wneuper/isa: test/Tools/isac/Knowledge/rational.sml@ee2a5f066e44

     1 (* Title: tests for rationals

     2    Author: Stefan Karnel

     3    Copyright (c) Stefan Karnel 2002

     4    Use is subject to license terms.

     6 12345678901234567890123456789012345678901234567890123456789012345678901234567890

     7         10        20        30        40        50        60        70        80

     8 LEGEND

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

    10 WN070906

    11    nonterm.SK   marks non-terminating examples

    12    ord.SK       PARTIALLY marks crucial ordering examples

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

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

    15 *)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

    75 (*poly_of_term is quite liberal:*)

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

    77   within a monomial are arbitrary*)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   101 val expT = HOLogic.realT

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

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

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

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

   106 ;

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

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

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

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

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

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

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

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

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

   117 ;

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

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

   120 ;

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

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

   123 ;

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

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

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

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

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

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

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

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

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

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

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

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

   137 ;

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

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

   140 ;

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

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

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

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

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

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

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

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

   150               (* n1    d1                   *)

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

   152               (* n2    d2                   *)

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

   154 if term2str t' =

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

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

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

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

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

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

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

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

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

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

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

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

   168             " +\n" ^

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

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

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

   172 ;

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

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

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

   177 val t = str2term "nothing / to_add";

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

   179 ;

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

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

   182 if term2str t' =

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

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

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

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

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

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

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

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

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

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

   194 ;

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

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

   197 ;

   198 val t = str2term "nothing / to_add";

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

   200 ;

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

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

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

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

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

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

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

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

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

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

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

   216 (* trace down until prepats are evaluated

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

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

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

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

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

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

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

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

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

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

   228     fun chk_prepat thy erls [] t = true

   229       | chk_prepat thy erls prepat t =

   230         let

   231           fun chk (pres, pat) =

   232             (let

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

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

   235              in

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

   237              end) handle _ => false

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

   239              | scan_ f (pp::pps) =

   240                if f pp then true else scan_ f pps;

   241         in scan_ chk prepat end;

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

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

   244       if chk_prepat thy erls prepat t

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

   246       else NONE;

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

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

   249   (thy, rrls, t);

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

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

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

   253           fun chk (pres, pat) =

   254             (let

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

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

   257              in

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

   259              end) handle _ => false

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

   261              | scan_ f (pp::pps) =

   262                if f pp then true else scan_ f pps;

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

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

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

   267 f;

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

   269 term2str t1 = "?r is_expanded";

   270 term2str t2 = "?s is_expanded";

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

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

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

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

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

   276 (*subst =

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

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

   279 ;

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

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

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

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

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

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

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

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

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

   289 rewrite__set_ thy (i+1) false;

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   319 if term2str t' =

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

   321   andalso terms2str asm = "[]"

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

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

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

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

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

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

   329 if term2str t' =

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

   331   andalso terms2str asm = "[]"

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

   334 val t = str2term

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

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

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

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

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

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

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

   342 ============ inhibit exn WN130826 TODO ========================================================*)

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

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

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

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

   349 "-------- WN";

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

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

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

   354 "-------- example 186a";

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

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

   357   is_expanded (str2term "x * y");

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

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

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

   362 "-------- example 186b";

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

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

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

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

   368 "-------- example 186c";

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

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

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

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

   374 (* !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! exception Div raised !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

   375   see --- fun rewrite_set_ downto fun gcd_poly ---

   376 "-------- example 187a";

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

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

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

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

   381 *)

   383 (* doesn't terminate !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

   384   see --- fun rewrite_set_ downto fun gcd_poly ---

   385 "-------- example 187b";

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

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

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

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

   390 *)

   392 (* doesn't terminate !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

   393   see --- fun rewrite_set_ downto fun gcd_poly ---

   394 "-------- example 187c";

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

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

   397 if (term2str t', terms2str asm) =

   398   ("3 * z ^^^ 3 / (5 * (y * x))", "[\"3 * (z * (y ^^^ 2 * x ^^^ 5)) ~= 0\"]")

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

   400 *)

   402 "-------- example 188a";

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

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

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

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

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

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

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

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

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

   414 "-------- example 188b";

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

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

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

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

   420 "-------- example 188c";

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

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

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

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

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

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

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

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

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

   432 "-------- example 190a";

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

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

   435 if (term2str t', terms2str asm) =

   436   ("(1 + 9 * a ^^^ 2) / (3 * a + 9 * a ^^^ 2)", "[\"3 * a + 9 * a ^^^ 2 ~= 0\"]")

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

   439 "-------- example 190c";

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

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

   442 if (term2str t', terms2str asm) =

   443   ("(1 + 3 * a + 9 * a ^^^ 2 + 27 * a ^^^ 3) /\n(3 * a + 18 * a ^^^ 2 + 27 * a ^^^ 3)", "[]")

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

   446 "-------- example 191a";

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

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

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

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

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

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

   454 "-------- example 191b";

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

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

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

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

   460 "-------- example 191c";

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

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

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

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

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

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

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

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

   471 "-------- example 192b";

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

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

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

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

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

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

   479 if (term2str t', terms2str asm) =

   480   ("(7 * x ^^^ 3 + -1 * x ^^^ 2 * y) / (7 * x * y ^^^ 2 + -1 * y ^^^ 3)", "[]")

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

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

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

   485 if (term2str t', terms2str asm) =

   486   ("(7 * x ^^^ 3 + -1 * x ^^^ 2 * y) / (7 * x * y ^^^ 2 + -1 * y ^^^ 3)", "[]")

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

   489 "-------- example 193a";

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

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

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

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

   495 "-------- example 193b";

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

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

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

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

   501 "-------- example 193c";

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

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

   504 if (term2str t', terms2str asm) =

   505   ("(2 * x + 10 * x ^^^ 2) / (1 + -5 * x)", "[\"1 + -5 * x ~= 0\"]")

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

   508 (*WN:*)

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

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

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

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

   514 (*========== inhibit exn WN130826 TODO =========================================================

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   540 if term2str t' =

   541   "(-1 + x) * (-1 + x) / ((1 + x) * (-1 + x)) +\n(1 + x) * (1 + x) / ((1 + x) * (-1 + x))"

   542   andalso terms2str asm = "[]"

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

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

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

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

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

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

   550 if term2str t' =

   551   "(-1 + x) * (-1 + x) / ((1 + x) * (-1 + x)) +\n(1 + x) * (1 + x) / ((1 + x) * (-1 + x))"

   552   andalso terms2str asm = "[]"

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

   555 val t = str2term

   556   "((1) / (2*x + 2)) + ((1) / (2*x + (-2))) + ((1) / ( x^^^2 + (-1)))+((1) / (x^^^2 + (-2)*x + 1))";

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

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

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

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

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

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

   565 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))";

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

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

   568 (term2str t3');

   569 (term2str t3'');

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

   572 term2str t4;

   573 term2str (hd(t5));

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

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

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

   580   term2str t';

   581   term2str t'';

   582   "(3 + x) * (3 - x) / ((3 - x) * (3 - x))";

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

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

   586 	    "(9 - x^^^2) / (9 - 6*x + x^^^2) + 1 / (3 - x)";

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

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

   589   term2str t';

   590   term2str t'';

   591   "(9 - x ^^^ 2) / ((3 - x) * (3 - x)) + 1 * (3 - x) / ((3 - x) * (3 - x))";

   592   "(4 + x) / (3 - x)";

   594 (*WN021016 added -----vv---*)

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

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

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

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

   599 		" * (9 - 6 * x + x ^^^ 2) / (9 - 6 * x + x ^^^ 2)" (*true*);

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

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

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

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

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

   606 		"(9 - x ^^^ 2) / (9 - 6 * x + x ^^^ 2)" (*true*);

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

   608 (*WN021016 added -----^^---*)

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

   610 val t = str2term "1 / a";

   611 val NONE =  cancel_p_ thy t;

   612 val NONE = rewrite_set_ thy false cancel_p t;

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

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

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

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

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

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

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

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

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

   624 term2str t' =

   625 "(-1 * x ^^^ 2 + y ^^^ 2) / ((-1 * x + y) * (-1 * x + y)) +\n1" ^

   626 " * (-1 * x + y) / ((-1 * x + y) * (-1 * x + y))" (*true*);

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   645  *** RATIONALS_TERM2POLY_EXCEPTION: Invalid Polynomial

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

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

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

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

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

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

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

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

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

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

   659 	    "(9 + (-1)* x^^^2)/(9 + (-1)* 6*x + x^^^2) + (1)/(3 + x)";

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

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

   662   term2str t';

   663   term2str t'';

   664   "(12 + 5 * x + x ^^^ 2) / (9 - x ^^^ 2)";

   665   "(12 + 5 * x + x ^^^ 2) / (9 + (-1) * x ^^^ 2)";

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

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

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

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

   672 val thy' = "Rational";

   673 val rls' = "cancel";

   674 val mp = "make_polynomial";

   676 writeln ("example 186:");

   677 writeln("a)");

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

   679 rewrite_set_ thy false cancel (str2term e186a');

   680 "@@@@@@@@@@@@@@";

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

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

   683   is_expanded (parse_rat "x * y");

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

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

   687 writeln("b)");

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

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

   690 writeln("c)");

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

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

   693     handle e => print_exn_G e;

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

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

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

   698 writeln ("example 187:");

   699 writeln("a)");

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

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

   702 writeln("b)");

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

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

   705 writeln("c)");

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

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

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

   709             "[\"3 * (z * (y ^^^ 2 * x ^^^ 5)) ~= 0\"]") then ()

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

   712 writeln "example 188:";

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

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

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

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

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

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

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

   721 writeln("b)");

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

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

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

   725 writeln("c)");

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

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

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

   730 val SOME (t,_) =

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

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

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

   735 writeln ("example 190:");

   736 writeln("c)");

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

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

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

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

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

   743 writeln ("example 191:");

   744 writeln("a)");

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

   746 (*WN.23.10.02-------

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

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

   749   false;

   750   is_expanded (parse_rat "x + y");

   751   true; -----------*)

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

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

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

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

   757 writeln("c)");

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

   759 (*WN.23.10.02-------

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

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

   762   false;

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

   764   false;

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

   766   true;------------*)

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

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

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

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

   773 writeln ("example 192:");

   774 writeln("b)");

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

   776 (*WN.23.10.02-------

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

   778 -------------------*)

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

   780 (*========== inhibit exn 110317 ================================================

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

   782 (*"(-1 * y * x ^^^ 2 + 7 * x ^^^ 3) / (-1 * y ^^^ 3 + 7 * x * y ^^^ 2)"WN050929*)

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

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

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

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

   787 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";

   788 ============ inhibit exn 110317 ==============================================*)

   790 writeln ("example 193:");

   791 writeln("a)");

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

   793 (*WN.23.10.02-------

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

   795 -------------------*)

   796 writeln("b)");

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

   798 (*WN.23.10.02-------

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

   800 writeln("c)");

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

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

   803 -------------------*)

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

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

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

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

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

   811 "-------- common_nominator_p ----------------------------";

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

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

   814 val rls' = "common_nominator_p";

   816 writeln ("example 204:");

   817 writeln("a)");

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

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

   820 writeln("b)");

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

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

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

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

   826 writeln ("example 205:");

   827 writeln("a)");

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

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

   830 writeln("b)");

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

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

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

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

   836 writeln ("example 206:");

   837 writeln("a)");

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

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

   840 writeln("b)");

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

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

   844 writeln ("example 207:");

   845 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)) ";

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

   848 writeln ("example 208:");

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

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

   852 writeln ("example 209:");

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

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

   856 writeln ("example 210:");

   857 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)) ";

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

   860 writeln ("example 211:");

   861 writeln("a)");

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

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

   864 writeln("b)");

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

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

   868 writeln ("example 212:");

   869 writeln("a)");

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

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

   872 writeln("b)");

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

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

   876 writeln ("example 213:");

