1 (* equational systems, minimal -- for use in Biegelinie
4 (c) due to copyright terms
7 theory EqSystem imports 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)
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)"
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)"
39 "[| some_of [bdv_1,bdv_2,bdv_3,bdv_4] occur_in b; Not (b=!=0) |]
40 ==> (a = b) = (a + -1*b = 0)"
42 "[| some_of [bdv_1,bdv_2,bdv_3,bdv_4] occur_in c |]
43 ==> (a = b + c) = (a + -1*c = b)"
45 "[| Not (some_of [bdv_1,bdv_2,bdv_3,bdv_4] occur_in a) |]
46 ==> (a + b = c) = (b = -1*a + c)"
51 "[| [] from_ [bdv_1,bdv_2,bdv_3,bdv_4] occur_exactly_in a; Not (a=!=0) |]
52 ==>(a * b = c) = (b = c / a)"
54 (*requires rew_ord for termination, eg. ord_simplify_Integral;
55 works for lists of any length, interestingly !?!*)
56 order_system_NxN: "[a,b] = [b,a]"
61 (** eval functions **)
63 (*certain variables of a given list occur _all_ in a term
64 args: all: ..variables, which are under consideration (eg. the bound vars)
65 vs: variables which must be in t,
66 and none of the others in all must be in t
67 t: the term under consideration
69 fun occur_exactly_in vs all t =
70 let fun occurs_in' a b = occurs_in b a
71 in foldl and_ (true, map (occurs_in' t) vs)
72 andalso not (foldl or_ (false, map (occurs_in' t)
73 (subtract op = vs all)))
76 (*("occur_exactly_in", ("EqSystem.occur'_exactly'_in",
77 eval_occur_exactly_in "#eval_occur_exactly_in_"))*)
78 fun eval_occur_exactly_in _ "EqSystem.occur'_exactly'_in"
79 (p as (Const ("EqSystem.occur'_exactly'_in",_)
81 if occur_exactly_in (isalist2list vs) (isalist2list all) t
82 then SOME ((term2str p) ^ " = True",
83 Trueprop $ (mk_equality (p, HOLogic.true_const)))
84 else SOME ((term2str p) ^ " = False",
85 Trueprop $ (mk_equality (p, HOLogic.false_const)))
86 | eval_occur_exactly_in _ _ _ _ = NONE;
89 overwritel (!calclist',
91 ("EqSystem.occur'_exactly'_in",
92 eval_occur_exactly_in "#eval_occur_exactly_in_"))
96 (** rewrite order 'ord_simplify_System' **)
98 (* order wrt. several linear (i.e. without exponents) variables "c","c_2",..
99 which leaves the monomials containing c, c_2,... at the end of an Integral
100 and puts the c, c_2,... rightmost within a monomial.
102 WN050906 this is a quick and dirty adaption of ord_make_polynomial_in,
103 which was most adequate, because it uses size_of_term*)
105 local (*. for simplify_System .*)
107 open Term; (* for type order = EQUAL | LESS | GREATER *)
109 fun pr_ord EQUAL = "EQUAL"
110 | pr_ord LESS = "LESS"
111 | pr_ord GREATER = "GREATER";
113 fun dest_hd' (Const (a, T)) = (((a, 0), T), 0)
114 | dest_hd' (Free (ccc, T)) =
116 "c"::[] => ((("|||||||||||||||||||||", 0), T), 1)(*greatest string WN*)
117 | "c"::"_"::_ => ((("|||||||||||||||||||||", 0), T), 1)
118 | _ => (((ccc, 0), T), 1))
119 | dest_hd' (Var v) = (v, 2)
120 | dest_hd' (Bound i) = ((("", i), dummyT), 3)
121 | dest_hd' (Abs (_, T, _)) = ((("", 0), T), 4);
123 fun size_of_term' (Free (ccc, _)) =
