1 (* Title: tests for rationals
3 Use is subject to license terms.
6 "-----------------------------------------------------------------------------";
7 "-----------------------------------------------------------------------------";
8 "table of contents -----------------------------------------------------------";
9 "-----------------------------------------------------------------------------";
10 "-------- fun poly_of_term ---------------------------------------------------";
11 "-------- fun is_poly --------------------------------------------------------";
12 "-------- fun term_of_poly ---------------------------------------------------";
13 "-------- integration lev.1 fun factout_p_ -----------------------------------";
14 "-------- integration lev.1 fun cancel_p_ ------------------------------------";
15 "-------- integration lev.1 fun common_nominator_p_ --------------------------";
16 "-------- integration lev.1 fun add_fraction_p_ ------------------------------";
17 "-------- and app_rev ...traced down from rewrite_set_ until prepats ---------";
18 "-------- fun rewrite_set_ cancel_p downto fun gcd_poly ----------------------";
19 "-------- rls norm_Rational downto fun gcd_poly ------------------------------";
20 "-------- rewrite_set_ cancel_p from: Mathematik 1 Schalk Reniets Verlag -----";
21 "-------- rewrite_set_ common_nominator_p from: Mathematik 1 Schalk ----------";
22 "-------- integration lev.1 -- lev.5: cancel_p_ & common_nominator_p_ --------";
23 "-------- reverse rewrite ----------------------------------------------------";
24 "-------- 'reverse-ruleset' cancel_p -----------------------------------------";
25 "-------- investigate rls norm_Rational --------------------------------------";
26 "-------- examples: rls norm_Rational ----------------------------------------";
27 "-------- numeral rationals -----------------------------";
28 "-------- cancellation ----------------------------------";
29 "-------- common denominator ----------------------------";
30 "-------- multiply and cancel ---------------------------";
31 "-------- common denominator and multiplication ---------";
32 "-------- double fractions ------------------------------";
33 "-------- crucial examples ------------------------------";
34 "-------- examples for Stefan Karnels thesis ------------";
35 "-------- me Schalk I No.186 ----------------------------";
36 "-------- interSteps ..Simp_Rat_Double_No-1.xml ---------";
37 "-------- interSteps ..Simp_Rat_Cancel_No-1.xml ---------";
38 "-------- investigate rulesets for cancel_p -------------";
39 "-------- investigate format of factout_ and factout_p_ -";
40 "-------- how to stepwise construct Scripts -------------";
41 "----------- get_denominator ----------------------------";
42 "--------- several errpats in complicated term -------------------";
43 "--------------------------------------------------------";
44 "-------- nonterminating cancel_p, norm_Rational 2002 ---";
45 "--------------------------------------------------------";
46 "--------------------------------------------------------";
49 "-------- fun poly_of_term ---------------------------------------------------";
50 "-------- fun poly_of_term ---------------------------------------------------";
51 "-------- fun poly_of_term ---------------------------------------------------";
52 val vs = "12 * x^^^3 * y^^^4 * z^^^6" |> str2term |> vars |> map free2str |> sort string_ord;
54 if poly_of_term vs (str2term "12::real") = SOME [(12, [0, 0, 0])]
55 then () else error "poly_of_term 1 changed";
56 if poly_of_term vs (str2term "x::real") = SOME [(1, [1, 0, 0])]
57 then () else error "poly_of_term 2 changed";
58 if poly_of_term vs (str2term "12 * x^^^3") = SOME [(12, [3, 0, 0])]
59 then () else error "poly_of_term 3 changed";
60 if poly_of_term vs (str2term "12 * x^^^3 * y^^^4 * z^^^6") = SOME [(12, [3, 4, 6])]
61 then () else error "poly_of_term 4 changed";
62 if poly_of_term vs (str2term "1 + 2 * x^^^3 * y^^^4 * z^^^6 + y") =
63 SOME [(1, [0, 0, 0]), (1, [0, 1, 0]), (2, [3, 4, 6])]
64 then () else error "poly_of_term 5 changed";
66 (*poly_of_term is quite liberal:*)
67 (*the coefficient may be somewhere, the order of variables and the parentheses
68 within a monomial are arbitrary*)
69 if poly_of_term vs (str2term "y^^^4 * (x^^^3 * 12 * z^^^6)") = SOME [(12, [3, 4, 6])]
70 then () else error "poly_of_term 6 changed";
72 (*there may even be more than 1 coefficient:*)
73 if poly_of_term vs (str2term "2 * y^^^4 * (x^^^3 * 6 * z^^^6)") = SOME [(12, [3, 4, 6])]
74 then () else error "poly_of_term 7 changed";
76 (*the order and the parentheses within monomials are arbitrary:*)
77 if poly_of_term vs (str2term "2 * x^^^3 * y^^^4 * z^^^6 + (7 * y^^^8 + 1)")
78 = SOME [(1, [0, 0, 0]), (7, [0, 8, 0]), (2, [3, 4, 6])]
79 then () else error "poly_of_term 8 changed";
81 "-------- fun is_poly --------------------------------------------------------";
82 "-------- fun is_poly --------------------------------------------------------";
83 "-------- fun is_poly --------------------------------------------------------";
84 if is_poly (str2term "2 * x^^^3 * y^^^4 * z^^^6 + 7 * y^^^8 + 1")
85 then () else error "is_poly 1 changed";
86 if not (is_poly (str2term "2 * (x^^^3 * y^^^4 * z^^^6 + 7) * y^^^8 + 1"))
87 then () else error "is_poly 2 changed";
89 "-------- fun term_of_poly ---------------------------------------------------";
90 "-------- fun term_of_poly ---------------------------------------------------";
91 "-------- fun term_of_poly ---------------------------------------------------";
92 val expT = HOLogic.realT
93 val Free (_, baseT) = (hd o vars o str2term) "12 * x^^^3 * y^^^4 * z^^^6";
94 val p = [(1, [0, 0, 0]), (7, [0, 8, 0]), (2, [3, 4, 5])]
95 val vs = ["x","y","z"]
96 (*precondition for [(c, es),...]: legth es = length vs*)
98 if term2str (term_of_poly baseT expT vs p) = "1 + 7 * y ^^^ 8 + 2 * x ^^^ 3 * y ^^^ 4 * z ^^^ 5"
99 then () else error "term_of_poly 1 changed";
101 "-------- integration lev.1 fun factout_p_ -----------------------------------";
102 "-------- integration lev.1 fun factout_p_ -----------------------------------";
103 "-------- integration lev.1 fun factout_p_ -----------------------------------";
104 val t = str2term "(x^^^2 + -1*y^^^2) / (x^^^2 + -1*x*y)"
105 val SOME (t', asm) = factout_p_ thy t;
106 if term2str t' = "(x + y) * (x + -1 * y) / (x * (x + -1 * y))"
107 then () else error ("factout_p_ term 1 changed: " ^ term2str t')
109 if terms2str asm = "[\"x ~= 0\",\"x + -1 * y ~= 0\"]"
110 then () else error "factout_p_ asm 1 changed"
112 val t = str2term "nothing + to_cancel ::real";
113 if NONE = factout_p_ thy t then () else error "factout_p_ doesn't report non-applicable";
115 val t = str2term "((3 * x^^^2 + 6 *x + 3) / (2*x + 2))";
116 val SOME (t', asm) = factout_p_ thy t;
117 if term2str t' = "(3 + 3 * x) * (1 + x) / (2 * (1 + x))" andalso
118 terms2str asm = "[\"1 + x ~= 0\"]"
119 then () else error "factout_p_ 1 changed";
121 "-------- integration lev.1 fun cancel_p_ ------------------------------------";
122 "-------- integration lev.1 fun cancel_p_ ------------------------------------";
123 "-------- integration lev.1 fun cancel_p_ ------------------------------------";
124 val t = str2term "(x^^^2 + -1*y^^^2) / (x^^^2 + -1*x*y)"
125 val SOME (t', asm) = cancel_p_ thy t;
126 if (term2str t', terms2str asm) = ("(x + y) / x", "[\"x ~= 0\"]")
127 then () else error ("cancel_p_ (t', asm) 1 changed: " ^ term2str t')
129 val t = str2term "nothing + to_cancel ::real";
130 if NONE = cancel_p_ thy t then () else error "cancel_p_ doesn't report non-applicable";
132 val t = str2term "((3 * x^^^2 + 6 *x + 3) / (2*x + 2))";
133 val SOME (t', asm) = cancel_p_ thy t;
134 if term2str t' = "(3 + 3 * x) / 2" andalso terms2str asm = "[]"
135 then () else error "cancel_p_ 1 changed";
137 "-------- integration lev.1 fun common_nominator_p_ --------------------------";
138 "-------- integration lev.1 fun common_nominator_p_ --------------------------";
139 "-------- integration lev.1 fun common_nominator_p_ --------------------------";
140 val t = str2term ("y / (a*x + b*x + c*x) " ^
144 val SOME (t', asm) = common_nominator_p_ thy t;
146 ("y * y / (x * ((a + b + c) * y)) " ^
147 (* n1 *d2'/ (c'* ( d1' *d2')) *)
148 "+ a * (a + b + c) / (x * ((a + b + c) * y))")
149 (* n2 * d1' / (c'* ( d1' *d2')) *)
150 then () else error "common_nominator_p_ term 1 changed";
151 if terms2str asm = "[\"a + b + c ~= 0\",\"y ~= 0\",\"x ~= 0\"]"
152 then () else error "common_nominator_p_ asm 1 changed"
154 "-------- example in mail Nipkow";
155 val t = str2term "x/(x^^^2 + -1*y^^^2) + y/(x^^^2 + -1*x*y)";
156 val SOME (t', asm) = common_nominator_p_ thy t;
157 if term2str t' = "x * x / " ^
158 "((x + -1 * y) * ((x + y) * x))" ^
161 "((x + -1 * y) * ((x + y) * x))"
162 then () else error "common_nominator_p_ term 2 changed"
164 if terms2str asm = "[\"x + y ~= 0\",\"x ~= 0\",\"x + -1 * y ~= 0\"]"
165 then () else error "common_nominator_p_ asm 2 changed"
167 "-------- example: applicable tested by SML code";
168 val t = str2term "nothing / to_add";
169 if NONE = common_nominator_p_ thy t then () else error "common_nominator_p_ term 3 changed";
171 val t = str2term "((x + (-1)) / (x + 1)) + ((x + 1) / (x + (-1)))";
172 val SOME (t', asm) = common_nominator_p_ thy t;
174 "(x + -1) * (-1 + x) / ((1 + x) * (-1 + x)) +\n(x + 1) * (1 + x) / ((1 + x) * (-1 + x))"
175 andalso terms2str asm = "[\"1 + x ~= 0\",\"-1 + x ~= 0\"]"
176 then () else error "common_nominator_p_ 3 changed";
178 "-------- integration lev.1 fun add_fraction_p_ ------------------------------";
179 "-------- integration lev.1 fun add_fraction_p_ ------------------------------";
180 "-------- integration lev.1 fun add_fraction_p_ ------------------------------";
181 val t = str2term "((x + (-1)) / (x + 1)) + ((x + 1) / (x + (-1)))";
182 val SOME (t', asm) = add_fraction_p_ thy t;
183 if term2str t' = "(2 + 2 * x ^^^ 2) / (-1 + x ^^^ 2)"
184 then () else error "add_fraction_p_ 3 changed";
186 if terms2str asm = "[\"-1 + x ^^^ 2 ~= 0\"]"
187 then () else error "add_fraction_p_ 3 changed";
189 val t = str2term "nothing / to_add";
190 if NONE = add_fraction_p_ thy t then () else error "add_fraction_p_ term 3 changed";
192 val t = str2term "((x + (-1)) / (x + 1)) + ((x + 1) / (x + (-1)))";
193 val SOME (t', asm) = add_fraction_p_ thy t;
194 if term2str t' = "(2 + 2 * x ^^^ 2) / (-1 + x ^^^ 2)" andalso
195 terms2str asm = "[\"-1 + x ^^^ 2 ~= 0\"]"
196 then () else error "add_fraction_p_ 3 changed";
198 "-------- and app_rev ...traced down from rewrite_set_ until prepats ---------";
199 "-------- and app_rev ...traced down from rewrite_set_ until prepats ---------";
200 "-------- and app_rev ...traced down from rewrite_set_ until prepats ---------";
201 (* trace down until prepats are evaluated
202 (which does not to work, because substitution is not done -- compare rew_sub!);
203 keep this sequence for the case, factout_p, cancel_p, common_nominator_p, add_fraction_p
204 (again) get prepat = [] changed to <>[]. *)
205 val t = str2term "(x^^^2 + -1*y^^^2) / (x^^^2 + -1*x*y)";
207 (*rewrite_set_ @{theory Isac} true cancel t = NONE; !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!*)
208 "~~~~~ fun rewrite_set_, args:"; val (thy, bool, rls, term) = (thy, false, cancel_p, t);
209 "~~~~~ fun rewrite__set_, args:"; val (thy, i, _, _, (rrls as Rrls _), t) =
210 (thy, 1, bool, [], rls, term);
211 (*val (t', asm, rew) = app_rev thy (i+1) rrls t; rew = false!!!!!!!!!!!!!!!!!!!!!*)
212 "~~~~~ and app_rev, args:"; val (thy, i, rrls, t) = (thy, (i+1), rrls, t);
213 fun chk_prepat thy erls [] t = true
214 | chk_prepat thy erls prepat t =
216 fun chk (pres, pat) =
218 val subst: Type.tyenv * Envir.tenv =
219 Pattern.match thy (pat, t) (Vartab.empty, Vartab.empty)
221 snd (eval__true thy (i + 1) (map (Envir.subst_term subst) pres) [] erls)
222 end) handle _ => false
223 fun scan_ f [] = false (*scan_ NEVER called by []*)
224 | scan_ f (pp::pps) =
225 if f pp then true else scan_ f pps;
226 in scan_ chk prepat end;
227 (* apply the normal_form of a rev-set *)
228 fun app_rev' thy (Rrls{erls,prepat,scr=Rfuns{normal_form,...},...}) t =
229 if chk_prepat thy erls prepat t
230 then ((*tracing("### app_rev': t = "^(term2str t));*) normal_form t)
232 (* val opt = app_rev' thy rrls t ..NONE!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!*)
233 "~~~~~ and app_rev', args:"; val (thy, (Rrls{erls,prepat,scr=Rfuns{normal_form,...},...}), t) =
235 (* chk_prepat thy erls prepat t = false!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!*)
236 (* app_sub thy i rrls t = false!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!*)
237 "~~~~~ fun chk_prepat, args:"; val (thy, erls, prepat, t) = (thy, erls, prepat, t);
238 fun chk (pres, pat) =
240 val subst: Type.tyenv * Envir.tenv =
241 Pattern.match thy (pat, t) (Vartab.empty, Vartab.empty)
243 snd (eval__true thy (i + 1) (map (Envir.subst_term subst) pres) [] erls)
244 end) handle _ => false
245 fun scan_ f [] = false (*scan_ NEVER called by []*)
246 | scan_ f (pp::pps) =
247 if f pp then true else scan_ f pps;
249 (*========== inhibit exn WN130823: prepat is empty ====================================
250 (* scan_ chk prepat = false!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!*)
251 "~~~~~ fun , args:"; val (f, (pp::pps)) = (chk, prepat);
253 val ([t1, t2], t) = pp;
254 term2str t1 = "?r is_expanded";
255 term2str t2 = "?s is_expanded";
256 term2str t = "?r / ?s";
257 (* f pp = false!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!*)
258 "~~~~~ fun chk, args:"; val (pres, pat) = (pp);
259 val subst: Type.tyenv * Envir.tenv =
260 Pattern.match thy (pat, t) (Vartab.empty, Vartab.empty)
262 ({}, {(("r", 0), ("real", Var (("r", 0), "real"))),
263 (("s", 0), ("real", Var (("s", 0), "real")))}*)
265 snd (eval__true thy (i + 1) (map (Envir.subst_term subst) pres) [] erls)
266 "~~~~~ fun eval__true, args:"; val (thy, i, asms, bdv, rls) =
267 (thy, (i + 1), (map (Envir.subst_term subst) pres), [], erls);
268 terms2str asms; (* = "[\"?r is_expanded\",\"?s is_expanded\"]"*)
269 asms = [@{term True}] orelse asms = []; (* = false*)
270 asms = [@{term False}] ; (* = false*)
271 "~~~~~ fun chk, args:"; val (indets, (a::asms)) = ([], asms);
272 bdv (*= []: _a list*);
273 val bdv : (term * term) list = [];
274 rewrite__set_ thy (i+1) false;
275 term2str a = "?r is_expanded"; (*hier m"usste doch der Numerator eingesetzt sein ??????????????*)
276 val SOME (Const ("HOL.False", _), []) = rewrite__set_ thy (i+1) false bdv rls a
277 ============ inhibit exn WN130823: prepat is empty ===================================*)
279 "-------- fun rewrite_set_ cancel_p downto fun gcd_poly ----------------------";
280 "-------- fun rewrite_set_ cancel_p downto fun gcd_poly ----------------------";
281 "-------- fun rewrite_set_ cancel_p downto fun gcd_poly ----------------------";
282 val t = str2term "(12 * x * y) / (8 * y^^^2 )";
283 (* "-------- example 187a": exception Div raised...
