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 \<up> instead of ^ in the respective theorem xxx in 2002
7 changed by: Richard Lang 020912
10 theory Poly imports Simplify begin
12 subsection \<open>remark on term-structure of polynomials\<close>
15 the code below reflects missing coordination between two authors:
16 * ML: built the equation solver; simple rule-sets, programs; better predicates for specifications.
17 * MG: built simplification of polynomials with AC rewriting by ML code
20 *** there are 5 kinds of expanded normalforms ***
22 [1] 'complete polynomial' (Komplettes Polynom), univariate
23 a_0 + a_1.x^1 +...+ a_n.x^n not (a_n = 0)
24 not (a_n = 0), some a_i may be zero (DON'T disappear),
25 variables in monomials lexicographically ordered and complete,
26 x written as 1*x^1, ...
27 [2] 'polynomial' (Polynom), univariate and multivariate
28 a_0 + a_1.x +...+ a_n.x^n not (a_n = 0)
29 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
30 not (a_n = 0), some a_i may be zero (ie. monomials disappear),
31 exponents and coefficients equal 1 are not (WN060904.TODO in cancel_p_)shown,
32 and variables in monomials are lexicographically ordered
33 examples: [1]: "1 + (-10) * x \<up> 1 + 25 * x \<up> 2"
34 [1]: "11 + 0 * x \<up> 1 + 1 * x \<up> 2"
35 [2]: "x + (-50) * x \<up> 3"
36 [2]: "(-1) * x * y \<up> 2 + 7 * x \<up> 3"
38 [3] 'expanded_term' (Ausmultiplizierter Term):
39 pull out unary minus to binary minus,
40 as frequently exercised in schools; other conditions for [2] hold however
41 examples: "a \<up> 2 - 2 * a * b + b \<up> 2"
42 "4 * x \<up> 2 - 9 * y \<up> 2"
43 [4] 'polynomial_in' (Polynom in):
44 polynomial in 1 variable with arbitrary coefficients
45 examples: "2 * x + (-50) * x \<up> 3" (poly in x)
46 "(u + v) + (2 * u \<up> 2) * a + (-u) * a \<up> 2 (poly in a)
47 [5] 'expanded_in' (Ausmultiplizierter Termin in):
48 analoguous to [3] with binary minus like [3]
49 examples: "2 * x - 50 * x \<up> 3" (expanded in x)
50 "(u + v) + (2 * u \<up> 2) * a - u * a \<up> 2 (expanded in a)
52 subsection \<open>consts definition for predicates in specifications\<close>
55 is_expanded_in :: "[real, real] => bool" ("_ is'_expanded'_in _")
56 is_poly_in :: "[real, real] => bool" ("_ is'_poly'_in _") (*RL DA *)
57 has_degree_in :: "[real, real] => real" ("_ has'_degree'_in _")(*RL DA *)
58 is_polyrat_in :: "[real, real] => bool" ("_ is'_polyrat'_in _")(*RL030626*)
60 is_multUnordered:: "real => bool" ("_ is'_multUnordered")
61 is_addUnordered :: "real => bool" ("_ is'_addUnordered") (*WN030618*)
62 is_polyexp :: "real => bool" ("_ is'_polyexp")
64 subsection \<open>theorems not yet adopted from Isabelle\<close>
65 axiomatization where (*.not contained in Isabelle2002,
66 stated as axioms, TODO: prove as theorems;
67 theorem-IDs 'xxxI' with \<up> instead of ^ in 'xxx' in Isabelle2002.*)
69 realpow_pow: "(a \<up> b) \<up> c = a \<up> (b * c)" and
70 realpow_addI: "r \<up> (n + m) = r \<up> n * r \<up> m" and
71 realpow_addI_assoc_l: "r \<up> n * (r \<up> m * s) = r \<up> (n + m) * s" and
72 realpow_addI_assoc_r: "s * r \<up> n * r \<up> m = s * r \<up> (n + m)" and
74 realpow_oneI: "r \<up> 1 = r" and
75 realpow_zeroI: "r \<up> 0 = 1" and
76 realpow_eq_oneI: "1 \<up> n = 1" and
77 realpow_multI: "(r * s) \<up> n = r \<up> n * s \<up> n" and
78 realpow_multI_poly: "[| r is_polyexp; s is_polyexp |] ==>
79 (r * s) \<up> n = r \<up> n * s \<up> n" and
80 realpow_minus_oneI: "(- 1) \<up> (2 * n) = 1" and
81 real_diff_0: "0 - x = - (x::real)" and
83 realpow_twoI: "r \<up> 2 = r * r" and
84 realpow_twoI_assoc_l: "r * (r * s) = r \<up> 2 * s" and
85 realpow_twoI_assoc_r: "s * r * r = s * r \<up> 2" and
86 realpow_two_atom: "r is_atom ==> r * r = r \<up> 2" and
87 realpow_plus_1: "r * r \<up> n = r \<up> (n + 1)" and
88 realpow_plus_1_assoc_l: "r * (r \<up> m * s) = r \<up> (1 + m) * s" and
89 realpow_plus_1_assoc_l2: "r \<up> m * (r * s) = r \<up> (1 + m) * s" and
90 realpow_plus_1_assoc_r: "s * r * r \<up> m = s * r \<up> (1 + m)" and
91 realpow_plus_1_atom: "r is_atom ==> r * r \<up> n = r \<up> (1 + n)" and
92 realpow_def_atom: "[| Not (r is_atom); 1 < n |]
93 ==> r \<up> n = r * r \<up> (n + -1)" and
94 realpow_addI_atom: "r is_atom ==> r \<up> n * r \<up> m = r \<up> (n + m)" and
97 realpow_minus_even: "n is_even ==> (- r) \<up> n = r \<up> n" and
98 realpow_minus_odd: "Not (n is_even) ==> (- r) \<up> n = -1 * r \<up> n" and
102 real_pp_binom_times: "(a + b)*(c + d) = a*c + a*d + b*c + b*d" and
103 real_pm_binom_times: "(a + b)*(c - d) = a*c - a*d + b*c - b*d" and
104 real_mp_binom_times: "(a - b)*(c + d) = a*c + a*d - b*c - b*d" and
105 real_mm_binom_times: "(a - b)*(c - d) = a*c - a*d - b*c + b*d" and
106 real_plus_binom_pow3: "(a + b) \<up> 3 = a \<up> 3 + 3*a \<up> 2*b + 3*a*b \<up> 2 + b \<up> 3" and
107 real_plus_binom_pow3_poly: "[| a is_polyexp; b is_polyexp |] ==>
108 (a + b) \<up> 3 = a \<up> 3 + 3*a \<up> 2*b + 3*a*b \<up> 2 + b \<up> 3" and
109 real_minus_binom_pow3: "(a - b) \<up> 3 = a \<up> 3 - 3*a \<up> 2*b + 3*a*b \<up> 2 - b \<up> 3" and
110 real_minus_binom_pow3_p: "(a + -1 * b) \<up> 3 = a \<up> 3 + -3*a \<up> 2*b + 3*a*b \<up> 2 +
112 (* real_plus_binom_pow: "[| n is_const; 3 < n |] ==>
113 (a + b) \<up> n = (a + b) * (a + b)\<up>(n - 1)" *)
114 real_plus_binom_pow4: "(a + b) \<up> 4 = (a \<up> 3 + 3*a \<up> 2*b + 3*a*b \<up> 2 + b \<up> 3)
116 real_plus_binom_pow4_poly: "[| a is_polyexp; b is_polyexp |] ==>
117 (a + b) \<up> 4 = (a \<up> 3 + 3*a \<up> 2*b + 3*a*b \<up> 2 + b \<up> 3)
119 real_plus_binom_pow5: "(a + b) \<up> 5 = (a \<up> 3 + 3*a \<up> 2*b + 3*a*b \<up> 2 + b \<up> 3)
120 *(a \<up> 2 + 2*a*b + b \<up> 2)" and
121 real_plus_binom_pow5_poly: "[| a is_polyexp; b is_polyexp |] ==>
122 (a + b) \<up> 5 = (a \<up> 3 + 3*a \<up> 2*b + 3*a*b \<up> 2
123 + b \<up> 3)*(a \<up> 2 + 2*a*b + b \<up> 2)" and
124 real_diff_plus: "a - b = a + -b" (*17.3.03: do_NOT_use*) and
125 real_diff_minus: "a - b = a + -1 * b" and
126 real_plus_binom_times: "(a + b)*(a + b) = a \<up> 2 + 2*a*b + b \<up> 2" and
127 real_minus_binom_times: "(a - b)*(a - b) = a \<up> 2 - 2*a*b + b \<up> 2" and
128 (*WN071229 changed for Schaerding -----vvv*)
129 (*real_plus_binom_pow2: "(a + b) \<up> 2 = a \<up> 2 + 2*a*b + b \<up> 2"*)
130 real_plus_binom_pow2: "(a + b) \<up> 2 = (a + b) * (a + b)" and
131 (*WN071229 changed for Schaerding -----\<up>*)
132 real_plus_binom_pow2_poly: "[| a is_polyexp; b is_polyexp |] ==>
133 (a + b) \<up> 2 = a \<up> 2 + 2*a*b + b \<up> 2" and
134 real_minus_binom_pow2: "(a - b) \<up> 2 = a \<up> 2 - 2*a*b + b \<up> 2" and
135 real_minus_binom_pow2_p: "(a - b) \<up> 2 = a \<up> 2 + -2*a*b + b \<up> 2" and
136 real_plus_minus_binom1: "(a + b)*(a - b) = a \<up> 2 - b \<up> 2" and
137 real_plus_minus_binom1_p: "(a + b)*(a - b) = a \<up> 2 + -1*b \<up> 2" and
138 real_plus_minus_binom1_p_p: "(a + b)*(a + -1 * b) = a \<up> 2 + -1*b \<up> 2" and
139 real_plus_minus_binom2: "(a - b)*(a + b) = a \<up> 2 - b \<up> 2" and
140 real_plus_minus_binom2_p: "(a - b)*(a + b) = a \<up> 2 + -1*b \<up> 2" and
141 real_plus_minus_binom2_p_p: "(a + -1 * b)*(a + b) = a \<up> 2 + -1*b \<up> 2" and
142 real_plus_binom_times1: "(a + 1*b)*(a + -1*b) = a \<up> 2 + -1*b \<up> 2" and
143 real_plus_binom_times2: "(a + -1*b)*(a + 1*b) = a \<up> 2 + -1*b \<up> 2" and
145 real_num_collect: "[| l is_const; m is_const |] ==>
146 l * n + m * n = (l + m) * n" and
147 (* FIXME.MG.0401: replace 'real_num_collect_assoc'
148 by 'real_num_collect_assoc_l' ... are equal, introduced by MG ! *)
149 real_num_collect_assoc: "[| l is_const; m is_const |] ==>
150 l * n + (m * n + k) = (l + m) * n + k" and
151 real_num_collect_assoc_l: "[| l is_const; m is_const |] ==>
152 l * n + (m * n + k) = (l + m)
154 real_num_collect_assoc_r: "[| l is_const; m is_const |] ==>
155 (k + m * n) + l * n = k + (l + m) * n" and
156 real_one_collect: "m is_const ==> n + m * n = (1 + m) * n" and
157 (* FIXME.MG.0401: replace 'real_one_collect_assoc'
158 by 'real_one_collect_assoc_l' ... are equal, introduced by MG ! *)
159 real_one_collect_assoc: "m is_const ==> n + (m * n + k) = (1 + m)* n + k" and
161 real_one_collect_assoc_l: "m is_const ==> n + (m * n + k) = (1 + m) * n + k" and
162 real_one_collect_assoc_r: "m is_const ==> (k + n) + m * n = k + (1 + m) * n" and
164 (* FIXME.MG.0401: replace 'real_mult_2_assoc'
165 by 'real_mult_2_assoc_l' ... are equal, introduced by MG ! *)
166 real_mult_2_assoc: "z1 + (z1 + k) = 2 * z1 + k" and
167 real_mult_2_assoc_l: "z1 + (z1 + k) = 2 * z1 + k" and