124 (case explode ccc of (*WN0510 hack for the bound variables*)
126 | "c"::"_"::is => 1000 * ((str2int o implode) is)
128 | size_of_term' (Abs (_,_,body)) = 1 + size_of_term' body
129 | size_of_term' (f$t) = size_of_term' f + size_of_term' t
130 | size_of_term' _ = 1;
132 fun Term_Ord.term_ord' pr thy (Abs (_, T, t), Abs(_, U, u)) = (* ~ term.ML *)
133 (case Term_Ord.term_ord' pr thy (t, u) of EQUAL => Term_Ord.typ_ord (T, U) | ord => ord)
134 | Term_Ord.term_ord' pr thy (t, u) =
137 val (f, ts) = strip_comb t and (g, us) = strip_comb u;
138 val _=writeln("t= f@ts= \""^
139 ((Syntax.string_of_term (thy2ctxt thy)) f)^"\" @ \"["^
140 (commas(map(Syntax.string_of_term (thy2ctxt thy)) ts))^"]\"");
141 val _=writeln("u= g@us= \""^
142 ((Syntax.string_of_term (thy2ctxt thy)) g)^"\" @ \"["^
143 (commas(map(Syntax.string_of_term (thy2ctxt thy)) us))^"]\"");
144 val _=writeln("size_of_term(t,u)= ("^
145 (string_of_int(size_of_term' t))^", "^
146 (string_of_int(size_of_term' u))^")");
147 val _=writeln("hd_ord(f,g) = "^((pr_ord o hd_ord)(f,g)));
148 val _=writeln("terms_ord(ts,us) = "^
149 ((pr_ord o terms_ord str false)(ts,us)));
150 val _=writeln("-------");
153 case int_ord (size_of_term' t, size_of_term' u) of
155 let val (f, ts) = strip_comb t and (g, us) = strip_comb u in
156 (case hd_ord (f, g) of EQUAL => (terms_ord str pr) (ts, us)
160 and hd_ord (f, g) = (* ~ term.ML *)
161 prod_ord (prod_ord Term_Ord.indexname_ord Term_Ord.typ_ord) int_ord (dest_hd' f,
163 and terms_ord str pr (ts, us) =
164 list_ord (term_ord' pr (assoc_thy "Isac"))(ts, us);
168 (*WN0510 for preliminary use in eval_order_system, see case-study mat-eng.tex
169 fun ord_simplify_System_rev (pr:bool) thy subst tu =
170 (term_ord' pr thy (Library.swap tu) = LESS);*)
173 fun ord_simplify_System (pr:bool) thy subst tu =
174 (term_ord' pr thy tu = LESS);
178 rew_ord' := overwritel (!rew_ord',
179 [("ord_simplify_System", ord_simplify_System false thy)
185 (*.adapted from 'order_add_mult_in' by just replacing the rew_ord.*)
186 val order_add_mult_System =
187 Rls{id = "order_add_mult_System", preconds = [],
188 rew_ord = ("ord_simplify_System",
189 ord_simplify_System false (theory "Integrate")),
190 erls = e_rls,srls = Erls, calc = [],
191 rules = [Thm ("real_mult_commute",num_str @{thm real_mult_commute}),
193 Thm ("real_mult_left_commute",num_str @{thm real_mult_left_commute}),
194 (*z1.0 * (z2.0 * z3.0) = z2.0 * (z1.0 * z3.0)*)
195 Thm ("real_mult_assoc",num_str @{thm real_mult_assoc}),
196 (*z1.0 * z2.0 * z3.0 = z1.0 * (z2.0 * z3.0)*)
197 Thm ("add_commute",num_str @{thm add_commute}),
199 Thm ("add_left_commute",num_str @{thm add_left_commute}),
200 (*x + (y + z) = y + (x + z)*)
201 Thm ("add_assoc",num_str @{thm add_assoc})
202 (*z1.0 + z2.0 + z3.0 = z1.0 + (z2.0 + z3.0)*)
206 (*.adapted from 'norm_Rational' by
207 #1 using 'ord_simplify_System' in 'order_add_mult_System'
208 #2 NOT using common_nominator_p .*)
209 val norm_System_noadd_fractions =
