(* Title: tests for rationals

    Author: Walther Neuper

    Use is subject to license terms.

 *)

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

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

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

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

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

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

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

 "-------- integration lev.1 fun factout_p_ -----------------------------------";

 "-------- integration lev.1 fun cancel_p_ ------------------------------------";

 "-------- integration lev.1 fun common_nominator_p_ --------------------------";

 "-------- integration lev.1 fun add_fraction_p_ ------------------------------";

 "-------- and app_rev ...traced down from rewrite_set_ until prepats ---------";

 "-------- fun rewrite_set_ cancel_p downto fun gcd_poly ----------------------";

 "-------- rls norm_Rational downto fun gcd_poly ------------------------------";

 "-------- rls norm_Rational downto fun add_fraction_p_ -----------------------";

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

 "-------- rewrite_set_ add_fractions_p from: Mathematik 1 Schalk -------------";

 "-------- integration lev.1 -- lev.5: cancel_p_ & add_fractions_p_ -----------";

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

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

 "-------- investigate rls norm_Rational --------------------------------------";

 "-------- examples: rls norm_Rational ----------------------------------------";

 "-------- rational numerals --------------------------------------------------";

 "-------- examples cancellation from: Mathematik 1 Schalk --------------------";

 "-------- examples common denominator from: Mathematik 1 Schalk --------------";

 "-------- examples multiply and cancel from: Mathematik 1 Schalk -------------";

 "-------- examples common denominator and multiplication from: Schalk --------";

 "-------- examples double fractions from: Mathematik 1 Schalk ----------------";

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

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

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

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

 "-------- fun eval_get_denominator -------------------------------------------";

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

 "-------- WN1309xx non-terminating rls norm_Rational -------------------------";

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

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

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

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

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

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

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

 then () else error "poly_of_term 1 changed";

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

 then () else error "poly_of_term 2 changed";

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

 then () else error "poly_of_term 3 changed";

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

 then () else error "poly_of_term 4 changed";

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

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

 then () else error "poly_of_term 5 changed";

 (*poly_of_term is quite liberal:*)

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

   within a monomial are arbitrary*)

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

 then () else error "poly_of_term 6 changed";

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

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

 then () else error "poly_of_term 7 changed";

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

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

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

 then () else error "poly_of_term 8 changed";

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

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

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

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

 then () else error "is_poly 1 changed";

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

 then () else error "is_poly 2 changed";

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

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

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

 val expT = HOLogic.realT

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

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

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

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

 ;

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

 then () else error "term_of_poly 1 changed";

 "-------- integration lev.1 fun factout_p_ -----------------------------------";

 "-------- integration lev.1 fun factout_p_ -----------------------------------";

 "-------- integration lev.1 fun factout_p_ -----------------------------------";

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

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

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

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

 ;

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

 then () else error "factout_p_ asm 1 changed"

 ;

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

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

 ;

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

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

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

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

 then () else error "factout_p_ 1 changed";

 "-------- integration lev.1 fun cancel_p_ ------------------------------------";

 "-------- integration lev.1 fun cancel_p_ ------------------------------------";

 "-------- integration lev.1 fun cancel_p_ ------------------------------------";

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

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

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

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

 ;

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

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

 ;

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

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

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

 then () else error "cancel_p_ 1 changed";

 "-------- integration lev.1 fun common_nominator_p_ --------------------------";

 "-------- integration lev.1 fun common_nominator_p_ --------------------------";

 "-------- integration lev.1 fun common_nominator_p_ --------------------------";

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

               (* n1    d1                   *)

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

               (* n2    d2                   *)

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

 if term2str t' =

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

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

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

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

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

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

 then () else error "common_nominator_p_ asm 1 changed"

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

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

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

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

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

             " +\n" ^ 

                  "y * (x + y) / " ^ 

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

 then () else error "common_nominator_p_ term 2 changed"

 ;

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

 then () else error "common_nominator_p_ asm 2 changed"

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

 val t = str2term "nothing / to_add";

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

 ;

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

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

 if term2str t' = 

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

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

 then () else error "common_nominator_p_ 3 changed";

 "-------- integration lev.1 fun add_fraction_p_ ------------------------------";

 "-------- integration lev.1 fun add_fraction_p_ ------------------------------";

 "-------- integration lev.1 fun add_fraction_p_ ------------------------------";

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

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

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

 then () else error "add_fraction_p_ 3 changed";

 ;

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

 then () else error "add_fraction_p_ 3 changed";

 ;

 val t = str2term "nothing / to_add";

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

 ;

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

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

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

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

 then () else error "add_fraction_p_ 3 changed";

 "-------- and app_rev ...traced down from rewrite_set_ until prepats ---------";

 "-------- and app_rev ...traced down from rewrite_set_ until prepats ---------";

 "-------- and app_rev ...traced down from rewrite_set_ until prepats ---------";

 (* trace down until prepats are evaluated 

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

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

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

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

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

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

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

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

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

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

     fun chk_prepat thy erls [] t = true

       | chk_prepat thy erls prepat t =

         let

           fun chk (pres, pat) =

             (let 

               val subst: Type.tyenv * Envir.tenv =

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

              in

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

              end) handle _ => false

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

              | scan_ f (pp::pps) =

                if f pp then true else scan_ f pps;

         in scan_ chk prepat end;

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

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

       if chk_prepat thy erls prepat t

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

       else NONE;

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

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

   (thy, rrls, t);

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

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

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

           fun chk (pres, pat) =

             (let 

               val subst: Type.tyenv * Envir.tenv =

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

              in

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

              end) handle _ => false

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

              | scan_ f (pp::pps) =

                if f pp then true else scan_ f pps;

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

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

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

 f;

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

 term2str t1 = "?r is_expanded";

 term2str t2 = "?s is_expanded";

 term2str t = "?r / ?s";

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

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

               val subst: Type.tyenv * Envir.tenv =

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

 (*subst = 

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

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

 ;

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

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

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

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

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

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

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

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

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

 rewrite__set_ thy (i+1) false;

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

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

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

 "-------- fun rewrite_set_ cancel_p downto fun gcd_poly ----------------------";

 "-------- fun rewrite_set_ cancel_p downto fun gcd_poly ----------------------";

 "-------- fun rewrite_set_ cancel_p downto fun gcd_poly ----------------------";

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

 (* "-------- example 187a": exception Div raised...

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

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

 (* "-------- example 187b": doesn't terminate...

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

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

 (* "-------- example 187c": doesn't terminate...

