wneuper/isa: test/Tools/isac/Knowledge/rational.sml@c9fbb8171a0a

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

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

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

   352   is_expanded (str2term "x * y");

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

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

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

   357 "-------- example 186b";

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

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

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

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

   363 "-------- example 186c";

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

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

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

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

   369 (* !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! exception Div raised !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

   370   see --- fun rewrite_set_ downto fun gcd_poly ---

   371 "-------- example 187a";

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

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

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

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

   376 *)

   378 (* doesn't terminate !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

   379   see --- fun rewrite_set_ downto fun gcd_poly ---

   380 "-------- example 187b";

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

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

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

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

   385 *)

   387 (* doesn't terminate !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

   388   see --- fun rewrite_set_ downto fun gcd_poly ---

   389 "-------- example 187c";

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

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

   392 if (term2str t', terms2str asm) =

   393   ("3 * z ^^^ 3 / (5 * (y * x))", "[\"3 * (z * (y ^^^ 2 * x ^^^ 5)) ~= 0\"]")

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

   395 *)

   397 "-------- example 188a";

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

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

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

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

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

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

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

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

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

   409 "-------- example 188b";

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

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

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

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

   415 "-------- example 188c";

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

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

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

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

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

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

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

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

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

   427 "-------- example 190a";

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

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

   430 if (term2str t', terms2str asm) =

   431   ("(1 + 9 * a ^^^ 2) / (3 * a + 9 * a ^^^ 2)", "[\"3 * a + 9 * a ^^^ 2 ~= 0\"]")

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

   434 "-------- example 190c";

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

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

   437 if (term2str t', terms2str asm) =

   438   ("(1 + 3 * a + 9 * a ^^^ 2 + 27 * a ^^^ 3) /\n(3 * a + 18 * a ^^^ 2 + 27 * a ^^^ 3)", "[]")

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

   441 "-------- example 191a";

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

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

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

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

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

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

   449 "-------- example 191b";

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

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

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

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

   455 "-------- example 191c";

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

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

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

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

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

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

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

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

   466 "-------- example 192b";

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

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

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

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

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

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

   474 if (term2str t', terms2str asm) =

   475   ("(7 * x ^^^ 3 + -1 * x ^^^ 2 * y) / (7 * x * y ^^^ 2 + -1 * y ^^^ 3)", "[]")

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

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

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

   480 if (term2str t', terms2str asm) =

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

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

   484 "-------- example 193a";

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

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

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

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

   490 "-------- example 193b";

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

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

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

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

   496 "-------- example 193c";

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

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

   499 if (term2str t', terms2str asm) =

   500   ("(2 * x + 10 * x ^^^ 2) / (1 + -5 * x)", "[\"1 + -5 * x ~= 0\"]")

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

   503 (*WN:*)

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

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

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

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

   509 (*========== inhibit exn WN130826 TODO =========================================================

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   535 if term2str t' =

   536   "(-1 + x) * (-1 + x) / ((1 + x) * (-1 + x)) +\n(1 + x) * (1 + x) / ((1 + x) * (-1 + x))"

   537   andalso terms2str asm = "[]"

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

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

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

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

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

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

   545 if term2str t' =

   546   "(-1 + x) * (-1 + x) / ((1 + x) * (-1 + x)) +\n(1 + x) * (1 + x) / ((1 + x) * (-1 + x))"

   547   andalso terms2str asm = "[]"

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

   550 val t = str2term

   551   "((1) / (2*x + 2)) + ((1) / (2*x + (-2))) + ((1) / ( x^^^2 + (-1)))+((1) / (x^^^2 + (-2)*x + 1))";

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

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

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

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

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

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

   560 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))";

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

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

   563 (term2str t3');

   564 (term2str t3'');

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

   567 term2str t4;

   568 term2str (hd(t5));

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

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

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

   575   term2str t';

   576   term2str t'';

   577   "(3 + x) * (3 - x) / ((3 - x) * (3 - x))";

   578   "(3 + x) / (3 - x)";

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

   581 	    "(9 - x^^^2) / (9 - 6*x + x^^^2) + 1 / (3 - x)";

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

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

   584   term2str t';

   585   term2str t'';

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

   587   "(4 + x) / (3 - x)";

   589 (*WN021016 added -----vv---*)

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

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

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

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

   594 		" * (9 - 6 * x + x ^^^ 2) / (9 - 6 * x + x ^^^ 2)" (*true*);

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

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

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

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

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

   601 		"(9 - x ^^^ 2) / (9 - 6 * x + x ^^^ 2)" (*true*);

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

   603 (*WN021016 added -----^^---*)

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

   605 val t = str2term "1 / a";

   606 val NONE =  cancel_p_ thy t;

   607 val NONE = rewrite_set_ thy false cancel_p t;

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

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

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

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

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

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

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

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

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

   619 term2str t' =

   620 "(-1 * x ^^^ 2 + y ^^^ 2) / ((-1 * x + y) * (-1 * x + y)) +\n1" ^

   621 " * (-1 * x + y) / ((-1 * x + y) * (-1 * x + y))" (*true*);

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   640  *** RATIONALS_TERM2POLY_EXCEPTION: Invalid Polynomial

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

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

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

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

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

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

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

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

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

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

   654 	    "(9 + (-1)* x^^^2)/(9 + (-1)* 6*x + x^^^2) + (1)/(3 + x)";

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

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

   657   term2str t';

   658   term2str t'';

   659   "(12 + 5 * x + x ^^^ 2) / (9 - x ^^^ 2)";

   660   "(12 + 5 * x + x ^^^ 2) / (9 + (-1) * x ^^^ 2)";

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

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

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

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

   667 val thy' = "Rational";

   668 val rls' = "cancel";

   669 val mp = "make_polynomial";

   671 writeln ("example 186:");

   672 writeln("a)");

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

   674 rewrite_set_ thy false cancel (str2term e186a');

   675 "@@@@@@@@@@@@@@";

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

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

   678   is_expanded (parse_rat "x * y");

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

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

   682 writeln("b)");

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

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

   685 writeln("c)");

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

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

   688     handle e => print_exn_G e;

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

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

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

   693 writeln ("example 187:");

   694 writeln("a)");

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

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

   697 writeln("b)");

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

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

   700 writeln("c)");

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

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

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

   704             "[\"3 * (z * (y ^^^ 2 * x ^^^ 5)) ~= 0\"]") then ()

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

   707 writeln "example 188:";

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

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

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

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

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

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

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

   716 writeln("b)");

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

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

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

   720 writeln("c)");

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

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

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

   725 val SOME (t,_) =

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

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

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

   730 writeln ("example 190:");

   731 writeln("c)");

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

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

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

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

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

   738 writeln ("example 191:");

   739 writeln("a)");

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

   741 (*WN.23.10.02-------

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

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

   744   false;

   745   is_expanded (parse_rat "x + y");

   746   true; -----------*)

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

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

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

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

   752 writeln("c)");

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

   754 (*WN.23.10.02-------

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

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

   757   false;

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

   759   false;

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

   761   true;------------*)

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

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

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

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

   768 writeln ("example 192:");

   769 writeln("b)");

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

   771 (*WN.23.10.02-------

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

   773 -------------------*)

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

   775 (*========== inhibit exn 110317 ================================================

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

   777 (*"(-1 * y * x ^^^ 2 + 7 * x ^^^ 3) / (-1 * y ^^^ 3 + 7 * x * y ^^^ 2)"WN050929*)

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

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

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

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

   782 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";

   783 ============ inhibit exn 110317 ==============================================*)

   785 writeln ("example 193:");

   786 writeln("a)");

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

   788 (*WN.23.10.02-------

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

   790 -------------------*)

   791 writeln("b)");

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

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

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

   795 writeln("c)");

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

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

   798 -------------------*)

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

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

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

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

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

   806 "-------- common_nominator_p ----------------------------";

   807 "-------- common_nominator_p ----------------------------";

   808 "-------- common_nominator_p ----------------------------";

   809 val rls' = "common_nominator_p";

   811 writeln ("example 204:");

   812 writeln("a)");

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

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

   815 writeln("b)");

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

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

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

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

   821 writeln ("example 205:");

   822 writeln("a)");

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

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

   825 writeln("b)");

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

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

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

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

   831 writeln ("example 206:");

   832 writeln("a)");

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

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

   835 writeln("b)");

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

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

   839 writeln ("example 207:");

   840 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)) ";

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

   843 writeln ("example 208:");

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

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

   847 writeln ("example 209:");

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

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

   851 writeln ("example 210:");

   852 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)) ";

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

   855 writeln ("example 211:");

   856 writeln("a)");

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

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

   859 writeln("b)");

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

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

   863 writeln ("example 212:");

   864 writeln("a)");

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

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

   867 writeln("b)");

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

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

   871 writeln ("example 213:");

   872 writeln("a)");

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

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

   875 writeln("b)");

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

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

   879 writeln ("example 214:");

   880 writeln("a)");

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

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

   883 writeln("b)");

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

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

   887 writeln ("example 216:");

   888 writeln("a)");

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

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

   891 writeln("b)");

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

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

   895 writeln ("example 217:");

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

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

   900 val rls' = "common_nominator";

   901 writeln ("example 218:");

   902 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))";

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

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

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

   907 writeln ("example 219:");

   908 writeln("a)");

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

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

   911 writeln("b)");

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

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

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

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

   917 writeln ("example 220:");

   918 writeln("a)");

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

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

   921 writeln("b)");

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

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

   925 writeln ("example 221:");

   926 writeln("a)");

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

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

   929 writeln("b)");

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

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

   933 writeln ("example 222:");

   934 writeln("a)");

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

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

   937 writeln("b)");

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

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

   941 writeln ("example 225:");

   942 writeln("a)");

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

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

   945 writeln("b)");

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

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

   949 writeln ("example 226:");

   950 writeln("a)");

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

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

   953 writeln("b)");

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

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

   957 writeln ("example 227:");

   958 writeln("a)");

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

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

   961 writeln("b)");

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

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

   965 writeln ("example 228:");

   966 writeln("a)");

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

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

   969 writeln("b)");

   970 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))";

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

   974 writeln ("example 229:");

   975 writeln("a)");

   976 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))";

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

   978 writeln("b)");

   979 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))";

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

   982 writeln ("example 230:");

   983 writeln("a)");

   984 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))";

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

   986 writeln("b)");

   987 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))";

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

   990 writeln ("example 231:");

   991 writeln("a)");

   992 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))";

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

   994 writeln("b)");

   995 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))";

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

   998 writeln ("example 232:");

   999 writeln("a)");

  1000 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))";

  1001 val e232a = the (rewrite_set thy' false "common_nominator" e232a');

  1002 writeln("b)");

  1003 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))";

  1004 val e232b = the (rewrite_set thy' false "common_nominator" e232b');

  1006 writeln ("example 233:");

  1007 writeln("a)");

  1008 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))";

  1009 val e233a = the (rewrite_set thy' false "common_nominator" e233a');

  1010 writeln("b)");

  1011 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))";

  1012 val e233b = the (rewrite_set thy' false "common_nominator" e233b');

  1014 writeln ("example 234:");

  1015 writeln("a)");

  1016 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))";

  1017 val e234a = the (rewrite_set thy' false "common_nominator" e234a');

  1018 writeln("b)");

  1019 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)) ";

