1 (* WN.020812: theorems in the Reals,
2 necessary for special rule sets, in addition to Isabelle2002.
3 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
4 !!! THIS IS THE _least_ NUMBER OF ADDITIONAL THEOREMS !!!
5 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
6 xxxI contain ^^^ instead of ^ in the respective theorem xxx in 2002
7 changed by: Richard Lang 020912
10 theory Poly imports Simplify begin
14 is'_expanded'_in :: "[real, real] => bool" ("_ is'_expanded'_in _")
15 is'_poly'_in :: "[real, real] => bool" ("_ is'_poly'_in _") (*RL DA *)
16 has'_degree'_in :: "[real, real] => real" ("_ has'_degree'_in _")(*RL DA *)
17 is'_polyrat'_in :: "[real, real] => bool" ("_ is'_polyrat'_in _")(*RL030626*)
19 is'_multUnordered:: "real => bool" ("_ is'_multUnordered")
20 is'_addUnordered :: "real => bool" ("_ is'_addUnordered") (*WN030618*)
21 is'_polyexp :: "real => bool" ("_ is'_polyexp")
26 ("((Script Expand'_binoms (_ =))//
29 (*-------------------- rules------------------------------------------------*)
30 axioms (*.not contained in Isabelle2002,
31 stated as axioms, TODO: prove as theorems;
32 theorem-IDs 'xxxI' with ^^^ instead of ^ in 'xxx' in Isabelle2002.*)
34 realpow_pow: "(a ^^^ b) ^^^ c = a ^^^ (b * c)"
35 realpow_addI: "r ^^^ (n + m) = r ^^^ n * r ^^^ m"
36 realpow_addI_assoc_l: "r ^^^ n * (r ^^^ m * s) = r ^^^ (n + m) * s"
37 realpow_addI_assoc_r: "s * r ^^^ n * r ^^^ m = s * r ^^^ (n + m)"
39 realpow_oneI: "r ^^^ 1 = r"
40 realpow_zeroI: "r ^^^ 0 = 1"
41 realpow_eq_oneI: "1 ^^^ n = 1"
42 realpow_multI: "(r * s) ^^^ n = r ^^^ n * s ^^^ n"
43 realpow_multI_poly: "[| r is_polyexp; s is_polyexp |] ==>
44 (r * s) ^^^ n = r ^^^ n * s ^^^ n"
45 realpow_minus_oneI: "-1 ^^^ (2 * n) = 1"
47 realpow_twoI: "r ^^^ 2 = r * r"
48 realpow_twoI_assoc_l: "r * (r * s) = r ^^^ 2 * s"
49 realpow_twoI_assoc_r: "s * r * r = s * r ^^^ 2"
50 realpow_two_atom: "r is_atom ==> r * r = r ^^^ 2"
51 realpow_plus_1: "r * r ^^^ n = r ^^^ (n + 1)"
52 realpow_plus_1_assoc_l: "r * (r ^^^ m * s) = r ^^^ (1 + m) * s"
53 realpow_plus_1_assoc_l2: "r ^^^ m * (r * s) = r ^^^ (1 + m) * s"
54 realpow_plus_1_assoc_r: "s * r * r ^^^ m = s * r ^^^ (1 + m)"
55 realpow_plus_1_atom: "r is_atom ==> r * r ^^^ n = r ^^^ (1 + n)"
56 realpow_def_atom: "[| Not (r is_atom); 1 < n |]
57 ==> r ^^^ n = r * r ^^^ (n + -1)"
58 realpow_addI_atom: "r is_atom ==> r ^^^ n * r ^^^ m = r ^^^ (n + m)"
61 realpow_minus_even: "n is_even ==> (- r) ^^^ n = r ^^^ n"
62 realpow_minus_odd: "Not (n is_even) ==> (- r) ^^^ n = -1 * r ^^^ n"
66 real_pp_binom_times: "(a + b)*(c + d) = a*c + a*d + b*c + b*d"
67 real_pm_binom_times: "(a + b)*(c - d) = a*c - a*d + b*c - b*d"
68 real_mp_binom_times: "(a - b)*(c + d) = a*c + a*d - b*c - b*d"
69 real_mm_binom_times: "(a - b)*(c - d) = a*c - a*d - b*c + b*d"
70 real_plus_binom_pow3: "(a + b)^^^3 = a^^^3 + 3*a^^^2*b + 3*a*b^^^2 + b^^^3"
71 real_plus_binom_pow3_poly: "[| a is_polyexp; b is_polyexp |] ==>
72 (a + b)^^^3 = a^^^3 + 3*a^^^2*b + 3*a*b^^^2 + b^^^3"
73 real_minus_binom_pow3: "(a - b)^^^3 = a^^^3 - 3*a^^^2*b + 3*a*b^^^2 - b^^^3"
74 real_minus_binom_pow3_p: "(a + -1 * b)^^^3 = a^^^3 + -3*a^^^2*b + 3*a*b^^^2 +
76 (* real_plus_binom_pow: "[| n is_const; 3 < n |] ==>
77 (a + b)^^^n = (a + b) * (a + b)^^^(n - 1)" *)
78 real_plus_binom_pow4: "(a + b)^^^4 = (a^^^3 + 3*a^^^2*b + 3*a*b^^^2 + b^^^3)
80 real_plus_binom_pow4_poly: "[| a is_polyexp; b is_polyexp |] ==>
81 (a + b)^^^4 = (a^^^3 + 3*a^^^2*b + 3*a*b^^^2 + b^^^3)
83 real_plus_binom_pow5: "(a + b)^^^5 = (a^^^3 + 3*a^^^2*b + 3*a*b^^^2 + b^^^3)
84 *(a^^^2 + 2*a*b + b^^^2)"
85 real_plus_binom_pow5_poly: "[| a is_polyexp; b is_polyexp |] ==>
86 (a + b)^^^5 = (a^^^3 + 3*a^^^2*b + 3*a*b^^^2
87 + b^^^3)*(a^^^2 + 2*a*b + b^^^2)"
88 real_diff_plus: "a - b = a + -b" (*17.3.03: do_NOT_use*)
89 real_diff_minus: "a - b = a + -1 * b"
90 real_plus_binom_times: "(a + b)*(a + b) = a^^^2 + 2*a*b + b^^^2"
91 real_minus_binom_times: "(a - b)*(a - b) = a^^^2 - 2*a*b + b^^^2"
92 (*WN071229 changed for Schaerding -----vvv*)
93 (*real_plus_binom_pow2: "(a + b)^^^2 = a^^^2 + 2*a*b + b^^^2"*)
94 real_plus_binom_pow2: "(a + b)^^^2 = (a + b) * (a + b)"
95 (*WN071229 changed for Schaerding -----^^^*)
96 real_plus_binom_pow2_poly: "[| a is_polyexp; b is_polyexp |] ==>
97 (a + b)^^^2 = a^^^2 + 2*a*b + b^^^2"
98 real_minus_binom_pow2: "(a - b)^^^2 = a^^^2 - 2*a*b + b^^^2"
99 real_minus_binom_pow2_p: "(a - b)^^^2 = a^^^2 + -2*a*b + b^^^2"
100 real_plus_minus_binom1: "(a + b)*(a - b) = a^^^2 - b^^^2"
101 real_plus_minus_binom1_p: "(a + b)*(a - b) = a^^^2 + -1*b^^^2"
102 real_plus_minus_binom1_p_p: "(a + b)*(a + -1 * b) = a^^^2 + -1*b^^^2"
103 real_plus_minus_binom2: "(a - b)*(a + b) = a^^^2 - b^^^2"
104 real_plus_minus_binom2_p: "(a - b)*(a + b) = a^^^2 + -1*b^^^2"
105 real_plus_minus_binom2_p_p: "(a + -1 * b)*(a + b) = a^^^2 + -1*b^^^2"
106 real_plus_binom_times1: "(a + 1*b)*(a + -1*b) = a^^^2 + -1*b^^^2"
107 real_plus_binom_times2: "(a + -1*b)*(a + 1*b) = a^^^2 + -1*b^^^2"
109 real_num_collect: "[| l is_const; m is_const |] ==>
110 l * n + m * n = (l + m) * n"
111 (* FIXME.MG.0401: replace 'real_num_collect_assoc'
112 by 'real_num_collect_assoc_l' ... are equal, introduced by MG ! *)
113 real_num_collect_assoc: "[| l is_const; m is_const |] ==>
114 l * n + (m * n + k) = (l + m) * n + k"
115 real_num_collect_assoc_l: "[| l is_const; m is_const |] ==>
116 l * n + (m * n + k) = (l + m)
118 real_num_collect_assoc_r: "[| l is_const; m is_const |] ==>
119 (k + m * n) + l * n = k + (l + m) * n"
120 real_one_collect: "m is_const ==> n + m * n = (1 + m) * n"
121 (* FIXME.MG.0401: replace 'real_one_collect_assoc'
122 by 'real_one_collect_assoc_l' ... are equal, introduced by MG ! *)
123 real_one_collect_assoc: "m is_const ==> n + (m * n + k) = (1 + m)* n + k"
125 real_one_collect_assoc_l: "m is_const ==> n + (m * n + k) = (1 + m) * n + k"
126 real_one_collect_assoc_r: "m is_const ==> (k + n) + m * n = k + (1 + m) * n"
128 (* FIXME.MG.0401: replace 'real_mult_2_assoc'
129 by 'real_mult_2_assoc_l' ... are equal, introduced by MG ! *)
130 real_mult_2_assoc: "z1 + (z1 + k) = 2 * z1 + k"
131 real_mult_2_assoc_l: "z1 + (z1 + k) = 2 * z1 + k"
132 real_mult_2_assoc_r: "(k + z1) + z1 = k + 2 * z1"
134 real_add_mult_distrib_poly: "w is_polyexp ==> (z1 + z2) * w = z1 * w + z2 * w"
135 real_add_mult_distrib2_poly:"w is_polyexp ==> w * (z1 + z2) = w * z1 + w * z2"
137 text {* remark on 'polynomials'
139 *** there are 5 kinds of expanded normalforms ***
141 [1] 'complete polynomial' (Komplettes Polynom), univariate
142 a_0 + a_1.x^1 +...+ a_n.x^n not (a_n = 0)
143 not (a_n = 0), some a_i may be zero (DON'T disappear),
144 variables in monomials lexicographically ordered and complete,
145 x written as 1*x^1, ...
