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 ("EqSystem.occur'_exactly'_in",_)
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 setup \<open>KEStore_Elems.add_calcs
83 ("EqSystem.occur'_exactly'_in",
84 eval_occur_exactly_in "#eval_occur_exactly_in_"))]\<close>
86 (** rewrite order 'ord_simplify_System' **)
88 (* order wrt. several linear (i.e. without exponents) variables "c","c_2",..
89 which leaves the monomials containing c, c_2,... at the end of an Integral
90 and puts the c, c_2,... rightmost within a monomial.
92 WN050906 this is a quick and dirty adaption of ord_make_polynomial_in,
93 which was most adequate, because it uses size_of_term*)
95 local (*. for simplify_System .*)
97 open Term; (* for type order = EQUAL | LESS | GREATER *)
99 fun pr_ord EQUAL = "EQUAL"
100 | pr_ord LESS = "LESS"
101 | pr_ord GREATER = "GREATER";
103 fun dest_hd' (Const (a, T)) = (((a, 0), T), 0)
104 | dest_hd' (Free (ccc, T)) =
105 (case Symbol.explode ccc of
106 "c"::[] => ((("|||||||||||||||||||||", 0), T), 1)(*greatest string WN*)
107 | "c"::"_"::_ => ((("|||||||||||||||||||||", 0), T), 1)
108 | _ => (((ccc, 0), T), 1))
109 | dest_hd' (Var v) = (v, 2)
110 | dest_hd' (Bound i) = ((("", i), dummyT), 3)
111 | dest_hd' (Abs (_, T, _)) = ((("", 0), T), 4);
113 fun size_of_term' (Free (ccc, _)) =
114 (case Symbol.explode ccc of (*WN0510 hack for the bound variables*)
116 | "c"::"_"::is => 1000 * ((TermC.int_of_str o implode) is)
118 | size_of_term' (Abs (_,_,body)) = 1 + size_of_term' body
119 | size_of_term' (f$t) = size_of_term' f + size_of_term' t
120 | size_of_term' _ = 1;
122 fun term_ord' pr thy (Abs (_, T, t), Abs(_, U, u)) = (* ~ term.ML *)
123 (case term_ord' pr thy (t, u) of EQUAL => Term_Ord.typ_ord (T, U) | ord => ord)
124 | term_ord' pr thy (t, u) =
128 val (f, ts) = strip_comb t and (g, us) = strip_comb u;
129 val _ = tracing ("t= f@ts= \"" ^ UnparseC.term_in_thy thy f ^ "\" @ \"[" ^
130 commas (map (UnparseC.term_in_thy thy) ts) ^ "]\"");
131 val _ = tracing ("u= g@us= \"" ^ UnparseC.term_in_thy thy g ^ "\" @ \"[" ^
132 commas (map (UnparseC.term_in_thy thy) us) ^ "]\"");
133 val _ = tracing ("size_of_term(t,u)= (" ^ string_of_int (size_of_term' t) ^ ", " ^
134 string_of_int (size_of_term' u) ^ ")");
135 val _ = tracing ("hd_ord(f,g) = " ^ ((pr_ord o hd_ord) (f,g)));
136 val _ = tracing ("terms_ord (ts,us) = " ^(pr_ord o terms_ord str false) (ts,us));
137 val _=tracing("-------");
140 case int_ord (size_of_term' t, size_of_term' u) of
142 let val (f, ts) = strip_comb t and (g, us) = strip_comb u
143 in (case hd_ord (f, g) of
144 EQUAL => (terms_ord str pr) (ts, us)
148 and hd_ord (f, g) = (* ~ term.ML *)
149 prod_ord (prod_ord Term_Ord.indexname_ord Term_Ord.typ_ord) int_ord (dest_hd' f, dest_hd' g)
150 and terms_ord str pr (ts, us) = list_ord (term_ord' pr (ThyC.get_theory "Isac_Knowledge"))(ts, us);
154 (*WN0510 for preliminary use in eval_order_system, see case-study mat-eng.tex
155 fun ord_simplify_System_rev (pr:bool) thy subst tu =
