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 (*stated as axioms, todo: prove as theorems
23 'bdv' is a constant handled on the meta-level
24 specifically as a 'bound variable' *)
26 commute_0_equality: "(0 = a) = (a = 0)" and
28 (*WN0510 see simliar rules 'isolate_' 'separate_' (by RL)
29 [bdv_1,bdv_2,bdv_3,bdv_4] work also for 2 and 3 bdvs, ugly !*)
31 "[| [] from [bdv_1,bdv_2,bdv_3,bdv_4] occur_exactly_in a |]
32 ==> (a + b = c) = (b = c + -1*a)" and
34 "[| some_of [bdv_1,bdv_2,bdv_3,bdv_4] occur_in b; Not (b=!=0) |]
35 ==> (a = b) = (a + -1*b = 0)" and
37 "[| some_of [bdv_1,bdv_2,bdv_3,bdv_4] occur_in c |]
38 ==> (a = b + c) = (a + -1*c = b)" and
40 "[| Not (some_of [bdv_1,bdv_2,bdv_3,bdv_4] occur_in a) |]
41 ==> (a + b = c) = (b = -1*a + c)" and
43 "[| [] from [bdv_1,bdv_2,bdv_3,bdv_4] occur_exactly_in a; Not (a=!=0) |]
44 ==>(a * b = c) = (b = c / a)"
45 axiomatization where (*..if replaced by "and" we get an error in
46 --- rewrite in [EqSystem,normalise,2x2] --- step "--- 3---";*)
47 order_system_NxN: "[a,b] = [b,a]"
48 (*requires rew_ord for termination, eg. ord_simplify_Integral;
49 works for lists of any length, interestingly !?!*)
54 (** eval functions **)
56 (*certain variables of a given list occur _all_ in a term
57 args: all: ..variables, which are under consideration (eg. the bound vars)
58 vs: variables which must be in t,
59 and none of the others in all must be in t
60 t: the term under consideration
62 fun occur_exactly_in vs all t =
63 let fun occurs_in' a b = Prog_Expr.occurs_in b a
64 in foldl and_ (true, map (occurs_in' t) vs)
65 andalso not (foldl or_ (false, map (occurs_in' t)
66 (subtract op = vs all)))
69 (*("occur_exactly_in", ("EqSystem.occur_exactly_in",
70 eval_occur_exactly_in "#eval_occur_exactly_in_") )*)
71 fun eval_occur_exactly_in _ "EqSystem.occur_exactly_in"
72 (p as (Const (\<^const_name>\<open>EqSystem.occur_exactly_in\<close>,_)
74 if occur_exactly_in (TermC.isalist2list vs) (TermC.isalist2list all) t
75 then SOME ((UnparseC.term p) ^ " = True",
76 HOLogic.Trueprop $ (TermC.mk_equality (p, @{term True})))
77 else SOME ((UnparseC.term p) ^ " = False",
78 HOLogic.Trueprop $ (TermC.mk_equality (p, @{term False})))
79 | eval_occur_exactly_in _ _ _ _ = NONE;
81 calculation occur_exactly_in = \<open>eval_occur_exactly_in "#eval_occur_exactly_in_"\<close>
84 (** rewrite order 'ord_simplify_System' **)
86 (* order wrt. several linear (i.e. without exponents) variables "c", "c_2",..
87 which leaves the monomials containing c, c_2,... at the end of an Integral
88 and puts the c, c_2,... rightmost within a monomial.
