thy t then () else error "add_fraction_p_ term 3 changed";

   200 ;

   201 val t = str2term "((x + (-1)) / (x + 1)) + ((x + 1) / (x + (-1)))";

   202 val SOME (t', asm) = add_fraction_p_ thy t;

   203 if term2str t' = "(2 + 2 * x ^^^ 2) / (-1 + x ^^^ 2)" andalso

   204   terms2str asm = "[\"-1 + x ^^^ 2 ~= 0\"]"

   205 then () else error "add_fraction_p_ 3 changed";

   207 "-------- TODO ---------------------------------------------------------------";

   208 "-------- TODO ---------------------------------------------------------------";

   209 "-------- TODO ---------------------------------------------------------------";

   213 "-------- and app_rev --------------------------------------------------------";

   214 "-------- and app_rev --------------------------------------------------------";

   215 "-------- and app_rev --------------------------------------------------------";

   216 (* trace down until prepats are evaluated

   217   (which does not to work, because substitution is not done -- compare rew_sub!);

   218   keep this sequence for the case, factout_p, cancel_p, common_nominator_p, add_fraction_p

   219   (again) get prepat = [] changed to <>[]. *)

   220 val t = str2term "(x^^^2 + -1*y^^^2) / (x^^^2 + -1*x*y)";

   222 (*rewrite_set_ @{theory Isac} true cancel t = NONE; !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!*)

   223 "~~~~~ fun rewrite_set_, args:"; val (thy, bool, rls, term) = (thy, false, cancel_p, t);

   224 "~~~~~ fun rewrite__set_, args:"; val (thy, i, _, _, (rrls as Rrls _), t) =

   225   (thy, 1, bool, [], rls, term);

   226 (*val (t', asm, rew) = app_rev thy (i+1) rrls t; rew = false!!!!!!!!!!!!!!!!!!!!!*)

   227 "~~~~~ and app_rev, args:"; val (thy, i, rrls, t) = (thy, (i+1), rrls, t);

   228     fun chk_prepat thy erls [] t = true

   229       | chk_prepat thy erls prepat t =

   230         let

   231           fun chk (pres, pat) =

   232             (let

   233               val subst: Type.tyenv * Envir.tenv =

   234                 Pattern.match thy (pat, t) (Vartab.empty, Vartab.empty)

   235              in

   236               snd (eval__true thy (i + 1) (map (Envir.subst_term subst) pres) [] erls)

   237              end) handle _ => false

   238            fun scan_ f [] = false (*scan_ NEVER called by []*)

   239              | scan_ f (pp::pps) =

   240                if f pp then true else scan_ f pps;

   241         in scan_ chk prepat end;

   242     (* apply the normal_form of a rev-set *)

   243     fun app_rev' thy (Rrls{erls,prepat,scr=Rfuns{normal_form,...},...}) t =

   244       if chk_prepat thy erls prepat t

   245       then ((*tracing("### app_rev': t = "^(term2str t));*) normal_form t)

   246       else NONE;

   247 (*  val opt = app_rev' thy rrls t  ..NONE!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!*)

   248 "~~~~~ and app_rev', args:"; val (thy, (Rrls{erls,prepat,scr=Rfuns{normal_form,...},...}), t) =

   249   (thy, rrls, t);

   250 (* chk_prepat thy erls prepat t = false!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!*)

   251 (* app_sub thy i rrls t = false!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!*)

   252 "~~~~~ fun chk_prepat, args:"; val (thy, erls, prepat, t) = (thy, erls, prepat, t);

   253           fun chk (pres, pat) =

   254             (let

   255               val subst: Type.tyenv * Envir.tenv =

   256                 Pattern.match thy (pat, t) (Vartab.empty, Vartab.empty)

   257              in

   258               snd (eval__true thy (i + 1) (map (Envir.subst_term subst) pres) [] erls)

   259              end) handle _ => false

   260            fun scan_ f [] = false (*scan_ NEVER called by []*)

   261              | scan_ f (pp::pps) =

   262                if f pp then true else scan_ f pps;

   264 (*========== inhibit exn WN130823: prepat is empty ====================================

   265 (* scan_ chk prepat = false!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!*)

   266 "~~~~~ fun , args:"; val (f, (pp::pps)) = (chk, prepat);

   267 f;

   268 val ([t1, t2], t) = pp;

   269 term2str t1 = "?r is_expanded";

   270 term2str t2 = "?s is_expanded";

   271 term2str t = "?r / ?s";

   272 (* f pp = false!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!*)

   273 "~~~~~ fun chk, args:"; val (pres, pat) = (pp);

   274               val subst: Type.tyenv * Envir.tenv =

   275                 Pattern.match thy (pat, t) (Vartab.empty, Vartab.empty)

   276 (*subst =

   277   ({}, {(("r", 0), ("real", Var (("r", 0), "real"))),

   278         (("s", 0), ("real", Var (("s", 0), "real")))}*)

   279 ;

   280               snd (eval__true thy (i + 1) (map (Envir.subst_term subst) pres) [] erls)

   281 "~~~~~ fun eval__true, args:"; val (thy, i, asms, bdv, rls) =

   282   (thy, (i + 1), (map (Envir.subst_term subst) pres), [], erls);

   283 terms2str asms;                         (* = "[\"?r is_expanded\",\"?s is_expanded\"]"*)

   284 asms = [@{term True}] orelse asms = []; (* = false*)

   285 asms = [@{term False}]                ; (* = false*)

   286 "~~~~~ fun chk, args:"; val (indets, (a::asms)) = ([], asms);

   287 bdv (*= []: _a list*);

   288 val bdv : (term * term) list = [];

   289 rewrite__set_ thy (i+1) false;

   290 term2str a = "?r is_expanded"; (*hier m"usste doch der Numerator eingesetzt sein ??????????????*)

   291 val SOME (Const ("HOL.False", _), []) = rewrite__set_ thy (i+1) false bdv rls a

   292 ============ inhibit exn WN130823: prepat is empty ===================================*)

   294 (*========== inhibit exn WN130824 TODO =======================================================

   295 "-------- external calculating functions test -----------";

   296 "-------- external calculating functions test -----------";

   297 "-------- external calculating functions test -----------";

   298 val t = str2term "((3 * x^^^2 + 6 *x + 3) / (2*x + 2))";

   299 val SOME (t', asm) = factout_p_ thy t;

   300 if term2str t' = "(3 + 3 * x) * (1 + x) / (2 * (1 + x))" andalso terms2str asm = "[]"

   301 then () else error "factout_p_ 1 changed";

   302 val SOME (t', asm) = cancel_p_ thy t;

   303 if term2str t' = "(3 + 3 * x) / 2" andalso terms2str asm = "[\"1 + x ~= 0\"]"

   304 then () else error "cancel_p_ 1 changed";

   306 val t = str2term "((-3 * x^^^2 + 6 *x - 3) / (2*x - 2))";

   307 val SOME (t', asm) = factout_(*p_*) thy t;

   308 if term2str t' = "(3 - 3 * x) * (-1 + x) / (2 * (-1 + x))" andalso terms2str asm = "[]"

   309 then () else error "factout_ 2 changed";

   310 val SOME (t', asm) = cancel_(*p_*) thy t;

   311 if term2str t' = "(3 - 3 * x) / 2" andalso terms2str asm = "[\"-1 + x ~= 0\"]"

   312 then () else error "cancel_ 2 changed";

   314 val t = str2term "((x + (-1)) / (x + 1)) + ((x + 1) / (x + (-1)))";

   315 val SOME (t', asm) = add_fraction_p_ thy t;

   316 if term2str t' = "(2 + 2 * x ^^^ 2) / (-1 + x ^^^ 2)" andalso terms2str asm = "[]"

   317 then () else error "add_fraction_p_ 3 changed";

   318 val SOME (t', asm) = common_nominator_p_ thy t;

   319 if term2str t' =

   320   "(-1 + x) * (-1 + x) / ((1 + x) * (-1 + x)) +\n(1 + x) * (1 + x) / ((1 + x) * (-1 + x))"

   321   andalso terms2str asm = "[]"

   322 then () else error "common_nominator_p_ 3 changed";

   324 val t = str2term "((x - 1) / (x + 1)) + ((x + 1) / (x - 1))";

   325 val SOME (t', asm) = add_fraction_ thy t;

   326 if term2str t' = "(2 + 2 * x ^^^ 2) / (-1 + x ^^^ 2)" andalso terms2str asm = "[]"

   327 then () else error "factout_ 4 changed";

   328 val SOME (t', asm) = common_nominator_ thy t;

   329 if term2str t' =

   330   "(-1 + x) * (-1 + x) / ((1 + x) * (-1 + x)) +\n(1 + x) * (1 + x) / ((1 + x) * (-1 + x))"

   331   andalso terms2str asm = "[]"

   332 then () else error "cancel_ 4 changed";

   334 val t = str2term

   335   "((1) / (2*x + 2)) + ((1) / (2*x + (-2))) + ((1) / ( x^^^2 + (-1)))+((1) / (x^^^2 + (-2)*x + 1))";

   336 val SOME (t', asm) = add_fraction_p_ thy t;

   337 if term2str t' = "x / (1 + -2 * x + x ^^^ 2)" andalso terms2str asm = "[\"1 + x ~= 0\"]"

   338 then () else error "add_fraction_p_ 5 changed";

   339 val SOME (t', asm) = norm_rational_ thy t;

   340 if term2str t' = "x / (1 + -2 * x + x ^^^ 2)" andalso terms2str asm = "[\"1 + x ~= 0\"]"

   341 then () else error "norm_rational_ 5 changed";

   344 "-------- cancel from: Mathematik 1 Schalk Reniets Verlag --------------------";

   345 "-------- cancel from: Mathematik 1 Schalk Reniets Verlag --------------------";

   346 "-------- cancel from: Mathematik 1 Schalk Reniets Verlag --------------------";

   347 val thy  = @{theory "Rational"};

   348 val thy' = "Rational";

   349 val rls' = "cancel";

   350 val mp = "make_polynomial";

   352 "-------- example 186a";

   353 val t = str2term "(14 * x * y) / (x * y)";

   354   is_expanded (str2term "14 * x * y");

   355   is_expanded (str2term "x * y");

   356 val SOME (t', asm) = rewrite_set_ thy false cancel t;

   357 if (term2str t', terms2str asm) = ("14 / 1", "[\"y * x ~= 0\"]")

   358 then () else error "rational.sml cancel Schalk 186a";

   360 "-------- example 186b";

   361 val t = str2term "(60 * a * b) / ( 15 * a  * b )";

   362 val SOME (t', asm) = rewrite_set_ thy false cancel t;

   363 if (term2str t', terms2str asm) = ("4 / 1", "[\"15 * (b * a) ~= 0\"]")

   364 then () else error "rational.sml cancel Schalk 186b";

   366 "-------- example 186c";

   367 val t = str2term "(144 * a^^^2 * b * c) / (12 * a * b * c)";

   368 val SOME (t', asm) = rewrite_set_ thy false cancel t;

   369 if (term2str t', terms2str asm) = ("12 * a / 1", "[\"12 * (c * (b * a)) ~= 0\"]")

   370 then () else error "rational.sml cancel Schalk 186c";

   372 "-------- example 187a";

   373 val t = str2term "(12 * x * y) / (8 * y^^^2 )";

   374 val SOME (t', asm) = rewrite_set_ thy false cancel t;

   375 if (term2str t', terms2str asm) = ("3 * x / (2 * y)", "[\"4 * y ~= 0\"]")

   376 then () else error "rational.sml cancel Schalk 187a";

   378 "-------- example 187b";

   379 val t = str2term "(8 * x^^^2 * y * z ) / (18 * x * y^^^2 * z )";

   380 val SOME (t', asm) = rewrite_set_ thy false cancel t;

   381 if (term2str t', terms2str asm) = ("4 * x / (9 * y)", "[\"2 * (z * (y * x)) ~= 0\"]")

   382 then () else error "rational.sml cancel Schalk 187b";

   384 "-------- example 187c";

   385 val t = str2term "(9 * x^^^5 * y^^^2 * z^^^4) / (15 * x^^^6 * y^^^3 * z )";

   386 val SOME (t', asm) = rewrite_set_ thy false cancel t;

   387 if (term2str t', terms2str asm) =

   388   ("3 * z ^^^ 3 / (5 * (y * x))", "[\"3 * (z * (y ^^^ 2 * x ^^^ 5)) ~= 0\"]")

   389 then () else error "rational.sml cancel Schalk 187c";

   391 "-------- example 188a";

   392 val t = str2term "(-8 + 8 * x) / (-9 + 9 * x)";

   393   is_expanded (str2term "8 * x + -8");

   394 val SOME (t', asm) = rewrite_set_ thy false cancel t;

   395 if (term2str t', terms2str asm) = ("8 / 9", "[\"-1 + x ~= 0\"]")

   396 then () else error "rational.sml cancel Schalk 188a";

   398 val t = str2term "(8*((-1) + x))/(9*((-1) + x))";

   399 val SOME (t,_) = rewrite_set_ thy false (*!!!!!!!!!!!!!!!!!!!!!!1*) make_polynomial t;

   400 if (term2str t', terms2str asm) = ("8 / 9", "[\"-1 + x ~= 0\"]")

   401 then () else error "rational.sml cancel Schalk make_polynomial 1";

   403 "-------- example 188b";

   404 val t = str2term "(-15 + 5 * x) / (-18 + 6 * x)";

   405 val SOME (t', asm) = rewrite_set_ thy false cancel t;

   406 if (term2str t', terms2str asm) = ("5 / 6", "[\"-3 + x ~= 0\"]")

   407 then () else error "rational.sml cancel Schalk 188b";

   409 "-------- example 188c";

   410 val t = str2term "(a + -1 * b) / (b + -1 * a)";

   411 val SOME (t', asm) = rewrite_set_ thy false  (*!!!!! NONE with cancel !!!!!*) cancel_p t;

   412 if (term2str t', terms2str asm) = ("-1 / 1", "[\"b + -1 * a ~= 0\"]")

   413 then () else error "rational.sml cancel Schalk 188c";

   415 is_expanded (str2term "a + -1 * b") = false (*!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!*);

   416 val t = str2term "((-1)*(b + (-1) * a))/(1*(b + (-1) * a))";

   417 val SOME (t', asm) = rewrite_set_ thy false make_polynomial t;

   418 if (term2str t', terms2str asm) = ("(a + -1 * b) / (-1 * a + b)", "[]")

   419 then () else error "rational.sml cancel Schalk make_polynomial 2";

   421 "-------- example 190a";

   422 val t = str2term "( 27 * a^^^3 + 9 * a^^^2 + 3 * a + 1 ) / ( 27 * a^^^3 + 18 * a^^^2 + 3 * a )";

   423 val SOME (t', asm) = rewrite_set_ thy false cancel t;

   424 if (term2str t', terms2str asm) =

   425   ("(1 + 9 * a ^^^ 2) / (3 * a + 9 * a ^^^ 2)", "[\"1 + 3 * a ~= 0\"]")

   426 then () else error "rational.sml cancel Schalk 190a";

   428 "-------- example 190c";

   429 val t = str2term "((1 + 9 * a ^^^ 2)*(1 + 3 * a))/((3 * a + 9 * a ^^^ 2)*(1 + 3 * a))";

   430 val SOME (t', asm) = rewrite_set_ thy false make_polynomial t;

   431 if (term2str t', terms2str asm) =

   432   ("(1 + 3 * a + 9 * a ^^^ 2 + 27 * a ^^^ 3) /\n(3 * a + 18 * a ^^^ 2 + 27 * a ^^^ 3)", "[]")

   433 then () else error "rational.sml make_polynomial Schalk 190c";

   435 "-------- example 191a";

   436 val t = str2term "( x^^^2 + -1 * y^^^2 ) / ( x + y )";

   437   is_expanded (str2term "x^^^2 + -1 * y^^^2") = false; (*!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!*)

   438   is_expanded (str2term "x + y") = true;

   439 val NONE (*!!!!!!!! WN.23.10.02 !!!!!!!!!!!!!!!*) = rewrite_set_ thy false cancel t;

   441 "-------- example 191b";

   442 val t = str2term "((x + (-1) * y)*(x + y))/((1)*(x + y))";

   443 val SOME (t', asm) = rewrite_set_ thy false make_polynomial t;

   444 if (term2str t', terms2str asm) = ("(x ^^^ 2 + -1 * y ^^^ 2) / (x + y)", "[]")

   445 then () else error "rational.sml make_polynomial Schalk 191b";

   447 "-------- example 191c";

   448 val t = str2term "( 9 * x^^^2 + -30 * x + 25 ) / ( 9 * x^^^2 + -25 )";

   449   is_expanded (str2term "9 * x^^^2 + -30 * x + 25") = false; (*!!!!WN.23.10.02!!!!!!!!!!*)

   450   is_expanded (str2term "25 + -30*x + 9*x^^^2") = false; (*!!!!!!!!WN.23.10.02!!!!!!!!!!*)

   451   is_expanded (str2term "-25 + 9*x^^^2") = true;

   453 val t = str2term "(((-5) + 3 * x)*((-5) + 3 * x))/((5 + 3 * x)*((-5) + 3 * x))";

   454 val SOME (t', asm) = rewrite_set_ thy false make_polynomial t;

   455 if (term2str t', terms2str asm) = ("(25 + -30 * x + 9 * x ^^^ 2) / (-25 + 9 * x ^^^ 2)", "[]")

   456 then () else error "rational.sml make_polynomial Schalk 191c";

   458 "-------- example 192b";

   459 val t = str2term "( 7 * x^^^3 + -1 * x^^^2 * y ) / ( 7 * x * y^^^2 + -1 *  y^^^3 )";

   460 val NONE (*!!!!!!!! WN.23.10.02 !!!!!!!!!!!!!!!*) = rewrite_set_ thy false cancel t;

   462 val t = str2term "((x ^^^ 2)*(7 * x + (-1) * y))/((y ^^^ 2)*(7 * x + (-1) * y))";

   463 val SOME (t', asm) = rewrite_set_ thy false make_polynomial t;

   464 if (term2str t', terms2str asm) =

   465   ("(7 * x ^^^ 3 + -1 * x ^^^ 2 * y) / (7 * x * y ^^^ 2 + -1 * y ^^^ 3)", "[]")

   466 then () else error "rational.sml make_polynomial Schalk 192b";

