Test_Isac works again, perfectly ..
# the same tests works as in 8df4b6196660 (the *child* of "Test_Isac works...")
# ..EXCEPT those marked with "exception Div raised"
# for general state of tests see Test_Isac section {* history of tests *}.
1 (* integration over the reals
4 (c) due to copyright terms
7 theory Integrate imports Diff begin
11 Integral :: "[real, real]=> real" ("Integral _ D _" 91)
12 (*new'_c :: "real => real" ("new'_c _" 66)*)
13 is'_f'_x :: "real => bool" ("_ is'_f'_x" 10)
15 (*descriptions in the related problems*)
16 integrateBy :: "real => una"
17 antiDerivative :: "real => una"
18 antiDerivativeName :: "(real => real) => una"
20 (*the CAS-command, eg. "Integrate (2*x^^^3, x)"*)
21 Integrate :: "[real * real] => real"
24 IntegrationScript :: "[real,real, real] => real"
25 ("((Script IntegrationScript (_ _ =))// (_))" 9)
26 NamedIntegrationScript :: "[real,real, real=>real, bool] => bool"
27 ("((Script NamedIntegrationScript (_ _ _=))// (_))" 9)
29 axioms(*axiomatization where*)
30 (*stated as axioms, todo: prove as theorems
31 'bdv' is a constant handled on the meta-level
32 specifically as a 'bound variable' *)
34 integral_const: "Not (bdv occurs_in u) ==> Integral u D bdv = u * bdv" (*and*)
35 integral_var: "Integral bdv D bdv = bdv ^^^ 2 / 2" (*and*)
37 integral_add: "Integral (u + v) D bdv =
38 (Integral u D bdv) + (Integral v D bdv)" (*and*)
39 integral_mult: "[| Not (bdv occurs_in u); bdv occurs_in v |] ==>
40 Integral (u * v) D bdv = u * (Integral v D bdv)" (*and*)
41 (*WN080222: this goes into sub-terms, too ...
42 call_for_new_c: "[| Not (matches (u + new_c v) a); Not (a is_f_x) |] ==>
45 integral_pow: "Integral bdv ^^^ n D bdv = bdv ^^^ (n+1) / (n + 1)"
50 (** eval functions **)
52 val c = Free ("c", HOLogic.realT);
53 (*.create a new unique variable 'c..' in a term; for use by Calc in a rls;
54 an alternative to do this would be '(Try (Calculate new_c_) (new_c es__))'
55 in the script; this will be possible if currying doesnt take the value
56 from a variable, but the value '(new_c es__)' itself.*)
59 case (Symbol.explode o id_of) var of
61 | "c"::"_"::is => (case (int_of_str o implode) is of
65 fun get_coeff c = case (Symbol.explode o id_of) c of
66 "c"::"_"::is => (the o int_of_str o implode) is
68 val cs = filter selc (vars term);
72 | [c] => Free ("c_2", HOLogic.realT)
74 let val max_coeff = maxl (map get_coeff cs)
75 in Free ("c_"^string_of_int (max_coeff + 1), HOLogic.realT) end
79 (*("new_c", ("Integrate.new'_c", eval_new_c "#new_c_"))*)
80 fun eval_new_c _ _ (p as (Const ("Integrate.new'_c",_) $ t)) _ =
81 SOME ((term2str p) ^ " = " ^ term2str (new_c p),
82 Trueprop $ (mk_equality (p, new_c p)))
83 | eval_new_c _ _ _ _ = NONE;
87 (*("add_new_c", ("Integrate.add'_new'_c", eval_add_new_c "#add_new_c_"))
88 add a new c to a term or a fun-equation;
89 this is _not in_ the term, because only applied to _whole_ term*)
90 fun eval_add_new_c (_:string) "Integrate.add'_new'_c" p (_:theory) =
91 let val p' = case p of
92 Const ("HOL.eq", T) $ lh $ rh =>
93 Const ("HOL.eq", T) $ lh $ mk_add rh (new_c rh)
94 | p => mk_add p (new_c p)
95 in SOME ((term2str p) ^ " = " ^ term2str p',
