1 (* theory collecting all knowledge
2 (predicates 'is_rootEq_in', 'is_sqrt_in', 'is_ratEq_in')
3 for PolynomialEquations.
4 alternative dependencies see @{theory "Isac_Knowledge"}
10 (c) by Richard Lang, 2003
13 theory PolyEq imports LinEq RootRatEq begin
15 (*-------------------- rules -------------------------------------------------*)
16 (* type real enforced by op " \<up> " *)
18 cancel_leading_coeff1: "Not (c =!= 0) ==> (a + b*bdv + c*bdv \<up> 2 = 0) =
19 (a/c + b/c*bdv + bdv \<up> 2 = 0)" and
20 cancel_leading_coeff2: "Not (c =!= 0) ==> (a - b*bdv + c*bdv \<up> 2 = 0) =
21 (a/c - b/c*bdv + bdv \<up> 2 = 0)" and
22 cancel_leading_coeff3: "Not (c =!= 0) ==> (a + b*bdv - c*bdv \<up> 2 = 0) =
23 (a/c + b/c*bdv - bdv \<up> 2 = 0)" and
25 cancel_leading_coeff4: "Not (c =!= 0) ==> (a + bdv + c*bdv \<up> 2 = 0) =
26 (a/c + 1/c*bdv + bdv \<up> 2 = 0)" and
27 cancel_leading_coeff5: "Not (c =!= 0) ==> (a - bdv + c*bdv \<up> 2 = 0) =
28 (a/c - 1/c*bdv + bdv \<up> 2 = 0)" and
29 cancel_leading_coeff6: "Not (c =!= 0) ==> (a + bdv - c*bdv \<up> 2 = 0) =
30 (a/c + 1/c*bdv - bdv \<up> 2 = 0)" and
32 cancel_leading_coeff7: "Not (c =!= 0) ==> ( b*bdv + c*bdv \<up> 2 = 0) =
33 ( b/c*bdv + bdv \<up> 2 = 0)" and
34 cancel_leading_coeff8: "Not (c =!= 0) ==> ( b*bdv - c*bdv \<up> 2 = 0) =
35 ( b/c*bdv - bdv \<up> 2 = 0)" and
37 cancel_leading_coeff9: "Not (c =!= 0) ==> ( bdv + c*bdv \<up> 2 = 0) =
38 ( 1/c*bdv + bdv \<up> 2 = 0)" and
39 cancel_leading_coeff10:"Not (c =!= 0) ==> ( bdv - c*bdv \<up> 2 = 0) =
40 ( 1/c*bdv - bdv \<up> 2 = 0)" and
42 cancel_leading_coeff11:"Not (c =!= 0) ==> (a + b*bdv \<up> 2 = 0) =
43 (a/b + bdv \<up> 2 = 0)" and
44 cancel_leading_coeff12:"Not (c =!= 0) ==> (a - b*bdv \<up> 2 = 0) =
45 (a/b - bdv \<up> 2 = 0)" and
46 cancel_leading_coeff13:"Not (c =!= 0) ==> ( b*bdv \<up> 2 = 0) =
47 ( bdv \<up> 2 = 0/b)" and
49 complete_square1: "(q + p*bdv + bdv \<up> 2 = 0) =
50 (q + (p/2 + bdv) \<up> 2 = (p/2) \<up> 2)" and
51 complete_square2: "( p*bdv + bdv \<up> 2 = 0) =
52 ( (p/2 + bdv) \<up> 2 = (p/2) \<up> 2)" and
53 complete_square3: "( bdv + bdv \<up> 2 = 0) =
54 ( (1/2 + bdv) \<up> 2 = (1/2) \<up> 2)" and
56 complete_square4: "(q - p*bdv + bdv \<up> 2 = 0) =
57 (q + (p/2 - bdv) \<up> 2 = (p/2) \<up> 2)" and
58 complete_square5: "(q + p*bdv - bdv \<up> 2 = 0) =
59 (q + (p/2 - bdv) \<up> 2 = (p/2) \<up> 2)" and
61 square_explicit1: "(a + b \<up> 2 = c) = ( b \<up> 2 = c - a)" and
62 square_explicit2: "(a - b \<up> 2 = c) = (-(b \<up> 2) = c - a)" and
64 (*bdv_explicit* required type constrain to real in --- (-8 - 2*x + x \<up> 2 = 0), by rewriting ---*)
65 bdv_explicit1: "(a + bdv = b) = (bdv = - a + (b::real))" and
66 bdv_explicit2: "(a - bdv = b) = ((-1)*bdv = - a + (b::real))" and
67 bdv_explicit3: "((-1)*bdv = b) = (bdv = (-1)*(b::real))" and
69 plus_leq: "(0 <= a + b) = ((-1)*b <= a)"(*Isa?*) and
70 minus_leq: "(0 <= a - b) = ( b <= a)"(*Isa?*) and
73 (*WN0509 compare LinEq.all_left "[|Not(b=!=0)|] ==> (a=b) = (a+(-1)*b=0)"*)
74 all_left: "[|Not(b=!=0)|] ==> (a = b) = (a - b = 0)" and
75 makex1_x: "a\<up>1 = a" and
76 real_assoc_1: "a+(b+c) = a+b+c" and
77 real_assoc_2: "a*(b*c) = a*b*c" and
79 (* ---- degree 0 ----*)
80 d0_true: "(0=0) = True" and
81 d0_false: "[|Not(bdv occurs_in a);Not(a=0)|] ==> (a=0) = False" and
82 (* ---- degree 1 ----*)
84 "[|Not(bdv occurs_in a)|] ==> (a + b*bdv = 0) = (b*bdv = (-1)*a)" and
86 "[|Not(bdv occurs_in a)|] ==> (a + bdv = 0) = ( bdv = (-1)*a)" and
88 "[|Not(b=0);Not(bdv occurs_in c)|] ==> (b*bdv = c) = (bdv = c/b)" and
89 (* ---- degree 2 ----*)
91 "[|Not(bdv occurs_in a)|] ==> (a + b*bdv \<up> 2=0) = (b*bdv \<up> 2= (-1)*a)" and
93 "[|Not(bdv occurs_in a)|] ==> (a + bdv \<up> 2=0) = ( bdv \<up> 2= (-1)*a)" and
95 "[|Not(b=0);Not(bdv occurs_in c)|] ==> (b*bdv \<up> 2=c) = (bdv \<up> 2=c/b)" and
97 d2_prescind1: "(a*bdv + b*bdv \<up> 2 = 0) = (bdv*(a +b*bdv)=0)" and
98 d2_prescind2: "(a*bdv + bdv \<up> 2 = 0) = (bdv*(a + bdv)=0)" and
99 d2_prescind3: "( bdv + b*bdv \<up> 2 = 0) = (bdv*(1+b*bdv)=0)" and
100 d2_prescind4: "( bdv + bdv \<up> 2 = 0) = (bdv*(1+ bdv)=0)" and
101 (* eliminate degree 2 *)
102 (* thm for neg arguments in sqroot have postfix _neg *)
103 d2_sqrt_equation1: "[|(0<=c);Not(bdv occurs_in c)|] ==>
104 (bdv \<up> 2=c) = ((bdv=sqrt c) | (bdv=(-1)*sqrt c ))" and
105 d2_sqrt_equation1_neg:
106 "[|(c<0);Not(bdv occurs_in c)|] ==> (bdv \<up> 2=c) = False" and
107 d2_sqrt_equation2: "(bdv \<up> 2=0) = (bdv=0)" and
108 d2_sqrt_equation3: "(b*bdv \<up> 2=0) = (bdv=0)"
109 axiomatization where (*AK..if replaced by "and" we get errors:
110 exception PTREE "nth _ []" raised
111 (line 783 of "/usr/local/isabisac/src/Tools/isac/Interpret/ctree.sml"):
112 'fun nth _ [] = raise PTREE "nth _ []"'
114 exception Bind raised
115 (line 1097 of "/usr/local/isabisac/test/Tools/isac/Frontend/interface.sml"):
116 'val (Form f, tac, asms) = pt_extract (pt, p);' *)
117 (* WN120315 these 2 thms need "::real", because no " \<up> " constrains type as
118 required in test --- rls d2_polyeq_bdv_only_simplify --- *)
119 d2_reduce_equation1: "(bdv*(a +b*bdv)=0) = ((bdv=0)|(a+b*bdv=(0::real)))" and
120 d2_reduce_equation2: "(bdv*(a + bdv)=0) = ((bdv=0)|(a+ bdv=(0::real)))"
122 axiomatization where (*..if replaced by "and" we get errors:
123 exception PTREE "nth _ []" raised
124 (line 783 of "/usr/local/isabisac/src/Tools/isac/Interpret/ctree.sml"):
125 'fun nth _ [] = raise PTREE "nth _ []"'
127 exception Bind raised
128 (line 1097 of "/usr/local/isabisac/test/Tools/isac/Frontend/interface.sml"):
129 'val (Form f, tac, asms) = pt_extract (pt, p);' *)
130 d2_pqformula1: "[|0<=p \<up> 2 - 4*q|] ==> (q+p*bdv+ bdv \<up> 2=0) =
131 ((bdv= (-1)*(p/2) + sqrt(p \<up> 2 - 4*q)/2)
132 | (bdv= (-1)*(p/2) - sqrt(p \<up> 2 - 4*q)/2))" and
133 d2_pqformula1_neg: "[|p \<up> 2 - 4*q<0|] ==> (q+p*bdv+ bdv \<up> 2=0) = False" and
134 d2_pqformula2: "[|0<=p \<up> 2 - 4*q|] ==> (q+p*bdv+1*bdv \<up> 2=0) =
135 ((bdv= (-1)*(p/2) + sqrt(p \<up> 2 - 4*q)/2)
136 | (bdv= (-1)*(p/2) - sqrt(p \<up> 2 - 4*q)/2))" and
137 d2_pqformula2_neg: "[|p \<up> 2 - 4*q<0|] ==> (q+p*bdv+1*bdv \<up> 2=0) = False" and
138 d2_pqformula3: "[|0<=1 - 4*q|] ==> (q+ bdv+ bdv \<up> 2=0) =
139 ((bdv= (-1)*(1/2) + sqrt(1 - 4*q)/2)
140 | (bdv= (-1)*(1/2) - sqrt(1 - 4*q)/2))" and
141 d2_pqformula3_neg: "[|1 - 4*q<0|] ==> (q+ bdv+ bdv \<up> 2=0) = False" and
142 d2_pqformula4: "[|0<=1 - 4*q|] ==> (q+ bdv+1*bdv \<up> 2=0) =
143 ((bdv= (-1)*(1/2) + sqrt(1 - 4*q)/2)
144 | (bdv= (-1)*(1/2) - sqrt(1 - 4*q)/2))" and
145 d2_pqformula4_neg: "[|1 - 4*q<0|] ==> (q+ bdv+1*bdv \<up> 2=0) = False" and
