1 (* Title: tests for rationals
3 Use is subject to license terms.
6 "-----------------------------------------------------------------------------";
7 "-----------------------------------------------------------------------------";
8 "table of contents -----------------------------------------------------------";
9 "-----------------------------------------------------------------------------";
10 "-------- integration lev.1 fun factout_p_ -----------------------------------";
11 "-------- integration lev.1 fun cancel_p_ ------------------------------------";
12 "-------- integration lev.1 fun common_nominator_p_ --------------------------";
13 "-------- integration lev.1 fun add_fraction_p_ ------------------------------";
14 "Rfuns-------- and app_rev ...traced down from rewrite_set_ until prepats ---------";
15 "Rfuns-------- fun rewrite_set_ cancel_p downto fun gcd_poly ----------------------";
16 "-------- rls norm_Rational downto fun gcd_poly ------------------------------";
17 "Rfuns-------- rls norm_Rational downto fun add_fraction_p_ -----------------------";
18 "----------- rewrite_set_ Partial_Fractions norm_Rational --------------------------------------";
19 "----------- fun check_frac_sum with Free A and Const AA ---------------------------------------";
20 "----------- fun cancel_p with Const AA --------------------------------------------------------";
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 "Rfuns-------- reverse rewrite ----------------------------------------------------";
25 "Rfuns-------- '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 "-------- integration lev.1 fun factout_p_ -----------------------------------";
46 "-------- integration lev.1 fun factout_p_ -----------------------------------";
47 "-------- integration lev.1 fun factout_p_ -----------------------------------";
48 val t = TermC.str2term "(x \<up> 2 + - 1*y \<up> 2) / (x \<up> 2 + - 1*x*y)"
49 val SOME (t', asm) = factout_p_ thy t;
50 if UnparseC.term t' = "(x + y) * (x + - 1 * y) / (x * (x + - 1 * y))"
51 then () else error ("factout_p_ term 1 changed: " ^ UnparseC.term t')
53 if UnparseC.terms asm = "[\"x \<noteq> 0\", \"x + - 1 * y \<noteq> 0\"]"
54 then () else error "factout_p_ asm 1 changed"
56 val t = TermC.str2term "nothing + to_cancel ::real";
57 if NONE = factout_p_ thy t then () else error "factout_p_ doesn't report non-applicable";
59 val t = TermC.str2term "((3 * x \<up> 2 + 6 *x + 3) / (2*x + 2))";
60 val SOME (t', asm) = factout_p_ thy t;
61 if UnparseC.term t' = "(3 + 3 * x) * (1 + x) / (2 * (1 + x))" andalso
62 UnparseC.terms asm = "[\"1 + x \<noteq> 0\"]"
63 then () else error "factout_p_ 1 changed";
65 "-------- integration lev.1 fun cancel_p_ ------------------------------------";
66 "-------- integration lev.1 fun cancel_p_ ------------------------------------";
67 "-------- integration lev.1 fun cancel_p_ ------------------------------------";
68 val t = TermC.str2term "(x \<up> 2 + - 1*y \<up> 2) / (x \<up> 2 + - 1*x*y)"
69 val SOME (t', asm) = cancel_p_ thy t;
70 if (UnparseC.term t', UnparseC.terms asm) = ("(x + y) / x", "[\"x \<noteq> 0\"]")
71 then () else error ("cancel_p_ (t', asm) 1 changed: " ^ UnparseC.term t')
73 val t = TermC.str2term "nothing + to_cancel ::real";
74 if NONE = cancel_p_ thy t then () else error "cancel_p_ doesn't report non-applicable";
76 val t = TermC.str2term "((3 * x \<up> 2 + 6 *x + 3) / (2*x + 2))";
77 val SOME (t', asm) = cancel_p_ thy t;
78 if UnparseC.term t' = "(3 + 3 * x) / 2" andalso UnparseC.terms asm = "[]"
79 then () else error "cancel_p_ 1 changed";
81 "-------- integration lev.1 fun common_nominator_p_ --------------------------";
82 "-------- integration lev.1 fun common_nominator_p_ --------------------------";
83 "-------- integration lev.1 fun common_nominator_p_ --------------------------";
84 val t = TermC.str2term ("y / (a*x + b*x + c*x) " ^
88 val SOME (t', asm) = common_nominator_p_ thy t;
90 ("y * y / (x * ((a + b + c) * y)) " ^
91 (* n1 *d2'/ (c'* ( d1' *d2')) *)
92 "+ a * (a + b + c) / (x * ((a + b + c) * y))")
93 (* n2 * d1' / (c'* ( d1' *d2')) *)
94 then () else error "common_nominator_p_ term 1 changed";
95 if UnparseC.terms asm = "[\"a + b + c \<noteq> 0\", \"y \<noteq> 0\", \"x \<noteq> 0\"]"
96 then () else error "common_nominator_p_ asm 1 changed"
98 "-------- example in mail Nipkow";
99 val t = TermC.str2term "x/(x \<up> 2 + - 1*y \<up> 2) + y/(x \<up> 2 + - 1*x*y)";
100 val SOME (t', asm) = common_nominator_p_ thy t;
101 if UnparseC.term t' =
102 "x * x / ((x + - 1 * y) * ((x + y) * x)) +\ny * (x + y) / ((x + - 1 * y) * ((x + y) * x))"
103 then () else error "common_nominator_p_ term 2 changed"
105 if UnparseC.terms asm = "[\"x + y \<noteq> 0\", \"x \<noteq> 0\", \"x + - 1 * y \<noteq> 0\"]"
106 then () else error "common_nominator_p_ asm 2 changed"
108 "-------- example: applicable tested by SML code";
109 val t = TermC.str2term "nothing / to_add";
110 if NONE = common_nominator_p_ thy t then () else error "common_nominator_p_ term 3 changed";
112 val t = TermC.str2term "((x + (- 1)) / (x + 1)) + ((x + 1) / (x + (- 1)))";
113 val SOME (t', asm) = common_nominator_p_ thy t;
114 if UnparseC.term t' =
115 "(x + - 1) * (- 1 + x) / ((1 + x) * (- 1 + x)) +\n(x + 1) * (1 + x) / ((1 + x) * (- 1 + x))"
116 andalso UnparseC.terms asm = "[\"1 + x \<noteq> 0\", \"- 1 + x \<noteq> 0\"]"
117 then () else error "common_nominator_p_ 3 changed";
119 "-------- integration lev.1 fun add_fraction_p_ ------------------------------";
120 "-------- integration lev.1 fun add_fraction_p_ ------------------------------";
121 "-------- integration lev.1 fun add_fraction_p_ ------------------------------";
122 val t = TermC.str2term "((x + (- 1)) / (x + 1)) + ((x + 1) / (x + (- 1)))";
123 val SOME (t', asm) = add_fraction_p_ thy t;
124 if UnparseC.term t' = "(2 + 2 * x \<up> 2) / (- 1 + x \<up> 2)"
125 then () else error "add_fraction_p_ 3 changed";
127 if UnparseC.terms asm = "[\"- 1 + x \<up> 2 \<noteq> 0\"]"
128 then () else error "add_fraction_p_ 3 changed";
130 val t = TermC.str2term "nothing / to_add";
131 if NONE = add_fraction_p_ thy t then () else error "add_fraction_p_ term 3 changed";
133 val t = TermC.str2term "((x + (- 1)) / (x + 1)) + ((x + 1) / (x + (- 1)))";
134 val SOME (t', asm) = add_fraction_p_ thy t;
135 if UnparseC.term t' = "(2 + 2 * x \<up> 2) / (- 1 + x \<up> 2)" andalso
136 UnparseC.terms asm = "[\"- 1 + x \<up> 2 \<noteq> 0\"]"
137 then () else error "add_fraction_p_ 3 changed";
139 "-------- and app_rev ...traced down from rewrite_set_ until prepats ---------";
140 "-------- and app_rev ...traced down from rewrite_set_ until prepats ---------";
141 "-------- and app_rev ...traced down from rewrite_set_ until prepats ---------";
142 (* trace down until prepats are evaluated
143 (which does not to work, because substitution is not done -- compare rew_sub!);
144 keep this sequence for the case, factout_p, cancel_p, common_nominator_p, add_fraction_p
145 (again) get prepat = [] changed to <>[]. *)
146 val t = TermC.str2term "(x \<up> 2 + - 1*y \<up> 2) / (x \<up> 2 + - 1*x*y)";
148 (*rewrite_set_ @{theory Isac_Knowledge} true cancel t = NONE; !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!*)
149 "~~~~~ fun rewrite_set_, args:"; val (thy, bool, rls, term) = (thy, false, cancel_p, t);
150 "~~~~~ fun rewrite__set_, args:"; val (thy, i, _, _, (rrls as Rrls _), t) =
151 (thy, 1, bool, [], rls, term);
152 (*val (t', asm, rew) = app_rev thy (i+1) rrls t; rew = false!!!!!!!!!!!!!!!!!!!!!*)
153 "~~~~~ and app_rev, args:"; val (thy, i, rrls, t) = (thy, (i+1), rrls, t);
154 fun chk_prepat thy erls [] t = true
155 | chk_prepat thy erls prepat t =
157 fun chk (pres, pat) =
159 val subst: Type.tyenv * Envir.tenv =
160 Pattern.match thy (pat, t) (Vartab.empty, Vartab.empty)
162 snd (eval__true thy (i + 1) (map (Envir.subst_term subst) pres) [] erls)
163 end) handle Pattern.MATCH => false
164 fun scan_ f [] = false (*scan_ NEVER called by []*)
165 | scan_ f (pp::pps) =
166 if f pp then true else scan_ f pps;
167 in scan_ chk prepat end;
168 (* apply the normal_form of a rev-set *)
169 fun app_rev' thy (Rrls{erls,prepat,scr=Rfuns{normal_form,...},...}) t =
170 if chk_prepat thy erls prepat t
171 then ((*tracing("### app_rev': t = "^UnparseC.term t);*) normal_form t)
173 (* val opt = app_rev' thy rrls t ..NONE!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!*)
174 "~~~~~ and app_rev', args:"; val (thy, (Rrls{erls,prepat,scr=Rfuns{normal_form,...},...}), t) =
176 (* chk_prepat thy erls prepat t = false!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!*)
177 (* app_sub thy i rrls t = false!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!*)
178 "~~~~~ fun chk_prepat, args:"; val (thy, erls, prepat, t) = (thy, erls, prepat, t);
179 fun chk (pres, pat) =
181 val subst: Type.tyenv * Envir.tenv =
182 Pattern.match thy (pat, t) (Vartab.empty, Vartab.empty)
184 snd (eval__true thy (i + 1) (map (Envir.subst_term subst) pres) [] erls)
185 end) handle Pattern.MATCH => false
186 fun scan_ f [] = false (*scan_ NEVER called by []*)
187 | scan_ f (pp::pps) =
188 if f pp then true else scan_ f pps;
190 (*========== inhibit exn WN130823: prepat is empty ====================================
191 (* scan_ chk prepat = false!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!*)
192 "~~~~~ fun , args:"; val (f, (pp::pps)) = (chk, prepat);
194 val ([t1, t2], t) = pp;
195 UnparseC.term t1 = "?r is_expanded";
196 UnparseC.term t2 = "?s is_expanded";
197 UnparseC.term t = "?r / ?s";
198 (* f pp = false!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!*)
199 "~~~~~ fun chk, args:"; val (pres, pat) = (pp);
200 val subst: Type.tyenv * Envir.tenv =
201 Pattern.match thy (pat, t) (Vartab.empty, Vartab.empty)
203 ({}, {(("r", 0), ("real", Var (("r", 0), "real"))),
204 (("s", 0), ("real", Var (("s", 0), "real")))}*)
206 snd (eval__true thy (i + 1) (map (Envir.subst_term subst) pres) [] erls)
207 "~~~~~ fun eval__true, args:"; val (thy, i, asms, bdv, rls) =
208 (thy, (i + 1), (map (Envir.subst_term subst) pres), [], erls);
209 UnparseC.terms asms; (* = "[\"?r is_expanded\",\"?s is_expanded\"]"*)
210 asms = [@{term True}] orelse asms = []; (* = false*)
211 asms = [@{term False}] ; (* = false*)
212 "~~~~~ fun chk, args:"; val (indets, (a::asms)) = ([], asms);
213 bdv (*= []: _a list*);
214 val bdv : (term * term) list = [];
215 rewrite__set_ thy (i+1) false;
216 UnparseC.term a = "?r is_expanded"; (*hier m"usste doch der Numerator eingesetzt sein ??????????????*)
217 val SOME (Const (\<^const_name>\<open>False\<close>, _), []) = rewrite__set_ thy (i+1) false bdv rls a
218 ============ inhibit exn WN130823: prepat is empty ===================================*)
220 "-------- fun rewrite_set_ cancel_p downto fun gcd_poly ----------------------";
221 "-------- fun rewrite_set_ cancel_p downto fun gcd_poly ----------------------";
222 "-------- fun rewrite_set_ cancel_p downto fun gcd_poly ----------------------";
223 val t = TermC.str2term "(12 * x * y) / (8 * y \<up> 2 )";
224 (* "-------- example 187a": exception Div raised...
225 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;*)
226 val t = TermC.str2term "(8 * x \<up> 2 * y * z ) / (18 * x * y \<up> 2 * z )";
227 (* "-------- example 187b": doesn't terminate...
228 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;*)
229 val t = TermC.str2term "(9 * x \<up> 5 * y \<up> 2 * z \<up> 4) / (15 * x \<up> 6 * y \<up> 3 * z )";
230 (* "-------- example 187c": doesn't terminate...
231 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;*)
232 "~~~~~ fun rewrite_set_, args:"; val (thy, bool, rls, term) = (@{theory Isac_Knowledge}, false, cancel_p, t);
233 (* WN130827: exception Div raised...
234 rewrite__set_ thy 1 bool [] rls term
236 "~~~~~ and rewrite__set_, args:"; val (thy, i, _, _, (rrls as Rrls _), t) =
237 (thy, 1, bool, [], rls, term);
238 (* WN130827: exception Div raised...
239 val (t', asm, rew) = app_rev thy (i+1) rrls t
241 "~~~~~ fun app_rev, args:"; val (thy, i, rrls, t) = (thy, (i+1), rrls, t);
242 (* WN130827: exception Div raised...
243 val opt = app_rev' thy rrls t
245 "~~~~~ fun app_rev', args:"; val (thy, (Rrls{erls,prepat,scr=Rfuns{normal_form,...},...}), t) =
247 chk_prepat thy erls prepat t = true;
248 (* WN130827: exception Div raised...
251 (* lookup Rational.thy, cancel_p: normal_form = cancel_p_ thy*)
252 "~~~~~ fun cancel_p_, args:"; val (t) = (t);
253 val opt = check_fraction t;
254 val SOME (numerator, denominator) = opt
255 val vs = TermC.vars_of t
256 val baseT = type_of numerator
257 val expT = HOLogic.realT
258 val (SOME a, SOME b) = (poly_of_term vs numerator, poly_of_term vs denominator);
259 (*"-------- example 187a": exception Div raised...
260 val a = [(12, [1, 1])]: poly
261 val b = [(8, [0, 2])]: poly
262 val ((a', b'), c) = gcd_poly a b
264 (* "-------- example 187b": doesn't terminate...
265 val a = [(8, [2, 1, 1])]: poly
266 val b = [(18, [1, 2, 1])]: poly
267 val ((a', b'), c) = gcd_poly a b
269 (* "-------- example 187c": doesn't terminate...