96 Trueprop $ (mk_equality (p, p')))
98 | eval_add_new_c _ _ _ _ = NONE;
101 (*("is_f_x", ("Integrate.is'_f'_x", eval_is_f_x "is_f_x_"))*)
102 fun eval_is_f_x _ _(p as (Const ("Integrate.is'_f'_x", _)
105 then SOME ((term2str p) ^ " = True",
106 Trueprop $ (mk_equality (p, @{term True})))
107 else SOME ((term2str p) ^ " = False",
108 Trueprop $ (mk_equality (p, @{term False})))
109 | eval_is_f_x _ _ _ _ = NONE;
111 calclist':= overwritel (!calclist',
112 [(*("new_c", ("Integrate.new'_c", eval_new_c "new_c_")),*)
113 ("add_new_c", ("Integrate.add'_new'_c", eval_add_new_c "add_new_c_")),
114 ("is_f_x", ("Integrate.is'_f'_x", eval_is_f_x "is_f_idextifier_"))
120 (*.rulesets for integration.*)
121 val integration_rules =
122 Rls {id="integration_rules", preconds = [],
123 rew_ord = ("termlessI",termlessI),
124 erls = Rls {id="conditions_in_integration_rules",
126 rew_ord = ("termlessI",termlessI),
128 srls = Erls, calc = [], errpatts = [],
129 rules = [(*for rewriting conditions in Thm's*)
130 Calc ("Atools.occurs'_in",
131 eval_occurs_in "#occurs_in_"),
132 Thm ("not_true",num_str @{thm not_true}),
133 Thm ("not_false",@{thm not_false})
136 srls = Erls, calc = [], errpatts = [],
138 Thm ("integral_const",num_str @{thm integral_const}),
139 Thm ("integral_var",num_str @{thm integral_var}),
140 Thm ("integral_add",num_str @{thm integral_add}),
141 Thm ("integral_mult",num_str @{thm integral_mult}),
142 Thm ("integral_pow",num_str @{thm integral_pow}),
143 Calc ("Groups.plus_class.plus", eval_binop "#add_")(*for n+1*)
149 Seq {id="add_new_c", preconds = [],
150 rew_ord = ("termlessI",termlessI),
151 erls = Rls {id="conditions_in_add_new_c",
153 rew_ord = ("termlessI",termlessI),
155 srls = Erls, calc = [], errpatts = [],
156 rules = [Calc ("Tools.matches", eval_matches""),
157 Calc ("Integrate.is'_f'_x",
158 eval_is_f_x "is_f_x_"),
159 Thm ("not_true",num_str @{thm not_true}),
160 Thm ("not_false",num_str @{thm not_false})
163 srls = Erls, calc = [], errpatts = [],
164 rules = [ (*Thm ("call_for_new_c", num_str @{thm call_for_new_c}),*)
165 Cal1 ("Integrate.add'_new'_c", eval_add_new_c "new_c_")
171 (*.rulesets for simplifying Integrals.*)
173 (*.for simplify_Integral adapted from 'norm_Rational_rls'.*)
174 val norm_Rational_rls_noadd_fractions =
175 Rls {id = "norm_Rational_rls_noadd_fractions", preconds = [],
176 rew_ord = ("dummy_ord",dummy_ord),
177 erls = norm_rat_erls, srls = Erls, calc = [], errpatts = [],
178 rules = [(*Rls_ add_fractions_p_rls,!!!*)
179 Rls_ (*rat_mult_div_pow original corrected WN051028*)
180 (Rls {id = "rat_mult_div_pow", preconds = [],
181 rew_ord = ("dummy_ord",dummy_ord),
182 erls = (*FIXME.WN051028 e_rls,*)
183 append_rls "e_rls-is_polyexp" e_rls
184 [Calc ("Poly.is'_polyexp",
185 eval_is_polyexp "")],
186 srls = Erls, calc = [], errpatts = [],
187 rules = [Thm ("rat_mult",num_str @{thm rat_mult}),
188 (*"?a / ?b * (?c / ?d) = ?a * ?c / (?b * ?d)"*)
189 Thm ("rat_mult_poly_l",num_str @{thm rat_mult_poly_l}),
190 (*"?c is_polyexp ==> ?c * (?a / ?b) = ?c * ?a / ?b"*)
191 Thm ("rat_mult_poly_r",num_str @{thm rat_mult_poly_r}),
192 (*"?c is_polyexp ==> ?a / ?b * ?c = ?a * ?c / ?b"*)
194 Thm ("real_divide_divide1_mg",
