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)
112 | dest_hd' _ = raise ERROR "dest_hd': uncovered case in fun.def.";
114 fun size_of_term' (Free (ccc, _)) =
115 (case Symbol.explode ccc of (*WN0510 hack for the bound variables*)
117 | "c"::"_"::is => 1000 * ((TermC.int_of_str o implode) is)
119 | size_of_term' (Abs (_,_,body)) = 1 + size_of_term' body
120 | size_of_term' (f$t) = size_of_term' f + size_of_term' t
121 | size_of_term' _ = 1;
123 fun term_ord' pr thy (Abs (_, T, t), Abs(_, U, u)) = (* ~ term.ML *)
124 (case term_ord' pr thy (t, u) of EQUAL => Term_Ord.typ_ord (T, U) | ord => ord)
125 | term_ord' pr thy (t, u) =
129 val (f, ts) = strip_comb t and (g, us) = strip_comb u;
130 val _ = tracing ("t= f@ts= \"" ^ UnparseC.term_in_thy thy f ^ "\" @ \"[" ^
131 commas (map (UnparseC.term_in_thy thy) ts) ^ "]\"");
132 val _ = tracing ("u= g@us= \"" ^ UnparseC.term_in_thy thy g ^ "\" @ \"[" ^
133 commas (map (UnparseC.term_in_thy thy) us) ^ "]\"");
134 val _ = tracing ("size_of_term(t,u)= (" ^ string_of_int (size_of_term' t) ^ ", " ^
135 string_of_int (size_of_term' u) ^ ")");
136 val _ = tracing ("hd_ord(f,g) = " ^ ((pr_ord o hd_ord) (f,g)));
137 val _ = tracing ("terms_ord (ts,us) = " ^(pr_ord o terms_ord str false) (ts,us));
138 val _=tracing("-------");
141 case int_ord (size_of_term' t, size_of_term' u) of
143 let val (f, ts) = strip_comb t and (g, us) = strip_comb u
144 in (case hd_ord (f, g) of
145 EQUAL => (terms_ord str pr) (ts, us)
149 and hd_ord (f, g) = (* ~ term.ML *)
150 prod_ord (prod_ord Term_Ord.indexname_ord Term_Ord.typ_ord) int_ord (dest_hd' f, dest_hd' g)
151 and terms_ord _ pr (ts, us) = list_ord (term_ord' pr (ThyC.get_theory "Isac_Knowledge"))(ts, us);
155 (*WN0510 for preliminary use in eval_order_system, see case-study mat-eng.tex
156 fun ord_simplify_System_rev (pr:bool) thy subst tu =
157 (term_ord' pr thy (Library.swap tu) = LESS);*)
160 fun ord_simplify_System (pr:bool) thy _(*subst*) tu =
161 (term_ord' pr thy tu = LESS);
165 Rewrite_Ord.rew_ord' := overwritel (! Rewrite_Ord.rew_ord',
166 [("ord_simplify_System", ord_simplify_System false thy)
172 (*.adapted from 'order_add_mult_in' by just replacing the rew_ord.*)
173 val order_add_mult_System =
174 Rule_Def.Repeat{id = "order_add_mult_System", preconds = [],
175 rew_ord = ("ord_simplify_System",
176 ord_simplify_System false @{theory "Integrate"}),
177 erls = Rule_Set.empty,srls = Rule_Set.Empty, calc = [], errpatts = [],
178 rules = [Rule.Thm ("mult.commute",ThmC.numerals_to_Free @{thm mult.commute}),
180 Rule.Thm ("real_mult_left_commute",ThmC.numerals_to_Free @{thm real_mult_left_commute}),
181 (*z1.0 * (z2.0 * z3.0) = z2.0 * (z1.0 * z3.0)*)
182 Rule.Thm ("mult.assoc",ThmC.numerals_to_Free @{thm mult.assoc}),
183 (*z1.0 * z2.0 * z3.0 = z1.0 * (z2.0 * z3.0)*)
184 Rule.Thm ("add.commute",ThmC.numerals_to_Free @{thm add.commute}),
186 Rule.Thm ("add.left_commute",ThmC.numerals_to_Free @{thm add.left_commute}),
187 (*x + (y + z) = y + (x + z)*)
188 Rule.Thm ("add.assoc",ThmC.numerals_to_Free @{thm add.assoc})
