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 !?!*)
52 (** eval functions **)
54 (*certain variables of a given list occur _all_ in a term
55 args: all: ..variables, which are under consideration (eg. the bound vars)
56 vs: variables which must be in t,
57 and none of the others in all must be in t
58 t: the term under consideration
60 fun occur_exactly_in vs all t =
61 let fun occurs_in' a b = Prog_Expr.occurs_in b a
62 in foldl and_ (true, map (occurs_in' t) vs)
63 andalso not (foldl or_ (false, map (occurs_in' t)
64 (subtract op = vs all)))
67 (*("occur_exactly_in", ("EqSystem.occur_exactly_in",
68 eval_occur_exactly_in "#eval_occur_exactly_in_") )*)
69 fun eval_occur_exactly_in _ "EqSystem.occur_exactly_in"
70 (p as (Const (\<^const_name>\<open>EqSystem.occur_exactly_in\<close>,_)
72 if occur_exactly_in (TermC.isalist2list vs) (TermC.isalist2list all) t
73 then SOME ((UnparseC.term p) ^ " = True",
74 HOLogic.Trueprop $ (TermC.mk_equality (p, @{term True})))
75 else SOME ((UnparseC.term p) ^ " = False",
76 HOLogic.Trueprop $ (TermC.mk_equality (p, @{term False})))
77 | eval_occur_exactly_in _ _ _ _ = NONE;
79 calculation occur_exactly_in = \<open>eval_occur_exactly_in "#eval_occur_exactly_in_"\<close>
82 (** rewrite order 'ord_simplify_System' **)
84 (* order wrt. several linear (i.e. without exponents) variables "c", "c_2",..
85 which leaves the monomials containing c, c_2,... at the end of an Integral
86 and puts the c, c_2,... rightmost within a monomial.
88 WN050906 this is a quick and dirty adaption of ord_make_polynomial_in,
89 which was most adequate, because it uses size_of_term*)
91 local (*. for simplify_System .*)
93 open Term; (* for type order = EQUAL | LESS | GREATER *)
95 fun pr_ord EQUAL = "EQUAL"
96 | pr_ord LESS = "LESS"
97 | pr_ord GREATER = "GREATER";
99 fun dest_hd' (Const (a, T)) = (((a, 0), T), 0)
100 | dest_hd' (Free (ccc, T)) =
101 (case Symbol.explode ccc of
102 "c"::[] => ((("|||||||||||||||||||||", 0), T), 1)(*greatest string WN*)
103 | "c"::"_"::_ => ((("|||||||||||||||||||||", 0), T), 1)
104 | _ => (((ccc, 0), T), 1))
105 | dest_hd' (Var v) = (v, 2)
106 | dest_hd' (Bound i) = ((("", i), dummyT), 3)
107 | dest_hd' (Abs (_, T, _)) = ((("", 0), T), 4)
108 | dest_hd' _ = raise ERROR "dest_hd': uncovered case in fun.def.";
110 fun size_of_term' (Free (ccc, _)) =
111 (case Symbol.explode ccc of (*WN0510 hack for the bound variables*)
113 | "c"::"_"::is => 1000 * ((TermC.int_of_str o implode) is)
115 | size_of_term' (Abs (_,_,body)) = 1 + size_of_term' body
116 | size_of_term' (f$t) = size_of_term' f + size_of_term' t
117 | size_of_term' _ = 1;
119 fun term_ord' pr thy (Abs (_, T, t), Abs(_, U, u)) = (* ~ term.ML *)
120 (case term_ord' pr thy (t, u) of EQUAL => Term_Ord.typ_ord (T, U) | ord => ord)
121 | term_ord' pr thy (t, u) =
125 val (f, ts) = strip_comb t and (g, us) = strip_comb u;
126 val _ = tracing ("t= f @ ts= \"" ^ UnparseC.term_in_thy thy f ^ "\" @ \"[" ^
127 commas (map (UnparseC.term_in_thy thy) ts) ^ "]\"");
128 val _ = tracing ("u= g @ us= \"" ^ UnparseC.term_in_thy thy g ^ "\" @ \"[" ^
129 commas (map (UnparseC.term_in_thy thy) us) ^ "]\"");
130 val _ = tracing ("size_of_term (t, u) = (" ^ string_of_int (size_of_term' t) ^ ", " ^
131 string_of_int (size_of_term' u) ^ ")");
132 val _ = tracing ("hd_ord (f, g) = " ^ ((pr_ord o hd_ord) (f, g)) );
133 (** )val _ = @{print tracing}{a = "hd_ord (f, g) = ", z = hd_ord (f, g)}( **)
134 val _ = tracing ("terms_ord (ts, us) = " ^(pr_ord o terms_ord str false) (ts,us));
