1 (*.eval_funs, rulesets, problems and methods concerning polynamials
2 authors: Matthias Goldgruber 2003
3 (c) due to copyright terms
5 use"../Knowledge/Poly.ML";
6 use"Knowledge/Poly.ML";
10 use_thy"Knowledge/Isac";
11 ****************************************************************.*)
13 (*.****************************************************************
14 remark on 'polynomials'
16 there are 5 kinds of expanded normalforms:
17 [1] 'complete polynomial' (Komplettes Polynom), univariate
18 a_0 + a_1.x^1 +...+ a_n.x^n not (a_n = 0)
19 not (a_n = 0), some a_i may be zero (DON'T disappear),
20 variables in monomials lexicographically ordered and complete,
21 x written as 1*x^1, ...
22 [2] 'polynomial' (Polynom), univariate and multivariate
23 a_0 + a_1.x +...+ a_n.x^n not (a_n = 0)
24 a_0 + a_1.x_1.x_2^n_12...x_m^n_1m +...+ a_n.x_1^n.x_2^n_n2...x_m^n_nm
25 not (a_n = 0), some a_i may be zero (ie. monomials disappear),
26 exponents and coefficients equal 1 are not (WN060904.TODO in cancel_p_)shown,
27 and variables in monomials are lexicographically ordered
28 examples: [1]: "1 + (-10) * x ^^^ 1 + 25 * x ^^^ 2"
29 [1]: "11 + 0 * x ^^^ 1 + 1 * x ^^^ 2"
30 [2]: "x + (-50) * x ^^^ 3"
31 [2]: "(-1) * x * y ^^^ 2 + 7 * x ^^^ 3"
33 [3] 'expanded_term' (Ausmultiplizierter Term):
34 pull out unary minus to binary minus,
35 as frequently exercised in schools; other conditions for [2] hold however
36 examples: "a ^^^ 2 - 2 * a * b + b ^^^ 2"
37 "4 * x ^^^ 2 - 9 * y ^^^ 2"
38 [4] 'polynomial_in' (Polynom in):
39 polynomial in 1 variable with arbitrary coefficients
40 examples: "2 * x + (-50) * x ^^^ 3" (poly in x)
41 "(u + v) + (2 * u ^^^ 2) * a + (-u) * a ^^^ 2 (poly in a)
42 [5] 'expanded_in' (Ausmultiplizierter Termin in):
43 analoguous to [3] with binary minus like [3]
44 examples: "2 * x - 50 * x ^^^ 3" (expanded in x)
45 "(u + v) + (2 * u ^^^ 2) * a - u * a ^^^ 2 (expanded in a)
46 *****************************************************************.*)
48 "******** Poly.ML begin ******************************************";
49 theory' := overwritel (!theory', [("Poly.thy",Poly.thy)]);
52 (* is_polyrat_in becomes true, if no bdv is in the denominator of a fraction*)
53 fun is_polyrat_in t v =
55 fun coeff_in c v = member op = (vars c) v;
56 fun finddivide (_ $ _ $ _ $ _) v = raise error("is_polyrat_in:")
57 (* at the moment there is no term like this, but ....*)
58 | finddivide (t as (Const ("HOL.divide",_) $ _ $ b)) v = not(coeff_in b v)
59 | finddivide (_ $ t1 $ t2) v = (finddivide t1 v) orelse (finddivide t2 v)
60 | finddivide (_ $ t1) v = (finddivide t1 v)
61 | finddivide _ _ = false;
66 fun eval_is_polyrat_in _ _ (p as (Const ("Poly.is'_polyrat'_in",_) $ t $ v)) _ =
67 if is_polyrat_in t v then
68 SOME ((term2str p) ^ " = True",
69 Trueprop $ (mk_equality (p, HOLogic.true_const)))
70 else SOME ((term2str p) ^ " = True",
71 Trueprop $ (mk_equality (p, HOLogic.false_const)))
72 | eval_is_polyrat_in _ _ _ _ = ((*writeln"### nichts matcht";*) NONE);
76 (*.a 'c is coefficient of v' if v does NOT occur in c.*)
77 fun coeff_in c v = not (member op = (vars c) v);
79 val v = (term_of o the o (parse thy)) "x";
80 val t = (term_of o the o (parse thy)) "1";
82 (*val it = true : bool*)
83 val t = (term_of o the o (parse thy)) "a*b+c";
85 (*val it = true : bool*)
86 val t = (term_of o the o (parse thy)) "a*x+c";
88 (*val it = false : bool*)
90 (*. a 'monomial t in variable v' is a term t with
91 either (1) v NOT existent in t, or (2) v contained in t,
93 if (2) then v is a factor on the very right, ev. with exponent.*)
94 fun factor_right_deg (*case 2*)
95 (t as Const ("op *",_) $ t1 $
96 (Const ("Atools.pow",_) $ vv $ Free (d,_))) v =
97 if ((vv = v) andalso (coeff_in t1 v)) then SOME (int_of_str' d) else NONE
99 (t as Const ("Atools.pow",_) $ vv $ Free (d,_)) v =
100 if (vv = v) then SOME (int_of_str' d) else NONE
101 | factor_right_deg (t as Const ("op *",_) $ t1 $ vv) v =
102 if ((vv = v) andalso (coeff_in t1 v))then SOME 1 else NONE
103 | factor_right_deg vv v =
104 if (vv = v) then SOME 1 else NONE;
105 fun mono_deg_in m v =
106 if coeff_in m v then (*case 1*) SOME 0
107 else factor_right_deg m v;
109 val v = (term_of o the o (parse thy)) "x";
110 val t = (term_of o the o (parse thy)) "(a*b+c)*x^^^7";
113 val t = (term_of o the o (parse thy)) "x^^^7";
116 val t = (term_of o the o (parse thy)) "(a*b+c)*x";
119 val t = (term_of o the o (parse thy)) "(a*b+x)*x";
122 val t = (term_of o the o (parse thy)) "x";
125 val t = (term_of o the o (parse thy)) "(a*b+c)";
128 val t = (term_of o the o (parse thy)) "ab - (a*b)*x";
132 fun expand_deg_in t v =
133 let fun edi ~1 ~1 (Const ("op +",_) $ t1 $ t2) =
134 (case mono_deg_in t2 v of (* $ is left associative*)
135 SOME d' => edi d' d' t1
137 | edi ~1 ~1 (Const ("op -",_) $ t1 $ t2) =
138 (case mono_deg_in t2 v of
139 SOME d' => edi d' d' t1
141 | edi d dmax (Const ("op -",_) $ t1 $ t2) =
142 (case mono_deg_in t2 v of
143 (*RL orelse ((d=0) andalso (d'=0)) need to handle 3+4-...4 +x*)
144 SOME d' => if ((d > d') orelse ((d=0) andalso (d'=0))) then edi d' dmax t1 else NONE
146 | edi d dmax (Const ("op +",_) $ t1 $ t2) =
147 (case mono_deg_in t2 v of
148 (*RL orelse ((d=0) andalso (d'=0)) need to handle 3+4-...4 +x*)
149 SOME d' => if ((d > d') orelse ((d=0) andalso (d'=0))) then edi d' dmax t1 else NONE
152 (case mono_deg_in t v of
155 | edi d dmax t = (*basecase last*)
156 (case mono_deg_in t v of
157 SOME d' => if ((d > d') orelse ((d=0) andalso (d'=0))) then SOME dmax else NONE
161 val v = (term_of o the o (parse thy)) "x";
162 val t = (term_of o the o (parse thy)) "a+b";
165 val t = (term_of o the o (parse thy)) "(a+b)*x";
168 val t = (term_of o the o (parse thy)) "a*b - (a+b)*x";
171 val t = (term_of o the o (parse thy)) "a*b + (a-b)*x";
174 val t = (term_of o the o (parse thy)) "a*b + (a+b)*x + x^^^2";
177 fun poly_deg_in t v =
178 let fun edi ~1 ~1 (Const ("op +",_) $ t1 $ t2) =
179 (case mono_deg_in t2 v of (* $ is left associative*)
180 SOME d' => edi d' d' t1
182 | edi d dmax (Const ("op +",_) $ t1 $ t2) =
183 (case mono_deg_in t2 v of
184 (*RL orelse ((d=0) andalso (d'=0)) need to handle 3+4-...4 +x*)
185 SOME d' => if ((d > d') orelse ((d=0) andalso (d'=0))) then edi d' dmax t1 else NONE
188 (case mono_deg_in t v of
191 | edi d dmax t = (*basecase last*)
192 (case mono_deg_in t v of
193 SOME d' => if ((d > d') orelse ((d=0) andalso (d'=0))) then SOME dmax else NONE
198 fun is_expanded_in t v =
199 case expand_deg_in t v of SOME _ => true | NONE => false;
201 case poly_deg_in t v of SOME _ => true | NONE => false;
202 fun has_degree_in t v =
203 case expand_deg_in t v of SOME d => d | NONE => ~1;
206 val v = (term_of o the o (parse thy)) "x";
207 val t = (term_of o the o (parse thy)) "a*b - (a+b)*x + x^^^2";
210 val t = (term_of o the o (parse thy)) "-8 - 2*x + x^^^2";
213 val t = (term_of o the o (parse thy)) "6 + 13*x + 6*x^^^2";
218 (*("is_expanded_in", ("Poly.is'_expanded'_in", eval_is_expanded_in ""))*)
219 fun eval_is_expanded_in _ _
220 (p as (Const ("Poly.is'_expanded'_in",_) $ t $ v)) _ =
221 if is_expanded_in t v
222 then SOME ((term2str p) ^ " = True",
223 Trueprop $ (mk_equality (p, HOLogic.true_const)))