156 (term_ord' pr thy (Library.swap tu) = LESS);*)
159 fun ord_simplify_System (pr:bool) thy subst tu =
160 (term_ord' pr thy tu = LESS);
164 Rewrite_Ord.rew_ord' := overwritel (! Rewrite_Ord.rew_ord',
165 [("ord_simplify_System", ord_simplify_System false thy)
171 (*.adapted from 'order_add_mult_in' by just replacing the rew_ord.*)
172 val order_add_mult_System =
173 Rule_Def.Repeat{id = "order_add_mult_System", preconds = [],
174 rew_ord = ("ord_simplify_System",
175 ord_simplify_System false @{theory "Integrate"}),
176 erls = Rule_Set.empty,srls = Rule_Set.Empty, calc = [], errpatts = [],
177 rules = [Rule.Thm ("mult.commute",ThmC.numerals_to_Free @{thm mult.commute}),
179 Rule.Thm ("real_mult_left_commute",ThmC.numerals_to_Free @{thm real_mult_left_commute}),
180 (*z1.0 * (z2.0 * z3.0) = z2.0 * (z1.0 * z3.0)*)
181 Rule.Thm ("mult.assoc",ThmC.numerals_to_Free @{thm mult.assoc}),
182 (*z1.0 * z2.0 * z3.0 = z1.0 * (z2.0 * z3.0)*)
183 Rule.Thm ("add.commute",ThmC.numerals_to_Free @{thm add.commute}),
185 Rule.Thm ("add.left_commute",ThmC.numerals_to_Free @{thm add.left_commute}),
186 (*x + (y + z) = y + (x + z)*)
187 Rule.Thm ("add.assoc",ThmC.numerals_to_Free @{thm add.assoc})
188 (*z1.0 + z2.0 + z3.0 = z1.0 + (z2.0 + z3.0)*)
190 scr = Rule.Empty_Prog};
193 (*.adapted from 'norm_Rational' by
194 #1 using 'ord_simplify_System' in 'order_add_mult_System'
195 #2 NOT using common_nominator_p .*)
196 val norm_System_noadd_fractions =
197 Rule_Def.Repeat {id = "norm_System_noadd_fractions", preconds = [],
198 rew_ord = ("dummy_ord",Rewrite_Ord.dummy_ord),
199 erls = norm_rat_erls, srls = Rule_Set.Empty, calc = [], errpatts = [],
200 rules = [(*sequence given by operator precedence*)
201 Rule.Rls_ discard_minus,
203 Rule.Rls_ rat_mult_divide,
205 Rule.Rls_ reduce_0_1_2,
206 Rule.Rls_ (*order_add_mult #1*) order_add_mult_System,
207 Rule.Rls_ collect_numerals,
208 (*Rule.Rls_ add_fractions_p, #2*)
211 scr = Rule.Empty_Prog
215 (*.adapted from 'norm_Rational' by
216 *1* using 'ord_simplify_System' in 'order_add_mult_System'.*)
218 Rule_Def.Repeat {id = "norm_System", preconds = [],
219 rew_ord = ("dummy_ord",Rewrite_Ord.dummy_ord),
220 erls = norm_rat_erls, srls = Rule_Set.Empty, calc = [], errpatts = [],
221 rules = [(*sequence given by operator precedence*)
222 Rule.Rls_ discard_minus,
224 Rule.Rls_ rat_mult_divide,
226 Rule.Rls_ reduce_0_1_2,
227 Rule.Rls_ (*order_add_mult *1*) order_add_mult_System,
228 Rule.Rls_ collect_numerals,
229 Rule.Rls_ add_fractions_p,
232 scr = Rule.Empty_Prog
236 (*.simplify an equational system BEFORE solving it such that parentheses are
237 ( ((u0*v0)*w0) + ( ((u1*v1)*w1) * c + ... +((u4*v4)*w4) * c_4 ) )
238 ATTENTION: works ONLY for bound variables c, c_1, c_2, c_3, c_4 :ATTENTION
239 This is a copy from 'make_ratpoly_in' with respective reductions:
240 *0* expand the term, ie. distribute * and / over +
241 *1* ord_simplify_System instead of termlessI
242 *2* no add_fractions_p (= common_nominator_p_rls !)
243 *3* discard_parentheses only for (.*(.*.))
