complete replacement of Rule.Thm/Eval by \<^rule_thm> and \<^rule_eval> in src/*
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 =
173 id = "order_add_mult_System", preconds = [],
174 rew_ord = ("ord_simplify_System", ord_simplify_System false @{theory "Integrate"}),
175 erls = Rule_Set.empty,srls = Rule_Set.Empty, calc = [], errpatts = [],
177 \<^rule_thm>\<open>mult.commute\<close>, (* z * w = w * z *)
178 \<^rule_thm>\<open>real_mult_left_commute\<close>, (*z1.0 * (z2.0 * z3.0) = z2.0 * (z1.0 * z3.0)*)
179 \<^rule_thm>\<open>mult.assoc\<close>, (*z1.0 * z2.0 * z3.0 = z1.0 * (z2.0 * z3.0)*)
180 \<^rule_thm>\<open>add.commute\<close>, (*z + w = w + z*)
181 \<^rule_thm>\<open>add.left_commute\<close>, (*x + (y + z) = y + (x + z)*)
182 \<^rule_thm>\<open>add.assoc\<close> ], (*z1.0 + z2.0 + z3.0 = z1.0 + (z2.0 + z3.0)*)
183 scr = Rule.Empty_Prog};
186 (*.adapted from 'norm_Rational' by
187 #1 using 'ord_simplify_System' in 'order_add_mult_System'
188 #2 NOT using common_nominator_p .*)
189 val norm_System_noadd_fractions =
190 Rule_Def.Repeat {id = "norm_System_noadd_fractions", preconds = [],
191 rew_ord = ("dummy_ord", Rewrite_Ord.dummy_ord),
192 erls = norm_rat_erls, srls = Rule_Set.Empty, calc = [], errpatts = [],
193 rules = [(*sequence given by operator precedence*)
194 Rule.Rls_ discard_minus,
196 Rule.Rls_ rat_mult_divide,
198 Rule.Rls_ reduce_0_1_2,
199 Rule.Rls_ (*order_add_mult #1*) order_add_mult_System,
200 Rule.Rls_ collect_numerals,
201 (*Rule.Rls_ add_fractions_p, #2*)
203 scr = Rule.Empty_Prog};
206 (*.adapted from 'norm_Rational' by
207 *1* using 'ord_simplify_System' in 'order_add_mult_System'.*)
209 Rule_Def.Repeat {id = "norm_System", preconds = [],
210 rew_ord = ("dummy_ord", Rewrite_Ord.dummy_ord),
211 erls = norm_rat_erls, srls = Rule_Set.Empty, calc = [], errpatts = [],
212 rules = [(*sequence given by operator precedence*)
213 Rule.Rls_ discard_minus,
215 Rule.Rls_ rat_mult_divide,
217 Rule.Rls_ reduce_0_1_2,
218 Rule.Rls_ (*order_add_mult *1*) order_add_mult_System,
219 Rule.Rls_ collect_numerals,
220 Rule.Rls_ add_fractions_p,
222 scr = Rule.Empty_Prog};
225 (*.simplify an equational system BEFORE solving it such that parentheses are
226 ( ((u0*v0)*w0) + ( ((u1*v1)*w1) * c + ... +((u4*v4)*w4) * c_4 ) )
227 ATTENTION: works ONLY for bound variables c, c_1, c_2, c_3, c_4 :ATTENTION
228 This is a copy from 'make_ratpoly_in' with respective reductions:
229 *0* expand the term, ie. distribute * and / over +
230 *1* ord_simplify_System instead of termlessI
231 *2* no add_fractions_p (= common_nominator_p_rls !)
232 *3* discard_parentheses only for (.*(.*.))
