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 "-------- rls norm_Rational downto fun add_fraction_p_ -----------------------";
21 "-------- rewrite_set_ cancel_p from: Mathematik 1 Schalk Reniets Verlag -----";
22 "-------- rewrite_set_ add_fractions_p from: Mathematik 1 Schalk -------------";
23 "-------- integration lev.1 -- lev.5: cancel_p_ & add_fractions_p_ -----------";
24 "-------- reverse rewrite ----------------------------------------------------";
25 "-------- 'reverse-ruleset' cancel_p -----------------------------------------";
26 "-------- investigate rls norm_Rational --------------------------------------";
27 "-------- examples: rls norm_Rational ----------------------------------------";
28 "-------- rational numerals --------------------------------------------------";
29 "-------- examples cancellation from: Mathematik 1 Schalk --------------------";
30 "-------- examples common denominator from: Mathematik 1 Schalk --------------";
31 "-------- examples multiply and cancel from: Mathematik 1 Schalk -------------";
32 "-------- examples common denominator and multiplication from: Schalk --------";
33 "-------- examples double fractions from: Mathematik 1 Schalk ----------------";
34 "-------- me Schalk I No.186 -------------------------------------------------";
35 "-------- interSteps ..Simp_Rat_Double_No-1.xml ------------------------------";
36 "-------- interSteps ..Simp_Rat_Cancel_No-1.xml ------------------------------";
37 "-------- investigate rulesets for cancel_p ----------------------------------";
38 "-------- fun eval_get_denominator -------------------------------------------";
39 "-------- several errpats in complicated term --------------------------------";
40 "-------- WN1309xx non-terminating rls norm_Rational -------------------------";
41 "-----------------------------------------------------------------------------";
42 "-----------------------------------------------------------------------------";
45 "-------- fun poly_of_term ---------------------------------------------------";
46 "-------- fun poly_of_term ---------------------------------------------------";
47 "-------- fun poly_of_term ---------------------------------------------------";
48 val vs = "12 * x^^^3 * y^^^4 * z^^^6" |> str2term |> vars |> map free2str |> sort string_ord;
50 if poly_of_term vs (str2term "12::real") = SOME [(12, [0, 0, 0])]
51 then () else error "poly_of_term 1 changed";
52 if poly_of_term vs (str2term "x::real") = SOME [(1, [1, 0, 0])]
53 then () else error "poly_of_term 2 changed";
54 if poly_of_term vs (str2term "12 * x^^^3") = SOME [(12, [3, 0, 0])]
55 then () else error "poly_of_term 3 changed";
56 if poly_of_term vs (str2term "12 * x^^^3 * y^^^4 * z^^^6") = SOME [(12, [3, 4, 6])]
57 then () else error "poly_of_term 4 changed";
58 if poly_of_term vs (str2term "1 + 2 * x^^^3 * y^^^4 * z^^^6 + y") =
59 SOME [(1, [0, 0, 0]), (1, [0, 1, 0]), (2, [3, 4, 6])]
60 then () else error "poly_of_term 5 changed";
62 (*poly_of_term is quite liberal:*)
63 (*the coefficient may be somewhere, the order of variables and the parentheses
64 within a monomial are arbitrary*)
65 if poly_of_term vs (str2term "y^^^4 * (x^^^3 * 12 * z^^^6)") = SOME [(12, [3, 4, 6])]
66 then () else error "poly_of_term 6 changed";
68 (*there may even be more than 1 coefficient:*)
69 if poly_of_term vs (str2term "2 * y^^^4 * (x^^^3 * 6 * z^^^6)") = SOME [(12, [3, 4, 6])]
70 then () else error "poly_of_term 7 changed";
72 (*the order and the parentheses within monomials are arbitrary:*)
73 if poly_of_term vs (str2term "2 * x^^^3 * y^^^4 * z^^^6 + (7 * y^^^8 + 1)")
74 = SOME [(1, [0, 0, 0]), (7, [0, 8, 0]), (2, [3, 4, 6])]
75 then () else error "poly_of_term 8 changed";
77 "-------- fun is_poly --------------------------------------------------------";
78 "-------- fun is_poly --------------------------------------------------------";
79 "-------- fun is_poly --------------------------------------------------------";
80 if is_poly (str2term "2 * x^^^3 * y^^^4 * z^^^6 + 7 * y^^^8 + 1")
81 then () else error "is_poly 1 changed";
82 if not (is_poly (str2term "2 * (x^^^3 * y^^^4 * z^^^6 + 7) * y^^^8 + 1"))
83 then () else error "is_poly 2 changed";
85 "-------- fun term_of_poly ---------------------------------------------------";
86 "-------- fun term_of_poly ---------------------------------------------------";
87 "-------- fun term_of_poly ---------------------------------------------------";
88 val expT = HOLogic.realT
89 val Free (_, baseT) = (hd o vars o str2term) "12 * x^^^3 * y^^^4 * z^^^6";
90 val p = [(1, [0, 0, 0]), (7, [0, 8, 0]), (2, [3, 4, 5])]
91 val vs = ["x","y","z"]
92 (*precondition for [(c, es),...]: legth es = length vs*)
94 if term2str (term_of_poly baseT expT vs p) = "1 + 7 * y ^^^ 8 + 2 * x ^^^ 3 * y ^^^ 4 * z ^^^ 5"
95 then () else error "term_of_poly 1 changed";
97 "-------- integration lev.1 fun factout_p_ -----------------------------------";
98 "-------- integration lev.1 fun factout_p_ -----------------------------------";
99 "-------- integration lev.1 fun factout_p_ -----------------------------------";
100 val t = str2term "(x^^^2 + -1*y^^^2) / (x^^^2 + -1*x*y)"
101 val SOME (t', asm) = factout_p_ thy t;
102 if term2str t' = "(x + y) * (x + -1 * y) / (x * (x + -1 * y))"
103 then () else error ("factout_p_ term 1 changed: " ^ term2str t')
105 if terms2str asm = "[\"x ~= 0\",\"x + -1 * y ~= 0\"]"
106 then () else error "factout_p_ asm 1 changed"
108 val t = str2term "nothing + to_cancel ::real";
109 if NONE = factout_p_ thy t then () else error "factout_p_ doesn't report non-applicable";
111 val t = str2term "((3 * x^^^2 + 6 *x + 3) / (2*x + 2))";
112 val SOME (t', asm) = factout_p_ thy t;
113 if term2str t' = "(3 + 3 * x) * (1 + x) / (2 * (1 + x))" andalso
114 terms2str asm = "[\"1 + x ~= 0\"]"
115 then () else error "factout_p_ 1 changed";
117 "-------- integration lev.1 fun cancel_p_ ------------------------------------";
118 "-------- integration lev.1 fun cancel_p_ ------------------------------------";
119 "-------- integration lev.1 fun cancel_p_ ------------------------------------";
120 val t = str2term "(x^^^2 + -1*y^^^2) / (x^^^2 + -1*x*y)"
121 val SOME (t', asm) = cancel_p_ thy t;
122 if (term2str t', terms2str asm) = ("(x + y) / x", "[\"x ~= 0\"]")
123 then () else error ("cancel_p_ (t', asm) 1 changed: " ^ term2str t')
125 val t = str2term "nothing + to_cancel ::real";
126 if NONE = cancel_p_ thy t then () else error "cancel_p_ doesn't report non-applicable";
128 val t = str2term "((3 * x^^^2 + 6 *x + 3) / (2*x + 2))";
129 val SOME (t', asm) = cancel_p_ thy t;
130 if term2str t' = "(3 + 3 * x) / 2" andalso terms2str asm = "[]"
131 then () else error "cancel_p_ 1 changed";
133 "-------- integration lev.1 fun common_nominator_p_ --------------------------";
134 "-------- integration lev.1 fun common_nominator_p_ --------------------------";
135 "-------- integration lev.1 fun common_nominator_p_ --------------------------";
136 val t = str2term ("y / (a*x + b*x + c*x) " ^
140 val SOME (t', asm) = common_nominator_p_ thy t;
142 ("y * y / (x * ((a + b + c) * y)) " ^
143 (* n1 *d2'/ (c'* ( d1' *d2')) *)
144 "+ a * (a + b + c) / (x * ((a + b + c) * y))")
145 (* n2 * d1' / (c'* ( d1' *d2')) *)
146 then () else error "common_nominator_p_ term 1 changed";
147 if terms2str asm = "[\"a + b + c ~= 0\",\"y ~= 0\",\"x ~= 0\"]"
148 then () else error "common_nominator_p_ asm 1 changed"
150 "-------- example in mail Nipkow";
151 val t = str2term "x/(x^^^2 + -1*y^^^2) + y/(x^^^2 + -1*x*y)";
152 val SOME (t', asm) = common_nominator_p_ thy t;
153 if term2str t' = "x * x / " ^
154 "((x + -1 * y) * ((x + y) * x))" ^
157 "((x + -1 * y) * ((x + y) * x))"
158 then () else error "common_nominator_p_ term 2 changed"
160 if terms2str asm = "[\"x + y ~= 0\",\"x ~= 0\",\"x + -1 * y ~= 0\"]"
161 then () else error "common_nominator_p_ asm 2 changed"
163 "-------- example: applicable tested by SML code";
164 val t = str2term "nothing / to_add";
165 if NONE = common_nominator_p_ thy t then () else error "common_nominator_p_ term 3 changed";
167 val t = str2term "((x + (-1)) / (x + 1)) + ((x + 1) / (x + (-1)))";
168 val SOME (t', asm) = common_nominator_p_ thy t;
170 "(x + -1) * (-1 + x) / ((1 + x) * (-1 + x)) +\n(x + 1) * (1 + x) / ((1 + x) * (-1 + x))"
171 andalso terms2str asm = "[\"1 + x ~= 0\",\"-1 + x ~= 0\"]"
172 then () else error "common_nominator_p_ 3 changed";
174 "-------- integration lev.1 fun add_fraction_p_ ------------------------------";
175 "-------- integration lev.1 fun add_fraction_p_ ------------------------------";
176 "-------- integration lev.1 fun add_fraction_p_ ------------------------------";
177 val t = str2term "((x + (-1)) / (x + 1)) + ((x + 1) / (x + (-1)))";
178 val SOME (t', asm) = add_fraction_p_ thy t;
179 if term2str t' = "(2 + 2 * x ^^^ 2) / (-1 + x ^^^ 2)"
180 then () else error "add_fraction_p_ 3 changed";
182 if terms2str asm = "[\"-1 + x ^^^ 2 ~= 0\"]"
183 then () else error "add_fraction_p_ 3 changed";
185 val t = str2term "nothing / to_add";
186 if NONE = add_fraction_p_ thy t then () else error "add_fraction_p_ term 3 changed";
188 val t = str2term "((x + (-1)) / (x + 1)) + ((x + 1) / (x + (-1)))";
189 val SOME (t', asm) = add_fraction_p_ thy t;
190 if term2str t' = "(2 + 2 * x ^^^ 2) / (-1 + x ^^^ 2)" andalso
191 terms2str asm = "[\"-1 + x ^^^ 2 ~= 0\"]"
192 then () else error "add_fraction_p_ 3 changed";
194 "-------- and app_rev ...traced down from rewrite_set_ until prepats ---------";
195 "-------- and app_rev ...traced down from rewrite_set_ until prepats ---------";
196 "-------- and app_rev ...traced down from rewrite_set_ until prepats ---------";
197 (* trace down until prepats are evaluated
198 (which does not to work, because substitution is not done -- compare rew_sub!);
199 keep this sequence for the case, factout_p, cancel_p, common_nominator_p, add_fraction_p
200 (again) get prepat = [] changed to <>[]. *)
201 val t = str2term "(x^^^2 + -1*y^^^2) / (x^^^2 + -1*x*y)";
203 (*rewrite_set_ @{theory Isac} true cancel t = NONE; !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!*)
204 "~~~~~ fun rewrite_set_, args:"; val (thy, bool, rls, term) = (thy, false, cancel_p, t);
205 "~~~~~ fun rewrite__set_, args:"; val (thy, i, _, _, (rrls as Rrls _), t) =
206 (thy, 1, bool, [], rls, term);
207 (*val (t', asm, rew) = app_rev thy (i+1) rrls t; rew = false!!!!!!!!!!!!!!!!!!!!!*)
208 "~~~~~ and app_rev, args:"; val (thy, i, rrls, t) = (thy, (i+1), rrls, t);
209 fun chk_prepat thy erls [] t = true
210 | chk_prepat thy erls prepat t =
212 fun chk (pres, pat) =
214 val subst: Type.tyenv * Envir.tenv =
215 Pattern.match thy (pat, t) (Vartab.empty, Vartab.empty)
217 snd (eval__true thy (i + 1) (map (Envir.subst_term subst) pres) [] erls)
218 end) handle _ => false
219 fun scan_ f [] = false (*scan_ NEVER called by []*)
220 | scan_ f (pp::pps) =
221 if f pp then true else scan_ f pps;
222 in scan_ chk prepat end;
223 (* apply the normal_form of a rev-set *)
224 fun app_rev' thy (Rrls{erls,prepat,scr=Rfuns{normal_form,...},...}) t =
225 if chk_prepat thy erls prepat t
226 then ((*tracing("### app_rev': t = "^term2str t);*) normal_form t)
228 (* val opt = app_rev' thy rrls t ..NONE!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!*)
229 "~~~~~ and app_rev', args:"; val (thy, (Rrls{erls,prepat,scr=Rfuns{normal_form,...},...}), t) =
231 (* chk_prepat thy erls prepat t = false!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!*)
232 (* app_sub thy i rrls t = false!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!*)
233 "~~~~~ fun chk_prepat, args:"; val (thy, erls, prepat, t) = (thy, erls, prepat, t);
234 fun chk (pres, pat) =
236 val subst: Type.tyenv * Envir.tenv =
237 Pattern.match thy (pat, t) (Vartab.empty, Vartab.empty)
239 snd (eval__true thy (i + 1) (map (Envir.subst_term subst) pres) [] erls)
240 end) handle _ => false
241 fun scan_ f [] = false (*scan_ NEVER called by []*)
242 | scan_ f (pp::pps) =
243 if f pp then true else scan_ f pps;
245 (*========== inhibit exn WN130823: prepat is empty ====================================
246 (* scan_ chk prepat = false!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!*)
247 "~~~~~ fun , args:"; val (f, (pp::pps)) = (chk, prepat);
249 val ([t1, t2], t) = pp;
250 term2str t1 = "?r is_expanded";
251 term2str t2 = "?s is_expanded";
252 term2str t = "?r / ?s";
253 (* f pp = false!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!*)
254 "~~~~~ fun chk, args:"; val (pres, pat) = (pp);
255 val subst: Type.tyenv * Envir.tenv =
256 Pattern.match thy (pat, t) (Vartab.empty, Vartab.empty)
258 ({}, {(("r", 0), ("real", Var (("r", 0), "real"))),
259 (("s", 0), ("real", Var (("s", 0), "real")))}*)
261 snd (eval__true thy (i + 1) (map (Envir.subst_term subst) pres) [] erls)
262 "~~~~~ fun eval__true, args:"; val (thy, i, asms, bdv, rls) =
263 (thy, (i + 1), (map (Envir.subst_term subst) pres), [], erls);
264 terms2str asms; (* = "[\"?r is_expanded\",\"?s is_expanded\"]"*)
265 asms = [@{term True}] orelse asms = []; (* = false*)
266 asms = [@{term False}] ; (* = false*)
267 "~~~~~ fun chk, args:"; val (indets, (a::asms)) = ([], asms);
268 bdv (*= []: _a list*);
269 val bdv : (term * term) list = [];
270 rewrite__set_ thy (i+1) false;
271 term2str a = "?r is_expanded"; (*hier m"usste doch der Numerator eingesetzt sein ??????????????*)
272 val SOME (Const ("HOL.False", _), []) = rewrite__set_ thy (i+1) false bdv rls a
273 ============ inhibit exn WN130823: prepat is empty ===================================*)
275 "-------- fun rewrite_set_ cancel_p downto fun gcd_poly ----------------------";
276 "-------- fun rewrite_set_ cancel_p downto fun gcd_poly ----------------------";
277 "-------- fun rewrite_set_ cancel_p downto fun gcd_poly ----------------------";
278 val t = str2term "(12 * x * y) / (8 * y^^^2 )";
279 (* "-------- example 187a": exception Div raised...