244 analoguous to simplify_Integral .*)
245 val simplify_System_parenthesized =
246 Rule_Set.Sequence {id = "simplify_System_parenthesized", preconds = []:term list,
247 rew_ord = ("dummy_ord", Rewrite_Ord.dummy_ord),
248 erls = Atools_erls, srls = Rule_Set.Empty, calc = [], errpatts = [],
249 rules = [Rule.Thm ("distrib_right",ThmC.numerals_to_Free @{thm distrib_right}),
250 (*"(?z1.0 + ?z2.0) * ?w = ?z1.0 * ?w + ?z2.0 * ?w"*)
251 Rule.Thm ("add_divide_distrib",ThmC.numerals_to_Free @{thm add_divide_distrib}),
252 (*"(?x + ?y) / ?z = ?x / ?z + ?y / ?z"*)
253 (*^^^^^ *0* ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^*)
254 Rule.Rls_ norm_Rational_noadd_fractions(**2**),
255 Rule.Rls_ (*order_add_mult_in*) norm_System_noadd_fractions (**1**),
256 Rule.Thm ("sym_mult.assoc",
257 ThmC.numerals_to_Free (@{thm mult.assoc} RS @{thm sym}))
258 (*Rule.Rls_ discard_parentheses *3**),
259 Rule.Rls_ collect_bdv, (*from make_polynomial_in WN051031 welldone?*)
260 Rule.Rls_ separate_bdv2,
261 Rule.Eval ("Rings.divide_class.divide", Prog_Expr.eval_cancel "#divide_e")
263 scr = Rule.Empty_Prog};
266 (*.simplify an equational system AFTER solving it;
267 This is a copy of 'make_ratpoly_in' with the differences
268 *1* ord_simplify_System instead of termlessI .*)
269 (*TODO.WN051031 ^^^^^^^^^^ should be in EACH rls contained *)
270 val simplify_System =
271 Rule_Set.Sequence {id = "simplify_System", preconds = []:term list,
272 rew_ord = ("dummy_ord", Rewrite_Ord.dummy_ord),
273 erls = Atools_erls, srls = Rule_Set.Empty, calc = [], errpatts = [],
274 rules = [Rule.Rls_ norm_Rational,
275 Rule.Rls_ (*order_add_mult_in*) norm_System (**1**),
276 Rule.Rls_ discard_parentheses,
277 Rule.Rls_ collect_bdv, (*from make_polynomial_in WN051031 welldone?*)
278 Rule.Rls_ separate_bdv2,
279 Rule.Eval ("Rings.divide_class.divide", Prog_Expr.eval_cancel "#divide_e")
281 scr = Rule.Empty_Prog};
283 val simplify_System =
284 Rule_Set.append_rules "simplify_System" simplify_System_parenthesized
285 [Rule.Thm ("sym_add.assoc",
286 ThmC.numerals_to_Free (@{thm add.assoc} RS @{thm sym}))];
291 Rule_Def.Repeat {id="isolate_bdvs", preconds = [],
292 rew_ord = ("e_rew_ord", Rewrite_Ord.e_rew_ord),
293 erls = Rule_Set.append_rules "erls_isolate_bdvs" Rule_Set.empty
294 [(Rule.Eval ("EqSystem.occur'_exactly'_in",
295 eval_occur_exactly_in
296 "#eval_occur_exactly_in_"))
298 srls = Rule_Set.Empty, calc = [], errpatts = [],
300 [Rule.Thm ("commute_0_equality", ThmC.numerals_to_Free @{thm commute_0_equality}),
301 Rule.Thm ("separate_bdvs_add", ThmC.numerals_to_Free @{thm separate_bdvs_add}),
302 Rule.Thm ("separate_bdvs_mult", ThmC.numerals_to_Free @{thm separate_bdvs_mult})],
303 scr = Rule.Empty_Prog};
306 val isolate_bdvs_4x4 =
307 Rule_Def.Repeat {id="isolate_bdvs_4x4", preconds = [],
308 rew_ord = ("e_rew_ord", Rewrite_Ord.e_rew_ord),
309 erls = Rule_Set.append_rules
310 "erls_isolate_bdvs_4x4" Rule_Set.empty
311 [Rule.Eval ("EqSystem.occur'_exactly'_in",
312 eval_occur_exactly_in "#eval_occur_exactly_in_"),
313 Rule.Eval ("Prog_Expr.ident", Prog_Expr.eval_ident "#ident_"),
314 Rule.Eval ("Prog_Expr.some'_occur'_in", Prog_Expr.eval_some_occur_in "#some_occur_in_"),
315 Rule.Thm ("not_true",ThmC.numerals_to_Free @{thm not_true}),
316 Rule.Thm ("not_false",ThmC.numerals_to_Free @{thm not_false})
