(* Title: tests for rationals

   Author: Stefan Karnel

   Copyright (c) Stefan Karnel 2002

   Use is subject to license terms.

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        10        20        30        40        50        60        70        80

LEGEND WN070906

   nonterm.SK   marks non-terminating examples

   ord.SK       PARTIALLY marks crucial ordering examples

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

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

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

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

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

"~~~~~BEGIN: decomment structure RationalI : RATIONALI ~~";

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

"-------- fun monom2term,  fun poly2term' ---------------";

"~~~~~END: decomment structure RationalI : RATIONALI ~~~~";

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

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

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

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

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

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

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

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

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

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

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

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

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

"~~~~~BEGIN: decomment structure RationalI : RATIONALI ~~";

(*.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    tests of internal functions: to make them work, 

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

(*##!!!

structure RationalI : RATIONALI =

struct

!!!##*)

(*##!!!

end;(*struct*)

open RationalI;

!!!##*)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~.*)

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

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

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

(*~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

writeln ("*********************   rational.sml - TESTS    *************************");

writeln ("***** divide tests *****");

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 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 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 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 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 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 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 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 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 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 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 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 error ("rational.sml: example failed");

writeln ("***** MV_CONTENT-TESTS *****");

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 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 error ("rational.sml: example failed");

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

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

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

writeln ("********************************************************");

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 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 monom2term,  fun poly2term' ---------------";

"-------- fun monom2term,  fun poly2term' ---------------";

"-------- fun monom2term,  fun poly2term' ---------------";

val t = monom2term ((3,[2,1,0]), ["c","b","a"](*reverse ???SK-*));

term2str t = "3 * (c ^^^ 2 * b)" (*true*);

val t = monom2term ((1,[1,0]), ["b","a"]);

term2str t = "b" (*true*);

val t = poly2term ([(1,[0,0,0]),(2,[1,0,0]),(3,[2,1,0]),(4,[3,2,1])], 

		   ["c","b","a"]);

term2str t = "1 + 2 * c + 3 * (c ^^^ 2 * b) + 4 * (c ^^^ 3 * (b ^^^ 2 * a))";

val t = poly2term ([(1,[1,0]),(1,[0,1])], ["b","a"]);

term2str t = "a + b" (*true*);

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~*)

"~~~~~END: decomment structure RationalI : RATIONALI ~~~~";

"~~~~~END: decomment structure RationalI : RATIONALI ~~~~";

"~~~~~END: decomment structure RationalI : RATIONALI ~~~~";

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

writeln ("***** mv_gcd-tests *****");

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 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 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 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 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 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 error ("rational.sml: example failed");

writeln ("***** kgv-tests *****");

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

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

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

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

writeln ("***** STEP_CANCEL_TESTS: *****");

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 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 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  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 ("Groups.times_class.times", [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)";

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

val t = str2term "(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' = "(9 - x ^^^ 2) / (9 - 6 * x + x ^^^ 2) +\n1" ^

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

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

val t = str2term "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' = "1 * (9 - 6 * x + x ^^^ 2) / (9 - 6 * x + x ^^^ 2) +\n" ^

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

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

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

(*WN030602 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 = str2term "(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' = "(y + x) * (y - x) / ((y - x) * (y - x))"(*true*);

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

val t = str2term "(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' =

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

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

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

val t = str2term "(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;

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

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

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

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

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

(*WN.16.10.02 WN070905 lexicographische Ordnung erhalten ! SK.ord*)

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

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

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

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

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

 *** RATIONALS_TERM2POLY_EXCEPTION: Invalid Polynomial 

WN.16.10.02 ?! + WN060831???SK4 

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

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

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

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

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

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

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

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

  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)";

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

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

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

val thy  = @{theory "Rational"};

val thy' = "Rational";

val rls' = "cancel";

val mp = "make_polynomial";

writeln ("example 186:");

writeln("a)");

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

rewrite_set_ thy false cancel (str2term e186a');

"@@@@@@@@@@@@@@";

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

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

  is_expanded (parse_rat "x * y");

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

else error "rational.sml cancel Schalk e186a";

writeln("b)");

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

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

writeln("c)");

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

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

    handle e => print_exn_G e;

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

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

else error "rational.sml cancel Schalk e186c";

writeln ("example 187:");

writeln("a)");

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

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

writeln("b)");

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

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

writeln("c)");

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

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

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

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

else error "rational.sml cancel Schalk e186d";

writeln "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 error "rational.sml: e188a new behaviour";

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

writeln("b)");

