(* Title: tests for rationals

   Author: Stefan Karnel

   Copyright (c) Stefan Karnel 2002

   Use is subject to license terms.

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LEGEND

WN120317.TODO postponed test/../ratinal,ratinal2.sml to joint work with dmeind 

WN070906

   nonterm.SK   marks non-terminating examples

   ord.SK       PARTIALLY marks crucial ordering examples

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

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

*)

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

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

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

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

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

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

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

"-------- fun factout_p_ -----------------------------------------------------";

"-------- fun common_nominator_p_ --------------------------------------------";

"-------- fun add_fraction_p_ ------------------------------------------------";

"-------- TODO ---------------------------------------------------------------";

"-------- and app_rev --------------------------------------------------------";

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

"----------- get_denominator ----------------------------";

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

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

"-------- nonterminating cancel_p, norm_Rational 2002 ---";

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

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

"-------- 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";

"-------- fun factout_p_ -----------------------------------------------------";

"-------- fun factout_p_ -----------------------------------------------------";

"-------- 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')

;

(*========== inhibit exn WN130822 asm different by Rational.thy | Rational2.thy ================

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

then () else error "factout_p_ asm 1 changed"

============ inhibit exn WN130822 asm different by Rational.thy | Rational2.thy ===============*)

;

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

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

"-------- fun common_nominator_p_ --------------------------------------------";

"-------- fun common_nominator_p_ --------------------------------------------";

"-------- 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";

(*========== inhibit exn WN130823 asm different Rational<-->2.thy ==============================

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

then () else error "common_nominator_p_ asm 1 changed"

============ inhibit exn WN130823 asm different Rational<-->2.thy =============================*)

"-------- 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"

;

(*========== inhibit exn WN130823 asm different Rational<-->2.thy ==============================

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

then () else error "common_nominator_p_ asm 2 changed"

============ inhibit exn WN130823 asm different Rational<-->2.thy =============================*)

"-------- 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";

"-------- fun add_fraction_p_ ------------------------------------------------";

"-------- fun add_fraction_p_ ------------------------------------------------";

"-------- 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";

(*========== inhibit exn WN130823 asm different Rational<-->2.thy ==============================

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

then () else error "add_fraction_p_ 3 changed";

============ inhibit exn WN130823 asm different Rational<-->2.thy =============================*)

"-------- TODO ---------------------------------------------------------------";

"-------- TODO ---------------------------------------------------------------";

"-------- TODO ---------------------------------------------------------------";

"-------- and app_rev --------------------------------------------------------";

"-------- and app_rev --------------------------------------------------------";

"-------- and app_rev --------------------------------------------------------";

(* 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 ===================================*)

(*========== inhibit exn WN130824 TODO =======================================================

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

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

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

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 = "[]"

then () else error "factout_p_ 1 changed";

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

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

then () else error "cancel_p_ 1 changed";

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 = "[]"

then () else error "factout_ 2 changed";

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

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

then () else error "cancel_ 2 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 = "[]"

then () else error "add_fraction_p_ 3 changed";

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

if term2str t' = 

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

  andalso terms2str asm = "[]"

then () else error "common_nominator_p_ 3 changed";

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

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

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

then () else error "factout_ 4 changed";

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

if term2str t' = 

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

  andalso terms2str asm = "[]"

then () else error "cancel_ 4 changed";

val t = str2term 

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

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

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

then () else error "add_fraction_p_ 5 changed";

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

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

then () else error "norm_rational_ 5 changed";

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

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

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

val thy  = @{theory "Rational"};

val thy' = "Rational";

val rls' = "cancel";

val mp = "make_polynomial";

"-------- 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 t;

if (term2str t', terms2str asm) = ("14 / 1", "[\"y * x ~= 0\"]")

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 t;

if (term2str t', terms2str asm) = ("4 / 1", "[\"15 * (b * a) ~= 0\"]")

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 t;

if (term2str t', terms2str asm) = ("12 * a / 1", "[\"12 * (c * (b * a)) ~= 0\"]")

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

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

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

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

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

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

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

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

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

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

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

"-------- 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 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 t;

if (term2str t', terms2str asm) = ("8 / 9", "[\"-1 + x ~= 0\"]")

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

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

val SOME (t,_) = rewrite_set_ thy false (*!!!!!!!!!!!!!!!!!!!!!!1*) make_polynomial t;

if (term2str t', terms2str asm) = ("8 / 9", "[\"-1 + x ~= 0\"]")

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 t;

if (term2str t', terms2str asm) = ("5 / 6", "[\"-3 + x ~= 0\"]")

