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 \<up> 3, x)"*)
21 Integrate :: "[real * real] => real"
24 (*stated as axioms, todo: prove as theorems
25 'bdv' is a constant handled on the meta-level
26 specifically as a 'bound variable' *)
28 (*Ambiguous input\<^here> produces 3 parse trees -----------------------------\\*)
29 integral_const: "Not (bdv occurs_in u) ==> Integral u D bdv = u * bdv" and
30 integral_var: "Integral bdv D bdv = bdv \<up> 2 / 2" and
32 integral_add: "Integral (u + v) D bdv =
33 (Integral u D bdv) + (Integral v D bdv)" and
34 integral_mult: "[| Not (bdv occurs_in u); bdv occurs_in v |] ==>
35 Integral (u * v) D bdv = u * (Integral v D bdv)" and
36 (*WN080222: this goes into sub-terms, too ...
37 call_for_new_c: "[| Not (matches (u + new_c v) a); Not (a is_f_x) |] ==>
40 integral_pow: "Integral bdv \<up> n D bdv = bdv \<up> (n+1) / (n + 1)"
41 (*Ambiguous input\<^here> produces 3 parse trees -----------------------------//*)
44 (** eval functions **)
46 val c = Free ("c", HOLogic.realT);
47 (*.create a new unique variable 'c..' in a term; for use by Rule.Eval in a rls;
48 an alternative to do this would be '(Try (Calculate new_c_) (new_c es__))'
49 in the script; this will be possible if currying doesnt take the value
50 from a variable, but the value '(new_c es__)' itself.*)
53 case (Symbol.explode o id_of) var of
55 | "c"::"_"::is => (case (TermC.int_opt_of_string o implode) is of
59 fun get_coeff c = case (Symbol.explode o id_of) c of
60 "c"::"_"::is => (the o TermC.int_opt_of_string o implode) is
62 val cs = filter selc (TermC.vars term);
66 | [_] => Free ("c_2", HOLogic.realT)
68 let val max_coeff = maxl (map get_coeff cs)
69 in Free ("c_"^string_of_int (max_coeff + 1), HOLogic.realT) end
73 (*("new_c", ("Integrate.new_c", eval_new_c "#new_c_"))*)
74 fun eval_new_c _ _ (p as (Const ("Integrate.new_c",_) $ t)) _ =
75 SOME ((UnparseC.term p) ^ " = " ^ UnparseC.term (new_c p),
76 Trueprop $ (mk_equality (p, new_c p)))
77 | eval_new_c _ _ _ _ = NONE;
81 (*("add_new_c", ("Integrate.add_new_c", eval_add_new_c "#add_new_c_"))
82 add a new c to a term or a fun-equation;
83 this is _not in_ the term, because only applied to _whole_ term*)
84 fun eval_add_new_c (_:string) "Integrate.add_new_c" p (_:theory) =
85 let val p' = case p of
86 Const ("HOL.eq", T) $ lh $ rh =>
87 Const ("HOL.eq", T) $ lh $ TermC.mk_add rh (new_c rh)
88 | p => TermC.mk_add p (new_c p)
89 in SOME ((UnparseC.term p) ^ " = " ^ UnparseC.term p',
90 HOLogic.Trueprop $ (TermC.mk_equality (p, p')))
92 | eval_add_new_c _ _ _ _ = NONE;
95 (*("is_f_x", ("Integrate.is_f_x", eval_is_f_x "is_f_x_"))*)
96 fun eval_is_f_x _ _(p as (Const ("Integrate.is_f_x", _)
