1 (* Title: tests for rationals
3 Copyright (c) Stefan Karnel 2002
4 Use is subject to license terms.
6 12345678901234567890123456789012345678901234567890123456789012345678901234567890
7 10 20 30 40 50 60 70 80
9 WN120317.TODO postponed test/../ratinal,ratinal2.sml to joint work with dmeind
11 nonterm.SK marks non-terminating examples
12 ord.SK PARTIALLY marks crucial ordering examples
13 *SK* of some (secondary) interest (on 070906)
14 WN060104 transfer examples marked with (*SR..*) to the exp-collection
17 "--------------------------------------------------------";
18 "--------------------------------------------------------";
19 "table of contents --------------------------------------";
20 "--------------------------------------------------------";
21 "~~~~~BEGIN: decomment structure RationalI : RATIONALI ~~";
22 "-------- ... missing WN060103 --------------------------";
23 "-------- fun monom2term, fun poly2term' ---------------";
24 "~~~~~END: decomment structure RationalI : RATIONALI ~~~~";
25 "-------- and app_rev --------------------------------------------------------";
26 "-------- external calculating functions test -----------";
27 "-------- cancel from: Mathematik 1 Schalk Reniets Verlag";
28 "-------- common_nominator_p ----------------------------";
29 "-------- reverse rewrite -------------------------------";
30 "-------- 'reverse-ruleset' cancel_p --------------------";
31 "-------- norm_Rational ---------------------------------";
32 "-------- numeral rationals -----------------------------";
33 "-------- cancellation ----------------------------------";
34 "-------- common denominator ----------------------------";
35 "-------- multiply and cancel ---------------------------";
36 "-------- common denominator and multiplication ---------";
37 "-------- double fractions ------------------------------";
38 "-------- crucial examples ------------------------------";
39 "-------- examples for Stefan Karnels thesis ------------";
40 "-------- me Schalk I No.186 ----------------------------";
41 "-------- interSteps ..Simp_Rat_Double_No-1.xml ---------";
42 "-------- interSteps ..Simp_Rat_Cancel_No-1.xml ---------";
43 "-------- investigate rulesets for cancel_p -------------";
44 "-------- investigate format of factout_ and factout_p_ -";
45 "-------- how to stepwise construct Scripts -------------";
46 "----------- get_denominator ----------------------------";
47 "--------- several errpats in complicated term -------------------";
48 "--------------------------------------------------------";
49 "-------- nonterminating cancel_p, norm_Rational 2002 ---";
50 "--------------------------------------------------------";
51 "--------------------------------------------------------";
54 "~~~~~BEGIN: decomment structure RationalI : RATIONALI ~~";
55 (*.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
56 tests of internal functions: to make them work,
57 out-comment (*!!!*) in knowledge/Rational.ML:
59 structure RationalI : RATIONALI =
66 !!!##*)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~.*)
68 "-------- ... missing WN060103 --------------------------";
69 "-------- ... missing WN060103 --------------------------";
70 "-------- ... missing WN060103 --------------------------";
71 (*~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
72 writeln ("********************* rational.sml - TESTS *************************");
73 writeln ("***** divide tests *****");
74 val mv_pquot1 = (#1(mv_division([(1,[1,1,1]),(1,[1,1,0]),(1,[1,0,1]),(1,[0,0,0])],[(1,[1,1,0]),(1,[0,0,0])],LEX_)));
75 (* result: [(1,[0,0,1]),(1,[0,0,0])] *)
76 if mv_pquot1=[(1,[0,0,1]),(1,[0,0,0])] then () else error ("rational.sml: example failed");
78 val mv_prest1 = (#2(mv_division([(1,[1,1,1]),(1,[1,1,0]),(1,[1,0,1]),(1,[0,0,0])],[(1,[1,1,0]),(1,[0,0,0])],LEX_)));
79 (* result: [(1,[1,0,1]),(~1,[0,0,1])] *)
80 if mv_prest1=[(1,[1,0,1]),(~1,[0,0,1])] then () else error ("rational.sml: example failed");
82 val mv_pquot2 = (#1(mv_division([(4,[2]),(8,[1]),(16,[0])],[(1,[1]),(1,[0])],LEX_)));
83 (* result: [(4,[1]),(4,[0])] *)
84 if mv_pquot2=[(4,[1]),(4,[0])] then () else error ("rational.sml: example failed");
86 val mv_prest2 = (#2(mv_division([(4,[2]),(8,[1]),(16,[0])],[(1,[1]),(1,[0])],LEX_)));
87 (* result: [(12,[0]] *)
88 if mv_prest2=[(12,[0])] then () else error ("rational.sml: example failed");
90 val mv_pquot3 = (#1(mv_division([(4,[2]),(~4,[0])],[(2,[1]),(2,[0])],LEX_)));
91 (* [(2,[1]),(~2,[0])] *)
92 if mv_pquot3=[(2,[1]),(~2,[0])] then () else error ("rational.sml: example failed");
94 val mv_prest3 = (#2(mv_division([(1,[2]),(~1,[0])],[(2,[1]),(2,[0])],LEX_)));
95 (* [(1,[2]),(~1,[0])] *)
96 if mv_prest3=[(1,[2]),(~1,[0])] then () else error ("rational.sml: example failed");
98 val mv_pquot4 = (#1(mv_division([(3,[1,1,1]),(4,[1,0,1]),(3,[0,0,1])],[(2,[1,0,0]),(4,[0,0,1])],LEX_)));
100 if mv_pquot4=[(1,[0,1,1])] then () else error ("rational.sml: example failed");
102 val mv_prest4 = (#2(mv_division([(3,[1,1,1]),(4,[1,0,1]),(3,[0,0,1])],[(2,[1,0,0]),(4,[0,0,1])],GGO_)));
103 (* [(1,[1,1,1]),(~4,[0,1,2]),(4,[1,0,1]),(3,[0,0,1])] *)
104 if mv_prest4 =[(1,[1,1,1]),(~4,[0,1,2]),(4,[1,0,1]),(3,[0,0,1])] then () else error ("rational.sml: example failed");
106 val mv_pquot5 = (#1(mv_division([(3,[1,1,1]),(4,[1,0,1]),(3,[0,0,1]),(6,[2,1,3]),(4,[0,4,1]),(1,[2,2,1])],[(1,[0,0,1])],LEX_)));
107 (* [(1,[2,2,0]),(6,[2,1,2]),(3,[1,1,0]),(4,[1,0,0]),(4,[0,4,0]),(3,[0,0,0])]*)
108 if mv_pquot5=[(1,[2,2,0]),(6,[2,1,2]),(3,[1,1,0]),(4,[1,0,0]),(4,[0,4,0]),(3,[0,0,0])] then () else error ("rational.sml: example failed");
110 val mv_prest5 = (#2(mv_division([(3,[1,1,1]),(4,[1,0,1]),(3,[0,0,1]),(6,[2,1,3]),(4,[0,4,1]),(1,[2,2,1])],[(1,[0,0,1])],LEX_)));
112 if mv_prest5=[] then () else error ("rational.sml: example failed");
114 (* (x^2 + 2(a+1)x + (a^2+2a+1)) / (x+a+1) = x+a+1 *)
115 val mv_pquot6 = (#1(mv_division([(1,[2,0,0]),(2,[1,1,0]),(2,[1,0,0]),(1,[0,2,0]),(2,[0,1,0]),(1,[0,0,0])],[(1,[1,0,0]),(1,[0,1,0]),(1,[0,0,0])],LEX_)));
116 if mv_pquot6=[(1,[1,0,0]),(1,[0,1,0]),(1,[0,0,0])] then () else error ("rational.sml: example failed");
118 val mv_prest6 = (#2(mv_division([(1,[2,0,0]),(2,[1,1,0]),(2,[1,0,0]),(1,[0,2,0]),(2,[0,1,0]),(1,[0,0,0])],[(1,[1,0,0]),(1,[0,1,0]),(1,[0,0,0])],LEX_)));
119 if mv_prest6=[] then () else error ("rational.sml: example failed");
122 writeln ("***** MV_CONTENT-TESTS *****");
123 val mv_cont1=mv_content([(1,[2,1]),(1,[2,0]),(1,[1,1]),(1,[1,0]),(1,[0,1]),(1,[0,0])]);
124 (* [(1,[0,1]),(1,[0,0])] *)
125 if mv_cont1=[(1,[0,1]),(1,[0,0])] then () else error ("rational.sml: example failed");
127 val mv_pp1=mv_pp([(1,[1,1]),(1,[1,0]),(1,[0,1]),(1,[0,0])]);
128 (*[(1,[1,0]),(1,[0,0])]*)
129 if mv_pp1=[(1,[1,0]),(1,[0,0])] then () else error ("rational.sml: example failed");
131 val mv_cont2=mv_content([(2,[1]),(4,[0])]);
133 if mv_cont2=[(2,[0])] then () else error ("rational.sml: example failed");
135 val mv_pp2=mv_pp([(2,[1]),(4,[0])]);
136 (* [(1,[1]),(2,[0])] *)
137 if mv_pp2=[(1,[1]),(2,[0])] then () else error ("rational.sml: example failed");
139 val mv_cont3=mv_content[(8,[2,1,1]),(12,[1,0,2]),(10,[2,2,0]),(16,[1,1,1])];
141 if mv_cont3=[(2,[0,0,0])] then () else error ("rational.sml: example failed");
143 val mv_pp3=mv_pp[(8,[2,1,1]),(12,[1,0,2]),(10,[2,2,0]),(16,[1,1,1])];
144 (* [(5,[2,2,0]),(4,[2,1,1]),(8,[1,1,1]),(6,[1,0,2])] *)
145 if mv_pp3=[(5,[2,2,0]),(4,[2,1,1]),(8,[1,1,1]),(6,[1,0,2])] then () else error ("rational.sml: example failed");
147 val mv_cont4=mv_content[(2,[2,1,0]),(3,[1,0,1]),(2,[1,1,0]),(3,[0,0,1])];
149 if mv_cont4=[(1,[0,0,0])] then () else error ("rational.sml: example failed");
151 val mv_pp4=mv_pp [(2,[2,1,0]),(3,[1,0,1]),(2,[1,1,0]),(3,[0,0,1])];
152 (* [(2,[2,1,0]),(2,[1,1,0]),(3,[1,0,1]),(3,[0,0,1])] *)
153 if mv_pp4=[(2,[2,1,0]),(2,[1,1,0]),(3,[1,0,1]),(3,[0,0,1])] then () else error ("rational.sml: example failed");
155 val con1=mv_content([(9,[2,0]),(15,[1,1]),(12,[1,0]),(6,[0,2]),(12,[0,1])]);
157 if con1=[(3,[0,0])] then () else error ("rational.sml: example failed");
159 val pp1=mv_pp([(9,[2,0]),(15,[1,1]),(12,[1,0]),(6,[0,2]),(12,[0,1])]);
160 (* [(3,[2,0]),(5,[1,1]),(4,[1,0]),(2,[0,2]),(4,[0,1])] *)
161 if pp1=[(3,[2,0]),(5,[1,1]),(4,[1,0]),(2,[0,2]),(4,[0,1])] then () else error ("rational.sml: example failed");
163 val con2=mv_content([(1,[2,0]),(1,[1,1]),(1,[1,0]),(1,[0,2]),(1,[0,1])]);
165 if con2=[(1,[0,0])] then () else error ("rational.sml: example failed");
167 val pp2 =mv_pp([(1,[2,0]),(1,[1,1]),(1,[1,0]),(1,[0,2]),(1,[0,1])]);
168 (* [(1,[2,0]),(1,[1,1]),(1,[1,0]),(1,[0,2]),(1,[0,1])] *)
169 if pp2=[(1,[2,0]),(1,[1,1]),(1,[1,0]),(1,[0,2]),(1,[0,1])] then () else error ("rational.sml: example failed");
171 val cont1 = mv_content [(1,[2,1,0]),(2,[2,1,0])];
173 if cont1=[(3,[0,1,0])] then () else error ("rational.sml: example failed");
175 val pp1 = mv_pp [(1,[2,1,0]),(2,[2,1,0])];
177 if pp1=[(1,[2,0,0])] then () else error ("rational.sml: example failed");
179 val cont2 = mv_content [(4,[1,2,0]),(2,[2,1,0])];
181 if cont2=[(2,[0,1,0])] then () else error ("rational.sml: example failed");
183 val pp2 = mv_pp [(4,[1,2,0]),(2,[2,1,0])];
184 (* [(1,[2,0,0]),(2,[1,1,0])] *)
185 if pp2=[(1,[2,0,0]),(2,[1,1,0])] then () else error ("rational.sml: example failed");
187 writeln ("********************************************************");
188 val cont3=mv_content [(65,[3,2,2]),(52,[3,2,1]),(26,[3,1,2]),(~95,[2,2,3]),(~76,[2,2,2]),(35,[2,2,1]),(28,[2,2,0]),(~38,[2,1,3]),(14,[2,1,1])];
189 if cont3=[(5,[0,2,1]),(4,[0,2,0]),(2,[0,1,1])] then () else error ("rational.sml: example failed");
190 val pp3=mv_pp [(65,[3,2,2]),(52,[3,2,1]),(26,[3,1,2]),(~95,[2,2,3]),(~76,[2,2,2]),(35,[2,2,1]),(28,[2,2,0]),(~38,[2,1,3]),(14,[2,1,1])];
193 "-------- fun monom2term, fun poly2term' ---------------";
194 "-------- fun monom2term, fun poly2term' ---------------";
195 "-------- fun monom2term, fun poly2term' ---------------";
196 val t = monom2term ((3,[2,1,0]), ["c","b","a"](*reverse ???SK-*));
197 term2str t = "3 * (c ^^^ 2 * b)" (*true*);
199 val t = monom2term ((1,[1,0]), ["b","a"]);
200 term2str t = "b" (*true*);
202 val t = poly2term ([(1,[0,0,0]),(2,[1,0,0]),(3,[2,1,0]),(4,[3,2,1])],
204 term2str t = "1 + 2 * c + 3 * (c ^^^ 2 * b) + 4 * (c ^^^ 3 * (b ^^^ 2 * a))";
206 val t = poly2term ([(1,[1,0]),(1,[0,1])], ["b","a"]);
207 term2str t = "a + b" (*true*);
209 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~*)
210 "~~~~~END: decomment structure RationalI : RATIONALI ~~~~";
211 "~~~~~END: decomment structure RationalI : RATIONALI ~~~~";
212 "~~~~~END: decomment structure RationalI : RATIONALI ~~~~";
215 fun parse_rat str = (term_of o the o (parse thy)) str;
217 writeln ("***** mv_gcd-tests *****");
218 val ggt1 = mv_gcd [(4,[2,2]),(8,[1,1]),(4,[0,0])] [(2,[1,1]),(2,[0,0])];
219 (* [(2,[1,1]),(2,[0,0])] *)
220 if ggt1=[(2,[1,1]),(2,[0,0])] then () else error ("rational.sml: example failed");
222 val ggt2 = mv_gcd [(8,[2,1,1]),(12,[1,0,2]),(10,[2,2,0]),(15,[1,1,1])] [(2,[2,1,0]),(3,[1,0,1]),(2,[1,1,0]),(3,[0,0,1])];
223 (* [(2,[1,1,0]),(3,[0,0,1])] *)
224 if ggt2=[(2,[1,1,0]),(3,[0,0,1])] then () else error ("rational.sml: example failed");
227 val ggt3 = mv_gcd [(1,[2,0]),(~2,[1,1]),(1,[0,2])] [(1,[1,0]),(~1,[0,1])];
228 (* [(1,[1,0]),(~1,[0,1])] *)
229 if ggt3=[(1,[1,0]),(~1,[0,1])] then () else error ("rational.sml: example failed");
232 val ggt4 = mv_gcd [(1,[2,1,0]),(2,[2,1,0])] [(5,[1,0,0])];
234 if ggt4=[(1,[1,0,0])] then () else error ("rational.sml: example failed");
237 val ggt5 = mv_gcd [(4,[2,0]),(~8,[1,1]),(4,[0,2])] [(1,[2,0]),(~1,[0,2])];
238 (* [(1,[1,0]),(~1,[0,1])] *)
239 if ggt5=[(1,[1,0]),(~1,[0,1])] then () else error ("rational.sml: example failed");
242 val ggt6 = mv_gcd [(10,[2,1,1]),(14,[1,1,0]),(3,[1,0,1]),(20,[1,2,1])] [(5,[1,1,1]),(7,[2,1,1])];
244 if ggt6=[(1,[1,0,0])] then () else error ("rational.sml: example failed");
246 writeln ("***** kgv-tests *****");
247 val kgv1=mv_lcm [(10,[])] [(15,[])];
249 if kgv1=[(30,[])] then () else error ("rational.sml: example failed");
251 val kgv2=mv_lcm [(1,[2,0]),(~2,[1,1]),(1,[0,2])] [(1,[1,0]),(~1,[0,1])];
252 (* [(1,[2,0]),(~2,[1,1]),(1,[0,2])] *)
253 if kgv2=[(1,[2,0]),(~2,[1,1]),(1,[0,2])] then () else error ("rational.sml: example failed");