   877 writeln("a)");

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

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

   880 writeln("b)");

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

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

   884 writeln ("example 214:");

   885 writeln("a)");

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

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

   888 writeln("b)");

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

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

   892 writeln ("example 216:");

   893 writeln("a)");

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

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

   896 writeln("b)");

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

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

   900 writeln ("example 217:");

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

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

   905 val rls' = "common_nominator";

   906 writeln ("example 218:");

   907 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))";

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

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

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

   912 writeln ("example 219:");

   913 writeln("a)");

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

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

   916 writeln("b)");

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

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

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

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

   922 writeln ("example 220:");

   923 writeln("a)");

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

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

   926 writeln("b)");

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

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

   930 writeln ("example 221:");

   931 writeln("a)");

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

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

   934 writeln("b)");

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

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

   938 writeln ("example 222:");

   939 writeln("a)");

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

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

   942 writeln("b)");

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

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

   946 writeln ("example 225:");

   947 writeln("a)");

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

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

   950 writeln("b)");

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

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

   954 writeln ("example 226:");

   955 writeln("a)");

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

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

   958 writeln("b)");

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

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

   962 writeln ("example 227:");

   963 writeln("a)");

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

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

   966 writeln("b)");

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

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

   970 writeln ("example 228:");

   971 writeln("a)");

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

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

   974 writeln("b)");

   975 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))";

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

   979 writeln ("example 229:");

   980 writeln("a)");

   981 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))";

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

   983 writeln("b)");

   984 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))";

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

   987 writeln ("example 230:");

   988 writeln("a)");

   989 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))";

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

   991 writeln("b)");

   992 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))";

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

   995 writeln ("example 231:");

   996 writeln("a)");

   997 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))";

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

   999 writeln("b)");

  1000 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))";

  1001 val e231b = the (rewrite_set thy' false "common_nominator" e231b');

  1003 writeln ("example 232:");

  1004 writeln("a)");

  1005 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))";

  1006 val e232a = the (rewrite_set thy' false "common_nominator" e232a');

  1007 writeln("b)");

  1008 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))";

  1009 val e232b = the (rewrite_set thy' false "common_nominator" e232b');

  1011 writeln ("example 233:");

  1012 writeln("a)");

  1013 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))";

  1014 val e233a = the (rewrite_set thy' false "common_nominator" e233a');

  1015 writeln("b)");

  1016 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))";

  1017 val e233b = the (rewrite_set thy' false "common_nominator" e233b');

  1019 writeln ("example 234:");

  1020 writeln("a)");

  1021 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))";

  1022 val e234a = the (rewrite_set thy' false "common_nominator" e234a');

  1023 writeln("b)");

  1024 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)) ";

  1025 val e234b = the (rewrite_set thy' false "common_nominator" e234b');

  1027 writeln ("example 235:");

  1028 writeln("a)");

  1029 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))";

  1030 val e235a = the (rewrite_set thy' false "common_nominator" e235a');

  1031 writeln("b)");

  1032 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)) ";

  1033 val e235b = the (rewrite_set thy' false "common_nominator" e235b');

  1035 writeln ("example 236:");

  1036 writeln("a)");

  1037 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))";

  1038 val e236a = the (rewrite_set thy' false "common_nominator" e236a');

  1039 writeln("b)");

  1040 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)) ";

  1041 val e236b = the (rewrite_set thy' false "common_nominator" e236b');

  1044 val rls' = "cancel";

  1045 writeln ("example heuberger:");

  1046 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)";

  1047 val eheu = the (rewrite_set thy' false "cancel" eheu');

  1049 val rls' = "common_nominator_p";

  1050 writeln ("example stiefel:");

  1051 val est1'="(7) / (-14) + (-2) / (4)";

  1052 val est1 = the (rewrite_set thy' false "common_nominator_p" est1');

  1053 if est1 = ("-1 / 1", "[]") then ()

  1054 else error "new behaviour in rational.sml: est1'";

  1056 val t = (term_of o the o (parse thy))

  1057 "(9 - x ^^^ 2) / (9 - 6 * x + x ^^^ 2)";

  1058 val SOME (t',_) = factout_ thy t;

  1059 if term2str t' = "(3 + x) * (3 - x) / ((3 - x) * (3 - x))" then ()

  1060 else error "rational.sml factout_ (9 - x ^^^ 2) / (9 - 6 * x + x ^^^ 2)";

  1063 "-------- reverse rewrite -------------------------------";

  1064 "-------- reverse rewrite -------------------------------";

  1065 "-------- reverse rewrite -------------------------------";

  1066 (*WN.28.8.02: tests for the 'reverse-rewrite' functions:

  1067   these are defined in Rationals.ML and stored in

  1068   the 'reverse-ruleset' cancel*)

  1070 (*the term for which reverse rewriting is demonstrated*)

  1071   val t = (term_of o the o (parse thy))

  1072 	      "(9 - x ^^^ 2) / (9 + 6 * x + x ^^^ 2)";

  1073   val Rrls {scr=Rfuns {init_state=ini,locate_rule=loc,

  1074   		       next_rule=nex,normal_form=nor,...},...} = cancel;

  1076 (*normal_form produces the result in ONE step*)

  1077   val SOME (t',_) = nor t;

  1078 if term2str t' = "(3 - x) / (3 + x)" then ()

  1079 else error "rational.sml normal_form (9 - x ^^^ 2) / (9 - 6 * x + x ^^^ 2)";

  1081 (*initialize the interpreter state used by the 'me'*)

  1082   val (t,_,revsets,_) = ini t;

  1084 (*find the rule 'r' to apply to term 't'*)

  1085   val SOME (r as (Thm (str, thm))) = nex revsets t;

  1086 if str = "sym_#power_Float ((3,0), (0,0)) __ ((2,0), (0,0))"

  1087    andalso string_of_thm thm =

  1088            (string_of_thm o make_thm o (cterm_of (@{theory "Isac"})))

  1089                (Trueprop $ (term_of o the o (parse thy)) "9 = 3 ^^^ 2") then ()

  1090 else error "rational.sml next_rule (9 - x ^^^ 2) / (9 - 6 * x + x ^^^ 2)";

  1091 (* before Isa02->09-2 was not checked automatically, ?was different?:

  1092 val SOME r = nex revsets t;

  1093 val r = Thm ("sym_#mult_2_3","6 = 2 * 3") : rule*)

  1095 (* check, if the rule 'r' applied by the user to 't' belongs to the ruleset;

  1096   if the rule is OK, the term resulting from applying the rule is returned,too;

  1097   there might be several rule applications inbetween,

  1098   which are listed after the head in reverse order *)

  1099   val (r, (t, asm))::_ = loc revsets t r;

  1100 if term2str t = "(9 - x ^^^ 2) / (3 ^^^ 2 + 6 * x + x ^^^ 2)" andalso asm = []

  1101 then ()

  1102 else error "rational.sml locate_rule (9 - x ^^^ 2) / (9 - 6 * x + x ^^^ 2)";

  1104 (* find the next rule to apply *)

  1105   val SOME (r as (Thm (str, thm))) = nex revsets t;

  1106 if str = "sym_#power_Float ((3,0), (0,0)) __ ((2,0), (0,0))" andalso

  1107    string_of_thm thm = (string_of_thm o make_thm o (cterm_of (@{theory "Isac"})))

  1108                 (Trueprop $ (term_of o the o (parse thy)) "9 = 3 ^^^ 2") then ()

  1109 else error "rational.sml next_rule (9 - x ^^^ 2) / (9 - 6 * x + x ^^^ 2)";

  1111 (*check the next rule*)

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

  1113 if term2str t = "(3 ^^^ 2 - x ^^^ 2) / (3 ^^^ 2 + 6 * x + x ^^^ 2)" then ()

  1114 else error "rational.sml locate_rule (9 - x ^^^ 2) / (9 - 6 * x + x ^^^ 2) II";

  1116 (*find and check the next rules, rewrite*)

  1117   val SOME r = nex revsets t;

  1118   val (r,(t,asm))::_ = loc revsets t r;

  1119 if term2str t = "(3 ^^^ 2 - x ^^^ 2) / (3 ^^^ 2 + 2 * 3 * x + x ^^^ 2)" then ()

  1120 else error "rational.sml locate_rule II";

  1122   val SOME r = nex revsets t;

  1123   val (r,(t,asm))::_ = loc revsets t r;

  1124 if term2str t = "(3 - x) * (3 + x) / (3 ^^^ 2 + 2 * 3 * x + x ^^^ 2)" then ()

  1125 else error "rational.sml next_rule II";

  1127   val SOME r = nex revsets t;

  1128   val (r,(t,asm))::_ = loc revsets t r;

  1129 if term2str t = "(3 - x) * (3 + x) / ((3 + x) * (3 + x))" then ()

  1130 else error "rational.sml next_rule III";

  1132   val SOME r = nex revsets t;

  1133   val (r, (t, asm)) :: _ = loc revsets t r;

  1134   val ss = term2str t;

  1135 if ss = "(3 - x) / (3 + x)" andalso terms2str asm = "[\"3 + x ~= 0\"]" then ()

  1136 else error "rational.sml: new behav. in rev-set cancel";

  1138 "-------- 'reverse-ruleset' cancel_p --------------------";

  1139 "-------- 'reverse-ruleset' cancel_p --------------------";

  1140 "-------- 'reverse-ruleset' cancel_p --------------------";

  1141 (*WN.11.9.02: the 'reverse-ruleset' cancel_p*)

  1143 (*the term for which reverse rewriting is demonstrated*)

  1144 val t = str2term "(9 + (-1)*x^^^2) / (9 + ((-6)*x + x^^^2))";

  1145 val Rrls {scr=Rfuns {init_state=ini,locate_rule=loc,

  1146 		       next_rule=nex,normal_form=nor,...},...} = cancel_p;

  1148 (*normal_form produces the result in ONE step*)

  1149 val SOME (t',_) = nor t;

  1150 term2str t' = "(3 + 1 * x) / (3 + -1 * x)";

  1152 (*initialize the interpreter state used by the 'me'*)

  1153 val SOME (t', asm) = cancel_p_ thy t;

  1154 term2str t' = "(3 + x) / (3 + -1 * x)" (*true*);

  1155 terms2str asm = "[\"3 + -1 * x ~= 0\"]" (*true*);

  1156 val (t,_,revsets,_) = ini t;

  1158 (* WN.10.10.02: dieser Fall terminiert nicht

  1159            (make_polynomial enth"alt zu viele rules)

  1160 WN060823 'init_state' requires rewriting on specified location in the term

  1161 print_depth 99; Rfuns; print_depth 3;

  1162 WN060831 cycling "sym_order_mult_rls_" "sym_mult_assoc"

  1163          as was with make_polynomial before ?!?*)

  1165 val SOME r = nex revsets t;

  1166 eq_Thm (r, Thm ("sym_#power_Float ((3,0), (0,0)) __ ((2,0), (0,0))",

  1167 		mk_thm thy "9 = 3 ^^^ 2"));

  1168 (*WN060831 *** id_of_thm

  1169            Exception- ERROR raised ...

  1170 val (r,(t,asm))::_ = loc revsets t r;

  1171 term2str t;

  1173   val SOME r = nex revsets t;

  1174   val (r,(t,asm))::_ = loc revsets t r;

  1175   term2str t;

  1176 *)

  1178 writeln "******************  all tests successfull  *************************";

  1182 (*WN.17.3.03 =========================================================vvv---*)

  1183 "-------- norm_Rational ---------------------------------";

  1184 "-------- norm_Rational ---------------------------------";

  1185 "-------- norm_Rational ---------------------------------";

  1186 val t = str2term "(3*x+5)/18 - x/2  - -(3*x - 2)/9 = 0";

  1187 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; term2str t';

  1188 if term2str t' = "1 / 18 = 0" then () else error "rational.sml 1";

