1 (* equational systems, minimal -- for use in Biegelinie
4 (c) due to copyright terms
7 theory EqSystem imports Integrate Rational Root begin
12 "[real list, real list, 'a] => bool" ("_ from _ occur'_exactly'_in _")
14 (*descriptions in the related problems*)
15 solveForVars :: "real list => toreall"
16 solution :: "bool list => toreall"
18 (*the CAS-command, eg. "solveSystem [x+y=1,y=2] [x,y]"*)
19 solveSystem :: "[bool list, real list] => bool list"
22 SolveSystemScript :: "[bool list, real list, bool list]
24 ("((Script SolveSystemScript (_ _ =))// (_))" 9)
26 axioms(*axiomatization where *)
27 (*stated as axioms, todo: prove as theorems
28 'bdv' is a constant handled on the meta-level
29 specifically as a 'bound variable' *)
31 commute_0_equality: "(0 = a) = (a = 0)" (*and*)
33 (*WN0510 see simliar rules 'isolate_' 'separate_' (by RL)
34 [bdv_1,bdv_2,bdv_3,bdv_4] work also for 2 and 3 bdvs, ugly !*)
36 "[| [] from [bdv_1,bdv_2,bdv_3,bdv_4] occur_exactly_in a |]
37 ==> (a + b = c) = (b = c + -1*a)" (*and*)
39 "[| some_of [bdv_1,bdv_2,bdv_3,bdv_4] occur_in b; Not (b=!=0) |]
40 ==> (a = b) = (a + -1*b = 0)" (*and*)
42 "[| some_of [bdv_1,bdv_2,bdv_3,bdv_4] occur_in c |]
43 ==> (a = b + c) = (a + -1*c = b)" (*and*)
45 "[| Not (some_of [bdv_1,bdv_2,bdv_3,bdv_4] occur_in a) |]
46 ==> (a + b = c) = (b = -1*a + c)" (*and*)
48 "[| [] from [bdv_1,bdv_2,bdv_3,bdv_4] occur_exactly_in a; Not (a=!=0) |]
49 ==>(a * b = c) = (b = c / a)"
50 axioms(*axiomatization where*) (*..if replaced by "and" we get an error in
51 --- rewrite in [EqSystem,normalize,2x2] --- step "--- 3---";*)
52 order_system_NxN: "[a,b] = [b,a]"
53 (*requires rew_ord for termination, eg. ord_simplify_Integral;
54 works for lists of any length, interestingly !?!*)
59 (** eval functions **)
61 (*certain variables of a given list occur _all_ in a term
62 args: all: ..variables, which are under consideration (eg. the bound vars)
63 vs: variables which must be in t,
64 and none of the others in all must be in t
65 t: the term under consideration
67 fun occur_exactly_in vs all t =
68 let fun occurs_in' a b = occurs_in b a
69 in foldl and_ (true, map (occurs_in' t) vs)
70 andalso not (foldl or_ (false, map (occurs_in' t)
71 (subtract op = vs all)))
74 (*("occur_exactly_in", ("EqSystem.occur'_exactly'_in",
75 eval_occur_exactly_in "#eval_occur_exactly_in_"))*)
76 fun eval_occur_exactly_in _ "EqSystem.occur'_exactly'_in"
77 (p as (Const ("EqSystem.occur'_exactly'_in",_)
79 if occur_exactly_in (isalist2list vs) (isalist2list all) t
80 then SOME ((term2str p) ^ " = True",
81 Trueprop $ (mk_equality (p, @{term True})))
82 else SOME ((term2str p) ^ " = False",
83 Trueprop $ (mk_equality (p, @{term False})))
84 | eval_occur_exactly_in _ _ _ _ = NONE;
87 overwritel (!calclist',
89 ("EqSystem.occur'_exactly'_in",
90 eval_occur_exactly_in "#eval_occur_exactly_in_"))
94 (** rewrite order 'ord_simplify_System' **)
96 (* order wrt. several linear (i.e. without exponents) variables "c","c_2",..
97 which leaves the monomials containing c, c_2,... at the end of an Integral
98 and puts the c, c_2,... rightmost within a monomial.
