more antiquotations for Isabelle/HOL consts/types, without change of semantics;
1 (* Title: tests for rationals
3 Use is subject to license terms.
6 "-----------------------------------------------------------------------------";
7 "-----------------------------------------------------------------------------";
8 "table of contents -----------------------------------------------------------";
9 "-----------------------------------------------------------------------------";
10 "-------- fun poly_of_term ---------------------------------------------------";
11 "-------- fun is_poly --------------------------------------------------------";
12 "-------- fun term_of_poly ---------------------------------------------------";
13 "-------- integration lev.1 fun factout_p_ -----------------------------------";
14 "-------- integration lev.1 fun cancel_p_ ------------------------------------";
15 "-------- integration lev.1 fun common_nominator_p_ --------------------------";
16 "-------- integration lev.1 fun add_fraction_p_ ------------------------------";
17 "Rfuns-------- and app_rev ...traced down from rewrite_set_ until prepats ---------";
18 "Rfuns-------- fun rewrite_set_ cancel_p downto fun gcd_poly ----------------------";
19 "-------- rls norm_Rational downto fun gcd_poly ------------------------------";
20 "Rfuns-------- rls norm_Rational downto fun add_fraction_p_ -----------------------";
21 "----------- rewrite_set_ Partial_Fractions norm_Rational --------------------------------------";
22 "----------- fun check_frac_sum with Free A and Const AA ---------------------------------------";
23 "----------- fun cancel_p with Const AA --------------------------------------------------------";
24 "-------- rewrite_set_ cancel_p from: Mathematik 1 Schalk Reniets Verlag -----";
25 "-------- rewrite_set_ add_fractions_p from: Mathematik 1 Schalk -------------";
26 "-------- integration lev.1 -- lev.5: cancel_p_ & add_fractions_p_ -----------";
27 "Rfuns-------- reverse rewrite ----------------------------------------------------";
28 "Rfuns-------- 'reverse-ruleset' cancel_p -----------------------------------------";
29 "-------- investigate rls norm_Rational --------------------------------------";
30 "-------- examples: rls norm_Rational ----------------------------------------";
31 "-------- rational numerals --------------------------------------------------";
32 "-------- examples cancellation from: Mathematik 1 Schalk --------------------";
33 "-------- examples common denominator from: Mathematik 1 Schalk --------------";
34 "-------- examples multiply and cancel from: Mathematik 1 Schalk -------------";
35 "-------- examples common denominator and multiplication from: Schalk --------";
36 "-------- examples double fractions from: Mathematik 1 Schalk ----------------";
37 "-------- me Schalk I No.186 -------------------------------------------------";
38 "-------- interSteps ..Simp_Rat_Double_No-1.xml ------------------------------";
39 "-------- interSteps ..Simp_Rat_Cancel_No-1.xml ------------------------------";
40 "-------- investigate rulesets for cancel_p ----------------------------------";
41 "-------- fun eval_get_denominator -------------------------------------------";
42 "-------- several errpats in complicated term --------------------------------";
43 "-------- WN1309xx non-terminating rls norm_Rational -------------------------";
44 "-----------------------------------------------------------------------------";
45 "-----------------------------------------------------------------------------";
48 "-------- fun poly_of_term ---------------------------------------------------";
49 "-------- fun poly_of_term ---------------------------------------------------";
50 "-------- fun poly_of_term ---------------------------------------------------";
51 val thy = @{theory Partial_Fractions};
52 val ctxt = Proof_Context.init_global @{theory}
53 val vs = TermC.vars_of (the (parseNEW ctxt "12 * x \<up> 3 * y \<up> 4 * z \<up> 6"));
55 if poly_of_term vs (TermC.str2term "12::real") = SOME [(12, [0, 0, 0])]
56 then () else error "poly_of_term 1 changed";
57 if poly_of_term vs (TermC.str2term "x::real") = SOME [(1, [1, 0, 0])]
58 then () else error "poly_of_term 2 changed";
59 if poly_of_term vs (TermC.str2term "12 * x \<up> 3") = SOME [(12, [3, 0, 0])]
60 then () else error "poly_of_term 3 changed";
61 if poly_of_term vs (TermC.str2term "12 * x \<up> 3 * y \<up> 4 * z \<up> 6") = SOME [(12, [3, 4, 6])]
62 then () else error "poly_of_term 4 changed";
63 if poly_of_term vs (TermC.str2term "1 + 2 * x \<up> 3 * y \<up> 4 * z \<up> 6 + y") =
64 SOME [(1, [0, 0, 0]), (1, [0, 1, 0]), (2, [3, 4, 6])]
65 then () else error "poly_of_term 5 changed";
67 (*poly_of_term is quite liberal:*)
68 (*the coefficient may be somewhere, the order of variables and the parentheses
69 within a monomial are arbitrary*)
70 if poly_of_term vs (TermC.str2term "y \<up> 4 * (x \<up> 3 * 12 * z \<up> 6)") = SOME [(12, [3, 4, 6])]
71 then () else error "poly_of_term 6 changed";
73 (*there may even be more than 1 coefficient:*)
74 if poly_of_term vs (TermC.str2term "2 * y \<up> 4 * (x \<up> 3 * 6 * z \<up> 6)") = SOME [(12, [3, 4, 6])]
75 then () else error "poly_of_term 7 changed";
77 (*the order and the parentheses within monomials are arbitrary:*)
78 if poly_of_term vs (TermC.str2term "2 * x \<up> 3 * y \<up> 4 * z \<up> 6 + (7 * y \<up> 8 + 1)")
79 = SOME [(1, [0, 0, 0]), (7, [0, 8, 0]), (2, [3, 4, 6])]
80 then () else error "poly_of_term 8 changed";
82 "-------- fun is_poly --------------------------------------------------------";
83 "-------- fun is_poly --------------------------------------------------------";
84 "-------- fun is_poly --------------------------------------------------------";
85 if is_poly (TermC.str2term "2 * x \<up> 3 * y \<up> 4 * z \<up> 6 + 7 * y \<up> 8 + 1")
86 then () else error "is_poly 1 changed";
87 if not (is_poly (TermC.str2term "2 * (x \<up> 3 * y \<up> 4 * z \<up> 6 + 7) * y \<up> 8 + 1"))
88 then () else error "is_poly 2 changed";
90 "-------- fun term_of_poly ---------------------------------------------------";
91 "-------- fun term_of_poly ---------------------------------------------------";
92 "-------- fun term_of_poly ---------------------------------------------------";
93 val expT = HOLogic.realT
94 val Free (_, baseT) = (hd o vars o TermC.str2term) "12 * x \<up> 3 * y \<up> 4 * z \<up> 6";
95 val p = [(1, [0, 0, 0]), (7, [0, 8, 0]), (2, [3, 4, 5])]
96 val vs = TermC.vars_of (the (parseNEW ctxt "12 * x \<up> 3 * y \<up> 4 * z \<up> 6"))
97 (*precondition for [(c, es),...]: legth es = length vs*)
99 if UnparseC.term (term_of_poly baseT expT vs p) = "1 + 7 * y \<up> 8 + 2 * x \<up> 3 * y \<up> 4 * z \<up> 5"
100 then () else error "term_of_poly 1 changed";
102 "-------- integration lev.1 fun factout_p_ -----------------------------------";
103 "-------- integration lev.1 fun factout_p_ -----------------------------------";
104 "-------- integration lev.1 fun factout_p_ -----------------------------------";
105 val t = TermC.str2term "(x \<up> 2 + -1*y \<up> 2) / (x \<up> 2 + -1*x*y)"
106 val SOME (t', asm) = factout_p_ thy t;
107 if UnparseC.term t' = "(x + y) * (x + -1 * y) / (x * (x + -1 * y))"
108 then () else error ("factout_p_ term 1 changed: " ^ UnparseC.term t')
110 if UnparseC.terms asm = "[\"x \<noteq> 0\",\"x + -1 * y \<noteq> 0\"]"
111 then () else error "factout_p_ asm 1 changed"
113 val t = TermC.str2term "nothing + to_cancel ::real";
114 if NONE = factout_p_ thy t then () else error "factout_p_ doesn't report non-applicable";
116 val t = TermC.str2term "((3 * x \<up> 2 + 6 *x + 3) / (2*x + 2))";
117 val SOME (t', asm) = factout_p_ thy t;
118 if UnparseC.term t' = "(3 + 3 * x) * (1 + x) / (2 * (1 + x))" andalso
119 UnparseC.terms asm = "[\"1 + x \<noteq> 0\"]"
120 then () else error "factout_p_ 1 changed";
122 "-------- integration lev.1 fun cancel_p_ ------------------------------------";
123 "-------- integration lev.1 fun cancel_p_ ------------------------------------";
124 "-------- integration lev.1 fun cancel_p_ ------------------------------------";
125 val t = TermC.str2term "(x \<up> 2 + -1*y \<up> 2) / (x \<up> 2 + -1*x*y)"
126 val SOME (t', asm) = cancel_p_ thy t;
127 if (UnparseC.term t', UnparseC.terms asm) = ("(x + y) / x", "[\"x \<noteq> 0\"]")
128 then () else error ("cancel_p_ (t', asm) 1 changed: " ^ UnparseC.term t')
130 val t = TermC.str2term "nothing + to_cancel ::real";
131 if NONE = cancel_p_ thy t then () else error "cancel_p_ doesn't report non-applicable";
133 val t = TermC.str2term "((3 * x \<up> 2 + 6 *x + 3) / (2*x + 2))";
134 val SOME (t', asm) = cancel_p_ thy t;
135 if UnparseC.term t' = "(3 + 3 * x) / 2" andalso UnparseC.terms asm = "[]"
136 then () else error "cancel_p_ 1 changed";
138 "-------- integration lev.1 fun common_nominator_p_ --------------------------";
139 "-------- integration lev.1 fun common_nominator_p_ --------------------------";
140 "-------- integration lev.1 fun common_nominator_p_ --------------------------";
141 val t = TermC.str2term ("y / (a*x + b*x + c*x) " ^
145 val SOME (t', asm) = common_nominator_p_ thy t;
146 if UnparseC.term t' =
147 ("y * y / (x * ((a + b + c) * y)) " ^
148 (* n1 *d2'/ (c'* ( d1' *d2')) *)
149 "+ a * (a + b + c) / (x * ((a + b + c) * y))")
150 (* n2 * d1' / (c'* ( d1' *d2')) *)
151 then () else error "common_nominator_p_ term 1 changed";
152 if UnparseC.terms asm = "[\"a + b + c \<noteq> 0\",\"y \<noteq> 0\",\"x \<noteq> 0\"]"
153 then () else error "common_nominator_p_ asm 1 changed"
155 "-------- example in mail Nipkow";
156 val t = TermC.str2term "x/(x \<up> 2 + -1*y \<up> 2) + y/(x \<up> 2 + -1*x*y)";
157 val SOME (t', asm) = common_nominator_p_ thy t;
158 if UnparseC.term t' = "x * x / " ^
159 "((x + -1 * y) * ((x + y) * x))" ^
162 "((x + -1 * y) * ((x + y) * x))"
163 then () else error "common_nominator_p_ term 2 changed"
165 if UnparseC.terms asm = "[\"x + y \<noteq> 0\",\"x \<noteq> 0\",\"x + -1 * y \<noteq> 0\"]"
166 then () else error "common_nominator_p_ asm 2 changed"
168 "-------- example: applicable tested by SML code";
169 val t = TermC.str2term "nothing / to_add";
170 if NONE = common_nominator_p_ thy t then () else error "common_nominator_p_ term 3 changed";
172 val t = TermC.str2term "((x + (-1)) / (x + 1)) + ((x + 1) / (x + (-1)))";
173 val SOME (t', asm) = common_nominator_p_ thy t;
174 if UnparseC.term t' =
175 "(x + -1) * (-1 + x) / ((1 + x) * (-1 + x)) +\n(x + 1) * (1 + x) / ((1 + x) * (-1 + x))"
176 andalso UnparseC.terms asm = "[\"1 + x \<noteq> 0\",\"-1 + x \<noteq> 0\"]"
177 then () else error "common_nominator_p_ 3 changed";
179 "-------- integration lev.1 fun add_fraction_p_ ------------------------------";
180 "-------- integration lev.1 fun add_fraction_p_ ------------------------------";
181 "-------- integration lev.1 fun add_fraction_p_ ------------------------------";
182 val t = TermC.str2term "((x + (-1)) / (x + 1)) + ((x + 1) / (x + (-1)))";
183 val SOME (t', asm) = add_fraction_p_ thy t;
184 if UnparseC.term t' = "(2 + 2 * x \<up> 2) / (-1 + x \<up> 2)"
185 then () else error "add_fraction_p_ 3 changed";
187 if UnparseC.terms asm = "[\"-1 + x \<up> 2 \<noteq> 0\"]"
188 then () else error "add_fraction_p_ 3 changed";
190 val t = TermC.str2term "nothing / to_add";
191 if NONE = add_fraction_p_ thy t then () else error "add_fraction_p_ term 3 changed";
193 val t = TermC.str2term "((x + (-1)) / (x + 1)) + ((x + 1) / (x + (-1)))";
194 val SOME (t', asm) = add_fraction_p_ thy t;
195 if UnparseC.term t' = "(2 + 2 * x \<up> 2) / (-1 + x \<up> 2)" andalso
196 UnparseC.terms asm = "[\"-1 + x \<up> 2 \<noteq> 0\"]"
197 then () else error "add_fraction_p_ 3 changed";
199 "-------- and app_rev ...traced down from rewrite_set_ until prepats ---------";
200 "-------- and app_rev ...traced down from rewrite_set_ until prepats ---------";
201 "-------- and app_rev ...traced down from rewrite_set_ until prepats ---------";
202 (* trace down until prepats are evaluated
203 (which does not to work, because substitution is not done -- compare rew_sub!);
204 keep this sequence for the case, factout_p, cancel_p, common_nominator_p, add_fraction_p
205 (again) get prepat = [] changed to <>[]. *)
206 val t = TermC.str2term "(x \<up> 2 + -1*y \<up> 2) / (x \<up> 2 + -1*x*y)";
208 (*rewrite_set_ @{theory Isac_Knowledge} true cancel t = NONE; !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!*)
209 "~~~~~ fun rewrite_set_, args:"; val (thy, bool, rls, term) = (thy, false, cancel_p, t);
210 "~~~~~ fun rewrite__set_, args:"; val (thy, i, _, _, (rrls as Rrls _), t) =
211 (thy, 1, bool, [], rls, term);
212 (*val (t', asm, rew) = app_rev thy (i+1) rrls t; rew = false!!!!!!!!!!!!!!!!!!!!!*)
213 "~~~~~ and app_rev, args:"; val (thy, i, rrls, t) = (thy, (i+1), rrls, t);
214 fun chk_prepat thy erls [] t = true
215 | chk_prepat thy erls prepat t =
217 fun chk (pres, pat) =
219 val subst: Type.tyenv * Envir.tenv =
220 Pattern.match thy (pat, t) (Vartab.empty, Vartab.empty)
222 snd (eval__true thy (i + 1) (map (Envir.subst_term subst) pres) [] erls)
223 end) handle Pattern.MATCH => false
224 fun scan_ f [] = false (*scan_ NEVER called by []*)
225 | scan_ f (pp::pps) =
226 if f pp then true else scan_ f pps;
227 in scan_ chk prepat end;
228 (* apply the normal_form of a rev-set *)
229 fun app_rev' thy (Rrls{erls,prepat,scr=Rfuns{normal_form,...},...}) t =
230 if chk_prepat thy erls prepat t
231 then ((*tracing("### app_rev': t = "^UnparseC.term t);*) normal_form t)
233 (* val opt = app_rev' thy rrls t ..NONE!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!*)
234 "~~~~~ and app_rev', args:"; val (thy, (Rrls{erls,prepat,scr=Rfuns{normal_form,...},...}), t) =
236 (* chk_prepat thy erls prepat t = false!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!*)
237 (* app_sub thy i rrls t = false!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!*)
238 "~~~~~ fun chk_prepat, args:"; val (thy, erls, prepat, t) = (thy, erls, prepat, t);
239 fun chk (pres, pat) =
241 val subst: Type.tyenv * Envir.tenv =
242 Pattern.match thy (pat, t) (Vartab.empty, Vartab.empty)
244 snd (eval__true thy (i + 1) (map (Envir.subst_term subst) pres) [] erls)
245 end) handle Pattern.MATCH => false
246 fun scan_ f [] = false (*scan_ NEVER called by []*)
247 | scan_ f (pp::pps) =
248 if f pp then true else scan_ f pps;
250 (*========== inhibit exn WN130823: prepat is empty ====================================
251 (* scan_ chk prepat = false!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!*)
252 "~~~~~ fun , args:"; val (f, (pp::pps)) = (chk, prepat);
254 val ([t1, t2], t) = pp;
255 UnparseC.term t1 = "?r is_expanded";
256 UnparseC.term t2 = "?s is_expanded";
257 UnparseC.term t = "?r / ?s";
258 (* f pp = false!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!*)
259 "~~~~~ fun chk, args:"; val (pres, pat) = (pp);
260 val subst: Type.tyenv * Envir.tenv =
261 Pattern.match thy (pat, t) (Vartab.empty, Vartab.empty)
263 ({}, {(("r", 0), ("real", Var (("r", 0), "real"))),
264 (("s", 0), ("real", Var (("s", 0), "real")))}*)
266 snd (eval__true thy (i + 1) (map (Envir.subst_term subst) pres) [] erls)
267 "~~~~~ fun eval__true, args:"; val (thy, i, asms, bdv, rls) =
268 (thy, (i + 1), (map (Envir.subst_term subst) pres), [], erls);
269 UnparseC.terms asms; (* = "[\"?r is_expanded\",\"?s is_expanded\"]"*)
270 asms = [@{term True}] orelse asms = []; (* = false*)
271 asms = [@{term False}] ; (* = false*)
272 "~~~~~ fun chk, args:"; val (indets, (a::asms)) = ([], asms);
273 bdv (*= []: _a list*);
274 val bdv : (term * term) list = [];
275 rewrite__set_ thy (i+1) false;
276 UnparseC.term a = "?r is_expanded"; (*hier m"usste doch der Numerator eingesetzt sein ??????????????*)
277 val SOME (Const (\<^const_name>\<open>False\<close>, _), []) = rewrite__set_ thy (i+1) false bdv rls a
278 ============ inhibit exn WN130823: prepat is empty ===================================*)
280 "-------- fun rewrite_set_ cancel_p downto fun gcd_poly ----------------------";
281 "-------- fun rewrite_set_ cancel_p downto fun gcd_poly ----------------------";
282 "-------- fun rewrite_set_ cancel_p downto fun gcd_poly ----------------------";
283 val t = TermC.str2term "(12 * x * y) / (8 * y \<up> 2 )";
284 (* "-------- example 187a": exception Div raised...