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

 "~~~~~ fun rewrite_set_, args:"; val (thy, bool, rls, term) = (@{theory Isac}, false, cancel_p, t);

 (* WN130827: exception Div raised...

 rewrite__set_ thy 1 bool [] rls term

 *)

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

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

 (* WN130827: exception Div raised...

 	val (t', asm, rew) = app_rev thy (i+1) rrls t

 *)

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

 (* WN130827: exception Div raised...

     val opt = app_rev' thy rrls t

 *)

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

   (thy, rrls, t);

 chk_prepat thy erls prepat t    = true;

 (* WN130827: exception Div raised...

 normal_form t

 *)

 (* lookup Rational.thy, cancel_p: normal_form = cancel_p_ thy*)

 "~~~~~ fun cancel_p_, args:"; val (t) = (t);

 val opt = check_fraction t;

 val SOME (numerator, denominator) = opt

         val vs = t |> vars |> map free2str |> sort string_ord

         val baseT = type_of numerator

         val expT = HOLogic.realT

 val (SOME a, SOME b) = (poly_of_term vs numerator, poly_of_term vs denominator);

 (*"-------- example 187a": exception Div raised...

 val a = [(12, [1, 1])]: poly

 val b = [(8, [0, 2])]: poly

               val ((a', b'), c) = gcd_poly a b

 *)

 (* "-------- example 187b": doesn't terminate...

 val a = [(8, [2, 1, 1])]: poly

 val b = [(18, [1, 2, 1])]: poly

               val ((a', b'), c) = gcd_poly a b

 *)

 (* "-------- example 187c": doesn't terminate...

 val a = [(9, [5, 2, 4])]: poly

 val b = [(15, [6, 3, 1])]: poly

               val ((a', b'), c) = gcd_poly a b

 *)

 "-------- rls norm_Rational downto fun gcd_poly ------------------------------";

 "-------- rls norm_Rational downto fun gcd_poly ------------------------------";

 "-------- rls norm_Rational downto fun gcd_poly ------------------------------";

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

 trace_rewrite := false (*true false*);

 (* trace stops with ...: (and then jEdit hangs)..

 rewrite_set_ thy false norm_Rational t;

 :

 ###  rls: cancel_p on: (-12 + 4 * y + 3 * x ^^^ 2 + -1 * (x ^^^ 2 * y)) /

 (-18 + -9 * x + 2 * y ^^^ 2 + x * y ^^^ 2)

 *)

 val t = str2term (*copy from above: "::real" is not required due to "^^^"*)

   ("(-12 + 4 * y + 3 * x ^^^ 2 + -1 * (x ^^^ 2 * y)) /" ^

   "(-18 + -9 * x + 2 * y ^^^ 2 + x * y ^^^ 2)");

 (*cancel_p_ thy t;

 exception Div raised*)

 "~~~~~ fun cancel_p_, args:"; val (t) = (t);

 val opt = check_fraction t;

 val SOME (numerator, denominator) = opt

         val vs = t |> vars |> map free2str |> sort string_ord

         val baseT = type_of numerator

         val expT = HOLogic.realT;

 (*default_print_depth 3; 999*)

 val (SOME a, SOME b) = (poly_of_term vs numerator, poly_of_term vs denominator);

 (*default_print_depth 3; 999*)

 (* does not terminate instead of returning ?:

         val ((a', b'), c) = gcd_poly a b

 val a = [(~12, [0, 0]), (3, [2, 0]), (4, [0, 1]), (~1, [2, 1])]: poly

 val b = [(~18, [0, 0]), (~9, [1, 0]), (2, [0, 2]), (1, [1, 2])]: poly

 *)

 "-------- rls norm_Rational downto fun add_fraction_p_ -----------------------";

 "-------- rls norm_Rational downto fun add_fraction_p_ -----------------------";

 "-------- rls norm_Rational downto fun add_fraction_p_ -----------------------";

 val thy =  @{theory Isac};

 "----- SK060904-2a non-termination of add_fraction_p_";

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

 		         " (-1 * a + b * x) / (a + b * x)      ");

 (* rewrite_set_ thy false norm_Rational t

 exception Div raised*)

 (* rewrite_set_ thy false add_fractions_p t;

 exception Div raised*)

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

   (@{theory Isac}, false, add_fractions_p, t);

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

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

 (* app_rev thy (i+1) rrls t;

 exception Div raised*)

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

     fun chk_prepat thy erls [] t = true

       | chk_prepat thy erls prepat t =

         let

           fun chk (pres, pat) =

             (let 

               val subst: Type.tyenv * Envir.tenv =

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

              in

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

              end) handle _ => false

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

              | scan_ f (pp::pps) =

                if f pp then true else scan_ f pps;

         in scan_ chk prepat end;

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

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

       if chk_prepat thy erls prepat t

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

       else NONE;

 (*  val opt = app_rev' thy rrls t;

 exception Div raised*)

 (*  val opt = app_rev' thy rrls t;

 exception Div raised*)

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

   (thy, rrls, t);

 chk_prepat thy erls prepat t = true       = true;

 (*normal_form t

 exception Div raised*)

 (* lookup Rational.thy, val add_fractions_p: normal_form = add_fraction_p_ thy*)

 (*add_fraction_p_ thy t

 exception Div raised*)

 "~~~~~ fun add_fraction_p_, args:"; val ((_: theory), t) = (thy, t);

 val SOME ((n1, d1), (n2, d2)) = check_frac_sum t;

 term2str n1; term2str d1; term2str n2; term2str d2;

       val vs = t |> vars |> map str_of_free_opt (* tolerate Var in simplification *)

         |> filter is_some |> map the |> sort string_ord;

 (*default_print_depth 3; 999*)

 val (SOME _, SOME a, SOME _, SOME b) =

   (poly_of_term vs n1, poly_of_term vs d1, poly_of_term vs n2, poly_of_term vs d2);

 (*default_print_depth 3; 999*)

 (*

 val a = [(1, [1, 0, 0]), (~1, [0, 1, 1])]: poly

 val b = [(1, [1, 0, 0]), (1, [0, 1, 1])]: poly

             val ((a', b'), c) = gcd_poly a b

 *)

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

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

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

 val thy  = @{theory "Rational"};

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

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

 if NONE = rewrite_set_ thy false cancel_p t then ()

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

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

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

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

   is_expanded (str2term "x * y");

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

 *)

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

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

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

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

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

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

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

 *)

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

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

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

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

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

 if (term2str t', terms2str asm) = 

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

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

 *)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

 if (term2str t', terms2str asm) = 

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

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

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

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

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

 if (term2str t', terms2str asm) = 

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

 if (term2str t', terms2str asm) = 

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

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

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

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

 if (term2str t', terms2str asm) = 

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

 if (term2str t', terms2str asm) = 

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

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

 (*WN:*)

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

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

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

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

 "-------- example heuberger";

 val t = str2term ("(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 SOME (t', asm) = rewrite_set_ thy false cancel_p t;

 if (term2str t', terms2str asm) = 

   ("(x ^^^ 3 + y) / (1 + 5 * x + x * y ^^^ 3)", "[\"1 + 5 * x + x * y ^^^ 3 ~= 0\"]")

 then () else error "rational.sml cancel_p heuberger";

 "-------- rewrite_set_ add_fractions_p from: Mathematik 1 Schalk -------------";

 "-------- rewrite_set_ add_fractions_p from: Mathematik 1 Schalk -------------";

 "-------- rewrite_set_ add_fractions_p from: Mathematik 1 Schalk -------------";

 (*deleted example 204 ... 236b at update Isabelle2012-->2013*)

 "-------- integration lev.1 -- lev.5: cancel_p_ & add_fractions_p_ -----------";

 "-------- integration lev.1 -- lev.5: cancel_p_ & add_fractions_p_ -----------";

 "-------- integration lev.1 -- lev.5: cancel_p_ & add_fractions_p_ -----------";

 val t = str2term ("123 = (a*x)/(b*x) + (c*x)/(d*x) + (e*x)/(f*x::real)");

 "-------- gcd_poly integration level 1: works on exact term";

 if NONE = cancel_p_ thy t then () else error "cancel_p_ works on exact fraction";

 if NONE = add_fraction_p_ thy t then () else error "add_fraction_p_ works on exact fraction";

 "-------- gcd_poly integration level 2: picks out ONE appropriate subterm";

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

 if term2str t' = "123 = a * x / (b * x) + c * x / (d * x) + e / f" 

 then () else error "level 2, rewrite_set_ cancel_p: changed";

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

 if term2str t' = "123 = (b * c * x + a * d * x) / (b * d * x) + e * x / (f * x)"

 then () else error "level 2, rewrite_set_ add_fractions_p: changed";

