1 (* Title: "Knowledge/rational-2.sml"
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 "-------- rewrite_set_ cancel_p from: Mathematik 1 Schalk Reniets Verlag -----";
21 "-------- rewrite_set_ add_fractions_p from: Mathematik 1 Schalk -------------";
22 "-------- integration lev.1 -- lev.5: cancel_p_ & add_fractions_p_ -----------";
23 "Rfuns-------- reverse rewrite ----------------------------------------------------";
24 "Rfuns-------- 'reverse-ruleset' cancel_p -----------------------------------------";
25 "-------- investigate rls norm_Rational --------------------------------------";
26 "-------- examples: rls norm_Rational ----------------------------------------";
27 "-------- rational numerals --------------------------------------------------";
28 "-------- examples cancellation from: Mathematik 1 Schalk --------------------";
29 "-------- examples common denominator from: Mathematik 1 Schalk --------------";
30 "-------- examples multiply and cancel from: Mathematik 1 Schalk -------------";
31 "-------- examples common denominator and multiplication from: Schalk --------";
32 "-------- examples double fractions from: Mathematik 1 Schalk ----------------";
33 "-------- me Schalk I No.186 -------------------------------------------------";
34 "-------- interSteps ..Simp_Rat_Double_No-1.xml ------------------------------";
35 "-------- interSteps ..Simp_Rat_Cancel_No-1.xml ------------------------------";
36 "-------- investigate rulesets for cancel_p ----------------------------------";
37 "-------- fun eval_get_denominator -------------------------------------------";
38 "-------- several errpats in complicated term --------------------------------";
39 "-------- WN1309xx non-terminating rls norm_Rational -------------------------";
40 "-----------------------------------------------------------------------------";
41 "-----------------------------------------------------------------------------";
44 "-------- integration lev.1 fun factout_p_ -----------------------------------";
45 "-------- integration lev.1 fun factout_p_ -----------------------------------";
46 "-------- integration lev.1 fun factout_p_ -----------------------------------";
47 val t = TermC.str2term "(x \<up> 2 + - 1*y \<up> 2) / (x \<up> 2 + - 1*x*y)"
48 val SOME (t', asm) = factout_p_ thy t;
49 if UnparseC.term t' = "(x + y) * (x + - 1 * y) / (x * (x + - 1 * y))"
50 then () else error ("factout_p_ term 1 changed: " ^ UnparseC.term t')
52 if UnparseC.terms asm = "[\"x \<noteq> 0\", \"x + - 1 * y \<noteq> 0\"]"
53 then () else error "factout_p_ asm 1 changed"
55 val t = TermC.str2term "nothing + to_cancel ::real";
56 if NONE = factout_p_ thy t then () else error "factout_p_ doesn't report non-applicable";
58 val t = TermC.str2term "((3 * x \<up> 2 + 6 *x + 3) / (2*x + 2))";
59 val SOME (t', asm) = factout_p_ thy t;
60 if UnparseC.term t' = "(3 + 3 * x) * (1 + x) / (2 * (1 + x))" andalso
61 UnparseC.terms asm = "[\"1 + x \<noteq> 0\"]"
62 then () else error "factout_p_ 1 changed";
64 "-------- integration lev.1 fun cancel_p_ ------------------------------------";
65 "-------- integration lev.1 fun cancel_p_ ------------------------------------";
66 "-------- integration lev.1 fun cancel_p_ ------------------------------------";
67 val t = TermC.str2term "(x \<up> 2 + - 1*y \<up> 2) / (x \<up> 2 + - 1*x*y)"
68 val SOME (t', asm) = cancel_p_ thy t;
69 if (UnparseC.term t', UnparseC.terms asm) = ("(x + y) / x", "[\"x \<noteq> 0\"]")
70 then () else error ("cancel_p_ (t', asm) 1 changed: " ^ UnparseC.term t')
72 val t = TermC.str2term "nothing + to_cancel ::real";
73 if NONE = cancel_p_ thy t then () else error "cancel_p_ doesn't report non-applicable";
75 val t = TermC.str2term "((3 * x \<up> 2 + 6 *x + 3) / (2*x + 2))";
76 val SOME (t', asm) = cancel_p_ thy t;
77 if UnparseC.term t' = "(3 + 3 * x) / 2" andalso UnparseC.terms asm = "[]"
78 then () else error "cancel_p_ 1 changed";
80 "-------- integration lev.1 fun common_nominator_p_ --------------------------";
81 "-------- integration lev.1 fun common_nominator_p_ --------------------------";
82 "-------- integration lev.1 fun common_nominator_p_ --------------------------";
83 val t = TermC.str2term ("y / (a*x + b*x + c*x) " ^
87 val SOME (t', asm) = common_nominator_p_ thy t;
89 ("y * y / (x * ((a + b + c) * y)) " ^
90 (* n1 *d2'/ (c'* ( d1' *d2')) *)
91 "+ a * (a + b + c) / (x * ((a + b + c) * y))")
92 (* n2 * d1' / (c'* ( d1' *d2')) *)
93 then () else error "common_nominator_p_ term 1 changed";
94 if UnparseC.terms asm = "[\"a + b + c \<noteq> 0\", \"y \<noteq> 0\", \"x \<noteq> 0\"]"
95 then () else error "common_nominator_p_ asm 1 changed"
97 "-------- example in mail Nipkow";
98 val t = TermC.str2term "x/(x \<up> 2 + - 1*y \<up> 2) + y/(x \<up> 2 + - 1*x*y)";
99 val SOME (t', asm) = common_nominator_p_ thy t;
100 if UnparseC.term t' =
101 "x * x / ((x + - 1 * y) * ((x + y) * x)) +\ny * (x + y) / ((x + - 1 * y) * ((x + y) * x))"
102 then () else error "common_nominator_p_ term 2 changed"
104 if UnparseC.terms asm = "[\"x + y \<noteq> 0\", \"x \<noteq> 0\", \"x + - 1 * y \<noteq> 0\"]"
105 then () else error "common_nominator_p_ asm 2 changed"
107 "-------- example: applicable tested by SML code";
108 val t = TermC.str2term "nothing / to_add";
109 if NONE = common_nominator_p_ thy t then () else error "common_nominator_p_ term 3 changed";
111 val t = TermC.str2term "((x + (- 1)) / (x + 1)) + ((x + 1) / (x + (- 1)))";
112 val SOME (t', asm) = common_nominator_p_ thy t;
113 if UnparseC.term t' =
114 "(x + - 1) * (- 1 + x) / ((1 + x) * (- 1 + x)) +\n(x + 1) * (1 + x) / ((1 + x) * (- 1 + x))"
115 andalso UnparseC.terms asm = "[\"1 + x \<noteq> 0\", \"- 1 + x \<noteq> 0\"]"
116 then () else error "common_nominator_p_ 3 changed";
118 "-------- integration lev.1 fun add_fraction_p_ ------------------------------";
119 "-------- integration lev.1 fun add_fraction_p_ ------------------------------";
120 "-------- integration lev.1 fun add_fraction_p_ ------------------------------";
121 val t = TermC.str2term "((x + (- 1)) / (x + 1)) + ((x + 1) / (x + (- 1)))";
122 val SOME (t', asm) = add_fraction_p_ thy t;
123 if UnparseC.term t' = "(2 + 2 * x \<up> 2) / (- 1 + x \<up> 2)"
124 then () else error "add_fraction_p_ 3 changed";
126 if UnparseC.terms asm = "[\"- 1 + x \<up> 2 \<noteq> 0\"]"
127 then () else error "add_fraction_p_ 3 changed";
129 val t = TermC.str2term "nothing / to_add";
130 if NONE = add_fraction_p_ thy t then () else error "add_fraction_p_ term 3 changed";
132 val t = TermC.str2term "((x + (- 1)) / (x + 1)) + ((x + 1) / (x + (- 1)))";
133 val SOME (t', asm) = add_fraction_p_ thy t;
134 if UnparseC.term t' = "(2 + 2 * x \<up> 2) / (- 1 + x \<up> 2)" andalso
135 UnparseC.terms asm = "[\"- 1 + x \<up> 2 \<noteq> 0\"]"
136 then () else error "add_fraction_p_ 3 changed";
138 "-------- and app_rev ...traced down from rewrite_set_ until prepats ---------";
139 "-------- and app_rev ...traced down from rewrite_set_ until prepats ---------";
140 "-------- and app_rev ...traced down from rewrite_set_ until prepats ---------";
141 (* trace down until prepats are evaluated
142 (which does not to work, because substitution is not done -- compare rew_sub!);
143 keep this sequence for the case, factout_p, cancel_p, common_nominator_p, add_fraction_p
144 (again) get prepat = [] changed to <>[]. *)
145 val t = TermC.str2term "(x \<up> 2 + - 1*y \<up> 2) / (x \<up> 2 + - 1*x*y)";
147 (*rewrite_set_ @{theory Isac_Knowledge} true cancel t = NONE; !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!*)
148 "~~~~~ fun rewrite_set_, args:"; val (thy, bool, rls, term) = (thy, false, cancel_p, t);
149 "~~~~~ fun rewrite__set_, args:"; val (thy, i, _, _, (rrls as Rrls _), t) =
150 (thy, 1, bool, [], rls, term);
151 (*val (t', asm, rew) = app_rev thy (i+1) rrls t; rew = false!!!!!!!!!!!!!!!!!!!!!*)
152 "~~~~~ and app_rev, args:"; val (thy, i, rrls, t) = (thy, (i+1), rrls, t);
153 fun chk_prepat thy erls [] t = true
154 | chk_prepat thy erls prepat t =
156 fun chk (pres, pat) =
158 val subst: Type.tyenv * Envir.tenv =
159 Pattern.match thy (pat, t) (Vartab.empty, Vartab.empty)
161 snd (eval__true thy (i + 1) (map (Envir.subst_term subst) pres) [] erls)
162 end) handle Pattern.MATCH => false
163 fun scan_ f [] = false (*scan_ NEVER called by []*)
164 | scan_ f (pp::pps) =
165 if f pp then true else scan_ f pps;
166 in scan_ chk prepat end;
167 (* apply the normal_form of a rev-set *)
168 fun app_rev' thy (Rrls{erls,prepat,scr=Rfuns{normal_form,...},...}) t =
169 if chk_prepat thy erls prepat t
170 then ((*tracing("### app_rev': t = "^UnparseC.term t);*) normal_form t)
172 (* val opt = app_rev' thy rrls t ..NONE!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!*)
173 "~~~~~ and app_rev', args:"; val (thy, (Rrls{erls,prepat,scr=Rfuns{normal_form,...},...}), t) =
175 (* chk_prepat thy erls prepat t = false!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!*)
176 (* app_sub thy i rrls t = false!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!*)
177 "~~~~~ fun chk_prepat, args:"; val (thy, erls, prepat, t) = (thy, erls, prepat, t);
178 fun chk (pres, pat) =
180 val subst: Type.tyenv * Envir.tenv =
181 Pattern.match thy (pat, t) (Vartab.empty, Vartab.empty)
183 snd (eval__true thy (i + 1) (map (Envir.subst_term subst) pres) [] erls)
184 end) handle Pattern.MATCH => false
185 fun scan_ f [] = false (*scan_ NEVER called by []*)
186 | scan_ f (pp::pps) =
187 if f pp then true else scan_ f pps;
189 (*========== inhibit exn WN130823: prepat is empty ====================================
190 (* scan_ chk prepat = false!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!*)
191 "~~~~~ fun , args:"; val (f, (pp::pps)) = (chk, prepat);
193 val ([t1, t2], t) = pp;
194 UnparseC.term t1 = "?r is_expanded";
195 UnparseC.term t2 = "?s is_expanded";
196 UnparseC.term t = "?r / ?s";
197 (* f pp = false!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!*)
198 "~~~~~ fun chk, args:"; val (pres, pat) = (pp);
199 val subst: Type.tyenv * Envir.tenv =
200 Pattern.match thy (pat, t) (Vartab.empty, Vartab.empty)
202 ({}, {(("r", 0), ("real", Var (("r", 0), "real"))),
203 (("s", 0), ("real", Var (("s", 0), "real")))}*)
205 snd (eval__true thy (i + 1) (map (Envir.subst_term subst) pres) [] erls)
206 "~~~~~ fun eval__true, args:"; val (thy, i, asms, bdv, rls) =
207 (thy, (i + 1), (map (Envir.subst_term subst) pres), [], erls);
208 UnparseC.terms asms; (* = "[\"?r is_expanded\",\"?s is_expanded\"]"*)
209 asms = [@{term True}] orelse asms = []; (* = false*)
210 asms = [@{term False}] ; (* = false*)
211 "~~~~~ fun chk, args:"; val (indets, (a::asms)) = ([], asms);
212 bdv (*= []: _a list*);
213 val bdv : (term * term) list = [];
214 rewrite__set_ thy (i+1) false;
215 UnparseC.term a = "?r is_expanded"; (*hier m"usste doch der Numerator eingesetzt sein ??????????????*)
216 val SOME (Const (\<^const_name>\<open>False\<close>, _), []) = rewrite__set_ thy (i+1) false bdv rls a
217 ============ inhibit exn WN130823: prepat is empty ===================================*)
219 "-------- fun rewrite_set_ cancel_p downto fun gcd_poly ----------------------";
220 "-------- fun rewrite_set_ cancel_p downto fun gcd_poly ----------------------";
221 "-------- fun rewrite_set_ cancel_p downto fun gcd_poly ----------------------";
222 val t = TermC.str2term "(12 * x * y) / (8 * y \<up> 2 )";
223 (* "-------- example 187a": exception Div raised...
224 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;*)
225 val t = TermC.str2term "(8 * x \<up> 2 * y * z ) / (18 * x * y \<up> 2 * z )";
226 (* "-------- example 187b": doesn't terminate...
227 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;*)
228 val t = TermC.str2term "(9 * x \<up> 5 * y \<up> 2 * z \<up> 4) / (15 * x \<up> 6 * y \<up> 3 * z )";
229 (* "-------- example 187c": doesn't terminate...
230 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;*)
231 "~~~~~ fun rewrite_set_, args:"; val (thy, bool, rls, term) = (@{theory Isac_Knowledge}, false, cancel_p, t);
232 (* WN130827: exception Div raised...
233 rewrite__set_ thy 1 bool [] rls term
235 "~~~~~ and rewrite__set_, args:"; val (thy, i, _, _, (rrls as Rrls _), t) =
236 (thy, 1, bool, [], rls, term);
237 (* WN130827: exception Div raised...