168 real_mult_2_assoc_r: "(k + z1) + z1 = k + 2 * z1" and
170 real_mult_left_commute: "z1 * (z2 * z3) = z2 * (z1 * z3)" and
171 real_mult_minus1: "-1 * z = - (z::real)" and
172 (*sym_real_mult_minus1 expands indefinitely without assumptions ...*)
173 real_mult_minus1_sym: "[| \<not>(matches (- 1 * x) z); \<not>(z is_atom) |] ==> - (z::real) = -1 * z" and
174 real_mult_2: "2 * z = z + (z::real)" and
176 real_add_mult_distrib_poly: "w is_polyexp ==> (z1 + z2) * w = z1 * w + z2 * w" and
177 real_add_mult_distrib2_poly:"w is_polyexp ==> w * (z1 + z2) = w * z1 + w * z2"
180 subsection \<open>auxiliary functions\<close>
183 [\<^const_name>\<open>plus\<close>, \<^const_name>\<open>minus\<close>,
184 \<^const_name>\<open>divide\<close>, \<^const_name>\<open>times\<close>,
185 \<^const_name>\<open>powr\<close>];
187 val int_ord_SAVE = int_ord;
188 (*for tests on rewrite orders*)
189 fun int_ord (i1, i2) =
190 (@{print} {a = "int_ord (" ^ string_of_int i1 ^ ", " ^ string_of_int i2 ^ ") = ", z = Int.compare (i1, i2)};
191 Int.compare (i1, i2));
192 (**)val int_ord = int_ord_SAVE; (*..outcomment for tests*)
194 subsubsection \<open>for predicates in specifications (ML)\<close>
196 (*--- auxiliary for is_expanded_in, is_poly_in, has_degree_in ---*)
197 (*. a "monomial t in variable v" is a term t with
198 either (1) v NOT existent in t, or (2) v contained in t,
200 if (2) then v is a factor on the very right, casually with exponent.*)
201 fun factor_right_deg (*case 2*)
202 (Const (\<^const_name>\<open>Groups.times_class.times\<close>, _) $
203 t1 $ (Const (\<^const_name>\<open>Transcendental.powr\<close>,_) $ vv $ num)) v =
204 if vv = v andalso not (Prog_Expr.occurs_in v t1) then SOME (snd (HOLogic.dest_number num))
206 | factor_right_deg (Const (\<^const_name>\<open>Transcendental.powr\<close>,_) $ vv $ num) v =
207 if (vv = v) then SOME (snd (HOLogic.dest_number num)) else NONE
209 | factor_right_deg (Const (\<^const_name>\<open>times\<close>,_) $ t1 $ vv) v =
210 if vv = v andalso not (Prog_Expr.occurs_in v t1) then SOME 1 else NONE
211 | factor_right_deg vv v =
212 if (vv = v) then SOME 1 else NONE;
213 fun mono_deg_in m v = (*case 1*)
214 if not (Prog_Expr.occurs_in v m) then (*case 1*) SOME 0 else factor_right_deg m v;
216 fun expand_deg_in t v =
218 fun edi ~1 ~1 (Const (\<^const_name>\<open>plus\<close>, _) $ t1 $ t2) =
219 (case mono_deg_in t2 v of (* $ is left associative*)
220 SOME d' => edi d' d' t1 | NONE => NONE)
221 | edi ~1 ~1 (Const (\<^const_name>\<open>minus\<close>, _) $ t1 $ t2) =
222 (case mono_deg_in t2 v of
223 SOME d' => edi d' d' t1 | NONE => NONE)
224 | edi d dmax (Const (\<^const_name>\<open>minus\<close>, _) $ t1 $ t2) =
225 (case mono_deg_in t2 v of (*(d = 0 andalso d' = 0) handle 3+4-...4 +x*)
226 SOME d' => if d > d' orelse (d = 0 andalso d' = 0) then edi d' dmax t1 else NONE
228 | edi d dmax (Const (\<^const_name>\<open>plus\<close>,_) $ t1 $ t2) =
229 (case mono_deg_in t2 v of
230 SOME d' => (*RL (d = 0 andalso d' = 0) need to handle 3+4-...4 +x*)
231 if d > d' orelse (d = 0 andalso d' = 0) then edi d' dmax t1 else NONE
234 (case mono_deg_in t v of d as SOME _ => d | NONE => NONE)
235 | edi d dmax t = (*basecase last*)
236 (case mono_deg_in t v of
237 SOME d' => if d > d' orelse (d = 0 andalso d' = 0) then SOME dmax else NONE
241 fun poly_deg_in t v =
243 fun edi ~1 ~1 (Const (\<^const_name>\<open>plus\<close>,_) $ t1 $ t2) =
244 (case mono_deg_in t2 v of (* $ is left associative *)
245 SOME d' => edi d' d' t1
247 | edi d dmax (Const (\<^const_name>\<open>plus\<close>,_) $ t1 $ t2) =
248 (case mono_deg_in t2 v of
249 SOME d' => (*RL (d = 0 andalso (d' = 0)) handle 3+4-...4 +x*)
250 if d > d' orelse (d = 0 andalso d' = 0) then edi d' dmax t1 else NONE
253 (case mono_deg_in t v of
256 | edi d dmax t = (* basecase last *)
257 (case mono_deg_in t v of
259 if d > d' orelse (d = 0 andalso d' = 0) then SOME dmax else NONE
264 subsubsection \<open>for hard-coded AC rewriting (MG)\<close>
266 (**. MG.03: make_polynomial_ ... uses SML-fun for ordering .**)
268 (*FIXME.0401: make SML-order local to make_polynomial(_) *)
269 (*FIXME.0401: replace 'make_polynomial'(old) by 'make_polynomial_'(MG) *)
270 (* Polynom --> List von Monomen *)
271 fun poly2list (Const (\<^const_name>\<open>plus\<close>,_) $ t1 $ t2) =
272 (poly2list t1) @ (poly2list t2)
275 (* Monom --> Liste von Variablen *)
276 fun monom2list (Const (\<^const_name>\<open>times\<close>,_) $ t1 $ t2) =
277 (monom2list t1) @ (monom2list t2)
278 | monom2list t = [t];
280 (* liefert Variablenname (String) einer Variablen und Basis bei Potenz *)
281 fun get_basStr (Const (\<^const_name>\<open>powr\<close>,_) $ Free (str, _) $ _) = str
282 | get_basStr (Const (\<^const_name>\<open>Transcendental.powr\<close>,_) $ n $ _) = TermC.to_string n
283 | get_basStr (Free (str, _)) = str
285 if TermC.is_num t then TermC.to_string t
286 else "|||"; (* gross gewichtet; für Brüche ect. *)
288 (* liefert Hochzahl (String) einer Variablen bzw Gewichtstring (zum Sortieren) *)
289 fun get_potStr (Const (\<^const_name>\<open>Transcendental.powr\<close>, _) $ Free _ $ Free (str, _)) = str
290 | get_potStr (Const (\<^const_name>\<open>Transcendental.powr\<close>, _) $ Free _ $ t) =
291 if TermC.is_num t then TermC.to_string t else "|||"
292 | get_potStr (Free _) = "---" (* keine Hochzahl --> kleinst gewichtet *)
293 | get_potStr _ = "||||||"; (* gross gewichtet; für Brüch ect. *)
295 (* Umgekehrte string_ord *)
296 val string_ord_rev = rev_order o string_ord;
298 (* Ordnung zum lexikographischen Vergleich zweier Variablen (oder Potenzen)
299 innerhalb eines Monomes:
300 - zuerst lexikographisch nach Variablenname
301 - wenn gleich: nach steigender Potenz *)
303 (@{print} {a = "var_ord ", a_b = "(" ^ UnparseC.term a ^ ", " ^ UnparseC.term b ^ ")",
304 sort_args = "(" ^ get_basStr a ^ ", " ^ get_potStr a ^ "), (" ^ get_basStr b ^ ", " ^ get_potStr b ^ ")"};
305 prod_ord string_ord string_ord
306 ((get_basStr a, get_potStr a), (get_basStr b, get_potStr b))
308 fun var_ord (a,b: term) =
309 prod_ord string_ord string_ord
310 ((get_basStr a, get_potStr a), (get_basStr b, get_potStr b));
312 (* Ordnung zum lexikographischen Vergleich zweier Variablen (oder Potenzen);
313 verwendet zum Sortieren von Monomen mittels Gesamtgradordnung:
314 - zuerst lexikographisch nach Variablenname
315 - wenn gleich: nach sinkender Potenz*)
316 fun var_ord_revPow (a, b: term) =
317 (@{print} {a = "var_ord_revPow ", at_bt = "(" ^ UnparseC.term a ^ ", " ^ UnparseC.term b ^ ")",
318 sort_args = "(" ^ get_basStr a ^ ", " ^ get_potStr a ^ "), (" ^ get_basStr b ^ ", " ^ get_potStr b ^ ")"};
319 prod_ord string_ord string_ord_rev
320 ((get_basStr a, get_potStr a), (get_basStr b, get_potStr b))
322 fun var_ord_revPow (a, b: term) =
323 prod_ord string_ord string_ord_rev
324 ((get_basStr a, get_potStr a), (get_basStr b, get_potStr b));
327 (* Ordnet ein Liste von Variablen (und Potenzen) lexikographisch *)
328 fun sort_varList ts =
329 (@{print} {a = "sort_varList", args = UnparseC.terms ts};
331 val sort_varList = sort var_ord;
333 (* Entfernet aeussersten Operator (Wurzel) aus einem Term und schreibt
334 Argumente in eine Liste *)
335 fun args u : term list =
337 fun stripc (f $ t, ts) = stripc (f, t::ts)
338 | stripc (t as Free _, ts) = (t::ts)
339 | stripc (_, ts) = ts
340 in stripc (u, []) end;
342 (* liefert True, falls der Term (Liste von Termen) nur Zahlen
343 (keine Variablen) enthaelt *)
344 fun filter_num ts = fold (curry and_) (map TermC.is_num ts) true
346 (* liefert True, falls der Term nur Zahlen (keine Variablen) enthaelt
347 dh. er ist ein numerischer Wert und entspricht einem Koeffizienten *)
348 fun is_nums t = filter_num [t];
350 (* Berechnet den Gesamtgrad eines Monoms *)
352 fun counter (n, []) = n
353 | counter (n, x :: xs) =
354 if (is_nums x) then counter (n, xs)
357 (Const (\<^const_name>\<open>Transcendental.powr\<close>, _) $ Free _ $ t) =>
359 then counter (t |> HOLogic.dest_number |> snd |> curry op + n, xs)
360 else counter (n + 1000, xs) (*FIXME.MG?!*)
361 | (Const (\<^const_name>\<open>numeral\<close>, _) $ num) =>
362 counter (n + 1 + HOLogic.dest_numeral num, xs)
363 | _ => counter (n + 1, xs)) (*FIXME.MG?! ... Brüche ect.*)
365 fun monom_degree l = counter (0, l)
368 (* wie Ordnung dict_ord (lexicographische Ordnung zweier Listen, mit Vergleich
369 der Listen-Elemente mit elem_ord) - Elemente die Bedingung cond erfuellen,
370 werden jedoch dabei ignoriert (uebersprungen) *)