90 WN050906 this is a quick and dirty adaption of ord_make_polynomial_in,
91 which was most adequate, because it uses size_of_term*)
93 local (*. for simplify_System .*)
95 open Term; (* for type order = EQUAL | LESS | GREATER *)
97 fun pr_ord EQUAL = "EQUAL"
98 | pr_ord LESS = "LESS"
99 | pr_ord GREATER = "GREATER";
101 fun dest_hd' (Const (a, T)) = (((a, 0), T), 0)
102 | dest_hd' (Free (ccc, T)) =
103 (case Symbol.explode ccc of
104 "c"::[] => ((("|||||||||||||||||||||", 0), T), 1)(*greatest string WN*)
105 | "c"::"_"::_ => ((("|||||||||||||||||||||", 0), T), 1)
106 | _ => (((ccc, 0), T), 1))
107 | dest_hd' (Var v) = (v, 2)
108 | dest_hd' (Bound i) = ((("", i), dummyT), 3)
109 | dest_hd' (Abs (_, T, _)) = ((("", 0), T), 4)
110 | dest_hd' _ = raise ERROR "dest_hd': uncovered case in fun.def.";
112 fun size_of_term' (Free (ccc, _)) =
113 (case Symbol.explode ccc of (*WN0510 hack for the bound variables*)
115 | "c"::"_"::is => 1000 * ((TermC.int_of_str o implode) is)
117 | size_of_term' (Abs (_,_,body)) = 1 + size_of_term' body
118 | size_of_term' (f$t) = size_of_term' f + size_of_term' t
119 | size_of_term' _ = 1;
121 fun term_ord' pr thy (Abs (_, T, t), Abs(_, U, u)) = (* ~ term.ML *)
122 (case term_ord' pr thy (t, u) of EQUAL => Term_Ord.typ_ord (T, U) | ord => ord)
123 | term_ord' pr thy (t, u) =
127 val (f, ts) = strip_comb t and (g, us) = strip_comb u;
128 val _ = tracing ("t= f@ts= \"" ^ UnparseC.term_in_thy thy f ^ "\" @ \"[" ^
129 commas (map (UnparseC.term_in_thy thy) ts) ^ "]\"");
130 val _ = tracing ("u= g@us= \"" ^ UnparseC.term_in_thy thy g ^ "\" @ \"[" ^
131 commas (map (UnparseC.term_in_thy thy) us) ^ "]\"");
132 val _ = tracing ("size_of_term(t,u)= (" ^ string_of_int (size_of_term' t) ^ ", " ^
133 string_of_int (size_of_term' u) ^ ")");
134 val _ = tracing ("hd_ord(f,g) = " ^ ((pr_ord o hd_ord) (f,g)));
135 val _ = tracing ("terms_ord (ts,us) = " ^(pr_ord o terms_ord str false) (ts,us));
136 val _=tracing("-------");
139 case int_ord (size_of_term' t, size_of_term' u) of
141 let val (f, ts) = strip_comb t and (g, us) = strip_comb u
142 in (case hd_ord (f, g) of
143 EQUAL => (terms_ord str pr) (ts, us)
147 and hd_ord (f, g) = (* ~ term.ML *)
148 prod_ord (prod_ord Term_Ord.indexname_ord Term_Ord.typ_ord) int_ord (dest_hd' f, dest_hd' g)
149 and terms_ord _ pr (ts, us) = list_ord (term_ord' pr (ThyC.get_theory "Isac_Knowledge"))(ts, us);
153 (*WN0510 for preliminary use in eval_order_system, see case-study mat-eng.tex
154 fun ord_simplify_System_rev (pr:bool) thy subst tu =
155 (term_ord' pr thy (Library.swap tu) = LESS);*)
158 fun ord_simplify_System (pr:bool) thy _(*subst*) (ts, us) =
159 (term_ord' pr thy (TermC.numerals_to_Free ts, TermC.numerals_to_Free us) = LESS);
163 Rewrite_Ord.rew_ord' := overwritel (! Rewrite_Ord.rew_ord',
164 [("ord_simplify_System", ord_simplify_System false \<^theory>)
170 (*.adapted from 'order_add_mult_in' by just replacing the rew_ord.*)
171 val order_add_mult_System =
172 Rule_Def.Repeat{id = "order_add_mult_System", preconds = [],
173 rew_ord = ("ord_simplify_System",
174 ord_simplify_System false @{theory "Integrate"}),
175 erls = Rule_Set.empty,srls = Rule_Set.Empty, calc = [], errpatts = [],
176 rules = [\<^rule_thm>\<open>mult.commute\<close>,
178 \<^rule_thm>\<open>real_mult_left_commute\<close>,