146 [2] 'polynomial' (Polynom), univariate and multivariate
147 a_0 + a_1.x +...+ a_n.x^n not (a_n = 0)
148 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
149 not (a_n = 0), some a_i may be zero (ie. monomials disappear),
150 exponents and coefficients equal 1 are not (WN060904.TODO in cancel_p_)shown,
151 and variables in monomials are lexicographically ordered
152 examples: [1]: "1 + (-10) * x ^^^ 1 + 25 * x ^^^ 2"
153 [1]: "11 + 0 * x ^^^ 1 + 1 * x ^^^ 2"
154 [2]: "x + (-50) * x ^^^ 3"
155 [2]: "(-1) * x * y ^^^ 2 + 7 * x ^^^ 3"
157 [3] 'expanded_term' (Ausmultiplizierter Term):
158 pull out unary minus to binary minus,
159 as frequently exercised in schools; other conditions for [2] hold however
160 examples: "a ^^^ 2 - 2 * a * b + b ^^^ 2"
161 "4 * x ^^^ 2 - 9 * y ^^^ 2"
162 [4] 'polynomial_in' (Polynom in):
163 polynomial in 1 variable with arbitrary coefficients
164 examples: "2 * x + (-50) * x ^^^ 3" (poly in x)
165 "(u + v) + (2 * u ^^^ 2) * a + (-u) * a ^^^ 2 (poly in a)
166 [5] 'expanded_in' (Ausmultiplizierter Termin in):
167 analoguous to [3] with binary minus like [3]
168 examples: "2 * x - 50 * x ^^^ 3" (expanded in x)
169 "(u + v) + (2 * u ^^^ 2) * a - u * a ^^^ 2 (expanded in a)
175 (* is_polyrat_in becomes true, if no bdv is in the denominator of a fraction*)
176 fun is_polyrat_in t v =
177 let fun coeff_in c v = member op = (vars c) v;
178 fun finddivide (_ $ _ $ _ $ _) v = raise error("is_polyrat_in:")
179 (* at the moment there is no term like this, but ....*)
180 | finddivide (t as (Const ("HOL.divide",_) $ _ $ b)) v =
182 | finddivide (_ $ t1 $ t2) v =
183 (finddivide t1 v) orelse (finddivide t2 v)
184 | finddivide (_ $ t1) v = (finddivide t1 v)
185 | finddivide _ _ = false;
186 in finddivide t v end;
188 fun eval_is_polyrat_in _ _(p as (Const ("Poly.is'_polyrat'_in",_) $ t $ v)) _ =
190 then SOME ((term2str p) ^ " = True",
191 Trueprop $ (mk_equality (p, HOLogic.true_const)))
192 else SOME ((term2str p) ^ " = True",
193 Trueprop $ (mk_equality (p, HOLogic.false_const)))
194 | eval_is_polyrat_in _ _ _ _ = ((*writeln"### no matches";*) NONE);
197 (*.a 'c is coefficient of v' if v does NOT occur in c.*)
198 fun coeff_in c v = not (member op = (vars c) v);
199 (* FIXME.WN100826 shift this into test--------------
200 val v = (term_of o the o (parse thy)) "x";
201 val t = (term_of o the o (parse thy)) "1";
203 (*val it = true : bool*)
204 val t = (term_of o the o (parse thy)) "a*b+c";
206 (*val it = true : bool*)
207 val t = (term_of o the o (parse thy)) "a*x+c";
209 (*val it = false : bool*)
210 ----------------------------------------------------*)
211 (*. a 'monomial t in variable v' is a term t with
212 either (1) v NOT existent in t, or (2) v contained in t,
214 if (2) then v is a factor on the very right, ev. with exponent.*)
215 fun factor_right_deg (*case 2*)
216 (t as Const ("op *",_) $ t1 $
217 (Const ("Atools.pow",_) $ vv $ Free (d,_))) v =
218 if ((vv = v) andalso (coeff_in t1 v)) then SOME (int_of_str' d) else NONE
219 | factor_right_deg (t as Const ("Atools.pow",_) $ vv $ Free (d,_)) v =
220 if (vv = v) then SOME (int_of_str' d) else NONE
221 | factor_right_deg (t as Const ("op *",_) $ t1 $ vv) v =
222 if ((vv = v) andalso (coeff_in t1 v))then SOME 1 else NONE
223 | factor_right_deg vv v =
224 if (vv = v) then SOME 1 else NONE;
225 fun mono_deg_in m v =
226 if coeff_in m v then (*case 1*) SOME 0
227 else factor_right_deg m v;
228 (* FIXME.WN100826 shift this into test-----------------------------
229 val v = (term_of o the o (parse thy)) "x";
230 val t = (term_of o the o (parse thy)) "(a*b+c)*x^^^7";
233 val t = (term_of o the o (parse thy)) "x^^^7";
236 val t = (term_of o the o (parse thy)) "(a*b+c)*x";
239 val t = (term_of o the o (parse thy)) "(a*b+x)*x";
242 val t = (term_of o the o (parse thy)) "x";
245 val t = (term_of o the o (parse thy)) "(a*b+c)";
248 val t = (term_of o the o (parse thy)) "ab - (a*b)*x";
251 ------------------------------------------------------------------*)
252 fun expand_deg_in t v =
253 let fun edi ~1 ~1 (Const ("op +",_) $ t1 $ t2) =
254 (case mono_deg_in t2 v of (* $ is left associative*)
255 SOME d' => edi d' d' t1
257 | edi ~1 ~1 (Const ("op -",_) $ t1 $ t2) =
258 (case mono_deg_in t2 v of
259 SOME d' => edi d' d' t1
261 | edi d dmax (Const ("op -",_) $ t1 $ t2) =
262 (case mono_deg_in t2 v of
263 (*RL orelse ((d=0) andalso (d'=0)) need to handle 3+4-...4 +x*)
264 SOME d' => if ((d > d') orelse ((d=0) andalso (d'=0)))
265 then edi d' dmax t1 else NONE
267 | edi d dmax (Const ("op +",_) $ t1 $ t2) =
268 (case mono_deg_in t2 v of
269 (*RL orelse ((d=0) andalso (d'=0)) need to handle 3+4-...4 +x*)
270 SOME d' => if ((d > d') orelse ((d=0) andalso (d'=0)))
271 then edi d' dmax t1 else NONE
273 | edi ~1 ~1 t = (case mono_deg_in t v of
276 | edi d dmax t = (*basecase last*)
277 (case mono_deg_in t v of
278 SOME d' => if ((d > d') orelse ((d=0) andalso (d'=0)))
279 then SOME dmax else NONE
282 (* FIXME.WN100826 shift this into test-----------------------------
283 val v = (term_of o the o (parse thy)) "x";
284 val t = (term_of o the o (parse thy)) "a+b";
287 val t = (term_of o the o (parse thy)) "(a+b)*x";
290 val t = (term_of o the o (parse thy)) "a*b - (a+b)*x";
293 val t = (term_of o the o (parse thy)) "a*b + (a-b)*x";
296 val t = (term_of o the o (parse thy)) "a*b + (a+b)*x + x^^^2";
298 -------------------------------------------------------------------*)
299 fun poly_deg_in t v =
300 let fun edi ~1 ~1 (Const ("op +",_) $ t1 $ t2) =
301 (case mono_deg_in t2 v of (* $ is left associative*)
302 SOME d' => edi d' d' t1
304 | edi d dmax (Const ("op +",_) $ t1 $ t2) =
305 (case mono_deg_in t2 v of
306 (*RL orelse ((d=0) andalso (d'=0)) need to handle 3+4-...4 +x*)
307 SOME d' => if ((d > d') orelse ((d=0) andalso (d'=0)))
308 then edi d' dmax t1 else NONE
310 | edi ~1 ~1 t = (case mono_deg_in t v of
313 | edi d dmax t = (*basecase last*)
314 (case mono_deg_in t v of
315 SOME d' => if ((d > d') orelse ((d=0) andalso (d'=0)))
316 then SOME dmax else NONE
321 fun is_expanded_in t v =
322 case expand_deg_in t v of SOME _ => true | NONE => false;
324 case poly_deg_in t v of SOME _ => true | NONE => false;
325 fun has_degree_in t v =
326 case expand_deg_in t v of SOME d => d | NONE => ~1;
328 (* FIXME.WN100826 shift this into test-----------------------------
329 val v = (term_of o the o (parse thy)) "x";
330 val t = (term_of o the o (parse thy)) "a*b - (a+b)*x + x^^^2";
333 val t = (term_of o the o (parse thy)) "-8 - 2*x + x^^^2";
336 val t = (term_of o the o (parse thy)) "6 + 13*x + 6*x^^^2";
339 -------------------------------------------------------------------*)
341 (*("is_expanded_in", ("Poly.is'_expanded'_in", eval_is_expanded_in ""))*)
342 fun eval_is_expanded_in _ _
343 (p as (Const ("Poly.is'_expanded'_in",_) $ t $ v)) _ =
344 if is_expanded_in t v
345 then SOME ((term2str p) ^ " = True",
346 Trueprop $ (mk_equality (p, HOLogic.true_const)))
347 else SOME ((term2str p) ^ " = True",
348 Trueprop $ (mk_equality (p, HOLogic.false_const)))
349 | eval_is_expanded_in _ _ _ _ = NONE;
351 val t = (term_of o the o (parse thy)) "(-8 - 2*x + x^^^2) is_expanded_in x";
352 val SOME (id, t') = eval_is_expanded_in 0 0 t 0;
353 (*val id = "Poly.is'_expanded'_in (-8 - 2 * x + x ^^^ 2) x = True"*)