371 fun dict_cond_ord _ _ ([], []) = (@{print} {a = "dict_cond_ord ([], [])"}; EQUAL)
372 | dict_cond_ord _ _ ([], _ :: _) = (@{print} {a = "dict_cond_ord ([], _ :: _)"}; LESS)
373 | dict_cond_ord _ _ (_ :: _, []) = (@{print} {a = "dict_cond_ord (_ :: _, [])"}; GREATER)
374 | dict_cond_ord elem_ord cond (x :: xs, y :: ys) =
375 (@{print} {a = "dict_cond_ord", args = "(" ^ UnparseC.terms (x :: xs) ^ ", " ^ UnparseC.terms (y :: ys) ^ ")",
376 is_nums = "(" ^ LibraryC.bool2str (cond x) ^ ", " ^ LibraryC.bool2str (cond y) ^ ")"};
377 case (cond x, cond y) of
379 (case elem_ord (x, y) of
380 EQUAL => dict_cond_ord elem_ord cond (xs, ys)
382 | (false, true) => dict_cond_ord elem_ord cond (x :: xs, ys)
383 | (true, false) => dict_cond_ord elem_ord cond (xs, y :: ys)
384 | (true, true) => dict_cond_ord elem_ord cond (xs, ys) );
385 fun dict_cond_ord _ _ ([], []) = EQUAL
386 | dict_cond_ord _ _ ([], _ :: _) = LESS
387 | dict_cond_ord _ _ (_ :: _, []) = GREATER
388 | dict_cond_ord elem_ord cond (x :: xs, y :: ys) =
389 (case (cond x, cond y) of
391 (case elem_ord (x, y) of
392 EQUAL => dict_cond_ord elem_ord cond (xs, ys)
394 | (false, true) => dict_cond_ord elem_ord cond (x :: xs, ys)
395 | (true, false) => dict_cond_ord elem_ord cond (xs, y :: ys)
396 | (true, true) => dict_cond_ord elem_ord cond (xs, ys) );
398 (* Gesamtgradordnung zum Vergleich von Monomen (Liste von Variablen/Potenzen):
399 zuerst nach Gesamtgrad, bei gleichem Gesamtgrad lexikographisch ordnen -
400 dabei werden Koeffizienten ignoriert (2*3*a \<up> 2*4*b gilt wie a \<up> 2*b) *)
401 fun degree_ord (xs, ys) =
402 prod_ord int_ord (dict_cond_ord var_ord_revPow is_nums)
403 ((monom_degree xs, xs), (monom_degree ys, ys));
405 fun hd_str str = substring (str, 0, 1);
406 fun tl_str str = substring (str, 1, (size str) - 1);
408 (* liefert nummerischen Koeffizienten eines Monoms oder NONE *)
409 fun get_koeff_of_mon [] = raise ERROR "get_koeff_of_mon: called with l = []"
410 | get_koeff_of_mon (x :: _) = if is_nums x then SOME x else NONE;
412 (* wandelt Koeffizient in (zum sortieren geeigneten) String um *)
413 fun koeff2ordStr (SOME t) =
416 if (t |> HOLogic.dest_number |> snd) < 0
417 then (t |> HOLogic.dest_number |> snd |> curry op * ~1 |> string_of_int) ^ "0" (* 3 < -3 *)
418 else (t |> HOLogic.dest_number |> snd |> string_of_int)
419 else "aaa" (* "num.Ausdruck" --> gross *)
420 | koeff2ordStr NONE = "---"; (* "kein Koeff" --> kleinste *)
422 (* Order zum Vergleich von Koeffizienten (strings):
423 "kein Koeff" < "0" < "1" < "-1" < "2" < "-2" < ... < "num.Ausdruck" *)
424 fun compare_koeff_ord (xs, ys) = string_ord
425 ((koeff2ordStr o get_koeff_of_mon) xs,
426 (koeff2ordStr o get_koeff_of_mon) ys);
428 (* Gesamtgradordnung degree_ord + Ordnen nach Koeffizienten falls EQUAL *)
429 fun koeff_degree_ord (xs, ys) =
430 prod_ord degree_ord compare_koeff_ord ((xs, xs), (ys, ys));
432 (* Ordnet ein Liste von Monomen (Monom = Liste von Variablen) mittels
434 val sort_monList = sort koeff_degree_ord;
436 (* Alternativ zu degree_ord koennte auch die viel einfachere und
437 kuerzere Ordnung simple_ord verwendet werden - ist aber nicht
438 fuer unsere Zwecke geeignet!
440 fun simple_ord (al,bl: term list) = dict_ord string_ord
441 (map get_basStr al, map get_basStr bl);
443 val sort_monList = sort simple_ord; *)
445 (* aus 2 Variablen wird eine Summe bzw ein Produkt erzeugt
446 (mit gewuenschtem Typen T) *)
447 fun plus T = Const (\<^const_name>\<open>plus\<close>, [T,T] ---> T);
448 fun mult T = Const (\<^const_name>\<open>times\<close>, [T,T] ---> T);
449 fun binop op_ t1 t2 = op_ $ t1 $ t2;
450 fun create_prod T (a,b) = binop (mult T) a b;
451 fun create_sum T (a,b) = binop (plus T) a b;
453 (* löscht letztes Element einer Liste *)
454 fun drop_last l = take ((length l)-1,l);
456 (* Liste von Variablen --> Monom *)
457 fun create_monom T vl = foldr (create_prod T) (drop_last vl, last_elem vl);
459 foldr bewirkt rechtslastige Klammerung des Monoms - ist notwendig, damit zwei
460 gleiche Monome zusammengefasst werden können (collect_numerals)!
461 zB: 2*(x*(y*z)) + 3*(x*(y*z)) --> (2+3)*(x*(y*z))*)
463 (* Liste von Monomen --> Polynom *)
464 fun create_polynom T ml = foldl (create_sum T) (hd ml, tl ml);
466 foldl bewirkt linkslastige Klammerung des Polynoms (der Summanten) -
467 bessere Darstellung, da keine Klammern sichtbar!
468 (und discard_parentheses in make_polynomial hat weniger zu tun) *)
470 (* sorts the variables (faktors) of an expanded polynomial lexicographical *)
471 fun sort_variables t =
473 val ll = map monom2list (poly2list t);
474 val lls = map sort_varList ll;
476 val ls = map (create_monom T) lls;
477 in create_polynom T ls end;
479 (* sorts the monoms of an expanded and variable-sorted polynomial
483 val ll = map monom2list (poly2list t);
484 val lls = sort_monList ll;
485 val T = Term.type_of t;
486 val ls = map (create_monom T) lls;
487 in create_polynom T ls end;
490 subsubsection \<open>rewrite order for hard-coded AC rewriting\<close>
492 local (*. for make_polynomial .*)
494 open Term; (* for type order = EQUAL | LESS | GREATER *)
496 fun pr_ord EQUAL = "EQUAL"
497 | pr_ord LESS = "LESS"
498 | pr_ord GREATER = "GREATER";
500 fun dest_hd' (Const (a, T)) = (* ~ term.ML *)
502 \<^const_name>\<open>powr\<close> => ((("|||||||||||||", 0), T), 0) (*WN greatest string*)
503 | _ => (((a, 0), T), 0))
504 | dest_hd' (Free (a, T)) = (((a, 0), T), 1)(*TODOO handle this as numeral, too? see EqSystem.thy*)
505 | dest_hd' (Var v) = (v, 2)
506 | dest_hd' (Bound i) = ((("", i), dummyT), 3)
507 | dest_hd' (Abs (_, T, _)) = ((("", 0), T), 4)
508 | dest_hd' t = raise TERM ("dest_hd'", [t]);
510 fun size_of_term' (Const(str,_) $ t) =
511 if \<^const_name>\<open>powr\<close>= str then 1000 + size_of_term' t else 1+size_of_term' t(*WN*)
512 | size_of_term' (Abs (_,_,body)) = 1 + size_of_term' body
513 | size_of_term' (f$t) = size_of_term' f + size_of_term' t
514 | size_of_term' _ = 1;
516 fun term_ord' pr thy (Abs (_, T, t), Abs(_, U, u)) = (* ~ term.ML *)
517 (case term_ord' pr thy (t, u) of EQUAL => Term_Ord.typ_ord (T, U) | ord => ord)
518 | term_ord' pr thy (t, u) =
521 val (f, ts) = strip_comb t and (g, us) = strip_comb u;
522 val _ = tracing ("t= f@ts= \"" ^ UnparseC.term_in_thy thy f ^ "\" @ \"[" ^
523 commas (map (UnparseC.term_in_thy thy) ts) ^ "]\"");
524 val _ = tracing("u= g@us= \"" ^ UnparseC.term_in_thy thy g ^ "\" @ \"[" ^
525 commas (map (UnparseC.term_in_thy thy) us) ^ "]\"");
526 val _ = tracing ("size_of_term(t,u)= (" ^ string_of_int (size_of_term' t) ^ ", " ^
527 string_of_int (size_of_term' u) ^ ")");
528 val _ = tracing ("hd_ord(f,g) = " ^ (pr_ord o hd_ord) (f,g));
529 val _ = tracing ("terms_ord(ts,us) = " ^ (pr_ord o terms_ord str false) (ts, us));
530 val _ = tracing ("-------");
533 case int_ord (size_of_term' t, size_of_term' u) of
535 let val (f, ts) = strip_comb t and (g, us) = strip_comb u in
536 (case hd_ord (f, g) of EQUAL => (terms_ord str pr) (ts, us)
540 and hd_ord (f, g) = (* ~ term.ML *)
541 prod_ord (prod_ord Term_Ord.indexname_ord Term_Ord.typ_ord) int_ord (dest_hd' f, dest_hd' g)
542 and terms_ord _ pr (ts, us) =
543 list_ord (term_ord' pr (ThyC.get_theory "Isac_Knowledge"))(ts, us);
547 fun ord_make_polynomial (pr:bool) thy (_: subst) (ts, us) =
548 (term_ord' pr thy(***) (TermC.numerals_to_Free ts, TermC.numerals_to_Free us) = LESS );
552 Rewrite_Ord.rew_ord' := overwritel (! Rewrite_Ord.rew_ord', (* TODO: make analogous to KEStore_Elems.add_mets *)
553 [("termlessI", termlessI), ("ord_make_polynomial", ord_make_polynomial false \<^theory>)]);
556 subsection \<open>predicates\<close>
557 subsubsection \<open>in specifications\<close>
559 (* is_polyrat_in becomes true, if no bdv is in the denominator of a fraction*)
560 fun is_polyrat_in t v =
562 fun finddivide (_ $ _ $ _ $ _) _ = raise ERROR("is_polyrat_in:")
563 (* at the moment there is no term like this, but ....*)
564 | finddivide (Const (\<^const_name>\<open>divide\<close>,_) $ _ $ b) v = not (Prog_Expr.occurs_in v b)
565 | finddivide (_ $ t1 $ t2) v = finddivide t1 v orelse finddivide t2 v
566 | finddivide (_ $ t1) v = finddivide t1 v
567 | finddivide _ _ = false;
568 in finddivide t v end;
570 fun is_expanded_in t v = case expand_deg_in t v of SOME _ => true | NONE => false;
571 fun is_poly_in t v = case poly_deg_in t v of SOME _ => true | NONE => false;
572 fun has_degree_in t v = case expand_deg_in t v of SOME d => d | NONE => ~1;
574 (*.the expression contains + - * ^ only ?