285 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;*)
286 val t = TermC.str2term "(8 * x \<up> 2 * y * z ) / (18 * x * y \<up> 2 * z )";
287 (* "-------- example 187b": doesn't terminate...
288 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;*)
289 val t = TermC.str2term "(9 * x \<up> 5 * y \<up> 2 * z \<up> 4) / (15 * x \<up> 6 * y \<up> 3 * z )";
290 (* "-------- example 187c": doesn't terminate...
291 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;*)
292 "~~~~~ fun rewrite_set_, args:"; val (thy, bool, rls, term) = (@{theory Isac_Knowledge}, false, cancel_p, t);
293 (* WN130827: exception Div raised...
294 rewrite__set_ thy 1 bool [] rls term
296 "~~~~~ and rewrite__set_, args:"; val (thy, i, _, _, (rrls as Rrls _), t) =
297 (thy, 1, bool, [], rls, term);
298 (* WN130827: exception Div raised...
299 val (t', asm, rew) = app_rev thy (i+1) rrls t
301 "~~~~~ fun app_rev, args:"; val (thy, i, rrls, t) = (thy, (i+1), rrls, t);
302 (* WN130827: exception Div raised...
303 val opt = app_rev' thy rrls t
305 "~~~~~ fun app_rev', args:"; val (thy, (Rrls{erls,prepat,scr=Rfuns{normal_form,...},...}), t) =
307 chk_prepat thy erls prepat t = true;
308 (* WN130827: exception Div raised...
311 (* lookup Rational.thy, cancel_p: normal_form = cancel_p_ thy*)
312 "~~~~~ fun cancel_p_, args:"; val (t) = (t);
313 val opt = check_fraction t;
314 val SOME (numerator, denominator) = opt
315 val vs = TermC.vars_of t
316 val baseT = type_of numerator
317 val expT = HOLogic.realT
318 val (SOME a, SOME b) = (poly_of_term vs numerator, poly_of_term vs denominator);
319 (*"-------- example 187a": exception Div raised...
320 val a = [(12, [1, 1])]: poly
321 val b = [(8, [0, 2])]: poly
322 val ((a', b'), c) = gcd_poly a b
324 (* "-------- example 187b": doesn't terminate...
325 val a = [(8, [2, 1, 1])]: poly
326 val b = [(18, [1, 2, 1])]: poly
327 val ((a', b'), c) = gcd_poly a b
329 (* "-------- example 187c": doesn't terminate...
330 val a = [(9, [5, 2, 4])]: poly
331 val b = [(15, [6, 3, 1])]: poly
332 val ((a', b'), c) = gcd_poly a b
335 "-------- rls norm_Rational downto fun gcd_poly ------------------------------";
336 "-------- rls norm_Rational downto fun gcd_poly ------------------------------";
337 "-------- rls norm_Rational downto fun gcd_poly ------------------------------";
338 val t = TermC.str2term "(x \<up> 2 - 4)*(3 - y) / ((y \<up> 2 - 9)*(2+x))";
339 Rewrite.trace_on := false (*true false*);
340 (* trace stops with ...: (and then jEdit hangs)..
341 rewrite_set_ thy false norm_Rational t;
343 ### rls: cancel_p on: (-12 + 4 * y + 3 * x \<up> 2 + -1 * (x \<up> 2 * y)) /
344 (-18 + -9 * x + 2 * y \<up> 2 + x * y \<up> 2)
346 val t = TermC.str2term (*copy from above: "::real" is not required due to " \<up> "*)
347 ("(-12 + 4 * y + 3 * x \<up> 2 + -1 * (x \<up> 2 * y)) /" ^
348 "(-18 + -9 * x + 2 * y \<up> 2 + x * y \<up> 2)");
350 exception Div raised*)
352 "~~~~~ fun cancel_p_, args:"; val (t) = (t);
353 val opt = check_fraction t;
354 val SOME (numerator, denominator) = opt
355 val vs = TermC.vars_of t
356 val baseT = type_of numerator
357 val expT = HOLogic.realT;
358 (*default_print_depth 3; 999*)
359 val (SOME a, SOME b) = (poly_of_term vs numerator, poly_of_term vs denominator);
360 (*default_print_depth 3; 999*)
361 (* does not terminate instead of returning ?:
362 val ((a', b'), c) = gcd_poly a b
363 val a = [(~12, [0, 0]), (3, [2, 0]), (4, [0, 1]), (~1, [2, 1])]: poly
364 val b = [(~18, [0, 0]), (~9, [1, 0]), (2, [0, 2]), (1, [1, 2])]: poly
367 "-------- rls norm_Rational downto fun add_fraction_p_ -----------------------";
368 "-------- rls norm_Rational downto fun add_fraction_p_ -----------------------";
369 "-------- rls norm_Rational downto fun add_fraction_p_ -----------------------";
370 val thy = @{theory Isac_Knowledge};
371 "----- SK060904-2a non-termination of add_fraction_p_";
372 val t = TermC.str2term (" (a + b * x) / (a + -1 * (b * x)) + " ^
373 " (-1 * a + b * x) / (a + b * x) ");
374 (* rewrite_set_ thy false norm_Rational t
375 exception Div raised*)
376 (* rewrite_set_ thy false add_fractions_p t;
377 exception Div raised*)
378 "~~~~~ fun rewrite_set_, args:"; val (thy, bool, rls, term) =
379 (@{theory Isac_Knowledge}, false, add_fractions_p, t);
380 "~~~~~ and rewrite__set_, args:"; val (thy, i, _, _, (rrls as Rrls _), t) =
381 (thy, 1, bool, [], rls, term);
382 (* app_rev thy (i+1) rrls t;
383 exception Div raised*)
384 "~~~~~ and app_rev, args:"; val (thy, i, rrls, t) = (thy, (i+1), rrls, t);
385 fun chk_prepat thy erls [] t = true
386 | chk_prepat thy erls prepat t =
388 fun chk (pres, pat) =
390 val subst: Type.tyenv * Envir.tenv =
391 Pattern.match thy (pat, t) (Vartab.empty, Vartab.empty)
393 snd (eval__true thy (i + 1) (map (Envir.subst_term subst) pres) [] erls)
394 end) handle Pattern.MATCH => false
395 fun scan_ f [] = false (*scan_ NEVER called by []*)
396 | scan_ f (pp::pps) =
397 if f pp then true else scan_ f pps;
398 in scan_ chk prepat end;
399 (* apply the normal_form of a rev-set *)
400 fun app_rev' thy (Rrls{erls,prepat,scr=Rfuns{normal_form,...},...}) t =
401 if chk_prepat thy erls prepat t
402 then ((*tracing("### app_rev': t = "^UnparseC.term t);*) normal_form t)
404 (* val opt = app_rev' thy rrls t;
405 exception Div raised*)
406 (* val opt = app_rev' thy rrls t;
407 exception Div raised*)
408 "~~~~~ and app_rev', args:"; val (thy, (Rrls{erls,prepat,scr=Rfuns{normal_form,...},...}), t) =
410 chk_prepat thy erls prepat t = true = true;
412 exception Div raised*)
413 (* lookup Rational.thy, val add_fractions_p: normal_form = add_fraction_p_ thy*)
414 (*add_fraction_p_ thy t
415 exception Div raised*)
416 "~~~~~ fun add_fraction_p_, args:"; val ((_: theory), t) = (thy, t);
417 val SOME ((n1, d1), (n2, d2)) = check_frac_sum t;
418 UnparseC.term n1; UnparseC.term d1; UnparseC.term n2; UnparseC.term d2;
419 val vs = TermC.vars_of t;
420 (*default_print_depth 3; 999*)
421 val (SOME _, SOME a, SOME _, SOME b) =
422 (poly_of_term vs n1, poly_of_term vs d1, poly_of_term vs n2, poly_of_term vs d2);
423 (*default_print_depth 3; 999*)
425 val a = [(1, [1, 0, 0]), (~1, [0, 1, 1])]: poly
426 val b = [(1, [1, 0, 0]), (1, [0, 1, 1])]: poly
427 val ((a', b'), c) = gcd_poly a b
430 "----------- fun check_frac_sum with Free A and Const AA ---------------------------------------";
431 "----------- fun check_frac_sum with Free A and Const AA ---------------------------------------";
432 "----------- fun check_frac_sum with Free A and Const AA ---------------------------------------";
433 val thy = @{theory Isac_Knowledge(*Partial_Fractions*)}
434 val ctxt = Proof_Context.init_global thy;
436 (*---------- (1) with Free A, B ----------------------------------------------------------------*)
437 val t = (the o (parseNEW ctxt)) "3 = A / 2 + A / 4 + (B / 2 + -1 * B / (2::real))";
438 (* required for applying thms in rewriting \<up> ^*)
439 (* we get details from here..*)
441 Rewrite.trace_on := false;
442 val SOME (t', _) = Rewrite.rewrite_set_ thy true add_fractions_p t;
443 Rewrite.trace_on := false;
445 add_fractions_p on: 3 = A / 2 + A / 4 + (B / 2 + -1 * B / 2) --> 3 = A / 2 + A / 4 + 0 / 2 *)
446 (* |||||||||||||||||||||||||||||||||||| *)
448 val t = (the o (parseNEW ctxt))(* ||||||||||||||||||||||||| GUESS 1 GUESS 1 GUESS 1 GUESS 1 *)
449 "A / 2 + A / 4 + (B / 2 + -1 * B / (2::real))";
450 "~~~~~ fun add_fraction_p_ , ad-hoc args:"; val (t) = (t);
451 val NONE = (*case*) check_frac_sum t (*of*)
454 add_fractions_p on: 3 = A / 2 + A / 4 + (B / 2 + -1 * B / 2) --> 3 = A / 2 + A / 4 + 0 / 2 *)
455 (* |||||||||||||||||||||||||||| *)
456 val t = (the o (parseNEW ctxt))(* ||||||||||||||||||||||||| GUESS 2 GUESS 2 GUESS 2 GUESS 2 *)
457 "A / 4 + (B / 2 + -1 * B / (2::real))";
458 "~~~~~ fun add_fraction_p_ , ad-hoc args:"; val (t) = (t);
459 val SOME ((n1, d1), (n2, d2)) = (*case*) check_frac_sum t (*of*);
460 (*+*)if (UnparseC.term n1, UnparseC.term d1) = ("A" , "4") andalso
461 (*+*) (UnparseC.term n2, UnparseC.term d2) = ("B / 2 + -1 * B / 2", "1")
462 (*+*)then () else error "check_frac_sum (A / 4 + (B / 2 + -1 * B / (2::real))) changed";
464 val vs = TermC.vars_of t;
465 val (SOME [(1, [1, 0])], SOME [(4, [0, 0])], NONE, SOME [(1, [0, 0])]) =
466 (*case*) (poly_of_term vs n1, poly_of_term vs d1, poly_of_term vs n2, poly_of_term vs d2) (*of*);
468 "~~~~~ fun poly_of_term , args:"; val (vs, t) = (vs, n1);
469 val SOME [(1, [xxx, 0])] = SOME [monom_of_term vs (1, replicate (length vs) 0) t];
470 (*+*)if xxx = 1 then () else error "monom_of_term changed"
472 "~~~~~ fun monom_of_term , args:"; val (vs, (c, es), (Free (id, _))) =
473 (vs, (1, replicate (length vs) 0), t);
474 case vs of [Free ("A", _), Free ("B", _)] =>
475 if c = 1 andalso id = "A"
476 then () else error "monom_of_term Free changed 1"
477 | _ => error "monom_of_term Free changed 2";
479 (*---------- (2) with Const AA, BB --------------------------------------------------------------*)
480 val t = (the o (parseNEW ctxt)) "3 = AA / 2 + AA / 4 + (BB / 2 + -1 * BB / 2)";
482 (* we get details from here..*)
484 Rewrite.trace_on := false;
485 val SOME (t', _) = Rewrite.rewrite_set_ thy true add_fractions_p t;
486 Rewrite.trace_on := false;
488 add_fractions_p on: 3 = A / 2 + A / 4 + (B / 2 + -1 * B / 2) --> 3 = A / 2 + A / 4 + 0 / 2 *)
489 (* |||||||||||||||||||||||||||||||||||| *)
490 val t = (the o (parseNEW ctxt))(* ||||||||||||||||||||||||| *)
491 "AA / 2 + AA / 4 + (BB / 2 + -1 * BB / 2)";
492 "~~~~~ fun add_fraction_p_ , ad-hoc args:"; val (t) = (t);
493 val NONE = (*case*) check_frac_sum t (*of*)
496 add_fractions_p on: 3 = A / 2 + A / 4 + (B / 2 + -1 * B / 2) --> 3 = A / 2 + A / 4 + 0 / 2 *)
497 (* |||||||||||||||||||||||||||| *)
498 val t = (the o (parseNEW ctxt))(* ||||||||||||||||||||||||| *)
499 "AA / 4 + (BB / 2 + -1 * BB / 2)";
500 "~~~~~ fun add_fraction_p_ , ad-hoc args:"; val (t) = (t);
501 val SOME ((n1, d1), (n2, d2)) = (*case*) check_frac_sum t (*of*);
502 (*+*)if (UnparseC.term n1, UnparseC.term d1) = ("AA" , "4") andalso
503 (*+*) (UnparseC.term n2, UnparseC.term d2) = ("BB / 2 + -1 * BB / 2", "1")
504 (*+*)then () else error "check_frac_sum (AA / 4 + (BB / 2 + -1 * BB / 2)) changed";
506 val vs = TermC.vars_of t;
507 val (SOME [(1, [1, 0])], SOME [(4, [0, 0])], NONE, SOME [(1, [0, 0])]) =
508 (*case*) (poly_of_term vs n1, poly_of_term vs d1, poly_of_term vs n2, poly_of_term vs d2) (*of*);
510 "~~~~~ fun poly_of_term , args:"; val (vs, t) = (vs, n1);
511 val SOME [(1, [xxx, 0])] = SOME [monom_of_term vs (1, replicate (length vs) 0) t];
512 (*+*)if xxx = 1 then () else error "monom_of_term changed"
514 "~~~~~ fun monom_of_term , args:"; val (vs, (c, es), (Const (id, _))) =
515 (vs, (1, replicate (length vs) 0), t);
516 case vs of [Const ("Partial_Fractions.AA", _), Const ("Partial_Fractions.BB", _)] =>
517 if c = 1 andalso id = "Partial_Fractions.AA"
518 then () else error "monom_of_term Const changed 1"
519 | _ => error "monom_of_term Const changed 2";
521 "----------- fun cancel_p with Const AA --------------------------------------------------------";
522 "----------- fun cancel_p with Const AA --------------------------------------------------------";