135 val _= tracing ("-------");
138 case int_ord (size_of_term' t, size_of_term' u) of
140 let val (f, ts) = strip_comb t and (g, us) = strip_comb u
141 in (case hd_ord (f, g) of
142 EQUAL => (terms_ord str pr) (ts, us)
146 and hd_ord (f, g) = (* ~ term.ML *)
147 prod_ord (prod_ord Term_Ord.indexname_ord Term_Ord.typ_ord) int_ord (dest_hd' f, dest_hd' g)
148 and terms_ord _ pr (ts, us) = list_ord (term_ord' pr (ThyC.get_theory "Isac_Knowledge"))(ts, us);
152 (*WN0510 for preliminary use in eval_order_system, see case-study mat-eng.tex
153 fun ord_simplify_System_rev (pr:bool) thy subst tu =
154 (term_ord' pr thy (Library.swap tu) = LESS);*)
157 fun ord_simplify_System (pr:bool) thy _(*subst*) (ts, us) =
158 (term_ord' pr thy (TermC.numerals_to_Free ts, TermC.numerals_to_Free us) = LESS);
162 Rewrite_Ord.rew_ord' := overwritel (! Rewrite_Ord.rew_ord',
163 [("ord_simplify_System", ord_simplify_System false \<^theory>)
169 (*.adapted from 'order_add_mult_in' by just replacing the rew_ord.*)
170 val order_add_mult_System =
172 id = "order_add_mult_System", preconds = [],
173 rew_ord = ("ord_simplify_System", ord_simplify_System false @{theory "Integrate"}),
174 erls = Rule_Set.empty,srls = Rule_Set.Empty, calc = [], errpatts = [],
176 \<^rule_thm>\<open>mult.commute\<close>, (* z * w = w * z *)
177 \<^rule_thm>\<open>real_mult_left_commute\<close>, (*z1.0 * (z2.0 * z3.0) = z2.0 * (z1.0 * z3.0)*)
178 \<^rule_thm>\<open>mult.assoc\<close>, (*z1.0 * z2.0 * z3.0 = z1.0 * (z2.0 * z3.0)*)
179 \<^rule_thm>\<open>add.commute\<close>, (*z + w = w + z*)
180 \<^rule_thm>\<open>add.left_commute\<close>, (*x + (y + z) = y + (x + z)*)
181 \<^rule_thm>\<open>add.assoc\<close> ], (*z1.0 + z2.0 + z3.0 = z1.0 + (z2.0 + z3.0)*)
182 scr = Rule.Empty_Prog};
185 (*.adapted from 'norm_Rational' by
186 #1 using 'ord_simplify_System' in 'order_add_mult_System'
187 #2 NOT using common_nominator_p .*)
188 val norm_System_noadd_fractions =
189 Rule_Def.Repeat {id = "norm_System_noadd_fractions", preconds = [],
190 rew_ord = ("dummy_ord", Rewrite_Ord.dummy_ord),
191 erls = norm_rat_erls, srls = Rule_Set.Empty, calc = [], errpatts = [],
192 rules = [(*sequence given by operator precedence*)
193 Rule.Rls_ discard_minus,
195 Rule.Rls_ rat_mult_divide,
197 Rule.Rls_ reduce_0_1_2,
198 Rule.Rls_ (*order_add_mult #1*) order_add_mult_System,
199 Rule.Rls_ collect_numerals,
200 (*Rule.Rls_ add_fractions_p, #2*)
202 scr = Rule.Empty_Prog};
205 (*.adapted from 'norm_Rational' by
206 *1* using 'ord_simplify_System' in 'order_add_mult_System'.*)
208 Rule_Def.Repeat {id = "norm_System", preconds = [],
209 rew_ord = ("dummy_ord", Rewrite_Ord.dummy_ord),
210 erls = norm_rat_erls, srls = Rule_Set.Empty, calc = [], errpatts = [],
211 rules = [(*sequence given by operator precedence*)
212 Rule.Rls_ discard_minus,
214 Rule.Rls_ rat_mult_divide,
216 Rule.Rls_ reduce_0_1_2,
217 Rule.Rls_ (*order_add_mult *1*) order_add_mult_System,
218 Rule.Rls_ collect_numerals,
219 Rule.Rls_ add_fractions_p,
221 scr = Rule.Empty_Prog};
224 (*.simplify an equational system BEFORE solving it such that parentheses are
225 ( ((u0*v0)*w0) + ( ((u1*v1)*w1) * c + ... +((u4*v4)*w4) * c_4 ) )
226 ATTENTION: works ONLY for bound variables c, c_1, c_2, c_3, c_4 :ATTENTION
227 This is a copy from 'make_ratpoly_in' with respective reductions:
228 *0* expand the term, ie. distribute * and / over +
229 *1* ord_simplify_System instead of termlessI
230 *2* no add_fractions_p (= common_nominator_p_rls !)