   468 (*WN050929, === inhibit exn 110317 ===*)

   469 val t = str2term "((x ^^^ 2)*(7 * x + (-1) * y))/((y ^^^ 2)*(7 * x + (-1) * y))";

   470 val SOME (t', asm) = rewrite_set_ thy false make_polynomial t;

   471 if (term2str t', terms2str asm) =

   472   ("(7 * x ^^^ 3 + -1 * x ^^^ 2 * y) / (7 * x * y ^^^ 2 + -1 * y ^^^ 3)", "[]")

   473 then () else error "rational.sml make_polynomial Schalk WN050929";

   475 "-------- example 193a";

   476 val t = str2term "( x^^^2 + -6 * x + 9 ) / ( x^^^2 + -9 )";

   477 val NONE (*!!!!!!!!!!!WN.23.10.02!!!!!!!!!!!!!11*) = rewrite_set_ thy false cancel t;

   479 "-------- example 193b";

   480 val t = str2term "( x^^^2 + -8 * x + 16 ) / ( 2 * x^^^2 + -32 )";

   481 val NONE (*!!!!!!!!!!!WN.23.10.02!!!!!!!!!!!!!11*) = rewrite_set_ thy false cancel t;

   483 "-------- example 193c";

   484 val t = str2term "( 2 * x + -50 * x^^^3 ) / ( 25 * x^^^2 + -10 * x + 1 )";

   485 val NONE (*!!!!!!!!!!!WN.23.10.02!!!!!!!!!!!!!11*) = rewrite_set_ thy false cancel t;

   487 (*WN:*)

   488 val t = str2term "(-25 + 9*x^^^2)/(5 + 3*x)";

   489 val SOME (t, asm) = rewrite_set_ thy false cancel t;

   490 if (term2str t', terms2str asm) =

   491   ("(7 * x ^^^ 3 + -1 * x ^^^ 2 * y) / (7 * x * y ^^^ 2 + -1 * y ^^^ 3)", "[\"5 + 3 * x ~= 0\"]")

   492 then () else error "rational.sml cancel WN 1";

   494 "-------- some old tests REVISE ! --------------------------------------------";

   495 "-------- some old tests REVISE ! --------------------------------------------";

   496 "-------- some old tests REVISE ! --------------------------------------------";

   497 (*WAS --- external calculating functions test ---*)

   498 val t = str2term "((3 * x^^^2 + 6 *x + 3) / (2*x + 2))";

   499 val SOME (t', asm) = factout_p_ thy t;

   500 if term2str t' = "(3 + 3 * x) * (1 + x) / (2 * (1 + x))" andalso terms2str asm = "[]"

   501 then () else error "factout_p_ 1 changed";

   502 val SOME (t', asm) = cancel_p_ thy t;

   503 if term2str t' = "(3 + 3 * x) / 2" andalso terms2str asm = "[\"1 + x ~= 0\"]"

   504 then () else error "cancel_p_ 1 changed";

   506 val t = str2term "((-3 * x^^^2 + 6 *x - 3) / (2*x - 2))";

   507 val SOME (t', asm) = factout_(*p_*) thy t;

   508 if term2str t' = "(3 - 3 * x) * (-1 + x) / (2 * (-1 + x))" andalso terms2str asm = "[]"

   509 then () else error "factout_ 2 changed";

   510 val SOME (t', asm) = cancel_(*p_*) thy t;

   511 if term2str t' = "(3 - 3 * x) / 2" andalso terms2str asm = "[\"-1 + x ~= 0\"]"

   512 then () else error "cancel_ 2 changed";

   514 val t = str2term "((x + (-1)) / (x + 1)) + ((x + 1) / (x + (-1)))";

   515 val SOME (t', asm) = add_fraction_p_ thy t;

   516 if term2str t' = "(2 + 2 * x ^^^ 2) / (-1 + x ^^^ 2)" andalso terms2str asm = "[]"

   517 then () else error "add_fraction_p_ 3 changed";

   518 val SOME (t', asm) = common_nominator_p_ thy t;

   519 if term2str t' =

   520   "(-1 + x) * (-1 + x) / ((1 + x) * (-1 + x)) +\n(1 + x) * (1 + x) / ((1 + x) * (-1 + x))"

   521   andalso terms2str asm = "[]"

   522 then () else error "common_nominator_p_ 3 changed";

   524 val t = str2term "((x - 1) / (x + 1)) + ((x + 1) / (x - 1))";

   525 val SOME (t', asm) = add_fraction_ thy t;

   526 if term2str t' = "(2 + 2 * x ^^^ 2) / (-1 + x ^^^ 2)" andalso terms2str asm = "[]"

   527 then () else error "factout_ 4 changed";

   528 val SOME (t', asm) = common_nominator_ thy t;

   529 if term2str t' =

   530   "(-1 + x) * (-1 + x) / ((1 + x) * (-1 + x)) +\n(1 + x) * (1 + x) / ((1 + x) * (-1 + x))"

   531   andalso terms2str asm = "[]"

   532 then () else error "cancel_ 4 changed";

   534 val t = str2term

   535   "((1) / (2*x + 2)) + ((1) / (2*x + (-2))) + ((1) / ( x^^^2 + (-1)))+((1) / (x^^^2 + (-2)*x + 1))";

   536 val SOME (t', asm) = add_fraction_p_ thy t;

   537 if term2str t' = "x / (1 + -2 * x + x ^^^ 2)" andalso terms2str asm = "[\"1 + x ~= 0\"]"

   538 then () else error "add_fraction_p_ 5 changed";

   539 val SOME (t', asm) = norm_rational_ thy t;

   540 if term2str t' = "x / (1 + -2 * x + x ^^^ 2)" andalso terms2str asm = "[\"1 + x ~= 0\"]"

   541 then () else error "norm_rational_ 5 changed";

   544 val t3 = (term_of o the o (parse thy)) "((1) / (2*x + 2)) + ((1) / (2*x - 2)) + ((1) / ( x^^^2 - 1))+((1) / (x^^^2 - 2 * x + 1))";

   545 val SOME (t3',_) = common_nominator_ thy t3;

   546 val SOME (t3'',_) = add_fraction_ thy t3;

   547 (term2str t3');

   548 (term2str t3'');

   550 val SOME(t4,t5) = norm_expanded_rat_ thy t3;

   551 term2str t4;

   552 term2str (hd(t5));

   556   val t=(term_of o the o (parse thy)) "(9 - x^^^2)/(9 - 6*x + x^^^2)";

   557   val SOME (t',_) = factout_ thy t;

   558   val SOME (t'',_) = cancel_ thy t;

   559   term2str t';

   560   term2str t'';

   561   "(3 + x) * (3 - x) / ((3 - x) * (3 - x))";

   562   "(3 + x) / (3 - x)";

   564   val t=(term_of o the o (parse thy))

   565 	    "(9 - x^^^2) / (9 - 6*x + x^^^2) + 1 / (3 - x)";

   566   val SOME (t',_) = common_nominator_ thy t;

   567   val SOME (t'',_) = add_fraction_ thy t;

   568   term2str t';

   569   term2str t'';

   570   "(9 - x ^^^ 2) / ((3 - x) * (3 - x)) + 1 * (3 - x) / ((3 - x) * (3 - x))";

   571   "(4 + x) / (3 - x)";

   573 (*WN021016 added -----vv---*)

   574 val t = str2term "(9 - x^^^2) / (9 - 6*x + x^^^2) + 1";

   575 val SOME (t',_) = common_nominator_ thy t;

   576 val SOME (t'',_) = add_fraction_ thy t;

   577 term2str t' = "(9 - x ^^^ 2) / (9 - 6 * x + x ^^^ 2) +\n1" ^

   578 		" * (9 - 6 * x + x ^^^ 2) / (9 - 6 * x + x ^^^ 2)" (*true*);

   579 term2str t'' = "6 / (3 - x)" (*true*);

   581 val t = str2term "1 + (9 - x^^^2) / (9 - 6*x + x^^^2)";

   582 val SOME (t',_) = common_nominator_ thy t;

   583 val SOME (t'',_) = add_fraction_ thy t;

   584 term2str t' = "1 * (9 - 6 * x + x ^^^ 2) / (9 - 6 * x + x ^^^ 2) +\n" ^

   585 		"(9 - x ^^^ 2) / (9 - 6 * x + x ^^^ 2)" (*true*);

   586 term2str t'' = "6 / (3 - x)" (*true*);

   587 (*WN021016 added -----^^---*)

   588 (*WN030602 added -----vv--- no rewrite -> NONE !*)

   589 val t = str2term "1 / a";

   590 val NONE =  cancel_p_ thy t;

   591 val NONE = rewrite_set_ thy false cancel_p t;

   592 (*WN.2.6.03 added -------^^---*)

   594 val t = str2term "(y^^^2 - x^^^2)/(y^^^2 - 2*y*x + x^^^2)";

   595 val SOME (t',_) = factout_ thy t;

   596 val SOME (t'',_) = cancel_ thy t;

   597 term2str t' = "(y + x) * (y - x) / ((y - x) * (y - x))"(*true*);

   598 term2str t'' = "(y + x) / (y - x)";

   600 val t = str2term "(y^^^2 - x^^^2)/(y^^^2 - 2*y*x + x^^^2) + 1 / (y - x)";

   601 val SOME (t',_) = common_nominator_ thy t;

   602 val SOME (t'',_) = add_fraction_ thy t;

   603 term2str t' =

   604 "(-1 * x ^^^ 2 + y ^^^ 2) / ((-1 * x + y) * (-1 * x + y)) +\n1" ^

   605 " * (-1 * x + y) / ((-1 * x + y) * (-1 * x + y))" (*true*);

   606 term2str t'' = "(-1 - x - y) / (x - y)" (*true*);

   608 val t = str2term "(x^^^2 - y^^^2)/(x^^^2 - 2*x*y + y^^^2) + 1 / (x - y)";

   609 val SOME (t',_) = common_nominator_ thy t;

   610 val SOME (t'',_) = add_fraction_ thy t;

   611 if term2str t' = "(-1 * y ^^^ 2 + x ^^^ 2) / ((-1 * y + x) * (-1 * y + x))" ^

   612 " +\n1 * (-1 * y + x) / ((-1 * y + x) * (-1 * y + x))" then ()

   613 else error "rational.sml lex-ord 1";

   614 if term2str t'' = "(-1 - y - x) / (y - x)" then ()

   615 else error "rational.sml lex-ord 2";

   616 (*WN.16.10.02 WN070905 lexicographische Ordnung erhalten *)

   619 val t = str2term "(x^^^2 - y^^^2)/(x^^^2 - 2*x*y + y^^^2)";

   620 val SOME (t',_) = norm_expanded_rat_ thy t;

   621 if term2str t' = "(x + y) / (x - y)" then ()

   622 else error "rational.sml term2poly: invalid x ^^^ 2 - y ^^^ 2 No.1";

   623 (*val SOME (t'',_) = norm_rational_ thy t;

   624  *** RATIONALS_TERM2POLY_EXCEPTION: Invalid Polynomial

   625 WN.16.10.02 ?! + WN060831???SK4

   626 WN070905 *** term2poly: invalid = x ^^^ 2 - y ^^^ 2*)

   629 val t = str2term "(9 - x^^^2)/(9 - 6*x + x^^^2) + (1)/(3 + x)";

   630 val SOME (t',_) = norm_expanded_rat_ thy t;

   631 if term2str t' = "(12 + 5 * x + x ^^^ 2) / (9 - x ^^^ 2)" then ()

   632 else error "rational.sml (9 - x^^^2)/(9 - 6*x + x^^^2) +...";

   633 (*val SOME (t'',_) = norm_rational_ thy t;

   634   *** RATIONALS_TERM2POLY_EXCEPTION: Invalid Polynomial WN.16.10.02 ?!

   635 WN070906 *** term2poly: invalid = 9 - x ^^^ 2 SK.term2poly*)

   637   val t=(term_of o the o (parse thy))

   638 	    "(9 + (-1)* x^^^2)/(9 + (-1)* 6*x + x^^^2) + (1)/(3 + x)";

   639   val SOME (t',_) = norm_expanded_rat_ thy t;

   640   val SOME (t'',_) = norm_rational_ thy t;

   641   term2str t';

   642   term2str t'';

   643   "(12 + 5 * x + x ^^^ 2) / (9 - x ^^^ 2)";

   644   "(12 + 5 * x + x ^^^ 2) / (9 + (-1) * x ^^^ 2)";

   647 "-------- cancel from: Mathematik 1 Schalk Reniets Verlag";

   648 "-------- cancel from: Mathematik 1 Schalk Reniets Verlag";

   649 "-------- cancel from: Mathematik 1 Schalk Reniets Verlag";

   650 val thy  = @{theory "Rational"};

   651 val thy' = "Rational";

   652 val rls' = "cancel";

   653 val mp = "make_polynomial";

   655 writeln ("example 186:");

   656 writeln("a)");

   657 val e186a'="(14 * x * y) / ( x * y )";(*SRC*)

   658 rewrite_set_ thy false cancel (str2term e186a');

   659 "@@@@@@@@@@@@@@";

   660 val e186a = the (rewrite_set thy' false "cancel" e186a');

   661   is_expanded (parse_rat "14 * x * y");

   662   is_expanded (parse_rat "x * y");

   663 if e186a = ("14 / 1", "[\"y * x ~= 0\"]") then ()

   664 else error "rational.sml cancel Schalk e186a";

   666 writeln("b)");

   667 val e186b'="(60 * a * b) / ( 15 * a  * b )";

   668 val e186b = the (rewrite_set thy' false "cancel" e186b');

   669 writeln("c)");

   670 val e186c'="(144 * a^^^2 * b * c) / (12 * a * b * c )";

   671 val e186c = (the (rewrite_set thy' false "cancel" e186c'))

   672     handle e => print_exn_G e;

   673 val t = (term_of o the o (parse thy)) e186c';

   674 if e186c = ("12 * a / 1", "[\"12 * (c * (b * a)) ~= 0\"]") then ()

   675 else error "rational.sml cancel Schalk e186c";

   677 writeln ("example 187:");

   678 writeln("a)");

   679 val e187a'="(12 * x * y) / (8 * y^^^2 )";(*SRC*)

   680 val e187a = the (rewrite_set thy' false "cancel" e187a');

   681 writeln("b)");

   682 val e187b'="(8 * x^^^2 * y * z ) / (18 * x * y^^^2 * z )";

   683 val e187b = the (rewrite_set thy' false "cancel" e187b');

   684 writeln("c)");

   685 val e187d'="(9 * x^^^5 * y^^^2 * z^^^4) / (15 * x^^^6 * y^^^3 * z )";(*SRC*)

   686 val e187d = the (rewrite_set thy' false "cancel" e187d');

   687 if e187d = ("3 * z ^^^ 3 / (5 * (y * x))",

   688             "[\"3 * (z * (y ^^^ 2 * x ^^^ 5)) ~= 0\"]") then ()

   689 else error "rational.sml cancel Schalk e186d";

   691 writeln "example 188:";

   692 val e188a'="(-8 + 8 * x) / (-9 + 9 * x)";(*SRC*)

   693 val e188a = the (rewrite_set thy' false "cancel" e188a');

   694   is_expanded (parse_rat "8 * x + -8");

   695 (* e188a = ("8 / 9",["not ((-1) + x = 0)"]) then () 13.3.03*)

   696 if e188a = ("8 / 9", "[\"-1 + x ~= 0\"]") then ()

   697 else error "rational.sml: e188a new behaviour";

   699 val SOME (t,_) = rewrite_set thy' false mp "(8*((-1) + x))/(9*((-1) + x))";

   700 writeln("b)");

   701 val e188b'="(-15 + 5 * x) / (-18 + 6 * x)";(*SRC*)

   702 val SOME (t, asm) = rewrite_set thy' false "cancel" e188b';

   703 t = "5 / 6" (*true*);

   704 writeln("c)");

   706 val e188c'="( a + -1 * b ) / ( b + -1 * a )";

   707 val e188c = the (rewrite_set thy' false "cancel_p" e188c');

   708 (*is_expanded (parse_rat "a + -1 * b");*)

   709 val SOME (t,_) =

   710     rewrite_set thy' false mp "((-1)*(b + (-1) * a))/(1*(b + (-1) * a))";

   711 if t= "(a + -1 * b) / (-1 * a + b)"  then()

   712 else error "rational.sml: e188c new behaviour";

   714 writeln ("example 190:");

   715 writeln("c)");

   716 val e190c'="( 27 * a^^^3 + 9 * a^^^2 + 3 * a + 1 ) / ( 27 * a^^^3 + 18 * a^^^2 + 3 * a )";

   717 val e190c = the (rewrite_set thy' false "cancel" e190c');

   718 val SOME (t,_) = rewrite_set thy' false mp "((1 + 9 * a ^^^ 2)*(1 + 3 * a))/((3 * a + 9 * a ^^^ 2)*(1 + 3 * a))";

   719 if t = "(1 + 3 * a + 9 * a ^^^ 2 + 27 * a ^^^ 3) /\n(3 * a + 18 * a ^^^ 2 + 27 * a ^^^ 3)" then ()

   720 else error "rational.sml: e190c new behaviour";

   722 writeln ("example 191:");

   723 writeln("a)");

   724 val e191a'="( x^^^2 + -1 * y^^^2 ) / ( x + y )";

   725 (*WN.23.10.02-------

   726 val e191a = the (rewrite_set thy' false "cancel" e191a');

   727   is_expanded (parse_rat "x^^^2 + -1 * y^^^2");

   728   false;

   729   is_expanded (parse_rat "x + y");

   730   true; -----------*)

   731 val SOME (t,_) = rewrite_set thy' false mp "((x + (-1) * y)*(x + y))/((1)*(x + y))";

   732 (* t="(x ^^^ 2 + -1 * y ^^^ 2) / (x + y)" then() WN.13.3.03*)

   733 if t="(x ^^^ 2 + -1 * y ^^^ 2) / (x + y)" then()

   734 else error "rational.sml: e191a new behaviour";

   736 writeln("c)");

   737 val e191c'="( 9 * x^^^2 + -30 * x + 25 ) / ( 9 * x^^^2 + -25 )";

   738 (*WN.23.10.02-------

   739 val e191c = the (rewrite_set thy' false "cancel" e191c');

   740   is_expanded (parse_rat "9 * x^^^2 + -30 * x + 25");