179 (*z1.0 * (z2.0 * z3.0) = z2.0 * (z1.0 * z3.0)*)
180 \<^rule_thm>\<open>mult.assoc\<close>,
181 (*z1.0 * z2.0 * z3.0 = z1.0 * (z2.0 * z3.0)*)
182 \<^rule_thm>\<open>add.commute\<close>,
184 \<^rule_thm>\<open>add.left_commute\<close>,
185 (*x + (y + z) = y + (x + z)*)
186 \<^rule_thm>\<open>add.assoc\<close>
187 (*z1.0 + z2.0 + z3.0 = z1.0 + (z2.0 + z3.0)*)
189 scr = Rule.Empty_Prog};
192 (*.adapted from 'norm_Rational' by
193 #1 using 'ord_simplify_System' in 'order_add_mult_System'
194 #2 NOT using common_nominator_p .*)
195 val norm_System_noadd_fractions =
196 Rule_Def.Repeat {id = "norm_System_noadd_fractions", preconds = [],
197 rew_ord = ("dummy_ord",Rewrite_Ord.dummy_ord),
198 erls = norm_rat_erls, srls = Rule_Set.Empty, calc = [], errpatts = [],
199 rules = [(*sequence given by operator precedence*)
200 Rule.Rls_ discard_minus,
202 Rule.Rls_ rat_mult_divide,
204 Rule.Rls_ reduce_0_1_2,
205 Rule.Rls_ (*order_add_mult #1*) order_add_mult_System,
206 Rule.Rls_ collect_numerals,
207 (*Rule.Rls_ add_fractions_p, #2*)
210 scr = Rule.Empty_Prog
214 (*.adapted from 'norm_Rational' by
215 *1* using 'ord_simplify_System' in 'order_add_mult_System'.*)
217 Rule_Def.Repeat {id = "norm_System", preconds = [],
218 rew_ord = ("dummy_ord",Rewrite_Ord.dummy_ord),
219 erls = norm_rat_erls, srls = Rule_Set.Empty, calc = [], errpatts = [],
220 rules = [(*sequence given by operator precedence*)
221 Rule.Rls_ discard_minus,
223 Rule.Rls_ rat_mult_divide,
225 Rule.Rls_ reduce_0_1_2,
226 Rule.Rls_ (*order_add_mult *1*) order_add_mult_System,
227 Rule.Rls_ collect_numerals,
228 Rule.Rls_ add_fractions_p,
231 scr = Rule.Empty_Prog
235 (*.simplify an equational system BEFORE solving it such that parentheses are
236 ( ((u0*v0)*w0) + ( ((u1*v1)*w1) * c + ... +((u4*v4)*w4) * c_4 ) )
237 ATTENTION: works ONLY for bound variables c, c_1, c_2, c_3, c_4 :ATTENTION
238 This is a copy from 'make_ratpoly_in' with respective reductions:
239 *0* expand the term, ie. distribute * and / over +
240 *1* ord_simplify_System instead of termlessI
241 *2* no add_fractions_p (= common_nominator_p_rls !)
242 *3* discard_parentheses only for (.*(.*.))
243 analoguous to simplify_Integral .*)
244 val simplify_System_parenthesized =
245 Rule_Set.Sequence {id = "simplify_System_parenthesized", preconds = []:term list,
246 rew_ord = ("dummy_ord", Rewrite_Ord.dummy_ord),
247 erls = Atools_erls, srls = Rule_Set.Empty, calc = [], errpatts = [],
248 rules = [\<^rule_thm>\<open>distrib_right\<close>,
249 (*"(?z1.0 + ?z2.0) * ?w = ?z1.0 * ?w + ?z2.0 * ?w"*)
250 \<^rule_thm>\<open>add_divide_distrib\<close>,
251 (*"(?x + ?y) / ?z = ?x / ?z + ?y / ?z"*)
252 (*^^^^^ *0* ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^*)
253 Rule.Rls_ norm_Rational_noadd_fractions(**2**),
254 Rule.Rls_ (*order_add_mult_in*) norm_System_noadd_fractions (**1**),
255 \<^rule_thm_sym>\<open>mult.assoc\<close>
256 (*Rule.Rls_ discard_parentheses *3**),
257 Rule.Rls_ collect_bdv, (*from make_polynomial_in WN051031 welldone?*)
258 Rule.Rls_ separate_bdv2,
259 \<^rule_eval>\<open>divide\<close> (Prog_Expr.eval_cancel "#divide_e")
261 scr = Rule.Empty_Prog};
264 (*.simplify an equational system AFTER solving it;
265 This is a copy of 'make_ratpoly_in' with the differences
