1 (* theory collecting all knowledge
2 (predicates 'is_rootEq_in', 'is_sqrt_in', 'is_ratEq_in')
3 for PolynomialEquations.
4 alternative dependencies see Isac.thy
10 (c) by Richard Lang, 2003
13 theory PolyEq imports LinEq RootRatEq begin
17 (*---------scripts--------------------------*)
20 bool list] => bool list"
21 ("((Script Complete'_square (_ _ =))//
26 bool list] => bool list"
27 ("((Script Normalize'_poly (_ _=))//
29 Solve'_d0'_polyeq'_equation
31 bool list] => bool list"
32 ("((Script Solve'_d0'_polyeq'_equation (_ _ =))//
34 Solve'_d1'_polyeq'_equation
36 bool list] => bool list"
37 ("((Script Solve'_d1'_polyeq'_equation (_ _ =))//
39 Solve'_d2'_polyeq'_equation
41 bool list] => bool list"
42 ("((Script Solve'_d2'_polyeq'_equation (_ _ =))//
44 Solve'_d2'_polyeq'_sqonly'_equation
46 bool list] => bool list"
47 ("((Script Solve'_d2'_polyeq'_sqonly'_equation (_ _ =))//
49 Solve'_d2'_polyeq'_bdvonly'_equation
51 bool list] => bool list"
52 ("((Script Solve'_d2'_polyeq'_bdvonly'_equation (_ _ =))//
54 Solve'_d2'_polyeq'_pq'_equation
56 bool list] => bool list"
57 ("((Script Solve'_d2'_polyeq'_pq'_equation (_ _ =))//
59 Solve'_d2'_polyeq'_abc'_equation
61 bool list] => bool list"
62 ("((Script Solve'_d2'_polyeq'_abc'_equation (_ _ =))//
64 Solve'_d3'_polyeq'_equation
66 bool list] => bool list"
67 ("((Script Solve'_d3'_polyeq'_equation (_ _ =))//
69 Solve'_d4'_polyeq'_equation
71 bool list] => bool list"
72 ("((Script Solve'_d4'_polyeq'_equation (_ _ =))//
76 bool list] => bool list"
77 ("((Script Biquadrat'_poly (_ _=))//
80 (*-------------------- rules -------------------------------------------------*)
83 cancel_leading_coeff1: "Not (c =!= 0) ==> (a + b*bdv + c*bdv^^^2 = 0) =
84 (a/c + b/c*bdv + bdv^^^2 = 0)"
85 cancel_leading_coeff2: "Not (c =!= 0) ==> (a - b*bdv + c*bdv^^^2 = 0) =
86 (a/c - b/c*bdv + bdv^^^2 = 0)"
87 cancel_leading_coeff3: "Not (c =!= 0) ==> (a + b*bdv - c*bdv^^^2 = 0) =
88 (a/c + b/c*bdv - bdv^^^2 = 0)"
90 cancel_leading_coeff4: "Not (c =!= 0) ==> (a + bdv + c*bdv^^^2 = 0) =
91 (a/c + 1/c*bdv + bdv^^^2 = 0)"
92 cancel_leading_coeff5: "Not (c =!= 0) ==> (a - bdv + c*bdv^^^2 = 0) =
93 (a/c - 1/c*bdv + bdv^^^2 = 0)"
94 cancel_leading_coeff6: "Not (c =!= 0) ==> (a + bdv - c*bdv^^^2 = 0) =
95 (a/c + 1/c*bdv - bdv^^^2 = 0)"
97 cancel_leading_coeff7: "Not (c =!= 0) ==> ( b*bdv + c*bdv^^^2 = 0) =
98 ( b/c*bdv + bdv^^^2 = 0)"
99 cancel_leading_coeff8: "Not (c =!= 0) ==> ( b*bdv - c*bdv^^^2 = 0) =
100 ( b/c*bdv - bdv^^^2 = 0)"
102 cancel_leading_coeff9: "Not (c =!= 0) ==> ( bdv + c*bdv^^^2 = 0) =
103 ( 1/c*bdv + bdv^^^2 = 0)"
104 cancel_leading_coeff10:"Not (c =!= 0) ==> ( bdv - c*bdv^^^2 = 0) =
105 ( 1/c*bdv - bdv^^^2 = 0)"
107 cancel_leading_coeff11:"Not (c =!= 0) ==> (a + b*bdv^^^2 = 0) =
109 cancel_leading_coeff12:"Not (c =!= 0) ==> (a - b*bdv^^^2 = 0) =
111 cancel_leading_coeff13:"Not (c =!= 0) ==> ( b*bdv^^^2 = 0) =
114 complete_square1: "(q + p*bdv + bdv^^^2 = 0) =
115 (q + (p/2 + bdv)^^^2 = (p/2)^^^2)"
116 complete_square2: "( p*bdv + bdv^^^2 = 0) =
117 ( (p/2 + bdv)^^^2 = (p/2)^^^2)"
118 complete_square3: "( bdv + bdv^^^2 = 0) =
119 ( (1/2 + bdv)^^^2 = (1/2)^^^2)"
121 complete_square4: "(q - p*bdv + bdv^^^2 = 0) =
122 (q + (p/2 - bdv)^^^2 = (p/2)^^^2)"
123 complete_square5: "(q + p*bdv - bdv^^^2 = 0) =
124 (q + (p/2 - bdv)^^^2 = (p/2)^^^2)"
126 square_explicit1: "(a + b^^^2 = c) = ( b^^^2 = c - a)"
127 square_explicit2: "(a - b^^^2 = c) = (-(b^^^2) = c - a)"
129 bdv_explicit1: "(a + bdv = b) = (bdv = - a + b)"
130 bdv_explicit2: "(a - bdv = b) = ((-1)*bdv = - a + b)"
131 bdv_explicit3: "((-1)*bdv = b) = (bdv = (-1)*b)"
133 plus_leq: "(0 <= a + b) = ((-1)*b <= a)"(*Isa?*)
134 minus_leq: "(0 <= a - b) = ( b <= a)"(*Isa?*)
137 (*WN0509 compare LinEq.all_left "[|Not(b=!=0)|] ==> (a=b) = (a+(-1)*b=0)"*)
138 all_left: "[|Not(b=!=0)|] ==> (a = b) = (a - b = 0)"
139 makex1_x: "a^^^1 = a"
140 real_assoc_1: "a+(b+c) = a+b+c"
141 real_assoc_2: "a*(b*c) = a*b*c"
143 (* ---- degree 0 ----*)
144 d0_true: "(0=0) = True"
145 d0_false: "[|Not(bdv occurs_in a);Not(a=0)|] ==> (a=0) = False"
146 (* ---- degree 1 ----*)
148 "[|Not(bdv occurs_in a)|] ==> (a + b*bdv = 0) = (b*bdv = (-1)*a)"
150 "[|Not(bdv occurs_in a)|] ==> (a + bdv = 0) = ( bdv = (-1)*a)"
152 "[|Not(b=0);Not(bdv occurs_in c)|] ==> (b*bdv = c) = (bdv = c/b)"
153 (* ---- degree 2 ----*)
155 "[|Not(bdv occurs_in a)|] ==> (a + b*bdv^^^2=0) = (b*bdv^^^2= (-1)*a)"
157 "[|Not(bdv occurs_in a)|] ==> (a + bdv^^^2=0) = ( bdv^^^2= (-1)*a)"
159 "[|Not(b=0);Not(bdv occurs_in c)|] ==> (b*bdv^^^2=c) = (bdv^^^2=c/b)"
161 d2_prescind1: "(a*bdv + b*bdv^^^2 = 0) = (bdv*(a +b*bdv)=0)"
162 d2_prescind2: "(a*bdv + bdv^^^2 = 0) = (bdv*(a + bdv)=0)"
163 d2_prescind3: "( bdv + b*bdv^^^2 = 0) = (bdv*(1+b*bdv)=0)"
164 d2_prescind4: "( bdv + bdv^^^2 = 0) = (bdv*(1+ bdv)=0)"
165 (* eliminate degree 2 *)
166 (* thm for neg arguments in sqroot have postfix _neg *)
167 d2_sqrt_equation1: "[|(0<=c);Not(bdv occurs_in c)|] ==>
168 (bdv^^^2=c) = ((bdv=sqrt c) | (bdv=(-1)*sqrt c ))"
169 d2_sqrt_equation1_neg:
170 "[|(c<0);Not(bdv occurs_in c)|] ==> (bdv^^^2=c) = False"
171 d2_sqrt_equation2: "(bdv^^^2=0) = (bdv=0)"
172 d2_sqrt_equation3: "(b*bdv^^^2=0) = (bdv=0)"
173 d2_reduce_equation1: "(bdv*(a +b*bdv)=0) = ((bdv=0)|(a+b*bdv=0))"
174 d2_reduce_equation2: "(bdv*(a + bdv)=0) = ((bdv=0)|(a+ bdv=0))"
175 d2_pqformula1: "[|0<=p^^^2 - 4*q|] ==> (q+p*bdv+ bdv^^^2=0) =
176 ((bdv= (-1)*(p/2) + sqrt(p^^^2 - 4*q)/2)
177 | (bdv= (-1)*(p/2) - sqrt(p^^^2 - 4*q)/2))"
178 d2_pqformula1_neg: "[|p^^^2 - 4*q<0|] ==> (q+p*bdv+ bdv^^^2=0) = False"
179 d2_pqformula2: "[|0<=p^^^2 - 4*q|] ==> (q+p*bdv+1*bdv^^^2=0) =
180 ((bdv= (-1)*(p/2) + sqrt(p^^^2 - 4*q)/2)
181 | (bdv= (-1)*(p/2) - sqrt(p^^^2 - 4*q)/2))"
182 d2_pqformula2_neg: "[|p^^^2 - 4*q<0|] ==> (q+p*bdv+1*bdv^^^2=0) = False"
183 d2_pqformula3: "[|0<=1 - 4*q|] ==> (q+ bdv+ bdv^^^2=0) =
184 ((bdv= (-1)*(1/2) + sqrt(1 - 4*q)/2)
185 | (bdv= (-1)*(1/2) - sqrt(1 - 4*q)/2))"
186 d2_pqformula3_neg: "[|1 - 4*q<0|] ==> (q+ bdv+ bdv^^^2=0) = False"
187 d2_pqformula4: "[|0<=1 - 4*q|] ==> (q+ bdv+1*bdv^^^2=0) =
188 ((bdv= (-1)*(1/2) + sqrt(1 - 4*q)/2)
189 | (bdv= (-1)*(1/2) - sqrt(1 - 4*q)/2))"
190 d2_pqformula4_neg: "[|1 - 4*q<0|] ==> (q+ bdv+1*bdv^^^2=0) = False"
191 d2_pqformula5: "[|0<=p^^^2 - 0|] ==> ( p*bdv+ bdv^^^2=0) =
192 ((bdv= (-1)*(p/2) + sqrt(p^^^2 - 0)/2)
193 | (bdv= (-1)*(p/2) - sqrt(p^^^2 - 0)/2))"
194 (* d2_pqformula5_neg not need p^2 never less zero in R *)
195 d2_pqformula6: "[|0<=p^^^2 - 0|] ==> ( p*bdv+1*bdv^^^2=0) =