  1020 val e234b = the (rewrite_set thy' false "common_nominator" e234b');

  1022 writeln ("example 235:");

  1023 writeln("a)");

  1024 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))";

  1025 val e235a = the (rewrite_set thy' false "common_nominator" e235a');

  1026 writeln("b)");

  1027 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)) ";

  1028 val e235b = the (rewrite_set thy' false "common_nominator" e235b');

  1030 writeln ("example 236:");

  1031 writeln("a)");

  1032 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))";

  1033 val e236a = the (rewrite_set thy' false "common_nominator" e236a');

  1034 writeln("b)");

  1035 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)) ";

  1036 val e236b = the (rewrite_set thy' false "common_nominator" e236b');

  1039 val rls' = "cancel";

  1040 writeln ("example heuberger:");

  1041 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)";

  1042 val eheu = the (rewrite_set thy' false "cancel" eheu');

  1044 val rls' = "common_nominator_p";

  1045 writeln ("example stiefel:");

  1046 val est1'="(7) / (-14) + (-2) / (4)";

  1047 val est1 = the (rewrite_set thy' false "common_nominator_p" est1');

  1048 if est1 = ("-1 / 1", "[]") then ()

  1049 else error "new behaviour in rational.sml: est1'";

  1051 val t = (term_of o the o (parse thy))

  1052 "(9 - x ^^^ 2) / (9 - 6 * x + x ^^^ 2)";

  1053 val SOME (t',_) = factout_ thy t;

  1054 if term2str t' = "(3 + x) * (3 - x) / ((3 - x) * (3 - x))" then ()

  1055 else error "rational.sml factout_ (9 - x ^^^ 2) / (9 - 6 * x + x ^^^ 2)";

  1058 "-------- reverse rewrite -------------------------------";

  1059 "-------- reverse rewrite -------------------------------";

  1060 "-------- reverse rewrite -------------------------------";

  1061 (*WN.28.8.02: tests for the 'reverse-rewrite' functions:

  1062   these are defined in Rationals.ML and stored in

  1063   the 'reverse-ruleset' cancel*)

  1065 (*the term for which reverse rewriting is demonstrated*)

  1066   val t = (term_of o the o (parse thy))

  1067 	      "(9 - x ^^^ 2) / (9 + 6 * x + x ^^^ 2)";

  1068   val Rrls {scr=Rfuns {init_state=ini,locate_rule=loc,

  1069   		       next_rule=nex,normal_form=nor,...},...} = cancel;

  1071 (*normal_form produces the result in ONE step*)

  1072   val SOME (t',_) = nor t;

  1073 if term2str t' = "(3 - x) / (3 + x)" then ()

  1074 else error "rational.sml normal_form (9 - x ^^^ 2) / (9 - 6 * x + x ^^^ 2)";

  1076 (*initialize the interpreter state used by the 'me'*)

  1077   val (t,_,revsets,_) = ini t;

  1079 (*find the rule 'r' to apply to term 't'*)

  1080   val SOME (r as (Thm (str, thm))) = nex revsets t;

  1081 if str = "sym_#power_Float ((3,0), (0,0)) __ ((2,0), (0,0))"

  1082    andalso string_of_thm thm =

  1083            (string_of_thm o make_thm o (cterm_of (@{theory "Isac"})))

  1084                (Trueprop $ (term_of o the o (parse thy)) "9 = 3 ^^^ 2") then ()

  1085 else error "rational.sml next_rule (9 - x ^^^ 2) / (9 - 6 * x + x ^^^ 2)";

  1086 (* before Isa02->09-2 was not checked automatically, ?was different?:

  1087 val SOME r = nex revsets t;

  1088 val r = Thm ("sym_#mult_2_3","6 = 2 * 3") : rule*)

  1090 (* check, if the rule 'r' applied by the user to 't' belongs to the ruleset;

  1091   if the rule is OK, the term resulting from applying the rule is returned,too;

  1092   there might be several rule applications inbetween,

  1093   which are listed after the head in reverse order *)

  1094   val (r, (t, asm))::_ = loc revsets t r;

  1095 if term2str t = "(9 - x ^^^ 2) / (3 ^^^ 2 + 6 * x + x ^^^ 2)" andalso asm = []

  1096 then ()

  1097 else error "rational.sml locate_rule (9 - x ^^^ 2) / (9 - 6 * x + x ^^^ 2)";

  1099 (* find the next rule to apply *)

  1100   val SOME (r as (Thm (str, thm))) = nex revsets t;

  1101 if str = "sym_#power_Float ((3,0), (0,0)) __ ((2,0), (0,0))" andalso

  1102    string_of_thm thm = (string_of_thm o make_thm o (cterm_of (@{theory "Isac"})))

  1103                 (Trueprop $ (term_of o the o (parse thy)) "9 = 3 ^^^ 2") then ()

  1104 else error "rational.sml next_rule (9 - x ^^^ 2) / (9 - 6 * x + x ^^^ 2)";

  1106 (*check the next rule*)

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

  1108 if term2str t = "(3 ^^^ 2 - x ^^^ 2) / (3 ^^^ 2 + 6 * x + x ^^^ 2)" then ()

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

  1111 (*find and check the next rules, rewrite*)

  1112   val SOME r = nex revsets t;

  1113   val (r,(t,asm))::_ = loc revsets t r;

  1114 if term2str t = "(3 ^^^ 2 - x ^^^ 2) / (3 ^^^ 2 + 2 * 3 * x + x ^^^ 2)" then ()

  1115 else error "rational.sml locate_rule II";

  1117   val SOME r = nex revsets t;

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

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

  1120 else error "rational.sml next_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 + x) * (3 + x))" then ()

  1125 else error "rational.sml next_rule III";

  1127   val SOME r = nex revsets t;

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

  1129   val ss = term2str t;

  1130 if ss = "(3 - x) / (3 + x)" andalso terms2str asm = "[\"3 + x ~= 0\"]" then ()

  1131 else error "rational.sml: new behav. in rev-set cancel";

  1133 "-------- 'reverse-ruleset' cancel_p --------------------";

  1134 "-------- 'reverse-ruleset' cancel_p --------------------";

  1135 "-------- 'reverse-ruleset' cancel_p --------------------";

  1136 (*WN.11.9.02: the 'reverse-ruleset' cancel_p*)

  1138 (*the term for which reverse rewriting is demonstrated*)

  1139 val t = str2term "(9 + (-1)*x^^^2) / (9 + ((-6)*x + x^^^2))";

  1140 val Rrls {scr=Rfuns {init_state=ini,locate_rule=loc,

  1141 		       next_rule=nex,normal_form=nor,...},...} = cancel_p;

  1143 (*normal_form produces the result in ONE step*)

  1144 val SOME (t',_) = nor t;

  1145 term2str t' = "(3 + 1 * x) / (3 + -1 * x)";

  1147 (*initialize the interpreter state used by the 'me'*)

  1148 val SOME (t', asm) = cancel_p_ thy t;

  1149 term2str t' = "(3 + x) / (3 + -1 * x)" (*true*);

  1150 terms2str asm = "[\"3 + -1 * x ~= 0\"]" (*true*);

  1151 val (t,_,revsets,_) = ini t;

  1153 (* WN.10.10.02: dieser Fall terminiert nicht

  1154            (make_polynomial enth"alt zu viele rules)

  1155 WN060823 'init_state' requires rewriting on specified location in the term

  1156 print_depth 99; Rfuns; print_depth 3;

  1157 WN060831 cycling "sym_order_mult_rls_" "sym_mult_assoc"

  1158          as was with make_polynomial before ?!?*)

  1160 val SOME r = nex revsets t;

  1161 eq_Thm (r, Thm ("sym_#power_Float ((3,0), (0,0)) __ ((2,0), (0,0))",

  1162 		mk_thm thy "9 = 3 ^^^ 2"));

  1163 (*WN060831 *** id_of_thm

  1164            Exception- ERROR raised ...

  1165 val (r,(t,asm))::_ = loc revsets t r;

  1166 term2str t;

  1168   val SOME r = nex revsets t;

  1169   val (r,(t,asm))::_ = loc revsets t r;

  1170   term2str t;

  1171 *)

  1173 writeln "******************  all tests successfull  *************************";

  1177 (*WN.17.3.03 =========================================================vvv---*)

  1178 "-------- norm_Rational ---------------------------------";

  1179 "-------- norm_Rational ---------------------------------";

  1180 "-------- norm_Rational ---------------------------------";

  1181 val t = str2term "(3*x+5)/18 - x/2  - -(3*x - 2)/9 = 0";

  1182 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; term2str t';

  1183 if term2str t' = "1 / 18 = 0" then () else error "rational.sml 1";

  1185 val t = str2term "(17*x - 51)/9 - (-(13*x - 3)/6) + 11 - (9*x - 7)/4 = 0";

  1186 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; term2str t';

  1187 if term2str t' = "(237 + 65 * x) / 36 = 0" then ()

  1188 else error "rational.sml 2";

  1190 val t = str2term "(1/2 + (5*x)/2)^^^2 - ((13*x)/2 - 5/2)^^^2 - (6*x)^^^2 + 29";

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

  1192 if term2str t' = "23 + 35 * x + -72 * x ^^^ 2" then ()

  1193 else error "rational.sml 3";

  1195 (*trace_rewrite:=true;*)

  1196 val t = str2term "Not (6*x is_atom)";

  1197 val SOME (t',_) = rewrite_set_ thy false powers_erls t; term2str t';

  1198 "HOL.True";

  1199 val t = str2term "1 < 2";

  1200 val SOME (t',_) = rewrite_set_ thy false powers_erls t; term2str t';

  1201 "HOL.True";

  1203 val t = str2term "(6*x)^^^2";

  1204 val SOME (t',_) = rewrite_ thy dummy_ord powers_erls false

  1205 			   (num_str @{thm realpow_def_atom}) t;

  1206 if term2str t' = "6 * x * (6 * x) ^^^ (2 + -1)" then ()

  1207 else error "rational.sml powers_erls (6*x)^^^2";

  1209 val t = str2term "-1 * (-2 * (5 / 2 * (13 * x / 2)))";

  1210 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; term2str t';

  1211 if term2str t' = "65 * x / 2" then () else error "rational.sml 4";

  1213 val t = str2term "1 - ((13*x)/2 - 5/2)^^^2";

  1214 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; term2str t';

  1215 (*bef.040209: if term2str t' = "(-21 + (130 * x + -169 * x ^^^ 2)) / 4"then()*)

  1216 if term2str t' = "(-21 + 130 * x + -169 * x ^^^ 2) / 4" then ()

  1217 else error "rational.sml 5";

  1219 (*SRAM Schalk I, p.92 Nr. 609a*)

  1220 val t = str2term "2*(3 - x/5)/3 - 4*(1 - x/3) - x/3 - 2*(x/2 - 1/4)/27 +5/54";

  1221 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; term2str t';

  1222 if term2str t' = "(-255 + 112 * x) / 135" then ()

  1223 else error "rational.sml 6";

  1225 (*SRAM Schalk I, p.92 Nr. 610c*)

  1226 val t = str2term "((x- 1)/(x+1) + 1) / ((x- 1)/(x+1) - (x+1)/(x- 1)) - 2";

  1227 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; term2str t';

  1228 if term2str t' = "(-3 + -1 * x) / 2" then () else error "rational.sml 7";