100 WN050906 this is a quick and dirty adaption of ord_make_polynomial_in,
101 which was most adequate, because it uses size_of_term*)
103 local (*. for simplify_System .*)
105 open Term; (* for type order = EQUAL | LESS | GREATER *)
107 fun pr_ord EQUAL = "EQUAL"
108 | pr_ord LESS = "LESS"
109 | pr_ord GREATER = "GREATER";
111 fun dest_hd' (Const (a, T)) = (((a, 0), T), 0)
112 | dest_hd' (Free (ccc, T)) =
113 (case Symbol.explode ccc of
114 "c"::[] => ((("|||||||||||||||||||||", 0), T), 1)(*greatest string WN*)
115 | "c"::"_"::_ => ((("|||||||||||||||||||||", 0), T), 1)
116 | _ => (((ccc, 0), T), 1))
117 | dest_hd' (Var v) = (v, 2)
118 | dest_hd' (Bound i) = ((("", i), dummyT), 3)
119 | dest_hd' (Abs (_, T, _)) = ((("", 0), T), 4);
121 fun size_of_term' (Free (ccc, _)) =
122 (case Symbol.explode ccc of (*WN0510 hack for the bound variables*)
124 | "c"::"_"::is => 1000 * ((str2int o implode) is)
126 | size_of_term' (Abs (_,_,body)) = 1 + size_of_term' body
127 | size_of_term' (f$t) = size_of_term' f + size_of_term' t
128 | size_of_term' _ = 1;
130 fun term_ord' pr thy (Abs (_, T, t), Abs(_, U, u)) = (* ~ term.ML *)
131 (case term_ord' pr thy (t, u) of EQUAL => Term_Ord.typ_ord (T, U) | ord => ord)
132 | term_ord' pr thy (t, u) =
136 val (f, ts) = strip_comb t and (g, us) = strip_comb u;
137 val _ = tracing ("t= f@ts= \"" ^ term_to_string''' thy f ^ "\" @ \"[" ^
138 commas (map (term_to_string''' thy) ts) ^ "]\"");
139 val _ = tracing ("u= g@us= \"" ^ term_to_string''' thy g ^ "\" @ \"[" ^
140 commas (map (term_to_string''' thy) us) ^ "]\"");
141 val _ = tracing ("size_of_term(t,u)= (" ^ string_of_int (size_of_term' t) ^ ", " ^
142 string_of_int (size_of_term' u) ^ ")");
143 val _ = tracing ("hd_ord(f,g) = " ^ ((pr_ord o hd_ord) (f,g)));
144 val _ = tracing ("terms_ord (ts,us) = " ^(pr_ord o terms_ord str false) (ts,us));
145 val _=tracing("-------");
148 case int_ord (size_of_term' t, size_of_term' u) of
150 let val (f, ts) = strip_comb t and (g, us) = strip_comb u
151 in (case hd_ord (f, g) of
152 EQUAL => (terms_ord str pr) (ts, us)
156 and hd_ord (f, g) = (* ~ term.ML *)
157 prod_ord (prod_ord Term_Ord.indexname_ord Term_Ord.typ_ord) int_ord (dest_hd' f, dest_hd' g)
158 and terms_ord str pr (ts, us) = list_ord (term_ord' pr (assoc_thy "Isac"))(ts, us);
162 (*WN0510 for preliminary use in eval_order_system, see case-study mat-eng.tex
163 fun ord_simplify_System_rev (pr:bool) thy subst tu =
164 (term_ord' pr thy (Library.swap tu) = LESS);*)
167 fun ord_simplify_System (pr:bool) thy subst tu =
168 (term_ord' pr thy tu = LESS);
172 rew_ord' := overwritel (!rew_ord',
173 [("ord_simplify_System", ord_simplify_System false thy)
179 (*.adapted from 'order_add_mult_in' by just replacing the rew_ord.*)
180 val order_add_mult_System =
181 Rls{id = "order_add_mult_System", preconds = [],
182 rew_ord = ("ord_simplify_System",
183 ord_simplify_System false @{theory "Integrate"}),
184 erls = e_rls,srls = Erls, calc = [], errpatts = [],
185 rules = [Thm ("mult_commute",num_str @{thm mult_commute}),
187 Thm ("real_mult_left_commute",num_str @{thm real_mult_left_commute}),
188 (*z1.0 * (z2.0 * z3.0) = z2.0 * (z1.0 * z3.0)*)
189 Thm ("mult_assoc",num_str @{thm mult_assoc}),