523 "----------- fun cancel_p with Const AA --------------------------------------------------------";
524 val thy = @{theory Partial_Fractions};
525 val ctxt = Proof_Context.init_global @{theory}
526 val SOME t = TermC.parseNEW ctxt "2 * AA / 2"; (* Const ("Free ("AA", "real") *)
528 val SOME (t', _) = rewrite_set_ thy true cancel_p t;
530 Const (\<^const_name>\<open>divide\<close>, _) $ Const ("Partial_Fractions.AA", _) $ Free ("1", _) => ()
531 | _ => error "WRONG rewrite_set_ cancel_p (2 * AA / 2) \<longrightarrow> AA changed";
533 "~~~~~ fun cancel_p , args:"; val (t) = (t);
534 val opt = check_fraction t
535 val SOME (numerator, denominator) = (*case*) opt (*of*);
537 if UnparseC.term numerator = "2 * AA" andalso UnparseC.term denominator = "2"
538 then () else error "check_fraction (2 * AA / 2) changed";
539 val vs = TermC.vars_of t;
541 [Const ("Partial_Fractions.AA", _)] => ()
542 | _ => error "rewrite_set_ cancel_p (2 * AA / 2) \<longrightarrow> AA/1 changed";
545 "-------- rewrite_set_ cancel_p from: Mathematik 1 Schalk Reniets Verlag -----";
546 "-------- rewrite_set_ cancel_p from: Mathematik 1 Schalk Reniets Verlag -----";
547 "-------- rewrite_set_ cancel_p from: Mathematik 1 Schalk Reniets Verlag -----";
548 val thy = @{theory "Rational"};
550 val t = TermC.str2term "(2 + -3 * x) / 9";
551 if NONE = rewrite_set_ thy false cancel_p t then ()
552 else error "rewrite_set_ cancel_p must return NONE, if the term cannot be cancelled";
554 "-------- example 186a";
555 val t = TermC.str2term "(14 * x * y) / (x * y)";
556 is_expanded (TermC.str2term "14 * x * y");
557 is_expanded (TermC.str2term "x * y");
558 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;
559 if (UnparseC.term t', UnparseC.terms asm) = ("14 / 1", "[]")
560 then () else error "rational.sml cancel Schalk 186a";
562 "-------- example 186b";
563 val t = TermC.str2term "(60 * a * b) / ( 15 * a * b )";
564 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;
565 if (UnparseC.term t', UnparseC.terms asm) = ("4 / 1", "[]")
566 then () else error "rational.sml cancel Schalk 186b";
568 "-------- example 186c";
569 val t = TermC.str2term "(144 * a \<up> 2 * b * c) / (12 * a * b * c)";
570 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;
571 if (UnparseC.term t', UnparseC.terms asm) = ("12 * a / 1", "[]")
572 then () else error "rational.sml cancel Schalk 186c";
574 (* !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! exception Div raised !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
575 see --- fun rewrite_set_ downto fun gcd_poly ---
576 "-------- example 187a";
577 val t = TermC.str2term "(12 * x * y) / (8 * y \<up> 2 )";
578 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;
579 if (UnparseC.term t', UnparseC.terms asm) = ("3 * x / (2 * y)", "[\"4 * y ~= 0\"]")
580 then () else error "rational.sml cancel Schalk 187a";
583 (* doesn't terminate !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
584 see --- fun rewrite_set_ downto fun gcd_poly ---
585 "-------- example 187b";
586 val t = TermC.str2term "(8 * x \<up> 2 * y * z ) / (18 * x * y \<up> 2 * z )";
587 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;
588 if (UnparseC.term t', UnparseC.terms asm) = ("4 * x / (9 * y)", "[\"2 * (z * (y * x)) ~= 0\"]")
589 then () else error "rational.sml cancel Schalk 187b";
592 (* doesn't terminate !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
593 see --- fun rewrite_set_ downto fun gcd_poly ---
594 "-------- example 187c";
595 val t = TermC.str2term "(9 * x \<up> 5 * y \<up> 2 * z \<up> 4) / (15 * x \<up> 6 * y \<up> 3 * z )";
596 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;
597 if (UnparseC.term t', UnparseC.terms asm) =
598 ("3 * z \<up> 3 / (5 * (y * x))", "[\"3 * (z * (y \<up> 2 * x \<up> 5)) ~= 0\"]")
599 then () else error "rational.sml cancel Schalk 187c";
602 "-------- example 188a";
603 val t = TermC.str2term "(-8 + 8 * x) / (-9 + 9 * x)";
604 is_expanded (TermC.str2term "8 * x + -8");
605 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;
606 if (UnparseC.term t', UnparseC.terms asm) = ("8 / 9", "[]")
607 then () else error "rational.sml cancel Schalk 188a";
609 val t = TermC.str2term "(8*((-1) + x))/(9*((-1) + x))";
610 val SOME (t, _) = rewrite_set_ thy false make_polynomial t;
611 if (UnparseC.term t', UnparseC.terms asm) = ("8 / 9", "[]")
612 then () else error "rational.sml cancel Schalk make_polynomial 1";
614 "-------- example 188b";
615 val t = TermC.str2term "(-15 + 5 * x) / (-18 + 6 * x)";
616 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;
617 if (UnparseC.term t', UnparseC.terms asm) = ("5 / 6", "[]")
618 then () else error "rational.sml cancel Schalk 188b";
620 "-------- example 188c";
621 val t = TermC.str2term "(a + -1 * b) / (b + -1 * a)";
622 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;
623 if (UnparseC.term t', UnparseC.terms asm) = ("-1 / 1", "[]")
624 then () else error "rational.sml cancel Schalk 188c";
626 is_expanded (TermC.str2term "a + -1 * b") = true;
627 val t = TermC.str2term "((-1)*(b + (-1) * a))/(1*(b + (-1) * a))";
628 val SOME (t', asm) = rewrite_set_ thy false make_polynomial t;
629 if (UnparseC.term t', UnparseC.terms asm) = ("(a + -1 * b) / (-1 * a + b)", "[]")
630 then () else error "rational.sml cancel Schalk make_polynomial 2";
632 "-------- example 190a";
633 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 )";
634 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;
635 if (UnparseC.term t', UnparseC.terms asm) =
636 ("(1 + 9 * a \<up> 2) / (3 * a + 9 * a \<up> 2)", "[\"3 * a + 9 * a \<up> 2 \<noteq> 0\"]")
637 then () else error "rational.sml cancel Schalk 190a";
639 "-------- example 190c";
640 val t = TermC.str2term "((1 + 9 * a \<up> 2)*(1 + 3 * a))/((3 * a + 9 * a \<up> 2)*(1 + 3 * a))";
641 val SOME (t', asm) = rewrite_set_ thy false make_polynomial t;
642 if (UnparseC.term t', UnparseC.terms asm) =
643 ("(1 + 3 * a + 9 * a \<up> 2 + 27 * a \<up> 3) /\n(3 * a + 18 * a \<up> 2 + 27 * a \<up> 3)", "[]")
644 then () else error "rational.sml make_polynomial Schalk 190c";
646 "-------- example 191a";
647 val t = TermC.str2term "( x \<up> 2 + -1 * y \<up> 2 ) / ( x + y )";
648 is_expanded (TermC.str2term "x \<up> 2 + -1 * y \<up> 2") = false; (*!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!*)
649 is_expanded (TermC.str2term "x + y") = true;
650 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;
651 if (UnparseC.term t', UnparseC.terms asm) = ("(x + -1 * y) / 1", "[]")
652 then () else error "rational.sml make_polynomial Schalk 191a";
654 "-------- example 191b";
655 val t = TermC.str2term "((x + (-1) * y)*(x + y))/((1)*(x + y))";
656 val SOME (t', asm) = rewrite_set_ thy false make_polynomial t;
657 if (UnparseC.term t', UnparseC.terms asm) = ("(x \<up> 2 + -1 * y \<up> 2) / (x + y)", "[]")
658 then () else error "rational.sml make_polynomial Schalk 191b";
660 "-------- example 191c";
661 val t = TermC.str2term "( 9 * x \<up> 2 + -30 * x + 25 ) / ( 9 * x \<up> 2 + -25 )";
662 is_expanded (TermC.str2term "9 * x \<up> 2 + -30 * x + 25") = true;
663 is_expanded (TermC.str2term "25 + -30*x + 9*x \<up> 2") = true;
664 is_expanded (TermC.str2term "-25 + 9*x \<up> 2") = true;
666 val t = TermC.str2term "(((-5) + 3 * x)*((-5) + 3 * x))/((5 + 3 * x)*((-5) + 3 * x))";
667 val SOME (t', asm) = rewrite_set_ thy false make_polynomial t;
668 if (UnparseC.term t', UnparseC.terms asm) = ("(25 + -30 * x + 9 * x \<up> 2) / (-25 + 9 * x \<up> 2)", "[]")
669 then () else error "rational.sml make_polynomial Schalk 191c";
671 "-------- example 192b";
672 val t = TermC.str2term "( 7 * x \<up> 3 + -1 * x \<up> 2 * y ) / ( 7 * x * y \<up> 2 + -1 * y \<up> 3 )";
673 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;
674 if (UnparseC.term t', UnparseC.terms asm) = ("x \<up> 2 / y \<up> 2", "[\"y \<up> 2 \<noteq> 0\"]")
675 then () else error "rational.sml cancel_p Schalk 192b";
677 val t = TermC.str2term "((x \<up> 2)*(7 * x + (-1) * y))/((y \<up> 2)*(7 * x + (-1) * y))";
678 val SOME (t', asm) = rewrite_set_ thy false make_polynomial t;
679 if (UnparseC.term t', UnparseC.terms asm) =
680 ("(7 * x \<up> 3 + -1 * x \<up> 2 * y) / (7 * x * y \<up> 2 + -1 * y \<up> 3)", "[]")
681 then () else error "rational.sml make_polynomial Schalk 192b";
683 val t = TermC.str2term "((x \<up> 2)*(7 * x + (-1) * y))/((y \<up> 2)*(7 * x + (-1) * y))";
684 val SOME (t', asm) = rewrite_set_ thy false make_polynomial t;
685 if (UnparseC.term t', UnparseC.terms asm) =
686 ("(7 * x \<up> 3 + -1 * x \<up> 2 * y) / (7 * x * y \<up> 2 + -1 * y \<up> 3)", "[]")
687 then () else error "rational.sml make_polynomial Schalk WN050929 not working";
689 "-------- example 193a";
690 val t = TermC.str2term "( x \<up> 2 + -6 * x + 9 ) / ( x \<up> 2 + -9 )";
691 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;
692 if (UnparseC.term t', UnparseC.terms asm) = ("(-3 + x) / (3 + x)", "[\"3 + x \<noteq> 0\"]")
693 then () else error "rational.sml cancel_p Schalk 193a";
695 "-------- example 193b";
696 val t = TermC.str2term "( x \<up> 2 + -8 * x + 16 ) / ( 2 * x \<up> 2 + -32 )";
697 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;
698 if (UnparseC.term t', UnparseC.terms asm) = ("(-4 + x) / (8 + 2 * x)", "[\"8 + 2 * x \<noteq> 0\"]")
699 then () else error "rational.sml cancel_p Schalk 193b";
701 "-------- example 193c";
702 val t = TermC.str2term "( 2 * x + -50 * x \<up> 3 ) / ( 25 * x \<up> 2 + -10 * x + 1 )";
703 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;
704 if (UnparseC.term t', UnparseC.terms asm) =
705 ("(2 * x + 10 * x \<up> 2) / (1 + -5 * x)", "[\"1 + -5 * x \<noteq> 0\"]")
706 then () else error "rational.sml cancel_p Schalk 193c";
709 val t = TermC.str2term "(-25 + 9*x \<up> 2)/(5 + 3*x)";
710 val SOME (t, asm) = rewrite_set_ thy false cancel_p t;
711 if (UnparseC.term t', UnparseC.terms asm) = ("(2 * x + 10 * x \<up> 2) / (1 + -5 * x)", "[]")
712 then () else error "rational.sml cancel WN 1";
714 "-------- example heuberger";
715 val t = TermC.str2term ("(x \<up> 4 + x * y + x \<up> 3 * y + y \<up> 2) / " ^
716 "(x + 5 * x \<up> 2 + y + 5 * x * y + x \<up> 2 * y \<up> 3 + x * y \<up> 4)");
717 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;
718 if (UnparseC.term t', UnparseC.terms asm) =
719 ("(x \<up> 3 + y) / (1 + 5 * x + x * y \<up> 3)", "[\"1 + 5 * x + x * y \<up> 3 \<noteq> 0\"]")
720 then () else error "rational.sml cancel_p heuberger";
722 "-------- rewrite_set_ add_fractions_p from: Mathematik 1 Schalk -------------";
723 "-------- rewrite_set_ add_fractions_p from: Mathematik 1 Schalk -------------";
724 "-------- rewrite_set_ add_fractions_p from: Mathematik 1 Schalk -------------";
725 (*deleted example 204 ... 236b at update Isabelle2012-->2013*)
727 "-------- integration lev.1 -- lev.5: cancel_p_ & add_fractions_p_ -----------";
728 "-------- integration lev.1 -- lev.5: cancel_p_ & add_fractions_p_ -----------";
729 "-------- integration lev.1 -- lev.5: cancel_p_ & add_fractions_p_ -----------";
730 val t = TermC.str2term ("123 = (a*x)/(b*x) + (c*x)/(d*x) + (e*x)/(f*x::real)");
