(*.tests for rationals

    author: Stefan Karnel

    Copyright (c) Stefan Karnel 2002

    Use is subject to license terms.

 use"../smltest/IsacKnowledge/rational.sml";

 use"rational.sml";

 .*)

 (******************************************************************

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

  # exp_IsacCore_Simp_Rat_Cancel.xml     from rational.sml    (*SRC*)  10 exp

  # exp_IsacCore_Simp_Rat_Add.xml        from rational.sml    (*SRA*)  11 exp

  # exp_IsacCore_Simp_Rat_Mult.xml	from rational.sml    (*SRM*)   5 exp

  # exp_IsacCore_Simp_Rat_AddMult.xml	from rational.sml    (*SRAM*) 11 exp

  # exp_IsacCore_Simp_Rat_Double.xml	from rational.sml    (*SRD*)  12 exp

 *******************************************************************)

 "-----------------------------------------------------------------";

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

 "-----------------------------------------------------------------";

 "-------- ... missing WN060103 -----------------------------------";

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

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

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

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

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

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

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

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

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

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

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

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

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

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

 "-----------------------------------------------------------------";

 "-----------------------------------------------------------------";

 "-----------------------------------------------------------------";

 (*. tests of internal functions: to make them work, 

     out-comment (*!!!*) in knowledge/Rational.ML:

 (*!!!

 structure RationalI : RATIONALI =

 struct

 !!!*)

 (*!!!

 end;(*struct*)

 open RationalI;

 !!!*)

 print("\n\n*********************   rational.sml - TESTS    *************************\n\n");

 print("\n\n***** divide tests *****\n");

 val mv_pquot1 = (#1(mv_division([(1,[1,1,1]),(1,[1,1,0]),(1,[1,0,1]),(1,[0,0,0])],[(1,[1,1,0]),(1,[0,0,0])],LEX_)));

 (* result: [(1,[0,0,1]),(1,[0,0,0])] *)

 if mv_pquot1=[(1,[0,0,1]),(1,[0,0,0])] then () else raise error ("rational.sml: example failed");

 val mv_prest1 = (#2(mv_division([(1,[1,1,1]),(1,[1,1,0]),(1,[1,0,1]),(1,[0,0,0])],[(1,[1,1,0]),(1,[0,0,0])],LEX_)));

 (* result: [(1,[1,0,1]),(~1,[0,0,1])] *)

 if mv_prest1=[(1,[1,0,1]),(~1,[0,0,1])] then () else raise error ("rational.sml: example failed");

 val mv_pquot2 = (#1(mv_division([(4,[2]),(8,[1]),(16,[0])],[(1,[1]),(1,[0])],LEX_)));

 (* result: [(4,[1]),(4,[0])] *)

 if mv_pquot2=[(4,[1]),(4,[0])] then () else raise error ("rational.sml: example failed");

 val mv_prest2 = (#2(mv_division([(4,[2]),(8,[1]),(16,[0])],[(1,[1]),(1,[0])],LEX_)));

 (* result: [(12,[0]] *)

 if mv_prest2=[(12,[0])] then () else raise error ("rational.sml: example failed");

 val mv_pquot3 = (#1(mv_division([(4,[2]),(~4,[0])],[(2,[1]),(2,[0])],LEX_)));

 (* [(2,[1]),(~2,[0])] *)

 if mv_pquot3=[(2,[1]),(~2,[0])] then () else raise error ("rational.sml: example failed");

 val mv_prest3 = (#2(mv_division([(1,[2]),(~1,[0])],[(2,[1]),(2,[0])],LEX_)));

 (* [(1,[2]),(~1,[0])] *)

 if mv_prest3=[(1,[2]),(~1,[0])] then () else raise error ("rational.sml: example failed");

 val mv_pquot4 = (#1(mv_division([(3,[1,1,1]),(4,[1,0,1]),(3,[0,0,1])],[(2,[1,0,0]),(4,[0,0,1])],LEX_)));

 (* [(1,[0,1,1])] *)

 if mv_pquot4=[(1,[0,1,1])] then () else raise error ("rational.sml: example failed");

 val mv_prest4 = (#2(mv_division([(3,[1,1,1]),(4,[1,0,1]),(3,[0,0,1])],[(2,[1,0,0]),(4,[0,0,1])],GGO_)));

 (* [(1,[1,1,1]),(~4,[0,1,2]),(4,[1,0,1]),(3,[0,0,1])] *)

 if mv_prest4 =[(1,[1,1,1]),(~4,[0,1,2]),(4,[1,0,1]),(3,[0,0,1])] then () else raise error ("rational.sml: example failed");

 val mv_pquot5 = (#1(mv_division([(3,[1,1,1]),(4,[1,0,1]),(3,[0,0,1]),(6,[2,1,3]),(4,[0,4,1]),(1,[2,2,1])],[(1,[0,0,1])],LEX_)));

 (* [(1,[2,2,0]),(6,[2,1,2]),(3,[1,1,0]),(4,[1,0,0]),(4,[0,4,0]),(3,[0,0,0])]*)

 if mv_pquot5=[(1,[2,2,0]),(6,[2,1,2]),(3,[1,1,0]),(4,[1,0,0]),(4,[0,4,0]),(3,[0,0,0])] then () else raise error ("rational.sml: example failed");

 val mv_prest5 = (#2(mv_division([(3,[1,1,1]),(4,[1,0,1]),(3,[0,0,1]),(6,[2,1,3]),(4,[0,4,1]),(1,[2,2,1])],[(1,[0,0,1])],LEX_)));

 (* [] *)

 if mv_prest5=[] then () else raise error ("rational.sml: example failed");

 (* (x^2 + 2(a+1)x + (a^2+2a+1)) / (x+a+1) = x+a+1 *)

 val mv_pquot6 = (#1(mv_division([(1,[2,0,0]),(2,[1,1,0]),(2,[1,0,0]),(1,[0,2,0]),(2,[0,1,0]),(1,[0,0,0])],[(1,[1,0,0]),(1,[0,1,0]),(1,[0,0,0])],LEX_)));

 if mv_pquot6=[(1,[1,0,0]),(1,[0,1,0]),(1,[0,0,0])] then () else raise error ("rational.sml: example failed");

 val mv_prest6 = (#2(mv_division([(1,[2,0,0]),(2,[1,1,0]),(2,[1,0,0]),(1,[0,2,0]),(2,[0,1,0]),(1,[0,0,0])],[(1,[1,0,0]),(1,[0,1,0]),(1,[0,0,0])],LEX_)));

 if mv_prest6=[] then () else raise error ("rational.sml: example failed");

 print("\n\n***** MV_CONTENT-TESTS *****\n");

 val mv_cont1=mv_content([(1,[2,1]),(1,[2,0]),(1,[1,1]),(1,[1,0]),(1,[0,1]),(1,[0,0])]);

 (* [(1,[0,1]),(1,[0,0])] *)

 if  mv_cont1=[(1,[0,1]),(1,[0,0])] then () else raise error ("rational.sml: example failed");

 val mv_pp1=mv_pp([(1,[1,1]),(1,[1,0]),(1,[0,1]),(1,[0,0])]);

 (*[(1,[1,0]),(1,[0,0])]*)

 if mv_pp1=[(1,[1,0]),(1,[0,0])] then () else raise error ("rational.sml: example failed");

 val mv_cont2=mv_content([(2,[1]),(4,[0])]);

 (* [(2,[0])] *)

 if mv_cont2=[(2,[0])] then () else raise error ("rational.sml: example failed");

 val mv_pp2=mv_pp([(2,[1]),(4,[0])]);

 (* [(1,[1]),(2,[0])] *)

 if mv_pp2=[(1,[1]),(2,[0])] then () else raise error ("rational.sml: example failed");

 val mv_cont3=mv_content[(8,[2,1,1]),(12,[1,0,2]),(10,[2,2,0]),(16,[1,1,1])];

 (* [(2,[0,0,0])] *)

 if mv_cont3=[(2,[0,0,0])] then () else raise error ("rational.sml: example failed");

 val mv_pp3=mv_pp[(8,[2,1,1]),(12,[1,0,2]),(10,[2,2,0]),(16,[1,1,1])];