190 (*z1.0 * z2.0 * z3.0 = z1.0 * (z2.0 * z3.0)*)
191 Thm ("add_commute",num_str @{thm add_commute}),
193 Thm ("add_left_commute",num_str @{thm add_left_commute}),
194 (*x + (y + z) = y + (x + z)*)
195 Thm ("add_assoc",num_str @{thm add_assoc})
196 (*z1.0 + z2.0 + z3.0 = z1.0 + (z2.0 + z3.0)*)
201 (*.adapted from 'norm_Rational' by
202 #1 using 'ord_simplify_System' in 'order_add_mult_System'
203 #2 NOT using common_nominator_p .*)
204 val norm_System_noadd_fractions =
205 Rls {id = "norm_System_noadd_fractions", preconds = [],
206 rew_ord = ("dummy_ord",dummy_ord),
207 erls = norm_rat_erls, srls = Erls, calc = [], errpatts = [],
208 rules = [(*sequence given by operator precedence*)
211 Rls_ rat_mult_divide,
214 Rls_ (*order_add_mult #1*) order_add_mult_System,
215 Rls_ collect_numerals,
216 (*Rls_ add_fractions_p, #2*)
219 scr = Prog ((term_of o the o (parse thy)) "empty_script")
223 (*.adapted from 'norm_Rational' by
224 *1* using 'ord_simplify_System' in 'order_add_mult_System'.*)
226 Rls {id = "norm_System", preconds = [],
227 rew_ord = ("dummy_ord",dummy_ord),
228 erls = norm_rat_erls, srls = Erls, calc = [], errpatts = [],
229 rules = [(*sequence given by operator precedence*)
232 Rls_ rat_mult_divide,
235 Rls_ (*order_add_mult *1*) order_add_mult_System,
236 Rls_ collect_numerals,
237 Rls_ add_fractions_p,
240 scr = Prog ((term_of o the o (parse thy)) "empty_script")
244 (*.simplify an equational system BEFORE solving it such that parentheses are
245 ( ((u0*v0)*w0) + ( ((u1*v1)*w1) * c + ... +((u4*v4)*w4) * c_4 ) )
246 ATTENTION: works ONLY for bound variables c, c_1, c_2, c_3, c_4 :ATTENTION
247 This is a copy from 'make_ratpoly_in' with respective reductions:
248 *0* expand the term, ie. distribute * and / over +
249 *1* ord_simplify_System instead of termlessI
250 *2* no add_fractions_p (= common_nominator_p_rls !)
251 *3* discard_parentheses only for (.*(.*.))
252 analoguous to simplify_Integral .*)
253 val simplify_System_parenthesized =
254 Seq {id = "simplify_System_parenthesized", preconds = []:term list,
255 rew_ord = ("dummy_ord", dummy_ord),
256 erls = Atools_erls, srls = Erls, calc = [], errpatts = [],
257 rules = [Thm ("distrib_right",num_str @{thm distrib_right}),
258 (*"(?z1.0 + ?z2.0) * ?w = ?z1.0 * ?w + ?z2.0 * ?w"*)
259 Thm ("add_divide_distrib",num_str @{thm add_divide_distrib}),
260 (*"(?x + ?y) / ?z = ?x / ?z + ?y / ?z"*)
261 (*^^^^^ *0* ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^*)
262 Rls_ norm_Rational_noadd_fractions(**2**),
263 Rls_ (*order_add_mult_in*) norm_System_noadd_fractions (**1**),
264 Thm ("sym_mult_assoc",
265 num_str (@{thm mult_assoc} RS @{thm sym}))
266 (*Rls_ discard_parentheses *3**),
267 Rls_ collect_bdv, (*from make_polynomial_in WN051031 welldone?*)
269 Calc ("Fields.inverse_class.divide" ,eval_cancel "#divide_e")
274 (*.simplify an equational system AFTER solving it;
275 This is a copy of 'make_ratpoly_in' with the differences
276 *1* ord_simplify_System instead of termlessI .*)
277 (*TODO.WN051031 ^^^^^^^^^^ should be in EACH rls contained *)
278 val simplify_System =
279 Seq {id = "simplify_System", preconds = []:term list,
280 rew_ord = ("dummy_ord", dummy_ord),