238 val (t', asm, rew) = app_rev thy (i+1) rrls t
240 "~~~~~ fun app_rev, args:"; val (thy, i, rrls, t) = (thy, (i+1), rrls, t);
241 (* WN130827: exception Div raised...
242 val opt = app_rev' thy rrls t
244 "~~~~~ fun app_rev', args:"; val (thy, (Rrls{erls,prepat,scr=Rfuns{normal_form,...},...}), t) =
246 chk_prepat thy erls prepat t = true;
247 (* WN130827: exception Div raised...
250 (* lookup Rational.thy, cancel_p: normal_form = cancel_p_ thy*)
251 "~~~~~ fun cancel_p_, args:"; val (t) = (t);
252 val opt = check_fraction t;
253 val SOME (numerator, denominator) = opt
254 val vs = TermC.vars_of t
255 val baseT = type_of numerator
256 val expT = HOLogic.realT
257 val (SOME a, SOME b) = (poly_of_term vs numerator, poly_of_term vs denominator);
258 (*"-------- example 187a": exception Div raised...
259 val a = [(12, [1, 1])]: poly
260 val b = [(8, [0, 2])]: poly
261 val ((a', b'), c) = gcd_poly a b
263 (* "-------- example 187b": doesn't terminate...
264 val a = [(8, [2, 1, 1])]: poly
265 val b = [(18, [1, 2, 1])]: poly
266 val ((a', b'), c) = gcd_poly a b
268 (* "-------- example 187c": doesn't terminate...
269 val a = [(9, [5, 2, 4])]: poly
270 val b = [(15, [6, 3, 1])]: poly
271 val ((a', b'), c) = gcd_poly a b
274 "-------- rls norm_Rational downto fun gcd_poly ------------------------------";
275 "-------- rls norm_Rational downto fun gcd_poly ------------------------------";
276 "-------- rls norm_Rational downto fun gcd_poly ------------------------------";
277 val t = TermC.str2term "(x \<up> 2 - 4)*(3 - y) / ((y \<up> 2 - 9)*(2+x))";
278 Rewrite.trace_on := false (*true false*);
279 (* trace stops with ...: (and then jEdit hangs)..
280 rewrite_set_ thy false norm_Rational t;
282 ### rls: cancel_p on: (- 12 + 4 * y + 3 * x \<up> 2 + - 1 * (x \<up> 2 * y)) /
283 (- 18 + -9 * x + 2 * y \<up> 2 + x * y \<up> 2)
285 val t = TermC.str2term (*copy from above: "::real" is not required due to " \<up> "*)
286 ("(- 12 + 4 * y + 3 * x \<up> 2 + - 1 * (x \<up> 2 * y)) /" ^
287 "(- 18 + -9 * x + 2 * y \<up> 2 + x * y \<up> 2)");
289 exception Div raised*)
291 "~~~~~ fun cancel_p_, args:"; val (t) = (t);
292 val opt = check_fraction t;
293 val SOME (numerator, denominator) = opt
294 val vs = TermC.vars_of t
295 val baseT = type_of numerator
296 val expT = HOLogic.realT;
297 (*default_print_depth 3; 999*)
298 val (SOME a, SOME b) = (poly_of_term vs numerator, poly_of_term vs denominator);
299 (*default_print_depth 3; 999*)
300 (* does not terminate instead of returning ?:
301 val ((a', b'), c) = gcd_poly a b
302 val a = [(~12, [0, 0]), (3, [2, 0]), (4, [0, 1]), (~1, [2, 1])]: poly
303 val b = [(~18, [0, 0]), (~9, [1, 0]), (2, [0, 2]), (1, [1, 2])]: poly
306 "-------- rls norm_Rational downto fun add_fraction_p_ -----------------------";
307 "-------- rls norm_Rational downto fun add_fraction_p_ -----------------------";
308 "-------- rls norm_Rational downto fun add_fraction_p_ -----------------------";
309 val thy = @{theory Isac_Knowledge};
310 "----- SK060904- 2a non-termination of add_fraction_p_";
311 val t = TermC.str2term (" (a + b * x) / (a + - 1 * (b * x)) + " ^
312 " (- 1 * a + b * x) / (a + b * x) ");
313 (* rewrite_set_ thy false norm_Rational t
314 exception Div raised*)
315 (* rewrite_set_ thy false add_fractions_p t;
316 exception Div raised*)
317 "~~~~~ fun rewrite_set_, args:"; val (thy, bool, rls, term) =
318 (@{theory Isac_Knowledge}, false, add_fractions_p, t);
319 "~~~~~ and rewrite__set_, args:"; val (thy, i, _, _, (rrls as Rrls _), t) =
320 (thy, 1, bool, [], rls, term);
321 (* app_rev thy (i+1) rrls t;
322 exception Div raised*)
323 "~~~~~ and app_rev, args:"; val (thy, i, rrls, t) = (thy, (i+1), rrls, t);
324 fun chk_prepat thy erls [] t = true
325 | chk_prepat thy erls prepat t =
327 fun chk (pres, pat) =
329 val subst: Type.tyenv * Envir.tenv =
330 Pattern.match thy (pat, t) (Vartab.empty, Vartab.empty)
332 snd (eval__true thy (i + 1) (map (Envir.subst_term subst) pres) [] erls)
333 end) handle Pattern.MATCH => false
334 fun scan_ f [] = false (*scan_ NEVER called by []*)
335 | scan_ f (pp::pps) =
336 if f pp then true else scan_ f pps;
337 in scan_ chk prepat end;
338 (* apply the normal_form of a rev-set *)
339 fun app_rev' thy (Rrls{erls,prepat,scr=Rfuns{normal_form,...},...}) t =
340 if chk_prepat thy erls prepat t
341 then ((*tracing("### app_rev': t = "^UnparseC.term t);*) normal_form t)
343 (* val opt = app_rev' thy rrls t;
344 exception Div raised*)
345 (* val opt = app_rev' thy rrls t;
346 exception Div raised*)
347 "~~~~~ and app_rev', args:"; val (thy, (Rrls{erls,prepat,scr=Rfuns{normal_form,...},...}), t) =
349 chk_prepat thy erls prepat t = true = true;
351 exception Div raised*)
352 (* lookup Rational.thy, val add_fractions_p: normal_form = add_fraction_p_ thy*)
353 (*add_fraction_p_ thy t
354 exception Div raised*)
355 "~~~~~ fun add_fraction_p_, args:"; val ((_: theory), t) = (thy, t);
356 val SOME ((n1, d1), (n2, d2)) = check_frac_sum t;
357 UnparseC.term n1; UnparseC.term d1; UnparseC.term n2; UnparseC.term d2;
358 val vs = TermC.vars_of t;
359 (*default_print_depth 3; 999*)
360 val (SOME _, SOME a, SOME _, SOME b) =
361 (poly_of_term vs n1, poly_of_term vs d1, poly_of_term vs n2, poly_of_term vs d2);
362 (*default_print_depth 3; 999*)
364 val a = [(1, [1, 0, 0]), (~1, [0, 1, 1])]: poly
365 val b = [(1, [1, 0, 0]), (1, [0, 1, 1])]: poly
366 val ((a', b'), c) = gcd_poly a b
369 "----------- fun check_frac_sum with Free A and Const AA ---------------------------------------";
370 "----------- fun check_frac_sum with Free A and Const AA ---------------------------------------";
371 "----------- fun check_frac_sum with Free A and Const AA ---------------------------------------";
372 val thy = @{theory Isac_Knowledge(*Partial_Fractions*)}
373 val ctxt = Proof_Context.init_global thy;
375 (*---------- (1) with Free A, B ----------------------------------------------------------------*)
376 val t = (the o (parseNEW ctxt)) "3 = A / 2 + A / 4 + (B / 2 + - 1 * B / (2::real))";
377 (* required for applying thms in rewriting \<up> ^*)
378 (* we get details from here..*)
380 Rewrite.trace_on := false; (*true false*)
381 val SOME (t', _) = Rewrite.rewrite_set_ thy true add_fractions_p t;
382 Rewrite.trace_on := false; (*true false*)
384 add_fractions_p on: 3 = A / 2 + A / 4 + (B / 2 + - 1 * B / 2) --> 3 = A / 2 + A / 4 + 0 / 2 *)
385 (* |||||||||||||||||||||||||||||||||||| *)
387 val t = (the o (parseNEW ctxt))(* ||||||||||||||||||||||||| GUESS 1 GUESS 1 GUESS 1 GUESS 1 *)
388 "A / 2 + A / 4 + (B / 2 + - 1 * B / (2::real))";
389 "~~~~~ fun add_fraction_p_ , ad-hoc args:"; val (t) = (t);
390 val NONE = (*case*) check_frac_sum t (*of*)
393 add_fractions_p on: 3 = A / 2 + A / 4 + (B / 2 + - 1 * B / 2) --> 3 = A / 2 + A / 4 + 0 / 2 *)
394 (* |||||||||||||||||||||||||||| *)
395 val t = (the o (parseNEW ctxt))(* ||||||||||||||||||||||||| GUESS 2 GUESS 2 GUESS 2 GUESS 2 *)
396 "A / 4 + (B / 2 + - 1 * B / (2::real))";
397 "~~~~~ fun add_fraction_p_ , ad-hoc args:"; val (t) = (t);
398 val SOME ((n1, d1), (n2, d2)) = (*case*) check_frac_sum t (*of*);
399 (*+*)if (UnparseC.term n1, UnparseC.term d1) = ("A" , "4") andalso
400 (*+*) (UnparseC.term n2, UnparseC.term d2) = ("B / 2 + - 1 * B / 2", "1")
401 (*+*)then () else error "check_frac_sum (A / 4 + (B / 2 + - 1 * B / (2::real))) changed";
403 val vs = TermC.vars_of t;
404 val (SOME [(1, [1, 0])], SOME [(4, [0, 0])], NONE, SOME [(1, [0, 0])]) =
405 (*case*) (poly_of_term vs n1, poly_of_term vs d1, poly_of_term vs n2, poly_of_term vs d2) (*of*);
407 "~~~~~ fun poly_of_term , args:"; val (vs, t) = (vs, n1);
408 val SOME [(1, [xxx, 0])] = SOME [monom_of_term vs (1, replicate (length vs) 0) t];
409 (*+*)if xxx = 1 then () else error "monom_of_term changed"
411 "~~~~~ fun monom_of_term , args:"; val (vs, (c, es), (Free (id, _))) =
412 (vs, (1, replicate (length vs) 0), t);
413 case vs of [Free ("A", _), Free ("B", _)] =>
414 if c = 1 andalso id = "A"
415 then () else error "monom_of_term Free changed 1"
416 | _ => error "monom_of_term Free changed 2";
418 (*---------- (2) with Const AA, BB --------------------------------------------------------------*)
419 val t = (the o (parseNEW ctxt)) "3 = AA / 2 + AA / 4 + (BB / 2 + - 1 * BB / 2)";
421 (* we get details from here..*)
423 Rewrite.trace_on := false; (*true false*)
424 val SOME (t', _) = Rewrite.rewrite_set_ thy true add_fractions_p t;
425 Rewrite.trace_on := false; (*true false*)
427 add_fractions_p on: 3 = A / 2 + A / 4 + (B / 2 + - 1 * B / 2) --> 3 = A / 2 + A / 4 + 0 / 2 *)
428 (* |||||||||||||||||||||||||||||||||||| *)
429 val t = (the o (parseNEW ctxt))(* ||||||||||||||||||||||||| *)
430 "AA / 2 + AA / 4 + (BB / 2 + - 1 * BB / 2)";
431 "~~~~~ fun add_fraction_p_ , ad-hoc args:"; val (t) = (t);
432 val NONE = (*case*) check_frac_sum t (*of*)
435 add_fractions_p on: 3 = A / 2 + A / 4 + (B / 2 + - 1 * B / 2) --> 3 = A / 2 + A / 4 + 0 / 2 *)
436 (* |||||||||||||||||||||||||||| *)
437 val t = TermC.str2term (* ||||||||||||||||||||||||| *)
438 "AA / 4 + (BB / 2 + - 1 * BB / 2)";
439 "~~~~~ fun add_fraction_p_ , ad-hoc args:"; val (t) = (t);
440 val SOME ((n1, d1), (n2, d2)) = (*case*) check_frac_sum t (*of*);
441 (*+*)if (UnparseC.term n1, UnparseC.term d1) = ("AA" , "4") andalso
442 (*+*) (UnparseC.term n2, UnparseC.term d2) = ("BB / 2 + - 1 * BB / 2", "1")
443 (*+*)then () else error "check_frac_sum (AA / 4 + (BB / 2 + - 1 * BB / 2)) changed";
445 val vs = TermC.vars_of t;
446 val (SOME [(1, [1, 0])], SOME [(4, [0, 0])], NONE, SOME [(1, [0, 0])]) =
447 (*case*) (poly_of_term vs n1, poly_of_term vs d1, poly_of_term vs n2, poly_of_term vs d2) (*of*);
448 "~~~~~ fun poly_of_term , args:"; val (vs, t) = (vs, n1);
449 val SOME [(1, [1, 0])] = SOME [monom_of_term vs (1, replicate (length vs) 0) t];
451 "~~~~~ fun monom_of_term , args:"; val (vs, (c, es), (Const (id, _))) =
452 (vs, (1, replicate (length vs) 0), t);
453 case vs of [Const (\<^const_name>\<open>AA\<close>, _), Const (\<^const_name>\<open>BB\<close>, _)] =>
454 if c = 1 andalso id = "Partial_Fractions.AA"
455 then () else error "monom_of_term Const changed 1"
456 | _ => error "monom_of_term Const changed 2";
459 "-------- rewrite_set_ cancel_p from: Mathematik 1 Schalk Reniets Verlag -----";
460 "-------- rewrite_set_ cancel_p from: Mathematik 1 Schalk Reniets Verlag -----";
461 "-------- rewrite_set_ cancel_p from: Mathematik 1 Schalk Reniets Verlag -----";
462 val thy = @{theory "Rational"};
464 val t = TermC.str2term "(2 + -3 * x) / 9";