189 (*z1.0 + z2.0 + z3.0 = z1.0 + (z2.0 + z3.0)*)
191 scr = Rule.Empty_Prog};
194 (*.adapted from 'norm_Rational' by
195 #1 using 'ord_simplify_System' in 'order_add_mult_System'
196 #2 NOT using common_nominator_p .*)
197 val norm_System_noadd_fractions =
198 Rule_Def.Repeat {id = "norm_System_noadd_fractions", preconds = [],
199 rew_ord = ("dummy_ord",Rewrite_Ord.dummy_ord),
200 erls = norm_rat_erls, srls = Rule_Set.Empty, calc = [], errpatts = [],
201 rules = [(*sequence given by operator precedence*)
202 Rule.Rls_ discard_minus,
204 Rule.Rls_ rat_mult_divide,
206 Rule.Rls_ reduce_0_1_2,
207 Rule.Rls_ (*order_add_mult #1*) order_add_mult_System,
208 Rule.Rls_ collect_numerals,
209 (*Rule.Rls_ add_fractions_p, #2*)
212 scr = Rule.Empty_Prog
216 (*.adapted from 'norm_Rational' by
217 *1* using 'ord_simplify_System' in 'order_add_mult_System'.*)
219 Rule_Def.Repeat {id = "norm_System", preconds = [],
220 rew_ord = ("dummy_ord",Rewrite_Ord.dummy_ord),
221 erls = norm_rat_erls, srls = Rule_Set.Empty, calc = [], errpatts = [],
222 rules = [(*sequence given by operator precedence*)
223 Rule.Rls_ discard_minus,
225 Rule.Rls_ rat_mult_divide,
227 Rule.Rls_ reduce_0_1_2,
228 Rule.Rls_ (*order_add_mult *1*) order_add_mult_System,
229 Rule.Rls_ collect_numerals,
230 Rule.Rls_ add_fractions_p,
233 scr = Rule.Empty_Prog
237 (*.simplify an equational system BEFORE solving it such that parentheses are
238 ( ((u0*v0)*w0) + ( ((u1*v1)*w1) * c + ... +((u4*v4)*w4) * c_4 ) )
239 ATTENTION: works ONLY for bound variables c, c_1, c_2, c_3, c_4 :ATTENTION
240 This is a copy from 'make_ratpoly_in' with respective reductions:
241 *0* expand the term, ie. distribute * and / over +
242 *1* ord_simplify_System instead of termlessI
243 *2* no add_fractions_p (= common_nominator_p_rls !)
244 *3* discard_parentheses only for (.*(.*.))
245 analoguous to simplify_Integral .*)
246 val simplify_System_parenthesized =
247 Rule_Set.Sequence {id = "simplify_System_parenthesized", preconds = []:term list,
248 rew_ord = ("dummy_ord", Rewrite_Ord.dummy_ord),
249 erls = Atools_erls, srls = Rule_Set.Empty, calc = [], errpatts = [],
250 rules = [Rule.Thm ("distrib_right",ThmC.numerals_to_Free @{thm distrib_right}),
251 (*"(?z1.0 + ?z2.0) * ?w = ?z1.0 * ?w + ?z2.0 * ?w"*)
252 Rule.Thm ("add_divide_distrib",ThmC.numerals_to_Free @{thm add_divide_distrib}),
253 (*"(?x + ?y) / ?z = ?x / ?z + ?y / ?z"*)
254 (*^^^^^ *0* ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^*)
255 Rule.Rls_ norm_Rational_noadd_fractions(**2**),
256 Rule.Rls_ (*order_add_mult_in*) norm_System_noadd_fractions (**1**),
257 Rule.Thm ("sym_mult.assoc",
258 ThmC.numerals_to_Free (@{thm mult.assoc} RS @{thm sym}))
259 (*Rule.Rls_ discard_parentheses *3**),
260 Rule.Rls_ collect_bdv, (*from make_polynomial_in WN051031 welldone?*)
261 Rule.Rls_ separate_bdv2,
262 Rule.Eval ("Rings.divide_class.divide", Prog_Expr.eval_cancel "#divide_e")
264 scr = Rule.Empty_Prog};
267 (*.simplify an equational system AFTER solving it;