 (* [(5,[2,2,0]),(4,[2,1,1]),(8,[1,1,1]),(6,[1,0,2])] *)

 if mv_pp3=[(5,[2,2,0]),(4,[2,1,1]),(8,[1,1,1]),(6,[1,0,2])] then () else raise error ("rational.sml: example failed");

 val mv_cont4=mv_content[(2,[2,1,0]),(3,[1,0,1]),(2,[1,1,0]),(3,[0,0,1])];

 (* [(1,[0,0,0])] *)

 if mv_cont4=[(1,[0,0,0])] then () else raise error ("rational.sml: example failed");

 val mv_pp4=mv_pp [(2,[2,1,0]),(3,[1,0,1]),(2,[1,1,0]),(3,[0,0,1])];

 (* [(2,[2,1,0]),(2,[1,1,0]),(3,[1,0,1]),(3,[0,0,1])] *)

 if mv_pp4=[(2,[2,1,0]),(2,[1,1,0]),(3,[1,0,1]),(3,[0,0,1])] then () else raise error ("rational.sml: example failed");

 val con1=mv_content([(9,[2,0]),(15,[1,1]),(12,[1,0]),(6,[0,2]),(12,[0,1])]);

 (* [(3,[0,0])] *)

 if con1=[(3,[0,0])] then () else raise error ("rational.sml: example failed");

 val pp1=mv_pp([(9,[2,0]),(15,[1,1]),(12,[1,0]),(6,[0,2]),(12,[0,1])]);

 (* [(3,[2,0]),(5,[1,1]),(4,[1,0]),(2,[0,2]),(4,[0,1])] *)

 if pp1=[(3,[2,0]),(5,[1,1]),(4,[1,0]),(2,[0,2]),(4,[0,1])] then () else raise error ("rational.sml: example failed");

 val con2=mv_content([(1,[2,0]),(1,[1,1]),(1,[1,0]),(1,[0,2]),(1,[0,1])]);

 (* [(1,[0,0])] *)

 if con2=[(1,[0,0])] then () else raise error ("rational.sml: example failed");

 val pp2 =mv_pp([(1,[2,0]),(1,[1,1]),(1,[1,0]),(1,[0,2]),(1,[0,1])]);

 (* [(1,[2,0]),(1,[1,1]),(1,[1,0]),(1,[0,2]),(1,[0,1])] *)

 if pp2=[(1,[2,0]),(1,[1,1]),(1,[1,0]),(1,[0,2]),(1,[0,1])] then () else raise error ("rational.sml: example failed");

 val cont1 = mv_content [(1,[2,1,0]),(2,[2,1,0])];

 (* [(3,[0,1,0])] *)

 if cont1=[(3,[0,1,0])] then () else raise error ("rational.sml: example failed");

 val pp1 = mv_pp [(1,[2,1,0]),(2,[2,1,0])];

 (* [(1,[2,0,0])] *)

 if pp1=[(1,[2,0,0])] then () else raise error ("rational.sml: example failed");

 val cont2 = mv_content [(4,[1,2,0]),(2,[2,1,0])];

 (* [(2,[0,1,0])] *)

 if cont2=[(2,[0,1,0])] then () else raise error ("rational.sml: example failed");

 val pp2 = mv_pp [(4,[1,2,0]),(2,[2,1,0])];

 (* [(1,[2,0,0]),(2,[1,1,0])] *)

 if pp2=[(1,[2,0,0]),(2,[1,1,0])] then () else raise error ("rational.sml: example failed");

 print("\n\n\n\n********************************************************\n\n");

 val cont3=mv_content [(65,[3,2,2]),(52,[3,2,1]),(26,[3,1,2]),(~95,[2,2,3]),(~76,[2,2,2]),(35,[2,2,1]),(28,[2,2,0]),(~38,[2,1,3]),(14,[2,1,1])];

 if cont3=[(5,[0,2,1]),(4,[0,2,0]),(2,[0,1,1])] then () else raise error ("rational.sml: example failed");

 val pp3=mv_pp [(65,[3,2,2]),(52,[3,2,1]),(26,[3,1,2]),(~95,[2,2,3]),(~76,[2,2,2]),(35,[2,2,1]),(28,[2,2,0]),(~38,[2,1,3]),(14,[2,1,1])];

 ---------------------------------------------------------------------------*)

 fun parse_rat str = (term_of o the o (parse thy)) str;

 print("\n\n***** mv_gcd-tests *****\n");

 val ggt1 = mv_gcd [(4,[2,2]),(8,[1,1]),(4,[0,0])] [(2,[1,1]),(2,[0,0])];

 (* [(2,[1,1]),(2,[0,0])] *)

 if ggt1=[(2,[1,1]),(2,[0,0])] then () else raise error ("rational.sml: example failed");

 val ggt2 = mv_gcd [(8,[2,1,1]),(12,[1,0,2]),(10,[2,2,0]),(15,[1,1,1])] [(2,[2,1,0]),(3,[1,0,1]),(2,[1,1,0]),(3,[0,0,1])];

 (* [(2,[1,1,0]),(3,[0,0,1])] *)

 if ggt2=[(2,[1,1,0]),(3,[0,0,1])] then () else raise error ("rational.sml: example failed");

 val ggt3 = mv_gcd [(1,[2,0]),(~2,[1,1]),(1,[0,2])] [(1,[1,0]),(~1,[0,1])];

 (* [(1,[1,0]),(~1,[0,1])] *)

 if ggt3=[(1,[1,0]),(~1,[0,1])] then () else raise error ("rational.sml: example failed");

 val ggt4 = mv_gcd [(1,[2,1,0]),(2,[2,1,0])] [(5,[1,0,0])];

 (* [(1,[1,0,0])] *)

 if ggt4=[(1,[1,0,0])] then () else raise error ("rational.sml: example failed");

 val ggt5 = mv_gcd [(4,[2,0]),(~8,[1,1]),(4,[0,2])] [(1,[2,0]),(~1,[0,2])];

 (* [(1,[1,0]),(~1,[0,1])] *)

 if ggt5=[(1,[1,0]),(~1,[0,1])] then () else raise error ("rational.sml: example failed");

 val ggt6 = mv_gcd [(10,[2,1,1]),(14,[1,1,0]),(3,[1,0,1]),(20,[1,2,1])] [(5,[1,1,1]),(7,[2,1,1])];

 (* [(1,[0,0,0])] *)

 if ggt6=[(1,[1,0,0])] then () else raise error ("rational.sml: example failed");

 print("\n\n***** kgv-tests *****\n");

 val kgv1=mv_lcm [(10,[])] [(15,[])];

 (* [(30,[])] *)

 if kgv1=[(30,[])] then () else raise error ("rational.sml: example failed");

 val kgv2=mv_lcm [(1,[2,0]),(~2,[1,1]),(1,[0,2])] [(1,[1,0]),(~1,[0,1])];

 (* [(1,[2,0]),(~2,[1,1]),(1,[0,2])] *)

 if kgv2=[(1,[2,0]),(~2,[1,1]),(1,[0,2])] then () else raise error ("rational.sml: example failed");

 val kgv3=mv_lcm [(4,[2,0]),(~8,[1,1]),(4,[0,2])] [(1,[2,0]),(~1,[0,2])];

 (* [(4,[3,0]),(~4,[2,1]),(~4,[1,2]),(4,[0,3])] *)

 if kgv3=[(4,[3,0]),(~4,[2,1]),(~4,[1,2]),(4,[0,3])] then () else raise error ("rational.sml: example failed");

 (*!!!--------

 print("\n\n***** STEP_CANCEL_TESTS: *****\n");

 val term2 = (term_of o the o (parse thy)) " (9 * a^^^2 * b) /  (6 * a * c)";

 val div2 = term2str (step_cancel term2);

 if div2 =  "3 * (a * b) * (3 * a) / (2 * c * (3 * a))" then () else raise error ("rational.sml: example failed");

 val term1 = (term_of o the o (parse thy)) "(10 * a^^^2 * b * c + 14 * a * b + 3 * a * c + 20 * a * b^^^2 * c) / a";

 val div1  = term2str(step_cancel term1);

 if div1 =  "(3 * c + 14 * b + 20 * (b ^^^ 2 * c) + 10 * (a * (b * c))) * a / (1 * a)" then () else raise error ("rational.sml: example failed");

 val term3 = (term_of o the o (parse thy)) "(10 * a^^^2 * b * c) / (1 * x * y * z) ";

 val div3 = term2str(step_cancel term3);

 if div3="10 * a ^^^ 2 * b * c / (1 * x * y * z)" then () else  raise error ("rational.sml: example failed");