266 *1* ord_simplify_System instead of termlessI .*)
267 (*TODO.WN051031 ^^^^^^^^^^ should be in EACH rls contained *)
268 val simplify_System =
269 Rule_Set.Sequence {id = "simplify_System", preconds = []:term list,
270 rew_ord = ("dummy_ord", Rewrite_Ord.dummy_ord),
271 erls = Atools_erls, srls = Rule_Set.Empty, calc = [], errpatts = [],
272 rules = [Rule.Rls_ norm_Rational,
273 Rule.Rls_ (*order_add_mult_in*) norm_System (**1**),
274 Rule.Rls_ discard_parentheses,
275 Rule.Rls_ collect_bdv, (*from make_polynomial_in WN051031 welldone?*)
276 Rule.Rls_ separate_bdv2,
277 \<^rule_eval>\<open>divide\<close> (Prog_Expr.eval_cancel "#divide_e")
279 scr = Rule.Empty_Prog};
281 val simplify_System =
282 Rule_Set.append_rules "simplify_System" simplify_System_parenthesized
283 [\<^rule_thm_sym>\<open>add.assoc\<close>];
288 Rule_Def.Repeat {id="isolate_bdvs", preconds = [],
289 rew_ord = ("e_rew_ord", Rewrite_Ord.e_rew_ord),
290 erls = Rule_Set.append_rules "erls_isolate_bdvs" Rule_Set.empty
291 [(\<^rule_eval>\<open>occur_exactly_in\<close> (eval_occur_exactly_in "#eval_occur_exactly_in_"))],
292 srls = Rule_Set.Empty, calc = [], errpatts = [],
294 [\<^rule_thm>\<open>commute_0_equality\<close>,
295 \<^rule_thm>\<open>separate_bdvs_add\<close>,
296 \<^rule_thm>\<open>separate_bdvs_mult\<close>],
297 scr = Rule.Empty_Prog};
300 val isolate_bdvs_4x4 =
301 Rule_Def.Repeat {id="isolate_bdvs_4x4", preconds = [],
302 rew_ord = ("e_rew_ord", Rewrite_Ord.e_rew_ord),
303 erls = Rule_Set.append_rules
304 "erls_isolate_bdvs_4x4" Rule_Set.empty
305 [\<^rule_eval>\<open>occur_exactly_in\<close> (eval_occur_exactly_in "#eval_occur_exactly_in_"),
306 \<^rule_eval>\<open>Prog_Expr.ident\<close> (Prog_Expr.eval_ident "#ident_"),
307 \<^rule_eval>\<open>Prog_Expr.some_occur_in\<close> (Prog_Expr.eval_some_occur_in "#some_occur_in_"),
308 \<^rule_thm>\<open>not_true\<close>,
309 \<^rule_thm>\<open>not_false\<close>
311 srls = Rule_Set.Empty, calc = [], errpatts = [],
312 rules = [\<^rule_thm>\<open>commute_0_equality\<close>,
313 \<^rule_thm>\<open>separate_bdvs0\<close>,
314 \<^rule_thm>\<open>separate_bdvs_add1\<close>,
315 \<^rule_thm>\<open>separate_bdvs_add2\<close>,
316 \<^rule_thm>\<open>separate_bdvs_mult\<close>
317 ], scr = Rule.Empty_Prog};
322 (*.order the equations in a system such, that a triangular system (if any)
323 appears as [..c_4 = .., ..., ..., ..c_1 + ..c_2 + ..c_3 ..c_4 = ..].*)
325 Rule_Def.Repeat {id="order_system", preconds = [],
326 rew_ord = ("ord_simplify_System",
327 ord_simplify_System false \<^theory>),
328 erls = Rule_Set.Empty, srls = Rule_Set.Empty, calc = [], errpatts = [],
329 rules = [\<^rule_thm>\<open>order_system_NxN\<close>
331 scr = Rule.Empty_Prog};
333 val prls_triangular =
334 Rule_Def.Repeat {id="prls_triangular", preconds = [],
335 rew_ord = ("e_rew_ord", Rewrite_Ord.e_rew_ord),
336 erls = Rule_Def.Repeat {id="erls_prls_triangular", preconds = [],
337 rew_ord = ("e_rew_ord", Rewrite_Ord.e_rew_ord),
338 erls = Rule_Set.Empty, srls = Rule_Set.Empty, calc = [], errpatts = [],
339 rules = [(*for precond NTH_CONS ...*)
340 \<^rule_eval>\<open>less\<close> (Prog_Expr.eval_equ "#less_"),
341 \<^rule_eval>\<open>plus\<close> (**)(eval_binop "#add_"),
342 Rule.Eval ("EqSystem.occur_exactly_in",