281 erls = Atools_erls, srls = Erls, calc = [], errpatts = [],
282 rules = [Rls_ norm_Rational,
283 Rls_ (*order_add_mult_in*) norm_System (**1**),
284 Rls_ discard_parentheses,
285 Rls_ collect_bdv, (*from make_polynomial_in WN051031 welldone?*)
287 Calc ("Fields.inverse_class.divide" ,eval_cancel "#divide_e")
291 val simplify_System =
292 append_rls "simplify_System" simplify_System_parenthesized
293 [Thm ("sym_add_assoc",
294 num_str (@{thm add_assoc} RS @{thm sym}))];
299 Rls {id="isolate_bdvs", preconds = [],
300 rew_ord = ("e_rew_ord", e_rew_ord),
301 erls = append_rls "erls_isolate_bdvs" e_rls
302 [(Calc ("EqSystem.occur'_exactly'_in",
303 eval_occur_exactly_in
304 "#eval_occur_exactly_in_"))
306 srls = Erls, calc = [], errpatts = [],
308 [Thm ("commute_0_equality", num_str @{thm commute_0_equality}),
309 Thm ("separate_bdvs_add", num_str @{thm separate_bdvs_add}),
310 Thm ("separate_bdvs_mult", num_str @{thm separate_bdvs_mult})],
314 val isolate_bdvs_4x4 =
315 Rls {id="isolate_bdvs_4x4", preconds = [],
316 rew_ord = ("e_rew_ord", e_rew_ord),
318 "erls_isolate_bdvs_4x4" e_rls
319 [Calc ("EqSystem.occur'_exactly'_in",
320 eval_occur_exactly_in "#eval_occur_exactly_in_"),
321 Calc ("Atools.ident",eval_ident "#ident_"),
322 Calc ("Atools.some'_occur'_in",
323 eval_some_occur_in "#some_occur_in_"),
324 Thm ("not_true",num_str @{thm not_true}),
325 Thm ("not_false",num_str @{thm not_false})
327 srls = Erls, calc = [], errpatts = [],
328 rules = [Thm ("commute_0_equality", num_str @{thm commute_0_equality}),
329 Thm ("separate_bdvs0", num_str @{thm separate_bdvs0}),
330 Thm ("separate_bdvs_add1", num_str @{thm separate_bdvs_add1}),
331 Thm ("separate_bdvs_add1", num_str @{thm separate_bdvs_add2}),
332 Thm ("separate_bdvs_mult", num_str @{thm separate_bdvs_mult})
338 (*.order the equations in a system such, that a triangular system (if any)
339 appears as [..c_4 = .., ..., ..., ..c_1 + ..c_2 + ..c_3 ..c_4 = ..].*)
341 Rls {id="order_system", preconds = [],
342 rew_ord = ("ord_simplify_System",
343 ord_simplify_System false thy),
344 erls = Erls, srls = Erls, calc = [], errpatts = [],
345 rules = [Thm ("order_system_NxN", num_str @{thm order_system_NxN})
349 val prls_triangular =
350 Rls {id="prls_triangular", preconds = [],
351 rew_ord = ("e_rew_ord", e_rew_ord),
352 erls = Rls {id="erls_prls_triangular", preconds = [],
353 rew_ord = ("e_rew_ord", e_rew_ord),
354 erls = Erls, srls = Erls, calc = [], errpatts = [],
355 rules = [(*for precond NTH_CONS ...*)
356 Calc ("Orderings.ord_class.less",eval_equ "#less_"),
357 Calc ("Groups.plus_class.plus", eval_binop "#add_")
358 (*immediately repeated rewrite pushes
359 '+' into precondition !*)
362 srls = Erls, calc = [], errpatts = [],
363 rules = [Thm ("NTH_CONS",num_str @{thm NTH_CONS}),
364 Calc ("Groups.plus_class.plus", eval_binop "#add_"),
365 Thm ("NTH_NIL",num_str @{thm NTH_NIL}),
366 Thm ("tl_Cons",num_str @{thm tl_Cons}),
367 Thm ("tl_Nil",num_str @{thm tl_Nil}),
368 Calc ("EqSystem.occur'_exactly'_in",
369 eval_occur_exactly_in
370 "#eval_occur_exactly_in_")
376 (*WN060914 quickly created for 4x4;
377 more similarity to prls_triangular desirable*)
378 val prls_triangular4 =
379 Rls {id="prls_triangular4", preconds = [],