284 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;*)
285 val t = str2term "(8 * x^^^2 * y * z ) / (18 * x * y^^^2 * z )";
286 (* "-------- example 187b": doesn't terminate...
287 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;*)
288 val t = str2term "(9 * x^^^5 * y^^^2 * z^^^4) / (15 * x^^^6 * y^^^3 * z )";
289 (* "-------- example 187c": doesn't terminate...
290 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;*)
291 "~~~~~ fun rewrite_set_, args:"; val (thy, bool, rls, term) = (@{theory Isac}, false, cancel_p, t);
292 (* WN130827: exception Div raised...
293 rewrite__set_ thy 1 bool [] rls term
295 "~~~~~ and rewrite__set_, args:"; val (thy, i, _, _, (rrls as Rrls _), t) =
296 (thy, 1, bool, [], rls, term);
297 (* WN130827: exception Div raised...
298 val (t', asm, rew) = app_rev thy (i+1) rrls t
300 "~~~~~ fun app_rev, args:"; val (thy, i, rrls, t) = (thy, (i+1), rrls, t);
301 (* WN130827: exception Div raised...
302 val opt = app_rev' thy rrls t
304 "~~~~~ fun app_rev', args:"; val (thy, (Rrls{erls,prepat,scr=Rfuns{normal_form,...},...}), t) =
306 chk_prepat thy erls prepat t = true;
307 (* WN130827: exception Div raised...
310 (* lookup Rational.thy, cancel_p: normal_form = cancel_p_ thy*)
311 "~~~~~ fun cancel_p_, args:"; val (t) = (t);
312 val opt = check_fraction t;
313 val SOME (numerator, denominator) = opt
314 val vs = t |> vars |> map free2str |> sort string_ord
315 val baseT = type_of numerator
316 val expT = HOLogic.realT
317 val (SOME a, SOME b) = (poly_of_term vs numerator, poly_of_term vs denominator);
318 (*"-------- example 187a": exception Div raised...
319 val a = [(12, [1, 1])]: poly
320 val b = [(8, [0, 2])]: poly
321 val ((a', b'), c) = gcd_poly a b
323 (* "-------- example 187b": doesn't terminate...
324 val a = [(8, [2, 1, 1])]: poly
325 val b = [(18, [1, 2, 1])]: poly
326 val ((a', b'), c) = gcd_poly a b
328 (* "-------- example 187c": doesn't terminate...
329 val a = [(9, [5, 2, 4])]: poly
330 val b = [(15, [6, 3, 1])]: poly
331 val ((a', b'), c) = gcd_poly a b
334 "-------- rls norm_Rational downto fun gcd_poly ------------------------------";
335 "-------- rls norm_Rational downto fun gcd_poly ------------------------------";
336 "-------- rls norm_Rational downto fun gcd_poly ------------------------------";
337 val t = str2term "(x^^^2 - 4)*(3 - y) / ((y^^^2 - 9)*(2+x))";
338 trace_rewrite := false (*true false*);
339 (* trace stops with ...: (and then jEdit hangs)..
340 rewrite_set_ thy false norm_Rational t;
342 ### rls: cancel_p on: (-12 + 4 * y + 3 * x ^^^ 2 + -1 * (x ^^^ 2 * y)) /
343 (-18 + -9 * x + 2 * y ^^^ 2 + x * y ^^^ 2)
345 val t = str2term (*copy from above: "::real" is not required due to "^^^"*)
346 ("(-12 + 4 * y + 3 * x ^^^ 2 + -1 * (x ^^^ 2 * y)) /" ^
347 "(-18 + -9 * x + 2 * y ^^^ 2 + x * y ^^^ 2)");
349 exception Div raised*)
351 "~~~~~ fun cancel_p_, args:"; val (t) = (t);
352 val opt = check_fraction t;
353 val SOME (numerator, denominator) = opt
354 val vs = t |> vars |> map free2str |> sort string_ord
355 val baseT = type_of numerator
356 val expT = HOLogic.realT;
357 print_depth 3; (*999*)
358 val (SOME a, SOME b) = (poly_of_term vs numerator, poly_of_term vs denominator);
359 print_depth 3; (*999*)
360 (* does not terminate instead of returning ?:
361 val ((a', b'), c) = gcd_poly a b
362 val a = [(~12, [0, 0]), (3, [2, 0]), (4, [0, 1]), (~1, [2, 1])]: poly
363 val b = [(~18, [0, 0]), (~9, [1, 0]), (2, [0, 2]), (1, [1, 2])]: poly
366 "-------- rewrite_set_ cancel_p from: Mathematik 1 Schalk Reniets Verlag -----";
367 "-------- rewrite_set_ cancel_p from: Mathematik 1 Schalk Reniets Verlag -----";
368 "-------- rewrite_set_ cancel_p from: Mathematik 1 Schalk Reniets Verlag -----";
369 val thy = @{theory "Rational"};
371 val t = str2term "(2 + -3 * x) / 9";
372 if NONE = rewrite_set_ thy false cancel_p t then ()
373 else error "rewrite_set_ cancel_p must return NONE, if the term cannot be cancelled";
375 "-------- example 186a";
376 val t = str2term "(14 * x * y) / (x * y)";
377 is_expanded (str2term "14 * x * y");
378 is_expanded (str2term "x * y");
379 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;
380 if (term2str t', terms2str asm) = ("14 / 1", "[]")
381 then () else error "rational.sml cancel Schalk 186a";
383 "-------- example 186b";
384 val t = str2term "(60 * a * b) / ( 15 * a * b )";
385 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;
386 if (term2str t', terms2str asm) = ("4 / 1", "[]")
387 then () else error "rational.sml cancel Schalk 186b";
389 "-------- example 186c";
390 val t = str2term "(144 * a^^^2 * b * c) / (12 * a * b * c)";
391 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;
392 if (term2str t', terms2str asm) = ("12 * a / 1", "[]")
393 then () else error "rational.sml cancel Schalk 186c";
395 (* !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! exception Div raised !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
396 see --- fun rewrite_set_ downto fun gcd_poly ---
397 "-------- example 187a";
398 val t = str2term "(12 * x * y) / (8 * y^^^2 )";
399 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;
400 if (term2str t', terms2str asm) = ("3 * x / (2 * y)", "[\"4 * y ~= 0\"]")
401 then () else error "rational.sml cancel Schalk 187a";
404 (* doesn't terminate !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
405 see --- fun rewrite_set_ downto fun gcd_poly ---
406 "-------- example 187b";
407 val t = str2term "(8 * x^^^2 * y * z ) / (18 * x * y^^^2 * z )";
408 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;
409 if (term2str t', terms2str asm) = ("4 * x / (9 * y)", "[\"2 * (z * (y * x)) ~= 0\"]")
410 then () else error "rational.sml cancel Schalk 187b";
413 (* doesn't terminate !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
414 see --- fun rewrite_set_ downto fun gcd_poly ---
415 "-------- example 187c";
416 val t = str2term "(9 * x^^^5 * y^^^2 * z^^^4) / (15 * x^^^6 * y^^^3 * z )";
417 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;
418 if (term2str t', terms2str asm) =
419 ("3 * z ^^^ 3 / (5 * (y * x))", "[\"3 * (z * (y ^^^ 2 * x ^^^ 5)) ~= 0\"]")
420 then () else error "rational.sml cancel Schalk 187c";
423 "-------- example 188a";
424 val t = str2term "(-8 + 8 * x) / (-9 + 9 * x)";
425 is_expanded (str2term "8 * x + -8");
426 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;
427 if (term2str t', terms2str asm) = ("8 / 9", "[]")
428 then () else error "rational.sml cancel Schalk 188a";
430 val t = str2term "(8*((-1) + x))/(9*((-1) + x))";
431 val SOME (t,_) = rewrite_set_ thy false make_polynomial t;
432 if (term2str t', terms2str asm) = ("8 / 9", "[]")
433 then () else error "rational.sml cancel Schalk make_polynomial 1";
435 "-------- example 188b";
436 val t = str2term "(-15 + 5 * x) / (-18 + 6 * x)";
437 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;
438 if (term2str t', terms2str asm) = ("5 / 6", "[]")
439 then () else error "rational.sml cancel Schalk 188b";
441 "-------- example 188c";
442 val t = str2term "(a + -1 * b) / (b + -1 * a)";
443 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;
444 if (term2str t', terms2str asm) = ("-1 / 1", "[]")
445 then () else error "rational.sml cancel Schalk 188c";
447 is_expanded (str2term "a + -1 * b") = true;
448 val t = str2term "((-1)*(b + (-1) * a))/(1*(b + (-1) * a))";
449 val SOME (t', asm) = rewrite_set_ thy false make_polynomial t;
450 if (term2str t', terms2str asm) = ("(a + -1 * b) / (-1 * a + b)", "[]")
451 then () else error "rational.sml cancel Schalk make_polynomial 2";
453 "-------- example 190a";
454 val t = str2term "( 27 * a^^^3 + 9 * a^^^2 + 3 * a + 1 ) / ( 27 * a^^^3 + 18 * a^^^2 + 3 * a )";
455 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;
456 if (term2str t', terms2str asm) =
457 ("(1 + 9 * a ^^^ 2) / (3 * a + 9 * a ^^^ 2)", "[\"3 * a + 9 * a ^^^ 2 ~= 0\"]")
458 then () else error "rational.sml cancel Schalk 190a";
460 "-------- example 190c";
461 val t = str2term "((1 + 9 * a ^^^ 2)*(1 + 3 * a))/((3 * a + 9 * a ^^^ 2)*(1 + 3 * a))";
462 val SOME (t', asm) = rewrite_set_ thy false make_polynomial t;
463 if (term2str t', terms2str asm) =
464 ("(1 + 3 * a + 9 * a ^^^ 2 + 27 * a ^^^ 3) /\n(3 * a + 18 * a ^^^ 2 + 27 * a ^^^ 3)", "[]")
465 then () else error "rational.sml make_polynomial Schalk 190c";
467 "-------- example 191a";
468 val t = str2term "( x^^^2 + -1 * y^^^2 ) / ( x + y )";
469 is_expanded (str2term "x^^^2 + -1 * y^^^2") = false; (*!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!*)
470 is_expanded (str2term "x + y") = true;
471 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;
472 if (term2str t', terms2str asm) = ("(x + -1 * y) / 1", "[]")
473 then () else error "rational.sml make_polynomial Schalk 191a";
475 "-------- example 191b";
476 val t = str2term "((x + (-1) * y)*(x + y))/((1)*(x + y))";
477 val SOME (t', asm) = rewrite_set_ thy false make_polynomial t;
478 if (term2str t', terms2str asm) = ("(x ^^^ 2 + -1 * y ^^^ 2) / (x + y)", "[]")
479 then () else error "rational.sml make_polynomial Schalk 191b";
481 "-------- example 191c";
482 val t = str2term "( 9 * x^^^2 + -30 * x + 25 ) / ( 9 * x^^^2 + -25 )";
483 is_expanded (str2term "9 * x^^^2 + -30 * x + 25") = true;
484 is_expanded (str2term "25 + -30*x + 9*x^^^2") = true;
485 is_expanded (str2term "-25 + 9*x^^^2") = true;
487 val t = str2term "(((-5) + 3 * x)*((-5) + 3 * x))/((5 + 3 * x)*((-5) + 3 * x))";
488 val SOME (t', asm) = rewrite_set_ thy false make_polynomial t;
489 if (term2str t', terms2str asm) = ("(25 + -30 * x + 9 * x ^^^ 2) / (-25 + 9 * x ^^^ 2)", "[]")
490 then () else error "rational.sml make_polynomial Schalk 191c";
492 "-------- example 192b";
493 val t = str2term "( 7 * x^^^3 + -1 * x^^^2 * y ) / ( 7 * x * y^^^2 + -1 * y^^^3 )";
494 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;
495 if (term2str t', terms2str asm) = ("x ^^^ 2 / y ^^^ 2", "[\"y ^^^ 2 ~= 0\"]")
496 then () else error "rational.sml cancel_p Schalk 192b";
498 val t = str2term "((x ^^^ 2)*(7 * x + (-1) * y))/((y ^^^ 2)*(7 * x + (-1) * y))";
499 val SOME (t', asm) = rewrite_set_ thy false make_polynomial t;
500 if (term2str t', terms2str asm) =
501 ("(7 * x ^^^ 3 + -1 * x ^^^ 2 * y) / (7 * x * y ^^^ 2 + -1 * y ^^^ 3)", "[]")
502 then () else error "rational.sml make_polynomial Schalk 192b";
504 val t = str2term "((x ^^^ 2)*(7 * x + (-1) * y))/((y ^^^ 2)*(7 * x + (-1) * y))";
505 val SOME (t', asm) = rewrite_set_ thy false make_polynomial t;
506 if (term2str t', terms2str asm) =
