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 ("EqSystem.occur_exactly_in",_)
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 setup \<open>KEStore_Elems.add_calcs
81 ("EqSystem.occur_exactly_in",
82 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*) tu =
159 (term_ord' pr thy tu = 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 ("mult.commute",ThmC.numerals_to_Free @{thm mult.commute}),
178 Rule.Thm ("real_mult_left_commute",ThmC.numerals_to_Free @{thm real_mult_left_commute}),
179 (*z1.0 * (z2.0 * z3.0) = z2.0 * (z1.0 * z3.0)*)
180 Rule.Thm ("mult.assoc",ThmC.numerals_to_Free @{thm mult.assoc}),
181 (*z1.0 * z2.0 * z3.0 = z1.0 * (z2.0 * z3.0)*)
182 Rule.Thm ("add.commute",ThmC.numerals_to_Free @{thm add.commute}),
184 Rule.Thm ("add.left_commute",ThmC.numerals_to_Free @{thm add.left_commute}),
185 (*x + (y + z) = y + (x + z)*)
186 Rule.Thm ("add.assoc",ThmC.numerals_to_Free @{thm add.assoc})
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 ("distrib_right",ThmC.numerals_to_Free @{thm distrib_right}),
249 (*"(?z1.0 + ?z2.0) * ?w = ?z1.0 * ?w + ?z2.0 * ?w"*)
250 Rule.Thm ("add_divide_distrib",ThmC.numerals_to_Free @{thm add_divide_distrib}),
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_mult.assoc",
256 ThmC.numerals_to_Free (@{thm mult.assoc} RS @{thm sym}))
257 (*Rule.Rls_ discard_parentheses *3**),
258 Rule.Rls_ collect_bdv, (*from make_polynomial_in WN051031 welldone?*)
259 Rule.Rls_ separate_bdv2,
260 \<^rule_eval>\<open>divide\<close> (Prog_Expr.eval_cancel "#divide_e")
262 scr = Rule.Empty_Prog};
265 (*.simplify an equational system AFTER solving it;
266 This is a copy of 'make_ratpoly_in' with the differences
267 *1* ord_simplify_System instead of termlessI .*)
268 (*TODO.WN051031 ^^^^^^^^^^ should be in EACH rls contained *)
269 val simplify_System =
270 Rule_Set.Sequence {id = "simplify_System", preconds = []:term list,
271 rew_ord = ("dummy_ord", Rewrite_Ord.dummy_ord),
272 erls = Atools_erls, srls = Rule_Set.Empty, calc = [], errpatts = [],
273 rules = [Rule.Rls_ norm_Rational,
274 Rule.Rls_ (*order_add_mult_in*) norm_System (**1**),
275 Rule.Rls_ discard_parentheses,
276 Rule.Rls_ collect_bdv, (*from make_polynomial_in WN051031 welldone?*)
277 Rule.Rls_ separate_bdv2,
278 \<^rule_eval>\<open>divide\<close> (Prog_Expr.eval_cancel "#divide_e")
280 scr = Rule.Empty_Prog};
282 val simplify_System =
283 Rule_Set.append_rules "simplify_System" simplify_System_parenthesized
284 [Rule.Thm ("sym_add.assoc",
285 ThmC.numerals_to_Free (@{thm add.assoc} RS @{thm sym}))];
290 Rule_Def.Repeat {id="isolate_bdvs", preconds = [],
291 rew_ord = ("e_rew_ord", Rewrite_Ord.e_rew_ord),