146 d2_pqformula5: "[|0<=p \<up> 2 - 0|] ==> ( p*bdv+ bdv \<up> 2=0) =
147 ((bdv= (-1)*(p/2) + sqrt(p \<up> 2 - 0)/2)
148 | (bdv= (-1)*(p/2) - sqrt(p \<up> 2 - 0)/2))" and
149 (* d2_pqformula5_neg not need p^2 never less zero in R *)
150 d2_pqformula6: "[|0<=p \<up> 2 - 0|] ==> ( p*bdv+1*bdv \<up> 2=0) =
151 ((bdv= (-1)*(p/2) + sqrt(p \<up> 2 - 0)/2)
152 | (bdv= (-1)*(p/2) - sqrt(p \<up> 2 - 0)/2))" and
153 (* d2_pqformula6_neg not need p^2 never less zero in R *)
154 d2_pqformula7: "[|0<=1 - 0|] ==> ( bdv+ bdv \<up> 2=0) =
155 ((bdv= (-1)*(1/2) + sqrt(1 - 0)/2)
156 | (bdv= (-1)*(1/2) - sqrt(1 - 0)/2))" and
157 (* d2_pqformula7_neg not need, because 1<0 ==> False*)
158 d2_pqformula8: "[|0<=1 - 0|] ==> ( bdv+1*bdv \<up> 2=0) =
159 ((bdv= (-1)*(1/2) + sqrt(1 - 0)/2)
160 | (bdv= (-1)*(1/2) - sqrt(1 - 0)/2))" and
161 (* d2_pqformula8_neg not need, because 1<0 ==> False*)
162 d2_pqformula9: "[|Not(bdv occurs_in q); 0<= (-1)*4*q|] ==>
163 (q+ 1*bdv \<up> 2=0) = ((bdv= 0 + sqrt(0 - 4*q)/2)
164 | (bdv= 0 - sqrt(0 - 4*q)/2))" and
166 "[|Not(bdv occurs_in q); (-1)*4*q<0|] ==> (q+ 1*bdv \<up> 2=0) = False" and
168 "[|Not(bdv occurs_in q); 0<= (-1)*4*q|] ==> (q+ bdv \<up> 2=0) =
169 ((bdv= 0 + sqrt(0 - 4*q)/2)
170 | (bdv= 0 - sqrt(0 - 4*q)/2))" and
172 "[|Not(bdv occurs_in q); (-1)*4*q<0|] ==> (q+ bdv \<up> 2=0) = False" and
174 "[|0<=b \<up> 2 - 4*a*c|] ==> (c + b*bdv+a*bdv \<up> 2=0) =
175 ((bdv=( -b + sqrt(b \<up> 2 - 4*a*c))/(2*a))
176 | (bdv=( -b - sqrt(b \<up> 2 - 4*a*c))/(2*a)))" and
178 "[|b \<up> 2 - 4*a*c<0|] ==> (c + b*bdv+a*bdv \<up> 2=0) = False" and
180 "[|0<=1 - 4*a*c|] ==> (c+ bdv+a*bdv \<up> 2=0) =
181 ((bdv=( -1 + sqrt(1 - 4*a*c))/(2*a))
182 | (bdv=( -1 - sqrt(1 - 4*a*c))/(2*a)))" and
184 "[|1 - 4*a*c<0|] ==> (c+ bdv+a*bdv \<up> 2=0) = False" and
186 "[|0<=b \<up> 2 - 4*1*c|] ==> (c + b*bdv+ bdv \<up> 2=0) =
187 ((bdv=( -b + sqrt(b \<up> 2 - 4*1*c))/(2*1))
188 | (bdv=( -b - sqrt(b \<up> 2 - 4*1*c))/(2*1)))" and
190 "[|b \<up> 2 - 4*1*c<0|] ==> (c + b*bdv+ bdv \<up> 2=0) = False" and
192 "[|0<=1 - 4*1*c|] ==> (c + bdv+ bdv \<up> 2=0) =
193 ((bdv=( -1 + sqrt(1 - 4*1*c))/(2*1))
194 | (bdv=( -1 - sqrt(1 - 4*1*c))/(2*1)))" and
196 "[|1 - 4*1*c<0|] ==> (c + bdv+ bdv \<up> 2=0) = False" and
198 "[|Not(bdv occurs_in c); 0<=0 - 4*a*c|] ==> (c + a*bdv \<up> 2=0) =
199 ((bdv=( 0 + sqrt(0 - 4*a*c))/(2*a))
200 | (bdv=( 0 - sqrt(0 - 4*a*c))/(2*a)))" and
202 "[|Not(bdv occurs_in c); 0 - 4*a*c<0|] ==> (c + a*bdv \<up> 2=0) = False" and
204 "[|Not(bdv occurs_in c); 0<=0 - 4*1*c|] ==> (c+ bdv \<up> 2=0) =
205 ((bdv=( 0 + sqrt(0 - 4*1*c))/(2*1))
206 | (bdv=( 0 - sqrt(0 - 4*1*c))/(2*1)))" and
208 "[|Not(bdv occurs_in c); 0 - 4*1*c<0|] ==> (c+ bdv \<up> 2=0) = False" and
210 "[|0<=b \<up> 2 - 0|] ==> ( b*bdv+a*bdv \<up> 2=0) =
211 ((bdv=( -b + sqrt(b \<up> 2 - 0))/(2*a))
212 | (bdv=( -b - sqrt(b \<up> 2 - 0))/(2*a)))" and
213 (* d2_abcformula7_neg not need b^2 never less zero in R *)
215 "[|0<=b \<up> 2 - 0|] ==> ( b*bdv+ bdv \<up> 2=0) =
216 ((bdv=( -b + sqrt(b \<up> 2 - 0))/(2*1))
217 | (bdv=( -b - sqrt(b \<up> 2 - 0))/(2*1)))" and
218 (* d2_abcformula8_neg not need b^2 never less zero in R *)
220 "[|0<=1 - 0|] ==> ( bdv+a*bdv \<up> 2=0) =
221 ((bdv=( -1 + sqrt(1 - 0))/(2*a))
222 | (bdv=( -1 - sqrt(1 - 0))/(2*a)))" and
223 (* d2_abcformula9_neg not need, because 1<0 ==> False*)
225 "[|0<=1 - 0|] ==> ( bdv+ bdv \<up> 2=0) =
226 ((bdv=( -1 + sqrt(1 - 0))/(2*1))
227 | (bdv=( -1 - sqrt(1 - 0))/(2*1)))" and
228 (* d2_abcformula10_neg not need, because 1<0 ==> False*)
231 (* ---- degree 3 ----*)
233 "(a*bdv + b*bdv \<up> 2 + c*bdv \<up> 3=0) = (bdv=0 | (a + b*bdv + c*bdv \<up> 2=0))" and
235 "( bdv + b*bdv \<up> 2 + c*bdv \<up> 3=0) = (bdv=0 | (1 + b*bdv + c*bdv \<up> 2=0))" and
237 "(a*bdv + bdv \<up> 2 + c*bdv \<up> 3=0) = (bdv=0 | (a + bdv + c*bdv \<up> 2=0))" and
239 "( bdv + bdv \<up> 2 + c*bdv \<up> 3=0) = (bdv=0 | (1 + bdv + c*bdv \<up> 2=0))" and
241 "(a*bdv + b*bdv \<up> 2 + bdv \<up> 3=0) = (bdv=0 | (a + b*bdv + bdv \<up> 2=0))" and
243 "( bdv + b*bdv \<up> 2 + bdv \<up> 3=0) = (bdv=0 | (1 + b*bdv + bdv \<up> 2=0))" and
245 "(a*bdv + bdv \<up> 2 + bdv \<up> 3=0) = (bdv=0 | (1 + bdv + bdv \<up> 2=0))" and
247 "( bdv + bdv \<up> 2 + bdv \<up> 3=0) = (bdv=0 | (1 + bdv + bdv \<up> 2=0))" and
249 "(a*bdv + c*bdv \<up> 3=0) = (bdv=0 | (a + c*bdv \<up> 2=0))" and
250 d3_reduce_equation10:
251 "( bdv + c*bdv \<up> 3=0) = (bdv=0 | (1 + c*bdv \<up> 2=0))" and
252 d3_reduce_equation11:
253 "(a*bdv + bdv \<up> 3=0) = (bdv=0 | (a + bdv \<up> 2=0))" and
254 d3_reduce_equation12:
255 "( bdv + bdv \<up> 3=0) = (bdv=0 | (1 + bdv \<up> 2=0))" and
256 d3_reduce_equation13:
257 "( b*bdv \<up> 2 + c*bdv \<up> 3=0) = (bdv=0 | ( b*bdv + c*bdv \<up> 2=0))" and
258 d3_reduce_equation14:
259 "( bdv \<up> 2 + c*bdv \<up> 3=0) = (bdv=0 | ( bdv + c*bdv \<up> 2=0))" and
260 d3_reduce_equation15:
261 "( b*bdv \<up> 2 + bdv \<up> 3=0) = (bdv=0 | ( b*bdv + bdv \<up> 2=0))" and
262 d3_reduce_equation16:
263 "( bdv \<up> 2 + bdv \<up> 3=0) = (bdv=0 | ( bdv + bdv \<up> 2=0))" and
265 "[|Not(bdv occurs_in a)|] ==> (a + b*bdv \<up> 3=0) = (b*bdv \<up> 3= (-1)*a)" and
267 "[|Not(bdv occurs_in a)|] ==> (a + bdv \<up> 3=0) = ( bdv \<up> 3= (-1)*a)" and
269 "[|Not(b=0);Not(bdv occurs_in a)|] ==> (b*bdv \<up> 3=c) = (bdv \<up> 3=c/b)" and
271 "(bdv \<up> 3=0) = (bdv=0)" and
273 "(bdv \<up> 3=c) = (bdv = nroot 3 c)" and
275 (* ---- degree 4 ----*)
276 (* RL03.FIXME es wir nicht getestet ob u>0 *)
278 "(c+b*bdv \<up> 2+a*bdv \<up> 4=0) =
279 ((a*u \<up> 2+b*u+c=0) & (bdv \<up> 2=u))" and
281 (* ---- 7.3.02 von Termorder ---- *)
283 bdv_collect_1: "l * bdv + m * bdv = (l + m) * bdv" and
284 bdv_collect_2: "bdv + m * bdv = (1 + m) * bdv" and
285 bdv_collect_3: "l * bdv + bdv = (l + 1) * bdv" and
287 (* bdv_collect_assoc0_1 "l * bdv + m * bdv + k = (l + m) * bdv + k"
288 bdv_collect_assoc0_2 "bdv + m * bdv + k = (1 + m) * bdv + k"
289 bdv_collect_assoc0_3 "l * bdv + bdv + k = (l + 1) * bdv + k"
291 bdv_collect_assoc1_1: "l * bdv + (m * bdv + k) = (l + m) * bdv + k" and
292 bdv_collect_assoc1_2: "bdv + (m * bdv + k) = (1 + m) * bdv + k" and
293 bdv_collect_assoc1_3: "l * bdv + (bdv + k) = (l + 1) * bdv + k" and
295 bdv_collect_assoc2_1: "k + l * bdv + m * bdv = k + (l + m) * bdv" and
296 bdv_collect_assoc2_2: "k + bdv + m * bdv = k + (1 + m) * bdv" and
297 bdv_collect_assoc2_3: "k + l * bdv + bdv = k + (l + 1) * bdv" and
300 bdv_n_collect_1: "l * bdv \<up> n + m * bdv \<up> n = (l + m) * bdv \<up> n" and
301 bdv_n_collect_2: " bdv \<up> n + m * bdv \<up> n = (1 + m) * bdv \<up> n" and