196 ((bdv= (-1)*(p/2) + sqrt(p^^^2 - 0)/2)
197 | (bdv= (-1)*(p/2) - sqrt(p^^^2 - 0)/2))"
198 (* d2_pqformula6_neg not need p^2 never less zero in R *)
199 d2_pqformula7: "[|0<=1 - 0|] ==> ( bdv+ bdv^^^2=0) =
200 ((bdv= (-1)*(1/2) + sqrt(1 - 0)/2)
201 | (bdv= (-1)*(1/2) - sqrt(1 - 0)/2))"
202 (* d2_pqformula7_neg not need, because 1<0 ==> False*)
203 d2_pqformula8: "[|0<=1 - 0|] ==> ( bdv+1*bdv^^^2=0) =
204 ((bdv= (-1)*(1/2) + sqrt(1 - 0)/2)
205 | (bdv= (-1)*(1/2) - sqrt(1 - 0)/2))"
206 (* d2_pqformula8_neg not need, because 1<0 ==> False*)
207 d2_pqformula9: "[|Not(bdv occurs_in q); 0<= (-1)*4*q|] ==>
208 (q+ 1*bdv^^^2=0) = ((bdv= 0 + sqrt(0 - 4*q)/2)
209 | (bdv= 0 - sqrt(0 - 4*q)/2))"
211 "[|Not(bdv occurs_in q); (-1)*4*q<0|] ==> (q+ 1*bdv^^^2=0) = False"
213 "[|Not(bdv occurs_in q); 0<= (-1)*4*q|] ==> (q+ bdv^^^2=0) =
214 ((bdv= 0 + sqrt(0 - 4*q)/2)
215 | (bdv= 0 - sqrt(0 - 4*q)/2))"
217 "[|Not(bdv occurs_in q); (-1)*4*q<0|] ==> (q+ bdv^^^2=0) = False"
219 "[|0<=b^^^2 - 4*a*c|] ==> (c + b*bdv+a*bdv^^^2=0) =
220 ((bdv=( -b + sqrt(b^^^2 - 4*a*c))/(2*a))
221 | (bdv=( -b - sqrt(b^^^2 - 4*a*c))/(2*a)))"
223 "[|b^^^2 - 4*a*c<0|] ==> (c + b*bdv+a*bdv^^^2=0) = False"
225 "[|0<=1 - 4*a*c|] ==> (c+ bdv+a*bdv^^^2=0) =
226 ((bdv=( -1 + sqrt(1 - 4*a*c))/(2*a))
227 | (bdv=( -1 - sqrt(1 - 4*a*c))/(2*a)))"
229 "[|1 - 4*a*c<0|] ==> (c+ bdv+a*bdv^^^2=0) = False"
231 "[|0<=b^^^2 - 4*1*c|] ==> (c + b*bdv+ bdv^^^2=0) =
232 ((bdv=( -b + sqrt(b^^^2 - 4*1*c))/(2*1))
233 | (bdv=( -b - sqrt(b^^^2 - 4*1*c))/(2*1)))"
235 "[|b^^^2 - 4*1*c<0|] ==> (c + b*bdv+ bdv^^^2=0) = False"
237 "[|0<=1 - 4*1*c|] ==> (c + bdv+ bdv^^^2=0) =
238 ((bdv=( -1 + sqrt(1 - 4*1*c))/(2*1))
239 | (bdv=( -1 - sqrt(1 - 4*1*c))/(2*1)))"
241 "[|1 - 4*1*c<0|] ==> (c + bdv+ bdv^^^2=0) = False"
243 "[|Not(bdv occurs_in c); 0<=0 - 4*a*c|] ==> (c + a*bdv^^^2=0) =
244 ((bdv=( 0 + sqrt(0 - 4*a*c))/(2*a))
245 | (bdv=( 0 - sqrt(0 - 4*a*c))/(2*a)))"
247 "[|Not(bdv occurs_in c); 0 - 4*a*c<0|] ==> (c + a*bdv^^^2=0) = False"
249 "[|Not(bdv occurs_in c); 0<=0 - 4*1*c|] ==> (c+ bdv^^^2=0) =
250 ((bdv=( 0 + sqrt(0 - 4*1*c))/(2*1))
251 | (bdv=( 0 - sqrt(0 - 4*1*c))/(2*1)))"
253 "[|Not(bdv occurs_in c); 0 - 4*1*c<0|] ==> (c+ bdv^^^2=0) = False"
255 "[|0<=b^^^2 - 0|] ==> ( b*bdv+a*bdv^^^2=0) =
256 ((bdv=( -b + sqrt(b^^^2 - 0))/(2*a))
257 | (bdv=( -b - sqrt(b^^^2 - 0))/(2*a)))"
258 (* d2_abcformula7_neg not need b^2 never less zero in R *)
260 "[|0<=b^^^2 - 0|] ==> ( b*bdv+ bdv^^^2=0) =
261 ((bdv=( -b + sqrt(b^^^2 - 0))/(2*1))
262 | (bdv=( -b - sqrt(b^^^2 - 0))/(2*1)))"
263 (* d2_abcformula8_neg not need b^2 never less zero in R *)
265 "[|0<=1 - 0|] ==> ( bdv+a*bdv^^^2=0) =
266 ((bdv=( -1 + sqrt(1 - 0))/(2*a))
267 | (bdv=( -1 - sqrt(1 - 0))/(2*a)))"
268 (* d2_abcformula9_neg not need, because 1<0 ==> False*)
270 "[|0<=1 - 0|] ==> ( bdv+ bdv^^^2=0) =
271 ((bdv=( -1 + sqrt(1 - 0))/(2*1))
272 | (bdv=( -1 - sqrt(1 - 0))/(2*1)))"
273 (* d2_abcformula10_neg not need, because 1<0 ==> False*)
275 (* ---- degree 3 ----*)
277 "(a*bdv + b*bdv^^^2 + c*bdv^^^3=0) = (bdv=0 | (a + b*bdv + c*bdv^^^2=0))"
279 "( bdv + b*bdv^^^2 + c*bdv^^^3=0) = (bdv=0 | (1 + b*bdv + c*bdv^^^2=0))"
281 "(a*bdv + bdv^^^2 + c*bdv^^^3=0) = (bdv=0 | (a + bdv + c*bdv^^^2=0))"
283 "( bdv + bdv^^^2 + c*bdv^^^3=0) = (bdv=0 | (1 + bdv + c*bdv^^^2=0))"
285 "(a*bdv + b*bdv^^^2 + bdv^^^3=0) = (bdv=0 | (a + b*bdv + bdv^^^2=0))"
287 "( bdv + b*bdv^^^2 + bdv^^^3=0) = (bdv=0 | (1 + b*bdv + bdv^^^2=0))"
289 "(a*bdv + bdv^^^2 + bdv^^^3=0) = (bdv=0 | (1 + bdv + bdv^^^2=0))"
291 "( bdv + bdv^^^2 + bdv^^^3=0) = (bdv=0 | (1 + bdv + bdv^^^2=0))"
293 "(a*bdv + c*bdv^^^3=0) = (bdv=0 | (a + c*bdv^^^2=0))"
294 d3_reduce_equation10:
295 "( bdv + c*bdv^^^3=0) = (bdv=0 | (1 + c*bdv^^^2=0))"
296 d3_reduce_equation11:
297 "(a*bdv + bdv^^^3=0) = (bdv=0 | (a + bdv^^^2=0))"
298 d3_reduce_equation12:
299 "( bdv + bdv^^^3=0) = (bdv=0 | (1 + bdv^^^2=0))"
300 d3_reduce_equation13:
301 "( b*bdv^^^2 + c*bdv^^^3=0) = (bdv=0 | ( b*bdv + c*bdv^^^2=0))"
302 d3_reduce_equation14:
303 "( bdv^^^2 + c*bdv^^^3=0) = (bdv=0 | ( bdv + c*bdv^^^2=0))"
304 d3_reduce_equation15:
305 "( b*bdv^^^2 + bdv^^^3=0) = (bdv=0 | ( b*bdv + bdv^^^2=0))"
306 d3_reduce_equation16:
307 "( bdv^^^2 + bdv^^^3=0) = (bdv=0 | ( bdv + bdv^^^2=0))"
309 "[|Not(bdv occurs_in a)|] ==> (a + b*bdv^^^3=0) = (b*bdv^^^3= (-1)*a)"
311 "[|Not(bdv occurs_in a)|] ==> (a + bdv^^^3=0) = ( bdv^^^3= (-1)*a)"
313 "[|Not(b=0);Not(bdv occurs_in a)|] ==> (b*bdv^^^3=c) = (bdv^^^3=c/b)"
315 "(bdv^^^3=0) = (bdv=0)"
317 "(bdv^^^3=c) = (bdv = nroot 3 c)"
319 (* ---- degree 4 ----*)
320 (* RL03.FIXME es wir nicht getestet ob u>0 *)
322 "(c+b*bdv^^^2+a*bdv^^^4=0) =
323 ((a*u^^^2+b*u+c=0) & (bdv^^^2=u))"
325 (* ---- 7.3.02 von Termorder ---- *)
327 bdv_collect_1: "l * bdv + m * bdv = (l + m) * bdv"
328 bdv_collect_2: "bdv + m * bdv = (1 + m) * bdv"
329 bdv_collect_3: "l * bdv + bdv = (l + 1) * bdv"
331 (* bdv_collect_assoc0_1 "l * bdv + m * bdv + k = (l + m) * bdv + k"
332 bdv_collect_assoc0_2 "bdv + m * bdv + k = (1 + m) * bdv + k"
333 bdv_collect_assoc0_3 "l * bdv + bdv + k = (l + 1) * bdv + k"
335 bdv_collect_assoc1_1:"l * bdv + (m * bdv + k) = (l + m) * bdv + k"
336 bdv_collect_assoc1_2:"bdv + (m * bdv + k) = (1 + m) * bdv + k"
337 bdv_collect_assoc1_3:"l * bdv + (bdv + k) = (l + 1) * bdv + k"
339 bdv_collect_assoc2_1:"k + l * bdv + m * bdv = k + (l + m) * bdv"
340 bdv_collect_assoc2_2:"k + bdv + m * bdv = k + (1 + m) * bdv"
341 bdv_collect_assoc2_3:"k + l * bdv + bdv = k + (l + 1) * bdv"
344 bdv_n_collect_1: "l * bdv^^^n + m * bdv^^^n = (l + m) * bdv^^^n"
345 bdv_n_collect_2: " bdv^^^n + m * bdv^^^n = (1 + m) * bdv^^^n"
346 bdv_n_collect_3: "l * bdv^^^n + bdv^^^n = (l + 1) * bdv^^^n" (*order!*)
348 bdv_n_collect_assoc1_1:"l * bdv^^^n + (m * bdv^^^n + k) = (l + m) * bdv^^^n + k"
349 bdv_n_collect_assoc1_2:"bdv^^^n + (m * bdv^^^n + k) = (1 + m) * bdv^^^n + k"
350 bdv_n_collect_assoc1_3:"l * bdv^^^n + (bdv^^^n + k) = (l + 1) * bdv^^^n + k"
352 bdv_n_collect_assoc2_1:"k + l * bdv^^^n + m * bdv^^^n = k + (l + m) * bdv^^^n"
353 bdv_n_collect_assoc2_2:"k + bdv^^^n + m * bdv^^^n = k + (1 + m) * bdv^^^n"
354 bdv_n_collect_assoc2_3:"k + l * bdv^^^n + bdv^^^n = k + (l + 1) * bdv^^^n"
357 real_minus_div: "- (a / b) = (-1 * a) / b"
359 separate_bdv: "(a * bdv) / b = (a / b) * bdv"
360 separate_bdv_n: "(a * bdv ^^^ n) / b = (a / b) * bdv ^^^ n"
361 separate_1_bdv: "bdv / b = (1 / b) * bdv"
362 separate_1_bdv_n: "bdv ^^^ n / b = (1 / b) * bdv ^^^ n"
367 (*-------------------------rulse-------------------------*)