231 *3* discard_parentheses only for (.*(.*.))
232 analoguous to simplify_Integral .*)
233 val simplify_System_parenthesized =
234 Rule_Set.Sequence {id = "simplify_System_parenthesized", preconds = []:term list,
235 rew_ord = ("dummy_ord", Rewrite_Ord.dummy_ord),
236 erls = Atools_erls, srls = Rule_Set.Empty, calc = [], errpatts = [],
238 \<^rule_thm>\<open>distrib_right\<close>, (*"(?z1.0 + ?z2.0) * ?w = ?z1.0 * ?w + ?z2.0 * ?w"*)
239 \<^rule_thm>\<open>add_divide_distrib\<close>, (*"(?x + ?y) / ?z = ?x / ?z + ?y / ?z"*)
240 Rule.Rls_ norm_Rational_noadd_fractions,
241 Rule.Rls_ (*order_add_mult_in*) norm_System_noadd_fractions,
242 \<^rule_thm_sym>\<open>mult.assoc\<close>,
243 Rule.Rls_ collect_bdv, (*from make_polynomial_in WN051031 welldone?*)
244 Rule.Rls_ separate_bdv2,
245 \<^rule_eval>\<open>divide\<close> (Prog_Expr.eval_cancel "#divide_e")],
246 scr = Rule.Empty_Prog};
249 (*.simplify an equational system AFTER solving it;
250 This is a copy of 'make_ratpoly_in' with the differences
251 *1* ord_simplify_System instead of termlessI .*)
252 (*TODO.WN051031 ^^^^^^^^^^ should be in EACH rls contained *)
253 val simplify_System =
254 Rule_Set.Sequence {id = "simplify_System", preconds = []:term list,
255 rew_ord = ("dummy_ord", Rewrite_Ord.dummy_ord),
256 erls = Atools_erls, srls = Rule_Set.Empty, calc = [], errpatts = [],
258 Rule.Rls_ norm_Rational,
259 Rule.Rls_ (*order_add_mult_in*) norm_System (**1**),
260 Rule.Rls_ discard_parentheses,
261 Rule.Rls_ collect_bdv, (*from make_polynomial_in WN051031 welldone?*)
262 Rule.Rls_ separate_bdv2,
263 \<^rule_eval>\<open>divide\<close> (Prog_Expr.eval_cancel "#divide_e")],
264 scr = Rule.Empty_Prog};
266 val simplify_System =
267 Rule_Set.append_rules "simplify_System" simplify_System_parenthesized
268 [\<^rule_thm_sym>\<open>add.assoc\<close>];
274 id="isolate_bdvs", preconds = [], rew_ord = ("e_rew_ord", Rewrite_Ord.e_rew_ord),
275 erls = Rule_Set.append_rules "erls_isolate_bdvs" Rule_Set.empty [
276 (\<^rule_eval>\<open>occur_exactly_in\<close> (eval_occur_exactly_in "#eval_occur_exactly_in_"))],
277 srls = Rule_Set.Empty, calc = [], errpatts = [],
279 \<^rule_thm>\<open>commute_0_equality\<close>,
280 \<^rule_thm>\<open>separate_bdvs_add\<close>,
281 \<^rule_thm>\<open>separate_bdvs_mult\<close>],
282 scr = Rule.Empty_Prog};
285 val isolate_bdvs_4x4 =
286 Rule_Def.Repeat {id="isolate_bdvs_4x4", preconds = [],
287 rew_ord = ("e_rew_ord", Rewrite_Ord.e_rew_ord),
288 erls = Rule_Set.append_rules "erls_isolate_bdvs_4x4" Rule_Set.empty [
289 \<^rule_eval>\<open>occur_exactly_in\<close> (eval_occur_exactly_in "#eval_occur_exactly_in_"),
290 \<^rule_eval>\<open>Prog_Expr.ident\<close> (Prog_Expr.eval_ident "#ident_"),
291 \<^rule_eval>\<open>Prog_Expr.some_occur_in\<close> (Prog_Expr.eval_some_occur_in "#some_occur_in_"),
292 \<^rule_thm>\<open>not_true\<close>,
293 \<^rule_thm>\<open>not_false\<close>],