   741   false;

   742   is_expanded (parse_rat "25 + -30*x + 9*x^^^2");

   743   false;

   744   is_expanded (parse_rat "-25 + 9*x^^^2");

   745   true;------------*)

   746 val SOME (t,_) = rewrite_set thy' false mp "(((-5) + 3 * x)*((-5) + 3 * x))/((5 + 3 * x)*((-5) + 3 * x))";

   747 (* t="(25 + ((-30) * x + 9 * x ^^^ 2)) / ((-25) + 9 * x ^^^ 2)"then() 13.3.03*)

   748 if t= "(25 + -30 * x + 9 * x ^^^ 2) / (-25 + 9 * x ^^^ 2)" then()

   749 else error "rational.sml: 'e191c' new behaviour";

   752 writeln ("example 192:");

   753 writeln("b)");

   754 val e192b'="( 7 * x^^^3 + -1 * x^^^2 * y ) / ( 7 * x * y^^^2 + -1 *  y^^^3 )";

   755 (*WN.23.10.02-------

   756 val e192b = the (rewrite_set thy' false "cancel" e192b');

   757 -------------------*)

   758 val SOME (t',_) = rewrite_set thy' false mp "((x ^^^ 2)*(7 * x + (-1) * y))/((y ^^^ 2)*(7 * x + (-1) * y))";

   759 (*========== inhibit exn 110317 ================================================

   760 if t' = "(7 * x ^^^ 3 + -1 * x ^^^ 2 * y) / (7 * x * y ^^^ 2 + -1 * y ^^^ 3)"

   761 (*"(-1 * y * x ^^^ 2 + 7 * x ^^^ 3) / (-1 * y ^^^ 3 + 7 * x * y ^^^ 2)"WN050929*)

   762 then () else error "rational.sml: 'e192b' new behaviour";

   763 (*^^^ works with MG's simplifier vvv*)

   764 val t =str2term"((x ^^^ 2)*(7 * x + (-1) * y))/((y ^^^ 2)*(7 * x + (-1) * y))";

   765 val SOME (t',_) = rewrite_set_ thy false make_polynomial t;

   766 if term2str t' = "(7 * x ^^^ 3 + -1 * x ^^^ 2 * y) / (7 * x * y ^^^ 2 + -1 * y ^^^ 3)" then () else error "rational.sml: 'e192b'MG new behaviour";

   767 ============ inhibit exn 110317 ==============================================*)

   769 writeln ("example 193:");

   770 writeln("a)");

   771 val e193a'="( x^^^2 + -6 * x + 9 ) / ( x^^^2 + -9 )";

   772 (*WN.23.10.02-------

   773 val e193a = the (rewrite_set thy' false "cancel" e193a');

   774 -------------------*)

   775 writeln("b)");

   776 val e193b'="( x^^^2 + -8 * x + 16 ) / ( 2 * x^^^2 + -32 )";

   777 (*WN.23.10.02-------

   778 val e193b = the (rewrite_set thy' false "cancel" e193b');

   779 writeln("c)");

   780 val e193c'="( 2 * x + -50 * x^^^3 ) / ( 25 * x^^^2 + -10 * x + 1 )";

   781 val SOME(t,_) = rewrite_set thy' false "cancel" e193c';

   782 -------------------*)

   784 val wn01 = "(-25 + 9*x^^^2)/(5 + 3*x)";

   785 val SOME (t, asm) = rewrite_set thy' false "cancel" wn01;

   786 (* t = "((-5) + 3 * x) / 1" then () WN.13.3.03*)

   787 if t = "(-5 + 3 * x) / 1" andalso asm = "[\"5 + 3 * x ~= 0\"]" then ()

   788 else error "rational.sml: new behav. in cancel wn01";

   790 "-------- common_nominator_p ----------------------------";

   791 "-------- common_nominator_p ----------------------------";

   792 "-------- common_nominator_p ----------------------------";

   793 val rls' = "common_nominator_p";

   795 writeln ("example 204:");

   796 writeln("a)");

   797 val e204a'="((5 * x) / 9) + ((3 * x) / 9) + (x / 9)";

   798 val e204a = the (rewrite_set thy' false "common_nominator_p" e204a');

   799 writeln("b)");

   800 val e204b'="5 / x + -3 / x + -1 / x";

   801 val e204b = the (rewrite_set thy' false "common_nominator_p" e204b');

   802 if e204b = ("1 / x", "[]") then ()

   803 else error "rational.sml common_nominator_p example e204b";

   805 writeln ("example 205:");

   806 writeln("a)");

   807 val e205a'="((4 * x + 7) / 8) + ((4 * x + 3) / 8)";

   808 val e205a = the (rewrite_set thy' false "common_nominator_p" e205a');

   809 writeln("b)");

   810 val e205b'="((5 * x + 2) / 3) + ((-2 * x + 1) / 3)";

   811 val e205b = the (rewrite_set thy' false "common_nominator_p" e205b');

   812 if e205b = ("(1 + x) / 1", "[]") then ()

   813 else error "rational.sml common_nominator_p example e204b";

   815 writeln ("example 206:");

   816 writeln("a)");

   817 val e206a'="((5 * x + 4) / (2 * x + -1)) + ((9 * x + 5) / (2 * x + -1))";

   818 val e206a = the (rewrite_set thy' false "common_nominator_p" e206a');

   819 writeln("b)");

   820 val e206b'="((17 * x + -23) / (5 * x + 4)) + ((-25 + -17 * x) / (5 * x + 4))";

   821 val e206b = the (rewrite_set thy' false "common_nominator_p" e206b');

   823 writeln ("example 207:");

   824 val e207'="((3 * x * y + 3 * y) / (x * y)) + ((5 * x * y + 7 * y) / (x * y)) + ((9 * x * y + -2 * y) / (x * y)) + ((x * y + 4 * y) / (x * y)) ";

   825 val e207 = the (rewrite_set thy' false "common_nominator_p" e207');

   827 writeln ("example 208:");

   828 val e208'="((3 * x + 2) / (x + 2)) + ((5 * x + -1) / (x + 2)) + ((-7 * x + -3) / (x + 2)) + ((-1 * x + -3) / (x + 2)) ";

   829 val e208 = the (rewrite_set thy' false "common_nominator_p" e208');

   831 writeln ("example 209:");

   832 val e209'="((3 * x + -7 * y + 3 * z) / (4)) + ((2 * x + 17 * y + 10 * z) / (4)) + ((-1 * x + 2 * y + z) / (4)) ";

   833 val e209 = the (rewrite_set thy' false "common_nominator_p" e209');

   835 writeln ("example 210:");

   836 val e210'="((2 * x + 3 +  -1 * x^^^2) / (5 * x)) + ((5 * x^^^2 + -2 * x + 1) / (5 * x)) + ((-3 * x^^^2 + -2 * x + 1) / (5 * x)) + ((-1 * x^^^2 + -3 * x + -5) / (5 * x)) ";

   837 val e210 = the (rewrite_set thy' false "common_nominator_p" e210');

   839 writeln ("example 211:");

   840 writeln("a)");

   841 val e211a'="((b) / (a + -1 * b)) + ((-1 * a) / (a + -1 * b))";

   842 val e211a = the (rewrite_set thy' false "common_nominator_p" e211a');

   843 writeln("b)");

   844 val e211b'="((b) / (b^^^2 + -1 * a^^^2)) + ((-1 * a) / (b^^^2 + -1 * a^^^2))";

   845 val e211b = the (rewrite_set thy' false "common_nominator_p" e211b');

   847 writeln ("example 212:");

   848 writeln("a)");

   849 val e212a'="((4) / (x)) + ((-3) / (y)) + -1";

   850 val e212a = the (rewrite_set thy' false "common_nominator_p" e212a');

   851 writeln("b)");

   852 val e212b'="((4) / (x)) + ((-5) / (y)) + ((6) / (x*y))";

   853 val e212b = the (rewrite_set thy' false "common_nominator_p" e212b');

   855 writeln ("example 213:");

   856 writeln("a)");

   857 val e213a'="((5 * x) / (3 * y^^^2)) + ((19 * z) / (6 * x * y)) +  ((-2 * x) / (3 * y^^^2)) + ((7 * y^^^2) / (6 * x^^^2)) ";

   858 val e213a = the (rewrite_set thy' false "common_nominator_p" e213a');

   859 writeln("b)");

   860 val e213b'="((2 * b) / (3 * a^^^2)) + ((3 * c) / (7 * a * b)) +  ((4 * b) / (3 * a^^^2)) + ((3 * a) / (7 * b^^^2))";

   861 val e213b = the (rewrite_set thy' false "common_nominator_p" e213b');

   863 writeln ("example 214:");

   864 writeln("a)");

   865 val e214a'="((3 * x + 2 * y + 2 * z) / (4)) + ((-5 * x + -3 * y) / (3)) + ((x + y + -2 * z) / (2))";

   866 val e214a = the (rewrite_set thy' false "common_nominator_p" e214a');

   867 writeln("b)");

   868 val e214b'="((5 * x + 2 * y + z) / (2)) + ((-7 * x + -3 * y) / (3)) + ((3 * x + 6 * y + -1 * z) / (12))";

   869 val e214b = the (rewrite_set thy' false "common_nominator_p" e214b');

   871 writeln ("example 216:");

   872 writeln("a)");

   873 val e216a'="((2 * b + 3 * c) / (a * c)) + ((3 * a + b) / (a * b)) + ((-2 * b^^^2 + -3 * a * c) / (a * b * c))";

   874 val e216a = the (rewrite_set thy' false "common_nominator_p" e216a');

   875 writeln("b)");

   876 val e216b'="((2 * a + 3 * b) / (b * c)) + ((3 * c + a) / (a * c)) + ((-2 * a^^^2 + -3 * b * c) / (a * b * c))";

   877 val e216b = the (rewrite_set thy' false "common_nominator_p" e216b');

   879 writeln ("example 217:");

   880 val e217'="((z + -1) / (z)) + ((3 * z ^^^2 + -6 * z + 5) / (z^^^2)) + ((-4 * z^^^3 + 7 * z^^^2 + -5 * z + 5) / (z^^^3))";

   881 val e217 = the (rewrite_set thy' false "common_nominator_p" e217');

   884 val rls' = "common_nominator";

   885 writeln ("example 218:");

   886 val e218'="((9 * a^^^3 - 5 * a^^^2 + 2 * a + 8) / (108 * a^^^4)) + ((-5 * a + 3 * a^^^2 + 4) / (8 * a^^^3)) + ((-261 * a^^^3 + 19 * a^^^2 + -112 * a + 16) / (216 * a^^^4))";

   887 val e218 = the (rewrite_set thy' false "common_nominator" e218');

   888 if e218 = ("(16 - 63 * a ^^^ 2 - 81 * a ^^^ 3) / (108 * a ^^^ 4)", "[]") then ()

   889 else error "rationa.sml common_nominator example e218";

   891 writeln ("example 219:");

   892 writeln("a)");

   893 val e219a'="((1) / (y + 1)) + ((1) / (y + 2)) + ((1) / (y + 3))";

   894 val e219a = the (rewrite_set thy' false "common_nominator" e219a');

   895 writeln("b)");

   896 val e219b'="((1) / (x + 1)) + ((1) / (x + 2)) + ((-2) / (x + 3))";

   897 val e219b = the (rewrite_set thy' false "common_nominator" e219b');

   898 if e219b = ("(5 + 3 * x) / (6 + 11 * x + 6 * x ^^^ 2 + x ^^^ 3)", "[]") then ()

   899 else error "rationa.sml common_nominator example e219b";

   901 writeln ("example 220:");

   902 writeln("a)");

   903 val e220a'="((17) / (5 * r + -2)) + ((-13) / (2 * r + 3)) + ((4) / (3 * r + -5))";

   904 val e220a = the (rewrite_set thy' false "common_nominator" e220a');

   905 writeln("b)");

   906 val e220b'="((20 * a) / (a + -3)) + ((-19 * a) / (a + -4)) + ((a) / (a + -5))";

   907 val e220b = the (rewrite_set thy' false "common_nominator" e220b');

   909 writeln ("example 221:");

   910 writeln("a)");

   911 val e221a'="((a + b) / (a + -1 * b)) + ((a + -1 * b) / (a + b))";

   912 val e221a = the (rewrite_set thy' false "common_nominator" e221a');

   913 writeln("b)");

   914 val e221b'="((x + -1 * y) / (x + y)) + ((x + y) / (x + -1 * y)) ";

   915 val e221b = the (rewrite_set thy' false "common_nominator" e221b');

   917 writeln ("example 222:");

   918 writeln("a)");

   919 val e222a'="((1 + -1 * x) / (1 + x)) + ((-1 + -1 * x) / (1 + -1 * x)) + ((4 * x) / (1 + -1 * x^^^2))";

   920 val e222a = the (rewrite_set thy' false "common_nominator" e222a');

   921 writeln("b)");

   922 val e222b'="((1 + x ) / (1 + -1 * x)) + ((-1 + x) / (1 + x)) + ((2 * x) / (1 + -1 * x^^^2))";

   923 val e222b = the (rewrite_set thy' false "common_nominator" e222b');

   925 writeln ("example 225:");

   926 writeln("a)");

   927 val e225a'="((6 * a) / (a^^^2 + -64)) + ((a + 2) / (2 * a + 16)) + ((-1) / (2))";

   928 val e225a = the (rewrite_set thy' false "common_nominator" e225a');

   929 writeln("b)");

   930 val e225b'="((a + 2 ) / (2 * a + 12)) + ((4 * a) / (a^^^2 + -36)) + ((-1) / (2))";

   931 val e225b = the (rewrite_set thy' false "common_nominator" e225b');

   933 writeln ("example 226:");

   934 writeln("a)");

   935 val e226a'="((35 * z) / (49 * z^^^2 + -4)) + -1 + ((14 * z + -1) / (14 * z + 4)) ";

   936 val e226a = the (rewrite_set thy' false "common_nominator" e226a');

   937 writeln("b)");

   938 val e226b'="((45 * a * b) / (25 * a^^^2 + -9 * b^^^2)) + ((20 * a + 3 * b) / (10 * a + 6 * b))  + -2";

   939 val e226b = the (rewrite_set thy' false "common_nominator" e226b');

   941 writeln ("example 227:");

   942 writeln("a)");

   943 val e227a'="((6 * z + 11) / (6 * z + 14)) + ((9 * z ) / (9 * z^^^2 + -49)) + -1 ";

   944 val e227a = the (rewrite_set thy' false "common_nominator" e227a');

   945 writeln("b)");

   946 val e227b'="((16 * a + 37 * b) / (4 * a + 10 * b)) + ((6 * a * b) / (4 * a^^^2 + -25 * b^^^2)) + -4 ";

   947 val e227b = the (rewrite_set thy' false "common_nominator" e227b');

   949 writeln ("example 228:");

   950 writeln("a)");

   951 val e228a'="((7 * a + 11) / (3 * a^^^2 + -3)) + ((-2 * a + -1) / (a^^^2 + -1 * a)) + ((-1) / (3 * a + 3))";

   952 val e228a = the (rewrite_set thy' false "common_nominator" e228a');

   953 writeln("b)");

   954 val e228b'="((11 * z + 2 * b) / (4 * b * z + -8 * b^^^2)) + ((-8 * z) / (z^^^2 + -4 * b^^^2)) + ((-9 * z + -2 * b) / (4 * b * z + 8 * b^^^2))";

   955 val e228b = the (rewrite_set thy' false "common_nominator" e228b');

   958 writeln ("example 229:");

   959 writeln("a)");

   960 val e229a'="((5 * x^^^2 + y) / (x + 2 * y)) + ((-8 * x^^^3 + 4 * x^^^2 * y + 3 * x * y) / (x^^^2 + -4 * y^^^2)) + ((3 * x^^^2 + -4 * y) / (x + -2 * y))";

   961 val e229a = the (rewrite_set thy' false "common_nominator" e229a');

   962 writeln("b)");

   963 val e229b'="((7 * x^^^2 + y) / (x + 3 * y)) + ((-24 * x^^^2 * y + 5 * x * y + 21 * y^^^2) / (x^^^2 + -9 * y^^^2)) + ((4 * x^^^2 + -6 * y) / (x + -3 * y))";

   964 val e229b = the (rewrite_set thy' false "common_nominator" e229b');

   966 writeln ("example 230:");

   967 writeln("a)");

   968 val e230a'="((5 * x^^^2 + y) / (2 * x + y)) + ((-16 * x^^^3 + 2 * x^^^2 * y + 6 * x * y) / (4 * x^^^2 + -1 * y^^^2)) + ((3 * x^^^2 + -4 * y) / (2 * x + -1 * y))";

   969 val e230a = the (rewrite_set thy' false "common_nominator" e230a');

   970 writeln("b)");

   971 val e230b'="((7 * x^^^2 + y) / (3 * x + y)) + ((-3 * x^^^3  + 15 * x * y + -7 * x^^^2 * y + 7 * y^^^2) / (9 * x^^^2 + -1 * y^^^2)) + ((4 * x^^^2 + -6 * y) / (3 * x + -1 * y))";

   972 val e230b = the (rewrite_set thy' false "common_nominator" e230b');

   974 writeln ("example 231:");

   975 writeln("a)");

   976 val e231a'="((2 * x + 5 * y) / (x)) + ((2 * x^^^3 + -5 * y^^^3 + 3 * x * y^^^2) / (x^^^3 + -2 * x^^^2 * y + x * y^^^2)) + ((-3 * x + -6 * y) / (x + -1 * y))";

   977 val e231a = the (rewrite_set thy' false "common_nominator" e231a');

   978 writeln("b)");

   979 val e231b'="((6 * x + 2 * y) / (x)) + ((6 * x^^^2 * y + -4 * x * y^^^2 + -2 * y^^^3) / (x^^^3 + -2 * x^^^2 * y + x * y^^^2)) + ((-5 * x + -3 * y) / (x + -1 * y))";

   980 val e231b = the (rewrite_set thy' false "common_nominator" e231b');

   982 writeln ("example 232:");

   983 writeln("a)");

   984 val e232a'="((2 * x + 3 * y) / (x)) + ((4 * x^^^3 + -1 * x * y^^^2 + -3 * y^^^3) / (x^^^3 + -2 * x^^^2 * y + x * y^^^2)) + ((-5 * x + -6 * y) / (x + -1 * y))";

