wneuper/isa: test/Tools/isac/Knowledge/rational.sml@0831a4a6ec8a

     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 "-------- integration lev.1 -- lev.5: cancel_p_ & common_nominator_p_ --------";

    34 "-------- reverse rewrite -------------------------------";

    35 "-------- 'reverse-ruleset' cancel_p --------------------";

    36 "-------- norm_Rational ---------------------------------";

    37 "-------- numeral rationals -----------------------------";

    38 "-------- cancellation ----------------------------------";

    39 "-------- common denominator ----------------------------";

    40 "-------- multiply and cancel ---------------------------";

    41 "-------- common denominator and multiplication ---------";

    42 "-------- double fractions ------------------------------";

    43 "-------- crucial examples ------------------------------";

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

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

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

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

    48 "-------- investigate rulesets for cancel_p -------------";

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

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

    51 "----------- get_denominator ----------------------------";

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

    53 "--------------------------------------------------------";

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

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

    56 "--------------------------------------------------------";

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

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

    61 "-------- fun poly_of_term ---------------------------------------------------";

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

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

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

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

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

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

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

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

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

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

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

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

    76 (*poly_of_term is quite liberal:*)

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

    78   within a monomial are arbitrary*)

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

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

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

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

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

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

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

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

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

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

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

    93 "-------- fun is_poly --------------------------------------------------------";

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

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

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

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

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

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

   101 "-------- fun term_of_poly ---------------------------------------------------";

   102 val expT = HOLogic.realT

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

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

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

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

   107 ;

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

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

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

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

   113 "-------- fun factout_p_ -----------------------------------------------------";

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

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

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

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

   118 ;

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

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

   121 ;

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

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

   124 ;

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

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

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

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

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

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

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

   133 "-------- fun cancel_p_ ------------------------------------------------------";

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

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

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

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

   138 ;

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

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

   141 ;

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

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

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

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

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

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

   149 "-------- fun common_nominator_p_ --------------------------------------------";

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

   151               (* n1    d1                   *)

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

   153               (* n2    d2                   *)

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

   155 if term2str t' =

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

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

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

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

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

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

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

   164 "-------- example in mail Nipkow";

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

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

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

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

   169             " +\n" ^

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

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

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

   173 ;

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

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

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

   178 val t = str2term "nothing / to_add";

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

   180 ;

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

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

   183 if term2str t' =

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

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

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

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

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

   190 "-------- fun add_fraction_p_ ------------------------------------------------";

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

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

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

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

   195 ;

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

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

   198 ;

   199 val t = str2term "nothing / to_add";

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

   201 ;

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

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

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

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

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

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

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

   210 "-------- TODO ---------------------------------------------------------------";

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

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

   216 "-------- and app_rev --------------------------------------------------------";

   217 (* trace down until prepats are evaluated

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

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

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

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

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

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

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

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

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

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

   229     fun chk_prepat thy erls [] t = true

   230       | chk_prepat thy erls prepat t =

   231         let

   232           fun chk (pres, pat) =

   233             (let

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

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

   236              in

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

   238              end) handle _ => false

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

   240              | scan_ f (pp::pps) =

   241                if f pp then true else scan_ f pps;

   242         in scan_ chk prepat end;

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

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

   245       if chk_prepat thy erls prepat t

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

   247       else NONE;

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

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

   250   (thy, rrls, t);

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

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

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

   254           fun chk (pres, pat) =

   255             (let

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

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

   258              in

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

   260              end) handle _ => false

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

   262              | scan_ f (pp::pps) =

   263                if f pp then true else scan_ f pps;

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

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

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

   268 f;

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

   270 term2str t1 = "?r is_expanded";

   271 term2str t2 = "?s is_expanded";

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

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

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

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

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

   277 (*subst =

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

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

   280 ;

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

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

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

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

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

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

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

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

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

   290 rewrite__set_ thy (i+1) false;

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

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

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

   295 (*========== inhibit exn WN130824 TODO =======================================================

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

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

   298 "-------- external calculating functions test -----------";

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   320 if term2str t' =

   321   "(-1 + x) * (-1 + x) / ((1 + x) * (-1 + x)) +\n(1 + x) * (1 + x) / ((1 + x) * (-1 + x))"

   322   andalso terms2str asm = "[]"

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

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

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

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

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

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

   330 if term2str t' =

   331   "(-1 + x) * (-1 + x) / ((1 + x) * (-1 + x)) +\n(1 + x) * (1 + x) / ((1 + x) * (-1 + x))"

   332   andalso terms2str asm = "[]"

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

   335 val t = str2term

   336   "((1) / (2*x + 2)) + ((1) / (2*x + (-2))) + ((1) / ( x^^^2 + (-1)))+((1) / (x^^^2 + (-2)*x + 1))";

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

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

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

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

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

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

   343 ============ inhibit exn WN130826 TODO ========================================================*)

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

   347 "-------- cancel_p from: Mathematik 1 Schalk Reniets Verlag ------------------";

   348 "-------- cancel_p from: Mathematik 1 Schalk Reniets Verlag ------------------";

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

   350 "-------- WN";

   351 val t = str2term "(2 + -3 * x) / 9";

   352 if NONE = rewrite_set_ thy false cancel_p t then ()

   353 else error "rewrite_set_ cancel_p must return NONE, if the term cannot be cancelled";

   355 "-------- example 186a";

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

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

   358   is_expanded (str2term "x * y");

   359 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;

   360 if (term2str t', terms2str asm) = ("14 / 1", "[]")

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

   363 "-------- example 186b";

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

   365 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;

   366 if (term2str t', terms2str asm) = ("4 / 1", "[]")

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

   369 "-------- example 186c";

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

   371 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;

   372 if (term2str t', terms2str asm) = ("12 * a / 1", "[]")

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

   375 (* !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! exception Div raised !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

   376   see --- fun rewrite_set_ downto fun gcd_poly ---

   377 "-------- example 187a";

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

   379 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;

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

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

   382 *)

   384 (* doesn't terminate !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

   385   see --- fun rewrite_set_ downto fun gcd_poly ---

   386 "-------- example 187b";

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

   388 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;

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

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

   391 *)

   393 (* doesn't terminate !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

   394   see --- fun rewrite_set_ downto fun gcd_poly ---

   395 "-------- example 187c";

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

   397 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;

   398 if (term2str t', terms2str asm) =

   399   ("3 * z ^^^ 3 / (5 * (y * x))", "[\"3 * (z * (y ^^^ 2 * x ^^^ 5)) ~= 0\"]")

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

   401 *)

   403 "-------- example 188a";

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

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

   406 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;

   407 if (term2str t', terms2str asm) = ("8 / 9", "[]")

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

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

   411 val SOME (t,_) = rewrite_set_ thy false make_polynomial t;

   412 if (term2str t', terms2str asm) = ("8 / 9", "[]")

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

   415 "-------- example 188b";

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

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

   418 if (term2str t', terms2str asm) = ("5 / 6", "[]")

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

   421 "-------- example 188c";

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

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

   424 if (term2str t', terms2str asm) = ("-1 / 1", "[]")

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

   427 is_expanded (str2term "a + -1 * b") = true;

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

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

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

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

   433 "-------- example 190a";

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

   435 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;

   436 if (term2str t', terms2str asm) =

   437   ("(1 + 9 * a ^^^ 2) / (3 * a + 9 * a ^^^ 2)", "[\"3 * a + 9 * a ^^^ 2 ~= 0\"]")

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

   440 "-------- example 190c";

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

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

   443 if (term2str t', terms2str asm) =

   444   ("(1 + 3 * a + 9 * a ^^^ 2 + 27 * a ^^^ 3) /\n(3 * a + 18 * a ^^^ 2 + 27 * a ^^^ 3)", "[]")

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

   447 "-------- example 191a";

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

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

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

   451 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;

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

   453 then () else error "rational.sml make_polynomial Schalk 191a";

   455 "-------- example 191b";

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

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

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

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

   461 "-------- example 191c";

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

   463   is_expanded (str2term "9 * x^^^2 + -30 * x + 25") = true;

   464   is_expanded (str2term "25 + -30*x + 9*x^^^2") = true;

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

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

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

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

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

   472 "-------- example 192b";

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

   474 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;

   475 if (term2str t', terms2str asm) = ("x ^^^ 2 / y ^^^ 2", "[\"y ^^^ 2 ~= 0\"]")

   476 then () else error "rational.sml cancel_p Schalk 192b";

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

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

   480 if (term2str t', terms2str asm) =

   481   ("(7 * x ^^^ 3 + -1 * x ^^^ 2 * y) / (7 * x * y ^^^ 2 + -1 * y ^^^ 3)", "[]")

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

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

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

   486 if (term2str t', terms2str asm) =

   487   ("(7 * x ^^^ 3 + -1 * x ^^^ 2 * y) / (7 * x * y ^^^ 2 + -1 * y ^^^ 3)", "[]")

   488 then () else error "rational.sml make_polynomial Schalk WN050929 not working";

   490 "-------- example 193a";

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

   492 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;

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

   494 then () else error "rational.sml cancel_p Schalk 193a";

   496 "-------- example 193b";

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

   498 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;

   499 if (term2str t', terms2str asm) = ("(-4 + x) / (8 + 2 * x)", "[\"8 + 2 * x ~= 0\"]")

   500 then () else error "rational.sml cancel_p Schalk 193b";

   502 "-------- example 193c";

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

   504 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;

   505 if (term2str t', terms2str asm) =

   506   ("(2 * x + 10 * x ^^^ 2) / (1 + -5 * x)", "[\"1 + -5 * x ~= 0\"]")

   507 then () else error "rational.sml cancel_p Schalk 193c";

   509 (*WN:*)

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

   511 val SOME (t, asm) = rewrite_set_ thy false cancel_p t;

   512 if (term2str t', terms2str asm) = ("(2 * x + 10 * x ^^^ 2) / (1 + -5 * x)", "[]")

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

   515 (*========== inhibit exn WN130826 TODO =========================================================

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   541 if term2str t' =

   542   "(-1 + x) * (-1 + x) / ((1 + x) * (-1 + x)) +\n(1 + x) * (1 + x) / ((1 + x) * (-1 + x))"

   543   andalso terms2str asm = "[]"

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

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

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

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

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

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

   551 if term2str t' =

   552   "(-1 + x) * (-1 + x) / ((1 + x) * (-1 + x)) +\n(1 + x) * (1 + x) / ((1 + x) * (-1 + x))"

   553   andalso terms2str asm = "[]"

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

   556 val t = str2term

   557   "((1) / (2*x + 2)) + ((1) / (2*x + (-2))) + ((1) / ( x^^^2 + (-1)))+((1) / (x^^^2 + (-2)*x + 1))";

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

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

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

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

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

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

   566 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))";

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

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

   569 (term2str t3');

   570 (term2str t3'');

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

   573 term2str t4;

   574 term2str (hd(t5));

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

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

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

   581   term2str t';

   582   term2str t'';

   583   "(3 + x) * (3 - x) / ((3 - x) * (3 - x))";

   584   "(3 + x) / (3 - x)";

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

   587 	    "(9 - x^^^2) / (9 - 6*x + x^^^2) + 1 / (3 - x)";

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

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

   590   term2str t';

   591   term2str t'';

   592   "(9 - x ^^^ 2) / ((3 - x) * (3 - x)) + 1 * (3 - x) / ((3 - x) * (3 - x))";

   593   "(4 + x) / (3 - x)";

   595 (*WN021016 added -----vv---*)

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

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

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

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

   600 		" * (9 - 6 * x + x ^^^ 2) / (9 - 6 * x + x ^^^ 2)" (*true*);

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

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

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

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

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

   607 		"(9 - x ^^^ 2) / (9 - 6 * x + x ^^^ 2)" (*true*);

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

   609 (*WN021016 added -----^^---*)

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

   611 val t = str2term "1 / a";

   612 val NONE =  cancel_p_ thy t;

   613 val NONE = rewrite_set_ thy false cancel_p t;

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

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

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

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

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

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

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

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

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

   625 term2str t' =

   626 "(-1 * x ^^^ 2 + y ^^^ 2) / ((-1 * x + y) * (-1 * x + y)) +\n1" ^

   627 " * (-1 * x + y) / ((-1 * x + y) * (-1 * x + y))" (*true*);

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   646  *** RATIONALS_TERM2POLY_EXCEPTION: Invalid Polynomial

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

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

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

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

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

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

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

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

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

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

   660 	    "(9 + (-1)* x^^^2)/(9 + (-1)* 6*x + x^^^2) + (1)/(3 + x)";

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

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

   663   term2str t';

   664   term2str t'';

   665   "(12 + 5 * x + x ^^^ 2) / (9 - x ^^^ 2)";

   666   "(12 + 5 * x + x ^^^ 2) / (9 + (-1) * x ^^^ 2)";

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

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

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

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

   673 val thy' = "Rational";

   674 val rls' = "cancel";

   675 val mp = "make_polynomial";

   677 writeln ("example 186:");

   678 writeln("a)");

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

   680 rewrite_set_ thy false cancel (str2term e186a');

   681 "@@@@@@@@@@@@@@";