280 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;*)
281 val t = str2term "(8 * x^^^2 * y * z ) / (18 * x * y^^^2 * z )";
282 (* "-------- example 187b": doesn't terminate...
283 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;*)
284 val t = str2term "(9 * x^^^5 * y^^^2 * z^^^4) / (15 * x^^^6 * y^^^3 * z )";
285 (* "-------- example 187c": doesn't terminate...
286 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;*)
287 "~~~~~ fun rewrite_set_, args:"; val (thy, bool, rls, term) = (@{theory Isac}, false, cancel_p, t);
288 (* WN130827: exception Div raised...
289 rewrite__set_ thy 1 bool [] rls term
291 "~~~~~ and rewrite__set_, args:"; val (thy, i, _, _, (rrls as Rrls _), t) =
292 (thy, 1, bool, [], rls, term);
293 (* WN130827: exception Div raised...
294 val (t', asm, rew) = app_rev thy (i+1) rrls t
296 "~~~~~ fun app_rev, args:"; val (thy, i, rrls, t) = (thy, (i+1), rrls, t);
297 (* WN130827: exception Div raised...
298 val opt = app_rev' thy rrls t
300 "~~~~~ fun app_rev', args:"; val (thy, (Rrls{erls,prepat,scr=Rfuns{normal_form,...},...}), t) =
302 chk_prepat thy erls prepat t = true;
303 (* WN130827: exception Div raised...
306 (* lookup Rational.thy, cancel_p: normal_form = cancel_p_ thy*)
307 "~~~~~ fun cancel_p_, args:"; val (t) = (t);
308 val opt = check_fraction t;
309 val SOME (numerator, denominator) = opt
310 val vs = t |> vars |> map free2str |> sort string_ord
311 val baseT = type_of numerator
312 val expT = HOLogic.realT
313 val (SOME a, SOME b) = (poly_of_term vs numerator, poly_of_term vs denominator);
314 (*"-------- example 187a": exception Div raised...
315 val a = [(12, [1, 1])]: poly
316 val b = [(8, [0, 2])]: poly
317 val ((a', b'), c) = gcd_poly a b
319 (* "-------- example 187b": doesn't terminate...
320 val a = [(8, [2, 1, 1])]: poly
321 val b = [(18, [1, 2, 1])]: poly
322 val ((a', b'), c) = gcd_poly a b
324 (* "-------- example 187c": doesn't terminate...
325 val a = [(9, [5, 2, 4])]: poly
326 val b = [(15, [6, 3, 1])]: poly
327 val ((a', b'), c) = gcd_poly a b
330 "-------- rls norm_Rational downto fun gcd_poly ------------------------------";
331 "-------- rls norm_Rational downto fun gcd_poly ------------------------------";
332 "-------- rls norm_Rational downto fun gcd_poly ------------------------------";
333 val t = str2term "(x^^^2 - 4)*(3 - y) / ((y^^^2 - 9)*(2+x))";
334 trace_rewrite := false (*true false*);
335 (* trace stops with ...: (and then jEdit hangs)..
336 rewrite_set_ thy false norm_Rational t;
338 ### rls: cancel_p on: (-12 + 4 * y + 3 * x ^^^ 2 + -1 * (x ^^^ 2 * y)) /
339 (-18 + -9 * x + 2 * y ^^^ 2 + x * y ^^^ 2)
341 val t = str2term (*copy from above: "::real" is not required due to "^^^"*)
342 ("(-12 + 4 * y + 3 * x ^^^ 2 + -1 * (x ^^^ 2 * y)) /" ^
343 "(-18 + -9 * x + 2 * y ^^^ 2 + x * y ^^^ 2)");
345 exception Div raised*)
347 "~~~~~ fun cancel_p_, args:"; val (t) = (t);
348 val opt = check_fraction t;
349 val SOME (numerator, denominator) = opt
350 val vs = t |> vars |> map free2str |> sort string_ord
351 val baseT = type_of numerator
352 val expT = HOLogic.realT;
353 print_depth 3; (*999*)
354 val (SOME a, SOME b) = (poly_of_term vs numerator, poly_of_term vs denominator);
355 print_depth 3; (*999*)
356 (* does not terminate instead of returning ?:
357 val ((a', b'), c) = gcd_poly a b
358 val a = [(~12, [0, 0]), (3, [2, 0]), (4, [0, 1]), (~1, [2, 1])]: poly
359 val b = [(~18, [0, 0]), (~9, [1, 0]), (2, [0, 2]), (1, [1, 2])]: poly
362 "-------- rls norm_Rational downto fun add_fraction_p_ -----------------------";
363 "-------- rls norm_Rational downto fun add_fraction_p_ -----------------------";
364 "-------- rls norm_Rational downto fun add_fraction_p_ -----------------------";
365 val thy = @{theory Isac};
366 "----- SK060904-2a non-termination of add_fraction_p_";
367 val t = str2term (" (a + b * x) / (a + -1 * (b * x)) + " ^
368 " (-1 * a + b * x) / (a + b * x) ");
369 (* rewrite_set_ thy false norm_Rational t
370 exception Div raised*)
371 (* rewrite_set_ thy false add_fractions_p t;
372 exception Div raised*)
373 "~~~~~ fun rewrite_set_, args:"; val (thy, bool, rls, term) =
374 (@{theory Isac}, false, add_fractions_p, t);
375 "~~~~~ and rewrite__set_, args:"; val (thy, i, _, _, (rrls as Rrls _), t) =
376 (thy, 1, bool, [], rls, term);
377 (* app_rev thy (i+1) rrls t;
378 exception Div raised*)
379 "~~~~~ and app_rev, args:"; val (thy, i, rrls, t) = (thy, (i+1), rrls, t);
380 fun chk_prepat thy erls [] t = true
381 | chk_prepat thy erls prepat t =
383 fun chk (pres, pat) =
385 val subst: Type.tyenv * Envir.tenv =
386 Pattern.match thy (pat, t) (Vartab.empty, Vartab.empty)
388 snd (eval__true thy (i + 1) (map (Envir.subst_term subst) pres) [] erls)
389 end) handle _ => false
390 fun scan_ f [] = false (*scan_ NEVER called by []*)
391 | scan_ f (pp::pps) =
392 if f pp then true else scan_ f pps;
393 in scan_ chk prepat end;
394 (* apply the normal_form of a rev-set *)
395 fun app_rev' thy (Rrls{erls,prepat,scr=Rfuns{normal_form,...},...}) t =
396 if chk_prepat thy erls prepat t
397 then ((*tracing("### app_rev': t = "^term2str t);*) normal_form t)
399 (* val opt = app_rev' thy rrls t;
400 exception Div raised*)
401 (* val opt = app_rev' thy rrls t;
402 exception Div raised*)
403 "~~~~~ and app_rev', args:"; val (thy, (Rrls{erls,prepat,scr=Rfuns{normal_form,...},...}), t) =
405 chk_prepat thy erls prepat t = true = true;
407 exception Div raised*)
408 (* lookup Rational.thy, val add_fractions_p: normal_form = add_fraction_p_ thy*)
409 (*add_fraction_p_ thy t
410 exception Div raised*)
411 "~~~~~ fun add_fraction_p_, args:"; val ((_: theory), t) = (thy, t);
412 val SOME ((n1, d1), (n2, d2)) = check_frac_sum t;
413 term2str n1; term2str d1; term2str n2; term2str d2;
414 val vs = t |> vars |> map str_of_free_opt (* tolerate Var in simplification *)
415 |> filter is_some |> map the |> sort string_ord;
416 print_depth 3; (*999*)
417 val (SOME _, SOME a, SOME _, SOME b) =
418 (poly_of_term vs n1, poly_of_term vs d1, poly_of_term vs n2, poly_of_term vs d2);
419 print_depth 3; (*999*)
421 val a = [(1, [1, 0, 0]), (~1, [0, 1, 1])]: poly
422 val b = [(1, [1, 0, 0]), (1, [0, 1, 1])]: poly
423 val ((a', b'), c) = gcd_poly a b
426 "-------- rewrite_set_ cancel_p from: Mathematik 1 Schalk Reniets Verlag -----";
427 "-------- rewrite_set_ cancel_p from: Mathematik 1 Schalk Reniets Verlag -----";
428 "-------- rewrite_set_ cancel_p from: Mathematik 1 Schalk Reniets Verlag -----";
429 val thy = @{theory "Rational"};
431 val t = str2term "(2 + -3 * x) / 9";
432 if NONE = rewrite_set_ thy false cancel_p t then ()
433 else error "rewrite_set_ cancel_p must return NONE, if the term cannot be cancelled";
435 "-------- example 186a";
436 val t = str2term "(14 * x * y) / (x * y)";
437 is_expanded (str2term "14 * x * y");
438 is_expanded (str2term "x * y");
439 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;
440 if (term2str t', terms2str asm) = ("14 / 1", "[]")
441 then () else error "rational.sml cancel Schalk 186a";
443 "-------- example 186b";
444 val t = str2term "(60 * a * b) / ( 15 * a * b )";
445 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;
446 if (term2str t', terms2str asm) = ("4 / 1", "[]")
447 then () else error "rational.sml cancel Schalk 186b";
449 "-------- example 186c";
450 val t = str2term "(144 * a^^^2 * b * c) / (12 * a * b * c)";
451 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;
452 if (term2str t', terms2str asm) = ("12 * a / 1", "[]")
453 then () else error "rational.sml cancel Schalk 186c";
455 (* !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! exception Div raised !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
456 see --- fun rewrite_set_ downto fun gcd_poly ---
457 "-------- example 187a";
458 val t = str2term "(12 * x * y) / (8 * y^^^2 )";
459 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;
460 if (term2str t', terms2str asm) = ("3 * x / (2 * y)", "[\"4 * y ~= 0\"]")
461 then () else error "rational.sml cancel Schalk 187a";
464 (* doesn't terminate !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
465 see --- fun rewrite_set_ downto fun gcd_poly ---
466 "-------- example 187b";
467 val t = str2term "(8 * x^^^2 * y * z ) / (18 * x * y^^^2 * z )";
468 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;
469 if (term2str t', terms2str asm) = ("4 * x / (9 * y)", "[\"2 * (z * (y * x)) ~= 0\"]")
470 then () else error "rational.sml cancel Schalk 187b";
473 (* doesn't terminate !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
474 see --- fun rewrite_set_ downto fun gcd_poly ---
475 "-------- example 187c";
476 val t = str2term "(9 * x^^^5 * y^^^2 * z^^^4) / (15 * x^^^6 * y^^^3 * z )";
477 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;
478 if (term2str t', terms2str asm) =
479 ("3 * z ^^^ 3 / (5 * (y * x))", "[\"3 * (z * (y ^^^ 2 * x ^^^ 5)) ~= 0\"]")
480 then () else error "rational.sml cancel Schalk 187c";
483 "-------- example 188a";
484 val t = str2term "(-8 + 8 * x) / (-9 + 9 * x)";
485 is_expanded (str2term "8 * x + -8");
486 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;
487 if (term2str t', terms2str asm) = ("8 / 9", "[]")
488 then () else error "rational.sml cancel Schalk 188a";
490 val t = str2term "(8*((-1) + x))/(9*((-1) + x))";
491 val SOME (t, _) = rewrite_set_ thy false make_polynomial t;
492 if (term2str t', terms2str asm) = ("8 / 9", "[]")
493 then () else error "rational.sml cancel Schalk make_polynomial 1";
495 "-------- example 188b";
496 val t = str2term "(-15 + 5 * x) / (-18 + 6 * x)";
497 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;
498 if (term2str t', terms2str asm) = ("5 / 6", "[]")
499 then () else error "rational.sml cancel Schalk 188b";
501 "-------- example 188c";
502 val t = str2term "(a + -1 * b) / (b + -1 * a)";
503 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;
504 if (term2str t', terms2str asm) = ("-1 / 1", "[]")
505 then () else error "rational.sml cancel Schalk 188c";
507 is_expanded (str2term "a + -1 * b") = true;
508 val t = str2term "((-1)*(b + (-1) * a))/(1*(b + (-1) * a))";
509 val SOME (t', asm) = rewrite_set_ thy false make_polynomial t;
510 if (term2str t', terms2str asm) = ("(a + -1 * b) / (-1 * a + b)", "[]")
511 then () else error "rational.sml cancel Schalk make_polynomial 2";
513 "-------- example 190a";
514 val t = str2term "( 27 * a^^^3 + 9 * a^^^2 + 3 * a + 1 ) / ( 27 * a^^^3 + 18 * a^^^2 + 3 * a )";