343 eval_occur_exactly_in "#eval_occur_exactly_in_")
344 (*immediately repeated rewrite pushes
345 '+' into precondition !*)
347 scr = Rule.Empty_Prog},
348 srls = Rule_Set.Empty, calc = [], errpatts = [],
349 rules = [\<^rule_thm>\<open>NTH_CONS\<close>,
350 \<^rule_eval>\<open>plus\<close> (**)(eval_binop "#add_"),
351 \<^rule_thm>\<open>NTH_NIL\<close>,
352 \<^rule_thm>\<open>tl_Cons\<close>,
353 \<^rule_thm>\<open>tl_Nil\<close>,
354 \<^rule_eval>\<open>occur_exactly_in\<close> (eval_occur_exactly_in "#eval_occur_exactly_in_")
356 scr = Rule.Empty_Prog};
360 (*WN060914 quickly created for 4x4;
361 more similarity to prls_triangular desirable*)
362 val prls_triangular4 =
363 Rule_Def.Repeat {id="prls_triangular4", preconds = [],
364 rew_ord = ("e_rew_ord", Rewrite_Ord.e_rew_ord),
365 erls = Rule_Def.Repeat {id="erls_prls_triangular4", preconds = [],
366 rew_ord = ("e_rew_ord", Rewrite_Ord.e_rew_ord),
367 erls = Rule_Set.Empty, srls = Rule_Set.Empty, calc = [], errpatts = [],
368 rules = [(*for precond NTH_CONS ...*)
369 \<^rule_eval>\<open>less\<close> (Prog_Expr.eval_equ "#less_"),
370 \<^rule_eval>\<open>plus\<close> (**)(eval_binop "#add_")
371 (*immediately repeated rewrite pushes
372 '+' into precondition !*)
374 scr = Rule.Empty_Prog},
375 srls = Rule_Set.Empty, calc = [], errpatts = [],
376 rules = [\<^rule_thm>\<open>NTH_CONS\<close>,
377 \<^rule_eval>\<open>plus\<close> (**)(eval_binop "#add_"),
378 \<^rule_thm>\<open>NTH_NIL\<close>,
379 \<^rule_thm>\<open>tl_Cons\<close>,
380 \<^rule_thm>\<open>tl_Nil\<close>,
381 \<^rule_eval>\<open>occur_exactly_in\<close> (eval_occur_exactly_in "#eval_occur_exactly_in_")
383 scr = Rule.Empty_Prog};
387 simplify_System_parenthesized = \<open>prep_rls' simplify_System_parenthesized\<close> and
388 simplify_System = \<open>prep_rls' simplify_System\<close> and
389 isolate_bdvs = \<open>prep_rls' isolate_bdvs\<close> and
390 isolate_bdvs_4x4 = \<open>prep_rls' isolate_bdvs_4x4\<close> and
391 order_system = \<open>prep_rls' order_system\<close> and
392 order_add_mult_System = \<open>prep_rls' order_add_mult_System\<close> and
393 norm_System_noadd_fractions = \<open>prep_rls' norm_System_noadd_fractions\<close> and
394 norm_System = \<open>prep_rls' norm_System\<close>
397 section \<open>Problems\<close>
399 problem pbl_equsys : "system" =
400 \<open>Rule_Set.append_rules "empty" Rule_Set.empty [(*for preds in where_*)]\<close>
401 CAS: "solveSystem e_s v_s"
402 Given: "equalities e_s" "solveForVars v_s"
403 Find: "solution ss'''" (*''' is copy-named*)
405 problem pbl_equsys_lin : "LINEAR/system" =
406 \<open>Rule_Set.append_rules "empty" Rule_Set.empty [(*for preds in where_*)]\<close>
407 CAS: "solveSystem e_s v_s"
408 Given: "equalities e_s" "solveForVars v_s"
409 (*TODO.WN050929 check linearity*)
410 Find: "solution ss'''"
412 problem pbl_equsys_lin_2x2: "2x2/LINEAR/system" =
413 \<open>Rule_Set.append_rules "prls_2x2_linear_system" Rule_Set.empty
414 [\<^rule_thm>\<open>LENGTH_CONS\<close>,
415 \<^rule_thm>\<open>LENGTH_NIL\<close>,
416 \<^rule_eval>\<open>plus\<close> (**)(eval_binop "#add_"),
417 \<^rule_eval>\<open>HOL.eq\<close> (Prog_Expr.eval_equal "#equal_")]\<close>
418 CAS: "solveSystem e_s v_s"
419 Given: "equalities e_s" "solveForVars v_s"
420 Where: "Length (e_s:: bool list) = 2" "Length v_s = 2"