292 erls = Rule_Set.append_rules "erls_isolate_bdvs" Rule_Set.empty
293 [(\<^rule_eval>\<open>occur_exactly_in\<close> (eval_occur_exactly_in "#eval_occur_exactly_in_"))],
294 srls = Rule_Set.Empty, calc = [], errpatts = [],
296 [Rule.Thm ("commute_0_equality", ThmC.numerals_to_Free @{thm commute_0_equality}),
297 Rule.Thm ("separate_bdvs_add", ThmC.numerals_to_Free @{thm separate_bdvs_add}),
298 Rule.Thm ("separate_bdvs_mult", ThmC.numerals_to_Free @{thm separate_bdvs_mult})],
299 scr = Rule.Empty_Prog};
302 val isolate_bdvs_4x4 =
303 Rule_Def.Repeat {id="isolate_bdvs_4x4", preconds = [],
304 rew_ord = ("e_rew_ord", Rewrite_Ord.e_rew_ord),
305 erls = Rule_Set.append_rules
306 "erls_isolate_bdvs_4x4" Rule_Set.empty
307 [\<^rule_eval>\<open>occur_exactly_in\<close> (eval_occur_exactly_in "#eval_occur_exactly_in_"),
308 \<^rule_eval>\<open>Prog_Expr.ident\<close> (Prog_Expr.eval_ident "#ident_"),
309 \<^rule_eval>\<open>Prog_Expr.some_occur_in\<close> (Prog_Expr.eval_some_occur_in "#some_occur_in_"),
310 Rule.Thm ("not_true",ThmC.numerals_to_Free @{thm not_true}),
311 Rule.Thm ("not_false",ThmC.numerals_to_Free @{thm not_false})
313 srls = Rule_Set.Empty, calc = [], errpatts = [],
314 rules = [Rule.Thm ("commute_0_equality", ThmC.numerals_to_Free @{thm commute_0_equality}),
315 Rule.Thm ("separate_bdvs0", ThmC.numerals_to_Free @{thm separate_bdvs0}),
316 Rule.Thm ("separate_bdvs_add1", ThmC.numerals_to_Free @{thm separate_bdvs_add1}),
317 Rule.Thm ("separate_bdvs_add1", ThmC.numerals_to_Free @{thm separate_bdvs_add2}),
318 Rule.Thm ("separate_bdvs_mult", ThmC.numerals_to_Free @{thm separate_bdvs_mult})
319 ], scr = Rule.Empty_Prog};
324 (*.order the equations in a system such, that a triangular system (if any)
325 appears as [..c_4 = .., ..., ..., ..c_1 + ..c_2 + ..c_3 ..c_4 = ..].*)
327 Rule_Def.Repeat {id="order_system", preconds = [],
328 rew_ord = ("ord_simplify_System",
329 ord_simplify_System false \<^theory>),
330 erls = Rule_Set.Empty, srls = Rule_Set.Empty, calc = [], errpatts = [],
331 rules = [Rule.Thm ("order_system_NxN", ThmC.numerals_to_Free @{thm order_system_NxN})
333 scr = Rule.Empty_Prog};
335 val prls_triangular =
336 Rule_Def.Repeat {id="prls_triangular", preconds = [],
337 rew_ord = ("e_rew_ord", Rewrite_Ord.e_rew_ord),
338 erls = Rule_Def.Repeat {id="erls_prls_triangular", preconds = [],
339 rew_ord = ("e_rew_ord", Rewrite_Ord.e_rew_ord),
340 erls = Rule_Set.Empty, srls = Rule_Set.Empty, calc = [], errpatts = [],
341 rules = [(*for precond NTH_CONS ...*)
342 \<^rule_eval>\<open>less\<close> (Prog_Expr.eval_equ "#less_"),
343 \<^rule_eval>\<open>plus\<close> (**)(eval_binop "#add_")