268 This is a copy of 'make_ratpoly_in' with the differences
269 *1* ord_simplify_System instead of termlessI .*)
270 (*TODO.WN051031 ^^^^^^^^^^ should be in EACH rls contained *)
271 val simplify_System =
272 Rule_Set.Sequence {id = "simplify_System", preconds = []:term list,
273 rew_ord = ("dummy_ord", Rewrite_Ord.dummy_ord),
274 erls = Atools_erls, srls = Rule_Set.Empty, calc = [], errpatts = [],
275 rules = [Rule.Rls_ norm_Rational,
276 Rule.Rls_ (*order_add_mult_in*) norm_System (**1**),
277 Rule.Rls_ discard_parentheses,
278 Rule.Rls_ collect_bdv, (*from make_polynomial_in WN051031 welldone?*)
279 Rule.Rls_ separate_bdv2,
280 Rule.Eval ("Rings.divide_class.divide", Prog_Expr.eval_cancel "#divide_e")
282 scr = Rule.Empty_Prog};
284 val simplify_System =
285 Rule_Set.append_rules "simplify_System" simplify_System_parenthesized
286 [Rule.Thm ("sym_add.assoc",
287 ThmC.numerals_to_Free (@{thm add.assoc} RS @{thm sym}))];
292 Rule_Def.Repeat {id="isolate_bdvs", preconds = [],
293 rew_ord = ("e_rew_ord", Rewrite_Ord.e_rew_ord),
294 erls = Rule_Set.append_rules "erls_isolate_bdvs" Rule_Set.empty
295 [(Rule.Eval ("EqSystem.occur'_exactly'_in",
296 eval_occur_exactly_in
297 "#eval_occur_exactly_in_"))
299 srls = Rule_Set.Empty, calc = [], errpatts = [],
301 [Rule.Thm ("commute_0_equality", ThmC.numerals_to_Free @{thm commute_0_equality}),
302 Rule.Thm ("separate_bdvs_add", ThmC.numerals_to_Free @{thm separate_bdvs_add}),
303 Rule.Thm ("separate_bdvs_mult", ThmC.numerals_to_Free @{thm separate_bdvs_mult})],
304 scr = Rule.Empty_Prog};
307 val isolate_bdvs_4x4 =
308 Rule_Def.Repeat {id="isolate_bdvs_4x4", preconds = [],
309 rew_ord = ("e_rew_ord", Rewrite_Ord.e_rew_ord),
310 erls = Rule_Set.append_rules
311 "erls_isolate_bdvs_4x4" Rule_Set.empty
312 [Rule.Eval ("EqSystem.occur'_exactly'_in",
313 eval_occur_exactly_in "#eval_occur_exactly_in_"),
314 Rule.Eval ("Prog_Expr.ident", Prog_Expr.eval_ident "#ident_"),
315 Rule.Eval ("Prog_Expr.some'_occur'_in", Prog_Expr.eval_some_occur_in "#some_occur_in_"),
316 Rule.Thm ("not_true",ThmC.numerals_to_Free @{thm not_true}),
317 Rule.Thm ("not_false",ThmC.numerals_to_Free @{thm not_false})
319 srls = Rule_Set.Empty, calc = [], errpatts = [],
320 rules = [Rule.Thm ("commute_0_equality", ThmC.numerals_to_Free @{thm commute_0_equality}),
321 Rule.Thm ("separate_bdvs0", ThmC.numerals_to_Free @{thm separate_bdvs0}),
322 Rule.Thm ("separate_bdvs_add1", ThmC.numerals_to_Free @{thm separate_bdvs_add1}),
323 Rule.Thm ("separate_bdvs_add1", ThmC.numerals_to_Free @{thm separate_bdvs_add2}),
324 Rule.Thm ("separate_bdvs_mult", ThmC.numerals_to_Free @{thm separate_bdvs_mult})
325 ], scr = Rule.Empty_Prog};
330 (*.order the equations in a system such, that a triangular system (if any)
331 appears as [..c_4 = .., ..., ..., ..c_1 + ..c_2 + ..c_3 ..c_4 = ..].*)
333 Rule_Def.Repeat {id="order_system", preconds = [],
334 rew_ord = ("ord_simplify_System",
335 ord_simplify_System false thy),
336 erls = Rule_Set.Empty, srls = Rule_Set.Empty, calc = [], errpatts = [],