210 Rls {id = "norm_System_noadd_fractions", preconds = [],
211 rew_ord = ("dummy_ord",dummy_ord),
212 erls = norm_rat_erls, srls = Erls, calc = [],
213 rules = [(*sequence given by operator precedence*)
216 Rls_ rat_mult_divide,
219 Rls_ (*order_add_mult #1*) order_add_mult_System,
220 Rls_ collect_numerals,
221 (*Rls_ add_fractions_p, #2*)
224 scr = Script ((term_of o the o (parse thy))
227 (*.adapted from 'norm_Rational' by
228 *1* using 'ord_simplify_System' in 'order_add_mult_System'.*)
230 Rls {id = "norm_System", preconds = [],
231 rew_ord = ("dummy_ord",dummy_ord),
232 erls = norm_rat_erls, srls = Erls, calc = [],
233 rules = [(*sequence given by operator precedence*)
236 Rls_ rat_mult_divide,
239 Rls_ (*order_add_mult *1*) order_add_mult_System,
240 Rls_ collect_numerals,
241 Rls_ add_fractions_p,
244 scr = Script ((term_of o the o (parse thy))
248 (*.simplify an equational system BEFORE solving it such that parentheses are
249 ( ((u0*v0)*w0) + ( ((u1*v1)*w1) * c + ... +((u4*v4)*w4) * c_4 ) )
250 ATTENTION: works ONLY for bound variables c, c_1, c_2, c_3, c_4 :ATTENTION
251 This is a copy from 'make_ratpoly_in' with respective reductions:
252 *0* expand the term, ie. distribute * and / over +
253 *1* ord_simplify_System instead of termlessI
254 *2* no add_fractions_p (= common_nominator_p_rls !)
255 *3* discard_parentheses only for (.*(.*.))
256 analoguous to simplify_Integral .*)
257 val simplify_System_parenthesized =
258 Seq {id = "simplify_System_parenthesized", preconds = []:term list,
259 rew_ord = ("dummy_ord", dummy_ord),
260 erls = Atools_erls, srls = Erls, calc = [],
261 rules = [Thm ("left_distrib",num_str @{thm left_distrib}),
262 (*"(?z1.0 + ?z2.0) * ?w = ?z1.0 * ?w + ?z2.0 * ?w"*)
263 Thm ("add_divide_distrib",num_str @{thm add_divide_distrib}),
264 (*"(?x + ?y) / ?z = ?x / ?z + ?y / ?z"*)
265 (*^^^^^ *0* ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^*)
266 Rls_ norm_Rational_noadd_fractions(**2**),
267 Rls_ (*order_add_mult_in*) norm_System_noadd_fractions (**1**),
268 Thm ("sym_real_mult_assoc",
269 num_str (@{thm real_mult_assoc} RS @{thm sym}))
270 (*Rls_ discard_parentheses *3**),
271 Rls_ collect_bdv, (*from make_polynomial_in WN051031 welldone?*)
273 Calc ("HOL.divide" ,eval_cancel "#divide_e")
277 (*.simplify an equational system AFTER solving it;
278 This is a copy of 'make_ratpoly_in' with the differences
279 *1* ord_simplify_System instead of termlessI .*)
280 (*TODO.WN051031 ^^^^^^^^^^ should be in EACH rls contained *)
281 val simplify_System =
282 Seq {id = "simplify_System", preconds = []:term list,
283 rew_ord = ("dummy_ord", dummy_ord),
284 erls = Atools_erls, srls = Erls, calc = [],
285 rules = [Rls_ norm_Rational,
286 Rls_ (*order_add_mult_in*) norm_System (**1**),
287 Rls_ discard_parentheses,
288 Rls_ collect_bdv, (*from make_polynomial_in WN051031 welldone?*)
290 Calc ("HOL.divide" ,eval_cancel "#divide_e")
294 val simplify_System =
295 append_rls "simplify_System" simplify_System_parenthesized