 --------------------------------------------------------------------------!!!*)

 (*-----versuche 13.3.03-----

  val t = str2term "1 - x^^^2 - 5 * x^^^5";

  val vs=(((map free2str) o vars) t);

  val Some ml = expanded2poly t vs;

  poly2term (ml, vs);

  poly2term'(rev(sort (mv_geq LEX_) (ml)),vs);

  poly2term'([(~5,[5]),(~1,[2]),(1,[0])], vs);

  monom2term((~5,[5]),vs);

  monom2term((~1,[2]),vs);

  val t' = monom2term((1,[0]),vs);(*uncaught exception LIST*)

  val (i,is) = (~1,[2]);

  val ttt = Const ("op *", [HOLogic.realT,HOLogic.realT]---> HOLogic.realT) $

 		   (Const ("uminus", HOLogic.realT --> HOLogic.realT) $

 		   Free ((str_of_int o abs) i, HOLogic.realT)) $

 		   powerproduct2term(is, vs);

  term2str ttt;

 -------versuche 13.3.03-----*)

  val t = str2term "1 - x^^^2 - 5 * x^^^5";

  val Some t' = expanded2polynomial t; term2str t';

 "1 + - 1 * x ^^^ 2 + - 5 * x ^^^ 5";

  val t = str2term "1 - x";

  val Some t' = expanded2polynomial t; term2str t';

 "1 + - 1 * x";

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

  val Some t' = expanded2polynomial t; term2str t';

 "1 + - 1 * x";

  val t = (term_of o the o (parse thy)) "1 + (-1) * x ^^^ 2 + (-5) * x ^^^5";

  val Some t' = polynomial2expanded t; term2str t';

 "1 - x ^^^ 2 - 5 * x ^^^ 5";

 " external calculating functions test ";

 " external calculating functions test ";

 " external calculating functions test ";

 val t1 = (term_of o the o (parse thy)) "((3 * x^^^2 + 6 *x + 3) / (2*x + 2))";

 val Some (t1',asm)= factout_p_ thy t1;

 term2str t1'; terms2str asm;

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

 "[]";

 val Some (t1',asm)= cancel_p_ thy t1;

 term2str t1'; terms2str asm;

 "(3 + 3 * x) / 2";

 "[\"1 + 1 * x ~= 0\"]";

 val t = (term_of o the o (parse thy)) "((-3 * x^^^2 + 6 *x - 3) / (2*x - 2))";

 val Some (t',asm)= cancel_ thy t; 

 term2str t'; terms2str asm;

 "(3 - 3 * x) / 2";

 "[\"-1 + x ~= 0\"]";

 val Some (t',asm)= factout_ thy t;

 term2str t'; terms2str asm;

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

 "[]";

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

 val Some (t',asm) = add_fraction_p_ thy t;

 term2str t'; terms2str asm;

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

 "[]";

 val Some (t',asm) = common_nominator_p_ thy t; 

 term2str t'; terms2str asm;

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

 "[]";

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

 val Some (t',asm) = add_fraction_ thy t;

 term2str t'; terms2str asm;

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

 "[]";

 val Some (t',asm) = common_nominator_ thy t; 

 term2str t'; terms2str asm;

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

 "[]";

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

 val Some (t',asm) = common_nominator_p_ thy t; 

 term2str t'; terms2str asm;

 "1 * (1 + -2 * x + 1 * x ^^^ 2) /\n((-1 + 1 * x) * (2 * ((-1 + 1 * x) * (1 + 1 * x)))) +\n(1 * (-1 + 1 * x ^^^ 2) /\n ((-1 + 1 * x) * (2 * ((-1 + 1 * x) * (1 + 1 * x)))) +\n (1 * (-2 + 2 * x) / ((-1 + 1 * x) * (2 * ((-1 + 1 * x) * (1 + 1 * x)))) +\n  1 * (#";                

 "[]";

 val Some (t',asm) = add_fraction_p_ thy t; 

 term2str t'; terms2str asm;

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

 "[\"1 + 1 * x ~= 0\"]";

 val Some(t',asm) = norm_rational_ thy t;

 term2str t'; terms2str asm;

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

 "[\"1 + 1 * x ~= 0\"]";

 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))";

 val Some (t3',_) = common_nominator_ thy t3; 

 val Some (t3'',_) = add_fraction_ thy t3; 

 (term2str t3'); 

 (term2str t3''); 

 val Some(t4,t5) = norm_expanded_rat_ thy t3;

 term2str t4;

 term2str (hd(t5));

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

   val Some (t',_) = factout_ thy t;

   val Some (t'',_) = cancel_ thy t;

   term2str t';

   term2str t'';

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

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

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

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

   val Some (t',_) = common_nominator_ thy t;

   val Some (t'',_) = add_fraction_ thy t;

   term2str t';

   term2str t'';

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

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

   (*WN.16.10.02 hinzugef"ugt -----vv---*)

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

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

   val Some (t',_) = common_nominator_ thy t;

   val Some (t'',_) = add_fraction_ thy t;

   term2str t';

   term2str t'';

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

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

   "6 / (3 - x)";

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

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

   val Some (t',_) = common_nominator_ thy t;

   val Some (t'',_) = add_fraction_ thy t;

   term2str t';

   term2str t'';

   "1 * (9 - 6 * x + x ^^^ 2) / (9 - 6 * x + x ^^^ 2) +\

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

   "6 / (3 - x)";

   (*WN.16.10.02 hinzugef"ugt -----^^---*)

   (*WN.2.6.03 added -------vv--- no rewrite -> None !*)

  val t = str2term "1 / a";

  val None =  cancel_p_ thy t;

  val None = rewrite_set_ thy false cancel_p t;

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

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

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

   val Some (t',_) = factout_ thy t;

   val Some (t'',_) = cancel_ thy t;

   term2str t';

   term2str t'';

   "(y + x) * (y - x) / ((y - x) * (y - x))";

   "(y + x) / (y - x)";

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

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

   val Some (t',_) = common_nominator_ thy t;

   val Some (t'',_) = add_fraction_ thy t;

   term2str t';

   term2str t'';

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

   \1 * ((-1) * x + y) / (((-1) * x + y) * ((-1) * x + y))";

   "((-1) - x - y) / (x - y)";

   (*WN.16.10.02     ^^^^^^^ Reihenfolge aus Angabe umgekehrt ?!*)

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

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

   val Some (t',_) = common_nominator_ thy t;

   val Some (t'',_) = add_fraction_ thy t;

   term2str t';

   term2str t'';

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

   \1 * ((-1) * y + x) / (((-1) * y + x) * ((-1) * y + x))";

   "((-1) - y - x) / (y - x)";

   (*WN.16.10.02     ^^^^^^^ lexicographische Ordnung ?!*)

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

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

   val Some (t',_) = norm_expanded_rat_ thy t;

   term2str t';

   "(y + x) / (y - x)";

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

   term2str t'';

   *** RATIONALS_TERM2POLY_EXCEPTION: Invalid Polynomial 

   WN.16.10.02 ?!*)

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

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

   val Some (t',_) = norm_expanded_rat_ thy t;

   term2str t';

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

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

   term2str t'';

   *** RATIONALS_TERM2POLY_EXCEPTION: Invalid Polynomial

   WN.16.10.02 ?!*)

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

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

   val Some (t',_) = norm_expanded_rat_ thy t;

   val Some (t'',_) = norm_rational_ thy t;

   term2str t';

   term2str t'';

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

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

 " examples from: Mathematik 1 Schalk Reniets Verlag ";

 " examples from: Mathematik 1 Schalk Reniets Verlag ";

 " examples from: Mathematik 1 Schalk Reniets Verlag ";

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

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

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

 val thy' = "Rational.thy";

 val rls' = "cancel";

 val mp = "make_polynomial";

 print("\n\nexample 186:\n");

 print("a)\n");

