(* tests for rationals

   Stefan Karnel

   2002

   use"../kbtest/rational.sml";

   use"kbtest/rational.sml";

   use"rational.sml";

*)

(*--------------------------------15.10.02---

(* tests*)

print("\n\n*********************   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 error ("Test 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 ("Test 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 ("Test 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 ("Test 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 ("Test 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 ("Test 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 ("Test 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 ("Test 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 ("Test 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 ("Test 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 ("Test 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 ("Test failed");

(* Exception tests *)

(* mv_division ([(1,[0,0,0])],[(0,[1,2,3])],LEX_); *)

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 error ("Test 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 ("Test failed");

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

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

if mv_cont2=[(2,[0])] then () else error ("Test failed");

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

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

if mv_pp2=[(1,[1]),(2,[0])] then () else error ("Test 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 ("Test 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 ("Test 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 ("Test 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 ("Test 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 ("Test 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 ("Test 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 ("Test 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 ("Test 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 ("Test 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 ("Test 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 ("Test 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 ("Test 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=[(1,[0,1,0])] then () else error ("Test 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])];

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

val ggt1 = gcd_poly [(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 ("Test failed");

val ggt2 = gcd_poly [(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 ("Test failed");

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

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

if ggt3=[(1,[1,0,0]),(~1,[0,0,1])] then () else error ("Test failed");

val ggt4 = gcd_poly [(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 ("Test failed");

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

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

if ggt5=[(1,[1,0,0]),(~1,[0,0,1])] then () else error ("Test failed");

val ggt6 = gcd_poly [(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 ("Test failed");

val ggt7 = gcd_poly [(~169,[4,3,3]),(273,[4,2,2]),(247,[3,3,4]),(~91,[3,3,2]),(78,[3,3,1]),(~399,[3,2,3]),(147,[3,2,1]),(~114,[2,3,2]),(42,[2,3,0])]  [(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])];

(* [(13,[3,1,1]),(~19,[2,1,2]),(7,[2,1,0])] *)

if ggt7=[(13,[3,1,1]),(~19,[2,1,2]),(7,[2,1,0])] then () else error ("Test failed");

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

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

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

if kgv1=[(30,[])] then () else error ("Test failed");

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

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

if kgv2=[(1,[2,0,0]),(~2,[1,0,1]),(1,[0,0,2])] then () else error ("Test failed");

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

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

if kgv3=[(4,[3,0,0]),(~4,[2,0,1]),(~4,[1,0,2]),(4,[0,0,3])] then () else error ("Test failed");

(*

print("***** TERM2POLY-TESTS *****\n"); 

val t0 = (term_of o the o (parse thy)) "3 * 4";

val t1 = (term_of o the o (parse thy)) "27";

val t11= (term_of o the o (parse thy)) "27 * c";

val t2 = (term_of o the o (parse thy)) "b";

val t23= (term_of o the o (parse thy)) "c^^^7";

val t24= (term_of o the o (parse thy)) "5 * c^^^7";

val t26= (term_of o the o (parse thy)) "a * b"; 

val t3 = (term_of o the o (parse thy)) "2  +  a";

val t4 = (term_of o the o (parse thy)) "b  +  a";

val t5 = (term_of o the o (parse thy)) "2  +  a^^^3";*)

val t6 = (term_of o the o (parse thy)) "5*a^^^2*b^^^3*c+4*a^^^4*b+2*a*c";

val t7 = (term_of o the o (parse thy)) "a-b";

(*

(the o term2poly) t0;

(the o term2poly) t1;

(the o term2poly) t11;

(the o term2poly) t2;

(the o term2poly) t23;

(the o term2poly) t24;

(the o term2poly) t26;

(the o term2poly) t3;

(the o term2poly) t4;

(the o term2poly) t5;

(the o term2poly) t6;

(the o term2poly) t7;*)

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   = step_cancel term2;

atomt div2; 

*)

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  = step_cancel term1;

(*if div1 =  (term_of o the o (parse thy)) "((10 * a * b * c + 14 * b + 3 * c + 20 * b^^^2 * c) * a) /// (1 * a)" then () else raise error ("Test failed");*)

val term2 = (term_of o the o (parse thy)) "(10 * a^^^2 * b * c + 14 * a * b + 3 * a * c + 20 * a * b^^^2 * c) /// (5 * a * b * c  +  7 * a^^^2 * b * c) ";

val div2  = step_cancel term2;

(*if div2 = ([(10,[1,1,1]),(20,[0,2,1]),(14,[0,1,0]),(3,[0,0,1])],[(1,[1,0,0])],[(7,[1,1,1]),(5,[0,1,1])]) then () else raise error ("Test failed");*)

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

val div3 = step_cancel term3;