296 [Thm ("sym_add_assoc",
297 num_str (@{thm add_assoc} RS @{thm sym}))];
301 Rls {id="isolate_bdvs", preconds = [],
302 rew_ord = ("e_rew_ord", e_rew_ord),
303 erls = append_rls "erls_isolate_bdvs" e_rls
304 [(Calc ("EqSystem.occur'_exactly'_in",
305 eval_occur_exactly_in
306 "#eval_occur_exactly_in_"))
308 srls = Erls, calc = [],
309 rules = [Thm ("commute_0_equality",
310 num_str @{commute_0_equality),
311 Thm ("separate_bdvs_add", num_str @{thm separate_bdvs_add}),
312 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 = [],
328 rules = [Thm ("commute_0_equality",
329 num_str @{commute_0_equality),
330 Thm ("separate_bdvs0", num_str @{thm separate_bdvs0}),
331 Thm ("separate_bdvs_add1", num_str @{thm separate_bdvs_add1}),
332 Thm ("separate_bdvs_add1", num_str @{thm separate_bdvs_add2}),
333 Thm ("separate_bdvs_mult", num_str @{thm separate_bdvs_mult})
336 (*.order the equations in a system such, that a triangular system (if any)
337 appears as [..c_4 = .., ..., ..., ..c_1 + ..c_2 + ..c_3 ..c_4 = ..].*)
339 Rls {id="order_system", preconds = [],
340 rew_ord = ("ord_simplify_System",
341 ord_simplify_System false thy),
342 erls = Erls, srls = Erls, calc = [],
343 rules = [Thm ("order_system_NxN", num_str @{thm order_system_NxN})
347 val prls_triangular =
348 Rls {id="prls_triangular", preconds = [],
349 rew_ord = ("e_rew_ord", e_rew_ord),
350 erls = Rls {id="erls_prls_triangular", preconds = [],
351 rew_ord = ("e_rew_ord", e_rew_ord),
352 erls = Erls, srls = Erls, calc = [],
353 rules = [(*for precond nth_Cons_ ...*)
354 Calc ("op <",eval_equ "#less_"),
355 Calc ("op +", eval_binop "#add_")
356 (*immediately repeated rewrite pushes
357 '+' into precondition !*)
360 srls = Erls, calc = [],
361 rules = [Thm ("nth_Cons_",num_str @{thm nth_Cons_}),
362 Calc ("op +", eval_binop "#add_"),
363 Thm ("nth_Nil_",num_str @{thm nth_Nil_}),
364 Thm ("tl_Cons",num_str @{thm tl_Cons}),
365 Thm ("tl_Nil",num_str @{thm tl_Nil}),
366 Calc ("EqSystem.occur'_exactly'_in",
367 eval_occur_exactly_in
368 "#eval_occur_exactly_in_")
372 (*WN060914 quickly created for 4x4;
373 more similarity to prls_triangular desirable*)
374 val prls_triangular4 =
375 Rls {id="prls_triangular4", preconds = [],
376 rew_ord = ("e_rew_ord", e_rew_ord),
377 erls = Rls {id="erls_prls_triangular4", preconds = [],
378 rew_ord = ("e_rew_ord", e_rew_ord),
379 erls = Erls, srls = Erls, calc = [],
380 rules = [(*for precond nth_Cons_ ...*)
381 Calc ("op <",eval_equ "#less_"),
382 Calc ("op +", eval_binop "#add_")
383 (*immediately repeated rewrite pushes
384 '+' into precondition !*)
387 srls = Erls, calc = [],
388 rules = [Thm ("nth_Cons_",num_str @{thm nth_Cons_}),
389 Calc ("op +", eval_binop "#add_"),
390 Thm ("nth_Nil_",num_str @{thm thm nth_Nil_}),
391 Thm ("tl_Cons",num_str @{thm tl_Cons}),
392 Thm ("tl_Nil",num_str @{thm tl_Nil}),
393 Calc ("EqSystem.occur'_exactly'_in",
394 eval_occur_exactly_in
395 "#eval_occur_exactly_in_")
400 overwritelthy @{theory}