575 this is weaker than 'is_polynomial' !.*)
576 fun is_polyexp (Free _) = true
577 | is_polyexp (Const _) = true (* potential danger: bdv is not considered *)
578 | is_polyexp (Const (\<^const_name>\<open>plus\<close>,_) $ Free _ $ num) =
579 if TermC.is_num num then true
580 else if TermC.is_variable num then true
582 | is_polyexp (Const (\<^const_name>\<open>plus\<close>, _) $ num $ Free _) =
583 if TermC.is_num num then true
584 else if TermC.is_variable num then true
586 | is_polyexp (Const (\<^const_name>\<open>minus\<close>, _) $ Free _ $ num) =
587 if TermC.is_num num then true
588 else if TermC.is_variable num then true
590 | is_polyexp (Const (\<^const_name>\<open>times\<close>, _) $ num $ Free _) =
591 if TermC.is_num num then true
592 else if TermC.is_variable num then true
594 | is_polyexp (Const (\<^const_name>\<open>Transcendental.powr\<close>,_) $ Free _ $ num) =
595 if TermC.is_num num then true
596 else if TermC.is_variable num then true
598 | is_polyexp (Const (\<^const_name>\<open>plus_class.plus\<close>,_) $ t1 $ t2) =
599 ((is_polyexp t1) andalso (is_polyexp t2))
600 | is_polyexp (Const (\<^const_name>\<open>Groups.minus_class.minus\<close>,_) $ t1 $ t2) =
601 ((is_polyexp t1) andalso (is_polyexp t2))
602 | is_polyexp (Const (\<^const_name>\<open>Groups.times_class.times\<close>,_) $ t1 $ t2) =
603 ((is_polyexp t1) andalso (is_polyexp t2))
604 | is_polyexp (Const (\<^const_name>\<open>Transcendental.powr\<close>,_) $ t1 $ t2) =
605 ((is_polyexp t1) andalso (is_polyexp t2))
606 | is_polyexp num = TermC.is_num num;
609 subsubsection \<open>for hard-coded AC rewriting\<close>
611 (* auch Klammerung muss übereinstimmen;
612 sort_variables klammert Produkte rechtslastig*)
613 fun is_multUnordered t = ((is_polyexp t) andalso not (t = sort_variables t));
615 fun is_addUnordered t = ((is_polyexp t) andalso not (t = sort_monoms t));
618 subsection \<open>evaluations functions\<close>
619 subsubsection \<open>for predicates\<close>
621 fun eval_is_polyrat_in _ _(p as (Const (\<^const_name>\<open>Poly.is_polyrat_in\<close>, _) $ t $ v)) _ =
623 then SOME ((UnparseC.term p) ^ " = True",
624 HOLogic.Trueprop $ (TermC.mk_equality (p, @{term True})))
625 else SOME ((UnparseC.term p) ^ " = True",
626 HOLogic.Trueprop $ (TermC.mk_equality (p, @{term False})))
627 | eval_is_polyrat_in _ _ _ _ = ((*tracing"### no matches";*) NONE);
629 (*("is_expanded_in", ("Poly.is_expanded_in", eval_is_expanded_in ""))*)
630 fun eval_is_expanded_in _ _
631 (p as (Const (\<^const_name>\<open>Poly.is_expanded_in\<close>, _) $ t $ v)) _ =
632 if is_expanded_in t v
633 then SOME ((UnparseC.term p) ^ " = True",
634 HOLogic.Trueprop $ (TermC.mk_equality (p, @{term True})))
635 else SOME ((UnparseC.term p) ^ " = True",
636 HOLogic.Trueprop $ (TermC.mk_equality (p, @{term False})))
637 | eval_is_expanded_in _ _ _ _ = NONE;
639 (*("is_poly_in", ("Poly.is_poly_in", eval_is_poly_in ""))*)
640 fun eval_is_poly_in _ _
641 (p as (Const (\<^const_name>\<open>Poly.is_poly_in\<close>, _) $ t $ v)) _ =
643 then SOME ((UnparseC.term p) ^ " = True",
644 HOLogic.Trueprop $ (TermC.mk_equality (p, @{term True})))
645 else SOME ((UnparseC.term p) ^ " = True",
646 HOLogic.Trueprop $ (TermC.mk_equality (p, @{term False})))
647 | eval_is_poly_in _ _ _ _ = NONE;
649 (*("has_degree_in", ("Poly.has_degree_in", eval_has_degree_in ""))*)
650 fun eval_has_degree_in _ _
651 (p as (Const (\<^const_name>\<open>Poly.has_degree_in\<close>, _) $ t $ v)) _ =
652 let val d = has_degree_in t v
653 val d' = TermC.term_of_num HOLogic.realT d
654 in SOME ((UnparseC.term p) ^ " = " ^ (string_of_int d),
655 HOLogic.Trueprop $ (TermC.mk_equality (p, d')))
657 | eval_has_degree_in _ _ _ _ = NONE;
659 (*("is_polyexp", ("Poly.is_polyexp", eval_is_polyexp ""))*)
660 fun eval_is_polyexp (thmid:string) _
661 (t as (Const (\<^const_name>\<open>Poly.is_polyexp\<close>, _) $ arg)) thy =
663 then SOME (TermC.mk_thmid thmid (UnparseC.term_in_thy thy arg) "",
664 HOLogic.Trueprop $ (TermC.mk_equality (t, @{term True})))
665 else SOME (TermC.mk_thmid thmid (UnparseC.term_in_thy thy arg) "",
666 HOLogic.Trueprop $ (TermC.mk_equality (t, @{term False})))
667 | eval_is_polyexp _ _ _ _ = NONE;
670 subsubsection \<open>for hard-coded AC rewriting\<close>
673 (*("is_addUnordered", ("Poly.is_addUnordered", eval_is_addUnordered ""))*)
674 fun eval_is_addUnordered (thmid:string) _
675 (t as (Const (\<^const_name>\<open>Poly.is_addUnordered\<close>, _) $ arg)) thy =
676 if is_addUnordered arg
677 then SOME (TermC.mk_thmid thmid (UnparseC.term_in_thy thy arg) "",
678 HOLogic.Trueprop $ (TermC.mk_equality (t, @{term True})))
679 else SOME (TermC.mk_thmid thmid (UnparseC.term_in_thy thy arg) "",
680 HOLogic.Trueprop $ (TermC.mk_equality (t, @{term False})))
681 | eval_is_addUnordered _ _ _ _ = NONE;
683 fun eval_is_multUnordered (thmid:string) _
684 (t as (Const (\<^const_name>\<open>Poly.is_multUnordered\<close>, _) $ arg)) thy =
685 if is_multUnordered arg
686 then SOME (TermC.mk_thmid thmid (UnparseC.term_in_thy thy arg) "",
687 HOLogic.Trueprop $ (TermC.mk_equality (t, @{term True})))
688 else SOME (TermC.mk_thmid thmid (UnparseC.term_in_thy thy arg) "",
689 HOLogic.Trueprop $ (TermC.mk_equality (t, @{term False})))
690 | eval_is_multUnordered _ _ _ _ = NONE;
692 calculation is_polyrat_in = \<open>eval_is_polyrat_in "#eval_is_polyrat_in"\<close>
693 calculation is_expanded_in = \<open>eval_is_expanded_in ""\<close>
694 calculation is_poly_in = \<open>eval_is_poly_in ""\<close>
695 calculation has_degree_in = \<open>eval_has_degree_in ""\<close>
696 calculation is_polyexp = \<open>eval_is_polyexp ""\<close>
697 calculation is_multUnordered = \<open>eval_is_multUnordered ""\<close>
698 calculation is_addUnordered = \<open>eval_is_addUnordered ""\<close>
700 subsection \<open>rule-sets\<close>
701 subsubsection \<open>without specific order\<close>
703 (* used only for merge *)
704 val calculate_Poly = Rule_Set.append_rules "calculate_PolyFIXXXME.not.impl." Rule_Set.empty [];
706 (*.for evaluation of conditions in rewrite rules.*)
707 val Poly_erls = Rule_Set.append_rules "Poly_erls" Atools_erls
708 [\<^rule_eval>\<open>HOL.eq\<close> (Prog_Expr.eval_equal "#equal_"),
709 \<^rule_thm>\<open>real_unari_minus\<close>,
710 \<^rule_eval>\<open>plus\<close> (eval_binop "#add_"),
711 \<^rule_eval>\<open>minus\<close> (eval_binop "#sub_"),
712 \<^rule_eval>\<open>times\<close> (eval_binop "#mult_"),
713 \<^rule_eval>\<open>powr\<close> (eval_binop "#power_")];
715 val poly_crls = Rule_Set.append_rules "poly_crls" Atools_crls
716 [\<^rule_eval>\<open>HOL.eq\<close> (Prog_Expr.eval_equal "#equal_"),
717 \<^rule_thm>\<open>real_unari_minus\<close>,
718 \<^rule_eval>\<open>plus\<close> (eval_binop "#add_"),
719 \<^rule_eval>\<open>minus\<close> (eval_binop "#sub_"),
720 \<^rule_eval>\<open>times\<close> (eval_binop "#mult_"),
721 \<^rule_eval>\<open>powr\<close> (eval_binop "#power_")];
725 Rule_Def.Repeat {id = "expand", preconds = [], rew_ord = ("dummy_ord", Rewrite_Ord.dummy_ord),
726 erls = Rule_Set.empty,srls = Rule_Set.Empty, calc = [], errpatts = [],
727 rules = [\<^rule_thm>\<open>distrib_right\<close>,
728 (*"(z1.0 + z2.0) * w = z1.0 * w + z2.0 * w"*)
729 \<^rule_thm>\<open>distrib_left\<close>
730 (*"w * (z1.0 + z2.0) = w * z1.0 + w * z2.0"*)
731 ], scr = Rule.Empty_Prog};
733 (* erls for calculate_Rational + etc *)
735 Rule_Def.Repeat {id = "powers_erls", preconds = [], rew_ord = ("dummy_ord",Rewrite_Ord.dummy_ord),
736 erls = Rule_Set.empty, srls = Rule_Set.Empty, calc = [], errpatts = [],
738 [Rule.Eval ("Prog_Expr.matches", Prog_Expr.eval_matches "#matches_"),
739 Rule.Eval ("Prog_Expr.is_atom", Prog_Expr.eval_is_atom "#is_atom_"),
740 Rule.Eval ("Prog_Expr.is_even", Prog_Expr.eval_is_even "#is_even_"),
741 Rule.Eval ("Orderings.ord_class.less", Prog_Expr.eval_equ "#less_"),
742 Rule.Thm ("not_false", @{thm not_false}),
743 Rule.Thm ("not_true", @{thm not_true}),
744 Rule.Eval ("Groups.plus_class.plus", (**)eval_binop "#add_")