224 else SOME ((term2str p) ^ " = True",
225 Trueprop $ (mk_equality (p, HOLogic.false_const)))
226 | eval_is_expanded_in _ _ _ _ = NONE;
228 val t = (term_of o the o (parse thy)) "(-8 - 2*x + x^^^2) is_expanded_in x";
229 val SOME (id, t') = eval_is_expanded_in 0 0 t 0;
230 (*val id = "Poly.is'_expanded'_in (-8 - 2 * x + x ^^^ 2) x = True"*)
232 (*val it = "Poly.is'_expanded'_in (-8 - 2 * x + x ^^^ 2) x = True"*)
234 (*("is_poly_in", ("Poly.is'_poly'_in", eval_is_poly_in ""))*)
235 fun eval_is_poly_in _ _
236 (p as (Const ("Poly.is'_poly'_in",_) $ t $ v)) _ =
238 then SOME ((term2str p) ^ " = True",
239 Trueprop $ (mk_equality (p, HOLogic.true_const)))
240 else SOME ((term2str p) ^ " = True",
241 Trueprop $ (mk_equality (p, HOLogic.false_const)))
242 | eval_is_poly_in _ _ _ _ = NONE;
244 val t = (term_of o the o (parse thy)) "(8 + 2*x + x^^^2) is_poly_in x";
245 val SOME (id, t') = eval_is_poly_in 0 0 t 0;
246 (*val id = "Poly.is'_poly'_in (8 + 2 * x + x ^^^ 2) x = True"*)
248 (*val it = "Poly.is'_poly'_in (8 + 2 * x + x ^^^ 2) x = True"*)
251 (*("has_degree_in", ("Poly.has'_degree'_in", eval_has_degree_in ""))*)
252 fun eval_has_degree_in _ _
253 (p as (Const ("Poly.has'_degree'_in",_) $ t $ v)) _ =
254 let val d = has_degree_in t v
255 val d' = term_of_num HOLogic.realT d
256 in SOME ((term2str p) ^ " = " ^ (string_of_int d),
257 Trueprop $ (mk_equality (p, d')))
259 | eval_has_degree_in _ _ _ _ = NONE;
261 > val t = (term_of o the o (parse thy)) "(-8 - 2*x + x^^^2) has_degree_in x";
262 > val SOME (id, t') = eval_has_degree_in 0 0 t 0;
263 val id = "Poly.has'_degree'_in (-8 - 2 * x + x ^^^ 2) x = 2" : string
265 val it = "Poly.has'_degree'_in (-8 - 2 * x + x ^^^ 2) x = 2" : string
270 append_rls "calculate_PolyFIXXXME.not.impl." e_rls
273 (*.for evaluation of conditions in rewrite rules.*)
275 append_rls "Poly_erls" Atools_erls
276 [ Calc ("op =",eval_equal "#equal_"),
277 Thm ("real_unari_minus",num_str real_unari_minus),
278 Calc ("op +",eval_binop "#add_"),
279 Calc ("op -",eval_binop "#sub_"),
280 Calc ("op *",eval_binop "#mult_"),
281 Calc ("Atools.pow" ,eval_binop "#power_")
285 append_rls "poly_crls" Atools_crls
286 [ Calc ("op =",eval_equal "#equal_"),
287 Thm ("real_unari_minus",num_str real_unari_minus),
288 Calc ("op +",eval_binop "#add_"),
289 Calc ("op -",eval_binop "#sub_"),
290 Calc ("op *",eval_binop "#mult_"),
291 Calc ("Atools.pow" ,eval_binop "#power_")
295 local (*. for make_polynomial .*)
297 open Term; (* for type order = EQUAL | LESS | GREATER *)
299 fun pr_ord EQUAL = "EQUAL"
300 | pr_ord LESS = "LESS"
301 | pr_ord GREATER = "GREATER";
303 fun dest_hd' (Const (a, T)) = (* ~ term.ML *)
305 "Atools.pow" => ((("|||||||||||||", 0), T), 0) (*WN greatest string*)
306 | _ => (((a, 0), T), 0))
307 | dest_hd' (Free (a, T)) = (((a, 0), T), 1)
308 | dest_hd' (Var v) = (v, 2)
309 | dest_hd' (Bound i) = ((("", i), dummyT), 3)
310 | dest_hd' (Abs (_, T, _)) = ((("", 0), T), 4);
312 fun get_order_pow (t $ (Free(order,_))) = (* RL FIXXXME:geht zufaellig?WN*)
313 (case int_of_str (order) of
316 | get_order_pow _ = 0;
318 fun size_of_term' (Const(str,_) $ t) =
319 if "Atools.pow"= str then 1000 + size_of_term' t else 1+size_of_term' t(*WN*)
320 | size_of_term' (Abs (_,_,body)) = 1 + size_of_term' body
321 | size_of_term' (f$t) = size_of_term' f + size_of_term' t
322 | size_of_term' _ = 1;
324 fun term_ord' pr thy (Abs (_, T, t), Abs(_, U, u)) = (* ~ term.ML *)
325 (case term_ord' pr thy (t, u) of EQUAL => typ_ord (T, U) | ord => ord)
326 | term_ord' pr thy (t, u) =
329 val (f, ts) = strip_comb t and (g, us) = strip_comb u;
330 val _=writeln("t= f@ts= \""^
331 ((Syntax.string_of_term (thy2ctxt thy)) f)^"\" @ \"["^
332 (commas(map(Syntax.string_of_term (thy2ctxt thy))ts))^"]\"");
333 val _=writeln("u= g@us= \""^
334 ((Syntax.string_of_term (thy2ctxt thy)) g)^"\" @ \"["^
335 (commas(map(Syntax.string_of_term (thy2ctxt thy))us))^"]\"");
336 val _=writeln("size_of_term(t,u)= ("^
337 (string_of_int(size_of_term' t))^", "^
338 (string_of_int(size_of_term' u))^")");
339 val _=writeln("hd_ord(f,g) = "^((pr_ord o hd_ord)(f,g)));
340 val _=writeln("terms_ord(ts,us) = "^
341 ((pr_ord o terms_ord str false)(ts,us)));
342 val _=writeln("-------");
345 case int_ord (size_of_term' t, size_of_term' u) of
347 let val (f, ts) = strip_comb t and (g, us) = strip_comb u in
348 (case hd_ord (f, g) of EQUAL => (terms_ord str pr) (ts, us)
352 and hd_ord (f, g) = (* ~ term.ML *)
353 prod_ord (prod_ord indexname_ord typ_ord) int_ord (dest_hd' f, dest_hd' g)
354 and terms_ord str pr (ts, us) =
355 list_ord (term_ord' pr (assoc_thy "Isac.thy"))(ts, us);
358 fun ord_make_polynomial (pr:bool) thy (_:subst) tu =
359 (term_ord' pr thy(***) tu = LESS );
364 rew_ord' := overwritel (!rew_ord',
365 [("termlessI", termlessI),
366 ("ord_make_polynomial", ord_make_polynomial false thy)
371 Rls{id = "expand", preconds = [],
372 rew_ord = ("dummy_ord", dummy_ord),
373 erls = e_rls,srls = Erls,
376 rules = [Thm ("real_add_mult_distrib" ,num_str real_add_mult_distrib),
377 (*"(z1.0 + z2.0) * w = z1.0 * w + z2.0 * w"*)
378 Thm ("real_add_mult_distrib2",num_str real_add_mult_distrib2)
379 (*"w * (z1.0 + z2.0) = w * z1.0 + w * z2.0"*)
380 ], scr = EmptyScr}:rls;
382 (*----------------- Begin: rulesets for make_polynomial_ -----------------
383 'rlsIDs' redefined by MG as 'rlsIDs_'
387 Rls{id = "discard_minus_", preconds = [],
388 rew_ord = ("dummy_ord", dummy_ord),
389 erls = e_rls,srls = Erls,
392 rules = [Thm ("real_diff_minus",num_str real_diff_minus),
393 (*"a - b = a + -1 * b"*)
394 Thm ("sym_real_mult_minus1",num_str (real_mult_minus1 RS sym))
396 ], scr = EmptyScr}:rls;
398 Rls{id = "expand_poly_", preconds = [],
399 rew_ord = ("dummy_ord", dummy_ord),
400 erls = e_rls,srls = Erls,
403 rules = [Thm ("real_plus_binom_pow4",num_str real_plus_binom_pow4),
404 (*"(a + b)^^^4 = ... "*)
405 Thm ("real_plus_binom_pow5",num_str real_plus_binom_pow5),
406 (*"(a + b)^^^5 = ... "*)
407 Thm ("real_plus_binom_pow3",num_str real_plus_binom_pow3),
408 (*"(a + b)^^^3 = a^^^3 + 3*a^^^2*b + 3*a*b^^^2 + b^^^3" *)
410 (*WN071229 changed/removed for Schaerding -----vvv*)
411 (*Thm ("real_plus_binom_pow2",num_str real_plus_binom_pow2),*)
412 (*"(a + b)^^^2 = a^^^2 + 2*a*b + b^^^2"*)
413 Thm ("real_plus_binom_pow2",num_str real_plus_binom_pow2),
414 (*"(a + b)^^^2 = (a + b) * (a + b)"*)
415 (*Thm ("real_plus_minus_binom1_p_p",
416 num_str real_plus_minus_binom1_p_p),*)
417 (*"(a + b)*(a + -1 * b) = a^^^2 + -1*b^^^2"*)
418 (*Thm ("real_plus_minus_binom2_p_p",
419 num_str real_plus_minus_binom2_p_p),*)
420 (*"(a + -1 * b)*(a + b) = a^^^2 + -1*b^^^2"*)
421 (*WN071229 changed/removed for Schaerding -----^^^*)
423 Thm ("real_add_mult_distrib" ,num_str real_add_mult_distrib),
424 (*"(z1.0 + z2.0) * w = z1.0 * w + z2.0 * w"*)
425 Thm ("real_add_mult_distrib2",num_str real_add_mult_distrib2),
426 (*"w * (z1.0 + z2.0) = w * z1.0 + w * z2.0"*)
428 Thm ("realpow_multI", num_str realpow_multI),
429 (*"(r * s) ^^^ n = r ^^^ n * s ^^^ n"*)
430 Thm ("realpow_pow",num_str realpow_pow)
431 (*"(a ^^^ b) ^^^ c = a ^^^ (b * c)"*)
432 ], scr = EmptyScr}:rls;
434 (*.the expression contains + - * ^ only ?