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

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

   684   is_expanded (parse_rat "x * y");

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

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

   688 writeln("b)");

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

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

   691 writeln("c)");

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

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

   694     handle e => print_exn_G e;

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

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

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

   699 writeln ("example 187:");

   700 writeln("a)");

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

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

   703 writeln("b)");

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

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

   706 writeln("c)");

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

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

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

   710             "[\"3 * (z * (y ^^^ 2 * x ^^^ 5)) ~= 0\"]") then ()

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

   713 writeln "example 188:";

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

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

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

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

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

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

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

   722 writeln("b)");

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

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

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

   726 writeln("c)");

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

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

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

   731 val SOME (t,_) =

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

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

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

   736 writeln ("example 190:");

   737 writeln("c)");

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

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

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

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

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

   744 writeln ("example 191:");

   745 writeln("a)");

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

   747 (*WN.23.10.02-------

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

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

   750   false;

   751   is_expanded (parse_rat "x + y");

   752   true; -----------*)

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

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

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

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

   758 writeln("c)");

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

   760 (*WN.23.10.02-------

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

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

   763   false;

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

   765   false;

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

   767   true;------------*)

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

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

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

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

   774 writeln ("example 192:");

   775 writeln("b)");

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

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

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

   779 -------------------*)

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

   781 (*========== inhibit exn 110317 ================================================

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

   783 (*"(-1 * y * x ^^^ 2 + 7 * x ^^^ 3) / (-1 * y ^^^ 3 + 7 * x * y ^^^ 2)"WN050929*)

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

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

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

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

   788 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";

   789 ============ inhibit exn 110317 ==============================================*)

   791 writeln ("example 193:");

   792 writeln("a)");

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

   794 (*WN.23.10.02-------

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

   796 -------------------*)

   797 writeln("b)");

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

   799 (*WN.23.10.02-------

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

   801 writeln("c)");

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

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

   804 -------------------*)

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

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

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

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

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

   812 "-------- common_nominator_p ----------------------------";

   813 "-------- common_nominator_p ----------------------------";

   814 "-------- common_nominator_p ----------------------------";

   815 val rls' = "common_nominator_p";

   817 writeln ("example 204:");

   818 writeln("a)");

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

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

   821 writeln("b)");

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

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

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

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

   827 writeln ("example 205:");

   828 writeln("a)");

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

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

   831 writeln("b)");

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

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

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

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

   837 writeln ("example 206:");

   838 writeln("a)");

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

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

   841 writeln("b)");

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

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

   845 writeln ("example 207:");

   846 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)) ";

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

   849 writeln ("example 208:");

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

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

   853 writeln ("example 209:");

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

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

   857 writeln ("example 210:");

   858 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)) ";

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

   861 writeln ("example 211:");

   862 writeln("a)");

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

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

   865 writeln("b)");

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

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

   869 writeln ("example 212:");

   870 writeln("a)");

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

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

   873 writeln("b)");

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

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

   877 writeln ("example 213:");

   878 writeln("a)");

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

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

   881 writeln("b)");

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

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

   885 writeln ("example 214:");

   886 writeln("a)");

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

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

   889 writeln("b)");

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

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

   893 writeln ("example 216:");

   894 writeln("a)");

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

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

   897 writeln("b)");

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

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

   901 writeln ("example 217:");

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

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

   906 val rls' = "common_nominator";

   907 writeln ("example 218:");

   908 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))";

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

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

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

   913 writeln ("example 219:");

   914 writeln("a)");

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

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

   917 writeln("b)");

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

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

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

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

   923 writeln ("example 220:");

   924 writeln("a)");

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

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

   927 writeln("b)");

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

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

   931 writeln ("example 221:");

   932 writeln("a)");

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

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

   935 writeln("b)");

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

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

   939 writeln ("example 222:");

   940 writeln("a)");

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

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

   943 writeln("b)");

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

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

   947 writeln ("example 225:");

   948 writeln("a)");

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

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

   951 writeln("b)");

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

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

   955 writeln ("example 226:");

   956 writeln("a)");

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

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

   959 writeln("b)");

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

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

   963 writeln ("example 227:");

   964 writeln("a)");

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

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

   967 writeln("b)");

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

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

   971 writeln ("example 228:");

   972 writeln("a)");

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

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

   975 writeln("b)");

   976 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))";

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

   980 writeln ("example 229:");

   981 writeln("a)");

   982 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))";

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

   984 writeln("b)");

   985 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))";

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

   988 writeln ("example 230:");

   989 writeln("a)");

   990 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))";

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

   992 writeln("b)");

   993 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))";

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

   996 writeln ("example 231:");

   997 writeln("a)");

   998 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))";

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

  1000 writeln("b)");

  1001 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))";

  1002 val e231b = the (rewrite_set thy' false "common_nominator" e231b');

  1004 writeln ("example 232:");

  1005 writeln("a)");

  1006 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))";

  1007 val e232a = the (rewrite_set thy' false "common_nominator" e232a');

  1008 writeln("b)");

  1009 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))";

  1010 val e232b = the (rewrite_set thy' false "common_nominator" e232b');

  1012 writeln ("example 233:");

  1013 writeln("a)");

  1014 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))";

  1015 val e233a = the (rewrite_set thy' false "common_nominator" e233a');

  1016 writeln("b)");

  1017 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))";

  1018 val e233b = the (rewrite_set thy' false "common_nominator" e233b');

  1020 writeln ("example 234:");

  1021 writeln("a)");

  1022 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))";

  1023 val e234a = the (rewrite_set thy' false "common_nominator" e234a');

  1024 writeln("b)");

  1025 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)) ";

  1026 val e234b = the (rewrite_set thy' false "common_nominator" e234b');

  1028 writeln ("example 235:");

  1029 writeln("a)");

  1030 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))";

  1031 val e235a = the (rewrite_set thy' false "common_nominator" e235a');

  1032 writeln("b)");

  1033 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)) ";

  1034 val e235b = the (rewrite_set thy' false "common_nominator" e235b');

  1036 writeln ("example 236:");

  1037 writeln("a)");

  1038 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))";

  1039 val e236a = the (rewrite_set thy' false "common_nominator" e236a');

  1040 writeln("b)");

  1041 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)) ";

  1042 val e236b = the (rewrite_set thy' false "common_nominator" e236b');

  1045 val rls' = "cancel";

  1046 writeln ("example heuberger:");

  1047 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)";

  1048 val eheu = the (rewrite_set thy' false "cancel" eheu');

  1050 val rls' = "common_nominator_p";

  1051 writeln ("example stiefel:");

  1052 val est1'="(7) / (-14) + (-2) / (4)";

  1053 val est1 = the (rewrite_set thy' false "common_nominator_p" est1');

  1054 if est1 = ("-1 / 1", "[]") then ()

  1055 else error "new behaviour in rational.sml: est1'";

  1057 val t = (term_of o the o (parse thy))

  1058 "(9 - x ^^^ 2) / (9 - 6 * x + x ^^^ 2)";

  1059 val SOME (t',_) = factout_ thy t;

  1060 if term2str t' = "(3 + x) * (3 - x) / ((3 - x) * (3 - x))" then ()

  1061 else error "rational.sml factout_ (9 - x ^^^ 2) / (9 - 6 * x + x ^^^ 2)";

  1063 "-------- integration lev.1 -- lev.5: cancel_p_ & common_nominator_p_ --------";

  1064 "-------- integration lev.1 -- lev.5: cancel_p_ & common_nominator_p_ --------";

  1065 "-------- integration lev.1 -- lev.5: cancel_p_ & common_nominator_p_ --------";

  1066 val t = str2term ("123 = (a*x)/(b*x) + (c*x)/(d*x) + (e*x)/(f*x::real)");

  1067 "-------- gcd_poly integration level 1: works on exact term";

  1068 if NONE = cancel_p_ thy t then () else error "cancel_p_ works on exact fraction";

  1069 if NONE = add_fraction_p_ thy t then () else error "add_fraction_p_ works on exact fraction";

  1071 "-------- gcd_poly integration level 2: picks out ONE appropriate subterm";

  1072 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;

  1073 if term2str t' = "123 = a * x / (b * x) + c * x / (d * x) + e / f"

  1074 then () else error "level 2, rewrite_set_ cancel_p: changed";

  1075 val SOME (t', asm) = rewrite_set_ thy false add_fractions_p t;

  1076 if term2str t' = "123 = (b * c * x + a * d * x) / (b * d * x) + e * x / (f * x)"

  1077 then () else error "level 2, rewrite_set_ add_fractions_p: changed";

  1079 "-------- gcd_poly integration level 3: rewrites all appropriate subterms";

  1080 val SOME (t', asm) = rewrite_set_ thy false cancel_p_rls t;

  1081 if term2str t' = "123 = a / b + c / d + e / f"

  1082 then () else error "level 3, rewrite_set_ cancel_p_rls: changed";

  1083 val SOME (t', asm) = rewrite_set_ thy false common_nominator_p_rls t; (*CREATE add_fractions_p_rls*)

  1084 if term2str t' = "123 = (b * d * e * x + b * c * f * x + a * d * f * x) / (b * d * f * x)"

  1085 then () else error "level 3, rewrite_set_ add_fractions_p_rls: changed";

  1087 "-------- gcd_poly integration level 4: iteration cancel_p -- add_fraction_p";

  1088 (* simpler variant *)

  1089 val testrls = append_rls "testrls" e_rls [Rls_ cancel_p, Rls_ add_fractions_p]

  1090 val SOME (t', asm) = rewrite_set_ thy false testrls t;

  1091 (*trace_rewrite := false;

  1092 #  rls: testrls on: 123 = a * x / (b * x) + c * x / (d * x) + e * x / (f * x)

  1093 ##  rls: cancel_p on: 123 = a * x / (b * x) + c * x / (d * x) + e * x / (f * x)

  1094 ##  rls: common_nominator_p on: 123 = a * x / (b * x) + c * x / (d * x) + e / f

  1095 ##  rls: cancel_p on: 123 = (b * c * x + a * d * x) / (b * d * x) + e / f

  1096 ##  rls: common_nominator_p on: 123 = (b * c + a * d) / (b * d) + e / f

  1097 ##  rls: cancel_p on: 123 = (b * d * e + b * c * f + a * d * f) / (b * d * f)

  1098 ##  rls: common_nominator_p on: 123 = (b * d * e + b * c * f + a * d * f) / (b * d * f) *)

  1099 if term2str t' = "123 = (b * d * e + b * c * f + a * d * f) / (b * d * f)"

  1100 then () else error "level 4, rewrite_set_ *_p: changed";

  1102 (* complicated variant *)

  1103 val testrls_rls = append_rls "testrls_rls" e_rls [Rls_ cancel_p_rls, Rls_ common_nominator_p_rls];

  1104 val SOME (t', asm) = rewrite_set_ thy false testrls_rls t;

  1105 (*#  rls: testrls_rls on: 123 = a * x / (b * x) + c * x / (d * x) + e * x / (f * x)

  1106 ##  rls: cancel_p_rls on: 123 = a * x / (b * x) + c * x / (d * x) + e * x / (f * x)

  1107 ###  rls: cancel_p on: 123 = a * x / (b * x) + c * x / (d * x) + e * x / (f * x)

  1108 ###  rls: cancel_p on: 123 = a * x / (b * x) + c * x / (d * x) + e / f

  1109 ###  rls: cancel_p on: 123 = a * x / (b * x) + c / d + e / f

  1110 ###  rls: cancel_p on: 123 = a / b + c / d + e / f

  1111 ##  rls: common_nominator_p_rls on: 123 = a / b + c / d + e / f

  1112 ###  rls: common_nominator_p on: 123 = a / b + c / d + e / f

  1113 ###  rls: common_nominator_p on: 123 = (b * c + a * d) / (b * d) + e / f

  1114 ###  rls: common_nominator_p on: 123 = (b * d * e + b * c * f + a * d * f) / (b * d * f)

  1115 ##  rls: cancel_p_rls on: 123 = (b * d * e + b * c * f + a * d * f) / (b * d * f)

  1116 ###  rls: cancel_p on: 123 = (b * d * e + b * c * f + a * d * f) / (b * d * f)

  1117 ##  rls: common_nominator_p_rls on: 123 = (b * d * e + b * c * f + a * d * f) / (b * d * f)

  1118 ###  rls: common_nominator_p on: 123 = (b * d * e + b * c * f + a * d * f) / (b * d * f) *)

  1119 if term2str t' = "123 = (b * d * e + b * c * f + a * d * f) / (b * d * f)"

  1120 then () else error "level 4, rewrite_set_ *_p_rls: changed"

  1122 "-------- gcd_poly integration level 5: cancel_p & add_fraction_p within norm_Rational";

  1123 val SOME (t', asm) = rewrite_set_ thy false norm_Rational t;

  1124 if term2str t' = "123 = (a * d * f + b * c * f + b * d * e) / (b * d * f)"

  1125 then () else error "level 5, rewrite_set_ norm_Rational: changed"

  1127 "-------- reverse rewrite -------------------------------";

  1128 "-------- reverse rewrite -------------------------------";

  1129 "-------- reverse rewrite -------------------------------";