507 ("(7 * x ^^^ 3 + -1 * x ^^^ 2 * y) / (7 * x * y ^^^ 2 + -1 * y ^^^ 3)", "[]")
508 then () else error "rational.sml make_polynomial Schalk WN050929 not working";
510 "-------- example 193a";
511 val t = str2term "( x^^^2 + -6 * x + 9 ) / ( x^^^2 + -9 )";
512 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;
513 if (term2str t', terms2str asm) = ("(-3 + x) / (3 + x)", "[\"3 + x ~= 0\"]")
514 then () else error "rational.sml cancel_p Schalk 193a";
516 "-------- example 193b";
517 val t = str2term "( x^^^2 + -8 * x + 16 ) / ( 2 * x^^^2 + -32 )";
518 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;
519 if (term2str t', terms2str asm) = ("(-4 + x) / (8 + 2 * x)", "[\"8 + 2 * x ~= 0\"]")
520 then () else error "rational.sml cancel_p Schalk 193b";
522 "-------- example 193c";
523 val t = str2term "( 2 * x + -50 * x^^^3 ) / ( 25 * x^^^2 + -10 * x + 1 )";
524 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;
525 if (term2str t', terms2str asm) =
526 ("(2 * x + 10 * x ^^^ 2) / (1 + -5 * x)", "[\"1 + -5 * x ~= 0\"]")
527 then () else error "rational.sml cancel_p Schalk 193c";
530 val t = str2term "(-25 + 9*x^^^2)/(5 + 3*x)";
531 val SOME (t, asm) = rewrite_set_ thy false cancel_p t;
532 if (term2str t', terms2str asm) = ("(2 * x + 10 * x ^^^ 2) / (1 + -5 * x)", "[]")
533 then () else error "rational.sml cancel WN 1";
535 "-------- example heuberger";
536 val t = str2term ("(x^^^4 + x * y + x^^^3 * y + y^^^2) / " ^
537 "(x + 5 * x^^^2 + y + 5 * x * y + x^^^2 * y^^^3 + x * y^^^4)");
538 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;
539 if (term2str t', terms2str asm) =
540 ("(x ^^^ 3 + y) / (1 + 5 * x + x * y ^^^ 3)", "[\"1 + 5 * x + x * y ^^^ 3 ~= 0\"]")
541 then () else error "rational.sml cancel_p heuberger";
543 "-------- rewrite_set_ common_nominator_p from: Mathematik 1 Schalk ----------";
544 "-------- rewrite_set_ common_nominator_p from: Mathematik 1 Schalk ----------";
545 "-------- rewrite_set_ common_nominator_p from: Mathematik 1 Schalk ----------";
546 (*deleted example 204 ... 236b at update Isabelle2012-->2013*)
548 "-------- integration lev.1 -- lev.5: cancel_p_ & common_nominator_p_ --------";
549 "-------- integration lev.1 -- lev.5: cancel_p_ & common_nominator_p_ --------";
550 "-------- integration lev.1 -- lev.5: cancel_p_ & common_nominator_p_ --------";
551 val t = str2term ("123 = (a*x)/(b*x) + (c*x)/(d*x) + (e*x)/(f*x::real)");
552 "-------- gcd_poly integration level 1: works on exact term";
553 if NONE = cancel_p_ thy t then () else error "cancel_p_ works on exact fraction";
554 if NONE = add_fraction_p_ thy t then () else error "add_fraction_p_ works on exact fraction";
556 "-------- gcd_poly integration level 2: picks out ONE appropriate subterm";
557 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;
558 if term2str t' = "123 = a * x / (b * x) + c * x / (d * x) + e / f"
559 then () else error "level 2, rewrite_set_ cancel_p: changed";
560 val SOME (t', asm) = rewrite_set_ thy false add_fractions_p t;
561 if term2str t' = "123 = (b * c * x + a * d * x) / (b * d * x) + e * x / (f * x)"
562 then () else error "level 2, rewrite_set_ add_fractions_p: changed";
564 "-------- gcd_poly integration level 3: rewrites all appropriate subterms";
565 val SOME (t', asm) = rewrite_set_ thy false cancel_p_rls t;
566 if term2str t' = "123 = a / b + c / d + e / f"
567 then () else error "level 3, rewrite_set_ cancel_p_rls: changed";
568 val SOME (t', asm) = rewrite_set_ thy false common_nominator_p_rls t; (*CREATE add_fractions_p_rls*)
569 if term2str t' = "123 = (b * d * e * x + b * c * f * x + a * d * f * x) / (b * d * f * x)"
570 then () else error "level 3, rewrite_set_ add_fractions_p_rls: changed";
572 "-------- gcd_poly integration level 4: iteration cancel_p -- add_fraction_p";
573 (* simpler variant *)
574 val testrls = append_rls "testrls" e_rls [Rls_ cancel_p, Rls_ add_fractions_p]
575 val SOME (t', asm) = rewrite_set_ thy false testrls t;
576 (*trace_rewrite := false;
577 # rls: testrls on: 123 = a * x / (b * x) + c * x / (d * x) + e * x / (f * x)
578 ## rls: cancel_p on: 123 = a * x / (b * x) + c * x / (d * x) + e * x / (f * x)
579 ## rls: common_nominator_p on: 123 = a * x / (b * x) + c * x / (d * x) + e / f
580 ## rls: cancel_p on: 123 = (b * c * x + a * d * x) / (b * d * x) + e / f
581 ## rls: common_nominator_p on: 123 = (b * c + a * d) / (b * d) + e / f
582 ## rls: cancel_p on: 123 = (b * d * e + b * c * f + a * d * f) / (b * d * f)
583 ## rls: common_nominator_p on: 123 = (b * d * e + b * c * f + a * d * f) / (b * d * f) *)
584 if term2str t' = "123 = (b * d * e + b * c * f + a * d * f) / (b * d * f)"
585 then () else error "level 4, rewrite_set_ *_p: changed";
587 (* complicated variant *)
588 val testrls_rls = append_rls "testrls_rls" e_rls [Rls_ cancel_p_rls, Rls_ common_nominator_p_rls];
589 val SOME (t', asm) = rewrite_set_ thy false testrls_rls t;
590 (*# rls: testrls_rls on: 123 = a * x / (b * x) + c * x / (d * x) + e * x / (f * x)
591 ## rls: cancel_p_rls on: 123 = a * x / (b * x) + c * x / (d * x) + e * x / (f * x)
592 ### rls: cancel_p on: 123 = a * x / (b * x) + c * x / (d * x) + e * x / (f * x)
593 ### rls: cancel_p on: 123 = a * x / (b * x) + c * x / (d * x) + e / f
594 ### rls: cancel_p on: 123 = a * x / (b * x) + c / d + e / f
595 ### rls: cancel_p on: 123 = a / b + c / d + e / f
596 ## rls: common_nominator_p_rls on: 123 = a / b + c / d + e / f
597 ### rls: common_nominator_p on: 123 = a / b + c / d + e / f
598 ### rls: common_nominator_p on: 123 = (b * c + a * d) / (b * d) + e / f
599 ### rls: common_nominator_p on: 123 = (b * d * e + b * c * f + a * d * f) / (b * d * f)
600 ## rls: cancel_p_rls on: 123 = (b * d * e + b * c * f + a * d * f) / (b * d * f)
601 ### rls: cancel_p on: 123 = (b * d * e + b * c * f + a * d * f) / (b * d * f)
602 ## rls: common_nominator_p_rls on: 123 = (b * d * e + b * c * f + a * d * f) / (b * d * f)
603 ### rls: common_nominator_p on: 123 = (b * d * e + b * c * f + a * d * f) / (b * d * f) *)
604 if term2str t' = "123 = (b * d * e + b * c * f + a * d * f) / (b * d * f)"
605 then () else error "level 4, rewrite_set_ *_p_rls: changed"
607 "-------- gcd_poly integration level 5: cancel_p & add_fraction_p within norm_Rational";
608 val SOME (t', asm) = rewrite_set_ thy false norm_Rational t;
609 if term2str t' = "123 = (a * d * f + b * c * f + b * d * e) / (b * d * f)"
610 then () else error "level 5, rewrite_set_ norm_Rational: changed"
612 "-------- reverse rewrite ----------------------------------------------------";
613 "-------- reverse rewrite ----------------------------------------------------";
614 "-------- reverse rewrite ----------------------------------------------------";
615 (** the term for which reverse rewriting is demonstrated **)
616 val t = str2term "(9 + -1 * x ^^^ 2) / (9 + 6 * x + x ^^^ 2)";
617 val Rrls {scr = Rfuns {init_state = ini, locate_rule = loc,
618 next_rule = nex, normal_form = nor, ...},...} = cancel_p;
620 (** normal_form produces the result in ONE step **)
621 val SOME (t',_) = nor t;
622 if term2str t' = "(3 + -1 * x) / (3 + x)" then ()
623 else error "rational.sml normal_form (9 - x ^^^ 2) / (9 - 6 * x + x ^^^ 2)";
625 (** initialize the interpreter state used by the 'me' **)
626 val (t, _, revsets, _) = ini t;
628 if length (hd revsets) = 11 then () else error "length of revset changed";
629 if (revsets |> nth 1 |> nth 1 |> id_of_thm) =
630 (@{thm realpow_twoI} |> string_of_thm |> thmID_of_derivation_name)
631 then () else error "first element of revset changed";
633 (revsets |> nth 1 |> nth 1 |> rule2str) = "Thm (\"realpow_twoI\",??.unknown)" andalso
634 (revsets |> nth 1 |> nth 2 |> rule2str) =
635 "Thm (\"sym_#power_Float ((3,0), (0,0)) __ ((2,0), (0,0))\",??.unknown)" andalso
636 (revsets |> nth 1 |> nth 3 |> rule2str) =
637 "Thm (\"sym_#mult_Float ((2,0), (0,0)) __ ((3,0), (0,0))\",??.unknown)" andalso
638 (revsets |> nth 1 |> nth 4 |> rule2str) =
639 "Thm (\"sym_#mult_Float ((~1,0), (0,0)) __ ((3,0), (0,0))\",??.unknown)" andalso
640 (revsets |> nth 1 |> nth 5 |> rule2str) =
641 "Thm (\"sym_#mult_Float ((3,0), (0,0)) __ ((3,0), (0,0))\",??.unknown)" andalso
642 (revsets |> nth 1 |> nth 6 |> rule2str) = "Rls_ (\"sym_order_mult_rls_\")" andalso
643 (revsets |> nth 1 |> nth 7 |> rule2str) =
644 "Thm (\"sym_mult_assoc\",Groups.semigroup_mult_class.mult_assoc)"
645 then () else error "first 7 elements in revset changed"
647 (** find the rule 'r' to apply to term 't' **)
648 (*/------- since cancel_ (accepted "-" between monomials) has been replaced by cancel_p_
649 for Isabelle2013, we don't get a working revset, but non-termination:
651 val SOME (r as (Thm (str, thm))) = nex revsets t;
653 ((3 * 3 + -1 * x * x) / (3 * 3 + 2 * 3 * x + x * x),
654 Rls_ ("sym_order_mult_rls_"), ((3 * 3 + -1 * (x * x)) / (3 * 3 + 2 * (3 * x) + x * x), []))","
655 ((3 * 3 + -1 * (x * x)) / (3 * 3 + 2 * (3 * x) + x * x),
656 Thm ("sym_mult_assoc",""), ((3 * 3 + -1 * (x * x)) / (3 * 3 + 2 * 3 * x + x * x), []))","
657 ((3 * 3 + -1 * (x * x)) / (3 * 3 + 2 * 3 * x + x * x),
658 Thm ("sym_mult_assoc",""), ((3 * 3 + -1 * x * x) / (3 * 3 + 2 * 3 * x + x * x), []))","
659 ((3 * 3 + -1 * x * x) / (3 * 3 + 2 * 3 * x + x * x), Rls_ ("sym_order_mult_rls_"), ((3 * 3 + -1 * (x * x)) / (3 * 3 + 2 * (3 * x) + x * x), []))","
662 Thm ("sym_#mult_2_3", "6 = 2 * 3")
663 ### Isabelle2009-2 for cancel_ (not cancel_p_):
664 if str = "sym_#power_Float ((3,0), (0,0)) __ ((2,0), (0,0))"
665 andalso string_of_thm thm =
666 (string_of_thm o make_thm o (cterm_of (@{theory "Isac"})))
667 (Trueprop $ (term_of o the o (parse thy)) "9 = 3 ^^^ 2") then ()
668 else error "rational.sml next_rule (9 - x ^^^ 2) / (9 - 6 * x + x ^^^ 2)";
669 \---------------------------------------------------------------------------------------/*)
671 (** check, if the rule 'r' applied by the user to 't' belongs to the ruleset;
672 if the rule is OK, the term resulting from applying the rule is returned,too;
673 there might be several rule applications inbetween,
674 which are listed after the head in reverse order **)
675 (*/-------------------------------------------- Isabelle2013: this gives "error id_of_thm";
676 we don't repair this, because interaction within "reverse rewriting" never worked properly:
678 val (r, (t, asm))::_ = loc revsets t r;
679 if term2str t = "(9 - x ^^^ 2) / (3 ^^^ 2 + 6 * x + x ^^^ 2)" andalso asm = []
680 then () else error "rational.sml locate_rule (9 - x ^^^ 2) / (9 - 6 * x + x ^^^ 2)";