  1190 val t = str2term "(17*x - 51)/9 - (-(13*x - 3)/6) + 11 - (9*x - 7)/4 = 0";

  1191 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; term2str t';

  1192 if term2str t' = "(237 + 65 * x) / 36 = 0" then ()

  1193 else error "rational.sml 2";

  1195 val t = str2term "(1/2 + (5*x)/2)^^^2 - ((13*x)/2 - 5/2)^^^2 - (6*x)^^^2 + 29";

  1196 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; term2str t';

  1197 if term2str t' = "23 + 35 * x + -72 * x ^^^ 2" then ()

  1198 else error "rational.sml 3";

  1200 (*trace_rewrite:=true;*)

  1201 val t = str2term "Not (6*x is_atom)";

  1202 val SOME (t',_) = rewrite_set_ thy false powers_erls t; term2str t';

  1203 "HOL.True";

  1204 val t = str2term "1 < 2";

  1205 val SOME (t',_) = rewrite_set_ thy false powers_erls t; term2str t';

  1206 "HOL.True";

  1208 val t = str2term "(6*x)^^^2";

  1209 val SOME (t',_) = rewrite_ thy dummy_ord powers_erls false

  1210 			   (num_str @{thm realpow_def_atom}) t;

  1211 if term2str t' = "6 * x * (6 * x) ^^^ (2 + -1)" then ()

  1212 else error "rational.sml powers_erls (6*x)^^^2";

  1214 val t = str2term "-1 * (-2 * (5 / 2 * (13 * x / 2)))";

  1215 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; term2str t';

  1216 if term2str t' = "65 * x / 2" then () else error "rational.sml 4";

  1218 val t = str2term "1 - ((13*x)/2 - 5/2)^^^2";

  1219 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; term2str t';

  1220 (*bef.040209: if term2str t' = "(-21 + (130 * x + -169 * x ^^^ 2)) / 4"then()*)

  1221 if term2str t' = "(-21 + 130 * x + -169 * x ^^^ 2) / 4" then ()

  1222 else error "rational.sml 5";

  1224 (*SRAM Schalk I, p.92 Nr. 609a*)

  1225 val t = str2term "2*(3 - x/5)/3 - 4*(1 - x/3) - x/3 - 2*(x/2 - 1/4)/27 +5/54";

  1226 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; term2str t';

  1227 if term2str t' = "(-255 + 112 * x) / 135" then ()

  1228 else error "rational.sml 6";

  1230 (*SRAM Schalk I, p.92 Nr. 610c*)

  1231 val t = str2term "((x- 1)/(x+1) + 1) / ((x- 1)/(x+1) - (x+1)/(x- 1)) - 2";

  1232 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; term2str t';

  1233 if term2str t' = "(-3 + -1 * x) / 2" then () else error "rational.sml 7";

  1235 (*SRAM Schalk I, p.92 Nr. 476a*)

  1236 val t = str2term "(x^^^2/(1 - x^^^2) + 1)/(x/(1 - x) + 1) * (1 + x)";

  1237 (*. a/b : c/d translated to a/b * d/c .*)

  1238 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; term2str t';

  1239 (*if term2str t' = "1 / 1" then () else error "rational.sml 8";3.6.03*)

  1240 if term2str t' = "1" then () else error "rational.sml 8";

  1242 (*............................vvv---TODO: sollte gehen mit poly_order *)

  1243 (*Schalk I, p.92 Nr. 472a*)

  1244 val t = str2term "((8*x^^^2 - 32*y^^^2)/(2*x + 4*y))/((4*x - 8*y)/(x + y))";

  1245 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; term2str t';

  1246 if term2str t' = "x + y" then () else error "rational.sml p.92 Nr. 472a";

  1248 (*Schalk I, p.70 Nr. 480b; a/b : c/d translated to a/b * d/c*)

  1249 val t = str2term ("((12*x*y/(9*x^^^2 - y^^^2))/" ^

  1250 		 "(1/(3*x - y)^^^2 - 1/(3*x + y)^^^2)) *" ^

  1251 		 "(1/(x - 5*y)^^^2 - 1/(x + 5*y)^^^2)/" ^

  1252 		 "(20*x*y/(x^^^2 - 25*y^^^2))");

  1253 (*... nicht simpl, zerlegt ...*)

  1254 val t = str2term ("((12*x*y/(9*x^^^2 - y^^^2))/" ^

  1255 		 "(1/(3*x - y)^^^2 - 1/(3*x + y)^^^2))");

  1256 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; term2str t';

  1257 "(-12 * (x * y ^^^ 3) + 108 * (x * (y * x ^^^ 2))) / (12 * (x * y))";

  1258 (*                             ~~~~~~~~~~ poly_order notwendig!*)

  1259 val t = str2term ("(1/(x - 5*y)^^^2 - 1/(x + 5*y)^^^2)/" ^

  1260 		 "(20*x*y/(x^^^2 - 25*y^^^2))");

  1261 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; term2str t';

  1262 if term2str t' = "1 / (x ^^^ 2 + -25 * y ^^^ 2)" then ()

  1263 else error "rational.sml norm_Rational 1 / (x ^^^ 2 + -25 * y ^^^ 2)";

  1265 "nonterm.SK6 ----- SK060904-2a non-termination of add_fraction_p_";

  1266 (*WN.2.6.03 from rlang.sml 56a

  1267 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)";

  1268 val NONE = rewrite_set_ thy false common_nominator_p t;

  1270 WN060831 nonterm.SK7

  1271 val t = str2term "(a + b * x) / (a + -1 * (b * x)) + (-1 * a + b * x) / (a + b * x)";

  1272 val NONE = add_fraction_p_ thy t;

  1273 *)

  1276 (* ------------------------------------------------------------------- *)

  1277 (*---------vvv------------ MG: ab 1.7.03 ----------------vvv-----------*)

  1278 (*                 Simplifier fuer beliebige Buchterme                 *)

  1279 (* ------------------------------------------------------------------- *)

  1280 (*----------------------- norm_Rational_mg ----------------------------*)

  1281 (* ------------------------------------------------------------------- *)

  1283 "-------- numeral rationals -----------------------------";

  1284 "-------- numeral rationals -----------------------------";

  1285 "-------- numeral rationals -----------------------------";

  1286 (*SRA Schalk I, p.40 Nr. 164b *)

  1287 val t = str2term "(47/6 - 76/9 + 13/4)/(35/12)";

  1288 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1289 term2str t;

  1290 if (term2str t) = "19 / 21" then ()

  1291 else error "rational.sml: diff.behav. in norm_Rational_mg 1";

  1293 (*SRA Schalk I, p.40 Nr. 166a *)

  1294 val t = str2term "((5/4)/(4+22/7) + 37/20)*(110/3 - 110/9 * 23/11)";

  1295 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1296 term2str t;

  1297 if (term2str t) = "45 / 2" then ()

  1298 else error "rational.sml: diff.behav. in norm_Rational_mg 2";

  1301 "-------- cancellation ----------------------------------";

  1302 "-------- cancellation ----------------------------------";

  1303 "-------- cancellation ----------------------------------";

  1304 (* e190c Stefan K.*)

  1305 val t = str2term

  1306 "((1 + 9 * a ^^^ 2)*(1 + 3 * a))/((3 * a + 9 * a ^^^ 2)*(1 + 3 * a))";

  1307 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1308 term2str t;

  1309 if (term2str t) =

  1310 "(1 + 9 * a ^^^ 2) / (3 * a + 9 * a ^^^ 2)"

  1311 then ()

  1312 else error "rational.sml: diff.behav. in norm_Rational_mg 3";

  1314 (* e192b Stefan K.*)

  1315 val t = str2term

  1316 "((x ^^^ 2)*(7 * x + (-1) * y))/((y ^^^ 2)*(7 * x + (-1) * y))";

  1317 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1318 term2str t;

  1319 if (term2str t) =

  1320 "x ^^^ 2 / y ^^^ 2"

  1321 then ()

  1322 else error "rational.sml: diff.behav. in norm_Rational_mg 4";

  1324 (*SRC Schalk I, p.66 Nr. 379c *)

  1325 val t = str2term

  1326 "(a - b)/(b - a)";

  1327 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1328 term2str t;

  1329 if (term2str t) =

  1330 "-1"

  1331 then ()

  1332 else error "rational.sml: diff.behav. in norm_Rational_mg 5";

  1334 (*SRC Schalk I, p.66 Nr. 380b *)

  1335 val t = str2term

  1336 "15*(3*x+3)*(4*x+9)/(12*(2*x+7)*(5*x+5))";

  1337 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1338 term2str t;

  1339 if (term2str t) =

  1340 "(27 + 12 * x) / (28 + 8 * x)"

  1341 then ()

  1342 else error "rational.sml: diff.behav. in norm_Rational_mg 6";

  1344 (*Schalk I, p.60 Nr. 215c *)

  1345 (* Falsches Ergebnis: rechnet lange und cancel_p kann nicht weiter krzen!!!*)

  1346 (* WN060831????MG1

  1347 val t = str2term "(a+b)^^^4*(x - y)/((x - y)^^^3*(a+b)^^^2)";

  1348 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1349 term2str t;

  1350 if (term2str t) =

  1351 "(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)"

  1352 then ()

  1353 else error "rational.sml: diff.behav. in norm_Rational_mg 7";

  1354 *)

  1355 (*val t = str2term

  1356 "(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)"

  1357 val SOME (t,_) = rewrite_set_ thy false cancel_p t;

  1358 term2str t;*)

  1359 (* uncaught exception nonexhaustive binding failure

  1360    raised at: stdIn:93.1-93.51 *)

  1362 (*Schalk I, p.66 Nr. 381a *)

  1363 (* ACHTUNG: rechnet ca. 2 Minuten !!! *)

  1364 (* WN060831???MG2

  1365 val t = str2term "18*(a+b)^^^3*(a - b)^^^2/(72*(a - b)^^^3*(a+b)^^^2)";

  1366 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1367 term2str t;

  1368 if (term2str t) =

  1369 "(a + b) / (4 * a + -4 * b)"

  1370 then () else error "rational.sml: diff.behav. in norm_Rational_mg 8";

  1371 *)

  1373 (*SRC Schalk I, p.66 Nr. 381b *)

  1374 val t = str2term

  1375 "(4*x^^^2 - 20*x + 25)/(2*x - 5)^^^3";

  1376 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1377 term2str t;

  1378 if (term2str t) =

  1379 "-1 / (5 + -2 * x)"

  1380 then ()

  1381 else error "rational.sml: diff.behav. in norm_Rational_mg 9";

  1383 (*SRC Schalk I, p.66 Nr. 381c *)

  1384 val t = str2term

  1385 "(27*a^^^3+9*a^^^2+3*a+1)/(27*a^^^3+18*a^^^2+3*a)";

  1386 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1387 term2str t;

  1388 if (term2str t) =

  1389 "(1 + 9 * a ^^^ 2) / (3 * a + 9 * a ^^^ 2)"

  1390 then ()

  1391 else error "rational.sml: diff.behav. in norm_Rational_mg 10";

  1393 (*SRC Schalk I, p.66 Nr. 383a *)

  1394 val t = str2term

  1395 "(5*a^^^2 - 5*a*b)/(a - b)^^^2";

  1396 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1397 term2str t;

  1398 if (term2str t) =

  1399 "5 * a / (a + -1 * b)"

  1400 then ()

  1401 else error "rational.sml: diff.behav. in norm_Rational_mg 11";

  1404 "-------- common denominator ----------------------------";

  1405 "-------- common denominator ----------------------------";

  1406 "-------- common denominator ----------------------------";

  1407 (*SRA Schalk I, p.67 Nr. 403a *)

  1408 val t = str2term

  1409 "4/x - 3/y - 1";

  1410 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1411 term2str t;

  1412 if (term2str t) =

  1413 "(-3 * x + 4 * y + -1 * x * y) / (x * y)"

  1414 then ()