   985 val e232a = the (rewrite_set thy' false "common_nominator" e232a');

   986 writeln("b)");

   987 val e232b'="((5 * x + 2 * y) / (x)) + ((2 * x^^^3 + -3 * x * y^^^2 + 3 * x^^^2 * y + -2 * y^^^3) / (x^^^3 + -2 * x^^^2 * y + x * y^^^2)) + ((-6 * x + -3 * y) / (x + -1 * y))";

   988 val e232b = the (rewrite_set thy' false "common_nominator" e232b');

   990 writeln ("example 233:");

   991 writeln("a)");

   992 val e233a'="((5 * x + 6 * y) / (x)) + ((5 * x * y^^^2 + -6 * y^^^3 + -2 * x^^^3 + 3 * x^^^2 * y) / (x^^^3 + -2 * x^^^2 * y + x * y^^^2)) + ((-2 * x + -3 * y) / (x + -1 * y))";

   993 val e233a = the (rewrite_set thy' false "common_nominator" e233a');

   994 writeln("b)");

   995 val e233b'="((6 * x + 5 * y) / (x)) + ((4 * x^^^2 * y + 3 * x * y^^^2 + -5 * y^^^3 + -2 * x^^^3) / (x^^^3 + -2 * x^^^2 * y + x * y^^^2)) + ((-3 * x + -2 * y) / (x + -1 * y))";

   996 val e233b = the (rewrite_set thy' false "common_nominator" e233b');

   998 writeln ("example 234:");

   999 writeln("a)");

  1000 val e234a'="((5 * a + b) / (2 * a * b + -2 * b^^^2)) + ((-3 * a + -1 * b) / (2 * a * b + 2 * b^^^2)) + ((-2 * a) / (a^^^2 + -1 * b^^^2))";

  1001 val e234a = the (rewrite_set thy' false "common_nominator" e234a');

  1002 writeln("b)");

  1003 val e234b'="((5 * a + 3 * b) / (6 * a * b + -18 * b^^^2)) + ((-3 * a + -3 * b) / (6 * a * b + 18 * b^^^2)) + ((-2 * a) / (a^^^2 + -9 * b^^^2)) ";

  1004 val e234b = the (rewrite_set thy' false "common_nominator" e234b');

  1006 writeln ("example 235:");

  1007 writeln("a)");

  1008 val e235a'="((10 * x + 3 * y) / (12 * x * y + -18 * y^^^2)) + ((-6 * x + -3 * y) / (12 * x * y + 18 * y^^^2)) + ((-4 * x) / (4 * x^^^2 + -9 * y^^^2))";

  1009 val e235a = the (rewrite_set thy' false "common_nominator" e235a');

  1010 writeln("b)");

  1011 val e235b'="((8 * a + b) / (4 * a * b + -2 * b^^^2)) + ((-4 * a + -1 * b) / (4 * a * b + 2 * b^^^2)) + ((-2 * a) / (4 * a^^^2 + -1 * b^^^2)) ";

  1012 val e235b = the (rewrite_set thy' false "common_nominator" e235b');

  1014 writeln ("example 236:");

  1015 writeln("a)");

  1016 val e236a'="((8 * a + 5 * b) / (20 * a * b + -50 * b^^^2)) + ((-4 * a + -5 * b) / (20 * a * b + 50 * b^^^2)) + ((-2 * a) / (4 * a^^^2 + -25 * b^^^2))";

  1017 val e236a = the (rewrite_set thy' false "common_nominator" e236a');

  1018 writeln("b)");

  1019 val e236b'="((24 * x + y) / (6 * x * y + -2 * y^^^2)) + ((-18 * x + -1 * y) / (6 * x * y + 2 * y^^^2)) + ((-15 * x) / (9 * x^^^2 + -1 * y^^^2)) ";

  1020 val e236b = the (rewrite_set thy' false "common_nominator" e236b');

  1023 val rls' = "cancel";

  1024 writeln ("example heuberger:");

  1025 val eheu'="(x^^^4 + x * y + x^^^3 * y + y^^^2) / (x + 5 * x^^^2 + y + 5 * x * y + x^^^2 * y^^^3 + x * y^^^4)";

  1026 val eheu = the (rewrite_set thy' false "cancel" eheu');

  1028 val rls' = "common_nominator_p";

  1029 writeln ("example stiefel:");

  1030 val est1'="(7) / (-14) + (-2) / (4)";

  1031 val est1 = the (rewrite_set thy' false "common_nominator_p" est1');

  1032 if est1 = ("-1 / 1", "[]") then ()

  1033 else error "new behaviour in rational.sml: est1'";

  1035 val t = (term_of o the o (parse thy))

  1036 "(9 - x ^^^ 2) / (9 - 6 * x + x ^^^ 2)";

  1037 val SOME (t',_) = factout_ thy t;

  1038 if term2str t' = "(3 + x) * (3 - x) / ((3 - x) * (3 - x))" then ()

  1039 else error "rational.sml factout_ (9 - x ^^^ 2) / (9 - 6 * x + x ^^^ 2)";

  1042 "-------- reverse rewrite -------------------------------";

  1043 "-------- reverse rewrite -------------------------------";

  1044 "-------- reverse rewrite -------------------------------";

  1045 (*WN.28.8.02: tests for the 'reverse-rewrite' functions:

  1046   these are defined in Rationals.ML and stored in

  1047   the 'reverse-ruleset' cancel*)

  1049 (*the term for which reverse rewriting is demonstrated*)

  1050   val t = (term_of o the o (parse thy))

  1051 	      "(9 - x ^^^ 2) / (9 + 6 * x + x ^^^ 2)";

  1052   val Rrls {scr=Rfuns {init_state=ini,locate_rule=loc,

  1053   		       next_rule=nex,normal_form=nor,...},...} = cancel;

  1055 (*normal_form produces the result in ONE step*)

  1056   val SOME (t',_) = nor t;

  1057 if term2str t' = "(3 - x) / (3 + x)" then ()

  1058 else error "rational.sml normal_form (9 - x ^^^ 2) / (9 - 6 * x + x ^^^ 2)";

  1060 (*initialize the interpreter state used by the 'me'*)

  1061   val (t,_,revsets,_) = ini t;

  1063 (*find the rule 'r' to apply to term 't'*)

  1064   val SOME (r as (Thm (str, thm))) = nex revsets t;

  1065 if str = "sym_#power_Float ((3,0), (0,0)) __ ((2,0), (0,0))"

  1066    andalso string_of_thm thm =

  1067            (string_of_thm o make_thm o (cterm_of (@{theory "Isac"})))

  1068                (Trueprop $ (term_of o the o (parse thy)) "9 = 3 ^^^ 2") then ()

  1069 else error "rational.sml next_rule (9 - x ^^^ 2) / (9 - 6 * x + x ^^^ 2)";

  1070 (* before Isa02->09-2 was not checked automatically, ?was different?:

  1071 val SOME r = nex revsets t;

  1072 val r = Thm ("sym_#mult_2_3","6 = 2 * 3") : rule*)

  1074 (* check, if the rule 'r' applied by the user to 't' belongs to the ruleset;

  1075   if the rule is OK, the term resulting from applying the rule is returned,too;

  1076   there might be several rule applications inbetween,

  1077   which are listed after the head in reverse order *)

  1078   val (r, (t, asm))::_ = loc revsets t r;

  1079 if term2str t = "(9 - x ^^^ 2) / (3 ^^^ 2 + 6 * x + x ^^^ 2)" andalso asm = []

  1080 then ()

  1081 else error "rational.sml locate_rule (9 - x ^^^ 2) / (9 - 6 * x + x ^^^ 2)";

  1083 (* find the next rule to apply *)

  1084   val SOME (r as (Thm (str, thm))) = nex revsets t;

  1085 if str = "sym_#power_Float ((3,0), (0,0)) __ ((2,0), (0,0))" andalso

  1086    string_of_thm thm = (string_of_thm o make_thm o (cterm_of (@{theory "Isac"})))

  1087                 (Trueprop $ (term_of o the o (parse thy)) "9 = 3 ^^^ 2") then ()

  1088 else error "rational.sml next_rule (9 - x ^^^ 2) / (9 - 6 * x + x ^^^ 2)";

  1090 (*check the next rule*)

  1091   val (r, (t, asm)) :: _ = loc revsets t r;

  1092 if term2str t = "(3 ^^^ 2 - x ^^^ 2) / (3 ^^^ 2 + 6 * x + x ^^^ 2)" then ()

  1093 else error "rational.sml locate_rule (9 - x ^^^ 2) / (9 - 6 * x + x ^^^ 2) II";

  1095 (*find and check the next rules, rewrite*)

  1096   val SOME r = nex revsets t;

  1097   val (r,(t,asm))::_ = loc revsets t r;

  1098 if term2str t = "(3 ^^^ 2 - x ^^^ 2) / (3 ^^^ 2 + 2 * 3 * x + x ^^^ 2)" then ()

  1099 else error "rational.sml locate_rule II";

  1101   val SOME r = nex revsets t;

  1102   val (r,(t,asm))::_ = loc revsets t r;

  1103 if term2str t = "(3 - x) * (3 + x) / (3 ^^^ 2 + 2 * 3 * x + x ^^^ 2)" then ()

  1104 else error "rational.sml next_rule II";

  1106   val SOME r = nex revsets t;

  1107   val (r,(t,asm))::_ = loc revsets t r;

  1108 if term2str t = "(3 - x) * (3 + x) / ((3 + x) * (3 + x))" then ()

  1109 else error "rational.sml next_rule III";

  1111   val SOME r = nex revsets t;

  1112   val (r, (t, asm)) :: _ = loc revsets t r;

  1113   val ss = term2str t;

  1114 if ss = "(3 - x) / (3 + x)" andalso terms2str asm = "[\"3 + x ~= 0\"]" then ()

  1115 else error "rational.sml: new behav. in rev-set cancel";

  1117 "-------- 'reverse-ruleset' cancel_p --------------------";

  1118 "-------- 'reverse-ruleset' cancel_p --------------------";

  1119 "-------- 'reverse-ruleset' cancel_p --------------------";

  1120 (*WN.11.9.02: the 'reverse-ruleset' cancel_p*)

  1122 (*the term for which reverse rewriting is demonstrated*)

  1123 val t = str2term "(9 + (-1)*x^^^2) / (9 + ((-6)*x + x^^^2))";

  1124 val Rrls {scr=Rfuns {init_state=ini,locate_rule=loc,

  1125 		       next_rule=nex,normal_form=nor,...},...} = cancel_p;

  1127 (*normal_form produces the result in ONE step*)

  1128 val SOME (t',_) = nor t;

  1129 term2str t' = "(3 + 1 * x) / (3 + -1 * x)";

  1131 (*initialize the interpreter state used by the 'me'*)

  1132 val SOME (t', asm) = cancel_p_ thy t;

  1133 term2str t' = "(3 + x) / (3 + -1 * x)" (*true*);

  1134 terms2str asm = "[\"3 + -1 * x ~= 0\"]" (*true*);

  1135 val (t,_,revsets,_) = ini t;

  1137 (* WN.10.10.02: dieser Fall terminiert nicht

  1138            (make_polynomial enth"alt zu viele rules)

  1139 WN060823 'init_state' requires rewriting on specified location in the term

  1140 print_depth 99; Rfuns; print_depth 3;

  1141 WN060831 cycling "sym_order_mult_rls_" "sym_mult_assoc"

  1142          as was with make_polynomial before ?!?*)

  1144 val SOME r = nex revsets t;

  1145 eq_Thm (r, Thm ("sym_#power_Float ((3,0), (0,0)) __ ((2,0), (0,0))",

  1146 		mk_thm thy "9 = 3 ^^^ 2"));

  1147 (*WN060831 *** id_of_thm

  1148            Exception- ERROR raised ...

  1149 val (r,(t,asm))::_ = loc revsets t r;

  1150 term2str t;

  1152   val SOME r = nex revsets t;

  1153   val (r,(t,asm))::_ = loc revsets t r;

  1154   term2str t;

  1155 *)

  1157 writeln "******************  all tests successfull  *************************";

  1161 (*WN.17.3.03 =========================================================vvv---*)

  1162 "-------- norm_Rational ---------------------------------";

  1163 "-------- norm_Rational ---------------------------------";

  1164 "-------- norm_Rational ---------------------------------";

  1165 val t = str2term "(3*x+5)/18 - x/2  - -(3*x - 2)/9 = 0";

  1166 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; term2str t';

  1167 if term2str t' = "1 / 18 = 0" then () else error "rational.sml 1";

  1169 val t = str2term "(17*x - 51)/9 - (-(13*x - 3)/6) + 11 - (9*x - 7)/4 = 0";

  1170 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; term2str t';

  1171 if term2str t' = "(237 + 65 * x) / 36 = 0" then ()

  1172 else error "rational.sml 2";

  1174 val t = str2term "(1/2 + (5*x)/2)^^^2 - ((13*x)/2 - 5/2)^^^2 - (6*x)^^^2 + 29";

  1175 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; term2str t';

  1176 if term2str t' = "23 + 35 * x + -72 * x ^^^ 2" then ()

  1177 else error "rational.sml 3";

  1179 (*trace_rewrite:=true;*)

  1180 val t = str2term "Not (6*x is_atom)";

  1181 val SOME (t',_) = rewrite_set_ thy false powers_erls t; term2str t';

  1182 "HOL.True";

  1183 val t = str2term "1 < 2";

  1184 val SOME (t',_) = rewrite_set_ thy false powers_erls t; term2str t';

  1185 "HOL.True";

  1187 val t = str2term "(6*x)^^^2";

  1188 val SOME (t',_) = rewrite_ thy dummy_ord powers_erls false

  1189 			   (num_str @{thm realpow_def_atom}) t;

  1190 if term2str t' = "6 * x * (6 * x) ^^^ (2 + -1)" then ()

  1191 else error "rational.sml powers_erls (6*x)^^^2";

  1193 val t = str2term "-1 * (-2 * (5 / 2 * (13 * x / 2)))";

  1194 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; term2str t';

  1195 if term2str t' = "65 * x / 2" then () else error "rational.sml 4";

  1197 val t = str2term "1 - ((13*x)/2 - 5/2)^^^2";

  1198 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; term2str t';

  1199 (*bef.040209: if term2str t' = "(-21 + (130 * x + -169 * x ^^^ 2)) / 4"then()*)

  1200 if term2str t' = "(-21 + 130 * x + -169 * x ^^^ 2) / 4" then ()

  1201 else error "rational.sml 5";

  1203 (*SRAM Schalk I, p.92 Nr. 609a*)

  1204 val t = str2term "2*(3 - x/5)/3 - 4*(1 - x/3) - x/3 - 2*(x/2 - 1/4)/27 +5/54";

  1205 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; term2str t';

  1206 if term2str t' = "(-255 + 112 * x) / 135" then ()

  1207 else error "rational.sml 6";

  1209 (*SRAM Schalk I, p.92 Nr. 610c*)

  1210 val t = str2term "((x- 1)/(x+1) + 1) / ((x- 1)/(x+1) - (x+1)/(x- 1)) - 2";

  1211 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; term2str t';

  1212 if term2str t' = "(-3 + -1 * x) / 2" then () else error "rational.sml 7";

  1214 (*SRAM Schalk I, p.92 Nr. 476a*)

  1215 val t = str2term "(x^^^2/(1 - x^^^2) + 1)/(x/(1 - x) + 1) * (1 + x)";

  1216 (*. a/b : c/d translated to a/b * d/c .*)

  1217 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; term2str t';

  1218 (*if term2str t' = "1 / 1" then () else error "rational.sml 8";3.6.03*)

  1219 if term2str t' = "1" then () else error "rational.sml 8";

  1221 (*............................vvv---TODO: sollte gehen mit poly_order *)

  1222 (*Schalk I, p.92 Nr. 472a*)

  1223 val t = str2term "((8*x^^^2 - 32*y^^^2)/(2*x + 4*y))/((4*x - 8*y)/(x + y))";

  1224 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; term2str t';

  1225 if term2str t' = "x + y" then () else error "rational.sml p.92 Nr. 472a";

  1227 (*Schalk I, p.70 Nr. 480b; a/b : c/d translated to a/b * d/c*)

  1228 val t = str2term ("((12*x*y/(9*x^^^2 - y^^^2))/" ^

  1229 		 "(1/(3*x - y)^^^2 - 1/(3*x + y)^^^2)) *" ^

  1230 		 "(1/(x - 5*y)^^^2 - 1/(x + 5*y)^^^2)/" ^

  1231 		 "(20*x*y/(x^^^2 - 25*y^^^2))");

  1232 (*... nicht simpl, zerlegt ...*)

  1233 val t = str2term ("((12*x*y/(9*x^^^2 - y^^^2))/" ^

  1234 		 "(1/(3*x - y)^^^2 - 1/(3*x + y)^^^2))");

  1235 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; term2str t';

  1236 "(-12 * (x * y ^^^ 3) + 108 * (x * (y * x ^^^ 2))) / (12 * (x * y))";

  1237 (*                             ~~~~~~~~~~ poly_order notwendig!*)

  1238 val t = str2term ("(1/(x - 5*y)^^^2 - 1/(x + 5*y)^^^2)/" ^

  1239 		 "(20*x*y/(x^^^2 - 25*y^^^2))");

  1240 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; term2str t';

  1241 if term2str t' = "1 / (x ^^^ 2 + -25 * y ^^^ 2)" then ()

  1242 else error "rational.sml norm_Rational 1 / (x ^^^ 2 + -25 * y ^^^ 2)";

  1244 "nonterm.SK6 ----- SK060904-2a non-termination of add_fraction_p_";

  1245 (*WN.2.6.03 from rlang.sml 56a

  1246 val t = str2term "(a + b * x) / (a + -1 * (b * x)) + (-1 * a + b * x) / (a + b * x) = 4 * (a * b) / (a ^^^ 2 + -1 * b ^^^ 2)";

  1247 val NONE = rewrite_set_ thy false common_nominator_p t;

  1249 WN060831 nonterm.SK7

  1250 val t = str2term "(a + b * x) / (a + -1 * (b * x)) + (-1 * a + b * x) / (a + b * x)";

  1251 val NONE = add_fraction_p_ thy t;

  1252 *)

  1255 (* ------------------------------------------------------------------- *)

  1256 (*---------vvv------------ MG: ab 1.7.03 ----------------vvv-----------*)