368 val PolyEq_prls = (*3.10.02:just the following order due to subterm evaluation*)
369 append_rls "PolyEq_prls" e_rls
370 [Calc ("Atools.ident",eval_ident "#ident_"),
371 Calc ("Tools.matches",eval_matches ""),
372 Calc ("Tools.lhs" ,eval_lhs ""),
373 Calc ("Tools.rhs" ,eval_rhs ""),
374 Calc ("Poly.is'_expanded'_in",eval_is_expanded_in ""),
375 Calc ("Poly.is'_poly'_in",eval_is_poly_in ""),
376 Calc ("Poly.has'_degree'_in",eval_has_degree_in ""),
377 Calc ("Poly.is'_polyrat'_in",eval_is_polyrat_in ""),
378 (*Calc ("Atools.occurs'_in",eval_occurs_in ""), *)
379 (*Calc ("Atools.is'_const",eval_const "#is_const_"),*)
380 Calc ("op =",eval_equal "#equal_"),
381 Calc ("RootEq.is'_rootTerm'_in",eval_is_rootTerm_in ""),
382 Calc ("RatEq.is'_ratequation'_in",eval_is_ratequation_in ""),
383 Thm ("not_true",num_str @{thm not_true}),
384 Thm ("not_false",num_str @{thm not_false}),
385 Thm ("and_true",num_str @{thm and_true}),
386 Thm ("and_false",num_str @{thm and_false}),
387 Thm ("or_true",num_str @{thm or_true}),
388 Thm ("or_false",num_str @{thm or_false})
392 merge_rls "PolyEq_erls" LinEq_erls
393 (append_rls "ops_preds" calculate_Rational
394 [Calc ("op =",eval_equal "#equal_"),
395 Thm ("plus_leq", num_str @{thm plus_leq}),
396 Thm ("minus_leq", num_str @{thm minus_leq}),
397 Thm ("rat_leq1", num_str @{thm rat_leq1}),
398 Thm ("rat_leq2", num_str @{thm rat_leq2}),
399 Thm ("rat_leq3", num_str @{thm rat_leq3})
403 merge_rls "PolyEq_crls" LinEq_crls
404 (append_rls "ops_preds" calculate_Rational
405 [Calc ("op =",eval_equal "#equal_"),
406 Thm ("plus_leq", num_str @{thm plus_leq}),
407 Thm ("minus_leq", num_str @{thm minus_leq}),
408 Thm ("rat_leq1", num_str @{thm rat_leq1}),
409 Thm ("rat_leq2", num_str @{thm rat_leq2}),
410 Thm ("rat_leq3", num_str @{thm rat_leq3})
413 val cancel_leading_coeff = prep_rls(
414 Rls {id = "cancel_leading_coeff", preconds = [],
415 rew_ord = ("e_rew_ord",e_rew_ord),
416 erls = PolyEq_erls, srls = Erls, calc = [], (*asm_thm = [],*)
417 rules = [Thm ("cancel_leading_coeff1",num_str @{thm cancel_leading_coeff1}),
418 Thm ("cancel_leading_coeff2",num_str @{thm cancel_leading_coeff2}),
419 Thm ("cancel_leading_coeff3",num_str @{thm cancel_leading_coeff3}),
420 Thm ("cancel_leading_coeff4",num_str @{thm cancel_leading_coeff4}),
421 Thm ("cancel_leading_coeff5",num_str @{thm cancel_leading_coeff5}),
422 Thm ("cancel_leading_coeff6",num_str @{thm cancel_leading_coeff6}),
423 Thm ("cancel_leading_coeff7",num_str @{thm cancel_leading_coeff7}),
424 Thm ("cancel_leading_coeff8",num_str @{thm cancel_leading_coeff8}),
425 Thm ("cancel_leading_coeff9",num_str @{thm cancel_leading_coeff9}),
426 Thm ("cancel_leading_coeff10",num_str @{thm cancel_leading_coeff10}),
427 Thm ("cancel_leading_coeff11",num_str @{thm cancel_leading_coeff11}),
428 Thm ("cancel_leading_coeff12",num_str @{thm cancel_leading_coeff12}),
429 Thm ("cancel_leading_coeff13",num_str @{thm cancel_leading_coeff13})
431 scr = Script ((term_of o the o (parse thy))
435 val complete_square = prep_rls(
436 Rls {id = "complete_square", preconds = [],
437 rew_ord = ("e_rew_ord",e_rew_ord),
438 erls = PolyEq_erls, srls = Erls, calc = [], (*asm_thm = [],*)
439 rules = [Thm ("complete_square1",num_str @{thm complete_square1}),
440 Thm ("complete_square2",num_str @{thm complete_square2}),
441 Thm ("complete_square3",num_str @{thm complete_square3}),
442 Thm ("complete_square4",num_str @{thm complete_square4}),
443 Thm ("complete_square5",num_str @{thm complete_square5})
445 scr = Script ((term_of o the o (parse thy))
449 val polyeq_simplify = prep_rls(
450 Rls {id = "polyeq_simplify", preconds = [],
451 rew_ord = ("termlessI",termlessI),
456 rules = [Thm ("real_assoc_1",num_str @{thm real_assoc_1}),
457 Thm ("real_assoc_2",num_str @{thm real_assoc_2}),
458 Thm ("real_diff_minus",num_str @{thm real_diff_minus}),
459 Thm ("real_unari_minus",num_str @{thm real_unari_minus}),
460 Thm ("realpow_multI",num_str @{thm realpow_multI}),
461 Calc ("op +",eval_binop "#add_"),
462 Calc ("op -",eval_binop "#sub_"),
463 Calc ("op *",eval_binop "#mult_"),
464 Calc ("HOL.divide", eval_cancel "#divide_e"),
465 Calc ("NthRoot.sqrt",eval_sqrt "#sqrt_"),
466 Calc ("Atools.pow" ,eval_binop "#power_"),
469 scr = Script ((term_of o the o (parse thy)) "empty_script")
472 ruleset' := overwritelthy @{theory} (!ruleset',
473 [("cancel_leading_coeff",cancel_leading_coeff),
474 ("complete_square",complete_square),
475 ("PolyEq_erls",PolyEq_erls),(*FIXXXME:del with rls.rls'*)
476 ("polyeq_simplify",polyeq_simplify)]);
479 (* ------------- polySolve ------------------ *)
481 (*isolate the bound variable in an d0 equation; 'bdv' is a meta-constant*)
482 val d0_polyeq_simplify = prep_rls(
483 Rls {id = "d0_polyeq_simplify", preconds = [],
484 rew_ord = ("e_rew_ord",e_rew_ord),
489 rules = [Thm("d0_true",num_str @{thm d0_true}),
490 Thm("d0_false",num_str @{thm d0_false})
492 scr = Script ((term_of o the o (parse thy)) "empty_script")
496 (*isolate the bound variable in an d1 equation; 'bdv' is a meta-constant*)
497 val d1_polyeq_simplify = prep_rls(
498 Rls {id = "d1_polyeq_simplify", preconds = [],
499 rew_ord = ("e_rew_ord",e_rew_ord),
503 (*asm_thm = [("d1_isolate_div","")],*)
505 Thm("d1_isolate_add1",num_str @{thm d1_isolate_add1}),
506 (* a+bx=0 -> bx=-a *)
507 Thm("d1_isolate_add2",num_str @{thm d1_isolate_add2}),
509 Thm("d1_isolate_div",num_str @{thm d1_isolate_div})
512 scr = Script ((term_of o the o (parse thy)) "empty_script")
516 (* isolate the bound variable in an d2 equation with bdv only;
517 'bdv' is a meta-constant*)
518 val d2_polyeq_bdv_only_simplify = prep_rls(
519 Rls {id = "d2_polyeq_bdv_only_simplify", preconds = [],
520 rew_ord = ("e_rew_ord",e_rew_ord),
524 (*asm_thm = [("d2_sqrt_equation1",""),("d2_sqrt_equation1_neg",""),
525 ("d2_isolate_div","")],*)
526 rules = [Thm("d2_prescind1",num_str @{thm d2_prescind1}),
527 (* ax+bx^2=0 -> x(a+bx)=0 *)
528 Thm("d2_prescind2",num_str @{thm d2_prescind2}),
529 (* ax+ x^2=0 -> x(a+ x)=0 *)
530 Thm("d2_prescind3",num_str @{thm d2_prescind3}),
531 (* x+bx^2=0 -> x(1+bx)=0 *)
532 Thm("d2_prescind4",num_str @{thm d2_prescind4}),
533 (* x+ x^2=0 -> x(1+ x)=0 *)
534 Thm("d2_sqrt_equation1",num_str @{thm d2_sqrt_equation1}),
535 (* x^2=c -> x=+-sqrt(c)*)
536 Thm("d2_sqrt_equation1_neg",num_str @{thm d2_sqrt_equation1_neg}),
537 (* [0<c] x^2=c -> [] *)
538 Thm("d2_sqrt_equation2",num_str @{thm d2_sqrt_equation2}),
540 Thm("d2_reduce_equation1",num_str @{thm d2_reduce_equation1}),
541 (* x(a+bx)=0 -> x=0 | a+bx=0*)