402 [("simplify_System_parenthesized", prep_rls simplify_System_parenthesized),
403 ("simplify_System", prep_rls simplify_System),
404 ("isolate_bdvs", prep_rls isolate_bdvs),
405 ("isolate_bdvs_4x4", prep_rls isolate_bdvs_4x4),
406 ("order_system", prep_rls order_system),
407 ("order_add_mult_System", prep_rls order_add_mult_System),
408 ("norm_System_noadd_fractions", prep_rls norm_System_noadd_fractions),
409 ("norm_System", prep_rls norm_System)
416 (prep_pbt thy "pbl_equsys" [] e_pblID
418 [("#Given" ,["equalities es_", "solveForVars vs_"]),
419 ("#Find" ,["solution ss___"](*___ is copy-named*))
421 append_rls "e_rls" e_rls [(*for preds in where_*)],
422 SOME "solveSystem es_ vs_",
425 (prep_pbt thy "pbl_equsys_lin" [] e_pblID
426 (["linear", "system"],
427 [("#Given" ,["equalities es_", "solveForVars vs_"]),
428 (*TODO.WN050929 check linearity*)
429 ("#Find" ,["solution ss___"])
431 append_rls "e_rls" e_rls [(*for preds in where_*)],
432 SOME "solveSystem es_ vs_",
435 (prep_pbt thy "pbl_equsys_lin_2x2" [] e_pblID
436 (["2x2", "linear", "system"],
437 (*~~~~~~~~~~~~~~~~~~~~~~~~~*)
438 [("#Given" ,["equalities es_", "solveForVars vs_"]),
439 ("#Where" ,["length_ (es_:: bool list) = 2", "length_ vs_ = 2"]),
440 ("#Find" ,["solution ss___"])
442 append_rls "prls_2x2_linear_system" e_rls
443 [Thm ("length_Cons_",num_str @{thm length_Cons_}),
444 Thm ("length_Nil_",num_str @{thm length_Nil_}),
445 Calc ("op +", eval_binop "#add_"),
446 Calc ("op =",eval_equal "#equal_")
448 SOME "solveSystem es_ vs_",
451 (prep_pbt thy "pbl_equsys_lin_2x2_tri" [] e_pblID
452 (["triangular", "2x2", "linear", "system"],
453 [("#Given" ,["equalities es_", "solveForVars vs_"]),
455 ["(tl vs_) from_ vs_ occur_exactly_in (nth_ 1 (es_::bool list))",
456 " vs_ from_ vs_ occur_exactly_in (nth_ 2 (es_::bool list))"]),
457 ("#Find" ,["solution ss___"])
460 SOME "solveSystem es_ vs_",
461 [["EqSystem","top_down_substitution","2x2"]]));
463 (prep_pbt thy "pbl_equsys_lin_2x2_norm" [] e_pblID
464 (["normalize", "2x2", "linear", "system"],
465 [("#Given" ,["equalities es_", "solveForVars vs_"]),
466 ("#Find" ,["solution ss___"])
468 append_rls "e_rls" e_rls [(*for preds in where_*)],
469 SOME "solveSystem es_ vs_",
470 [["EqSystem","normalize","2x2"]]));
472 (prep_pbt thy "pbl_equsys_lin_3x3" [] e_pblID
473 (["3x3", "linear", "system"],
474 (*~~~~~~~~~~~~~~~~~~~~~~~~~*)
475 [("#Given" ,["equalities es_", "solveForVars vs_"]),
476 ("#Where" ,["length_ (es_:: bool list) = 3", "length_ vs_ = 3"]),
477 ("#Find" ,["solution ss___"])
479 append_rls "prls_3x3_linear_system" e_rls
480 [Thm ("length_Cons_",num_str @{thm length_Cons_}),
481 Thm ("length_Nil_",num_str @{thm length_Nil_}),
482 Calc ("op +", eval_binop "#add_"),
483 Calc ("op =",eval_equal "#equal_")
485 SOME "solveSystem es_ vs_",
488 (prep_pbt thy "pbl_equsys_lin_4x4" [] e_pblID
489 (["4x4", "linear", "system"],
490 (*~~~~~~~~~~~~~~~~~~~~~~~~~*)
491 [("#Given" ,["equalities es_", "solveForVars vs_"]),