746 scr = Rule.Empty_Prog
751 Rule_Def.Repeat {id = "discard_minus", preconds = [], rew_ord = ("dummy_ord", Rewrite_Ord.dummy_ord),
752 erls = powers_erls, srls = Rule_Set.Empty, calc = [], errpatts = [],
754 [\<^rule_thm>\<open>real_diff_minus\<close>,
755 (*"a - b = a + -1 * b"*)
756 Rule.Thm ("real_mult_minus1_sym", (@{thm real_mult_minus1_sym}))
757 (*"\<not>(z is_const) ==> - (z::real) = -1 * z"*)],
758 scr = Rule.Empty_Prog};
761 Rule_Def.Repeat{id = "expand_poly_", preconds = [],
762 rew_ord = ("dummy_ord", Rewrite_Ord.dummy_ord),
763 erls = powers_erls, srls = Rule_Set.Empty,
764 calc = [], errpatts = [],
766 [\<^rule_thm>\<open>real_plus_binom_pow4\<close>,
767 (*"(a + b) \<up> 4 = ... "*)
768 \<^rule_thm>\<open>real_plus_binom_pow5\<close>,
769 (*"(a + b) \<up> 5 = ... "*)
770 \<^rule_thm>\<open>real_plus_binom_pow3\<close>,
771 (*"(a + b) \<up> 3 = a \<up> 3 + 3*a \<up> 2*b + 3*a*b \<up> 2 + b \<up> 3" *)
772 (*WN071229 changed/removed for Schaerding -----vvv*)
773 (*\<^rule_thm>\<open>real_plus_binom_pow2\<close>,*)
774 (*"(a + b) \<up> 2 = a \<up> 2 + 2*a*b + b \<up> 2"*)
775 \<^rule_thm>\<open>real_plus_binom_pow2\<close>,
776 (*"(a + b) \<up> 2 = (a + b) * (a + b)"*)
777 (*\<^rule_thm>\<open>real_plus_minus_binom1_p_p\<close>,*)
778 (*"(a + b)*(a + -1 * b) = a \<up> 2 + -1*b \<up> 2"*)
779 (*\<^rule_thm>\<open>real_plus_minus_binom2_p_p\<close>,*)
780 (*"(a + -1 * b)*(a + b) = a \<up> 2 + -1*b \<up> 2"*)
781 (*WN071229 changed/removed for Schaerding -----\<up>*)
783 \<^rule_thm>\<open>distrib_right\<close>,
784 (*"(z1.0 + z2.0) * w = z1.0 * w + z2.0 * w"*)
785 \<^rule_thm>\<open>distrib_left\<close>,
786 (*"w * (z1.0 + z2.0) = w * z1.0 + w * z2.0"*)
788 \<^rule_thm>\<open>realpow_multI\<close>,
789 (*"(r * s) \<up> n = r \<up> n * s \<up> n"*)
790 \<^rule_thm>\<open>realpow_pow\<close>,
791 (*"(a \<up> b) \<up> c = a \<up> (b * c)"*)
793 Rule.Thm ("realpow_minus_even", @{thm realpow_minus_even}),
794 (*"n is_even ==> (- r) \<up> n = r \<up> n"*)
795 Rule.Thm ("realpow_minus_odd", @{thm realpow_minus_odd})
796 (*"Not (n is_even) ==> (- r) \<up> n = -1 * r \<up> n"*)
798 ], scr = Rule.Empty_Prog};
800 val expand_poly_rat_ =
801 Rule_Def.Repeat{id = "expand_poly_rat_", preconds = [],
802 rew_ord = ("dummy_ord", Rewrite_Ord.dummy_ord),
803 erls = Rule_Set.append_rules "Rule_Set.empty-expand_poly_rat_" Rule_Set.empty
804 [Rule.Eval ("Poly.is_polyexp", eval_is_polyexp ""),
805 Rule.Eval ("Prog_Expr.is_even", Prog_Expr.eval_is_even "#is_even_"),
806 Rule.Thm ("not_false", @{thm not_false}),
807 Rule.Thm ("not_true", @{thm not_true})
809 srls = Rule_Set.Empty,
810 calc = [], errpatts = [],
812 [\<^rule_thm>\<open>real_plus_binom_pow4_poly\<close>,
813 (*"[| a is_polyexp; b is_polyexp |] ==> (a + b) \<up> 4 = ... "*)
814 \<^rule_thm>\<open>real_plus_binom_pow5_poly\<close>,
815 (*"[| a is_polyexp; b is_polyexp |] ==> (a + b) \<up> 5 = ... "*)
816 \<^rule_thm>\<open>real_plus_binom_pow2_poly\<close>,
817 (*"[| a is_polyexp; b is_polyexp |] ==>
818 (a + b) \<up> 2 = a \<up> 2 + 2*a*b + b \<up> 2"*)
819 \<^rule_thm>\<open>real_plus_binom_pow3_poly\<close>,
820 (*"[| a is_polyexp; b is_polyexp |] ==>
821 (a + b) \<up> 3 = a \<up> 3 + 3*a \<up> 2*b + 3*a*b \<up> 2 + b \<up> 3" *)
822 \<^rule_thm>\<open>real_plus_minus_binom1_p_p\<close>,
823 (*"(a + b)*(a + -1 * b) = a \<up> 2 + -1*b \<up> 2"*)
824 \<^rule_thm>\<open>real_plus_minus_binom2_p_p\<close>,
825 (*"(a + -1 * b)*(a + b) = a \<up> 2 + -1*b \<up> 2"*)
827 \<^rule_thm>\<open>real_add_mult_distrib_poly\<close>,
828 (*"w is_polyexp ==> (z1.0 + z2.0) * w = z1.0 * w + z2.0 * w"*)
829 \<^rule_thm>\<open>real_add_mult_distrib2_poly\<close>,
830 (*"w is_polyexp ==> w * (z1.0 + z2.0) = w * z1.0 + w * z2.0"*)
832 \<^rule_thm>\<open>realpow_multI_poly\<close>,
833 (*"[| r is_polyexp; s is_polyexp |] ==>
834 (r * s) \<up> n = r \<up> n * s \<up> n"*)
835 Rule.Thm ("realpow_pow", @{thm realpow_pow}),
836 (*"(a \<up> b) \<up> c = a \<up> (b * c)"*)
837 Rule.Thm ("realpow_minus_even", @{thm realpow_minus_even}),
838 (*"n is_even ==> (- r) \<up> n = r \<up> n"*)
839 Rule.Thm ("realpow_minus_odd", @{thm realpow_minus_odd})
840 (*"\<not> (n is_even) ==> (- r) \<up> n = -1 * r \<up> n"*)
841 ], scr = Rule.Empty_Prog};
843 val simplify_power_ =
844 Rule_Def.Repeat{id = "simplify_power_", preconds = [],
845 rew_ord = ("dummy_ord", Rewrite_Ord.dummy_ord),
846 erls = Rule_Set.empty, srls = Rule_Set.Empty,
847 calc = [], errpatts = [],
848 rules = [(*MG: Reihenfolge der folgenden 2 Rule.Thm muss so bleiben, wegen
849 a*(a*a) --> a*a \<up> 2 und nicht a*(a*a) --> a \<up> 2*a *)
850 \<^rule_thm_sym>\<open>realpow_twoI\<close>,
851 (*"r * r = r \<up> 2"*)
852 \<^rule_thm>\<open>realpow_twoI_assoc_l\<close>,
853 (*"r * (r * s) = r \<up> 2 * s"*)
855 \<^rule_thm>\<open>realpow_plus_1\<close>,
856 (*"r * r \<up> n = r \<up> (n + 1)"*)
857 \<^rule_thm>\<open>realpow_plus_1_assoc_l\<close>,
858 (*"r * (r \<up> m * s) = r \<up> (1 + m) * s"*)
859 (*MG 9.7.03: neues Rule.Thm wegen a*(a*(a*b)) --> a \<up> 2*(a*b) *)
860 \<^rule_thm>\<open>realpow_plus_1_assoc_l2\<close>,
861 (*"r \<up> m * (r * s) = r \<up> (1 + m) * s"*)
863 \<^rule_thm_sym>\<open>realpow_addI\<close>,
864 (*"r \<up> n * r \<up> m = r \<up> (n + m)"*)
865 \<^rule_thm>\<open>realpow_addI_assoc_l\<close>,
866 (*"r \<up> n * (r \<up> m * s) = r \<up> (n + m) * s"*)
868 (* ist in expand_poly - wird hier aber auch gebraucht, wegen:
869 "r * r = r \<up> 2" wenn r=a \<up> b*)
870 \<^rule_thm>\<open>realpow_pow\<close>
871 (*"(a \<up> b) \<up> c = a \<up> (b * c)"*)
872 ], scr = Rule.Empty_Prog};
874 val calc_add_mult_pow_ =
875 Rule_Def.Repeat{id = "calc_add_mult_pow_", preconds = [],
876 rew_ord = ("dummy_ord", Rewrite_Ord.dummy_ord),
877 erls = Atools_erls(*erls3.4.03*),srls = Rule_Set.Empty,
878 calc = [("PLUS" , (\<^const_name>\<open>plus\<close>, eval_binop "#add_")),
879 ("TIMES" , (\<^const_name>\<open>times\<close>, eval_binop "#mult_")),
880 ("POWER", (\<^const_name>\<open>powr\<close>, eval_binop "#power_"))
883 rules = [\<^rule_eval>\<open>plus\<close> (eval_binop "#add_"),
884 \<^rule_eval>\<open>times\<close> (eval_binop "#mult_"),
885 \<^rule_eval>\<open>powr\<close> (eval_binop "#power_")
886 ], scr = Rule.Empty_Prog};
888 val reduce_012_mult_ =
889 Rule_Def.Repeat{id = "reduce_012_mult_", preconds = [],
890 rew_ord = ("dummy_ord", Rewrite_Ord.dummy_ord),
891 erls = Rule_Set.empty,srls = Rule_Set.Empty,
892 calc = [], errpatts = [],
893 rules = [(* MG: folgende Rule.Thm müssen hier stehen bleiben: *)
894 \<^rule_thm>\<open>mult_1_right\<close>,
895 (*"z * 1 = z"*) (*wegen "a * b * b \<up> (-1) + a"*)
896 \<^rule_thm>\<open>realpow_zeroI\<close>,
897 (*"r \<up> 0 = 1"*) (*wegen "a*a \<up> (-1)*c + b + c"*)
898 \<^rule_thm>\<open>realpow_oneI\<close>,
900 \<^rule_thm>\<open>realpow_eq_oneI\<close>
902 ], scr = Rule.Empty_Prog};
904 val collect_numerals_ =
905 Rule_Def.Repeat{id = "collect_numerals_", preconds = [],
906 rew_ord = ("dummy_ord", Rewrite_Ord.dummy_ord),
907 erls = Atools_erls, srls = Rule_Set.Empty,
908 calc = [("PLUS" , (\<^const_name>\<open>plus\<close>, eval_binop "#add_"))
911 [\<^rule_thm>\<open>real_num_collect\<close>,
912 (*"[| l is_const; m is_const |]==>l * n + m * n = (l + m) * n"*)
913 \<^rule_thm>\<open>real_num_collect_assoc_r\<close>,
914 (*"[| l is_const; m is_const |] ==> \
915 \(k + m * n) + l * n = k + (l + m)*n"*)
916 \<^rule_thm>\<open>real_one_collect\<close>,
917 (*"m is_const ==> n + m * n = (1 + m) * n"*)
918 \<^rule_thm>\<open>real_one_collect_assoc_r\<close>,
919 (*"m is_const ==> (k + n) + m * n = k + (m + 1) * n"*)
921 \<^rule_eval>\<open>plus\<close> (eval_binop "#add_"),
923 (*MG: Reihenfolge der folgenden 2 Rule.Thm muss so bleiben, wegen
924 (a+a)+a --> a + 2*a --> 3*a and not (a+a)+a --> 2*a + a *)
925 \<^rule_thm>\<open>real_mult_2_assoc_r\<close>,
926 (*"(k + z1) + z1 = k + 2 * z1"*)
927 \<^rule_thm_sym>\<open>real_mult_2\<close>
928 (*"z1 + z1 = 2 * z1"*)
929 ], scr = Rule.Empty_Prog};
932 Rule_Def.Repeat{id = "reduce_012_", preconds = [],