255 val kgv3=mv_lcm [(4,[2,0]),(~8,[1,1]),(4,[0,2])] [(1,[2,0]),(~1,[0,2])];
256 (* [(4,[3,0]),(~4,[2,1]),(~4,[1,2]),(4,[0,3])] *)
257 if kgv3=[(4,[3,0]),(~4,[2,1]),(~4,[1,2]),(4,[0,3])] then () else error ("rational.sml: example failed");
260 writeln ("***** STEP_CANCEL_TESTS: *****");
262 val term2 = (term_of o the o (parse thy)) " (9 * a^^^2 * b) / (6 * a * c)";
263 val div2 = term2str (step_cancel term2);
264 if div2 = "3 * (a * b) * (3 * a) / (2 * c * (3 * a))" then () else error ("rational.sml: example failed");
267 val term1 = (term_of o the o (parse thy)) "(10 * a^^^2 * b * c + 14 * a * b + 3 * a * c + 20 * a * b^^^2 * c) / a";
268 val div1 = term2str(step_cancel term1);
269 if div1 = "(3 * c + 14 * b + 20 * (b ^^^ 2 * c) + 10 * (a * (b * c))) * a / (1 * a)" then () else error ("rational.sml: example failed");
271 val term3 = (term_of o the o (parse thy)) "(10 * a^^^2 * b * c) / (1 * x * y * z) ";
272 val div3 = term2str(step_cancel term3);
273 if div3="10 * a ^^^ 2 * b * c / (1 * x * y * z)" then () else error ("rational.sml: example failed");
275 --------------------------------------------------------------------------!!!*)
277 (*-----versuche 13.3.03-----
278 val t = str2term "1 - x^^^2 - 5 * x^^^5";
279 val vs=(((map free2str) o vars) t);
280 val SOME ml = expanded2poly t vs;
282 poly2term'(rev(sort (mv_geq LEX_) (ml)),vs);
283 poly2term'([(~5,[5]),(~1,[2]),(1,[0])], vs);
284 monom2term((~5,[5]),vs);
285 monom2term((~1,[2]),vs);
286 val t' = monom2term((1,[0]),vs);(*uncaught exception LIST*)
288 val (i,is) = (~1,[2]);
289 val ttt = Const ("Groups.times_class.times", [HOLogic.realT,HOLogic.realT]---> HOLogic.realT) $
290 (Const ("uminus", HOLogic.realT --> HOLogic.realT) $
291 Free ((str_of_int o abs) i, HOLogic.realT)) $
292 powerproduct2term(is, vs);
294 -------versuche 13.3.03-----*)
296 val t = str2term "1 - x^^^2 - 5 * x^^^5";
297 val SOME t' = expanded2polynomial t; term2str t';
298 "1 + - 1 * x ^^^ 2 + - 5 * x ^^^ 5";
299 val t = str2term "1 - x";
300 val SOME t' = expanded2polynomial t; term2str t';
302 val t = str2term "1 + (-1) * x";
303 val SOME t' = expanded2polynomial t; term2str t';
305 val t = (term_of o the o (parse thy)) "1 + (-1) * x ^^^ 2 + (-5) * x ^^^5";
306 val SOME t' = polynomial2expanded t; term2str t';
307 "1 - x ^^^ 2 - 5 * x ^^^ 5";
309 "-------- and app_rev --------------------------------------------------------";
310 "-------- and app_rev --------------------------------------------------------";
311 "-------- and app_rev --------------------------------------------------------";
312 (* trace down until prepats are evaluated
313 (which does not to work, because substitution is not done -- compare rew_sub!);
314 keep this sequence for the case, factout_p, cancel_p, common_nominator_p, add_fraction_p
315 (again) get prepat = [] changed to <>[]. *)
316 val t = str2term "(x^^^2 + -1*y^^^2) / (x^^^2 + -1*x*y)";
318 (*rewrite_set_ @{theory Isac} true cancel t = NONE; !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!*)
319 "~~~~~ fun rewrite_set_, args:"; val (thy, bool, rls, term) = (thy, false, cancel_p, t);
320 "~~~~~ fun rewrite__set_, args:"; val (thy, i, _, _, (rrls as Rrls _), t) =
321 (thy, 1, bool, [], rls, term);
322 (*val (t', asm, rew) = app_rev thy (i+1) rrls t; rew = false!!!!!!!!!!!!!!!!!!!!!*)
323 "~~~~~ fun app_rev, args:"; val (thy, i, rrls, t) = (thy, (i+1), rrls, t);
324 fun chk_prepat thy erls [] t = true
325 | chk_prepat thy erls prepat t =
327 fun chk (pres, pat) =
329 val subst: Type.tyenv * Envir.tenv =
330 Pattern.match thy (pat, t) (Vartab.empty, Vartab.empty)
332 snd (eval__true thy (i + 1) (map (Envir.subst_term subst) pres) [] erls)
333 end) handle _ => false
334 fun scan_ f [] = false (*scan_ NEVER called by []*)
335 | scan_ f (pp::pps) =
336 if f pp then true else scan_ f pps;
337 in scan_ chk prepat end;
338 (* apply the normal_form of a rev-set *)
339 fun app_rev' thy (Rrls{erls,prepat,scr=Rfuns{normal_form,...},...}) t =
340 if chk_prepat thy erls prepat t
341 then ((*tracing("### app_rev': t = "^(term2str t));*) normal_form t)
343 (* val opt = app_rev' thy rrls t ..NONE!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!*)
344 "~~~~~ and app_rev', args:"; val (thy, (Rrls{erls,prepat,scr=Rfuns{normal_form,...},...}), t) =
346 (* chk_prepat thy erls prepat t = false!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!*)
347 (* app_sub thy i rrls t = false!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!*)
348 "~~~~~ fun chk_prepat, args:"; val (thy, erls, prepat, t) = (thy, erls, prepat, t);
349 fun chk (pres, pat) =
351 val subst: Type.tyenv * Envir.tenv =
352 Pattern.match thy (pat, t) (Vartab.empty, Vartab.empty)
354 snd (eval__true thy (i + 1) (map (Envir.subst_term subst) pres) [] erls)
355 end) handle _ => false
356 fun scan_ f [] = false (*scan_ NEVER called by []*)
357 | scan_ f (pp::pps) =
358 if f pp then true else scan_ f pps;
360 (*========== inhibit exn WN130823: (pp::pps) = [(*prepat*)] ====================================
361 (* scan_ chk prepat = false!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!*)
362 "~~~~~ fun , args:"; val (f, (pp::pps)) = (chk, prepat);
364 val ([t1, t2], t) = pp;
365 term2str t1 = "?r is_expanded";
366 term2str t2 = "?s is_expanded";
367 term2str t = "?r / ?s";
368 (* f pp = false!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!*)
369 "~~~~~ fun chk, args:"; val (pres, pat) = (pp);
370 val subst: Type.tyenv * Envir.tenv =
371 Pattern.match thy (pat, t) (Vartab.empty, Vartab.empty)
373 ({}, {(("r", 0), ("real", Var (("r", 0), "real"))),
374 (("s", 0), ("real", Var (("s", 0), "real")))}*)
376 snd (eval__true thy (i + 1) (map (Envir.subst_term subst) pres) [] erls)
377 "~~~~~ fun eval__true, args:"; val (thy, i, asms, bdv, rls) =
378 (thy, (i + 1), (map (Envir.subst_term subst) pres), [], erls);
379 terms2str asms; (* = "[\"?r is_expanded\",\"?s is_expanded\"]"*)
380 asms = [@{term True}] orelse asms = []; (* = false*)
381 asms = [@{term False}] ; (* = false*)
382 "~~~~~ fun chk, args:"; val (indets, (a::asms)) = ([], asms);
383 bdv (*= []: _a list*);
384 val bdv : (term * term) list = [];
385 rewrite__set_ thy (i+1) false;
386 term2str a = "?r is_expanded"; (*hier m"usste doch der Numerator eingesetzt sein ??????????????*)
387 val SOME (Const ("HOL.False", _), []) = rewrite__set_ thy (i+1) false bdv rls a
388 ============ inhibit exn WN130823: (pp::pps) = [(*prepat*)] ===================================*)
390 "-------- external calculating functions test -----------";
391 "-------- external calculating functions test -----------";
392 "-------- external calculating functions test -----------";
393 val t1 = (term_of o the o (parse thy)) "((3 * x^^^2 + 6 *x + 3) / (2*x + 2))";
394 val SOME (t1',asm)= factout_p_ thy t1;
395 term2str t1'; terms2str asm;
396 "(3 + 3 * x) * (1 + 1 * x) / (2 * (1 + 1 * x))";
398 val SOME (t1',asm)= cancel_p_ thy t1;
399 term2str t1'; terms2str asm;
401 "[\"1 + 1 * x ~= 0\"]";
403 val t = (term_of o the o (parse thy)) "((-3 * x^^^2 + 6 *x - 3) / (2*x - 2))";
404 val SOME (t',asm)= cancel_ thy t;
405 term2str t'; terms2str asm;
408 val SOME (t',asm)= factout_ thy t;
409 term2str t'; terms2str asm;
410 "(3 - 3 * x) * (-1 + x) / (2 * (-1 + x))";
413 val t = str2term "((x+ (-1)) / (x + 1)) + ((x + 1) / (x + (-1)))";
414 val SOME (t',asm) = add_fraction_p_ thy t;
415 term2str t'; terms2str asm;
416 "(2 + 2 * x ^^^ 2) / (-1 + 1 * x ^^^ 2)";
418 val SOME (t',asm) = common_nominator_p_ thy t;
419 term2str t'; terms2str asm;
420 "(-1 + 1 * x) * (-1 + 1 * x) / ((1 + 1 * x) * (-1 + 1 * x)) +\n(1 + 1 * x) * (1 + 1 * x) / ((1 + 1 * x) * (-1 + 1 * x))";
423 val t = str2term "((x - 1) / (x + 1)) + ((x + 1) / (x - 1))";
424 val SOME (t',asm) = add_fraction_ thy t;
425 term2str t'; terms2str asm;
426 "(2 + 2 * x ^^^ 2) / (-1 + x ^^^ 2)";
428 val SOME (t',asm) = common_nominator_ thy t;
429 term2str t'; terms2str asm;
430 "(-1 + x) * (-1 + x) / ((1 + x) * (-1 + x)) +\n(1 + x) * (1 + x) / ((1 + x) * (-1 + x))";
433 val t = str2term "((1) / (2*x + 2)) + ((1) / (2*x + (-2))) + ((1) / ( x^^^2 + (-1)))+((1) / (x^^^2 + (-2)*x + 1))";
434 val SOME (t',asm) = common_nominator_p_ thy t;
435 term2str t'; terms2str asm;
436 "1 * (1 + -2 * x + 1 * x ^^^ 2) /\n((-1 + 1 * x) * (2 * ((-1 + 1 * x) * (1 + 1 * x)))) +\n(1 * (-1 + 1 * x ^^^ 2) /\n ((-1 + 1 * x) * (2 * ((-1 + 1 * x) * (1 + 1 * x)))) +\n (1 * (-2 + 2 * x) / ((-1 + 1 * x) * (2 * ((-1 + 1 * x) * (1 + 1 * x)))) +\n 1 * (#";
438 val SOME (t',asm) = add_fraction_p_ thy t;
439 term2str t'; terms2str asm;
440 "1 * x / (1 + -2 * x + 1 * x ^^^ 2)";
441 "[\"1 + 1 * x ~= 0\"]";
442 val SOME(t',asm) = norm_rational_ thy t;
443 term2str t'; terms2str asm;
444 "1 * x / (1 + -2 * x + 1 * x ^^^ 2)";
445 "[\"1 + 1 * x ~= 0\"]";
447 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))";
448 val SOME (t3',_) = common_nominator_ thy t3;
449 val SOME (t3'',_) = add_fraction_ thy t3;
453 val SOME(t4,t5) = norm_expanded_rat_ thy t3;
459 val t=(term_of o the o (parse thy)) "(9 - x^^^2)/(9 - 6*x + x^^^2)";
460 val SOME (t',_) = factout_ thy t;
461 val SOME (t'',_) = cancel_ thy t;
464 "(3 + x) * (3 - x) / ((3 - x) * (3 - x))";
467 val t=(term_of o the o (parse thy))
468 "(9 - x^^^2) / (9 - 6*x + x^^^2) + 1 / (3 - x)";
469 val SOME (t',_) = common_nominator_ thy t;
470 val SOME (t'',_) = add_fraction_ thy t;
473 "(9 - x ^^^ 2) / ((3 - x) * (3 - x)) + 1 * (3 - x) / ((3 - x) * (3 - x))";
476 (*WN021016 added -----vv---*)
477 val t = str2term "(9 - x^^^2) / (9 - 6*x + x^^^2) + 1";
478 val SOME (t',_) = common_nominator_ thy t;
479 val SOME (t'',_) = add_fraction_ thy t;
480 term2str t' = "(9 - x ^^^ 2) / (9 - 6 * x + x ^^^ 2) +\n1" ^
481 " * (9 - 6 * x + x ^^^ 2) / (9 - 6 * x + x ^^^ 2)" (*true*);
482 term2str t'' = "6 / (3 - x)" (*true*);
484 val t = str2term "1 + (9 - x^^^2) / (9 - 6*x + x^^^2)";
485 val SOME (t',_) = common_nominator_ thy t;
486 val SOME (t'',_) = add_fraction_ thy t;
487 term2str t' = "1 * (9 - 6 * x + x ^^^ 2) / (9 - 6 * x + x ^^^ 2) +\n" ^
488 "(9 - x ^^^ 2) / (9 - 6 * x + x ^^^ 2)" (*true*);
489 term2str t'' = "6 / (3 - x)" (*true*);
490 (*WN021016 added -----^^---*)
491 (*WN030602 added -----vv--- no rewrite -> NONE !*)
492 val t = str2term "1 / a";
493 val NONE = cancel_p_ thy t;
494 val NONE = rewrite_set_ thy false cancel_p t;
495 (*WN.2.6.03 added -------^^---*)
497 val t = str2term "(y^^^2 - x^^^2)/(y^^^2 - 2*y*x + x^^^2)";
498 val SOME (t',_) = factout_ thy t;
499 val SOME (t'',_) = cancel_ thy t;
500 term2str t' = "(y + x) * (y - x) / ((y - x) * (y - x))"(*true*);
501 term2str t'' = "(y + x) / (y - x)";
503 val t = str2term "(y^^^2 - x^^^2)/(y^^^2 - 2*y*x + x^^^2) + 1 / (y - x)";
504 val SOME (t',_) = common_nominator_ thy t;
505 val SOME (t'',_) = add_fraction_ thy t;
507 "(-1 * x ^^^ 2 + y ^^^ 2) / ((-1 * x + y) * (-1 * x + y)) +\n1" ^
508 " * (-1 * x + y) / ((-1 * x + y) * (-1 * x + y))" (*true*);
509 term2str t'' = "(-1 - x - y) / (x - y)" (*true*);
511 val t = str2term "(x^^^2 - y^^^2)/(x^^^2 - 2*x*y + y^^^2) + 1 / (x - y)";
512 val SOME (t',_) = common_nominator_ thy t;
513 val SOME (t'',_) = add_fraction_ thy t;
514 if term2str t' = "(-1 * y ^^^ 2 + x ^^^ 2) / ((-1 * y + x) * (-1 * y + x))" ^
515 " +\n1 * (-1 * y + x) / ((-1 * y + x) * (-1 * y + x))" then ()
516 else error "rational.sml lex-ord 1";
517 if term2str t'' = "(-1 - y - x) / (y - x)" then ()
518 else error "rational.sml lex-ord 2";
519 (*WN.16.10.02 WN070905 lexicographische Ordnung erhalten ! SK.ord*)
522 val t = str2term "(x^^^2 - y^^^2)/(x^^^2 - 2*x*y + y^^^2)";
523 val SOME (t',_) = norm_expanded_rat_ thy t;
524 if term2str t' = "(x + y) / (x - y)" then ()
525 else error "rational.sml term2poly: invalid x ^^^ 2 - y ^^^ 2 No.1";
526 (*val SOME (t'',_) = norm_rational_ thy t;
527 *** RATIONALS_TERM2POLY_EXCEPTION: Invalid Polynomial
528 WN.16.10.02 ?! + WN060831???SK4
529 WN070905 *** term2poly: invalid = x ^^^ 2 - y ^^^ 2*)
532 val t = str2term "(9 - x^^^2)/(9 - 6*x + x^^^2) + (1)/(3 + x)";
533 val SOME (t',_) = norm_expanded_rat_ thy t;
534 if term2str t' = "(12 + 5 * x + x ^^^ 2) / (9 - x ^^^ 2)" then ()
535 else error "rational.sml (9 - x^^^2)/(9 - 6*x + x^^^2) +...";
536 (*val SOME (t'',_) = norm_rational_ thy t;
537 *** RATIONALS_TERM2POLY_EXCEPTION: Invalid Polynomial WN.16.10.02 ?!