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  (*!!!!! NONE with cancel !!!!!*) cancel_p t;

if (term2str t', terms2str asm) = ("-1 / 1", "[\"b + -1 * a ~= 0\"]")

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

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

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 t;

if (term2str t', terms2str asm) = 

  ("(1 + 9 * a ^^^ 2) / (3 * a + 9 * a ^^^ 2)", "[\"1 + 3 * a ~= 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 NONE (*!!!!!!!! WN.23.10.02 !!!!!!!!!!!!!!!*) = rewrite_set_ thy false cancel t;

"-------- 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") = false; (*!!!!WN.23.10.02!!!!!!!!!!*)

  is_expanded (str2term "25 + -30*x + 9*x^^^2") = false; (*!!!!!!!!WN.23.10.02!!!!!!!!!!*)

  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 NONE (*!!!!!!!! WN.23.10.02 !!!!!!!!!!!!!!!*) = rewrite_set_ thy false cancel t;

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";

(*WN050929, === inhibit exn 110317 ===*)

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";

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

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

val NONE (*!!!!!!!!!!!WN.23.10.02!!!!!!!!!!!!!11*) = rewrite_set_ thy false cancel t;

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

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

val NONE (*!!!!!!!!!!!WN.23.10.02!!!!!!!!!!!!!11*) = rewrite_set_ thy false cancel t;

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

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

val NONE (*!!!!!!!!!!!WN.23.10.02!!!!!!!!!!!!!11*) = rewrite_set_ thy false cancel t;

(*WN:*)

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

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

if (term2str t', terms2str asm) = 

  ("(7 * x ^^^ 3 + -1 * x ^^^ 2 * y) / (7 * x * y ^^^ 2 + -1 * y ^^^ 3)", "[\"5 + 3 * x ~= 0\"]")

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

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

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

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

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

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 = "[]"

then () else error "factout_p_ 1 changed";

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

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

then () else error "cancel_p_ 1 changed";

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 = "[]"

then () else error "factout_ 2 changed";

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

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

then () else error "cancel_ 2 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 = "[]"

then () else error "add_fraction_p_ 3 changed";

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

if term2str t' = 

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

  andalso terms2str asm = "[]"

then () else error "common_nominator_p_ 3 changed";

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

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

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

then () else error "factout_ 4 changed";

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

if term2str t' = 

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

  andalso terms2str asm = "[]"

then () else error "cancel_ 4 changed";

val t = str2term 

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

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

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

then () else error "add_fraction_p_ 5 changed";

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

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

then () else error "norm_rational_ 5 changed";

val t3 = (term_of o the o (parse thy)) "((1) / (2*x + 2)) + ((1) / (2*x - 2)) + ((1) / ( x^^^2 - 1))+((1) / (x^^^2 - 2 * x + 1))";

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

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

(term2str t3'); 

(term2str t3''); 

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

term2str t4;

term2str (hd(t5));

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

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

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

  term2str t';

  term2str t'';

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

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

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

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

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

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

  term2str t';

  term2str t'';

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

val t = str2term "1 / a";

val NONE =  cancel_p_ thy t;

val NONE = rewrite_set_ thy false cancel_p t;

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

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

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

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

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

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

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

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

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

term2str t' =

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

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

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

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

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

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

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

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

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

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

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

(*WN.16.10.02 WN070905 lexicographische Ordnung erhalten *)

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

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

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

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

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

 *** RATIONALS_TERM2POLY_EXCEPTION: Invalid Polynomial 

WN.16.10.02 ?! + WN060831???SK4 

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

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

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

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

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

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

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

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

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

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

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

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

  term2str t';

  term2str t'';

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

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

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

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

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

val thy  = @{theory "Rational"};

val thy' = "Rational";

val rls' = "cancel";

val mp = "make_polynomial";

writeln ("example 186:");

writeln("a)");

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

rewrite_set_ thy false cancel (str2term e186a');

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

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

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

  is_expanded (parse_rat "x * y");

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

else error "rational.sml cancel Schalk e186a";

writeln("b)");

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

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

writeln("c)");

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

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

    handle e => print_exn_G e;

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

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

else error "rational.sml cancel Schalk e186c";

writeln ("example 187:");

writeln("a)");

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

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

writeln("b)");

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

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

writeln("c)");

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

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

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

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

else error "rational.sml cancel Schalk e186d";

writeln "example 188:";

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

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

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

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

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

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

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

writeln("b)");

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

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

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

writeln("c)");

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

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

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

val SOME (t,_) = 

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

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

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

writeln ("example 190:");

writeln("c)");

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

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

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

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

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

writeln ("example 191:");

writeln("a)");