  1130 (*WN.28.8.02: tests for the 'reverse-rewrite' functions:

  1131   these are defined in Rationals.ML and stored in

  1132   the 'reverse-ruleset' cancel*)

  1134 (*the term for which reverse rewriting is demonstrated*)

  1135   val t = (term_of o the o (parse thy))

  1136 	      "(9 - x ^^^ 2) / (9 + 6 * x + x ^^^ 2)";

  1137   val Rrls {scr=Rfuns {init_state=ini,locate_rule=loc,

  1138   		       next_rule=nex,normal_form=nor,...},...} = cancel;

  1140 (*normal_form produces the result in ONE step*)

  1141   val SOME (t',_) = nor t;

  1142 if term2str t' = "(3 - x) / (3 + x)" then ()

  1143 else error "rational.sml normal_form (9 - x ^^^ 2) / (9 - 6 * x + x ^^^ 2)";

  1145 (*initialize the interpreter state used by the 'me'*)

  1146   val (t,_,revsets,_) = ini t;

  1148 (*find the rule 'r' to apply to term 't'*)

  1149   val SOME (r as (Thm (str, thm))) = nex revsets t;

  1150 if str = "sym_#power_Float ((3,0), (0,0)) __ ((2,0), (0,0))"

  1151    andalso string_of_thm thm =

  1152            (string_of_thm o make_thm o (cterm_of (@{theory "Isac"})))

  1153                (Trueprop $ (term_of o the o (parse thy)) "9 = 3 ^^^ 2") then ()

  1154 else error "rational.sml next_rule (9 - x ^^^ 2) / (9 - 6 * x + x ^^^ 2)";

  1155 (* before Isa02->09-2 was not checked automatically, ?was different?:

  1156 val SOME r = nex revsets t;

  1157 val r = Thm ("sym_#mult_2_3","6 = 2 * 3") : rule*)

  1159 (* check, if the rule 'r' applied by the user to 't' belongs to the ruleset;

  1160   if the rule is OK, the term resulting from applying the rule is returned,too;

  1161   there might be several rule applications inbetween,

  1162   which are listed after the head in reverse order *)

  1163   val (r, (t, asm))::_ = loc revsets t r;

  1164 if term2str t = "(9 - x ^^^ 2) / (3 ^^^ 2 + 6 * x + x ^^^ 2)" andalso asm = []

  1165 then ()

  1166 else error "rational.sml locate_rule (9 - x ^^^ 2) / (9 - 6 * x + x ^^^ 2)";

  1168 (* find the next rule to apply *)

  1169   val SOME (r as (Thm (str, thm))) = nex revsets t;

  1170 if str = "sym_#power_Float ((3,0), (0,0)) __ ((2,0), (0,0))" andalso

  1171    string_of_thm thm = (string_of_thm o make_thm o (cterm_of (@{theory "Isac"})))

  1172                 (Trueprop $ (term_of o the o (parse thy)) "9 = 3 ^^^ 2") then ()

  1173 else error "rational.sml next_rule (9 - x ^^^ 2) / (9 - 6 * x + x ^^^ 2)";

  1175 (*check the next rule*)

  1176   val (r, (t, asm)) :: _ = loc revsets t r;

  1177 if term2str t = "(3 ^^^ 2 - x ^^^ 2) / (3 ^^^ 2 + 6 * x + x ^^^ 2)" then ()

  1178 else error "rational.sml locate_rule (9 - x ^^^ 2) / (9 - 6 * x + x ^^^ 2) II";

  1180 (*find and check the next rules, rewrite*)

  1181   val SOME r = nex revsets t;

  1182   val (r,(t,asm))::_ = loc revsets t r;

  1183 if term2str t = "(3 ^^^ 2 - x ^^^ 2) / (3 ^^^ 2 + 2 * 3 * x + x ^^^ 2)" then ()

  1184 else error "rational.sml locate_rule II";

  1186   val SOME r = nex revsets t;

  1187   val (r,(t,asm))::_ = loc revsets t r;

  1188 if term2str t = "(3 - x) * (3 + x) / (3 ^^^ 2 + 2 * 3 * x + x ^^^ 2)" then ()

  1189 else error "rational.sml next_rule II";

  1191   val SOME r = nex revsets t;

  1192   val (r,(t,asm))::_ = loc revsets t r;

  1193 if term2str t = "(3 - x) * (3 + x) / ((3 + x) * (3 + x))" then ()

  1194 else error "rational.sml next_rule III";

  1196   val SOME r = nex revsets t;

  1197   val (r, (t, asm)) :: _ = loc revsets t r;

  1198   val ss = term2str t;

  1199 if ss = "(3 - x) / (3 + x)" andalso terms2str asm = "[\"3 + x ~= 0\"]" then ()

  1200 else error "rational.sml: new behav. in rev-set cancel";

  1202 "-------- 'reverse-ruleset' cancel_p --------------------";

  1203 "-------- 'reverse-ruleset' cancel_p --------------------";

  1204 "-------- 'reverse-ruleset' cancel_p --------------------";

  1205 (*WN.11.9.02: the 'reverse-ruleset' cancel_p*)

  1207 (*the term for which reverse rewriting is demonstrated*)

  1208 val t = str2term "(9 + (-1)*x^^^2) / (9 + ((-6)*x + x^^^2))";

  1209 val Rrls {scr=Rfuns {init_state=ini,locate_rule=loc,

  1210 		       next_rule=nex,normal_form=nor,...},...} = cancel_p;

  1212 (*normal_form produces the result in ONE step*)

  1213 val SOME (t',_) = nor t;

  1214 term2str t' = "(3 + 1 * x) / (3 + -1 * x)";

  1216 (*initialize the interpreter state used by the 'me'*)

  1217 val SOME (t', asm) = cancel_p_ thy t;

  1218 term2str t' = "(3 + x) / (3 + -1 * x)" (*true*);

  1219 terms2str asm = "[\"3 + -1 * x ~= 0\"]" (*true*);

  1220 val (t,_,revsets,_) = ini t;

  1222 (* WN.10.10.02: dieser Fall terminiert nicht

  1223            (make_polynomial enth"alt zu viele rules)

  1224 WN060823 'init_state' requires rewriting on specified location in the term

  1225 print_depth 99; Rfuns; print_depth 3;

  1226 WN060831 cycling "sym_order_mult_rls_" "sym_mult_assoc"

  1227          as was with make_polynomial before ?!?*)

  1229 val SOME r = nex revsets t;

  1230 eq_Thm (r, Thm ("sym_#power_Float ((3,0), (0,0)) __ ((2,0), (0,0))",

  1231 		mk_thm thy "9 = 3 ^^^ 2"));

  1232 (*WN060831 *** id_of_thm

  1233            Exception- ERROR raised ...

  1234 val (r,(t,asm))::_ = loc revsets t r;

  1235 term2str t;

  1237   val SOME r = nex revsets t;

  1238   val (r,(t,asm))::_ = loc revsets t r;

  1239   term2str t;

  1240 *)

  1242 writeln "******************  all tests successfull  *************************";

  1246 (*WN.17.3.03 =========================================================vvv---*)

  1247 "-------- norm_Rational ---------------------------------";

  1248 "-------- norm_Rational ---------------------------------";

  1249 "-------- norm_Rational ---------------------------------";

  1250 val t = str2term "(3*x+5)/18 - x/2  - -(3*x - 2)/9 = 0";

  1251 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; term2str t';

  1252 if term2str t' = "1 / 18 = 0" then () else error "rational.sml 1";

  1254 val t = str2term "(17*x - 51)/9 - (-(13*x - 3)/6) + 11 - (9*x - 7)/4 = 0";

  1255 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; term2str t';

  1256 if term2str t' = "(237 + 65 * x) / 36 = 0" then ()

  1257 else error "rational.sml 2";

  1259 val t = str2term "(1/2 + (5*x)/2)^^^2 - ((13*x)/2 - 5/2)^^^2 - (6*x)^^^2 + 29";

  1260 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; term2str t';

  1261 if term2str t' = "23 + 35 * x + -72 * x ^^^ 2" then ()

  1262 else error "rational.sml 3";

  1264 (*trace_rewrite:=true;*)

  1265 val t = str2term "Not (6*x is_atom)";

  1266 val SOME (t',_) = rewrite_set_ thy false powers_erls t; term2str t';

  1267 "HOL.True";

  1268 val t = str2term "1 < 2";

  1269 val SOME (t',_) = rewrite_set_ thy false powers_erls t; term2str t';

  1270 "HOL.True";

  1272 val t = str2term "(6*x)^^^2";

  1273 val SOME (t',_) = rewrite_ thy dummy_ord powers_erls false

  1274 			   (num_str @{thm realpow_def_atom}) t;

  1275 if term2str t' = "6 * x * (6 * x) ^^^ (2 + -1)" then ()

  1276 else error "rational.sml powers_erls (6*x)^^^2";

  1278 val t = str2term "-1 * (-2 * (5 / 2 * (13 * x / 2)))";

  1279 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; term2str t';

  1280 if term2str t' = "65 * x / 2" then () else error "rational.sml 4";

  1282 val t = str2term "1 - ((13*x)/2 - 5/2)^^^2";

  1283 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; term2str t';

  1284 (*bef.040209: if term2str t' = "(-21 + (130 * x + -169 * x ^^^ 2)) / 4"then()*)

  1285 if term2str t' = "(-21 + 130 * x + -169 * x ^^^ 2) / 4" then ()

  1286 else error "rational.sml 5";

  1288 (*SRAM Schalk I, p.92 Nr. 609a*)

  1289 val t = str2term "2*(3 - x/5)/3 - 4*(1 - x/3) - x/3 - 2*(x/2 - 1/4)/27 +5/54";

  1290 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; term2str t';

  1291 if term2str t' = "(-255 + 112 * x) / 135" then ()

  1292 else error "rational.sml 6";

  1294 (*SRAM Schalk I, p.92 Nr. 610c*)

  1295 val t = str2term "((x- 1)/(x+1) + 1) / ((x- 1)/(x+1) - (x+1)/(x- 1)) - 2";

  1296 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; term2str t';

  1297 if term2str t' = "(-3 + -1 * x) / 2" then () else error "rational.sml 7";

  1299 (*SRAM Schalk I, p.92 Nr. 476a*)

  1300 val t = str2term "(x^^^2/(1 - x^^^2) + 1)/(x/(1 - x) + 1) * (1 + x)";

  1301 (*. a/b : c/d translated to a/b * d/c .*)

  1302 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; term2str t';

  1303 (*if term2str t' = "1 / 1" then () else error "rational.sml 8";3.6.03*)

  1304 if term2str t' = "1" then () else error "rational.sml 8";

  1306 (*............................vvv---TODO: sollte gehen mit poly_order *)

  1307 (*Schalk I, p.92 Nr. 472a*)

  1308 val t = str2term "((8*x^^^2 - 32*y^^^2)/(2*x + 4*y))/((4*x - 8*y)/(x + y))";

  1309 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; term2str t';

  1310 if term2str t' = "x + y" then () else error "rational.sml p.92 Nr. 472a";

  1312 (*Schalk I, p.70 Nr. 480b; a/b : c/d translated to a/b * d/c*)

  1313 val t = str2term ("((12*x*y/(9*x^^^2 - y^^^2))/" ^

  1314 		 "(1/(3*x - y)^^^2 - 1/(3*x + y)^^^2)) *" ^

  1315 		 "(1/(x - 5*y)^^^2 - 1/(x + 5*y)^^^2)/" ^

  1316 		 "(20*x*y/(x^^^2 - 25*y^^^2))");

  1317 (*... nicht simpl, zerlegt ...*)

  1318 val t = str2term ("((12*x*y/(9*x^^^2 - y^^^2))/" ^

  1319 		 "(1/(3*x - y)^^^2 - 1/(3*x + y)^^^2))");

  1320 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; term2str t';

  1321 "(-12 * (x * y ^^^ 3) + 108 * (x * (y * x ^^^ 2))) / (12 * (x * y))";

  1322 (*                             ~~~~~~~~~~ poly_order notwendig!*)

  1323 val t = str2term ("(1/(x - 5*y)^^^2 - 1/(x + 5*y)^^^2)/" ^

  1324 		 "(20*x*y/(x^^^2 - 25*y^^^2))");

  1325 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; term2str t';

  1326 if term2str t' = "1 / (x ^^^ 2 + -25 * y ^^^ 2)" then ()

  1327 else error "rational.sml norm_Rational 1 / (x ^^^ 2 + -25 * y ^^^ 2)";

  1329 "nonterm.SK6 ----- SK060904-2a non-termination of add_fraction_p_";

  1330 (*WN.2.6.03 from rlang.sml 56a

  1331 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)";

  1332 val NONE = rewrite_set_ thy false common_nominator_p t;

  1334 WN060831 nonterm.SK7

  1335 val t = str2term "(a + b * x) / (a + -1 * (b * x)) + (-1 * a + b * x) / (a + b * x)";

  1336 val NONE = add_fraction_p_ thy t;

  1337 *)

  1340 (* ------------------------------------------------------------------- *)

