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 "^^^" *)
18 cancel_leading_coeff1: "Not (c =!= 0) ==> (a + b*bdv + c*bdv^^^2 = 0) =
19 (a/c + b/c*bdv + bdv^^^2 = 0)" and
20 cancel_leading_coeff2: "Not (c =!= 0) ==> (a - b*bdv + c*bdv^^^2 = 0) =
21 (a/c - b/c*bdv + bdv^^^2 = 0)" and
22 cancel_leading_coeff3: "Not (c =!= 0) ==> (a + b*bdv - c*bdv^^^2 = 0) =
23 (a/c + b/c*bdv - bdv^^^2 = 0)" and
25 cancel_leading_coeff4: "Not (c =!= 0) ==> (a + bdv + c*bdv^^^2 = 0) =
26 (a/c + 1/c*bdv + bdv^^^2 = 0)" and
27 cancel_leading_coeff5: "Not (c =!= 0) ==> (a - bdv + c*bdv^^^2 = 0) =
28 (a/c - 1/c*bdv + bdv^^^2 = 0)" and
29 cancel_leading_coeff6: "Not (c =!= 0) ==> (a + bdv - c*bdv^^^2 = 0) =
30 (a/c + 1/c*bdv - bdv^^^2 = 0)" and
32 cancel_leading_coeff7: "Not (c =!= 0) ==> ( b*bdv + c*bdv^^^2 = 0) =
33 ( b/c*bdv + bdv^^^2 = 0)" and
34 cancel_leading_coeff8: "Not (c =!= 0) ==> ( b*bdv - c*bdv^^^2 = 0) =
35 ( b/c*bdv - bdv^^^2 = 0)" and
37 cancel_leading_coeff9: "Not (c =!= 0) ==> ( bdv + c*bdv^^^2 = 0) =
38 ( 1/c*bdv + bdv^^^2 = 0)" and
39 cancel_leading_coeff10:"Not (c =!= 0) ==> ( bdv - c*bdv^^^2 = 0) =
40 ( 1/c*bdv - bdv^^^2 = 0)" and
42 cancel_leading_coeff11:"Not (c =!= 0) ==> (a + b*bdv^^^2 = 0) =
43 (a/b + bdv^^^2 = 0)" and
44 cancel_leading_coeff12:"Not (c =!= 0) ==> (a - b*bdv^^^2 = 0) =
45 (a/b - bdv^^^2 = 0)" and
46 cancel_leading_coeff13:"Not (c =!= 0) ==> ( b*bdv^^^2 = 0) =
49 complete_square1: "(q + p*bdv + bdv^^^2 = 0) =
50 (q + (p/2 + bdv)^^^2 = (p/2)^^^2)" and
51 complete_square2: "( p*bdv + bdv^^^2 = 0) =
52 ( (p/2 + bdv)^^^2 = (p/2)^^^2)" and
53 complete_square3: "( bdv + bdv^^^2 = 0) =
54 ( (1/2 + bdv)^^^2 = (1/2)^^^2)" and
56 complete_square4: "(q - p*bdv + bdv^^^2 = 0) =
57 (q + (p/2 - bdv)^^^2 = (p/2)^^^2)" and
58 complete_square5: "(q + p*bdv - bdv^^^2 = 0) =
59 (q + (p/2 - bdv)^^^2 = (p/2)^^^2)" and
61 square_explicit1: "(a + b^^^2 = c) = ( b^^^2 = c - a)" and
62 square_explicit2: "(a - b^^^2 = c) = (-(b^^^2) = c - a)" and
64 (*bdv_explicit* required type constrain to real in --- (-8 - 2*x + x^^^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^^^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^^^2=0) = (b*bdv^^^2= (-1)*a)" and
93 "[|Not(bdv occurs_in a)|] ==> (a + bdv^^^2=0) = ( bdv^^^2= (-1)*a)" and
95 "[|Not(b=0);Not(bdv occurs_in c)|] ==> (b*bdv^^^2=c) = (bdv^^^2=c/b)" and
97 d2_prescind1: "(a*bdv + b*bdv^^^2 = 0) = (bdv*(a +b*bdv)=0)" and
98 d2_prescind2: "(a*bdv + bdv^^^2 = 0) = (bdv*(a + bdv)=0)" and
99 d2_prescind3: "( bdv + b*bdv^^^2 = 0) = (bdv*(1+b*bdv)=0)" and
100 d2_prescind4: "( bdv + bdv^^^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^^^2=c) = ((bdv=sqrt c) | (bdv=(-1)*sqrt c ))" and
105 d2_sqrt_equation1_neg:
106 "[|(c<0);Not(bdv occurs_in c)|] ==> (bdv^^^2=c) = False" and
107 d2_sqrt_equation2: "(bdv^^^2=0) = (bdv=0)" and
108 d2_sqrt_equation3: "(b*bdv^^^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 "^^^" 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^^^2 - 4*q|] ==> (q+p*bdv+ bdv^^^2=0) =
131 ((bdv= (-1)*(p/2) + sqrt(p^^^2 - 4*q)/2)
132 | (bdv= (-1)*(p/2) - sqrt(p^^^2 - 4*q)/2))" and
133 d2_pqformula1_neg: "[|p^^^2 - 4*q<0|] ==> (q+p*bdv+ bdv^^^2=0) = False" and
134 d2_pqformula2: "[|0<=p^^^2 - 4*q|] ==> (q+p*bdv+1*bdv^^^2=0) =
135 ((bdv= (-1)*(p/2) + sqrt(p^^^2 - 4*q)/2)
136 | (bdv= (-1)*(p/2) - sqrt(p^^^2 - 4*q)/2))" and
137 d2_pqformula2_neg: "[|p^^^2 - 4*q<0|] ==> (q+p*bdv+1*bdv^^^2=0) = False" and
138 d2_pqformula3: "[|0<=1 - 4*q|] ==> (q+ bdv+ bdv^^^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^^^2=0) = False" and
142 d2_pqformula4: "[|0<=1 - 4*q|] ==> (q+ bdv+1*bdv^^^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^^^2=0) = False" and
146 d2_pqformula5: "[|0<=p^^^2 - 0|] ==> ( p*bdv+ bdv^^^2=0) =
147 ((bdv= (-1)*(p/2) + sqrt(p^^^2 - 0)/2)
148 | (bdv= (-1)*(p/2) - sqrt(p^^^2 - 0)/2))" and
149 (* d2_pqformula5_neg not need p^2 never less zero in R *)
150 d2_pqformula6: "[|0<=p^^^2 - 0|] ==> ( p*bdv+1*bdv^^^2=0) =
151 ((bdv= (-1)*(p/2) + sqrt(p^^^2 - 0)/2)
152 | (bdv= (-1)*(p/2) - sqrt(p^^^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^^^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^^^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^^^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^^^2=0) = False" and
168 "[|Not(bdv occurs_in q); 0<= (-1)*4*q|] ==> (q+ bdv^^^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^^^2=0) = False" and
174 "[|0<=b^^^2 - 4*a*c|] ==> (c + b*bdv+a*bdv^^^2=0) =
175 ((bdv=( -b + sqrt(b^^^2 - 4*a*c))/(2*a))
176 | (bdv=( -b - sqrt(b^^^2 - 4*a*c))/(2*a)))" and
178 "[|b^^^2 - 4*a*c<0|] ==> (c + b*bdv+a*bdv^^^2=0) = False" and
180 "[|0<=1 - 4*a*c|] ==> (c+ bdv+a*bdv^^^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^^^2=0) = False" and
186 "[|0<=b^^^2 - 4*1*c|] ==> (c + b*bdv+ bdv^^^2=0) =
187 ((bdv=( -b + sqrt(b^^^2 - 4*1*c))/(2*1))
188 | (bdv=( -b - sqrt(b^^^2 - 4*1*c))/(2*1)))" and
190 "[|b^^^2 - 4*1*c<0|] ==> (c + b*bdv+ bdv^^^2=0) = False" and
192 "[|0<=1 - 4*1*c|] ==> (c + bdv+ bdv^^^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^^^2=0) = False" and
198 "[|Not(bdv occurs_in c); 0<=0 - 4*a*c|] ==> (c + a*bdv^^^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^^^2=0) = False" and
204 "[|Not(bdv occurs_in c); 0<=0 - 4*1*c|] ==> (c+ bdv^^^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^^^2=0) = False" and
210 "[|0<=b^^^2 - 0|] ==> ( b*bdv+a*bdv^^^2=0) =
211 ((bdv=( -b + sqrt(b^^^2 - 0))/(2*a))
212 | (bdv=( -b - sqrt(b^^^2 - 0))/(2*a)))" and
213 (* d2_abcformula7_neg not need b^2 never less zero in R *)
215 "[|0<=b^^^2 - 0|] ==> ( b*bdv+ bdv^^^2=0) =
216 ((bdv=( -b + sqrt(b^^^2 - 0))/(2*1))
217 | (bdv=( -b - sqrt(b^^^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^^^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^^^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^^^2 + c*bdv^^^3=0) = (bdv=0 | (a + b*bdv + c*bdv^^^2=0))" and
235 "( bdv + b*bdv^^^2 + c*bdv^^^3=0) = (bdv=0 | (1 + b*bdv + c*bdv^^^2=0))" and
237 "(a*bdv + bdv^^^2 + c*bdv^^^3=0) = (bdv=0 | (a + bdv + c*bdv^^^2=0))" and
239 "( bdv + bdv^^^2 + c*bdv^^^3=0) = (bdv=0 | (1 + bdv + c*bdv^^^2=0))" and
241 "(a*bdv + b*bdv^^^2 + bdv^^^3=0) = (bdv=0 | (a + b*bdv + bdv^^^2=0))" and
243 "( bdv + b*bdv^^^2 + bdv^^^3=0) = (bdv=0 | (1 + b*bdv + bdv^^^2=0))" and