682 (* find the next rule to apply *)
683 val SOME (r as (Thm (str, thm))) = nex revsets t;
684 if str = "sym_#power_Float ((3,0), (0,0)) __ ((2,0), (0,0))" andalso
685 string_of_thm thm = (string_of_thm o make_thm o (cterm_of (@{theory "Isac"})))
686 (Trueprop $ (term_of o the o (parse thy)) "9 = 3 ^^^ 2") then ()
687 else error "rational.sml next_rule (9 - x ^^^ 2) / (9 - 6 * x + x ^^^ 2)";
689 (*check the next rule*)
690 val (r, (t, asm)) :: _ = loc revsets t r;
691 if term2str t = "(3 ^^^ 2 - x ^^^ 2) / (3 ^^^ 2 + 6 * x + x ^^^ 2)" then ()
692 else error "rational.sml locate_rule (9 - x ^^^ 2) / (9 - 6 * x + x ^^^ 2) II";
694 (*find and check the next rules, rewrite*)
695 val SOME r = nex revsets t;
696 val (r,(t,asm))::_ = loc revsets t r;
697 if term2str t = "(3 ^^^ 2 - x ^^^ 2) / (3 ^^^ 2 + 2 * 3 * x + x ^^^ 2)" then ()
698 else error "rational.sml locate_rule II";
700 val SOME r = nex revsets t;
701 val (r,(t,asm))::_ = loc revsets t r;
702 if term2str t = "(3 - x) * (3 + x) / (3 ^^^ 2 + 2 * 3 * x + x ^^^ 2)" then ()
703 else error "rational.sml next_rule II";
705 val SOME r = nex revsets t;
706 val (r,(t,asm))::_ = loc revsets t r;
707 if term2str t = "(3 - x) * (3 + x) / ((3 + x) * (3 + x))" then ()
708 else error "rational.sml next_rule III";
710 val SOME r = nex revsets t;
711 val (r, (t, asm)) :: _ = loc revsets t r;
713 if ss = "(3 - x) / (3 + x)" andalso terms2str asm = "[\"3 + x ~= 0\"]" then ()
714 else error "rational.sml: new behav. in rev-set cancel";
715 \--------------------------------------------------------------------------------------/*)
717 "-------- 'reverse-ruleset' cancel_p -----------------------------------------";
718 "-------- 'reverse-ruleset' cancel_p -----------------------------------------";
719 "-------- 'reverse-ruleset' cancel_p -----------------------------------------";
720 (*Isabelle2013: the example below shows, why "reverse rewriting" only worked for
723 (*the term for which reverse rewriting is demonstrated*)
724 val t = str2term "(9 + (-1)*x^^^2) / (9 + ((-6)*x + x^^^2))";
725 val Rrls {scr=Rfuns {init_state=ini,locate_rule=loc,
726 next_rule=nex,normal_form=nor,...},...} = cancel_p;
728 (*normal_form produces the result in ONE step*)
729 val SOME (t',_) = nor t;
730 term2str t' = "(3 + 1 * x) / (3 + -1 * x)";
732 (*initialize the interpreter state used by the 'me'*)
733 val SOME (t', asm) = cancel_p_ thy t;
734 term2str t' = "(3 + x) / (3 + -1 * x)" (*true*);
735 terms2str asm = "[\"3 + -1 * x ~= 0\"]" (*true*);
736 val (t,_,revsets,_) = ini t;
738 (* WN.10.10.02: dieser Fall terminiert nicht
739 (make_polynomial enth"alt zu viele rules)
740 WN060823 'init_state' requires rewriting on specified location in the term
741 print_depth 99; Rfuns; print_depth 3;
742 WN060831 cycling "sym_order_mult_rls_" "sym_mult_assoc"
743 as was with make_polynomial before ?!?*)
745 val SOME r = nex revsets t;
746 eq_Thm (r, Thm ("sym_#power_Float ((3,0), (0,0)) __ ((2,0), (0,0))",
747 mk_thm thy "9 = 3 ^^^ 2"));
748 (*WN060831 *** id_of_thm
749 Exception- ERROR raised ...
750 val (r,(t,asm))::_ = loc revsets t r;
753 val SOME r = nex revsets t;
754 val (r,(t,asm))::_ = loc revsets t r;
758 (*========== inhibit exn WN130824 TODO =======================================================
759 "-------- examples: rls norm_Rational ----------------------------------------";
760 "-------- examples: rls norm_Rational ----------------------------------------";
761 "-------- examples: rls norm_Rational ----------------------------------------";
762 val t = str2term "(3*x+5)/18 - x/2 - -(3*x - 2)/9 = 0";
763 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; term2str t';
764 if term2str t' = "1 / 18 = 0" then () else error "rational.sml 1";
766 val t = str2term "(17*x - 51)/9 - (-(13*x - 3)/6) + 11 - (9*x - 7)/4 = 0";
767 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; term2str t';
768 if term2str t' = "(237 + 65 * x) / 36 = 0" then ()
769 else error "rational.sml 2";
771 val t = str2term "(1/2 + (5*x)/2)^^^2 - ((13*x)/2 - 5/2)^^^2 - (6*x)^^^2 + 29";
772 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; term2str t';
773 if term2str t' = "23 + 35 * x + -72 * x ^^^ 2" then ()
774 else error "rational.sml 3";
776 (*trace_rewrite:=true;*)
777 val t = str2term "Not (6*x is_atom)";
778 val SOME (t',_) = rewrite_set_ thy false powers_erls t; term2str t';
780 val t = str2term "1 < 2";
781 val SOME (t',_) = rewrite_set_ thy false powers_erls t; term2str t';
784 val t = str2term "(6*x)^^^2";
785 val SOME (t',_) = rewrite_ thy dummy_ord powers_erls false
786 (num_str @{thm realpow_def_atom}) t;
787 if term2str t' = "6 * x * (6 * x) ^^^ (2 + -1)" then ()
788 else error "rational.sml powers_erls (6*x)^^^2";
790 val t = str2term "-1 * (-2 * (5 / 2 * (13 * x / 2)))";
791 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; term2str t';
792 if term2str t' = "65 * x / 2" then () else error "rational.sml 4";
794 val t = str2term "1 - ((13*x)/2 - 5/2)^^^2";
795 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; term2str t';
796 (*bef.040209: if term2str t' = "(-21 + (130 * x + -169 * x ^^^ 2)) / 4"then()*)
797 if term2str t' = "(-21 + 130 * x + -169 * x ^^^ 2) / 4" then ()
798 else error "rational.sml 5";
800 (*SRAM Schalk I, p.92 Nr. 609a*)
801 val t = str2term "2*(3 - x/5)/3 - 4*(1 - x/3) - x/3 - 2*(x/2 - 1/4)/27 +5/54";
802 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; term2str t';
803 if term2str t' = "(-255 + 112 * x) / 135" then ()
804 else error "rational.sml 6";
806 (*SRAM Schalk I, p.92 Nr. 610c*)
807 val t = str2term "((x- 1)/(x+1) + 1) / ((x- 1)/(x+1) - (x+1)/(x- 1)) - 2";
808 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; term2str t';
809 if term2str t' = "(-3 + -1 * x) / 2" then () else error "rational.sml 7";
811 (*SRAM Schalk I, p.92 Nr. 476a*)
812 val t = str2term "(x^^^2/(1 - x^^^2) + 1)/(x/(1 - x) + 1) * (1 + x)";
813 (*. a/b : c/d translated to a/b * d/c .*)
814 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; term2str t';
815 (*if term2str t' = "1 / 1" then () else error "rational.sml 8";3.6.03*)
816 if term2str t' = "1" then () else error "rational.sml 8";
818 (*............................vvv---TODO: sollte gehen mit poly_order *)
819 (*Schalk I, p.92 Nr. 472a*)
820 val t = str2term "((8*x^^^2 - 32*y^^^2)/(2*x + 4*y))/((4*x - 8*y)/(x + y))";
821 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; term2str t';
822 if term2str t' = "x + y" then () else error "rational.sml p.92 Nr. 472a";
824 (*Schalk I, p.70 Nr. 480b; a/b : c/d translated to a/b * d/c*)
825 val t = str2term ("((12*x*y/(9*x^^^2 - y^^^2))/" ^
826 "(1/(3*x - y)^^^2 - 1/(3*x + y)^^^2)) *" ^
827 "(1/(x - 5*y)^^^2 - 1/(x + 5*y)^^^2)/" ^
828 "(20*x*y/(x^^^2 - 25*y^^^2))");
829 (*... nicht simpl, zerlegt ...*)
830 val t = str2term ("((12*x*y/(9*x^^^2 - y^^^2))/" ^
831 "(1/(3*x - y)^^^2 - 1/(3*x + y)^^^2))");
832 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; term2str t';
833 "(-12 * (x * y ^^^ 3) + 108 * (x * (y * x ^^^ 2))) / (12 * (x * y))";
834 (* ~~~~~~~~~~ poly_order notwendig!*)
835 val t = str2term ("(1/(x - 5*y)^^^2 - 1/(x + 5*y)^^^2)/" ^
836 "(20*x*y/(x^^^2 - 25*y^^^2))");
837 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; term2str t';
838 if term2str t' = "1 / (x ^^^ 2 + -25 * y ^^^ 2)" then ()
839 else error "rational.sml norm_Rational 1 / (x ^^^ 2 + -25 * y ^^^ 2)";
841 "nonterm.SK6 ----- SK060904-2a non-termination of add_fraction_p_";
842 (*WN.2.6.03 from rlang.sml 56a
843 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)";
844 val NONE = rewrite_set_ thy false common_nominator_p t;
847 val t = str2term "(a + b * x) / (a + -1 * (b * x)) + (-1 * a + b * x) / (a + b * x)";
848 val NONE = add_fraction_p_ thy t;
852 (* ------------------------------------------------------------------- *)
853 (*---------vvv------------ MG: ab 1.7.03 ----------------vvv-----------*)
854 (* Simplifier fuer beliebige Buchterme *)
855 (* ------------------------------------------------------------------- *)
856 (*----------------------- norm_Rational_mg ----------------------------*)
857 (* ------------------------------------------------------------------- *)
859 "-------- numeral rationals -----------------------------";
860 "-------- numeral rationals -----------------------------";
861 "-------- numeral rationals -----------------------------";
862 (*SRA Schalk I, p.40 Nr. 164b *)
863 val t = str2term "(47/6 - 76/9 + 13/4)/(35/12)";
864 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
866 if (term2str t) = "19 / 21" then ()
867 else error "rational.sml: diff.behav. in norm_Rational_mg 1";
869 (*SRA Schalk I, p.40 Nr. 166a *)
870 val t = str2term "((5/4)/(4+22/7) + 37/20)*(110/3 - 110/9 * 23/11)";
871 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
873 if (term2str t) = "45 / 2" then ()
874 else error "rational.sml: diff.behav. in norm_Rational_mg 2";
877 "-------- cancellation ----------------------------------";
878 "-------- cancellation ----------------------------------";
879 "-------- cancellation ----------------------------------";
882 "((1 + 9 * a ^^^ 2)*(1 + 3 * a))/((3 * a + 9 * a ^^^ 2)*(1 + 3 * a))";
883 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
886 "(1 + 9 * a ^^^ 2) / (3 * a + 9 * a ^^^ 2)"
888 else error "rational.sml: diff.behav. in norm_Rational_mg 3";
892 "((x ^^^ 2)*(7 * x + (-1) * y))/((y ^^^ 2)*(7 * x + (-1) * y))";
893 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
898 else error "rational.sml: diff.behav. in norm_Rational_mg 4";
900 (*SRC Schalk I, p.66 Nr. 379c *)
903 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
908 else error "rational.sml: diff.behav. in norm_Rational_mg 5";
910 (*SRC Schalk I, p.66 Nr. 380b *)
912 "15*(3*x+3)*(4*x+9)/(12*(2*x+7)*(5*x+5))";
913 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
916 "(27 + 12 * x) / (28 + 8 * x)"
918 else error "rational.sml: diff.behav. in norm_Rational_mg 6";
920 (*Schalk I, p.60 Nr. 215c *)
921 (* Falsches Ergebnis: rechnet lange und cancel_p kann nicht weiter krzen!!!*)
923 val t = str2term "(a+b)^^^4*(x - y)/((x - y)^^^3*(a+b)^^^2)";
924 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
927 "(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)"
929 else error "rational.sml: diff.behav. in norm_Rational_mg 7";
932 "(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)"
933 val SOME (t,_) = rewrite_set_ thy false cancel_p t;
935 (* uncaught exception nonexhaustive binding failure
936 raised at: stdIn:93.1-93.51 *)