(*val mul1=(term_of o the o (parse thy)) "(5*a*b*c+4*a*b+2*a*c)"; 

val mul2=(the (term2poly((term_of o the o (parse thy)) "13*a^^^2*b*c+7*a*b-19*a*b*c^^^2")));

val mul3=(the (term2poly((term_of o the o (parse thy)) "6*a*b^^^2-13*a^^^2*b^^^2*c^^^2+21*a^^^2*b*c")));

val t1=mv_mul(mul1,mul2,LEX_);

val t2=mv_mul(mul3,mul2,LEX_);

val div3=step_cancel t1 t2;

if div3=([(5,[0,1,1]),(4,[0,1,0]),(2,[0,0,1])],[(13,[3,1,1]),(~19,[2,1,2]),(7,[2,1,0])],[(~13,[1,2,2]),(21,[1,1,1]),(6,[0,2,0])]) then () else error ("Test failed");*)

print("\n\n***** all tests successfull ;-) ******\n\n");

val thy = Rational.thy;

val rls = Prls {func=cancel};

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

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

val (t,asm) = the (rewrite_set_ thy eval_rls false rls t);

val thy' = "Rational";

val rls' = "cancel";

val t' = "(1 + 1 * a ^^^ 1)///(-2 + 2 * a ^^^ 2)";

val (t',asm') = the (rewrite_set thy' "eval_rls" false rls' t');

(*if t' = "1 /// (-2 + 2 * a)" then ()

else error "tests/rationals.sml(1): new behaviour";*)

val thy' = "Rational";

val rls' = "cancel";

val t' = "(10 * a^^^2 * b * c + 14 * a * b + 3 * a * c + 20 * a * b^^^2 * c) /// (5 * a * b * c  +  7 * a^^^2 * b * c) ";

val (t',asm') = the (rewrite_set thy' "eval_rls" false rls' t');

val thy' = "Rational";

val rls' = "cancel";

val t' = "(a^^^2 * b  + 2 * a * b +  b ) /// ( a^^^2   - 1 )";

val (t',asm') = the (rewrite_set thy' "eval_rls" false rls' t');

(*

val term2 = (term_of o the o (parse thy))  "(a^^^2 * b  + 2 * a * b +  b ) /// ( a^^^2   - 1 )";

val div2  = direct_cancel term2;

val t = (term_of o the o (parse thy)) "(1 + 1 * a ^^^ 1)///(-2 + 2 * a ^^^ 2) = 0";*)

(*WN folgendes aus examples he"uberkopiert ...*)

(* examples from:

   Mathematik 1

   Schalk 

   Reniets Verlag *)

val thy' = "Rational";

val rls' = "cancel";

val mp = "make_polynomial";

(* page 63 *)

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

print("a)\n");

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

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

print("b)\n");

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

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

print("c)\n");

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

val e186c = (the (rewrite_set thy' "rational_erls" false rls' 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 )";

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

print("b)\n");

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

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

print("c)\n");

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

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

print("\n\nexample 188:\n");

print("a)\n");

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

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

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

if t="((-8) + 8 * x) / ((-9) + 9 * x)"then()

else error "rationals.sml: e188a new behaviour";

print("b)\n");

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

val e188b = the (rewrite_set thy' "rational_erls" false rls' e188b');

print("c)\n");

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

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

val SOME (t,_) = rewrite_set thy' "rational_erls" false mp "((-1)*(b + (-1) * a))/(1*(b + (-1) * a))";

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

else error "rationals.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' "rational_erls" false rls' e190c');

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

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

(*TERMORDER               ~~~~~~~       ~~~~~~~*)

else error "rationals.sml: e190c new behaviour";

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

print("a)\n");

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

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

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

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

else error "rationals.sml: e191a new behaviour";

print("c)\n");

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

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

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

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

else error "rationals.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 )";

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

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

if t="(7 * x ^^^ 3 + -1 * (y * x ^^^ 2)) / (-1 * y ^^^ 3 + 7 * (x * y ^^^ 2))"then()

(*TERMORDER                ~~~~~*)

else error "rationals.sml: 'e192b' new behaviour";

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

print("a)\n");

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

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

print("b)\n");

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

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

print("c)\n");

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

val SOME(t,_) = rewrite_set thy' "rational_erls" false rls' e193c';

--------------------------------15.10.02---*)

(*---------- WN: 10.9.02:

ML> val e204a = the (rewrite_set thy' "rational_erls" false rls' e204a');

*** RATIONALS_STEP_CANCEL_EXCEPTION: Invalid fraction                           

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

print("a)\n");

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

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

print("b)\n");

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

val e204b = the (rewrite_set thy' "rational_erls" false rls' 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' "rational_erls" false rls' e205a');

print("b)\n");

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

val e205b = the (rewrite_set thy' "rational_erls" false rls' 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' "rational_erls" false rls' e206a'); 

print("b)\n");