294 srls = Rule_Set.Empty, calc = [], errpatts = [],
296 \<^rule_thm>\<open>commute_0_equality\<close>,
297 \<^rule_thm>\<open>separate_bdvs0\<close>,
298 \<^rule_thm>\<open>separate_bdvs_add1\<close>,
299 \<^rule_thm>\<open>separate_bdvs_add2\<close>,
300 \<^rule_thm>\<open>separate_bdvs_mult\<close>],
301 scr = Rule.Empty_Prog};
306 (*.order the equations in a system such, that a triangular system (if any)
307 appears as [..c_4 = .., ..., ..., ..c_1 + ..c_2 + ..c_3 ..c_4 = ..].*)
309 Rule_Def.Repeat {id="order_system", preconds = [],
310 rew_ord = ("ord_simplify_System", ord_simplify_System false \<^theory>),
311 erls = Rule_Set.Empty, srls = Rule_Set.Empty, calc = [], errpatts = [],
313 \<^rule_thm>\<open>order_system_NxN\<close>],
314 scr = Rule.Empty_Prog};
316 val prls_triangular =
318 id="prls_triangular", preconds = [], rew_ord = ("e_rew_ord", Rewrite_Ord.e_rew_ord),
319 erls = Rule_Def.Repeat {
320 id="erls_prls_triangular", preconds = [], rew_ord = ("e_rew_ord", Rewrite_Ord.e_rew_ord),
321 erls = Rule_Set.Empty, srls = Rule_Set.Empty, calc = [], errpatts = [],
322 rules = [(*for precond NTH_CONS ...*)
323 \<^rule_eval>\<open>less\<close> (Prog_Expr.eval_equ "#less_"),
324 \<^rule_eval>\<open>plus\<close> (**)(eval_binop "#add_"),
325 \<^rule_eval>\<open>occur_exactly_in\<close> (eval_occur_exactly_in "#eval_occur_exactly_in_")],
326 (*immediately repeated rewrite pushes '+' into precondition !*)
327 scr = Rule.Empty_Prog},
328 srls = Rule_Set.Empty, calc = [], errpatts = [],
330 \<^rule_thm>\<open>NTH_CONS\<close>,
331 \<^rule_eval>\<open>plus\<close> (**)(eval_binop "#add_"),
332 \<^rule_thm>\<open>NTH_NIL\<close>,
333 \<^rule_thm>\<open>tl_Cons\<close>,
334 \<^rule_thm>\<open>tl_Nil\<close>,
335 \<^rule_eval>\<open>occur_exactly_in\<close> (eval_occur_exactly_in "#eval_occur_exactly_in_")],
336 scr = Rule.Empty_Prog};
340 (*WN060914 quickly created for 4x4;
341 more similarity to prls_triangular desirable*)
342 val prls_triangular4 =
344 id="prls_triangular4", preconds = [], rew_ord = ("e_rew_ord", Rewrite_Ord.e_rew_ord),
345 erls = Rule_Def.Repeat {
346 id="erls_prls_triangular4", preconds = [], rew_ord = ("e_rew_ord", Rewrite_Ord.e_rew_ord),
347 erls = Rule_Set.Empty, srls = Rule_Set.Empty, calc = [], errpatts = [],
348 rules = [(*for precond NTH_CONS ...*)
349 \<^rule_eval>\<open>less\<close> (Prog_Expr.eval_equ "#less_"),
350 \<^rule_eval>\<open>plus\<close> (**)(eval_binop "#add_")],
351 (*immediately repeated rewrite pushes '+' into precondition !*)
352 scr = Rule.Empty_Prog},
353 srls = Rule_Set.Empty, calc = [], errpatts = [],
355 \<^rule_thm>\<open>NTH_CONS\<close>,
356 \<^rule_eval>\<open>plus\<close> (**)(eval_binop "#add_"),
357 \<^rule_thm>\<open>NTH_NIL\<close>,
358 \<^rule_thm>\<open>tl_Cons\<close>,
359 \<^rule_thm>\<open>tl_Nil\<close>,
360 \<^rule_eval>\<open>occur_exactly_in\<close> (eval_occur_exactly_in "#eval_occur_exactly_in_")],