421 Find: "solution ss'''"
423 problem pbl_equsys_lin_2x2_tri : "triangular/2x2/LINEAR/system" =
424 \<open>prls_triangular\<close>
425 Method: "EqSystem/top_down_substitution/2x2"
426 CAS: "solveSystem e_s v_s"
427 Given: "equalities e_s" "solveForVars v_s"
429 "(tl v_s) from v_s occur_exactly_in (NTH 1 (e_s::bool list))"
430 " v_s from v_s occur_exactly_in (NTH 2 (e_s::bool list))"
431 Find: "solution ss'''"
433 problem pbl_equsys_lin_2x2_norm : "normalise/2x2/LINEAR/system" =
434 \<open>Rule_Set.append_rules "empty" Rule_Set.empty [(*for preds in where_*)]\<close>
435 Method: "EqSystem/normalise/2x2"
436 CAS: "solveSystem e_s v_s"
437 Given: "equalities e_s" "solveForVars v_s"
438 Find: "solution ss'''"
440 problem pbl_equsys_lin_3x3 : "3x3/LINEAR/system" =
441 \<open>Rule_Set.append_rules "prls_3x3_linear_system" Rule_Set.empty
442 [\<^rule_thm>\<open>LENGTH_CONS\<close>,
443 \<^rule_thm>\<open>LENGTH_NIL\<close>,
444 \<^rule_eval>\<open>plus\<close> (**)(eval_binop "#add_"),
445 \<^rule_eval>\<open>HOL.eq\<close> (Prog_Expr.eval_equal "#equal_")]\<close>
446 CAS: "solveSystem e_s v_s"
447 Given: "equalities e_s" "solveForVars v_s"
448 Where: "Length (e_s:: bool list) = 3" "Length v_s = 3"
449 Find: "solution ss'''"
451 problem pbl_equsys_lin_4x4 : "4x4/LINEAR/system" =
452 \<open>Rule_Set.append_rules "prls_4x4_linear_system" Rule_Set.empty
453 [\<^rule_thm>\<open>LENGTH_CONS\<close>,
454 \<^rule_thm>\<open>LENGTH_NIL\<close>,
455 \<^rule_eval>\<open>plus\<close> (**)(eval_binop "#add_"),
456 \<^rule_eval>\<open>HOL.eq\<close> (Prog_Expr.eval_equal "#equal_")]\<close>
457 CAS: "solveSystem e_s v_s"
458 Given: "equalities e_s" "solveForVars v_s"
459 Where: "Length (e_s:: bool list) = 4" "Length v_s = 4"
460 Find: "solution ss'''"
462 problem pbl_equsys_lin_4x4_tri : "triangular/4x4/LINEAR/system" =
463 \<open>Rule_Set.append_rules "prls_tri_4x4_lin_sys" prls_triangular
464 [\<^rule_eval>\<open>Prog_Expr.occurs_in\<close> (Prog_Expr.eval_occurs_in "")]\<close>
465 Method: "EqSystem/top_down_substitution/4x4"
466 CAS: "solveSystem e_s v_s"
467 Given: "equalities e_s" "solveForVars v_s"
468 Where: (*accepts missing variables up to diagional form*)
469 "(NTH 1 (v_s::real list)) occurs_in (NTH 1 (e_s::bool list))"
470 "(NTH 2 (v_s::real list)) occurs_in (NTH 2 (e_s::bool list))"
471 "(NTH 3 (v_s::real list)) occurs_in (NTH 3 (e_s::bool list))"
472 "(NTH 4 (v_s::real list)) occurs_in (NTH 4 (e_s::bool list))"
473 Find: "solution ss'''"
475 problem pbl_equsys_lin_4x4_norm : "normalise/4x4/LINEAR/system" =
476 \<open>Rule_Set.append_rules "empty" Rule_Set.empty [(*for preds in where_*)]\<close>
477 Method: "EqSystem/normalise/4x4"
478 CAS: "solveSystem e_s v_s"
479 Given: "equalities e_s" "solveForVars v_s"
480 (*Length is checked 1 level above*)
481 Find: "solution ss'''"
484 (*this is for NTH only*)
485 val srls = Rule_Def.Repeat {id="srls_normalise_4x4",
487 rew_ord = ("termlessI",termlessI),
488 erls = Rule_Set.append_rules "erls_in_srls_IntegrierenUnd.." Rule_Set.empty
489 [(*for asm in NTH_CONS ...*)
490 \<^rule_eval>\<open>less\<close> (Prog_Expr.eval_equ "#less_"),
491 (*2nd NTH_CONS pushes n+-1 into asms*)
492 \<^rule_eval>\<open>plus\<close> (**)(eval_binop "#add_")
494 srls = Rule_Set.Empty, calc = [], errpatts = [],