731 "-------- gcd_poly integration level 1: works on exact term";
732 if NONE = cancel_p_ thy t then () else error "cancel_p_ works on exact fraction";
733 if NONE = add_fraction_p_ thy t then () else error "add_fraction_p_ works on exact fraction";
735 "-------- gcd_poly integration level 2: picks out ONE appropriate subterm";
736 val SOME (t', asm) = rewrite_set_ thy false cancel_p t;
737 if UnparseC.term t' = "123 = a * x / (b * x) + c * x / (d * x) + e / f"
738 then () else error "level 2, rewrite_set_ cancel_p: changed";
739 val SOME (t', asm) = rewrite_set_ thy false add_fractions_p t;
740 if UnparseC.term t' = "123 = (b * c * x + a * d * x) / (b * d * x) + e * x / (f * x)"
741 then () else error "level 2, rewrite_set_ add_fractions_p: changed";
743 "-------- gcd_poly integration level 3: rewrites all appropriate subterms";
744 val SOME (t', asm) = rewrite_set_ thy false cancel_p_rls t;
745 if UnparseC.term t' = "123 = a / b + c / d + e / f"
746 then () else error "level 3, rewrite_set_ cancel_p_rls: changed";
747 val SOME (t', asm) = rewrite_set_ thy false add_fractions_p_rls t; (*CREATE add_fractions_p_rls*)
748 if UnparseC.term t' = "123 = (b * d * e * x + b * c * f * x + a * d * f * x) / (b * d * f * x)"
749 then () else error "level 3, rewrite_set_ add_fractions_p_rls: changed";
751 "-------- gcd_poly integration level 4: iteration cancel_p -- add_fraction_p";
752 (* simpler variant *)
753 val testrls = Rule_Set.append_rules "testrls" Rule_Set.empty [Rls_ cancel_p, Rls_ add_fractions_p]
754 val SOME (t', asm) = rewrite_set_ thy false testrls t;
755 (*Rewrite.trace_on := false;
756 # rls: testrls on: 123 = a * x / (b * x) + c * x / (d * x) + e * x / (f * x)
757 ## rls: cancel_p on: 123 = a * x / (b * x) + c * x / (d * x) + e * x / (f * x)
758 ## rls: add_fractions_p on: 123 = a * x / (b * x) + c * x / (d * x) + e / f
759 ## rls: cancel_p on: 123 = (b * c * x + a * d * x) / (b * d * x) + e / f
760 ## rls: add_fractions_p on: 123 = (b * c + a * d) / (b * d) + e / f
761 ## rls: cancel_p on: 123 = (b * d * e + b * c * f + a * d * f) / (b * d * f)
762 ## rls: add_fractions_p on: 123 = (b * d * e + b * c * f + a * d * f) / (b * d * f) *)
763 if UnparseC.term t' = "123 = (b * d * e + b * c * f + a * d * f) / (b * d * f)"
764 then () else error "level 4, rewrite_set_ *_p: changed";
766 (* complicated variant *)
767 val testrls_rls = Rule_Set.append_rules "testrls_rls" Rule_Set.empty [Rls_ cancel_p_rls, Rls_ add_fractions_p_rls];
768 val SOME (t', asm) = rewrite_set_ thy false testrls_rls t;
769 (*# rls: testrls_rls on: 123 = a * x / (b * x) + c * x / (d * x) + e * x / (f * x)
770 ## rls: cancel_p_rls on: 123 = a * x / (b * x) + c * x / (d * x) + e * x / (f * x)
771 ### rls: cancel_p on: 123 = a * x / (b * x) + c * x / (d * x) + e * x / (f * x)
772 ### rls: cancel_p on: 123 = a * x / (b * x) + c * x / (d * x) + e / f
773 ### rls: cancel_p on: 123 = a * x / (b * x) + c / d + e / f
774 ### rls: cancel_p on: 123 = a / b + c / d + e / f
775 ## rls: add_fractions_p_rls on: 123 = a / b + c / d + e / f
776 ### rls: add_fractions_p on: 123 = a / b + c / d + e / f
777 ### rls: add_fractions_p on: 123 = (b * c + a * d) / (b * d) + e / f
778 ### rls: add_fractions_p on: 123 = (b * d * e + b * c * f + a * d * f) / (b * d * f)
779 ## rls: cancel_p_rls on: 123 = (b * d * e + b * c * f + a * d * f) / (b * d * f)
780 ### rls: cancel_p on: 123 = (b * d * e + b * c * f + a * d * f) / (b * d * f)
781 ## rls: add_fractions_p_rls on: 123 = (b * d * e + b * c * f + a * d * f) / (b * d * f)
782 ### rls: add_fractions_p on: 123 = (b * d * e + b * c * f + a * d * f) / (b * d * f) *)
783 if UnparseC.term t' = "123 = (b * d * e + b * c * f + a * d * f) / (b * d * f)"
784 then () else error "level 4, rewrite_set_ *_p_rls: changed"
786 "-------- gcd_poly integration level 5: cancel_p & add_fraction_p within norm_Rational";
787 val SOME (t', asm) = rewrite_set_ thy false norm_Rational t;
788 if UnparseC.term t' = "123 = (a * d * f + b * c * f + b * d * e) / (b * d * f)"
789 then () else error "level 5, rewrite_set_ norm_Rational: changed"
791 "-------- reverse rewrite ----------------------------------------------------";
792 "-------- reverse rewrite ----------------------------------------------------";
793 "-------- reverse rewrite ----------------------------------------------------";
794 (** the term for which reverse rewriting is demonstrated **)
795 val t = TermC.str2term "(9 + -1 * x \<up> 2) / (9 + 6 * x + x \<up> 2)";
796 val Rrls {scr = Rfuns {init_state = ini, locate_rule = loc,
797 next_rule = nex, normal_form = nor, ...},...} = cancel_p;
799 (** normal_form produces the result in ONE step **)
800 val SOME (t',_) = nor t;
801 if UnparseC.term t' = "(3 + -1 * x) / (3 + x)" then ()
802 else error "rational.sml normal_form (9 - x \<up> 2) / (9 - 6 * x + x \<up> 2)";
804 (** initialize the interpreter state used by the 'me' **)
805 val (t, _, revsets, _) = ini t;
807 if length (hd revsets) = 11 then () else error "length of revset changed";
808 if (revsets |> nth 1 |> nth 1 |> id_of_thm) =
809 (@{thm realpow_twoI} |> Thm.get_name_hint |> ThmC.cut_id)
810 then () else error "first element of revset changed";
812 (revsets |> nth 1 |> nth 1 |> Rule.to_string) = "Thm (\"realpow_twoI\",?r1 \<up> 2 = ?r1 * ?r1)" andalso
813 (revsets |> nth 1 |> nth 2 |> Rule.to_string) = "Thm (\"#: 9 = 3 \<up> 2\",9 = 3 \<up> 2)" andalso
814 (revsets |> nth 1 |> nth 3 |> Rule.to_string) = "Thm (\"#: 6 * x = 2 * (3 * x)\",6 * x = 2 * (3 * x))"
816 (revsets |> nth 1 |> nth 4 |> Rule.to_string) = "Thm (\"#: -3 * x = -1 * (3 * x)\",-3 * x = -1 * (3 * x))"
818 (revsets |> nth 1 |> nth 5 |> Rule.to_string) = "Thm (\"#: 9 = 3 * 3\",9 = 3 * 3)" andalso
819 (revsets |> nth 1 |> nth 6 |> Rule.to_string) = "Rls_ (\"sym_order_mult_rls_\")" andalso
820 (revsets |> nth 1 |> nth 7 |> Rule.to_string) =
821 "Thm (\"sym_mult.assoc\",?a * (?b * ?c) = ?a * ?b * ?c)"
822 then () else error "first 7 elements in revset changed"
824 (** find the rule 'r' to apply to term 't' **)
825 (*/------- WN1309: since cancel_ (accepted "-" between monomials) has been replaced by cancel_p_
826 for Isabelle2013, we don't get a working revset, but non-termination:
828 val SOME (r as (Thm (str, thm))) = nex revsets t;
830 ((3 * 3 + -1 * x * x) / (3 * 3 + 2 * 3 * x + x * x),
831 Rls_ ("sym_order_mult_rls_"), ((3 * 3 + -1 * (x * x)) / (3 * 3 + 2 * (3 * x) + x * x), []))", "
832 ((3 * 3 + -1 * (x * x)) / (3 * 3 + 2 * (3 * x) + x * x),
833 Thm ("sym_mult.assoc", ""), ((3 * 3 + -1 * (x * x)) / (3 * 3 + 2 * 3 * x + x * x), []))", "
834 ((3 * 3 + -1 * (x * x)) / (3 * 3 + 2 * 3 * x + x * x),
835 Thm ("sym_mult.assoc", ""), ((3 * 3 + -1 * x * x) / (3 * 3 + 2 * 3 * x + x * x), []))", "
836 ((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), []))", "
839 Thm ("sym_#mult_2_3", "6 = 2 * 3")
840 ### Isabelle2009-2 for cancel_ (not cancel_p_):
841 if str = "sym_#power_Float ((3,0), (0,0)) __ ((2,0), (0,0))"
842 andalso ThmC.string_of_thm thm =
843 (string_of_thm (Thm.make_thm @{theory "Isac_Knowledge"}
844 (Trueprop $ (Thm.term_of o the o (TermC.parse thy)) "9 = 3 \<up> 2"))) then ()
845 else error "rational.sml next_rule (9 - x \<up> 2) / (9 - 6 * x + x \<up> 2)";
846 \---------------------------------------------------------------------------------------/*)
848 (** check, if the rule 'r' applied by the user to 't' belongs to the ruleset;
849 if the rule is OK, the term resulting from applying the rule is returned,too;
850 there might be several rule applications inbetween,
851 which are listed after the head in reverse order **)
852 (*/-------------------------------------------- Isabelle2013: this gives "error id_of_thm";
853 we don't repair this, because interaction within "reverse rewriting" never worked properly:
855 val (r, (t, asm))::_ = loc revsets t r;
856 if UnparseC.term t = "(9 - x \<up> 2) / (3 \<up> 2 + 6 * x + x \<up> 2)" andalso asm = []
857 then () else error "rational.sml locate_rule (9 - x \<up> 2) / (9 - 6 * x + x \<up> 2)";
859 (* find the next rule to apply *)
860 val SOME (r as (Thm (str, thm))) = nex revsets t;
861 if str = "sym_#power_Float ((3,0), (0,0)) __ ((2,0), (0,0))" andalso
862 ThmC.string_of_thm thm = (string_of_thm (ThmC_Def.make_thm @{theory "Isac_Knowledge"}
863 (Trueprop $ (Thm.term_of o the o (TermC.parse thy)) "9 = 3 \<up> 2"))) then ()
864 else error "rational.sml next_rule (9 - x \<up> 2) / (9 - 6 * x + x \<up> 2)";
866 (*check the next rule*)
867 val (r, (t, asm)) :: _ = loc revsets t r;
868 if UnparseC.term t = "(3 \<up> 2 - x \<up> 2) / (3 \<up> 2 + 6 * x + x \<up> 2)" then ()
869 else error "rational.sml locate_rule (9 - x \<up> 2) / (9 - 6 * x + x \<up> 2) II";
871 (*find and check the next rules, rewrite*)
872 val SOME r = nex revsets t;
873 val (r,(t,asm))::_ = loc revsets t r;
874 if UnparseC.term t = "(3 \<up> 2 - x \<up> 2) / (3 \<up> 2 + 2 * 3 * x + x \<up> 2)" then ()
875 else error "rational.sml locate_rule II";
877 val SOME r = nex revsets t;
878 val (r,(t,asm))::_ = loc revsets t r;
879 if UnparseC.term t = "(3 - x) * (3 + x) / (3 \<up> 2 + 2 * 3 * x + x \<up> 2)" then ()
880 else error "rational.sml next_rule II";
882 val SOME r = nex revsets t;
883 val (r,(t,asm))::_ = loc revsets t r;
884 if UnparseC.term t = "(3 - x) * (3 + x) / ((3 + x) * (3 + x))" then ()
885 else error "rational.sml next_rule III";
887 val SOME r = nex revsets t;
888 val (r, (t, asm)) :: _ = loc revsets t r;
889 val ss = UnparseC.term t;
890 if ss = "(3 - x) / (3 + x)" andalso UnparseC.terms asm = "[\"3 + x ~= 0\"]" then ()
891 else error "rational.sml: new behav. in rev-set cancel";
892 \--------------------------------------------------------------------------------------/*)
894 "-------- 'reverse-ruleset' cancel_p -----------------------------------------";
895 "-------- 'reverse-ruleset' cancel_p -----------------------------------------";
896 "-------- 'reverse-ruleset' cancel_p -----------------------------------------";
897 (*WN130909: the example below shows, why "reverse rewriting" only worked for
900 (*the term for which reverse rewriting is demonstrated*)
901 val t = TermC.str2term "(9 + (-1)*x \<up> 2) / (9 + ((-6)*x + x \<up> 2))";
902 val Rrls {scr=Rfuns {init_state=ini,locate_rule=loc,
903 next_rule=nex,normal_form=nor,...},...} = cancel_p;
905 (*normal_form produces the result in ONE step*)
906 val SOME (t',_) = nor t;
907 UnparseC.term t' = "(3 + 1 * x) / (3 + -1 * x)";
909 (*initialize the interpreter state used by the 'me'*)
910 val SOME (t', asm) = cancel_p_ thy t;
911 UnparseC.term t' = "(3 + x) / (3 + -1 * x)" (*true*);
912 UnparseC.terms asm = "[\"3 + -1 * x ~= 0\"]" (*true*);
913 val (t,_,revsets,_) = ini t;
915 (* WN.10.10.02: dieser Fall terminiert nicht
916 (make_polynomial enth"alt zu viele rules)
917 WN060823 'init_state' requires rewriting on specified location in the term
918 default_print_depth 99; Rfuns; default_print_depth 3;
919 WN060831 cycling "sym_order_mult_rls_" "sym_mult.assoc"
920 as was with make_polynomial before ?!?*)
922 val SOME r = nex revsets t;
923 eq_Thm (r, Thm ("sym_#power_Float ((3,0), (0,0)) __ ((2,0), (0,0))",
924 mk_thm thy "9 = 3 \<up> 2"));