99 then SOME ((UnparseC.term p) ^ " = True",
100 HOLogic.Trueprop $ (TermC.mk_equality (p, @{term True})))
101 else SOME ((UnparseC.term p) ^ " = False",
102 HOLogic.Trueprop $ (TermC.mk_equality (p, @{term False})))
103 | eval_is_f_x _ _ _ _ = NONE;
105 setup \<open>KEStore_Elems.add_calcs
106 [("add_new_c", ("Integrate.add_new_c", eval_add_new_c "add_new_c_")),
107 ("is_f_x", ("Integrate.is_f_x", eval_is_f_x "is_f_idextifier_"))]\<close>
111 (*.rulesets for integration.*)
112 val integration_rules =
113 Rule_Def.Repeat {id="integration_rules", preconds = [],
114 rew_ord = ("termlessI",termlessI),
115 erls = Rule_Def.Repeat {id="conditions_in_integration_rules",
117 rew_ord = ("termlessI",termlessI),
118 erls = Rule_Set.Empty,
119 srls = Rule_Set.Empty, calc = [], errpatts = [],
120 rules = [(*for rewriting conditions in Thm's*)
121 \<^rule_eval>\<open>Prog_Expr.occurs_in\<close> (Prog_Expr.eval_occurs_in "#occurs_in_"),
122 Rule.Thm ("not_true", ThmC.numerals_to_Free @{thm not_true}),
123 Rule.Thm ("not_false",@{thm not_false})
125 scr = Rule.Empty_Prog},
126 srls = Rule_Set.Empty, calc = [], errpatts = [],
128 Rule.Thm ("integral_const", ThmC.numerals_to_Free @{thm integral_const}),
129 Rule.Thm ("integral_var", ThmC.numerals_to_Free @{thm integral_var}),
130 Rule.Thm ("integral_add", ThmC.numerals_to_Free @{thm integral_add}),
131 Rule.Thm ("integral_mult", ThmC.numerals_to_Free @{thm integral_mult}),
132 Rule.Thm ("integral_pow", ThmC.numerals_to_Free @{thm integral_pow}),
133 \<^rule_eval>\<open>plus\<close> (**)(eval_binop "#add_")(*for n+1*)
135 scr = Rule.Empty_Prog};
139 Rule_Set.Sequence {id="add_new_c", preconds = [],
140 rew_ord = ("termlessI",termlessI),
141 erls = Rule_Def.Repeat {id="conditions_in_add_new_c",
143 rew_ord = ("termlessI",termlessI),
144 erls = Rule_Set.Empty,
145 srls = Rule_Set.Empty, calc = [], errpatts = [],
146 rules = [\<^rule_eval>\<open>Prog_Expr.matches\<close> (Prog_Expr.eval_matches""),
147 \<^rule_eval>\<open>Integrate.is_f_x\<close> (eval_is_f_x "is_f_x_"),
148 Rule.Thm ("not_true", ThmC.numerals_to_Free @{thm not_true}),
149 Rule.Thm ("not_false", ThmC.numerals_to_Free @{thm not_false})
151 scr = Rule.Empty_Prog},
152 srls = Rule_Set.Empty, calc = [], errpatts = [],
153 rules = [ (*Rule.Thm ("call_for_new_c", ThmC.numerals_to_Free @{thm call_for_new_c}),*)
154 Rule.Cal1 ("Integrate.add_new_c", eval_add_new_c "new_c_")
156 scr = Rule.Empty_Prog};
160 (*.rulesets for simplifying Integrals.*)
162 (*.for simplify_Integral adapted from 'norm_Rational_rls'.*)
163 val norm_Rational_rls_noadd_fractions =
164 Rule_Def.Repeat {id = "norm_Rational_rls_noadd_fractions", preconds = [],