  1415 else error "rational.sml: diff.behav. in norm_Rational_mg 12";

  1417 (*SRA Schalk I, p.67 Nr. 407b *)

  1418 val t = str2term

  1419 "(2*a+3*b)/(b*c) + (3*c+a)/(a*c) - (2*a^^^2+3*b*c)/(a*b*c)";

  1420 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1421 term2str t;

  1422 if (term2str t) =

  1423 "4 / c"

  1424 then ()

  1425 else error "rational.sml: diff.behav. in norm_Rational_mg 13";

  1427 (*SRA Schalk I, p.67 Nr. 410b *)

  1428 val t = str2term

  1429 "1/(x+1) + 1/(x+2) - 2/(x+3)";

  1430 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1431 term2str t;

  1432 if (term2str t) =

  1433 "(5 + 3 * x) / (6 + 11 * x + 6 * x ^^^ 2 + x ^^^ 3)"

  1434 then ()

  1435 else error "rational.sml: diff.behav. in norm_Rational_mg 14";

  1437 (*SRA Schalk I, p.67 Nr. 413b *)

  1438 val t = str2term

  1439 "(1+x)/(1 - x) - (1 - x)/(1+x) + 2*x/(1 - x^^^2)";

  1440 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1441 term2str t;

  1442 if (term2str t) =

  1443 "6 * x / (1 + -1 * x ^^^ 2)"

  1444 then ()

  1445 else error "rational.sml: diff.behav. in norm_Rational_mg 15";

  1447 (*SRA Schalk I, p.68 Nr. 414a *)

  1448 val t = str2term

  1449 "(x + 2)/(x - 1) + (x - 3)/(x - 2) - (x + 1)/((x - 1)*(x - 2))";

  1450 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1451 term2str t;

  1452 if (term2str t) =

  1453 "(-2 + -5 * x + 2 * x ^^^ 2) / (2 + -3 * x + x ^^^ 2)"

  1454 then ()

  1455 else error "rational.sml: diff.behav. in norm_Rational_mg 16";

  1457 (*SRA Schalk I, p.68 Nr. 423a *)

  1458 val t = str2term

  1459 "(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)";

  1460 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1461 term2str t;

  1462 if (term2str t) =

  1463 "1"

  1464 then ()

  1465 else error "rational.sml: diff.behav. in norm_Rational_mg 17";

  1467 (*SRA Schalk I, p.68 Nr. 428b *)

  1468 val t = str2term

  1469 "1/(a - b)^^^2 + 1/(a+b)^^^2 - 2/(a^^^2 - b^^^2) - 4*(b^^^2 - 1)/(a^^^2 - b^^^2)^^^2";

  1470 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1471 term2str t;

  1472 if (term2str t) =

  1473 "4 / (a ^^^ 4 + -2 * a ^^^ 2 * b ^^^ 2 + b ^^^ 4)"

  1474 then ()

  1475 else error "rational.sml: diff.behav. in norm_Rational_mg 18";

  1477 (*SRA Schalk I, p.68 Nr. 430b *)

  1478 val t = str2term

  1479 "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";

  1480 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1481 term2str t;

  1482 if (term2str t) =

  1483 "a + 3 * b"

  1484 then ()

  1485 else error "rational.sml: diff.behav. in norm_Rational_mg 19";

  1488 (*SRA Schalk I, p.68 Nr. 432 *)

  1489 val t = str2term

  1490 "(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)";

  1491 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1492 term2str t;

  1493 if (term2str t) =

  1494 "0"

  1495 then ()

  1496 else error "rational.sml: diff.behav. in norm_Rational_mg 20";

  1498 (*Eigenes*)

  1499 val t = str2term

  1500 "3*a/(a*b) + x/y";

  1501 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1502 term2str t;

  1503 if (term2str t) =

  1504 "(3 * y + b * x) / (b * y)"

  1505 then ()

  1506 else error "rational.sml: diff.behav. in norm_Rational_mg 21";

  1509 "-------- multiply and cancel ---------------------------";

  1510 "-------- multiply and cancel ---------------------------";

  1511 "-------- multiply and cancel ---------------------------";

  1512 (*SRM Schalk I, p.68 Nr. 436a *)

  1513 val t = str2term

  1514 "3*(x+y)/(15*(x - y)) * 25*(x - y)^^^2/(18*(x+y)^^^2)";

  1515 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1516 term2str t;

  1517 if (term2str t) =

  1518 "(5 * x + -5 * y) / (18 * x + 18 * y)"

  1519 then ()

  1520 else error "rational.sml: diff.behav. in norm_Rational_mg 22";

  1522 (*SRM.test Schalk I, p.68 Nr. 436b *)

  1523 (*WN060420???MG3 crashes with method 'simplify' in

  1524   IsacCore > Simplification > Rational Terms > Multiplication > No.2*)

  1525 val t = str2term "5*a*(a - b)^^^2*(a + b)^^^3/(7*b*(a - b)^^^3) * 7*b/(a + b)^^^3";

  1526 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1527 term2str t;

  1528 if (term2str t) =

  1529 "5 * a / (a + -1 * b)"

  1530 then ()

  1531 else error "rational.sml: diff.behav. in norm_Rational_mg 23";

  1533 (*Schalk I, p.68 Nr. 437a *)

  1534 val t = str2term "(3*a - 4*b)/(4*c+3*e) * (3*a+4*b)/(9*a^^^2 - 16*b^^^2)";

  1535 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1536 if (term2str t) = "1 / (4 * c + 3 * e)" then ()

  1537 else error "rational.sml: diff.behav. in norm_Rational_mg 24";

  1539 "----- S.K. corrected non-termination 060904";

  1540 val t = str2term "(3*a - 4*b) * (3*a+4*b)/((4*c+3*e)*(9*a^^^2 - 16*b^^^2))";

  1541 val SOME (t',_) = rewrite_set_ thy false make_polynomial t;

  1542 term2str t';

  1543 if term2str t' =

  1544   "(9 * a ^^^ 2 + -16 * b ^^^ 2) /\n(36 * a ^^^ 2 * c + 27 * a ^^^ 2 * e + -64 * b ^^^ 2 * c +\n -48 * b ^^^ 2 * e)"

  1545 (*"(9 * a ^^^ 2 + -16 * b ^^^ 2) / (36 * a ^^^ 2 * c + 27 * a ^^^ 2 * e + -64 * b ^^^ 2 * c + -48 * b ^^^ 2 * e)"*)

  1546 then () else error "rational.sml: S.K.8..corrected 060904-6";

  1548 "----- S.K. corrected non-termination of cancel_p_";

  1549 val t'' = str2term ("(9 * a ^^^ 2 + -16 * b ^^^ 2) /" ^

  1550 "(36 * a^^^2 * c + (27 * a^^^2 * e + (-64 * b^^^2 * c + -48 * b^^^2 * e)))");

  1551 val SOME (t',_) = rewrite_set_ thy false cancel_p t'';

  1552 if term2str t' = "1 / (4 * c + 3 * e)" then ()

  1553 else error "rational.sml: diff.behav. in cancel_p S.K.8";

  1555 (*Schalk I, p.68 Nr. 437b *)

  1556 (* nonterm.SK9 loops: cancel_p kann nicht weiter kuerzen!!! *)

  1557 val t'' = str2term "(a + b)/(x^^^2 - y^^^2) * ((x - y)^^^2/(a^^^2 - b^^^2))";

  1558 (* val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t'';

  1559    *)

  1561 "================delete===";

  1562 (*a casual output from above*)

  1563 val t = str2term

  1564 "(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)";

  1565 (* WN060831 nonterm.SK10

  1566 val SOME (t,_) = rewrite_set_ thy false cancel_p t;

  1567 term2str t;

  1568 *)

  1570 (*SRM Schalk I, p.68 Nr. 438a *)

  1571 val t = str2term

  1572 "x*y/(x*y - y^^^2)*(x^^^2 - x*y)";

  1573 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1574 term2str t;

  1575 if (term2str t) =

  1576 "x ^^^ 2"

  1577 then ()

  1578 else error "rational.sml: diff.behav. in norm_Rational_mg 24";

  1580 (*SRM Schalk I, p.68 Nr. 439b *)

  1581 val t = str2term

  1582 "(4*x^^^2+4*x+1)*((x^^^2 - 2*x^^^3)/(4*x^^^2+2*x))";

  1583 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1584 term2str t;

  1585 if (term2str t) =

  1586 "(x + -4 * x ^^^ 3) / 2"

  1587 then ()

  1588 else error "rational.sml: diff.behav. in norm_Rational_mg 25";

  1590 (*SRM Schalk I, p.68 Nr. 440a *)

  1591 val t = str2term

  1592 "(x^^^2 - 2*x)/(x^^^2 - 3*x) * (x - 3)^^^2/(x^^^2 - 4)";

  1593 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1594 term2str t;

  1595 if (term2str t) =

  1596 "(-3 + x) / (2 + x)"

  1597 then ()

  1598 else error "rational.sml: diff.behav. in norm_Rational_mg 26";

  1600 "----- Schalk I, p.68 Nr. 440b SK11 works since 0707xx";

  1601 val t = str2term

  1602 "(a^^^3 - 9*a)/(a^^^3*b - a*b^^^3)*(a^^^2*b+a*b^^^2)/(a+3)";

  1603 val SOME (t,_) = rewrite_set_ thy false norm_Rational t;

  1604 if term2str t = "(-3 * a + a ^^^ 2) / (a + -1 * b)" then ()

  1605 else error "rational.sml: diff.behav. in norm_Rational 27";

  1607 "----- SK12 works since 0707xx";

  1608 val t = str2term "(a^^^3 - 9*a)*(a^^^2*b+a*b^^^2)/((a^^^3*b - a*b^^^3)*(a+3))";

  1609 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; term2str t';

  1610 if term2str t' = "(-3 * a + a ^^^ 2) / (a + -1 * b)" then ()

  1611 else error "rational.sml: diff.behav. in norm_Rational 28";

  1614 "-------- common denominator and multiplication ---------";

  1615 "-------- common denominator and multiplication ---------";

  1616 "-------- common denominator and multiplication ---------";

  1617 (*SRAM Schalk I, p.69 Nr. 441b *)

  1618 val t = str2term "(4*a/3 + 3*b^^^2/a^^^3 + b/(4*a))*(4*b/(3*a))";

  1619 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1620 term2str t;

  1621 if (term2str t) =

  1622   "(36 * b ^^^ 3 + 3 * a ^^^ 2 * b ^^^ 2 + 16 * a ^^^ 4 * b) / (9 * a ^^^ 4)"

  1623 then () else error "rational.sml: diff.behav. in norm_Rational_mg 28";

  1625 (*SRAM Schalk I, p.69 Nr. 442b *)

  1626 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)";

  1627 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1628 term2str t;

  1629 if (term2str t) =

  1630 "5 * x ^^^ 2 / (a * b ^^^ 3 * c)"

  1631 then () else error "rational.sml: diff.behav. in norm_Rational_mg 29";

  1633 (*SRAM Schalk I, p.69 Nr. 443b *)

  1634 val t = str2term "(a/2 + b/3)*(b/3 - a/2)";

  1635 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1636 term2str t;

  1637 if (term2str t) =

  1638 "(-9 * a ^^^ 2 + 4 * b ^^^ 2) / 36"

  1639 then () else error "rational.sml: diff.behav. in norm_Rational_mg 30";

  1641 (*SRAM Schalk I, p.69 Nr. 445b *)

  1642 val t = str2term "(a^^^2/9 + 2*a/(3*b) + 4/b^^^2)*(a/3 - 2/b) + 8/b^^^3";

  1643 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1644 term2str t;

  1645 if (term2str t) =

  1646 "a ^^^ 3 / 27"

  1647 then () else error "rational.sml: diff.behav. in norm_Rational_mg 31";

  1649 (*SRAM Schalk I, p.69 Nr. 446b *)