245 "(a*bdv + bdv^^^2 + bdv^^^3=0) = (bdv=0 | (1 + bdv + bdv^^^2=0))" and
247 "( bdv + bdv^^^2 + bdv^^^3=0) = (bdv=0 | (1 + bdv + bdv^^^2=0))" and
249 "(a*bdv + c*bdv^^^3=0) = (bdv=0 | (a + c*bdv^^^2=0))" and
250 d3_reduce_equation10:
251 "( bdv + c*bdv^^^3=0) = (bdv=0 | (1 + c*bdv^^^2=0))" and
252 d3_reduce_equation11:
253 "(a*bdv + bdv^^^3=0) = (bdv=0 | (a + bdv^^^2=0))" and
254 d3_reduce_equation12:
255 "( bdv + bdv^^^3=0) = (bdv=0 | (1 + bdv^^^2=0))" and
256 d3_reduce_equation13:
257 "( b*bdv^^^2 + c*bdv^^^3=0) = (bdv=0 | ( b*bdv + c*bdv^^^2=0))" and
258 d3_reduce_equation14:
259 "( bdv^^^2 + c*bdv^^^3=0) = (bdv=0 | ( bdv + c*bdv^^^2=0))" and
260 d3_reduce_equation15:
261 "( b*bdv^^^2 + bdv^^^3=0) = (bdv=0 | ( b*bdv + bdv^^^2=0))" and
262 d3_reduce_equation16:
263 "( bdv^^^2 + bdv^^^3=0) = (bdv=0 | ( bdv + bdv^^^2=0))" and
265 "[|Not(bdv occurs_in a)|] ==> (a + b*bdv^^^3=0) = (b*bdv^^^3= (-1)*a)" and
267 "[|Not(bdv occurs_in a)|] ==> (a + bdv^^^3=0) = ( bdv^^^3= (-1)*a)" and
269 "[|Not(b=0);Not(bdv occurs_in a)|] ==> (b*bdv^^^3=c) = (bdv^^^3=c/b)" and
271 "(bdv^^^3=0) = (bdv=0)" and
273 "(bdv^^^3=c) = (bdv = nroot 3 c)" and
275 (* ---- degree 4 ----*)
276 (* RL03.FIXME es wir nicht getestet ob u>0 *)
278 "(c+b*bdv^^^2+a*bdv^^^4=0) =
279 ((a*u^^^2+b*u+c=0) & (bdv^^^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^^^n + m * bdv^^^n = (l + m) * bdv^^^n" and
301 bdv_n_collect_2: " bdv^^^n + m * bdv^^^n = (1 + m) * bdv^^^n" and
302 bdv_n_collect_3: "l * bdv^^^n + bdv^^^n = (l + 1) * bdv^^^n" (*order!*) and
304 bdv_n_collect_assoc1_1:
305 "l * bdv^^^n + (m * bdv^^^n + k) = (l + m) * bdv^^^n + k" and
306 bdv_n_collect_assoc1_2: "bdv^^^n + (m * bdv^^^n + k) = (1 + m) * bdv^^^n + k" and
307 bdv_n_collect_assoc1_3: "l * bdv^^^n + (bdv^^^n + k) = (l + 1) * bdv^^^n + k" and
309 bdv_n_collect_assoc2_1: "k + l * bdv^^^n + m * bdv^^^n = k +(l + m) * bdv^^^n" and
310 bdv_n_collect_assoc2_2: "k + bdv^^^n + m * bdv^^^n = k + (1 + m) * bdv^^^n" and
311 bdv_n_collect_assoc2_3: "k + l * bdv^^^n + bdv^^^n = k + (l + 1) * bdv^^^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 ^^^ n) / b = (a / b) * bdv ^^^ n" and
318 separate_1_bdv: "bdv / b = (1 / b) * (bdv::real)" and
319 separate_1_bdv_n: "bdv ^^^ n / b = (1 / b) * bdv ^^^ 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.Num_Calc ("Prog_Expr.ident", Prog_Expr.eval_ident "#ident_"),
328 Rule.Num_Calc ("Prog_Expr.matches", Prog_Expr.eval_matches ""),
329 Rule.Num_Calc ("Prog_Expr.lhs", Prog_Expr.eval_lhs ""),
330 Rule.Num_Calc ("Prog_Expr.rhs", Prog_Expr.eval_rhs ""),
331 Rule.Num_Calc ("Poly.is'_expanded'_in", eval_is_expanded_in ""),
332 Rule.Num_Calc ("Poly.is'_poly'_in", eval_is_poly_in ""),
333 Rule.Num_Calc ("Poly.has'_degree'_in", eval_has_degree_in ""),
334 Rule.Num_Calc ("Poly.is'_polyrat'_in", eval_is_polyrat_in ""),
335 (*Rule.Num_Calc ("Prog_Expr.occurs'_in", Prog_Expr.eval_occurs_in ""), *)
336 (*Rule.Num_Calc ("Prog_Expr.is'_const", Prog_Expr.eval_const "#is_const_"),*)
337 Rule.Num_Calc ("HOL.eq", Prog_Expr.eval_equal "#equal_"),
338 Rule.Num_Calc ("RootEq.is'_rootTerm'_in", eval_is_rootTerm_in ""),
339 Rule.Num_Calc ("RatEq.is'_ratequation'_in", eval_is_ratequation_in ""),
340 Rule.Thm ("not_true",TermC.num_str @{thm not_true}),
341 Rule.Thm ("not_false",TermC.num_str @{thm not_false}),
342 Rule.Thm ("and_true",TermC.num_str @{thm and_true}),
343 Rule.Thm ("and_false",TermC.num_str @{thm and_false}),
344 Rule.Thm ("or_true",TermC.num_str @{thm or_true}),
345 Rule.Thm ("or_false",TermC.num_str @{thm or_false})
349 Rule_Set.merge "PolyEq_erls" LinEq_erls
350 (Rule_Set.append_rules "ops_preds" calculate_Rational
351 [Rule.Num_Calc ("HOL.eq", Prog_Expr.eval_equal "#equal_"),
352 Rule.Thm ("plus_leq", TermC.num_str @{thm plus_leq}),
353 Rule.Thm ("minus_leq", TermC.num_str @{thm minus_leq}),
354 Rule.Thm ("rat_leq1", TermC.num_str @{thm rat_leq1}),
355 Rule.Thm ("rat_leq2", TermC.num_str @{thm rat_leq2}),
356 Rule.Thm ("rat_leq3", TermC.num_str @{thm rat_leq3})
360 Rule_Set.merge "PolyEq_crls" LinEq_crls
361 (Rule_Set.append_rules "ops_preds" calculate_Rational
362 [Rule.Num_Calc ("HOL.eq", Prog_Expr.eval_equal "#equal_"),
363 Rule.Thm ("plus_leq", TermC.num_str @{thm plus_leq}),
364 Rule.Thm ("minus_leq", TermC.num_str @{thm minus_leq}),
365 Rule.Thm ("rat_leq1", TermC.num_str @{thm rat_leq1}),
366 Rule.Thm ("rat_leq2", TermC.num_str @{thm rat_leq2}),
367 Rule.Thm ("rat_leq3", TermC.num_str @{thm rat_leq3})
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 ("cancel_leading_coeff1",TermC.num_str @{thm cancel_leading_coeff1}),
376 Rule.Thm ("cancel_leading_coeff2",TermC.num_str @{thm cancel_leading_coeff2}),
377 Rule.Thm ("cancel_leading_coeff3",TermC.num_str @{thm cancel_leading_coeff3}),
378 Rule.Thm ("cancel_leading_coeff4",TermC.num_str @{thm cancel_leading_coeff4}),
379 Rule.Thm ("cancel_leading_coeff5",TermC.num_str @{thm cancel_leading_coeff5}),
380 Rule.Thm ("cancel_leading_coeff6",TermC.num_str @{thm cancel_leading_coeff6}),
381 Rule.Thm ("cancel_leading_coeff7",TermC.num_str @{thm cancel_leading_coeff7}),
382 Rule.Thm ("cancel_leading_coeff8",TermC.num_str @{thm cancel_leading_coeff8}),
383 Rule.Thm ("cancel_leading_coeff9",TermC.num_str @{thm cancel_leading_coeff9}),
384 Rule.Thm ("cancel_leading_coeff10",TermC.num_str @{thm cancel_leading_coeff10}),
385 Rule.Thm ("cancel_leading_coeff11",TermC.num_str @{thm cancel_leading_coeff11}),
386 Rule.Thm ("cancel_leading_coeff12",TermC.num_str @{thm cancel_leading_coeff12}),
387 Rule.Thm ("cancel_leading_coeff13",TermC.num_str @{thm cancel_leading_coeff13})
388 ],scr = Rule.EmptyScr});
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 ("complete_square1",TermC.num_str @{thm complete_square1}),
398 Rule.Thm ("complete_square2",TermC.num_str @{thm complete_square2}),
399 Rule.Thm ("complete_square3",TermC.num_str @{thm complete_square3}),
400 Rule.Thm ("complete_square4",TermC.num_str @{thm complete_square4}),
401 Rule.Thm ("complete_square5",TermC.num_str @{thm complete_square5})
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 ("real_assoc_1",TermC.num_str @{thm real_assoc_1}),
413 Rule.Thm ("real_assoc_2",TermC.num_str @{thm real_assoc_2}),
414 Rule.Thm ("real_diff_minus",TermC.num_str @{thm real_diff_minus}),