302 bdv_n_collect_3: "l * bdv \<up> n + bdv \<up> n = (l + 1) * bdv \<up> n" (*order!*) and
304 bdv_n_collect_assoc1_1:
305 "l * bdv \<up> n + (m * bdv \<up> n + k) = (l + m) * bdv \<up> n + k" and
306 bdv_n_collect_assoc1_2: "bdv \<up> n + (m * bdv \<up> n + k) = (1 + m) * bdv \<up> n + k" and
307 bdv_n_collect_assoc1_3: "l * bdv \<up> n + (bdv \<up> n + k) = (l + 1) * bdv \<up> n + k" and
309 bdv_n_collect_assoc2_1: "k + l * bdv \<up> n + m * bdv \<up> n = k +(l + m) * bdv \<up> n" and
310 bdv_n_collect_assoc2_2: "k + bdv \<up> n + m * bdv \<up> n = k + (1 + m) * bdv \<up> n" and
311 bdv_n_collect_assoc2_3: "k + l * bdv \<up> n + bdv \<up> n = k + (l + 1) * bdv \<up> n" and
314 real_minus_div: "- (a / b) = (-1 * a) / b" and
316 separate_bdv: "(a * bdv) / b = (a / b) * (bdv::real)" and
317 separate_bdv_n: "(a * bdv \<up> n) / b = (a / b) * bdv \<up> n" and
318 separate_1_bdv: "bdv / b = (1 / b) * (bdv::real)" and
319 separate_1_bdv_n: "bdv \<up> n / b = (1 / b) * bdv \<up> n"
324 (*-------------------------rulse-------------------------*)
325 val PolyEq_prls = (*3.10.02:just the following order due to subterm evaluation*)
326 Rule_Set.append_rules "PolyEq_prls" Rule_Set.empty
327 [Rule.Eval ("Prog_Expr.ident", Prog_Expr.eval_ident "#ident_"),
328 Rule.Eval ("Prog_Expr.matches", Prog_Expr.eval_matches "#matches_"),
329 Rule.Eval ("Prog_Expr.lhs", Prog_Expr.eval_lhs ""),
330 Rule.Eval ("Prog_Expr.rhs", Prog_Expr.eval_rhs ""),
331 Rule.Eval ("Poly.is_expanded_in", eval_is_expanded_in ""),
332 Rule.Eval ("Poly.is_poly_in", eval_is_poly_in ""),
333 Rule.Eval ("Poly.has_degree_in", eval_has_degree_in ""),
334 Rule.Eval ("Poly.is_polyrat_in", eval_is_polyrat_in ""),
335 (*Rule.Eval ("Prog_Expr.occurs_in", Prog_Expr.eval_occurs_in ""), *)
336 (*Rule.Eval ("Prog_Expr.is_const", Prog_Expr.eval_const "#is_const_"),*)
337 Rule.Eval ("HOL.eq", Prog_Expr.eval_equal "#equal_"),
338 Rule.Eval ("RootEq.is_rootTerm_in", eval_is_rootTerm_in ""),
339 Rule.Eval ("RatEq.is_ratequation_in", eval_is_ratequation_in ""),
340 Rule.Thm ("not_true", @{thm not_true}),
341 Rule.Thm ("not_false", @{thm not_false}),
342 Rule.Thm ("and_true", @{thm and_true}),
343 Rule.Thm ("and_false", @{thm and_false}),
344 Rule.Thm ("or_true", @{thm or_true}),
345 Rule.Thm ("or_false", @{thm or_false})
349 Rule_Set.merge "PolyEq_erls" LinEq_erls
350 (Rule_Set.append_rules "ops_preds" calculate_Rational
351 [\<^rule_eval>\<open>HOL.eq\<close> (Prog_Expr.eval_equal "#equal_"),
352 \<^rule_thm>\<open>plus_leq\<close>,
353 \<^rule_thm>\<open>minus_leq\<close>,
354 \<^rule_thm>\<open>rat_leq1\<close>,
355 \<^rule_thm>\<open>rat_leq2\<close>,
356 \<^rule_thm>\<open>rat_leq3\<close>
360 Rule_Set.merge "PolyEq_crls" LinEq_crls
361 (Rule_Set.append_rules "ops_preds" calculate_Rational
362 [\<^rule_eval>\<open>HOL.eq\<close> (Prog_Expr.eval_equal "#equal_"),
363 \<^rule_thm>\<open>plus_leq\<close>,
364 \<^rule_thm>\<open>minus_leq\<close>,
365 \<^rule_thm>\<open>rat_leq1\<close>,
366 \<^rule_thm>\<open>rat_leq2\<close>,
367 \<^rule_thm>\<open>rat_leq3\<close>
370 val cancel_leading_coeff = prep_rls'(
371 Rule_Def.Repeat {id = "cancel_leading_coeff", preconds = [],
372 rew_ord = ("e_rew_ord",Rewrite_Ord.e_rew_ord),
373 erls = PolyEq_erls, srls = Rule_Set.Empty, calc = [], errpatts = [],
375 [\<^rule_thm>\<open>cancel_leading_coeff1\<close>,
376 \<^rule_thm>\<open>cancel_leading_coeff2\<close>,
377 \<^rule_thm>\<open>cancel_leading_coeff3\<close>,
378 \<^rule_thm>\<open>cancel_leading_coeff4\<close>,
379 \<^rule_thm>\<open>cancel_leading_coeff5\<close>,
380 \<^rule_thm>\<open>cancel_leading_coeff6\<close>,
381 \<^rule_thm>\<open>cancel_leading_coeff7\<close>,
382 \<^rule_thm>\<open>cancel_leading_coeff8\<close>,
383 \<^rule_thm>\<open>cancel_leading_coeff9\<close>,
384 \<^rule_thm>\<open>cancel_leading_coeff10\<close>,
385 \<^rule_thm>\<open>cancel_leading_coeff11\<close>,
386 \<^rule_thm>\<open>cancel_leading_coeff12\<close>,
387 \<^rule_thm>\<open>cancel_leading_coeff13\<close>
388 ],scr = Rule.Empty_Prog});
390 val prep_rls' = Auto_Prog.prep_rls @{theory};
393 val complete_square = prep_rls'(
394 Rule_Def.Repeat {id = "complete_square", preconds = [],
395 rew_ord = ("e_rew_ord",Rewrite_Ord.e_rew_ord),
396 erls = PolyEq_erls, srls = Rule_Set.Empty, calc = [], errpatts = [],
397 rules = [\<^rule_thm>\<open>complete_square1\<close>,
398 \<^rule_thm>\<open>complete_square2\<close>,
399 \<^rule_thm>\<open>complete_square3\<close>,
400 \<^rule_thm>\<open>complete_square4\<close>,
401 \<^rule_thm>\<open>complete_square5\<close>
403 scr = Rule.Empty_Prog
406 val polyeq_simplify = prep_rls'(
407 Rule_Def.Repeat {id = "polyeq_simplify", preconds = [],
408 rew_ord = ("termlessI",termlessI),
410 srls = Rule_Set.Empty,
411 calc = [], errpatts = [],
412 rules = [\<^rule_thm>\<open>real_assoc_1\<close>,
413 \<^rule_thm>\<open>real_assoc_2\<close>,
414 \<^rule_thm>\<open>real_diff_minus\<close>,
415 \<^rule_thm>\<open>real_unari_minus\<close>,
416 \<^rule_thm>\<open>realpow_multI\<close>,
417 \<^rule_eval>\<open>plus\<close> (**)(eval_binop "#add_"),
418 \<^rule_eval>\<open>minus\<close> (**)(eval_binop "#sub_"),
419 \<^rule_eval>\<open>times\<close> (**)(eval_binop "#mult_"),
420 \<^rule_eval>\<open>divide\<close> (Prog_Expr.eval_cancel "#divide_e"),
421 \<^rule_eval>\<open>sqrt\<close> (eval_sqrt "#sqrt_"),
422 \<^rule_eval>\<open>powr\<close> (**)(eval_binop "#power_"),
425 scr = Rule.Empty_Prog
429 cancel_leading_coeff = cancel_leading_coeff and
430 complete_square = complete_square and
431 PolyEq_erls = PolyEq_erls and
432 polyeq_simplify = polyeq_simplify
435 (* ------------- polySolve ------------------ *)
437 (*isolate the bound variable in an d0 equation; 'bdv' is a meta-constant*)
438 val d0_polyeq_simplify = prep_rls'(
439 Rule_Def.Repeat {id = "d0_polyeq_simplify", preconds = [],
440 rew_ord = ("e_rew_ord",Rewrite_Ord.e_rew_ord),
442 srls = Rule_Set.Empty,
443 calc = [], errpatts = [],
444 rules = [\<^rule_thm>\<open>d0_true\<close>, \<^rule_thm>\<open>d0_false\<close>],
445 scr = Rule.Empty_Prog
449 (*isolate the bound variable in an d1 equation; 'bdv' is a meta-constant*)
450 val d1_polyeq_simplify = prep_rls'(
451 Rule_Def.Repeat {id = "d1_polyeq_simplify", preconds = [],
452 rew_ord = ("e_rew_ord",Rewrite_Ord.e_rew_ord),
454 srls = Rule_Set.Empty,
455 calc = [], errpatts = [],
457 \<^rule_thm>\<open>d1_isolate_add1\<close>,
458 (* a+bx=0 -> bx=-a *)
459 \<^rule_thm>\<open>d1_isolate_add2\<close>,
461 \<^rule_thm>\<open>d1_isolate_div\<close>