542 Thm("d2_reduce_equation2",num_str @{thm d2_reduce_equation2}),
543 (* x(a+ x)=0 -> x=0 | a+ x=0*)
544 Thm("d2_isolate_div",num_str @{thm d2_isolate_div})
545 (* bx^2=c -> x^2=c/b*)
547 scr = Script ((term_of o the o (parse thy)) "empty_script")
550 (* isolate the bound variable in an d2 equation with sqrt only;
551 'bdv' is a meta-constant*)
552 val d2_polyeq_sq_only_simplify = prep_rls(
553 Rls {id = "d2_polyeq_sq_only_simplify", preconds = [],
554 rew_ord = ("e_rew_ord",e_rew_ord),
558 (*asm_thm = [("d2_sqrt_equation1",""),("d2_sqrt_equation1_neg",""),
559 ("d2_isolate_div","")],*)
560 rules = [Thm("d2_isolate_add1",num_str @{thm d2_isolate_add1}),
561 (* a+ bx^2=0 -> bx^2=(-1)a*)
562 Thm("d2_isolate_add2",num_str @{thm d2_isolate_add2}),
563 (* a+ x^2=0 -> x^2=(-1)a*)
564 Thm("d2_sqrt_equation2",num_str @{thm d2_sqrt_equation2}),
566 Thm("d2_sqrt_equation1",num_str @{thm d2_sqrt_equation1}),
567 (* x^2=c -> x=+-sqrt(c)*)
568 Thm("d2_sqrt_equation1_neg",num_str @{thm d2_sqrt_equation1_neg}),
569 (* [c<0] x^2=c -> x=[] *)
570 Thm("d2_isolate_div",num_str @{thm d2_isolate_div})
571 (* bx^2=c -> x^2=c/b*)
573 scr = Script ((term_of o the o (parse thy)) "empty_script")
576 (* isolate the bound variable in an d2 equation with pqFormula;
577 'bdv' is a meta-constant*)
578 val d2_polyeq_pqFormula_simplify = prep_rls(
579 Rls {id = "d2_polyeq_pqFormula_simplify", preconds = [],
580 rew_ord = ("e_rew_ord",e_rew_ord), erls = PolyEq_erls,
581 srls = Erls, calc = [],
582 rules = [Thm("d2_pqformula1",num_str @{thm d2_pqformula1}),
584 Thm("d2_pqformula1_neg",num_str @{thm d2_pqformula1_neg}),
586 Thm("d2_pqformula2",num_str @{thm d2_pqformula2}),
588 Thm("d2_pqformula2_neg",num_str @{thm d2_pqformula2_neg}),
590 Thm("d2_pqformula3",num_str @{thm d2_pqformula3}),
592 Thm("d2_pqformula3_neg",num_str @{thm d2_pqformula3_neg}),
594 Thm("d2_pqformula4",num_str @{thm d2_pqformula4}),
596 Thm("d2_pqformula4_neg",num_str @{thm d2_pqformula4_neg}),
598 Thm("d2_pqformula5",num_str @{thm d2_pqformula5}),
600 Thm("d2_pqformula6",num_str @{thm d2_pqformula6}),
602 Thm("d2_pqformula7",num_str @{thm d2_pqformula7}),
604 Thm("d2_pqformula8",num_str @{thm d2_pqformula8}),
606 Thm("d2_pqformula9",num_str @{thm d2_pqformula9}),
608 Thm("d2_pqformula9_neg",num_str @{thm d2_pqformula9_neg}),
610 Thm("d2_pqformula10",num_str @{thm d2_pqformula10}),
612 Thm("d2_pqformula10_neg",num_str @{thm d2_pqformula10_neg}),
614 Thm("d2_sqrt_equation2",num_str @{thm d2_sqrt_equation2}),
616 Thm("d2_sqrt_equation3",num_str @{thm d2_sqrt_equation3})
619 scr = Script ((term_of o the o (parse thy)) "empty_script")
622 (* isolate the bound variable in an d2 equation with abcFormula;
623 'bdv' is a meta-constant*)
624 val d2_polyeq_abcFormula_simplify = prep_rls(
625 Rls {id = "d2_polyeq_abcFormula_simplify", preconds = [],
626 rew_ord = ("e_rew_ord",e_rew_ord), erls = PolyEq_erls,
627 srls = Erls, calc = [],
628 rules = [Thm("d2_abcformula1",num_str @{thm d2_abcformula1}),
630 Thm("d2_abcformula1_neg",num_str @{thm d2_abcformula1_neg}),
632 Thm("d2_abcformula2",num_str @{thm d2_abcformula2}),
634 Thm("d2_abcformula2_neg",num_str @{thm d2_abcformula2_neg}),
636 Thm("d2_abcformula3",num_str @{thm d2_abcformula3}),
638 Thm("d2_abcformula3_neg",num_str @{thm d2_abcformula3_neg}),
640 Thm("d2_abcformula4",num_str @{thm d2_abcformula4}),
642 Thm("d2_abcformula4_neg",num_str @{thm d2_abcformula4_neg}),
644 Thm("d2_abcformula5",num_str @{thm d2_abcformula5}),
646 Thm("d2_abcformula5_neg",num_str @{thm d2_abcformula5_neg}),
648 Thm("d2_abcformula6",num_str @{thm d2_abcformula6}),
650 Thm("d2_abcformula6_neg",num_str @{thm d2_abcformula6_neg}),
652 Thm("d2_abcformula7",num_str @{thm d2_abcformula7}),
654 Thm("d2_abcformula8",num_str @{thm d2_abcformula8}),
656 Thm("d2_abcformula9",num_str @{thm d2_abcformula9}),
658 Thm("d2_abcformula10",num_str @{thm d2_abcformula10}),
660 Thm("d2_sqrt_equation2",num_str @{thm d2_sqrt_equation2}),
662 Thm("d2_sqrt_equation3",num_str @{thm d2_sqrt_equation3})
665 scr = Script ((term_of o the o (parse thy)) "empty_script")
668 (* isolate the bound variable in an d2 equation;
669 'bdv' is a meta-constant*)
670 val d2_polyeq_simplify = prep_rls(
671 Rls {id = "d2_polyeq_simplify", preconds = [],
672 rew_ord = ("e_rew_ord",e_rew_ord), erls = PolyEq_erls,
673 srls = Erls, calc = [],
674 rules = [Thm("d2_pqformula1",num_str @{thm d2_pqformula1}),
676 Thm("d2_pqformula1_neg",num_str @{thm d2_pqformula1_neg}),
678 Thm("d2_pqformula2",num_str @{thm d2_pqformula2}),
680 Thm("d2_pqformula2_neg",num_str @{thm d2_pqformula2_neg}),
682 Thm("d2_pqformula3",num_str @{thm d2_pqformula3}),
684 Thm("d2_pqformula3_neg",num_str @{thm d2_pqformula3_neg}),
686 Thm("d2_pqformula4",num_str @{thm d2_pqformula4}),
688 Thm("d2_pqformula4_neg",num_str @{thm d2_pqformula4_neg}),
690 Thm("d2_abcformula1",num_str @{thm d2_abcformula1}),
692 Thm("d2_abcformula1_neg",num_str @{thm d2_abcformula1_neg}),
694 Thm("d2_abcformula2",num_str @{thm d2_abcformula2}),
696 Thm("d2_abcformula2_neg",num_str @{thm d2_abcformula2_neg}),
698 Thm("d2_prescind1",num_str @{thm d2_prescind1}),
699 (* ax+bx^2=0 -> x(a+bx)=0 *)
700 Thm("d2_prescind2",num_str @{thm d2_prescind2}),
701 (* ax+ x^2=0 -> x(a+ x)=0 *)
702 Thm("d2_prescind3",num_str @{thm d2_prescind3}),
703 (* x+bx^2=0 -> x(1+bx)=0 *)
704 Thm("d2_prescind4",num_str @{thm d2_prescind4}),
705 (* x+ x^2=0 -> x(1+ x)=0 *)
706 Thm("d2_isolate_add1",num_str @{thm d2_isolate_add1}),
707 (* a+ bx^2=0 -> bx^2=(-1)a*)
708 Thm("d2_isolate_add2",num_str @{thm d2_isolate_add2}),
709 (* a+ x^2=0 -> x^2=(-1)a*)
710 Thm("d2_sqrt_equation1",num_str @{thm d2_sqrt_equation1}),
711 (* x^2=c -> x=+-sqrt(c)*)
712 Thm("d2_sqrt_equation1_neg",num_str @{thm d2_sqrt_equation1_neg}),
713 (* [c<0] x^2=c -> x=[]*)
714 Thm("d2_sqrt_equation2",num_str @{thm d2_sqrt_equation2}),
716 Thm("d2_reduce_equation1",num_str @{thm d2_reduce_equation1}),
717 (* x(a+bx)=0 -> x=0 | a+bx=0*)
718 Thm("d2_reduce_equation2",num_str @{thm d2_reduce_equation2}),
719 (* x(a+ x)=0 -> x=0 | a+ x=0*)
720 Thm("d2_isolate_div",num_str @{thm d2_isolate_div})
721 (* bx^2=c -> x^2=c/b*)
723 scr = Script ((term_of o the o (parse thy)) "empty_script")
727 (* isolate the bound variable in an d3 equation; 'bdv' is a meta-constant *)
728 val d3_polyeq_simplify = prep_rls(
729 Rls {id = "d3_polyeq_simplify", preconds = [],
730 rew_ord = ("e_rew_ord",e_rew_ord), erls = PolyEq_erls,
731 srls = Erls, calc = [],
733 [Thm("d3_reduce_equation1",num_str @{thm d3_reduce_equation1}),