435 this is weaker than 'is_polynomial' !.*)
436 fun is_polyexp (Free _) = true
437 | is_polyexp (Const ("op +",_) $ Free _ $ Free _) = true
438 | is_polyexp (Const ("op -",_) $ Free _ $ Free _) = true
439 | is_polyexp (Const ("op *",_) $ Free _ $ Free _) = true
440 | is_polyexp (Const ("Atools.pow",_) $ Free _ $ Free _) = true
441 | is_polyexp (Const ("op +",_) $ t1 $ t2) =
442 ((is_polyexp t1) andalso (is_polyexp t2))
443 | is_polyexp (Const ("op -",_) $ t1 $ t2) =
444 ((is_polyexp t1) andalso (is_polyexp t2))
445 | is_polyexp (Const ("op *",_) $ t1 $ t2) =
446 ((is_polyexp t1) andalso (is_polyexp t2))
447 | is_polyexp (Const ("Atools.pow",_) $ t1 $ t2) =
448 ((is_polyexp t1) andalso (is_polyexp t2))
449 | is_polyexp _ = false;
451 (*("is_polyexp", ("Poly.is'_polyexp", eval_is_polyexp ""))*)
452 fun eval_is_polyexp (thmid:string) _
453 (t as (Const("Poly.is'_polyexp", _) $ arg)) thy =
455 then SOME (mk_thmid thmid ""
456 ((Syntax.string_of_term (thy2ctxt thy)) arg) "",
457 Trueprop $ (mk_equality (t, HOLogic.true_const)))
458 else SOME (mk_thmid thmid ""
459 ((Syntax.string_of_term (thy2ctxt thy)) arg) "",
460 Trueprop $ (mk_equality (t, HOLogic.false_const)))
461 | eval_is_polyexp _ _ _ _ = NONE;
463 val expand_poly_rat_ =
464 Rls{id = "expand_poly_rat_", preconds = [],
465 rew_ord = ("dummy_ord", dummy_ord),
466 erls = append_rls "e_rls-is_polyexp" e_rls
467 [Calc ("Poly.is'_polyexp", eval_is_polyexp "")
472 rules = [Thm ("real_plus_binom_pow4_poly",num_str real_plus_binom_pow4_poly),
473 (*"[| a is_polyexp; b is_polyexp |] ==> (a + b)^^^4 = ... "*)
474 Thm ("real_plus_binom_pow5_poly",num_str real_plus_binom_pow5_poly),
475 (*"[| a is_polyexp; b is_polyexp |] ==> (a + b)^^^5 = ... "*)
476 Thm ("real_plus_binom_pow2_poly",num_str real_plus_binom_pow2_poly),
477 (*"[| a is_polyexp; b is_polyexp |] ==>
478 (a + b)^^^2 = a^^^2 + 2*a*b + b^^^2"*)
479 Thm ("real_plus_binom_pow3_poly",num_str real_plus_binom_pow3_poly),
480 (*"[| a is_polyexp; b is_polyexp |] ==>
481 (a + b)^^^3 = a^^^3 + 3*a^^^2*b + 3*a*b^^^2 + b^^^3" *)
482 Thm ("real_plus_minus_binom1_p_p",num_str real_plus_minus_binom1_p_p),
483 (*"(a + b)*(a + -1 * b) = a^^^2 + -1*b^^^2"*)
484 Thm ("real_plus_minus_binom2_p_p",num_str real_plus_minus_binom2_p_p),
485 (*"(a + -1 * b)*(a + b) = a^^^2 + -1*b^^^2"*)
487 Thm ("real_add_mult_distrib_poly" ,num_str real_add_mult_distrib_poly),
488 (*"w is_polyexp ==> (z1.0 + z2.0) * w = z1.0 * w + z2.0 * w"*)
489 Thm ("real_add_mult_distrib2_poly",num_str real_add_mult_distrib2_poly),
490 (*"w is_polyexp ==> w * (z1.0 + z2.0) = w * z1.0 + w * z2.0"*)
492 Thm ("realpow_multI_poly", num_str realpow_multI_poly),
493 (*"[| r is_polyexp; s is_polyexp |] ==>
494 (r * s) ^^^ n = r ^^^ n * s ^^^ n"*)
495 Thm ("realpow_pow",num_str realpow_pow)
496 (*"(a ^^^ b) ^^^ c = a ^^^ (b * c)"*)
497 ], scr = EmptyScr}:rls;
499 val simplify_power_ =
500 Rls{id = "simplify_power_", preconds = [],
501 rew_ord = ("dummy_ord", dummy_ord),
502 erls = e_rls, srls = Erls,
505 rules = [(*MG: Reihenfolge der folgenden 2 Thm muss so bleiben, wegen
506 a*(a*a) --> a*a^^^2 und nicht a*(a*a) --> a^^^2*a *)
507 Thm ("sym_realpow_twoI",num_str (realpow_twoI RS sym)),
508 (*"r * r = r ^^^ 2"*)
509 Thm ("realpow_twoI_assoc_l",num_str realpow_twoI_assoc_l),
510 (*"r * (r * s) = r ^^^ 2 * s"*)
512 Thm ("realpow_plus_1",num_str realpow_plus_1),
513 (*"r * r ^^^ n = r ^^^ (n + 1)"*)
514 Thm ("realpow_plus_1_assoc_l", num_str realpow_plus_1_assoc_l),
515 (*"r * (r ^^^ m * s) = r ^^^ (1 + m) * s"*)
516 (*MG 9.7.03: neues Thm wegen a*(a*(a*b)) --> a^^^2*(a*b) *)
517 Thm ("realpow_plus_1_assoc_l2", num_str realpow_plus_1_assoc_l2),
518 (*"r ^^^ m * (r * s) = r ^^^ (1 + m) * s"*)
520 Thm ("sym_realpow_addI",num_str (realpow_addI RS sym)),
521 (*"r ^^^ n * r ^^^ m = r ^^^ (n + m)"*)
522 Thm ("realpow_addI_assoc_l", num_str realpow_addI_assoc_l),
523 (*"r ^^^ n * (r ^^^ m * s) = r ^^^ (n + m) * s"*)
525 (* ist in expand_poly - wird hier aber auch gebraucht, wegen:
526 "r * r = r ^^^ 2" wenn r=a^^^b*)
527 Thm ("realpow_pow",num_str realpow_pow)
528 (*"(a ^^^ b) ^^^ c = a ^^^ (b * c)"*)
529 ], scr = EmptyScr}:rls;
531 val calc_add_mult_pow_ =
532 Rls{id = "calc_add_mult_pow_", preconds = [],
533 rew_ord = ("dummy_ord", dummy_ord),
534 erls = Atools_erls(*erls3.4.03*),srls = Erls,
535 calc = [("PLUS" , ("op +", eval_binop "#add_")),
536 ("TIMES" , ("op *", eval_binop "#mult_")),
537 ("POWER", ("Atools.pow", eval_binop "#power_"))
540 rules = [Calc ("op +", eval_binop "#add_"),
541 Calc ("op *", eval_binop "#mult_"),
542 Calc ("Atools.pow", eval_binop "#power_")
543 ], scr = EmptyScr}:rls;
545 val reduce_012_mult_ =
546 Rls{id = "reduce_012_mult_", preconds = [],
547 rew_ord = ("dummy_ord", dummy_ord),
548 erls = e_rls,srls = Erls,
551 rules = [(* MG: folgende Thm müssen hier stehen bleiben: *)
552 Thm ("real_mult_1_right",num_str real_mult_1_right),
553 (*"z * 1 = z"*) (*wegen "a * b * b^^^(-1) + a"*)
554 Thm ("realpow_zeroI",num_str realpow_zeroI),
555 (*"r ^^^ 0 = 1"*) (*wegen "a*a^^^(-1)*c + b + c"*)
556 Thm ("realpow_oneI",num_str realpow_oneI),
558 Thm ("realpow_eq_oneI",num_str realpow_eq_oneI)
560 ], scr = EmptyScr}:rls;
562 val collect_numerals_ =
563 Rls{id = "collect_numerals_", preconds = [],
564 rew_ord = ("dummy_ord", dummy_ord),
565 erls = Atools_erls, srls = Erls,
566 calc = [("PLUS" , ("op +", eval_binop "#add_"))
568 rules = [Thm ("real_num_collect",num_str real_num_collect),
569 (*"[| l is_const; m is_const |]==>l * n + m * n = (l + m) * n"*)
570 Thm ("real_num_collect_assoc_r",num_str real_num_collect_assoc_r),
571 (*"[| l is_const; m is_const |] ==> \
572 \(k + m * n) + l * n = k + (l + m)*n"*)
573 Thm ("real_one_collect",num_str real_one_collect),
574 (*"m is_const ==> n + m * n = (1 + m) * n"*)
575 Thm ("real_one_collect_assoc_r",num_str real_one_collect_assoc_r),
576 (*"m is_const ==> (k + n) + m * n = k + (m + 1) * n"*)
578 Calc ("op +", eval_binop "#add_"),
580 (*MG: Reihenfolge der folgenden 2 Thm muss so bleiben, wegen
581 (a+a)+a --> a + 2*a --> 3*a and not (a+a)+a --> 2*a + a *)
582 Thm ("real_mult_2_assoc_r",num_str real_mult_2_assoc_r),
583 (*"(k + z1) + z1 = k + 2 * z1"*)
584 Thm ("sym_real_mult_2",num_str (real_mult_2 RS sym))
585 (*"z1 + z1 = 2 * z1"*)
587 ], scr = EmptyScr}:rls;
590 Rls{id = "reduce_012_", preconds = [],
591 rew_ord = ("dummy_ord", dummy_ord),
592 erls = e_rls,srls = Erls,
595 rules = [Thm ("real_mult_1",num_str real_mult_1),
597 Thm ("real_mult_0",num_str real_mult_0),
599 Thm ("real_mult_0_right",num_str real_mult_0_right),
601 Thm ("real_add_zero_left",num_str real_add_zero_left),
603 Thm ("real_add_zero_right",num_str real_add_zero_right),
604 (*"z + 0 = z"*) (*wegen a+b-b --> a+(1-1)*b --> a+0 --> a*)
606 (*Thm ("realpow_oneI",num_str realpow_oneI)*)
608 Thm ("real_0_divide",num_str real_0_divide)(*WN060914*)
610 ], scr = EmptyScr}:rls;
612 (*ein Hilfs-'ruleset' (benutzt das leere 'ruleset')*)
613 val discard_parentheses_ =
614 append_rls "discard_parentheses_" e_rls
615 [Thm ("sym_real_mult_assoc", num_str (real_mult_assoc RS sym))
616 (*"?z1.1 * (?z2.1 * ?z3.1) = ?z1.1 * ?z2.1 * ?z3.1"*)
617 (*Thm ("sym_real_add_assoc",num_str (real_add_assoc RS sym))*)
618 (*"?z1.1 + (?z2.1 + ?z3.1) = ?z1.1 + ?z2.1 + ?z3.1"*)
621 (*----------------- End: rulesets for make_polynomial_ -----------------*)
623 (*MG.0401 ev. for use in rls with ordered rewriting ?