 "-------- gcd_poly integration level 3: rewrites all appropriate subterms";

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

 if term2str t' = "123 = a / b + c / d + e / f"

 then () else error "level 3, rewrite_set_ cancel_p_rls: changed";

 val SOME (t', asm) = rewrite_set_ thy false add_fractions_p_rls t; (*CREATE add_fractions_p_rls*)

 if term2str t' = "123 = (b * d * e * x + b * c * f * x + a * d * f * x) / (b * d * f * x)"

 then () else error "level 3, rewrite_set_ add_fractions_p_rls: changed";

 "-------- gcd_poly integration level 4: iteration cancel_p -- add_fraction_p";

 (* simpler variant *)

 val testrls = append_rls "testrls" e_rls [Rls_ cancel_p, Rls_ add_fractions_p]

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

 (*trace_rewrite := false;

 #  rls: testrls on: 123 = a * x / (b * x) + c * x / (d * x) + e * x / (f * x) 

 ##  rls: cancel_p on: 123 = a * x / (b * x) + c * x / (d * x) + e * x / (f * x) 

 ##  rls: add_fractions_p on: 123 = a * x / (b * x) + c * x / (d * x) + e / f 

 ##  rls: cancel_p on: 123 = (b * c * x + a * d * x) / (b * d * x) + e / f 

 ##  rls: add_fractions_p on: 123 = (b * c + a * d) / (b * d) + e / f 

 ##  rls: cancel_p on: 123 = (b * d * e + b * c * f + a * d * f) / (b * d * f) 

 ##  rls: add_fractions_p on: 123 = (b * d * e + b * c * f + a * d * f) / (b * d * f) *)

 if term2str t' = "123 = (b * d * e + b * c * f + a * d * f) / (b * d * f)"

 then () else error "level 4, rewrite_set_ *_p: changed";

 (* complicated variant *)

 val testrls_rls = append_rls "testrls_rls" e_rls [Rls_ cancel_p_rls, Rls_ add_fractions_p_rls];

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

 (*#  rls: testrls_rls on: 123 = a * x / (b * x) + c * x / (d * x) + e * x / (f * x) 

 ##  rls: cancel_p_rls on: 123 = a * x / (b * x) + c * x / (d * x) + e * x / (f * x) 

 ###  rls: cancel_p on: 123 = a * x / (b * x) + c * x / (d * x) + e * x / (f * x) 

 ###  rls: cancel_p on: 123 = a * x / (b * x) + c * x / (d * x) + e / f 

 ###  rls: cancel_p on: 123 = a * x / (b * x) + c / d + e / f 

 ###  rls: cancel_p on: 123 = a / b + c / d + e / f 

 ##  rls: add_fractions_p_rls on: 123 = a / b + c / d + e / f 

 ###  rls: add_fractions_p on: 123 = a / b + c / d + e / f 

 ###  rls: add_fractions_p on: 123 = (b * c + a * d) / (b * d) + e / f 

 ###  rls: add_fractions_p on: 123 = (b * d * e + b * c * f + a * d * f) / (b * d * f) 

 ##  rls: cancel_p_rls on: 123 = (b * d * e + b * c * f + a * d * f) / (b * d * f) 

 ###  rls: cancel_p on: 123 = (b * d * e + b * c * f + a * d * f) / (b * d * f) 

 ##  rls: add_fractions_p_rls on: 123 = (b * d * e + b * c * f + a * d * f) / (b * d * f) 

 ###  rls: add_fractions_p on: 123 = (b * d * e + b * c * f + a * d * f) / (b * d * f) *)

 if term2str t' = "123 = (b * d * e + b * c * f + a * d * f) / (b * d * f)"

 then () else error "level 4, rewrite_set_ *_p_rls: changed"

 "-------- gcd_poly integration level 5: cancel_p & add_fraction_p within norm_Rational";

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

 if term2str t' = "123 = (a * d * f + b * c * f + b * d * e) / (b * d * f)"

 then () else error "level 5, rewrite_set_ norm_Rational: changed"

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

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

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

 (** 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;

 if term2str t' = "(3 + -1 * 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;

 if length (hd revsets) = 11 then () else error "length of revset changed";

 if (revsets |> nth 1 |> nth 1 |> id_of_thm) = 

   (@{thm realpow_twoI} |> Thm.get_name_hint |> thmID_of_derivation_name)

 then () else error "first element of revset changed";

 if

 (revsets |> nth 1 |> nth 1 |> rule2str) = "Thm (\"realpow_twoI\",?r1 ^^^ 2 = ?r1 * ?r1)" andalso

 (revsets |> nth 1 |> nth 2 |> rule2str) = "Thm (\"#: 9 = 3 ^^^ 2\",9 = 3 ^^^ 2)" andalso

 (revsets |> nth 1 |> nth 3 |> rule2str) = "Thm (\"#: 6 * x = 2 * (3 * x)\",6 * x = 2 * (3 * x))" 

 andalso

 (revsets |> nth 1 |> nth 4 |> rule2str) = "Thm (\"#: -3 * x = -1 * (3 * x)\",-3 * x = -1 * (3 * x))" 

 andalso

 (revsets |> nth 1 |> nth 5 |> rule2str) = "Thm (\"#: 9 = 3 * 3\",9 = 3 * 3)" andalso

 (revsets |> nth 1 |> nth 6 |> rule2str) = "Rls_ (\"sym_order_mult_rls_\")" andalso

 (revsets |> nth 1 |> nth 7 |> rule2str) = 

   "Thm (\"sym_mult_assoc\",?a * (?b * ?c) = ?a * ?b * ?c)"

 then () else error "first 7 elements in revset changed"

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

 (*/------- WN1309: since cancel_ (accepted "-" between monomials) has been replaced by cancel_p_ 

   for Isabelle2013, we don't get a working revset, but non-termination:

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

   :

 ((3 * 3 + -1 * x * x) / (3 * 3 + 2 * 3 * x + x * x), 

   Rls_ ("sym_order_mult_rls_"), ((3 * 3 + -1 * (x * x)) / (3 * 3 + 2 * (3 * x) + x * x), []))","

 ((3 * 3 + -1 * (x * x)) / (3 * 3 + 2 * (3 * x) + x * x), 

   Thm ("sym_mult_assoc",""), ((3 * 3 + -1 * (x * x)) / (3 * 3 + 2 * 3 * x + x * x), []))","

 ((3 * 3 + -1 * (x * x)) / (3 * 3 + 2 * 3 * x + x * x), 

   Thm ("sym_mult_assoc",""), ((3 * 3 + -1 * x * x) / (3 * 3 + 2 * 3 * x + x * x), []))","

 ((3 * 3 + -1 * x * x) / (3 * 3 + 2 * 3 * x + x * x), Rls_ ("sym_order_mult_rls_"), ((3 * 3 + -1 * (x * x)) / (3 * 3 + 2 * (3 * x) + x * x), []))","

  :

 ### Isabelle2002:

   Thm ("sym_#mult_2_3", "6 = 2 * 3")

 ### Isabelle2009-2 for cancel_ (not cancel_p_):

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

    andalso string_of_thm thm = 

            (string_of_thm o Thm.make_thm o (Thm.global_cterm_of (@{theory "Isac"})))

                (Trueprop $ (Thm.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, 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 **)

 (*/-------------------------------------------- Isabelle2013: this gives "error id_of_thm";

   we don't repair this, because interaction within "reverse rewriting" never worked properly:

   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 Thm.make_thm o (Thm.global_cterm_of (@{theory "Isac"})))

                 (Trueprop $ (Thm.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 -----------------------------------------";