344 (*immediately repeated rewrite pushes
345 '+' into precondition !*)
347 scr = Rule.Empty_Prog},
348 srls = Rule_Set.Empty, calc = [], errpatts = [],
349 rules = [Rule.Thm ("NTH_CONS",ThmC.numerals_to_Free @{thm NTH_CONS}),
350 \<^rule_eval>\<open>plus\<close> (**)(eval_binop "#add_"),
351 Rule.Thm ("NTH_NIL",ThmC.numerals_to_Free @{thm NTH_NIL}),
352 Rule.Thm ("tl_Cons",ThmC.numerals_to_Free @{thm tl_Cons}),
353 Rule.Thm ("tl_Nil",ThmC.numerals_to_Free @{thm tl_Nil}),
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 ("NTH_CONS",ThmC.numerals_to_Free @{thm NTH_CONS}),
377 \<^rule_eval>\<open>plus\<close> (**)(eval_binop "#add_"),
378 Rule.Thm ("NTH_NIL",ThmC.numerals_to_Free @{thm NTH_NIL}),
379 Rule.Thm ("tl_Cons",ThmC.numerals_to_Free @{thm tl_Cons}),
380 Rule.Thm ("tl_Nil",ThmC.numerals_to_Free @{thm tl_Nil}),
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 setup \<open>KEStore_Elems.add_pbts
400 [(Problem.prep_input @{theory} "pbl_equsys" [] Problem.id_empty
402 [("#Given" ,["equalities e_s", "solveForVars v_s"]),
403 ("#Find" ,["solution ss'''"](*''' is copy-named*))],
404 Rule_Set.append_rules "empty" Rule_Set.empty [(*for preds in where_*)], SOME "solveSystem e_s v_s", [])),
405 (Problem.prep_input @{theory} "pbl_equsys_lin" [] Problem.id_empty
406 (["LINEAR", "system"],
407 [("#Given" ,["equalities e_s", "solveForVars v_s"]),
408 (*TODO.WN050929 check linearity*)
409 ("#Find" ,["solution ss'''"])],
410 Rule_Set.append_rules "empty" Rule_Set.empty [(*for preds in where_*)], SOME "solveSystem e_s v_s", [])),
411 (Problem.prep_input @{theory} "pbl_equsys_lin_2x2" [] Problem.id_empty
412 (["2x2", "LINEAR", "system"],
413 (*~~~~~~~~~~~~~~~~~~~~~~~~~*)
414 [("#Given" ,["equalities e_s", "solveForVars v_s"]),
415 ("#Where" ,["Length (e_s:: bool list) = 2", "Length v_s = 2"]),
416 ("#Find" ,["solution ss'''"])],
417 Rule_Set.append_rules "prls_2x2_linear_system" Rule_Set.empty
418 [Rule.Thm ("LENGTH_CONS",ThmC.numerals_to_Free @{thm LENGTH_CONS}),
419 Rule.Thm ("LENGTH_NIL",ThmC.numerals_to_Free @{thm LENGTH_NIL}),
420 \<^rule_eval>\<open>plus\<close> (**)(eval_binop "#add_"),
421 \<^rule_eval>\<open>HOL.eq\<close> (Prog_Expr.eval_equal "#equal_")],
422 SOME "solveSystem e_s v_s", [])),
423 (Problem.prep_input @{theory} "pbl_equsys_lin_2x2_tri" [] Problem.id_empty
424 (["triangular", "2x2", "LINEAR", "system"],
425 [("#Given" ,["equalities e_s", "solveForVars v_s"]),
427 ["(tl v_s) from v_s occur_exactly_in (NTH 1 (e_s::bool list))",
428 " v_s from v_s occur_exactly_in (NTH 2 (e_s::bool list))"]),
429 ("#Find" ,["solution ss'''"])],
430 prls_triangular, SOME "solveSystem e_s v_s", [["EqSystem", "top_down_substitution", "2x2"]])),