355 (*val it = "Poly.is'_expanded'_in (-8 - 2 * x + x ^^^ 2) x = True"*)
358 (*("is_poly_in", ("Poly.is'_poly'_in", eval_is_poly_in ""))*)
359 fun eval_is_poly_in _ _
360 (p as (Const ("Poly.is'_poly'_in",_) $ t $ v)) _ =
362 then SOME ((term2str p) ^ " = True",
363 Trueprop $ (mk_equality (p, HOLogic.true_const)))
364 else SOME ((term2str p) ^ " = True",
365 Trueprop $ (mk_equality (p, HOLogic.false_const)))
366 | eval_is_poly_in _ _ _ _ = NONE;
368 val t = (term_of o the o (parse thy)) "(8 + 2*x + x^^^2) is_poly_in x";
369 val SOME (id, t') = eval_is_poly_in 0 0 t 0;
370 (*val id = "Poly.is'_poly'_in (8 + 2 * x + x ^^^ 2) x = True"*)
372 (*val it = "Poly.is'_poly'_in (8 + 2 * x + x ^^^ 2) x = True"*)
375 (*("has_degree_in", ("Poly.has'_degree'_in", eval_has_degree_in ""))*)
376 fun eval_has_degree_in _ _
377 (p as (Const ("Poly.has'_degree'_in",_) $ t $ v)) _ =
378 let val d = has_degree_in t v
379 val d' = term_of_num HOLogic.realT d
380 in SOME ((term2str p) ^ " = " ^ (string_of_int d),
381 Trueprop $ (mk_equality (p, d')))
383 | eval_has_degree_in _ _ _ _ = NONE;
385 > val t = (term_of o the o (parse thy)) "(-8 - 2*x + x^^^2) has_degree_in x";
386 > val SOME (id, t') = eval_has_degree_in 0 0 t 0;
387 val id = "Poly.has'_degree'_in (-8 - 2 * x + x ^^^ 2) x = 2" : string
389 val it = "Poly.has'_degree'_in (-8 - 2 * x + x ^^^ 2) x = 2" : string
394 append_rls "calculate_PolyFIXXXME.not.impl." e_rls
397 (*.for evaluation of conditions in rewrite rules.*)
399 append_rls "Poly_erls" Atools_erls
400 [ Calc ("op =",eval_equal "#equal_"),
401 Thm ("real_unari_minus",num_str @{thm real_unari_minus}),
402 Calc ("op +",eval_binop "#add_"),
403 Calc ("op -",eval_binop "#sub_"),
404 Calc ("op *",eval_binop "#mult_"),
405 Calc ("Atools.pow" ,eval_binop "#power_")
409 append_rls "poly_crls" Atools_crls
410 [ Calc ("op =",eval_equal "#equal_"),
411 Thm ("real_unari_minus",num_str @{thm real_unari_minus}),
412 Calc ("op +",eval_binop "#add_"),
413 Calc ("op -",eval_binop "#sub_"),
414 Calc ("op *",eval_binop "#mult_"),
415 Calc ("Atools.pow" ,eval_binop "#power_")
418 local (*. for make_polynomial .*)
420 open Term; (* for type order = EQUAL | LESS | GREATER *)
422 fun pr_ord EQUAL = "EQUAL"
423 | pr_ord LESS = "LESS"
424 | pr_ord GREATER = "GREATER";
426 fun dest_hd' (Const (a, T)) = (* ~ term.ML *)
428 "Atools.pow" => ((("|||||||||||||", 0), T), 0) (*WN greatest string*)
429 | _ => (((a, 0), T), 0))
430 | dest_hd' (Free (a, T)) = (((a, 0), T), 1)
431 | dest_hd' (Var v) = (v, 2)
432 | dest_hd' (Bound i) = ((("", i), dummyT), 3)
433 | dest_hd' (Abs (_, T, _)) = ((("", 0), T), 4);
435 fun get_order_pow (t $ (Free(order,_))) = (* RL FIXXXME:geht zufaellig?WN*)
436 (case int_of_str (order) of
439 | get_order_pow _ = 0;
441 fun size_of_term' (Const(str,_) $ t) =
442 if "Atools.pow"= str then 1000 + size_of_term' t else 1+size_of_term' t(*WN*)
443 | size_of_term' (Abs (_,_,body)) = 1 + size_of_term' body
444 | size_of_term' (f$t) = size_of_term' f + size_of_term' t
445 | size_of_term' _ = 1;
447 fun term_ord' pr thy (Abs (_, T, t), Abs(_, U, u)) = (* ~ term.ML *)
448 (case term_ord' pr thy (t, u) of EQUAL => Term_Ord.typ_ord (T, U) | ord => ord)
449 | term_ord' pr thy (t, u) =
452 val (f, ts) = strip_comb t and (g, us) = strip_comb u;
453 val _=writeln("t= f@ts= \""^
454 ((Syntax.string_of_term (thy2ctxt thy)) f)^"\" @ \"["^
455 (commas(map(Syntax.string_of_term (thy2ctxt thy))ts))^"]\"");
456 val _=writeln("u= g@us= \""^
457 ((Syntax.string_of_term (thy2ctxt thy)) g)^"\" @ \"["^
458 (commas(map(Syntax.string_of_term (thy2ctxt thy))us))^"]\"");
459 val _=writeln("size_of_term(t,u)= ("^
460 (string_of_int(size_of_term' t))^", "^
461 (string_of_int(size_of_term' u))^")");
462 val _=writeln("hd_ord(f,g) = "^((pr_ord o hd_ord)(f,g)));
463 val _=writeln("terms_ord(ts,us) = "^
464 ((pr_ord o terms_ord str false)(ts,us)));
465 val _=writeln("-------");
468 case int_ord (size_of_term' t, size_of_term' u) of
470 let val (f, ts) = strip_comb t and (g, us) = strip_comb u in
471 (case hd_ord (f, g) of EQUAL => (terms_ord str pr) (ts, us)
475 and hd_ord (f, g) = (* ~ term.ML *)
476 prod_ord (prod_ord Term_Ord.indexname_ord Term_Ord.typ_ord) int_ord (dest_hd' f, dest_hd' g)
477 and terms_ord str pr (ts, us) =
478 list_ord (term_ord' pr (assoc_thy "Isac.thy"))(ts, us);
481 fun ord_make_polynomial (pr:bool) thy (_:subst) tu =
482 (term_ord' pr thy(***) tu = LESS );
487 rew_ord' := overwritel (!rew_ord',
488 [("termlessI", termlessI),
489 ("ord_make_polynomial", ord_make_polynomial false thy)
494 Rls{id = "expand", preconds = [], rew_ord = ("dummy_ord", dummy_ord),
495 erls = e_rls,srls = Erls, calc = [],
496 rules = [Thm ("left_distrib" ,num_str @{thm left_distrib}),
497 (*"(z1.0 + z2.0) * w = z1.0 * w + z2.0 * w"*)
498 Thm ("right_distrib",num_str @{thm right_distrib})
499 (*"w * (z1.0 + z2.0) = w * z1.0 + w * z2.0"*)
500 ], scr = EmptyScr}:rls;
502 (*----------------- Begin: rulesets for make_polynomial_ -----------------
503 'rlsIDs' redefined by MG as 'rlsIDs_'
507 Rls{id = "discard_minus1", preconds = [],
508 rew_ord = ("dummy_ord", dummy_ord),
509 erls = e_rls,srls = Erls,
512 rules = [Thm ("real_diff_minus",num_str @{thm real_diff_minus}),
513 (*"a - b = a + -1 * b"*)
514 Thm ("sym_real_mult_minus1",
515 num_str (@{thm real_mult_minus1} RS @{thm sym}))
517 ], scr = EmptyScr}:rls;
519 Rls{id = "expand_poly_", preconds = [],
520 rew_ord = ("dummy_ord", dummy_ord),
521 erls = e_rls,srls = Erls,
524 rules = [Thm ("real_plus_binom_pow4",num_str @{thm real_plus_binom_pow4}),
525 (*"(a + b)^^^4 = ... "*)
526 Thm ("real_plus_binom_pow5",num_str @{thm real_plus_binom_pow5}),
527 (*"(a + b)^^^5 = ... "*)
528 Thm ("real_plus_binom_pow3",num_str @{thm real_plus_binom_pow3}),
529 (*"(a + b)^^^3 = a^^^3 + 3*a^^^2*b + 3*a*b^^^2 + b^^^3" *)
530 (*WN071229 changed/removed for Schaerding -----vvv*)
531 (*Thm ("real_plus_binom_pow2",num_str @{thm real_plus_binom_pow2}),*)
532 (*"(a + b)^^^2 = a^^^2 + 2*a*b + b^^^2"*)
533 Thm ("real_plus_binom_pow2",num_str @{thm real_plus_binom_pow2}),
534 (*"(a + b)^^^2 = (a + b) * (a + b)"*)
535 (*Thm ("real_plus_minus_binom1_p_p",
536 num_str @{thm real_plus_minus_binom1_p_p}),*)
537 (*"(a + b)*(a + -1 * b) = a^^^2 + -1*b^^^2"*)
538 (*Thm ("real_plus_minus_binom2_p_p",
539 num_str @{thm real_plus_minus_binom2_p_p}),*)
540 (*"(a + -1 * b)*(a + b) = a^^^2 + -1*b^^^2"*)
541 (*WN071229 changed/removed for Schaerding -----^^^*)
543 Thm ("left_distrib" ,num_str @{thm left_distrib}),
544 (*"(z1.0 + z2.0) * w = z1.0 * w + z2.0 * w"*)
545 Thm ("right_distrib",num_str @{thm right_distrib}),
546 (*"w * (z1.0 + z2.0) = w * z1.0 + w * z2.0"*)
548 Thm ("realpow_multI", num_str @{thm realpow_multI}),
549 (*"(r * s) ^^^ n = r ^^^ n * s ^^^ n"*)
550 Thm ("realpow_pow",num_str @{thm realpow_pow})
551 (*"(a ^^^ b) ^^^ c = a ^^^ (b * c)"*)
552 ], scr = EmptyScr}:rls;
554 (*.the expression contains + - * ^ only ?
555 this is weaker than 'is_polynomial' !.*)
556 fun is_polyexp (Free _) = true
557 | is_polyexp (Const ("op +",_) $ Free _ $ Free _) = true
558 | is_polyexp (Const ("op -",_) $ Free _ $ Free _) = true
559 | is_polyexp (Const ("op *",_) $ Free _ $ Free _) = true
560 | is_polyexp (Const ("Atools.pow",_) $ Free _ $ Free _) = true
561 | is_polyexp (Const ("op +",_) $ t1 $ t2) =