  1230 (*SRAM Schalk I, p.92 Nr. 476a*)

  1231 val t = str2term "(x^^^2/(1 - x^^^2) + 1)/(x/(1 - x) + 1) * (1 + x)";

  1232 (*. a/b : c/d translated to a/b * d/c .*)

  1233 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; term2str t';

  1234 (*if term2str t' = "1 / 1" then () else error "rational.sml 8";3.6.03*)

  1235 if term2str t' = "1" then () else error "rational.sml 8";

  1237 (*............................vvv---TODO: sollte gehen mit poly_order *)

  1238 (*Schalk I, p.92 Nr. 472a*)

  1239 val t = str2term "((8*x^^^2 - 32*y^^^2)/(2*x + 4*y))/((4*x - 8*y)/(x + y))";

  1240 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; term2str t';

  1241 if term2str t' = "x + y" then () else error "rational.sml p.92 Nr. 472a";

  1243 (*Schalk I, p.70 Nr. 480b; a/b : c/d translated to a/b * d/c*)

  1244 val t = str2term ("((12*x*y/(9*x^^^2 - y^^^2))/" ^

  1245 		 "(1/(3*x - y)^^^2 - 1/(3*x + y)^^^2)) *" ^

  1246 		 "(1/(x - 5*y)^^^2 - 1/(x + 5*y)^^^2)/" ^

  1247 		 "(20*x*y/(x^^^2 - 25*y^^^2))");

  1248 (*... nicht simpl, zerlegt ...*)

  1249 val t = str2term ("((12*x*y/(9*x^^^2 - y^^^2))/" ^

  1250 		 "(1/(3*x - y)^^^2 - 1/(3*x + y)^^^2))");

  1251 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; term2str t';

  1252 "(-12 * (x * y ^^^ 3) + 108 * (x * (y * x ^^^ 2))) / (12 * (x * y))";

  1253 (*                             ~~~~~~~~~~ poly_order notwendig!*)

  1254 val t = str2term ("(1/(x - 5*y)^^^2 - 1/(x + 5*y)^^^2)/" ^

  1255 		 "(20*x*y/(x^^^2 - 25*y^^^2))");

  1256 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; term2str t';

  1257 if term2str t' = "1 / (x ^^^ 2 + -25 * y ^^^ 2)" then ()

  1258 else error "rational.sml norm_Rational 1 / (x ^^^ 2 + -25 * y ^^^ 2)";

  1260 "nonterm.SK6 ----- SK060904-2a non-termination of add_fraction_p_";

  1261 (*WN.2.6.03 from rlang.sml 56a

  1262 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)";

  1263 val NONE = rewrite_set_ thy false common_nominator_p t;

  1265 WN060831 nonterm.SK7

  1266 val t = str2term "(a + b * x) / (a + -1 * (b * x)) + (-1 * a + b * x) / (a + b * x)";

  1267 val NONE = add_fraction_p_ thy t;

  1268 *)

  1271 (* ------------------------------------------------------------------- *)

  1272 (*---------vvv------------ MG: ab 1.7.03 ----------------vvv-----------*)

  1273 (*                 Simplifier fuer beliebige Buchterme                 *)

  1274 (* ------------------------------------------------------------------- *)

  1275 (*----------------------- norm_Rational_mg ----------------------------*)

  1276 (* ------------------------------------------------------------------- *)

  1278 "-------- numeral rationals -----------------------------";

  1279 "-------- numeral rationals -----------------------------";

  1280 "-------- numeral rationals -----------------------------";

  1281 (*SRA Schalk I, p.40 Nr. 164b *)

  1282 val t = str2term "(47/6 - 76/9 + 13/4)/(35/12)";

  1283 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1284 term2str t;

  1285 if (term2str t) = "19 / 21" then ()

  1286 else error "rational.sml: diff.behav. in norm_Rational_mg 1";

  1288 (*SRA Schalk I, p.40 Nr. 166a *)

  1289 val t = str2term "((5/4)/(4+22/7) + 37/20)*(110/3 - 110/9 * 23/11)";

  1290 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1291 term2str t;

  1292 if (term2str t) = "45 / 2" then ()

  1293 else error "rational.sml: diff.behav. in norm_Rational_mg 2";

  1296 "-------- cancellation ----------------------------------";

  1297 "-------- cancellation ----------------------------------";

  1298 "-------- cancellation ----------------------------------";

  1299 (* e190c Stefan K.*)

  1300 val t = str2term

  1301 "((1 + 9 * a ^^^ 2)*(1 + 3 * a))/((3 * a + 9 * a ^^^ 2)*(1 + 3 * a))";

  1302 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1303 term2str t;

  1304 if (term2str t) =

  1305 "(1 + 9 * a ^^^ 2) / (3 * a + 9 * a ^^^ 2)"

  1306 then ()

  1307 else error "rational.sml: diff.behav. in norm_Rational_mg 3";

  1309 (* e192b Stefan K.*)

  1310 val t = str2term

  1311 "((x ^^^ 2)*(7 * x + (-1) * y))/((y ^^^ 2)*(7 * x + (-1) * y))";

  1312 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1313 term2str t;

  1314 if (term2str t) =

  1315 "x ^^^ 2 / y ^^^ 2"

  1316 then ()

  1317 else error "rational.sml: diff.behav. in norm_Rational_mg 4";

  1319 (*SRC Schalk I, p.66 Nr. 379c *)

  1320 val t = str2term

  1321 "(a - b)/(b - a)";

  1322 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1323 term2str t;

  1324 if (term2str t) =

  1325 "-1"

  1326 then ()

  1327 else error "rational.sml: diff.behav. in norm_Rational_mg 5";

  1329 (*SRC Schalk I, p.66 Nr. 380b *)

  1330 val t = str2term

  1331 "15*(3*x+3)*(4*x+9)/(12*(2*x+7)*(5*x+5))";

  1332 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1333 term2str t;

  1334 if (term2str t) =

  1335 "(27 + 12 * x) / (28 + 8 * x)"

  1336 then ()

  1337 else error "rational.sml: diff.behav. in norm_Rational_mg 6";

  1339 (*Schalk I, p.60 Nr. 215c *)

  1340 (* Falsches Ergebnis: rechnet lange und cancel_p kann nicht weiter krzen!!!*)

  1341 (* WN060831????MG1

  1342 val t = str2term "(a+b)^^^4*(x - y)/((x - y)^^^3*(a+b)^^^2)";

  1343 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1344 term2str t;

  1345 if (term2str t) =

  1346 "(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)"

  1347 then ()

  1348 else error "rational.sml: diff.behav. in norm_Rational_mg 7";

  1349 *)

  1350 (*val t = str2term

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

  1353 term2str t;*)

  1354 (* uncaught exception nonexhaustive binding failure

  1355    raised at: stdIn:93.1-93.51 *)

  1357 (*Schalk I, p.66 Nr. 381a *)

  1358 (* ACHTUNG: rechnet ca. 2 Minuten !!! *)

  1359 (* WN060831???MG2

  1360 val t = str2term "18*(a+b)^^^3*(a - b)^^^2/(72*(a - b)^^^3*(a+b)^^^2)";

  1361 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1362 term2str t;

  1363 if (term2str t) =

  1364 "(a + b) / (4 * a + -4 * b)"

  1365 then () else error "rational.sml: diff.behav. in norm_Rational_mg 8";

  1366 *)

  1368 (*SRC Schalk I, p.66 Nr. 381b *)

  1369 val t = str2term

  1370 "(4*x^^^2 - 20*x + 25)/(2*x - 5)^^^3";

  1371 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1372 term2str t;

  1373 if (term2str t) =

  1374 "-1 / (5 + -2 * x)"

  1375 then ()

  1376 else error "rational.sml: diff.behav. in norm_Rational_mg 9";

  1378 (*SRC Schalk I, p.66 Nr. 381c *)

  1379 val t = str2term

  1380 "(27*a^^^3+9*a^^^2+3*a+1)/(27*a^^^3+18*a^^^2+3*a)";

  1381 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1382 term2str t;

  1383 if (term2str t) =

  1384 "(1 + 9 * a ^^^ 2) / (3 * a + 9 * a ^^^ 2)"

  1385 then ()

  1386 else error "rational.sml: diff.behav. in norm_Rational_mg 10";

  1388 (*SRC Schalk I, p.66 Nr. 383a *)

  1389 val t = str2term

  1390 "(5*a^^^2 - 5*a*b)/(a - b)^^^2";

  1391 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1392 term2str t;

  1393 if (term2str t) =

  1394 "5 * a / (a + -1 * b)"

  1395 then ()

  1396 else error "rational.sml: diff.behav. in norm_Rational_mg 11";

  1399 "-------- common denominator ----------------------------";

  1400 "-------- common denominator ----------------------------";

  1401 "-------- common denominator ----------------------------";

  1402 (*SRA Schalk I, p.67 Nr. 403a *)

  1403 val t = str2term

  1404 "4/x - 3/y - 1";

  1405 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1406 term2str t;

  1407 if (term2str t) =

  1408 "(-3 * x + 4 * y + -1 * x * y) / (x * y)"

  1409 then ()

  1410 else error "rational.sml: diff.behav. in norm_Rational_mg 12";

  1412 (*SRA Schalk I, p.67 Nr. 407b *)

  1413 val t = str2term

  1414 "(2*a+3*b)/(b*c) + (3*c+a)/(a*c) - (2*a^^^2+3*b*c)/(a*b*c)";

  1415 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1416 term2str t;

  1417 if (term2str t) =

  1418 "4 / c"

  1419 then ()

  1420 else error "rational.sml: diff.behav. in norm_Rational_mg 13";

  1422 (*SRA Schalk I, p.67 Nr. 410b *)

  1423 val t = str2term

  1424 "1/(x+1) + 1/(x+2) - 2/(x+3)";

  1425 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1426 term2str t;

  1427 if (term2str t) =

  1428 "(5 + 3 * x) / (6 + 11 * x + 6 * x ^^^ 2 + x ^^^ 3)"

  1429 then ()

  1430 else error "rational.sml: diff.behav. in norm_Rational_mg 14";

  1432 (*SRA Schalk I, p.67 Nr. 413b *)

  1433 val t = str2term

  1434 "(1+x)/(1 - x) - (1 - x)/(1+x) + 2*x/(1 - x^^^2)";

  1435 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1436 term2str t;

  1437 if (term2str t) =

  1438 "6 * x / (1 + -1 * x ^^^ 2)"

  1439 then ()

  1440 else error "rational.sml: diff.behav. in norm_Rational_mg 15";

  1442 (*SRA Schalk I, p.68 Nr. 414a *)

  1443 val t = str2term

  1444 "(x + 2)/(x - 1) + (x - 3)/(x - 2) - (x + 1)/((x - 1)*(x - 2))";

  1445 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1446 term2str t;

  1447 if (term2str t) =

  1448 "(-2 + -5 * x + 2 * x ^^^ 2) / (2 + -3 * x + x ^^^ 2)"

  1449 then ()

  1450 else error "rational.sml: diff.behav. in norm_Rational_mg 16";

  1452 (*SRA Schalk I, p.68 Nr. 423a *)

  1453 val t = str2term

  1454 "(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)";

  1455 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1456 term2str t;

  1457 if (term2str t) =

  1458 "1"

  1459 then ()