465 if NONE = rewrite_set_ thy false cancel_p t then ()
466 else error "rewrite_set_ cancel_p must return NONE, if the term cannot be cancelled";
468 "-------- example 186a";
469 val t = TermC.str2term "(14 * x * y) / (x * y)";
470 is_expanded (TermC.str2term "14 * x * y");
471 is_expanded (TermC.str2term "x * y");
472 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;
473 if (UnparseC.term t', UnparseC.terms asm) = ("14 / 1", "[]")
474 then () else error "rational.sml cancel Schalk 186a";
476 "-------- example 186b";
477 val t = TermC.str2term "(60 * a * b) / ( 15 * a * b )";
478 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;
479 if (UnparseC.term t', UnparseC.terms asm) = ("4 / 1", "[]")
480 then () else error "rational.sml cancel Schalk 186b";
482 "-------- example 186c";
483 val t = TermC.str2term "(144 * a \<up> 2 * b * c) / (12 * a * b * c)";
484 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;
485 if (UnparseC.term t', UnparseC.terms asm) = ("12 * a / 1", "[]")
486 then () else error "rational.sml cancel Schalk 186c";
488 (* !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! exception Div raised !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
489 see --- fun rewrite_set_ downto fun gcd_poly ---
490 "-------- example 187a";
491 val t = TermC.str2term "(12 * x * y) / (8 * y \<up> 2 )";
492 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;
493 if (UnparseC.term t', UnparseC.terms asm) = ("3 * x / (2 * y)", "[\"4 * y ~= 0\"]")
494 then () else error "rational.sml cancel Schalk 187a";
497 (* doesn't terminate !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
498 see --- fun rewrite_set_ downto fun gcd_poly ---
499 "-------- example 187b";
500 val t = TermC.str2term "(8 * x \<up> 2 * y * z ) / (18 * x * y \<up> 2 * z )";
501 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;
502 if (UnparseC.term t', UnparseC.terms asm) = ("4 * x / (9 * y)", "[\"2 * (z * (y * x)) ~= 0\"]")
503 then () else error "rational.sml cancel Schalk 187b";
506 (* doesn't terminate !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
507 see --- fun rewrite_set_ downto fun gcd_poly ---
508 "-------- example 187c";
509 val t = TermC.str2term "(9 * x \<up> 5 * y \<up> 2 * z \<up> 4) / (15 * x \<up> 6 * y \<up> 3 * z )";
510 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;
511 if (UnparseC.term t', UnparseC.terms asm) =
512 ("3 * z \<up> 3 / (5 * (y * x))", "[\"3 * (z * (y \<up> 2 * x \<up> 5)) ~= 0\"]")
513 then () else error "rational.sml cancel Schalk 187c";
516 "-------- example 188a";
517 val t = TermC.str2term "(-8 + 8 * x) / (-9 + 9 * x)";
518 is_expanded (TermC.str2term "8 * x + -8");
519 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;
520 if (UnparseC.term t', UnparseC.terms asm) = ("8 / 9", "[]")
521 then () else error "rational.sml cancel Schalk 188a";
523 val t = TermC.str2term "(8*((- 1) + x))/(9*((- 1) + x))";
524 val SOME (t, _) = rewrite_set_ thy false make_polynomial t;
525 if (UnparseC.term t', UnparseC.terms asm) = ("8 / 9", "[]")
526 then () else error "rational.sml cancel Schalk make_polynomial 1";
528 "-------- example 188b";
529 val t = TermC.str2term "(- 15 + 5 * x) / (- 18 + 6 * x)";
530 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;
531 if (UnparseC.term t', UnparseC.terms asm) = ("5 / 6", "[]")
532 then () else error "rational.sml cancel Schalk 188b";
534 "-------- example 188c";
535 val t = TermC.str2term "(a + - 1 * b) / (b + - 1 * a)";
536 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;
537 if (UnparseC.term t', UnparseC.terms asm) = ("- 1 / 1", "[]")
538 then () else error "rational.sml cancel Schalk 188c";
540 is_expanded (TermC.str2term "a + - 1 * b") = true;
541 val t = TermC.str2term "((- 1)*(b + (- 1) * a))/(1*(b + (- 1) * a))";
542 val SOME (t', asm) = rewrite_set_ thy false make_polynomial t;
543 if (UnparseC.term t', UnparseC.terms asm) = ("(a + - 1 * b) / (- 1 * a + b)", "[]")
544 then () else error "rational.sml cancel Schalk make_polynomial 2";
546 "-------- example 190a";
547 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 )";
548 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;
549 if (UnparseC.term t', UnparseC.terms asm) =
550 ("(1 + 9 * a \<up> 2) / (3 * a + 9 * a \<up> 2)", "[\"3 * a + 9 * a \<up> 2 \<noteq> 0\"]")
551 then () else error "rational.sml cancel Schalk 190a";
553 "-------- example 190c";
554 val t = TermC.str2term "((1 + 9 * a \<up> 2)*(1 + 3 * a))/((3 * a + 9 * a \<up> 2)*(1 + 3 * a))";
555 val SOME (t', asm) = rewrite_set_ thy false make_polynomial t;
556 if (UnparseC.term t', UnparseC.terms asm) =
557 ("(1 + 3 * a + 9 * a \<up> 2 + 27 * a \<up> 3) /\n(3 * a + 18 * a \<up> 2 + 27 * a \<up> 3)", "[]")
558 then () else error "rational.sml make_polynomial Schalk 190c";
560 "-------- example 191a";
561 val t = TermC.str2term "( x \<up> 2 + - 1 * y \<up> 2 ) / ( x + y )";
562 is_expanded (TermC.str2term "x \<up> 2 + - 1 * y \<up> 2") = false; (*!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!*)
563 is_expanded (TermC.str2term "x + y") = true;
564 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;
565 if (UnparseC.term t', UnparseC.terms asm) = ("(x + - 1 * y) / 1", "[]")
566 then () else error "rational.sml make_polynomial Schalk 191a";
568 "-------- example 191b";
569 val t = TermC.str2term "((x + (- 1) * y)*(x + y))/((1)*(x + y))";
570 val SOME (t', asm) = rewrite_set_ thy false make_polynomial t;
571 if (UnparseC.term t', UnparseC.terms asm) = ("(x \<up> 2 + - 1 * y \<up> 2) / (x + y)", "[]")
572 then () else error "rational.sml make_polynomial Schalk 191b";
574 "-------- example 191c";
575 val t = TermC.str2term "( 9 * x \<up> 2 + -30 * x + 25 ) / ( 9 * x \<up> 2 + - 25 )";
576 is_expanded (TermC.str2term "9 * x \<up> 2 + -30 * x + 25") = true;
577 is_expanded (TermC.str2term "25 + -30*x + 9*x \<up> 2") = true;
578 is_expanded (TermC.str2term "- 25 + 9*x \<up> 2") = true;
580 val t = TermC.str2term "(((-5) + 3 * x)*((-5) + 3 * x))/((5 + 3 * x)*((-5) + 3 * x))";
581 val SOME (t', asm) = rewrite_set_ thy false make_polynomial t;
582 if (UnparseC.term t', UnparseC.terms asm) = ("(25 + - 30 * x + 9 * x \<up> 2) / (- 25 + 9 * x \<up> 2)", "[]")
583 then () else error "rational.sml make_polynomial Schalk 191c";
585 "-------- example 192b";
586 val t = TermC.str2term "( 7 * x \<up> 3 + - 1 * x \<up> 2 * y ) / ( 7 * x * y \<up> 2 + - 1 * y \<up> 3 )";
587 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;
588 if (UnparseC.term t', UnparseC.terms asm) = ("x \<up> 2 / y \<up> 2", "[\"y \<up> 2 \<noteq> 0\"]")
589 then () else error "rational.sml cancel_p Schalk 192b";
591 val t = TermC.str2term "((x \<up> 2)*(7 * x + (- 1) * y))/((y \<up> 2)*(7 * x + (- 1) * y))";
592 val SOME (t', asm) = rewrite_set_ thy false make_polynomial t;
593 if (UnparseC.term t', UnparseC.terms asm) =
594 ("(7 * x \<up> 3 + - 1 * x \<up> 2 * y) /\n(7 * x * y \<up> 2 + - 1 * y \<up> 3)", "[]")
595 then () else error "rational.sml make_polynomial Schalk 192b";
597 val t = TermC.str2term "((x \<up> 2)*(7 * x + (- 1) * y))/((y \<up> 2)*(7 * x + (- 1) * y))";
598 val SOME (t', asm) = rewrite_set_ thy false make_polynomial t;
599 if (UnparseC.term t', UnparseC.terms asm) =
600 ("(7 * x \<up> 3 + - 1 * x \<up> 2 * y) /\n(7 * x * y \<up> 2 + - 1 * y \<up> 3)", "[]")
601 then () else error "rational.sml make_polynomial Schalk WN050929 not working";
603 "-------- example 193a";
604 val t = TermC.str2term "( x \<up> 2 + -6 * x + 9 ) / ( x \<up> 2 + -9 )";
605 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;
606 if (UnparseC.term t', UnparseC.terms asm) = ("(- 3 + x) / (3 + x)", "[\"3 + x \<noteq> 0\"]")
607 then () else error "rational.sml cancel_p Schalk 193a";
609 "-------- example 193b";
610 val t = TermC.str2term "( x \<up> 2 + -8 * x + 16 ) / ( 2 * x \<up> 2 + -32 )";
611 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;
612 if (UnparseC.term t', UnparseC.terms asm) = ("(- 4 + x) / (8 + 2 * x)", "[\"8 + 2 * x \<noteq> 0\"]")
613 then () else error "rational.sml cancel_p Schalk 193b";
615 "-------- example 193c";
616 val t = TermC.str2term "( 2 * x + -50 * x \<up> 3 ) / ( 25 * x \<up> 2 + - 10 * x + 1 )";
617 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;
618 if (UnparseC.term t', UnparseC.terms asm) =
619 ("(2 * x + 10 * x \<up> 2) / (1 + - 5 * x)", "[\"1 + - 5 * x \<noteq> 0\"]")
620 then () else error "rational.sml cancel_p Schalk 193c";
622 (*WN: improved with new numerals*)
623 val t = TermC.str2term "(- 25 + 9*x \<up> 2)/(5 + 3*x)";
624 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;
625 if (UnparseC.term t', UnparseC.terms asm) = ("(- 5 + 3 * x) / 1", "[]")
626 then () else error "rational.sml cancel WN 1";
628 "-------- example heuberger";
629 val t = TermC.str2term ("(x \<up> 4 + x * y + x \<up> 3 * y + y \<up> 2) / " ^
630 "(x + 5 * x \<up> 2 + y + 5 * x * y + x \<up> 2 * y \<up> 3 + x * y \<up> 4)");
631 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;
632 if (UnparseC.term t', UnparseC.terms asm) =
633 ("(x \<up> 3 + y) / (1 + 5 * x + x * y \<up> 3)", "[\"1 + 5 * x + x * y \<up> 3 \<noteq> 0\"]")
634 then () else error "rational.sml cancel_p heuberger";
636 "-------- rewrite_set_ add_fractions_p from: Mathematik 1 Schalk -------------";
637 "-------- rewrite_set_ add_fractions_p from: Mathematik 1 Schalk -------------";
638 "-------- rewrite_set_ add_fractions_p from: Mathematik 1 Schalk -------------";
639 (*deleted example 204 ... 236b at update Isabelle2012-->2013*)
641 "-------- integration lev.1 -- lev.5: cancel_p_ & add_fractions_p_ -----------";
642 "-------- integration lev.1 -- lev.5: cancel_p_ & add_fractions_p_ -----------";
643 "-------- integration lev.1 -- lev.5: cancel_p_ & add_fractions_p_ -----------";
644 val t = TermC.str2term ("123 = (a*x)/(b*x) + (c*x)/(d*x) + (e*x)/(f*x::real)");
645 "-------- gcd_poly integration level 1: works on exact term";
646 if NONE = cancel_p_ thy t then () else error "cancel_p_ works on exact fraction";
647 if NONE = add_fraction_p_ thy t then () else error "add_fraction_p_ works on exact fraction";
649 "-------- gcd_poly integration level 2: picks out ONE appropriate subterm";
650 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;
651 if UnparseC.term t' = "123 = a * x / (b * x) + c * x / (d * x) + e / f"
652 then () else error "level 2, rewrite_set_ cancel_p: changed";
653 val SOME (t', asm) = rewrite_set_ thy false add_fractions_p t;
654 if UnparseC.term t' = "123 = (b * c * x + a * d * x) / (b * d * x) + e * x / (f * x)"
655 then () else error "level 2, rewrite_set_ add_fractions_p: changed";