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

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

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

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

  false;

  is_expanded (parse_rat "x + y");

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

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

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

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

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

writeln("c)");

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

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

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

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

  false;

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

  false;

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

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

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

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

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

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

writeln ("example 192:");

writeln("b)");

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

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

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

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

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

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

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

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

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

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

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

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

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

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

writeln ("example 193:");

writeln("a)");

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

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

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

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

writeln("b)");

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

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

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

writeln("c)");

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

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

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

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

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

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

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

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

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

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

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

val rls' = "common_nominator_p";

writeln ("example 204:");

writeln("a)");

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

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

writeln("b)");

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

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

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

else error "rational.sml common_nominator_p example e204b";

writeln ("example 205:");

writeln("a)");

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

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

writeln("b)");

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

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

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

else error "rational.sml common_nominator_p example e204b";

writeln ("example 206:");

writeln("a)");

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

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

writeln("b)");

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

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

writeln ("example 207:");

val e207'="((3 * x * y + 3 * y) / (x * y)) + ((5 * x * y + 7 * y) / (x * y)) + ((9 * x * y + -2 * y) / (x * y)) + ((x * y + 4 * y) / (x * y)) ";

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

writeln ("example 208:");

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

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

writeln ("example 209:");

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

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

writeln ("example 210:");

val e210'="((2 * x + 3 +  -1 * x^^^2) / (5 * x)) + ((5 * x^^^2 + -2 * x + 1) / (5 * x)) + ((-3 * x^^^2 + -2 * x + 1) / (5 * x)) + ((-1 * x^^^2 + -3 * x + -5) / (5 * x)) ";

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

writeln ("example 211:");

writeln("a)"); 

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

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

writeln("b)");

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

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

writeln ("example 212:");

writeln("a)");

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

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

writeln("b)");

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

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

writeln ("example 213:");

writeln("a)"); 

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

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

writeln("b)"); 

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

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

writeln ("example 214:");

writeln("a)");

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

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

writeln("b)");

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

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

writeln ("example 216:");

writeln("a)"); 

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

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

writeln("b)");

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

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

writeln ("example 217:");

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

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

val rls' = "common_nominator";

writeln ("example 218:"); 

val e218'="((9 * a^^^3 - 5 * a^^^2 + 2 * a + 8) / (108 * a^^^4)) + ((-5 * a + 3 * a^^^2 + 4) / (8 * a^^^3)) + ((-261 * a^^^3 + 19 * a^^^2 + -112 * a + 16) / (216 * a^^^4))";

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

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

else error "rationa.sml common_nominator example e218";

writeln ("example 219:");

writeln("a)");

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

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

writeln("b)");

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

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

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

else error "rationa.sml common_nominator example e219b";

writeln ("example 220:");

writeln("a)");

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

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

writeln("b)");

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

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

writeln ("example 221:");

writeln("a)");

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

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

writeln("b)");

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

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

writeln ("example 222:");

writeln("a)");

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

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

writeln("b)");

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

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

writeln ("example 225:");

writeln("a)");

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

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

writeln("b)");

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

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

writeln ("example 226:");

writeln("a)");

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

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

writeln("b)"); 

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

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

writeln ("example 227:");

writeln("a)");

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

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

writeln("b)");

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

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

writeln ("example 228:");

writeln("a)");

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

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

writeln("b)");

val e228b'="((11 * z + 2 * b) / (4 * b * z + -8 * b^^^2)) + ((-8 * z) / (z^^^2 + -4 * b^^^2)) + ((-9 * z + -2 * b) / (4 * b * z + 8 * b^^^2))";

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

writeln ("example 229:");

writeln("a)");

val e229a'="((5 * x^^^2 + y) / (x + 2 * y)) + ((-8 * x^^^3 + 4 * x^^^2 * y + 3 * x * y) / (x^^^2 + -4 * y^^^2)) + ((3 * x^^^2 + -4 * y) / (x + -2 * y))";

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

writeln("b)");

val e229b'="((7 * x^^^2 + y) / (x + 3 * y)) + ((-24 * x^^^2 * y + 5 * x * y + 21 * y^^^2) / (x^^^2 + -9 * y^^^2)) + ((4 * x^^^2 + -6 * y) / (x + -3 * y))"; 

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

writeln ("example 230:");

writeln("a)"); 

val e230a'="((5 * x^^^2 + y) / (2 * x + y)) + ((-16 * x^^^3 + 2 * x^^^2 * y + 6 * x * y) / (4 * x^^^2 + -1 * y^^^2)) + ((3 * x^^^2 + -4 * y) / (2 * x + -1 * y))";