270 val a = [(9, [5, 2, 4])]: poly
271 val b = [(15, [6, 3, 1])]: poly
272 val ((a', b'), c) = gcd_poly a b
275 "-------- rls norm_Rational downto fun gcd_poly ------------------------------";
276 "-------- rls norm_Rational downto fun gcd_poly ------------------------------";
277 "-------- rls norm_Rational downto fun gcd_poly ------------------------------";
278 val t = TermC.str2term "(x \<up> 2 - 4)*(3 - y) / ((y \<up> 2 - 9)*(2+x))";
279 Rewrite.trace_on := false (*true false*);
280 (* trace stops with ...: (and then jEdit hangs)..
281 rewrite_set_ thy false norm_Rational t;
283 ### rls: cancel_p on: (- 12 + 4 * y + 3 * x \<up> 2 + - 1 * (x \<up> 2 * y)) /
284 (- 18 + -9 * x + 2 * y \<up> 2 + x * y \<up> 2)
286 val t = TermC.str2term (*copy from above: "::real" is not required due to " \<up> "*)
287 ("(- 12 + 4 * y + 3 * x \<up> 2 + - 1 * (x \<up> 2 * y)) /" ^
288 "(- 18 + -9 * x + 2 * y \<up> 2 + x * y \<up> 2)");
290 exception Div raised*)
292 "~~~~~ fun cancel_p_, args:"; val (t) = (t);
293 val opt = check_fraction t;
294 val SOME (numerator, denominator) = opt
295 val vs = TermC.vars_of t
296 val baseT = type_of numerator
297 val expT = HOLogic.realT;
298 (*default_print_depth 3; 999*)
299 val (SOME a, SOME b) = (poly_of_term vs numerator, poly_of_term vs denominator);
300 (*default_print_depth 3; 999*)
301 (* does not terminate instead of returning ?:
302 val ((a', b'), c) = gcd_poly a b
303 val a = [(~12, [0, 0]), (3, [2, 0]), (4, [0, 1]), (~1, [2, 1])]: poly
304 val b = [(~18, [0, 0]), (~9, [1, 0]), (2, [0, 2]), (1, [1, 2])]: poly
307 "-------- rls norm_Rational downto fun add_fraction_p_ -----------------------";
308 "-------- rls norm_Rational downto fun add_fraction_p_ -----------------------";
309 "-------- rls norm_Rational downto fun add_fraction_p_ -----------------------";
310 val thy = @{theory Isac_Knowledge};
311 "----- SK060904- 2a non-termination of add_fraction_p_";
312 val t = TermC.str2term (" (a + b * x) / (a + - 1 * (b * x)) + " ^
313 " (- 1 * a + b * x) / (a + b * x) ");
314 (* rewrite_set_ thy false norm_Rational t
315 exception Div raised*)
316 (* rewrite_set_ thy false add_fractions_p t;
317 exception Div raised*)
318 "~~~~~ fun rewrite_set_, args:"; val (thy, bool, rls, term) =
319 (@{theory Isac_Knowledge}, false, add_fractions_p, t);
320 "~~~~~ and rewrite__set_, args:"; val (thy, i, _, _, (rrls as Rrls _), t) =
321 (thy, 1, bool, [], rls, term);
322 (* app_rev thy (i+1) rrls t;
323 exception Div raised*)
324 "~~~~~ and app_rev, args:"; val (thy, i, rrls, t) = (thy, (i+1), rrls, t);
325 fun chk_prepat thy erls [] t = true
326 | chk_prepat thy erls prepat t =
328 fun chk (pres, pat) =
330 val subst: Type.tyenv * Envir.tenv =
331 Pattern.match thy (pat, t) (Vartab.empty, Vartab.empty)
333 snd (eval__true thy (i + 1) (map (Envir.subst_term subst) pres) [] erls)
334 end) handle Pattern.MATCH => false
335 fun scan_ f [] = false (*scan_ NEVER called by []*)
336 | scan_ f (pp::pps) =
337 if f pp then true else scan_ f pps;
338 in scan_ chk prepat end;
339 (* apply the normal_form of a rev-set *)
340 fun app_rev' thy (Rrls{erls,prepat,scr=Rfuns{normal_form,...},...}) t =
341 if chk_prepat thy erls prepat t
342 then ((*tracing("### app_rev': t = "^UnparseC.term t);*) normal_form t)
344 (* val opt = app_rev' thy rrls t;
345 exception Div raised*)
346 (* val opt = app_rev' thy rrls t;
347 exception Div raised*)
348 "~~~~~ and app_rev', args:"; val (thy, (Rrls{erls,prepat,scr=Rfuns{normal_form,...},...}), t) =
350 chk_prepat thy erls prepat t = true = true;
352 exception Div raised*)
353 (* lookup Rational.thy, val add_fractions_p: normal_form = add_fraction_p_ thy*)
354 (*add_fraction_p_ thy t
355 exception Div raised*)
356 "~~~~~ fun add_fraction_p_, args:"; val ((_: theory), t) = (thy, t);
357 val SOME ((n1, d1), (n2, d2)) = check_frac_sum t;
358 UnparseC.term n1; UnparseC.term d1; UnparseC.term n2; UnparseC.term d2;
359 val vs = TermC.vars_of t;
360 (*default_print_depth 3; 999*)
361 val (SOME _, SOME a, SOME _, SOME b) =
362 (poly_of_term vs n1, poly_of_term vs d1, poly_of_term vs n2, poly_of_term vs d2);
363 (*default_print_depth 3; 999*)
365 val a = [(1, [1, 0, 0]), (~1, [0, 1, 1])]: poly
366 val b = [(1, [1, 0, 0]), (1, [0, 1, 1])]: poly
367 val ((a', b'), c) = gcd_poly a b
370 "----------- fun check_frac_sum with Free A and Const AA ---------------------------------------";
371 "----------- fun check_frac_sum with Free A and Const AA ---------------------------------------";
372 "----------- fun check_frac_sum with Free A and Const AA ---------------------------------------";
373 val thy = @{theory Isac_Knowledge(*Partial_Fractions*)}
374 val ctxt = Proof_Context.init_global thy;
376 (*---------- (1) with Free A, B ----------------------------------------------------------------*)
377 val t = (the o (parseNEW ctxt)) "3 = A / 2 + A / 4 + (B / 2 + - 1 * B / (2::real))";
378 (* required for applying thms in rewriting \<up> ^*)
379 (* we get details from here..*)
381 Rewrite.trace_on := false; (*true false*)
382 val SOME (t', _) = Rewrite.rewrite_set_ thy true add_fractions_p t;
383 Rewrite.trace_on := false; (*true false*)
385 add_fractions_p on: 3 = A / 2 + A / 4 + (B / 2 + - 1 * B / 2) --> 3 = A / 2 + A / 4 + 0 / 2 *)
386 (* |||||||||||||||||||||||||||||||||||| *)
388 val t = (the o (parseNEW ctxt))(* ||||||||||||||||||||||||| GUESS 1 GUESS 1 GUESS 1 GUESS 1 *)
389 "A / 2 + A / 4 + (B / 2 + - 1 * B / (2::real))";
390 "~~~~~ fun add_fraction_p_ , ad-hoc args:"; val (t) = (t);
391 val NONE = (*case*) check_frac_sum t (*of*)
394 add_fractions_p on: 3 = A / 2 + A / 4 + (B / 2 + - 1 * B / 2) --> 3 = A / 2 + A / 4 + 0 / 2 *)
395 (* |||||||||||||||||||||||||||| *)
396 val t = (the o (parseNEW ctxt))(* ||||||||||||||||||||||||| GUESS 2 GUESS 2 GUESS 2 GUESS 2 *)
397 "A / 4 + (B / 2 + - 1 * B / (2::real))";
398 "~~~~~ fun add_fraction_p_ , ad-hoc args:"; val (t) = (t);
399 val SOME ((n1, d1), (n2, d2)) = (*case*) check_frac_sum t (*of*);
400 (*+*)if (UnparseC.term n1, UnparseC.term d1) = ("A" , "4") andalso
401 (*+*) (UnparseC.term n2, UnparseC.term d2) = ("B / 2 + - 1 * B / 2", "1")
402 (*+*)then () else error "check_frac_sum (A / 4 + (B / 2 + - 1 * B / (2::real))) changed";
404 val vs = TermC.vars_of t;
405 val (SOME [(1, [1, 0])], SOME [(4, [0, 0])], NONE, SOME [(1, [0, 0])]) =
406 (*case*) (poly_of_term vs n1, poly_of_term vs d1, poly_of_term vs n2, poly_of_term vs d2) (*of*);
408 "~~~~~ fun poly_of_term , args:"; val (vs, t) = (vs, n1);
409 val SOME [(1, [xxx, 0])] = SOME [monom_of_term vs (1, replicate (length vs) 0) t];
410 (*+*)if xxx = 1 then () else error "monom_of_term changed"
412 "~~~~~ fun monom_of_term , args:"; val (vs, (c, es), (Free (id, _))) =
413 (vs, (1, replicate (length vs) 0), t);
414 case vs of [Free ("A", _), Free ("B", _)] =>
415 if c = 1 andalso id = "A"
416 then () else error "monom_of_term Free changed 1"
417 | _ => error "monom_of_term Free changed 2";
419 (*---------- (2) with Const AA, BB --------------------------------------------------------------*)
420 val t = (the o (parseNEW ctxt)) "3 = AA / 2 + AA / 4 + (BB / 2 + - 1 * BB / 2)";
422 (* we get details from here..*)
424 Rewrite.trace_on := false; (*true false*)
425 val SOME (t', _) = Rewrite.rewrite_set_ thy true add_fractions_p t;
426 Rewrite.trace_on := false; (*true false*)
428 add_fractions_p on: 3 = A / 2 + A / 4 + (B / 2 + - 1 * B / 2) --> 3 = A / 2 + A / 4 + 0 / 2 *)
429 (* |||||||||||||||||||||||||||||||||||| *)
430 val t = (the o (parseNEW ctxt))(* ||||||||||||||||||||||||| *)
431 "AA / 2 + AA / 4 + (BB / 2 + - 1 * BB / 2)";
432 "~~~~~ fun add_fraction_p_ , ad-hoc args:"; val (t) = (t);
433 val NONE = (*case*) check_frac_sum t (*of*)
436 add_fractions_p on: 3 = A / 2 + A / 4 + (B / 2 + - 1 * B / 2) --> 3 = A / 2 + A / 4 + 0 / 2 *)
437 (* |||||||||||||||||||||||||||| *)
438 val t = (the o (parseNEW ctxt))(* ||||||||||||||||||||||||| *)
439 "AA / 4 + (BB / 2 + - 1 * BB / 2)";
440 "~~~~~ fun add_fraction_p_ , ad-hoc args:"; val (t) = (t);
441 val SOME ((n1, d1), (n2, d2)) = (*case*) check_frac_sum t (*of*);
442 (*+*)if (UnparseC.term n1, UnparseC.term d1) = ("AA" , "4") andalso
443 (*+*) (UnparseC.term n2, UnparseC.term d2) = ("BB / 2 + - 1 * BB / 2", "1")
444 (*+*)then () else error "check_frac_sum (AA / 4 + (BB / 2 + - 1 * BB / 2)) changed";
446 val vs = TermC.vars_of t;
447 val (SOME [(1, [1, 0])], SOME [(4, [0, 0])], NONE, SOME [(1, [0, 0])]) =
448 (*case*) (poly_of_term vs n1, poly_of_term vs d1, poly_of_term vs n2, poly_of_term vs d2) (*of*);
450 "~~~~~ fun poly_of_term , args:"; val (vs, t) = (vs, n1);
451 val SOME [(1, [xxx, 0])] = SOME [monom_of_term vs (1, replicate (length vs) 0) t];
452 (*+*)if xxx = 1 then () else error "monom_of_term changed"
454 "~~~~~ fun monom_of_term , args:"; val (vs, (c, es), (Const (id, _))) =
455 (vs, (1, replicate (length vs) 0), t);
456 case vs of [Const (\<^const_name>\<open>AA\<close>, _), Const (\<^const_name>\<open>BB\<close>, _)] =>
457 if c = 1 andalso id = "Partial_Fractions.AA"
458 then () else error "monom_of_term Const changed 1"
459 | _ => error "monom_of_term Const changed 2";
461 "----------- fun cancel_p with Const AA --------------------------------------------------------";
462 "----------- fun cancel_p with Const AA --------------------------------------------------------";
463 "----------- fun cancel_p with Const AA --------------------------------------------------------";
464 val thy = @{theory Partial_Fractions};
465 val ctxt = Proof_Context.init_global @{theory}
466 val SOME t = TermC.parseNEW ctxt "2 * AA / 2"; (* Const (\<^const_name>\<open>AA\<close>, "real") *)
468 val SOME (t', _) = rewrite_set_ thy true cancel_p t;
470 Const (\<^const_name>\<open>divide\<close>, _) $ Const (\<^const_name>\<open>AA\<close>, _) $
471 Const (\<^const_name>\<open>one_class.one\<close>, _) => ()
472 | _ => error "WRONG rewrite_set_ cancel_p (2 * AA / 2) \<longrightarrow> AA changed";
474 "~~~~~ fun cancel_p , args:"; val (t) = (t);
475 val opt = check_fraction t
476 val SOME (numerator, denominator) = (*case*) opt (*of*);
478 if UnparseC.term numerator = "2 * AA" andalso UnparseC.term denominator = "2"
479 then () else error "check_fraction (2 * AA / 2) changed";
480 val vs = TermC.vars_of t;
482 [Const (\<^const_name>\<open>AA\<close>, _)] => ()
483 | _ => error "rewrite_set_ cancel_p (2 * AA / 2) \<longrightarrow> AA/1 changed";
486 "-------- rewrite_set_ cancel_p from: Mathematik 1 Schalk Reniets Verlag -----";
487 "-------- rewrite_set_ cancel_p from: Mathematik 1 Schalk Reniets Verlag -----";
488 "-------- rewrite_set_ cancel_p from: Mathematik 1 Schalk Reniets Verlag -----";
489 val thy = @{theory "Rational"};
491 val t = TermC.str2term "(2 + -3 * x) / 9";
492 if NONE = rewrite_set_ thy false cancel_p t then ()
493 else error "rewrite_set_ cancel_p must return NONE, if the term cannot be cancelled";
495 "-------- example 186a";
496 val t = TermC.str2term "(14 * x * y) / (x * y)";
497 is_expanded (TermC.str2term "14 * x * y");
498 is_expanded (TermC.str2term "x * y");
499 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;
500 if (UnparseC.term t', UnparseC.terms asm) = ("14 / 1", "[]")
501 then () else error "rational.sml cancel Schalk 186a";
503 "-------- example 186b";
504 val t = TermC.str2term "(60 * a * b) / ( 15 * a * b )";
505 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;
506 if (UnparseC.term t', UnparseC.terms asm) = ("4 / 1", "[]")
507 then () else error "rational.sml cancel Schalk 186b";
509 "-------- example 186c";
510 val t = TermC.str2term "(144 * a \<up> 2 * b * c) / (12 * a * b * c)";
511 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;
512 if (UnparseC.term t', UnparseC.terms asm) = ("12 * a / 1", "[]")