337 rules = [Rule.Thm ("order_system_NxN", ThmC.numerals_to_Free @{thm order_system_NxN})
339 scr = Rule.Empty_Prog};
341 val prls_triangular =
342 Rule_Def.Repeat {id="prls_triangular", preconds = [],
343 rew_ord = ("e_rew_ord", Rewrite_Ord.e_rew_ord),
344 erls = Rule_Def.Repeat {id="erls_prls_triangular", preconds = [],
345 rew_ord = ("e_rew_ord", Rewrite_Ord.e_rew_ord),
346 erls = Rule_Set.Empty, srls = Rule_Set.Empty, calc = [], errpatts = [],
347 rules = [(*for precond NTH_CONS ...*)
348 Rule.Eval ("Orderings.ord_class.less", Prog_Expr.eval_equ "#less_"),
349 Rule.Eval ("Groups.plus_class.plus", (**)eval_binop "#add_")
350 (*immediately repeated rewrite pushes
351 '+' into precondition !*)
353 scr = Rule.Empty_Prog},
354 srls = Rule_Set.Empty, calc = [], errpatts = [],
355 rules = [Rule.Thm ("NTH_CONS",ThmC.numerals_to_Free @{thm NTH_CONS}),
356 Rule.Eval ("Groups.plus_class.plus", (**)eval_binop "#add_"),
357 Rule.Thm ("NTH_NIL",ThmC.numerals_to_Free @{thm NTH_NIL}),
358 Rule.Thm ("tl_Cons",ThmC.numerals_to_Free @{thm tl_Cons}),
359 Rule.Thm ("tl_Nil",ThmC.numerals_to_Free @{thm tl_Nil}),
360 Rule.Eval ("EqSystem.occur'_exactly'_in",
361 eval_occur_exactly_in
362 "#eval_occur_exactly_in_")
364 scr = Rule.Empty_Prog};
368 (*WN060914 quickly created for 4x4;
369 more similarity to prls_triangular desirable*)
370 val prls_triangular4 =
371 Rule_Def.Repeat {id="prls_triangular4", preconds = [],
372 rew_ord = ("e_rew_ord", Rewrite_Ord.e_rew_ord),
373 erls = Rule_Def.Repeat {id="erls_prls_triangular4", preconds = [],
374 rew_ord = ("e_rew_ord", Rewrite_Ord.e_rew_ord),
375 erls = Rule_Set.Empty, srls = Rule_Set.Empty, calc = [], errpatts = [],
376 rules = [(*for precond NTH_CONS ...*)
377 Rule.Eval ("Orderings.ord_class.less", Prog_Expr.eval_equ "#less_"),
378 Rule.Eval ("Groups.plus_class.plus", (**)eval_binop "#add_")
379 (*immediately repeated rewrite pushes
380 '+' into precondition !*)
382 scr = Rule.Empty_Prog},
383 srls = Rule_Set.Empty, calc = [], errpatts = [],
384 rules = [Rule.Thm ("NTH_CONS",ThmC.numerals_to_Free @{thm NTH_CONS}),
385 Rule.Eval ("Groups.plus_class.plus", (**)eval_binop "#add_"),
386 Rule.Thm ("NTH_NIL",ThmC.numerals_to_Free @{thm NTH_NIL}),
387 Rule.Thm ("tl_Cons",ThmC.numerals_to_Free @{thm tl_Cons}),
388 Rule.Thm ("tl_Nil",ThmC.numerals_to_Free @{thm tl_Nil}),
389 Rule.Eval ("EqSystem.occur'_exactly'_in",
390 eval_occur_exactly_in
391 "#eval_occur_exactly_in_")
393 scr = Rule.Empty_Prog};
396 setup \<open>KEStore_Elems.add_rlss
397 [("simplify_System_parenthesized",
398 (Context.theory_name @{theory}, prep_rls' simplify_System_parenthesized)),
399 ("simplify_System", (Context.theory_name @{theory}, prep_rls' simplify_System)),
400 ("isolate_bdvs", (Context.theory_name @{theory}, prep_rls' isolate_bdvs)),
401 ("isolate_bdvs_4x4", (Context.theory_name @{theory}, prep_rls' isolate_bdvs_4x4)),
402 ("order_system", (Context.theory_name @{theory}, prep_rls' order_system)),
403 ("order_add_mult_System", (Context.theory_name @{theory}, prep_rls' order_add_mult_System)),