464 scr = Rule.Empty_Prog
468 subsection \<open>degree 2\<close>
470 (* isolate the bound variable in an d2 equation with bdv only;
471 "bdv" is a meta-constant substituted for the "x" below by isac's rewriter. *)
472 val d2_polyeq_bdv_only_simplify = prep_rls'(
473 Rule_Def.Repeat {id = "d2_polyeq_bdv_only_simplify", preconds = [], rew_ord = ("e_rew_ord",Rewrite_Ord.e_rew_ord),
474 erls = PolyEq_erls, srls = Rule_Set.Empty, calc = [], errpatts = [],
476 [\<^rule_thm>\<open>d2_prescind1\<close>, (* ax+bx^2=0 -> x(a+bx)=0 *)
477 \<^rule_thm>\<open>d2_prescind2\<close>, (* ax+ x^2=0 -> x(a+ x)=0 *)
478 \<^rule_thm>\<open>d2_prescind3\<close>, (* x+bx^2=0 -> x(1+bx)=0 *)
479 \<^rule_thm>\<open>d2_prescind4\<close>, (* x+ x^2=0 -> x(1+ x)=0 *)
480 \<^rule_thm>\<open>d2_sqrt_equation1\<close>, (* x^2=c -> x=+-sqrt(c) *)
481 \<^rule_thm>\<open>d2_sqrt_equation1_neg\<close>, (* [0<c] x^2=c -> []*)
482 \<^rule_thm>\<open>d2_sqrt_equation2\<close>, (* x^2=0 -> x=0 *)
483 \<^rule_thm>\<open>d2_reduce_equation1\<close>,(* x(a+bx)=0 -> x=0 |a+bx=0*)
484 \<^rule_thm>\<open>d2_reduce_equation2\<close>,(* x(a+ x)=0 -> x=0 |a+ x=0*)
485 \<^rule_thm>\<open>d2_isolate_div\<close> (* bx^2=c -> x^2=c/b *)
487 scr = Rule.Empty_Prog
491 (* isolate the bound variable in an d2 equation with sqrt only;
492 'bdv' is a meta-constant*)
493 val d2_polyeq_sq_only_simplify = prep_rls'(
494 Rule_Def.Repeat {id = "d2_polyeq_sq_only_simplify", preconds = [],
495 rew_ord = ("e_rew_ord",Rewrite_Ord.e_rew_ord),
497 srls = Rule_Set.Empty,
498 calc = [], errpatts = [],
499 (*asm_thm = [("d2_sqrt_equation1", ""),("d2_sqrt_equation1_neg", ""),
500 ("d2_isolate_div", "")],*)
501 rules = [\<^rule_thm>\<open>d2_isolate_add1\<close>,
502 (* a+ bx^2=0 -> bx^2=(-1)a*)
503 \<^rule_thm>\<open>d2_isolate_add2\<close>,
504 (* a+ x^2=0 -> x^2=(-1)a*)
505 \<^rule_thm>\<open>d2_sqrt_equation2\<close>,
507 \<^rule_thm>\<open>d2_sqrt_equation1\<close>,
508 (* x^2=c -> x=+-sqrt(c)*)
509 \<^rule_thm>\<open>d2_sqrt_equation1_neg\<close>,
510 (* [c<0] x^2=c -> x=[] *)
511 \<^rule_thm>\<open>d2_isolate_div\<close>
512 (* bx^2=c -> x^2=c/b*)
514 scr = Rule.Empty_Prog
518 (* isolate the bound variable in an d2 equation with pqFormula;
519 'bdv' is a meta-constant*)
520 val d2_polyeq_pqFormula_simplify = prep_rls'(
521 Rule_Def.Repeat {id = "d2_polyeq_pqFormula_simplify", preconds = [],
522 rew_ord = ("e_rew_ord",Rewrite_Ord.e_rew_ord), erls = PolyEq_erls,
523 srls = Rule_Set.Empty, calc = [], errpatts = [],
524 rules = [\<^rule_thm>\<open>d2_pqformula1\<close>,
526 \<^rule_thm>\<open>d2_pqformula1_neg\<close>,
528 \<^rule_thm>\<open>d2_pqformula2\<close>,
530 \<^rule_thm>\<open>d2_pqformula2_neg\<close>,
532 \<^rule_thm>\<open>d2_pqformula3\<close>,
534 \<^rule_thm>\<open>d2_pqformula3_neg\<close>,
536 \<^rule_thm>\<open>d2_pqformula4\<close>,
538 \<^rule_thm>\<open>d2_pqformula4_neg\<close>,
540 \<^rule_thm>\<open>d2_pqformula5\<close>,
542 \<^rule_thm>\<open>d2_pqformula6\<close>,
544 \<^rule_thm>\<open>d2_pqformula7\<close>,
546 \<^rule_thm>\<open>d2_pqformula8\<close>,
548 \<^rule_thm>\<open>d2_pqformula9\<close>,
550 \<^rule_thm>\<open>d2_pqformula9_neg\<close>,
552 \<^rule_thm>\<open>d2_pqformula10\<close>,
554 \<^rule_thm>\<open>d2_pqformula10_neg\<close>,
556 \<^rule_thm>\<open>d2_sqrt_equation2\<close>,
558 \<^rule_thm>\<open>d2_sqrt_equation3\<close>
560 ],scr = Rule.Empty_Prog
564 (* isolate the bound variable in an d2 equation with abcFormula;
565 'bdv' is a meta-constant*)
566 val d2_polyeq_abcFormula_simplify = prep_rls'(
567 Rule_Def.Repeat {id = "d2_polyeq_abcFormula_simplify", preconds = [],
568 rew_ord = ("e_rew_ord",Rewrite_Ord.e_rew_ord), erls = PolyEq_erls,
569 srls = Rule_Set.Empty, calc = [], errpatts = [],
570 rules = [\<^rule_thm>\<open>d2_abcformula1\<close>,
572 \<^rule_thm>\<open>d2_abcformula1_neg\<close>,
574 \<^rule_thm>\<open>d2_abcformula2\<close>,
576 \<^rule_thm>\<open>d2_abcformula2_neg\<close>,
578 \<^rule_thm>\<open>d2_abcformula3\<close>,
580 \<^rule_thm>\<open>d2_abcformula3_neg\<close>,
582 \<^rule_thm>\<open>d2_abcformula4\<close>,
584 \<^rule_thm>\<open>d2_abcformula4_neg\<close>,
586 \<^rule_thm>\<open>d2_abcformula5\<close>,
588 \<^rule_thm>\<open>d2_abcformula5_neg\<close>,
590 \<^rule_thm>\<open>d2_abcformula6\<close>,
592 \<^rule_thm>\<open>d2_abcformula6_neg\<close>,
594 \<^rule_thm>\<open>d2_abcformula7\<close>,
596 \<^rule_thm>\<open>d2_abcformula8\<close>,
598 \<^rule_thm>\<open>d2_abcformula9\<close>,
600 \<^rule_thm>\<open>d2_abcformula10\<close>,
602 \<^rule_thm>\<open>d2_sqrt_equation2\<close>,
604 \<^rule_thm>\<open>d2_sqrt_equation3\<close>
607 scr = Rule.Empty_Prog
612 (* isolate the bound variable in an d2 equation;
613 'bdv' is a meta-constant*)
614 val d2_polyeq_simplify = prep_rls'(
615 Rule_Def.Repeat {id = "d2_polyeq_simplify", preconds = [],
616 rew_ord = ("e_rew_ord",Rewrite_Ord.e_rew_ord), erls = PolyEq_erls,
617 srls = Rule_Set.Empty, calc = [], errpatts = [],
618 rules = [\<^rule_thm>\<open>d2_pqformula1\<close>,
620 \<^rule_thm>\<open>d2_pqformula1_neg\<close>,
622 \<^rule_thm>\<open>d2_pqformula2\<close>,
624 \<^rule_thm>\<open>d2_pqformula2_neg\<close>,
626 \<^rule_thm>\<open>d2_pqformula3\<close>,
628 \<^rule_thm>\<open>d2_pqformula3_neg\<close>,
630 \<^rule_thm>\<open>d2_pqformula4\<close>,
632 \<^rule_thm>\<open>d2_pqformula4_neg\<close>,
634 \<^rule_thm>\<open>d2_abcformula1\<close>,
636 \<^rule_thm>\<open>d2_abcformula1_neg\<close>,
638 \<^rule_thm>\<open>d2_abcformula2\<close>,
640 \<^rule_thm>\<open>d2_abcformula2_neg\<close>,
642 \<^rule_thm>\<open>d2_prescind1\<close>,
643 (* ax+bx^2=0 -> x(a+bx)=0 *)
644 \<^rule_thm>\<open>d2_prescind2\<close>,
645 (* ax+ x^2=0 -> x(a+ x)=0 *)
646 \<^rule_thm>\<open>d2_prescind3\<close>,
647 (* x+bx^2=0 -> x(1+bx)=0 *)
648 \<^rule_thm>\<open>d2_prescind4\<close>,
649 (* x+ x^2=0 -> x(1+ x)=0 *)
650 \<^rule_thm>\<open>d2_isolate_add1\<close>,
651 (* a+ bx^2=0 -> bx^2=(-1)a*)
652 \<^rule_thm>\<open>d2_isolate_add2\<close>,
653 (* a+ x^2=0 -> x^2=(-1)a*)
654 \<^rule_thm>\<open>d2_sqrt_equation1\<close>,
655 (* x^2=c -> x=+-sqrt(c)*)
656 \<^rule_thm>\<open>d2_sqrt_equation1_neg\<close>,
657 (* [c<0] x^2=c -> x=[]*)
658 \<^rule_thm>\<open>d2_sqrt_equation2\<close>,
660 \<^rule_thm>\<open>d2_reduce_equation1\<close>,
661 (* x(a+bx)=0 -> x=0 | a+bx=0*)
662 \<^rule_thm>\<open>d2_reduce_equation2\<close>,
663 (* x(a+ x)=0 -> x=0 | a+ x=0*)
664 \<^rule_thm>\<open>d2_isolate_div\<close>
665 (* bx^2=c -> x^2=c/b*)
667 scr = Rule.Empty_Prog
673 (* isolate the bound variable in an d3 equation; 'bdv' is a meta-constant *)