  1257 (*                 Simplifier fuer beliebige Buchterme                 *)

  1258 (* ------------------------------------------------------------------- *)

  1259 (*----------------------- norm_Rational_mg ----------------------------*)

  1260 (* ------------------------------------------------------------------- *)

  1262 "-------- numeral rationals -----------------------------";

  1263 "-------- numeral rationals -----------------------------";

  1264 "-------- numeral rationals -----------------------------";

  1265 (*SRA Schalk I, p.40 Nr. 164b *)

  1266 val t = str2term "(47/6 - 76/9 + 13/4)/(35/12)";

  1267 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1268 term2str t;

  1269 if (term2str t) = "19 / 21" then ()

  1270 else error "rational.sml: diff.behav. in norm_Rational_mg 1";

  1272 (*SRA Schalk I, p.40 Nr. 166a *)

  1273 val t = str2term "((5/4)/(4+22/7) + 37/20)*(110/3 - 110/9 * 23/11)";

  1274 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1275 term2str t;

  1276 if (term2str t) = "45 / 2" then ()

  1277 else error "rational.sml: diff.behav. in norm_Rational_mg 2";

  1280 "-------- cancellation ----------------------------------";

  1281 "-------- cancellation ----------------------------------";

  1282 "-------- cancellation ----------------------------------";

  1283 (* e190c Stefan K.*)

  1284 val t = str2term

  1285 "((1 + 9 * a ^^^ 2)*(1 + 3 * a))/((3 * a + 9 * a ^^^ 2)*(1 + 3 * a))";

  1286 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1287 term2str t;

  1288 if (term2str t) =

  1289 "(1 + 9 * a ^^^ 2) / (3 * a + 9 * a ^^^ 2)"

  1290 then ()

  1291 else error "rational.sml: diff.behav. in norm_Rational_mg 3";

  1293 (* e192b Stefan K.*)

  1294 val t = str2term

  1295 "((x ^^^ 2)*(7 * x + (-1) * y))/((y ^^^ 2)*(7 * x + (-1) * y))";

  1296 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1297 term2str t;

  1298 if (term2str t) =

  1299 "x ^^^ 2 / y ^^^ 2"

  1300 then ()

  1301 else error "rational.sml: diff.behav. in norm_Rational_mg 4";

  1303 (*SRC Schalk I, p.66 Nr. 379c *)

  1304 val t = str2term

  1305 "(a - b)/(b - a)";

  1306 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1307 term2str t;

  1308 if (term2str t) =

  1309 "-1"

  1310 then ()

  1311 else error "rational.sml: diff.behav. in norm_Rational_mg 5";

  1313 (*SRC Schalk I, p.66 Nr. 380b *)

  1314 val t = str2term

  1315 "15*(3*x+3)*(4*x+9)/(12*(2*x+7)*(5*x+5))";

  1316 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1317 term2str t;

  1318 if (term2str t) =

  1319 "(27 + 12 * x) / (28 + 8 * x)"

  1320 then ()

  1321 else error "rational.sml: diff.behav. in norm_Rational_mg 6";

  1323 (*Schalk I, p.60 Nr. 215c *)

  1324 (* Falsches Ergebnis: rechnet lange und cancel_p kann nicht weiter krzen!!!*)

  1325 (* WN060831????MG1

  1326 val t = str2term "(a+b)^^^4*(x - y)/((x - y)^^^3*(a+b)^^^2)";

  1327 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1328 term2str t;

  1329 if (term2str t) =

  1330 "(a ^^^ 4 * x + -1 * a ^^^ 4 * y + 4 * a ^^^ 3 * b * x + -4 * a ^^^ 3 * b * y + 6 * a ^^^ 2 * b ^^^ 2 * x + -6 * a ^^^ 2 * b ^^^ 2 * y + 4 * a * b ^^^ 3 * x + -4 * a * b ^^^ 3 * y + b ^^^ 4 * x + -1 * b ^^^ 4 * y) /(a ^^^ 2 * x ^^^ 3 + -3 * a ^^^ 2 * x ^^^ 2 * y + 3 * a ^^^ 2 * x * y ^^^ 2 + -1 * a ^^^ 2 * y ^^^ 3 + 2 * a * b * x ^^^ 3 + -6 * a * b * x ^^^ 2 * y + 6 * a * b * x * y ^^^ 2 + -2 * a * b * y ^^^ 3 + b ^^^ 2 * x ^^^ 3 + -3 * b ^^^ 2 * x ^^^ 2 * y + 3 * b ^^^ 2 * x * y ^^^ 2 + -1 * b ^^^ 2 * y ^^^ 3)"

  1331 then ()

  1332 else error "rational.sml: diff.behav. in norm_Rational_mg 7";

  1333 *)

  1334 (*val t = str2term

  1335 "(a ^^^ 4 * x + -1 * a ^^^ 4 * y + 4 * a ^^^ 3 * b * x + -4 * a ^^^ 3 * b * y + 6 * a ^^^ 2 * b ^^^ 2 * x + -6 * a ^^^ 2 * b ^^^ 2 * y + 4 * a * b ^^^ 3 * x + -4 * a * b ^^^ 3 * y + b ^^^ 4 * x + -1 * b ^^^ 4 * y) /(a ^^^ 2 * x ^^^ 3 + -3 * a ^^^ 2 * x ^^^ 2 * y + 3 * a ^^^ 2 * x * y ^^^ 2 + -1 * a ^^^ 2 * y ^^^ 3 + 2 * a * b * x ^^^ 3 + -6 * a * b * x ^^^ 2 * y + 6 * a * b * x * y ^^^ 2 + -2 * a * b * y ^^^ 3 + b ^^^ 2 * x ^^^ 3 + -3 * b ^^^ 2 * x ^^^ 2 * y + 3 * b ^^^ 2 * x * y ^^^ 2 + -1 * b ^^^ 2 * y ^^^ 3)"

  1336 val SOME (t,_) = rewrite_set_ thy false cancel_p t;

  1337 term2str t;*)

  1338 (* uncaught exception nonexhaustive binding failure

  1339    raised at: stdIn:93.1-93.51 *)

  1341 (*Schalk I, p.66 Nr. 381a *)

  1342 (* ACHTUNG: rechnet ca. 2 Minuten !!! *)

  1343 (* WN060831???MG2

  1344 val t = str2term "18*(a+b)^^^3*(a - b)^^^2/(72*(a - b)^^^3*(a+b)^^^2)";

  1345 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1346 term2str t;

  1347 if (term2str t) =

  1348 "(a + b) / (4 * a + -4 * b)"

  1349 then () else error "rational.sml: diff.behav. in norm_Rational_mg 8";

  1350 *)

  1352 (*SRC Schalk I, p.66 Nr. 381b *)

  1353 val t = str2term

  1354 "(4*x^^^2 - 20*x + 25)/(2*x - 5)^^^3";

  1355 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1356 term2str t;

  1357 if (term2str t) =

  1358 "-1 / (5 + -2 * x)"

  1359 then ()

  1360 else error "rational.sml: diff.behav. in norm_Rational_mg 9";

  1362 (*SRC Schalk I, p.66 Nr. 381c *)

  1363 val t = str2term

  1364 "(27*a^^^3+9*a^^^2+3*a+1)/(27*a^^^3+18*a^^^2+3*a)";

  1365 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1366 term2str t;

  1367 if (term2str t) =

  1368 "(1 + 9 * a ^^^ 2) / (3 * a + 9 * a ^^^ 2)"

  1369 then ()

  1370 else error "rational.sml: diff.behav. in norm_Rational_mg 10";

  1372 (*SRC Schalk I, p.66 Nr. 383a *)

  1373 val t = str2term

  1374 "(5*a^^^2 - 5*a*b)/(a - b)^^^2";

  1375 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1376 term2str t;

  1377 if (term2str t) =

  1378 "5 * a / (a + -1 * b)"

  1379 then ()

  1380 else error "rational.sml: diff.behav. in norm_Rational_mg 11";

  1383 "-------- common denominator ----------------------------";

  1384 "-------- common denominator ----------------------------";

  1385 "-------- common denominator ----------------------------";

  1386 (*SRA Schalk I, p.67 Nr. 403a *)

  1387 val t = str2term

  1388 "4/x - 3/y - 1";

  1389 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1390 term2str t;

  1391 if (term2str t) =

  1392 "(-3 * x + 4 * y + -1 * x * y) / (x * y)"

  1393 then ()

  1394 else error "rational.sml: diff.behav. in norm_Rational_mg 12";

  1396 (*SRA Schalk I, p.67 Nr. 407b *)

  1397 val t = str2term

  1398 "(2*a+3*b)/(b*c) + (3*c+a)/(a*c) - (2*a^^^2+3*b*c)/(a*b*c)";

  1399 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1400 term2str t;

  1401 if (term2str t) =

  1402 "4 / c"

  1403 then ()

  1404 else error "rational.sml: diff.behav. in norm_Rational_mg 13";

  1406 (*SRA Schalk I, p.67 Nr. 410b *)

  1407 val t = str2term

  1408 "1/(x+1) + 1/(x+2) - 2/(x+3)";

  1409 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1410 term2str t;

  1411 if (term2str t) =

  1412 "(5 + 3 * x) / (6 + 11 * x + 6 * x ^^^ 2 + x ^^^ 3)"

  1413 then ()

  1414 else error "rational.sml: diff.behav. in norm_Rational_mg 14";

  1416 (*SRA Schalk I, p.67 Nr. 413b *)

  1417 val t = str2term

  1418 "(1+x)/(1 - x) - (1 - x)/(1+x) + 2*x/(1 - x^^^2)";

  1419 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1420 term2str t;

  1421 if (term2str t) =

  1422 "6 * x / (1 + -1 * x ^^^ 2)"

  1423 then ()

  1424 else error "rational.sml: diff.behav. in norm_Rational_mg 15";

  1426 (*SRA Schalk I, p.68 Nr. 414a *)

  1427 val t = str2term

  1428 "(x + 2)/(x - 1) + (x - 3)/(x - 2) - (x + 1)/((x - 1)*(x - 2))";

  1429 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1430 term2str t;

  1431 if (term2str t) =

  1432 "(-2 + -5 * x + 2 * x ^^^ 2) / (2 + -3 * x + x ^^^ 2)"

  1433 then ()

  1434 else error "rational.sml: diff.behav. in norm_Rational_mg 16";

  1436 (*SRA Schalk I, p.68 Nr. 423a *)

  1437 val t = str2term

  1438 "(2*x+3*y)/x + (4*x^^^3 - x*y^^^2 - 3*y^^^3)/(x^^^3 - 2*x^^^2*y+x*y^^^2) - (5*x+6*y)/(x - y)";

  1439 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1440 term2str t;

  1441 if (term2str t) =

  1442 "1"

  1443 then ()

  1444 else error "rational.sml: diff.behav. in norm_Rational_mg 17";

  1446 (*SRA Schalk I, p.68 Nr. 428b *)

  1447 val t = str2term

  1448 "1/(a - b)^^^2 + 1/(a+b)^^^2 - 2/(a^^^2 - b^^^2) - 4*(b^^^2 - 1)/(a^^^2 - b^^^2)^^^2";

  1449 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1450 term2str t;

  1451 if (term2str t) =

  1452 "4 / (a ^^^ 4 + -2 * a ^^^ 2 * b ^^^ 2 + b ^^^ 4)"

  1453 then ()

  1454 else error "rational.sml: diff.behav. in norm_Rational_mg 18";

  1456 (*SRA Schalk I, p.68 Nr. 430b *)

  1457 val t = str2term

  1458 "a^^^2/(a - 3*b) - 108*a*b^^^3/((a+3*b)*(a^^^2 - 9*b^^^2)) - 9*b^^^2*(a - 3*b)/(a+3*b)^^^2";

  1459 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1460 term2str t;

  1461 if (term2str t) =

  1462 "a + 3 * b"

  1463 then ()

  1464 else error "rational.sml: diff.behav. in norm_Rational_mg 19";

  1467 (*SRA Schalk I, p.68 Nr. 432 *)

  1468 val t = str2term

  1469 "(a^^^2+a*b)/(a^^^2 - b^^^2) - (b^^^2 - a*b)/(b^^^2 - a^^^2) + a^^^2*(a - b)/(a^^^3 - a^^^2*b) - 2*a*(a^^^2 - b^^^2)/(a^^^3 - a*b^^^2) - 2*b^^^2/(a^^^2 - b^^^2)";

  1470 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1471 term2str t;

  1472 if (term2str t) =

  1473 "0"

  1474 then ()

  1475 else error "rational.sml: diff.behav. in norm_Rational_mg 20";

  1477 (*Eigenes*)

  1478 val t = str2term

  1479 "3*a/(a*b) + x/y";

  1480 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1481 term2str t;

  1482 if (term2str t) =

  1483 "(3 * y + b * x) / (b * y)"

  1484 then ()

  1485 else error "rational.sml: diff.behav. in norm_Rational_mg 21";

  1488 "-------- multiply and cancel ---------------------------";

  1489 "-------- multiply and cancel ---------------------------";

  1490 "-------- multiply and cancel ---------------------------";

  1491 (*SRM Schalk I, p.68 Nr. 436a *)

  1492 val t = str2term

  1493 "3*(x+y)/(15*(x - y)) * 25*(x - y)^^^2/(18*(x+y)^^^2)";

  1494 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1495 term2str t;

  1496 if (term2str t) =

  1497 "(5 * x + -5 * y) / (18 * x + 18 * y)"

  1498 then ()

  1499 else error "rational.sml: diff.behav. in norm_Rational_mg 22";

  1501 (*SRM.test Schalk I, p.68 Nr. 436b *)

  1502 (*WN060420???MG3 crashes with method 'simplify' in

  1503   IsacCore > Simplification > Rational Terms > Multiplication > No.2*)

  1504 val t = str2term "5*a*(a - b)^^^2*(a + b)^^^3/(7*b*(a - b)^^^3) * 7*b/(a + b)^^^3";

  1505 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1506 term2str t;

  1507 if (term2str t) =

  1508 "5 * a / (a + -1 * b)"

  1509 then ()

  1510 else error "rational.sml: diff.behav. in norm_Rational_mg 23";

  1512 (*Schalk I, p.68 Nr. 437a *)

  1513 val t = str2term "(3*a - 4*b)/(4*c+3*e) * (3*a+4*b)/(9*a^^^2 - 16*b^^^2)";

  1514 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1515 if (term2str t) = "1 / (4 * c + 3 * e)" then ()

  1516 else error "rational.sml: diff.behav. in norm_Rational_mg 24";

  1518 "----- S.K. corrected non-termination 060904";

  1519 val t = str2term "(3*a - 4*b) * (3*a+4*b)/((4*c+3*e)*(9*a^^^2 - 16*b^^^2))";

  1520 val SOME (t',_) = rewrite_set_ thy false make_polynomial t;

  1521 term2str t';

  1522 if term2str t' =

  1523   "(9 * a ^^^ 2 + -16 * b ^^^ 2) /\n(36 * a ^^^ 2 * c + 27 * a ^^^ 2 * e + -64 * b ^^^ 2 * c +\n -48 * b ^^^ 2 * e)"

  1524 (*"(9 * a ^^^ 2 + -16 * b ^^^ 2) / (36 * a ^^^ 2 * c + 27 * a ^^^ 2 * e + -64 * b ^^^ 2 * c + -48 * b ^^^ 2 * e)"*)

  1525 then () else error "rational.sml: S.K.8..corrected 060904-6";

  1527 "----- S.K. corrected non-termination of cancel_p_";

  1528 val t'' = str2term ("(9 * a ^^^ 2 + -16 * b ^^^ 2) /" ^

  1529 "(36 * a^^^2 * c + (27 * a^^^2 * e + (-64 * b^^^2 * c + -48 * b^^^2 * e)))");

  1530 val SOME (t',_) = rewrite_set_ thy false cancel_p t'';

  1531 if term2str t' = "1 / (4 * c + 3 * e)" then ()

  1532 else error "rational.sml: diff.behav. in cancel_p S.K.8";

  1534 (*Schalk I, p.68 Nr. 437b *)

  1535 (* nonterm.SK9 loops: cancel_p kann nicht weiter kuerzen!!! *)

  1536 val t'' = str2term "(a + b)/(x^^^2 - y^^^2) * ((x - y)^^^2/(a^^^2 - b^^^2))";

  1537 (* val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t'';

  1538    *)

  1540 "================delete===";

  1541 (*a casual output from above*)

  1542 val t = str2term

  1543 "(a * x ^^^ 2 + -2 * a * x * y + a * y ^^^ 2 + b * x ^^^ 2 + -2 * b * x * y + b * y ^^^ 2) /(a ^^^ 2 * x ^^^ 2 + -1 * a ^^^ 2 * y ^^^ 2 + -1 * b ^^^ 2 * x ^^^ 2 + b ^^^ 2 * y ^^^ 2)";

  1544 (* WN060831 nonterm.SK10

  1545 val SOME (t,_) = rewrite_set_ thy false cancel_p t;

  1546 term2str t;

  1547 *)

  1549 (*SRM Schalk I, p.68 Nr. 438a *)

  1550 val t = str2term

  1551 "x*y/(x*y - y^^^2)*(x^^^2 - x*y)";

  1552 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1553 term2str t;

  1554 if (term2str t) =

  1555 "x ^^^ 2"

  1556 then ()

  1557 else error "rational.sml: diff.behav. in norm_Rational_mg 24";

  1559 (*SRM Schalk I, p.68 Nr. 439b *)

  1560 val t = str2term

  1561 "(4*x^^^2+4*x+1)*((x^^^2 - 2*x^^^3)/(4*x^^^2+2*x))";

  1562 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1563 term2str t;

  1564 if (term2str t) =

  1565 "(x + -4 * x ^^^ 3) / 2"

  1566 then ()

  1567 else error "rational.sml: diff.behav. in norm_Rational_mg 25";

  1569 (*SRM Schalk I, p.68 Nr. 440a *)

  1570 val t = str2term

  1571 "(x^^^2 - 2*x)/(x^^^2 - 3*x) * (x - 3)^^^2/(x^^^2 - 4)";

  1572 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1573 term2str t;

  1574 if (term2str t) =

  1575 "(-3 + x) / (2 + x)"

  1576 then ()

  1577 else error "rational.sml: diff.behav. in norm_Rational_mg 26";