 (*WN130909: the example below shows, why "reverse rewriting" only worked for

   special cases.*)

 (*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

 default_print_depth 99; Rfuns; default_print_depth 3;

 WN060831 cycling "sym_order_mult_rls_" "sym_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;

 *)                    

 "-------- examples: rls norm_Rational ----------------------------------------";

 "-------- examples: rls norm_Rational ----------------------------------------";

 "-------- examples: rls 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";

 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 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';

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

 else 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';

 "HOL.True";

 val t = str2term "1 < 2";

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

 "HOL.True";

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

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

 			   (num_str @{thm realpow_def_atom}) t;

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

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

 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 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';

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

 else 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 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 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" then () else error "rational.sml 8";

 (*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';

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

 (*Schalk I, p.70 Nr. 480b: SEE rational.sml --- nonterminating rls norm_Rational ---*)

 (*WN130910 add_fractions_p exception Div raised + history:

 ### WN.2.6.03 from rlang.sml 56a 

 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 add_fractions_p t;

 THE ERROR ALREADY OCCURS IN THIS PART:

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

 val NONE = add_fraction_p_ thy t;

 SEE Test_Some.thy: section {* add_fractions_p downto exception Div raised ===

 *)

 "-------- rational numerals --------------------------------------------------";

 "-------- rational numerals --------------------------------------------------";

 "-------- rational numerals --------------------------------------------------";

 (*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 t;

 term2str t;

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

 else error "rational.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 t;

 term2str t;

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

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

 "-------- examples cancellation from: Mathematik 1 Schalk --------------------";

 "-------- examples cancellation from: Mathematik 1 Schalk --------------------";

 "-------- examples cancellation from: Mathematik 1 Schalk --------------------";

 (* 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 t;

 if term2str t = "(1 + 9 * a ^^^ 2) / (3 * a + 9 * a ^^^ 2)"

 then () else error "rational.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 t;

 if term2str t = "x ^^^ 2 / y ^^^ 2"

 then () else error "rational.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 t;

 term2str t;

 if term2str t = "-1"

 then () else error "rational.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 t;

 if term2str t = "(27 + 12 * x) / (28 + 8 * x)"

 then () else error "rational.sml: diff.behav. in norm_Rational_mg 6";

 (*Schalk I, p.60 Nr. 215c: was not cancelled with Isabelle2002 *)

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

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

 if term2str t = "(a ^^^ 2 + 2 * a * b + b ^^^ 2) / (x ^^^ 2 + -2 * x * y + y ^^^ 2)"

 then () else error "rational.sml: diff.behav. in norm_Rational_mg 7";

 (*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 t;

 if term2str t = "1 / (-5 + 2 * x)"

 then () else error "rational.sml: diff.behav. in norm_Rational_mg 9";

 (* 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 t;

 if term2str t =  "(1 + 9 * a ^^^ 2) / (3 * a + 9 * a ^^^ 2)"

 then () else error "rational.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 t;

 if term2str t = "x ^^^ 2 / y ^^^ 2"

 then () else error "rational.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 t;

 if term2str t = "-1"

 then () else error "rational.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 t;

 if term2str t = "(27 + 12 * x) / (28 + 8 * x)"

 then () else error "rational.sml: diff.behav. in norm_Rational_mg 6";

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

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

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

 if term2str t = "(a ^^^ 2 + 2 * a * b + b ^^^ 2) / (x ^^^ 2 + -2 * x * y + y ^^^ 2)"

 then () else error "Schalk I, p.60 Nr. 215c: with Isabelle2002 cancellation incomplete, changed";

 (* extreme example from somewhere *)

 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;

 if term2str t = "(a ^^^ 2 + 2 * a * b + b ^^^ 2) / (x ^^^ 2 + -2 * x * y + y ^^^ 2)"

 then () else error "with Isabelle2002: NONE -- now SOME changed";

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

 (* ATTENTION: here the rls is very slow. In Isabelle2002 this required 2 min *)

 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 t;

 if term2str t = "(a + b) / (4 * a + -4 * b)"

 then () else error "rational.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 t;

 if term2str t = "1 / (-5 + 2 * x)"

 then () else error "rational.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 t;

 if term2str t = "(1 + 9 * a ^^^ 2) / (3 * a + 9 * a ^^^ 2)"

 then () else error "rational.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 t;

 if term2str t = "-5 * a / (-1 * a + b)"

 then () else error "rational.sml: diff.behav. in norm_Rational_mg 11";

 "----- NOT TERMINATING ?: worked before 0707xx";

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

 (* WN130911 "exception Div raised" by 

   cancel_p_ thy (str2term ("(-1 + -1 * b + a ^^^ 2 + a ^^^ 2 * b) /" ^

                            "(-1 + -1 * a + b ^^^ 2 + a * b ^^^ 2)"))

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

 if term2str t = "(1 + -1 * a) / (1 + -1 * b)" then ()

 else error "rational.sml MG tests 3e";

 *)

 "-------- examples common denominator from: Mathematik 1 Schalk --------------";

 "-------- examples common denominator from: Mathematik 1 Schalk --------------";

 "-------- examples common denominator from: Mathematik 1 Schalk --------------";

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

 val t = str2term "4/x - 3/y - 1";

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

 if term2str t = "(-3 * x + 4 * y + -1 * x * y) / (x * y)"

 then () else error "rational.sml: diff.behav. in norm_Rational_mg 12";

 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 t;

 if term2str t = "4 / c"

 then () else error "rational.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)";

 (* WN130911 non-termination due to non-termination of

   cancel_p_ thy (str2term "(5 + 3 * x) / (6 + 11 * x + 6 * x ^^^ 2 + x ^^^ 3)")

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

 if term2str t = "(5 + 3 * x) / (6 + 11 * x + 6 * x ^^^ 2 + x ^^^ 3)"

 then () else error "rational.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 t;

 if term2str t = "6 * x / (1 + -1 * x ^^^ 2)"

 then () else error "rational.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 t;

 if term2str t ="(-2 + -5 * x + 2 * x ^^^ 2) / (2 + -3 * x + x ^^^ 2)"

 then () else error "rational.sml: diff.behav. in norm_Rational_mg 16";

 (*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";

 (* WN130911 non-termination due to non-termination of

   cancel_p_ thy (str2term "(4 + -4 * b ^^^ 2) / (a ^^^ 4 + -2 * (a ^^^ 2 * b ^^^ 2) + b ^^^ 4)")

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

 if term2str t = "4 / (a ^^^ 4 + -2 * a ^^^ 2 * b ^^^ 2 + b ^^^ 4)"

 then () else error "rational.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 t;

 if term2str t = "a + 3 * b"

 then () else error "rational.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 t;

 term2str t;

 if term2str t = "0"

 then () else error "rational.sml: diff.behav. in norm_Rational_mg 20";

 (* some example *)

 val t = str2term "3*a / (a*b)  +  x/y";

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

 if term2str t = "(3 * y + b * x) / (b * y)"

 then () else error "rational.sml: diff.behav. in norm_Rational_mg 21";

 "-------- examples multiply and cancel from: Mathematik 1 Schalk -------------";

 "-------- examples multiply and cancel from: Mathematik 1 Schalk -------------";

 "-------- examples multiply and cancel from: Mathematik 1 Schalk -------------";

 (*------- 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 t;

 if term2str t = "(-5 * x + 5 * y) / (-18 * x + -18 * y)"

 then () else error "rational.sml: diff.behav. in norm_Rational_mg 22";

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

 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 t;

 if term2str t = "5 * a / (a + -1 * b)"

 then () else error "rational.sml: diff.behav. in norm_Rational_mg 23";