538 WN070906 *** term2poly: invalid = 9 - x ^^^ 2 SK.term2poly*)
540 val t=(term_of o the o (parse thy))
541 "(9 + (-1)* x^^^2)/(9 + (-1)* 6*x + x^^^2) + (1)/(3 + x)";
542 val SOME (t',_) = norm_expanded_rat_ thy t;
543 val SOME (t'',_) = norm_rational_ thy t;
546 "(12 + 5 * x + x ^^^ 2) / (9 - x ^^^ 2)";
547 "(12 + 5 * x + x ^^^ 2) / (9 + (-1) * x ^^^ 2)";
550 "-------- cancel from: Mathematik 1 Schalk Reniets Verlag";
551 "-------- cancel from: Mathematik 1 Schalk Reniets Verlag";
552 "-------- cancel from: Mathematik 1 Schalk Reniets Verlag";
553 val thy = @{theory "Rational"};
554 val thy' = "Rational";
556 val mp = "make_polynomial";
558 writeln ("example 186:");
560 val e186a'="(14 * x * y) / ( x * y )";(*SRC*)
561 rewrite_set_ thy false cancel (str2term e186a');
563 val e186a = the (rewrite_set thy' false "cancel" e186a');
564 is_expanded (parse_rat "14 * x * y");
565 is_expanded (parse_rat "x * y");
566 if e186a = ("14 / 1", "[\"y * x ~= 0\"]") then ()
567 else error "rational.sml cancel Schalk e186a";
570 val e186b'="(60 * a * b) / ( 15 * a * b )";
571 val e186b = the (rewrite_set thy' false "cancel" e186b');
573 val e186c'="(144 * a^^^2 * b * c) / (12 * a * b * c )";
574 val e186c = (the (rewrite_set thy' false "cancel" e186c'))
575 handle e => print_exn_G e;
576 val t = (term_of o the o (parse thy)) e186c';
577 if e186c = ("12 * a / 1", "[\"12 * (c * (b * a)) ~= 0\"]") then ()
578 else error "rational.sml cancel Schalk e186c";
580 writeln ("example 187:");
582 val e187a'="(12 * x * y) / (8 * y^^^2 )";(*SRC*)
583 val e187a = the (rewrite_set thy' false "cancel" e187a');
585 val e187b'="(8 * x^^^2 * y * z ) / (18 * x * y^^^2 * z )";
586 val e187b = the (rewrite_set thy' false "cancel" e187b');
588 val e187d'="(9 * x^^^5 * y^^^2 * z^^^4) / (15 * x^^^6 * y^^^3 * z )";(*SRC*)
589 val e187d = the (rewrite_set thy' false "cancel" e187d');
590 if e187d = ("3 * z ^^^ 3 / (5 * (y * x))",
591 "[\"3 * (z * (y ^^^ 2 * x ^^^ 5)) ~= 0\"]") then ()
592 else error "rational.sml cancel Schalk e186d";
594 writeln "example 188:";
595 val e188a'="(-8 + 8 * x) / (-9 + 9 * x)";(*SRC*)
596 val e188a = the (rewrite_set thy' false "cancel" e188a');
597 is_expanded (parse_rat "8 * x + -8");
598 (* e188a = ("8 / 9",["not ((-1) + x = 0)"]) then () 13.3.03*)
599 if e188a = ("8 / 9", "[\"-1 + x ~= 0\"]") then ()
600 else error "rational.sml: e188a new behaviour";
602 val SOME (t,_) = rewrite_set thy' false mp "(8*((-1) + x))/(9*((-1) + x))";
604 val e188b'="(-15 + 5 * x) / (-18 + 6 * x)";(*SRC*)
605 val SOME (t, asm) = rewrite_set thy' false "cancel" e188b';
606 t = "5 / 6" (*true*);
609 val e188c'="( a + -1 * b ) / ( b + -1 * a )";
610 val e188c = the (rewrite_set thy' false "cancel_p" e188c');
611 (*is_expanded (parse_rat "a + -1 * b");*)
613 rewrite_set thy' false mp "((-1)*(b + (-1) * a))/(1*(b + (-1) * a))";
614 if t= "(a + -1 * b) / (-1 * a + b)" then()
615 else error "rational.sml: e188c new behaviour";
617 writeln ("example 190:");
619 val e190c'="( 27 * a^^^3 + 9 * a^^^2 + 3 * a + 1 ) / ( 27 * a^^^3 + 18 * a^^^2 + 3 * a )";
620 val e190c = the (rewrite_set thy' false "cancel" e190c');
621 val SOME (t,_) = rewrite_set thy' false mp "((1 + 9 * a ^^^ 2)*(1 + 3 * a))/((3 * a + 9 * a ^^^ 2)*(1 + 3 * a))";
622 if t = "(1 + 3 * a + 9 * a ^^^ 2 + 27 * a ^^^ 3) /\n(3 * a + 18 * a ^^^ 2 + 27 * a ^^^ 3)" then ()
623 else error "rational.sml: e190c new behaviour";
625 writeln ("example 191:");
627 val e191a'="( x^^^2 + -1 * y^^^2 ) / ( x + y )";
629 val e191a = the (rewrite_set thy' false "cancel" e191a');
630 is_expanded (parse_rat "x^^^2 + -1 * y^^^2");
632 is_expanded (parse_rat "x + y");
634 val SOME (t,_) = rewrite_set thy' false mp "((x + (-1) * y)*(x + y))/((1)*(x + y))";
635 (* t="(x ^^^ 2 + -1 * y ^^^ 2) / (x + y)" then() WN.13.3.03*)
636 if t="(x ^^^ 2 + -1 * y ^^^ 2) / (x + y)" then()
637 else error "rational.sml: e191a new behaviour";
640 val e191c'="( 9 * x^^^2 + -30 * x + 25 ) / ( 9 * x^^^2 + -25 )";
642 val e191c = the (rewrite_set thy' false "cancel" e191c');
643 is_expanded (parse_rat "9 * x^^^2 + -30 * x + 25");
645 is_expanded (parse_rat "25 + -30*x + 9*x^^^2");
647 is_expanded (parse_rat "-25 + 9*x^^^2");
649 val SOME (t,_) = rewrite_set thy' false mp "(((-5) + 3 * x)*((-5) + 3 * x))/((5 + 3 * x)*((-5) + 3 * x))";
650 (* t="(25 + ((-30) * x + 9 * x ^^^ 2)) / ((-25) + 9 * x ^^^ 2)"then() 13.3.03*)
651 if t= "(25 + -30 * x + 9 * x ^^^ 2) / (-25 + 9 * x ^^^ 2)" then()
652 else error "rational.sml: 'e191c' new behaviour";
655 writeln ("example 192:");
657 val e192b'="( 7 * x^^^3 + -1 * x^^^2 * y ) / ( 7 * x * y^^^2 + -1 * y^^^3 )";
659 val e192b = the (rewrite_set thy' false "cancel" e192b');
660 -------------------*)
661 val SOME (t',_) = rewrite_set thy' false mp "((x ^^^ 2)*(7 * x + (-1) * y))/((y ^^^ 2)*(7 * x + (-1) * y))";
662 (*========== inhibit exn 110317 ================================================
663 if t' = "(7 * x ^^^ 3 + -1 * x ^^^ 2 * y) / (7 * x * y ^^^ 2 + -1 * y ^^^ 3)"
664 (*"(-1 * y * x ^^^ 2 + 7 * x ^^^ 3) / (-1 * y ^^^ 3 + 7 * x * y ^^^ 2)"WN050929*)
665 then () else error "rational.sml: 'e192b' new behaviour";
666 (*^^^ works with MG's simplifier vvv*)
667 val t =str2term"((x ^^^ 2)*(7 * x + (-1) * y))/((y ^^^ 2)*(7 * x + (-1) * y))";
668 val SOME (t',_) = rewrite_set_ thy false make_polynomial t;
669 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";
670 ============ inhibit exn 110317 ==============================================*)
672 writeln ("example 193:");
674 val e193a'="( x^^^2 + -6 * x + 9 ) / ( x^^^2 + -9 )";
676 val e193a = the (rewrite_set thy' false "cancel" e193a');
677 -------------------*)
679 val e193b'="( x^^^2 + -8 * x + 16 ) / ( 2 * x^^^2 + -32 )";
681 val e193b = the (rewrite_set thy' false "cancel" e193b');
683 val e193c'="( 2 * x + -50 * x^^^3 ) / ( 25 * x^^^2 + -10 * x + 1 )";
684 val SOME(t,_) = rewrite_set thy' false "cancel" e193c';
685 -------------------*)
687 val wn01 = "(-25 + 9*x^^^2)/(5 + 3*x)";
688 val SOME (t, asm) = rewrite_set thy' false "cancel" wn01;
689 (* t = "((-5) + 3 * x) / 1" then () WN.13.3.03*)
690 if t = "(-5 + 3 * x) / 1" andalso asm = "[\"5 + 3 * x ~= 0\"]" then ()
691 else error "rational.sml: new behav. in cancel wn01";
693 "-------- common_nominator_p ----------------------------";
694 "-------- common_nominator_p ----------------------------";
695 "-------- common_nominator_p ----------------------------";
696 val rls' = "common_nominator_p";
698 writeln ("example 204:");
700 val e204a'="((5 * x) / 9) + ((3 * x) / 9) + (x / 9)";
701 val e204a = the (rewrite_set thy' false "common_nominator_p" e204a');
703 val e204b'="5 / x + -3 / x + -1 / x";
704 val e204b = the (rewrite_set thy' false "common_nominator_p" e204b');
705 if e204b = ("1 / x", "[]") then ()
706 else error "rational.sml common_nominator_p example e204b";
708 writeln ("example 205:");
710 val e205a'="((4 * x + 7) / 8) + ((4 * x + 3) / 8)";
711 val e205a = the (rewrite_set thy' false "common_nominator_p" e205a');
713 val e205b'="((5 * x + 2) / 3) + ((-2 * x + 1) / 3)";
714 val e205b = the (rewrite_set thy' false "common_nominator_p" e205b');
715 if e205b = ("(1 + x) / 1", "[]") then ()
716 else error "rational.sml common_nominator_p example e204b";
718 writeln ("example 206:");
720 val e206a'="((5 * x + 4) / (2 * x + -1)) + ((9 * x + 5) / (2 * x + -1))";
721 val e206a = the (rewrite_set thy' false "common_nominator_p" e206a');
723 val e206b'="((17 * x + -23) / (5 * x + 4)) + ((-25 + -17 * x) / (5 * x + 4))";
724 val e206b = the (rewrite_set thy' false "common_nominator_p" e206b');
726 writeln ("example 207:");
727 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)) ";
728 val e207 = the (rewrite_set thy' false "common_nominator_p" e207');
730 writeln ("example 208:");
731 val e208'="((3 * x + 2) / (x + 2)) + ((5 * x + -1) / (x + 2)) + ((-7 * x + -3) / (x + 2)) + ((-1 * x + -3) / (x + 2)) ";
732 val e208 = the (rewrite_set thy' false "common_nominator_p" e208');
734 writeln ("example 209:");
735 val e209'="((3 * x + -7 * y + 3 * z) / (4)) + ((2 * x + 17 * y + 10 * z) / (4)) + ((-1 * x + 2 * y + z) / (4)) ";
736 val e209 = the (rewrite_set thy' false "common_nominator_p" e209');
738 writeln ("example 210:");
739 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)) ";
740 val e210 = the (rewrite_set thy' false "common_nominator_p" e210');
742 writeln ("example 211:");
744 val e211a'="((b) / (a + -1 * b)) + ((-1 * a) / (a + -1 * b))";
745 val e211a = the (rewrite_set thy' false "common_nominator_p" e211a');
747 val e211b'="((b) / (b^^^2 + -1 * a^^^2)) + ((-1 * a) / (b^^^2 + -1 * a^^^2))";
748 val e211b = the (rewrite_set thy' false "common_nominator_p" e211b');
750 writeln ("example 212:");
752 val e212a'="((4) / (x)) + ((-3) / (y)) + -1";
753 val e212a = the (rewrite_set thy' false "common_nominator_p" e212a');
755 val e212b'="((4) / (x)) + ((-5) / (y)) + ((6) / (x*y))";
756 val e212b = the (rewrite_set thy' false "common_nominator_p" e212b');
758 writeln ("example 213:");
760 val e213a'="((5 * x) / (3 * y^^^2)) + ((19 * z) / (6 * x * y)) + ((-2 * x) / (3 * y^^^2)) + ((7 * y^^^2) / (6 * x^^^2)) ";
761 val e213a = the (rewrite_set thy' false "common_nominator_p" e213a');
763 val e213b'="((2 * b) / (3 * a^^^2)) + ((3 * c) / (7 * a * b)) + ((4 * b) / (3 * a^^^2)) + ((3 * a) / (7 * b^^^2))";
764 val e213b = the (rewrite_set thy' false "common_nominator_p" e213b');
766 writeln ("example 214:");
768 val e214a'="((3 * x + 2 * y + 2 * z) / (4)) + ((-5 * x + -3 * y) / (3)) + ((x + y + -2 * z) / (2))";
769 val e214a = the (rewrite_set thy' false "common_nominator_p" e214a');
771 val e214b'="((5 * x + 2 * y + z) / (2)) + ((-7 * x + -3 * y) / (3)) + ((3 * x + 6 * y + -1 * z) / (12))";
772 val e214b = the (rewrite_set thy' false "common_nominator_p" e214b');
774 writeln ("example 216:");
776 val e216a'="((2 * b + 3 * c) / (a * c)) + ((3 * a + b) / (a * b)) + ((-2 * b^^^2 + -3 * a * c) / (a * b * c))";
777 val e216a = the (rewrite_set thy' false "common_nominator_p" e216a');
779 val e216b'="((2 * a + 3 * b) / (b * c)) + ((3 * c + a) / (a * c)) + ((-2 * a^^^2 + -3 * b * c) / (a * b * c))";
780 val e216b = the (rewrite_set thy' false "common_nominator_p" e216b');
782 writeln ("example 217:");
783 val e217'="((z + -1) / (z)) + ((3 * z ^^^2 + -6 * z + 5) / (z^^^2)) + ((-4 * z^^^3 + 7 * z^^^2 + -5 * z + 5) / (z^^^3))";
784 val e217 = the (rewrite_set thy' false "common_nominator_p" e217');
787 val rls' = "common_nominator";
788 writeln ("example 218:");
789 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))";
790 val e218 = the (rewrite_set thy' false "common_nominator" e218');
791 if e218 = ("(16 - 63 * a ^^^ 2 - 81 * a ^^^ 3) / (108 * a ^^^ 4)", "[]") then ()
792 else error "rationa.sml common_nominator example e218";
794 writeln ("example 219:");