674 val d3_polyeq_simplify = prep_rls'(
675 Rule_Def.Repeat {id = "d3_polyeq_simplify", preconds = [],
676 rew_ord = ("e_rew_ord",Rewrite_Ord.e_rew_ord), erls = PolyEq_erls,
677 srls = Rule_Set.Empty, calc = [], errpatts = [],
679 [\<^rule_thm>\<open>d3_reduce_equation1\<close>,
680 (*a*bdv + b*bdv \<up> 2 + c*bdv \<up> 3=0) =
681 (bdv=0 | (a + b*bdv + c*bdv \<up> 2=0)*)
682 \<^rule_thm>\<open>d3_reduce_equation2\<close>,
683 (* bdv + b*bdv \<up> 2 + c*bdv \<up> 3=0) =
684 (bdv=0 | (1 + b*bdv + c*bdv \<up> 2=0)*)
685 \<^rule_thm>\<open>d3_reduce_equation3\<close>,
686 (*a*bdv + bdv \<up> 2 + c*bdv \<up> 3=0) =
687 (bdv=0 | (a + bdv + c*bdv \<up> 2=0)*)
688 \<^rule_thm>\<open>d3_reduce_equation4\<close>,
689 (* bdv + bdv \<up> 2 + c*bdv \<up> 3=0) =
690 (bdv=0 | (1 + bdv + c*bdv \<up> 2=0)*)
691 \<^rule_thm>\<open>d3_reduce_equation5\<close>,
692 (*a*bdv + b*bdv \<up> 2 + bdv \<up> 3=0) =
693 (bdv=0 | (a + b*bdv + bdv \<up> 2=0)*)
694 \<^rule_thm>\<open>d3_reduce_equation6\<close>,
695 (* bdv + b*bdv \<up> 2 + bdv \<up> 3=0) =
696 (bdv=0 | (1 + b*bdv + bdv \<up> 2=0)*)
697 \<^rule_thm>\<open>d3_reduce_equation7\<close>,
698 (*a*bdv + bdv \<up> 2 + bdv \<up> 3=0) =
699 (bdv=0 | (1 + bdv + bdv \<up> 2=0)*)
700 \<^rule_thm>\<open>d3_reduce_equation8\<close>,
701 (* bdv + bdv \<up> 2 + bdv \<up> 3=0) =
702 (bdv=0 | (1 + bdv + bdv \<up> 2=0)*)
703 \<^rule_thm>\<open>d3_reduce_equation9\<close>,
704 (*a*bdv + c*bdv \<up> 3=0) =
705 (bdv=0 | (a + c*bdv \<up> 2=0)*)
706 \<^rule_thm>\<open>d3_reduce_equation10\<close>,
707 (* bdv + c*bdv \<up> 3=0) =
708 (bdv=0 | (1 + c*bdv \<up> 2=0)*)
709 \<^rule_thm>\<open>d3_reduce_equation11\<close>,
710 (*a*bdv + bdv \<up> 3=0) =
711 (bdv=0 | (a + bdv \<up> 2=0)*)
712 \<^rule_thm>\<open>d3_reduce_equation12\<close>,
713 (* bdv + bdv \<up> 3=0) =
714 (bdv=0 | (1 + bdv \<up> 2=0)*)
715 \<^rule_thm>\<open>d3_reduce_equation13\<close>,
716 (* b*bdv \<up> 2 + c*bdv \<up> 3=0) =
717 (bdv=0 | ( b*bdv + c*bdv \<up> 2=0)*)
718 \<^rule_thm>\<open>d3_reduce_equation14\<close>,
719 (* bdv \<up> 2 + c*bdv \<up> 3=0) =
720 (bdv=0 | ( bdv + c*bdv \<up> 2=0)*)
721 \<^rule_thm>\<open>d3_reduce_equation15\<close>,
722 (* b*bdv \<up> 2 + bdv \<up> 3=0) =
723 (bdv=0 | ( b*bdv + bdv \<up> 2=0)*)
724 \<^rule_thm>\<open>d3_reduce_equation16\<close>,
725 (* bdv \<up> 2 + bdv \<up> 3=0) =
726 (bdv=0 | ( bdv + bdv \<up> 2=0)*)
727 \<^rule_thm>\<open>d3_isolate_add1\<close>,
728 (*[|Not(bdv occurs_in a)|] ==> (a + b*bdv \<up> 3=0) =
729 (bdv=0 | (b*bdv \<up> 3=a)*)
730 \<^rule_thm>\<open>d3_isolate_add2\<close>,
731 (*[|Not(bdv occurs_in a)|] ==> (a + bdv \<up> 3=0) =
732 (bdv=0 | ( bdv \<up> 3=a)*)
733 \<^rule_thm>\<open>d3_isolate_div\<close>,
734 (*[|Not(b=0)|] ==> (b*bdv \<up> 3=c) = (bdv \<up> 3=c/b*)
735 \<^rule_thm>\<open>d3_root_equation2\<close>,
736 (*(bdv \<up> 3=0) = (bdv=0) *)
737 \<^rule_thm>\<open>d3_root_equation1\<close>
738 (*bdv \<up> 3=c) = (bdv = nroot 3 c*)
740 scr = Rule.Empty_Prog
746 (*isolate the bound variable in an d4 equation; 'bdv' is a meta-constant*)
747 val d4_polyeq_simplify = prep_rls'(
748 Rule_Def.Repeat {id = "d4_polyeq_simplify", preconds = [],
749 rew_ord = ("e_rew_ord",Rewrite_Ord.e_rew_ord), erls = PolyEq_erls,
750 srls = Rule_Set.Empty, calc = [], errpatts = [],
752 [\<^rule_thm>\<open>d4_sub_u1\<close>
753 (* ax^4+bx^2+c=0 -> x=+-sqrt(ax^2+bx^+c) *)
755 scr = Rule.Empty_Prog
759 d0_polyeq_simplify = d0_polyeq_simplify and
760 d1_polyeq_simplify = d1_polyeq_simplify and
761 d2_polyeq_simplify = d2_polyeq_simplify and
762 d2_polyeq_bdv_only_simplify = d2_polyeq_bdv_only_simplify and
763 d2_polyeq_sq_only_simplify = d2_polyeq_sq_only_simplify and
765 d2_polyeq_pqFormula_simplify = d2_polyeq_pqFormula_simplify and
766 d2_polyeq_abcFormula_simplify = d2_polyeq_abcFormula_simplify and
767 d3_polyeq_simplify = d3_polyeq_simplify and
768 d4_polyeq_simplify = d4_polyeq_simplify
770 problem pbl_equ_univ_poly : "polynomial/univariate/equation" =
771 \<open>PolyEq_prls\<close>
772 CAS: "solve (e_e::bool, v_v)"
773 Given: "equality e_e" "solveFor v_v"
775 "~((e_e::bool) is_ratequation_in (v_v::real))"
776 "~((lhs e_e) is_rootTerm_in (v_v::real))"
777 "~((rhs e_e) is_rootTerm_in (v_v::real))"
778 Find: "solutions v_v'i'"
781 problem pbl_equ_univ_poly_deg0 : "degree_0/polynomial/univariate/equation" =
782 \<open>PolyEq_prls\<close>
783 Method: "PolyEq/solve_d0_polyeq_equation"
784 CAS: "solve (e_e::bool, v_v)"
785 Given: "equality e_e" "solveFor v_v"
787 "matches (?a = 0) e_e"
788 "(lhs e_e) is_poly_in v_v"
789 "((lhs e_e) has_degree_in v_v ) = 0"
790 Find: "solutions v_v'i'"
793 problem pbl_equ_univ_poly_deg1 : "degree_1/polynomial/univariate/equation" =
794 \<open>PolyEq_prls\<close>
795 Method: "PolyEq/solve_d1_polyeq_equation"
796 CAS: "solve (e_e::bool, v_v)"
797 Given: "equality e_e" "solveFor v_v"
799 "matches (?a = 0) e_e"
800 "(lhs e_e) is_poly_in v_v"
801 "((lhs e_e) has_degree_in v_v ) = 1"
802 Find: "solutions v_v'i'"
805 problem pbl_equ_univ_poly_deg2 : "degree_2/polynomial/univariate/equation" =
806 \<open>PolyEq_prls\<close>
807 Method: "PolyEq/solve_d2_polyeq_equation"
808 CAS: "solve (e_e::bool, v_v)"
809 Given: "equality e_e" "solveFor v_v"
811 "matches (?a = 0) e_e"
812 "(lhs e_e) is_poly_in v_v "
813 "((lhs e_e) has_degree_in v_v ) = 2"
814 Find: "solutions v_v'i'"
816 problem pbl_equ_univ_poly_deg2_sqonly : "sq_only/degree_2/polynomial/univariate/equation" =
817 \<open>PolyEq_prls\<close>
818 Method: "PolyEq/solve_d2_polyeq_sqonly_equation"
819 CAS: "solve (e_e::bool, v_v)"
820 Given: "equality e_e" "solveFor v_v"
822 "matches ( ?a + ?v_ \<up> 2 = 0) e_e |
823 matches ( ?a + ?b*?v_ \<up> 2 = 0) e_e |
824 matches ( ?v_ \<up> 2 = 0) e_e |
825 matches ( ?b*?v_ \<up> 2 = 0) e_e"
826 "Not (matches (?a + ?v_ + ?v_ \<up> 2 = 0) e_e) &
827 Not (matches (?a + ?b*?v_ + ?v_ \<up> 2 = 0) e_e) &
828 Not (matches (?a + ?v_ + ?c*?v_ \<up> 2 = 0) e_e) &
829 Not (matches (?a + ?b*?v_ + ?c*?v_ \<up> 2 = 0) e_e) &
830 Not (matches ( ?v_ + ?v_ \<up> 2 = 0) e_e) &
831 Not (matches ( ?b*?v_ + ?v_ \<up> 2 = 0) e_e) &
832 Not (matches ( ?v_ + ?c*?v_ \<up> 2 = 0) e_e) &
833 Not (matches ( ?b*?v_ + ?c*?v_ \<up> 2 = 0) e_e)"
834 Find: "solutions v_v'i'"
836 problem pbl_equ_univ_poly_deg2_bdvonly : "bdv_only/degree_2/polynomial/univariate/equation" =
837 \<open>PolyEq_prls\<close>
838 Method: "PolyEq/solve_d2_polyeq_bdvonly_equation"
839 CAS: "solve (e_e::bool, v_v)"
840 Given: "equality e_e" "solveFor v_v"
842 "matches (?a*?v_ + ?v_ \<up> 2 = 0) e_e |
843 matches ( ?v_ + ?v_ \<up> 2 = 0) e_e |