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

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

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

   is_expanded (parse_rat "x * y");

 print("b)\n");

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

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

 print("c)\n");

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

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

     handle e => print_exn e;

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

 atomt t;

 print("\n\nexample 187:\n");

 print("a)\n");

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

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

 print("b)\n");

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

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

 print("c)\n");

 val e187c'="(9 * x^^^5 * y^^^2 * z^^^4) / (15 * x^^^6 * y^^^3 * z )";(*SRC*)

 val e187c = the (rewrite_set thy' false "cancel" e187c');

 "example 188:";

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

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

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

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

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

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

 val Some (t,_) = rewrite_set thy' false mp 

 			     "(8*((-1) + x))/(9*((-1) + x))";

 print("b)\n");

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

 val Some (t,_) = rewrite_set thy' false "cancel" e188b';

 print("c)\n");

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

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

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

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

   false; -----------!!!*)

 val Some (t,_) = 

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

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

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

 print("\n\nexample 190:\n");

 print("c)\n");

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

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

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

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

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

 print("\n\nexample 191:\n");

 print("a)\n");

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

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

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

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

   false;

   is_expanded (parse_rat "x + y");

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

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

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

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

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

 print("c)\n");

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

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

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

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

   false;

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

   false;

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

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

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

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

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

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

 print("\n\nexample 192:\n");

 print("b)\n");

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

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

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

 -------------------*)

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

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

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

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

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

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

 val Some (t',_) = rewrite_set_ Isac.thy false make_polynomial t;

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

 print("\n\nexample 193:\n");

 print("a)\n");

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

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

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

 -------------------*)

 print("b)\n");

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

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

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

 print("c)\n");

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

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

 -------------------*)

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

 val Some (t,_) = rewrite_set thy' false "cancel" wn01;

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

 if t = "(-5 + 3 * x) / 1" then ()

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

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

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

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

 val rls' = "common_nominator_p";

 print("\n\nexample 204:\n");

 print("a)\n");

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

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

 print("b)\n");

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

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

 print("\n\nexample 205:\n");

 print("a)\n");

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

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

 print("b)\n");

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

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

 print("\n\nexample 206:\n");

 print("a)\n");

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

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

 print("b)\n");

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

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

 print("\n\nexample 207:\n");

 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)) ";

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

 print("\n\nexample 208:\n");

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

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

 print("\n\nexample 209:\n");

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

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

 print("\n\nexample 210:\n");

 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)) ";

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

 print("\n\nexample 211:\n");

 print("a)\n"); 

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

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

 print("b)\n");

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

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

 print("\n\nexample 212:\n");

 print("a)\n");

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

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

 print("b)\n");

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

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

 print("\n\nexample 213:\n");

 print("a)\n"); 

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

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

 print("b)\n"); 

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

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

 print("\n\nexample 214:\n");

 print("a)\n");

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

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

 print("b)\n");

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

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

 print("\n\nexample 216:\n");

 print("a)\n"); 

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

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

 print("b)\n");

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

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

 print("\n\nexample 217:\n");

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

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

 val rls' = "common_nominator";

 print("\n\nexample 218:\n"); 

 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))";

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

 print("\n\nexample 219:\n");

 print("a)\n");

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

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

 print("b)\n");

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

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

 print("\n\nexample 220:\n");

 print("a)\n");

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

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

 print("b)\n");

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

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

 print("\n\nexample 221:\n");

 print("a)\n");

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

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

 print("b)\n");

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

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

 print("\n\nexample 222:\n");

 print("a)\n");

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

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

 print("b)\n");

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

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

 print("\n\nexample 225:\n");

 print("a)\n");

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

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

 print("b)\n");

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

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

 print("\n\nexample 226:\n");

 print("a)\n");

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

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

 print("b)\n"); 

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

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

 print("\n\nexample 227:\n");

 print("a)\n");

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

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

 print("b)\n");

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

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

 print("\n\nexample 228:\n");

 print("a)\n");

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

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

 print("b)\n");

 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))";

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

 print("\n\nexample 229:\n");

 print("a)\n");

 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))";

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

 print("b)\n");

 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))"; 

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

 print("\n\nexample 230:\n");

 print("a)\n"); 

 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))";

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

 print("b)\n");

 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))";

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

 print("\n\nexample 231:\n");

 print("a)\n");

 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))";

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

 print("b)\n");

 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))";

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

 print("\n\nexample 232:\n");

 print("a)\n");

 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))";

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

 print("b)\n");

 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))";

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

 print("\n\nexample 233:\n");

 print("a)\n");

 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))";

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

 print("b)\n");

 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))";

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

 print("\n\nexample 234:\n");

 print("a)\n");

 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))";

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

 print("b)\n"); 

 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)) ";

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

 print("\n\nexample 235:\n");

 print("a)\n");

 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))";

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

 print("b)\n"); 

 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)) ";

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

 print("\n\nexample 236:\n");

 print("a)\n"); 

 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))";

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

 print("b)\n");   

 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)) ";

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

 val rls' = "cancel";

 print("\n\nexample heuberger:\n");

 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)";

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

 val rls' = "common_nominator_p";

 print("\n\nexample stiefel:\n");

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

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

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

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

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

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

 val Some (t',_) = factout_ thy t;

 term2str t';

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

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

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

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

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

   these are defined in Rationals.ML and stored in 

   the 'reverse-ruleset' cancel*)

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

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

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

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

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

 (*normal_form produces the result in ONE step*)

   val Some (t',_) = nor t;

   term2str t';

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

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

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

   val Some r = nex revsets t;

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

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

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

   there might be several rule applications inbetween,

   which are listed after the head in reverse order*)

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

   term2str t;

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

 (*find the next rule to apply*)

   val Some r = nex revsets t;

   (*val r = Thm ("sym_#power_3_2","9 = 3 ^^^ 2") : rule*)

 (*check the next rule*)

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

   term2str t;

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

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

   val Some r = nex revsets t;

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

   term2str t;

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

   val Some r = nex revsets t;

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

   term2str t;

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

   val Some r = nex revsets t;

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

   term2str t;

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

   val Some r = nex revsets t;

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

   val ss = term2str t;

   if ss = "(3 - x) / (3 + x)" then ()

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

   terms2str asm; 

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

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

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

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

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

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

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

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

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

   (*normal_form produces the result in ONE step*)

   val Some (t',_) = nor t; term2str t';

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

 val Some (t', asm) = cancel_p_ thy t; term2str t'; terms2str asm;

 (* WN.14.3.03: rewrite__ Thm | Calc | missing: Rls_  FIXXXXME --------

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

 ---------------------------------------------------------------------*)

 (*

  val [rs] = revsets;

  filter_out (eq_Thms ["sym_real_add_zero_left",

 		      "sym_real_mult_0",

 		      "sym_real_mult_1"]) rs;

  10.10.02: dieser Fall terminiert nicht 

            (make_polynomial enth"alt zu viele rules)

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

 print_depth 99; Rfuns; print_depth 3;

   val Some r = nex revsets t;

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

   term2str t;

   val Some r = nex revsets t;

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

   term2str t;

  ------ revset ----------------------------------------------------

 /// [Thm ("sym_real_add_zero_left","?z = 0 + ?z"),

 ///  Thm ("sym_real_mult_0","0 = 0 * ?z"),

 !    Thm ("sym_#mult_2_(-3)","(-6) * x = 2 * ((-3) * x)"),

 !    Thm ("sym_#add_(-3)_3","0 = (-3) + 3"),

 ?    Thm ("sym_real_num_collect_assoc",

        "[| ?l is_const; ?m is_const |]