925 (*WN060831 *** id_of_thm
926 Exception- ERROR raised ...
927 val (r,(t,asm))::_ = loc revsets t r;
930 val SOME r = nex revsets t;
931 val (r,(t,asm))::_ = loc revsets t r;
935 "-------- examples: rls norm_Rational ----------------------------------------";
936 "-------- examples: rls norm_Rational ----------------------------------------";
937 "-------- examples: rls norm_Rational ----------------------------------------";
938 val t = TermC.str2term "(3*x+5)/18 - x/2 - -(3*x - 2)/9 = 0";
939 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; UnparseC.term t';
940 if UnparseC.term t' = "1 / 18 = 0" then () else error "rational.sml 1";
942 val t = TermC.str2term "(17*x - 51)/9 - (-(13*x - 3)/6) + 11 - (9*x - 7)/4 = 0";
943 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; UnparseC.term t';
944 if UnparseC.term t' = "(237 + 65 * x) / 36 = 0" then ()
945 else error "rational.sml 2";
947 val t = TermC.str2term "(1/2 + (5*x)/2) \<up> 2 - ((13*x)/2 - 5/2) \<up> 2 - (6*x) \<up> 2 + 29";
948 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; UnparseC.term t';
949 if UnparseC.term t' = "23 + 35 * x + -72 * x \<up> 2" then ()
950 else error "rational.sml 3";
952 (*Rewrite.trace_on:=true;*)
953 val t = TermC.str2term "Not (6*x is_atom)";
954 val SOME (t',_) = rewrite_set_ thy false powers_erls t; UnparseC.term t';
955 \<^const_name>\<open>True\<close>;
956 val t = TermC.str2term "1 < 2";
957 val SOME (t',_) = rewrite_set_ thy false powers_erls t; UnparseC.term t';
958 \<^const_name>\<open>True\<close>;
960 val t = TermC.str2term "(6*x) \<up> 2";
961 val SOME (t',_) = rewrite_ thy dummy_ord powers_erls false
962 (ThmC.numerals_to_Free @{thm realpow_def_atom}) t;
963 if UnparseC.term t' = "6 * x * (6 * x) \<up> (2 + -1)" then ()
964 else error "rational.sml powers_erls (6*x) \<up> 2";
966 val t = TermC.str2term "-1 * (-2 * (5 / 2 * (13 * x / 2)))";
967 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; UnparseC.term t';
968 if UnparseC.term t' = "65 * x / 2" then () else error "rational.sml 4";
970 val t = TermC.str2term "1 - ((13*x)/2 - 5/2) \<up> 2";
971 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; UnparseC.term t';
972 if UnparseC.term t' = "(-21 + 130 * x + -169 * x \<up> 2) / 4" then ()
973 else error "rational.sml 5";
975 (*SRAM Schalk I, p.92 Nr. 609a*)
976 val t = TermC.str2term "2*(3 - x/5)/3 - 4*(1 - x/3) - x/3 - 2*(x/2 - 1/4)/27 +5/54";
977 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; UnparseC.term t';
978 if UnparseC.term t' = "(-255 + 112 * x) / 135" then ()
979 else error "rational.sml 6";
981 (*SRAM Schalk I, p.92 Nr. 610c*)
982 val t = TermC.str2term "((x- 1)/(x+1) + 1) / ((x- 1)/(x+1) - (x+1)/(x- 1)) - 2";
983 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; UnparseC.term t';
984 if UnparseC.term t' = "(3 + x) / -2" then () else error "rational.sml 7";
986 (*SRAM Schalk I, p.92 Nr. 476a*)
987 val t = TermC.str2term "(x \<up> 2/(1 - x \<up> 2) + 1)/(x/(1 - x) + 1) * (1 + x)";
988 (*. a/b : c/d translated to a/b * d/c .*)
989 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; UnparseC.term t';
990 if UnparseC.term t' = "1" then () else error "rational.sml 8";
992 (*Schalk I, p.92 Nr. 472a*)
993 val t = TermC.str2term "((8*x \<up> 2 - 32*y \<up> 2)/(2*x + 4*y))/((4*x - 8*y)/(x + y))";
994 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; UnparseC.term t';
995 if UnparseC.term t' = "x + y" then () else error "rational.sml p.92 Nr. 472a";
997 (*Schalk I, p.70 Nr. 480b: SEE rational.sml --- nonterminating rls norm_Rational ---*)
999 (*WN130910 add_fractions_p exception Div raised + history:
1000 ### WN.2.6.03 from rlang.sml 56a
1001 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)";
1002 val NONE = rewrite_set_ thy false add_fractions_p t;
1004 THE ERROR ALREADY OCCURS IN THIS PART:
1005 val t = TermC.str2term "(a + b * x) / (a + -1 * (b * x)) + (-1 * a + b * x) / (a + b * x)";
1006 val NONE = add_fraction_p_ thy t;
1008 SEE Test_Some.thy: section {* add_fractions_p downto exception Div raised ===
1011 "-------- rational numerals --------------------------------------------------";
1012 "-------- rational numerals --------------------------------------------------";
1013 "-------- rational numerals --------------------------------------------------";
1014 (*SRA Schalk I, p.40 Nr. 164b *)
1015 val t = TermC.str2term "(47/6 - 76/9 + 13/4)/(35/12)";
1016 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1018 if UnparseC.term t = "19 / 21" then ()
1019 else error "rational.sml: diff.behav. in norm_Rational_mg 1";
1021 (*SRA Schalk I, p.40 Nr. 166a *)
1022 val t = TermC.str2term "((5/4)/(4+22/7) + 37/20)*(110/3 - 110/9 * 23/11)";
1023 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1025 if UnparseC.term t = "45 / 2" then ()
1026 else error "rational.sml: diff.behav. in norm_Rational_mg 2";
1028 "-------- examples cancellation from: Mathematik 1 Schalk --------------------";
1029 "-------- examples cancellation from: Mathematik 1 Schalk --------------------";
1030 "-------- examples cancellation from: Mathematik 1 Schalk --------------------";
1031 (* e190c Stefan K.*)
1032 val t = TermC.str2term "((1 + 9*a \<up> 2) * (1 + 3*a)) / ((3*a + 9*a \<up> 2) * (1 + 3*a))";
1033 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1034 if UnparseC.term t = "(1 + 9 * a \<up> 2) / (3 * a + 9 * a \<up> 2)"
1035 then () else error "rational.sml: diff.behav. in norm_Rational_mg 3";
1037 (* e192b Stefan K.*)
1038 val t = TermC.str2term "(x \<up> 2 * (7*x + (-1)*y)) / (y \<up> 2 * (7*x + (-1)*y))";
1039 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1040 if UnparseC.term t = "x \<up> 2 / y \<up> 2"
1041 then () else error "rational.sml: diff.behav. in norm_Rational_mg 4";
1043 (*SRC Schalk I, p.66 Nr. 379c *)
1044 val t = TermC.str2term "(a - b)/(b - a)";
1045 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1047 if UnparseC.term t = "-1"
1048 then () else error "rational.sml: diff.behav. in norm_Rational_mg 5";
1050 (*SRC Schalk I, p.66 Nr. 380b *)
1051 val t = TermC.str2term "15*(3*x + 3) * (4*x + 9) / (12*(2*x + 7) * (5*x + 5))";
1052 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1053 if UnparseC.term t = "(27 + 12 * x) / (28 + 8 * x)"
1054 then () else error "rational.sml: diff.behav. in norm_Rational_mg 6";
1056 (*Schalk I, p.60 Nr. 215c: was not cancelled with Isabelle2002 *)
1057 val t = TermC.str2term "(a + b) \<up> 4 * (x - y) / ((x - y) \<up> 3 * (a + b) \<up> 2)";
1058 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1059 if UnparseC.term t = "(a \<up> 2 + 2 * a * b + b \<up> 2) / (x \<up> 2 + -2 * x * y + y \<up> 2)"
1060 then () else error "rational.sml: diff.behav. in norm_Rational_mg 7";
1062 (*SRC Schalk I, p.66 Nr. 381b *)
1063 val t = TermC.str2term
1064 "(4*x \<up> 2 - 20*x + 25)/(2*x - 5) \<up> 3";
1065 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1066 if UnparseC.term t = "1 / (-5 + 2 * x)"
1067 then () else error "rational.sml: diff.behav. in norm_Rational_mg 9";
1069 (* e190c Stefan K.*)
1070 val t = TermC.str2term "((1 + 9*a \<up> 2) * (1 + 3*a)) / ((3*a + 9*a \<up> 2) * (1 + 3 * a))";
1071 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1072 if UnparseC.term t = "(1 + 9 * a \<up> 2) / (3 * a + 9 * a \<up> 2)"
1073 then () else error "rational.sml: diff.behav. in norm_Rational_mg 3";
1075 (* e192b Stefan K.*)
1076 val t = TermC.str2term "(x \<up> 2 * (7*x + (-1)*y)) / (y \<up> 2 * (7*x + (-1)*y))";
1077 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1078 if UnparseC.term t = "x \<up> 2 / y \<up> 2"
1079 then () else error "rational.sml: diff.behav. in norm_Rational_mg 4";
1081 (*SRC Schalk I, p.66 Nr. 379c *)
1082 val t = TermC.str2term "(a - b) / (b - a)";
1083 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1084 if UnparseC.term t = "-1"
1085 then () else error "rational.sml: diff.behav. in norm_Rational_mg 5";
1087 (*SRC Schalk I, p.66 Nr. 380b *)
1088 val t = TermC.str2term "15*(3*x + 3) * (4*x + 9) / (12*(2*x + 7) * (5*x + 5))";
1089 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1090 if UnparseC.term t = "(27 + 12 * x) / (28 + 8 * x)"
1091 then () else error "rational.sml: diff.behav. in norm_Rational_mg 6";
1093 (*Schalk I, p.60 Nr. 215c *)
1094 val t = TermC.str2term "(a + b) \<up> 4 * (x - y) / ((x - y) \<up> 3 * (a + b) \<up> 2)";
1095 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1096 if UnparseC.term t = "(a \<up> 2 + 2 * a * b + b \<up> 2) / (x \<up> 2 + -2 * x * y + y \<up> 2)"
1097 then () else error "Schalk I, p.60 Nr. 215c: with Isabelle2002 cancellation incomplete, changed";
1099 (* extreme example from somewhere *)
1100 val t = TermC.str2term
1101 ("(a \<up> 4 * x + -1*a \<up> 4 * y + 4*a \<up> 3 * b * x + -4*a \<up> 3 * b * y + " ^
1102 "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 + " ^
1103 "b \<up> 4 * x + -1*b \<up> 4 * y) " ^
1104 " / (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 + " ^
1105 "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 + " ^
1106 "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)")
1107 val SOME (t, _) = rewrite_set_ thy false cancel_p t;
1108 if UnparseC.term t = "(a \<up> 2 + 2 * a * b + b \<up> 2) / (x \<up> 2 + -2 * x * y + y \<up> 2)"
1109 then () else error "with Isabelle2002: NONE -- now SOME changed";
1111 (*Schalk I, p.66 Nr. 381a *)
1112 (* ATTENTION: here the rls is very slow. In Isabelle2002 this required 2 min *)
1113 val t = TermC.str2term "18*(a + b) \<up> 3 * (a - b) \<up> 2 / (72*(a - b) \<up> 3 * (a + b) \<up> 2)";
1114 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1115 if UnparseC.term t = "(a + b) / (4 * a + -4 * b)"
1116 then () else error "rational.sml: diff.behav. in norm_Rational_mg 8";
1118 (*SRC Schalk I, p.66 Nr. 381b *)
1119 val t = TermC.str2term "(4*x \<up> 2 - 20*x + 25) / (2*x - 5) \<up> 3";
1120 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1121 if UnparseC.term t = "1 / (-5 + 2 * x)"
1122 then () else error "rational.sml: diff.behav. in norm_Rational_mg 9";
1124 (*SRC Schalk I, p.66 Nr. 381c *)
1125 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)";
1126 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1127 if UnparseC.term t = "(1 + 9 * a \<up> 2) / (3 * a + 9 * a \<up> 2)"
1128 then () else error "rational.sml: diff.behav. in norm_Rational_mg 10";
1130 (*SRC Schalk I, p.66 Nr. 383a *)
1131 val t = TermC.str2term "(5*a \<up> 2 - 5*a*b) / (a - b) \<up> 2";
1132 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1133 if UnparseC.term t = "-5 * a / (-1 * a + b)"
1134 then () else error "rational.sml: diff.behav. in norm_Rational_mg 11";
1136 "----- NOT TERMINATING ?: worked before 0707xx";
1137 val t = TermC.str2term "(a \<up> 2 - 1)*(b + 1) / ((b \<up> 2 - 1)*(a+1))";
1138 (* WN130911 "exception Div raised" by
1139 cancel_p_ thy (TermC.str2term ("(-1 + -1 * b + a \<up> 2 + a \<up> 2 * b) /" ^
1140 "(-1 + -1 * a + b \<up> 2 + a * b \<up> 2)"))
1142 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1143 if UnparseC.term t = "(1 + -1 * a) / (1 + -1 * b)" then ()
1144 else error "rational.sml MG tests 3e";
1147 "-------- examples common denominator from: Mathematik 1 Schalk --------------";
1148 "-------- examples common denominator from: Mathematik 1 Schalk --------------";
1149 "-------- examples common denominator from: Mathematik 1 Schalk --------------";
1150 (*SRA Schalk I, p.67 Nr. 403a *)
1151 val t = TermC.str2term "4/x - 3/y - 1";
1152 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1153 if UnparseC.term t = "(-3 * x + 4 * y + -1 * x * y) / (x * y)"
1154 then () else error "rational.sml: diff.behav. in norm_Rational_mg 12";
1156 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)";
1157 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1158 if UnparseC.term t = "4 / c"
1159 then () else error "rational.sml: diff.behav. in norm_Rational_mg 13";
1161 (*SRA Schalk I, p.67 Nr. 410b *)
1162 val t = TermC.str2term "1/(x+1) + 1/(x+2) - 2/(x+3)";
1163 (* WN130911 non-termination due to non-termination of
1164 cancel_p_ thy (TermC.str2term "(5 + 3 * x) / (6 + 11 * x + 6 * x \<up> 2 + x \<up> 3)")
1166 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1167 if UnparseC.term t = "(5 + 3 * x) / (6 + 11 * x + 6 * x \<up> 2 + x \<up> 3)"
1168 then () else error "rational.sml: diff.behav. in norm_Rational_mg 14";
1171 (*SRA Schalk I, p.67 Nr. 413b *)
1172 val t = TermC.str2term "(1 + x)/(1 - x) - (1 - x)/(1 + x) + 2*x/(1 - x \<up> 2)";
1173 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1174 if UnparseC.term t = "6 * x / (1 + -1 * x \<up> 2)"
1175 then () else error "rational.sml: diff.behav. in norm_Rational_mg 15";
1177 (*SRA Schalk I, p.68 Nr. 414a *)
1178 val t = TermC.str2term "(x + 2)/(x - 1) + (x - 3)/(x - 2) - (x + 1)/((x - 1)*(x - 2))";
1179 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1180 if UnparseC.term t ="(-2 + -5 * x + 2 * x \<up> 2) / (2 + -3 * x + x \<up> 2)"
1181 then () else error "rational.sml: diff.behav. in norm_Rational_mg 16";
1183 (*SRA Schalk I, p.68 Nr. 428b *)
1184 val t = TermC.str2term
1185 "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";
1186 (* WN130911 non-termination due to non-termination of
1187 cancel_p_ thy (TermC.str2term "(4 + -4 * b \<up> 2) / (a \<up> 4 + -2 * (a \<up> 2 * b \<up> 2) + b \<up> 4)")
1189 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1190 if UnparseC.term t = "4 / (a \<up> 4 + -2 * a \<up> 2 * b \<up> 2 + b \<up> 4)"
1191 then () else error "rational.sml: diff.behav. in norm_Rational_mg 18";
1194 (*SRA Schalk I, p.68 Nr. 430b *)
1195 val t = TermC.str2term
1196 "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";
1197 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1198 if UnparseC.term t = "a + 3 * b"
1199 then () else error "rational.sml: diff.behav. in norm_Rational_mg 19";
1201 (*SRA Schalk I, p.68 Nr. 432 *)
1202 val t = TermC.str2term