492 ("#Where" ,["length_ (es_:: bool list) = 4", "length_ vs_ = 4"]),
493 ("#Find" ,["solution ss___"])
495 append_rls "prls_4x4_linear_system" e_rls
496 [Thm ("length_Cons_",num_str @{thm length_Cons_}),
497 Thm ("length_Nil_",num_str @{thm length_Nil_}),
498 Calc ("op +", eval_binop "#add_"),
499 Calc ("op =",eval_equal "#equal_")
501 SOME "solveSystem es_ vs_",
504 (prep_pbt thy "pbl_equsys_lin_4x4_tri" [] e_pblID
505 (["triangular", "4x4", "linear", "system"],
506 [("#Given" ,["equalities es_", "solveForVars vs_"]),
507 ("#Where" , (*accepts missing variables up to diagional form*)
508 ["(nth_ 1 (vs_::real list)) occurs_in (nth_ 1 (es_::bool list))",
509 "(nth_ 2 (vs_::real list)) occurs_in (nth_ 2 (es_::bool list))",
510 "(nth_ 3 (vs_::real list)) occurs_in (nth_ 3 (es_::bool list))",
511 "(nth_ 4 (vs_::real list)) occurs_in (nth_ 4 (es_::bool list))"
513 ("#Find" ,["solution ss___"])
515 append_rls "prls_tri_4x4_lin_sys" prls_triangular
516 [Calc ("Atools.occurs'_in",eval_occurs_in "")],
517 SOME "solveSystem es_ vs_",
518 [["EqSystem","top_down_substitution","4x4"]]));
521 (prep_pbt thy "pbl_equsys_lin_4x4_norm" [] e_pblID
522 (["normalize", "4x4", "linear", "system"],
523 [("#Given" ,["equalities es_", "solveForVars vs_"]),
524 (*length_ is checked 1 level above*)
525 ("#Find" ,["solution ss___"])
527 append_rls "e_rls" e_rls [(*for preds in where_*)],
528 SOME "solveSystem es_ vs_",
529 [["EqSystem","normalize","4x4"]]));
538 (prep_met thy "met_eqsys" [] e_metID
541 {rew_ord'="tless_true", rls' = Erls, calc = [],
542 srls = Erls, prls = Erls, crls = Erls, nrls = Erls},
546 (prep_met thy "met_eqsys_topdown" [] e_metID
547 (["EqSystem","top_down_substitution"],
549 {rew_ord'="tless_true", rls' = Erls, calc = [],
550 srls = Erls, prls = Erls, crls = Erls, nrls = Erls},
554 (prep_met thy "met_eqsys_topdown_2x2" [] e_metID
555 (["EqSystem","top_down_substitution","2x2"],
556 [("#Given" ,["equalities es_", "solveForVars vs_"]),
558 ["(tl vs_) from_ vs_ occur_exactly_in (nth_ 1 (es_::bool list))",
559 " vs_ from_ vs_ occur_exactly_in (nth_ 2 (es_::bool list))"]),
560 ("#Find" ,["solution ss___"])
562 {rew_ord'="ord_simplify_System", rls' = Erls, calc = [],
563 srls = append_rls "srls_top_down_2x2" e_rls
564 [Thm ("hd_thm",num_str @{thm hd_thm}),
565 Thm ("tl_Cons",num_str @{thm tl_Cons}),
566 Thm ("tl_Nil",num_str @{thm tl_Nil})
568 prls = prls_triangular, crls = Erls, nrls = Erls},
569 "Script SolveSystemScript (es_::bool list) (vs_::real list) = " ^
570 " (let e1__ = Take (hd es_); " ^
571 " e1__ = ((Try (Rewrite_Set_Inst [(bdv_1, hd vs_),(bdv_2, hd (tl vs_))]" ^
572 " isolate_bdvs False)) @@ " ^
573 " (Try (Rewrite_Set_Inst [(bdv_1, hd vs_),(bdv_2, hd (tl vs_))]" ^
574 " simplify_System False))) e1__; " ^
575 " e2__ = Take (hd (tl es_)); " ^
576 " e2__ = ((Substitute [e1__]) @@ " ^
577 " (Try (Rewrite_Set_Inst [(bdv_1, hd vs_),(bdv_2, hd (tl vs_))]" ^
578 " simplify_System_parenthesized False)) @@ " ^