562 ((is_polyexp t1) andalso (is_polyexp t2))
563 | is_polyexp (Const ("op -",_) $ t1 $ t2) =
564 ((is_polyexp t1) andalso (is_polyexp t2))
565 | is_polyexp (Const ("op *",_) $ t1 $ t2) =
566 ((is_polyexp t1) andalso (is_polyexp t2))
567 | is_polyexp (Const ("Atools.pow",_) $ t1 $ t2) =
568 ((is_polyexp t1) andalso (is_polyexp t2))
569 | is_polyexp _ = false;
571 (*("is_polyexp", ("Poly.is'_polyexp", eval_is_polyexp ""))*)
572 fun eval_is_polyexp (thmid:string) _
573 (t as (Const("Poly.is'_polyexp", _) $ arg)) thy =
575 then SOME (mk_thmid thmid ""
576 ((Syntax.string_of_term (thy2ctxt thy)) arg) "",
577 Trueprop $ (mk_equality (t, HOLogic.true_const)))
578 else SOME (mk_thmid thmid ""
579 ((Syntax.string_of_term (thy2ctxt thy)) arg) "",
580 Trueprop $ (mk_equality (t, HOLogic.false_const)))
581 | eval_is_polyexp _ _ _ _ = NONE;
583 val expand_poly_rat_ =
584 Rls{id = "expand_poly_rat_", preconds = [],
585 rew_ord = ("dummy_ord", dummy_ord),
586 erls = append_rls "e_rls-is_polyexp" e_rls
587 [Calc ("Poly.is'_polyexp", eval_is_polyexp "")
593 [Thm ("real_plus_binom_pow4_poly", num_str @{thm real_plus_binom_pow4_poly}),
594 (*"[| a is_polyexp; b is_polyexp |] ==> (a + b)^^^4 = ... "*)
595 Thm ("real_plus_binom_pow5_poly", num_str @{thm real_plus_binom_pow5_poly}),
596 (*"[| a is_polyexp; b is_polyexp |] ==> (a + b)^^^5 = ... "*)
597 Thm ("real_plus_binom_pow2_poly",num_str @{thm real_plus_binom_pow2_poly}),
598 (*"[| a is_polyexp; b is_polyexp |] ==>
599 (a + b)^^^2 = a^^^2 + 2*a*b + b^^^2"*)
600 Thm ("real_plus_binom_pow3_poly",num_str @{thm real_plus_binom_pow3_poly}),
601 (*"[| a is_polyexp; b is_polyexp |] ==>
602 (a + b)^^^3 = a^^^3 + 3*a^^^2*b + 3*a*b^^^2 + b^^^3" *)
603 Thm ("real_plus_minus_binom1_p_p",num_str @{thm real_plus_minus_binom1_p_p}),
604 (*"(a + b)*(a + -1 * b) = a^^^2 + -1*b^^^2"*)
605 Thm ("real_plus_minus_binom2_p_p",num_str @{thm real_plus_minus_binom2_p_p}),
606 (*"(a + -1 * b)*(a + b) = a^^^2 + -1*b^^^2"*)
608 Thm ("real_add_mult_distrib_poly",
609 num_str @{thm real_add_mult_distrib_poly}),
610 (*"w is_polyexp ==> (z1.0 + z2.0) * w = z1.0 * w + z2.0 * w"*)
611 Thm("real_add_mult_distrib2_poly",
612 num_str @{thm real_add_mult_distrib2_poly}),
613 (*"w is_polyexp ==> w * (z1.0 + z2.0) = w * z1.0 + w * z2.0"*)
615 Thm ("realpow_multI_poly", num_str @{thm realpow_multI_poly}),
616 (*"[| r is_polyexp; s is_polyexp |] ==>
617 (r * s) ^^^ n = r ^^^ n * s ^^^ n"*)
618 Thm ("realpow_pow",num_str @{thm realpow_pow})
619 (*"(a ^^^ b) ^^^ c = a ^^^ (b * c)"*)
620 ], scr = EmptyScr}:rls;
622 val simplify_power_ =
623 Rls{id = "simplify_power_", preconds = [],
624 rew_ord = ("dummy_ord", dummy_ord),
625 erls = e_rls, srls = Erls,
628 rules = [(*MG: Reihenfolge der folgenden 2 Thm muss so bleiben, wegen
629 a*(a*a) --> a*a^^^2 und nicht a*(a*a) --> a^^^2*a *)
630 Thm ("sym_realpow_twoI",
631 num_str (@{thm realpow_twoI} RS @{thm sym})),
632 (*"r * r = r ^^^ 2"*)
633 Thm ("realpow_twoI_assoc_l",num_str @{thm realpow_twoI_assoc_l}),
634 (*"r * (r * s) = r ^^^ 2 * s"*)
636 Thm ("realpow_plus_1",num_str @{thm realpow_plus_1}),
637 (*"r * r ^^^ n = r ^^^ (n + 1)"*)
638 Thm ("realpow_plus_1_assoc_l",
639 num_str @{thm realpow_plus_1_assoc_l}),
640 (*"r * (r ^^^ m * s) = r ^^^ (1 + m) * s"*)
641 (*MG 9.7.03: neues Thm wegen a*(a*(a*b)) --> a^^^2*(a*b) *)
642 Thm ("realpow_plus_1_assoc_l2",
643 num_str @{thm realpow_plus_1_assoc_l2}),
644 (*"r ^^^ m * (r * s) = r ^^^ (1 + m) * s"*)
646 Thm ("sym_realpow_addI",
647 num_str (@{thm realpow_addI} RS @{thm sym})),
648 (*"r ^^^ n * r ^^^ m = r ^^^ (n + m)"*)
649 Thm ("realpow_addI_assoc_l",num_str @{thm realpow_addI_assoc_l}),
650 (*"r ^^^ n * (r ^^^ m * s) = r ^^^ (n + m) * s"*)
652 (* ist in expand_poly - wird hier aber auch gebraucht, wegen:
653 "r * r = r ^^^ 2" wenn r=a^^^b*)
654 Thm ("realpow_pow",num_str @{thm realpow_pow})
655 (*"(a ^^^ b) ^^^ c = a ^^^ (b * c)"*)
656 ], scr = EmptyScr}:rls;
658 val calc_add_mult_pow_ =
659 Rls{id = "calc_add_mult_pow_", preconds = [],
660 rew_ord = ("dummy_ord", dummy_ord),
661 erls = Atools_erls(*erls3.4.03*),srls = Erls,
662 calc = [("PLUS" , ("op +", eval_binop "#add_")),
663 ("TIMES" , ("op *", eval_binop "#mult_")),
664 ("POWER", ("Atools.pow", eval_binop "#power_"))
667 rules = [Calc ("op +", eval_binop "#add_"),
668 Calc ("op *", eval_binop "#mult_"),
669 Calc ("Atools.pow", eval_binop "#power_")
670 ], scr = EmptyScr}:rls;
672 val reduce_012_mult_ =
673 Rls{id = "reduce_012_mult_", preconds = [],
674 rew_ord = ("dummy_ord", dummy_ord),
675 erls = e_rls,srls = Erls,
678 rules = [(* MG: folgende Thm müssen hier stehen bleiben: *)
679 Thm ("mult_1_right",num_str @{thm mult_1_right}),
680 (*"z * 1 = z"*) (*wegen "a * b * b^^^(-1) + a"*)
681 Thm ("realpow_zeroI",num_str @{thm realpow_zeroI}),
682 (*"r ^^^ 0 = 1"*) (*wegen "a*a^^^(-1)*c + b + c"*)
683 Thm ("realpow_oneI",num_str @{thm realpow_oneI}),
685 Thm ("realpow_eq_oneI",num_str @{thm realpow_eq_oneI})
687 ], scr = EmptyScr}:rls;
689 val collect_numerals_ =
690 Rls{id = "collect_numerals_", preconds = [],
691 rew_ord = ("dummy_ord", dummy_ord),
692 erls = Atools_erls, srls = Erls,
693 calc = [("PLUS" , ("op +", eval_binop "#add_"))
696 [Thm ("real_num_collect",num_str @{thm real_num_collect}),
697 (*"[| l is_const; m is_const |]==>l * n + m * n = (l + m) * n"*)
698 Thm ("real_num_collect_assoc_r",num_str @{thm real_num_collect_assoc_r}),
699 (*"[| l is_const; m is_const |] ==> \
700 \(k + m * n) + l * n = k + (l + m)*n"*)
701 Thm ("real_one_collect",num_str @{thm real_one_collect}),
702 (*"m is_const ==> n + m * n = (1 + m) * n"*)
703 Thm ("real_one_collect_assoc_r",num_str @{thm real_one_collect_assoc_r}),
704 (*"m is_const ==> (k + n) + m * n = k + (m + 1) * n"*)
706 Calc ("op +", eval_binop "#add_"),
708 (*MG: Reihenfolge der folgenden 2 Thm muss so bleiben, wegen
709 (a+a)+a --> a + 2*a --> 3*a and not (a+a)+a --> 2*a + a *)
710 Thm ("real_mult_2_assoc_r",num_str @{thm real_mult_2_assoc_r}),
711 (*"(k + z1) + z1 = k + 2 * z1"*)
712 Thm ("sym_real_mult_2",num_str (@{thm real_mult_2} RS @{thm sym}))
713 (*"z1 + z1 = 2 * z1"*)
714 ], scr = EmptyScr}:rls;
717 Rls{id = "reduce_012_", preconds = [],
718 rew_ord = ("dummy_ord", dummy_ord),
719 erls = e_rls,srls = Erls, calc = [],
720 rules = [Thm ("mult_1_left",num_str @{thm mult_1_left}),
722 Thm ("mult_zero_left",num_str @{thm mult_zero_left}),
724 Thm ("mult_zero_right",num_str @{thm mult_zero_right}),
726 Thm ("add_0_left",num_str @{thm add_0_left}),
728 Thm ("add_0_right",num_str @{thm add_0_right}),
729 (*"z + 0 = z"*) (*wegen a+b-b --> a+(1-1)*b --> a+0 --> a*)
731 (*Thm ("realpow_oneI",num_str @{thm realpow_oneI})*)
733 Thm ("divide_zero_left",num_str @{thm divide_zero_left})(*WN060914*)
735 ], scr = EmptyScr}:rls;
737 (*ein Hilfs-'ruleset' (benutzt das leere 'ruleset')*)
738 val discard_parentheses1 =
739 append_rls "discard_parentheses1" e_rls
740 [Thm ("sym_real_mult_assoc",
741 num_str (@{thm real_mult_assoc} RS @{thm sym}))
742 (*"?z1.1 * (?z2.1 * ?z3.1) = ?z1.1 * ?z2.1 * ?z3.1"*)
743 (*Thm ("sym_add_assoc",
744 num_str (@{thm add_assoc} RS @{thm sym}))*)
745 (*"?z1.1 + (?z2.1 + ?z3.1) = ?z1.1 + ?z2.1 + ?z3.1"*)
748 (*----------------- End: rulesets for make_polynomial_ -----------------*)
750 (*MG.0401 ev. for use in rls with ordered rewriting ?