318 srls = Rule_Set.Empty, calc = [], errpatts = [],
319 rules = [Rule.Thm ("commute_0_equality", ThmC.numerals_to_Free @{thm commute_0_equality}),
320 Rule.Thm ("separate_bdvs0", ThmC.numerals_to_Free @{thm separate_bdvs0}),
321 Rule.Thm ("separate_bdvs_add1", ThmC.numerals_to_Free @{thm separate_bdvs_add1}),
322 Rule.Thm ("separate_bdvs_add1", ThmC.numerals_to_Free @{thm separate_bdvs_add2}),
323 Rule.Thm ("separate_bdvs_mult", ThmC.numerals_to_Free @{thm separate_bdvs_mult})
324 ], scr = Rule.Empty_Prog};
329 (*.order the equations in a system such, that a triangular system (if any)
330 appears as [..c_4 = .., ..., ..., ..c_1 + ..c_2 + ..c_3 ..c_4 = ..].*)
332 Rule_Def.Repeat {id="order_system", preconds = [],
333 rew_ord = ("ord_simplify_System",
334 ord_simplify_System false thy),
335 erls = Rule_Set.Empty, srls = Rule_Set.Empty, calc = [], errpatts = [],
336 rules = [Rule.Thm ("order_system_NxN", ThmC.numerals_to_Free @{thm order_system_NxN})
338 scr = Rule.Empty_Prog};
340 val prls_triangular =
341 Rule_Def.Repeat {id="prls_triangular", preconds = [],
342 rew_ord = ("e_rew_ord", Rewrite_Ord.e_rew_ord),
343 erls = Rule_Def.Repeat {id="erls_prls_triangular", preconds = [],
344 rew_ord = ("e_rew_ord", Rewrite_Ord.e_rew_ord),
345 erls = Rule_Set.Empty, srls = Rule_Set.Empty, calc = [], errpatts = [],
346 rules = [(*for precond NTH_CONS ...*)
347 Rule.Eval ("Orderings.ord_class.less", Prog_Expr.eval_equ "#less_"),
348 Rule.Eval ("Groups.plus_class.plus", (**)eval_binop "#add_")
349 (*immediately repeated rewrite pushes
350 '+' into precondition !*)
352 scr = Rule.Empty_Prog},
353 srls = Rule_Set.Empty, calc = [], errpatts = [],
354 rules = [Rule.Thm ("NTH_CONS",ThmC.numerals_to_Free @{thm NTH_CONS}),
355 Rule.Eval ("Groups.plus_class.plus", (**)eval_binop "#add_"),
356 Rule.Thm ("NTH_NIL",ThmC.numerals_to_Free @{thm NTH_NIL}),
357 Rule.Thm ("tl_Cons",ThmC.numerals_to_Free @{thm tl_Cons}),
358 Rule.Thm ("tl_Nil",ThmC.numerals_to_Free @{thm tl_Nil}),
359 Rule.Eval ("EqSystem.occur'_exactly'_in",
360 eval_occur_exactly_in
361 "#eval_occur_exactly_in_")
363 scr = Rule.Empty_Prog};
367 (*WN060914 quickly created for 4x4;
368 more similarity to prls_triangular desirable*)
369 val prls_triangular4 =
370 Rule_Def.Repeat {id="prls_triangular4", preconds = [],
371 rew_ord = ("e_rew_ord", Rewrite_Ord.e_rew_ord),
372 erls = Rule_Def.Repeat {id="erls_prls_triangular4", preconds = [],
373 rew_ord = ("e_rew_ord", Rewrite_Ord.e_rew_ord),
374 erls = Rule_Set.Empty, srls = Rule_Set.Empty, calc = [], errpatts = [],
375 rules = [(*for precond NTH_CONS ...*)
376 Rule.Eval ("Orderings.ord_class.less", Prog_Expr.eval_equ "#less_"),
377 Rule.Eval ("Groups.plus_class.plus", (**)eval_binop "#add_")
378 (*immediately repeated rewrite pushes
379 '+' into precondition !*)
381 scr = Rule.Empty_Prog},
382 srls = Rule_Set.Empty, calc = [], errpatts = [],
383 rules = [Rule.Thm ("NTH_CONS",ThmC.numerals_to_Free @{thm NTH_CONS}),
384 Rule.Eval ("Groups.plus_class.plus", (**)eval_binop "#add_"),
385 Rule.Thm ("NTH_NIL",ThmC.numerals_to_Free @{thm NTH_NIL}),
386 Rule.Thm ("tl_Cons",ThmC.numerals_to_Free @{thm tl_Cons}),
387 Rule.Thm ("tl_Nil",ThmC.numerals_to_Free @{thm tl_Nil}),
388 Rule.Eval ("EqSystem.occur'_exactly'_in",
389 eval_occur_exactly_in
390 "#eval_occur_exactly_in_")