657 "-------- gcd_poly integration level 3: rewrites all appropriate subterms";
658 val SOME (t', asm) = rewrite_set_ thy false cancel_p_rls t;
659 if UnparseC.term t' = "123 = a / b + c / d + e / f"
660 then () else error "level 3, rewrite_set_ cancel_p_rls: changed";
661 val SOME (t', asm) = rewrite_set_ thy false add_fractions_p_rls t; (*CREATE add_fractions_p_rls*)
662 if UnparseC.term t' = "123 = (b * d * e * x + b * c * f * x + a * d * f * x) / (b * d * f * x)"
663 then () else error "level 3, rewrite_set_ add_fractions_p_rls: changed";
665 "-------- gcd_poly integration level 4: iteration cancel_p -- add_fraction_p";
666 (* simpler variant *)
667 val testrls = Rule_Set.append_rules "testrls" Rule_Set.empty [Rls_ cancel_p, Rls_ add_fractions_p]
668 val SOME (t', asm) = rewrite_set_ thy false testrls t;
669 (*Rewrite.trace_on := false; (*true false*)
670 # rls: testrls on: 123 = a * x / (b * x) + c * x / (d * x) + e * x / (f * x)
671 ## rls: cancel_p on: 123 = a * x / (b * x) + c * x / (d * x) + e * x / (f * x)
672 ## rls: add_fractions_p on: 123 = a * x / (b * x) + c * x / (d * x) + e / f
673 ## rls: cancel_p on: 123 = (b * c * x + a * d * x) / (b * d * x) + e / f
674 ## rls: add_fractions_p on: 123 = (b * c + a * d) / (b * d) + e / f
675 ## rls: cancel_p on: 123 = (b * d * e + b * c * f + a * d * f) / (b * d * f)
676 ## rls: add_fractions_p on: 123 = (b * d * e + b * c * f + a * d * f) / (b * d * f) *)
677 if UnparseC.term t' = "123 = (b * d * e + b * c * f + a * d * f) / (b * d * f)"
678 then () else error "level 4, rewrite_set_ *_p: changed";
680 (* complicated variant *)
681 val testrls_rls = Rule_Set.append_rules "testrls_rls" Rule_Set.empty [Rls_ cancel_p_rls, Rls_ add_fractions_p_rls];
682 val SOME (t', asm) = rewrite_set_ thy false testrls_rls t;
683 (*# rls: testrls_rls on: 123 = a * x / (b * x) + c * x / (d * x) + e * x / (f * x)
684 ## rls: cancel_p_rls on: 123 = a * x / (b * x) + c * x / (d * x) + e * x / (f * x)
685 ### rls: cancel_p on: 123 = a * x / (b * x) + c * x / (d * x) + e * x / (f * x)
686 ### rls: cancel_p on: 123 = a * x / (b * x) + c * x / (d * x) + e / f
687 ### rls: cancel_p on: 123 = a * x / (b * x) + c / d + e / f
688 ### rls: cancel_p on: 123 = a / b + c / d + e / f
689 ## rls: add_fractions_p_rls on: 123 = a / b + c / d + e / f
690 ### rls: add_fractions_p on: 123 = a / b + c / d + e / f
691 ### rls: add_fractions_p on: 123 = (b * c + a * d) / (b * d) + e / f
692 ### rls: add_fractions_p on: 123 = (b * d * e + b * c * f + a * d * f) / (b * d * f)
693 ## rls: cancel_p_rls on: 123 = (b * d * e + b * c * f + a * d * f) / (b * d * f)
694 ### rls: cancel_p on: 123 = (b * d * e + b * c * f + a * d * f) / (b * d * f)
695 ## rls: add_fractions_p_rls on: 123 = (b * d * e + b * c * f + a * d * f) / (b * d * f)
696 ### rls: add_fractions_p on: 123 = (b * d * e + b * c * f + a * d * f) / (b * d * f) *)
697 if UnparseC.term t' = "123 = (b * d * e + b * c * f + a * d * f) / (b * d * f)"
698 then () else error "level 4, rewrite_set_ *_p_rls: changed"
700 "-------- gcd_poly integration level 5: cancel_p & add_fraction_p within norm_Rational";
701 val SOME (t', asm) = rewrite_set_ thy false norm_Rational t;
702 if UnparseC.term t' = "123 = (a * d * f + b * c * f + b * d * e) / (b * d * f)"
703 then () else error "level 5, rewrite_set_ norm_Rational: changed"
705 "-------- reverse rewrite ----------------------------------------------------";
706 "-------- reverse rewrite ----------------------------------------------------";
707 "-------- reverse rewrite ----------------------------------------------------";
708 (** the term for which reverse rewriting is demonstrated **)
709 val t = TermC.str2term "(9 + - 1 * x \<up> 2) / (9 + 6 * x + x \<up> 2)";
710 val Rrls {scr = Rfuns {init_state = ini, locate_rule = loc,
711 next_rule = nex, normal_form = nor, ...},...} = cancel_p;
713 (** normal_form produces the result in ONE step **)
714 val SOME (t', _) = nor t;
715 if UnparseC.term t' = "(3 + - 1 * x) / (3 + x)" then ()
716 else error "rational.sml normal_form (9 - x \<up> 2) / (9 - 6 * x + x \<up> 2)";
718 (** initialize the interpreter state used by the 'me' **)
719 val (t, _, revsets, _) = ini t;
721 if length (hd revsets) = 11 then () else error "length of revset changed";
722 if (revsets |> nth 1 |> nth 1 |> Rule.id) =
723 (@{thm realpow_twoI} |> Thm.get_name_hint |> ThmC.cut_id)
724 then () else error "first element of revset changed";
726 (revsets |> nth 1 |> nth 1 |> Rule.to_string) = "Thm (\"realpow_twoI\", ?r1 \<up> 2 = ?r1 * ?r1)" andalso
727 (revsets |> nth 1 |> nth 2 |> Rule.to_string) = "Thm (\"#: 9 = 3 \<up> 2\", 9 = 3 \<up> 2)" andalso
728 (revsets |> nth 1 |> nth 3 |> Rule.to_string) = "Thm (\"#: 6 * x = 2 * (3 * x)\", 6 * x = 2 * (3 * x))"
730 (revsets |> nth 1 |> nth 4 |> Rule.to_string) = "Thm (\"#: - 3 * x = - 1 * (3 * x)\", - 3 * x = - 1 * (3 * x))"
732 (revsets |> nth 1 |> nth 5 |> Rule.to_string) = "Thm (\"#: 9 = 3 * 3\", 9 = 3 * 3)" andalso
733 (revsets |> nth 1 |> nth 6 |> Rule.to_string) = "Rls_ (\"sym_order_mult_rls_\")" andalso
734 (revsets |> nth 1 |> nth 7 |> Rule.to_string) =
735 "Thm (\"sym_mult.assoc\", ?a * (?b * ?c) = ?a * ?b * ?c)"
736 then () else error "first 7 elements in revset changed"
738 (** find the rule 'r' to apply to term 't' **)
739 (*/------- WN1309: since cancel_ (accepted "-" between monomials) has been replaced by cancel_p_
740 for Isabelle2013, we don't get a working revset, but non-termination:
742 val SOME (r as (Thm (str, thm))) = nex revsets t;
744 ((3 * 3 + - 1 * x * x) / (3 * 3 + 2 * 3 * x + x * x),
745 Rls_ ("sym_order_mult_rls_"), ((3 * 3 + - 1 * (x * x)) / (3 * 3 + 2 * (3 * x) + x * x), []))", "
746 ((3 * 3 + - 1 * (x * x)) / (3 * 3 + 2 * (3 * x) + x * x),
747 Thm ("sym_mult.assoc", ""), ((3 * 3 + - 1 * (x * x)) / (3 * 3 + 2 * 3 * x + x * x), []))", "
748 ((3 * 3 + - 1 * (x * x)) / (3 * 3 + 2 * 3 * x + x * x),
749 Thm ("sym_mult.assoc", ""), ((3 * 3 + - 1 * x * x) / (3 * 3 + 2 * 3 * x + x * x), []))", "
750 ((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), []))", "
753 Thm ("sym_#mult_2_3", "6 = 2 * 3")
754 ### Isabelle2009- 2 for cancel_ (not cancel_p_):
755 if str = "sym_#power_Float ((3,0), (0,0)) __ ((2,0), (0,0))"
756 andalso ThmC.string_of_thm thm =
757 (string_of_thm (Thm.make_thm @{theory "Isac_Knowledge"}
758 (Trueprop $ (Thm.term_of o the o (TermC.parse thy)) "9 = 3 \<up> 2"))) then ()
759 else error "rational.sml next_rule (9 - x \<up> 2) / (9 - 6 * x + x \<up> 2)";
760 \---------------------------------------------------------------------------------------/*)
762 (** check, if the rule 'r' applied by the user to 't' belongs to the ruleset;
763 if the rule is OK, the term resulting from applying the rule is returned,too;
764 there might be several rule applications inbetween,
765 which are listed after the head in reverse order **)
766 (*/-------------------------------------------- Isabelle2013: this gives "error id_of_thm";
767 we don't repair this, because interaction within "reverse rewriting" never worked properly:
769 val (r, (t, asm))::_ = loc revsets t r;
770 if UnparseC.term t = "(9 - x \<up> 2) / (3 \<up> 2 + 6 * x + x \<up> 2)" andalso asm = []
771 then () else error "rational.sml locate_rule (9 - x \<up> 2) / (9 - 6 * x + x \<up> 2)";
773 (* find the next rule to apply *)
774 val SOME (r as (Thm (str, thm))) = nex revsets t;
775 if str = "sym_#power_Float ((3,0), (0,0)) __ ((2,0), (0,0))" andalso
776 ThmC.string_of_thm thm = (string_of_thm (ThmC_Def.make_thm @{theory "Isac_Knowledge"}
777 (Trueprop $ (Thm.term_of o the o (TermC.parse thy)) "9 = 3 \<up> 2"))) then ()
778 else error "rational.sml next_rule (9 - x \<up> 2) / (9 - 6 * x + x \<up> 2)";
780 (*check the next rule*)
781 val (r, (t, asm)) :: _ = loc revsets t r;
782 if UnparseC.term t = "(3 \<up> 2 - x \<up> 2) / (3 \<up> 2 + 6 * x + x \<up> 2)" then ()
783 else error "rational.sml locate_rule (9 - x \<up> 2) / (9 - 6 * x + x \<up> 2) II";
785 (*find and check the next rules, rewrite*)
786 val SOME r = nex revsets t;
787 val (r,(t,asm))::_ = loc revsets t r;
788 if UnparseC.term t = "(3 \<up> 2 - x \<up> 2) / (3 \<up> 2 + 2 * 3 * x + x \<up> 2)" then ()
789 else error "rational.sml locate_rule II";
791 val SOME r = nex revsets t;
792 val (r,(t,asm))::_ = loc revsets t r;
793 if UnparseC.term t = "(3 - x) * (3 + x) / (3 \<up> 2 + 2 * 3 * x + x \<up> 2)" then ()
794 else error "rational.sml next_rule II";
796 val SOME r = nex revsets t;
797 val (r,(t,asm))::_ = loc revsets t r;
798 if UnparseC.term t = "(3 - x) * (3 + x) / ((3 + x) * (3 + x))" then ()
799 else error "rational.sml next_rule III";
801 val SOME r = nex revsets t;
802 val (r, (t, asm)) :: _ = loc revsets t r;
803 val ss = UnparseC.term t;
804 if ss = "(3 - x) / (3 + x)" andalso UnparseC.terms asm = "[\"3 + x ~= 0\"]" then ()
805 else error "rational.sml: new behav. in rev-set cancel";
806 \--------------------------------------------------------------------------------------/*)
808 "-------- 'reverse-ruleset' cancel_p -----------------------------------------";
809 "-------- 'reverse-ruleset' cancel_p -----------------------------------------";
810 "-------- 'reverse-ruleset' cancel_p -----------------------------------------";
811 (*WN130909: the example below shows, why "reverse rewriting" only worked for
814 (*the term for which reverse rewriting is demonstrated*)
815 val t = TermC.str2term "(9 + (- 1)*x \<up> 2) / (9 + ((-6)*x + x \<up> 2))";
816 val Rrls {scr=Rfuns {init_state=ini,locate_rule=loc,
817 next_rule=nex,normal_form=nor,...},...} = cancel_p;
819 (*normal_form produces the result in ONE step*)
820 val SOME (t', _) = nor t;
821 if UnparseC.term t' = "(3 + x) / (3 + - 1 * x)"
822 then () else error "cancel_p normal_form CHANGED";;
824 (*initialize the interpreter state used by the 'me'*)
825 val SOME (t', asm) = cancel_p_ thy t;
826 if (UnparseC.term t', UnparseC.terms asm) = ("(3 + x) / (3 + - 1 * x)", "[\"3 + - 1 * x \<noteq> 0\"]")
827 then () else error "cancel_p CHANGED";;
829 val (t,_,revsets,_) = ini t;
831 (* WN.10.10.02: dieser Fall terminiert nicht
832 (make_polynomial enth"alt zu viele rules)
833 WN060823 'init_state' requires rewriting on specified location in the term
834 default_print_depth 99; Rfuns; default_print_depth 3;
835 WN060831 cycling "sym_order_mult_rls_" "sym_mult.assoc"
836 as was with make_polynomial before ?!?* )
838 val SOME r = nex revsets t;
839 eq_Thm (r, Thm ("sym_#power_Float ((3,0), (0,0)) __ ((2,0), (0,0))",
840 mk_thm thy "9 = 3 \<up> 2"));