  1341 (*---------vvv------------ MG: ab 1.7.03 ----------------vvv-----------*)

  1342 (*                 Simplifier fuer beliebige Buchterme                 *)

  1343 (* ------------------------------------------------------------------- *)

  1344 (*----------------------- norm_Rational_mg ----------------------------*)

  1345 (* ------------------------------------------------------------------- *)

  1347 "-------- numeral rationals -----------------------------";

  1348 "-------- numeral rationals -----------------------------";

  1349 "-------- numeral rationals -----------------------------";

  1350 (*SRA Schalk I, p.40 Nr. 164b *)

  1351 val t = str2term "(47/6 - 76/9 + 13/4)/(35/12)";

  1352 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1353 term2str t;

  1354 if (term2str t) = "19 / 21" then ()

  1355 else error "rational.sml: diff.behav. in norm_Rational_mg 1";

  1357 (*SRA Schalk I, p.40 Nr. 166a *)

  1358 val t = str2term "((5/4)/(4+22/7) + 37/20)*(110/3 - 110/9 * 23/11)";

  1359 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1360 term2str t;

  1361 if (term2str t) = "45 / 2" then ()

  1362 else error "rational.sml: diff.behav. in norm_Rational_mg 2";

  1365 "-------- cancellation ----------------------------------";

  1366 "-------- cancellation ----------------------------------";

  1367 "-------- cancellation ----------------------------------";

  1368 (* e190c Stefan K.*)

  1369 val t = str2term

  1370 "((1 + 9 * a ^^^ 2)*(1 + 3 * a))/((3 * a + 9 * a ^^^ 2)*(1 + 3 * a))";

  1371 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1372 term2str t;

  1373 if (term2str t) =

  1374 "(1 + 9 * a ^^^ 2) / (3 * a + 9 * a ^^^ 2)"

  1375 then ()

  1376 else error "rational.sml: diff.behav. in norm_Rational_mg 3";

  1378 (* e192b Stefan K.*)

  1379 val t = str2term

  1380 "((x ^^^ 2)*(7 * x + (-1) * y))/((y ^^^ 2)*(7 * x + (-1) * y))";

  1381 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1382 term2str t;

  1383 if (term2str t) =

  1384 "x ^^^ 2 / y ^^^ 2"

  1385 then ()

  1386 else error "rational.sml: diff.behav. in norm_Rational_mg 4";

  1388 (*SRC Schalk I, p.66 Nr. 379c *)

  1389 val t = str2term

  1390 "(a - b)/(b - a)";

  1391 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1392 term2str t;

  1393 if (term2str t) =

  1394 "-1"

  1395 then ()

  1396 else error "rational.sml: diff.behav. in norm_Rational_mg 5";

  1398 (*SRC Schalk I, p.66 Nr. 380b *)

  1399 val t = str2term

  1400 "15*(3*x+3)*(4*x+9)/(12*(2*x+7)*(5*x+5))";

  1401 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1402 term2str t;

  1403 if (term2str t) =

  1404 "(27 + 12 * x) / (28 + 8 * x)"

  1405 then ()

  1406 else error "rational.sml: diff.behav. in norm_Rational_mg 6";

  1408 (*Schalk I, p.60 Nr. 215c *)

  1409 (* Falsches Ergebnis: rechnet lange und cancel_p kann nicht weiter krzen!!!*)

  1410 (* WN060831????MG1

  1411 val t = str2term "(a+b)^^^4*(x - y)/((x - y)^^^3*(a+b)^^^2)";

  1412 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1413 term2str t;

  1414 if (term2str t) =

  1415 "(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)"

  1416 then ()

  1417 else error "rational.sml: diff.behav. in norm_Rational_mg 7";

  1418 *)

  1419 (*val t = str2term

  1420 "(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)"

  1421 val SOME (t,_) = rewrite_set_ thy false cancel_p t;

  1422 term2str t;*)

  1423 (* uncaught exception nonexhaustive binding failure

  1424    raised at: stdIn:93.1-93.51 *)

  1426 (*Schalk I, p.66 Nr. 381a *)

  1427 (* ACHTUNG: rechnet ca. 2 Minuten !!! *)

  1428 (* WN060831???MG2

  1429 val t = str2term "18*(a+b)^^^3*(a - b)^^^2/(72*(a - b)^^^3*(a+b)^^^2)";

  1430 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1431 term2str t;

  1432 if (term2str t) =

  1433 "(a + b) / (4 * a + -4 * b)"

  1434 then () else error "rational.sml: diff.behav. in norm_Rational_mg 8";

  1435 *)

  1437 (*SRC Schalk I, p.66 Nr. 381b *)

  1438 val t = str2term

  1439 "(4*x^^^2 - 20*x + 25)/(2*x - 5)^^^3";

  1440 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1441 term2str t;

  1442 if (term2str t) =

  1443 "-1 / (5 + -2 * x)"

  1444 then ()

  1445 else error "rational.sml: diff.behav. in norm_Rational_mg 9";

  1447 (*SRC Schalk I, p.66 Nr. 381c *)

  1448 val t = str2term

  1449 "(27*a^^^3+9*a^^^2+3*a+1)/(27*a^^^3+18*a^^^2+3*a)";

  1450 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1451 term2str t;

  1452 if (term2str t) =

  1453 "(1 + 9 * a ^^^ 2) / (3 * a + 9 * a ^^^ 2)"

  1454 then ()

  1455 else error "rational.sml: diff.behav. in norm_Rational_mg 10";

  1457 (*SRC Schalk I, p.66 Nr. 383a *)

  1458 val t = str2term

  1459 "(5*a^^^2 - 5*a*b)/(a - b)^^^2";

  1460 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1461 term2str t;

  1462 if (term2str t) =

  1463 "5 * a / (a + -1 * b)"

  1464 then ()

  1465 else error "rational.sml: diff.behav. in norm_Rational_mg 11";

  1468 "-------- common denominator ----------------------------";

  1469 "-------- common denominator ----------------------------";

  1470 "-------- common denominator ----------------------------";

  1471 (*SRA Schalk I, p.67 Nr. 403a *)

  1472 val t = str2term

  1473 "4/x - 3/y - 1";

  1474 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1475 term2str t;

  1476 if (term2str t) =

  1477 "(-3 * x + 4 * y + -1 * x * y) / (x * y)"

  1478 then ()

  1479 else error "rational.sml: diff.behav. in norm_Rational_mg 12";

  1481 (*SRA Schalk I, p.67 Nr. 407b *)

  1482 val t = str2term

  1483 "(2*a+3*b)/(b*c) + (3*c+a)/(a*c) - (2*a^^^2+3*b*c)/(a*b*c)";

  1484 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1485 term2str t;

  1486 if (term2str t) =

  1487 "4 / c"

  1488 then ()

  1489 else error "rational.sml: diff.behav. in norm_Rational_mg 13";

  1491 (*SRA Schalk I, p.67 Nr. 410b *)

  1492 val t = str2term

  1493 "1/(x+1) + 1/(x+2) - 2/(x+3)";

  1494 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1495 term2str t;

  1496 if (term2str t) =

  1497 "(5 + 3 * x) / (6 + 11 * x + 6 * x ^^^ 2 + x ^^^ 3)"

  1498 then ()

  1499 else error "rational.sml: diff.behav. in norm_Rational_mg 14";

  1501 (*SRA Schalk I, p.67 Nr. 413b *)

  1502 val t = str2term

  1503 "(1+x)/(1 - x) - (1 - x)/(1+x) + 2*x/(1 - x^^^2)";

  1504 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1505 term2str t;

  1506 if (term2str t) =

  1507 "6 * x / (1 + -1 * x ^^^ 2)"

  1508 then ()

  1509 else error "rational.sml: diff.behav. in norm_Rational_mg 15";

  1511 (*SRA Schalk I, p.68 Nr. 414a *)

  1512 val t = str2term

  1513 "(x + 2)/(x - 1) + (x - 3)/(x - 2) - (x + 1)/((x - 1)*(x - 2))";

  1514 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1515 term2str t;

  1516 if (term2str t) =

  1517 "(-2 + -5 * x + 2 * x ^^^ 2) / (2 + -3 * x + x ^^^ 2)"

  1518 then ()

  1519 else error "rational.sml: diff.behav. in norm_Rational_mg 16";

  1521 (*SRA Schalk I, p.68 Nr. 423a *)

  1522 val t = str2term

  1523 "(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)";

  1524 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1525 term2str t;

  1526 if (term2str t) =

  1527 "1"

  1528 then ()

  1529 else error "rational.sml: diff.behav. in norm_Rational_mg 17";

  1531 (*SRA Schalk I, p.68 Nr. 428b *)

  1532 val t = str2term

  1533 "1/(a - b)^^^2 + 1/(a+b)^^^2 - 2/(a^^^2 - b^^^2) - 4*(b^^^2 - 1)/(a^^^2 - b^^^2)^^^2";

  1534 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1535 term2str t;

  1536 if (term2str t) =

  1537 "4 / (a ^^^ 4 + -2 * a ^^^ 2 * b ^^^ 2 + b ^^^ 4)"

  1538 then ()

  1539 else error "rational.sml: diff.behav. in norm_Rational_mg 18";

  1541 (*SRA Schalk I, p.68 Nr. 430b *)

  1542 val t = str2term

  1543 "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";

  1544 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1545 term2str t;

  1546 if (term2str t) =

  1547 "a + 3 * b"

  1548 then ()

  1549 else error "rational.sml: diff.behav. in norm_Rational_mg 19";

  1552 (*SRA Schalk I, p.68 Nr. 432 *)

  1553 val t = str2term

  1554 "(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)";

  1555 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1556 term2str t;

  1557 if (term2str t) =

  1558 "0"

  1559 then ()

  1560 else error "rational.sml: diff.behav. in norm_Rational_mg 20";

  1562 (*Eigenes*)

  1563 val t = str2term

  1564 "3*a/(a*b) + x/y";

  1565 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1566 term2str t;

  1567 if (term2str t) =

  1568 "(3 * y + b * x) / (b * y)"

  1569 then ()

  1570 else error "rational.sml: diff.behav. in norm_Rational_mg 21";

  1573 "-------- multiply and cancel ---------------------------";

  1574 "-------- multiply and cancel ---------------------------";

  1575 "-------- multiply and cancel ---------------------------";

  1576 (*SRM Schalk I, p.68 Nr. 436a *)

  1577 val t = str2term

  1578 "3*(x+y)/(15*(x - y)) * 25*(x - y)^^^2/(18*(x+y)^^^2)";

  1579 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1580 term2str t;

  1581 if (term2str t) =

  1582 "(5 * x + -5 * y) / (18 * x + 18 * y)"

  1583 then ()

  1584 else error "rational.sml: diff.behav. in norm_Rational_mg 22";

  1586 (*SRM.test Schalk I, p.68 Nr. 436b *)

  1587 (*WN060420???MG3 crashes with method 'simplify' in

  1588   IsacCore > Simplification > Rational Terms > Multiplication > No.2*)

  1589 val t = str2term "5*a*(a - b)^^^2*(a + b)^^^3/(7*b*(a - b)^^^3) * 7*b/(a + b)^^^3";

  1590 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1591 term2str t;

  1592 if (term2str t) =

  1593 "5 * a / (a + -1 * b)"

  1594 then ()

  1595 else error "rational.sml: diff.behav. in norm_Rational_mg 23";

  1597 (*Schalk I, p.68 Nr. 437a *)

  1598 val t = str2term "(3*a - 4*b)/(4*c+3*e) * (3*a+4*b)/(9*a^^^2 - 16*b^^^2)";

  1599 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1600 if (term2str t) = "1 / (4 * c + 3 * e)" then ()

  1601 else error "rational.sml: diff.behav. in norm_Rational_mg 24";

  1603 "----- S.K. corrected non-termination 060904";

  1604 val t = str2term "(3*a - 4*b) * (3*a+4*b)/((4*c+3*e)*(9*a^^^2 - 16*b^^^2))";

  1605 val SOME (t',_) = rewrite_set_ thy false make_polynomial t;

  1606 term2str t';

  1607 if term2str t' =

  1608   "(9 * a ^^^ 2 + -16 * b ^^^ 2) /\n(36 * a ^^^ 2 * c + 27 * a ^^^ 2 * e + -64 * b ^^^ 2 * c +\n -48 * b ^^^ 2 * e)"

  1609 (*"(9 * a ^^^ 2 + -16 * b ^^^ 2) / (36 * a ^^^ 2 * c + 27 * a ^^^ 2 * e + -64 * b ^^^ 2 * c + -48 * b ^^^ 2 * e)"*)

  1610 then () else error "rational.sml: S.K.8..corrected 060904-6";

  1612 "----- S.K. corrected non-termination of cancel_p_";

  1613 val t'' = str2term ("(9 * a ^^^ 2 + -16 * b ^^^ 2) /" ^

  1614 "(36 * a^^^2 * c + (27 * a^^^2 * e + (-64 * b^^^2 * c + -48 * b^^^2 * e)))");

  1615 val SOME (t',_) = rewrite_set_ thy false cancel_p t'';

  1616 if term2str t' = "1 / (4 * c + 3 * e)" then ()