  1460 else error "rational.sml: diff.behav. in norm_Rational_mg 17";

  1462 (*SRA Schalk I, p.68 Nr. 428b *)

  1463 val t = str2term

  1464 "1/(a - b)^^^2 + 1/(a+b)^^^2 - 2/(a^^^2 - b^^^2) - 4*(b^^^2 - 1)/(a^^^2 - b^^^2)^^^2";

  1465 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1466 term2str t;

  1467 if (term2str t) =

  1468 "4 / (a ^^^ 4 + -2 * a ^^^ 2 * b ^^^ 2 + b ^^^ 4)"

  1469 then ()

  1470 else error "rational.sml: diff.behav. in norm_Rational_mg 18";

  1472 (*SRA Schalk I, p.68 Nr. 430b *)

  1473 val t = str2term

  1474 "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";

  1475 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1476 term2str t;

  1477 if (term2str t) =

  1478 "a + 3 * b"

  1479 then ()

  1480 else error "rational.sml: diff.behav. in norm_Rational_mg 19";

  1483 (*SRA Schalk I, p.68 Nr. 432 *)

  1484 val t = str2term

  1485 "(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)";

  1486 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1487 term2str t;

  1488 if (term2str t) =

  1489 "0"

  1490 then ()

  1491 else error "rational.sml: diff.behav. in norm_Rational_mg 20";

  1493 (*Eigenes*)

  1494 val t = str2term

  1495 "3*a/(a*b) + x/y";

  1496 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1497 term2str t;

  1498 if (term2str t) =

  1499 "(3 * y + b * x) / (b * y)"

  1500 then ()

  1501 else error "rational.sml: diff.behav. in norm_Rational_mg 21";

  1504 "-------- multiply and cancel ---------------------------";

  1505 "-------- multiply and cancel ---------------------------";

  1506 "-------- multiply and cancel ---------------------------";

  1507 (*SRM Schalk I, p.68 Nr. 436a *)

  1508 val t = str2term

  1509 "3*(x+y)/(15*(x - y)) * 25*(x - y)^^^2/(18*(x+y)^^^2)";

  1510 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1511 term2str t;

  1512 if (term2str t) =

  1513 "(5 * x + -5 * y) / (18 * x + 18 * y)"

  1514 then ()

  1515 else error "rational.sml: diff.behav. in norm_Rational_mg 22";

  1517 (*SRM.test Schalk I, p.68 Nr. 436b *)

  1518 (*WN060420???MG3 crashes with method 'simplify' in

  1519   IsacCore > Simplification > Rational Terms > Multiplication > No.2*)

  1520 val t = str2term "5*a*(a - b)^^^2*(a + b)^^^3/(7*b*(a - b)^^^3) * 7*b/(a + b)^^^3";

  1521 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1522 term2str t;

  1523 if (term2str t) =

  1524 "5 * a / (a + -1 * b)"

  1525 then ()

  1526 else error "rational.sml: diff.behav. in norm_Rational_mg 23";

  1528 (*Schalk I, p.68 Nr. 437a *)

  1529 val t = str2term "(3*a - 4*b)/(4*c+3*e) * (3*a+4*b)/(9*a^^^2 - 16*b^^^2)";

  1530 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1531 if (term2str t) = "1 / (4 * c + 3 * e)" then ()

  1532 else error "rational.sml: diff.behav. in norm_Rational_mg 24";

  1534 "----- S.K. corrected non-termination 060904";

  1535 val t = str2term "(3*a - 4*b) * (3*a+4*b)/((4*c+3*e)*(9*a^^^2 - 16*b^^^2))";

  1536 val SOME (t',_) = rewrite_set_ thy false make_polynomial t;

  1537 term2str t';

  1538 if term2str t' =

  1539   "(9 * a ^^^ 2 + -16 * b ^^^ 2) /\n(36 * a ^^^ 2 * c + 27 * a ^^^ 2 * e + -64 * b ^^^ 2 * c +\n -48 * b ^^^ 2 * e)"

  1540 (*"(9 * a ^^^ 2 + -16 * b ^^^ 2) / (36 * a ^^^ 2 * c + 27 * a ^^^ 2 * e + -64 * b ^^^ 2 * c + -48 * b ^^^ 2 * e)"*)

  1541 then () else error "rational.sml: S.K.8..corrected 060904-6";

  1543 "----- S.K. corrected non-termination of cancel_p_";

  1544 val t'' = str2term ("(9 * a ^^^ 2 + -16 * b ^^^ 2) /" ^

  1545 "(36 * a^^^2 * c + (27 * a^^^2 * e + (-64 * b^^^2 * c + -48 * b^^^2 * e)))");

  1546 val SOME (t',_) = rewrite_set_ thy false cancel_p t'';

  1547 if term2str t' = "1 / (4 * c + 3 * e)" then ()

  1548 else error "rational.sml: diff.behav. in cancel_p S.K.8";

  1550 (*Schalk I, p.68 Nr. 437b *)

  1551 (* nonterm.SK9 loops: cancel_p kann nicht weiter kuerzen!!! *)

  1552 val t'' = str2term "(a + b)/(x^^^2 - y^^^2) * ((x - y)^^^2/(a^^^2 - b^^^2))";

  1553 (* val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t'';

  1554    *)

  1556 "================delete===";

  1557 (*a casual output from above*)

  1558 val t = str2term

  1559 "(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)";

  1560 (* WN060831 nonterm.SK10

  1561 val SOME (t,_) = rewrite_set_ thy false cancel_p t;

  1562 term2str t;

  1563 *)

  1565 (*SRM Schalk I, p.68 Nr. 438a *)

  1566 val t = str2term

  1567 "x*y/(x*y - y^^^2)*(x^^^2 - x*y)";

  1568 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1569 term2str t;

  1570 if (term2str t) =

  1571 "x ^^^ 2"

  1572 then ()

  1573 else error "rational.sml: diff.behav. in norm_Rational_mg 24";

  1575 (*SRM Schalk I, p.68 Nr. 439b *)

  1576 val t = str2term

  1577 "(4*x^^^2+4*x+1)*((x^^^2 - 2*x^^^3)/(4*x^^^2+2*x))";

  1578 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1579 term2str t;

  1580 if (term2str t) =

  1581 "(x + -4 * x ^^^ 3) / 2"

  1582 then ()

  1583 else error "rational.sml: diff.behav. in norm_Rational_mg 25";

  1585 (*SRM Schalk I, p.68 Nr. 440a *)

  1586 val t = str2term

  1587 "(x^^^2 - 2*x)/(x^^^2 - 3*x) * (x - 3)^^^2/(x^^^2 - 4)";

  1588 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1589 term2str t;

  1590 if (term2str t) =

  1591 "(-3 + x) / (2 + x)"

  1592 then ()

  1593 else error "rational.sml: diff.behav. in norm_Rational_mg 26";

  1595 "----- Schalk I, p.68 Nr. 440b SK11 works since 0707xx";

  1596 val t = str2term

  1597 "(a^^^3 - 9*a)/(a^^^3*b - a*b^^^3)*(a^^^2*b+a*b^^^2)/(a+3)";

  1598 val SOME (t,_) = rewrite_set_ thy false norm_Rational t;

  1599 if term2str t = "(-3 * a + a ^^^ 2) / (a + -1 * b)" then ()

  1600 else error "rational.sml: diff.behav. in norm_Rational 27";

  1602 "----- SK12 works since 0707xx";

  1603 val t = str2term "(a^^^3 - 9*a)*(a^^^2*b+a*b^^^2)/((a^^^3*b - a*b^^^3)*(a+3))";

  1604 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; term2str t';

  1605 if term2str t' = "(-3 * a + a ^^^ 2) / (a + -1 * b)" then ()

  1606 else error "rational.sml: diff.behav. in norm_Rational 28";

  1609 "-------- common denominator and multiplication ---------";

  1610 "-------- common denominator and multiplication ---------";

  1611 "-------- common denominator and multiplication ---------";

  1612 (*SRAM Schalk I, p.69 Nr. 441b *)

  1613 val t = str2term "(4*a/3 + 3*b^^^2/a^^^3 + b/(4*a))*(4*b/(3*a))";

  1614 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1615 term2str t;

  1616 if (term2str t) =

  1617   "(36 * b ^^^ 3 + 3 * a ^^^ 2 * b ^^^ 2 + 16 * a ^^^ 4 * b) / (9 * a ^^^ 4)"

  1618 then () else error "rational.sml: diff.behav. in norm_Rational_mg 28";

  1620 (*SRAM Schalk I, p.69 Nr. 442b *)

  1621 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)";

  1622 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1623 term2str t;

  1624 if (term2str t) =

  1625 "5 * x ^^^ 2 / (a * b ^^^ 3 * c)"

  1626 then () else error "rational.sml: diff.behav. in norm_Rational_mg 29";

  1628 (*SRAM Schalk I, p.69 Nr. 443b *)

  1629 val t = str2term "(a/2 + b/3)*(b/3 - a/2)";

  1630 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1631 term2str t;

  1632 if (term2str t) =

  1633 "(-9 * a ^^^ 2 + 4 * b ^^^ 2) / 36"

  1634 then () else error "rational.sml: diff.behav. in norm_Rational_mg 30";

  1636 (*SRAM Schalk I, p.69 Nr. 445b *)

  1637 val t = str2term "(a^^^2/9 + 2*a/(3*b) + 4/b^^^2)*(a/3 - 2/b) + 8/b^^^3";

  1638 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1639 term2str t;

  1640 if (term2str t) =

  1641 "a ^^^ 3 / 27"

  1642 then () else error "rational.sml: diff.behav. in norm_Rational_mg 31";

  1644 (*SRAM Schalk I, p.69 Nr. 446b *)

  1645 val t = str2term "(x/(5*x + 4*y) - y/(5*x - 4*y) + 1)*(25*x^^^2 - 16*y^^^2)";

  1646 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1647 term2str t;

  1648 if (term2str t) =

  1649 "30 * x ^^^ 2 + -9 * x * y + -20 * y ^^^ 2"

  1650 then ()

  1651 else error "rational.sml: diff.behav. in norm_Rational_mg 32";

  1653 (*SRAM Schalk I, p.69 Nr. 449a *)(*Achtung: rechnet ca 8 Sekunden*)

  1654 val t = str2term

  1655 "(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)";

  1656 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1657 term2str t;

  1658 if (term2str t) = "(-81 * x ^^^ 4 + 16 * x ^^^ 8 * y ^^^ 4) / (81 * y ^^^ 8)"

  1659 then () else error "rational.sml: diff.behav. in norm_Rational_mg 33";

  1662 (*SRAM Schalk I, p.69 Nr. 450a *)

  1663 val t = str2term

  1664 "(4*x/(3*y)+2*y/(3*x))^^^2 - (2*y/(3*x) - 2*x/y)*(2*y/(3*x)+2*x/y)";

  1665 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1666 term2str t;

  1667 if (term2str t) =

  1668 "(52 * x ^^^ 2 + 16 * y ^^^ 2) / (9 * y ^^^ 2)"

  1669 then ()

  1670 else error "rational.sml: diff.behav. in norm_Rational_mg 34";

  1673 "-------- double fractions ------------------------------";

  1674 "-------- double fractions ------------------------------";

  1675 "-------- double fractions ------------------------------";