933 rew_ord = ("dummy_ord", Rewrite_Ord.dummy_ord),
934 erls = Rule_Set.empty,srls = Rule_Set.Empty, calc = [], errpatts = [],
935 rules = [\<^rule_thm>\<open>mult_1_left\<close>,
937 \<^rule_thm>\<open>mult_zero_left\<close>,
939 \<^rule_thm>\<open>mult_zero_right\<close>,
941 \<^rule_thm>\<open>add_0_left\<close>,
943 \<^rule_thm>\<open>add_0_right\<close>,
944 (*"z + 0 = z"*) (*wegen a+b-b --> a+(1-1)*b --> a+0 --> a*)
946 (*\<^rule_thm>\<open>realpow_oneI\<close>*)
947 (*"?r \<up> 1 = ?r"*)
948 \<^rule_thm>\<open>division_ring_divide_zero\<close>
950 ], scr = Rule.Empty_Prog};
952 val discard_parentheses1 =
953 Rule_Set.append_rules "discard_parentheses1" Rule_Set.empty
954 [\<^rule_thm_sym>\<open>mult.assoc\<close>
955 (*"?z1.1 * (?z2.1 * ?z3.1) = ?z1.1 * ?z2.1 * ?z3.1"*)
956 (*\<^rule_thm_sym>\<open>add.assoc\<close>*)
957 (*"?z1.1 + (?z2.1 + ?z3.1) = ?z1.1 + ?z2.1 + ?z3.1"*)
961 Rule_Def.Repeat{id = "expand_poly", preconds = [],
962 rew_ord = ("dummy_ord", Rewrite_Ord.dummy_ord),
963 erls = powers_erls, srls = Rule_Set.Empty, calc = [], errpatts = [],
965 [Rule.Thm ("distrib_right" , @{thm distrib_right}),
966 (*"(z1.0 + z2.0) * w = z1.0 * w + z2.0 * w"*)
967 \<^rule_thm>\<open>distrib_left\<close>,
968 (*"w * (z1.0 + z2.0) = w * z1.0 + w * z2.0"*)
969 (*Rule.Thm ("distrib_right1", @{thm distrib_right}1),
970 ....... 18.3.03 undefined???*)
972 \<^rule_thm>\<open>real_plus_binom_pow2\<close>,
973 (*"(a + b) \<up> 2 = a \<up> 2 + 2*a*b + b \<up> 2"*)
974 \<^rule_thm>\<open>real_minus_binom_pow2_p\<close>,
975 (*"(a - b) \<up> 2 = a \<up> 2 + -2*a*b + b \<up> 2"*)
976 \<^rule_thm>\<open>real_plus_minus_binom1_p\<close>,
977 (*"(a + b)*(a - b) = a \<up> 2 + -1*b \<up> 2"*)
978 \<^rule_thm>\<open>real_plus_minus_binom2_p\<close>,
979 (*"(a - b)*(a + b) = a \<up> 2 + -1*b \<up> 2"*)
981 \<^rule_thm>\<open>minus_minus\<close>,
983 \<^rule_thm>\<open>real_diff_minus\<close>,
984 (*"a - b = a + -1 * b"*)
985 Rule.Thm ("real_mult_minus1_sym", (@{thm real_mult_minus1_sym}))
986 (*"\<not>(z is_const) ==> - (z::real) = -1 * z"*)
988 (*\<^rule_thm>\<open>real_minus_add_distrib\<close>,*)
989 (*"- (?x + ?y) = - ?x + - ?y"*)
990 (*\<^rule_thm>\<open>real_diff_plus\<close>*)
992 ], scr = Rule.Empty_Prog};
995 Rule_Def.Repeat{id = "simplify_power", preconds = [],
996 rew_ord = ("dummy_ord", Rewrite_Ord.dummy_ord),
997 erls = Rule_Set.empty, srls = Rule_Set.Empty,
998 calc = [], errpatts = [],
999 rules = [\<^rule_thm>\<open>realpow_multI\<close>,
1000 (*"(r * s) \<up> n = r \<up> n * s \<up> n"*)
1002 \<^rule_thm_sym>\<open>realpow_twoI\<close>,
1003 (*"r1 * r1 = r1 \<up> 2"*)
1004 \<^rule_thm>\<open>realpow_plus_1\<close>,
1005 (*"r * r \<up> n = r \<up> (n + 1)"*)
1006 \<^rule_thm>\<open>realpow_pow\<close>,
1007 (*"(a \<up> b) \<up> c = a \<up> (b * c)"*)
1008 \<^rule_thm_sym>\<open>realpow_addI\<close>,
1009 (*"r \<up> n * r \<up> m = r \<up> (n + m)"*)
1010 \<^rule_thm>\<open>realpow_oneI\<close>,
1012 \<^rule_thm>\<open>realpow_eq_oneI\<close>
1014 ], scr = Rule.Empty_Prog};
1016 val collect_numerals =
1017 Rule_Def.Repeat{id = "collect_numerals", preconds = [],
1018 rew_ord = ("dummy_ord", Rewrite_Ord.dummy_ord),
1019 erls = Atools_erls(*erls3.4.03*),srls = Rule_Set.Empty,
1020 calc = [("PLUS" , (\<^const_name>\<open>plus\<close>, eval_binop "#add_")),
1021 ("TIMES" , (\<^const_name>\<open>times\<close>, eval_binop "#mult_")),
1022 ("POWER", (\<^const_name>\<open>powr\<close>, eval_binop "#power_"))
1024 rules = [\<^rule_thm>\<open>real_num_collect\<close>,
1025 (*"[| l is_const; m is_const |]==>l * n + m * n = (l + m) * n"*)
1026 \<^rule_thm>\<open>real_num_collect_assoc\<close>,
1027 (*"[| l is_const; m is_const |] ==>
1028 l * n + (m * n + k) = (l + m) * n + k"*)
1029 \<^rule_thm>\<open>real_one_collect\<close>,
1030 (*"m is_const ==> n + m * n = (1 + m) * n"*)
1031 \<^rule_thm>\<open>real_one_collect_assoc\<close>,
1032 (*"m is_const ==> k + (n + m * n) = k + (1 + m) * n"*)
1033 \<^rule_eval>\<open>plus\<close> (eval_binop "#add_"),
1034 \<^rule_eval>\<open>times\<close> (eval_binop "#mult_"),
1035 \<^rule_eval>\<open>powr\<close> (eval_binop "#power_")
1036 ], scr = Rule.Empty_Prog};
1038 Rule_Def.Repeat{id = "reduce_012", preconds = [],
1039 rew_ord = ("dummy_ord", Rewrite_Ord.dummy_ord),
1040 erls = Rule_Set.empty,srls = Rule_Set.Empty,
1041 calc = [], errpatts = [],
1042 rules = [\<^rule_thm>\<open>mult_1_left\<close>,
1044 (*\<^rule_thm>\<open>real_mult_minus1\<close>,14.3.03*)
1046 Rule.Thm ("minus_mult_left", (@{thm minus_mult_left} RS @{thm sym})),
1047 (*- (?x * ?y) = "- ?x * ?y"*)
1048 (*\<^rule_thm>\<open>real_minus_mult_cancel\<close>,
1049 (*"- ?x * - ?y = ?x * ?y"*)---*)
1050 \<^rule_thm>\<open>mult_zero_left\<close>,
1052 \<^rule_thm>\<open>add_0_left\<close>,
1054 \<^rule_thm>\<open>right_minus\<close>,
1056 \<^rule_thm_sym>\<open>real_mult_2\<close>,
1057 (*"z1 + z1 = 2 * z1"*)
1058 \<^rule_thm>\<open>real_mult_2_assoc\<close>
1059 (*"z1 + (z1 + k) = 2 * z1 + k"*)
1060 ], scr = Rule.Empty_Prog};
1062 val discard_parentheses =
1063 Rule_Set.append_rules "discard_parentheses" Rule_Set.empty
1064 [\<^rule_thm_sym>\<open>mult.assoc\<close>, \<^rule_thm_sym>\<open>add.assoc\<close>];
1067 subsubsection \<open>hard-coded AC rewriting\<close>
1069 (*MG.0401: termorders for multivariate polys dropped due to principal problems:
1070 (total-degree-)ordering of monoms NOT possible with size_of_term GIVEN*)
1071 val order_add_mult =
1072 Rule_Def.Repeat{id = "order_add_mult", preconds = [],
1073 rew_ord = ("ord_make_polynomial",ord_make_polynomial false \<^theory>),
1074 erls = Rule_Set.empty,srls = Rule_Set.Empty,
1075 calc = [], errpatts = [],
1076 rules = [\<^rule_thm>\<open>mult.commute\<close>,
1078 \<^rule_thm>\<open>real_mult_left_commute\<close>,
1079 (*z1.0 * (z2.0 * z3.0) = z2.0 * (z1.0 * z3.0)*)
1080 \<^rule_thm>\<open>mult.assoc\<close>,
1081 (*z1.0 * z2.0 * z3.0 = z1.0 * (z2.0 * z3.0)*)
1082 \<^rule_thm>\<open>add.commute\<close>,
1084 \<^rule_thm>\<open>add.left_commute\<close>,
1085 (*x + (y + z) = y + (x + z)*)
1086 \<^rule_thm>\<open>add.assoc\<close>
1087 (*z1.0 + z2.0 + z3.0 = z1.0 + (z2.0 + z3.0)*)
1088 ], scr = Rule.Empty_Prog};
1089 (*MG.0401: termorders for multivariate polys dropped due to principal problems:
1090 (total-degree-)ordering of monoms NOT possible with size_of_term GIVEN*)
1092 Rule_Def.Repeat{id = "order_mult", preconds = [],
1093 rew_ord = ("ord_make_polynomial",ord_make_polynomial false \<^theory>),
1094 erls = Rule_Set.empty,srls = Rule_Set.Empty,
1095 calc = [], errpatts = [],
1096 rules = [\<^rule_thm>\<open>mult.commute\<close>,
1098 \<^rule_thm>\<open>real_mult_left_commute\<close>,
1099 (*z1.0 * (z2.0 * z3.0) = z2.0 * (z1.0 * z3.0)*)
1100 \<^rule_thm>\<open>mult.assoc\<close>
1101 (*z1.0 * z2.0 * z3.0 = z1.0 * (z2.0 * z3.0)*)
1102 ], scr = Rule.Empty_Prog};
1105 fun attach_form (_: Rule.rule list list) (_: term) (_: term) = (*still missing*)
1106 []:(Rule.rule * (term * term list)) list;
1107 fun init_state (_: term) = Rule_Set.e_rrlsstate;
1108 fun locate_rule (_: Rule.rule list list) (_: term) (_: Rule.rule) =
1109 ([]:(Rule.rule * (term * term list)) list);
1110 fun next_rule (_: Rule.rule list list) (_: term) = (NONE: Rule.rule option);
1111 fun normal_form t = SOME (sort_variables t, []: term list);
1114 Rule_Set.Rrls {id = "order_mult_",
1116 (* ?p matched with the current term gives an environment,
1117 which evaluates (the instantiated) "?p is_multUnordered" to true *)
1118 [([TermC.parse_patt \<^theory> "?p is_multUnordered"],
1119 TermC.parse_patt \<^theory> "?p :: real")],
1120 rew_ord = ("dummy_ord", Rewrite_Ord.dummy_ord),
1121 erls = Rule_Set.append_rules "Rule_Set.empty-is_multUnordered" Rule_Set.empty
1122 [\<^rule_eval>\<open>is_multUnordered\<close> (eval_is_multUnordered "")],
1123 calc = [("PLUS" , (\<^const_name>\<open>plus\<close>, eval_binop "#add_")),
1124 ("TIMES" , (\<^const_name>\<open>times\<close>, eval_binop "#mult_")),