734 (*a*bdv + b*bdv^^^2 + c*bdv^^^3=0) =
735 (bdv=0 | (a + b*bdv + c*bdv^^^2=0)*)
736 Thm("d3_reduce_equation2",num_str @{thm d3_reduce_equation2}),
737 (* bdv + b*bdv^^^2 + c*bdv^^^3=0) =
738 (bdv=0 | (1 + b*bdv + c*bdv^^^2=0)*)
739 Thm("d3_reduce_equation3",num_str @{thm d3_reduce_equation3}),
740 (*a*bdv + bdv^^^2 + c*bdv^^^3=0) =
741 (bdv=0 | (a + bdv + c*bdv^^^2=0)*)
742 Thm("d3_reduce_equation4",num_str @{thm d3_reduce_equation4}),
743 (* bdv + bdv^^^2 + c*bdv^^^3=0) =
744 (bdv=0 | (1 + bdv + c*bdv^^^2=0)*)
745 Thm("d3_reduce_equation5",num_str @{thm d3_reduce_equation5}),
746 (*a*bdv + b*bdv^^^2 + bdv^^^3=0) =
747 (bdv=0 | (a + b*bdv + bdv^^^2=0)*)
748 Thm("d3_reduce_equation6",num_str @{thm d3_reduce_equation6}),
749 (* bdv + b*bdv^^^2 + bdv^^^3=0) =
750 (bdv=0 | (1 + b*bdv + bdv^^^2=0)*)
751 Thm("d3_reduce_equation7",num_str @{thm d3_reduce_equation7}),
752 (*a*bdv + bdv^^^2 + bdv^^^3=0) =
753 (bdv=0 | (1 + bdv + bdv^^^2=0)*)
754 Thm("d3_reduce_equation8",num_str @{thm d3_reduce_equation8}),
755 (* bdv + bdv^^^2 + bdv^^^3=0) =
756 (bdv=0 | (1 + bdv + bdv^^^2=0)*)
757 Thm("d3_reduce_equation9",num_str @{thm d3_reduce_equation9}),
758 (*a*bdv + c*bdv^^^3=0) =
759 (bdv=0 | (a + c*bdv^^^2=0)*)
760 Thm("d3_reduce_equation10",num_str @{thm d3_reduce_equation10}),
761 (* bdv + c*bdv^^^3=0) =
762 (bdv=0 | (1 + c*bdv^^^2=0)*)
763 Thm("d3_reduce_equation11",num_str @{thm d3_reduce_equation11}),
764 (*a*bdv + bdv^^^3=0) =
765 (bdv=0 | (a + bdv^^^2=0)*)
766 Thm("d3_reduce_equation12",num_str @{thm d3_reduce_equation12}),
767 (* bdv + bdv^^^3=0) =
768 (bdv=0 | (1 + bdv^^^2=0)*)
769 Thm("d3_reduce_equation13",num_str @{thm d3_reduce_equation13}),
770 (* b*bdv^^^2 + c*bdv^^^3=0) =
771 (bdv=0 | ( b*bdv + c*bdv^^^2=0)*)
772 Thm("d3_reduce_equation14",num_str @{thm d3_reduce_equation14}),
773 (* bdv^^^2 + c*bdv^^^3=0) =
774 (bdv=0 | ( bdv + c*bdv^^^2=0)*)
775 Thm("d3_reduce_equation15",num_str @{thm d3_reduce_equation15}),
776 (* b*bdv^^^2 + bdv^^^3=0) =
777 (bdv=0 | ( b*bdv + bdv^^^2=0)*)
778 Thm("d3_reduce_equation16",num_str @{thm d3_reduce_equation16}),
779 (* bdv^^^2 + bdv^^^3=0) =
780 (bdv=0 | ( bdv + bdv^^^2=0)*)
781 Thm("d3_isolate_add1",num_str @{thm d3_isolate_add1}),
782 (*[|Not(bdv occurs_in a)|] ==> (a + b*bdv^^^3=0) =
783 (bdv=0 | (b*bdv^^^3=a)*)
784 Thm("d3_isolate_add2",num_str @{thm d3_isolate_add2}),
785 (*[|Not(bdv occurs_in a)|] ==> (a + bdv^^^3=0) =
786 (bdv=0 | ( bdv^^^3=a)*)
787 Thm("d3_isolate_div",num_str @{thm d3_isolate_div}),
788 (*[|Not(b=0)|] ==> (b*bdv^^^3=c) = (bdv^^^3=c/b*)
789 Thm("d3_root_equation2",num_str @{thm d3_root_equation2}),
790 (*(bdv^^^3=0) = (bdv=0) *)
791 Thm("d3_root_equation1",num_str @{thm d3_root_equation1})
792 (*bdv^^^3=c) = (bdv = nroot 3 c*)
794 scr = Script ((term_of o the o (parse thy)) "empty_script")
798 (*isolate the bound variable in an d4 equation; 'bdv' is a meta-constant*)
799 val d4_polyeq_simplify = prep_rls(
800 Rls {id = "d4_polyeq_simplify", preconds = [],
801 rew_ord = ("e_rew_ord",e_rew_ord), erls = PolyEq_erls,
802 srls = Erls, calc = [],
804 [Thm("d4_sub_u1",num_str @{thm d4_sub_u1)
805 (* ax^4+bx^2+c=0 -> x=+-sqrt(ax^2+bx^+c) *)
807 scr = Script ((term_of o the o (parse thy)) "empty_script")
811 overwritelthy @{theory}
813 [("d0_polyeq_simplify", d0_polyeq_simplify),
814 ("d1_polyeq_simplify", d1_polyeq_simplify),
815 ("d2_polyeq_simplify", d2_polyeq_simplify),
816 ("d2_polyeq_bdv_only_simplify", d2_polyeq_bdv_only_simplify),
817 ("d2_polyeq_sq_only_simplify", d2_polyeq_sq_only_simplify),
818 ("d2_polyeq_pqFormula_simplify", d2_polyeq_pqFormula_simplify),
819 ("d2_polyeq_abcFormula_simplify",
820 d2_polyeq_abcFormula_simplify),
821 ("d3_polyeq_simplify", d3_polyeq_simplify),
822 ("d4_polyeq_simplify", d4_polyeq_simplify)
825 (*------------------------problems------------------------*)
827 (get_pbt ["degree_2","polynomial","univariate","equation"]);
831 (*-------------------------poly-----------------------*)
833 (prep_pbt thy "pbl_equ_univ_poly" [] e_pblID
834 (["polynomial","univariate","equation"],
835 [("#Given" ,["equality e_e","solveFor v_v"]),
836 ("#Where" ,["~((e_e::bool) is_ratequation_in (v_v::real))",
837 "~((lhs e_e) is_rootTerm_in (v_v::real))",
838 "~((rhs e_e) is_rootTerm_in (v_v::real))"]),
839 ("#Find" ,["solutions v_i"])
841 PolyEq_prls, SOME "solve (e_e::bool, v_v)",
845 (prep_pbt thy "pbl_equ_univ_poly_deg0" [] e_pblID
846 (["degree_0","polynomial","univariate","equation"],
847 [("#Given" ,["equality e_e","solveFor v_v"]),
848 ("#Where" ,["matches (?a = 0) e_e",
849 "(lhs e_e) is_poly_in v_v",
850 "((lhs e_e) has_degree_in v_v ) = 0"
852 ("#Find" ,["solutions v_i"])
854 PolyEq_prls, SOME "solve (e_e::bool, v_v)",
855 [["PolyEq","solve_d0_polyeq_equation"]]));
859 (prep_pbt thy "pbl_equ_univ_poly_deg1" [] e_pblID
860 (["degree_1","polynomial","univariate","equation"],
861 [("#Given" ,["equality e_e","solveFor v_v"]),
862 ("#Where" ,["matches (?a = 0) e_e",
863 "(lhs e_e) is_poly_in v_v",
864 "((lhs e_e) has_degree_in v_v ) = 1"
866 ("#Find" ,["solutions v_i"])
868 PolyEq_prls, SOME "solve (e_e::bool, v_v)",
869 [["PolyEq","solve_d1_polyeq_equation"]]));
873 (prep_pbt thy "pbl_equ_univ_poly_deg2" [] e_pblID
874 (["degree_2","polynomial","univariate","equation"],
875 [("#Given" ,["equality e_e","solveFor v_v"]),
876 ("#Where" ,["matches (?a = 0) e_e",
877 "(lhs e_e) is_poly_in v_v ",
878 "((lhs e_e) has_degree_in v_v ) = 2"]),
879 ("#Find" ,["solutions v_i"])
881 PolyEq_prls, SOME "solve (e_e::bool, v_v)",
882 [["PolyEq","solve_d2_polyeq_equation"]]));
885 (prep_pbt thy "pbl_equ_univ_poly_deg2_sqonly" [] e_pblID
886 (["sq_only","degree_2","polynomial","univariate","equation"],
887 [("#Given" ,["equality e_e","solveFor v_v"]),
888 ("#Where" ,["matches ( ?a + ?v_^^^2 = 0) e_e | " ^
889 "matches ( ?a + ?b*?v_^^^2 = 0) e_e | " ^
890 "matches ( ?v_^^^2 = 0) e_e | " ^
891 "matches ( ?b*?v_^^^2 = 0) e_e" ,
892 "Not (matches (?a + ?v_ + ?v_^^^2 = 0) e_e) &" ^
893 "Not (matches (?a + ?b*?v_ + ?v_^^^2 = 0) e_e) &" ^
894 "Not (matches (?a + ?v_ + ?c*?v_^^^2 = 0) e_e) &" ^
895 "Not (matches (?a + ?b*?v_ + ?c*?v_^^^2 = 0) e_e) &" ^
896 "Not (matches ( ?v_ + ?v_^^^2 = 0) e_e) &" ^
897 "Not (matches ( ?b*?v_ + ?v_^^^2 = 0) e_e) &" ^
898 "Not (matches ( ?v_ + ?c*?v_^^^2 = 0) e_e) &" ^
899 "Not (matches ( ?b*?v_ + ?c*?v_^^^2 = 0) e_e)"]),
900 ("#Find" ,["solutions v_i"])
902 PolyEq_prls, SOME "solve (e_e::bool, v_v)",
903 [["PolyEq","solve_d2_polyeq_sqonly_equation"]]));
906 (prep_pbt thy "pbl_equ_univ_poly_deg2_bdvonly" [] e_pblID