415 Rule.Thm ("real_unari_minus",TermC.num_str @{thm real_unari_minus}),
416 Rule.Thm ("realpow_multI",TermC.num_str @{thm realpow_multI}),
417 Rule.Num_Calc ("Groups.plus_class.plus", (**)eval_binop "#add_"),
418 Rule.Num_Calc ("Groups.minus_class.minus", (**)eval_binop "#sub_"),
419 Rule.Num_Calc ("Groups.times_class.times", (**)eval_binop "#mult_"),
420 Rule.Num_Calc ("Rings.divide_class.divide", Prog_Expr.eval_cancel "#divide_e"),
421 Rule.Num_Calc ("NthRoot.sqrt", eval_sqrt "#sqrt_"),
422 Rule.Num_Calc ("Prog_Expr.pow" , (**)eval_binop "#power_"),
428 setup \<open>KEStore_Elems.add_rlss
429 [("cancel_leading_coeff", (Context.theory_name @{theory}, cancel_leading_coeff)),
430 ("complete_square", (Context.theory_name @{theory}, complete_square)),
431 ("PolyEq_erls", (Context.theory_name @{theory}, PolyEq_erls)),
432 ("polyeq_simplify", (Context.theory_name @{theory}, polyeq_simplify))]\<close>
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("d0_true",TermC.num_str @{thm d0_true}),
445 Rule.Thm("d0_false",TermC.num_str @{thm d0_false})
451 (*isolate the bound variable in an d1 equation; 'bdv' is a meta-constant*)
452 val d1_polyeq_simplify = prep_rls'(
453 Rule_Def.Repeat {id = "d1_polyeq_simplify", preconds = [],
454 rew_ord = ("e_rew_ord",Rewrite_Ord.e_rew_ord),
456 srls = Rule_Set.Empty,
457 calc = [], errpatts = [],
459 Rule.Thm("d1_isolate_add1",TermC.num_str @{thm d1_isolate_add1}),
460 (* a+bx=0 -> bx=-a *)
461 Rule.Thm("d1_isolate_add2",TermC.num_str @{thm d1_isolate_add2}),
463 Rule.Thm("d1_isolate_div",TermC.num_str @{thm d1_isolate_div})
470 subsection \<open>degree 2\<close>
472 (* isolate the bound variable in an d2 equation with bdv only;
473 "bdv" is a meta-constant substituted for the "x" below by isac's rewriter. *)
474 val d2_polyeq_bdv_only_simplify = prep_rls'(
475 Rule_Def.Repeat {id = "d2_polyeq_bdv_only_simplify", preconds = [], rew_ord = ("e_rew_ord",Rewrite_Ord.e_rew_ord),
476 erls = PolyEq_erls, srls = Rule_Set.Empty, calc = [], errpatts = [],
478 [Rule.Thm ("d2_prescind1", TermC.num_str @{thm d2_prescind1}), (* ax+bx^2=0 -> x(a+bx)=0 *)
479 Rule.Thm ("d2_prescind2", TermC.num_str @{thm d2_prescind2}), (* ax+ x^2=0 -> x(a+ x)=0 *)
480 Rule.Thm ("d2_prescind3", TermC.num_str @{thm d2_prescind3}), (* x+bx^2=0 -> x(1+bx)=0 *)
481 Rule.Thm ("d2_prescind4", TermC.num_str @{thm d2_prescind4}), (* x+ x^2=0 -> x(1+ x)=0 *)
482 Rule.Thm ("d2_sqrt_equation1", TermC.num_str @{thm d2_sqrt_equation1}), (* x^2=c -> x=+-sqrt(c) *)
483 Rule.Thm ("d2_sqrt_equation1_neg", TermC.num_str @{thm d2_sqrt_equation1_neg}), (* [0<c] x^2=c -> []*)
484 Rule.Thm ("d2_sqrt_equation2", TermC.num_str @{thm d2_sqrt_equation2}), (* x^2=0 -> x=0 *)
485 Rule.Thm ("d2_reduce_equation1", TermC.num_str @{thm d2_reduce_equation1}),(* x(a+bx)=0 -> x=0 |a+bx=0*)
486 Rule.Thm ("d2_reduce_equation2", TermC.num_str @{thm d2_reduce_equation2}),(* x(a+ x)=0 -> x=0 |a+ x=0*)
487 Rule.Thm ("d2_isolate_div", TermC.num_str @{thm d2_isolate_div}) (* bx^2=c -> x^2=c/b *)
493 (* isolate the bound variable in an d2 equation with sqrt only;
494 'bdv' is a meta-constant*)
495 val d2_polyeq_sq_only_simplify = prep_rls'(
496 Rule_Def.Repeat {id = "d2_polyeq_sq_only_simplify", preconds = [],
497 rew_ord = ("e_rew_ord",Rewrite_Ord.e_rew_ord),
499 srls = Rule_Set.Empty,
500 calc = [], errpatts = [],
501 (*asm_thm = [("d2_sqrt_equation1",""),("d2_sqrt_equation1_neg",""),
502 ("d2_isolate_div","")],*)
503 rules = [Rule.Thm("d2_isolate_add1",TermC.num_str @{thm d2_isolate_add1}),
504 (* a+ bx^2=0 -> bx^2=(-1)a*)
505 Rule.Thm("d2_isolate_add2",TermC.num_str @{thm d2_isolate_add2}),
506 (* a+ x^2=0 -> x^2=(-1)a*)
507 Rule.Thm("d2_sqrt_equation2",TermC.num_str @{thm d2_sqrt_equation2}),
509 Rule.Thm("d2_sqrt_equation1",TermC.num_str @{thm d2_sqrt_equation1}),
510 (* x^2=c -> x=+-sqrt(c)*)
511 Rule.Thm("d2_sqrt_equation1_neg",TermC.num_str @{thm d2_sqrt_equation1_neg}),
512 (* [c<0] x^2=c -> x=[] *)
513 Rule.Thm("d2_isolate_div",TermC.num_str @{thm d2_isolate_div})
514 (* bx^2=c -> x^2=c/b*)
520 (* isolate the bound variable in an d2 equation with pqFormula;
521 'bdv' is a meta-constant*)
522 val d2_polyeq_pqFormula_simplify = prep_rls'(
523 Rule_Def.Repeat {id = "d2_polyeq_pqFormula_simplify", preconds = [],
524 rew_ord = ("e_rew_ord",Rewrite_Ord.e_rew_ord), erls = PolyEq_erls,
525 srls = Rule_Set.Empty, calc = [], errpatts = [],
526 rules = [Rule.Thm("d2_pqformula1",TermC.num_str @{thm d2_pqformula1}),
528 Rule.Thm("d2_pqformula1_neg",TermC.num_str @{thm d2_pqformula1_neg}),
530 Rule.Thm("d2_pqformula2",TermC.num_str @{thm d2_pqformula2}),
532 Rule.Thm("d2_pqformula2_neg",TermC.num_str @{thm d2_pqformula2_neg}),
534 Rule.Thm("d2_pqformula3",TermC.num_str @{thm d2_pqformula3}),
536 Rule.Thm("d2_pqformula3_neg",TermC.num_str @{thm d2_pqformula3_neg}),
538 Rule.Thm("d2_pqformula4",TermC.num_str @{thm d2_pqformula4}),
540 Rule.Thm("d2_pqformula4_neg",TermC.num_str @{thm d2_pqformula4_neg}),
542 Rule.Thm("d2_pqformula5",TermC.num_str @{thm d2_pqformula5}),
544 Rule.Thm("d2_pqformula6",TermC.num_str @{thm d2_pqformula6}),
546 Rule.Thm("d2_pqformula7",TermC.num_str @{thm d2_pqformula7}),
548 Rule.Thm("d2_pqformula8",TermC.num_str @{thm d2_pqformula8}),
550 Rule.Thm("d2_pqformula9",TermC.num_str @{thm d2_pqformula9}),
552 Rule.Thm("d2_pqformula9_neg",TermC.num_str @{thm d2_pqformula9_neg}),
554 Rule.Thm("d2_pqformula10",TermC.num_str @{thm d2_pqformula10}),
556 Rule.Thm("d2_pqformula10_neg",TermC.num_str @{thm d2_pqformula10_neg}),
558 Rule.Thm("d2_sqrt_equation2",TermC.num_str @{thm d2_sqrt_equation2}),
560 Rule.Thm("d2_sqrt_equation3",TermC.num_str @{thm d2_sqrt_equation3})
562 ],scr = Rule.EmptyScr
566 (* isolate the bound variable in an d2 equation with abcFormula;
567 'bdv' is a meta-constant*)
568 val d2_polyeq_abcFormula_simplify = prep_rls'(
569 Rule_Def.Repeat {id = "d2_polyeq_abcFormula_simplify", preconds = [],
570 rew_ord = ("e_rew_ord",Rewrite_Ord.e_rew_ord), erls = PolyEq_erls,
571 srls = Rule_Set.Empty, calc = [], errpatts = [],
572 rules = [Rule.Thm("d2_abcformula1",TermC.num_str @{thm d2_abcformula1}),
574 Rule.Thm("d2_abcformula1_neg",TermC.num_str @{thm d2_abcformula1_neg}),
576 Rule.Thm("d2_abcformula2",TermC.num_str @{thm d2_abcformula2}),