579 " (Try (Rewrite_Set_Inst [(bdv_1, hd vs_),(bdv_2, hd (tl vs_))]" ^
580 " isolate_bdvs False)) @@ " ^
581 " (Try (Rewrite_Set_Inst [(bdv_1, hd vs_),(bdv_2, hd (tl vs_))]" ^
582 " simplify_System False))) e2__; " ^
583 " es__ = Take [e1__, e2__] " ^
584 " in (Try (Rewrite_Set order_system False)) es__)"
585 (*---------------------------------------------------------------------------
586 this script does NOT separate the equations as abolve,
587 but it does not yet work due to preliminary script-interpreter,
588 see eqsystem.sml 'script [EqSystem,top_down_substitution,2x2] Vers.2'
590 "Script SolveSystemScript (es_::bool list) (vs_::real list) = " ^
591 " (let es__ = Take es_; " ^
592 " e1__ = hd es__; " ^
593 " e2__ = hd (tl es__); " ^
594 " es__ = [e1__, Substitute [e1__] e2__] " ^
595 " in ((Try (Rewrite_Set_Inst [(bdv_1, hd vs_),(bdv_2, hd (tl vs_))]" ^
596 " simplify_System_parenthesized False)) @@ " ^
597 " (Try (Rewrite_Set_Inst [(bdv_1, hd vs_),(bdv_2, hd (tl vs_))] " ^
598 " isolate_bdvs False)) @@ " ^
599 " (Try (Rewrite_Set_Inst [(bdv_1, hd vs_),(bdv_2, hd (tl vs_))]" ^
600 " simplify_System False))) es__)"
601 ---------------------------------------------------------------------------*)
604 (prep_met thy "met_eqsys_norm" [] e_metID
605 (["EqSystem","normalize"],
607 {rew_ord'="tless_true", rls' = Erls, calc = [],
608 srls = Erls, prls = Erls, crls = Erls, nrls = Erls},
612 (prep_met thy "met_eqsys_norm_2x2" [] e_metID
613 (["EqSystem","normalize","2x2"],
614 [("#Given" ,["equalities es_", "solveForVars vs_"]),
615 ("#Find" ,["solution ss___"])],
616 {rew_ord'="tless_true", rls' = Erls, calc = [],
617 srls = append_rls "srls_normalize_2x2" e_rls
618 [Thm ("hd_thm",num_str @{thm hd_thm}),
619 Thm ("tl_Cons",num_str @{thm tl_Cons}),
620 Thm ("tl_Nil",num_str @{thm tl_Nil})
622 prls = Erls, crls = Erls, nrls = Erls},
623 "Script SolveSystemScript (es_::bool list) (vs_::real list) = " ^
624 " (let es__ = ((Try (Rewrite_Set norm_Rational False)) @@ " ^
625 " (Try (Rewrite_Set_Inst [(bdv_1, hd vs_),(bdv_2, hd (tl vs_))]" ^
626 " simplify_System_parenthesized False)) @@ " ^
627 " (Try (Rewrite_Set_Inst [(bdv_1, hd vs_),(bdv_2, hd (tl vs_))]" ^
628 " isolate_bdvs False)) @@ " ^
629 " (Try (Rewrite_Set_Inst [(bdv_1, hd vs_),(bdv_2, hd (tl vs_))]" ^
630 " simplify_System_parenthesized False)) @@ " ^
631 " (Try (Rewrite_Set order_system False))) es_ " ^
632 " in (SubProblem (EqSystem_,[linear,system],[no_met]) " ^
633 " [BOOL_LIST es__, REAL_LIST vs_]))"
636 (*this is for nth_ only*)
637 val srls = Rls {id="srls_normalize_4x4",
639 rew_ord = ("termlessI",termlessI),
640 erls = append_rls "erls_in_srls_IntegrierenUnd.." e_rls
641 [(*for asm in nth_Cons_ ...*)
642 Calc ("op <",eval_equ "#less_"),
643 (*2nd nth_Cons_ pushes n+-1 into asms*)
644 Calc("op +", eval_binop "#add_")
646 srls = Erls, calc = [],
647 rules = [Thm ("nth_Cons_",num_str @{thm nth_Cons_}),
648 Calc("op +", eval_binop "#add_"),
649 Thm ("nth_Nil_",num_str @{thm nth_Nil_})],