431 (Problem.prep_input @{theory} "pbl_equsys_lin_2x2_norm" [] Problem.id_empty
432 (["normalise", "2x2", "LINEAR", "system"],
433 [("#Given" ,["equalities e_s", "solveForVars v_s"]),
434 ("#Find" ,["solution ss'''"])],
435 Rule_Set.append_rules "empty" Rule_Set.empty [(*for preds in where_*)],
436 SOME "solveSystem e_s v_s",
437 [["EqSystem", "normalise", "2x2"]])),
438 (Problem.prep_input @{theory} "pbl_equsys_lin_3x3" [] Problem.id_empty
439 (["3x3", "LINEAR", "system"],
440 (*~~~~~~~~~~~~~~~~~~~~~~~~~*)
441 [("#Given" ,["equalities e_s", "solveForVars v_s"]),
442 ("#Where" ,["Length (e_s:: bool list) = 3", "Length v_s = 3"]),
443 ("#Find" ,["solution ss'''"])],
444 Rule_Set.append_rules "prls_3x3_linear_system" Rule_Set.empty
445 [Rule.Thm ("LENGTH_CONS",ThmC.numerals_to_Free @{thm LENGTH_CONS}),
446 Rule.Thm ("LENGTH_NIL",ThmC.numerals_to_Free @{thm LENGTH_NIL}),
447 \<^rule_eval>\<open>plus\<close> (**)(eval_binop "#add_"),
448 \<^rule_eval>\<open>HOL.eq\<close> (Prog_Expr.eval_equal "#equal_")],
449 SOME "solveSystem e_s v_s", [])),
450 (Problem.prep_input @{theory} "pbl_equsys_lin_4x4" [] Problem.id_empty
451 (["4x4", "LINEAR", "system"],
452 (*~~~~~~~~~~~~~~~~~~~~~~~~~*)
453 [("#Given" ,["equalities e_s", "solveForVars v_s"]),
454 ("#Where" ,["Length (e_s:: bool list) = 4", "Length v_s = 4"]),
455 ("#Find" ,["solution ss'''"])],
456 Rule_Set.append_rules "prls_4x4_linear_system" Rule_Set.empty
457 [Rule.Thm ("LENGTH_CONS",ThmC.numerals_to_Free @{thm LENGTH_CONS}),
458 Rule.Thm ("LENGTH_NIL",ThmC.numerals_to_Free @{thm LENGTH_NIL}),
459 \<^rule_eval>\<open>plus\<close> (**)(eval_binop "#add_"),
460 \<^rule_eval>\<open>HOL.eq\<close> (Prog_Expr.eval_equal "#equal_")],
461 SOME "solveSystem e_s v_s", [])),
462 (Problem.prep_input @{theory} "pbl_equsys_lin_4x4_tri" [] Problem.id_empty
463 (["triangular", "4x4", "LINEAR", "system"],
464 [("#Given" ,["equalities e_s", "solveForVars v_s"]),
465 ("#Where" , (*accepts missing variables up to diagional form*)
466 ["(NTH 1 (v_s::real list)) occurs_in (NTH 1 (e_s::bool list))",
467 "(NTH 2 (v_s::real list)) occurs_in (NTH 2 (e_s::bool list))",
468 "(NTH 3 (v_s::real list)) occurs_in (NTH 3 (e_s::bool list))",
469 "(NTH 4 (v_s::real list)) occurs_in (NTH 4 (e_s::bool list))"]),
470 ("#Find" ,["solution ss'''"])],
471 Rule_Set.append_rules "prls_tri_4x4_lin_sys" prls_triangular
472 [\<^rule_eval>\<open>Prog_Expr.occurs_in\<close> (Prog_Expr.eval_occurs_in "")],
473 SOME "solveSystem e_s v_s",
474 [["EqSystem", "top_down_substitution", "4x4"]])),
475 (Problem.prep_input @{theory} "pbl_equsys_lin_4x4_norm" [] Problem.id_empty
476 (["normalise", "4x4", "LINEAR", "system"],