515 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;
516 if (term2str t', terms2str asm) =
517 ("(1 + 9 * a ^^^ 2) / (3 * a + 9 * a ^^^ 2)", "[\"3 * a + 9 * a ^^^ 2 ~= 0\"]")
518 then () else error "rational.sml cancel Schalk 190a";
520 "-------- example 190c";
521 val t = str2term "((1 + 9 * a ^^^ 2)*(1 + 3 * a))/((3 * a + 9 * a ^^^ 2)*(1 + 3 * a))";
522 val SOME (t', asm) = rewrite_set_ thy false make_polynomial t;
523 if (term2str t', terms2str asm) =
524 ("(1 + 3 * a + 9 * a ^^^ 2 + 27 * a ^^^ 3) /\n(3 * a + 18 * a ^^^ 2 + 27 * a ^^^ 3)", "[]")
525 then () else error "rational.sml make_polynomial Schalk 190c";
527 "-------- example 191a";
528 val t = str2term "( x^^^2 + -1 * y^^^2 ) / ( x + y )";
529 is_expanded (str2term "x^^^2 + -1 * y^^^2") = false; (*!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!*)
530 is_expanded (str2term "x + y") = true;
531 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;
532 if (term2str t', terms2str asm) = ("(x + -1 * y) / 1", "[]")
533 then () else error "rational.sml make_polynomial Schalk 191a";
535 "-------- example 191b";
536 val t = str2term "((x + (-1) * y)*(x + y))/((1)*(x + y))";
537 val SOME (t', asm) = rewrite_set_ thy false make_polynomial t;
538 if (term2str t', terms2str asm) = ("(x ^^^ 2 + -1 * y ^^^ 2) / (x + y)", "[]")
539 then () else error "rational.sml make_polynomial Schalk 191b";
541 "-------- example 191c";
542 val t = str2term "( 9 * x^^^2 + -30 * x + 25 ) / ( 9 * x^^^2 + -25 )";
543 is_expanded (str2term "9 * x^^^2 + -30 * x + 25") = true;
544 is_expanded (str2term "25 + -30*x + 9*x^^^2") = true;
545 is_expanded (str2term "-25 + 9*x^^^2") = true;
547 val t = str2term "(((-5) + 3 * x)*((-5) + 3 * x))/((5 + 3 * x)*((-5) + 3 * x))";
548 val SOME (t', asm) = rewrite_set_ thy false make_polynomial t;
549 if (term2str t', terms2str asm) = ("(25 + -30 * x + 9 * x ^^^ 2) / (-25 + 9 * x ^^^ 2)", "[]")
550 then () else error "rational.sml make_polynomial Schalk 191c";
552 "-------- example 192b";
553 val t = str2term "( 7 * x^^^3 + -1 * x^^^2 * y ) / ( 7 * x * y^^^2 + -1 * y^^^3 )";
554 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;
555 if (term2str t', terms2str asm) = ("x ^^^ 2 / y ^^^ 2", "[\"y ^^^ 2 ~= 0\"]")
556 then () else error "rational.sml cancel_p Schalk 192b";
558 val t = str2term "((x ^^^ 2)*(7 * x + (-1) * y))/((y ^^^ 2)*(7 * x + (-1) * y))";
559 val SOME (t', asm) = rewrite_set_ thy false make_polynomial t;
560 if (term2str t', terms2str asm) =
561 ("(7 * x ^^^ 3 + -1 * x ^^^ 2 * y) / (7 * x * y ^^^ 2 + -1 * y ^^^ 3)", "[]")
562 then () else error "rational.sml make_polynomial Schalk 192b";
564 val t = str2term "((x ^^^ 2)*(7 * x + (-1) * y))/((y ^^^ 2)*(7 * x + (-1) * y))";
565 val SOME (t', asm) = rewrite_set_ thy false make_polynomial t;
566 if (term2str t', terms2str asm) =
567 ("(7 * x ^^^ 3 + -1 * x ^^^ 2 * y) / (7 * x * y ^^^ 2 + -1 * y ^^^ 3)", "[]")
568 then () else error "rational.sml make_polynomial Schalk WN050929 not working";
570 "-------- example 193a";
571 val t = str2term "( x^^^2 + -6 * x + 9 ) / ( x^^^2 + -9 )";
572 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;
573 if (term2str t', terms2str asm) = ("(-3 + x) / (3 + x)", "[\"3 + x ~= 0\"]")
574 then () else error "rational.sml cancel_p Schalk 193a";
576 "-------- example 193b";
577 val t = str2term "( x^^^2 + -8 * x + 16 ) / ( 2 * x^^^2 + -32 )";
578 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;
579 if (term2str t', terms2str asm) = ("(-4 + x) / (8 + 2 * x)", "[\"8 + 2 * x ~= 0\"]")
580 then () else error "rational.sml cancel_p Schalk 193b";
582 "-------- example 193c";
583 val t = str2term "( 2 * x + -50 * x^^^3 ) / ( 25 * x^^^2 + -10 * x + 1 )";
584 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;
585 if (term2str t', terms2str asm) =
586 ("(2 * x + 10 * x ^^^ 2) / (1 + -5 * x)", "[\"1 + -5 * x ~= 0\"]")
587 then () else error "rational.sml cancel_p Schalk 193c";
590 val t = str2term "(-25 + 9*x^^^2)/(5 + 3*x)";
591 val SOME (t, asm) = rewrite_set_ thy false cancel_p t;
592 if (term2str t', terms2str asm) = ("(2 * x + 10 * x ^^^ 2) / (1 + -5 * x)", "[]")
593 then () else error "rational.sml cancel WN 1";
595 "-------- example heuberger";
596 val t = str2term ("(x^^^4 + x * y + x^^^3 * y + y^^^2) / " ^
597 "(x + 5 * x^^^2 + y + 5 * x * y + x^^^2 * y^^^3 + x * y^^^4)");
598 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;
599 if (term2str t', terms2str asm) =
600 ("(x ^^^ 3 + y) / (1 + 5 * x + x * y ^^^ 3)", "[\"1 + 5 * x + x * y ^^^ 3 ~= 0\"]")
601 then () else error "rational.sml cancel_p heuberger";
603 "-------- rewrite_set_ add_fractions_p from: Mathematik 1 Schalk -------------";
604 "-------- rewrite_set_ add_fractions_p from: Mathematik 1 Schalk -------------";
605 "-------- rewrite_set_ add_fractions_p from: Mathematik 1 Schalk -------------";
606 (*deleted example 204 ... 236b at update Isabelle2012-->2013*)
608 "-------- integration lev.1 -- lev.5: cancel_p_ & add_fractions_p_ -----------";
609 "-------- integration lev.1 -- lev.5: cancel_p_ & add_fractions_p_ -----------";
610 "-------- integration lev.1 -- lev.5: cancel_p_ & add_fractions_p_ -----------";
611 val t = str2term ("123 = (a*x)/(b*x) + (c*x)/(d*x) + (e*x)/(f*x::real)");
612 "-------- gcd_poly integration level 1: works on exact term";
613 if NONE = cancel_p_ thy t then () else error "cancel_p_ works on exact fraction";
614 if NONE = add_fraction_p_ thy t then () else error "add_fraction_p_ works on exact fraction";
616 "-------- gcd_poly integration level 2: picks out ONE appropriate subterm";
617 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;
618 if term2str t' = "123 = a * x / (b * x) + c * x / (d * x) + e / f"
619 then () else error "level 2, rewrite_set_ cancel_p: changed";
620 val SOME (t', asm) = rewrite_set_ thy false add_fractions_p t;
621 if term2str t' = "123 = (b * c * x + a * d * x) / (b * d * x) + e * x / (f * x)"
622 then () else error "level 2, rewrite_set_ add_fractions_p: changed";
624 "-------- gcd_poly integration level 3: rewrites all appropriate subterms";
625 val SOME (t', asm) = rewrite_set_ thy false cancel_p_rls t;
626 if term2str t' = "123 = a / b + c / d + e / f"
627 then () else error "level 3, rewrite_set_ cancel_p_rls: changed";
628 val SOME (t', asm) = rewrite_set_ thy false add_fractions_p_rls t; (*CREATE add_fractions_p_rls*)
629 if term2str t' = "123 = (b * d * e * x + b * c * f * x + a * d * f * x) / (b * d * f * x)"
630 then () else error "level 3, rewrite_set_ add_fractions_p_rls: changed";
632 "-------- gcd_poly integration level 4: iteration cancel_p -- add_fraction_p";
633 (* simpler variant *)
634 val testrls = append_rls "testrls" e_rls [Rls_ cancel_p, Rls_ add_fractions_p]
635 val SOME (t', asm) = rewrite_set_ thy false testrls t;
636 (*trace_rewrite := false;
637 # rls: testrls on: 123 = a * x / (b * x) + c * x / (d * x) + e * x / (f * x)
638 ## rls: cancel_p on: 123 = a * x / (b * x) + c * x / (d * x) + e * x / (f * x)
639 ## rls: add_fractions_p on: 123 = a * x / (b * x) + c * x / (d * x) + e / f
640 ## rls: cancel_p on: 123 = (b * c * x + a * d * x) / (b * d * x) + e / f
641 ## rls: add_fractions_p on: 123 = (b * c + a * d) / (b * d) + e / f
642 ## rls: cancel_p on: 123 = (b * d * e + b * c * f + a * d * f) / (b * d * f)
643 ## rls: add_fractions_p on: 123 = (b * d * e + b * c * f + a * d * f) / (b * d * f) *)
644 if term2str t' = "123 = (b * d * e + b * c * f + a * d * f) / (b * d * f)"
645 then () else error "level 4, rewrite_set_ *_p: changed";
647 (* complicated variant *)
648 val testrls_rls = append_rls "testrls_rls" e_rls [Rls_ cancel_p_rls, Rls_ add_fractions_p_rls];
649 val SOME (t', asm) = rewrite_set_ thy false testrls_rls t;
650 (*# rls: testrls_rls on: 123 = a * x / (b * x) + c * x / (d * x) + e * x / (f * x)
651 ## rls: cancel_p_rls on: 123 = a * x / (b * x) + c * x / (d * x) + e * x / (f * x)
652 ### rls: cancel_p on: 123 = a * x / (b * x) + c * x / (d * x) + e * x / (f * x)
653 ### rls: cancel_p on: 123 = a * x / (b * x) + c * x / (d * x) + e / f
654 ### rls: cancel_p on: 123 = a * x / (b * x) + c / d + e / f
655 ### rls: cancel_p on: 123 = a / b + c / d + e / f
656 ## rls: add_fractions_p_rls on: 123 = a / b + c / d + e / f
657 ### rls: add_fractions_p on: 123 = a / b + c / d + e / f
658 ### rls: add_fractions_p on: 123 = (b * c + a * d) / (b * d) + e / f
659 ### rls: add_fractions_p on: 123 = (b * d * e + b * c * f + a * d * f) / (b * d * f)
660 ## rls: cancel_p_rls on: 123 = (b * d * e + b * c * f + a * d * f) / (b * d * f)
661 ### rls: cancel_p on: 123 = (b * d * e + b * c * f + a * d * f) / (b * d * f)
662 ## rls: add_fractions_p_rls on: 123 = (b * d * e + b * c * f + a * d * f) / (b * d * f)
663 ### rls: add_fractions_p on: 123 = (b * d * e + b * c * f + a * d * f) / (b * d * f) *)
664 if term2str t' = "123 = (b * d * e + b * c * f + a * d * f) / (b * d * f)"
665 then () else error "level 4, rewrite_set_ *_p_rls: changed"
667 "-------- gcd_poly integration level 5: cancel_p & add_fraction_p within norm_Rational";
668 val SOME (t', asm) = rewrite_set_ thy false norm_Rational t;
669 if term2str t' = "123 = (a * d * f + b * c * f + b * d * e) / (b * d * f)"
670 then () else error "level 5, rewrite_set_ norm_Rational: changed"
672 "-------- reverse rewrite ----------------------------------------------------";
673 "-------- reverse rewrite ----------------------------------------------------";
674 "-------- reverse rewrite ----------------------------------------------------";
675 (** the term for which reverse rewriting is demonstrated **)
676 val t = str2term "(9 + -1 * x ^^^ 2) / (9 + 6 * x + x ^^^ 2)";
677 val Rrls {scr = Rfuns {init_state = ini, locate_rule = loc,
678 next_rule = nex, normal_form = nor, ...},...} = cancel_p;
680 (** normal_form produces the result in ONE step **)
681 val SOME (t',_) = nor t;
682 if term2str t' = "(3 + -1 * x) / (3 + x)" then ()
683 else error "rational.sml normal_form (9 - x ^^^ 2) / (9 - 6 * x + x ^^^ 2)";
685 (** initialize the interpreter state used by the 'me' **)
686 val (t, _, revsets, _) = ini t;
688 if length (hd revsets) = 11 then () else error "length of revset changed";
689 if (revsets |> nth 1 |> nth 1 |> id_of_thm) =
690 (@{thm realpow_twoI} |> string_of_thm |> thmID_of_derivation_name)
691 then () else error "first element of revset changed";
693 (revsets |> nth 1 |> nth 1 |> rule2str) = "Thm (\"realpow_twoI\",??.unknown)" andalso
694 (revsets |> nth 1 |> nth 2 |> rule2str) =