233 analoguous to simplify_Integral .*)
234 val simplify_System_parenthesized =
235 Rule_Set.Sequence {id = "simplify_System_parenthesized", preconds = []:term list,
236 rew_ord = ("dummy_ord", Rewrite_Ord.dummy_ord),
237 erls = Atools_erls, srls = Rule_Set.Empty, calc = [], errpatts = [],
239 \<^rule_thm>\<open>distrib_right\<close>, (*"(?z1.0 + ?z2.0) * ?w = ?z1.0 * ?w + ?z2.0 * ?w"*)
240 \<^rule_thm>\<open>add_divide_distrib\<close>, (*"(?x + ?y) / ?z = ?x / ?z + ?y / ?z"*)
241 Rule.Rls_ norm_Rational_noadd_fractions,
242 Rule.Rls_ (*order_add_mult_in*) norm_System_noadd_fractions,
243 \<^rule_thm_sym>\<open>mult.assoc\<close>,
244 Rule.Rls_ collect_bdv, (*from make_polynomial_in WN051031 welldone?*)
245 Rule.Rls_ separate_bdv2,
246 \<^rule_eval>\<open>divide\<close> (Prog_Expr.eval_cancel "#divide_e")],
247 scr = Rule.Empty_Prog};
250 (*.simplify an equational system AFTER solving it;
251 This is a copy of 'make_ratpoly_in' with the differences
252 *1* ord_simplify_System instead of termlessI .*)
253 (*TODO.WN051031 ^^^^^^^^^^ should be in EACH rls contained *)
254 val simplify_System =
255 Rule_Set.Sequence {id = "simplify_System", preconds = []:term list,
256 rew_ord = ("dummy_ord", Rewrite_Ord.dummy_ord),
257 erls = Atools_erls, srls = Rule_Set.Empty, calc = [], errpatts = [],
259 Rule.Rls_ norm_Rational,
260 Rule.Rls_ (*order_add_mult_in*) norm_System (**1**),
261 Rule.Rls_ discard_parentheses,
262 Rule.Rls_ collect_bdv, (*from make_polynomial_in WN051031 welldone?*)
263 Rule.Rls_ separate_bdv2,
264 \<^rule_eval>\<open>divide\<close> (Prog_Expr.eval_cancel "#divide_e")],
265 scr = Rule.Empty_Prog};
267 val simplify_System =
268 Rule_Set.append_rules "simplify_System" simplify_System_parenthesized
269 [\<^rule_thm_sym>\<open>add.assoc\<close>];
275 id="isolate_bdvs", preconds = [], rew_ord = ("e_rew_ord", Rewrite_Ord.e_rew_ord),
276 erls = Rule_Set.append_rules "erls_isolate_bdvs" Rule_Set.empty [
277 (\<^rule_eval>\<open>occur_exactly_in\<close> (eval_occur_exactly_in "#eval_occur_exactly_in_"))],
278 srls = Rule_Set.Empty, calc = [], errpatts = [],
280 \<^rule_thm>\<open>commute_0_equality\<close>,
281 \<^rule_thm>\<open>separate_bdvs_add\<close>,
282 \<^rule_thm>\<open>separate_bdvs_mult\<close>],
283 scr = Rule.Empty_Prog};
286 val isolate_bdvs_4x4 =
287 Rule_Def.Repeat {id="isolate_bdvs_4x4", preconds = [],
288 rew_ord = ("e_rew_ord", Rewrite_Ord.e_rew_ord),
289 erls = Rule_Set.append_rules "erls_isolate_bdvs_4x4" Rule_Set.empty [
290 \<^rule_eval>\<open>occur_exactly_in\<close> (eval_occur_exactly_in "#eval_occur_exactly_in_"),
291 \<^rule_eval>\<open>Prog_Expr.ident\<close> (Prog_Expr.eval_ident "#ident_"),
292 \<^rule_eval>\<open>Prog_Expr.some_occur_in\<close> (Prog_Expr.eval_some_occur_in "#some_occur_in_"),
293 \<^rule_thm>\<open>not_true\<close>,
294 \<^rule_thm>\<open>not_false\<close>],