907 (["bdv_only","degree_2","polynomial","univariate","equation"],
908 [("#Given" ,["equality e_e","solveFor v_v"]),
909 ("#Where" ,["matches (?a*?v_ + ?v_^^^2 = 0) e_e | " ^
910 "matches ( ?v_ + ?v_^^^2 = 0) e_e | " ^
911 "matches ( ?v_ + ?b*?v_^^^2 = 0) e_e | " ^
912 "matches (?a*?v_ + ?b*?v_^^^2 = 0) e_e | " ^
913 "matches ( ?v_^^^2 = 0) e_e | " ^
914 "matches ( ?b*?v_^^^2 = 0) e_e "]),
915 ("#Find" ,["solutions v_i"])
917 PolyEq_prls, SOME "solve (e_e::bool, v_v)",
918 [["PolyEq","solve_d2_polyeq_bdvonly_equation"]]));
921 (prep_pbt thy "pbl_equ_univ_poly_deg2_pq" [] e_pblID
922 (["pqFormula","degree_2","polynomial","univariate","equation"],
923 [("#Given" ,["equality e_e","solveFor v_v"]),
924 ("#Where" ,["matches (?a + 1*?v_^^^2 = 0) e_e | " ^
925 "matches (?a + ?v_^^^2 = 0) e_e"]),
926 ("#Find" ,["solutions v_i"])
928 PolyEq_prls, SOME "solve (e_e::bool, v_v)",
929 [["PolyEq","solve_d2_polyeq_pq_equation"]]));
932 (prep_pbt thy "pbl_equ_univ_poly_deg2_abc" [] e_pblID
933 (["abcFormula","degree_2","polynomial","univariate","equation"],
934 [("#Given" ,["equality e_e","solveFor v_v"]),
935 ("#Where" ,["matches (?a + ?v_^^^2 = 0) e_e | " ^
936 "matches (?a + ?b*?v_^^^2 = 0) e_e"]),
937 ("#Find" ,["solutions v_i"])
939 PolyEq_prls, SOME "solve (e_e::bool, v_v)",
940 [["PolyEq","solve_d2_polyeq_abc_equation"]]));
944 (prep_pbt thy "pbl_equ_univ_poly_deg3" [] e_pblID
945 (["degree_3","polynomial","univariate","equation"],
946 [("#Given" ,["equality e_e","solveFor v_v"]),
947 ("#Where" ,["matches (?a = 0) e_e",
948 "(lhs e_e) is_poly_in v_v ",
949 "((lhs e_e) has_degree_in v_v) = 3"]),
950 ("#Find" ,["solutions v_i"])
952 PolyEq_prls, SOME "solve (e_e::bool, v_v)",
953 [["PolyEq","solve_d3_polyeq_equation"]]));
957 (prep_pbt thy "pbl_equ_univ_poly_deg4" [] e_pblID
958 (["degree_4","polynomial","univariate","equation"],
959 [("#Given" ,["equality e_e","solveFor v_v"]),
960 ("#Where" ,["matches (?a = 0) e_e",
961 "(lhs e_e) is_poly_in v_v ",
962 "((lhs e_e) has_degree_in v_v) = 4"]),
963 ("#Find" ,["solutions v_i"])
965 PolyEq_prls, SOME "solve (e_e::bool, v_v)",
966 [(*["PolyEq","solve_d4_polyeq_equation"]*)]));
968 (*--- normalize ---*)
970 (prep_pbt thy "pbl_equ_univ_poly_norm" [] e_pblID
971 (["normalize","polynomial","univariate","equation"],
972 [("#Given" ,["equality e_e","solveFor v_v"]),
973 ("#Where" ,["(Not((matches (?a = 0 ) e_e ))) |" ^
974 "(Not(((lhs e_e) is_poly_in v_v)))"]),
975 ("#Find" ,["solutions v_i"])
977 PolyEq_prls, SOME "solve (e_e::bool, v_v)",
978 [["PolyEq","normalize_poly"]]));
979 (*-------------------------expanded-----------------------*)
981 (prep_pbt thy "pbl_equ_univ_expand" [] e_pblID
982 (["expanded","univariate","equation"],
983 [("#Given" ,["equality e_e","solveFor v_v"]),
984 ("#Where" ,["matches (?a = 0) e_e",
985 "(lhs e_e) is_expanded_in v_v "]),
986 ("#Find" ,["solutions v_i"])
988 PolyEq_prls, SOME "solve (e_e::bool, v_v)",
993 (prep_pbt thy "pbl_equ_univ_expand_deg2" [] e_pblID
994 (["degree_2","expanded","univariate","equation"],
995 [("#Given" ,["equality e_e","solveFor v_v"]),
996 ("#Where" ,["((lhs e_e) has_degree_in v_v) = 2"]),
997 ("#Find" ,["solutions v_i"])
999 PolyEq_prls, SOME "solve (e_e::bool, v_v)",
1000 [["PolyEq","complete_square"]]));
1003 "-------------------------methods-----------------------";
1005 (prep_met thy "met_polyeq" [] e_metID
1008 {rew_ord'="tless_true",rls'=Atools_erls,calc = [], srls = e_rls, prls=e_rls,
1009 crls=PolyEq_crls, nrls=norm_Rational}, "empty_script"));
1012 (prep_met thy "met_polyeq_norm" [] e_metID
1013 (["PolyEq","normalize_poly"],
1014 [("#Given" ,["equality e_e","solveFor v_v"]),
1015 ("#Where" ,["(Not((matches (?a = 0 ) e_e ))) |" ^
1016 "(Not(((lhs e_e) is_poly_in v_v)))"]),
1017 ("#Find" ,["solutions v_i"])
1019 {rew_ord'="termlessI",
1024 crls=PolyEq_crls, nrls=norm_Rational
1025 "Script Normalize_poly (e_e::bool) (v_v::real) = " ^
1026 "(let e_e =((Try (Rewrite all_left False)) @@ " ^
1027 " (Try (Repeat (Rewrite makex1_x False))) @@ " ^
1028 " (Try (Repeat (Rewrite_Set expand_binoms False))) @@ " ^
1029 " (Try (Repeat (Rewrite_Set_Inst [(bdv,v_::real)] " ^
1030 " make_ratpoly_in False))) @@ " ^
1031 " (Try (Repeat (Rewrite_Set polyeq_simplify False)))) e_e " ^
1032 " in (SubProblem (PolyEq_,[polynomial,univariate,equation], " ^
1033 " [no_met]) [BOOL e_e, REAL v_]))"
1037 (prep_met thy "met_polyeq_d0" [] e_metID
1038 (["PolyEq","solve_d0_polyeq_equation"],
1039 [("#Given" ,["equality e_e","solveFor v_v"]),
1040 ("#Where" ,["(lhs e_e) is_poly_in v_v ",
1041 "((lhs e_e) has_degree_in v_v) = 0"]),
1042 ("#Find" ,["solutions v_i"])
1044 {rew_ord'="termlessI",
1048 calc=[("sqrt", ("NthRoot.sqrt", eval_sqrt "#sqrt_"))],
1049 crls=PolyEq_crls, nrls=norm_Rational},
1050 "Script Solve_d0_polyeq_equation (e_e::bool) (v_v::real) = " ^
1051 "(let e_e = ((Try (Rewrite_Set_Inst [(bdv,v_::real)] " ^
1052 " d0_polyeq_simplify False))) e_e " ^
1053 " in ((Or_to_List e_e)::bool list))"
1057 (prep_met thy "met_polyeq_d1" [] e_metID
1058 (["PolyEq","solve_d1_polyeq_equation"],
1059 [("#Given" ,["equality e_e","solveFor v_v"]),
1060 ("#Where" ,["(lhs e_e) is_poly_in v_v ",
1061 "((lhs e_e) has_degree_in v_v) = 1"]),
1062 ("#Find" ,["solutions v_i"])
1064 {rew_ord'="termlessI",
1068 calc=[("sqrt", ("NthRoot.sqrt", eval_sqrt "#sqrt_"))],
1069 crls=PolyEq_crls, nrls=norm_Rational(*,
1070 (* asm_rls=["d1_polyeq_simplify"],*)
1072 asm_thm=[("d1_isolate_div","")]*)},
1073 "Script Solve_d1_polyeq_equation (e_e::bool) (v_v::real) = " ^
1074 "(let e_e = ((Try (Rewrite_Set_Inst [(bdv,v_::real)] " ^
1075 " d1_polyeq_simplify True)) @@ " ^
1076 " (Try (Rewrite_Set polyeq_simplify False)) @@ " ^
1077 " (Try (Rewrite_Set norm_Rational_parenthesized False))) e_;" ^
1078 " (L_::bool list) = ((Or_to_List e_e)::bool list) " ^
1079 " in Check_elementwise L_ {(v_v::real). Assumptions} )"
1083 (prep_met thy "met_polyeq_d22" [] e_metID
1084 (["PolyEq","solve_d2_polyeq_equation"],
1085 [("#Given" ,["equality e_e","solveFor v_v"]),
1086 ("#Where" ,["(lhs e_e) is_poly_in v_v ",
1087 "((lhs e_e) has_degree_in v_v) = 2"]),
1088 ("#Find" ,["solutions v_i"])
1090 {rew_ord'="termlessI",
1094 calc=[("sqrt", ("NthRoot.sqrt", eval_sqrt "#sqrt_"))],
1095 crls=PolyEq_crls, nrls=norm_Rational},
1096 "Script Solve_d2_polyeq_equation (e_e::bool) (v_v::real) = " ^
1097 " (let e_e = ((Try (Rewrite_Set_Inst [(bdv,v_::real)] " ^
1098 " d2_polyeq_simplify True)) @@ " ^
1099 " (Try (Rewrite_Set polyeq_simplify False)) @@ " ^
1100 " (Try (Rewrite_Set_Inst [(bdv,v_::real)] " ^