1125 ("DIVIDE", (\<^const_name>\<open>divide\<close>, Prog_Expr.eval_cancel "#divide_e")),
1126 ("POWER" , (\<^const_name>\<open>powr\<close>, eval_binop "#power_"))],
1128 scr = Rule.Rfuns {init_state = init_state,
1129 normal_form = normal_form,
1130 locate_rule = locate_rule,
1131 next_rule = next_rule,
1132 attach_form = attach_form}};
1133 val order_mult_rls_ =
1134 Rule_Def.Repeat {id = "order_mult_rls_", preconds = [],
1135 rew_ord = ("dummy_ord", Rewrite_Ord.dummy_ord),
1136 erls = Rule_Set.empty,srls = Rule_Set.Empty,
1137 calc = [], errpatts = [],
1138 rules = [Rule.Rls_ order_mult_
1139 ], scr = Rule.Empty_Prog};
1143 fun attach_form (_: Rule.rule list list) (_: term) (_: term) = (*still missing*)
1144 []: (Rule.rule * (term * term list)) list;
1145 fun init_state (_: term) = Rule_Set.e_rrlsstate;
1146 fun locate_rule (_: Rule.rule list list) (_: term) (_: Rule.rule) =
1147 ([]: (Rule.rule * (term * term list)) list);
1148 fun next_rule (_: Rule.rule list list) (_: term) = (NONE: Rule.rule option);
1149 fun normal_form t = SOME (sort_monoms t,[]: term list);
1152 Rule_Set.Rrls {id = "order_add_",
1153 prepat = (*WN.18.6.03 Preconditions und Pattern,
1154 die beide passen muessen, damit das Rule_Set.Rrls angewandt wird*)
1155 [([TermC.parse_patt @{theory} "?p is_addUnordered"],
1156 TermC.parse_patt @{theory} "?p :: real"
1157 (*WN.18.6.03 also KEIN pattern, dieses erzeugt nur das Environment
1158 fuer die Evaluation der Precondition "p is_addUnordered"*))],
1159 rew_ord = ("dummy_ord", Rewrite_Ord.dummy_ord),
1160 erls = Rule_Set.append_rules "Rule_Set.empty-is_addUnordered" Rule_Set.empty(*MG: poly_erls*)
1161 [\<^rule_eval>\<open>is_addUnordered\<close> (eval_is_addUnordered "")],
1162 calc = [("PLUS" ,(\<^const_name>\<open>plus\<close>, eval_binop "#add_")),
1163 ("TIMES" ,(\<^const_name>\<open>times\<close>, eval_binop "#mult_")),
1164 ("DIVIDE",(\<^const_name>\<open>divide\<close>, Prog_Expr.eval_cancel "#divide_e")),
1165 ("POWER" ,(\<^const_name>\<open>powr\<close> , eval_binop "#power_"))],
1167 scr = Rule.Rfuns {init_state = init_state,
1168 normal_form = normal_form,
1169 locate_rule = locate_rule,
1170 next_rule = next_rule,
1171 attach_form = attach_form}};
1173 val order_add_rls_ =
1174 Rule_Def.Repeat {id = "order_add_rls_", preconds = [],
1175 rew_ord = ("dummy_ord", Rewrite_Ord.dummy_ord),
1176 erls = Rule_Set.empty,srls = Rule_Set.Empty,
1177 calc = [], errpatts = [],
1178 rules = [Rule.Rls_ order_add_
1179 ], scr = Rule.Empty_Prog};
1182 text \<open>rule-set make_polynomial also named norm_Poly:
1183 Rewrite order has not been implemented properly; the order is better in
1184 make_polynomial_in (coded in SML).
1185 Notes on state of development:
1186 \# surprise 2006: test --- norm_Poly NOT COMPLETE ---
1187 \# migration Isabelle2002 --> 2011 weakened the rule set, see test
1188 --- Matthias Goldgruber 2003 rewrite orders ---, raise ERROR "ord_make_polynomial_in #16b"
1191 (*. see MG-DA.p.52ff .*)
1192 val make_polynomial(*MG.03, overwrites version from above,
1193 previously 'make_polynomial_'*) =
1194 Rule_Set.Sequence {id = "make_polynomial", preconds = []:term list,
1195 rew_ord = ("dummy_ord", Rewrite_Ord.dummy_ord),
1196 erls = Atools_erls, srls = Rule_Set.Empty,calc = [], errpatts = [],
1197 rules = [Rule.Rls_ discard_minus,
1198 Rule.Rls_ expand_poly_,
1199 \<^rule_eval>\<open>times\<close> (eval_binop "#mult_"),
1200 Rule.Rls_ order_mult_rls_,
1201 Rule.Rls_ simplify_power_,
1202 Rule.Rls_ calc_add_mult_pow_,
1203 Rule.Rls_ reduce_012_mult_,
1204 Rule.Rls_ order_add_rls_,
1205 Rule.Rls_ collect_numerals_,
1206 Rule.Rls_ reduce_012_,
1207 Rule.Rls_ discard_parentheses1
1209 scr = Rule.Empty_Prog
1213 val norm_Poly(*=make_polynomial*) =
1214 Rule_Set.Sequence {id = "norm_Poly", preconds = []:term list,
1215 rew_ord = ("dummy_ord", Rewrite_Ord.dummy_ord),
1216 erls = Atools_erls, srls = Rule_Set.Empty, calc = [], errpatts = [],
1217 rules = [Rule.Rls_ discard_minus,
1218 Rule.Rls_ expand_poly_,
1219 \<^rule_eval>\<open>times\<close> (eval_binop "#mult_"),
1220 Rule.Rls_ order_mult_rls_,
1221 Rule.Rls_ simplify_power_,
1222 Rule.Rls_ calc_add_mult_pow_,
1223 Rule.Rls_ reduce_012_mult_,
1224 Rule.Rls_ order_add_rls_,
1225 Rule.Rls_ collect_numerals_,
1226 Rule.Rls_ reduce_012_,
1227 Rule.Rls_ discard_parentheses1
1229 scr = Rule.Empty_Prog
1233 (* MG:03 Like make_polynomial_ but without Rule.Rls_ discard_parentheses1
1234 and expand_poly_rat_ instead of expand_poly_, see MG-DA.p.56ff*)
1235 (* MG necessary for termination of norm_Rational(*_mg*) in Rational.ML*)
1236 val make_rat_poly_with_parentheses =
1237 Rule_Set.Sequence{id = "make_rat_poly_with_parentheses", preconds = []:term list,
1238 rew_ord = ("dummy_ord", Rewrite_Ord.dummy_ord),
1239 erls = Atools_erls, srls = Rule_Set.Empty, calc = [], errpatts = [],
1240 rules = [Rule.Rls_ discard_minus,
1241 Rule.Rls_ expand_poly_rat_,(*ignors rationals*)
1242 \<^rule_eval>\<open>times\<close> (eval_binop "#mult_"),
1243 Rule.Rls_ order_mult_rls_,
1244 Rule.Rls_ simplify_power_,
1245 Rule.Rls_ calc_add_mult_pow_,
1246 Rule.Rls_ reduce_012_mult_,
1247 Rule.Rls_ order_add_rls_,
1248 Rule.Rls_ collect_numerals_,
1249 Rule.Rls_ reduce_012_
1250 (*Rule.Rls_ discard_parentheses1 *)
1252 scr = Rule.Empty_Prog
1256 (*.a minimal ruleset for reverse rewriting of factions [2];
1257 compare expand_binoms.*)
1259 Rule_Set.Sequence{id = "rev_rew_p", preconds = [], rew_ord = ("termlessI",termlessI),
1260 erls = Atools_erls, srls = Rule_Set.Empty,
1261 calc = [(*("PLUS" , (\<^const_name>\<open>plus\<close>, eval_binop "#add_")),
1262 ("TIMES" , (\<^const_name>\<open>times\<close>, eval_binop "#mult_")),
1263 ("POWER", (\<^const_name>\<open>powr\<close>, eval_binop "#power_"))*)
1265 rules = [\<^rule_thm>\<open>real_plus_binom_times\<close>,
1266 (*"(a + b)*(a + b) = a ^ 2 + 2 * a * b + b ^ 2*)
1267 \<^rule_thm>\<open>real_plus_binom_times1\<close>,
1268 (*"(a + 1*b)*(a + -1*b) = a \<up> 2 + -1*b \<up> 2"*)
1269 \<^rule_thm>\<open>real_plus_binom_times2\<close>,
1270 (*"(a + -1*b)*(a + 1*b) = a \<up> 2 + -1*b \<up> 2"*)
1272 \<^rule_thm>\<open>mult_1_left\<close>,(*"1 * z = z"*)
1274 \<^rule_thm>\<open>distrib_right\<close>,
1275 (*"(z1.0 + z2.0) * w = z1.0 * w + z2.0 * w"*)
1276 \<^rule_thm>\<open>distrib_left\<close>,
1277 (*"w * (z1.0 + z2.0) = w * z1.0 + w * z2.0"*)
1279 \<^rule_thm>\<open>mult.assoc\<close>,
1280 (*"?z1.1 * ?z2.1 * ?z3. =1 ?z1.1 * (?z2.1 * ?z3.1)"*)
1281 Rule.Rls_ order_mult_rls_,
1282 (*Rule.Rls_ order_add_rls_,*)
1284 \<^rule_eval>\<open>plus\<close> (eval_binop "#add_"),
1285 \<^rule_eval>\<open>times\<close> (eval_binop "#mult_"),
1286 \<^rule_eval>\<open>powr\<close> (eval_binop "#power_"),
1288 \<^rule_thm_sym>\<open>realpow_twoI\<close>,
1289 (*"r1 * r1 = r1 \<up> 2"*)
1290 \<^rule_thm_sym>\<open>real_mult_2\<close>,
1291 (*"z1 + z1 = 2 * z1"*)
1292 \<^rule_thm>\<open>real_mult_2_assoc\<close>,
1293 (*"z1 + (z1 + k) = 2 * z1 + k"*)
1295 \<^rule_thm>\<open>real_num_collect\<close>,
1296 (*"[| l is_const; m is_const |]==>l * n + m * n = (l + m) * n"*)
1297 \<^rule_thm>\<open>real_num_collect_assoc\<close>,
1298 (*"[| l is_const; m is_const |] ==>
1299 l * n + (m * n + k) = (l + m) * n + k"*)
1300 \<^rule_thm>\<open>real_one_collect\<close>,
1301 (*"m is_const ==> n + m * n = (1 + m) * n"*)
1302 \<^rule_thm>\<open>real_one_collect_assoc\<close>,
1303 (*"m is_const ==> k + (n + m * n) = k + (1 + m) * n"*)
1305 \<^rule_thm>\<open>realpow_multI\<close>,
1306 (*"(r * s) \<up> n = r \<up> n * s \<up> n"*)
1308 \<^rule_eval>\<open>plus\<close> (eval_binop "#add_"),
1309 \<^rule_eval>\<open>times\<close> (eval_binop "#mult_"),
1310 \<^rule_eval>\<open>powr\<close> (eval_binop "#power_"),
1312 \<^rule_thm>\<open>mult_1_left\<close>,(*"1 * z = z"*)
1313 \<^rule_thm>\<open>mult_zero_left\<close>,(*"0 * z = 0"*)
1314 \<^rule_thm>\<open>add_0_left\<close>(*0 + z = z*)
1316 (*Rule.Rls_ order_add_rls_*)
1319 scr = Rule.Empty_Prog};
1322 subsection \<open>rule-sets with explicit program for intermediate steps\<close>