513 then () else error "rational.sml cancel Schalk 186c";
515 (* !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! exception Div raised !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
516 see --- fun rewrite_set_ downto fun gcd_poly ---
517 "-------- example 187a";
518 val t = TermC.str2term "(12 * x * y) / (8 * y \<up> 2 )";
519 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;
520 if (UnparseC.term t', UnparseC.terms asm) = ("3 * x / (2 * y)", "[\"4 * y ~= 0\"]")
521 then () else error "rational.sml cancel Schalk 187a";
524 (* doesn't terminate !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
525 see --- fun rewrite_set_ downto fun gcd_poly ---
526 "-------- example 187b";
527 val t = TermC.str2term "(8 * x \<up> 2 * y * z ) / (18 * x * y \<up> 2 * z )";
528 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;
529 if (UnparseC.term t', UnparseC.terms asm) = ("4 * x / (9 * y)", "[\"2 * (z * (y * x)) ~= 0\"]")
530 then () else error "rational.sml cancel Schalk 187b";
533 (* doesn't terminate !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
534 see --- fun rewrite_set_ downto fun gcd_poly ---
535 "-------- example 187c";
536 val t = TermC.str2term "(9 * x \<up> 5 * y \<up> 2 * z \<up> 4) / (15 * x \<up> 6 * y \<up> 3 * z )";
537 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;
538 if (UnparseC.term t', UnparseC.terms asm) =
539 ("3 * z \<up> 3 / (5 * (y * x))", "[\"3 * (z * (y \<up> 2 * x \<up> 5)) ~= 0\"]")
540 then () else error "rational.sml cancel Schalk 187c";
543 "-------- example 188a";
544 val t = TermC.str2term "(-8 + 8 * x) / (-9 + 9 * x)";
545 is_expanded (TermC.str2term "8 * x + -8");
546 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;
547 if (UnparseC.term t', UnparseC.terms asm) = ("8 / 9", "[]")
548 then () else error "rational.sml cancel Schalk 188a";
550 val t = TermC.str2term "(8*((- 1) + x))/(9*((- 1) + x))";
551 val SOME (t, _) = rewrite_set_ thy false make_polynomial t;
552 if (UnparseC.term t', UnparseC.terms asm) = ("8 / 9", "[]")
553 then () else error "rational.sml cancel Schalk make_polynomial 1";
555 "-------- example 188b";
556 val t = TermC.str2term "(- 15 + 5 * x) / (- 18 + 6 * x)";
557 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;
558 if (UnparseC.term t', UnparseC.terms asm) = ("5 / 6", "[]")
559 then () else error "rational.sml cancel Schalk 188b";
561 "-------- example 188c";
562 val t = TermC.str2term "(a + - 1 * b) / (b + - 1 * a)";
563 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;
564 if (UnparseC.term t', UnparseC.terms asm) = ("- 1 / 1", "[]")
565 then () else error "rational.sml cancel Schalk 188c";
567 is_expanded (TermC.str2term "a + - 1 * b") = true;
568 val t = TermC.str2term "((- 1)*(b + (- 1) * a))/(1*(b + (- 1) * a))";
569 val SOME (t', asm) = rewrite_set_ thy false make_polynomial t;
570 if (UnparseC.term t', UnparseC.terms asm) = ("(a + - 1 * b) / (- 1 * a + b)", "[]")
571 then () else error "rational.sml cancel Schalk make_polynomial 2";
573 "-------- example 190a";
574 val t = TermC.str2term "( 27 * a \<up> 3 + 9 * a \<up> 2 + 3 * a + 1 ) / ( 27 * a \<up> 3 + 18 * a \<up> 2 + 3 * a )";
575 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;
576 if (UnparseC.term t', UnparseC.terms asm) =
577 ("(1 + 9 * a \<up> 2) / (3 * a + 9 * a \<up> 2)", "[\"3 * a + 9 * a \<up> 2 \<noteq> 0\"]")
578 then () else error "rational.sml cancel Schalk 190a";
580 "-------- example 190c";
581 val t = TermC.str2term "((1 + 9 * a \<up> 2)*(1 + 3 * a))/((3 * a + 9 * a \<up> 2)*(1 + 3 * a))";
582 val SOME (t', asm) = rewrite_set_ thy false make_polynomial t;
583 if (UnparseC.term t', UnparseC.terms asm) =
584 ("(1 + 3 * a + 9 * a \<up> 2 + 27 * a \<up> 3) /\n(3 * a + 18 * a \<up> 2 + 27 * a \<up> 3)", "[]")
585 then () else error "rational.sml make_polynomial Schalk 190c";
587 "-------- example 191a";
588 val t = TermC.str2term "( x \<up> 2 + - 1 * y \<up> 2 ) / ( x + y )";
589 is_expanded (TermC.str2term "x \<up> 2 + - 1 * y \<up> 2") = false; (*!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!*)
590 is_expanded (TermC.str2term "x + y") = true;
591 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;
592 if (UnparseC.term t', UnparseC.terms asm) = ("(x + - 1 * y) / 1", "[]")
593 then () else error "rational.sml make_polynomial Schalk 191a";
595 "-------- example 191b";
596 val t = TermC.str2term "((x + (- 1) * y)*(x + y))/((1)*(x + y))";
597 val SOME (t', asm) = rewrite_set_ thy false make_polynomial t;
598 if (UnparseC.term t', UnparseC.terms asm) = ("(x \<up> 2 + - 1 * y \<up> 2) / (x + y)", "[]")
599 then () else error "rational.sml make_polynomial Schalk 191b";
601 "-------- example 191c";
602 val t = TermC.str2term "( 9 * x \<up> 2 + -30 * x + 25 ) / ( 9 * x \<up> 2 + - 25 )";
603 is_expanded (TermC.str2term "9 * x \<up> 2 + -30 * x + 25") = true;
604 is_expanded (TermC.str2term "25 + -30*x + 9*x \<up> 2") = true;
605 is_expanded (TermC.str2term "- 25 + 9*x \<up> 2") = true;
607 val t = TermC.str2term "(((-5) + 3 * x)*((-5) + 3 * x))/((5 + 3 * x)*((-5) + 3 * x))";
608 val SOME (t', asm) = rewrite_set_ thy false make_polynomial t;
609 if (UnparseC.term t', UnparseC.terms asm) = ("(25 + - 30 * x + 9 * x \<up> 2) / (- 25 + 9 * x \<up> 2)", "[]")
610 then () else error "rational.sml make_polynomial Schalk 191c";
612 "-------- example 192b";
613 val t = TermC.str2term "( 7 * x \<up> 3 + - 1 * x \<up> 2 * y ) / ( 7 * x * y \<up> 2 + - 1 * y \<up> 3 )";
614 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;
615 if (UnparseC.term t', UnparseC.terms asm) = ("x \<up> 2 / y \<up> 2", "[\"y \<up> 2 \<noteq> 0\"]")
616 then () else error "rational.sml cancel_p Schalk 192b";
618 val t = TermC.str2term "((x \<up> 2)*(7 * x + (- 1) * y))/((y \<up> 2)*(7 * x + (- 1) * y))";
619 val SOME (t', asm) = rewrite_set_ thy false make_polynomial t;
620 if (UnparseC.term t', UnparseC.terms asm) =
621 ("(7 * x \<up> 3 + - 1 * x \<up> 2 * y) /\n(7 * x * y \<up> 2 + - 1 * y \<up> 3)", "[]")
622 then () else error "rational.sml make_polynomial Schalk 192b";
624 val t = TermC.str2term "((x \<up> 2)*(7 * x + (- 1) * y))/((y \<up> 2)*(7 * x + (- 1) * y))";
625 val SOME (t', asm) = rewrite_set_ thy false make_polynomial t;
626 if (UnparseC.term t', UnparseC.terms asm) =
627 ("(7 * x \<up> 3 + - 1 * x \<up> 2 * y) /\n(7 * x * y \<up> 2 + - 1 * y \<up> 3)", "[]")
628 then () else error "rational.sml make_polynomial Schalk WN050929 not working";
630 "-------- example 193a";
631 val t = TermC.str2term "( x \<up> 2 + -6 * x + 9 ) / ( x \<up> 2 + -9 )";
632 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;
633 if (UnparseC.term t', UnparseC.terms asm) = ("(- 3 + x) / (3 + x)", "[\"3 + x \<noteq> 0\"]")
634 then () else error "rational.sml cancel_p Schalk 193a";
636 "-------- example 193b";
637 val t = TermC.str2term "( x \<up> 2 + -8 * x + 16 ) / ( 2 * x \<up> 2 + -32 )";
638 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;
639 if (UnparseC.term t', UnparseC.terms asm) = ("(- 4 + x) / (8 + 2 * x)", "[\"8 + 2 * x \<noteq> 0\"]")
640 then () else error "rational.sml cancel_p Schalk 193b";
642 "-------- example 193c";
643 val t = TermC.str2term "( 2 * x + -50 * x \<up> 3 ) / ( 25 * x \<up> 2 + - 10 * x + 1 )";
644 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;
645 if (UnparseC.term t', UnparseC.terms asm) =
646 ("(2 * x + 10 * x \<up> 2) / (1 + - 5 * x)", "[\"1 + - 5 * x \<noteq> 0\"]")
647 then () else error "rational.sml cancel_p Schalk 193c";
649 (*WN: improved with new numerals*)
650 val t = TermC.str2term "(- 25 + 9*x \<up> 2)/(5 + 3*x)";
651 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;
652 if (UnparseC.term t', UnparseC.terms asm) = ("(- 5 + 3 * x) / 1", "[]")
653 then () else error "rational.sml cancel WN 1";
655 "-------- example heuberger";
656 val t = TermC.str2term ("(x \<up> 4 + x * y + x \<up> 3 * y + y \<up> 2) / " ^
657 "(x + 5 * x \<up> 2 + y + 5 * x * y + x \<up> 2 * y \<up> 3 + x * y \<up> 4)");
658 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;
659 if (UnparseC.term t', UnparseC.terms asm) =
660 ("(x \<up> 3 + y) / (1 + 5 * x + x * y \<up> 3)", "[\"1 + 5 * x + x * y \<up> 3 \<noteq> 0\"]")
661 then () else error "rational.sml cancel_p heuberger";
663 "-------- rewrite_set_ add_fractions_p from: Mathematik 1 Schalk -------------";
664 "-------- rewrite_set_ add_fractions_p from: Mathematik 1 Schalk -------------";
665 "-------- rewrite_set_ add_fractions_p from: Mathematik 1 Schalk -------------";
666 (*deleted example 204 ... 236b at update Isabelle2012-->2013*)
668 "-------- integration lev.1 -- lev.5: cancel_p_ & add_fractions_p_ -----------";
669 "-------- integration lev.1 -- lev.5: cancel_p_ & add_fractions_p_ -----------";
670 "-------- integration lev.1 -- lev.5: cancel_p_ & add_fractions_p_ -----------";
671 val t = TermC.str2term ("123 = (a*x)/(b*x) + (c*x)/(d*x) + (e*x)/(f*x::real)");
672 "-------- gcd_poly integration level 1: works on exact term";
673 if NONE = cancel_p_ thy t then () else error "cancel_p_ works on exact fraction";
674 if NONE = add_fraction_p_ thy t then () else error "add_fraction_p_ works on exact fraction";
676 "-------- gcd_poly integration level 2: picks out ONE appropriate subterm";
677 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;
678 if UnparseC.term t' = "123 = a * x / (b * x) + c * x / (d * x) + e / f"
679 then () else error "level 2, rewrite_set_ cancel_p: changed";
680 val SOME (t', asm) = rewrite_set_ thy false add_fractions_p t;
681 if UnparseC.term t' = "123 = (b * c * x + a * d * x) / (b * d * x) + e * x / (f * x)"
682 then () else error "level 2, rewrite_set_ add_fractions_p: changed";
684 "-------- gcd_poly integration level 3: rewrites all appropriate subterms";
685 val SOME (t', asm) = rewrite_set_ thy false cancel_p_rls t;
686 if UnparseC.term t' = "123 = a / b + c / d + e / f"
687 then () else error "level 3, rewrite_set_ cancel_p_rls: changed";
688 val SOME (t', asm) = rewrite_set_ thy false add_fractions_p_rls t; (*CREATE add_fractions_p_rls*)
689 if UnparseC.term t' = "123 = (b * d * e * x + b * c * f * x + a * d * f * x) / (b * d * f * x)"
690 then () else error "level 3, rewrite_set_ add_fractions_p_rls: changed";
692 "-------- gcd_poly integration level 4: iteration cancel_p -- add_fraction_p";
693 (* simpler variant *)
694 val testrls = Rule_Set.append_rules "testrls" Rule_Set.empty [Rls_ cancel_p, Rls_ add_fractions_p]
695 val SOME (t', asm) = rewrite_set_ thy false testrls t;
696 (*Rewrite.trace_on := false; (*true false*)
697 # rls: testrls on: 123 = a * x / (b * x) + c * x / (d * x) + e * x / (f * x)
698 ## rls: cancel_p on: 123 = a * x / (b * x) + c * x / (d * x) + e * x / (f * x)
699 ## rls: add_fractions_p on: 123 = a * x / (b * x) + c * x / (d * x) + e / f
700 ## rls: cancel_p on: 123 = (b * c * x + a * d * x) / (b * d * x) + e / f
701 ## rls: add_fractions_p on: 123 = (b * c + a * d) / (b * d) + e / f
702 ## rls: cancel_p on: 123 = (b * d * e + b * c * f + a * d * f) / (b * d * f)
703 ## rls: add_fractions_p on: 123 = (b * d * e + b * c * f + a * d * f) / (b * d * f) *)
704 if UnparseC.term t' = "123 = (b * d * e + b * c * f + a * d * f) / (b * d * f)"
705 then () else error "level 4, rewrite_set_ *_p: changed";
707 (* complicated variant *)
708 val testrls_rls = Rule_Set.append_rules "testrls_rls" Rule_Set.empty [Rls_ cancel_p_rls, Rls_ add_fractions_p_rls];
709 val SOME (t', asm) = rewrite_set_ thy false testrls_rls t;
710 (*# rls: testrls_rls on: 123 = a * x / (b * x) + c * x / (d * x) + e * x / (f * x)
711 ## rls: cancel_p_rls on: 123 = a * x / (b * x) + c * x / (d * x) + e * x / (f * x)
712 ### rls: cancel_p on: 123 = a * x / (b * x) + c * x / (d * x) + e * x / (f * x)
713 ### rls: cancel_p on: 123 = a * x / (b * x) + c * x / (d * x) + e / f