392 scr = Rule.Empty_Prog};
395 setup \<open>KEStore_Elems.add_rlss
396 [("simplify_System_parenthesized",
397 (Context.theory_name @{theory}, prep_rls' simplify_System_parenthesized)),
398 ("simplify_System", (Context.theory_name @{theory}, prep_rls' simplify_System)),
399 ("isolate_bdvs", (Context.theory_name @{theory}, prep_rls' isolate_bdvs)),
400 ("isolate_bdvs_4x4", (Context.theory_name @{theory}, prep_rls' isolate_bdvs_4x4)),
401 ("order_system", (Context.theory_name @{theory}, prep_rls' order_system)),
402 ("order_add_mult_System", (Context.theory_name @{theory}, prep_rls' order_add_mult_System)),
403 ("norm_System_noadd_fractions",
404 (Context.theory_name @{theory}, prep_rls' norm_System_noadd_fractions)),
405 ("norm_System", (Context.theory_name @{theory}, prep_rls' norm_System))]\<close>
408 setup \<open>KEStore_Elems.add_pbts
409 [(Specify.prep_pbt thy "pbl_equsys" [] Problem.id_empty
411 [("#Given" ,["equalities e_s", "solveForVars v_s"]),
412 ("#Find" ,["solution ss'''"](*''' is copy-named*))],
413 Rule_Set.append_rules "empty" Rule_Set.empty [(*for preds in where_*)], SOME "solveSystem e_s v_s", [])),
414 (Specify.prep_pbt thy "pbl_equsys_lin" [] Problem.id_empty
415 (["LINEAR", "system"],
416 [("#Given" ,["equalities e_s", "solveForVars v_s"]),
417 (*TODO.WN050929 check linearity*)
418 ("#Find" ,["solution ss'''"])],
419 Rule_Set.append_rules "empty" Rule_Set.empty [(*for preds in where_*)], SOME "solveSystem e_s v_s", [])),
420 (Specify.prep_pbt thy "pbl_equsys_lin_2x2" [] Problem.id_empty
421 (["2x2", "LINEAR", "system"],
422 (*~~~~~~~~~~~~~~~~~~~~~~~~~*)
423 [("#Given" ,["equalities e_s", "solveForVars v_s"]),
424 ("#Where" ,["LENGTH (e_s:: bool list) = 2", "LENGTH v_s = 2"]),
425 ("#Find" ,["solution ss'''"])],
426 Rule_Set.append_rules "prls_2x2_linear_system" Rule_Set.empty
427 [Rule.Thm ("LENGTH_CONS",ThmC.numerals_to_Free @{thm LENGTH_CONS}),
428 Rule.Thm ("LENGTH_NIL",ThmC.numerals_to_Free @{thm LENGTH_NIL}),
429 Rule.Eval ("Groups.plus_class.plus", (**)eval_binop "#add_"),
430 Rule.Eval ("HOL.eq", Prog_Expr.eval_equal "#equal_")],
431 SOME "solveSystem e_s v_s", [])),
432 (Specify.prep_pbt thy "pbl_equsys_lin_2x2_tri" [] Problem.id_empty
433 (["triangular", "2x2", "LINEAR", "system"],
434 [("#Given" ,["equalities e_s", "solveForVars v_s"]),
436 ["(tl v_s) from v_s occur_exactly_in (NTH 1 (e_s::bool list))",
437 " v_s from v_s occur_exactly_in (NTH 2 (e_s::bool list))"]),
438 ("#Find" ,["solution ss'''"])],
439 prls_triangular, SOME "solveSystem e_s v_s", [["EqSystem","top_down_substitution","2x2"]])),
440 (Specify.prep_pbt thy "pbl_equsys_lin_2x2_norm" [] Problem.id_empty
441 (["normalise", "2x2", "LINEAR", "system"],
442 [("#Given" ,["equalities e_s", "solveForVars v_s"]),
443 ("#Find" ,["solution ss'''"])],
444 Rule_Set.append_rules "empty" Rule_Set.empty [(*for preds in where_*)],
445 SOME "solveSystem e_s v_s",
446 [["EqSystem","normalise","2x2"]])),
447 (Specify.prep_pbt thy "pbl_equsys_lin_3x3" [] Problem.id_empty
448 (["3x3", "LINEAR", "system"],
449 (*~~~~~~~~~~~~~~~~~~~~~~~~~*)
450 [("#Given" ,["equalities e_s", "solveForVars v_s"]),
451 ("#Where" ,["LENGTH (e_s:: bool list) = 3", "LENGTH v_s = 3"]),
452 ("#Find" ,["solution ss'''"])],
453 Rule_Set.append_rules "prls_3x3_linear_system" Rule_Set.empty