624 val collect_numerals_left =
625 Rls{id = "collect_numerals", preconds = [],
626 rew_ord = ("dummy_ord", dummy_ord),
627 erls = Atools_erls(*erls3.4.03*),srls = Erls,
628 calc = [("PLUS" , ("op +", eval_binop "#add_")),
629 ("TIMES" , ("op *", eval_binop "#mult_")),
630 ("POWER", ("Atools.pow", eval_binop "#power_"))
633 rules = [Thm ("real_num_collect",num_str real_num_collect),
634 (*"[| l is_const; m is_const |]==>l * n + m * n = (l + m) * n"*)
635 Thm ("real_num_collect_assoc",num_str real_num_collect_assoc),
636 (*"[| l is_const; m is_const |] ==>
637 l * n + (m * n + k) = (l + m) * n + k"*)
638 Thm ("real_one_collect",num_str real_one_collect),
639 (*"m is_const ==> n + m * n = (1 + m) * n"*)
640 Thm ("real_one_collect_assoc",num_str real_one_collect_assoc),
641 (*"m is_const ==> n + (m * n + k) = (1 + m) * n + k"*)
643 Calc ("op +", eval_binop "#add_"),
645 (*MG am 2.5.03: 2 Theoreme aus reduce_012 hierher verschoben*)
646 Thm ("sym_real_mult_2",num_str (real_mult_2 RS sym)),
647 (*"z1 + z1 = 2 * z1"*)
648 Thm ("real_mult_2_assoc",num_str real_mult_2_assoc)
649 (*"z1 + (z1 + k) = 2 * z1 + k"*)
650 ], scr = EmptyScr}:rls;*)
653 Rls{id = "expand_poly", preconds = [],
654 rew_ord = ("dummy_ord", dummy_ord),
655 erls = e_rls,srls = Erls,
658 rules = [Thm ("real_add_mult_distrib" ,num_str real_add_mult_distrib),
659 (*"(z1.0 + z2.0) * w = z1.0 * w + z2.0 * w"*)
660 Thm ("real_add_mult_distrib2",num_str real_add_mult_distrib2),
661 (*"w * (z1.0 + z2.0) = w * z1.0 + w * z2.0"*)
662 (*Thm ("real_add_mult_distrib1",num_str real_add_mult_distrib1),
663 ....... 18.3.03 undefined???*)
665 Thm ("real_plus_binom_pow2",num_str real_plus_binom_pow2),
666 (*"(a + b)^^^2 = a^^^2 + 2*a*b + b^^^2"*)
667 Thm ("real_minus_binom_pow2_p",num_str real_minus_binom_pow2_p),
668 (*"(a - b)^^^2 = a^^^2 + -2*a*b + b^^^2"*)
669 Thm ("real_plus_minus_binom1_p",
670 num_str real_plus_minus_binom1_p),
671 (*"(a + b)*(a - b) = a^^^2 + -1*b^^^2"*)
672 Thm ("real_plus_minus_binom2_p",
673 num_str real_plus_minus_binom2_p),
674 (*"(a - b)*(a + b) = a^^^2 + -1*b^^^2"*)
676 Thm ("real_minus_minus",num_str real_minus_minus),
678 Thm ("real_diff_minus",num_str real_diff_minus),
679 (*"a - b = a + -1 * b"*)
680 Thm ("sym_real_mult_minus1",num_str (real_mult_minus1 RS sym))
686 (*Thm ("real_minus_add_distrib",
687 num_str real_minus_add_distrib),*)
688 (*"- (?x + ?y) = - ?x + - ?y"*)
689 (*Thm ("real_diff_plus",num_str real_diff_plus)*)
691 ], scr = EmptyScr}:rls;
693 Rls{id = "simplify_power", preconds = [],
694 rew_ord = ("dummy_ord", dummy_ord),
695 erls = e_rls, srls = Erls,
698 rules = [Thm ("realpow_multI", num_str realpow_multI),
699 (*"(r * s) ^^^ n = r ^^^ n * s ^^^ n"*)
701 Thm ("sym_realpow_twoI",num_str (realpow_twoI RS sym)),
702 (*"r1 * r1 = r1 ^^^ 2"*)
703 Thm ("realpow_plus_1",num_str realpow_plus_1),
704 (*"r * r ^^^ n = r ^^^ (n + 1)"*)
705 Thm ("realpow_pow",num_str realpow_pow),
706 (*"(a ^^^ b) ^^^ c = a ^^^ (b * c)"*)
707 Thm ("sym_realpow_addI",num_str (realpow_addI RS sym)),
708 (*"r ^^^ n * r ^^^ m = r ^^^ (n + m)"*)
709 Thm ("realpow_oneI",num_str realpow_oneI),
711 Thm ("realpow_eq_oneI",num_str realpow_eq_oneI)
713 ], scr = EmptyScr}:rls;
714 (*MG.0401: termorders for multivariate polys dropped due to principal problems:
715 (total-degree-)ordering of monoms NOT possible with size_of_term GIVEN*)
717 Rls{id = "order_add_mult", preconds = [],
718 rew_ord = ("ord_make_polynomial",ord_make_polynomial false Poly.thy),
719 erls = e_rls,srls = Erls,
722 rules = [Thm ("real_mult_commute",num_str real_mult_commute),
724 Thm ("real_mult_left_commute",num_str real_mult_left_commute),
725 (*z1.0 * (z2.0 * z3.0) = z2.0 * (z1.0 * z3.0)*)
726 Thm ("real_mult_assoc",num_str real_mult_assoc),
727 (*z1.0 * z2.0 * z3.0 = z1.0 * (z2.0 * z3.0)*)
728 Thm ("real_add_commute",num_str real_add_commute),
730 Thm ("real_add_left_commute",num_str real_add_left_commute),
731 (*x + (y + z) = y + (x + z)*)
732 Thm ("real_add_assoc",num_str real_add_assoc)
733 (*z1.0 + z2.0 + z3.0 = z1.0 + (z2.0 + z3.0)*)
734 ], scr = EmptyScr}:rls;
735 (*MG.0401: termorders for multivariate polys dropped due to principal problems:
736 (total-degree-)ordering of monoms NOT possible with size_of_term GIVEN*)
738 Rls{id = "order_mult", preconds = [],
739 rew_ord = ("ord_make_polynomial",ord_make_polynomial false Poly.thy),
740 erls = e_rls,srls = Erls,
743 rules = [Thm ("real_mult_commute",num_str real_mult_commute),
745 Thm ("real_mult_left_commute",num_str real_mult_left_commute),
746 (*z1.0 * (z2.0 * z3.0) = z2.0 * (z1.0 * z3.0)*)
747 Thm ("real_mult_assoc",num_str real_mult_assoc)
748 (*z1.0 * z2.0 * z3.0 = z1.0 * (z2.0 * z3.0)*)
749 ], scr = EmptyScr}:rls;
750 val collect_numerals =
751 Rls{id = "collect_numerals", preconds = [],
752 rew_ord = ("dummy_ord", dummy_ord),
753 erls = Atools_erls(*erls3.4.03*),srls = Erls,
754 calc = [("PLUS" , ("op +", eval_binop "#add_")),
755 ("TIMES" , ("op *", eval_binop "#mult_")),
756 ("POWER", ("Atools.pow", eval_binop "#power_"))
759 rules = [Thm ("real_num_collect",num_str real_num_collect),
760 (*"[| l is_const; m is_const |]==>l * n + m * n = (l + m) * n"*)
761 Thm ("real_num_collect_assoc",num_str real_num_collect_assoc),
762 (*"[| l is_const; m is_const |] ==>
763 l * n + (m * n + k) = (l + m) * n + k"*)
764 Thm ("real_one_collect",num_str real_one_collect),
765 (*"m is_const ==> n + m * n = (1 + m) * n"*)
766 Thm ("real_one_collect_assoc",num_str real_one_collect_assoc),
767 (*"m is_const ==> k + (n + m * n) = k + (1 + m) * n"*)
768 Calc ("op +", eval_binop "#add_"),
769 Calc ("op *", eval_binop "#mult_"),
770 Calc ("Atools.pow", eval_binop "#power_")
771 ], scr = EmptyScr}:rls;
773 Rls{id = "reduce_012", preconds = [],
774 rew_ord = ("dummy_ord", dummy_ord),
775 erls = e_rls,srls = Erls,
778 rules = [Thm ("real_mult_1",num_str real_mult_1),
780 (*Thm ("real_mult_minus1",num_str real_mult_minus1),14.3.03*)
782 Thm ("sym_real_mult_minus_eq1",
783 num_str (real_mult_minus_eq1 RS sym)),
784 (*- (?x * ?y) = "- ?x * ?y"*)
785 (*Thm ("real_minus_mult_cancel",num_str real_minus_mult_cancel),
786 (*"- ?x * - ?y = ?x * ?y"*)---*)
787 Thm ("real_mult_0",num_str real_mult_0),
789 Thm ("real_add_zero_left",num_str real_add_zero_left),
791 Thm ("real_add_minus",num_str real_add_minus),
793 Thm ("sym_real_mult_2",num_str (real_mult_2 RS sym)),
794 (*"z1 + z1 = 2 * z1"*)
795 Thm ("real_mult_2_assoc",num_str real_mult_2_assoc)
796 (*"z1 + (z1 + k) = 2 * z1 + k"*)
797 ], scr = EmptyScr}:rls;
798 (*ein Hilfs-'ruleset' (benutzt das leere 'ruleset')*)
799 val discard_parentheses =
800 append_rls "discard_parentheses" e_rls
801 [Thm ("sym_real_mult_assoc", num_str (real_mult_assoc RS sym)),
802 Thm ("sym_real_add_assoc",num_str (real_add_assoc RS sym))];