695 "Thm (\"sym_#power_Float ((3,0), (0,0)) __ ((2,0), (0,0))\",??.unknown)" andalso
696 (revsets |> nth 1 |> nth 3 |> rule2str) =
697 "Thm (\"sym_#mult_Float ((2,0), (0,0)) __ ((3,0), (0,0))\",??.unknown)" andalso
698 (revsets |> nth 1 |> nth 4 |> rule2str) =
699 "Thm (\"sym_#mult_Float ((~1,0), (0,0)) __ ((3,0), (0,0))\",??.unknown)" andalso
700 (revsets |> nth 1 |> nth 5 |> rule2str) =
701 "Thm (\"sym_#mult_Float ((3,0), (0,0)) __ ((3,0), (0,0))\",??.unknown)" andalso
702 (revsets |> nth 1 |> nth 6 |> rule2str) = "Rls_ (\"sym_order_mult_rls_\")" andalso
703 (revsets |> nth 1 |> nth 7 |> rule2str) =
704 "Thm (\"sym_mult_assoc\",Groups.semigroup_mult_class.mult_assoc)"
705 then () else error "first 7 elements in revset changed"
707 (** find the rule 'r' to apply to term 't' **)
708 (*/------- WN1309: since cancel_ (accepted "-" between monomials) has been replaced by cancel_p_
709 for Isabelle2013, we don't get a working revset, but non-termination:
711 val SOME (r as (Thm (str, thm))) = nex revsets t;
713 ((3 * 3 + -1 * x * x) / (3 * 3 + 2 * 3 * x + x * x),
714 Rls_ ("sym_order_mult_rls_"), ((3 * 3 + -1 * (x * x)) / (3 * 3 + 2 * (3 * x) + x * x), []))","
715 ((3 * 3 + -1 * (x * x)) / (3 * 3 + 2 * (3 * x) + x * x),
716 Thm ("sym_mult_assoc",""), ((3 * 3 + -1 * (x * x)) / (3 * 3 + 2 * 3 * x + x * x), []))","
717 ((3 * 3 + -1 * (x * x)) / (3 * 3 + 2 * 3 * x + x * x),
718 Thm ("sym_mult_assoc",""), ((3 * 3 + -1 * x * x) / (3 * 3 + 2 * 3 * x + x * x), []))","
719 ((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), []))","
722 Thm ("sym_#mult_2_3", "6 = 2 * 3")
723 ### Isabelle2009-2 for cancel_ (not cancel_p_):
724 if str = "sym_#power_Float ((3,0), (0,0)) __ ((2,0), (0,0))"
725 andalso string_of_thm thm =
726 (string_of_thm o make_thm o (cterm_of (@{theory "Isac"})))
727 (Trueprop $ (term_of o the o (parse thy)) "9 = 3 ^^^ 2") then ()
728 else error "rational.sml next_rule (9 - x ^^^ 2) / (9 - 6 * x + x ^^^ 2)";
729 \---------------------------------------------------------------------------------------/*)
731 (** check, if the rule 'r' applied by the user to 't' belongs to the ruleset;
732 if the rule is OK, the term resulting from applying the rule is returned,too;
733 there might be several rule applications inbetween,
734 which are listed after the head in reverse order **)
735 (*/-------------------------------------------- Isabelle2013: this gives "error id_of_thm";
736 we don't repair this, because interaction within "reverse rewriting" never worked properly:
738 val (r, (t, asm))::_ = loc revsets t r;
739 if term2str t = "(9 - x ^^^ 2) / (3 ^^^ 2 + 6 * x + x ^^^ 2)" andalso asm = []
740 then () else error "rational.sml locate_rule (9 - x ^^^ 2) / (9 - 6 * x + x ^^^ 2)";
742 (* find the next rule to apply *)
743 val SOME (r as (Thm (str, thm))) = nex revsets t;
744 if str = "sym_#power_Float ((3,0), (0,0)) __ ((2,0), (0,0))" andalso
745 string_of_thm thm = (string_of_thm o make_thm o (cterm_of (@{theory "Isac"})))
746 (Trueprop $ (term_of o the o (parse thy)) "9 = 3 ^^^ 2") then ()
747 else error "rational.sml next_rule (9 - x ^^^ 2) / (9 - 6 * x + x ^^^ 2)";
749 (*check the next rule*)
750 val (r, (t, asm)) :: _ = loc revsets t r;
751 if term2str t = "(3 ^^^ 2 - x ^^^ 2) / (3 ^^^ 2 + 6 * x + x ^^^ 2)" then ()
752 else error "rational.sml locate_rule (9 - x ^^^ 2) / (9 - 6 * x + x ^^^ 2) II";
754 (*find and check the next rules, rewrite*)
755 val SOME r = nex revsets t;
756 val (r,(t,asm))::_ = loc revsets t r;
757 if term2str t = "(3 ^^^ 2 - x ^^^ 2) / (3 ^^^ 2 + 2 * 3 * x + x ^^^ 2)" then ()
758 else error "rational.sml locate_rule II";
760 val SOME r = nex revsets t;
761 val (r,(t,asm))::_ = loc revsets t r;
762 if term2str t = "(3 - x) * (3 + x) / (3 ^^^ 2 + 2 * 3 * x + x ^^^ 2)" then ()
763 else error "rational.sml next_rule II";
765 val SOME r = nex revsets t;
766 val (r,(t,asm))::_ = loc revsets t r;
767 if term2str t = "(3 - x) * (3 + x) / ((3 + x) * (3 + x))" then ()
768 else error "rational.sml next_rule III";
770 val SOME r = nex revsets t;
771 val (r, (t, asm)) :: _ = loc revsets t r;
773 if ss = "(3 - x) / (3 + x)" andalso terms2str asm = "[\"3 + x ~= 0\"]" then ()
774 else error "rational.sml: new behav. in rev-set cancel";
775 \--------------------------------------------------------------------------------------/*)
777 "-------- 'reverse-ruleset' cancel_p -----------------------------------------";
778 "-------- 'reverse-ruleset' cancel_p -----------------------------------------";
779 "-------- 'reverse-ruleset' cancel_p -----------------------------------------";
780 (*WN130909: the example below shows, why "reverse rewriting" only worked for
783 (*the term for which reverse rewriting is demonstrated*)
784 val t = str2term "(9 + (-1)*x^^^2) / (9 + ((-6)*x + x^^^2))";
785 val Rrls {scr=Rfuns {init_state=ini,locate_rule=loc,
786 next_rule=nex,normal_form=nor,...},...} = cancel_p;
788 (*normal_form produces the result in ONE step*)
789 val SOME (t',_) = nor t;
790 term2str t' = "(3 + 1 * x) / (3 + -1 * x)";
792 (*initialize the interpreter state used by the 'me'*)
793 val SOME (t', asm) = cancel_p_ thy t;
794 term2str t' = "(3 + x) / (3 + -1 * x)" (*true*);
795 terms2str asm = "[\"3 + -1 * x ~= 0\"]" (*true*);
796 val (t,_,revsets,_) = ini t;
798 (* WN.10.10.02: dieser Fall terminiert nicht
799 (make_polynomial enth"alt zu viele rules)
800 WN060823 'init_state' requires rewriting on specified location in the term
801 print_depth 99; Rfuns; print_depth 3;
802 WN060831 cycling "sym_order_mult_rls_" "sym_mult_assoc"
803 as was with make_polynomial before ?!?*)
805 val SOME r = nex revsets t;
806 eq_Thm (r, Thm ("sym_#power_Float ((3,0), (0,0)) __ ((2,0), (0,0))",
807 mk_thm thy "9 = 3 ^^^ 2"));
808 (*WN060831 *** id_of_thm
809 Exception- ERROR raised ...
810 val (r,(t,asm))::_ = loc revsets t r;
813 val SOME r = nex revsets t;
814 val (r,(t,asm))::_ = loc revsets t r;
818 "-------- examples: rls norm_Rational ----------------------------------------";
819 "-------- examples: rls norm_Rational ----------------------------------------";
820 "-------- examples: rls norm_Rational ----------------------------------------";
821 val t = str2term "(3*x+5)/18 - x/2 - -(3*x - 2)/9 = 0";
822 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; term2str t';
823 if term2str t' = "1 / 18 = 0" then () else error "rational.sml 1";
825 val t = str2term "(17*x - 51)/9 - (-(13*x - 3)/6) + 11 - (9*x - 7)/4 = 0";
826 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; term2str t';
827 if term2str t' = "(237 + 65 * x) / 36 = 0" then ()
828 else error "rational.sml 2";
830 val t = str2term "(1/2 + (5*x)/2)^^^2 - ((13*x)/2 - 5/2)^^^2 - (6*x)^^^2 + 29";
831 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; term2str t';
832 if term2str t' = "23 + 35 * x + -72 * x ^^^ 2" then ()
833 else error "rational.sml 3";
835 (*trace_rewrite:=true;*)
836 val t = str2term "Not (6*x is_atom)";
837 val SOME (t',_) = rewrite_set_ thy false powers_erls t; term2str t';
839 val t = str2term "1 < 2";
840 val SOME (t',_) = rewrite_set_ thy false powers_erls t; term2str t';
843 val t = str2term "(6*x)^^^2";
844 val SOME (t',_) = rewrite_ thy dummy_ord powers_erls false
845 (num_str @{thm realpow_def_atom}) t;
846 if term2str t' = "6 * x * (6 * x) ^^^ (2 + -1)" then ()
847 else error "rational.sml powers_erls (6*x)^^^2";
849 val t = str2term "-1 * (-2 * (5 / 2 * (13 * x / 2)))";
850 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; term2str t';
851 if term2str t' = "65 * x / 2" then () else error "rational.sml 4";
853 val t = str2term "1 - ((13*x)/2 - 5/2)^^^2";
854 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; term2str t';
855 if term2str t' = "(-21 + 130 * x + -169 * x ^^^ 2) / 4" then ()
856 else error "rational.sml 5";
858 (*SRAM Schalk I, p.92 Nr. 609a*)
859 val t = str2term "2*(3 - x/5)/3 - 4*(1 - x/3) - x/3 - 2*(x/2 - 1/4)/27 +5/54";
860 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; term2str t';
861 if term2str t' = "(-255 + 112 * x) / 135" then ()
862 else error "rational.sml 6";
864 (*SRAM Schalk I, p.92 Nr. 610c*)
865 val t = str2term "((x- 1)/(x+1) + 1) / ((x- 1)/(x+1) - (x+1)/(x- 1)) - 2";
866 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; term2str t';
867 if term2str t' = "(-3 + -1 * x) / 2" then () else error "rational.sml 7";
869 (*SRAM Schalk I, p.92 Nr. 476a*)
870 val t = str2term "(x^^^2/(1 - x^^^2) + 1)/(x/(1 - x) + 1) * (1 + x)";
871 (*. a/b : c/d translated to a/b * d/c .*)
872 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; term2str t';
873 if term2str t' = "1" then () else error "rational.sml 8";
875 (*Schalk I, p.92 Nr. 472a*)
876 val t = str2term "((8*x^^^2 - 32*y^^^2)/(2*x + 4*y))/((4*x - 8*y)/(x + y))";
877 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; term2str t';
878 if term2str t' = "x + y" then () else error "rational.sml p.92 Nr. 472a";
880 (*Schalk I, p.70 Nr. 480b: SEE rational.sml --- nonterminating rls norm_Rational ---*)
882 (*WN130910 add_fractions_p exception Div raised + history:
883 ### WN.2.6.03 from rlang.sml 56a
884 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)";
885 val NONE = rewrite_set_ thy false add_fractions_p t;
887 THE ERROR ALREADY OCCURS IN THIS PART:
888 val t = str2term "(a + b * x) / (a + -1 * (b * x)) + (-1 * a + b * x) / (a + b * x)";
889 val NONE = add_fraction_p_ thy t;
891 SEE Test_Some.thy: section {* add_fractions_p downto exception Div raised ===
894 "-------- rational numerals --------------------------------------------------";
895 "-------- rational numerals --------------------------------------------------";
896 "-------- rational numerals --------------------------------------------------";
897 (*SRA Schalk I, p.40 Nr. 164b *)
898 val t = str2term "(47/6 - 76/9 + 13/4)/(35/12)";
899 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
901 if term2str t = "19 / 21" then ()
902 else error "rational.sml: diff.behav. in norm_Rational_mg 1";
904 (*SRA Schalk I, p.40 Nr. 166a *)
905 val t = str2term "((5/4)/(4+22/7) + 37/20)*(110/3 - 110/9 * 23/11)";
906 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
908 if term2str t = "45 / 2" then ()
909 else error "rational.sml: diff.behav. in norm_Rational_mg 2";
911 "-------- examples cancellation from: Mathematik 1 Schalk --------------------";
912 "-------- examples cancellation from: Mathematik 1 Schalk --------------------";