  1650 val t = str2term "(x/(5*x + 4*y) - y/(5*x - 4*y) + 1)*(25*x^^^2 - 16*y^^^2)";

  1651 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1652 term2str t;

  1653 if (term2str t) =

  1654 "30 * x ^^^ 2 + -9 * x * y + -20 * y ^^^ 2"

  1655 then ()

  1656 else error "rational.sml: diff.behav. in norm_Rational_mg 32";

  1658 (*SRAM Schalk I, p.69 Nr. 449a *)(*Achtung: rechnet ca 8 Sekunden*)

  1659 val t = str2term

  1660 "(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)";

  1661 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1662 term2str t;

  1663 if (term2str t) = "(-81 * x ^^^ 4 + 16 * x ^^^ 8 * y ^^^ 4) / (81 * y ^^^ 8)"

  1664 then () else error "rational.sml: diff.behav. in norm_Rational_mg 33";

  1667 (*SRAM Schalk I, p.69 Nr. 450a *)

  1668 val t = str2term

  1669 "(4*x/(3*y)+2*y/(3*x))^^^2 - (2*y/(3*x) - 2*x/y)*(2*y/(3*x)+2*x/y)";

  1670 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1671 term2str t;

  1672 if (term2str t) =

  1673 "(52 * x ^^^ 2 + 16 * y ^^^ 2) / (9 * y ^^^ 2)"

  1674 then ()

  1675 else error "rational.sml: diff.behav. in norm_Rational_mg 34";

  1678 "-------- double fractions ------------------------------";

  1679 "-------- double fractions ------------------------------";

  1680 "-------- double fractions ------------------------------";

  1681 (*SRD Schalk I, p.69 Nr. 454b *)

  1682 val t = str2term

  1683 "((2 - x)/(2*a)) / (2*a/(x - 2))";

  1684 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1685 term2str t;

  1686 if (term2str t) =

  1687 "(-4 + 4 * x + -1 * x ^^^ 2) / (4 * a ^^^ 2)"

  1688 then ()

  1689 else error "rational.sml: diff.behav. in norm_Rational_mg 35";

  1691 (*SRD Schalk I, p.69 Nr. 455a *)

  1692 val t = str2term

  1693 "(a^^^2 + 1)/(a^^^2 - 1) / ((a+1)/(a - 1))";

  1694 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1695 term2str t;

  1696 if (term2str t) =

  1697 "(1 + a ^^^ 2) / (1 + 2 * a + a ^^^ 2)" then ()

  1698 else error "rational.sml: diff.behav. in norm_Rational_mg 36";

  1701 "----- Schalk I, p.69 Nr. 455b";

  1702 val t = str2term "(x^^^2 - 4)/(y^^^2 - 9)/((2+x)/(3 - y))";

  1703 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1704 if term2str t = "(2 + -1 * x) / (3 + y)" then ()

  1705 else error "rational.sml: diff.behav. in norm_Rational_mg 37";

  1707 "----- SK060904-1a non-termination of cancel_p_ ?: worked before 0707xx";

  1708 val t = str2term "(x^^^2 - 4)*(3 - y)/((y^^^2 - 9)*(2+x))";

  1709 val SOME (t,_) = rewrite_set_ thy false norm_Rational t;

  1710 if term2str t = "(2 + -1 * x) / (3 + y)" then ()

  1711 else error "rational.sml: diff.behav. in norm_Rational_mg 37b";

  1713 "----- ?: worked before 0707xx";

  1714 val t = str2term "(3 + -1 * y) / (-9 + y ^^^ 2)";

  1715 val SOME (t,_) = rewrite_set_ thy false norm_Rational t;

  1716 if term2str t = "-1 / (3 + y)" then ()

  1717 else error "rational.sml: -1 / (3 + y) norm_Rational";

  1719 (*SRD Schalk I, p.69 Nr. 456b *)

  1720 val t = str2term

  1721 "(b^^^3 - b^^^2)/(b^^^2+b)/(b^^^2 - 1)";

  1722 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1723 term2str t;

  1724 if (term2str t) = "b / (1 + 2 * b + b ^^^ 2)" then ()

  1725 else error "rational.sml: diff.behav. in norm_Rational_mg 38";

  1727 (*SRD Schalk I, p.69 Nr. 457b *)

  1728 val t = str2term

  1729 "(16*a^^^2 - 9*b^^^2)/(2*a+3*a*b) / ((4*a+3*b)/(4*a^^^2 - 9*a^^^2*b^^^2))";

  1730 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1731 term2str t;

  1732 if (term2str t) =

  1733   "8 * a ^^^ 2 + -6 * a * b + -12 * a ^^^ 2 * b + 9 * a * b ^^^ 2"

  1734 then () else error "rational.sml: diff.behav. in norm_Rational_mg 39";

  1736 "----- Schalk I, p.69 Nr. 458b works since 0707";

  1737 val t = str2term

  1738 "(2*a^^^2*x - a^^^2)/(a*x - b*x) / (b^^^2*(2*x - 1)/(x*(a - b)))";

  1739 val SOME (t,_) = rewrite_set_ thy false norm_Rational t;

  1740 if term2str t = "a ^^^ 2 / b ^^^ 2" then ()

  1741 else error "rational.sml: diff.behav. in norm_Rational_mg 39b";

  1743 (*SRD Schalk I, p.69 Nr. 459b *)

  1744 val t = str2term "(a^^^2 - b^^^2)/(a*b) / (4*(a+b)^^^2/a)";

  1745 val SOME (t,_) = rewrite_set_ thy false norm_Rational t;

  1746 if term2str t = "(a + -1 * b) / (4 * a * b + 4 * b ^^^ 2)" then ()

  1747 else error "rational.sml: diff.behav. in norm_Rational_mg 41";

  1750 (*Schalk I, p.69 Nr. 460b nonterm.SK

  1751 val t = str2term

  1752 "(9*(x^^^2 - 8*x+16)/(4*(y^^^2 - 2*y+1)))/((3*x - 12)/(16*y - 16))";

  1753 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1754 if term2str t =

  1755 then ()

  1756 else error "rational.sml: diff.behav. in norm_Rational_mg 42";

  1758 val t = str2term

  1759 "9*(x^^^2 - 8*x+16)*(16*y - 16)/(4*(y^^^2 - 2*y+1)*(3*x - 12))";

  1760 val SOME (t',_) = rewrite_set_ thy false norm_Rational t;

  1761 ... non terminating.

  1762 val SOME (t',_) = rewrite_set_ thy false make_polynomial t;

  1763 "(-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)";

  1764 val SOME (t,_) = rewrite_set_ thy false cancel_p t';

  1765 ... non terminating.*)

  1767 (*SRD Schalk I, p.70 Nr. 472a *)

  1768 val t = str2term ("((8*x^^^2 - 32*y^^^2)/(2*x + 4*y))/" ^

  1769 		 "((4*x - 8*y)/(x + y))");

  1770 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1771 term2str t;

  1772 if (term2str t) =

  1773 "x + y"

  1774 then ()

  1775 else error "rational.sml: diff.behav. in norm_Rational_mg 43";

  1778 (*----------------------------------------------------------------------*)

  1779 (*---------------------- Einfache Dppelbrche --------------------------*)

  1780 (*----------------------------------------------------------------------*)

  1782 (*SRD Schalk I, p.69 Nr. 461a *)

  1783 val t = str2term

  1784 "(2/(x+3) + 2/(x - 3)) / (8*x/(x^^^2 - 9))";

  1785 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1786 term2str t;

  1787 if (term2str t) =

  1788 "1 / 2"

  1789 then ()

  1790 else error "rational.sml: diff.behav. in norm_Rational_mg 44";

  1792 (*SRD Schalk I, p.69 Nr. 464b *)

  1793 val t = str2term

  1794 "(a - a/(a - 2)) / (a + a/(a - 2))";

  1795 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1796 term2str t;

  1797 if (term2str t) =

  1798 "(3 + -1 * a) / (1 + -1 * a)"

  1799 then ()

  1800 else error "rational.sml: diff.behav. in norm_Rational_mg 45";

  1802 (*SRD Schalk I, p.69 Nr. 465b *)

  1803 val t = str2term

  1804 "((x+3*y)/9 + (4*y^^^2 - 9*z^^^2)/(16*x)) /(x/9+y/6+z/4)";

  1805 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1806 term2str t;

  1807 if (term2str t) =

  1808 "(4 * x + 6 * y + -9 * z) / (4 * x)"

  1809 then ()

  1810 else error "rational.sml: diff.behav. in norm_Rational_mg 46";

  1812 (*SRD Schalk I, p.69 Nr. 466b *)

  1813 val t = str2term

  1814 "((1 - 7*(x - 2)/(x^^^2 - 4)) / (6/(x+2))) / (3/(x+5)+30/(x^^^2 - 25))";

  1815 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1816 term2str t;

  1817 if (term2str t) =

  1818 "(25 + -10 * x + x ^^^ 2) / 18"

  1819 then ()

  1820 else error "rational.sml: diff.behav. in norm_Rational_mg 47";

  1822 (*SRD Schalk I, p.70 Nr. 469 *)

  1823 val t = str2term

  1824 "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))";

  1825 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1826 term2str t;

  1827 if (term2str t) =

  1828 "3 * b ^^^ 3 / (2 * a + -2 * b)"

  1829 then ()

  1830 else error "rational.sml: diff.behav. in norm_Rational_mg 48";

  1832 (*----------------------------------------------------------------------*)

  1833 (*---------------------- Mehrfache Dppelbrueche ------------------------*)

  1834 (*----------------------------------------------------------------------*)

  1836 (*SRD.test Schalk I, p.70 Nr. 476b *) (* Rechenzeit: 10 sec *)

  1837 (*WN060419 crashes with method 'simplify' ????SK*)

  1838 val t = str2term

  1839 "((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)";

  1840 val SOME (t,_) = rewrite_set_ thy false norm_Rational t;

  1841 if term2str t = "1 / (a ^^^ 2 + -1 * b ^^^ 2)" then ()

  1842 else error "rational.sml: diff.behav. in norm_Rational_mg 49";

  1844 "----- Schalk I, p.70 Nr. 477a";

  1845 (* MG Achtung: terme explodieren; Bsp zu komplex;

  1846    L???ung sollte (ziemlich grosser) Faktorisierter Ausdruck sein

  1847 val t = str2term "b*y/(b - 2*y)/((b^^^2 - y^^^2)/(b+2*y)) /" ^

  1848 		 "(b^^^2*y+b*y^^^2)*(a+x)^^^2/((b^^^2 - 4*y^^^2)*(a+2*x)^^^2)";

  1849 val SOME (t',_) = rewrite_set_ thy false norm_Rational t;

  1852 val t = str2term "b*y*(b+2*y)*(b^^^2 - 4*y^^^2)*(a+2*x)^^^2 / " ^

  1853 		 "((b - 2*y)*(b^^^2 - y^^^2)*(b^^^2*y+b*y^^^2)*(a+x)^^^2)";

  1854 val SOME (t,_) = rewrite_set_ thy false make_polynomial t;

  1855 ????SK ???MG*)

  1858 "----- Schalk I, p.70 Nr. 478b ----- Rechenzeit: 5 sec";

  1859 val t = str2term ("(a - (a*b+b^^^2)/(a+b))/(b+(a - b)/(1+(a+b)/(a - b))) / " ^

  1860 		 "((a - a^^^2/(a+b))/(a+(a*b)/(a - b)))");

  1861 val SOME (t',_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1862 if term2str t' =

  1863 "(2 * a ^^^ 3 + 2 * a ^^^ 2 * b) / (a ^^^ 2 * b + b ^^^ 3)"

  1864 then ()

  1865 else error "rational.sml: diff.behav. in norm_Rational_mg 51";

  1867 (* TODO.new_c:WN050820 STOP_REW_SUB introduced gave ...