   	==> (?l + ?m) * ?n + ?k = ?l * ?n + (?m * ?n + ?k)"),

 OK   Thm ("sym_real_mult_2_assoc","2 * ?z1.0 + ?k = ?z1.0 + (?z1.0 + ?k)"),

 OK   Thm ("sym_real_add_left_commute","?y + (?x + ?z) = ?x + (?y + ?z)"),

 ///  Thm ("sym_real_mult_1","?z = 1 * ?z"),

 !    Thm ("sym_#power_3_2","9 = 3 ^^^ 2"),

 !    Thm ("sym_#mult_-1_-1","1 * x ^^^ 2 = -1 * (-1 * x ^^^ 2)"),

 !    Thm ("sym_#mult_-1_3","(-3) * x = -1 * (3 * x)"),

 OK   Thm ("realpow_twoI","?r1 ^^^ 2 = ?r1 * ?r1"  [.]),

 OK   Thm ("sym_real_add_assoc",

       "?z1.0 + (?z2.0 + ?z3.0) = ?z1.0 + ?z2.0 + ?z3.0"),

 OK   Thm

      ("sym_real_mult_assoc","?z1.0 * (?z2.0 * ?z3.0) = ?z1.0 * ?z2.0 * ?z3.0"),

 OK   Thm ("sym_real_mult_left_commute",

       "?z2.0 * (?z1.0 * ?z3.0) = ?z1.0 * (?z2.0 * ?z3.0)"),

 OK   Thm ("sym_real_mult_commute","?w * ?z = ?z * ?w"),

 ?    Thm ("sym_real_add_mult_distrib2",

       "?w * ?z1.0 + ?w * ?z2.0 = ?w * (?z1.0 + ?z2.0)"),

 ?    Thm ("sym_real_add_mult_distrib",

       "?z1.0 * ?w + ?z2.0 * ?w = (?z1.0 + ?z2.0) * ?w"),

 OK   Thm ("real_mult_div_cancel2","?k ~= 0 ==> ?m * ?k / (?n * ?k) = ?m / ?n")]

  -------- revset ---------------------------------------------------- 

   val t = (term_of o the o (parse thy)) "(-6) * x";

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

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

   val thm = (mk_thm thy "(-6) * x = 2 * ((-3) * x)") 

       handle e => print_exn e;

   val Some (t',_) = rewrite_ thy e_rew_ord e_rls false thm t;     

   term2str t';

 ----------------------------------------------------------------------*)

 print "\n\n\n******************  all tests successfull  *************************\n";

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

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

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

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

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

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

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

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

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

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

 else raise error "rational.sml 2";

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

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

 (*before 040209:if term2str t' = "(23 + (35 * x + -72 * x ^^^ 2)) / 1"then()*) 

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

 else raise error "rational.sml 3";

 trace_rewrite:=true;

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

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

 "True";

 val t = str2term "1 < 2";

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

 "True";

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

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

 			   (num_str realpow_def_atom) t;

 term2str t';

 trace_rewrite:=false;

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

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

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

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

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

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

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

 else raise error "rational.sml 5";

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

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

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

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

 else raise error "rational.sml 6";

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

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

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

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

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

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

 		 \ (1 + x)";(*. a/b : c/d translated to a/b * d/c .*)

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

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

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

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

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

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

 		 \((4*x - 8*y)/(x + y))";

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

 "(-32 * y ^^^ 3 +\n (8 * x ^^^ 3 + (-32 * (x * y ^^^ 2) + 8 * (y * x ^^^ 2)))) /\n(-32 * y ^^^ 2 + 8 * x ^^^ 2)";

 (*###########################statt "x + y" poly_order notwendig!*) 

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

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

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

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

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

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

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

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

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

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

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

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

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

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

 "(-500 * (x * y ^^^ 3) /\n (x ^^^ 4 + (625 * y ^^^ 4 + -50 * (x ^^^ 2 * y ^^^ 2))) +\n 20 * (x * (y * x ^^^ 2)) /\n (x ^^^ 4 + (625 * y ^^^ 4 + -50 * (x ^^^ 2 * y ^^^ 2)))) /\n(20 * (x * y))";

 trace_rewrite:=true;

 trace_rewrite:=false;

 (*WN.17.3.03 =========================================================^^^---*)

 (*WN.2.6.03 from rlang.sml 56a =======================================vvv---*)

 (*... takes uncredible much time:*)

 (*

 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)";

 val None = rewrite_set_ thy false common_nominator_p t;

 writeln

 "----- rational.sml: add_fraction_p_ throws overflow exn ! -----\n\

 \----- rational.sml: add_fraction_p_ throws overflow exn ! -----\n\

 \----- rational.sml: add_fraction_p_ throws overflow exn ! -----\n\

 \----- rational.sml: add_fraction_p_ throws overflow exn ! -----\n\

 \----- rational.sml: add_fraction_p_ throws overflow exn ! -----\n";

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

 val None = add_fraction_p_ thy t; 

 *)

 (*

 GC #1.48.88.223.2775.290178:   (0 ms)

 GC #1.48.88.223.2776.290363:   (0 ms)

 GC #1.48.88.223.2777.290712:   (0 ms)

 GC #1.48.88.223.2778.291102:   (0 ms)

 GC #1.48.88.223.2779.291684:   (0 ms)

 GC #1.48.88.223.2780.292389:   (0 ms)

 GC #1.48.88.223.2781.293163:   (0 ms)

 GC #1.48.88.223.2782.294133:   (0 ms)

 GC #1.48.88.223.2783.295181:   (0 ms)

 *)

 (*WN.2.6.03 from rlang.sml 56a =======================================^^^---*)

 (* 25.08.03:  Durchlauf(rechen)zeit: 1 min *)

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

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

 (*                  Simplifier für beliebige Buchterme                 *) 

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

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

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

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

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

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

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

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

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

 term2str t;

 if (term2str t) = 

 "19 / 21"

 then ()

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

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

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

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

 term2str t;

 if (term2str t) = 

 "45 / 2"

 then ()

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

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

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

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

 (* e190c Stefan K.*)

 val t = str2term

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

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

 term2str t;

 if (term2str t) = 

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

 then ()

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

 (* e192b Stefan K.*)

 val t = str2term

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

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

 term2str t;

 if (term2str t) = 

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

 then ()

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

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

 val t = str2term 

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

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

 term2str t;

 if (term2str t) =

 "-1"

 then ()

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

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

 val t = str2term 

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

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

 term2str t;

 if (term2str t) =

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

 then ()

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

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

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

 (*val t = str2term 

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

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

 term2str t;

 if (term2str t) =

 "(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)"

 then ()

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

 *)

 (*val t = str2term 

 "(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)"

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

 term2str t;*)

 (* uncaught exception nonexhaustive binding failure

    raised at: stdIn:93.1-93.51 *)

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

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

 (*val t = str2term 

 "18*(a+b)^^^3*(a - b)^^^2/(72*(a - b)^^^3*(a+b)^^^2)";

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

 term2str t;

 if (term2str t) =

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

 then ()

 else raise error "rational.sml.sml: diff.behav. in norm_Rational_mg 8";

 *)

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

 val t = str2term 

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

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

 term2str t;

 if (term2str t) =

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

 then ()

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

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

 val t = str2term 

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

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

 term2str t;

 if (term2str t) =

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

 then ()

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

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

 val t = str2term 

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

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

 term2str t;

 if (term2str t) =

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

 then ()

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

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

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

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

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

 val t = str2term 

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

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

 term2str t;

 if (term2str t) =

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

 then ()

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

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

 val t = str2term 

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

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

 term2str t;

 if (term2str t) =

 "4 / c"

 then ()

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

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

 val t = str2term 

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

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

 term2str t;

 if (term2str t) =

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

 then ()

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

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

 val t = str2term 

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

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

 term2str t;

 if (term2str t) =

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

 then ()

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

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

 val t = str2term 

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

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

 term2str t;

 if (term2str t) =

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

 then ()

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

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

 val t = str2term 

 "(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)";

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

 term2str t;

 if (term2str t) =

 "1"

 then ()

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

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

 val t = str2term 

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

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

 term2str t;

 if (term2str t) =

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

 then ()

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

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

 val t = str2term 

 "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";

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

 term2str t;

 if (term2str t) =

 "a + 3 * b"

 then ()

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

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

 val t = str2term 

 "(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)";

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

 term2str t;

 if (term2str t) =

 "0"

 then ()

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

 (*Eigenes*)

 val t = str2term 

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

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

 term2str t;

 if (term2str t) =

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

 then ()

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

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

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

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

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

 val t = str2term 

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

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

 term2str t;

 if (term2str t) =

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

 then ()

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

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

 (*WN060420 crashes with method 'simplify' in 

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

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

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

 term2str t;

 if (term2str t) =

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

 then ()

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

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

 (* SK loops: rechnet ewig; cancel_p hängt sich auf...

 val t = str2term 

 "(3*a - 4*b)/(4*c+3*e) * (3*a+4*b)/(9*a^^^2 - 16*b^^^2)";

 val Some (t,_) = 

 rewrite_set_ thy false norm_Rational(*_mg*) t;

 term2str t;

 SK loops

 val t = str2term 

 "(3*a - 4*b) * (3*a+4*b)/((4*c+3*e)*(9*a^^^2 - 16*b^^^2))";

 val Some (t,_) = 

 rewrite_set_ thy false make_polynomial t;

 term2str t;

 "(9 * a ^^^ 2 + -16 * b ^^^ 2) /

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

 val Some (t,_) = 

 rewrite_set_ thy false cancel_p t;

 term2str t;

 *)

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

 (* SK loops, Falsches Ergebnis: cancel_p kann nicht weiter kürzen!!!