404 ("norm_System_noadd_fractions",
405 (Context.theory_name @{theory}, prep_rls' norm_System_noadd_fractions)),
406 ("norm_System", (Context.theory_name @{theory}, prep_rls' norm_System))]\<close>
409 section \<open>Problems\<close>
411 setup \<open>KEStore_Elems.add_pbts
412 [(Problem.prep_input thy "pbl_equsys" [] Problem.id_empty
414 [("#Given" ,["equalities e_s", "solveForVars v_s"]),
415 ("#Find" ,["solution ss'''"](*''' is copy-named*))],
416 Rule_Set.append_rules "empty" Rule_Set.empty [(*for preds in where_*)], SOME "solveSystem e_s v_s", [])),
417 (Problem.prep_input thy "pbl_equsys_lin" [] Problem.id_empty
418 (["LINEAR", "system"],
419 [("#Given" ,["equalities e_s", "solveForVars v_s"]),
420 (*TODO.WN050929 check linearity*)
421 ("#Find" ,["solution ss'''"])],
422 Rule_Set.append_rules "empty" Rule_Set.empty [(*for preds in where_*)], SOME "solveSystem e_s v_s", [])),
423 (Problem.prep_input thy "pbl_equsys_lin_2x2" [] Problem.id_empty
424 (["2x2", "LINEAR", "system"],
425 (*~~~~~~~~~~~~~~~~~~~~~~~~~*)
426 [("#Given" ,["equalities e_s", "solveForVars v_s"]),
427 ("#Where" ,["Length (e_s:: bool list) = 2", "Length v_s = 2"]),
428 ("#Find" ,["solution ss'''"])],
429 Rule_Set.append_rules "prls_2x2_linear_system" Rule_Set.empty
430 [Rule.Thm ("LENGTH_CONS",ThmC.numerals_to_Free @{thm LENGTH_CONS}),
431 Rule.Thm ("LENGTH_NIL",ThmC.numerals_to_Free @{thm LENGTH_NIL}),
432 Rule.Eval ("Groups.plus_class.plus", (**)eval_binop "#add_"),
433 Rule.Eval ("HOL.eq", Prog_Expr.eval_equal "#equal_")],
434 SOME "solveSystem e_s v_s", [])),
435 (Problem.prep_input thy "pbl_equsys_lin_2x2_tri" [] Problem.id_empty
436 (["triangular", "2x2", "LINEAR", "system"],
437 [("#Given" ,["equalities e_s", "solveForVars v_s"]),
439 ["(tl v_s) from v_s occur_exactly_in (NTH 1 (e_s::bool list))",
440 " v_s from v_s occur_exactly_in (NTH 2 (e_s::bool list))"]),
441 ("#Find" ,["solution ss'''"])],
442 prls_triangular, SOME "solveSystem e_s v_s", [["EqSystem", "top_down_substitution", "2x2"]])),
443 (Problem.prep_input thy "pbl_equsys_lin_2x2_norm" [] Problem.id_empty
444 (["normalise", "2x2", "LINEAR", "system"],
445 [("#Given" ,["equalities e_s", "solveForVars v_s"]),
446 ("#Find" ,["solution ss'''"])],
447 Rule_Set.append_rules "empty" Rule_Set.empty [(*for preds in where_*)],
448 SOME "solveSystem e_s v_s",
449 [["EqSystem", "normalise", "2x2"]])),
450 (Problem.prep_input thy "pbl_equsys_lin_3x3" [] Problem.id_empty
451 (["3x3", "LINEAR", "system"],
452 (*~~~~~~~~~~~~~~~~~~~~~~~~~*)
453 [("#Given" ,["equalities e_s", "solveForVars v_s"]),
454 ("#Where" ,["Length (e_s:: bool list) = 3", "Length v_s = 3"]),
455 ("#Find" ,["solution ss'''"])],
456 Rule_Set.append_rules "prls_3x3_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 ("Groups.plus_class.plus", (**)eval_binop "#add_"),
460 Rule.Eval ("HOL.eq", Prog_Expr.eval_equal "#equal_")],
461 SOME "solveSystem e_s v_s", [])),
462 (Problem.prep_input thy "pbl_equsys_lin_4x4" [] Problem.id_empty
463 (["4x4", "LINEAR", "system"],
464 (*~~~~~~~~~~~~~~~~~~~~~~~~~*)