  1868 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 ()

  1869 else error "rational.sml: works again";

  1870 re-outcommented with TODO.new_c: cvs before 071227, 11:50*)

  1874 (*Schalk I, p.70 Nr. 480a *)

  1875 (* Achtung: rechnet ewig; cancel_p kann nicht krzen: WN060831 nonterm.SK00

  1876 val t = str2term

  1877 "(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)))";

  1878 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1879 term2str t;

  1880 if (term2str t) =

  1882 then ()

  1883 else error "rational.sml: diff.behav. in norm_Rational_mg 52";

  1885 (*MG Berechne Zwischenergebnisse: WN060831 nonterm.SK00*)

  1886 val t = str2term

  1887 "(1/x+1/y+1/z)/(1/x - 1/y - 1/z)";

  1888 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1889 term2str t;

  1890 "(x ^^^ 2 * y ^^^ 2 * z + x ^^^ 2 * y * z ^^^ 2 + x * y ^^^ 2 * z ^^^ 2) /

  1891 (-1 * x ^^^ 2 * y ^^^ 2 * z + -1 * x ^^^ 2 * y * z ^^^ 2 + x * y ^^^ 2 * z ^^^ 2)";

  1892 val t = str2term

  1893 "2*x^^^2/(x^^^2 - z^^^2)/(x/(x+z)+x/(x - z))";

  1894 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1895 term2str t;

  1896 "1"

  1898 (* SK 1. Ausdruck kann nicht weiter gekrzt werden; cancel_p !!!*)

  1899 ###  rls: cancel_p on:

  1900 (x ^^^ 2 * (y ^^^ 2 * z) + x ^^^ 2 * (y * z ^^^ 2) + x * (y ^^^ 2 * z ^^^ 2)) /

  1901 (-1 * (x ^^^ 2 * (y ^^^ 2 * z)) + -1 * (x ^^^ 2 * (y * z ^^^ 2)) + x * (y ^^^ 2 * z ^^^ 2))

  1902 GC #3.61.81.101.197.17503:   (0 ms)

  1903 *** RATIONALS_DIRECT_CANCEL_EXCEPTION: Invalid fraction

  1905 val t = str2term

  1906 "(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))";

  1907 val SOME (t,_) = rewrite_set_ thy false cancel_p t;

  1908 term2str t;

  1909 (*uncaught exception nonexhaustive binding failure*)

  1911 (* Das kann er aber krzen !!????: *)

  1912 val t = str2term

  1913 "(x^^^2 * (y^^^2 * z) +  x * (y^^^2 * z^^^2)) / (-1 * (x^^^2 * (y * z^^^2)) + x * (y^^^2 * z^^^2))";

  1914 val SOME (t,_) = rewrite_set_ thy false cancel_p t;

  1915 term2str t;

  1916 "(-1 * (y * x) + -1 * (z * y)) / (1 * (z * x) + -1 * (z * y))";

  1917 *)

  1920 "-------- crucial examples ------------------------------";

  1921 "-------- crucial examples ------------------------------";

  1922 "-------- crucial examples ------------------------------";

  1923 (*Schalk I, p.60 Nr. 215d *)

  1924 (* Achtung: rechnet ewig ...

  1925 val t = str2term "(a-b)^^^3 * (x+y)^^^4 / ((x+y)^^^2 * (a-b)^^^5)";

  1926 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1927 term2str t; noterm.SK

  1928 *)

  1930 (* Kein Wunder, denn Z???ler und Nenner extra als Polynom dargestellt ergibt:*)

  1931 (*

  1932 val t = str2term "(a-b)^^^3 * (x+y)^^^4";

  1933 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1934 term2str t;

  1935 "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";

  1936 val t = str2term "((x+y)^^^2 * (a-b)^^^5)";

  1937 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1938 term2str t;

  1939 "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";

  1940 *)

  1941 (*anscheinend macht dem Rechner das Krzen diese Bruches keinen Spass mehr ...*)

  1943 (*--------------------------------------------------------------------*)

  1944 (*Schalk I, p.70 Nr. 480b

  1945 val t = str2term "((12*x*y/(9*x^^^2 - y^^^2))/" ^

  1946 		 "(1/(3*x - y)^^^2 - 1/(3*x + y)^^^2)) *" ^

  1947 		 "(1/(x - 5*y)^^^2 - 1/(x + 5*y)^^^2)/" ^

  1948 		 "(20*x*y/(x^^^2 - 25*y^^^2))";

  1949 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1950 SK.nonterm

  1951 Kann nicht weiter vereinfacht werden !!!!?? *)

  1953 (*--------------------------------------------------------------------*)

  1954 "---- MGs test set";

  1955 val t = str2term " (1 + x^^^5) / (y + x) + x^^^3 / x ";

  1956 val SOME (t,_) = rewrite_set_ thy false common_nominator_p t;

  1957 if term2str t = "(1 + x ^^^ 3 + x ^^^ 5 + y * x ^^^ 2) / (x + y)" then()

  1958 else error "";

  1960 (*--------------------------------------------------------------------*)

  1961 (* cancel_p liefert nicht immer Polynomnormalform (2): WN060831???SK3b

  1962    ---> Sortierung FALSCH !!  *)

  1963 val t = str2term "b^^^3 * a^^^5/a ";

  1964 val SOME (t,_) = rewrite_set_ thy false cancel_p t;

  1965 term2str t;

  1966 "1 * (a^^^4 * b^^^3) / 1"; (*OK*)

  1968 val t = str2term "b^^^3 * a^^^5/b ";

  1969 val SOME (t,_) = rewrite_set_ thy false cancel_p t;

  1970 term2str t;

  1971 "1 * (b^^^2 * a^^^5) / 1"; (*cancel_p sortiert hier falsch um!*)

  1973 (* Problem liegt NICHT bei ord_make_polynomial! (siehe folgende Bsple) *)

  1974 (*

  1975 val x = str2term "x"; val bdv = str2term "bdv";

  1976 val t1 = str2term "b^^^2 * a^^^5";

  1977 val t2 = str2term "a^^^5 * b^^^2 ";

  1978 ord_make_polynomial false Rational.thy [(x,bdv)] (t1,t2); (*false*)

  1979 *)

  1980 (* ==> "b^^^2 * a^^^5" > "a^^^5 * b^^^2 " ... OK!*)

  1982 (*--------------------------------------------------------------------*)

  1983 (* cancel_p liefert nicht immer Polynomnormalform (2): WN060831???SK3c

  1984    ---> erzeugt berflssige "1 * ..."

  1986 val t = str2term "-1 / (3 + y)";

  1987 (*~~         *)

  1988 val SOME (t,_) = rewrite_set_ thy false cancel_p t;

  1989 term2str t;

  1990 "-1 / (3 + 1 * y)";

  1991 (********* Das ist das PROBLEM !!!!!!!??? *******************)

  1992 (* -1 im Z???ler der Angabe verursacht das Problem !*)

  1993 *)

  1995 (* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - *)

  1996 "----- MGs test set";

  1997 val t = str2term "(a^^^2 + -1)/(a+1)";

  1998 val SOME (t',_) = rewrite_set_ thy false cancel_p t;

  1999 if term2str t' = "(-1 + a) / 1" then ()

  2000 else error "rational.sml MG tests 3d";

  2002 "----- NOT TERMINATING ?: worked before 0707xx";

  2003 val t = str2term "(a^^^2 - 1)*(b + 1) / ((b^^^2 - 1)*(a+1))";

  2004 val SOME (t'',_) = rewrite_set_ thy false norm_Rational t;

  2005 if term2str t'' = "(1 + -1 * a) / (1 + -1 * b)" then ()

  2006 else error "rational.sml MG tests 3e";

  2008 "----- corrected SK060905";

  2009 val t = str2term "(4*x^^^2 - 20*x + 25)/(2*x - 5)^^^3";

  2010 val SOME (t',_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  2011 if term2str t' = "-1 / (5 + -2 * x)" then ()

  2012 else error "rational.sml corrected SK060905";

  2015 "-------- examples for Stefan Karnels thesis ------------";

  2016 "-------- examples for Stefan Karnels thesis ------------";

  2017 "-------- examples for Stefan Karnels thesis ------------";

  2018 (*SRAM Schalk I, p.69 Nr. 442b --- abgewandelt*)

  2019 val t = str2term

  2020 "(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))";

  2021 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  2022 term2str t;

  2023 if (term2str t) =

  2024 "5 * x ^^^ 2 / (b ^^^ 3 * c)"

  2025 then ()

  2026 else error "rational.sml: diff.behav. in norm_Rational_mg 53";

  2029 "-------- me Schalk I No.186 ----------------------------";

  2030 "-------- me Schalk I No.186 ----------------------------";

  2031 "-------- me Schalk I No.186 ----------------------------";

  2032 val fmz = ["Term ((14 * x * y) / ( x * y ))",

  2033 	   "normalform N"];

  2034 val (dI',pI',mI') =

  2035   ("Rational",["rational","simplification"],

  2036    ["simplification","of_rationals"]);

  2037 val p = e_pos'; val c = [];

  2038 val (p,_,f,nxt,_,pt) = CalcTreeTEST [(fmz, (dI',pI',mI'))];

  2039 val (p,_,f,nxt,_,pt) = me nxt p c pt;

  2040 val (p,_,f,nxt,_,pt) = me nxt p c pt;

  2041 val (p,_,f,nxt,_,pt) = me nxt p c pt;

  2042 val (p,_,f,nxt,_,pt) = me nxt p c pt;

  2043 val (p,_,f,nxt,_,pt) = me nxt p c pt;

  2044 val (p,_,f,nxt,_,pt) = me nxt p c pt;

  2045 val (p,_,f,nxt,_,pt) = me nxt p c pt;

  2046 val (p,_,f,nxt,_,pt) = me nxt p c pt; f2str f;

  2047 val (p,_,f,nxt,_,pt) = me nxt p c pt; f2str f;

  2048 val (p,_,f,nxt,_,pt) = me nxt p c pt; f2str f;(*++ for explicit script*)

  2049 val (p,_,f,nxt,_,pt) = me nxt p c pt; f2str f;(*++ for explicit script*)

  2050 case (f2str f, nxt) of

  2051     ("14", ("End_Proof'", _)) => ()

  2052   | _ => error "rational.sml diff.behav. in me Schalk I No.186";

  2055 "-------- interSteps ..Simp_Rat_Double_No-1.xml ---------";

  2056 "-------- interSteps ..Simp_Rat_Double_No-1.xml ---------";

  2057 "-------- interSteps ..Simp_Rat_Double_No-1.xml ---------";

  2058 states:=[];

  2059 CalcTree

  2060 [(["Term (((2 - x)/(2*a)) / (2*a/(x - 2)))", "normalform N"],

  2061   ("Rational",["rational","simplification"],

  2062   ["simplification","of_rationals"]))];

  2063 Iterator 1;

  2064 moveActiveRoot 1;

  2065 autoCalculate 1 CompleteCalc;

  2066 val ((pt,p),_) = get_calc 1; show_pt pt;

  2068 interSteps 1 ([1],Res);

  2069 val ((pt,p),_) = get_calc 1; show_pt pt;

  2072 "-------- interSteps ..Simp_Rat_Cancel_No-1.xml ---------";

  2073 "-------- interSteps ..Simp_Rat_Cancel_No-1.xml ---------";

  2074 "-------- interSteps ..Simp_Rat_Cancel_No-1.xml ---------";

  2075 states:=[];

  2076 CalcTree

  2077 [(["Term ((a^2 + -1*b^2) / (a^2 + -2*a*b + b^2))", "normalform N"],

  2078   ("Rational",["rational","simplification"],

  2079   ["simplification","of_rationals"]))];

  2080 Iterator 1;

  2081 moveActiveRoot 1;

  2082 autoCalculate 1 CompleteCalc;

  2083 val ((pt,p),_) = get_calc 1; show_pt pt;

  2084 (*========== inhibit exn 110314 ================================================

  2085 (*with explicit script done already... and removed [1,..] at below...