380 rew_ord = ("e_rew_ord", e_rew_ord),
381 erls = Rls {id="erls_prls_triangular4", preconds = [],
382 rew_ord = ("e_rew_ord", e_rew_ord),
383 erls = Erls, srls = Erls, calc = [], errpatts = [],
384 rules = [(*for precond NTH_CONS ...*)
385 Calc ("Orderings.ord_class.less",eval_equ "#less_"),
386 Calc ("Groups.plus_class.plus", eval_binop "#add_")
387 (*immediately repeated rewrite pushes
388 '+' into precondition !*)
391 srls = Erls, calc = [], errpatts = [],
392 rules = [Thm ("NTH_CONS",num_str @{thm NTH_CONS}),
393 Calc ("Groups.plus_class.plus", eval_binop "#add_"),
394 Thm ("NTH_NIL",num_str @{thm NTH_NIL}),
395 Thm ("tl_Cons",num_str @{thm tl_Cons}),
396 Thm ("tl_Nil",num_str @{thm tl_Nil}),
397 Calc ("EqSystem.occur'_exactly'_in",
398 eval_occur_exactly_in
399 "#eval_occur_exactly_in_")
406 overwritelthy @{theory}
408 [("simplify_System_parenthesized", prep_rls simplify_System_parenthesized),
409 ("simplify_System", prep_rls simplify_System),
410 ("isolate_bdvs", prep_rls isolate_bdvs),
411 ("isolate_bdvs_4x4", prep_rls isolate_bdvs_4x4),
412 ("order_system", prep_rls order_system),
413 ("order_add_mult_System", prep_rls order_add_mult_System),
414 ("norm_System_noadd_fractions", prep_rls norm_System_noadd_fractions),
415 ("norm_System", prep_rls norm_System)
423 (prep_pbt thy "pbl_equsys" [] e_pblID
425 [("#Given" ,["equalities e_s", "solveForVars v_s"]),
426 ("#Find" ,["solution ss'''"](*''' is copy-named*))
428 append_rls "e_rls" e_rls [(*for preds in where_*)],
429 SOME "solveSystem e_s v_s",
432 (prep_pbt thy "pbl_equsys_lin" [] e_pblID
433 (["linear", "system"],
434 [("#Given" ,["equalities e_s", "solveForVars v_s"]),
435 (*TODO.WN050929 check linearity*)
436 ("#Find" ,["solution ss'''"])
438 append_rls "e_rls" e_rls [(*for preds in where_*)],
439 SOME "solveSystem e_s v_s",
442 (prep_pbt thy "pbl_equsys_lin_2x2" [] e_pblID
443 (["2x2", "linear", "system"],
444 (*~~~~~~~~~~~~~~~~~~~~~~~~~*)
445 [("#Given" ,["equalities e_s", "solveForVars v_s"]),
446 ("#Where" ,["LENGTH (e_s:: bool list) = 2", "LENGTH v_s = 2"]),
447 ("#Find" ,["solution ss'''"])
449 append_rls "prls_2x2_linear_system" e_rls
450 [Thm ("LENGTH_CONS",num_str @{thm LENGTH_CONS}),
451 Thm ("LENGTH_NIL",num_str @{thm LENGTH_NIL}),
452 Calc ("Groups.plus_class.plus", eval_binop "#add_"),
453 Calc ("HOL.eq",eval_equal "#equal_")
455 SOME "solveSystem e_s v_s",
460 (prep_pbt thy "pbl_equsys_lin_2x2_tri" [] e_pblID
461 (["triangular", "2x2", "linear", "system"],
462 [("#Given" ,["equalities e_s", "solveForVars v_s"]),
464 ["(tl v_s) from v_s occur_exactly_in (NTH 1 (e_s::bool list))",
465 " v_s from v_s occur_exactly_in (NTH 2 (e_s::bool list))"]),
466 ("#Find" ,["solution ss'''"])
469 SOME "solveSystem e_s v_s",
470 [["EqSystem","top_down_substitution","2x2"]]));
472 (prep_pbt thy "pbl_equsys_lin_2x2_norm" [] e_pblID
473 (["normalize", "2x2", "linear", "system"],
474 [("#Given" ,["equalities e_s", "solveForVars v_s"]),
475 ("#Find" ,["solution ss'''"])
477 append_rls "e_rls" e_rls [(*for preds in where_*)],
478 SOME "solveSystem e_s v_s",
479 [["EqSystem","normalize","2x2"]]));
481 (prep_pbt thy "pbl_equsys_lin_3x3" [] e_pblID
482 (["3x3", "linear", "system"],
483 (*~~~~~~~~~~~~~~~~~~~~~~~~~*)