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

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

 (* raises an exception for unclear reasons:

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

 :

 ###  rls: cancel_p on: (9 * a ^^^ 2 + -16 * b ^^^ 2) / (4 * c + 3 * e) /

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

 exception Div raised

 BUT

 val t = str2term 

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

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

 NONE = cancel_p_ thy t;

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

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

 *)

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

 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;

 if term2str t = 

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

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

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

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

 val t'' = str2term ("(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'';

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

 then () else error "rational.sml: diff.behav. in cancel_p S.K.8";

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

 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 t;

 :

 ####  rls: cancel_p on: (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) 

 exception Div raised

 *)

 (*------- 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 t;

 if term2str t = "x ^^^ 2"

 then () else error "rational.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 t;

 if term2str t = "(x + -4 * x ^^^ 3) / 2"

 then () else error "rational.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 t;

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

 then () else error "rational.sml: diff.behav. in norm_Rational_mg 26";

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

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

 (* WN130911 non-termination for unclear reasons:

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

 ... ENDS WITH THIS TRACE:

 :

 ###  rls: cancel_p on: (-9 * (a ^^^ 3 * b) + -9 * (a ^^^ 2 * b ^^^ 2) + a ^^^ 5 * b +

  a ^^^ 4 * b ^^^ 2) /

 (a ^^^ 3 * b + -1 * (a * b ^^^ 3)) /

 (3 + a)

 BUT THIS IS CORRECTLY RECOGNISED 

 val t = str2term 

   ("(-9 * (a^^^3 * b) + -9 * (a^^^2 * b^^^2) + a^^^5 * b + a^^^4 * b^^^2)  /" ^

   "(a^^^3 * b + -1 * (a * b^^^3))  /  (3 + (a::real))");

 AS

 NONE = cancel_p_ thy t;

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

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

 *)

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

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

 (* WN130911 non-termination due to non-termination of

   cancel_p_ thy (str2term "(4 + -4 * b ^^^ 2) / (a ^^^ 4 + -2 * (a ^^^ 2 * b ^^^ 2) + b ^^^ 4)")

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

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

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

 *)

 "-------- examples common denominator and multiplication from: Schalk --------";

 "-------- examples common denominator and multiplication from: Schalk --------";

 "-------- examples common denominator and multiplication from: Schalk --------";

 (*------- 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 t;

 if term2str t = "(36 * b ^^^ 3 + 3 * a ^^^ 2 * b ^^^ 2 + 16 * a ^^^ 4 * b) / (9 * a ^^^ 4)"

 then () else error "rational.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 t;

 if term2str t = "5 * x ^^^ 2 / (a * b ^^^ 3 * c)"

 then () else error "rational.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 t;

 if term2str t = "(-9 * a ^^^ 2 + 4 * b ^^^ 2) / 36"

 then () else error "rational.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 t;

 if term2str t = "a ^^^ 3 / 27"

 then () else error "rational.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 t;

 if term2str t = "30 * x ^^^ 2 + -9 * x * y + -20 * y ^^^ 2"

 then () else error "rational.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 t;

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

 then () else error "rational.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 t;

 if term2str t = "(52 * x ^^^ 2 + 16 * y ^^^ 2) / (9 * y ^^^ 2)"

 then () else error "rational.sml: diff.behav. in norm_Rational_mg 34";

 (*------- 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 t;

 if term2str t = "5 * x ^^^ 2 / (b ^^^ 3 * c)"

 then () else error "rational.sml: diff.behav. in norm_Rational_mg 53";

 "-------- examples double fractions from: Mathematik 1 Schalk ----------------";

 "-------- examples double fractions from: Mathematik 1 Schalk ----------------";

 "-------- examples double fractions from: Mathematik 1 Schalk ----------------";

 "----- 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 t;

 if term2str t = "(-4 + 4 * x + -1 * x ^^^ 2) / (4 * a ^^^ 2)"

 then () else error "rational.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 t;

 if term2str t = "(1 + a ^^^ 2) / (1 + 2 * a + a ^^^ 2)" then ()

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

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

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

 (* WN130911 non-termination due to non-termination of

   cancel_p_ thy (str2term ("(-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 norm_Rational t;

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

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

 *)

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

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

 (* WN130911 non-termination due to non-termination of

   cancel_p_ thy (str2term ("(-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 norm_Rational t;

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

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

 *)

 "----- ?: worked before 0707xx";

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

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

 if term2str t = "-1 / (3 + y)"

 then () else error "rational.sml: -1 / (3 + y) norm_Rational";

 "----- 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 t;

 if term2str t = "b / (1 + 2 * b + b ^^^ 2)"

 then () else error "rational.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 t;

 if term2str t = "8 * a ^^^ 2 + -6 * a * b + -12 * a ^^^ 2 * b + 9 * a * b ^^^ 2"

 then () else error "rational.sml: diff.behav. in norm_Rational_mg 39";

 "----- Schalk I, p.69 Nr. 458b works since 0707";

 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 t;

 :

 ###  rls: cancel_p on: (-1 * a ^^^ 2 + 2 * (a ^^^ 2 * x)) / (a * x + -1 * (b * x)) /

 ((-1 * b ^^^ 2 + 2 * (b ^^^ 2 * x)) / (a * x + -1 * (b * x))) 

 exception Div raised

 BUT

 val t = str2term 

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

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

 NONE = cancel_p_ thy t;

 if term2str t = "a ^^^ 2 / b ^^^ 2" then ()

 else error "rational.sml: diff.behav. in norm_Rational_mg 39b";

 *)

 "----- 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 t;

 if term2str t = "(a + -1 * b) / (4 * a * b + 4 * b ^^^ 2)" then ()

 else error "rational.sml: diff.behav. in norm_Rational_mg 41";

 "----- Schalk I, p.69 Nr. 460b nonterm.SK";

 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 t;

 exception Div raised

 BUT

 val t = str2term 

   ("(144 + -72 * x + 9 * x ^^^ 2) / (4 + -8 * y + 4 * y ^^^ 2) /" ^

   "((-12 + 3 * x) / (-16 + 16 * y))");

 NONE = cancel_p_ thy t;

 if term2str t = !!!!!!!!!!!!!!!!!!!!!!!!!

 then () else error "rational.sml: diff.behav. in norm_Rational_mg 42";

 *)

 "----- some variant of the above; was non-terminating before";

 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 norm_Rational t;

 if term2str t = "(48 + -12 * x) / (1 + -1 * y)"

 then () else error "some variant of the above; was non-terminating before";

 "----- 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 t;

 if term2str t = "x + y"

 then () else error "rational.sml: diff.behav. in norm_Rational_mg 43";

 "----- 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 t;

 if term2str t = "(2 * a ^^^ 3 + 2 * a ^^^ 2 * b) / (a ^^^ 2 * b + b ^^^ 3)"

 then () else error "rational.sml: diff.behav. in norm_Rational_mg 51";

 (*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 t;

 if term2str t = "1 / 2"

 then () else error "rational.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 t;

 if term2str t = "(-3 + a) / (-1 + a)"

 then () else error "rational.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)";

 (* WN130911 non-termination due to non-termination of

   cancel_p_ thy (str2term 

     ("("(576 * x ^^^ 2 + 1728 * (x * y) + 1296 * y ^^^ 2 + -2916 * z ^^^ 2) /" ^

       "(576 * x ^^^ 2 + 864 * (x * y) + 1296 * (x * z))"))