477 [("#Given" ,["equalities e_s", "solveForVars v_s"]),
478 (*Length is checked 1 level above*)
479 ("#Find" ,["solution ss'''"])],
480 Rule_Set.append_rules "empty" Rule_Set.empty [(*for preds in where_*)],
481 SOME "solveSystem e_s v_s",
482 [["EqSystem", "normalise", "4x4"]]))]\<close>
485 (*this is for NTH only*)
486 val srls = Rule_Def.Repeat {id="srls_normalise_4x4",
488 rew_ord = ("termlessI",termlessI),
489 erls = Rule_Set.append_rules "erls_in_srls_IntegrierenUnd.." Rule_Set.empty
490 [(*for asm in NTH_CONS ...*)
491 \<^rule_eval>\<open>less\<close> (Prog_Expr.eval_equ "#less_"),
492 (*2nd NTH_CONS pushes n+-1 into asms*)
493 \<^rule_eval>\<open>plus\<close> (**)(eval_binop "#add_")
495 srls = Rule_Set.Empty, calc = [], errpatts = [],
496 rules = [Rule.Thm ("NTH_CONS",ThmC.numerals_to_Free @{thm NTH_CONS}),
497 \<^rule_eval>\<open>plus\<close> (**)(eval_binop "#add_"),
498 Rule.Thm ("NTH_NIL",ThmC.numerals_to_Free @{thm NTH_NIL})],
499 scr = Rule.Empty_Prog};
502 section \<open>Methods\<close>
504 setup \<open>KEStore_Elems.add_mets
505 [MethodC.prep_input @{theory} "met_eqsys" [] MethodC.id_empty
507 {rew_ord'="tless_true", rls' = Rule_Set.Empty, calc = [], srls = Rule_Set.Empty, prls = Rule_Set.Empty, crls = Rule_Set.Empty,
508 errpats = [], nrls = Rule_Set.Empty},
510 MethodC.prep_input @{theory} "met_eqsys_topdown" [] MethodC.id_empty
511 (["EqSystem", "top_down_substitution"], [],
512 {rew_ord'="tless_true", rls' = Rule_Set.Empty, calc = [], srls = Rule_Set.Empty, prls = Rule_Set.Empty, crls = Rule_Set.Empty,
513 errpats = [], nrls = Rule_Set.Empty},
517 partial_function (tailrec) solve_system :: "bool list => real list => bool list"
519 "solve_system e_s v_s = (
523 (Try (Rewrite_Set_Inst [(''bdv_1'', hd v_s), (''bdv_2'', hd (tl v_s))] ''isolate_bdvs'')) #>
524 (Try (Rewrite_Set_Inst [(''bdv_1'', hd v_s), (''bdv_2'', hd (tl v_s))] ''simplify_System''))
526 e_2 = Take (hd (tl e_s));
528 (Substitute [e_1]) #>
529 (Try (Rewrite_Set_Inst [(''bdv_1'', hd v_s), (''bdv_2'', hd (tl v_s))] ''simplify_System_parenthesized'' )) #>
530 (Try (Rewrite_Set_Inst [(''bdv_1'', hd v_s), (''bdv_2'', hd (tl v_s))] ''isolate_bdvs'' )) #>
531 (Try (Rewrite_Set_Inst [(''bdv_1'', hd v_s), (''bdv_2'', hd (tl v_s))] ''simplify_System'' ))
533 e__s = Take [e_1, e_2]
535 Try (Rewrite_Set ''order_system'' ) e__s) "
536 setup \<open>KEStore_Elems.add_mets
537 [MethodC.prep_input @{theory} "met_eqsys_topdown_2x2" [] MethodC.id_empty
538 (["EqSystem", "top_down_substitution", "2x2"],
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'''"])],
544 {rew_ord'="ord_simplify_System", rls' = Rule_Set.Empty, calc = [],