1203 ("(a \<up> 2 + a*b) / (a \<up> 2 - b \<up> 2) - (b \<up> 2 - a*b) / (b \<up> 2 - a \<up> 2) + " ^
1204 "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) - " ^
1205 "2*b \<up> 2 / (a \<up> 2 - b \<up> 2)");
1206 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1207 if UnparseC.term t = (*"0" ..isabisac15 | Isabelle2017..*) "0 / (a \<up> 2 + -1 * b \<up> 2)"
1208 then () else error "rational.sml: diff.behav. in norm_Rational_mg 20";
1211 val t = TermC.str2term "3*a / (a*b) + x/y";
1212 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1213 if UnparseC.term t = "(3 * y + b * x) / (b * y)"
1214 then () else error "rational.sml: diff.behav. in norm_Rational_mg 21";
1216 "-------- examples multiply and cancel from: Mathematik 1 Schalk -------------";
1217 "-------- examples multiply and cancel from: Mathematik 1 Schalk -------------";
1218 "-------- examples multiply and cancel from: Mathematik 1 Schalk -------------";
1219 (*------- SRM Schalk I, p.68 Nr. 436a *)
1220 val t = TermC.str2term "3*(x+y) / (15*(x - y)) * 25*(x - y) \<up> 2 / (18*(x + y) \<up> 2)";
1221 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1222 if UnparseC.term t = "(-5 * x + 5 * y) / (-18 * x + -18 * y)"
1223 then () else error "rational.sml: diff.behav. in norm_Rational_mg 22";
1225 (*------- SRM.test Schalk I, p.68 Nr. 436b *)
1226 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";
1227 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1228 if UnparseC.term t = "5 * a / (a + -1 * b)"
1229 then () else error "rational.sml: diff.behav. in norm_Rational_mg 23";
1231 (*------- Schalk I, p.68 Nr. 437a *)
1232 val t = TermC.str2term "(3*a - 4*b) / (4*c+3*e) * (3*a+4*b)/(9*a \<up> 2 - 16*b \<up> 2)";
1233 (* raises an exception for unclear reasons:
1234 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1236 ### rls: cancel_p on: (9 * a \<up> 2 + -16 * b \<up> 2) / (4 * c + 3 * e) /
1237 (9 * a \<up> 2 + -16 * b \<up> 2)
1238 exception Div raised
1241 val t = TermC.str2term
1242 ("(9 * a \<up> 2 + -16 * b \<up> 2) / (4 * c + 3 * e) /" ^
1243 "(9 * a \<up> 2 + -16 * b \<up> 2)");
1244 NONE = cancel_p_ thy t;
1246 if UnparseC.term t = "1 / (4 * c + 3 * e)" then ()
1247 else error "rational.sml: diff.behav. in norm_Rational_mg 24";
1250 "----- S.K. corrected non-termination 060904";
1251 val t = TermC.str2term "(3*a - 4*b) * (3*a+4*b)/((4*c+3*e)*(9*a \<up> 2 - 16*b \<up> 2))";
1252 val SOME (t, _) = rewrite_set_ thy false make_polynomial t;
1253 if UnparseC.term t =
1254 "(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)"
1255 (*"(9 * a \<up> 2 + -16 * b \<up> 2) / (36 * a \<up> 2 * c + 27 * a \<up> 2 * e + -64 * b \<up> 2 * c + -48 * b \<up> 2 * e)"*)
1256 then () else error "rational.sml: S.K.8..corrected 060904-6";
1258 "----- S.K. corrected non-termination of cancel_p_";
1259 val t'' = TermC.str2term ("(9 * a \<up> 2 + -16 * b \<up> 2) /" ^
1260 "(36 * a \<up> 2 * c + (27 * a \<up> 2 * e + (-64 * b \<up> 2 * c + -48 * b \<up> 2 * e)))");
1261 (* /--- DOES NOT TERMINATE AT TRANSITION isabisac15 --> Isabelle2017 --------------------------\
1262 val SOME (t',_) = rewrite_set_ thy false cancel_p t'';
1263 if UnparseC.term t' = "1 / (4 * c + 3 * e)"
1264 then () else error "rational.sml: diff.behav. in cancel_p S.K.8";
1265 \--- DOES NOT TERMINATE AT TRANSITION isabisac15 --> Isabelle2017 --------------------------/*)
1267 (*------- Schalk I, p.68 Nr. 437b*)
1268 val t = TermC.str2term "(a + b)/(x \<up> 2 - y \<up> 2) * ((x - y) \<up> 2/(a \<up> 2 - b \<up> 2))";
1269 (*val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1271 #### rls: cancel_p on: (a * x \<up> 2 + -2 * (a * (x * y)) + a * y \<up> 2 + b * x \<up> 2 +
1272 -2 * (b * (x * y)) +
1274 (a \<up> 2 * x \<up> 2 + -1 * (a \<up> 2 * y \<up> 2) + -1 * (b \<up> 2 * x \<up> 2) +
1275 b \<up> 2 * y \<up> 2)
1276 exception Div raised
1279 (*------- SRM Schalk I, p.68 Nr. 438a *)
1280 val t = TermC.str2term "x*y / (x*y - y \<up> 2) * (x \<up> 2 - x*y)";
1281 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1282 if UnparseC.term t = "x \<up> 2"
1283 then () else error "rational.sml: diff.behav. in norm_Rational_mg 24";
1285 (*------- SRM Schalk I, p.68 Nr. 439b *)
1286 val t = TermC.str2term "(4*x \<up> 2 + 4*x + 1) * ((x \<up> 2 - 2*x \<up> 3) / (4*x \<up> 2 + 2*x))";
1287 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1288 if UnparseC.term t = "(x + -4 * x \<up> 3) / 2"
1289 then () else error "rational.sml: diff.behav. in norm_Rational_mg 25";
1291 (*------- SRM Schalk I, p.68 Nr. 440a *)
1292 val t = TermC.str2term "(x \<up> 2 - 2*x) / (x \<up> 2 - 3*x) * (x - 3) \<up> 2 / (x \<up> 2 - 4)";
1293 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1294 if UnparseC.term t = "(-3 + x) / (2 + x)"
1295 then () else error "rational.sml: diff.behav. in norm_Rational_mg 26";
1297 "----- Schalk I, p.68 Nr. 440b SK11 works since 0707xx";
1298 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)";
1299 (* WN130911 non-termination for unclear reasons:
1300 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1302 ... ENDS WITH THIS TRACE:
1304 ### rls: cancel_p on: (-9 * (a \<up> 3 * b) + -9 * (a \<up> 2 * b \<up> 2) + a \<up> 5 * b +
1305 a \<up> 4 * b \<up> 2) /
1306 (a \<up> 3 * b + -1 * (a * b \<up> 3)) /
1308 BUT THIS IS CORRECTLY RECOGNISED
1309 val t = TermC.str2term
1310 ("(-9 * (a \<up> 3 * b) + -9 * (a \<up> 2 * b \<up> 2) + a \<up> 5 * b + a \<up> 4 * b \<up> 2) /" ^
1311 "(a \<up> 3 * b + -1 * (a * b \<up> 3)) / (3 + (a::real))");
1313 NONE = cancel_p_ thy t;
1315 if UnparseC.term t = "(-3 * a + a \<up> 2) / (a + -1 * b)" then ()
1316 else error "rational.sml: diff.behav. in norm_Rational 27";
1319 "----- SK12 works since 0707xx";
1320 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))";
1321 (* WN130911 non-termination due to non-termination of
1322 cancel_p_ thy (TermC.str2term "(4 + -4 * b \<up> 2) / (a \<up> 4 + -2 * (a \<up> 2 * b \<up> 2) + b \<up> 4)")
1324 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1325 if UnparseC.term t' = "(-3 * a + a \<up> 2) / (a + -1 * b)" then ()
1326 else error "rational.sml: diff.behav. in norm_Rational 28";
1329 "-------- examples common denominator and multiplication from: Schalk --------";
1330 "-------- examples common denominator and multiplication from: Schalk --------";
1331 "-------- examples common denominator and multiplication from: Schalk --------";
1332 (*------- SRAM Schalk I, p.69 Nr. 441b *)
1333 val t = TermC.str2term "(4*a/3 + 3*b \<up> 2/a \<up> 3 + b/(4*a))*(4*b/(3*a))";
1334 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1335 if UnparseC.term t = "(36 * b \<up> 3 + 3 * a \<up> 2 * b \<up> 2 + 16 * a \<up> 4 * b) / (9 * a \<up> 4)"
1336 then () else error "rational.sml: diff.behav. in norm_Rational_mg 28";
1338 (*------- SRAM Schalk I, p.69 Nr. 442b *)
1339 val t = TermC.str2term ("(15*a \<up> 2/x \<up> 3 - 5*b \<up> 4/x \<up> 2 + 25*c \<up> 2/x) * " ^
1340 "(x \<up> 3/(5*a*b \<up> 3*c \<up> 3)) + 1/c \<up> 3 * (b*x/a - 3*a/b \<up> 3)");
1341 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1342 if UnparseC.term t = "5 * x \<up> 2 / (a * b \<up> 3 * c)"
1343 then () else error "rational.sml: diff.behav. in norm_Rational_mg 29";
1345 (*------- SRAM Schalk I, p.69 Nr. 443b *)
1346 val t = TermC.str2term "(a/2 + b/3) * (b/3 - a/2)";
1347 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1348 if UnparseC.term t = "(-9 * a \<up> 2 + 4 * b \<up> 2) / 36"
1349 then () else error "rational.sml: diff.behav. in norm_Rational_mg 30";
1351 (*------- SRAM Schalk I, p.69 Nr. 445b *)
1352 val t = TermC.str2term "(a \<up> 2/9 + 2*a/(3*b) + 4/b \<up> 2)*(a/3 - 2/b) + 8/b \<up> 3";
1353 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1354 if UnparseC.term t = "a \<up> 3 / 27"
1355 then () else error "rational.sml: diff.behav. in norm_Rational_mg 31";
1357 (*------- SRAM Schalk I, p.69 Nr. 446b *)
1358 val t = TermC.str2term "(x/(5*x + 4*y) - y/(5*x - 4*y) + 1)*(25*x \<up> 2 - 16*y \<up> 2)";
1359 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1360 if UnparseC.term t = (*"30 * x \<up> 2 + -9 * x * y + -20 * y \<up> 2" ..isabisac15 | Isabelle2017..*)
1361 "(-30 * x \<up> 2 + 9 * x * y + 20 * y \<up> 2) / -1"
1362 then () else error "rational.sml: diff.behav. in norm_Rational_mg 32";
1364 (*------- SRAM Schalk I, p.69 Nr. 449a *)(*Achtung: rechnet ca 8 Sekunden*)
1365 val t = TermC.str2term
1366 "(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)";
1367 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1368 if UnparseC.term t = "(-81 * x \<up> 4 + 16 * x \<up> 8 * y \<up> 4) / (81 * y \<up> 8)"
1369 then () else error "rational.sml: diff.behav. in norm_Rational_mg 33";
1371 (*------- SRAM Schalk I, p.69 Nr. 450a *)
1372 val t = TermC.str2term
1373 "(4*x/(3*y)+2*y/(3*x)) \<up> 2 - (2*y/(3*x) - 2*x/y)*(2*y/(3*x)+2*x/y)";
1374 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1375 if UnparseC.term t = "(52 * x \<up> 2 + 16 * y \<up> 2) / (9 * y \<up> 2)"
1376 then () else error "rational.sml: diff.behav. in norm_Rational_mg 34";
1378 (*------- SRAM Schalk I, p.69 Nr. 442b --- abgewandelt*)
1379 val t = TermC.str2term
1380 ("(15*a \<up> 4/(a*x \<up> 3) - 5*a*((b \<up> 4 - 5*c \<up> 2*x) / x \<up> 2)) * " ^
1381 "(x \<up> 3/(5*a*b \<up> 3*c \<up> 3)) + a/c \<up> 3 * (x*(b/a) - 3*b*(a/b \<up> 4))");
1382 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1383 if UnparseC.term t = "5 * x \<up> 2 / (b \<up> 3 * c)"
1384 then () else error "rational.sml: diff.behav. in norm_Rational_mg 53";
1387 "-------- examples double fractions from: Mathematik 1 Schalk ----------------";
1388 "-------- examples double fractions from: Mathematik 1 Schalk ----------------";
1389 "-------- examples double fractions from: Mathematik 1 Schalk ----------------";
1390 "----- SRD Schalk I, p.69 Nr. 454b";
1391 val t = TermC.str2term "((2 - x)/(2*a)) / (2*a/(x - 2))";
1392 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1393 if UnparseC.term t = "(-4 + 4 * x + -1 * x \<up> 2) / (4 * a \<up> 2)"
1394 then () else error "rational.sml: diff.behav. in norm_Rational_mg 35";
1396 "----- SRD Schalk I, p.69 Nr. 455a";
1397 val t = TermC.str2term "(a \<up> 2 + 1)/(a \<up> 2 - 1) / ((a+1)/(a - 1))";
1398 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1399 if UnparseC.term t = "(1 + a \<up> 2) / (1 + 2 * a + a \<up> 2)" then ()
1400 else error "rational.sml: diff.behav. in norm_Rational_mg 36";
1402 "----- Schalk I, p.69 Nr. 455b";
1403 val t = TermC.str2term "(x \<up> 2 - 4)/(y \<up> 2 - 9)/((2+x)/(3 - y))";
1404 (* WN130911 non-termination due to non-termination of
1405 cancel_p_ thy (TermC.str2term ("(-12 + 4 * y + 3 * x \<up> 2 + -1 * (x \<up> 2 * y)) /" ^
1406 "(-18 + -9 * x + 2 * y \<up> 2 + x * y \<up> 2)"))
1408 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1409 if UnparseC.term t = "(2 + -1 * x) / (3 + y)" then ()
1410 else error "rational.sml: diff.behav. in norm_Rational_mg 37";
1413 "----- SK060904-1a non-termination of cancel_p_ ?: worked before 0707xx";
1414 val t = TermC.str2term "(x \<up> 2 - 4)*(3 - y) / ((y \<up> 2 - 9)*(2+x))";
1415 (* WN130911 non-termination due to non-termination of
1416 cancel_p_ thy (TermC.str2term ("(-12 + 4 * y + 3 * x \<up> 2 + -1 * (x \<up> 2 * y)) /" ^
1417 "(-18 + -9 * x + 2 * y \<up> 2 + x * y \<up> 2)"))
1419 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1420 if UnparseC.term t = "(2 + -1 * x) / (3 + y)" then ()
1421 else error "rational.sml: diff.behav. in norm_Rational_mg 37b";
1424 "----- ?: worked before 0707xx";
1425 val t = TermC.str2term "(3 + -1 * y) / (-9 + y \<up> 2)";
1426 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1427 if UnparseC.term t = "-1 / (3 + y)"
1428 then () else error "rational.sml: -1 / (3 + y) norm_Rational";
1430 "----- SRD Schalk I, p.69 Nr. 456b";
1431 val t = TermC.str2term "(b \<up> 3 - b \<up> 2) / (b \<up> 2+b) / (b \<up> 2 - 1)";
1432 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1433 if UnparseC.term t = "b / (1 + 2 * b + b \<up> 2)"
1434 then () else error "rational.sml: diff.behav. in norm_Rational_mg 38";
1436 "----- SRD Schalk I, p.69 Nr. 457b";
1437 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))";
1438 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1439 if UnparseC.term t = "8 * a \<up> 2 + -6 * a * b + -12 * a \<up> 2 * b + 9 * a * b \<up> 2"
1440 then () else error "rational.sml: diff.behav. in norm_Rational_mg 39";
1442 "----- Schalk I, p.69 Nr. 458b works since 0707";
1443 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)))";
1444 (*val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1446 ### rls: cancel_p on: (-1 * a \<up> 2 + 2 * (a \<up> 2 * x)) / (a * x + -1 * (b * x)) /
1447 ((-1 * b \<up> 2 + 2 * (b \<up> 2 * x)) / (a * x + -1 * (b * x)))
1448 exception Div raised
1451 val t = TermC.str2term
1452 ("(-1 * a \<up> 2 + 2 * (a \<up> 2 * x)) / (a * x + -1 * (b * x)) /" ^
1453 "((-1 * b \<up> 2 + 2 * (b \<up> 2 * x)) / (a * x + -1 * (b * x)))");
1454 NONE = cancel_p_ thy t;
1456 if UnparseC.term t = "a \<up> 2 / b \<up> 2" then ()
1457 else error "rational.sml: diff.behav. in norm_Rational_mg 39b";
1460 "----- SRD Schalk I, p.69 Nr. 459b";
1461 val t = TermC.str2term "(a \<up> 2 - b \<up> 2)/(a*b) / (4*(a+b) \<up> 2/a)";
1462 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1463 if UnparseC.term t = "(a + -1 * b) / (4 * a * b + 4 * b \<up> 2)" then ()
1464 else error "rational.sml: diff.behav. in norm_Rational_mg 41";
1466 "----- Schalk I, p.69 Nr. 460b nonterm.SK";
1467 val t = TermC.str2term "(9*(x \<up> 2 - 8*x + 16) / (4*(y \<up> 2 - 2*y + 1))) / ((3*x - 12) / (16*y - 16))";
1468 (*val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1469 exception Div raised
1472 val t = TermC.str2term
1473 ("(144 + -72 * x + 9 * x \<up> 2) / (4 + -8 * y + 4 * y \<up> 2) /" ^
1474 "((-12 + 3 * x) / (-16 + 16 * y))");
1475 NONE = cancel_p_ thy t;
1477 if UnparseC.term t = !!!!!!!!!!!!!!!!!!!!!!!!!