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

 if term2str t = "(4 * x + 6 * y + -9 * z) / (4 * x)"

 then () else error "rational.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 t;

 if term2str t = "(25 + -10 * x + x ^^^ 2) / 18"

 then () else error "rational.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 t;

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

 then () else error "rational.sml: diff.behav. in norm_Rational_mg 48";

 "-------- 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",["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;

 val (p,_,f,nxt,_,pt) = me nxt p c pt; f2str f;(*++ for explicit script*)

 val (p,_,f,nxt,_,pt) = me nxt p c pt; f2str f;(*++ for explicit script*)

 case (f2str f, nxt) of

     ("14", ("End_Proof'", _)) => ()

   | _ => 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 ------------------------------";

 reset_states ();

 CalcTree [(["Term (((2 - x)/(2*a)) / (2*a/(x - 2)))", "normalform N"], 

   ("Rational", ["rational", "simplification"], ["simplification", "of_rationals"]))];

 Iterator 1;

 moveActiveRoot 1;

 autoCalculate 1 CompleteCalc;

 val ((pt, p), _) = get_calc 1; 

 (*

 show_pt pt;

 [

 (([], Frm), Simplify ((2 - x) / (2 * a) / (2 * a / (x - 2)))),

 (([1], Frm), (2 - x) / (2 * a) / (2 * a / (x - 2))),

 (([1], Res), (2 + -1 * x) / (2 * a) / (2 * a / (x + -1 * 2))),

 (([2], Res), (2 + -1 * x) / (2 * a) / (2 * a / (-2 + x))),

 (([3], Res), (2 + -1 * x) * (-2 + x) / (2 * a * (2 * a))),

 (([4], Res), (-4 + 4 * x + -1 * x ^^^ 2) / (4 * a ^^^ 2)),

 (([], Res), (-4 + 4 * x + -1 * x ^^^ 2) / (4 * a ^^^ 2))] 

 *)

 interSteps 1 ([1], Res);

 val ((pt, p), _) = get_calc 1; 

 (*show_pt pt;

 [

 (([], Frm), Simplify ((2 - x) / (2 * a) / (2 * a / (x - 2)))),

 (([1], Frm), (2 - x) / (2 * a) / (2 * a / (x - 2))),

 (([1,1], Frm), (2 - x) / (2 * a) / (2 * a / (x - 2))),

 (([1,1], Res), (2 - x) / (2 * a) / (2 * a / (x + -1 * 2))),

 (([1,2], Res), (2 + -1 * x) / (2 * a) / (2 * a / (x + -1 * 2))),

 (([1], Res), (2 + -1 * x) / (2 * a) / (2 * a / (x + -1 * 2))),

 (([2], Res), (2 + -1 * x) / (2 * a) / (2 * a / (-2 + x))),

 (([3], Res), (2 + -1 * x) * (-2 + x) / (2 * a * (2 * a))),

 (([4], Res), (-4 + 4 * x + -1 * x ^^^ 2) / (4 * a ^^^ 2)),

 (([], Res), (-4 + 4 * x + -1 * x ^^^ 2) / (4 * a ^^^ 2))] 

 *)

 val (t, asm) = get_obj g_result pt [1, 1];

 if term2str t = "(2 - x) / (2 * a) / (2 * a / (x + -1 * 2))" andalso terms2str asm = "[]"

 then () else error "2nd interSteps ..Simp_Rat_Double_No-1 changed on [1, 1]";

 val (t, asm) = get_obj g_result pt [1, 2];

 if term2str t = "(2 + -1 * x) / (2 * a) / (2 * a / (x + -1 * 2))" andalso terms2str asm = "[]"

 then () else error "3rd interSteps ..Simp_Rat_Double_No-1 changed on [1, 2]";

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

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

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

 reset_states ();

 CalcTree [(["Term ((a^2 + -1*b^2) / (a^2 + -2*a*b + b^2))", "normalform N"], 

   ("Rational", ["rational", "simplification"], ["simplification", "of_rationals"]))];

 Iterator 1;

 moveActiveRoot 1;

 autoCalculate 1 CompleteCalc;

 val ((pt, p), _) = get_calc 1;

 (*show_pt pt;

 [

 (([], Frm), Simplify ((a ^^^ 2 + -1 * b ^^^ 2) / (a ^^^ 2 + -2 * a * b + b ^^^ 2))),

 (([1], Frm), (a ^^^ 2 + -1 * b ^^^ 2) / (a ^^^ 2 + -2 * a * b + b ^^^ 2)),

 (([1], Res), (a ^^^ 2 + -1 * b ^^^ 2) / (a ^^^ 2 + -2 * (a * b) + b ^^^ 2)),

 (([2], Res), (a + b) / (a + -1 * b)),

 (([], Res), (a + b) / (a + -1 * b))] 

 *)

 interSteps 1 ([2], Res);

 val ((pt, p), _) = get_calc 1;

 (*show_pt pt;

 [

 (([], Frm), Simplify ((a ^^^ 2 + -1 * b ^^^ 2) / (a ^^^ 2 + -2 * a * b + b ^^^ 2))),

 (([1], Frm), (a ^^^ 2 + -1 * b ^^^ 2) / (a ^^^ 2 + -2 * a * b + b ^^^ 2)),

 (([1], Res), (a ^^^ 2 + -1 * b ^^^ 2) / (a ^^^ 2 + -2 * (a * b) + b ^^^ 2)),

 (([2,1], Frm), (a ^^^ 2 + -1 * b ^^^ 2) / (a ^^^ 2 + -2 * (a * b) + b ^^^ 2)),

 (([2,1], Res), (a + b) / (a + -1 * b)),

 (([2], Res), (a + b) / (a + -1 * b)),

 (([], Res), (a + b) / (a + -1 * b))] 

 *)

 interSteps 1 ([2,1],Res);

 val ((pt, p), _) = get_calc 1; 

 (*show_pt pt;

 [

 (([], Frm), Simplify ((a ^^^ 2 + -1 * b ^^^ 2) / (a ^^^ 2 + -2 * a * b + b ^^^ 2))),