545 srls = Rule_Set.append_rules "srls_top_down_2x2" Rule_Set.empty
546 [Rule.Thm ("hd_thm",ThmC.numerals_to_Free @{thm hd_thm}),
547 Rule.Thm ("tl_Cons",ThmC.numerals_to_Free @{thm tl_Cons}),
548 Rule.Thm ("tl_Nil",ThmC.numerals_to_Free @{thm tl_Nil})],
549 prls = prls_triangular, crls = Rule_Set.Empty, errpats = [], nrls = Rule_Set.Empty},
550 @{thm solve_system.simps})]
552 setup \<open>KEStore_Elems.add_mets
553 [MethodC.prep_input @{theory} "met_eqsys_norm" [] MethodC.id_empty
554 (["EqSystem", "normalise"], [],
555 {rew_ord'="tless_true", rls' = Rule_Set.Empty, calc = [], srls = Rule_Set.Empty, prls = Rule_Set.Empty, crls = Rule_Set.Empty,
556 errpats = [], nrls = Rule_Set.Empty},
560 partial_function (tailrec) solve_system2 :: "bool list => real list => bool list"
562 "solve_system2 e_s v_s = (
565 (Try (Rewrite_Set ''norm_Rational'' )) #>
566 (Try (Rewrite_Set_Inst [(''bdv_1'', hd v_s), (''bdv_2'', hd (tl v_s))] ''simplify_System_parenthesized'' )) #>
567 (Try (Rewrite_Set_Inst [(''bdv_1'', hd v_s), (''bdv_2'', hd (tl v_s))] ''isolate_bdvs'' )) #>
568 (Try (Rewrite_Set_Inst [(''bdv_1'', hd v_s), (''bdv_2'', hd (tl v_s))] ''simplify_System_parenthesized'' )) #>
569 (Try (Rewrite_Set ''order_system'' ))
572 SubProblem (''EqSystem'', [''LINEAR'', ''system''], [''no_met''])
573 [BOOL_LIST e__s, REAL_LIST v_s])"
574 setup \<open>KEStore_Elems.add_mets
575 [MethodC.prep_input @{theory} "met_eqsys_norm_2x2" [] MethodC.id_empty
576 (["EqSystem", "normalise", "2x2"],
577 [("#Given" ,["equalities e_s", "solveForVars v_s"]),
578 ("#Find" ,["solution ss'''"])],
579 {rew_ord'="tless_true", rls' = Rule_Set.Empty, calc = [],
580 srls = Rule_Set.append_rules "srls_normalise_2x2" Rule_Set.empty
581 [Rule.Thm ("hd_thm",ThmC.numerals_to_Free @{thm hd_thm}),
582 Rule.Thm ("tl_Cons",ThmC.numerals_to_Free @{thm tl_Cons}),
583 Rule.Thm ("tl_Nil",ThmC.numerals_to_Free @{thm tl_Nil})],
584 prls = Rule_Set.Empty, crls = Rule_Set.Empty, errpats = [], nrls = Rule_Set.Empty},
585 @{thm solve_system2.simps})]
588 partial_function (tailrec) solve_system3 :: "bool list => real list => bool list"
590 "solve_system3 e_s v_s = (
593 (Try (Rewrite_Set ''norm_Rational'' )) #>
594 (Repeat (Rewrite ''commute_0_equality'' )) #>
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_3'', NTH 4 v_s )] ''simplify_System_parenthesized'' )) #>
597 (Try (Rewrite_Set_Inst [(''bdv_1'', NTH 1 v_s), (''bdv_2'', NTH 2 v_s ),
598 (''bdv_3'', NTH 3 v_s), (''bdv_3'', NTH 4 v_s )] ''isolate_bdvs_4x4'' )) #>
599 (Try (Rewrite_Set_Inst [(''bdv_1'', NTH 1 v_s), (''bdv_2'', NTH 2 v_s ),
600 (''bdv_3'', NTH 3 v_s), (''bdv_3'', NTH 4 v_s )] ''simplify_System_parenthesized'' )) #>