913 "-------- examples cancellation from: Mathematik 1 Schalk --------------------";
915 val t = str2term "((1 + 9*a^^^2) * (1 + 3*a)) / ((3*a + 9*a^^^2) * (1 + 3*a))";
916 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
917 if term2str t = "(1 + 9 * a ^^^ 2) / (3 * a + 9 * a ^^^ 2)"
918 then () else error "rational.sml: diff.behav. in norm_Rational_mg 3";
921 val t = str2term "(x^^^2 * (7*x + (-1)*y)) / (y^^^2 * (7*x + (-1)*y))";
922 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
923 if term2str t = "x ^^^ 2 / y ^^^ 2"
924 then () else error "rational.sml: diff.behav. in norm_Rational_mg 4";
926 (*SRC Schalk I, p.66 Nr. 379c *)
927 val t = str2term "(a - b)/(b - a)";
928 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
931 then () else error "rational.sml: diff.behav. in norm_Rational_mg 5";
933 (*SRC Schalk I, p.66 Nr. 380b *)
934 val t = str2term "15*(3*x + 3) * (4*x + 9) / (12*(2*x + 7) * (5*x + 5))";
935 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
936 if term2str t = "(27 + 12 * x) / (28 + 8 * x)"
937 then () else error "rational.sml: diff.behav. in norm_Rational_mg 6";
939 (*Schalk I, p.60 Nr. 215c: was not cancelled with Isabelle2002 *)
940 val t = str2term "(a + b)^^^4 * (x - y) / ((x - y)^^^3 * (a + b)^^^2)";
941 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
942 if term2str t = "(a ^^^ 2 + 2 * a * b + b ^^^ 2) / (x ^^^ 2 + -2 * x * y + y ^^^ 2)"
943 then () else error "rational.sml: diff.behav. in norm_Rational_mg 7";
945 (*SRC Schalk I, p.66 Nr. 381b *)
947 "(4*x^^^2 - 20*x + 25)/(2*x - 5)^^^3";
948 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
949 if term2str t = "1 / (-5 + 2 * x)"
950 then () else error "rational.sml: diff.behav. in norm_Rational_mg 9";
953 val t = str2term "((1 + 9*a^^^2) * (1 + 3*a)) / ((3*a + 9*a^^^2) * (1 + 3 * a))";
954 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
955 if term2str t = "(1 + 9 * a ^^^ 2) / (3 * a + 9 * a ^^^ 2)"
956 then () else error "rational.sml: diff.behav. in norm_Rational_mg 3";
959 val t = str2term "(x^^^2 * (7*x + (-1)*y)) / (y^^^2 * (7*x + (-1)*y))";
960 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
961 if term2str t = "x ^^^ 2 / y ^^^ 2"
962 then () else error "rational.sml: diff.behav. in norm_Rational_mg 4";
964 (*SRC Schalk I, p.66 Nr. 379c *)
965 val t = str2term "(a - b) / (b - a)";
966 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
968 then () else error "rational.sml: diff.behav. in norm_Rational_mg 5";
970 (*SRC Schalk I, p.66 Nr. 380b *)
971 val t = str2term "15*(3*x + 3) * (4*x + 9) / (12*(2*x + 7) * (5*x + 5))";
972 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
973 if term2str t = "(27 + 12 * x) / (28 + 8 * x)"
974 then () else error "rational.sml: diff.behav. in norm_Rational_mg 6";
976 (*Schalk I, p.60 Nr. 215c *)
977 val t = str2term "(a + b)^^^4 * (x - y) / ((x - y)^^^3 * (a + b)^^^2)";
978 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
979 if term2str t = "(a ^^^ 2 + 2 * a * b + b ^^^ 2) / (x ^^^ 2 + -2 * x * y + y ^^^ 2)"
980 then () else error "Schalk I, p.60 Nr. 215c: with Isabelle2002 cancellation incomplete, changed";
982 (* extreme example from somewhere *)
984 ("(a^^^4 * x + -1*a^^^4 * y + 4*a^^^3 * b * x + -4*a^^^3 * b * y + " ^
985 "6*a^^^2 * b^^^2 * x + -6*a^^^2 * b^^^2 * y + 4*a * b^^^3 * x + -4*a * b^^^3 * y + " ^
986 "b^^^4 * x + -1*b^^^4 * y) " ^
987 " / (a^^^2 * x^^^3 + -3*a^^^2 * x^^^2 * y + 3*a^^^2 * x * y^^^2 + -1*a^^^2 * y^^^3 + " ^
988 "2*a * b * x^^^3 + -6*a * b * x^^^2 * y + 6*a * b * x * y^^^2 + -2*a * b * y^^^3 + " ^
989 "b^^^2 * x^^^3 + -3*b^^^2 * x^^^2 * y + 3*b^^^2 * x * y^^^2 + -1*b ^^^ 2 * y ^^^ 3)")
990 val SOME (t, _) = rewrite_set_ thy false cancel_p t;
991 if term2str t = "(a ^^^ 2 + 2 * a * b + b ^^^ 2) / (x ^^^ 2 + -2 * x * y + y ^^^ 2)"
992 then () else error "with Isabelle2002: NONE -- now SOME changed";
994 (*Schalk I, p.66 Nr. 381a *)
995 (* ATTENTION: here the rls is very slow. In Isabelle2002 this required 2 min *)
996 val t = str2term "18*(a + b)^^^3 * (a - b)^^^2 / (72*(a - b)^^^3 * (a + b)^^^2)";
997 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
998 if term2str t = "(a + b) / (4 * a + -4 * b)"
999 then () else error "rational.sml: diff.behav. in norm_Rational_mg 8";
1001 (*SRC Schalk I, p.66 Nr. 381b *)
1002 val t = str2term "(4*x^^^2 - 20*x + 25) / (2*x - 5)^^^3";
1003 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1004 if term2str t = "1 / (-5 + 2 * x)"
1005 then () else error "rational.sml: diff.behav. in norm_Rational_mg 9";
1007 (*SRC Schalk I, p.66 Nr. 381c *)
1008 val t = str2term "(27*a^^^3 + 9*a^^^2+3*a+1) / (27*a^^^3 + 18*a^^^2+3*a)";
1009 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1010 if term2str t = "(1 + 9 * a ^^^ 2) / (3 * a + 9 * a ^^^ 2)"
1011 then () else error "rational.sml: diff.behav. in norm_Rational_mg 10";
1013 (*SRC Schalk I, p.66 Nr. 383a *)
1014 val t = str2term "(5*a^^^2 - 5*a*b) / (a - b)^^^2";
1015 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1016 if term2str t = "-5 * a / (-1 * a + b)"
1017 then () else error "rational.sml: diff.behav. in norm_Rational_mg 11";
1019 "----- NOT TERMINATING ?: worked before 0707xx";
1020 val t = str2term "(a^^^2 - 1)*(b + 1) / ((b^^^2 - 1)*(a+1))";
1021 (* WN130911 "exception Div raised" by
1022 cancel_p_ thy (str2term ("(-1 + -1 * b + a ^^^ 2 + a ^^^ 2 * b) /" ^
1023 "(-1 + -1 * a + b ^^^ 2 + a * b ^^^ 2)"))
1025 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1026 if term2str t = "(1 + -1 * a) / (1 + -1 * b)" then ()
1027 else error "rational.sml MG tests 3e";
1030 "-------- examples common denominator from: Mathematik 1 Schalk --------------";
1031 "-------- examples common denominator from: Mathematik 1 Schalk --------------";
1032 "-------- examples common denominator from: Mathematik 1 Schalk --------------";
1033 (*SRA Schalk I, p.67 Nr. 403a *)
1034 val t = str2term "4/x - 3/y - 1";
1035 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1036 if term2str t = "(-3 * x + 4 * y + -1 * x * y) / (x * y)"
1037 then () else error "rational.sml: diff.behav. in norm_Rational_mg 12";
1039 val t = str2term "(2*a+3*b)/(b*c) + (3*c+a)/(a*c) - (2*a^^^2+3*b*c)/(a*b*c)";
1040 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1041 if term2str t = "4 / c"
1042 then () else error "rational.sml: diff.behav. in norm_Rational_mg 13";
1044 (*SRA Schalk I, p.67 Nr. 410b *)
1045 val t = str2term "1/(x+1) + 1/(x+2) - 2/(x+3)";
1046 (* WN130911 non-termination due to non-termination of
1047 cancel_p_ thy (str2term "(5 + 3 * x) / (6 + 11 * x + 6 * x ^^^ 2 + x ^^^ 3)")
1049 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1050 if term2str t = "(5 + 3 * x) / (6 + 11 * x + 6 * x ^^^ 2 + x ^^^ 3)"
1051 then () else error "rational.sml: diff.behav. in norm_Rational_mg 14";
1054 (*SRA Schalk I, p.67 Nr. 413b *)
1055 val t = str2term "(1 + x)/(1 - x) - (1 - x)/(1 + x) + 2*x/(1 - x^^^2)";
1056 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1057 if term2str t = "6 * x / (1 + -1 * x ^^^ 2)"
1058 then () else error "rational.sml: diff.behav. in norm_Rational_mg 15";
1060 (*SRA Schalk I, p.68 Nr. 414a *)
1061 val t = str2term "(x + 2)/(x - 1) + (x - 3)/(x - 2) - (x + 1)/((x - 1)*(x - 2))";
1062 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1063 if term2str t ="(-2 + -5 * x + 2 * x ^^^ 2) / (2 + -3 * x + x ^^^ 2)"
1064 then () else error "rational.sml: diff.behav. in norm_Rational_mg 16";
1066 (*SRA Schalk I, p.68 Nr. 428b *)
1068 "1/(a - b)^^^2 + 1/(a + b)^^^2 - 2/(a^^^2 - b^^^2) - 4*(b^^^2 - 1)/(a^^^2 - b^^^2)^^^2";
1069 (* WN130911 non-termination due to non-termination of
1070 cancel_p_ thy (str2term "(4 + -4 * b ^^^ 2) / (a ^^^ 4 + -2 * (a ^^^ 2 * b ^^^ 2) + b ^^^ 4)")
1072 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1073 if term2str t = "4 / (a ^^^ 4 + -2 * a ^^^ 2 * b ^^^ 2 + b ^^^ 4)"
1074 then () else error "rational.sml: diff.behav. in norm_Rational_mg 18";
1077 (*SRA Schalk I, p.68 Nr. 430b *)
1079 "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";
1080 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1081 if term2str t = "a + 3 * b"
1082 then () else error "rational.sml: diff.behav. in norm_Rational_mg 19";
1084 (*SRA Schalk I, p.68 Nr. 432 *)
1086 ("(a^^^2 + a*b) / (a^^^2 - b^^^2) - (b^^^2 - a*b) / (b^^^2 - a^^^2) + " ^
1087 "a^^^2*(a - b) / (a^^^3 - a^^^2*b) - 2*a*(a^^^2 - b^^^2) / (a^^^3 - a*b^^^2) - " ^
1088 "2*b^^^2 / (a^^^2 - b^^^2)");
1089 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1092 then () else error "rational.sml: diff.behav. in norm_Rational_mg 20";
1095 val t = str2term "3*a / (a*b) + x/y";
1096 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1097 if term2str t = "(3 * y + b * x) / (b * y)"
1098 then () else error "rational.sml: diff.behav. in norm_Rational_mg 21";
1100 "-------- examples multiply and cancel from: Mathematik 1 Schalk -------------";
1101 "-------- examples multiply and cancel from: Mathematik 1 Schalk -------------";
1102 "-------- examples multiply and cancel from: Mathematik 1 Schalk -------------";
1103 (*------- SRM Schalk I, p.68 Nr. 436a *)
1104 val t = str2term "3*(x+y) / (15*(x - y)) * 25*(x - y)^^^2 / (18*(x + y)^^^2)";
1105 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1106 if term2str t = "(-5 * x + 5 * y) / (-18 * x + -18 * y)"
1107 then () else error "rational.sml: diff.behav. in norm_Rational_mg 22";
1109 (*------- SRM.test Schalk I, p.68 Nr. 436b *)
1110 val t = str2term "5*a*(a - b)^^^2*(a + b)^^^3/(7*b*(a - b)^^^3) * 7*b/(a + b)^^^3";
1111 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1112 if term2str t = "5 * a / (a + -1 * b)"
1113 then () else error "rational.sml: diff.behav. in norm_Rational_mg 23";
1115 (*------- Schalk I, p.68 Nr. 437a *)
1116 val t = str2term "(3*a - 4*b) / (4*c+3*e) * (3*a+4*b)/(9*a^^^2 - 16*b^^^2)";
1117 (* raises an exception for unclear reasons:
1118 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1120 ### rls: cancel_p on: (9 * a ^^^ 2 + -16 * b ^^^ 2) / (4 * c + 3 * e) /
1121 (9 * a ^^^ 2 + -16 * b ^^^ 2)
1122 exception Div raised
1126 ("(9 * a ^^^ 2 + -16 * b ^^^ 2) / (4 * c + 3 * e) /" ^
1127 "(9 * a ^^^ 2 + -16 * b ^^^ 2)");
1128 NONE = cancel_p_ thy t;
1130 if term2str t = "1 / (4 * c + 3 * e)" then ()
1131 else error "rational.sml: diff.behav. in norm_Rational_mg 24";
1134 "----- S.K. corrected non-termination 060904";
1135 val t = str2term "(3*a - 4*b) * (3*a+4*b)/((4*c+3*e)*(9*a^^^2 - 16*b^^^2))";
1136 val SOME (t, _) = rewrite_set_ thy false make_polynomial t;