652 (prep_met thy "met_eqsys_norm_4x4" [] e_metID
653 (["EqSystem","normalize","4x4"],
654 [("#Given" ,["equalities es_", "solveForVars vs_"]),
655 ("#Find" ,["solution ss___"])],
656 {rew_ord'="tless_true", rls' = Erls, calc = [],
657 srls = append_rls "srls_normalize_4x4" srls
658 [Thm ("hd_thm",num_str @{thm hd_thm}),
659 Thm ("tl_Cons",num_str @{thm tl_Cons}),
660 Thm ("tl_Nil",num_str @{thm tl_Nil})
662 prls = Erls, crls = Erls, nrls = Erls},
663 (*GOON met ["EqSystem","normalize","4x4"] @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@*)
664 "Script SolveSystemScript (es_::bool list) (vs_::real list) = " ^
666 " ((Try (Rewrite_Set norm_Rational False)) @@ " ^
667 " (Repeat (Rewrite commute_0_equality False)) @@ " ^
668 " (Try (Rewrite_Set_Inst [(bdv_1, nth_ 1 vs_),(bdv_2, nth_ 2 vs_ ), " ^
669 " (bdv_3, nth_ 3 vs_),(bdv_3, nth_ 4 vs_ )] " ^
670 " simplify_System_parenthesized False)) @@ " ^
671 " (Try (Rewrite_Set_Inst [(bdv_1, nth_ 1 vs_),(bdv_2, nth_ 2 vs_ ), " ^
672 " (bdv_3, nth_ 3 vs_),(bdv_3, nth_ 4 vs_ )] " ^
673 " isolate_bdvs_4x4 False)) @@ " ^
674 " (Try (Rewrite_Set_Inst [(bdv_1, nth_ 1 vs_),(bdv_2, nth_ 2 vs_ ), " ^
675 " (bdv_3, nth_ 3 vs_),(bdv_3, nth_ 4 vs_ )] " ^
676 " simplify_System_parenthesized False)) @@ " ^
677 " (Try (Rewrite_Set order_system False))) es_ " ^
678 " in (SubProblem (EqSystem_,[linear,system],[no_met]) " ^
679 " [BOOL_LIST es__, REAL_LIST vs_]))"
682 (prep_met thy "met_eqsys_topdown_4x4" [] e_metID
683 (["EqSystem","top_down_substitution","4x4"],
684 [("#Given" ,["equalities es_", "solveForVars vs_"]),
685 ("#Where" , (*accepts missing variables up to diagonal form*)
686 ["(nth_ 1 (vs_::real list)) occurs_in (nth_ 1 (es_::bool list))",
687 "(nth_ 2 (vs_::real list)) occurs_in (nth_ 2 (es_::bool list))",
688 "(nth_ 3 (vs_::real list)) occurs_in (nth_ 3 (es_::bool list))",
689 "(nth_ 4 (vs_::real list)) occurs_in (nth_ 4 (es_::bool list))"
691 ("#Find" ,["solution ss___"])
693 {rew_ord'="ord_simplify_System", rls' = Erls, calc = [],
694 srls = append_rls "srls_top_down_4x4" srls [],
695 prls = append_rls "prls_tri_4x4_lin_sys" prls_triangular
696 [Calc ("Atools.occurs'_in",eval_occurs_in "")],
697 crls = Erls, nrls = Erls},
698 (*FIXXXXME.WN060916: this script works ONLY for exp 7.79 @@@@@@@@@@@@@@@@@@@@*)
699 "Script SolveSystemScript (es_::bool list) (vs_::real list) = " ^
700 " (let e1_ = nth_ 1 es_; " ^
701 " e2_ = Take (nth_ 2 es_); " ^
702 " e2_ = ((Substitute [e1_]) @@ " ^
703 " (Try (Rewrite_Set_Inst [(bdv_1,nth_ 1 vs_),(bdv_2,nth_ 2 vs_)," ^
704 " (bdv_3,nth_ 3 vs_),(bdv_4,nth_ 4 vs_)]" ^
705 " simplify_System_parenthesized False)) @@ " ^
706 " (Try (Rewrite_Set_Inst [(bdv_1,nth_ 1 vs_),(bdv_2,nth_ 2 vs_)," ^
707 " (bdv_3,nth_ 3 vs_),(bdv_4,nth_ 4 vs_)]" ^
708 " isolate_bdvs False)) @@ " ^
709 " (Try (Rewrite_Set_Inst [(bdv_1,nth_ 1 vs_),(bdv_2,nth_ 2 vs_)," ^
710 " (bdv_3,nth_ 3 vs_),(bdv_4,nth_ 4 vs_)]" ^
711 " norm_Rational False))) e2_ " ^
712 " in [e1_, e2_, nth_ 3 es_, nth_ 4 es_])"