796 val e219a'="((1) / (y + 1)) + ((1) / (y + 2)) + ((1) / (y + 3))";
797 val e219a = the (rewrite_set thy' false "common_nominator" e219a');
799 val e219b'="((1) / (x + 1)) + ((1) / (x + 2)) + ((-2) / (x + 3))";
800 val e219b = the (rewrite_set thy' false "common_nominator" e219b');
801 if e219b = ("(5 + 3 * x) / (6 + 11 * x + 6 * x ^^^ 2 + x ^^^ 3)", "[]") then ()
802 else error "rationa.sml common_nominator example e219b";
804 writeln ("example 220:");
806 val e220a'="((17) / (5 * r + -2)) + ((-13) / (2 * r + 3)) + ((4) / (3 * r + -5))";
807 val e220a = the (rewrite_set thy' false "common_nominator" e220a');
809 val e220b'="((20 * a) / (a + -3)) + ((-19 * a) / (a + -4)) + ((a) / (a + -5))";
810 val e220b = the (rewrite_set thy' false "common_nominator" e220b');
812 writeln ("example 221:");
814 val e221a'="((a + b) / (a + -1 * b)) + ((a + -1 * b) / (a + b))";
815 val e221a = the (rewrite_set thy' false "common_nominator" e221a');
817 val e221b'="((x + -1 * y) / (x + y)) + ((x + y) / (x + -1 * y)) ";
818 val e221b = the (rewrite_set thy' false "common_nominator" e221b');
820 writeln ("example 222:");
822 val e222a'="((1 + -1 * x) / (1 + x)) + ((-1 + -1 * x) / (1 + -1 * x)) + ((4 * x) / (1 + -1 * x^^^2))";
823 val e222a = the (rewrite_set thy' false "common_nominator" e222a');
825 val e222b'="((1 + x ) / (1 + -1 * x)) + ((-1 + x) / (1 + x)) + ((2 * x) / (1 + -1 * x^^^2))";
826 val e222b = the (rewrite_set thy' false "common_nominator" e222b');
828 writeln ("example 225:");
830 val e225a'="((6 * a) / (a^^^2 + -64)) + ((a + 2) / (2 * a + 16)) + ((-1) / (2))";
831 val e225a = the (rewrite_set thy' false "common_nominator" e225a');
833 val e225b'="((a + 2 ) / (2 * a + 12)) + ((4 * a) / (a^^^2 + -36)) + ((-1) / (2))";
834 val e225b = the (rewrite_set thy' false "common_nominator" e225b');
836 writeln ("example 226:");
838 val e226a'="((35 * z) / (49 * z^^^2 + -4)) + -1 + ((14 * z + -1) / (14 * z + 4)) ";
839 val e226a = the (rewrite_set thy' false "common_nominator" e226a');
841 val e226b'="((45 * a * b) / (25 * a^^^2 + -9 * b^^^2)) + ((20 * a + 3 * b) / (10 * a + 6 * b)) + -2";
842 val e226b = the (rewrite_set thy' false "common_nominator" e226b');
844 writeln ("example 227:");
846 val e227a'="((6 * z + 11) / (6 * z + 14)) + ((9 * z ) / (9 * z^^^2 + -49)) + -1 ";
847 val e227a = the (rewrite_set thy' false "common_nominator" e227a');
849 val e227b'="((16 * a + 37 * b) / (4 * a + 10 * b)) + ((6 * a * b) / (4 * a^^^2 + -25 * b^^^2)) + -4 ";
850 val e227b = the (rewrite_set thy' false "common_nominator" e227b');
852 writeln ("example 228:");
854 val e228a'="((7 * a + 11) / (3 * a^^^2 + -3)) + ((-2 * a + -1) / (a^^^2 + -1 * a)) + ((-1) / (3 * a + 3))";
855 val e228a = the (rewrite_set thy' false "common_nominator" e228a');
857 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))";
858 val e228b = the (rewrite_set thy' false "common_nominator" e228b');
861 writeln ("example 229:");
863 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))";
864 val e229a = the (rewrite_set thy' false "common_nominator" e229a');
866 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))";
867 val e229b = the (rewrite_set thy' false "common_nominator" e229b');
869 writeln ("example 230:");
871 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))";
872 val e230a = the (rewrite_set thy' false "common_nominator" e230a');
874 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))";
875 val e230b = the (rewrite_set thy' false "common_nominator" e230b');
877 writeln ("example 231:");
879 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))";
880 val e231a = the (rewrite_set thy' false "common_nominator" e231a');
882 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))";
883 val e231b = the (rewrite_set thy' false "common_nominator" e231b');
885 writeln ("example 232:");
887 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))";
888 val e232a = the (rewrite_set thy' false "common_nominator" e232a');
890 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))";
891 val e232b = the (rewrite_set thy' false "common_nominator" e232b');
893 writeln ("example 233:");
895 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))";
896 val e233a = the (rewrite_set thy' false "common_nominator" e233a');
898 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))";
899 val e233b = the (rewrite_set thy' false "common_nominator" e233b');
901 writeln ("example 234:");
903 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))";
904 val e234a = the (rewrite_set thy' false "common_nominator" e234a');
906 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)) ";
907 val e234b = the (rewrite_set thy' false "common_nominator" e234b');
909 writeln ("example 235:");
911 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))";
912 val e235a = the (rewrite_set thy' false "common_nominator" e235a');
914 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)) ";
915 val e235b = the (rewrite_set thy' false "common_nominator" e235b');
917 writeln ("example 236:");
919 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))";
920 val e236a = the (rewrite_set thy' false "common_nominator" e236a');
922 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)) ";
923 val e236b = the (rewrite_set thy' false "common_nominator" e236b');
927 writeln ("example heuberger:");
928 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)";
929 val eheu = the (rewrite_set thy' false "cancel" eheu');
931 val rls' = "common_nominator_p";
932 writeln ("example stiefel:");
933 val est1'="(7) / (-14) + (-2) / (4)";
934 val est1 = the (rewrite_set thy' false "common_nominator_p" est1');
935 if est1 = ("-1 / 1", "[]") then ()
936 else error "new behaviour in rational.sml: est1'";
938 val t = (term_of o the o (parse thy))
939 "(9 - x ^^^ 2) / (9 - 6 * x + x ^^^ 2)";
940 val SOME (t',_) = factout_ thy t;
941 if term2str t' = "(3 + x) * (3 - x) / ((3 - x) * (3 - x))" then ()
942 else error "rational.sml factout_ (9 - x ^^^ 2) / (9 - 6 * x + x ^^^ 2)";
945 "-------- reverse rewrite -------------------------------";
946 "-------- reverse rewrite -------------------------------";
947 "-------- reverse rewrite -------------------------------";
948 (*WN.28.8.02: tests for the 'reverse-rewrite' functions:
949 these are defined in Rationals.ML and stored in
950 the 'reverse-ruleset' cancel*)
952 (*the term for which reverse rewriting is demonstrated*)
953 val t = (term_of o the o (parse thy))
954 "(9 - x ^^^ 2) / (9 + 6 * x + x ^^^ 2)";
955 val Rrls {scr=Rfuns {init_state=ini,locate_rule=loc,
956 next_rule=nex,normal_form=nor,...},...} = cancel;
958 (*normal_form produces the result in ONE step*)
959 val SOME (t',_) = nor t;
960 if term2str t' = "(3 - x) / (3 + x)" then ()
961 else error "rational.sml normal_form (9 - x ^^^ 2) / (9 - 6 * x + x ^^^ 2)";
963 (*initialize the interpreter state used by the 'me'*)
964 val (t,_,revsets,_) = ini t;
966 (*find the rule 'r' to apply to term 't'*)
967 val SOME (r as (Thm (str, thm))) = nex revsets t;
968 if str = "sym_#power_Float ((3,0), (0,0)) __ ((2,0), (0,0))"
969 andalso string_of_thm thm =
970 (string_of_thm o make_thm o (cterm_of (@{theory "Isac"})))
971 (Trueprop $ (term_of o the o (parse thy)) "9 = 3 ^^^ 2") then ()
972 else error "rational.sml next_rule (9 - x ^^^ 2) / (9 - 6 * x + x ^^^ 2)";
973 (* before Isa02->09-2 was not checked automatically, ?was different?:
974 val SOME r = nex revsets t;
975 val r = Thm ("sym_#mult_2_3","6 = 2 * 3") : rule*)
977 (* check, if the rule 'r' applied by the user to 't' belongs to the ruleset;
978 if the rule is OK, the term resulting from applying the rule is returned,too;
979 there might be several rule applications inbetween,
980 which are listed after the head in reverse order *)
981 val (r, (t, asm))::_ = loc revsets t r;
982 if term2str t = "(9 - x ^^^ 2) / (3 ^^^ 2 + 6 * x + x ^^^ 2)" andalso asm = []
984 else error "rational.sml locate_rule (9 - x ^^^ 2) / (9 - 6 * x + x ^^^ 2)";
986 (* find the next rule to apply *)
987 val SOME (r as (Thm (str, thm))) = nex revsets t;
988 if str = "sym_#power_Float ((3,0), (0,0)) __ ((2,0), (0,0))" andalso
989 string_of_thm thm = (string_of_thm o make_thm o (cterm_of (@{theory "Isac"})))
990 (Trueprop $ (term_of o the o (parse thy)) "9 = 3 ^^^ 2") then ()
991 else error "rational.sml next_rule (9 - x ^^^ 2) / (9 - 6 * x + x ^^^ 2)";
993 (*check the next rule*)
994 val (r, (t, asm)) :: _ = loc revsets t r;
995 if term2str t = "(3 ^^^ 2 - x ^^^ 2) / (3 ^^^ 2 + 6 * x + x ^^^ 2)" then ()
996 else error "rational.sml locate_rule (9 - x ^^^ 2) / (9 - 6 * x + x ^^^ 2) II";
998 (*find and check the next rules, rewrite*)
999 val SOME r = nex revsets t;
1000 val (r,(t,asm))::_ = loc revsets t r;
1001 if term2str t = "(3 ^^^ 2 - x ^^^ 2) / (3 ^^^ 2 + 2 * 3 * x + x ^^^ 2)" then ()
1002 else error "rational.sml locate_rule II";
1004 val SOME r = nex revsets t;
1005 val (r,(t,asm))::_ = loc revsets t r;
1006 if term2str t = "(3 - x) * (3 + x) / (3 ^^^ 2 + 2 * 3 * x + x ^^^ 2)" then ()
1007 else error "rational.sml next_rule II";
1009 val SOME r = nex revsets t;
1010 val (r,(t,asm))::_ = loc revsets t r;
1011 if term2str t = "(3 - x) * (3 + x) / ((3 + x) * (3 + x))" then ()
1012 else error "rational.sml next_rule III";
1014 val SOME r = nex revsets t;
1015 val (r, (t, asm)) :: _ = loc revsets t r;
1016 val ss = term2str t;
1017 if ss = "(3 - x) / (3 + x)" andalso terms2str asm = "[\"3 + x ~= 0\"]" then ()
1018 else error "rational.sml: new behav. in rev-set cancel";
1020 "-------- 'reverse-ruleset' cancel_p --------------------";
1021 "-------- 'reverse-ruleset' cancel_p --------------------";
1022 "-------- 'reverse-ruleset' cancel_p --------------------";
1023 (*WN.11.9.02: the 'reverse-ruleset' cancel_p*)
1025 (*the term for which reverse rewriting is demonstrated*)
1026 val t = str2term "(9 + (-1)*x^^^2) / (9 + ((-6)*x + x^^^2))";
1027 val Rrls {scr=Rfuns {init_state=ini,locate_rule=loc,
1028 next_rule=nex,normal_form=nor,...},...} = cancel_p;
1030 (*normal_form produces the result in ONE step*)
1031 val SOME (t',_) = nor t;
1032 term2str t' = "(3 + 1 * x) / (3 + -1 * x)";
1034 (*initialize the interpreter state used by the 'me'*)
1035 val SOME (t', asm) = cancel_p_ thy t;
1036 term2str t' = "(3 + x) / (3 + -1 * x)" (*true*);
1037 terms2str asm = "[\"3 + -1 * x ~= 0\"]" (*true*);
1038 val (t,_,revsets,_) = ini t;
1040 (* WN.10.10.02: dieser Fall terminiert nicht
1041 (make_polynomial enth"alt zu viele rules)
1042 WN060823 'init_state' requires rewriting on specified location in the term
1043 print_depth 99; Rfuns; print_depth 3;
1044 WN060831 cycling "sym_order_mult_rls_" "sym_mult_assoc"
1045 as was with make_polynomial before ?!?*)
1047 val SOME r = nex revsets t;
1048 eq_Thm (r, Thm ("sym_#power_Float ((3,0), (0,0)) __ ((2,0), (0,0))",
1049 mk_thm thy "9 = 3 ^^^ 2"));