841 ( *WN060831 *** id_of_thm
842 Exception- ERROR raised ...
843 val (r,(t,asm))::_ = loc revsets t r;
846 val SOME r = nex revsets t;
847 val (r,(t,asm))::_ = loc revsets t r;
851 "-------- examples: rls norm_Rational ----------------------------------------";
852 "-------- examples: rls norm_Rational ----------------------------------------";
853 "-------- examples: rls norm_Rational ----------------------------------------";
854 Rewrite.trace_on := false; (*true false*)
856 val t = TermC.str2term "Not (6*x is_atom)";
857 val SOME (t',_) = rewrite_set_ thy false powers_erls t; UnparseC.term t';
859 val t = TermC.str2term "1 < 2";
860 val SOME (t',_) = rewrite_set_ thy false powers_erls t; UnparseC.term t';
863 val t = TermC.str2term "(6*x) \<up> 2";
864 val SOME (t',_) = rewrite_ thy dummy_ord powers_erls false @{thm realpow_def_atom} t;
865 if UnparseC.term t' = "6 * x * (6 * x) \<up> (2 + - 1)" then ()
866 else error "rational.sml powers_erls (6*x) \<up> 2";
868 val t = TermC.str2term "- 1 * (- 2 * (5 / 2 * (13 * x / 2)))";
869 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; UnparseC.term t';
870 if UnparseC.term t' = "65 * x / 2" then () else error "rational.sml 4";
872 val t = TermC.str2term "1 - ((13*x)/2 - 5/2) \<up> 2";
873 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; UnparseC.term t';
874 if UnparseC.term t' = "(- 21 + 130 * x + - 169 * x \<up> 2) / 4" then ()
875 else error "rational.sml 5";
877 (*SRAM Schalk I, p.92 Nr. 609a*)
878 val t = TermC.str2term "2*(3 - x/5)/3 - 4*(1 - x/3) - x/3 - 2*(x/2 - 1/4)/27 +5/54";
879 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; UnparseC.term t';
880 if UnparseC.term t' = "(- 255 + 112 * x) / 135" then ()
881 else error "rational.sml 6";
883 (*SRAM Schalk I, p.92 Nr. 610c*)
884 val t = TermC.str2term "((x- 1)/(x+1) + 1) / ((x- 1)/(x+1) - (x+1)/(x- 1)) - 2";
885 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; UnparseC.term t';
886 if UnparseC.term t' = "(3 + x) / - 2" then () else error "rational.sml 7";
888 (*SRAM Schalk I, p.92 Nr. 476a*)
889 val t = TermC.str2term "(x \<up> 2/(1 - x \<up> 2) + 1)/(x/(1 - x) + 1) * (1 + x)";
890 (*. a/b : c/d translated to a/b * d/c .*)
891 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; UnparseC.term t';
892 if UnparseC.term t' = "1" then () else error "rational.sml 8";
894 (*Schalk I, p.92 Nr. 472a*)
895 val t = TermC.str2term "((8*x \<up> 2 - 32*y \<up> 2)/(2*x + 4*y))/((4*x - 8*y)/(x + y))";
896 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; UnparseC.term t';
897 if UnparseC.term t' = "x + y" then () else error "rational.sml p.92 Nr. 472a";
899 (*Schalk I, p.70 Nr. 480b: SEE rational.sml --- nonterminating rls norm_Rational ---*)
901 (*WN130910 add_fractions_p exception Div raised + history:
902 ### WN.2.6.03 from rlang.sml 56a
903 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)";
904 val NONE = rewrite_set_ thy false add_fractions_p t;
906 THE ERROR ALREADY OCCURS IN THIS PART:
907 val t = TermC.str2term "(a + b * x) / (a + - 1 * (b * x)) + (- 1 * a + b * x) / (a + b * x)";
908 val NONE = add_fraction_p_ thy t;
910 SEE Test_Some.thy: section {* add_fractions_p downto exception Div raised ===
913 "-------- rational numerals --------------------------------------------------";
914 "-------- rational numerals --------------------------------------------------";
915 "-------- rational numerals --------------------------------------------------";
916 (*SRA Schalk I, p.40 Nr. 164b *)
917 val t = TermC.str2term "(47/6 - 76/9 + 13/4)/(35/12)";
918 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
919 if UnparseC.term t = "19 / 21" then ()
920 else error "rational.sml: diff.behav. in norm_Rational_mg 1";
922 (*SRA Schalk I, p.40 Nr. 166a *)
923 val t = TermC.str2term "((5/4)/(4+22/7) + 37/20)*(110/3 - 110/9 * 23/11)";
924 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
925 if UnparseC.term t = "45 / 2" then ()
926 else error "rational.sml: diff.behav. in norm_Rational_mg 2";
928 "-------- examples cancellation from: Mathematik 1 Schalk --------------------";
929 "-------- examples cancellation from: Mathematik 1 Schalk --------------------";
930 "-------- examples cancellation from: Mathematik 1 Schalk --------------------";
932 val t = TermC.str2term "((1 + 9*a \<up> 2) * (1 + 3*a)) / ((3*a + 9*a \<up> 2) * (1 + 3*a))";
933 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
934 if UnparseC.term t = "(1 + 9 * a \<up> 2) / (3 * a + 9 * a \<up> 2)"
935 then () else error "rational.sml: diff.behav. in norm_Rational_mg 3";
938 val t = TermC.str2term "(x \<up> 2 * (7*x + (- 1)*y)) / (y \<up> 2 * (7*x + (- 1)*y))";
939 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
940 if UnparseC.term t = "x \<up> 2 / y \<up> 2"
941 then () else error "rational.sml: diff.behav. in norm_Rational_mg 4";
943 (*SRC Schalk I, p.66 Nr. 379c *)
944 val t = TermC.str2term "(a - b)/(b - a)";
945 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
946 if UnparseC.term t = "- 1"
947 then () else error "rational.sml: diff.behav. in norm_Rational_mg 5";
949 (*SRC Schalk I, p.66 Nr. 380b *)
950 val t = TermC.str2term "15*(3*x + 3) * (4*x + 9) / (12*(2*x + 7) * (5*x + 5))";
951 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
952 if UnparseC.term t = "(27 + 12 * x) / (28 + 8 * x)"
953 then () else error "rational.sml: diff.behav. in norm_Rational_mg 6";
956 val t = TermC.str2term "((1 + 9*a \<up> 2) * (1 + 3*a)) / ((3*a + 9*a \<up> 2) * (1 + 3 * a))";
957 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
958 if UnparseC.term t = "(1 + 9 * a \<up> 2) / (3 * a + 9 * a \<up> 2)"
959 then () else error "rational.sml: diff.behav. in norm_Rational_mg 3";
962 val t = TermC.str2term "(x \<up> 2 * (7*x + (- 1)*y)) / (y \<up> 2 * (7*x + (- 1)*y))";
963 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
964 if UnparseC.term t = "x \<up> 2 / y \<up> 2"
965 then () else error "rational.sml: diff.behav. in norm_Rational_mg 4";
967 (*SRC Schalk I, p.66 Nr. 379c *)
968 val t = TermC.str2term "(a - b) / (b - a)";
969 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
970 if UnparseC.term t = "- 1"
971 then () else error "rational.sml: diff.behav. in norm_Rational_mg 5";
973 (*SRC Schalk I, p.66 Nr. 380b *)
974 val t = TermC.str2term "15*(3*x + 3) * (4*x + 9) / (12*(2*x + 7) * (5*x + 5))";
975 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
976 if UnparseC.term t = "(27 + 12 * x) / (28 + 8 * x)"
977 then () else error "rational.sml: diff.behav. in norm_Rational_mg 6";
979 (* extreme example from somewhere *)
980 val t = TermC.str2term
981 ("(a \<up> 4 * x + - 1*a \<up> 4 * y + 4*a \<up> 3 * b * x + -4*a \<up> 3 * b * y + " ^
982 "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 + " ^
983 "b \<up> 4 * x + - 1*b \<up> 4 * y) " ^
984 " / (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 + " ^
985 "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 + " ^
986 "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)")
987 val SOME (t, _) = rewrite_set_ thy false cancel_p t;
988 if UnparseC.term t = "(a \<up> 2 + 2 * a * b + b \<up> 2) / (x \<up> 2 + - 2 * x * y + y \<up> 2)"
989 then () else error "with Isabelle2002: NONE -- now SOME changed";
991 (*Schalk I, p.66 Nr. 381a *)
992 (* ATTENTION: here the rls is very slow. In Isabelle2002 this required 2 min *)
993 val t = TermC.str2term "18*(a + b) \<up> 3 * (a - b) \<up> 2 / (72*(a - b) \<up> 3 * (a + b) \<up> 2)";
994 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
995 if UnparseC.term t = "(a + b) / (4 * a + - 4 * b)"
996 then () else error "rational.sml: diff.behav. in norm_Rational_mg 8";
998 (*SRC Schalk I, p.66 Nr. 381b *)
999 val t = TermC.str2term "(4*x \<up> 2 - 20*x + 25) / (2*x - 5) \<up> 3";
1000 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1001 if UnparseC.term t = "1 / (- 5 + 2 * x)"
1002 then () else error "rational.sml: diff.behav. in norm_Rational_mg 9";
1004 (*SRC Schalk I, p.66 Nr. 381c *)
1005 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)";
1006 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1007 if UnparseC.term t = "(1 + 9 * a \<up> 2) / (3 * a + 9 * a \<up> 2)"
1008 then () else error "rational.sml: diff.behav. in norm_Rational_mg 10";
1010 (*SRC Schalk I, p.66 Nr. 383a *)
1011 val t = TermC.str2term "(5*a \<up> 2 - 5*a*b) / (a - b) \<up> 2";
1012 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1013 if UnparseC.term t = "- 5 * a / (- 1 * a + b)"
1014 then () else error "rational.sml: diff.behav. in norm_Rational_mg 11";
1016 "----- NOT TERMINATING ?: worked before 0707xx";
1017 val t = TermC.str2term "(a \<up> 2 - 1)*(b + 1) / ((b \<up> 2 - 1)*(a+1))";
1018 (* WN130911 "exception Div raised" by
1019 cancel_p_ thy (TermC.str2term ("(- 1 + - 1 * b + a \<up> 2 + a \<up> 2 * b) /" ^
1020 "(- 1 + - 1 * a + b \<up> 2 + a * b \<up> 2)"))
1022 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1023 if UnparseC.term t = "(1 + - 1 * a) / (1 + - 1 * b)" then ()
1024 else error "rational.sml MG tests 3e";
1027 "-------- examples common denominator from: Mathematik 1 Schalk --------------";
1028 "-------- examples common denominator from: Mathematik 1 Schalk --------------";
1029 "-------- examples common denominator from: Mathematik 1 Schalk --------------";
1030 (*SRA Schalk I, p.67 Nr. 403a *)
1031 val t = TermC.str2term "4/x - 3/y - 1";
1032 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1033 if UnparseC.term t = "(- 3 * x + 4 * y + - 1 * x * y) / (x * y)"
1034 then () else error "rational.sml: diff.behav. in norm_Rational_mg 12";
1036 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)";
1037 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1038 if UnparseC.term t = "4 / c"
1039 then () else error "rational.sml: diff.behav. in norm_Rational_mg 13";
1041 (*SRA Schalk I, p.67 Nr. 410b *)
1042 val t = TermC.str2term "1/(x+1) + 1/(x+2) - 2/(x+3)";
1043 (* WN130911 non-termination due to non-termination of
1044 cancel_p_ thy (TermC.str2term "(5 + 3 * x) / (6 + 11 * x + 6 * x \<up> 2 + x \<up> 3)")
1046 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1047 if UnparseC.term t = "(5 + 3 * x) / (6 + 11 * x + 6 * x \<up> 2 + x \<up> 3)"
1048 then () else error "rational.sml: diff.behav. in norm_Rational_mg 14";
1051 (*SRA Schalk I, p.67 Nr. 413b *)
1052 val t = TermC.str2term "(1 + x)/(1 - x) - (1 - x)/(1 + x) + 2*x/(1 - x \<up> 2)";
1053 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1054 if UnparseC.term t = "6 * x / (1 + - 1 * x \<up> 2)"
1055 then () else error "rational.sml: diff.behav. in norm_Rational_mg 15";
1057 (*SRA Schalk I, p.68 Nr. 414a *)
1058 val t = TermC.str2term "(x + 2)/(x - 1) + (x - 3)/(x - 2) - (x + 1)/((x - 1)*(x - 2))";
1059 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1060 if UnparseC.term t ="(- 2 + - 5 * x + 2 * x \<up> 2) / (2 + - 3 * x + x \<up> 2)"
1061 then () else error "rational.sml: diff.behav. in norm_Rational_mg 16";
1063 (*SRA Schalk I, p.68 Nr. 428b *)
1064 val t = TermC.str2term
1065 "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";
1066 (* WN130911 non-termination due to non-termination of
1067 cancel_p_ thy (TermC.str2term "(4 + -4 * b \<up> 2) / (a \<up> 4 + - 2 * (a \<up> 2 * b \<up> 2) + b \<up> 4)")
1069 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1070 if UnparseC.term t = "4 / (a \<up> 4 + - 2 * a \<up> 2 * b \<up> 2 + b \<up> 4)"
1071 then () else error "rational.sml: diff.behav. in norm_Rational_mg 18";
1074 (*SRA Schalk I, p.68 Nr. 430b *)
1075 val t = TermC.str2term
1076 "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";
1077 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1078 if UnparseC.term t = "a + 3 * b"
1079 then () else error "rational.sml: diff.behav. in norm_Rational_mg 19";
1081 (*SRA Schalk I, p.68 Nr. 432 *)
1082 val t = TermC.str2term
1083 ("(a \<up> 2 + a*b) / (a \<up> 2 - b \<up> 2) - (b \<up> 2 - a*b) / (b \<up> 2 - a \<up> 2) + " ^
1084 "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) - " ^
1085 "2*b \<up> 2 / (a \<up> 2 - b \<up> 2)");
1086 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1087 if UnparseC.term t = (*"0" ..isabisac15 | Isabelle2017..*) "0 / (a \<up> 2 + - 1 * b \<up> 2)"
1088 then () else error "rational.sml: diff.behav. in norm_Rational_mg 20";
1091 val t = TermC.str2term "3*a / (a*b) + x/y";
1092 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1093 if UnparseC.term t = "(3 * y + b * x) / (b * y)"
1094 then () else error "rational.sml: diff.behav. in norm_Rational_mg 21";
1097 "-------- examples multiply and cancel from: Mathematik 1 Schalk -------------";
1098 "-------- examples multiply and cancel from: Mathematik 1 Schalk -------------";
1099 "-------- examples multiply and cancel from: Mathematik 1 Schalk -------------";
1100 (*------- SRM Schalk I, p.68 Nr. 436a *)