1138 "(9 * a ^^^ 2 + -16 * b ^^^ 2) /\n(36 * a ^^^ 2 * c + 27 * a ^^^ 2 * e + -64 * b ^^^ 2 * c +\n -48 * b ^^^ 2 * e)"
1139 (*"(9 * a ^^^ 2 + -16 * b ^^^ 2) / (36 * a ^^^ 2 * c + 27 * a ^^^ 2 * e + -64 * b ^^^ 2 * c + -48 * b ^^^ 2 * e)"*)
1140 then () else error "rational.sml: S.K.8..corrected 060904-6";
1142 "----- S.K. corrected non-termination of cancel_p_";
1143 val t'' = str2term ("(9 * a ^^^ 2 + -16 * b ^^^ 2) /" ^
1144 "(36 * a^^^2 * c + (27 * a^^^2 * e + (-64 * b^^^2 * c + -48 * b^^^2 * e)))");
1145 val SOME (t',_) = rewrite_set_ thy false cancel_p t'';
1146 if term2str t' = "1 / (4 * c + 3 * e)"
1147 then () else error "rational.sml: diff.behav. in cancel_p S.K.8";
1149 (*------- Schalk I, p.68 Nr. 437b*)
1150 val t = str2term "(a + b)/(x^^^2 - y^^^2) * ((x - y)^^^2/(a^^^2 - b^^^2))";
1151 (*val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1153 #### rls: cancel_p on: (a * x ^^^ 2 + -2 * (a * (x * y)) + a * y ^^^ 2 + b * x ^^^ 2 +
1154 -2 * (b * (x * y)) +
1156 (a ^^^ 2 * x ^^^ 2 + -1 * (a ^^^ 2 * y ^^^ 2) + -1 * (b ^^^ 2 * x ^^^ 2) +
1158 exception Div raised
1161 (*------- SRM Schalk I, p.68 Nr. 438a *)
1162 val t = str2term "x*y / (x*y - y^^^2) * (x^^^2 - x*y)";
1163 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1164 if term2str t = "x ^^^ 2"
1165 then () else error "rational.sml: diff.behav. in norm_Rational_mg 24";
1167 (*------- SRM Schalk I, p.68 Nr. 439b *)
1168 val t = str2term "(4*x^^^2 + 4*x + 1) * ((x^^^2 - 2*x^^^3) / (4*x^^^2 + 2*x))";
1169 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1170 if term2str t = "(x + -4 * x ^^^ 3) / 2"
1171 then () else error "rational.sml: diff.behav. in norm_Rational_mg 25";
1173 (*------- SRM Schalk I, p.68 Nr. 440a *)
1174 val t = str2term "(x^^^2 - 2*x) / (x^^^2 - 3*x) * (x - 3)^^^2 / (x^^^2 - 4)";
1175 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1176 if term2str t = "(-3 + x) / (2 + x)"
1177 then () else error "rational.sml: diff.behav. in norm_Rational_mg 26";
1179 "----- Schalk I, p.68 Nr. 440b SK11 works since 0707xx";
1180 val t = str2term "(a^^^3 - 9*a) / (a^^^3*b - a*b^^^3) * (a^^^2*b + a*b^^^2) / (a+3)";
1181 (* WN130911 non-termination for unclear reasons:
1182 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1184 ... ENDS WITH THIS TRACE:
1186 ### rls: cancel_p on: (-9 * (a ^^^ 3 * b) + -9 * (a ^^^ 2 * b ^^^ 2) + a ^^^ 5 * b +
1187 a ^^^ 4 * b ^^^ 2) /
1188 (a ^^^ 3 * b + -1 * (a * b ^^^ 3)) /
1190 BUT THIS IS CORRECTLY RECOGNISED
1192 ("(-9 * (a^^^3 * b) + -9 * (a^^^2 * b^^^2) + a^^^5 * b + a^^^4 * b^^^2) /" ^
1193 "(a^^^3 * b + -1 * (a * b^^^3)) / (3 + (a::real))");
1195 NONE = cancel_p_ thy t;
1197 if term2str t = "(-3 * a + a ^^^ 2) / (a + -1 * b)" then ()
1198 else error "rational.sml: diff.behav. in norm_Rational 27";
1201 "----- SK12 works since 0707xx";
1202 val t = str2term "(a^^^3 - 9*a) * (a^^^2*b+a*b^^^2) / ((a^^^3*b - a*b^^^3) * (a+3))";
1203 (* WN130911 non-termination due to non-termination of
1204 cancel_p_ thy (str2term "(4 + -4 * b ^^^ 2) / (a ^^^ 4 + -2 * (a ^^^ 2 * b ^^^ 2) + b ^^^ 4)")
1206 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1207 if term2str t' = "(-3 * a + a ^^^ 2) / (a + -1 * b)" then ()
1208 else error "rational.sml: diff.behav. in norm_Rational 28";
1211 "-------- examples common denominator and multiplication from: Schalk --------";
1212 "-------- examples common denominator and multiplication from: Schalk --------";
1213 "-------- examples common denominator and multiplication from: Schalk --------";
1214 (*------- SRAM Schalk I, p.69 Nr. 441b *)
1215 val t = str2term "(4*a/3 + 3*b^^^2/a^^^3 + b/(4*a))*(4*b/(3*a))";
1216 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1217 if term2str t = "(36 * b ^^^ 3 + 3 * a ^^^ 2 * b ^^^ 2 + 16 * a ^^^ 4 * b) / (9 * a ^^^ 4)"
1218 then () else error "rational.sml: diff.behav. in norm_Rational_mg 28";
1220 (*------- SRAM Schalk I, p.69 Nr. 442b *)
1221 val t = str2term ("(15*a^^^2/x^^^3 - 5*b^^^4/x^^^2 + 25*c^^^2/x) * " ^
1222 "(x^^^3/(5*a*b^^^3*c^^^3)) + 1/c^^^3 * (b*x/a - 3*a/b^^^3)");
1223 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1224 if term2str t = "5 * x ^^^ 2 / (a * b ^^^ 3 * c)"
1225 then () else error "rational.sml: diff.behav. in norm_Rational_mg 29";
1227 (*------- SRAM Schalk I, p.69 Nr. 443b *)
1228 val t = str2term "(a/2 + b/3) * (b/3 - a/2)";
1229 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1230 if term2str t = "(-9 * a ^^^ 2 + 4 * b ^^^ 2) / 36"
1231 then () else error "rational.sml: diff.behav. in norm_Rational_mg 30";
1233 (*------- SRAM Schalk I, p.69 Nr. 445b *)
1234 val t = str2term "(a^^^2/9 + 2*a/(3*b) + 4/b^^^2)*(a/3 - 2/b) + 8/b^^^3";
1235 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1236 if term2str t = "a ^^^ 3 / 27"
1237 then () else error "rational.sml: diff.behav. in norm_Rational_mg 31";
1239 (*------- SRAM Schalk I, p.69 Nr. 446b *)
1240 val t = str2term "(x/(5*x + 4*y) - y/(5*x - 4*y) + 1)*(25*x^^^2 - 16*y^^^2)";
1241 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1242 if term2str t = "30 * x ^^^ 2 + -9 * x * y + -20 * y ^^^ 2"
1243 then () else error "rational.sml: diff.behav. in norm_Rational_mg 32";
1245 (*------- SRAM Schalk I, p.69 Nr. 449a *)(*Achtung: rechnet ca 8 Sekunden*)
1247 "(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)";
1248 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1249 if term2str t = "(-81 * x ^^^ 4 + 16 * x ^^^ 8 * y ^^^ 4) / (81 * y ^^^ 8)"
1250 then () else error "rational.sml: diff.behav. in norm_Rational_mg 33";
1252 (*------- SRAM Schalk I, p.69 Nr. 450a *)
1254 "(4*x/(3*y)+2*y/(3*x))^^^2 - (2*y/(3*x) - 2*x/y)*(2*y/(3*x)+2*x/y)";
1255 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1256 if term2str t = "(52 * x ^^^ 2 + 16 * y ^^^ 2) / (9 * y ^^^ 2)"
1257 then () else error "rational.sml: diff.behav. in norm_Rational_mg 34";
1259 (*------- SRAM Schalk I, p.69 Nr. 442b --- abgewandelt*)
1261 ("(15*a^^^4/(a*x^^^3) - 5*a*((b^^^4 - 5*c^^^2*x) / x^^^2)) * " ^
1262 "(x^^^3/(5*a*b^^^3*c^^^3)) + a/c^^^3 * (x*(b/a) - 3*b*(a/b^^^4))");
1263 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1264 if term2str t = "5 * x ^^^ 2 / (b ^^^ 3 * c)"
1265 then () else error "rational.sml: diff.behav. in norm_Rational_mg 53";
1268 "-------- examples double fractions from: Mathematik 1 Schalk ----------------";
1269 "-------- examples double fractions from: Mathematik 1 Schalk ----------------";
1270 "-------- examples double fractions from: Mathematik 1 Schalk ----------------";
1271 "----- SRD Schalk I, p.69 Nr. 454b";
1272 val t = str2term "((2 - x)/(2*a)) / (2*a/(x - 2))";
1273 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1274 if term2str t = "(-4 + 4 * x + -1 * x ^^^ 2) / (4 * a ^^^ 2)"
1275 then () else error "rational.sml: diff.behav. in norm_Rational_mg 35";
1277 "----- SRD Schalk I, p.69 Nr. 455a";
1278 val t = str2term "(a^^^2 + 1)/(a^^^2 - 1) / ((a+1)/(a - 1))";
1279 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1280 if term2str t = "(1 + a ^^^ 2) / (1 + 2 * a + a ^^^ 2)" then ()
1281 else error "rational.sml: diff.behav. in norm_Rational_mg 36";
1283 "----- Schalk I, p.69 Nr. 455b";
1284 val t = str2term "(x^^^2 - 4)/(y^^^2 - 9)/((2+x)/(3 - y))";
1285 (* WN130911 non-termination due to non-termination of
1286 cancel_p_ thy (str2term ("(-12 + 4 * y + 3 * x ^^^ 2 + -1 * (x ^^^ 2 * y)) /" ^
1287 "(-18 + -9 * x + 2 * y ^^^ 2 + x * y ^^^ 2)"))
1289 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1290 if term2str t = "(2 + -1 * x) / (3 + y)" then ()
1291 else error "rational.sml: diff.behav. in norm_Rational_mg 37";
1294 "----- SK060904-1a non-termination of cancel_p_ ?: worked before 0707xx";
1295 val t = str2term "(x^^^2 - 4)*(3 - y) / ((y^^^2 - 9)*(2+x))";
1296 (* WN130911 non-termination due to non-termination of
1297 cancel_p_ thy (str2term ("(-12 + 4 * y + 3 * x ^^^ 2 + -1 * (x ^^^ 2 * y)) /" ^
1298 "(-18 + -9 * x + 2 * y ^^^ 2 + x * y ^^^ 2)"))
1300 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1301 if term2str t = "(2 + -1 * x) / (3 + y)" then ()
1302 else error "rational.sml: diff.behav. in norm_Rational_mg 37b";
1305 "----- ?: worked before 0707xx";
1306 val t = str2term "(3 + -1 * y) / (-9 + y ^^^ 2)";
1307 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1308 if term2str t = "-1 / (3 + y)"
1309 then () else error "rational.sml: -1 / (3 + y) norm_Rational";
1311 "----- SRD Schalk I, p.69 Nr. 456b";
1312 val t = str2term "(b^^^3 - b^^^2) / (b^^^2+b) / (b^^^2 - 1)";
1313 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1314 if term2str t = "b / (1 + 2 * b + b ^^^ 2)"
1315 then () else error "rational.sml: diff.behav. in norm_Rational_mg 38";
1317 "----- SRD Schalk I, p.69 Nr. 457b";
1318 val t = str2term "(16*a^^^2 - 9*b^^^2)/(2*a+3*a*b) / ((4*a+3*b)/(4*a^^^2 - 9*a^^^2*b^^^2))";
1319 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1320 if term2str t = "8 * a ^^^ 2 + -6 * a * b + -12 * a ^^^ 2 * b + 9 * a * b ^^^ 2"
1321 then () else error "rational.sml: diff.behav. in norm_Rational_mg 39";
1323 "----- Schalk I, p.69 Nr. 458b works since 0707";
1324 val t = str2term "(2*a^^^2*x - a^^^2) / (a*x - b*x) / (b^^^2*(2*x - 1) / (x*(a - b)))";
1325 (*val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1327 ### rls: cancel_p on: (-1 * a ^^^ 2 + 2 * (a ^^^ 2 * x)) / (a * x + -1 * (b * x)) /
1328 ((-1 * b ^^^ 2 + 2 * (b ^^^ 2 * x)) / (a * x + -1 * (b * x)))
1329 exception Div raised
1333 ("(-1 * a ^^^ 2 + 2 * (a ^^^ 2 * x)) / (a * x + -1 * (b * x)) /" ^
1334 "((-1 * b ^^^ 2 + 2 * (b ^^^ 2 * x)) / (a * x + -1 * (b * x)))");
1335 NONE = cancel_p_ thy t;
1337 if term2str t = "a ^^^ 2 / b ^^^ 2" then ()
1338 else error "rational.sml: diff.behav. in norm_Rational_mg 39b";
1341 "----- SRD Schalk I, p.69 Nr. 459b";
1342 val t = str2term "(a^^^2 - b^^^2)/(a*b) / (4*(a+b)^^^2/a)";
1343 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1344 if term2str t = "(a + -1 * b) / (4 * a * b + 4 * b ^^^ 2)" then ()
1345 else error "rational.sml: diff.behav. in norm_Rational_mg 41";
1347 "----- Schalk I, p.69 Nr. 460b nonterm.SK";
1348 val t = str2term "(9*(x^^^2 - 8*x + 16) / (4*(y^^^2 - 2*y + 1))) / ((3*x - 12) / (16*y - 16))";
1349 (*val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1350 exception Div raised
1354 ("(144 + -72 * x + 9 * x ^^^ 2) / (4 + -8 * y + 4 * y ^^^ 2) /" ^
1355 "((-12 + 3 * x) / (-16 + 16 * y))");
1356 NONE = cancel_p_ thy t;
1358 if term2str t = !!!!!!!!!!!!!!!!!!!!!!!!!