  1676 (*SRD Schalk I, p.69 Nr. 454b *)

  1677 val t = str2term

  1678 "((2 - x)/(2*a)) / (2*a/(x - 2))";

  1679 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1680 term2str t;

  1681 if (term2str t) =

  1682 "(-4 + 4 * x + -1 * x ^^^ 2) / (4 * a ^^^ 2)"

  1683 then ()

  1684 else error "rational.sml: diff.behav. in norm_Rational_mg 35";

  1686 (*SRD Schalk I, p.69 Nr. 455a *)

  1687 val t = str2term

  1688 "(a^^^2 + 1)/(a^^^2 - 1) / ((a+1)/(a - 1))";

  1689 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1690 term2str t;

  1691 if (term2str t) =

  1692 "(1 + a ^^^ 2) / (1 + 2 * a + a ^^^ 2)" then ()

  1693 else error "rational.sml: diff.behav. in norm_Rational_mg 36";

  1696 "----- Schalk I, p.69 Nr. 455b";

  1697 val t = str2term "(x^^^2 - 4)/(y^^^2 - 9)/((2+x)/(3 - y))";

  1698 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1699 if term2str t = "(2 + -1 * x) / (3 + y)" then ()

  1700 else error "rational.sml: diff.behav. in norm_Rational_mg 37";

  1702 "----- SK060904-1a non-termination of cancel_p_ ?: worked before 0707xx";

  1703 val t = str2term "(x^^^2 - 4)*(3 - y)/((y^^^2 - 9)*(2+x))";

  1704 val SOME (t,_) = rewrite_set_ thy false norm_Rational t;

  1705 if term2str t = "(2 + -1 * x) / (3 + y)" then ()

  1706 else error "rational.sml: diff.behav. in norm_Rational_mg 37b";

  1708 "----- ?: worked before 0707xx";

  1709 val t = str2term "(3 + -1 * y) / (-9 + y ^^^ 2)";

  1710 val SOME (t,_) = rewrite_set_ thy false norm_Rational t;

  1711 if term2str t = "-1 / (3 + y)" then ()

  1712 else error "rational.sml: -1 / (3 + y) norm_Rational";

  1714 (*SRD Schalk I, p.69 Nr. 456b *)

  1715 val t = str2term

  1716 "(b^^^3 - b^^^2)/(b^^^2+b)/(b^^^2 - 1)";

  1717 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1718 term2str t;

  1719 if (term2str t) = "b / (1 + 2 * b + b ^^^ 2)" then ()

  1720 else error "rational.sml: diff.behav. in norm_Rational_mg 38";

  1722 (*SRD Schalk I, p.69 Nr. 457b *)

  1723 val t = str2term

  1724 "(16*a^^^2 - 9*b^^^2)/(2*a+3*a*b) / ((4*a+3*b)/(4*a^^^2 - 9*a^^^2*b^^^2))";

  1725 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1726 term2str t;

  1727 if (term2str t) =

  1728   "8 * a ^^^ 2 + -6 * a * b + -12 * a ^^^ 2 * b + 9 * a * b ^^^ 2"

  1729 then () else error "rational.sml: diff.behav. in norm_Rational_mg 39";

  1731 "----- Schalk I, p.69 Nr. 458b works since 0707";

  1732 val t = str2term

  1733 "(2*a^^^2*x - a^^^2)/(a*x - b*x) / (b^^^2*(2*x - 1)/(x*(a - b)))";

  1734 val SOME (t,_) = rewrite_set_ thy false norm_Rational t;

  1735 if term2str t = "a ^^^ 2 / b ^^^ 2" then ()

  1736 else error "rational.sml: diff.behav. in norm_Rational_mg 39b";

  1738 (*SRD Schalk I, p.69 Nr. 459b *)

  1739 val t = str2term "(a^^^2 - b^^^2)/(a*b) / (4*(a+b)^^^2/a)";

  1740 val SOME (t,_) = rewrite_set_ thy false norm_Rational t;

  1741 if term2str t = "(a + -1 * b) / (4 * a * b + 4 * b ^^^ 2)" then ()

  1742 else error "rational.sml: diff.behav. in norm_Rational_mg 41";

  1745 (*Schalk I, p.69 Nr. 460b nonterm.SK

  1746 val t = str2term

  1747 "(9*(x^^^2 - 8*x+16)/(4*(y^^^2 - 2*y+1)))/((3*x - 12)/(16*y - 16))";

  1748 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1749 if term2str t =

  1750 then ()

  1751 else error "rational.sml: diff.behav. in norm_Rational_mg 42";

  1753 val t = str2term

  1754 "9*(x^^^2 - 8*x+16)*(16*y - 16)/(4*(y^^^2 - 2*y+1)*(3*x - 12))";

  1755 val SOME (t',_) = rewrite_set_ thy false norm_Rational t;

  1756 ... non terminating.

  1757 val SOME (t',_) = rewrite_set_ thy false make_polynomial t;

  1758 "(-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)";

  1759 val SOME (t,_) = rewrite_set_ thy false cancel_p t';

  1760 ... non terminating.*)

  1762 (*SRD Schalk I, p.70 Nr. 472a *)

  1763 val t = str2term ("((8*x^^^2 - 32*y^^^2)/(2*x + 4*y))/" ^

  1764 		 "((4*x - 8*y)/(x + y))");

  1765 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1766 term2str t;

  1767 if (term2str t) =

  1768 "x + y"

  1769 then ()

  1770 else error "rational.sml: diff.behav. in norm_Rational_mg 43";

  1773 (*----------------------------------------------------------------------*)

  1774 (*---------------------- Einfache Dppelbrche --------------------------*)

  1775 (*----------------------------------------------------------------------*)

  1777 (*SRD Schalk I, p.69 Nr. 461a *)

  1778 val t = str2term

  1779 "(2/(x+3) + 2/(x - 3)) / (8*x/(x^^^2 - 9))";

  1780 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1781 term2str t;

  1782 if (term2str t) =

  1783 "1 / 2"

  1784 then ()

  1785 else error "rational.sml: diff.behav. in norm_Rational_mg 44";

  1787 (*SRD Schalk I, p.69 Nr. 464b *)

  1788 val t = str2term

  1789 "(a - a/(a - 2)) / (a + a/(a - 2))";

  1790 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1791 term2str t;

  1792 if (term2str t) =

  1793 "(3 + -1 * a) / (1 + -1 * a)"

  1794 then ()

  1795 else error "rational.sml: diff.behav. in norm_Rational_mg 45";

  1797 (*SRD Schalk I, p.69 Nr. 465b *)

  1798 val t = str2term

  1799 "((x+3*y)/9 + (4*y^^^2 - 9*z^^^2)/(16*x)) /(x/9+y/6+z/4)";

  1800 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1801 term2str t;

  1802 if (term2str t) =

  1803 "(4 * x + 6 * y + -9 * z) / (4 * x)"

  1804 then ()

  1805 else error "rational.sml: diff.behav. in norm_Rational_mg 46";

  1807 (*SRD Schalk I, p.69 Nr. 466b *)

  1808 val t = str2term

  1809 "((1 - 7*(x - 2)/(x^^^2 - 4)) / (6/(x+2))) / (3/(x+5)+30/(x^^^2 - 25))";

  1810 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1811 term2str t;

  1812 if (term2str t) =

  1813 "(25 + -10 * x + x ^^^ 2) / 18"

  1814 then ()

  1815 else error "rational.sml: diff.behav. in norm_Rational_mg 47";

  1817 (*SRD Schalk I, p.70 Nr. 469 *)

  1818 val t = str2term

  1819 "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))";

  1820 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1821 term2str t;

  1822 if (term2str t) =

  1823 "3 * b ^^^ 3 / (2 * a + -2 * b)"

  1824 then ()

  1825 else error "rational.sml: diff.behav. in norm_Rational_mg 48";

  1827 (*----------------------------------------------------------------------*)

  1828 (*---------------------- Mehrfache Dppelbrueche ------------------------*)

  1829 (*----------------------------------------------------------------------*)

  1831 (*SRD.test Schalk I, p.70 Nr. 476b *) (* Rechenzeit: 10 sec *)

  1832 (*WN060419 crashes with method 'simplify' ????SK*)

  1833 val t = str2term

  1834 "((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)";

  1835 val SOME (t,_) = rewrite_set_ thy false norm_Rational t;

  1836 if term2str t = "1 / (a ^^^ 2 + -1 * b ^^^ 2)" then ()

  1837 else error "rational.sml: diff.behav. in norm_Rational_mg 49";

  1839 "----- Schalk I, p.70 Nr. 477a";

  1840 (* MG Achtung: terme explodieren; Bsp zu komplex;

  1841    L???ung sollte (ziemlich grosser) Faktorisierter Ausdruck sein

  1842 val t = str2term "b*y/(b - 2*y)/((b^^^2 - y^^^2)/(b+2*y)) /" ^

  1843 		 "(b^^^2*y+b*y^^^2)*(a+x)^^^2/((b^^^2 - 4*y^^^2)*(a+2*x)^^^2)";

  1844 val SOME (t',_) = rewrite_set_ thy false norm_Rational t;

  1847 val t = str2term "b*y*(b+2*y)*(b^^^2 - 4*y^^^2)*(a+2*x)^^^2 / " ^

  1848 		 "((b - 2*y)*(b^^^2 - y^^^2)*(b^^^2*y+b*y^^^2)*(a+x)^^^2)";

  1849 val SOME (t,_) = rewrite_set_ thy false make_polynomial t;

  1850 ????SK ???MG*)

  1853 "----- Schalk I, p.70 Nr. 478b ----- Rechenzeit: 5 sec";

  1854 val t = str2term ("(a - (a*b+b^^^2)/(a+b))/(b+(a - b)/(1+(a+b)/(a - b))) / " ^

  1855 		 "((a - a^^^2/(a+b))/(a+(a*b)/(a - b)))");

  1856 val SOME (t',_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1857 if term2str t' =

  1858 "(2 * a ^^^ 3 + 2 * a ^^^ 2 * b) / (a ^^^ 2 * b + b ^^^ 3)"

  1859 then ()

  1860 else error "rational.sml: diff.behav. in norm_Rational_mg 51";

  1862 (* TODO.new_c:WN050820 STOP_REW_SUB introduced gave ...