454 [Rule.Thm ("LENGTH_CONS",ThmC.numerals_to_Free @{thm LENGTH_CONS}),
455 Rule.Thm ("LENGTH_NIL",ThmC.numerals_to_Free @{thm LENGTH_NIL}),
456 Rule.Eval ("Groups.plus_class.plus", (**)eval_binop "#add_"),
457 Rule.Eval ("HOL.eq", Prog_Expr.eval_equal "#equal_")],
458 SOME "solveSystem e_s v_s", [])),
459 (Specify.prep_pbt thy "pbl_equsys_lin_4x4" [] Problem.id_empty
460 (["4x4", "LINEAR", "system"],
461 (*~~~~~~~~~~~~~~~~~~~~~~~~~*)
462 [("#Given" ,["equalities e_s", "solveForVars v_s"]),
463 ("#Where" ,["LENGTH (e_s:: bool list) = 4", "LENGTH v_s = 4"]),
464 ("#Find" ,["solution ss'''"])],
465 Rule_Set.append_rules "prls_4x4_linear_system" Rule_Set.empty
466 [Rule.Thm ("LENGTH_CONS",ThmC.numerals_to_Free @{thm LENGTH_CONS}),
467 Rule.Thm ("LENGTH_NIL",ThmC.numerals_to_Free @{thm LENGTH_NIL}),
468 Rule.Eval ("Groups.plus_class.plus", (**)eval_binop "#add_"),
469 Rule.Eval ("HOL.eq", Prog_Expr.eval_equal "#equal_")],
470 SOME "solveSystem e_s v_s", [])),
471 (Specify.prep_pbt thy "pbl_equsys_lin_4x4_tri" [] Problem.id_empty
472 (["triangular", "4x4", "LINEAR", "system"],
473 [("#Given" ,["equalities e_s", "solveForVars v_s"]),
474 ("#Where" , (*accepts missing variables up to diagional form*)
475 ["(NTH 1 (v_s::real list)) occurs_in (NTH 1 (e_s::bool list))",
476 "(NTH 2 (v_s::real list)) occurs_in (NTH 2 (e_s::bool list))",
477 "(NTH 3 (v_s::real list)) occurs_in (NTH 3 (e_s::bool list))",
478 "(NTH 4 (v_s::real list)) occurs_in (NTH 4 (e_s::bool list))"]),
479 ("#Find" ,["solution ss'''"])],
480 Rule_Set.append_rules "prls_tri_4x4_lin_sys" prls_triangular
481 [Rule.Eval ("Prog_Expr.occurs'_in", Prog_Expr.eval_occurs_in "")],
482 SOME "solveSystem e_s v_s",
483 [["EqSystem","top_down_substitution","4x4"]])),
484 (Specify.prep_pbt thy "pbl_equsys_lin_4x4_norm" [] Problem.id_empty
485 (["normalise", "4x4", "LINEAR", "system"],
486 [("#Given" ,["equalities e_s", "solveForVars v_s"]),
487 (*LENGTH is checked 1 level above*)
488 ("#Find" ,["solution ss'''"])],
489 Rule_Set.append_rules "empty" Rule_Set.empty [(*for preds in where_*)],
490 SOME "solveSystem e_s v_s",
491 [["EqSystem","normalise","4x4"]]))]\<close>
494 (*this is for NTH only*)
495 val srls = Rule_Def.Repeat {id="srls_normalise_4x4",
497 rew_ord = ("termlessI",termlessI),
498 erls = Rule_Set.append_rules "erls_in_srls_IntegrierenUnd.." Rule_Set.empty
499 [(*for asm in NTH_CONS ...*)
500 Rule.Eval ("Orderings.ord_class.less", Prog_Expr.eval_equ "#less_"),
501 (*2nd NTH_CONS pushes n+-1 into asms*)
502 Rule.Eval("Groups.plus_class.plus", (**)eval_binop "#add_")
504 srls = Rule_Set.Empty, calc = [], errpatts = [],
505 rules = [Rule.Thm ("NTH_CONS",ThmC.numerals_to_Free @{thm NTH_CONS}),
506 Rule.Eval("Groups.plus_class.plus", (**)eval_binop "#add_"),
507 Rule.Thm ("NTH_NIL",ThmC.numerals_to_Free @{thm NTH_NIL})],
508 scr = Rule.Empty_Prog};
512 setup \<open>KEStore_Elems.add_mets
513 [Specify.prep_met thy "met_eqsys" [] Method.id_empty
515 {rew_ord'="tless_true", rls' = Rule_Set.Empty, calc = [], srls = Rule_Set.Empty, prls = Rule_Set.Empty, crls = Rule_Set.Empty,
516 errpats = [], nrls = Rule_Set.Empty},
518 Specify.prep_met thy "met_eqsys_topdown" [] Method.id_empty
519 (["EqSystem","top_down_substitution"], [],
520 {rew_ord'="tless_true", rls' = Rule_Set.Empty, calc = [], srls = Rule_Set.Empty, prls = Rule_Set.Empty, crls = Rule_Set.Empty,