601 (Try (Rewrite_Set ''order_system''))
604 SubProblem (''EqSystem'', [''LINEAR'', ''system''], [''no_met''])
605 [BOOL_LIST e__s, REAL_LIST v_s])"
606 setup \<open>KEStore_Elems.add_mets
607 [MethodC.prep_input @{theory} "met_eqsys_norm_4x4" [] MethodC.id_empty
608 (["EqSystem", "normalise", "4x4"],
609 [("#Given" ,["equalities e_s", "solveForVars v_s"]),
610 ("#Find" ,["solution ss'''"])],
611 {rew_ord'="tless_true", rls' = Rule_Set.Empty, calc = [],
612 srls = Rule_Set.append_rules "srls_normalise_4x4" srls
613 [Rule.Thm ("hd_thm",ThmC.numerals_to_Free @{thm hd_thm}),
614 Rule.Thm ("tl_Cons",ThmC.numerals_to_Free @{thm tl_Cons}),
615 Rule.Thm ("tl_Nil",ThmC.numerals_to_Free @{thm tl_Nil})],
616 prls = Rule_Set.Empty, crls = Rule_Set.Empty, errpats = [], nrls = Rule_Set.Empty},
617 (*STOPPED.WN06? met ["EqSystem", "normalise", "4x4"] #>#>#>#>#>#>#>#>#>#>#>#>#>@*)
618 @{thm solve_system3.simps})]
621 partial_function (tailrec) solve_system4 :: "bool list => real list => bool list"
623 "solve_system4 e_s v_s = (
626 e_2 = Take (NTH 2 e_s);
628 (Substitute [e_1]) #>
629 (Try (Rewrite_Set_Inst [(''bdv_1'',NTH 1 v_s), (''bdv_2'',NTH 2 v_s),
630 (''bdv_3'',NTH 3 v_s), (''bdv_4'',NTH 4 v_s)] ''simplify_System_parenthesized'' )) #>
631 (Try (Rewrite_Set_Inst [(''bdv_1'',NTH 1 v_s), (''bdv_2'',NTH 2 v_s),
632 (''bdv_3'',NTH 3 v_s), (''bdv_4'',NTH 4 v_s)] ''isolate_bdvs'' )) #>
633 (Try (Rewrite_Set_Inst [(''bdv_1'',NTH 1 v_s), (''bdv_2'',NTH 2 v_s),
634 (''bdv_3'',NTH 3 v_s), (''bdv_4'',NTH 4 v_s)] ''norm_Rational'' ))
637 [e_1, e_2, NTH 3 e_s, NTH 4 e_s])"
638 setup \<open>KEStore_Elems.add_mets
639 [MethodC.prep_input @{theory} "met_eqsys_topdown_4x4" [] MethodC.id_empty
640 (["EqSystem", "top_down_substitution", "4x4"],
641 [("#Given" ,["equalities e_s", "solveForVars v_s"]),
642 ("#Where" , (*accepts missing variables up to diagonal form*)
643 ["(NTH 1 (v_s::real list)) occurs_in (NTH 1 (e_s::bool list))",
644 "(NTH 2 (v_s::real list)) occurs_in (NTH 2 (e_s::bool list))",
645 "(NTH 3 (v_s::real list)) occurs_in (NTH 3 (e_s::bool list))",
646 "(NTH 4 (v_s::real list)) occurs_in (NTH 4 (e_s::bool list))"]),
647 ("#Find", ["solution ss'''"])],
648 {rew_ord'="ord_simplify_System", rls' = Rule_Set.Empty, calc = [],
649 srls = Rule_Set.append_rules "srls_top_down_4x4" srls [],
650 prls = Rule_Set.append_rules "prls_tri_4x4_lin_sys" prls_triangular
651 [\<^rule_eval>\<open>Prog_Expr.occurs_in\<close> (Prog_Expr.eval_occurs_in "")],
652 crls = Rule_Set.Empty, errpats = [], nrls = Rule_Set.Empty},
653 (*FIXXXXME.WN060916: this script works ONLY for exp 7.79 #>#>#>#>#>#>#>#>#>#>*)
654 @{thm solve_system4.simps})]