751 val collect_numerals_left =
752 Rls{id = "collect_numerals", preconds = [],
753 rew_ord = ("dummy_ord", dummy_ord),
754 erls = Atools_erls(*erls3.4.03*),srls = Erls,
755 calc = [("PLUS" , ("op +", eval_binop "#add_")),
756 ("TIMES" , ("op *", eval_binop "#mult_")),
757 ("POWER", ("Atools.pow", eval_binop "#power_"))
760 rules = [Thm ("real_num_collect",num_str @{thm real_num_collect}),
761 (*"[| l is_const; m is_const |]==>l * n + m * n = (l + m) * n"*)
762 Thm ("real_num_collect_assoc",num_str @{thm real_num_collect_assoc}),
763 (*"[| l is_const; m is_const |] ==>
764 l * n + (m * n + k) = (l + m) * n + k"*)
765 Thm ("real_one_collect",num_str @{thm real_one_collect}),
766 (*"m is_const ==> n + m * n = (1 + m) * n"*)
767 Thm ("real_one_collect_assoc",num_str @{thm real_one_collect_assoc}),
768 (*"m is_const ==> n + (m * n + k) = (1 + m) * n + k"*)
770 Calc ("op +", eval_binop "#add_"),
772 (*MG am 2.5.03: 2 Theoreme aus reduce_012 hierher verschoben*)
773 Thm ("sym_real_mult_2",
774 num_str (@{thm real_mult_2} RS @{thm sym})),
775 (*"z1 + z1 = 2 * z1"*)
776 Thm ("real_mult_2_assoc",num_str @{thm real_mult_2_assoc})
777 (*"z1 + (z1 + k) = 2 * z1 + k"*)
778 ], scr = EmptyScr}:rls;*)
781 Rls{id = "expand_poly", preconds = [],
782 rew_ord = ("dummy_ord", dummy_ord),
783 erls = e_rls,srls = Erls,
786 rules = [Thm ("left_distrib" ,num_str @{thm left_distrib}),
787 (*"(z1.0 + z2.0) * w = z1.0 * w + z2.0 * w"*)
788 Thm ("right_distrib",num_str @{thm right_distrib}),
789 (*"w * (z1.0 + z2.0) = w * z1.0 + w * z2.0"*)
790 (*Thm ("left_distrib1",num_str @{thm left_distrib}1),
791 ....... 18.3.03 undefined???*)
793 Thm ("real_plus_binom_pow2",num_str @{thm real_plus_binom_pow2}),
794 (*"(a + b)^^^2 = a^^^2 + 2*a*b + b^^^2"*)
795 Thm ("real_minus_binom_pow2_p",num_str @{thm real_minus_binom_pow2_p}),
796 (*"(a - b)^^^2 = a^^^2 + -2*a*b + b^^^2"*)
797 Thm ("real_plus_minus_binom1_p",
798 num_str @{thm real_plus_minus_binom1_p}),
799 (*"(a + b)*(a - b) = a^^^2 + -1*b^^^2"*)
800 Thm ("real_plus_minus_binom2_p",
801 num_str @{thm real_plus_minus_binom2_p}),
802 (*"(a - b)*(a + b) = a^^^2 + -1*b^^^2"*)
804 Thm ("minus_minus",num_str @{thm minus_minus}),
806 Thm ("real_diff_minus",num_str @{thm real_diff_minus}),
807 (*"a - b = a + -1 * b"*)
808 Thm ("sym_real_mult_minus1",
809 num_str (@{thm real_mult_minus1} RS @{thm sym}))
812 (*Thm ("real_minus_add_distrib",
813 num_str @{thm real_minus_add_distrib}),*)
814 (*"- (?x + ?y) = - ?x + - ?y"*)
815 (*Thm ("real_diff_plus",num_str @{thm real_diff_plus})*)
817 ], scr = EmptyScr}:rls;
820 Rls{id = "simplify_power", preconds = [],
821 rew_ord = ("dummy_ord", dummy_ord),
822 erls = e_rls, srls = Erls,
825 rules = [Thm ("realpow_multI", num_str @{thm realpow_multI}),
826 (*"(r * s) ^^^ n = r ^^^ n * s ^^^ n"*)
828 Thm ("sym_realpow_twoI",
829 num_str( @{thm realpow_twoI} RS @{thm sym})),
830 (*"r1 * r1 = r1 ^^^ 2"*)
831 Thm ("realpow_plus_1",num_str @{thm realpow_plus_1}),
832 (*"r * r ^^^ n = r ^^^ (n + 1)"*)
833 Thm ("realpow_pow",num_str @{thm realpow_pow}),
834 (*"(a ^^^ b) ^^^ c = a ^^^ (b * c)"*)
835 Thm ("sym_realpow_addI",
836 num_str (@{thm realpow_addI} RS @{thm sym})),
837 (*"r ^^^ n * r ^^^ m = r ^^^ (n + m)"*)
838 Thm ("realpow_oneI",num_str @{thm realpow_oneI}),
840 Thm ("realpow_eq_oneI",num_str @{thm realpow_eq_oneI})
842 ], scr = EmptyScr}:rls;
843 (*MG.0401: termorders for multivariate polys dropped due to principal problems:
844 (total-degree-)ordering of monoms NOT possible with size_of_term GIVEN*)
846 Rls{id = "order_add_mult", preconds = [],
847 rew_ord = ("ord_make_polynomial",ord_make_polynomial false thy),
848 erls = e_rls,srls = Erls,
851 rules = [Thm ("real_mult_commute",num_str @{thm real_mult_commute}),
853 Thm ("real_mult_left_commute",num_str @{thm real_mult_left_commute}),
854 (*z1.0 * (z2.0 * z3.0) = z2.0 * (z1.0 * z3.0)*)
855 Thm ("real_mult_assoc",num_str @{thm real_mult_assoc}),
856 (*z1.0 * z2.0 * z3.0 = z1.0 * (z2.0 * z3.0)*)
857 Thm ("add_commute",num_str @{thm add_commute}),
859 Thm ("add_left_commute",num_str @{thm add_left_commute}),
860 (*x + (y + z) = y + (x + z)*)
861 Thm ("add_assoc",num_str @{thm add_assoc})
862 (*z1.0 + z2.0 + z3.0 = z1.0 + (z2.0 + z3.0)*)
863 ], scr = EmptyScr}:rls;
864 (*MG.0401: termorders for multivariate polys dropped due to principal problems:
865 (total-degree-)ordering of monoms NOT possible with size_of_term GIVEN*)
867 Rls{id = "order_mult", preconds = [],
868 rew_ord = ("ord_make_polynomial",ord_make_polynomial false thy),
869 erls = e_rls,srls = Erls,
872 rules = [Thm ("real_mult_commute",num_str @{thm real_mult_commute}),
874 Thm ("real_mult_left_commute",num_str @{thm real_mult_left_commute}),
875 (*z1.0 * (z2.0 * z3.0) = z2.0 * (z1.0 * z3.0)*)
876 Thm ("real_mult_assoc",num_str @{thm real_mult_assoc})
877 (*z1.0 * z2.0 * z3.0 = z1.0 * (z2.0 * z3.0)*)
878 ], scr = EmptyScr}:rls;
879 val collect_numerals =
880 Rls{id = "collect_numerals", preconds = [],
881 rew_ord = ("dummy_ord", dummy_ord),
882 erls = Atools_erls(*erls3.4.03*),srls = Erls,
883 calc = [("PLUS" , ("op +", eval_binop "#add_")),
884 ("TIMES" , ("op *", eval_binop "#mult_")),
885 ("POWER", ("Atools.pow", eval_binop "#power_"))
888 rules = [Thm ("real_num_collect",num_str @{thm real_num_collect}),
889 (*"[| l is_const; m is_const |]==>l * n + m * n = (l + m) * n"*)
890 Thm ("real_num_collect_assoc",num_str @{thm real_num_collect_assoc}),
891 (*"[| l is_const; m is_const |] ==>
892 l * n + (m * n + k) = (l + m) * n + k"*)
893 Thm ("real_one_collect",num_str @{thm real_one_collect}),
894 (*"m is_const ==> n + m * n = (1 + m) * n"*)
895 Thm ("real_one_collect_assoc",num_str @{thm real_one_collect_assoc}),
896 (*"m is_const ==> k + (n + m * n) = k + (1 + m) * n"*)
897 Calc ("op +", eval_binop "#add_"),
898 Calc ("op *", eval_binop "#mult_"),
899 Calc ("Atools.pow", eval_binop "#power_")
900 ], scr = EmptyScr}:rls;
902 Rls{id = "reduce_012", preconds = [],
903 rew_ord = ("dummy_ord", dummy_ord),
904 erls = e_rls,srls = Erls,
907 rules = [Thm ("mult_1_left",num_str @{thm mult_1_left}),
909 (*Thm ("real_mult_minus1",num_str @{thm real_mult_minus1}),14.3.03*)
911 Thm ("minus_mult_left",
912 num_str (@{thm minus_mult_left} RS @{thm sym})),
913 (*- (?x * ?y) = "- ?x * ?y"*)
914 (*Thm ("real_minus_mult_cancel",
915 num_str @{thm real_minus_mult_cancel}),
916 (*"- ?x * - ?y = ?x * ?y"*)---*)
917 Thm ("mult_zero_left",num_str @{thm mult_zero_left}),
919 Thm ("add_0_left",num_str @{thm add_0_left}),
921 Thm ("right_minus",num_str @{thm right_minus}),
923 Thm ("sym_real_mult_2",
924 num_str (@{thm real_mult_2} RS @{thm sym})),
925 (*"z1 + z1 = 2 * z1"*)
926 Thm ("real_mult_2_assoc",num_str @{thm real_mult_2_assoc})
927 (*"z1 + (z1 + k) = 2 * z1 + k"*)
928 ], scr = EmptyScr}:rls;
929 (*ein Hilfs-'ruleset' (benutzt das leere 'ruleset')*)
930 val discard_parentheses =
931 append_rls "discard_parentheses" e_rls
932 [Thm ("sym_real_mult_assoc",
933 num_str (@{thm real_mult_assoc} RS @{thm sym})),
934 Thm ("sym_add_assoc",
935 num_str (@{thm add_assoc} RS @{thm sym}))];
937 val scr_make_polynomial =
938 "Script Expand_binoms t_t = " ^
940 "((Try (Repeat (Rewrite real_diff_minus False))) @@ " ^
942 " (Try (Repeat (Rewrite left_distrib False))) @@ " ^
943 " (Try (Repeat (Rewrite right_distrib False))) @@ " ^
944 " (Try (Repeat (Rewrite left_diff_distrib False))) @@ " ^
945 " (Try (Repeat (Rewrite right_diff_distrib False))) @@ " ^
947 " (Try (Repeat (Rewrite mult_1_left False))) @@ " ^
948 " (Try (Repeat (Rewrite mult_zero_left False))) @@ " ^
949 " (Try (Repeat (Rewrite add_0_left False))) @@ " ^
951 " (Try (Repeat (Rewrite real_mult_commute False))) @@ " ^
952 " (Try (Repeat (Rewrite real_mult_left_commute False))) @@ " ^
953 " (Try (Repeat (Rewrite real_mult_assoc False))) @@ " ^
954 " (Try (Repeat (Rewrite add_commute False))) @@ " ^
955 " (Try (Repeat (Rewrite add_left_commute False))) @@ " ^
956 " (Try (Repeat (Rewrite add_assoc False))) @@ " ^
958 " (Try (Repeat (Rewrite sym_realpow_twoI False))) @@ " ^
959 " (Try (Repeat (Rewrite realpow_plus_1 False))) @@ " ^
960 " (Try (Repeat (Rewrite sym_real_mult_2 False))) @@ " ^
961 " (Try (Repeat (Rewrite real_mult_2_assoc False))) @@ " ^
963 " (Try (Repeat (Rewrite real_num_collect False))) @@ " ^
964 " (Try (Repeat (Rewrite real_num_collect_assoc False))) @@ " ^
966 " (Try (Repeat (Rewrite real_one_collect False))) @@ " ^
967 " (Try (Repeat (Rewrite real_one_collect_assoc False))) @@ " ^
969 " (Try (Repeat (Calculate PLUS ))) @@ " ^