1101 " d1_polyeq_simplify True)) @@ " ^
1102 " (Try (Rewrite_Set polyeq_simplify False)) @@ " ^
1103 " (Try (Rewrite_Set norm_Rational_parenthesized False))) e_;" ^
1104 " (L_::bool list) = ((Or_to_List e_e)::bool list) " ^
1105 " in Check_elementwise L_ {(v_v::real). Assumptions} )"
1109 (prep_met thy "met_polyeq_d2_bdvonly" [] e_metID
1110 (["PolyEq","solve_d2_polyeq_bdvonly_equation"],
1111 [("#Given" ,["equality e_e","solveFor v_v"]),
1112 ("#Where" ,["(lhs e_e) is_poly_in v_v ",
1113 "((lhs e_e) has_degree_in v_v) = 2"]),
1114 ("#Find" ,["solutions v_i"])
1116 {rew_ord'="termlessI",
1120 calc=[("sqrt", ("NthRoot.sqrt", eval_sqrt "#sqrt_"))],
1121 crls=PolyEq_crls, nrls=norm_Rational},
1122 "Script Solve_d2_polyeq_bdvonly_equation (e_e::bool) (v_v::real) =" ^
1123 " (let e_e = ((Try (Rewrite_Set_Inst [(bdv,v_::real)] " ^
1124 " d2_polyeq_bdv_only_simplify True)) @@ " ^
1125 " (Try (Rewrite_Set polyeq_simplify False)) @@ " ^
1126 " (Try (Rewrite_Set_Inst [(bdv,v_::real)] " ^
1127 " d1_polyeq_simplify True)) @@ " ^
1128 " (Try (Rewrite_Set polyeq_simplify False)) @@ " ^
1129 " (Try (Rewrite_Set norm_Rational_parenthesized False))) e_;" ^
1130 " (L_::bool list) = ((Or_to_List e_e)::bool list) " ^
1131 " in Check_elementwise L_ {(v_v::real). Assumptions} )"
1135 (prep_met thy "met_polyeq_d2_sqonly" [] e_metID
1136 (["PolyEq","solve_d2_polyeq_sqonly_equation"],
1137 [("#Given" ,["equality e_e","solveFor v_v"]),
1138 ("#Where" ,["(lhs e_e) is_poly_in v_v ",
1139 "((lhs e_e) has_degree_in v_v) = 2"]),
1140 ("#Find" ,["solutions v_i"])
1142 {rew_ord'="termlessI",
1146 calc=[("sqrt", ("NthRoot.sqrt", eval_sqrt "#sqrt_"))],
1147 crls=PolyEq_crls, nrls=norm_Rational},
1148 "Script Solve_d2_polyeq_sqonly_equation (e_e::bool) (v_v::real) =" ^
1149 " (let e_e = ((Try (Rewrite_Set_Inst [(bdv,v_::real)] " ^
1150 " d2_polyeq_sq_only_simplify True)) @@ " ^
1151 " (Try (Rewrite_Set polyeq_simplify False)) @@ " ^
1152 " (Try (Rewrite_Set norm_Rational_parenthesized False))) e_; " ^
1153 " (L_::bool list) = ((Or_to_List e_e)::bool list) " ^
1154 " in Check_elementwise L_ {(v_v::real). Assumptions} )"
1158 (prep_met thy "met_polyeq_d2_pq" [] e_metID
1159 (["PolyEq","solve_d2_polyeq_pq_equation"],
1160 [("#Given" ,["equality e_e","solveFor v_v"]),
1161 ("#Where" ,["(lhs e_e) is_poly_in v_v ",
1162 "((lhs e_e) has_degree_in v_v) = 2"]),
1163 ("#Find" ,["solutions v_i"])
1165 {rew_ord'="termlessI",
1169 calc=[("sqrt", ("NthRoot.sqrt", eval_sqrt "#sqrt_"))],
1170 crls=PolyEq_crls, nrls=norm_Rational},
1171 "Script Solve_d2_polyeq_pq_equation (e_e::bool) (v_v::real) = " ^
1172 " (let e_e = ((Try (Rewrite_Set_Inst [(bdv,v_::real)] " ^
1173 " d2_polyeq_pqFormula_simplify True)) @@ " ^
1174 " (Try (Rewrite_Set polyeq_simplify False)) @@ " ^
1175 " (Try (Rewrite_Set norm_Rational_parenthesized False))) e_;" ^
1176 " (L_::bool list) = ((Or_to_List e_e)::bool list) " ^
1177 " in Check_elementwise L_ {(v_v::real). Assumptions} )"
1181 (prep_met thy "met_polyeq_d2_abc" [] e_metID
1182 (["PolyEq","solve_d2_polyeq_abc_equation"],
1183 [("#Given" ,["equality e_e","solveFor v_v"]),
1184 ("#Where" ,["(lhs e_e) is_poly_in v_v ",
1185 "((lhs e_e) has_degree_in v_v) = 2"]),
1186 ("#Find" ,["solutions v_i"])
1188 {rew_ord'="termlessI",
1192 calc=[("sqrt", ("NthRoot.sqrt", eval_sqrt "#sqrt_"))],
1193 crls=PolyEq_crls, nrls=norm_Rational},
1194 "Script Solve_d2_polyeq_abc_equation (e_e::bool) (v_v::real) = " ^
1195 " (let e_e = ((Try (Rewrite_Set_Inst [(bdv,v_::real)] " ^
1196 " d2_polyeq_abcFormula_simplify True)) @@ " ^
1197 " (Try (Rewrite_Set polyeq_simplify False)) @@ " ^
1198 " (Try (Rewrite_Set norm_Rational_parenthesized False))) e_;" ^
1199 " (L_::bool list) = ((Or_to_List e_e)::bool list) " ^
1200 " in Check_elementwise L_ {(v_v::real). Assumptions} )"
1204 (prep_met thy "met_polyeq_d3" [] e_metID
1205 (["PolyEq","solve_d3_polyeq_equation"],
1206 [("#Given" ,["equality e_e","solveFor v_v"]),
1207 ("#Where" ,["(lhs e_e) is_poly_in v_v ",
1208 "((lhs e_e) has_degree_in v_v) = 3"]),
1209 ("#Find" ,["solutions v_i"])
1211 {rew_ord'="termlessI",
1215 calc=[("sqrt", ("NthRoot.sqrt", eval_sqrt "#sqrt_"))],
1216 crls=PolyEq_crls, nrls=norm_Rational},
1217 "Script Solve_d3_polyeq_equation (e_e::bool) (v_v::real) = " ^
1218 " (let e_e = ((Try (Rewrite_Set_Inst [(bdv,v_::real)] " ^
1219 " d3_polyeq_simplify True)) @@ " ^
1220 " (Try (Rewrite_Set polyeq_simplify False)) @@ " ^
1221 " (Try (Rewrite_Set_Inst [(bdv,v_::real)] " ^
1222 " d2_polyeq_simplify True)) @@ " ^
1223 " (Try (Rewrite_Set polyeq_simplify False)) @@ " ^
1224 " (Try (Rewrite_Set_Inst [(bdv,v_::real)] " ^
1225 " d1_polyeq_simplify True)) @@ " ^
1226 " (Try (Rewrite_Set polyeq_simplify False)) @@ " ^
1227 " (Try (Rewrite_Set norm_Rational_parenthesized False))) e_;" ^
1228 " (L_::bool list) = ((Or_to_List e_e)::bool list) " ^
1229 " in Check_elementwise L_ {(v_v::real). Assumptions} )"
1232 (*.solves all expanded (ie. normalized) terms of degree 2.*)
1233 (*Oct.02 restriction: 'eval_true 0 =< discriminant' ony for integer values
1234 by 'PolyEq_erls'; restricted until Float.thy is implemented*)
1236 (prep_met thy "met_polyeq_complsq" [] e_metID
1237 (["PolyEq","complete_square"],
1238 [("#Given" ,["equality e_e","solveFor v_v"]),
1239 ("#Where" ,["matches (?a = 0) e_e",
1240 "((lhs e_e) has_degree_in v_v) = 2"]),
1241 ("#Find" ,["solutions v_i"])
1243 {rew_ord'="termlessI",rls'=PolyEq_erls,srls=e_rls,prls=PolyEq_prls,
1244 calc=[("sqrt", ("NthRoot.sqrt", eval_sqrt "#sqrt_"))],
1245 crls=PolyEq_crls, nrls=norm_Rational},
1246 "Script Complete_square (e_e::bool) (v_v::real) = " ^
1247 "(let e_e = ((Try (Rewrite_Set_Inst [(bdv,v_)] cancel_leading_coeff True))" ^
1248 " @@ (Try (Rewrite_Set_Inst [(bdv,v_)] complete_square True)) " ^
1249 " @@ (Try (Rewrite square_explicit1 False)) " ^
1250 " @@ (Try (Rewrite square_explicit2 False)) " ^
1251 " @@ (Rewrite root_plus_minus True) " ^
1252 " @@ (Try (Repeat (Rewrite_Inst [(bdv,v_)] bdv_explicit1 False))) " ^
1253 " @@ (Try (Repeat (Rewrite_Inst [(bdv,v_)] bdv_explicit2 False))) " ^
1254 " @@ (Try (Repeat " ^
1255 " (Rewrite_Inst [(bdv,v_)] bdv_explicit3 False))) " ^
1256 " @@ (Try (Rewrite_Set calculate_RootRat False)) " ^
1257 " @@ (Try (Repeat (Calculate SQRT)))) e_e " ^
1258 " in ((Or_to_List e_e)::bool list))"
1262 (* termorder hacked by MG *)
1263 local (*. for make_polynomial_in .*)
1265 open Term; (* for type order = EQUAL | LESS | GREATER *)
1267 fun pr_ord EQUAL = "EQUAL"
1268 | pr_ord LESS = "LESS"