165 rew_ord = ("dummy_ord",Rewrite_Ord.dummy_ord),
166 erls = norm_rat_erls, srls = Rule_Set.Empty, calc = [], errpatts = [],
167 rules = [(*Rule.Rls_ add_fractions_p_rls,!!!*)
168 Rule.Rls_ (*rat_mult_div_pow original corrected WN051028*)
169 (Rule_Def.Repeat {id = "rat_mult_div_pow", preconds = [],
170 rew_ord = ("dummy_ord",Rewrite_Ord.dummy_ord),
171 erls = (*FIXME.WN051028 Rule_Set.empty,*)
172 Rule_Set.append_rules "Rule_Set.empty-is_polyexp" Rule_Set.empty
173 [\<^rule_eval>\<open>is_polyexp\<close> (eval_is_polyexp "")],
174 srls = Rule_Set.Empty, calc = [], errpatts = [],
175 rules = [Rule.Thm ("rat_mult", ThmC.numerals_to_Free @{thm rat_mult}),
176 (*"?a / ?b * (?c / ?d) = ?a * ?c / (?b * ?d)"*)
177 Rule.Thm ("rat_mult_poly_l", ThmC.numerals_to_Free @{thm rat_mult_poly_l}),
178 (*"?c is_polyexp ==> ?c * (?a / ?b) = ?c * ?a / ?b"*)
179 Rule.Thm ("rat_mult_poly_r", ThmC.numerals_to_Free @{thm rat_mult_poly_r}),
180 (*"?c is_polyexp ==> ?a / ?b * ?c = ?a * ?c / ?b"*)
182 Rule.Thm ("real_divide_divide1_mg",
183 ThmC.numerals_to_Free @{thm real_divide_divide1_mg}),
184 (*"y ~= 0 ==> (u / v) / (y / z) = (u * z) / (y * v)"*)
185 Rule.Thm ("divide_divide_eq_right",
186 ThmC.numerals_to_Free @{thm divide_divide_eq_right}),
187 (*"?x / (?y / ?z) = ?x * ?z / ?y"*)
188 Rule.Thm ("divide_divide_eq_left",
189 ThmC.numerals_to_Free @{thm divide_divide_eq_left}),
190 (*"?x / ?y / ?z = ?x / (?y * ?z)"*)
191 \<^rule_eval>\<open>divide\<close> (Prog_Expr.eval_cancel "#divide_e"),
193 Rule.Thm ("rat_power", ThmC.numerals_to_Free @{thm rat_power})
194 (*"(?a / ?b) \<up> ?n = ?a \<up> ?n / ?b \<up> ?n"*)
196 scr = Rule.Empty_Prog
198 Rule.Rls_ make_rat_poly_with_parentheses,
199 Rule.Rls_ cancel_p_rls,(*FIXME:cancel_p does NOT order sometimes*)
200 Rule.Rls_ rat_reduce_1
202 scr = Rule.Empty_Prog
205 (*.for simplify_Integral adapted from 'norm_Rational'.*)
206 val norm_Rational_noadd_fractions =
207 Rule_Set.Sequence {id = "norm_Rational_noadd_fractions", preconds = [],
208 rew_ord = ("dummy_ord",Rewrite_Ord.dummy_ord),
209 erls = norm_rat_erls, srls = Rule_Set.Empty, calc = [], errpatts = [],
210 rules = [Rule.Rls_ discard_minus,
211 Rule.Rls_ rat_mult_poly,(* removes double fractions like a/b/c *)
212 Rule.Rls_ make_rat_poly_with_parentheses, (*WN0510 also in(#)below*)
213 Rule.Rls_ cancel_p_rls, (*FIXME.MG:cancel_p does NOT order sometim*)
214 Rule.Rls_ norm_Rational_rls_noadd_fractions,(* the main rls (#) *)
215 Rule.Rls_ discard_parentheses1 (* mult only *)
217 scr = Rule.Empty_Prog
220 (*.simplify terms before and after Integration such that
221 ..a.x^2/2 + b.x^3/3.. is made to ..a/2.x^2 + b/3.x^3.. (and NO
222 common denominator as done by norm_Rational or make_ratpoly_in.