295 srls = Rule_Set.Empty, calc = [], errpatts = [],
297 \<^rule_thm>\<open>commute_0_equality\<close>,
298 \<^rule_thm>\<open>separate_bdvs0\<close>,
299 \<^rule_thm>\<open>separate_bdvs_add1\<close>,
300 \<^rule_thm>\<open>separate_bdvs_add2\<close>,
301 \<^rule_thm>\<open>separate_bdvs_mult\<close>],
302 scr = Rule.Empty_Prog};
307 (*.order the equations in a system such, that a triangular system (if any)
308 appears as [..c_4 = .., ..., ..., ..c_1 + ..c_2 + ..c_3 ..c_4 = ..].*)
310 Rule_Def.Repeat {id="order_system", preconds = [],
311 rew_ord = ("ord_simplify_System", ord_simplify_System false \<^theory>),
312 erls = Rule_Set.Empty, srls = Rule_Set.Empty, calc = [], errpatts = [],
314 \<^rule_thm>\<open>order_system_NxN\<close>],
315 scr = Rule.Empty_Prog};
317 val prls_triangular =
319 id="prls_triangular", preconds = [], rew_ord = ("e_rew_ord", Rewrite_Ord.e_rew_ord),
320 erls = Rule_Def.Repeat {
321 id="erls_prls_triangular", preconds = [], rew_ord = ("e_rew_ord", Rewrite_Ord.e_rew_ord),
322 erls = Rule_Set.Empty, srls = Rule_Set.Empty, calc = [], errpatts = [],
323 rules = [(*for precond NTH_CONS ...*)
324 \<^rule_eval>\<open>less\<close> (Prog_Expr.eval_equ "#less_"),
325 \<^rule_eval>\<open>plus\<close> (**)(eval_binop "#add_"),
326 \<^rule_eval>\<open>occur_exactly_in\<close> (eval_occur_exactly_in "#eval_occur_exactly_in_")],
327 (*immediately repeated rewrite pushes '+' into precondition !*)
328 scr = Rule.Empty_Prog},
329 srls = Rule_Set.Empty, calc = [], errpatts = [],
331 \<^rule_thm>\<open>NTH_CONS\<close>,
332 \<^rule_eval>\<open>plus\<close> (**)(eval_binop "#add_"),
333 \<^rule_thm>\<open>NTH_NIL\<close>,
334 \<^rule_thm>\<open>tl_Cons\<close>,
335 \<^rule_thm>\<open>tl_Nil\<close>,
336 \<^rule_eval>\<open>occur_exactly_in\<close> (eval_occur_exactly_in "#eval_occur_exactly_in_")],
337 scr = Rule.Empty_Prog};
341 (*WN060914 quickly created for 4x4;
342 more similarity to prls_triangular desirable*)
343 val prls_triangular4 =
345 id="prls_triangular4", preconds = [], rew_ord = ("e_rew_ord", Rewrite_Ord.e_rew_ord),
346 erls = Rule_Def.Repeat {
347 id="erls_prls_triangular4", preconds = [], rew_ord = ("e_rew_ord", Rewrite_Ord.e_rew_ord),
348 erls = Rule_Set.Empty, srls = Rule_Set.Empty, calc = [], errpatts = [],
349 rules = [(*for precond NTH_CONS ...*)
350 \<^rule_eval>\<open>less\<close> (Prog_Expr.eval_equ "#less_"),
351 \<^rule_eval>\<open>plus\<close> (**)(eval_binop "#add_")],
352 (*immediately repeated rewrite pushes '+' into precondition !*)
353 scr = Rule.Empty_Prog},
354 srls = Rule_Set.Empty, calc = [], errpatts = [],
356 \<^rule_thm>\<open>NTH_CONS\<close>,
357 \<^rule_eval>\<open>plus\<close> (**)(eval_binop "#add_"),
358 \<^rule_thm>\<open>NTH_NIL\<close>,
359 \<^rule_thm>\<open>tl_Cons\<close>,
360 \<^rule_thm>\<open>tl_Nil\<close>,
361 \<^rule_eval>\<open>occur_exactly_in\<close> (eval_occur_exactly_in "#eval_occur_exactly_in_")],