970 " (Try (Repeat (Calculate TIMES ))) @@ " ^
971 " (Try (Repeat (Calculate POWER)))) " ^
974 (*version used by MG.02/03, overwritten by version AG in 04 below
975 val make_polynomial = prep_rls(
976 Seq{id = "make_polynomial", preconds = []:term list,
977 rew_ord = ("dummy_ord", dummy_ord),
978 erls = Atools_erls, srls = Erls,
979 calc = [],(*asm_thm = [],*)
980 rules = [Rls_ expand_poly,
982 Rls_ simplify_power, (*realpow_eq_oneI, eg. x^1 --> x *)
983 Rls_ collect_numerals, (*eg. x^(2+ -1) --> x^1 *)
985 Thm ("realpow_oneI",num_str @{thm realpow_oneI}),(*in --^*)
986 Rls_ discard_parentheses
991 val scr_expand_binoms =
992 "Script Expand_binoms t_t =" ^
994 "((Try (Repeat (Rewrite real_plus_binom_pow2 False))) @@ " ^
995 " (Try (Repeat (Rewrite real_plus_binom_times False))) @@ " ^
996 " (Try (Repeat (Rewrite real_minus_binom_pow2 False))) @@ " ^
997 " (Try (Repeat (Rewrite real_minus_binom_times False))) @@ " ^
998 " (Try (Repeat (Rewrite real_plus_minus_binom1 False))) @@ " ^
999 " (Try (Repeat (Rewrite real_plus_minus_binom2 False))) @@ " ^
1001 " (Try (Repeat (Rewrite mult_1_left False))) @@ " ^
1002 " (Try (Repeat (Rewrite mult_zero_left False))) @@ " ^
1003 " (Try (Repeat (Rewrite add_0_left False))) @@ " ^
1005 " (Try (Repeat (Calculate PLUS ))) @@ " ^
1006 " (Try (Repeat (Calculate TIMES ))) @@ " ^
1007 " (Try (Repeat (Calculate POWER))) @@ " ^
1009 " (Try (Repeat (Rewrite sym_realpow_twoI False))) @@ " ^
1010 " (Try (Repeat (Rewrite realpow_plus_1 False))) @@ " ^
1011 " (Try (Repeat (Rewrite sym_real_mult_2 False))) @@ " ^
1012 " (Try (Repeat (Rewrite real_mult_2_assoc False))) @@ " ^
1014 " (Try (Repeat (Rewrite real_num_collect False))) @@ " ^
1015 " (Try (Repeat (Rewrite real_num_collect_assoc False))) @@ " ^
1017 " (Try (Repeat (Rewrite real_one_collect False))) @@ " ^
1018 " (Try (Repeat (Rewrite real_one_collect_assoc False))) @@ " ^
1020 " (Try (Repeat (Calculate PLUS ))) @@ " ^
1021 " (Try (Repeat (Calculate TIMES ))) @@ " ^
1022 " (Try (Repeat (Calculate POWER)))) " ^
1026 Rls{id = "expand_binoms", preconds = [], rew_ord = ("termlessI",termlessI),
1027 erls = Atools_erls, srls = Erls,
1028 calc = [("PLUS" , ("op +", eval_binop "#add_")),
1029 ("TIMES" , ("op *", eval_binop "#mult_")),
1030 ("POWER", ("Atools.pow", eval_binop "#power_"))
1032 rules = [Thm ("real_plus_binom_pow2",
1033 num_str @{thm real_plus_binom_pow2}),
1034 (*"(a + b) ^^^ 2 = a ^^^ 2 + 2 * a * b + b ^^^ 2"*)
1035 Thm ("real_plus_binom_times",
1036 num_str @{thm real_plus_binom_times}),
1037 (*"(a + b)*(a + b) = ...*)
1038 Thm ("real_minus_binom_pow2",
1039 num_str @{thm real_minus_binom_pow2}),
1040 (*"(a - b) ^^^ 2 = a ^^^ 2 - 2 * a * b + b ^^^ 2"*)
1041 Thm ("real_minus_binom_times",
1042 num_str @{thm real_minus_binom_times}),
1043 (*"(a - b)*(a - b) = ...*)
1044 Thm ("real_plus_minus_binom1",
1045 num_str @{thm real_plus_minus_binom1}),
1046 (*"(a + b) * (a - b) = a ^^^ 2 - b ^^^ 2"*)
1047 Thm ("real_plus_minus_binom2",
1048 num_str @{thm real_plus_minus_binom2}),
1049 (*"(a - b) * (a + b) = a ^^^ 2 - b ^^^ 2"*)
1051 Thm ("real_pp_binom_times",num_str @{thm real_pp_binom_times}),
1052 (*(a + b)*(c + d) = a*c + a*d + b*c + b*d*)
1053 Thm ("real_pm_binom_times",num_str @{thm real_pm_binom_times}),
1054 (*(a + b)*(c - d) = a*c - a*d + b*c - b*d*)
1055 Thm ("real_mp_binom_times",num_str @{thm real_mp_binom_times}),
1056 (*(a - b)*(c + d) = a*c + a*d - b*c - b*d*)
1057 Thm ("real_mm_binom_times",num_str @{thm real_mm_binom_times}),
1058 (*(a - b)*(c - d) = a*c - a*d - b*c + b*d*)
1059 Thm ("realpow_multI",num_str @{thm realpow_multI}),
1060 (*(a*b)^^^n = a^^^n * b^^^n*)
1061 Thm ("real_plus_binom_pow3",num_str @{thm real_plus_binom_pow3}),
1062 (* (a + b)^^^3 = a^^^3 + 3*a^^^2*b + 3*a*b^^^2 + b^^^3 *)
1063 Thm ("real_minus_binom_pow3",
1064 num_str @{thm real_minus_binom_pow3}),
1065 (* (a - b)^^^3 = a^^^3 - 3*a^^^2*b + 3*a*b^^^2 - b^^^3 *)
1068 (*Thm ("left_distrib" ,num_str @{thm left_distrib}),
1069 (*"(z1.0 + z2.0) * w = z1.0 * w + z2.0 * w"*)
1070 Thm ("right_distrib",num_str @{thm right_distrib}),
1071 (*"w * (z1.0 + z2.0) = w * z1.0 + w * z2.0"*)
1072 Thm ("left_diff_distrib" ,num_str @{thm left_diff_distrib}),
1073 (*"(z1.0 - z2.0) * w = z1.0 * w - z2.0 * w"*)
1074 Thm ("left_diff_distrib2",num_str @{thm left_diff_distrib2}),
1075 (*"w * (z1.0 - z2.0) = w * z1.0 - w * z2.0"*)
1077 Thm ("mult_1_left",num_str @{thm mult_1_left}),
1079 Thm ("mult_zero_left",num_str @{thm mult_zero_left}),
1081 Thm ("add_0_left",num_str @{thm add_0_left}),(*"0 + z = z"*)
1083 Calc ("op +", eval_binop "#add_"),
1084 Calc ("op *", eval_binop "#mult_"),
1085 Calc ("Atools.pow", eval_binop "#power_"),
1086 (*Thm ("real_mult_commute",num_str @{thm real_mult_commute}),
1088 Thm ("real_mult_left_commute",
1089 num_str @{thm real_mult_left_commute}),
1090 Thm ("real_mult_assoc",num_str @{thm real_mult_assoc}),
1091 Thm ("add_commute",num_str @{thm add_commute}),
1092 Thm ("add_left_commute",num_str @{thm add_left_commute}),
1093 Thm ("add_assoc",num_str @{thm add_assoc}),
1095 Thm ("sym_realpow_twoI",
1096 num_str (@{thm realpow_twoI} RS @{thm sym})),
1097 (*"r1 * r1 = r1 ^^^ 2"*)
1098 Thm ("realpow_plus_1",num_str @{thm realpow_plus_1}),
1099 (*"r * r ^^^ n = r ^^^ (n + 1)"*)
1100 (*Thm ("sym_real_mult_2",
1101 num_str (@{thm real_mult_2} RS @{thm sym})),
1102 (*"z1 + z1 = 2 * z1"*)*)
1103 Thm ("real_mult_2_assoc",num_str @{thm real_mult_2_assoc}),
1104 (*"z1 + (z1 + k) = 2 * z1 + k"*)
1106 Thm ("real_num_collect",num_str @{thm real_num_collect}),
1107 (*"[| l is_const; m is_const |] ==>l * n + m * n = (l + m) * n"*)
1108 Thm ("real_num_collect_assoc",
1109 num_str @{thm real_num_collect_assoc}),
1110 (*"[| l is_const; m is_const |] ==>
1111 l * n + (m * n + k) = (l + m) * n + k"*)
1112 Thm ("real_one_collect",num_str @{thm real_one_collect}),
1113 (*"m is_const ==> n + m * n = (1 + m) * n"*)
1114 Thm ("real_one_collect_assoc",
1115 num_str @{thm real_one_collect_assoc}),
1116 (*"m is_const ==> k + (n + m * n) = k + (1 + m) * n"*)
1118 Calc ("op +", eval_binop "#add_"),
1119 Calc ("op *", eval_binop "#mult_"),
1120 Calc ("Atools.pow", eval_binop "#power_")
1122 scr = Script ((term_of o the o (parse thy)) scr_expand_binoms)
1126 (**. MG.03: make_polynomial_ ... uses SML-fun for ordering .**)
1128 (*FIXME.0401: make SML-order local to make_polynomial(_) *)
1129 (*FIXME.0401: replace 'make_polynomial'(old) by 'make_polynomial_'(MG) *)
1130 (* Polynom --> List von Monomen *)
1131 fun poly2list (Const ("op +",_) $ t1 $ t2) =
1132 (poly2list t1) @ (poly2list t2)
1133 | poly2list t = [t];
1135 (* Monom --> Liste von Variablen *)
1136 fun monom2list (Const ("op *",_) $ t1 $ t2) =
1137 (monom2list t1) @ (monom2list t2)
1138 | monom2list t = [t];
1140 (* liefert Variablenname (String) einer Variablen und Basis bei Potenz *)
1141 fun get_basStr (Const ("Atools.pow",_) $ Free (str, _) $ _) = str
1142 | get_basStr (Free (str, _)) = str
1143 | get_basStr t = "|||"; (* gross gewichtet; für Brüch ect. *)
1145 raise error("get_basStr: called with t= "^(term2str t));*)
1147 (* liefert Hochzahl (String) einer Variablen bzw Gewichtstring (zum Sortieren) *)
1148 fun get_potStr (Const ("Atools.pow",_) $ Free _ $ Free (str, _)) = str
1149 | get_potStr (Const ("Atools.pow",_) $ Free _ $ _ ) = "|||" (* gross gewichtet *)
1150 | get_potStr (Free (str, _)) = "---" (* keine Hochzahl --> kleinst gewichtet *)
1151 | get_potStr t = "||||||"; (* gross gewichtet; für Brüch ect. *)
1153 raise error("get_potStr: called with t= "^(term2str t));*)
1155 (* Umgekehrte string_ord *)
1156 val string_ord_rev = rev_order o string_ord;
1158 (* Ordnung zum lexikographischen Vergleich zweier Variablen (oder Potenzen)
1159 innerhalb eines Monomes:
1160 - zuerst lexikographisch nach Variablenname
1161 - wenn gleich: nach steigender Potenz *)
1162 fun var_ord (a,b: term) = prod_ord string_ord string_ord
1163 ((get_basStr a, get_potStr a), (get_basStr b, get_potStr b));
1165 (* Ordnung zum lexikographischen Vergleich zweier Variablen (oder Potenzen);
1166 verwendet zum Sortieren von Monomen mittels Gesamtgradordnung:
1167 - zuerst lexikographisch nach Variablenname
1168 - wenn gleich: nach sinkender Potenz*)
1169 fun var_ord_revPow (a,b: term) = prod_ord string_ord string_ord_rev
1170 ((get_basStr a, get_potStr a), (get_basStr b, get_potStr b));
1173 (* Ordnet ein Liste von Variablen (und Potenzen) lexikographisch *)
1174 val sort_varList = sort var_ord;
1176 (* Entfernet aeussersten Operator (Wurzel) aus einem Term und schreibt