1101 val t = TermC.str2term "3*(x+y) / (15*(x - y)) * 25*(x - y) \<up> 2 / (18*(x + y) \<up> 2)";
1102 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1103 if UnparseC.term t = "(- 5 * x + 5 * y) / (- 18 * x + - 18 * y)"
1104 then () else error "rational.sml: diff.behav. in norm_Rational_mg 22";
1106 (*------- SRM.test Schalk I, p.68 Nr. 436b *)
1107 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";
1108 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1109 if UnparseC.term t = "5 * a / (a + - 1 * b)"
1110 then () else error "rational.sml: diff.behav. in norm_Rational_mg 23";
1112 (*------- Schalk I, p.68 Nr. 437a *)
1113 val t = TermC.str2term "(3*a - 4*b) / (4*c+3*e) * (3*a+4*b)/(9*a \<up> 2 - 16*b \<up> 2)";
1114 (* raises an exception for unclear reasons:
1115 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1117 ### rls: cancel_p on: (9 * a \<up> 2 + - 16 * b \<up> 2) / (4 * c + 3 * e) /
1118 (9 * a \<up> 2 + - 16 * b \<up> 2)
1119 exception Div raised
1122 val t = TermC.str2term
1123 ("(9 * a \<up> 2 + - 16 * b \<up> 2) / (4 * c + 3 * e) /" ^
1124 "(9 * a \<up> 2 + - 16 * b \<up> 2)");
1125 NONE = cancel_p_ thy t;
1127 if UnparseC.term t = "1 / (4 * c + 3 * e)" then ()
1128 else error "rational.sml: diff.behav. in norm_Rational_mg 24";
1131 "----- S.K. corrected non-termination 060904";
1132 val t = TermC.str2term "(3*a - 4*b) * (3*a+4*b)/((4*c+3*e)*(9*a \<up> 2 - 16*b \<up> 2))";
1133 val SOME (t, _) = rewrite_set_ thy false make_polynomial t;
1134 if UnparseC.term t =
1135 "(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)"
1136 then () else error "rational.sml: S.K.8..corrected 060904-6";
1138 "----- S.K. corrected non-termination of cancel_p_";
1139 val t'' = TermC.str2term ("(9 * a \<up> 2 + - 16 * b \<up> 2) /" ^
1140 "(36 * a \<up> 2 * c + (27 * a \<up> 2 * e + (-64 * b \<up> 2 * c + -48 * b \<up> 2 * e)))");
1141 (* /--- DOES NOT TERMINATE AT TRANSITION isabisac15 --> Isabelle2017 --------------------------\
1142 val SOME (t',_) = rewrite_set_ thy false cancel_p t'';
1143 if UnparseC.term t' = "1 / (4 * c + 3 * e)"
1144 then () else error "rational.sml: diff.behav. in cancel_p S.K.8";
1145 \--- DOES NOT TERMINATE AT TRANSITION isabisac15 --> Isabelle2017 --------------------------/*)
1147 (*------- Schalk I, p.68 Nr. 437b*)
1148 val t = TermC.str2term "(a + b)/(x \<up> 2 - y \<up> 2) * ((x - y) \<up> 2/(a \<up> 2 - b \<up> 2))";
1149 (*val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1151 #### rls: cancel_p on: (a * x \<up> 2 + - 2 * (a * (x * y)) + a * y \<up> 2 + b * x \<up> 2 +
1152 - 2 * (b * (x * y)) +
1154 (a \<up> 2 * x \<up> 2 + - 1 * (a \<up> 2 * y \<up> 2) + - 1 * (b \<up> 2 * x \<up> 2) +
1155 b \<up> 2 * y \<up> 2)
1156 exception Div raised
1159 (*------- SRM Schalk I, p.68 Nr. 438a *)
1160 val t = TermC.str2term "x*y / (x*y - y \<up> 2) * (x \<up> 2 - x*y)";
1161 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1162 if UnparseC.term t = "x \<up> 2"
1163 then () else error "rational.sml: diff.behav. in norm_Rational_mg 24";
1165 (*------- SRM Schalk I, p.68 Nr. 439b *)
1166 val t = TermC.str2term "(4*x \<up> 2 + 4*x + 1) * ((x \<up> 2 - 2*x \<up> 3) / (4*x \<up> 2 + 2*x))";
1167 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1168 if UnparseC.term t = "(x + - 4 * x \<up> 3) / 2"
1169 then () else error "rational.sml: diff.behav. in norm_Rational_mg 25";
1171 (*------- SRM Schalk I, p.68 Nr. 440a *)
1172 val t = TermC.str2term "(x \<up> 2 - 2*x) / (x \<up> 2 - 3*x) * (x - 3) \<up> 2 / (x \<up> 2 - 4)";
1173 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1174 if UnparseC.term t = "(- 3 + x) / (2 + x)"
1175 then () else error "rational.sml: diff.behav. in norm_Rational_mg 26";
1177 "----- Schalk I, p.68 Nr. 440b SK11 works since 0707xx";
1178 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)";
1179 (* WN130911 non-termination for unclear reasons:
1180 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1182 ... ENDS WITH THIS TRACE:
1184 ### rls: cancel_p on: (-9 * (a \<up> 3 * b) + -9 * (a \<up> 2 * b \<up> 2) + a \<up> 5 * b +
1185 a \<up> 4 * b \<up> 2) /
1186 (a \<up> 3 * b + - 1 * (a * b \<up> 3)) /
1188 BUT THIS IS CORRECTLY RECOGNISED
1189 val t = TermC.str2term
1190 ("(-9 * (a \<up> 3 * b) + -9 * (a \<up> 2 * b \<up> 2) + a \<up> 5 * b + a \<up> 4 * b \<up> 2) /" ^
1191 "(a \<up> 3 * b + - 1 * (a * b \<up> 3)) / (3 + (a::real))");
1193 NONE = cancel_p_ thy t;
1195 if UnparseC.term t = "(-3 * a + a \<up> 2) / (a + - 1 * b)" then ()
1196 else error "rational.sml: diff.behav. in norm_Rational 27";
1199 "----- SK12 works since 0707xx";
1200 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))";
1201 (* WN130911 non-termination due to non-termination of
1202 cancel_p_ thy (TermC.str2term "(4 + -4 * b \<up> 2) / (a \<up> 4 + - 2 * (a \<up> 2 * b \<up> 2) + b \<up> 4)")
1204 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1205 if UnparseC.term t' = "(-3 * a + a \<up> 2) / (a + - 1 * b)" then ()
1206 else error "rational.sml: diff.behav. in norm_Rational 28";
1209 "-------- examples common denominator and multiplication from: Schalk --------";
1210 "-------- examples common denominator and multiplication from: Schalk --------";
1211 "-------- examples common denominator and multiplication from: Schalk --------";
1212 (*------- SRAM Schalk I, p.69 Nr. 441b *)
1213 val t = TermC.str2term "(4*a/3 + 3*b \<up> 2/a \<up> 3 + b/(4*a))*(4*b/(3*a))";
1214 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1215 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)"
1216 then () else error "rational.sml: diff.behav. in norm_Rational_mg 28";
1218 (*------- SRAM Schalk I, p.69 Nr. 442b *)
1219 val t = TermC.str2term ("(15*a \<up> 2/x \<up> 3 - 5*b \<up> 4/x \<up> 2 + 25*c \<up> 2/x) * " ^
1220 "(x \<up> 3/(5*a*b \<up> 3*c \<up> 3)) + 1/c \<up> 3 * (b*x/a - 3*a/b \<up> 3)");
1221 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1222 if UnparseC.term t = "5 * x \<up> 2 / (a * b \<up> 3 * c)"
1223 then () else error "rational.sml: diff.behav. in norm_Rational_mg 29";
1225 (*------- SRAM Schalk I, p.69 Nr. 443b *)
1226 val t = TermC.str2term "(a/2 + b/3) * (b/3 - a/2)";
1227 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1228 if UnparseC.term t = "(- 9 * a \<up> 2 + 4 * b \<up> 2) / 36"
1229 then () else error "rational.sml: diff.behav. in norm_Rational_mg 30";
1231 (*------- SRAM Schalk I, p.69 Nr. 445b *)
1232 val t = TermC.str2term "(a \<up> 2/9 + 2*a/(3*b) + 4/b \<up> 2)*(a/3 - 2/b) + 8/b \<up> 3";
1233 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1234 if UnparseC.term t = "a \<up> 3 / 27"
1235 then () else error "rational.sml: diff.behav. in norm_Rational_mg 31";
1237 (*------- SRAM Schalk I, p.69 Nr. 446b *)
1238 val t = TermC.str2term "(x/(5*x + 4*y) - y/(5*x - 4*y) + 1)*(25*x \<up> 2 - 16*y \<up> 2)";
1239 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1240 if UnparseC.term t = (*"30 * x \<up> 2 + -9 * x * y + - 20 * y \<up> 2" ..isabisac15 | Isabelle2017..*)
1241 "(- 30 * x \<up> 2 + 9 * x * y + 20 * y \<up> 2) / - 1"
1242 then () else error "rational.sml: diff.behav. in norm_Rational_mg 32";
1244 (*------- SRAM Schalk I, p.69 Nr. 449a *)(*Achtung: rechnet ca 8 Sekunden*)
1245 val t = TermC.str2term
1246 "(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)";
1247 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1248 if UnparseC.term t = "(- 81 * x \<up> 4 + 16 * x \<up> 8 * y \<up> 4) / (81 * y \<up> 8)"
1249 then () else error "rational.sml: diff.behav. in norm_Rational_mg 33";
1251 (*------- SRAM Schalk I, p.69 Nr. 450a *)
1252 val t = TermC.str2term
1253 "(4*x/(3*y)+2*y/(3*x)) \<up> 2 - (2*y/(3*x) - 2*x/y)*(2*y/(3*x)+2*x/y)";
1254 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1255 if UnparseC.term t = "(52 * x \<up> 2 + 16 * y \<up> 2) / (9 * y \<up> 2)"
1256 then () else error "rational.sml: diff.behav. in norm_Rational_mg 34";
1258 (*------- SRAM Schalk I, p.69 Nr. 442b --- abgewandelt*)
1259 val t = TermC.str2term
1260 ("(15*a \<up> 4/(a*x \<up> 3) - 5*a*((b \<up> 4 - 5*c \<up> 2*x) / x \<up> 2)) * " ^
1261 "(x \<up> 3/(5*a*b \<up> 3*c \<up> 3)) + a/c \<up> 3 * (x*(b/a) - 3*b*(a/b \<up> 4))");
1262 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1263 if UnparseC.term t = "5 * x \<up> 2 / (b \<up> 3 * c)"
1264 then () else error "rational.sml: diff.behav. in norm_Rational_mg 53";
1267 "-------- examples double fractions from: Mathematik 1 Schalk ----------------";
1268 "-------- examples double fractions from: Mathematik 1 Schalk ----------------";
1269 "-------- examples double fractions from: Mathematik 1 Schalk ----------------";
1270 "----- SRD Schalk I, p.69 Nr. 454b";
1271 val t = TermC.str2term "((2 - x)/(2*a)) / (2*a/(x - 2))";
1272 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1273 if UnparseC.term t = "(- 4 + 4 * x + - 1 * x \<up> 2) / (4 * a \<up> 2)"
1274 then () else error "rational.sml: diff.behav. in norm_Rational_mg 35";
1276 "----- SRD Schalk I, p.69 Nr. 455a";
1277 val t = TermC.str2term "(a \<up> 2 + 1)/(a \<up> 2 - 1) / ((a+1)/(a - 1))";
1278 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1279 if UnparseC.term t = "(1 + a \<up> 2) / (1 + 2 * a + a \<up> 2)" then ()
1280 else error "rational.sml: diff.behav. in norm_Rational_mg 36";
1282 "----- Schalk I, p.69 Nr. 455b";
1283 val t = TermC.str2term "(x \<up> 2 - 4)/(y \<up> 2 - 9)/((2+x)/(3 - y))";
1284 (* WN130911 non-termination due to non-termination of
1285 cancel_p_ thy (TermC.str2term ("(- 12 + 4 * y + 3 * x \<up> 2 + - 1 * (x \<up> 2 * y)) /" ^
1286 "(- 18 + -9 * x + 2 * y \<up> 2 + x * y \<up> 2)"))
1288 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1289 if UnparseC.term t = "(2 + - 1 * x) / (3 + y)" then ()
1290 else error "rational.sml: diff.behav. in norm_Rational_mg 37";
1293 "----- SK060904- 1a non-termination of cancel_p_ ?: worked before 0707xx";
1294 val t = TermC.str2term "(x \<up> 2 - 4)*(3 - y) / ((y \<up> 2 - 9)*(2+x))";
1295 (* WN130911 non-termination due to non-termination of
1296 cancel_p_ thy (TermC.str2term ("(- 12 + 4 * y + 3 * x \<up> 2 + - 1 * (x \<up> 2 * y)) /" ^
1297 "(- 18 + -9 * x + 2 * y \<up> 2 + x * y \<up> 2)"))
1299 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1300 if UnparseC.term t = "(2 + - 1 * x) / (3 + y)" then ()
1301 else error "rational.sml: diff.behav. in norm_Rational_mg 37b";
1304 "----- ?: worked before 0707xx";
1305 val t = TermC.str2term "(3 + - 1 * y) / (-9 + y \<up> 2)";
1306 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1307 if UnparseC.term t = "- 1 / (3 + y)"
1308 then () else error "rational.sml: - 1 / (3 + y) norm_Rational";
1310 "----- SRD Schalk I, p.69 Nr. 456b";
1311 val t = TermC.str2term "(b \<up> 3 - b \<up> 2) / (b \<up> 2+b) / (b \<up> 2 - 1)";
1312 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1313 if UnparseC.term t = "b / (1 + 2 * b + b \<up> 2)"
1314 then () else error "rational.sml: diff.behav. in norm_Rational_mg 38";
1316 "----- SRD Schalk I, p.69 Nr. 457b";
1317 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))";
1318 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1319 if UnparseC.term t = "8 * a \<up> 2 + - 6 * a * b + - 12 * a \<up> 2 * b + 9 * a * b \<up> 2"
1320 then () else error "rational.sml: diff.behav. in norm_Rational_mg 39";
1322 "----- Schalk I, p.69 Nr. 458b works since 0707";
1323 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)))";
1324 (*val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1326 ### rls: cancel_p on: (- 1 * a \<up> 2 + 2 * (a \<up> 2 * x)) / (a * x + - 1 * (b * x)) /
1327 ((- 1 * b \<up> 2 + 2 * (b \<up> 2 * x)) / (a * x + - 1 * (b * x)))
1328 exception Div raised
1331 val t = TermC.str2term
1332 ("(- 1 * a \<up> 2 + 2 * (a \<up> 2 * x)) / (a * x + - 1 * (b * x)) /" ^
1333 "((- 1 * b \<up> 2 + 2 * (b \<up> 2 * x)) / (a * x + - 1 * (b * x)))");
1334 NONE = cancel_p_ thy t;
1336 if UnparseC.term t = "a \<up> 2 / b \<up> 2" then ()
1337 else error "rational.sml: diff.behav. in norm_Rational_mg 39b";
1340 "----- SRD Schalk I, p.69 Nr. 459b";
1341 val t = TermC.str2term "(a \<up> 2 - b \<up> 2)/(a*b) / (4*(a+b) \<up> 2/a)";
1342 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1343 if UnparseC.term t = "(a + - 1 * b) / (4 * a * b + 4 * b \<up> 2)" then ()
1344 else error "rational.sml: diff.behav. in norm_Rational_mg 41";
1346 "----- Schalk I, p.69 Nr. 460b nonterm.SK";
1347 val t = TermC.str2term "(9*(x \<up> 2 - 8*x + 16) / (4*(y \<up> 2 - 2*y + 1))) / ((3*x - 12) / (16*y - 16))";
1348 (*val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1349 exception Div raised
1352 val t = TermC.str2term
1353 ("(144 + -72 * x + 9 * x \<up> 2) / (4 + -8 * y + 4 * y \<up> 2) /" ^
1354 "((- 12 + 3 * x) / (- 16 + 16 * y))");
1355 NONE = cancel_p_ thy t;
1357 if UnparseC.term t = !!!!!!!!!!!!!!!!!!!!!!!!!