  1617 else error "rational.sml: diff.behav. in cancel_p S.K.8";

  1619 (*Schalk I, p.68 Nr. 437b *)

  1620 (* nonterm.SK9 loops: cancel_p kann nicht weiter kuerzen!!! *)

  1621 val t'' = str2term "(a + b)/(x^^^2 - y^^^2) * ((x - y)^^^2/(a^^^2 - b^^^2))";

  1622 (* val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t'';

  1623    *)

  1625 "================delete===";

  1626 (*a casual output from above*)

  1627 val t = str2term

  1628 "(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)";

  1629 (* WN060831 nonterm.SK10

  1630 val SOME (t,_) = rewrite_set_ thy false cancel_p t;

  1631 term2str t;

  1632 *)

  1634 (*SRM Schalk I, p.68 Nr. 438a *)

  1635 val t = str2term

  1636 "x*y/(x*y - y^^^2)*(x^^^2 - x*y)";

  1637 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1638 term2str t;

  1639 if (term2str t) =

  1640 "x ^^^ 2"

  1641 then ()

  1642 else error "rational.sml: diff.behav. in norm_Rational_mg 24";

  1644 (*SRM Schalk I, p.68 Nr. 439b *)

  1645 val t = str2term

  1646 "(4*x^^^2+4*x+1)*((x^^^2 - 2*x^^^3)/(4*x^^^2+2*x))";

  1647 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1648 term2str t;

  1649 if (term2str t) =

  1650 "(x + -4 * x ^^^ 3) / 2"

  1651 then ()

  1652 else error "rational.sml: diff.behav. in norm_Rational_mg 25";

  1654 (*SRM Schalk I, p.68 Nr. 440a *)

  1655 val t = str2term

  1656 "(x^^^2 - 2*x)/(x^^^2 - 3*x) * (x - 3)^^^2/(x^^^2 - 4)";

  1657 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1658 term2str t;

  1659 if (term2str t) =

  1660 "(-3 + x) / (2 + x)"

  1661 then ()

  1662 else error "rational.sml: diff.behav. in norm_Rational_mg 26";

  1664 "----- Schalk I, p.68 Nr. 440b SK11 works since 0707xx";

  1665 val t = str2term

  1666 "(a^^^3 - 9*a)/(a^^^3*b - a*b^^^3)*(a^^^2*b+a*b^^^2)/(a+3)";

  1667 val SOME (t,_) = rewrite_set_ thy false norm_Rational t;

  1668 if term2str t = "(-3 * a + a ^^^ 2) / (a + -1 * b)" then ()

  1669 else error "rational.sml: diff.behav. in norm_Rational 27";

  1671 "----- SK12 works since 0707xx";

  1672 val t = str2term "(a^^^3 - 9*a)*(a^^^2*b+a*b^^^2)/((a^^^3*b - a*b^^^3)*(a+3))";

  1673 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; term2str t';

  1674 if term2str t' = "(-3 * a + a ^^^ 2) / (a + -1 * b)" then ()

  1675 else error "rational.sml: diff.behav. in norm_Rational 28";

  1678 "-------- common denominator and multiplication ---------";

  1679 "-------- common denominator and multiplication ---------";

  1680 "-------- common denominator and multiplication ---------";

  1681 (*SRAM Schalk I, p.69 Nr. 441b *)

  1682 val t = str2term "(4*a/3 + 3*b^^^2/a^^^3 + b/(4*a))*(4*b/(3*a))";

  1683 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1684 term2str t;

  1685 if (term2str t) =

  1686   "(36 * b ^^^ 3 + 3 * a ^^^ 2 * b ^^^ 2 + 16 * a ^^^ 4 * b) / (9 * a ^^^ 4)"

  1687 then () else error "rational.sml: diff.behav. in norm_Rational_mg 28";

  1689 (*SRAM Schalk I, p.69 Nr. 442b *)

  1690 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)";

  1691 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1692 term2str t;

  1693 if (term2str t) =

  1694 "5 * x ^^^ 2 / (a * b ^^^ 3 * c)"

  1695 then () else error "rational.sml: diff.behav. in norm_Rational_mg 29";

  1697 (*SRAM Schalk I, p.69 Nr. 443b *)

  1698 val t = str2term "(a/2 + b/3)*(b/3 - a/2)";

  1699 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1700 term2str t;

  1701 if (term2str t) =

  1702 "(-9 * a ^^^ 2 + 4 * b ^^^ 2) / 36"

  1703 then () else error "rational.sml: diff.behav. in norm_Rational_mg 30";

  1705 (*SRAM Schalk I, p.69 Nr. 445b *)

  1706 val t = str2term "(a^^^2/9 + 2*a/(3*b) + 4/b^^^2)*(a/3 - 2/b) + 8/b^^^3";

  1707 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1708 term2str t;

  1709 if (term2str t) =

  1710 "a ^^^ 3 / 27"

  1711 then () else error "rational.sml: diff.behav. in norm_Rational_mg 31";

  1713 (*SRAM Schalk I, p.69 Nr. 446b *)

  1714 val t = str2term "(x/(5*x + 4*y) - y/(5*x - 4*y) + 1)*(25*x^^^2 - 16*y^^^2)";

  1715 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1716 term2str t;

  1717 if (term2str t) =

  1718 "30 * x ^^^ 2 + -9 * x * y + -20 * y ^^^ 2"

  1719 then ()

  1720 else error "rational.sml: diff.behav. in norm_Rational_mg 32";

  1722 (*SRAM Schalk I, p.69 Nr. 449a *)(*Achtung: rechnet ca 8 Sekunden*)

  1723 val t = str2term

  1724 "(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)";

  1725 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1726 term2str t;

  1727 if (term2str t) = "(-81 * x ^^^ 4 + 16 * x ^^^ 8 * y ^^^ 4) / (81 * y ^^^ 8)"

  1728 then () else error "rational.sml: diff.behav. in norm_Rational_mg 33";

  1731 (*SRAM Schalk I, p.69 Nr. 450a *)

  1732 val t = str2term

  1733 "(4*x/(3*y)+2*y/(3*x))^^^2 - (2*y/(3*x) - 2*x/y)*(2*y/(3*x)+2*x/y)";

  1734 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1735 term2str t;

  1736 if (term2str t) =

  1737 "(52 * x ^^^ 2 + 16 * y ^^^ 2) / (9 * y ^^^ 2)"

  1738 then ()

  1739 else error "rational.sml: diff.behav. in norm_Rational_mg 34";

  1742 "-------- double fractions ------------------------------";

  1743 "-------- double fractions ------------------------------";

  1744 "-------- double fractions ------------------------------";

  1745 (*SRD Schalk I, p.69 Nr. 454b *)

  1746 val t = str2term

  1747 "((2 - x)/(2*a)) / (2*a/(x - 2))";

  1748 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1749 term2str t;

  1750 if (term2str t) =

  1751 "(-4 + 4 * x + -1 * x ^^^ 2) / (4 * a ^^^ 2)"

  1752 then ()

  1753 else error "rational.sml: diff.behav. in norm_Rational_mg 35";

  1755 (*SRD Schalk I, p.69 Nr. 455a *)

  1756 val t = str2term

  1757 "(a^^^2 + 1)/(a^^^2 - 1) / ((a+1)/(a - 1))";

  1758 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1759 term2str t;

  1760 if (term2str t) =

  1761 "(1 + a ^^^ 2) / (1 + 2 * a + a ^^^ 2)" then ()

  1762 else error "rational.sml: diff.behav. in norm_Rational_mg 36";

  1765 "----- Schalk I, p.69 Nr. 455b";

  1766 val t = str2term "(x^^^2 - 4)/(y^^^2 - 9)/((2+x)/(3 - y))";

  1767 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1768 if term2str t = "(2 + -1 * x) / (3 + y)" then ()

  1769 else error "rational.sml: diff.behav. in norm_Rational_mg 37";

  1771 "----- SK060904-1a non-termination of cancel_p_ ?: worked before 0707xx";

  1772 val t = str2term "(x^^^2 - 4)*(3 - y)/((y^^^2 - 9)*(2+x))";

  1773 val SOME (t,_) = rewrite_set_ thy false norm_Rational t;

  1774 if term2str t = "(2 + -1 * x) / (3 + y)" then ()

  1775 else error "rational.sml: diff.behav. in norm_Rational_mg 37b";

  1777 "----- ?: worked before 0707xx";

  1778 val t = str2term "(3 + -1 * y) / (-9 + y ^^^ 2)";

  1779 val SOME (t,_) = rewrite_set_ thy false norm_Rational t;

  1780 if term2str t = "-1 / (3 + y)" then ()

  1781 else error "rational.sml: -1 / (3 + y) norm_Rational";

  1783 (*SRD Schalk I, p.69 Nr. 456b *)

  1784 val t = str2term

  1785 "(b^^^3 - b^^^2)/(b^^^2+b)/(b^^^2 - 1)";

  1786 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1787 term2str t;

  1788 if (term2str t) = "b / (1 + 2 * b + b ^^^ 2)" then ()

  1789 else error "rational.sml: diff.behav. in norm_Rational_mg 38";

  1791 (*SRD Schalk I, p.69 Nr. 457b *)

  1792 val t = str2term

  1793 "(16*a^^^2 - 9*b^^^2)/(2*a+3*a*b) / ((4*a+3*b)/(4*a^^^2 - 9*a^^^2*b^^^2))";

  1794 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1795 term2str t;

  1796 if (term2str t) =

  1797   "8 * a ^^^ 2 + -6 * a * b + -12 * a ^^^ 2 * b + 9 * a * b ^^^ 2"

  1798 then () else error "rational.sml: diff.behav. in norm_Rational_mg 39";

  1800 "----- Schalk I, p.69 Nr. 458b works since 0707";

  1801 val t = str2term

  1802 "(2*a^^^2*x - a^^^2)/(a*x - b*x) / (b^^^2*(2*x - 1)/(x*(a - b)))";

  1803 val SOME (t,_) = rewrite_set_ thy false norm_Rational t;

  1804 if term2str t = "a ^^^ 2 / b ^^^ 2" then ()

  1805 else error "rational.sml: diff.behav. in norm_Rational_mg 39b";

  1807 (*SRD Schalk I, p.69 Nr. 459b *)

  1808 val t = str2term "(a^^^2 - b^^^2)/(a*b) / (4*(a+b)^^^2/a)";

  1809 val SOME (t,_) = rewrite_set_ thy false norm_Rational t;

  1810 if term2str t = "(a + -1 * b) / (4 * a * b + 4 * b ^^^ 2)" then ()

  1811 else error "rational.sml: diff.behav. in norm_Rational_mg 41";

  1814 (*Schalk I, p.69 Nr. 460b nonterm.SK

  1815 val t = str2term

  1816 "(9*(x^^^2 - 8*x+16)/(4*(y^^^2 - 2*y+1)))/((3*x - 12)/(16*y - 16))";

  1817 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1818 if term2str t =

  1819 then ()

  1820 else error "rational.sml: diff.behav. in norm_Rational_mg 42";

  1822 val t = str2term

  1823 "9*(x^^^2 - 8*x+16)*(16*y - 16)/(4*(y^^^2 - 2*y+1)*(3*x - 12))";

  1824 val SOME (t',_) = rewrite_set_ thy false norm_Rational t;

  1825 ... non terminating.