  1863 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 ()

  1864 else error "rational.sml: works again";

  1865 re-outcommented with TODO.new_c: cvs before 071227, 11:50*)

  1869 (*Schalk I, p.70 Nr. 480a *)

  1870 (* Achtung: rechnet ewig; cancel_p kann nicht krzen: WN060831 nonterm.SK00

  1871 val t = str2term

  1872 "(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)))";

  1873 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1874 term2str t;

  1875 if (term2str t) =

  1877 then ()

  1878 else error "rational.sml: diff.behav. in norm_Rational_mg 52";

  1880 (*MG Berechne Zwischenergebnisse: WN060831 nonterm.SK00*)

  1881 val t = str2term

  1882 "(1/x+1/y+1/z)/(1/x - 1/y - 1/z)";

  1883 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1884 term2str t;

  1885 "(x ^^^ 2 * y ^^^ 2 * z + x ^^^ 2 * y * z ^^^ 2 + x * y ^^^ 2 * z ^^^ 2) /

  1886 (-1 * x ^^^ 2 * y ^^^ 2 * z + -1 * x ^^^ 2 * y * z ^^^ 2 + x * y ^^^ 2 * z ^^^ 2)";

  1887 val t = str2term

  1888 "2*x^^^2/(x^^^2 - z^^^2)/(x/(x+z)+x/(x - z))";

  1889 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1890 term2str t;

  1891 "1"

  1893 (* SK 1. Ausdruck kann nicht weiter gekrzt werden; cancel_p !!!*)

  1894 ###  rls: cancel_p on:

  1895 (x ^^^ 2 * (y ^^^ 2 * z) + x ^^^ 2 * (y * z ^^^ 2) + x * (y ^^^ 2 * z ^^^ 2)) /

  1896 (-1 * (x ^^^ 2 * (y ^^^ 2 * z)) + -1 * (x ^^^ 2 * (y * z ^^^ 2)) + x * (y ^^^ 2 * z ^^^ 2))

  1897 GC #3.61.81.101.197.17503:   (0 ms)

  1898 *** RATIONALS_DIRECT_CANCEL_EXCEPTION: Invalid fraction

  1900 val t = str2term

  1901 "(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))";

  1902 val SOME (t,_) = rewrite_set_ thy false cancel_p t;

  1903 term2str t;

  1904 (*uncaught exception nonexhaustive binding failure*)

  1906 (* Das kann er aber krzen !!????: *)

  1907 val t = str2term

  1908 "(x^^^2 * (y^^^2 * z) +  x * (y^^^2 * z^^^2)) / (-1 * (x^^^2 * (y * z^^^2)) + x * (y^^^2 * z^^^2))";

  1909 val SOME (t,_) = rewrite_set_ thy false cancel_p t;

  1910 term2str t;

  1911 "(-1 * (y * x) + -1 * (z * y)) / (1 * (z * x) + -1 * (z * y))";

  1912 *)

  1915 "-------- crucial examples ------------------------------";

  1916 "-------- crucial examples ------------------------------";

  1917 "-------- crucial examples ------------------------------";

  1918 (*Schalk I, p.60 Nr. 215d *)

  1919 (* Achtung: rechnet ewig ...

  1920 val t = str2term "(a-b)^^^3 * (x+y)^^^4 / ((x+y)^^^2 * (a-b)^^^5)";

  1921 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1922 term2str t; noterm.SK

  1923 *)

  1925 (* Kein Wunder, denn Z???ler und Nenner extra als Polynom dargestellt ergibt:*)

  1926 (*

  1927 val t = str2term "(a-b)^^^3 * (x+y)^^^4";

  1928 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1929 term2str t;

  1930 "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";

  1931 val t = str2term "((x+y)^^^2 * (a-b)^^^5)";

  1932 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1933 term2str t;

  1934 "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";

  1935 *)

  1936 (*anscheinend macht dem Rechner das Krzen diese Bruches keinen Spass mehr ...*)

  1938 (*--------------------------------------------------------------------*)

  1939 (*Schalk I, p.70 Nr. 480b

  1940 val t = str2term "((12*x*y/(9*x^^^2 - y^^^2))/" ^

  1941 		 "(1/(3*x - y)^^^2 - 1/(3*x + y)^^^2)) *" ^

  1942 		 "(1/(x - 5*y)^^^2 - 1/(x + 5*y)^^^2)/" ^

  1943 		 "(20*x*y/(x^^^2 - 25*y^^^2))";

  1944 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  1945 SK.nonterm

  1946 Kann nicht weiter vereinfacht werden !!!!?? *)

  1948 (*--------------------------------------------------------------------*)

  1949 "---- MGs test set";

  1950 val t = str2term " (1 + x^^^5) / (y + x) + x^^^3 / x ";

  1951 val SOME (t,_) = rewrite_set_ thy false common_nominator_p t;

  1952 if term2str t = "(1 + x ^^^ 3 + x ^^^ 5 + y * x ^^^ 2) / (x + y)" then()

  1953 else error "";

  1955 (*--------------------------------------------------------------------*)

  1956 (* cancel_p liefert nicht immer Polynomnormalform (2): WN060831???SK3b

  1957    ---> Sortierung FALSCH !!  *)

  1958 val t = str2term "b^^^3 * a^^^5/a ";

  1959 val SOME (t,_) = rewrite_set_ thy false cancel_p t;

  1960 term2str t;

  1961 "1 * (a^^^4 * b^^^3) / 1"; (*OK*)

  1963 val t = str2term "b^^^3 * a^^^5/b ";

  1964 val SOME (t,_) = rewrite_set_ thy false cancel_p t;

  1965 term2str t;

  1966 "1 * (b^^^2 * a^^^5) / 1"; (*cancel_p sortiert hier falsch um!*)

  1968 (* Problem liegt NICHT bei ord_make_polynomial! (siehe folgende Bsple) *)

  1969 (*

  1970 val x = str2term "x"; val bdv = str2term "bdv";

  1971 val t1 = str2term "b^^^2 * a^^^5";

  1972 val t2 = str2term "a^^^5 * b^^^2 ";

  1973 ord_make_polynomial false Rational.thy [(x,bdv)] (t1,t2); (*false*)

  1974 *)

  1975 (* ==> "b^^^2 * a^^^5" > "a^^^5 * b^^^2 " ... OK!*)

  1977 (*--------------------------------------------------------------------*)

  1978 (* cancel_p liefert nicht immer Polynomnormalform (2): WN060831???SK3c

  1979    ---> erzeugt berflssige "1 * ..."

  1981 val t = str2term "-1 / (3 + y)";

  1982 (*~~         *)

  1983 val SOME (t,_) = rewrite_set_ thy false cancel_p t;

  1984 term2str t;

  1985 "-1 / (3 + 1 * y)";

  1986 (********* Das ist das PROBLEM !!!!!!!??? *******************)

  1987 (* -1 im Z???ler der Angabe verursacht das Problem !*)

  1988 *)

  1990 (* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - *)

  1991 "----- MGs test set";

  1992 val t = str2term "(a^^^2 + -1)/(a+1)";

  1993 val SOME (t',_) = rewrite_set_ thy false cancel_p t;

  1994 if term2str t' = "(-1 + a) / 1" then ()

  1995 else error "rational.sml MG tests 3d";

  1997 "----- NOT TERMINATING ?: worked before 0707xx";

  1998 val t = str2term "(a^^^2 - 1)*(b + 1) / ((b^^^2 - 1)*(a+1))";

  1999 val SOME (t'',_) = rewrite_set_ thy false norm_Rational t;

  2000 if term2str t'' = "(1 + -1 * a) / (1 + -1 * b)" then ()

  2001 else error "rational.sml MG tests 3e";

  2003 "----- corrected SK060905";

  2004 val t = str2term "(4*x^^^2 - 20*x + 25)/(2*x - 5)^^^3";

  2005 val SOME (t',_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  2006 if term2str t' = "-1 / (5 + -2 * x)" then ()

  2007 else error "rational.sml corrected SK060905";

  2010 "-------- examples for Stefan Karnels thesis ------------";

  2011 "-------- examples for Stefan Karnels thesis ------------";

  2012 "-------- examples for Stefan Karnels thesis ------------";

  2013 (*SRAM Schalk I, p.69 Nr. 442b --- abgewandelt*)

  2014 val t = str2term

  2015 "(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))";

  2016 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  2017 term2str t;

  2018 if (term2str t) =

  2019 "5 * x ^^^ 2 / (b ^^^ 3 * c)"

  2020 then ()

  2021 else error "rational.sml: diff.behav. in norm_Rational_mg 53";

  2024 "-------- me Schalk I No.186 ----------------------------";

  2025 "-------- me Schalk I No.186 ----------------------------";

  2026 "-------- me Schalk I No.186 ----------------------------";

  2027 val fmz = ["Term ((14 * x * y) / ( x * y ))",

  2028 	   "normalform N"];

  2029 val (dI',pI',mI') =

  2030   ("Rational",["rational","simplification"],

  2031    ["simplification","of_rationals"]);

  2032 val p = e_pos'; val c = [];

  2033 val (p,_,f,nxt,_,pt) = CalcTreeTEST [(fmz, (dI',pI',mI'))];

  2034 val (p,_,f,nxt,_,pt) = me nxt p c pt;

  2035 val (p,_,f,nxt,_,pt) = me nxt p c pt;

  2036 val (p,_,f,nxt,_,pt) = me nxt p c pt;

  2037 val (p,_,f,nxt,_,pt) = me nxt p c pt;

  2038 val (p,_,f,nxt,_,pt) = me nxt p c pt;

  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; f2str f;

  2042 val (p,_,f,nxt,_,pt) = me nxt p c pt; f2str f;

  2043 val (p,_,f,nxt,_,pt) = me nxt p c pt; f2str f;(*++ for explicit script*)

  2044 val (p,_,f,nxt,_,pt) = me nxt p c pt; f2str f;(*++ for explicit script*)

  2045 case (f2str f, nxt) of

  2046     ("14", ("End_Proof'", _)) => ()

  2047   | _ => error "rational.sml diff.behav. in me Schalk I No.186";

  2050 "-------- interSteps ..Simp_Rat_Double_No-1.xml ---------";

  2051 "-------- interSteps ..Simp_Rat_Double_No-1.xml ---------";

  2052 "-------- interSteps ..Simp_Rat_Double_No-1.xml ---------";

  2053 states:=[];

  2054 CalcTree

  2055 [(["Term (((2 - x)/(2*a)) / (2*a/(x - 2)))", "normalform N"],

  2056   ("Rational",["rational","simplification"],

  2057   ["simplification","of_rationals"]))];

  2058 Iterator 1;

  2059 moveActiveRoot 1;

  2060 autoCalculate 1 CompleteCalc;

  2061 val ((pt,p),_) = get_calc 1; show_pt pt;

  2063 interSteps 1 ([1],Res);

  2064 val ((pt,p),_) = get_calc 1; show_pt pt;

  2067 "-------- interSteps ..Simp_Rat_Cancel_No-1.xml ---------";

  2068 "-------- interSteps ..Simp_Rat_Cancel_No-1.xml ---------";

  2069 "-------- interSteps ..Simp_Rat_Cancel_No-1.xml ---------";

  2070 states:=[];

  2071 CalcTree

  2072 [(["Term ((a^2 + -1*b^2) / (a^2 + -2*a*b + b^2))", "normalform N"],

  2073   ("Rational",["rational","simplification"],

  2074   ["simplification","of_rationals"]))];

  2075 Iterator 1;

  2076 moveActiveRoot 1;

  2077 autoCalculate 1 CompleteCalc;

  2078 val ((pt,p),_) = get_calc 1; show_pt pt;

  2079 (*========== inhibit exn 110314 ================================================

  2080 (*with explicit script done already... and removed [1,..] at below...