938 (*Schalk I, p.66 Nr. 381a *)
939 (* ACHTUNG: rechnet ca. 2 Minuten !!! *)
941 val t = str2term "18*(a+b)^^^3*(a - b)^^^2/(72*(a - b)^^^3*(a+b)^^^2)";
942 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
945 "(a + b) / (4 * a + -4 * b)"
946 then () else error "rational.sml: diff.behav. in norm_Rational_mg 8";
949 (*SRC Schalk I, p.66 Nr. 381b *)
951 "(4*x^^^2 - 20*x + 25)/(2*x - 5)^^^3";
952 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
957 else error "rational.sml: diff.behav. in norm_Rational_mg 9";
959 (*SRC Schalk I, p.66 Nr. 381c *)
961 "(27*a^^^3+9*a^^^2+3*a+1)/(27*a^^^3+18*a^^^2+3*a)";
962 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
965 "(1 + 9 * a ^^^ 2) / (3 * a + 9 * a ^^^ 2)"
967 else error "rational.sml: diff.behav. in norm_Rational_mg 10";
969 (*SRC Schalk I, p.66 Nr. 383a *)
971 "(5*a^^^2 - 5*a*b)/(a - b)^^^2";
972 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
975 "5 * a / (a + -1 * b)"
977 else error "rational.sml: diff.behav. in norm_Rational_mg 11";
980 "-------- common denominator ----------------------------";
981 "-------- common denominator ----------------------------";
982 "-------- common denominator ----------------------------";
983 (*SRA Schalk I, p.67 Nr. 403a *)
986 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
989 "(-3 * x + 4 * y + -1 * x * y) / (x * y)"
991 else error "rational.sml: diff.behav. in norm_Rational_mg 12";
993 (*SRA Schalk I, p.67 Nr. 407b *)
995 "(2*a+3*b)/(b*c) + (3*c+a)/(a*c) - (2*a^^^2+3*b*c)/(a*b*c)";
996 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1001 else error "rational.sml: diff.behav. in norm_Rational_mg 13";
1003 (*SRA Schalk I, p.67 Nr. 410b *)
1005 "1/(x+1) + 1/(x+2) - 2/(x+3)";
1006 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1009 "(5 + 3 * x) / (6 + 11 * x + 6 * x ^^^ 2 + x ^^^ 3)"
1011 else error "rational.sml: diff.behav. in norm_Rational_mg 14";
1013 (*SRA Schalk I, p.67 Nr. 413b *)
1015 "(1+x)/(1 - x) - (1 - x)/(1+x) + 2*x/(1 - x^^^2)";
1016 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1019 "6 * x / (1 + -1 * x ^^^ 2)"
1021 else error "rational.sml: diff.behav. in norm_Rational_mg 15";
1023 (*SRA Schalk I, p.68 Nr. 414a *)
1025 "(x + 2)/(x - 1) + (x - 3)/(x - 2) - (x + 1)/((x - 1)*(x - 2))";
1026 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1029 "(-2 + -5 * x + 2 * x ^^^ 2) / (2 + -3 * x + x ^^^ 2)"
1031 else error "rational.sml: diff.behav. in norm_Rational_mg 16";
1033 (*SRA Schalk I, p.68 Nr. 423a *)
1035 "(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)";
1036 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1041 else error "rational.sml: diff.behav. in norm_Rational_mg 17";
1043 (*SRA Schalk I, p.68 Nr. 428b *)
1045 "1/(a - b)^^^2 + 1/(a+b)^^^2 - 2/(a^^^2 - b^^^2) - 4*(b^^^2 - 1)/(a^^^2 - b^^^2)^^^2";
1046 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1049 "4 / (a ^^^ 4 + -2 * a ^^^ 2 * b ^^^ 2 + b ^^^ 4)"
1051 else error "rational.sml: diff.behav. in norm_Rational_mg 18";
1053 (*SRA Schalk I, p.68 Nr. 430b *)
1055 "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";
1056 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1061 else error "rational.sml: diff.behav. in norm_Rational_mg 19";
1064 (*SRA Schalk I, p.68 Nr. 432 *)
1066 "(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)";
1067 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1072 else error "rational.sml: diff.behav. in norm_Rational_mg 20";
1077 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1080 "(3 * y + b * x) / (b * y)"
1082 else error "rational.sml: diff.behav. in norm_Rational_mg 21";
1085 "-------- multiply and cancel ---------------------------";
1086 "-------- multiply and cancel ---------------------------";
1087 "-------- multiply and cancel ---------------------------";
1088 (*SRM Schalk I, p.68 Nr. 436a *)
1090 "3*(x+y)/(15*(x - y)) * 25*(x - y)^^^2/(18*(x+y)^^^2)";
1091 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1094 "(5 * x + -5 * y) / (18 * x + 18 * y)"
1096 else error "rational.sml: diff.behav. in norm_Rational_mg 22";
1098 (*SRM.test Schalk I, p.68 Nr. 436b *)
1099 (*WN060420???MG3 crashes with method 'simplify' in
1100 IsacCore > Simplification > Rational Terms > Multiplication > No.2*)
1101 val t = str2term "5*a*(a - b)^^^2*(a + b)^^^3/(7*b*(a - b)^^^3) * 7*b/(a + b)^^^3";
1102 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1105 "5 * a / (a + -1 * b)"
1107 else error "rational.sml: diff.behav. in norm_Rational_mg 23";
1109 (*Schalk I, p.68 Nr. 437a *)
1110 val t = str2term "(3*a - 4*b)/(4*c+3*e) * (3*a+4*b)/(9*a^^^2 - 16*b^^^2)";
1111 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1112 if (term2str t) = "1 / (4 * c + 3 * e)" then ()
1113 else error "rational.sml: diff.behav. in norm_Rational_mg 24";
1115 "----- S.K. corrected non-termination 060904";
1116 val t = str2term "(3*a - 4*b) * (3*a+4*b)/((4*c+3*e)*(9*a^^^2 - 16*b^^^2))";
1117 val SOME (t',_) = rewrite_set_ thy false make_polynomial t;
1120 "(9 * a ^^^ 2 + -16 * b ^^^ 2) /\n(36 * a ^^^ 2 * c + 27 * a ^^^ 2 * e + -64 * b ^^^ 2 * c +\n -48 * b ^^^ 2 * e)"
1121 (*"(9 * a ^^^ 2 + -16 * b ^^^ 2) / (36 * a ^^^ 2 * c + 27 * a ^^^ 2 * e + -64 * b ^^^ 2 * c + -48 * b ^^^ 2 * e)"*)
1122 then () else error "rational.sml: S.K.8..corrected 060904-6";
1124 "----- S.K. corrected non-termination of cancel_p_";
1125 val t'' = str2term ("(9 * a ^^^ 2 + -16 * b ^^^ 2) /" ^
1126 "(36 * a^^^2 * c + (27 * a^^^2 * e + (-64 * b^^^2 * c + -48 * b^^^2 * e)))");
1127 val SOME (t',_) = rewrite_set_ thy false cancel_p t'';
1128 if term2str t' = "1 / (4 * c + 3 * e)" then ()
1129 else error "rational.sml: diff.behav. in cancel_p S.K.8";
1131 (*Schalk I, p.68 Nr. 437b *)
1132 (* nonterm.SK9 loops: cancel_p kann nicht weiter kuerzen!!! *)
1133 val t'' = str2term "(a + b)/(x^^^2 - y^^^2) * ((x - y)^^^2/(a^^^2 - b^^^2))";
1134 (* val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t'';
1137 "================delete===";
1138 (*a casual output from above*)
1140 "(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)";
1141 (* WN060831 nonterm.SK10
1142 val SOME (t,_) = rewrite_set_ thy false cancel_p t;
1146 (*SRM Schalk I, p.68 Nr. 438a *)
1148 "x*y/(x*y - y^^^2)*(x^^^2 - x*y)";
1149 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1154 else error "rational.sml: diff.behav. in norm_Rational_mg 24";
1156 (*SRM Schalk I, p.68 Nr. 439b *)
1158 "(4*x^^^2+4*x+1)*((x^^^2 - 2*x^^^3)/(4*x^^^2+2*x))";
1159 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1162 "(x + -4 * x ^^^ 3) / 2"
1164 else error "rational.sml: diff.behav. in norm_Rational_mg 25";
1166 (*SRM Schalk I, p.68 Nr. 440a *)
1168 "(x^^^2 - 2*x)/(x^^^2 - 3*x) * (x - 3)^^^2/(x^^^2 - 4)";
1169 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1172 "(-3 + x) / (2 + x)"
1174 else error "rational.sml: diff.behav. in norm_Rational_mg 26";
1176 "----- Schalk I, p.68 Nr. 440b SK11 works since 0707xx";
1178 "(a^^^3 - 9*a)/(a^^^3*b - a*b^^^3)*(a^^^2*b+a*b^^^2)/(a+3)";
1179 val SOME (t,_) = rewrite_set_ thy false norm_Rational t;
1180 if term2str t = "(-3 * a + a ^^^ 2) / (a + -1 * b)" then ()
1181 else error "rational.sml: diff.behav. in norm_Rational 27";
1183 "----- SK12 works since 0707xx";
1184 val t = str2term "(a^^^3 - 9*a)*(a^^^2*b+a*b^^^2)/((a^^^3*b - a*b^^^3)*(a+3))";
1185 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; term2str t';
1186 if term2str t' = "(-3 * a + a ^^^ 2) / (a + -1 * b)" then ()
1187 else error "rational.sml: diff.behav. in norm_Rational 28";
1190 "-------- common denominator and multiplication ---------";
1191 "-------- common denominator and multiplication ---------";
1192 "-------- common denominator and multiplication ---------";
1193 (*SRAM Schalk I, p.69 Nr. 441b *)
1194 val t = str2term "(4*a/3 + 3*b^^^2/a^^^3 + b/(4*a))*(4*b/(3*a))";
1195 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1198 "(36 * b ^^^ 3 + 3 * a ^^^ 2 * b ^^^ 2 + 16 * a ^^^ 4 * b) / (9 * a ^^^ 4)"
1199 then () else error "rational.sml: diff.behav. in norm_Rational_mg 28";
1201 (*SRAM Schalk I, p.69 Nr. 442b *)
1202 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)";
1203 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1206 "5 * x ^^^ 2 / (a * b ^^^ 3 * c)"
1207 then () else error "rational.sml: diff.behav. in norm_Rational_mg 29";
1209 (*SRAM Schalk I, p.69 Nr. 443b *)
1210 val t = str2term "(a/2 + b/3)*(b/3 - a/2)";
1211 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1214 "(-9 * a ^^^ 2 + 4 * b ^^^ 2) / 36"
1215 then () else error "rational.sml: diff.behav. in norm_Rational_mg 30";
1217 (*SRAM Schalk I, p.69 Nr. 445b *)
1218 val t = str2term "(a^^^2/9 + 2*a/(3*b) + 4/b^^^2)*(a/3 - 2/b) + 8/b^^^3";
1219 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1223 then () else error "rational.sml: diff.behav. in norm_Rational_mg 31";
1225 (*SRAM Schalk I, p.69 Nr. 446b *)
1226 val t = str2term "(x/(5*x + 4*y) - y/(5*x - 4*y) + 1)*(25*x^^^2 - 16*y^^^2)";
1227 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1230 "30 * x ^^^ 2 + -9 * x * y + -20 * y ^^^ 2"
1232 else error "rational.sml: diff.behav. in norm_Rational_mg 32";
1234 (*SRAM Schalk I, p.69 Nr. 449a *)(*Achtung: rechnet ca 8 Sekunden*)
1236 "(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)";
1237 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1239 if (term2str t) = "(-81 * x ^^^ 4 + 16 * x ^^^ 8 * y ^^^ 4) / (81 * y ^^^ 8)"
1240 then () else error "rational.sml: diff.behav. in norm_Rational_mg 33";
1243 (*SRAM Schalk I, p.69 Nr. 450a *)
1245 "(4*x/(3*y)+2*y/(3*x))^^^2 - (2*y/(3*x) - 2*x/y)*(2*y/(3*x)+2*x/y)";
1246 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1249 "(52 * x ^^^ 2 + 16 * y ^^^ 2) / (9 * y ^^^ 2)"
1251 else error "rational.sml: diff.behav. in norm_Rational_mg 34";
1254 "-------- double fractions ------------------------------";
1255 "-------- double fractions ------------------------------";
1256 "-------- double fractions ------------------------------";
1257 (*SRD Schalk I, p.69 Nr. 454b *)
1259 "((2 - x)/(2*a)) / (2*a/(x - 2))";
1260 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1263 "(-4 + 4 * x + -1 * x ^^^ 2) / (4 * a ^^^ 2)"
1265 else error "rational.sml: diff.behav. in norm_Rational_mg 35";
1267 (*SRD Schalk I, p.69 Nr. 455a *)
1269 "(a^^^2 + 1)/(a^^^2 - 1) / ((a+1)/(a - 1))";
1270 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1273 "(1 + a ^^^ 2) / (1 + 2 * a + a ^^^ 2)" then ()
1274 else error "rational.sml: diff.behav. in norm_Rational_mg 36";
1277 "----- Schalk I, p.69 Nr. 455b";