1177 Argumente in eine Liste *)
1178 fun args u : term list =
1179 let fun stripc (f$t, ts) = stripc (f, t::ts)
1180 | stripc (t as Free _, ts) = (t::ts)
1181 | stripc (_, ts) = ts
1182 in stripc (u, []) end;
1184 (* liefert True, falls der Term (Liste von Termen) nur Zahlen
1185 (keine Variablen) enthaelt *)
1186 fun filter_num [] = true
1187 | filter_num [Free x] = if (is_num (Free x)) then true
1189 | filter_num ((Free _)::_) = false
1191 (filter_num o (filter_out is_num) o flat o (map args)) ts;
1193 (* liefert True, falls der Term nur Zahlen (keine Variablen) enthaelt
1194 dh. er ist ein numerischer Wert und entspricht einem Koeffizienten *)
1195 fun is_nums t = filter_num [t];
1197 (* Berechnet den Gesamtgrad eines Monoms *)
1199 fun counter (n, []) = n
1200 | counter (n, x :: xs) =
1205 (Const ("Atools.pow", _) $ Free (str_b, _) $ Free (str_h, T)) =>
1206 if (is_nums (Free (str_h, T))) then
1207 counter (n + (the (int_of_str str_h)), xs)
1208 else counter (n + 1000, xs) (*FIXME.MG?!*)
1209 | (Const ("Atools.pow", _) $ Free (str_b, _) $ _ ) =>
1210 counter (n + 1000, xs) (*FIXME.MG?!*)
1211 | (Free (str, _)) => counter (n + 1, xs)
1212 (*| _ => raise error("monom_degree: called with factor: "^(term2str x)))*)
1213 | _ => counter (n + 10000, xs)) (*FIXME.MG?! ... Brüche ect.*)
1215 fun monom_degree l = counter (0, l)
1218 (* wie Ordnung dict_ord (lexicographische Ordnung zweier Listen, mit Vergleich
1219 der Listen-Elemente mit elem_ord) - Elemente die Bedingung cond erfuellen,
1220 werden jedoch dabei ignoriert (uebersprungen) *)
1221 fun dict_cond_ord _ _ ([], []) = EQUAL
1222 | dict_cond_ord _ _ ([], _ :: _) = LESS
1223 | dict_cond_ord _ _ (_ :: _, []) = GREATER
1224 | dict_cond_ord elem_ord cond (x :: xs, y :: ys) =
1225 (case (cond x, cond y) of
1226 (false, false) => (case elem_ord (x, y) of
1227 EQUAL => dict_cond_ord elem_ord cond (xs, ys)
1229 | (false, true) => dict_cond_ord elem_ord cond (x :: xs, ys)
1230 | (true, false) => dict_cond_ord elem_ord cond (xs, y :: ys)
1231 | (true, true) => dict_cond_ord elem_ord cond (xs, ys) );
1233 (* Gesamtgradordnung zum Vergleich von Monomen (Liste von Variablen/Potenzen):
1234 zuerst nach Gesamtgrad, bei gleichem Gesamtgrad lexikographisch ordnen -
1235 dabei werden Koeffizienten ignoriert (2*3*a^^^2*4*b gilt wie a^^^2*b) *)
1236 fun degree_ord (xs, ys) =
1237 prod_ord int_ord (dict_cond_ord var_ord_revPow is_nums)
1238 ((monom_degree xs, xs), (monom_degree ys, ys));
1240 fun hd_str str = substring (str, 0, 1);
1241 fun tl_str str = substring (str, 1, (size str) - 1);
1243 (* liefert nummerischen Koeffizienten eines Monoms oder NONE *)
1244 fun get_koeff_of_mon [] = raise error("get_koeff_of_mon: called with l = []")
1245 | get_koeff_of_mon (l as x::xs) = if is_nums x then SOME x
1248 (* wandelt Koeffizient in (zum sortieren geeigneten) String um *)
1249 fun koeff2ordStr (SOME x) = (case x of
1251 if (hd_str str) = "-" then (tl_str str)^"0" (* 3 < -3 *)
1253 | _ => "aaa") (* "num.Ausdruck" --> gross *)
1254 | koeff2ordStr NONE = "---"; (* "kein Koeff" --> kleinste *)
1256 (* Order zum Vergleich von Koeffizienten (strings):
1257 "kein Koeff" < "0" < "1" < "-1" < "2" < "-2" < ... < "num.Ausdruck" *)
1258 fun compare_koeff_ord (xs, ys) =
1259 string_ord ((koeff2ordStr o get_koeff_of_mon) xs,
1260 (koeff2ordStr o get_koeff_of_mon) ys);
1262 (* Gesamtgradordnung degree_ord + Ordnen nach Koeffizienten falls EQUAL *)
1263 fun koeff_degree_ord (xs, ys) =
1264 prod_ord degree_ord compare_koeff_ord ((xs, xs), (ys, ys));
1266 (* Ordnet ein Liste von Monomen (Monom = Liste von Variablen) mittels
1267 Gesamtgradordnung *)
1268 val sort_monList = sort koeff_degree_ord;
1270 (* Alternativ zu degree_ord koennte auch die viel einfachere und
1271 kuerzere Ordnung simple_ord verwendet werden - ist aber nicht
1272 fuer unsere Zwecke geeignet!
1274 fun simple_ord (al,bl: term list) = dict_ord string_ord
1275 (map get_basStr al, map get_basStr bl);
1277 val sort_monList = sort simple_ord; *)
1279 (* aus 2 Variablen wird eine Summe bzw ein Produkt erzeugt
1280 (mit gewuenschtem Typen T) *)
1281 fun plus T = Const ("op +", [T,T] ---> T);
1282 fun mult T = Const ("op *", [T,T] ---> T);
1283 fun binop op_ t1 t2 = op_ $ t1 $ t2;
1284 fun create_prod T (a,b) = binop (mult T) a b;
1285 fun create_sum T (a,b) = binop (plus T) a b;
1287 (* löscht letztes Element einer Liste *)
1288 fun drop_last l = take ((length l)-1,l);
1290 (* Liste von Variablen --> Monom *)
1291 fun create_monom T vl = foldr (create_prod T) (drop_last vl, last_elem vl);
1293 foldr bewirkt rechtslastige Klammerung des Monoms - ist notwendig, damit zwei
1294 gleiche Monome zusammengefasst werden können (collect_numerals)!
1295 zB: 2*(x*(y*z)) + 3*(x*(y*z)) --> (2+3)*(x*(y*z))*)
1297 (* Liste von Monomen --> Polynom *)
1298 fun create_polynom T ml = foldl (create_sum T) (hd ml, tl ml);
1300 foldl bewirkt linkslastige Klammerung des Polynoms (der Summanten) -
1301 bessere Darstellung, da keine Klammern sichtbar!
1302 (und discard_parentheses in make_polynomial hat weniger zu tun) *)
1304 (* sorts the variables (faktors) of an expanded polynomial lexicographical *)
1305 fun sort_variables t =
1307 val ll = map monom2list (poly2list t);
1308 val lls = map sort_varList ll;
1310 val ls = map (create_monom T) lls;
1311 in create_polynom T ls end;
1313 (* sorts the monoms of an expanded and variable-sorted polynomial
1317 val ll = map monom2list (poly2list t);
1318 val lls = sort_monList ll;
1320 val ls = map (create_monom T) lls;
1321 in create_polynom T ls end;
1323 (* auch Klammerung muss übereinstimmen;
1324 sort_variables klammert Produkte rechtslastig*)
1325 fun is_multUnordered t = ((is_polyexp t) andalso not (t = sort_variables t));
1327 fun eval_is_multUnordered (thmid:string) _
1328 (t as (Const("Poly.is'_multUnordered", _) $ arg)) thy =
1329 if is_multUnordered arg
1330 then SOME (mk_thmid thmid ""
1331 ((Syntax.string_of_term (thy2ctxt thy)) arg) "",
1332 Trueprop $ (mk_equality (t, HOLogic.true_const)))
1333 else SOME (mk_thmid thmid ""
1334 ((Syntax.string_of_term (thy2ctxt thy)) arg) "",
1335 Trueprop $ (mk_equality (t, HOLogic.false_const)))
1336 | eval_is_multUnordered _ _ _ _ = NONE;
1339 fun attach_form (_:rule list list) (_:term) (_:term) = (*still missing*)
1340 []:(rule * (term * term list)) list;
1341 fun init_state (_:term) = e_rrlsstate;
1342 fun locate_rule (_:rule list list) (_:term) (_:rule) =
1343 ([]:(rule * (term * term list)) list);
1344 fun next_rule (_:rule list list) (_:term) = (NONE:rule option);
1345 fun normal_form t = SOME (sort_variables t,[]:term list);
1348 Rrls {id = "order_mult_",
1350 [([(term_of o the o (parse thy)) "p is_multUnordered"],
1351 parse_patt thy "?p" )],
1352 rew_ord = ("dummy_ord", dummy_ord),
1353 erls = append_rls "e_rls-is_multUnordered" e_rls(*MG: poly_erls*)
1354 [Calc ("Poly.is'_multUnordered",
1355 eval_is_multUnordered "")],
1356 calc = [("PLUS" , ("op +" , eval_binop "#add_")),
1357 ("TIMES" , ("op *" , eval_binop "#mult_")),
1358 ("DIVIDE", ("HOL.divide", eval_cancel "#divide_")),
1359 ("POWER" , ("Atools.pow", eval_binop "#power_"))],
1360 scr=Rfuns {init_state = init_state,
1361 normal_form = normal_form,
1362 locate_rule = locate_rule,
1363 next_rule = next_rule,
1364 attach_form = attach_form}};
1365 val order_mult_rls_ =
1366 Rls{id = "order_mult_rls_", preconds = [],
1367 rew_ord = ("dummy_ord", dummy_ord),
1368 erls = e_rls,srls = Erls,
1370 rules = [Rls_ order_mult_
1371 ], scr = EmptyScr}:rls;
1373 fun is_addUnordered t = ((is_polyexp t) andalso not (t = sort_monoms t));
1376 (*("is_addUnordered", ("Poly.is'_addUnordered", eval_is_addUnordered ""))*)
1377 fun eval_is_addUnordered (thmid:string) _
1378 (t as (Const("Poly.is'_addUnordered", _) $ arg)) thy =
1379 if is_addUnordered arg
1380 then SOME (mk_thmid thmid ""
1381 ((Syntax.string_of_term (thy2ctxt thy)) arg) "",
1382 Trueprop $ (mk_equality (t, HOLogic.true_const)))
1383 else SOME (mk_thmid thmid ""
1384 ((Syntax.string_of_term (thy2ctxt thy)) arg) "",
1385 Trueprop $ (mk_equality (t, HOLogic.false_const)))
1386 | eval_is_addUnordered _ _ _ _ = NONE;
1388 fun attach_form (_:rule list list) (_:term) (_:term) = (*still missing*)
1389 []:(rule * (term * term list)) list;
1390 fun init_state (_:term) = e_rrlsstate;
1391 fun locate_rule (_:rule list list) (_:term) (_:rule) =
1392 ([]:(rule * (term * term list)) list);
1393 fun next_rule (_:rule list list) (_:term) = (NONE:rule option);