521 errpats = [], nrls = Rule_Set.Empty},
525 partial_function (tailrec) solve_system :: "bool list => real list => bool list"
527 "solve_system e_s v_s = (
531 (Try (Rewrite_Set_Inst [(''bdv_1'', hd v_s), (''bdv_2'', hd (tl v_s))] ''isolate_bdvs'')) #>
532 (Try (Rewrite_Set_Inst [(''bdv_1'', hd v_s), (''bdv_2'', hd (tl v_s))] ''simplify_System''))
534 e_2 = Take (hd (tl e_s));
536 (Substitute [e_1]) #>
537 (Try (Rewrite_Set_Inst [(''bdv_1'', hd v_s), (''bdv_2'', hd (tl v_s))] ''simplify_System_parenthesized'' )) #>
538 (Try (Rewrite_Set_Inst [(''bdv_1'', hd v_s), (''bdv_2'', hd (tl v_s))] ''isolate_bdvs'' )) #>
539 (Try (Rewrite_Set_Inst [(''bdv_1'', hd v_s), (''bdv_2'', hd (tl v_s))] ''simplify_System'' ))
541 e__s = Take [e_1, e_2]
543 Try (Rewrite_Set ''order_system'' ) e__s) "
544 setup \<open>KEStore_Elems.add_mets
545 [Specify.prep_met thy "met_eqsys_topdown_2x2" [] Method.id_empty
546 (["EqSystem", "top_down_substitution", "2x2"],
547 [("#Given", ["equalities e_s", "solveForVars v_s"]),
549 ["(tl v_s) from v_s occur_exactly_in (NTH 1 (e_s::bool list))",
550 " v_s from v_s occur_exactly_in (NTH 2 (e_s::bool list))"]),
551 ("#Find" ,["solution ss'''"])],
552 {rew_ord'="ord_simplify_System", rls' = Rule_Set.Empty, calc = [],
553 srls = Rule_Set.append_rules "srls_top_down_2x2" Rule_Set.empty
554 [Rule.Thm ("hd_thm",ThmC.numerals_to_Free @{thm hd_thm}),
555 Rule.Thm ("tl_Cons",ThmC.numerals_to_Free @{thm tl_Cons}),
556 Rule.Thm ("tl_Nil",ThmC.numerals_to_Free @{thm tl_Nil})],
557 prls = prls_triangular, crls = Rule_Set.Empty, errpats = [], nrls = Rule_Set.Empty},
558 @{thm solve_system.simps})]
560 setup \<open>KEStore_Elems.add_mets
561 [Specify.prep_met thy "met_eqsys_norm" [] Method.id_empty
562 (["EqSystem", "normalise"], [],
563 {rew_ord'="tless_true", rls' = Rule_Set.Empty, calc = [], srls = Rule_Set.Empty, prls = Rule_Set.Empty, crls = Rule_Set.Empty,
564 errpats = [], nrls = Rule_Set.Empty},
568 partial_function (tailrec) solve_system2 :: "bool list => real list => bool list"
570 "solve_system2 e_s v_s = (
573 (Try (Rewrite_Set ''norm_Rational'' )) #>
574 (Try (Rewrite_Set_Inst [(''bdv_1'', hd v_s), (''bdv_2'', hd (tl v_s))] ''simplify_System_parenthesized'' )) #>
575 (Try (Rewrite_Set_Inst [(''bdv_1'', hd v_s), (''bdv_2'', hd (tl v_s))] ''isolate_bdvs'' )) #>
576 (Try (Rewrite_Set_Inst [(''bdv_1'', hd v_s), (''bdv_2'', hd (tl v_s))] ''simplify_System_parenthesized'' )) #>
577 (Try (Rewrite_Set ''order_system'' ))
580 SubProblem (''EqSystem'', [''LINEAR'', ''system''], [''no_met''])
581 [BOOL_LIST e__s, REAL_LIST v_s])"
582 setup \<open>KEStore_Elems.add_mets
583 [Specify.prep_met thy "met_eqsys_norm_2x2" [] Method.id_empty
584 (["EqSystem","normalise","2x2"],
585 [("#Given" ,["equalities e_s", "solveForVars v_s"]),
586 ("#Find" ,["solution ss'''"])],
587 {rew_ord'="tless_true", rls' = Rule_Set.Empty, calc = [],
588 srls = Rule_Set.append_rules "srls_normalise_2x2" Rule_Set.empty
589 [Rule.Thm ("hd_thm",ThmC.numerals_to_Free @{thm hd_thm}),
590 Rule.Thm ("tl_Cons",ThmC.numerals_to_Free @{thm tl_Cons}),
591 Rule.Thm ("tl_Nil",ThmC.numerals_to_Free @{thm tl_Nil})],
592 prls = Rule_Set.Empty, crls = Rule_Set.Empty, errpats = [], nrls = Rule_Set.Empty},
593 @{thm solve_system2.simps})]