1358 then () else error "rational.sml: diff.behav. in norm_Rational_mg 42";
1361 "----- some variant of the above; was non-terminating before";
1362 val t = TermC.str2term "9*(x \<up> 2 - 8*x+16)*(16*y - 16)/(4*(y \<up> 2 - 2*y+1)*(3*x - 12))";
1363 val SOME (t , _) = rewrite_set_ thy false norm_Rational t;
1364 if UnparseC.term t = "(48 + - 12 * x) / (1 + - 1 * y)"
1365 then () else error "some variant of the above; was non-terminating before";
1367 "----- SRD Schalk I, p.70 Nr. 472a";
1368 val t = TermC.str2term ("((8*x \<up> 2 - 32*y \<up> 2) / (2*x + 4*y)) / ((4*x - 8*y) / (x + y))");
1369 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1370 if UnparseC.term t = "x + y"
1371 then () else error "rational.sml: diff.behav. in norm_Rational_mg 43";
1373 "----- Schalk I, p.70 Nr. 478b ----- Rechenzeit: 5 sec";
1374 val t = TermC.str2term ("(a - (a*b + b \<up> 2)/(a+b))/(b+(a - b)/(1+(a+b)/(a - b))) / " ^
1375 "((a - a \<up> 2/(a+b))/(a+(a*b)/(a - b)))");
1376 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1377 if UnparseC.term t = "(2 * a \<up> 3 + 2 * a \<up> 2 * b) / (a \<up> 2 * b + b \<up> 3)"
1378 then () else error "rational.sml: diff.behav. in norm_Rational_mg 51";
1380 (*SRD Schalk I, p.69 Nr. 461a *)
1381 val t = TermC.str2term "(2/(x+3) + 2/(x - 3)) / (8*x/(x \<up> 2 - 9))";
1382 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1383 if UnparseC.term t = "1 / 2"
1384 then () else error "rational.sml: diff.behav. in norm_Rational_mg 44";
1386 (*SRD Schalk I, p.69 Nr. 464b *)
1387 val t = TermC.str2term "(a - a/(a - 2)) / (a + a/(a - 2))";
1388 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1389 if UnparseC.term t = "(- 3 + a) / (- 1 + a)"
1390 then () else error "rational.sml: diff.behav. in norm_Rational_mg 45";
1392 (*SRD Schalk I, p.69 Nr. 465b *)
1393 val t = TermC.str2term "((x+3*y)/9 + (4*y \<up> 2 - 9*z \<up> 2)/(16*x)) / (x/9 + y/6 + z/4)";
1394 (* WN130911 non-termination due to non-termination of
1395 cancel_p_ thy (TermC.str2term
1396 ("("(576 * x \<up> 2 + 1728 * (x * y) + 1296 * y \<up> 2 + - 2916 * z \<up> 2) /" ^
1397 "(576 * x \<up> 2 + 864 * (x * y) + 1296 * (x * z))"))
1399 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1400 if UnparseC.term t = "(4 * x + 6 * y + -9 * z) / (4 * x)"
1401 then () else error "rational.sml: diff.behav. in norm_Rational_mg 46";
1404 (*SRD Schalk I, p.69 Nr. 466b *)
1405 val t = TermC.str2term "((1 - 7*(x - 2)/(x \<up> 2 - 4)) / (6/(x+2))) / (3/(x+5)+30/(x \<up> 2 - 25))";
1406 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1407 if UnparseC.term t = "(25 + - 10 * x + x \<up> 2) / 18"
1408 then () else error "rational.sml: diff.behav. in norm_Rational_mg 47";
1410 (*SRD Schalk I, p.70 Nr. 469 *)
1411 val t = TermC.str2term ("3*b \<up> 2 / (4*a \<up> 2 - 8*a*b + 4*b \<up> 2) / " ^
1412 "(a / (a \<up> 2*b - b \<up> 3) + (a - b) / (4*a*b \<up> 2 + 4*b \<up> 3) - 1 / (4*b \<up> 2))");
1413 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1414 if UnparseC.term t = "- 3 * b \<up> 3 / (- 2 * a + 2 * b)"
1415 then () else error "rational.sml: diff.behav. in norm_Rational_mg 48";
1418 "-------- me Schalk I No.186 -------------------------------------------------";
1419 "-------- me Schalk I No.186 -------------------------------------------------";
1420 "-------- me Schalk I No.186 -------------------------------------------------";
1421 val fmz = ["Term ((14 * x * y) / ( x * y ))", "normalform N"];
1423 ("Rational",["rational", "simplification"],
1424 ["simplification", "of_rationals"]);
1425 val p = e_pos'; val c = [];
1426 val (p,_,f,nxt,_,pt) = CalcTreeTEST [(fmz, (dI',pI',mI'))];
1427 val (p,_,f,nxt,_,pt) = me nxt p c pt;
1428 val (p,_,f,nxt,_,pt) = me nxt p c pt;
1429 val (p,_,f,nxt,_,pt) = me nxt p c pt;
1430 val (p,_,f,nxt,_,pt) = me nxt p c pt;
1431 val (p,_,f,nxt,_,pt) = me nxt p c pt;
1432 val (p,_,f,nxt,_,pt) = me nxt p c pt;
1433 val (p,_,f,nxt,_,pt) = me nxt p c pt;
1434 val (p,_,f,nxt,_,pt) = me nxt p c pt; f2str f;
1435 val (p,_,f,nxt,_,pt) = me nxt p c pt; f2str f;
1436 val (p,_,f,nxt,_,pt) = me nxt p c pt; f2str f;(*++ for explicit script*)
1437 val (p,_,f,nxt,_,pt) = me nxt p c pt; f2str f;(*++ for explicit script*)
1438 case (f2str f, nxt) of
1439 ("14", End_Proof') => ()
1440 | _ => error "rational.sml diff.behav. in me Schalk I No.186";
1443 "-------- interSteps ..Simp_Rat_Double_No- 1.xml ------------------------------";
1444 "-------- interSteps ..Simp_Rat_Double_No- 1.xml ------------------------------";
1445 "-------- interSteps ..Simp_Rat_Double_No- 1.xml ------------------------------";
1447 CalcTree [(["Term (((2 - x)/(2*a)) / (2*a/(x - 2)))", "normalform N"],
1448 ("Rational", ["rational", "simplification"], ["simplification", "of_rationals"]))];
1451 autoCalculate 1 CompleteCalc;
1452 val ((pt, p), _) = get_calc 1;
1454 Test_Tool.show_pt pt;
1456 (([], Frm), Simplify ((2 - x) / (2 * a) / (2 * a / (x - 2)))),
1457 (([1], Frm), (2 - x) / (2 * a) / (2 * a / (x - 2))),
1458 (([1], Res), (2 + - 1 * x) / (2 * a) / (2 * a / (x + - 1 * 2))),
1459 (([2], Res), (2 + - 1 * x) / (2 * a) / (2 * a / (- 2 + x))),
1460 (([3], Res), (2 + - 1 * x) * (- 2 + x) / (2 * a * (2 * a))),
1461 (([4], Res), (-4 + 4 * x + - 1 * x \<up> 2) / (4 * a \<up> 2)),
1462 (([], Res), (-4 + 4 * x + - 1 * x \<up> 2) / (4 * a \<up> 2))]
1464 interSteps 1 ([1], Res);
1465 val ((pt, p), _) = get_calc 1;
1466 (*Test_Tool.show_pt pt;
1468 (([], Frm), Simplify ((2 - x) / (2 * a) / (2 * a / (x - 2)))),
1469 (([1], Frm), (2 - x) / (2 * a) / (2 * a / (x - 2))),
1470 (([1,1], Frm), (2 - x) / (2 * a) / (2 * a / (x - 2))),
1471 (([1,1], Res), (2 - x) / (2 * a) / (2 * a / (x + - 1 * 2))),
1472 (([1,2], Res), (2 + - 1 * x) / (2 * a) / (2 * a / (x + - 1 * 2))),
1473 (([1], Res), (2 + - 1 * x) / (2 * a) / (2 * a / (x + - 1 * 2))),
1474 (([2], Res), (2 + - 1 * x) / (2 * a) / (2 * a / (- 2 + x))),
1475 (([3], Res), (2 + - 1 * x) * (- 2 + x) / (2 * a * (2 * a))),
1476 (([4], Res), (-4 + 4 * x + - 1 * x \<up> 2) / (4 * a \<up> 2)),
1477 (([], Res), (-4 + 4 * x + - 1 * x \<up> 2) / (4 * a \<up> 2))]
1479 val (t, asm) = get_obj g_result pt [1, 1];
1480 if UnparseC.term t = "(2 - x) / (2 * a) / (2 * a / (x + - 1 * 2))" andalso UnparseC.terms asm = "[]"
1481 then () else error "2nd interSteps ..Simp_Rat_Double_No- 1 changed on [1, 1]";
1482 val (t, asm) = get_obj g_result pt [1, 2];
1483 if UnparseC.term t = "(2 + - 1 * x) / (2 * a) / (2 * a / (x + - 1 * 2))" andalso UnparseC.terms asm = "[]"
1484 then () else error "3rd interSteps ..Simp_Rat_Double_No- 1 changed on [1, 2]";
1488 "-------- interSteps ..Simp_Rat_Cancel_No- 1.xml ------------------------------";
1489 "-------- interSteps ..Simp_Rat_Cancel_No- 1.xml ------------------------------";
1490 "-------- interSteps ..Simp_Rat_Cancel_No- 1.xml ------------------------------";
1492 CalcTree [(["Term ((a \<up> 2 + - 1*b \<up> 2) / (a \<up> 2 + - 2*a*b + b \<up> 2))", "normalform N"],
1493 ("Rational", ["rational", "simplification"], ["simplification", "of_rationals"]))];
1496 autoCalculate 1 CompleteCalc;
1497 val ((pt, p), _) = get_calc 1;
1498 (*Test_Tool.show_pt pt;
1500 (([], Frm), Simplify ((a \<up> 2 + - 1 * b \<up> 2) / (a \<up> 2 + - 2 * a * b + b \<up> 2))),
1501 (([1], Frm), (a \<up> 2 + - 1 * b \<up> 2) / (a \<up> 2 + - 2 * a * b + b \<up> 2)),
1502 (([1], Res), (a \<up> 2 + - 1 * b \<up> 2) / (a \<up> 2 + - 2 * (a * b) + b \<up> 2)),
1503 (([2], Res), (a + b) / (a + - 1 * b)),
1504 (([], Res), (a + b) / (a + - 1 * b))]
1506 interSteps 1 ([2], Res);
1507 val ((pt, p), _) = get_calc 1;
1508 (*Test_Tool.show_pt pt;
1510 (([], Frm), Simplify ((a \<up> 2 + - 1 * b \<up> 2) / (a \<up> 2 + - 2 * a * b + b \<up> 2))),
1511 (([1], Frm), (a \<up> 2 + - 1 * b \<up> 2) / (a \<up> 2 + - 2 * a * b + b \<up> 2)),
1512 (([1], Res), (a \<up> 2 + - 1 * b \<up> 2) / (a \<up> 2 + - 2 * (a * b) + b \<up> 2)),
1513 (([2,1], Frm), (a \<up> 2 + - 1 * b \<up> 2) / (a \<up> 2 + - 2 * (a * b) + b \<up> 2)),
1514 (([2,1], Res), (a + b) / (a + - 1 * b)),
1515 (([2], Res), (a + b) / (a + - 1 * b)),
1516 (([], Res), (a + b) / (a + - 1 * b))]
1518 interSteps 1 ([2,1],Res);
1519 val ((pt, p), _) = get_calc 1;
1520 (*Test_Tool.show_pt pt;
1522 (([], Frm), Simplify ((a \<up> 2 + - 1 * b \<up> 2) / (a \<up> 2 + - 2 * a * b + b \<up> 2))),
1523 (([1], Frm), (a \<up> 2 + - 1 * b \<up> 2) / (a \<up> 2 + - 2 * a * b + b \<up> 2)),
1524 (([1], Res), (a \<up> 2 + - 1 * b \<up> 2) / (a \<up> 2 + - 2 * (a * b) + b \<up> 2)),
1525 (([2,1], Frm), (a \<up> 2 + - 1 * b \<up> 2) / (a \<up> 2 + - 2 * (a * b) + b \<up> 2)),
1526 (([2,1,1], Frm), (a \<up> 2 + - 1 * b \<up> 2) / (a \<up> 2 + - 2 * (a * b) + b \<up> 2)),
1527 (([2,1,1], Res), (a \<up> 2 + - 1 * (a * b) + a * b + - 1 * b \<up> 2) /
1528 (a \<up> 2 + - 2 * (a * b) + 1 * b \<up> 2)),
1529 (([2,1,2], Res), (a \<up> 2 + - 1 * (a * b) + a * b + - 1 * b \<up> 2) /
1530 (a \<up> 2 + - 2 * (a * b) + - 1 \<up> 2 * b \<up> 2)),
1531 (([2,1,3], Res), (a \<up> 2 + - 1 * (a * b) + a * b + - 1 * b \<up> 2) /
1532 (a \<up> 2 + - 2 * (a * b) + (- 1 * b) \<up> 2)),
1533 (([2,1,4], Res), (a * a + - 1 * (a * b) + a * b + - 1 * b \<up> 2) /
1534 (a \<up> 2 + - 2 * (a * b) + (- 1 * b) \<up> 2)),
1535 (([2,1,5], Res), (a * a + - 1 * (a * b) + a * b + - 1 * (b * b)) /
1536 (a \<up> 2 + - 2 * (a * b) + (- 1 * b) \<up> 2)),
1537 (([2,1,6], Res), (a * a + - 1 * (a * b) + a * b + - 1 * (b * b)) /
1538 (a \<up> 2 + - 1 * (2 * (a * b)) + (- 1 * b) \<up> 2)),
1539 (([2,1,7], Res), (a * a + a * (- 1 * b) + (b * a + b * (- 1 * b))) /
1540 (a \<up> 2 + 2 * (a * (- 1 * b)) + (- 1 * b) \<up> 2)),
1541 (([2,1,8], Res), (a * a + a * (- 1 * b) + (b * a + b * (- 1 * b))) /
1542 (a \<up> 2 + 2 * a * (- 1 * b) + (- 1 * b) \<up> 2)),
1543 (([2,1,9], Res), (a * (a + - 1 * b) + (b * a + b * (- 1 * b))) /
1544 (a \<up> 2 + 2 * a * (- 1 * b) + (- 1 * b) \<up> 2)),
1545 (([2,1,10], Res), (a * (a + - 1 * b) + b * (a + - 1 * b)) /
1546 (a \<up> 2 + 2 * a * (- 1 * b) + (- 1 * b) \<up> 2)),
1547 (([2,1,11], Res), (a + b) * (a + - 1 * b) / (a \<up> 2 + 2 * a * (- 1 * b) + (- 1 * b) \<up> 2)),
1548 (([2,1,12], Res), (a + b) * (a + - 1 * b) / ((a + - 1 * b) * (a + - 1 * b))),
1549 (([2,1,13], Res), (a + b) / (a + - 1 * b)),
1550 (([2,1], Res), (a + b) / (a + - 1 * b)),
1551 (([2], Res), (a + b) / (a + - 1 * b)),
1552 (([], Res), (a + b) / (a + - 1 * b))]
1554 val newnds = children (get_nd pt [2,1]) (*see "fun detailrls"*);