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

writeln("b)");

val e230b'="((7 * x^^^2 + y) / (3 * x + y)) + ((-3 * x^^^3  + 15 * x * y + -7 * x^^^2 * y + 7 * y^^^2) / (9 * x^^^2 + -1 * y^^^2)) + ((4 * x^^^2 + -6 * y) / (3 * x + -1 * y))";

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

writeln ("example 231:");

writeln("a)");

val e231a'="((2 * x + 5 * y) / (x)) + ((2 * x^^^3 + -5 * y^^^3 + 3 * x * y^^^2) / (x^^^3 + -2 * x^^^2 * y + x * y^^^2)) + ((-3 * x + -6 * y) / (x + -1 * y))";

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

writeln("b)");

val e231b'="((6 * x + 2 * y) / (x)) + ((6 * x^^^2 * y + -4 * x * y^^^2 + -2 * y^^^3) / (x^^^3 + -2 * x^^^2 * y + x * y^^^2)) + ((-5 * x + -3 * y) / (x + -1 * y))";

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

writeln ("example 232:");

writeln("a)");

val e232a'="((2 * x + 3 * y) / (x)) + ((4 * x^^^3 + -1 * x * y^^^2 + -3 * y^^^3) / (x^^^3 + -2 * x^^^2 * y + x * y^^^2)) + ((-5 * x + -6 * y) / (x + -1 * y))";

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

writeln("b)");

val e232b'="((5 * x + 2 * y) / (x)) + ((2 * x^^^3 + -3 * x * y^^^2 + 3 * x^^^2 * y + -2 * y^^^3) / (x^^^3 + -2 * x^^^2 * y + x * y^^^2)) + ((-6 * x + -3 * y) / (x + -1 * y))";

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

writeln ("example 233:");

writeln("a)");

val e233a'="((5 * x + 6 * y) / (x)) + ((5 * x * y^^^2 + -6 * y^^^3 + -2 * x^^^3 + 3 * x^^^2 * y) / (x^^^3 + -2 * x^^^2 * y + x * y^^^2)) + ((-2 * x + -3 * y) / (x + -1 * y))";

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

writeln("b)");

val e233b'="((6 * x + 5 * y) / (x)) + ((4 * x^^^2 * y + 3 * x * y^^^2 + -5 * y^^^3 + -2 * x^^^3) / (x^^^3 + -2 * x^^^2 * y + x * y^^^2)) + ((-3 * x + -2 * y) / (x + -1 * y))";

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

writeln ("example 234:");

writeln("a)");

val e234a'="((5 * a + b) / (2 * a * b + -2 * b^^^2)) + ((-3 * a + -1 * b) / (2 * a * b + 2 * b^^^2)) + ((-2 * a) / (a^^^2 + -1 * b^^^2))";

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

writeln("b)"); 

val e234b'="((5 * a + 3 * b) / (6 * a * b + -18 * b^^^2)) + ((-3 * a + -3 * b) / (6 * a * b + 18 * b^^^2)) + ((-2 * a) / (a^^^2 + -9 * b^^^2)) ";

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

writeln ("example 235:");

writeln("a)");

val e235a'="((10 * x + 3 * y) / (12 * x * y + -18 * y^^^2)) + ((-6 * x + -3 * y) / (12 * x * y + 18 * y^^^2)) + ((-4 * x) / (4 * x^^^2 + -9 * y^^^2))";

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

writeln("b)"); 

val e235b'="((8 * a + b) / (4 * a * b + -2 * b^^^2)) + ((-4 * a + -1 * b) / (4 * a * b + 2 * b^^^2)) + ((-2 * a) / (4 * a^^^2 + -1 * b^^^2)) ";

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

writeln ("example 236:");

writeln("a)"); 

val e236a'="((8 * a + 5 * b) / (20 * a * b + -50 * b^^^2)) + ((-4 * a + -5 * b) / (20 * a * b + 50 * b^^^2)) + ((-2 * a) / (4 * a^^^2 + -25 * b^^^2))";

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

writeln("b)");   

val e236b'="((24 * x + y) / (6 * x * y + -2 * y^^^2)) + ((-18 * x + -1 * y) / (6 * x * y + 2 * y^^^2)) + ((-15 * x) / (9 * x^^^2 + -1 * y^^^2)) ";

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

val rls' = "cancel";

writeln ("example heuberger:");

val eheu'="(x^^^4 + x * y + x^^^3 * y + y^^^2) / (x + 5 * x^^^2 + y + 5 * x * y + x^^^2 * y^^^3 + x * y^^^4)";