1050 (*WN060831 *** id_of_thm
1051 Exception- ERROR raised ...
1052 val (r,(t,asm))::_ = loc revsets t r;
1055 val SOME r = nex revsets t;
1056 val (r,(t,asm))::_ = loc revsets t r;
1060 writeln "****************** all tests successfull *************************";
1064 (*WN.17.3.03 =========================================================vvv---*)
1065 "-------- norm_Rational ---------------------------------";
1066 "-------- norm_Rational ---------------------------------";
1067 "-------- norm_Rational ---------------------------------";
1068 val t = str2term "(3*x+5)/18 - x/2 - -(3*x - 2)/9 = 0";
1069 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; term2str t';
1070 if term2str t' = "1 / 18 = 0" then () else error "rational.sml 1";
1072 val t = str2term "(17*x - 51)/9 - (-(13*x - 3)/6) + 11 - (9*x - 7)/4 = 0";
1073 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; term2str t';
1074 if term2str t' = "(237 + 65 * x) / 36 = 0" then ()
1075 else error "rational.sml 2";
1077 val t = str2term "(1/2 + (5*x)/2)^^^2 - ((13*x)/2 - 5/2)^^^2 - (6*x)^^^2 + 29";
1078 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; term2str t';
1079 if term2str t' = "23 + 35 * x + -72 * x ^^^ 2" then ()
1080 else error "rational.sml 3";
1082 (*trace_rewrite:=true;*)
1083 val t = str2term "Not (6*x is_atom)";
1084 val SOME (t',_) = rewrite_set_ thy false powers_erls t; term2str t';
1086 val t = str2term "1 < 2";
1087 val SOME (t',_) = rewrite_set_ thy false powers_erls t; term2str t';
1090 val t = str2term "(6*x)^^^2";
1091 val SOME (t',_) = rewrite_ thy dummy_ord powers_erls false
1092 (num_str @{thm realpow_def_atom}) t;
1093 if term2str t' = "6 * x * (6 * x) ^^^ (2 + -1)" then ()
1094 else error "rational.sml powers_erls (6*x)^^^2";
1096 val t = str2term "-1 * (-2 * (5 / 2 * (13 * x / 2)))";
1097 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; term2str t';
1098 if term2str t' = "65 * x / 2" then () else error "rational.sml 4";
1100 val t = str2term "1 - ((13*x)/2 - 5/2)^^^2";
1101 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; term2str t';
1102 (*bef.040209: if term2str t' = "(-21 + (130 * x + -169 * x ^^^ 2)) / 4"then()*)
1103 if term2str t' = "(-21 + 130 * x + -169 * x ^^^ 2) / 4" then ()
1104 else error "rational.sml 5";
1106 (*SRAM Schalk I, p.92 Nr. 609a*)
1107 val t = str2term "2*(3 - x/5)/3 - 4*(1 - x/3) - x/3 - 2*(x/2 - 1/4)/27 +5/54";
1108 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; term2str t';
1109 if term2str t' = "(-255 + 112 * x) / 135" then ()
1110 else error "rational.sml 6";
1112 (*SRAM Schalk I, p.92 Nr. 610c*)
1113 val t = str2term "((x- 1)/(x+1) + 1) / ((x- 1)/(x+1) - (x+1)/(x- 1)) - 2";
1114 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; term2str t';
1115 if term2str t' = "(-3 + -1 * x) / 2" then () else error "rational.sml 7";
1117 (*SRAM Schalk I, p.92 Nr. 476a*)
1118 val t = str2term "(x^^^2/(1 - x^^^2) + 1)/(x/(1 - x) + 1) * (1 + x)";
1119 (*. a/b : c/d translated to a/b * d/c .*)
1120 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; term2str t';
1121 (*if term2str t' = "1 / 1" then () else error "rational.sml 8";3.6.03*)
1122 if term2str t' = "1" then () else error "rational.sml 8";
1124 (*............................vvv---TODO: sollte gehen mit poly_order *)
1125 (*Schalk I, p.92 Nr. 472a*)
1126 val t = str2term "((8*x^^^2 - 32*y^^^2)/(2*x + 4*y))/((4*x - 8*y)/(x + y))";
1127 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; term2str t';
1128 if term2str t' = "x + y" then () else error "rational.sml p.92 Nr. 472a";
1130 (*Schalk I, p.70 Nr. 480b; a/b : c/d translated to a/b * d/c*)
1131 val t = str2term ("((12*x*y/(9*x^^^2 - y^^^2))/" ^
1132 "(1/(3*x - y)^^^2 - 1/(3*x + y)^^^2)) *" ^
1133 "(1/(x - 5*y)^^^2 - 1/(x + 5*y)^^^2)/" ^
1134 "(20*x*y/(x^^^2 - 25*y^^^2))");
1135 (*... nicht simpl, zerlegt ...*)
1136 val t = str2term ("((12*x*y/(9*x^^^2 - y^^^2))/" ^
1137 "(1/(3*x - y)^^^2 - 1/(3*x + y)^^^2))");
1138 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; term2str t';
1139 "(-12 * (x * y ^^^ 3) + 108 * (x * (y * x ^^^ 2))) / (12 * (x * y))";
1140 (* ~~~~~~~~~~ poly_order notwendig!*)
1141 val t = str2term ("(1/(x - 5*y)^^^2 - 1/(x + 5*y)^^^2)/" ^
1142 "(20*x*y/(x^^^2 - 25*y^^^2))");
1143 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; term2str t';
1144 if term2str t' = "1 / (x ^^^ 2 + -25 * y ^^^ 2)" then ()
1145 else error "rational.sml norm_Rational 1 / (x ^^^ 2 + -25 * y ^^^ 2)";
1147 "nonterm.SK6 ----- SK060904-2a non-termination of add_fraction_p_";
1148 (*WN.2.6.03 from rlang.sml 56a
1149 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)";
1150 val NONE = rewrite_set_ thy false common_nominator_p t;
1152 WN060831 nonterm.SK7
1153 val t = str2term "(a + b * x) / (a + -1 * (b * x)) + (-1 * a + b * x) / (a + b * x)";
1154 val NONE = add_fraction_p_ thy t;
1158 (* ------------------------------------------------------------------- *)
1159 (*---------vvv------------ MG: ab 1.7.03 ----------------vvv-----------*)
1160 (* Simplifier fuer beliebige Buchterme *)
1161 (* ------------------------------------------------------------------- *)
1162 (*----------------------- norm_Rational_mg ----------------------------*)
1163 (* ------------------------------------------------------------------- *)
1165 "-------- numeral rationals -----------------------------";
1166 "-------- numeral rationals -----------------------------";
1167 "-------- numeral rationals -----------------------------";
1168 (*SRA Schalk I, p.40 Nr. 164b *)
1169 val t = str2term "(47/6 - 76/9 + 13/4)/(35/12)";
1170 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1172 if (term2str t) = "19 / 21" then ()
1173 else error "rational.sml: diff.behav. in norm_Rational_mg 1";
1175 (*SRA Schalk I, p.40 Nr. 166a *)
1176 val t = str2term "((5/4)/(4+22/7) + 37/20)*(110/3 - 110/9 * 23/11)";
1177 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1179 if (term2str t) = "45 / 2" then ()
1180 else error "rational.sml: diff.behav. in norm_Rational_mg 2";
1183 "-------- cancellation ----------------------------------";
1184 "-------- cancellation ----------------------------------";
1185 "-------- cancellation ----------------------------------";
1186 (* e190c Stefan K.*)
1188 "((1 + 9 * a ^^^ 2)*(1 + 3 * a))/((3 * a + 9 * a ^^^ 2)*(1 + 3 * a))";
1189 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1192 "(1 + 9 * a ^^^ 2) / (3 * a + 9 * a ^^^ 2)"
1194 else error "rational.sml: diff.behav. in norm_Rational_mg 3";
1196 (* e192b Stefan K.*)
1198 "((x ^^^ 2)*(7 * x + (-1) * y))/((y ^^^ 2)*(7 * x + (-1) * y))";
1199 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1204 else error "rational.sml: diff.behav. in norm_Rational_mg 4";
1206 (*SRC Schalk I, p.66 Nr. 379c *)
1209 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1214 else error "rational.sml: diff.behav. in norm_Rational_mg 5";
1216 (*SRC Schalk I, p.66 Nr. 380b *)
1218 "15*(3*x+3)*(4*x+9)/(12*(2*x+7)*(5*x+5))";
1219 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1222 "(27 + 12 * x) / (28 + 8 * x)"
1224 else error "rational.sml: diff.behav. in norm_Rational_mg 6";
1226 (*Schalk I, p.60 Nr. 215c *)
1227 (* Falsches Ergebnis: rechnet lange und cancel_p kann nicht weiter krzen!!!*)
1229 val t = str2term "(a+b)^^^4*(x - y)/((x - y)^^^3*(a+b)^^^2)";
1230 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1233 "(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)"
1235 else error "rational.sml: diff.behav. in norm_Rational_mg 7";
1238 "(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)"
1239 val SOME (t,_) = rewrite_set_ thy false cancel_p t;
1241 (* uncaught exception nonexhaustive binding failure
1242 raised at: stdIn:93.1-93.51 *)
1244 (*Schalk I, p.66 Nr. 381a *)
1245 (* ACHTUNG: rechnet ca. 2 Minuten !!! *)
1247 val t = str2term "18*(a+b)^^^3*(a - b)^^^2/(72*(a - b)^^^3*(a+b)^^^2)";
1248 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1251 "(a + b) / (4 * a + -4 * b)"
1252 then () else error "rational.sml: diff.behav. in norm_Rational_mg 8";
1255 (*SRC Schalk I, p.66 Nr. 381b *)
1257 "(4*x^^^2 - 20*x + 25)/(2*x - 5)^^^3";
1258 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1263 else error "rational.sml: diff.behav. in norm_Rational_mg 9";
1265 (*SRC Schalk I, p.66 Nr. 381c *)
1267 "(27*a^^^3+9*a^^^2+3*a+1)/(27*a^^^3+18*a^^^2+3*a)";
1268 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1271 "(1 + 9 * a ^^^ 2) / (3 * a + 9 * a ^^^ 2)"
1273 else error "rational.sml: diff.behav. in norm_Rational_mg 10";
1275 (*SRC Schalk I, p.66 Nr. 383a *)
1277 "(5*a^^^2 - 5*a*b)/(a - b)^^^2";
1278 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1281 "5 * a / (a + -1 * b)"
1283 else error "rational.sml: diff.behav. in norm_Rational_mg 11";
1286 "-------- common denominator ----------------------------";
1287 "-------- common denominator ----------------------------";
1288 "-------- common denominator ----------------------------";
1289 (*SRA Schalk I, p.67 Nr. 403a *)
1292 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1295 "(-3 * x + 4 * y + -1 * x * y) / (x * y)"
1297 else error "rational.sml: diff.behav. in norm_Rational_mg 12";
1299 (*SRA Schalk I, p.67 Nr. 407b *)
1301 "(2*a+3*b)/(b*c) + (3*c+a)/(a*c) - (2*a^^^2+3*b*c)/(a*b*c)";
1302 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1307 else error "rational.sml: diff.behav. in norm_Rational_mg 13";
1309 (*SRA Schalk I, p.67 Nr. 410b *)
1311 "1/(x+1) + 1/(x+2) - 2/(x+3)";
1312 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1315 "(5 + 3 * x) / (6 + 11 * x + 6 * x ^^^ 2 + x ^^^ 3)"
1317 else error "rational.sml: diff.behav. in norm_Rational_mg 14";
1319 (*SRA Schalk I, p.67 Nr. 413b *)
1321 "(1+x)/(1 - x) - (1 - x)/(1+x) + 2*x/(1 - x^^^2)";
1322 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1325 "6 * x / (1 + -1 * x ^^^ 2)"
1327 else error "rational.sml: diff.behav. in norm_Rational_mg 15";
1329 (*SRA Schalk I, p.68 Nr. 414a *)
1331 "(x + 2)/(x - 1) + (x - 3)/(x - 2) - (x + 1)/((x - 1)*(x - 2))";
1332 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1335 "(-2 + -5 * x + 2 * x ^^^ 2) / (2 + -3 * x + x ^^^ 2)"
1337 else error "rational.sml: diff.behav. in norm_Rational_mg 16";
1339 (*SRA Schalk I, p.68 Nr. 423a *)
1341 "(2*x+3*y)/x + (4*x^^^3 - x*y^^^2 - 3*y^^^3)/(x^^^3 - 2*x^^^2*y+x*y^^^2) - (5*x+6*y)/(x - y)";
1342 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1347 else error "rational.sml: diff.behav. in norm_Rational_mg 17";
1349 (*SRA Schalk I, p.68 Nr. 428b *)
1351 "1/(a - b)^^^2 + 1/(a+b)^^^2 - 2/(a^^^2 - b^^^2) - 4*(b^^^2 - 1)/(a^^^2 - b^^^2)^^^2";
1352 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1355 "4 / (a ^^^ 4 + -2 * a ^^^ 2 * b ^^^ 2 + b ^^^ 4)"
1357 else error "rational.sml: diff.behav. in norm_Rational_mg 18";
1359 (*SRA Schalk I, p.68 Nr. 430b *)
1361 "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";
1362 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1367 else error "rational.sml: diff.behav. in norm_Rational_mg 19";
1370 (*SRA Schalk I, p.68 Nr. 432 *)
1372 "(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)";
1373 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1378 else error "rational.sml: diff.behav. in norm_Rational_mg 20";
1383 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1386 "(3 * y + b * x) / (b * y)"
1388 else error "rational.sml: diff.behav. in norm_Rational_mg 21";
1391 "-------- multiply and cancel ---------------------------";
1392 "-------- multiply and cancel ---------------------------";
1393 "-------- multiply and cancel ---------------------------";
1394 (*SRM Schalk I, p.68 Nr. 436a *)
1396 "3*(x+y)/(15*(x - y)) * 25*(x - y)^^^2/(18*(x+y)^^^2)";
1397 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1400 "(5 * x + -5 * y) / (18 * x + 18 * y)"
1402 else error "rational.sml: diff.behav. in norm_Rational_mg 22";
1404 (*SRM.test Schalk I, p.68 Nr. 436b *)
1405 (*WN060420???MG3 crashes with method 'simplify' in
1406 IsacCore > Simplification > Rational Terms > Multiplication > No.2*)
1407 val t = str2term "5*a*(a - b)^^^2*(a + b)^^^3/(7*b*(a - b)^^^3) * 7*b/(a + b)^^^3";
1408 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1411 "5 * a / (a + -1 * b)"
1413 else error "rational.sml: diff.behav. in norm_Rational_mg 23";