  2086 interSteps 1 ([1],Res);

  2087 val ((pt,p),_) = get_calc 1; show_pt pt;

  2088 *)

  2089 interSteps 1 ([2],Res);

  2090 val ((pt,p),_) = get_calc 1; show_pt pt;

  2092 interSteps 1 ([2,1],Res);

  2093 val ((pt,p),_) = get_calc 1; show_pt pt;

  2094 val newnds = children (get_nd pt [2,1]) (*see "fun detailrls"*);

  2095 (*if length newnds = 12 then () WN060905*)

  2096 if length newnds = 13 then ()

  2097 else error "rational.sml: interSteps cancel_p rev_rew_p";

  2099 val p = ([2,1,9],Res);

  2100 getTactic 1 p;

  2101 val (_, tac, _) = pt_extract (pt, p);

  2102 (*case tac of SOME (Rewrite ("sym_real_plus_binom_times1", _)) => ()

  2103 WN060905*)

  2104 case tac of SOME (Rewrite ("sym_real_add_mult_distrib2", _)) => ()

  2105 | _ => error "rational.sml: getTactic, sym_real_plus_binom_times1";

  2106 ============ inhibit exn 110314 ==============================================*)

  2109 "-------- investigate rulesets for cancel_p -------------";

  2110 "-------- investigate rulesets for cancel_p -------------";

  2111 "-------- investigate rulesets for cancel_p -------------";

  2112 val thy = @{theory "Rational"};

  2113 "---------------- (a^^^2 + -1*b^^^2) / (a^^^2 + -2*a*b + b^^^2)";

  2114 val t = str2term "(a^^^2 + -1*b^^^2) / (a^^^2 + -2*a*b + b^^^2)";

  2115 val tt = str2term "(1 * a + 1 * b) * (1 * a + -1 * b)"(*numerator only*);

  2116 "----- with rewrite_set_";

  2117 val SOME (tt',asm) = rewrite_set_ thy false make_polynomial tt;

  2118 term2str tt'= "a ^^^ 2 + -1 * b ^^^ 2" (*true*);

  2119 val tt = str2term "((1 * a + -1 * b) * (1 * a + -1 * b))"(*denominator only*);

  2120 val SOME (tt',asm) = rewrite_set_ thy false make_polynomial tt;

  2121 term2str tt' = "a ^^^ 2 + -2 * a * b + b ^^^ 2" (*true*);

  2123 "----- with make_deriv";

  2124 val SOME (tt, _) = factout_p_ thy t; term2str tt =

  2125 "(1 * a + 1 * b) * (1 * a + -1 * b) / ((1 * a + -1 * b) * (1 * a + -1 * b))";

  2126 (*

  2127 "--- with ruleset as before 060829";

  2128 val {rules, rew_ord=(_,ro),...} =

  2129     rep_rls (assoc_rls "make_polynomial");

  2130 val der = make_deriv thy Atools_erls rules ro NONE tt;

  2131 print_depth 99; map (term2str o #1) der; print_depth 3;

  2132 print_depth 99; map (rule2str o #2) der; print_depth 3;

  2133 ... did not terminate*)

  2134 "--- with simpler ruleset";

  2135 val {rules, rew_ord=(_,ro),...} =

  2136     rep_rls (assoc_rls "rev_rew_p");

  2137 val der = make_deriv thy Atools_erls rules ro NONE tt;

  2138 print_depth 99; writeln (deriv2str der); print_depth 3;

  2140 print_depth 99; map (term2str o #1) der; print_depth 3;

  2141 "...,(-1 * b ^^^ 2 + a ^^^ 2) / (-2 * (a * b) + a ^^^ 2 + (-1 * b) ^^^ 2) ]";

  2142 print_depth 99; map (rule2str o #2) der; print_depth 3;

  2143 print_depth 99; map (term2str o #1 o #3) der; print_depth 3;

  2145 val der = make_deriv thy Atools_erls rules ro NONE

  2146 		     (str2term "(1 * a + 1 * b) * (1 * a + -1 * b)");

  2147 print_depth 99; writeln (deriv2str der); print_depth 3;

  2149 val {rules, rew_ord=(_,ro),...} =

  2150     rep_rls (assoc_rls "rev_rew_p");

  2151 val der = make_deriv thy Atools_erls rules ro NONE

  2152 		     (str2term "(1 * a + -1 * b) * (1 * a + -1 * b)");

  2153 print_depth 99; writeln (deriv2str der); print_depth 3;

  2154 print_depth 99; map (term2str o #1) der; print_depth 3;

  2155 (*WN060829 ...postponed*)

  2158 "-------- investigate format of factout_ and factout_p_ -";

  2159 "-------- investigate format of factout_ and factout_p_ -";

  2160 "-------- investigate format of factout_ and factout_p_ -";

  2161 val {rules, rew_ord = (_,ro),...} = rep_rls (assoc_rls "make_polynomial");

  2162 val (thy, eval_rls) = (@{theory "Rational"}, Atools_erls)(*see 'fun init_state'*);

  2163 val Rrls {scr = Rfuns {init_state,...},...} = assoc_rls "cancel_p";

  2165 "----- see Rational.ML, local cancel_p, fun init_state";

  2166 val t = str2term "(a^^^2 + (-1)*b^^^2) / (a^^^2 + (-2)*a*b + b^^^2)";

  2167 val SOME (t',_) = factout_p_ thy t; term2str t';

  2168 (*

  2169 val rtas = reverse_deriv thy eval_rls rules ro NONE t';

  2170 writeln(trtas2str rst);

  2171 *)

  2174 "----- see Rational.ML, local cancel_p, fun init_state";

  2175 val t = str2term "a^^^2 / a";

  2176 val SOME (t',_) = factout_p_ thy t;

  2177 term2str t' = "a * a / (1 * a)" (*true*);

  2178 (*... can be canceled with

  2179 real_mult_div_cancel2 ?k ~= 0 ==> ?m * ?k / (?n * ?k) = ?m / ?n"*)

  2180 (* sml/ME/rewtools.sml:

  2181 val rtas = reverse_deriv thy Atools_erls rules ro NONE t';

  2182 writeln (deri2str rtas);

  2183 *)

  2186 "-------- SK 060904 ----------------------------------------------";

  2187 "----- order on polynomials -- input + output";

  2188 val thy = @{theory "Isac"};

  2189 val t = str2term "(a + -1 * b) / (-1 * a + b)";

  2190 val SOME (t', _) = factout_p_ thy t; term2str t';

  2191 val SOME (t', _) = cancel_p_ thy t; term2str t';

  2193 val t = str2term "a*b*c*d / (d*e*f*g)";

  2194 val SOME (t', _) = cancel_p_ thy t; term2str t';

  2196 val t = str2term "a*(b*(c*d)) / (b*(e*(f*g)))";

  2197 val SOME (t', _) = cancel_p_ thy t; term2str t';

  2198 (*???order.SK  ???*)

  2200 "----- SK060904-1a non-termination of cancel_p_ ? worked before 0707xx";

  2201 val t = str2term "(x^^^2 - 4)*(3 - y) / ((y^^^2 - 9)*(2+x))";

  2202 val SOME (t',_) = rewrite_set_ thy false norm_Rational t;

  2203 if term2str t' = "(2 + -1 * x) / (3 + y)" then ()

  2204 else error "rational.sml SK060904-1a worked since 0707xx";

  2206 "----- SK060904-1b non-termination of cancel_p_ ... worked before 0707xx";

  2207 val t = str2term ("(9 * a ^^^ 2 + -16 * b ^^^ 2) /" ^

  2208 "(36 * a^^^2 * c + (27 * a^^^2 * e + (-64 * b^^^2 * c + -48 * b^^^2 * e)))");

  2209 val SOME (t',_) = cancel_p_ thy t;

  2210 if term2str t' = "1 / (4 * c + 3 * e)" then ()

  2211 else error "rational.sml SK060904-1b";

  2214 "----- SK060904-2a non-termination of add_fraction_p_";

  2215 val t = str2term (" (a + b * x) / (a + -1 * (b * x)) +  " ^

  2216 		  " (-1 * a + b * x) / (a + b * x)      ");

  2217 (* nonterm.SK

  2218 val SOME (t',_) = rewrite_set_ thy false common_nominator_p t;

  2220 common_nominator_p_ thy t;

  2221 " (a + b * x)*(a + b * x) / ((a + -1 * (b * x))*(a + -1 * (b * x))) +  " ^

  2222 " (-1 * a + b * x)*(a + -1 * (b * x)) / ((a + b * x)*(-1 * a + b * x)) ";

  2224 add_fraction_p_ thy t;

  2225 " ((a + b * x)*(a + b * x)  +  (-1 * a + b * x)*(a + -1 * (b * x))) /" ^

  2226 " ((a + b * x)*(-1 * a + b * x))                                     ";

  2227 *)

  2230 "-------- how to stepwise construct Scripts -------------";

  2231 "-------- how to stepwise construct Scripts -------------";

  2232 "-------- how to stepwise construct Scripts -------------";

  2233 val thy = @{theory "Rational"};

  2234 (*no trailing _*)

  2235 val p1 = parse thy (

  2236 "Script SimplifyScript (t_t::real) =                             " ^

  2237 "  (Rewrite_Set discard_minus False                   " ^

  2238 "   t_t)");

  2240 (*required (): (Rewrite_Set discard_minus False)*)

  2241 val p2 = parse thy (

  2242 "Script SimplifyScript (t_t::real) =                             " ^

  2243 "  (Rewrite_Set discard_minus False                   " ^

  2244 "   t_t)");

  2246 val p3 = parse thy (

  2247 "Script SimplifyScript (t_t::real) =                             " ^

  2248 "  ((Rewrite_Set discard_minus False)                   " ^

  2249 "   t_t)");

  2251 val p4 = parse thy (

  2252 "Script SimplifyScript (t_t::real) =                             " ^

  2253 "  ((Rewrite_Set discard_minus False)                   " ^

  2254 "   t_t)");

  2256 val p5 = parse thy (

  2257 "Script SimplifyScript (t_t::real) =                             " ^

  2258 "  ((Try (Rewrite_Set discard_minus False)                   " ^

  2259 "    Try (Rewrite_Set discard_parentheses False))               " ^

  2260 "   t_t)");

  2262 val p6 = parse thy (

  2263 "Script SimplifyScript (t_t::real) =                             " ^

  2264 "  ((Try (Rewrite_Set discard_minus False) @@                   " ^

  2265 "    Try (Rewrite_Set rat_mult_poly False) @@                    " ^

  2266 "    Try (Rewrite_Set make_rat_poly_with_parentheses False) @@   " ^

  2267 "    Try (Rewrite_Set cancel_p_rls False) @@                     " ^

  2268 "    (Repeat                                                     " ^

  2269 "     ((Try (Rewrite_Set common_nominator_p_rls False) @@        " ^

  2270 "       Try (Rewrite_Set rat_mult_div_pow False) @@              " ^

  2271 "       Try (Rewrite_Set make_rat_poly_with_parentheses False) @@" ^

  2272 "       Try (Rewrite_Set cancel_p_rls False) @@                  " ^

  2273 "       Try (Rewrite_Set rat_reduce_1 False)))) @@               " ^

  2274 "    Try (Rewrite_Set discard_parentheses False))               " ^

  2275 "   t_t)"

  2276 );

  2278 "----------- get_denominator ----------------------------";

  2279 "----------- get_denominator ----------------------------";

  2280 "----------- get_denominator ----------------------------";

  2281 val thy = @{theory Isac};

  2282 val t = term_of (the (parse thy "get_denominator ((a +x)/b)"));

  2283 val SOME (_, t') = eval_get_denominator "" 0 t thy;

  2284 if term2str t' = "get_denominator ((a + x) / b) = b" then ()

  2285 else error "get_denominator ((a + x) / b) = b"

  2288 "--------- several errpats in complicated term -------------------";