  1826 val SOME (t',_) = rewrite_set_ thy false make_polynomial t;

  1827 "(-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)";

  1828 val SOME (t,_) = rewrite_set_ thy false cancel_p t';

  1829 ... non terminating.*)

  1831 (*SRD Schalk I, p.70 Nr. 472a *)

  1832 val t = str2term ("((8*x^^^2 - 32*y^^^2)/(2*x + 4*y))/" ^

  1833 		 "((4*x - 8*y)/(x + y))");

  1834 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1835 term2str t;

  1836 if (term2str t) =

  1837 "x + y"

  1838 then ()

  1839 else error "rational.sml: diff.behav. in norm_Rational_mg 43";

  1842 (*----------------------------------------------------------------------*)

  1843 (*---------------------- Einfache Dppelbrche --------------------------*)

  1844 (*----------------------------------------------------------------------*)

  1846 (*SRD Schalk I, p.69 Nr. 461a *)

  1847 val t = str2term

  1848 "(2/(x+3) + 2/(x - 3)) / (8*x/(x^^^2 - 9))";

  1849 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1850 term2str t;

  1851 if (term2str t) =

  1852 "1 / 2"

  1853 then ()

  1854 else error "rational.sml: diff.behav. in norm_Rational_mg 44";

  1856 (*SRD Schalk I, p.69 Nr. 464b *)

  1857 val t = str2term

  1858 "(a - a/(a - 2)) / (a + a/(a - 2))";

  1859 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1860 term2str t;

  1861 if (term2str t) =

  1862 "(3 + -1 * a) / (1 + -1 * a)"

  1863 then ()

  1864 else error "rational.sml: diff.behav. in norm_Rational_mg 45";

  1866 (*SRD Schalk I, p.69 Nr. 465b *)

  1867 val t = str2term

  1868 "((x+3*y)/9 + (4*y^^^2 - 9*z^^^2)/(16*x)) /(x/9+y/6+z/4)";

  1869 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1870 term2str t;

  1871 if (term2str t) =

  1872 "(4 * x + 6 * y + -9 * z) / (4 * x)"

  1873 then ()

  1874 else error "rational.sml: diff.behav. in norm_Rational_mg 46";

  1876 (*SRD Schalk I, p.69 Nr. 466b *)

  1877 val t = str2term

  1878 "((1 - 7*(x - 2)/(x^^^2 - 4)) / (6/(x+2))) / (3/(x+5)+30/(x^^^2 - 25))";

  1879 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1880 term2str t;

  1881 if (term2str t) =

  1882 "(25 + -10 * x + x ^^^ 2) / 18"

  1883 then ()

  1884 else error "rational.sml: diff.behav. in norm_Rational_mg 47";

  1886 (*SRD Schalk I, p.70 Nr. 469 *)

  1887 val t = str2term

  1888 "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))";

  1889 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1890 term2str t;

  1891 if (term2str t) =

  1892 "3 * b ^^^ 3 / (2 * a + -2 * b)"

  1893 then ()

  1894 else error "rational.sml: diff.behav. in norm_Rational_mg 48";

  1896 (*----------------------------------------------------------------------*)

  1897 (*---------------------- Mehrfache Dppelbrueche ------------------------*)

  1898 (*----------------------------------------------------------------------*)

  1900 (*SRD.test Schalk I, p.70 Nr. 476b *) (* Rechenzeit: 10 sec *)

  1901 (*WN060419 crashes with method 'simplify' ????SK*)

  1902 val t = str2term

  1903 "((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)";

  1904 val SOME (t,_) = rewrite_set_ thy false norm_Rational t;

  1905 if term2str t = "1 / (a ^^^ 2 + -1 * b ^^^ 2)" then ()

  1906 else error "rational.sml: diff.behav. in norm_Rational_mg 49";

  1908 "----- Schalk I, p.70 Nr. 477a";

  1909 (* MG Achtung: terme explodieren; Bsp zu komplex;

  1910    L???ung sollte (ziemlich grosser) Faktorisierter Ausdruck sein

  1911 val t = str2term "b*y/(b - 2*y)/((b^^^2 - y^^^2)/(b+2*y)) /" ^

  1912 		 "(b^^^2*y+b*y^^^2)*(a+x)^^^2/((b^^^2 - 4*y^^^2)*(a+2*x)^^^2)";

  1913 val SOME (t',_) = rewrite_set_ thy false norm_Rational t;

  1916 val t = str2term "b*y*(b+2*y)*(b^^^2 - 4*y^^^2)*(a+2*x)^^^2 / " ^

  1917 		 "((b - 2*y)*(b^^^2 - y^^^2)*(b^^^2*y+b*y^^^2)*(a+x)^^^2)";

  1918 val SOME (t,_) = rewrite_set_ thy false make_polynomial t;

  1919 ????SK ???MG*)

  1922 "----- Schalk I, p.70 Nr. 478b ----- Rechenzeit: 5 sec";

  1923 val t = str2term ("(a - (a*b+b^^^2)/(a+b))/(b+(a - b)/(1+(a+b)/(a - b))) / " ^

  1924 		 "((a - a^^^2/(a+b))/(a+(a*b)/(a - b)))");

  1925 val SOME (t',_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1926 if term2str t' =

  1927 "(2 * a ^^^ 3 + 2 * a ^^^ 2 * b) / (a ^^^ 2 * b + b ^^^ 3)"

  1928 then ()

  1929 else error "rational.sml: diff.behav. in norm_Rational_mg 51";

  1931 (* TODO.new_c:WN050820 STOP_REW_SUB introduced gave ...

  1932 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 ()

  1933 else error "rational.sml: works again";

  1934 re-outcommented with TODO.new_c: cvs before 071227, 11:50*)

  1938 (*Schalk I, p.70 Nr. 480a *)

  1939 (* Achtung: rechnet ewig; cancel_p kann nicht krzen: WN060831 nonterm.SK00

  1940 val t = str2term

  1941 "(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)))";

  1942 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1943 term2str t;

  1944 if (term2str t) =

  1946 then ()

  1947 else error "rational.sml: diff.behav. in norm_Rational_mg 52";

  1949 (*MG Berechne Zwischenergebnisse: WN060831 nonterm.SK00*)

  1950 val t = str2term

  1951 "(1/x+1/y+1/z)/(1/x - 1/y - 1/z)";

  1952 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1953 term2str t;

  1954 "(x ^^^ 2 * y ^^^ 2 * z + x ^^^ 2 * y * z ^^^ 2 + x * y ^^^ 2 * z ^^^ 2) /

  1955 (-1 * x ^^^ 2 * y ^^^ 2 * z + -1 * x ^^^ 2 * y * z ^^^ 2 + x * y ^^^ 2 * z ^^^ 2)";

  1956 val t = str2term

  1957 "2*x^^^2/(x^^^2 - z^^^2)/(x/(x+z)+x/(x - z))";

  1958 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1959 term2str t;

  1960 "1"

  1962 (* SK 1. Ausdruck kann nicht weiter gekrzt werden; cancel_p !!!*)

  1963 ###  rls: cancel_p on:

  1964 (x ^^^ 2 * (y ^^^ 2 * z) + x ^^^ 2 * (y * z ^^^ 2) + x * (y ^^^ 2 * z ^^^ 2)) /

  1965 (-1 * (x ^^^ 2 * (y ^^^ 2 * z)) + -1 * (x ^^^ 2 * (y * z ^^^ 2)) + x * (y ^^^ 2 * z ^^^ 2))

  1966 GC #3.61.81.101.197.17503:   (0 ms)

  1967 *** RATIONALS_DIRECT_CANCEL_EXCEPTION: Invalid fraction

  1969 val t = str2term

  1970 "(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))";

  1971 val SOME (t,_) = rewrite_set_ thy false cancel_p t;

  1972 term2str t;

  1973 (*uncaught exception nonexhaustive binding failure*)

  1975 (* Das kann er aber krzen !!????: *)

  1976 val t = str2term

  1977 "(x^^^2 * (y^^^2 * z) +  x * (y^^^2 * z^^^2)) / (-1 * (x^^^2 * (y * z^^^2)) + x * (y^^^2 * z^^^2))";

  1978 val SOME (t,_) = rewrite_set_ thy false cancel_p t;

  1979 term2str t;

  1980 "(-1 * (y * x) + -1 * (z * y)) / (1 * (z * x) + -1 * (z * y))";

  1981 *)

  1984 "-------- crucial examples ------------------------------";

  1985 "-------- crucial examples ------------------------------";

  1986 "-------- crucial examples ------------------------------";

  1987 (*Schalk I, p.60 Nr. 215d *)

  1988 (* Achtung: rechnet ewig ...

  1989 val t = str2term "(a-b)^^^3 * (x+y)^^^4 / ((x+y)^^^2 * (a-b)^^^5)";

  1990 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1991 term2str t; noterm.SK

  1992 *)

  1994 (* Kein Wunder, denn Z???ler und Nenner extra als Polynom dargestellt ergibt:*)

  1995 (*

  1996 val t = str2term "(a-b)^^^3 * (x+y)^^^4";

  1997 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1998 term2str t;

  1999 "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";

  2000 val t = str2term "((x+y)^^^2 * (a-b)^^^5)";

  2001 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  2002 term2str t;

  2003 "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";

  2004 *)

  2005 (*anscheinend macht dem Rechner das Krzen diese Bruches keinen Spass mehr ...*)

  2007 (*--------------------------------------------------------------------*)

  2008 (*Schalk I, p.70 Nr. 480b

  2009 val t = str2term "((12*x*y/(9*x^^^2 - y^^^2))/" ^

  2010 		 "(1/(3*x - y)^^^2 - 1/(3*x + y)^^^2)) *" ^

  2011 		 "(1/(x - 5*y)^^^2 - 1/(x + 5*y)^^^2)/" ^

  2012 		 "(20*x*y/(x^^^2 - 25*y^^^2))";

  2013 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  2014 SK.nonterm

  2015 Kann nicht weiter vereinfacht werden !!!!?? *)

  2017 (*--------------------------------------------------------------------*)

  2018 "---- MGs test set";

  2019 val t = str2term " (1 + x^^^5) / (y + x) + x^^^3 / x ";

  2020 val SOME (t,_) = rewrite_set_ thy false common_nominator_p t;

  2021 if term2str t = "(1 + x ^^^ 3 + x ^^^ 5 + y * x ^^^ 2) / (x + y)" then()

  2022 else error "";

  2024 (*--------------------------------------------------------------------*)

  2025 (* cancel_p liefert nicht immer Polynomnormalform (2): WN060831???SK3b

  2026    ---> Sortierung FALSCH !!  *)

  2027 val t = str2term "b^^^3 * a^^^5/a ";

  2028 val SOME (t,_) = rewrite_set_ thy false cancel_p t;

  2029 term2str t;

  2030 "1 * (a^^^4 * b^^^3) / 1"; (*OK*)

  2032 val t = str2term "b^^^3 * a^^^5/b ";

  2033 val SOME (t,_) = rewrite_set_ thy false cancel_p t;

  2034 term2str t;

  2035 "1 * (b^^^2 * a^^^5) / 1"; (*cancel_p sortiert hier falsch um!*)

  2037 (* Problem liegt NICHT bei ord_make_polynomial! (siehe folgende Bsple) *)

  2038 (*

  2039 val x = str2term "x"; val bdv = str2term "bdv";

  2040 val t1 = str2term "b^^^2 * a^^^5";

  2041 val t2 = str2term "a^^^5 * b^^^2 ";

  2042 ord_make_polynomial false Rational.thy [(x,bdv)] (t1,t2); (*false*)

  2043 *)

  2044 (* ==> "b^^^2 * a^^^5" > "a^^^5 * b^^^2 " ... OK!*)

  2046 (*--------------------------------------------------------------------*)

  2047 (* cancel_p liefert nicht immer Polynomnormalform (2): WN060831???SK3c

  2048    ---> erzeugt berflssige "1 * ..."

  2050 val t = str2term "-1 / (3 + y)";

  2051 (*~~         *)

  2052 val SOME (t,_) = rewrite_set_ thy false cancel_p t;

  2053 term2str t;

  2054 "-1 / (3 + 1 * y)";

  2055 (********* Das ist das PROBLEM !!!!!!!??? *******************)

  2056 (* -1 im Z???ler der Angabe verursacht das Problem !*)

  2057 *)

  2059 (* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - *)

  2060 "----- MGs test set";

  2061 val t = str2term "(a^^^2 + -1)/(a+1)";

  2062 val SOME (t',_) = rewrite_set_ thy false cancel_p t;

  2063 if term2str t' = "(-1 + a) / 1" then ()

  2064 else error "rational.sml MG tests 3d";

  2066 "----- NOT TERMINATING ?: worked before 0707xx";

  2067 val t = str2term "(a^^^2 - 1)*(b + 1) / ((b^^^2 - 1)*(a+1))";

  2068 val SOME (t'',_) = rewrite_set_ thy false norm_Rational t;

  2069 if term2str t'' = "(1 + -1 * a) / (1 + -1 * b)" then ()

  2070 else error "rational.sml MG tests 3e";

  2072 "----- corrected SK060905";

  2073 val t = str2term "(4*x^^^2 - 20*x + 25)/(2*x - 5)^^^3";

  2074 val SOME (t',_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  2075 if term2str t' = "-1 / (5 + -2 * x)" then ()

  2076 else error "rational.sml corrected SK060905";

  2079 "-------- examples for Stefan Karnels thesis ------------";

  2080 "-------- examples for Stefan Karnels thesis ------------";

  2081 "-------- examples for Stefan Karnels thesis ------------";

  2082 (*SRAM Schalk I, p.69 Nr. 442b --- abgewandelt*)

  2083 val t = str2term

  2084 "(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))";

  2085 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  2086 term2str t;

  2087 if (term2str t) =

  2088 "5 * x ^^^ 2 / (b ^^^ 3 * c)"

  2089 then ()

  2090 else error "rational.sml: diff.behav. in norm_Rational_mg 53";

  2093 "-------- me Schalk I No.186 ----------------------------";

  2094 "-------- me Schalk I No.186 ----------------------------";

  2095 "-------- me Schalk I No.186 ----------------------------";

  2096 val fmz = ["Term ((14 * x * y) / ( x * y ))",

  2097 	   "normalform N"];

  2098 val (dI',pI',mI') =

  2099   ("Rational",["rational","simplification"],

  2100    ["simplification","of_rationals"]);

  2101 val p = e_pos'; val c = [];

  2102 val (p,_,f,nxt,_,pt) = CalcTreeTEST [(fmz, (dI',pI',mI'))];

  2103 val (p,_,f,nxt,_,pt) = me nxt p c pt;

  2104 val (p,_,f,nxt,_,pt) = me nxt p c pt;

  2105 val (p,_,f,nxt,_,pt) = me nxt p c pt;

  2106 val (p,_,f,nxt,_,pt) = me nxt p c pt;