1478 then () else error "rational.sml: diff.behav. in norm_Rational_mg 42";
1481 "----- some variant of the above; was non-terminating before";
1482 val t = TermC.str2term "9*(x \<up> 2 - 8*x+16)*(16*y - 16)/(4*(y \<up> 2 - 2*y+1)*(3*x - 12))";
1483 val SOME (t , _) = rewrite_set_ thy false norm_Rational t;
1484 if UnparseC.term t = "(48 + -12 * x) / (1 + -1 * y)"
1485 then () else error "some variant of the above; was non-terminating before";
1487 "----- SRD Schalk I, p.70 Nr. 472a";
1488 val t = TermC.str2term ("((8*x \<up> 2 - 32*y \<up> 2) / (2*x + 4*y)) / ((4*x - 8*y) / (x + y))");
1489 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1490 if UnparseC.term t = "x + y"
1491 then () else error "rational.sml: diff.behav. in norm_Rational_mg 43";
1493 "----- Schalk I, p.70 Nr. 478b ----- Rechenzeit: 5 sec";
1494 val t = TermC.str2term ("(a - (a*b + b \<up> 2)/(a+b))/(b+(a - b)/(1+(a+b)/(a - b))) / " ^
1495 "((a - a \<up> 2/(a+b))/(a+(a*b)/(a - b)))");
1496 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1497 if UnparseC.term t = "(2 * a \<up> 3 + 2 * a \<up> 2 * b) / (a \<up> 2 * b + b \<up> 3)"
1498 then () else error "rational.sml: diff.behav. in norm_Rational_mg 51";
1500 (*SRD Schalk I, p.69 Nr. 461a *)
1501 val t = TermC.str2term "(2/(x+3) + 2/(x - 3)) / (8*x/(x \<up> 2 - 9))";
1502 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1503 if UnparseC.term t = "1 / 2"
1504 then () else error "rational.sml: diff.behav. in norm_Rational_mg 44";
1506 (*SRD Schalk I, p.69 Nr. 464b *)
1507 val t = TermC.str2term "(a - a/(a - 2)) / (a + a/(a - 2))";
1508 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1509 if UnparseC.term t = "(-3 + a) / (-1 + a)"
1510 then () else error "rational.sml: diff.behav. in norm_Rational_mg 45";
1512 (*SRD Schalk I, p.69 Nr. 465b *)
1513 val t = TermC.str2term "((x+3*y)/9 + (4*y \<up> 2 - 9*z \<up> 2)/(16*x)) / (x/9 + y/6 + z/4)";
1514 (* WN130911 non-termination due to non-termination of
1515 cancel_p_ thy (TermC.str2term
1516 ("("(576 * x \<up> 2 + 1728 * (x * y) + 1296 * y \<up> 2 + -2916 * z \<up> 2) /" ^
1517 "(576 * x \<up> 2 + 864 * (x * y) + 1296 * (x * z))"))
1519 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1520 if UnparseC.term t = "(4 * x + 6 * y + -9 * z) / (4 * x)"
1521 then () else error "rational.sml: diff.behav. in norm_Rational_mg 46";
1524 (*SRD Schalk I, p.69 Nr. 466b *)
1525 val t = TermC.str2term "((1 - 7*(x - 2)/(x \<up> 2 - 4)) / (6/(x+2))) / (3/(x+5)+30/(x \<up> 2 - 25))";
1526 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1527 if UnparseC.term t = "(25 + -10 * x + x \<up> 2) / 18"
1528 then () else error "rational.sml: diff.behav. in norm_Rational_mg 47";
1530 (*SRD Schalk I, p.70 Nr. 469 *)
1531 val t = TermC.str2term ("3*b \<up> 2 / (4*a \<up> 2 - 8*a*b + 4*b \<up> 2) / " ^
1532 "(a / (a \<up> 2*b - b \<up> 3) + (a - b) / (4*a*b \<up> 2 + 4*b \<up> 3) - 1 / (4*b \<up> 2))");
1533 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1534 if UnparseC.term t = "-3 * b \<up> 3 / (-2 * a + 2 * b)"
1535 then () else error "rational.sml: diff.behav. in norm_Rational_mg 48";
1537 "-------- me Schalk I No.186 -------------------------------------------------";
1538 "-------- me Schalk I No.186 -------------------------------------------------";
1539 "-------- me Schalk I No.186 -------------------------------------------------";
1540 val fmz = ["Term ((14 * x * y) / ( x * y ))", "normalform N"];
1542 ("Rational",["rational", "simplification"],
1543 ["simplification", "of_rationals"]);
1544 val p = e_pos'; val c = [];
1545 val (p,_,f,nxt,_,pt) = CalcTreeTEST [(fmz, (dI',pI',mI'))];
1546 val (p,_,f,nxt,_,pt) = me nxt p c pt;
1547 val (p,_,f,nxt,_,pt) = me nxt p c pt;
1548 val (p,_,f,nxt,_,pt) = me nxt p c pt;
1549 val (p,_,f,nxt,_,pt) = me nxt p c pt;
1550 val (p,_,f,nxt,_,pt) = me nxt p c pt;
1551 val (p,_,f,nxt,_,pt) = me nxt p c pt;
1552 val (p,_,f,nxt,_,pt) = me nxt p c pt;
1553 val (p,_,f,nxt,_,pt) = me nxt p c pt; f2str f;
1554 val (p,_,f,nxt,_,pt) = me nxt p c pt; f2str f;
1555 val (p,_,f,nxt,_,pt) = me nxt p c pt; f2str f;(*++ for explicit script*)
1556 val (p,_,f,nxt,_,pt) = me nxt p c pt; f2str f;(*++ for explicit script*)
1557 case (f2str f, nxt) of
1558 ("14", ("End_Proof'", _)) => ()
1559 | _ => error "rational.sml diff.behav. in me Schalk I No.186";
1561 "-------- interSteps ..Simp_Rat_Double_No-1.xml ------------------------------";
1562 "-------- interSteps ..Simp_Rat_Double_No-1.xml ------------------------------";
1563 "-------- interSteps ..Simp_Rat_Double_No-1.xml ------------------------------";
1565 CalcTree [(["Term (((2 - x)/(2*a)) / (2*a/(x - 2)))", "normalform N"],
1566 ("Rational", ["rational", "simplification"], ["simplification", "of_rationals"]))];
1569 autoCalculate 1 CompleteCalc;
1570 val ((pt, p), _) = get_calc 1;
1572 Test_Tool.show_pt pt;
1574 (([], Frm), Simplify ((2 - x) / (2 * a) / (2 * a / (x - 2)))),
1575 (([1], Frm), (2 - x) / (2 * a) / (2 * a / (x - 2))),
1576 (([1], Res), (2 + -1 * x) / (2 * a) / (2 * a / (x + -1 * 2))),
1577 (([2], Res), (2 + -1 * x) / (2 * a) / (2 * a / (-2 + x))),
1578 (([3], Res), (2 + -1 * x) * (-2 + x) / (2 * a * (2 * a))),
1579 (([4], Res), (-4 + 4 * x + -1 * x \<up> 2) / (4 * a \<up> 2)),
1580 (([], Res), (-4 + 4 * x + -1 * x \<up> 2) / (4 * a \<up> 2))]
1582 interSteps 1 ([1], Res);
1583 val ((pt, p), _) = get_calc 1;
1584 (*Test_Tool.show_pt pt;
1586 (([], Frm), Simplify ((2 - x) / (2 * a) / (2 * a / (x - 2)))),
1587 (([1], Frm), (2 - x) / (2 * a) / (2 * a / (x - 2))),
1588 (([1,1], Frm), (2 - x) / (2 * a) / (2 * a / (x - 2))),
1589 (([1,1], Res), (2 - x) / (2 * a) / (2 * a / (x + -1 * 2))),
1590 (([1,2], Res), (2 + -1 * x) / (2 * a) / (2 * a / (x + -1 * 2))),
1591 (([1], Res), (2 + -1 * x) / (2 * a) / (2 * a / (x + -1 * 2))),
1592 (([2], Res), (2 + -1 * x) / (2 * a) / (2 * a / (-2 + x))),
1593 (([3], Res), (2 + -1 * x) * (-2 + x) / (2 * a * (2 * a))),
1594 (([4], Res), (-4 + 4 * x + -1 * x \<up> 2) / (4 * a \<up> 2)),
1595 (([], Res), (-4 + 4 * x + -1 * x \<up> 2) / (4 * a \<up> 2))]
1597 val (t, asm) = get_obj g_result pt [1, 1];
1598 if UnparseC.term t = "(2 - x) / (2 * a) / (2 * a / (x + -1 * 2))" andalso UnparseC.terms asm = "[]"
1599 then () else error "2nd interSteps ..Simp_Rat_Double_No-1 changed on [1, 1]";
1600 val (t, asm) = get_obj g_result pt [1, 2];
1601 if UnparseC.term t = "(2 + -1 * x) / (2 * a) / (2 * a / (x + -1 * 2))" andalso UnparseC.terms asm = "[]"
1602 then () else error "3rd interSteps ..Simp_Rat_Double_No-1 changed on [1, 2]";
1605 "-------- interSteps ..Simp_Rat_Cancel_No-1.xml ------------------------------";
1606 "-------- interSteps ..Simp_Rat_Cancel_No-1.xml ------------------------------";
1607 "-------- interSteps ..Simp_Rat_Cancel_No-1.xml ------------------------------";
1609 CalcTree [(["Term ((a^2 + -1*b^2) / (a^2 + -2*a*b + b^2))", "normalform N"],
1610 ("Rational", ["rational", "simplification"], ["simplification", "of_rationals"]))];
1613 autoCalculate 1 CompleteCalc;
1614 val ((pt, p), _) = get_calc 1;
1615 (*Test_Tool.show_pt pt;
1617 (([], Frm), Simplify ((a \<up> 2 + -1 * b \<up> 2) / (a \<up> 2 + -2 * a * b + b \<up> 2))),
1618 (([1], Frm), (a \<up> 2 + -1 * b \<up> 2) / (a \<up> 2 + -2 * a * b + b \<up> 2)),
1619 (([1], Res), (a \<up> 2 + -1 * b \<up> 2) / (a \<up> 2 + -2 * (a * b) + b \<up> 2)),
1620 (([2], Res), (a + b) / (a + -1 * b)),
1621 (([], Res), (a + b) / (a + -1 * b))]
1623 interSteps 1 ([2], Res);
1624 val ((pt, p), _) = get_calc 1;
1625 (*Test_Tool.show_pt pt;
1627 (([], Frm), Simplify ((a \<up> 2 + -1 * b \<up> 2) / (a \<up> 2 + -2 * a * b + b \<up> 2))),
1628 (([1], Frm), (a \<up> 2 + -1 * b \<up> 2) / (a \<up> 2 + -2 * a * b + b \<up> 2)),
1629 (([1], Res), (a \<up> 2 + -1 * b \<up> 2) / (a \<up> 2 + -2 * (a * b) + b \<up> 2)),
1630 (([2,1], Frm), (a \<up> 2 + -1 * b \<up> 2) / (a \<up> 2 + -2 * (a * b) + b \<up> 2)),
1631 (([2,1], Res), (a + b) / (a + -1 * b)),
1632 (([2], Res), (a + b) / (a + -1 * b)),
1633 (([], Res), (a + b) / (a + -1 * b))]
1635 interSteps 1 ([2,1],Res);
1636 val ((pt, p), _) = get_calc 1;
1637 (*Test_Tool.show_pt pt;
1639 (([], Frm), Simplify ((a \<up> 2 + -1 * b \<up> 2) / (a \<up> 2 + -2 * a * b + b \<up> 2))),
1640 (([1], Frm), (a \<up> 2 + -1 * b \<up> 2) / (a \<up> 2 + -2 * a * b + b \<up> 2)),
1641 (([1], Res), (a \<up> 2 + -1 * b \<up> 2) / (a \<up> 2 + -2 * (a * b) + b \<up> 2)),
1642 (([2,1], Frm), (a \<up> 2 + -1 * b \<up> 2) / (a \<up> 2 + -2 * (a * b) + b \<up> 2)),
1643 (([2,1,1], Frm), (a \<up> 2 + -1 * b \<up> 2) / (a \<up> 2 + -2 * (a * b) + b \<up> 2)),
1644 (([2,1,1], Res), (a \<up> 2 + -1 * (a * b) + a * b + -1 * b \<up> 2) /
1645 (a \<up> 2 + -2 * (a * b) + 1 * b \<up> 2)),
1646 (([2,1,2], Res), (a \<up> 2 + -1 * (a * b) + a * b + -1 * b \<up> 2) /
1647 (a \<up> 2 + -2 * (a * b) + -1 \<up> 2 * b \<up> 2)),
1648 (([2,1,3], Res), (a \<up> 2 + -1 * (a * b) + a * b + -1 * b \<up> 2) /
1649 (a \<up> 2 + -2 * (a * b) + (-1 * b) \<up> 2)),
1650 (([2,1,4], Res), (a * a + -1 * (a * b) + a * b + -1 * b \<up> 2) /
1651 (a \<up> 2 + -2 * (a * b) + (-1 * b) \<up> 2)),
1652 (([2,1,5], Res), (a * a + -1 * (a * b) + a * b + -1 * (b * b)) /
1653 (a \<up> 2 + -2 * (a * b) + (-1 * b) \<up> 2)),
1654 (([2,1,6], Res), (a * a + -1 * (a * b) + a * b + -1 * (b * b)) /
1655 (a \<up> 2 + -1 * (2 * (a * b)) + (-1 * b) \<up> 2)),
1656 (([2,1,7], Res), (a * a + a * (-1 * b) + (b * a + b * (-1 * b))) /
1657 (a \<up> 2 + 2 * (a * (-1 * b)) + (-1 * b) \<up> 2)),
1658 (([2,1,8], Res), (a * a + a * (-1 * b) + (b * a + b * (-1 * b))) /
1659 (a \<up> 2 + 2 * a * (-1 * b) + (-1 * b) \<up> 2)),
1660 (([2,1,9], Res), (a * (a + -1 * b) + (b * a + b * (-1 * b))) /
1661 (a \<up> 2 + 2 * a * (-1 * b) + (-1 * b) \<up> 2)),
1662 (([2,1,10], Res), (a * (a + -1 * b) + b * (a + -1 * b)) /
1663 (a \<up> 2 + 2 * a * (-1 * b) + (-1 * b) \<up> 2)),
1664 (([2,1,11], Res), (a + b) * (a + -1 * b) / (a \<up> 2 + 2 * a * (-1 * b) + (-1 * b) \<up> 2)),
1665 (([2,1,12], Res), (a + b) * (a + -1 * b) / ((a + -1 * b) * (a + -1 * b))),
1666 (([2,1,13], Res), (a + b) / (a + -1 * b)),
1667 (([2,1], Res), (a + b) / (a + -1 * b)),
1668 (([2], Res), (a + b) / (a + -1 * b)),
1669 (([], Res), (a + b) / (a + -1 * b))]
1671 val newnds = children (get_nd pt [2,1]) (*see "fun detailrls"*);
1672 if length newnds = 13 then () else error "rational.sml: interSteps cancel_p rev_rew_p";