578 Rule.Thm("d2_abcformula2_neg",TermC.num_str @{thm d2_abcformula2_neg}),
580 Rule.Thm("d2_abcformula3",TermC.num_str @{thm d2_abcformula3}),
582 Rule.Thm("d2_abcformula3_neg",TermC.num_str @{thm d2_abcformula3_neg}),
584 Rule.Thm("d2_abcformula4",TermC.num_str @{thm d2_abcformula4}),
586 Rule.Thm("d2_abcformula4_neg",TermC.num_str @{thm d2_abcformula4_neg}),
588 Rule.Thm("d2_abcformula5",TermC.num_str @{thm d2_abcformula5}),
590 Rule.Thm("d2_abcformula5_neg",TermC.num_str @{thm d2_abcformula5_neg}),
592 Rule.Thm("d2_abcformula6",TermC.num_str @{thm d2_abcformula6}),
594 Rule.Thm("d2_abcformula6_neg",TermC.num_str @{thm d2_abcformula6_neg}),
596 Rule.Thm("d2_abcformula7",TermC.num_str @{thm d2_abcformula7}),
598 Rule.Thm("d2_abcformula8",TermC.num_str @{thm d2_abcformula8}),
600 Rule.Thm("d2_abcformula9",TermC.num_str @{thm d2_abcformula9}),
602 Rule.Thm("d2_abcformula10",TermC.num_str @{thm d2_abcformula10}),
604 Rule.Thm("d2_sqrt_equation2",TermC.num_str @{thm d2_sqrt_equation2}),
606 Rule.Thm("d2_sqrt_equation3",TermC.num_str @{thm d2_sqrt_equation3})
614 (* isolate the bound variable in an d2 equation;
615 'bdv' is a meta-constant*)
616 val d2_polyeq_simplify = prep_rls'(
617 Rule_Def.Repeat {id = "d2_polyeq_simplify", preconds = [],
618 rew_ord = ("e_rew_ord",Rewrite_Ord.e_rew_ord), erls = PolyEq_erls,
619 srls = Rule_Set.Empty, calc = [], errpatts = [],
620 rules = [Rule.Thm("d2_pqformula1",TermC.num_str @{thm d2_pqformula1}),
622 Rule.Thm("d2_pqformula1_neg",TermC.num_str @{thm d2_pqformula1_neg}),
624 Rule.Thm("d2_pqformula2",TermC.num_str @{thm d2_pqformula2}),
626 Rule.Thm("d2_pqformula2_neg",TermC.num_str @{thm d2_pqformula2_neg}),
628 Rule.Thm("d2_pqformula3",TermC.num_str @{thm d2_pqformula3}),
630 Rule.Thm("d2_pqformula3_neg",TermC.num_str @{thm d2_pqformula3_neg}),
632 Rule.Thm("d2_pqformula4",TermC.num_str @{thm d2_pqformula4}),
634 Rule.Thm("d2_pqformula4_neg",TermC.num_str @{thm d2_pqformula4_neg}),
636 Rule.Thm("d2_abcformula1",TermC.num_str @{thm d2_abcformula1}),
638 Rule.Thm("d2_abcformula1_neg",TermC.num_str @{thm d2_abcformula1_neg}),
640 Rule.Thm("d2_abcformula2",TermC.num_str @{thm d2_abcformula2}),
642 Rule.Thm("d2_abcformula2_neg",TermC.num_str @{thm d2_abcformula2_neg}),
644 Rule.Thm("d2_prescind1",TermC.num_str @{thm d2_prescind1}),
645 (* ax+bx^2=0 -> x(a+bx)=0 *)
646 Rule.Thm("d2_prescind2",TermC.num_str @{thm d2_prescind2}),
647 (* ax+ x^2=0 -> x(a+ x)=0 *)
648 Rule.Thm("d2_prescind3",TermC.num_str @{thm d2_prescind3}),
649 (* x+bx^2=0 -> x(1+bx)=0 *)
650 Rule.Thm("d2_prescind4",TermC.num_str @{thm d2_prescind4}),
651 (* x+ x^2=0 -> x(1+ x)=0 *)
652 Rule.Thm("d2_isolate_add1",TermC.num_str @{thm d2_isolate_add1}),
653 (* a+ bx^2=0 -> bx^2=(-1)a*)
654 Rule.Thm("d2_isolate_add2",TermC.num_str @{thm d2_isolate_add2}),
655 (* a+ x^2=0 -> x^2=(-1)a*)
656 Rule.Thm("d2_sqrt_equation1",TermC.num_str @{thm d2_sqrt_equation1}),
657 (* x^2=c -> x=+-sqrt(c)*)
658 Rule.Thm("d2_sqrt_equation1_neg",TermC.num_str @{thm d2_sqrt_equation1_neg}),
659 (* [c<0] x^2=c -> x=[]*)
660 Rule.Thm("d2_sqrt_equation2",TermC.num_str @{thm d2_sqrt_equation2}),
662 Rule.Thm("d2_reduce_equation1",TermC.num_str @{thm d2_reduce_equation1}),
663 (* x(a+bx)=0 -> x=0 | a+bx=0*)
664 Rule.Thm("d2_reduce_equation2",TermC.num_str @{thm d2_reduce_equation2}),
665 (* x(a+ x)=0 -> x=0 | a+ x=0*)
666 Rule.Thm("d2_isolate_div",TermC.num_str @{thm d2_isolate_div})
667 (* bx^2=c -> x^2=c/b*)
675 (* isolate the bound variable in an d3 equation; 'bdv' is a meta-constant *)
676 val d3_polyeq_simplify = prep_rls'(
677 Rule_Def.Repeat {id = "d3_polyeq_simplify", preconds = [],
678 rew_ord = ("e_rew_ord",Rewrite_Ord.e_rew_ord), erls = PolyEq_erls,
679 srls = Rule_Set.Empty, calc = [], errpatts = [],
681 [Rule.Thm("d3_reduce_equation1",TermC.num_str @{thm d3_reduce_equation1}),
682 (*a*bdv + b*bdv^^^2 + c*bdv^^^3=0) =
683 (bdv=0 | (a + b*bdv + c*bdv^^^2=0)*)
684 Rule.Thm("d3_reduce_equation2",TermC.num_str @{thm d3_reduce_equation2}),
685 (* bdv + b*bdv^^^2 + c*bdv^^^3=0) =
686 (bdv=0 | (1 + b*bdv + c*bdv^^^2=0)*)
687 Rule.Thm("d3_reduce_equation3",TermC.num_str @{thm d3_reduce_equation3}),
688 (*a*bdv + bdv^^^2 + c*bdv^^^3=0) =
689 (bdv=0 | (a + bdv + c*bdv^^^2=0)*)
690 Rule.Thm("d3_reduce_equation4",TermC.num_str @{thm d3_reduce_equation4}),
691 (* bdv + bdv^^^2 + c*bdv^^^3=0) =
692 (bdv=0 | (1 + bdv + c*bdv^^^2=0)*)
693 Rule.Thm("d3_reduce_equation5",TermC.num_str @{thm d3_reduce_equation5}),
694 (*a*bdv + b*bdv^^^2 + bdv^^^3=0) =
695 (bdv=0 | (a + b*bdv + bdv^^^2=0)*)
696 Rule.Thm("d3_reduce_equation6",TermC.num_str @{thm d3_reduce_equation6}),
697 (* bdv + b*bdv^^^2 + bdv^^^3=0) =
698 (bdv=0 | (1 + b*bdv + bdv^^^2=0)*)
699 Rule.Thm("d3_reduce_equation7",TermC.num_str @{thm d3_reduce_equation7}),
700 (*a*bdv + bdv^^^2 + bdv^^^3=0) =
701 (bdv=0 | (1 + bdv + bdv^^^2=0)*)
702 Rule.Thm("d3_reduce_equation8",TermC.num_str @{thm d3_reduce_equation8}),
703 (* bdv + bdv^^^2 + bdv^^^3=0) =
704 (bdv=0 | (1 + bdv + bdv^^^2=0)*)
705 Rule.Thm("d3_reduce_equation9",TermC.num_str @{thm d3_reduce_equation9}),
706 (*a*bdv + c*bdv^^^3=0) =
707 (bdv=0 | (a + c*bdv^^^2=0)*)
708 Rule.Thm("d3_reduce_equation10",TermC.num_str @{thm d3_reduce_equation10}),
709 (* bdv + c*bdv^^^3=0) =
710 (bdv=0 | (1 + c*bdv^^^2=0)*)
711 Rule.Thm("d3_reduce_equation11",TermC.num_str @{thm d3_reduce_equation11}),
712 (*a*bdv + bdv^^^3=0) =
713 (bdv=0 | (a + bdv^^^2=0)*)
714 Rule.Thm("d3_reduce_equation12",TermC.num_str @{thm d3_reduce_equation12}),
715 (* bdv + bdv^^^3=0) =
716 (bdv=0 | (1 + bdv^^^2=0)*)
717 Rule.Thm("d3_reduce_equation13",TermC.num_str @{thm d3_reduce_equation13}),
718 (* b*bdv^^^2 + c*bdv^^^3=0) =
719 (bdv=0 | ( b*bdv + c*bdv^^^2=0)*)
720 Rule.Thm("d3_reduce_equation14",TermC.num_str @{thm d3_reduce_equation14}),
721 (* bdv^^^2 + c*bdv^^^3=0) =
722 (bdv=0 | ( bdv + c*bdv^^^2=0)*)
723 Rule.Thm("d3_reduce_equation15",TermC.num_str @{thm d3_reduce_equation15}),
724 (* b*bdv^^^2 + bdv^^^3=0) =
725 (bdv=0 | ( b*bdv + bdv^^^2=0)*)
726 Rule.Thm("d3_reduce_equation16",TermC.num_str @{thm d3_reduce_equation16}),
727 (* bdv^^^2 + bdv^^^3=0) =
728 (bdv=0 | ( bdv + bdv^^^2=0)*)
729 Rule.Thm("d3_isolate_add1",TermC.num_str @{thm d3_isolate_add1}),
730 (*[|Not(bdv occurs_in a)|] ==> (a + b*bdv^^^3=0) =
731 (bdv=0 | (b*bdv^^^3=a)*)