362 scr = Rule.Empty_Prog};
366 simplify_System_parenthesized = \<open>prep_rls' simplify_System_parenthesized\<close> and
367 simplify_System = \<open>prep_rls' simplify_System\<close> and
368 isolate_bdvs = \<open>prep_rls' isolate_bdvs\<close> and
369 isolate_bdvs_4x4 = \<open>prep_rls' isolate_bdvs_4x4\<close> and
370 order_system = \<open>prep_rls' order_system\<close> and
371 order_add_mult_System = \<open>prep_rls' order_add_mult_System\<close> and
372 norm_System_noadd_fractions = \<open>prep_rls' norm_System_noadd_fractions\<close> and
373 norm_System = \<open>prep_rls' norm_System\<close>
376 section \<open>Problems\<close>
378 problem pbl_equsys : "system" =
379 \<open>Rule_Set.append_rules "empty" Rule_Set.empty [(*for preds in where_*)]\<close>
380 CAS: "solveSystem e_s v_s"
381 Given: "equalities e_s" "solveForVars v_s"
382 Find: "solution ss'''" (*''' is copy-named*)
384 problem pbl_equsys_lin : "LINEAR/system" =
385 \<open>Rule_Set.append_rules "empty" Rule_Set.empty [(*for preds in where_*)]\<close>
386 CAS: "solveSystem e_s v_s"
387 Given: "equalities e_s" "solveForVars v_s"
388 (*TODO.WN050929 check linearity*)
389 Find: "solution ss'''"
391 problem pbl_equsys_lin_2x2: "2x2/LINEAR/system" =
392 \<open>Rule_Set.append_rules "prls_2x2_linear_system" Rule_Set.empty
393 [\<^rule_thm>\<open>LENGTH_CONS\<close>,
394 \<^rule_thm>\<open>LENGTH_NIL\<close>,
395 \<^rule_eval>\<open>plus\<close> (**)(eval_binop "#add_"),
396 \<^rule_eval>\<open>HOL.eq\<close> (Prog_Expr.eval_equal "#equal_")]\<close>
397 CAS: "solveSystem e_s v_s"
398 Given: "equalities e_s" "solveForVars v_s"
399 Where: "Length (e_s:: bool list) = 2" "Length v_s = 2"
400 Find: "solution ss'''"
402 problem pbl_equsys_lin_2x2_tri : "triangular/2x2/LINEAR/system" =
403 \<open>prls_triangular\<close>
404 Method: "EqSystem/top_down_substitution/2x2"
405 CAS: "solveSystem e_s v_s"
406 Given: "equalities e_s" "solveForVars v_s"
408 "(tl v_s) from v_s occur_exactly_in (NTH 1 (e_s::bool list))"
409 " v_s from v_s occur_exactly_in (NTH 2 (e_s::bool list))"
410 Find: "solution ss'''"
412 problem pbl_equsys_lin_2x2_norm : "normalise/2x2/LINEAR/system" =
413 \<open>Rule_Set.append_rules "empty" Rule_Set.empty [(*for preds in where_*)]\<close>
414 Method: "EqSystem/normalise/2x2"
415 CAS: "solveSystem e_s v_s"
416 Given: "equalities e_s" "solveForVars v_s"
417 Find: "solution ss'''"
419 problem pbl_equsys_lin_3x3 : "3x3/LINEAR/system" =
420 \<open>Rule_Set.append_rules "prls_3x3_linear_system" Rule_Set.empty
421 [\<^rule_thm>\<open>LENGTH_CONS\<close>,
422 \<^rule_thm>\<open>LENGTH_NIL\<close>,
423 \<^rule_eval>\<open>plus\<close> (**)(eval_binop "#add_"),
424 \<^rule_eval>\<open>HOL.eq\<close> (Prog_Expr.eval_equal "#equal_")]\<close>
425 CAS: "solveSystem e_s v_s"
426 Given: "equalities e_s" "solveForVars v_s"
427 Where: "Length (e_s:: bool list) = 3" "Length v_s = 3"