223 This is a copy from 'make_ratpoly_in' with respective reduction of rules and
224 *1* expand the term, ie. distribute * and / over +
227 Rule_Set.append_rules "separate_bdv2"
229 [Rule.Thm ("separate_bdv", ThmC.numerals_to_Free @{thm separate_bdv}),
230 (*"?a * ?bdv / ?b = ?a / ?b * ?bdv"*)
231 Rule.Thm ("separate_bdv_n", ThmC.numerals_to_Free @{thm separate_bdv_n}),
232 Rule.Thm ("separate_1_bdv", ThmC.numerals_to_Free @{thm separate_1_bdv}),
233 (*"?bdv / ?b = (1 / ?b) * ?bdv"*)
234 Rule.Thm ("separate_1_bdv_n", ThmC.numerals_to_Free @{thm separate_1_bdv_n})(*,
235 (*"?bdv \<up> ?n / ?b = 1 / ?b * ?bdv \<up> ?n"*)
236 *****Rule.Thm ("add_divide_distrib",
237 ***** ThmC.numerals_to_Free @{thm add_divide_distrib})
238 (*"(?x + ?y) / ?z = ?x / ?z + ?y / ?z"*)----------*)
240 val simplify_Integral =
241 Rule_Set.Sequence {id = "simplify_Integral", preconds = []:term list,
242 rew_ord = ("dummy_ord", Rewrite_Ord.dummy_ord),
243 erls = Atools_erls, srls = Rule_Set.Empty,
244 calc = [], errpatts = [],
245 rules = [Rule.Thm ("distrib_right", ThmC.numerals_to_Free @{thm distrib_right}),
246 (*"(?z1.0 + ?z2.0) * ?w = ?z1.0 * ?w + ?z2.0 * ?w"*)
247 Rule.Thm ("add_divide_distrib", ThmC.numerals_to_Free @{thm add_divide_distrib}),
248 (*"(?x + ?y) / ?z = ?x / ?z + ?y / ?z"*)
249 (*^^^^^ *1* ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^*)
250 Rule.Rls_ norm_Rational_noadd_fractions,
251 Rule.Rls_ order_add_mult_in,
252 Rule.Rls_ discard_parentheses,
253 (*Rule.Rls_ collect_bdv, from make_polynomial_in*)
254 Rule.Rls_ separate_bdv2,
255 \<^rule_eval>\<open>divide\<close> (Prog_Expr.eval_cancel "#divide_e")
257 scr = Rule.Empty_Prog};
260 (*simplify terms before and after Integration such that
261 ..a.x^2/2 + b.x^3/3.. is made to ..a/2.x^2 + b/3.x^3.. (and NO
262 common denominator as done by norm_Rational or make_ratpoly_in.
263 This is a copy from 'make_polynomial_in' with insertions from
265 THIS IS KEPT FOR COMPARISON ............................................
266 * val simplify_Integral = prep_rls'(
267 * Rule_Set.Sequence {id = "", preconds = []:term list,
268 * rew_ord = ("dummy_ord", Rewrite_Ord.dummy_ord),
269 * erls = Atools_erls, srls = Rule_Set.Empty,
270 * calc = [], (*asm_thm = [],*)
271 * rules = [Rule.Rls_ expand_poly,
272 * Rule.Rls_ order_add_mult_in,
273 * Rule.Rls_ simplify_power,
274 * Rule.Rls_ collect_numerals,
275 * Rule.Rls_ reduce_012,
276 * Rule.Thm ("realpow_oneI", ThmC.numerals_to_Free @{thm realpow_oneI}),
277 * Rule.Rls_ discard_parentheses,
278 * Rule.Rls_ collect_bdv,
279 * (*below inserted from 'make_ratpoly_in'*)
280 * Rule.Rls_ (Rule_Set.append_rules "separate_bdv"
282 * [Rule.Thm ("separate_bdv", ThmC.numerals_to_Free @{thm separate_bdv}),
283 * (*"?a * ?bdv / ?b = ?a / ?b * ?bdv"*)
284 * Rule.Thm ("separate_bdv_n", ThmC.numerals_to_Free @{thm separate_bdv_n}),
285 * Rule.Thm ("separate_1_bdv", ThmC.numerals_to_Free @{thm separate_1_bdv}),
286 * (*"?bdv / ?b = (1 / ?b) * ?bdv"*)
287 * Rule.Thm ("separate_1_bdv_n", ThmC.numerals_to_Free @{thm separate_1_bdv_n})(*,
288 * (*"?bdv \<up> ?n / ?b = 1 / ?b * ?bdv \<up> ?n"*)
289 * Rule.Thm ("add_divide_distrib",
290 * ThmC.numerals_to_Free @{thm add_divide_distrib})
291 * (*"(?x + ?y) / ?z = ?x / ?z + ?y / ?z"*)*)
293 * \<^rule_eval>\<open>divide\<close> (eval_cancel "#divide_e")
295 * scr = Rule.Empty_Prog