361 scr = Rule.Empty_Prog};
365 simplify_System_parenthesized = \<open>prep_rls' simplify_System_parenthesized\<close> and
366 simplify_System = \<open>prep_rls' simplify_System\<close> and
367 isolate_bdvs = \<open>prep_rls' isolate_bdvs\<close> and
368 isolate_bdvs_4x4 = \<open>prep_rls' isolate_bdvs_4x4\<close> and
369 order_system = \<open>prep_rls' order_system\<close> and
370 order_add_mult_System = \<open>prep_rls' order_add_mult_System\<close> and
371 norm_System_noadd_fractions = \<open>prep_rls' norm_System_noadd_fractions\<close> and
372 norm_System = \<open>prep_rls' norm_System\<close>
375 section \<open>Problems\<close>
377 problem pbl_equsys : "system" =
378 \<open>Rule_Set.append_rules "empty" Rule_Set.empty [(*for preds in where_*)]\<close>
379 CAS: "solveSystem e_s v_s"
380 Given: "equalities e_s" "solveForVars v_s"
381 Find: "solution ss'''" (*''' is copy-named*)
383 problem pbl_equsys_lin : "LINEAR/system" =
384 \<open>Rule_Set.append_rules "empty" Rule_Set.empty [(*for preds in where_*)]\<close>
385 CAS: "solveSystem e_s v_s"
386 Given: "equalities e_s" "solveForVars v_s"
387 (*TODO.WN050929 check linearity*)
388 Find: "solution ss'''"
390 problem pbl_equsys_lin_2x2: "2x2/LINEAR/system" =
391 \<open>Rule_Set.append_rules "prls_2x2_linear_system" Rule_Set.empty
392 [\<^rule_thm>\<open>LENGTH_CONS\<close>,
393 \<^rule_thm>\<open>LENGTH_NIL\<close>,
394 \<^rule_eval>\<open>plus\<close> (**)(eval_binop "#add_"),
395 \<^rule_eval>\<open>HOL.eq\<close> (Prog_Expr.eval_equal "#equal_")]\<close>
396 CAS: "solveSystem e_s v_s"
397 Given: "equalities e_s" "solveForVars v_s"
398 Where: "Length (e_s:: bool list) = 2" "Length v_s = 2"
399 Find: "solution ss'''"
401 problem pbl_equsys_lin_2x2_tri : "triangular/2x2/LINEAR/system" =
402 \<open>prls_triangular\<close>
403 Method: "EqSystem/top_down_substitution/2x2"
404 CAS: "solveSystem e_s v_s"
405 Given: "equalities e_s" "solveForVars v_s"
407 "(tl v_s) from v_s occur_exactly_in (NTH 1 (e_s::bool list))"
408 " v_s from v_s occur_exactly_in (NTH 2 (e_s::bool list))"
409 Find: "solution ss'''"
411 problem pbl_equsys_lin_2x2_norm : "normalise/2x2/LINEAR/system" =
412 \<open>Rule_Set.append_rules "empty" Rule_Set.empty [(*for preds in where_*)]\<close>
413 Method: "EqSystem/normalise/2x2"
414 CAS: "solveSystem e_s v_s"
415 Given: "equalities e_s" "solveForVars v_s"
416 Find: "solution ss'''"
418 problem pbl_equsys_lin_3x3 : "3x3/LINEAR/system" =
419 \<open>Rule_Set.append_rules "prls_3x3_linear_system" Rule_Set.empty
420 [\<^rule_thm>\<open>LENGTH_CONS\<close>,
421 \<^rule_thm>\<open>LENGTH_NIL\<close>,
422 \<^rule_eval>\<open>plus\<close> (**)(eval_binop "#add_"),
423 \<^rule_eval>\<open>HOL.eq\<close> (Prog_Expr.eval_equal "#equal_")]\<close>
424 CAS: "solveSystem e_s v_s"
425 Given: "equalities e_s" "solveForVars v_s"