495 rules = [\<^rule_thm>\<open>NTH_CONS\<close>,
496 \<^rule_eval>\<open>plus\<close> (**)(eval_binop "#add_"),
497 \<^rule_thm>\<open>NTH_NIL\<close>],
498 scr = Rule.Empty_Prog};
501 section \<open>Methods\<close>
503 method met_eqsys : "EqSystem" =
504 \<open>{rew_ord'="tless_true", rls' = Rule_Set.Empty, calc = [], srls = Rule_Set.Empty, prls = Rule_Set.Empty, crls = Rule_Set.Empty,
505 errpats = [], nrls = Rule_Set.Empty}\<close>
507 method met_eqsys_topdown : "EqSystem/top_down_substitution" =
508 \<open>{rew_ord'="tless_true", rls' = Rule_Set.Empty, calc = [], srls = Rule_Set.Empty, prls = Rule_Set.Empty, crls = Rule_Set.Empty,
509 errpats = [], nrls = Rule_Set.Empty}\<close>
511 partial_function (tailrec) solve_system :: "bool list => real list => bool list"
513 "solve_system e_s v_s = (
517 (Try (Rewrite_Set_Inst [(''bdv_1'', hd v_s), (''bdv_2'', hd (tl v_s))] ''isolate_bdvs'')) #>
518 (Try (Rewrite_Set_Inst [(''bdv_1'', hd v_s), (''bdv_2'', hd (tl v_s))] ''simplify_System''))
520 e_2 = Take (hd (tl e_s));
522 (Substitute [e_1]) #>
523 (Try (Rewrite_Set_Inst [(''bdv_1'', hd v_s), (''bdv_2'', hd (tl v_s))] ''simplify_System_parenthesized'' )) #>
524 (Try (Rewrite_Set_Inst [(''bdv_1'', hd v_s), (''bdv_2'', hd (tl v_s))] ''isolate_bdvs'' )) #>
525 (Try (Rewrite_Set_Inst [(''bdv_1'', hd v_s), (''bdv_2'', hd (tl v_s))] ''simplify_System'' ))
527 e__s = Take [e_1, e_2]
529 Try (Rewrite_Set ''order_system'' ) e__s) "
531 method met_eqsys_topdown_2x2 : "EqSystem/top_down_substitution/2x2" =
532 \<open>{rew_ord'="ord_simplify_System", rls' = Rule_Set.Empty, calc = [],
533 srls = Rule_Set.append_rules "srls_top_down_2x2" Rule_Set.empty
534 [\<^rule_thm>\<open>hd_thm\<close>,
535 \<^rule_thm>\<open>tl_Cons\<close>,
536 \<^rule_thm>\<open>tl_Nil\<close>],
537 prls = prls_triangular, crls = Rule_Set.Empty, errpats = [], nrls = Rule_Set.Empty}\<close>
538 Program: solve_system.simps
539 Given: "equalities e_s" "solveForVars v_s"
541 "(tl v_s) from v_s occur_exactly_in (NTH 1 (e_s::bool list))"
542 " v_s from v_s occur_exactly_in (NTH 2 (e_s::bool list))"
543 Find: "solution ss'''"
545 method met_eqsys_norm : "EqSystem/normalise" =
546 \<open>{rew_ord'="tless_true", rls' = Rule_Set.Empty, calc = [], srls = Rule_Set.Empty, prls = Rule_Set.Empty, crls = Rule_Set.Empty,
547 errpats = [], nrls = Rule_Set.Empty}\<close>
549 partial_function (tailrec) solve_system2 :: "bool list => real list => bool list"
551 "solve_system2 e_s v_s = (
554 (Try (Rewrite_Set ''norm_Rational'' )) #>
555 (Try (Rewrite_Set_Inst [(''bdv_1'', hd v_s), (''bdv_2'', hd (tl v_s))] ''simplify_System_parenthesized'' )) #>
556 (Try (Rewrite_Set_Inst [(''bdv_1'', hd v_s), (''bdv_2'', hd (tl v_s))] ''isolate_bdvs'' )) #>
557 (Try (Rewrite_Set_Inst [(''bdv_1'', hd v_s), (''bdv_2'', hd (tl v_s))] ''simplify_System_parenthesized'' )) #>
558 (Try (Rewrite_Set ''order_system'' ))
561 SubProblem (''EqSystem'', [''LINEAR'', ''system''], [''no_met''])
562 [BOOL_LIST e__s, REAL_LIST v_s])"
564 method met_eqsys_norm_2x2 : "EqSystem/normalise/2x2" =
565 \<open>{rew_ord'="tless_true", rls' = Rule_Set.Empty, calc = [],
566 srls = Rule_Set.append_rules "srls_normalise_2x2" Rule_Set.empty
567 [\<^rule_thm>\<open>hd_thm\<close>,
568 \<^rule_thm>\<open>tl_Cons\<close>,