297 .......................................................................*)
300 Rule_Set.Sequence {id="integration", preconds = [],
301 rew_ord = ("termlessI",termlessI),
302 erls = Rule_Def.Repeat {id="conditions_in_integration",
304 rew_ord = ("termlessI",termlessI),
305 erls = Rule_Set.Empty,
306 srls = Rule_Set.Empty, calc = [], errpatts = [],
308 scr = Rule.Empty_Prog},
309 srls = Rule_Set.Empty, calc = [], errpatts = [],
310 rules = [ Rule.Rls_ integration_rules,
312 Rule.Rls_ simplify_Integral
314 scr = Rule.Empty_Prog};
316 val prep_rls' = Auto_Prog.prep_rls @{theory};
319 integration_rules = \<open>prep_rls' integration_rules\<close> and
320 add_new_c = \<open>prep_rls' add_new_c\<close> and
321 simplify_Integral = \<open>prep_rls' simplify_Integral\<close> and
322 integration = \<open>prep_rls' integration\<close> and
323 separate_bdv2 = \<open>prep_rls' separate_bdv2\<close> and
324 norm_Rational_noadd_fractions = \<open>prep_rls' norm_Rational_noadd_fractions\<close> and
325 norm_Rational_rls_noadd_fractions = \<open>prep_rls' norm_Rational_rls_noadd_fractions\<close>
328 setup \<open>KEStore_Elems.add_pbts
329 [(Problem.prep_input @{theory} "pbl_fun_integ" [] Problem.id_empty
330 (["integrate", "function"],
331 [("#Given" ,["functionTerm f_f", "integrateBy v_v"]),
332 ("#Find" ,["antiDerivative F_F"])],
333 Rule_Set.append_rules "empty" Rule_Set.empty [(*for preds in where_*)],
334 SOME "Integrate (f_f, v_v)",
335 [["diff", "integration"]])),
336 (*here "named" is used differently from Differentiation"*)
337 (Problem.prep_input @{theory} "pbl_fun_integ_nam" [] Problem.id_empty
338 (["named", "integrate", "function"],
339 [("#Given" ,["functionTerm f_f", "integrateBy v_v"]),
340 ("#Find" ,["antiDerivativeName F_F"])],
341 Rule_Set.append_rules "empty" Rule_Set.empty [(*for preds in where_*)],
342 SOME "Integrate (f_f, v_v)",
343 [["diff", "integration", "named"]]))]\<close>
347 partial_function (tailrec) integrate :: "real \<Rightarrow> real \<Rightarrow> real"
349 "integrate f_f v_v = (
351 t_t = Take (Integral f_f D v_v)
353 (Rewrite_Set_Inst [(''bdv'', v_v)] ''integration'') t_t)"
354 setup \<open>KEStore_Elems.add_mets
355 [MethodC.prep_input @{theory} "met_diffint" [] MethodC.id_empty
356 (["diff", "integration"],
357 [("#Given" ,["functionTerm f_f", "integrateBy v_v"]), ("#Find" ,["antiDerivative F_F"])],
358 {rew_ord'="tless_true", rls'=Atools_erls, calc = [], srls = Rule_Set.empty, prls=Rule_Set.empty,
359 crls = Atools_erls, errpats = [], nrls = Rule_Set.empty},
360 @{thm integrate.simps})]
363 partial_function (tailrec) intergrate_named :: "real \<Rightarrow> real \<Rightarrow> (real \<Rightarrow> real) \<Rightarrow> bool"
365 "intergrate_named f_f v_v F_F =(
367 t_t = Take (F_F v_v = Integral f_f D v_v)
369 (Try (Rewrite_Set_Inst [(''bdv'', v_v)] ''simplify_Integral'')) #>
370 (Rewrite_Set_Inst [(''bdv'', v_v)] ''integration'')
372 setup \<open>KEStore_Elems.add_mets
373 [MethodC.prep_input @{theory} "met_diffint_named" [] MethodC.id_empty
374 (["diff", "integration", "named"],
375 [("#Given" ,["functionTerm f_f", "integrateBy v_v"]),
376 ("#Find" ,["antiDerivativeName F_F"])],
377 {rew_ord'="tless_true", rls'=Atools_erls, calc = [], srls = Rule_Set.empty, prls=Rule_Set.empty,
378 crls = Atools_erls, errpats = [], nrls = Rule_Set.empty},
379 @{thm intergrate_named.simps})]