  2107 val (p,_,f,nxt,_,pt) = me nxt p c pt;

  2108 val (p,_,f,nxt,_,pt) = me nxt p c pt;

  2109 val (p,_,f,nxt,_,pt) = me nxt p c pt;

  2110 val (p,_,f,nxt,_,pt) = me nxt p c pt; f2str f;

  2111 val (p,_,f,nxt,_,pt) = me nxt p c pt; f2str f;

  2112 val (p,_,f,nxt,_,pt) = me nxt p c pt; f2str f;(*++ for explicit script*)

  2113 val (p,_,f,nxt,_,pt) = me nxt p c pt; f2str f;(*++ for explicit script*)

  2114 case (f2str f, nxt) of

  2115     ("14", ("End_Proof'", _)) => ()

  2116   | _ => error "rational.sml diff.behav. in me Schalk I No.186";

  2119 "-------- interSteps ..Simp_Rat_Double_No-1.xml ---------";

  2120 "-------- interSteps ..Simp_Rat_Double_No-1.xml ---------";

  2121 "-------- interSteps ..Simp_Rat_Double_No-1.xml ---------";

  2122 states:=[];

  2123 CalcTree

  2124 [(["Term (((2 - x)/(2*a)) / (2*a/(x - 2)))", "normalform N"],

  2125   ("Rational",["rational","simplification"],

  2126   ["simplification","of_rationals"]))];

  2127 Iterator 1;

  2128 moveActiveRoot 1;

  2129 autoCalculate 1 CompleteCalc;

  2130 val ((pt,p),_) = get_calc 1; show_pt pt;

  2132 interSteps 1 ([1],Res);

  2133 val ((pt,p),_) = get_calc 1; show_pt pt;

  2136 "-------- interSteps ..Simp_Rat_Cancel_No-1.xml ---------";

  2137 "-------- interSteps ..Simp_Rat_Cancel_No-1.xml ---------";

  2138 "-------- interSteps ..Simp_Rat_Cancel_No-1.xml ---------";

  2139 states:=[];

  2140 CalcTree

  2141 [(["Term ((a^2 + -1*b^2) / (a^2 + -2*a*b + b^2))", "normalform N"],

  2142   ("Rational",["rational","simplification"],

  2143   ["simplification","of_rationals"]))];

  2144 Iterator 1;

  2145 moveActiveRoot 1;

  2146 autoCalculate 1 CompleteCalc;

  2147 val ((pt,p),_) = get_calc 1; show_pt pt;

  2148 (*========== inhibit exn 110314 ================================================

  2149 (*with explicit script done already... and removed [1,..] at below...

  2150 interSteps 1 ([1],Res);

  2151 val ((pt,p),_) = get_calc 1; show_pt pt;

  2152 *)

  2153 interSteps 1 ([2],Res);

  2154 val ((pt,p),_) = get_calc 1; show_pt pt;

  2156 interSteps 1 ([2,1],Res);

  2157 val ((pt,p),_) = get_calc 1; show_pt pt;

  2158 val newnds = children (get_nd pt [2,1]) (*see "fun detailrls"*);

  2159 (*if length newnds = 12 then () WN060905*)

  2160 if length newnds = 13 then ()

  2161 else error "rational.sml: interSteps cancel_p rev_rew_p";

  2163 val p = ([2,1,9],Res);

  2164 getTactic 1 p;

  2165 val (_, tac, _) = pt_extract (pt, p);

  2166 (*case tac of SOME (Rewrite ("sym_real_plus_binom_times1", _)) => ()

  2167 WN060905*)

  2168 case tac of SOME (Rewrite ("sym_real_add_mult_distrib2", _)) => ()

  2169 | _ => error "rational.sml: getTactic, sym_real_plus_binom_times1";

  2170 ============ inhibit exn 110314 ==============================================*)

  2173 "-------- investigate rulesets for cancel_p -------------";

  2174 "-------- investigate rulesets for cancel_p -------------";

  2175 "-------- investigate rulesets for cancel_p -------------";

  2176 val thy = @{theory "Rational"};

  2177 "---------------- (a^^^2 + -1*b^^^2) / (a^^^2 + -2*a*b + b^^^2)";

  2178 val t = str2term "(a^^^2 + -1*b^^^2) / (a^^^2 + -2*a*b + b^^^2)";

  2179 val tt = str2term "(1 * a + 1 * b) * (1 * a + -1 * b)"(*numerator only*);

  2180 "----- with rewrite_set_";

  2181 val SOME (tt',asm) = rewrite_set_ thy false make_polynomial tt;

  2182 term2str tt'= "a ^^^ 2 + -1 * b ^^^ 2" (*true*);

  2183 val tt = str2term "((1 * a + -1 * b) * (1 * a + -1 * b))"(*denominator only*);

  2184 val SOME (tt',asm) = rewrite_set_ thy false make_polynomial tt;

  2185 term2str tt' = "a ^^^ 2 + -2 * a * b + b ^^^ 2" (*true*);

  2187 "----- with make_deriv";

  2188 val SOME (tt, _) = factout_p_ thy t; term2str tt =

  2189 "(1 * a + 1 * b) * (1 * a + -1 * b) / ((1 * a + -1 * b) * (1 * a + -1 * b))";

  2190 (*

  2191 "--- with ruleset as before 060829";

  2192 val {rules, rew_ord=(_,ro),...} =

  2193     rep_rls (assoc_rls "make_polynomial");

  2194 val der = make_deriv thy Atools_erls rules ro NONE tt;

  2195 print_depth 99; map (term2str o #1) der; print_depth 3;

  2196 print_depth 99; map (rule2str o #2) der; print_depth 3;

  2197 ... did not terminate*)

  2198 "--- with simpler ruleset";

  2199 val {rules, rew_ord=(_,ro),...} =

  2200     rep_rls (assoc_rls "rev_rew_p");

  2201 val der = make_deriv thy Atools_erls rules ro NONE tt;

  2202 print_depth 99; writeln (deriv2str der); print_depth 3;

  2204 print_depth 99; map (term2str o #1) der; print_depth 3;

  2205 "...,(-1 * b ^^^ 2 + a ^^^ 2) / (-2 * (a * b) + a ^^^ 2 + (-1 * b) ^^^ 2) ]";

  2206 print_depth 99; map (rule2str o #2) der; print_depth 3;

  2207 print_depth 99; map (term2str o #1 o #3) der; print_depth 3;

  2209 val der = make_deriv thy Atools_erls rules ro NONE

  2210 		     (str2term "(1 * a + 1 * b) * (1 * a + -1 * b)");

  2211 print_depth 99; writeln (deriv2str der); print_depth 3;

  2213 val {rules, rew_ord=(_,ro),...} =

  2214     rep_rls (assoc_rls "rev_rew_p");

  2215 val der = make_deriv thy Atools_erls rules ro NONE

  2216 		     (str2term "(1 * a + -1 * b) * (1 * a + -1 * b)");

  2217 print_depth 99; writeln (deriv2str der); print_depth 3;

  2218 print_depth 99; map (term2str o #1) der; print_depth 3;

  2219 (*WN060829 ...postponed*)

  2222 "-------- investigate format of factout_ and factout_p_ -";

  2223 "-------- investigate format of factout_ and factout_p_ -";

  2224 "-------- investigate format of factout_ and factout_p_ -";

  2225 val {rules, rew_ord = (_,ro),...} = rep_rls (assoc_rls "make_polynomial");

  2226 val (thy, eval_rls) = (@{theory "Rational"}, Atools_erls)(*see 'fun init_state'*);

  2227 val Rrls {scr = Rfuns {init_state,...},...} = assoc_rls "cancel_p";

  2229 "----- see Rational.ML, local cancel_p, fun init_state";

  2230 val t = str2term "(a^^^2 + (-1)*b^^^2) / (a^^^2 + (-2)*a*b + b^^^2)";

  2231 val SOME (t',_) = factout_p_ thy t; term2str t';

  2232 (*

  2233 val rtas = reverse_deriv thy eval_rls rules ro NONE t';

  2234 writeln(trtas2str rst);

  2235 *)

  2238 "----- see Rational.ML, local cancel_p, fun init_state";

  2239 val t = str2term "a^^^2 / a";

  2240 val SOME (t',_) = factout_p_ thy t;

  2241 term2str t' = "a * a / (1 * a)" (*true*);

  2242 (*... can be canceled with

  2243 real_mult_div_cancel2 ?k ~= 0 ==> ?m * ?k / (?n * ?k) = ?m / ?n"*)

  2244 (* sml/ME/rewtools.sml:

  2245 val rtas = reverse_deriv thy Atools_erls rules ro NONE t';

  2246 writeln (deri2str rtas);

  2247 *)

  2250 "-------- SK 060904 ----------------------------------------------";

  2251 "----- order on polynomials -- input + output";

  2252 val thy = @{theory "Isac"};

  2253 val t = str2term "(a + -1 * b) / (-1 * a + b)";

  2254 val SOME (t', _) = factout_p_ thy t; term2str t';

  2255 val SOME (t', _) = cancel_p_ thy t; term2str t';

  2257 val t = str2term "a*b*c*d / (d*e*f*g)";

  2258 val SOME (t', _) = cancel_p_ thy t; term2str t';

  2260 val t = str2term "a*(b*(c*d)) / (b*(e*(f*g)))";

  2261 val SOME (t', _) = cancel_p_ thy t; term2str t';

  2262 (*???order.SK  ???*)

  2264 "----- SK060904-1a non-termination of cancel_p_ ? worked before 0707xx";

  2265 val t = str2term "(x^^^2 - 4)*(3 - y) / ((y^^^2 - 9)*(2+x))";

  2266 val SOME (t',_) = rewrite_set_ thy false norm_Rational t;

  2267 if term2str t' = "(2 + -1 * x) / (3 + y)" then ()

  2268 else error "rational.sml SK060904-1a worked since 0707xx";

  2270 "----- SK060904-1b non-termination of cancel_p_ ... worked before 0707xx";

  2271 val t = str2term ("(9 * a ^^^ 2 + -16 * b ^^^ 2) /" ^

  2272 "(36 * a^^^2 * c + (27 * a^^^2 * e + (-64 * b^^^2 * c + -48 * b^^^2 * e)))");

  2273 val SOME (t',_) = cancel_p_ thy t;

  2274 if term2str t' = "1 / (4 * c + 3 * e)" then ()

  2275 else error "rational.sml SK060904-1b";

  2278 "----- SK060904-2a non-termination of add_fraction_p_";

  2279 val t = str2term (" (a + b * x) / (a + -1 * (b * x)) +  " ^

  2280 		  " (-1 * a + b * x) / (a + b * x)      ");

  2281 (* nonterm.SK

  2282 val SOME (t',_) = rewrite_set_ thy false common_nominator_p t;

  2284 common_nominator_p_ thy t;

  2285 " (a + b * x)*(a + b * x) / ((a + -1 * (b * x))*(a + -1 * (b * x))) +  " ^

  2286 " (-1 * a + b * x)*(a + -1 * (b * x)) / ((a + b * x)*(-1 * a + b * x)) ";

  2288 add_fraction_p_ thy t;

  2289 " ((a + b * x)*(a + b * x)  +  (-1 * a + b * x)*(a + -1 * (b * x))) /" ^

  2290 " ((a + b * x)*(-1 * a + b * x))                                     ";

  2291 *)

  2294 "-------- how to stepwise construct Scripts -------------";

  2295 "-------- how to stepwise construct Scripts -------------";

  2296 "-------- how to stepwise construct Scripts -------------";

  2297 val thy = @{theory "Rational"};

  2298 (*no trailing _*)

  2299 val p1 = parse thy (

  2300 "Script SimplifyScript (t_t::real) =                             " ^

  2301 "  (Rewrite_Set discard_minus False                   " ^

  2302 "   t_t)");

  2304 (*required (): (Rewrite_Set discard_minus False)*)

  2305 val p2 = parse thy (

  2306 "Script SimplifyScript (t_t::real) =                             " ^

  2307 "  (Rewrite_Set discard_minus False                   " ^

  2308 "   t_t)");

  2310 val p3 = parse thy (

  2311 "Script SimplifyScript (t_t::real) =                             " ^

  2312 "  ((Rewrite_Set discard_minus False)                   " ^

  2313 "   t_t)");

  2315 val p4 = parse thy (

  2316 "Script SimplifyScript (t_t::real) =                             " ^

  2317 "  ((Rewrite_Set discard_minus False)                   " ^

  2318 "   t_t)");

  2320 val p5 = parse thy (

  2321 "Script SimplifyScript (t_t::real) =                             " ^

  2322 "  ((Try (Rewrite_Set discard_minus False)                   " ^

  2323 "    Try (Rewrite_Set discard_parentheses False))               " ^

  2324 "   t_t)");