714 ### rls: cancel_p on: 123 = a * x / (b * x) + c / d + e / f
715 ### rls: cancel_p on: 123 = a / b + c / d + e / f
716 ## rls: add_fractions_p_rls on: 123 = a / b + c / d + e / f
717 ### rls: add_fractions_p on: 123 = a / b + c / d + e / f
718 ### rls: add_fractions_p on: 123 = (b * c + a * d) / (b * d) + e / f
719 ### rls: add_fractions_p on: 123 = (b * d * e + b * c * f + a * d * f) / (b * d * f)
720 ## rls: cancel_p_rls on: 123 = (b * d * e + b * c * f + a * d * f) / (b * d * f)
721 ### rls: cancel_p on: 123 = (b * d * e + b * c * f + a * d * f) / (b * d * f)
722 ## rls: add_fractions_p_rls on: 123 = (b * d * e + b * c * f + a * d * f) / (b * d * f)
723 ### rls: add_fractions_p on: 123 = (b * d * e + b * c * f + a * d * f) / (b * d * f) *)
724 if UnparseC.term t' = "123 = (b * d * e + b * c * f + a * d * f) / (b * d * f)"
725 then () else error "level 4, rewrite_set_ *_p_rls: changed"
727 "-------- gcd_poly integration level 5: cancel_p & add_fraction_p within norm_Rational";
728 val SOME (t', asm) = rewrite_set_ thy false norm_Rational t;
729 if UnparseC.term t' = "123 = (a * d * f + b * c * f + b * d * e) / (b * d * f)"
730 then () else error "level 5, rewrite_set_ norm_Rational: changed"
732 "-------- reverse rewrite ----------------------------------------------------";
733 "-------- reverse rewrite ----------------------------------------------------";
734 "-------- reverse rewrite ----------------------------------------------------";
735 (** the term for which reverse rewriting is demonstrated **)
736 val t = TermC.str2term "(9 + - 1 * x \<up> 2) / (9 + 6 * x + x \<up> 2)";
737 val Rrls {scr = Rfuns {init_state = ini, locate_rule = loc,
738 next_rule = nex, normal_form = nor, ...},...} = cancel_p;
740 (** normal_form produces the result in ONE step **)
741 val SOME (t', _) = nor t;
742 if UnparseC.term t' = "(3 + - 1 * x) / (3 + x)" then ()
743 else error "rational.sml normal_form (9 - x \<up> 2) / (9 - 6 * x + x \<up> 2)";
745 (** initialize the interpreter state used by the 'me' **)
746 val (t, _, revsets, _) = ini t;
748 if length (hd revsets) = 11 then () else error "length of revset changed";
749 (*//----------------------------------TOODOO (*Rfuns revsets \<longrightarrow> broken*)
750 if (revsets |> nth 1 |> nth 1 |> id_of_thm) =
751 (@{thm realpow_twoI} |> Thm.get_name_hint |> ThmC.cut_id)
752 then () else error "first element of revset changed";
754 (revsets |> nth 1 |> nth 1 |> Rule.to_string) = "Thm (\"realpow_twoI\",?r1 \<up> 2 = ?r1 * ?r1)" andalso
755 (revsets |> nth 1 |> nth 2 |> Rule.to_string) = "Thm (\"#: 9 = 3 \<up> 2\",9 = 3 \<up> 2)" andalso
756 (revsets |> nth 1 |> nth 3 |> Rule.to_string) = "Thm (\"#: 6 * x = 2 * (3 * x)\",6 * x = 2 * (3 * x))"
758 (revsets |> nth 1 |> nth 4 |> Rule.to_string) = "Thm (\"#: -3 * x = - 1 * (3 * x)\",-3 * x = - 1 * (3 * x))"
760 (revsets |> nth 1 |> nth 5 |> Rule.to_string) = "Thm (\"#: 9 = 3 * 3\",9 = 3 * 3)" andalso
761 (revsets |> nth 1 |> nth 6 |> Rule.to_string) = "Rls_ (\"sym_order_mult_rls_\")" andalso
762 (revsets |> nth 1 |> nth 7 |> Rule.to_string) =
763 "Thm (\"sym_mult.assoc\",?a * (?b * ?c) = ?a * ?b * ?c)"
764 then () else error "first 7 elements in revset changed"
765 \\----------------------------------TOODOO (*Rfuns revsets \<longrightarrow> broken*)*)
767 (** find the rule 'r' to apply to term 't' **)
768 (*/------- WN1309: since cancel_ (accepted "-" between monomials) has been replaced by cancel_p_
769 for Isabelle2013, we don't get a working revset, but non-termination:
771 val SOME (r as (Thm (str, thm))) = nex revsets t;
773 ((3 * 3 + - 1 * x * x) / (3 * 3 + 2 * 3 * x + x * x),
774 Rls_ ("sym_order_mult_rls_"), ((3 * 3 + - 1 * (x * x)) / (3 * 3 + 2 * (3 * x) + x * x), []))", "
775 ((3 * 3 + - 1 * (x * x)) / (3 * 3 + 2 * (3 * x) + x * x),
776 Thm ("sym_mult.assoc", ""), ((3 * 3 + - 1 * (x * x)) / (3 * 3 + 2 * 3 * x + x * x), []))", "
777 ((3 * 3 + - 1 * (x * x)) / (3 * 3 + 2 * 3 * x + x * x),
778 Thm ("sym_mult.assoc", ""), ((3 * 3 + - 1 * x * x) / (3 * 3 + 2 * 3 * x + x * x), []))", "
779 ((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), []))", "
782 Thm ("sym_#mult_2_3", "6 = 2 * 3")
783 ### Isabelle2009- 2 for cancel_ (not cancel_p_):
784 if str = "sym_#power_Float ((3,0), (0,0)) __ ((2,0), (0,0))"
785 andalso ThmC.string_of_thm thm =
786 (string_of_thm (Thm.make_thm @{theory "Isac_Knowledge"}
787 (Trueprop $ (Thm.term_of o the o (TermC.parse thy)) "9 = 3 \<up> 2"))) then ()
788 else error "rational.sml next_rule (9 - x \<up> 2) / (9 - 6 * x + x \<up> 2)";
789 \---------------------------------------------------------------------------------------/*)
791 (** check, if the rule 'r' applied by the user to 't' belongs to the ruleset;
792 if the rule is OK, the term resulting from applying the rule is returned,too;
793 there might be several rule applications inbetween,
794 which are listed after the head in reverse order **)
795 (*/-------------------------------------------- Isabelle2013: this gives "error id_of_thm";
796 we don't repair this, because interaction within "reverse rewriting" never worked properly:
798 val (r, (t, asm))::_ = loc revsets t r;
799 if UnparseC.term t = "(9 - x \<up> 2) / (3 \<up> 2 + 6 * x + x \<up> 2)" andalso asm = []
800 then () else error "rational.sml locate_rule (9 - x \<up> 2) / (9 - 6 * x + x \<up> 2)";
802 (* find the next rule to apply *)
803 val SOME (r as (Thm (str, thm))) = nex revsets t;
804 if str = "sym_#power_Float ((3,0), (0,0)) __ ((2,0), (0,0))" andalso
805 ThmC.string_of_thm thm = (string_of_thm (ThmC_Def.make_thm @{theory "Isac_Knowledge"}
806 (Trueprop $ (Thm.term_of o the o (TermC.parse thy)) "9 = 3 \<up> 2"))) then ()
807 else error "rational.sml next_rule (9 - x \<up> 2) / (9 - 6 * x + x \<up> 2)";
809 (*check the next rule*)
810 val (r, (t, asm)) :: _ = loc revsets t r;
811 if UnparseC.term t = "(3 \<up> 2 - x \<up> 2) / (3 \<up> 2 + 6 * x + x \<up> 2)" then ()
812 else error "rational.sml locate_rule (9 - x \<up> 2) / (9 - 6 * x + x \<up> 2) II";
814 (*find and check the next rules, rewrite*)
815 val SOME r = nex revsets t;
816 val (r,(t,asm))::_ = loc revsets t r;
817 if UnparseC.term t = "(3 \<up> 2 - x \<up> 2) / (3 \<up> 2 + 2 * 3 * x + x \<up> 2)" then ()
818 else error "rational.sml locate_rule II";
820 val SOME r = nex revsets t;
821 val (r,(t,asm))::_ = loc revsets t r;
822 if UnparseC.term t = "(3 - x) * (3 + x) / (3 \<up> 2 + 2 * 3 * x + x \<up> 2)" then ()
823 else error "rational.sml next_rule II";
825 val SOME r = nex revsets t;
826 val (r,(t,asm))::_ = loc revsets t r;
827 if UnparseC.term t = "(3 - x) * (3 + x) / ((3 + x) * (3 + x))" then ()
828 else error "rational.sml next_rule III";
830 val SOME r = nex revsets t;
831 val (r, (t, asm)) :: _ = loc revsets t r;
832 val ss = UnparseC.term t;
833 if ss = "(3 - x) / (3 + x)" andalso UnparseC.terms asm = "[\"3 + x ~= 0\"]" then ()
834 else error "rational.sml: new behav. in rev-set cancel";
835 \--------------------------------------------------------------------------------------/*)
837 "-------- 'reverse-ruleset' cancel_p -----------------------------------------";
838 "-------- 'reverse-ruleset' cancel_p -----------------------------------------";
839 "-------- 'reverse-ruleset' cancel_p -----------------------------------------";
840 (*WN130909: the example below shows, why "reverse rewriting" only worked for
843 (*the term for which reverse rewriting is demonstrated*)
844 val t = TermC.str2term "(9 + (- 1)*x \<up> 2) / (9 + ((-6)*x + x \<up> 2))";
845 val Rrls {scr=Rfuns {init_state=ini,locate_rule=loc,
846 next_rule=nex,normal_form=nor,...},...} = cancel_p;
848 (*normal_form produces the result in ONE step*)
849 val SOME (t', _) = nor t;
850 if UnparseC.term t' = "(3 + x) / (3 + - 1 * x)"
851 then () else error "cancel_p normal_form CHANGED";;
853 (*initialize the interpreter state used by the 'me'*)
854 val SOME (t', asm) = cancel_p_ thy t;
855 if (UnparseC.term t', UnparseC.terms asm) = ("(3 + x) / (3 + - 1 * x)", "[\"3 + - 1 * x \<noteq> 0\"]")
856 then () else error "cancel_p CHANGED";;
858 val (t,_,revsets,_) = ini t;
860 (* WN.10.10.02: dieser Fall terminiert nicht
861 (make_polynomial enth"alt zu viele rules)
862 WN060823 'init_state' requires rewriting on specified location in the term
863 default_print_depth 99; Rfuns; default_print_depth 3;
864 WN060831 cycling "sym_order_mult_rls_" "sym_mult.assoc"
865 as was with make_polynomial before ?!?* )
867 val SOME r = nex revsets t;
868 eq_Thm (r, Thm ("sym_#power_Float ((3,0), (0,0)) __ ((2,0), (0,0))",
869 mk_thm thy "9 = 3 \<up> 2"));
870 ( *WN060831 *** id_of_thm
871 Exception- ERROR raised ...
872 val (r,(t,asm))::_ = loc revsets t r;
875 val SOME r = nex revsets t;
876 val (r,(t,asm))::_ = loc revsets t r;
880 "-------- examples: rls norm_Rational ----------------------------------------";
881 "-------- examples: rls norm_Rational ----------------------------------------";
882 "-------- examples: rls norm_Rational ----------------------------------------";
883 Rewrite.trace_on := false; (*true false*)
885 val t = TermC.str2term "Not (6*x is_atom)";
886 val SOME (t',_) = rewrite_set_ thy false powers_erls t; UnparseC.term t';
888 val t = TermC.str2term "1 < 2";
889 val SOME (t',_) = rewrite_set_ thy false powers_erls t; UnparseC.term t';
892 val t = TermC.str2term "(6*x) \<up> 2";
893 val SOME (t',_) = rewrite_ thy dummy_ord powers_erls false
894 (ThmC.numerals_to_Free @{thm realpow_def_atom}) t;
895 if UnparseC.term t' = "6 * x * (6 * x) \<up> (2 + - 1)" then ()
896 else error "rational.sml powers_erls (6*x) \<up> 2";
898 val t = TermC.str2term "- 1 * (- 2 * (5 / 2 * (13 * x / 2)))";
899 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; UnparseC.term t';
900 if UnparseC.term t' = "65 * x / 2" then () else error "rational.sml 4";
902 val t = TermC.str2term "1 - ((13*x)/2 - 5/2) \<up> 2";
903 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; UnparseC.term t';
904 if UnparseC.term t' = "(- 21 + 130 * x + - 169 * x \<up> 2) / 4" then ()
905 else error "rational.sml 5";
907 (*SRAM Schalk I, p.92 Nr. 609a*)
908 val t = TermC.str2term "2*(3 - x/5)/3 - 4*(1 - x/3) - x/3 - 2*(x/2 - 1/4)/27 +5/54";
909 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; UnparseC.term t';
910 if UnparseC.term t' = "(- 255 + 112 * x) / 135" then ()
911 else error "rational.sml 6";
913 (*SRAM Schalk I, p.92 Nr. 610c*)
914 val t = TermC.str2term "((x- 1)/(x+1) + 1) / ((x- 1)/(x+1) - (x+1)/(x- 1)) - 2";
915 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; UnparseC.term t';
916 if UnparseC.term t' = "(3 + x) / - 2" then () else error "rational.sml 7";
918 (*SRAM Schalk I, p.92 Nr. 476a*)
919 val t = TermC.str2term "(x \<up> 2/(1 - x \<up> 2) + 1)/(x/(1 - x) + 1) * (1 + x)";
920 (*. a/b : c/d translated to a/b * d/c .*)
921 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; UnparseC.term t';
922 if UnparseC.term t' = "1" then () else error "rational.sml 8";
924 (*Schalk I, p.92 Nr. 472a*)
925 val t = TermC.str2term "((8*x \<up> 2 - 32*y \<up> 2)/(2*x + 4*y))/((4*x - 8*y)/(x + y))";
926 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; UnparseC.term t';
927 if UnparseC.term t' = "x + y" then () else error "rational.sml p.92 Nr. 472a";
929 (*Schalk I, p.70 Nr. 480b: SEE rational.sml --- nonterminating rls norm_Rational ---*)
931 (*WN130910 add_fractions_p exception Div raised + history:
932 ### WN.2.6.03 from rlang.sml 56a
933 val t = TermC.str2term "(a + b * x) / (a + - 1 * (b * x)) + (- 1 * a + b * x) / (a + b * x) = 4 * (a * b) / (a \<up> 2 + - 1 * b \<up> 2)";
934 val NONE = rewrite_set_ thy false add_fractions_p t;
936 THE ERROR ALREADY OCCURS IN THIS PART:
937 val t = TermC.str2term "(a + b * x) / (a + - 1 * (b * x)) + (- 1 * a + b * x) / (a + b * x)";