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

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

 term2str t;

 "(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)";

 *)

 (*

 val t = str2term 

 "(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)";

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

 term2str t;

 uncaught exception nonexhaustive binding failure

   raised at: stdIn:270.1-270.51

 *)

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

 val t = str2term 

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

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

 term2str t;

 if (term2str t) =

 "x ^^^ 2"

 then ()

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

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

 val t = str2term 

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

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

 term2str t;

 if (term2str t) =

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

 then ()

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

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

 val t = str2term 

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

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

 term2str t;

 if (term2str t) =

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

 then ()

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

 (*Schalk I, p.68 Nr. 440b *)

 (* Achtung: rechnet ewig; cancel_p hängt sich auf...

 val t = str2term 

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

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

 term2str t;

 if (term2str t) =

 then ()

 else raise error "rational.sml.sml: diff.behav. in norm_Rational_mg 27";

 val t = str2term 

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

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

 term2str t;

 "(-9 * a ^^^ 3 * b + -9 * a ^^^ 2 * b ^^^ 2 + a ^^^ 5 * b + a ^^^ 4 * b ^^^ 2)/

 (3 * a ^^^ 3 * b + -3 * a * b ^^^ 3 + a ^^^ 4 * b + -1 * a ^^^ 2 * b ^^^ 3)";

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

 term2str t;

 *)

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

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

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

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

 (*--------- Gemeinsamer Nenner und Multiplikation von Brüchen ----------*)

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

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

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

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

 term2str t;

 if (term2str t) =

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

 then ()

 else raise error "rational.sml.sml: diff.behav. in norm_Rational_mg 28";

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

 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)";

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

 term2str t;

 if (term2str t) =

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

 then ()

 else raise error "rational.sml.sml: diff.behav. in norm_Rational_mg 29";

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

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

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

 term2str t;

 if (term2str t) =

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

 then ()

 else raise error "rational.sml.sml: diff.behav. in norm_Rational_mg 30";

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

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

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

 term2str t;

 if (term2str t) =

 "a ^^^ 3 / 27"

 then ()

 else raise error "rational.sml.sml: diff.behav. in norm_Rational_mg 31";

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

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

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

 term2str t;

 if (term2str t) =

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

 then ()

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

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

 val t = str2term 

 "(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)";

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

 term2str t;

 if (term2str t) =

 "(-81 * x ^^^ 4 + 16 * x ^^^ 8 * y ^^^ 4) / (81 * y ^^^ 8)"

 then ()

 else raise error "rational.sml.sml: diff.behav. in norm_Rational_mg 33";

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

 val t = str2term 

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

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

 term2str t;

 if (term2str t) =

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

 then ()

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

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

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

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

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

 val t = str2term 

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

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

 term2str t;

 if (term2str t) = 

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

 then ()

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

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

 val t = str2term 

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

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

 term2str t;

 if (term2str t) = 

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

 then ()

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

 (*Schalk I, p.69 Nr. 455b *)

 (* Achtung: Endlosschleife

 val t = str2term 

 "(x^^^2 - 4)/(y^^^2 - 9)/((2+x)/(3 - y))";

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

 term2str t;

 if (term2str t) = 

 then ()

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

 val t = str2term 

 "(x^^^2 - 4)*(3 - y)/((y^^^2 - 9)*(2+x))";

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

 term2str t;

 "(-12 + 4 * y + 3 * x ^^^ 2 + -1 * x ^^^ 2 * y) /

 (-18 + -9 * x + 2 * y ^^^ 2 + x * y ^^^ 2)"

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

 term2str t;

 (* Achtung: rechnet ewig; cancel_p hängt sich auf...*)

 val t = str2term 

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

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

 term2str t;

 (*ENDLOSSCHLEIFE!!!*)

 val t = str2term 

 "-1 / (3 + y)";

 (*~~         *)

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

 term2str t;

 "-1 / (3 + 1 * y)";

 (********* Das ist das PROBLEM !!!!!!!??? *******************)

 (* -1 im Zähler der Angabe verursacht das Problem !*)

 *)

 (*SRD Schalk I, p.69 Nr. 456b *)

 val t = str2term 

 "(b^^^3 - b^^^2)/(b^^^2+b)/(b^^^2 - 1)";

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

 term2str t;

 if (term2str t) = 

 "b / (1 + 2 * b + b ^^^ 2)"

 then ()

 else raise error "rational.sml.sml: diff.behav. in norm_Rational_mg 38";

 (*SRD Schalk I, p.69 Nr. 457b *)

 val t = str2term 

 "(16*a^^^2 - 9*b^^^2)/(2*a+3*a*b) / ((4*a+3*b)/(4*a^^^2 - 9*a^^^2*b^^^2))";

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

 term2str t;

 if (term2str t) = 

 "8 * a ^^^ 2 + -6 * a * b + -12 * a ^^^ 2 * b + 9 * a * b ^^^ 2"

 then ()

 else raise error "rational.sml.sml: diff.behav. in norm_Rational_mg 39";

 (*Schalk I, p.69 Nr. 458b *)

 (* Achtung: rechnet ewig; cancel_p hängt sich auf...

 val t = str2term 

 "(2*a^^^2*x - a^^^2)/(a*x - b*x) / (b^^^2*(2*x - 1)/(x*(a - b)))";

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

 term2str t;

 if (term2str t) = 

 then ()

 else raise error "rational.sml.sml: diff.behav. in norm_Rational_mg 40";

 *)

 (*SRD Schalk I, p.69 Nr. 459b *)

 val t = str2term 

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

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

 term2str t;

 if (term2str t) = 

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

 then ()

 else raise error "rational.sml.sml: diff.behav. in norm_Rational_mg 41";

 (*Schalk I, p.69 Nr. 460b *)

 (* Achtung: rechnet ewig; cancel_p hängt sich auf...

 val t = str2term 

 "(9*(x^^^2 - 8*x+16)/(4*(y^^^2 - 2*y+1)))/((3*x - 12)/(16*y - 16))";

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

 term2str t;

 if (term2str t) = 

 then ()

 else raise error "rational.sml.sml: diff.behav. in norm_Rational_mg 42";

 val t = str2term 

 "9*(x^^^2 - 8*x+16)*(16*y - 16)/(4*(y^^^2 - 2*y+1)*(3*x - 12))";

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

 term2str t;

 "(-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)";

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

 term2str t;

 *)

 (*SRD Schalk I, p.70 Nr. 472a *)

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

 		 \((4*x - 8*y)/(x + y))";

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

 term2str t;

 if (term2str t) = 

 "x + y"

 then ()

 else raise error "rational.sml.sml: diff.behav. in norm_Rational_mg 43";

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

 (*---------------------- Einfache Dppelbrüche --------------------------*)

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

 (*SRD Schalk I, p.69 Nr. 461a *)

 val t = str2term 

 "(2/(x+3) + 2/(x - 3)) / (8*x/(x^^^2 - 9))";

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

 term2str t;

 if (term2str t) = 

 "1 / 2"

 then ()

 else raise error "rational.sml.sml: diff.behav. in norm_Rational_mg 44";

 (*SRD Schalk I, p.69 Nr. 464b *)

 val t = str2term 

 "(a - a/(a - 2)) / (a + a/(a - 2))";