  2326 val p6 = parse thy (

  2327 "Script SimplifyScript (t_t::real) =                             " ^

  2328 "  ((Try (Rewrite_Set discard_minus False) @@                   " ^

  2329 "    Try (Rewrite_Set rat_mult_poly False) @@                    " ^

  2330 "    Try (Rewrite_Set make_rat_poly_with_parentheses False) @@   " ^

  2331 "    Try (Rewrite_Set cancel_p_rls False) @@                     " ^

  2332 "    (Repeat                                                     " ^

  2333 "     ((Try (Rewrite_Set common_nominator_p_rls False) @@        " ^

  2334 "       Try (Rewrite_Set rat_mult_div_pow False) @@              " ^

  2335 "       Try (Rewrite_Set make_rat_poly_with_parentheses False) @@" ^

  2336 "       Try (Rewrite_Set cancel_p_rls False) @@                  " ^

  2337 "       Try (Rewrite_Set rat_reduce_1 False)))) @@               " ^

  2338 "    Try (Rewrite_Set discard_parentheses False))               " ^

  2339 "   t_t)"

  2340 );

  2342 "----------- get_denominator ----------------------------";

  2343 "----------- get_denominator ----------------------------";

  2344 "----------- get_denominator ----------------------------";

  2345 val thy = @{theory Isac};

  2346 val t = term_of (the (parse thy "get_denominator ((a +x)/b)"));

  2347 val SOME (_, t') = eval_get_denominator "" 0 t thy;

  2348 if term2str t' = "get_denominator ((a + x) / b) = b" then ()

  2349 else error "get_denominator ((a + x) / b) = b"

  2352 "--------- several errpats in complicated term -------------------";

  2353 "--------- several errpats in complicated term -------------------";

  2354 "--------- several errpats in complicated term -------------------";

  2355 (*TODO: instead of Gabriella's example here (27.Jul.12) find a simpler one*)

  2356 states:=[];

  2357 CalcTree

  2358 [(["Term ((5*b + 25)/(a^2 - b^2) * (a - b)/(5*b))", "normalform N"],

  2359   ("Rational",["rational","simplification"], ["simplification","of_rationals"]))];

  2360 Iterator 1;

  2361 moveActiveRoot 1;

  2362 autoCalculate 1 CompleteCalc;

  2363 val ((pt,p),_) = get_calc 1; show_pt pt;

  2365 "-------- nonterminating cancel_p, norm_Rational 2002 ---";

  2366 "-------- nonterminating cancel_p, norm_Rational 2002 ---";

  2367 "-------- nonterminating cancel_p, norm_Rational 2002 ---";

  2368 (*------------------------------------------------------------------------------------\

  2369 "--- WN121204: searched rational.sml for "nonterm", added new numbering" ^

  2370 "              and thoroughly tested with this numbering subsequently";

  2371 "--- 1 ---";

  2372 WN.2.6.03 from rlang.sml 56a

  2373 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)";

  2374 val NONE = rewrite_set_ thy false common_nominator_p t;

  2375 "--- 2 ---";

  2376 WN060831 nonterm.SK7

  2377 val t = str2term "(a + b * x) / (a + -1 * (b * x)) + (-1 * a + b * x) / (a + b * x)";

  2378 val NONE = add_fraction_p_ thy t;

  2379 "--- 3 ---";

  2380 nonterm.SK9 loops: cancel_p kann nicht weiter kuerzen!!! *)

  2381 val t'' = str2term "(a + b)/(x^^^2 - y^^^2) * ((x - y)^^^2/(a^^^2 - b^^^2))";

  2382 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t'';

  2383 "--- 4 ---";

  2384 val t = str2term

  2385 "(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)";

  2386 WN060831 nonterm.SK10

  2387 val SOME (t,_) = rewrite_set_ thy false cancel_p t;

  2388 term2str t;

  2389 "--- 5 ---";

  2390 val t = str2term

  2391 "(9*(x^^^2 - 8*x+16)/(4*(y^^^2 - 2*y+1)))/((3*x - 12)/(16*y - 16))";

  2392 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  2393 if term2str t =

  2394 then ()

  2395 else error "rational.sml: diff.behav. in norm_Rational_mg 42";

  2396 "--- 6 ---";

  2397 val t = str2term

  2398 "9*(x^^^2 - 8*x+16)*(16*y - 16)/(4*(y^^^2 - 2*y+1)*(3*x - 12))";

  2399 val SOME (t',_) = rewrite_set_ thy false norm_Rational t;

  2400 ... non terminating.

  2401 "--- 7 ---";

  2402 val SOME (t',_) = rewrite_set_ thy false make_polynomial t;

  2403 "(-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)";

  2404 val SOME (t,_) = rewrite_set_ thy false cancel_p t';

  2405 "--- 8 ---";

  2406 val t = str2term "(a-b)^^^3 * (x+y)^^^4 / ((x+y)^^^2 * (a-b)^^^5)";

  2407 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  2408 "--- 9 ---";

  2409 val t = str2term "((12*x*y/(9*x^^^2 - y^^^2))/" ^

  2410 		 "(1/(3*x - y)^^^2 - 1/(3*x + y)^^^2)) *" ^

  2411 		 "(1/(x - 5*y)^^^2 - 1/(x + 5*y)^^^2)/" ^

  2412 		 "(20*x*y/(x^^^2 - 25*y^^^2))";

  2413 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  2414 "--- 10 ---";

  2415 SK060904-2a non-termination of add_fraction_p_";

  2416 val t = str2term (" (a + b * x) / (a + -1 * (b * x)) +  " ^

  2417 		  " (-1 * a + b * x) / (a + b * x)      ");

  2418 (* nonterm.SK

  2419 val SOME (t',_) = rewrite_set_ thy false common_nominator_p t;

  2420 "--- 11 ---";

  2421 common_nominator_p_ thy t;

  2422 " (a + b * x)*(a + b * x) / ((a + -1 * (b * x))*(a + -1 * (b * x))) +  " ^

  2423 " (-1 * a + b * x)*(a + -1 * (b * x)) / ((a + b * x)*(-1 * a + b * x)) ";

  2424 "--- 12 ---";

  2425 add_fraction_p_ thy t;

  2426 " ((a + b * x)*(a + b * x)  +  (-1 * a + b * x)*(a + -1 * (b * x))) /" ^

  2427 " ((a + b * x)*(-1 * a + b * x))                                     ";

  2428 --------------------------------------------------------------------------------------/*)

  2430 (*------------------------------------------------------------------------------------\

  2431 "WN121205: thoroughly tested with the above numbering.";

  2432 "  errors in cancel_p: --- 4,5,6,7.";

  2433 "  error in common_nominator_p, common_nominator_p_: --- 10; 1,2?.";

  2434 "  errors caused by ruleset norm_Rational: --- 8,9.";

  2435 trace_rewrite := true;

  2437 "--- 1 ---: non-terminating with ### add_fract: done t22 ";

  2438 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)";

  2439 trace_rewrite:= true;

  2440 rewrite_set_ thy false common_nominator_p t;

  2442 "--- 2 ---: non-terminating with ### add_fract: done t22 ";

  2443 val t = str2term "(a + b * x) / (a + -1 * (b * x)) + (-1 * a + b * x) / (a + b * x)";

  2444 add_fraction_p_ thy t;

  2446 "--- 3 ---: norm_Rational calls Rrls cancel_p with non-normalised polys";

  2447 val t = str2term "(a + b)/(x^^^2 - y^^^2) * ((x - y)^^^2/(a^^^2 - b^^^2))";

  2448 rewrite_set_ thy false norm_Rational t;

  2449 (*tracing end...####  rls: cancel_p on:

  2450 (a * x ^^^ 2 + -2 * (a * (x * y)) + a * y ^^^ 2 + b * x ^^^ 2 + -2 * (b * (x * y)) + b * y ^^^ 2) /

  2451 (a ^^^ 2 * x ^^^ 2 + -1 * (a ^^^ 2 * y ^^^ 2) + -1 * (b ^^^ 2 * x ^^^ 2) + b ^^^ 2 * y ^^^ 2) *)

  2453 "--- 4 ---: non-terminating with Rrls cancel_p on plausible input";

  2454 val t = str2term (

  2455 "(a * x ^^^ 2 + -2 * a * x * y + a * y ^^^ 2 + b * x ^^^ 2 + -2 * b * x * y + b * y ^^^ 2) /"^

  2456 "(a ^^^ 2 * x ^^^ 2 + -1 * a ^^^ 2 * y ^^^ 2 + -1 * b ^^^ 2 * x ^^^ 2 + b ^^^ 2 * y ^^^ 2)");

  2457 rewrite_set_ thy false cancel_p t;

  2458 (*#  rls: cancel_p on:

  2459 (a * x ^^^ 2 + -2 * a * x * y + a * y ^^^ 2 + b * x ^^^ 2 + -2 * b * x * y + b * y ^^^ 2) /

  2460 (a ^^^ 2 * x ^^^ 2 + -1 * a ^^^ 2 * y ^^^ 2 + -1 * b ^^^ 2 * x ^^^ 2 + b ^^^ 2 * y ^^^ 2) *)

  2462 "--- 5 ---: non-terminating with Rrls cancel_p on plausible input";

  2463 val t = str2term "(9*(x^^^2 - 8*x+16)/(4*(y^^^2 - 2*y+1)))/((3*x - 12)/(16*y - 16))";

  2464 rewrite_set_ thy false norm_Rational t;

  2465 (*####  rls: cancel_p on:

  2466 (2304 + -1152 * x + -2304 * y + 144 * x ^^^ 2 + 1152 * (x * y) + -144 * (x ^^^ 2 * y)) /

  2467 (48 + -12 * x + -96 * y + 24 * (x * y) + 48 * y ^^^ 2 + -12 * (x * y ^^^ 2))  *)

  2469 "--- 6 ---: non-terminating with Rrls cancel_p on plausible input";

  2470 val t = str2term

  2471 "9*(x^^^2 - 8*x+16)*(16*y - 16)/(4*(y^^^2 - 2*y+1)*(3*x - 12))";

  2472 rewrite_set_ thy false norm_Rational t;

  2473 (*###  rls: cancel_p on: (-2304 + 1152 * x + 2304 * y + -144 * x ^^^ 2 + -1152 * (x * y) +

  2474  144 * (x ^^^ 2 * y)) /

  2475 (-48 + 12 * x + 96 * y + -24 * (x * y) + -48 * y ^^^ 2 + 12 * (x * y ^^^ 2)) *)

  2477 "--- 7 ---: non-terminating with Rrls cancel_p on plausible input";

  2478 val t' = str2term (

  2479 "(-2304 + 1152 * x + 2304 * y + -144 * x ^^^ 2 + -1152 * x * y + 144 * x ^^^ 2 * y) /"^

  2480 "(-48 + 12 * x + 96 * y + -24 * x * y + -48 * y ^^^ 2 + 12 * x * y ^^^ 2)");

  2481 rewrite_set_ thy false cancel_p t';

  2483 "--- 8 ---: bottom of output cannot be looked ad (due to looping?)";

  2484 val t = str2term "(a-b)^^^3 * (x+y)^^^4 / ((x+y)^^^2 * (a-b)^^^5)";

  2485 val SOME (t,_) = rewrite_set_ thy false norm_Rational t;

  2487 "--- 9 ---: probably error in norm_Rational";

  2488 val t = str2term (

  2489 "((12*x*y / (9*x^^^2 - y^^^2)) / (1/(3*x - y)^^^2 - 1/(3*x + y)^^^2)) *" ^

  2490 		"(1/(x - 5*y)^^^2 - 1/(x + 5*y)^^^2) / (20*x*y/(x^^^2 - 25*y^^^2))");

  2491 rewrite_set_ thy false norm_Rational t;

  2492 (*####  rls: cancel_p on: (19440 * (x ^^^ 8 * y ^^^ 2) + -490320 * (x ^^^ 6 * y ^^^ 4) +

  2493  108240 * (x ^^^ 4 * y ^^^ 6) +

  2494  -6000 * (x ^^^ 2 * y ^^^ 8)) /

  2495 (2160 * (x ^^^ 8 * y ^^^ 2) + -108240 * (x ^^^ 6 * y ^^^ 4) +

  2496  1362000 * (x ^^^ 4 * y ^^^ 6) +

  2497  -150000 * (x ^^^ 2 * y ^^^ 8)) *)

  2499 "--- 10 ---: non-terminating with ### add_fract: done t22: error in common_nominator_p ";

  2500 val t = str2term (" (a + b * x) / (a + -1 * (b * x)) + (-1 * a + b * x) / (a + b * x)");

  2501 rewrite_set_ thy false common_nominator_p t; (*### add_fract: done t22 *)

  2502 common_nominator_p_ thy t;                   (*loops without output*)

  2503 "--- reformulated 10:";

  2504 val t = str2term "(a + -1 * (b * x)) / (a + b * x)";

  2505 rewrite_set_ thy false cancel_p t = NONE;

  2506 "--- 11 ---";

  2507 "--- 12 ---";

  2508 "...both are to be considered after common_nominator_p_ is improved";

  2509 --------------------------------------------------------------------------------------/*)

  2510 ============ inhibit exn WN130824 TODO ======================================================*)

author	Walther Neuper <neuper@ist.tugraz.at>
	Mon, 02 Sep 2013 15:17:34 +0200
changeset 52100	0831a4a6ec8a
parent 52096	ee2a5f066e44
child 52101	c3f399ce32af
permissions	-rw-r--r--