484 [("#Given" ,["equalities e_s", "solveForVars v_s"]),
485 ("#Where" ,["LENGTH (e_s:: bool list) = 3", "LENGTH v_s = 3"]),
486 ("#Find" ,["solution ss'''"])
488 append_rls "prls_3x3_linear_system" e_rls
489 [Thm ("LENGTH_CONS",num_str @{thm LENGTH_CONS}),
490 Thm ("LENGTH_NIL",num_str @{thm LENGTH_NIL}),
491 Calc ("Groups.plus_class.plus", eval_binop "#add_"),
492 Calc ("HOL.eq",eval_equal "#equal_")
494 SOME "solveSystem e_s v_s",
497 (prep_pbt thy "pbl_equsys_lin_4x4" [] e_pblID
498 (["4x4", "linear", "system"],
499 (*~~~~~~~~~~~~~~~~~~~~~~~~~*)
500 [("#Given" ,["equalities e_s", "solveForVars v_s"]),
501 ("#Where" ,["LENGTH (e_s:: bool list) = 4", "LENGTH v_s = 4"]),
502 ("#Find" ,["solution ss'''"])
504 append_rls "prls_4x4_linear_system" e_rls
505 [Thm ("LENGTH_CONS",num_str @{thm LENGTH_CONS}),
506 Thm ("LENGTH_NIL",num_str @{thm LENGTH_NIL}),
507 Calc ("Groups.plus_class.plus", eval_binop "#add_"),
508 Calc ("HOL.eq",eval_equal "#equal_")
510 SOME "solveSystem e_s v_s",
515 (prep_pbt thy "pbl_equsys_lin_4x4_tri" [] e_pblID
516 (["triangular", "4x4", "linear", "system"],
517 [("#Given" ,["equalities e_s", "solveForVars v_s"]),
518 ("#Where" , (*accepts missing variables up to diagional form*)
519 ["(NTH 1 (v_s::real list)) occurs_in (NTH 1 (e_s::bool list))",
520 "(NTH 2 (v_s::real list)) occurs_in (NTH 2 (e_s::bool list))",
521 "(NTH 3 (v_s::real list)) occurs_in (NTH 3 (e_s::bool list))",
522 "(NTH 4 (v_s::real list)) occurs_in (NTH 4 (e_s::bool list))"
524 ("#Find" ,["solution ss'''"])
526 append_rls "prls_tri_4x4_lin_sys" prls_triangular
527 [Calc ("Atools.occurs'_in",eval_occurs_in "")],
528 SOME "solveSystem e_s v_s",
529 [["EqSystem","top_down_substitution","4x4"]]));
533 (prep_pbt thy "pbl_equsys_lin_4x4_norm" [] e_pblID
534 (["normalize", "4x4", "linear", "system"],
535 [("#Given" ,["equalities e_s", "solveForVars v_s"]),
536 (*LENGTH is checked 1 level above*)
537 ("#Find" ,["solution ss'''"])
539 append_rls "e_rls" e_rls [(*for preds in where_*)],
540 SOME "solveSystem e_s v_s",
541 [["EqSystem","normalize","4x4"]]));
552 (prep_met thy "met_eqsys" [] e_metID
555 {rew_ord'="tless_true", rls' = Erls, calc = [],
556 srls = Erls, prls = Erls, crls = Erls, errpats = [], nrls = Erls},
560 (prep_met thy "met_eqsys_topdown" [] e_metID
561 (["EqSystem","top_down_substitution"],
563 {rew_ord'="tless_true", rls' = Erls, calc = [],
564 srls = Erls, prls = Erls, crls = Erls, errpats = [], nrls = Erls},
570 (prep_met thy "met_eqsys_topdown_2x2" [] e_metID
571 (["EqSystem","top_down_substitution","2x2"],
572 [("#Given" ,["equalities e_s", "solveForVars v_s"]),
574 ["(tl v_s) from v_s occur_exactly_in (NTH 1 (e_s::bool list))",
575 " v_s from v_s occur_exactly_in (NTH 2 (e_s::bool list))"]),
576 ("#Find" ,["solution ss'''"])
578 {rew_ord'="ord_simplify_System", rls' = Erls, calc = [],
579 srls = append_rls "srls_top_down_2x2" e_rls
580 [Thm ("hd_thm",num_str @{thm hd_thm}),
581 Thm ("tl_Cons",num_str @{thm tl_Cons}),
582 Thm ("tl_Nil",num_str @{thm tl_Nil})
584 prls = prls_triangular, crls = Erls, errpats = [], nrls = Erls},
585 "Script SolveSystemScript (e_s::bool list) (v_s::real list) = " ^
586 " (let e_1 = Take (hd e_s); " ^