1415 (*Schalk I, p.68 Nr. 437a *)
1416 val t = str2term "(3*a - 4*b)/(4*c+3*e) * (3*a+4*b)/(9*a^^^2 - 16*b^^^2)";
1417 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1418 if (term2str t) = "1 / (4 * c + 3 * e)" then ()
1419 else error "rational.sml: diff.behav. in norm_Rational_mg 24";
1421 "----- S.K. corrected non-termination 060904";
1422 val t = str2term "(3*a - 4*b) * (3*a+4*b)/((4*c+3*e)*(9*a^^^2 - 16*b^^^2))";
1423 val SOME (t',_) = rewrite_set_ thy false make_polynomial t;
1426 "(9 * a ^^^ 2 + -16 * b ^^^ 2) /\n(36 * a ^^^ 2 * c + 27 * a ^^^ 2 * e + -64 * b ^^^ 2 * c +\n -48 * b ^^^ 2 * e)"
1427 (*"(9 * a ^^^ 2 + -16 * b ^^^ 2) / (36 * a ^^^ 2 * c + 27 * a ^^^ 2 * e + -64 * b ^^^ 2 * c + -48 * b ^^^ 2 * e)"*)
1428 then () else error "rational.sml: S.K.8..corrected 060904-6";
1430 "----- S.K. corrected non-termination of cancel_p_";
1431 val t'' = str2term ("(9 * a ^^^ 2 + -16 * b ^^^ 2) /" ^
1432 "(36 * a^^^2 * c + (27 * a^^^2 * e + (-64 * b^^^2 * c + -48 * b^^^2 * e)))");
1433 val SOME (t',_) = rewrite_set_ thy false cancel_p t'';
1434 if term2str t' = "1 / (4 * c + 3 * e)" then ()
1435 else error "rational.sml: diff.behav. in cancel_p S.K.8";
1437 (*Schalk I, p.68 Nr. 437b *)
1438 (* nonterm.SK9 loops: cancel_p kann nicht weiter kuerzen!!! *)
1439 val t'' = str2term "(a + b)/(x^^^2 - y^^^2) * ((x - y)^^^2/(a^^^2 - b^^^2))";
1440 (* val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t'';
1443 "================delete===";
1444 (*a casual output from above*)
1446 "(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)";
1447 (* WN060831 nonterm.SK10
1448 val SOME (t,_) = rewrite_set_ thy false cancel_p t;
1452 (*SRM Schalk I, p.68 Nr. 438a *)
1454 "x*y/(x*y - y^^^2)*(x^^^2 - x*y)";
1455 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1460 else error "rational.sml: diff.behav. in norm_Rational_mg 24";
1462 (*SRM Schalk I, p.68 Nr. 439b *)
1464 "(4*x^^^2+4*x+1)*((x^^^2 - 2*x^^^3)/(4*x^^^2+2*x))";
1465 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1468 "(x + -4 * x ^^^ 3) / 2"
1470 else error "rational.sml: diff.behav. in norm_Rational_mg 25";
1472 (*SRM Schalk I, p.68 Nr. 440a *)
1474 "(x^^^2 - 2*x)/(x^^^2 - 3*x) * (x - 3)^^^2/(x^^^2 - 4)";
1475 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1478 "(-3 + x) / (2 + x)"
1480 else error "rational.sml: diff.behav. in norm_Rational_mg 26";
1482 "----- Schalk I, p.68 Nr. 440b SK11 works since 0707xx";
1484 "(a^^^3 - 9*a)/(a^^^3*b - a*b^^^3)*(a^^^2*b+a*b^^^2)/(a+3)";
1485 val SOME (t,_) = rewrite_set_ thy false norm_Rational t;
1486 if term2str t = "(-3 * a + a ^^^ 2) / (a + -1 * b)" then ()
1487 else error "rational.sml: diff.behav. in norm_Rational 27";
1489 "----- SK12 works since 0707xx";
1490 val t = str2term "(a^^^3 - 9*a)*(a^^^2*b+a*b^^^2)/((a^^^3*b - a*b^^^3)*(a+3))";
1491 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; term2str t';
1492 if term2str t' = "(-3 * a + a ^^^ 2) / (a + -1 * b)" then ()
1493 else error "rational.sml: diff.behav. in norm_Rational 28";
1496 "-------- common denominator and multiplication ---------";
1497 "-------- common denominator and multiplication ---------";
1498 "-------- common denominator and multiplication ---------";
1499 (*SRAM Schalk I, p.69 Nr. 441b *)
1500 val t = str2term "(4*a/3 + 3*b^^^2/a^^^3 + b/(4*a))*(4*b/(3*a))";
1501 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1504 "(36 * b ^^^ 3 + 3 * a ^^^ 2 * b ^^^ 2 + 16 * a ^^^ 4 * b) / (9 * a ^^^ 4)"
1505 then () else error "rational.sml: diff.behav. in norm_Rational_mg 28";
1507 (*SRAM Schalk I, p.69 Nr. 442b *)
1508 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)";
1509 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1512 "5 * x ^^^ 2 / (a * b ^^^ 3 * c)"
1513 then () else error "rational.sml: diff.behav. in norm_Rational_mg 29";
1515 (*SRAM Schalk I, p.69 Nr. 443b *)
1516 val t = str2term "(a/2 + b/3)*(b/3 - a/2)";
1517 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1520 "(-9 * a ^^^ 2 + 4 * b ^^^ 2) / 36"
1521 then () else error "rational.sml: diff.behav. in norm_Rational_mg 30";
1523 (*SRAM Schalk I, p.69 Nr. 445b *)
1524 val t = str2term "(a^^^2/9 + 2*a/(3*b) + 4/b^^^2)*(a/3 - 2/b) + 8/b^^^3";
1525 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1529 then () else error "rational.sml: diff.behav. in norm_Rational_mg 31";
1531 (*SRAM Schalk I, p.69 Nr. 446b *)
1532 val t = str2term "(x/(5*x + 4*y) - y/(5*x - 4*y) + 1)*(25*x^^^2 - 16*y^^^2)";
1533 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1536 "30 * x ^^^ 2 + -9 * x * y + -20 * y ^^^ 2"
1538 else error "rational.sml: diff.behav. in norm_Rational_mg 32";
1540 (*SRAM Schalk I, p.69 Nr. 449a *)(*Achtung: rechnet ca 8 Sekunden*)
1542 "(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)";
1543 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1545 if (term2str t) = "(-81 * x ^^^ 4 + 16 * x ^^^ 8 * y ^^^ 4) / (81 * y ^^^ 8)"
1546 then () else error "rational.sml: diff.behav. in norm_Rational_mg 33";
1549 (*SRAM Schalk I, p.69 Nr. 450a *)
1551 "(4*x/(3*y)+2*y/(3*x))^^^2 - (2*y/(3*x) - 2*x/y)*(2*y/(3*x)+2*x/y)";
1552 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1555 "(52 * x ^^^ 2 + 16 * y ^^^ 2) / (9 * y ^^^ 2)"
1557 else error "rational.sml: diff.behav. in norm_Rational_mg 34";
1560 "-------- double fractions ------------------------------";
1561 "-------- double fractions ------------------------------";
1562 "-------- double fractions ------------------------------";
1563 (*SRD Schalk I, p.69 Nr. 454b *)
1565 "((2 - x)/(2*a)) / (2*a/(x - 2))";
1566 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1569 "(-4 + 4 * x + -1 * x ^^^ 2) / (4 * a ^^^ 2)"
1571 else error "rational.sml: diff.behav. in norm_Rational_mg 35";
1573 (*SRD Schalk I, p.69 Nr. 455a *)
1575 "(a^^^2 + 1)/(a^^^2 - 1) / ((a+1)/(a - 1))";
1576 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1579 "(1 + a ^^^ 2) / (1 + 2 * a + a ^^^ 2)" then ()
1580 else error "rational.sml: diff.behav. in norm_Rational_mg 36";
1583 "----- Schalk I, p.69 Nr. 455b";
1584 val t = str2term "(x^^^2 - 4)/(y^^^2 - 9)/((2+x)/(3 - y))";
1585 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1586 if term2str t = "(2 + -1 * x) / (3 + y)" then ()
1587 else error "rational.sml: diff.behav. in norm_Rational_mg 37";
1589 "----- SK060904-1a non-termination of cancel_p_ ?: worked before 0707xx";
1590 val t = str2term "(x^^^2 - 4)*(3 - y)/((y^^^2 - 9)*(2+x))";
1591 val SOME (t,_) = rewrite_set_ thy false norm_Rational t;
1592 if term2str t = "(2 + -1 * x) / (3 + y)" then ()
1593 else error "rational.sml: diff.behav. in norm_Rational_mg 37b";
1595 "----- ?: worked before 0707xx";
1596 val t = str2term "(3 + -1 * y) / (-9 + y ^^^ 2)";
1597 val SOME (t,_) = rewrite_set_ thy false norm_Rational t;
1598 if term2str t = "-1 / (3 + y)" then ()
1599 else error "rational.sml: -1 / (3 + y) norm_Rational";
1601 (*SRD Schalk I, p.69 Nr. 456b *)
1603 "(b^^^3 - b^^^2)/(b^^^2+b)/(b^^^2 - 1)";
1604 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1606 if (term2str t) = "b / (1 + 2 * b + b ^^^ 2)" then ()
1607 else error "rational.sml: diff.behav. in norm_Rational_mg 38";
1609 (*SRD Schalk I, p.69 Nr. 457b *)
1611 "(16*a^^^2 - 9*b^^^2)/(2*a+3*a*b) / ((4*a+3*b)/(4*a^^^2 - 9*a^^^2*b^^^2))";
1612 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1615 "8 * a ^^^ 2 + -6 * a * b + -12 * a ^^^ 2 * b + 9 * a * b ^^^ 2"
1616 then () else error "rational.sml: diff.behav. in norm_Rational_mg 39";
1618 "----- Schalk I, p.69 Nr. 458b works since 0707";
1620 "(2*a^^^2*x - a^^^2)/(a*x - b*x) / (b^^^2*(2*x - 1)/(x*(a - b)))";
1621 val SOME (t,_) = rewrite_set_ thy false norm_Rational t;
1622 if term2str t = "a ^^^ 2 / b ^^^ 2" then ()
1623 else error "rational.sml: diff.behav. in norm_Rational_mg 39b";
1625 (*SRD Schalk I, p.69 Nr. 459b *)
1626 val t = str2term "(a^^^2 - b^^^2)/(a*b) / (4*(a+b)^^^2/a)";
1627 val SOME (t,_) = rewrite_set_ thy false norm_Rational t;
1628 if term2str t = "(a + -1 * b) / (4 * a * b + 4 * b ^^^ 2)" then ()
1629 else error "rational.sml: diff.behav. in norm_Rational_mg 41";
1632 (*Schalk I, p.69 Nr. 460b nonterm.SK
1634 "(9*(x^^^2 - 8*x+16)/(4*(y^^^2 - 2*y+1)))/((3*x - 12)/(16*y - 16))";
1635 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1638 else error "rational.sml: diff.behav. in norm_Rational_mg 42";
1641 "9*(x^^^2 - 8*x+16)*(16*y - 16)/(4*(y^^^2 - 2*y+1)*(3*x - 12))";
1642 val SOME (t',_) = rewrite_set_ thy false norm_Rational t;
1643 ... non terminating.
1644 val SOME (t',_) = rewrite_set_ thy false make_polynomial t;
1645 "(-2304 + 1152 * x + 2304 * y + -144 * x ^^^ 2 + -1152 * x * y + 144 * x ^^^ 2 * y) /(-48 + 12 * x + 96 * y + -24 * x * y + -48 * y ^^^ 2 + 12 * x * y ^^^ 2)";
1646 val SOME (t,_) = rewrite_set_ thy false cancel_p t';
1647 ... non terminating.*)
1649 (*SRD Schalk I, p.70 Nr. 472a *)
1650 val t = str2term ("((8*x^^^2 - 32*y^^^2)/(2*x + 4*y))/" ^
1651 "((4*x - 8*y)/(x + y))");
1652 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1657 else error "rational.sml: diff.behav. in norm_Rational_mg 43";
1660 (*----------------------------------------------------------------------*)
1661 (*---------------------- Einfache Dppelbrche --------------------------*)
1662 (*----------------------------------------------------------------------*)
1664 (*SRD Schalk I, p.69 Nr. 461a *)
1666 "(2/(x+3) + 2/(x - 3)) / (8*x/(x^^^2 - 9))";
1667 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1672 else error "rational.sml: diff.behav. in norm_Rational_mg 44";
1674 (*SRD Schalk I, p.69 Nr. 464b *)
1676 "(a - a/(a - 2)) / (a + a/(a - 2))";
1677 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1680 "(3 + -1 * a) / (1 + -1 * a)"
1682 else error "rational.sml: diff.behav. in norm_Rational_mg 45";
1684 (*SRD Schalk I, p.69 Nr. 465b *)
1686 "((x+3*y)/9 + (4*y^^^2 - 9*z^^^2)/(16*x)) /(x/9+y/6+z/4)";
1687 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1690 "(4 * x + 6 * y + -9 * z) / (4 * x)"
1692 else error "rational.sml: diff.behav. in norm_Rational_mg 46";
1694 (*SRD Schalk I, p.69 Nr. 466b *)
1696 "((1 - 7*(x - 2)/(x^^^2 - 4)) / (6/(x+2))) / (3/(x+5)+30/(x^^^2 - 25))";
1697 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1700 "(25 + -10 * x + x ^^^ 2) / 18"
1702 else error "rational.sml: diff.behav. in norm_Rational_mg 47";
1704 (*SRD Schalk I, p.70 Nr. 469 *)
1706 "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))";
1707 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1710 "3 * b ^^^ 3 / (2 * a + -2 * b)"
1712 else error "rational.sml: diff.behav. in norm_Rational_mg 48";
1714 (*----------------------------------------------------------------------*)
1715 (*---------------------- Mehrfache Dppelbrueche ------------------------*)
1716 (*----------------------------------------------------------------------*)
1718 (*SRD.test Schalk I, p.70 Nr. 476b *) (* Rechenzeit: 10 sec *)
1719 (*WN060419 crashes with method 'simplify' ????SK*)
1721 "((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)";
1722 val SOME (t,_) = rewrite_set_ thy false norm_Rational t;
1723 if term2str t = "1 / (a ^^^ 2 + -1 * b ^^^ 2)" then ()
1724 else error "rational.sml: diff.behav. in norm_Rational_mg 49";
1726 "----- Schalk I, p.70 Nr. 477a";
1727 (* MG Achtung: terme explodieren; Bsp zu komplex;
1728 L???ung sollte (ziemlich grosser) Faktorisierter Ausdruck sein
1729 val t = str2term "b*y/(b - 2*y)/((b^^^2 - y^^^2)/(b+2*y)) /" ^
1730 "(b^^^2*y+b*y^^^2)*(a+x)^^^2/((b^^^2 - 4*y^^^2)*(a+2*x)^^^2)";
1731 val SOME (t',_) = rewrite_set_ thy false norm_Rational t;
1734 val t = str2term "b*y*(b+2*y)*(b^^^2 - 4*y^^^2)*(a+2*x)^^^2 / " ^
1735 "((b - 2*y)*(b^^^2 - y^^^2)*(b^^^2*y+b*y^^^2)*(a+x)^^^2)";
1736 val SOME (t,_) = rewrite_set_ thy false make_polynomial t;
1740 "----- Schalk I, p.70 Nr. 478b ----- Rechenzeit: 5 sec";
1741 val t = str2term ("(a - (a*b+b^^^2)/(a+b))/(b+(a - b)/(1+(a+b)/(a - b))) / " ^
1742 "((a - a^^^2/(a+b))/(a+(a*b)/(a - b)))");
1743 val SOME (t',_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1745 "(2 * a ^^^ 3 + 2 * a ^^^ 2 * b) / (a ^^^ 2 * b + b ^^^ 3)"
1747 else error "rational.sml: diff.behav. in norm_Rational_mg 51";
1749 (* TODO.new_c:WN050820 STOP_REW_SUB introduced gave ...