1278 val t = str2term "(x^^^2 - 4)/(y^^^2 - 9)/((2+x)/(3 - y))";
1279 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1280 if term2str t = "(2 + -1 * x) / (3 + y)" then ()
1281 else error "rational.sml: diff.behav. in norm_Rational_mg 37";
1283 "----- SK060904-1a non-termination of cancel_p_ ?: worked before 0707xx";
1284 val t = str2term "(x^^^2 - 4)*(3 - y)/((y^^^2 - 9)*(2+x))";
1285 val SOME (t,_) = rewrite_set_ thy false norm_Rational t;
1286 if term2str t = "(2 + -1 * x) / (3 + y)" then ()
1287 else error "rational.sml: diff.behav. in norm_Rational_mg 37b";
1289 "----- ?: worked before 0707xx";
1290 val t = str2term "(3 + -1 * y) / (-9 + y ^^^ 2)";
1291 val SOME (t,_) = rewrite_set_ thy false norm_Rational t;
1292 if term2str t = "-1 / (3 + y)" then ()
1293 else error "rational.sml: -1 / (3 + y) norm_Rational";
1295 (*SRD Schalk I, p.69 Nr. 456b *)
1297 "(b^^^3 - b^^^2)/(b^^^2+b)/(b^^^2 - 1)";
1298 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1300 if (term2str t) = "b / (1 + 2 * b + b ^^^ 2)" then ()
1301 else error "rational.sml: diff.behav. in norm_Rational_mg 38";
1303 (*SRD Schalk I, p.69 Nr. 457b *)
1305 "(16*a^^^2 - 9*b^^^2)/(2*a+3*a*b) / ((4*a+3*b)/(4*a^^^2 - 9*a^^^2*b^^^2))";
1306 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1309 "8 * a ^^^ 2 + -6 * a * b + -12 * a ^^^ 2 * b + 9 * a * b ^^^ 2"
1310 then () else error "rational.sml: diff.behav. in norm_Rational_mg 39";
1312 "----- Schalk I, p.69 Nr. 458b works since 0707";
1314 "(2*a^^^2*x - a^^^2)/(a*x - b*x) / (b^^^2*(2*x - 1)/(x*(a - b)))";
1315 val SOME (t,_) = rewrite_set_ thy false norm_Rational t;
1316 if term2str t = "a ^^^ 2 / b ^^^ 2" then ()
1317 else error "rational.sml: diff.behav. in norm_Rational_mg 39b";
1319 (*SRD Schalk I, p.69 Nr. 459b *)
1320 val t = str2term "(a^^^2 - b^^^2)/(a*b) / (4*(a+b)^^^2/a)";
1321 val SOME (t,_) = rewrite_set_ thy false norm_Rational t;
1322 if term2str t = "(a + -1 * b) / (4 * a * b + 4 * b ^^^ 2)" then ()
1323 else error "rational.sml: diff.behav. in norm_Rational_mg 41";
1326 (*Schalk I, p.69 Nr. 460b nonterm.SK
1328 "(9*(x^^^2 - 8*x+16)/(4*(y^^^2 - 2*y+1)))/((3*x - 12)/(16*y - 16))";
1329 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1332 else error "rational.sml: diff.behav. in norm_Rational_mg 42";
1335 "9*(x^^^2 - 8*x+16)*(16*y - 16)/(4*(y^^^2 - 2*y+1)*(3*x - 12))";
1336 val SOME (t',_) = rewrite_set_ thy false norm_Rational t;
1337 ... non terminating.
1338 val SOME (t',_) = rewrite_set_ thy false make_polynomial t;
1339 "(-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)";
1340 val SOME (t,_) = rewrite_set_ thy false cancel_p t';
1341 ... non terminating.*)
1343 (*SRD Schalk I, p.70 Nr. 472a *)
1344 val t = str2term ("((8*x^^^2 - 32*y^^^2)/(2*x + 4*y))/" ^
1345 "((4*x - 8*y)/(x + y))");
1346 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1351 else error "rational.sml: diff.behav. in norm_Rational_mg 43";
1354 (*----------------------------------------------------------------------*)
1355 (*---------------------- Einfache Dppelbrche --------------------------*)
1356 (*----------------------------------------------------------------------*)
1358 (*SRD Schalk I, p.69 Nr. 461a *)
1360 "(2/(x+3) + 2/(x - 3)) / (8*x/(x^^^2 - 9))";
1361 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1366 else error "rational.sml: diff.behav. in norm_Rational_mg 44";
1368 (*SRD Schalk I, p.69 Nr. 464b *)
1370 "(a - a/(a - 2)) / (a + a/(a - 2))";
1371 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1374 "(3 + -1 * a) / (1 + -1 * a)"
1376 else error "rational.sml: diff.behav. in norm_Rational_mg 45";
1378 (*SRD Schalk I, p.69 Nr. 465b *)
1380 "((x+3*y)/9 + (4*y^^^2 - 9*z^^^2)/(16*x)) /(x/9+y/6+z/4)";
1381 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1384 "(4 * x + 6 * y + -9 * z) / (4 * x)"
1386 else error "rational.sml: diff.behav. in norm_Rational_mg 46";
1388 (*SRD Schalk I, p.69 Nr. 466b *)
1390 "((1 - 7*(x - 2)/(x^^^2 - 4)) / (6/(x+2))) / (3/(x+5)+30/(x^^^2 - 25))";
1391 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1394 "(25 + -10 * x + x ^^^ 2) / 18"
1396 else error "rational.sml: diff.behav. in norm_Rational_mg 47";
1398 (*SRD Schalk I, p.70 Nr. 469 *)
1400 "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))";
1401 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1404 "3 * b ^^^ 3 / (2 * a + -2 * b)"
1406 else error "rational.sml: diff.behav. in norm_Rational_mg 48";
1408 (*----------------------------------------------------------------------*)
1409 (*---------------------- Mehrfache Dppelbrueche ------------------------*)
1410 (*----------------------------------------------------------------------*)
1412 (*SRD.test Schalk I, p.70 Nr. 476b *) (* Rechenzeit: 10 sec *)
1413 (*WN060419 crashes with method 'simplify' ????SK*)
1415 "((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)";
1416 val SOME (t,_) = rewrite_set_ thy false norm_Rational t;
1417 if term2str t = "1 / (a ^^^ 2 + -1 * b ^^^ 2)" then ()
1418 else error "rational.sml: diff.behav. in norm_Rational_mg 49";
1420 "----- Schalk I, p.70 Nr. 477a";
1421 (* MG Achtung: terme explodieren; Bsp zu komplex;
1422 L???ung sollte (ziemlich grosser) Faktorisierter Ausdruck sein
1423 val t = str2term "b*y/(b - 2*y)/((b^^^2 - y^^^2)/(b+2*y)) /" ^
1424 "(b^^^2*y+b*y^^^2)*(a+x)^^^2/((b^^^2 - 4*y^^^2)*(a+2*x)^^^2)";
1425 val SOME (t',_) = rewrite_set_ thy false norm_Rational t;
1428 val t = str2term "b*y*(b+2*y)*(b^^^2 - 4*y^^^2)*(a+2*x)^^^2 / " ^
1429 "((b - 2*y)*(b^^^2 - y^^^2)*(b^^^2*y+b*y^^^2)*(a+x)^^^2)";
1430 val SOME (t,_) = rewrite_set_ thy false make_polynomial t;
1434 "----- Schalk I, p.70 Nr. 478b ----- Rechenzeit: 5 sec";
1435 val t = str2term ("(a - (a*b+b^^^2)/(a+b))/(b+(a - b)/(1+(a+b)/(a - b))) / " ^
1436 "((a - a^^^2/(a+b))/(a+(a*b)/(a - b)))");
1437 val SOME (t',_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1439 "(2 * a ^^^ 3 + 2 * a ^^^ 2 * b) / (a ^^^ 2 * b + b ^^^ 3)"
1441 else error "rational.sml: diff.behav. in norm_Rational_mg 51";
1443 (* TODO.new_c:WN050820 STOP_REW_SUB introduced gave ...
1444 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 ()
1445 else error "rational.sml: works again";
1446 re-outcommented with TODO.new_c: cvs before 071227, 11:50*)
1450 (*Schalk I, p.70 Nr. 480a *)
1451 (* Achtung: rechnet ewig; cancel_p kann nicht krzen: WN060831 nonterm.SK00
1453 "(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)))";
1454 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1459 else error "rational.sml: diff.behav. in norm_Rational_mg 52";
1461 (*MG Berechne Zwischenergebnisse: WN060831 nonterm.SK00*)
1463 "(1/x+1/y+1/z)/(1/x - 1/y - 1/z)";
1464 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1466 "(x ^^^ 2 * y ^^^ 2 * z + x ^^^ 2 * y * z ^^^ 2 + x * y ^^^ 2 * z ^^^ 2) /
1467 (-1 * x ^^^ 2 * y ^^^ 2 * z + -1 * x ^^^ 2 * y * z ^^^ 2 + x * y ^^^ 2 * z ^^^ 2)";
1469 "2*x^^^2/(x^^^2 - z^^^2)/(x/(x+z)+x/(x - z))";
1470 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1474 (* SK 1. Ausdruck kann nicht weiter gekrzt werden; cancel_p !!!*)
1475 ### rls: cancel_p on:
1476 (x ^^^ 2 * (y ^^^ 2 * z) + x ^^^ 2 * (y * z ^^^ 2) + x * (y ^^^ 2 * z ^^^ 2)) /
1477 (-1 * (x ^^^ 2 * (y ^^^ 2 * z)) + -1 * (x ^^^ 2 * (y * z ^^^ 2)) + x * (y ^^^ 2 * z ^^^ 2))
1478 GC #3.61.81.101.197.17503: (0 ms)
1479 *** RATIONALS_DIRECT_CANCEL_EXCEPTION: Invalid fraction
1482 "(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))";
1483 val SOME (t,_) = rewrite_set_ thy false cancel_p t;
1485 (*uncaught exception nonexhaustive binding failure*)
1487 (* Das kann er aber krzen !!????: *)
1489 "(x^^^2 * (y^^^2 * z) + x * (y^^^2 * z^^^2)) / (-1 * (x^^^2 * (y * z^^^2)) + x * (y^^^2 * z^^^2))";
1490 val SOME (t,_) = rewrite_set_ thy false cancel_p t;
1492 "(-1 * (y * x) + -1 * (z * y)) / (1 * (z * x) + -1 * (z * y))";
1496 "-------- crucial examples ------------------------------";
1497 "-------- crucial examples ------------------------------";
1498 "-------- crucial examples ------------------------------";
1499 (*Schalk I, p.60 Nr. 215d *)
1500 (* Achtung: rechnet ewig ...
1501 val t = str2term "(a-b)^^^3 * (x+y)^^^4 / ((x+y)^^^2 * (a-b)^^^5)";
1502 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1503 term2str t; noterm.SK
1506 (* Kein Wunder, denn Z???ler und Nenner extra als Polynom dargestellt ergibt:*)
1508 val t = str2term "(a-b)^^^3 * (x+y)^^^4";
1509 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1511 "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";
1512 val t = str2term "((x+y)^^^2 * (a-b)^^^5)";
1513 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1515 "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";
1517 (*anscheinend macht dem Rechner das Krzen diese Bruches keinen Spass mehr ...*)
1519 (*--------------------------------------------------------------------*)
1520 (*Schalk I, p.70 Nr. 480b
1521 val t = str2term "((12*x*y/(9*x^^^2 - y^^^2))/" ^
1522 "(1/(3*x - y)^^^2 - 1/(3*x + y)^^^2)) *" ^
1523 "(1/(x - 5*y)^^^2 - 1/(x + 5*y)^^^2)/" ^
1524 "(20*x*y/(x^^^2 - 25*y^^^2))";
1525 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1527 Kann nicht weiter vereinfacht werden !!!!?? *)
1529 (*--------------------------------------------------------------------*)
1530 "---- MGs test set";
1531 val t = str2term " (1 + x^^^5) / (y + x) + x^^^3 / x ";
1532 val SOME (t,_) = rewrite_set_ thy false common_nominator_p t;
1533 if term2str t = "(1 + x ^^^ 3 + x ^^^ 5 + y * x ^^^ 2) / (x + y)" then()
1536 (*--------------------------------------------------------------------*)
1537 (* cancel_p liefert nicht immer Polynomnormalform (2): WN060831???SK3b
1538 ---> Sortierung FALSCH !! *)
1539 val t = str2term "b^^^3 * a^^^5/a ";
1540 val SOME (t,_) = rewrite_set_ thy false cancel_p t;
1542 "1 * (a^^^4 * b^^^3) / 1"; (*OK*)
1544 val t = str2term "b^^^3 * a^^^5/b ";
1545 val SOME (t,_) = rewrite_set_ thy false cancel_p t;
1547 "1 * (b^^^2 * a^^^5) / 1"; (*cancel_p sortiert hier falsch um!*)
1549 (* Problem liegt NICHT bei ord_make_polynomial! (siehe folgende Bsple) *)
1551 val x = str2term "x"; val bdv = str2term "bdv";
1552 val t1 = str2term "b^^^2 * a^^^5";
1553 val t2 = str2term "a^^^5 * b^^^2 ";
1554 ord_make_polynomial false Rational.thy [(x,bdv)] (t1,t2); (*false*)
1556 (* ==> "b^^^2 * a^^^5" > "a^^^5 * b^^^2 " ... OK!*)
1558 (*--------------------------------------------------------------------*)
1559 (* cancel_p liefert nicht immer Polynomnormalform (2): WN060831???SK3c
1560 ---> erzeugt berflssige "1 * ..."