732 Rule.Thm("d3_isolate_add2",TermC.num_str @{thm d3_isolate_add2}),
733 (*[|Not(bdv occurs_in a)|] ==> (a + bdv^^^3=0) =
734 (bdv=0 | ( bdv^^^3=a)*)
735 Rule.Thm("d3_isolate_div",TermC.num_str @{thm d3_isolate_div}),
736 (*[|Not(b=0)|] ==> (b*bdv^^^3=c) = (bdv^^^3=c/b*)
737 Rule.Thm("d3_root_equation2",TermC.num_str @{thm d3_root_equation2}),
738 (*(bdv^^^3=0) = (bdv=0) *)
739 Rule.Thm("d3_root_equation1",TermC.num_str @{thm d3_root_equation1})
740 (*bdv^^^3=c) = (bdv = nroot 3 c*)
748 (*isolate the bound variable in an d4 equation; 'bdv' is a meta-constant*)
749 val d4_polyeq_simplify = prep_rls'(
750 Rule_Def.Repeat {id = "d4_polyeq_simplify", preconds = [],
751 rew_ord = ("e_rew_ord",Rewrite_Ord.e_rew_ord), erls = PolyEq_erls,
752 srls = Rule_Set.Empty, calc = [], errpatts = [],
754 [Rule.Thm("d4_sub_u1",TermC.num_str @{thm d4_sub_u1})
755 (* ax^4+bx^2+c=0 -> x=+-sqrt(ax^2+bx^+c) *)
760 setup \<open>KEStore_Elems.add_rlss
761 [("d0_polyeq_simplify", (Context.theory_name @{theory}, d0_polyeq_simplify)),
762 ("d1_polyeq_simplify", (Context.theory_name @{theory}, d1_polyeq_simplify)),
763 ("d2_polyeq_simplify", (Context.theory_name @{theory}, d2_polyeq_simplify)),
764 ("d2_polyeq_bdv_only_simplify", (Context.theory_name @{theory}, d2_polyeq_bdv_only_simplify)),
765 ("d2_polyeq_sq_only_simplify", (Context.theory_name @{theory}, d2_polyeq_sq_only_simplify)),
767 ("d2_polyeq_pqFormula_simplify",
768 (Context.theory_name @{theory}, d2_polyeq_pqFormula_simplify)),
769 ("d2_polyeq_abcFormula_simplify",
770 (Context.theory_name @{theory}, d2_polyeq_abcFormula_simplify)),
771 ("d3_polyeq_simplify", (Context.theory_name @{theory}, d3_polyeq_simplify)),
772 ("d4_polyeq_simplify", (Context.theory_name @{theory}, d4_polyeq_simplify))]\<close>
774 (*------------------------problems------------------------*)
776 (get_pbt ["degree_2","polynomial","univariate","equation"]);
780 setup \<open>KEStore_Elems.add_pbts
781 [(Specify.prep_pbt thy "pbl_equ_univ_poly" [] Celem.e_pblID
782 (["polynomial","univariate","equation"],
783 [("#Given" ,["equality e_e","solveFor v_v"]),
784 ("#Where" ,["~((e_e::bool) is_ratequation_in (v_v::real))",
785 "~((lhs e_e) is_rootTerm_in (v_v::real))",
786 "~((rhs e_e) is_rootTerm_in (v_v::real))"]),
787 ("#Find" ,["solutions v_v'i'"])],
788 PolyEq_prls, SOME "solve (e_e::bool, v_v)", [])),
790 (Specify.prep_pbt thy "pbl_equ_univ_poly_deg0" [] Celem.e_pblID
791 (["degree_0","polynomial","univariate","equation"],
792 [("#Given" ,["equality e_e","solveFor v_v"]),
793 ("#Where" ,["matches (?a = 0) e_e",
794 "(lhs e_e) is_poly_in v_v",
795 "((lhs e_e) has_degree_in v_v ) = 0"]),
796 ("#Find" ,["solutions v_v'i'"])],
797 PolyEq_prls, SOME "solve (e_e::bool, v_v)", [["PolyEq","solve_d0_polyeq_equation"]])),
799 (Specify.prep_pbt thy "pbl_equ_univ_poly_deg1" [] Celem.e_pblID
800 (["degree_1","polynomial","univariate","equation"],
801 [("#Given" ,["equality e_e","solveFor v_v"]),
802 ("#Where" ,["matches (?a = 0) e_e",
803 "(lhs e_e) is_poly_in v_v",
804 "((lhs e_e) has_degree_in v_v ) = 1"]),
805 ("#Find" ,["solutions v_v'i'"])],
806 PolyEq_prls, SOME "solve (e_e::bool, v_v)", [["PolyEq","solve_d1_polyeq_equation"]])),
808 (Specify.prep_pbt thy "pbl_equ_univ_poly_deg2" [] Celem.e_pblID
809 (["degree_2","polynomial","univariate","equation"],
810 [("#Given" ,["equality e_e","solveFor v_v"]),
811 ("#Where" ,["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'"])],
815 PolyEq_prls, SOME "solve (e_e::bool, v_v)", [["PolyEq","solve_d2_polyeq_equation"]])),
816 (Specify.prep_pbt thy "pbl_equ_univ_poly_deg2_sqonly" [] Celem.e_pblID
817 (["sq_only","degree_2","polynomial","univariate","equation"],
818 [("#Given" ,["equality e_e","solveFor v_v"]),
819 ("#Where" ,["matches ( ?a + ?v_^^^2 = 0) e_e | " ^
820 "matches ( ?a + ?b*?v_^^^2 = 0) e_e | " ^
821 "matches ( ?v_^^^2 = 0) e_e | " ^
822 "matches ( ?b*?v_^^^2 = 0) e_e" ,
823 "Not (matches (?a + ?v_ + ?v_^^^2 = 0) e_e) &" ^
824 "Not (matches (?a + ?b*?v_ + ?v_^^^2 = 0) e_e) &" ^
825 "Not (matches (?a + ?v_ + ?c*?v_^^^2 = 0) e_e) &" ^
826 "Not (matches (?a + ?b*?v_ + ?c*?v_^^^2 = 0) e_e) &" ^
827 "Not (matches ( ?v_ + ?v_^^^2 = 0) e_e) &" ^
828 "Not (matches ( ?b*?v_ + ?v_^^^2 = 0) e_e) &" ^
829 "Not (matches ( ?v_ + ?c*?v_^^^2 = 0) e_e) &" ^
830 "Not (matches ( ?b*?v_ + ?c*?v_^^^2 = 0) e_e)"]),
831 ("#Find" ,["solutions v_v'i'"])],
832 PolyEq_prls, SOME "solve (e_e::bool, v_v)",
833 [["PolyEq","solve_d2_polyeq_sqonly_equation"]])),
834 (Specify.prep_pbt thy "pbl_equ_univ_poly_deg2_bdvonly" [] Celem.e_pblID
835 (["bdv_only","degree_2","polynomial","univariate","equation"],
836 [("#Given", ["equality e_e","solveFor v_v"]),
837 ("#Where", ["matches (?a*?v_ + ?v_^^^2 = 0) e_e | " ^
838 "matches ( ?v_ + ?v_^^^2 = 0) e_e | " ^
839 "matches ( ?v_ + ?b*?v_^^^2 = 0) e_e | " ^
840 "matches (?a*?v_ + ?b*?v_^^^2 = 0) e_e | " ^
841 "matches ( ?v_^^^2 = 0) e_e | " ^
842 "matches ( ?b*?v_^^^2 = 0) e_e "]),
843 ("#Find", ["solutions v_v'i'"])],
844 PolyEq_prls, SOME "solve (e_e::bool, v_v)",
845 [["PolyEq","solve_d2_polyeq_bdvonly_equation"]])),
846 (Specify.prep_pbt thy "pbl_equ_univ_poly_deg2_pq" [] Celem.e_pblID
847 (["pqFormula","degree_2","polynomial","univariate","equation"],
848 [("#Given", ["equality e_e","solveFor v_v"]),
849 ("#Where", ["matches (?a + 1*?v_^^^2 = 0) e_e | " ^
850 "matches (?a + ?v_^^^2 = 0) e_e"]),
851 ("#Find", ["solutions v_v'i'"])],
852 PolyEq_prls, SOME "solve (e_e::bool, v_v)", [["PolyEq","solve_d2_polyeq_pq_equation"]])),
853 (Specify.prep_pbt thy "pbl_equ_univ_poly_deg2_abc" [] Celem.e_pblID
854 (["abcFormula","degree_2","polynomial","univariate","equation"],
855 [("#Given", ["equality e_e","solveFor v_v"]),
856 ("#Where", ["matches (?a + ?v_^^^2 = 0) e_e | " ^
857 "matches (?a + ?b*?v_^^^2 = 0) e_e"]),
858 ("#Find", ["solutions v_v'i'"])],
859 PolyEq_prls, SOME "solve (e_e::bool, v_v)", [["PolyEq","solve_d2_polyeq_abc_equation"]])),
861 (Specify.prep_pbt thy "pbl_equ_univ_poly_deg3" [] Celem.e_pblID
862 (["degree_3","polynomial","univariate","equation"],
863 [("#Given", ["equality e_e","solveFor v_v"]),
864 ("#Where", ["matches (?a = 0) e_e",
865 "(lhs e_e) is_poly_in v_v ",
866 "((lhs e_e) has_degree_in v_v) = 3"]),
867 ("#Find", ["solutions v_v'i'"])],