1555 if length newnds = 13 then () else error "rational.sml: interSteps cancel_p rev_rew_p";
1557 val p = ([2,1,9],Res);
1559 val (_, tac, _) = ME_Misc.pt_extract (pt, p);
1560 case tac of SOME (Rewrite ("sym_distrib_left", _)) => ()
1561 | _ => error "rational.sml: getTactic, sym_real_plus_binom_times1";
1564 "-------- investigate rulesets for cancel_p ----------------------------------";
1565 "-------- investigate rulesets for cancel_p ----------------------------------";
1566 "-------- investigate rulesets for cancel_p ----------------------------------";
1567 val thy = @{theory "Rational"};
1568 val t = TermC.str2term "(a \<up> 2 + - 1*b \<up> 2) / (a \<up> 2 + - 2*a*b + b \<up> 2)";
1569 val tt = TermC.str2term "(1 * a + 1 * b) * (1 * a + - 1 * b)"(*numerator only*);
1571 "----- with rewrite_set_";
1572 val SOME (tt',asm) = rewrite_set_ thy false make_polynomial tt;
1573 if UnparseC.term tt'= "a \<up> 2 + - 1 * b \<up> 2" then () else error "rls chancel_p 1";
1574 val tt = TermC.str2term "((1 * a + - 1 * b) * (1 * a + - 1 * b))"(*denominator only*);
1575 val SOME (tt',asm) = rewrite_set_ thy false make_polynomial tt;
1576 if UnparseC.term tt' = "a \<up> 2 + - 2 * a * b + b \<up> 2" then () else error "rls chancel_p 2";
1578 "----- with Derive.do_one; WN1130912 not investigated further, will be discontinued";
1579 val SOME (tt, _) = factout_p_ thy t;
1580 if UnparseC.term tt = "(a + b) * (a + - 1 * b) / ((a + - 1 * b) * (a + - 1 * b))"
1581 then () else error "rls chancel_p 3";
1583 "--- with simpler ruleset";
1584 val {rules, rew_ord= (_, ro), ...} = Rule_Set.rep (assoc_rls "rev_rew_p");
1585 val der = Derive.do_one thy Atools_erls rules ro NONE tt;
1586 if length der = 12 then () else error "WN1130912 rls chancel_p 4";
1587 (*default_print_depth 99;*) writeln (Derive.deriv2str der); (*default_print_depth 3;*)
1589 (*default_print_depth 99;*) map (UnparseC.term o #1) der; (*default_print_depth 3;*)
1590 "...,(- 1 * b \<up> 2 + a \<up> 2) / (- 2 * (a * b) + a \<up> 2 + (- 1 * b) \<up> 2) ]";
1591 (*default_print_depth 99;*) map (Rule.to_string o #2) der; (*default_print_depth 3;*)
1592 (*default_print_depth 99;*) map (UnparseC.term o #1 o #3) der; (*default_print_depth 3;*)
1594 val der = Derive.do_one thy Atools_erls rules ro NONE
1595 (TermC.str2term "(1 * a + 1 * b) * (1 * a + - 1 * b)");
1596 (*default_print_depth 99;*) writeln (Derive.deriv2str der); (*default_print_depth 3;*)
1598 val {rules, rew_ord=(_,ro),...} = Rule_Set.rep (assoc_rls "rev_rew_p");
1599 val der = Derive.do_one thy Atools_erls rules ro NONE
1600 (TermC.str2term "(1 * a + - 1 * b) * (1 * a + - 1 * b)");
1601 (*default_print_depth 99;*) writeln (Derive.deriv2str der); (*default_print_depth 3;*)
1602 (*default_print_depth 99;*) map (UnparseC.term o #1) der; (*default_print_depth 3;*)
1603 (*WN060829 ...postponed*)
1606 "-------- fun eval_get_denominator -------------------------------------------";
1607 "-------- fun eval_get_denominator -------------------------------------------";
1608 "-------- fun eval_get_denominator -------------------------------------------";
1609 val thy = @{theory Isac_Knowledge};
1610 val t = Thm.term_of (the (TermC.parse thy "get_denominator ((a +x)/b)"));
1611 val SOME (_, t') = eval_get_denominator "" 0 t thy;
1612 if UnparseC.term t' = "get_denominator ((a + x) / b) = b"
1613 then () else error "get_denominator ((a + x) / b) = b"
1616 "-------- several errpats in complicated term --------------------------------";
1617 "-------- several errpats in complicated term --------------------------------";
1618 "-------- several errpats in complicated term --------------------------------";
1619 (*WN12xxxx TODO: instead of Gabriella's example here (27.Jul.12) find a simpler one
1620 WN130912: kept this test, although not clear what for*)
1622 CalcTree [(["Term ((5*b + 25)/(a^2 - b^2) * (a - b)/(5*b))", "normalform N"],
1623 ("Rational", ["rational", "simplification"], ["simplification", "of_rationals"]))];
1626 autoCalculate 1 CompleteCalc;
1627 val ((pt, p), _) = get_calc 1;
1628 (*Test_Tool.show_pt pt;
1630 (([], Frm), Simplify ((5 * b + 25) / (a \<up> 2 - b \<up> 2) * (a - b) / (5 * b))),
1631 (([1], Frm), (5 * b + 25) / (a \<up> 2 - b \<up> 2) * (a - b) / (5 * b)),
1632 (([1], Res), (5 * b + 25) / (a \<up> 2 + - 1 * b \<up> 2) * (a + - 1 * b) / (5 * b)),
1633 (([2], Res), (5 * b + 25) * (a + - 1 * b) / (a \<up> 2 + - 1 * b \<up> 2) / (5 * b)),
1634 (([3], Res), (25 * a + - 25 * b + 5 * (a * b) + -5 * b \<up> 2) / (a \<up> 2 + - 1 * b \<up> 2) /
1636 (([4], Res), (25 + 5 * b) / (a + b) / (5 * b)),
1637 (([5], Res), (25 + 5 * b) / ((a + b) * (5 * b))),
1638 (([6], Res), (25 + 5 * b) / (5 * (a * b) + 5 * b \<up> 2)),
1639 (([7], Res), (5 + b) / (a * b + b \<up> 2)),
1640 (([], Res), (5 + b) / (a * b + b \<up> 2))] *)
1643 "-------- WN1309xx non-terminating rls norm_Rational -------------------------";
1644 "-------- WN1309xx non-terminating rls norm_Rational -------------------------";
1645 "-------- WN1309xx non-terminating rls norm_Rational -------------------------";
1646 (*------- Schalk I, p.70 Nr. 480b; a/b : c/d translated to a/b * d/c*)
1647 val t = TermC.str2term
1648 ("((12*x*y / (9*x \<up> 2 - y \<up> 2)) / (1 / (3*x - y) \<up> 2 - 1 / (3*x + y) \<up> 2)) * " ^
1649 "((1/(x - 5*y) \<up> 2 - 1/(x + 5*y) \<up> 2) / (20*x*y / (x \<up> 2 - 25*y \<up> 2)))");
1651 (*1st factor separately simplified *)
1652 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))";
1653 val SOME (t', _) = rewrite_set_ thy false norm_Rational t;
1654 if UnparseC.term t' = "(- 9 * x \<up> 2 + y \<up> 2) / - 1" then () else error "Nr. 480b lhs changed";
1655 (*2nd factor separately simplified *)
1656 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)))";
1657 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; UnparseC.term t';
1658 if UnparseC.term t' = "- 1 / (- 1 * x \<up> 2 + 25 * y \<up> 2)" then () else error "Nr. 480b rhs changed";
1660 "-------- Schalk I, p.70 Nr. 477a: terms are exploding ?!?";
1661 val t = TermC.str2term ("b*y/(b - 2*y)/((b \<up> 2 - y \<up> 2)/(b+2*y)) /" ^
1662 "(b \<up> 2*y + b*y \<up> 2) * (a+x) \<up> 2 / ((b \<up> 2 - 4*y \<up> 2) * (a+2*x) \<up> 2)");
1663 (*val SOME (t',_) = rewrite_set_ thy false norm_Rational t;
1665 ### rls: cancel_p on: (a \<up> 2 * (b * y) + 2 * (a * (b * (x * y))) + b * (x \<up> 2 * y)) /
1667 ((b \<up> 2 + - 1 * y \<up> 2) / (b + 2 * y)) /
1668 (b \<up> 2 * y + b * y \<up> 2) /
1669 (a \<up> 2 * b \<up> 2 + -4 * (a \<up> 2 * y \<up> 2) + 4 * (a * (b \<up> 2 * x)) +
1670 - 16 * (a * (x * y \<up> 2)) +
1671 4 * (b \<up> 2 * x \<up> 2) +
1672 - 16 * (x \<up> 2 * y \<up> 2))
1673 exception Div raised
1676 val t = TermC.str2term
1677 ("(a \<up> 2 * (b * y) + 2 * (a * (b * (x * y))) + b * (x \<up> 2 * y)) /" ^
1679 "((b \<up> 2 + - 1 * y \<up> 2) / (b + 2 * y)) /" ^
1680 "(b \<up> 2 * y + b * y \<up> 2) /" ^
1681 "(a \<up> 2 * b \<up> 2 + -4 * (a \<up> 2 * y \<up> 2) + 4 * (a * (b \<up> 2 * x)) +" ^
1682 "- 16 * (a * (x * y \<up> 2)) +" ^
1683 "4 * (b \<up> 2 * x \<up> 2) +" ^
1684 "- 16 * (x \<up> 2 * y \<up> 2))");
1685 NONE = cancel_p_ thy t;
1688 (*------- Schalk I, p.70 Nr. 476b in 2003 this worked using 10 sec. *)
1689 val t = TermC.str2term
1690 ("((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) / " ^
1691 "((a \<up> 2 + b \<up> 2) \<up> 2 / (a + b) \<up> 2)");
1692 (* Rewrite.trace_on := true; (*true false*)
1693 rewrite_set_ thy false norm_Rational t;
1695 #### rls: cancel_p on: (2 * (a \<up> 7 * b) + 4 * (a \<up> 6 * b \<up> 2) + 6 * (a \<up> 5 * b \<up> 3) +
1696 8 * (a \<up> 4 * b \<up> 4) +
1697 6 * (a \<up> 3 * b \<up> 5) +
1698 4 * (a \<up> 2 * b \<up> 6) +
1699 2 * (a * b \<up> 7)) /
1700 (2 * (a \<up> 9 * b) + 4 * (a \<up> 8 * b \<up> 2) +
1701 2 * (2 * (a \<up> 7 * b \<up> 3)) +
1702 4 * (a \<up> 6 * b \<up> 4) +
1703 -4 * (a \<up> 4 * b \<up> 6) +
1704 -4 * (a \<up> 3 * b \<up> 7) +
1705 -4 * (a \<up> 2 * b \<up> 8) +
1706 - 2 * (a * b \<up> 9))
1708 if UnparseC.term t = "1 / (a \<up> 2 + - 1 * b \<up> 2)" then ()
1709 else error "rational.sml: diff.behav. in norm_Rational_mg 49";
1712 "-------- Schalk I, p.70 Nr. 480a: terms are exploding ?!?";
1713 val t = TermC.str2term ("(1/x + 1/y + 1/z) / (1/x - 1/y - 1/z) / " ^
1714 "(2*x \<up> 2 / (x \<up> 2 - z \<up> 2) / (x / (x + z) + x / (x - z)))");
1715 (* Rewrite.trace_on := true; (*true false*)
1716 rewrite_set_ thy false norm_Rational t;
1718 #### rls: cancel_p on: (2 * (x \<up> 6 * (y \<up> 2 * z)) + 2 * (x \<up> 6 * (y * z \<up> 2)) +
1719 2 * (x \<up> 5 * (y \<up> 2 * z \<up> 2)) +
1720 - 2 * (x \<up> 4 * (y \<up> 2 * z \<up> 3)) +
1721 - 2 * (x \<up> 4 * (y * z \<up> 4)) +
1722 - 2 * (x \<up> 3 * (y \<up> 2 * z \<up> 4))) /
1723 (- 2 * (x \<up> 6 * (y \<up> 2 * z)) + - 2 * (x \<up> 6 * (y * z \<up> 2)) +
1724 2 * (x \<up> 5 * (y \<up> 2 * z \<up> 2)) +
1725 2 * (x \<up> 4 * (y \<up> 2 * z \<up> 3)) +
1726 2 * (x \<up> 4 * (y * z \<up> 4)) +
1727 - 2 * (x \<up> 3 * (y \<up> 2 * z \<up> 4)))
1730 "-------- Schalk I, p.60 Nr. 215d: terms are exploding, internal loop does not terminate";
1731 val t = TermC.str2term "(a-b) \<up> 3 * (x+y) \<up> 4 / ((x+y) \<up> 2 * (a-b) \<up> 5)";
1732 (* Kein Wunder, denn Z???ler und Nenner extra als Polynom dargestellt ergibt:
1734 val t = TermC.str2term "(a-b) \<up> 3 * (x+y) \<up> 4";
1735 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1737 "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";
1738 val t = TermC.str2term "((x+y) \<up> 2 * (a-b) \<up> 5)";
1739 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1741 "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";
1743 anscheinend macht dem Rechner das Krzen diese Bruches keinen Spass mehr ...*)
1745 "-------- Schalk I, p.70 Nr. 480b: terms are exploding, Rewrite.trace_on stops at";
1746 val t = TermC.str2term ("((12*x*y/(9*x \<up> 2 - y \<up> 2))/" ^
1747 "(1/(3*x - y) \<up> 2 - 1/(3*x + y) \<up> 2)) *" ^
1748 "(1/(x - 5*y) \<up> 2 - 1/(x + 5*y) \<up> 2)/" ^
1749 "(20*x*y/(x \<up> 2 - 25*y \<up> 2))");
1750 (*val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1752 #### rls: cancel_p on: (19440 * (x \<up> 8 * y \<up> 2) + -490320 * (x \<up> 6 * y \<up> 4) +
1753 108240 * (x \<up> 4 * y \<up> 6) +
1754 -6000 * (x \<up> 2 * y \<up> 8)) /
1755 (2160 * (x \<up> 8 * y \<up> 2) + - 108240 * (x \<up> 6 * y \<up> 4) +
1756 1362000 * (x \<up> 4 * y \<up> 6) +
1757 - 150000 * (x \<up> 2 * y \<up> 8))