1394 fun normal_form t = SOME (sort_monoms t,[]:term list);
1397 Rrls {id = "order_add_",
1398 prepat = (*WN.18.6.03 Preconditions und Pattern,
1399 die beide passen muessen, damit das Rrls angewandt wird*)
1400 [([(term_of o the o (parse thy)) "p is_addUnordered"],
1402 (*WN.18.6.03 also KEIN pattern, dieses erzeugt nur das Environment
1403 fuer die Evaluation der Precondition "p is_addUnordered"*))],
1404 rew_ord = ("dummy_ord", dummy_ord),
1405 erls = append_rls "e_rls-is_addUnordered" e_rls(*MG: poly_erls*)
1406 [Calc ("Poly.is'_addUnordered", eval_is_addUnordered "")
1407 (*WN.18.6.03 definiert in thy,
1408 evaluiert prepat*)],
1409 calc = [("PLUS" ,("op +" ,eval_binop "#add_")),
1410 ("TIMES" ,("op *" ,eval_binop "#mult_")),
1411 ("DIVIDE" ,("HOL.divide" ,eval_cancel "#divide_")),
1412 ("POWER" ,("Atools.pow" ,eval_binop "#power_"))],
1414 scr=Rfuns {init_state = init_state,
1415 normal_form = normal_form,
1416 locate_rule = locate_rule,
1417 next_rule = next_rule,
1418 attach_form = attach_form}};
1420 val order_add_rls_ =
1421 Rls{id = "order_add_rls_", preconds = [],
1422 rew_ord = ("dummy_ord", dummy_ord),
1423 erls = e_rls,srls = Erls,
1426 rules = [Rls_ order_add_
1427 ], scr = EmptyScr}:rls;
1429 (*. see MG-DA.p.52ff .*)
1430 val make_polynomial(*MG.03, overwrites version from above,
1431 previously 'make_polynomial_'*) =
1432 Seq {id = "make_polynomial", preconds = []:term list,
1433 rew_ord = ("dummy_ord", dummy_ord),
1434 erls = Atools_erls, srls = Erls,calc = [],
1435 rules = [Rls_ discard_minus1,
1437 Calc ("op *", eval_binop "#mult_"),
1438 Rls_ order_mult_rls_,
1439 Rls_ simplify_power_,
1440 Rls_ calc_add_mult_pow_,
1441 Rls_ reduce_012_mult_,
1442 Rls_ order_add_rls_,
1443 Rls_ collect_numerals_,
1445 Rls_ discard_parentheses1
1449 val norm_Poly(*=make_polynomial*) =
1450 Seq {id = "norm_Poly", preconds = []:term list,
1451 rew_ord = ("dummy_ord", dummy_ord),
1452 erls = Atools_erls, srls = Erls, calc = [],
1453 rules = [Rls_ discard_minus1,
1455 Calc ("op *", eval_binop "#mult_"),
1456 Rls_ order_mult_rls_,
1457 Rls_ simplify_power_,
1458 Rls_ calc_add_mult_pow_,
1459 Rls_ reduce_012_mult_,
1460 Rls_ order_add_rls_,
1461 Rls_ collect_numerals_,
1463 Rls_ discard_parentheses1
1468 (* MG:03 Like make_polynomial_ but without Rls_ discard_parentheses1
1469 and expand_poly_rat_ instead of expand_poly_, see MG-DA.p.56ff*)
1470 (* MG necessary for termination of norm_Rational(*_mg*) in Rational.ML*)
1471 val make_rat_poly_with_parentheses =
1472 Seq{id = "make_rat_poly_with_parentheses", preconds = []:term list,
1473 rew_ord = ("dummy_ord", dummy_ord),
1474 erls = Atools_erls, srls = Erls, calc = [],
1475 rules = [Rls_ discard_minus1,
1476 Rls_ expand_poly_rat_,(*ignors rationals*)
1477 Calc ("op *", eval_binop "#mult_"),
1478 Rls_ order_mult_rls_,
1479 Rls_ simplify_power_,
1480 Rls_ calc_add_mult_pow_,
1481 Rls_ reduce_012_mult_,
1482 Rls_ order_add_rls_,
1483 Rls_ collect_numerals_,
1485 (*Rls_ discard_parentheses1 *)
1490 (*.a minimal ruleset for reverse rewriting of factions [2];
1491 compare expand_binoms.*)
1493 Seq{id = "reverse_rewriting", preconds = [], rew_ord = ("termlessI",termlessI),
1494 erls = Atools_erls, srls = Erls,
1495 calc = [(*("PLUS" , ("op +", eval_binop "#add_")),
1496 ("TIMES" , ("op *", eval_binop "#mult_")),
1497 ("POWER", ("Atools.pow", eval_binop "#power_"))*)
1499 rules = [Thm ("real_plus_binom_times" ,num_str @{thm real_plus_binom_times}),
1500 (*"(a + b)*(a + b) = a ^ 2 + 2 * a * b + b ^ 2*)
1501 Thm ("real_plus_binom_times1" ,num_str @{thm real_plus_binom_times1}),
1502 (*"(a + 1*b)*(a + -1*b) = a^^^2 + -1*b^^^2"*)
1503 Thm ("real_plus_binom_times2" ,num_str @{thm real_plus_binom_times2}),
1504 (*"(a + -1*b)*(a + 1*b) = a^^^2 + -1*b^^^2"*)
1506 Thm ("mult_1_left",num_str @{thm mult_1_left}),(*"1 * z = z"*)
1508 Thm ("left_distrib" ,num_str @{thm left_distrib}),
1509 (*"(z1.0 + z2.0) * w = z1.0 * w + z2.0 * w"*)
1510 Thm ("right_distrib",num_str @{thm right_distrib}),
1511 (*"w * (z1.0 + z2.0) = w * z1.0 + w * z2.0"*)
1513 Thm ("real_mult_assoc", num_str @{thm real_mult_assoc}),
1514 (*"?z1.1 * ?z2.1 * ?z3. =1 ?z1.1 * (?z2.1 * ?z3.1)"*)
1515 Rls_ order_mult_rls_,
1516 (*Rls_ order_add_rls_,*)
1518 Calc ("op +", eval_binop "#add_"),
1519 Calc ("op *", eval_binop "#mult_"),
1520 Calc ("Atools.pow", eval_binop "#power_"),
1522 Thm ("sym_realpow_twoI",
1523 num_str (@{thm realpow_twoI} RS @{thm sym})),
1524 (*"r1 * r1 = r1 ^^^ 2"*)
1525 Thm ("sym_real_mult_2",
1526 num_str (@{thm real_mult_2} RS @{thm sym})),
1527 (*"z1 + z1 = 2 * z1"*)
1528 Thm ("real_mult_2_assoc",num_str @{thm real_mult_2_assoc}),
1529 (*"z1 + (z1 + k) = 2 * z1 + k"*)
1531 Thm ("real_num_collect",num_str @{thm real_num_collect}),
1532 (*"[| l is_const; m is_const |]==>l * n + m * n = (l + m) * n"*)
1533 Thm ("real_num_collect_assoc",num_str @{thm real_num_collect_assoc}),
1534 (*"[| l is_const; m is_const |] ==>
1535 l * n + (m * n + k) = (l + m) * n + k"*)
1536 Thm ("real_one_collect",num_str @{thm real_one_collect}),
1537 (*"m is_const ==> n + m * n = (1 + m) * n"*)
1538 Thm ("real_one_collect_assoc",num_str @{thm real_one_collect_assoc}),
1539 (*"m is_const ==> k + (n + m * n) = k + (1 + m) * n"*)
1541 Thm ("realpow_multI", num_str @{thm realpow_multI}),
1542 (*"(r * s) ^^^ n = r ^^^ n * s ^^^ n"*)
1544 Calc ("op +", eval_binop "#add_"),
1545 Calc ("op *", eval_binop "#mult_"),
1546 Calc ("Atools.pow", eval_binop "#power_"),
1548 Thm ("mult_1_left",num_str @{thm mult_1_left}),(*"1 * z = z"*)
1549 Thm ("mult_zero_left",num_str @{thm mult_zero_left}),(*"0 * z = 0"*)
1550 Thm ("add_0_left",num_str @{thm add_0_left})(*0 + z = z*)
1552 (*Rls_ order_add_rls_*)
1555 scr = EmptyScr}:rls;
1558 overwritelthy @{theory} (!ruleset',
1559 [("norm_Poly", prep_rls norm_Poly),
1560 ("Poly_erls",Poly_erls)(*FIXXXME:del with rls.rls'*),
1561 ("expand", prep_rls expand),
1562 ("expand_poly", prep_rls expand_poly),
1563 ("simplify_power", prep_rls simplify_power),
1564 ("order_add_mult", prep_rls order_add_mult),
1565 ("collect_numerals", prep_rls collect_numerals),
1566 ("collect_numerals_", prep_rls collect_numerals_),
1567 ("reduce_012", prep_rls reduce_012),
1568 ("discard_parentheses", prep_rls discard_parentheses),
1569 ("make_polynomial", prep_rls make_polynomial),
1570 ("expand_binoms", prep_rls expand_binoms),
1571 ("rev_rew_p", prep_rls rev_rew_p),
1572 ("discard_minus1", prep_rls discard_minus1),
1573 ("expand_poly_", prep_rls expand_poly_),
1574 ("expand_poly_rat_", prep_rls expand_poly_rat_),
1575 ("simplify_power_", prep_rls simplify_power_),
1576 ("calc_add_mult_pow_", prep_rls calc_add_mult_pow_),
1577 ("reduce_012_mult_", prep_rls reduce_012_mult_),
1578 ("reduce_012_", prep_rls reduce_012_),
1579 ("discard_parentheses1",prep_rls discard_parentheses1),
1580 ("order_mult_rls_", prep_rls order_mult_rls_),
1581 ("order_add_rls_", prep_rls order_add_rls_),
1582 ("make_rat_poly_with_parentheses",
1583 prep_rls make_rat_poly_with_parentheses)
1586 calclist':= overwritel (!calclist',
1587 [("is_polyrat_in", ("Poly.is'_polyrat'_in",
1588 eval_is_polyrat_in "#eval_is_polyrat_in")),
1589 ("is_expanded_in", ("Poly.is'_expanded'_in", eval_is_expanded_in "")),
1590 ("is_poly_in", ("Poly.is'_poly'_in", eval_is_poly_in "")),
1591 ("has_degree_in", ("Poly.has'_degree'_in", eval_has_degree_in "")),
1592 ("is_polyexp", ("Poly.is'_polyexp", eval_is_polyexp "")),
1593 ("is_multUnordered", ("Poly.is'_multUnordered", eval_is_multUnordered"")),
1594 ("is_addUnordered", ("Poly.is'_addUnordered", eval_is_addUnordered ""))
1597 val ------------------------------------------------------ = "11111";
1602 (prep_pbt thy "pbl_simp_poly" [] e_pblID
1603 (["polynomial","simplification"],
1604 [("#Given" ,["TERM t_t"]),
1605 ("#Where" ,["t_t is_polyexp"]),
1606 ("#Find" ,["normalform n_n"])
1608 append_rls "e_rls" e_rls [(*for preds in where_*)
1609 Calc ("Poly.is'_polyexp", eval_is_polyexp "")],
1610 SOME "Simplify t_t",
1611 [["simplification","for_polynomials"]]));
1616 (prep_met thy "met_simp_poly" [] e_metID
1617 (["simplification","for_polynomials"],
1618 [("#Given" ,["TERM t_t"]),
1619 ("#Where" ,["t_t is_polyexp"]),
1620 ("#Find" ,["normalform n_n"])
1622 {rew_ord'="tless_true",
1626 prls = append_rls "simplification_for_polynomials_prls" e_rls
1627 [(*for preds in where_*)
1628 Calc ("Poly.is'_polyexp",eval_is_polyexp"")],
1629 crls = e_rls, nrls = norm_Poly},
1630 "Script SimplifyScript (t_t::real) = " ^
1631 " ((Rewrite_Set norm_Poly False) t_t)"