868 PolyEq_prls, SOME "solve (e_e::bool, v_v)", [["PolyEq","solve_d3_polyeq_equation"]])),
870 (Specify.prep_pbt thy "pbl_equ_univ_poly_deg4" [] Celem.e_pblID
871 (["degree_4","polynomial","univariate","equation"],
872 [("#Given", ["equality e_e","solveFor v_v"]),
873 ("#Where", ["matches (?a = 0) e_e",
874 "(lhs e_e) is_poly_in v_v ",
875 "((lhs e_e) has_degree_in v_v) = 4"]),
876 ("#Find", ["solutions v_v'i'"])],
877 PolyEq_prls, SOME "solve (e_e::bool, v_v)", [(*["PolyEq","solve_d4_polyeq_equation"]*)])),
878 (*--- normalise ---*)
879 (Specify.prep_pbt thy "pbl_equ_univ_poly_norm" [] Celem.e_pblID
880 (["normalise","polynomial","univariate","equation"],
881 [("#Given", ["equality e_e","solveFor v_v"]),
882 ("#Where", ["(Not((matches (?a = 0 ) e_e ))) |" ^
883 "(Not(((lhs e_e) is_poly_in v_v)))"]),
884 ("#Find", ["solutions v_v'i'"])],
885 PolyEq_prls, SOME "solve (e_e::bool, v_v)", [["PolyEq", "normalise_poly"]])),
886 (*-------------------------expanded-----------------------*)
887 (Specify.prep_pbt thy "pbl_equ_univ_expand" [] Celem.e_pblID
888 (["expanded","univariate","equation"],
889 [("#Given", ["equality e_e","solveFor v_v"]),
890 ("#Where", ["matches (?a = 0) e_e",
891 "(lhs e_e) is_expanded_in v_v "]),
892 ("#Find", ["solutions v_v'i'"])],
893 PolyEq_prls, SOME "solve (e_e::bool, v_v)", [])),
895 (Specify.prep_pbt thy "pbl_equ_univ_expand_deg2" [] Celem.e_pblID
896 (["degree_2","expanded","univariate","equation"],
897 [("#Given", ["equality e_e","solveFor v_v"]),
898 ("#Where", ["((lhs e_e) has_degree_in v_v) = 2"]),
899 ("#Find", ["solutions v_v'i'"])],
900 PolyEq_prls, SOME "solve (e_e::bool, v_v)", [["PolyEq","complete_square"]]))]\<close>
902 text \<open>"-------------------------methods-----------------------"\<close>
903 setup \<open>KEStore_Elems.add_mets
904 [Specify.prep_met thy "met_polyeq" [] Celem.e_metID
906 {rew_ord'="tless_true",rls'=Atools_erls,calc = [], srls = Rule_Set.empty, prls=Rule_Set.empty,
907 crls=PolyEq_crls, errpats = [], nrls = norm_Rational},
911 partial_function (tailrec) normalize_poly_eq :: "bool \<Rightarrow> real \<Rightarrow> bool"
913 "normalize_poly_eq e_e v_v = (
916 (Try (Rewrite ''all_left'')) #>
917 (Try (Repeat (Rewrite ''makex1_x''))) #>
918 (Try (Repeat (Rewrite_Set ''expand_binoms''))) #>
919 (Try (Repeat (Rewrite_Set_Inst [(''bdv'', v_v)] ''make_ratpoly_in''))) #>
920 (Try (Repeat (Rewrite_Set ''polyeq_simplify''))) ) e_e
922 SubProblem (''PolyEq'', [''polynomial'', ''univariate'', ''equation''], [''no_met''])
923 [BOOL e_e, REAL v_v])"
924 setup \<open>KEStore_Elems.add_mets
925 [Specify.prep_met thy "met_polyeq_norm" [] Celem.e_metID
926 (["PolyEq", "normalise_poly"],
927 [("#Given" ,["equality e_e","solveFor v_v"]),
928 ("#Where" ,["(Not((matches (?a = 0 ) e_e ))) | (Not(((lhs e_e) is_poly_in v_v)))"]),
929 ("#Find" ,["solutions v_v'i'"])],
930 {rew_ord'="termlessI", rls'=PolyEq_erls, srls=Rule_Set.empty, prls=PolyEq_prls, calc=[],
931 crls=PolyEq_crls, errpats = [], nrls = norm_Rational},
932 @{thm normalize_poly_eq.simps})]
935 partial_function (tailrec) solve_poly_equ :: "bool \<Rightarrow> real \<Rightarrow> bool list"
937 "solve_poly_equ e_e v_v = (
939 e_e = (Try (Rewrite_Set_Inst [(''bdv'', v_v)] ''d0_polyeq_simplify'')) e_e
942 setup \<open>KEStore_Elems.add_mets
943 [Specify.prep_met thy "met_polyeq_d0" [] Celem.e_metID
944 (["PolyEq","solve_d0_polyeq_equation"],
945 [("#Given" ,["equality e_e","solveFor v_v"]),
946 ("#Where" ,["(lhs e_e) is_poly_in v_v ", "((lhs e_e) has_degree_in v_v) = 0"]),
947 ("#Find" ,["solutions v_v'i'"])],
948 {rew_ord'="termlessI", rls'=PolyEq_erls, srls=Rule_Set.empty, prls=PolyEq_prls,
949 calc=[("sqrt", ("NthRoot.sqrt", eval_sqrt "#sqrt_"))], crls=PolyEq_crls, errpats = [],
950 nrls = norm_Rational},
951 @{thm solve_poly_equ.simps})]
954 partial_function (tailrec) solve_poly_eq1 :: "bool \<Rightarrow> real \<Rightarrow> bool list"
956 "solve_poly_eq1 e_e v_v = (
959 (Try (Rewrite_Set_Inst [(''bdv'', v_v)] ''d1_polyeq_simplify'')) #>
960 (Try (Rewrite_Set ''polyeq_simplify'')) #>
961 (Try (Rewrite_Set ''norm_Rational_parenthesized'')) ) e_e;
964 Check_elementwise L_L {(v_v::real). Assumptions})"
965 setup \<open>KEStore_Elems.add_mets
966 [Specify.prep_met thy "met_polyeq_d1" [] Celem.e_metID
967 (["PolyEq","solve_d1_polyeq_equation"],
968 [("#Given" ,["equality e_e","solveFor v_v"]),
969 ("#Where" ,["(lhs e_e) is_poly_in v_v ", "((lhs e_e) has_degree_in v_v) = 1"]),
970 ("#Find" ,["solutions v_v'i'"])],
971 {rew_ord'="termlessI", rls'=PolyEq_erls, srls=Rule_Set.empty, prls=PolyEq_prls,
972 calc=[("sqrt", ("NthRoot.sqrt", eval_sqrt "#sqrt_"))], crls=PolyEq_crls, errpats = [],
973 nrls = norm_Rational},
974 @{thm solve_poly_eq1.simps})]
977 partial_function (tailrec) solve_poly_equ2 :: "bool \<Rightarrow> real \<Rightarrow> bool list"
979 "solve_poly_equ2 e_e v_v = (
982 (Try (Rewrite_Set_Inst [(''bdv'', v_v)] ''d2_polyeq_simplify'')) #>
983 (Try (Rewrite_Set ''polyeq_simplify'')) #>
984 (Try (Rewrite_Set_Inst [(''bdv'', v_v)] ''d1_polyeq_simplify'')) #>
985 (Try (Rewrite_Set ''polyeq_simplify'')) #>
986 (Try (Rewrite_Set ''norm_Rational_parenthesized'')) ) e_e;
989 Check_elementwise L_L {(v_v::real). Assumptions})"
990 setup \<open>KEStore_Elems.add_mets
991 [Specify.prep_met thy "met_polyeq_d22" [] Celem.e_metID
992 (["PolyEq","solve_d2_polyeq_equation"],
993 [("#Given" ,["equality e_e","solveFor v_v"]),
994 ("#Where" ,["(lhs e_e) is_poly_in v_v ", "((lhs e_e) has_degree_in v_v) = 2"]),
995 ("#Find" ,["solutions v_v'i'"])],
996 {rew_ord'="termlessI", rls'=PolyEq_erls, srls=Rule_Set.empty, prls=PolyEq_prls,
997 calc=[("sqrt", ("NthRoot.sqrt", eval_sqrt "#sqrt_"))], crls=PolyEq_crls, errpats = [],
998 nrls = norm_Rational},
999 @{thm solve_poly_equ2.simps})]
1002 partial_function (tailrec) solve_poly_equ0 :: "bool \<Rightarrow> real \<Rightarrow> bool list"
1004 "solve_poly_equ0 e_e v_v = (
1007 (Try (Rewrite_Set_Inst [(''bdv'', v_v)] ''d2_polyeq_bdv_only_simplify'')) #>
1008 (Try (Rewrite_Set ''polyeq_simplify'')) #>
1009 (Try (Rewrite_Set_Inst [(''bdv'',v_v::real)] ''d1_polyeq_simplify'')) #>
1010 (Try (Rewrite_Set ''polyeq_simplify'')) #>
1011 (Try (Rewrite_Set ''norm_Rational_parenthesized'')) ) e_e;
1012 L_L = Or_to_List e_e
1014 Check_elementwise L_L {(v_v::real). Assumptions})"