  2081 interSteps 1 ([1],Res);

  2082 val ((pt,p),_) = get_calc 1; show_pt pt;

  2083 *)

  2084 interSteps 1 ([2],Res);

  2085 val ((pt,p),_) = get_calc 1; show_pt pt;

  2087 interSteps 1 ([2,1],Res);

  2088 val ((pt,p),_) = get_calc 1; show_pt pt;

  2089 val newnds = children (get_nd pt [2,1]) (*see "fun detailrls"*);

  2090 (*if length newnds = 12 then () WN060905*)

  2091 if length newnds = 13 then ()

  2092 else error "rational.sml: interSteps cancel_p rev_rew_p";

  2094 val p = ([2,1,9],Res);

  2095 getTactic 1 p;

  2096 val (_, tac, _) = pt_extract (pt, p);

  2097 (*case tac of SOME (Rewrite ("sym_real_plus_binom_times1", _)) => ()

  2098 WN060905*)

  2099 case tac of SOME (Rewrite ("sym_real_add_mult_distrib2", _)) => ()

  2100 | _ => error "rational.sml: getTactic, sym_real_plus_binom_times1";

  2101 ============ inhibit exn 110314 ==============================================*)

  2104 "-------- investigate rulesets for cancel_p -------------";

  2105 "-------- investigate rulesets for cancel_p -------------";

  2106 "-------- investigate rulesets for cancel_p -------------";

  2107 val thy = @{theory "Rational"};

  2108 "---------------- (a^^^2 + -1*b^^^2) / (a^^^2 + -2*a*b + b^^^2)";

  2109 val t = str2term "(a^^^2 + -1*b^^^2) / (a^^^2 + -2*a*b + b^^^2)";

  2110 val tt = str2term "(1 * a + 1 * b) * (1 * a + -1 * b)"(*numerator only*);

  2111 "----- with rewrite_set_";

  2112 val SOME (tt',asm) = rewrite_set_ thy false make_polynomial tt;

  2113 term2str tt'= "a ^^^ 2 + -1 * b ^^^ 2" (*true*);

  2114 val tt = str2term "((1 * a + -1 * b) * (1 * a + -1 * b))"(*denominator only*);

  2115 val SOME (tt',asm) = rewrite_set_ thy false make_polynomial tt;

  2116 term2str tt' = "a ^^^ 2 + -2 * a * b + b ^^^ 2" (*true*);

  2118 "----- with make_deriv";

  2119 val SOME (tt, _) = factout_p_ thy t; term2str tt =

  2120 "(1 * a + 1 * b) * (1 * a + -1 * b) / ((1 * a + -1 * b) * (1 * a + -1 * b))";

  2121 (*

  2122 "--- with ruleset as before 060829";

  2123 val {rules, rew_ord=(_,ro),...} =

  2124     rep_rls (assoc_rls "make_polynomial");

  2125 val der = make_deriv thy Atools_erls rules ro NONE tt;

  2126 print_depth 99; map (term2str o #1) der; print_depth 3;

  2127 print_depth 99; map (rule2str o #2) der; print_depth 3;

  2128 ... did not terminate*)

  2129 "--- with simpler ruleset";

  2130 val {rules, rew_ord=(_,ro),...} =

  2131     rep_rls (assoc_rls "rev_rew_p");

  2132 val der = make_deriv thy Atools_erls rules ro NONE tt;

  2133 print_depth 99; writeln (deriv2str der); print_depth 3;

  2135 print_depth 99; map (term2str o #1) der; print_depth 3;

  2136 "...,(-1 * b ^^^ 2 + a ^^^ 2) / (-2 * (a * b) + a ^^^ 2 + (-1 * b) ^^^ 2) ]";

  2137 print_depth 99; map (rule2str o #2) der; print_depth 3;

  2138 print_depth 99; map (term2str o #1 o #3) der; print_depth 3;

  2140 val der = make_deriv thy Atools_erls rules ro NONE

  2141 		     (str2term "(1 * a + 1 * b) * (1 * a + -1 * b)");

  2142 print_depth 99; writeln (deriv2str der); print_depth 3;

  2144 val {rules, rew_ord=(_,ro),...} =

  2145     rep_rls (assoc_rls "rev_rew_p");

  2146 val der = make_deriv thy Atools_erls rules ro NONE

  2147 		     (str2term "(1 * a + -1 * b) * (1 * a + -1 * b)");

  2148 print_depth 99; writeln (deriv2str der); print_depth 3;

  2149 print_depth 99; map (term2str o #1) der; print_depth 3;

  2150 (*WN060829 ...postponed*)

  2153 "-------- investigate format of factout_ and factout_p_ -";

  2154 "-------- investigate format of factout_ and factout_p_ -";

  2155 "-------- investigate format of factout_ and factout_p_ -";

  2156 val {rules, rew_ord = (_,ro),...} = rep_rls (assoc_rls "make_polynomial");

  2157 val (thy, eval_rls) = (@{theory "Rational"}, Atools_erls)(*see 'fun init_state'*);

  2158 val Rrls {scr = Rfuns {init_state,...},...} = assoc_rls "cancel_p";

  2160 "----- see Rational.ML, local cancel_p, fun init_state";

  2161 val t = str2term "(a^^^2 + (-1)*b^^^2) / (a^^^2 + (-2)*a*b + b^^^2)";

  2162 val SOME (t',_) = factout_p_ thy t; term2str t';

  2163 (*

  2164 val rtas = reverse_deriv thy eval_rls rules ro NONE t';

  2165 writeln(trtas2str rst);

  2166 *)

  2169 "----- see Rational.ML, local cancel_p, fun init_state";

  2170 val t = str2term "a^^^2 / a";

  2171 val SOME (t',_) = factout_p_ thy t;

  2172 term2str t' = "a * a / (1 * a)" (*true*);

  2173 (*... can be canceled with

  2174 real_mult_div_cancel2 ?k ~= 0 ==> ?m * ?k / (?n * ?k) = ?m / ?n"*)

  2175 (* sml/ME/rewtools.sml:

  2176 val rtas = reverse_deriv thy Atools_erls rules ro NONE t';

  2177 writeln (deri2str rtas);

  2178 *)

  2181 "-------- SK 060904 ----------------------------------------------";

  2182 "----- order on polynomials -- input + output";

  2183 val thy = @{theory "Isac"};

  2184 val t = str2term "(a + -1 * b) / (-1 * a + b)";

  2185 val SOME (t', _) = factout_p_ thy t; term2str t';

  2186 val SOME (t', _) = cancel_p_ thy t; term2str t';

  2188 val t = str2term "a*b*c*d / (d*e*f*g)";

  2189 val SOME (t', _) = cancel_p_ thy t; term2str t';

  2191 val t = str2term "a*(b*(c*d)) / (b*(e*(f*g)))";

  2192 val SOME (t', _) = cancel_p_ thy t; term2str t';

  2193 (*???order.SK  ???*)

  2195 "----- SK060904-1a non-termination of cancel_p_ ? worked before 0707xx";

  2196 val t = str2term "(x^^^2 - 4)*(3 - y) / ((y^^^2 - 9)*(2+x))";

  2197 val SOME (t',_) = rewrite_set_ thy false norm_Rational t;

  2198 if term2str t' = "(2 + -1 * x) / (3 + y)" then ()

  2199 else error "rational.sml SK060904-1a worked since 0707xx";

  2201 "----- SK060904-1b non-termination of cancel_p_ ... worked before 0707xx";

  2202 val t = str2term ("(9 * a ^^^ 2 + -16 * b ^^^ 2) /" ^

  2203 "(36 * a^^^2 * c + (27 * a^^^2 * e + (-64 * b^^^2 * c + -48 * b^^^2 * e)))");

  2204 val SOME (t',_) = cancel_p_ thy t;

  2205 if term2str t' = "1 / (4 * c + 3 * e)" then ()

  2206 else error "rational.sml SK060904-1b";

  2209 "----- SK060904-2a non-termination of add_fraction_p_";

  2210 val t = str2term (" (a + b * x) / (a + -1 * (b * x)) +  " ^

  2211 		  " (-1 * a + b * x) / (a + b * x)      ");

  2212 (* nonterm.SK

  2213 val SOME (t',_) = rewrite_set_ thy false common_nominator_p t;

  2215 common_nominator_p_ thy t;

  2216 " (a + b * x)*(a + b * x) / ((a + -1 * (b * x))*(a + -1 * (b * x))) +  " ^

  2217 " (-1 * a + b * x)*(a + -1 * (b * x)) / ((a + b * x)*(-1 * a + b * x)) ";

  2219 add_fraction_p_ thy t;

  2220 " ((a + b * x)*(a + b * x)  +  (-1 * a + b * x)*(a + -1 * (b * x))) /" ^

  2221 " ((a + b * x)*(-1 * a + b * x))                                     ";

  2222 *)

  2225 "-------- how to stepwise construct Scripts -------------";

  2226 "-------- how to stepwise construct Scripts -------------";

  2227 "-------- how to stepwise construct Scripts -------------";

  2228 val thy = @{theory "Rational"};

  2229 (*no trailing _*)

  2230 val p1 = parse thy (

  2231 "Script SimplifyScript (t_t::real) =                             " ^

  2232 "  (Rewrite_Set discard_minus False                   " ^

  2233 "   t_t)");

  2235 (*required (): (Rewrite_Set discard_minus False)*)

  2236 val p2 = parse thy (

  2237 "Script SimplifyScript (t_t::real) =                             " ^

  2238 "  (Rewrite_Set discard_minus False                   " ^

  2239 "   t_t)");

  2241 val p3 = parse thy (

  2242 "Script SimplifyScript (t_t::real) =                             " ^

  2243 "  ((Rewrite_Set discard_minus False)                   " ^

  2244 "   t_t)");

  2246 val p4 = parse thy (

  2247 "Script SimplifyScript (t_t::real) =                             " ^

  2248 "  ((Rewrite_Set discard_minus False)                   " ^

  2249 "   t_t)");

  2251 val p5 = parse thy (

  2252 "Script SimplifyScript (t_t::real) =                             " ^

  2253 "  ((Try (Rewrite_Set discard_minus False)                   " ^

  2254 "    Try (Rewrite_Set discard_parentheses False))               " ^

  2255 "   t_t)");

  2257 val p6 = parse thy (

  2258 "Script SimplifyScript (t_t::real) =                             " ^

  2259 "  ((Try (Rewrite_Set discard_minus False) @@                   " ^

  2260 "    Try (Rewrite_Set rat_mult_poly False) @@                    " ^

  2261 "    Try (Rewrite_Set make_rat_poly_with_parentheses False) @@   " ^

  2262 "    Try (Rewrite_Set cancel_p_rls False) @@                     " ^

  2263 "    (Repeat                                                     " ^

  2264 "     ((Try (Rewrite_Set common_nominator_p_rls False) @@        " ^

  2265 "       Try (Rewrite_Set rat_mult_div_pow False) @@              " ^

  2266 "       Try (Rewrite_Set make_rat_poly_with_parentheses False) @@" ^

  2267 "       Try (Rewrite_Set cancel_p_rls False) @@                  " ^

  2268 "       Try (Rewrite_Set rat_reduce_1 False)))) @@               " ^

  2269 "    Try (Rewrite_Set discard_parentheses False))               " ^

  2270 "   t_t)"

  2271 );

  2273 "----------- get_denominator ----------------------------";

  2274 "----------- get_denominator ----------------------------";

  2275 "----------- get_denominator ----------------------------";

  2276 val thy = @{theory Isac};

  2277 val t = term_of (the (parse thy "get_denominator ((a +x)/b)"));

  2278 val SOME (_, t') = eval_get_denominator "" 0 t thy;

  2279 if term2str t' = "get_denominator ((a + x) / b) = b" then ()

  2280 else error "get_denominator ((a + x) / b) = b"

  2283 "--------- several errpats in complicated term -------------------";