1750 if term2str t' = "(a ^^^ 4 + -1 * a ^^^ 2 * b ^^^ 2) /(a * b * (b + (a * (a + -1 * b) + -1 * b * (a + -1 * b)) / (2 * a)) * (a + -1 * b))" then ()
1751 else error "rational.sml: works again";
1752 re-outcommented with TODO.new_c: cvs before 071227, 11:50*)
1756 (*Schalk I, p.70 Nr. 480a *)
1757 (* Achtung: rechnet ewig; cancel_p kann nicht krzen: WN060831 nonterm.SK00
1759 "(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)))";
1760 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1765 else error "rational.sml: diff.behav. in norm_Rational_mg 52";
1767 (*MG Berechne Zwischenergebnisse: WN060831 nonterm.SK00*)
1769 "(1/x+1/y+1/z)/(1/x - 1/y - 1/z)";
1770 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1772 "(x ^^^ 2 * y ^^^ 2 * z + x ^^^ 2 * y * z ^^^ 2 + x * y ^^^ 2 * z ^^^ 2) /
1773 (-1 * x ^^^ 2 * y ^^^ 2 * z + -1 * x ^^^ 2 * y * z ^^^ 2 + x * y ^^^ 2 * z ^^^ 2)";
1775 "2*x^^^2/(x^^^2 - z^^^2)/(x/(x+z)+x/(x - z))";
1776 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1780 (* SK 1. Ausdruck kann nicht weiter gekrzt werden; cancel_p !!!*)
1781 ### rls: cancel_p on:
1782 (x ^^^ 2 * (y ^^^ 2 * z) + x ^^^ 2 * (y * z ^^^ 2) + x * (y ^^^ 2 * z ^^^ 2)) /
1783 (-1 * (x ^^^ 2 * (y ^^^ 2 * z)) + -1 * (x ^^^ 2 * (y * z ^^^ 2)) + x * (y ^^^ 2 * z ^^^ 2))
1784 GC #3.61.81.101.197.17503: (0 ms)
1785 *** RATIONALS_DIRECT_CANCEL_EXCEPTION: Invalid fraction
1788 "(x^^^2 * (y^^^2 * z) + x^^^2 * (y * z^^^2) + x * (y^^^2 * z^^^2)) / (-1 * (x^^^2 * (y^^^2 * z)) + -1 * (x^^^2 * (y * z^^^2)) + x * (y^^^2 * z^^^2))";
1789 val SOME (t,_) = rewrite_set_ thy false cancel_p t;
1791 (*uncaught exception nonexhaustive binding failure*)
1793 (* Das kann er aber krzen !!????: *)
1795 "(x^^^2 * (y^^^2 * z) + x * (y^^^2 * z^^^2)) / (-1 * (x^^^2 * (y * z^^^2)) + x * (y^^^2 * z^^^2))";
1796 val SOME (t,_) = rewrite_set_ thy false cancel_p t;
1798 "(-1 * (y * x) + -1 * (z * y)) / (1 * (z * x) + -1 * (z * y))";
1802 "-------- crucial examples ------------------------------";
1803 "-------- crucial examples ------------------------------";
1804 "-------- crucial examples ------------------------------";
1805 (*Schalk I, p.60 Nr. 215d *)
1806 (* Achtung: rechnet ewig ...
1807 val t = str2term "(a-b)^^^3 * (x+y)^^^4 / ((x+y)^^^2 * (a-b)^^^5)";
1808 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1809 term2str t; noterm.SK
1812 (* Kein Wunder, denn Z???ler und Nenner extra als Polynom dargestellt ergibt:*)
1814 val t = str2term "(a-b)^^^3 * (x+y)^^^4";
1815 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1817 "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";
1818 val t = str2term "((x+y)^^^2 * (a-b)^^^5)";
1819 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1821 "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";
1823 (*anscheinend macht dem Rechner das Krzen diese Bruches keinen Spass mehr ...*)
1825 (*--------------------------------------------------------------------*)
1826 (*Schalk I, p.70 Nr. 480b
1827 val t = str2term "((12*x*y/(9*x^^^2 - y^^^2))/" ^
1828 "(1/(3*x - y)^^^2 - 1/(3*x + y)^^^2)) *" ^
1829 "(1/(x - 5*y)^^^2 - 1/(x + 5*y)^^^2)/" ^
1830 "(20*x*y/(x^^^2 - 25*y^^^2))";
1831 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1833 Kann nicht weiter vereinfacht werden !!!!?? *)
1835 (*--------------------------------------------------------------------*)
1836 "---- MGs test set";
1837 val t = str2term " (1 + x^^^5) / (y + x) + x^^^3 / x ";
1838 val SOME (t,_) = rewrite_set_ thy false common_nominator_p t;
1839 if term2str t = "(1 + x ^^^ 3 + x ^^^ 5 + y * x ^^^ 2) / (x + y)" then()
1842 (*--------------------------------------------------------------------*)
1843 (* cancel_p liefert nicht immer Polynomnormalform (2): WN060831???SK3b
1844 ---> Sortierung FALSCH !! *)
1845 val t = str2term "b^^^3 * a^^^5/a ";
1846 val SOME (t,_) = rewrite_set_ thy false cancel_p t;
1848 "1 * (a^^^4 * b^^^3) / 1"; (*OK*)
1850 val t = str2term "b^^^3 * a^^^5/b ";
1851 val SOME (t,_) = rewrite_set_ thy false cancel_p t;
1853 "1 * (b^^^2 * a^^^5) / 1"; (*cancel_p sortiert hier falsch um!*)
1855 (* Problem liegt NICHT bei ord_make_polynomial! (siehe folgende Bsple) *)
1857 val x = str2term "x"; val bdv = str2term "bdv";
1858 val t1 = str2term "b^^^2 * a^^^5";
1859 val t2 = str2term "a^^^5 * b^^^2 ";
1860 ord_make_polynomial false Rational.thy [(x,bdv)] (t1,t2); (*false*)
1862 (* ==> "b^^^2 * a^^^5" > "a^^^5 * b^^^2 " ... OK!*)
1864 (*--------------------------------------------------------------------*)
1865 (* cancel_p liefert nicht immer Polynomnormalform (2): WN060831???SK3c
1866 ---> erzeugt berflssige "1 * ..."
1868 val t = str2term "-1 / (3 + y)";
1870 val SOME (t,_) = rewrite_set_ thy false cancel_p t;
1873 (********* Das ist das PROBLEM !!!!!!!??? *******************)
1874 (* -1 im Z???ler der Angabe verursacht das Problem !*)
1877 (* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - *)
1878 "----- MGs test set";
1879 val t = str2term "(a^^^2 + -1)/(a+1)";
1880 val SOME (t',_) = rewrite_set_ thy false cancel_p t;
1881 if term2str t' = "(-1 + a) / 1" then ()
1882 else error "rational.sml MG tests 3d";
1884 "----- NOT TERMINATING ?: worked before 0707xx";
1885 val t = str2term "(a^^^2 - 1)*(b + 1) / ((b^^^2 - 1)*(a+1))";
1886 val SOME (t'',_) = rewrite_set_ thy false norm_Rational t;
1887 if term2str t'' = "(1 + -1 * a) / (1 + -1 * b)" then ()
1888 else error "rational.sml MG tests 3e";
1890 "----- corrected SK060905";
1891 val t = str2term "(4*x^^^2 - 20*x + 25)/(2*x - 5)^^^3";
1892 val SOME (t',_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1893 if term2str t' = "-1 / (5 + -2 * x)" then ()
1894 else error "rational.sml corrected SK060905";
1897 "-------- examples for Stefan Karnels thesis ------------";
1898 "-------- examples for Stefan Karnels thesis ------------";
1899 "-------- examples for Stefan Karnels thesis ------------";
1900 (*SRAM Schalk I, p.69 Nr. 442b --- abgewandelt*)
1902 "(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))";
1903 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1906 "5 * x ^^^ 2 / (b ^^^ 3 * c)"
1908 else error "rational.sml: diff.behav. in norm_Rational_mg 53";
1911 "-------- me Schalk I No.186 ----------------------------";
1912 "-------- me Schalk I No.186 ----------------------------";
1913 "-------- me Schalk I No.186 ----------------------------";
1914 val fmz = ["Term ((14 * x * y) / ( x * y ))",
1917 ("Rational",["rational","simplification"],
1918 ["simplification","of_rationals"]);
1919 val p = e_pos'; val c = [];
1920 val (p,_,f,nxt,_,pt) = CalcTreeTEST [(fmz, (dI',pI',mI'))];
1921 val (p,_,f,nxt,_,pt) = me nxt p c pt;
1922 val (p,_,f,nxt,_,pt) = me nxt p c pt;
1923 val (p,_,f,nxt,_,pt) = me nxt p c pt;
1924 val (p,_,f,nxt,_,pt) = me nxt p c pt;
1925 val (p,_,f,nxt,_,pt) = me nxt p c pt;
1926 val (p,_,f,nxt,_,pt) = me nxt p c pt;
1927 val (p,_,f,nxt,_,pt) = me nxt p c pt;
1928 val (p,_,f,nxt,_,pt) = me nxt p c pt; f2str f;
1929 val (p,_,f,nxt,_,pt) = me nxt p c pt; f2str f;
1930 val (p,_,f,nxt,_,pt) = me nxt p c pt; f2str f;(*++ for explicit script*)
1931 val (p,_,f,nxt,_,pt) = me nxt p c pt; f2str f;(*++ for explicit script*)
1932 case (f2str f, nxt) of
1933 ("14", ("End_Proof'", _)) => ()
1934 | _ => error "rational.sml diff.behav. in me Schalk I No.186";
1937 "-------- interSteps ..Simp_Rat_Double_No-1.xml ---------";
1938 "-------- interSteps ..Simp_Rat_Double_No-1.xml ---------";
1939 "-------- interSteps ..Simp_Rat_Double_No-1.xml ---------";
1942 [(["Term (((2 - x)/(2*a)) / (2*a/(x - 2)))", "normalform N"],
1943 ("Rational",["rational","simplification"],
1944 ["simplification","of_rationals"]))];
1947 autoCalculate 1 CompleteCalc;
1948 val ((pt,p),_) = get_calc 1; show_pt pt;
1950 interSteps 1 ([1],Res);
1951 val ((pt,p),_) = get_calc 1; show_pt pt;
1954 "-------- interSteps ..Simp_Rat_Cancel_No-1.xml ---------";
1955 "-------- interSteps ..Simp_Rat_Cancel_No-1.xml ---------";
1956 "-------- interSteps ..Simp_Rat_Cancel_No-1.xml ---------";
1959 [(["Term ((a^2 + -1*b^2) / (a^2 + -2*a*b + b^2))", "normalform N"],
1960 ("Rational",["rational","simplification"],
1961 ["simplification","of_rationals"]))];
1964 autoCalculate 1 CompleteCalc;
1965 val ((pt,p),_) = get_calc 1; show_pt pt;
1966 (*========== inhibit exn 110314 ================================================
1967 (*with explicit script done already... and removed [1,..] at below...
1968 interSteps 1 ([1],Res);
1969 val ((pt,p),_) = get_calc 1; show_pt pt;
1971 interSteps 1 ([2],Res);
1972 val ((pt,p),_) = get_calc 1; show_pt pt;
1974 interSteps 1 ([2,1],Res);
1975 val ((pt,p),_) = get_calc 1; show_pt pt;
1976 val newnds = children (get_nd pt [2,1]) (*see "fun detailrls"*);
1977 (*if length newnds = 12 then () WN060905*)
1978 if length newnds = 13 then ()
1979 else error "rational.sml: interSteps cancel_p rev_rew_p";
1981 val p = ([2,1,9],Res);
1983 val (_, tac, _) = pt_extract (pt, p);
1984 (*case tac of SOME (Rewrite ("sym_real_plus_binom_times1", _)) => ()
1986 case tac of SOME (Rewrite ("sym_real_add_mult_distrib2", _)) => ()
1987 | _ => error "rational.sml: getTactic, sym_real_plus_binom_times1";
1988 ============ inhibit exn 110314 ==============================================*)
1991 "-------- investigate rulesets for cancel_p -------------";
1992 "-------- investigate rulesets for cancel_p -------------";
1993 "-------- investigate rulesets for cancel_p -------------";
1994 val thy = @{theory "Rational"};
1995 "---------------- (a^^^2 + -1*b^^^2) / (a^^^2 + -2*a*b + b^^^2)";
1996 val t = str2term "(a^^^2 + -1*b^^^2) / (a^^^2 + -2*a*b + b^^^2)";
1997 val tt = str2term "(1 * a + 1 * b) * (1 * a + -1 * b)"(*numerator only*);
1998 "----- with rewrite_set_";
1999 val SOME (tt',asm) = rewrite_set_ thy false make_polynomial tt;
2000 term2str tt'= "a ^^^ 2 + -1 * b ^^^ 2" (*true*);
2001 val tt = str2term "((1 * a + -1 * b) * (1 * a + -1 * b))"(*denominator only*);
2002 val SOME (tt',asm) = rewrite_set_ thy false make_polynomial tt;
2003 term2str tt' = "a ^^^ 2 + -2 * a * b + b ^^^ 2" (*true*);
2005 "----- with make_deriv";
2006 val SOME (tt, _) = factout_p_ thy t; term2str tt =
2007 "(1 * a + 1 * b) * (1 * a + -1 * b) / ((1 * a + -1 * b) * (1 * a + -1 * b))";
2009 "--- with ruleset as before 060829";
2010 val {rules, rew_ord=(_,ro),...} =
2011 rep_rls (assoc_rls "make_polynomial");
2012 val der = make_deriv thy Atools_erls rules ro NONE tt;
2013 print_depth 99; map (term2str o #1) der; print_depth 3;
2014 print_depth 99; map (rule2str o #2) der; print_depth 3;
2015 ... did not terminate*)
2016 "--- with simpler ruleset";
2017 val {rules, rew_ord=(_,ro),...} =
2018 rep_rls (assoc_rls "rev_rew_p");
2019 val der = make_deriv thy Atools_erls rules ro NONE tt;
2020 print_depth 99; writeln (deriv2str der); print_depth 3;
2022 print_depth 99; map (term2str o #1) der; print_depth 3;
2023 "...,(-1 * b ^^^ 2 + a ^^^ 2) / (-2 * (a * b) + a ^^^ 2 + (-1 * b) ^^^ 2) ]";
2024 print_depth 99; map (rule2str o #2) der; print_depth 3;
2025 print_depth 99; map (term2str o #1 o #3) der; print_depth 3;
2027 val der = make_deriv thy Atools_erls rules ro NONE
2028 (str2term "(1 * a + 1 * b) * (1 * a + -1 * b)");
2029 print_depth 99; writeln (deriv2str der); print_depth 3;
2031 val {rules, rew_ord=(_,ro),...} =
2032 rep_rls (assoc_rls "rev_rew_p");
2033 val der = make_deriv thy Atools_erls rules ro NONE
2034 (str2term "(1 * a + -1 * b) * (1 * a + -1 * b)");
2035 print_depth 99; writeln (deriv2str der); print_depth 3;
2036 print_depth 99; map (term2str o #1) der; print_depth 3;
2037 (*WN060829 ...postponed*)
2040 "-------- investigate format of factout_ and factout_p_ -";
2041 "-------- investigate format of factout_ and factout_p_ -";
2042 "-------- investigate format of factout_ and factout_p_ -";
2043 val {rules, rew_ord = (_,ro),...} = rep_rls (assoc_rls "make_polynomial");
2044 val (thy, eval_rls) = (@{theory "Rational"}, Atools_erls)(*see 'fun init_state'*);
2045 val Rrls {scr = Rfuns {init_state,...},...} = assoc_rls "cancel_p";
2047 "----- see Rational.ML, local cancel_p, fun init_state";