195 num_str @{thm real_divide_divide1_mg}),
196 (*"y ~= 0 ==> (u / v) / (y / z) = (u * z) / (y * v)"*)
197 Thm ("divide_divide_eq_right",
198 num_str @{thm divide_divide_eq_right}),
199 (*"?x / (?y / ?z) = ?x * ?z / ?y"*)
200 Thm ("divide_divide_eq_left",
201 num_str @{thm divide_divide_eq_left}),
202 (*"?x / ?y / ?z = ?x / (?y * ?z)"*)
203 Calc ("Fields.inverse_class.divide" ,eval_cancel "#divide_e"),
205 Thm ("rat_power", num_str @{thm rat_power})
206 (*"(?a / ?b) ^^^ ?n = ?a ^^^ ?n / ?b ^^^ ?n"*)
208 scr = Prog ((term_of o the o (parse thy)) "empty_script")
210 Rls_ make_rat_poly_with_parentheses,
211 Rls_ cancel_p_rls,(*FIXME:cancel_p does NOT order sometimes*)
214 scr = Prog ((term_of o the o (parse thy)) "empty_script")
217 (*.for simplify_Integral adapted from 'norm_Rational'.*)
218 val norm_Rational_noadd_fractions =
219 Seq {id = "norm_Rational_noadd_fractions", preconds = [],
220 rew_ord = ("dummy_ord",dummy_ord),
221 erls = norm_rat_erls, srls = Erls, calc = [], errpatts = [],
222 rules = [Rls_ discard_minus,
223 Rls_ rat_mult_poly,(* removes double fractions like a/b/c *)
224 Rls_ make_rat_poly_with_parentheses, (*WN0510 also in(#)below*)
225 Rls_ cancel_p_rls, (*FIXME.MG:cancel_p does NOT order sometim*)
226 Rls_ norm_Rational_rls_noadd_fractions,(* the main rls (#) *)
227 Rls_ discard_parentheses1 (* mult only *)
229 scr = Prog ((term_of o the o (parse thy)) "empty_script")
232 (*.simplify terms before and after Integration such that
233 ..a.x^2/2 + b.x^3/3.. is made to ..a/2.x^2 + b/3.x^3.. (and NO
234 common denominator as done by norm_Rational or make_ratpoly_in.
235 This is a copy from 'make_ratpoly_in' with respective reduction of rules and
236 *1* expand the term, ie. distribute * and / over +
239 append_rls "separate_bdv2"
241 [Thm ("separate_bdv", num_str @{thm separate_bdv}),
242 (*"?a * ?bdv / ?b = ?a / ?b * ?bdv"*)
243 Thm ("separate_bdv_n", num_str @{thm separate_bdv_n}),
244 Thm ("separate_1_bdv", num_str @{thm separate_1_bdv}),
245 (*"?bdv / ?b = (1 / ?b) * ?bdv"*)
246 Thm ("separate_1_bdv_n", num_str @{thm separate_1_bdv_n})(*,
247 (*"?bdv ^^^ ?n / ?b = 1 / ?b * ?bdv ^^^ ?n"*)
248 *****Thm ("add_divide_distrib",
249 *****num_str @{thm add_divide_distrib})
250 (*"(?x + ?y) / ?z = ?x / ?z + ?y / ?z"*)----------*)
252 val simplify_Integral =
253 Seq {id = "simplify_Integral", preconds = []:term list,
254 rew_ord = ("dummy_ord", dummy_ord),
255 erls = Atools_erls, srls = Erls,
256 calc = [], errpatts = [],
257 rules = [Thm ("distrib_right",num_str @{thm distrib_right}),
258 (*"(?z1.0 + ?z2.0) * ?w = ?z1.0 * ?w + ?z2.0 * ?w"*)
259 Thm ("add_divide_distrib",num_str @{thm add_divide_distrib}),
260 (*"(?x + ?y) / ?z = ?x / ?z + ?y / ?z"*)
261 (*^^^^^ *1* ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^*)
262 Rls_ norm_Rational_noadd_fractions,
263 Rls_ order_add_mult_in,
264 Rls_ discard_parentheses,
265 (*Rls_ collect_bdv, from make_polynomial_in*)
267 Calc ("Fields.inverse_class.divide" ,eval_cancel "#divide_e")
272 (*simplify terms before and after Integration such that
273 ..a.x^2/2 + b.x^3/3.. is made to ..a/2.x^2 + b/3.x^3.. (and NO
274 common denominator as done by norm_Rational or make_ratpoly_in.