428 Find: "solution ss'''"
430 problem pbl_equsys_lin_4x4 : "4x4/LINEAR/system" =
431 \<open>Rule_Set.append_rules "prls_4x4_linear_system" Rule_Set.empty
432 [\<^rule_thm>\<open>LENGTH_CONS\<close>,
433 \<^rule_thm>\<open>LENGTH_NIL\<close>,
434 \<^rule_eval>\<open>plus\<close> (**)(eval_binop "#add_"),
435 \<^rule_eval>\<open>HOL.eq\<close> (Prog_Expr.eval_equal "#equal_")]\<close>
436 CAS: "solveSystem e_s v_s"
437 Given: "equalities e_s" "solveForVars v_s"
438 Where: "Length (e_s:: bool list) = 4" "Length v_s = 4"
439 Find: "solution ss'''"
441 problem pbl_equsys_lin_4x4_tri : "triangular/4x4/LINEAR/system" =
442 \<open>Rule_Set.append_rules "prls_tri_4x4_lin_sys" prls_triangular
443 [\<^rule_eval>\<open>Prog_Expr.occurs_in\<close> (Prog_Expr.eval_occurs_in "")]\<close>
444 Method: "EqSystem/top_down_substitution/4x4"
445 CAS: "solveSystem e_s v_s"
446 Given: "equalities e_s" "solveForVars v_s"
447 Where: (*accepts missing variables up to diagional form*)
448 "(NTH 1 (v_s::real list)) occurs_in (NTH 1 (e_s::bool list))"
449 "(NTH 2 (v_s::real list)) occurs_in (NTH 2 (e_s::bool list))"
450 "(NTH 3 (v_s::real list)) occurs_in (NTH 3 (e_s::bool list))"
451 "(NTH 4 (v_s::real list)) occurs_in (NTH 4 (e_s::bool list))"
452 Find: "solution ss'''"
454 problem pbl_equsys_lin_4x4_norm : "normalise/4x4/LINEAR/system" =
455 \<open>Rule_Set.append_rules "empty" Rule_Set.empty [(*for preds in where_*)]\<close>
456 Method: "EqSystem/normalise/4x4"
457 CAS: "solveSystem e_s v_s"
458 Given: "equalities e_s" "solveForVars v_s"
459 (*Length is checked 1 level above*)
460 Find: "solution ss'''"
463 (*this is for NTH only*)
464 val srls = Rule_Def.Repeat {id="srls_normalise_4x4",
466 rew_ord = ("termlessI",termlessI),
467 erls = Rule_Set.append_rules "erls_in_srls_IntegrierenUnd.." Rule_Set.empty
468 [(*for asm in NTH_CONS ...*)
469 \<^rule_eval>\<open>less\<close> (Prog_Expr.eval_equ "#less_"),
470 (*2nd NTH_CONS pushes n+-1 into asms*)
471 \<^rule_eval>\<open>plus\<close> (**)(eval_binop "#add_")
473 srls = Rule_Set.Empty, calc = [], errpatts = [],
474 rules = [\<^rule_thm>\<open>NTH_CONS\<close>,
475 \<^rule_eval>\<open>plus\<close> (**)(eval_binop "#add_"),
476 \<^rule_thm>\<open>NTH_NIL\<close>],
477 scr = Rule.Empty_Prog};
480 section \<open>Methods\<close>
482 method met_eqsys : "EqSystem" =
483 \<open>{rew_ord'="tless_true", rls' = Rule_Set.Empty, calc = [], srls = Rule_Set.Empty, prls = Rule_Set.Empty, crls = Rule_Set.Empty,
484 errpats = [], nrls = Rule_Set.Empty}\<close>
486 method met_eqsys_topdown : "EqSystem/top_down_substitution" =
487 \<open>{rew_ord'="tless_true", rls' = Rule_Set.Empty, calc = [], srls = Rule_Set.Empty, prls = Rule_Set.Empty, crls = Rule_Set.Empty,
488 errpats = [], nrls = Rule_Set.Empty}\<close>
490 partial_function (tailrec) solve_system :: "bool list => real list => bool list"
492 "solve_system e_s v_s = (