  1579 "----- Schalk I, p.68 Nr. 440b SK11 works since 0707xx";

  1580 val t = str2term

  1581 "(a^^^3 - 9*a)/(a^^^3*b - a*b^^^3)*(a^^^2*b+a*b^^^2)/(a+3)";

  1582 val SOME (t,_) = rewrite_set_ thy false norm_Rational t;

  1583 if term2str t = "(-3 * a + a ^^^ 2) / (a + -1 * b)" then ()

  1584 else error "rational.sml: diff.behav. in norm_Rational 27";

  1586 "----- SK12 works since 0707xx";

  1587 val t = str2term "(a^^^3 - 9*a)*(a^^^2*b+a*b^^^2)/((a^^^3*b - a*b^^^3)*(a+3))";

  1588 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; term2str t';

  1589 if term2str t' = "(-3 * a + a ^^^ 2) / (a + -1 * b)" then ()

  1590 else error "rational.sml: diff.behav. in norm_Rational 28";

  1593 "-------- common denominator and multiplication ---------";

  1594 "-------- common denominator and multiplication ---------";

  1595 "-------- common denominator and multiplication ---------";

  1596 (*SRAM Schalk I, p.69 Nr. 441b *)

  1597 val t = str2term "(4*a/3 + 3*b^^^2/a^^^3 + b/(4*a))*(4*b/(3*a))";

  1598 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1599 term2str t;

  1600 if (term2str t) =

  1601   "(36 * b ^^^ 3 + 3 * a ^^^ 2 * b ^^^ 2 + 16 * a ^^^ 4 * b) / (9 * a ^^^ 4)"

  1602 then () else error "rational.sml: diff.behav. in norm_Rational_mg 28";

  1604 (*SRAM Schalk I, p.69 Nr. 442b *)

  1605 val t = str2term "(15*a^^^2/x^^^3 - 5*b^^^4/x^^^2 + 25*c^^^2/x)*(x^^^3/(5*a*b^^^3*c^^^3)) + 1/c^^^3 * (b*x/a - 3*a/b^^^3)";

  1606 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1607 term2str t;

  1608 if (term2str t) =

  1609 "5 * x ^^^ 2 / (a * b ^^^ 3 * c)"

  1610 then () else error "rational.sml: diff.behav. in norm_Rational_mg 29";

  1612 (*SRAM Schalk I, p.69 Nr. 443b *)

  1613 val t = str2term "(a/2 + b/3)*(b/3 - a/2)";

  1614 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1615 term2str t;

  1616 if (term2str t) =

  1617 "(-9 * a ^^^ 2 + 4 * b ^^^ 2) / 36"

  1618 then () else error "rational.sml: diff.behav. in norm_Rational_mg 30";

  1620 (*SRAM Schalk I, p.69 Nr. 445b *)

  1621 val t = str2term "(a^^^2/9 + 2*a/(3*b) + 4/b^^^2)*(a/3 - 2/b) + 8/b^^^3";

  1622 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1623 term2str t;

  1624 if (term2str t) =

  1625 "a ^^^ 3 / 27"

  1626 then () else error "rational.sml: diff.behav. in norm_Rational_mg 31";

  1628 (*SRAM Schalk I, p.69 Nr. 446b *)

  1629 val t = str2term "(x/(5*x + 4*y) - y/(5*x - 4*y) + 1)*(25*x^^^2 - 16*y^^^2)";

  1630 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1631 term2str t;

  1632 if (term2str t) =

  1633 "30 * x ^^^ 2 + -9 * x * y + -20 * y ^^^ 2"

  1634 then ()

  1635 else error "rational.sml: diff.behav. in norm_Rational_mg 32";

  1637 (*SRAM Schalk I, p.69 Nr. 449a *)(*Achtung: rechnet ca 8 Sekunden*)

  1638 val t = str2term

  1639 "(2*x^^^2/(3*y)+x/y^^^2)*(4*x^^^4/(9*y^^^2)+x^^^2/y^^^4)*(2*x^^^2/(3*y) - x/y^^^2)";

  1640 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1641 term2str t;

  1642 if (term2str t) = "(-81 * x ^^^ 4 + 16 * x ^^^ 8 * y ^^^ 4) / (81 * y ^^^ 8)"

  1643 then () else error "rational.sml: diff.behav. in norm_Rational_mg 33";

  1646 (*SRAM Schalk I, p.69 Nr. 450a *)

  1647 val t = str2term

  1648 "(4*x/(3*y)+2*y/(3*x))^^^2 - (2*y/(3*x) - 2*x/y)*(2*y/(3*x)+2*x/y)";

  1649 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1650 term2str t;

  1651 if (term2str t) =

  1652 "(52 * x ^^^ 2 + 16 * y ^^^ 2) / (9 * y ^^^ 2)"

  1653 then ()

  1654 else error "rational.sml: diff.behav. in norm_Rational_mg 34";

  1657 "-------- double fractions ------------------------------";

  1658 "-------- double fractions ------------------------------";

  1659 "-------- double fractions ------------------------------";

  1660 (*SRD Schalk I, p.69 Nr. 454b *)

  1661 val t = str2term

  1662 "((2 - x)/(2*a)) / (2*a/(x - 2))";

  1663 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1664 term2str t;

  1665 if (term2str t) =

  1666 "(-4 + 4 * x + -1 * x ^^^ 2) / (4 * a ^^^ 2)"

  1667 then ()

  1668 else error "rational.sml: diff.behav. in norm_Rational_mg 35";

  1670 (*SRD Schalk I, p.69 Nr. 455a *)

  1671 val t = str2term

  1672 "(a^^^2 + 1)/(a^^^2 - 1) / ((a+1)/(a - 1))";

  1673 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1674 term2str t;

  1675 if (term2str t) =

  1676 "(1 + a ^^^ 2) / (1 + 2 * a + a ^^^ 2)" then ()

  1677 else error "rational.sml: diff.behav. in norm_Rational_mg 36";

  1680 "----- Schalk I, p.69 Nr. 455b";

  1681 val t = str2term "(x^^^2 - 4)/(y^^^2 - 9)/((2+x)/(3 - y))";

  1682 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1683 if term2str t = "(2 + -1 * x) / (3 + y)" then ()

  1684 else error "rational.sml: diff.behav. in norm_Rational_mg 37";

  1686 "----- SK060904-1a non-termination of cancel_p_ ?: worked before 0707xx";

  1687 val t = str2term "(x^^^2 - 4)*(3 - y)/((y^^^2 - 9)*(2+x))";

  1688 val SOME (t,_) = rewrite_set_ thy false norm_Rational t;

  1689 if term2str t = "(2 + -1 * x) / (3 + y)" then ()

  1690 else error "rational.sml: diff.behav. in norm_Rational_mg 37b";

  1692 "----- ?: worked before 0707xx";

  1693 val t = str2term "(3 + -1 * y) / (-9 + y ^^^ 2)";

  1694 val SOME (t,_) = rewrite_set_ thy false norm_Rational t;

  1695 if term2str t = "-1 / (3 + y)" then ()

  1696 else error "rational.sml: -1 / (3 + y) norm_Rational";

  1698 (*SRD Schalk I, p.69 Nr. 456b *)

  1699 val t = str2term

  1700 "(b^^^3 - b^^^2)/(b^^^2+b)/(b^^^2 - 1)";

  1701 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1702 term2str t;

  1703 if (term2str t) = "b / (1 + 2 * b + b ^^^ 2)" then ()

  1704 else error "rational.sml: diff.behav. in norm_Rational_mg 38";

  1706 (*SRD Schalk I, p.69 Nr. 457b *)

  1707 val t = str2term

  1708 "(16*a^^^2 - 9*b^^^2)/(2*a+3*a*b) / ((4*a+3*b)/(4*a^^^2 - 9*a^^^2*b^^^2))";

  1709 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1710 term2str t;

  1711 if (term2str t) =

  1712   "8 * a ^^^ 2 + -6 * a * b + -12 * a ^^^ 2 * b + 9 * a * b ^^^ 2"

  1713 then () else error "rational.sml: diff.behav. in norm_Rational_mg 39";

  1715 "----- Schalk I, p.69 Nr. 458b works since 0707";

  1716 val t = str2term

  1717 "(2*a^^^2*x - a^^^2)/(a*x - b*x) / (b^^^2*(2*x - 1)/(x*(a - b)))";

  1718 val SOME (t,_) = rewrite_set_ thy false norm_Rational t;

  1719 if term2str t = "a ^^^ 2 / b ^^^ 2" then ()

  1720 else error "rational.sml: diff.behav. in norm_Rational_mg 39b";

  1722 (*SRD Schalk I, p.69 Nr. 459b *)

  1723 val t = str2term "(a^^^2 - b^^^2)/(a*b) / (4*(a+b)^^^2/a)";

  1724 val SOME (t,_) = rewrite_set_ thy false norm_Rational t;

  1725 if term2str t = "(a + -1 * b) / (4 * a * b + 4 * b ^^^ 2)" then ()

  1726 else error "rational.sml: diff.behav. in norm_Rational_mg 41";

  1729 (*Schalk I, p.69 Nr. 460b nonterm.SK

  1730 val t = str2term

  1731 "(9*(x^^^2 - 8*x+16)/(4*(y^^^2 - 2*y+1)))/((3*x - 12)/(16*y - 16))";

  1732 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1733 if term2str t =

  1734 then ()

  1735 else error "rational.sml: diff.behav. in norm_Rational_mg 42";

  1737 val t = str2term

  1738 "9*(x^^^2 - 8*x+16)*(16*y - 16)/(4*(y^^^2 - 2*y+1)*(3*x - 12))";

  1739 val SOME (t',_) = rewrite_set_ thy false norm_Rational t;

  1740 ... non terminating.

  1741 val SOME (t',_) = rewrite_set_ thy false make_polynomial t;

  1742 "(-2304 + 1152 * x + 2304 * y + -144 * x ^^^ 2 + -1152 * x * y + 144 * x ^^^ 2 * y) /(-48 + 12 * x + 96 * y + -24 * x * y + -48 * y ^^^ 2 + 12 * x * y ^^^ 2)";

  1743 val SOME (t,_) = rewrite_set_ thy false cancel_p t';

  1744 ... non terminating.*)

  1746 (*SRD Schalk I, p.70 Nr. 472a *)

  1747 val t = str2term ("((8*x^^^2 - 32*y^^^2)/(2*x + 4*y))/" ^

  1748 		 "((4*x - 8*y)/(x + y))");

  1749 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1750 term2str t;

  1751 if (term2str t) =

  1752 "x + y"

  1753 then ()

  1754 else error "rational.sml: diff.behav. in norm_Rational_mg 43";

  1757 (*----------------------------------------------------------------------*)

  1758 (*---------------------- Einfache Dppelbrche --------------------------*)

  1759 (*----------------------------------------------------------------------*)

  1761 (*SRD Schalk I, p.69 Nr. 461a *)

  1762 val t = str2term

  1763 "(2/(x+3) + 2/(x - 3)) / (8*x/(x^^^2 - 9))";

  1764 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1765 term2str t;

  1766 if (term2str t) =

  1767 "1 / 2"

  1768 then ()

  1769 else error "rational.sml: diff.behav. in norm_Rational_mg 44";

  1771 (*SRD Schalk I, p.69 Nr. 464b *)

  1772 val t = str2term

  1773 "(a - a/(a - 2)) / (a + a/(a - 2))";

  1774 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1775 term2str t;

  1776 if (term2str t) =

  1777 "(3 + -1 * a) / (1 + -1 * a)"

  1778 then ()

  1779 else error "rational.sml: diff.behav. in norm_Rational_mg 45";

  1781 (*SRD Schalk I, p.69 Nr. 465b *)

  1782 val t = str2term

  1783 "((x+3*y)/9 + (4*y^^^2 - 9*z^^^2)/(16*x)) /(x/9+y/6+z/4)";

  1784 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1785 term2str t;

  1786 if (term2str t) =

  1787 "(4 * x + 6 * y + -9 * z) / (4 * x)"

  1788 then ()

  1789 else error "rational.sml: diff.behav. in norm_Rational_mg 46";

  1791 (*SRD Schalk I, p.69 Nr. 466b *)

  1792 val t = str2term

  1793 "((1 - 7*(x - 2)/(x^^^2 - 4)) / (6/(x+2))) / (3/(x+5)+30/(x^^^2 - 25))";

  1794 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1795 term2str t;

  1796 if (term2str t) =

  1797 "(25 + -10 * x + x ^^^ 2) / 18"

  1798 then ()

  1799 else error "rational.sml: diff.behav. in norm_Rational_mg 47";

  1801 (*SRD Schalk I, p.70 Nr. 469 *)

  1802 val t = str2term

  1803 "3*b^^^2/(4*a^^^2 - 8*a*b + 4*b^^^2)/(a/(a^^^2*b - b^^^3) + (a - b)/(4*a*b^^^2+4*b^^^3) - 1/(4*b^^^2))";

  1804 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1805 term2str t;

  1806 if (term2str t) =

  1807 "3 * b ^^^ 3 / (2 * a + -2 * b)"

  1808 then ()

  1809 else error "rational.sml: diff.behav. in norm_Rational_mg 48";

  1811 (*----------------------------------------------------------------------*)

  1812 (*---------------------- Mehrfache Dppelbrueche ------------------------*)

  1813 (*----------------------------------------------------------------------*)

  1815 (*SRD.test Schalk I, p.70 Nr. 476b *) (* Rechenzeit: 10 sec *)

  1816 (*WN060419 crashes with method 'simplify' ????SK*)

  1817 val t = str2term

  1818 "((a^^^2 - b^^^2)/(2*a*b)+2*a*b/(a^^^2 - b^^^2))/((a^^^2+b^^^2)/(2*a*b)+1) / ((a^^^2+b^^^2)^^^2/(a+b)^^^2)";

  1819 val SOME (t,_) = rewrite_set_ thy false norm_Rational t;

  1820 if term2str t = "1 / (a ^^^ 2 + -1 * b ^^^ 2)" then ()

  1821 else error "rational.sml: diff.behav. in norm_Rational_mg 49";

  1823 "----- Schalk I, p.70 Nr. 477a";

  1824 (* MG Achtung: terme explodieren; Bsp zu komplex;

  1825    L???ung sollte (ziemlich grosser) Faktorisierter Ausdruck sein

  1826 val t = str2term "b*y/(b - 2*y)/((b^^^2 - y^^^2)/(b+2*y)) /" ^

  1827 		 "(b^^^2*y+b*y^^^2)*(a+x)^^^2/((b^^^2 - 4*y^^^2)*(a+2*x)^^^2)";

  1828 val SOME (t',_) = rewrite_set_ thy false norm_Rational t;

  1831 val t = str2term "b*y*(b+2*y)*(b^^^2 - 4*y^^^2)*(a+2*x)^^^2 / " ^

  1832 		 "((b - 2*y)*(b^^^2 - y^^^2)*(b^^^2*y+b*y^^^2)*(a+x)^^^2)";

  1833 val SOME (t,_) = rewrite_set_ thy false make_polynomial t;

  1834 ????SK ???MG*)

  1837 "----- Schalk I, p.70 Nr. 478b ----- Rechenzeit: 5 sec";

  1838 val t = str2term ("(a - (a*b+b^^^2)/(a+b))/(b+(a - b)/(1+(a+b)/(a - b))) / " ^

  1839 		 "((a - a^^^2/(a+b))/(a+(a*b)/(a - b)))");

  1840 val SOME (t',_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1841 if term2str t' =

  1842 "(2 * a ^^^ 3 + 2 * a ^^^ 2 * b) / (a ^^^ 2 * b + b ^^^ 3)"

  1843 then ()

  1844 else error "rational.sml: diff.behav. in norm_Rational_mg 51";

  1846 (* TODO.new_c:WN050820 STOP_REW_SUB introduced gave ...

  1847 if term2str t' = "(a ^^^ 4 + -1 * a ^^^ 2 * b ^^^ 2) /(a * b * (b + (a * (a + -1 * b) + -1 * b * (a + -1 * b)) / (2 * a)) * (a + -1 * b))" then ()

  1848 else error "rational.sml: works again";

  1849 re-outcommented with TODO.new_c: cvs before 071227, 11:50*)

  1853 (*Schalk I, p.70 Nr. 480a *)

  1854 (* Achtung: rechnet ewig; cancel_p kann nicht krzen: WN060831 nonterm.SK00

  1855 val t = str2term

  1856 "(1/x+1/y+1/z)/(1/x - 1/y - 1/z) / (2*x^^^2/(x^^^2 - z^^^2)/(x/(x+z)+x/(x - z)))";

  1857 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1858 term2str t;

  1859 if (term2str t) =

  1861 then ()

  1862 else error "rational.sml: diff.behav. in norm_Rational_mg 52";

  1864 (*MG Berechne Zwischenergebnisse: WN060831 nonterm.SK00*)

  1865 val t = str2term

  1866 "(1/x+1/y+1/z)/(1/x - 1/y - 1/z)";

  1867 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1868 term2str t;

  1869 "(x ^^^ 2 * y ^^^ 2 * z + x ^^^ 2 * y * z ^^^ 2 + x * y ^^^ 2 * z ^^^ 2) /

  1870 (-1 * x ^^^ 2 * y ^^^ 2 * z + -1 * x ^^^ 2 * y * z ^^^ 2 + x * y ^^^ 2 * z ^^^ 2)";

  1871 val t = str2term

  1872 "2*x^^^2/(x^^^2 - z^^^2)/(x/(x+z)+x/(x - z))";

  1873 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1874 term2str t;

  1875 "1"

  1877 (* SK 1. Ausdruck kann nicht weiter gekrzt werden; cancel_p !!!*)

  1878 ###  rls: cancel_p on:

  1879 (x ^^^ 2 * (y ^^^ 2 * z) + x ^^^ 2 * (y * z ^^^ 2) + x * (y ^^^ 2 * z ^^^ 2)) /

  1880 (-1 * (x ^^^ 2 * (y ^^^ 2 * z)) + -1 * (x ^^^ 2 * (y * z ^^^ 2)) + x * (y ^^^ 2 * z ^^^ 2))

  1881 GC #3.61.81.101.197.17503:   (0 ms)

  1882 *** RATIONALS_DIRECT_CANCEL_EXCEPTION: Invalid fraction

  1884 val t = str2term

  1885 "(x^^^2 * (y^^^2 * z) + x^^^2 * (y * z^^^2) + x * (y^^^2 * z^^^2)) / (-1 * (x^^^2 * (y^^^2 * z)) + -1 * (x^^^2 * (y * z^^^2)) + x * (y^^^2 * z^^^2))";

  1886 val SOME (t,_) = rewrite_set_ thy false cancel_p t;

  1887 term2str t;

  1888 (*uncaught exception nonexhaustive binding failure*)

  1890 (* Das kann er aber krzen !!????: *)

  1891 val t = str2term

  1892 "(x^^^2 * (y^^^2 * z) +  x * (y^^^2 * z^^^2)) / (-1 * (x^^^2 * (y * z^^^2)) + x * (y^^^2 * z^^^2))";

  1893 val SOME (t,_) = rewrite_set_ thy false cancel_p t;

  1894 term2str t;

  1895 "(-1 * (y * x) + -1 * (z * y)) / (1 * (z * x) + -1 * (z * y))";

  1896 *)

  1899 "-------- crucial examples ------------------------------";

  1900 "-------- crucial examples ------------------------------";

  1901 "-------- crucial examples ------------------------------";

  1902 (*Schalk I, p.60 Nr. 215d *)

  1903 (* Achtung: rechnet ewig ...