804 val scr_make_polynomial =
805 "Script Expand_binoms t_ =\
807 \((Try (Repeat (Rewrite real_diff_minus False))) @@ \
809 \ (Try (Repeat (Rewrite real_add_mult_distrib False))) @@ \
810 \ (Try (Repeat (Rewrite real_add_mult_distrib2 False))) @@ \
811 \ (Try (Repeat (Rewrite real_diff_mult_distrib False))) @@ \
812 \ (Try (Repeat (Rewrite real_diff_mult_distrib2 False))) @@ \
814 \ (Try (Repeat (Rewrite real_mult_1 False))) @@ \
815 \ (Try (Repeat (Rewrite real_mult_0 False))) @@ \
816 \ (Try (Repeat (Rewrite real_add_zero_left False))) @@ \
818 \ (Try (Repeat (Rewrite real_mult_commute False))) @@ \
819 \ (Try (Repeat (Rewrite real_mult_left_commute False))) @@ \
820 \ (Try (Repeat (Rewrite real_mult_assoc False))) @@ \
821 \ (Try (Repeat (Rewrite real_add_commute False))) @@ \
822 \ (Try (Repeat (Rewrite real_add_left_commute False))) @@ \
823 \ (Try (Repeat (Rewrite real_add_assoc False))) @@ \
825 \ (Try (Repeat (Rewrite sym_realpow_twoI False))) @@ \
826 \ (Try (Repeat (Rewrite realpow_plus_1 False))) @@ \
827 \ (Try (Repeat (Rewrite sym_real_mult_2 False))) @@ \
828 \ (Try (Repeat (Rewrite real_mult_2_assoc False))) @@ \
830 \ (Try (Repeat (Rewrite real_num_collect False))) @@ \
831 \ (Try (Repeat (Rewrite real_num_collect_assoc False))) @@ \
833 \ (Try (Repeat (Rewrite real_one_collect False))) @@ \
834 \ (Try (Repeat (Rewrite real_one_collect_assoc False))) @@ \
836 \ (Try (Repeat (Calculate plus ))) @@ \
837 \ (Try (Repeat (Calculate times ))) @@ \
838 \ (Try (Repeat (Calculate power_)))) \
841 (*version used by MG.02/03, overwritten by version AG in 04 below
842 val make_polynomial = prep_rls(
843 Seq{id = "make_polynomial", preconds = []:term list,
844 rew_ord = ("dummy_ord", dummy_ord),
845 erls = Atools_erls, srls = Erls,
846 calc = [],(*asm_thm = [],*)
847 rules = [Rls_ expand_poly,
849 Rls_ simplify_power, (*realpow_eq_oneI, eg. x^1 --> x *)
850 Rls_ collect_numerals, (*eg. x^(2+ -1) --> x^1 *)
852 Thm ("realpow_oneI",num_str realpow_oneI),(*in --^*)
853 Rls_ discard_parentheses
858 val scr_expand_binoms =
859 "Script Expand_binoms t_ =\
861 \((Try (Repeat (Rewrite real_plus_binom_pow2 False))) @@ \
862 \ (Try (Repeat (Rewrite real_plus_binom_times False))) @@ \
863 \ (Try (Repeat (Rewrite real_minus_binom_pow2 False))) @@ \
864 \ (Try (Repeat (Rewrite real_minus_binom_times False))) @@ \
865 \ (Try (Repeat (Rewrite real_plus_minus_binom1 False))) @@ \
866 \ (Try (Repeat (Rewrite real_plus_minus_binom2 False))) @@ \
868 \ (Try (Repeat (Rewrite real_mult_1 False))) @@ \
869 \ (Try (Repeat (Rewrite real_mult_0 False))) @@ \
870 \ (Try (Repeat (Rewrite real_add_zero_left False))) @@ \
872 \ (Try (Repeat (Calculate plus ))) @@ \
873 \ (Try (Repeat (Calculate times ))) @@ \
874 \ (Try (Repeat (Calculate power_))) @@ \
876 \ (Try (Repeat (Rewrite sym_realpow_twoI False))) @@ \
877 \ (Try (Repeat (Rewrite realpow_plus_1 False))) @@ \
878 \ (Try (Repeat (Rewrite sym_real_mult_2 False))) @@ \
879 \ (Try (Repeat (Rewrite real_mult_2_assoc False))) @@ \
881 \ (Try (Repeat (Rewrite real_num_collect False))) @@ \
882 \ (Try (Repeat (Rewrite real_num_collect_assoc False))) @@ \
884 \ (Try (Repeat (Rewrite real_one_collect False))) @@ \
885 \ (Try (Repeat (Rewrite real_one_collect_assoc False))) @@ \
887 \ (Try (Repeat (Calculate plus ))) @@ \
888 \ (Try (Repeat (Calculate times ))) @@ \
889 \ (Try (Repeat (Calculate power_)))) \
893 Rls{id = "expand_binoms", preconds = [], rew_ord = ("termlessI",termlessI),
894 erls = Atools_erls, srls = Erls,
895 calc = [("PLUS" , ("op +", eval_binop "#add_")),
896 ("TIMES" , ("op *", eval_binop "#mult_")),
897 ("POWER", ("Atools.pow", eval_binop "#power_"))
900 rules = [Thm ("real_plus_binom_pow2" ,num_str real_plus_binom_pow2),
901 (*"(a + b) ^^^ 2 = a ^^^ 2 + 2 * a * b + b ^^^ 2"*)
902 Thm ("real_plus_binom_times" ,num_str real_plus_binom_times),
903 (*"(a + b)*(a + b) = ...*)
904 Thm ("real_minus_binom_pow2" ,num_str real_minus_binom_pow2),
905 (*"(a - b) ^^^ 2 = a ^^^ 2 - 2 * a * b + b ^^^ 2"*)
906 Thm ("real_minus_binom_times",num_str real_minus_binom_times),
907 (*"(a - b)*(a - b) = ...*)
908 Thm ("real_plus_minus_binom1",num_str real_plus_minus_binom1),
909 (*"(a + b) * (a - b) = a ^^^ 2 - b ^^^ 2"*)
910 Thm ("real_plus_minus_binom2",num_str real_plus_minus_binom2),
911 (*"(a - b) * (a + b) = a ^^^ 2 - b ^^^ 2"*)
913 Thm ("real_pp_binom_times",num_str real_pp_binom_times),
914 (*(a + b)*(c + d) = a*c + a*d + b*c + b*d*)
915 Thm ("real_pm_binom_times",num_str real_pm_binom_times),
916 (*(a + b)*(c - d) = a*c - a*d + b*c - b*d*)
917 Thm ("real_mp_binom_times",num_str real_mp_binom_times),
918 (*(a - b)*(c + d) = a*c + a*d - b*c - b*d*)
919 Thm ("real_mm_binom_times",num_str real_mm_binom_times),
920 (*(a - b)*(c - d) = a*c - a*d - b*c + b*d*)
921 Thm ("realpow_multI",num_str realpow_multI),
922 (*(a*b)^^^n = a^^^n * b^^^n*)
923 Thm ("real_plus_binom_pow3",num_str real_plus_binom_pow3),
924 (* (a + b)^^^3 = a^^^3 + 3*a^^^2*b + 3*a*b^^^2 + b^^^3 *)
925 Thm ("real_minus_binom_pow3",num_str real_minus_binom_pow3),
926 (* (a - b)^^^3 = a^^^3 - 3*a^^^2*b + 3*a*b^^^2 - b^^^3 *)
929 (* Thm ("real_add_mult_distrib" ,num_str real_add_mult_distrib),
930 (*"(z1.0 + z2.0) * w = z1.0 * w + z2.0 * w"*)
931 Thm ("real_add_mult_distrib2",num_str real_add_mult_distrib2),
932 (*"w * (z1.0 + z2.0) = w * z1.0 + w * z2.0"*)
933 Thm ("real_diff_mult_distrib" ,num_str real_diff_mult_distrib),
934 (*"(z1.0 - z2.0) * w = z1.0 * w - z2.0 * w"*)
935 Thm ("real_diff_mult_distrib2",num_str real_diff_mult_distrib2),
936 (*"w * (z1.0 - z2.0) = w * z1.0 - w * z2.0"*)
939 Thm ("real_mult_1",num_str real_mult_1), (*"1 * z = z"*)
940 Thm ("real_mult_0",num_str real_mult_0), (*"0 * z = 0"*)
941 Thm ("real_add_zero_left",num_str real_add_zero_left),(*"0 + z = z"*)
943 Calc ("op +", eval_binop "#add_"),
944 Calc ("op *", eval_binop "#mult_"),
945 Calc ("Atools.pow", eval_binop "#power_"),
947 Thm ("real_mult_commute",num_str real_mult_commute), (*AC-rewriting*)
948 Thm ("real_mult_left_commute",num_str real_mult_left_commute), (**)
949 Thm ("real_mult_assoc",num_str real_mult_assoc), (**)
950 Thm ("real_add_commute",num_str real_add_commute), (**)
951 Thm ("real_add_left_commute",num_str real_add_left_commute), (**)
952 Thm ("real_add_assoc",num_str real_add_assoc), (**)
955 Thm ("sym_realpow_twoI",num_str (realpow_twoI RS sym)),
956 (*"r1 * r1 = r1 ^^^ 2"*)
957 Thm ("realpow_plus_1",num_str realpow_plus_1),
958 (*"r * r ^^^ n = r ^^^ (n + 1)"*)
959 (*Thm ("sym_real_mult_2",num_str (real_mult_2 RS sym)),
960 (*"z1 + z1 = 2 * z1"*)*)
961 Thm ("real_mult_2_assoc",num_str real_mult_2_assoc),
962 (*"z1 + (z1 + k) = 2 * z1 + k"*)
964 Thm ("real_num_collect",num_str real_num_collect),
965 (*"[| l is_const; m is_const |] ==> l * n + m * n = (l + m) * n"*)