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

 term2str t;

 if (term2str t) = 

 "(3 + -1 * a) / (1 + -1 * a)"

 then ()

 else raise error "rational.sml.sml: diff.behav. in norm_Rational_mg 45";

 (*SRD Schalk I, p.69 Nr. 465b *)

 val t = str2term 

 "((x+3*y)/9 + (4*y^^^2 - 9*z^^^2)/(16*x)) /(x/9+y/6+z/4)";

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

 term2str t;

 if (term2str t) = 

 "(4 * x + 6 * y + -9 * z) / (4 * x)"

 then ()

 else raise error "rational.sml.sml: diff.behav. in norm_Rational_mg 46";

 (*SRD Schalk I, p.69 Nr. 466b *)

 val t = str2term 

 "((1 - 7*(x - 2)/(x^^^2 - 4)) / (6/(x+2))) / (3/(x+5)+30/(x^^^2 - 25))";

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

 term2str t;

 if (term2str t) = 

 "(25 + -10 * x + x ^^^ 2) / 18"

 then ()

 else raise error "rational.sml.sml: diff.behav. in norm_Rational_mg 47";

 (*SRD Schalk I, p.70 Nr. 469 *)

 val t = str2term 

 "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))";

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

 term2str t;

 if (term2str t) = 

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

 then ()

 else raise error "rational.sml.sml: diff.behav. in norm_Rational_mg 48";

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

 (*---------------------- Mehrfache Dppelbrüche -------------------------*)

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

 (*SRD.test Schalk I, p.70 Nr. 476b *) (* Rechenzeit: 10 sec *)

 (*WN060419 crashes with method 'simplify'*)

 val t = str2term 

 "((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)";

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

 term2str t;

 if (term2str t) = 

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

 then ()

 else raise error "rational.sml.sml: diff.behav. in norm_Rational_mg 49";

 (*Schalk I, p.70 Nr. 477a *)

 (* Achtung: rechnet ewig; Bsp zu komplex; 

    Lösung sollte (ziemlich grosser) Faktorisierter Ausdruck sein 

 val t = str2term 

 "b*y/(b - 2*y)/((b^^^2 - y^^^2)/(b+2*y)) / (b^^^2*y+b*y^^^2)*(a+x)^^^2/((b^^^2 - 4*y^^^2)*(a+2*x)^^^2)";

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

 term2str t;

 if (term2str t) = 

 then ()

 else raise error "rational.sml.sml: diff.behav. in norm_Rational_mg 50";

 val t = str2term 

 "b*y*(b+2*y)*(b^^^2 - 4*y^^^2)*(a+2*x)^^^2 / ((b - 2*y)*(b^^^2 - y^^^2)*(b^^^2*y+b*y^^^2)*(a+x)^^^2)";

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

 term2str t;

 "(a ^^^ 2 * b ^^^ 4 * y + 2 * a ^^^ 2 * b ^^^ 3 * y ^^^ 2 +\n -4 * a ^^^ 2 * b ^^^ 2 * y ^^^ 3 +\n -8 * a ^^^ 2 * b * y ^^^ 4 +\n 4 * a * b ^^^ 4 * x * y +\n 8 * a * b ^^^ 3 * x * y ^^^ 2 +\n -16 * a * b ^^^ 2 * x * y ^^^ 3 +\n -32 * a * b * x * y ^^^ 4 +\n 4 * b ^^^ 4 * x ^^^ 2 * y +\n 8 * b ^^^ 3 * x ^^^ 2 * y ^^^ 2 +\n -16 * b ^^^ 2 * x ^^^ 2 * y ^^^ 3 +\n -32 * b * x ^^^ 2 * y ^^^ 4) 

 /\n(a ^^^ 2 * b ^^^ 5 * y + -1 * a ^^^ 2 * b ^^^ 4 * y ^^^ 2 +\n -3 * a ^^^ 2 * b ^^^ 3 * y ^^^ 3 +\n a ^^^ 2 * b ^^^ 2 * y ^^^ 4 +\n 2 * a ^^^ 2 * b * y ^^^ 5 +\n 2 * a * b ^^^ 5 * x * y +\n -2 * a * b ^^^ 4 * x * y ^^^ 2 +\n -6 * a * b ^^^ 3 * x * y ^^^ 3 +\n 2 * a * b ^^^ 2 * x * y ^^^ 4 +\n 4 * a * b * x * y ^^^ 5 +\n b ^^^ 5 * x ^^^ 2 * y +\n -1 * b ^^^ 4 * x ^^^ 2 * y ^^^ 2 +\n -3 * b ^^^ 3 * x ^^^ 2 * y ^^^ 3 +\n b ^^^ 2 * x ^^^ 2 * y ^^^ 4 +\n 2 * b * x ^^^ 2 * y ^^^ 5)";

 *)

 (*Schalk I, p.70 Nr. 478b *) (* Rechenzeit: 5 sec *)

 val t = str2term 

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

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

 term2str t;

 (* WN050820 was OK until STOP_REW_SUB introduced -- the ONLY change !!!

 if (term2str t) = 

 "(2 * a ^^^ 3 + 2 * a ^^^ 2 * b) / (a ^^^ 2 * b + b ^^^ 3)"

 then ()

 else raise error "rational.sml.sml: diff.behav. in norm_Rational_mg 51";

 -----------------------------------------------------------------------*)

 (*Schalk I, p.70 Nr. 480a *)

 (* SK Achtung: rechnet ewig; cancel_p kann nicht kürzen: 

 val t = str2term 

 "(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)))";

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

 term2str t;

 if (term2str t) = 

 then ()

 else raise error "rational.sml.sml: diff.behav. in norm_Rational_mg 52";

 (* Berechne Zwischenergebnisse:*)

 val t = str2term 

 "(1/x+1/y+1/z)/(1/x - 1/y - 1/z)";

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

 term2str t;

 "(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)";

 val t = str2term 

 "2*x^^^2/(x^^^2 - z^^^2)/(x/(x+z)+x/(x - z))";

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

 term2str t;

 "1"

 (* SK 1. Ausdruck kann nicht weiter gekürzt werden; cancel_p !!!*)

 ###  rls: cancel_p on: 

 (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))

 GC #3.61.81.101.197.17503:   (0 ms)

 *** RATIONALS_DIRECT_CANCEL_EXCEPTION: Invalid fraction

 val t = str2term 

 "(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))";

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

 term2str t;

 (*uncaught exception nonexhaustive binding failure*)

 (* Das kann er aber kürzen !!???: *)

 val t = str2term 

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

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

 term2str t;

 "(-1 * (y * x) + -1 * (z * y)) / (1 * (z * x) + -1 * (z * y))";

 *)

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

 (*----------------------- Problem-Beispiele --------------------------*)

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

 (*Schalk I, p.60 Nr. 215d *)

 (* Achtung: rechnet ewig ...

 val t = str2term "(a-b)^^^3 * (x+y)^^^4 / ((x+y)^^^2 * (a-b)^^^5)";

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

 term2str t;

 *)

 (* Kein Wunder, denn Zähler und Nenner extra als Polynom dargestellt ergibt:*)

 (*

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

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

 term2str t;

 "a^^^3 * x^^^4 + 4 * a^^^3 * x^^^3 * y +\n6 * a^^^3 * x^^^2 * y^^^2 +\n4 * a^^^3 * x * y^^^3 +\na^^^3 * y^^^4 +\n-3 * a^^^2 * b * x^^^4 +\n-12 * a^^^2 * b * x^^^3 * y +\n-18 * a^^^2 * b * x^^^2 * y^^^2 +\n-12 * a^^^2 * b * x * y^^^3 +\n-3 * a^^^2 * b * y^^^4 +\n3 * a * b^^^2 * x^^^4 +\n12 * a * b^^^2 * x^^^3 * y +\n18 * a * b^^^2 * x^^^2 * y^^^2 +\n12 * a * b^^^2 * x * y^^^3 +\n3 * a * b^^^2 * y^^^4 +\n-1 * b^^^3 * x^^^4 +\n-4 * b^^^3 * x^^^3 * y +\n-6 * b^^^3 * x^^^2 * y^^^2 +\n-4 * b^^^3 * x * y^^^3 +\n-1 * b^^^3 * y^^^4";

 val t = str2term "((x+y)^^^2 * (a-b)^^^5)";