465 [("#Given" ,["equalities e_s", "solveForVars v_s"]),
466 ("#Where" ,["Length (e_s:: bool list) = 4", "Length v_s = 4"]),
467 ("#Find" ,["solution ss'''"])],
468 Rule_Set.append_rules "prls_4x4_linear_system" Rule_Set.empty
469 [Rule.Thm ("LENGTH_CONS",ThmC.numerals_to_Free @{thm LENGTH_CONS}),
470 Rule.Thm ("LENGTH_NIL",ThmC.numerals_to_Free @{thm LENGTH_NIL}),
471 Rule.Eval ("Groups.plus_class.plus", (**)eval_binop "#add_"),
472 Rule.Eval ("HOL.eq", Prog_Expr.eval_equal "#equal_")],
473 SOME "solveSystem e_s v_s", [])),
474 (Problem.prep_input thy "pbl_equsys_lin_4x4_tri" [] Problem.id_empty
475 (["triangular", "4x4", "LINEAR", "system"],
476 [("#Given" ,["equalities e_s", "solveForVars v_s"]),
477 ("#Where" , (*accepts missing variables up to diagional form*)
478 ["(NTH 1 (v_s::real list)) occurs_in (NTH 1 (e_s::bool list))",
479 "(NTH 2 (v_s::real list)) occurs_in (NTH 2 (e_s::bool list))",
480 "(NTH 3 (v_s::real list)) occurs_in (NTH 3 (e_s::bool list))",
481 "(NTH 4 (v_s::real list)) occurs_in (NTH 4 (e_s::bool list))"]),
482 ("#Find" ,["solution ss'''"])],
483 Rule_Set.append_rules "prls_tri_4x4_lin_sys" prls_triangular
484 [Rule.Eval ("Prog_Expr.occurs'_in", Prog_Expr.eval_occurs_in "")],
485 SOME "solveSystem e_s v_s",
486 [["EqSystem", "top_down_substitution", "4x4"]])),
487 (Problem.prep_input thy "pbl_equsys_lin_4x4_norm" [] Problem.id_empty
488 (["normalise", "4x4", "LINEAR", "system"],
489 [("#Given" ,["equalities e_s", "solveForVars v_s"]),
490 (*Length is checked 1 level above*)
491 ("#Find" ,["solution ss'''"])],
492 Rule_Set.append_rules "empty" Rule_Set.empty [(*for preds in where_*)],
493 SOME "solveSystem e_s v_s",
494 [["EqSystem", "normalise", "4x4"]]))]\<close>
497 (*this is for NTH only*)
498 val srls = Rule_Def.Repeat {id="srls_normalise_4x4",
500 rew_ord = ("termlessI",termlessI),
501 erls = Rule_Set.append_rules "erls_in_srls_IntegrierenUnd.." Rule_Set.empty
502 [(*for asm in NTH_CONS ...*)
503 Rule.Eval ("Orderings.ord_class.less", Prog_Expr.eval_equ "#less_"),
504 (*2nd NTH_CONS pushes n+-1 into asms*)
505 Rule.Eval("Groups.plus_class.plus", (**)eval_binop "#add_")
507 srls = Rule_Set.Empty, calc = [], errpatts = [],
508 rules = [Rule.Thm ("NTH_CONS",ThmC.numerals_to_Free @{thm NTH_CONS}),
509 Rule.Eval("Groups.plus_class.plus", (**)eval_binop "#add_"),
510 Rule.Thm ("NTH_NIL",ThmC.numerals_to_Free @{thm NTH_NIL})],
511 scr = Rule.Empty_Prog};
514 section \<open>Methods\<close>
516 setup \<open>KEStore_Elems.add_mets
517 [MethodC.prep_input thy "met_eqsys" [] MethodC.id_empty
519 {rew_ord'="tless_true", rls' = Rule_Set.Empty, calc = [], srls = Rule_Set.Empty, prls = Rule_Set.Empty, crls = Rule_Set.Empty,
520 errpats = [], nrls = Rule_Set.Empty},
522 MethodC.prep_input thy "met_eqsys_topdown" [] MethodC.id_empty
523 (["EqSystem", "top_down_substitution"], [],
524 {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},
529 partial_function (tailrec) solve_system :: "bool list => real list => bool list"
531 "solve_system e_s v_s = (