426 Where: "Length (e_s:: bool list) = 3" "Length v_s = 3"
427 Find: "solution ss'''"
429 problem pbl_equsys_lin_4x4 : "4x4/LINEAR/system" =
430 \<open>Rule_Set.append_rules "prls_4x4_linear_system" Rule_Set.empty
431 [\<^rule_thm>\<open>LENGTH_CONS\<close>,
432 \<^rule_thm>\<open>LENGTH_NIL\<close>,
433 \<^rule_eval>\<open>plus\<close> (**)(eval_binop "#add_"),
434 \<^rule_eval>\<open>HOL.eq\<close> (Prog_Expr.eval_equal "#equal_")]\<close>
435 CAS: "solveSystem e_s v_s"
436 Given: "equalities e_s" "solveForVars v_s"
437 Where: "Length (e_s:: bool list) = 4" "Length v_s = 4"
438 Find: "solution ss'''"
440 problem pbl_equsys_lin_4x4_tri : "triangular/4x4/LINEAR/system" =
441 \<open>Rule_Set.append_rules "prls_tri_4x4_lin_sys" prls_triangular
442 [\<^rule_eval>\<open>Prog_Expr.occurs_in\<close> (Prog_Expr.eval_occurs_in "")]\<close>
443 Method: "EqSystem/top_down_substitution/4x4"
444 CAS: "solveSystem e_s v_s"
445 Given: "equalities e_s" "solveForVars v_s"
446 Where: (*accepts missing variables up to diagional form*)
447 "(NTH 1 (v_s::real list)) occurs_in (NTH 1 (e_s::bool list))"
448 "(NTH 2 (v_s::real list)) occurs_in (NTH 2 (e_s::bool list))"
449 "(NTH 3 (v_s::real list)) occurs_in (NTH 3 (e_s::bool list))"
450 "(NTH 4 (v_s::real list)) occurs_in (NTH 4 (e_s::bool list))"
451 Find: "solution ss'''"
453 problem pbl_equsys_lin_4x4_norm : "normalise/4x4/LINEAR/system" =
454 \<open>Rule_Set.append_rules "empty" Rule_Set.empty [(*for preds in where_*)]\<close>
455 Method: "EqSystem/normalise/4x4"
456 CAS: "solveSystem e_s v_s"
457 Given: "equalities e_s" "solveForVars v_s"
458 (*Length is checked 1 level above*)
459 Find: "solution ss'''"
462 (*this is for NTH only*)
463 val srls = Rule_Def.Repeat {id="srls_normalise_4x4",
465 rew_ord = ("termlessI",termlessI),
466 erls = Rule_Set.append_rules "erls_in_srls_IntegrierenUnd.." Rule_Set.empty
467 [(*for asm in NTH_CONS ...*)
468 \<^rule_eval>\<open>less\<close> (Prog_Expr.eval_equ "#less_"),
469 (*2nd NTH_CONS pushes n+-1 into asms*)
470 \<^rule_eval>\<open>plus\<close> (**)(eval_binop "#add_")
472 srls = Rule_Set.Empty, calc = [], errpatts = [],
473 rules = [\<^rule_thm>\<open>NTH_CONS\<close>,
474 \<^rule_eval>\<open>plus\<close> (**)(eval_binop "#add_"),
475 \<^rule_thm>\<open>NTH_NIL\<close>],
476 scr = Rule.Empty_Prog};
479 section \<open>Methods\<close>
481 method met_eqsys : "EqSystem" =
482 \<open>{rew_ord'="tless_true", rls' = Rule_Set.Empty, calc = [], srls = Rule_Set.Empty, prls = Rule_Set.Empty, crls = Rule_Set.Empty,
483 errpats = [], nrls = Rule_Set.Empty}\<close>
485 method met_eqsys_topdown : "EqSystem/top_down_substitution" =
486 \<open>{rew_ord'="tless_true", rls' = Rule_Set.Empty, calc = [], srls = Rule_Set.Empty, prls = Rule_Set.Empty, crls = Rule_Set.Empty,
487 errpats = [], nrls = Rule_Set.Empty}\<close>
489 partial_function (tailrec) solve_system :: "bool list => real list => bool list"