1323 partial_function (tailrec) expand_binoms_2 :: "real \<Rightarrow> real"
1325 "expand_binoms_2 term = (
1327 (Try (Repeat (Rewrite ''real_plus_binom_pow2''))) #>
1328 (Try (Repeat (Rewrite ''real_plus_binom_times''))) #>
1329 (Try (Repeat (Rewrite ''real_minus_binom_pow2''))) #>
1330 (Try (Repeat (Rewrite ''real_minus_binom_times''))) #>
1331 (Try (Repeat (Rewrite ''real_plus_minus_binom1''))) #>
1332 (Try (Repeat (Rewrite ''real_plus_minus_binom2''))) #>
1334 (Try (Repeat (Rewrite ''mult_1_left''))) #>
1335 (Try (Repeat (Rewrite ''mult_zero_left''))) #>
1336 (Try (Repeat (Rewrite ''add_0_left''))) #>
1338 (Try (Repeat (Calculate ''PLUS''))) #>
1339 (Try (Repeat (Calculate ''TIMES''))) #>
1340 (Try (Repeat (Calculate ''POWER''))) #>
1342 (Try (Repeat (Rewrite ''sym_realpow_twoI''))) #>
1343 (Try (Repeat (Rewrite ''realpow_plus_1''))) #>
1344 (Try (Repeat (Rewrite ''sym_real_mult_2''))) #>
1345 (Try (Repeat (Rewrite ''real_mult_2_assoc''))) #>
1347 (Try (Repeat (Rewrite ''real_num_collect''))) #>
1348 (Try (Repeat (Rewrite ''real_num_collect_assoc''))) #>
1350 (Try (Repeat (Rewrite ''real_one_collect''))) #>
1351 (Try (Repeat (Rewrite ''real_one_collect_assoc''))) #>
1353 (Try (Repeat (Calculate ''PLUS''))) #>
1354 (Try (Repeat (Calculate ''TIMES''))) #>
1355 (Try (Repeat (Calculate ''POWER''))))
1359 Rule_Def.Repeat{id = "expand_binoms", preconds = [], rew_ord = ("termlessI",termlessI),
1360 erls = Atools_erls, srls = Rule_Set.Empty,
1361 calc = [("PLUS" , (\<^const_name>\<open>plus\<close>, eval_binop "#add_")),
1362 ("TIMES" , (\<^const_name>\<open>times\<close>, eval_binop "#mult_")),
1363 ("POWER", (\<^const_name>\<open>powr\<close>, eval_binop "#power_"))
1365 rules = [\<^rule_thm>\<open>real_plus_binom_pow2\<close>,
1366 (*"(a + b) \<up> 2 = a \<up> 2 + 2 * a * b + b \<up> 2"*)
1367 \<^rule_thm>\<open>real_plus_binom_times\<close>,
1368 (*"(a + b)*(a + b) = ...*)
1369 \<^rule_thm>\<open>real_minus_binom_pow2\<close>,
1370 (*"(a - b) \<up> 2 = a \<up> 2 - 2 * a * b + b \<up> 2"*)
1371 \<^rule_thm>\<open>real_minus_binom_times\<close>,
1372 (*"(a - b)*(a - b) = ...*)
1373 \<^rule_thm>\<open>real_plus_minus_binom1\<close>,
1374 (*"(a + b) * (a - b) = a \<up> 2 - b \<up> 2"*)
1375 \<^rule_thm>\<open>real_plus_minus_binom2\<close>,
1376 (*"(a - b) * (a + b) = a \<up> 2 - b \<up> 2"*)
1378 \<^rule_thm>\<open>real_pp_binom_times\<close>,
1379 (*(a + b)*(c + d) = a*c + a*d + b*c + b*d*)
1380 \<^rule_thm>\<open>real_pm_binom_times\<close>,
1381 (*(a + b)*(c - d) = a*c - a*d + b*c - b*d*)
1382 \<^rule_thm>\<open>real_mp_binom_times\<close>,
1383 (*(a - b)*(c + d) = a*c + a*d - b*c - b*d*)
1384 \<^rule_thm>\<open>real_mm_binom_times\<close>,
1385 (*(a - b)*(c - d) = a*c - a*d - b*c + b*d*)
1386 \<^rule_thm>\<open>realpow_multI\<close>,
1387 (*(a*b) \<up> n = a \<up> n * b \<up> n*)
1388 \<^rule_thm>\<open>real_plus_binom_pow3\<close>,
1389 (* (a + b) \<up> 3 = a \<up> 3 + 3*a \<up> 2*b + 3*a*b \<up> 2 + b \<up> 3 *)
1390 \<^rule_thm>\<open>real_minus_binom_pow3\<close>,
1391 (* (a - b) \<up> 3 = a \<up> 3 - 3*a \<up> 2*b + 3*a*b \<up> 2 - b \<up> 3 *)
1394 (*\<^rule_thm>\<open>distrib_right\<close>,
1395 (*"(z1.0 + z2.0) * w = z1.0 * w + z2.0 * w"*)
1396 \<^rule_thm>\<open>distrib_left\<close>,
1397 (*"w * (z1.0 + z2.0) = w * z1.0 + w * z2.0"*)
1398 \<^rule_thm>\<open>left_diff_distrib\<close>,
1399 (*"(z1.0 - z2.0) * w = z1.0 * w - z2.0 * w"*)
1400 \<^rule_thm>\<open>right_diff_distrib\<close>,
1401 (*"w * (z1.0 - z2.0) = w * z1.0 - w * z2.0"*)
1403 \<^rule_thm>\<open>mult_1_left\<close>,
1405 \<^rule_thm>\<open>mult_zero_left\<close>,
1407 \<^rule_thm>\<open>add_0_left\<close>,(*"0 + z = z"*)
1409 \<^rule_eval>\<open>plus\<close> (eval_binop "#add_"),
1410 \<^rule_eval>\<open>times\<close> (eval_binop "#mult_"),
1411 \<^rule_eval>\<open>powr\<close> (eval_binop "#power_"),
1412 (*\<^rule_thm>\<open>mult.commute\<close>,
1414 \<^rule_thm>\<open>real_mult_left_commute\<close>,
1415 \<^rule_thm>\<open>mult.assoc\<close>,
1416 \<^rule_thm>\<open>add.commute\<close>,
1417 \<^rule_thm>\<open>add.left_commute\<close>,
1418 \<^rule_thm>\<open>add.assoc\<close>,
1420 \<^rule_thm_sym>\<open>realpow_twoI\<close>,
1421 (*"r1 * r1 = r1 \<up> 2"*)
1422 \<^rule_thm>\<open>realpow_plus_1\<close>,
1423 (*"r * r \<up> n = r \<up> (n + 1)"*)
1424 (*\<^rule_thm_sym>\<open>real_mult_2\<close>,
1425 (*"z1 + z1 = 2 * z1"*)*)
1426 \<^rule_thm>\<open>real_mult_2_assoc\<close>,
1427 (*"z1 + (z1 + k) = 2 * z1 + k"*)
1429 \<^rule_thm>\<open>real_num_collect\<close>,
1430 (*"[| l is_const; m is_const |] ==>l * n + m * n = (l + m) * n"*)
1431 \<^rule_thm>\<open>real_num_collect_assoc\<close>,
1432 (*"[| l is_const; m is_const |] ==>
1433 l * n + (m * n + k) = (l + m) * n + k"*)
1434 \<^rule_thm>\<open>real_one_collect\<close>,
1435 (*"m is_const ==> n + m * n = (1 + m) * n"*)
1436 \<^rule_thm>\<open>real_one_collect_assoc\<close>,
1437 (*"m is_const ==> k + (n + m * n) = k + (1 + m) * n"*)
1439 \<^rule_eval>\<open>plus\<close> (eval_binop "#add_"),
1440 \<^rule_eval>\<open>times\<close> (eval_binop "#mult_"),
1441 \<^rule_eval>\<open>powr\<close> (eval_binop "#power_")
1443 scr = Rule.Prog (Program.prep_program @{thm expand_binoms_2.simps})
1447 subsection \<open>add to Know_Store\<close>
1448 subsubsection \<open>rule-sets\<close>
1449 ML \<open>val prep_rls' = Auto_Prog.prep_rls @{theory}\<close>
1452 norm_Poly = \<open>prep_rls' norm_Poly\<close> and
1453 Poly_erls = \<open>prep_rls' Poly_erls\<close> (*FIXXXME:del with rls.rls'*) and
1454 expand = \<open>prep_rls' expand\<close> and
1455 expand_poly = \<open>prep_rls' expand_poly\<close> and
1456 simplify_power = \<open>prep_rls' simplify_power\<close> and
1458 order_add_mult = \<open>prep_rls' order_add_mult\<close> and
1459 collect_numerals = \<open>prep_rls' collect_numerals\<close> and
1460 collect_numerals_= \<open>prep_rls' collect_numerals_\<close> and
1461 reduce_012 = \<open>prep_rls' reduce_012\<close> and
1462 discard_parentheses = \<open>prep_rls' discard_parentheses\<close> and
1464 make_polynomial = \<open>prep_rls' make_polynomial\<close> and
1465 expand_binoms = \<open>prep_rls' expand_binoms\<close> and
1466 rev_rew_p = \<open>prep_rls' rev_rew_p\<close> and
1467 discard_minus = \<open>prep_rls' discard_minus\<close> and
1468 expand_poly_ = \<open>prep_rls' expand_poly_\<close> and
1470 expand_poly_rat_ = \<open>prep_rls' expand_poly_rat_\<close> and
1471 simplify_power_ = \<open>prep_rls' simplify_power_\<close> and
1472 calc_add_mult_pow_ = \<open>prep_rls' calc_add_mult_pow_\<close> and
1473 reduce_012_mult_ = \<open>prep_rls' reduce_012_mult_\<close> and
1474 reduce_012_ = \<open>prep_rls' reduce_012_\<close> and
1476 discard_parentheses1 = \<open>prep_rls' discard_parentheses1\<close> and
1477 order_mult_rls_ = \<open>prep_rls' order_mult_rls_\<close> and
1478 order_add_rls_ = \<open>prep_rls' order_add_rls_\<close> and
1479 make_rat_poly_with_parentheses = \<open>prep_rls' make_rat_poly_with_parentheses\<close>
1481 subsection \<open>problems\<close>
1483 problem pbl_simp_poly : "polynomial/simplification" =
1484 \<open>Rule_Set.append_rules "empty" Rule_Set.empty [(*for preds in where_*)
1485 \<^rule_eval>\<open>is_polyexp\<close> (eval_is_polyexp "")]\<close>
1486 Method: "simplification/for_polynomials"
1489 Where: "t_t is_polyexp"
1490 Find: "normalform n_n"
1492 subsection \<open>methods\<close>
1494 partial_function (tailrec) simplify :: "real \<Rightarrow> real"
1496 "simplify term = ((Rewrite_Set ''norm_Poly'') term)"
1498 method met_simp_poly : "simplification/for_polynomials" =
1499 \<open>{rew_ord'="tless_true", rls' = Rule_Set.empty, calc = [], srls = Rule_Set.empty,
1500 prls = Rule_Set.append_rules "simplification_for_polynomials_prls" Rule_Set.empty
1501 [(*for preds in where_*) \<^rule_eval>\<open>is_polyexp\<close> (eval_is_polyexp"")],
1502 crls = Rule_Set.empty, errpats = [], nrls = norm_Poly}\<close>
1503 Program: simplify.simps
1505 Where: "t_t is_polyexp"
1506 Find: "normalform n_n"