275 This is a copy from 'make_polynomial_in' with insertions from
277 THIS IS KEPT FOR COMPARISON ............................................
278 * val simplify_Integral = prep_rls(
279 * Seq {id = "", preconds = []:term list,
280 * rew_ord = ("dummy_ord", dummy_ord),
281 * erls = Atools_erls, srls = Erls,
282 * calc = [], (*asm_thm = [],*)
283 * rules = [Rls_ expand_poly,
284 * Rls_ order_add_mult_in,
285 * Rls_ simplify_power,
286 * Rls_ collect_numerals,
288 * Thm ("realpow_oneI",num_str @{thm realpow_oneI}),
289 * Rls_ discard_parentheses,
291 * (*below inserted from 'make_ratpoly_in'*)
292 * Rls_ (append_rls "separate_bdv"
294 * [Thm ("separate_bdv", num_str @{thm separate_bdv}),
295 * (*"?a * ?bdv / ?b = ?a / ?b * ?bdv"*)
296 * Thm ("separate_bdv_n", num_str @{thm separate_bdv_n}),
297 * Thm ("separate_1_bdv", num_str @{thm separate_1_bdv}),
298 * (*"?bdv / ?b = (1 / ?b) * ?bdv"*)
299 * Thm ("separate_1_bdv_n", num_str @{thm separate_1_bdv_n})(*,
300 * (*"?bdv ^^^ ?n / ?b = 1 / ?b * ?bdv ^^^ ?n"*)
301 * Thm ("add_divide_distrib",
302 * num_str @{thm add_divide_distrib})
303 * (*"(?x + ?y) / ?z = ?x / ?z + ?y / ?z"*)*)
305 * Calc ("Fields.inverse_class.divide" ,eval_cancel "#divide_e")
309 .......................................................................*)
312 Seq {id="integration", preconds = [],
313 rew_ord = ("termlessI",termlessI),
314 erls = Rls {id="conditions_in_integration",
316 rew_ord = ("termlessI",termlessI),
318 srls = Erls, calc = [], errpatts = [],
321 srls = Erls, calc = [], errpatts = [],
322 rules = [ Rls_ integration_rules,
324 Rls_ simplify_Integral
328 overwritelthy @{theory} (!ruleset',
329 [("integration_rules", prep_rls integration_rules),
330 ("add_new_c", prep_rls add_new_c),
331 ("simplify_Integral", prep_rls simplify_Integral),
332 ("integration", prep_rls integration),
333 ("separate_bdv2", separate_bdv2),
334 ("norm_Rational_noadd_fractions", norm_Rational_noadd_fractions),
335 ("norm_Rational_rls_noadd_fractions",
336 norm_Rational_rls_noadd_fractions)
344 (prep_pbt thy "pbl_fun_integ" [] e_pblID
345 (["integrate","function"],
346 [("#Given" ,["functionTerm f_f", "integrateBy v_v"]),
347 ("#Find" ,["antiDerivative F_F"])
349 append_rls "e_rls" e_rls [(*for preds in where_*)],
350 SOME "Integrate (f_f, v_v)",
351 [["diff","integration"]]));
353 (*here "named" is used differently from Differentiation"*)
355 (prep_pbt thy "pbl_fun_integ_nam" [] e_pblID
356 (["named","integrate","function"],
357 [("#Given" ,["functionTerm f_f", "integrateBy v_v"]),
358 ("#Find" ,["antiDerivativeName F_F"])
360 append_rls "e_rls" e_rls [(*for preds in where_*)],
361 SOME "Integrate (f_f, v_v)",
362 [["diff","integration","named"]]));
369 (prep_met thy "met_diffint" [] e_metID
370 (["diff","integration"],
371 [("#Given" ,["functionTerm f_f", "integrateBy v_v"]),
372 ("#Find" ,["antiDerivative F_F"])
374 {rew_ord'="tless_true", rls'=Atools_erls, calc = [],
377 crls = Atools_erls, errpats = [], nrls = e_rls},
378 "Script IntegrationScript (f_f::real) (v_v::real) = " ^
379 " (let t_t = Take (Integral f_f D v_v) " ^
380 " in (Rewrite_Set_Inst [(bdv,v_v)] integration False) (t_t::real))"
386 (prep_met thy "met_diffint_named" [] e_metID
387 (["diff","integration","named"],
388 [("#Given" ,["functionTerm f_f", "integrateBy v_v"]),
389 ("#Find" ,["antiDerivativeName F_F"])
391 {rew_ord'="tless_true", rls'=Atools_erls, calc = [],
394 crls = Atools_erls, errpats = [], nrls = e_rls},
395 "Script NamedIntegrationScript (f_f::real) (v_v::real) (F_F::real=>real) = " ^
396 " (let t_t = Take (F_F v_v = Integral f_f D v_v) " ^
397 " in ((Try (Rewrite_Set_Inst [(bdv,v_v)] simplify_Integral False)) @@ " ^
398 " (Rewrite_Set_Inst [(bdv,v_v)] integration False)) t_t) "