569 \<^rule_thm>\<open>tl_Nil\<close>],
570 prls = Rule_Set.Empty, crls = Rule_Set.Empty, errpats = [], nrls = Rule_Set.Empty}\<close>
571 Program: solve_system2.simps
572 Given: "equalities e_s" "solveForVars v_s"
573 Find: "solution ss'''"
575 partial_function (tailrec) solve_system3 :: "bool list => real list => bool list"
577 "solve_system3 e_s v_s = (
580 (Try (Rewrite_Set ''norm_Rational'' )) #>
581 (Repeat (Rewrite ''commute_0_equality'' )) #>
582 (Try (Rewrite_Set_Inst [(''bdv_1'', NTH 1 v_s), (''bdv_2'', NTH 2 v_s ),
583 (''bdv_3'', NTH 3 v_s), (''bdv_3'', NTH 4 v_s )] ''simplify_System_parenthesized'' )) #>
584 (Try (Rewrite_Set_Inst [(''bdv_1'', NTH 1 v_s), (''bdv_2'', NTH 2 v_s ),
585 (''bdv_3'', NTH 3 v_s), (''bdv_3'', NTH 4 v_s )] ''isolate_bdvs_4x4'' )) #>
586 (Try (Rewrite_Set_Inst [(''bdv_1'', NTH 1 v_s), (''bdv_2'', NTH 2 v_s ),
587 (''bdv_3'', NTH 3 v_s), (''bdv_3'', NTH 4 v_s )] ''simplify_System_parenthesized'' )) #>
588 (Try (Rewrite_Set ''order_system''))
591 SubProblem (''EqSystem'', [''LINEAR'', ''system''], [''no_met''])
592 [BOOL_LIST e__s, REAL_LIST v_s])"
594 method met_eqsys_norm_4x4 : "EqSystem/normalise/4x4" =
595 \<open>{rew_ord'="tless_true", rls' = Rule_Set.Empty, calc = [],
597 Rule_Set.append_rules "srls_normalise_4x4" srls
598 [\<^rule_thm>\<open>hd_thm\<close>, \<^rule_thm>\<open>tl_Cons\<close>, \<^rule_thm>\<open>tl_Nil\<close>],
599 prls = Rule_Set.Empty, crls = Rule_Set.Empty, errpats = [], nrls = Rule_Set.Empty}\<close>
600 Program: solve_system3.simps
601 Given: "equalities e_s" "solveForVars v_s"
602 Find: "solution ss'''"
603 (*STOPPED.WN06? met ["EqSystem", "normalise", "4x4"] #>#>#>#>#>#>#>#>#>#>#>#>#>@*)
605 partial_function (tailrec) solve_system4 :: "bool list => real list => bool list"
607 "solve_system4 e_s v_s = (
610 e_2 = Take (NTH 2 e_s);
612 (Substitute [e_1]) #>
613 (Try (Rewrite_Set_Inst [(''bdv_1'',NTH 1 v_s), (''bdv_2'',NTH 2 v_s),
614 (''bdv_3'',NTH 3 v_s), (''bdv_4'',NTH 4 v_s)] ''simplify_System_parenthesized'' )) #>
615 (Try (Rewrite_Set_Inst [(''bdv_1'',NTH 1 v_s), (''bdv_2'',NTH 2 v_s),
616 (''bdv_3'',NTH 3 v_s), (''bdv_4'',NTH 4 v_s)] ''isolate_bdvs'' )) #>
617 (Try (Rewrite_Set_Inst [(''bdv_1'',NTH 1 v_s), (''bdv_2'',NTH 2 v_s),
618 (''bdv_3'',NTH 3 v_s), (''bdv_4'',NTH 4 v_s)] ''norm_Rational'' ))
621 [e_1, e_2, NTH 3 e_s, NTH 4 e_s])"
623 method met_eqsys_topdown_4x4 : "EqSystem/top_down_substitution/4x4" =
624 \<open>{rew_ord'="ord_simplify_System", rls' = Rule_Set.Empty, calc = [],
625 srls = Rule_Set.append_rules "srls_top_down_4x4" srls [],
626 prls = Rule_Set.append_rules "prls_tri_4x4_lin_sys" prls_triangular
627 [\<^rule_eval>\<open>Prog_Expr.occurs_in\<close> (Prog_Expr.eval_occurs_in "")],
628 crls = Rule_Set.Empty, errpats = [], nrls = Rule_Set.Empty}\<close>
629 (*FIXXXXME.WN060916: this script works ONLY for exp 7.79 #>#>#>#>#>#>#>#>#>#>*)
630 Program: solve_system4.simps
631 Given: "equalities e_s" "solveForVars v_s"
632 Where: (*accepts missing variables up to diagonal form*)
633 "(NTH 1 (v_s::real list)) occurs_in (NTH 1 (e_s::bool list))"
634 "(NTH 2 (v_s::real list)) occurs_in (NTH 2 (e_s::bool list))"
635 "(NTH 3 (v_s::real list)) occurs_in (NTH 3 (e_s::bool list))"
636 "(NTH 4 (v_s::real list)) occurs_in (NTH 4 (e_s::bool list))"
637 Find: "solution ss'''"