 (([1], Frm), (a ^^^ 2 + -1 * b ^^^ 2) / (a ^^^ 2 + -2 * a * b + b ^^^ 2)),

 (([1], Res), (a ^^^ 2 + -1 * b ^^^ 2) / (a ^^^ 2 + -2 * (a * b) + b ^^^ 2)),

 (([2,1], Frm), (a ^^^ 2 + -1 * b ^^^ 2) / (a ^^^ 2 + -2 * (a * b) + b ^^^ 2)),

 (([2,1,1], Frm), (a ^^^ 2 + -1 * b ^^^ 2) / (a ^^^ 2 + -2 * (a * b) + b ^^^ 2)),

 (([2,1,1], Res), (a ^^^ 2 + -1 * (a * b) + a * b + -1 * b ^^^ 2) /

 (a ^^^ 2 + -2 * (a * b) + 1 * b ^^^ 2)),

 (([2,1,2], Res), (a ^^^ 2 + -1 * (a * b) + a * b + -1 * b ^^^ 2) /

 (a ^^^ 2 + -2 * (a * b) + -1 ^^^ 2 * b ^^^ 2)),

 (([2,1,3], Res), (a ^^^ 2 + -1 * (a * b) + a * b + -1 * b ^^^ 2) /

 (a ^^^ 2 + -2 * (a * b) + (-1 * b) ^^^ 2)),

 (([2,1,4], Res), (a * a + -1 * (a * b) + a * b + -1 * b ^^^ 2) /

 (a ^^^ 2 + -2 * (a * b) + (-1 * b) ^^^ 2)),

 (([2,1,5], Res), (a * a + -1 * (a * b) + a * b + -1 * (b * b)) /

 (a ^^^ 2 + -2 * (a * b) + (-1 * b) ^^^ 2)),

 (([2,1,6], Res), (a * a + -1 * (a * b) + a * b + -1 * (b * b)) /

 (a ^^^ 2 + -1 * (2 * (a * b)) + (-1 * b) ^^^ 2)),

 (([2,1,7], Res), (a * a + a * (-1 * b) + (b * a + b * (-1 * b))) /

 (a ^^^ 2 + 2 * (a * (-1 * b)) + (-1 * b) ^^^ 2)),

 (([2,1,8], Res), (a * a + a * (-1 * b) + (b * a + b * (-1 * b))) /

 (a ^^^ 2 + 2 * a * (-1 * b) + (-1 * b) ^^^ 2)),

 (([2,1,9], Res), (a * (a + -1 * b) + (b * a + b * (-1 * b))) /

 (a ^^^ 2 + 2 * a * (-1 * b) + (-1 * b) ^^^ 2)),

 (([2,1,10], Res), (a * (a + -1 * b) + b * (a + -1 * b)) /

 (a ^^^ 2 + 2 * a * (-1 * b) + (-1 * b) ^^^ 2)),

 (([2,1,11], Res), (a + b) * (a + -1 * b) / (a ^^^ 2 + 2 * a * (-1 * b) + (-1 * b) ^^^ 2)),

 (([2,1,12], Res), (a + b) * (a + -1 * b) / ((a + -1 * b) * (a + -1 * b))),

 (([2,1,13], Res), (a + b) / (a + -1 * b)),

 (([2,1], Res), (a + b) / (a + -1 * b)),

 (([2], Res), (a + b) / (a + -1 * b)),

 (([], Res), (a + b) / (a + -1 * b))] 

 *)

 val newnds = children (get_nd pt [2,1]) (*see "fun detailrls"*);

 if length newnds = 13 then () else error "rational.sml: interSteps cancel_p rev_rew_p";

 val p = ([2,1,9],Res);

 getTactic 1 p;

 val (_, tac, _) = pt_extract (pt, p);

 case tac of SOME (Rewrite ("sym_distrib_left", _)) => ()

 | _ => error "rational.sml: getTactic, sym_real_plus_binom_times1";

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

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

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

 val thy = @{theory "Rational"};

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

 val tt = str2term "(1 * a + 1 * b) * (1 * a + -1 * b)"(*numerator only*);

 "----- with rewrite_set_";

 val SOME (tt',asm) = rewrite_set_ thy false make_polynomial tt;

 if term2str tt'= "a ^^^ 2 + -1 * b ^^^ 2" then () else error "rls chancel_p 1";

 val tt = str2term "((1 * a + -1 * b) * (1 * a + -1 * b))"(*denominator only*);

 val SOME (tt',asm) = rewrite_set_ thy false make_polynomial tt;

 if term2str tt' = "a ^^^ 2 + -2 * a * b + b ^^^ 2" then () else error "rls chancel_p 2";

 "----- with make_deriv; WN1130912 not investigated further, will be discontinued";

 val SOME (tt, _) = factout_p_ thy t; 

 if term2str tt = "(a + b) * (a + -1 * b) / ((a + -1 * b) * (a + -1 * b))"

 then () else error "rls chancel_p 3";

 term2str tt = "(1 * a + 1 * b) * (1 * a + -1 * b) / ((1 * a + -1 * b) * (1 * a + -1 * b))";

 "--- with simpler ruleset";

 val {rules, rew_ord= (_, ro), ...} = rep_rls (assoc_rls "rev_rew_p");

 val der = make_deriv thy Atools_erls rules ro NONE tt;

 if length der = 12 then () else error "WN1130912 rls chancel_p 4";

 (*default_print_depth 99;*) writeln (deriv2str der); (*default_print_depth 3;*)

 (*default_print_depth 99;*) map (term2str o #1) der; (*default_print_depth 3;*)

 "...,(-1 * b ^^^ 2 + a ^^^ 2) / (-2 * (a * b) + a ^^^ 2 + (-1 * b) ^^^ 2) ]";

 (*default_print_depth 99;*) map (rule2str o #2) der; (*default_print_depth 3;*)

 (*default_print_depth 99;*) map (term2str o #1 o #3) der; (*default_print_depth 3;*)

 val der = make_deriv thy Atools_erls rules ro NONE 

 	(str2term "(1 * a + 1 * b) * (1 * a + -1 * b)");

 (*default_print_depth 99;*) writeln (deriv2str der); (*default_print_depth 3;*)

 val {rules, rew_ord=(_,ro),...} = rep_rls (assoc_rls "rev_rew_p");

 val der = make_deriv thy Atools_erls rules ro NONE 

 	(str2term "(1 * a + -1 * b) * (1 * a + -1 * b)");

 (*default_print_depth 99;*) writeln (deriv2str der); (*default_print_depth 3;*)

 (*default_print_depth 99;*) map (term2str o #1) der; (*default_print_depth 3;*)

 (*WN060829 ...postponed*)

 "-------- fun eval_get_denominator -------------------------------------------";

 "-------- fun eval_get_denominator -------------------------------------------";

 "-------- fun eval_get_denominator -------------------------------------------";

 val thy = @{theory Isac};

 val t = Thm.term_of (the (parse thy "get_denominator ((a +x)/b)"));

 val SOME (_, t') = eval_get_denominator "" 0 t thy;

 if term2str t' = "get_denominator ((a + x) / b) = b"

 then () else error "get_denominator ((a + x) / b) = b"

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

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

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

 (*WN12xxxx TODO: instead of Gabriella's example here (27.Jul.12) find a simpler one

   WN130912: kept this test, although not clear what for*)

 reset_states ();

 CalcTree [(["Term ((5*b + 25)/(a^2 - b^2) * (a - b)/(5*b))", "normalform N"], 

   ("Rational", ["rational", "simplification"], ["simplification", "of_rationals"]))];

 Iterator 1;

 moveActiveRoot 1;

 autoCalculate 1 CompleteCalc;

 val ((pt, p), _) = get_calc 1;

 (*show_pt pt;

 [

 (([], Frm), Simplify ((5 * b + 25) / (a ^^^ 2 - b ^^^ 2) * (a - b) / (5 * b))),

 (([1], Frm), (5 * b + 25) / (a ^^^ 2 - b ^^^ 2) * (a - b) / (5 * b)),

 (([1], Res), (5 * b + 25) / (a ^^^ 2 + -1 * b ^^^ 2) * (a + -1 * b) / (5 * b)),

 (([2], Res), (5 * b + 25) * (a + -1 * b) / (a ^^^ 2 + -1 * b ^^^ 2) / (5 * b)),

 (([3], Res), (25 * a + -25 * b + 5 * (a * b) + -5 * b ^^^ 2) / (a ^^^ 2 + -1 * b ^^^ 2) /

 (5 * b)),

 (([4], Res), (25 + 5 * b) / (a + b) / (5 * b)),

 (([5], Res), (25 + 5 * b) / ((a + b) * (5 * b))),

 (([6], Res), (25 + 5 * b) / (5 * (a * b) + 5 * b ^^^ 2)),

 (([7], Res), (5 + b) / (a * b + b ^^^ 2)),

 (([], Res), (5 + b) / (a * b + b ^^^ 2))] *)

 "-------- WN1309xx non-terminating rls norm_Rational -------------------------";

 "-------- WN1309xx non-terminating rls norm_Rational -------------------------";

 "-------- WN1309xx non-terminating rls norm_Rational -------------------------";