496 (Try (Rewrite_Set_Inst [(''bdv_1'', hd v_s), (''bdv_2'', hd (tl v_s))] ''isolate_bdvs'')) #>
497 (Try (Rewrite_Set_Inst [(''bdv_1'', hd v_s), (''bdv_2'', hd (tl v_s))] ''simplify_System''))
499 e_2 = Take (hd (tl e_s));
501 (Substitute [e_1]) #>
502 (Try (Rewrite_Set_Inst [(''bdv_1'', hd v_s), (''bdv_2'', hd (tl v_s))] ''simplify_System_parenthesized'' )) #>
503 (Try (Rewrite_Set_Inst [(''bdv_1'', hd v_s), (''bdv_2'', hd (tl v_s))] ''isolate_bdvs'' )) #>
504 (Try (Rewrite_Set_Inst [(''bdv_1'', hd v_s), (''bdv_2'', hd (tl v_s))] ''simplify_System'' ))
506 e__s = Take [e_1, e_2]
508 Try (Rewrite_Set ''order_system'' ) e__s) "
510 method met_eqsys_topdown_2x2 : "EqSystem/top_down_substitution/2x2" =
511 \<open>{rew_ord'="ord_simplify_System", rls' = Rule_Set.Empty, calc = [],
512 srls = Rule_Set.append_rules "srls_top_down_2x2" Rule_Set.empty
513 [\<^rule_thm>\<open>hd_thm\<close>,
514 \<^rule_thm>\<open>tl_Cons\<close>,
515 \<^rule_thm>\<open>tl_Nil\<close>],
516 prls = prls_triangular, crls = Rule_Set.Empty, errpats = [], nrls = Rule_Set.Empty}\<close>
517 Program: solve_system.simps
518 Given: "equalities e_s" "solveForVars v_s"
520 "(tl v_s) from v_s occur_exactly_in (NTH 1 (e_s::bool list))"
521 " v_s from v_s occur_exactly_in (NTH 2 (e_s::bool list))"
522 Find: "solution ss'''"
524 method met_eqsys_norm : "EqSystem/normalise" =
525 \<open>{rew_ord'="tless_true", rls' = Rule_Set.Empty, calc = [], srls = Rule_Set.Empty, prls = Rule_Set.Empty, crls = Rule_Set.Empty,
526 errpats = [], nrls = Rule_Set.Empty}\<close>
528 partial_function (tailrec) solve_system2 :: "bool list => real list => bool list"
530 "solve_system2 e_s v_s = (
533 (Try (Rewrite_Set ''norm_Rational'' )) #>
534 (Try (Rewrite_Set_Inst [(''bdv_1'', hd v_s), (''bdv_2'', hd (tl v_s))] ''simplify_System_parenthesized'' )) #>
535 (Try (Rewrite_Set_Inst [(''bdv_1'', hd v_s), (''bdv_2'', hd (tl v_s))] ''isolate_bdvs'' )) #>
536 (Try (Rewrite_Set_Inst [(''bdv_1'', hd v_s), (''bdv_2'', hd (tl v_s))] ''simplify_System_parenthesized'' )) #>
537 (Try (Rewrite_Set ''order_system'' ))
540 SubProblem (''EqSystem'', [''LINEAR'', ''system''], [''no_met''])
541 [BOOL_LIST e__s, REAL_LIST v_s])"
543 method met_eqsys_norm_2x2 : "EqSystem/normalise/2x2" =
544 \<open>{rew_ord'="tless_true", rls' = Rule_Set.Empty, calc = [],
545 srls = Rule_Set.append_rules "srls_normalise_2x2" Rule_Set.empty
546 [\<^rule_thm>\<open>hd_thm\<close>,
547 \<^rule_thm>\<open>tl_Cons\<close>,
548 \<^rule_thm>\<open>tl_Nil\<close>],
549 prls = Rule_Set.Empty, crls = Rule_Set.Empty, errpats = [], nrls = Rule_Set.Empty}\<close>
550 Program: solve_system2.simps
551 Given: "equalities e_s" "solveForVars v_s"
552 Find: "solution ss'''"
554 partial_function (tailrec) solve_system3 :: "bool list => real list => bool list"
556 "solve_system3 e_s v_s = (
559 (Try (Rewrite_Set ''norm_Rational'' )) #>