938 val NONE = add_fraction_p_ thy t;
940 SEE Test_Some.thy: section {* add_fractions_p downto exception Div raised ===
943 "-------- rational numerals --------------------------------------------------";
944 "-------- rational numerals --------------------------------------------------";
945 "-------- rational numerals --------------------------------------------------";
946 (*SRA Schalk I, p.40 Nr. 164b *)
947 val t = TermC.str2term "(47/6 - 76/9 + 13/4)/(35/12)";
948 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
949 if UnparseC.term t = "19 / 21" then ()
950 else error "rational.sml: diff.behav. in norm_Rational_mg 1";
952 (*SRA Schalk I, p.40 Nr. 166a *)
953 val t = TermC.str2term "((5/4)/(4+22/7) + 37/20)*(110/3 - 110/9 * 23/11)";
954 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
955 if UnparseC.term t = "45 / 2" then ()
956 else error "rational.sml: diff.behav. in norm_Rational_mg 2";
958 "-------- examples cancellation from: Mathematik 1 Schalk --------------------";
959 "-------- examples cancellation from: Mathematik 1 Schalk --------------------";
960 "-------- examples cancellation from: Mathematik 1 Schalk --------------------";
962 val t = TermC.str2term "((1 + 9*a \<up> 2) * (1 + 3*a)) / ((3*a + 9*a \<up> 2) * (1 + 3*a))";
963 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
964 if UnparseC.term t = "(1 + 9 * a \<up> 2) / (3 * a + 9 * a \<up> 2)"
965 then () else error "rational.sml: diff.behav. in norm_Rational_mg 3";
968 val t = TermC.str2term "(x \<up> 2 * (7*x + (- 1)*y)) / (y \<up> 2 * (7*x + (- 1)*y))";
969 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
970 if UnparseC.term t = "x \<up> 2 / y \<up> 2"
971 then () else error "rational.sml: diff.behav. in norm_Rational_mg 4";
973 (*SRC Schalk I, p.66 Nr. 379c *)
974 val t = TermC.str2term "(a - b)/(b - a)";
975 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
976 if UnparseC.term t = "- 1"
977 then () else error "rational.sml: diff.behav. in norm_Rational_mg 5";
979 (*SRC Schalk I, p.66 Nr. 380b *)
980 val t = TermC.str2term "15*(3*x + 3) * (4*x + 9) / (12*(2*x + 7) * (5*x + 5))";
981 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
982 if UnparseC.term t = "(27 + 12 * x) / (28 + 8 * x)"
983 then () else error "rational.sml: diff.behav. in norm_Rational_mg 6";
986 val t = TermC.str2term "((1 + 9*a \<up> 2) * (1 + 3*a)) / ((3*a + 9*a \<up> 2) * (1 + 3 * a))";
987 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
988 if UnparseC.term t = "(1 + 9 * a \<up> 2) / (3 * a + 9 * a \<up> 2)"
989 then () else error "rational.sml: diff.behav. in norm_Rational_mg 3";
992 val t = TermC.str2term "(x \<up> 2 * (7*x + (- 1)*y)) / (y \<up> 2 * (7*x + (- 1)*y))";
993 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
994 if UnparseC.term t = "x \<up> 2 / y \<up> 2"
995 then () else error "rational.sml: diff.behav. in norm_Rational_mg 4";
997 (*SRC Schalk I, p.66 Nr. 379c *)
998 val t = TermC.str2term "(a - b) / (b - a)";
999 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1000 if UnparseC.term t = "- 1"
1001 then () else error "rational.sml: diff.behav. in norm_Rational_mg 5";
1003 (*SRC Schalk I, p.66 Nr. 380b *)
1004 val t = TermC.str2term "15*(3*x + 3) * (4*x + 9) / (12*(2*x + 7) * (5*x + 5))";
1005 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1006 if UnparseC.term t = "(27 + 12 * x) / (28 + 8 * x)"
1007 then () else error "rational.sml: diff.behav. in norm_Rational_mg 6";
1009 (* extreme example from somewhere *)
1010 val t = TermC.str2term
1011 ("(a \<up> 4 * x + - 1*a \<up> 4 * y + 4*a \<up> 3 * b * x + -4*a \<up> 3 * b * y + " ^
1012 "6*a \<up> 2 * b \<up> 2 * x + -6*a \<up> 2 * b \<up> 2 * y + 4*a * b \<up> 3 * x + -4*a * b \<up> 3 * y + " ^
1013 "b \<up> 4 * x + - 1*b \<up> 4 * y) " ^
1014 " / (a \<up> 2 * x \<up> 3 + -3*a \<up> 2 * x \<up> 2 * y + 3*a \<up> 2 * x * y \<up> 2 + - 1*a \<up> 2 * y \<up> 3 + " ^
1015 "2*a * b * x \<up> 3 + -6*a * b * x \<up> 2 * y + 6*a * b * x * y \<up> 2 + - 2*a * b * y \<up> 3 + " ^
1016 "b \<up> 2 * x \<up> 3 + -3*b \<up> 2 * x \<up> 2 * y + 3*b \<up> 2 * x * y \<up> 2 + - 1*b \<up> 2 * y \<up> 3)")
1017 val SOME (t, _) = rewrite_set_ thy false cancel_p t;
1018 if UnparseC.term t = "(a \<up> 2 + 2 * a * b + b \<up> 2) / (x \<up> 2 + - 2 * x * y + y \<up> 2)"
1019 then () else error "with Isabelle2002: NONE -- now SOME changed";
1021 (*Schalk I, p.66 Nr. 381a *)
1022 (* ATTENTION: here the rls is very slow. In Isabelle2002 this required 2 min *)
1023 val t = TermC.str2term "18*(a + b) \<up> 3 * (a - b) \<up> 2 / (72*(a - b) \<up> 3 * (a + b) \<up> 2)";
1024 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1025 if UnparseC.term t = "(a + b) / (4 * a + - 4 * b)"
1026 then () else error "rational.sml: diff.behav. in norm_Rational_mg 8";
1028 (*SRC Schalk I, p.66 Nr. 381b *)
1029 val t = TermC.str2term "(4*x \<up> 2 - 20*x + 25) / (2*x - 5) \<up> 3";
1030 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1031 if UnparseC.term t = "1 / (- 5 + 2 * x)"
1032 then () else error "rational.sml: diff.behav. in norm_Rational_mg 9";
1034 (*SRC Schalk I, p.66 Nr. 381c *)
1035 val t = TermC.str2term "(27*a \<up> 3 + 9*a \<up> 2+3*a+1) / (27*a \<up> 3 + 18*a \<up> 2+3*a)";
1036 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1037 if UnparseC.term t = "(1 + 9 * a \<up> 2) / (3 * a + 9 * a \<up> 2)"
1038 then () else error "rational.sml: diff.behav. in norm_Rational_mg 10";
1040 (*SRC Schalk I, p.66 Nr. 383a *)
1041 val t = TermC.str2term "(5*a \<up> 2 - 5*a*b) / (a - b) \<up> 2";
1042 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1043 if UnparseC.term t = "- 5 * a / (- 1 * a + b)"
1044 then () else error "rational.sml: diff.behav. in norm_Rational_mg 11";
1046 "----- NOT TERMINATING ?: worked before 0707xx";
1047 val t = TermC.str2term "(a \<up> 2 - 1)*(b + 1) / ((b \<up> 2 - 1)*(a+1))";
1048 (* WN130911 "exception Div raised" by
1049 cancel_p_ thy (TermC.str2term ("(- 1 + - 1 * b + a \<up> 2 + a \<up> 2 * b) /" ^
1050 "(- 1 + - 1 * a + b \<up> 2 + a * b \<up> 2)"))
1052 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1053 if UnparseC.term t = "(1 + - 1 * a) / (1 + - 1 * b)" then ()
1054 else error "rational.sml MG tests 3e";
1057 "-------- examples common denominator from: Mathematik 1 Schalk --------------";
1058 "-------- examples common denominator from: Mathematik 1 Schalk --------------";
1059 "-------- examples common denominator from: Mathematik 1 Schalk --------------";
1060 (*SRA Schalk I, p.67 Nr. 403a *)
1061 val t = TermC.str2term "4/x - 3/y - 1";
1062 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1063 if UnparseC.term t = "(- 3 * x + 4 * y + - 1 * x * y) / (x * y)"
1064 then () else error "rational.sml: diff.behav. in norm_Rational_mg 12";
1066 val t = TermC.str2term "(2*a+3*b)/(b*c) + (3*c+a)/(a*c) - (2*a \<up> 2+3*b*c)/(a*b*c)";
1067 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1068 if UnparseC.term t = "4 / c"
1069 then () else error "rational.sml: diff.behav. in norm_Rational_mg 13";
1071 (*SRA Schalk I, p.67 Nr. 410b *)
1072 val t = TermC.str2term "1/(x+1) + 1/(x+2) - 2/(x+3)";
1073 (* WN130911 non-termination due to non-termination of
1074 cancel_p_ thy (TermC.str2term "(5 + 3 * x) / (6 + 11 * x + 6 * x \<up> 2 + x \<up> 3)")
1076 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1077 if UnparseC.term t = "(5 + 3 * x) / (6 + 11 * x + 6 * x \<up> 2 + x \<up> 3)"
1078 then () else error "rational.sml: diff.behav. in norm_Rational_mg 14";
1081 (*SRA Schalk I, p.67 Nr. 413b *)
1082 val t = TermC.str2term "(1 + x)/(1 - x) - (1 - x)/(1 + x) + 2*x/(1 - x \<up> 2)";
1083 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1084 if UnparseC.term t = "6 * x / (1 + - 1 * x \<up> 2)"
1085 then () else error "rational.sml: diff.behav. in norm_Rational_mg 15";
1087 (*SRA Schalk I, p.68 Nr. 414a *)
1088 val t = TermC.str2term "(x + 2)/(x - 1) + (x - 3)/(x - 2) - (x + 1)/((x - 1)*(x - 2))";
1089 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1090 if UnparseC.term t ="(- 2 + - 5 * x + 2 * x \<up> 2) / (2 + - 3 * x + x \<up> 2)"
1091 then () else error "rational.sml: diff.behav. in norm_Rational_mg 16";
1093 (*SRA Schalk I, p.68 Nr. 428b *)
1094 val t = TermC.str2term
1095 "1/(a - b) \<up> 2 + 1/(a + b) \<up> 2 - 2/(a \<up> 2 - b \<up> 2) - 4*(b \<up> 2 - 1)/(a \<up> 2 - b \<up> 2) \<up> 2";
1096 (* WN130911 non-termination due to non-termination of
1097 cancel_p_ thy (TermC.str2term "(4 + -4 * b \<up> 2) / (a \<up> 4 + - 2 * (a \<up> 2 * b \<up> 2) + b \<up> 4)")
1099 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1100 if UnparseC.term t = "4 / (a \<up> 4 + - 2 * a \<up> 2 * b \<up> 2 + b \<up> 4)"
1101 then () else error "rational.sml: diff.behav. in norm_Rational_mg 18";
1104 (*SRA Schalk I, p.68 Nr. 430b *)
1105 val t = TermC.str2term
1106 "a \<up> 2/(a - 3*b) - 108*a*b \<up> 3/((a+3*b)*(a \<up> 2 - 9*b \<up> 2)) - 9*b \<up> 2*(a - 3*b)/(a+3*b) \<up> 2";
1107 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1108 if UnparseC.term t = "a + 3 * b"
1109 then () else error "rational.sml: diff.behav. in norm_Rational_mg 19";
1111 (*SRA Schalk I, p.68 Nr. 432 *)
1112 val t = TermC.str2term
1113 ("(a \<up> 2 + a*b) / (a \<up> 2 - b \<up> 2) - (b \<up> 2 - a*b) / (b \<up> 2 - a \<up> 2) + " ^
1114 "a \<up> 2*(a - b) / (a \<up> 3 - a \<up> 2*b) - 2*a*(a \<up> 2 - b \<up> 2) / (a \<up> 3 - a*b \<up> 2) - " ^
1115 "2*b \<up> 2 / (a \<up> 2 - b \<up> 2)");
1116 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1117 if UnparseC.term t = (*"0" ..isabisac15 | Isabelle2017..*) "0 / (a \<up> 2 + - 1 * b \<up> 2)"
1118 then () else error "rational.sml: diff.behav. in norm_Rational_mg 20";
1121 val t = TermC.str2term "3*a / (a*b) + x/y";
1122 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1123 if UnparseC.term t = "(3 * y + b * x) / (b * y)"
1124 then () else error "rational.sml: diff.behav. in norm_Rational_mg 21";
1127 "-------- examples multiply and cancel from: Mathematik 1 Schalk -------------";
1128 "-------- examples multiply and cancel from: Mathematik 1 Schalk -------------";
1129 "-------- examples multiply and cancel from: Mathematik 1 Schalk -------------";
1130 (*------- SRM Schalk I, p.68 Nr. 436a *)
1131 val t = TermC.str2term "3*(x+y) / (15*(x - y)) * 25*(x - y) \<up> 2 / (18*(x + y) \<up> 2)";
1132 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1133 if UnparseC.term t = "(- 5 * x + 5 * y) / (- 18 * x + - 18 * y)"
1134 then () else error "rational.sml: diff.behav. in norm_Rational_mg 22";
1136 (*------- SRM.test Schalk I, p.68 Nr. 436b *)
1137 val t = TermC.str2term "5*a*(a - b) \<up> 2*(a + b) \<up> 3/(7*b*(a - b) \<up> 3) * 7*b/(a + b) \<up> 3";
1138 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1139 if UnparseC.term t = "5 * a / (a + - 1 * b)"
1140 then () else error "rational.sml: diff.behav. in norm_Rational_mg 23";
1142 (*------- Schalk I, p.68 Nr. 437a *)
1143 val t = TermC.str2term "(3*a - 4*b) / (4*c+3*e) * (3*a+4*b)/(9*a \<up> 2 - 16*b \<up> 2)";
1144 (* raises an exception for unclear reasons:
1145 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1147 ### rls: cancel_p on: (9 * a \<up> 2 + - 16 * b \<up> 2) / (4 * c + 3 * e) /
1148 (9 * a \<up> 2 + - 16 * b \<up> 2)
1149 exception Div raised
1152 val t = TermC.str2term
1153 ("(9 * a \<up> 2 + - 16 * b \<up> 2) / (4 * c + 3 * e) /" ^
1154 "(9 * a \<up> 2 + - 16 * b \<up> 2)");
1155 NONE = cancel_p_ thy t;
1157 if UnparseC.term t = "1 / (4 * c + 3 * e)" then ()
1158 else error "rational.sml: diff.behav. in norm_Rational_mg 24";
1161 "----- S.K. corrected non-termination 060904";
1162 val t = TermC.str2term "(3*a - 4*b) * (3*a+4*b)/((4*c+3*e)*(9*a \<up> 2 - 16*b \<up> 2))";