  1904 val t = str2term "(a-b)^^^3 * (x+y)^^^4 / ((x+y)^^^2 * (a-b)^^^5)";

  1905 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1906 term2str t; noterm.SK

  1907 *)

  1909 (* Kein Wunder, denn Z???ler und Nenner extra als Polynom dargestellt ergibt:*)

  1910 (*

  1911 val t = str2term "(a-b)^^^3 * (x+y)^^^4";

  1912 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1913 term2str t;

  1914 "a^^^3 * x^^^4 + 4 * a^^^3 * x^^^3 * y +6 * a^^^3 * x^^^2 * y^^^2 +4 * a^^^3 * x * y^^^3 +a^^^3 * y^^^4 +-3 * a^^^2 * b * x^^^4 +-12 * a^^^2 * b * x^^^3 * y +-18 * a^^^2 * b * x^^^2 * y^^^2 +-12 * a^^^2 * b * x * y^^^3 +-3 * a^^^2 * b * y^^^4 +3 * a * b^^^2 * x^^^4 +12 * a * b^^^2 * x^^^3 * y +18 * a * b^^^2 * x^^^2 * y^^^2 +12 * a * b^^^2 * x * y^^^3 +3 * a * b^^^2 * y^^^4 +-1 * b^^^3 * x^^^4 +-4 * b^^^3 * x^^^3 * y +-6 * b^^^3 * x^^^2 * y^^^2 +-4 * b^^^3 * x * y^^^3 +-1 * b^^^3 * y^^^4";

  1915 val t = str2term "((x+y)^^^2 * (a-b)^^^5)";

  1916 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1917 term2str t;

  1918 "a^^^5 * x^^^2 + 2 * a^^^5 * x * y + a^^^5 * y^^^2 +-5 * a^^^4 * b * x^^^2 +-10 * a^^^4 * b * x * y +-5 * a^^^4 * b * y^^^2 +10 * a^^^3 * b^^^2 * x^^^2 +20 * a^^^3 * b^^^2 * x * y +10 * a^^^3 * b^^^2 * y^^^2 +-10 * a^^^2 * b^^^3 * x^^^2 +-20 * a^^^2 * b^^^3 * x * y +-10 * a^^^2 * b^^^3 * y^^^2 +5 * a * b^^^4 * x^^^2 +10 * a * b^^^4 * x * y +5 * a * b^^^4 * y^^^2 +-1 * b^^^5 * x^^^2 +-2 * b^^^5 * x * y +-1 * b^^^5 * y^^^2";

  1919 *)

  1920 (*anscheinend macht dem Rechner das Krzen diese Bruches keinen Spass mehr ...*)

  1922 (*--------------------------------------------------------------------*)

  1923 (*Schalk I, p.70 Nr. 480b

  1924 val t = str2term "((12*x*y/(9*x^^^2 - y^^^2))/" ^

  1925 		 "(1/(3*x - y)^^^2 - 1/(3*x + y)^^^2)) *" ^

  1926 		 "(1/(x - 5*y)^^^2 - 1/(x + 5*y)^^^2)/" ^

  1927 		 "(20*x*y/(x^^^2 - 25*y^^^2))";

  1928 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1929 SK.nonterm

  1930 Kann nicht weiter vereinfacht werden !!!!?? *)

  1932 (*--------------------------------------------------------------------*)

  1933 "---- MGs test set";

  1934 val t = str2term " (1 + x^^^5) / (y + x) + x^^^3 / x ";

  1935 val SOME (t,_) = rewrite_set_ thy false common_nominator_p t;

  1936 if term2str t = "(1 + x ^^^ 3 + x ^^^ 5 + y * x ^^^ 2) / (x + y)" then()

  1937 else error "";

  1939 (*--------------------------------------------------------------------*)

  1940 (* cancel_p liefert nicht immer Polynomnormalform (2): WN060831???SK3b

  1941    ---> Sortierung FALSCH !!  *)

  1942 val t = str2term "b^^^3 * a^^^5/a ";

  1943 val SOME (t,_) = rewrite_set_ thy false cancel_p t;

  1944 term2str t;

  1945 "1 * (a^^^4 * b^^^3) / 1"; (*OK*)

  1947 val t = str2term "b^^^3 * a^^^5/b ";

  1948 val SOME (t,_) = rewrite_set_ thy false cancel_p t;

  1949 term2str t;

  1950 "1 * (b^^^2 * a^^^5) / 1"; (*cancel_p sortiert hier falsch um!*)

  1952 (* Problem liegt NICHT bei ord_make_polynomial! (siehe folgende Bsple) *)

  1953 (*

  1954 val x = str2term "x"; val bdv = str2term "bdv";

  1955 val t1 = str2term "b^^^2 * a^^^5";

  1956 val t2 = str2term "a^^^5 * b^^^2 ";

  1957 ord_make_polynomial false Rational.thy [(x,bdv)] (t1,t2); (*false*)

  1958 *)

  1959 (* ==> "b^^^2 * a^^^5" > "a^^^5 * b^^^2 " ... OK!*)

  1961 (*--------------------------------------------------------------------*)

  1962 (* cancel_p liefert nicht immer Polynomnormalform (2): WN060831???SK3c

  1963    ---> erzeugt berflssige "1 * ..."

  1965 val t = str2term "-1 / (3 + y)";

  1966 (*~~         *)

  1967 val SOME (t,_) = rewrite_set_ thy false cancel_p t;

  1968 term2str t;

  1969 "-1 / (3 + 1 * y)";

  1970 (********* Das ist das PROBLEM !!!!!!!??? *******************)

  1971 (* -1 im Z???ler der Angabe verursacht das Problem !*)

  1972 *)

  1974 (* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - *)

  1975 "----- MGs test set";

  1976 val t = str2term "(a^^^2 + -1)/(a+1)";

  1977 val SOME (t',_) = rewrite_set_ thy false cancel_p t;

  1978 if term2str t' = "(-1 + a) / 1" then ()

  1979 else error "rational.sml MG tests 3d";

  1981 "----- NOT TERMINATING ?: worked before 0707xx";

  1982 val t = str2term "(a^^^2 - 1)*(b + 1) / ((b^^^2 - 1)*(a+1))";

  1983 val SOME (t'',_) = rewrite_set_ thy false norm_Rational t;

  1984 if term2str t'' = "(1 + -1 * a) / (1 + -1 * b)" then ()

  1985 else error "rational.sml MG tests 3e";

  1987 "----- corrected SK060905";

  1988 val t = str2term "(4*x^^^2 - 20*x + 25)/(2*x - 5)^^^3";

  1989 val SOME (t',_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1990 if term2str t' = "-1 / (5 + -2 * x)" then ()

  1991 else error "rational.sml corrected SK060905";

  1994 "-------- examples for Stefan Karnels thesis ------------";

  1995 "-------- examples for Stefan Karnels thesis ------------";

  1996 "-------- examples for Stefan Karnels thesis ------------";

  1997 (*SRAM Schalk I, p.69 Nr. 442b --- abgewandelt*)

  1998 val t = str2term

  1999 "(15*a^^^4/(a*x^^^3) - 5*a*((b^^^4 - 5*c^^^2*x)/x^^^2))*(x^^^3/(5*a*b^^^3*c^^^3)) + a/c^^^3 * (x*(b/a) - 3*b*(a/b^^^4))";

  2000 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  2001 term2str t;

  2002 if (term2str t) =

  2003 "5 * x ^^^ 2 / (b ^^^ 3 * c)"

  2004 then ()

  2005 else error "rational.sml: diff.behav. in norm_Rational_mg 53";

  2008 "-------- me Schalk I No.186 ----------------------------";

  2009 "-------- me Schalk I No.186 ----------------------------";

  2010 "-------- me Schalk I No.186 ----------------------------";

  2011 val fmz = ["Term ((14 * x * y) / ( x * y ))",

  2012 	   "normalform N"];

  2013 val (dI',pI',mI') =

  2014   ("Rational",["rational","simplification"],

  2015    ["simplification","of_rationals"]);

  2016 val p = e_pos'; val c = [];

  2017 val (p,_,f,nxt,_,pt) = CalcTreeTEST [(fmz, (dI',pI',mI'))];

  2018 val (p,_,f,nxt,_,pt) = me nxt p c pt;

  2019 val (p,_,f,nxt,_,pt) = me nxt p c pt;

  2020 val (p,_,f,nxt,_,pt) = me nxt p c pt;

  2021 val (p,_,f,nxt,_,pt) = me nxt p c pt;

  2022 val (p,_,f,nxt,_,pt) = me nxt p c pt;

  2023 val (p,_,f,nxt,_,pt) = me nxt p c pt;

  2024 val (p,_,f,nxt,_,pt) = me nxt p c pt;

  2025 val (p,_,f,nxt,_,pt) = me nxt p c pt; f2str f;

  2026 val (p,_,f,nxt,_,pt) = me nxt p c pt; f2str f;

  2027 val (p,_,f,nxt,_,pt) = me nxt p c pt; f2str f;(*++ for explicit script*)

  2028 val (p,_,f,nxt,_,pt) = me nxt p c pt; f2str f;(*++ for explicit script*)

  2029 case (f2str f, nxt) of

  2030     ("14", ("End_Proof'", _)) => ()

  2031   | _ => error "rational.sml diff.behav. in me Schalk I No.186";

  2034 "-------- interSteps ..Simp_Rat_Double_No-1.xml ---------";

  2035 "-------- interSteps ..Simp_Rat_Double_No-1.xml ---------";

  2036 "-------- interSteps ..Simp_Rat_Double_No-1.xml ---------";

  2037 states:=[];

  2038 CalcTree

  2039 [(["Term (((2 - x)/(2*a)) / (2*a/(x - 2)))", "normalform N"],

  2040   ("Rational",["rational","simplification"],

  2041   ["simplification","of_rationals"]))];

  2042 Iterator 1;

  2043 moveActiveRoot 1;

  2044 autoCalculate 1 CompleteCalc;

  2045 val ((pt,p),_) = get_calc 1; show_pt pt;

  2047 interSteps 1 ([1],Res);

  2048 val ((pt,p),_) = get_calc 1; show_pt pt;

  2051 "-------- interSteps ..Simp_Rat_Cancel_No-1.xml ---------";

  2052 "-------- interSteps ..Simp_Rat_Cancel_No-1.xml ---------";

  2053 "-------- interSteps ..Simp_Rat_Cancel_No-1.xml ---------";

  2054 states:=[];

  2055 CalcTree

  2056 [(["Term ((a^2 + -1*b^2) / (a^2 + -2*a*b + b^2))", "normalform N"],

  2057   ("Rational",["rational","simplification"],

  2058   ["simplification","of_rationals"]))];

  2059 Iterator 1;

  2060 moveActiveRoot 1;

  2061 autoCalculate 1 CompleteCalc;

  2062 val ((pt,p),_) = get_calc 1; show_pt pt;

  2063 (*========== inhibit exn 110314 ================================================

  2064 (*with explicit script done already... and removed [1,..] at below...

  2065 interSteps 1 ([1],Res);

  2066 val ((pt,p),_) = get_calc 1; show_pt pt;

  2067 *)

  2068 interSteps 1 ([2],Res);

  2069 val ((pt,p),_) = get_calc 1; show_pt pt;

  2071 interSteps 1 ([2,1],Res);

  2072 val ((pt,p),_) = get_calc 1; show_pt pt;

  2073 val newnds = children (get_nd pt [2,1]) (*see "fun detailrls"*);

  2074 (*if length newnds = 12 then () WN060905*)

  2075 if length newnds = 13 then ()

  2076 else error "rational.sml: interSteps cancel_p rev_rew_p";

  2078 val p = ([2,1,9],Res);

  2079 getTactic 1 p;

  2080 val (_, tac, _) = pt_extract (pt, p);

  2081 (*case tac of SOME (Rewrite ("sym_real_plus_binom_times1", _)) => ()

  2082 WN060905*)

  2083 case tac of SOME (Rewrite ("sym_real_add_mult_distrib2", _)) => ()

  2084 | _ => error "rational.sml: getTactic, sym_real_plus_binom_times1";

  2085 ============ inhibit exn 110314 ==============================================*)

  2088 "-------- investigate rulesets for cancel_p -------------";

  2089 "-------- investigate rulesets for cancel_p -------------";

  2090 "-------- investigate rulesets for cancel_p -------------";

  2091 val thy = @{theory "Rational"};

  2092 "---------------- (a^^^2 + -1*b^^^2) / (a^^^2 + -2*a*b + b^^^2)";

  2093 val t = str2term "(a^^^2 + -1*b^^^2) / (a^^^2 + -2*a*b + b^^^2)";

  2094 val tt = str2term "(1 * a + 1 * b) * (1 * a + -1 * b)"(*numerator only*);

  2095 "----- with rewrite_set_";

  2096 val SOME (tt',asm) = rewrite_set_ thy false make_polynomial tt;

  2097 term2str tt'= "a ^^^ 2 + -1 * b ^^^ 2" (*true*);

  2098 val tt = str2term "((1 * a + -1 * b) * (1 * a + -1 * b))"(*denominator only*);

  2099 val SOME (tt',asm) = rewrite_set_ thy false make_polynomial tt;

  2100 term2str tt' = "a ^^^ 2 + -2 * a * b + b ^^^ 2" (*true*);

  2102 "----- with make_deriv";

  2103 val SOME (tt, _) = factout_p_ thy t; term2str tt =

  2104 "(1 * a + 1 * b) * (1 * a + -1 * b) / ((1 * a + -1 * b) * (1 * a + -1 * b))";

  2105 (*

  2106 "--- with ruleset as before 060829";

  2107 val {rules, rew_ord=(_,ro),...} =

  2108     rep_rls (assoc_rls "make_polynomial");

  2109 val der = make_deriv thy Atools_erls rules ro NONE tt;

  2110 print_depth 99; map (term2str o #1) der; print_depth 3;

  2111 print_depth 99; map (rule2str o #2) der; print_depth 3;

  2112 ... did not terminate*)

  2113 "--- with simpler ruleset";

  2114 val {rules, rew_ord=(_,ro),...} =

  2115     rep_rls (assoc_rls "rev_rew_p");

  2116 val der = make_deriv thy Atools_erls rules ro NONE tt;

  2117 print_depth 99; writeln (deriv2str der); print_depth 3;

  2119 print_depth 99; map (term2str o #1) der; print_depth 3;

  2120 "...,(-1 * b ^^^ 2 + a ^^^ 2) / (-2 * (a * b) + a ^^^ 2 + (-1 * b) ^^^ 2) ]";

  2121 print_depth 99; map (rule2str o #2) der; print_depth 3;

  2122 print_depth 99; map (term2str o #1 o #3) der; print_depth 3;

  2124 val der = make_deriv thy Atools_erls rules ro NONE

  2125 		     (str2term "(1 * a + 1 * b) * (1 * a + -1 * b)");

  2126 print_depth 99; writeln (deriv2str der); print_depth 3;

  2128 val {rules, rew_ord=(_,ro),...} =

  2129     rep_rls (assoc_rls "rev_rew_p");

  2130 val der = make_deriv thy Atools_erls rules ro NONE

  2131 		     (str2term "(1 * a + -1 * b) * (1 * a + -1 * b)");

  2132 print_depth 99; writeln (deriv2str der); print_depth 3;

  2133 print_depth 99; map (term2str o #1) der; print_depth 3;

  2134 (*WN060829 ...postponed*)

  2137 "-------- investigate format of factout_ and factout_p_ -";

  2138 "-------- investigate format of factout_ and factout_p_ -";

  2139 "-------- investigate format of factout_ and factout_p_ -";

  2140 val {rules, rew_ord = (_,ro),...} = rep_rls (assoc_rls "make_polynomial");

  2141 val (thy, eval_rls) = (@{theory "Rational"}, Atools_erls)(*see 'fun init_state'*);

  2142 val Rrls {scr = Rfuns {init_state,...},...} = assoc_rls "cancel_p";

  2144 "----- see Rational.ML, local cancel_p, fun init_state";

  2145 val t = str2term "(a^^^2 + (-1)*b^^^2) / (a^^^2 + (-2)*a*b + b^^^2)";

  2146 val SOME (t',_) = factout_p_ thy t; term2str t';

  2147 (*

  2148 val rtas = reverse_deriv thy eval_rls rules ro NONE t';

  2149 writeln(trtas2str rst);

  2150 *)

  2153 "----- see Rational.ML, local cancel_p, fun init_state";

  2154 val t = str2term "a^^^2 / a";

  2155 val SOME (t',_) = factout_p_ thy t;

  2156 term2str t' = "a * a / (1 * a)" (*true*);

  2157 (*... can be canceled with

  2158 real_mult_div_cancel2 ?k ~= 0 ==> ?m * ?k / (?n * ?k) = ?m / ?n"*)

  2159 (* sml/ME/rewtools.sml:

  2160 val rtas = reverse_deriv thy Atools_erls rules ro NONE t';

  2161 writeln (deri2str rtas);

  2162 *)

  2165 "-------- SK 060904 ----------------------------------------------";

  2166 "----- order on polynomials -- input + output";

  2167 val thy = @{theory "Isac"};

  2168 val t = str2term "(a + -1 * b) / (-1 * a + b)";