596 partial_function (tailrec) solve_system3 :: "bool list => real list => bool list"
598 "solve_system3 e_s v_s = (
601 (Try (Rewrite_Set ''norm_Rational'' )) #>
602 (Repeat (Rewrite ''commute_0_equality'' )) #>
603 (Try (Rewrite_Set_Inst [(''bdv_1'', NTH 1 v_s), (''bdv_2'', NTH 2 v_s ),
604 (''bdv_3'', NTH 3 v_s), (''bdv_3'', NTH 4 v_s )] ''simplify_System_parenthesized'' )) #>
605 (Try (Rewrite_Set_Inst [(''bdv_1'', NTH 1 v_s), (''bdv_2'', NTH 2 v_s ),
606 (''bdv_3'', NTH 3 v_s), (''bdv_3'', NTH 4 v_s )] ''isolate_bdvs_4x4'' )) #>
607 (Try (Rewrite_Set_Inst [(''bdv_1'', NTH 1 v_s), (''bdv_2'', NTH 2 v_s ),
608 (''bdv_3'', NTH 3 v_s), (''bdv_3'', NTH 4 v_s )] ''simplify_System_parenthesized'' )) #>
609 (Try (Rewrite_Set ''order_system''))
612 SubProblem (''EqSystem'', [''LINEAR'', ''system''], [''no_met''])
613 [BOOL_LIST e__s, REAL_LIST v_s])"
614 setup \<open>KEStore_Elems.add_mets
615 [Specify.prep_met thy "met_eqsys_norm_4x4" [] Method.id_empty
616 (["EqSystem","normalise","4x4"],
617 [("#Given" ,["equalities e_s", "solveForVars v_s"]),
618 ("#Find" ,["solution ss'''"])],
619 {rew_ord'="tless_true", rls' = Rule_Set.Empty, calc = [],
620 srls = Rule_Set.append_rules "srls_normalise_4x4" srls
621 [Rule.Thm ("hd_thm",ThmC.numerals_to_Free @{thm hd_thm}),
622 Rule.Thm ("tl_Cons",ThmC.numerals_to_Free @{thm tl_Cons}),
623 Rule.Thm ("tl_Nil",ThmC.numerals_to_Free @{thm tl_Nil})],
624 prls = Rule_Set.Empty, crls = Rule_Set.Empty, errpats = [], nrls = Rule_Set.Empty},
625 (*STOPPED.WN06? met ["EqSystem","normalise","4x4"] #>#>#>#>#>#>#>#>#>#>#>#>#>@*)
626 @{thm solve_system3.simps})]
629 partial_function (tailrec) solve_system4 :: "bool list => real list => bool list"
631 "solve_system4 e_s v_s = (
634 e_2 = Take (NTH 2 e_s);
636 (Substitute [e_1]) #>
637 (Try (Rewrite_Set_Inst [(''bdv_1'',NTH 1 v_s), (''bdv_2'',NTH 2 v_s),
638 (''bdv_3'',NTH 3 v_s), (''bdv_4'',NTH 4 v_s)] ''simplify_System_parenthesized'' )) #>
639 (Try (Rewrite_Set_Inst [(''bdv_1'',NTH 1 v_s), (''bdv_2'',NTH 2 v_s),
640 (''bdv_3'',NTH 3 v_s), (''bdv_4'',NTH 4 v_s)] ''isolate_bdvs'' )) #>
641 (Try (Rewrite_Set_Inst [(''bdv_1'',NTH 1 v_s), (''bdv_2'',NTH 2 v_s),
642 (''bdv_3'',NTH 3 v_s), (''bdv_4'',NTH 4 v_s)] ''norm_Rational'' ))
645 [e_1, e_2, NTH 3 e_s, NTH 4 e_s])"
646 setup \<open>KEStore_Elems.add_mets
647 [Specify.prep_met thy "met_eqsys_topdown_4x4" [] Method.id_empty
648 (["EqSystem","top_down_substitution","4x4"],
649 [("#Given" ,["equalities e_s", "solveForVars v_s"]),
650 ("#Where" , (*accepts missing variables up to diagonal form*)
651 ["(NTH 1 (v_s::real list)) occurs_in (NTH 1 (e_s::bool list))",
652 "(NTH 2 (v_s::real list)) occurs_in (NTH 2 (e_s::bool list))",
653 "(NTH 3 (v_s::real list)) occurs_in (NTH 3 (e_s::bool list))",
654 "(NTH 4 (v_s::real list)) occurs_in (NTH 4 (e_s::bool list))"]),
655 ("#Find", ["solution ss'''"])],
656 {rew_ord'="ord_simplify_System", rls' = Rule_Set.Empty, calc = [],
657 srls = Rule_Set.append_rules "srls_top_down_4x4" srls [],
658 prls = Rule_Set.append_rules "prls_tri_4x4_lin_sys" prls_triangular
659 [Rule.Eval ("Prog_Expr.occurs'_in", Prog_Expr.eval_occurs_in "")],
660 crls = Rule_Set.Empty, errpats = [], nrls = Rule_Set.Empty},
661 (*FIXXXXME.WN060916: this script works ONLY for exp 7.79 #>#>#>#>#>#>#>#>#>#>*)
662 @{thm solve_system4.simps})]