 (*------- 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)))");

 (*1st factor separately simplified *)

 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; 

 if term2str t' = "(-9 * x ^^^ 2 + y ^^^ 2) / -1" then () else error "Nr. 480b lhs changed";

 (*2nd factor separately simplified *)

 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';

 if term2str t' = "-1 / (-1 * x ^^^ 2 + 25 * y ^^^ 2)" then () else error "Nr. 480b rhs changed";

 "-------- Schalk I, p.70 Nr. 477a: terms are exploding ?!?";

 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 t;

 :

 ###  rls: cancel_p on: (a ^^^ 2 * (b * y) + 2 * (a * (b * (x * y))) + b * (x ^^^ 2 * y)) /

 (b + -2 * y) /

 ((b ^^^ 2 + -1 * y ^^^ 2) / (b + 2 * y)) /

 (b ^^^ 2 * y + b * y ^^^ 2) /

 (a ^^^ 2 * b ^^^ 2 + -4 * (a ^^^ 2 * y ^^^ 2) + 4 * (a * (b ^^^ 2 * x)) +

  -16 * (a * (x * y ^^^ 2)) +

  4 * (b ^^^ 2 * x ^^^ 2) +

  -16 * (x ^^^ 2 * y ^^^ 2)) 

 exception Div raised

 BUT

 val t = str2term 

   ("(a ^^^ 2 * (b * y) + 2 * (a * (b * (x * y))) + b * (x ^^^ 2 * y)) /" ^

   "(b + -2 * y) /" ^

   "((b ^^^ 2 + -1 * y ^^^ 2) / (b + 2 * y)) /" ^

   "(b ^^^ 2 * y + b * y ^^^ 2) /" ^

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

   "-16 * (a * (x * y ^^^ 2)) +" ^

   "4 * (b ^^^ 2 * x ^^^ 2) +" ^

   "-16 * (x ^^^ 2 * y ^^^ 2))");

 NONE = cancel_p_ thy t;

 *)

 (*------- Schalk I, p.70 Nr. 476b in 2003 this worked using 10 sec. *)

 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)");

 (*

 trace_rewrite := true;

 rewrite_set_ thy false norm_Rational t;

 :

 ####  rls: cancel_p on: (2 * (a ^^^ 7 * b) + 4 * (a ^^^ 6 * b ^^^ 2) + 6 * (a ^^^ 5 * b ^^^ 3) +

  8 * (a ^^^ 4 * b ^^^ 4) +

  6 * (a ^^^ 3 * b ^^^ 5) +

  4 * (a ^^^ 2 * b ^^^ 6) +

  2 * (a * b ^^^ 7)) /

 (2 * (a ^^^ 9 * b) + 4 * (a ^^^ 8 * b ^^^ 2) +

  2 * (2 * (a ^^^ 7 * b ^^^ 3)) +

  4 * (a ^^^ 6 * b ^^^ 4) +

  -4 * (a ^^^ 4 * b ^^^ 6) +

  -4 * (a ^^^ 3 * b ^^^ 7) +

  -4 * (a ^^^ 2 * b ^^^ 8) +

  -2 * (a * b ^^^ 9))

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

 else error "rational.sml: diff.behav. in norm_Rational_mg 49";

 *)

 "-------- Schalk I, p.70 Nr. 480a: terms are exploding ?!?";

 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)))");

 (*

 trace_rewrite := true;

 rewrite_set_ thy false norm_Rational t;

 :

 ####  rls: cancel_p on: (2 * (x ^^^ 6 * (y ^^^ 2 * z)) + 2 * (x ^^^ 6 * (y * z ^^^ 2)) +

  2 * (x ^^^ 5 * (y ^^^ 2 * z ^^^ 2)) +

  -2 * (x ^^^ 4 * (y ^^^ 2 * z ^^^ 3)) +

  -2 * (x ^^^ 4 * (y * z ^^^ 4)) +

  -2 * (x ^^^ 3 * (y ^^^ 2 * z ^^^ 4))) /

 (-2 * (x ^^^ 6 * (y ^^^ 2 * z)) + -2 * (x ^^^ 6 * (y * z ^^^ 2)) +

  2 * (x ^^^ 5 * (y ^^^ 2 * z ^^^ 2)) +

  2 * (x ^^^ 4 * (y ^^^ 2 * z ^^^ 3)) +

  2 * (x ^^^ 4 * (y * z ^^^ 4)) +

  -2 * (x ^^^ 3 * (y ^^^ 2 * z ^^^ 4)))

 *)

 "-------- Schalk I, p.60 Nr. 215d: terms are exploding, internal loop does not terminate";

 val t = str2term "(a-b)^^^3 * (x+y)^^^4 / ((x+y)^^^2 * (a-b)^^^5)";

 (* Kein Wunder, denn Z???ler 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 t;

 term2str t;

 "a^^^3 * x^^^4 + 4 * a^^^3 * x^^^3 * y +6 * a^^^3 * x^^^2 * y^^^2 +4 * a^^^3 * x * y^^^3 +a^^^3 * y^^^4 +-3 * a^^^2 * b * x^^^4 +-12 * a^^^2 * b * x^^^3 * y +-18 * a^^^2 * b * x^^^2 * y^^^2 +-12 * a^^^2 * b * x * y^^^3 +-3 * a^^^2 * b * y^^^4 +3 * a * b^^^2 * x^^^4 +12 * a * b^^^2 * x^^^3 * y +18 * a * b^^^2 * x^^^2 * y^^^2 +12 * a * b^^^2 * x * y^^^3 +3 * a * b^^^2 * y^^^4 +-1 * b^^^3 * x^^^4 +-4 * b^^^3 * x^^^3 * y +-6 * b^^^3 * x^^^2 * y^^^2 +-4 * b^^^3 * x * y^^^3 +-1 * b^^^3 * y^^^4";

 val t = str2term "((x+y)^^^2 * (a-b)^^^5)";

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

 term2str t;

 "a^^^5 * x^^^2 + 2 * a^^^5 * x * y + a^^^5 * y^^^2 +-5 * a^^^4 * b * x^^^2 +-10 * a^^^4 * b * x * y +-5 * a^^^4 * b * y^^^2 +10 * a^^^3 * b^^^2 * x^^^2 +20 * a^^^3 * b^^^2 * x * y +10 * a^^^3 * b^^^2 * y^^^2 +-10 * a^^^2 * b^^^3 * x^^^2 +-20 * a^^^2 * b^^^3 * x * y +-10 * a^^^2 * b^^^3 * y^^^2 +5 * a * b^^^4 * x^^^2 +10 * a * b^^^4 * x * y +5 * a * b^^^4 * y^^^2 +-1 * b^^^5 * x^^^2 +-2 * b^^^5 * x * y +-1 * b^^^5 * y^^^2";

 anscheinend macht dem Rechner das Krzen diese Bruches keinen Spass mehr ...*)

 "-------- Schalk I, p.70 Nr. 480b: terms are exploding, trace_rewrite stops at";

 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 t;

 :

 ####  rls: cancel_p on: (19440 * (x ^^^ 8 * y ^^^ 2) + -490320 * (x ^^^ 6 * y ^^^ 4) +

  108240 * (x ^^^ 4 * y ^^^ 6) +

  -6000 * (x ^^^ 2 * y ^^^ 8)) /

 (2160 * (x ^^^ 8 * y ^^^ 2) + -108240 * (x ^^^ 6 * y ^^^ 4) +

  1362000 * (x ^^^ 4 * y ^^^ 6) +

  -150000 * (x ^^^ 2 * y ^^^ 8))

 *)