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

 term2str t;

 "a^^^5 * x^^^2 + 2 * a^^^5 * x * y + a^^^5 * y^^^2 +\n-5 * a^^^4 * b * x^^^2 +\n-10 * a^^^4 * b * x * y +\n-5 * a^^^4 * b * y^^^2 +\n10 * a^^^3 * b^^^2 * x^^^2 +\n20 * a^^^3 * b^^^2 * x * y +\n10 * a^^^3 * b^^^2 * y^^^2 +\n-10 * a^^^2 * b^^^3 * x^^^2 +\n-20 * a^^^2 * b^^^3 * x * y +\n-10 * a^^^2 * b^^^3 * y^^^2 +\n5 * a * b^^^4 * x^^^2 +\n10 * a * b^^^4 * x * y +\n5 * a * b^^^4 * y^^^2 +\n-1 * b^^^5 * x^^^2 +\n-2 * b^^^5 * x * y +\n-1 * b^^^5 * y^^^2";

 *)

 (*anscheinend macht dem Rechner das Kürzen diese Bruches keinen Spass mehr ...*)

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

 (*Schalk I, p.70 Nr. 480b *)

 (*

 trace_rewrite:=false; (*sonst dauert Berechnung sehr lange!*)

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

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

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

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

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

 term2str t;

 "((180 * x^^^4 * y + 675 * x^^^3 * y^^^2 + 1105 * x^^^2 * y^^^3 +\n  -75 * x * y^^^4 +\n  -125 * y^^^5) /\n (x^^^3 + 5 * x^^^2 * y + -25 * x * y^^^2 + -125 * y^^^3) +\n (225 * x^^^2 * y^^^2 + -25 * y^^^4) /\n (x^^^2 + 10 * x * y + 25 * y^^^2)) /\n(20 * x * y)";

 *)

 (* Kann nicht weiter vereinfacht werden !!!!?? *)

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

 (*Seltsames in common_nominator_p: *)

 (*

 val t = str2term " (1 + x^^^5) / (x + y) + x^^^3 / x ";

 (*                                  ~~~~~  !!! *)

 val Some (t,_) = rewrite_set_ thy false common_nominator_p t;

 term2str t;

 *)

 (*### add_fraction_p_ called

 uncaught exception nonexhaustive binding failure

   raised at: stdIn:175.1-175.61*)

 val t = str2term " (1 + x^^^5) / (y + x) + x^^^3 / x ";

 (*                                  ~~~~~  !!! *)

 val Some (t,_) = rewrite_set_ thy false common_nominator_p t;

 term2str t;

 "(1 + 1 * x^^^3 + 1 * x^^^5 + 1 * (y * x^^^2)) / (1 * x + 1 * y)";

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

 (* cancel_p liefert nicht immer Polynomnormalform (2):

    ---> Sortierung FALSCH !!  *)

 val t = str2term

  "b^^^3 * a^^^5/a ";

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

 term2str t;

 "1 * (a^^^4 * b^^^3) / 1"; (*OK*)

 val t = str2term

  "b^^^3 * a^^^5/b ";

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

 term2str t;

 "1 * (b^^^2 * a^^^5) / 1"; (*cancel_p sortiert hier falsch um!*)

 (* Problem liegt NICHT bei ord_make_polynomial! (siehe folgende Bsple) *)

 (*

 val x = str2term "x"; val bdv = str2term "bdv";

 val t1 = str2term "b^^^2 * a^^^5";

 val t2 = str2term "a^^^5 * b^^^2 ";

 ord_make_polynomial false Rational.thy [(x,bdv)] (t1,t2); (*false*)

 *)

 (* ==> "b^^^2 * a^^^5" > "a^^^5 * b^^^2 " ... OK!*)

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

 (* cancel_p liefert nicht immer Polynomnormalform (2):

    ---> erzeugt überflüssige "1 * ..."

 val t = str2term 

 "-1 / (3 + y)";

 (*~~         *)

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

 term2str t;

 "-1 / (3 + 1 * y)";

 (********* Das ist das PROBLEM !!!!!!!??? *******************)

 (* -1 im Zähler der Angabe verursacht das Problem !*)

 *)

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

 (*Eigenes*)

 (* Achtung: Endlosschleife bei cancel_p:

 val t = str2term 

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

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

 term2str t;

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

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

 term2str t;

 "(-1 + 1 * a) / 1"; (*OK*)

 val t = str2term 

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

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

 term2str t;

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

 (-1 + -1 * a + b^^^2 + a * b^^^2)"

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

 term2str t;

 (* Endlosschleife*)

 *)

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

 (*                      EHEMALIG (MG 2004) FEHLER                     *)

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

 (* cancel_p liefert nicht immer Polynomnormalform (2):

    ---> Minus (binärer Operator) wird erzeugt !!!  *)

 (*val t = str2term 

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

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

 term2str t;

 "-1 / (5 - 2 * x)";*)

 (*      ~~~        *)

 (*aber wenn im Zähler kein Minus tritt Problem nicht auf !!??:*)

 (*val t = str2term 

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

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

 term2str t;

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

 (* ACHTUNG: Endlosschleife !!! *)

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

 (*val t = str2term 

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

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

 *)

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

 "--------------------------------------------------------------------";

 "----------------------- Muster-Beispiele fuer DA -------------------";

 "--------------------------------------------------------------------";

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

 val t = str2term 

 "(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))";

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

 term2str t;

 if (term2str t) =

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

 then ()

 else raise error "rational.sml.sml: diff.behav. in norm_Rational_mg 53";

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

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

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

 val fmz = ["term ((14 * x * y) / ( x * y ))",

 	   "normalform N"];

 val (dI',pI',mI') =

   ("Rational.thy",["rational","simplification"],

    ["simplification","of_rationals"]);

 val p = e_pos'; val c = []; 

 val (p,_,f,nxt,_,pt) = CalcTreeTEST [(fmz, (dI',pI',mI'))];

 val (p,_,f,nxt,_,pt) = me nxt p c pt;

 val (p,_,f,nxt,_,pt) = me nxt p c pt;

 val (p,_,f,nxt,_,pt) = me nxt p c pt;

 val (p,_,f,nxt,_,pt) = me nxt p c pt;

 val (p,_,f,nxt,_,pt) = me nxt p c pt;

 val (p,_,f,nxt,_,pt) = me nxt p c pt;

 val (p,_,f,nxt,_,pt) = me nxt p c pt;

 val (p,_,f,nxt,_,pt) = me nxt p c pt; f2str f;

 val (p,_,f,nxt,_,pt) = me nxt p c pt; f2str f;

 if f2str f = "14" then ()

 else raise error "rational.sml diff.behav. in me Schalk I No.186";

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

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

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

 states:=[];

 CalcTree

 [(["term (((2 - x)/(2*a)) / (2*a/(x - 2)))", "normalform N"], 

   ("Rational.thy",["rational","simplification"],

   ["simplification","of_rationals"]))];

 Iterator 1;

 moveActiveRoot 1;

 autoCalculate 1 CompleteCalc;

 val ((pt,p),_) = get_calc 1; show_pt pt;

 interSteps 1 ([1],Res);

 val ((pt,p),_) = get_calc 1; show_pt pt;

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

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

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

 states:=[];

 CalcTree

 [(["term ((14 * x * y) / ( x * y ))", "normalform N"], 

   ("Rational.thy",["rational","simplification"],

   ["simplification","of_rationals"]))];

 Iterator 1;

 moveActiveRoot 1;

 autoCalculate 1 CompleteCalc;

 val ((pt,p),_) = get_calc 1; show_pt pt;

 interSteps 1 ([1],Res);

 val ((pt,p),_) = get_calc 1; show_pt pt;

 interSteps 1 ([1,2],Res);

 val ((pt,p),_) = get_calc 1; show_pt pt;

 (*WN060823 cancel_p_ init_state does not terminate, see Rational.ML

 interSteps 1 ([1,2,1],Res);

 val ((pt,p),_) = get_calc 1; show_pt pt;

 *)