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

val e206b = the (rewrite_set thy' "rational_erls" false rls' 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' "rational_erls" false rls' 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' "rational_erls" false rls' 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' "rational_erls" false rls' 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' "rational_erls" false rls' 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' "rational_erls" false rls' 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' "rational_erls" false rls' e211b');

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

print("a)\n");

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

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

print("b)\n");

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

val e212b = the (rewrite_set thy' "rational_erls" false rls' 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' "rational_erls" false rls' 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' "rational_erls" false rls' 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' "rational_erls" false rls' 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' "rational_erls" false rls' 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' "rational_erls" false rls' 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' "rational_erls" false rls' 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' "rational_erls" false rls' e217'); 

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' "rational_erls" false rls' 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' "rational_erls" false rls' e219a');

print("b)\n");

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

val e219b = the (rewrite_set thy' "rational_erls" false rls' 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' "rational_erls" false rls' e220a');

print("b)\n");

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

val e220b = the (rewrite_set thy' "rational_erls" false rls' 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' "rational_erls" false rls' e221a');

print("b)\n");

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

val e221b = the (rewrite_set thy' "rational_erls" false rls' 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' "rational_erls" false rls' 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' "rational_erls" false rls' 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' "rational_erls" false rls' e225a');

print("b)\n");

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

val e225b = the (rewrite_set thy' "rational_erls" false rls' 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' "rational_erls" false rls' 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' "rational_erls" false rls' 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' "rational_erls" false rls' 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' "rational_erls" false rls' 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' "rational_erls" false rls' 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' "rational_erls" false rls' 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' "rational_erls" false rls' 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' "rational_erls" false rls' 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' "rational_erls" false rls' 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' "rational_erls" false rls' 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' "rational_erls" false rls' 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' "rational_erls" false rls' 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' "rational_erls" false rls' 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' "rational_erls" false rls' 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' "rational_erls" false rls' 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' "rational_erls" false rls' 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' "rational_erls" false rls' 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' "rational_erls" false rls' 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' "rational_erls" false rls' 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' "rational_erls" false rls' 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' "rational_erls" false rls' 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' "rational_erls" false rls' e236b');  

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' "rational_erls" false rls' eheu');

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

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

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

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

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

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

(*

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

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

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

   term2str t';

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

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

*)

(*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 thead 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 error "rational.sml: new behav. in rev-set cancel";

  terms2str asm; 

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

  (*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;

  (*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;

(*

 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_poly enth"alt zu viele rules)

  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';

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

(* SK: Testbeispiele --- WN kopiert Rational.ML -> rational.sml-----

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

val SOME (t1',rest)= cancel_ thy t1;

val SOME (t1'',_)= factor_out_gcd_ thy t1;

print(term2str t1'^" + Einschr\"ankung: "^term2str (hd(rest)));

term2str t1'';

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

val SOME (t1',_)= cancel_ thy t1;

val SOME (t1'',_)= factor_expanded_ thy t1;

term2str t1';

term2str t1'';

val t2 = (term_of o the o (parse thy)) "((x+ (-1)) / (x + 1)) + ((x + 1) / (x + (-1)))";

val SOME (t2',_) = add_fractions_ thy t2;

val SOME (t2'',_) = common_nominators_ thy t2; 

term2str t2';

term2str t2'';

val t2 = (term_of o the o (parse thy)) "((x - 1) / (x + 1)) + ((x + 1) / (x - 1))";

val SOME (t2',_) = add_expanded_frac_ thy t2;

val SOME (t2'',_) = common_expanded_nom_ thy t2; 

term2str t2';

term2str t2'';

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_nominators_ thy t3; 

val SOME (t3'',_) = add_fractions_ thy t3; 

(term2str t3');

(term2str t3'');

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_expanded_nom_ thy t3; 

val SOME (t3'',_) = add_expanded_frac_ thy t3; 

(term2str t3');

(term2str t3'');

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

(*

val SOME(t4,t5) = norm_rational_ thy t3;

term2str t4;

term2str (hd(t5));*)

(*val test1 = (term_of o the o (parse thy)) "1 - x^^^2 - 5 * x^^^5";

val test2 = (term_of o the o (parse thy)) "1 + (-1) * x ^^^ 2 + (-5) * x ^^^ 5";

val test2 = (term_of o the o (parse thy)) "1 - x";

val test2 = (term_of o the o (parse thy)) "1 + (-1) * x";

term2str(expanded2term(test1));

term2str(term2expanded(test2)); *)

(* WN kopiert 16.10.02 Rational.ML -> rational.sml-----vvv---*)

  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 -----^^---*)

  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)";

(* WN kopiert 16.10.02 Rational.ML -> rational.sml-----^^^---*)