1359 then () else error "rational.sml: diff.behav. in norm_Rational_mg 42";
1362 "----- some variant of the above; was non-terminating before";
1363 val t = str2term "9*(x^^^2 - 8*x+16)*(16*y - 16)/(4*(y^^^2 - 2*y+1)*(3*x - 12))";
1364 val SOME (t , _) = rewrite_set_ thy false norm_Rational t;
1365 if term2str t = "(48 + -12 * x) / (1 + -1 * y)"
1366 then () else error "some variant of the above; was non-terminating before";
1368 "----- SRD Schalk I, p.70 Nr. 472a";
1369 val t = str2term ("((8*x^^^2 - 32*y^^^2) / (2*x + 4*y)) / ((4*x - 8*y) / (x + y))");
1370 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1371 if term2str t = "x + y"
1372 then () else error "rational.sml: diff.behav. in norm_Rational_mg 43";
1374 "----- Schalk I, p.70 Nr. 478b ----- Rechenzeit: 5 sec";
1375 val t = str2term ("(a - (a*b + b^^^2)/(a+b))/(b+(a - b)/(1+(a+b)/(a - b))) / " ^
1376 "((a - a^^^2/(a+b))/(a+(a*b)/(a - b)))");
1377 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1378 if term2str t = "(2 * a ^^^ 3 + 2 * a ^^^ 2 * b) / (a ^^^ 2 * b + b ^^^ 3)"
1379 then () else error "rational.sml: diff.behav. in norm_Rational_mg 51";
1381 (*SRD Schalk I, p.69 Nr. 461a *)
1382 val t = str2term "(2/(x+3) + 2/(x - 3)) / (8*x/(x^^^2 - 9))";
1383 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1384 if term2str t = "1 / 2"
1385 then () else error "rational.sml: diff.behav. in norm_Rational_mg 44";
1387 (*SRD Schalk I, p.69 Nr. 464b *)
1388 val t = str2term "(a - a/(a - 2)) / (a + a/(a - 2))";
1389 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1390 if term2str t = "(-3 + a) / (-1 + a)"
1391 then () else error "rational.sml: diff.behav. in norm_Rational_mg 45";
1393 (*SRD Schalk I, p.69 Nr. 465b *)
1394 val t = str2term "((x+3*y)/9 + (4*y^^^2 - 9*z^^^2)/(16*x)) / (x/9 + y/6 + z/4)";
1395 (* WN130911 non-termination due to non-termination of
1396 cancel_p_ thy (str2term
1397 ("("(576 * x ^^^ 2 + 1728 * (x * y) + 1296 * y ^^^ 2 + -2916 * z ^^^ 2) /" ^
1398 "(576 * x ^^^ 2 + 864 * (x * y) + 1296 * (x * z))"))
1400 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1401 if term2str t = "(4 * x + 6 * y + -9 * z) / (4 * x)"
1402 then () else error "rational.sml: diff.behav. in norm_Rational_mg 46";
1405 (*SRD Schalk I, p.69 Nr. 466b *)
1406 val t = str2term "((1 - 7*(x - 2)/(x^^^2 - 4)) / (6/(x+2))) / (3/(x+5)+30/(x^^^2 - 25))";
1407 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1408 if term2str t = "(25 + -10 * x + x ^^^ 2) / 18"
1409 then () else error "rational.sml: diff.behav. in norm_Rational_mg 47";
1411 (*SRD Schalk I, p.70 Nr. 469 *)
1412 val t = str2term ("3*b^^^2 / (4*a^^^2 - 8*a*b + 4*b^^^2) / " ^
1413 "(a / (a^^^2*b - b^^^3) + (a - b) / (4*a*b^^^2 + 4*b^^^3) - 1 / (4*b^^^2))");
1414 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1415 if term2str t = "-3 * b ^^^ 3 / (-2 * a + 2 * b)"
1416 then () else error "rational.sml: diff.behav. in norm_Rational_mg 48";
1418 "-------- me Schalk I No.186 -------------------------------------------------";
1419 "-------- me Schalk I No.186 -------------------------------------------------";
1420 "-------- me Schalk I No.186 -------------------------------------------------";
1421 val fmz = ["Term ((14 * x * y) / ( x * y ))", "normalform N"];
1423 ("Rational",["rational","simplification"],
1424 ["simplification","of_rationals"]);
1425 val p = e_pos'; val c = [];
1426 val (p,_,f,nxt,_,pt) = CalcTreeTEST [(fmz, (dI',pI',mI'))];
1427 val (p,_,f,nxt,_,pt) = me nxt p c pt;
1428 val (p,_,f,nxt,_,pt) = me nxt p c pt;
1429 val (p,_,f,nxt,_,pt) = me nxt p c pt;
1430 val (p,_,f,nxt,_,pt) = me nxt p c pt;
1431 val (p,_,f,nxt,_,pt) = me nxt p c pt;
1432 val (p,_,f,nxt,_,pt) = me nxt p c pt;
1433 val (p,_,f,nxt,_,pt) = me nxt p c pt;
1434 val (p,_,f,nxt,_,pt) = me nxt p c pt; f2str f;
1435 val (p,_,f,nxt,_,pt) = me nxt p c pt; f2str f;
1436 val (p,_,f,nxt,_,pt) = me nxt p c pt; f2str f;(*++ for explicit script*)
1437 val (p,_,f,nxt,_,pt) = me nxt p c pt; f2str f;(*++ for explicit script*)
1438 case (f2str f, nxt) of
1439 ("14", ("End_Proof'", _)) => ()
1440 | _ => error "rational.sml diff.behav. in me Schalk I No.186";
1442 "-------- interSteps ..Simp_Rat_Double_No-1.xml ------------------------------";
1443 "-------- interSteps ..Simp_Rat_Double_No-1.xml ------------------------------";
1444 "-------- interSteps ..Simp_Rat_Double_No-1.xml ------------------------------";
1446 CalcTree [(["Term (((2 - x)/(2*a)) / (2*a/(x - 2)))", "normalform N"],
1447 ("Rational", ["rational", "simplification"], ["simplification", "of_rationals"]))];
1450 autoCalculate 1 CompleteCalc;
1451 val ((pt, p), _) = get_calc 1;
1455 (([], Frm), Simplify ((2 - x) / (2 * a) / (2 * a / (x - 2)))),
1456 (([1], Frm), (2 - x) / (2 * a) / (2 * a / (x - 2))),
1457 (([1], Res), (2 + -1 * x) / (2 * a) / (2 * a / (x + -1 * 2))),
1458 (([2], Res), (2 + -1 * x) / (2 * a) / (2 * a / (-2 + x))),
1459 (([3], Res), (2 + -1 * x) * (-2 + x) / (2 * a * (2 * a))),
1460 (([4], Res), (-4 + 4 * x + -1 * x ^^^ 2) / (4 * a ^^^ 2)),
1461 (([], Res), (-4 + 4 * x + -1 * x ^^^ 2) / (4 * a ^^^ 2))]
1463 interSteps 1 ([1], Res);
1464 val ((pt, p), _) = get_calc 1;
1467 (([], Frm), Simplify ((2 - x) / (2 * a) / (2 * a / (x - 2)))),
1468 (([1], Frm), (2 - x) / (2 * a) / (2 * a / (x - 2))),
1469 (([1,1], Frm), (2 - x) / (2 * a) / (2 * a / (x - 2))),
1470 (([1,1], Res), (2 - x) / (2 * a) / (2 * a / (x + -1 * 2))),
1471 (([1,2], Res), (2 + -1 * x) / (2 * a) / (2 * a / (x + -1 * 2))),
1472 (([1], Res), (2 + -1 * x) / (2 * a) / (2 * a / (x + -1 * 2))),
1473 (([2], Res), (2 + -1 * x) / (2 * a) / (2 * a / (-2 + x))),
1474 (([3], Res), (2 + -1 * x) * (-2 + x) / (2 * a * (2 * a))),
1475 (([4], Res), (-4 + 4 * x + -1 * x ^^^ 2) / (4 * a ^^^ 2)),
1476 (([], Res), (-4 + 4 * x + -1 * x ^^^ 2) / (4 * a ^^^ 2))]
1478 val (t, asm) = get_obj g_result pt [1, 1];
1479 if term2str t = "(2 - x) / (2 * a) / (2 * a / (x + -1 * 2))" andalso terms2str asm = "[]"
1480 then () else error "2nd interSteps ..Simp_Rat_Double_No-1 changed on [1, 1]";
1481 val (t, asm) = get_obj g_result pt [1, 2];
1482 if term2str t = "(2 + -1 * x) / (2 * a) / (2 * a / (x + -1 * 2))" andalso terms2str asm = "[]"
1483 then () else error "3rd interSteps ..Simp_Rat_Double_No-1 changed on [1, 2]";
1486 "-------- interSteps ..Simp_Rat_Cancel_No-1.xml ------------------------------";
1487 "-------- interSteps ..Simp_Rat_Cancel_No-1.xml ------------------------------";
1488 "-------- interSteps ..Simp_Rat_Cancel_No-1.xml ------------------------------";
1490 CalcTree [(["Term ((a^2 + -1*b^2) / (a^2 + -2*a*b + b^2))", "normalform N"],
1491 ("Rational", ["rational", "simplification"], ["simplification", "of_rationals"]))];
1494 autoCalculate 1 CompleteCalc;
1495 val ((pt, p), _) = get_calc 1;
1498 (([], Frm), Simplify ((a ^^^ 2 + -1 * b ^^^ 2) / (a ^^^ 2 + -2 * a * b + b ^^^ 2))),
1499 (([1], Frm), (a ^^^ 2 + -1 * b ^^^ 2) / (a ^^^ 2 + -2 * a * b + b ^^^ 2)),
1500 (([1], Res), (a ^^^ 2 + -1 * b ^^^ 2) / (a ^^^ 2 + -2 * (a * b) + b ^^^ 2)),
1501 (([2], Res), (a + b) / (a + -1 * b)),
1502 (([], Res), (a + b) / (a + -1 * b))]
1504 interSteps 1 ([2], Res);
1505 val ((pt, p), _) = get_calc 1;
1508 (([], Frm), Simplify ((a ^^^ 2 + -1 * b ^^^ 2) / (a ^^^ 2 + -2 * a * b + b ^^^ 2))),
1509 (([1], Frm), (a ^^^ 2 + -1 * b ^^^ 2) / (a ^^^ 2 + -2 * a * b + b ^^^ 2)),
1510 (([1], Res), (a ^^^ 2 + -1 * b ^^^ 2) / (a ^^^ 2 + -2 * (a * b) + b ^^^ 2)),
1511 (([2,1], Frm), (a ^^^ 2 + -1 * b ^^^ 2) / (a ^^^ 2 + -2 * (a * b) + b ^^^ 2)),
1512 (([2,1], Res), (a + b) / (a + -1 * b)),
1513 (([2], Res), (a + b) / (a + -1 * b)),
1514 (([], Res), (a + b) / (a + -1 * b))]
1516 interSteps 1 ([2,1],Res);
1517 val ((pt, p), _) = get_calc 1;
1520 (([], Frm), Simplify ((a ^^^ 2 + -1 * b ^^^ 2) / (a ^^^ 2 + -2 * a * b + b ^^^ 2))),
1521 (([1], Frm), (a ^^^ 2 + -1 * b ^^^ 2) / (a ^^^ 2 + -2 * a * b + b ^^^ 2)),
1522 (([1], Res), (a ^^^ 2 + -1 * b ^^^ 2) / (a ^^^ 2 + -2 * (a * b) + b ^^^ 2)),
1523 (([2,1], Frm), (a ^^^ 2 + -1 * b ^^^ 2) / (a ^^^ 2 + -2 * (a * b) + b ^^^ 2)),
1524 (([2,1,1], Frm), (a ^^^ 2 + -1 * b ^^^ 2) / (a ^^^ 2 + -2 * (a * b) + b ^^^ 2)),
1525 (([2,1,1], Res), (a ^^^ 2 + -1 * (a * b) + a * b + -1 * b ^^^ 2) /
1526 (a ^^^ 2 + -2 * (a * b) + 1 * b ^^^ 2)),
1527 (([2,1,2], Res), (a ^^^ 2 + -1 * (a * b) + a * b + -1 * b ^^^ 2) /
1528 (a ^^^ 2 + -2 * (a * b) + -1 ^^^ 2 * b ^^^ 2)),
1529 (([2,1,3], Res), (a ^^^ 2 + -1 * (a * b) + a * b + -1 * b ^^^ 2) /
1530 (a ^^^ 2 + -2 * (a * b) + (-1 * b) ^^^ 2)),
1531 (([2,1,4], Res), (a * a + -1 * (a * b) + a * b + -1 * b ^^^ 2) /
1532 (a ^^^ 2 + -2 * (a * b) + (-1 * b) ^^^ 2)),
1533 (([2,1,5], Res), (a * a + -1 * (a * b) + a * b + -1 * (b * b)) /
1534 (a ^^^ 2 + -2 * (a * b) + (-1 * b) ^^^ 2)),
1535 (([2,1,6], Res), (a * a + -1 * (a * b) + a * b + -1 * (b * b)) /
1536 (a ^^^ 2 + -1 * (2 * (a * b)) + (-1 * b) ^^^ 2)),
1537 (([2,1,7], Res), (a * a + a * (-1 * b) + (b * a + b * (-1 * b))) /
1538 (a ^^^ 2 + 2 * (a * (-1 * b)) + (-1 * b) ^^^ 2)),
1539 (([2,1,8], Res), (a * a + a * (-1 * b) + (b * a + b * (-1 * b))) /
1540 (a ^^^ 2 + 2 * a * (-1 * b) + (-1 * b) ^^^ 2)),
1541 (([2,1,9], Res), (a * (a + -1 * b) + (b * a + b * (-1 * b))) /
1542 (a ^^^ 2 + 2 * a * (-1 * b) + (-1 * b) ^^^ 2)),
1543 (([2,1,10], Res), (a * (a + -1 * b) + b * (a + -1 * b)) /
1544 (a ^^^ 2 + 2 * a * (-1 * b) + (-1 * b) ^^^ 2)),
1545 (([2,1,11], Res), (a + b) * (a + -1 * b) / (a ^^^ 2 + 2 * a * (-1 * b) + (-1 * b) ^^^ 2)),
1546 (([2,1,12], Res), (a + b) * (a + -1 * b) / ((a + -1 * b) * (a + -1 * b))),
1547 (([2,1,13], Res), (a + b) / (a + -1 * b)),
1548 (([2,1], Res), (a + b) / (a + -1 * b)),
1549 (([2], Res), (a + b) / (a + -1 * b)),
1550 (([], Res), (a + b) / (a + -1 * b))]