1015 setup \<open>KEStore_Elems.add_mets
1016 [Specify.prep_met thy "met_polyeq_d2_bdvonly" [] Celem.e_metID
1017 (["PolyEq","solve_d2_polyeq_bdvonly_equation"],
1018 [("#Given" ,["equality e_e","solveFor v_v"]),
1019 ("#Where" ,["(lhs e_e) is_poly_in v_v ", "((lhs e_e) has_degree_in v_v) = 2"]),
1020 ("#Find" ,["solutions v_v'i'"])],
1021 {rew_ord'="termlessI", rls'=PolyEq_erls, srls=Rule_Set.empty, prls=PolyEq_prls,
1022 calc=[("sqrt", ("NthRoot.sqrt", eval_sqrt "#sqrt_"))], crls=PolyEq_crls, errpats = [],
1023 nrls = norm_Rational},
1024 @{thm solve_poly_equ0.simps})]
1027 partial_function (tailrec) solve_poly_equ_sqrt :: "bool \<Rightarrow> real \<Rightarrow> bool list"
1029 "solve_poly_equ_sqrt e_e v_v = (
1032 (Try (Rewrite_Set_Inst [(''bdv'', v_v)] ''d2_polyeq_sq_only_simplify'')) #>
1033 (Try (Rewrite_Set ''polyeq_simplify'')) #>
1034 (Try (Rewrite_Set ''norm_Rational_parenthesized'')) ) e_e;
1035 L_L = Or_to_List e_e
1037 Check_elementwise L_L {(v_v::real). Assumptions})"
1038 setup \<open>KEStore_Elems.add_mets
1039 [Specify.prep_met thy "met_polyeq_d2_sqonly" [] Celem.e_metID
1040 (["PolyEq","solve_d2_polyeq_sqonly_equation"],
1041 [("#Given" ,["equality e_e","solveFor v_v"]),
1042 ("#Where" ,["(lhs e_e) is_poly_in v_v ", "((lhs e_e) has_degree_in v_v) = 2"]),
1043 ("#Find" ,["solutions v_v'i'"])],
1044 {rew_ord'="termlessI", rls'=PolyEq_erls, srls=Rule_Set.empty, prls=PolyEq_prls,
1045 calc=[("sqrt", ("NthRoot.sqrt", eval_sqrt "#sqrt_"))], crls=PolyEq_crls, errpats = [],
1046 nrls = norm_Rational},
1047 @{thm solve_poly_equ_sqrt.simps})]
1050 partial_function (tailrec) solve_poly_equ_pq :: "bool \<Rightarrow> real \<Rightarrow> bool list"
1052 "solve_poly_equ_pq e_e v_v = (
1055 (Try (Rewrite_Set_Inst [(''bdv'', v_v)] ''d2_polyeq_pqFormula_simplify'')) #>
1056 (Try (Rewrite_Set ''polyeq_simplify'')) #>
1057 (Try (Rewrite_Set ''norm_Rational_parenthesized'')) ) e_e;
1058 L_L = Or_to_List e_e
1060 Check_elementwise L_L {(v_v::real). Assumptions})"
1061 setup \<open>KEStore_Elems.add_mets
1062 [Specify.prep_met thy "met_polyeq_d2_pq" [] Celem.e_metID
1063 (["PolyEq","solve_d2_polyeq_pq_equation"],
1064 [("#Given" ,["equality e_e","solveFor v_v"]),
1065 ("#Where" ,["(lhs e_e) is_poly_in v_v ", "((lhs e_e) has_degree_in v_v) = 2"]),
1066 ("#Find" ,["solutions v_v'i'"])],
1067 {rew_ord'="termlessI", rls'=PolyEq_erls, srls=Rule_Set.empty, prls=PolyEq_prls,
1068 calc=[("sqrt", ("NthRoot.sqrt", eval_sqrt "#sqrt_"))], crls=PolyEq_crls, errpats = [],
1069 nrls = norm_Rational},
1070 @{thm solve_poly_equ_pq.simps})]
1073 partial_function (tailrec) solve_poly_equ_abc :: "bool \<Rightarrow> real \<Rightarrow> bool list"
1075 "solve_poly_equ_abc e_e v_v = (
1078 (Try (Rewrite_Set_Inst [(''bdv'', v_v)] ''d2_polyeq_abcFormula_simplify'')) #>
1079 (Try (Rewrite_Set ''polyeq_simplify'')) #>
1080 (Try (Rewrite_Set ''norm_Rational_parenthesized'')) ) e_e;
1081 L_L = Or_to_List e_e
1082 in Check_elementwise L_L {(v_v::real). Assumptions})"
1083 setup \<open>KEStore_Elems.add_mets
1084 [Specify.prep_met thy "met_polyeq_d2_abc" [] Celem.e_metID
1085 (["PolyEq","solve_d2_polyeq_abc_equation"],
1086 [("#Given" ,["equality e_e","solveFor v_v"]),
1087 ("#Where" ,["(lhs e_e) is_poly_in v_v ", "((lhs e_e) has_degree_in v_v) = 2"]),
1088 ("#Find" ,["solutions v_v'i'"])],
1089 {rew_ord'="termlessI", rls'=PolyEq_erls,srls=Rule_Set.empty, prls=PolyEq_prls,
1090 calc=[("sqrt", ("NthRoot.sqrt", eval_sqrt "#sqrt_"))], crls=PolyEq_crls, errpats = [],
1091 nrls = norm_Rational},
1092 @{thm solve_poly_equ_abc.simps})]
1095 partial_function (tailrec) solve_poly_equ3 :: "bool \<Rightarrow> real \<Rightarrow> bool list"
1097 "solve_poly_equ3 e_e v_v = (
1100 (Try (Rewrite_Set_Inst [(''bdv'', v_v)] ''d3_polyeq_simplify'')) #>
1101 (Try (Rewrite_Set ''polyeq_simplify'')) #>
1102 (Try (Rewrite_Set_Inst [(''bdv'', v_v)] ''d2_polyeq_simplify'')) #>
1103 (Try (Rewrite_Set ''polyeq_simplify'')) #>
1104 (Try (Rewrite_Set_Inst [(''bdv'',v_v)] ''d1_polyeq_simplify'')) #>
1105 (Try (Rewrite_Set ''polyeq_simplify'')) #>
1106 (Try (Rewrite_Set ''norm_Rational_parenthesized''))) e_e;
1107 L_L = Or_to_List e_e
1109 Check_elementwise L_L {(v_v::real). Assumptions})"
1110 setup \<open>KEStore_Elems.add_mets
1111 [Specify.prep_met thy "met_polyeq_d3" [] Celem.e_metID
1112 (["PolyEq","solve_d3_polyeq_equation"],
1113 [("#Given" ,["equality e_e","solveFor v_v"]),
1114 ("#Where" ,["(lhs e_e) is_poly_in v_v ", "((lhs e_e) has_degree_in v_v) = 3"]),
1115 ("#Find" ,["solutions v_v'i'"])],
1116 {rew_ord'="termlessI", rls'=PolyEq_erls, srls=Rule_Set.empty, prls=PolyEq_prls,
1117 calc=[("sqrt", ("NthRoot.sqrt", eval_sqrt "#sqrt_"))], crls=PolyEq_crls, errpats = [],
1118 nrls = norm_Rational},
1119 @{thm solve_poly_equ3.simps})]
1121 (*.solves all expanded (ie. normalised) terms of degree 2.*)
1122 (*Oct.02 restriction: 'eval_true 0 =< discriminant' ony for integer values
1123 by 'PolyEq_erls'; restricted until Float.thy is implemented*)
1124 partial_function (tailrec) solve_by_completing_square :: "bool \<Rightarrow> real \<Rightarrow> bool list"
1126 "solve_by_completing_square e_e v_v = (
1128 (Try (Rewrite_Set_Inst [(''bdv'', v_v)] ''cancel_leading_coeff'')) #>
1129 (Try (Rewrite_Set_Inst [(''bdv'', v_v)] ''complete_square'')) #>
1130 (Try (Rewrite ''square_explicit1'')) #>
1131 (Try (Rewrite ''square_explicit2'')) #>
1132 (Rewrite ''root_plus_minus'') #>
1133 (Try (Repeat (Rewrite_Inst [(''bdv'', v_v)] ''bdv_explicit1''))) #>
1134 (Try (Repeat (Rewrite_Inst [(''bdv'', v_v)] ''bdv_explicit2''))) #>
1135 (Try (Repeat (Rewrite_Inst [(''bdv'', v_v)] ''bdv_explicit3''))) #>
1136 (Try (Rewrite_Set ''calculate_RootRat'')) #>
1137 (Try (Repeat (Calculate ''SQRT'')))) e_e
1140 setup \<open>KEStore_Elems.add_mets
1141 [Specify.prep_met thy "met_polyeq_complsq" [] Celem.e_metID
1142 (["PolyEq","complete_square"],
1143 [("#Given" ,["equality e_e","solveFor v_v"]),
1144 ("#Where" ,["matches (?a = 0) e_e", "((lhs e_e) has_degree_in v_v) = 2"]),
1145 ("#Find" ,["solutions v_v'i'"])],
1146 {rew_ord'="termlessI",rls'=PolyEq_erls,srls=Rule_Set.empty,prls=PolyEq_prls,
1147 calc=[("sqrt", ("NthRoot.sqrt", eval_sqrt "#sqrt_"))], crls=PolyEq_crls, errpats = [],
1148 nrls = norm_Rational},
1149 @{thm solve_by_completing_square.simps})]