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

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

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

writeln("c)");

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

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

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

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 error "rational.sml: e188c new behaviour";

writeln ("example 190:");

writeln("c)");

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 error "rational.sml: e190c new behaviour";

writeln ("example 191:");

writeln("a)");

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 error "rational.sml: e191a new behaviour";

writeln("c)");

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

writeln ("example 192:");

writeln("b)");

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 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_ thy false make_polynomial t;

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";

writeln ("example 193:");

writeln("a)");

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

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

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

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

writeln("b)");

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

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

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

writeln("c)");

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, asm) = rewrite_set thy' false "cancel" wn01;

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

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

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

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

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

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

val rls' = "common_nominator_p";

writeln ("example 204:");

writeln("a)");

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

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

writeln("b)");

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

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

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

else error "rational.sml common_nominator_p example e204b";

writeln ("example 205:");

writeln("a)");

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

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

writeln("b)");

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

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

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

else error "rational.sml common_nominator_p example e204b";

writeln ("example 206:");

writeln("a)");

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'); 

writeln("b)");

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');

writeln ("example 207:");

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'); 

writeln ("example 208:");

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'); 

writeln ("example 209:");

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'); 

writeln ("example 210:");

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'); 

writeln ("example 211:");

writeln("a)"); 

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

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

writeln("b)");

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');

writeln ("example 212:");

writeln("a)");

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

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

writeln("b)");

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

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

writeln ("example 213:");

writeln("a)"); 

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'); 

writeln("b)"); 

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');

writeln ("example 214:");

writeln("a)");

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'); 

writeln("b)");

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');

writeln ("example 216:");

writeln("a)"); 

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');  

writeln("b)");

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');

writeln ("example 217:");

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";

writeln ("example 218:"); 

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'); 

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

else error "rationa.sml common_nominator example e218";

writeln ("example 219:");

writeln("a)");

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

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

writeln("b)");

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

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

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

else error "rationa.sml common_nominator example e219b";

writeln ("example 220:");

writeln("a)");

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

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

writeln("b)");

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

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

writeln ("example 221:");

writeln("a)");

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

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

writeln("b)");

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

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

writeln ("example 222:");

writeln("a)");

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');

writeln("b)");

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'); 

writeln ("example 225:");

writeln("a)");

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

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

writeln("b)");

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

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

writeln ("example 226:");

writeln("a)");

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');

writeln("b)"); 

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');  

writeln ("example 227:");

writeln("a)");

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');

writeln("b)");

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'); 

writeln ("example 228:");

writeln("a)");

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'); 

writeln("b)");

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');  

writeln ("example 229:");

writeln("a)");

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'); 

writeln("b)");

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'); 

writeln ("example 230:");

writeln("a)"); 

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');

writeln("b)");

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');

writeln ("example 231:");

writeln("a)");

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'); 

writeln("b)");

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');

writeln ("example 232:");

writeln("a)");

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'); 

writeln("b)");

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');

writeln ("example 233:");

writeln("a)");

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'); 

writeln("b)");

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');

writeln ("example 234:");

writeln("a)");

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'); 

writeln("b)"); 

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');  

writeln ("example 235:");

writeln("a)");

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'); 

writeln("b)"); 

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');  

writeln ("example 236:");

writeln("a)"); 

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');  

writeln("b)");   

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";

writeln ("example heuberger:");

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";

writeln ("example stiefel:");

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

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

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

else 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;

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

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

"-------- 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;

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

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

(*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 as (Thm (str, thm))) = nex revsets t;

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

   andalso string_of_thm thm = 

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

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

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

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

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;

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

then () 

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

(* find the next rule to apply *)

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

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

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

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

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

(*check the next rule*)

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

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

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

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

  val SOME r = nex revsets t;

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

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

else error "rational.sml locate_rule II";

  val SOME r = nex revsets t;

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

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

else error "rational.sml next_rule II";

  val SOME r = nex revsets t;

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

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

else error "rational.sml next_rule III";

  val SOME r = nex revsets t;

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

  val ss = term2str t;

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

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

"-------- '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 = str2term "(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' = "(3 + 1 * x) / (3 + -1 * x)";

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

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

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

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

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

(* WN.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;

WN060831 cycling "sym_order_mult_rls_" "sym_real_mult_assoc"

         as was with make_polynomial before ?!?*)

val SOME r = nex revsets t;

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

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

(*WN060831 *** id_of_thm

           Exception- ERROR raised ...

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;

*)

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

(*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 error "rational.sml 1";