844 matches ( ?v_ + ?b*?v_ \<up> 2 = 0) e_e |
845 matches (?a*?v_ + ?b*?v_ \<up> 2 = 0) e_e |
846 matches ( ?v_ \<up> 2 = 0) e_e |
847 matches ( ?b*?v_ \<up> 2 = 0) e_e "
848 Find: "solutions v_v'i'"
850 problem pbl_equ_univ_poly_deg2_pq : "pqFormula/degree_2/polynomial/univariate/equation" =
851 \<open>PolyEq_prls\<close>
852 Method: "PolyEq/solve_d2_polyeq_pq_equation"
853 CAS: "solve (e_e::bool, v_v)"
854 Given: "equality e_e" "solveFor v_v"
856 "matches (?a + 1*?v_ \<up> 2 = 0) e_e |
857 matches (?a + ?v_ \<up> 2 = 0) e_e"
858 Find: "solutions v_v'i'"
860 problem pbl_equ_univ_poly_deg2_abc : "abcFormula/degree_2/polynomial/univariate/equation" =
861 \<open>PolyEq_prls\<close>
862 Method: "PolyEq/solve_d2_polyeq_abc_equation"
863 CAS: "solve (e_e::bool, v_v)"
864 Given: "equality e_e" "solveFor v_v"
866 "matches (?a + ?v_ \<up> 2 = 0) e_e |
867 matches (?a + ?b*?v_ \<up> 2 = 0) e_e"
868 Find: "solutions v_v'i'"
871 problem pbl_equ_univ_poly_deg3 : "degree_3/polynomial/univariate/equation" =
872 \<open>PolyEq_prls\<close>
873 Method: "PolyEq/solve_d3_polyeq_equation"
874 CAS: "solve (e_e::bool, v_v)"
875 Given: "equality e_e" "solveFor v_v"
877 "matches (?a = 0) e_e"
878 "(lhs e_e) is_poly_in v_v"
879 "((lhs e_e) has_degree_in v_v) = 3"
880 Find: "solutions v_v'i'"
883 problem pbl_equ_univ_poly_deg4 : "degree_4/polynomial/univariate/equation" =
884 \<open>PolyEq_prls\<close>
885 (*Method: "PolyEq/solve_d4_polyeq_equation"*)
886 CAS: "solve (e_e::bool, v_v)"
887 Given: "equality e_e" "solveFor v_v"
889 "matches (?a = 0) e_e"
890 "(lhs e_e) is_poly_in v_v"
891 "((lhs e_e) has_degree_in v_v) = 4"
892 Find: "solutions v_v'i'"
894 (*--- normalise ---*)
895 problem pbl_equ_univ_poly_norm : "normalise/polynomial/univariate/equation" =
896 \<open>PolyEq_prls\<close>
897 Method: "PolyEq/normalise_poly"
898 CAS: "solve (e_e::bool, v_v)"
899 Given: "equality e_e" "solveFor v_v"
901 "(Not((matches (?a = 0 ) e_e ))) |
902 (Not(((lhs e_e) is_poly_in v_v)))"
903 Find: "solutions v_v'i'"
905 (*-------------------------expanded-----------------------*)
906 problem "pbl_equ_univ_expand" : "expanded/univariate/equation" =
907 \<open>PolyEq_prls\<close>
908 CAS: "solve (e_e::bool, v_v)"
909 Given: "equality e_e" "solveFor v_v"
911 "matches (?a = 0) e_e"
912 "(lhs e_e) is_expanded_in v_v "
913 Find: "solutions v_v'i'"
916 problem pbl_equ_univ_expand_deg2 : "degree_2/expanded/univariate/equation" =
917 \<open>PolyEq_prls\<close>
918 Method: "PolyEq/complete_square"
919 CAS: "solve (e_e::bool, v_v)"
920 Given: "equality e_e" "solveFor v_v"
921 Where: "((lhs e_e) has_degree_in v_v) = 2"
922 Find: "solutions v_v'i'"
924 text \<open>"-------------------------methods-----------------------"\<close>
926 method met_polyeq : "PolyEq" =
927 \<open>{rew_ord'="tless_true",rls'=Atools_erls,calc = [], srls = Rule_Set.empty, prls=Rule_Set.empty,
928 crls=PolyEq_crls, errpats = [], nrls = norm_Rational}\<close>
930 partial_function (tailrec) normalize_poly_eq :: "bool \<Rightarrow> real \<Rightarrow> bool"
932 "normalize_poly_eq e_e v_v = (
935 (Try (Rewrite ''all_left'')) #>
936 (Try (Repeat (Rewrite ''makex1_x''))) #>
937 (Try (Repeat (Rewrite_Set ''expand_binoms''))) #>
938 (Try (Repeat (Rewrite_Set_Inst [(''bdv'', v_v)] ''make_ratpoly_in''))) #>
939 (Try (Repeat (Rewrite_Set ''polyeq_simplify''))) ) e_e
941 SubProblem (''PolyEq'', [''polynomial'', ''univariate'', ''equation''], [''no_met''])
942 [BOOL e_e, REAL v_v])"
944 method met_polyeq_norm : "PolyEq/normalise_poly" =
945 \<open>{rew_ord'="termlessI", rls'=PolyEq_erls, srls=Rule_Set.empty, prls=PolyEq_prls, calc=[],
946 crls=PolyEq_crls, errpats = [], nrls = norm_Rational}\<close>
947 Program: normalize_poly_eq.simps
948 Given: "equality e_e" "solveFor v_v"
949 Where: "(Not((matches (?a = 0 ) e_e ))) | (Not(((lhs e_e) is_poly_in v_v)))"
950 Find: "solutions v_v'i'"
952 partial_function (tailrec) solve_poly_equ :: "bool \<Rightarrow> real \<Rightarrow> bool list"
954 "solve_poly_equ e_e v_v = (
956 e_e = (Try (Rewrite_Set_Inst [(''bdv'', v_v)] ''d0_polyeq_simplify'')) e_e
960 method met_polyeq_d0 : "PolyEq/solve_d0_polyeq_equation" =
961 \<open>{rew_ord'="termlessI", rls'=PolyEq_erls, srls=Rule_Set.empty, prls=PolyEq_prls,
962 calc=[("sqrt", (\<^const_name>\<open>sqrt\<close>, eval_sqrt "#sqrt_"))], crls=PolyEq_crls, errpats = [],
963 nrls = norm_Rational}\<close>
964 Program: solve_poly_equ.simps
965 Given: "equality e_e" "solveFor v_v"
966 Where: "(lhs e_e) is_poly_in v_v " "((lhs e_e) has_degree_in v_v) = 0"
967 Find: "solutions v_v'i'"
969 partial_function (tailrec) solve_poly_eq1 :: "bool \<Rightarrow> real \<Rightarrow> bool list"
971 "solve_poly_eq1 e_e v_v = (
974 (Try (Rewrite_Set_Inst [(''bdv'', v_v)] ''d1_polyeq_simplify'')) #>
975 (Try (Rewrite_Set ''polyeq_simplify'')) #>
976 (Try (Rewrite_Set ''norm_Rational_parenthesized'')) ) e_e;
979 Check_elementwise L_L {(v_v::real). Assumptions})"
981 method met_polyeq_d1 : "PolyEq/solve_d1_polyeq_equation" =
982 \<open>{rew_ord'="termlessI", rls'=PolyEq_erls, srls=Rule_Set.empty, prls=PolyEq_prls,
983 calc=[("sqrt", (\<^const_name>\<open>sqrt\<close>, eval_sqrt "#sqrt_"))], crls=PolyEq_crls, errpats = [],
984 nrls = norm_Rational}\<close>
985 Program: solve_poly_eq1.simps
986 Given: "equality e_e" "solveFor v_v"
987 Where: "(lhs e_e) is_poly_in v_v" "((lhs e_e) has_degree_in v_v) = 1"
988 Find: "solutions v_v'i'"
990 partial_function (tailrec) solve_poly_equ2 :: "bool \<Rightarrow> real \<Rightarrow> bool list"
992 "solve_poly_equ2 e_e v_v = (
995 (Try (Rewrite_Set_Inst [(''bdv'', v_v)] ''d2_polyeq_simplify'')) #>
996 (Try (Rewrite_Set ''polyeq_simplify'')) #>
997 (Try (Rewrite_Set_Inst [(''bdv'', v_v)] ''d1_polyeq_simplify'')) #>
998 (Try (Rewrite_Set ''polyeq_simplify'')) #>
999 (Try (Rewrite_Set ''norm_Rational_parenthesized'')) ) e_e;
1000 L_L = Or_to_List e_e
1002 Check_elementwise L_L {(v_v::real). Assumptions})"
1004 method met_polyeq_d22 : "PolyEq/solve_d2_polyeq_equation" =
1005 \<open>{rew_ord'="termlessI", rls'=PolyEq_erls, srls=Rule_Set.empty, prls=PolyEq_prls,
1006 calc=[("sqrt", (\<^const_name>\<open>sqrt\<close>, eval_sqrt "#sqrt_"))], crls=PolyEq_crls, errpats = [],
1007 nrls = norm_Rational}\<close>
1008 Program: solve_poly_equ2.simps
1009 Given: "equality e_e" "solveFor v_v"
1010 Where: "(lhs e_e) is_poly_in v_v " "((lhs e_e) has_degree_in v_v) = 2"
1011 Find: "solutions v_v'i'"
1013 partial_function (tailrec) solve_poly_equ0 :: "bool \<Rightarrow> real \<Rightarrow> bool list"
1015 "solve_poly_equ0 e_e v_v = (