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''))
538 e_2 = Take (hd (tl e_s));
540 (Substitute [e_1]) #>
541 (Try (Rewrite_Set_Inst [(''bdv_1'', hd v_s), (''bdv_2'', hd (tl v_s))] ''simplify_System_parenthesized'' )) #>
542 (Try (Rewrite_Set_Inst [(''bdv_1'', hd v_s), (''bdv_2'', hd (tl v_s))] ''isolate_bdvs'' )) #>
543 (Try (Rewrite_Set_Inst [(''bdv_1'', hd v_s), (''bdv_2'', hd (tl v_s))] ''simplify_System'' ))
545 e__s = Take [e_1, e_2]
547 Try (Rewrite_Set ''order_system'' ) e__s) "
548 setup \<open>KEStore_Elems.add_mets
549 [MethodC.prep_input thy "met_eqsys_topdown_2x2" [] MethodC.id_empty
550 (["EqSystem", "top_down_substitution", "2x2"],
551 [("#Given", ["equalities e_s", "solveForVars v_s"]),
553 ["(tl v_s) from v_s occur_exactly_in (NTH 1 (e_s::bool list))",
554 " v_s from v_s occur_exactly_in (NTH 2 (e_s::bool list))"]),
555 ("#Find" ,["solution ss'''"])],
556 {rew_ord'="ord_simplify_System", rls' = Rule_Set.Empty, calc = [],
557 srls = Rule_Set.append_rules "srls_top_down_2x2" Rule_Set.empty
558 [Rule.Thm ("hd_thm",ThmC.numerals_to_Free @{thm hd_thm}),
559 Rule.Thm ("tl_Cons",ThmC.numerals_to_Free @{thm tl_Cons}),
560 Rule.Thm ("tl_Nil",ThmC.numerals_to_Free @{thm tl_Nil})],
561 prls = prls_triangular, crls = Rule_Set.Empty, errpats = [], nrls = Rule_Set.Empty},
562 @{thm solve_system.simps})]
564 setup \<open>KEStore_Elems.add_mets
565 [MethodC.prep_input thy "met_eqsys_norm" [] MethodC.id_empty
566 (["EqSystem", "normalise"], [],
567 {rew_ord'="tless_true", rls' = Rule_Set.Empty, calc = [], srls = Rule_Set.Empty, prls = Rule_Set.Empty, crls = Rule_Set.Empty,
568 errpats = [], nrls = Rule_Set.Empty},
572 partial_function (tailrec) solve_system2 :: "bool list => real list => bool list"
574 "solve_system2 e_s v_s = (
577 (Try (Rewrite_Set ''norm_Rational'' )) #>
578 (Try (Rewrite_Set_Inst [(''bdv_1'', hd v_s), (''bdv_2'', hd (tl v_s))] ''simplify_System_parenthesized'' )) #>
579 (Try (Rewrite_Set_Inst [(''bdv_1'', hd v_s), (''bdv_2'', hd (tl v_s))] ''isolate_bdvs'' )) #>
580 (Try (Rewrite_Set_Inst [(''bdv_1'', hd v_s), (''bdv_2'', hd (tl v_s))] ''simplify_System_parenthesized'' )) #>
581 (Try (Rewrite_Set ''order_system'' ))
584 SubProblem (''EqSystem'', [''LINEAR'', ''system''], [''no_met''])
585 [BOOL_LIST e__s, REAL_LIST v_s])"
586 setup \<open>KEStore_Elems.add_mets
587 [MethodC.prep_input thy "met_eqsys_norm_2x2" [] MethodC.id_empty
588 (["EqSystem", "normalise", "2x2"],
589 [("#Given" ,["equalities e_s", "solveForVars v_s"]),
590 ("#Find" ,["solution ss'''"])],
591 {rew_ord'="tless_true", rls' = Rule_Set.Empty, calc = [],
592 srls = Rule_Set.append_rules "srls_normalise_2x2" Rule_Set.empty
593 [Rule.Thm ("hd_thm",ThmC.numerals_to_Free @{thm hd_thm}),
594 Rule.Thm ("tl_Cons",ThmC.numerals_to_Free @{thm tl_Cons}),
595 Rule.Thm ("tl_Nil",ThmC.numerals_to_Free @{thm tl_Nil})],
596 prls = Rule_Set.Empty, crls = Rule_Set.Empty, errpats = [], nrls = Rule_Set.Empty},
597 @{thm solve_system2.simps})]
600 partial_function (tailrec) solve_system3 :: "bool list => real list => bool list"