2048 val t = str2term "(a^^^2 + (-1)*b^^^2) / (a^^^2 + (-2)*a*b + b^^^2)";
2049 val SOME (t',_) = factout_p_ thy t; term2str t';
2051 val rtas = reverse_deriv thy eval_rls rules ro NONE t';
2052 writeln(trtas2str rst);
2056 "----- see Rational.ML, local cancel_p, fun init_state";
2057 val t = str2term "a^^^2 / a";
2058 val SOME (t',_) = factout_p_ thy t;
2059 term2str t' = "a * a / (1 * a)" (*true*);
2060 (*... can be canceled with
2061 real_mult_div_cancel2 ?k ~= 0 ==> ?m * ?k / (?n * ?k) = ?m / ?n"*)
2062 (* sml/ME/rewtools.sml:
2063 val rtas = reverse_deriv thy Atools_erls rules ro NONE t';
2064 writeln (deri2str rtas);
2068 "-------- SK 060904 ----------------------------------------------";
2069 "----- order on polynomials -- input + output";
2070 val thy = @{theory "Isac"};
2071 val t = str2term "(a + -1 * b) / (-1 * a + b)";
2072 val SOME (t', _) = factout_p_ thy t; term2str t';
2073 val SOME (t', _) = cancel_p_ thy t; term2str t';
2075 val t = str2term "a*b*c*d / (d*e*f*g)";
2076 val SOME (t', _) = cancel_p_ thy t; term2str t';
2078 val t = str2term "a*(b*(c*d)) / (b*(e*(f*g)))";
2079 val SOME (t', _) = cancel_p_ thy t; term2str t';
2082 "----- SK060904-1a non-termination of cancel_p_ ? worked before 0707xx";
2083 val t = str2term "(x^^^2 - 4)*(3 - y) / ((y^^^2 - 9)*(2+x))";
2084 val SOME (t',_) = rewrite_set_ thy false norm_Rational t;
2085 if term2str t' = "(2 + -1 * x) / (3 + y)" then ()
2086 else error "rational.sml SK060904-1a worked since 0707xx";
2088 "----- SK060904-1b non-termination of cancel_p_ ... worked before 0707xx";
2089 val t = str2term ("(9 * a ^^^ 2 + -16 * b ^^^ 2) /" ^
2090 "(36 * a^^^2 * c + (27 * a^^^2 * e + (-64 * b^^^2 * c + -48 * b^^^2 * e)))");
2091 val SOME (t',_) = cancel_p_ thy t;
2092 if term2str t' = "1 / (4 * c + 3 * e)" then ()
2093 else error "rational.sml SK060904-1b";
2096 "----- SK060904-2a non-termination of add_fraction_p_";
2097 val t = str2term (" (a + b * x) / (a + -1 * (b * x)) + " ^
2098 " (-1 * a + b * x) / (a + b * x) ");
2100 val SOME (t',_) = rewrite_set_ thy false common_nominator_p t;
2102 common_nominator_p_ thy t;
2103 " (a + b * x)*(a + b * x) / ((a + -1 * (b * x))*(a + -1 * (b * x))) + " ^
2104 " (-1 * a + b * x)*(a + -1 * (b * x)) / ((a + b * x)*(-1 * a + b * x)) ";
2106 add_fraction_p_ thy t;
2107 " ((a + b * x)*(a + b * x) + (-1 * a + b * x)*(a + -1 * (b * x))) /" ^
2108 " ((a + b * x)*(-1 * a + b * x)) ";
2112 "-------- how to stepwise construct Scripts -------------";
2113 "-------- how to stepwise construct Scripts -------------";
2114 "-------- how to stepwise construct Scripts -------------";
2115 val thy = @{theory "Rational"};
2117 val p1 = parse thy (
2118 "Script SimplifyScript (t_t::real) = " ^
2119 " (Rewrite_Set discard_minus False " ^
2122 (*required (): (Rewrite_Set discard_minus False)*)
2123 val p2 = parse thy (
2124 "Script SimplifyScript (t_t::real) = " ^
2125 " (Rewrite_Set discard_minus False " ^
2128 val p3 = parse thy (
2129 "Script SimplifyScript (t_t::real) = " ^
2130 " ((Rewrite_Set discard_minus False) " ^
2133 val p4 = parse thy (
2134 "Script SimplifyScript (t_t::real) = " ^
2135 " ((Rewrite_Set discard_minus False) " ^
2138 val p5 = parse thy (
2139 "Script SimplifyScript (t_t::real) = " ^
2140 " ((Try (Rewrite_Set discard_minus False) " ^
2141 " Try (Rewrite_Set discard_parentheses False)) " ^
2144 val p6 = parse thy (
2145 "Script SimplifyScript (t_t::real) = " ^
2146 " ((Try (Rewrite_Set discard_minus False) @@ " ^
2147 " Try (Rewrite_Set rat_mult_poly False) @@ " ^
2148 " Try (Rewrite_Set make_rat_poly_with_parentheses False) @@ " ^
2149 " Try (Rewrite_Set cancel_p_rls False) @@ " ^
2151 " ((Try (Rewrite_Set common_nominator_p_rls False) @@ " ^
2152 " Try (Rewrite_Set rat_mult_div_pow False) @@ " ^
2153 " Try (Rewrite_Set make_rat_poly_with_parentheses False) @@" ^
2154 " Try (Rewrite_Set cancel_p_rls False) @@ " ^
2155 " Try (Rewrite_Set rat_reduce_1 False)))) @@ " ^
2156 " Try (Rewrite_Set discard_parentheses False)) " ^
2160 "----------- get_denominator ----------------------------";
2161 "----------- get_denominator ----------------------------";
2162 "----------- get_denominator ----------------------------";
2163 val thy = @{theory Isac};
2164 val t = term_of (the (parse thy "get_denominator ((a +x)/b)"));
2165 val SOME (_, t') = eval_get_denominator "" 0 t thy;
2166 if term2str t' = "get_denominator ((a + x) / b) = b" then ()
2167 else error "get_denominator ((a + x) / b) = b"
2170 "--------- several errpats in complicated term -------------------";
2171 "--------- several errpats in complicated term -------------------";
2172 "--------- several errpats in complicated term -------------------";
2173 (*TODO: instead of Gabriella's example here (27.Jul.12) find a simpler one*)
2176 [(["Term ((5*b + 25)/(a^2 - b^2) * (a - b)/(5*b))", "normalform N"],
2177 ("Rational",["rational","simplification"], ["simplification","of_rationals"]))];
2180 autoCalculate 1 CompleteCalc;
2181 val ((pt,p),_) = get_calc 1; show_pt pt;
2183 "-------- nonterminating cancel_p, norm_Rational 2002 ---";
2184 "-------- nonterminating cancel_p, norm_Rational 2002 ---";
2185 "-------- nonterminating cancel_p, norm_Rational 2002 ---";
2186 (*------------------------------------------------------------------------------------\
2187 "--- WN121204: searched rational.sml for "nonterm", added new numbering" ^
2188 " and thoroughly tested with this numbering subsequently";
2190 WN.2.6.03 from rlang.sml 56a
2191 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)";
2192 val NONE = rewrite_set_ thy false common_nominator_p t;
2194 WN060831 nonterm.SK7
2195 val t = str2term "(a + b * x) / (a + -1 * (b * x)) + (-1 * a + b * x) / (a + b * x)";
2196 val NONE = add_fraction_p_ thy t;
2198 nonterm.SK9 loops: cancel_p kann nicht weiter kuerzen!!! *)
2199 val t'' = str2term "(a + b)/(x^^^2 - y^^^2) * ((x - y)^^^2/(a^^^2 - b^^^2))";
2200 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t'';
2203 "(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)";
2204 WN060831 nonterm.SK10
2205 val SOME (t,_) = rewrite_set_ thy false cancel_p t;
2209 "(9*(x^^^2 - 8*x+16)/(4*(y^^^2 - 2*y+1)))/((3*x - 12)/(16*y - 16))";
2210 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
2213 else error "rational.sml: diff.behav. in norm_Rational_mg 42";
2216 "9*(x^^^2 - 8*x+16)*(16*y - 16)/(4*(y^^^2 - 2*y+1)*(3*x - 12))";
2217 val SOME (t',_) = rewrite_set_ thy false norm_Rational t;
2218 ... non terminating.
2220 val SOME (t',_) = rewrite_set_ thy false make_polynomial t;
2221 "(-2304 + 1152 * x + 2304 * y + -144 * x ^^^ 2 + -1152 * x * y + 144 * x ^^^ 2 * y) /(-48 + 12 * x + 96 * y + -24 * x * y + -48 * y ^^^ 2 + 12 * x * y ^^^ 2)";
2222 val SOME (t,_) = rewrite_set_ thy false cancel_p t';
2224 val t = str2term "(a-b)^^^3 * (x+y)^^^4 / ((x+y)^^^2 * (a-b)^^^5)";
2225 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
2227 val t = str2term "((12*x*y/(9*x^^^2 - y^^^2))/" ^
2228 "(1/(3*x - y)^^^2 - 1/(3*x + y)^^^2)) *" ^
2229 "(1/(x - 5*y)^^^2 - 1/(x + 5*y)^^^2)/" ^
2230 "(20*x*y/(x^^^2 - 25*y^^^2))";
2231 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
2233 SK060904-2a non-termination of add_fraction_p_";
2234 val t = str2term (" (a + b * x) / (a + -1 * (b * x)) + " ^
2235 " (-1 * a + b * x) / (a + b * x) ");
2237 val SOME (t',_) = rewrite_set_ thy false common_nominator_p t;
2239 common_nominator_p_ thy t;
2240 " (a + b * x)*(a + b * x) / ((a + -1 * (b * x))*(a + -1 * (b * x))) + " ^
2241 " (-1 * a + b * x)*(a + -1 * (b * x)) / ((a + b * x)*(-1 * a + b * x)) ";
2243 add_fraction_p_ thy t;
2244 " ((a + b * x)*(a + b * x) + (-1 * a + b * x)*(a + -1 * (b * x))) /" ^
2245 " ((a + b * x)*(-1 * a + b * x)) ";
2246 --------------------------------------------------------------------------------------/*)
2248 (*------------------------------------------------------------------------------------\
2249 "WN121205: thoroughly tested with the above numbering.";
2250 " errors in cancel_p: --- 4,5,6,7.";
2251 " error in common_nominator_p, common_nominator_p_: --- 10; 1,2?.";
2252 " errors caused by ruleset norm_Rational: --- 8,9.";
2253 trace_rewrite := true;
2255 "--- 1 ---: non-terminating with ### add_fract: done t22 ";
2256 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)";
2257 trace_rewrite:= true;
2258 rewrite_set_ thy false common_nominator_p t;
2260 "--- 2 ---: non-terminating with ### add_fract: done t22 ";
2261 val t = str2term "(a + b * x) / (a + -1 * (b * x)) + (-1 * a + b * x) / (a + b * x)";
2262 add_fraction_p_ thy t;
2264 "--- 3 ---: norm_Rational calls Rrls cancel_p with non-normalised polys";
2265 val t = str2term "(a + b)/(x^^^2 - y^^^2) * ((x - y)^^^2/(a^^^2 - b^^^2))";
2266 rewrite_set_ thy false norm_Rational t;
2267 (*tracing end...#### rls: cancel_p on:
2268 (a * x ^^^ 2 + -2 * (a * (x * y)) + a * y ^^^ 2 + b * x ^^^ 2 + -2 * (b * (x * y)) + b * y ^^^ 2) /
2269 (a ^^^ 2 * x ^^^ 2 + -1 * (a ^^^ 2 * y ^^^ 2) + -1 * (b ^^^ 2 * x ^^^ 2) + b ^^^ 2 * y ^^^ 2) *)
2271 "--- 4 ---: non-terminating with Rrls cancel_p on plausible input";
2273 "(a * x ^^^ 2 + -2 * a * x * y + a * y ^^^ 2 + b * x ^^^ 2 + -2 * b * x * y + b * y ^^^ 2) /"^
2274 "(a ^^^ 2 * x ^^^ 2 + -1 * a ^^^ 2 * y ^^^ 2 + -1 * b ^^^ 2 * x ^^^ 2 + b ^^^ 2 * y ^^^ 2)");
2275 rewrite_set_ thy false cancel_p t;
2276 (*# rls: cancel_p on:
2277 (a * x ^^^ 2 + -2 * a * x * y + a * y ^^^ 2 + b * x ^^^ 2 + -2 * b * x * y + b * y ^^^ 2) /
2278 (a ^^^ 2 * x ^^^ 2 + -1 * a ^^^ 2 * y ^^^ 2 + -1 * b ^^^ 2 * x ^^^ 2 + b ^^^ 2 * y ^^^ 2) *)
2280 "--- 5 ---: non-terminating with Rrls cancel_p on plausible input";
2281 val t = str2term "(9*(x^^^2 - 8*x+16)/(4*(y^^^2 - 2*y+1)))/((3*x - 12)/(16*y - 16))";
2282 rewrite_set_ thy false norm_Rational t;
2283 (*#### rls: cancel_p on:
2284 (2304 + -1152 * x + -2304 * y + 144 * x ^^^ 2 + 1152 * (x * y) + -144 * (x ^^^ 2 * y)) /
2285 (48 + -12 * x + -96 * y + 24 * (x * y) + 48 * y ^^^ 2 + -12 * (x * y ^^^ 2)) *)
2287 "--- 6 ---: non-terminating with Rrls cancel_p on plausible input";
2289 "9*(x^^^2 - 8*x+16)*(16*y - 16)/(4*(y^^^2 - 2*y+1)*(3*x - 12))";
2290 rewrite_set_ thy false norm_Rational t;
2291 (*### rls: cancel_p on: (-2304 + 1152 * x + 2304 * y + -144 * x ^^^ 2 + -1152 * (x * y) +
2292 144 * (x ^^^ 2 * y)) /
2293 (-48 + 12 * x + 96 * y + -24 * (x * y) + -48 * y ^^^ 2 + 12 * (x * y ^^^ 2)) *)
2295 "--- 7 ---: non-terminating with Rrls cancel_p on plausible input";
2297 "(-2304 + 1152 * x + 2304 * y + -144 * x ^^^ 2 + -1152 * x * y + 144 * x ^^^ 2 * y) /"^
2298 "(-48 + 12 * x + 96 * y + -24 * x * y + -48 * y ^^^ 2 + 12 * x * y ^^^ 2)");
2299 rewrite_set_ thy false cancel_p t';
2301 "--- 8 ---: bottom of output cannot be looked ad (due to looping?)";
2302 val t = str2term "(a-b)^^^3 * (x+y)^^^4 / ((x+y)^^^2 * (a-b)^^^5)";
2303 val SOME (t,_) = rewrite_set_ thy false norm_Rational t;
2305 "--- 9 ---: probably error in norm_Rational";
2307 "((12*x*y / (9*x^^^2 - y^^^2)) / (1/(3*x - y)^^^2 - 1/(3*x + y)^^^2)) *" ^
2308 "(1/(x - 5*y)^^^2 - 1/(x + 5*y)^^^2) / (20*x*y/(x^^^2 - 25*y^^^2))");
2309 rewrite_set_ thy false norm_Rational t;
2310 (*#### rls: cancel_p on: (19440 * (x ^^^ 8 * y ^^^ 2) + -490320 * (x ^^^ 6 * y ^^^ 4) +
2311 108240 * (x ^^^ 4 * y ^^^ 6) +
2312 -6000 * (x ^^^ 2 * y ^^^ 8)) /
2313 (2160 * (x ^^^ 8 * y ^^^ 2) + -108240 * (x ^^^ 6 * y ^^^ 4) +
2314 1362000 * (x ^^^ 4 * y ^^^ 6) +
2315 -150000 * (x ^^^ 2 * y ^^^ 8)) *)
2317 "--- 10 ---: non-terminating with ### add_fract: done t22: error in common_nominator_p ";
2318 val t = str2term (" (a + b * x) / (a + -1 * (b * x)) + (-1 * a + b * x) / (a + b * x)");
2319 rewrite_set_ thy false common_nominator_p t; (*### add_fract: done t22 *)
2320 common_nominator_p_ thy t; (*loops without output*)
2321 "--- reformulated 10:";
2322 val t = str2term "(a + -1 * (b * x)) / (a + b * x)";
2323 rewrite_set_ thy false cancel_p t = NONE;
2326 "...both are to be considered after common_nominator_p_ is improved";
2327 --------------------------------------------------------------------------------------/*)