587 " e_1 = ((Try (Rewrite_Set_Inst [(bdv_1, hd v_s),(bdv_2, hd (tl v_s))]" ^
588 " isolate_bdvs False)) @@ " ^
589 " (Try (Rewrite_Set_Inst [(bdv_1, hd v_s),(bdv_2, hd (tl v_s))]" ^
590 " simplify_System False))) e_1; " ^
591 " e_2 = Take (hd (tl e_s)); " ^
592 " e_2 = ((Substitute [e_1]) @@ " ^
593 " (Try (Rewrite_Set_Inst [(bdv_1, hd v_s),(bdv_2, hd (tl v_s))]" ^
594 " simplify_System_parenthesized False)) @@ " ^
595 " (Try (Rewrite_Set_Inst [(bdv_1, hd v_s),(bdv_2, hd (tl v_s))]" ^
596 " isolate_bdvs False)) @@ " ^
597 " (Try (Rewrite_Set_Inst [(bdv_1, hd v_s),(bdv_2, hd (tl v_s))]" ^
598 " simplify_System False))) e_2; " ^
599 " e__s = Take [e_1, e_2] " ^
600 " in (Try (Rewrite_Set order_system False)) e__s)"
601 (*---------------------------------------------------------------------------
602 this script does NOT separate the equations as above,
603 but it does not yet work due to preliminary script-interpreter,
604 see eqsystem.sml 'script [EqSystem,top_down_substitution,2x2] Vers.2'
606 "Script SolveSystemScript (e_s::bool list) (v_s::real list) = " ^
607 " (let e__s = Take e_s; " ^
609 " e_2 = hd (tl e__s); " ^
610 " e__s = [e_1, Substitute [e_1] e_2] " ^
611 " in ((Try (Rewrite_Set_Inst [(bdv_1, hd v_s),(bdv_2, hd (tl v_s))]" ^
612 " simplify_System_parenthesized False)) @@ " ^
613 " (Try (Rewrite_Set_Inst [(bdv_1, hd v_s),(bdv_2, hd (tl v_s))] " ^
614 " isolate_bdvs False)) @@ " ^
615 " (Try (Rewrite_Set_Inst [(bdv_1, hd v_s),(bdv_2, hd (tl v_s))]" ^
616 " simplify_System False))) e__s)"
617 ---------------------------------------------------------------------------*)
622 (prep_met thy "met_eqsys_norm" [] e_metID
623 (["EqSystem","normalize"],
625 {rew_ord'="tless_true", rls' = Erls, calc = [],
626 srls = Erls, prls = Erls, crls = Erls, errpats = [], nrls = Erls},
632 (prep_met thy "met_eqsys_norm_2x2" [] e_metID
633 (["EqSystem","normalize","2x2"],
634 [("#Given" ,["equalities e_s", "solveForVars v_s"]),
635 ("#Find" ,["solution ss'''"])],
636 {rew_ord'="tless_true", rls' = Erls, calc = [],
637 srls = append_rls "srls_normalize_2x2" e_rls
638 [Thm ("hd_thm",num_str @{thm hd_thm}),
639 Thm ("tl_Cons",num_str @{thm tl_Cons}),
640 Thm ("tl_Nil",num_str @{thm tl_Nil})
642 prls = Erls, crls = Erls, errpats = [], nrls = Erls},
643 "Script SolveSystemScript (e_s::bool list) (v_s::real list) = " ^
644 " (let e__s = ((Try (Rewrite_Set norm_Rational False)) @@ " ^
645 " (Try (Rewrite_Set_Inst [(bdv_1, hd v_s),(bdv_2, hd (tl v_s))]" ^
646 " simplify_System_parenthesized False)) @@ " ^
647 " (Try (Rewrite_Set_Inst [(bdv_1, hd v_s),(bdv_2, hd (tl v_s))]" ^
648 " isolate_bdvs False)) @@ " ^
649 " (Try (Rewrite_Set_Inst [(bdv_1, hd v_s),(bdv_2, hd (tl v_s))]" ^
650 " simplify_System_parenthesized False)) @@ " ^
651 " (Try (Rewrite_Set order_system False))) e_s " ^
652 " in (SubProblem (EqSystem',[linear,system],[no_met]) " ^
653 " [BOOL_LIST e__s, REAL_LIST v_s]))"
658 (*this is for NTH only*)
659 val srls = Rls {id="srls_normalize_4x4",
661 rew_ord = ("termlessI",termlessI),
662 erls = append_rls "erls_in_srls_IntegrierenUnd.." e_rls
663 [(*for asm in NTH_CONS ...*)
664 Calc ("Orderings.ord_class.less",eval_equ "#less_"),
665 (*2nd NTH_CONS pushes n+-1 into asms*)