  2289 "--------- several errpats in complicated term -------------------";

  2290 "--------- several errpats in complicated term -------------------";

  2291 (*TODO: instead of Gabriella's example here (27.Jul.12) find a simpler one*)

  2292 states:=[];

  2293 CalcTree

  2294 [(["Term ((5*b + 25)/(a^2 - b^2) * (a - b)/(5*b))", "normalform N"],

  2295   ("Rational",["rational","simplification"], ["simplification","of_rationals"]))];

  2296 Iterator 1;

  2297 moveActiveRoot 1;

  2298 autoCalculate 1 CompleteCalc;

  2299 val ((pt,p),_) = get_calc 1; show_pt pt;

  2301 "-------- nonterminating cancel_p, norm_Rational 2002 ---";

  2302 "-------- nonterminating cancel_p, norm_Rational 2002 ---";

  2303 "-------- nonterminating cancel_p, norm_Rational 2002 ---";

  2304 (*------------------------------------------------------------------------------------\

  2305 "--- WN121204: searched rational.sml for "nonterm", added new numbering" ^

  2306 "              and thoroughly tested with this numbering subsequently";

  2307 "--- 1 ---";

  2308 WN.2.6.03 from rlang.sml 56a

  2309 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)";

  2310 val NONE = rewrite_set_ thy false common_nominator_p t;

  2311 "--- 2 ---";

  2312 WN060831 nonterm.SK7

  2313 val t = str2term "(a + b * x) / (a + -1 * (b * x)) + (-1 * a + b * x) / (a + b * x)";

  2314 val NONE = add_fraction_p_ thy t;

  2315 "--- 3 ---";

  2316 nonterm.SK9 loops: cancel_p kann nicht weiter kuerzen!!! *)

  2317 val t'' = str2term "(a + b)/(x^^^2 - y^^^2) * ((x - y)^^^2/(a^^^2 - b^^^2))";

  2318 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t'';

  2319 "--- 4 ---";

  2320 val t = str2term

  2321 "(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)";

  2322 WN060831 nonterm.SK10

  2323 val SOME (t,_) = rewrite_set_ thy false cancel_p t;

  2324 term2str t;

  2325 "--- 5 ---";

  2326 val t = str2term

  2327 "(9*(x^^^2 - 8*x+16)/(4*(y^^^2 - 2*y+1)))/((3*x - 12)/(16*y - 16))";

  2328 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  2329 if term2str t =

  2330 then ()

  2331 else error "rational.sml: diff.behav. in norm_Rational_mg 42";

  2332 "--- 6 ---";

  2333 val t = str2term

  2334 "9*(x^^^2 - 8*x+16)*(16*y - 16)/(4*(y^^^2 - 2*y+1)*(3*x - 12))";

  2335 val SOME (t',_) = rewrite_set_ thy false norm_Rational t;

  2336 ... non terminating.

  2337 "--- 7 ---";

  2338 val SOME (t',_) = rewrite_set_ thy false make_polynomial t;

  2339 "(-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)";

  2340 val SOME (t,_) = rewrite_set_ thy false cancel_p t';

  2341 "--- 8 ---";

  2342 val t = str2term "(a-b)^^^3 * (x+y)^^^4 / ((x+y)^^^2 * (a-b)^^^5)";

  2343 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  2344 "--- 9 ---";

  2345 val t = str2term "((12*x*y/(9*x^^^2 - y^^^2))/" ^

  2346 		 "(1/(3*x - y)^^^2 - 1/(3*x + y)^^^2)) *" ^

  2347 		 "(1/(x - 5*y)^^^2 - 1/(x + 5*y)^^^2)/" ^

  2348 		 "(20*x*y/(x^^^2 - 25*y^^^2))";

  2349 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  2350 "--- 10 ---";

  2351 SK060904-2a non-termination of add_fraction_p_";

  2352 val t = str2term (" (a + b * x) / (a + -1 * (b * x)) +  " ^

  2353 		  " (-1 * a + b * x) / (a + b * x)      ");

  2354 (* nonterm.SK

  2355 val SOME (t',_) = rewrite_set_ thy false common_nominator_p t;

  2356 "--- 11 ---";

  2357 common_nominator_p_ thy t;

  2358 " (a + b * x)*(a + b * x) / ((a + -1 * (b * x))*(a + -1 * (b * x))) +  " ^

  2359 " (-1 * a + b * x)*(a + -1 * (b * x)) / ((a + b * x)*(-1 * a + b * x)) ";

  2360 "--- 12 ---";

  2361 add_fraction_p_ thy t;

  2362 " ((a + b * x)*(a + b * x)  +  (-1 * a + b * x)*(a + -1 * (b * x))) /" ^

  2363 " ((a + b * x)*(-1 * a + b * x))                                     ";

  2364 --------------------------------------------------------------------------------------/*)

  2366 (*------------------------------------------------------------------------------------\

  2367 "WN121205: thoroughly tested with the above numbering.";

  2368 "  errors in cancel_p: --- 4,5,6,7.";

  2369 "  error in common_nominator_p, common_nominator_p_: --- 10; 1,2?.";

  2370 "  errors caused by ruleset norm_Rational: --- 8,9.";

  2371 trace_rewrite := true;

  2373 "--- 1 ---: non-terminating with ### add_fract: done t22 ";

  2374 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)";

  2375 trace_rewrite:= true;

  2376 rewrite_set_ thy false common_nominator_p t;

  2378 "--- 2 ---: non-terminating with ### add_fract: done t22 ";

  2379 val t = str2term "(a + b * x) / (a + -1 * (b * x)) + (-1 * a + b * x) / (a + b * x)";

  2380 add_fraction_p_ thy t;

  2382 "--- 3 ---: norm_Rational calls Rrls cancel_p with non-normalised polys";

  2383 val t = str2term "(a + b)/(x^^^2 - y^^^2) * ((x - y)^^^2/(a^^^2 - b^^^2))";

  2384 rewrite_set_ thy false norm_Rational t;

  2385 (*tracing end...####  rls: cancel_p on:

  2386 (a * x ^^^ 2 + -2 * (a * (x * y)) + a * y ^^^ 2 + b * x ^^^ 2 + -2 * (b * (x * y)) + b * y ^^^ 2) /

  2387 (a ^^^ 2 * x ^^^ 2 + -1 * (a ^^^ 2 * y ^^^ 2) + -1 * (b ^^^ 2 * x ^^^ 2) + b ^^^ 2 * y ^^^ 2) *)

  2389 "--- 4 ---: non-terminating with Rrls cancel_p on plausible input";

  2390 val t = str2term (

  2391 "(a * x ^^^ 2 + -2 * a * x * y + a * y ^^^ 2 + b * x ^^^ 2 + -2 * b * x * y + b * y ^^^ 2) /"^

  2392 "(a ^^^ 2 * x ^^^ 2 + -1 * a ^^^ 2 * y ^^^ 2 + -1 * b ^^^ 2 * x ^^^ 2 + b ^^^ 2 * y ^^^ 2)");

  2393 rewrite_set_ thy false cancel_p t;

  2394 (*#  rls: cancel_p on:

  2395 (a * x ^^^ 2 + -2 * a * x * y + a * y ^^^ 2 + b * x ^^^ 2 + -2 * b * x * y + b * y ^^^ 2) /

  2396 (a ^^^ 2 * x ^^^ 2 + -1 * a ^^^ 2 * y ^^^ 2 + -1 * b ^^^ 2 * x ^^^ 2 + b ^^^ 2 * y ^^^ 2) *)

  2398 "--- 5 ---: non-terminating with Rrls cancel_p on plausible input";

  2399 val t = str2term "(9*(x^^^2 - 8*x+16)/(4*(y^^^2 - 2*y+1)))/((3*x - 12)/(16*y - 16))";

  2400 rewrite_set_ thy false norm_Rational t;

  2401 (*####  rls: cancel_p on:

  2402 (2304 + -1152 * x + -2304 * y + 144 * x ^^^ 2 + 1152 * (x * y) + -144 * (x ^^^ 2 * y)) /

  2403 (48 + -12 * x + -96 * y + 24 * (x * y) + 48 * y ^^^ 2 + -12 * (x * y ^^^ 2))  *)

  2405 "--- 6 ---: non-terminating with Rrls cancel_p on plausible input";

  2406 val t = str2term

  2407 "9*(x^^^2 - 8*x+16)*(16*y - 16)/(4*(y^^^2 - 2*y+1)*(3*x - 12))";

  2408 rewrite_set_ thy false norm_Rational t;

  2409 (*###  rls: cancel_p on: (-2304 + 1152 * x + 2304 * y + -144 * x ^^^ 2 + -1152 * (x * y) +

  2410  144 * (x ^^^ 2 * y)) /

  2411 (-48 + 12 * x + 96 * y + -24 * (x * y) + -48 * y ^^^ 2 + 12 * (x * y ^^^ 2)) *)

  2413 "--- 7 ---: non-terminating with Rrls cancel_p on plausible input";

  2414 val t' = str2term (

  2415 "(-2304 + 1152 * x + 2304 * y + -144 * x ^^^ 2 + -1152 * x * y + 144 * x ^^^ 2 * y) /"^

  2416 "(-48 + 12 * x + 96 * y + -24 * x * y + -48 * y ^^^ 2 + 12 * x * y ^^^ 2)");

  2417 rewrite_set_ thy false cancel_p t';

  2419 "--- 8 ---: bottom of output cannot be looked ad (due to looping?)";

  2420 val t = str2term "(a-b)^^^3 * (x+y)^^^4 / ((x+y)^^^2 * (a-b)^^^5)";

  2421 val SOME (t,_) = rewrite_set_ thy false norm_Rational t;

  2423 "--- 9 ---: probably error in norm_Rational";

  2424 val t = str2term (

  2425 "((12*x*y / (9*x^^^2 - y^^^2)) / (1/(3*x - y)^^^2 - 1/(3*x + y)^^^2)) *" ^

  2426 		"(1/(x - 5*y)^^^2 - 1/(x + 5*y)^^^2) / (20*x*y/(x^^^2 - 25*y^^^2))");

  2427 rewrite_set_ thy false norm_Rational t;

  2428 (*####  rls: cancel_p on: (19440 * (x ^^^ 8 * y ^^^ 2) + -490320 * (x ^^^ 6 * y ^^^ 4) +

  2429  108240 * (x ^^^ 4 * y ^^^ 6) +

  2430  -6000 * (x ^^^ 2 * y ^^^ 8)) /

  2431 (2160 * (x ^^^ 8 * y ^^^ 2) + -108240 * (x ^^^ 6 * y ^^^ 4) +

  2432  1362000 * (x ^^^ 4 * y ^^^ 6) +

  2433  -150000 * (x ^^^ 2 * y ^^^ 8)) *)

  2435 "--- 10 ---: non-terminating with ### add_fract: done t22: error in common_nominator_p ";

  2436 val t = str2term (" (a + b * x) / (a + -1 * (b * x)) + (-1 * a + b * x) / (a + b * x)");

  2437 rewrite_set_ thy false common_nominator_p t; (*### add_fract: done t22 *)

  2438 common_nominator_p_ thy t;                   (*loops without output*)

  2439 "--- reformulated 10:";

  2440 val t = str2term "(a + -1 * (b * x)) / (a + b * x)";

  2441 rewrite_set_ thy false cancel_p t = NONE;

  2442 "--- 11 ---";

  2443 "--- 12 ---";

  2444 "...both are to be considered after common_nominator_p_ is improved";

  2445 --------------------------------------------------------------------------------------/*)

  2446 ============ inhibit exn WN130824 TODO ======================================================*)

author	Walther Neuper <neuper@ist.tugraz.at>
	Fri, 30 Aug 2013 13:43:30 +0200
changeset 52096	ee2a5f066e44
parent 52095	c9fbb8171a0a
child 52100	0831a4a6ec8a
permissions	-rw-r--r--