1269 | pr_ord GREATER = "GREATER";
1271 fun dest_hd' x (Const (a, T)) = (((a, 0), T), 0)
1272 | dest_hd' x (t as Free (a, T)) =
1273 if x = t then ((("|||||||||||||", 0), T), 0) (*WN*)
1274 else (((a, 0), T), 1)
1275 | dest_hd' x (Var v) = (v, 2)
1276 | dest_hd' x (Bound i) = ((("", i), dummyT), 3)
1277 | dest_hd' x (Abs (_, T, _)) = ((("", 0), T), 4);
1279 fun size_of_term' x (Const ("Atools.pow",_) $ Free (var,_) $ Free (pot,_)) =
1282 (if xstr = var then 1000*(the (int_of_str pot)) else 3)
1283 | _ => raise error ("size_of_term' called with subst = "^
1285 | size_of_term' x (Free (subst,_)) =
1287 (Free (xstr,_)) => (if xstr = subst then 1000 else 1)
1288 | _ => raise error ("size_of_term' called with subst = "^
1290 | size_of_term' x (Abs (_,_,body)) = 1 + size_of_term' x body
1291 | size_of_term' x (f$t) = size_of_term' x f + size_of_term' x t
1292 | size_of_term' x _ = 1;
1295 fun Term_Ord.term_ord' x pr thy (Abs (_, T, t), Abs(_, U, u)) = (* ~ term.ML *)
1296 (case Term_Ord.term_ord' x pr thy (t, u) of EQUAL => Term_Ord.typ_ord (T, U) | ord => ord)
1297 | Term_Ord.term_ord' x pr thy (t, u) =
1300 val (f, ts) = strip_comb t and (g, us) = strip_comb u;
1301 val _=writeln("t= f@ts= \""^
1302 ((Syntax.string_of_term (thy2ctxt thy)) f)^"\" @ \"["^
1303 (commas(map(string_of_cterm o cterm_of(sign_of thy)) ts))^"]\"");
1304 val _=writeln("u= g@us= \""^
1305 ((Syntax.string_of_term (thy2ctxt thy)) g)^"\" @ \"["^
1306 (commas(map(string_of_cterm o cterm_of(sign_of thy)) us))^"]\"");
1307 val _=writeln("size_of_term(t,u)= ("^
1308 (string_of_int(size_of_term' x t))^", "^
1309 (string_of_int(size_of_term' x u))^")");
1310 val _=writeln("hd_ord(f,g) = "^((pr_ord o (hd_ord x))(f,g)));
1311 val _=writeln("terms_ord(ts,us) = "^
1312 ((pr_ord o (terms_ord x) str false)(ts,us)));
1313 val _=writeln("-------");
1316 case int_ord (size_of_term' x t, size_of_term' x u) of
1318 let val (f, ts) = strip_comb t and (g, us) = strip_comb u in
1319 (case hd_ord x (f, g) of EQUAL => (terms_ord x str pr) (ts, us)
1323 and hd_ord x (f, g) = (* ~ term.ML *)
1324 prod_ord (prod_ord indexname_ord Term_Ord.typ_ord) int_ord (dest_hd' x f,
1326 and terms_ord x str pr (ts, us) =
1327 list_ord (term_ord' x pr (assoc_thy "Isac.thy"))(ts, us);
1330 fun ord_make_polynomial_in (pr:bool) thy subst tu =
1332 (* val _=writeln("*** subs variable is: "^(subst2str subst)); *)
1335 (_,x)::_ => (term_ord' x pr thy tu = LESS)
1336 | _ => raise error ("ord_make_polynomial_in called with subst = "^
1341 val order_add_mult_in = prep_rls(
1342 Rls{id = "order_add_mult_in", preconds = [],
1343 rew_ord = ("ord_make_polynomial_in",
1344 ord_make_polynomial_in false Poly.thy),
1345 erls = e_rls,srls = Erls,
1348 rules = [Thm ("real_mult_commute",num_str @{thm real_mult_commute}),
1350 Thm ("real_mult_left_commute",num_str @{thm real_mult_left_commute}),
1351 (*z1.0 * (z2.0 * z3.0) = z2.0 * (z1.0 * z3.0)*)
1352 Thm ("real_mult_assoc",num_str @{thm real_mult_assoc}),
1353 (*z1.0 * z2.0 * z3.0 = z1.0 * (z2.0 * z3.0)*)
1354 Thm ("add_commute",num_str @{thm add_commute}),
1356 Thm ("add_left_commute",num_str @{thm add_left_commute}),
1357 (*x + (y + z) = y + (x + z)*)
1358 Thm ("add_assoc",num_str @{thm add_assoc})
1359 (*z1.0 + z2.0 + z3.0 = z1.0 + (z2.0 + z3.0)*)
1360 ], scr = EmptyScr}:rls);
1362 val collect_bdv = prep_rls(
1363 Rls{id = "collect_bdv", preconds = [],
1364 rew_ord = ("dummy_ord", dummy_ord),
1365 erls = e_rls,srls = Erls,
1368 rules = [Thm ("bdv_collect_1",num_str @{thm bdv_collect_1}),
1369 Thm ("bdv_collect_2",num_str @{thm bdv_collect_2}),
1370 Thm ("bdv_collect_3",num_str @{thm bdv_collect_3}),
1372 Thm ("bdv_collect_assoc1_1",num_str @{thm bdv_collect_assoc1_1}),
1373 Thm ("bdv_collect_assoc1_2",num_str @{thm bdv_collect_assoc1_2}),
1374 Thm ("bdv_collect_assoc1_3",num_str @{thm bdv_collect_assoc1_3}),
1376 Thm ("bdv_collect_assoc2_1",num_str @{thm bdv_collect_assoc2_1}),
1377 Thm ("bdv_collect_assoc2_2",num_str @{thm bdv_collect_assoc2_2}),
1378 Thm ("bdv_collect_assoc2_3",num_str @{thm bdv_collect_assoc2_3}),
1381 Thm ("bdv_n_collect_1",num_str @{thm bdv_n_collect_1}),
1382 Thm ("bdv_n_collect_2",num_str @{thm bdv_n_collect_2}),
1383 Thm ("bdv_n_collect_3",num_str @{thm bdv_n_collect_3}),
1385 Thm ("bdv_n_collect_assoc1_1",num_str @{thm bdv_n_collect_assoc1_1}),
1386 Thm ("bdv_n_collect_assoc1_2",num_str @{thm bdv_n_collect_assoc1_2}),
1387 Thm ("bdv_n_collect_assoc1_3",num_str @{thm bdv_n_collect_assoc1_3}),
1389 Thm ("bdv_n_collect_assoc2_1",num_str @{thm bdv_n_collect_assoc2_1}),
1390 Thm ("bdv_n_collect_assoc2_2",num_str @{thm bdv_n_collect_assoc2_2}),
1391 Thm ("bdv_n_collect_assoc2_3",num_str @{thm bdv_n_collect_assoc2_3)
1392 ], scr = EmptyScr}:rls);
1394 (*.transforms an arbitrary term without roots to a polynomial [4]
1395 according to knowledge/Poly.sml.*)
1396 val make_polynomial_in = prep_rls(
1397 Seq {id = "make_polynomial_in", preconds = []:term list,
1398 rew_ord = ("dummy_ord", dummy_ord),
1399 erls = Atools_erls, srls = Erls,
1400 calc = [], (*asm_thm = [],*)
1401 rules = [Rls_ expand_poly,
1402 Rls_ order_add_mult_in,
1403 Rls_ simplify_power,
1404 Rls_ collect_numerals,
1406 Thm ("realpow_oneI",num_str @{thm realpow_oneI}),
1407 Rls_ discard_parentheses,
1414 append_rls "separate_bdvs"
1416 [Thm ("separate_bdv", num_str @{separate_bdv}),
1417 (*"?a * ?bdv / ?b = ?a / ?b * ?bdv"*)
1418 Thm ("separate_bdv_n", num_str @{separate_bdv_n}),
1419 Thm ("separate_1_bdv", num_str @{separate_1_bdv}),
1420 (*"?bdv / ?b = (1 / ?b) * ?bdv"*)
1421 Thm ("separate_1_bdv_n", num_str @{separate_1_bdv_n}),
1422 (*"?bdv ^^^ ?n / ?b = 1 / ?b * ?bdv ^^^ ?n"*)
1423 Thm ("nadd_divide_distrib",
1424 num_str @{thm nadd_divide_distrib})
1425 (*"(?x + ?y) / ?z = ?x / ?z + ?y / ?z"
1426 WN051031 DOES NOT BELONG TO HERE*)
1428 val make_ratpoly_in = prep_rls(
1429 Seq {id = "make_ratpoly_in", preconds = []:term list,
1430 rew_ord = ("dummy_ord", dummy_ord),
1431 erls = Atools_erls, srls = Erls,
1432 calc = [], (*asm_thm = [],*)
1433 rules = [Rls_ norm_Rational,
1434 Rls_ order_add_mult_in,
1435 Rls_ discard_parentheses,
1437 (* Rls_ rearrange_assoc, WN060916 why does cancel_p not work?*)
1439 (*Calc ("HOL.divide" ,eval_cancel "#divide_e") too weak!*)
1441 scr = EmptyScr}:rls);
1444 ruleset' := overwritelthy @{theory} (!ruleset',
1445 [("order_add_mult_in", order_add_mult_in),
1446 ("collect_bdv", collect_bdv),
1447 ("make_polynomial_in", make_polynomial_in),
1448 ("make_ratpoly_in", make_ratpoly_in),
1449 ("separate_bdvs", separate_bdvs)