1018 (Try (Rewrite_Set_Inst [(''bdv'', v_v)] ''d2_polyeq_bdv_only_simplify'')) #>
1019 (Try (Rewrite_Set ''polyeq_simplify'')) #>
1020 (Try (Rewrite_Set_Inst [(''bdv'',v_v::real)] ''d1_polyeq_simplify'')) #>
1021 (Try (Rewrite_Set ''polyeq_simplify'')) #>
1022 (Try (Rewrite_Set ''norm_Rational_parenthesized'')) ) e_e;
1023 L_L = Or_to_List e_e
1025 Check_elementwise L_L {(v_v::real). Assumptions})"
1027 method met_polyeq_d2_bdvonly : "PolyEq/solve_d2_polyeq_bdvonly_equation" =
1028 \<open>{rew_ord'="termlessI", rls'=PolyEq_erls, srls=Rule_Set.empty, prls=PolyEq_prls,
1029 calc=[("sqrt", (\<^const_name>\<open>sqrt\<close>, eval_sqrt "#sqrt_"))], crls=PolyEq_crls, errpats = [],
1030 nrls = norm_Rational}\<close>
1031 Program: solve_poly_equ0.simps
1032 Given: "equality e_e" "solveFor v_v"
1033 Where: "(lhs e_e) is_poly_in v_v " "((lhs e_e) has_degree_in v_v) = 2"
1034 Find: "solutions v_v'i'"
1036 partial_function (tailrec) solve_poly_equ_sqrt :: "bool \<Rightarrow> real \<Rightarrow> bool list"
1038 "solve_poly_equ_sqrt e_e v_v = (
1041 (Try (Rewrite_Set_Inst [(''bdv'', v_v)] ''d2_polyeq_sq_only_simplify'')) #>
1042 (Try (Rewrite_Set ''polyeq_simplify'')) #>
1043 (Try (Rewrite_Set ''norm_Rational_parenthesized'')) ) e_e;
1044 L_L = Or_to_List e_e
1046 Check_elementwise L_L {(v_v::real). Assumptions})"
1048 method met_polyeq_d2_sqonly : "PolyEq/solve_d2_polyeq_sqonly_equation" =
1049 \<open>{rew_ord'="termlessI", rls'=PolyEq_erls, srls=Rule_Set.empty, prls=PolyEq_prls,
1050 calc=[("sqrt", (\<^const_name>\<open>sqrt\<close>, eval_sqrt "#sqrt_"))], crls=PolyEq_crls, errpats = [],
1051 nrls = norm_Rational}\<close>
1052 Program: solve_poly_equ_sqrt.simps
1053 Given: "equality e_e" "solveFor v_v"
1054 Where: "(lhs e_e) is_poly_in v_v " "((lhs e_e) has_degree_in v_v) = 2"
1055 Find: "solutions v_v'i'"
1057 partial_function (tailrec) solve_poly_equ_pq :: "bool \<Rightarrow> real \<Rightarrow> bool list"
1059 "solve_poly_equ_pq e_e v_v = (
1062 (Try (Rewrite_Set_Inst [(''bdv'', v_v)] ''d2_polyeq_pqFormula_simplify'')) #>
1063 (Try (Rewrite_Set ''polyeq_simplify'')) #>
1064 (Try (Rewrite_Set ''norm_Rational_parenthesized'')) ) e_e;
1065 L_L = Or_to_List e_e
1067 Check_elementwise L_L {(v_v::real). Assumptions})"
1069 method met_polyeq_d2_pq : "PolyEq/solve_d2_polyeq_pq_equation" =
1070 \<open>{rew_ord'="termlessI", rls'=PolyEq_erls, srls=Rule_Set.empty, prls=PolyEq_prls,
1071 calc=[("sqrt", (\<^const_name>\<open>sqrt\<close>, eval_sqrt "#sqrt_"))], crls=PolyEq_crls, errpats = [],
1072 nrls = norm_Rational}\<close>
1073 Program: solve_poly_equ_pq.simps
1074 Given: "equality e_e" "solveFor v_v"
1075 Where: "(lhs e_e) is_poly_in v_v " "((lhs e_e) has_degree_in v_v) = 2"
1076 Find: "solutions v_v'i'"
1078 partial_function (tailrec) solve_poly_equ_abc :: "bool \<Rightarrow> real \<Rightarrow> bool list"
1080 "solve_poly_equ_abc e_e v_v = (
1083 (Try (Rewrite_Set_Inst [(''bdv'', v_v)] ''d2_polyeq_abcFormula_simplify'')) #>
1084 (Try (Rewrite_Set ''polyeq_simplify'')) #>
1085 (Try (Rewrite_Set ''norm_Rational_parenthesized'')) ) e_e;
1086 L_L = Or_to_List e_e
1087 in Check_elementwise L_L {(v_v::real). Assumptions})"
1089 method met_polyeq_d2_abc : "PolyEq/solve_d2_polyeq_abc_equation" =
1090 \<open>{rew_ord'="termlessI", rls'=PolyEq_erls,srls=Rule_Set.empty, prls=PolyEq_prls,
1091 calc=[("sqrt", (\<^const_name>\<open>sqrt\<close>, eval_sqrt "#sqrt_"))], crls=PolyEq_crls, errpats = [],
1092 nrls = norm_Rational}\<close>
1093 Program: solve_poly_equ_abc.simps
1094 Given: "equality e_e" "solveFor v_v"
1095 Where: "(lhs e_e) is_poly_in v_v " "((lhs e_e) has_degree_in v_v) = 2"
1096 Find: "solutions v_v'i'"
1098 partial_function (tailrec) solve_poly_equ3 :: "bool \<Rightarrow> real \<Rightarrow> bool list"
1100 "solve_poly_equ3 e_e v_v = (
1103 (Try (Rewrite_Set_Inst [(''bdv'', v_v)] ''d3_polyeq_simplify'')) #>
1104 (Try (Rewrite_Set ''polyeq_simplify'')) #>
1105 (Try (Rewrite_Set_Inst [(''bdv'', v_v)] ''d2_polyeq_simplify'')) #>
1106 (Try (Rewrite_Set ''polyeq_simplify'')) #>
1107 (Try (Rewrite_Set_Inst [(''bdv'',v_v)] ''d1_polyeq_simplify'')) #>
1108 (Try (Rewrite_Set ''polyeq_simplify'')) #>
1109 (Try (Rewrite_Set ''norm_Rational_parenthesized''))) e_e;
1110 L_L = Or_to_List e_e
1112 Check_elementwise L_L {(v_v::real). Assumptions})"
1114 method met_polyeq_d3 : "PolyEq/solve_d3_polyeq_equation" =
1115 \<open>{rew_ord'="termlessI", rls'=PolyEq_erls, srls=Rule_Set.empty, prls=PolyEq_prls,
1116 calc=[("sqrt", (\<^const_name>\<open>sqrt\<close>, eval_sqrt "#sqrt_"))], crls=PolyEq_crls, errpats = [],
1117 nrls = norm_Rational}\<close>
1118 Program: solve_poly_equ3.simps
1119 Given: "equality e_e" "solveFor v_v"
1120 Where: "(lhs e_e) is_poly_in v_v " "((lhs e_e) has_degree_in v_v) = 3"
1121 Find: "solutions v_v'i'"
1123 (*.solves all expanded (ie. normalised) terms of degree 2.*)
1124 (*Oct.02 restriction: 'eval_true 0 =< discriminant' ony for integer values
1125 by 'PolyEq_erls'; restricted until Float.thy is implemented*)
1126 partial_function (tailrec) solve_by_completing_square :: "bool \<Rightarrow> real \<Rightarrow> bool list"
1128 "solve_by_completing_square e_e v_v = (
1130 (Try (Rewrite_Set_Inst [(''bdv'', v_v)] ''cancel_leading_coeff'')) #>
1131 (Try (Rewrite_Set_Inst [(''bdv'', v_v)] ''complete_square'')) #>
1132 (Try (Rewrite ''square_explicit1'')) #>
1133 (Try (Rewrite ''square_explicit2'')) #>
1134 (Rewrite ''root_plus_minus'') #>
1135 (Try (Repeat (Rewrite_Inst [(''bdv'', v_v)] ''bdv_explicit1''))) #>
1136 (Try (Repeat (Rewrite_Inst [(''bdv'', v_v)] ''bdv_explicit2''))) #>
1137 (Try (Repeat (Rewrite_Inst [(''bdv'', v_v)] ''bdv_explicit3''))) #>
1138 (Try (Rewrite_Set ''calculate_RootRat'')) #>
1139 (Try (Repeat (Calculate ''SQRT'')))) e_e
1143 method met_polyeq_complsq : "PolyEq/complete_square" =
1144 \<open>{rew_ord'="termlessI",rls'=PolyEq_erls,srls=Rule_Set.empty,prls=PolyEq_prls,
1145 calc=[("sqrt", (\<^const_name>\<open>sqrt\<close>, eval_sqrt "#sqrt_"))], crls=PolyEq_crls, errpats = [],
1146 nrls = norm_Rational}\<close>
1147 Program: solve_by_completing_square.simps
1148 Given: "equality e_e" "solveFor v_v"
1149 Where: "matches (?a = 0) e_e" "((lhs e_e) has_degree_in v_v) = 2"
1150 Find: "solutions v_v'i'"
1154 (* termorder hacked by MG *)
1155 local (*. for make_polynomial_in .*)
1157 open Term; (* for type order = EQUAL | LESS | GREATER *)
1159 fun pr_ord EQUAL = "EQUAL"
1160 | pr_ord LESS = "LESS"
1161 | pr_ord GREATER = "GREATER";
1163 fun dest_hd' _ (Const (a, T)) = (((a, 0), T), 0)
1164 | dest_hd' x (t as Free (a, T)) =