966 Thm ("real_num_collect_assoc",num_str real_num_collect_assoc),
967 (*"[| l is_const; m is_const |] ==> l * n + (m * n + k) = (l + m) * n + k"*)
968 Thm ("real_one_collect",num_str real_one_collect),
969 (*"m is_const ==> n + m * n = (1 + m) * n"*)
970 Thm ("real_one_collect_assoc",num_str real_one_collect_assoc),
971 (*"m is_const ==> k + (n + m * n) = k + (1 + m) * n"*)
973 Calc ("op +", eval_binop "#add_"),
974 Calc ("op *", eval_binop "#mult_"),
975 Calc ("Atools.pow", eval_binop "#power_")
977 scr = Script ((term_of o the o (parse thy)) scr_expand_binoms)
981 "******* Poly.ML end ******* ...RL";
984 (**. MG.03: make_polynomial_ ... uses SML-fun for ordering .**)
986 (*FIXME.0401: make SML-order local to make_polynomial(_) *)
987 (*FIXME.0401: replace 'make_polynomial'(old) by 'make_polynomial_'(MG) *)
988 (* Polynom --> List von Monomen *)
989 fun poly2list (Const ("op +",_) $ t1 $ t2) =
990 (poly2list t1) @ (poly2list t2)
993 (* Monom --> Liste von Variablen *)
994 fun monom2list (Const ("op *",_) $ t1 $ t2) =
995 (monom2list t1) @ (monom2list t2)
996 | monom2list t = [t];
998 (* liefert Variablenname (String) einer Variablen und Basis bei Potenz *)
999 fun get_basStr (Const ("Atools.pow",_) $ Free (str, _) $ _) = str
1000 | get_basStr (Free (str, _)) = str
1001 | get_basStr t = "|||"; (* gross gewichtet; für Brüch ect. *)
1003 raise error("get_basStr: called with t= "^(term2str t));*)
1005 (* liefert Hochzahl (String) einer Variablen bzw Gewichtstring (zum Sortieren) *)
1006 fun get_potStr (Const ("Atools.pow",_) $ Free _ $ Free (str, _)) = str
1007 | get_potStr (Const ("Atools.pow",_) $ Free _ $ _ ) = "|||" (* gross gewichtet *)
1008 | get_potStr (Free (str, _)) = "---" (* keine Hochzahl --> kleinst gewichtet *)
1009 | get_potStr t = "||||||"; (* gross gewichtet; für Brüch ect. *)
1011 raise error("get_potStr: called with t= "^(term2str t));*)
1013 (* Umgekehrte string_ord *)
1014 val string_ord_rev = rev_order o string_ord;
1016 (* Ordnung zum lexikographischen Vergleich zweier Variablen (oder Potenzen)
1017 innerhalb eines Monomes:
1018 - zuerst lexikographisch nach Variablenname
1019 - wenn gleich: nach steigender Potenz *)
1020 fun var_ord (a,b: term) = prod_ord string_ord string_ord
1021 ((get_basStr a, get_potStr a), (get_basStr b, get_potStr b));
1023 (* Ordnung zum lexikographischen Vergleich zweier Variablen (oder Potenzen);
1024 verwendet zum Sortieren von Monomen mittels Gesamtgradordnung:
1025 - zuerst lexikographisch nach Variablenname
1026 - wenn gleich: nach sinkender Potenz*)
1027 fun var_ord_revPow (a,b: term) = prod_ord string_ord string_ord_rev
1028 ((get_basStr a, get_potStr a), (get_basStr b, get_potStr b));
1031 (* Ordnet ein Liste von Variablen (und Potenzen) lexikographisch *)
1032 val sort_varList = sort var_ord;
1034 (* Entfernet aeussersten Operator (Wurzel) aus einem Term und schreibt
1035 Argumente in eine Liste *)
1036 fun args u : term list =
1037 let fun stripc (f$t, ts) = stripc (f, t::ts)
1038 | stripc (t as Free _, ts) = (t::ts)
1039 | stripc (_, ts) = ts
1040 in stripc (u, []) end;
1042 (* liefert True, falls der Term (Liste von Termen) nur Zahlen
1043 (keine Variablen) enthaelt *)
1044 fun filter_num [] = true
1045 | filter_num [Free x] = if (is_num (Free x)) then true
1047 | filter_num ((Free _)::_) = false
1049 (filter_num o (filter_out is_num) o flat o (map args)) ts;
1051 (* liefert True, falls der Term nur Zahlen (keine Variablen) enthaelt
1052 dh. er ist ein numerischer Wert und entspricht einem Koeffizienten *)
1053 fun is_nums t = filter_num [t];
1055 (* Berechnet den Gesamtgrad eines Monoms *)
1057 fun counter (n, []) = n
1058 | counter (n, x :: xs) =
1063 (Const ("Atools.pow", _) $ Free (str_b, _) $ Free (str_h, T)) =>
1064 if (is_nums (Free (str_h, T))) then
1065 counter (n + (the (int_of_str str_h)), xs)
1066 else counter (n + 1000, xs) (*FIXME.MG?!*)
1067 | (Const ("Atools.pow", _) $ Free (str_b, _) $ _ ) =>
1068 counter (n + 1000, xs) (*FIXME.MG?!*)
1069 | (Free (str, _)) => counter (n + 1, xs)
1070 (*| _ => raise error("monom_degree: called with factor: "^(term2str x)))*)
1071 | _ => counter (n + 10000, xs)) (*FIXME.MG?! ... Brüche ect.*)
1073 fun monom_degree l = counter (0, l)
1076 (* wie Ordnung dict_ord (lexicographische Ordnung zweier Listen, mit Vergleich
1077 der Listen-Elemente mit elem_ord) - Elemente die Bedingung cond erfuellen,
1078 werden jedoch dabei ignoriert (uebersprungen) *)
1079 fun dict_cond_ord _ _ ([], []) = EQUAL
1080 | dict_cond_ord _ _ ([], _ :: _) = LESS
1081 | dict_cond_ord _ _ (_ :: _, []) = GREATER
1082 | dict_cond_ord elem_ord cond (x :: xs, y :: ys) =
1083 (case (cond x, cond y) of
1084 (false, false) => (case elem_ord (x, y) of
1085 EQUAL => dict_cond_ord elem_ord cond (xs, ys)
1087 | (false, true) => dict_cond_ord elem_ord cond (x :: xs, ys)
1088 | (true, false) => dict_cond_ord elem_ord cond (xs, y :: ys)
1089 | (true, true) => dict_cond_ord elem_ord cond (xs, ys) );
1091 (* Gesamtgradordnung zum Vergleich von Monomen (Liste von Variablen/Potenzen):
1092 zuerst nach Gesamtgrad, bei gleichem Gesamtgrad lexikographisch ordnen -
1093 dabei werden Koeffizienten ignoriert (2*3*a^^^2*4*b gilt wie a^^^2*b) *)
1094 fun degree_ord (xs, ys) =
1095 prod_ord int_ord (dict_cond_ord var_ord_revPow is_nums)
1096 ((monom_degree xs, xs), (monom_degree ys, ys));
1098 fun hd_str str = substring (str, 0, 1);
1099 fun tl_str str = substring (str, 1, (size str) - 1);
1101 (* liefert nummerischen Koeffizienten eines Monoms oder NONE *)
1102 fun get_koeff_of_mon [] = raise error("get_koeff_of_mon: called with l = []")
1103 | get_koeff_of_mon (l as x::xs) = if is_nums x then SOME x
1106 (* wandelt Koeffizient in (zum sortieren geeigneten) String um *)
1107 fun koeff2ordStr (SOME x) = (case x of
1109 if (hd_str str) = "-" then (tl_str str)^"0" (* 3 < -3 *)
1111 | _ => "aaa") (* "num.Ausdruck" --> gross *)
1112 | koeff2ordStr NONE = "---"; (* "kein Koeff" --> kleinste *)
1114 (* Order zum Vergleich von Koeffizienten (strings):
1115 "kein Koeff" < "0" < "1" < "-1" < "2" < "-2" < ... < "num.Ausdruck" *)
1116 fun compare_koeff_ord (xs, ys) =
1117 string_ord ((koeff2ordStr o get_koeff_of_mon) xs,
1118 (koeff2ordStr o get_koeff_of_mon) ys);
1120 (* Gesamtgradordnung degree_ord + Ordnen nach Koeffizienten falls EQUAL *)
1121 fun koeff_degree_ord (xs, ys) =
1122 prod_ord degree_ord compare_koeff_ord ((xs, xs), (ys, ys));
1124 (* Ordnet ein Liste von Monomen (Monom = Liste von Variablen) mittels
1125 Gesamtgradordnung *)
1126 val sort_monList = sort koeff_degree_ord;
1128 (* Alternativ zu degree_ord koennte auch die viel einfachere und
1129 kuerzere Ordnung simple_ord verwendet werden - ist aber nicht
1130 fuer unsere Zwecke geeignet!