491 "solve_system e_s v_s = (
495 (Try (Rewrite_Set_Inst [(''bdv_1'', hd v_s), (''bdv_2'', hd (tl v_s))] ''isolate_bdvs'')) #>
496 (Try (Rewrite_Set_Inst [(''bdv_1'', hd v_s), (''bdv_2'', hd (tl v_s))] ''simplify_System''))
498 e_2 = Take (hd (tl e_s));
500 (Substitute [e_1]) #>
501 (Try (Rewrite_Set_Inst [(''bdv_1'', hd v_s), (''bdv_2'', hd (tl v_s))] ''simplify_System_parenthesized'' )) #>
502 (Try (Rewrite_Set_Inst [(''bdv_1'', hd v_s), (''bdv_2'', hd (tl v_s))] ''isolate_bdvs'' )) #>
503 (Try (Rewrite_Set_Inst [(''bdv_1'', hd v_s), (''bdv_2'', hd (tl v_s))] ''simplify_System'' ))
505 e__s = Take [e_1, e_2]
507 Try (Rewrite_Set ''order_system'' ) e__s) "
509 method met_eqsys_topdown_2x2 : "EqSystem/top_down_substitution/2x2" =
510 \<open>{rew_ord'="ord_simplify_System", rls' = Rule_Set.Empty, calc = [],
511 srls = Rule_Set.append_rules "srls_top_down_2x2" Rule_Set.empty
512 [\<^rule_thm>\<open>hd_thm\<close>,
513 \<^rule_thm>\<open>tl_Cons\<close>,
514 \<^rule_thm>\<open>tl_Nil\<close>],
515 prls = prls_triangular, crls = Rule_Set.Empty, errpats = [], nrls = Rule_Set.Empty}\<close>
516 Program: solve_system.simps
517 Given: "equalities e_s" "solveForVars v_s"
519 "(tl v_s) from v_s occur_exactly_in (NTH 1 (e_s::bool list))"
520 " v_s from v_s occur_exactly_in (NTH 2 (e_s::bool list))"
521 Find: "solution ss'''"
523 method met_eqsys_norm : "EqSystem/normalise" =
524 \<open>{rew_ord'="tless_true", rls' = Rule_Set.Empty, calc = [], srls = Rule_Set.Empty, prls = Rule_Set.Empty, crls = Rule_Set.Empty,
525 errpats = [], nrls = Rule_Set.Empty}\<close>
527 partial_function (tailrec) solve_system2 :: "bool list => real list => bool list"
529 "solve_system2 e_s v_s = (
532 (Try (Rewrite_Set ''norm_Rational'' )) #>
533 (Try (Rewrite_Set_Inst [(''bdv_1'', hd v_s), (''bdv_2'', hd (tl v_s))] ''simplify_System_parenthesized'' )) #>
534 (Try (Rewrite_Set_Inst [(''bdv_1'', hd v_s), (''bdv_2'', hd (tl v_s))] ''isolate_bdvs'' )) #>
535 (Try (Rewrite_Set_Inst [(''bdv_1'', hd v_s), (''bdv_2'', hd (tl v_s))] ''simplify_System_parenthesized'' )) #>
536 (Try (Rewrite_Set ''order_system'' ))
539 SubProblem (''EqSystem'', [''LINEAR'', ''system''], [''no_met''])
540 [BOOL_LIST e__s, REAL_LIST v_s])"
542 method met_eqsys_norm_2x2 : "EqSystem/normalise/2x2" =
543 \<open>{rew_ord'="tless_true", rls' = Rule_Set.Empty, calc = [],
544 srls = Rule_Set.append_rules "srls_normalise_2x2" Rule_Set.empty
545 [\<^rule_thm>\<open>hd_thm\<close>,
546 \<^rule_thm>\<open>tl_Cons\<close>,
547 \<^rule_thm>\<open>tl_Nil\<close>],
548 prls = Rule_Set.Empty, crls = Rule_Set.Empty, errpats = [], nrls = Rule_Set.Empty}\<close>
549 Program: solve_system2.simps
550 Given: "equalities e_s" "solveForVars v_s"
551 Find: "solution ss'''"
553 partial_function (tailrec) solve_system3 :: "bool list => real list => bool list"
555 "solve_system3 e_s v_s = (
558 (Try (Rewrite_Set ''norm_Rational'' )) #>