  2284 "--------- several errpats in complicated term -------------------";

  2285 "--------- several errpats in complicated term -------------------";

  2286 (*TODO: instead of Gabriella's example here (27.Jul.12) find a simpler one*)

  2287 states:=[];

  2288 CalcTree

  2289 [(["Term ((5*b + 25)/(a^2 - b^2) * (a - b)/(5*b))", "normalform N"],

  2290   ("Rational",["rational","simplification"], ["simplification","of_rationals"]))];

  2291 Iterator 1;

  2292 moveActiveRoot 1;

  2293 autoCalculate 1 CompleteCalc;

  2294 val ((pt,p),_) = get_calc 1; show_pt pt;

  2296 "-------- nonterminating cancel_p, norm_Rational 2002 ---";

  2297 "-------- nonterminating cancel_p, norm_Rational 2002 ---";

  2298 "-------- nonterminating cancel_p, norm_Rational 2002 ---";

  2299 (*------------------------------------------------------------------------------------\

  2300 "--- WN121204: searched rational.sml for "nonterm", added new numbering" ^

  2301 "              and thoroughly tested with this numbering subsequently";

  2302 "--- 1 ---";

  2303 WN.2.6.03 from rlang.sml 56a

  2304 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)";

  2305 val NONE = rewrite_set_ thy false common_nominator_p t;

  2306 "--- 2 ---";

  2307 WN060831 nonterm.SK7

  2308 val t = str2term "(a + b * x) / (a + -1 * (b * x)) + (-1 * a + b * x) / (a + b * x)";

  2309 val NONE = add_fraction_p_ thy t;

  2310 "--- 3 ---";

  2311 nonterm.SK9 loops: cancel_p kann nicht weiter kuerzen!!! *)

  2312 val t'' = str2term "(a + b)/(x^^^2 - y^^^2) * ((x - y)^^^2/(a^^^2 - b^^^2))";

  2313 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t'';

  2314 "--- 4 ---";

  2315 val t = str2term

  2316 "(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)";

  2317 WN060831 nonterm.SK10

  2318 val SOME (t,_) = rewrite_set_ thy false cancel_p t;

  2319 term2str t;

  2320 "--- 5 ---";

  2321 val t = str2term

  2322 "(9*(x^^^2 - 8*x+16)/(4*(y^^^2 - 2*y+1)))/((3*x - 12)/(16*y - 16))";

  2323 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  2324 if term2str t =

  2325 then ()

  2326 else error "rational.sml: diff.behav. in norm_Rational_mg 42";

  2327 "--- 6 ---";

  2328 val t = str2term

  2329 "9*(x^^^2 - 8*x+16)*(16*y - 16)/(4*(y^^^2 - 2*y+1)*(3*x - 12))";

  2330 val SOME (t',_) = rewrite_set_ thy false norm_Rational t;

  2331 ... non terminating.

  2332 "--- 7 ---";

  2333 val SOME (t',_) = rewrite_set_ thy false make_polynomial t;

  2334 "(-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)";

  2335 val SOME (t,_) = rewrite_set_ thy false cancel_p t';

  2336 "--- 8 ---";

  2337 val t = str2term "(a-b)^^^3 * (x+y)^^^4 / ((x+y)^^^2 * (a-b)^^^5)";

  2338 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  2339 "--- 9 ---";

  2340 val t = str2term "((12*x*y/(9*x^^^2 - y^^^2))/" ^

  2341 		 "(1/(3*x - y)^^^2 - 1/(3*x + y)^^^2)) *" ^

  2342 		 "(1/(x - 5*y)^^^2 - 1/(x + 5*y)^^^2)/" ^

  2343 		 "(20*x*y/(x^^^2 - 25*y^^^2))";

  2344 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;

  2345 "--- 10 ---";

  2346 SK060904-2a non-termination of add_fraction_p_";

  2347 val t = str2term (" (a + b * x) / (a + -1 * (b * x)) +  " ^

  2348 		  " (-1 * a + b * x) / (a + b * x)      ");

  2349 (* nonterm.SK

  2350 val SOME (t',_) = rewrite_set_ thy false common_nominator_p t;

  2351 "--- 11 ---";

  2352 common_nominator_p_ thy t;

  2353 " (a + b * x)*(a + b * x) / ((a + -1 * (b * x))*(a + -1 * (b * x))) +  " ^

  2354 " (-1 * a + b * x)*(a + -1 * (b * x)) / ((a + b * x)*(-1 * a + b * x)) ";

  2355 "--- 12 ---";

  2356 add_fraction_p_ thy t;

  2357 " ((a + b * x)*(a + b * x)  +  (-1 * a + b * x)*(a + -1 * (b * x))) /" ^

  2358 " ((a + b * x)*(-1 * a + b * x))                                     ";

  2359 --------------------------------------------------------------------------------------/*)

  2361 (*------------------------------------------------------------------------------------\

  2362 "WN121205: thoroughly tested with the above numbering.";

  2363 "  errors in cancel_p: --- 4,5,6,7.";

  2364 "  error in common_nominator_p, common_nominator_p_: --- 10; 1,2?.";

  2365 "  errors caused by ruleset norm_Rational: --- 8,9.";

  2366 trace_rewrite := true;

  2368 "--- 1 ---: non-terminating with ### add_fract: done t22 ";

  2369 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)";

  2370 trace_rewrite:= true;

  2371 rewrite_set_ thy false common_nominator_p t;

  2373 "--- 2 ---: 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)";

  2375 add_fraction_p_ thy t;

  2377 "--- 3 ---: norm_Rational calls Rrls cancel_p with non-normalised polys";

  2378 val t = str2term "(a + b)/(x^^^2 - y^^^2) * ((x - y)^^^2/(a^^^2 - b^^^2))";

  2379 rewrite_set_ thy false norm_Rational t;

  2380 (*tracing end...####  rls: cancel_p on:

  2381 (a * x ^^^ 2 + -2 * (a * (x * y)) + a * y ^^^ 2 + b * x ^^^ 2 + -2 * (b * (x * y)) + b * y ^^^ 2) /

  2382 (a ^^^ 2 * x ^^^ 2 + -1 * (a ^^^ 2 * y ^^^ 2) + -1 * (b ^^^ 2 * x ^^^ 2) + b ^^^ 2 * y ^^^ 2) *)

  2384 "--- 4 ---: non-terminating with Rrls cancel_p on plausible input";

  2385 val t = str2term (

  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)");

  2388 rewrite_set_ thy false cancel_p t;

  2389 (*#  rls: cancel_p on:

  2390 (a * x ^^^ 2 + -2 * a * x * y + a * y ^^^ 2 + b * x ^^^ 2 + -2 * b * x * y + b * y ^^^ 2) /

  2391 (a ^^^ 2 * x ^^^ 2 + -1 * a ^^^ 2 * y ^^^ 2 + -1 * b ^^^ 2 * x ^^^ 2 + b ^^^ 2 * y ^^^ 2) *)

  2393 "--- 5 ---: non-terminating with Rrls cancel_p on plausible input";

  2394 val t = str2term "(9*(x^^^2 - 8*x+16)/(4*(y^^^2 - 2*y+1)))/((3*x - 12)/(16*y - 16))";

  2395 rewrite_set_ thy false norm_Rational t;

  2396 (*####  rls: cancel_p on:

  2397 (2304 + -1152 * x + -2304 * y + 144 * x ^^^ 2 + 1152 * (x * y) + -144 * (x ^^^ 2 * y)) /

  2398 (48 + -12 * x + -96 * y + 24 * (x * y) + 48 * y ^^^ 2 + -12 * (x * y ^^^ 2))  *)

  2400 "--- 6 ---: non-terminating with Rrls cancel_p on plausible input";

  2401 val t = str2term

  2402 "9*(x^^^2 - 8*x+16)*(16*y - 16)/(4*(y^^^2 - 2*y+1)*(3*x - 12))";

  2403 rewrite_set_ thy false norm_Rational t;

  2404 (*###  rls: cancel_p on: (-2304 + 1152 * x + 2304 * y + -144 * x ^^^ 2 + -1152 * (x * y) +

  2405  144 * (x ^^^ 2 * y)) /

  2406 (-48 + 12 * x + 96 * y + -24 * (x * y) + -48 * y ^^^ 2 + 12 * (x * y ^^^ 2)) *)

  2408 "--- 7 ---: non-terminating with Rrls cancel_p on plausible input";

  2409 val t' = str2term (

  2410 "(-2304 + 1152 * x + 2304 * y + -144 * x ^^^ 2 + -1152 * x * y + 144 * x ^^^ 2 * y) /"^

  2411 "(-48 + 12 * x + 96 * y + -24 * x * y + -48 * y ^^^ 2 + 12 * x * y ^^^ 2)");

  2412 rewrite_set_ thy false cancel_p t';

  2414 "--- 8 ---: bottom of output cannot be looked ad (due to looping?)";

  2415 val t = str2term "(a-b)^^^3 * (x+y)^^^4 / ((x+y)^^^2 * (a-b)^^^5)";

  2416 val SOME (t,_) = rewrite_set_ thy false norm_Rational t;

  2418 "--- 9 ---: probably error in norm_Rational";

  2419 val t = str2term (

  2420 "((12*x*y / (9*x^^^2 - y^^^2)) / (1/(3*x - y)^^^2 - 1/(3*x + y)^^^2)) *" ^

  2421 		"(1/(x - 5*y)^^^2 - 1/(x + 5*y)^^^2) / (20*x*y/(x^^^2 - 25*y^^^2))");

  2422 rewrite_set_ thy false norm_Rational t;

  2423 (*####  rls: cancel_p on: (19440 * (x ^^^ 8 * y ^^^ 2) + -490320 * (x ^^^ 6 * y ^^^ 4) +

  2424  108240 * (x ^^^ 4 * y ^^^ 6) +

  2425  -6000 * (x ^^^ 2 * y ^^^ 8)) /

  2426 (2160 * (x ^^^ 8 * y ^^^ 2) + -108240 * (x ^^^ 6 * y ^^^ 4) +

  2427  1362000 * (x ^^^ 4 * y ^^^ 6) +

  2428  -150000 * (x ^^^ 2 * y ^^^ 8)) *)

  2430 "--- 10 ---: non-terminating with ### add_fract: done t22: error in common_nominator_p ";

  2431 val t = str2term (" (a + b * x) / (a + -1 * (b * x)) + (-1 * a + b * x) / (a + b * x)");

  2432 rewrite_set_ thy false common_nominator_p t; (*### add_fract: done t22 *)

  2433 common_nominator_p_ thy t;                   (*loops without output*)

  2434 "--- reformulated 10:";

  2435 val t = str2term "(a + -1 * (b * x)) / (a + b * x)";

  2436 rewrite_set_ thy false cancel_p t = NONE;

  2437 "--- 11 ---";

  2438 "--- 12 ---";

  2439 "...both are to be considered after common_nominator_p_ is improved";

  2440 --------------------------------------------------------------------------------------/*)

  2441 ============ inhibit exn WN130824 TODO ======================================================*)

author	Walther Neuper <neuper@ist.tugraz.at>
	Wed, 28 Aug 2013 11:48:37 +0200
changeset 52095	c9fbb8171a0a
parent 52094	61cccc3f2f56
child 52096	ee2a5f066e44
permissions	-rw-r--r--