1132 fun simple_ord (al,bl: term list) = dict_ord string_ord
1133 (map get_basStr al, map get_basStr bl);
1135 val sort_monList = sort simple_ord; *)
1137 (* aus 2 Variablen wird eine Summe bzw ein Produkt erzeugt
1138 (mit gewuenschtem Typen T) *)
1139 fun plus T = Const ("op +", [T,T] ---> T);
1140 fun mult T = Const ("op *", [T,T] ---> T);
1141 fun binop op_ t1 t2 = op_ $ t1 $ t2;
1142 fun create_prod T (a,b) = binop (mult T) a b;
1143 fun create_sum T (a,b) = binop (plus T) a b;
1145 (* löscht letztes Element einer Liste *)
1146 fun drop_last l = take ((length l)-1,l);
1148 (* Liste von Variablen --> Monom *)
1149 fun create_monom T vl = foldr (create_prod T) (drop_last vl, last_elem vl);
1151 foldr bewirkt rechtslastige Klammerung des Monoms - ist notwendig, damit zwei
1152 gleiche Monome zusammengefasst werden können (collect_numerals)!
1153 zB: 2*(x*(y*z)) + 3*(x*(y*z)) --> (2+3)*(x*(y*z))*)
1155 (* Liste von Monomen --> Polynom *)
1156 fun create_polynom T ml = foldl (create_sum T) (hd ml, tl ml);
1158 foldl bewirkt linkslastige Klammerung des Polynoms (der Summanten) -
1159 bessere Darstellung, da keine Klammern sichtbar!
1160 (und discard_parentheses in make_polynomial hat weniger zu tun) *)
1162 (* sorts the variables (faktors) of an expanded polynomial lexicographical *)
1163 fun sort_variables t =
1165 val ll = map monom2list (poly2list t);
1166 val lls = map sort_varList ll;
1168 val ls = map (create_monom T) lls;
1169 in create_polynom T ls end;
1171 (* sorts the monoms of an expanded and variable-sorted polynomial
1175 val ll = map monom2list (poly2list t);
1176 val lls = sort_monList ll;
1178 val ls = map (create_monom T) lls;
1179 in create_polynom T ls end;
1181 (* auch Klammerung muss übereinstimmen;
1182 sort_variables klammert Produkte rechtslastig*)
1183 fun is_multUnordered t = ((is_polyexp t) andalso not (t = sort_variables t));
1185 fun eval_is_multUnordered (thmid:string) _
1186 (t as (Const("Poly.is'_multUnordered", _) $ arg)) thy =
1187 if is_multUnordered arg
1188 then SOME (mk_thmid thmid ""
1189 ((Syntax.string_of_term (thy2ctxt thy)) arg) "",
1190 Trueprop $ (mk_equality (t, HOLogic.true_const)))
1191 else SOME (mk_thmid thmid ""
1192 ((Syntax.string_of_term (thy2ctxt thy)) arg) "",
1193 Trueprop $ (mk_equality (t, HOLogic.false_const)))
1194 | eval_is_multUnordered _ _ _ _ = NONE;
1197 fun attach_form (_:rule list list) (_:term) (_:term) = (*still missing*)
1198 []:(rule * (term * term list)) list;
1199 fun init_state (_:term) = e_rrlsstate;
1200 fun locate_rule (_:rule list list) (_:term) (_:rule) =
1201 ([]:(rule * (term * term list)) list);
1202 fun next_rule (_:rule list list) (_:term) = (NONE:rule option);
1203 fun normal_form t = SOME (sort_variables t,[]:term list);
1206 Rrls {id = "order_mult_",
1208 [([(term_of o the o (parse thy)) "p is_multUnordered"],
1209 (term_of o the o (parse thy)) "?p" )],
1210 rew_ord = ("dummy_ord", dummy_ord),
1211 erls = append_rls "e_rls-is_multUnordered" e_rls(*MG: poly_erls*)
1212 [Calc ("Poly.is'_multUnordered", eval_is_multUnordered "")
1214 calc = [("PLUS" ,("op +" ,eval_binop "#add_")),
1215 ("TIMES" ,("op *" ,eval_binop "#mult_")),
1216 ("DIVIDE" ,("HOL.divide" ,eval_cancel "#divide_")),
1217 ("POWER" ,("Atools.pow" ,eval_binop "#power_"))],
1219 scr=Rfuns {init_state = init_state,
1220 normal_form = normal_form,
1221 locate_rule = locate_rule,
1222 next_rule = next_rule,
1223 attach_form = attach_form}};
1225 val order_mult_rls_ =
1226 Rls{id = "order_mult_rls_", preconds = [],
1227 rew_ord = ("dummy_ord", dummy_ord),
1228 erls = e_rls,srls = Erls,
1231 rules = [Rls_ order_mult_
1232 ], scr = EmptyScr}:rls;
1234 fun is_addUnordered t = ((is_polyexp t) andalso not (t = sort_monoms t));
1237 (*("is_addUnordered", ("Poly.is'_addUnordered", eval_is_addUnordered ""))*)
1238 fun eval_is_addUnordered (thmid:string) _
1239 (t as (Const("Poly.is'_addUnordered", _) $ arg)) thy =
1240 if is_addUnordered arg
1241 then SOME (mk_thmid thmid ""
1242 ((Syntax.string_of_term (thy2ctxt thy)) arg) "",
1243 Trueprop $ (mk_equality (t, HOLogic.true_const)))
1244 else SOME (mk_thmid thmid ""
1245 ((Syntax.string_of_term (thy2ctxt thy)) arg) "",
1246 Trueprop $ (mk_equality (t, HOLogic.false_const)))
1247 | eval_is_addUnordered _ _ _ _ = NONE;
1249 fun attach_form (_:rule list list) (_:term) (_:term) = (*still missing*)
1250 []:(rule * (term * term list)) list;
1251 fun init_state (_:term) = e_rrlsstate;
1252 fun locate_rule (_:rule list list) (_:term) (_:rule) =
1253 ([]:(rule * (term * term list)) list);
1254 fun next_rule (_:rule list list) (_:term) = (NONE:rule option);
1255 fun normal_form t = SOME (sort_monoms t,[]:term list);
1258 Rrls {id = "order_add_",
1259 prepat = (*WN.18.6.03 Preconditions und Pattern,
1260 die beide passen muessen, damit das Rrls angewandt wird*)
1261 [([(term_of o the o (parse thy)) "p is_addUnordered"],
1262 (term_of o the o (parse thy)) "?p"
1263 (*WN.18.6.03 also KEIN pattern, dieses erzeugt nur das Environment
1264 fuer die Evaluation der Precondition "p is_addUnordered"*))],
1265 rew_ord = ("dummy_ord", dummy_ord),
1266 erls = append_rls "e_rls-is_addUnordered" e_rls(*MG: poly_erls*)
1267 [Calc ("Poly.is'_addUnordered", eval_is_addUnordered "")
1268 (*WN.18.6.03 definiert in Poly.thy,
1269 evaluiert prepat*)],
1270 calc = [("PLUS" ,("op +" ,eval_binop "#add_")),
1271 ("TIMES" ,("op *" ,eval_binop "#mult_")),
1272 ("DIVIDE" ,("HOL.divide" ,eval_cancel "#divide_")),
1273 ("POWER" ,("Atools.pow" ,eval_binop "#power_"))],
1275 scr=Rfuns {init_state = init_state,
1276 normal_form = normal_form,
1277 locate_rule = locate_rule,
1278 next_rule = next_rule,
1279 attach_form = attach_form}};
1281 val order_add_rls_ =
1282 Rls{id = "order_add_rls_", preconds = [],
1283 rew_ord = ("dummy_ord", dummy_ord),
1284 erls = e_rls,srls = Erls,
1287 rules = [Rls_ order_add_
1288 ], scr = EmptyScr}:rls;
1290 (*. see MG-DA.p.52ff .*)
1291 val make_polynomial(*MG.03, overwrites version from above,
1292 previously 'make_polynomial_'*) =
1293 Seq {id = "make_polynomial", preconds = []:term list,
1294 rew_ord = ("dummy_ord", dummy_ord),
1295 erls = Atools_erls, srls = Erls,calc = [],
1296 rules = [Rls_ discard_minus_,
1298 Calc ("op *", eval_binop "#mult_"),
1299 Rls_ order_mult_rls_,
1300 Rls_ simplify_power_,
1301 Rls_ calc_add_mult_pow_,
1302 Rls_ reduce_012_mult_,
1303 Rls_ order_add_rls_,
1304 Rls_ collect_numerals_,
1306 Rls_ discard_parentheses_
1310 val norm_Poly(*=make_polynomial*) =