1562 val t = str2term "-1 / (3 + y)";
1564 val SOME (t,_) = rewrite_set_ thy false cancel_p t;
1567 (********* Das ist das PROBLEM !!!!!!!??? *******************)
1568 (* -1 im Z???ler der Angabe verursacht das Problem !*)
1571 (* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - *)
1572 "----- MGs test set";
1573 val t = str2term "(a^^^2 + -1)/(a+1)";
1574 val SOME (t',_) = rewrite_set_ thy false cancel_p t;
1575 if term2str t' = "(-1 + a) / 1" then ()
1576 else error "rational.sml MG tests 3d";
1578 "----- NOT TERMINATING ?: worked before 0707xx";
1579 val t = str2term "(a^^^2 - 1)*(b + 1) / ((b^^^2 - 1)*(a+1))";
1580 val SOME (t'',_) = rewrite_set_ thy false norm_Rational t;
1581 if term2str t'' = "(1 + -1 * a) / (1 + -1 * b)" then ()
1582 else error "rational.sml MG tests 3e";
1584 "----- corrected SK060905";
1585 val t = str2term "(4*x^^^2 - 20*x + 25)/(2*x - 5)^^^3";
1586 val SOME (t',_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1587 if term2str t' = "-1 / (5 + -2 * x)" then ()
1588 else error "rational.sml corrected SK060905";
1591 "-------- examples for Stefan Karnels thesis ------------";
1592 "-------- examples for Stefan Karnels thesis ------------";
1593 "-------- examples for Stefan Karnels thesis ------------";
1594 (*SRAM Schalk I, p.69 Nr. 442b --- abgewandelt*)
1596 "(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))";
1597 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1600 "5 * x ^^^ 2 / (b ^^^ 3 * c)"
1602 else error "rational.sml: diff.behav. in norm_Rational_mg 53";
1605 "-------- me Schalk I No.186 ----------------------------";
1606 "-------- me Schalk I No.186 ----------------------------";
1607 "-------- me Schalk I No.186 ----------------------------";
1608 val fmz = ["Term ((14 * x * y) / ( x * y ))",
1611 ("Rational",["rational","simplification"],
1612 ["simplification","of_rationals"]);
1613 val p = e_pos'; val c = [];
1614 val (p,_,f,nxt,_,pt) = CalcTreeTEST [(fmz, (dI',pI',mI'))];
1615 val (p,_,f,nxt,_,pt) = me nxt p c pt;
1616 val (p,_,f,nxt,_,pt) = me nxt p c pt;
1617 val (p,_,f,nxt,_,pt) = me nxt p c pt;
1618 val (p,_,f,nxt,_,pt) = me nxt p c pt;
1619 val (p,_,f,nxt,_,pt) = me nxt p c pt;
1620 val (p,_,f,nxt,_,pt) = me nxt p c pt;
1621 val (p,_,f,nxt,_,pt) = me nxt p c pt;
1622 val (p,_,f,nxt,_,pt) = me nxt p c pt; f2str f;
1623 val (p,_,f,nxt,_,pt) = me nxt p c pt; f2str f;
1624 val (p,_,f,nxt,_,pt) = me nxt p c pt; f2str f;(*++ for explicit script*)
1625 val (p,_,f,nxt,_,pt) = me nxt p c pt; f2str f;(*++ for explicit script*)
1626 case (f2str f, nxt) of
1627 ("14", ("End_Proof'", _)) => ()
1628 | _ => error "rational.sml diff.behav. in me Schalk I No.186";
1631 "-------- interSteps ..Simp_Rat_Double_No-1.xml ---------";
1632 "-------- interSteps ..Simp_Rat_Double_No-1.xml ---------";
1633 "-------- interSteps ..Simp_Rat_Double_No-1.xml ---------";
1636 [(["Term (((2 - x)/(2*a)) / (2*a/(x - 2)))", "normalform N"],
1637 ("Rational",["rational","simplification"],
1638 ["simplification","of_rationals"]))];
1641 autoCalculate 1 CompleteCalc;
1642 val ((pt,p),_) = get_calc 1; show_pt pt;
1644 interSteps 1 ([1],Res);
1645 val ((pt,p),_) = get_calc 1; show_pt pt;
1648 "-------- interSteps ..Simp_Rat_Cancel_No-1.xml ---------";
1649 "-------- interSteps ..Simp_Rat_Cancel_No-1.xml ---------";
1650 "-------- interSteps ..Simp_Rat_Cancel_No-1.xml ---------";
1653 [(["Term ((a^2 + -1*b^2) / (a^2 + -2*a*b + b^2))", "normalform N"],
1654 ("Rational",["rational","simplification"],
1655 ["simplification","of_rationals"]))];
1658 autoCalculate 1 CompleteCalc;
1659 val ((pt,p),_) = get_calc 1; show_pt pt;
1660 (*========== inhibit exn 110314 ================================================
1661 (*with explicit script done already... and removed [1,..] at below...
1662 interSteps 1 ([1],Res);
1663 val ((pt,p),_) = get_calc 1; show_pt pt;
1665 interSteps 1 ([2],Res);
1666 val ((pt,p),_) = get_calc 1; show_pt pt;
1668 interSteps 1 ([2,1],Res);
1669 val ((pt,p),_) = get_calc 1; show_pt pt;
1670 val newnds = children (get_nd pt [2,1]) (*see "fun detailrls"*);
1671 (*if length newnds = 12 then () WN060905*)
1672 if length newnds = 13 then ()
1673 else error "rational.sml: interSteps cancel_p rev_rew_p";
1675 val p = ([2,1,9],Res);
1677 val (_, tac, _) = pt_extract (pt, p);
1678 (*case tac of SOME (Rewrite ("sym_real_plus_binom_times1", _)) => ()
1680 case tac of SOME (Rewrite ("sym_real_add_mult_distrib2", _)) => ()
1681 | _ => error "rational.sml: getTactic, sym_real_plus_binom_times1";
1682 ============ inhibit exn 110314 ==============================================*)
1685 "-------- investigate rulesets for cancel_p -------------";
1686 "-------- investigate rulesets for cancel_p -------------";
1687 "-------- investigate rulesets for cancel_p -------------";
1688 val thy = @{theory "Rational"};
1689 "---------------- (a^^^2 + -1*b^^^2) / (a^^^2 + -2*a*b + b^^^2)";
1690 val t = str2term "(a^^^2 + -1*b^^^2) / (a^^^2 + -2*a*b + b^^^2)";
1691 val tt = str2term "(1 * a + 1 * b) * (1 * a + -1 * b)"(*numerator only*);
1692 "----- with rewrite_set_";
1693 val SOME (tt',asm) = rewrite_set_ thy false make_polynomial tt;
1694 term2str tt'= "a ^^^ 2 + -1 * b ^^^ 2" (*true*);
1695 val tt = str2term "((1 * a + -1 * b) * (1 * a + -1 * b))"(*denominator only*);
1696 val SOME (tt',asm) = rewrite_set_ thy false make_polynomial tt;
1697 term2str tt' = "a ^^^ 2 + -2 * a * b + b ^^^ 2" (*true*);
1699 "----- with make_deriv";
1700 val SOME (tt, _) = factout_p_ thy t; term2str tt =
1701 "(1 * a + 1 * b) * (1 * a + -1 * b) / ((1 * a + -1 * b) * (1 * a + -1 * b))";
1703 "--- with ruleset as before 060829";
1704 val {rules, rew_ord=(_,ro),...} =
1705 rep_rls (assoc_rls "make_polynomial");
1706 val der = make_deriv thy Atools_erls rules ro NONE tt;
1707 print_depth 99; map (term2str o #1) der; print_depth 3;
1708 print_depth 99; map (rule2str o #2) der; print_depth 3;
1709 ... did not terminate*)
1710 "--- with simpler ruleset";
1711 val {rules, rew_ord=(_,ro),...} =
1712 rep_rls (assoc_rls "rev_rew_p");
1713 val der = make_deriv thy Atools_erls rules ro NONE tt;
1714 print_depth 99; writeln (deriv2str der); print_depth 3;
1716 print_depth 99; map (term2str o #1) der; print_depth 3;
1717 "...,(-1 * b ^^^ 2 + a ^^^ 2) / (-2 * (a * b) + a ^^^ 2 + (-1 * b) ^^^ 2) ]";
1718 print_depth 99; map (rule2str o #2) der; print_depth 3;
1719 print_depth 99; map (term2str o #1 o #3) der; print_depth 3;
1721 val der = make_deriv thy Atools_erls rules ro NONE
1722 (str2term "(1 * a + 1 * b) * (1 * a + -1 * b)");
1723 print_depth 99; writeln (deriv2str der); print_depth 3;
1725 val {rules, rew_ord=(_,ro),...} =
1726 rep_rls (assoc_rls "rev_rew_p");
1727 val der = make_deriv thy Atools_erls rules ro NONE
1728 (str2term "(1 * a + -1 * b) * (1 * a + -1 * b)");
1729 print_depth 99; writeln (deriv2str der); print_depth 3;
1730 print_depth 99; map (term2str o #1) der; print_depth 3;
1731 (*WN060829 ...postponed*)
1734 "-------- investigate format of factout_ and factout_p_ -";
1735 "-------- investigate format of factout_ and factout_p_ -";
1736 "-------- investigate format of factout_ and factout_p_ -";
1737 val {rules, rew_ord = (_,ro),...} = rep_rls (assoc_rls "make_polynomial");
1738 val (thy, eval_rls) = (@{theory "Rational"}, Atools_erls)(*see 'fun init_state'*);
1739 val Rrls {scr = Rfuns {init_state,...},...} = assoc_rls "cancel_p";
1741 "----- see Rational.ML, local cancel_p, fun init_state";
1742 val t = str2term "(a^^^2 + (-1)*b^^^2) / (a^^^2 + (-2)*a*b + b^^^2)";
1743 val SOME (t',_) = factout_p_ thy t; term2str t';
1745 val rtas = reverse_deriv thy eval_rls rules ro NONE t';
1746 writeln(trtas2str rst);
1750 "----- see Rational.ML, local cancel_p, fun init_state";
1751 val t = str2term "a^^^2 / a";
1752 val SOME (t',_) = factout_p_ thy t;
1753 term2str t' = "a * a / (1 * a)" (*true*);
1754 (*... can be canceled with
1755 real_mult_div_cancel2 ?k ~= 0 ==> ?m * ?k / (?n * ?k) = ?m / ?n"*)
1756 (* sml/ME/rewtools.sml:
1757 val rtas = reverse_deriv thy Atools_erls rules ro NONE t';
1758 writeln (deri2str rtas);
1762 "-------- SK 060904 ----------------------------------------------";
1763 "----- order on polynomials -- input + output";
1764 val thy = @{theory "Isac"};
1765 val t = str2term "(a + -1 * b) / (-1 * a + b)";
1766 val SOME (t', _) = factout_p_ thy t; term2str t';
1767 val SOME (t', _) = cancel_p_ thy t; term2str t';
1769 val t = str2term "a*b*c*d / (d*e*f*g)";
1770 val SOME (t', _) = cancel_p_ thy t; term2str t';
1772 val t = str2term "a*(b*(c*d)) / (b*(e*(f*g)))";
1773 val SOME (t', _) = cancel_p_ thy t; term2str t';
1776 "----- SK060904-1a non-termination of cancel_p_ ? worked before 0707xx";
1777 val t = str2term "(x^^^2 - 4)*(3 - y) / ((y^^^2 - 9)*(2+x))";
1778 val SOME (t',_) = rewrite_set_ thy false norm_Rational t;
1779 if term2str t' = "(2 + -1 * x) / (3 + y)" then ()
1780 else error "rational.sml SK060904-1a worked since 0707xx";
1782 "----- SK060904-1b non-termination of cancel_p_ ... worked before 0707xx";
1783 val t = str2term ("(9 * a ^^^ 2 + -16 * b ^^^ 2) /" ^
1784 "(36 * a^^^2 * c + (27 * a^^^2 * e + (-64 * b^^^2 * c + -48 * b^^^2 * e)))");
1785 val SOME (t',_) = cancel_p_ thy t;
1786 if term2str t' = "1 / (4 * c + 3 * e)" then ()
1787 else error "rational.sml SK060904-1b";
1790 "----- SK060904-2a non-termination of add_fraction_p_";
1791 val t = str2term (" (a + b * x) / (a + -1 * (b * x)) + " ^
1792 " (-1 * a + b * x) / (a + b * x) ");
1794 val SOME (t',_) = rewrite_set_ thy false common_nominator_p t;
1796 common_nominator_p_ thy t;
1797 " (a + b * x)*(a + b * x) / ((a + -1 * (b * x))*(a + -1 * (b * x))) + " ^
1798 " (-1 * a + b * x)*(a + -1 * (b * x)) / ((a + b * x)*(-1 * a + b * x)) ";
1800 add_fraction_p_ thy t;