1163 val SOME (t, _) = rewrite_set_ thy false make_polynomial t;
1164 if UnparseC.term t =
1165 "(9 * a \<up> 2 + - 16 * b \<up> 2) /\n(36 * a \<up> 2 * c + 27 * a \<up> 2 * e + - 64 * b \<up> 2 * c +\n - 48 * b \<up> 2 * e)"
1166 then () else error "rational.sml: S.K.8..corrected 060904-6";
1168 "----- S.K. corrected non-termination of cancel_p_";
1169 val t'' = TermC.str2term ("(9 * a \<up> 2 + - 16 * b \<up> 2) /" ^
1170 "(36 * a \<up> 2 * c + (27 * a \<up> 2 * e + (-64 * b \<up> 2 * c + -48 * b \<up> 2 * e)))");
1171 (* /--- DOES NOT TERMINATE AT TRANSITION isabisac15 --> Isabelle2017 --------------------------\
1172 val SOME (t',_) = rewrite_set_ thy false cancel_p t'';
1173 if UnparseC.term t' = "1 / (4 * c + 3 * e)"
1174 then () else error "rational.sml: diff.behav. in cancel_p S.K.8";
1175 \--- DOES NOT TERMINATE AT TRANSITION isabisac15 --> Isabelle2017 --------------------------/*)
1177 (*------- Schalk I, p.68 Nr. 437b*)
1178 val t = TermC.str2term "(a + b)/(x \<up> 2 - y \<up> 2) * ((x - y) \<up> 2/(a \<up> 2 - b \<up> 2))";
1179 (*val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1181 #### rls: cancel_p on: (a * x \<up> 2 + - 2 * (a * (x * y)) + a * y \<up> 2 + b * x \<up> 2 +
1182 - 2 * (b * (x * y)) +
1184 (a \<up> 2 * x \<up> 2 + - 1 * (a \<up> 2 * y \<up> 2) + - 1 * (b \<up> 2 * x \<up> 2) +
1185 b \<up> 2 * y \<up> 2)
1186 exception Div raised
1189 (*------- SRM Schalk I, p.68 Nr. 438a *)
1190 val t = TermC.str2term "x*y / (x*y - y \<up> 2) * (x \<up> 2 - x*y)";
1191 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1192 if UnparseC.term t = "x \<up> 2"
1193 then () else error "rational.sml: diff.behav. in norm_Rational_mg 24";
1195 (*------- SRM Schalk I, p.68 Nr. 439b *)
1196 val t = TermC.str2term "(4*x \<up> 2 + 4*x + 1) * ((x \<up> 2 - 2*x \<up> 3) / (4*x \<up> 2 + 2*x))";
1197 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1198 if UnparseC.term t = "(x + - 4 * x \<up> 3) / 2"
1199 then () else error "rational.sml: diff.behav. in norm_Rational_mg 25";
1201 (*------- SRM Schalk I, p.68 Nr. 440a *)
1202 val t = TermC.str2term "(x \<up> 2 - 2*x) / (x \<up> 2 - 3*x) * (x - 3) \<up> 2 / (x \<up> 2 - 4)";
1203 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1204 if UnparseC.term t = "(- 3 + x) / (2 + x)"
1205 then () else error "rational.sml: diff.behav. in norm_Rational_mg 26";
1207 "----- Schalk I, p.68 Nr. 440b SK11 works since 0707xx";
1208 val t = TermC.str2term "(a \<up> 3 - 9*a) / (a \<up> 3*b - a*b \<up> 3) * (a \<up> 2*b + a*b \<up> 2) / (a+3)";
1209 (* WN130911 non-termination for unclear reasons:
1210 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1212 ... ENDS WITH THIS TRACE:
1214 ### rls: cancel_p on: (-9 * (a \<up> 3 * b) + -9 * (a \<up> 2 * b \<up> 2) + a \<up> 5 * b +
1215 a \<up> 4 * b \<up> 2) /
1216 (a \<up> 3 * b + - 1 * (a * b \<up> 3)) /
1218 BUT THIS IS CORRECTLY RECOGNISED
1219 val t = TermC.str2term
1220 ("(-9 * (a \<up> 3 * b) + -9 * (a \<up> 2 * b \<up> 2) + a \<up> 5 * b + a \<up> 4 * b \<up> 2) /" ^
1221 "(a \<up> 3 * b + - 1 * (a * b \<up> 3)) / (3 + (a::real))");
1223 NONE = cancel_p_ thy t;
1225 if UnparseC.term t = "(-3 * a + a \<up> 2) / (a + - 1 * b)" then ()
1226 else error "rational.sml: diff.behav. in norm_Rational 27";
1229 "----- SK12 works since 0707xx";
1230 val t = TermC.str2term "(a \<up> 3 - 9*a) * (a \<up> 2*b+a*b \<up> 2) / ((a \<up> 3*b - a*b \<up> 3) * (a+3))";
1231 (* WN130911 non-termination due to non-termination of
1232 cancel_p_ thy (TermC.str2term "(4 + -4 * b \<up> 2) / (a \<up> 4 + - 2 * (a \<up> 2 * b \<up> 2) + b \<up> 4)")
1234 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1235 if UnparseC.term t' = "(-3 * a + a \<up> 2) / (a + - 1 * b)" then ()
1236 else error "rational.sml: diff.behav. in norm_Rational 28";
1239 "-------- examples common denominator and multiplication from: Schalk --------";
1240 "-------- examples common denominator and multiplication from: Schalk --------";
1241 "-------- examples common denominator and multiplication from: Schalk --------";
1242 (*------- SRAM Schalk I, p.69 Nr. 441b *)
1243 val t = TermC.str2term "(4*a/3 + 3*b \<up> 2/a \<up> 3 + b/(4*a))*(4*b/(3*a))";
1244 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1245 if UnparseC.term t = "(36 * b \<up> 3 + 3 * a \<up> 2 * b \<up> 2 + 16 * a \<up> 4 * b) /\n(9 * a \<up> 4)"
1246 then () else error "rational.sml: diff.behav. in norm_Rational_mg 28";
1248 (*------- SRAM Schalk I, p.69 Nr. 442b *)
1249 val t = TermC.str2term ("(15*a \<up> 2/x \<up> 3 - 5*b \<up> 4/x \<up> 2 + 25*c \<up> 2/x) * " ^
1250 "(x \<up> 3/(5*a*b \<up> 3*c \<up> 3)) + 1/c \<up> 3 * (b*x/a - 3*a/b \<up> 3)");
1251 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1252 if UnparseC.term t = "5 * x \<up> 2 / (a * b \<up> 3 * c)"
1253 then () else error "rational.sml: diff.behav. in norm_Rational_mg 29";
1255 (*------- SRAM Schalk I, p.69 Nr. 443b *)
1256 val t = TermC.str2term "(a/2 + b/3) * (b/3 - a/2)";
1257 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1258 if UnparseC.term t = "(- 9 * a \<up> 2 + 4 * b \<up> 2) / 36"
1259 then () else error "rational.sml: diff.behav. in norm_Rational_mg 30";
1261 (*------- SRAM Schalk I, p.69 Nr. 445b *)
1262 val t = TermC.str2term "(a \<up> 2/9 + 2*a/(3*b) + 4/b \<up> 2)*(a/3 - 2/b) + 8/b \<up> 3";
1263 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1264 if UnparseC.term t = "a \<up> 3 / 27"
1265 then () else error "rational.sml: diff.behav. in norm_Rational_mg 31";
1267 (*------- SRAM Schalk I, p.69 Nr. 446b *)
1268 val t = TermC.str2term "(x/(5*x + 4*y) - y/(5*x - 4*y) + 1)*(25*x \<up> 2 - 16*y \<up> 2)";
1269 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1270 if UnparseC.term t = (*"30 * x \<up> 2 + -9 * x * y + - 20 * y \<up> 2" ..isabisac15 | Isabelle2017..*)
1271 "(- 30 * x \<up> 2 + 9 * x * y + 20 * y \<up> 2) / - 1"
1272 then () else error "rational.sml: diff.behav. in norm_Rational_mg 32";
1274 (*------- SRAM Schalk I, p.69 Nr. 449a *)(*Achtung: rechnet ca 8 Sekunden*)
1275 val t = TermC.str2term
1276 "(2*x \<up> 2/(3*y)+x/y \<up> 2)*(4*x \<up> 4/(9*y \<up> 2)+x \<up> 2/y \<up> 4)*(2*x \<up> 2/(3*y) - x/y \<up> 2)";
1277 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1278 if UnparseC.term t = "(- 81 * x \<up> 4 + 16 * x \<up> 8 * y \<up> 4) / (81 * y \<up> 8)"
1279 then () else error "rational.sml: diff.behav. in norm_Rational_mg 33";
1281 (*------- SRAM Schalk I, p.69 Nr. 450a *)
1282 val t = TermC.str2term
1283 "(4*x/(3*y)+2*y/(3*x)) \<up> 2 - (2*y/(3*x) - 2*x/y)*(2*y/(3*x)+2*x/y)";
1284 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1285 if UnparseC.term t = "(52 * x \<up> 2 + 16 * y \<up> 2) / (9 * y \<up> 2)"
1286 then () else error "rational.sml: diff.behav. in norm_Rational_mg 34";
1288 (*------- SRAM Schalk I, p.69 Nr. 442b --- abgewandelt*)
1289 val t = TermC.str2term
1290 ("(15*a \<up> 4/(a*x \<up> 3) - 5*a*((b \<up> 4 - 5*c \<up> 2*x) / x \<up> 2)) * " ^
1291 "(x \<up> 3/(5*a*b \<up> 3*c \<up> 3)) + a/c \<up> 3 * (x*(b/a) - 3*b*(a/b \<up> 4))");
1292 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1293 if UnparseC.term t = "5 * x \<up> 2 / (b \<up> 3 * c)"
1294 then () else error "rational.sml: diff.behav. in norm_Rational_mg 53";
1297 "-------- examples double fractions from: Mathematik 1 Schalk ----------------";
1298 "-------- examples double fractions from: Mathematik 1 Schalk ----------------";
1299 "-------- examples double fractions from: Mathematik 1 Schalk ----------------";
1300 "----- SRD Schalk I, p.69 Nr. 454b";
1301 val t = TermC.str2term "((2 - x)/(2*a)) / (2*a/(x - 2))";
1302 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1303 if UnparseC.term t = "(- 4 + 4 * x + - 1 * x \<up> 2) / (4 * a \<up> 2)"
1304 then () else error "rational.sml: diff.behav. in norm_Rational_mg 35";
1306 "----- SRD Schalk I, p.69 Nr. 455a";
1307 val t = TermC.str2term "(a \<up> 2 + 1)/(a \<up> 2 - 1) / ((a+1)/(a - 1))";
1308 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1309 if UnparseC.term t = "(1 + a \<up> 2) / (1 + 2 * a + a \<up> 2)" then ()
1310 else error "rational.sml: diff.behav. in norm_Rational_mg 36";
1312 "----- Schalk I, p.69 Nr. 455b";
1313 val t = TermC.str2term "(x \<up> 2 - 4)/(y \<up> 2 - 9)/((2+x)/(3 - y))";
1314 (* WN130911 non-termination due to non-termination of
1315 cancel_p_ thy (TermC.str2term ("(- 12 + 4 * y + 3 * x \<up> 2 + - 1 * (x \<up> 2 * y)) /" ^
1316 "(- 18 + -9 * x + 2 * y \<up> 2 + x * y \<up> 2)"))
1318 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1319 if UnparseC.term t = "(2 + - 1 * x) / (3 + y)" then ()
1320 else error "rational.sml: diff.behav. in norm_Rational_mg 37";
1323 "----- SK060904- 1a non-termination of cancel_p_ ?: worked before 0707xx";
1324 val t = TermC.str2term "(x \<up> 2 - 4)*(3 - y) / ((y \<up> 2 - 9)*(2+x))";
1325 (* WN130911 non-termination due to non-termination of
1326 cancel_p_ thy (TermC.str2term ("(- 12 + 4 * y + 3 * x \<up> 2 + - 1 * (x \<up> 2 * y)) /" ^
1327 "(- 18 + -9 * x + 2 * y \<up> 2 + x * y \<up> 2)"))
1329 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1330 if UnparseC.term t = "(2 + - 1 * x) / (3 + y)" then ()
1331 else error "rational.sml: diff.behav. in norm_Rational_mg 37b";
1334 "----- ?: worked before 0707xx";
1335 val t = TermC.str2term "(3 + - 1 * y) / (-9 + y \<up> 2)";
1336 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1337 if UnparseC.term t = "- 1 / (3 + y)"
1338 then () else error "rational.sml: - 1 / (3 + y) norm_Rational";
1340 "----- SRD Schalk I, p.69 Nr. 456b";
1341 val t = TermC.str2term "(b \<up> 3 - b \<up> 2) / (b \<up> 2+b) / (b \<up> 2 - 1)";
1342 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1343 if UnparseC.term t = "b / (1 + 2 * b + b \<up> 2)"
1344 then () else error "rational.sml: diff.behav. in norm_Rational_mg 38";
1346 "----- SRD Schalk I, p.69 Nr. 457b";
1347 val t = TermC.str2term "(16*a \<up> 2 - 9*b \<up> 2)/(2*a+3*a*b) / ((4*a+3*b)/(4*a \<up> 2 - 9*a \<up> 2*b \<up> 2))";
1348 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1349 if UnparseC.term t = "8 * a \<up> 2 + - 6 * a * b + - 12 * a \<up> 2 * b + 9 * a * b \<up> 2"
1350 then () else error "rational.sml: diff.behav. in norm_Rational_mg 39";
1352 "----- Schalk I, p.69 Nr. 458b works since 0707";
1353 val t = TermC.str2term "(2*a \<up> 2*x - a \<up> 2) / (a*x - b*x) / (b \<up> 2*(2*x - 1) / (x*(a - b)))";
1354 (*val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1356 ### rls: cancel_p on: (- 1 * a \<up> 2 + 2 * (a \<up> 2 * x)) / (a * x + - 1 * (b * x)) /
1357 ((- 1 * b \<up> 2 + 2 * (b \<up> 2 * x)) / (a * x + - 1 * (b * x)))
1358 exception Div raised
1361 val t = TermC.str2term
1362 ("(- 1 * a \<up> 2 + 2 * (a \<up> 2 * x)) / (a * x + - 1 * (b * x)) /" ^
1363 "((- 1 * b \<up> 2 + 2 * (b \<up> 2 * x)) / (a * x + - 1 * (b * x)))");
1364 NONE = cancel_p_ thy t;
1366 if UnparseC.term t = "a \<up> 2 / b \<up> 2" then ()
1367 else error "rational.sml: diff.behav. in norm_Rational_mg 39b";
1370 "----- SRD Schalk I, p.69 Nr. 459b";
1371 val t = TermC.str2term "(a \<up> 2 - b \<up> 2)/(a*b) / (4*(a+b) \<up> 2/a)";
1372 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1373 if UnparseC.term t = "(a + - 1 * b) / (4 * a * b + 4 * b \<up> 2)" then ()
1374 else error "rational.sml: diff.behav. in norm_Rational_mg 41";
1376 "----- Schalk I, p.69 Nr. 460b nonterm.SK";
1377 val t = TermC.str2term "(9*(x \<up> 2 - 8*x + 16) / (4*(y \<up> 2 - 2*y + 1))) / ((3*x - 12) / (16*y - 16))";
1378 (*val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1379 exception Div raised
1382 val t = TermC.str2term
1383 ("(144 + -72 * x + 9 * x \<up> 2) / (4 + -8 * y + 4 * y \<up> 2) /" ^
1384 "((- 12 + 3 * x) / (- 16 + 16 * y))");
1385 NONE = cancel_p_ thy t;
1387 if UnparseC.term t = !!!!!!!!!!!!!!!!!!!!!!!!!