560 (Repeat (Rewrite ''commute_0_equality'' )) #>
561 (Try (Rewrite_Set_Inst [(''bdv_1'', NTH 1 v_s), (''bdv_2'', NTH 2 v_s ),
562 (''bdv_3'', NTH 3 v_s), (''bdv_3'', NTH 4 v_s )] ''simplify_System_parenthesized'' )) #>
563 (Try (Rewrite_Set_Inst [(''bdv_1'', NTH 1 v_s), (''bdv_2'', NTH 2 v_s ),
564 (''bdv_3'', NTH 3 v_s), (''bdv_3'', NTH 4 v_s )] ''isolate_bdvs_4x4'' )) #>
565 (Try (Rewrite_Set_Inst [(''bdv_1'', NTH 1 v_s), (''bdv_2'', NTH 2 v_s ),
566 (''bdv_3'', NTH 3 v_s), (''bdv_3'', NTH 4 v_s )] ''simplify_System_parenthesized'' )) #>
567 (Try (Rewrite_Set ''order_system''))
570 SubProblem (''EqSystem'', [''LINEAR'', ''system''], [''no_met''])
571 [BOOL_LIST e__s, REAL_LIST v_s])"
573 method met_eqsys_norm_4x4 : "EqSystem/normalise/4x4" =
574 \<open>{rew_ord'="tless_true", rls' = Rule_Set.Empty, calc = [],
576 Rule_Set.append_rules "srls_normalise_4x4" srls
577 [\<^rule_thm>\<open>hd_thm\<close>, \<^rule_thm>\<open>tl_Cons\<close>, \<^rule_thm>\<open>tl_Nil\<close>],
578 prls = Rule_Set.Empty, crls = Rule_Set.Empty, errpats = [], nrls = Rule_Set.Empty}\<close>
579 Program: solve_system3.simps
580 Given: "equalities e_s" "solveForVars v_s"
581 Find: "solution ss'''"
582 (*STOPPED.WN06? met ["EqSystem", "normalise", "4x4"] #>#>#>#>#>#>#>#>#>#>#>#>#>@*)
584 partial_function (tailrec) solve_system4 :: "bool list => real list => bool list"
586 "solve_system4 e_s v_s = (
589 e_2 = Take (NTH 2 e_s);
591 (Substitute [e_1]) #>
592 (Try (Rewrite_Set_Inst [(''bdv_1'',NTH 1 v_s), (''bdv_2'',NTH 2 v_s),
593 (''bdv_3'',NTH 3 v_s), (''bdv_4'',NTH 4 v_s)] ''simplify_System_parenthesized'' )) #>
594 (Try (Rewrite_Set_Inst [(''bdv_1'',NTH 1 v_s), (''bdv_2'',NTH 2 v_s),
595 (''bdv_3'',NTH 3 v_s), (''bdv_4'',NTH 4 v_s)] ''isolate_bdvs'' )) #>
596 (Try (Rewrite_Set_Inst [(''bdv_1'',NTH 1 v_s), (''bdv_2'',NTH 2 v_s),
597 (''bdv_3'',NTH 3 v_s), (''bdv_4'',NTH 4 v_s)] ''norm_Rational'' ))
600 [e_1, e_2, NTH 3 e_s, NTH 4 e_s])"
602 method met_eqsys_topdown_4x4 : "EqSystem/top_down_substitution/4x4" =
603 \<open>{rew_ord'="ord_simplify_System", rls' = Rule_Set.Empty, calc = [],
604 srls = Rule_Set.append_rules "srls_top_down_4x4" srls [],
605 prls = Rule_Set.append_rules "prls_tri_4x4_lin_sys" prls_triangular
606 [\<^rule_eval>\<open>Prog_Expr.occurs_in\<close> (Prog_Expr.eval_occurs_in "")],
607 crls = Rule_Set.Empty, errpats = [], nrls = Rule_Set.Empty}\<close>
608 (*FIXXXXME.WN060916: this script works ONLY for exp 7.79 #>#>#>#>#>#>#>#>#>#>*)
609 Program: solve_system4.simps
610 Given: "equalities e_s" "solveForVars v_s"
611 Where: (*accepts missing variables up to diagonal form*)
612 "(NTH 1 (v_s::real list)) occurs_in (NTH 1 (e_s::bool list))"
613 "(NTH 2 (v_s::real list)) occurs_in (NTH 2 (e_s::bool list))"
614 "(NTH 3 (v_s::real list)) occurs_in (NTH 3 (e_s::bool list))"
615 "(NTH 4 (v_s::real list)) occurs_in (NTH 4 (e_s::bool list))"
616 Find: "solution ss'''"