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' x (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' x (Var v) = (v, 2)
1168 | dest_hd' x (Bound i) = ((("", i), dummyT), 3)
1169 | dest_hd' x (Abs (_, T, _)) = ((("", 0), T), 4);
1171 fun size_of_term' x (Const ("Prog_Expr.pow",_) $ Free (var,_) $ Free (pot,_)) =
1174 (if xstr = var then 1000*(the (TermC.int_of_str_opt pot)) else 3)
1175 | _ => error ("size_of_term' called with subst = "^
1177 | size_of_term' x (Free (subst,_)) =
1179 (Free (xstr,_)) => (if xstr = subst then 1000 else 1)
1180 | _ => error ("size_of_term' called with subst = "^
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' x _ = 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= \"" ^ Rule.term_to_string''' thy f ^ "\" @ \"[" ^
1194 commas (map (Rule.term_to_string''' thy) ts) ^ "]\"");
1195 val _ = tracing ("u= g@us= \"" ^ Rule.term_to_string''' thy g ^ "\" @ \"[" ^
1196 commas(map (Rule.term_to_string''' 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 str pr (ts, us) =
1217 list_ord (term_ord' x pr (Celem.assoc_thy "Isac_Knowledge"))(ts, us);
1221 fun ord_make_polynomial_in (pr:bool) thy subst tu =
1223 (* val _=tracing("*** subs variable is: "^(Env.subst2str subst)); *)
1226 (_,x)::_ => (term_ord' x pr thy tu = LESS)
1227 | _ => error ("ord_make_polynomial_in called with subst = "^
1228 (Env.subst2str subst))
1234 val order_add_mult_in = prep_rls'(
1235 Rule_Def.Repeat{id = "order_add_mult_in", preconds = [],
1236 rew_ord = ("ord_make_polynomial_in",
1237 ord_make_polynomial_in false @{theory "Poly"}),
1238 erls = Rule_Set.empty,srls = Rule_Set.Empty,
1239 calc = [], errpatts = [],
1240 rules = [Rule.Thm ("mult_commute",TermC.num_str @{thm mult.commute}),
1242 Rule.Thm ("real_mult_left_commute",TermC.num_str @{thm real_mult_left_commute}),
1243 (*z1.0 * (z2.0 * z3.0) = z2.0 * (z1.0 * z3.0)*)
1244 Rule.Thm ("mult_assoc",TermC.num_str @{thm mult.assoc}),
1245 (*z1.0 * z2.0 * z3.0 = z1.0 * (z2.0 * z3.0)*)
1246 Rule.Thm ("add_commute",TermC.num_str @{thm add.commute}),
1248 Rule.Thm ("add_left_commute",TermC.num_str @{thm add.left_commute}),
1249 (*x + (y + z) = y + (x + z)*)
1250 Rule.Thm ("add_assoc",TermC.num_str @{thm add.assoc})
1251 (*z1.0 + z2.0 + z3.0 = z1.0 + (z2.0 + z3.0)*)
1252 ], scr = Rule.EmptyScr});
1256 val collect_bdv = prep_rls'(
1257 Rule_Def.Repeat{id = "collect_bdv", preconds = [],
1258 rew_ord = ("dummy_ord", Rewrite_Ord.dummy_ord),
1259 erls = Rule_Set.empty,srls = Rule_Set.Empty,
1260 calc = [], errpatts = [],
1261 rules = [Rule.Thm ("bdv_collect_1",TermC.num_str @{thm bdv_collect_1}),
1262 Rule.Thm ("bdv_collect_2",TermC.num_str @{thm bdv_collect_2}),
1263 Rule.Thm ("bdv_collect_3",TermC.num_str @{thm bdv_collect_3}),
1265 Rule.Thm ("bdv_collect_assoc1_1",TermC.num_str @{thm bdv_collect_assoc1_1}),
1266 Rule.Thm ("bdv_collect_assoc1_2",TermC.num_str @{thm bdv_collect_assoc1_2}),
1267 Rule.Thm ("bdv_collect_assoc1_3",TermC.num_str @{thm bdv_collect_assoc1_3}),
1269 Rule.Thm ("bdv_collect_assoc2_1",TermC.num_str @{thm bdv_collect_assoc2_1}),
1270 Rule.Thm ("bdv_collect_assoc2_2",TermC.num_str @{thm bdv_collect_assoc2_2}),
1271 Rule.Thm ("bdv_collect_assoc2_3",TermC.num_str @{thm bdv_collect_assoc2_3}),
1274 Rule.Thm ("bdv_n_collect_1",TermC.num_str @{thm bdv_n_collect_1}),
1275 Rule.Thm ("bdv_n_collect_2",TermC.num_str @{thm bdv_n_collect_2}),
1276 Rule.Thm ("bdv_n_collect_3",TermC.num_str @{thm bdv_n_collect_3}),
1278 Rule.Thm ("bdv_n_collect_assoc1_1",TermC.num_str @{thm bdv_n_collect_assoc1_1}),
1279 Rule.Thm ("bdv_n_collect_assoc1_2",TermC.num_str @{thm bdv_n_collect_assoc1_2}),
1280 Rule.Thm ("bdv_n_collect_assoc1_3",TermC.num_str @{thm bdv_n_collect_assoc1_3}),
1282 Rule.Thm ("bdv_n_collect_assoc2_1",TermC.num_str @{thm bdv_n_collect_assoc2_1}),
1283 Rule.Thm ("bdv_n_collect_assoc2_2",TermC.num_str @{thm bdv_n_collect_assoc2_2}),
1284 Rule.Thm ("bdv_n_collect_assoc2_3",TermC.num_str @{thm bdv_n_collect_assoc2_3})
1285 ], scr = Rule.EmptyScr});
1289 (*.transforms an arbitrary term without roots to a polynomial [4]
1290 according to knowledge/Poly.sml.*)
1291 val make_polynomial_in = prep_rls'(
1292 Rule_Set.Seqence {id = "make_polynomial_in", preconds = []:term list,
1293 rew_ord = ("dummy_ord", Rewrite_Ord.dummy_ord),
1294 erls = Atools_erls, srls = Rule_Set.Empty,
1295 calc = [], errpatts = [],
1296 rules = [Rule.Rls_ expand_poly,
1297 Rule.Rls_ order_add_mult_in,
1298 Rule.Rls_ simplify_power,
1299 Rule.Rls_ collect_numerals,
1300 Rule.Rls_ reduce_012,
1301 Rule.Thm ("realpow_oneI",TermC.num_str @{thm realpow_oneI}),
1302 Rule.Rls_ discard_parentheses,
1303 Rule.Rls_ collect_bdv
1311 Rule_Set.append_rules "separate_bdvs"
1313 [Rule.Thm ("separate_bdv", TermC.num_str @{thm separate_bdv}),
1314 (*"?a * ?bdv / ?b = ?a / ?b * ?bdv"*)
1315 Rule.Thm ("separate_bdv_n", TermC.num_str @{thm separate_bdv_n}),
1316 Rule.Thm ("separate_1_bdv", TermC.num_str @{thm separate_1_bdv}),
1317 (*"?bdv / ?b = (1 / ?b) * ?bdv"*)
1318 Rule.Thm ("separate_1_bdv_n", TermC.num_str @{thm separate_1_bdv_n}),
1319 (*"?bdv ^^^ ?n / ?b = 1 / ?b * ?bdv ^^^ ?n"*)
1320 Rule.Thm ("add_divide_distrib",
1321 TermC.num_str @{thm add_divide_distrib})
1322 (*"(?x + ?y) / ?z = ?x / ?z + ?y / ?z"
1323 WN051031 DOES NOT BELONG TO HERE*)
1327 val make_ratpoly_in = prep_rls'(
1328 Rule_Set.Seqence {id = "make_ratpoly_in", preconds = []:term list,
1329 rew_ord = ("dummy_ord", Rewrite_Ord.dummy_ord),
1330 erls = Atools_erls, srls = Rule_Set.Empty,
1331 calc = [], errpatts = [],
1332 rules = [Rule.Rls_ norm_Rational,
1333 Rule.Rls_ order_add_mult_in,
1334 Rule.Rls_ discard_parentheses,
1335 Rule.Rls_ separate_bdvs,
1336 (* Rule.Rls_ rearrange_assoc, WN060916 why does cancel_p not work?*)
1338 (*Rule.Num_Calc ("Rings.divide_class.divide" , eval_cancel "#divide_e") too weak!*)
1340 scr = Rule.EmptyScr});
1342 setup \<open>KEStore_Elems.add_rlss
1343 [("order_add_mult_in", (Context.theory_name @{theory}, order_add_mult_in)),
1344 ("collect_bdv", (Context.theory_name @{theory}, collect_bdv)),
1345 ("make_polynomial_in", (Context.theory_name @{theory}, make_polynomial_in)),
1346 ("make_ratpoly_in", (Context.theory_name @{theory}, make_ratpoly_in)),
1347 ("separate_bdvs", (Context.theory_name @{theory}, separate_bdvs))]\<close>