  2169 val SOME (t', _) = factout_p_ thy t; term2str t';

  2170 val SOME (t', _) = cancel_p_ thy t; term2str t';

  2172 val t = str2term "a*b*c*d / (d*e*f*g)";

  2173 val SOME (t', _) = cancel_p_ thy t; term2str t';

  2175 val t = str2term "a*(b*(c*d)) / (b*(e*(f*g)))";

  2176 val SOME (t', _) = cancel_p_ thy t; term2str t';

  2177 (*???order.SK  ???*)

  2179 "----- SK060904-1a non-termination of cancel_p_ ? worked before 0707xx";

  2180 val t = str2term "(x^^^2 - 4)*(3 - y) / ((y^^^2 - 9)*(2+x))";

  2181 val SOME (t',_) = rewrite_set_ thy false norm_Rational t;

  2182 if term2str t' = "(2 + -1 * x) / (3 + y)" then ()

  2183 else error "rational.sml SK060904-1a worked since 0707xx";

  2185 "----- SK060904-1b non-termination of cancel_p_ ... worked before 0707xx";

  2186 val t = str2term ("(9 * a ^^^ 2 + -16 * b ^^^ 2) /" ^

  2187 "(36 * a^^^2 * c + (27 * a^^^2 * e + (-64 * b^^^2 * c + -48 * b^^^2 * e)))");

  2188 val SOME (t',_) = cancel_p_ thy t;

  2189 if term2str t' = "1 / (4 * c + 3 * e)" then ()

  2190 else error "rational.sml SK060904-1b";

  2193 "----- SK060904-2a non-termination of add_fraction_p_";

  2194 val t = str2term (" (a + b * x) / (a + -1 * (b * x)) +  " ^

  2195 		  " (-1 * a + b * x) / (a + b * x)      ");

  2196 (* nonterm.SK

  2197 val SOME (t',_) = rewrite_set_ thy false common_nominator_p t;

  2199 common_nominator_p_ thy t;

  2200 " (a + b * x)*(a + b * x) / ((a + -1 * (b * x))*(a + -1 * (b * x))) +  " ^

  2201 " (-1 * a + b * x)*(a + -1 * (b * x)) / ((a + b * x)*(-1 * a + b * x)) ";

  2203 add_fraction_p_ thy t;

  2204 " ((a + b * x)*(a + b * x)  +  (-1 * a + b * x)*(a + -1 * (b * x))) /" ^

  2205 " ((a + b * x)*(-1 * a + b * x))                                     ";

  2206 *)

  2209 "-------- how to stepwise construct Scripts -------------";

  2210 "-------- how to stepwise construct Scripts -------------";

  2211 "-------- how to stepwise construct Scripts -------------";

  2212 val thy = @{theory "Rational"};

  2213 (*no trailing _*)

  2214 val p1 = parse thy (

  2215 "Script SimplifyScript (t_t::real) =                             " ^

  2216 "  (Rewrite_Set discard_minus False                   " ^

  2217 "   t_t)");

  2219 (*required (): (Rewrite_Set discard_minus False)*)

  2220 val p2 = parse thy (

  2221 "Script SimplifyScript (t_t::real) =                             " ^

  2222 "  (Rewrite_Set discard_minus False                   " ^

  2223 "   t_t)");

  2225 val p3 = parse thy (

  2226 "Script SimplifyScript (t_t::real) =                             " ^

  2227 "  ((Rewrite_Set discard_minus False)                   " ^

  2228 "   t_t)");

  2230 val p4 = parse thy (

  2231 "Script SimplifyScript (t_t::real) =                             " ^

  2232 "  ((Rewrite_Set discard_minus False)                   " ^

  2233 "   t_t)");

  2235 val p5 = parse thy (

  2236 "Script SimplifyScript (t_t::real) =                             " ^

  2237 "  ((Try (Rewrite_Set discard_minus False)                   " ^

  2238 "    Try (Rewrite_Set discard_parentheses False))               " ^

  2239 "   t_t)");

  2241 val p6 = parse thy (

  2242 "Script SimplifyScript (t_t::real) =                             " ^

  2243 "  ((Try (Rewrite_Set discard_minus False) @@                   " ^

  2244 "    Try (Rewrite_Set rat_mult_poly False) @@                    " ^

  2245 "    Try (Rewrite_Set make_rat_poly_with_parentheses False) @@   " ^

  2246 "    Try (Rewrite_Set cancel_p_rls False) @@                     " ^

  2247 "    (Repeat                                                     " ^

  2248 "     ((Try (Rewrite_Set common_nominator_p_rls False) @@        " ^

  2249 "       Try (Rewrite_Set rat_mult_div_pow False) @@              " ^

  2250 "       Try (Rewrite_Set make_rat_poly_with_parentheses False) @@" ^

  2251 "       Try (Rewrite_Set cancel_p_rls False) @@                  " ^

  2252 "       Try (Rewrite_Set rat_reduce_1 False)))) @@               " ^

  2253 "    Try (Rewrite_Set discard_parentheses False))               " ^

  2254 "   t_t)"

  2255 );

  2257 "----------- get_denominator ----------------------------";

  2258 "----------- get_denominator ----------------------------";

  2259 "----------- get_denominator ----------------------------";

  2260 val thy = @{theory Isac};

  2261 val t = term_of (the (parse thy "get_denominator ((a +x)/b)"));

  2262 val SOME (_, t') = eval_get_denominator "" 0 t thy;

  2263 if term2str t' = "get_denominator ((a + x) / b) = b" then ()

  2264 else error "get_denominator ((a + x) / b) = b"

  2267 "--------- several errpats in complicated term -------------------";

  2268 "--------- several errpats in complicated term -------------------";

  2269 "--------- several errpats in complicated term -------------------";

  2270 (*TODO: instead of Gabriella's example here (27.Jul.12) find a simpler one*)

  2271 states:=[];

  2272 CalcTree

  2273 [(["Term ((5*b + 25)/(a^2 - b^2) * (a - b)/(5*b))", "normalform N"],

  2274   ("Rational",["rational","simplification"], ["simplification","of_rationals"]))];

  2275 Iterator 1;

  2276 moveActiveRoot 1;

  2277 autoCalculate 1 CompleteCalc;

  2278 val ((pt,p),_) = get_calc 1; show_pt pt;

  2280 "-------- nonterminating cancel_p, norm_Rational 2002 ---";

  2281 "-------- nonterminating cancel_p, norm_Rational 2002 ---";

  2282 "-------- nonterminating cancel_p, norm_Rational 2002 ---";

  2283 (*------------------------------------------------------------------------------------\

  2284 "--- WN121204: searched rational.sml for "nonterm", added new numbering" ^

  2285 "              and thoroughly tested with this numbering subsequently";

  2286 "--- 1 ---";

  2287 WN.2.6.03 from rlang.sml 56a

  2288 val t = str2term "(a + b * x) / (a + -1 * (b * x)) + (-1 * a + b * x) / (a + b * x) = 4 * (a * b) / (a ^^^ 2 + -1 * b ^^^ 2)";

  2289 val NONE = rewrite_set_ thy false common_nominator_p t;

  2290 "--- 2 ---";

  2291 WN060831 nonterm.SK7

  2292 val t = str2term "(a + b * x) / (a + -1 * (b * x)) + (-1 * a + b * x) / (a + b * x)";

  2293 val NONE = add_fraction_p_ thy t;

  2294 "--- 3 ---";

  2295 nonterm.SK9 loops: cancel_p kann nicht weiter kuerzen!!! *)

  2296 val t'' = str2term "(a + b)/(x^^^2 - y^^^2) * ((x - y)^^^2/(a^^^2 - b^^^2))";

  2297 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t'';

  2298 "--- 4 ---";

  2299 val t = str2term

  2300 "(a * x ^^^ 2 + -2 * a * x * y + a * y ^^^ 2 + b * x ^^^ 2 + -2 * b * x * y + b * y ^^^ 2) /(a ^^^ 2 * x ^^^ 2 + -1 * a ^^^ 2 * y ^^^ 2 + -1 * b ^^^ 2 * x ^^^ 2 + b ^^^ 2 * y ^^^ 2)";

  2301 WN060831 nonterm.SK10

  2302 val SOME (t,_) = rewrite_set_ thy false cancel_p t;

  2303 term2str t;

  2304 "--- 5 ---";

  2305 val t = str2term

  2306 "(9*(x^^^2 - 8*x+16)/(4*(y^^^2 - 2*y+1)))/((3*x - 12)/(16*y - 16))";

  2307 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  2308 if term2str t =

  2309 then ()

  2310 else error "rational.sml: diff.behav. in norm_Rational_mg 42";

  2311 "--- 6 ---";

  2312 val t = str2term

  2313 "9*(x^^^2 - 8*x+16)*(16*y - 16)/(4*(y^^^2 - 2*y+1)*(3*x - 12))";

  2314 val SOME (t',_) = rewrite_set_ thy false norm_Rational t;

  2315 ... non terminating.

  2316 "--- 7 ---";

  2317 val SOME (t',_) = rewrite_set_ thy false make_polynomial t;

  2318 "(-2304 + 1152 * x + 2304 * y + -144 * x ^^^ 2 + -1152 * x * y + 144 * x ^^^ 2 * y) /(-48 + 12 * x + 96 * y + -24 * x * y + -48 * y ^^^ 2 + 12 * x * y ^^^ 2)";

  2319 val SOME (t,_) = rewrite_set_ thy false cancel_p t';

  2320 "--- 8 ---";

  2321 val t = str2term "(a-b)^^^3 * (x+y)^^^4 / ((x+y)^^^2 * (a-b)^^^5)";

  2322 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  2323 "--- 9 ---";

  2324 val t = str2term "((12*x*y/(9*x^^^2 - y^^^2))/" ^

  2325 		 "(1/(3*x - y)^^^2 - 1/(3*x + y)^^^2)) *" ^

  2326 		 "(1/(x - 5*y)^^^2 - 1/(x + 5*y)^^^2)/" ^

  2327 		 "(20*x*y/(x^^^2 - 25*y^^^2))";

  2328 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  2329 "--- 10 ---";

  2330 SK060904-2a non-termination of add_fraction_p_";

  2331 val t = str2term (" (a + b * x) / (a + -1 * (b * x)) +  " ^

  2332 		  " (-1 * a + b * x) / (a + b * x)      ");

  2333 (* nonterm.SK

  2334 val SOME (t',_) = rewrite_set_ thy false common_nominator_p t;

  2335 "--- 11 ---";

  2336 common_nominator_p_ thy t;

  2337 " (a + b * x)*(a + b * x) / ((a + -1 * (b * x))*(a + -1 * (b * x))) +  " ^

  2338 " (-1 * a + b * x)*(a + -1 * (b * x)) / ((a + b * x)*(-1 * a + b * x)) ";

  2339 "--- 12 ---";

  2340 add_fraction_p_ thy t;

  2341 " ((a + b * x)*(a + b * x)  +  (-1 * a + b * x)*(a + -1 * (b * x))) /" ^

  2342 " ((a + b * x)*(-1 * a + b * x))                                     ";

  2343 --------------------------------------------------------------------------------------/*)

  2345 (*------------------------------------------------------------------------------------\

  2346 "WN121205: thoroughly tested with the above numbering.";

  2347 "  errors in cancel_p: --- 4,5,6,7.";

  2348 "  error in common_nominator_p, common_nominator_p_: --- 10; 1,2?.";

  2349 "  errors caused by ruleset norm_Rational: --- 8,9.";

  2350 trace_rewrite := true;

  2352 "--- 1 ---: non-terminating with ### add_fract: done t22 ";

  2353 val t = str2term "(a + b * x) / (a + -1 * (b * x)) + (-1 * a + b * x) / (a + b * x) = 4 * (a * b) / (a ^^^ 2 + -1 * b ^^^ 2)";

  2354 trace_rewrite:= true;

  2355 rewrite_set_ thy false common_nominator_p t;

  2357 "--- 2 ---: non-terminating with ### add_fract: done t22 ";

  2358 val t = str2term "(a + b * x) / (a + -1 * (b * x)) + (-1 * a + b * x) / (a + b * x)";

  2359 add_fraction_p_ thy t;

  2361 "--- 3 ---: norm_Rational calls Rrls cancel_p with non-normalised polys";

  2362 val t = str2term "(a + b)/(x^^^2 - y^^^2) * ((x - y)^^^2/(a^^^2 - b^^^2))";

  2363 rewrite_set_ thy false norm_Rational t;

  2364 (*tracing end...####  rls: cancel_p on:

  2365 (a * x ^^^ 2 + -2 * (a * (x * y)) + a * y ^^^ 2 + b * x ^^^ 2 + -2 * (b * (x * y)) + b * y ^^^ 2) /

  2366 (a ^^^ 2 * x ^^^ 2 + -1 * (a ^^^ 2 * y ^^^ 2) + -1 * (b ^^^ 2 * x ^^^ 2) + b ^^^ 2 * y ^^^ 2) *)

  2368 "--- 4 ---: non-terminating with Rrls cancel_p on plausible input";

  2369 val t = str2term (

  2370 "(a * x ^^^ 2 + -2 * a * x * y + a * y ^^^ 2 + b * x ^^^ 2 + -2 * b * x * y + b * y ^^^ 2) /"^

  2371 "(a ^^^ 2 * x ^^^ 2 + -1 * a ^^^ 2 * y ^^^ 2 + -1 * b ^^^ 2 * x ^^^ 2 + b ^^^ 2 * y ^^^ 2)");

  2372 rewrite_set_ thy false cancel_p t;

  2373 (*#  rls: cancel_p on:

  2374 (a * x ^^^ 2 + -2 * a * x * y + a * y ^^^ 2 + b * x ^^^ 2 + -2 * b * x * y + b * y ^^^ 2) /

  2375 (a ^^^ 2 * x ^^^ 2 + -1 * a ^^^ 2 * y ^^^ 2 + -1 * b ^^^ 2 * x ^^^ 2 + b ^^^ 2 * y ^^^ 2) *)

  2377 "--- 5 ---: non-terminating with Rrls cancel_p on plausible input";

  2378 val t = str2term "(9*(x^^^2 - 8*x+16)/(4*(y^^^2 - 2*y+1)))/((3*x - 12)/(16*y - 16))";

  2379 rewrite_set_ thy false norm_Rational t;

  2380 (*####  rls: cancel_p on:

  2381 (2304 + -1152 * x + -2304 * y + 144 * x ^^^ 2 + 1152 * (x * y) + -144 * (x ^^^ 2 * y)) /

  2382 (48 + -12 * x + -96 * y + 24 * (x * y) + 48 * y ^^^ 2 + -12 * (x * y ^^^ 2))  *)

  2384 "--- 6 ---: non-terminating with Rrls cancel_p on plausible input";

  2385 val t = str2term

  2386 "9*(x^^^2 - 8*x+16)*(16*y - 16)/(4*(y^^^2 - 2*y+1)*(3*x - 12))";

  2387 rewrite_set_ thy false norm_Rational t;

  2388 (*###  rls: cancel_p on: (-2304 + 1152 * x + 2304 * y + -144 * x ^^^ 2 + -1152 * (x * y) +

  2389  144 * (x ^^^ 2 * y)) /

  2390 (-48 + 12 * x + 96 * y + -24 * (x * y) + -48 * y ^^^ 2 + 12 * (x * y ^^^ 2)) *)

  2392 "--- 7 ---: non-terminating with Rrls cancel_p on plausible input";

  2393 val t' = str2term (

  2394 "(-2304 + 1152 * x + 2304 * y + -144 * x ^^^ 2 + -1152 * x * y + 144 * x ^^^ 2 * y) /"^

  2395 "(-48 + 12 * x + 96 * y + -24 * x * y + -48 * y ^^^ 2 + 12 * x * y ^^^ 2)");

  2396 rewrite_set_ thy false cancel_p t';

  2398 "--- 8 ---: bottom of output cannot be looked ad (due to looping?)";

  2399 val t = str2term "(a-b)^^^3 * (x+y)^^^4 / ((x+y)^^^2 * (a-b)^^^5)";

  2400 val SOME (t,_) = rewrite_set_ thy false norm_Rational t;

  2402 "--- 9 ---: probably error in norm_Rational";

  2403 val t = str2term (

  2404 "((12*x*y / (9*x^^^2 - y^^^2)) / (1/(3*x - y)^^^2 - 1/(3*x + y)^^^2)) *" ^

  2405 		"(1/(x - 5*y)^^^2 - 1/(x + 5*y)^^^2) / (20*x*y/(x^^^2 - 25*y^^^2))");

  2406 rewrite_set_ thy false norm_Rational t;

  2407 (*####  rls: cancel_p on: (19440 * (x ^^^ 8 * y ^^^ 2) + -490320 * (x ^^^ 6 * y ^^^ 4) +

  2408  108240 * (x ^^^ 4 * y ^^^ 6) +

  2409  -6000 * (x ^^^ 2 * y ^^^ 8)) /

  2410 (2160 * (x ^^^ 8 * y ^^^ 2) + -108240 * (x ^^^ 6 * y ^^^ 4) +

  2411  1362000 * (x ^^^ 4 * y ^^^ 6) +

  2412  -150000 * (x ^^^ 2 * y ^^^ 8)) *)

  2414 "--- 10 ---: non-terminating with ### add_fract: done t22: error in common_nominator_p ";

  2415 val t = str2term (" (a + b * x) / (a + -1 * (b * x)) + (-1 * a + b * x) / (a + b * x)");

  2416 rewrite_set_ thy false common_nominator_p t; (*### add_fract: done t22 *)

  2417 common_nominator_p_ thy t;                   (*loops without output*)

  2418 "--- reformulated 10:";

  2419 val t = str2term "(a + -1 * (b * x)) / (a + b * x)";

  2420 rewrite_set_ thy false cancel_p t = NONE;

  2421 "--- 11 ---";

  2422 "--- 12 ---";

  2423 "...both are to be considered after common_nominator_p_ is improved";

  2424 --------------------------------------------------------------------------------------/*)

  2425 ============ inhibit exn WN130824 TODO ======================================================*)

author	Walther Neuper <neuper@ist.tugraz.at>
	Wed, 28 Aug 2013 11:02:03 +0200
changeset 52094	61cccc3f2f56
parent 52093	1010f90b4d36
child 52095	c9fbb8171a0a
permissions	-rw-r--r--