559 (Repeat (Rewrite ''commute_0_equality'' )) #>
560 (Try (Rewrite_Set_Inst [(''bdv_1'', NTH 1 v_s), (''bdv_2'', NTH 2 v_s ),
561 (''bdv_3'', NTH 3 v_s), (''bdv_3'', NTH 4 v_s )] ''simplify_System_parenthesized'' )) #>
562 (Try (Rewrite_Set_Inst [(''bdv_1'', NTH 1 v_s), (''bdv_2'', NTH 2 v_s ),
563 (''bdv_3'', NTH 3 v_s), (''bdv_3'', NTH 4 v_s )] ''isolate_bdvs_4x4'' )) #>
564 (Try (Rewrite_Set_Inst [(''bdv_1'', NTH 1 v_s), (''bdv_2'', NTH 2 v_s ),
565 (''bdv_3'', NTH 3 v_s), (''bdv_3'', NTH 4 v_s )] ''simplify_System_parenthesized'' )) #>
566 (Try (Rewrite_Set ''order_system''))
569 SubProblem (''EqSystem'', [''LINEAR'', ''system''], [''no_met''])
570 [BOOL_LIST e__s, REAL_LIST v_s])"
572 method met_eqsys_norm_4x4 : "EqSystem/normalise/4x4" =
573 \<open>{rew_ord'="tless_true", rls' = Rule_Set.Empty, calc = [],
575 Rule_Set.append_rules "srls_normalise_4x4" srls
576 [\<^rule_thm>\<open>hd_thm\<close>, \<^rule_thm>\<open>tl_Cons\<close>, \<^rule_thm>\<open>tl_Nil\<close>],
577 prls = Rule_Set.Empty, crls = Rule_Set.Empty, errpats = [], nrls = Rule_Set.Empty}\<close>
578 Program: solve_system3.simps
579 Given: "equalities e_s" "solveForVars v_s"
580 Find: "solution ss'''"
581 (*STOPPED.WN06? met ["EqSystem", "normalise", "4x4"] #>#>#>#>#>#>#>#>#>#>#>#>#>@*)
583 partial_function (tailrec) solve_system4 :: "bool list => real list => bool list"
585 "solve_system4 e_s v_s = (
588 e_2 = Take (NTH 2 e_s);
590 (Substitute [e_1]) #>
591 (Try (Rewrite_Set_Inst [(''bdv_1'',NTH 1 v_s), (''bdv_2'',NTH 2 v_s),
592 (''bdv_3'',NTH 3 v_s), (''bdv_4'',NTH 4 v_s)] ''simplify_System_parenthesized'' )) #>
593 (Try (Rewrite_Set_Inst [(''bdv_1'',NTH 1 v_s), (''bdv_2'',NTH 2 v_s),
594 (''bdv_3'',NTH 3 v_s), (''bdv_4'',NTH 4 v_s)] ''isolate_bdvs'' )) #>
595 (Try (Rewrite_Set_Inst [(''bdv_1'',NTH 1 v_s), (''bdv_2'',NTH 2 v_s),
596 (''bdv_3'',NTH 3 v_s), (''bdv_4'',NTH 4 v_s)] ''norm_Rational'' ))
599 [e_1, e_2, NTH 3 e_s, NTH 4 e_s])"
601 method met_eqsys_topdown_4x4 : "EqSystem/top_down_substitution/4x4" =
602 \<open>{rew_ord'="ord_simplify_System", rls' = Rule_Set.Empty, calc = [],
603 srls = Rule_Set.append_rules "srls_top_down_4x4" srls [],
604 prls = Rule_Set.append_rules "prls_tri_4x4_lin_sys" prls_triangular
605 [\<^rule_eval>\<open>Prog_Expr.occurs_in\<close> (Prog_Expr.eval_occurs_in "")],
606 crls = Rule_Set.Empty, errpats = [], nrls = Rule_Set.Empty}\<close>
607 (*FIXXXXME.WN060916: this script works ONLY for exp 7.79 #>#>#>#>#>#>#>#>#>#>*)
608 Program: solve_system4.simps
609 Given: "equalities e_s" "solveForVars v_s"
610 Where: (*accepts missing variables up to diagonal form*)
611 "(NTH 1 (v_s::real list)) occurs_in (NTH 1 (e_s::bool list))"
612 "(NTH 2 (v_s::real list)) occurs_in (NTH 2 (e_s::bool list))"
613 "(NTH 3 (v_s::real list)) occurs_in (NTH 3 (e_s::bool list))"
614 "(NTH 4 (v_s::real list)) occurs_in (NTH 4 (e_s::bool list))"
615 Find: "solution ss'''"