666 Calc("Groups.plus_class.plus", eval_binop "#add_")
668 srls = Erls, calc = [], errpatts = [],
669 rules = [Thm ("NTH_CONS",num_str @{thm NTH_CONS}),
670 Calc("Groups.plus_class.plus", eval_binop "#add_"),
671 Thm ("NTH_NIL",num_str @{thm NTH_NIL})],
675 (prep_met thy "met_eqsys_norm_4x4" [] e_metID
676 (["EqSystem","normalize","4x4"],
677 [("#Given" ,["equalities e_s", "solveForVars v_s"]),
678 ("#Find" ,["solution ss'''"])],
679 {rew_ord'="tless_true", rls' = Erls, calc = [],
680 srls = append_rls "srls_normalize_4x4" srls
681 [Thm ("hd_thm",num_str @{thm hd_thm}),
682 Thm ("tl_Cons",num_str @{thm tl_Cons}),
683 Thm ("tl_Nil",num_str @{thm tl_Nil})
685 prls = Erls, crls = Erls, errpats = [], nrls = Erls},
686 (*STOPPED.WN06? met ["EqSystem","normalize","4x4"] @@@@@@@@@@@@@@@@@@@@@@@@@@@*)
687 "Script SolveSystemScript (e_s::bool list) (v_s::real list) = " ^
689 " ((Try (Rewrite_Set norm_Rational False)) @@ " ^
690 " (Repeat (Rewrite commute_0_equality False)) @@ " ^
691 " (Try (Rewrite_Set_Inst [(bdv_1, NTH 1 v_s),(bdv_2, NTH 2 v_s ), " ^
692 " (bdv_3, NTH 3 v_s),(bdv_3, NTH 4 v_s )] " ^
693 " simplify_System_parenthesized False)) @@ " ^
694 " (Try (Rewrite_Set_Inst [(bdv_1, NTH 1 v_s),(bdv_2, NTH 2 v_s ), " ^
695 " (bdv_3, NTH 3 v_s),(bdv_3, NTH 4 v_s )] " ^
696 " isolate_bdvs_4x4 False)) @@ " ^
697 " (Try (Rewrite_Set_Inst [(bdv_1, NTH 1 v_s),(bdv_2, NTH 2 v_s ), " ^
698 " (bdv_3, NTH 3 v_s),(bdv_3, NTH 4 v_s )] " ^
699 " simplify_System_parenthesized False)) @@ " ^
700 " (Try (Rewrite_Set order_system False))) e_s " ^
701 " in (SubProblem (EqSystem',[linear,system],[no_met]) " ^
702 " [BOOL_LIST e__s, REAL_LIST v_s]))"
706 (prep_met thy "met_eqsys_topdown_4x4" [] e_metID
707 (["EqSystem","top_down_substitution","4x4"],
708 [("#Given" ,["equalities e_s", "solveForVars v_s"]),
709 ("#Where" , (*accepts missing variables up to diagonal form*)
710 ["(NTH 1 (v_s::real list)) occurs_in (NTH 1 (e_s::bool list))",
711 "(NTH 2 (v_s::real list)) occurs_in (NTH 2 (e_s::bool list))",
712 "(NTH 3 (v_s::real list)) occurs_in (NTH 3 (e_s::bool list))",
713 "(NTH 4 (v_s::real list)) occurs_in (NTH 4 (e_s::bool list))"
715 ("#Find" ,["solution ss'''"])
717 {rew_ord'="ord_simplify_System", rls' = Erls, calc = [],
718 srls = append_rls "srls_top_down_4x4" srls [],
719 prls = append_rls "prls_tri_4x4_lin_sys" prls_triangular
720 [Calc ("Atools.occurs'_in",eval_occurs_in "")],
721 crls = Erls, errpats = [], nrls = Erls},
722 (*FIXXXXME.WN060916: this script works ONLY for exp 7.79 @@@@@@@@@@@@@@@@@@@@*)
723 "Script SolveSystemScript (e_s::bool list) (v_s::real list) = " ^
724 " (let e_1 = NTH 1 e_s; " ^
725 " e_2 = Take (NTH 2 e_s); " ^
726 " e_2 = ((Substitute [e_1]) @@ " ^
727 " (Try (Rewrite_Set_Inst [(bdv_1,NTH 1 v_s),(bdv_2,NTH 2 v_s)," ^
728 " (bdv_3,NTH 3 v_s),(bdv_4,NTH 4 v_s)]" ^
729 " simplify_System_parenthesized False)) @@ " ^
730 " (Try (Rewrite_Set_Inst [(bdv_1,NTH 1 v_s),(bdv_2,NTH 2 v_s)," ^
731 " (bdv_3,NTH 3 v_s),(bdv_4,NTH 4 v_s)]" ^
732 " isolate_bdvs False)) @@ " ^
733 " (Try (Rewrite_Set_Inst [(bdv_1,NTH 1 v_s),(bdv_2,NTH 2 v_s)," ^
734 " (bdv_3,NTH 3 v_s),(bdv_4,NTH 4 v_s)]" ^
735 " norm_Rational False))) e_2 " ^
736 " in [e_1, e_2, NTH 3 e_s, NTH 4 e_s])"