1311 Seq {id = "norm_Poly", preconds = []:term list,
1312 rew_ord = ("dummy_ord", dummy_ord),
1313 erls = Atools_erls, srls = Erls, calc = [],
1314 rules = [Rls_ discard_minus_,
1316 Calc ("op *", eval_binop "#mult_"),
1317 Rls_ order_mult_rls_,
1318 Rls_ simplify_power_,
1319 Rls_ calc_add_mult_pow_,
1320 Rls_ reduce_012_mult_,
1321 Rls_ order_add_rls_,
1322 Rls_ collect_numerals_,
1324 Rls_ discard_parentheses_
1329 (* MG:03 Like make_polynomial_ but without Rls_ discard_parentheses_
1330 and expand_poly_rat_ instead of expand_poly_, see MG-DA.p.56ff*)
1331 (* MG necessary for termination of norm_Rational(*_mg*) in Rational.ML*)
1332 val make_rat_poly_with_parentheses =
1333 Seq{id = "make_rat_poly_with_parentheses", preconds = []:term list,
1334 rew_ord = ("dummy_ord", dummy_ord),
1335 erls = Atools_erls, srls = Erls, calc = [],
1336 rules = [Rls_ discard_minus_,
1337 Rls_ expand_poly_rat_,(*ignors rationals*)
1338 Calc ("op *", eval_binop "#mult_"),
1339 Rls_ order_mult_rls_,
1340 Rls_ simplify_power_,
1341 Rls_ calc_add_mult_pow_,
1342 Rls_ reduce_012_mult_,
1343 Rls_ order_add_rls_,
1344 Rls_ collect_numerals_,
1346 (*Rls_ discard_parentheses_ *)
1351 (*.a minimal ruleset for reverse rewriting of factions [2];
1352 compare expand_binoms.*)
1354 Seq{id = "reverse_rewriting", preconds = [], rew_ord = ("termlessI",termlessI),
1355 erls = Atools_erls, srls = Erls,
1356 calc = [(*("PLUS" , ("op +", eval_binop "#add_")),
1357 ("TIMES" , ("op *", eval_binop "#mult_")),
1358 ("POWER", ("Atools.pow", eval_binop "#power_"))*)
1360 rules = [Thm ("real_plus_binom_times" ,num_str real_plus_binom_times),
1361 (*"(a + b)*(a + b) = a ^ 2 + 2 * a * b + b ^ 2*)
1362 Thm ("real_plus_binom_times1" ,num_str real_plus_binom_times1),
1363 (*"(a + 1*b)*(a + -1*b) = a^^^2 + -1*b^^^2"*)
1364 Thm ("real_plus_binom_times2" ,num_str real_plus_binom_times2),
1365 (*"(a + -1*b)*(a + 1*b) = a^^^2 + -1*b^^^2"*)
1367 Thm ("real_mult_1",num_str real_mult_1),(*"1 * z = z"*)
1369 Thm ("real_add_mult_distrib" ,num_str real_add_mult_distrib),
1370 (*"(z1.0 + z2.0) * w = z1.0 * w + z2.0 * w"*)
1371 Thm ("real_add_mult_distrib2",num_str real_add_mult_distrib2),
1372 (*"w * (z1.0 + z2.0) = w * z1.0 + w * z2.0"*)
1374 Thm ("real_mult_assoc", num_str real_mult_assoc),
1375 (*"?z1.1 * ?z2.1 * ?z3. =1 ?z1.1 * (?z2.1 * ?z3.1)"*)
1376 Rls_ order_mult_rls_,
1377 (*Rls_ order_add_rls_,*)
1379 Calc ("op +", eval_binop "#add_"),
1380 Calc ("op *", eval_binop "#mult_"),
1381 Calc ("Atools.pow", eval_binop "#power_"),
1383 Thm ("sym_realpow_twoI",num_str (realpow_twoI RS sym)),
1384 (*"r1 * r1 = r1 ^^^ 2"*)
1385 Thm ("sym_real_mult_2",num_str (real_mult_2 RS sym)),
1386 (*"z1 + z1 = 2 * z1"*)
1387 Thm ("real_mult_2_assoc",num_str real_mult_2_assoc),
1388 (*"z1 + (z1 + k) = 2 * z1 + k"*)
1390 Thm ("real_num_collect",num_str real_num_collect),
1391 (*"[| l is_const; m is_const |]==>l * n + m * n = (l + m) * n"*)
1392 Thm ("real_num_collect_assoc",num_str real_num_collect_assoc),
1393 (*"[| l is_const; m is_const |] ==>
1394 l * n + (m * n + k) = (l + m) * n + k"*)
1395 Thm ("real_one_collect",num_str real_one_collect),
1396 (*"m is_const ==> n + m * n = (1 + m) * n"*)
1397 Thm ("real_one_collect_assoc",num_str real_one_collect_assoc),
1398 (*"m is_const ==> k + (n + m * n) = k + (1 + m) * n"*)
1400 Thm ("realpow_multI", num_str realpow_multI),
1401 (*"(r * s) ^^^ n = r ^^^ n * s ^^^ n"*)
1403 Calc ("op +", eval_binop "#add_"),
1404 Calc ("op *", eval_binop "#mult_"),
1405 Calc ("Atools.pow", eval_binop "#power_"),
1407 Thm ("real_mult_1",num_str real_mult_1),(*"1 * z = z"*)
1408 Thm ("real_mult_0",num_str real_mult_0),(*"0 * z = 0"*)
1409 Thm ("real_add_zero_left",num_str real_add_zero_left)(*0 + z = z*)
1411 (*Rls_ order_add_rls_*)
1414 scr = EmptyScr}:rls;
1417 overwritelthy thy (!ruleset',
1418 [("norm_Poly", prep_rls norm_Poly),
1419 ("Poly_erls",Poly_erls)(*FIXXXME:del with rls.rls'*),
1420 ("expand", prep_rls expand),
1421 ("expand_poly", prep_rls expand_poly),
1422 ("simplify_power", prep_rls simplify_power),
1423 ("order_add_mult", prep_rls order_add_mult),
1424 ("collect_numerals", prep_rls collect_numerals),
1425 ("collect_numerals_", prep_rls collect_numerals_),
1426 ("reduce_012", prep_rls reduce_012),
1427 ("discard_parentheses", prep_rls discard_parentheses),
1428 ("make_polynomial", prep_rls make_polynomial),
1429 ("expand_binoms", prep_rls expand_binoms),
1430 ("rev_rew_p", prep_rls rev_rew_p),
1431 ("discard_minus_", prep_rls discard_minus_),
1432 ("expand_poly_", prep_rls expand_poly_),
1433 ("expand_poly_rat_", prep_rls expand_poly_rat_),
1434 ("simplify_power_", prep_rls simplify_power_),
1435 ("calc_add_mult_pow_", prep_rls calc_add_mult_pow_),
1436 ("reduce_012_mult_", prep_rls reduce_012_mult_),
1437 ("reduce_012_", prep_rls reduce_012_),
1438 ("discard_parentheses_",prep_rls discard_parentheses_),
1439 ("order_mult_rls_", prep_rls order_mult_rls_),
1440 ("order_add_rls_", prep_rls order_add_rls_),
1441 ("make_rat_poly_with_parentheses",
1442 prep_rls make_rat_poly_with_parentheses)
1449 calclist':= overwritel (!calclist',
1450 [("is_polyrat_in", ("Poly.is'_polyrat'_in",
1451 eval_is_polyrat_in "#eval_is_polyrat_in")),
1452 ("is_expanded_in", ("Poly.is'_expanded'_in", eval_is_expanded_in "")),
1453 ("is_poly_in", ("Poly.is'_poly'_in", eval_is_poly_in "")),
1454 ("has_degree_in", ("Poly.has'_degree'_in", eval_has_degree_in "")),
1455 ("is_polyexp", ("Poly.is'_polyexp", eval_is_polyexp "")),
1456 ("is_multUnordered", ("Poly.is'_multUnordered", eval_is_multUnordered"")),
1457 ("is_addUnordered", ("Poly.is'_addUnordered", eval_is_addUnordered ""))
1464 (prep_pbt Poly.thy "pbl_simp_poly" [] e_pblID
1465 (["polynomial","simplification"],
1466 [("#Given" ,["term t_"]),
1467 ("#Where" ,["t_ is_polyexp"]),
1468 ("#Find" ,["normalform n_"])
1470 append_rls "e_rls" e_rls [(*for preds in where_*)
1471 Calc ("Poly.is'_polyexp", eval_is_polyexp "")],
1473 [["simplification","for_polynomials"]]));
1479 (prep_met Poly.thy "met_simp_poly" [] e_metID
1480 (["simplification","for_polynomials"],
1481 [("#Given" ,["term t_"]),
1482 ("#Where" ,["t_ is_polyexp"]),
1483 ("#Find" ,["normalform n_"])
1485 {rew_ord'="tless_true",
1489 prls = append_rls "simplification_for_polynomials_prls" e_rls
1490 [(*for preds in where_*)
1491 Calc ("Poly.is'_polyexp",eval_is_polyexp"")],
1492 crls = e_rls, nrls = norm_Poly},
1493 "Script SimplifyScript (t_::real) = \
1494 \ ((Rewrite_Set norm_Poly False) t_)"