1674 val p = ([2,1,9],Res);
1676 val (_, tac, _) = ME_Misc.pt_extract (pt, p);
1677 case tac of SOME (Rewrite ("sym_distrib_left", _)) => ()
1678 | _ => error "rational.sml: getTactic, sym_real_plus_binom_times1";
1681 "-------- investigate rulesets for cancel_p ----------------------------------";
1682 "-------- investigate rulesets for cancel_p ----------------------------------";
1683 "-------- investigate rulesets for cancel_p ----------------------------------";
1684 val thy = @{theory "Rational"};
1685 val t = TermC.str2term "(a \<up> 2 + -1*b \<up> 2) / (a \<up> 2 + -2*a*b + b \<up> 2)";
1686 val tt = TermC.str2term "(1 * a + 1 * b) * (1 * a + -1 * b)"(*numerator only*);
1688 "----- with rewrite_set_";
1689 val SOME (tt',asm) = rewrite_set_ thy false make_polynomial tt;
1690 if UnparseC.term tt'= "a \<up> 2 + -1 * b \<up> 2" then () else error "rls chancel_p 1";
1691 val tt = TermC.str2term "((1 * a + -1 * b) * (1 * a + -1 * b))"(*denominator only*);
1692 val SOME (tt',asm) = rewrite_set_ thy false make_polynomial tt;
1693 if UnparseC.term tt' = "a \<up> 2 + -2 * a * b + b \<up> 2" then () else error "rls chancel_p 2";
1695 "----- with .make_deriv; WN1130912 not investigated further, will be discontinued";
1696 val SOME (tt, _) = factout_p_ thy t;
1697 if UnparseC.term tt = "(a + b) * (a + -1 * b) / ((a + -1 * b) * (a + -1 * b))"
1698 then () else error "rls chancel_p 3";
1699 UnparseC.term tt = "(1 * a + 1 * b) * (1 * a + -1 * b) / ((1 * a + -1 * b) * (1 * a + -1 * b))";
1701 "--- with simpler ruleset";
1702 val {rules, rew_ord= (_, ro), ...} = Rule_Set.rep (assoc_rls "rev_rew_p");
1703 val der = .make_deriv thy Atools_erls rules ro NONE tt;
1704 if length der = 12 then () else error "WN1130912 rls chancel_p 4";
1705 (*default_print_depth 99;*) writeln (deriv2str der); (*default_print_depth 3;*)
1707 (*default_print_depth 99;*) map (UnparseC.term o #1) der; (*default_print_depth 3;*)
1708 "...,(-1 * b \<up> 2 + a \<up> 2) / (-2 * (a * b) + a \<up> 2 + (-1 * b) \<up> 2) ]";
1709 (*default_print_depth 99;*) map (Rule.to_string o #2) der; (*default_print_depth 3;*)
1710 (*default_print_depth 99;*) map (UnparseC.term o #1 o #3) der; (*default_print_depth 3;*)
1712 val der = .make_deriv thy Atools_erls rules ro NONE
1713 (TermC.str2term "(1 * a + 1 * b) * (1 * a + -1 * b)");
1714 (*default_print_depth 99;*) writeln (deriv2str der); (*default_print_depth 3;*)
1716 val {rules, rew_ord=(_,ro),...} = Rule_Set.rep (assoc_rls "rev_rew_p");
1717 val der = .make_deriv thy Atools_erls rules ro NONE
1718 (TermC.str2term "(1 * a + -1 * b) * (1 * a + -1 * b)");
1719 (*default_print_depth 99;*) writeln (deriv2str der); (*default_print_depth 3;*)
1720 (*default_print_depth 99;*) map (UnparseC.term o #1) der; (*default_print_depth 3;*)
1721 (*WN060829 ...postponed*)
1724 "-------- fun eval_get_denominator -------------------------------------------";
1725 "-------- fun eval_get_denominator -------------------------------------------";
1726 "-------- fun eval_get_denominator -------------------------------------------";
1727 val thy = @{theory Isac_Knowledge};
1728 val t = Thm.term_of (the (TermC.parse thy "get_denominator ((a +x)/b)"));
1729 val SOME (_, t') = eval_get_denominator "" 0 t thy;
1730 if UnparseC.term t' = "get_denominator ((a + x) / b) = b"
1731 then () else error "get_denominator ((a + x) / b) = b"
1734 "-------- several errpats in complicated term --------------------------------";
1735 "-------- several errpats in complicated term --------------------------------";
1736 "-------- several errpats in complicated term --------------------------------";
1737 (*WN12xxxx TODO: instead of Gabriella's example here (27.Jul.12) find a simpler one
1738 WN130912: kept this test, although not clear what for*)
1740 CalcTree [(["Term ((5*b + 25)/(a^2 - b^2) * (a - b)/(5*b))", "normalform N"],
1741 ("Rational", ["rational", "simplification"], ["simplification", "of_rationals"]))];
1744 autoCalculate 1 CompleteCalc;
1745 val ((pt, p), _) = get_calc 1;
1746 (*Test_Tool.show_pt pt;
1748 (([], Frm), Simplify ((5 * b + 25) / (a \<up> 2 - b \<up> 2) * (a - b) / (5 * b))),
1749 (([1], Frm), (5 * b + 25) / (a \<up> 2 - b \<up> 2) * (a - b) / (5 * b)),
1750 (([1], Res), (5 * b + 25) / (a \<up> 2 + -1 * b \<up> 2) * (a + -1 * b) / (5 * b)),
1751 (([2], Res), (5 * b + 25) * (a + -1 * b) / (a \<up> 2 + -1 * b \<up> 2) / (5 * b)),
1752 (([3], Res), (25 * a + -25 * b + 5 * (a * b) + -5 * b \<up> 2) / (a \<up> 2 + -1 * b \<up> 2) /
1754 (([4], Res), (25 + 5 * b) / (a + b) / (5 * b)),
1755 (([5], Res), (25 + 5 * b) / ((a + b) * (5 * b))),
1756 (([6], Res), (25 + 5 * b) / (5 * (a * b) + 5 * b \<up> 2)),
1757 (([7], Res), (5 + b) / (a * b + b \<up> 2)),
1758 (([], Res), (5 + b) / (a * b + b \<up> 2))] *)
1761 "-------- WN1309xx non-terminating rls norm_Rational -------------------------";
1762 "-------- WN1309xx non-terminating rls norm_Rational -------------------------";
1763 "-------- WN1309xx non-terminating rls norm_Rational -------------------------";
1764 (*------- Schalk I, p.70 Nr. 480b; a/b : c/d translated to a/b * d/c*)
1765 val t = TermC.str2term
1766 ("((12*x*y / (9*x \<up> 2 - y \<up> 2)) / (1 / (3*x - y) \<up> 2 - 1 / (3*x + y) \<up> 2)) * " ^
1767 "((1/(x - 5*y) \<up> 2 - 1/(x + 5*y) \<up> 2) / (20*x*y / (x \<up> 2 - 25*y \<up> 2)))");
1769 (*1st factor separately simplified *)
1770 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))";
1771 val SOME (t', _) = rewrite_set_ thy false norm_Rational t;
1772 if UnparseC.term t' = "(-9 * x \<up> 2 + y \<up> 2) / -1" then () else error "Nr. 480b lhs changed";
1773 (*2nd factor separately simplified *)
1774 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)))";
1775 val SOME (t',_) = rewrite_set_ thy false norm_Rational t; UnparseC.term t';
1776 if UnparseC.term t' = "-1 / (-1 * x \<up> 2 + 25 * y \<up> 2)" then () else error "Nr. 480b rhs changed";
1778 "-------- Schalk I, p.70 Nr. 477a: terms are exploding ?!?";
1779 val t = TermC.str2term ("b*y/(b - 2*y)/((b \<up> 2 - y \<up> 2)/(b+2*y)) /" ^
1780 "(b \<up> 2*y + b*y \<up> 2) * (a+x) \<up> 2 / ((b \<up> 2 - 4*y \<up> 2) * (a+2*x) \<up> 2)");
1781 (*val SOME (t',_) = rewrite_set_ thy false norm_Rational t;
1783 ### rls: cancel_p on: (a \<up> 2 * (b * y) + 2 * (a * (b * (x * y))) + b * (x \<up> 2 * y)) /
1785 ((b \<up> 2 + -1 * y \<up> 2) / (b + 2 * y)) /
1786 (b \<up> 2 * y + b * y \<up> 2) /
1787 (a \<up> 2 * b \<up> 2 + -4 * (a \<up> 2 * y \<up> 2) + 4 * (a * (b \<up> 2 * x)) +
1788 -16 * (a * (x * y \<up> 2)) +
1789 4 * (b \<up> 2 * x \<up> 2) +
1790 -16 * (x \<up> 2 * y \<up> 2))
1791 exception Div raised
1794 val t = TermC.str2term
1795 ("(a \<up> 2 * (b * y) + 2 * (a * (b * (x * y))) + b * (x \<up> 2 * y)) /" ^
1797 "((b \<up> 2 + -1 * y \<up> 2) / (b + 2 * y)) /" ^
1798 "(b \<up> 2 * y + b * y \<up> 2) /" ^
1799 "(a \<up> 2 * b \<up> 2 + -4 * (a \<up> 2 * y \<up> 2) + 4 * (a * (b \<up> 2 * x)) +" ^
1800 "-16 * (a * (x * y \<up> 2)) +" ^
1801 "4 * (b \<up> 2 * x \<up> 2) +" ^
1802 "-16 * (x \<up> 2 * y \<up> 2))");
1803 NONE = cancel_p_ thy t;
1806 (*------- Schalk I, p.70 Nr. 476b in 2003 this worked using 10 sec. *)
1807 val t = TermC.str2term
1808 ("((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) / " ^
1809 "((a \<up> 2 + b \<up> 2) \<up> 2 / (a + b) \<up> 2)");
1810 (* Rewrite.trace_on := true;
1811 rewrite_set_ thy false norm_Rational t;
1813 #### rls: cancel_p on: (2 * (a \<up> 7 * b) + 4 * (a \<up> 6 * b \<up> 2) + 6 * (a \<up> 5 * b \<up> 3) +
1814 8 * (a \<up> 4 * b \<up> 4) +
1815 6 * (a \<up> 3 * b \<up> 5) +
1816 4 * (a \<up> 2 * b \<up> 6) +
1817 2 * (a * b \<up> 7)) /
1818 (2 * (a \<up> 9 * b) + 4 * (a \<up> 8 * b \<up> 2) +
1819 2 * (2 * (a \<up> 7 * b \<up> 3)) +
1820 4 * (a \<up> 6 * b \<up> 4) +
1821 -4 * (a \<up> 4 * b \<up> 6) +
1822 -4 * (a \<up> 3 * b \<up> 7) +
1823 -4 * (a \<up> 2 * b \<up> 8) +
1824 -2 * (a * b \<up> 9))
1826 if UnparseC.term t = "1 / (a \<up> 2 + -1 * b \<up> 2)" then ()
1827 else error "rational.sml: diff.behav. in norm_Rational_mg 49";
1830 "-------- Schalk I, p.70 Nr. 480a: terms are exploding ?!?";
1831 val t = TermC.str2term ("(1/x + 1/y + 1/z) / (1/x - 1/y - 1/z) / " ^
1832 "(2*x \<up> 2 / (x \<up> 2 - z \<up> 2) / (x / (x + z) + x / (x - z)))");
1833 (* Rewrite.trace_on := true;
1834 rewrite_set_ thy false norm_Rational t;
1836 #### rls: cancel_p on: (2 * (x \<up> 6 * (y \<up> 2 * z)) + 2 * (x \<up> 6 * (y * z \<up> 2)) +
1837 2 * (x \<up> 5 * (y \<up> 2 * z \<up> 2)) +
1838 -2 * (x \<up> 4 * (y \<up> 2 * z \<up> 3)) +
1839 -2 * (x \<up> 4 * (y * z \<up> 4)) +
1840 -2 * (x \<up> 3 * (y \<up> 2 * z \<up> 4))) /
1841 (-2 * (x \<up> 6 * (y \<up> 2 * z)) + -2 * (x \<up> 6 * (y * z \<up> 2)) +
1842 2 * (x \<up> 5 * (y \<up> 2 * z \<up> 2)) +
1843 2 * (x \<up> 4 * (y \<up> 2 * z \<up> 3)) +
1844 2 * (x \<up> 4 * (y * z \<up> 4)) +
1845 -2 * (x \<up> 3 * (y \<up> 2 * z \<up> 4)))
1848 "-------- Schalk I, p.60 Nr. 215d: terms are exploding, internal loop does not terminate";
1849 val t = TermC.str2term "(a-b) \<up> 3 * (x+y) \<up> 4 / ((x+y) \<up> 2 * (a-b) \<up> 5)";
1850 (* Kein Wunder, denn Z???ler und Nenner extra als Polynom dargestellt ergibt:
1852 val t = TermC.str2term "(a-b) \<up> 3 * (x+y) \<up> 4";
1853 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1855 "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";
1856 val t = TermC.str2term "((x+y) \<up> 2 * (a-b) \<up> 5)";
1857 val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1859 "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";
1861 anscheinend macht dem Rechner das Krzen diese Bruches keinen Spass mehr ...*)
1863 "-------- Schalk I, p.70 Nr. 480b: terms are exploding, Rewrite.trace_on stops at";
1864 val t = TermC.str2term ("((12*x*y/(9*x \<up> 2 - y \<up> 2))/" ^
1865 "(1/(3*x - y) \<up> 2 - 1/(3*x + y) \<up> 2)) *" ^
1866 "(1/(x - 5*y) \<up> 2 - 1/(x + 5*y) \<up> 2)/" ^
1867 "(20*x*y/(x \<up> 2 - 25*y \<up> 2))");
1868 (*val SOME (t, _) = rewrite_set_ thy false norm_Rational t;
1870 #### rls: cancel_p on: (19440 * (x \<up> 8 * y \<up> 2) + -490320 * (x \<up> 6 * y \<up> 4) +
1871 108240 * (x \<up> 4 * y \<up> 6) +
1872 -6000 * (x \<up> 2 * y \<up> 8)) /
1873 (2160 * (x \<up> 8 * y \<up> 2) + -108240 * (x \<up> 6 * y \<up> 4) +
1874 1362000 * (x \<up> 4 * y \<up> 6) +
1875 -150000 * (x \<up> 2 * y \<up> 8))