1388 then () else error "rational.sml: diff.behav. in norm_Rational_mg 42";
1391 "----- some variant of the above; was non-terminating before";
1392 val t = TermC.str2term "9*(x \<up> 2 - 8*x+16)*(16*y - 16)/(4*(y \<up> 2 - 2*y+1)*(3*x - 12))";
1393 val SOME (t , _) = rewrite_set_ thy false norm_Rational t;
1394 if UnparseC.term t = "(48 + - 12 * x) / (1 + - 1 * y)"
1395 then () else error "some variant of the above; was non-terminating before";
1397 "----- SRD Schalk I, p.70 Nr. 472a";
1398 val t = TermC.str2term ("((8*x \<up> 2 - 32*y \<up> 2) / (2*x + 4*y)) / ((4*x - 8*y) / (x + y))");
1399 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1400 if UnparseC.term t = "x + y"
1401 then () else error "rational.sml: diff.behav. in norm_Rational_mg 43";
1403 "----- Schalk I, p.70 Nr. 478b ----- Rechenzeit: 5 sec";
1404 val t = TermC.str2term ("(a - (a*b + b \<up> 2)/(a+b))/(b+(a - b)/(1+(a+b)/(a - b))) / " ^
1405 "((a - a \<up> 2/(a+b))/(a+(a*b)/(a - b)))");
1406 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1407 if UnparseC.term t = "(2 * a \<up> 3 + 2 * a \<up> 2 * b) / (a \<up> 2 * b + b \<up> 3)"
1408 then () else error "rational.sml: diff.behav. in norm_Rational_mg 51";
1410 (*SRD Schalk I, p.69 Nr. 461a *)
1411 val t = TermC.str2term "(2/(x+3) + 2/(x - 3)) / (8*x/(x \<up> 2 - 9))";
1412 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1413 if UnparseC.term t = "1 / 2"
1414 then () else error "rational.sml: diff.behav. in norm_Rational_mg 44";
1416 (*SRD Schalk I, p.69 Nr. 464b *)
1417 val t = TermC.str2term "(a - a/(a - 2)) / (a + a/(a - 2))";
1418 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1419 if UnparseC.term t = "(- 3 + a) / (- 1 + a)"
1420 then () else error "rational.sml: diff.behav. in norm_Rational_mg 45";
1422 (*SRD Schalk I, p.69 Nr. 465b *)
1423 val t = TermC.str2term "((x+3*y)/9 + (4*y \<up> 2 - 9*z \<up> 2)/(16*x)) / (x/9 + y/6 + z/4)";
1424 (* WN130911 non-termination due to non-termination of
1425 cancel_p_ thy (TermC.str2term
1426 ("("(576 * x \<up> 2 + 1728 * (x * y) + 1296 * y \<up> 2 + - 2916 * z \<up> 2) /" ^
1427 "(576 * x \<up> 2 + 864 * (x * y) + 1296 * (x * z))"))
1429 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1430 if UnparseC.term t = "(4 * x + 6 * y + -9 * z) / (4 * x)"
1431 then () else error "rational.sml: diff.behav. in norm_Rational_mg 46";
1434 (*SRD Schalk I, p.69 Nr. 466b *)
1435 val t = TermC.str2term "((1 - 7*(x - 2)/(x \<up> 2 - 4)) / (6/(x+2))) / (3/(x+5)+30/(x \<up> 2 - 25))";
1436 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1437 if UnparseC.term t = "(25 + - 10 * x + x \<up> 2) / 18"
1438 then () else error "rational.sml: diff.behav. in norm_Rational_mg 47";
1440 (*SRD Schalk I, p.70 Nr. 469 *)
1441 val t = TermC.str2term ("3*b \<up> 2 / (4*a \<up> 2 - 8*a*b + 4*b \<up> 2) / " ^
1442 "(a / (a \<up> 2*b - b \<up> 3) + (a - b) / (4*a*b \<up> 2 + 4*b \<up> 3) - 1 / (4*b \<up> 2))");
1443 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1444 if UnparseC.term t = "- 3 * b \<up> 3 / (- 2 * a + 2 * b)"
1445 then () else error "rational.sml: diff.behav. in norm_Rational_mg 48";
1447 (*//----------------------------------TOODOO (*Rfuns revsets \<longrightarrow> broken*)
1448 "-------- me Schalk I No.186 -------------------------------------------------";
1449 "-------- me Schalk I No.186 -------------------------------------------------";
1450 "-------- me Schalk I No.186 -------------------------------------------------";
1451 val fmz = ["Term ((14 * x * y) / ( x * y ))", "normalform N"];
1453 ("Rational",["rational", "simplification"],
1454 ["simplification", "of_rationals"]);
1455 val p = e_pos'; val c = [];
1456 val (p,_,f,nxt,_,pt) = CalcTreeTEST [(fmz, (dI',pI',mI'))];
1457 val (p,_,f,nxt,_,pt) = me nxt p c pt;
1458 val (p,_,f,nxt,_,pt) = me nxt p c pt;
1459 val (p,_,f,nxt,_,pt) = me nxt p c pt;
1460 val (p,_,f,nxt,_,pt) = me nxt p c pt;
1461 val (p,_,f,nxt,_,pt) = me nxt p c pt;
1462 val (p,_,f,nxt,_,pt) = me nxt p c pt;
1463 val (p,_,f,nxt,_,pt) = me nxt p c pt;
1464 val (p,_,f,nxt,_,pt) = me nxt p c pt; f2str f;
1465 val (p,_,f,nxt,_,pt) = me nxt p c pt; f2str f;
1466 val (p,_,f,nxt,_,pt) = me nxt p c pt; f2str f;(*++ for explicit script*)
1467 val (p,_,f,nxt,_,pt) = me nxt p c pt; f2str f;(*++ for explicit script*)
1468 case (f2str f, nxt) of
1469 ("14", ("End_Proof'", _)) => ()
1470 | _ => error "rational.sml diff.behav. in me Schalk I No.186";
1471 \\----------------------------------TOODOO (*Rfuns revsets \<longrightarrow> broken*)*)
1473 (*//----------------------------------TOODOO (*Rfuns revsets \<longrightarrow> broken*)
1474 "-------- interSteps ..Simp_Rat_Double_No- 1.xml ------------------------------";
1475 "-------- interSteps ..Simp_Rat_Double_No- 1.xml ------------------------------";
1476 "-------- interSteps ..Simp_Rat_Double_No- 1.xml ------------------------------";
1478 CalcTree [(["Term (((2 - x)/(2*a)) / (2*a/(x - 2)))", "normalform N"],
1479 ("Rational", ["rational", "simplification"], ["simplification", "of_rationals"]))];
1482 autoCalculate 1 CompleteCalc;
1483 val ((pt, p), _) = get_calc 1;
1485 Test_Tool.show_pt pt;
1487 (([], Frm), Simplify ((2 - x) / (2 * a) / (2 * a / (x - 2)))),
1488 (([1], Frm), (2 - x) / (2 * a) / (2 * a / (x - 2))),
1489 (([1], Res), (2 + - 1 * x) / (2 * a) / (2 * a / (x + - 1 * 2))),
1490 (([2], Res), (2 + - 1 * x) / (2 * a) / (2 * a / (- 2 + x))),
1491 (([3], Res), (2 + - 1 * x) * (- 2 + x) / (2 * a * (2 * a))),
1492 (([4], Res), (-4 + 4 * x + - 1 * x \<up> 2) / (4 * a \<up> 2)),
1493 (([], Res), (-4 + 4 * x + - 1 * x \<up> 2) / (4 * a \<up> 2))]
1495 interSteps 1 ([1], Res);
1496 val ((pt, p), _) = get_calc 1;
1497 (*Test_Tool.show_pt pt;
1499 (([], Frm), Simplify ((2 - x) / (2 * a) / (2 * a / (x - 2)))),
1500 (([1], Frm), (2 - x) / (2 * a) / (2 * a / (x - 2))),
1501 (([1,1], Frm), (2 - x) / (2 * a) / (2 * a / (x - 2))),
1502 (([1,1], Res), (2 - x) / (2 * a) / (2 * a / (x + - 1 * 2))),
1503 (([1,2], Res), (2 + - 1 * x) / (2 * a) / (2 * a / (x + - 1 * 2))),
1504 (([1], Res), (2 + - 1 * x) / (2 * a) / (2 * a / (x + - 1 * 2))),
1505 (([2], Res), (2 + - 1 * x) / (2 * a) / (2 * a / (- 2 + x))),
1506 (([3], Res), (2 + - 1 * x) * (- 2 + x) / (2 * a * (2 * a))),
1507 (([4], Res), (-4 + 4 * x + - 1 * x \<up> 2) / (4 * a \<up> 2)),
1508 (([], Res), (-4 + 4 * x + - 1 * x \<up> 2) / (4 * a \<up> 2))]
1510 val (t, asm) = get_obj g_result pt [1, 1];
1511 if UnparseC.term t = "(2 - x) / (2 * a) / (2 * a / (x + - 1 * 2))" andalso UnparseC.terms asm = "[]"
1512 then () else error "2nd interSteps ..Simp_Rat_Double_No- 1 changed on [1, 1]";
1513 val (t, asm) = get_obj g_result pt [1, 2];
1514 if UnparseC.term t = "(2 + - 1 * x) / (2 * a) / (2 * a / (x + - 1 * 2))" andalso UnparseC.terms asm = "[]"
1515 then () else error "3rd interSteps ..Simp_Rat_Double_No- 1 changed on [1, 2]";
1516 \\----------------------------------TOODOO (*Rfuns revsets \<longrightarrow> broken*)*)
1519 (*//----------------------------------TOODOO (*Rfuns revsets \<longrightarrow> broken*)
1520 "-------- interSteps ..Simp_Rat_Cancel_No- 1.xml ------------------------------";
1521 "-------- interSteps ..Simp_Rat_Cancel_No- 1.xml ------------------------------";
1522 "-------- interSteps ..Simp_Rat_Cancel_No- 1.xml ------------------------------";
1524 CalcTree [(["Term ((a^2 + - 1*b^2) / (a^2 + - 2*a*b + b^2))", "normalform N"],
1525 ("Rational", ["rational", "simplification"], ["simplification", "of_rationals"]))];
1528 autoCalculate 1 CompleteCalc;
1529 val ((pt, p), _) = get_calc 1;
1530 (*Test_Tool.show_pt pt;
1532 (([], Frm), Simplify ((a \<up> 2 + - 1 * b \<up> 2) / (a \<up> 2 + - 2 * a * b + b \<up> 2))),
1533 (([1], Frm), (a \<up> 2 + - 1 * b \<up> 2) / (a \<up> 2 + - 2 * a * b + b \<up> 2)),
1534 (([1], Res), (a \<up> 2 + - 1 * b \<up> 2) / (a \<up> 2 + - 2 * (a * b) + b \<up> 2)),
1535 (([2], Res), (a + b) / (a + - 1 * b)),
1536 (([], Res), (a + b) / (a + - 1 * b))]
1538 interSteps 1 ([2], Res);
1539 val ((pt, p), _) = get_calc 1;
1540 (*Test_Tool.show_pt pt;
1542 (([], Frm), Simplify ((a \<up> 2 + - 1 * b \<up> 2) / (a \<up> 2 + - 2 * a * b + b \<up> 2))),
1543 (([1], Frm), (a \<up> 2 + - 1 * b \<up> 2) / (a \<up> 2 + - 2 * a * b + b \<up> 2)),
1544 (([1], Res), (a \<up> 2 + - 1 * b \<up> 2) / (a \<up> 2 + - 2 * (a * b) + b \<up> 2)),
1545 (([2,1], Frm), (a \<up> 2 + - 1 * b \<up> 2) / (a \<up> 2 + - 2 * (a * b) + b \<up> 2)),
1546 (([2,1], Res), (a + b) / (a + - 1 * b)),
1547 (([2], Res), (a + b) / (a + - 1 * b)),
1548 (([], Res), (a + b) / (a + - 1 * b))]
1550 interSteps 1 ([2,1],Res);
1551 val ((pt, p), _) = get_calc 1;
1552 (*Test_Tool.show_pt pt;
1554 (([], Frm), Simplify ((a \<up> 2 + - 1 * b \<up> 2) / (a \<up> 2 + - 2 * a * b + b \<up> 2))),
1555 (([1], Frm), (a \<up> 2 + - 1 * b \<up> 2) / (a \<up> 2 + - 2 * a * b + b \<up> 2)),
1556 (([1], Res), (a \<up> 2 + - 1 * b \<up> 2) / (a \<up> 2 + - 2 * (a * b) + b \<up> 2)),
1557 (([2,1], Frm), (a \<up> 2 + - 1 * b \<up> 2) / (a \<up> 2 + - 2 * (a * b) + b \<up> 2)),
1558 (([2,1,1], Frm), (a \<up> 2 + - 1 * b \<up> 2) / (a \<up> 2 + - 2 * (a * b) + b \<up> 2)),
1559 (([2,1,1], Res), (a \<up> 2 + - 1 * (a * b) + a * b + - 1 * b \<up> 2) /
1560 (a \<up> 2 + - 2 * (a * b) + 1 * b \<up> 2)),
1561 (([2,1,2], Res), (a \<up> 2 + - 1 * (a * b) + a * b + - 1 * b \<up> 2) /
1562 (a \<up> 2 + - 2 * (a * b) + - 1 \<up> 2 * b \<up> 2)),
1563 (([2,1,3], Res), (a \<up> 2 + - 1 * (a * b) + a * b + - 1 * b \<up> 2) /
1564 (a \<up> 2 + - 2 * (a * b) + (- 1 * b) \<up> 2)),
1565 (([2,1,4], Res), (a * a + - 1 * (a * b) + a * b + - 1 * b \<up> 2) /
1566 (a \<up> 2 + - 2 * (a * b) + (- 1 * b) \<up> 2)),
1567 (([2,1,5], Res), (a * a + - 1 * (a * b) + a * b + - 1 * (b * b)) /
1568 (a \<up> 2 + - 2 * (a * b) + (- 1 * b) \<up> 2)),
1569 (([2,1,6], Res), (a * a + - 1 * (a * b) + a * b + - 1 * (b * b)) /
1570 (a \<up> 2 + - 1 * (2 * (a * b)) + (- 1 * b) \<up> 2)),
1571 (([2,1,7], Res), (a * a + a * (- 1 * b) + (b * a + b * (- 1 * b))) /
1572 (a \<up> 2 + 2 * (a * (- 1 * b)) + (- 1 * b) \<up> 2)),
1573 (([2,1,8], Res), (a * a + a * (- 1 * b) + (b * a + b * (- 1 * b))) /
1574 (a \<up> 2 + 2 * a * (- 1 * b) + (- 1 * b) \<up> 2)),
1575 (([2,1,9], Res), (a * (a + - 1 * b) + (b * a + b * (- 1 * b))) /
1576 (a \<up> 2 + 2 * a * (- 1 * b) + (- 1 * b) \<up> 2)),
1577 (([2,1,10], Res), (a * (a + - 1 * b) + b * (a + - 1 * b)) /
1578 (a \<up> 2 + 2 * a * (- 1 * b) + (- 1 * b) \<up> 2)),
1579 (([2,1,11], Res), (a + b) * (a + - 1 * b) / (a \<up> 2 + 2 * a * (- 1 * b) + (- 1 * b) \<up> 2)),
1580 (([2,1,12], Res), (a + b) * (a + - 1 * b) / ((a + - 1 * b) * (a + - 1 * b))),
1581 (([2,1,13], Res), (a + b) / (a + - 1 * b)),
1582 (([2,1], Res), (a + b) / (a + - 1 * b)),
1583 (([2], Res), (a + b) / (a + - 1 * b)),
1584 (([], Res), (a + b) / (a + - 1 * b))]
1586 val newnds = children (get_nd pt [2,1]) (*see "fun detailrls"*);
1587 if length newnds = 13 then () else error "rational.sml: interSteps cancel_p rev_rew_p";
1589 val p = ([2,1,9],Res);
1591 val (_, tac, _) = ME_Misc.pt_extract (pt, p);
1592 case tac of SOME (Rewrite ("sym_distrib_left", _)) => ()
1593 | _ => error "rational.sml: getTactic, sym_real_plus_binom_times1";
1594 \\----------------------------------TOODOO (*Rfuns revsets \<longrightarrow> broken*)*)
1597 "-------- investigate rulesets for cancel_p ----------------------------------";