1801 " ((a + b * x)*(a + b * x) + (-1 * a + b * x)*(a + -1 * (b * x))) /" ^
1802 " ((a + b * x)*(-1 * a + b * x)) ";
1806 "-------- how to stepwise construct Scripts -------------";
1807 "-------- how to stepwise construct Scripts -------------";
1808 "-------- how to stepwise construct Scripts -------------";
1809 val thy = @{theory "Rational"};
1811 val p1 = parse thy (
1812 "Script SimplifyScript (t_t::real) = " ^
1813 " (Rewrite_Set discard_minus False " ^
1816 (*required (): (Rewrite_Set discard_minus False)*)
1817 val p2 = parse thy (
1818 "Script SimplifyScript (t_t::real) = " ^
1819 " (Rewrite_Set discard_minus False " ^
1822 val p3 = parse thy (
1823 "Script SimplifyScript (t_t::real) = " ^
1824 " ((Rewrite_Set discard_minus False) " ^
1827 val p4 = parse thy (
1828 "Script SimplifyScript (t_t::real) = " ^
1829 " ((Rewrite_Set discard_minus False) " ^
1832 val p5 = parse thy (
1833 "Script SimplifyScript (t_t::real) = " ^
1834 " ((Try (Rewrite_Set discard_minus False) " ^
1835 " Try (Rewrite_Set discard_parentheses False)) " ^
1838 val p6 = parse thy (
1839 "Script SimplifyScript (t_t::real) = " ^
1840 " ((Try (Rewrite_Set discard_minus False) @@ " ^
1841 " Try (Rewrite_Set rat_mult_poly False) @@ " ^
1842 " Try (Rewrite_Set make_rat_poly_with_parentheses False) @@ " ^
1843 " Try (Rewrite_Set cancel_p_rls False) @@ " ^
1845 " ((Try (Rewrite_Set common_nominator_p_rls False) @@ " ^
1846 " Try (Rewrite_Set rat_mult_div_pow False) @@ " ^
1847 " Try (Rewrite_Set make_rat_poly_with_parentheses False) @@" ^
1848 " Try (Rewrite_Set cancel_p_rls False) @@ " ^
1849 " Try (Rewrite_Set rat_reduce_1 False)))) @@ " ^
1850 " Try (Rewrite_Set discard_parentheses False)) " ^
1854 "----------- get_denominator ----------------------------";
1855 "----------- get_denominator ----------------------------";
1856 "----------- get_denominator ----------------------------";
1857 val thy = @{theory Isac};
1858 val t = term_of (the (parse thy "get_denominator ((a +x)/b)"));
1859 val SOME (_, t') = eval_get_denominator "" 0 t thy;
1860 if term2str t' = "get_denominator ((a + x) / b) = b" then ()
1861 else error "get_denominator ((a + x) / b) = b"
1864 "--------- several errpats in complicated term -------------------";
1865 "--------- several errpats in complicated term -------------------";
1866 "--------- several errpats in complicated term -------------------";
1867 (*TODO: instead of Gabriella's example here (27.Jul.12) find a simpler one*)
1870 [(["Term ((5*b + 25)/(a^2 - b^2) * (a - b)/(5*b))", "normalform N"],
1871 ("Rational",["rational","simplification"], ["simplification","of_rationals"]))];
1874 autoCalculate 1 CompleteCalc;
1875 val ((pt,p),_) = get_calc 1; show_pt pt;
1877 "-------- nonterminating cancel_p, norm_Rational 2002 ---";
1878 "-------- nonterminating cancel_p, norm_Rational 2002 ---";
1879 "-------- nonterminating cancel_p, norm_Rational 2002 ---";
1880 (*------------------------------------------------------------------------------------\
1881 "--- WN121204: searched rational.sml for "nonterm", added new numbering" ^
1882 " and thoroughly tested with this numbering subsequently";
1884 WN.2.6.03 from rlang.sml 56a
1885 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)";
1886 val NONE = rewrite_set_ thy false common_nominator_p t;
1888 WN060831 nonterm.SK7
1889 val t = str2term "(a + b * x) / (a + -1 * (b * x)) + (-1 * a + b * x) / (a + b * x)";
1890 val NONE = add_fraction_p_ thy t;
1892 nonterm.SK9 loops: cancel_p kann nicht weiter kuerzen!!! *)
1893 val t'' = str2term "(a + b)/(x^^^2 - y^^^2) * ((x - y)^^^2/(a^^^2 - b^^^2))";
1894 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t'';
1897 "(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)";
1898 WN060831 nonterm.SK10
1899 val SOME (t,_) = rewrite_set_ thy false cancel_p t;
1903 "(9*(x^^^2 - 8*x+16)/(4*(y^^^2 - 2*y+1)))/((3*x - 12)/(16*y - 16))";
1904 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1907 else error "rational.sml: diff.behav. in norm_Rational_mg 42";
1910 "9*(x^^^2 - 8*x+16)*(16*y - 16)/(4*(y^^^2 - 2*y+1)*(3*x - 12))";
1911 val SOME (t',_) = rewrite_set_ thy false norm_Rational t;
1912 ... non terminating.
1914 val SOME (t',_) = rewrite_set_ thy false make_polynomial t;
1915 "(-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)";
1916 val SOME (t,_) = rewrite_set_ thy false cancel_p t';
1918 val t = str2term "(a-b)^^^3 * (x+y)^^^4 / ((x+y)^^^2 * (a-b)^^^5)";
1919 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1921 val t = str2term "((12*x*y/(9*x^^^2 - y^^^2))/" ^
1922 "(1/(3*x - y)^^^2 - 1/(3*x + y)^^^2)) *" ^
1923 "(1/(x - 5*y)^^^2 - 1/(x + 5*y)^^^2)/" ^
1924 "(20*x*y/(x^^^2 - 25*y^^^2))";
1925 val SOME (t,_) = rewrite_set_ thy false norm_Rational(*_mg*) t;
1927 SK060904-2a non-termination of add_fraction_p_";
1928 val t = str2term (" (a + b * x) / (a + -1 * (b * x)) + " ^
1929 " (-1 * a + b * x) / (a + b * x) ");
1930 (* nonterm.SK = WN130830
1931 val SOME (t',_) = rewrite_set_ thy false common_nominator_p t;
1933 common_nominator_p_ thy t;
1934 " (a + b * x)*(a + b * x) / ((a + -1 * (b * x))*(a + -1 * (b * x))) + " ^
1935 " (-1 * a + b * x)*(a + -1 * (b * x)) / ((a + b * x)*(-1 * a + b * x)) ";
1937 add_fraction_p_ thy t;
1938 " ((a + b * x)*(a + b * x) + (-1 * a + b * x)*(a + -1 * (b * x))) /" ^
1939 " ((a + b * x)*(-1 * a + b * x)) ";
1940 --------------------------------------------------------------------------------------/*)
1942 (*------------------------------------------------------------------------------------\
1943 "WN121205: thoroughly tested with the above numbering.";
1944 " errors in cancel_p: --- 4,5,6,7.";
1945 " error in common_nominator_p, common_nominator_p_: --- 10; 1,2?.";
1946 " errors caused by ruleset norm_Rational: --- 8,9.";
1947 trace_rewrite := false;
1949 "--- 1 ---: non-terminating with ### add_fract: done t22 "; = WN130830
1950 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)";
1951 trace_rewrite := false;
1952 rewrite_set_ thy false common_nominator_p t;
1954 "--- 2 ---: non-terminating with ### add_fract: done t22 ";
1955 val t = str2term "(a + b * x) / (a + -1 * (b * x)) + (-1 * a + b * x) / (a + b * x)";
1956 add_fraction_p_ thy t;
1958 "--- 3 ---: norm_Rational calls Rrls cancel_p with non-normalised polys";
1959 val t = str2term "(a + b)/(x^^^2 - y^^^2) * ((x - y)^^^2/(a^^^2 - b^^^2))";
1960 rewrite_set_ thy false norm_Rational t;
1961 (*tracing end...#### rls: cancel_p on:
1962 (a * x ^^^ 2 + -2 * (a * (x * y)) + a * y ^^^ 2 + b * x ^^^ 2 + -2 * (b * (x * y)) + b * y ^^^ 2) /
1963 (a ^^^ 2 * x ^^^ 2 + -1 * (a ^^^ 2 * y ^^^ 2) + -1 * (b ^^^ 2 * x ^^^ 2) + b ^^^ 2 * y ^^^ 2) *)
1965 "--- 4 ---: non-terminating with Rrls cancel_p on plausible input";
1967 "(a * x ^^^ 2 + -2 * a * x * y + a * y ^^^ 2 + b * x ^^^ 2 + -2 * b * x * y + b * y ^^^ 2) /"^
1968 "(a ^^^ 2 * x ^^^ 2 + -1 * a ^^^ 2 * y ^^^ 2 + -1 * b ^^^ 2 * x ^^^ 2 + b ^^^ 2 * y ^^^ 2)");
1969 rewrite_set_ thy false cancel_p t;
1970 (*# rls: cancel_p on:
1971 (a * x ^^^ 2 + -2 * a * x * y + a * y ^^^ 2 + b * x ^^^ 2 + -2 * b * x * y + b * y ^^^ 2) /
1972 (a ^^^ 2 * x ^^^ 2 + -1 * a ^^^ 2 * y ^^^ 2 + -1 * b ^^^ 2 * x ^^^ 2 + b ^^^ 2 * y ^^^ 2) *)
1974 "--- 5 ---: non-terminating with Rrls cancel_p on plausible input";
1975 val t = str2term "(9*(x^^^2 - 8*x+16)/(4*(y^^^2 - 2*y+1)))/((3*x - 12)/(16*y - 16))";
1976 rewrite_set_ thy false norm_Rational t;
1977 (*#### rls: cancel_p on:
1978 (2304 + -1152 * x + -2304 * y + 144 * x ^^^ 2 + 1152 * (x * y) + -144 * (x ^^^ 2 * y)) /
1979 (48 + -12 * x + -96 * y + 24 * (x * y) + 48 * y ^^^ 2 + -12 * (x * y ^^^ 2)) *)
1981 "--- 6 ---: non-terminating with Rrls cancel_p on plausible input";
1983 "9*(x^^^2 - 8*x+16)*(16*y - 16)/(4*(y^^^2 - 2*y+1)*(3*x - 12))";
1984 rewrite_set_ thy false norm_Rational t;
1985 (*### rls: cancel_p on: (-2304 + 1152 * x + 2304 * y + -144 * x ^^^ 2 + -1152 * (x * y) +
1986 144 * (x ^^^ 2 * y)) /
1987 (-48 + 12 * x + 96 * y + -24 * (x * y) + -48 * y ^^^ 2 + 12 * (x * y ^^^ 2)) *)
1989 "--- 7 ---: non-terminating with Rrls cancel_p on plausible input";
1991 "(-2304 + 1152 * x + 2304 * y + -144 * x ^^^ 2 + -1152 * x * y + 144 * x ^^^ 2 * y) /"^
1992 "(-48 + 12 * x + 96 * y + -24 * x * y + -48 * y ^^^ 2 + 12 * x * y ^^^ 2)");
1993 rewrite_set_ thy false cancel_p t';
1995 "--- 8 ---: bottom of output cannot be looked ad (due to looping?)";
1996 val t = str2term "(a-b)^^^3 * (x+y)^^^4 / ((x+y)^^^2 * (a-b)^^^5)";
1997 val SOME (t,_) = rewrite_set_ thy false norm_Rational t;
1999 "--- 9 ---: probably error in norm_Rational";
2001 "((12*x*y / (9*x^^^2 - y^^^2)) / (1/(3*x - y)^^^2 - 1/(3*x + y)^^^2)) *" ^
2002 "(1/(x - 5*y)^^^2 - 1/(x + 5*y)^^^2) / (20*x*y/(x^^^2 - 25*y^^^2))");
2003 rewrite_set_ thy false norm_Rational t;
2004 (*#### rls: cancel_p on: (19440 * (x ^^^ 8 * y ^^^ 2) + -490320 * (x ^^^ 6 * y ^^^ 4) +
2005 108240 * (x ^^^ 4 * y ^^^ 6) +
2006 -6000 * (x ^^^ 2 * y ^^^ 8)) /
2007 (2160 * (x ^^^ 8 * y ^^^ 2) + -108240 * (x ^^^ 6 * y ^^^ 4) +
2008 1362000 * (x ^^^ 4 * y ^^^ 6) +
2009 -150000 * (x ^^^ 2 * y ^^^ 8)) *)
2011 "--- 10 ---: non-terminating with ### add_fract: done t22: error in common_nominator_p ";
2012 val t = str2term (" (a + b * x) / (a + -1 * (b * x)) + (-1 * a + b * x) / (a + b * x)");
2013 rewrite_set_ thy false common_nominator_p t; (*### add_fract: done t22 *)
2014 common_nominator_p_ thy t; (*loops without output*)
2015 "--- reformulated 10:";
2016 val t = str2term "(a + -1 * (b * x)) / (a + b * x)";
2017 rewrite_set_ thy false cancel_p t = NONE;
2020 "...both are to be considered after common_nominator_p_ is improved";
2021 --------------------------------------------------------------------------------------/*)
2022 ============ inhibit exn WN130824 TODO ======================================================*)