1165 if x = t then ((("|||||||||||||", 0), T), 0) (*WN*)
1166 else (((a, 0), T), 1)
1167 | dest_hd' _ (Var v) = (v, 2)
1168 | dest_hd' _ (Bound i) = ((("", i), dummyT), 3)
1169 | dest_hd' _ (Abs (_, T, _)) = ((("", 0), T), 4)
1170 | dest_hd' _ _ = raise ERROR "dest_hd': uncovered case in fun.def.";
1172 fun size_of_term' x (Const (\<^const_name>\<open>Transcendental.powr\<close>,_) $
1173 Free (var, _) $ Const (\<^const_name>\<open>numeral\<close>, _) $ pot) =
1176 (if xstr = var then 1000 * (HOLogic.dest_numeral pot) else 3)
1177 | _ => raise ERROR ("size_of_term' called with subst = " ^ UnparseC.term x))
1178 | size_of_term' x (Free (subst, _)) =
1180 (Free (xstr, _)) => (if xstr = subst then 1000 else 1)
1181 | _ => raise ERROR ("size_of_term' called with subst = " ^ UnparseC.term x))
1182 | size_of_term' x (Abs (_, _, body)) = 1 + size_of_term' x body
1183 | size_of_term' x (f$t) = size_of_term' x f + size_of_term' x t
1184 | size_of_term' _ _ = 1;
1186 fun term_ord' x pr thy (Abs (_, T, t), Abs(_, U, u)) = (* ~ term.ML *)
1187 (case term_ord' x pr thy (t, u) of EQUAL => Term_Ord.typ_ord (T, U) | ord => ord)
1188 | term_ord' x pr thy (t, u) =
1192 val (f, ts) = strip_comb t and (g, us) = strip_comb u;
1193 val _ = tracing ("t= f@ts= \"" ^ UnparseC.term_in_thy thy f ^ "\" @ \"[" ^
1194 commas (map (UnparseC.term_in_thy thy) ts) ^ "]\"");
1195 val _ = tracing ("u= g@us= \"" ^ UnparseC.term_in_thy thy g ^ "\" @ \"[" ^
1196 commas(map (UnparseC.term_in_thy thy) us) ^ "]\"");
1197 val _ = tracing ("size_of_term(t,u)= (" ^ string_of_int (size_of_term' x t) ^ ", " ^
1198 string_of_int (size_of_term' x u) ^ ")");
1199 val _ = tracing ("hd_ord(f,g) = " ^ (pr_ord o (hd_ord x)) (f,g));
1200 val _ = tracing ("terms_ord(ts,us) = " ^ (pr_ord o (terms_ord x) str false) (ts, us));
1201 val _ = tracing ("-------");
1204 case int_ord (size_of_term' x t, size_of_term' x u) of
1206 let val (f, ts) = strip_comb t and (g, us) = strip_comb u
1208 (case hd_ord x (f, g) of
1209 EQUAL => (terms_ord x str pr) (ts, us)
1213 and hd_ord x (f, g) = (* ~ term.ML *)
1214 prod_ord (prod_ord Term_Ord.indexname_ord Term_Ord.typ_ord)
1215 int_ord (dest_hd' x f, dest_hd' x g)
1216 and terms_ord x _ pr (ts, us) =
1217 list_ord (term_ord' x pr (ThyC.get_theory "Isac_Knowledge"))(ts, us);
1221 fun ord_make_polynomial_in (pr:bool) thy subst (ts, us) =
1222 ((** )tracing ("*** subs variable is: " ^ (Env.subst2str subst)); ( **)
1224 (_, x) :: _ => (term_ord' x pr thy (TermC.numerals_to_Free ts, TermC.numerals_to_Free us) = LESS)
1225 | _ => raise ERROR ("ord_make_polynomial_in called with subst = " ^ Env.subst2str subst))
1231 val order_add_mult_in = prep_rls'(
1232 Rule_Def.Repeat{id = "order_add_mult_in", preconds = [],
1233 rew_ord = ("ord_make_polynomial_in", ord_make_polynomial_in false @{theory "Poly"}),
1234 erls = Rule_Set.empty,srls = Rule_Set.Empty,
1235 calc = [], errpatts = [],
1236 rules = [\<^rule_thm>\<open>mult.commute\<close>,
1238 \<^rule_thm>\<open>real_mult_left_commute\<close>,
1239 (*z1.0 * (z2.0 * z3.0) = z2.0 * (z1.0 * z3.0)*)
1240 \<^rule_thm>\<open>mult.assoc\<close>,
1241 (*z1.0 * z2.0 * z3.0 = z1.0 * (z2.0 * z3.0)*)
1242 \<^rule_thm>\<open>add.commute\<close>,
1244 \<^rule_thm>\<open>add.left_commute\<close>,
1245 (*x + (y + z) = y + (x + z)*)
1246 \<^rule_thm>\<open>add.assoc\<close>
1247 (*z1.0 + z2.0 + z3.0 = z1.0 + (z2.0 + z3.0)*)
1248 ], scr = Rule.Empty_Prog});
1252 val collect_bdv = prep_rls'(
1253 Rule_Def.Repeat{id = "collect_bdv", preconds = [],
1254 rew_ord = ("dummy_ord", Rewrite_Ord.dummy_ord),
1255 erls = Rule_Set.empty,srls = Rule_Set.Empty,
1256 calc = [], errpatts = [],
1257 rules = [\<^rule_thm>\<open>bdv_collect_1\<close>,
1258 \<^rule_thm>\<open>bdv_collect_2\<close>,
1259 \<^rule_thm>\<open>bdv_collect_3\<close>,
1261 \<^rule_thm>\<open>bdv_collect_assoc1_1\<close>,
1262 \<^rule_thm>\<open>bdv_collect_assoc1_2\<close>,
1263 \<^rule_thm>\<open>bdv_collect_assoc1_3\<close>,
1265 \<^rule_thm>\<open>bdv_collect_assoc2_1\<close>,
1266 \<^rule_thm>\<open>bdv_collect_assoc2_2\<close>,
1267 \<^rule_thm>\<open>bdv_collect_assoc2_3\<close>,
1270 \<^rule_thm>\<open>bdv_n_collect_1\<close>,
1271 \<^rule_thm>\<open>bdv_n_collect_2\<close>,
1272 \<^rule_thm>\<open>bdv_n_collect_3\<close>,
1274 \<^rule_thm>\<open>bdv_n_collect_assoc1_1\<close>,
1275 \<^rule_thm>\<open>bdv_n_collect_assoc1_2\<close>,
1276 \<^rule_thm>\<open>bdv_n_collect_assoc1_3\<close>,
1278 \<^rule_thm>\<open>bdv_n_collect_assoc2_1\<close>,
1279 \<^rule_thm>\<open>bdv_n_collect_assoc2_2\<close>,
1280 \<^rule_thm>\<open>bdv_n_collect_assoc2_3\<close>
1281 ], scr = Rule.Empty_Prog});
1285 (*.transforms an arbitrary term without roots to a polynomial [4]
1286 according to knowledge/Poly.sml.*)
1287 val make_polynomial_in = prep_rls'(
1288 Rule_Set.Sequence {id = "make_polynomial_in", preconds = []:term list,
1289 rew_ord = ("dummy_ord", Rewrite_Ord.dummy_ord),
1290 erls = Atools_erls, srls = Rule_Set.Empty,
1291 calc = [], errpatts = [],
1292 rules = [Rule.Rls_ expand_poly,
1293 Rule.Rls_ order_add_mult_in,
1294 Rule.Rls_ simplify_power,
1295 Rule.Rls_ collect_numerals,
1296 Rule.Rls_ reduce_012,
1297 \<^rule_thm>\<open>realpow_oneI\<close>,
1298 Rule.Rls_ discard_parentheses,
1299 Rule.Rls_ collect_bdv
1301 scr = Rule.Empty_Prog
1307 Rule_Set.append_rules "separate_bdvs"
1309 [\<^rule_thm>\<open>separate_bdv\<close>,
1310 (*"?a * ?bdv / ?b = ?a / ?b * ?bdv"*)
1311 \<^rule_thm>\<open>separate_bdv_n\<close>,
1312 \<^rule_thm>\<open>separate_1_bdv\<close>,
1313 (*"?bdv / ?b = (1 / ?b) * ?bdv"*)
1314 \<^rule_thm>\<open>separate_1_bdv_n\<close>,
1315 (*"?bdv \<up> ?n / ?b = 1 / ?b * ?bdv \<up> ?n"*)
1316 \<^rule_thm>\<open>add_divide_distrib\<close>
1317 (*"(?x + ?y) / ?z = ?x / ?z + ?y / ?z"
1318 WN051031 DOES NOT BELONG TO HERE*)
1322 val make_ratpoly_in = prep_rls'(
1323 Rule_Set.Sequence {id = "make_ratpoly_in", preconds = []:term list,
1324 rew_ord = ("dummy_ord", Rewrite_Ord.dummy_ord),
1325 erls = Atools_erls, srls = Rule_Set.Empty,
1326 calc = [], errpatts = [],
1327 rules = [Rule.Rls_ norm_Rational,
1328 Rule.Rls_ order_add_mult_in,
1329 Rule.Rls_ discard_parentheses,
1330 Rule.Rls_ separate_bdvs,
1331 (* Rule.Rls_ rearrange_assoc, WN060916 why does cancel_p not work?*)
1333 (*\<^rule_eval>\<open>divide\<close> (eval_cancel "#divide_e") too weak!*)
1335 scr = Rule.Empty_Prog});
1338 order_add_mult_in = order_add_mult_in and
1339 collect_bdv = collect_bdv and
1340 make_polynomial_in = make_polynomial_in and
1341 make_ratpoly_in = make_ratpoly_in and
1342 separate_bdvs = separate_bdvs