602 "solve_system3 e_s v_s = (
605 (Try (Rewrite_Set ''norm_Rational'' )) #>
606 (Repeat (Rewrite ''commute_0_equality'' )) #>
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_Inst [(''bdv_1'', NTH 1 v_s), (''bdv_2'', NTH 2 v_s ),
610 (''bdv_3'', NTH 3 v_s), (''bdv_3'', NTH 4 v_s )] ''isolate_bdvs_4x4'' )) #>
611 (Try (Rewrite_Set_Inst [(''bdv_1'', NTH 1 v_s), (''bdv_2'', NTH 2 v_s ),
612 (''bdv_3'', NTH 3 v_s), (''bdv_3'', NTH 4 v_s )] ''simplify_System_parenthesized'' )) #>
613 (Try (Rewrite_Set ''order_system''))
616 SubProblem (''EqSystem'', [''LINEAR'', ''system''], [''no_met''])
617 [BOOL_LIST e__s, REAL_LIST v_s])"
618 setup \<open>KEStore_Elems.add_mets
619 [MethodC.prep_input thy "met_eqsys_norm_4x4" [] MethodC.id_empty
620 (["EqSystem", "normalise", "4x4"],
621 [("#Given" ,["equalities e_s", "solveForVars v_s"]),
622 ("#Find" ,["solution ss'''"])],
623 {rew_ord'="tless_true", rls' = Rule_Set.Empty, calc = [],
624 srls = Rule_Set.append_rules "srls_normalise_4x4" srls
625 [Rule.Thm ("hd_thm",ThmC.numerals_to_Free @{thm hd_thm}),
626 Rule.Thm ("tl_Cons",ThmC.numerals_to_Free @{thm tl_Cons}),
627 Rule.Thm ("tl_Nil",ThmC.numerals_to_Free @{thm tl_Nil})],
628 prls = Rule_Set.Empty, crls = Rule_Set.Empty, errpats = [], nrls = Rule_Set.Empty},
629 (*STOPPED.WN06? met ["EqSystem", "normalise", "4x4"] #>#>#>#>#>#>#>#>#>#>#>#>#>@*)
630 @{thm solve_system3.simps})]
633 partial_function (tailrec) solve_system4 :: "bool list => real list => bool list"
635 "solve_system4 e_s v_s = (
638 e_2 = Take (NTH 2 e_s);
640 (Substitute [e_1]) #>
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)] ''simplify_System_parenthesized'' )) #>
643 (Try (Rewrite_Set_Inst [(''bdv_1'',NTH 1 v_s), (''bdv_2'',NTH 2 v_s),
644 (''bdv_3'',NTH 3 v_s), (''bdv_4'',NTH 4 v_s)] ''isolate_bdvs'' )) #>
645 (Try (Rewrite_Set_Inst [(''bdv_1'',NTH 1 v_s), (''bdv_2'',NTH 2 v_s),
646 (''bdv_3'',NTH 3 v_s), (''bdv_4'',NTH 4 v_s)] ''norm_Rational'' ))
649 [e_1, e_2, NTH 3 e_s, NTH 4 e_s])"
650 setup \<open>KEStore_Elems.add_mets
651 [MethodC.prep_input thy "met_eqsys_topdown_4x4" [] MethodC.id_empty
652 (["EqSystem", "top_down_substitution", "4x4"],
653 [("#Given" ,["equalities e_s", "solveForVars v_s"]),
654 ("#Where" , (*accepts missing variables up to diagonal form*)
655 ["(NTH 1 (v_s::real list)) occurs_in (NTH 1 (e_s::bool list))",
656 "(NTH 2 (v_s::real list)) occurs_in (NTH 2 (e_s::bool list))",
657 "(NTH 3 (v_s::real list)) occurs_in (NTH 3 (e_s::bool list))",
658 "(NTH 4 (v_s::real list)) occurs_in (NTH 4 (e_s::bool list))"]),
659 ("#Find", ["solution ss'''"])],
660 {rew_ord'="ord_simplify_System", rls' = Rule_Set.Empty, calc = [],
661 srls = Rule_Set.append_rules "srls_top_down_4x4" srls [],
662 prls = Rule_Set.append_rules "prls_tri_4x4_lin_sys" prls_triangular
663 [Rule.Eval ("Prog_Expr.occurs'_in", Prog_Expr.eval_occurs_in "")],
664 crls = Rule_Set.Empty, errpats = [], nrls = Rule_Set.Empty},
665 (*FIXXXXME.WN060916: this script works ONLY for exp 7.79 #>#>#>#>#>#>#>#>#>#>*)
666 @{thm solve_system4.simps})]