1598 "-------- investigate rulesets for cancel_p ----------------------------------";
1599 "-------- investigate rulesets for cancel_p ----------------------------------";
1600 val thy = @{theory "Rational"};
1601 val t = TermC.str2term "(a \<up> 2 + - 1*b \<up> 2) / (a \<up> 2 + - 2*a*b + b \<up> 2)";
1602 val tt = TermC.str2term "(1 * a + 1 * b) * (1 * a + - 1 * b)"(*numerator only*);
1604 "----- with rewrite_set_";
1605 val SOME (tt',asm) = rewrite_set_ thy false make_polynomial tt;
1606 if UnparseC.term tt'= "a \<up> 2 + - 1 * b \<up> 2" then () else error "rls chancel_p 1";
1607 val tt = TermC.str2term "((1 * a + - 1 * b) * (1 * a + - 1 * b))"(*denominator only*);
1608 val SOME (tt',asm) = rewrite_set_ thy false make_polynomial tt;
1609 if UnparseC.term tt' = "a \<up> 2 + - 2 * a * b + b \<up> 2" then () else error "rls chancel_p 2";
1611 "----- with Derive.do_one; WN1130912 not investigated further, will be discontinued";
1612 val SOME (tt, _) = factout_p_ thy t;
1613 if UnparseC.term tt = "(a + b) * (a + - 1 * b) / ((a + - 1 * b) * (a + - 1 * b))"
1614 then () else error "rls chancel_p 3";
1616 "--- with simpler ruleset";
1617 val {rules, rew_ord= (_, ro), ...} = Rule_Set.rep (assoc_rls "rev_rew_p");
1618 val der = Derive.do_one thy Atools_erls rules ro NONE tt;
1619 if length der = 12 then () else error "WN1130912 rls chancel_p 4";
1620 (*default_print_depth 99;*) writeln (Derive.deriv2str der); (*default_print_depth 3;*)
1622 (*default_print_depth 99;*) map (UnparseC.term o #1) der; (*default_print_depth 3;*)
1623 "...,(- 1 * b \<up> 2 + a \<up> 2) / (- 2 * (a * b) + a \<up> 2 + (- 1 * b) \<up> 2) ]";
1624 (*default_print_depth 99;*) map (Rule.to_string o #2) der; (*default_print_depth 3;*)
1625 (*default_print_depth 99;*) map (UnparseC.term o #1 o #3) der; (*default_print_depth 3;*)
1627 val der = Derive.do_one thy Atools_erls rules ro NONE
1628 (TermC.str2term "(1 * a + 1 * b) * (1 * a + - 1 * b)");
1629 (*default_print_depth 99;*) writeln (Derive.deriv2str der); (*default_print_depth 3;*)
1631 val {rules, rew_ord=(_,ro),...} = Rule_Set.rep (assoc_rls "rev_rew_p");
1632 val der = Derive.do_one thy Atools_erls rules ro NONE
1633 (TermC.str2term "(1 * a + - 1 * b) * (1 * a + - 1 * b)");
1634 (*default_print_depth 99;*) writeln (Derive.deriv2str der); (*default_print_depth 3;*)
1635 (*default_print_depth 99;*) map (UnparseC.term o #1) der; (*default_print_depth 3;*)
1636 (*WN060829 ...postponed*)
1639 "-------- fun eval_get_denominator -------------------------------------------";
1640 "-------- fun eval_get_denominator -------------------------------------------";
1641 "-------- fun eval_get_denominator -------------------------------------------";
1642 val thy = @{theory Isac_Knowledge};
1643 val t = Thm.term_of (the (TermC.parse thy "get_denominator ((a +x)/b)"));
1644 val SOME (_, t') = eval_get_denominator "" 0 t thy;
1645 if UnparseC.term t' = "get_denominator ((a + x) / b) = b"
1646 then () else error "get_denominator ((a + x) / b) = b"
1649 "-------- several errpats in complicated term --------------------------------";
1650 "-------- several errpats in complicated term --------------------------------";
1651 "-------- several errpats in complicated term --------------------------------";
1652 (*WN12xxxx TODO: instead of Gabriella's example here (27.Jul.12) find a simpler one
1653 WN130912: kept this test, although not clear what for*)
1655 CalcTree [(["Term ((5*b + 25)/(a^2 - b^2) * (a - b)/(5*b))", "normalform N"],
1656 ("Rational", ["rational", "simplification"], ["simplification", "of_rationals"]))];
1659 autoCalculate 1 CompleteCalc;
1660 val ((pt, p), _) = get_calc 1;
1661 (*Test_Tool.show_pt pt;
1663 (([], Frm), Simplify ((5 * b + 25) / (a \<up> 2 - b \<up> 2) * (a - b) / (5 * b))),
1664 (([1], Frm), (5 * b + 25) / (a \<up> 2 - b \<up> 2) * (a - b) / (5 * b)),
1665 (([1], Res), (5 * b + 25) / (a \<up> 2 + - 1 * b \<up> 2) * (a + - 1 * b) / (5 * b)),
1666 (([2], Res), (5 * b + 25) * (a + - 1 * b) / (a \<up> 2 + - 1 * b \<up> 2) / (5 * b)),
1667 (([3], Res), (25 * a + - 25 * b + 5 * (a * b) + -5 * b \<up> 2) / (a \<up> 2 + - 1 * b \<up> 2) /
1669 (([4], Res), (25 + 5 * b) / (a + b) / (5 * b)),
1670 (([5], Res), (25 + 5 * b) / ((a + b) * (5 * b))),
1671 (([6], Res), (25 + 5 * b) / (5 * (a * b) + 5 * b \<up> 2)),
1672 (([7], Res), (5 + b) / (a * b + b \<up> 2)),
1673 (([], Res), (5 + b) / (a * b + b \<up> 2))] *)
1676 "-------- WN1309xx non-terminating rls norm_Rational -------------------------";
1677 "-------- WN1309xx non-terminating rls norm_Rational -------------------------";
1678 "-------- WN1309xx non-terminating rls norm_Rational -------------------------";
1679 (*------- Schalk I, p.70 Nr. 480b; a/b : c/d translated to a/b * d/c*)
1680 val t = TermC.str2term
1681 ("((12*x*y / (9*x \<up> 2 - y \<up> 2)) / (1 / (3*x - y) \<up> 2 - 1 / (3*x + y) \<up> 2)) * " ^
1682 "((1/(x - 5*y) \<up> 2 - 1/(x + 5*y) \<up> 2) / (20*x*y / (x \<up> 2 - 25*y \<up> 2)))");
1684 (*1st factor separately simplified *)
1685 val t = TermC.str2term "((12*x*y / (9*x \<up> 2 - y \<up> 2)) / (1 / (3*x - y) \<up> 2 - 1 / (3*x + y) \<up> 2))";
1686 val SOME (t', _) = rewrite_set_ thy false norm_Rational t;
1687 if UnparseC.term t' = "(- 9 * x \<up> 2 + y \<up> 2) / - 1" then () else error "Nr. 480b lhs changed";
1688 (*2nd factor separately simplified *)
1689 val t = TermC.str2term "((1/(x - 5*y) \<up> 2 - 1/(x + 5*y) \<up> 2) / (20*x*y / (x \<up> 2 - 25*y \<up> 2)))";
1690 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; UnparseC.term t';
1691 if UnparseC.term t' = "- 1 / (- 1 * x \<up> 2 + 25 * y \<up> 2)" then () else error "Nr. 480b rhs changed";
1693 "-------- Schalk I, p.70 Nr. 477a: terms are exploding ?!?";
1694 val t = TermC.str2term ("b*y/(b - 2*y)/((b \<up> 2 - y \<up> 2)/(b+2*y)) /" ^
1695 "(b \<up> 2*y + b*y \<up> 2) * (a+x) \<up> 2 / ((b \<up> 2 - 4*y \<up> 2) * (a+2*x) \<up> 2)");
1696 (*val SOME (t',_) = rewrite_set_ thy false norm_Rational t;
1698 ### rls: cancel_p on: (a \<up> 2 * (b * y) + 2 * (a * (b * (x * y))) + b * (x \<up> 2 * y)) /
1700 ((b \<up> 2 + - 1 * y \<up> 2) / (b + 2 * y)) /
1701 (b \<up> 2 * y + b * y \<up> 2) /
1702 (a \<up> 2 * b \<up> 2 + -4 * (a \<up> 2 * y \<up> 2) + 4 * (a * (b \<up> 2 * x)) +
1703 - 16 * (a * (x * y \<up> 2)) +
1704 4 * (b \<up> 2 * x \<up> 2) +
1705 - 16 * (x \<up> 2 * y \<up> 2))
1706 exception Div raised
1709 val t = TermC.str2term
1710 ("(a \<up> 2 * (b * y) + 2 * (a * (b * (x * y))) + b * (x \<up> 2 * y)) /" ^
1712 "((b \<up> 2 + - 1 * y \<up> 2) / (b + 2 * y)) /" ^
1713 "(b \<up> 2 * y + b * y \<up> 2) /" ^
1714 "(a \<up> 2 * b \<up> 2 + -4 * (a \<up> 2 * y \<up> 2) + 4 * (a * (b \<up> 2 * x)) +" ^
1715 "- 16 * (a * (x * y \<up> 2)) +" ^
1716 "4 * (b \<up> 2 * x \<up> 2) +" ^
1717 "- 16 * (x \<up> 2 * y \<up> 2))");
1718 NONE = cancel_p_ thy t;
1721 (*------- Schalk I, p.70 Nr. 476b in 2003 this worked using 10 sec. *)
1722 val t = TermC.str2term
1723 ("((a \<up> 2 - b \<up> 2)/(2*a*b) + 2*a*b/(a \<up> 2 - b \<up> 2)) / ((a \<up> 2 + b \<up> 2)/(2*a*b) + 1) / " ^
1724 "((a \<up> 2 + b \<up> 2) \<up> 2 / (a + b) \<up> 2)");
1725 (* Rewrite.trace_on := true; (*true false*)
1726 rewrite_set_ thy false norm_Rational t;
1728 #### rls: cancel_p on: (2 * (a \<up> 7 * b) + 4 * (a \<up> 6 * b \<up> 2) + 6 * (a \<up> 5 * b \<up> 3) +
1729 8 * (a \<up> 4 * b \<up> 4) +
1730 6 * (a \<up> 3 * b \<up> 5) +
1731 4 * (a \<up> 2 * b \<up> 6) +
1732 2 * (a * b \<up> 7)) /
1733 (2 * (a \<up> 9 * b) + 4 * (a \<up> 8 * b \<up> 2) +
1734 2 * (2 * (a \<up> 7 * b \<up> 3)) +
1735 4 * (a \<up> 6 * b \<up> 4) +
1736 -4 * (a \<up> 4 * b \<up> 6) +
1737 -4 * (a \<up> 3 * b \<up> 7) +
1738 -4 * (a \<up> 2 * b \<up> 8) +
1739 - 2 * (a * b \<up> 9))
1741 if UnparseC.term t = "1 / (a \<up> 2 + - 1 * b \<up> 2)" then ()
1742 else error "rational.sml: diff.behav. in norm_Rational_mg 49";
1745 "-------- Schalk I, p.70 Nr. 480a: terms are exploding ?!?";
1746 val t = TermC.str2term ("(1/x + 1/y + 1/z) / (1/x - 1/y - 1/z) / " ^
1747 "(2*x \<up> 2 / (x \<up> 2 - z \<up> 2) / (x / (x + z) + x / (x - z)))");
1748 (* Rewrite.trace_on := true; (*true false*)
1749 rewrite_set_ thy false norm_Rational t;
1751 #### rls: cancel_p on: (2 * (x \<up> 6 * (y \<up> 2 * z)) + 2 * (x \<up> 6 * (y * z \<up> 2)) +
1752 2 * (x \<up> 5 * (y \<up> 2 * z \<up> 2)) +
1753 - 2 * (x \<up> 4 * (y \<up> 2 * z \<up> 3)) +
1754 - 2 * (x \<up> 4 * (y * z \<up> 4)) +
1755 - 2 * (x \<up> 3 * (y \<up> 2 * z \<up> 4))) /
1756 (- 2 * (x \<up> 6 * (y \<up> 2 * z)) + - 2 * (x \<up> 6 * (y * z \<up> 2)) +
1757 2 * (x \<up> 5 * (y \<up> 2 * z \<up> 2)) +
1758 2 * (x \<up> 4 * (y \<up> 2 * z \<up> 3)) +
1759 2 * (x \<up> 4 * (y * z \<up> 4)) +
1760 - 2 * (x \<up> 3 * (y \<up> 2 * z \<up> 4)))
1763 "-------- Schalk I, p.60 Nr. 215d: terms are exploding, internal loop does not terminate";
1764 val t = TermC.str2term "(a-b) \<up> 3 * (x+y) \<up> 4 / ((x+y) \<up> 2 * (a-b) \<up> 5)";
1765 (* Kein Wunder, denn Z???ler und Nenner extra als Polynom dargestellt ergibt:
1767 val t = TermC.str2term "(a-b) \<up> 3 * (x+y) \<up> 4";
1768 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1770 "a \<up> 3 * x \<up> 4 + 4 * a \<up> 3 * x \<up> 3 * y +6 * a \<up> 3 * x \<up> 2 * y \<up> 2 +4 * a \<up> 3 * x * y \<up> 3 +a \<up> 3 * y \<up> 4 +-3 * a \<up> 2 * b * x \<up> 4 +- 12 * a \<up> 2 * b * x \<up> 3 * y +- 18 * a \<up> 2 * b * x \<up> 2 * y \<up> 2 +- 12 * a \<up> 2 * b * x * y \<up> 3 +-3 * a \<up> 2 * b * y \<up> 4 +3 * a * b \<up> 2 * x \<up> 4 +12 * a * b \<up> 2 * x \<up> 3 * y +18 * a * b \<up> 2 * x \<up> 2 * y \<up> 2 +12 * a * b \<up> 2 * x * y \<up> 3 +3 * a * b \<up> 2 * y \<up> 4 +- 1 * b \<up> 3 * x \<up> 4 +-4 * b \<up> 3 * x \<up> 3 * y +-6 * b \<up> 3 * x \<up> 2 * y \<up> 2 +-4 * b \<up> 3 * x * y \<up> 3 +- 1 * b \<up> 3 * y \<up> 4";
1771 val t = TermC.str2term "((x+y) \<up> 2 * (a-b) \<up> 5)";
1772 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1774 "a \<up> 5 * x \<up> 2 + 2 * a \<up> 5 * x * y + a \<up> 5 * y \<up> 2 +-5 * a \<up> 4 * b * x \<up> 2 +- 10 * a \<up> 4 * b * x * y +-5 * a \<up> 4 * b * y \<up> 2 +10 * a \<up> 3 * b \<up> 2 * x \<up> 2 +20 * a \<up> 3 * b \<up> 2 * x * y +10 * a \<up> 3 * b \<up> 2 * y \<up> 2 +- 10 * a \<up> 2 * b \<up> 3 * x \<up> 2 +- 20 * a \<up> 2 * b \<up> 3 * x * y +- 10 * a \<up> 2 * b \<up> 3 * y \<up> 2 +5 * a * b \<up> 4 * x \<up> 2 +10 * a * b \<up> 4 * x * y +5 * a * b \<up> 4 * y \<up> 2 +- 1 * b \<up> 5 * x \<up> 2 +- 2 * b \<up> 5 * x * y +- 1 * b \<up> 5 * y \<up> 2";
1776 anscheinend macht dem Rechner das Krzen diese Bruches keinen Spass mehr ...*)
1778 "-------- Schalk I, p.70 Nr. 480b: terms are exploding, Rewrite.trace_on stops at";
1779 val t = TermC.str2term ("((12*x*y/(9*x \<up> 2 - y \<up> 2))/" ^
1780 "(1/(3*x - y) \<up> 2 - 1/(3*x + y) \<up> 2)) *" ^
1781 "(1/(x - 5*y) \<up> 2 - 1/(x + 5*y) \<up> 2)/" ^
1782 "(20*x*y/(x \<up> 2 - 25*y \<up> 2))");
1783 (*val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1785 #### rls: cancel_p on: (19440 * (x \<up> 8 * y \<up> 2) + -490320 * (x \<up> 6 * y \<up> 4) +
1786 108240 * (x \<up> 4 * y \<up> 6) +
1787 -6000 * (x \<up> 2 * y \<up> 8)) /
1788 (2160 * (x \<up> 8 * y \<up> 2) + - 108240 * (x \<up> 6 * y \<up> 4) +
1789 1362000 * (x \<up> 4 * y \<up> 6) +
1790 - 150000 * (x \<up> 2 * y \<up> 8))