1552 val newnds = children (get_nd pt [2,1]) (*see "fun detailrls"*);
1553 if length newnds = 13 then () else error "rational.sml: interSteps cancel_p rev_rew_p";
1555 val p = ([2,1,9],Res);
1557 val (_, tac, _) = pt_extract (pt, p);
1558 case tac of SOME (Rewrite ("sym_distrib_left", _)) => ()
1559 | _ => error "rational.sml: getTactic, sym_real_plus_binom_times1";
1562 "-------- investigate rulesets for cancel_p ----------------------------------";
1563 "-------- investigate rulesets for cancel_p ----------------------------------";
1564 "-------- investigate rulesets for cancel_p ----------------------------------";
1565 val thy = @{theory "Rational"};
1566 val t = str2term "(a^^^2 + -1*b^^^2) / (a^^^2 + -2*a*b + b^^^2)";
1567 val tt = str2term "(1 * a + 1 * b) * (1 * a + -1 * b)"(*numerator only*);
1569 "----- with rewrite_set_";
1570 val SOME (tt',asm) = rewrite_set_ thy false make_polynomial tt;
1571 if term2str tt'= "a ^^^ 2 + -1 * b ^^^ 2" then () else error "rls chancel_p 1";
1572 val tt = str2term "((1 * a + -1 * b) * (1 * a + -1 * b))"(*denominator only*);
1573 val SOME (tt',asm) = rewrite_set_ thy false make_polynomial tt;
1574 if term2str tt' = "a ^^^ 2 + -2 * a * b + b ^^^ 2" then () else error "rls chancel_p 2";
1576 "----- with make_deriv; WN1130912 not investigated further, will be discontinued";
1577 val SOME (tt, _) = factout_p_ thy t;
1578 if term2str tt = "(a + b) * (a + -1 * b) / ((a + -1 * b) * (a + -1 * b))"
1579 then () else error "rls chancel_p 3";
1580 term2str tt = "(1 * a + 1 * b) * (1 * a + -1 * b) / ((1 * a + -1 * b) * (1 * a + -1 * b))";
1582 "--- with simpler ruleset";
1583 val {rules, rew_ord= (_, ro), ...} = rep_rls (assoc_rls "rev_rew_p");
1584 val der = make_deriv thy Atools_erls rules ro NONE tt;
1585 if length der = 12 then () else error "WN1130912 rls chancel_p 4";
1586 print_depth 99; writeln (deriv2str der); print_depth 3;
1588 print_depth 99; map (term2str o #1) der; print_depth 3;
1589 "...,(-1 * b ^^^ 2 + a ^^^ 2) / (-2 * (a * b) + a ^^^ 2 + (-1 * b) ^^^ 2) ]";
1590 print_depth 99; map (rule2str o #2) der; print_depth 3;
1591 print_depth 99; map (term2str o #1 o #3) der; print_depth 3;
1593 val der = make_deriv thy Atools_erls rules ro NONE
1594 (str2term "(1 * a + 1 * b) * (1 * a + -1 * b)");
1595 print_depth 99; writeln (deriv2str der); print_depth 3;
1597 val {rules, rew_ord=(_,ro),...} = rep_rls (assoc_rls "rev_rew_p");
1598 val der = make_deriv thy Atools_erls rules ro NONE
1599 (str2term "(1 * a + -1 * b) * (1 * a + -1 * b)");
1600 print_depth 99; writeln (deriv2str der); print_depth 3;
1601 print_depth 99; map (term2str o #1) der; print_depth 3;
1602 (*WN060829 ...postponed*)
1605 "-------- fun eval_get_denominator -------------------------------------------";
1606 "-------- fun eval_get_denominator -------------------------------------------";
1607 "-------- fun eval_get_denominator -------------------------------------------";
1608 val thy = @{theory Isac};
1609 val t = term_of (the (parse thy "get_denominator ((a +x)/b)"));
1610 val SOME (_, t') = eval_get_denominator "" 0 t thy;
1611 if term2str t' = "get_denominator ((a + x) / b) = b"
1612 then () else error "get_denominator ((a + x) / b) = b"
1615 "-------- several errpats in complicated term --------------------------------";
1616 "-------- several errpats in complicated term --------------------------------";
1617 "-------- several errpats in complicated term --------------------------------";
1618 (*WN12xxxx TODO: instead of Gabriella's example here (27.Jul.12) find a simpler one
1619 WN130912: kept this test, although not clear what for*)
1621 CalcTree [(["Term ((5*b + 25)/(a^2 - b^2) * (a - b)/(5*b))", "normalform N"],
1622 ("Rational", ["rational", "simplification"], ["simplification", "of_rationals"]))];
1625 autoCalculate 1 CompleteCalc;
1626 val ((pt, p), _) = get_calc 1;
1629 (([], Frm), Simplify ((5 * b + 25) / (a ^^^ 2 - b ^^^ 2) * (a - b) / (5 * b))),
1630 (([1], Frm), (5 * b + 25) / (a ^^^ 2 - b ^^^ 2) * (a - b) / (5 * b)),
1631 (([1], Res), (5 * b + 25) / (a ^^^ 2 + -1 * b ^^^ 2) * (a + -1 * b) / (5 * b)),
1632 (([2], Res), (5 * b + 25) * (a + -1 * b) / (a ^^^ 2 + -1 * b ^^^ 2) / (5 * b)),
1633 (([3], Res), (25 * a + -25 * b + 5 * (a * b) + -5 * b ^^^ 2) / (a ^^^ 2 + -1 * b ^^^ 2) /
1635 (([4], Res), (25 + 5 * b) / (a + b) / (5 * b)),
1636 (([5], Res), (25 + 5 * b) / ((a + b) * (5 * b))),
1637 (([6], Res), (25 + 5 * b) / (5 * (a * b) + 5 * b ^^^ 2)),
1638 (([7], Res), (5 + b) / (a * b + b ^^^ 2)),
1639 (([], Res), (5 + b) / (a * b + b ^^^ 2))] *)
1642 "-------- WN1309xx non-terminating rls norm_Rational -------------------------";
1643 "-------- WN1309xx non-terminating rls norm_Rational -------------------------";
1644 "-------- WN1309xx non-terminating rls norm_Rational -------------------------";
1645 (*------- Schalk I, p.70 Nr. 480b; a/b : c/d translated to a/b * d/c*)
1647 ("((12*x*y / (9*x^^^2 - y^^^2)) / (1 / (3*x - y)^^^2 - 1 / (3*x + y)^^^2)) * " ^
1648 "((1/(x - 5*y)^^^2 - 1/(x + 5*y)^^^2) / (20*x*y / (x^^^2 - 25*y^^^2)))");
1650 (*1st factor separately simplified *)
1651 val t = str2term "((12*x*y / (9*x^^^2 - y^^^2)) / (1 / (3*x - y)^^^2 - 1 / (3*x + y)^^^2))";
1652 val SOME (t', _) = rewrite_set_ thy false norm_Rational t;
1653 if term2str t' = "(-9 * x ^^^ 2 + y ^^^ 2) / -1" then () else error "Nr. 480b lhs changed";
1654 (*2nd factor separately simplified *)
1655 val t = str2term "((1/(x - 5*y)^^^2 - 1/(x + 5*y)^^^2) / (20*x*y / (x^^^2 - 25*y^^^2)))";
1656 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; term2str t';
1657 if term2str t' = "-1 / (-1 * x ^^^ 2 + 25 * y ^^^ 2)" then () else error "Nr. 480b rhs changed";
1659 "-------- Schalk I, p.70 Nr. 477a: terms are exploding ?!?";
1660 val t = str2term ("b*y/(b - 2*y)/((b^^^2 - y^^^2)/(b+2*y)) /" ^
1661 "(b^^^2*y + b*y^^^2) * (a+x)^^^2 / ((b^^^2 - 4*y^^^2) * (a+2*x)^^^2)");
1662 (*val SOME (t',_) = rewrite_set_ thy false norm_Rational t;
1664 ### rls: cancel_p on: (a ^^^ 2 * (b * y) + 2 * (a * (b * (x * y))) + b * (x ^^^ 2 * y)) /
1666 ((b ^^^ 2 + -1 * y ^^^ 2) / (b + 2 * y)) /
1667 (b ^^^ 2 * y + b * y ^^^ 2) /
1668 (a ^^^ 2 * b ^^^ 2 + -4 * (a ^^^ 2 * y ^^^ 2) + 4 * (a * (b ^^^ 2 * x)) +
1669 -16 * (a * (x * y ^^^ 2)) +
1670 4 * (b ^^^ 2 * x ^^^ 2) +
1671 -16 * (x ^^^ 2 * y ^^^ 2))
1672 exception Div raised
1676 ("(a ^^^ 2 * (b * y) + 2 * (a * (b * (x * y))) + b * (x ^^^ 2 * y)) /" ^
1678 "((b ^^^ 2 + -1 * y ^^^ 2) / (b + 2 * y)) /" ^
1679 "(b ^^^ 2 * y + b * y ^^^ 2) /" ^
1680 "(a ^^^ 2 * b ^^^ 2 + -4 * (a ^^^ 2 * y ^^^ 2) + 4 * (a * (b ^^^ 2 * x)) +" ^
1681 "-16 * (a * (x * y ^^^ 2)) +" ^
1682 "4 * (b ^^^ 2 * x ^^^ 2) +" ^
1683 "-16 * (x ^^^ 2 * y ^^^ 2))");
1684 NONE = cancel_p_ thy t;
1687 (*------- Schalk I, p.70 Nr. 476b in 2003 this worked using 10 sec. *)
1689 ("((a^^^2 - b^^^2)/(2*a*b) + 2*a*b/(a^^^2 - b^^^2)) / ((a^^^2 + b^^^2)/(2*a*b) + 1) / " ^
1690 "((a^^^2 + b^^^2)^^^2 / (a + b)^^^2)");
1692 trace_rewrite := true;
1693 rewrite_set_ thy false norm_Rational t;
1695 #### rls: cancel_p on: (2 * (a ^^^ 7 * b) + 4 * (a ^^^ 6 * b ^^^ 2) + 6 * (a ^^^ 5 * b ^^^ 3) +
1696 8 * (a ^^^ 4 * b ^^^ 4) +
1697 6 * (a ^^^ 3 * b ^^^ 5) +
1698 4 * (a ^^^ 2 * b ^^^ 6) +
1699 2 * (a * b ^^^ 7)) /
1700 (2 * (a ^^^ 9 * b) + 4 * (a ^^^ 8 * b ^^^ 2) +
1701 2 * (2 * (a ^^^ 7 * b ^^^ 3)) +
1702 4 * (a ^^^ 6 * b ^^^ 4) +
1703 -4 * (a ^^^ 4 * b ^^^ 6) +
1704 -4 * (a ^^^ 3 * b ^^^ 7) +
1705 -4 * (a ^^^ 2 * b ^^^ 8) +
1708 if term2str t = "1 / (a ^^^ 2 + -1 * b ^^^ 2)" then ()
1709 else error "rational.sml: diff.behav. in norm_Rational_mg 49";
1712 "-------- Schalk I, p.70 Nr. 480a: terms are exploding ?!?";
1713 val t = str2term ("(1/x + 1/y + 1/z) / (1/x - 1/y - 1/z) / " ^
1714 "(2*x^^^2 / (x^^^2 - z^^^2) / (x / (x + z) + x / (x - z)))");
1716 trace_rewrite := true;
1717 rewrite_set_ thy false norm_Rational t;
1719 #### rls: cancel_p on: (2 * (x ^^^ 6 * (y ^^^ 2 * z)) + 2 * (x ^^^ 6 * (y * z ^^^ 2)) +
1720 2 * (x ^^^ 5 * (y ^^^ 2 * z ^^^ 2)) +
1721 -2 * (x ^^^ 4 * (y ^^^ 2 * z ^^^ 3)) +
1722 -2 * (x ^^^ 4 * (y * z ^^^ 4)) +
1723 -2 * (x ^^^ 3 * (y ^^^ 2 * z ^^^ 4))) /
1724 (-2 * (x ^^^ 6 * (y ^^^ 2 * z)) + -2 * (x ^^^ 6 * (y * z ^^^ 2)) +
1725 2 * (x ^^^ 5 * (y ^^^ 2 * z ^^^ 2)) +
1726 2 * (x ^^^ 4 * (y ^^^ 2 * z ^^^ 3)) +
1727 2 * (x ^^^ 4 * (y * z ^^^ 4)) +
1728 -2 * (x ^^^ 3 * (y ^^^ 2 * z ^^^ 4)))
1731 "-------- Schalk I, p.60 Nr. 215d: terms are exploding, internal loop does not terminate";
1732 val t = str2term "(a-b)^^^3 * (x+y)^^^4 / ((x+y)^^^2 * (a-b)^^^5)";
1733 (* Kein Wunder, denn Z???ler und Nenner extra als Polynom dargestellt ergibt:
1735 val t = str2term "(a-b)^^^3 * (x+y)^^^4";
1736 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1738 "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";
1739 val t = str2term "((x+y)^^^2 * (a-b)^^^5)";
1740 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1742 "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";
1744 anscheinend macht dem Rechner das Krzen diese Bruches keinen Spass mehr ...*)
1746 "-------- Schalk I, p.70 Nr. 480b: terms are exploding, trace_rewrite stops at";
1747 val t = str2term ("((12*x*y/(9*x^^^2 - y^^^2))/" ^
1748 "(1/(3*x - y)^^^2 - 1/(3*x + y)^^^2)) *" ^
1749 "(1/(x - 5*y)^^^2 - 1/(x + 5*y)^^^2)/" ^
1750 "(20*x*y/(x^^^2 - 25*y^^^2))");
1751 (*val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1753 #### rls: cancel_p on: (19440 * (x ^^^ 8 * y ^^^ 2) + -490320 * (x ^^^ 6 * y ^^^ 4) +
1754 108240 * (x ^^^ 4 * y ^^^ 6) +
1755 -6000 * (x ^^^ 2 * y ^^^ 8)) /
1756 (2160 * (x ^^^ 8 * y ^^^ 2) + -108240 * (x ^^^ 6 * y ^^^ 4) +
1757 1362000 * (x ^^^ 4 * y ^^^ 6) +
1758 -150000 * (x ^^^ 2 * y ^^^ 8))